Communications In Mathematical Physics - Volume 263

Commun. Math. Phys. 263, 1–19 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1491-6 Communications in Mathe...

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Commun. Math. Phys. 263, 1–19 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1491-6

Communications in

Mathematical Physics

Characterization and ‘Source-Receiver’ Continuation of Seismic Reflection Data Maarten V. de Hoop1 , Gunther Uhlmann2, 1

Center for Computational and Applied Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907, USA. E-mail: [email protected] 2 Department of Mathematics, University of Washington, Seattle, WA 98195-4350, USA. E-mail: [email protected] Received: 29 June 2004 / Accepted: 3 October 2005 Published online: 26 January 2006 – © Springer-Verlag 2006

Abstract: In reflection seismology one places sources and receivers on the Earth’s surface. The source generates elastic waves in the subsurface, that are reflected where the medium properties, stiffness and density, vary discontinuously. In the field, often, there are obstructions to collect seismic data for all source-receiver pairs desirable or needed for data processing and application of inverse scattering methods. Typically, data are measured on the Earth’s surface. We employ the term data continuation to describe the act of computing data that have not been collected in the field. Seismic data are commonly modeled by a scattering operator developed in a high-frequency, single scattering approximation. We initially focus on the determination of the range of the forward scattering operator that models the singular part of the data in the mentioned approximation. This encompasses the analysis of the properties of, and the construction of, a minimal elliptic projector that projects a space of distributions on the data acquisition manifold to the range of the mentioned scattering operator. This projector can be directly used for the purpose of seismic data continuation, and is derived from the global parametrix of a homogeneous pseudodifferential equation the solution of which coincides with the range of the scattering operator. We illustrate the data continuation by a numerical example.

1. Introduction In reflection seismology one places sources and receivers on the Earth’s surface. The source generates elastic waves in the subsurface, that are reflected where the medium properties, stiffness and density, vary discontinuously. Seismic data collected in the field are often not ideal for data processing and application of inverse scattering methods. Typically, data are measured on the Earth’s (two-dimensional) surface; the location of the

This research was supported in part under NSF CMG grant EAR-0417891. Partly supported by a John Simon Guggenheim fellowship.

2

M.V. de Hoop, G. Uhlmann

receiver relative to the source can be coordinated by offset and azimuth. We employ the term data continuation to describe the act of computing data that have not been collected in the field. Special cases of data continuation are the so-called ‘transformation to zero offset’ (derived from what seismologists call Dip MoveOut [7] to generate data at zero offsets, and ‘transformation to common azimuth’ (derived from what seismologists call Azimuth MoveOut [2]) to generate data at a fixed, prescribed azimuth. Data continuation can also play the role of ‘forward extrapolation’ [9] in a data regularization scheme. Seismic data are commonly modeled by a scattering operator developed in a highfrequency single scattering approximation. In this approximation one assumes that the medium is described by a singular contrast superimposed on a smooth background. Under geological constraints, often, the contrast is a conormal distribution. Initially, we focus on the determination of the range of the forward scattering operator that models the singular part of the data in the single scattering approximation. This encompasses the analysis of the properties of, and the construction of, a minimal elliptic projector that projects the space of distributions on the acquisition manifold to the range of the mentioned scattering operator. This projector can be directly used for the purpose of seismic data continuation, and is derived from the global parametrix of a homogeneous pseudodifferential equation the solution of which coincides with the range of the scattering operator. Through characterization of features in the data, applications of data continuation extend to survey design (i.e. the design of the acquisition geometry describing the locations of the source-receiver pairs). The range of the scattering operator can also be used as a criterion for muting the data for features that are undesirable for the purpose of imaging the data (such as multiple scattered waves). The notion of data continuation has been introduced in exploration seismology quite some time ago. As early as in 1982, Bolondi et al. [3] came up with the idea of describing data offset continuation and Dip MoveOut in the form of solving a partial differential equation. Their approach, built on the approach of Deregowski and Rocca [7], is valid in homogeneous media for acoustic waves while their partial differential operator is approximate only. An ‘exact’ partial differential equation for space dimension n = 2 that addresses mentioned offset continuation was later derived by Goldin [11]. In this application it is implicitly used that the kernel of the associated partial differential operator determines the range of the operator that models the singular part of seismic data – in the single scattering approximation. The operator can be written in the form of a generalized Radon transform. Heuristically, the procedure and analysis presented in this paper can be thought of as a generalization from two to higher dimensions, from acoustic to elastic, and from homogeneous to heterogeneous media, of Goldin’s ‘offset continuation’ equation approach. Let the data be denoted by d = d(s, r, t), where s denotes source position, r receiver position, t the time, while (s, r, t) ∈ Y and Y denotes the acquisition manifold. Let Y ⊂ R2n−1 . We introduce the map κ: (s, r, t) → (z, tn , h), z = 21 (r + s), h = 21 (r − s), tn = tn (h, t) =

t2 −

4h2 , v2

where, v is the acoustic wave speed. Let r be the pull back of d by the inverse of this map, r = (κ −1 )∗ d. The singular support of r can be parametrized by (z, h) according to (z, Tn (z, h), h) with Tn (z, h) = tn (h, T (z, h)), in which the function T (z, h) denotes the traveltime of a particular reflection in the data; in seismological terms, the function

Characterization of Seismic Reflection Data

3

Tn is the traveltime ‘after Normal MoveOut correction’. Goldin’s equation is of the form (n = 2) 2 ∂2 ∂2 ∂ (1) − 2 . +h P r = 0, P := tn ∂tn ∂h ∂h2 ∂z This equation is supplemented with the initial conditions r(z, tn , h)|h=h0 ,

∂r (z, tn , h)|h=h0 . ∂h

The first initial condition represents what seismologists call a post-normal-moveout constant-offset section at half offset h0 ; the second initial condition is the first-order derivative of post-normal-moveout section at half offset h0 . Goldin’s equation is not exact in the sense that it does not account for the symbols of the reflection operators associated with the reflectors in the subsurface. The notion of data continuation has also been introduced and exploited in helical x-ray transmission tomography (CT) [22]. Consider a flat area detector, which is contained in the plane described by Cartesian coordinates (u, v). Let R denote the radius of the helix and 2πh its pitch. Let λ denote the angle describing rotation of the cone vertex. The axial shift of the assembly of the x-ray source and the detector is denoted by ζ . The data are denoted by g = g(u, v, λ, ζ ). In this case, John’s equation [19] describing the range of the Radon transform is used. John’s equation for the x-ray transform in dimension 3 is given by R2

∂ 2g ∂ 2g ∂g ∂ 2g ∂ 2g ∂ 2g − 2u + (R 2 + u2 ) =R − Rh + uv 2 . ∂u∂ζ ∂v ∂u∂v ∂λ∂v ∂v∂ζ ∂v

(2)

This equation is supplemented with the initial conditions g(u, v, λ, ζ )|ζ =0 that are measured. (The standard form of John’s equation is much simpler than (2).) Gel’fand and Graev [10] have generalized John’s result to k-planes in Rn . John’s (and Goldin’s) partial differential equation in higher space dimension is second order and of ultrahyperbolic type. The seismic forward scattering operator is a Fourier integral operator and can be identified with a generalized Radon transform [1, 5, 23]. We characterize seismic data by analyzing the range of the forward scattering operator. This range coincides with the kernel of a self-adjoint, second-order pseudodifferential operator, P , derived from annihilators, Pi , of the data, d, Pi d = 0, P = Pi2 . (3) i

Let Q denote the global parametrix of P . The mentioned elliptic minimal projector then follows to be π = I − QP + smoothing operator

(4)

and provides the Fourier integral operator for continuing the singular part of the data. The annihilators are functionally dependent on the background medium, and hence can be used to form a criterion to estimate it. This estimation is known to seismologists as

4


‘velocity analysis’ and can be formulated as a reflection tomography problem. Thus, data continuation and reflection tomography, and imaging, are intimately connected. The results presented in this paper are based on the work by Guillemin and Uhlmann [15]. Here, we speak of ‘data’ continuation rather than ‘offset’ continuation, because our approach continues data in sources and receivers and not only in offset due to the heterogeneity of the subsurface we can allow. 2. Modeling of Seismic Data in the Single Scattering Approximation The propagation and scattering of seismic waves is governed by the elastic wave equation, which is written in the form Wil ul = fi ,

(5)

where ul =

ρ(x)(displacement)l ,

fi = √

1 (volume force density)i , ρ(x)

(6)

∂ cij kl (x) ∂ . ∂xj ρ(x) ∂xk

(7)

and Wil = δil

∂2 + Ail + l.o.t., ∂t 2

Ail = −

Here, x ∈ Rn and the subscripts i, j, k, l ∈ {1, . . . , n}; ρ is the density of mass while cij kl denotes the stiffnesss tensor. The system of partial differential equations is assumed to be of principal type. It supports different wave types (modes), one ‘compressional’ and n − 1 ‘shear’. We label the modes by M, N, . . . . For waves in mode M, singularities are propagated along bicharacteristics, that are determined by Hamilton’s equations generated by a Hamiltonian BM , ,

dt = 1, dλ

dξ ∂ = − BM (x, ξ ) , dλ ∂x

dτ = 0. dλ

∂ dx = BM (x, ξ ) dλ ∂ξ

(8)

The BM follow from the diagonalization of the principal symbol matrix of Ail , as the square roots of its eigenvalues. Clearly, the solution may be parameterized by t. We denote the solution of (8) subject to and initial values (x0 , ξ0 ) at t = 0 by (xM (x0 , ξ0 , t), ξM (x0 , ξ0 , t)). In the contrast formulation the total value of the medium parameters ρ, cij kl is written as the sum of a smooth background constituent ρ(x), cij kl (x) and a singular perturbation δρ(x), δcij kl (x), viz. ρ + δρ, cij kl + δcij kl . This decomposition induces a perturbation of Wil (cf. (7)), δWil = δil

δρ(x) ∂ 2 ∂ δcij kl (x) ∂ − . 2 ρ(x) ∂t ∂xj ρ(x) ∂xk

The scattered field, in the single scattering approximation, satisfies Wil δul = −δWil ul .


5

Data are measurements of the scattered wave field δu, which we relate here to the Green’s function perturbation: They are assumed to be representable by δGMN ( x, x , t) for ( x, x , t) in some acquisition manifold, which contains the receiver and source points and time. Let y → ( x (y), x (y), t (y)) be a coordinate transformation, such that y = (y , y ) and the acquisition manifold, Y say, is given by y = 0. We assume that the dimension of y is 2 + c, where c is the codimension of the acquisition geometry. For example, for marine acquisition in seismic reflection data, c = 1, while also in global seismology for many, but not all, regions c = 1 – seismologists recognize this as lack of ‘azimuthal’ coverage. An example of c = 2 is provided by the common-‘offset’ acquisition geometry. In this framework, the data are modeled by δρ(x) δcij kl (x) x (y , 0), x (y , 0), t (y , 0)). (9) , → δGMN ( ρ(x) ρ(x) ξ0 ), We investigate the propagation of singularities by this mapping. Let τ = ∓BM (x0 , and ξ0 , ± t), x = xN (x0 , ξ0 , ± t), x = xM (x0 , ξ = ξM (x0 , ξ0 , ± t), ξ = ξN (x0 , ξ0 , ± t).

t = t + t,

ξ0 , ξ0 , t, t), η(x0 , ξ0 , ξ0 , t, t)) by transforming ( x, x , t + t, ξ , ξ, τ) We then obtain (y(x0 , to (y, η) coordinates. We invoke the following assumptions that concern scattering over π and rays grazing the acquisition manifold, Assumption 1. There are no elements (y , 0, η , η ) with (y , η ) ∈ T ∗ Y \0 such that there is a direct bicharacteristic from ( x (y , 0), ξ (y , 0, η , η )) to ( x (y , 0), − ξ (y , 0, η , η )) with arrival time t (y , 0). Assumption 2. The matrix ∂y has maximal rank. ∂(x0 , ξ0 , ξ0 , t, t)

(10)

The propagation of singularities by (9) is governed by the canonical relation ξ0 , ξ0 , t, t), η (x0 , ξ0 , ξ0 , t, t); x0 , ξ0 + ξ0 ) | , (11) MN = {(y (x0 , ∗ ∗ BM (x0 , ξ0 ) = BN (x0 , ξ0 ) = ∓τ, y (x0 , ξ0 , ξ0 , t, t) = 0} ⊂ T Y \0 × T X\0. The condition y (x0 , ξ0 , ξ0 , t, t) = 0 determines the traveltimes t for given (x0 , ξ0 ) and t for given (x0 , ξ0 ). Following Maslov and Fedoriuk [21], we choose coordinates for MN of the form (yI , x0 , ηJ ),

(12)

where I ∪ J is a partition of {1, . . . , 2n − 1 − c}, with associated generating function SMN = SMN (yI , x0 , ηJ ). The phase function in these coordinates becomes MN =

MN (y , x0 , ηJ ). Let τ = 21 ( τ + τ ) and τ¯ = τ − τ . The map (x0 , ξ0 , ξ0 , t, t) → (x0 , yI , y , ηJ , τ¯ )

(13)

6


is bijective. Thus, for y = 0 and τ¯ = 0 we can express ( ξ0 , ξ0 ) as functions of (yI , x0 , ηJ ). The amplitude associated with MN , to leading order, can be written in the form |bMN (yI , x0 , ηJ )| ∂( x, x , t) −1/2 ∂(x0 , ξ0 , ξ0 , t, t ) 1/2 1 − n+1+c 4 . (14) = (2π ) det ∂(y , y ) det ∂(x , y , y , η , τ¯ ) 2 4τ 0 I y =0,τ¯ =0 J

δcij kl , are described by conormal distributions. We consider the We assume that δρ ρ ρ case of a single interface, and a jump discontinuity in (δρ, δcij kl ) across this interface. Let κ : Rn → Rn , x → z be a coordinate transformation such that the interface is given by zn = 0. The corresponding cotangent vector is denoted by ζ , and transforms −1 t according to ζi (x, ξ ) = (( ∂κ ∂x ) )ij ξj ; the z form coordinates on the manifold X, and δc we write z = (z , zn ). We introduce the distributions (δρ, ij kl ) by pull back with κ: δρ(κ(x)) = δρ(x),

δc ij kl (κ(x)) = δcij kl (x).

(15)

Then ∂ ∂z = n ρ + l.o.t., δρ ∂x ∂x

ρ =

∂ δρ, ∂zn ∂ δ c

where ρ contains a factor δ(zn (x)), and similarly for ∂xij kl . Substituting (15) into the integral over X representing the high-frequency Born approximation for scattered waves, and integrating by parts, then yields an oscillatory integral representation in which wMN;0 (yI , x, ηJ )

δcij kl (x) δρ(x) + wMN ;ij kl (yI , x, ηJ ) , ρ(x) ρ(x)

where w stands for the contrast-source radiation patterns derived from the pseudodifferential operators that diagonalize the elastodynamic system of equations [23], has been replaced by ∂zn δ(zn (x)). 2iτ RMN (yI , x, ηJ ) ∂x n Here we use that τ will be one of the components of ηJ . Also (· · · ) ∂z ∂x δ(zn (x)) ∂zn dx = zn =0 (· · · ) det ∂x ∂z ∂x dz becomes the Euclidean surface integral over the surface or manifold zn = 0. Theorem 1 [23]. Suppose Assumptions 1, and 2 are satisfied microlocally for the relevant part of the data. Let MN (y, x, ηJ ) and bMN (yI , x, ηJ ) be the phase function and amplitude introduced above. Then the mapping ∂zn refl F : ∂x δ(zn (x)) → GMN (y),


7

where

− |J2 | − 3n−1−c 4 Grefl (y) = (2π ) (2iτ (ηJ ) bMN (yI , x, ηJ )RMN (yI , x, ηJ ) + l.o.t.) MN X ∂zn × (16) ∂x δ(zn (x)) exp[i MN (y, x, ηJ )] dxdηJ , defines a Fourier integral operator with canonical relation MN and of order n−1+c −1. 4 F models seismic reflection data. In the Kirchhoff approximation, one can identify the principal part of RMN with the plane-wave reflection coefficient: Using (13) we find the (x, ξ0 , ξ0 ) associated with (yI , x, ηJ ). A reflection from an incident N -mode with covector ξ0 into a scattered M-mode with covector ξ0 takes place, at x, if the frequencies are equal and ξ0 + ξ0 is in the wavefront set of δ(zn (x)). Given ξ0 , ξ0 one can identify the down- and upgoing modes µ(M), ν(N ) relative to the interface, and define (at least to highest order) the reflection coefficient at x, RMN = Rµ(M),ν(N) (z (x), ζ (x, ξ0 ), τ ) if zn (x) = 0,

(17)

see De Hoop and Bleistein [4] and Stolk and De Hoop [23]. The Kirchhoff approximation requires the following assumption Assumption 3. There are no rays tangent to the interface zn = 0, i.e. elements in MN associated with (x(z , 0), ξ0 (z , 0, ζ , 0)) or with (x(z , 0), ξ0 (z , 0, ζ , 0)) (cf. (11)). For a treatment of reflection and transmission of waves in the elastic case, using microlocal analysis, see Taylor [24]; for the acoustic case, see also Hansen [16]. Examples ∂zn of conormal distributions, ∂x δ(zn (x)), in the Earth sciences the reflections off which are observed, include the core-mantle boundary, thermal and chemical boundary layers in the deep mantle, fault zones, and geological interfaces in sedimentary basins.

3. Extension of the Scattering Operator For simplicity of notation, from here on, we drop the subscripts MN and consider a single mode pair. In the single scattering approximation, subject to restriction to the acquisition manifold Y , the singular part of the medium parameters is a function of n variables, while the data are a function of 2n − 1 − c variables. Here, we discuss the extension of the scattering operator to act on distributions of 2n − 1(−c) variables, equal to the number of degrees of freedom in the acquisition.

3.1. The wavefront set of seismic data . The wavefront set of the modeled data is not arbitrary. This is a consequence of the fact that data consist of multiple experiments designed to provide a degree of redundancy, which we explain here. Assumption 4 (Guillemin [13]). The projection πY of on T ∗ Y \0 is an embedding.

8


This assumption is known as the Bolker condition. It admits the presence of caustics. Because is a canonical relation that projects submersively on the subsurface variables (x, ξ ) (using that the operator Wil is of principal type), the projection of (11) on T ∗ Y \0 is immersive [17, Lemma 25.3.6 and (25.3.4)]. In fact, only the injectivity part of the Bolker condition needs to be verified. The image L of πY is locally a coisotropic submanifold of T ∗ Y \0. Hence, for each (y, η) ∈ L, (T(y,η) L)⊥ ⊂ T(y,η) L. Setting V(y,η) = (T(y,η) L)⊥ , the vector bundle V → L whose fiber at (y, η) is (T(y,η) L)⊥ , is an integrable subbundle of T L. Applying [14, Prop. 8.1], from the Bolker condition it follows that L satisfies their Axiom F: the foliation of L associated with V is fibrating, i.e. there exists a C ∞ Hausdorff manifold X and a smooth fiber map L → X whose fibers are the connected leaves of the foliation defined by V . We choose coordinates revealing the mentioned fibration. Since the projection πX of on T ∗ X\0 is submersive, we can choose (x, ξ ) as the first 2n local coordinates on ; the remaining dim Y − n = n − 1 − c coordinates are denoted by e ∈ E, E being a manifold itself. The sets X (x, ξ ) = const. are the isotropic fibers of the fibration of Hörmander [18], Theorem 21.2.6, see also Theorem 21.2.4. Duistermaat [8] calls them characteristic strips (see Theorem 3.6.2). Also, ν = ξ −1 ξ is then identified as the migration dip. The wavefront set of the data is contained in L and is a union of such fibers. The map πX πY−1 : L → X is a canonical isotropic fibration, known to seismologists as map migration. We consider again the canonical relation and suppose that Assumption 4 is satisfied. We define as the mapping πY πX−1 , : (x, ξ, e) → (y(x, ξ, e), η(x, ξ, e)) : T ∗ X\0 × E → T ∗ Y \0, which is known to seismologists as map demigration. This map conserves the symplectic form of T ∗ X\0. Indeed, let σY denote the fundamental symplectic form on T ∗ Y \0. We consider the vector fields over an open subset of L with components wxi = ∂(y,η) ∂xi and similarly for wξi and wei . Then σY (wxi , wxj ) = σY (wξi , wξj ) = 0, σY (wξi , wxj ) = δij , σY (wei , wxj ) = σY (wei , wξj ) = σY (wei , wej ) = 0.

(18)

The (x, ξ, e) are ‘symplectic coordinates’ on the projection L of on T ∗ Y \0. In the following lemma, we extend these coordinates to symplectic coordinates on an open neighborhood of L, which is a manifestation of Darboux’s theorem stating that T ∗ Y can be covered with symplectic local charts. Lemma 1. Let L be an embedded coisotropic submanifold of T ∗ Y \0, with coordinates (x, ξ, e) such that (18) holds. Denote L (y, η) = (x, ξ, e). We can find a homogeneous canonical map G from an open part of T ∗ (X × E)\0 to an open neighborhood of L in T ∗ Y \0, such that G(x, e, ξ, ε = 0) = (x, ξ, e).

3.2. An invertible Fourier integral operator. Let M be the canonical relation associated with the map G we introduced in Lemma 1, i.e. M = {(G(x, e, ξ, ε); x, e, ξ, ε)} ⊂ T ∗ Y \0 × T ∗ (X × E)\0.


9

We now construct a Maslov-type phase function for M that is directly related to a phase function for . Suppose (yI , x, ηJ ) are suitable coordinates for , at ε = 0. For ε small, the constant-ε subset of M allows the same set of coordinates, thus we can use coordinates (yI , ηJ , x, ε) on M. Now there is (see Theorem 4.21 in Maslov and Fedoriuk [21]) a function S(yI , x, ηJ , ε), called the generating function, such that M is given by yJ = −

∂S , ∂ηJ

∂S ξ = − , ∂x

ηI =

∂S , ∂yI

e=

∂S . ∂ε

(19)

Thus a phase function for M is given by (y, x, e, ηJ , ε) = S(yI , x, ηJ , ε) + ηJ , yJ − ε, e.

(20)

A phase function for then follows as

(y, x, ηJ ) = (y, x, ∂S ∂ε |ε=0 , ηJ , 0) = S(yI , x, ηJ , 0) + ηJ , yJ . We introduce the amplitude b(yI , x, ηJ , ε) on M such that b(yI , x, ηJ , ε = 0) coincides with the amplitude in Theorem 1. To leading order, ∂ b=0 ∂ε because the coordinates εi are in involution. We construct a mapping from the reflectivity function to seismic data, extending the mapping from contrast to data. This is done by applying the results of Sect. 3.1 to the Kirchhoff modeling formula (16). We apply the change of coordinates on from (yI , x, ηJ ) to (x, ξ, e) to the symbol RMN and write now RMN = R(x, ξ, e). To highest order, R does not depend on ξ and is simply a function of (x, e). Theorem 2 [23]. Suppose microlocally that Assumptions 3 (no grazing rays at any interface), 1 (no scattering over π), 2 (transversality), and 4 (Bolker condition) are satisfied. Let H be the Fourier integral operator, H: E (X × E) → D (Y ), with canonical relation given by the extended map G : (x, ξ, e, ε) → (y, η) constructed in Sect. 3.1, and with amplitude to highest order given by (2π )n/2 2iτ (ηJ )b(yI , x, ηJ , ε) expressible in terms of the coordinates (x, e, ξ, ε). Then the data, in both Born and Kirchhoff approximations, can be modeled by H acting on a distribution r(x, e) of the form r(x, e) = R(x, Dx , e) c(x),

(21)

where R stands for a smooth e-family of pseudodifferential operators and c ∈ E (X). For n the Kirchhoff approximation the distribution c equals ∂z ∂x δ(zn (x)), while the principal symbol of the pseudodifferential operator R equals R(x, e), so to highest order ∂zn (22) r(x, e) = R(x, e) ∂x δ(zn (x)).

10


For the Born approximation the function r(x, e) is given by a pseudodifferential opera−1 (w tor R with principal symbol (2iτ MN ;0 (x, ξ, e), wMN;ij kl (x, ξ, e)), acting

(x, ξ, e)) on a distribution c given by

δcij kl δρ ρ , ρ

, so to highest order

δcij kl (x) δρ(x) r(x, e) = (2iτ (x, Dx , e))−1 wMN ;0 (x, Dx , e) +wMN;ij kl (x, Dx , e) . ρ(x) ρ(x) The operator H is invertible. Remark 1. Microlocally, we have obtained the following diagram (suggested by Symes, personal communication) H

E (X × E) −→ D (Y ) R(x, Dx , e) ↑ ↑ Id E (X)

F

−→

(23)

D (Y )

We note that R(x, Dx , e) is of order 0. H −1 maps data into what seismologists call common-image-point gathers (the integral over ε replaces the notion of beamforming; e plays the role of scattering angle and azimuth). 4. A Procedure for Data Continuation 4.1. The range of the scattering operator. If n − 1 − c > 0, there is a redundancy in the data parametrized by the variable e. The redundancy in the data manifests itself as a redundancy in images of the subsurface from these data. A smooth background is considered ‘acceptable’ if the data are contained in the range of F (or H ). If a smooth background is acceptable, then applying the operator H −1 of Theorem 2 to the data results in a reflectivity distribution r(x, e), the singular support (in x) of which does not depend on e. One way to measure the agreement in singular supports between images of reflectivity r(x, e) parametrized by e is by taking a derivative with respect to e. Taking (21) as the point of departure, we find that ∂ ∂R ∂R R(x, Dx , e) − (x, Dx , e) r(x, e) = R(x, Dx , e), (x, Dx , e) c(x). (24) ∂e ∂e ∂e Hence, microlocally where R(x, Dx , e) is elliptic, ∂R ∂ R(x, Dx , e) − (x, Dx , e) ∂e ∂e ∂R (x, Dx , e) R(x, Dx , e)−1 r(x, e) = 0 − R(x, Dx , e), ∂e

(25)

to all orders. We observe that the first operator acting on distributions in (x, e) in the sum is of order 1, the second operator is of order 0, while the third operator is of order −1. Falling back on (22) we exploit that, up to leading order, the operator R acts as a multiplication by R(x, e). Clearly, ∂zn ∂R R(x, e), (x, e) ∂x δ(zn (x)) = 0, ∂e


11

cf. (24). Substituting (21) into (25) reveals that the operator in between parentheses on ∂ the left-hand side equals (R(x, e) ∂e − ∂R ∂e (x, e)) up to the leading two orders. Hence, R(x, e)

∂ ∂R − (x, e) r(x, e) = 0 ∂e ∂e

(26)

up to the highest two orders. Conjugating the operator in between parentheses in (26), or in (25), with the invertible Fourier integral operator H , we obtain a pseudodifferential operator on D (Y ) [23] Lemma 2. Let the pseudodifferential operators Pi (y, Dy ) : D (Y ) → D (Y ) of order 1 be given by the composition ∂ ∂R Pi (y, Dy ) = H R(x, e) − (x, e) H −1 , i = 1, . . . , n − 1 − c . ∂ei ∂ei r

Then for Kirchhoff data d(y) modeled by F , we have to the highest two orders, Pi (y, Dy )d(y) = 0, i = 1, . . . , n − 1 − c.

(27)

Microlocally, for values of e where R(x, e) = 0, the operator Pi (y , Dy ) can be modeled after (25) such that (27) is valid to all orders. The principal part of the symbol of Pi is denoted by pi , while the next order term in the symbol’s polyhomogeneous expansion is denoted by pi;0 . The subprincipal symbols (which show up naturally in the Weyl calculus of symbols), ci , of the annihilators are 2p i then given by ci := pi;0 + 2i j ∂y∂j ∂η . j Remark 2. The wavefront set of the data is contained in L = πY ( ), which, in analogy with the eikonal equation, is also the submanifold of T ∗ Y \0 defined by pi (y, η) = 0,

i = 1, . . . , n − 1 − c,

(28)

where pi is the principal symbol of Pi as before, and is of codimension 2 [2(n − 1) − c + 1] − (3n − 1 − c) = n − 1 − c, which is also the dimension of the covector ε. n−1

The operator F in Theorem 1 is continuous H 2 (X) → L2 (Y ). We now define the operator projection, π : L2 (Y ) → L2 (Y ), onto the range of the scattering operator F . Microlocally, π 2 = π. Since Assumption 4 is satisfied, using [14, Prop. 8.3], π is an elliptic minimal projector. By [14, Theorem 6.6], the kernel of 2 P = P12 + · · · + Pn−1−c

(29)

is identical with the range of π . More precisely, let Q denote the global parametrix of P , then, by [14, Theorem 6.7], π = I − QP + smoothing operator.

(30)

12


4.2. A global parametrix. The construction of a global parametrix, Q, for an operator of the type P is given by Guillemin and Uhlmann [15]. A natural parametrix for P would 1 have as principal symbol 2 . However, this expression becomes singu2 p1 + · · · + pn−1−c lar at the set {p1 = · · · = pn−1−c = 0}. A class of operators, containing pseudodifferential operators with singular symbols, was introduced by Guillemin and Uhlmann [15]. The wavefront set of the kernels of these operators consist of two Langrangian manifolds, 0 and 1 say, intersecting cleanly in a submanifold of given codimension. In our case, 0 is the diagonal diag(T ∗ Y \0) in T ∗ Y \0 × T ∗ Y \0, while 1 is the fiber X

product L × L. The Lagrangian submanifold 1 ⊂ T ∗ Y \0 × T ∗ Y \0 precisely consists of points on the joint flowout from diag(T ∗ Y \0) ∩ {p1 = · · · = pn−1−c = 0} by the Hamiltonian flows of the Hp1 , . . . , Hpn−1−c , where Hpi denotes the Hamiltonian field associated with the function pi . The flowout is described by the solution to the Hamilton systems with parameters ei , ∂yj ∂pi = (y, η), ∂ei ∂ηj

∂ηj ∂pi =− (y, η), ∂ei ∂yj

1 ≤ i, j ≤ n − 1 − c.

(31)

The Lagrangian submanifolds 0 and 1 intersect cleanly in a submanifold of codimension n − 1 − c, see Remark 2. Guillemin and Uhlmann’s construction relies on the introduction of the space of distributional half densities, I p,l (Y ×Y ; 0 , 1 ), defining a class of Fourier integral operators with singular symbols, with the properties ∩l I p,l (Y × Y ; 0 , 1 ) = I p (Y × Y, 1 ) (defining standard Fourier integral operators with canonical relation 1 ) for p fixed, and ∩p I p,l (Y × Y ; 0 , 1 ) = C0∞ (Y × Y ) for l fixed. Viewing the Schwartz kernel of the identity (I ) as an element of I 0,0 (Y × Y ; 0 , 1 ), Guillemin and Uhlmann’s recursive construction results in QP = I − π + R, where the kernel of π belongs to ∩l I p,l (Y × Y ; 0 , 1 ), and R is a smoothing operator with kernel in ∩p I p,l (Y × Y ; 0 , 1 ). Here, we discuss the properties of Q. We observe that 1 ◦ 1 = 1 . The elliptic minimal projector π , introduced in the previous subsection, is a Fourier integral operator with canonical relation 1 , (31)

L −→ L ↓ ↓ X −→ X

(32)

The wavefront set of Q is contained in 0 ∪ 1 . Q is a pseudodifferential operator on 1 0 \( 0 ∩ 1 ) and its principal symbol there is given by σ0 = 2 up 2 p1 + · · · + pn−1−c to Maslov factors and half densities. Q is a Fourier integral operator on 1 \( 0 ∩ 1 ). Its principal symbol, σ1 , solves the transport equation n−1−c

iHpi − ci

2

σ1 = σπ

i=1

on 1 \( 0 ∩ 1 ), where σπ denotes the symbol of π . (We return to the evaluation of σπ in Sect. 6.) The expression between parentheses is an elliptic differential operator of


13

order 2 on each fiber of L. The equation is Laplace’s equation in every leaf of the foliation generated by the commuting vector fields Hpi , i = 1, . . . , n − 1 − c. The principal symbol σ1 has a conormal singularity at 0 ∩ 1 , expressible by an appropriate Fourier transform of the singularity of σ0 , see [15, (5.14)].

4.3. Data continuation. We apply the results of the previous subsections to the problem of source-receiver continuation of seismic data: Seismic data are commonly measured on an open subset of the manifold of all possible observations. Continuation of these data from the open subset to the full acquisition manifold is desired for various data processing procedures, including imaging – in the seismic literature this continuation is referred to as the ‘forward extrapolation’ step within data regularization [9], or ‘data healing’. Theorem 3. Suppose u is a distribution belonging to the range of the scattering operator F . Let χ = χ (y, Dy ) be a pseudodifferential operator of order 0 that acts as a cutoff in phase space T ∗ Y \0. Assume that we observe u0 = χ u in accordance with the constraints of the acquisition geometry. Suppose χ is elliptic on a leaf of the foliation of L, then WF(π u0 ) intersected with this leaf is equal to WF(u) intersected with the same leaf. In this case, π heals the data on this leaf. Proof. We observe that u = πv for some v. Then u0 = (χ π) v.

(33)

Because π 2 u = u, it is natural to investigate πu0 , i.e. πu0 = (πχ π) v.

(34)

If χ is elliptic on a leaf of the foliation of L, then (π − π χ π ) v = 0, or u − π u0 = 0, microlocally on this leaf. This implies the statement in the theorem. We implement π by making use of the following observation. In view of the Bolker condition, Assumption 4, the composition F ∗ F is an elliptic pseudodifferential operator of order n − 1. Let denote the parametrix for F ∗ F . The operator F F ∗ belongs to Guillemin and Sternberg’s algebra RL [14] of Fourier integral operators with canonical relation 1 [6]. Clearly, (F F ∗ )2 = F F ∗ microlocally, while I − F ∗ F = I − F ∗ F is the orthogonal projection onto the kernel of F ∗ F . Indeed, F F ∗ is precisely an elliptic minimal projector [14, Proof of Thm. 8.3] of the type introduced in Sect. 4. The symbol of this operator follows by the standard composition calculus. Following the composition of F with F ∗ in F F ∗ , we represent the canonical relation 1 as the composition of canonical relations with ∗ . Remark 3. The transformation to zero offset (TZO) of seismic data, which is derived from Dip MoveOut, can be expressed in the form R0 π , where R0 is the restriction of distributions on Y to an acquisition manifold with coinciding sources and receivers: In this case, y = (y , y ) with y = ( 21 (s + r), t) and y = 21 (r − s) if (s, r, t) are the original local coordinates on Y . Assumption 2, subject to this substitution, guarantees that the composition, R0 π, is again a Fourier integral operator.

14


5. Goldin’s Equation Revisited In context of the simplest seismic scattering theory, in a background that essentially is constant, the following simplications are made. To begin with, the source, s, and receiver, r, points in Y are assumed to be contained in a flat surface, Rn−1 , while e is initially replaced by half source-receiver offset h = 21 (r − s) ∈ Rn−1 . Thus y is replaced by (s, r, t); we write η = (σ, ρ, τ ). Essentially, we assume that the rays between reflector and acquisition surface are straight, see Fig. 1. We repeat the NMO correction, 4h2 κ : (s, r, t) → (z, tn , h), z = 21 (r + s), h = 21 (r − s), tn = t 2 − 2 , v of the introduction. (The subscript n refers in this section to normal moveout.) Here, v could be thought of as the so-called NMO velocity, which can be introduced for ‘pure mode’ scattering (i.e. M = N ), even in the anisotropic media under consideration here [12]; Goldin, however, restricts his analysis to an isotropic medium and compressional waves. We will reproduce Goldin’s result here in the context of our analysis subject to the substitutions n = 2 and c = 0 (and m = 1). NMO correction applied to the data yields (κ −1 )∗ d. Including a so-called geometrical spreading correction, a multiplication by time t, then leads to the map d(s, r, t) → ((κ −1 )∗ (t d))(z, tn , h)

(35)

that replaces (H −1 d)(x, e); the point x has attained coordinates (z, tn ). The outcome is of the form r(z, tn , h) = Rn (z, h) cn (tn − Tn (z, h)).

Fig. 1. Geometry underlying the annihilator symbol for constant coefficients

(36)


15

Equation (36) replaces (22). The reflection time, T (s, r, x) maps under κ according to 4h2 T (s, r, x) → Tn (z, h) = (T (z − h, z + h, x))2 − 2 . v We observe that, in the simplication considered, Rn is independent of tn , while cn is not only a function of (z, tn ) but also of h. Hence, a simple derivative of r with respect to h, motivated by (26), would not yield a vanishing outcome up to leading order. Instead, it is possible to construct a candidate operator P1 , acting on the data, directly. In Fig. 1 we introduce angles α and γ ; in fact, (x, α, γ ) can be identified with (x, ν, e). Using simple trigonometric identities (including the law of sines) and the geometry in Fig. 1 α (observing that the total length of the reflected ray is vt = (r − s) cos sin γ with 2γ denoting the scattering angle and α denoting the incidence angle of the zero-offset ray at the surface), it follows that τ

(r − s)2 2 −1 (σ − ρ)−2 (r − s) t −τ σρ = 0 (37) p1 (s, r, t, σ, ρ, τ ) = t + v2 v2 defines the points in L (Remark 2) in the simplification under consideration; p1 is v dependent. Applying the coordinate transformation implied by κ to this symbol, and multiplying the result by frequency τ , yields p (z, tn , h, ζ, τn , ε) = (τp1 )(κ(s, r, t), ((κ )−1 )t (σ, ρ, τ ))

(38)

p (z, tn , h, ζ, τn , ε) = −tn τn ε + h (ζ 2 − ε 2 ),

(39)

or

which defines the principal symbol of an operator P ; we observe that p is v independent. We recover Goldin’s equation, P (z, tn , h, Dz , Dtn , Dh )r = 0,

2 ∂2 ∂2 ∂ P (z, tn , h, Dz , Dtn , Dh ) = tn − 2 , −h ∂tn ∂h ∂z2 ∂h

(40)

which is valid up to highest order. It would be valid up to the next order if we had not applied the geometrical spreading correction in (35). Accounting for this correction leads to a subprincipal symbol contribution: 2 ∂ ∂2 ∂ ∂2 P (z, tn , h, Dz , Dtn , Dh ) := tn −h . − − 2 2 ∂tn ∂h ∂z ∂h ∂h Through the coordinate transformation implied by κ, we obtain a subprincipal symbol contribution to the operator with principal symbol p1 (cf. (37)). Note that the operator P is of second order unlike the operator annihilating r in (26) which is of first order. However, in (38) we introduced a multiplication by τ , raising the order by one. A first-order operator derived from P in (40) follows to be P (z, tn , h, Dz , Dtn , Dh ) =

∂ ∂h

−1

h tn

∂2 ∂2 − ∂h2 ∂z2

+

∂ . ∂tn

16


We write down the Hamiltonian system describing the flowout as in (31); there is only one such system since n − c − 1 = 1. We use the first-order symbol, p (z, tn , h, ζ, τn , ε) = τn − εthn (ζ 2 − ε 2 ) (we omitted a factor i), so that 2hζ ∂z =− ∂e εtn

,

∂ζ = 0, ∂e

∂tn =1 ∂e

,

∂τn h = − 2 (ζ 2 − ε 2 ), ∂e εtn

∂h 2h ζ2 h , =− 2 + ∂e ε tn tn

(41)

∂ε 1 (ζ 2 − ε 2 ). = ∂e εtn

We set e = tn , and eliminate τn . To this end, we introduce the slowness vectors ζ and ε according to ζ = τn ζ and ε = τn ε. Substituting τn = εthn (ζ 2 − ε 2 ) (using that p = 0), and the equation for

∂τn ∂e ,

then yields the system

2h ζ ∂z =− ∂tn ε tn

,

∂ ζ = ∂tn

ζ , tn

∂h h 1 =− + , ∂tn ε tn

∂ ε = ∂tn

ζ2 . ε tn

(42)

We note that tn and h are directly related to one another. Indeed, let the zero-offset reflecton time be given by T0 = T (z0 , z0 , x) = Tn (z0 , 0). Then, for given z, h Tn = 2 . T0 (h − (z − z0 )2 )1/2 Along bicharacteristics, h v2 =− ε tn 4 sin2 α

(43)

is invariant (i.e. its derivative with respect to e is zero). We can convert tn to half scattering angle γ – keeping (x, α) fixed – according to the relation cos α cos γ Tn = , T0 (cos2 α − sin2 γ )1/2

T0 =

2V . v

(44)

We discuss in as much the kernel of P determines the range of F under the simplication (‘straight rays’) under consideration. The range is described by wavefields of the form −1 cos2 γ + V C ¯ r(x, γ ) δ(t − T ), T = T (s, r, x), (45) d(s, r, t) = vt cos γ obtained after preprocessing d for time signature (a 2.5D correction) and source or receiver radiation characteristics, applying an appropriate pseudodifferential operator. In (45), V = 21 vT (z, z, x) denotes the length of the zero-offset ray, and C = x 2 (x1 ) cos3 α denotes the curvature of the reflector, see Fig. 1. We have assumed that the reflecting


17

interface can be described by the graph (x1 , x 2 (x1 )). (If the interface is the zero level ∂z2 set of z2 = z2 (x1 , x2 ) then we assume that ∂x = 0.) Equation (45) is the outcome of 2 a stationary phase calculation of the scattered field in the Kirchhoff approximation. We ¯ r, t) and obtain set r(x, γ ) ≡ 1 and apply (κ −1 )∗ to d(s, r¯ (z, tn , h) = An (z, h) δ(tn − Tn ), Tn = Tn (z, h), −1 cos2 γ + V C t , An = vt cos γ tn

(46)

because | dtdtn | = ttn . Expression (46) is indeed of the form (36). To verify whether the wavefields in (45), via (46), coincide with functions in the kern nel of P , up to leading order, we first notice that by derivation p (z, Tn , h, −iτn ∂T ∂z , τn , n −iτn ∂T ∂h ) = 0. Secondly, we consider the transport equation derived from (40), which is given by Tn − 2h

∂Tn ∂h

∂Tn ∂An ∂An + 2h + hAn ∂h ∂z ∂z

∂ 2 Tn ∂ 2 Tn − 2 ∂z ∂h2

= 0.

n ∂Tn The velocity vector associated with a ray or characteristic is given by ( ∂T ∂z , ∂h ). Thus, along a characteristic, the transport equation becomes

−

1 dAn ∂Tn −1 ∂ 2 Tn ∂ 2 Tn = 0, + h Tn − An dTn ∂h ∂z2 ∂h2

(47)

where we made use of (42). We change variables according to (44), with 1 dTn sin2 α sin γ . =− Tn dγ (cos2 α − sin2 γ ) cos γ Furthermore, using the ray geometry, we find the identity Tn

∂ 2 Tn ∂ 2 Tn − ∂z2 ∂h2

∂ 2T cos2 γ sin2 α + V C cos2 γ =4 T = 4 + . ∂s∂r v2 v 2 cos2 γ + V C

(48)

Substituting identity (48) and invariant (43) into (47), applying the change of variables (44), leads to the equation, −

1 dAn + An dγ

1 1 − (cos2 α − sin2 γ ) cos2 γ + V C

sin γ cos γ = 0.

(49)

This equation can be directly integrated to yield solutions for An of the type (46). We conclude that the kernel of P generates wavefields of the type (45) which comprise the range of the scattering operator subject to processing for time signature and setting r(x, γ ) ≡ 1.

18


6. Numerical Example The minimal elliptic projector π is a Fourier integral operator and is directly implementable and applicable to data. This is the subject of this section. Indeed, given a smooth background model, we can construct a minimal elliptic projector for data continuation by operator composition, F F ∗ . On the other hand, however, in Sect. 4 we showed that, given the annihilators of the data (in practice, just their principal parts), we can construct the global parametrix Q, from which the elliptic minimal projector follows. This procedure is related to what Guillemin and Sternberg call relative geometrical quantization. We include an example to confirm the computability of our result. The algorithm used is designed and explained in [20]. In our example, n = 2 and χ is replaced by a smooth cutoff ψY ; the cutoff restricts the data to the set {(s, r, t) | s, r ∈ Rn−1 , r −s > h0 , t ∈ (0, T )}. The goal is to continue the data to an acquisition manifold with the constraint

r − s > h0 removed. Elastic-wave data were simulated over a model illustrated in Fig. 2 (top). The P-wave velocities are shown in grey scale; a low-velocity Gaussian lens was inserted (white-to-grey). The continuation is illustrated for the P-wave constituents even though the simulated data contained S waves as well. By selecting the vertical

Fig. 2. A numerical example of data continuation: The top figure is the isotropic P-wave velocity model used in the reflection data simulation, containing a Gaussian (low velocity) lens (in white). The bottom two figures both show a shot record (receiver location versus time) with source location at the black vertical line in top figure; the left record shows the outcome of continuation (the data in between the two vertical lines was missing) while the right record shows the original simulated data


19

displacement component, most energy in the wavefield can be attributed to P waves. For one value of s the synthetic data as a function of r and t are shown in Fig. 2 (bottom, right). We set T = 3s and h0 = 500m. The input to the continuation (u0 ) were the data with r values in between the black vertical lines removed Fig. 2 (bottom, left). The result of the continuation is plotted in between the black vertical lines of the same figure and should be compared with Fig. 2 (bottom, right). References 1. Beylkin, G.: The inversion problem and applications of the generalized Radon transform. Comm. Pure Appl. Math. XXXVII, 579–599 (1984) 2. Biondi, B., Fomel, S., Chemingui, N.: Azimuth moveout for 3-D prestack imaging. Geophysics 63, 574–588 (1998) 3. Bolondi, G., Loinger, E., Rocca, F.: Offset continuation of seismic sections. Geoph. Prosp. 30, 813–828 (1982) 4. De Hoop, M.V., Bleistein, N.: Generalized radon transform inversions for reflectivity in anisotropic elastic media. Inverse Problems 13, 669–690 (1997) 5. De Hoop, M.V., Brandsberg-Dahl, S.: Maslov asymptotic extension of generalized Radon transform inversion in anisotropic elastic media: A least-squares approach. Inverse Problems 16, 519–562 (2000) 6. De Hoop, M.V., Malcolm, A.E., Le Rousseau, J.H.: Seismic wavefield ‘continuation’ in the single scattering approximation: A framework for Dip and Azimuth MoveOut. Can. Appl. Math. Q. 10, 199–238 (2002) 7. Deregowski, S.G., Rocca, F.: Geometrical optics and wave theory of constant offset sections in layered media. Geoph. Prosp. 29, 374–406 (1981) 8. Duistermaat, J.J.: Fourier integral operators. Boston: Birkhäuser, 1996 9. Fomel, S.: Theory of differential offset continuation. Geophysics 68, 718–732 (2003) 10. Gel’fand, I.M., Graev, M.I.: Complexes of straight lines in the space Cn . Funct. Anal. Appl. 2, 39–52 (1968) 11. Goldin, S.: Superposition and continuation of transformations used in seismic migration. Russ. Geol. and Geophys. 35, 131–145 (1994) 12. Grechka, V., Tsvankin, I., Cohen, J.K.: Generalized Dix equation and analytic treatment of normalmoveout velocity for anisotropic media. Geoph. Prosp. 47, 117–148 (1999) 13. Guillemin, V.: In: Pseudodifferential operators and applications (Notre Dame, Ind., 1984), Chapter “On some results of Gel’fand in integral geometry”, Providence, RI: Amer. Math. Soc., 1985, pp. 149–155 14. Guillemin, V., Sternberg, S.: Some problems in integral geometry and some related problems in microlocal analysis. Amer. J. of Math. 101, 915–955 (1979) 15. Guillemin, V., Uhlmann, G.: Oscillatory integrals with singular symbols. Duke Math. J. 48, 251–267 (1981) 16. Hansen, S.: Solution of a hyperbolic inverse problem by linearization. Commun. Par. Differ. Eqs. 16, 291–309 (1991) 17. Hörmander, L.: The analysis of linear partial differential operators. Volume IV. Berlin: SpringerVerlag, 1985 18. Hörmander, L.: The analysis of linear partial differential operators. Volume III. Berlin: SpringerVerlag, 1985 19. John, F.: The ultrahyperbolic differential equation with four independent variables. Duke Math. J. 4, 300–322 (1938) 20. Malcolm, A.E., De Hoop, M.V., Le Rousseau, J.H.: The applicability of DMO/AMO in the presence of caustics. Geophysics 70, 51 (2005) 21. Maslov, V.P., Fedoriuk, M.V.: Semi-classical approximation in quantum mechanics. Dordrecht: Reidel Publishing Company, 1981 22. Patch, S.K.: Computation of unmeasured third-generation VCT views from measured views. IEEE Trans. Med. Imaging 21, 801–813 (2002) 23. Stolk, C.C., De Hoop, M.V.: Microlocal analysis of seismic inverse scattering in anisotropic, elastic media. Comm. Pure Appl. Math. 55, 261–301 (2002) 24. Taylor, M.E.: Reflection of singularities of solutions to systems of differential equations. Comm. Pure Appl. Math. 28, 457–478 (1975) Communicated by P. Constantin


Communications in


Mott Law as Lower Bound for a Random Walk in a Random Environment A. Faggionato1 , H. Schulz-Baldes2 , D. Spehner3 1 2 3

Weierstrass Institut für Angewandte Analysis und Stochastic, 10117 Berlin, Germany Institut für Mathematik, Technische Universität Berlin, 10623 Berlin, Germany Fachbereich Physik, Universität Duisburg-Essen, 45117 Essen, Germany

Received: 21 July 2004 / Accepted: 16 September 2005 Published online: 24 January 2006 – © Springer-Verlag 2006

Abstract: We consider a random walk on the support of an ergodic stationary simple point process on Rd , d ≥ 2, which satisfies a mixing condition w.r.t. the translations or has a strictly positive density uniformly on large enough cubes. Furthermore the point process is furnished with independent random bounded energy marks. The transition rates of the random walk decay exponentially in the jump distances and depend on the energies through a factor of the Boltzmann-type. This is an effective model for the phonon-induced hopping of electrons in disordered solids within the regime of strong Anderson localization. We show that the rescaled random walk converges to a Brownian motion whose diffusion coefficient is bounded below by Mott’s law for the variable range hopping conductivity at zero frequency. The proof of the lower bound involves estimates for the supercritical regime of an associated site percolation problem. 1. Introduction 1.1. Main Result. Let us directly describe the model and the main results of this work, deferring a discussion of the underlying physics to the next section. Suppose given an infinite countable set of random points {xj } ⊂ Rd distributed according to some ergodic stationary simple point process. One can identify this set with the simple counting measure ξˆ = j δxj having {xj } as its support, and then write x ∈ ξˆ if x ∈ {xj }. The σ -algebra B(Nˆ ) on the space Nˆ of counting measures on Rd is generated by the family of subsets {ξˆ ∈ Nˆ : ξˆ (B) = n}, where B ⊂ Rd is Borel and n ∈ N. The distribution Pˆ of the point process is a probability on the measure space (Nˆ , B(Nˆ )). It is stationary and ergodic w.r.t. the translations x → x + y of Rd . In the sequel, we need to impose boundedness of some κth moment defined by ρκ := EPˆ ξˆ (C1 )κ , (1) ˆ Then ρ = ρ1 is the so–called where C1 = [− 21 , 21 ]d and EPˆ is the expectation w.r.t. P. intensity of the process.

22

A. Faggionato, H. Schulz-Baldes, D. Spehner

To each xj is associated a random energy mark Exj ∈ [−1, 1]. These marks are drawn independently and identically according to a probability measure ν. Again, {(xj , Exj )} is naturally identified with an element ξ of the space N of counting measures on Rd × [−1, 1], and the distribution P of the marked process is a measure on (N , B(N )) (with B(N ) defined similarly to B(Nˆ )). The distribution P is said to be the ν–randomization of Pˆ [Kal]. It is stationary and ergodic w.r.t. Rd –translations. In order to assure that {xj } contains the origin, we consider the measurable subset N0 = {ξ ∈ N : ξ({0} × [−1, 1]) = 1} furnished with the σ -algebra B(N0 ) = {A ∩ N0 : A ∈ B(N )}. The random environment is given by a configuration ξ ∈ N0 randomly chosen along the Palm distribution P0 associated to P. Roughly, one can think of P0 as the probability on (N0 , B(N0 )) obtained by conditioning P to the event N0 (see Sect. 2). Note that almost each environment is a simple counting measure, and therefore it can be identified with its support as we will do in what follows. For a fixed environment ξ ≡ {(xj , Exj )} ∈ N0 let us consider a continuous-time random walk over the points {xj } starting at the origin x = 0 with transition rates from x ∈ ξˆ to y ∈ ξˆ given by cx,y (ξ ) := exp − |x − y| − β(|Ex − Ey | + |Ex | + |Ey |) , x = y , (2) where β > 0 is the inverse temperature. More precisely, let ξ = D([0, ∞), supp(ξˆ )) be the space of right-continuous paths on the support of ξˆ having left limits, endowed ξ ξ with the Skorohod topology [Bil]. Let us write (Xt )t≥0 for a generic element of ξ . If P0 denotes the distribution on (ξ , B(ξ )) of the above random walk starting at the origin, ξ ξ ξ ξ then the set of stationary transition probabilities pt (y|x) := P0 (Xs+t = y|Xs = x), x, y ∈ ξˆ , t ≥ 0, s > 0 satisfy the following conditions for small values of t [Bre]: ξ

(C1) pt (y|x) = cx,y (ξ ) t + o(t) if x = y; ξ (C2) pt (x|x) = 1−λx (ξ ) t +o(t) with λx (ξ ) := y∈ξˆ cx,y (ξ ), where cx,x (ξ ) := 0. It is verified in Appendix A that, provided that ρ2 < ∞, no explosions occur and thus the random walk is well-defined for P0 –almost all ξ . Our main interest concerns the long time asymptotics of the random walk and the diffusion matrix D defined by 1 ξ (a · Da) = lim a ∈ Rd , EP0 EPξ (Xt · a)2 , (3) t→∞ t 0 where (a · b) denotes the scalar product of the vectors a and b in Rd . The main results of the work are (i) the existence of the limit (3) in any dimension d ≥ 1 as well as the convergence of the (diffusively rescaled) random walk to a Brownian motion with finite covariance matrix D ≥ 0; (ii) a quantitative lower bound on D in dimension d ≥ 2 under given assumptions on the energy distribution ν and either one of the following two technical hypotheses. Let denote the Lebesgue measure and CN = [−N/2, N/2]d . Given A ⊂ Rd , let FA be the σ –subalgebra in B(Nˆ ) generated by the random variables ξˆ (B) with B ⊂ A and B ∈ B(Rd ). (H1) Pˆ admits a lower bound ρ > 0 on the point density: ξˆ (CN ) ≥ ρ (CN ), with ρ and N0 independent on ξˆ .

ˆ ∀ N ≥ N0 , P-a.s. ,

(4)

Mott Law as Lower Bound for a Random Walk in a Random Environment

23

(H2) Pˆ satisfies the following mixing condition: there exists a function h : R+ → R+ with h(r) ≤ c(1 + r 2d+7+δ )−1 for some c, δ > 0 such that for any r2 ≥ r1 > 1, ˆ ˆ ˆ ∀ A ∈ FCr1 , P-a.s. (5) ≤ r1d r2d−1 h(r2 − r1 ) , P(A|FRd \Cr ) − P(A) 2

We feel that Hypotheses (H1) and (H2) cover nearly all interesting examples see however Example 2 below. The uniform lower bound (H1) holds in the case of random and quasiperiodic tilings and, more generally, the so-called Delone sets [BHZ]. The type of mixing condition (H2) is inspired by decorrelation estimates holding for Gibbs measures of spin systems in a high temperature phase [Mar]. It is satisfied for a stationary Poisson point process as well as for point processes with finite range correlations. Due ˆ (H2) implies that Pˆ is a mixing, and, in particular, ergodic point to the stationarity of P, process (see [DV, Chap. 10]). We can now state more precisely the above-mentioned results. Theorem 1. Let Pˆ be the distribution of an ergodic stationary simple point process on Rd , let P be the distribution of its ν–randomization with a probability measure ν on [−1, 1], and let P0 be the Palm distribution associated to P. Assume that ρ12 < ∞ and that δ(xj ,Ej ) ⇒ ξ = Sx ξ := δ(xj −x,Ej ) ∀ x ∈ Rd \ {0}, P a.s. (6) ξ= j

j

Condition (6) is automatically satisfied if ν is not a Dirac measure. Then: ξ

(i) The limit in (3) exists and the rescaled process Y ξ,ε = (εXtε−2 )t≥0 defined on ξ

(ξ , P0 ) converges weakly in P0 -probability as ε → 0 to a Brownian motion W D with covariance matrix D. Namely, for any bounded continuous function F on the path space D([0, ∞), Rd ) endowed with the Skorohod topology, → E F WD in P0 -probability . EPξ F Y ξ,ε 0

(ii) Suppose d ≥ 2 and let either (H1) or (H2) be satisfied. Furthermore, suppose that there are some positive constants α, c0 such that, for any 0 < E ≤ 1,

Then

ν([−E, E]) ≥ c0 E 1+α .

(7)

d(α+1) α+1 D ≥ c1 β − α+1+d exp −c2 β α+1+d 1d ,

(8)

where 1d is the d × d identity matrix and c1 and c2 are some positive β-independent constants. The important factor in the lower bound (8) is the exponential factor and not the power law in front of it (on which we comment below though). Based on the following heuristics due to Mott [Mot, SE], we expect that the expression in the exponential in (8) captures the good asymptotic behavior of ln D in the low temperature limit β ↑ ∞ if ν([−E, E]) ∼ c0 E 1+α as E ↓ 0. Indeed, as β becomes larger, the rates (2) fluctuate widely with (x, y) because of the exponential energy factor. The low temperature limit effectively selects only jumps between points with energies in a small interval [−E(β), E(β)] shrinking to zero as β ↑ ∞. Assuming that D is determined by those

24


jumps with the largest rate, one obtains directly the characteristic exponential factor on the r.h.s. of (8) by maximizing these rates for a fixed temperature under the constraint that the mean density of points xj with energies in [−E(β), E(β)] is equal to ρ ν([−E(β), E(β)]) ∼ c0 ρE(β)1+α . One speaks of variable range hopping since the characteristic mean distance |x − y| between sites with optimal jump rates varies heavily with the temperature. A crucial (and physically reasonable, as discussed below) element of this argument is the independence of the energies Ex . The selection of the points {xj } with energies in the window [−E(β), E(β)] then corresponds mathematically to a p-thinning with p = ν([−E(β), E(β)]). It is a well-known fact (see e.g. [Kal, Theorem 16.19]) that an adequate rescaling of the p-thinning of a stationary point process converges in the limit p ↓ 0 (corresponding here to β ↑ ∞) to a stationary Poisson point process (PPP). Hence one might call the stationary PPP the normal form of a model leading Mott’s law, namely the exponential factor on the r.h.s. of (8), and we believe that proving the upper bound corresponding to (8) should therefore be most simple for the PPP. In dimension d = 1, a different behavior of D is expected [LB] and this will not be considered here. Note that statement (i) does not necessarily imply that the motion of the particle is diffusive at large time, since it could happen that D = 0. (α+1)(2−d)

The preexponential factor in (8) can be improved to β (1+α+d) by means of formal scaling arguments on the formulas in Sects. 4 to 6. As we are not sure that this is optimal and we do not control the constant c2 in (8) anyway, we choose not to develop this improvement in detail.

1.2. Physical discussion. Our main motivation for studying the above model comes from its importance for phonon-assisted hopping conduction [SE] in disordered solids in which the Fermi level (set equal to 0 above) lies in a region of strong Anderson localization. This means that the electron Hamiltonian has exponentially localized quantum eigenstates with localization centers xj if the corresponding energies Exj are close to the Fermi level. The DC conductivity of such materials would vanish if it were not for the lattice vibrations (phonons) at nonzero temperature. They induce transitions between the localized eigenstates, the rate of which can be calculated from first principle by means of the Fermi golden rule [MA, SE]. In the variable range hopping regime at low temperature, the Markov and adiabatic (or rotating wave) approximations can be used to treat quantum mechanically the electron-phonon coupling [Spe]. Coherences between electronic eigenstates with different energies decay very rapidly under the resulting dissipative electronic dynamics and one can show that the hopping DC conductivity of the disordered solid coincides with the conductivity associated with a Markov jump process on the set of localization centers {xj }, hence justifying the use of a model of classical mechanics [BRSW]. Because Pauli blocking due to Fermi statistics of the electrons has to be taken into account, this leads to a rather complicated exclusion process (e.g. [Qua, FM]). If, however, the blocking is treated in an effective medium (or mean field) approximation, one obtains a family of independent random walks with rates which are given by (2) in the limit β ↑ ∞ [MA, AHL]. Let us discuss the remaining aspects of the model. The stationarity of the underlying simple point process {xj } simply reflects that the material is homogeneous, while the independence of the energy marks is compatible with Poisson level statistics, which is a general rough indicator for the localization regime and has been proven to hold for an Anderson model [Min]. The exponent α allows to model a possible Coulomb pseudogap in the density of states [SE].


25

Having in mind the Einstein relation between the conductivity and the diffusion coefficient (which can be stated as a theorem for a number of models [Spo]), the lower bound (8) gives a lower bound on the hopping DC condutivity. In the above materials, the DC conductivity shows experimentally Mott’s law, namely a low-temperature behavior which is well approximated by the exponential factor in the r.h.s. of (8) with α = 0, as predicted by Mott [Mot] based on the optimization argument discussed above. In certain materials having a Coulomb pseudogap in the density of state, Mott’s law with α = d −1 is observed, as predicted by Efros and Shklovskii [EF]. A first convincing justification of Mott’s argument was given by Ambegoakar, Halperin and Langer [AHL], who first reduced the hopping model to a related random resistor network, in a manner similar to the work of Miller and Abrahams [MA], and then pointed out that the constant c2 in (8) can be estimated using percolation theory [SE]. Our proof of the lower bound (8) is inspired by this work. Let us also mention that the low frequency AC conductivity (response to an oscillating electric field) in disordered solids has recently been studied within a quantum-mechanical one-body approximation in [KLP]. Here the energy necessary for a jump between localized states comes from a resonance at the frequency of the external electric field rather than a phonon. It leads to another well-known formula for the conductivity which is also due to Mott. 1.3. Overview. Let us develop the main ideas of the proof of Theorem 1, leaving precise statements and their proofs to the following sections. The model described above is a random walk in a random environment. A main tool used in this work is the contribution of De Masi, Ferrari, Goldstein and Wick [DFGW] which is based on prior work by Kipnis and Varadhan [KV]. They construct a new Markov process, called the environment viewed from the particle, which allows to translate the homogeneity of the medium ξ into properties of the random walk. In Sect. 3, we argue that Xt has finite moments ξ w.r.t. P0 (dξ )P0 (Proposition 1) and study the generator of the process environment viewed from the particle when the initial environment is chosen according to the Palm distribution P0 (Propositions 2 and 3), thus allowing to apply the general Theorem 2.2 of [DFGW] to deduce the existence of the limit (3). The convergence to a Brownian motion stated in Theorem 1 also follows, but this could have been obtained (avoiding an analysis of the infinitesimal generator) by applying Theorem 17.1 of [Bil] and Theorem 2.1 of [DFGW]. The results of [DFGW] also lead to a variational formula for the diffusion matrix D (Theorem 2 below). The main virtue of this formula is that it allows to bound D from below through bounds on the transition rates. The first step in proving Theorem 1(ii) is to define a new random walk with transition rates bounded above by the rates (2). This is done in Sect. 4 in the following way. For a fixed configuration ξ ∈ N0 of the environment, consider the set {xjc } = {xj : |Exj | ≤ Ec } of all random points having energies inside a given energy window [−Ec , Ec ] with 0 < Ec ≤ 1. The distribution Pˆ c of these points is obtained from P by a δc -thinning with δc = ν([−Ec , Ec ]). Given a cut-off distance rc > 0, consider the random walk on supp(ξˆ ) with the transition rates cˆx,y (ξ ) = χ (|x −y| ≤ rc )χ (|Ex | ≤ Ec )χ (|Ey | ≤ Ec ), where χ is the characteristic function. Since we want this random walk to have a strictly positive diffusion coefficient in the limit Ec → 0, one must choose rc such that the mean number of points xj with energies in [−Ec , Ec ] inside a ball of radius rc is larger than an Ec -independent constant c3 > 0. This mean number is equal to c4 δc rcd and is larger than c5 Ec1+α rcd by assumption (7), where c4 and c5 are constants depending on ρ and d −(1+α)/d only. Hence rc = c6 Ec . It is shown in Proposition 5 that the diffusion matrix of

26


the new random walk is equal to δc D(rc , Ec ), where D(rc , Ec ) is the diffusion matrix of a random walk on {xjc } with energy-independent transition rates χ (|x − y| ≤ rc ). By the monotonicity of D in the jump rates and since cx,y (ξ ) ≥ exp(−rc − 4βEc ) cˆx,y (ξ ), one gets using also assumption (7) and the constant c0 therein: D ≥ c0 Ec1+α e−rc −4βEc D(rc , Ec ) .

(9)

In Sect. 5, a lower bound on D(rc , Ec ) is obtained by considering periodic approximants (in the limit of large periods) as in [DFGW]. The diffusion coefficient of these approximants can be computed as the resistance of a random resistor network. The resistance of the random resistor network is bounded by invoking estimates from percolation theory in Sect. 6, hence showing that, if rc is large enough, D(rc , Ec ) > c7 1d , where −(1+α)/d c7 > 0 is independent on Ec , β. Recalling that rc = c6 Ec , an optimization w.r.t. d Ec of the right member of (9) then yields Ec = c8 β − 1+α+d and thus the lower bound (8). Let us note that this optimization is the same as in the Mott argument discussed above and that Ec ↓ 0 and rc ↑ ∞ as β ↑ ∞. Moreover, our optimized lower bound results from a critical resistor network roughly approximating the one appearing in [AHL]. The paper is organized as follows. In Sect. 2 we recall some definitions and results about point processes and state some technical results needed later on. The statements (i) and (ii) of Theorem 1 are proven in Sect. 3 and in Sects. 4 to 6, respectively. In Appendix A we show that the continuous-time random walk in the random environment is well defined by verifying the absence of explosion phenomena. Appendix B contains some technical proofs about the Palm measure. Appendix C is devoted to the proof of Proposition 1. 2. The Random Environment In this section, we recall some properties of point processes (for more details, see [DV, FKAS, MKM, Kal, Tho]). In the sequel, given a topological space X, B(X) will denote the σ -algebra of Borel subsets of X. Given a set A, |A| will denote its cardinality. Moreover, given a probability measure µ, we write Eµ for the corresponding expectation. 2.1. Stationary simple marked point processes. Given a bounded complete separable metric space K, consider the space N := N (Rd × K) of all counting measures ξ on Rd × K, i.e. integer-valued measures such that ξ(B × K) < ∞ for any bounded set B ∈ B(Rd ). One can show that ξ ∈ N if and only if ξ = j δ(xj ,kj ) where δ is the Dirac measure and {(xj , kj )} is a countable family of (not necessarily distinct) points in Rd × K with at most finitely many points in any bounded set. Then kj is called the mark at xj . Given ξ ∈ N , we write ξˆ ∈ N (Rd ) for the counting measure on Rd defined by ξˆ (B) = ξ(B × K) for any B ∈ B(Rd ). Given x ∈ Rd , we write x ∈ ξˆ whenever x ∈ supp(ξˆ ). If ξˆ ({x}) ≤ 1 for any x ∈ Rd , we say that ξ ∈ N is simple and write kxj := kj for any xj ∈ ξˆ . A metric on N can be defined in the following way [MKM, Sect. 1.15]. Fix an element d k ∗ ∈ K. Denote by Br (x, k) and Br the open balls in R × K of radius r > 0 centred ∗ on (x, k) and on (0, k ), respectively. Let ξ = i∈I δ(xi ,ki ) and ξ = j ∈J δ(xj ,kj ) be elements of N , where I , J are countable sets. Then ξ and ξ are close to each other if any point (xi , ki ) contained in Bn is close to a point (xj , kj ) for arbitrary large n, up


27

to “boundary effects”. More precisely, given a positive integer n, let dn (ξ, ξ ) be the infimum over all ε > 0 such that there is a one-to-one map f from a (possibly empty) subset D of I into a subset of J with the properties: (i) supp(ξ ) ∩ Bn−ε ⊂ {(xi , ki ) : i ∈ D}; (ii) supp(ξ ) ∩ Bn−ε ⊂ {(xj , kj ) : j ∈ f (D)}; (iii) (xf (i) , kf (i) ) ∈ Bε (xi , ki ) for i ∈ D. −n

One can show that dN (ξ, ξ ) = ∞ n=1 2 dn (ξ, ξ ) is a bounded metric on N and for this metric N is complete and separable. Moreover, the sets {ξ ∈ N : ξ(B) = n}, B ∈ B(Rd × K), n ∈ N, generate theBorel σ -algebra B(N ) and dN generates the coarsest topology such that ξ ∈ N → ξ(dx, dk) f (x, k) is continuous for any continuous function f ≥ 0 on Rd × K with bounded support. Finally, by choosing different reference points k ∗ one obtains equivalent metrics. A marked point process on Rd with marks in K is then a measurable map from a probability space into N . We denote by P its distribution (a probability measure on (N , B(N ))). We say that the process is simple if P-almost all ξ ∈ N are simple. The translations on Rd extend naturally to Rd × K by Sx : (y, k) → (x + y, k). This induces an action S of the translation group Rd on N by (Sx ξ )(B) = ξ(Sx B), where B ∈ B(Rd × K) and x ∈ Rd . For simple counting measures, δ(y−x,ky ) . Sx ξ = y∈ξˆ

A marked point process is said to be stationary if P(A) = P(Sx A) for all x ∈ Rd , A ∈ B(N ), and (space) ergodic if the σ -algebra of translation invariant sets is trivial, i.e., if A ∈ B(N ) satisfies Sx A = A for all x ∈ Rd then P(A) ∈ {0, 1}. Due to [DV, Prop. 10.1.IV], if P is stationary and gives no weight to the trivial measure without any point (which will be assumed here), then P ξ ∈ N : supp(ξˆ ) = ∞ = 1 . (10) The marked point processes studied in this work are obtained by the procedure of randomization, which we recall now together with the related notion of thinning (see [Kal]). ˆ be a stationary simple point process (SSPP) on Rd , ν be a probability measure Let ˆ is the stationary simple marked on [−1, 1] and p ∈ [0, 1]. The ν–randomization of point process (SSMPP) ν obtained by assigning to each realization ξˆ = i∈I δxi of ˆ the measure ξ = i∈I δ(xi ,Ei ) , where {Ei }i∈I are independent identically distributed ˆ of ˆ is the SSPP on random variables having distribution ν. Finally, the p–thinning p d R obtained by assigning to each realization ξˆ the measure i∈I Pi δxi , where {Pi }i∈I are independent Bernoulli variables with Prob(Pi = 1) = p and Prob(Pi = 0) = 1 − p. ˆ p are examples of stationary cluster processes, also Both the point processes ν and called homogeneous cluster fields (see [DV, Chap. 8] and [MKM, Chap. 10]). In particular, ergodicity is conserved by ν–randomization and p–thinning ([DV, Prop. 10.3.IX] and [MKM, Prop. 11.1.4]). To conclude, let us give a few examples. Example 1. A Poisson point process (PPP) appears, as already discussed, naturally as limit distribution of thinnings. Given a measure µ on X, with X equal to Rd or Rd × [−1, 1], the PPP on X with intensity measure µ is defined by the two conditions (i) for any B ∈ B(X), ξ(B) is a Poisson random variable with expectation µ(B); (ii) for any

28


disjoint sets B1 , . . . , Bn ∈ B(X), ξ(B1 ), . . . , ξ(Bn ) are independent. A PPP on Rd is stationary if and only if its intensity measure µ is proportional to the Lebesgue measure, µ = ρ dx. In such a case it is an ergodic process satisfying the hypothesis (H2) of Theorem 1 and all moments ρκ , κ > 0 in (1) are finite. Its p–thinning is the PPP on Rd with intensity pρ while its ν–randomization is the PPP on Rd × [−1, 1] with intensity measure ρ dx ⊗ ν. Example 2. Let us associate to the uniformly distributed random variable y in the unit cube C1 the point measure ξˆ = z∈Zd δz+y . The corresponding point process is an ergodic SSPP satisfying ρκ = 1 for any κ > 0. Although this process satisfies (H1), the SSMPP obtained from it via p-thinning and ν-randomization does not and does also not satisfy (H2). However, Theorem 1(ii) is still valid for this SSMPP, as can be checked by restricting the analysis of Sect. 6 to regions which are unions of boxes of the form z + [0, 1)d , z ∈ Zd and using the independence of ξˆ (A) and ξˆ (B) when A, B are disjoint unions of such boxes. Other examples of ergodic SSMPP can be obtained by means of SSPP with short– range correlations (see [DV, Exercise 10.3.4]). Of particular relevance for solid state physics are point processes associated to random or quasiperiodic tilings [BHZ], which satisfy the hypothesis (H1) of Theorem 1. 2.2. The Palm distribution. In what follows, it will always be assumed that Pˆ and P are defined as in Theorem 1 and that (6) is satisfied if ν is a Dirac measure. In order to shorten notations, we will write N and Nˆ for N (Rd ×[−1, 1]) and N (Rd ), respectively. We would like now to “pick up at random” a point among {xj } and take it as the origin. One thus looks at the following borelian subset of N :

N0 := ξ ∈ N : 0 ∈ ξˆ . Since N0 is closed, it defines a bounded complete separable metric space. Note that x ∈ ξˆ if and only if Sx ξ ∈ N0 . The Palm distribution P0 on N0 associated to P is now defined as follows. map G from N into N (Rd × N0 ) given by Consider the measurable ∗ ∗ ξ → ξ = x∈ξˆ δ(x,Sx ξ ) . Let P = G∗ P be the distribution of the marked point prod cess on R × N0 with mark space N0 , namely P ∗ is the image under G of the probability measure P on N . It is easy to show that G ◦Sx = Sx∗ ◦G for x ∈ Rd , where Sx∗ is the action ∗ on Rd × N0 of the translations given by (y, ξ ) → (y + x, ξ ). As a result, P is also stationary. Then, for any fixed A ∈ B(N0 ), the measure µA (B) = P ∗ (dξ ∗ ) ξ ∗ (B ×A) on Rd is translation invariant and thus proportional to the Lebesgue measure. This implies that, for any N > 0 and any A ∈ B(N0 ), 1 ∗ ∗ ∗ CP (A) := P (dξ ) ξ (C1 × A) = d P ∗ (dξ ∗ ) ξ ∗ (CN × A). N N (Rd ×N0 ) N (Rd ×N0 ) The Palm distribution associated to P is the probability measure P0 on N0 obtained from CP by normalization, namely, P0 = ρ −1 CP , where ρ is defined in (1). Thus, for any N > 0, 1 1 P0 (A) := ξˆ (dx) χA (Sx ξ ) , P(dξ ) (11) ρ Nd N CN


29

where χA is the characteristic function on the Borel set A ⊂ N0 . One can show [FKAS, Theorem 1.2.8] that for any nonnegative measurable function f on Rd × N0 , 1 dx P0 (dξ ) f (x, ξ ) = P(dξ ) (12) ξˆ (dx) f (x, Sx ξ ) , ρ N Rd N0 Rd which is used in [DV] as the definition of P0 . Similarly, there is a Palm distribution Pˆ 0 on Nˆ 0 := {ξˆ ∈ Nˆ : 0 ∈ ξˆ } associated to the distribution Pˆ of a SSPP on Rd . It is known that the Palm distribution of a stationary PPP on Rd with distribution Pˆ (Example 1 above) is the convolution Pˆ 0 = Pˆ ∗ δδ0 of Pˆ with the Dirac measure at ξˆ = δ0 (i.e. Pˆ 0 is simply obtained by adding a point at the origin). The Palm distribution of a PPP on Rd ×[−1, 1] with intensity measure ρ dx ⊗ν is the convolution P0 = P ∗ζ , where ζ is the distribution of a marked point process obtained by ν–randomization of δδ0 . The Palm distribution associated to the SSPP in Example 2 is Pˆ 0 = δx∈Zd δx . Its ν–randomization is the Palm distribution of the ν–randomization of Example 2. We collect in the lemma below a number of results on the Palm distribution which will be needed in the sequel. Their proofs are given in Appendix B. Lemma 1. (i) Let k : N0 ×N0 → R be a measurable function such that ξˆ (dx) |k(ξ, Sx ξ )| and ξˆ (dx) |k(Sx ξ, ξ )| are in L1 (N0 , P0 ). Then ˆ P0 (dξ ) ξˆ (dx) k(Sx ξ, ξ ) . P0 (dξ ) ξ (dx) k(ξ, Sx ξ ) = (ii) Let ∈ B(N ) be such that Sx = for all x ∈ Rd . Then P() = 1 if and only if P0 (0 ) = 1 with 0 = ∩ N0 . (iii) Let P be ergodic and A, B ∈ B(N0 ) be such that B ⊂ A, P0 (A \ B) = 0 and Sx ξ ∈ A for any ξ ∈ B and any x ∈ ξˆ . Then P0 (A) ∈ {0, 1}. (iv) Let Aj ∈ B(Rd ) for j = 1, . . . , n. Then   n n c c n+1   ˆ ˆ ≤ EP ξˆ (A˜ j )n+1 , (13) EP 0 ξ (Aj ) + EP ξ (C1 ) ρ ρ j =1

j =1

where A˜ j := ∪x∈C1 Aj + x and c is a positive constant depending on n. Remark 1. Here we point out a simple geometric property of point measures ξ within the set W := ξ ∈ N0 : Sx ξ = ξ ∀x ∈ Rd \ {0} , (14) which will be fundamental in order to apply the methods developed in [KV] and [DFGW]. Let us consider a sequence {xn }n≥0 of elements in supp(ξˆ ) with x0 = 0 and set ξn := Sxn ξ . The ξn can be thought of as the environment viewed from the point xn . Due to the definition of W, {xn }n≥0 can be recovered from {ξn }n∈N by means of the identities xn+1 − xn = (ξn , ξn+1 ), where the function : W × N0 →

n ∈ N,

Rd

is defined as x if ξ

= Sx ξ , (ξ , ξ

) := 0 otherwise .

Note that, by Lemma 1(ii), condition (6) is equivalent to P0 (W) = 1.

(15)

30


3. Variational Formula The main object of this section is to show the following result, implying Theorem 1(i). Theorem 2. Let P satisfy the assumptions of Theorem 1(i). Then the limit (3) exists and D is given by the variational formula 2 P0 (dξ ) ξˆ (dx)c0,x (ξ ) a · x +∇x f (ξ ) , a ∈ Rd ,(16) (a · D a) = inf f ∈L∞ (N0 ,P0 )

with ∇x f (ξ ) := f (Sx ξ ) − f (ξ ) . ξ

(17) ξ

Moreover, the rescaled process Y ξ,ε := (εXtε−2 )t≥0 defined on (ξ , P0 ) converges weakly in P0 –probability as ε → 0 to a Brownian motion W D with covariance matrix D. The proof is based on the theory of Ref. [KV] and [DFGW] and, in particular, Theorem 2.2 of [DFGW]. Because of the geometric disorder and the possibility of jumps between any of the random points, the application of this general theorem to our model is technically considerably more involved than in the case of the lattice model with jumps to nearest neighbors studied in [DFGW, Sect. 4]. As a preamble, let us state a result on ξ the process Xt proven in Appendix C which will be used several times below. Proposition 1. Let P satisfy ρκ < ∞ for some integer κ > 3. Then, given t > 0 and 0 < γ < κ − 3, ξ EP0 EPξ |Xt |γ < ∞ . 0

Remark 2. From the variational formula of the diffusion matrix D given in Theorem 2 one can easily prove (see e.g. [DFGW]) that D is a multiple of the identity whenever P is isotropic (i.e., it is invariant under all rotations by π/2 in a coordinate plane). In this case, the arguments leading to a lower bound on D are slightly simpler (and can be easily adapted to the general case). Therefore, in order to simplify the discussion and without loss of generality, in the last Sects. 5 and 6 we will assume P to be isotropic. 3.1. The result of De Masi, Ferrari, Goldstein and Wick. A main idea in [DFGW] is to study the process (SXξ ξ )t≥0 with values in the space N0 of the environment cont

ξ

figurations, instead of the random walk (Xt )t≥0 . This process is called the process ξ environment viewed from the particle. It is defined on the probability space (ξ , P0 ), with ξ = D([0, ∞), supp(ξˆ )). Let Pξ be its distribution on the path space := D([0, ∞), N0 ) (endowed as usual with the Skorohod topology). A generic element of will be denoted by ξ = (ξt )t≥0 . Let us set P := P0 (dξ )Pξ . The environment process is the process (ξt )t≥0 defined on the probability space (, P) with distribution P. This is a continuous–time jump Markov process with initial measure P0 and transition probabilities P(ξs+t = ξ | ξs = ξ ) = Pξ (ξt = ξ ) =: pt (ξ |ξ )

∀ s, t ≥ 0


with, for any ξ ∈ W,

pt (ξ |ξ ) =

if ξ = Sx ξ for some x ∈ ξˆ , otherwise .

ξ

pt (x|0) 0

31

(18)

For any time t ≥ 0, let us introduce the random variable Xt : → Rd defined by Xt (ξ ) := s (ξ ) , (19) s∈[0,t]

where

s (ξ ) :=

x 0

if ξs = Sx ξs − otherwise

and the sum runs over all jump times s for which s (ξ ) = 0. Note that {X[s,t] := Xt − Xs : t > s ≥ 0} defines an antisymmetric additive covariant family of random variables as defined in [DFGW], and Xt has paths in D([0, ∞), Rd ). The crucial link to the dynamics of a particle in a fixed environment is now the following: due to Remark 1, for any ξ ∈ W, the distribution of the process (Xt )t≥0 defined on (, Pξ ) is equal to ξ the distribution P0 of the randomwalk on supp(ξˆ ) (naturally embedded in Rd ) starting at the origin. Recalling that P = P0 (dξ ) Pξ , this implies ξ EP0 EPξ (Xt · a)2 = EP (Xt · a)2 , (20) 0

which gives a way to calculate the diffusion matrix D from the distribution P on . In order to apply Theorem 2.2 of [DFGW], it is enough to verify the following hypothesis: (a) the environment process is reversible and ergodic; (b) the random variables X[s,t] , 0 ≤ s < t are in L1 (, P); (c) the mean forward velocity exists: ϕ(ξ ) := L2 −lim t↓0

(d) the martingale Xt −

t 0

1 EPξ (Xt ) . t

(21)

ds ϕ(ξs ) is in L2 (, P).

Let us assume ρ12 < ∞. Then, statement (a) will be proved in Proposition 2, Subsect. 3.3. The statement (b) follows from Proposition 1. The L2 -convergence in (c) will be proved in Subsect. 3.4 (Proposition 4), where we also show the L2 -convergence in the following formula defining the mean square displacement matrix ψ(ξ ): 1 (a · ψ(ξ )a) := L2 −lim EPξ (a · Xt )2 . (22) t↓0 t The last point (d) is a consequence of Proposition 1 assuring that Xt ∈ L2 (, P) and t the fact that 0 ds ϕ(ξs ) ∈ L2 (N0 , P0 ), which can be proved by means of the Cauchy–Schwarz inequality, the stationarity of P following from (a), and the property ϕ ∈ L2 (N , P0 ).

32


Once hypotheses (a)-(d) have been verified, one can invoke [DFGW, Theorem 2.2 and Remark 4, p. 802] to conclude that limit (3) exists and that the rescaled random walk Y ξ,ε converges weakly in P0 –probability to the Brownian motion W D with covariance matrix D given by (3), and that D is moreover given by ∞ dt ϕ · a , et L ϕ · a , (23) (a · Da) = EP0 (a · ψa) − 2 P0

0

where L is the generator of the environment process and the integral on the r.h.s. is finite. Formula (16) can be deduced from the expressions of L, ϕ and ψ established in the following subsections (Propositions 3 and 4) by using a known general result on self-adjoint operators stated in (47) below. 3.2. Preliminaries. Before starting to prove the above-mentioned statements (a)-(d), let us fix some notations and recall some general facts about jump Markov processes. In what follows, given a complete separable metric space Z we denote by F(Z) the family of bounded Borel functions on Z and, given a (not necessarily finite) interval I ⊂ R, we denote by D(I, Z) the space of right continuous paths z = (zt )t∈I , zt ∈ Z, having left limits. The path space D(I, Z) is endowed with the Skorohod topology [Bil] which is the natural choice for the study of jump Markov processes. For a time s ≥ 0, the time translation τs is defined as τs : D([0, ∞), Z) → D([0, ∞), Z),

(τs z)t := zt+s .

Moreover, given 0 ≤ a < b, we denote by R[a,b] the function R[a,b] : D([0, ∞), Z) → D([a, b], Z),

(R[a,b] z)t := lim za+b−t−δ . δ↓0

R[a,b] z is the time–reflection of (zt )t∈[a,b] w.r.t. the middle point of [a, b], and it can naturally be extended to paths on [0, a + b]. A continuous–time Markov process with path in D([0, ∞), Z) and distribution p is called stationary if Ep (F ) = Ep (F ◦τs ) for all s ≥ 0 and for any bounded Borel function F on D([0, ∞), Z). It is called reversible if Ep (F ) = Ep (F ◦ R[a,b] ) for all b > a ≥ 0 and any bounded Borel function F on D([0, ∞), Z) such that F (z) depends only on (zt )t∈[a,b] . Thanks to the Markov property, one can show that stationarity is equivalent to Ep f (z0 ) = Ep f (zs ) , ∀ s ≥ 0 , ∀ f ∈ F(Z), (24) while reversibility is equivalent to Ep f (z0 )g(zs ) = Ep g(z0 )f (zs ) ,

∀ s ≥ 0 , ∀ f, g ∈ F(Z) .

(25)

In particular, stationarity follows from reversibility. process is called Finally, the Markov (time) ergodic if p(A) ∈ {0, 1} whenever A ∈ B D([0, ∞), Z) is time-shift invariant, i.e. A = τs A for all s ≥ 0. Recall that if the Markov process is stationary then it can be extended to a Markov process with path space D(R, Z) and the resulting distribution is univocally determined (this follows from Kolmogorov’s extension theorem and the regularity of paths). Now stationarity, reversibility and ergodicity of the extended process are defined as above by means of τs , s ∈ R, and R[a,b] , −∞ < a < b < ∞. Then one can check that these properties are preserved by extension (for what concerns ergodicity,


33

see in particular [Ros, Chapter 15, p. 96–97]). Therefore our definitions coincide with those in [DFGW]. All the above definitions and remarks can be extended in a natural way to discrete–time Markov processes (with path space Z N ). Moreover, in the discrete case, stationarity and reversibility are equivalent respectively to (24) and (25) with s = 1. We conclude this section recalling the standard construction of the continuous–time random walk satisfying conditions (C1) and (C2) in the Introduction. We first note that these conditions are meaningful for P0 –almost all ξ if EP0 (λ0 ) < ∞. In fact, due to the bound λx (ξ ) ≤ e4β e|x| λ0 (ξ ), one can infer from EP0 (λ0 ) < ∞ that λx (ξ ) < ∞ for any x ∈ ξˆ , P0 a.s. We note that the condition EP0 (λ0 ) < ∞ is equivalent to ρ2 < ∞ due to the following lemma: Lemma 2. For any positive integer k, EP0 (λk0 ) < ∞ if and only if ρ k+1 < ∞. Proof. Note that for suitable positive constants c1 , c2 one has ξˆ (C1 + z)e−|z| ≤ λ0 (ξ ) ≤ c2 ξˆ (C1 + z)e−|z| , c1 z∈Zd

P0 -a.s.

z∈Zd

Next let us expand the k th power of these inequalities. By applying Lemma 1(iv) and using the stationarity of P, one gets that EP0 (λk0 ) < ∞ if ρ k+1 < ∞. Suppose now that EP0 (λk0 ) < ∞. Then the above expansion in k th power implies that EP0 (ξˆ (C1 )k ) < ∞. Since due to (11), 2d 2d k ˆ P(dξ ) ξˆ (dx) ξˆ (C1 + x)k ≥ EP (ξˆ (C1/2 )k+1 ) , EP0 (ξ (C1 ) ) = ρ N ρ C1/2 one concludes that EP (ξˆ (C1/2 )k+1 ) < ∞, which is equivalent to ρk+1 < ∞.

The construction of the continuous–time random walk follows standard references (e.g. [Bre, Chap. 15] and [Kal, Chap. 12]) and can be described roughly as follows: After arriving at site y ∈ ξˆ , the particle waits an exponential time with parameter λy (ξ ) and then jumps to another site z ∈ ξˆ with probability pξ (z|y) :=

cy,z (ξ ) . λy (ξ )

(26)

More precisely, consider ξ ∈ N0 such that 0 < λz (ξ ) < ∞ for any z ∈ ξˆ and set ξ ˜ ξ is denoted by X˜ nξ ˜ ξ := supp(ξˆ ) N . A generic path in . Given x ∈ ξˆ , let P˜ x n≥0 ˜ ξ of a discrete–time random walk on supp(ξˆ ) starting in x and be the distribution on having transition probabilities p ξ (z|y). Let , Q be another probability space where ξ the random variables Tn,z , z ∈ ξˆ , n ∈ N, are independent and exponentially distributed ξ with parameter λz (ξ ), namely Q Tz,n > t ) = exp −λz (ξ )t . On the probability space ˜ ξ × , P˜ xξ ⊗ Q) define the following functions: ( ξ

ξ

ξ ξ 0,X˜ 0

R0 := 0 ; Rn := T ξ

+T

ξ ξ 1,X˜ 1

+ ··· + T ξ

ξ ξ n−1,X˜ n−1 ξ

n∗ (t) := n if Rn ≤ t < Rn+1 .

if

n≥1,

34

A. Faggionato, H. Schulz-Baldes, D. Spehner ξ

ξ

Note that n∗ (t) is well posed for any t ≥ 0 only if limn↑∞ Rn = ∞. If P˜ xξ ⊗ Q lim Rnξ = ∞ = 1 , n↑∞

(27)

ξ ξ )t≥0 , defined P˜ x ⊗Q–almost ξ n∗ (t)

then, due to [Bre, Theorem 15.37], the random walk ( X˜

everywhere, is a jump Markov process whose distribution satisfies the infinitesimal conξ ditions (C1) and (C2). The condition limn↑∞ Rn = ∞ assures that no explosion phenomenon takes place, notably only finitely many jumps can occur in finite time intervals. In Appendix A we prove that (27) is verified if ρ2 < ∞. 3.3. The environment viewed from the particle. The process environment viewed from the particle and the environment process have been introduced in Sect. 3.1. Given t > 0, we write n∗ (t) for the function on the path space = D ([0, ∞), N0 ) associating to each ξ ∈ the corresponding number of jumps in the time interval [0, t]. Motivated by further applications, it is convenient to consider also the discrete–time versions of the above processes. Consider the discrete-time Markov process SX˜ ξ ξ n≥0 defined on n ˜ := N N and denote a generic path ˜ ξ , P˜ ξ , call P˜ ξ its distribution on the path space 0 0 ˜ by (ξn )n≥0 . Such a Markov process can be thought of as the environment viewed in ξ from the particle performing the discrete–time random walk with distribution P˜ 0 . Let us point out a few properties of the distribution P˜ ξ . First, we remark that due to the covariant relations cz,y (Sx ξ ) = cz+x,y+x (ξ ) , λy (Sx ξ ) = λy+x (ξ ), (28) ˜ ξ , P˜ ξ ) and the process (λ0 (ξn ))n∈N defined on the process λX˜ ξ (ξ ) n∈N defined on ( 0 n ˜ P˜ ξ ) have the same distribution. Moreover, due to Remark 1, if ξ ∈ W, then the (, ˜ P˜ ξ ) by ζ0 = 0 and process (ζn )n∈N defined on (, ζn =

n−1

(ξk , ξk+1 ) , ∀ n ≥ 1 ,

k=0

˜ ξ with distribution P˜ ξ . Finally, it is conwhere (ξ, ξ ) is given by (15), has paths in 0 venient to consider a suitable average of the distributions P˜ ξ . To this aim, let Q0 be the probability measure on N0 defined as Q0 (dξ ) :=

λ0 (ξ ) P0 (dξ ) , EP0 (λ0 )

Q0 (dξ )P˜ ξ . If ξ ∈ W, the transition probabilities are λ−1 if ξ = Sx ξ ,

0 (ξ )c0,x (ξ ) p(ξ |ξ ) := P˜ ξn+1 = ξ |ξn = ξ = 0 otherwise .

and set P˜ :=

Note that, due to (28) and the symmetry of the jump rates (2), λ0 (ξ )p(ξ |ξ ) = λ0 (ξ )p(ξ |ξ ).


35

Proposition 2. Let ρ2 < ∞. Then the process (ξt )t≥0 defined on (, P) is reversible, i.e. EP f (ξ0 )g(ξt ) = EP g(ξ0 )f (ξt )

∀ f, g ∈ F(N0 ) , ∀ t > 0 ,

(29)

and is (time) ergodic if P is ergodic. Similarly, the discrete-time Markov process (ξn )n≥0 ˜ is reversible and is (time) ergodic. ˜ P) defined on (, Having at our disposal Lemma 1, the proof follows modifying arguments of e.g. [DFGW]. Proof. We give the proof for the continuous–time process, the discrete–time case being similar. We first verify the symmetric property pt (ξ |ξ ) = pt (ξ |ξ ). Actually, thanks to the construction of the dynamics given in Sect. 3.2, one can show that for any positive integer n and any ξ = ξ (0) , ξ (1) , . . . , ξ (n−1) , ξ (n) = ξ ∈ N0 , Pξ n∗ (t) = n, ξR1 =ξ (1) , . . . , ξRn =ξ (n) =Pξ n∗ (t) = n, ξR1 = ξ (n−1) , . . . , ξRn =ξ (0) , where, given ξ ∈ , R1 (ξ ) < R2 (ξ ) < . . . denote the jump times of the path ξ . Next, given f, g ∈ F(N0 ) one gets by applying Lemma 1(i) and using pt (ξ |ξ ) = pt (ξ |ξ ) that P0 (dξ ) ξˆ (dx) pt (Sx ξ |ξ ) f (ξ )g(Sx ξ ) = P0 (dξ ) ξˆ (dx) pt (Sx ξ |ξ ) f (Sx ξ )g(ξ ), (30) which is equivalent to (29). Hence P is reversible. Due to Corollary 5 in [Ros, Chap. IV], in order to prove ergodicity it is enough to show that P0 (A) ∈ {0, 1} if A ∈ B(N0 ) has the following property: Pξ (ξt ∈ A) = χA (ξ ) for P0 –almost all ξ . Given such a set A, then there exists a Borel subset B ⊂ A such that P0 (A \ B) = 0 and Pξ (ξt ∈ A) = 1 for any ξ ∈ B. Fix ξ ∈ B and x ∈ ξˆ , then Pξ (ξt = Sx ξ, ξt ∈ A) = Pξ (ξt = Sx ξ ) > 0 (the last bound follows from the positivity of the jump rates). Hence Sx ξ ∈ A. Lemma 1(iii) implies that P0 (A) ∈ {0, 1}, thus allowing to conclude the proof. Let P fulfill the assumption of Proposition 2. Then,

(Tt f )(ξ ) := EPξ f (ξt ) =

ξˆ (dx) pt (Sx ξ |ξ ) f (Sx ξ ) ,

P0 a.s.

(31)

defines a strongly continuous contraction semigroup on L2 (N0 , P0 ) (Markov semigroup). Actually, (i) Tt : L2 (N0 , P0 ) → L2 (N0 , P0 ) is self-adjoint by (29) and is a contraction by the Cauchy-Schwarz inequality and the stationarity of P; (ii) Tt+s = Tt Ts follows from the Markov nature of the process; (iii) the continuity follows from the following argument: first observe that it is enough to prove the continuity of Tt f at t = 0 for f ∈ L∞ (N0 , P0 ), which is obtained from the dominated convergence theorem and ξ the estimate |(Tt f − f )(ξ )| ≤ 2f ∞ (1 − pt (0|0)). Let us denote by L the generator of the Markov semigroup (Tt )t≥0 and by D(L) ⊂ L2 (N0 , P0 ) its domain.

36


Proposition 3. Let P satisfy ρ4 < ∞. Then L is nonpositive and self–adjoint with core L∞ (N0 , P0 ). For any f ∈ L∞ (N0 , P0 ), one has for P0 -a.e. ξ , (32) (Lf )(ξ ) = ξˆ (dx) c0,x (ξ ) ∇x f (ξ ) , where ∇x f is defined in (17), and, moreover, 1 P0 (dξ ) ξˆ (dx) c0,x (ξ ) (∇x f (ξ ))2 . f, (−L)f P0 = 2

(33)

Proof. The self-adjointness of L follows from [RS, Vol.2, Theorem X.1]. Actually, (i) L is closed as a generator of a strongly continuous semigroup [RS, Vol.2, Chap. X.8]; (ii) L is symmetric because Tt is self-adjoint; (iii) the spectrum of L is included in (−∞, 0] by contractivity of the semigroup. Note that (iii) also implies that L is non-positive. We use the abbreviation Lp for Lp = Lp (N0 , P0 ), p = 2 or ∞. For any f ∈ L∞ , denote by f the function defined by the r.h.s. of (32). Due to Lemma 2, EP0 (λ20 ) < ∞ and in particular 2 P0 (dξ )(f )(ξ ) ≤ 4 f 2∞ EP0 λ20 < ∞ , thus implying that : L∞ → L2 is a well-defined operator. We claim that L2 − lim t↓0

Tt f − f = f , t

∀ f ∈ L∞ .

(34)

Note that (34) implies that L∞ ⊂ D(L) and Lf = f for all f ∈ L∞ . Since moreover Tt is a contraction and Tt L∞ ⊂ L∞ , it then follows from [RS, Vol.2, Theorem X.49] that L∞ is a core for L and L is the closure of . Finally, using (30) in the limit t → 0, by straightforward computations (33) can be derived from (32). Let us now prove (34). We assume ξ ∈ W and we set, for ξ = ξ , pt,1 (ξ |ξ ) := Pξ (ξt = ξ , n∗ (t) = 1) = pt (ξ |ξ ) − Pξ (ξt = ξ , n∗ (t) ≥ 2) . Thanks to the construction of the dynamics described in Sect. 3.2 and due to the estimate 1 − e−u ≤ u, u ≥ 0, one has for any x ∈ ξˆ and x = 0, ξ ξ ξ pt,1 (Sx ξ |ξ ) ≤ P˜ 0 ⊗ Q(X˜ 1 = x, T0,0 ≤ t) = pξ (x|0)(1 − e−λ0 (ξ )t ) ≤ c0,x (ξ ) t .(35) Let f ∈ L∞ . In view also of (31) and ξˆ (dx) pt (Sx ξ |ξ ) = 1, ˆ Tt f − f − t f (ξ ) = ξ (dx) f (Sx ξ ) − f (ξ ) pt (Sx ξ |ξ ) − c0,x (ξ ) t ≤ 2 f ∞ ξˆ (dx) pt (Sx ξ |ξ ) − pt,1 (Sx ξ |ξ ) {x=0} + ξˆ (dx) −pt,1 (Sx ξ |ξ ) + c0,x (ξ ) t . {x=0}

The first integral in the second line can be bounded by Pξ (n∗ (t) ≥ 2). The second integral equals − Pξ ( n∗ (t) = 1 ) + λ0 (ξ ) t = −1 + e−λ0 (ξ )t + λ0 (ξ ) t + Pξ (n∗ (t) ≥ 2) .


37

By collecting the above estimates, we get 2 32 f 2∞ 1 2 P T ≤ E f − f − tf E (n (t) ≥ 2) t ∗ P P ξ 0 0 t2 t2 2 2 8f ∞ −λ0 t + E + λ t −1 + e . (36) 0 P 0 t2 By using the estimate (e−u − 1 + u)2 ≤ u3 /2 for u ≥ 0 and the finiteness of EP0 (λ30 ), it is easy to check that the second term in the r.h.s. tends to zero as t → 0. In order to bound the first term, we observe that ξ ξ ξ Pξ n∗ (t) ≥ 2 ≤ P˜ 0 ⊗ Q(T0,0 ≤ t, T ˜ ξ ≤ t) 1,X1 −λ0 (ξ )t = 1−e ξˆ (dx) p(Sx ξ |ξ ) 1 − e−λ0 (Sx ξ )t . Due to the estimate 1 − e−u ≤ u, this implies the bound 2 Pξ (n∗ (t) ≥ 2) ≤ t λ0 (ξ ) ξˆ (dx) p(Sx ξ |ξ )λ0 (Sx ξ ) = t 2 λ0 (ξ ) EP˜ ξ λ0 (ξ1 ) (. 37) Due also to the estimate 1 − e−u ≤ 1, it is also true that Pξ (n∗ (t) ≥ 2) ≤ t EP˜ ξ λ0 (ξ1 ) .

(38)

˜ one obtains By multiplying the last two inequalities, and using the stationarity of P, 2 1 EP0 Pξ2 (n∗ (t) ≥ 2) ≤ t EP0 λ0 (ξ ) EP˜ ξ λ0 (ξ1 ) 2 t ≤ t EP0 λ0 (ξ )EP˜ ξ λ20 (ξ1 ) = t EP0 λ30 , thus implying that the first term on the r.h.s. of (36) goes to 0 as t → 0.

3.4. Mean forward velocity and infinitesimal square displacement. Proposition 4. Let P satisfy ρ12 < ∞ and let ϕ be the Rd -valued function on N0 and ψ be the function on N0 with values in the real symmetric d × d matrices, respectively defined by ˆ ϕ(ξ ) = ξ (dx) c0,x (ξ ) x , (a · ψ(ξ )a) = ξˆ (dx) c0,x (ξ ) (a · x)2 . (39) (i) ϕ(ξ ) is in L2 (N0 , P0 ) and is equal to the mean forward velocity given by the convergent L2 -strong limit (21). (ii) (a · ψ(ξ )a) is in L2 (N0 , P0 ) and is equal to the infinitesimal mean square displacement given by the convergent L2 -strong limit (22). We point out that ϕ(ξ ) and ψ(ξ ) are well defined for P0 almost all ξ since ρ2 < ∞ (see for example the proof of Lemma 2).

38


Proof. (i) One has 2 2 1 2 P P0 (dξ ) EPξ Xt χ n∗ (t) = 1 − t ϕ(ξ ) (dξ ) E (X ) − t ϕ(ξ ) ≤ 0 Pξ t 2 2 t t 2 2 + 2 P0 (dξ ) EPξ Xt χ n∗ (t) ≥ 2 . (40) t We first show that the first term on the r.h.s. vanishes as t → 0. Using the same notation as in the proof of Proposition 3 and invoking (35), 2 EP0 EPξ Xt χ n∗ (t) = 1 −t ϕ(ξ ) 2 ˆ ξ (dx) pt,1 (Sx ξ |ξ )−t c0,x (ξ ) x = EP0 {x=0}

is bounded according to the Cauchy-Schwarz inequality by EP0 ξˆ (dx) −pt,1 (Sx ξ |ξ ) + t c0,x (ξ ) {x=0}

{y=0}

ξˆ (dy) −pt,1 (Sy ξ |ξ ) + t c0,y (ξ ) |y|2 .

(41)

Let us denote by I1 (ξ ) and I2 (ξ ) the (non negative) integrals over ξˆ (dx) and ξˆ (dy) respectively. Using the identities of the proof of Proposition 3, the inequality 0 ≤ −1 + e−u + u ≤ u2 , u ≥ 0, and (37), we deduce I1 (ξ ) = −1 + e−tλ0 (ξ ) + tλ0 (ξ ) + Pξ (n∗ (t) ≥ 2) ≤ t 2 λ0 (ξ )2 + t 2 λ0 (ξ ) EP˜ ξ (λ0 (ξ1 )). Moreover, I2 (ξ ) ≤ t ξˆ (dy)c0,y (ξ )|y|2 . Hence (41) is bounded by t 3 EP0 λ20 (ξ ) ξˆ (dy) c0,y (ξ )|y|2 +EP0 λ0 (ξ )EP˜ ξ (λ0 (ξ1 )) ξˆ (dy) c0,y (ξ )|y|2 . As long as ρ4 < ∞, the first expression can be bounded by applying Lemma 1(iv) (see the argument leading to Lemma 2). A short calculation shows that the second expression equals P0 (dξ ) ξˆ (dx) c0,x (ξ ) ξˆ (dz) cx,z (ξ ) ξˆ (dy) c0,y (ξ ) |y|2 and is therefore bounded if ρ4 < ∞ (again by means of Lemma 1(iv)). Resuming the results obtained so far, one gets 1 EP Xt χ n∗ (t) = 1 − t ϕ(ξ )2 = O(t) . P (dξ ) (42) 0 ξ t2 We now turn to the second term in (40). By Proposition 1, EP0 (EPξ (|Xt |γ )) < ∞ as long as 0 < γ < κ − 3 whenever ρκ < ∞ for κ integer. By applying twice the Hölder inequality, if γ > 2, γ γ2 2 2γ −2 1− γ2 EP0 EPξ Xt χ n∗ (t) ≥ 2 ≤ EP0 EPξ Xt EP0 Pξ n∗ (t) ≥ 2 γ −2 .


39

Let us take (38) to the power γ /(γ − 2), multiply the result by (37). This yields 2γ −2 3γ −4 3γ −4 3γ −4 2γ −2 γ −2 γ −2 . EP0 Pξ n∗ (t) ≥ 2 γ −2 ≤ t γ −2 EP0 λ0 (ξ ) EP˜ λ0 (ξ1 ) = t γ −2 EP0 λ0 ξ

Hence, by Lemma 2, if ρκ < ∞ is satisfied for integer κ > (4γ − 6)/(γ − 2) and γ < k − 3, there is a finite constant C > 0 such that 3γ −4 2 (43) EP0 EPξ Xt χ n∗ (t) ≥ 2 ≤ C t γ . One concludes the proof by choosing γ > 4 and by combining (40), (42) and (43), as long as κ > 7. (ii) One follows the same strategy. The first term in the equation corresponding to (40) can be dealt with in exactly the same way. In the argument for the second term, |Xt | is replaced by |Xt |2 so that one needs 2γ < κ − 3, hence κ > 11. 3.5. Proof of Theorem 2. Since all conditions (a)-(d) of Subsect. 3.1 have been checked in the preceding subsections, as already pointed out, one can invoke [DFGW, Theorem 2.2] to conclude that the limit (3) exists and that the rescaled random walk Y ξ,ε converges weakly in P0 –probability to the Brownian motion W D . We can now also derive the variational formula (16) from the general expression (23). Let us first quote some general results concerning self–adjoint operators. Let (, µ) be a probability space and denote by . , . µ and by .µ the scalar product and the norm on H = L2 (, µ). Let L : D(L) → H be a nonpositive self–adjoint operator with (dense) domain D(L) ⊂ H and assume C ⊂ D(L) is a core of L. The space of D(|L|1/2 ) ∩ (Ker(L))⊥ H1 is the completion 1/2 1/2 under the norm f 1 := |L| f µ for f ∈ D(|L| ), while the dual H−1 of H1 −1/2 ) = Ran(|L|1/2 ) under . , . µ can be identified with the completion −1/2 of D(|L| ϕ µ for ϕ ∈ D(|L|−1/2 ). Given under the .−1 -norm defined as ϕ−1 := |L| ϕ ∈ H ∩ H−1 , the dual norm ϕ−1 admits several useful characterizations: ϕ2−1 =

|ϕ, f µ |2 = f 21 f ∈H1 ∩H sup

sup f ∈C ∩(Ker(L))⊥

|ϕ, f µ |2 , f 21

(44)

where the last identity results from the fact that C is a core for L. Moreover, the identity ϕ2−1 = sup 2 ϕ, f µ − f, (−L)f µ (45) f ∈C

is obtained by using the nonlinearity in f of the expression in the r.h.s. of (45) and observing that ϕ ∈ (Ker(L))⊥ . Finally, it follows from spectral calculus that ∞ dt ϕ, et L ϕµ . (46) ϕ2−1 = 0

In what follows, we extend the definition of · −1 to the whole space H by setting ϕ−1 := ∞ whenever ϕ ∈ H and ϕ ∈ H−1 . Thanks to this choice, identities (44), (45) and (46) are true for all ϕ ∈ H. Invoking (45) and (46), one obtains ∞ dt ϕ · a , et L ϕ · a = sup 2 ϕ · a , f P0 − f, (−L)f P0 . 0

P0

f ∈L∞ (N0 ,P0 )

(47) Using (33), (39) and Lemma 1(i), a short calculation starting from (23) yields (16).

40


4. Bound by Cut-off on the Transition Rates This section and the next ones are devoted to the proof of Theorem 1(ii). In particular, ˆ P and ν satisfy the conditions of Theorem 1(ii) although many partial we assume that P, results are true under much weaker conditions. The variational formula (16) is particularly suited in order to derive bounds on the diffusion matrix D. For example, due to the monotonicity of the jump rates cx,y (ξ ) in the inverse temperature β, one deduces that the diffusion matrix is a non-increasing function of β. The aim of this section is to obtain more quantitative bounds. Given an energy 0 ≤ Ec ≤ 1, we define the map c : N → Nˆ := N (Rd ) as follows: c (ξ ) (A) := ξ(A × [−Ec , Ec ]) , A ∈ B(Rd ) . (48) d Note that Pˆ c := P ◦ −1 c is the distribution of a point process on R with finite intensity ˆ ˆ ρc := EPˆ c ( ξ (C1 ) ) ≤ EP ( ξ (C1 ) ) = ρ, and in general

EPˆ c ( ξˆ (C1 )κ ) ≤ ρκ ,

∀ κ > 0.

(49)

In what follows, we assume that ρc > 0. It can readily be checked that Pˆ c is an ergodic SSPP on Rd . We write Pˆ 0c for the Palm distribution associated to Pˆ c . Note that the distribution Pˆ c is obtained from Pˆ by δc –thinning with δc := ν([−Ec , Ec ]). Thus, ρc = δc ρ. The relation between the Palm distributions P0 and Pˆ 0c is described in the following lemma. Lemma 3. For any Borel set A ∈ B(Nˆ 0 ) one has Pˆ 0c (A) = ρ ρc−1 P0 ( |E0 | ≤ Ec , c (ξ ) ∈ A ). Proof. The assertion is proven by comparing the two following identities obtained from (11): 1 c c ˆ ˆ ˆ P0 (A) = P (d ξ ) ξˆ (dx)χA (Sx ξˆ ) , ρc Nˆ C1 1 P0 ( |E0 | ≤ Ec , c (ξ ) ∈ A ) = P(dξ ) ξˆ (dx) χ (|Ex | ≤ Ec ) χA (c (Sx ξ )) ρ N C1 1 = P(dξ ) c (ξ ) (dx) χA Sx (c (ξ )) . ρ N C1 Proposition 5. Fix a distance rc > 0 and an energy 0 ≤ Ec ≤ 1 and let Pˆ 0c be as above. Moreover, define ϕc (ξˆ ) := ξˆ (dx) cˆ0,x x , (a · ψc (ξˆ )a) := ξˆ (dx) cˆ0,x (a · x)2 (50) as functions on Nˆ 0 , where cˆ0,x := χ (|x| ≤ rc ). Then the diffusion matrix D for the ξ process (Xt )t≥0 in Theorem 2 admits the following lower bound D ≥

ρc −rc −4 β Ec e Dc (rc , Ec ) , ρ


41

where

(a · Dc (rc , Ec ) a) := EPˆ c (a · ψc a) 0

∞

− 2

dt ϕc · a , et Lc ϕc · a ˆ c , (51) P0

0

and Lc is the unique self–adjoint operator on L2 (Nˆ 0 , Pˆ 0c ) such that (Lc f )(ξˆ ) =

ξˆ (dx) cˆ0,x ∇x f (ξˆ ) ,

∀ f ∈ L∞ (Nˆ 0 , Pˆ 0c ) .

(52)

One can prove by the same arguments used in the proof of Proposition 3 that Lc is well-defined and self–adjoint. Let Pˆ c and Pˆ cˆ be the probability measures on the path ξ

ˆ := D( [0, ∞), Nˆ 0 ) associated to the Markov process with generator Lc and space initial distribution Pˆ 0c and δξˆ , respectively, with ξˆ ∈ Nˆ 0 . One can prove that these Markov processes are well–defined (in particular, Pˆ c is well–defined for Pˆ c –almost all ξˆ ) ξˆ

0

and exhibit a realization as jump processes by means of the same arguments used in Sect. 3.2 (note that, for a suitable positive constant c, ξˆ (dx)cˆ0,x ≤ c λ0 (ξ ) for any ξ ∈ N0 , thus allowing to exclude explosion phenomena from the results of Appendix A). Finally, ˆ Xt (ξˆ ) is defined as in (19). given ξˆ ∈ , Proof. Note that c0,x (ξ ) ≥ e−rc −4 β Ec c˜0,x (ξ ) , where c˜x,y (ξ ) := χ |Ex | ≤ Ec , |Ey | ≤ Ec , |x − y| ≤ rc )

,

x, y ∈ ξˆ .

Then (16) implies that (a · Da) ≥ e−rc −4 β Ec g(a), where 2 g(a) := inf P0 (dξ ) ξˆ (dx) c˜0,x (ξ ) a · x + ∇x f (ξ ) ≥0. f ∈L∞ (N0 ,P0 )

By the same arguments used in the proof of Proposition 3 one can show that there is a unique self–adjoint operator L˜ on L2 (N0 , P0 ) such that ˜ (Lf )(ξ ) := ξˆ (dx) c˜0,x (ξ ) ∇x f (ξ ) , ∀ f ∈ L∞ (N0 , P0 ). Moreover, L∞ (N0 , P0 ) is a core of L˜ and ˜ P = 1 P0 (dξ ) ξˆ (dx)c˜0,x (ξ ) (∇x f (ξ ))2 , f, (−L)f 0 2 Next let us introduce the functions ϕ(ξ ˜ ) = ξˆ (dx) c˜0,x (ξ ) x ,

˜ )a) = (a · ψ(ξ

∀f ∈ L∞ (N0 , P0 ).(53)

ξˆ (dx) c˜0,x (ξ ) (a · x)2 .

42


Then we obtain by means of straightforward computations and the identities (45), (46) and (53) that ˜ P ˜ − 2 supf ∈L∞ (N0 ,P0 ) 2 ϕ˜ · a, f P0 − f, (−L)f g(a) = EP0 (a · ψa) 0 ∞ ˜ ˜ = EP0 (a · ψa) − 2 0 dt ϕ˜ · a , et L ϕ˜ · a . P0

At this point, in order to get (51), it is enough to show that ˜ = δc EPˆ c (a · ψc a) , EP0 (a · ψa) 0

and

˜ ϕ˜ · a , et L ϕ˜ · a

P0

= δc ϕc · a , et Lc ϕc · a ˆ c . P0

This can be derived from Lemma 3 and the following identities, where c is defined by (48): ψ˜ = χ (|E0 | ≤ Ec ) ψc ◦ c , ϕ˜ = χ (|E0 | ≤ Ec ) ϕc ◦ c , ˜ ◦ c ) = χ (|E0 | ≤ Ec ) (Lc f ) ◦ c . L(f 5. Periodic Approximants and Resistor Networks In this section, we compare Dc (rc , Ec ) to the diffusion coefficient of adequately defined periodic approximants, which then in turn can be calculated as the conductance of a random resistor network as in [DFGW]. There have been numerous works on periodic approximants; a recent one containing further references is [Owh]. 5.1. Random walk on a periodized medium. Let us choose a given direction in Rd , say, the direction parallel to the axis of the first coordinate. Given a fixed configuration ξˆ ∈ Nˆ and N > rc , we define the following subsets of Rd ξˆ

QN : = supp(ξˆ ) ∩ Cˇ 2N , ξˆ

ξˆ

+ − V N : = QN ∪ N ∪ N ,

± N := Zd ∩ {x : x (1) = ±N, |x (j ) | < N for j = 2, . . . , d}, ξˆ ±

BN

ξˆ

± := QN ∩ BN ,

− + := {x ∈ Cˇ 2N : x (1) ∈ (−N, −N + rc ]} and BN := where Cˇ 2N := (−N, N )d , BN (1) ˇ {x ∈ C2N : x ∈ [N − rc , N )}. ξˆ

ξˆ

ξˆ

ξˆ

Next let us introduce a graph (V N , E N ) with set of vertices V N and set of edges E N . ξˆ

ξˆ

Two vertices x, y ∈ QN are connected by a non-oriented edge {x, y} ∈ E N if and only ξˆ +

ξˆ −

if |x − y| ≤ rc ; moreover, all vertices x ∈ BN (respectively x ∈ BN ) are connected ξˆ

+ − ± to all y ∈ N (respectively y ∈ N ) by an edge {x, y} ∈ E N and the points of N are not connected between themselves.

Mott Law as Lower Bound for a Random Walk in a Random Environment ξˆ

ξˆ

ξˆ

43 ξˆ

We now define another graph (VN , EN ) obtained from (V N , E N ) by identifying the vertices x− = (−N, x (2) , . . . , x (d) ) ξˆ

x+ = (N, x (2) , . . . , x (d) ) .

and

ξˆ

Let us write π : V N → VN for the identification map on the sets of vertices. Hence ξˆ

ξˆ

ξˆ

− + π(N ) = π(N ) and π restricted to QN is the identity map. The set VN = π(V N ) − ) is represents the medium periodized along the first coordinate. A vertex y ∈ π(N ξˆ +

ξˆ −

ξˆ

connected to all vertices x ∈ BN ∪ BN by an edge of EN . ξˆ

ξˆ

Now a continuous–time random walk with state space VN and infinitesimal generator

LN is given by

ξˆ LN f (x) =

ξˆ

c({x, y}) f (y) − f (x) ,

ξˆ

∀ x ∈ VN ,

ξˆ

y∈VN : {x,y}∈EN ξˆ

where the bond-dependent transition rates c({x, y}) are defined for any {x, y} ∈ EN by  ξˆ 1 if x, y ∈ QN , (54) c({x, y}) = − −  1− if x ∈ π(N ) or y ∈ π(N ). | | N

ξˆ

ξˆ

ξˆ

Clearly the generator LN is symmetric w.r.t. the uniform distribution mN on VN given by 1 ξˆ mN = ˆ δx . V ξ ξˆ x∈VN

N

ξˆ

ξˆ

Hence the Markov process with generator LN and initial distribution mN is reversible. Note that it is not ergodic, however, if there are more than one cluster (equivalence class of edges). In the latter case, the ergodic measures are the uniform distributions on a given cluster and this is sufficient for the present purposes. ξˆ ξˆ ξˆ We write PN (respectively PN,x ) for the probability on the path space N = D [0, ∞), ξˆ ξˆ VN associated to the random walk with initial distribution mN (respectively δx ) and ξˆ

generator LN .

ξˆ

Let us introduce an antisymmetric function d1 (x, y) on VN such that  ξˆ (1) (1)  if x, y ∈ QN ,  y − x ξˆ − d1 (x, y) = y (1) + N if y ∈ QN , y (1) < 0, x ∈ π(N ),    (1) ξˆ − if y ∈ QN , y (1) > 0, x ∈ π(N ) . y −N Finally, given t ≥ 0, we define the random variable (1)ξˆ d1 (ωs− , ωs ) , XN,t (ω) = s∈[0,t] : ωs =ωs−

44

A. Faggionato, H. Schulz-Baldes, D. Spehner ξˆ

where (ωs )s≥0 ∈ N . It is the sum of position increments along the first coordinate (1)ξˆ

axis for all jumps occurring in the time interval [0, t]. Clearly, XN,t gives rise to a time-covariant and antisymmetric family so that, as in Sect. 3, [DFGW, Theorem 2.2] can be used in order to deduce the following result. Proposition 6. Given N ∈ N, N > rc , and ξˆ ∈ Nˆ , lim

t↑∞

(1)ξˆ 1 ξˆ E ξˆ (XN,t )2 = DN , P t N ξˆ

where the diffusion coefficient DN is finite and given by ∞ ξˆ ξˆ ξˆ ξˆ ξˆ ξˆ dt ϕN , et LN ϕN DN = mN ψN − 2 0

ξˆ

ξˆ

,

ξˆ

mN

(55)

ξˆ

with ψN , ϕN (scalar) functions on VN defined as ξˆ ξˆ c({x, y}) d1 (x, y)2 , ϕN (x)= ψN (x)= ξˆ y : {y,x}∈EN

c({x, y}) d1 (x, y).

(56)

ξˆ y : {y,x}∈EN

5.2. Link to periodized medium. Here we show that the diffusion matrix (51) can be bounded from below in terms of the average of the diffusion coefficient associated to the periodized random media. Our proof follows the arguments of [DFGW, Prop. 4.13], but additional technical problems are related to the randomness of geometry (absence of any lattice structure) and possible (albeit integrable) singularities of the mean forward velocity and infinitesimal mean square displacement. Proposition 7. Suppose that for 1 ≤ p ≤ 8 lim

N↑∞

ρc (C2N ) = 1 ˆξ (C2N ) + a2N

in Lp ( Nˆ , Pˆ c ) ,

(57)

± where ρc := EPˆ c (ξˆ (C1 )) and a2N := |N | = (2N − 1)d−1 . Then, for any t > 0, ˆ ˆ ξ ξ (58) lim EPˆ c mN ψN = EPˆ c ψc(11) , 0 N↑∞ ξˆ ξˆ ξˆ lim EPˆ c ϕN , et LN ϕN ξˆ = ϕc(1) , et Lc ϕc(1) ˆ c , (59) (11)

P0

mN

N↑∞

(1)

where ψc and ϕc are the first diagonal matrix element of the matrix ψc and the first component of the vector ϕc introduced in (50), respectively. Since Dc (rc , Ec ) is given by (51) and is a multiple of the identity (cf. Remark 2), the identities (58) and (59) combined with Fatou’s Lemma immediately imply: Corollary 1. Under the same hypothesis as above, Dc (rc , Ec ) ≥

ξˆ lim sup EPˆ c DN N↑∞

where 1d is the d × d identity matrix.

1d ,

(60)


45

Before giving the proof, let us comment on its assumptions. In Sect. 6 we will show that condition (57) is always satisfied. Due to (49), ρp < ∞ implies EPˆ c ( ξˆ (C1 )p ) < ∞ for any p > 0. As Pˆ c is ergodic, this implies the following ergodic theorem, an extension of [DV, Theorem 10.2]. We recall that a convex averaging sequence of sets {An } in Rd is a sequence of convex sets such that An ⊂ An+1 and An contains a ball of radius rn with rn → ∞ as n → ∞. Lemma 4. Suppose that ρp < ∞, p ≥ 1. Then, given a convex averaging sequence of Borel sets {An } in Rd , ξˆ (An ) → 1 ρc (An )


ξˆ (An ) → 1 ρc (An )

and

Pˆ c -a.s.

We will also need a bound on EPˆ c ((ξˆ (An )/ (An ))p ), uniformly in n, for a sequence of sets that does not satisfy the assumptions of Lemma 4. To this aim we note that, given a Borel set B ⊂ Rd which is a union of k non-overlapping cubes of side 1, one has p ∀ p ≥ 1. (61) EPˆ c ξˆ (B)/k ≤ EPˆ c ξˆ (C1 )p ≤ ρp , This follows from the stationarity of Pˆ c and the convexity of the function f (x) = x p , x ≥ 0. Proof of Proposition 7. Without loss of generality, we assume rc = 1. Note that, since Pˆ is stationary with finite intensity ρ1 , one has P-a.s. ξˆ (∂Ck ) = 0 for all k ∈ N. In what follows we hence may assume ξ to be as such, thus allowing to simplify notation since C2N ∩ supp(ξˆ ) = Cˇ 2N ∩ supp(ξˆ ). A key observation in order to prove (58) and (59) is the following identity, valid for any nonnegative measurable function h defined on Nˆ 0 . It follows easily from (12): EPˆ c ∀ B ∈ B(Rd ). (62) ξˆ (dx)h(Sx ξˆ ) = ρc (B) EPˆ c (h), B

0

From this identity we can deduce for any h ∈ L2 (N0 , Pˆ 0c ) that 1 lim EPˆ c ξˆ (dx)h(Sx ξˆ ) = EPˆ c (h) . (63) 0 N↑∞ ξˆ (C2N ) + a2N C2N −2 In fact, due to (62), it is enough to show that 1 1 ˆ ˆ EPˆ c ξ (dx)h(Sx ξ ) ↓ 0 , as N ↑ ∞. − ξˆ (C2N )+a2N ρc (C2N−2 ) C2N−2 (64) By applying twice the Cauchy-Schwarz inequality and by invoking (62), we obtain 2 l.h.s. of (64) ρ (C 2 ξˆ (C 2N−2 ) 2N−2 ) c ≤ EPˆ c −1 ˆξ (C2N ) + a2N ρc2 (C2N−2 )2 2 1 EPˆ c ξˆ (dx)h(Sx ξˆ ) ξˆ (C2N−2 ) C2N −2 ρ (C 2 ξˆ (C c 2N−2 ) 2N−2 ) −1 ≤ EPˆ c EPˆ c (h2 ) . 0 ˆξ (C2N ) + a2N ρc (C2N−2 )

46


At this point, (64) follows by applying the Cauchy-Schwarz inequality to the first expectation above and then applying (61) and the limit (57) for p = 4. ξˆ

N,

ξˆ

Let now hN be a function on VN such that for some constant c > 0 independent of ξˆ |hN (x)|

≤ c

 ξˆ (B1 (x))

if x ∈ QN ,

 |BN |

otherwise ,

ξˆ

ξˆ

a2N

ξˆ

ξˆ −

ξˆ +

ξˆ

where BN = BN ∪ BN and B1 (x) is the closed unit ball centered in x. Note that ψN ξˆ

and ϕN satisfy this inequality. We claim that the mean boundary contribution vanishes in the limit: 1 ξˆ lim EPˆ c for 1 ≤ p ≤ 4. (65) |hN (x)|p = 0 , N↑∞ ξˆ (C2N ) + a2N ξˆ ξˆ x∈VN \QN −1

In fact, the sum in (65) can be bounded by ξˆ

p

c a2N

|BN |p p a2N

+ cp

p

ξˆ (B1 (x))

(66)

.

ξˆ ξˆ x∈QN \QN −1

By the Cauchy-Schwarz inequality EPˆ c

a2N ξˆ (C2N ) + a2N

ξˆ

|BN |p

1

≤ E 2ˆ c

p

P

a2N

2 a2N

(ξˆ (C2N ) + a2N )2

1

E 2ˆ c

ξˆ

|BN |2p

P

2p

.

a2N

The first factor on the r.h.s. is negligible as N ↑ ∞ because of the limit (57) for p = 2, while the second factor is bounded, uniformly in N , because of (61). For the second summand in (66), we use twice the Cauchy-Schwarz inequality and invoke (62) to deduce 1 p 1 ξˆ (C2N \ C2N−2 ) EPˆ c ≤ E 2ˆ c ξˆ B1 (x) P (ξˆ (C2N ) + a2N )2 ξˆ (C2N ) + a2N ξˆ ξˆ x∈QN \QN −1 1

×E 2ˆ c P

1 ξˆ (C2N \ C2N−2 )

p 2 ξˆ B1 (x)

ξˆ

ξˆ

x∈QN \QN −1

1 1 2 2p ξˆ (C2N \ C2N−2 ) ≤ ρc (C2N \ C2N−2 )EPˆ c . E 2ˆ c ξˆ B1 (0) P0 (ξˆ (C2N ) + a2N )2 The last factor is bounded by hypothesis, the first one converges to 0 as N ↑ ∞ because of Lemma 4 and (57). ξˆ ξˆ (11) In order to prove (58) observe that ψc (Sx ξˆ ) = ψN (x) if x ∈ QN−1 . Therefore we can write 1 1 ξˆ ξˆ ξˆ mN (ψN ) = ξˆ (dx)ψc(11) (Sx ξˆ )+ ψN (x) . ξˆ (C2N )+a2N C2N−2 ξˆ (C2N )+a2N ξˆ

x∈VN \C2N−2

Mott Law as Lower Bound for a Random Walk in a Random Environment ξˆ

47

ξˆ

Now (58) follows easily from (63) and (65) with hN := ψN . Note that by the same arguments one can prove ˆ ˆ

ξ ξ = EPˆ c (|ϕc(1) |p ) < ∞ , 1 ≤ p ≤ 4 , (67) lim EPˆ c mN |ϕN (x)|p N↑∞

0

which will be useful below. In order to prove (59), we fix 0 < α < 1 and set M = 2N − 2[N α ], where [N α ] denotes the integer part of N α . Moreover, we define the hitting times ξˆ

ξˆ

τN (ω) = inf {s ≥ 0 : ωs ∈ C2N−2 } ,

ξˆ

ω = (ωs )s≥0 ∈ N = D([0, ∞), VN ).(68)

ξˆ ξˆ Recall the definitions of the distribution Pˆ cˆ , PN,x and PN given in Sects. 4 and 5.1. ξ ξˆ ξˆ ξˆ Thanks to the identity (et LN ϕN )(x) = E ξˆ ϕN (ωt ) , we can write PN,x

ξˆ

ξˆ

ξˆ

EPˆ c ϕN , et LN ϕN where

ξˆ mN

ξˆ ξˆ ξˆ = EPˆ c A1,N + A2,N + A3,N ,

ξˆ ξˆ ξˆ ξˆ A1,N = mN χ (x ∈ CM ) ϕN (x) E ξˆ ϕN (ωt ) , PN,x ˆξ ˆξ ˆξ ξˆ ξˆ A2,N = mN χ (x ∈ CM ) ϕN (x) E ξˆ χ (τN ≤ t) ϕN (ωt ) , PN,x ξˆ ξˆ ξˆ ξˆ ξˆ A3,N = mN χ (x ∈ CM ) ϕN (x) E ξˆ χ (τN > t) ϕN (ωt ) . PN,x

Then (59) follows from ξˆ lim EPˆ c A1,N = 0, N↑∞ ξˆ lim EPˆ c A2,N = 0,

N↑∞

ξˆ lim EPˆ c A3,N = ϕc(1) , et Lc ϕc(1) Pˆ c . 0 N↑∞

(69)

Let us first prove the first limit in (69). By several applications of Cauchy-Schwarz ξˆ ξˆ ξˆ inequality and due to the identity PN = mN (dx)PN,x , we get 1 1 E ˆ c Aξˆ ≤ E 2 mξˆ V ξˆ \ CM E 2 mξˆ ϕ ξˆ (x)2 E 1,N N N N N ˆc ˆc P

P 1 2

ξˆ

ξˆ

≤ E ˆ c mN VN \ CM P 1 2

= E ˆc P

ξˆ ξˆ mN VN

\ CM

P

ξˆ PN,x

ξˆ

ϕN (ωt )2

1 ˆ 41 ξ ξˆ E 4ˆ c mN ϕN (x)4 E ˆc E

P 1 2

E ˆc P

ξˆ mN

ξˆ ϕN (x)4 ξˆ

P

ξˆ PN

ξˆ

ϕN (ωt )4

, ξˆ

where the last identity follows from the stationarity of LN w.r.t. mN . Due to the dominated convergence theorem, the first expectation on the r.h.s. goes to 0, while the second expectation is bounded due to (67).

48


In order to prove the second limit in (69), we apply twice the Cauchy-Schwarz ξˆ inequality in order to obtain the bound EPˆ c A2,N by 1 ˆ ˆ 1 ξˆ 1 ξˆ

ξ ξ ξˆ ξˆ E 2ˆ c mN ϕN (x)2 E 4ˆ c E ξˆ ϕN (ωt )4 E 4ˆ c mN χ (x ∈ CM )PN,x (τN ≤ t) P

P

P

PN

(70) Again, because of stationarity and (67), the first two factors on the r.h.s. are bounded while the last one converges to 0 due to Lemma 5 below. ˆ = D([0, ∞), Nˆ 0 ) Finally we prove the last limit in (69). To this aim, given ξˆ ∈ and x ∈ CM , we set

τN,x (ξˆ ) = inf s ≥ 0 : x + Xs (ξˆ ) ∈ C2N−2 , (71) where Xs (ξˆ ) is defined as in (19). Note that for x ∈ CM ∩ supp(ξˆ ), ξˆ

ϕN (x) = ϕc(1) (Sx ξˆ ),

E

ξˆ PN,x

ξˆ ξˆ χ (τN > t) ϕN (ωt ) = EPˆ c χ (τN,x > t) ϕc(1) (ξˆt ) . Sx ξˆ

Therefore ˆ! ξˆ "

ξ . EPˆ c A3,N = EPˆ c mN χ (x ∈ CM ) ϕc(1) (Sx ξˆ ) EPˆ c χ (τN,x > t) ϕc(1) (ξˆt ) Sx ξˆ On the other hand, by applying the Cauchy-Schwarz inequality as in (70) and due to Lemma 5, we obtain ˆ! "

ξ = 0. lim EPˆ c mN χ (x ∈ CM ) |ϕc(1) (Sx ξˆ )| EPˆ c χ (τN,x ≤ t) |ϕc(1) (ξˆt )| N ↑∞

Sx ξˆ

The last two identities imply ˆ! ξˆ "

ξ lim EPˆ c A3,N = lim EPˆ c mN χ (x ∈ CM ) ϕc(1) (Sx ξˆ ) EPˆ c ϕc(1) (ξˆt ) . N ↑∞ N↑∞ Sx ξˆ (72) Observe now that (63) remains valid if the integral is performed on CM in place of C2N −2 (the arguments used in the proof there work also in this case) and the function h(ξˆ ) is defined as h(ξˆ ) = ϕc(1) (ξˆ ) EPˆ c ϕc(1) (ξˆt ) = ϕc(1) (ξˆ ) et Lc ϕc(1) (ξˆ ) . ξˆ

Note that h ∈ L2 (Nˆ 0 , Pˆ 0c ). Therefore we can conclude that the r.h.s. of (72) is equal to (1) (1) ϕc , et Lc ϕc Pˆ c . 0

ξˆ

Lemma 5. Let τN and τN,x be defined as in (68) and (71), and let M = 2N − 2[N α ]. Then ˆ! "

ξ ξˆ ξˆ lim EPˆ c mN χ (x ∈ CM ) PN,x τN ≤ t = 0, (73) N↑∞ ˆ! "

ξ = 0. (74) lim EPˆ c mN χ (x ∈ CM ) Pˆ c ˆ τN,x ≤ t N↑∞

Sx ξ


49

Proof. One can check by a coupling argument that the two expectations in (73) and (74) coincide: for each N ∈ N+ , ξˆ ∈ Nˆ and x ∈ CM ∩ supp(ξˆ ), one can define a probability ξˆ ˆ such that measure µ on × N

ξˆ

ˆ = P (A), µ(A × ) N,x

ξˆ

µ(N × B) = Pˆ c ˆ (B), Sx ξ

ξˆ

ˆ ∀A ∈ B(N ), ∀B ∈ B(),

ξˆ ξˆ and such that, µ almost surely, τN (ω) = τN,x (ξˆ ) and ωs = x+Xs (ξˆ ) for any 0 ≤ s < τN . ξˆ ξˆ Such a coupling µ implies Pˆ c ˆ τN,x ≤ t = PN,x τN ≤ t . Thus we need to prove Sx ξ only (73). Moreover, without loss of generality, we assume rc = 1. To this aim let us cover C2N−2 \ CM by disjoint cubes C1,i of side 1, i ∈ I , so that C2N−2 \ CM = ∪i∈I C1,i (the boundaries of these cubes are suitably chosen for them to be disjoint). Finally, given a positive integer n, we set

I∗n = {(l1 , . . . , ln ) ∈ I n : lj = lk if j = k}. ξˆ

For paths ω such that τN (ω) < ∞, let us define k = k(ω) as the number of different ξˆ

cubes C1,i , i ∈ I , visited by the particle in the time interval [0, τN (ω) ) and more k over we define by induction (i1 , . . . , ik ) ∈ I∗k , (x1 , . . . , xk ) ∈ C2N−2 \ CM with xj ∈ C1,ij ∀j : 1 ≤ j ≤ k, and (t1 , . . . , tk ) as follows: Let x1 be the first point reached in C2N −2 \ CM and t1 be the time spent in x1 before jumping away. The index i1 is characterized by the requirement that x1 ∈ C1,i1 . Suppose now that i1 , . . . , ij , x1 , . . . , xj and t1 , . . . , tj have been defined and that j < k. Then xj +1 is the first point ξˆ in C2N −2 \ CM ∪ C1,i1 ∪ · · · ∪ C1,ij visited during the time interval [0, τN (ω) ) and tj +1 is the time spent at xj +1 during such a first visit. Moreover, ij +1 is such that xj +1 ∈ C1,ij +1 . ξˆ

Now let Ti , i ∈ I and ξˆ ∈ Nˆ , be a family of independent exponential random variξˆ

ables (all independent from the above random objects) and such that Ti has parameter #1,i ), where ξˆ C #1,i = {y ∈ Rd : dist(y, C1,i ) ≤ 1 }. C Since, given ξˆ , k and (x1 , . . . , xk ), tj (1 ≤ j ≤ k) are independent exponential variables #1,ij and since k ≥ kmin := [N α ] − 1, we and tj has parameter not larger than ξˆ C obtain ˆ! "

ξ ξˆ ξˆ EPˆ c mN χ (x ∈ CM ) PN,x τN ≤ t =

|I |

n=kmin l∈I∗n

≤

|I | n=kmin l∈I∗n

ˆ! ξ EPˆ c mN χ (x∈CM ) y∈

$n

"

ξˆ ξˆ PN,x τN ≤ t, k = n, xl = yl , 1 ≤ l ≤ n

ξˆ j =1 C1,lj ∩VN

ˆ! " ξ ξˆ EPˆ c mN χ (x ∈ CM ) PN,x k = n, i1 = l1 , . . . , in = ln

ξˆ

ξˆ × Prob Tl1 + · · · + Tln ≤ t ,

(75)

50


where the last inequality follows from the bound ξˆ ξˆ ξˆ ξˆ PN,x τN ≤ t | k = n, x1 = y1 , . . . , xn = yn ≤ Prob Tl1 + · · · + Tln ≤ t . In order to estimate the probability in the r.h.s., we use an argument to that similar #1 ) , where of the proof of Proposition 1 in Appendix C. Let us define m := EPˆ c ξˆ (C #1 = {y ∈ Rd : dist(y, C1 ) ≤ 1}. Given κ > 0 and l ∈ I∗n as above, we define C A = A(κ, l) as follows

#1,lj > κ m > n . A = ξˆ ∈ Nˆ : j : 1 ≤ j ≤ n and ξˆ C 2 Then, by the Chebyshev inequality and the stationarity of Pˆ c , 2 #1,lj > κ m ≤ 2 Pˆ c ξˆ (C #1 ) > κ m → 0, Pˆ c A ≤ EPˆ c j : 1 ≤ j ≤ n and ξˆ C n as κ → ∞. Note that the complement Ac of A can be written as

#1,lj ≤ κ m ≥ n Ac = ξˆ ∈ Nˆ : j : 1 ≤ j ≤ n and ξˆ C , 2 ∗ where [n/2]∗ is defined as n/2 for n even and as (n + 1)/2 for n odd. If ξˆ ∈ Ac then at ξˆ ξˆ least n2 ∗ of the exponential variables Tl1 , . . . ,Tln have parameter not larger than κ m. Then, by a coupling argument (e.g. Appendix C), we get for all ξˆ ∈ Ac , ξˆ ξˆ Prob Tl1 + · · · + Tln ≤ t ≤ e−κ mt

∞ r=[n/2]∗

(κ m t)r =: φ(κ, n) . r!

Due to the above estimates and since n ≥ kmin = [N α ] − 1, we get #1 ) > κ m + φ(κ, N α ) . r.h.s. of (75) ≤ 2 Pˆ c ξˆ (C The lemma follows by taking first the limit N ↑ ∞ and then the limit κ ↑ ∞.

5.3. Random resistor networks. We conclude this section by pointing out that the diffuξˆ

sion coefficient DN of the periodized medium can be expressed in terms of the effective ξˆ

ξˆ

conductance of the graph (V N , E N ) when assigning suitable bond conductances. More ξˆ

ξˆ

precisely, consider the electrical network given by the graph (V N , E N ), where the bond ξˆ

{x, y} ∈ E N has conductivity c({π(x), π(y)}) with c({·, ·}) defined in (54). Then, the ξˆ

− to effective conductance GN of this network is defined as the current flowing from N + − + N when a unit potential difference between N to N is imposed. It can be calculated from Ohm’s law and the Kirchhoff rule as follows. Let the electrical potential V (x) − + vanish on the left border N , be equal to 1 on the right border N , and satisfy:

c({π(x), π(y)}) V (y) − V (x) = 0 for any ξˆ

y : {y,x}∈E N

ξˆ

x ∈ QN .


51

Then the effective conductance is given by the current flowing through the surfaces {x ∈ [−N, N]d : x (1) = ±N }: ξˆ

GN =

1 − V (x) .

V (x) =

ξˆ − x∈BN

(76)

ξˆ + x∈BN

ξˆ

By a well-known analogy it is linked to the diffusion coefficient DN (see e.g. [DFGW, Prop. 4.15] for a similar proof): Proposition 8. One has ξˆ

DN =

8 N2 ξˆ |V N |

ξˆ

GN .

(77)

6. Percolation Estimates Let us set Fr := FRd \Cr and recall that ρc = ρ δc with δc = ν([−Ec , Ec ]).

6.1. Point density estimates. Here we show how the ergodic properties of Lemma 4 combined with the hypothesis (H1) or (H2) imply (57). Proposition 9. Suppose that ρ8 < ∞ and that the hypothesis (H1) or (H2) holds. For 1 ≤ p ≤ 8, lim

N↑∞

ρc (CN ) = 1 ˆξ (CN ) + aN


(78)

where aN = (N − 1)d−1 . We will first prove the following criterion. Lemma 6. Property (78) holds if one has, for some 0 < ρ < ρ, lim N p Pˆ ξˆ (CN ) ≤ ρ N d = 0 .

N↑∞

(79)

Proof. We first check that (79) implies that, for some 0 < ρ

< ρ δc , lim N p Pˆ c ξˆ (CN ) ≤ ρ

N d = 0 .

N↑∞

(80)

If δc = 1, this is clearly true so let us suppose that 0 < δc < 1. Set δ˜c = 1 − δc . If Cjk denotes the binomial coefficient, we have

52


N d ] [ρ c ˆ

d ˆ ˆ ξˆ (CN ) = k) P ξ (CN ) ≤ ρ N = P(

k=0 ∞

+

k=[ρ

N d ]+1 [ρ N d ]

≤

k

ˆ ξˆ (CN ) = k) P(

j =k−[ρ

N d ] k

ˆ ξˆ (CN ) = k)+ sup P(

k>[ρ N d ]

k=0

j k−j Cjk δ˜c δc

Cjk δ˜c δc

j k−j

j =k−[ρ

N d ]

≤ Pˆ ξˆ (CN ) ≤ ρ N d +exp(−c[ρ N d ](δc − ρ

/ρ )2 ), where the last inequality, given ρ

< δc ρ , follows from a standard large deviation type estimate for Bernoulli variables with some c > 0. Multiplying by N p , (79) thus implies (80). Now set AN = {ξˆ : ξˆ (CN ) ≤ ρ

N d }. Then, for some c > 0 independent of N , p ρc (CN ) p fN (ξˆ ) := − 1 ≤ c ρc N p χAN (ξˆ ) + fN (ξˆ ) χAcN (ξˆ ) . ξˆ (CN ) + aN Integrating w.r.t. Pˆ c , the first term vanishes in the limit N ↑ ∞ because of (80). For the second, let us first note that Lemma 4 implies that limN↑0 fN χAcN = 0 holds Pˆ c -a.s. Furthermore, |fN χAcN | ≤ c

< ∞ uniformly in N so that the dominated convergence theorem assures that limN↑0 EPˆ c (fN χAcN ) = 0. Proof of Proposition 9. Due to Lemma 6 we only need to show that (79) is satisfied for some ρ < ρ. This is trivially true if (H1) holds. Hence let us consider the case where (H2) holds. This implies E ˆ (f | Fr ) − E ˆ (f ) ≤ f ∞ r d r d−1 h(r2 − r1 ) , ˆ P-a.s. , (81) 2 1 2 P P where f is a bounded FCr1 –measurable function.

Let C1i denote the unit cube centered at i ∈ Zd and Cˇ 1i be the interior of C1i . Let j IN ⊂ Zd be such that CN = ∪i∈IN C1i and Cˇ 1i ∩ Cˇ 1 = ∅ if i = j . Hence |IN | = N d . ˜ ˜ Given M > 0, set Y˜i (ξˆ ) = min{ξˆ (Cˇ 1i ), M 2 } and Yi = Yi − EPˆ (Yi ). Note that Yi is centered, FC i –measurable and Yi ∞ ≤ M. We choose M large enough so that 1 ρ˜ := E ˆ (Y˜i ) > ρ which is possible because limM↑∞ E ˆ (Y˜i ) = ρ > ρ . Now P

ξˆ (CN ) ≤ ρ N d

⊂

  

i∈IN

 

P

    Yi (ξˆ ) ≥ (ρ˜ − ρ )N d . Y˜i (ξˆ ) ≤ ρ N d ⊂    i∈IN

Hence it is sufficient to show that, for a > 0,   Yi ≥ aN d  = 0 . lim N p Pˆ  N↑∞

i∈IN

(82)


By the Chebyshev inequality, one has for any even q ∈ N:   1 Pˆ  Yi ≥ aN d  ≤ q dq EPˆ Yi1 · · · Yiq . a N

53

(83)

i1 ,... ,iq ∈IN

i∈IN

We will now bound the sum in the r.h.s. of (83). Let us define the norm x = max{|x (k) | : 1 ≤ k ≤ d} on Rd (recall that x (k) is the k th component of x) and introduce the notation i = (i1 , . . . , iq ), I N = (IN )q , and rj (i) = min{ij − ik : k = 1, . . . , q, k = j }. If r1 (i) = · · · = rN (i) = 0, i.e., if each point appears at least twice in (i1 , . . . , iq ), then use the bound EPˆ (Yi1 · · · Yiq ) ≤ M q . The number of i ∈ I N satisfying this property is at most cN dq/2 (here and below c is a varying constant depending only on d and on q). i Suppose now that, say, r1 (i) = r ≥ 1. Then the open cubes Cˇ 1i2 , . . . , Cˇ 1q are contained i1 in A := Rd − C2r−1 and thus Yi2 , . . . , Yiq are FA -measurable. Using the conditional expectation, (81) and the fact that Yi1 is centered and Yij ∞ ≤ M, EPˆ (Yi1 · · · Yiq ) ≤ M q−1 EPˆ EPˆ (Yi1 |FA ) ≤ M q h(2r − 2) (2r − 1)d−1 . Note that EPˆ (Yi1 · · · Yiq ) is invariant under permutations of the indices i1 , . . . , iq . Hence

EPˆ (Yi1 · · · Yiq ) ≤ c

N

EPˆ (Yi1 · · · Yiq )

r=0 i∈K N (r)

i∈I N

≤ c M q N dq/2 + c M q

N

h(2r − 2) (2r − 1)d−1 |K N (r)| ,

r=1

where K N (r) = {i ∈ I N : r1 (i) = r, r2 (i) ≤ r, . . . , rq (i) ≤ r}. One has |K N (r)| ≤ c N dq/2 r dq/2−1 .

(84)

In fact, on the set of points i1 , . . . , iq (treated as distinguishable) let us define a graph structure by connecting two points i ∼ j with a bond whenever i − j ≤ r. We call G(i) the resulting graph. Note that each connected component of G(i) has cardinality at least 2 whenever i ∈ K N (r), therefore G(i) has at most q/2 connected components. We claim that, given 1 ≤ l ≤ q/2, {i ∈ K (r) : G(i) has l connected components} ≤ c(N/r)dl r dq−1 . (85) N In order to prove (85), suppose that the connected component containing i1 has cardinality k1 , while the other components have cardinality k2 , . . . , kl respectively. Each component can be built by first choosing one of its points in IN (there are N d possible choices), then its neighboring points w.r.t. ∼ (for each such neighboring point there are at most cr d possible choices) and then iteratively adding neighboring points w.r.t. ∼. Therefore, the j th component can be built in at most cN d r d(kj −1) ways. If j = 1, since i1 has a neighboring point at distance exactly r, the upper bound can be improved by cN d r d−1+d(k1 −2) . Summing over all possible k1 , . . . , kl such that k1 + · · · + kl = q, one gets (85). Since r ≤ N, (85) implies |K N (r)| ≤

q/2 l=1

c(N/r)dl r dq−1 ≤ c(N/r)dq/2 r dq−1 ,

54


thus concluding the proof of (84). It implies ∞ q dq/2 1 + c

EPˆ (Yi1 · · · Yiq ) ≤ M N h(2r − 2) r dq/2+d−2

.

(86)

r=1

i∈I N

Provided that dq > 2p and the sum over r converges, that is, if dq/2 − d ≤ 8, we get the result (79) by combining (82), (83), and (86). Choosing for q the smallest even integer larger than 16/d, (79) is true for 1 ≤ p ≤ 8 and dq/2 − d ≤ 8 as required. 6.2. Domination. Due to Proposition 9, we may apply the results of Sect. 5 so that combining with Proposition 3,   2 8 N ˆ ξ D ≥ ν([−Ec , Ec ]) e−rc −4βEc lim sup EPˆ c  ˆ GN  . (87) ξ N→∞ |V N | ξˆ

rc from below, we will discretize the In order to bound the conductance GN for N space Rd using cubes of appropriate size and spacing. Given r2 ≥ r1 > 0, let us then consider the following functions on Nˆ : σj (ξˆ ) := χ ξˆ (Cr1 + r2 j ) > 0 , j ∈ Zd . (88) They form a random field = (σj )j ∈Zd on the probability space (Nˆ , Pˆ c ). If Pˆ is a PPP, the σj are independent random variables. For a process with finite range correlations, this independence can also be assured by an adequate choice of r1 and r2 , but in general the σj are correlated. The side length r1 and spacing r2 are going to be chosen of order O(rc ) in such a way that all points of neighboring cubes have an euclidean distance less ξˆ

ξˆ

than rc and they are thus connected by an edge of the graph (V N , E N ). Next note that the σj take values in {0, 1}. We shall consider the associated site percolation problem with bonds between nearest neighbors only [Gri]. For this purpose, p we shall compare with a random field Z p = (zj )j ∈Zd of independent and identically p p distributed random variables with Prob(zj = 1) = p and Prob(zj = 0) = 1 − p. In this independent case, it is well-known that there is a critical probability pc (d) ∈ (0, 1) such that, if p > pc (d), there is almost surely a unique infinite cluster, while for p < pc (d) there is almost surely none [Gri]. We will need somewhat finer estimates for the supercritical regime. Let |.| denote the Euclidean norm in Rd . A left-right crossing (LR-crossp ing) with length k − 1 of C2N of a configuration (zj )j ∈Zd is a sequence of distinct points p y1 , . . . , yk in C2N ∩ Zd such that |yi − yi+1 | = 1 for 1 ≤ i < k, zyi = 1 for 1 ≤ i ≤ k, (1) (1) (1) (s) (s) y1 = −N , yk = N, −N < yi < N for 1 < i < k, and finally yi = yj for any s ≥ 3 and for 1 ≤ i < j ≤ k. Two crossings are called disjoint if all the involved yj ’s are distinct. In the same way, one defines disjoint LR-crossings for (σj )j ∈Zd . Note that this definition of LR-crossings for d ≥ 3 uses LR-crossings in 2-dimensional slices only. For the random field Z p , the techniques of [Gri, Sect. 2.6 and 11.3] transposed to site percolation imply that, if p > pc (2), there are positive constants a = a(p), b = b(p), and c = c(p) such that for all N ∈ N+ , Prob Z p has less than bN d−1 disjoint LR–crossings in C2N ≤ c e−a N . (89)


55

In order to transpose this result on Z p to one for , we will use the concept of stochastic dominance [Gri, Sect. 7.4]. One writes ≥st Z p whenever EPˆ c (f ()) ≥ EProb (f (Z p )) ,

(90)

for any bounded, increasing, measurable function f : {0, 1}Z → R (recall that a function is increasing if f ((zj )j ∈Zd ) ≥ f ((zj )j ∈Zd ) whenever zj ≥ zj for all j ∈ Zd ). As the event on the l.h.s. of (89) is decreasing, ≥st Z p with p > pc (2) implies that for all N ∈ N+ , Pˆ c (σj )j ∈Zd has less than bN d−1 disjoint LR–crossings in C2N ≤ c e−a N . (91) d

Moreover, let us call the configurations ξˆ in the set on the l.h.s. N -bad, those in the complementary set N-good. For every N –good ξˆ , let us fix a set of at least configuration bN d−1 disjoint LR–crossings in C2N for σj (ξˆ ) j ∈Zd and denote it CN (ξˆ ). Given an LR–crossing γ in C2N , we write L(γ ) for its length. Note that, since the LR–crossings are self–avoiding, L(γ ) = |supp(γ )| − 1 for all γ ∈ CN (ξˆ ). Moreover, since paths in CN (ξˆ ) are disjoint and have support in C2N ∩ Zd , γ ∈CN (ξˆ ) |supp(γ )| ≤ (2N + 1)d . The above estimates imply that γ ∈CN (ξˆ ) L(γ ) ≤ (2N + 1)d ≤ (4N )d . In particular, due to the Jensen inequality, for any N–good configuration ξˆ ,

1 |CN (ξˆ )|2 b2 N d−2 . ≥ ≥ L(γ ) 4d γ ∈CN (ξˆ ) L(γ ) ˆ γ ∈CN (ξ )

(92)

This will allow us to prove a lower bound on (87). Hence we need the following criterion for domination. Lemma 7. ≥st Z p holds with r1 = r, r2 = 2r if Pˆ and r > 0 satisfy the following: There exists ρ > 0 such that r d ν([−Ec , Ec ]) ≥ −

ln(p/2) , ρ

(93)

and 3p Pˆ ξˆ (Cr ) < ρ r d F2 r ≤ 1 − , 2

ˆ P–a.s.

(94)

Proof. The proof is based on the following criterion [Gri, Sect. 7.4]: if for any finite subset J of Zd , i ∈ Zd \ J and zj ∈ {0, 1} for j ∈ J satisfying Pˆ c (σj = zj ∀j ∈ J ) > 0, one has Pˆ c (σi = 1 | σj = zj ∀ j ∈ J ) ≥ p ,

(95)

then ≥st Z p . Hence let J, i, zj be as above and set δ˜c := 1 − δc and J0 := {j ∈ J : zj = 0} as well as J1 := {j ∈ J : zj = 1}. Moreover, given k ∈ NJ0 and s ∈ NJ+1 , let W (k, s) := ξˆ ∈ Nˆ : ξˆ (Cr + 2rj ) = kj ∀j ∈ J0 , ξˆ (Cr + 2rj ) = sj ∀j ∈ J1 .

56


Then Pˆ c (σi = 0 | σj = zj ∀j ∈ J ) n$ $ ˜ ˜kj j ∈J (1 − δ˜csj ) ˆ ˆ J n∈N P ξ (Cr + 2ri) = n , W (k, s) δc j ∈J0 δc k∈NJ0 1 s∈N+1 = . $ kj $ sj ˜ ˜ ˆ W (k, s) J1 P j ∈J0 δc j ∈J1 (1 − δc ) k∈NJ0 s∈N+

Within this, we can, moreover, replace Pˆ ξˆ (Cr + 2ri) = n , W (k, s) = Pˆ ξˆ (Cr + 2ri) = n | W (k, s) Pˆ W (k, s) . Finally, note that W (k, s) ∈ FA , where A = Rd \ C2r + 2ri . As δ˜c ≤ e−δc , we obtain the following bound

d Pˆ ξˆ (Cr + 2ri) = n | W (k, s) δ˜cn ≤ Pˆ ξˆ (Cr + 2ri) < ρ r d | W (k, s) + e−δc ρ r . n∈N

ˆ (93) and (94) imply (95). Due to the stationarity of P,

6.3. Proof of Theorem 1(ii). We fix p > pc (2) and ρ < ρ. Then, given Ec , we choose rc such that (93) is satisfied, i.e. rc = c(Ecα+1 )−1/d for some constant c. As rc ↑ ∞ in the limit of low temperature, we can next check that the condition (94) also holds. This is trivial for a process with a uniform lower bound (4) on the point density. For a mixing point process satisfying (5), one has Pˆ ξˆ (Cr ) < ρ r d F2r ≤ Pˆ ξˆ (Cr ) < ρ r d + r d (2r)d−1 h(r) , Pˆ − a.s. Due to the hypothesis on h, the second term converges to 0 in the limit r ↑ ∞. If ρ < ρ, the first one can be bounded by the Chebychev inequality: ξˆ (C ) ξˆ (C ) 1 r r

d

ˆ ξˆ (Cr ) ≤ ρ r ) ≤ Pˆ ˆ P( P(dξ ) − ρ > ρ − ρ ≤ − ρ .

(Cr ) (Cr ) ρ−ρ By Lemma 4, the expression on the r.h.s. can be made arbitrarily small by choosing r sufficiently large, thus implying that (94) is satisfied for r sufficiently large. In conclusion, due to Lemma 7, (91) holds for r large enough, i.e. temperature low enough. We fix such a value r satisfying (91) and call it rp . Consider the variables (σj )j ∈Zd defined for r1 = rp , r2 = 2rp and choose rc = 1 (d+8) 2 rp . This assures that, if neighboring sites j and j in Zd have σj (ξˆ ) = σj (ξˆ ) = 1, then Crp + 2j rp and Crp + 2j rp contain each a point and these points are separated by a distance less than rc . Two neighboring sites j and j in Zd such that σj (ξˆ ) = σj (ξˆ ) = 1 define a bond of the site percolation problem. To such a bond one can associate (at least) two points x ∈ supp ξˆ ∩ (Crp + 2j rp ) and y ∈ supp ξˆ ∩ (Crp + 2j rp ) separated by a distance less than rc . Given N integer, we define Nˆ := max n ∈ N : Crp + 2rp j ⊂ C2[rp N] , ∀j ∈ C2n ∩ Zd .


57

Note that Nˆ = O(N ). If j, j ∈ C2Nˆ ∩ Zd , then the above associated points x and y are ξˆ

ξˆ

linked by an edge of the graph (V [rp N] , E [rp N] ) defined in Sect. 5.1. Each LR-crossing of C2Nˆ for the site percolation problem gives in a natural way a connected path of edges ξˆ

ξˆ

± of the graph (V [rp N] , E [rp N] ) which connects the boundary faces N .

ξˆ ˆ For a N–good configuration ξˆ , we now bound the conductance G[rp N] from below. For

ξˆ + ˆ− , N }, this purpose, let us consider the random resistor network with vertices Q[rp N] ∪{ˆ N ξˆ

ξˆ

where unit conductances are put on all edges in E [rp N] with vertices in Q[rp N] as well as ξˆ ±

± and all points of B[rp N] . This new network between the two added boundary points ˆ N is obtained from the one of Sect. 5.1 upon placing superconducting wires between all + − + and [r so that they can be identified with a single point ˆ N and vertices of [r p N] p N]

ξˆ − − . The conductance gN of this new network (defined as the current flowing from ˆ N ˆ N + to ˆ N when a unit potential difference is imposed between these two points) is precisely ξˆ

± have the same potential (0 or 1 respectively) equal to G[rp N] because all points of [r p N] ξˆ ±

and each has links to all points of B[rp N] with equal conductances summing up to 1. ξˆ

In order to bound gN from below, we now invoke Rayleigh’s monotonicity law which states that eliminating links (i.e. conductances) from the network always lowers its conductance. ˆ For a given N-good configuration ξˆ , we cut all links but those belonging to the family + − and ˆ N is of disjoint paths associated to CNˆ (ξˆ ). Each of these paths γ connecting ˆ N self-avoiding and hence has a conductance bounded below by 1/L(γ ). As all the paths ξˆ of C ˆ (ξˆ ) are disjoint and they are connecting ˆ + and ˆ − in parallel, g is the sum of N

N

N

N

ξˆ

the conductances of all paths and it follows from (92) that gN ≥ c(b)N d−2 for some positive constant c(b) depending on b. We therefore deduce that     2 2  [rp N ]   [rp N]  ξˆ EPˆ c  ˆ G[rp N]  ≥ c(b) EPˆ c  ˆ N d−2 χ (ξˆ is Nˆ –good ) . ξ ξ |V [rp N] | |V [rp N] |

Due to (91) and Proposition 9 the r.h.s. converges to a positive value. Combining this with the estimate (87) we obtain − α+1 d

D ≥ C ν([−Ec , Ec ]) e−rc −4βEc ≥ C Ec1+α exp(−cEc

− 4βEc ) ,

where C and C are positive constants. Optimizing the exponent leads to Ec = c β − α+1+d which completes the proof. d

A. Proof that the Random Walk is Well-Defined Proposition 10. Let P be ergodic with ρ2 < ∞. Then for P0 –almost all ξ ∈ N0 and for ξ all x ∈ ξˆ , there exists a unique probability measure Px on ξ = D([0, ∞), supp(ξˆ )) of

58

A. Faggionato, H. Schulz-Baldes, D. Spehner ξ

a continuous–time random walk starting at x whose transition probabilities pt (y|x) := ξ ξ ξ Px (Xs+t = y|Xs = x), x, y ∈ ξˆ , t ≥ 0, s ≥ 0 satisfy the infinitesimal conditions (C1) and (C2). Proof. The uniqueness follows from [Bre, Chap. 15]. In order to prove existence, due to the construction described in Sect. 3.2, we only need to prove (27) for P0 –almost all ξ and for any x ∈ ξˆ . According to [Bre, Prop. 15.43], condition (27) is implied by the following one: P˜ xξ

∞ n=0

1 =∞ = 1. λX˜ ξ (ξ )

(96)

n

Due to the identity ξ P˜ 0

∞ n=1

1 ξ 1 = ∞ X˜ 1 = x = P˜ xξ =∞ , λX˜ ξ (ξ ) λX˜ ξ (ξ ) ∞

n=0

n

∀ x ∈ ξˆ ,

n

the proof will be completed if we can show (96) for x = 0 and P0 –almost all ξ and, in particular, if we can show ∞ ∞ 1 1 ξ P˜ =∞ = Q0 (dξ ) P˜ 0 =∞ =1, λ0 (ξn ) λX˜ ξ (ξ ) n=0

n=0

n

˜ P˜ , and Q0 are defined in Sect. 3.3. Due to Proposition 2, P˜ where the distributions P, 0 is ergodic and therefore, according to ergodic theory (see [Ros, Chap. IV]), ξ

N 1 1 1 1 = = EQ0 , N↑∞ N λ0 (ξn ) λ0 EP0 (λ0 )

lim

˜ P-almost surely,

n=0

thus allowing to conclude the proof.

Remark 3. Explosions are excluded if supx∈ξˆ λx (ξ ) < ∞ (in such a case (96) is always true), but this simple criterion is typically not satisfied in our case. For instance, for a PPP e−|x−y| ≥ e−4β−1 sup ξˆ (C1 + x) = ∞, P0 -a.s. sup λx (ξ ) ≥ e−4β sup x∈ξˆ

x∈ξˆ y∈ξˆ ,|y−x|≤1

x∈ξˆ

B. Proof of Lemma 1 Note that the statements (ii) and (iii) of Lemma 1 are proved in [FKAS, Corollary 1.2.11 and Theorem 1.3.9] in dimension d = 1. The proof below is valid for any dimension d. Proof of Lemma 1. (i) Let h(ξ, ξ ) := k(ξ, ξ ) − k(ξ , ξ ). By the definition (11) of the Palm distribution P0 , ∀N > 0, ∀A ∈ B(Rd ) and for any non negative measurable function f , 1 ˆ (dy) P(dξ ) ξ ξˆ (dx)f (Sy ξ, Sx ξ ) . P0 (dξ ) ξˆ (dx)f (ξ, Sx ξ )= ρN d A CN A+y (97)


59

The antisymmetry of h(ξ, ξ ) and the identity above imply 1 ˆ (dy) P(dξ ) ξ ξˆ (dx)h(Sy ξ, Sx ξ ).(98) P0 (dξ ) ξˆ (dx) h(ξ, Sx ξ )= ρN d Rd CN Rd \CN Let us split the last integral into two integrals over Rd \ CN+√N and over CN+√N \ CN . Using (97) again, 1 ˆ ˆ P(dξ ) ξ, S ξ ) ξ (dy) ξ (dx)h(S y x ρN d CN Rd \CN +√N ≤ P0 (dξ ) ξˆ (dx) |k(ξ, Sx ξ )| + |k(Sx ξ, ξ )| , Rd \C√N

which converges to zero as N → ∞ by the dominated convergence theorem. The same holds for 1 ˆ (dy) ˆ (dx) h(Sy ξ, Sx ξ ) , P(dξ ) ξ ξ ρN d CN C √ \CN N+ N

since, due to (97), it can be bounded by 1 ˆ P(dξ ) ξ (dx) ξˆ (dy) |k(Sy ξ, Sx ξ )| + |k(Sx ξ, Sy ξ )| √ d ρN d CN + N \CN R √ d d (N + N ) − N P0 (dξ ) ξˆ (dy) |k(Sy ξ, ξ )| + |k(ξ, Sy ξ )| . = d d N R Letting N → ∞ in (98) leads to the result. (ii) Since ∈ B(N ) is translation invariant, one has χ0 (Sx ξ ) = χ (ξ ) for all ξ ∈ N and x ∈ ξˆ . The above remark together with (11) gives 1 1 P(dξ ) ξˆ (C1 ) . ξˆ (dx)χ0 (Sx ξ ) = P(dξ ) P0 (0 ) = ρ ρ C1 Comparing with (1), this yields P0 (0 ) = 1 if P() = 1. Reciprocally, always due to (1), if P0 (0 ) = 1, one gets ξˆ (C1 ) = 0 for P–almost all ξ ∈ N \ , and by translation invariance ξ = 0 for P–almost all ξ ∈ N \ , thus * implying that P() = 1. (iii) Let us suppose that P0 (A) = P0 (B) > 0 and set := x∈Rd Sx B. This is a translation-invariant Borel subset of N (see Lemma 8) and B ⊂ ∩ N0 ⊂ A. In particular, P() ∈ {0, 1} by the ergodicity of P. Since χB (Sy ξ ) ≤ χ (ξ ) for all ξ ∈ N and y ∈ Rd , it follows from (11) that 1 1 ξˆ (dy) χB (Sy ξ ) ≤ P(dξ ) P(ξ )ξˆ (C1 ) . P0 (B) = ρ N ρ C1 Therefore, P() = 0 would imply that P0 (B) = 0, in contradiction with our assumption. Thus P() = 1. But ∩ N0 ⊂ A, therefore the statement follows from (ii). (iv) The thesis follows by observing that (11) implies k k 1 1 ˆ ˆ ˆ EP0 P(dξ ) ξ (dx) P(dξ )ξˆ (C1 ) ξ (Aj ) = ξ (Aj +x) ≤ ξˆ (A˜ j ) ρ N ρ N C 1 j =1 j =1 j =1 k

60


k+1 and by applying the estimate a1 · · · ak+1 ≤ c(k +1) (a1k+1 +· · ·+ak+1 ), a1 , . . . , ak+1 ≥ 0. * Lemma 8. Let A ∈ B(N0 ). Then x∈Rd Sx A ∈ B(N ).

Proof. Let us introduce the following lexicographic ordering on Rd : x ≺ y if and only if either |x| < |y| or |x| = |y| and there is k, 1 ≤ k ≤ d, such that x (k) < y (k) and x (l) = y (l) for l < k (here x (k) is the k th component of the vector x). Given ξˆ ∈ Nˆ , one can then order the support of ξˆ according to ≺: {y1 (ξˆ ), y2 (ξˆ ), . . . , yN (ξˆ )} if N := ξˆ (Rd ) < ∞ , ˆ supp(ξ ) = otherwise , {yj (ξˆ )}j ∈N+ where yj ≺ yk whenever j < k. For any n ∈ N, let xn : Nˆ → Rd then be defined as yn (ξˆ ) if n ≤ ξˆ (Rd ) , xn (ξˆ ) = yN (ξˆ ) if n > N := ξˆ (Rd ) . Using an adequate family of finite disjoint covers of Rd and the fact that ξˆ ∈ Nˆ → ξˆ (B) is a Borel function for every Borel set B ⊂ Rd , one can verify that xn is a Borel function for each n. Moreover, supp(ξˆ ) = {xn (ξˆ ) : n ∈ N} for all ξˆ ∈ Nˆ . Due to the definition of the Borel sets in N and Nˆ , the map π : N → Nˆ given by π(ξ ) = ξˆ is Borel, and by [MKM, Sect. 6.1] the function F : Rd × N → N given by F (x, ξ ) = Sx ξ is even continuous. Hence we conclude that Hn (ξ ) := F xn (ξˆ ), ξ = Sxn (ξˆ ) ξ , Hn : N → N0 , function. Now is a Borel function. Its restriction Hˆ n : N0 → N0 is then also * a Borel −1 (A) is a Borel ˆ given a Borel subset A of N0 , we conclude that (A) := ∞ H n=1 n subset in N0 . One can check that (A) = {ξ : ξ = Sx η for some η ∈ A and x ∈ ηˆ }. Since N0 is a Borel subset of N , it follows that (A) is a Borel subset of N as is H1−1 (A) since H1 is a Borel function. The identity + Sx A , H1−1 (A) = x∈Rd

now completes the proof. C. Proof of Proposition 1

Proof of Proposition 1. Due to the construction of the dynamics given in Sect. 3.2, ξ ξ EP0 EPξ |Xt |γ = EP0 EP˜ ξ ⊗Q |X˜ ξ |γ . 0

0

n∗ (t)


61

Let p, q > 1 be such that 1/p + 1/q = 1. Due to the Hölder inequality, ξ EP0 EP˜ ξ ⊗Q |X˜ ξ

n∗ (t)

0

≤

|γ

∞

=

ξ ξ EP0 EP˜ ξ ⊗Q |X˜ nξ |γ χ n∗ (t) ≥ 1 χ n∗ (t) = n 0

n=1

∞

1 1 ξ q p ξ EP0 EP˜ ξ ⊗Q |X˜ nξ |γ q χ n∗ (t) ≥ 1 EP0 P˜ 0ξ ⊗ Q (n∗ (t) = n) . 0

n=1 ξ

ξ ξ 0,X˜ 0

Clearly, n∗ (t) ≥ 1 means T

≤ t. It then follows from the estimate 1 − e−u ≤ u,

u ≥ 0, that ξ EP˜ ξ ⊗Q |X˜ nξ |γ q χ n∗ (t) ≥ 1 = 1 − e−λ0 (ξ )t EP˜ ξ |X˜ nξ |γ q ≤ λ0 (ξ )t EP˜ ξ |X˜ nξ |γ q . 0

0

0

(99) We then obtain ∞ ξ EP0 EPξ |Xt |γ ≤C 0

1/q Q0 (dξ )EP˜ ξ |X˜ nξ |γ q 0

n=1

1/p ξ ξ ˜ P0 (dξ )P0 ⊗ Q n∗ (t)=n ,

(100)

with C = [t EP0 (λ0 )]1/q . We claim that there is a (time-independent) constant C > 0 such that (101) Q0 (dξ ) EP˜ ξ |X˜ nξ |γ q ≤ C nγ q . 0

To show this, let us note first that, given X˜ 0 = 0, by another application of the Hölder inequality, ξ

n−1 ξ γ q ξ n−1 ξ γ q ξ γ q−1 X˜ X˜ ˜ ˜ ˜ ξ γ q , X = − X ≤ n n m m+1 m+1 − Xm m=0

m=0

where it has been assumed that γ q > 1. One can derive from the stationarity of P˜ and Remark 1 that ξ ξ γ q ξ γ q ˜ ˜ = Q0 (dξ ) EP˜ ξ X˜ 1 := C

Q0 (dξ ) EP˜ ξ Xn+1 − Xn 0

0

for any n ∈ N. One concludes the proof of (101) by checking that C is finite. Actually, by (26), EP0 (λ0 ) C is equal to |x| P0 (dξ ) ξˆ (dx) e− 2 , P0 (dξ ) ξˆ (dx) c0,x (ξ )|x|γ q ≤ c for a suitable constant c. The r.h.s. can be bounded by means of Lemma 1(iv) and the same argument leading to Lemma 2. In view of (100) and (101), the proposition will be proved if we can show that the ξ ξ expectation EP0 (P˜ 0 ⊗ Q(n∗ (t) = n)) converges to zero more rapidly than n−(γ +1)p as

62


n → ∞. Let us fix 0 < α < 1. We will show that, if l > 0 is such that EP0 (λl+1 0 ) < ∞, then ξ ξ EP0 P˜ 0 ⊗ Q n∗ (t) = n = O(n−αl ) .

(102)

To this end, let us first make a general observation. Let λ > 0 and let T1 , . . . , Tk be independent exponential variables on some probability space (, µ), with parameters λ1 , . . . , λk ≤ λ. Define the random variables Tj := (λj /λ)Tj , j = 1, . . . , k. These are independent identically distributed exponential variables with parameter λ. As Tj ≤ Tj , this shows that ∞ (λt)k (λt)j +k µ T1 + · · · + Tk ≤ t ≤ µ T1 + · · · + Tk ≤ t = e−λt ≤ . (j + k)! k! j =0

(103) ξ In order to proceed, for all ξ ∈ N0 , let us set Bn := x ∈ ξˆ : λx (ξ ) ≤ nα as well as Aξn :=

n ξ ˜ ξ : ∃ J ⊂ In , |J | > , X˜ ξ ∈ Bnξ ∀ j ∈ J , X˜ k )k≥0 ∈ j 2

ξ ξ where In := {0, . . . , n − 1} and |J | is the cardinality of J . We write P˜ 0 ⊗ Q n∗ (t) = n = gn (ξ ) + hn (ξ ) with

gn (ξ ) := P˜ 0 ⊗ Q ξ

ξ n∗ (t) = n ∩ Aξn ,

hn (ξ ) := P˜ 0 ⊗ Q ξ

ξ n∗ (t) = n ∩ (Aξn )c .

ξ ξ We first estimate gn . Obviously {n∗ (t) = n} is contained in { j ∈J T

j,X˜ j

ξ

≤ t}. As a

result, gn (ξ ) ≤

χ xj ∈ Bnξ ∀ j ∈ J χ xi ∈ / Bnξ ∀ i ∈ In \ J

J ⊂In ,|J |>n/2 x0 ,... ,xn−1 ∈ξˆ

ξ ξ ξ ξ P˜ 0 X˜ 0 = x0 , . . . , X˜ n−1 = xn−1 Q Tj,xj ≤ t

≤

max

k=[n/2]+1,... ,n−1

(nα t)k

k!

j ∈J

.

√ Thanks to the Stirling formula k! ∼ k k e−k 2πk as k → ∞, the last expression can be bounded by a constant times (2 e t)n/2 n−n(1−α)/2 and is thus exponentially small. We now turn to EP0 (hn ), n ≥ 1. Clearly, ξ P˜ 0

Aξn

n−1 c 2 ξ 2 ξ ≤ EP˜ ξ χ X˜ 0 ∈ / Bnξ +· · ·+χ X˜ n−1 ∈ / Bnξ = EP˜ ξ λ0 (ξm ) > nα . n 0 n m=0


63

By Proposition 2 and invoking Chebyshev’s inequality, one obtains for any l > 0, c c ξ ξ ξ EP0 hn ≤ P0 (dξ )P˜ 0 ⊗ Q n∗ (t) ≥ 1 ∩ Aξn ≤ t P0 (dξ ) λ0 (ξ )P˜ 0 Aξn ≤

n−1 2t P0 (dξ ) λ0 (ξ ) EP˜ ξ λ0 (ξm ) > nα = 2t EP0 λ0 χ (λ0 > nα ) n m=0

2t ≤ αl EP0 λl+1 , 0 n where the second inequality follows from the same argument leading to (99) and the ˜ This proves (102). We may now choose equality follows from the stationarity of P. p = α −1 > 1 arbitrarily close to 1 so that γ q > 1 and such that one may take for l the smallest integer strictly greater than γ + 1. For such a choice the sum (100) converges. We can now invoke Lemma 2 to get the result. ˇ y, B. Derrida, P. A. Ferrari, D. Gabrielli, A. Acknowledgement. We would like to thank A. Bovier, J. Cern´ Ramirez and R. Siegmund-Schultze for very useful comments. The work was supported by the SFB 288, SFB/TR 12 and the Dutch-German Bilateral Research Group “Mathematics of random spatial models from physics and biology".

References [AHL]

Ambegoakar, V., Halperin, B.I., Langer, J.S.: Hopping Conductivity in Disordered Systems. Phys, Rev B 4, 2612–2620 (1971) [BRSW] Bellissard, J., Rebolledo, R., Spehner, D., von Waldenfels, W.: In preparation [BHZ] Bellissard, J., Hermann, D., Zarrouati, M.: Hull of Aperiodic Solids and Gap Labelling Theorems. In: Directions in Mathematical Quasicrystals, M.B. Baake, R.V. Moody, eds., CRM Monograph Series, Volume 13, Providence, RI: Amer. Math.Soc., (2000) 207–259 [Bil] Billingsley, P.: Convergence of Probability Measures. New York: Wiley, 1968 [BS] Bolthausen, E., Sznitman, A.-S.: Ten lectures on random media. DMV Seminar 32 Basel: Birkhäuser, 2002 [Bre] Breiman, L.: Probability. Reading, MA: Addison–Wesley, 1953 [DV] Daley, D.J., Vere–Jones, D.: An Introduction to the Theory of Point Processes. New York: Springer, 1988 [DFGW] De Masi, A., Ferrari, P.A., Goldstein, S., Wick, W.D.: An Invariance Principle for Reversible Markov Processes. Applications to Random Motions in Random Environments. J. Stat. Phys. 55, 787–855 (1989) [EF] Efros, A.L., Shklovskii, B.I.: Coulomb gap and low temperature conductivity of disordered systems. J. Phys. C: Solid State Phys. 8, L49–L51 (1975) [FM] Faggionato, A., Martinelli, F.: Hydrodynamic limit of a disordered lattice gas. Probab. Theory Related Fields 127, 535–608 (2003) [FKAS] Franken, P., König, D., Arndt, U., Schmidt, V.: Queues and Point Processes. Berlin: Akadamie-Verlag, 1981 [Gri] Grimmett, G.: Percolation. Second Edition, Grundlehren 321, Berlin: Springer, 1999 [KV] Kipnis, C., Varadhan, S.R.S.: Central Limit Theorem for Additive Functionals of Reversible Markov Processes and Applications to Simple Exclusion. Commun. Math. Phys. 104, 1–19 (1986) [Kal] Kallenberg, O.: Foundations of Modern Probability. Second Edition, New York: Springer-Verlag, 2001 [KLP] Kirsch, W., Lenoble, O., Pastur, L.: On the Mott formula for the a.c. conductivity and binary correlators in the strong localization regime of disordered systems. J. Phys. A: Math. Gen. 36, 12157–12180 (2003) [LB] Ladieu, F., Bouchaud, J.-P.: Conductance statistics in small GaAs:Si wires at low temperatures: I. Theoretical analysis: truncated quantum fluctuations in insulating wires. J. Phys. I France 3, 2311–2320 (1993) [Mar] Martinelli, F.: Lectures on Glauber dynamics for discrete spin models. Lecture Notes in Mathematics, Vol. 1717, Berlin-Heidelberg-Newyork: Springer, 2000

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A. Faggionato, H. Schulz-Baldes, D. Spehner Matthes, K., Kerstan, J., Mecke, J.: Infinitely Divisible Point Processes. Wiley Series in Probability and Mathematical Physics, Newyork: Wiley, 1978 Meester, R., Roy, R.: Continuum Percolation. Cambridge: Cambridge University Press, 1996 Miller, A., Abrahams, E.: Impurity Conduction at Low Concentrations. Phys. Rev. 120, 745– 755 (1960) Minami, N.: Local fluctuation of the spectrum of a multidimensional Anderson tight binding model. Commun. Math. Phys. 177, 709–725 (1996) Mott, N.F.: J. Non-Crystal. Solids 1, 1 (1968); N. F. Mott, Phil. Mag 19, 835 (1969); Mott, N.F., Davis, E.A.: Electronic Processes in Non-Crystaline Materials. New York: Oxford University Press, 1979 Owhadi, H.: Approximation of the effective conductivity of ergodic media by periodization. Probab. Theory Related Fields 125, 225–258, (2003) Quastel, J.: Diffusion in Disordered Media. In: Funaki, T., Woyczinky, W., eds., Proceedings on stochastic method for nonlinear P.D.E., IMA volumes in Mathematics 77, New York: Springer Verlag, 1995, pp. 65–79 Reed, M., Simon, B.: Methods of Modern Mathematical Physics I-IV. San Diego: Academic Press, 1980 Rosenblatt, M.: Markov Processes. Structure and Asymptotic Behavior. Grundlehren 184, Berlin: Springer, 1971 Shklovskii, B., Efros, A.L.: Electronic Properties of Doped Semiconductors. Berlin: Springer, 1984 Spehner, D.: Contributions a` la théorie du transport e´ lectronique dissipatif dans les solides apériodiques. PhD Thesis, Toulouse, 2000 Spohn, H.: Large Scale Dynamics of Interacting Particles. Berlin: Springer, 1991 Thorisson, H.: Coupling, Stationarity, and Regeneration. New York: Springer, 2000

Communicated by M. Aizenman


Communications in


A Hopf Bundle Over a Quantum Four-Sphere from the Symplectic Group Giovanni Landi1 , Chiara Pagani2 , Cesare Reina2 1 2

Dipartimento di Matematica e Informatica, Università di Trieste, Via A.Valerio 12/1, 34127 Trieste, Italy, and I.N.F.N., Sezione di Napoli, Napoli, Italy. E-mail: [email protected] S.I.S.S.A. International School for Advanced Studies, Via Beirut 2-4, 34014 Trieste, Italy. E-mail: [email protected]; [email protected]

Received: 7 September 2004 / Accepted: 16 August 2005 Published online: 24 January 2006 – © Springer-Verlag 2006

Abstract: We construct a quantum version of the SU (2) Hopf bundle S 7 → S 4 . The quantum sphere Sq7 arises from the symplectic group Spq (2) and a quantum 4-sphere Sq4 is obtained via a suitable self-adjoint idempotent p whose entries generate the algebra A(Sq4 ) of polynomial functions over it. This projection determines a deformation of an (anti-)instanton bundle over the classical sphere S 4 . We compute the fundamental Khomology class of Sq4 and pair it with the class of p in the K-theory getting the value −1 for the topological charge. There is a right coaction of SUq (2) on Sq7 such that the algebra A(Sq7 ) is a non-trivial quantum principal bundle over A(Sq4 ) with structure quantum group A(SUq (2)).

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . 2. Odd Spheres from Quantum Symplectic Groups . 2.1 The quantum groups Spq (N, C) and Spq (n) 2.2 The odd symplectic spheres . . . . . . . . . 2.3 The symplectic 7-sphere Sq7 . . . . . . . . . 3. The Principal Bundle A(Sq4 ) → A(Sq7 ) . . . . . . 3.1 The quantum sphere Sq4 . . . . . . . . . . . 3.2 The SUq (2)-coaction . . . . . . . . . . . . 4. Representations of the Algebra A(Sq4 ) . . . . . . 4.1 The representation β . . . . . . . . . . . . 4.2 The representation σ . . . . . . . . . . . . 5. The Index Pairings . . . . . . . . . . . . . . . . 6. Quantum Principal Bundle Structure . . . . . . .

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66 67 67 68 70 72 73 76 79 79 79 80 82

66

G. Landi, C. Pagani, C. Reina

6.1 The associated bundle and the coequivariant maps . . . . . . . . . . . A. The Classical Hopf Fibration S 7 → S 4 . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85 86 88

1. Introduction In this paper we study yet another example of how “quantization removes degeneracy” by constructing a new quantum version of the Hopf bundle S 7 → S 4 . This is the first outcome of our attempt to generalize to the quantum case the ADHM construction of SU (2) instantons together with their moduli spaces. The q-monopole on two dimensional quantum spheres has been constructed in [8] more than a decade ago. There the general notion of a quantum principal bundle with quantum differential calculi, from a geometrical point of view was also introduced. With universal differential calculi, this notion was later realised to be equivalent to the one of Hopf-Galois extension (see e.g. [14]). An analogous construction for q-instantons and their principal bundles has been an open problem ever since. A step in this direction was taken in [3] resulting in a bundle which is only a coalgebra extension [4]. Here we present a quantum principal instanton bundle which is a honest Hopf-Galois extension. One advantage is that non-universal calculi may be constructed on the bundle, as opposite to the case of a coalgebra bundle where there is not such a possibility. In analogy with the classical case [1], it is natural to start with the quantum version of the (compact) symplectic groups A(Spq (n)), i.e. the Hopf algebras generated by matrix j elements Ti ’s with commutation rules coming from the R matrix of the C-series [25]. These quantum groups have comodule-subalgebras A(Sq4n−1 ) yielding deformations of the algebras of polynomials over the spheres S 4n−1 , which give more examples of the general construction of quantum homogeneous spaces [8]. The relevant case for us is n = 2, i.e. the symplectic quantum 7-sphere A(Sq7 ), which is generated by the matrix elements of the first and the last column of T . Indeed, as 4 1 . A similar conjugation occurs for the elements of the middle we will see, T i ∝ T4−i columns, but contrary to what happens at q = 1, they do not generate a subalgebra. The algebra A(Sq7 ) is the quantum version of the homogeneous space Sp(2)/Sp(1) and the injection A(Sq7 ) → A(Spq (2)) is a quantum principal bundle with “structure Hopf algebra” A(Spq (1)). Most importantly, we show that Sq7 is the total space of a quantum SUq (2) principal bundle over a quantum 4-sphere Sq4 . Unlike the previous construction, this is obviously not a quantum homogeneous structure. The algebra A(Sq4 ) is constructed as the subalgebra of A(Sq7 ) generated by the matrix elements of a self-adjoint projection p which generalizes the anti-instanton of charge −1. This projection will be of the form vv ∗ with v a 4 × 2 matrix whose entries are made out of generators of A(Sq7 ). The naive generalization of the classical case produces a subalgebra with extra generators which vanish at q = 1. Luckily enough, there is just one alternative choice of v which gives the right number of generators of an algebra which deforms the algebra of polynomial functions of S 4 . At q = 1 this gives a projection which is gauge equivalent to the standard one. This good choice becomes even better because there is a natural coaction of SUq (2) on A(Sq7 ) with coinvariant algebra A(Sq4 ) and the injection A(Sq4 ) → A(Sq7 ) turns out to be a faithfully flat A(SUq (2))-Hopf-Galois extension.

A Hopf Bundle Over a Quantum Four-Sphere from the Symplectic Group

67

Finally, we set up the stage to compute the charge of our projection and to prove the non-triviality of our principal bundle. Following a general strategy of noncommutative index theorem [10], we construct representations of the algebra A(Sq4 ) and the corresponding K-homology. The analogue of the fundamental class of S 4 is given by a non-trivial Fredholm module µ. The natural coupling between µ and the projection p is computed via the pairing of the corresponding Chern characters ch∗ (µ) ∈ H C ∗ [A(Sq4 )] and ch∗ (p) ∈ H C∗ [A(Sq4 )] in cyclic cohomology and homology respectively [10]. As expected the result of this pairing, which is an integer in principle being the index of a Fredholm operator, is actually −1 and therefore the bundle is non-trivial. Clearly the example presented in this paper is very special and limited, since it is just a particular anti-instanton of charge −1. Indeed our construction is based on the requirement that the matrix v giving the projection is linear in the generators of A(Sq7 ) and such that v ∗ v = 1. This is false even classically at generic moduli and generic charge, except for the case considered here (and for a similar construction for the case of charge 1). A more elaborate strategy is needed to tackle the general case. 2. Odd Spheres from Quantum Symplectic Groups We recall the construction of quantum spheres associated with the compact real form of the quantum symplectic groups Spq (N, C) (N = 2n), the latter being given in [25]. Later we shall specialize to the case N = 4 and the corresponding 7-sphere will provide the ‘total space’ of our quantum Hopf bundle. 2.1. The quantum groups Spq (N, C) and Spq (n). The algebra A(Spq (N, C)) is the associative noncommutative algebra generated over the ring of Laurent polynomials Cq := C[q, q −1 ] by the entries Ti j , i, j = 1, . . . , N of a matrix T which satisfy RTT equations: R T1 T2 = T2 T1 R ,

T1 = T ⊗ 1 ,

T2 = 1 ⊗ T .

In components (T ⊗ 1)ij kl = Ti k δj l . Here the relevant N 2 × N 2 matrix R is the one for the CN series and has the form [25], R=q

N

ei i ⊗ e i i +

ei i ⊗ ej j + q −1

i,j =1

i=1

+(q − q −1 )

N i=j,j

N i,j =1

i>j

ei j ⊗ ej i − (q − q −1 )

N

ei i ⊗ e i i

i=1 N

q ρi −ρj εi εj ei j ⊗ ei j ,

i,j =1

i>j

where i = N + 1 − i ; ei j ∈ Mn (C) are the elementary matrices, i.e. (ej i )kl = δj l δ ik ; εi = 1, for i = 1, . . . , n ; εi = −1, for i = n + 1, . . . , N ; (ρ1 , . . . , ρN ) = (n, n − 1, . . . , 1, −1, . . . , −n).

(1)

68


The symplectic group structure comes from the matrix Ci j = q ρj εi δij by imposing the additional relations T CT t C −1 = CT t C −1 T = 1. The Hopf algebra co-structures (, ε, S) of the quantum group Spq (N, C) are given by .

(T ) = T ⊗ T ,

ε(T ) = I ,

S(T ) = CT t C −1 .

In components the antipode explicitly reads

S(T )i j = −q ρi +ρj εi εj Tj i .

(2)

At q = 1 the Hopf algebra Spq (N, C) reduces to the algebra of polynomial functions over the symplectic group Sp(N, C). The compact real form A(Spq (n)) of the quantum group A(Spq (N, C)) is given by taking q ∈ R and the anti-involution [25] T = S(T )t = C t T (C −1 )t .

(3)

2.2. The odd symplectic spheres. Let us denote xi = Ti N ,

v j = S(T )N j ,

i, j = 1, . . . , N .

As we will show, these generators give subalgebras of A(Spq (N, C)). With the natural involution (3), the algebra generated by the {xi , v j } can be thought of as the algebra A(Sq4n−1 ) of polynomial functions on a quantum sphere of ‘dimension’ 4n − 1. From here on, whenever no confusion arises, the sum over repeated indexes is understood. In components the RTT equations are given by Rij kp Tk r Tp s = Tj p Ti m Rmp rs .

(4)

Hence Rij kl Tk r = Tj p Ti m Rmp rs S(T )s l , and in turn S(T )p j Rij kl = Ti a Rap rs S(T )s l S(T )r k , so that S(T )a i S(T )p j Rij kl = Rap rs S(T )s l S(T )r k .

(5)

Conversely, if we multiply Rij kp Tk r = Tj l Ti m Rml rs S(T )s p on the left by S(T ) we have S(T )l j Rij kp Tk r = Ti m Rml rs S(T )s p .

(6)

We shall use Eqs. (4), (5) and (6) to describe the algebra generated by the xi ’s and by the v i ’s.


69

The algebra Cq [xi ] From (4) with r = s = N we have Rij kp xk xp = Tj p Ti m Rmp NN .

(7)

Since the only element Rmp NN ∝ em N ⊗ ep N (m, p ≤ N ) which is different from zero is RNN NN = q, it follows that Rij kp xk xp = q xj xi ,

(8)

and the elements xi ’s give an algebra with commutation relations xi xj = qxj xi ,

i < j, i = j ,

xi xi = q −2 xi xi + (q −2 − 1)

i−1

q ρi −ρk εi εk xk xk ,

i < i .

(9)

k=1

The algebra Cq [v i ] Putting a = p = N in Eq. (5), we get v i v j Rij kl = RNN rs S(T )s l S(T )r k . The sum on the r.h.s. reduces to RNN NN S(T )N l S(T )N k and the v i ’s give an algebra with commutation relations v l v k Rlk j i = qv i v j .

(10)

Explicitly v i v j = q −1 v j v i , i i

i < j, i = j ,

2 i i

v v = q v v + (q − 1) 2

N

q ρk −ρi εk εi v k v k ,

i < i .

(11)

k=i +1

The algebra Cq [xi , v j ]. Finally, for l = r = N Eq. (6) reads: v j Rij kp xk = Ti m RmN Ns S(T )s p . Once more, the only term in R of the form em N ⊗ eN s (m ≤ N ) is eN N ⊗ eN N and therefore v j Rij kp xk = q xi v p .

(12)

Explicitly the mixed commutation rules for the algebra Cq [xi , v j ] read, xi v i = v i xi + (1 − q −2 )

i−1 k=1

i

v k xk + (1 − q −2 )q ρi −ρi v i xi , if i>i

i

xi v = q −2 v xi , xi v j = q −1 v j x i ,

i = j

and

i < j ,

xi v j = q −1 v j x i + (q −2 − 1)q ρi −ρj εi εj v i xj ,

i = j

and

i > j . (13)

70


The quantum spheres Sq4n−1 . Let us observe that with the anti-involution (3) we have the identification v i = S(T )N i = x¯ i . The subalgebra A(Sq4n−1 ) of A(Spq (n)) generated by {xi , v i = x¯ i , i = 1, . . . , 2n} is the algebra of polynomial functions on a sphere. Indeed N S(T )T = I ⇒ S(T )N i Ti N = δN = 1, i.e.

x¯ i xi = 1 .

(14)

i

Furthermore, the restriction of the comultiplication is a natural left coaction L : A(Sq4n−1 ) −→ A(Spq (n)) ⊗ A(Sq4n−1 ) . The fact that L is an algebra map then implies that A(Sq4n−1 ) is a comodule algebra over A(Spq (n)). At q = 1 this algebra reduces to the algebra of polynomial functions over the spheres S 4n−1 as homogeneous spaces of the symplectic group Sp(n): S 4n−1 = Sp(n)/Sp(n−1). 2.3. The symplectic 7-sphere Sq7 . The algebra A(Sq7 ) is generated by the elements xi = Ti 4 and x¯ i = S(T )4 i = q 2+ρi εi Ti 1 , for i = 1, . . . , 4. From S(T ) T = 1 we have the sphere relation 4i=1 x¯ i xi = 1. Since we shall systematically use them in the following, we shall explicitly give the commutation relations among the generators. From (9), the algebra of the xi ’s is given by x1 x2 = qx2 x1 , x1 x3 = qx3 x1 , x2 x4 = qx4 x2 , x3 x4 = qx4 x3 , x4 x1 = q −2 x1 x4 , x3 x2 = q −2 x2 x3 + q −2 (q −1 − q)x1 x4 ,

(15)

together with their conjugates (given in (11)). We have also the commutation relations between the xi and the x¯ j deduced from (12): x1 x¯ 1 x1 x¯ 3 x2 x¯ 2 x2 x¯ 3 x2 x¯ 4 x3 x¯ 3 x3 x¯ 4 x4 x¯ 4

= x¯ 1 x1 , x1 x¯ 2 = q −1 x¯ 2 x1 , −1 3 = q x¯ x1 , x1 x¯ 4 = q −2 x¯ 4 x1 , = x¯ 2 x2 + (1 − q −2 )x¯ 1 x1 , = q −2 x¯ 3 x2 , = q −1 x¯ 4 x2 + q −1 (q −2 − 1)x¯ 3 x1 , = x¯ 3 x3 + (1 − q −2 )[x¯ 1 x1 + (1 + q −2 )x¯ 2 x2 ] , = q −1 x¯ 4 x3 + (1 − q −2 )q −3 x¯ 2 x1 , = x¯ 4 x4 + (1 − q −2 )[(1 + q −4 )x¯ 1 x1 + x¯ 2 x2 + x¯ 3 x3 ] ,

(16)

again with their conjugates. Next we show that the algebra A(Sq7 ) can be realized as the subalgebra of A(Spq (2)) generated by the coinvariants under the right-coaction of A(Spq (1)), in complete analogy with the classical homogeneous space Sp(2)/Sp(1) S 7 .


71

Lemma 1. The two-sided *-ideal in A(Spq (2)) generated as Iq = {T1 1 − 1, T4 4 − 1, T1 2 , T1 3 , T1 4 , T2 1 , T2 4 , T3 1 , T3 4 , T4 1 , T4 2 , T4 3 } with the involution (3) is a Hopf ideal.

Proof. Since S(T )i j ∝ Tj i , S(Iq ) ⊆ Iq which also proves that Iq is a *-ideal. One easily shows that ε(Iq ) = 0 and (Iq ) ⊆ Iq ⊗ A(Spq (2)) + A(Spq (2)) ⊗ Iq .

Proposition 1. The Hopf algebra Bq := A(Spq (2))/Iq is isomorphic to the coordinate algebra A(SUq 2 (2)) ∼ = A(Spq (1)). Proof. Using T = S(T )t and setting T2 2 = α, T3 2 = γ , the algebra Bq can be described as the algebra generated by the entries of the matrix 

1 0 T = 0 0

0 0 α −q 2 γ¯ γ α¯ 0 0

 0 0 . 0 1

(17)

The commutation relations deduced from RTT equations (4) read: α γ¯ = q 2 γ¯ α , αγ = q 2 γ α , γ γ¯ = γ¯ γ , αα ¯ + γ¯ γ = 1 ; α α¯ + q 4 γ γ¯ = 1 .

(18)

Hence, as an algebra Bq is isomorphic to the algebra A(SUq 2 (2)). Furthermore, the restriction of the coproduct of A(Spq (2)) to Bq endows the latter with a coalgebra . structure, (T ) = T ⊗ T , which is the same as the one of A(SUq 2 (2)). We can conclude that also as a Hopf algebra, Bq is isomorphic to the Hopf algebra A(SUq 2 (2)) ∼ = A(Spq (1)).

Proposition 2. The algebra A(Sq7 ) ⊂ A(Spq (2)) is the algebra of coinvariants with respect to the natural right coaction .

R : A(Spq (2)) → A(Spq (2)) ⊗ A(Spq (1)) ;

.

R (T ) = T ⊗ T .

Proof. It is straightforward to show that the generators of the algebra A(Sq7 ) are coinvariants: R (xi ) = R (Ti4 ) = xi ⊗ 1 ; R (x¯ i ) = −q 2+ρi εi R (Ti1 ) = x¯ i ⊗ 1, thus the algebra A(Sq7 ) is made of coinvariants. There are no other coinvariants of degree one since each row of the submatrix of T made out of the two central columns is a fundamental comodule under the coaction of SUq 2 (2). Other coinvariants arising at higher even degree are of the form (Ti2 Ti3 − q 2 Ti3 Ti2 )n ; thanks to the commutation relations of A(Spq (2)), one checks these belong to A(Sq7 ) as well. It is an easy computation to check that similar expressions involving elements from different rows cannot be coinvariant.

72


The previous construction is one more example of the general construction of a quantum principal bundle over a quantum homogeneous space [8]. The latter is the datum of a Hopf quotient π : A(G) → A(K) with the right coaction of A(K) on A(G) given by the reduced coproduct R := (id ⊗ π ), where is the coproduct of A(G). The subalgebra B ⊂ A(G) made of the coinvariants with respect to R is called a quantum homogeneous space. To prove that it is a quantum principal bundle one needs some more assumptions (see Lemma 5.2 of [8]). In our case A(G) = A(Spq (2)), A(K) = A(Spq (1)) with π(T ) = T . We will prove in Sect. 6 that the resulting inclusion B = A(Sq7 ) → A(Spq (2)) is indeed a Hopf Galois extension and hence a quantum principal bundle. 3. The Principal Bundle A(Sq4 ) → A(Sq7 ) The fundamental step of this paper is to make the sphere Sq7 itself into the total space of a quantum principal bundle over a deformed 4-sphere. Unlike what we saw in the previous section, this is not a quantum homogeneous space construction and it is not obvious that such a bundle exists at all. Nonetheless the notion of quantum bundle is more general and one only needs that the total space algebra is a comodule algebra over an Hopf algebra with additional suitable properties. The notion of quantum principle bundle, as said, is encoded in the one of Hopf-Galois extension (see e.g. [8, 14]). Let us recall some relevant definitions [20] (see also [22]). Recall that we work over the field k = C. Definition 1. Let H be a Hopf algebra and P a right H -comodule algebra with multiplication m : P ⊗ P → P and coaction R : P → P ⊗ H . Let B ⊆ P be the subalgebra of coinvariants, i.e. B = {p ∈ P | R (p) = p ⊗ 1}. The extension B ⊆ P is called an H Hopf-Galois extension if the canonical map χ : P ⊗B P −→ P ⊗ H , χ := (m ⊗ id) ◦ (id ⊗B R ) ,

p ⊗B p → χ (p ⊗B p) = p p(0) ⊗ p(1) (19)

is bijective. We use Sweedler-like notation R p = p(0) ⊗ p(1) . The canonical map is left P -linear and right H -colinear and is a morphism (an isomorphism for Hopf-Galois extensions) of left P -modules and right H -comodules. It is also clear that P is both a left and a right B-module. The injectivity of the canonical map dualizes the condition of a group action X×G → X to be free: if α is the map α : X × G → X ×M X, (x, g) → (x, x · g) then α ∗ = χ with P , H the algebras of functions on X, G respectively and the action is free if and only if α is injective. Here M := X/G is the space of orbits with projection map π : X → M, π(x · g) = π(x), for all x ∈ X, g ∈ G. Furthermore, α is surjective if and only if for all x ∈ X, the fibre π −1 (π(x)) of π(x) is equal to the residue class x · G, that is, if and only if G acts transitively on the fibres of π . In differential geometry a principle bundle is more than just a free and effective action of a Lie group. In our example, thanks to the fact that the “structure group” is SUq (2), from Th. I of [28] further nice properties can be established. We shall elaborate more on these points later on in Sect. 6 . The first natural step would be to construct a map from Sq7 into a deformation of the Stieffel variety of unitary frames of 2-planes in C4 to parallel the classical construction


73

as recalled in Appendix A. The naive choice we have is to take as generators the elements of two (conjugate) columns of the matrix T . We are actually forced to take the first and the last columns of the matrix T because the other choice (i.e. the second and the third columns) does not yield a subalgebra since commutation relations of their elements will involve elements from the other two columns. If we set   x1 x¯ 4  q −1 x¯ 3 x2   v= (20)  −q −3 x¯ 2 x3  , −4 1 −q x¯ x4 we have v ∗ v = I2 and the matrix p = v v ∗ is a self-adjoint idempotent, i.e. p = p ∗ = p2 . At q = 1 the entries of p are invariant for the natural action of SU (2) on S 7 and generate the algebra of polynomials on S 4 . This fails to be the case at generic q due to the occurrence of extra generators, e.g. p14 = (1 − q −2 )x1 x¯ 4 ,

p23 = (1 − q −2 )x2 x¯ 3 ,

(21)

which vanish at q = 1. 3.1. The quantum sphere Sq4 . These facts indicate that the naive quantum analogue of the quaternionic projective line as a homogeneous space of Spq (2) has not the right number of generators. Rather surprisingly, we shall anyhow be able to select another subalgebra of A(Sq7 ) which is a deformation of the algebra of polynomials on S 4 having the same number of generators. These generators come from a better choice of a projection. On the free module E := C4 ⊗ A(Sq7 ) we consider the hermitian structure given by h(|ξ1 , |ξ2 ) =

4

j j ξ¯1 ξ2 .

j =1

To every element |ξ ∈ E one associates an element ξ | in the dual module E ∗ by the pairing ξ | (|η) := ξ |η = h(|ξ , |η). Guided by the classical construction which we present in Appendix A, we shall look for two elements |φ1 , |φ2 in E with the property that φ1 |φ1 = 1 ,

φ2 |φ2 = 1 ,

φ1 |φ2 = 0 .

As a consequence, the matrix valued function defined by p := |φ1 φ1 | + |φ2 φ2 | ,

(22)

is a self-adjoint idempotent (a projection). In principle, p ∈ Mat4 (A(Sq7 )), but we can choose |φ1 , |φ2 in such a way that the entries of p will generate a subalgebra A(Sq4 ) of A(Sq7 ) which is a deformation of the algebra of polynomial functions on the 4-sphere S 4 . The two elements |φ1 , |φ2 will be obtained in two steps as follows.

74


i Firstly we write the relation 1 = x¯ xi in terms of the quadratic elements x¯ 1 x1 , 2 3 4 x2 x¯ , x¯ x3 , x4 x¯ by using the commutation relations of Sect. 2.3. We have that 1 = q −6 x¯ 1 x1 + q −2 x2 x¯ 2 + q −2 x¯ 3 x3 + x4 x¯ 4 . Then we take, |φ1 = (q −3 x1 , −q −1 x¯ 2 , q −1 x3 , −x¯ 4 )t ,

(23)

(t denoting transposition) which is such that φ1 |φ1 = 1. Next, we write 1 = x¯ i xi as a function of the quadratic elements x1 x¯ 1 , x¯ 2 x2 , x3 x¯ 3 , x¯ 4 x4 : 1 = q −2 x1 x¯ 1 + q −4 x¯ 2 x2 + x3 x¯ 3 + x¯ 4 x4 . By taking, |φ2 = (±q −2 x2 , ±q −1 x¯ 1 , ±x4 , ±x¯ 3 )t we get φ2 |φ2 = 1. The signs will be chosen in order to have also the orthogonality φ1 |φ2 = 0; for |φ2 = (q −2 x2 , q −1 x¯ 1 , −x4 , −x¯ 3 )t this is satisfied. The matrix



q −3 x1  −q −1 x¯ 2 v = (|φ1 , |φ2 ) =   q −1 x3 −x¯ 4

 q −2 x2 q −1 x¯ 1   . −x4  −x¯ 3

(24)

(25)

is such that v ∗ v = 1 and hence p = vv ∗ is a self-adjoint projection. Proposition 3. The entries of the projection p = vv ∗ , with v given in (25), generate a subalgebra of A(Sq7 ) which is a deformation of the algebra of polynomial functions on the 4-sphere S 4 . Proof. Let us compute explicitly the components of the projection p and their commutation relations. 1. The diagonal elements are given by p11 = q −6 x1 x¯ 1 + q −4 x2 x¯ 2 , p22 = q −2 x¯ 2 x2 + q −2 x¯ 1 x1 , p33 = q −2 x3 x¯ 3 + x4 x¯ 4 , p44 = x¯ 4 x4 + x¯ 3 x3 , and satisfy the relation q −2 p11 + q 2 p22 + p33 + p44 = 2 .

(26)

Only one of the pii ’s is independent; indeed by using the commutation relations and the equation x¯ i xi = 1, we can rewrite the pii ’s in terms of t := p22 , as p11 = q −2 t , p22 = t , p33 = 1 − q −4 t , p44 = 1 − q 2 t . Equation (26) is easily verified. Notice that t is self-adjoint: t¯ = t.

(27)


75

2. As in the classical case, the elements p12 , p34 (and their conjugates) vanish: p12 = −q −4 x1 x2 + q −3 x2 x1 = 0 ,

p34 = −q −1 x3 x4 + x4 x3 = 0 .

3. The remaining elements are given by p13 = q −4 x1 x¯ 3 − q −2 x2 x¯ 4 , p23 = −q −2 x¯ 2 x¯ 3 − q −1 x¯ 1 x¯ 4 ,

p14 = −q −3 x1 x4 − q −2 x2 x3 , p24 = q −1 x¯ 2 x4 − q −1 x¯ 1 x3 ,

with pj i = p¯ ij when j > i. By using the commutation relations of A(Sq7 ), one finds that only two of these are independent. We take them to be p13 and p14 ; one finds p23 = q −2 p¯ 14 and p24 = −q 2 p¯ 13 . Finally, we also have the sphere relation, 2 2 2 (q 6 − q 8 )p11 + p22 + p44 + q 4 (p13 p31 + p14 p41 ) + q 2 (p24 p42 + p23 p32 ) 2

= x¯ i xi = 1 .

(28)

Summing up, together with t = p22 , we set a := p13 and b := p14 . Then the projection p takes the following form  −2  q t0 a b     −q 2 a¯  t q −2 b¯ 0   p= (29)  .   q −2 b 1 − q −4 t 0  a¯    1 − q 2t b¯ −q 2 a 0 By construction p∗ = p and this means that t¯ = t, as observed, and that a, ¯ b¯ are con2 jugate to a, b respectively. Also, by construction p = p; this property gives the easiest way to compute the commutation relations between the generators. One finds, ab = q 4 ba , ab ¯ = ba¯ , ta = q −2 at , tb = q 4 bt ,

(30)

together with their conjugates, and sphere relations a a¯ + bb¯ = q −2 t (1 − q −2 t) , ¯ = (1 − q −4 )t 2 . bb¯ − q −4 bb

¯ = t (1 − t) , q 4 aa ¯ + q −4 bb

It is straightforward to check also the relation (28).

(31)

¯ t ¯ b, b, We define the algebra A(Sq4 ) to be the algebra generated by the elements a, a, with the commutation relations (30) and (31). For q = 1 it reduces to the algebra of polynomial functions on the sphere S 4 . Otherwise, we can limit ourselves to |q| < 1, because the map q → q −1 , yields an isomorphic algebra.

a → q 2 a, ¯

¯ b → q −2 b,

t → q −2 t

76


At q = 1, the projection p in (29) is conjugate to the classical one given in Appendix A by the matrix diag[1, −1, 1, 1] (up to a renaming of the generators). Our sphere Sq4 seems to be different from the one constructed in [3]. Two of our generators commute and most importantly, it does not come from a deformation of a subgroup (let alone coisotropic) of Sp(2). However, at the continuous level these two quantum spheres are the same since the C ∗ -algebra completion of both polynomial algebras is the minimal unitization K ⊕ CI of the compact operators on an infinite dimensional separable Hilbert space, a property shared with Podle´s standard sphere as well [24]. This fact will be derived in Sect. 4 when we study the representations of the algebra A(Sq4 ). 3.2. The SUq (2)-coaction. We now give a coaction of the quantum group SUq (2) on the sphere Sq7 . This coaction will be used later in Sect. 6 when analyzing the quantum principle bundle structure. Let us observe that the two pairs of generators (x1 , x2 ), (x3 , x4 ) both yield a quantum plane, x1 x2 = qx2 x1 , x3 x4 = qx4 x3 ,

x¯ 1 x¯ 2 = q −1 x¯ 2 x¯ 1 , x¯ 3 x¯ 4 = q −1 x¯ 4 x¯ 3 .

Then we shall look for a right-coaction of SUq (2) on the rows of the matrix v in (25). Other pairs of generators yield quantum planes but the only choice which gives a projection with the right number of generators is the one given above. The defining matrix of the quantum group SUq (2) reads α −q γ¯ (32) γ α¯ with commutation relations [30], αγ = qγ α , α γ¯ = q γ¯ α , α α¯ + q 2 γ¯ γ = 1 , αα ¯ + γ¯ γ = 1 .

γ γ¯ = γ¯ γ ,

We define a coaction of SUq (2) on the matrix (25) by,   −3 q x1 q −2 x2  −q −1 x¯ 2 q −1 x¯ 1  . α −q γ¯ ⊗ δR (v) :=  .  q −1 x3 −x4  γ α¯ −x¯ 4 −x¯ 3

(33)

(34)

We shall prove presently that this coaction comes from a coaction of A(SUq (2)) on the sphere algebra A(Sq7 ). For the moment we remark that, by its form in (34) the entries of the projection p = vv ∗ are automatically coinvariants, a fact that we shall also prove explicitly in the following. On the generators, the coaction (34) is given explicitly by δR (x1 ) = x1 ⊗ α + q x2 ⊗ γ δR (x2 ) = −x1 ⊗ γ¯ + x2 ⊗ α¯ δR (x3 ) = x3 ⊗ α − q x4 ⊗ γ δR (x4 ) = x3 ⊗ γ¯ + x4 ⊗ α¯ ,

, δR (x¯ 1 ) = q x¯ 2 ⊗ γ¯ + x¯ 1 ⊗ α¯ = δR (x1 ) , , δR (x¯ 2 ) = x¯ 2 ⊗ α − x¯ 1 ⊗ γ = δR (x2 ) , , δR (x¯ 3 ) = −q x¯ 4 ⊗ γ¯ + x¯ 3 ⊗ α¯ = δR (x3 ) , δR (x¯ 4 ) = x¯ 4 ⊗ α + x¯ 3 ⊗ γ = δR (x4 ),

(35)


77

from which it is also clear its compatibility with the anti-involution, i.e. δR (x¯ i ) = δR (xi ). The map δR in (35) extends as an algebra homomorphism to the whole of A(Sq7 ). Then, as alluded to before, we have the following Proposition 4. The coaction (35) is a right coaction of the quantum group SUq (2) on the 7-sphere Sq7 , δR : A(Sq7 ) −→ A(Sq7 ) ⊗ A(SUq (2)) .

(36)

Proof. By using the commutation relations of A(SUq (2)) in (33), a lengthy but easy computation gives that the commutation relations of A(Sq7 ) are preserved. This fact also shows that extending δR as an algebra homomorphism yields a consistent coaction.

Proposition 5. The algebra A(Sq4 ) is the algebra of coinvariants under the coaction defined in (35). Proof. We have to show that A(Sq4 ) = {f ∈ A(Sq7 ) | δR (f ) = f ⊗ 1}. By using the commutation relations of A(Sq7 ) and those of A(SUq (2)), we first prove explicitly that the generators of A(Sq4 ) are coinvariants: δR (a) = q −4 δR (x1 )δR (x¯ 3 ) − q −2 δR (x2 )δR (x¯ 4 ) = q −4 x1 x¯ 3 ⊗ (α α¯ + q 2 γ¯ γ ) − q −2 x2 x¯ 4 ⊗ (γ γ¯ + αα) ¯ = (q −4 x1 x¯ 3 − q −2 x2 x¯ 4 ) ⊗ 1 = a ⊗ 1, δR (b) = −q −3 δR (x1 )δR (x4 ) − q −2 δR (x2 )δR (x3 ) = −q −3 x1 x4 ⊗ (α α¯ + q 2 γ¯ γ ) − q −2 x2 x3 ⊗ (γ γ¯ + αα) ¯ = −(q −3 x1 x4 + q −2 x2 x3 ) ⊗ 1 = b ⊗ 1, δR (t) = q −2 δR (x¯ 2 )δR (x2 ) + q −2 δR (x¯ 1 )δR (x1 ) = q −2 x¯ 2 x2 ⊗ (α α¯ + q 2 γ¯ γ ) + q −2 x¯ 1 x1 ⊗ (γ γ¯ + αα) ¯ −2 2 −2 1 = (q x¯ x2 + q x¯ x1 ) ⊗ 1 = t ⊗ 1. By construction the coaction is compatible with the anti-involution so that ¯ = δR (b) = b¯ ⊗ 1. ¯ = δR (a) = a¯ ⊗ 1, δR (b) δR (a) In fact, this only shows that A(Sq4 ) is made of coinvariants but does not rule out the possibility of other coinvariants not in A(Sq4 ). However this does not happen for the following reason. From Eq. (35) it is clear that w1 ∈ {x1 , x3 , x¯ 2 , x¯ 4 } (respectively w−1 ∈ {x2 , x4 , x¯ 1 , x¯ 3 }) are weight vectors of weight 1 (resp. −1) in the fundamental comodule of SUq (2). It follows that the only possible coinvariants are of the form (w1 w−1 − qw−1 w1 )n . When n = 1 these are just the generators of A(Sq4 ).

Remark 1. The last part of the proof above is also related to the quantum Plücker coordinates. For every 2 × 2 matrix of (25), let us define the determinant by det

a11 a12 a21 a22

:= a11 a22 − q a12 a21 .

(37)

78


(Note that a12 , a21 do not commute and so in the previous formula the ordering between them is fixed.) Let mij be the minors of (25) obtained by considering the i, j rows. Then m12 = q 2 p11 = t , m13 = p14 = b , m14 = −q p13 = −q a , m23 = p24 = −q 2 a¯ , m24 = −q p23 = −q −1 b¯ , m34 = −q p33 = q −3 t − q .

(38)

At q = 1, these give the classical Plücker coordinates [1]. The right coaction of SUq (2) on the 7-sphere Sq7 can be written as  α −γ¯ 0 .  qγ α ¯ 0 δR (x1 , x2 , x3 , x4 ) = (x1 , x2 , x3 , x4 ) ⊗  0 0 α 0 0 −qγ

 0 0 , γ¯  α¯

(39)

together with δR (x¯i ) = δR (xi ). In the block-diagonal matrix which appears in (39) the second copy is given by SUq (2) while the first one is twisted as α −γ¯ 1 0 α γ¯ 1 0 = . qγ α¯ 0 −1 −qγ α¯ 0 −1 A similar phenomenon occurs in [3]. Remark 2. It is also interesting to observe that δR (v ∗ v) = v ∗ v ⊗ 1 = 1 ⊗ 1 . Indeed, δR (φ1 |φ1 ) = δR (q −6 x¯ 1 x1 + q −2 x2 x¯ 2 + q −2 x¯ 3 x3 + x4 x¯ 4 ) = (−q −5 x¯ 2 x1 + q −2 x1 x¯ 2 + q −1 x¯ 4 x3 − x3 x¯ 4 ) ⊗ γ¯ α +(q −4 x¯ 2 x2 + q −2 x1 x¯ 1 + x¯ 4 x4 + x3 x¯ 3 ) ⊗ γ¯ γ +(q −6 x¯ 1 x1 + q −2 x2 x¯ 2 + q −2 x¯ 3 x3 + x4 x¯ 4 ) ⊗ αα ¯ ¯ +(−q −5 x¯ 1 x2 + q −2 x2 x¯ 1 + q −1 x¯ 3 x4 − x4 x¯ 3 ) ⊗ αγ = φ2 |φ1 ⊗ γ¯ α + φ2 |φ2 ⊗ γ¯ γ + φ1 |φ1 ⊗ αα ¯ + φ1 |φ2 ⊗ αγ ¯ = 1 ⊗ (γ¯ γ + αα) ¯ =1⊗1, δR (φ2 |φ2 ) = δR (q −2 x1 x¯ 1 + q −4 x¯ 2 x2 + x3 x¯ 3 + x¯ 4 x4 ) = (q −4 x¯ 2 x1 − q −1 x1 x¯ 2 − x¯ 4 x3 + qx3 x¯ 4 ) ⊗ α γ¯ +(q −4 x¯ 2 x2 + q −2 x1 x¯ 1 + x¯ 4 x4 + x3 x¯ 3 ) ⊗ α α¯ +(q −4 x¯ 1 x1 + x2 x¯ 2 + x¯ 3 x3 + q 2 x4 x¯ 4 ) ⊗ γ γ¯ +(q −4 x¯ 1 x2 − q −1 x2 x¯ 1 − x¯ 3 x4 + qx4 x¯ 3 ) ⊗ γ α¯ 2 φ1 |φ1 ⊗ γ γ¯ −q φ1 |φ2 ⊗ γ α¯ = −q φ2 |φ1 ⊗ α γ¯ +φ2 |φ2 ⊗ α α+q ¯ 2 = 1 ⊗ (α α¯ + q γ γ¯ ) = 1 ⊗ 1 , δR (φ1 |φ2 ) = q −5 δR (x¯ 1 )δR (x2 ) − q −2 δR (x2 )δR (x¯ 1 ) −q −1 δR (x¯ 3 )δR (x4 ) + δR (x4 )δR (x¯ 3 ) = 0, since δR defines a coaction on Sq7 and so preserves its commutation relations.


79

4. Representations of the Algebra A(Sq4 ) Let us now construct irreducible ∗-representations of A(Sq4 ) as bounded operators on a separable Hilbert space H. For the moment, we denote in the same way the elements of the algebra and their images as operators in the given representation. As mentioned before, since q → q −1 gives an isomorphic algebra, we can restrict ourselves to |q| < 1. We will consider the representations which are t-finite [19], i.e. such that the eigenvectors of t span H. Since the self-adjoint operator t must be bounded due to the spherical relations, from ¯ it follows that the spectrum should the commutation relations ta = q −2 at, t b¯ = q −4 bt, 2k ¯ be of the form λq and a, b (resp. a, ¯ b) act as rising (resp. lowering) operators on the eigenvectors of t. Then boundedness implies the existence of a highest weight vector, i.e. there exists a vector |0, 0 such that t |0, 0 = t00 |0, 0 , a |0, 0 = 0, b¯ |0, 0 = 0 . By evaluating

q 4 aa ¯

+ bb¯ =

(1 − q −4 t)t

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on |0, 0 we have

(1 − q −4 t00 )t00 = 0. According to the values of the eigenvalue t00 we have two representations. 4.1. The representation β. The first representation, that we call β, is obtained for t00 = 0. Then, t |0, 0 = 0 implies t = 0. Moreover, using the commutation relations (30) and (31), it follows that this representation is the trivial one t = 0, a = 0, b = 0 ,

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the representation Hilbert space being just C; of course, β(1) = 1. 4.2. The representation σ . The second representation, that we call σ, is obtained for t00 = q 4 . This is infinite dimensional. We take the set |m, n = Nmn a¯ m bn |0, 0 with n, m ∈ N, to be an orthonormal basis of the representation Hilbert space H, with N00 = 1 and Nmn ∈ R the normalizations, to be computed below. Then t |m, n = tmn |m, n , a¯ |m, n = amn |m + 1, n , b |m, n = bmn |m, n + 1 . By requiring that we have a ∗-representation we have also that a |m, n = am−1,n |m − 1, n , b¯ |m, n = bm,n−1 |m, n − 1 , with the following recursion relations: am,n±1 = q ±2 am,n ,

bm±1,n = q ±2 bm,n ,

bm,n = q 2 a2n+1,m .

By explicit computation, we find tm,n = q 2m+4n+4 , 1 −1 am,n = Nmn Nm+1,n = (1 − q 2m+2 ) 2 q m+2n+1 ,

−1 = (1 − q 4n+4 ) 2 q 2(m+n+2) . bm,n = Nmn Nm,n+1 1

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80


In conclusion we have the following action: t |m, n = q 2m+4n+4 |m, n , a¯ |m, n = (1 − q

2m+2

1 2

) q

(43)

m+2n+1

|m + 1, n ,

1 2

a |m, n = (1 − q 2m ) q m+2n |m − 1, n , 1

b |m, n = (1 − q 4n+4 ) 2 q 2(m+n+2) |m, n + 1 , b¯ |m, n = (1 − q 4n ) 2 q 2(m+n+1) |m, n − 1 . 1

It is straightforward to check that all the defining relations (30) and (31) are satisfied. In this representation the algebra generators are all trace class: Tr(t) = q 4

m

Tr(|a|) = q

q 2m

q 4n =

n

1 (1 − q 2m+2 ) 2 q m+2n = m,n

≤

q4 (1 − q 2 )(1 − q 4 )

,

1 q (1 − q 2m+2 ) 2 q m 1 − q2 m

q m q q = , 1 − q2 m (1 − q)(1 − q 2 )

Tr(|b|) = q 4

1

(1 − q 4n+4 ) 2 q 2(n+m) =

m,n

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1 q4 (1 − q 4n+4 ) 2 q 2n 1 − q2 n

q 4 2n q4 ≤ q = . 1 − q2 n (1 − q 2 )2 From the sequence of Schatten ideals in the algebra of compact operators one know [29] that the norm closure of trace class operators gives the ideal of compact operators K. As a consequence, the closure of A(Sq4 ) is the C ∗ -algebra C(Sq4 ) = K ⊕ CI. 5. The Index Pairings The ‘defining’ self-adjoint idempotent p in (29) determines a class in the K-theory of Sq4 , i.e. [p] ∈ K0 [C(Sq4 )]. A way to prove its nontriviality is by pairing it with a nontrivial element in the dual K-homology, i.e. with (the class of) a nontrivial Fredholm module [µ] ∈ K 0 [C(Sq4 )]. In fact, in order to compute the pairing of K-theory with K-homology, it is more convenient to first compute the corresponding Chern characters in the cyclic homology ch∗ (p) ∈ H C∗ [A(Sq4 )] and cyclic cohomology ch∗ (µ) ∈ H C ∗ [A(Sq4 )] respectively, and then use the pairing between cyclic homology and cohomology [10]. Like it happens for the q-monopole [14], to compute the pairing and to prove the nontriviality of the bundle it is enough to consider H C0 [A(Sq4 )] and dually to take a suitable trace of the projector. The Chern character of the projection p in (29) has a component in degree zero ch0 (p) ∈ H C0 [A(Sq4 )] simply given by the matrix trace, ch0 (p) := tr(p) = 2 − q −4 (1 − q 2 )(1 − q 4 ) t ∈ A(Sq4 ).

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81

The higher degree parts of ch∗ (p) are obtained via the periodicity operator S; not needing them here we shall not dwell more upon this point and refer to [10] for the relevant details. As mentioned, the K-homology of an involutive algebra A is given in terms of homotopy classes of Fredholm modules. In the present situation we are dealing with a 1-summable Fredholm module [µ] ∈ K 0 [C(Sq4 )]. This is in contrast to the fact that the analogous element of K0 (S 4 ) for the undeformed sphere is given by a 4-summable Fredholm module, being the fundamental class of S 4 . The Fredholm module µ := (H, , γ ) is constructed as follows. The Hilbert space is H = Hσ ⊕ Hσ and the representation is = σ ⊕ β. Here σ is the representation of A(Sq4 ) introduced in (43) and β given in (41) is trivially extended to Hσ . The grading operator is 1 0 γ = . 0 −1 The corresponding Chern character ch∗ (µ) of the class of this Fredholm module has a component in degree 0, ch0 (µ) ∈ H C 0 [A(Sq2n )]. From the general construction [10], the element ch0 (µev ) is the trace τ 1 (x) := Tr (γ (x)) = Tr (σ (x) − β(x)) .

(46)

The operator σ (x) − β(x) is always trace class. Obviously τ 1 (1) = 0. The higher degree parts of ch∗ (µev ) can again be obtained via a periodicity operator. A similar construction of the class [µ] and the corresponding Chern character were given in [21] for quantum two and three dimensional spheres. We are ready to compute the pairing: [µ], [p] := ch0 (µ), ch0 (p) = −q −4 (1 − q 2 )(1 − q 4 ) τ 1 (t) = −q −4 (1 − q 2 )(1 − q 4 ) Tr(t) = −q −4 (1 − q 2 )(1 − q 4 )q 4 (1 − q 2 )−1 (1 − q 4 )−1 = −1 .

(47)

This result shows also that the right A(Sq4 )-module p[A(Sq4 )4 ] is not free. Indeed, any free module is represented in K0 [C(Sq4 )] by the idempotent 1, and since [µ], [1] = 0, the evaluation of [µ] on any free module always gives zero. We can extract the ‘trivial’ element in the K-homology K 0 [C(Sq4 )] of the quantum sphere Sq4 and use it to measure the ‘rank’ of the idempotent p. This generator corresponds to the trivial generator of the K-homology K0 (S 4 ) of the classical sphere S 4 . The latter (classical) generator is the image of the generator of the K-homology of a point by the functorial map K∗ (ι) : K0 (∗) → K0 (S N ), where ι : ∗ → S N is the inclusion of a point into the sphere. Now, the quantum sphere Sq4 has just one ‘classical point’, i.e. the 1-dimensional representation β constructed in Sect. 4.1. The corresponding 1summable Fredholm module [ε] ∈ K 0 [C(Sq4 )] is easily described: the Hilbert space is C with representation β; the grading operator is γ = 1. Then the degree 0 component ch0 (ε) ∈ H C 0 [A(Sq2n )] of the corresponding Chern character is the trace given by the representation itself (since it is a homomorphism to a commutative algebra), τ 0 (x) = β(x) ,

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82


and vanishes on all the generators whereas τ 0 (1) = 1. Not surprisingly, the pairing with the class of the idempotent p is [ε], [p] := τ 0 (ch0 (p)) = β(2) = 2 .

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6. Quantum Principal Bundle Structure Recall that if H is a Hopf algebra and P a right H -comodule algebra with multiplication m : P ⊗ P → P and coaction R : P → P ⊗ H and B ⊆ P is the subalgebra of coinvariants, the extension B ⊆ P is H Hopf-Galois if the canonical map χ : P ⊗B P −→ P ⊗ H , p ⊗B p → χ (p ⊗B p) = p p(0) ⊗ p(1) ,

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is bijective. As mentioned, for us a quantum principle bundle will be the same as a Hopf-Galois extension. For quantum structure groups which are cosemisimple and have bijective antipodes, as is the case for SUq (2), Th. I of [28] grants further nice properties. In particular the surjectivity of the canonical map implies bijectivity and faithfully flatness of the extension. Moreover, an additional useful result [26] is that the map χ is surjective whenever, for any generator h of H , the element 1 ⊗ h is in its image. This follows from the left P -linearity -colinearity of the map χ . Indeed, and right H let h, k be two elements of H and pi ⊗ pi , qj ⊗ qj ∈ P ⊗ P be such that χ( pi ⊗B pi ) = 1 ⊗ h, χ ( qj ⊗B qj ) = 1 ⊗ k. Then χ ( pi qj ⊗B qj pi ) = 1 ⊗ kh, that is 1 ⊗ kh is in the image of χ. But, since the map χ is left P -linear, this implies its surjectivity. Definition 2. Let P be a bimodule over the ring B. any Given two elements |ξ1 and |ξ2 . m in the free module E = C ⊗ P , we shall define ξ1 ⊗B ξ2 ∈ P ⊗B P by

.

ξ1 ⊗B ξ2 :=

m

j j ξ¯1 ⊗B ξ2 .

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j =1

. Analogously, one can define quantities ξ1 ⊗ ξ2 ∈ P ⊗ P with the same formula as above and tensor products taken over the ground field C. Proposition 6. The extension A(Sq7 ) ⊂ A(Spq (2)) is a faithfully flat A(Spq (1))-HopfGalois extension. Proof. Now P = A(Spq (2)), H = A(Spq (1)) and B = A(Sq7 ) and the coaction R of H is given just before Prop. 2. Since A(Spq (1)) A(SUq 2 (2)) has a bijective antipode and is cosemisimple ([19], Chap. 11), from the general considerations given above in order to show the bijectivity of the canonical map χ : A(Spq (2)) ⊗A(Sq7 ) A(Spq (2)) −→ A(Spq (2)) ⊗ A(Spq (1)) , it is enough ¯ γ¯ of A(Spq (1)) in (17) are in its image. to show that all generators α, γ , α, Let T 2 , T 3 be the second and third columns of the defining matrix T of Spq (2). We shall think of them as elements of the free module C4 ⊗ A(Spq (2)). Obviously,


83

T i |T j = δ ij . Recalling that A(Spq (2)) is both a left and right A(Sq7 )-module and using Def. 2, we have that   . .   T 2 ⊗A(Sq7 ) T 2 T 2 ⊗A(Sq7 ) T 3 α −q 2 γ¯   .   .  χ  = 1 ⊗  . . 3 2 3 3 γ α ¯ T ⊗A(Sq7 ) T T ⊗A(Sq7 ) T

Indeed,

. 2 χ ( T 2 ⊗A(Sq7 ) T 2 ) = T i R Ti2 = T 2 |T 2 ⊗ α + T 2 |T 3 ⊗ γ = 1 ⊗ α , . 3 χ ( T 3 ⊗A(Sq7 ) T 2 ) = T i R Ti2 = T 3 |T 2 ⊗ α + T 3 |T 3 ⊗ γ = 1 ⊗ γ ;

a similar computation giving the other two generators. Proposition 7. extension.

The extension A(Sq4 )

⊂

A(Sq7 ) is a faithfully flat A(SUq (2))-Hopf-Galois

Proof. Now P = A(Sq7 ), H = A(SUq (2)) and B = A(Sq4 ) and the coaction δR of H is given in Prop. 4. As already mentioned A(SUq (2)) has a bijective antipode and is cosemisimple, then as before in order to show the bijectivity of the canonical map χ : A(Sq7 ) ⊗A(Sq4 ) A(Sq7 ) −→ A(Sq7 ) ⊗ A(SUq (2)) , we have to show that all generators α, γ , α, ¯ γ¯ of A(SUq (2)) in (32) are in its image. Recalling that A(Sq7 ) is both a left and right A(Sq4 )-module and using Def. 2, we have that   . .   φ1 ⊗A(Sq4 ) φ1 φ1 ⊗A(Sq4 ) φ2 α −q γ¯   .    , χ  = 1 ⊗  . . γ α ¯ φ2 ⊗A(Sq4 ) φ1 φ2 ⊗A(Sq4 ) φ2 where |φ1 , |φ2 are the two vectors introduced in Eqs. (23) and (24). Indeed

. χ ( φ1 ⊗A(Sq4 ) φ1 ) = χ q −6 x¯ 1 ⊗A(Sq4 ) x1 + q −2 x2 ⊗A(Sq4 ) x¯ 2 +q −2 x¯ 3 ⊗A(Sq4 ) x3 + x4 ⊗A(Sq4 ) x¯ 4 = q −6 x¯ 1 δR (x1 ) + q −2 x2 δR (x¯ 2 ) + q −2 x¯ 3 δR (x3 ) + x4 δR (x¯ 4 ) = q −6 x¯ 1 x1 ⊗ α + q −5 x¯ 1 x2 ⊗ γ + q −2 x2 x¯ 2 ⊗ α − q −2 x2 x¯ 1 ⊗ γ +q −2 x¯ 3 x3 ⊗ α − q −1 x¯ 3 x4 ⊗ γ + x4 x¯ 4 ⊗ α + x4 x¯ 3 ⊗ γ = φ1 |φ1 ⊗ α = 1 ⊗ α ,

. χ( φ2 ⊗A(Sq4 ) φ1 ) = q −5 x¯ 2 δR (x1 ) − q −2 x1 δR (x¯ 2 ) − q −1 x¯ 4 δR (x3 ) + x3 δR (x¯ 4 )

= q −5 x¯ 2 x1 ⊗ α + q −4 x¯ 2 x2 ⊗ γ − q −2 x1 x¯ 2 ⊗ α + q −2 x1 x¯ 1 ⊗ γ −q −1 x¯ 4 x3 ⊗ α + x¯ 4 x4 ⊗ γ + x3 x¯ 4 ⊗ α + x3 x¯ 3 ⊗ γ = φ2 |φ1 ⊗ α + φ2 |φ2 ⊗ γ = 1 ⊗ γ , with similar computations for the other generators.

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It was proven in [4] that the bundle constructed in [3] is a coalgebra Galois extension [9, 6]. The fact that our bundle A(Sq4 ) ⊂ A(Sq7 ) is Hopf-Galois shows also that these two bundles cannot be the same. On our extension A(Sq4 ) ⊂ A(Sq7 ) there is a strong connection. Indeed a H -HopfGalois extension B ⊆ P for which H is cosemisimple and has a bijective antipode is also equivariantly projective, that is there exists a left B-linear right H -colinear splitting s : P → B ⊗ P of the multiplication map m : B ⊗ P → P , m ◦ s = idP [27]. Such a map characterizes the so-called strong connection. Constructing a strong connection is an alternative way to prove that one has a Hopf Galois extension [12, 13]. In particular, if H has an invertible antipode S, an equivalent description of a strong connection can be given in terms of a map : H → P ⊗ P satisfying a list of conditions [17, 7] (see also [15, 5]). We denote by the coproduct on H with Sweedler notation (h) = h(1) ⊗ h(2) , by δ : P → P ⊗ H the right-comodule structure on P with notation δp = p(0) ⊗ p(1) , and δl : P → H ⊗ P is the induced left H -comodule structure of P defined by δl (p) = S −1 (p(1) ) ⊗ p(0) . Then, for the map one requires that (1) = 1 ⊗ 1 and that for all h ∈ H , χ ((h)) = 1 ⊗ h , (h(1) ) ⊗ h(2) = (id ⊗ δ) ◦ (h) , h(1) ⊗ (h(2) ) = (δl ⊗ id) ◦ (h).

(52)

The splitting s of the multiplication map is then given by s :P →B ⊗P ,

p → p(0) (p(1) ) .

Now, if g, h ∈ H are such that (g) = g 1 ⊗ g 2 and (h) = h1 ⊗ h2 satisfy condition (52) so does (gh) defined by (gh) := h1 g 1 ⊗ g 2 h2 .

(53)

If H has a PBW basis [18], this fact can be used to iteratively construct once one knows its value on the generators of H . For H = A(SUq (2)), with generators, α, γ , α¯ and γ¯ , the PBW basis is given by α k γ l γ¯ m , with k, l, m ∈ {0, 1, 2, . . . } and γ k γ¯ l α¯ m , with k, l ∈ {0, 1, 2, . . . } and m ∈ {1, 2, . . . } [30]. Then, for our extension A(Sq4 ) ⊂ A(Sq7 ) the map can be constructed as follows. Firstly, we put (1) = 1 ⊗ 1. Then, on the generators we set . . (α) := φ1 ⊗ φ1 , (α) ¯ := φ2 ⊗ φ2 , . . (γ ) := φ2 ⊗ φ1 , (γ¯ ) := −q −1 φ1 ⊗ φ2 . These expressions for satisfy all the properties (52): Firstly, χ ((α)) = 1 ⊗ α follows from the proof of Prop. 7. Then, (id ⊗ δ) ◦ (α) = q −6 x¯ 1 ⊗ δx1 + q −2 x2 ⊗ δ x¯ 2 + q −2 x¯ 3 ⊗ δx3 + x4 ⊗ δ x¯ 4 . . = φ1 ⊗ φ 1 ⊗ α + φ 1 ⊗ φ 2 ⊗ γ = (α) ⊗ α − q(γ¯ ) ⊗ γ = (α(1) ) ⊗ α(2) .


85

Moreover (δl ⊗ id) ◦ (α) = q −6 (α ⊗ x¯ 1 − q 2 γ¯ ⊗ x¯ 2 ) ⊗ x1 + q −2 (q γ¯ ⊗ x1 + α ⊗ x2 ) ⊗ x¯ 2 +q −2 (q 2 γ¯ ⊗ x¯ 4 + α ⊗ x¯ 3 ) ⊗ x3 + (−q γ¯ ⊗ x3 + α ⊗ x4 ) ⊗ x¯ 4 . . = α ⊗ φ1 ⊗ φ1 − q γ¯ ⊗ φ2 ⊗ φ1 = α ⊗ (α) − q γ¯ ⊗ (γ ) = α(1) ⊗ (α(2) ) . Similar computations can be carried for γ , α¯ and γ¯ . That an iterative procedure constructed by using (53) on the PBW basis leads to a well defined on the whole of H = A(SUq (2)) will be proven in the forthcoming paper [23] where other elaborations coming from the existence of a strong connection will be presented as well. 6.1. The associated bundle and the coequivariant maps. We now give some elements of the theory of associated quantum vector bundles [8] (see also [11]). Let B ⊂ P be a H -Galois extension with R the coaction of H on P . Let ρ : V → H ⊗ V be a corepresentation of H with V a finite dimensional vector space. A coequivariant map is an element ϕ in P ⊗ V with the property that (R ⊗ id)ϕ = (id ⊗ (S ⊗ id) ◦ ρ)ϕ

(54)

where S is the antipode of H . The collection ρ (P , V ) of coequivariant maps is a right and left B-module. The algebraic analogue of bundle nontriviality is translated in the fact that the HopfGalois extension B ⊂ P is not cleft. On the other hand, it is know that for a cleft Hopf-Galois extension, the module of coequivariant maps ρ (P , V ) is isomorphic to the free module of coinvariant maps 0 (P , V ) = B ⊗ V [8, 14]. For our A(SUq (2))-Hopf-Galois extension A(Sq4 ) ⊂ A(Sq7 ), let ρ1 : C2 → C2 ⊗ A(SUq (2)) be the fundamental corepresentation of A(SUq (2)) with 1 (A(Sq7 ), C2 ) the right A(Sq4 )-module of corresponding coequivariant maps. Now, the projection p in (29) determines a quantum vector bundle over Sq4 whose module of section is p[A(Sq4 )4 ], which is clearly a right A(Sq4 )-module. The following proposition in straightforward Proposition 8. The modules E := p[A(Sq4 )4 ] and 1 (A(Sq7 ), C2 ) are isomorphic as right A(Sq4 )-modules. Proof. Remember that p = vv ∗ with v in (25). The element p(F ) ∈ E, with F = (f1 , f2 , f3 , f4 )t , corresponds to the equivariant map v ∗ F ∈ 1 (A(Sq7 ), C2 ).

We expect that a similar construction extends to every irreducible corepresentation of A(SUq (2)) by means of suitable projections giving the corresponding associated bundles [23]. Proposition 9. The Hopf-Galois extension A(Sq4 ) ⊂ A(Sq7 ) is not cleft. Proof. As mentioned, the cleftness of the extension does imply that all modules of coequivariant maps are free. On the other hand, the nontriviality of the pairing (47) between the defining projection p in (29) and the Fredholm module µ constructed in Sect. 5 also shows that the module p[A(Sq4 )4 ] ρ (A(Sq7 ), C2 ) is not free.

86


Acknowledgements. We thank the referee for many useful comments and suggestions. We are grateful to Tomasz Brzeziński and Piotr M. Hajac for several important remarks on a previous version of the compuscript. Also, Eli Hawkins, Walter van Suijlekom, Marco Tarlini are thanked for very useful discussions.

A. The Classical Hopf Fibration S 7 → S 4 We shall review the classical construction of the basic anti-instanton bundle over the four dimensional sphere S 4 in a ‘noncommutative parlance’ following [16]. This has been useful in the main text for our construction of the quantum deformation of the Hopf bundle. We write the generic element of the group SU (2) as w1 w2 w= . (55) −w¯ 2 w¯ 1 The SU (2) principal fibration SU (2) → S 7 → S 4 over the sphere S 4 is explicitly real4 2 ized as follows. The total space is S 7 = {z = (z1 , z2 , z3 , z4 ) ∈ C4 , i=1 |zi | = 1} , with right diagonal action   w1 w2 0 0  −w¯ 2 w¯ 1 0 0  S 7 × SU (2) → S 7 , z · w := (z1 , z2 , z3 , z4 )  . (56) 0 0 w1 w 2  0 0 −w¯ 2 w¯ 1 The bundle projection π : S 7 → S 4 is just the Hopf projection and it can be explicitly given as π(z1 , z2 , z3 , z4 ) := (x, α, β) with x = |z1 |2 + |z2 |2 − |z3 |2 − |z4 |2 = −1 + 2(|z1 |2 + |z2 |2 ) = 1 − 2(|z3 |2 + |z4 |2 ) , α = 2(z1 z¯ 3 + z2 z¯ 4 ) , β = 2(−z1 z4 + z2 z3 ) . (57) One checks that |α|2 + |β|2 + x 2 = ( 4i=1 |zi |2 )2 = 1 . We need the rank 2 complex vector bundle E associated with the defining left representation ρ of SU (2) on C2 . The quickest way to get this is to identify S 7 with the unit sphere in the 2-dimensional quaternionic (right) H-module H2 and S 4 with the projective line P1 (H), i.e. the set of equivalence classes (w1 , w2 )t (w1 , w2 )t λ with (w1 , w2 ) ∈ S 7 and λ ∈ Sp(1) SU (2). Identifying H C2 , the vector (w1 , w2 )t ∈ S 7 reads   z 1 z2  −¯z z¯  v =  2 1. (58) z3 z4 −¯z4 z¯ 3 This is actually a map from S 7 to the Stieffel variety of frames for E. In particular, notice that the two vectors |ψ1 , |ψ2 given by the columns of v are orthonormal, indeed v ∗ v = I2 . As a consequence, p := vv ∗ = |ψ1 ψ1 | + |ψ2 ψ2 | is a self-adjoint idempotent (a projector), p2 = p, p ∗ = p. Of course p is SU (2) invariant and hence its entries are functions on S 4 rather than S 7 . An explicit computation yields   1+x 0 α β 1  0 1 + x −β¯ α¯   , p=  (59)  α ¯ −β 1 − x 0  2 β¯ α 0 1−x


87

where (x, α, β) are the coordinates (57) on S 4 . Then p ∈ Mat4 (C ∞ (S 4 , C)) is of rank 2 by construction. The matrix v in (58) is a particular example of the matrices v given in [1], for n = 1, k = 1, C0 = 0, C1 = 1, D0 = 1, D1 = 0. This gives the (anti-)instanton of charge −1 centered at the origin and with unit scale. The only difference is that here we identify C4 with H2 as a right H-module. This notwithstanding, the projections constructed in the two formalisms actually coincide. Finally recall that, as mentioned already, the classical limit of our quantum projection (29) is conjugate to (59). The canonical connection associated with the projector, ∇ := p ◦ d : ∞ (S 4 , E) → ∞ (S 4 , E) ⊗C ∞ (S 4 ,C) 1 (S 4 , C),

(60)

corresponds to a Lie-algebra valued (su(2)) 1-form A on S 7 whose matrix components are given by Aij = ψi |dψj ,

i, j = 1, 2 .

(61)

This connection can be used to compute the Chern character of the bundle. Out of the curvature of the connection ∇ 2 = p(dp)2 one has the Chern 2-form and 4-form given respectively by 1 tr(p(dp)2 ) , 2πi 1 C2 (p) := − 2 [tr(p(dp)4 ) − C1 (p)C1 (p)] , 8π C1 (p) := −

(62)

with the trace tr just an ordinary matrix trace. It turns out that the 2-form p(dp)2 has vanishing trace so that C1 (p) = 0. As for the second Chern class, a straightforward calculation shows that 1 [(x0 dx4 − x4 dx0 )(dξ )3 + 3dx0 dx4 ξ (dξ )2 ] 32π 2 3 = − 2 [x0 dx1 dx2 dx3 dx4 + cyclic permutations] 8π 3 = − 2 d(vol(S 4 )) . 8π

C2 (p) = −

(63)

The second Chern number is then given by c2 (p) =

S4

C2 (p) = −

3 8π 2

S4

d(vol(S 4 )) = −

3 8 2 π = −1 . 8π 2 3

(64)

The connection A in (61) is (anti-)self-dual, i.e. its curvature FA := dA + A ∧ A satisfies (anti-)self-duality equations, ∗H FA = −FA , with ∗H the Hodge map of the canonical (round) metric on the sphere S 4 . It is indeed the basic Yang-Mills anti-instanton found in [2].

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References 1. Atiyah, M.: The geometry of Yang-Mills fields. Lezioni Fermiane. Accademia Nazionale dei Lincei e Scuola Normale Superiore, Pisa 1979 2. Belavin, A., Polyakov, A., Schwartz, A., Tyupkin, Y.: Pseudoparticles solutions of the Yang-Mills equations. Phys. Lett. 58 B, 85–87 (1975) 3. Bonechi, F., Ciccoli, N., Tarlini, M.: Noncommutative instantons on the 4-sphere from quantum groups. Commun. Math. Phys. 226, 419–432 (2002) 4. Bonechi, F., Ciccoli, N., D¸abrowski, L., Tarlini, L.M.: Bijectivity of the canonical map for the non-commutative instanton bundle. J. Geom. Phys. 51, 71–81 (2004) 5. Brzeziński, T., D¸abrowski, L., Zielinski, B.: Hopf fibration and monopole connection over the contact quantum spheres. J. Geom. Phys. 50, 345–359 (2004) 6. Brzeziński, T., Hajac, P.M.: Coalgebra extensions and algebra coextensions of Galois type. Commun. Algebra 27, 1347–1368 (1999) 7. Brzeziński, T., Hajac, P.M.: The Chern-Galois character. C. R. Acad. Sci. Paris, Ser. I 333, 113–116 (2004) 8. Brzeziński, T., Majid, S.: Quantum group gauge theory on quantum spaces. Commun. Math. Phys. 157, 591–638 (1993) Erratum 167, 235 (1995) 9. Brzeziński, T., Majid, S.: Coalgebra Bundles. Commun. Math. Phys. 191, 467–492 (1998) 10. Connes, A.: Noncommutative geometry. London-New York: Academic Press, 1994 11. Durdevich, M.: Geometry of quantum principal bundles I. Commun. Math. Phys. 175, 427–521 (1996); Geometry of quantum principal bundles II. Rev. Math. Phys. 9, 531–607 (1997) 12. D¸abrowski, L., Grosse, H., Hajac, P.M.: Strong connections and Chern-Connes pairing in the HopfGalois theory. Commun. Math. Phys. 206, 247–264 (1999) 13. Hajac, P.M.: Strong connections on quantum principal bundles. Commun. Math. Phys. 182, 579–617 (1996) 14. Hajac, P.M., Majid, S.: Projective module description of the q-monopole. Commun. Math. Phys. 206, 247–264 (1999) 15. Hajac, P.M., Matthes, R., Szymański, W.: A locally trivial quantum Hopf fibration. http:// arXiv.org/list/math.QA/0112317, 2001; to appear in Algebra and Representation Theory 16. Landi, G.: Deconstructing monopoles and instantons. Rev. Math. Phys. 12, 1367–1390 (2000) 17. Majid, S.: Quantum and braided group Riemannian geometry. J. Geom. Phys. 30, 113–146 (1999) 18. Kassel, C.: Quantum groups. Berlin-Heidelberg-New York: Springer 1995 19. Klimyk,A., Schmüdgen, K.: Quantum groups and their representations. Berlin-Heidelberg: Springer Verlag, 1997 20. Kreimer, H.F., Takeuchi, M.: Hopf algebras and Galois extensions of an algebra. Indiana Univ. Math. J. 30, 675–692 (1981) 21. Masuda, T., Nakagami, Y., Watanabe, J.: Noncommutative differential geometry on the quantum SU (2). I:An algebraic viewpoint. K-Theory 4, 157–180 (1990); Noncommutative differential geometry on the quantum two sphere of P.Podle´s. I: An algebraic viewpoint. K-Theory 5, 151–175 (1991) 22. Montgomery, S.: Hopf algebras and their actions on rings. Providence, RI: AMS 1993 23. Pagani, C.: In preparation 24. Podle´s, P.: Quantum spheres. Lett. Math. Phys. 14, 193–202 (1987) 25. Reshetikhin, N.Yu., Takhtadzhyan, L.A., Faddeev, L.D.: Quantization of Lie groups and Lie algebras. Leningrad Math. J. 1, 193–225 (1990) 26. Schauenburg, P.: Bi-Galois objects over Taft algebras. Israel J. Math. 115, 101–123 (2000) 27. Schauenburg, P., Schneider, H.: Galois type extensions of noncommutative algebras. In preparation 28. Schneider, H.: Principal homogeneous spaces for arbitrary Hopf algebras. Israel J. Math. 72, 167–195 (1990) 29. Simon, B.: Trace ideals and their applications. Cambridge: Cambridge Univ. Press, 1979 30. Woronowicz, S.L.: Twisted SU(2) group. An example of a noncommutative differential calculus. Publ. Res. Inst. Math. Sci. 23, 117–181 (1987) Communicated by L. Takhtajan


Communications in


Setting the Quantum Integrand of M-Theory Daniel S. Freed1 , Gregory W. Moore2 1

Department of Mathematics, University of Texas at Austin, 1 University Station, Austin, TX, 78712-0257, USA. E-mail: [email protected] 2 Department of Physics, Rutgers University, Piscataway, NJ 08855-0849, USA. E-mail: [email protected] Received: 21 September 2004 / Accepted: 26 January 2005 Published online: 24 January 2006 – © Springer-Verlag 2006

Abstract: In anomaly-free quantum field theories the integrand in the bosonic functional integral—the exponential of the effective action after integrating out fermions—is often defined only up to a phase without an additional choice. We term this choice “setting the quantum integrand”. In the low-energy approximation to M-theory the E8 -model for the C-field allows us to set the quantum integrand using geometric index theory. We derive mathematical results of independent interest about pfaffians of Dirac operators in 8k + 3 dimensions, both on closed manifolds and manifolds with boundary. These theorems are used to set the quantum integrand of M-theory for closed manifolds and for compact manifolds with either temporal (global) or spatial (local) boundary conditions. In particular, we show that M-theory makes sense on arbitrary 11-manifolds with spatial boundary, generalizing the construction of heterotic M-theory on cylinders.

Contents 1. Determinants, Pfaffians, and η-Invariants . . . . . . . . . 1.1. Determinant line bundle . . . . . . . . . . . . . . 1.2. Odd dimensions . . . . . . . . . . . . . . . . . . . 1.3. 8k + 3 dimensions . . . . . . . . . . . . . . . . . . 1.4. ζ -functions . . . . . . . . . . . . . . . . . . . . . 2. M-theory Action on Closed Manifolds . . . . . . . . . . 3. (8k + 3)-Dimensional Manifolds with Boundary . . . . 3.1. Generalities . . . . . . . . . . . . . . . . . . . . . 3.2. Global boundary conditions . . . . . . . . . . . . . 3.3. Local boundary conditions . . . . . . . . . . . . . 4. M-Theory Action on Compact Manifolds with Boundary

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The work of D.F. is supported in part by NSF grant DMS-0305505. The work of G.M. is supported in part by DOE grant DE-FG02-96ER40949

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4.1. Actions and anomalies . . . . . . . . . . . . . . . 4.2. Temporal boundary conditions . . . . . . . . . . . 4.3. Spatial boundary conditions . . . . . . . . . . . . 5. Further Discussion . . . . . . . . . . . . . . . . . . . . 5.1. The E8 -model . . . . . . . . . . . . . . . . . . . . 5.2. M2-branes . . . . . . . . . . . . . . . . . . . . . . 5.3. Boundary values of C-fields . . . . . . . . . . . . 5.4. Temporal boundaries and the Hamiltonian anomaly 5.5. Topological terms . . . . . . . . . . . . . . . . . . Appendix A: The Gravitino Path Integral . . . . . . . . . . . A.1. The gravitino theory . . . . . . . . . . . . . . . . . A.2. Local analysis of the equations of motion . . . . . A.3. Partition function . . . . . . . . . . . . . . . . . . A.4. Boundary conditions for ghosts . . . . . . . . . . . Appendix B: Quaternionic Fredholms and Pfaffians . . . . .

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The low-energy approximation to M-theory is a refinement of classical 11-dimensional supergravity. It has a simple field content: a metric g, a 3-form gauge potential C, and a gravitino. The M-theory action contains rather subtle “Chern-Simons” terms which, on a topologically nontrivial manifold Y , raise delicate issues in the definition of the (exponentiated) action. Some aspects of the problem were resolved by Witten [W1]. The key ingredients are: a quantization law for C and a background magnetic current induced by the fourth Stiefel-Whitney class of the underlying manifold; an expression for the exponentiated Chern-Simons terms using an E8 gauge field and an associated Dirac operator in 12 dimensions; and finally a sign ambiguity in the gravitino partition function. In [DFM] the link to E8 was used to construct a model for the C-field and define precisely the action, assuming that the metric g is fixed. The present paper gives a complete treatment of the M-theory action as a function of both C and g. Furthermore, we treat manifolds with boundary. The boundary may have several components and each component is interpreted either as a fixed time slice (temporal boundary) or a boundary in space (spatial boundary). We do not mix temporal and spatial boundary conditions. Our discussion of spatial boundaries in §4.3 generalizes the case Y = X × [0, 1], where X is a closed 10-manifold, which was described in the work of Horava and Witten [HW1, HW2]. Our analysis here makes it clear that the anomaly cancellation is local. (As emphasized in [BM] the locality of anomaly cancellation in the Horava-Witten model is far from obvious.) In particular, we show that there is no topological obstruction to formulating M-theory on an 11-manifold with an arbitrary number of boundary components, provided an independent E8 super-Yang-Mills multiplet is present on each component. The analysis here is more than a cancellation of anomalies in M-theory. Already in [W1] Witten showed that there is a nontrivial Green-Schwarz mechanism canceling global anomalies on closed 11-manifolds. We go further and show that the anomaly is canceled canonically. This is a crucial distinction for the following reason. The absence of anomalies is a necessary condition for a quantum theory to be well-defined, but the cancellation mechanism depends on physically measurable choices. Put differently, there are undetermined phases if the configuration space of bosonic fields is not connected. As we explain quite generally in §4.1, the exponentiated effective action after integrating out fermionic fields is naturally a section of a hermitian line bundle with covariant derivative over the space of bosonic fields. The absence of anomalies means

Setting the Quantum Integrand of M-Theory

91

that the line bundle is geometrically trivializable, i.e., the covariant derivative is flat with no holonomy. If there are no anomalies then global trivializations exist, and a choice of trivialization determines the integrand of the bosonic functional integral. When we make such a choice we say we have set the quantum integrand. The general uniqueness question for settings of the quantum integrand is discussed in §5.5. Our main result is that in M-theory there is a canonical choice of trivialization, thus a canonical setting of the quantum integrand of M-theory. The procedure by which we set the quantum integrand of M-theory is, as we have mentioned, an example of the GreenSchwarz mechanism. Quite generally, by the Green-Schwarz mechanism we mean that setting the quantum integrand involves a trivialization of the tensor product of two line bundles with covariant derivative, one coming from integration over fermionic fields and the other from the simultaneous presence of electric and magnetic current; see [F2, F3 , Part 3 ] for a general discussion. The integral over fermionic fields is a section of a pfaffian line bundle. In this paper we use the E8 -model for the C-field to define the exponentiated electric coupling. This has the advantage that the associated line bundle with covariant derivative is defined by Atiyah-Patodi-Singer η-invariants associated to the E8 -gauge fields. With this model, then, we can analyze both line bundles in the context of standard invariants of geometric index theory and explicitly write down the trivialization which sets the quantum integrand. The mathematical results we apply to M-theory are given in §1 for closed manifolds and in §3 for manifolds with boundary. Determinant and pfaffian line bundles are usually considered for families of Dirac operators on even dimensional manifolds, but our interest here is in the odd dimensional case. As we explain in §1.2 there is a second natural real line bundle with covariant derivative in odd dimensions, defined using the exponentiated η-invariant, and it is isomorphic to the determinant line bundle (Proposition 1.16). This isomorphism is equivalent to a trivialization of the tensor product—the trivialization needed for the physics—since the second line bundle is real. This isomorphism induces a real structure on the determinant line bundle in odd dimensions. Also, it induces a nonflat complex trivialization of the determinant line bundle, so gives a definition of the determinant of the Dirac operator in odd dimensions as a complex number [S]; see Remark 1.20. This definition is often used in the physics literature, and is arrived at with Pauli-Villars regularization [R1, R2, ADM]. However, the definition as an element of the determinant line is more fundamental. There is an important refinement (Proposition 1.31) to pfaffians in dimensions 3, 11, 19, . . . which includes the dimensions of interest in M-theory: 11 for the bulk and 3 for M2-branes (§5.2). This refinement is topological in a sense made precise in Appendix B (Proposition B.2). We take up the generalization of this isomorphism to Dirac operators on manifolds with boundary in Sect. 3. Most often considered in the geometric index theory literature are boundary conditions of global type, which in the physics correspond to a temporal boundary. The generalization of the basic theorem to this case is straightforward (§3.2). Local boundary conditions arise in the physics from spatial boundaries, but because they do not exist for every Dirac operator they are less studied. The generalization to this case is more subtle and (in general dimensions) is the subject of the forthcoming thesis of Matthew Scholl. The applications to M-theory on manifolds with boundary appear as Theorem 4.16 (temporal boundary) and Conjecture 4.35 (spatial boundary). Our treatment falls short by not defining precisely the partition function of the Rarita-Schwinger (gravitino) field. The definition commonly used in the literature seems

92


ill-defined due to singularities related to the zeromodes of bosonic ghosts for supersymmetry transformations. Moreover, the derivation of the standard expression in terms of pfaffians of Dirac operators assumes an off-shell formulation of supergravity, something which is lacking in the 10- and 11-dimensional cases. Nevertheless, we take the standard expression as motivation for the line bundle with covariant derivative of which the Rarita-Schwinger partition function is a section. We present a derivation of the standard formula in Appendix A, mostly to motivate the local boundary conditions for the ghost fields which are used in Sect. 4.3. The precise definition of the Rarita-Schwinger partition function is a general issue which we leave to future work. Another issue we do not confront is the dependence of the covariant derivative on the Rarita-Schwinger line bundle on background fluxes. Nontrivial dependence can in principle arise from terms of the form ψGψ in the supergravity action. (There are additional terms of a similar nature in heterotic M-theory.) We believe the above issues will not drastically alter the discussion we give, which is based on the simple assumption that the Rarita-Schwinger partition function is a section of the line bundle in Eq. (2.2), equipped with the standard covariant derivative. Some general issues of independent interest arose during our investigations. One concerns the definition of anomalies and the setting of quantum integrands for manifolds with temporal boundaries. This forms part of the discussion in §4.1 and is elaborated in §5.4 where we relate it to the Hamiltonian interpretation of anomalies. There are interesting mathematical questions which underlie that discussion, but they are not treated here. Another issue concerns boundary values for fields with automorphisms, such as gauge fields. Then the boundary condition includes a choice of isomorphism (for example, see [FQ] where gauge theories with finite gauge groups are treated carefully), and this shows up in the physics as certain phases, such as θ-angles. In §5.3 we indicate how this works for the C-field in M-theory. 1. Determinants, Pfaffians, and η-Invariants The geometry of determinant line bundles was developed in [Q, BF]; see [F1] for a survey. In §1.1 we recall the main points. Our discussion is phrased in general terms and applies in arbitrary dimensions. In odd dimensions Clifford multiplication by the volume form induces a real structure on the determinant line bundle, which we explain in §1.2 by introducing a manifestly real line bundle associated to the η-invariant [APS] and proving it is isomorphic to the determinant line bundle. In §1.3 we prove a refinement in dimensions 8k + 3 (k ∈ Z≥0 ) coming from the quaternionic structure. Some details about ζ -functions are addressed in §1.4.

1.1. Determinant line bundle. Definition 1.1. Let T be a smooth manifold. A geometric family of Dirac operators parametrized by T consists of: (i) a Riemannian manifold Y → T ; that is, a fiber bundle Y → T , a metric on the relative tangent bundle T (Y/T ) → Y, and a horizontal distribution H on Y (thus H ⊕ T (Y/T ) = T Y); and (ii) a bundle M = M 0 ⊕M 1 → Y of complex Z/2Z-graded Cliff(Y/T )-modules with compatible metric and covariant derivative.

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93

The metric and horizontal distribution determine a Levi-Civita covariant derivative on T (Y/T ) → Y. The Riemannian metrics determine a bundle Cliff(Y/T ) → Y of (finite dimensional) Clifford algebras: the fiber at y ∈ Y is the real Clifford algebra of the relative cotangent space Ty∗ (Y/T ). The Clifford module structure on M is given as a map γ : T ∗ (Y/T ) −→ End(M)

(1.2)

which obeys the Clifford relation γ (θ1 )γ (θ2 ) + γ (θ2 )γ (θ1 ) = −2θ1 , θ2 ,

θ1 , θ2 ∈ Ty∗ (Y/T ),

y ∈ Y. (1.3)

We ask that the image consist of odd skew-adjoint transformations. The compatibility in the last line of Definition 1.1 also requires that (1.2) be flat. For each t ∈ T the Dirac operator Dt = γ ◦ ∇ is defined on sections of M Y → Yt . It is odd relative to the t Z/2Z-grading.1 To illustrate the notation let T be a point, so D is a Dirac operator on a single manifold Y . Suppose first that dim Y = 2m is even and Y is spin. Then for the standard Dirac operator M = S is the bundle of Z/2Z-graded spinors with homogeneous components S 0 , S 1 of complex rank 2m−1 , the bundles of chiral spinors. The Dirac operator interchanges the chirality of homogeneous spinor fields. The covariant derivative on S is induced from the Levi-Civita covariant derivative. If dim Y = 2m + 1 is odd, then we usually say that spinors are ungraded: there is no chirality. In the Z/2Z-graded setup we can take each of S 0 , S 1 to be the ungraded spinor bundle of complex rank 2m . This is compatible with the observation that for any Z/2Z-graded Cliff(Y )-module M → Y Clifford multiplication by the volume form provides an isomorphism M 0 → M 1 if dim Y is odd; see the next section for consequences. For Dirac operators with coefficients in a vector bundle E → Y take M = S ⊗ E. We occasionally denote this Dirac operator as ‘DM ’. In the application to field theory the parameter space is typically an infinite dimensional space B of all bosonic fields; from this point of view we study the pullback by a map T → B. Given a geometric family of Dirac operators parametrized by T let H = H0 ⊕ H1 −→ T

be the Hilbert space bundle whose fiber at t ∈ T is the space of L2 sections of M Y → t Yt . Assume each fiber Yt is closed, i.e., compact without boundary. Then the Dirac operator Dt extends to an odd self-adjoint operator on Ht , and so Dt2 to an even selfadjoint operator on Ht . The spectrum of Dt2 is nonnegative, discrete, has no accumulation points, and the eigenspaces are graded finite dimensional subspaces of Ht . Furthermore, if λ2 > 0 is an eigenvalue of Dt2 , then Dt /λ is an isometry from the even component of the λ2 -eigenspace to the odd component. Define spec0 (Dt2 ) to be the spectrum of Dt2 restricted to Ht0 . There is a canonical open cover {Ua }a≥0 of T : Ua = t ∈ T : a ∈ / spec0 (Dt2 ) . (1.4) 1 We remark that the Z/2Z-grading on M is not the physics grading of bosonic and fermionic fields. In our exposition here sections of M—for example, spinor fields—are treated as ordinary commuting fields. When we turn to the physics applications in §2 we use the proper action for fermionic fields.

94


On Ua we introduce the Z/2Z-graded vector bundle H(a) = H0 (a) ⊕ H1 (a) −→ Ua

(1.5)

whose fiber H(a)t at t ∈ T is the sum of the eigenspaces of for eigenvalues less than a. Then H(a) is smooth of finite rank, with constant rank on each component of Ua . Furthermore, the geometric data induces a metric and covariant derivative on H(a). The Dirac operator D restricts to an operator D(a) on H(a). Global geometric invariants of Dirac operators are constructed by patching invariants on Ua . Recall that the determinant line Det E of a finite dimensional ungraded vector space E is its highest exterior power. A linear map S : E0 → E1 between vector spaces of the same dimension has a determinant det S ∈ Hom(Det E0 , Det E1 ) ∼ (1.6) = Det E1 ⊗ (Det E0 )∗ Dt2

which is the induced map on the highest exterior power. The line which appears in (1.6) is the determinant line of the Z/2Z-graded vector space E0 ⊕E1 . It is natural to grade Det E by dim E. For our purposes we take the grading to lie in Z/2Z rather than Z. Returning to the family of Dirac operators, define the line bundle Det H(a) = Det H1 (a) ⊗ Det H0 (a)∗ −→ Ua . For b > a we set H(a, b) = H0 (a, b) ⊕ H1 (a, b) −→ Ua ∩ Ub whose fiber at t is the sum of the eigenspaces of Dt2 for eigenvalues between a and b. There is a canonical isomorphism Det H(a) ⊗ Det H(a, b) −→ Det H(b)

on Ua ∩ Ub

(1.7)

and a canonical nonzero det D(a, b) of Det H(a, b), where D(a, b)t : H0 (a, b)t → H1 (a, b)t is the restriction of the “chiral” Dirac operator Dt0 : Ht0 → Ht1 . From (1.7) we obtain the patching isomorphism section2

Det H(a) −→ Det H(b)

on Ua ∩ Ub ,

(1.8)

and a cocycle identity on Ua ∩Ub ∩Uc , whence a global smooth line bundle Det D → T . Furthermore, the sections det D(a) of Det H(a), defined analogously to det D(a, b), patch to a smooth section det D of Det D → T . The patching isomorphism (1.8) preserves the Z/2Z grading of the determinant line: the parity of Det Dt is the parity of index Dt0 . The metric and covariant derivative on H(a) induce a metric and covariant derivative on Det H(a), but these are not preserved by (1.8). Modify the metric and covariant derivative to obtain invariance under patching: multiply the metric on Det H(a)t by λ2 (1.9) a> 0 define η(α)t [s] = sign(λ − α)|λ|−s − sign(α) · # spec0 (ωt Dt ) ∩ {0} λ∈spec0 (ωt Dt ) \ {0}

and set η(α)t to be the value of the meromorphic continuation of η(α)t [s] at s = 0. For α < β we have η(β)t η(α)t on Vα ∩ Vβ . (1.12) = − # λ ∈ spec0 (ωt Dt ) : α < λ < β 2 2 We use (1.12) to construct two invariants. First, let T denote the group of unit norm complex numbers. Then

η(α) τ (α) = exp 2πi : Vα −→ T (1.13) 2 is invariant under patching, so defines a global function τD : T −→ T.

(1.14)

96


Second, we use the integers in the last term of (1.12) to patch a principal Z-bundle on T : the fiber at t ∈ T is

α 0},

, 1

e−2ikL c(t, k) = O

k

,

k ∈ D3 = {k | Im k < 0, Im k 3 > 0}.

(4.30)

In the case of the mKdV-I equation, the given (linearly well-posed) boundary conditions are g0 (t), h0 (t), and h1 (t), and we are looking for expressions for g1 (t), g2 (t), and h2 (t). Substitute (4.26) into (4.29) and rewrite the resulting equation in the form:

t

t 3 (0) (0) 2ikL 8ik 3 τ −e F1 (t, 2τ − t)e dτ + F1 (t, 2τ − t)e8ik τ dτ 0 0

t 3 3 (1) +ik F1 (t, 2τ − t)e8ik τ dτ = G1 (t, k) + G2 (t, k) + e8ik t c(t, k), (4.31) 0

where

t

t 3 3 (1) (2) G1 (t, k) = e2ikL ik F1 (t, 2τ − t)e8ik τ dτ + k 2 F1 (t, 2τ − t)e8ik τ dτ 0 0

t 3 (2) −k 2 F1 (t, 2τ − t)e8ik τ dτ, (4.32) 0

t

t 3 3 G2 (t, k) = 2e2ikL F2 (t, t − 2τ, k)e8ik τ dτ F1 (t, 2τ − t, k)e8ik τ dτ 0 0

t

t 3 8ik 3 τ −2 F2 (t, t − 2τ, k)e dτ F1 (t, 2τ − t, k)e8ik τ dτ. (4.33) 0

0

Let D = {k : 0 < arg k < π/3}. Considering (4.31) in D as well as replacing k by 2π i Ek and by E 2 k in (4.31), where E = e 3 , we obtain three equations, which are valid for k ∈ D. These equations can be written in the vector form as follows: 3

E(k)U (t, k) = H1 (t, k) + H2 (t, k) + e8ik t Hc (t, k),

k ∈ D,

(4.34)

Integrable Nonlinear Evolution Equations on a Finite Interval

163

where (j = 1, 2)  1 1 −e2ikL , 1 E E(k) = −e2iEkL 2 kL 2 kL −2iE 2 −2iE E e −1 e   t (0) 8ik 3 τ F1 (t, 2τ − t)e dτ     0 t   3τ (0) 8ik  U (t, k) =  F1 (t, 2τ − t)e dτ  ,   0 t   (1) 8ik 3 τ ik F1 (t, 2τ − t)e dτ 



 Gj (t, k) , Gj (t, Ek) Hj (t, k) =  2 kL −2iE 2 Gj (t, E k) e  c(t, k) . c(t, Ek) Hc (t, k) =  2 kL −2iE 2 c(t, E k) e 

0

¯ Notice that det E(k) → 1 − E = 0 as |k| → ∞, k ∈ D. 3

Multiply (4.34) by diag{k 2 , k 2 , −ik}E −1 (k)e−8ik t , 0 < t < t, and integrate along the contour ∂D (0) , which is the boundary of D deformed (in its finite part) to pass above the zeros of det E(k). Then (4.30) implies that the term containing Hc vanishes. In order to evaluate the other terms we will use the following identities (see, e.g., [2]):

t π 3

k2 α(τ )e8ik (τ −t ) dτ dk = (4.35) α(t ), 12 ∂D (0) 0

t 3

km α(τ )e8ik (τ −t ) dτ dk ∂D (0) 0

t 1 m 8ik 3 (τ −t )

= k α(τ )e dτ − α(t ) dk, (4.36) 8ik 3 ∂D (0) 0 where m = 3, 4 and α(τ ) is a smooth function for 0 < τ < t. Then the integration by parts together with Jordan’s lemma show that one can pass to the limit as t → t in the right-hand side of (4.36). Applying (4.36) to the integral term containing H1 one obtains

3

diag{k 2 , k 2 , −ik}E −1 (k)H1 (t, k)e−8ik t dk ∂D (0)



   ˜ 1 (t, t , k) G k2 0 0 ˜ 1 (t, t , Ek)  0 k 2 0  E −1 (k)   dk, G = (0) 2 ∂D kL −2iE

2 ˜ 0 0 −ik G1 (t, t , E k) e

(4.37)

where

˜ 1 (t, t , k) = e2ikL ik G 0

+k

t

2

t

0 t

−k 2 0

(1)

F1 (t, 2τ − t)e8ik

3 (τ −t )

(2)

3 (τ −t )

(2)

3 (τ −t )

F1 (t, 2τ − t)e8ik F1 (t, 2τ − t)e8ik

dτ −

1 (1) F (t, 2t − t) 8k 2 1

dτ −

1 (2) F (t, 2t − t) 8ik 1

dτ +

1 (2) F (t, 2t − t). (4.38) 8ik 1

164

A. Boutet de Monvel, A.S. Fokas, D. Shepelsky

Applying (4.35) to the integral in the left-hand side of (4.34) we arrive at the equation   (0) F1 (t, 2t − t)  (0)  F1 (t, 2t − t) (1) F1 (t, 2t − t)  2    ˜ 1 (t, t , k)

G k 0 0 12 ˜ 1 (t, t , Ek)  0 k 2 0  E −1 (k)   dk G = π ∂D (0) 0 0 −ik 2 ˜ 1 (t, t , E 2 k) e−2iE kL G  2   

G2 (t, k) k 0 0 12  0 k 2 0  E −1 (k)   e−8ik 3 t dk. (4.39) G2 (t, Ek) + 2 π ∂D (0) 0 0 −ik e−2iE kL G2 (t, E 2 k) Evaluating this equation at t = t and using (4.27) and (4.28) we find the following equations for h2 (t), g1 (t), and g2 (t):   ˜ 1 (t, k)

G 1 12i ˜ 1 (t, Ek)  dk G g1 (t) = g0 (t)N2 (t, t) − k E −1 (k)  3 2 π ∂D (0) 2 kL −2iE 2 ˜ 1 (t, E k) G e  

G2 (t, k) 12i  e−8ik 3 t dk, G2 (t, Ek) − k E −1 (k)  2 3 π ∂D (0) e−2iE kL G2 (t, E 2 k) 1 g2 (t) = 2λg03 (t) + g0 (t)M2 (t, t) + g1 (t)N2 (t, t) 2   ˜ 1 (t, k)

G 24 ˜ 1 (t, Ek)  dk G − k 2 E −1 (k)  2 π ∂D (0) 2 kL −2iE 2 ˜ 1 (t, E k) G e  

G2 (t, k) 24  e−8ik 3 t dk, G2 (t, Ek) − k 2 E −1 (k)  2 2 π ∂D (0) e−2iE kL G2 (t, E 2 k) 1 h2 (t) = 2λh30 (t) + h0 (t)M2 (t, t) + h1 (t)N2 (t, t) 2   ˜ 1 (t, k)

G 24 ˜ 1 (t, Ek)  dk G − k 2 E −1 (k)  1 π ∂D (0) 2 kL −2iE 2 ˜ 1 (t, E k) G e  

G2 (t, k) 24  e−8ik 3 t dk, G2 (t, Ek) − k 2 E −1 (k)  (4.40) 2 1 π ∂D (0) e−2iE kL G2 (t, E 2 k) where E −1 (k) j , j = 1, 2, 3, denotes the j th row of E −1 (k) and t 3 1 2ikL ˜ G1 (t, k) = e ik M1 (t, 2τ − t) − h0 (t)N2 (t, 2τ − t) e8ik (τ −t) dτ 2 0 1 1 − 2 h1 (t) + h0 (t)N2 (t, t) 8k 16k 2


t

1 h0 (t) 4ik 0

t 1 3 −k 2 N1 (t, 2τ − t)e8ik (τ −t) dτ + g0 (t). 4ik 0

+k 2

N1 (t, 2τ − t)e8ik

3 (τ −t)

165

dτ −

(4.41)

˜ 1 (t, k), G2 (t, k), N2 (t, t), M2 (t, t), N2 (t, t), and M2 (t, t) involved in The functions G (4.40) can be expressed in terms of (t, k) and (t, k), see Definitions 3.2 and 3.3. Indeed, (3.6) and (3.9) together with (4.2) give

t

1 3 1 (t, k)e8ik t , 2 0

t 1 3 3 F1 (t, 2τ − t, k)e8ik τ dτ = 1 (t, k)e8ik t , 2 0

t ¯ 8ik 3 τ dτ = 1 (2 (t, k) ¯ − 1), F2 (t, t − 2τ, k)e 2 0

t ¯ 8ik 3 τ dτ = 1 (2 (t, k) ¯ − 1). F2 (t, t − 2τ, k)e 2 0 3

F1 (t, 2τ − t, k)e8ik τ dτ =

(4.42) (4.43) (4.44) (4.45)

Hence, G2 (t, k) can be written as follows: G2 (t, k) =

1 8ik 3 t 2ikL ¯ − 1)1 (t, k) − (2 (t, k) ¯ − 1)1 (t, k) . (2 (t, k) e e 2

(4.46)

Since the exponentials in (4.42) depend on k only through k 3 , supplementing (4.42) with the two equations obtained from (4.42) by replacing k with Ek and E 2 k and taking into account (4.27) we obtain a linear system of equations, the solution of which gives 

 ! t (2) 3 k 2 0 F1 (t, 2τ − t, k)e8ik τ dτ  ! t (1)  3  ik 0 F1 (t, 2τ − t, k)e8ik τ dτ  ! t (0) 8ik 3 τ dτ 0 F1 (t, 2τ − t, k)e   2 (t, E 2 k) (t, k) + E (t, Ek) + E 1 1 1 1 3 = e8ik t 1 (t, k) + E 2 1 (t, Ek) + E1 (t, E 2 k) . 6 1 (t, k) + 1 (t, Ek) + 1 (t, E 2 k)

(4.47)

Similarly, supplementing (4.44) with the two equations obtained from (4.44) by replacing k with Ek and E 2 k and taking into account (4.27) we find 

 ! t (2) 3 k 2 0 F2 (t, t − 2τ, k)e8ik τ dτ  ! t (1)  3  ik 0 F2 (t, t − 2τ, k)e8ik τ dτ  ! t (0) 3 8ik τ dτ 0 F2 (t, t − 2τ, k)e   2 2 2 (t, Ek) + E 2 (t, E k) 1 2 (t, k) + E = 2 (t, k) + E 2 2 (t, Ek) + E2 (t, E 2 k) . 6 1 (t, k) + 1 (t, Ek) + 1 (t, E 2 k) − 3

(4.48)

166

A. Boutet de Monvel, A.S. Fokas, D. Shepelsky (1)

(2)

Using (4.35) and the expressions for F2 (t, s) and F2 (t, s) in terms of Nj (t, s), j = 1, 2 and M2 (t, s), see (4.27), from (4.48) we conclude that

2 N2 (t, t) = 2 (t, k) + E2 (t, Ek) + E 2 2 (t, E 2 k) dk, (4.49) π ∂D (0)

2i M2 (t, t) = −λg02 (t) − k 2 (t, k) + E 2 2 (t, Ek) + E2 (t, E 2 k) dk. π ∂D (0) Similarly,

2 (t, k) + E2 (t, Ek) + E 2 2 (t, E 2 k) dk, (4.50) ∂D (0)

2i M2 (t, t) = −λg02 (t) − k 2 (t, k) + E 2 2 (t, Ek) + E2 (t, E 2 k) dk. π ∂D (0) N2 (t, t) =

2 π

˜ 1: Substituting (4.47) and (4.50) into (4.41) we obtain the following expression for G ˜ 1 (t, k) = e2ikL 1 e8ik 3 t 1 (t, k) − 1 (t, Ek) − 1 (t, E 2 k) − 1 h1 (t) G 3 8k 2

1 1 − h0 (t) 2 (t, ζ ) + E2 (t, Eζ ) + E 2 2 (t, E 2 ζ ) dζ h0 (t)+ 2 4ik 8πk ∂D (0) 1 8ik 3 t 1 − e (4.51) 1 (t, k) + E1 (t, Ek) + E 2 1 (t, E 2 k) + g0 (t). 6 4ik Using (4.46), (4.51), (4.49), and (4.50) in (4.40) we obtain the equations for g1 , g2 , and h2 in terms of and . These equations, together with (3.5) and the similar equation for (t, k) constitute a system of four nonlinear ODEs for 1 , 2 , 1 , and 2 . mKdV-II. The integral representations for A and B are the same as in (4.26)–(4.28) but with g0 and g2 replaced by −g0 and −g2 , respectively. Similarly, the integral representations for A and B are the same as in the case of the mKdV-I, with h0 and h2 replaced by −h0 and −h2 , respectively. The global relation (4.29) and relations (4.30) become e−2ikL A(t, k)B(t, k) − B(t, k)A(t, k) = e8ik t c(t, k), 3

and

1

c(t, k) = O

k ∈ D1 ∪ D3 , (4.52)

, k ∈ D1 = {k | Im k < 0, Im k 3 > 0}, 1 e2ikL c(t, k) = O , k ∈ D3 = {k | Im k > 0, Im k 3 > 0}, (4.53) k respectively. Let g0 (t), g1 (t), and h0 (t) be the given boundary conditions. Then the analysis of the global relation consists in finding equations for g2 , h1 , and h2 . Substitute (4.26) into (4.52) and rewrite the resulting equation in the form:

t

t 3 3 (0) (1) −e−2ikL F1 (t, 2τ − t)e8ik τ dτ − e−2ikL ik F1 (t, 2τ − t)e8ik τ dτ 0 0

t 3 (0) 8ik 3 τ + F1 (t, 2τ − t)e dτ = G1 (t, k) + G2 (t, k) + e8ik t c(t, k), (4.54) 0

k


167

where

t

t 3 3 (2) (1) F1 (t, 2τ − t)e8ik τ dτ − ik F1 (t, 2τ − t)e8ik τ dτ G1 (t, k) = e−2ikL k 2 0 0

t 3 (2) −k 2 F1 (t, 2τ − t)e8ik τ dτ, (4.55) 0

t

t 3 3 G2 (t, k) = 2e−2ikL F2 (t, t − 2τ, k)e8ik τ dτ F1 (t, 2τ − t, k)e8ik τ dτ 0 0

t

t 3 3 −2 F2 (t, t − 2τ, k)e8ik τ dτ F1 (t, 2τ − t, k)e8ik τ dτ. (4.56) 0

0

Let, as above, D = {k : 0 < arg k < π/3}. Considering (4.54) in D as well as replacing 2π i k by Ek and by E 2 k in (4.54), where E = e 3 , we get three equations valid for k ∈ D, which can be written in the vector form as follows: 3

E(k)U (t, k) = H1 (t, k) + H2 (t, k) + e8ik t Hc (t, k),

k ∈ D,

(4.57)

where (j = 1, 2)  −1 e2ikL −E e2iEkL , E(k) =  2 kL 2 kL −2iE 2 −2iE −e −E e 1  t  (0) 8ik 3 τ F (t, 2τ − t)e dτ   1  0 t    (1) 8ik 3 τ , U (t, k) =  ik F (t, 2τ − t)e dτ   1  0t    3 (0) F1 (t, 2τ − t)e8ik τ dτ 

−1 −1

 e2ikL Gj (t, k) Hj (t, k) = e2iEkL Gj (t, Ek) , Gj (t, E 2 k) 

 e2ikL c(t, k) Hc (t, k) = e2iEkL c(t, Ek) . c(t, E 2 k) 

0

¯ Notice that det E(k) → E − 1 = 0 as |k| → ∞, k ∈ D. 3t

2 2 −1 −8ik Multiply (4.57) by diag{k , −ik, k }E (k)e , 0 < t < t, and integrate over (0) the contour ∂D , which is the boundary of D deformed (in its finite part) to pass above the zeros of det E(k). Then (4.53) implies that the term containing Hc vanishes, and the resulting equation takes the form 

   2  2ikL  (0) ˜ 1 (t, t , k)

F1 (t, 2t − t) e G k 0 0 12  (1)   0 k 2 0  E −1 (k) e2iEkL G ˜ 1 (t, t , Ek) dk F1 (t, 2t − t) = (0) π ∂D (0) ˜ 1 (t, t , E 2 k) 0 0 −ik G F1 (t, 2t − t)   2  2ikL 

G2 (t, k) e k 0 0 12  0 k 2 0 E −1 (k)e2iEkL G2 (t, Ek)e−8ik 3 t dk, + π ∂D (0) 0 0 −ik G2 (t, E 2 k) (4.58)

168


where

−2ikL ˜ G1 (t, t , k) = e k2 0

t

−ik 0

−k

t

(2)

F1 (t, 2τ − t)e8ik

(1)

F1 (t, 2τ − t)e8ik

2 0

t

(2)

3 (τ −t )

F1 (t, 2τ − t)e8ik

3 (τ −t )

dτ +

3 (τ −t )

dτ −

1 (2) F (t, 2t − t) 8ik 1

1 (1) F (t, 2t − t) 8k 2 1

dτ +

1 (2) F (t, 2t − t). 8ik 1 (4.59)

Evaluating this equation at t = t and using (4.27) and (4.28) we find the following equations for g2 (t), h1 (t), and h2 (t): 

 ˜ 1 (t, k) e2ikL G 1 12i ˜ 1 (t, Ek) g1 (t) = − g0 (t)N2 (t, t) − k E −1 (k) e2iEkL G 3 2 π ∂D (0) ˜ G1 (t, E 2 k)  2ikL 

G2 (t, k) e 12i 3 − k E −1 (k) e2iEkL G2 (t, Ek) e−8ik t dk, 2 π ∂D (0) G2 (t, E 2 k)  2ikL  ˜ 1 (t, k)

e G 1 12i ˜ 1 (t, Ek) dk h1 (t) = − h0 (t)N2 (t, t) − k E −1 (k) e2iEkL G 2 2 π ∂D (0) ˜ 1 (t, E 2 k) G   2ikL

G2 (t, k) e 12i 3 − k E −1 (k) e2iEkL G2 (t, Ek) e−8ik t dk, 3 π ∂D (0) G2 (t, E 2 k) 1 h2 (t) = 2λh30 (t) + h0 (t)M2 (t, t) − h1 (t)N2 (t, t) 2  2ikL  ˜ 1 (t, k)

e G 24 ˜ 1 (t, Ek) dk k 2 E −1 (k) 1 e2iEkL G + π ∂D (0) ˜ G1 (t, E 2 k)   2ikL

G2 (t, k) e 24 3 + k 2 E −1 (k) 1 e2iEkL G2 (t, Ek) e−8ik t dk, (4.60) (0) π ∂D G2 (t, E 2 k)

where t 1 3 ˜ 1 (t, k) = e−2ikL k 2 G h0 (t) N1 (t, 2τ − t)e8ik (τ −t) dτ + 4ik 0

t 3 1 1 −ik M1 (t, 2τ − t) + g0 (t)N2 (t, 2τ − t) e8ik (τ −t) dτ + 2 g1 (t) 2 8k 0

t 1 1 3 + g0 (t). g0 (t)N2 (t, t) − k 2 N1 (t, 2τ − t)e8ik (τ −t) dτ − 16k 2 4ik 0 (4.61)


169

Now one can express the functions involved in (4.60) in terms of and . The formulas for N2 (t, t), M2 (t, t), N2 (t, t), and M2 (t, t) have the same form as in the case of mKdV-I, (4.49) and (4.50), whereas 1 3 ¯ − 1)1 (t, k) − (2 (t, k) ¯ − 1)1 (t, k) (4.62) G2 (t, k) = e8ik t e−2ikL (2 (t, k) 2 and

˜ 1 (t, k) = e−2ikL 1 e8ik 3 t 1 (t, k) + E1 (t, Ek) + E 2 1 (t, E 2 k) + 1 h0 (t) G 6 4ik 1 8ik 3 t 1 1 − e 1 (t, k) − 1 (t, Ek) − 1 (t, E 2 k) + 2 g1 (t) − g0 (t) 3 8k 4ik

1 + g0 (t) (4.63) 2 (t, ζ ) + E2 (t, Eζ ) + E 2 2 (t, E 2 ζ ) dζ. 2 8π k ∂D (0)

5. Conclusions We have presented a general method for the analysis of initial boundary value problems for nonlinear integrable evolution equations on the finite interval and have applied this method to the sine-Gordon and the two mKdV equations. In particular: 1. Given the Dirichlet data for the sG equation, q(0, t) = g0 (t) and q(L, t) = g1 (t), we have characterized the Neumann boundary values qx (0, t) = g1 (t) and qx (L, t) = h1 (t) through a system of nonlinear ODEs for the functions 1 , 2 , 1 , and 2 . The functions 1 and 2 satisfy Eqs. (3.5), the functions 1 and 2 satisfy similar equations, and the Neumann boundary values are given by Eqs. (4.21) and (4.25). Similarly, given the boundary data q(0, t) = g0 (t), q(L, t) = h0 (t), qx (L, t) = h1 (t) for the mKdV-I equation (q(0, t) = g0 (t), qx (0, t) = g1 (t), q(L, t) = h0 (t) for the mKdV-II equation), we have characterized the boundary values qx (0, t) = g1 (t), qxx (0, t) = g2 (t), qxx (L, t) = h2 (t) (qxx (0, t) = g2 (t), qx (L, t) = h1 (t), qxx (L, t) = h2 (t), respectively) through a system of nonlinear ODEs. 2. Given the initial conditions q(x, 0) = q0 (x) (q(x, 0) = q0 (x) and qt (x, 0) = q1 (x) for the sG equation) we have defined {a(k), b(k)}, see Definition 3.1. Given {gl (t)}n−1 0 we have defined {A(k), B(k)}, we have defined {A(k), B(k)}, and given {hl (t)}n−1 0 see Definitions 3.2 and 3.3. 3. Given {a(k), b(k), A(k), B(k), A(k), B(k)} we have defined a Riemann-Hilbert problem for M(x, t, k), and then we have defined q(x, t) in terms of M. We have shown that q(x, t) solves the given nonlinear equation and that q(x, 0) = q0 (x) ∂xl q(0, t) = gl (t),

(and qt (x, 0) = q1 (x) for sG), ∂xl q(L, t) = hl (t), 0 ≤ l ≤ n − 1,

see Theorem 3.1. The most difficult step of this method is the analysis of the global relation coupling the spectral functions. Generally, this leads to a system of nonlinear ODEs. For integrable evolution PDEs on the half-line, there exist particular boundary conditions, the so-called linearizable boundary conditions, for which this nonlinear system can be avoided: the global relation yields directly S(k) in terms of s(k) and the prescribed boundary conditions [21, 15, 3]. Different aspects of linearizable boundary conditions have been studied

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by a number of authors, see, for example, [28, 29, 25, 1]. The analysis of linearizable boundary conditions on a finite domain will be presented elsewhere. Here we only note that x-periodic boundary conditions belong to the linearizable class. In this case S(k) = SL (k) and the global relation simplifies. The analysis of this simplified global relation, together with the results presented in this paper, yields a new formalism for the solution of this classical problem. The main advantage of the inverse scattering method, in comparison with the standard PDE techniques, is that it yields explicit asymptotic results. Indeed, using the inverse scattering method, the solution of the Cauchy problem on the line for an integrable nonlinear PDE can be expressed through the solution of a matrix Riemann-Hilbert problem which has a jump matrix involving an exponential (x, t)-dependence. By making use of the Deift-Zhou method [11] (which is a nonlinear version of the classical steepest descent method), it is possible to compute explicitly the long time behavior of the solution. Furthermore, using a nontrivial extension of the Deift-Zhou method [7], it is also possible to compute the small dispersion limit of the solution. Neither of these two important asymptotic results can be obtained by standard PDE techniques. An important feature of the method of [12] is that it yields the solution of the given initial boundary value problem in terms of a matrix Riemann-Hilbert problem which also involves a jump matrix with an exponential (x, t)-dependence. The curve along which this jump matrix is defined, is now more complicated, but this does not pose any additional difficulties. Thus, it is again possible to obtain explicit asymptotic results. Indeed, for problems on the half-line, the long time asymptotics for decreasing and for time-periodic boundary conditions is obtained in [17–19, 4] (see also [21] and [3]). Furthermore, the zero dispersion limit of the NLS equation is computed in [26]. For problems on the interval, it is again possible to study the asymptotic properties of the solution. In particular, it should be possible to study the small dispersion limit. Another important feature of the method of [12] is that it characterizes the generalized Dirichlet-to-Neumann map. For example, for the Dirichlet problem for the NLS equation on the half-line, the method of [12] yields qx (0, t) in terms of q(x, 0) and q(0, t). Actually, it is shown in [2] and [16] that qx (0, t) can be expressed explicitly through the solution of a system of nonlinear ODEs uniquely defined in terms of q(x, 0) and q(0, t). This is the first time in the literature that such an explicit result is obtained for a nonlinear evolution PDE. In this paper we have presented similar results for initial boundary problems on the interval. For example, for the case of the Dirichlet problem for the sG equation, Eqs. (4.21) and (4.25) express qx (0, t) and qx (L, t) in terms of a system of four nonlinear ODEs which is uniquely defined in terms of q(x, 0), qt (x, 0), q(0, t), and q(L, t). Such explicit results cannot be obtained by the standard PDE techniques. Different approaches to initial-boundary value problems for soliton equations are presented in [8–10], where the analyticity properties of the scattering matrix for the x-equation of an associated Lax pair are used to obtain either an integro-differential evolution equation or a nonlinear Riemann–Hilbert problem for this scattering matrix. In [24], the boundary value problem for the nonlinear Schrödinger equation on an interval is transformed into a Cauchy problem for periodic profiles, for which the algebro-geometric tools of the finite gap method are applied giving a system of nonlinear ordinary differential equations with algebraic right-hand sides for the time evolution of the associated spectral data. The analysis of initial-boundary value problems for linear evolution PDEs on the finite interval shows that, while there exists discrete spectrum for the Dirichlet problem of dispersive PDEs involving second order derivatives, typical boundary value problems


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for PDEs involving third order derivatives do not possess discrete spectrum [22]. Since the algebro-geometric approach to nonlinear evolution PDEs is based on the nonlinearisation of the discrete spectrum, the above suggests that such an approach for the finite interval may be appropriate for the nonlinear Schrödinger, but not for the mKdV and the KdV. References 1. Adler, V.E., Gürel, B., Gürses, M., Habibullin, I.: Boundary conditions for integrable equations. J. Phys. A 30, 3505–3513 (1997) 2. Boutet de Monvel, A., Fokas, A.S., Shepelsky, D.: Analysis of the global relation for the nonlinear Schrödinger equation on the half-line. Lett. Math. Phys. 65, 199–212 (2003) 3. Boutet de Monvel, A., Fokas, A.S., Shepelsky, D.: The mKdV equation on the half-line. J. Inst. Math. Jussieu 3(2), 139–164 (2004) 4. Boutet de Monvel, A., Kotlyarov, V.: Generation of asymptotic solitons of the nonlinear Schrödinger equation by boundary data. J. Math. Phys. 44(8), 3185–3215 (2003) 5. Boutet de Monvel, A., Shepelsky, D.: The modified KdV equation on a finite interval. C. R. Math. Acad. Sci. Paris 337(8), 517–522 (2003) 6. Boutet de Monvel, A., Shepelsky, D.: Initial boundary value problem for the mKdV equation on a finite interval. Ann. Inst. Fourier (Grenoble) 54(5), 1477–1495 (2004) 7. Deift, P., Venakides, S., Zhou, X.: New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems. Intl. Math. Res. Notices 1997, No. 6, pp. 286–299 8. Degasperis, A., Manakov, S.V., Santini, P.M.: On the initial-boundary value problems for soliton equations. JETP Letters 74(10), 481–485 (2001) 9. Degasperis, A., Manakov, S.V., Santini, P.M.: Initial-boundary problems for linear and soliton PDEs. Theoret. and Math. Phys. 133(2), 1475–1489 (2002) 10. Degasperis, A., Manakov, S.V., Santini, P.M.: Integrable and nonintegrable initial boundary value problems for soliton equations. J. Nonlinear Math. Phys. 12, 228–243 (2005) 11. Deift, P., Zhou, X.: A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation. Ann. Math. (2) 137, 295–368 (1993) 12. Fokas, A.S.: A unified transform method for solving linear and certain nonlinear PDEs. Proc. Roy. Soc. London Ser. A 453, 1411–1443 (1997) 13. Fokas, A.S.: On the integrability of linear and nonlinear partial differential equations. J. Math. Phys. 41, 4188–4237 (2000) 14. Fokas, A.S.: Two dimensional linear PDEs in a convex polygon. Proc. Roy. Soc. London Ser. A 457, 371–393 (2001) 15. Fokas, A.S.: Integrable nonlinear evolution equations on the half-line. Commun. Math. Phys. 230, 1–39 (2002) 16. Fokas, A.S.: The generalized Dirichlet-to-Neumann map for certain nonlinear evolution PDEs. Commun. Pure Appl. Math. 58, 639–670 (2005) 17. Fokas, A.S., Its, A.R.: An initial-boundary value problem for the sine-Gordon equation. Theor. Math. Physics 92, 388–403 (1992) 18. Fokas, A.S., Its, A.R.: An initial-boundary value problem for the Korteweg-de Vries equation. Math. Comput. Simul. 37, 293–321 (1994) 19. Fokas, A.S., Its, A.R.: The linearization of the initial-boundary value problem of the nonlinear Schrödinger equation. SIAM J. Math. Anal. 27, 738–764 (1996) 20. Fokas, A.S., Its, A.R.: The nonlinear Schrödinger equation on the interval. J. Phys. A 37, 6091–6114 (2004) 21. Fokas, A.S., Its, A.R., Sung, L.Y.: The nonlinear Schrödinger equation on the half-line. Nonlinearity 18, 1771–1822 (2005) 22. Fokas, A.S., Pelloni, B.: A transform method for evolution PDEs on the interval. IMA J. Appl. Math. 70(4), 564–587(2005) 23. Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura, R.M.: Methods for solving the Korteweg–de Vries equation. Phys. Rev. Lett. 19, 1095–1097 (1967) 24. Grinevich, P.G., Santini, P.M.: The initial-boundary value problem on the interval for the nonlinear Schrödinger equation. The algebro-geometric approach. I. In: V.M. Buchstaber, I.M.Krichever, (eds.), Geometry, Topology, and Mathematical Physics: S.P. Novikov Seminar 2001-2003, Volume 212 of AMS Translations Ser. 2, Providence, R.I.: Amer. Math. Soc., 2004, pp. 157–178 25. Habibullin, I.T.: Bäcklund transformation and integrable boundary-initial value problems. In: Nonlinear world, Volume 1 (Kiev, 1989), River Edge, N.J.: World Sci. Publishing, 1990, pp. 130–138

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26. Kamvissis, S.: Semiclassical nonlinear Schrödinger on the half line. J. Math. Phys. 44, 5849–5868 (2003) 27. Lax, P.D.: Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math. 21, 467–490 (1968) 28. Sklyanin, E.K.: Boundary conditions for integrable equations. Funct. Anal. Appl. 21, 86–87 (1987) 29. Tarasov, V.O.: An boundary value problem for the nonlinear Schrödinger equation. Zap. Nauchn. Sem. LOMI 169, 151–165 (1988); [transl.: J. Soviet Math. 54, 958–967 (1991)] 30. Zakharov, V.E., Shabat, A.B.: A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I and II. Funct. Anal. Appl. 8, 226–235 (1974) and 13, 166–174(1979) 31. Zhou, X.: The Riemann-Hilbert problem and inverse scattering. SIAM J. Math. Anal. 20, 966–986 (1989) 32. Zhou, X.: Inverse scattering transform for systems with rational spectral dependence. J. Differ. Eqs. 115, 277–303 (1995) Communicated by P. Constantin


Communications in


Fermionic Quantization and Configuration Spaces for the Skyrme and Faddeev-Hopf Models Dave Auckly1, , Martin Speight 2, 1 2

Department of Mathematics, Kansas State University, Manhattan, Kansas 66506, USA School of Mathematics, University of Leeds, Leeds LS2 9JT, England

Received: 1 November 2004 / Accepted: 12 July 2005 Published online: 26 January 2006 – © Springer-Verlag 2006

Abstract: The fundamental group and rational cohomology of the configuration spaces of the Skyrme and Faddeev-Hopf models are computed. Physical space is taken to be a compact oriented 3-manifold, either with or without a marked point representing an end at infinity. For the Skyrme model, the codomain is any Lie group, while for the FaddeevHopf model it is S 2 . It is determined when the topology of configuration space permits fermionic and isospinorial quantization of the solitons of the model within generalizations of the frameworks of Finkelstein-Rubinstein and Sorkin. Fermionic quantization of Skyrmions is possible only if the target group contains a symplectic or special unitary factor, while fermionic quantization of Hopfions is always possible. Geometric interpretations of the results are given. 1. Introduction The most straightforward procedure for quantizing a Lagrangian dynamical system with configuration space Q is to specify the quantum state by a wavefunction ψ : Q → C, but many other procedures are possible [30]. One may take ψ to be a section of a complex line bundle over Q. Clearly we recover the original quantization if the bundle is trivial. Depending on the topology of Q (H 2 (Q, Z)) there may be many inequivalent line bundles, giving rise to quantization ambiguity. Quantization ambiguity can be used to generate fermionic or bosonic quantizations of the same classical system. The example of a charged particle under the influence of a magnetic monopole studied by Dirac [12, 13] clearly demonstrates the utility of complex line bundles for analyzing quantum dynamics. See also, the discussion in [36] and [7, Chap.6]. In the case of a Lagrangian field theory supporting topological solitons, configuration space is typically the space of (sufficiently regular) maps from some 3-manifold

The first author was partially supported by NSF grant DMS-0204651. The second author was partially supported by EPSRC grant GR/R66982/01.

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(representing physical space) to some target manifold. A famous example of this is the Skyrme model with target space SU(N ), N ≥ 2 and physical space R3 . Here fermionic quantization is phenomenologically crucial since the solitons are taken to represent protons and neutrons. Recall the distinction between bosons and fermions: a macroscopic ensemble of identical bosons behaves statistically as if arbitrarily many particles can lie in the same state, while a macroscopic ensemble of identical fermions behaves as if no two particles can lie in the same state. Photons are examples of particles with bosonic statistics and electrons are examples of particles with fermionic statistics. There are several theoretical models of particle statistics. In quantum mechanics, the wavefunction representing a multiparticle state is symmetric under exchange of any pair of identical bosons, and antisymmetric under exchange of any pair of identical fermions. In conventional perturbative quantum field theory, commuting fields are used to represent bosons and anti-commuting fields are used to represent fermions. More precisely, bosons have commuting creation operators and fermions have anti-commuting creation operators. However, there are times when fermions may arise within a field theory with purely bosonic fundamental fields. This phenomenon is called emergent fermionicity, and it relies crucially on the topological properties of the underlying configuration space of the model. Analogous instances of emergent fermionicity in quantum mechanical (rather than field theoretic) settings are described in [10, 29, 36] and [7, Chap. 7]. When Skyrme originally proposed his model, it was not clear how fermionic quantization of the solitons could be achieved, a fundamental gap which he acknowledged [31]. The possibility of fermionically quantizing the Skyrme model was first demonstrated (for N = 2) by Finkelstein and Rubinstein [15]. Full consistency of their quantization procedure was established by Giulini [16]. The case N ≥ 3 was dealt with in a rather different way by Witten [35] at the cost of introducing a new term into the Skyrme action. This was a crucial development, since the N = 3 model is particularly phenomenologically favoured. Although the approaches of Finkelstein-Rubinstein and Witten appear quite different, they can be treated in a common framework, as demonstrated by Sorkin [32]. See [1] and [7] for exposition about where the Skyrme model fits into modern physics. Also see [2], and [28] for a discussion of fermionic quantization of SU(N ) valued Skyrme fields. We will review the Finkelstein-Rubinstein and Sorkin models of particle statistics in Sect. 5 below, and describe the obvious generalization of their models when the domain of the soliton is something other than R3 . Spin is a property of a particle state associated with how it transforms under spatial rotations. Let us briefly review the usual mathematical models of spin. There are two general situations. In one, space admits a global rotational symmetry, in the other, it does not. When physical space is R3 , so space-time is the usual Minkowski space, the classical rotational symmetries induce quantum symmetries that are representations of the Spin group. These irreducible representations are labeled by half integers (one half of the number of boxes in a Young diagram for a representation of SU(2).) The integral representations are honest representations of the rotation group, but the fractional ones are not. The spin of a particle is the half integer labelling the SU(2) representation under which its wavefunction transforms. If the spin is not an integer, the particle is said to be spinorial. When physical space is not R3 , one can instead consider the bundle of frames over spacetime (vierbeins). Spin may then be modeled by the action of the rotation group on these frames [9]. The easiest case in this direction is when space-time admits a spin structure. This reconciliation of spin with the possibility of a curved space time was an important discovery in the last century. The situation is parallel in nonlinear models. In

Quantization and Configuration Spaces for Skyrme and Faddeev-Hopf Models

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the easiest example of a model with solitons incorporating spin, the configuration space is not R3 , but maps defined on R3 . The rotation group acts on such maps by precomposition. Sorkin described a model for the spin of solitons which generalizes to any maps defined on any domain. We cover this in Sect. 5. Although spin and particle statistics have completely different conceptual origins, there are strong connexions between the two. The spin-statistics theorem asserts, in the context of axiomatic quantum field theory, that particles are fermionic if and only if they are spinorial. Said differently, any particle with fractional spin is a fermion, and any particle with integral spin is a boson. Analogous spin-statistics theorems have been found for solitons also. Such a theorem was proved for SU(2) Skyrmions on R3 by Finkelstein and Rubinstein [15], and for arbitrary textures on R3 by Sorkin [32], using only the topology of configuration space. By a texture, we mean that the field must approach a constant limiting value at spatial infinity, in contrast to, say, monopoles and vortices. So pervasive is the link between fermionicity and spinoriality that the two are often elided. For example, Ramadas argued that it was possible to fermionically quantize SU(N ) Skyrmions on R3 because it was possible to spinorially quantize them [28]. Isospin is the conserved quantity analogous to spin associated with an internal rotational symmetry. As in the simplest model of spin, a particle’s isospin is a half integer labelling the representation of the quantum symmetry group corresponding to the classical internal SO(3) symmetry. In all the models we consider, the target space has a natural SO(3) action, so it will make sense to determine whether these models admit isospinorial quantization in the usual sense. Krusch [24] has shown that SU(2) Skyrmions are spinorial if and only if they are isospinorial, which is in good agreement with nuclear phenomenology, since they represent bound states of nucleons. Recall that nucleon is the collective term for the proton and neutron. Both have spin and isospin 1/2 but are in different eigenstates of the 3rd component of isospin: the proton has I3 = 1/2, and the neutron has I3 = −1/2. In general, the integrality of spin is unrelated to that of isospin. Strange hadrons can have half-integer spin and integer isospin (and vice-versa). For example, the -baryon has isospin 1, but spin 1/2, and the K mesons have isospin 1/2 and spin 1. One would hope, therefore, that the correlation found by Krusch fails in the SU(N ) Skyrme model with N > 2, since this is supposed to model low energy QCD with more than two light quark flavours, and should therefore be able to accommodate the more exotic baryons. The mathematical reader unfamiliar with spin, isospin, strangeness, etc. may find the book by Halzen and Martin [18] and the comprehensive listing of particles and their properties in [26] helpful. Emergent fermionicity, like (iso)spinoriality, can often be incorporated into a quantum system by exploiting the possibility of differences between the classical and quantum symmetries of the space of quantum states [7, Chap. 7]. A spinning top is a well known example of this. The classical symmetry group is SO(3), while the quantum symmetry group for some quantizations is SU(2), [10, 29]. An electron in the field of a magnetic monopole is also a good example, [7, Chap. 7] and [36]. We emphasize, however, that emergent fermionicity does not depend on any symmetry assumptions. In fact, the model of particle statistics that we mainly consider (Sorkin’s model) depends only on the topology of the configuration space. The purpose of this paper is to determine the quantization ambiguity for a wide class of field theories supporting topological solitons of texture type in 3 spatial dimensions. We will allow (the one point compactification of) physical space to be any compact, oriented 3-manifold M and target space to be any Lie group G, or the 2-sphere. The results use only the topology of configuration space and are completely independent of

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the dynamical details of the field theory. They cover, therefore, the Faddev-Hopf and general Skyrme models on any orientable domain. Our main mathematical results will be the computation of the fundamental group and the rational or real cohomology ring of Q. (The universal coefficient theorem implies that the rational dimension of the rational cohomology is equal to the real dimension of the real cohomology. Homotopy theorists tend to express results using rational coefficients and physicists tend to use real or complex coefficients.) We shall see that quantization ambiguity, as described by H 2 (Q, Z), may be reconstructed from these data. We also give geometric interpretations of the algebraic results which are useful for purposes of visualization. We then determine under what circumstances the quantization ambiguity allows for consistent fermionic quantization of Skyrmions and Hopfions within the frameworks of Finkelstein-Rubinstein and Sorkin. We finally discuss the spinorial and isospinorial quantization of these models. The main motivation for this work was to test the phenomenon of emergent fermionicity (i.e. fermionic solitons in a bosonic theory) in the Skyrme model to see, in particular, whether it survives the generalization from domain R3 to domain M. Our philosophy is that a concept in field theory which cannot be properly formulated on any oriented domain should not be considered fundamental. In fact, we shall see indications that emergent fermionicity is insensitive to the topology of M, but depends crucially on the topology of the target space. 2. Notation and Statement of Results Recall that topologically distinct complex line bundles over a topological space Q are classified by H 2 (Q, Z). Note that Q need have no differentiable structure to make sense of this: we can define c1 (L) ∈ H 2 (Q, Z) corresponding to bundle L directly in terms of the transition functions of L rather than thinking of it as the curvature of a unitary connexion on L [30]. So this classification applies in the cases of interest. The free part of H 2 (Q, Z) is determined by H 2 (Q, R), while its torsion is isomorphic to the torsion of H1 (Q). For any topological space, H1 (Q) is isomorphic to the abelianization of π1 (Q). The bundle classification problem is solved, therefore, once we know π1 (Q) and H 2 (Q, R). In this section we will define the configuration spaces that we consider, set up notation and state our main topological results. There are in fact many different but related configuration spaces that we could consider (for example spaces of free maps versus spaces of base pointed maps) and several different possibilities depending on whether the domain is connected, etc. We give clean statements of our results for special cases in this section, and describe how to obtain the most general results in the next section. The next section will also include some specific examples. Of course homotopy theorists have studied the algebraic topology of spaces of maps. The paper by Federer gives a spectral sequence whose limit group is a sum of composition factors of homotopy groups for a space of based maps [14]. We do not need a way to compute – we actually need the computations, and this is what is contained here. Let M be a compact, oriented 3-manifold and G be any Lie group. Then the first configuration space we consider is either FreeMaps(M, G), the space of continuous maps M → G, or GM , the subset of FreeMaps(M, G) consisting of those maps which send a chosen basepoint x0 ∈ M to 1 ∈ G. We will address configuration spaces of S 2 -valued maps later in this section and paper. Both FreeMaps(M, G) and GM are given the compact open topology. In practice some Sobolev topology depending on the energy functional is probably appropriate. The issue of checking the algebraic topology


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arguments given in this paper for classes of Sobolev maps is interesting. See [6] for a discussion of the correct setting and arguments generalizing the labels of the path components of these configuration spaces for Sobolev maps. The space FreeMaps(M, G) is appropriate to the G-valued Skyrme model on a genuinely compact domain, while GM is appropriate to the case where physical space Mˆ is noncompact but has a connected end at infinity which, for finite energy maps, may be regarded as a single point x0 in the ˆ one point compactification M of M. M The space G splits into disjoint path components which are labeled by certain cohomology classes on M [4]. Let (GM )0 be the identity component of the topological group GM , that is, the path component containing the constant map u(x) = 1. In physical terms (GM )0 is the vacuum sector of the model. Then (GM )0 is a normal subgroup of GM whose cosets are precisely the other path components. The set of path components itself has a natural group structure. As a set, the space of path components of the based maps is given by the following proposition. Proposition 1 (Auckly-Kapitanski). Let G be a compact, connected Lie group and M be a connected, closed 3-manifold. The set of path components of GM is GM /(GM )0 ∼ = H 3 (M; π3 (G)) × H 1 (M; H1 (G)). The reason the above proposition only describes the set of path components is that Auckly and Kapitanski only establish an exact sequence 0 → H 3 (M; π3 (G)) → GM /(GM )0 → H 1 (M; H1 (G)) → 0. To understand the group structure on the set of path components one would have to understand a bit more about this sequence (e.g. does it split?). Every path component of GM is homeomorphic to (GM )0 since u(x) → u(x)−1 u(x) is a homeomorphism M M u (G )0 → (G )0 . Our first result computes the fundamental group of the configuration space of based G-valued maps. Theorem 2. If M is a closed, connected, orientable 3-manifold, and G is any Lie group, then π1 (GM ) ∼ = Zs2 ⊕ H 2 (M; π3 (G)). Here s is the number of symplectic factors in the Lie algebra of G. Our next result gives the whole real cohomology ring H ∗ ((GM )0 , R), including its multiplicative structure. This, of course, includes the required computation of H 2 ((GM )0 , R). Similarly to Yang-Mills theory, there is a µ map, µ : Hd (M; R) ⊗ H j (G; R) → H j −d (GM ; R), and the cohomology ring is generated (as an algebra) by the images of this map. To state the theorem we do not need the definition of this µ map, but the definition may be found in Subsect. 4.3 of Sect. 4, in particular, Eq. (4.1). Theorem 3. Let G be a compact, simply-connected, simple Lie group. The cohomology ring of any of these groups is a free graded-commutative unital algebra over R generated by degree k elements xk for certain values of k (and with at most one exception at most one generator for any given degree). The values of k depend on the group and are

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listed in Table 3 in Sect. 3. Let M be a closed, connected, orientable 3-manifold. The cohomology ring H ∗ ((GM )0 ; R) is the free graded-commutative unital algebra over R generated by the elements µ(jd ⊗ xk ), where jd form a basis for Hd (M; R) for d > 0 and k − d > 0. The examples in the next section best illuminate the details of the above theorem. Turning to the Faddeev-Hopf model, the configuration space of interest is either the space of free S 2 -valued maps FreeMaps(M, S 2 ), or (S 2 )M , the space of based continuous maps M → S 2 . One can analyze FreeMaps(M, S 2 ) in terms of (S 2 )M by making use of the natural fibration π

(S 2 )M → FreeMaps(M, S 2 ) → S 2 ,

π : u(x) → u(x0 ).

The fundamental cohomology class (orientation class), µS 2 ∈ H 2 (S 2 , Z) plays an important role in the description of the mapping spaces of S 2 -valued maps. The path components of (S 2 )M were determined by Pontrjagin [27]: Theorem 4 (Pontrjagin). Let M be a closed, connected, oriented 3-manifold, and µS 2 be a generator of H 2 (S 2 ; Z) ∼ = Z. To any based map ϕ from M to S 2 one may asso∗ ciate the cohomology class, ϕ µS 2 ∈ H 2 (M; Z). Every second cohomology class may be obtained from some map and any two maps with different cohomology classes lie in distinct path components of (S 2 )M . Furthermore, the set of path components corresponding to a cohomology class, α ∈ H 2 (M) is in bijective correspondence with H 3 (M)/(2α H 1 (M)). A discussion of this theorem in the setting of the Faddeev model may be found in [5] and [6]. Let (S 2 )M 0 denote the vacuum sector, that is the path component of the constant 2 M map, and (S 2 )M ϕ denote the path component containing ϕ. Since (S ) is not a topological group, there is no reason to expect all its path components to be homeomorphic. 2 M We will prove, however that two components (S 2 )M ϕ and (S )ψ are homeomorphic if ∗ ∗ ϕ µS 2 = ψ µ S 2 : ∼ 2 M Theorem 5. Let ϕ ∈ (S 2 )M such that ϕ ∗ µS 2 = ψ ∗ µS 2 . Then (S 2 )M ϕ = (S )ψ . Moreover, the fundamental group of any component can be computed, as follows. Theorem 6. Let M be closed, connected and orientable. For any ϕ ∈ (S 2 )M , the fundamental group of (S 2 )M ϕ is given by 2 ∗ ∼ π1 ((S 2 )M ϕ ) = Z2 ⊕ H (M; Z) ⊕ ker(2ϕ µS 2 ).

Here 2ϕ ∗ µS 2 : H 1 (M; Z) → H 3 (M; Z) is the usual map given by the cup product. There is a general relationship between the fundamental group of the configuration space of based S 2 -valued maps and the corresponding configuration space of free maps. It implies the following result for the fundamental group of the space of free S 2 -valued maps. Theorem 7. We have π1 (FreeMaps(M, S 2 )ϕ ) ∼ = Z2 ⊕ H 2 (M; Z)/ 2ϕ ∗ µS 2

⊕ ker(2ϕ ∗ µS 2 ).


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To complete the classification of complex line bundles over (S 2 )M ϕ one also needs 2 2 M the second cohomology H ((S )ϕ ; R), which can be extracted from the above theorem and the following computation of the cohomology ring H ∗ ((S 2 )M ϕ , R): Theorem 8. Let M be closed, connected and orientable, let ϕ : M → S 2 , let jd form a basis for Hd (M; R) for d < 3, and let {αk } for a basis for ker(2ϕ ∗ µS 2 ∪) : H 1 (M; Z) → H 3 (M; Z)). The cohomology ring H ∗ ((S 2 )M ϕ ; R) is the free graded-commutative unital algebra over R generated by the elements αk and µ(jd ⊗ x), where x ∈ H 3 (Sp1 ; Z) is the orientation class. The classes αk have degree 1 and µ(jd ⊗ x) have degree 3 − d. We can compute the cohomology of the space of free S 2 -valued maps using the following theorem. p,q

Theorem 9. There is a spectral sequence with E2 = H p (S 2 ; R)⊗H q ((S 2 )M ϕ ; R) con∗ 2 verging to H (FreeMaps(M, S )ϕ ; R). The second differential is given by d2 µ( (2) ⊗ x) = 2ϕ ∗ µS 2 []µS 2 with d2 of any other generator trivial. All higher differentials are trivial as well. In order to compare the classical and quantum isospin symmetries, we will use the following theorem due to Gottlieb [17]. It is based on earlier work of Hattori andYoshida [19]. Theorem 10 (Gottlieb). Let L → X be a complex line bundle over a locally compact space. An action of a compact connected Lie group on X, say ρ : X × G → X, lifts to a bundle action on L if and only if two obstructions vanish. The first obstruction is the pullback of the first Chern class, L∗x0 c1 (L) ∈ H 2 (G; Z). Here Lx0 is the map induced by applying the group action to the base point. The second obstruction lives in H 1 (X; H 1 (G; Z)). We have taken the liberty of radically changing the notation from the original theorem, and we have only stated the result for line bundles. The actual theorem is stated for principal torus bundles. Since our configuration spaces are not locally compact, we should point out that we will use one direction of this theorem by restricting to a locally compact equivariant subset. In the other direction, we will just outline a construction of the lifted action. Our main physical conclusions are: C1 In these models, there is a portion of quantization ambiguity that depends only on the codomain and is completely independent of the topology of the domain. This allows for the possibility that emergent fermionicity may only depend on the target. C2 It is possible to quantize G-valued solitons fermionically (with odd exhange statistics) if and only if the Lie algebra contains a symplectic (Cn ) or special unitary (An ) factor. C3 It is possible to quantize G-valued solitons with fractional isospin when the Lie algebra of G contains a symplectic (Cn ) or special unitary (An ) factor. C4 It is not possible to quantize G-valued solitons with fractional isospin when the Lie algebra does not contain such a factor. C5 It is always possible to choose a quantization of these systems with integral isospin (however such might not be consistent with other constraints on the model). C6 It is always possible to quantize S 2 -valued solitons with fractional isospin and odd exchange statistics.

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The rest of this paper is structured as follows. In Sect. 3 we describe how to reduce the description of the topology of general G-valued and S 2 -valued mapping spaces to the theorems listed in this section. We also provide several illustrative examples. In Sect. 4 we review the Pontrjagin-Thom construction and describe geometric interpretations of some of our results using the Pontrjagin-Thom construction. Physical applications, particularly the possibility of consistent fermionic quantization of Skyrmions, are discussed in Sect. 5. Finally, Sect. 6 contains the proofs of our results. 3. Preliminary Reductions and Examples We begin this section with a collection of observations that allows one to reduce questions about the topology of various mapping spaces of G-valued and S 2 -valued maps to the theorems listed in the previous section. Many of these observations will reduce a more general mapping space to a product of special mapping spaces, or put such spaces into fibrations. These reductions ensure that our results are valid for arbitrary closed, orientable 3-manifolds, and valid for any Lie group. It follows directly from the definition of π1 that π1 (X × Y ) ∼ = π1 (X) × π1 (Y ). The cohomology of a product is described by the Künneth theorem, see [33]. For real coefficients it takes the simple form, H ∗ (X × Y ) ∼ = H ∗ (X) ⊗ H ∗ (Y ). The cohomology ring of a disjoint union of spaces is the direct sum of the corresponding cohomology rings, ∗ i. e. H ∗ (⊥ ⊥Xν ; A) = H (Xν ; A). Recall that a fibration is a map with the covering homotopy property, see for example [33]. Given a fibration F → E → B, there is an induced long exact sequence of homotopy groups, . . . → πk+1 (B) → πk (F ) → πk (E) → πk (B) → . . ., see [33]. By itself this sequence is not enough to determine the fundamental group of a term in a fibration from the other terms. However, combined with a bit of information about the twisting in the bundle it will be enough information. One can also relate the cohomology rings of the terms in a fibration. This is accomplished by the Serre spectral sequence, see [33]. ∼ FreeMaps(Xν , Y ) and Y ⊥⊥Xν = ∼ Y X0 × Reduction 1. We have, FreeMaps(⊥⊥Xν , Y ) = ν =0 FreeMaps(Xν , Y ), where X0 is the component of X containing the base point. It follows that there is no loss of generality in assuming that M is connected. Likewise there is no loss of generality in assuming that the target is connected because of the following reduction. Reduction 2. We have, FreeMaps(X, ⊥⊥Yν ) ∼ = ⊥⊥FreeMaps(X, Yν ) and assuming X is connected, Y X = Y0X , where Y0 is the component containing the base point. Both FreeMaps(M, G) and GM are topological groups under pointwise multiplication. In fact FreeMaps(M, G) ∼ = GM G, the isomorphism being u(x) → (u(x)u(x0 )−1 , u(x0 )), which is clearly a homeomorphism FreeMaps(M, G) → GM × G. It is thus straightforward to deduce π1 (FreeMaps(M, G) and H ∗ (FreeMaps(M, G), R) from π1 (GM ) and H ∗ (GM , Z). Note that the based case includes the standard choice Mˆ = R3 . Reduction 3. We have FreeMaps(M, G) ∼ = GM × G. In the same way, we can reduce the free maps case to the based case for S 2 -valued maps. In this case we only obtain a fibration. See Lemmas 18, 20 and 21. Reduction 4. We have a fibration, (S 2 )M → FreeMaps(M, S 2 ) → S 2 , π0 (Freemaps(M, S 2 )) = π0 ((S 2 )M ), and π1 (FreeMaps(M, S 2 )ϕ ) = π1 ((S 2 )M ϕ ).


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The relevant information about the twisting in this fibration, as far as the fundamental group detects it, is given in the proof of Theorem 7 contained in Subsect. 6.3. For cohomology, the information is encoded in the second differential of the associated spectral sequence. Returning to the case of group-valued maps we know by the CartanMalcev-Iwasawa theorem that any connected Lie group is homeomorphic to a product G = K × Rn , where K is compact [22]. Reduction 5. If X and Y are homotopy equivalent to X and Y respectively, then (Y )X is homotopy equivalent to Y X . In particular we have GM K M .

Recall that every path component of GM is homeomorphic to (GM )0 We may therefore consider only the vacuum sector (GM )0 , without loss of generality. We shall see that things are very different for the Faddeev-Hopf configuration space, where we must keep track of which path component we are studying. ˜ is the universal covering group of its identity comReduction 6. If G is a Lie group, G ponent and M is a 3-manifold, then ˜ M )0 ∼ (G = (GM )0 . Proof. Without loss of generality we may assume that G is connected. We have the exact sequence, ˜ M → GM → H 1 (M; H1 (G)) → 0, 1→G from [4]. The exactness follows from the unique path lifting property of covers at the first term, the lifting criteria for maps to the universal cover at the center term, and induction ˜ M maps to the on the skeleton of M at the last term. Clearly, the identity component of G M identity component of G . By the above sequence, this map is injective. Any element ˜ M , say of (GM )0 , say u, maps to 0 in H 1 (M; H1 (G)), so is the image of some map in G u. ˜ Using the homotopy lifting property of covering spaces, we may lift the homotopy of ˜ M )0 . u to a constant map, to a homotopy of u˜ to a constant map and conclude that u˜ ∈ (G M M ˜ It follows that the map, (G )0 → (G )0 is a homeomorphism. Reduction 7. The universal covering group of any compact Lie group is a product of Rm with a finite number of compact, simple, simply-connected factors [25]. Furthermore, X X Yν ∼ = FreeMaps(X, Yν ). = Yν and FreeMaps(X, Yν )) ∼ We have therefore reduced to the case of closed, connected, orientable M and compact, simple, simply-connected Lie groups. Recall from Proposition 1 that the path components of a configuration space of group-valued maps depend on the fundamental group of the group. The fundamental group of any Lie group is a discrete subgroup of the center of the universal covering group. The center of such a group is just the product of the centers of the factors. All compact, simple, simply-connected Lie groups are listed together with their center and rational cohomology in Table 1. Some comments about Table 1 are in order at this point. The cohomology of a compact, simple, simply-connected Lie group is a free unital graded-commutative algebra. Each generator in the table is labeled with its degree. Thus Q[x3 , x5 ] is not the ring of all polynomials in x3 and x5 since x32 = x52 = 0 and x3 x5 = −x5 x3 by graded-commutativity. The last generator of the cohomology ring of Dn is labeled with a y instead of an x

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D. Auckly, M. Speight Table 1. Simple groups group, G

center, Z(G)

H ∗ (G; Q)

An = SU(n + 1), n ≥ 2 Bn = Spin(2n + 1), n ≥ 3 Cn = Sp(n), n ≥ 1 Dn = Spin(2n), n ≥ 4

Zn+1 Z2 Z2 Z2 ⊕ Z2 for n ≡2 0 Z4 for n ≡2 1 Z3 Z2 0 0 0

Q[x3 , x5 , . . . x2n+1 ] Q[x3 , x7 , . . . x4n−1 ] Q[x3 , x7 , . . . x4n−1 ] Q[x3 , x7 , . . . x4n−5 , y2n−1 ]

E6 E7 E8 F4 G2

Q[x3 , x9 , x11 , x15 , x17 , x23 ] Q[x3 , x11 , x15 , x19 , x23 , x27 , x35 ] Q[x3 , x15 , x23 , x27 , x35 , x39 , x47 , x59 ] Q[x3 , x11 , x15 , x23 ] Q[x3 , x11 ]

because there are two generators in degree 2n − 1 when n is even. As usual, SU(k) is the set of special unitary matrices, that is complex matrices with unit determinant satisfying, A∗ A = I . The symplectic groups, Sp(k), consist of the quaternionic matrices satisfying A∗ A = I , and the special orthogonal groups, SO(k) consist of the real matrices with unit determinant satisfying A∗ A = I . The spin groups, Spin(k) are the universal covering groups of the special orthogonal groups. The definitions of the exceptional groups may be found in [3]. The following isomorphisms hold, SU(2) ∼ = Sp(1) ∼ = Spin(3), ∼ ∼ Spin(5) = Sp(2), and Spin(6) = SU(4), [3]. We will need some homotopy groups of Lie groups. Recall that the higher homotopy groups of a space are isomorphic to the higher homotopy groups of the universal cover of the space, and the higher homotopy groups take products to products. We have π3 (G) ∼ = Z for any of the simple G, and π4 (Sp(n)) ∼ = Z2 and π4 (G) = 0 for all other simple groups [25]. This is the reason we grouped the simple groups as we did. Note in particular that we are calling SU(2) a symplectic group. 3.1. Examples. In this subsection, we present two examples that suffice to illustrate all seven reductions described earlier. Example 1. For our first example, we take M = (S 2 × S 1 )⊥⊥RP 3 and a b G = Sp(2) × ∈ GL(2, R) . 0 c We take ((1, 0, 0), (1, 0)) ∈ S 2 × S 1 as the base point in M. In this example, neither the domain nor codomain is connected (G/G0 ∼ = Z2 × Z2 ). In addition, the group is not reductive. We also see exactly what is meant by the number of symplectic factors in the Lie algebra: it is just the number of Cn factors in the Lie algebra of the maximal compact subgroup of the identity component of G. This example requires Reductions 1, 2, 3, and 5. To analyze the topology of the spaces of free and based maps, it suffices to understand maps from S 2 × S 1 and RP 2 into the identity component, G0 (Reductions 1, 2 and 3). In fact, we may replace G0 with Sp(2) (Reduction 5). Proposition 1 implies 3 2 1 that π0 (Sp(2)S ×S ) = Z and π0 (Sp(2)RP ) = Z, so π0 (FreeMaps(M, G)) = Z42 × Z2 1 2 and π0 (GM ) = Z22 × Z2 . Similarly, Theorem 2 implies that π1 (Sp(2)S ×S ) = Z2 ⊕ Z 3 and π1 (Sp(2)RP ) = Z2 ⊕ Z2 , so π1 (FreeMaps(M, G)) = π1 (GM ) = Z32 ⊕ Z. Turning to the cohomology, we know that H ∗ (Sp(2); R) is the free graded-commutative unital algebra generated by x3 and x7 . Graded-commutative means xy =


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(−1)|x||y| yx. It follows that any term with repeated factors is zero. We can list the generators of the groups in each degree. In the expression below we list the generators left to right from degree 0 with each degree separated by vertical lines: H ∗ (Sp(2); R) = |1|0|0|x3 |0|0|0|x7 |0|0|x3 x7 |. The product structure is apparent. Theorem 3 tells us that H ∗ ((Sp(2))S0 ×S ; R) is the free unital graded-commutative algebra generated by µ([S 2 ×pt]⊗x3 )(1) , µ([pt×S 1 ]⊗x3 )(2) , µ([S 2 × S 1 ] ⊗ x7 )(4) , µ([S 2 × pt] ⊗ x7 )(5) , and µ([pt × S 1 ] ⊗ x7 )(6) . Here we have included the degree of the generator as a superscript. In the same way we see that P 3 ; R) is the free unital graded-commutative algebra (FUGCA) generH ∗ ((Sp(2))R 0 ated by µ([RP 3 ] ⊗ x7 )(4) . Using the reductions and the Künneth theorem we see that H ∗ ((GM )0 ; R) is the FUGCA generated by µ([S 2 × pt] ⊗ x3 )(1) , µ([pt × S 1 ] ⊗ x3 )(2) , x3 , µ([S 2 × S 1 ] ⊗ x7 )(4) , µ([RP 3 ] ⊗ x7 )(4) , µ([S 2 × pt] ⊗ x7 )(5) , µ([pt × S 1 ] ⊗ x7 )(6) , x7 . Notice that this is not finitely generated as a vector space even though it is finitely generated as an algebra. This is because it is possible to have repeated even degree factors. The vector space in each degree is still finite dimensional. The cohomology ring space of based maps is just the direct of the configuration ∗ M sum, H ∗ (GM ; R) = π0 (GM ) H ((G )0 ; R). Notice that it is infinitely generated as an algebra. The cohomology of the identity component will usually be the important thing. Using the reductions, we see that the identity component of the space of free maps is up to homotopy just the product, FreeMaps(M, G)0 = GM × Sp(2), so the cohomology ring H ∗ (FreeMaps(M, G)0 ; R) is obtained from H ∗ ((GM )0 ; R) by adjoining new generators in degrees 3 and 7, say y3 and y7 . Thus H 2 ((GM )0 ; Z) ∼ = H 2 (FreeMaps(M, G)0 ; Z) ∼ = Z ⊕ Z32 . 2

1

Example 2. For this example, we take M = T 3 #L(m, 1), G1 = SO(8) and G = U(2) × SO(8). Recall that the lens space L(m, 1) is the quotient Sp(1)/Zm , where we view Zm as the mth roots of unity in S 1 ⊂ Sp(1). In this example we will need to use Reductions 6 and 7. The unitary group is isomorphic to Sp(1) ×Z2 S 1 , where Z2 is viewed as the diagonal subgroup, ±(1, 1). The universal covering group of SO(8) is Spin(8). It follows that G1 and G are connected, G has universal covering group Sp(1) × R × Spin(8), and the fundamental groups are π1 (Spin(8)) = Z2 and π1 (G) = Z ⊕ Z2 . The group G has two simple factors, one of which is symplectic. The integral cohomology of M is given by H 1 (M; Z) ∼ = Z3 and H 2 (M; Z) ∼ = Z3 ⊕ Zm . The universal coefficient theorem and Proposition 1 imply π0 (GM ) = π (FreeMaps(M, G1 )) = Z × Z42 if m is even, Z × Z32 0 1 if m is odd, and π0 (GM ) = π0 (FreeMaps(M, G)) = Z5 × Z42 if m is even and Z5 × Z32 3 if m is odd. Theorem 2 implies that π1 (GM 1 ) = Z ⊕ Zm , π1 (FreeMaps(M, G1 )) = 3 M 6 2 Z ⊕ Zm ⊕ Z2 , π1 (G ) = Z ⊕ Zm ⊕ Z2 and π1 (FreeMaps(M, G)) = Z7 ⊕ Z2m ⊕ Z22 . Turning once again to cohomology, we see from Theorem 3 that H ∗ ((U(2)M )0 ; R) is the FUGCA generated by µ([T 2 × pt] ⊗ x3 )(1) , µ([S 1 × pt × S 1 ] ⊗ x3 )(1) , µ([pt × T 2 ] ⊗ x3 )(1) , µ([S 1 × pt] ⊗ x3 )(2) , µ([pt × S 1 pt] ⊗ x3 )(2) , and µ([pt × S 1 ] ⊗ x3 )(2) . 2 (1) 1 Also, H ∗ ((GM 1 )0 ; R) is the FUGCA generated by µ([T × pt] ⊗ y3 ) , µ([S × 1 (1) 2 (1) 1 (2) 1 pt × S ] ⊗ y3 ) , µ([pt × T ] ⊗ y3 ) , µ([S × pt] ⊗ y3 ) , µ([pt × S pt] ⊗ y3 )(2) , µ([pt × S 1 ] ⊗ y3 )(2) , µ([T 3 ] ⊗ y7 )(4) , µ([T 3 ] ⊗ z7 )(4) , µ([T 2 × pt] ⊗ y7 )(5) , µ([S 1 × pt × S 1 ] ⊗ y7 )(5) , µ([pt × T 2 ] ⊗ y7 )(5) , µ([S 1 × pt] ⊗ y7 )(6) , µ([pt × S 1 pt] ⊗ y7 )(6) , µ([pt×S 1 ]⊗y7 )(6) , µ([T 2 ×pt]⊗z7 )(5) , µ([S 1 ×pt×S 1 ]⊗z7 )(5) , µ([pt×T 2 ]⊗z7 )(5) , µ([S 1 × pt] ⊗ z7 )(6) , µ([pt × S 1 pt] ⊗ z7 )(6) , µ([pt × S 1 ] ⊗ z7 )(6) , µ([T 3 ] ⊗ y11 )(8) ,

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µ([T 2 × pt] ⊗ y11 )(9) , µ([S 1 × pt × S 1 ] ⊗ y11 )(9) , µ([pt × T 2 ] ⊗ y11 )(9) , µ([S 1 × pt] ⊗ y11 )(10) , µ([pt × S 1 pt] ⊗ y11 )(10) , and µ([pt × S 1 ] ⊗ y11 )(10) . Therefore, H ∗ ((GM )0 ; R) is the FUGCA generated by all of the generators listed for the two previous algebras. We changed the notation for the generators of the cohomology of the Lie groups as needed. To get to the cohomology of the identity component of the space of free maps, we would just have to add generators for the cohomology of the group G0 to this list. In general the cohomology of a connected Lie group is the same as the cohomology of the maximal compact subgroup, and every compact Lie group has a finite cover that is a product of simple, simply-connected, compact Lie groups and a torus. In this case, we need to add generators, t1 , u3 , w3 , u7 , v7 , and u11 . 2 ∼ 6 ∼ 6 Thus, H 2 ((GM 1 )0 ; Z) = Z ⊕ Zm , H (FreeMaps(G1 , M)0 ; Z) = Z ⊕ Zm ⊕ Z2 , H 2 ((GM )0 ; Z) ∼ = Z21 ⊕ Z2m ⊕ Z2 , and H 2 (FreeMaps(G, M)0 ; Z) ∼ = Z21 ⊕ Z2m ⊕ Z22 . 2 We can also analyze the topology of the space of S -valued maps with domain M. The path components of (S 2 )M agree with the path components of FreeMaps(M, S 2 ) (Reduction 4) and are given by Theorem 4. Let ϕ0 : M → S 2 be the constant map and let ϕ3 : M → S 2 be the map constructed as the composition of the map M → T 3 (collapse the L(m, 1)), the projection T 3 → T 2 , and a degree three map T 2 → S 2 . According to Theorem 6 and Theorem 7, we have ∼ 6 π1 (FreeMaps(M, S 2 )ϕ0 ) = π1 ((S 2 )M ϕ0 ) = Z ⊕ Zm ⊕ Z2 , ∼ 5 π1 ((S 2 )M and ϕ3 ) = Z ⊕ Zm ⊕ Z2 , 2 4 ∼ π1 (FreeMaps(M, S )ϕ3 ) = Z ⊕ Zm ⊕ Z6 ⊕ Z2 . Using Theorem 8 we can write out generators for the cohomology. The cohomology, 2 (1) 1 1 (1) H ∗ ((S 2 )M ϕ0 ; R) is the FGCUA generated by P D([T × pt]) , P D([S × pt × S ]) , P D([pt × T 2 ])(1) , µ([T 2 × pt] ⊗ x)(1) , µ([S 1 × pt × S 1 ] ⊗ x)(1) , µ([pt × T 2 ] ⊗ x)(1) , µ([S 1 × pt] ⊗ x)(2) , µ([pt × S 1 pt] ⊗ x)(2) , and µ([pt × S 1 ] ⊗ x)(2) , where PD denotes Poincaré dual. 1 1 (1) Similarly, H ∗ ((S 2 )M ϕ3 ; R) is the FGCUA generated by P D([S ×pt×S ]) , P D([pt× 2 (1) 2 (1) 1 1 (1) 2 T ]) , µ([T × pt] ⊗ x) , µ([S × pt × S ] ⊗ x) , µ([pt × T ] ⊗ x)(1) , µ([S 1 × pt] ⊗ x)(2) , µ([pt × S 1 pt] ⊗ x)(2) , and µ([pt × S 1 ] ⊗ x)(2) . The reason why there is no generator corresponding to P D([T 2 × pt])(1) in the ϕ3 cohomology is that it is not in the kernel since 2ϕ3∗ µS 2 P D([T 2 × pt])(1) = 6µM . We can use Theorem 9 to compute the cohomology of the space of free maps. In the component with ϕ0 we notice that the second differential is trivial because ϕ0∗ µS 2 = 0. It follows that H ∗ (FreeMaps(M, S 2 )ϕ0 ; R) is the graded-commutative, unital algebra generated by P D([T 2 ×pt])(1) , P D([S 1 ×pt×S 1 ])(1) , P D([pt×T 2 ])(1) , µ([T 2 ×pt]⊗x)(1) , µ([S 1 × pt × S 1 ] ⊗ x)(1) , µ([pt × T 2 ] ⊗ x)(1) , µ([S 1 × pt] ⊗ x)(2) , µ([pt × S 1 pt] ⊗ x)(2) , µ([pt × S 1 ] ⊗ x)(2) , and µS 2 . Notice that this algebra is not free. It is subject to the single relation, µ2S 2 = 0. 2 In the component containing ϕ3 all of the generators of H ∗ ((S 2 )M ϕ3 ; R) except µ([T × pt] ⊗ x)(1) survive to H ∗ (FreeMaps(M, S 2 )ϕ3 ; R) because they are in the kernel of d2 . However, d2 µ([T 2 × pt] ⊗ x)(1) = 6µS 2 so µS 2 does not survive and H ∗ (FreemapsM, Sϕ23 ; R) is the FUGCA generated by P D([S 1 × pt × S 1 ])(1) , P D([pt × T 2 ])(1) , µ([S 1 × pt × S 1 ] ⊗ x)(1) , µ([pt × T 2 ] ⊗ x)(1) , µ([S 1 × pt] ⊗ x)(2) , µ([pt × S 1 pt] ⊗ x)(2) , and µ([pt × S 1 ] ⊗ x)(2) . 2 2 ∼ 18 ∼ 19 Thus H 2 ((S 2 )M ϕ0 ; Z) = Z ⊕ Zm ⊕ Z2 , H (FreeMaps(M, S )ϕ0 ; Z) = Z ⊕ Zm ⊕ 2 2 M 13 2 2 Z2 , H ((S )ϕ3 ; Z) ∼ = Z ⊕Zm ⊕Z2 , and H (FreeMaps(M, S )ϕ3 ; Z) ∼ = Z9 ⊕Zm ⊕Z2 .


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4. Geometric Interpretations We will follow the folklore maxim: think with intersection theory and prove with cohomology. The combination of Poincaré duality and the Pontrjagin-Thom construction gives a powerful tool for visualizing results in algebraic topology. If W is an oriented n-dimensional homology manifold, Poincaré duality is the isomorphism H k (W ) ∼ = Hn−k (W ). It is tempting to think of the k th cohomology as the dual of the k th homology. This is not far from the truth. The universal coefficient theorem is the split exact sequence 0 → Ext1Z (Hk−1 (W ; Z), A) → H k (W ; A) → HomZ (Hk (W ; Z), A) → 0. Putting this together, we see that every degree k cohomology class corresponds to a unique (n − k)-cycle (codimension k homology cycle), and the image of the cocycle applied to a k-cycle is the weighted number of intersection points with the corresponding (n−k)-cycle. For field coefficients this is the entire story since there is no torsion and the Ext group vanishes. With other coefficients, this gives the correct answer up to torsion. The Pontrjagin-Thom construction associates a framed codimension k submanifold of W to any map W → S k . The associated submanifold is just the inverse image of a regular point. This is well defined up to a framed cobordism. The framing is the inverse image of a standard frame. Going the other way, a framed submanifold produces a map W → S k defined via the exponential map on fibers of a tubular neighborhood of the submanifold and as the constant map outside of the neighborhood. We will take this up in greater detail later in this section. Before addressing the topology of our configuration spaces, we need to understand the cohomology of Lie groups. A number of different approaches may be utilized to compute the real cohomology of a compact Lie group: H-space methods, equivariant Morse theory, the Leray-Serre spectral sequence, Hodge theory. The cohomology is a free graded-commutative algebra over R. Recall that this means that xy = (−1)deg(x)deg(y) yx. For our purposes, the spectral sequence and Hodge theory are the two most important. The fibration SU(N ) → SU(N + 1) → S 2N+1 may be used to compute the cohomology of SU(N ), and we will use it and other similar fibrations to compute the cohomology of various configuration spaces. According to Hodge theory, the real cohomology is isomorphic to the collection of harmonic forms. Any compact Lie group admits an Ad-invariant innerproduct on the Lie algebra obtained by averaging any innerproduct over the group, or as the Killing form, X, Y = −Tr(ad(X)ad(Y )) in the semisimple case. Such an innerproduct induces a biinvariant metric on the group. With respect to this metric, the space of harmonic forms is isomorphic to the space of Ad-invariant forms on the Lie algebra. Any harmonic form induces a form on the Lie algebra by restriction and any Ad invariant form on the Lie algebra induces a harmonic form via left translation. In the case of SU(N ), these forms may be described as products of the elements, xj = Tr((u−1 du)j ). In some applications it might be appropriate to include a normalizing constant so that the integral of each of these forms on an associated primitive homology class is 1. 4.1. Components of GM . For simplicity, we will just consider geometric descriptions of G-valued maps for the compact, simple, simply-connected Lie groups. By applying the Pontrjagin-Thom construction, we will obtain a correspondence between homotopy classes of based maps M → G and finite collections of signed points in M. This may be used to give a geometric interpretation of Proposition 1. In physical terms, the signed points may be thought of as particles and anti-particles in the theory.

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To use the Pontrjagin-Thom construction in this setting we need a special basis for H∗ (G; R). By the universal coefficient theorem, there are (2k + 1)-cycles β2k+1 in H2k+1 (G; R) dual to x2k+1 . Assuming that the generators x2k+1 are suitably normalized, we may assume that the β2k+1 are integral classes, i.e. images of elements of the form β2k+1 for β2k+1 ∈ H2k+1 (G; Z). We will often use notation from de Rham theory to denote the analogous constructions in singular, or cellular theory. For example, the evaluation pairing between cohomology and homology is called the cap product. It is

usually denoted, x ∩ β or x[β]. The cap product corresponds to integration ( β x) in de Rham theory. By Poincaré duality, we can identify each cocycle x2k+1 with a codimension (2k + 1)-cycle F in G so that the image of any (2k + 1)-chain c2k+1 under x2k+1 is precisely the algebraic intersection number of F and c2k+1 . Hence, each compact, simple, simply-connected Lie group contains a codimension 3 cycle F Poincaré dual to x3 , which intersects β3 algebraically in one positively oriented point. We will shortly describe these codimension 3 cycles in greater detail, but we first describe how these cycles may be used to determine the path components of the configuration space. Assume for now that the cycle F has a trivial normal bundle. We will justify this assumption later. (Throughout this paper we will use normal bundles, open and closed tubular neighborhoods and the relation between them via the exponential map without explicitly writing the map. If ⊂ M then ν ⊂ T M, will denote the normal bundle and N ⊂ M will denote the closed tubular neighborhood.) Fix a trivialization of the normal bundle. Using this trivialization, we may associate a finite collection of signed points to any generic based map, u : M → G. To such a map we associate the collection of points, u−1 (F ). Such a point is positively oriented if the push forward of an oriented frame at the point has the same orientation as the trivialization of the normal bundle at the image. Conversely, to any finite collection of signed points we may associate a based map, u : M → G. Using a positively or negatively oriented frame at each point, we construct a diffeomorphism from the closed tubular neighborhood of each point to the 3-disk of radius π in the space of purely imaginary quaternions, sp(1). Via the exponential map, exp : sp(1) → Sp(1) given by, exp(x) = cos(|x|) + sin(|x|) |x| x we define a map from the closed tubular neighborhood of the points to Sp(1). This map may be extended to the whole 3-manifold by sending points in the complement of the neighborhood to −1. We next modify the map by multiplying by −1, so that the base point will be 1. Finally, we notice that the class, β3 is represented by a homomorphic image of Sp(1) in any Lie group. For the classical groups, this homomorphism is just the standard inclusion, Sp(1) = SU(2) → SU(n + 1), Sp(1) = Spin(3) → Spin(n), or Sp(1) → Sp(n). The homomorphism for each exceptional group is described in [4]. This matches exactly with the statement of Proposition 1. In the case we are considering here, H1 (G; Z) = 0, ˜ Z)) ∼ ˜ Z)) is and an element of H 3 (M; π3 (G)) ∼ = H 3 (M; H3 (G; = H 3 (M; H g−3 (G; just a machine that eats a 3-cycle in M, i.e. [M], and spits out a machine that eats a codimension 3-cycle in G, i.e. F , and spits out an integer. If G is not simple, there will be independent codimension 3-cycles for each simple factor, and one could interpret the intersection number with each cycle as a different type of particle (soliton). If G were not simply connected, the element of H 1 (M; H1 (G0 )) would be the obvious one, and one obtains the element of H 3 (M; π3 (G)) from a modification of the map into G that ˜ lifts to G. It is not difficult to describe the cycles β2k+1 and F for SU(n+1). Recall that the suspension of a pointed topological space is SX = X × [0, 1]/(X × {0, 1} ∪ {p0 } × [0, 1]). This may be visualized as the product X × S 1 with the circle above the marked point in X and the copy of X above a marked point in S 1 collapsed to a point. Identify CP k


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with U(k + 1)/(U(1) × U(k)) and define β2k+1 : SCP k → SU(n + 1) by β2k+1 ([A, t]) = [A, eπit ⊕ e−πit Ik ]com ⊕ In−k . Here, [A, B]com = ABA−1 B −1 is the usual commutator in a group. The normalization constants of x2k+1 would ensure that β2k+1 x2k+1 = 1. The values of these constants for k = 1 have been computed in [4]. We do not need these constants for this present work. The value of the normalization constants for k = 2 would, for example, be important if one wished to add a Wess-Zumino term to the Skyrme Lagrangian. The multiplication on a Lie group may be used to endow the homology of the Lie group with a unital, graded-commutative algebra structure, and the cohomology with a comultiplication. The homology product is given by (σ : → G) · (σ : → G) := (σ σ : × → G) and the comultipication on cohomology is dual to this. The multiplication and comultiplication give H ∗ (G; R) the structure of a Hopf algebra. It is exactly in this context that Hopf algebras were first defined. Using this algebra structure, we may give an explicit description of the Poincaré duality isomorphism. Any product of generators, xj in H ∗ (G; R) is sent to the element of H∗ (G; R) obtained from the product, nk=1 β2k+1 by removing the corresponding βj . In particular, F = nk=2 β2k+1 is the cycle

Poincaré dual to x3 . Geometrically, Poincaré duality is described by the equation, ω = #(P D(ω) ∩ ). Since SCP k is not a manifold, some words about our interpretation of the normal bundle to F are in order at this point. For SU(2) we may take F = {−1}. This is a codimension 3 submanifold, so there are no problems. Recall that CP k − CP k−1 is homeomorphic to R2k . It follows that the subset of F , call it F0 , obtained from the product of the SCP k − SCP k−1 is a codimension 3 cell properly embedded in SU(n + 1) − (F − F0 ). Since F − F0 has codimension 5, we may assume, using general position, that any map of a 3-manifold into SU(n + 1) avoids F − F0 . As F0 is contractible, it has a trivial normal bundle, justifying our assumption at the beginning of this description. 4.2. The fundamental group of GM . The Pontrjagin-Thom construction may also be used to understand the isomorphism, φ : π1 (GM ) → Zs2 ⊕ H 2 (M; π3 (G)), asserted in Theorem 2. A loop in (GM )0 based at the constant map u(x) = 1, may be regarded as a based map γ : SM → G. The identifications in the suspension provide a particularly nice way to summarize all of the constraints on γ imposed by the base points. We will use the same notation for the map, γ : M × [0, 1] → G obtained from γ by composition with the natural projection. The inverse image γ −1 (F ) with framing obtained by pulling back the trivialization of ν(F ) may be associated to γ . Conversely, given a framed link in (M − p0 ) × (0, 1) one may construct an element of π1 (GM ). Using the framing, each fiber of the closed tubular neighborhood to the link may be identified with the disk of radius π in sp(1). As before −1 times the exponential map may be used to construct a map, γ : SM → G representing an element of π1 (GM ). It is now possible to describe the geometric content of the isomorphism in Theorem 2. For a class of loops [γ ] ∈ (GM )0 , let φ(γ ) = (φ1 (γ ), φ2 (γ )). Restrict attention to the case of simply-connected G, and make the identifications, π3 (G) ∼ = H3 (G; Z) ∼ = H g−3 (G; Z). An element of H 2 (M; π3 (G)) may be interpreted as a function that associates an integer to a surface in M, say , and a codimension 3 cycle in G, say F . Set

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φ2 (γ )(, F ) = #( × [0, 1] ∩ γ −1 (F )). Note that γ −1 (F ) inherits an orientation from the framing and orientation on M. Using Poincaré duality this may be said in a different way. The homology class of γ −1 (F ) in (M − p0 ) × (0, 1) projects to an element of H1 (M) dual to the element associated to φ2 (γ ). The first component of the isomorphism counts the parity of the number of twists in the framing. Consider the framing in greater detail. Using a spin structure on M we associate a canonical framing to any oriented 1-dimensional submanifold of (M − p0 ) × (0, 1). See Proposition 14 in the proofs section. For now restrict attention to null-homologous submanifolds. Let be an oriented 2-dimensional submanifold of (M −N (p0 ))×(0, 1) with non-trivial boundary. The normal bundle to inherits an orientation from the orientations on (M − p0 ) × (0, 1) and . Oriented 2-plane bundles are classified by the second cohomology. Since H 2 (; Z) = 0, the normal bundle is trivial. Let (e1 , e2 ) be an oriented trivialization of this bundle. Let e3 ∈ (T |∂ ) be the outward unit normal. The canonical framing on ∂ is (e1 , e2 , e3 ). Given a second framing, (f1 , f2 , f3 ) on ∂ and an orientation preserving parameterization of the boundary, we obtain an element A ∈ π1 (GL+ (3, R)) = π1 (SO(3)) ∼ = Z2 satisfying (f1 , f2 , f3 ) = (e1 , e2 , e3 )A. This is 3 the origin of the first component of the isomorphism. The generator of π1 (Sp(1)S ) ∼ = Z2 is represented by, x1 −λ¯ x¯2 γ : (λ, x1 , x2 ) → , λx2 x¯1 having identified S 3 with the unit sphere in C2 (so |x1 |2 + |x2 |2 = 1), S 1 ∼ = U(1) and Sp(1) ∼ = SU(2). The image of γ under the obvious inclusion ι :SU(2) →SU(3), that is, ι(U ) = diag(U, 1), is homotopically trivial, as can be seen by constructing an explicit homotopy between it and ι ◦ γ (1, ·). First note that any SU(3) matrix is uniquely determined by its first two columns, which must be an orthonormal pair. For all t ∈ [0, 1], let µt (λ) = tλ + 1 − t (so µ1 = id and µ0 = 1) and define     x1 −λ¯ x¯2 , e :=  µt (λ)x2 v :=  x¯1  , v⊥ := v − (e† v)e. 2 0 1 − |µt (λ)| x2 Then

v⊥ (t, λ, x1 , x2 ) → e, ,∗ |v⊥ |

is the required homotopy between ι ◦ γ (t = 1) and the trivial loop based at ι : S 3 →SU(3). It is straightforward to check that e and v are never parallel (so the map is well defined), that (t, λ, 1, 0) → I3 for all t, λ (this is a homotopy through loops of based maps S 3 →SU(3)) and that (t, 1, x1 , x2 ) → ι(x1 , x2 ) for all t (each loop is based at ι). The homomorphic image of Sp(1) is contained in a standardly embedded SU(3) in each of the exceptional groups and the classical groups SU(n + 1), n ≥ 2, and Spin(N ), N ≥ 7, [4]. This is the reason why the Z2 factors only correspond to the symplectic factors of the Lie group. The following figures show some loops in the configuration spaces. For the first two figures, the horizontal direction represents the interval direction in M × [0, 1]. The disks represent the x − y plane in a coordinate chart in M, and we suppress the z direction due to lack of space. Figure 1 shows two copies of a typical loop representing an element in a


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symplectic Z2 factor. Only the first vector of the framing is shown in Fig. 1. The second vector is obtained by taking the cross product with the tangent vector to the curve in the displayed slice, and the final vector is the z-direction. It is easy to see that the left copy may be deformed into the right copy. We describe the left copy as follows: a particle and antiparticle are born; the particle undergoes a full rotation; the two particles then annihilate. The right copy may be described as follows: a first particle-antiparticle pair is born; a second pair is born; the two particles exchange positions without rotating; the first particle and second antiparticle annihilate; the remaining pair annihilates. Notice that there are two ways a pair of particles can exchange positions. Representing the particles by people in a room, the two people may step sideways forwards/backwards and sideways following diametrically opposite points on a circle always facing the back of the room. This is the exchange without rotating described in Fig. 1. This exchange 3 is non-trivial in π1 (Sp(1)S ). The second way a pair of people may change positions is to walk around a circle at diametrically opposite points always facing the direction that they walk to end up facing the opposite direction that they started. This second change of position is actually homotopically trivial. Since the framed links in Fig. 1 avoid the slices, M × {0, 1}, they represent a loop based at the constant identity map. It is possible to describe a framing without drawing any normal vectors at all. The first vector may be taken perpendicular to the plane of the figure, the second vector may be obtained from the cross product with the tangent vector, and the third vector may be taken to be the suppressed z-direction. The framing obtained by following this convention is called the black-board framing. We use the blackboard framing in Fig. 2. The Pontrjagin-Thom construction may also be used to visualize loops in other components of the configuration space. Figure 2 shows a loop in the degree 2 component of the space of maps from M to Sp(1). We can also use the Pontrjagin-Thom construction to draw figures of homotopies between loops in configuration space. Figure 3 displays a homotopy between the loop corresponding to a canonically framed unknot and the constant loop. In this figure, the horizontal direction represents the second interval factor of M ×[0, 1]×[0, 1], the direction out of the page represents the first interval factor, the vertical direction represents

Fig. 1. The rotation or exchange loop

Fig. 2. The degree 2 exchange loop

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Fig. 3. The contraction of a canonically framed contractible link

the x direction, and the y and z directions are suppressed. The framing is given by the normal vector to the hemisphere, the y direction and the z direction. 4.3. Cohomology of GM . We now turn to a description of the real cohomology of GM . We will use the slant product to associate a cohomology class on GM to a pair consisting of a homology class on M and a cohomology class on G. Recall that the slant product is a map H n (X × Y ; A) ⊗ Hk (X; B) → H n−k (Y ; A ⊗ B), [33]. In addition the universal coefficient theorem allows us to identify H k (GM ; R) with Hom(Hk (GM ; Z), R). Let σ : → M be a singular chain representing a homology class in Hd (M; R) = Hd (M; Z)⊗ R (instead of viewing singular chains as linear combinations of singular simplices, we will combine them together and view a singular chain as a map of a special polytope into the space), and let xj be a cohomology class in H j (G; R). To define the image of the mu map, µ( ⊗ xj ), let u : F → GM be a singular chain representing an element in Hd−j (GM ). This induces a natural singular chain u : M × F → G. The pullback ∗ j produces u xj ∈ H (M × F ; R). The formal definition of the mu map is then, u∗ xj /)[F ]. µ( ⊗ xj )(u) := (

(4.1)

Writing this in notation from the de Rham model of cohomology may help to clarify the definitions. In principle one could construct a homology theory based on smooth chains and make the following rigorous. The µ map produces a (j − d)-cocycle in GM from a d-cycle in M and a j -cocycle in G. On the level of chains, let ed : D d → M be a d-cell, and xj be a closed j -form on G. Given a singular simplex, u : j −d → GM , let u : M × j −d → G be the natural map and write d u ∗ xj . µ(e ⊗ xj )(u) = D d ×j −d

Using the product formula for the boundary, ∂(D d × j −d+1 ) = (∂D d ) × j −d+1 + (−1)d D d × ∂j −d+1 , we can get a simple formula for the coboundary of the image of an element under the µ-map. Let v : j −d+1 → GM , be a singular simplex, then δ(µ(ed ⊗ xj ))(v) =

j −d+1

(−1)k

k=0

∗

D d ×j −d

(v ◦ f k ) xj =

D d ×∂j −d+1

v ∗ xj


= (−1)d+1 =

v ∗ xj + (−1)d

(∂D d )×j −d+1 d+1 (−1) µ((∂ed ) ⊗ xj )(v).

∂(D d ×j −d+1 )

191

v ∗ xj (4.2)

We used Stokes’s theorem in the last line. It follows that µ is well defined at the level of homology. Theorem 3 asserts that H ∗ (GM 0 ; R) is a finitely generated algebra with generators µ(jd ⊗ xk ), where {jd } and {xk } are bases for H∗ (M; R) and H ∗ (G; R) respectively. The multiplication on H ∗ (GM ; R) is given by the cup product. Recall that this is defined at the level of cochains by, (α β)(w) = α( k w)β(w ), where α is a k-cocycle, β is a -cocycle, w is a (k + )-singular simplex and k w is the front k-face and w is the back -face [33]. Note that is graded-commutative, that is, α β = (−1)k β α. It is instructive to understand some classes that do not appear as generators. One might expect µ(pt ⊗ xj ) to be a generator in degree j . However, since GM consists of based maps, the induced map u : M × F → G arising from a chain u : F → GM restricts to a constant map on pt × F . It follows that µ(pt ⊗ xj ) = 0. There would be an analogous class if we considered the cohomology of the space of free maps. Turning to the other end of the spectrum, one might expect to see classes of the form µ(M ⊗ x3 ) in degree zero. Such certainly could not appear in the cohomology of the identity component GM 0 . In fact we stated our theorem for the identity component because the argument leading to generators of the form µ( ⊗ x3 ) breaks down when is a 3-cycle and x3 is a 3-cocycle. The argument starts by considering maps of spheres into the group G, and then assembles the cohomology of these mapping spaces (which are denoted by k G) into the cohomology of GM . The path fibration is used to compute the cohomology of the k G. The fibration leading to the cohomology of 3 G does not have a simply connected base and this is the break down. See Lemma 15. Finally one might expect to see classes of the form µ( ⊗ xj ∪ xk ). It will turn out in the course of the proof (Lemma 15) that such classes vanish. Up to this point, our geometric descriptions of the algebraic topology of configuration spaces have been simpler than we had any right to expect. We were able to describe the space of path components and the fundamental group of the configuration space of maps from an orientable 3-manifold into an arbitrary simply-connected Lie group by just considering subgroups isomorphic to Sp(1). This will not hold for all homotopy invariants of GM . The main object of interest to us is the second cohomology of the configuration space with integral coefficients, because this classifies the complex line bundles over the configuration space (the quantization ambiguity). It is possible to describe one second cohomology class on Sp(n)M in terms of Sp(1) geometry. However we need to pass to SU(3) subgroups to get at the second cohomology in general. Before considering these geometric representatives of the second cohomology, briefly recall the definition of the Ext groups. Given R-modules A and B, pick a free resolution of A say → C2 → C1 → C0 → A. The k th Ext group is just defined to be the k th homology of the complex Hom(C∗ , B), i.e. ExtkR (A, B) = Hk (Hom(C∗ , B)). When R is a PID (principal ideal domain) every R-module has a free resolution of the form, 0 → C1 → C0 → A. Given such a resolution one obtains the exact sequence, 0 → Hom(A, B) → Hom(C0 , B) → Hom(C1 , B) → Ext1R (A, B) → 0,

(4.3)

and all higher Ext groups vanish. We will always take R = Z and drop the ground ring from the notation. Based on the above exact sequence, we say that the Ext groups

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measure the failure of Hom to be exact, i.e. take exact sequences to exact sequences. The Ext group may also be identified with the collection of extensions of A by B [33]. By the universal coefficient theorem H 2 (GM ; Z) ∼ = Ext1 (H1 (GM ; Z), Z) ⊕ Hom(H2 (GM ; Z), Z), 2 M ∼ H (G ; R) = Ext1 (H1 (GM ; Z), R) ⊕ Hom(H2 (GM ; Z), R). Now for all A, Ext1 (A, R) = 0, so Hom(H2 (GM ; Z), Z) is a free abelian group of rank b2 = dimR H 2 (GM ; R). In addition, Ext1 (A, Z) is just the torsion subgroup of A and H1 (GM ; Z) ∼ = π1ab (GM ) = π1 (GM ). Hence H 2 (GM ; Z) ∼ = Zb2 ⊕ Tor(π1 (GM )), where π1 (GM ) and the Betti number b2 may be obtained from Theorems 2 and 3. We will use the universal coefficient theorem and Ext groups to describe some cohomology classes of our configuration spaces. There is a natural Z2 contained in the fundamental group of the configuration space for any group with a symplectic factor. This Z2 is generated by the exchange loop. Wrapping twice around the exchange loop is the boundary of a disk in the configuration space. Since RP 2 is the result of identifying the points on the boundary of a disk via a degree 2 map, one expects to find an RP 2 embedded into any of the Skyrme configuration spaces with a symplectic factor. In [32], 3 R. Sorkin describes an embedding, fstat : RP 2 → SU(n + 1)S . He also describes an 3 embedding, fspin : RP 3 → SU(n + 1)S . He further shows that fspin restricted to the RP 2 subspace is homotopic to fstat . Using the map, M → M (3) /M (2) ∼ = S 3 , these M (k) induce maps into SU(n + 1) . Here M is the k skeleton of M with respect to some CW structure. In fact, using the inclusion of Sp(1) = SU(2) into any simply-connected simple Lie group one obtains maps from RP 2 and RP 3 into any configuration space of Lie group valued maps. This is most interesting when the map factors through a symplectic factor. We briefly recall Sorkin’s elegant construction. Describe RP 2 as the 2-sphere with antipodal points identified. By the addition of particle antiparticle pairs, we may assume that there are two particles in a coordinate chart. We may place the particles at antipodal points of a sphere in a coordinate chart using frames parallel to the coordinate directions. The map obtained from these frames using the Pontrjagin-Thom construction is fstat . The projective space, RP 3 is homeomorphic to the rotation group SO(3). The map fspin may be described by using SO(3) to rotate a single frame and then applying the Pontrjagin-Thom construction. Sorkin includes a second unaffected particle in his description of fspin to make the comparison with fstat easier. 3 A degree one map M → S 3 (which always exists) induces a map GS → GM . The 3 space GS is typically denoted 3 G. If the Lie algebra of the maximal compact subgroup admits a symplectic factor, then we have an interesting map Sp(1) → G which induces a map 3 Sp(1) → 3 G. We will see in the course of our proofs that on the level of π1 or H1 these maps give a sequence of injections, H1 (RP 2 ; Z) → H1 (RP 3 ; Z) → H1 (3 Sp(1); Z) → H1 (3 G; Z) → H1 (GM ; Z). The universal coefficient theorem implies that there is a Z2 factor in the second cohomology of GM when G contains a symplectic factor. In fact, in this case we see that twice the exchange loop is a generator of the 1-dimensional boundaries. This means we can define a homomorphism from B1 (the 1-dimensional boundaries) to Z taking twice the


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exchange loop to 1. The cocycle defined by following the boundary map from 2-chains by this homomorphism generates the Z2 in H 2 (GM ; Z). We see that this class evaluates nontrivially on the Sorkin RP 2 . When G has unitary factors, there will be infinite cyclic factors in the second cohomology of GM . This is nicely explained by a construction of Ramadas, [28]. Ramadas constructs a map, S 2 × S 3 → SU(3). This construction goes as follows. He first defines a map K : SU(2) → SU(2) by 2 1 a b −b¯ 2 a . K( ¯ ) = (|a|4 + |b|4 )− 2 b a¯ b2 a¯ 2 ¯ σ : S 1 \SU(2)× This map satisfies, K(diag(λ, λ)A) = K(A)diag(λ2 , λ¯ 2 ). Finally define SU(2) → SU(3) by σ ([A], B) = diag(1, K(A))diag(ABA∗ , 1)diag(1, K(A)∗ ). ¯ of SU(2). It is well known that Here we are viewing S 1 as the subgroup diag(λ, λ) 1 2 3 ∼ ∼ S \SU(2) = S and SU(2) = S . The map σ : S 2 × S 3 → SU(3) induces a map σ : S 2 → 3 SU(3). Ramadas shows that this map generates H2 (3 SU(3); Z) ∼ = Z. Combining with the degree one map from M and the inclusion into a special unitary factor of G, we obtain a map S 2 → GM generating an infinite cyclic factor of H2 (GM ; Z). By the universal coefficient theorem a map from H2 (GM ; Z) to Z taking this generator to 1 is a cohomology class in H 2 (GM ; Z). Clearly this class evaluates non-trivially on this S 2 . If G does not have a symplectic or special unitary factor, then there is no reason to expect any elements of the second cohomology. In fact under this hypothesis, H 2 (3 G; Z) = 0. It is worth mentioning how these maps behave in general. The third homotopy group of any Lie group is generated by homomorphic images of Sp(1). Each time one of these generators is contained in a symplectic factor, we get a Z2 in the second cohomology detected by a Sorkin RP 2 . When one of these factors is not contained in a symplectic factor, it is contained in a copy of SU(3). This kills the Z2 factor in π1 as explained above in Subsect. 4.2. If the SU(3) is contained in a special unitary factor, the Sorkin map RP 2 → Sp(1)M → SU(3)M → GM pulls back the second cohomology class described by Ramadas (and extended to arbitrary M and G with special unitary factor as above) to the generator of H 2 (RP 2 ; Z) ∼ = Z2 . (Ramadas proves that the generator of H 2 (3 SU(3); Z) pulls back to the generator of H 2 (RP 2 ; Z) and the rest follows from our proofs.) If this SU(3) is not contained in a special unitary factor, it follows from our proofs that the second homology class associated to S 2 → GM bounds, so there is no associated cohomology class. 2 M 4.4. Components of (S 2 )M ϕ . The picture of the components of (S )ϕ arising from the Pontrjagin-Thom construction and Poincaré duality is quite nice. The inverse image of a regular value in S 2 is Poincaré dual to ϕ ∗ µS 2 . The number of twists in the framing of a second map with the same pull-back is the element of H 3 (M; Z)/ 2ϕ ∗ µS 2 . This is very similar to the description of elements of the fundamental group of GM when G has symplectic factors. We give three examples to clarify this. Identify S 2 with CP 1 and consider the maps. We have ϕ1 , ϕ1 , ϕ3 : CP 1 × S 1 → CP 1 given by,

ϕ1 ([z : w], λ)=[z : w],

ϕ1 ([z : w], λ)=[λz : w],

and

ϕ3 ([z : w], λ)=[z3 : w3 ]. (4.4)

194


We can view CP 1 ×S 1 as S 2 ×[0, 1] (a spherical shell) with the inner and outer (S 2 ×{0} and S 2 ×{1}) spheres identified. Using this convention and the framing conventions from Subsect. 4.2, we have displayed the framed 1-manifolds arising as the inverse images of a regular value in Fig. 4. It may appear that there is a well defined twist number associated to a S 2 -valued map. However, there is a homeomorphism of CP 1 × S 1 twisting the 2-sphere (such a map is given by ([z : w], λ) → ([λz : w], λ)). This will change the number of twists in a framing, but will not change the relative number of twists. The reason why this relative number of twists is only at most well defined modulo twice the divisibility of the cohomology class ϕ ∗ µS 2 is demonstrated for ϕ1 in Fig. 5. 2 M 4.5. Fundamental group of (S 2 )M ϕ . An element of π1 ((S )ϕ ) is represented by a map, 1 2 γ : M × S → S . The inverse image of a regular value is a 2-dimensional submanifold, say . This defines an element of H 1 (M; Z) as follows. To any 1-cycle in M, say σ , we associate the intersection number of and σ ×S 1 . Since our loop is in the path component of ϕ, the surface is parallel to the ϕ-inverse image of a regular value. This implies that our element of H 1 (M; Z) is in the kernel of the map, 2ϕ ∗ µS 2 : H 1 (M; Z) → H 3 (M; Z). Given any element of this kernel, we can define a loop in (S 2 )M ϕ via the q-map defined in Sect. 6.3. There is a map from u : M × S 1 → Sp(1) that may be used to change this new loop back into γ . The remaining homotopy invariants of γ are just those of u as described in Subsect. 4.2.

5. Physical Consequences As explained in Sect. 3, the configuration space of the Skyrme model with arbitrary target group is homotopy equivalent to the configuration space of a collection of uncoupled Skyrme fields each taking values in a compact, simply connected, simple Lie group. We will therefore assume, throughout this section that G is compact, simply connected and simple. In this case, by Proposition 1, the path components of GM are labelled by H 3 (M; Z) ∼ = Z, identified with the baryon number B of the configuration. This identification has already been justified by consideration of the Pontrjagin-Thom construction. Let us denote the baryon number B sector by QB .

Fig. 4. Pontrjagin-Thom representatives of the S 2 -valued maps ϕ1 , ϕ1 and ϕ3

Fig. 5. Introducing 2d twists


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We first recall how Finkelstein and Rubinstein introduced fermionicity to the Skyrme B , model [15]. The idea is that the quantum state is specified by a wavefunction on Q the universal cover of QB , rather than QB itself. By the uniqueness of lifts, there is a B by deck transformations. Let π : Q B → QB denote the natural action of π1 (QB ) on Q covering projection, λ ∈ π1 (QB ) and Dλ be the associated deck transformation. Since all points in π −1 (u) are physically indistinguishable, we must impose the constraint |ψ(Dλ q)| = |ψ(q)| B and λ ∈ π1 (QB ). This leaves us the B → C, for all q ∈ Q on the wavefunction ψ : Q freedom to assign phases to the deck transformations, that is, the remaining quantization ambiguity consists of a choice of U (1) representation of π1 (QB ). The possibility of fermionic quantization arises if the two-Skyrmion exchange loop in Q2 is noncontractible with even order: we can then choose a representation which assigns this loop the phase −1. In this case our wavefunction aquires a minus sign under Skyrmion interchange. Clearly, the Finkelstein-Rubinstein model could apply to any sigma model with a configuration space admiting non-trivial elements of the fundamental group representing the exchange of identical particles. In particular, the domain does not have to be R3 . Note we have insisted that the wavefunction ψ have support on a single path component QB , because baryon number is conserved in nature, so transitions which change B have zero probability. It seems, then, that the choice of representation of π1 (QB ) can be made independently for each B, but in fact there is a strong consistency requirement between the representations associated with the various components. Recall that all the sectors are homeomorphic and that given any u ∈ QB one obtains a homeomorphism Q0 → QB by pointwise multiplication by u. Hence, to each u ∈ QB there is associated an isomorphism π1 (Q0 ) → π1 (QB ), so one has a map QB → Iso(π1 (Q0 ), π1 (QB )). Since QB is connected and π1 is discrete, this map is constant, that is, there is a canonical isomorphism π1 (Q0 ) → π1 (QB ), which may be obtained by pointwise multiplication by any charge B configuration. Having chosen a representation of π1 (Q0 ), we obtain canonical representations of π1 (QB ) for all other B. Physically, we are demanding that the phase introduced by transporting a configuration around a closed loop should be independent of the presence of static Skyrmions held remote from the loop. This places nontrivial consistency conditions, if we are to obtain a genuinely fermionic quantization. In particular, the loop in Q2B consisting of the exchange of a pair of identical charge B Skyrmions must be assigned the phase (−1)B , since a charge B Skyrmion represents a bound state of B nucleons, which is a fermion for B odd and a boson for B even. The Finkelstein-Rubinstein formalism can be used to give a consistent fermionic quantization of the Skyrme model on any domain M if G = Sp(n), but not for any of the other simple target groups. In this case, Theorem 2 tells us that π1 (QB ) ≡ Z2 ⊕ H1 (M), and we can choose (and fix) a U (1) representation which maps the generator of Z2 to (−1). The generator of the Z2 -factor in the baryon number zero component is exactly the rotation–exchange loop as may be seen in the proof of Proposition 13 in the next section. To see that this assigns phase (−1) to the 2-Skyrmion exchange loop, we may consider the Pontrjagin-Thom representative of the loop. This is a framed 1-cycle in S 1 × M depicted in Fig. 6. It is framed-cobordant to the representative of the loop in which one of the Skyrmions remains static, while the other rotates through 2π about its center. Figure 6 gives a sketch of the cobordism. The horizontal direction represents the

196


Fig. 6. The cobordism between Skyrmion rotation and Skyrmion exchange

loop parameter (“time”), the vertical direction represents M and the direction into the page represents the cobordism parameter. The framing has been omitted, and the start and end 1-cycles of the cobordism have been repeated, for clarity. Note that the apparent self intersection of the cobordism (along the dashed line) is an artifact of the pictorial projection from 5 dimensions to 3. Hence, the exchange loop in Q2 is homotopic to the loop represented by one static Skyrmion and one Skyrmion that undergoes a full rotation. To identify the phase assigned to this homotopy class, we must transfer the loops to Q0 by adding a pair of anti-Skyrmions, as depicted in Fig. 7. This changes each configuration by multiplying by a fixed charge −2 configuration which is 1 outside a small ball – precisely one of the homeomorphisms discussed above. The figure may be described thus: the exchange loop is homotopic to the rotation loop with an extra static 1-Skyrmion lump (far left) which is transferred to the vacuum sector by adding a stationary pair of anti-Skyrmions (2nd box). This loop is homotopic to the charge 0 rotation loop of Fig. 1, via the sequence of moves shown. The orientations on the curves indicate how to assign a framing via the blackboard framing convention. The resulting Pontrjagin-Thom representative is framed cobordant to the charge 0 exchange loop described in Sect. 4.2, which, as explained, generates the Z2 factor in π1 (Q0 ). Hence, the loop along which two identical 1-Skyrmions are exchanged (without rotating) around a contractible path in M is assigned the phase (−1). Exchange of higher charge Skyrmions may be treated by considering composites of B unit Skyrmions, as depicted for B = 2 in Fig. 8. The loop may be deformed into one with four distinct single exchange events (surrounded by dashed boxes). Each of these may be replaced by a pair of uncrossed strands, one of which has a 2π twist, using the homotopy described in Figs. 1 and 6 in each box. Since each strand has an even number of twists, this is homotopic to the constant loop. Hence it must be assigned the phase

~

~

Fig. 7. Mapping the Skyrmion exchange loop into the vacuum sector Q0


197

Fig. 8. Exchange of baryon number 2 Skyrmions is contractible in Q4

(+1). The argument clearly generalizes: given an exchange loop of a pair of charge B composites, one may isolate B 2 single exchange events, each of which can be replaced by a single twist in one of the uncrossed strands. It is easy to see that the twists may be distributed so that every strand except at most one has an even number of twists. Hence if B is even, this last strand also has an even number of twists and the loop is necessarily contractible. If B is odd, the last strand has an odd number of twists, so the loop is homotopic to the loop where 2B − 1 Skyrmions remain static and one Skyrmion executes a 2π twist. Adding 2B anti-Skyrmions, this loop is identified with the baryon number 0 exchange loop and hence receives a phase of (−1). Finkelstein and Rubinstein also model spin in this framework. In this model spin is determined by the phase associated to the rotation loop. As we saw in the previous section the rotation and exchange loops agree up to homotopy confirming the spin statistics theorem in this model. This is essentially the observation that the exchange loop is homotopic to the 2π rotation loop (Fig. 6). Note that throughout the above discussion we have used only a local version of exchange to model particle statistics, and verify the spin-statistics correlation. This definition of particle statistics and spin makes sense on general M, even without an action of SO(3), because the exchange and rotation loops have support over a single coordinate chart, so we have a local notion of rotating a Skyrmion. Things become much more subtle when a loop has a Pontrjagin-Thom representative which projects to a nontrivial cycle in M. It should not be surprising that it requires a spin structure on M to specify whether the constituent Skyrmions of such a loop undergo an even or odd number of rotations. It was precisely the generalization from R3 to general spaces that motivated the definition of spin structures in the first place. Notice that by changing the spin structure, we can interpret a loop as either having an even or odd number of rotations, so one must fix a spin structure on space before discussing spin. (This is similar to the reason, discussed in Sect. 4.4 above, why the secondary invariant for path components of S 2 -valued maps is only a relative invariant.) Even in the simple case of quantization of many point particles on a topologically nontrivial domain, the statistical type (boson, fermion or something more exotic) of the particles is usually taken to be determined only by their exchange behavior around trivial loops in M [21]. It may be more reasonable to require that spin be determined by the behavior of locally supported rotations, but to insist that the statistical type be consistent under any particle exchange. It follows from Proposition 13 that the notion of an exchange or rotation loop around a contractible loop is well defined independent of the choice of spin structure. This is just the image of π4 (G). That the parity of a rotation around a non-contractible loop is determined by a spin structure is explained in Proposition 14 in the next section. In fact, the Finkelstein-Rubinstein quantization scheme remains consistently fermionic in this extended sense provided that the correct representation into U(1) is chosen. As with more traditional models of spin, a spin structure on the domain will be required. When the domain has non-trivial first cohomology with Z2 coefficients there are many

198


spin structures to choose from. Selecting a spin structure produces an isomorphism of π1 (QB ) with Z2 ⊕H1 (M). The required representation is just projection onto the Z2 factor. Exchange around a (possibly) non-contractible simple closed curve in space means that two identical solitons start at antipodal points on the curve and each one moves without rotating half way around the curve to exchange places with the other soliton. The notion of moving without rotating is where the spin structure enters. We will define this after describing the representation of an exchange displayed in Fig. 9. Each rectangle in this figure represents a slice of a cobordism. The horizontal direction represents time, the vertical direction represents space, and the thick lines are the world lines of the solitons. The top and bottom of each rectangle are to be identified to make each slice a cylinder representing the curve cross time. We can imagine different spin structures obtained by identifying the top and bottom in the straightforward way or by putting a full twist before making the identification. The first slice is just the exchange around the curve. One of the loops makes a lefthand rotation followed by a righthand rotation, but this wobble is the same as no rotation at all. Adding a ribbon between the non-rotating soliton and the right rotation of the bottom soliton produces the second slice. This slice may be described as one soliton making a full left rotation in a fixed location while a second soliton traverses once around the curve without rotation. This slice homotopes to the third slice, and a second ribbon gives the fourth slice. The fourth slice may be described as follows. The vertical S-curve represents the birth of a Skyrmion-anti-Skyrmion pair after which the Skyrmion and anti-Skyrmion move in opposite directions around the curve until they collide and annihilate. The horizontal lines are two (nearly) static Skyrmions, and the figure eight curve is a contractible (left) rotation loop. By definition, the exchange is non-rotating with respect to the spin structure if the Z2 representation of the vertical S-curve resulting from a baryon number 1 exchange is trivial. We now see that the general exchange is consistent because the two horizontal lines contribute nothing to the representation, the S-curve contributes nothing, and the baryon number B contractible rotation loop contributes (−1)B as described previously and seen in Fig. 8. As will be seen in Sect. 6.3, there is a close connexion between Sp(1)M and (S 2 )M . This allows us to transfer the Finkelstein-Rubinstein construction of a fermionic quantization scheme for Sp(N ) valued Skyrmions to the Faddeev-Hopf model. Recall that here, unless H 2 (M; Z) = 0, the path components of Q are not labelled simply by an integer, but rather are separated by an invariant α ∈ H 2 (M), and a relative invariant c ∈ H 3 (M)/2α H 1 (M). Configurations with α = 0 necessarily have support which wraps around a nontrivial cycle in M. They therefore lack one of the key point-like characteristics of conventional solitons: they are not homotopic to arbitrarily highly localized configurations. Such configurations are intrinsically tied to some topological “defect” in physical space, and so are somewhat exotic. We therefore mainly restrict our attention to configurations with α = 0. As with Skyrmions, these configurations are labelled by B ∈Z∼ = H 3 (M; Z) which we identify with the Hopfion number. This is the relative

~

~

Fig. 9. Sum of a loop with the rotation loop

~


199

invariant between the given configuration and the constant map. It turns out that all the α = 0 sectors, QB , are homeomorphic (Theorem 5) and, further, that QB is homotopy equivalent to (Sp(1))M 0 , the vacuum sector of the classical Skyrme model. The homotopy equivalence is given by the fibration f ∗ of Lemma 20. This may be used to map the charge zero Skyrmion exchange loop to a charge 2 Hopfion exchange loop, generating a Z2 factor in π1 (Q2 ). It follows that Hopfions can be quantized fermionically within the Finkelstein-Rubinstein scheme. In the more exotic configurations with α = 0, the relative invariant takes values in Z/(2dZ) ∼ = H 3 (M; Z)/(2α H 1 (M; Z)). So even though the Hopfion number is not defined in this case, the parity of the Hopfion number is well defined and the Finkelstein-Rubinstein formalism still yields a consistently fermionic quantization scheme. We return now to the Skyrme model, but in the case where G is not Sp(N ) but is rather SU(N ), Spin or exceptional. In this case the Skyrmion exchange loop is contractible, so must be assigned phase (+1) in the Finkelstein-Rubinstein quantization scheme so that only bosonic quantization is possible in that framework. To proceed, one may take the wavefunction to be a section of a complex line bundle over QB equipped with a unitary connexion. Parallel transport with respect to the connexion associates phases to closed loops in QB in a way that one might hope will mimic fermionic behaviour. The problem with this is that the holonomy of a loop is not (for non-flat connexions) homotopy invariant, so the phase assigned to an exchange loop will depend on the fine detail of how the exchange is transacted. To get around this, Sorkin introduced a purely topological definition of statistical type and spin for solitons defined on R3 . His definition extends immediately to solitons defined on arbitrary domains. We review these definitions next. Recall Sorkin’s definition of fstat : RP 2 → Q2 : choose a sphere in R3 and associate to each antipodal pair of points on this sphere the charge 2 configuration with Pontrjagin-Thom representation given by that pair of points, framed by the coordinate basis vectors. The subscript stat in fstat refers to statistics. There is an associated homomor∗ : H 2 (Q ) → H 2 (RP 2 ) ∼ Z given by pullback. According to Sorkin’s phism fstat = 2 2 definition, the quantization wherein the wavefunction is a section of the line bundle over ∗ (c) = 1, bosonic otherwise, Q associated with class c ∈ H 2 (Q; Z) is fermionic if fstat ∗ (c) represents the [32]. Thinking of fstat as an inclusion map, the pulled-back class fstat Chern class of the restriction of the bundle associated to c over Q2 to the subset RP 2 . The intuition behind this definition is that if there was a unitary connection on the bundle with parallel transport equal to (−1) around exchange loops, then the restriction to the bundle to such a RP 2 would have to be non-trivial. This definition generalizes to solitons defined on arbitrary domains by analogous maps from RP 2 into the configuration space based on embeddings of S 2 in the domain. To make sense of the framing, one must pick a trivialization of the tangent bundle of the domain restricted to the S 2 . Up to homotopy there is a unique such framing. The elementary Sorkin maps will be the ones associated with sufficiently small spheres that lie in a single coordinate chart. To model spin for solitons with domain R3 , Sorkin considers the action of SO(3) on the configuration space given by precomposition with any field. The orbit of a basic soliton with Pontrjagin-Thom representative given by one point and an arbitrary frame has representatives obtained by rotates of the frame. This rotation may be performed on any isolated lump in any component of configuration space to define a map fspin : SO(3) → QB . Sorkin defines the quantization associated to a class c ∈ H 2 (QB ; Z) to ∗ c ∈ H 2 (SO(3); Z) ∼ Z is non-trivial, [32]. be spinorial if and only if the pull-back fspin = 2 This definition generalizes immediately to solitons on arbitrary domains. The intuition behind this definition is clearly explained in the paper of Ramadas, [28]. The idea is

200


that the classical SO(3) symmetry of QB lifts to a quantum Sp(1) = SU(2) = Spin(3) symmetry that descends to an SO(3) action on the space of quantum states if and only ∗ c ∈ H 2 (SO(3); Z) is trivial. In the case where the domain is not if the pull-back fspin 3 R , there is no SO(3) symmetry, but there is still a SO(3) orbit of any single soliton, obtained by rotating the framing of a single-point Pontrjagin-Thom representative, so one still has a local notion of what it means to rotate a soliton. One may still define a map fspin : SO(3) → QB and define the quantization corresponding to class c to be spinorial ∗ c = 0, though there is no corresponding statement about quantum if and only if fspin symmetries. This should be contrasted with the case of isospin, which we discuss later in this section. Sorkin proves a version of the spin statistics theorem when the domain is R3 . Recall that rotations may be represented by vectors along the axis of the rotation with magnitude equal to the angle of rotation. A one-half rotation in one direction is equivalent to a one-half rotation in the opposite direction. This gives a natural inclusion of RP 2 into SO(3) as the set of one-half rotations. This inclusion induces an isomorphism on cohomology, ι∗ : H 2 (SO(3); Z) → H 2 (RP 2 ; Z). Sorkin’s version of the spin statistics ∗ ∗ . There is one slightly stronger version of the spin theorem states that ι∗ fspin = fstat statistics correspondence that one may hope for when the domain is arbitrary. We will discuss this later in this section. Ramadas proved that the Sorkin definition of statistical type and spinoriality were strict generalizations of the Finkelstein-Rubinstein definition when the target group is SU(N ). The statement works as follows. One first notices that the universal coefficient theorem gives an isomorphism H 2 (Q; Z) ∼ = Hom(H2 (Q; Z), Z) ⊕ Ext1 (H1 (Q; Z), Z) ∼ = Hom(H2 (Q; Z), Z) ⊕ Ext1 (π1 (Q), Z). When fermionic quantization is possible in the framework of Finkelstein-Rubinstein, the exchange loop is an element of order 2 in π1 (Q). Ramadas shows that the corresponding element of H 2 (Q; Z) pulls back to the non-trivial element of H 2 (SO(3); Z) under fspin (when the target is SU(2) so that fspin is defined). More precisely, he shows several 3 3 things. He shows that H 2 (SU(N )S ; Z) ∼ = Z for N > 2 and H 2 (SU(2)S ; Z) ∼ = Z2 . He shows that the inclusion SU(N ) → SU(N + 1) induces an isomorphism 3

3

H 2 (SU(N + 1)S ; Z) → H 2 (SU(N )S ; Z) for N > 2 and a surjection for N = 2. The N > 2 case follows from the fibration SU(N ) → SU(N + 1) → S 2N+1 . The N = 2 case follows from the four term exact sequence induced by the Ext functor together with several ingeniously defined maps, 3 3 see [28]. Since H 2 (SU(2)S ; Z) ∼ = Z2 , the exchange loop in π1 (SU(2)S ) corresponds 3 to the generator of H 2 (SU(2)S ; Z) under the universal coefficient isomorphism. This class pulls back to the generator of H 2 (SO(3); Z) under fspin . Thus, when it is possible to quantize fermionically in the Finkelstein-Rubinstein framework, the exchange loop is an element of order 2, so it corresponds to a cohomology class which pulls back non-trivially under fspin and fstat . Hence it is possible to quantize fermionically in the Sorkin framework, also. Now turn to the case of an arbitrary domain and compact, simply connected, simple target group. Given an arbitrary domain, M, we can construct a degree one map to S 3 by collapsing the 2-skeleton. This map induces, via precomposition, a map between the


201

3

corresponding configuration spaces, ιM : GS → GM . (Given a soliton configuration on S 3 define one on M by mapping points in M to points in S 3 , then follow the configuration into the target.) The induced map on cohomology is surjective, so there is a portion of the quantization ambiguity (H 2 (QB ; Z)) that depends only on the codomain and is completely independent of the domain. This confirms our first physical conclusion, C1. Recall that we call H 2 (QB ; Z) the quantization ambiguity because line bundles are classified by elements of H 2 (QB ; Z) and wave functions are sections of such bundles. By Theorems 2 and 3 this cohomology is H 2 (QB , Z) = Zb2 (QB ) ⊕ Tor(H1 (M)) ⊕ π4 (G), where b2 (QB ) = b1 (M) + 1 for G = SU(N ), N ≥ 3, and b2 (QB ) = b1 (M) for G = Spin(N ), N ≥ 7, G = Sp(N ), N ≥ 1, or G exceptional. Here bk (X) denotes the k th Betti number of X, that is, dimR Hk (X; R). Also π4 (G) = Z2 for G = Sp(N ), N ≥ 1, and π4 (G) = 0 otherwise. Notice that the elementary Sorkin maps factor through 3 ιM : GS → GM . Use fEstat and fEspin to denote the elementary Sorkin maps defined on an arbitrary domain. By definition we have, fEstat = ιM ◦ fstat and fEspin = ιM ◦ fspin . 3 If G = Spin(N ), N ≥ 7, or exceptional, then H 2 (GS ; Z) = 0 so ι∗M is trivial. Hence, fermionic quantization is impossible in these cases. The inclusions Sp(N ) → Sp(N +1) 3 induce maps on the configuration spaces Sp(N )S that induce isomorphisms on cohomology, and Sp(1) ∼ = SU(2), so we have reduced to the special unitary case. If G = SU(N ), 3 2 N ≥ 3, then H (GS ; Z) = Z and ι∗M maps Tor(H1 (M)) and all the generators µ(1k ⊗ x3 ) to 0, and µ([M]⊗x5 ) to µ([S 3 ]⊗x5 ). Since the map on cohomology induced by ιM is surjective, fermionic quantization in the generalization of the sense of Sorkin is possible over an arbitrary domain if and only if it is possible for domain S 3 . Combined with the 3 result of Ramadas that fspin : H 2 (GS ; Z) → H 2 (SO(3); Z) surjects and Sorkin’s spin ∗ = ι∗ f statistics correlation that fstat spin , one obtains fstat (mµ([M] ⊗ x5 )) = m ∈ Z2 , so quantization on the bundle represented by the class mµ([M] ⊗ x5 ) is fermionic if and only if m is odd. Our second physical conclusion, C2, establishing necessary and sufficient conditions for the existence of fermionic quantizations follows from these comments. There are consistency conditions that one would like to check with regard to the generalized Sorkin model of particle statistics. Since QB is connected and H 2 (Q0 , Z) is discrete, there is a canonical isomorphism H 2 (Q0 ) → H 2 (QB ), so a class c ∈ H 2 (Q0 ; Z) defines a fixed class over each sector QB . As in the discussion of the Finkelstein-Rubinstein model of particle statistics, we would like to know that Baryon number B lumps ∗ c = 0). We in QB are all bosonic when the one lump class in Q1 is bosonic (fEstat would also like to know that Baryon number B lumps in QB are bosonic or fermionic ∗ c = 1). according to the parity of B when the one lump class in Q1 is fermionic (fEstat The statistical type of a Baryon number B lump may be defined via a generalization of the Sorkin map in which a pair of Baryon number B lumps is placed at antipodal points on a sphere. With this definition, cobordism arguments similar to those given in the Finkelstein-Rubinstein case show that this model of particle statistics is indeed consistent. As in the Finkelstein-Rubinstein case, the spin of a particle will just be determined by a local picture, and the statistical type may be based on non-local exchanges of identical particles. A non-local exchange will be defined by a generalization of the Sorkin map associated to an arbitrary embedded 2-sphere, say S → M. Denote the associated map by, fS:stat : RP 2 → QB . There are two cases: either S separates the domain, so

202


M − S = M1 ∪ M2 , or S does not separate. If the 2-sphere in M separates, we can define a degree one map p : M → S 3 by collapsing the relative 2-skeleta of M 1 and M 2 . As in the case of the elementary Sorkin map, we obtain fS:stat = fstat ◦ p , where 3 p : GS → GM . It follows that this model of particle statistics is consistent with these non-local exchanges. If the sphere does not separate, then there is a simple path from one side of the sphere to the other side of the sphere. A tubular neighborhood of the union of this path and the sphere is homeomorphic to a punctured S 2 × S 1 . We may construct a degree one projection from M to S 2 × S 1 by collapsing the complement of this tubular neighborhood. This intertwines the Sorkin map defined using the non-separating sphere with the Sorkin map defined using S 2 × {1} ⊂ S 2 × S 1 . If we knew the following conjecture, then this model of particle statistics would be consistent in this larger sense. As it is, we know that it satisfies the stronger consistency condition in the typical case where the domain does not contain a non-separating sphere. Conjecture. fS∗2 ×{1}:stat µ([S 2 × S 1 ] ⊗ x 5 ) = 0. To discuss isospin, we recall some standard facts about extending group actions on a be the associated princonfiguration space. Given a complex line bundle over Q, let Q cipal U(1) bundle. If a group acts on Q it is possible to construct an extension of by so that the projection to intertwines the two actions. The extension U(1) that acts on Q may be defined as equivalence classes of paths in , see [8]. The quantum symmetry group is a subgroup of this extension. When = SO(3) the possible U(1) extensions are SO(3) × U(1) and U(2). These correspond to integral and fractional isospin respectively when SO(3) acts as rotations on the target. Recall that every compact, simply connected, simple Lie group G has a Sp(1) subgroup. We define the isospin action on G to be the adjoint action of this Sp(1) subgroup. This coincides with the usual definition if G = Sp(1). Of course, we can always take a trivial line bundle over Q, so any of our configuration spaces admit quantizations with integral isospin, confirming our fifth physical conclusion, C5. To justify our remaining physical conclusions about isospin, we review the required 3 constructions. The Sorkin map SO(3) → SU(2)S is the map obtained by the isospin action. To see this, notice that we can rotate the frame in the Pontrjagin-Thom representative by either rotating the domain or by rotating the codomain. When a class in H 2 (GM ; Z) pulls back to the generator of H 2 (RP 2 ; Z) under the Sorkin map, we claim that the associated quantization has fractional isospin. Assume otherwise so the extension of the rotation group is SO(3)×U(1). This means that the SO(3) subgroup is a lift of the SO(3) action on the configuration space to the bundle over the space. Restricting this to the image of SO(3) under fspin we obtain a contradiction from Theorem 10. Since such classes exist whenever the configuration space admits a fermionic quantization, we obtain our third physical conclusion, C3. To show that quantizations with fractional isospin are not possible when the group does not have a symplectic or special unitary factor, one must just follow through the construction of the extension given in [8] to see that the resulting extension is trivial. This establishes our fourth conclusion, C4. As we noted earlier, the relation between the configuration space of Sp(1)-valued maps and S 2 -valued maps implies that it is always possible to fermionically quantize Hopfions. Since it is possible to quantize Sp(1)-valued solitons with fractional isospin, the same relation implies that it is possible to quantize S 2 -valued solitons with fractional isospin. This is our sixth physical conclusion, C6.


203

6. Proofs We begin by recalling some basic homotopy and homology theory [33]. For pointed spaces X, Y , let [X, Y ] denote the set of based homotopy classes of maps X → Y . There is a distinguished element 0 in [X, Y ], namely the class of the constant map. Given a map f : X → X , there is for each Y a natural map f ∗ : [X , Y ] → [X, Y ] defined by composition. We define ker f ∗ ⊂ [X , Y ] to be the inverse image of the null g

f

class 0 ∈ [X, Y ]. A sequence of maps X → X → X is coexact if ker f ∗ = Im g ∗ for every choice of codomain Y . Longer sequences of maps are coexact if every constituent triple is coexact. Note that this makes sense even in the absence of group structure. If Y happens to be a Lie group G, as it will be for us, then [X, G] inherits a group structure by pointwise multiplication, f ∗ and g ∗ are homomorphisms, and the sequence g∗

f∗

[X , G] → [X , G] → [X, G] is exact in the usual sense. In the following, we will make extensive use of the following standard result [33]: Proposition 11. If X is a CW complex and A ⊂ X is a subcomplex then there is an infinite coexact sequence, A →X → X/A → SA → SX → S(X/A) → · · · → S n A → S n X → S n (X/A) → · · ·, where S n denotes iterated suspension. The proofs will use several naturally defined homomorphisms. Any map f : X → Y defines homomorphisms f∗ : Hk (X) → Hk (Y ) which depend on f only up to homotopy. Hence, one has natural maps Hk : [X, Y ] → Hom(Hk (X), Hk (Y )). There is a natural (Hurewicz) homomorphism Hur k : πk (X) → Hk (X) sending each map S k → X to the push-forward of the fundamental class via the map in X. If X is (k − 1)-connected then Hur k is an isomorphism. There is also a natural isomorphism Suspk : Hk (SX) → Hk−1 (X) relating the homologies of X and SX. We may now prove a preliminary lemma that is used in the computation of both the fundamental group and the real cohomology. This lemma is the place where we use the assumption that the domain is orientable. This lemma was used in [4] as well. Note that SX (k) denotes the suspension of the k-skeleton of X. The k-skeleton of the suspension of X will always be denoted (SX)(k) . Lemma 12. For a closed, connected, orientable 3-manifold, and simply-connected, compact Lie group the map, [SM, G] → [SM (2) , G] induced by inclusion is surjective. Proof. Start with a cell decomposition of M with exactly one 0-cell and exactly one 3-cell. The sequences, M/M (2)

∂-

SM (2)

- SM,

and SM (1)

- SM (2)

q S(M (2) /M (1) ),

204


are coexact by Proposition 11 with X = M, A = M (2) and X = M (2) , A = M (1) respectively. Hence, the sequences, ∂∗ [SM, G] - [SM (2) , G] - [M/M (2) , G], and q∗ [S(M (2) /M (1) ), G] - [SM (2) , G] - [SM (1) , G] = 0, are exact. The group [SM (1) , G] is trivial because G is 2-connected. Hence q ∗ is sur(3) (2) is homeomorphic to D 3 /S 2 . Under this identification, jective. The space M /M x ∂(x) = f (3) |x| , |x| for x ∈ D 3 , where f (3) : S 2 → M (2) is the attaching map for the 3-cell. Now we can construct the following commutative diagram: [S(M (2) /M (1) ), G]

∂∗ ◦ q∗

- [M (3) /M (2) , G]

H3 Hom(H3 (S(M

H3

(2)

? /M (1) )), H3 (G))

Hom(H3 (M

(3)

∗ ◦ Hur 3 ∗ Susp−1 3

Hom(H2 (M

(2)

? /M (2) ), H3 (G)) Hur 3 ∗

? /M (1) ), π3 (G))

? δ(3) Hom(H3 (M /M (2) ), π3 (G)).

Note that both maps H3 are isomorphisms, and Hur 3 : π3 (G) → H3 (G) is an isomorphism because G is 2-connected, so the vertical maps are all isomorphisms. We may therefore identify ∂ ∗ ◦q ∗ with the map δ. But δ is the coboundary map in the CW cochain complex for H ∗ (M; π3 (G)). Since M is orientable, this coboundary is trivial. Since q ∗ is surjective, we conclude that ∂ ∗ = 0. 6.1. The fundamental group of the Skyrme configuration space. We saw in section 3 that it was sufficient to study simple, simply-connected Lie groups. We begin our study of the fundamental group by showing that the fundamental group of the configuration space fits into a short exact sequence in this case. Proposition 13. If M is a closed, connected, orientable 3-manifold and G is a simple, simply-connected Lie group, then 0 → π4 (G) → π1 (GM ) → H 2 (M; π3 (G)) → 0 is an exact sequence of abelian groups. The maps in this sequence will be defined in the course of the proof. Proof. We have π1 (GM ) = [S 1 , GM ] ∼ = [SM, G], where SM is the suspension of M. The sequence (SM)(3) → SM → SM/(SM)(3)


205

is coexact by Proposition 11 with X = SM, A = (SM)(3) , and hence induces the exact sequence of groups: [SM/(SM)(3) , G] → [SM, G] → [(SM)(3) , G].

(6.1)

Noting that (SM)(3) = SM (2) , Lemma 12 implies that the last map in the above sequence is surjective. This will become the exact sequence we seek, after suitable identifications. First we identify the third group in sequence 6.1 with the second cohomology of M, in similar fashion to the proof of Lemma 12. From Proposition 11, we have exact sequences, q∗ [S((SM)(2) /(SM)(1) ), G] - [S((SM)(2) ), G] - [S((SM)(1) ), G] (X = (SM)(2) , A = (SM)(1) ) and ∂∗ [S((SM)(2) ), G] - [(SM)(3) /(SM)(2) , G] - [(SM)(3) , G], (X = (SM)(3) , A = (SM)(2) ). Since G is 2-connected, [S((SM)(1) ), G] = 0, and q ∗ is surjective. Thus coker(∂ ∗ ◦ q ∗ ) = coker(∂ ∗ ) = [(SM)(3) , G]. Using the Hurewicz and suspension isomorphisms as before, we may identify ∂ ∗ ◦ q ∗ with a coboundary map in the CW cochain complex for H ∗ (M; π3 (G)): [S((SM)(2) /(SM)(1) ), G]

Hom(H2 ((SM)

(2)

? /(SM)(1) ), π3 (G))

∂∗ ◦ q∗

δ

- [(SM)(3) /(SM)(2) , G]

? - Hom(H3 ((SM)(3) /(SM)(2) ), π3 (G)).

Now, [(SM)(3) , G] ∼ = coker(∂ ∗ ◦ q ∗ ) ∼ = coker(δ) ∼ = H 3 (SM; π3 (G)) ∼ = H 2 (M; π3 (G)). The first group in sequence (6.1) is π4 (G) because SM/(SM)(3) ∼ = S 4 . For the nonsymplectic groups π4 (G) = 0, and we are done. For the higher symplectic groups, the fibration Sp(n) → Sp(n + 1) → S 4n+3 induces a fibration (Sp(n))M → (Sp(n + 1))M → (S 4n+3 )M . The homotopy exact sequence of this fibration reads π2 ((S 4n+3 )M ) → π1 ((Sp(n))M ) → π1 ((Sp(n + 1))M ) → π1 ((S 4n+3 )M ). Now πk ((S 4n+3 )M ) = [S k , (S 4n+3 )M ] ∼ = [S k M, S 4n+3 ]. For k = 1, 2 these groups are trivial since S 4n+3 is 5-connected. Hence π1 (Sp(n + 1)M ) ∼ = π1 (Sp(n)M ) for all n ≥ 1, so the proposition reduces to showing that the first map in sequence (6.1) is injective for G = Sp(1). In the special case of M ∼ = S 3 the exchange loop depicted in Fig. 1 3 represents the generator of π4 (Sp(1)) ∼ = π1 ((Sp(1))S ). Our final task is to show that

206


the image of this generator under push forward by the collapsing map M → M/M (2) is non-trivial in π1 ((Sp(1))M ). Proceed indirectly and assume that there is a homotopy between the constant loop and the exchange loop, say H : M × [0, 1] × [0, 1] → Sp(1). Set = H −1 (−1). = ∩ where Now glue the homotopy from Fig. 3 to this homotopy and let defined over is the hemisphere in Fig. 3. The trivialization of the normal bundle to defined over do not match. The and the trivialization of the normal bundle to discrepency is the generator of π1 (SO(3)). It follows that the second Stiefel-Whitney )) ∈ H 2 (M × [0, 1] × [0, 1]; π1 (SO(3))) is non-trivial [34]. However, class, w2 (N ( the Whitney product formula yields, )) = w(N( )) w(T ) w(N ( = w(T ⊕ N ( )) = w(T (M × [0, 1] × [0, 1])| ) = 1. ) = 1 because the Stiefel-Whitney class Here w is the total Stiefel-Whitney class, w(T of any orientable surface is 1, and T (M × [0, 1] × [0, 1])| is trivial. This contradiction establishes the proposition. ←

It is well known that a split exact sequence of abelian groups, 0 → K → G → H → 0, induces an isomorphism, K ⊕H ∼ = G. The following proposition will establish such a splitting, and therefore, complete our computation of the fundamental group of the Skyrme configuration spaces. The proof will require surgery descriptions of 3-manifolds, so we recall what this means. Given a framed link, say L, (i.e. 1-dimensional submanifold with trivialized normal bundle or identification of a closed tubular neighborhood with ⊥⊥S 1 × D 2 ) in S 3 = ∂D 4 , we define a 4-manifold by D 4 ∪⊥⊥S 1 ×D 2 D 2 × D 2 . The boundary of this 4-manifold is said to be the 3-manifold obtained by surgery on L. It is denoted, SL3 . Proposition 14. The sequence, 0 → π4 (G) → π1 (GM ) → H 2 (M; π3 (G)) → 0 splits, and there is a splitting associated to each spin structure on M. Proof. As we saw at the end of the proof of the previous proposition, it is sufficient to check the result for G = Sp(1). Since the three dimensional Spin cobordism group is trivial, every 3-manifold is surgery on a framed link with even self-linking numbers [23]. Such a surgery description induces a Spin structure in M. Let M = SL3 be such a surgery description, orient the link and let {µj }cj =1 be the positively oriented meridians to the components of the link. These meridians generate H1 (M) ∼ = H 2 (M; π3 (Sp(1))). This last isomorphism is Poincaré duality. Define a splitting by: 1 M s : H1 (M) → π1 ((Sp(1)) ); s(µj ) = P T µj × , canonical framing . 2 Here P T represents the Pontrjagin-Thom construction and the canonical framing is constructed as follows. The first vector is chosen to be the 0-framing on µj considered as an unknot in S 3 . The second vector is obtained by taking the cross product of the tangent vector with the first vector, and the third vector is just the direction of the interval. We will now check that this map respects the relations in H1 (M). Let QL = (nj k ) be the


207

linking matrix so that, H1 (M) = µj |nj k µj = 0 . We are using the summation convention in this description. The 2-cycle representing the relation, nj k µj = 0 may be constructed from a Seifert surface to the j th component of the link, when this component is viewed as a knot in S 3 . Let j denote this Seifert surface. The desired 2-cycle is then ◦ j = (j − N (L)) ∪ σj . Here σj is the surface in S 1 × D 2 with njj meridians depicted j is exactly the relation, nj k µj = 0. The framon the left in Fig. 10. The boundary of ing on each copy of µk for k = j induced from this surface agrees with the 0-framing. j may be extended to a The framing on each copy of µj is −sign(njj ). The surface, surface in M × [0, 1] × [0, 1] by adding a collar of the boundary in the direction of the second interval followed by one band for each pair of the µj as depicted on the right of Fig. 10. The resulting surface has a canonical framing, and the corresponding homotopy given by the Pontrjagin-Thom construction homotopes the loop corresponding to the relation to a loop corresponding to a ±2-framed unlink. Such a loop is null-homotopic, as required. We remark that the Spin structures on M correspond to H 1 (M; Z2 ). In addition, the splittings of Z2 → π1 (Sp(1)M ) → H 2 (M; Z) corresponds to the group cohomology, H 1 (H 2 (M; Z); Z2 ) ∼ = H 1 (H1 (M; Z); Z2 ) ∼ = H 1 (M; Z2 ). The last isomorphism is because the 2-skeleton of M is the 2-skeleton of a K(H1(M;Z),1). A combination of Propositions 13 and 14 together with Reductions 5, 6 and 7 and the corollary of the universal coefficient theorem that H ∗ (X; A⊕B) ∼ = H ∗ (X; A)⊕H ∗ (X; B) give Theorem 2. 6.2. Cohomology of Skyrme configuration spaces. As we have seen we may restrict our attention to compact, simple, simply-connected G. Recall the cohomology classes, xj , and the µ-map defined in Sect. 4. Throughout this section we will take the coefficients of any homology or cohomology to be the real numbers unless noted to the contrary. To compute the cohomology of GM we will use the cofibrations M (k) → M (k+1) → M (k+1) /M (k) and the fact that M (k+1) /M (k) is a bouquet of spheres to reduce the problem to the case where the domain is a sphere. Briefly recall the computation of the cohomology of the loop spaces. These are well known results, but we sketch a proof because this explains why the classes µ(, xj xk ) k are trivial. As usual let k G = GS denote the k-iterated loop space. We have the following lemma.

1

1

n

jj

Fig. 10. The 2-cycle in the proof of Proposition 14

208


Lemma 15. The cohomology rings of the first loop groups are given by, H ∗ (G) = R[µ([S 1 ] ⊗ xj )], H ∗ (2 G) = R[µ([S 2 ] ⊗ xj )], and H ∗ (30 G) = R[µ([S 3 ] ⊗ xj ), j > 3]. Proof. Recall that the path space, P G is contractible, and fits into a fibration, G → p,q P G → G. The Serre spectral sequence of this fibration has E2 = H p (G; H q (G)). Since G is simply-connected, the coefficient system is untwisted. Since P G is contractible all classes of positive degree have to die at some point in this spectral sequence. By location we know that all differentials of x3 vanish, so there must be some class in H 2 (G) mapping to x3 . The class µ([S 1 ] ⊗ x3 ) is one such class, and is the only class that there can be without having something elselive to the limit group of the spectral sequence. Notice that classes of the form x3 xjk are images of classes of the form µ([S 1 ] ⊗ x3 ) xjk , so we have killed all classes with a factor of x3 . In the same way, we can kill terms with a factor of the next xj . We conclude that H ∗ (G) = R[µ([S 1 ] ⊗ xj )]. Repeating the argument with the fibration, 2 G → P G → G we obtain, H ∗ (2 G) = R[µ([S 2 ] ⊗ xj )]. This time the coefficient system is untwisted because π1 (G) ∼ = π2 (G) = 0. We need to adjust the argument a bit at the next stage because π0 (3 G) ∼ = π1 (2 G) ∼ = 3 ∼ π3 (G) = Z. This shows that the path components of G may be labeled by the integers. Each component is homeomorphic to the identity component since 3 G is a topological group. In this case, we have no guarantee that the coefficient system is untwisted, so we will use a different approach that will be useful again in Sect. 6.4. Let 2 G denote the universal cover of 2 G and let 3 G denote the identity component of 0 2G → 2 G that may be used to obtain 3 G. These fit into a fibration, 3 G → P 0

H ∗ (30 G) = R[µ([S 3 ] ⊗ xj ), j > 3]. We will use equivariant cohomology to compute 2 G. the cohomology of Recall that any Lie group, say , acts properly on a contractible space called the total space of the universal bundle. This space is denoted E. The quotient of this by is the classifying space B. Let X be a space (i.e. a space with a action) and consider the space X := E × X. The cohomology of the space X is called the equivariant cohomology of X. It is denoted by H∗ (X). When the action on X is free and proper (as it is in our case), we have a fibration X → X/ obtained by ignoring the E component in the definition of X . The fiber of this fibration is just E which is contractible, so the spectral sequence of the fibration implies that the cohomology of X/ is isomorphic to the equivariant cohomology of X. By ignoring the X component in the definition of X we obtain a fibration X → B that may be used to relate the equivariant cohomology of X to the cohomology of X. 2 G and = π (2 G) ∼ If we apply these ideas with X = = Z, we obtain a spec1 tral sequence that may be used to show that the cohomology of 2˜ G is generated by µ([S 2 ] ⊗ xj ) for j > 3. This then plugs in to give the stated result for 30 G. Returning to the situation of a 3-manifold domain, let M have a cell decomposition with one 0-cell (p0 ) several 1-cells (er ) several 2-cells (fs ) and one 3-cell ([M]). Since GX∨Y = GX × GY , and M (k+1) /M (k) is a bouquet of spheres we have, H ∗ (GM ) = R[µ(er ⊗ xj )], (2) (1) H ∗ (GM /M ) = R[µ(fs ⊗ xj )], (1)

M (3) /M (2)

H ∗ (G0

) = R[µ([M] ⊗ xj ), j > 0].


209 (2)

The next lemma assembles these facts into the cohomology of GM . Lemma 16. If r1 form a basis for H1 (M), and s2 form a basis for H2 (M) we have, H ∗ (GM ) = R[µ(r1 ⊗ xj ), µ(s2 ⊗ xk )]. (2)

Proof. The cofibration M (1) → M (2) → M (2) /M (1) leads to a fibration, GM

(2) /M (1)

→ GM

(2)

(1)

→ GM .

(1)

Since G is 2-connected, π1 (GM ) = 0, so the coefficients in the cohomology appearing in the second term of the Serre spectral sequence are untwisted. We have, E2∗,∗ = H ∗ (GM ; H ∗ (GM /M )) (1) (2) (1) = H ∗ (GM ) ⊗ H ∗ (GM /M ) ∼ = R[µ(er ⊗ xj ), µ(fs ⊗ xk )]. (1)

(2)

(1)

To go further we need to understand the differentials in this spectral sequence. Since j −1,0 µ(er ⊗ xj ) ∈ E2 , we have dk µ(er ⊗ xj ) = 0 for all k. We will show that d µ(fs ⊗ xk ) = 0 for < k − 1 and dk−1 µ(fs ⊗ xk ) = −µ((∂fs ) ⊗ xk ) from which the result (1) will follow. The multiplication on G induces a ring structure on the homology of GM (2) (1) and GM /M . Using the homology spectral sequence of the path fibrations, one may (1) j −1 → GM show that these homology groups are generated by cycles er βj : r (2) (1) and fs βk : sk−2 → GM /M dual to µ(er ⊗ xj ) and µ(fs ⊗ xk ) respectively. The product in the homology ring of GX is given by β · β : × → GX with β · β (x, y)(p) = β(x)(p)β (y)(p). The computation (4.2) in Sect. 4 shows that the differential of our spectral sequence is given by (d µ(fs ⊗ xk ))(β ⊗ β ) = − w ∗ xk , (∂fs )××

where w(p, x, y) = β(x)(p)β (y)(π(p)) and π : M (2) → M (2) /M (1) is the canonical projection. We are using the integral as a suggestive notation for the cap product. We see that the map w factors through (∂fs ) × × point. When < k − 1, (∂fs ) × × point has dimension less than k, so the differential is trivial. For = k − 1, this reduces to the claimed result. We remark that the above lemma is a valid computation of the cohomology of GK when K is any connected 2-complex. To go up to the next and final stage we need to (2) (3) (2) analyze the action of the fundamental group of GM on the cohomology of GM /M . This will require Lemma 12 from the beginning of this section. This is the place in the cohomology computation where we use the fact that M is orientable. Let us review the situation for general fibrations first. If F → E → B is a fibration and γ : [0, 1] → B represents an element of the fundamental group of B one can define a map, : F × [0, 1] → B by (x, t) = γ (t). The map, 0 : F → B lifts to the inclusion, F → E. By the homotopy lifting property, there is a lift, : F × [0, 1] → E. The restriction, 1 : F → F induces a map on the cohomology of F . This is how the fundamental group of the base of a fibration acts on the cohomology of the fiber. The following lemma shows that the action of the fundamental group on the cohomology of the fiber of our final fibration is trivial.

210


Lemma 17. If M is an orientable 3-manifold and G is a compact, simply-connected Lie (2) (3) (2) group then the action of π1 (GM ) on H ∗ (GM /M ) is trivial. (2)

Proof. Let γ : SM (2) → G represent an element of π1 (GM ). By Lemma 12, this (3) (2) extends to a map, ζ : SM (3) → G. Now define : GM /M × [0, 1] → GM by (u, t)(x) = u([x]) ζ ([x, t]). The upper triangle of the following diagram commutes because ζ ([x, 0]) = 1. The lower triangle commutes because, ζ |SM (2) = γ, GM

(3) /M (2)

× {0}

- GM

? G

M (3) /M (2)

× [0, 1]

? - GM (2)

Thus is an appropriate lift. Since u([x]) = 1 for x ∈ M (2) , 1 is the identity map and the action on the cohomology of the fiber is trivial. We can now complete the proof of Theorem 2. The cofibration M (2) → M (3) → M (3) /M (2) leads to a fibration, GM

(3) /M (3)

→ GM

(3)

(2)

→ GM .

By the previous lemma, the coefficients in the cohomology appearing in the second term of the Serre spectral sequence are untwisted. Using Lemma 16, we have, E2∗,∗ = H ∗ (GM ; H ∗ (GM /M )) = H ∗ (GM ) ⊗ H ∗ (GM ∼ = R[µ(r1 ⊗ xj ), µ(s2 ⊗ xk ), µ([M] ⊗ x ), > 0]. (2)

(3)

(2)

(2)

(3) /M (2)

)

Repeating the argument from Lemma 16 with computation (4.2), we see that all of the differentials of this spectral sequence vanish. This completes our computation of the cohomology of the Skyrme configuration space. 6.3. The fundamental group of Faddeev-Hopf configuration spaces. In this subsection we compute the fundamental group of the Faddeev-Hopf configuration space, (S 2 )M . Recall (Theorem 4) that the path components (S 2 )M ϕ (where ϕ is any representative ∗ of the component) fall into families labelled by ϕ µS 2 ∈ H 2 (M; Z), where µS 2 is a generator of H 2 (S 2 ; Z), and that components within a given family are labelled by α ∈ H 3 (M; Z)/2ϕ ∗ µS 2 H 1 (M; Z). To analyze the Faddeev-Hopf configuration space in more detail we will further exploit its natural relationship with the classical (G = SU(2) = Sp(1)) Skyrme configuration space. These ideas were concurrently introduced in [5]. We identify S 2 with the unit purely imaginary quaternions, and S 1 with the unit complex numbers. The quotient, Sp(1)/S 1 is homeomorphic to S 2 , with an explicit homeomorphism given by [q] → qiq ∗ . Our main tool will be the map q : S 2 × S 1 → Sp(1),

q(x, λ) = qλq ∗ ,

It is not difficult to verify the following properties of q:

where x = qiq ∗ .

(6.2)


1. 2. 3. 4. 5. 6.

211

It is well defined and smooth. q(x, λ1 λ2 ) = q(x, λ1 )q(x, λ2 ). q−1 (1) = S 2 × {1}. q(x, λ)x(q(x, λ))∗ = x. q(x, ·) : S 1 → {q|qxq ∗ = x} is a diffeomorphism. deg(q) = 2 with the standard “outer normal first” orientations.

For example writing x = qiq ∗ , the fourth property may be verified as q(x, λ)x(q(x, λ))∗ = qλq ∗ qiq ∗ qλ∗ q ∗ = x. We will let λx denote the inverse to q(x, ·). We will also use the related maps, ρ : S 2 × Sp(1) × S 1 → S 2 × Sp(1) and f : S 2 × Sp(1) → S 2 × S 2 defined by ρ(x, y, λ) = (x, yq(x, λ)) and f (x, q) = (x, qxq ∗ ). Properties (2) and (3) show that ρ is a free right action. Properties (4) and (5) show that f is a principal fibration with action ρ. As our first application of these maps we show that the evaluation map is a fibration. Lemma 18. The evaluation map, evp0 : FreeMap(M, S 2 ) → S 2 , given by evp0 (ϕ) = ϕ(p0 ) is a fibration. Proof. We just need to construct the diagonal map in the following diagram. h FreeMap(M, S 2 )

X × {0}

H

i

? X × [0, 1]

evp0 ? - S2

H

If we define the horizontal maps in the following diagram by µ(x, t) = (h(x)(p0 ), H (x, t)) and ν(x) = (h(x)(p0 ), 1), then the existence of the diagonal map will follow because f is a fibration. X × {0} i ? X × [0, 1]

ν 2 S × Sp(1) µ

f

? - S2 × S2 µ

The desired map is just, H (x, t)(p) = µ2 (x, t)h(x)(p)(µ2 (x, t))∗ .

Clearly the fiber of this fibration is (S 2 )M . Recall that we are using (S 2 )M ϕ to denote the ϕ-component of the space of based maps. In [5] these ideas were used to give a new proof of Pontrjagin’s homotopy classification of maps from a 3-manifold to S 2 . The following lemma comes from that paper. A second proof of this lemma may be found in [6]. Lemma 19 (Auckly-Kapitanski). There exists a map u : M → Sp(1) such that ψ : M → S 2 and ϕ : M → S 2 are related by ψ = uϕu∗ if and only if ψ ∗ µS 2 = ϕ ∗ µS 2 .

212


Theorem 5 follows directly from this lemma. Assuming ψ ∗ µS 2 = ϕ ∗ µS 2 , define a 2 M ∗ map F : (S 2 )M ϕ → (S )ψ by F (ξ ) = uξ u . This is clearly well defined because any map homotopic to ϕ will be mapped to a map homotopic to ψ under F . There is a well-defined inverse given by F −1 (ζ ) = u∗ ζ u. We have a fibration relating the identity component of the Skyrme configuration space to any component of the Faddeev-Hopf configuration space. Lemma 20. The map induced by f , f∗

2 M {ϕ} × Sp(1)M 0 → {ϕ} × (S )ϕ

is a fibration. Proof. Once again we just need to construct the diagonal map in a diagram. hSp(1)M 0

X × {0} i

H

? X × [0, 1]

f∗

? - (S 2 )M ϕ H

So once again we consider a second diagram. M × X × {0} i ? M × X × [0, 1]

ν 2 S × Sp(1) µ

f

? - S2 × S2 µ

Here the horizontal maps are given by ν(p, x) = (ϕ(p), h(x)(p)) and µ(p, x, t) = (ϕ(p), H (x, t)(p)). The diagonal lift exists because f is a fibration. We need to use Property (5) of q to adjust the base points. Let x0 be the basepoint of S 2 and define the desired lift by H (x, t)(p) = µ2 (p, x, t)q(ϕ(p), λx0 (µ2 (p, x, t)−1 ). This completes the proof.

By Property (5) of q, we see that any element of the fiber of the above fibration may be written in the form q(ϕ, λ) for some map λ : M → S 1 . Since q(ϕ, λ) is null homotopic, its degree must be zero. By Property (6) of q, this implies that λ∗ µS 1 must be in the kernel of the cup product 2ϕ ∗ µS 2 . Conversely, given any map λ with λ∗ µS 1 ∈ ker(2ϕ ∗ µS 2 ) we get an element of the fiber. Recall that the components of the space of maps from M to S 1 correspond to H 1 (M; Z) and each component is homeomorphic to the identity component which is homeomorphic to RM which is con∗ tractible. It follows that up to homotopy Sp(1)M 0 is a regular ker(2ϕ µS 2 ) cover of


213

∗ (S 2 )M ϕ (the fiber is homotopy equivalent to ker(2ϕ µS 2 )). The homotopy sequence of the fibration then gives us the following sequence: ∗ 0 → π1 (Sp(1)M ) → π1 ((S 2 )M ϕ ) → ker(2ϕ µS 2 ) → 0.

(6.3)

Since we do not already know that π1 ((S 2 )M ϕ ) is abelian, we not only need to show that the sequence splits, we also need to show that the image of the splitting commutes with the image of π1 (Sp(1)M ). This is the content of the following lemma. This lemma will complete the proof of Theorem 6. Lemma 21. The sequence (6.3) splits and the image of the splitting commutes with the image of π1 (Sp(1)M ). Proof. Given θ ∈ ker(2ϕ ∗ µS 2 ) define a corresponding map λθ : M → S 1 in the usual p

θ

way, λθ (p) = e p0 . This induces a map, qθ : M → Sp(1) by qθ (p) = q(ϕ(p), λθ (p)). We compute the degree as follows:

deg(qθ ) = M qθ∗ µSp(1)

= 2 M ϕ ∗ µS 2 ∧ λ∗θ µS 1 = 2 M ϕ ∗ µS 2 ∧ λ∗θ = 0. It follows that there is a homotopy, H θ with H θ (0) = 1 and H θ (1) = qθ . Define the splitting by sending θ to Hθ ∈ π1 ((S 2 )ϕ ) given by Hθ (t)(p) = f (ϕ(p), H θ (t)(p)). To see that the image of this splitting commutes with the image of π1 (Sp(1)M ), let 2t γ : [0, 1] → Sp(1)M be a loop and define maps δ1 and δ2 by δ1 (t, s) = s+1 for 1 t ≤ 2 (s + 1), δ1 (t, s) = 1 otherwise and δ2 (t, s) = 1 − δ1 (1 − t, s). We see that f (ϕ, (γ ◦ δ1 ) · (H θ ◦ δ2 )) is a homotopy between f (ϕ, γ ) ∗ f (ϕ, H θ ) and f (ϕ, γ · H θ ). Likewise, f (ϕ, (γ ◦ δ2 ) · (H θ ◦ δ1 )) is a homotopy between f (ϕ, H θ ) ∗ f (ϕ, γ ) and f (ϕ, γ · H θ ). To prove Theorem 7, notice that we have a left S 1 action on (S 2 )M ϕ given by z · ψ := zψz∗ . We claim that the fibration, FreeMap(M, S 2 )ϕ → S 2 , is just the fiber bundle with associated principal bundle Sp(1) → S 2 and fiber (S 2 )M ϕ . In fact, the map Sp(1) ×S 1 2 , M) given by [q, ψ] → qψq ∗ is the desired isomorphism. (S 2 )M → FreeMaps(S ϕ Now consider the homotopy exact sequence of the fibration, FreeMap(M, S 2 )ϕ → S 2 , 2 → π2 (S 2 ) → π1 ((S 2 )M ϕ ) → π1 (FreeMaps(S , M)ϕ ) → 0.

It follows that π1 (FreeMaps(S 2 , M)ϕ ) is just the quotient of π1 ((S 2 )M ϕ ) by the image of π2 (S 2 ). The next lemma identifies this image, to complete the proof of Theorem 7. Lemma 22. The image of the map from π2 is the subgroup of H 2 (M; Z) < π1 ((S 2 )M ϕ ) generated by 2ϕ ∗ µS 2 . Proof. Recall that the map from π2 of the base to π1 of the fiber is defined by taking a map of a disk into the base to the restriction to the boundary of a lift of the disk to the total space. Since the boundary of the disk maps to the base point, the restriction to the boundary of the lift lies in the fiber. The homotopy exact sequence of the fibration, Sp(1) → S 2 implies that the disk representing a generator of π2 (S 2 ) lifts to a disk with boundary generating the fundamental group of the fiber S 1 . This lift, say γ to Sp(1)

214


2 ∼ gives a lift D 2 → Sp(1) ×S 1 (S 2 )M γ (z), 1]. ϕ = FreeMaps(M, S ) defined by z → [ Restricted to the boundary, this map is just z → [z, ϕ] = [1, zϕz∗ ]. It follows that the image of π2 (S 2 ) is just the subgroup generated by the loop zϕz∗ . We now just have to trace this loop through the proof of the isomorphism, 2 ∗ ∼ π1 ((S 2 )M ϕ ) = Z2 ⊕ H (M; Z) ⊕ ker(2ϕ µS 2 ).

The projection to ker(2ϕ ∗ µS 2 ) was defined by taking a lift of each map in the M 1-parameter family representing the loop in π1 ((S 2 )M ϕ ) to Sp(1)0 and comparing the maps at the beginning and end. In our case the entire path consistently lifts to the ∗ path γϕ : S 1 → Sp(1)M 0 given by γϕ (z) = zq(ϕ, z ). It follows that the component in ∗ ker(2ϕ µS 2 ) is zero. A loop such as γϕ naturally defines a map, γ¯ϕ : M ×S 1 → Sp(1). The image of our loop in H 2 (M; Z) is just γ¯ϕ∗ µSp(1) /[pt × S 1 ]. In notation reminis

cent of differential forms this would be pt×S 1 γ¯ϕ∗ µSp(1) . In order to evaluate this, we write γ¯ϕ as the composition of the map (ϕ, idS 1 ) : M × S 1 → S 2 × S 1 and the map q˜ : S 2 × S 1 → Sp(1) given by q˜ (x, z) = zq(x, z∗ ). This latter map is then expressed as the composition of (q, pr∗2 ) : S 2 × S 1 → Sp(1) × S 1 and the map Sp(1) × S 1 → Sp(1) given by (u, λ) → λu. The form µSp(1) pulls back to µSp(1) 1 under the first map, and this pulls back to 2µS 2 µS 1 under the first factor of q˜ since q has degree two. In particular q˜ has degree two as well. We can now complete this computation to see that our loop projects to 2ϕ ∗ µS 2 in H 2 (M; Z). To complete the proof, we need to compute the projection of our loop in the Z2 -factor. The projection to Z2 is defined by multiplying the inverse of our map by the image of 2ϕ ∗ µS 2 under the splitting H 2 → π1 and taking the framing of the inverse image of a regular value. The equivalence classes of framings may be identified with Z2 since the inverse image is homologically trivial. Alternatively we may compare the framing coming from our map to the framing of the map coming from the splitting. The image under the splitting of 2ϕ ∗ µS 2 is, of course, just two times the image of ϕ ∗ µS 2 under the splitting. The inverse image coming from our map is just two copies of the inverse image of a frame under the map (ϕ, idS 1 ) : M ×S 1 → S 2 ×S 1 . This means that the projection is even, so zero in Z2 . 6.4. The cohomology of Faddeev-Hopf configuration spaces. In order to compute the M 2 M cohomology of (S 2 )M ϕ we will use the fibration, Sp(1)0 → (S )ϕ . The fiber of this 1 M ∗ fibration is just ⊥⊥α∈K (S )α , where K = ker(2ϕ µS 2 ∪). Up to homotopy, the fiber is just K. In fact we can assume that the fiber is exactly K if we first take the quotient by (S 1 )M 0 (which is contractible by Reduction 6). It is slightly tricky to use a spectral sequence to compute the cohomology of the base of a fibration, so we will use equivariant cohomology to recast the problem. Recall that any Lie group acts properly on a contractible space called the total space of the universal bundle. In our case, this space is denoted EK. The quotient of this by K is the classifying space BK. Let X be a K space (i.e. a space with a K action) and consider the space XK := EK ×K X. We 1 M will be interested in the situation when X = Sp(1)M 0 (or rather this divided by (S )0 , but this has the same homotopy type). The cohomology of the space XK is called the equivariant cohomology of X. It is denoted by HK∗ (X). When the K action on X is free and proper (as it is in our case), we have a fibration XK → X/K obtained by ignoring the EK component in the definition of XK . The fiber of this fibration is just EK which is contractible, so the spectral sequence of the fibration implies that the cohomology of X/K is isomorphic to the equivariant cohomology of X. By ignoring the X component


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in the definition of XK we obtain a fibration XK → BK which may be used to compute the equivariant cohomology of X. Since H 1 (M; Z) is a free abelian group, the kernel K is as well. It follows that we may take EK to be Rn with n equal to the rank of K and with K acting by translations. It follows that BK is just an n-torus, and we have a spectral sequence with E2 term, p,q ˜ M )) converging to the cohomology of (S 2 )M . Clearly the E2 = H p (T n ; H q (Sp(1) ϕ 0 fundamental group of T n is just K. To compute the action of K on H ∗ (Sp(1)M 0 ), let M 1 ∗ q λ : M → S satisfy λ µS 1 ∈ K, and µ( ⊗ x) ∈ H (Sp(1)0 ) with σ : → M. Let u : q → Sp(1)M 0 be a singular q-simplex and let m : Sp(1) × Sp(1) → Sp(1) be the multiplication. Then we have,

(λ∗ µS 1 · µ( ⊗ x))(u) = ×q m( u, q(ϕ, λ) ◦ (σ, 1))∗ x ∗ x + R∗ u∗ x = ×q (σ, 1)∗ q(ϕ, λ)∗ L u q(ϕ,λ)◦(σ,1)

= ×q u∗ x = µ( ⊗ x)(u). Thus the fundamental group of the base acts trivially on the cohomology of the fiber. Because this fibration has an associated principal fibration with discrete group, all of the higher differentials vanish, and we obtain Theorem 8. Theorem 9 will follow from considerations of a general fiber bundle with structure group S 1 and one computation. Let P → X be a principal S 1 bundle with simply-connected base and let τ : S 1 × F → F be a left action. The Serre spectral sequence of the p,q fibration E = P ×S 1 F → X has E2 = H p (X; H q (F ; R)) and second differential ∗ 1 d2 ω = c1 (P ) ∪ τ ω/[S ]. In our case, the principal bundle is Sp(1) → S 2 . It follows immediately that the coefficient system in the E2 term of the Serre spectral sequence is untwisted, and that the only non-trivial differential is the d2 differential. In this case, the first Chern class is µS 2 . The action that we consider is the map τ : S 1 × (S 2 )M ϕ given by τ (z, u) = zuz∗ . In fact, we only need to consider the effect of this action on terms coming from Sp(1)M 0 . This is because the action is trivial on the classes coming from BK. This can be seen by considering a map from (S 2 )M ϕ to BK. However, the easiest way to see this is first to compute the cohomology of the fiber bundle with fiber Sp(1)M 0 , and then recognize that, up to homotopy, the total space of this bundle is a regular Kcover of FreeMaps(M, S 2 )ϕ . Either way, we need to compute the second differential M ˜ (ϕ, z)u. coming from the action, τ0 : S 1 × Sp(1)M 0 → Sp(1)0 given by τ0 (z, u) = q M Let u : F → Sp(1)0 be a singular chain and compute τ0∗ µ( ⊗ x)/[S 1 ] (u) =

S 1 ××F

∗

u ◦ (σ, prF ) m ◦ (˜q(ϕ ◦ σ, prS 1 ),

x.

Here m : Sp(1) × Sp(1) → Sp(1) is multiplication, and the rest of the maps are as in the definition of µ( ⊗ x) in line (4.1). This vanishes for dimensional reasons when is a 1-cycle (ϕ ◦ σ would push it forward to a 1-cycle in S 2 ). When is a 2-cycle, we use the that q˜ : S 2 × S 1 → Sp(1) has degree two to conclude ∗product rule and the fact 1 ∗ that τ0 µ( ⊗ x)/[S ] = 2ϕ µS 2 []. This completes the proof of our last theorem. Acknowledgement. We would like to thank Louis Crane, Steffen Krusch and Larry Weaver for helpful conversations about particle physics.

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References 1. Aitchison, I.J.R.: Effective Lagrangians and soliton physics I: derivative expansion, and decoupling. Acta Phys. Polon. B18:3, 191–205 (1987) 2. Aitchison, I.J.R.: Berry phases, magnetic monopoles, and Wess-Zumino terms or how the Skyrmion got its spin. Acta Phys. Polon. B18:3, 207–235 (1987) 3. Adams, J.F.: Lectures on exceptional Lie groups. Chicago and London: The University of Chicago Press, 1996 4. Auckly, D., Kapitanski, L.: Holonomy and Skyrme’s Model. Commun. Math. Phys. 240, 97–122 (2003) 5. Auckly, D., Kapitanski, L.: Analysis of the Faddeev Model. Commun. Math. Phys. 256, 611–620 (2005) 6. Auckly, D., Kapitanski, L.: The Pontrjagin-Hopf invariants for Sobolev maps. In progress 7. Balachandran, A., Marmo, G., Skagerstam, B., Stern, A.: Classical topology and quantum states. New Jersey: World Scientific, 1991 8. Balachandran, A., Marmo, G., Simoni, A., Sparno, G.: Quantum bundles and their symmetries. Int. J. Mod. Phys. A 7:8, 1641–1667 (1982) 9. Birrell, N., Davies, P.: Quantum fields in curved space. Cambridge: Cambridge University Press, 1982 ¨ 10. Bopp, F., Haag, Z.: Uber die möglichkeit von Spinmodellen. Zeits. fur Natur. 5a, 644 (1950) 11. Bröcker, Th., tom Dieck, T.: Representation of compact Lie groups. New York: Springer-Verlag, 1985 12. Dirac, P.: Quantized singularities in the electromagnetic field. Proc. Roy. Soc. London A133, 60–72 (1931) 13. Dirac, P.: The theory of magnetic poles. Phys. Rev. 74, 817–830 (1948) 14. Federer, H: A study of function spaces by spectral sequences. Ann. of Math. 61, 340–361 (1956) 15. Finkelstein, D., Rubinstein, J.: Connection between spin statisitics and kinks. J. Math. Phys. 9, 1762–1779 (1968) 16. Giulini, D.: On the possibility of spinorial quantization in the Skyrme model. Mod. Phys. Lett. A8, 1917–1924 (1993) 17. Gottlieb, D.: Lifting actions in fibrations. In: Geometric applications of homotopy theory I. Lecture Notes in Mathematics 657, Berlin-Heidelberg-New york: Springer, 1978, pp. 217–253 18. Halzen, F., Martin, A.: Quarks and Leptons: an introductory course in modern particle physics. New York: John Wiley and Sons, 1984 19. Hattori, A., Yoshida, T.: Lifting group actions in fiber bundles. Japan J. Math. 24, 13–25 (1976) 20. Helgason, S.: Differential geometry, Lie groups and symmetric spaces. Providence, Rhode Island: Amer. Math. Soc., 2001 21. Imbo, T.D., Shah Imbo, C., Sudarshan, E.C.G.: Identical particles, exotic statistics and braid groups. Phys. Lett. B 234, 103–107 (1990) 22. Iwasawa, K.: On some types of topological groups. Ann. of Math. 50:3, 507–558 (1949) 23. Kirby, R.C.: The topology of 4-manifolds. Lecture Notes in Mathematics 1374, New York: SpringerVerlag, 1985 24. Krusch, S.: Homotopy of rational maps and quantization of Skyrmions. Annals Phys. 304, 103–127 (2003) 25. Mimura, M., Toda, H.: Topology of Lie groups, I and II. Transl. Amer. Math. Soc., Providence, Rhode Island: Amer. Math. Soc., 1991 26. Particle data group: Review of particle physics. http://pdg.lbl.gov 27. Pontrjagin, L.: A classification of mappings of the 3-dimensional complex into the 2-dimensional sphere. Mat. Sbornik N.S. 9:51, 331–363 (1941) 28. Ramadas, T.R.: The Wess-Zumino term and fermionic solitons. Commun. Math. Phys. 93, 355–365 (1984) 29. Schulman, L.: A path integral for spin. Phys. Rev. 176:5, 1558–1569 (1968) 30. Simms, D.J., Woodhouse, N.M.J.: Geometric quantization. Lecture Notes in Physics 53, Berlin: Springer-Verlag, 1977 31. Skyrme, T.H.R.: A unified field theory of mesons and baryons. Nucl. Phys. 31, 555–569 (1962) 32. Sorkin, R.: A general relation between kink-exchange and kink-rotation. Commun. Math. Phys. 115, 421–434 (1988) 33. Spanier, E.H.: Algebraic topology. New York: Springer, 1966 34. Steenrod, N.: The topology of fiber bundles. Princeton: Princeton University Press, 1951 35. Witten, E.: Current algebra, baryons and quark confinement. Nucl. Phys. B233, 433–444 (1983) 36. Zaccaria, F., Sudarshan, E., Nilsson, J., Mukunda, N., Marmo, G., Balachandran, A.: Universal unfolding of Hamiltonian systems: from symplectic structures to fibre bundles. Phys. Rev. D27, 2327–2340 (1983) Communicated by L. Takhtajan


Communications in


Unitary Representations of Super Lie Groups and Applications to the Classification and Multiplet Structure of Super Particles C. Carmeli1 , G. Cassinelli1 , A. Toigo1 , V.S. Varadarajan2 1 2

Dipartimento di Fisica, Università di Genova, I.N.F.N., Sezione di Genova, Via Dodecaneso 33, 16146 Genova, Italy. E-mail: [email protected]; [email protected]; [email protected] Department of Mathematics, University of California at Los Angeles, Box 951555, Los Angeles, CA 90095-1555, USA. E-mail: [email protected]

Received: 23 December 2004 / Accepted: 9 May 2005 Published online: 24 January 2006 – © Springer-Verlag 2006

Abstract: It is well known that the category of super Lie groups (SLG) is equivalent to the category of super Harish-Chandra pairs (SHCP). Using this equivalence, we define the category of unitary representations (UR’s) of a super Lie group. We give an extension of the classical inducing construction and Mackey imprimitivity theorem to this setting. We use our results to classify the irreducible unitary representations of semidirect products of super translation groups by classical Lie groups, in particular of the super Poincaré groups in arbitrary dimension and signature. Finally we compare our results with those in the physical literature on the structure and classification of super multiplets. 1. Introduction The classification of free relativistic super particles in SUSY quantum mechanics is well known (see for example [SS74, FSZ81]). It is based on the technique of little (super)groups which, in the classical context, goes back to Wigner and Mackey. However the treatments of this question in the physics literature make the implicit assumption that the technique of little groups remains valid in the SUSY set up without any changes; in particular no attempt is made in currently available treatments to exhibit the SUSY transformations explicitly for the super particle. The aim of this paper is to remedy this situation by laying a precise mathematical foundation for the theory of unitary representations of super Lie groups, and then to apply it to the case of the super Poincaré groups. In the process of doing this we clarify and extend the results in the physical literature to Minkowskian spacetimes of arbitrary dimension D ≥ 4 and N -extended supersymmetry for arbitrary N ≥ 1. Super Lie groups differ from classical Lie groups in a fundamental way: one should think of them as group-valued functors rather than groups. As long as only finite dimensional representations are being considered, there is no difficulty in adapting the functorial theory to study representations. However, this marriage of the functorial

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approach with representation theory requires modifications when one considers infinite dimensional representations. Our entire approach is based on two observations. The first is to view a super Lie group as a Harish-Chandra pair, namely a pair (G0 , ᒄ) where G0 is a classical Lie group, ᒄ is a super Lie algebra which is a G0 -module, Lie(G0 ) = ᒄ0 , and the action of ᒄ0 on ᒄ is the differential of the action of G0 . That this is a valid starting point is justified by the result [DM99] that the category of super Lie groups is equivalent to the category of Harish-Chandra pairs. This point of view leads naturally to define a unitary representation of a super Lie group (G0 , ᒄ) as a classical unitary representation of G0 together with a compatible infinitesimal unitary action of ᒄ. This is in fact very close to the approach of the physicists. The source of our second observation is more technical and is the fact, which is a consequence of the commutation rules, that the operators corresponding to the odd elements of ᒄ are in general unbounded and so care is needed to work with them. Our second observation is in fact a basic result of this paper, namely, that the commutation rules and the symmetry requirements that are implicit in a supersymmetric theory force the unbounded odd operators to be well behaved and lead to an essentially unique way to define a unitary representation of a super Lie group. Of course this aspect was not treated in the physical literature, not only because representations of the big super Lie groups were not considered, but even more, because only finite dimensional super representations of the little groups were considered, where unbounded phenomena obviously do not occur. Our treatment has the additional feature that it is able to handle the construction of super particles with infinite spin also. In the first section of the paper we treat the foundations of the theory of unitary representations of a super Lie group based on these two observations. The basic result is Proposition 2 which asserts that the odd operators are essentially self adjoint on their domain and that the representation of the Harish-Chandra pair, extended to the space of C ∞ vectors of the representation of G0 , is unique. This result is essential to everything we do and shows that the formal aspects of representations of super Lie groups already control all their analytic aspects. The second section discusses the imprimitivity theorem in the super context. Here an important assumption is made, namely that the super homogeneous space on which we have the system of imprimitivity is a purely even manifold, or equivalently, the sub super Lie group (H0 , ᒅ) defining the system of imprimitivity has the same odd dimension as the ambient super Lie group, i.e., ᒅ1 = ᒄ1 . This restriction, although severe, is entirely adequate for treating super Poincaré groups and more generally a wide class of super semi direct products. The main result is Theorem 2 which asserts that the inducing functor is an equivalence of categories from the category of unitary representations of (H0 , ᒅ) to the category of super systems of imprimitivity based on G0 /H0 . These two sections complete the foundational aspects of this paper. The third section is concerned with applications and bringing our treatment as close as possible to the ones in the literature. We consider a super semi direct product of a super translation group with a classical Lie group L0 acting on it. No special assumption is made about the action of L0 on ᒄ1 , so that the class of super Lie groups considered is vastly larger than the ones treated in the physical literature, where this action is always assumed to be spinorial. If T0 is the vector group which is the even part of the super translation group, the theory developed in §§3,4 leads to the result that the irreducible representations of (G0 , ᒄ) are in one-one correspondence with certain L0 -orbits in the dual T0∗ of T0 together with certain irreducible representations of the super Lie groups (little groups) which are stabilizers of the points in the orbits. This is the super version of the classical little group method (Theorem 3).

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It is from this point on that the SUSY theory acquires its own distinctive flavor. In the first place, unlike the classical situation, Theorem 3 stipulates that not all orbits are allowed, only those belonging to a suitable subset T0+ . We shall call these orbits admissible. These are the orbits where the little super group admits an irreducible unitary representation which restricts to a character of T0 . These representations will be called admissible. These orbits satisfy a positivity condition which we interpret as the condition of positivity of energy. This condition is therefore necessary for admissibility. However it requires some effort to show that it is also sufficient for admissibility, and then to determine all the irreducible unitary representations of the little super Lie group at λ (Theorem 4). The road to Theorem 4 is somewhat complicated. Let λ ∈ T0∗ be fixed. The classical stabilizer of λ is T0 Lλ0 , where Lλ0 is the classical little group at λ, namely the stabilizer of λ in L0 . The super stabilizer of λ is the super Lie group S λ defined by S λ = (T0 Lλ0 , ᒄλ ),

ᒄλ = ᒑ0 ⊕ ᒉ0λ ⊕ ᒄ1 .

(By convention, the Lie algebra of a Lie group is denoted by the corresponding gothic letter.) Given λ, there is an associated Lλ0 -invariant quadratic form λ on ᒄ1 which will be nonnegative definite; the nonnegativity of λ is the positive energy condition mentioned earlier. This form need not be strictly positive, but one can pass to the quotient ᒄ1λ of ᒄ1 by its radical, and obtain a positive definite quadratic vector space on which the classical part Lλ0 of the little group operates. So we obtain a map jλ : Lλ0 −→ O(ᒄ1λ ). Now Lλ0 need not be connected and so jλ need not map into SO(ᒄ1λ ). Also, since we are not making any assumption about the action of L0 on ᒄ1 , it is quite possible that dim(ᒄ1λ ) could be odd. We introduce the subgroup j −1 (SO(ᒄ1λ )) if jλ (Lλ0 ) ⊂ SO(ᒄ1λ ) and dim(ᒄ1λ ) is even λ L00 = λλ L0 otherwise. Then Lλ00 is either the whole of Lλ0 or a (normal) subgroup of index 2 in it. Theorem 4 asserts that there is a functorial map r −→ θrλ which is an equivalence of categories from the category of unitary projective representations r of Lλ00 corresponding to a certain canonical multiplier µλ , to the category of admissible unitary representations of the super group S λ . In particular, the admissible irreducible unitary representations of S λ correspond one-one to irreducible unitary µλ -representations of Lλ00 . We shall now explain how this correspondence is set up. The odd operators of the super representation of the little group are obtained from a self adjoint representation of the Clifford algebra of ᒄ1λ . Since any such representation is a multiple of an essentially unique one, we begin with an irreducible self adjoint representation. Let us call it τλ . In the space of τλ one can define in an essentially unique manner a projective representation κλ of Lλ0 that intertwines τλ and its transforms by elements of Lλ0 : κλ (t)τλ (X)κλ (t)−1 = τλ (tX).

(∗)

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The class of the multiplier µλ in H 2 (Lλ0 , T) is then uniquely determined. We can normalize µλ so that it takes only ±1-values. Starting from any unitary µλ -representation r of Lλ00 we build the µλ -representation Ind(r) of Lλ0 induced by r. Then the admissible unitary representation θrλ of S λ corresponding to r is given by θrλ = eiλ(t) Ind(r) ⊗ κλ , 1 ⊗ τλ ; the fact that µλ is ±1-valued implies that θrλ is an ordinary rather than a projective representation. The representation rλ of the super Lie group (G0 , ᒄ) induced by θrλ from S λ is then irreducible if r is irreducible. Theorem 5 asserts that all the irreducible unitary representations of the super Lie group are parametrized bijectively as above by the irreducible representations of Lλ00 , thus giving the super version of the classical theory. The representation κλ is therefore at the heart of the theory of irreducible unitary representations of super semi direct products. It is finite dimensional; in fact it is the lift via jλ to Lλ0 of the spin representation of the quadratic vector space. It is dependent only on λ. Clearly, the representation Ind(r) ⊗ κλ of Lλ0 will not in general be irreducible even if r is. The irreducible constituents of Ind(r)⊗κλ then define, via the classical procedure of inducing, irreducible unitary representations of G0 which are the constituents of the even part of the full super representation. This family of irreducible representations of G0 is the multiplet of the irreducible representation of the super semi direct product in question. If µλ is trivial, we can choose r to be trivial, and the corresponding multiplet is the fundamental multiplet. Thus κλ determines the entire correspondence in a simple manner. In §4.3 we discuss the case of the super Poincaré groups. We consider spacetimes of Minkowski signature and of arbitrary dimension D ≥ 4 together with N -extended supersymmetry for arbitrary N ≥ 1. In this case, the groups Lλ0 are all connected and Lλ0 = Lλ00 . Moreover, the multiplier µλ becomes trivial, so that κλ becomes an ordinary representation (Lemma 13). Hence θrλ = eiλ(t) r ⊗ κλ , 1 ⊗ τλ . Thus in this case we finally reach the conclusion that the super particles are parametrized by the admissible orbits and irreducible unitary representations of the stabilizers of the classical little groups, exactly as in the classical theory. The positive energy condition λ ≥ 0 becomes just that λ, which we replace by p to display the fact that it is a momentum vector, lies in the closure of the forward light cone. Thus the orbits of imaginary mass are excluded by supersymmetry (Theorem 6). Our approach enables us to handle super particles with infinite spin in the same manner as those with finite spin because of the result that the odd operators of the little group are bounded (Lemma 10). Finally, in §4.4 we give the explicit determination of κp when the spacetime has arbitrary dimension D ≥ 4 and we have N -extended supersymmetry. The results in the physical literature are in general only for D = 4. The literature on supersymmetric representation theory is very extensive. We have been particularly influenced by [SS74, Fer01, Fer03, FSZ81]. In [DP85, DP86, DP87]


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Dobrev et al. discuss representations of superconformal groups induced from a maximal parabolic subgroup. The papers [SS74] gave the structure of κp for N = 1 supersymmetry, while [FSZ81] gave a very complete treatment of the structure of κp in the case of extended supersymmetry when D = 4, including the case of central charges. 2. Super Lie Groups and Their Unitary Representations 2.1. Super Hilbert spaces. All sesquilinear forms are linear in the first argument and conjugate linear in the second. We use the usual terminology of super geometry as in [DM99, Var04]. A super Hilbert space (SHS) is a super vector space H = H0 ⊕ H1 over C with a scalar product (· , ·) such that H is a Hilbert space under (· , ·), and Hi (i = 0, 1) are mutually orthogonal closed linear subspaces. If we define   if x and y are of opposite parity 0 x, y = (x, y) if x and y are even  i(x, y) if x and y are odd then x, y is an even super Hermitian form with y, x = (−1)p(x)p(y) x, y , x, x > 0(x = 0 even), i −1 x, x > 0(x = 0 odd). If T (H → H) is a bounded linear operator, we denote by T ∗ its Hilbert space adjoint and by T † its super adjoint given by T x, y = (−1)p(T )p(x) x, T † y . Clearly T † is bounded, p(T ) = p(T † ), and T † = T ∗ or −iT ∗ according as T is even or odd. For unbounded T we define T † in terms of T ∗ by the above formula. These definitions are equally consistent if we use −i in place of i. But our convention is as above. 2.2. SUSY quantum mechanics. In SUSY quantum mechanics in a SHS H, it is usual to stipulate that the Hamiltonian H = X 2 , where X is an odd operator [Wit82]; it is customary to argue that this implies that H ≥ 0 (positivity of energy); but this is true only if we know that H is essentially self adjoint on the domain of X 2 . We shall now prove two lemmas which clarify this situation and will play a crucial role when we consider systems with a super Lie group of symmetries. If A is a linear operator on H, we denote by D (A) its domain. We always assume that D (A) is dense in H, and then refer to it as densely defined. We write A ≺ B if D(A) ⊂ D(B) and B restricts to A on D(A); A is symmetric iff A ≺ A∗ , and then A has a closure A. A ≺ B ⇒ B ∗ ≺ A∗ . If A is symmetric we say that it is essentially self adjoint if A is self adjoint; in this case A∗ = A. If A is symmetric and B is a symmetric extension of A, then A ≺ B ≺ A∗ ; in fact A ≺ A∗ and A ≺ B ≺ B ∗ , and so B ∗ ≺ A∗ and A ≺ B ≺ B ∗ ≺ A∗ . If A is self adjoint and L ⊂ D(A), we say that L is a core for A if A is the closure of its restriction to L. A vector ψ ∈ H is analytic for a symmetric operator H if ψ ∈ D(H n ) for all n and the series n t n (n!)−1 ||H n ψ|| < ∞ for some t > 0. It is a well known result of Nelson [Nel59] that if D ⊂ D(H ) and contains a dense set of analytic vectors, then H is essentially self adjoint on D. In this case ψ ∈ D(H ) is analytic for H if and only if t −→ eitH ψ is analytic in t ∈ R. If A is self adjoint, then A2 , defined on the domain D(A2 ) = {ψ ψ, Aψ ∈ D(A)}, is self adjoint; this is well known and follows easily from the spectral theorem.

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Lemma 1. Let H be a self adjoint operator on H and U (t) = eitH the corresponding one parameter unitary group. Let B ⊂ D(H ) be a dense U -invariant linear subspace. We then have the following: (i) B is a core for H . (ii) Let X be a symmetric operator with B ⊂ D(X) such that XB ⊂ D(X) and 2 X2 B = H|B . Then X|B is essentially self adjoint, X|B = X and X = H . In particular, H ≥ 0, D(H ) ⊂ D(X). Finally, these results are valid if we only assume that B is invariant under H and contains a dense set of analytic vectors. Proof. Let H1 = H B . We must show that if L(λ)(λ ∈ C) is the subspace of ψ such that H1∗ ψ = λψ, then L(λ) = 0 if (λ) = 0. Now L(λ) is a closed subspace. Moreover, as H1 is invariant under U , so is H1∗ and so L(λ) is invariant under U also. So the vectors in L(λ) that are C ∞ for U are dense in L(λ) and so it is enough to prove that 0 is the only C ∞ vector in L(λ). But H = H ∗ ≺ H1∗ while the C ∞ vectors for U are all in D(H ), and so if ψ is a C ∞ vector in L(λ), H ψ = H1∗ ψ = λψ. This is a contradiction since H is self adjoint and so all its eigenvalues are real. This proves (i). Let X1 = X B . Clearly, X1 is symmetric on B. It is enough to show that X1 is 2

essentially self adjoint and X1 = H , since in this case X1 ≺ X ≺ X1∗ = X1 and hence X = X1 . We have X12 = H1 . So H ≥ 0 on B and hence H ≥ 0 by (i). Again it is a question of showing that for λ ∈ C with (λ) = 0, we must have M(λ) = 0, where M(λ) is the eigenspace for X1∗ for the eigenvalue λ. Let ψ ∈ M(λ). Now, for ϕ ∈ B, (X12 ϕ, ψ) = (X1 ϕ, X1∗ ψ) = λ(X1 ϕ, ψ) = λ (ϕ, ψ) = (ϕ, λ2 ψ). 2

Hence ψ ∈ D((X12 )∗ ) and (X12 )∗ ψ = λ2 ψ. But X12 = H1 and so (X12 )∗ = H1∗ = H by (i). So H ψ = λ2 ψ. Hence λ2 is real and ≥ 0. This contradicts the fact that (λ) = 0. 2 2 2 Furthermore, X12 ≺ X1 and so X1 = (X1 )∗ ≺ (X12 )∗ = H1∗ = H . On the other 2

2

2

2

hand, as X1 is closed, H = H 1 = X12 ≺ X1 . So H = X1 = X . This means that D(H ) ⊂ D(X). Finally, let us assume that H B ⊂ B and that B contains a dense set of analytic vectors for H . Clearly B is a core for H . If ψ ∈ B is analytic for H we have X 2n ψ = H n ψ ∈ B and X2n+1 ψ ∈ D(X) by assumption, and ||X n ψ||2 = |(H n ψ, ψ)| ≤ ||ψ||||H n ψ|| ≤ M n n! for some M > 0 and all n. Thus ψ is analytic for X and so its essential self adjointness is a consequence of the theorem of Nelson. The rest of the argument is unchanged. Lemma 2. Let A be a self adjoint operator in H. Let M be a smooth (resp. analytic) manifold and f (M −→ H) a smooth (resp. analytic) map. We assume that (i) f (M) ⊂ D(A2 ) and (ii) A2 f : m −→ A2 f (m) is a smooth (resp. analytic) map of M into H. Then Af : m −→ Af (m) is a smooth (resp. analytic) map of M into H. Moreover, if E is any smooth differential operator on M, (Ef )(m) ∈ D(A2 ) for all m ∈ M, and E(A2 f ) = A2 Ef,

E(Af ) = AEf.


223

Proof. It is standard that if g(M −→ H) is smooth (resp. analytic) and L is a bounded linear operator on H, then Lg is a smooth (resp. analytic) map. We have Aψ = A(I + A2 )−1 (I + A2 )ψ,

ψ ∈ D(A2 ).

Moreover A(I + A2 )−1 is a bounded operator. Now (I + A2 )f is smooth (resp. analytic) and so it is immediate from the above that Af is smooth (resp. analytic). For the last part we assume that M is an open set in Rn since the result is clearly local. Let x i (1 ≤ i ≤ n) be the coordinates and let ∂j = ∂/∂x j , ∂ α = ∂1α1 . . . ∂nαn , where α = (α1 , . . . , αn ). It is enough to prove that (∂ α f )(m) ∈ D(A2 ),

A2 ∂ α f = ∂ α (A2 f ),

A∂ α f = ∂ α (Af ).

We begin with a simple observation. Since (A2 ψ, ψ) = ||Aψ||2 for all ψ ∈ D(A2 ), it follows that whenever ψn ∈ D(A2 ) and (ψn ) and (A2 ψn ) are Cauchy sequences, then (Aψn ) is also a Cauchy sequence; moreover, if ψ = limn ψn , then ψ ∈ D(A2 ) and A2 ψ = limn A2 ψn , Aψ = limn Aψn . This said, we shall prove the above formulae by induction on |α| = α1 + · · · + αn . Assume them for |α| ≤ r and fix j, 1 ≤ j ≤ n. Let g = ∂ α f, |α| = r. Let gh (x) =

1 1 g(x , . . . , x j + h, . . . , x n ) − g(x 1 , . . . , x n ) h

(h is in j th place).

Then, as h → 0, gh (x) → ∂j ∂ α f (x) while A2 gh (x) = (∂ α A2 f )h (x) → ∂j ∂ α A2 f (x), and Agh (x) = (∂ α Af )h (x) → ∂j ∂ α Af (x). From the observation made above we have ∂j ∂ α f (x) ∈ D(A2 ) and A2 and A map it respectively into ∂j ∂ α A2 f (x) and ∂j ∂ α Af (x). Definition 1. For self adjoint operators L, X with L bounded, we write L ↔ X to mean that L commutes with the spectral projections of X. Lemma 3. Let X be a self adjoint operator and B a dense subspace of H which is a core for X such that XB ⊂ B. If L is a bounded self adjoint operator such that LB ⊂ B, then the following are equivalent: (i) LX = XL on B (ii) LX = XL on D(X) (this carries with it the inclusion LD(X) ⊂ D(X)) (iii) L ↔ X. In this case eitL X = XeitL for all t ∈ R. Proof. (i) ⇐⇒ (ii). Let b ∈ D(X). Then there is a sequence (bn ) in B such that bn → b, Xbn → Xb. Then XLbn = LXbn → LXb. Since Lbn → Lb we infer that Lb ∈ D(X) and XLb = LXb. This proves (i) ⇒ (ii). The reverse implication is trivial. n itL Xb = (ii) ⇒ (iii). We have Ln Xb = XL bn for alln b ∈ D(X), nitL≥ 1. So e itL n n b, XsN → e Xb. n ((it) /n!)XL b. If sN = n≤N ((it) /n!)L b, then sN → e So eitL b ∈ D(X) and XeitL b = eitL Xb. If U (t) = eitL , this means that U (t)XU (t)−1 = X and so, by the uniqueness of the spectral resolution of X, U (t) commutes with the spectral projections of X. But then L ↔ X. (iii) ⇒ (i). Under (iii) we have U (t)XU (t)−1 = X or XU (t)b = U (t)Xb for b ∈ D(X), U (t)b being in D(X) for all t. Let bt = (it)−1 (U (t)b − b). Then Xbt = (it)−1 (U (t)Xb − Xb). Hence, as t → 0, bt → Lb while Xbt → LXb. Hence Lb ∈ D(X) and XLb = LXb. Thus we have (ii), hence (i).

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2.3. Unitary representations of super Lie groups. We take the point of view [DM99] that a super Lie group (SLG) is a super Harish-Chandra pair (G0 , ᒄ) that is a pair (G0 , ᒄ), where G0 is a classical Lie group, ᒄ is a super Lie algebra which is a G0 -module, Lie(G0 ) = ᒄ0 , and the action of ᒄ0 on ᒄ is the differential of the action of G0 . The notion of morphisms between two super Lie groups in the above sense is the obvious one from which it is easy to see what is meant by a finite dimensional representation of a SLG (G0 , ᒄ): it is a triple (π0 , π, V ), where π0 is an even representation of G0 in a super vector space V of finite dimension over C, i.e., a representation such that π0 (g) is even for all g ∈ G0 ; π is a representation of the super Lie algebra ᒄ in V such that π ᒄ = dπ0 ; and 0

π(gX) = π0 (g)π(X)π0 (g)−1 ,

g ∈ G 0 , X ∈ ᒄ1 .

If V is a SHS and π(X)† = −π(X) for all X ∈ ᒄ, we say that (π0 , π, V ) is a unitary representation (UR) of the SLG (G0 , ᒄ). The condition on π is equivalent to saying that π0 is a unitary representation of G0 in the usual sense and π(X)∗ = −iπ(X) for all X ∈ ᒄ1 . It is then clear that a finite dimensional UR of (G0 , ᒄ) is a triple (π0 , π, V ), where (a) π0 is an even unitary representation of G0 and is a SHS V ; (b) π is a linear map of ᒄ1 into the space ᒄᒉ(V )1 of odd endomorphisms of V with π(X)∗ = −iπ(X) for all X ∈ ᒄ1 ; (c) dπ0 ([X, Y ]) = π(X)π(Y ) + π(Y )π(X) for X, Y ∈ ᒄ1 ; (d) π(g0 X) = π0 (g0 )π(X)π0 (g0 )−1 for X ∈ ᒄ1 , g0 ∈ G0 . Let ζ = e−iπ/4 ,

ρ(X) = ζ π(X).

Then, we may replace π(X) by ρ(X) for X ∈ ᒄ1 ; the condition (b) becomes the requirement that ρ(X) is self adjoint for all X ∈ ᒄ1 , while the commutation rule in condition (c) becomes −idπ0 ([X, Y ]) = ρ(X)ρ(Y ) + ρ(Y )ρ(X),

X, Y ∈ ᒄ1 .

If we want to extend this definition to the infinite dimensional context it is necessary to take into account the fact that the π(X) for X ∈ ᒄ1 will in general be unbounded; indeed, from (c) above we find that dπ0 ([X, X]) = 2π(X)2 , and as dπ0 typically takes elements of ᒄ0 into unbounded operators, the π(X) cannot be bounded. So the concept of a UR of a SLG in the infinite dimensional case must be formulated with greater care to take into account the domains of definition of the π(X) for X ∈ ᒄ1 . In the physics literature this aspect is generally ignored. We shall prove below that contrary to what one may expect, the domain restrictions can be formulated with great freedom, and the formal and rigorous pictures are essentially the same. In particular, the concept of a UR of a super Lie group is a very stable one and allows great flexibility of handling. If V is a super vector space (not necessarily finite dimensional), we write End(V ) for the super algebra of all endomorphisms of V . If π0 is a unitary representation of G0 in a Hilbert space H, we write C ∞ (π0 ) for the subspace of differentiable vectors in H for π0 . We denote by C ω (π0 ) the subspace of analytic vectors of π0 . Here we recall that a vector v ∈ H is called a differentiable vector (resp. analytic vector) for π0 if the map g → π0 (g)v is smooth (resp. analytic). If H is a SHS and π0 is even, then C ∞ (π0 ) and C ω (π0 ) are π0 -invariant dense linear super subspaces. We also need the following fact


225

which is standard but we shall give a partial proof because the argument will be used again later. Lemma 4. C ∞ (π0 ) and C ω (π0 ) are stable under dπ0 (ᒄ0 ). For any Z ∈ ᒄ0 , idπ0 (Z) is essentially self adjoint both on C ∞ (π0 ) and on C ω (π0 ); moreover, for any Z1 , . . . , Zr ∈ ᒄ0 and ψ ∈ C ∞ (π0 ), (resp. ψ ∈ C ω (π0 )) the map g −→ dπ0 (Z1 ) . . . dπ0 (Zr )π0 (g)ψ is C ∞ (resp. analytic). Proof. We prove only the second statement. That idπ0 (Z) is essentially self adjoint on C ∞ (π0 ) and on C ω (π0 ) is immediate from Lemma 1. Using the adjoint representation we have, for any Z ∈ ᒄ0 , gZ = i ci (g)Wi for g ∈ G0 , where the ci are analytic functions on G0 and Wi ∈ ᒄ0 . Hence, as dπ0 (Z)π0 (g) = π0 (g)dπ0 (g −1 Z), we can write dπ0 (Z1 ) · · · dπ0 (Zr )π0 (g)ψ as a linear combination with analytic coefficients of π0 (g)dπ0 (R1 ) · · · dπ0 (Rr )ψ for suitable Rj ∈ ᒄ0 . The result is then immediate. Definition 2. A unitary representation (UR) of a SLG (G0 , ᒄ) is a triple (π0 , ρ, H), H a SHS, with the following properties: (a) π0 is an even UR of G0 in H; (b) ρ(X −→ ρ(X)) is a linear map of ᒄ1 into End(C ∞ (π0 ))1 such that (i) ρ(g0 X) = π0 (g0 )ρ(X)π0 (g0 )−1 (X ∈ ᒄ1 , g0 ∈ G0 ), (ii) ρ(X) with domain C ∞ (π0 ) is symmetric for all X ∈ ᒄ1 , (iii) −idπ0 ([X, Y ]) = ρ(X)ρ(Y ) + ρ(Y )ρ(X) (X, Y ∈ ᒄ1 ) on C ∞ (π0 ). Proposition 1. If (π0 , ρ, H) is a UR of the SLG (G0 , ᒄ), then ρ(X) with domain C ∞ (π0 ) is essentially self adjoint for all X ∈ ᒄ1 . Moreover π : X0 + X1 −→ dπ0 (X0 ) + ζ −1 ρ(X1 )

(Xi ∈ ᒄi )

is a representation of ᒄ in C ∞ (π0 ). Proof. Let Z = (1/2)[X, X]. We apply Lemma 1 with U (t) = π0 (exp tZ) = eitH , B = C ∞ (π0 ). Then H = −idπ0 (Z) = ρ(X)2 on C ∞ (π0 ). We conclude that H and ρ(X) are 2 essentially self adjoint on C ∞ (π0 ) and that H = ρ(X) ; in particular, H ≥ 0. For the second assertion the only non-obvious statement is that for Z ∈ ᒄ0 , X ∈ ᒄ1 , ψ ∈ C ∞ (π0 ), ρ([Z, X])ψ = dπ0 (Z)ρ(X)ψ − ρ(X)dπ0 (Z)ψ. Let gt = exp(tZ) and let (Xk ) be a basis for ᒄ1 . Then gX = k ck (g)Xk , where the ck are smooth functions on G0 . So

π0 (gt )ρ(X)ψ = ρ(gt X)π0 (gt )ψ = ck (gt )ρ(Xk )π0 (gt )ψ. k

Now g −→ π0 (g)ψ is a smooth map into C ∞ (π0 ). On the other hand, if Hk = −(1/2)[Xk , Xk ], we have idπ0 (Hk ) = ρ(Xk )2 on C ∞ (π0 ), so π0 (g)ψ ∈ D(ρ(Xk )2 ), and by Lemma 4, ρ(Xk )2 π0 (g)ψ = idπ0 (Hk )π0 (g)ψ is smooth in g. Lemma 2 now applies and shows that ρ(Xk )π0 (gt )ψ is smooth in t and

d dt

ρ(Xk )π0 (gt )ψ = ρ(Xk )dπ0 (Z)ψ. t=0

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Hence dπ0 (Z)ρ(X)ψ =

ck (1)ρ(Xk )dπ0 (Z)ψ +

k

Since [Z, X] =

(Zck )(1)ρ(Xk )ψ.

k

(Zck )(1)Xk , k

X=

ck (1)Xk

k

the right side is equal to ρ(X)dπ0 (Z)ψ + ρ([Z, X])ψ. Remark 1. For Z such that exp tZ represents time translation, H is the energy operator, and so is positive in the supersymmetric model. We shall now show that one may replace C ∞ (π0 ) by a more or less arbitrary domain without changing the content of the definition. This shows that the concept of a UR of a SLG is a viable one even in the infinite dimensional context. Let us consider a system (π0 , ρ, B, H) with the following properties: (a) B is a dense super linear subspace of H invariant under π0 and B ⊂ D(dπ0 (Z)) for all Z ∈ [ᒄ1 , ᒄ1 ]; (b) (ρ(X))X∈ᒄ1 is a set of linear operators such that: (i) ρ(X) is symmetric for all X ∈ ᒄ1 , (ii) B ⊂ D(ρ(X)) for all X ∈ ᒄ1 , (iii) ρ(X)Bi ⊂ Hi+1 (mod 2) for all X ∈ ᒄ1 , (iv) ρ(aX + bY ) = aρ(X) + bρ(Y ) on B for X, Y ∈ ᒄ1 and a, b scalars, (v) π0 (g)ρ(X)π0 (g)−1 = ρ(gX) on B for all g ∈ G0 , X ∈ ᒄ1 , (vi) ρ(X)B ⊂ D(ρ(Y )) for all X, Y ∈ ᒄ1 , and −idπ0 ([X, Y ]) = ρ(X)ρ(Y ) + ρ(Y )ρ(X) on B. Proposition 2. Let (π0 , ρ, B, H) be as above. We then have the following: (a) For any X ∈ ᒄ1 , ρ(X) is essentially self adjoint and C ∞ (π0 ) ⊂ D(ρ(X)). (b) Let ρ(X) = ρ(X) C ∞ (π ) for X ∈ ᒄ1 . Then (π0 , ρ, H) is a UR of the SLG (G0 , ᒄ). 0

, ρ , H)

If (π0 is a UR of the SLG (G0 , ᒄ), such that B ⊂ D(ρ (X)) and ρ (X) restricts to ρ(X) on B for all X ∈ ᒄ1 , then ρ = ρ. Proof. Let X ∈ ᒄ1 . By assumption B is invariant under the one parameter unitary group generated by H = −(1/2)idπ0 ([X, X]) while H = ρ(X)2 on B. So, by Lemma 1, ρ(X) is essentially self adjoint, ρ(X) = ρ(X)|B , H = (ρ(X))2 , and D(H ) ⊂ D(ρ(X)). Since C ∞ (π0 ) ⊂ D(H ), we have proved (a). Let us now prove (b). If a is scalar and X ∈ ᒄ, ρ(aX) = aρ(X) follows from item (iv) and the fact that ρ(X) = ρ(X)|B . For the additivity of ρ, let X, Y ∈ ᒄ1 . Then ρ(X + Y ) is essentially self adjoint and its closure restricts to ρ(X + Y ) on C ∞ (π0 ). Then, viewing ρ(X) + ρ(Y ) as a symmetric operator defined on the intersection of the domains of the two operators (which includes C ∞ (π0 )), we see that ρ(X) + ρ(Y ) is


227

a symmetric extension of ρ(X)|B + ρ(Y )|B = ρ(X + Y )|B . But as ρ(X + Y )|B is essentially self adjoint, we have, by the remark made earlier, ρ(X) + ρ(Y ) ≺ ρ(X + Y )|B = ρ(X + Y ). Restricting both of these operators to C ∞ (π0 ) we find that ρ(X + Y ) = ρ(X) + ρ(Y ). From the relation π0 (g)ρ(X)π0 (g)−1 = ρ(gX) on B follows π0 (g)ρ(X)π0 (g)−1 = ρ(gX). The key step is now to prove that for any X ∈ ᒄ1 , ρ(X) maps C ∞ (π0 ) into itself. Fix X ∈ ᒄ1 , ψ ∈ C ∞ (π0 ). Now π0 (g)ρ(X)ψ = ρ(gX)π0 (g)ψ.

So, writing gX = k ck (g)Xk , where the ck are smooth functions on G0 and the (Xk ) a basis for ᒄ1 , we have, remembering the linearity of ρ on C ∞ (π0 ),

π0 (g)ρ(X)ψ = ck (g)ρ(Xk )π0 (g)ψ. k

It is thus enough to show that g −→ ρ(Xk )π0 (g)ψ is smooth. If Hk = −[Xk , Xk ]/2, we 2 know from Lemma 4 that π0 (g)ψ lies in D(Hk ) and idπ0 (Hk )π0 (g)ψ = ρ(Xk ) π0 (g)ψ is smooth in g. Lemma 2 now shows that ρ(Xk )π0 (g)ψ is smooth in g. It remains only to show that for X, Y ∈ ᒄ1 we have −idπ0 ([X, Y ]) = ρ(X)ρ(Y ) + ρ(Y )ρ(X) on C ∞ (π0 ). But, the right side is ρ(X + Y )2 − ρ(X)2 − ρ(Y )2 while the left side is the restriction of (−i/2)dπ0 ([X + Y, X + Y ]) + (i/2)dπ0 ([X, X]) + (i/2)dπ0 ([Y, Y ]) to C ∞ (π0 ), and so we are done. We must show the uniqueness of ρ. Let ρ have the required properties also. Then ρ (X) is essentially self adjoint on C ∞ (π0 ) and B is a core for its closure, by Lemma 1. Hence ρ (X) = ρ(X). The proof is complete. We shall now prove a variant of the above result involving analytic vectors. Proposition 3. (i) If (π0 , ρ, H) is a UR of the SLG (G0 , ᒄ), then ρ(X) maps C ω (π0 ) into itself for all X ∈ ᒄ1 , so that π, as in Proposition (1), is a representation of ᒄ in C ∞ (π0 ). (ii) Let G0 be connected. Let π0 be an even unitary representation of G0 and B ⊂ C ω (π0 ) a dense linear super subspace. Let π be a representation of ᒄ in B such that π(Z) ≺ dπ0 (Z) for Z ∈ ᒄ0 and ρ(X) = ζ π(X) is symmetric for X ∈ ᒄ1 . Then, for each X ∈ ᒄ1 , ρ(X) is essentially self adjoint on B and C ∞ (π0 ) ⊂ D(ρ(X)). If ρ(X) is the restriction of ρ(X) to C ∞ (π0 ), then (π0 , ρ, H) is a UR of the SLG (G0 , ᒄ) and is the unique one in the following sense: if (π0 , ρ , H) is a UR with B ⊂ D(ρ (X)) and ρ (X) |B = ρ(X) for all X ∈ ᒄ1 , then ρ = ρ. Proof. (i) This is proved as its C ∞ analogue in the proof of Proposition 2, using the analytic parts of Lemmas 2 and 4. (ii) The proof that ρ(X) for X ∈ ᒄ1 is essentially self adjoint with D(ρ(X)) ⊃ C ∞ (π0 ) follows as before from (the analytic part of) Lemma 1. The same goes for the linearity of ρ.

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We shall now show that for X ∈ ᒄ1 , g ∈ G0 ,

π0 (g −1 )ρ(X)π0 (g) = ρ(g −1 X).

(1)

Write g −1 X = k ck (g)Xk , where (Xk ) is a basis for ᒄ1 and the ck are analytic functions on G0 . We begin by showing that for all ψ ∈ B, ρ(X)π0 (g)ψ = π0 (g)ρ(g −1 X)ψ. Now π0 (g)ρ(g −1 X)ψ =

(2)

ck (g)π0 (g)ρ(Xk )ψ.

k

We argue as in Proposition 2 to conclude, using Lemmas 2 and 4, that the function ρ(X)π0 (g)ψ is analytic in g and its derivatives can be calculated explicitly. It is also clear that the right side is analytic in g since ρ(Xk )ψ ∈ B for all k. So, as G0 is connected, it is enough to prove that the two sides in (2) have all derivatives equal at g = 1. This comes down to showing that for any integer n ≥ 0 and any Z ∈ ᒄ0 ,

n ρ(X)dπ0 (Z)n ψ = (Z r ck )(1)dπ0 (Z)n−r ρ(Xk )ψ. (3) r k,r

Let λ be the representation of G0 on ᒄ1 and write λ again for dλ. Then, taking gt = exp(tZ),

λ(gt−1 )(X) = ck (gt )Xk , k

from which we get, on differentiating n times with respect to t at t = 0,

λ(−Z)r (X) = (Z r ck )(1)Xk . k

Hence the right side of (3) becomes

n dπ0 (Z)n−r ρ(λ(−Z)r (X))ψ. r r On the other hand, from the fact that π is a representation of ᒄ in B we get ρ(X)dπ0 (Z) = dπ0 (Z)ρ(X) + ρ(λ(−Z)(X)) on B. Iterating this we get, on B, ρ(X)dπ0 (Z)n =

n r

r

dπ0 (Z)n−r ρ(λ(−Z)r (X))

which gives (2). But then (1) follows from (2) by a simple closure argument. Using (1), the proof that ρ(X) maps C ∞ (π0 ) into itself is the same as Proposition 2. The proof of the relation −idπ0 ([X, Y ]) = ρ(X)ρ(Y ) + ρ(Y )ρ(X) for X, Y ∈ ᒄ1 is also the same. The proof is complete.


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2.4. The category of unitary representations of a super Lie group. If = (π0 , ρ, H) and = (π0 , ρ , H ) are two UR’s of a SLG (G0 , ᒄ), a morphism A : −→ is an even bounded linear operator from H to H such that A intertwines π0 , ρ and π0 , ρ ; notice that as soon as A intertwines π0 and π0 , it maps C ∞ (π0 ) into C ∞ (π0 ), and so the requirement that it intertwine ρ and ρ makes sense. An isomorphism is then a morphism A such that A−1 is a bounded operator; in this case A is a linear isomorphism of C ∞ (π0 ) with C ∞ (π0 ) intertwining ρ and ρ . If A is unitary we then speak of unitary equivalence of and . It is easily checked that equivalence implies unitary equivalence, just as in the classical case. is a subrepresentation of if H is a closed graded subspace of H invariant under π0 and ρ, and π0 (resp. ρ ) is the restriction of π0 (resp. ρ) to H (resp. C ∞ (π0 ) ∩ H ). The UR is said to be irreducible if there is no proper nonzero closed graded subspace H that defines a subrepresentation. If is a nonzero proper subrepresentation of , and H is H ⊥ , it follows from the self adjointness of ρ(X) for X ∈ ᒄ1 that H ∩ C ∞ (π0 ) is invariant under all ρ(X)(X ∈ ᒄ1 ); then the restrictions of π0 , ρ to H define a subrepresentation such that = ⊕ in an obvious manner. Lemma 5. is irreducible if and only if Hom(, ) = C. Proof. If splits as above, then the orthogonal projection H −→ H is a nonscalar element of Hom(, ). Conversely, suppose that is irreducible and A ∈ Hom(, ). Then A∗ ∈ Hom(, ) also and so, to prove that A is a scalar we may suppose that A is self adjoint. Let P be the spectral measure of A. Clearly all the P (E) are even. Then P commutes with π0 and so P (E) leaves C ∞ (π0 ) invariant for all Borel sets E. Moreover, by Lemma 3, the relation Aρ(X) = ρ(X)A on C ∞ (π0 ) implies that P (E) ↔ ρ(X) for all E and x ∈ ᒄ1 , and hence that P (E)ρ(X) = ρ(X)P (E) on C ∞ (π0 ) for all E, X. The range of P (E) thus defines a subrepresentation and so P (E) = 0 or I . Since this is true for all E, A must be a scalar. Lemma 6. Let (R0 , ᒏ) be a SLG and (θ, ρ θ ) a UR of it, in a Hilbert space L. Let P X be the spectral measure of ρ θ (X), X ∈ ᒏ1 . Then the following properties of a closed linear subspace M of L are equivalent: (i) M is stable under θ and M∞ = C ∞ (θ ) ∩ M is stable under all ρ θ (X), (X ∈ ᒏ1 ) (ii) M is stable under θ and all the spectral projections PFX (Borel F ⊂ R). In particular, (θ, ρ θ ) is irreducible if and only if L is irreducible under θ and all PFX . Proof. Follows from Lemma 3 applied to the orthogonal projection L : L −→ M. Indeed, suppose that M is a closed linear subspace of L stable under θ . Then L maps L∞ onto M∞ . By Lemma 3 L commutes with ρ θ (X) on L∞ if and only if L ↔ ρ θ (X); this is the same as saying that P X stabilizes M. 3. Induced Representations of Super Lie Groups, Super Systems of Imprimitivity, and the Super Imprimitivity Theorem 3.1. Smooth structure of the classical induced representation and its system of imprimitivity. Let G0 be a unimodular Lie group and H0 a closed Lie subgroup. We write = G0 /H0 and assume that has a G0 -invariant measure; one can easily modify the treatment below to avoid these assumptions. We write x → x for the natural map from G0 to and dx for a choice of the invariant measure on . For any UR σ of H0 in a Hilbert space K one has the representation π of G0 induced by σ . One may

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take π as acting in the Hilbert space H of (equivalence classes) of Borel functions f from G0 to K such that (i) f (xξ ) = σ (ξ )−1 f (x) for all x ∈ G0 , ξ ∈ H0 , and (ii) ||f ||2H := |f (x)|2K dx < ∞. Here |f (x)|K is the norm of f (x) in K, and the function x → |f (x)|2K is defined on so that it makes sense to integrate it on . Let P be the natural projection valued measure on H defined as follows: for any Borel set E ⊂ the projection P (E) is the operator f → χE f , where χE is the characteristic function of E. Then (π, H, P ) is the classical system of imprimitivity (SI) associated to the UR σ of H0 . In our case G0 is a Lie group and it is better to work with a smooth version of π; its structure is determined by a well known theorem of Dixmier-Malliavin in a manner that will be explained below. We begin with a standard but technical lemma that says that certain integrals containing a parameter are smooth. Lemma 7. Let M, N be smooth manifolds, dn a smooth measure on N , and B a separable Banach space with norm |·|. Let F : M × N −→ B be a map with the following properties: (i) For each n ∈ N , m → F (m, n) is smooth. (ii) If A ⊂ M is an open set with compact closure, and G is any derivative of F with respect to m, there is a gA ∈ L1 (N, dn) such that |G(m, n)| ≤ gA (n) for all m ∈ A, n ∈ N . Then f (m) = F (m, n)dn N

exists for all m and f is a smooth map of M into B. Proof. It is a question of proving that the integrals |G(m, n)|dn N

converge uniformly when m varies in an open subset A of M with compact closure. But the integrand is majorized by gA which is integrable on N and so the uniform convergence is clear. We also observe that any f ∈ H lies in Lp,loc (G0 ) for p = 1, 2, i.e., θ(x)|f (x)|2K is integrable on G0 for any continuous compactly supported scalar function θ ≥ 0. In fact θ (x)|f (x)|2K dx = θ (xξ )|f (xξ )|2K dξ dx = θ(x)|f (x)|2K dx < ∞, G0

H0

where θ (x) = G0 θ (xξ )dξ . In H we have the space C ∞ (π ) of smooth vectors for π . We also have its Garding subspace, the subspace spanned by all vectors π(α)h, where α ∈ Cc∞ (G0 ) and h ∈ H. We have −1 (π(α)h)(z) = α(x)h(x z)dx = α(zt −1 )h(t)dt (z ∈ G0 ). G0

G0

The integrals exist because h is locally L2 on G0 as mentioned above. Since h ∈ L1,loc (G0 ), α ∈ Cc∞ (G0 ), the conditions of Lemma 7 are met and so π(α)h is smooth. Thus all elements of the Garding space are smooth functions. But the Dixmier-Malliavin theorem asserts that the Garding space is exactly the same as C ∞ (π ) [DM78]. Thus all


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elements of C ∞ (π ) are smooth functions from G0 to K. This is the key point that leads to the smooth versions of the induced representation and the SI at the classical level. Let us define B as the space of all functions f from G0 to K such that (i) f is smooth and f (xξ ) = σ (ξ )−1 f (x) for all x ∈ G0 , ξ ∈ H0 , (ii) f has compact support mod H0 . Let Cc∞ (π ) be the subspace of all elements of C ∞ (π ) with compact support mod H0 . Proposition 4. B has the following properties: (i)B = Cc∞ (π ), (ii) B is dense in H, (iii) f (x) ∈ C ∞ (σ ) for all x ∈ G0 , (iv) B is stable under dπ . Proof. (i) Let f ∈ B. To show that f ∈ C ∞ (π ) it is enough to show that for any u ∈ H the map x → (π(x −1 )f, u)H is smooth in x. Now (π(x −1 )f, u)H = (f (xy), u(y))K dy.

Since |u|K is locally L1 on X and f is smooth, the conditions of Lemma 7 are met. We have B ⊂ Cc∞ (π ). The reverse inclusion is immediate from the Dixmier-Malliavin theorem, as remarked above. (ii) It is enough to prove that any h ∈ H with compact support mod H0 is in the closure of B. We know that π(α)h → h as α ∈ Cc∞ (G0 ) goes suitably to the delta function at the identity of G0 . But π(α)h is smooth and has compact support mod H0 because h has the same property, so that π(α)h ∈ B. (iii) Fix x ∈ G0 . Since σ (ξ )f (x) = f (xξ −1 ) for ξ ∈ H0 it is clear that f (x) ∈ C ∞ (σ ). (iv) Let f ∈ B, Z ∈ ᒄ0 . Then (dπ(Z)f )(x) = (d/dt)t=0 f (exp(−tZ)x) is smooth and we are done.

We refer to (π, B) as the smooth representation induced by σ . We shall also define the smooth version of the SI. For any u ∈ Cc∞ () let M(u) be the bounded operator on H which is multiplication by u. Then M(u) leaves B invariant and M : u → M(u) is a ∗-representation of the ∗-algebra Cc∞ () in H. It is natural to refer to (π, B, M) as the smooth system of imprimitivity associated to σ . Observe that f ∈ C ∞ (π ) has compact support mod H0 if and only if there is some u ∈ Cc∞ () such that f = M(u)f . Proposition 4 shows that B is thus determined intrinsically by the SI associated to σ . The passage from (π, H, P ) to (π, B, M) is thus functorial and is a categorical equivalence. Thus we are justified in working just with smooth SI’s. It is easy to see that the assignment that takes σ to the associated smooth SI is functorial. Indeed, let R be a morphism from σ to σ , i.e., R is a bounded operator from K to K intertwining σ and σ . We then define TR = T (H −→ H ) by (TR f )(x) = Rf (x)(x ∈ G0 ). It is then immediate that TR is a morphism from the (smooth) SI associated to σ to the (smooth) SI associated to σ . This functor is an equivalence of categories. To verify this one must show that every morphism between the two SI’s is of this form. This is of course classical but we sketch the argument depending on the following lemma which will be essentially used in the super context also. ∞ Lemma 8. Suppose f ∈ B and f (1) = 0. Then we can find ui ∈ Cc (), gi ∈ B such that (i) ui (1) = 0 for all i, (ii) we have f = i ui gi .

Proof. If f vanishes in a neighborhood of 1, we can choose u ∈ Cc∞ () such that u = 0 in a neighborhood of 1 and f = uf . The result is thus true for f . Let f ∈ B be arbitrary

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but vanishing at 1. Let ᒗ be a linear subspace of ᒄ0 = Lie (G0 ) complementary to ᒅ0 = Lie (H0 ). Then there is a sufficiently small r > 0 such that if ᒗr = {Z ∈ ᒗ | |Z| < r}, |·| being a norm on ᒗ, the map ᒗr × H0 −→ G0 ,

(Z, ξ ) −→ exp Z·ξ

is a diffeomorphism onto an open set G1 = G1 H0 of G0 . We transfer f from G1 to a function, denoted by ϕ on ᒗr × H0 . We have ϕ(0, ξ ) = 0, and ϕ(Z, ξ ξ ) = σ (ξ )−1 ϕ(Z, ξ ) for ξ ∈ H0 . If ti (1 ≤ i ≤ k) are the linear coordinates on ᒗ, ϕ(Z, ξ ) =

1

ti (Z)

(∂ϕ/∂ti )(sZ, ξ )ds. 0

i

1 The functions ψi (Z, ξ ) = 0 (∂ϕ/∂ti )(sZ, ξ )ds are smooth by Lemma 7 while ψi (Z, ξ ξ ) = σ (ξ )−1 ψi (Z, ξ ) for ξ ∈ H0 . So, going back to G1 we can write ∞ f = i ti hi , where ti are now in C (G1 ), right invariant under H0 and vanishing at 1, while the hi are smooth and satisfy hi (xξ ) = σ (ξ )−1 hi (x) for x ∈ G1 , ξ ∈ H0 . ∞ If u ∈ C c () is such that u is 1 in a neighborhood of 1 and supp (u) ⊂ G1 , then 2 ∞ u f = i ui gi where ui = uti ∈ Cc (), ui (1) = 0, and gi = uhi ∈ B. Since 2 2 2 f = u f + (1 − u )f and (1 − u )f = 0 in a neighborhood of 1, we are done. We can now determine all the morphisms from H to H . Let T be a morphism H −→ H . Then, as T commutes with multiplications by elements of Cc∞ (), it maps B to B . Moreover, for the same reason, the above lemma shows that if f ∈ B and f (1) = 0, then (Tf )(1) = 0. So the map R : f (1) −→ (Tf )(1)

(f ∈ B)

is well defined. From the fact that T intertwines π and π we obtain that (Tf )(x) = Rf (x) for all x ∈ G0 . To complete the proof we must show two things: (1) R is defined on all of C ∞ (σ ) and (2) R is bounded. For (1), let v ∈ C ∞ (σ ). In the earlier notation, if u ∈ Cc∞ (G0 ) is 1 in 1 and has support contained in G1 , then h : (exp Z, ξ ) → u(exp Z)σ (ξ )−1 v is in B and h(1) = v. For proving (2), let the constant C > 0 be such that (g ∈ H ).

||T g||H ≤ C||g||H

Then, taking g = u1/2 f for f ∈ B and u ≥ 0 in Cc∞ (), we get

u(x)|Rf (x)|K dx ≤ C 2

u(x)|f (x)|2K dx

for all f ∈ B and u ≥ 0 in Cc∞ (). So |Rf (x)|K ≤ C|f (x)|K for almost all x. As f and Rf = Tf are continuous this inequality is valid for all x, in particular for x = 1, proving that R is bounded.


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3.2. Representations induced from a special sub super Lie group. It is now our purpose to extend this smooth classical theory to the super context. A SLG (H0 , ᒅ) is a sub super Lie group of the SLG (G0 , ᒄ) if H0 ⊂ G0 , ᒅ ⊂ ᒄ, and the action of H0 on ᒅ is the restriction of the action of H0 (as a subgroup of G0 ) on ᒄ. We shall always suppose that H0 is closed in G0 . The sub SLG (H0 , ᒅ) is called special if ᒅ has the same odd part as ᒄ, i.e., ᒅ1 = ᒄ1 . In this case the super homogeneous space associated is purely even and coincides with = G0 /H0 . As in §3.1 we shall assume that admits an invariant measure although it is not difficult to modify the treatment to avoid this assumption. Both conditions are satisfied in the case of the super Poincaré groups and their variants. We start with a UR (σ, ρ σ , K) of (H0 , ᒅ) and associate to it the smooth induced representation (π, B) of the classical group G0 . In our case K is a SHS and so H becomes a SHS in a natural manner, the parity subspaces being the subspaces where f takes its values in the corresponding parity subspace of K. π is an even UR. We shall now define the operators ρ π (X) for X ∈ ᒄ1 as follows: (ρ π (X)f )(x) = ρ σ (x −1 X)f (x)

(f ∈ B).

Since the values of f are in C ∞ (σ ) the right side is well defined. In order to prove that the definition gives us an odd operator on B we need a lemma. Lemma 9. [ᒄ1 , ᒄ1 ] ⊂ ᒅ0 and is stable under G0 . In particular it is an ideal in ᒄ0 . Proof. For g ∈ G0 , Y, Y ∈ ᒄ1 , we have g[Y, Y ] = [gY, gY ] ∈ [ᒄ1 , ᒄ1 ]. Since ᒅ0 ⊕ ᒄ1 is a super Lie algebra, [ᒄ1 , ᒄ1 ] ⊂ ᒅ0 . Proposition 5. ρ π (X) is an odd linear map B −→ B for all X ∈ ᒄ1 . Moreover ρ π (X) is local, i.e., supp(ρ π (X)f ) ⊂ supp(f ) for f ∈ B. Finally, if Z ∈ ᒅ0 , we have −dσ (Z)f (g) = (Zf )(g). Proof. The support relation is trivial. Further, for x ∈ G0 , ξ ∈ H0 , (ρ π (X)f )(xξ ) = ρ σ (ξ −1 x −1 X)f (xξ ) = σ (ξ )−1 ρ σ (x −1 X)σ (ξ )f (xξ ) = σ (ξ )−1 (ρ π (X)f )(x). σ −1 It is thus a question of proving that g → ρ (g X)f (g) is smooth. If (Xk ) is a basis for ᒄ1 , g −1 X = k ck (g)Xk , where the ck are smooth functions and so it is enough to prove that g → ρ σ (Y )f (g) is smooth for any Y ∈ ᒄ, f ∈ B. We use Lemma 2. If Z = (1/2)[Y, Y ], we have ρ σ (Y )2 f (g) = −idσ (Z)f (g), and we need only show that −dσ (Z)f (g) is smooth in g. But Z ∈ ᒅ0 and f (g exp tZ) = σ (exp(−tZ))f (g) so that −dσ (Z)f (g) = (Zf )(g) is clearly smooth in g. Note that this argument applies to any Z ∈ ᒅ0 , giving the last assertion.

Proposition 6. (π, ρ σ , B) is a UR of the SLG (G0 , ᒄ). Proof. The symmetry of ρ π (X) and the relations ρ π (yX) = π(y)ρ π (X)π(y)−1 follow immediately from the corresponding relations for ρ σ . Suppose now that X, Y ∈ ᒄ1 .

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Then (ρ π (X)ρ π (Y )f )(x) = ρ σ (x −1 X)ρ σ (x −1 Y )f (x). Hence ((ρ π (X)ρ π (Y ) + ρ π (Y )ρ π (X))f )(x) = −idσ (x −1 [X, Y ])f (x) = i(x −1 [X, Y ]f )(x)( Proposition 5 ) = i(d/dt)t=0 f (x(x −1 exp t[X, Y ]x)) = i(d/dt)t=0 f (exp t[X, Y ]x) = i(d/dt)t=0 (π(exp(−t[X, Y ])f )(x) = −i(dπ([X, Y ])f )(x). This proves the proposition.

We refer to (π, ρ π , H) as the UR of the SLG (G0 , ᒄ) induced by the UR (σ, ρ σ , K) of (H0 , ᒅ), and to (π, ρ π , B) as the corresponding smooth induced UR. Write P for the natural projection valued measure in H based on : for any Borel E ⊂ , P (E) is the operator in H of multiplication by χE , the characteristic function of E. Recall the definition of ↔ before Lemma 3. Proposition 7. For X ∈ ᒄ1 , and u ∈ Cc∞ (), we have M(u) ↔ ρ π (X). Furthermore P (E) ↔ ρ π (X) for Borel E ⊂ . Proof. It is standard that a bounded operator commutes with all P (E) if and only if it commutes with all M(u) for u ∈ Cc∞ (). It is thus enough to prove that M(u) ↔ ρ π (X) for all u, X. On B we have M(u)ρ π (X) = ρ π (X)M(u) trivially from the definitions, and so we are done in view of Lemma 3. Theorem 1. The assignment that takes (σ, ρ σ ) to (π, ρ π , B, M) is a fully faithful functor.

Proof. Let R be a morphism intertwining (σ, ρ σ ) and (σ , ρ σ ), and let T : B −→ B be associated to R such that (Tf )(x) = Rf (x). It is then immediate that T intertwines ρ π and ρ π . Conversely, if T is a morphism between the induced systems, from the classical discussion following Lemma 8 we know that (Tf )(x) = Rf (x) for a bounded even operator R intertwining σ and σ . Since T intertwines ρ π and ρ π we conclude that R must intertwine ρ σ and ρ σ . 3.3. Super systems of imprimitivity and the super imprimitivity theorem. A super system of imprimitivity (SSI) based on is a collection (π, ρ π , H, P ), where (π, ρ π , H) is a UR of the SLG (G0 , ᒄ), (π, H, P ) is a classical system of imprimitivity, π, P are both even, and ρ π (X) ↔ P (E) for all X ∈ ᒄ1 and Borel E ⊂ . Let (π, ρ π , H) be the induced representation defined in §3.2 and let P be the projection valued measure introduced above. Proposition 7 shows that (π, ρ π , H, P ) is a SSI based on . We call this the SSI induced by (σ, ρ σ ). Theorem 2 (Super imprimitivity theorem). The assignment that takes (σ, ρ σ ) to (π, ρ π , H, P ) is an equivalence of categories from the category of UR ’s of the special sub SLG (H0 , ᒅ) to the category of SSI ’s based on . Proof. Let us first prove that any SSI of the SLG (G0 , ᒄ) is induced from a UR of the SLG (H0 , ᒅ). We may assume, in view of the classical imprimitivity theorem that π is


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the representation induced by a UR σ of H0 in K and that π acts by left translations on H. By assumption ρ π (X) leaves C ∞ (π ) invariant. We claim that it leaves Cc∞ (π ) also invariant. Indeed, let f ∈ Cc∞ (π ); then there is u ∈ Cc∞ () such that f = uf . On the other hand, by Lemma 3, ρ π (X)M(u) = M(u)ρ π (X) so that uf ∈ D(ρ π (X)) and ρ π (X)(uf ) = uρ π (X)f . Since uf = f this comes to ρ π (X)f = uρ π (X)f , showing that ρ π (X)f ∈ Cc∞ (π ). Thus the ρ π (X) leave B invariant and commute with all M(u) there. In other words we may work with the smooth SSI. By Lemma 8 the map f (1) −→ (ρ π (X)f )(1) is well defined and so, as in §3.1 we can define a map ρ σ (X) : C ∞ (σ ) −→ C ∞ (σ ) by ρ σ (X)v = (ρ π (X)f )(1),

f (1) = v,

f ∈ B.

Then, for f ∈ B, x ∈ G0 , (ρ π (X)f )(x) = (π(x −1 )ρ π (X)f )(1) = (ρ π (x −1 X)π(x −1 )f )(1) = ρ σ (x −1 X)(π(x −1 )f )(1) = ρ σ (x −1 X)f (x). If we now prove that (σ, ρ σ , K) is a UR of the SLG (H0 , ᒅ), we are done. This is completely formal. Covariance with respect to H0 . For f ∈ B, ξ ∈ H0 , ρ σ (ξ X)f (1) = (ρ π (ξ X)f )(1) = (π(ξ )ρ π (X)π(ξ −1 )f )(1) = (ρ π (X)π(ξ −1 )f )(ξ −1 ) = σ (ξ )(ρ π (X)π(ξ −1 )f )(1) = σ (ξ )ρ σ (X)(π(ξ −1 )f )(1) = σ (ξ )ρ σ (X)σ (ξ )−1 f (1). Odd commutators. Let X, Y ∈ ᒄ1 = ᒅ1 so that Z = [X, Y ] ∈ ᒅ0 . We have [ρ π (X), ρ π (Y )]f = −idπ([X.Y ])f for all f ∈ B. Now, i(−dπ(Z)f )(1) = i(d/dt)t=0 f (exp tZ) = i(d/dt)t=0 σ (exp(−tZ))f (1) = −idσ (Z)f (1). On the other hand, (ρ π (X)ρ π (Y )f )(1) = ρ σ (X)ρ σ (Y )f (1) so that the left side of (∗), evaluated at 1, becomes [ρ σ (X), ρ σ (Y )]f (1).

(∗)

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Thus [ρ σ (X), ρ σ (Y )]f (1) = −idσ (Z)f (1). Symmetry. From the symmetry of the ρ π (X) we have, for all f, g ∈ B, a, b ∈

Cc∞ (),

(ρ π (X)(af ), bg)H = (af, ρ π (X)(bg))H . This means that (ρ σ (x −1 X)f (x), g(x))K a(x)b(x)dx = (f (x), ρ σ (x −1 X)g(x))K a(x)b(x)dx. Since a and b are arbitrary we conclude that (ρ σ (x −1 X)f (x), g(x))K = (f (x), ρ σ (x −1 X)g(x))K for almost all x. All functions in sight are continuous and so this relation is true for all x. The evaluation at 1 gives the symmetry of ρ σ (X) on C ∞ (σ ). This proves that (σ, ρ σ , K) is a UR of the SLG (H0 , ᒅ) and that the corresponding induced SSI is the one we started with. To complete the proof we must show that the set of morphisms of the induced SSI’s is in canonical bijection with the set of morphisms of the inducing UR’s of the sub SLG in question. Let (π, ρ π , H, P ) and (π , ρ π , H , P ) be the SSI’s induced by (σ, ρ σ ) and (σ , ρ σ ) respectively. For any morphism R from (σ, ρ σ ) to (σ , ρ σ ) let T be as in Theorem 1. Then T extends uniquely to a bounded even operator from H to H , and the relations T M(u) = M (u)T for all u ∈ Cc∞ () imply that T P (E) = P (E)T for all Borel E ⊂ . Hence T is a morphism from (π, ρ π , H, P ) to (π , ρ π , H , P ). It is clear that the assignment R −→ T is functorial. To complete the proof we must show that any morphism T from (π, ρ π , H, P ) to (π , ρ π , H , P ) is of this form for a unique R. But T must take B = Cc∞ (π0 ) to B = Cc∞ (π0 ) and commute with the actions of Cc∞ (). Hence T is a morphism from (π, ρ π , B, M) to (π , ρ π , B , M ). Theorem 1 now implies that T arises from a unique morphism of (σ, ρ σ ) to (σ , ρ σ ). This finishes the proof of Theorem 2. 4. Representations of Super Semidirect Products and Super Poincaré Groups 4.1. Super semidirect products and their irreducible unitary representations. We start with a classical semidirect product G0 = T0 × L0 , where T0 is a vector space of finite dimension over R, the translation group, and L0 is a closed unimodular subgroup of GL(T0 ) acting on T0 naturally. For any Lie group the corresponding gothic letter denotes its Lie algebra. In applications L0 is usually an orthogonal group of Minkowskian signature, or its 2-fold cover, the corresponding spin group. By a super semidirect product (SSDP) we mean a SLG (G0 , ᒄ), where T0 acts trivially on ᒄ1 and [ᒄ1 , ᒄ1 ] ⊂ ᒑ0 . Clearly ᒑ := ᒑ0 ⊕ ᒄ1 is also a super Lie algebra, and (T0 , ᒑ) is a SLG called the super translation group. For any closed subgroup S0 ⊂ L0 , H0 = T0 S0 is a closed subgroup of G0 , ᒅ = ᒅ0 ⊕ ᒄ1 is a super Lie algebra, where ᒅ0 = ᒑ0 ⊕ ᒐ0 is the Lie algebra of H0 . Notice that (H0 , ᒅ) is a special sub SLG of (G0 , ᒄ). We begin by showing that the irreducible UR’s of (G0 , ᒄ) are in natural bijection with the irreducible UR’s of


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suitable special sub SLG’s of the form (H0 , ᒅ) with the property that the translations act as scalars. For brevity we shall write S = (G0 , ᒄ), T = (T0 , ᒑ). The action of L0 on T0 induces an action on the dual T0∗ of T0 . We assume that this action is regular, i.e., the orbits are all locally closed. By the well known theorem of Effros this implies that if Q is any projection valued measure on T0∗ such that QE = 0 or the identity operator I for any invariant Borel subset E of T0∗ , then Q is necessarily concentrated on a single orbit. This is precisely the condition under which the classical method of little groups of Frobenius-Mackey-Wigner works. For any λ ∈ T0∗ let Lλ0 be the stabilizer of λ in L0 and let ᒄλ = ᒑ0 ⊕ ᒉ0λ ⊕ ᒄ1 . The SLG (T0 Lλ0 , ᒄλ ) will be denoted by S λ . We shall call it the little super group at λ. It is a special sub SLG of (G0 , ᒄ). Two λ’s are called equivalent if they are in the same L0 -orbit. If θ is a UR of the classical group T0 L0 and O is an orbit in T0∗ , its spectrum is said to be in O if the spectral measure (via the SNAG theorem) of the restriction of θ to T0 is supported by O. Given λ ∈ T0∗ , a UR (σ, ρ σ ) of S λ is λ-admissible if σ (t) = eiλ(t) I for t ∈ T0 . λ itself is called admissible if there is an irreducible UR which is λ-admissible. It is obvious that the property of being admissible is preserved under the action of L0 . Let T0+ = λ ∈ T0∗ λ admissible . Then T0+ is an invariant subset of T0∗ . Theorem 3. The spectrum of every irreducible UR of the SLG (G0 , ᒄ) is in some orbit in T0+ . For each orbit in T0+ and choice of λ in that orbit, the assignment that takes a λ-admissible UR γ := (σ, ρ σ ) of S λ into the UR U γ of (G0 , ᒄ) induced by it, is a functor which is an equivalence of categories between the category of the λ-admissible UR’s of S λ and the category of UR’s of (G0 , ᒄ) with their spectra in that orbit. Varying λ in that orbit changes the functor into an equivalent one. In particular this functor gives a bijection between the respective sets of equivalence classes of irreducible UR’s. Proof. Notice first of all that since T0 acts trivially on ᒄ1 , π0 (t) commutes with ρ π (X) on C ∞ (π0 ) for all t ∈ T0 , X ∈ ᒄ1 . Hence PE ↔ ρ π (X) for all Borel E ⊂ , X ∈ ᒄ1 . For the first statement, let E be an invariant Borel subset of T0∗ . Let P be the spectral measure of the restriction of π to T0 . Then PE commutes with π, ρ π (X). So, if (π, ρ π ) is irreducible, PE = 0 or I . Hence P is concentrated in some orbit O, i.e., PO = I . The system (π, ρ π ) is clearly equivalent to (π, ρ π , P ) since P and the restriction of π to T0 generate the same algebra. If λ ∈ O and Lλ0 is the stabilizer of λ in L0 , we can transfer P from O to a projection valued measure P ∗ on L0 /Lλ0 = T0 L0 /T0 Lλ0 . So (π, ρ π ) is equivalent to the SSI (π, ρ π , P ∗ ). The rest of the theorem is an immediate consequence of Theorem 2. The fact that σ (t) = eiλ(t) I for t ∈ T0 is classical. Indeed, in the smooth model for π treated in §3.2, the fact that the spectrum of π is contained in the orbit −1 of λ implies that (π(t)f )(x) = eiλ(x tx) f (x) for all f ∈ B, t ∈ T0 , x ∈ G0 . Hence −1 iλ(t) −1 f (t ) = e f (1) while f (t ) = σ (t)f (1). So σ (t) = eiλ(t) I . Remark 2. In the classical theory all orbits of L0 are allowed and an additional argument of the positivity of energy is needed to single out the physically occurring representations. In SUSY theories as exemplified by Theorem 3, a restriction is already present: only orbits in T0+ are permitted. We shall prove in the next section that T0+ may be interpreted precisely as the set of all positive energy representations.

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4.2. Determination of the admissible orbits. Product structure of the representations of the little super groups. We fix λ ∈ T0+ and let (σ, ρ σ ) be a λ-admissible irreducible UR of S λ . Clearly −idσ (Z) = λ(Z)I

(Z ∈ ᒑ0 ).

Define λ (X1 , X2 ) = (1/2)λ([X1 , X2 ])

(X1 , X2 ∈ ᒄ1 ).

Then, on C ∞ (σ ), [ρ σ (X1 ), ρ σ (X2 )] = λ([X1 , X2 ])I = 2λ (X1 , X2 )I. Clearly λ is a symmetric bilinear form on ᒄ1 × ᒄ1 . Let Qλ (X) = λ (X, X) = (1/2)λ([X, X]). Then Qλ is invariant under Lλ0 because for X1 , X2 ∈ ᒄ1 , h ∈ L1 , [ρ σ (hX1 ), ρ σ (hX2 )] = σ (h)[ρ σ (X1 ), ρ σ (X2 )]σ (h)−1 = 2λ (X1 , X2 ). Now ρ σ (X)2 = Qλ (X)I

(X ∈ ᒄ1 ).

Since ρ σ (X) is essentially self adjoint on C ∞ (σ ), it is immediate that Qλ (X) ≥ 0. We thus obtain the necessary condition for admissibility: Qλ (X) = λ (X, X) ≥ 0

(X ∈ ᒄ1 ).

In the remainder of this subsection we shall show that the condition that λ ≥ 0, which we refer to as the positive energy condition, is also sufficient to ensure that λ is admissible. We will then find all the λ-admissible irreducible UR’s of S λ . It will follow in the next section that if the super Lie group (G0 , ᒄ1 ) is a super Poincaré group, the condition λ ≥ 0 expresses precisely the positivity of the energy. This is the reason for our describing this condition in the general case also as the positive energy condition. From now on we fix λ such that λ ≥ 0. Lemma 10. For any admissible UR (σ, ρ σ ) of S λ , ρ σ (X) is a bounded self adjoint operator for X ∈ ᒄ1 , and ρ σ (X)2 = Qλ (X)I . Moreover, Qλ ≥ 0 and is invariant under Lλ0 . Proof. We have, for X ∈ ᒄ1 , ψ ∈ C ∞ (σ ), |ρ σ (X)ψ|2K = (ρ σ (X)2 ψ, ψ) = Qλ (X)2 |ψ|2K which proves the lemma.


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This lemma suggests we study the following situation. Let W be a finite dimensional real vector space and let q be a nonnegative quadratic form on W , i.e., q(w) ≥ 0 for w ∈ W . Let ϕ be the corresponding symmetric bilinear form (q(w) = ϕ(w, w)). Let C be the real algebra generated by W with the relations w 2 = q(w)1(w ∈ W ). If q is nondegenerate, i.e., positive definite, this is the Clifford algebra associated to the quadratic vector space (W, q). If q = 0 it is just the exterior algebra over W . If (wi )1≤i≤n is a basis for W such that ϕ(wi , wj ) = εi δij with εi = 0 or 1 according as i ≤ a or > a, then C is the algebra generated by the wi with the relations wi wj + wj wi = 2εi δij . Let W0 be the radical of q, i.e., W0 = {w0 |ϕ(w0 , w) = 0 for all w ∈ W }; in the above notation W0 is spanned by the wi for i ≤ a. If W ∼ = W/W0 and q ∼ , ϕ ∼ are the corresponding objects induced on W ∼ , q ∼ is positive definite, and so we have the usual Clifford algebra C ∼ generated by (W ∼ , q ∼ ) with W ∼ ⊂ C ∼ . The natural map W −→ W ∼ extends uniquely to a morphism C −→ C ∼ which is clearly surjective. We claim that its kernel is the ideal C0 in C generated by W0 . Indeed, let I be this kernel. If s ∈ I , s is a linear combination of elements wI wJ , where wI is a product wi1 . . . wir (i1< · · · < ir ≤ a) and wJ is a product wj1 . . . wjs (a < j1 < · · · < js ); hence, s ≡ J cJ wJ mod C0 , and as the image of this element in C ∼ is 0, cJ = 0 for all J because the images of the wJ are linearly independent in C ∼ . Hence s ∈ C0 , proving our claim. A representation θ of C by bounded operators in a SHS K is called self adjoint (SA) if θ (w) is odd and self adjoint for all w ∈ W . θ can be viewed as a representation of the complexification C ⊗ C of C; a representation of C ⊗ C arises in this manner from a SA representation of C if and only if it maps elements of W into odd operators and takes complex conjugates to adjoints. Also we wish to stress that irreducibility is in the graded sense. Lemma 11. (i) If τ is a SA representation of C in K, then τ = 0 on C0 and so it is the lift of a SA representation τ ∼ of C ∼ . (ii) There exist irreducible SA representations τ of C; these are finite dimensional, unique if dim(W ∼ ) is odd, and unique up to parity reversal if dim(W ∼ ) is even. (iii) Let τ be an irreducible SA representation of C in a SHS L and let θ be any SA representation of C in a SHS R. Then R K ⊗ L, where K is a SHS and θ (a) = 1 ⊗ τ (a) for all a ∈ C; moreover, if dim(W ∼ ) is odd, we can choose K to be purely even. Proof. (i) If w ∈ W0 , then τ (w0 )2 = q(w0 )I = 0 and so, τ (w0 ) itself must be 0 since it is self adjoint. (ii) In view of (i) we may assume that W0 = 0 so that q is positive definite. Case I. dim(W ) = 2m. Select an ON basis a1 , b1 , . . . , am , bm for W . Let ej = (1/2)(aj + ibj ), fj = (1/2)(aj − ibj ). Then ϕ(ej , ek ) = ϕ(fj , fk ) = 0 while ϕ(ej , fk ) = (1/2)δj k . Then C ⊗ C is generated by the ej , fk with the relations ej ek + ek ej = fj fk + fk fj = 0,

ej fk + fk ej = δj k .

We now set up the standard “Schrödinger” representation of C ⊗ C. The representation acts on the SHS L = (U ), where U is a Hilbert space of dimension m and the grading on L is the Z2 -grading induced by the usual Z-grading of (U ). Let (uj )1≤j ≤m be an ON basis for U . We define τ (ej )f = uj ∧ f,

τ (fj )f = ∂(uj )(f )

(f ∈ (U )),

∂(u) for any u ∈ U being the odd derivation on (U ) such that ∂(u)v = 2(v, u) (here (·, ·) is the scalar product in (U ) extending the scalar product of U ). It is standard

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that τ is an irreducible representation of C ⊗ C. The vector 1 is called the Clifford vacuum. We shall now verify that τ is SA. Since aj = ej + fj , bj = −i(ej − fj ), we need to verify that τ (fj ) = τ (ej )∗ for all j , ∗ denoting adjoints. For any subset K = {k1 < · · · < kr } ⊂ {1, 2, . . . , m} we write uK = uk1 ∧ · · · ∧ ukr . Then we should verify that (uj ∧ uK , uL ) = (uK , ∂(uj )uL )

(K, L ⊂ {1, 2, . . . , m}).

Write K = {k1 , . . . , kr }, L = {1 , . . . , s }, where k1 < · · · < kr , 1 < · · · < s . We assume that j = a for some a and K = L \ {a }, as otherwise both sides are 0. Then K = {1 , . . . , a−1 , a+1 , . . . , s } (note that r = s − 1). But then both sides are equal to (−1)a−1 . From the general theory of Clifford algebras we know that if τ is another irreducible SA representation of C, then either τ ≈ τ or else τ ≈ τ , where is the parity reversal map and we write ≈ for linear (not necessarily unitary) equivalence. So it remains to show that ≈ implies unitary equivalence which we write . This is standard since the linear equivalence preserves self adjointness. Indeed, if R : τ1 −→ τ2 is an even linear isomorphism, then R ∗ R is an even automorphism of τ1 and so R ∗ R = a 2 I , where a is a scalar which is > 0. Then U = a −1 R is an even unitary isomorphism τ1 τ2 . Also for use in the odd case to be treated next, we note that τ is irreducible in the ungraded sense since its image is the full endomorphism algebra of L. Case II. dim(W ) = 2m + 1. It is enough to construct an irreducible SA representation as it will be unique up to linear, and hence unitary, equivalence. Let a0 , a1 , . . . , a2m be an ON basis for W . Write xj = ia0 aj (1 ≤ j ≤ 2m), x0 = i m a0 a1 . . . a2m . Then x02 = 1, xj xk + xk xj = 2δj k (j, k = 1, 2, . . . , 2m). Moreover x0 commutes with all aj and hence with all xj . The xj generate a Clifford algebra over R corresponding to a positive definite quadratic form and so there is an irreducible ungraded representation τ + of it in an ungraded Hilbert space L+ such that τ + (xj ) is self adjoint for all j = 1, 2, . . . , 2m (cf. the remark above). Within C⊗C the xj generate C ⊗ C + so that τ + is a representation of C ⊗ C + in L+ such that iτ + (a0 aj ) is self adjoint for all j . We now take 01 + + + + . L = L ⊕ L , τ = τ ⊕ τ , τ (x0 ) = 10 Here L is given the Z2 -grading such that the first and second copies of L+ are the even and odd parts. It is clear that τ is an irreducible representation of C ⊗ C. We wish to show that τ (ar ) is odd and self adjoint for 0 ≤ r ≤ 2m. But this follows from the fact that the τ (xr ) are self adjoint, τ (xj ) are even, and τ (x0 ) is odd, in view of the formulae a0 = i m x0 x1 . . . x2m ,

aj = −ia0 xj .

This finishes the proof of (ii). (iii) Let now θ be a SA representation of C in a SHS R of possibly infinite dimension. For any homogeneous ψ ∈ R the cyclic subspace θ (C)ψ is finite dimensional, hence closed, graded and is θ -stable; moreover by the SA nature of θ , for any graded invariant subspace its orthogonal complement is also graded and invariant. Hence we can write R = ⊕α Rα , where the sum is direct and each Rα is graded, invariant, and irreducible. Let L be a SHS on which we have an irreducible SA representation of C. If dim(W )∼ is even we can thus write R = (M0 ⊗ L) ⊕ (M1 ⊗ L), where the Mj are even Hilbert spaces; if dim(W ∼ ) is odd we can write R = K ⊗ L, where K is an even Hilbert space.


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In the first case, since M1 ⊗ L = M1 ⊗ L, we have R = K ⊗ L, where K is a SHS with K0 = M0 , K1 = M1 . For studying the question of admissibility of λ we need a second ingredient. Let H be a not necessarily connected Lie group and let us be given a morphism j : H −→ O(W ∼ ) so that H acts on W ∼ preserving the quadratic form on W ∼ . We wish to find out when there is a UR κ of H , possibly projective, and preferably, but not necessarily, even, in the space of the irreducible SA representation τ ∼ , such that κ(t)τ ∼ (w)κ(t)−1 = τ ∼ (tw)

(t ∈ H, w ∈ W ∼ ).

(∗)

For h ∈ O(W ∼ ), let τh∼ (w) = τ ∼ (hw)

(w ∈ W ∼ ).

Then τh∼ is also an irreducible SA representation of C ∼ and so we can find a unitary operator K(h) such that τh∼ (w) = K(h)τ ∼ (w)K(h)−1

(w ∈ W ∼ ).

If dim W ∼ is even, τ ∼ is irreducible even as an ungraded representation, and so K(h) will be unique up to a phase; it will be even or odd according as τh∼ τ ∼ or τh∼ τ ∼ , where is parity reversal. If dim W ∼ is odd, τ ∼ is irreducible only as a graded representation and so we also need to require K(h) to be an even operator in order that it is uniquely determined up to a phase. With this additional requirement in the odd dimensional case, we then see that in both cases the class of κ as a projective UR of H is uniquely determined, i.e., the class of its multiplier µ in H 2 (H, T) is fixed. In the following, we shall show that µ can be chosen to be ±1-valued, and examine the structure of κ more closely. We begin with some preparation (see [Del99, Var04]). Let C ∼× be the group of invertible elements in C ∼ . Define the full Clifford group as follows: = x ∈ C ∼× ∩ (C ∼+ ∪ C ∼− ) | xW ∼ x −1 ⊂ W ∼ . We have a homomorphism α : −→ O(W∼ ) given by α(x)w = (−1)p(x) xwx −1 for all w ∈ W ∼ , p(x) being 0 or 1 according as x ∈ C ∼+ or x ∈ C ∼− . Let β be the principal antiautomorphism of C ∼+ ; then xβ(x) ∈ R× for all x ∈ , and we write G for the kernel of the homomorphism x → xβ(x) of into R× . Since W ∼ is a positive definite quadratic space, we have an exact sequence α

1 −→ {±1} −→ G−→O(W ∼ ) −→ 1. For dim W ∼ ≥ 2, the connected component G0 of G is contained in C ∼+ and coincides with Spin(W ∼ ). Lemma 12. Let τ ∼ be a SA irreducible representation of C ∼ . We then have the following. (i) τ ∼ restricts to a unitary representation of G. (ii) The operator τ ∼ (x) is even or odd according as x ∈ G0 or x ∈ G\G0 . (iii) τ ∼ (x)τ ∼ (w)τ ∼ (x)−1 = (−1)p(x) τ ∼ (α(x)(w)) for x ∈ G, w ∈ W ∼ .

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Proof. Each x ∈ G is expressible in the form x = cv1 . . . vr , where vi are unit vectors in W ∼ and c ∈ {±1}. Since τ ∼ (vi ) is odd, the parity of τ ∼ (x) is the same as x. Moreover, since the τ ∼ (vi ) are self adjoint, τ ∼ (x) τ ∼ (x)∗ = c2 τ ∼ (v1 ) . . . τ ∼ (vr ) τ ∼ (vr ) . . . τ ∼ (v1 ) = I. This proves (i). (ii) and (iii) are obvious.

We now consider two cases. Case I. j (H ) ⊂ SO(W ∼ ). Let ζ be a Borel map of SO(W ∼ ) into Spin(W ∼ ) which is a right inverse of α(Spin(W ∼ ) −→ SO(W ∼ )) with ζ (1) = 1. Then ζ (xy) = ±ζ (x)ζ (y) for x, y ∈ SO(W ∼ ), and so κH = τ ∼ ◦ ζ ◦ j is an even projective UR of H satisfying (∗) with a ±-valued multiplier µH . Since ζ (1) = 1 it follows that µH is normalized, i.e., µH (h, 1) = µH (1, h) = 1

(h ∈ H ).

Clearly, the class of µH in H 2 (H, Z2 ) is trivial if and only if j : H → SO(W ∼ ) can be lifted to a morphism j : H → Spin(W ∼ ). In particular, this happens if H is connected and simply connected. Suppose now H is connected but j does not exist. We can then find a two-fold cover H ∼ of H with a covering map p(H ∼ −→ H ) such that j (H → SO(W ∼ )) lifts to a morphism j ∼ (H ∼ → Spin(W ∼ )), and if ξ is the nontrivial element in ker p, then j ∼ (ξ ) = −1. Lemma 13. If j maps H into SO(W ∼ ), there is a projectively unique even projective UR κ of H satisfying (∗), with a normalized ±1-valued multiplier µ. If H is connected, for κ to be an ordinary even representation (which will be unique up to multiplication by a character of H ) it is necessary and sufficient that either (i) j (H → SO(W ∼ )) can be lifted to Spin(W ∼ ) or (ii) there exists a character χ of H ∼ such that χ (ξ ) = −1. In particular, if H = A × T , where A is simply connected and T is a torus, then κ is an ordinary even unitary representation. Proof. The first statement has already been proved. We next prove the sufficiency part of the second statement. Sufficiency of (i) has already been observed. To see that (ii) is sufficient, note that κ ∼ = τ ∼ ◦ j ∼ is an even UR of H ∼ satisfying (∗); one can clearly replace κ ∼ by κ ∼ χ without destroying (∗). As κ ∼ (ξ ) = −1, we have (κ ∼ χ )(ξ ) = 1, and so it is immediate that κ ∼ χ descends to H. We leave the necessity part to the reader; it will not be used in the sequel. The statement for H = A × T will follow if we show that H ∼ has a character χ as in condition (ii). We have H ∼ = A × T ∼ , T ∼ being the double cover of T , and ξ = (1, t), with t = 1, t 2 = 1. There exists a character χ of T ∼ such that χ (t) = −1, and such a character can be extended to H ∼ by making it trivial on A. Case II. j (H ) ⊂ SO(W ∼ ). Let H0 = j −1 (SO(W ∼ )). Then H0 is a normal subgroup of H of index 2. We must distinguish two subcases.


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Case II.a. dim(W ∼ ) is even. Let ζ0 be a Borel right inverse of α(G0 −→ SO(W ∼ )) with ζ0 (1) = 1. Fix a unit vector v0 ∈ W ∼ and let r0 = −α(v0 ). Since α(v0 ) is the reflection in the hyperplane orthogonal to v0 and dim(W ∼ ) is even, we see that r0 ∈ O(W ∼ ) \ SO(W ∼ ). We then define a map ζ (O(W ∼ ) → G) by ζ0 (h) if h ∈ SO(W ∼ ) ζ (h) = ζ0 (h0 )v0 if h = h0 r0 , h0 ∈ SO(W ∼ ). Once again we have ζ (h1 h2 ) = ±ζ (h1 )ζ (h2 ). Define κH = τ ∼ ◦ ζ ◦ j. Then κH is a projective UR of H satisfying (∗) with a ±-valued normalized multiplier µH . But κH is not an even representation; elements of H \H0 map into odd unitary operators in the space of τ ∼ . We shall call such a representation of H graded with respect to H0 , or simply graded. We have thus proved the following. Lemma 14. If j (H ) ⊂ SO(W ∼ ), and dim(W ∼ ) is even, and if we define H0 = j −1 (SO(W ∼ )), then there is a projective UR κH , graded with respect to H0 , and satisfying (∗) with a ±-valued normalized multiplier µH . Before we take up the case when dim(W ∼ ) is odd, we shall describe how the projective graded representations of H are constructed. This is a very general situation and so we shall work with a locally compact second countable group A and a closed subgroup A0 of index 2; A0 is automatically normal and we write A1 = A \ A0 . Gradedness is with respect to A0 . We fix a ±-valued multiplier µ for A which is normalized. For brevity a representation will mean a unitary µ-representation. Moreover, with a slight abuse of language a µ-representation of A0 will mean a µ|A0 ×A0 -representation of A0 . If Rg is a graded representation in a SHS H, R the corresponding ungraded representation, and Pj is the orthogonal projection H −→ Hj , we associate to R the projection valued measure P on A/A0 , where PA0 = P0 and PA1 = P1 . Then the condition that Rg is graded is exactly the same as saying that (R, P ) is a system of imprimitivity for A based on A/A0 . Conversely, given a system of imprimitivity (R, P ) for A based on A/A0 , let us define the grading for H = H(R) by Hj = range of PAj (j = 0, 1); then R becomes a graded representation. Moreover for graded representations Rg , Rg , we have Hom(Rg , Rg ) = Hom((R, P ), (R , P )). In other words, the category of systems of imprimitivity for A based on A/A0 and the category of representations of A graded with respect to A0 are equivalent naturally. For any µ-representation r of A0 in a purely even Hilbert space H(r), let Rr := IndA A0 r be the representation of A induced by r. We recall that Rr acts in the Hilbert space H(Rr ) of all (equivalence classes of Borel) functions f (A → H(r)) such that for each α ∈ A0 , f (αa) = µ(α, a)r(α)f (a) for almost all a ∈ A; and (Rr (a)f )(y) = µ(y, a)f (ya)

(a, y ∈ A).

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The space H(Rr ) is naturally graded by defining H(Rr )j = {f ∈ H(Rr ) | supp(f ) ⊂ Aj }

(j = 0, 1).

r for Rr treated as a It is then obvious that Rr is a graded µ-representation. We write R graded representation. These remarks suggest the following lemma. r be the Lemma 15. For any unitary µ-representation r of A0 let Rr = Ind(r) and let R r is an equivalence graded µ-representation defined by Rr . Then the assignment r → R from the category of unitary µ-representations of A0 to the category of unitary graded µ-representations of A. Proof. Let us first assume that µ = 1. Then we are dealing with UR’s and the above remarks imply the lemma in view of the classical imprimitivity theorem. When µ is not 1 we go to the central extension A∼ of A by Z2 defined by µ. Recall that A∼ = A ×µ Z2 with multiplication defined by (a, ξ )(a , ξ ) = (aa , ξ ξ µ(a, a ))

(a, a ∈ A, ξ, ξ ∈ Z2 ).

∼ ∼ (We must give to A∼ the Weil topology.) Then A∼ 0 = A0 ×µ Z2 and A /A0 = A/A0 . ∼ ∼ The µ-representations R of A are in natural bijection with UR’s R of A such that R ∼ is nontrivial on Z2 by the correspondence

R ∼ (a, ξ ) = ξ R(a),

R(a) = R ∼ (a, 1).

The assignment R → R ∼ is an equivalence of categories. Analogous considerations hold for µ-representations r of A0 and UR’s r ∼ of A∼ 0 which are nontrivial on Z2 . The lemma would now follow if we establish two things: (a) For any unitary µ-representation r of A0 , and Rr = Ind(r), we have Rr∼ Ind(r ∼ ), ∼ and (b) If ρ is a UR of A∼ 0 and Ind(ρ) = R for some µ-representation R of A, then ρ = r ∼ for some µ-representation r of A0 . To prove (a) we set up the map f −→ f ∼ from H(Rr∼ ) to H(Ind(r ∼ )) by

f ∼ (a, ξ ) = f (a)ξ. It is an easy calculation that this is an isomorphism of Rr∼ with H(Ind(r ∼ )) that intertwines the two projection valued measures on A/A0 and A∼ /A∼ 0 ≈ A/A0 . To prove (b) we have only to check that ρ(1, ξ ) = ξ ; this however is a straightforward calculation. Remark 3. Given a graded µ-representation R of A, let r be the µ-representation of A0 defined by r(α) = R(α) H(R) (α ∈ A0 ). 0

It is then easy to show that R Rr . In fact it is enough to verify this (as before) when µ = 1. In this simple situation this is well known.


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We now resume our discussion and treat the odd dimensional case. Case II.b. dim(W ∼ ) is odd. We shall exhibit a projective even UR κ of H satisfying (∗). We refer back to the construction of τ ∼ in Lemma 11. Then 0 1 S= −1 0 is an odd unitary operator such that S 2 = −1 and τ ∼ (x)S = (−1)p(x) Sτ ∼ (x) for all x ∈ C ∼ . Let γ (O(W ∼ ) → G) be a Borel right inverse of α. Then, it is easily checked that if h ∈ H0 (τ ∼ ◦ γ ◦ j ) (h) κH (h) = (τ ∼ ◦ γ ◦ j ) (h) S if h ∈ H \H0 has the required properties. We now return to our original setting. First the graded representations of H are obtained by taking H = A, H0 = A0 in the foregoing discussion. Let ᒄ1λ = ᒄ1 /rad λ . We write Cλ for the algebra generated by ᒄ1 with the relations X 2 = Qλ (X)1 for all X ∈ ᒄ1 . We have a map jλ : Lλ0 −→ O(ᒄ1λ ). We take in the preceding theory q = Qλ ,

ϕ = λ ,

H = Lλ0 ,

W ∼ = ᒄ1λ ,

j = jλ ,

C ∼ = Cλ∼ .

Furthermore let κλ = κ,

µλ = µ,

τλ∼ = τ ∼ ,

τλ = lift of τ ∼ to Cλ .

Then µλ is a normalized multiplier for Lλ0 which we can choose to be ±-valued, κλ is a µλ -representation (unitary) of Lλ0 in the space of τλ , and κλ (t)τλ (X)κλ (t)−1 = τλ (tX)

(t ∈ Lλ0 ).

Moreover, κλ is graded if and only if jλ (Lλ0 ) ⊂ SO(ᒄ1λ ) and dim(ᒄ1λ ) is even, otherwise κλ is even. Finally, let j −1 (SO(ᒄ1λ )) if jλ (Lλ0 ) ⊂ SO(ᒄ1λ ) and dim(ᒄ1λ ) is even λ L00 = λλ L0 otherwise. r be the For any unitary µλ -representation r of Lλ00 in an even Hilbert space Kλ , let R λ λ unitary µλ - representation of L0 induced by r, which is graded if jλ (L0 ) ⊂ SO(ᒄ1λ ) and dim(ᒄ1λ ) is even, and is just r in all other cases. Theorem 4. Let λ be such that λ ≥ 0. Then λ is admissible, i.e., λ ∈ T0+ . For a fixed such λ let τλ be an irreducible SA representation of Cλ in a SHS Lλ and κλ the unitary µλ -representation of Lλ0 in Lλ associated to τλ as above. For any unitary µλ -represenr be the unitary µλ - representation of tation r of Lλ00 in an even Hilbert space Kλ , let R Lλ0 defined as above, and let θrλ = (σrλ , ρλσ )

246


be the UR of the little SLG S λ , where, for X ∈ ᒄ1 , h ∈ Lλ0 , t ∈ T0 , σrλ (th) = eiλ(t) σrλ (h)

and r (h) ⊗ κλ (h), (h) = R σrλ

ρλσ (X) = 1 ⊗ τλ (X)

(X ∈ ᒄ1 ).

Then θrλ is an admissible UR of S λ . The assignment r −→ θrλ is functorial, commutes with direct sums, and is an equivalence of categories from the category of unitary µλ representations of Lλ00 to the category of admissible UR’s of the little super group S λ . If Lλ0 is connected and satisfies either of the conditions of Lemma 13, then r −→ θrλ is an equivalence from the category of even UR’s of Lλ0 into the category of admissible UR’s of S λ . Proof. Once κλ is fixed, the assignment r −→ θrλ is clearly functorial (although it depends on κλ ). If dim(ᒄ1λ ) is even, a morphism M : : Rr1 −→ Rr2 and hence to the morr1 −→ r2 obviously gives rise to the morphism M ⊗ 1 from θr1 λ to θr2 λ . Conversely, if T is a bounded even operator commuting phism M with 1 ⊗ τλ , it is immediate (since the τλ (X) generate the full super algebra of endor1 ) → H(R r2 ) morphisms of Lλ ) that T must be of the form M ⊗ 1, where M : H(R is a bounded even operator. If now T intertwines Rr1 ⊗ κλ and Rr2 ⊗ κλ , then M must r1 , R r2 ) ≈ Hom(r1 , r2 ). Thus r −→ θrλ is a fully faithful functor. If belong to Hom(R dim(ᒄ1λ ) is odd, we can choose Kλ to be purely even (see Lemma 11). If T is a bounded even operator commuting with 1 ⊗ τλ , we use the fact that it commutes with + τ (a) 0 01 1⊗ and 1 ⊗ 10 0 τ + (a) (in the notation of Lemma 11) to conclude, via an argument similar to the one used in the even dimensional case, that T is of the form M ⊗ 1. Arguing as before we conclude that M ∈ Hom(r1 , r2 ). Thus r → θrλ is a fully faithful functor in this case also. It remains to show that every admissible UR of S λ is of the form θrλ . Let θ be an admissible UR of S λ in H. Then θ = (ξ, τ ), where ξ is an even UR of T0 Lλ0 which restricts to eiλ I on T0 , τ is a SA representation of Cλ related to ξ as usual. We may then assume by Lemma 11 that H = K ⊗ Lλ and τ = 1 ⊗ τλ . If dim(ᒄ1λ ) is odd, we choose K purely even. Then 1 ⊗ τλ (hX) = ξ(h)[1 ⊗ τλ (X)]ξ(h)−1 . But the same relation is true if we replace ξ by 1 ⊗ κλ . So if ξ1 = [1 ⊗ κλ ]−1 ξ , then ξ1 (h) is even or odd according to the grading of κλ (h), and commutes with 1 ⊗ τλ . Hence ξ1 is of the form R ⊗ 1 for a Borel map R from Lλ0 into the unitary group of K. Thus ξ(h) = [1 ⊗ κλ (h)][R (h) ⊗ 1]. The two factors on the right side of this equation commute; the left side is an even UR and the first factor on the right is a unitary µλ -representation of Lλ0 , which is graded or even according to dim(ᒄ1λ ) even or dim(ᒄ1λ ) odd. So R is a µ−1 λ = µλ -representation of Lλ0 in K, which is graded or even according to dim(ᒄ1λ ) even or dim(ᒄ1λ ) odd. This finishes the proof.


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Remark 4. If λ = 0, then λ = 0, Lλ = 0, and θr0 = (r, 0). Remark 5. For the super Poincaré groups we shall see in the next subsection that the situation is much simpler and Lλ0 is always connected and satisfies the conditions of Lemma 13. Combining Theorems 3 and 4 we obtain the following theorem. Let rλ be the UR of (G0 , ᒄ) induced by θrλ as described in Theorem 4. Theorem 5. Let λ be such that λ ≥ 0. The assignment that takes r to the UR rλ is an equivalence of categories from the category of unitary µλ -representations of Lλ00 to the category of UR’s of (G0 , ᒄ) whose spectra are contained in the orbit of λ. In particular, for r irreducible, rλ is irreducible, and every irreducible UR of (G0 , ᒄ) is obtained in this way. If the conditions of Lemma 13 are satisfied, then the r’s come from the category of UR’s of Lλ0 . In the case of super Poincaré groups (see Remark 5 above), rλ induced by θrλ represents a superparticle. In general the UR πrλ of T0 L0 contained in rλ will not be an irreducible UR of G0 . Its decomposition into irreducibles gives the multiplet that the UR of S determines. This is of course the set of irreducible UR’s Urλj of G0 induced by the rλj , where the rλj are the irreducible UR’s of Lλ0 contained in r ⊗ κλ : r ⊗ κλ = rλj , πrλ = Urλj . The set (rλj ) thus defines the multiplet. For r trivial the corresponding multiplet is called fundamental. 4.3. The case of the super Poincaré groups. We shall now specialize the entire theory to the case when (G0 , ᒄ) is a super Poincaré group (SPG). This means that the following conditions are satisfied. (a) T0 = R1,D−1 is the D-dimensional Minkowski space of signature (1, D − 1) with D ≥ 4; the Minkowski bilinear form is x, x = x0 x0 − j xj xj . (b) L0 = Spin(1, D − 1). (c) ᒄ1 is a real spinorial module for L0 , i.e., is a direct sum of spin representations over C. (d) For any 0 = X ∈ ᒄ1 , and any x ∈ T0 lying in the interior + of the forward light cone , we have [X, X], x > 0. If in (c) ᒄ1 is the sum of N real irreducible spin modules of L0 , we say we are in the context of N-extended supersymmetry. Sometimes N refers to the number of irreducible components over C. In (d) = {x | x, x ≥ 0, x0 ≥ 0},

+ = {x | x, x > 0, x0 > 0}.

In the case when D = 4 and ᒄ1 is the Majorana spinor, the condition (d) is automatic (one may have to change the sign of the odd commutators to achieve this); in the general case, as we shall see below, it ensures that only positive energy representations are allowed. We identify T0∗ with R1,D−1 by the pairing x, p = x0 p0 − j xj pj . The dual action of L0 is then the original action. The orbit structure of T0∗ is classical.

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Lemma 16. (i) Let V be a finite dimensional real vector space with a nondegenerate quadratic form and let V1 be a subspace of V on which the quadratic form remains nondegenerate. Then the spin representations of Spin(V ) restrict on Spin(V1 ) to direct sums of spin representations of Spin(V1 ). (ii) Suppose V = R1,D−1 . Let p ∈ V be such that p, p = ± m2 = 0 and V1 = p ⊥ . Then V1 is a quadratic subspace, the stabilizer p L0 of p in Spin(V ) is precisely Spin(V1 ), and it is Spin(D − 1) for p, p = m2 and Spin(1, D − 2) for p, p = −m2 = 0. Proof. (i) Let C, C1 be the Clifford algebras of V and V1 . Then C1+ ⊂ C + and hence, as the spin groups are imbedded in the even parts of the Clifford algebras, we have Spin(V1 ) ⊂ Spin(V ). Now the spin modules are precisely the modules for the even parts of the corresponding Clifford algebras and so, as these algebras are semisimple, the decomposition of the spin module of Spin(V ), viewed as an irreducible module for C + , into irreducible modules for C1+ under restriction to C1+ , gives the decomposition of the restriction of the original spin module to Spin(V1 ). See [Del99, Var04]. (ii) Choose an orthogonal basis (eα )0≤α≤D−1 such that e0 , e0 = −ej , ej = 1 for 1 ≤ j ≤ D − 1. It is easy to see that we can move p to either (m, 0, . . . , 0) or (0, m, 0, . . . , 0) by L0 and so we may assume that p is in one of these two positions. For u ∈ C(V )+ it is then a straightforward matter to verify that up = pu if and only if u ∈ C(V1 )+ . From the characterization of the spin group ([Del99, Var04]) it is now p clear that L0 = Spin(V1 ). Lemma 17. Let M be a connected real semisimple Lie group whose universal cover does not have a compact factor, i.e., the Lie algebra of M does not have a factor Lie algebra whose group is compact. Then M has no nontrivial morphisms into any compact Lie group, and hence no nontrivial finite dimensional UR’s. Proof. We may assume that M is simply connected. If such a morphism exists we have a nontrivial morphism ᒊ −→ ᒈ, where ᒊ is the Lie algebra of M and ᒈ is the Lie algebra of a compact Lie group. Let ᑾ be the kernel of this Lie algebra morphism. Then ᑾ is an ideal of ᒊ different from ᒊ, and so we can write ᒊ as ᑾ × ᒈ , where ᒈ is also an ideal and is non-zero; moreover, the map from ᒈ to ᒈ is injective. ᒈ is semisimple and admits an invariant negative definite form (the restriction from the Cartan- Killing form of ᒈ), and so its associated simply connected group K is compact. If A is the simply connected group for ᑾ, we have M = A × K , showing that M admits a compact factor, contrary to hypothesis. Corollary 1. If V is a quadratic vector space of signature (p, q), p, q > 0, p + q ≥ 3, then Spin(V ) does not have any nontrivial map into a compact Lie group. Proof. The Lie algebra is semisimple and the simple factors are not compact. Lemma 18. We have = {p | p ≥ 0}, i.e., for any p ∈ R1,D−1 , p ≥ 0 ⇐⇒ p0 ≥ 0, p, p ≥ 0. Moreover, p0 > 0, p, p > 0 ⇒ p > 0.


249

Proof. For 0 = X ∈ ᒄ1 we have [X, X], x > 0 for all x ∈ + and hence the inequality is true with ≥ 0 replacing > 0 for x ∈ . Hence 2p (X, X) = [X, X], p ≥ 0 if p ∈ . So ⊂ {p | p ≥ 0}. We shall show next that {p | p ≥ 0} ⊂ {p | p, p ≥ 0}. Suppose on the contrary that p ≥ 0 but p, p < 0. Since p is invariant under p p p L0 which is connected, we have a map L0 −→ SO(ᒄ1p ). Then L0 = Spin(V1 ) = p Spin(1, D − 2) by Lemma 16, and Corollary 1 shows that L0 has no nontrivial morp p phisms into any compact Lie group. Hence L0 acts trivially on ᒄ1p . Since L0 is a semip simple group, ᒄ1p can be lifted to an L0 -invariant subspace of ᒄ1 . Hence, if ᒄ1p = 0, p the action of L0 on ᒄ1 must have non-zero trivial submodules. However, by Lemma 16, the spin modules of Spin(V ) restrict on Spin(V1 ) to direct sums of spin modules of the smaller group and there is no trivial module in this decomposition. Hence ᒄ1p = 0, i.e., p = 0. Hence p vanishes on [ᒄ1 , ᒄ1 ]. Now [ᒄ1 , ᒄ1 ] is stable under L0 and non-zero, and so must be the whole of ᒑ0 . So p = 0, a contradiction. To finish the proof we should prove that if p ≥ 0 then p0 ≥ 0. Otherwise p0 < 0 and so −p ∈ and so from what we have already proved, we have −p = −p ≥ 0. Hence p = 0. But then as before p = 0, a contradiction. Finally, if p0 > 0 and p, p > 0, then p > 0 by definition of the SPG structure. This completes the proof. Theorem 6. Let S = (G0 , ᒄ) be a SPG. Then all stabilizers are connected and T0+ = {p | p ≥ 0} = . p

Moreover, κp is an even UR of L0 , and the irreducible UR’s of S whose spectra are in p the orbit of p are in natural bijection with the irreducible UR’s of L0 . The correspondp ing multiplet is then the set of irreducible UR’s parametrized by the irreducibles of L0 p occurring in the decomposition of α ⊗ κp as a UR of L0 . Proof. In view of Theorem 4 and Lemma 18 we have T0+ = . For p ∈ , the stabilizers p are all known classically. If p, p > 0, L0 = Spin(D − 1); if p, p = 0 but p0 > 0, p p then L0 = RD−2 × Spin(D − 2); and for p = 0, L0 = L0 . So, except when D = 4 and p is non-zero and is in the zero mass orbit, the stabilizer is connected and simply p connected, thus κp is an even UR of L0 by Lemma 13. But in the exceptional case, p L0 = R2 × S 1 , where S 1 is the circle, and Lemma 13 is again applicable. This finishes the proof. 4.4. Determination of κp and the structure of the multiplets. Examples. We have seen that the multiplet defined by the super particle αp is parametrized by the set of irrep ducible UR’s of L0 that occur in the decomposition of α ⊗ κp . Clearly it is desirable to determine κp as explicitly as possible. We shall do this in what follows. To determine κp the following lemma is useful. (W, q) is a positive definite quadratic vector space and ϕ is the bilinear form of q. C(W ) is the Clifford algebra of W

250


and H is a connected Lie group with a morphism H −→ SO(W ). τ is an irreducible SA representation of C(W ) and κ is a UR such that κ(t)τ (u)κ(t)−1 = τ (tu) for all u ∈ W, t ∈ H . We write ≈ for equivalence after multiplying by a suitable character. Lemma 19. Suppose that dim(W ) = 2m is even and WC := C ⊗R W has an isotropic subspace E of dimension m stable under H . Let η be the action of H on (E) extending its action on E. Then κ ≈ (E) ≈ (E ∗ )

(E ∗ is the complex conjugate of E).

Proof. Clearly E ∗ is also isotropic and H -stable. E ∩ E ∗ = 0 as otherwise E ∩ E ∗ ∩ W will be a non-zero isotropic subspace of W . So WC = E ⊕ E ∗ . We write τ for the representation of C(W ) in (E), where τ (u)(x) = u ∧ x,

τ (v)(x) = ∂(v)(x)

(u ∈ E, v ∈ E ∗ , x ∈ (E)).

Here ∂(v) is the odd derivation taking x ∈ E to 2ϕ(x, v). It is then routine to show that η(t)τ (u)η(t)−1 = τ (tu)

(u ∈ E ∪ E ∗ ).

Now τ is equivalent to τ and so we can transfer η to an action, written again as η, of H in the space of τ satisfying the above relation with respect to τ . It is not necessary that η be unitary. But we can normalize it to be a UR, namely κ(t) = | det(η(t))|−1/ dim(τ ) η(t). Remark 6. It is easy to give an independent argument that (E) ≈ (E ∗ ). For the unitary group U(E) of E let r be the representation on r (E), and let be their direct sum; then a simple calculation of the characters on the diagonal group shows that r ∗ det−1 ⊗n−r . Hence ∗ det−1 ⊗, showing that ∗ ≈ . It is then immediate that this result remains true for any group which acts unitarily on E. Corollary 2. The conditions of the above lemma are met if W = A ⊕ B, where A, B are orthogonal submodules for H which are equivalent. Moreover κ ≈ (E) (E ∗ ) (A) (B). Proof. Take ON bases (aj ), (bj ) for A and B respectively so that the map aj → bj is an isomorphism of H -modules. If E is the span of the ej = aj + ibj , it is easy to check that E is isotropic, and is a module for H which is equivalent to A and B. We now assume that for some r ≥ 3 we have a map H −→ Spin(r) −→ Spin(W ), where the first map is surjective, and H acts on W through Spin(W ). Further let the representation of Spin(r) on W be spinorial. We write σr for the (complex) spin representation of Spin(r) if r is odd and σr± for the (complex) spin representations of Spin(r) if r is even. Likewise we write sr , sr± for the real irreducible spin modules. Note that dim(W ) must be even.


251

Lemma 20. Let the representation of Spin(r) on W be spinorial. Let n be the number of real irreducible constituents of W as a module for Spin(r), and, when r is even, let n± be the number of irreducible constituents of real or quaternionic type. We then have the following determination of κ: r mod 8

κ

0(n± even ) 1, 7(n even) 2, 6 3, 5 4

+ − − ((n+ /2)σ r ⊕ (n /2)σr ) (n/2)σ r nσr+ ≈ nσr− nσr n+ σr+ ⊕ n− σr−

Proof. This is a routine application of the lemma and corollary above if we note the following facts: r ≡ 0 : Here σr± = sr± and W = n+ sr+ + n− sr− . r ≡ 1, 7 : Here σr = sr , W = nsr . r ≡ 2, 6 : Over C, sr becomes σr+ ⊕ σr− while σr± do not admit a non-zero invariant form. So WC = E ⊕ E ∗ , where E = nσr+ , E ∗ = nσr− , and q is zero on E. r ≡ 3, 5 : sr is quaternionic, W = nsr , WC = 2nσr and σr does not admit an invariant symmetric form. r ≡ 4 : sr± are quaternionic and σr± do not admit a non-zero invariant symmetric form; WC = E ⊕ E ∗ , where E = n+ σr+ + n− σr− and q is zero on E. In deriving these the reader should use the results in [Del99] and [Var04] on the reality of the complex spin modules and the theory of invariant forms for them. 4.4.1. Super Poincaré group associated to R1,3 : N=1 supersymmetry. Here T0 = R1,3 , L0 = SL(2, C)R , where the suffix R means that the complex group is viewed as a real Lie group. Let ᒐ = 2 ⊕ 2, 2 being the holomorphic representation of L0 in C2 and 2 its complex conjugate. Thus we identify ᒐ with C2 ⊕ C2 and introduce the conjugation on ᒐ given by (u, v) = (v, u). The action (u, v) → (gu, gv) of L0 (g is the complex conjugate of g) commutes with the conjugation and so defines the real form ᒐR invariant under L0 (Majorana spinor). We take ᒑ0 to be the space of 2 × 2 Hermitian matrices and the action of L0 on it as g, A → gAg T . For (ui , ui ) ∈ ᒐR (i = 1, 2) we put 1 (u1 u2 T + u2 u1 T ). 2 Then ᒄ = ᒄ0 ⊕ ᒄ1 with ᒄ0 = ᒑ0 ⊕ ᒉ0 , ᒄ1 = ᒐR is a super Lie algebra and (T0 L0 , ᒄ) is the SLG with which we are concerned. a0 + a3 a1 − ia2 1,3 Here R ᒑ0 by the map a → ha = ; ᒑ0 ᒑ∗0 with p ∈ ᒑ0 a1 + ia2 a0 − a3 viewed as the linear form a → a, p = a0 p0 − a1 p1 − a2 p2 − a3 p3 . Then [(u1 , u1 ), (u2 , u2 )] =

Qp ((u, u)) =

1 T u hpˇ u, 4

pˇ = (p0 , −p1 , −p2 , −p3 ). p

I: p0 > 0, m2 = p, p > 0. We take p = mI so that L0 = SU(2). Take E = {(u, 0)}, E ∗ = {(0, u)}. Then we are in the set up of Lemma 19. Then 2D j ⊕D j +1/2 ⊕ D j −1/2 (j ≥ 1/2) 0 1/2 j κp = (E) 2D ⊕ D , D ⊗ (E) = 2D 0 ⊕D 1/2 (j = 0).

252


Thus the multiplet with mass m has the same mass m and spins {j, j, j + 1/2, j − 1/2}(j > 0) . {0, 0, 1/2}(j = 0)

00 II: p0 > 0, p, p = 0. Here we take p = (1, 0, 0, −1), hp = . Then 02 a0 p L0 = . The characters χn/2 : a → a n (n ∈ Z) are viewed as characters of ca p L0 . Here Qp ((u, u)) = 41 uT hpˇ u = |u1 |2 . The radical of p is the span of (e2 , e2 ) and (ie2 , −ie2 ), e1 , e2 being the standard basis of C2 . We identify ᒄ1p with the span of (e1 , e1 ) and (ie1 , −ie1 ). We now apply Lemma 19 with E = C(e1 , 0) which carries the character defined by χ1/2 ; then (E) = χ0 ⊕ χ1/2 ,

χn/2 ⊗ (E) = χn/2 ⊕ χ(n+1)/2 .

The multiplet is {n/2, (n + 1)/2}. These results go back to [SS74]. 4.4.2. Extended supersymmetry. Here the SLG has still the Poincaré group as its even part but ᒄ1 is the sum of N > 1 copies of ᒐR . It is known ([Del99, Var04])that one can identify ᒄ1 with the direct sum ᒐN R of N copies of ᒐR in such a way that for the odd commutators we have

[(s1 , s2 , . . . , sN ), (s1 , s2 , . . . , sN )] = [si , si ]1 , 1≤i≤N

so that Qλ ((s1 , . . . , sN )) =

Q1λ ((si , si )).

1≤i≤N

Here the index 1 means the [ , ] and Q for the case N = 1 discussed above. Let E N = NE 1 . I: p0 > 0, m2 = p, p > 0. Then we apply Lemma 19 with E = E N so that κp = (N D 1/2 ). The decomposition of the exterior algebra of N D 1/2 is tedious but there is no difficulty in principle. We have

κp = cNr D r/2 cNr > 0, cNN = 1. 0≤r≤N

Then j + N/2 is the maximum value of r for which D r occurs in D j ⊗ (N D 1/2 ). The multiplet defined by the super particle of mass m is thus {j − N/2, j − N/2 + 1/2, . . . , j + N/2 − 1/2, j + N/2} (j ≥ N/2) . {0, 1/2, . . . , j + N/2 − 1/2, j + N/2} (0 ≤ j < N/2) II: p0 > 0, m = 0. Here

N κλ = (N χ1/2 ) = χr/2 . r 0≤r≤N

The multiplet of the super particle has the helicity content {r/2, (r + 1)/2, . . . , (r + N )/2}.


253

4.4.3. Super particles of infinite spin. The little groups for zero mass have irreducible p UR’s which are infinite dimensional. Since L0 is also a semidirect product its irreducible UR’s can be determined by the usual method. The orbits of S 1 in C (which is identified with its dual) are the circles {|a| = r} for r > 0 and the stabilizers of the points are all the same, the group {±1}. The irreducible UR’s of infinite dimension can then be parametrized as {αr,± }. Now (E) = χ0 ⊕ χ1/2 and an easy calculation gives αr,± ⊗ (E) = αr,+ ⊕ αr,− . The particles in the multiplet with mass 0 corresponding to spin (r, ±) consist of both types of infinite spin with the same r. 4.4.4. Super Poincaré groups of Minkowski super spacetimes of arbitrary dimension. p Let T0 = R1,D−1 . We first determine κp in the massive case. Here L0 = Spin(D − 1) p and the form p is strictly positive definite. So ᒄ1p = ᒄ1 and L0 acts on it by restriction, hence spinorially by Lemma 16. So Lemma 20 applies at once. It only remains to determine n, n± in terms of the corresponding N, N ± for ᒄ1 viewed as a module for p L0 . Notation is as in Lemma 20, and res is restriction to L0 ; r = D − 1. This is done p by writing ᒄ1 as a sum of the sD and determining the restrictions of the sD to L0 by dimension counting. We again omit the details but refer the reader to [Del99, Var04]. Proposition 8. When p0 > 0, m2 = p, p > 0, κp , the fundamental multiplet of the super particle of mass m, is given according to the following table: D mod 8

res (sD )

κp

0 1(N = 2k) 2(N = 2k) 3 4 5 6 7

2sD−1 + − sD−1 + sD−1 sD−1 sD−1 sD−1 + − sD−1 + sD−1 sD−1 2sD−1

N σD−1 + − kσD−1 ⊕ kσD−1 kσD−1 ± N σD−1 N σD−1 + − N σD−1 ⊕ N σD−1 N σD−1 ) ± (2N )σD−1 )

We now extend these results to the case when p has zero mass. Let V = R1,D−1 (D ≥ 4) and p = 0 a null vector in V . Let ej be the standard basis vectors for V so that (e0 , e0 ) = −(ej , ej ) = 1 for j = 1, . . . , D − 1. We may assume that p = e0 + e1 . Let V1 be the span of ej (2 ≤ j ≤ D − 1). The signature of V1 is (0, D − 2). Then p we have the flag 0 ⊂ Rp ⊂ p ⊥ ⊂ V left stable by the stabilizer L0 of p in L0 . The quadratic form on p⊥ has Rp as its radical and so induces a nondegenerate form on V1 := p ⊥ /Rp. Write L1 = Spin(V1 ). Note that V1 V1 . p We have a map x → x from L0 to L1 where, for v ∈ p ⊥ with image v ∈ V1 , p x v = (xv) . It is known that this is surjective and its kernel T1 := T0 is isomorphic p ⊥ to V1 canonically: for x ∈ T0 , the vector xe0 − e0 ∈ p , and the map that sends p x to the image t (x) of xe0 − e0 in V1 is well defined and is an isomorphism of T0 p with V1 . The map x → (t (x), x ) is an isomorphism of L0 with the semidirect product V1 × L1 . The Lie algebra of the big spin group L0 has the er es (r < s) as basis and it is p a simple calculation that the Lie algebra of L0 has as basis tj = (e0 + e1 )ej (2 ≤ j ≤

254


D − 1), er es (2 ≤ r < s ≤ D − 1) with the tj forming a basis of the Lie algebra of T1 . The er es (2 ≤ r < s ≤ D − 1) span a Lie subalgebra of the Lie algebra of L0 and the p p corresponding subgroup H ⊂ L0 is such that L0 T1 × H . For all of this see [Var04], pp. 36-37. p We shall now determine the structure of the restriction to L0 of the irreducible spin representation(s) of L, over R as well as over C. Since this may not be known widely we give some details. We begin with some preliminary remarks. p Let U be any finite dimensional complex L0 -module. Write U = ⊕χ Uχ , where Uχ , for any character (not necessarily unitary) χ of T1 , is the subspace of all elements u ∈ U such that (t − χ (t))m u = 0 for sufficiently large m. The action of L1 permutes the Uχ , and so, since L1 has no finite nontrivial orbit in the space of characters of T1 , it follows that the spectrum of T1 consists only of the trivial character, i.e., T1 acts unipotently. In particular U1 , the subspace of T1 -invariant elements of U , is = 0, an assertion which is then valid for real modules also. It follows that we have a strictly increasing filtration (Ui )i≥1 , where Ui+1 is the preimage in U of (U/Ui )1 . In particular, if U is semisimple, U = U1 . Lemma 21. Let W be an irreducible real or complex spin module for L0 . Let W1 be the subspace of all elements of W fixed by T1 and W 1 := W/W1 . We then have the following: p

(i) 0 = W1 = W , W1 is the unique proper non-zero L0 -submodule of W , and T1 acts trivially also on W 1 . p p (ii) W1 and W 1 are both irreducible L0 -modules on which L0 /T1 L1 acts as a spin module. (iii) The exact sequence 0 −→ W1 −→ W −→ W/W1 −→ 0 does not split. (iv) Over R, W, W1 , W 1 are all of the same type. If dim(V ) is odd, W1 W 1 . Let dim(V ) be even; then, over C, W1 , W 1 are the two irreducible spin modules for L1 ; over R, W1 W 1 when W is of complex type, namely, when D ≡ 0, 4 mod 8; otherwise, the modules W1 , W 1 are the two irreducible modules of L1 (which are either real or quaternionic). Proof. We first work over C. Let C be the Clifford algebra of V . The key point is that W1 = W . Suppose W1 = W . Then tj = 0 on W for all j . If D is odd, C + is a full matrix algebra and so all of its modules are faithful, giving a contradiction. Let D be even and W one of the spin modules for L0 . We know that inner automorphism by the invertible odd element e2 changes W to the other spin module. But as e2 tj e2−1 = tj for j > 2 and −t2 for j = 2, it follows that tj = 0 on the other spin module also. Hence tj = 0 in the irreducible module for the full Clifford algebra C. Now C is isomorphic to a full matrix algebra and so its modules are faithful, giving again tj = 0, a contradiction. p Let (Wi ) be the strictly increasing flag of L0 -modules, with T1 acting trivially on each Wi+1 /Wi , defined by the previous discussion. Let m be such that Wm = W . p Clearly m ≥ 2. On the other hand, the element −1 of L0 lies in L0 and as it acts as −1 on W , it acts as −1 on all the Wi+1 /Wi . Hence dim(Wi+1 /Wi ) ≥ dim(σD−2 ) (see Lemma 6.8.1 of [Var04]), and there is equality if and only if Wi+1 /Wi σD−2 . Since dim(σD ) = 2 dim(σD−2 ), we see at once that m = 2 and that both W1 and W/W1 are


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p

irreducible L0 -modules which are spin modules for L1 . The exact sequence in (iii) cannot split, as otherwise T1 will be trivial on all of W . Suppose now U is a non-zero proper p L0 -submodule of W . Then dim(U ) = dim(W1 ) for the same dimensional argument as above, and so U is irreducible, thus T1 is trivial on it, showing that U = W1 . We have thus proved (i)–(iii). We now prove (iv). There is nothing to prove when D is odd since there is only one p ± spin module. Suppose D is even. Let us again write σD−2 for the L0 -modules obtained p by lifting the irreducible spin modules of L1 to L0 . We consider two cases. ± D ≡ 0 , 4 mod 8 . In this case σD± are self dual while σD−2 are dual to each other. It + + is not restrictive to assume W = σD and W1 = σD−2 . We have the quotient map W = σD+ −→ W 1 = σ ; and σ is to be determined. Writing σ for the dual of σ , we get σ ⊂ (σD+ ) σD+ , so that, by the uniqueness of the submodule proved above, + − . Hence σ = σD−2 . σ = σD−2 ± are self dual. Again, supD ≡ 2 , 6 mod 8 . Now σD± are dual to each other while σD−2 + + pose W = σD and W1 = σD−2 . The above argument then gives σ ⊂ σD− . On the other hand, the inner automorphism by e2 transforms σD+ into σD− , and the subspace of σD+ fixed + by T1 into the corresponding subspace of σD− , while at the same time changing σD−2 − + + − into σD−2 . Hence it changes the inclusion σD−2 ⊂ σD into the inclusion σD−2 ⊂ σD− . − − . Dualizing, this gives σ = σD−2 once again. This finishes Hence we have σ = σD−2 the proof of the lemma over C.

We now work over R. Since both V and V1 have the same signature D −2, it is immediate that the real spin modules for L0 and L1 are of the same type. As an L0 -module, the complexification WC of W is either irreducible or is a direct sum U ⊕ U , where U is a complex spin module for L0 . In the first case W is of real type and the lemma follows from the lemma for the complex spin modules. In the second case W is of quaternionic or complex type according as U and U are equivalent or not. Complex type. We have WC = U ⊕ Z, where Z = U is the complex conjugate of U . We have U σD+ , Z σD− . Since CW1 = U1 ⊕ Z1 it is clear that 0 = W1 = W . The real irreducible spin modules of L1 have dimension 2D/2−1 and so we find that dim(W1 ) = 2D/2−1 and W1 , W 1 are both irreducible; they are equivalent as they are of complex type. A non-zero proper submodule R of W then has dimension 2D/2−1 and so must be irreducible. Hence either R = W1 or W = W1 ⊕ R. But then W = W1 , a contradiction. The same argument shows that the exact sequence in the lemma does not split. Quaternionic type. For proving (i)–(iii) of the lemma the argument is the same as in the complex type, except that Z U . We now check (iv). The case of odd dimension is obvi+ + + ous. So let D be even and W1 sD−2 . Then CW1 2σD−2 so that U1 Z1 σD−2 . − − − 1 1 1 1 Then U Z σD−2 and so CW 2σD−2 . Hence W sD−2 . The lemma is completely proved. Remark 7. Since we are interested in the quotient W 1 rather than W1 below, we change ± ± ∓ our convention slightly; for W = sD we write W 1 = sD−2 and W1 = sD−2 . We now come to the discussion of the structure of κp when p is in a massless orbit. Proposition 9. Let p0 > 0, p, p = 0. Then rad (p ) = ᒄT1 1 , the subspace of elements of ᒄ1 fixed by T1 . Moreover T1 acts trivially on ᒄ1p , rad (p ) ᒄ1p except in the

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cases D ≡ 2, 6 mod 8 when rad (p ) and ᒄ1p are dual to each other, and L0 /T1 L1 acts spinorially on rad p and ᒄ1p . In all cases dim(rad (p )) = dim(ᒄ1p ) = (1/2) dim(ᒄ1 ). For ᒄ1p as well as the associated κp the results are as in the following table: D mod 8

ᒄ1p

κp

0, 4 2(N ± = 2n± ) 6 1, 3(N = 2n) 5, 7

N sD−2 + − N + sD−2 + N − sD−2 + − + − N sD−2 + N sD−2 N sD−2 N sD−2

± N σD−2 + − n+ σD−2 + n− σD−2 + − − N + σD−2 + N σD−2 nσD−2 N σD−2

Proof. We have ᒄ1 = ⊕1≤i≤N ᒅi , where the ᒅi are real irreducible spin modules and + . Let ᒏ = rad and ᒏ [ᒅi , ᒅj ] = 0 for i = j while [X, X], q > 0 for all q ∈ p p ip the radical of the restriction of p to ᒅi . Since Qp (X) = i Qp (Xi ), where Xi is the component of X in ᒅi , it follows that ᒏp = ⊕i ᒏip . We now claim that ᒏip = (ᒅi )1 , namely, p the subspace of elements of ᒅi fixed by T1 . Since ᒏip is a L0 -submodule it suffices, in view of the lemma above, to show that 0 = ᒏip = ᒅi . If ᒏip were 0, p would be strictly p positive definite on ᒅi , and hence the action of L0 will have an invariant positive definite p quadratic form. So the action of L0 on ᒅi will be semisimple, implying that T1 will act trivially on ᒅi . This is impossible, since, by the preceding lemma, ᒅi = (ᒅi )1 . If ᒏip = ᒅi , then p = 0 on ᒅi , and this will imply that p = 0. Thus ᒏip = (ᒅi )1 , hence ᒏp = (ᒄ1 )1 . The other assertions except the table are now clear. For the table we need to observe that ᒄ1p = ⊕i ᒅi /ᒏip and that ᒏip ᒅi /ᒏip except when D ≡ 2, 6 mod 8; in these cases, the two modules are the two real or quaternionic spin modules which are dual to each other. The table is worked out in a similar manner to Proposition 8. We omit the details. Remark 8. The result that the dimension of ᒄ1p has 1/2 the dimension of ᒄ1 extends the known calculations when D = 4 (see [FSZ81]). 4.4.5. The role of the R-group in classifying the states of κp . In the case of N -extended p supersymmetry we have two groups acting on ᒄ1 : L0 , the even part of the little super group at p, and the R-group ([Del99, Var04]) R. Their actions commute and they both leave the quadratic form Qp invariant. In the massive case we have a map p

L0 × R −→ Spin(ᒄ1p ) so that one can speak of the restriction κp of the spin representation of Spin(ᒄ1p ) to p p L0 × R. The same is true in the massless case except we have to replace L0 by a twofold cover of it. It is thus desirable to not just determine κp as we have done but actually p determine this representation κp of L0 ×R. We have not done this but there is no difficulty in principle. However, when D = 4, we have a beautiful formula [FSZ81]. To describe this, assume that we are in the massive case. We first remark that ᒄ1 HN ⊗H S0 , where S0 is the quaternionic irreducible of SU(2) of dimension 4. Thus the R-group is the unitary group U(N, H). Over C we thus have ᒄ1C C2N ⊗ C2 , where the R-group is the p symplectic group Sp(2N, C) acting on the first factor and L0 SU(2) acts as D 1/2 on the second factor. The irreducible representations of Sp(2N, C)×SU(2) are outer tensor products of irreducibles a of the first factor and b of the second factor, written as (a, b).


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Let k denote the irreducible of dimension k of SU(2) and [2N ]k denote the irreducible representation of the symplectic group in the space of traceless antisymmetric tensors of rank k over C2N ; by convention for k = 0 this is the trivial representation and for k = 1 it is the vector representation. Then κp ([2N ]0 , N + 1) + ([2N ]1 , N) + . . . ([2N ]k , N + 1 − k) + . . . ([2N ]N , 1). To see how this follows from our theory note that C2N ⊗ e1 is a subspace satisfying the conditions of Lemma 19 for the symplectic group and so κp (C2N ). It is known that

(N + 1 − k)[2N ]k . (C2N ) 0≤k≤N

On the other hand we know that the representations of SU(2) in κp are precisely the N + 1 − k(0 ≤ k ≤ N ). The formula for κp is now immediate. In the massless case the R-group becomes U(N ) and

κp ((N − k)/2, [N ]k ), 0≤k≤N

where r/2 denotes the character denoted earlier by χr and [N ]k is the irreducible representation of U(N ) defined on the space k (CN ). We omit the proof which is similar. Acknowledgement. G. C. gratefully acknowledges a grant of the Università di Genova that has made possible a visit to Los Angeles during which some of the work for this paper has been done. He thanks Professor V. S. Varadarajan for his warm hospitality. V. S. V. would like to thank Professor Giuseppe Marmo and INFN, Naples, Professor Sergio Ferrara and CERN, Geneva, and Professors Enrico Beltrametti and Gianni Cassinelli and INFN, Genoa, for their hospitality during the summers of 2003 and 2004, during which most of the work for this paper was done. We are grateful to Professor Pierre Deligne of the Institute for Advanced Study, Princeton, NJ, for his interest in our work and for his comments which have improved the paper.

References [Del99] [DM78] [DM99] [DP85] [DP86]

[DP87] [Fer01] [Fer03] [FSZ81] [Nel59]

Deligne, P.: Notes on spinors. In: Quantum fields and strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997), Providence, RI: Amer. Math. Soc., 1999, pp. 99–135 Dixmier, J., Malliavin, P.: Factorisations de fonctions et de vecteurs indéfiniment différentiables. Bull. Sci. Math. (2), 102(4), 307–330 (1978) Deligne, P., Morgan, J.W.: Notes on supersymmetry (following Joseph Bernstein). In: Quantum fields and strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997), Providence, RI: Amer. Math. Soc., 1999, pp. 41–97 Dobrev, V.K., Petkova, V.B.: All positive energy unitary irreducible representations of extended conformal supersymmetry. Phys. Lett. B 162(1–3), 127–132 (1985) Dobrev, V.K., Petkova, V.B.: All positive energy unitary irreducible representations of the extended conformal superalgebra. In: Conformal groups and related symmetries: physical results and mathematical background (Clausthal-Zellerfeld, 1985), Vol. 261 of Lecture Notes in Phys., Berlin: Springer, 1986, pp. 300–308 Dobrev, V.K., Petkova, V.B.: Group-theoretical approach to extended conformal supersymmetry: function space realization and invariant differential operators. Fortschr. Phys. 35(7), 537–572 (1987) Ferrara, S.: UCLA lectures on supersymmetry. 2001 Ferrara, S.: UCLA lectures on supersymmetry. 2003 Ferrara, S., Savoy, C.A., Zumino, B.: General massive multiplets in extended supersymmetry. Phys. Lett. B 100(5), 393–398 (1981) Nelson, E.: Analytic vectors. Ann. Math. (2), 70, 572–615 (1959)

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Salam, A., Strathdee, J.: Unitary representations of super-gauge symmetries. Nucl Phys. B80, 499–505 (1974) [Var04] Varadarajan, V.S.: Supersymmetry for mathematicians: an introduction, Vol. 11 of Courant Lecture Notes in Mathematics. New York: New York University Courant Institute of Mathematical Sciences, 2004 [Wit82] Witten, E.: Supersymmetry and Morse theory. J. Differ. Geom. 17(4), 661–692 (1983) Communicated by Y. Kawahigashi


Communications in


Sufficiency in Quantum Statistical Inference Anna Jenˇcová1, , Dénes Petz2, 1 2

Mathematical Institute of the Slovak Academy of Sciences, Stefanikova 49, Bratislava, Slovakia. E-mail: jenca@mat,savba.sk Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, POB 127, 1364 Budapest, Hungary. E-mail: [email protected]

Received: 27 December 2004 / Accepted: 22 September 2005 Published online: 26 January 2006 – © Springer-Verlag 2006

Abstract: This paper attempts to develop a theory of sufficiency in the setting of noncommutative algebras parallel to the ideas in classical mathematical statistics. Sufficiency of a coarse-graining means that all information is extracted about the mutual relation of a given family of states. In the paper sufficient coarse-grainings are characterized in several equivalent ways and the non-commutative analogue of the factorization theorem is obtained. As an application we discuss exponential families. Our factorization theorem also implies two further important results, previously known only in finite Hilbert space dimension, but proved here in generality: the Koashi-Imoto theorem on maps leaving a family of states invariant, and the characterization of the general form of states in the equality case of strong subadditivity. 1. Introduction A quantum mechanical system is described by a C*-algebra, the dynamical variables (or observables) correspond to the self-adjoint elements and the physical states of the system are modelled by the normalized positive functionals of the algebra, see [4, 5]. The evolution of the system M can be described in the Heisenberg picture in which an observable A ∈ M moves into α(A), where α is a linear transformation. α is an automorphism in the case of the time evolution of a closed system but it could be the irreversible evolution of an open system. The Schrödinger picture is dual, it gives the transformation of the states, the state ϕ ∈ M∗ moves into ϕ ◦ α. The algebra of a quantum system is typically non-commutative but the mathematical formalism supports commutative algebras as well. A simple measurement is usually modelled by a family of pairwise orthogonal projections, or more generally, by a partition of unity, (Ei )ni=1 . Since all Ei are supposed to be positive and i Ei = I , β : Cn → M, (z1 , z2 , . . . , zn ) → i zi Ei gives a Supported by the EU Research Training Network Quantum Probability with Applications to Physics, Information Theory and Biology and Center of Excellence SAS Physics of Information I/2/2005. Supported by the Hungarian grant OTKA T032662

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positive unital mapping from the commutative C*-algebra Cn to the non-commutative algebra M. Every positive unital mapping occurs in this way. The essential concept in quantum information theory is the state transformation which is affine and the dual of a positive unital mapping. All these and several other situations justify the study of positive unital mappings between C*-algebras from a quantum statistical viewpoint. If the algebra M is “small” and N is “large”, and the mapping α : M → N sends the state ϕ of the system of interest to the state ϕ ◦ α at our disposal, then loss of information takes place and the problem of statistical inference is to reconstruct the real state from partial information. In this paper we mostly consider parametric statistical models, a parametric family S := {ϕθ : θ ∈ } of states is given and on the basis of the partial information the correct value of the parameter should be decided. If the partial information is the outcome of a measurement, then we have statistical inference in the very strong sense. However, there are “more quantum” situations, to decide between quantum states on the basis of quantum data, see Example 4 below. The problem we discuss is not the procedure of the decision about the true state of the system but the circumstances under which this is perfectly possible. The paper is organized as follows. In the next section we summarize the relevant basic concepts both in classical statistics and in the non-commutative framework. The first part of Sect. 3 is about sufficient subalgebras, or subsystems of a quantum system. The second part is devoted to sufficient coarse-grainings. Most of the result of this section has been known in a more restricted situation of faithful states, see Chap. 9 in [12]. The importance of the multiplicative domain of a completely positive mapping is emphasized here. This concept allows us to give a sufficient subalgebra determined by a sufficient coarse-graining. The quantum factorization theorem of Sect. 4 is the main result of the paper. The factorization of the states corresponds to a special structure of the algebras and the sufficient coarse-grainings. We use the properties of the von Neumann entropy and of the modular group to prove this result in some infinite dimensional situations (where the essential condition is the finiteness of the von Neumann entropy). The factorization implies a generalization of the Koashi-Imoto Theorem [7]. In Sect. 5 the equality case in the strong subadditivity of the von Neumann entropy is discussed in a possibly infinite dimensional framework and the factorization result is applied. 2. Preliminaries In this paper, C*-algebras always have a unit I . Given a C*-algebra M, a state ϕ of M is a linear function M → C such that ϕ(I ) = 1 = ϕ. (Note that the second condition is equivalent to the positivity of ϕ.) The books [4, 5] – among many others – explain the basic facts about C*-algebras. The class of finite dimensional full matrix algebras form a small and algebraically rather trivial subclass of C*-algebras, but from the view-point of non-commutative statistics, almost all ideas and concepts appear in this setting. A matrix algebra Mn (C) admits a canonical trace Tr and all states are described by their densities with respect to Tr. The correspondence is given by ϕ(A) = TrDϕ A (A ∈ Mn (C)) and we can simply identify the functional ϕ by the density Dϕ . Note that the density is a positive (semi-definite) matrix of trace 1. Let M and N be C*-algebras. Recall that 2-positivity of α : M → N means that α(A) α(B) AB ≥0 if ≥0 α(C) α(D) CD

Sufficiency in quantum statistical inference

261

for 2×2 matrices with operator entries. It is well-known that a 2-positive unit-preserving mapping α satisfies the Schwarz inequality α(A∗ A) ≥ α(A)∗ α(A).

(1)

A 2-positive unital mapping between C*-algebras will be called coarse-graining. Here are two fundamental examples. Example 1. Let X be a finite set and N be a C*-algebra. Assume that for each x ∈ X a positive operator E(x) ∈ N is given and x E(x) = I . In quantum mechanics such a setting is a model for a measurement with values in X . The space C(X ) of function on X is a C*-algebra and the partition of unity E induces a coarse-graining α : C(X ) → N given by α(f ) = x f (x)E(x). Therefore a coarsegraining defined on a commutative algebra is an equivalent way to give a measurement. (Note that the condition of 2-positivity is automatically fulfilled on a commutative algebra.) Example 2. Let M be the algebra of all bounded operators acting on a Hilbert space H and let N be the infinite tensor product M ⊗ M ⊗ . . .. (To understand the essence of the example one does not need the very formal definition of the infinite tensor product.) If γ denotes the right shift on N , then we can define a sequence αn of coarse-grainings M → N: 1 αn (A) := A + γ (A) + · · · + γ n−1 (A) . n αn is the quantum analogue of the sample mean. Let (Xi , Ai , µi ) be a measure space (i = 1, 2). Recall that a positive linear map M : L∞ (X1 , A1 , µ1 ) → L∞ (X2 , A2 , µ2 ) is called a Markov operator if it satisfies M1 = 1 and fn 0 implies Mfn 0. For mappings defined between von Neumann algebras, the monotone continuity is called normality. In the case that M and N are von Neumann algebras, a coarse-graining M → N will be always supposed to be normal. Our concept of coarse-graining is the analogue of the Markov operator. We mostly mean that a coarse-graining transforms observables to observables corresponding to the Heisenberg picture and in this case we assume that it is unit preserving. The dual of such a mapping acts on states or on density matrices and it will be called coarse-graining as well. We recall some well-known results from mathematical statistics, see [24] for details. Let (X, A) be a measurable space and let P = {Pθ : θ ∈ } be a set of probability measures on (X, A). A sub-σ -algebra A0 ⊂ A is sufficient for P if for all A ∈ A, there is an A0 -measurable function fA such that for all θ, fA = Pθ (A|A0 ) Pθ − almost everywhere, that is,

Pθ (A ∩ A0 ) =

fA dPθ

(2)

A0

for all A0 ∈ A0 and for all θ. It is clear from this definition that if A0 is sufficient then for all Pθ there is a common version of the conditional expectation Eθ [g|A0 ] for any measurable step function g, or, more generally, for any function g ∈ ∩θ∈ L1 (X, A, Pθ ). In the most important case, the family P is dominated, that is there is a σ -finite measure µ such that Pθ is absolutely continuous with respect to µ for all θ, this will be denoted by P 0 centered at x. We shall drop writing either x, or R in the notation of the sphere (ball) in the particular cases when either x = 0, or R = 1. For a fixed M > 1 we define the spherical shell A(M) := [k ∈ Rd∗ : M −1 ≤ |k| ≤ M] in the k-space, and A(M) := Rd × A(M) in the whole phase space. Given a vector v ∈ Rd∗ we denote by vˆ := v/|v| ∈ Sd−1 the unit vector in the direction of v. For any set A we shall denote by Ac its complement. For any non-negative integers p, q, r, positive times T > T∗ ≥ 0 and a function G : [T∗ , T ] × R2d ∗ → R that has p, q and r derivatives in the respective variables we define β γ [T∗ ,T ] := sup |∂tα ∂x ∂k G(t, x, k)|. (2.1) Gp,q,r (t,x,k)∈[T∗ ,T ]×R2d

The summation range covers all integers 0 ≤ α ≤ p and all integer valued multiindices |β| ≤ q and |γ | ≤ r. In the special case when T∗ = 0, T = +∞ we write p,q,r [0,+∞) ([0, +∞) × R2d Gp,q,r = Gp,q,r . We denote by Cb ∗ ) the space of all functions G with Gp,q,r < +∞. We shall also consider spaces of bounded and a suitable p p,q d number of times continuously differentiable functions Cb (R2d ∗ ) and Cb (R∗ ) with the respective norms · p,q and · p . 2.2. The background Hamiltonian. We assume that the background Hamiltonian H0 (k) is isotropic, that is, it depends only on k = |k|, and is uniform in space. Moreover, we assume that H0 : [0, +∞) → R is a strictly increasing function satisfying H0 (0) ≥ 0 and such that it is of C 3 -class of regularity in (0, +∞) with H0 (k) > 0 for all k > 0, and let h∗ (M):=

max

k∈[M −1 ,M]

(H0 (k)+|H0

(k)| + |H0

(k)|),

h∗ (M):=

min

k∈[M −1 ,M]

H0 (k). (2.2)

Two examples of such Hamiltonians are the quantum Hamiltonian H0 (k) = k 2 /2 and the acoustic wave Hamiltonian H0 (k) = c0 k. 2.3. The random medium. Let (, , P) be a probability space, and let E denote the expectation with respect to P. We denote by XLp () the Lp -norm of a given random variable X : → R, p ∈ [1, +∞]. Let H1 : Rd ×[0, +∞)× → R be a random field that is measurable and strictly stationary in the first variable. This means that for any shift x ∈ Rd and a collection of points k1 , . . . , kn ∈ [0, +∞), x1 , . . . , xn ∈ Rd the laws of (H1 (x1 + x, k1 ), . . . , H1 (xn + x, kn )) and (H1 (x1 , k1 ), . . . , H1 (xn , kn )) are identical. In addition, we assume that EH1 (x, k) = 0 for all k ≥ 0, x ∈ Rd , the realizations of H1 (x, k) are P–a.s. C 2 -smooth in (x, k) ∈ Rd × (0, +∞) and they satisfy Di,j (M):=max

ess-sup

|α|=i (x,k,ω)∈Rd ×[M −1 ,M]×

j

|∂xα ∂k H1 (x, k; ω)| < +∞,

i, j = 0, 1, 2. (2.3)

280

T. Komorowski, L. Ryzhik

˜ We define D(M) := 0≤i+j ≤2 Di,j (M). We suppose further that the random field is strongly mixing in the uniform sense. More precisely, for any R > 0 we let CRi and CRe be the σ –algebras generated by random variables H1 (x, k) with k ∈ [0, +∞), x ∈ BR and x ∈ BcR respectively. The uniform mixing coefficient between the σ –algebras is e ], φ(ρ) := sup[ |P(B) − P(B|A)| : R > 0, A ∈ CRi , B ∈ CR+ρ

for all ρ > 0. We suppose that φ(ρ) decays faster than any power: for each p > 0, hp := sup ρ p φ(ρ) < +∞.

(2.4)

ρ≥0

The two-point spatial correlation function of the random field H1 is R(y, k) := E[H1 (y, k)H1 (0, k)]. Note that (2.4) implies that for each p > 0, hp (M) :=

4

sup

d −1 i=0 |α|=i (y,k)∈R ×[M ,M]

(1+|y|2 )p/2 |∂yα R(y, k)| < +∞,

M > 0. (2.5)

We also assume that the correlation function R(y, l) is of the C ∞ -class for a fixed l > 0, is sufficiently smooth in l, and that for any fixed l > 0, ˆ R(k, l) does not vanish identically on any hyperplane Hp = {k : (k · p) = 0}. (2.6)

ˆ Here R(k, l) = R(x, l) exp(−ik · x)dx is the power spectrum of H1 . The above assumptions are satisfied, for example, if H1 (x, k) = c1 (x)h(k), where c1 (x) is a stationary uniformly mixing random field with a smooth correlation function, and h(k) is a smooth deterministic function. 2.4. Certain path-spaces. For fixed integers d, m ≥ 1 we let C d,m := C([0, +∞); Rd × Rm ∗ ): we shall omit the subscripts in the notation of the path space if m = d. We define (X(t), K(t)) : C d,m → Rd ×Rm ∗ as the canonical mapping (X(t; π ), K(t; π )) := π(t), π ∈ C d,m and also let θs (π )(·) := π(· + s) be the standard shift transformation. For any u ≤ v denote by Mvu the σ -algebra of subsets of C generated by (X(t), K(t)), t ∈ [u, v]. We write Mv := Mv0 and M for the σ algebra of Borel subsets of C. It coincides with the smallest σ –algebra that contains all Mt , t ≥ 0. (0) (0) ˜ For a given M > 0 and δ ∈ (0, δ∗ (M)] we Let δ∗ (M) := H0 M −1 /(2D(M)). let −1 √ √ 1 −1 −1 ˜ ˜ . H0 H0 −2 δ D(M) Mδ := max H0 (H0 (M)+2 δ D(M)), M (2.7) (0)

We select δ∗ (M) ∈ (0, δ∗ (M)) in such a way that Mδ < 2M for all δ ∈ (0, δ∗ (M)). For a particle that is governed by the Hamiltonian flow generated by Hδ (x, k) we have Mδ−1 ≤ |K(t)| ≤ Mδ for all t provided that K(0) ∈ A(M). Accordingly, we define C(T , δ) as the set of paths π ∈ C so that both (2Mδ )−1 ≤ |K(t)| ≤ 2Mδ , and

Diffusion in a Weakly Random Hamiltonian Flow

281

t √ X(t) − X(u) − H (K(s))K(s)ds ≤ D(2M ˆ ˜ δ ) δ(t − u), for all 0 ≤ u < t ≤ T . 0 u

In the case when δ = 1, or T = +∞ we shall write simply C(T ), or C(δ) respectively. 2.5. The main results. Let the function φδ (t, x, k) satisfy the Liouville equation ∂φ δ + ∇x Hδ (x, k) · ∇k φ δ − ∇k Hδ (x, k) · ∇x φ δ = 0, ∂t φ δ (0, x, k) = φ0 (δx, k).

(2.8)

We assume that the initial data φ0 (x, k) is a compactly supported function four times differentiable in k, twice differentiable in x whose support is contained inside a spherical shell A(M) = {(x, k) : M −1 < |k| < M} for some M > 1. Let us define the diffusion matrix Dmn (l, l) for l ∈ Sd−1 and l > 0 by

1 ∞ ∂ 2 R(H0 (l)sl, l) ds Dmn (l, l) = − 2 −∞ ∂xn ∂xm

∞ 2 1 ∂ R(sl, l) =− ds, m, n = 1, . . . , d. (2.9) 2H0 (l) −∞ ∂xn ∂xm Then we have the following result. Theorem 2.1. Assume that d ≥ 3. Let φ δ be the solution of (2.8) and let φ¯ ∈ Cb1,1,2 ([0, +∞); R2d ∗ ) satisfy d ¯ ∂ ∂ φ¯ ˆ k) ∂ φ + H (k) kˆ · ∇x φ, ¯ Dmn (k, = 0 ∂t ∂kn m,n=1 ∂km (2.10) ¯ x, k) = φ0 (x, k). φ(0, Suppose that M > 1. Then, there exist two constants C, α0 > 0 such that for all T ≥ 1 and all compact sets K ⊂ A(M) we have δ t x Eφ ¯ x, k) ≤ CT (1 + φ0 1,4 )δ α0 . (2.11) , , k − φ(t, sup δ δ (t,x,k)∈[0,T ]×K Remark 2.2. We shall denote by C, C1 , . . . , α0 , α1 , . . . , γ0 , γ1 , . . . throughout this article generic positive constants. Unless specified otherwise the constants denoted this way shall depend neither on δ, nor on T . In the statement of the results appearing throughout the paper we will always assume, unless stated otherwise, that the parameter T ≥ 1 Remark 2.3. Classical results of the theory of stochastic differential equations, see e.g. Theorem 6 of Chapter 2, p. 176 and Corollary 4 of Chapter 3, p. 303 of [6], imply that there exists a unique solution to the Cauchy problem (2.10) that belongs to the class Cb1,1,2 ([0, +∞) × R2d ∗ ). This solution admits a probabilistic representation using the law of a time homogeneous diffusion Qx,k whose Kolmogorov equation is given by (2.10), see Sect. 3.6 below. Note that

282

T. Komorowski, L. Ryzhik d

ˆ k)kˆm = − Dnm (k,

m=1

d m=1

=−

d m=1

1 2H0 (k) 1 2H0 (k)

∞ −∞

∞ −∞

ˆ k) ∂ 2 R(s k, kˆm ds ∂xn ∂xm ˆ k) d ∂R(s k, ds = 0, ds ∂xn

and thus the K-process generated by (2.10) is indeed a diffusion process on a sphere Sd−1 k , or, equivalently, Eqs. (2.10) for different values of k are decoupled. Assumption (2.6) implies the following. Proposition 2.4. The matrix D(l, l) has rank d − 1 for each l ∈ Sd−1 and l > 0. The proof is the same as that of Proposition 4.3 in [1]. It can be shown, using the argument given on pp. 122–123 of ibid., that, under assumption (2.6), Eq. (2.10) is hypoelliptic on the manifold Rd × Sd−1 for each k > 0. k We also show that solutions of (2.10) converge in the long time limit to the solutions of the spatial diffusion equation. More, precisely, we have the following re2 , x/γ , k), where φ¯ satisfies (2.10) with an initial data ¯ sult. Let φ¯ γ (t, x, k) = φ(t/γ φ¯ γ (0, t, x, k) = φ0 (γ x, k) and let w(t, x, k) be the solution of the spatial diffusion equation: d ∂w 2w = amn (k) ∂x∂n ∂x , m ∂t m,n=1

(2.12)

w(0, x, k) = φ¯ 0 (x, k), with the averaged initial data φ¯ 0 (x, k) =

1

d−1 Sd−1

φ0 (x, kl)d(l).

Here d(l) is the surface measure on the unit sphere Sd−1 and n is the area of the n-dimensional unit sphere. The diffusion matrix A(k) := [anm (k)] in (2.12) is given explicitly as

1 anm (k) = H (k)ln χm (kl)d(l), (2.13) d−1 Sd−1 0 where l = (l1 , . . . , ld ). The functions χj appearing above are the mean-zero solutions of d m,n=1

∂ ∂km

ˆ k) Dmn (k,

∂χj ∂kn

= −H0 (k)kˆj .

(2.14)

Note that Eqs. (2.14) for χm are elliptic on each sphere Sd−1 k . This follows from the fact that the equations for each such sphere are all decoupled and Proposition 2.4. (1) (1) Also note that the matrix A(k) is symmetric. Indeed, let c1 = (c1 , . . . , cd ), c2 = (2) (2) (i) d (c1 , . . . , cd ) ∈ Rd be fixed vectors and let χci := m=1 cm χm , i = 1, 2. Since the matrix D is symmetric we have


(A(k)c1 , c2 )Rd = − =− =

1 d−1 1

d d−1 m,n=1 S

d

χc1 (k)

283

∂ ∂km

∂ χc1 (l) d ∂l m R

ˆ k) Dmn (k,

∂χc2 (k) ˆ d(k) ∂kn

dl ∂χc2 (l) ˆ Dmn (l, |l|) δ(k − |l|) d−1 ∂ln |l|

d−1 m,n=1

1 ˆ k)∇χc1 (k k), ˆ ∇χc2 (k k)) ˆ Rd d(k)=(c ˆ (D(k, 1 , A(k)c2 )Rd .

d−1 Sd−1

(2.15) ˆ k)kˆ = 0. MoreThe last but one equality holds after integration by parts because D(k, over, substituting c1 = c2 we obtain that

1 ˆ k)∇χc1 (k k), ˆ ∇χc1 (k k)) ˆ Rd d(k) ˆ ≥ 0. (A(k)c1 , c1 )Rd = (D(k, d−1 Sd−1 k (2.16) In fact, the above inequality holds in the strict sense. This can be seen as follows. Since ˆ k) is one-dimensional and consists for each kˆ ∈ Sd−1 the null-space of the matrix D(k, ˆ in order for (A(k)c1 , c1 )Rd to vanish one needs that the of the vectors parallel to k, ˆ is parallel to kˆ for all k. ˆ This, however, together with (2.14), would gradient ∇χc1 (k k) d−1 imply that kˆ · c1 = 0 for all kˆ ∈ S , which is impossible. The following theorem holds. Theorem 2.5. For every 0 < T∗ < T < +∞ the re-scaled solution φ¯ γ (t, x, k) = 2 , x/γ , k) of (2.10) converges as γ → 0+ in C([T , T ]; L∞ (R2d )) to w(t, x, k). ¯ φ(t/γ ∗ Moreover, there exists a constant C > 0 so that we have √ w(t, ·) − φ¯ γ (t, ·)0,0 ≤ C γ T + γ φ0 1,1 (2.17) for all T∗ ≤ t ≤ T . Remark 2.6. In fact, as it will become apparent in the course of the proof, we have a stronger result, namely T∗ can be made to vanish as γ → 0+ . For instance, we can choose T∗ = γ 3/2 , see (4.16). The proof of Theorem 2.5 is based on some classical asymptotic expansions and is quite straightforward. As an immediate corollary of Theorems 2.1 and 2.5 we obtain the following result, which is the main result of this paper. Theorem 2.7. Assume that d ≥ 3, T∗ > 0 and M > 1. Let φδ be solution of (2.8) with the initial data φδ (0, x, k) = φ0 (δ 1+α x, k) and let w(t, ¯ x) be the solution of the diffusion equation (2.12) with the initial data w(0, x, k) = φ¯ 0 (x, k). Then, there exists α0 > 0 and a constant C > 0 so that for all 0 ≤ α < α0 , T∗ ≤ T and all compact sets K ⊂ A(M) we have: w(t, x, k) − Eφ¯ δ (t, x, k) ≤ CT δ α0 −α , sup (2.18) (t,x,k)∈[T∗ ,T ]×K

where φ¯ δ (t, x, k) := φδ t/δ 1+2α , x/δ 1+α , k .

284


Theorem 2.7 shows that the movement of a particle in a weakly random quenched Hamiltonian is, indeed, approximated by a Brownian motion in the long time-large space limit, at least for times T δ −α0 . In fact, according to Remark 2.6 we can allow T∗ to vanish as δ → 0 choosing T∗ = δ 3α/2 . The estimate (2.18) shall still be valid then, provided that T is not too small, e.g. one can assume that T ≥ 1. In the isotropic case when R = R(|x|, k) we may simplify the above expressions for the diffusion matrices [Dmn ] and [amn ]. In that case we have

∞ 2 ˆ k) ∂ R(H0 (k)s k, ˆ k) = − 1 ds Dmn (k, 2 −∞ ∂xn ∂xm

∞ kn km

kn km R (H0 (k)s, k) =− R (H (k)s, k) + δ − ds nm 0 k2 k2 H0 (k)s 0

∞

R (s, k) k n km 1 , ds δnm − 2 =−

H0 (k) 0 s k ˆ k)] has the form so that the matrix [Dmn (k, ˆ k) = D0 (k) I − kˆ ⊗ kˆ , D0 (k) = − D(k,

1 H0 (k)

∞ 0

R (s, k) ds. s

In that case the functions χj are given explicitly by ˆ k) = − χj (k,

(H0 (k))2 k 2 kˆj , D¯ 0 (k) = − (d − 1)D¯ 0 (k)

0

∞

R (s, k) ds s

and

(H0 (k))3 k 2 (H0 (k))3 k 2 ˆ = anm (k) = kˆn kˆm d(k) δnm . d−1 (d − 1)D¯ 0 (k) Sd−1 d(d − 1)D¯ 0 (k) 2.6. A formal derivation of the momentum diffusion. We now recall how the diffusion operator in (2.10) can be derived in a quick formal way. We represent the solution of (2.8) as φ δ (t, x, k) = ψ δ (δt, δx, k) and write an asymptotic multiple scale expansion for ψ δ , √ x x ¯ x, k) + δφ1 t, x, , k + δφ2 t, x, , k + . . . . (2.19) ψ δ (t, x, k) = φ(t, δ δ We assume formally that the leading order term φ¯ is deterministic and independent of the fast variable z = x/δ. We insert this expansion into (2.8) and obtain in the order O δ −1/2 : ∇z H1 (z, k) · ∇k φ¯ − H0 (k)kˆ · ∇z φ1 = 0.

(2.20)

Let θ 1 be a small positive regularization parameter that will be later sent to zero, and consider a regularized version of (2.20): 1

¯ − kˆ · ∇z φ1 + θφ1 = 0.

∇z H1 (z, k) · ∇k φ H0 (k)


285

Its solution is 1 φ1 (t, x, z, k) = −

H0 (k)

∞ 0

d ˆ k) ∂ φ(t, ¯ x, k) −θs ∂H1 (z + s k, e ds. ∂zm ∂km

(2.21)

m=1

The next order equation becomes upon averaging ∂ φ¯ ∂H1 (z, k) ˆ ¯ =E k · ∇z φ1 − E (∇z H1 (z, k) · ∇k φ1 ) + H0 (k)kˆ · ∇x φ. ∂t ∂k (2.22) The first two terms on the right hand side above may be computed explicitly using expression (2.21) for φ1 : ∂H1 (z, k) ˆ E k · ∇z φ1 − E (∇z H1 (z, k) · ∇k φ1 ) ∂k  

∞ d ˆ k) ∂ φ(t, ¯ x, k) −θs 1 ∂H (z, k) ∂ ∂H (z + s k, 1 1 = −E  e ds  kˆm ∂k ∂zm H0 (k) 0 ∂zn ∂kn m,n=1  

∞ d ˆ k) ∂ φ(t, ¯ x, k) −θs 1 ∂H (z, k) ∂ ∂H (z + s k, 1 1 +E  e ds  . ∂zm ∂km H0 (k) 0 ∂zn ∂kn m,n=1

Using spatial stationarity of H1 (z, k) we may rewrite the above as  

∞ d ˆ ¯ 1 ∂H1 (z, k) ˆ ∂ ∂H1 (z + s k, k) ∂ φ(t, x, k) −θs −E  e ds  km ∂k ∂zm H0 (k) 0 ∂zn ∂kn m,n=1  

∞ d ˆ ¯ 1 ∂H1 (z + s k, k) ∂ φ(t, x, k) −θs ∂ ∂ −E  H1 (z, k) e ds  ∂zm ∂km H0 (k) 0 ∂zn ∂kn m,n=1

∞ d ˆ k) ∂ φ(t, ¯ x, k) −θs 1 ∂ ∂ 2 H1 (z + s k, =− E H1 (z, k) e ds ∂km H0 (k) 0 ∂zn ∂zm ∂kn m,n=1

∞ 2 d ˆ k) ∂ φ(t, ¯ x, k) −θs 1 ∂ ∂ R(s k, =− e ds ∂km H0 (k) 0 ∂xn ∂xm ∂kn m,n=1

∞ 2 d ˆ k) ∂ φ(t, ¯ x, k) 1 ∂ 1 ∂ R(s k, →− ds , as θ → 0+ . 2 ∂km H0 (k) −∞ ∂xn ∂xm ∂kn m,n=1

We insert the above expression into (2.22) and obtain d ∂ ∂ φ¯ ∂ φ¯ ˆ = Dnm (k, k) + H0 (k)kˆ · ∇x φ¯ ∂t ∂kn ∂km

(2.23)

m,n=1

ˆ k) as in (2.9). Observe that (2.23) is nothing but (2.10). with the diffusion matrix D(k, However, the naive asymptotic expansion (2.19) may not be justified. The rigorous proof presented in the next section is based on a quite different method.

286


3. From the Liouville Equation to the Momentum Diffusion. Estimation of the Convergence Rates: Proof of Theorem 2.1 3.1. Outline of the proof. The basic idea of the proof of Theorem 2.1 is a modification of that of [1, 9]. We consider the trajectories corresponding to the Liouville equation (2.8) and introduce a stopping time, called τδ , that, among others, prevents near selfintersection of trajectories. This fact ensures that until the stopping time occurs the particle is “exploring a new territory” and, thanks to the strong mixing properties of the medium, “memory effects” are lost. Therefore, roughly speaking, until the stopping time the process is approximately characterized by the Markov property. Furthermore, since the amplitude of the random Hamiltonian is not strong enough to destroy the continuity of its path, it becomes a diffusion in the limit, as δ → 0. We introduce also an augmented process that follows the trajectories of the Hamiltonian flow until the stopping time τδ and becomes a diffusion after t = τδ . We show that the law of the augmented process is close to the law of a diffusion, see Proposition 3.4, with an explicit error bound. We also prove that the stopping time tends to infinity as δ → 0, once again with the error bound that is proved in Theorem 3.6. The combination of these two results allows us to estimate the difference between the solutions of the Liouville and the diffusion equations in a rather straightforward manner (see Sect. 3.9): they are close until the stopping time as the law of the diffusion is always close to that of the augmented process, while the latter coincides with the true process until τδ . On the other hand, the fact that τδ → ∞ as δ → 0 shows that with a large probability the augmented process is close to the true process. This combination finishes the proof. 3.2. The random characteristics corresponding to (2.8). Consider the motion of a particle governed by a Hamiltonian system of equations  (δ)  dz(δ) (t;x,k) z (t;x,k) (δ) (t; x, k)  = (∇ H ) , m  k δ dt δ      (δ) (δ) dm (t;x,k) (3.1) = − √1 (∇z Hδ ) z (t;x,k) , m(δ) (t; x, k) dt δ δ        z(δ) (0; x, k) = x, m(δ) (0; x, k) = k,  √ where the Hamiltonian Hδ (x, k) := H0 (k) + δH1 (x, k), k = |k|. The trajectories of (3.1) are the characteristics of the Liouville equation (2.8). The hypotheses made in Sect. 2 imply that the trajectory (z(δ) (t; x, k), m(δ) (t; x, k)) necessarily lies in C(T , δ) for each T > 0, δ ∈ (0, δ∗ (M)], provided that the initial data (x, k) ∈ A(M). Indeed, it follows √ from the Hamiltonian structure of (3.1) that the Hamiltonian Hδ (x, m) = H0 (m) + δH1 (x, m) must be conserved along the trajectory. Hence, the definition (2.7) implies that Mδ−1 ≤ |m(δ) (·; x, k)| ≤ Mδ . We denote by Qδs,x,k (·) the law over C of the process corresponding to (3.1) starting at t = s from (x, k) (this law is actually supported in C(δ)). We shall omit writing the subscript s when it equals to 0. 3.3. The stopping times. We now define the stopping time τδ , described in Sect. 3.1, that prevents the trajectories of (3.1) to have near self-intersections (recall that the intent of the stopping time is to prevent any “memory effects” of the trajectories). As we have already mentioned, we will later show that the probability of the event [ τδ < T ] for a fixed T > 0 goes to zero, as δ → 0.


287

Let 0 < 1 < 2 < 1/2, 3 ∈ (0, 1/2 − 2 ), 4 ∈ (1/2, 1 − 1 − 2 ) be small positive constants that will be further determined later and set N = [δ −1 ],

p = [δ −2 ],

q = p [δ −3 ],

N1 = Np [δ −4 ].

(3.2)

We will specify additional restrictions on the constants j as the need for such constraints arises. However, the basic requirement is that i , i ∈ {1, 2, 3} should be sufficiently small and 4 is bigger than 1/2, less than one and can be made as close to one as we would need it. It is important that 1 < 2 so that N p when δ 1. We introduce the (p) following (Mt )t≥0 –stopping times. Let tk := kp −1 be a mesh of times, and π ∈ C be a path. We define the “violent turn” stopping time (p) (p) Sδ (π ) := inf t ≥ 0 : for some k ≥ 0 we have t ∈ tk , tk+1 and

1 1 ˆ (p) ) · K(t) ˆ ˆ ˆ t (p) − 1 · K(t) K(t ≤ 1 − ≤ 1 − , or K , k−1 k N N1 N (3.3) ˆ ˆ where by convention we set K(−1/p) := K(0). Note that with the above choice of 4 (p) (p) ˆ ˆ we have K tk − 1/N1 · K(tk ) > 1 − 1/N , Qδx,k -a.s., provided that δ ∈ (0, δ0 ] and δ0 is sufficiently small. We adopt in (3.3) a customary convention that the infimum of an empty set equals +∞. The stopping time Sδ is triggered when the trajectory performs a sudden turn – this is undesirable as the trajectory may then return to the region it has already visited and create correlations with the past. For each t ≥ 0, we denote by Xt (π ) := X (s; π ) the trace of the spatial com0≤s≤t

ponent of the path π up to time t, and by Xt (q; π) := [x : dist (x, Xt (π )) ≤ 1/q] a tubular region around the path. We introduce the stopping time

(p) (p) Uδ (π ) := inf t ≥ 0 : ∃ k ≥ 1 and t ∈ [tk , tk+1 ) for which X(t) ∈ Xt (p) (q) . k−1

(3.4) It is associated with the return of the X component of the trajectory to the tube around its past – this is again an undesirable way to create correlations with the past. Finally, we set the stopping time τδ (π ) := Sδ (π ) ∧ Uδ (π ) ∧ δ −1 .

(3.5)

The last term appearing on the right hand side of (3.5) ensures that τδ < +∞, Qδx,k -a.s. 3.4. The cut-off functions and the corresponding dynamics. Let M > 1 be fixed and p, q, N, N1 be the positive integers defined in Sect. 3.3. We define now several auxiliary functions that will be used to introduce the cut-offs in the dynamics. These cut-offs will ensure that the particle moving under the modified dynamics will avoid self-intersections, will have no violent turns and the changes of its momentum will be under control. In addition, up to the stopping time τδ the motion of the particle will coincide with the motion under the original Hamiltonian flow. Let a1 = 2 and a2 = 3/2. The functions ψj : Rd × Sd−1 → [0, 1], j = 1, 2 are of C ∞ class and satisfy

288


ψj (k, l) =

   1,

if kˆ · l ≥ 1−1/N

and Mδ−1 ≤ |k| ≤ Mδ

  0,

if kˆ · l ≤ 1−aj /N,

or |k|≤(2Mδ )−1 ,

(3.6) or |k| ≥ 2Mδ .

One can construct ψj in such a way that for arbitrary nonnegative integers m, n it is possible to find a constant Cm,n for which ψj m,n ≤ Cm,n N m+n . The cut-off function (p) (p) (p) (p) for t ∈ [tk , tk+1 ) and k≥1, ψ1 k, Kˆ tk−1 ψ2 k, Kˆ tk −1/N1 (t, k; π ):= (p) ˆ ψ2 (k, K(0)) for t∈[0, t ), 1

(3.7) will allow us to control the direction of the particle motion over each interval of the partition as well as not to allow the trajectory to escape to the regions where the change of the size of the velocity can be uncontrollable. Let φ : Rd × Rd → [0, 1] be a function of the C ∞ class that satisfies φ(y, x) = 1, when |y − x| ≥ 1/(2q) and φ(y, x) = 0, when |y − x| ≤ 1/(3q). Again, in this case we can construct φ in such a way that φm,n ≤ Cq m+n for arbitrary integers m, n and a suitably chosen constant C. Let h∗ := [4h∗ (M)] + 1 and q∗ := qh∗ , cf. (2.2). The function φk : Rd × C → [0, 1] for a fixed path π is given by l φk (y; π) = . (3.8) φ y, X q∗ (p) 0≤l/q∗ ≤tk−1

We set (t, y; π) :=

  1,  φk (y; π),

(p)

if 0 ≤ t < t1 (p)

if tk

(p)

≤ t < tk+1 .

(3.9)

The function shall be used to modify the dynamics of the particle in order to avoid a possibility of near self-intersections of its trajectory. For a given t ≥ 0, (y, k) ∈ R2d ∗ and π ∈ C let us denote (t, y, k; π ) := (t, k; π ) (t, y; π ) . The following lemma can be verified by a direct calculation. Lemma 3.1. Let (β1 , β2 ) be a multi-index with nonnegative integer valued components, m = |β1 | + |β2 |. There exists a constant C depending only on m and M such that β β |∂y 1 ∂k 2 (t, y, k; π )| ≤ CT |β1 | q 2|β1 | N |β2 | for all t ∈ [0, T ], (y, k) ∈ A(2M), π ∈ C. Finally, let us set Fδ (t, y, l; π, ω) = (t, δy, l; π)∇y H1 (y, |l|; ω) .

(3.10)

Note that according to Lemma 3.1 we obtain that |∂y 1 ∂k 2 (t, δy, l; π)| ≤ CT |β1 | δ |β1 |[1−2(2 +3 )] N |β2 | β

β

(3.11)

for all t ∈ [0, T ], (y, k) ∈ A(2M), π ∈ C. For a fixed (x, k) ∈ R2d ∗ , δ > 0 and ω ∈ we consider the modified particle dynamics with the cut-off that is described by the stochastic process (y (δ) (t; x, k, ω), l (δ) (t; x, k, ω))t≥0 whose paths are the solutions of the following equation:


289

 (δ) (δ)  √  d y (δ) (t;x,k) y (t;x,k) (δ)  = H0 (|l (δ) (t; x, k)|)+ δ ∂l H1 , |l (t; x, k)| lˆ (t; x, k),   dt δ   (δ) (3.12) y (δ) (t;x,k) (δ) d l (t;x,k) (δ) (·; x, k), l (δ) (·; x, k) √1 Fδ t, = − , l (t; x, k); y   dt δ δ      (δ) y (0; x, k)=x, l (δ) (0; x, k) = k. (δ) (δ) ˜ We will denote by Q x,k the law of the modified process (y (·; x, k), l (·; x, k)) over (δ) C for a given δ > 0 and by E˜ x,k the corresponding expectation. We assume that the initial momentum k ∈ A(M). From the construction of the cut-offs we immediately conclude that (δ)

2 (δ) (δ) (p) lˆ (t) · lˆ (tk−1 ) ≥ 1− , N

(p)

(p)

t ∈ [tk−1 , tk+1 ),

∀ k ≥ 0.

(3.13)

3.5. Some consequences of the mixing assumption. For any t ≥ 0 we denote by Ft the σ -algebra generated by (y (δ) (s), l (δ) (s)), s ≤ t. Here we suppress, for the sake of abbreviation, writing the initial data in the notation of the trajectory. In this section we 2 assume that M > 1 is fixed, X1 , X2 : (R × Rd × Rd )2 → R are certain continuous functions, Z is a random variable and g1 , g2 are Rd ×[M −1 , M]-valued random vectors. We suppose further that Z, g1 , g2 , are Ft -measurable, while X˜ 1 , X˜ 2 are random fields of the form j j j . X˜ i (x, k) = Xi ∂k H1 (x, k), ∇x ∂k H1 (x, k), ∇x2 ∂k H1 (x, k) j =0,1

For i = 1, 2 we denote gi := (gi , gi ), where gi ∈ Rd and gi ∈ [M −1 , M]. We also let U (θ1 , θ2 ) := E X˜ 1 (θ1 )X˜ 2 (θ2 ) , θ1 , θ2 ∈ Rd × [M −1 , M]. (3.14) (1)

(2)

(1)

(2)

The following mixing lemma is useful in formalizing the “memory loss effect” and can be proved in the same way as Lemmas 5.2 and 5.3 of [1]. Lemma 3.2. (i) Assume that r, t ≥ 0 and (δ) (1) y (u) r inf gi − ≥ , u≤t δ δ

(3.15)

P–a.s. on the set Z = 0 for i = 1, 2. Then, we have r X1 L∞ X2 L∞ ZL1 () . E X˜ 1 (g1 )X˜ 2 (g2 )Z − E [U (g1 , g2 )Z] ≤ 2φ 2δ (3.16) (ii) Let EX˜ 1 (0, k) = 0 for all k ∈ [M −1 , M]. Furthermore, we assume that g2 satisfies (3.15), (δ) (1) y (u) r + r1 inf g1 − (3.17) ≥ u≤t δ δ

290


and |g1 − g2 | ≥ r1 δ −1 for some r1 ≥ 0, P-a.s. on the event Z = 0. Then, we have (1)

(1)

r φ 1/2 E X˜ 1 (g1 )X˜ 2 (g2 ) Z − E [U (g1 , g2 )Z] ≤ Cφ 1/2 2δ r 1 X1 L∞ X2 L∞ ZL1 () × 2δ

(3.18)

for some absolute constant C > 0. Here the function U is given by (3.14). 3.6. The momentum diffusion. Let k(t) be a diffusion, starting at k ∈ Rd∗ at t = 0, with the generator of the form LF (k) =

d

ˆ |k|)∂k2 ,k F (k) + Dmn (k, m n

m,n=1

=

d

d

ˆ |k|)∂km F (k) Em (k,

m=1

ˆ |k|)∂kn F (k) , ∂km Dm,n (k,

F ∈ C0∞ (Rd∗ ).

(3.19)

m,n=1

Here the diffusion matrix is given by (2.9) and the drift vector is ˆ l) = − Em (k,

d ˆ l) 1 +∞ ∂ 3 R(s k, s ds,

H0 (l)l ∂xm ∂xn2 0

m = 1, . . . , d.

n=1

Let Qx,k be the law of the process (x(t), k(t)) that starts at t = 0 from (x, k) given t ˆ by x(t) = x + 0 H0 (|k(s)|)k(s)ds, where k(t) is the diffusion described by (3.19). This process is a degenerate diffusion whose generator is given by ˜ (x, k) = Lk F (x, k) + H0 (|k|) kˆ · ∇x F (x, k), LF

F ∈ Cc∞ (R2d ∗ ).

(3.20)

Here the notation Lk stresses that the operator L defined in (3.19) acts on the respective function in the k variable. We denote by Mx,k the expectation corresponding to the path measure Qx,k .

3.7. The augmented process. The following construction of the augmentation of path measures has been carried out in Sect. 6.1 of [16]. Let s ≥ 0 be fixed and π ∈ C. Then, according to Lemma 6.1.1 of ibid. there exists a unique probability measure that is denoted by δπ ⊗s QX(s),K(s) , such that for any pair of events A ∈ Ms , B ∈ M we have δπ ⊗s QX(s),K(s) [A] = 1A (π ) and δπ ⊗s QX(s),K(s) [θs (B)] = QX(s),K(s) [B]. The following result is a direct consequence of Theorem 6.2.1 of [16]. (δ)

Proposition 3.3. There exists a unique probability measure Rx,k on C such that (δ)

(δ)

Rx,k [A] := Qx,k [A] for all A ∈ Mτδ and the regular conditional probability dis(δ)

tribution of Rx,k [ · |Mτδ ] is given by δπ ⊗τδ (π) QX(τδ (π)),K(τδ (π)) , π ∈ C. This measure (δ)

shall be also denoted by Qx,k ⊗τδ QX(τδ ),K(τδ ) .


291

Note that for any (x, k) ∈ A(M) and A ∈ Mτδ we have ˜ [A], Rx,k [A] = Qx,k [A] = Q x,k (δ)

(δ)

(δ)

(3.21)

that is, the law of the augmented process coincides with that of the true process, and of the modified process with the cut-offs until the stopping time τδ . Hence, accord(δ) ing to the uniqueness part of Proposition 3.3, in such a case Qx,k ⊗τδ QX(τδ ),K(τδ ) = (δ) (δ) ˜ ⊗τδ QX(τδ ),K(τδ ) . We denote by E the expectation with respect to the augmented Q x,k

x,k

(δ)

(δ)

measure described by the above proposition. Let also Rx,k,π , Ex,k,π denote the respective (δ)

conditional law and expectation obtained by conditioning Rx,k on Mτδ . The following proposition is of crucial importance for us, as it shows that the law of the augmented process is close to that of the momentum diffusion as δ → 0. To abbreviate the notation we let

t Nt (G) := G(t, X(t), K(t)) − G(0, X(0), K(0)) −

˜ (∂ + L)G(, X(), K())) d

0

for any G ∈ Cb1,1,3 ([0, +∞) × R2d ∗ ) and t ≥ 0. n Proposition 3.4. Suppose that (x, k) ∈ A(M) and ζ ∈ Cb ((R2d ∗ ) ) is nonnegative. Let γ0 ∈ (0, 1/2) and let 0 ≤ t1 < · · · < tn ≤ T∗ ≤ t < v ≤ T . We assume further that v − t ≥ δ γ0 . Then, there exist constants γ1 , C such that for any function G ∈ C 1,1,3 ([T∗ , T ] × R2d ∗ ) we have ! (δ) [T∗ ,T ] 2 (δ) ˜ T Ex,k ζ . (3.22) Ex,k [Nv (G) − Nt (G)] ζ˜ ≤ Cδ γ1 (v − t)G1,1,3

Here ζ˜ (π ) := ζ (X(t1 ), K(t1 ), . . . , X(tn ), K(tn )), π ∈ C(T , δ). The choice of the constants γ1 , C does not depend on (x, k), δ ∈ (0, 1], ζ , times t1 , . . . , tn , T∗ , T , v, t, or the function G. Proof. Let 0 = s0 ≤ s1 ≤ . . . ≤ sn ≤ t and B1 , . . . , Bn ∈ B(R2d ∗ ) be Borel sets. We denote A0 := C and for any k ∈ {1, . . . , n}, s ≤ sk we define the events Ak := [π : (X(s1 ), K(s1 )) ∈ B1 , . . . , (X(sk ), K(sk )) ∈ Bk ], and their shifted counterparts (s)

Ak := [π : (X(sk − s), K(sk − s)) ∈ Bk , . . . , (X(sn − s), K(sn − s)) ∈ Bn ]. 1,1,2 ([0, +∞) × R2d ) we let For (x, k) ∈ R2d ∗ , π ∈ C and G ∈ C ∗

"t G(t, x, k; π ) := H0 (|k|) kˆ · ∇x G(t, x, k) + 2 (t, X(t), K(t); π )Lk G(t, x, k) L −(t, X(t), K(t); π) ×

d m,n=1

ˆ |k|)∂kn G(t, x, k) ∂Km (t, X(t), K(t); π )Dm,n (k,

292


and "t (G) := G(t, X(t), K(t))−G(0, X(0), K(0))− N

t

" )G(, X(), K(); π ) d. (∂ + L

0

It follows from the definition of the stopping time τδ (π ) and the cut-off function that ∇K (t, X(t), K(t); π) = 0, t ∈ [0, τδ (π )], hence ˜ "t G(t, X(t), K(t); π) = LG(t, X(t), K(t); π ), t ∈ [0, τδ (π )]. L We need the following result. n Lemma 3.5. Suppose that (x, k) ∈ A(M) and ζ ∈ Cb ((R2d ∗ ) ) is nonnegative. Let

γ0 ∈ (0, 1), 0 ≤ t1 < · · · < tn ≤ T∗ ≤ t 0 such that for any function G ∈ C 1,1,3 ([T∗ , T ] × R2d ∗ ) we have ! ˜ (δ) " "t (G)]ζ˜ ≤ C δ γ1 (u − t)G[T∗ ,T ] T 2 E˜ (δ) ζ˜ . (3.23) Ex,k [Nu (G) − N 1,1,3 x,k

The choice of the constants γ1 , C does not depend on (x, k), δ ∈ (0, 1], times t1 , . . . , tn , T∗ , T , t, u, or function G. The proof of this lemma follows very closely the argument presented in Sect. 5.3 of [1] and we postpone it until the Appendix. In the meantime we apply this result to conclude the proof of Proposition 3.4. We write (δ)

Ex,k,π [Nv (G) − Nv∧τδ (π) (G), An ] =

n−1

(τ (π))

δ 1[sp ,sp+1 ) (τδ (π ))1Ap (π )MX(τδ (π)),K(τδ (π)) [Nv−τδ (π) (G), Ap+1 ]

p=0

+1[sn ,v) (τδ (π ))1An (π )MX(τδ (π)),K(τδ (π)) [Nv−τδ (π) (G)].

(3.24)

When τδ (π ) ∈ [sp , sp+1 ) we obviously have (τ (π))

(τ (π))

δ δ ] = MX(τδ (π)),K(τδ (π)) [Nt−τδ (π) (G), Ap+1 ] MX(τδ (π)),K(τδ (π)) [Nv−τδ (π) (G), Ap+1

and MX(τδ (π)),K(τδ (π)) [Nv−τδ (π) (G)] = 0. Hence the left hand side of (3.24) equals n−1

(τ (π))

δ 1[sp ,sp+1 ) (τδ (π ))1Ap (π )MX(τδ (π)),K(τδ (π)) [Nt−τδ (π) (G), Ap+1 ]

p=0 (δ)

= Ex,k,π [Nt (G) − Nt∧τδ (π) (G), An ].

(3.25)

We conclude from (3.24), (3.25) that (δ)

(δ)

Ex,k,π [Nv (G), An ] = Ex,k,π [Nv∧τδ (π) (G) + Nt (G) − Nt∧τδ (π) (G), An ] (δ)

= Ex,k,π [N(v∧τδ (π))∨t (G), An ],

(3.26)


293

and therefore

(δ) (δ) (δ) Ex,k [Nv (G), An ] = Ex,k Ex,k,π [N(v∧τδ (π))∨t (G), An ] $ (δ) (δ) # = Ex,k Ex,k,π N(v∧τδ (π))∨t (G), An , τδ (π ) ≤ t $ (δ) (δ) # +Ex,k Ex,k,π N(v∧τδ (π))∨t (G), An , τδ (π ) > t . (3.27) (δ)

The first term on the utmost right hand side of (3.27) equals Ex,k [Nt (G), An , τδ ≤ t], $ (δ) # while the second one equals E˜ N(v∧τδ )∨t (G), B . Here B := An ∩ [τδ > t] is an x,k

Mt -measurable event. Suppose that γ0 ∈ (γ0 + 1/2, 1) and let L := [δ −γ0 ] be yet another mesh size parameter. We define σ := L−1 [([L(v ∧ τδ )] + 2) ∨ ([Lt] + 2)] and note that (δ) E˜ x,k [Nσ (G), B] =

[Lv]+2 p=[Lt]+2

p (δ) E˜ x,k Np/L (G), B, σ = . L

(3.28)

Representing the event [σ = p/L] as the difference of [σ ≥ p/L] and [σ ≥ (p + 1)/L] (note that [σ ≥ ([Lv] + 3)/L] = ∅) and grouping the terms of the sum that correspond to the same index p we obtain that the right-hand side of (3.28) equals

$ [Lv]+2 p+1 (δ) # (δ) ˜ ˜ Ex,k Np+1/L (G) − Np/L (G), B, σ ≥ . Ex,k N([Lt]+2)/L (G), B + L p=[Lt]+2

(3.29) Since the event B ∩ [σ ≥ (p + 1)/L] is M(p−1)/L -measurable, from Lemma 3.5 we conclude that the absolute value of each term appearing under the summation sign in

˜ (δ) [B] which implies (3.29) can be estimated by C G1,1,3 δ γ1 L−1 Q x,k # $ ˜ (δ) (δ) Ex,k [Nσ (G), B] − E˜ x,k N([Lt]+2)L−1 (G), B [Lv] + 1 − [Lt] . L A direct calculation using formulas (3.1) allows us to conclude also that both |Nσ (G) − [T∗ ,T ] γ0 −1/2 δ . N(v∧τδ )∨t (G)| and |N([Lt]+2)L−1 (G) − Nt (G)| are estimated by CG1,1,3 Hence, (since γ0 > 1/2 + γ0 ) $ ˜ (δ) # (δ) Ex,k N(v∧τδ )∨t (G), B − E˜ x,k [Nt (G), B] $ (δ) # ≤ E˜ x,k Nσ (G) − N(v∧τδ )∨t (G), B $ (δ) (δ) # + E˜ x,k [Nσ (G), B] − E˜ x,k N([Lt]+2)L−1 (G), B $ (δ) # + E˜ x,k N([Lt]+2)L−1 (G) − Nt , B

[T∗ ,T ] 2 ˜ ≤ C δ γ1 G1,1,3 T Qx,k [B] (δ)

[T∗ ,T ] 2 ˜ T Qx,k [B] (v − t) ∨ δ γ0 ≤ Cδ γ1 G1,1,3 (δ)

(3.30)

294


for a certain constant C > 0 and γ1 := min[γ0 − γ0 − 1/2, γ1 ]. From (3.27), (3.30) and the observation just below (3.27), we obtain (δ) [T∗ ,T ] 2 (δ) T Rx,k [An ] (v − t) ∨ δ γ0 Ex,k [Nv (G) − Nt (G), An ] ≤ Cδ γ1 G1,1,3 for a certain constant C > 0 and the conclusion of Proposition 3.4 follows.

3.8. An estimate of the stopping time. The purpose of this section is to prove the fol(δ) lowing estimate for Rx,k [τδ < T ]. Theorem 3.6. Assume that the dimension d ≥ 3. Then, one can choose 1 , 2 , 3 , 4 in such a way that there exist constants C, γ > 0 for which (δ)

Rx,k [ τδ < T ] ≤ Cδ γ T ,

∀ δ ∈ (0, 1], T ≥ 1, (x, k) ∈ A(M).

(3.31)

Proof. With no loss of generality we can assume that δ −1 > T , since otherwise (3.31) holds with C = γ = 1. We obviously have then [ τδ < T ] = [ Uδ ≤ τδ , Uδ < T ] ∪ [ Sδ ≤ τδ , Sδ < T ]

(3.32)

with the stopping times Sδ and Uδ defined in (3.3) and (3.4). Let us denote the first and second event appearing on the right hand side of (3.32) by A(δ) and B(δ) respectively. (δ) To show that (3.32) holds we prove that the Rx,k probabilities of both events can be estimated by Cδ γ T for some C, γ > 0: see (3.40), (3.41) and (3.45). (δ)

3.8.1. An estimate of Rx,k [A(δ)]. The first step towards obtaining the desired estimate will be to replace the event A(δ) whose definition involves a stopping time by an event C(δ) whose definition depends only on deterministic times, see (3.33) below. Next we use the estimate (3.22) of Proposition 3.4 for an appropriately chosen function G to (δ) ˜ by an easier problem of reduce the question of bounding the Rx,k probability of A(δ) estimating its Qx,k probability (Qx,k corresponds to a degenerate diffusion determined by (3.20)). The latter is achieved by using bounds on heat kernels corresponding to hypoelliptic diffusions due to Kusuoka and Stroock. We assume in √this section to simplify the notation and without any loss of generality ˜ that h∗ (2M) + δ D(2M) ≤ 1. Note that then

j i 3 q ˜ A(δ) ⊂ A(δ) := X −X ≤ : 1 ≤ i ≤ j ≤ [T q], |i −j | ≥ q q q p (3.33) and thus

j i 3 ≤ [T q] max X q −X q ≤ q : & q 1 ≤ i ≤ j ≤ [T q], |i − j |≥ . p %

(δ) Rx,k [A(δ)]

2

(δ) Rx,k

(3.34)

Suppose that f (δ) : Rd → [0, 1] is a C ∞ –regular function that satisfies f (x) = 1, if |x| ≤ 4/q and f (δ) (x) = 0, if |x| ≥ 5/q. We assume furthermore that i, j are positive integers such that (j − i)/q ∈ [0, 1] and f (δ) 3 ≤ 2q 3 . For any x0 ∈ Rd and i/q ≤ t ≤ j/q define


295

j Gj (t, x, k; x0 ) := Mx,k f (δ) X − t − x0 . q for (x, k) ∈ A(Mδ ) and extend it elsewhere in such a way that it satisfies ∂t Gj (t, x, k; x0 ) + L˜ Gj (t, x, k; x0 ) = 0,

i/q ≤ t ≤ j/q,

where L˜ is a generator of a certain diffusion with C ∞ bounded coefficients that agree with the coefficients of L˜ on A(Mδ ). Hence, using Proposition 3.4 with v = j/q and t = i/q (note that v − t ≥ 1/p ≥ δ 2 and 2 ∈ (0, 1/2)), we obtain that there exists γ1 > 0 such that

i/q (δ) j i i i (δ) E X − x 0 − Gj ,X ,K ; x0 M x,k f q q q q j −i [i/q,j/q] ≤C Gj (·, ·, ·; x0 )1,1,3 T 2 δ γ1 , ∀ δ ∈ (0, 1]. (3.35) q According to [15], Theorem 2.58, p. 53, we have [i/q,j/q]

Gj (·, ·, ·; x0 )1,1,3

≤ Cf (δ) 3 ≤ Cq 3 ≤ Cδ −3(2 +3 ) ,

j ∈ {0, . . . , [qT ]}. (3.36)

Hence combining (3.35) and (3.36) we obtain that the left hand side of (3.35) is less q than, or equal to C δ γ1 −3(2 +3 ) for all δ ∈ (0, 1]. Let now i0 = j − so that 1 ≤ i ≤ p i0 ≤ j ≤ [T q]. We have

j j i 3 i (δ) (δ) (δ) Rx,k X f X −X ≤ ≤ E − X x,k q q q q q  

  i /q j  (δ)  (δ) (δ) 0 = Ex,k Ex,k f X (3.37) −y M .   q y=X(i/q)

According to (3.35) and (3.36) we can estimate the utmost right-hand side of (3.37) by % & 1 (δ) d sup Mx,k f X − y : x, y ∈ R , k ∈ A(2M) + C δ γ1 −3(2 +3 ) T 2 . p x,y,k (3.38) To estimate the first term in (3.38) we use the following. Lemma 3.7. Let p, q be as in (3.2). Then, there exist positive constants C1 , C2 and C3 such that for all x, y ∈ Rd , k ∈ A(2M), j ∈ {1, . . . , [pT ]}, δ ∈ (0, 1] we have

C2 5 j p −C3 p Qx,k X − y ≤ ≤ C1 . (3.39) +e p q qd We postpone the proof of the lemma for a moment in order to finish the estimate of (δ) Rx,k [A(δ)]. Using (3.39) we obtain that the expression in (3.38) can be estimated by

296


C1

* ) p C2 −C3 p + C δ γ1 −3(2 +3 ) T 2 ≤ C1 δ (d−C2 )2 +d3 + exp −C3 δ −2 +e qd +C δ γ1 −3(2 +3 ) T 2 .

Hence, from (3.34), we obtain that * ) (δ) Rx,k [A(δ)] ≤ [T q]2 C1 δ (d−C2 )2 +d3+ exp −C3 δ−2 +C δ γ1 −3(2 +3 ) T 2 * ) ≤ CT 2 δ (d−2−C2 )2+(d−2)3 + δ−2(2 +3 ) exp −C3 δ −2 + δ γ1 −5(2 +3 ) T 2 ≤ Cδ γ2 T 4

(3.40)

for γ2 := min[(d − 2 − C2 )2 + (d − 2)3 , γ1 − 5(2 + 3 )] > 0, provided that 2 + 3 < γ1 /5 and 2 ∈ (0, (d − 2)3 /(C2 + 2 − d)). Here with no loss of generality we have assumed that C2 + 2 > d. Recall also that d ≥ 3. Now suppose that γ3 ∈ (0, γ2 ). Consider two cases: T 3 < δ −γ3 and T 3 ≥ δ −γ3 . In the first one, the utmost right-hand side of (3.40) can be bounded from above by Cδ γ2 −γ3 T . In the second we have a trivial bound of the left side by δ γ3 /3 T . We have proved therefore that (δ)

Rx,k [A(δ)] ≤ Cδ γ T

(3.41)

for some C, γ > 0 independent of δ and T . Proof of Lemma 3.7. We prove this lemma by induction on j . First, we verify it for j = 1. Without any loss of generality we may suppose that k = (k1 , . . . , kd ) and kd > (4dMδ )−1 . Let D˜ mn : Rd−1 → R, m, n = 1, . . . , d − 1, E˜ m : Rd−1 → R, m = 1, . . . , d be given by + + D˜ pq (l) := Dpq (k −1 l, k −1 k 2 − l 2 , k), E˜ p (l) := Ep (k −1 l, k −1 k 2 − l 2 , k), √ : k −1 k 2 − l 2 > (4dMδ )−1 ], l = |l|. These functions are when l ∈ Z := [l ∈ Bd−1 k C ∞ smooth and bounded together with all their derivatives. Note also that the matrix ˜ = [D˜ mn ] is symmetric and Dξ ˜ · ξ ≥ λ0 |ξ |2 for all ξ ∈ Rd−1 and a certain λ0 > 0. The D projection K(t) = (K1 (t), . . . , Kd (t)) of the canonical path process (X(t; √ π ), K(t; π )) considered over the probability space (C, M, Qx,k0 ), where k0 := (l, k 2 − l 2 ), with l ∈ Z, is a diffusion whose generator equals L, see (3.19). It can be easily seen that (K1 (t), . . . , Kd−1 (t))t≥0 , is then a diffusion starting at l, whose generator N is of the form N F (l) :=

d−1

Xp2 F (l) +

p=1

d−1

aq (l)∂lq F (l),

F ∈ C0∞ (Rd−1 ),

q=1

where aq (l), q = 1, . . . , d − 1 are certain C ∞ -functions and Xp (l) :=

d−1

1/2 D˜ pq (l)∂lq ,

p = 1, . . . , d − 1.

q=1

The (d − 1) × (d − 1) matrix [D˜ pq (l)] is non-degenerate when l ∈ Z. Let 1/2

N˜ F (l, x) :=

d−1 p=1

X˜ p2 F (l, x) + X˜ 0 F (l, x),

F ∈ C0∞ (Rd−1 × Rd ),

(3.42)


297

where X˜ 0 is a C ∞ –smooth extension of the field

X0 (l) :=

d−1 d−1 H0 (k) H (k) + 2 lq ∂xq + 0 k − l 2 ∂xd + aq (l)∂lq , k k q=1

l ∈ Z.

q=1

It can be shown, by the same type of argument as that given on pp. 122–123 of [1], that for each (x, l), with l ∈ Z, the linear space spanned at that point by the fields belonging to the Lie algebra generated by [X0 , X1 ], . . . , [X0 , Xd−1 ], X1 , . . . , Xd−1 is of dimension 2d − 1. One can also guarantee that the extensions X˜ 0 , . . . , X˜ d−1 satisfy the same condition. We shall denote the respective extension of N by the same symbol. ˜ x,l0 be the path measures supported on C d−1 and Set l0 := (k1 , . . . , kd−1 ). Let Rl0 , R d,d−1 C respectively (see Sect. 2.4) that solve the martingale problems corresponding to the generators N and N˜ with the respective initial conditions at t = 0 given by l0 and (x, l0 ). Let r(t, x − y, l1 , l2 ), t ∈ (0, +∞), x, y ∈ Rd , l1 , l2 ∈ Rd−1 be the transition ˜ x,l0 . Using Corollary 3.25, p. 22 of [10] we of probability density that corresponds to R have that for some constants C, m > 0, r (t, x − y, k, l) ≤ Ct −m ,

provided that |x − y| ≤ 1, |k|, |l| ≤ 2M, t ∈ (0, 1]. (3.43)

Denote by τZ (π ) the exit time of a path π ∈ C d−1 from the set Z. For any π ∈ C d,d−1 we set also τ˜Z (π ) = τZ (K(·; π)). Let S : Bd−1 → Sd−1 be given by k k + S(l) := (l1 , . . . , ld−1 , k 2 − l 2 ),

l = (l1 , . . . , ld−1 ) ∈ Bd−1 k , l := |l|

˜ )(t) := (X(t; π ), S ◦ K(t; π )), t ≥ 0. For and let S˜ : C d,d−1 → C be given by S(π ˜ x,l0 [S˜ −1 (A)] = Qx,S(l0 ) [A]. Since the event [|X (1/p) − y| ≤ any A ∈ Mτ˜Z we have R 5/q]∩[τ˜Z ≥ 1/p] is Mτ˜Z –measurable we have for δ so small that q ≥ 5 and δ < δ∗ (M):

5 5 1 1 1 1 ˜ Qx,k X − y ≤ ≤ Rx,l0 X − y ≤ , τ˜Z ≥ + Rl0 τZ < p q p q p p d 4 ≤ C ω¯ d p m + Ce−C3 p . (3.44) q Here ω¯ d denotes the volume of Bd . To obtain the last inequality we have used (3.43) and an estimate for non-degenerate diffusions stating that Rl0 [τZ < 1/p] < Ce−C3 p for some constants C, C3 > 0 depending only on d and λ0 , see e.g. (2.1) p. 87 of [16]. Inequality (3.44) implies easily (3.39) for j = 1 with C1 = m. To finish the induction argument assume that (3.39) holds for a certain j . We show that it holds for j + 1 with the same constants C1 , C2 and C3 > 0. The latter follows easily from the

298


Chapman-Kolmogorov equation, since

5 j +1 Qx,k X − y ≤ = p q

5 j Qz,m X − y ≤ p q

Rd ×Sd−1 k

1 ×Q , x, k, dz, dm p C2 induction assumpt. p 1 −C3 p Q , x, k, dy, dl ≤ C1 +e qd p C2 p = C1 +e−C3 p , qd and the formula (3.39) for j + 1 follows. Here Q(t, x, k, ·, ·) is the transition of probability corresponding to the path measure Qx,k . (δ)

3.8.2. An estimate of Rx,k [B(δ)]. We start with a simple observation concerning the Hölder regularity of the K component of any path π ∈ B(δ). Let us denote ρ := 2Mδ−1 N −1/2 and (p) D := π ∈ C(T , δ) : |K(t) − K(s)| ≥ ρ for some k s.t. tk ≤ T (p) (p) (p) (p) and t ∈ [tk , tk+1 ], s∈[tk−1 , tk ] , where Mδ has been defined in (2.7) and N in (3.2). Suppose that π ∈ B(δ), then we can (p) (p) (p) (p) ˆ ˆ find t ∈ [tk , tk+1 ], s ∈ [tk−1 , tk ] for which K(t) · K(s) ≤ 1 − 1/N . This, however, implies that |K(t) − K(s)|2 ≥

1 ˆ 2 2 ˆ |K(t) − K(s)| ≥ 2 , 2 Mδ Mδ N (δ)

thus π ∈ D. Hence the desired estimate of Rx,k [B(δ)] follows from the following lemma. Lemma 3.8. Under the assumptions of Theorem 3.6 there exist C, γ > 0 such that (δ)

Rx,k [D] ≤ CT δ γ ,

∀ δ ∈ (0, 1], T ≥ 1, (x, k) ∈ A(M).

(3.45)

Proof. We define the following events:

2 F1 := |K(t) − K(s)| ≥ ρ for some s, t ∈ [0, T ], 0 < t − s < , t ≤ τδ , p

2 F2 := |K(t) − K(s)| ≥ ρ for some s, t ∈ [0, T ], 0 < t − s < , s ≥ τδ , p

ρ 2 F3 := |K(τδ ) − K(s)| ≥ for some s ∈ [0, T ], 0 < τδ − s < , τδ ≤ T , 2 p

ρ 2 F4 := |K(τδ ) − K(t)| ≥ for some t ∈ [0, T ], 0 < t − τδ < . 2 p


Observe that D ⊂

4

299

Fi . Note that F1 , F3 are Mτδ –measurable, hence

i=1

˜ [Fi ], Rx,k [Fi ] = Q x,k (δ)

(δ)

i = 1, 3.

(3.46)

On the other hand for i = 2, 4 we have

(δ) ˜ (δ) (dπ ), Rx,k [Fi ] = QX(τδ (π)),K(τδ (π)) [Fi,π ]Q x,k where for a given π ∈ C,

2 , F2,π := |K(t)−K(s)|≥ρ for some s, t ∈ [0, (T − τδ (π )) ∧ 0], 0 < t − s < p

ρ 2 F4,π := |K(0) − K(t)| ≥ for some t ∈ [0, (T − τδ (π )) ∧ 0], 0 < t < . 2 p Since all Fi , i = 1, 3 and Fi,π , i = 2, 4, π ∈ C are contained in the event

2 ρ for some s, t ∈ [0, T ], 0 < t − s < , F := |K(t) − K(s)| ≥ 2 p (3.45) would follow if we show that there exist C > 0 and γ > 0 for which ˜ (δ) [F ] ≤ CT δ γ for all (x, k) ∈ A(M) Q x,k

(3.47)

Qx,k [F ] ≤ CT δ γ for all (x, k) ∈ A(Mδ ).

(3.48)

and

The estimate (3.48) follows from elementary properties of diffusions, see e.g. (2.46) p. 47 of [15]. We carry on with the proof of (3.47). The argument is analogous to the

proof of Theorem 1.4.6 of [16]. Let L be a multiple of p such that L := [δ −γ0 ], where (L) γ0 ∈ (1/2, 1) is to be specified even further later on. Let also sk := k/L, k = 0, 1, . . . . We define now the stopping times τk (π ) that determine the times at which the K component of the path π performs k-th oscillation of size ρ/8. Let τ0 (π ) := 0 and for any k ≥ 0, ρ (L) (L) , τk+1 (π ) := inf sk ≥ τk (π ) : |K(sk ) − K(τk (π ))| ≥ 8 with the convention that τn+1 = +∞ when τn = +∞, or when the respective event is impossible. Let N# := min[n : τn+1 > T ] and δ ∗ := min[τn − τn−1 : n = 1, . . . , N# ]. Then, for a sufficiently small δ0 and δ ∈ (0, δ0 ) we have F ⊂ [δ ∗ ≤ 1/p] so we only ˜ (δ) probability of the latter event. need to estimate Q x,k Let f : Rd → [0, 1] be a function of Cc∞ (Rd ) class such that f (0) ≡ 1, when |k| ≤ ρ/16 and f (k) ≡ 0, when |k| ≥ ρ/8. Let also fl (·) := f (· − l) for any l ∈ Rd . Note that according to Lemma 3.5 we can choose constants Aρ , C > 0, where C is independent of ρ, in such a way that Aρ < CT 2 ρ −3 and the random sequence

N +1 (δ) l MN/L + Aρ N , N ≥ 0 (3.49) := E˜ x,k fl K SN L L

300


˜ (δ) –submartingale with respect to the filtration MN/L is a Q for all l with |l| ∈ x,k N≥0 −1 ((3Mδ ) , 3Mδ ) provided that δ is sufficiently small. We can decompose

˜ (δ) δ ∗ ≤ 2 ≤ Q ˜ (δ) δ ∗ ≤ 2 , N# > [δ −α ] ˜ (δ) δ ∗ ≤ 2 , N# ≤ [δ −α ] + Q Q x,k x,k x,k p p p −α

[δ ] (δ) 2 ˜ ˜ (δ) [N# > [δ −α ]], ≤ Q (3.50) +Q x,k τi − τi−1 ≤ x,k p i=1

where α > 0 is to be determined later. We will show that δ −α δ 1/2(1 +2 ) (δ) −α T ˜ Qx,k [N# > [δ ]] ≤ Ce 1 − 2 and

˜ (δ) τn+1 − τn ≤ [Lδ2 ]/L Mτn ≤ Cδ γ T 2 , Q x,k

(3.51)

(3.52)

for 0 < γ < min[2 − 31 /2, γ0 − (1 + 1 )/2]. From (3.49), (3.50) (3.51) and (3.52) we further conclude that δ −α

1/2(1 +2 ) δ 1 (δ) ∗ 2 γ −α T ˜ + Ce 1 − (3.53) ≤ CT δ Q x,k δ ≤ p 2 for some C > 0, independent of δ ∈ (0, 1] and T ≥ T0 , provided that we choose α ∈ (1/2(1 +2 ), γ ). This is possible if min[2 −31 /2, γ0 −(1+1 )/2] > (1 +2 )/2, which is true if we assume 2 > 101 > 0 and 1 > γ0 > (1+2 )/2+1 . Now, by the argument made after (3.40) we can always replace the first term on the right side of (3.53) by CT δ γ1 . We can also assume that the second term on the right hand side of (3.53) is less than or equal to CT δ γ1 . This can be Let β := α−1/2(1 +2 ). The term in ques) seen as follows. * tion is bounded by C exp T − C1 δ −β with C1 := inf ρ∈(0,1] ρ −1 log (1 − ρ/2)−1 . For ) * * ) δ −β ≥ 2T /C1 we get that exp T − C1 δ −β is less than or equal to exp −C1 δ −β /2 , while for δ −β < 2T /C1 the left side of (3.53) is obviously less than 2T δ β /C1 . In both cases we can find a bound as claimed. This proves (3.47) and hence the proof of Lemma 3.8 will be complete if we prove (3.51) and (3.52). ˜ (δ) , π ∈ C denote the family of the regular conditional probability To this end, let Q x,k,π ˜ (δ) [ · | Mτn ]. Then, there exists a Mτn measurable, distributions that corresponds to Q x,k

d ˜ null Q x,k probability event Z such that for each π ∈ Z and each l ∈ R∗ the random sequence (δ)

l l SN,π := SN 1[0,N/L] (τn (π )),

N ≥0

˜ (δ) . Let Tn,π := τn+1 ∧(τn (π )+2[Lδ ]/L), is an MN/L N≥0 submartingale under Q x,k,π where ∈ (0, 1) is a constant to be chosen later on. We can choose the event Z in such a way that ˜ (δ) [Tn,π ≥ τn (π )] = 1, Q x,k,π

∀ π ∈ Z.

(3.54)


301

K(τ (π)) Let SÑ,π := SN,πn , then the submartingale property of SÑ,π

N≥0

and (3.54) imply

that (δ) (δ) E˜ x,k,π S˜LTn,π ,π ≥ E˜ x,k,π S˜Lτn (π),π = 1 + Aρ τn (π ),

√

(3.55)

provided that γ0 ≥ (1 + 1 )/2. The latter condition assures that ρ ≥ C/(L δ) so that K(t) does not change by more than ρ during the time 1/L. In consequence of (3.55) we have

1 (δ) ˜ Ex,k,π fK(τn (π)) K Tn,π + (3.56) + 2Aρ δ ≥ 1, L as Tn,π − τn (π ) ≤ 2[Lδ ε ]/L. Since C fK(τ (π)) K Tn,π + 1 − fK(τn (π)) K Tn,π ≤ n Lρδ 1/2 L we obtain from (3.56) $ (δ) # 2Aρ δ ≥ E˜ x,k,π 1 − fK(τn (π)) K Tn,π −

C , Lρδ 1/2

so in particular

C [Lδ ] ˜ (δ) 1 − f ≥ E ≤ τ (π ) + , τ (K (τ )) n+1 n+1 n K(τn (π)) x,k,π Lρδ 1/2 L

[Lδ ] ˜ (δ) =Q . (3.57) x,k,π τn+1 ≤ τn (π ) + L We have shown, therefore, that

2 ˜ (δ) τn+1−τn ≤ [Lδ ] Mτn ≤ CT δ + C Q x,k L ρ3 Lρδ 1/2 2Aρ δ +

≤ C(δ −31 /2 T 2 + δ γ0 −(1+1 )/2 ) ≤ Cδ γ1 T 2

(3.58)

for γ1 < min[ − 31 /2, γ0 − (1 + 1 )/2] and some constant assume that T 2 δ γ1 /2 ≤ 1. If otherwise, we can always write

C > 0. We can always ˜ (δ) [F ] ≤ T δ γ /4 and Q x,k (3.47) follows. In particular, selecting := (1 + 2 )/2, one concludes from (3.58) that (1 +2 )/2 ] [Lδ (1 +2 )/2 (δ) (δ) τ −δ τ Mn ˜ E˜ x,k [exp{−(τn+1 −τn )}|M n ] ≤ e Q x,k τn+1 − τn ≥ L [Lδ (1 +2 )/2 ] τn (δ) ˜ + Qx,k τn+1 − τn ≤ M L (3.58) (1 +2 )/2 (1 +2 )/2 ≤ e−δ + C 1 − e−δ δ γ /2

δ (1 +2 )/2 , (3.59) 2 provided that δ is sufficiently small. From (3.59) one concludes easily, see e.g. Lemma 1.4.5, p. 38 of [16], that (3.51) holds. On the other hand, taking = 2 in (3.58) we obtain (3.52) with 0 < γ < min[2 − 31 /2, γ0 − (1 + 1 )/2]. Hence the proof of Lemma 3.8 is now complete. < 1−

302


3.9. The estimation of the convergence rate. The proof of Theorem 2.1. Recall that φδ , φ¯ satisfy (2.8), (2.10), respectively, with the initial condition φ0 . We start with the following lemma. Lemma 3.9. Assume that φ0 satisfies the hypotheses formulated in Sect. 2.5. Then, ¯ [0,T ] ≤ φ0 0,0 , φ 0,0,0

d

¯ [0,T ] ≤ φ0 1,0 . ∂xi φ 0,0,0

(3.60)

i=1

Furthermore, there exists a constant C > 0 such that for all T ≥ 1, ¯ [0,T ] ≤ Cφ0 1,2 . ∂t φ 0,0,0

(3.61)

In addition, for any nonnegative integer valued multi-index γ = (α1 , α2 , α3 ) satisfying |γ | ≤ 3 we have d i1 ,i2 ,i3 =1

γ

∂ki

1 ,ki2 ,ki3

¯ [0,T ] ≤ CT |γ | φ0 1,4 . φ 0,0,0

(3.62)

Proof. The first estimate of (3.60) is a consequence of the maximum principle for the solution of (2.10) while the second one follows directly from differentiating (2.10) with respect to x and applying again the maximum principle. To obtain the estimates (3.61) and (3.62) we note first that the application of the operator L˜ to both sides of (2.10) and ˜ 0 L∞ (A(M)) ¯ x, ·)L∞ (A(M)) ≤ Lφ the maximum principle leads to the estimate L˜ φ(t, for all t ≥ 0, hence we conclude bound (3.61). In fact, thanks to the already proven estimate (3.60) we conclude that ¯ x, ·)L∞ (A(M)) ≤ Cφ0 1,2 for some constant C > 0 and all (t, x) ∈ [0, +∞) × Lφ(t, in the proof of Lemma 3.7. Define S : Z × [M −1 , M] → A(M) Rd . Let Z be as √ as S(l, k) := (l, k 2 − l 2 ), where l = |l|. Let also ψ(l, k) = φ¯ ◦ S(l, k). We have ¯ ◦ S(l, k) = N ψ(l, k), see (3.42). The Lp estimates for elliptic partial differential (Lk φ) equations, see e.g. Theorem 9.13, p. 239 of [7] allow us to estimate ψW 2,p (Z) ≤ C(ψLp (Z) + N ψLp (Z) ) ≤ Cφ0 1,2 . Choosing p sufficiently large we obtain that i ∂li ψL∞ (Z) ≤ Cφ0 1,2 , which ¯ ·)L∞ (S(Z)) ≤ Cφ0 1,2 . Obviously, one can find in fact implies that D(·)∇k φ(t, a covering of A(M) with charts corresponding to different choices of the components of k being projected onto the hyperplane Rd−1 and we obtain in that way that ˆ k) ¯ ·)L∞ (A(M)) ≤ Cφ0 1,2 for all t ≥ 0. Since the rank of the matrix D(k, D(·)∇k φ(t, equals d − 1, with the kernel spanned by the vector k, we obtain in that way the L∞ estimates of directional derivatives in any direction perpendicular to k. We still need to obtain the L∞ bound on the derivative in the direction k, denoted by ∂n := k1 ∂k1 + . . . + kd ∂kd . To that purpose we apply ∂n to both sides of (2.10) and after a straightforward calculation ˜ n φ¯ − 2Lk φ¯ + L1 φ¯ + H

(k)kˆ · ∇x φ, ¯ where we get ∂t ∂n φ¯ = L∂ 0 L1 φ¯ :=

d m,n=1

∂ ∂km

¯ ˆ k) ∂ φ ∂k Dmn (k, ∂kn

.


303

ˆ k)kˆ = 0 implies that ∂k D(k, ˆ k)kˆ = 0, hence L1 φ(t, ¯ ·)L∞ (A(M)) ≤ Note that D(k, ¯ L∞ (A(M)) are bounded, hence Cφ0 1,2 . We already know that Lk φ¯ and ∇x φ ¯ ·)L∞ (A(M)) ≤ Cφ0 1,2 T for t ∈ [0, T ]. We have shown therefore that ∂n φ(t, ¯ φ(t, ·)1,1 ≤ Cφ0 1,2 T for t ∈ [0, T ]. The above procedure can be iterated in order to obtain the estimates of the suprema of derivatives of the higher order.

Proof of Theorem 2.1. Let u ∈ [δ γ0 , T ], where we assume that γ0 (as in the statement of ¯ Lemma 3.5) belongs to the interval (1/2, 1). Substituting for G(t, x, k) := φ(u−t, x, k),

γ 0 ζ ≡ 1 into (3.22) we obtain (taking v = u, t = δ ) (δ) γ0

γ0

γ0

E˜ ¯ x,k φ0 (X(u), K(u)) − φ(u − δ , X(δ ), K(δ ))

− δ

" )G(, X(), K()) d ≤ CG[0,T ] δ γ1 T 2 , (∂ + L 1,1,3 γ0

u

∀ δ ∈ (0, 1]. (3.63)

˜ –a.s. for some Using the fact that |X(δ γ0 ) − x| ≤ Cδ γ0 , |K(δ γ0 ) − k| ≤ Cδ γ0 −1/2 , Q x,k deterministic constant C > 0, cf. (3.12), and Lemma 3.9 we obtain that there exist constants C, γ > 0 such that  

u (δ) E˜ φ0 (X(u), K(u)) − φ(u, "  ¯ x, k) − (∂ + L )G(, X(), K()) d x,k (δ)

0

≤

] γ 2 CG[0,T 1,1,3 δ T ,

δ ∈ (0, 1], T ≥ 1, u ∈ [0, T ].

We have however (δ) # $ E ¯ φ (X(u), K(u)) − φ(u, x, k), τ ≥ T δ x,k 0 (δ) # $ ¯ = E˜ x,k φ0 (X(u), K(u)) − φ(u, x, k), τδ ≥ T (3.64) ] γ 2 [0,T ] ˜ (δ) ≤ CG[0,T 1,1,3 δ T + 2φ0 0,0 + T G1,1,2 Qx,k [τδ < T ].

(3.64)

(3.65)

˜ (δ) [τδ < T ] = Using Mτδ measurability of the event [τδ < T ] we obtain that Q x,k (δ)

Rx,k [ τδ < T ] and by virtue of Theorem 3.6 we can estimate the right hand side of (3.65) by ] γ 2 [0,T ] Lemma 3.9 γ CG[0,T δ T + Cδ T 2φ + T G ≤ Cδ γ T 5 . 0 0,0 1,1,3 1,1,2 On the other hand, the expression under the absolute value on the utmost left-hand side of (3.65) equals $ $ (δ) # (δ) # ¯ ¯ Ex,k φ0 (X(u), K(u)) − φ(u, x, k) − Ex,k φ0 (X(u), K(u)) − φ(u, x, k), τδ < T .

304


The second term can be estimated by (3.31)

(δ)

2φ0 0,0 Rx,k [τδ < T ] ≤ Cδ γ φ0 0,0 T , by virtue of Theorem 3.6. Since u x (δ) Eφδ , , k = Eφ0 (z(δ) (u; x, k), m(δ) (u; x, k)) = Ex,k φ0 (X(u), K(u)) δ δ we conclude from the above that the left-hand side of (2.11) can be estimated by Cδ γ φ0 1,4 T 5 for some constants C, γ > 0 independent of δ > 0, T ≥ 1. The bound appearing on the right-hand side of (2.11) can be now concluded by the same argument as the one used after (3.40). 4. Momentum Diffusion to Spatial Diffusion: Proof of Theorem 2.5 We show in this section that solutions of the momentum diffusion equation (2.10) in the long-time, large space limit converge to the solutions of the spatial diffusion equation 2 , x/γ , k), ¯ (2.12). We first recall the setup of Theorem 2.5. Let φ¯ γ (t, x, k) = φ(t/γ where φ¯ satisfies (2.10) and let w(t, x, k) be the solution of the spatial diffusion equation (2.12). In order to prove Theorem 2.5 we need to show that the re-scaled solution φγ (t, x, k) converges as γ → 0 in the space C([0, T ]; L∞ (A(M))) to w(t, x, k), so that w(t) − φ¯ γ (t)L∞ (A(M)) ≤ C γ T + γ 1/2 φ0 2,0 , 0 ≤ t ≤ T . (4.1) Proof of Theorem 2.5. The proof is quite standard. We present it for the sake of completeness and convenience to the reader. The function φ¯ γ is the unique Cb1,1,2 ([0, +∞), R2d ∗ )solution to d ∂ φ¯ γ ∂ φ¯ γ ∂ 2 ˆ k) γ Dmn (k, + γ H0 (k)kˆ · ∇x φ¯ γ , = ∂t ∂k ∂k m n m,n=1 (4.2) ¯ φγ (0, x, k) = φ0 (x, k), see Remark 2.3. We represent φ¯ γ as φ¯ γ = w + γ w1 + γ 2 w2 + R.

(4.3)

Here w is the solution of the diffusion equation (2.12), the correctors w1 and w2 will be constructed explicitly, and the remainder R will be shown to be small. The first corrector of the equation w1 is the unique solution of zero mean over each sphere Sd−1 k d m,n=1

∂ ∂km

ˆ k) ∂w1 Dmn (k, ∂kn

= −H0 (k)kˆ · ∇x w.

(4.4)

It has an explicit form w1 (t, x, k) =

d j =1

χj (k)

∂w(t, x, k) ∂xj

(4.5)


305

with the functions χj defined in (2.14). The second order corrector w2 is the unique zero mean over each sphere Sd−1 solution of the equation k d m,n=1

∂ ∂km

ˆ k) ∂w2 Dmn (k, ∂kn

=

∂w − H0 (k)kˆ · ∇x w1 . ∂t

(4.6)

Note that the expression on the right hand side of (4.6) is of zero mean since thanks to (2.12) and equality (2.13) we have

1 ∂w H (k)l · ∇x w1 d(l). = d−1 Sd−1 0 ∂t Equations (4.4) and (4.6) for various values of k = |k| are decoupled. As a consequence of this fact and the regularity properties for solutions of elliptic equations on a sphere we have that w1 , w2 belong to C([0, T ]; L∞ (A(M))). More explicitly, we may represent the function w2 as d

w2 (t, x, k) =

ψj l (k)

j,l=1

∂ 2 w(t, x, k) . ∂xj ∂xl

The functions ψj m (k) satisfy d m,n=1

∂ ∂km

ˆ k) Dmn (k,

∂ψj l ∂kn

= −H0 (k)kˆj χl (k) + aj l (k).

(4.7)

A unique mean-zero, bounded solution of (4.7) exists according to the Fredholm alternative combined with the regularity properties for solutions of (4.7) on each sphere Sd−1 k . With the above definitions of w, w1 , w2 , Eq. (4.2) for φ¯ γ implies that the remainder R in (4.3) satisfies ∂R ∂w1 ∂w2 + γ3 + γ4 − γ H0 (k)kˆ · ∇x R − γ 3 H0 (k)kˆ · ∇x w2 ∂t ∂t ∂t d ∂ ˆ k) ∂R . Dmn (k, = ∂km ∂kn

γ2

m,n=1

We re-write this equation in the form

∂ ∂R ˆ k) ∂R = f, − γ1 H0 (k)kˆ · ∇x R − γ12 dm,n=1 Dmn (k, ∂t ∂km ∂kn 2 R(0, x, k) = φ0 (x, k) − φ¯ 0 (x, k) − γ w1 (0, x, k) − γ w2 (0, x, k),

(4.8)

ˆ x w2 . Here, as before, R is understood as the where f := −γ ∂t w1 −γ 2 ∂t w2 −γ H0 (k)k·∇ 1,1,2 unique solution to (4.8) that belongs to Cb ([0, +∞), R2d ∗ ). We may split R = R1 +R2 according to the initial data and forcing in the equation: R1 satisfies d 1 1 ∂ ∂R1 ˆ k) ∂R1 = f, Dmn (k, − H0 (k)kˆ · ∇x R1 − 2 ∂t γ γ ∂k ∂kn m,n=1 m (4.9) R1 (0, x, k) = −γ w1 (0, x, k) − γ 2 w2 (0, x, k),

306


and the initial time boundary layer corrector R2 satisfies d ∂R2 1 1 ∂ ˆ k) ∂R2 = 0, Dmn (k, − H0 (k)kˆ · ∇x R2 − 2 ∂t γ γ ∂km ∂kn m,n=1

(4.10)

R2 (0, x, k) = φ0 (x, k) − φ¯ 0 (x, k). Using the probabilistic representation for the solution of (4.10) as well as the regularity of w1 and w2 we obtain that R1 (t)L∞ (A(M)) ≤ Cγ T , 0 ≤ t ≤ T .

(4.11)

γ

To obtain the bound for R2 we consider R2 (t, x, k) := R2 (γ 3/2 t, x, k). This function satisfies γ γ d ∂R2 ∂R2 ∂ γ 1

1/2 ˆ k) − γ H0 (k)kˆ · ∇x R2 − γ 1/2 m,n=1 Dmn (k, = 0, ∂t ∂km ∂kn R2 (0, x, k) = φ0 (x, k) − φ¯ 0 (x, k). γ

We also define R˜ 2 , the solution of γ

d γ ∂ R˜ 2 1 ∂ − 1/2 ∂t γ ∂km m,n=1

∂ R˜ ˆ k) 2 Dmn (k, ∂kn

γ

= 0, (4.12)

γ R˜ 2 (0, x, k) = φ0 (x, k) − φ¯ 0 (x, k).

The uniform ellipticity of the right hand side of (4.12) on each sphere Sd−1 implies, see k γ ˜ e.g. Proposition 13.3, p. 55 of [17], that the function R2 satisfies the decay estimate on each sphere Cγ (d−1)/4 Cγ (d−1)/4 γ R˜ 2 (t)L∞ (Sd−1 ) ≤ (d−1)/2 φ0 L1 (Sd−1 ) ≤ (d−1)/2 φ0 L∞ (Sd−1 ) (4.13) k k t t for t ∈ [0, T ] and, similarly, Cγ (d−1)/4 γ ∇x R˜ 2 (t)L∞ (Sd−1 ) ≤ (d−1)/2 φ0 1,0 . k t γ γ Furthermore, the difference q γ = R2 − R˜ 2 satisfies

d 1 ∂ ∂q γ ∂q γ γ 1/2

γ ˆ ˆ −γ H0 (k)k · ∇x q − 1/2 Dmn (k, k) = γ 1/2 H0 (k)kˆ · ∇x R˜ 2 , ∂t γ ∂km ∂kn m,n=1

q γ (0, x, k) = 0. We conclude, using the probabilistic representation of the solution of (4.14), that q γ (t)L∞ (A(M)) ≤ Cγ 1/2 tφ0 1,0 ,

(4.14)


307

and thus γ

R2 (γ 3/2 )L∞ (A(M)) ≤ R2 (1)L∞ (A(M)) + q γ (1)L∞ (A(M)) ≤ C γ (d−1)/4 φ0 0,0 + γ 1/2 φ0 1,0 . The maximum principle for (4.10) implies that we have the above estimate: R2 (t)L∞ (A(M)) ≤ C γ (d−1)/4 φ0 0,0 + γ 1/2 φ0 1,0 , t ≥ γ 3/2 .

(4.15)

Combining (4.3), (4.11) and (4.15) we conclude that w(t)− φ¯ γ (t)L∞ (A(M))≤C γ T +γ (d−1)/4+γ 1/2 φ0 1,0 , γ 3/2 ≤ t ≤ T , (4.16) and thus (4.1) follows, as d ≥ 3. This finishes the proof of Theorem 2.5.

5. The Spatial Diffusion of Wave Energy In this section we consider an application of the previous results to the random geometrical optics regime of propagation of acoustic waves. We show that when the wave length is much shorter than the correlation length of the random medium, there exist temporal and spatial scales where the energy density of the wave undergoes the spatial diffusion. We start with the wave equation in dimension d ≥ 3, 1 ∂ 2φ − φ = 0, c2 (x) ∂t 2

(5.1)

√ and assume that the wave speed has the form c(x) = c0 + δc1 (x). Here c0 > 0 is the constant sound speed of the uniform background medium, while the small parameter δ 1 measures the strength of the mean zero random perturbation c1 . Rescaling the spatial and temporal variables x = x /δ and t = t /δ we obtain (after dropping the primes) Eq. (5.1) with a rapidly fluctuating wave speed x √ cδ (x) = c0 + δc1 . (5.2) δ It is convenient to rewrite (5.1) as the system of acoustic equations for the “pressure” p = φt /c and “acoustic velocity” u = −∇φ: ∂u + ∇ (cδ (x)p) = 0, ∂t ∂p + cδ (x)∇ · u = 0. ∂t

(5.3)

We will denote for brevity v = (u, p) ∈ Rd+1 and write (5.3) in the more general form of a first order linear symmetric hyperbolic system. To do so we introduce symmetric matrices Aδ and D j defined by Aδ (x) = diag(1, 1, 1, cδ (x)), and D j = ej ⊗ ed+1 + ed+1 ⊗ ej , j = 1, . . . , d. (5.4) Here em ∈ Rd+1 is the standard orthonormal basis: (em )k = δmk .

308


We consider the initial data for (5.3) as a mixture of states. Let S be a measure space equipped with a non-negative finite measure µ. A typical example is that the initial data is random, S is the state space and µ is the corresponding probability measure. We assume that for each parameter ζ ∈ S and ε, δ > 0 the initial data is given by vεδ (0, x; ζ ) := (−ε∇φ0ε (x), 1/cδ (x)φ˙ 0ε (x)) and vεδ (t, x; ζ ) solves the system of equations d ∂vεδ ∂ Aδ (x)D j j Aδ (x)vεδ (x) = 0. + ∂t ∂x

(5.5)

j =1

The set of initial data is assumed to form an ε-oscillatory and compact at infinity family [5] as ε → 0. By the above we mean that for any continuous, compactly supported function ϕ : Rd → R we have

2 , δ lim lim sup |ϕvε | dk → 0 and lim lim sup |vεδ |2 dx → 0 R→+∞ ε→0+

R→+∞ ε→0+

|k|≥R/ε

|x|≥R

for a fixed realization ζ ∈ S of the initial data and each δ > 0. In the regime of geometric acoustics the scale ε of oscillations of the initial data is much smaller than the correlation length δ of the medium: ε δ 1. The dispersion matrix for (5.5) is P0δ (x, k) = i

d

Aδ (x)kj D j Aδ (x) = i

j =1

d

cδ (x)kj D j

j =1

= icδ (x) k˜ ⊗ ed+1 + ed+1 ⊗ k˜ ,

(5.6)

where k˜ = dj =1 kj ej . The self-adjoint matrix (−iP0δ ) has an eigenvalue H0 = 0 of the multiplicity d − 1, and two simple eigenvalues H±δ (x, k) = ±cδ (x)|k|.

(5.7)

Its eigenvectors are b0m

=

⊥ km ,0

1 , m = 1, . . . , d − 1; b± = √ 2

k˜ ± ed+1 , |k|

(5.8)

⊥ ∈ Rd is the orthonormal basis of vectors orthogonal to k. where km The (d + 1) × (d + 1) Wigner matrix of a mixture of solutions of (5.5) is defined by

1 εy εy δ Wε (t, x, k) = eik·y vεδ (t, x − ; ζ ) ⊗ vεδ∗ (t, x + ; ζ )dyµ(dζ ). (2π )d 2 2

Rd S

(5.9) It is well-known, see [5, 11, 13], that for each fixed δ > 0 (and even without introduction of a mixture of states) when Wεδ (0, x, k) converges weakly in S (Rd × Rd ), as ε → 0, to W0 (x, k) = u0+ (x, k)b+ (k) ⊗ b+ (k) + u0− (x, k)b− (k) ⊗ b− (k),

(5.10)


309

then Wεδ (t) converges weakly in S (Rd × Rd ) to U δ (t, x, k) = uδ+ (t, x, k)b+ (k) ⊗ b+ (k) + uδ− (t, x, k)b− (k) ⊗ b− (k). (δ)

The scalar amplitudes u± satisfy the Liouville equations:   ∂t uδ± + ∇k H±δ · ∇x uδ± − ∇x H±δ · ∇k uδ± = 0, 

(5.11)

uδ± (0, x, k) = u0± (x, k).

These equations are of the form (2.8), written in the macroscopic variables, with the Hamiltonian given by (5.7). One may obtain an L2 -error estimate for this convergence when a mixture of states is introduced, as in (5.9). In order to make the scale separation ε δ 1 precise we define the set ! Kµ := (ε, δ) : | ln ε|−2/3+µ ≤ δ ≤ 1 . The parameter µ is a fixed number in the interval (0, 2/3). The following proposition has been proved in Theorem 3.2 of [1], using straightforward if tedious asymptotic expansions. Proposition 5.1. Let the acoustic speed cδ (x) be of the form (5.2) and such that the Hamiltonian Hδ (x) given by (5.7) satisfies assumptions (2.3). We assume that the Wigner transform Wεδ satisfies Wεδ (0, x, k) → W0 (x, k) strongly in L2 (Rd × Rd ) as Kµ (ε, δ) → 0. (5.12) We also assume that the limit W0 ∈ Cc2 (R2d ∗ ) with a support that satisfies supp W0 (x, k) ⊆ A(M)

(5.13)

for some M > 0. Moreover, we assume that the initial limit Wigner transform W0 is of the form W0 (x, k) =

u0q (x, k)q (k), q (k) = bq (k) ⊗ bq (k).

(5.14)

q=±

Let U δ (t, x, k) =

uδp (t, x, k)p (k), where the functions uδp satisfy the Liouville

p=±

equations (5.11). Then there exists a constant C1 > 0 that is independent of δ so that 3/2 3/2 Wεδ (t, x, k) − U δ (t, x, k)2 ≤ C(δ) εW0 H 2 eC1 t/δ + ε 2 W0 H 3 eC1 t/δ +Wεδ (0) − W0 2 ,

(5.15)

where C(δ) is a rational function of δ with deterministic coefficients that may depend on the constant M > 0 in the bound (5.13) on the support of W0 .

310


The Liouville equations (5.11) are of the form (2.8). Therefore, one may pass to the limit δ → 0 in (5.11) using Theorem 2.1 and conclude that Euδ± converge to the respective solutions of d ∂ u¯ ± ∂ ∂ u¯ ± 2 ˆ = |k| Dmn (k) ± c0 kˆ · ∇x u¯ ± (5.16) ∂t ∂km ∂kn m,n=1

ˆ = [Dmn (k)] ˆ is with the initial conditions as in (5.11). Here the diffusion matrix D(k) given by

ˆ 1 ∞ ∂ 2 R(c0 s k) ˆ Dmn (k) = − ds, (5.17) 2 −∞ ∂xn ∂xm where R(x) is the correlation function of the random field c1 (x): E [c1 (z)c1 (x + z)] = R(x). Furthermore, it follows from Theorem 2.7 that there exists α0 > 0 so that solutions of (5.11) with the initial data of the form uδ± (0, x, k) = u0± (δ α x, k) with 0 < α < α0 , converge in the long time limit to the solutions of the spatial diffusion equation. More precisely, in that case the function u¯ δ (t, x, k) = uδ+ (t/δ 2α , x/δ α , k) (and similarly for uδ− ) converges as δ → 0 to w(t, x, k) – the solution of the spatial diffusion equation d ∂w ∂ 2w amn (k) = , ∂t ∂xn ∂xm m,n=1

w(0, x, k) = u¯ 0+ (x; k) :=

(5.18)

1 u0 (x, kl)d(l) d−1 Sd−1 +

with the diffusion matrix amn given by:

c0 anm (k) = ln χm (kl)d(l), d−1 Sd−1

(5.19)

and the functions χj above are the mean-zero solutions of d m,n=1

∂ ∂km

ˆ k 2 Dmn (k)

∂χj ∂kn

= −c0 kˆj .

(5.20)

Theorems 2.1, 2.5 and 2.7 allow us to make the passage to the limit ε, δ, γ → 0 rigorous. The assumption that ε δ γ is formalized as follows. We let ! Kµ,ρ := (ε, δ, γ ) : δ ≥ | ln ε|−2/3+µ and γ ≥ δ ρ , ± 3 2d with 0 < µ < 2/3, ρ ∈ (0, 1). Suppose also that u± 0 ∈ Cc (R∗ ) and supp u0 ⊆ A(M). Let

W 0 (x, k) := u0+ (x, k)b+ (k) ⊗ b+ (k) + u0− (x, k)b− (k) ⊗ b− (k),

(5.21)

and W (t, x, k) := w+ (t, x; k)b+ (k) ⊗ b+ (k) + w− (t, x; k)b− (k) ⊗ b− (k).

(5.22)

Our main result regarding the diffusion of wave energy can be stated as follows.


311

Theorem 5.2. Assume that the dimension d ≥ 3 and M ≥ 1 are fixed. Suppose for some 0 < µ < 2/3, ρ ∈ (0, 1) we have, with W 0 as in (5.21) and Wεδ defined by (5.9), 2

EW δ 0, x , k − W 0 (x, k) dxdk → 0, as (ε, δ, γ ) → 0 and (ε, δ, γ ) ∈ Kµ,ρ . ε γ R2d

Then, there exists ρ1 ∈ (0, ρ] such that for any T > T∗ > 0 we have 2

t x δ sup EWε γ 2 , γ , k − W (t, x, k) dxdk t∈[T∗ ,T ] → 0, as (ε, δ, γ ) → 0 and (ε, δ, γ ) ∈ Kµ,ρ1 . Here W (t, x, k) is of the form (5.22) with the functions w± that satisfy (5.18) with the initial data w± (0, x, k) = u¯ 0± (x, k). The proof follows immediately from Theorems 2.1, 2.5 and 2.7 as well as Proposition 5.1. A. The Proof of Lemma 3.5 Given s ≥ σ > 0, π ∈ C we define the linear approximation of the trajectory ˆ ) L(σ, s; π) := X(σ ) + (s − σ )H0 (K(σ ))K(σ

(A.1)

and for any v ∈ [0, 1] let R(v, σ, s; π) := (1 − v)L(σ, s; π ) + vX(s).

(A.2)

The following simple fact can be verified by a direct calculation, see Lemma 5.4 of [1]. Proposition A.1. Suppose that s ≥ σ ≥ 0 and π ∈ C(δ). Then,

s

˜ ˆ ˆ )|dρ. |X(s) − L(σ, s; π )| ≤ D(2M − H0 (K(σ ))K(σ δ ) δ(s − σ ) + |H0 (K(ρ))K(ρ) √

σ

We obtain from Proposition A.1 for each s ≥ σ an error for the first-order approximation of the trajectory √ C(s − σ )2 ˜ , |z(δ) (s) − l (δ) (σ, s)| ≤ D(2M √ δ ) δ(s − σ ) + 2 δ

δ ∈ (0, δ∗ (M)].

ˆ (δ) (σ ) is the linear approximation between the Here l (δ) (σ, s) := z(δ) (σ ) + (s − σ )m times σ and s and C :=

sup δ∈(0,δ∗ (M)]

˜ (Mδ h∗0 (Mδ ) + h˜ ∗0 (Mδ ))D(2M δ ).

With no loss of generality we may assume that x = 0 and that there exists k ≥ 0 such (p) (p) that u, t ∈ [tk , tk+1 ). We shall omit the initial condition in the notation of the solution to (3.12). Throughout this argument we use Proposition A.1 with σ (s) := s − δ 1−γA for some γA ∈ (0, 1/16 ∧ (1 − 4 )),

s ∈ [t, u].

(A.3)

312


The aforementioned proposition proves that for this choice of σ we have |L(δ) (σ, s) − y (δ) (s)| ≤ CA δ 3/2−2γA ,

∀ δ ∈ (0, 1].

(A.4)

Throughout this section we denote ζ˜ = ζ (y (δ) (t1 ), l (t1 ), . . . , y (δ) (tn ), l (tn )). We assume first that G ∈ C 2 (Rd∗ ) and G2 < +∞. Note that d u y (δ) (s) (δ) 1 (δ) (δ) (δ) G(l (u))−G(l (t)) = −√ , l (s) ds. ∂j G(l (s))Fj,δ s, δ δ (δ)

(δ)

j =1 t

(A.5) We can rewrite then (A.5) in the form I (1) + I (2) + I (3) , where d u y (δ) (s) (δ) 1 (1) (δ) I := − √ ∂j G(l (σ ))Fj,δ s, , l (σ ) ds, δ δ I

(2)

1 := δ

j =1 t d u s

i,j =1 t

×Fi,δ I (3)

∂j G(l

(ρ))∂i Fj,δ

y (δ) (s) (δ) s, , l (ρ) δ

σ

y (δ) (ρ) (δ) ρ, , l (ρ) ds dρ, δ

u s

d 1 := δ i,j =1 t

×Fi,δ

(δ)

2 ∂i,j G(l (δ) (ρ))Fj,δ

y (δ) (s) (δ) s, , l (ρ) δ

σ

y (δ) (ρ) (δ) ρ, , l (ρ) ds dρ, δ

and σ is given by (A.3). Each of these terms will be estimated separately below. A.1. The term E[I (1) ζ˜ ]. The term I (1) can be rewritten in the form J (1) + J (2) , where d u L(δ) (σ, s) (δ) 1 (1) (δ) J := − √ ∂j G(l (σ ))Fj,δ s, , l (σ ) ds, δ δ j =1 t

and J (2) := −

1 δ 3/2

d u 1

∂j G(l (δ) (σ ))∂yi Fj,δ

i,j =1 t 0 (δ) (δ) ×(yi (s) − Li (σ, s)) ds dv,

R (δ) (v, σ, s) (δ) s, , l (σ ) δ (A.6)

where, see (A.1) and (A.2), L(δ) (σ, s) = L(σ, s; y (δ) (·), l (δ) (·)), R (δ) (σ, s) = R(σ, s; y (δ) (·), l (δ) (·)). We use part (i) of Lemma 3.2 to handle the term E[J (1) ζ˜ ]. Let X˜ 1 (x, k) = −∂xi H1 (x, k), X˜ 2 (x, k) ≡ 1, (p) Z = tk , L(δ) (σ, s), l (δ) (σ ) ∂j G(l (δ) (σ ))ζ˜ ,


313

and g1 = (L(δ) (σ, s)δ −1 , |l (δ) (σ )|). Note that g1 and Z are both Fσ measurable. We need (p) to verify (3.15). Suppose therefore that Z = 0. For ρ ∈ [0, tk−1 ] we have |L(δ) (σ, s) − y (δ) (ρ)| ≥ (4q)−1 , provided that CA δ 3/2−2γA < 1/(12q), which holds for sufficiently small δ, since our assumptions on the exponents 2 , 3 , γA (namely that 2 , 3 < 1/8, (p) γA < 1/8) guarantee that 2 + 3 < 3/4 − γA /2. For ρ ∈ [tk−1 , σ ] we have (δ) (p) tk−1 (L(δ) (σ, s) − y (δ) (ρ)) · lˆ (δ) (δ) (p) tk−1 ≥ (s − σ )H0 (|l (δ) (σ )|) lˆ (σ ) · lˆ

σ √ y (δ) (ρ1 ) (δ) (δ) (δ) (p)

(δ) + H0 (|l (ρ1 )|) + δ ∂l H1 tk−1 dρ1 , |l (ρ1 )| lˆ (ρ1 ) · lˆ δ ρ √ (3.13) 2 2 ˜ ≥ (s − σ )h∗ (2Mδ ) 1 − + h∗ (2Mδ ) − δ D(2M δ ) (s − ρ) 1 − N N 2 ≥ (s − σ )h∗ (2Mδ ) 1 − , (A.7) N provided that δ ∈ (0, δ0 ] and δ0 is sufficiently small. We see from (A.7) that (3.15) is satisfied with r = (1 − 2/N) h∗ (2Mδ )δ 1−γA . Using Lemma 3.2 we estimate

u D(2M ˜ δ) (1) s − σ (1) ˜ ds G1 E[ζ˜ ] φ CA E[J ζ ] ≤ √ δ δ t (2) (1) −1/2 ≤ CA G1 E[ζ˜ ]δ φ CA δ −γA (u − t) ≤ CA G1 E[ζ˜ ]δ(u − t), (3)

(A.8)

(3)

and CA exists by virtue of assumption (2.4). On the other hand, the term J (2) defined (2) (2) by (A.6) may be written as J (2) = J1 + J2 ,where d u (δ) (σ, s) L 1 (2) J1 := − 3/2 , l (δ) (σ ) ∂j G(l (δ) (σ ))∂yi Fj,δ s, δ δ i,j =1 t

(δ)

(δ)

×(yi (s) − Li (σ, s)) ds and (2) J2

:= −

1 δ 5/2

u 1 1 d i,j,k=1 t

0 0 (δ)

∂y2i ,yk Fj,δ (δ)

R (δ) (θ v, σ, s) (δ) s, , l (σ ) v δ (δ)

(δ)

×∂j G(l (δ) (σ ))(yi (s) − Li (σ, s))(yk (s) − Lk (σ, s)) ds dv dθ. (A.9) (2)

The term involving J2 have then

may be handled easily with the help of (A.4) and (3.11). We

(2) (4) ˜ −5/2 3−4γA 2 δ T |E[J2 ζ˜ ]| ≤ CA D(2M δ )E[ζ˜ ]G1 (u − t)δ

≤ CA δ 1/2−4γA T 2 (u − t)E[ζ˜ ]G1 . (5)

(A.10)

314

T. Komorowski, L. Ryzhik (2)

In order to estimate the term corresponding to J1 (2) J1,1

:= −

1 δ 3/2

d u s i,j =1 t

×(s − ρ1 )

(2)

we write J1

(2)

(2)

= J1,1 + J1,2 , where

∂j G(l

(δ)

(σ ))∂yi Fj,δ

L(δ) (σ, s) (δ) , l (σ ) s, δ

σ

d (δ) (δ) H0 (|l (ρ1 )|) lî (ρ1 ) ds dρ1 dρ1

(A.11)

and (2) J1,2

d u s L(δ) (σ, s) (δ) 1 (δ) := − ∂j G(l (σ ))∂yi Fj,δ s, , l (σ ) δ δ i,j =1 t σ y (δ) (ρ) (δ) (δ) ×∂l H1 , |l (ρ)| lî (ρ) ds dρ, δ

with d (δ) (δ) (δ) (δ) (δ) H0 (|l (ρ1 )|) lî (ρ1 ) = H0

(|l (δ) (ρ1 )|) (lˆ (ρ1 ), l˙ (ρ1 ))Rd lî (ρ1 ) dρ1

d (δ) (δ) (δ) (δ) +H0 (|l (δ) (ρ1 )|)|l (δ) (ρ1 )|−1 li (ρ1 )−(lˆ (ρ1 ), l˙ (ρ1 ))Rd lî (ρ1 ) . dρ1 (A.12) (2)

(2)

(2)

(2)

(2)

We deal with J1,2 first. It may be split as J1,2 = J1,2,1 + J1,2,2 + J1,2,3 , where (2) J1,2,1

(2) J1,2,2

d u s L(δ) (σ, s) (δ) 1 (δ) := − ∂j G(l (σ ))∂yi Fj,δ s, , l (σ ) δ δ i,j =1 t σ L(δ) (σ, ρ) (δ) (δ) ×∂l H1 (A.13) , |l (σ )| lî (σ ) ds dρ, δ

d u s 1 L(δ) (σ, s) (δ) 1 (δ) , l (σ ) := − 2 ∂j G(l (σ ))∂yi Fj,δ s, δ δ i,j =1 t σ 0 R (δ) (v, σ, ρ) (δ) (δ) (δ) (δ) ×(∂yi ∂l H1 ) , |l (ρ)| (yi (ρ) − Li (σ, ρ)) lî (ρ) ds dρ dv δ

and (2) J1,2,3

d u s ρ L(δ) (σ, s) (δ) 1 (δ) , l (σ ) := − ∂j G(l (σ ))∂yi Fj,δ s, δ δ i,j =1 t σ σ d L(δ) (σ, s) (δ) (δ) × ∂l H 1 , |l (ρ1 )| lî (ρ1 ) ds dρ dρ1 . dρ1 δ


315

By virtue of (A.4), definition (3.10) and (3.11) we obtain easily that |E[J1,2,2 ζ˜ ]| ≤ CA δ 1/2−3γA G1 T (u − t)Eζ˜ . (2)

(6)

(A.14) (2)

The same argument and equality (A.12) also allow us to estimate |E[J1,2,3 ζ ]| by the right-hand side of (A.14). Using Lemma 3.1 and the definition (3.10) we conclude that there exists a constant (7) CA > 0 independent of δ such that (δ) (δ) ∂y Fj,δ s, L (σ, s) , l (δ) (σ ) − t (p), L(δ)(σ, s), l (δ)(σ ) ∂ 2 H1 L (σ, s) , |l (δ) (σ )| yi ,yj k i δ δ ≤ CA(7) δT ,

i, j = 1, . . . , d.

Therefore, we can write d u s 1 (p) (2) E[J ˜ E ∂j G(l (δ) (σ )) tk , L(δ) (σ, s), l (δ) (σ ) 1,2,1 ζ ] + δ i,j =1 t σ (δ) (σ, s) (δ) (σ, ρ) L L (δ) × ∂y2i ,yj H1 , |l (δ) (σ )| ∂l H1 , |l (δ) (σ )| lî (σ ) ζ˜ ds dρ δ δ ≤ CA δ 1−γA (u − t)G1 T Eζ˜ . (8)

(A.15)

We apply now part (ii) of Lemma 3.2 with (p) (δ) Z = ∂j G(l (δ) (σ )) tk , L(δ) (σ, s), l (δ) (σ ) lî (σ ) ζ˜ , X˜ 1 (x, k) := ∂x2i ,xj H1 (x, k), X˜ 2 (x) := ∂k H1 (x, k), L(δ) (σ, s) (δ) L(δ) (σ, ρ) (δ) g1 := , |l (σ )| , g2 := , |l (σ )| , δ δ (9)

r = CA (ρ − σ ),

(9)

r1 = CA (s − ρ).

We conclude that d u s (p) (δ) (δ) (δ) E J (2) ζ + 1 E ∂ G(l (σ )) t , L (σ, s), l (σ ) j 1,2,1 k δ i,j =1 t σ L(δ) (σ, s) − L(δ) (σ, ρ) (δ) (δ) 2 ×∂yi ,yj R1 , |l (σ )| lî (σ )ζ˜ ds dρ δ (10)

C (8) ≤ CA δ 1−γA (u − t)G1 T Eζ˜ + A G1 E[ζ˜ ] δ (9) (9)

u s C C (ρ − σ ) (s − ρ) A A × φ 1/2 φ 1/2 ds dρ, 2δ 2δ t

σ

(A.16)

316


where R1 (y, k) := E[H1 (y, k)∂k H1 (0, k)],

(y, k) ∈ Rd × [0, +∞).

(A.17)

We can use assumption (2.4) to estimate the second term on the right hand side of (A.16) (11) e.g. by CA δ(u − t)G1 Eζ˜ . The second term appearing on the left hand side of (A.16) equals  d u  1 (p) E ∂j G(l (δ) (σ )) tk , L(δ) (σ, s), l (δ) (σ ) (δ)

 H0 (|l (σ )|) j =1 t  s  

 d s − ρ (δ) × − ∂yj R1 H0 (|l (δ) (σ )|) lˆ (σ ), |l (δ) (σ )| dρ  ζ˜ ds, (A.18)  dρ δ σ

and integrating over dρ we obtain that it equals −

d j =1 t

u

E

∂j G(l (δ) (σ )) H0 (|l (δ) (σ )|)

(p) tk , L(δ) (σ, s), l (δ) (σ )

∂yj R1 0, |l (σ )| ζ˜ (δ)

ds

d u ∂j G(l (δ) (σ )) (p) (δ) (δ) (δ) −γA ˆ (δ) tk , L (σ, s), l (σ ) ∂yj R1 δ E (δ) l (σ ), |l (σ )| ζ˜ ds. + H0 (|l (σ )|) j =1 t

(A.19) (12)

By virtue of (2.5) the second term appearing in (A.19) is bounded e.g. by CA (12) t)G1 Eζ˜ for some constant CA > 0, thus we have shown that d u ∂j G(l (δ) (σ )) (p) (δ) (δ) E[J (2) ζ ]− E , L (σ, s), l (σ ) ∂yj R1 0, |l (δ) (σ )| t 1,2,1 k (δ)

H0 (|l (σ )|) j =1 t ≤ CA(13) δ 1−γA (u − t)G1 T Eζ˜ .

δ(u −

˜ζ ds

(A.20) (2)

Let us consider the term corresponding to J1,1 , cf. (A.11). Note that according to (A.12) (2)

(2)

(2)

and (3.12) we have J1,1 = J1,1,1 + J1,1,2 , where (2) J1,1,1

with

d u s L(δ) (σ, s) (δ) 1 (δ) , l (σ ) := − 2 ∂j G(l (σ ))∂yi Fj,δ s, δ δ i,j =1 t σ y (δ) (ρ1 ) (δ) ×(s − ρ1 )i ρ1 , , l (σ ) ds dρ1 , δ

i (ρ, y, l) := |l|−1 H0 (|l|) ˆl, Fδ (ρ, y, l) d li − Fi,δ (ρ, y, l) R

ˆ −H0 (|l|) l, Fδ (ρ, y, l) d lî , R


317

while (2) J1,1,2

d u s ρ1 L(δ) (σ, s) (δ) 1 (δ) := − 2 ∂j G(l (σ ))∂yi Fj,δ s, , l (σ ) δ δ i,j =1 t σ σ d y (δ) (ρ1 ) (δ) ×(s − ρ1 ) i ρ1 , (A.21) , l (ρ2 ) ds dρ1 dρ2 . dρ2 δ

Note that | dρd 2 i | ≤ CA δ −1/2 for some constant CA (14)

(14)

> 0. A straightforward com-

(2) |E[J1,1,2 ζ ]|

(15)

putation, using (A.3) and Lemma 3.1, shows that ≤ CA δ 1/2−3γA (u − t)G1 T E[ζ˜ ]. An application of (A.4), in the same fashion as it was done in the calcu(2) (2) lations concerning the terms E[J1,2,2 ζ ] and E[J1,2,3 ζ ], yields that d u s (δ) (σ, s) L 1 (2) (δ) (δ) E[J , l (σ ) (s−ρ1 )E ∂j G(l (σ ))∂yi Fj,δ s, 1,1,1 ζ ]+ 2 δ δ i,j =1 t σ L(δ) (σ, ρ1 ) (δ) (16) × i ρ1 , , l (σ ) ζ˜ ds dρ1 ≤ CA δ 1/2−4γA (u − t)G1 T E[ζ˜ ]. δ (A.22) For j = 1, . . . , d we let Vj (y, y , l) :=

d

H0

(|l|) − H0 (|l|) ∂y3i ,yj ,yk R(y − y , |l|)lî lˆk

i,k=1

+

d

H0 (|l|)|l|−1 ∂y3i ,yi ,yj R(y − y , |l|),

i=1

and also (t, y, y , l; π) := (t, y, l; π)(t, y , l; π ),

(A.23)

t ≥ 0, y, y ∈ Rd , l ∈ Rd∗ , π ∈ C, P := L(δ) (σ, s), L(δ) (σ, ρ1 ), l (δ) (σ ) , Pδ := δ −1 L(δ) (σ, s), δ −1 L(δ) (σ, ρ1 ), l (δ) (σ ) and (s) := (s, y (δ) (s), l (δ) (s); y (δ) (·), l (δ) (·)). Applying Lemma 3.1 and part ii) of Lemma 3.2, as in (A.15) and (A.16), we conclude that the difference between the second term on the left-hand side of (A.22) and d u s 1 (δ) ˜ ds dρ1 , ζ (s − ρ )E ∂ G(l (σ ))(σ, P )V (P ) 1 j j δ δ2 j =1 t

σ

(A.24)

318


(17) (1) can be estimated by CA δ γA (u − t)G1 E[ζ˜ ] for some γA > 0. Using the fact that (22)

|l (δ) (ρ) − l (δ) (σ )| ≤ CA δ 1/2−γA ,

ρ ∈ [σ, s],

(A.25)

estimate (A.4) and Lemma 3.1 we can argue that 2 (18) (σ, P ) − (s) ≤ CA (δ 1/2−γA −1 + δ 1/2−2(γA +2 +3 ) T ). We conclude therefore that the magnitude of the difference between the expression in (A.24) and     s

d u 1 2 E ∂j G(l (δ) (σ )) (s)  (s − ρ1 )Vj (Pδ ) dρ1  ζ˜  ds, (A.26) δ2 j =1 t

σ (2)

(19) (2) can be estimated by CA δ γA (u − t)G1 T E[ζ˜ ] for some γA > 0. Using shorthand (δ) notation Q(σ ) := H0 (|l (δ) (σ )|) lˆ (σ ) we can write the integral from σ to s appearing above as being equal to 

s d 1

(δ)

(δ)  (s − ρ ) (|l (σ )|) − H (|l (σ )|) H 1 0 0 δ2 s−δ 1−γA

i,k=1

s − ρ1 (δ) (δ) Q(σ ), |l (δ) (σ )| lî (σ )lˆk (σ ) + H0 (|l (δ) (σ )|)|l (δ) (σ )|−1 δ d s − ρ1 3 (δ) × ∂yi ,yi ,yj R dρ1 , Q(σ ), |l (σ )| δ ×∂y3i ,yj ,yk R

i=1

which upon the change of variables ρ1 := (s − ρ1 )/δ is equal to δ −γA

0

 ρ1 

d

H0

(|l (δ) (σ )|) − H0 (|l (δ) (σ )|)

i,k=1

(δ) (δ) ×∂y3i ,yj ,yk R ρ1 Q(σ ), |l (δ) (σ )| lî (σ )lˆk (σ ) + H0 (|l (δ) (σ )|)|l (δ) (σ )|−1

d

 ∂y3i ,yi ,yj R ρ1 Q(σ ), |l (δ) (σ )|  dρ1 . (A.27)

i=1

Using the fact that d

(δ) ∂y3i ,yj ,yk R ρ1 Q(σ ), |l (δ) (σ )| lˆk (σ )

k=1

=

d 2 (δ) R ρ Q(σ ), |l (σ )| ∂ 1 ,y y i j H0 (|l (δ) (σ )|) dρ1 1


319

we obtain, upon integrating by parts in the first term on the right-hand side of (A.27), that this expression equals H0 (|l (δ) (σ )|)−1 H0

(|l (δ) (σ )|) − H0 (|l (δ) (σ )|)  d δ −γA ∂y2 ,y R δ −γA Q(σ ), |l (δ) (σ )| lˆ(δ) (σ ) × i i j i=1 δ −γA

−



 (δ) ∂y2i ,yj R ρ1 Q(σ ), |l (δ) (σ )| lî (σ ) dρ1 

(A.28)

0 δ −γA

+H0 (|l (δ) (σ )|)|l (δ) (σ )|−1

ρ1 ∂y3i ,yi ,yj R ρ1 Q(σ ), |l (δ) (σ )| dρ1 .

0

(A.29) Note that ∇R(0, l) = 0 and d

(δ) ∂y2i ,yj R ρ1 Q(σ ), |l (δ) (σ )| lî (σ ) =

i=1

d (δ) ∂ R ρ Q(σ ), |l (σ )| . yj 1 H0 (|l (δ) (σ )|) dρ1 1

We obtain therefore that the expression in (A.28) equals H0 (|l (δ) (σ )|)−1 H0

(|l (δ) (σ )|) − H0 (|l (δ) (σ )|)  d (δ) × δ −γA ∂y2 ,y R δ −γA Q(σ ), |l (δ) (σ )| lˆ (σ ) i

i

j

i=1



−H0 (|l (δ) (σ )|)−1 ∂yj R δ −γA Q(σ ), |l (δ) (σ )| 

+H0 (|l (δ) (σ )|)|l (δ) (σ )|−1

−γA d δ

ρ1 ∂y3i ,yi ,yj R ρ1 Q(σ ), |l (δ) (σ )| dρ1 .

i=1 0

(A.30) Recalling assumption (2.5) we conclude that the expressions corresponding to the first (3) (3) two terms appearing in (A.30) are of order of magnitude O(δ γA ) for some γA > 0. Summarizing work done in this section, we have shown that     d u  2 (δ) (δ) E I (1) −  ˜ C (l (σ )) (s)∂ G(l (σ )) ds ζ j j   j =1 t

(20)

≤ CA δ

(4) γA

(u − t)G1 T 2 Eζ˜

(A.31)

320


(4)

for some constants CA , γA > 0 and (cf. (A.17)) Cj (l) := Ej (ˆl, |l|) +

∂yj R1 (0, |l|)

, H0 (|l|) +∞ d H0 (k) ρ1 ∂y3i ,yi ,yj R ρ1 H0 (k)ˆl, k dρ1 , Ej (ˆl, k) := − k

j = 1, . . . , d.

i=1 0

A.1.1. The terms E[I (2) ζ˜ ] and E[I (3) ζ˜ ]. The calculations concerning these terms essentially follow the respective steps performed in the previous section so we only highlight their main points. First, we note that the difference between E[I (2) ζ˜ ] and

(δ) d u s 1 y (s) (δ) y (δ) (ρ) (δ) E ∂j G(l (δ) (σ ))∂i Fj,δ s, , l (σ ) Fi,δ ρ, , l (σ ) ζ˜ ds dρ δ i,j =1 δ δ t

σ

(A.32) (5)

(21) is less than, or equal to CA δ γA (u − t)G1 E[ζ˜ ], cf. (A.25). Next we note that (A.32) equals

d u s (δ) (σ, s) 1 L E ∂j G(l (δ) (σ ))∂i Fj,δ s, , l (δ) (σ ) δ δ i,j =1 t σ L(δ) (σ, ρ) (δ) ×Fi,δ ρ, , l (σ ) ζ˜ ds dρ δ

u s 1 d 1 R (δ) (v, σ, s) (δ) (δ) + 2 E ∂j G(l (σ ))∂i ∂yk Fj,δ s, , l (σ ) δ δ i,j,k=1 t σ 0 L(δ) (σ, ρ) (δ) (δ) (δ) ˜ × Fi,δ ρ, , l (σ ) (yk (s) − Lk (σ, s))ζ ds dρ dv δ

u s 1 d 1 y (δ) (s) (δ) (δ) + 2 E ∂j G(l (σ ))∂i Fj,δ s, , l (σ ) δ δ i,j,k=1 t σ 0 R (δ) (v, σ, ρ) (δ) (δ) (δ) ˜ × ∂yk Fi,δ ρ, , l (σ ) (yk (ρ) − Lk (σ, ρ))ζ ds dρ dv. δ (A.33) A straightforward argument using Lemma 3.1 and (A.4) shows that both the second and (23) third terms of (A.33) can be estimated by CA δ 1/2−(6γA +1 ) (u − t)G1 T 2 E[ζ˜ ]. The first term, on the other hand, can be handled with the help of part ii) of Lemma 3.2 in (2) the same fashion as we have dealt with the term J1,2,1 , given by (A.13) of Sect. A.1, and


321

we obtain that     d u  2 (δ) (δ) (δ) (δ) E I (2) −  ˜ (|l (σ )|) (s)+J (s; y (·), l (·))(s) ∂ G(l (σ )) ds ζ D j j j   j =1 t

≤

(24) (6) CA δ γA (u − t)G1 T E[ζ˜ ].

(A.34)

Here Dj (l) :=

∂yj R2 (0,l) H0 (l)

R2 (y, l) := E[∂l H1 (y, l)H1 (0, l)],

,

Jj (s; y (δ) (·), l (δ) (·)) := − i (s) := ∂li

d

(δ)

i (s)Di,j (lˆ (σ ), |l (δ) (σ )|),

(A.35)

i=1 (s, y (δ) (s), l (δ) (s); y (δ) (·), l (δ) (·)).

Finally, concerning the limit of E[I (3) ζ˜ ], another application of (A.4) yields (25) (7) E[I (3) ζ˜ ] − I ≤ CA δ γA (u − t)G1 E[ζ˜ ],

(A.36)

where 1 I := δ

E t

σ

u s

2 ∂i,j G(l (δ) (σ ))Fj,δ

×Fi,δ ρ,

L(δ) (σ, s) (δ) s, , l (σ ) δ

L(δ) (σ, ρ) (δ) , l (σ ) ζ˜ ds dρ. δ

Then, we can use part ii) of Lemma 3.2 in order to obtain d u (δ) I − ˆ (σ ), |l (δ) (σ )|)2 (s)∂ 2 G(l (δ) (σ )) ds D ( l i,j i,j i,j =1 t

≤

(26) (8) CA δ γA (u − t)G2 T E[ζ˜ ].

(A.37)

Next we replace the argument σ , in formulas (A.31), (A.34) and (A.36), by s. This can be done thanks to estimate (A.25) and the assumption on the regularity of the random field H1 (·, ·). In order to make this approximation work we will be forced to use the third derivative of G(·). Finally (cf. (A.17), (A.35)) note that ∇y R1 (0, l) + ∇y R2 (0, l) = ∇

E[∂l H1 (y, l)H1 (y, l)] y y=0

= 0.

Hence we conclude that the assertion of the lemma holds for any function G ∈ C 3 (Rd∗ ) satisfying G3 < +∞. Generalization to an arbitrary G ∈ Cb1,1,3 ([0, +∞) × R2d ∗ )

322


is fairly standard. Let r0 be any positive integer and consider sk := t + kr0−1 (u − t), k = 0, . . . , r0 . Then ! E [G(u, y (δ) (u), l (δ) (u)) − G(t, y (δ) (t), l (δ) (t))]ζ˜ = =

r 0 −1 k=0 r 0 −1

! E [G(sk+1 , y (δ) (sk+1 ), l (δ) (sk+1 )) − G(sk , y (δ) (sk ), l (δ) (sk ))]ζ˜ . E [G(sk , y (δ) (sk ), l (δ) (sk+1 )) − G(sk , y (δ) (sk ), l (δ) (sk ))]ζ˜

k=0 r 0 −1

+

!

! E [G(sk+1 , y (δ) (sk+1 ), l (δ) (sk )) − G(sk , y (δ) (sk ), l (δ) (sk ))]ζ˜ . (A.38)

k=0

Using the already proven part of the lemma we obtain r −1 0 ! "sk+1 (G(sk , y (δ) (sk ), · )) − N "sk (G(sk , y (δ) (sk ), · ))]ζ˜ E [N k=0

(9)

≤ CA δ γA (u − t)G1,1,3 T 2 Eζ˜ . (27)

(A.39)

On the other hand, the second term on the right-hand side of (A.38) equals sk+1 r 0 −1

E

H0 (|l (δ) (ρ)|) +

√

δ∂l H1

k=0 sk

y (δ) (ρ) (δ) , |l (ρ)| δ

(δ)

×lˆ (ρ) · ∇y G(ρ, y (δ) (ρ), l (δ) (sk )) + ∂ρ G(ρ, y (δ) (ρ), l (δ) (sk )) ζ˜

dρ.

(A.40)

The conclusion of the lemma for an arbitrary function G ∈ Cb1,1,3 ([0, +∞) × R2d ∗ ) is an easy consequence of (A.38)–(A.40) upon passing to the limit with r0 → +∞. Acknowledgement. The research of TK was partially supported by KBN grant 2PO3A03123. The work of LR was partially supported by an NSF grant DMS-0203537, an ONR grant N00014-02-1-0089 and an Alfred P. Sloan Fellowship.

References 1. Bal, G., Komorowski, T., Ryzhik, L.: Self-averaging of Wigner Transforms in Random Media. Commun. Math. Phys. 242, 81–135 (2003) 2. Dürr, D., Goldstein, S., Lebowitz, J.: Asymptotic motion of a classical particle in a random potential in two dimensions: Landau model. Commun. Math. Phys. 113, 209–230 (1987) 3. Erdös, L.,Yau, H.T.: Linear Boltzmann equation as the weak coupling limit of a random Schrödinger Equation. Commun. Pure Appl. Math. 53, 667–735 (2000) 4. Erdös, L., Salmhofer, M., Yau, H.T.: Quantum diffusion of the random Schrödinger evolution in the scaling limit. http://arxiv.org/list/math-ph/0502025, 2005 5. Gérard, P., Markowich, P.A., Mauser, N.J., Poupaud, F.: Homogenization limits and Wigner transforms. Commun. Pure Appl. Math., 50, 323–380 (1997)


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6. Gikhman, I.I., Skorochod, A.V.: Theory of stochastic processes. Vol. 3, Berlin: Springer Verlag, 1974 7. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Berlin: Springer Verlag, 1998 8. Kesten, H., Papanicolaou, G.: A Limit Theorem For Turbulent Diffusion. Commun. Math. Phys. 65, 97–128 (1979) 9. Kesten, H., Papanicolaou, G.C.: A Limit Theorem for Stochastic Acceleration. Commun. Math. Phys. 78, 19–63 (1980) 10. Kusuoka, S., Stroock, D.: Applications of the Malliavin calculus, Part II. J. Fac. Sci. Univ. Tokyo, Sect. IA, Math, 32, 1–76 (1985) 11. Lions, P.-L., Paul, T.: Sur les mesures de Wigner. Rev. Mat. Iberoamericana, 9, 553–618 (1993) 12. Lukkarinen, J., Spohn, H.: Kinetic limit for wave propagation in a random medium. http://arxiv.org/list/math-ph/0505075, 2005 13. Ryzhik, L., Papanicolaou, G., Keller, J.: Transport equations for elastic and other waves in random media. Wave Motion 24, 327–370 (1996) 14. Spohn, H.: Derivation of the transport equation for electrons moving through random impurities. J. Stat. Phys. 17, 385–412 (1977) 15. Strook, D.: An Introduction to the Analysis of Paths on a Riemannian Manifold. Math. Surv. and Monographs 74, Providence, RI:Amer.Math.Soc., 2000 16. Strook, D., Varadhan, S.R.S.: Multidimensional Diffusion Processes. Berlin, Heidelberg, New York: Springer-Verlag, 1979 17. Taylor, M.: Partial differential equations. Vol. 2, New York: Springer-Verlag, 1996 Communicated by P. Constantin


Communications in


Quantum Variance and Ergodicity for the Baker’s Map M. Degli Esposti1 , S. Nonnenmacher2 , B. Winn1, 1

Dipartimento di Matematica, Università di Bologna, 40127 Bologna, Italy. E-mail: [email protected] 2 Service de Physique Théorique, CEA/DSM/PhT Unité de recherche associée au CNRS, CEA/Saclay, 91191 Gif-sur-Yvette cédex, France. E-mail: [email protected] Received: 15 December 2004/Accepted: 15 March 2005 Published online: 3 February 2006 – © Springer-Verlag 2006

Abstract: We prove an Egorov theorem, or quantum-classical correspondence, for the quantised baker’s map, valid up to the Ehrenfest time. This yields a logarithmic upper bound for the decay of the quantum variance, and, as a corollary, a quantum ergodic theorem for this map.

1. Introduction The correspondence principle of quantum mechanics suggests that in the classical limit the behaviour of quantum systems reproduces that of the system’s classical dynamics. It is becoming clear that to understand this process fully represents a challenge not only to methods of semiclassical analysis, but also the modern theory of dynamical systems. For a broad class of smooth Hamiltonian systems it has been proved that if the system is ergodic, then, in the classical limit, almost all eigenfunctions of the corresponding quantum mechanical Hamiltonian operator become equidistributed with respect to the natural measure (Liouville) over the energy shell. This is the content of the so-called ˇ Zel1, CdV, HMR]. quantum ergodicity theorem [Sni, This mathematical result, even if it can be considered quite mild from the physical point of view, still constitutes one of the few rigorous results concerning the properties of quantum eigenfunctions in the classical limit, and it still leaves open the possible existence of exceptional subsequences of eigenstates which might converge to other invariant measures. In the last few years a certain number of works have explored this mathematically and physically interesting issue. While exceptional subsequences can be present for some hyperbolic systems with extremely high quantum degeneracies [FDBN], it is believed that they do not exist for a typical chaotic system (by chaotic, we generally mean that the system is ergodic and mixing). The uniqueness of the classical limit for the Present address: Department of Mathematics, Texas A&M University, College Station, TX 778433368, USA. E-mail: [email protected]

326

M. Degli Esposti, S. Nonnenmacher, B. Winn

quantum diagonal matrix elements is called quantum unique ergodicity (QUE) [RudSar, Sar1]. There have been interesting recent results in this direction for Hecke eigenstates of the Laplacian on compact arithmetic surfaces [Lin], using methods which combine rigidity properties of semi-classical measures with purely dynamical systems theory. The model studied in the present paper is not a Hamiltonian flow, but rather a discretetime symplectic map on the 2-dimensional torus phase space. In the case of quantised hyperbolic automorphisms of the 2-torus (“quantum cat maps”), QUE has been proven along a subsequence of Planck’s constants [DEGI, KR2], and for a certain class of eigenstates (also called “Hecke” eigenstates) [KR1] without restricting Planck’s constant. QUE has also been proved in the case of some uniquely ergodic maps [MR, Ros]. Quantum (possibly non-unique) ergodicity has been shown for some ergodic maps which are smooth by parts, with discontinuities on a set of zero Lebesgue measure [DBDE, MO’K, DE+ ]. Discontinuities generally produce diffraction effects at the quantum level, which need to be taken care of (this problem also appears in the case of Euclidean billiards with non-smooth boundaries [GL, ZZ]). Most proofs of quantum ergodicity consist of showing that the quantum variance defined below (Eq. (1.1)) vanishes in the classical limit. To state our results we now turn to the specific dynamics considered in the present article. We take as classical dynamical system the baker’s map on T2 , the 2-dimensional torus [AA]. For any even positive integer N ∈ 2N (N is the inverse of Planck’s constant h), this map can be quantised into a unitary operator (propagator) Bˆ N acting on an N dimensional Hilbert space. The quantum variance measures the average equidistribution ˆ of the eigenfunctions {ϕN,j }N−1 j =0 of BN : S2 (a, N ) :=

N−1 2 1 W , Op (a)ϕ − a(q, p) dqdp . ϕN,j N,j N 2 N T

(1.1)

j =0

Here a is some smooth function (observable) on T2 and OpW N (·) is the Weyl quantisation mapping a classical observable to a corresponding quantum operator. The quantised baker’s map (or some variant of it) is a well-studied example in the physics literature ˙ which motivated our on quantum chaology [BV, Sa, SaVo, O’CTH, Lak, Kap, ALPZ], desire to provide rigorous proofs for both the quantum-classical correspondence and quantum ergodicity. In this paper we prove a logarithmic upper bound on the decay of the quantum variance (see Theorem 1 below), which implies quantum ergodicity as a byproduct (Corollary 2). A similar upper bound was first obtained by Zelditch [Zel2] in the case of the geodesic flow of a compact negatively curved Riemannian manifold, and was generalized by Robert [Rob] to more general ergodic Hamiltonian systems. Both are using some control on the rate of classical ergodicity (Zelditch also proved similar upper bounds for higher moments of the matrix elements). The main semiclassical ingredient needed for all proofs of quantum ergodicity is some control on the correspondence between quantum and classical evolutions of observables, namely some Egorov estimate. As for billiard flows [Fa], such a correspondence can only hold for observables supported away from the set of discontinuities. We establish this correspondence for the quantum baker’s map in Sect. 5.2, generalizing previous results [DBDE] for a subclass of observables (an Egorov theorem was already proven in [RubSal] for a different quantisation of the baker’s map). Some related results can be found in [BGP, BR] for the case of smooth Hamiltonian systems. To obtain this Egorov estimate, we study the propagation of coherent states (Gaussian wavepackets): they provide a convenient way to “avoid” the set of

Quantum Variance and Ergodicity for the Baker’s Map

327

discontinuities. The correspondence will hold up to times of the order of the Ehrenfest time log N (1.2) TE (N ) := log 2 (here log 2 is the positive Lyapunov exponent of the classical baker’s map). Equipped with this estimate, one could apply the general results of [MO’K] to prove that the quantum variance semiclassically vanishes. We prefer to generalise the method of [Schu2] (applied to smooth maps or flows) to our discontinuous baker’s map. This method, inspired by some earlier heuristic calculations [FP, Wil, EFK+ ], yields a logarithmic upper bound for the variance. It relies on the decay of classical correlations (mixing property), which is related, yet not equivalent, with the control on the rate of ergodicity used in [Zel2, Rob]. Our main result is the following theorem. Theorem 1. For any observable a ∈ C ∞ (T2 ), there is a constant C(a) depending only on a, such that the quantum variance over the eigenstates of Bˆ N satisfies: C(a) . ∀N ∈ 2N, S2 (a, N ) ≤ log N We believe that this method can be extended to any piecewise linear map satisfying a fast mixing. We also can speculate that the method would work for non-linear piecewise-smooth maps, although in that case the propagation of coherent states should be analysed in more detail (see Remark 2). The upper bound in Theorem 1 seems far from being sharp. The heuristic calculations in [FP, Wil, EFK+ ] suggest that the quantum variance decays like V (a) N −1 , where the prefactor V (a) is the classical variance of the observable a, appearing in the central limit theorem. This has been conjectured to be the true decay rate for a “generic” Anosov system. The decay of quantum variance has been studied numerically in [EFK+ ] for the baker’s map and [BSS] for Euclidean billiards; in both cases, the results seem to be compatible with a decay N −1 ; however, a discrepancy of around 10% was noted between the observed and conjectured prefactors. This was attributed to the low values of N (or energy in the case of billiards) considered. A more recent numerical study of a chaotic billiard, at higher energies, still reveals some (smaller) deviations from the conjectured law [Bar], leaving open the possibility of a decay N −γ with γ = 1. A decay of the form V˜ (a) N −1 (with an explicit factor V˜ (a)) could be rigorously proven for two particular Anosov systems, using their rich arithmetic structure [KR1, LS, RuSo]. In both cases, the prefactor V˜ (a) generally differs from the classical variance V (a), which is attributed to the arithmetic properties of the systems, which potentially makes them “non-generic”. Algebraic decays have also been proven for some uniquely ergodic (non-hyperbolic) maps [MR, Ros], by pushing the Egorov property to times of order O(N ). The rigorous investigation of the quantum variance thus remains an important open problem in quantum chaology [Sar2]. Quantum ergodicity follows from Theorem 1 as a corollary: N→∞

Corollary 2. For each N ∈ 2N there exists a subset JN ⊂ {1, . . . , N}, with #JNN −−−−→ 1, such that for any a ∈ C ∞ (T2 ) and any sequence (jN ∈ JN )N∈2N , (a)ϕ = a(x)dx. (1.3) lim ϕN,jN , OpW N,j N N N→∞

T2

328


This generalises a result of [DBDE] to any observable a ∈ C ∞ (T2 ) (previously only observables of the form a = a(q) could be handled). The restriction to a subset JN is the “almost all” clarification in quantum ergodicity. The paper is organised as follows. In Sect. 2 we briefly describe the classical baker’s map on T2 . In Sect. 3, we recall how this map can be quantised [BV] into an N × N unitary matrix. We then describe the action of the quantised baker map on coherent states (Proposition 5). This is the first step towards the Egorov estimates proven in Sect. 5 (Theorems 12 and 13, which shows the correspondence up to the Ehrenfest time). The first part of that section (Subsect. 5.1) compares the Weyl and anti-Wick quantisations, for observables which become more singular when N grows. This technical step is necessary to obtain Egorov estimates for times log N . In the final section, we implement the method of [Schu2] to the quantum baker’s map, using our Egorov estimates up to logarithmic times, and prove Theorem 1. 2. The Classical Baker’s Map The baker’s map1 is the prototype model for discontinuous hyperbolic systems, and it has been extensively studied in the literature. Standard results may be found in [AA], while the exponential mixing property was analyzed by [Has], and also derives from the results of [Ch]. Here, for the sake of fixing notations, we restrict ourself to recalling the very basic definitions and properties, referring the reader to the above references for more details concerning the ergodic properties of the map. We identify the torus T2 with the square [0, 1) × [0, 1). The first (horizontal) coordinate q represents the “position”, while the second (vertical) represents the “momentum”. In our notations, x = (q, p) will always represent a phase space point, either on R2 or on its quotient T2 . The baker’s map is defined as the following piecewise linear bijective transformation on T2 : (2q, p/2), if q ∈ [0, 1/2), (2.1) B(q, p) = (q , p ) = (2q − 1, (p + 1)/2), if q ∈ [1/2, 1). The transformation is discontinuous on the following subset of T2 : S1 := {p = 0} ∪ {q = 0} ∪ {q = 1/2},

(2.2)

and smooth everywhere else. If we consider iterates of the map, the discontinuity set becomes larger: for any n ∈ N, the map B n is piecewise linear, and discontinuous on the set n −1 2 j q= n , Sn := {p = 0} ∪ 2 j =0

B −n

while its inverse is discontinuous on the set S−n obtained from Sn by exchanging the q and p coordinates. Clearly, the discontinuity set becomes dense in T2 as |n| → ∞. The map is area preserving and uniformly hyperbolic outside the discontinuity set, with constant Lyapunov exponents ± log 2 and positive topological entropy (see below). The stable (resp. unstable) manifold is made of vertical (resp. horizontal) segments. A nice feature of this map lies in a simple symbolic coding for its orbits. Each real number q ∈ [0, 1) can be associated with a binary expansion 1 The name refers to the cutting and stretching mechanism in the dynamics of the map which is reminiscent of the procedure for making bread. Hence we write the word “baker” with a lower case “b”.


q = · 0 1 2 . . .

329

(i ∈ {0, 1}).

This representation is one-to-one if we forbid expansions of the form ·0 1 . . . 111 . . . . Using the same representation for the p-coordinate: p = · −1 −2 . . . , a point x = (q, p) ∈ T2 can be represented by the doubly-infinite sequence x = . . . −2 −1 · 0 1 . . . . Then, one can easily check that the baker’s map acts on this representation as a symbolic shift: B(. . . −2 −1 · 0 1 . . . ) = . . . −2 −1 0 · 1 . . . .

(2.3)

From this symbolic representation, one gets the Kolmogorov-Sinai entropy of the map with respect to the Lebesgue measure, hKS = log 2, as well as exponential mixing properties [Ch, Has]: there exists > 0 and C > 0 such that, for any smooth observables a, b on T2 , the correlation function a(x) b(B −n x) dx − a(x) dx b(x) dx (2.4) Kab (n) := T2

T2

T2

is bounded as |Kab (n)| ≤ C a C 1 b C 1 e−|n| .

(2.5)

According to [Has], one can take for any number smaller than log 2. 3. Quantised Baker’s Map The quantisation of the 2-torus phase space is now well-known and we refer the reader to [DEG], here describing only the important facts. The quantisation of an area-preserving map on the torus is less straightforward, and in general it contains some arbitrariness. The quantisation of linear symplectomorphisms of the 2-torus (or “generalised Arnold cat maps”) was first considered in [HB], and the case of nonlinear perturbations of cat maps was treated in [BdM+ ] (quantum ergodicity was proven for these maps in [BDB]). The scheme we present below, specific for the baker’s map, was introduced in [BV]. We start by defining the quantum Hilbert space associated to the torus phase space. For any ∈ (0, 1], we consider the quantum translations (elements of the Heisenberg ˆ 1 p)/ ˆ , v ∈ R2 , acting on L2 (R) and by extension on S (R). We group) Tˆv = ei(v2 q−v then define the space of distributions H = {ψ ∈ S (R), Tˆ(1,0) ψ = Tˆ(0,1) ψ = ψ}. These are distributions ψ(q) which are Z-periodic, and such that their -Fourier transform ∞ dq (Fˆ ψ)(p) := (3.1) ψ(q) e−iqp/ √ 2π −∞ is also Z-periodic.

330


One easily shows that this space is nontrivial iff (2π )−1 = N ∈ N, which we will always assume from now on. This space can be obtained as the image of L2 (R) through the “projector” (−1)Nm1 m2 Tˆm = (3.2) Tˆ0,m2 Tˆm1 ,0 . PˆT2 = m2 ∈Z

m∈Z2

m1 ∈Z

H = HN then forms an N -dimensional vector space of distributions, admitting a “position representation”

N−1 N−1 1 j ψ(q) = √ + ν =: ψj δ q − ψj qj (q), N N j =0 ν∈Z j =0

(3.3)

where each coefficient ψj ∈ C. Here we have denoted by {qj }N−1 j =0 the canonical (“position”) basis for HN . This space can be naturally equipped with the Hermitian inner product: qj , qk = δj k ⇒ ψ, ω :=

N−1

ψj ω j .

(3.4)

j =0

Since HN is the image of S(R) through the “projector” (3.2), any state ψ ∈ HN can be constructed by projecting some Schwartz function (q). The decomposition on the RHS of (3.2) suggests that we may first periodicise in the q-direction, obtaining a periodic function C (q); such a wavefunction describes a state living in the cylinder phase space C = T × R. The torus state ψ(q) is finally obtained by periodicising C in the Fourier variable; equivalently, the N components of ψ in the basis {qj } are obtained by sampling this function at the points qj = Nj : j 1 , 0 ≤ j < N. (3.5) ψj = √ C N N The -Fourier transform Fˆ (seen as a linear operator on S (R)) leaves the space HN invariant. On the basis {qj }, it acts as an N × N unitary matrix FˆN called the “discrete Fourier transform”: 1 (3.6) (FˆN )kj = √ e−2iπkj/N , k, j = 0, . . . , N − 1. N Fˆ quantises the rotation by −π/2 around the origin, F (q0 , p0 ) = (p0 , −q0 ). As a result, FˆN maps the “position basis” {qj } onto the “momentum basis” {pj }: pj =

N−1

(FˆN−1 )kj qk .

k=0

The quantised baker’s map Bˆ N was introduced by Balazs and Voros [BV]. They require N to be an even integer, and prescribe the following matrix in the basis {qj }:

FˆN/2 0 −1 ˆ ˆ ˆ ˆ . (3.7) with BN,mix := BN := (FN ) BN,mix , 0 FˆN/2


331

This definition was slightly modified by Saraceno [Sa], in order to restore the parity symmetry of the classical map. Although we will concentrate on the map (3.7), all our results also apply to this modified setting.

3.1. Notations. Since we will be dealing with quantities depending on Planck’s constant N (plus possibly other parameters), all asymptotic notations will refer to the classical limit N → ∞. The notations A = O(B) and A B both mean that there exists a constant c such that for any N ≥ 1, |A(N )| ≤ c|B(N)|. Writing A = Or (B) and A r B means that the constant c depends on the parameter r. Similarly A = o(B) and A 0, the C j -norm is defined as

f C j :=

∂ γ f C 0 .

0≤|γ |≤j γ

γ

Here γ = (γ1 , γ2 ) ∈ N20 denotes the multiindex of differentiation: ∂ γ = ∂q 1 ∂p2 , and |γ | := γ1 + γ2 . Because we want to consider large time evolution, namely times n log N , we need to consider (smooth) functions which depend on Planck’s constant 1/N . Indeed, starting from a given smooth function a, its evolution a ◦ B −n fluctuates more and more strongly along the vertical direction, while it is smoother and smoother along the horizontal one as n → ∞ (assuming a is supported away from the discontinuity set Sn ). For this reason, we introduce the following spaces of functions [DS, Chap. 7]: Definition 1. For any α = (α1 , α2 ) ∈ R2+ , we call Sα (T2 ) the space of N -dependent smooth functions f = f (·, N ) such that, for any multiindex γ ∈ N20 , the quantity

∂ γ f (·, N ) C 0 N α·γ N∈N

Cα,γ (f ) := sup

is finite (here α · γ = α1 γ1 + α2 γ2 ). The seminorms Cα,γ (γ ∈ N20 ) endow Sα (T2 ) with the structure of a Fréchet space.

332


4. Coherent States on T2 Our proof of the quantum-classical correspondence will use coherent states on T2 . Below we define them, and collect some useful properties. More comprehensive details and proofs may be found in [Fo, Per, LV, BDB, BonDB]. We define a (plane) coherent state at the origin with squeezing σ > 0 through its wavefunction 0,σ ∈ S(R) (we will always omit the indication of -dependence): σ 1/4 σ q 2 e− 2 . (4.1) 0,σ (q) := π The (plane) coherent state at the point x = (q0 , p0 ) ∈ R2 is obtained by applying a quantum translation Tˆx to the state above, which yields: σ 1/4 p0 q0 p0 q −σ (q−q0 )2 x,σ (q) := e−i 2 ei e 2 π = (2N σ )1/4 e−πiNq0 p0 +2πiNp0 q−σ Nπ(q−q0 ) . 2

(In the second line, we took = (2πN )−1 , as is required if we want to project on the torus). From here we obtain a coherent state on the cylinder by periodicising along the q-axis: x,σ,C (q) := x,σ (q + ν). (4.2) ν∈Z

Finally, the coherent state on the torus is obtained by further periodicising in the Fourier variable, or equivalently by sampling this cylinder wavefunction: its coefficients in the canonical basis read 1 ψx,σ,T2 j = √ x,σ,C (j/N ), j = 0, . . . , N − 1. (4.3) N One can check that ψx+m,σ,T2 ∝ ψx,σ,T2 for any m ∈ Z2 : up to a phase, the state ψx,σ,T2 depends on the projection on T2 of the point x. In the classical limit, it will often be useful to approximate a torus (or cylinder) coherent state by the corresponding planar one: Lemma 3. Let q0 ∈ (δ, 1 − δ) for some 0 < δ < 1/2. Then in the classical limit: 2 (4.4) ∀q ∈ [0, 1), x,σ,C (q) = x,σ (q) + O (σ N )1/4 e−πNσ δ . The error estimate is uniform for σ N ≥ 1. Proof. Extracting the ν = 0 term in (4.2), one gets 2 ∀q ∈ [0, 1), x,σ,C (q) = x,σ (q) + O (σ N )1/4 e−πσ N min{|q−q0 +ν| :ν=0} . Now, if q0 ∈ (δ, 1 − δ), one has |q − q0 | ≤ 1 − δ, so that ∀ν = 0,

|q − q0 − ν| ≥ |ν| − |q − q0 | ≥ 1 − |q − q0 | ≥ δ.

The next lemma describes how a torus coherent state transforms under the application of the discrete Fourier transform.


333

Lemma 4. For any x = (q0 , p0 ) ∈ R2 , let F x := (p0 , −q0 ) denote its rotation by −π/2 around the origin. Then ∀N ≥ 1, ∀σ > 0,

FˆN ψx,σ,T2 = ψF x,1/σ,T2 .

(4.5)

Proof. The plane coherent states, which are Gaussian wavefunctions, are obviously covariant through the Fourier transform Fˆ : a straightforward computation shows that ∀x ∈ R2 ,

Fˆ ψx,σ = ψF x,1/σ .

When (2π) = N −1 , we apply the projector (3.2) to both sides of this inequality, and remember that Fˆ acts on HN as the matrix FˆN : this means PˆT2 Fˆ = FˆN PˆT2 , so the above covariance is carried over to the torus coherent states. 4.1. Action of Bˆ N on coherent states. We assume N to be an even integer, and apply the matrix Bˆ N to the coherent state ψx,σ,T2 , seen as an N -component vector in the basis {qj }. We get nice results if the point x = (q0 , p0 ) is “far enough” from the singularity set S1 (in this case Bx is well-defined). More precisely, we define the following subsets of T2 : Definition 2. For any 0 < δ < 1/4 and 0 < γ < 1/2, let D1,δ,γ := (q, p) ∈ T2 , q ∈ (δ, 1/2 − δ) ∪ (1/2 + δ, 1 − δ), p ∈ (γ , 1 − γ ) . (4.6) The evolution of coherent states will be simple for states localised in this set. Proposition 5. For some parameters δ, γ (which may depend on N ), we consider points x = (q0 , p0 ) ∈ T2 in the set D1,δ,γ . We associate to these points the phase  0, if q0 ∈ (δ, 1/2 − δ), (4.7) (x) = p0 + 1 q0 + , if q0 ∈ (1/2 + δ, 1 − δ). 2 We assume that the squeezing σ may also depend on N , remaining in the interval σ ∈ [1/N, N ]. From δ, γ , σ we form the parameter θ = θ (δ, γ , σ ) := min(σ δ 2 , γ 2 /σ ).

(4.8)

Then, in the semiclassical limit, the coherent state ψx,σ,T2 evolves almost covariantly through the quantum baker’s map:

Bˆ N ψx,σ,T2 − eiπ(x) ψBx,σ/4,T2 HN = O(N 3/4 σ 1/4 e−πNθ ).

(4.9)

The implied constant is uniform with respect to δ, γ , σ ∈ [1/N, N ], and the point x ∈ D1,δ,γ . We notice that the exponential in the above remainder will be small only if θ >> 1/N , which requires both σ >> 1/N and σ 0, ∀x = (q0 , p0 ) ∈ R , p0 +1 i Sˆ1, x,σ = e 2 (q0 + 2 ) S1 x,σ/4 . The approximate covariance stated in Proposition 5 is therefore a microlocal version of this exact global covariance. Remark 2. The fact that the error is exponentially small is due to the piecewise-linear character of the map B. Indeed, for a nonlinear area-preserving map M on T2 , coherent states are also transformed covariantly through Mˆ N , but the error term is in general of order O(N x 3 ), where x is the “maximal width” of the coherent state (here x = max(σ, σ −1 ) N −1/2 ) [Schu1]. Moreover, in general the squeezing σ takes values in the complex half-plane {Re (σ ) > 0}: the reason why we can here restrict ourselves to the positive real line is due to the orientation of the baker’s dynamics. Proof of Proposition 5. Since we already know that FˆN acts covariantly on coherent states, we only need to analyse the action of Bˆ N,mix (Eq. (3.7)). We first consider a coherent state in the “left” strip (δ, 1/2 −δ)×(γ , 1−γ ) of D1,δ,γ . In this case, the “relevant” coefficients of Bˆ N,mix ψx,σ,T2 are in the interval 0 ≤ m < N2 :

N/2−1 j 1 ˆ ˆ =√ (FN/2 )mj x,σ,C . BN,mix ψx,σ,T2 m N N j =0

(4.10)

From the formula (3.6), we have for all 0 ≤ j, m < N/2: √ (FˆN/2 )mj = 2 (FˆN )2m j . Since q0 ∈ (δ, 1/2 − δ), for any N/2 ≤ j one has j/N − q0 ≥ δ; using Lemma 3, we obtain

j 2 ∀j ∈ {N/2, . . . , N − 1}, x,σ,C = O (σ N )1/4 e−πNσ δ . (4.11) N We can therefore extend the range of summation in (4.10) to j ∈ {0, . . . , N − 1}, incurring only an exponentially small error: √ N−1 2 Bˆ N,mix ψx,σ,T2 = 2 (FˆN )2m j ψx,σ,T2 + O((σ N )1/4 e−πNσ δ ) m

j

j =0

√ = 2 ψF x,1/σ,T2

2m

+ O((σ N )1/4 e−πNσ δ ).

In the last step, we have used the covariance property of Lemma 4.

2

(4.12)


335

Since p0 ∈ (γ , 1 − γ ) and N/σ ≥ 1, it follows from Lemma 3 and simple manipulations of plane coherent states that for all q ∈ [0, 1/2), √ √ 2 2 F x,1/σ,C (2q) = 2 F x,1/σ (2q) + O (N/σ )1/4 e−πNγ /σ 2 = (p0 /2,−2q0 ),4/σ (q) + O (N/σ )1/4 e−πNγ /σ 2 = (p0 /2,−2q0 ),4/σ,C (q) + O (N/σ )1/4 e−πNγ /σ . The identity (p0 /2, −2q0 ) = F Bx (valid for x in the left strip) inserted in (4.12) yields for all m ∈ {0, . . . , N/2 − 1}, Bˆ N,mix ψx,σ,T2 = ψFBx,4/σ,T2 m + O((σ N )1/4 e−πNθ ) m

(4.13)

(θ is defined in (4.8), and we used the assumption σ N > 1 to simplify the remainder). The remaining coefficients N/2 ≤ m < N are bounded using (4.11): 1 Bˆ N,mix ψx,σ,T2 =√ m N

N−1

(FˆN/2 )m j x,σ,C

j =N/2

j N

= O((σ N )1/4 e−πσ Nδ ). 2

(4.14) On the other hand, Lemma 3 shows that the coefficients ψFBx,4/σ,T2 m for N/2 ≤ m < N are bounded from above by the same RHS. Hence, Eq. (4.13) holds for all m = 0, . . . , N − 1.√ A norm estimate is obtained by multiplying this component-wise estimate by a factor N. We now apply the inverse Fourier transform and Lemma 4, to obtain the part of the theorem dealing with coherent states in the left strip of D1,δ,γ . A similar computation treats the case of coherent states in the right strip of D1,δ,γ . The large components of ψx,σ,T2 are in the interval j ≥ N/2, so the second block of Bˆ N,mix is relevant. The analogue to (4.13) reads, for m ∈ {N/2, . . . , N − 1}:

2 2m ˆ BN,mix ψx,σ,T2 = F x,1/σ − 1 + O((σ N )1/4 e−πNθ ). (4.15) m N N Proceeding as before, we identify for all q ∈ [1/2, 1), √ p0 +1 2 F x,1/σ (2q − 1) = eπiN(q0 + 2 ) ((p0 +1)/2,−(2q0 −1)),4/σ (q) = eπiN(q0 +

p0 +1 2 )

FBx,4/σ,C (q) + O((N/σ )1/4 e−πNγ

2 /σ

). (4.16)

Applying the inverse Fourier transform we obtain the second part of the theorem.

336


5. Egorov Property Our objective in this section is to control the evolution of quantum observables through Bˆ N , in terms of the corresponding classical evolution. Namely, we want to prove an Egorov theorem of the type N→∞

−n n

Bˆ N OpN (a) Bˆ N − OpN (a ◦ B −n ) −−−−→ 0.

(5.1)

Here OpN (a) is some quantisation of an observable a ∈ C ∞ (T2 ). As explained in the introduction, to avoid the diffraction problems due to the discontinuities of B, we will require the function a to be supported away from the set Sn of discontinuities of B n . Otherwise, a ◦ B −n may be discontinuous, and already its quantisation poses some problems. An Egorov theorem has been proven in [RubSal] for a different quantisation of the baker’s map, also using coherent states. In [DBDE, Cor. 17] an Egorov theorem was obtained for Bˆ N , but valid only for observables of the form a(q) (or a(p), depending on the direction of time) and restricting the observables to a “good” subspace of HN of dimension N − o(N ). Since we control the evolution of coherent states through Bˆ N (Proposition 5), it is natural to use a quantisation defined in terms of coherent states, namely the anti-Wick quantisation [Per] (see Definition 4 below). However, because the quasi-covariance (4.9) connects a squeezing σ to a squeezing σ/4, it will be necessary to relate the correspondAW,σ/4 ing quantisations OpAW,σ and OpN to one another. This will be done in the next N subsection, by using the Weyl quantisation as a reference. Besides, we want to control the correspondence (5.1) uniformly with respect to the time n. We already noticed that for n >> 1, an observable a supported away from Sn needs to fluctuate quite strongly along the q-direction, while its dependence in the p variable may remain mild. Likewise, a ◦ B −n , supported away from S−n , will strongly fluctuate along the p-direction. All results in this section will be stated for two classes of observables: – general functions f ∈ C ∞ (T2 ), without any indication on how f depends on N . This yields a Egorov theorem valid for time |n| ≤ ( 16 − ) TE (with > 0 fixed), which will suffice to prove Theorem 1 (TE = TE (N ) is the Ehrenfest time (1.2)). – functions f ∈ Sα (T2 ) for some α ∈ R2+ with |α| < 1 (see Definition 1). Here we use more sophisticated methods in order to push the Egorov theorem up to the times |n| ≤ (1 − )TE .

5.1. Weyl vs. anti-Wick quantizations on T2 . In this subsection, we define and compare the Weyl and anti-Wick quantisations on the torus. The main result is Proposition 8, which precisely estimates the discrepancies between these quantisations, in the classical limit. We start by recalling the definition of the Weyl quantisation on the torus [BDB, DEG]. Definition 3. Any function f ∈ C ∞ (T2 ) can be Fourier expanded as follows: f = f˜(k) ek , where ek (x) := e2πix∧k = e2πi(qk2 −pk1 ) . k∈Z2


337

The Weyl quantisation of this function is the following operator: OpW N (f ) :=

f˜(k) T (k),

where T (k) := Tˆhk .

(5.2)

k∈Z2

We use the same notations for translation operators T (k) acting on either HN or L2 (R); R2 (f ). in the latter case, the Weyl-quantised operator will be denoted by OpW, N The operators {T (k) ; k ∈ Z2 } acting on L2 (R) form an independent set of unitary operators. On the other hand, on HN these operators satisfy T (k + N m) = (−1)k∧m T (k). Hence, defining ZN := {−N/2, . . . , N/2 − 1}, the set {T (k), k ∈ Z2N } forms a basis of the space of operators on HN . This basis is orthonormal with respect to the Hilbert-Schmidt scalar product (3.8). The Weyl quantisations on L2 (R) and HN satisfy the following inequality [BDB, Lemma 3.9]: ∀f ∈ C ∞ (T2 ),

∀N ∈ N,

W,R

OpW (f ) B(L2 (R)) . (5.3) N (f ) B(HN ) ≤ OpN 2

This will allow us to use results pertaining to the Weyl quantisation of bounded functions on the plane (see the proof of Lemma 9). We now define a family of anti-Wick quantisations. Definition 4. For any squeezing σ > 0, the anti-Wick quantisation of a function f ∈ L1 (T2 ) is the operator OpAW,σ (f ) on HN defined as: N ∀φ, φ ∈ HN ,

φ, OpAW,σ (f ) φ := N N

T2

f (x) φ, ψx,σ,T2 ψx,σ,T2 , φ dx. (5.4)

Both Weyl and anti-Wick quantisations map a real observable onto a Hermitian operator. As opposed to the Weyl quantisation, the anti-Wick quantisation enjoys the important property of positivity. Namely, if the function a is nonnegative, then for any N, σ , the operator OpAW,σ (a) is positive. N These quantisations will be easy to compare once we have expressed the anti-Wick quantisation in terms of the Weyl one [BonDB]. Lemma 6. Using the quadratic form Qσ (k) := σ k12 + σ −1 k22 , one has the following expression for the anti-Wick quantisation: ∀f ∈ L1 (T2 ),

OpAW,σ (f ) = N

π

f˜(k) e− 2N Qσ (k) T (k).

(5.5)

k∈Z2 Equivalently, OpAW,σ (f ) = OpW N (f ), where the function f is obtained by convolution N of f (on R2 ) with the Gaussian kernel

KN,σ (x) := 2N e−2πNQσ (x) .

(5.6)

338


Proof. To prove this lemma, it is sufficient to show that for any k 0 ∈ Z2 , the anti-Wick quantisation on HN of the Fourier mode ek 0 (x) reads: π

(ek 0 ) = e− 2N Qσ (k 0 ) T (k 0 ). OpAW,σ N

(5.7)

This formula has been proven in [BonDB, Lemma 2.3 (ii)], yet we give here its proof for completeness. The idea is to decompose OpAW,σ (ek 0 ) in the basis {T (k), k ∈ Z2N }, N using the Hilbert-Schmidt scalar product (3.8). That is, we need to compute

T (k), OpAW,σ (e ) k0 = N

T2

ek 0 (x) ψx,σ,T2 , T (k)† ψx,σ,T2 dx.

(5.8)

The overlaps between torus coherent states derive from the overlaps between plane coherent states, which are simple Gaussian integrals: y,σ , x,σ R2 = ei

∀x, y ∈ R2 ,

y∧x 2

0,σ , Tˆx−y 0,σ R2 = ei

y∧x 2

e−

Qσ (x−y) 4

.

Using the projector (3.2), we get ψx,σ,T2 , Tˆk/N ψx,σ,T2 =

(−1)Nm1 m2 x,σ , Tˆk/N Tˆm x,σ R2

m∈Z2

=

(−1)Nm1 m2 +m∧k e2iπ(x∧(k+Nm)) e−

πN 2 Qσ (m+k/N)

.

m∈Z2

We insert this expression in the RHS of (5.8) (and remember that N is even):

πN T (k), OpAW,σ (e ) = δk 0 ,k+Nm (−1)m∧k e− 2 Qσ (m+k/N) . k 0 N m∈Z2

This expression vanishes unless k = k 1 , the unique element of Z2N such that k 1 = k 0 + N m1 for some m1 ∈ Z2 . The orthonormality of the basis {T (k) : k ∈ Z2N } gives that OpAW,σ (ek 0 ) = (−1)m1 ∧k 1 e− N

πN 2 Qσ (k 0 )

T (k 1 ) = e−

πN 2 Qσ (k 0 )

T (k 0 ).

A simple property of these quantisations is the semi-classical behaviour of the traces of quantized observables: Lemma 7. For any integer M ≥ 3, ∞

∀f ∈ C (T ), 2

f M 1 C W . Tr(OpN (f )) = f (x) dx + OM M 2 N N T

(5.9)

For the anti-Wick quantisation, we have: ∀f ∈ L (T ), 1

2

πN 1 AW,σ Tr(OpN (f )) = f (x) dx + O( f L1 e− 2 N T2

min(σ,1/σ )

).

(5.10)


339

Proof. The first identity uses the fact that on the space HN , 1 1 if k = N m for some m ∈ Z2 , Tr T (k) = 0 otherwise. N The error term in (5.9) is bounded above by coefficients of a smooth function satisfy ∀M ≥ 1,

Now, the Fourier

f C M . (1 + |k|)M

(5.11)

|f˜(k)| M

∀k ∈ Z2 ,

˜

m∈Z2 \{0} |f (N m)|.

Using this upper bound (with M ≥ 3) in the above sum yields (5.9). In the anti-Wick case, each term |f˜(N m)| ≤ f L1 of the sum is multiplied by πN πN 2 e− 2 Qσ (m) ≤ e− 2 min(σ,1/σ )|m| , which yields (5.10). We will now compare the Weyl and anti-Wick quantisations in the operator norm. We give two estimates, corresponding to the two classes of functions described in the introduction of this section. Proposition 8. I) For any f ∈ C ∞ (T2 ) and σ > 0, AW,σ (f ) f C 5

OpW N (f ) − OpN

max{σ, σ −1 } . N

(5.12)

Here σ may depend arbitrarily on N . II) Let α ∈ R2+ , |α| < 1 and assume that σ > 0 may depend on N such that the quantity α (N, σ ) := max

N 2α1 −1 σ

, N α1 +α2 −1 , σ N 2α2 −1

(5.13)

goes to zero as N → ∞. Then there exists a seminorm Nα on the space Sα (T2 ) such that, for any f = f (·, N ) ∈ Sα (T2 ), one has: ∀N ≥ 1,

(f (·, N )) − OpW

OpAW,σ N (f (·, N )) Nα (f ) α (N, σ ). (5.14) N

Remark 3. The effective “small parameter” α (N, σ ) will be small as N → ∞ only if three conditions are simultaneously satisfied: – |α| = α1 + α2 < 1, – N 2α1 0,

W,R2

Op

(f ) ≤ C

1 γ1 ,γ2 =0

∂ γ f C 0 (R2 ) βγ1 +(1−β)γ2 .

(2πN )−1

In the case = we apply this bound to a function f ∈ Sα (T2 ), selecting β = (β, 1 − β) such that β ≥ α: we then obtain the upper bound of (5.16) for R2 OpW, (f ). The inequality (5.3) shows that this bound applies as well to the Weyl N operator on HN . 2

We thank N. Anantharaman for pointing out to us this scaling argument.


341

Equipped with this lemma, we can now prove the second part of Proposition 8. From the Taylor expansion |f (x + y) − f (x) − (y · ∇)f (x)| ≤

1 max (y · ∇)2 f (z), z = x + ty 2 0≤t≤1

and Lemma 6, one easily checks that for any f ∈ C ∞ (T2 ),

f − f C 0 ≤

1 1 2

∂q f C 0 + 2 ∂q ∂p f C 0 + σ ∂p2 f C 0 . 8πN σ

Since differentiation commutes with convolution, one controls all derivatives: for all γ ∈ N20 ,

∂ γ (f − f ) C 0 ≤

1 1 γ +(2,0)

∂ f C 0 + 2 ∂ γ +(1,1) f C 0 + σ ∂ γ +(0,2) f C 0 . 8πN σ (5.19)

For f = f (·, N ) ∈ Sα (T2 ), this estimate implies:

∂ γ (f − f ) C 0 ≤ N α·γ

N 2α1 −1 σ

Cα,γ +(2,0) (f )

+N α1 +α2 −1 Cα,γ +(1,1) (f ) + σ N 2α2 −1 Cα,γ +(0,2) (f ) ≤ N α·γ α (N, σ ) Cα,γ +(2,0) (f ) +Cα,γ +(1,1) (f ) + Cα,γ +(0,2) (f ) . (5.20)

Here we used the parameter α (N, σ ) defined in (5.13). This shows that2 the function 1 f f ,rem (·, N ) := α (N,σ (·, N ) − f (·, N) is also an element of Sα (T ), with semi) norms dominated by seminorms of f . Applying Lemma 9 to that function and taking any β ≥ α, |β| = 1, we get

OpAW,σ (f (·, N)) − OpW N (f (·, N)) α (N, σ ) N

1

Cα,γ +γ (f ).

|γ |≤2 γ1 ,γ2 =0

The seminorm stated in the theorem can therefore be defined as Nα (f ) :=

1

|γ |≤2 γ1 ,γ2 =0

Cα,γ +γ (f ).

(5.21)

342


5.2. Egorov estimates for the baker’s map. We now turn to the proof of the Egorov property (5.1). Let us start with the case n = 1. We assume that a is supported in the set D1,δ,γ defined in Eq. (4.6), away from the discontinuity set S1 of B. Proposition 10. Let 0 < δ < 1/4 and 0 < γ < 1/2. Assume that the support of a ∈ C ∞ (T2 ) is contained in D1,δ,γ . Then, in the classical limit, −1 (a) Bˆ N − OpN

Bˆ N OpAW,σ N

AW,σ/4

(a ◦ B −1 ) a C 0 N 5/4 σ 1/4 e−πNθ ,

uniformly with respect to δ, γ , σ ∈ [1/N, N ]. Here we took as before θ = min(σ δ 2 , γ 2 /σ ). Proof. For any normalised state φ ∈ HN , we consider the matrix element ˆ −1 φ = N (a) B a(x) φ, Bˆ N ψx,σ,T2 Bˆ N ψx,σ,T2 , φ dx. φ, Bˆ N OpAW,σ N N T2

(5.22)

Using the quasi-covariance of coherent states localised in D1,δ,γ (Proposition 5) and applying the Cauchy-Schwarz inequality, the RHS reads a(x) φ, ψBx,σ/4,T2 ψBx,σ/4,T2 , φ dx + O( a C 0 N 5/4 σ 1/4 e−πNθ ). (5.23) N T2

The remainder is uniform with respect to the state φ. Through the variable substitution x = B −1 (y), this gives −1 (a)Bˆ N φ = φ, OpN φ, Bˆ N OpAW,σ N

AW,σ/4

(a ◦ B −1 )φ + O( a C 0 N 5/4 σ 1/4 e−πNθ ). (5.24)

Since the operators on both sides are self-adjoint, this identity implies the norm estimate of the proposition. Remark 4. Here we used the property that the linear local dynamics is the same at each point x ∈ T2 \S1 (expansion by a factor 2 along the horizontal, contraction by 1/2 along the vertical). Were this not the case, the state Bˆ N ψx,σ,T2 would be close to a coherent state at the point Bx, but with a squeezing depending on the point x. Integrating over x, we would get an anti-Wick quantisation of a ◦ B −1 with x-dependent squeezing, the analysis of which would be more complicated (see [Schu1, Chap. 4] for a discussion of such quantisations). We now generalise to n > 1. We assume that a is supported away from the set Sn of discontinuities of B n . More precisely, for some δ ∈ (0, 2−n−1 ) and γ ∈ (0, 1/2), we define the following open set, generalizing (4.6): k 2 Dn,δ,γ := (q, p) ∈ T , ∀k ∈ Z, q − n > δ, p ∈ (γ , 1 − γ ) . 2 The evolution of the sets Dn,δ,γ through B satisfies: ∀j ∈ {0, . . . , n − 1},

B j Dn,δ,γ ⊂ Dn−j,2j δ,γ /2j .

(5.25)

This is illustrated for n = 2, j = 1 in Fig. 5.1. If a is supported in Dn,δ,γ , then the support of a ◦ B −j is contained in Dn−j,2j δ,γ /2j ⊂ D1,2j δ,γ /2j . So for each 0 ≤ j < n, we can apply Proposition 10 to the observable a ◦ B −j , replacing the parameters δ, γ , σ by their corresponding values at time j ; we find that the parameter θ is independent of j . The triangle inequality then yields:


343

B

γ

γ 2

δ

2δ

Fig. 5.1. The action of the map B. On the left we show the set D2,δ,γ (shaded) and on the right is its image under the action of B

Corollary 11. Let n > 0 and for some δ ∈ (0, 2−n−1 ), γ ∈ (0, 1/2), let a ∈ C ∞ (T2 ) have support in Dn,δ,γ . Then, as N → ∞, AW,σ/4n

−n n

Bˆ N (a) Bˆ N − OpN OpAW,σ N

(a ◦ B −n ) a C 0 N 5/4 σ 1/4 e−πNθ .

(5.26)

This estimate is uniform with respect to n, the parameters δ, γ in the above ranges and n the squeezing σ ∈ [ 4N , N ]. Remark 5. The requirement N θ >> 1, together with the allowed ranges for δ, γ , impose n the restriction 4N 1, where TE is the Ehrenfest time (1.2). We can reach times n ∼ TE (1 − ) (with > 0 fixed) by taking the parameters δ = 2−n−2 N −1+ , γ 1, σ N 1− : in that case, the argument of the exponential in the RHS of Eq. (5.26) satisfies πN θ N , so that the RHS decays in the classical limit. We wish to obtain Egorov theorems where both terms correspond to a quantisation with the same parameter σ , or the Weyl quantisation. To do so, we will use Proposition 8 to replace the anti-Wick quantisations by the Weyl quantisation. Using the first statement of that proposition, we easily obtain the following Egorov theorem: Theorem 12. Let n > 0 and for some δ ∈ (0, 2−n−1 ), γ ∈ (0, 1/2), let a ∈ C ∞ (T2 ) have support in Dn,δ,γ . Then, in the limit N → ∞, and for any squeezing parameter n σ ∈ [ 4N , N], we have n W −n 5/4 1/4 −πNθ ˆ −n OpW σ e

Bˆ N N (a) BN − OpN (a ◦ B ) a C 0 N

n σ 4 1

a ◦ B −n C 5 . (5.27) max(σ, σ −1 ) a C 5 + max n , + 4 σ N

The implied constants are uniform in n, σ, δ, γ .

344


If n, δ, γ and the observable a supported on Dn,δ,γ are independent of N , the RHS semi-classically converges to zero if we simply take σ = 1. This is the “finite-time” Egorov theorem. On the other hand, if we let n grow with N , the function a needs to change with N as well (at least because its support needs to change). In the next subsection we construct a specific family of functions {an }n≥1 , each one supported away from Sn , and compute the estimate (5.27) for this family. Remark 6. The same estimate holds if we replace n by −n on the LHS of (5.27), and replace σ by σ −1 on the RHS, including the definition of θ . Now, the function a must be supported in the set D−n,δ,γ obtained from Dn,δ,γ by exchanging the roles of q and p. Indeed, using the unitarity of Bˆ N , we may interpret the estimate (4.9) as the quasicovariant evolution of the coherent state ψy,σ ,T2 (where y = Bx, σ = σ/4) into the state ψB −1 y,4σ ,T2 , and the rest of the proof identically follows. 5.3. Egorov estimates for truncated observables. 5.3.1. A family of admissible functions. For future purposes (see the proof of Theorem 1 in the next section), and in order to understand better the bound (5.27), we explicitly construct a sequence of functions {an }n≥0 , each function being supported away from Sn . This sequence is simply obtained by taking the products of a fixed observable a ∈ C ∞ (T2 ) with cutoff functions χδ,n , which we now describe. Definition 5. For some 0 < δ < 1/4, we consider a Z-periodic function χ˜ δ ∈ C ∞ (R) which vanishes for x ∈ [−δ, δ] mod Z and takes value 1 for x ∈ [2δ, 1 − 2δ] mod Z. For any n ≥ 0, we then define the following cutoff functions on T2 : χδ,n (x) := χ˜ δ (2n q) χ˜ δ (p), χδ,−n (x) := χ˜ δ (2n p) χ˜ δ (q). For any n ∈ Z, we split the observable a ∈ C ∞ (T2 ) into its “good part” an (x) := a(x) χδ,n (x) and its “bad part” anbad (x) = a(x) (1 − χδ,n (x)). One easily checks that an is supported on Dn,δ/2n ,δ , while anbad is supported on a neighbourhood of Sn of area O(δ). In light of Remark 6 we can, without loss of generality, consider only times n > 0. For any multiindex γ ∈ N20 , we have

∂ γ an C 0 γ a C |γ | 2nγ1 δ −|γ | .

(5.28)

When evolving an through the map B, the derivatives grow along p and decrease along q; after n iterations, an ◦ B −n is still smooth, and

∂ γ (an ◦ B −n ) C 0 γ a C |γ | 2nγ2 δ −|γ | .

(5.29)

These estimates show that the C 5 -norms of an and an ◦ B −n (appearing on the RHS of Eq. (5.27)) are both of order 25n /δ 5 . With our conventions, the parameter θ appearing


345

δ n in the RHS of (5.26) reads θ = max(σ,4 n /σ ) . We maximise it by selecting σ = 2 . With this choice, the upper bound (5.27) reads 2

n W −n 5/4 n/4 −πNδ 2 /2n ˆ −n

Bˆ N OpW 2 e N (an ) BN − OpN (an ◦ B ) a C 0 N

26n a C 5 . (5.30) N δ5 Using Remark 6, the same estimate holds if we replace n by −n on the LHS. The last term of the RHS in (5.30) can semiclassically vanish only if |n| < T6E . This time window, although not optimal (see the following subsection), will be sufficient to prove Theorem 1 in Sect. 6. Before that, in the last part of this section we will sharpen this estimate by using the second part of Proposition 8: this will allow us to prove a Egorov property up to times |n| ≤ (1 − )TE , for any > 0. +

5.3.2. Optimised Egorov estimates. In this subsection we prove the following “optimal” Egorov theorem. Theorem 13. Choose > 0 arbitrarily small, and consider any observable a ∈ C ∞ (T2 ). For any N ≥ 1 and n ∈ Z, construct the “good part” an of that observable using Definition 5 with a width δ(N ) ≥ min(N −/4 , 1/10). Then, the following Egorov estimate holds: there exists C > 0 (independent of a, ) and N() > 0 such that for any N ≥ N () and any time |n| ≤ (1 − )TE ,

a C 4 n W −n 3/2 −πN /2 ˆ −n OpW e + /2 . (5.31)

Bˆ N N (an ) BN − OpN (an ◦ B ) ≤ C a C 0 N N Proof. We only treat the case n ≥ 0, finally invoking the time-reversal symmetry as in Remark 6. We consider > 0 fixed, and define N () through the equation N ()−/4 = 1/10. We then take N ≥ N () and consider any positive time n ≤ (1 − )TE . The improvement over Theorem 12 will be a sharper bound for the norms OpAW,σ N AW,σ/4n

−n (an ) − OpW (an ◦ B −n ) − OpW N (an ) and OpN N (an ◦ B ) . Using the rescaled n −/4 time t = TE and the property δ(N) ≥ N , the bound (5.28) on derivatives of an reads:

∂ γ an C 0 γ a C |γ | 2nγ1 N 4 |γ | = a C |γ | N tγ1 N 4 |γ | . Thus, the derivatives of an scale as those of an N -dependent function in the space Sα t (T2 ), where α t := (t + /4, /4). As in the former subsection, we must take σ = 2n = N t to minimise the remainder. The second part of Proposition 8 applied to a function in Sα t (T2 ) yields a “small parameter” α t (N, 2n ) = N t+/2−1 , so that the difference between the two quantisations of an is bounded as n

AW,2

OpW (an ) a C 4 N t+/2−1 . N (an ) − OpN −n

AW,2 −n Similar considerations using (5.29) show that OpW (a ◦ B −n ) N (a ◦ B ) − OpN is bounded by the same quantity. The argument of the exponential in Eq. (5.26) takes the value Nθ = Nδ 2 /2n ≥ N 1−t−/2 , so that the full estimate reads: n W −n 3/2 −πN ˆ −n

Bˆ N OpW e N (an ) BN − OpN (an ◦ B ) a C 0 N

1−t−/2

+

a C 4 . N 1−t−/2

346


We obtain the bound (5.31) uniform in n by noticing that for the time window we consider, N 1−t−/2 ≥ N /2 . Our reason for believing that this estimate is “optimal” lies in Remark 5: we evolve states which stay away from the discontinuity set S1 along their evolution. Since any state satisfies qp 21 due to Heisenberg’s uncertainty principle, and q doubles at each time step, it is impossible for such a state to remain away from S1 during a time window larger than TE . Besides, at the time TE the “good part” an oscillates on a scale ≈ in the q direction, so it behaves more like a Fourier integral operator than an observable (pseudo-differential operator). 6. Quantum Ergodicity For any even N , we denote by {ϕN,j } the eigenvectors of Bˆ N (if some eigenvalues happen to be degenerate, which seems to be ruled out by numerical simulations, take an arbitrary ∞ 2 orthonormal eigenbasis). Let us consider a fixed real-valued observable a ∈ C (T ) satisfying T2 a(x) dx = 0. Quantum ergodicity follows if we prove that the quantum variance S2 (a, N ) =

N 1 2 N→∞ |ϕN,j , OpW −−−→ 0. N (a) ϕN,j | − N

(6.1)

j =1

One method to prove this limit for our quantised baker’s map would be to apply the methods of [MO’K]: one only needs the Egorov property (Theorem 12) for finite times n, and the classical ergodicity of B. However, this method seems unable to give information about the rate of decay of the variance. In order to prove the upper bound stated in Theorem 1, we will rather adapt the method used in [Zel2, Schu2] to our discontinuous map. This method requires the correlation functions of the classical map to decay sufficiently fast, which is the case here (Eq. 2.5). Proof of Theorem 1. To begin with, we consider the function

1 − cos x g(x) := 2 x2 and its Fourier transform ∞ 2π(1 − |k|), for −1 ≤ k ≤ 1, −2πikx g(k) ˆ = g(x) e dx = 0, elsewhere. −∞ For any T ≥ 1, we use it to construct the following periodic function: fT (θ ) := g(T (θ + m)). m∈Z

fT admits the Fourier decomposition fT (θ ) = k∈Z fˆT (k) e2πikθ , where |k| 2π 1 − for −T ≤ k ≤ T , T T fˆT (k) = 0 for |k| > T . Using this function, one may easily prove the following lemma [Schu2].

(6.2)


347

Lemma 14. With notations described above, for any even N ≥ 2 and T ≥ 1 one has 1 W −n n ˆ S2 (a, N ) ≤ OpW (a) B fˆT (n) Tr OpN (a) Bˆ N N N . N n∈Z

Notice that the terms in the sum on the RHS vanish for |n| > T . Proof. Let {ϕj } be the eigenbasis of Bˆ N , with Bˆ N ϕj = e2πiθj ϕj . Then one has N−1 −n n W 2 ˆ ˆ = Tr OpW (a) B Op (a) B e2πin(θk −θj ) |OpW N N N (a)ϕj , ϕk | . N N j,k=0

Multiplying by fˆT (n) and summing over n, we get, −n n W ˆ ˆ fˆT (n) Tr OpW (a) B Op (a) B N N N N n∈Z

=

N−1

2 fT (θk − θj ) |OpW N (a)ϕj , ϕk |

j,k=0

=

N−1

2 fT (0) |OpW N (a)ϕj , ϕj | +

j =0

2 fT (θk − θj ) |OpW N (a)ϕj , ϕk |

j =k

≥ N S2 (a, N ). The final inequality follows from the positivity of fT and the property fT (0) ≥ 1.

To prove the theorem we will estimate the traces appearing in Lemma 14. Due to the support properties of fˆT , only the terms with n ∈ [−T , T ] will be needed. We take the time T depending on N , precisely as T = T (N) :=

TE , 11

where TE is the Ehrenfest time (1.2). For each n ∈ Z ∩ [−T , T ], we will apply the Egorov Theorem 12. We first decompose a into a “good” part an and “bad” part anbad , as described in Definition 5: a = an + anbad ,

an := a.χδ,n .

(6.3)

We let the width δ > 0 depend on N as δ (log N )−1 . Therefore, for any n ∈ [−T , T ] |n| we will have 2δ N 1/10 . As a result, the bounds (5.28) for the derivatives of an read: ∀n ∈ Z ∩ [−T , T ],

|γ |

∂ γ an C 0 γ a C |γ | N 10 .

(6.4)

Furthermore, the same bounds are satisfied by the derivatives of anbad and an ◦ B −n . We decompose the traces of Lemma 14 according to the splitting (6.3): W W ˆn ˆ −n = Tr OpW ˆn ˆ −n Tr OpW N (a) BN OpN (a) BN N (a) BN OpN (an ) BN n W bad ˆ −n ˆ +Tr OpW (6.5) (a) B Op (a ) B N N n N N .

348


bad The second term in the RHS will be controlled by replacing OpW N (an ) by its anti-Wick quantisation: AW,1 bad ˆ −n W bad ˆ −n ˆn ˆn Tr OpW (an ) BN + RN (n) . = Tr OpW N (a) BN OpN (an ) BN N (a) BN OpN

(6.6) The remainder RN (n) is dealt with using part I of Proposition 8, together with the bounds (6.4) applied to anbad : AW,1 bad W bad (an )

RN (n) ≤ OpW N (a) OpN (an ) − OpN

anbad C 5

a C 5 . (6.7) OpW N (a) N N 1/2 AW,1 bad ˆ −n ˆn In order to compute Tr OpW (an ) BN , we split the function anbad into N (a) BN OpN OpW N (a)

bad − a bad , where a bad ≥ 0. We then use the its positive and negative parts, anbad = an,+ n,− n,± following (standard) linear algebra lemma to estimate the trace:

Lemma 15. Let A, B be self-adjoint operators on HN , and assume B is positive. Then |Tr(AB)| ≤ A Tr(B).

(6.8)

bad ) is positive, this lemma yields: Since the anti-Wick operator OpAW,1 (an,+ N AW,1 bad AW,1 bad ˆ −n ˆn (an,+ ) BN ≤ OpW (an,+ ) , Tr OpW N (a) BN OpN N (a) Tr OpN bad by a bad . By linearity and a bad + a bad = |a bad |, we get and similarly by replacing an,+ n,− n,+ n,− n AW,1 bad ˆ −n AW,1 ˆn (an ) BN ≤ OpW (|anbad |) . Tr OpW N (a) BN OpN N (a) Tr OpN From Eq. (5.10), the trace on the RHS is equal to N · anbad L1 (T2 ) 1 + O(e−πN/2 ) . Since anbad is supported on a neighbourhood of Sn of area O(δ), its L1 norm is of order O(δ a C 0 ). Using the Calderón-Vaillancourt estimate OpW N (a) ≤ C a C 2 , we have thus proven the following bound for the second term in (6.5):

1 W

a C 5 n bad ˆ −n δ

a . (6.9)

a Tr OpN (a) Bˆ N OpW (a ) B + 2 0 C C N n N N N 1/2

We now estimate the first term in (6.5). We write W −n ˆn W ˆ −n = Tr OpW Tr OpW N (a) BN OpN (an )BN N (a)OpN (an ◦ B ) + RN (n) ,

(6.10)

and control the remainder RN (n) with the Egorov estimate (5.30), remembering that n ≤ TE /11: 26n a C 5 5/4 n/4 −πNδ 2 /2n (a)

a N 2 e +

RN (n) OpW 0 C N N δ5 2

a C 5 . (6.11) N 2/5 The following lemma (proved in [MO’K, Lemma 3.1]) will allow us to replace the quantum product by a classical one.


349

Lemma 16. There exists C > 0 such that, for any pair a, b ∈ C ∞ (T2 ), ∀N ≥ 1,

W W

OpW N (a) OpN (b) − OpN (ab) ≤ C

a C 4 b C 4 . N

(6.12)

Using this lemma and the bounds (6.4), we get W −n W −n Tr OpW a(a (a) Op (a ◦ B ) = Tr Op ◦ B ) + R (n) , n n N N N N

a C 4

an C 4 an ◦ B −n C 4 . (6.13) 1/5 N N −n To finally estimate the trace of OpW N a(an ◦ B ) , we use Eq. (5.9) together with the estimates (6.4): a 2 3 1 W C −n a(an ◦ B −n )(x) dx + O . Tr OpN a(an ◦ B ) = N N2 T2 2

RN (n)

with

It remains to compute the integral on the RHS. We split it in two integrals, according to an = a − anbad . The second integral can be bounded by bad −n a(x) an (B x) dx ≤ a C 0 anbad L1 a 2C 0 δ, (6.14) T2

while the first one reads

T2

a(x) a(B −n x) dx = Ka a (n).

(6.15)

This integral is the classical autocorrelation function for the observable a(x), a purely classical quantity. At this point we must use the dynamical properties of the classical baker’s map B, namely its fast mixing properties (see the end of Sect. 2): for some > 0, the autocorrelation decays (when n → ∞) as Ka a (n) a 2C 1 e−|n| . Collecting all terms and using the properties of the function fˆT , Lemma 14 finally yields the following upper bound: 1 S2 (a, N ) a 2C 5 |fˆT (n)| e−|n| + δ + 1/5 N n∈[−T ,T ] 1 a 2C 5 +δ . T Since we took T log N and δ (log N )−1 , this concludes the proof of Theorem 1. Proof of Corollary 2. We start by picking an observable a ∈ C ∞ (T2 ), assuming a(x) N→∞

dx = 0. For any decreasing sequence α(N) −−−−→ 0, Chebychev’s inequality yields an upper bound on the number of eigenvectors of Bˆ N for which |ϕN,j , OpW N (a)ϕN,j | > α(N): # j ∈ {1, . . . , N} : |ϕN,j , OpW S2 (a, N ) N (a) ϕN,j | > α(N ) ≤ . (6.16) N α(N )2

350


From Theorem 1, if we take α(N ) >> (log N )−1/2 , the above fraction converges to zero. Defining JN (a) as the complement of the set in the above numerator, we obtain a sequence of subsets JN (a) ⊂ {1, . . . , N} satisfying #JNN(a) → 1, such that the eigenstates ϕN,jN with jN ∈ JN (a) satisfy (1.3). Using a standard diagonal argument [CdV, HMR, Zel1], one can then extract subsets JN ⊂ {1, . . . , N} independent of the observable a ∈ C ∞ (T2 ), with #JNN → 1, such that (1.3) is satisfied for any a ∈ C ∞ (T2 ) if one takes jN ∈ JN . Acknowledgement. We are grateful to R. Schubert for communicating to us his results [Schu2] prior to publication, and for interesting comments. We also thank S. De Bièvre, M. Saraceno, N. Anantharaman, A. Martinez and S. Graffi for interesting discussions and comments. This work has been partially supported by the European Commission under the Research Training Network (Mathematical Aspects of Quantum Chaos) HPRN-CT-2000-00103 of the IHP Programme.

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Faure, F., Nonnenmacher, S., De Bièvre, S.: Scarred eigenstates for quantum cat maps of minimal periods. Commun. Math. Phys. 239, 449–492 (2003) [Fo] Folland, G.B.: Harmonic analysis in phase space, The Annals of Mathematics Studies 122, Princeton, NJ: Princeton University Press, 1989 ´ Ergodic properties of eigenfunctions for the Dirichlet problem. [GL] Gérard, P., Leichtnam, E.: Duke Math. J. 71, 559–607 (1993) [HB] Hannay, J.H., Berry, M.V.: Quantisation of linear maps on the torus—Fresnel diffraction by a periodic grating. Physica D 1, 267–290 (1980) [Has] Hasegawa, H.H., Saphir, W.C.: Unitarity and irreversibility in chaotic systems. Phys. Rev. A 46, 7401–7423 (1992) [HMR] Helffer, B., Martinez, A., Robert, D.: Ergodicité et limite semi-classique. Commun. Math. Phys. 109, 313–326 (1987) [Kap] Kaplan, L., Heller, E.J.: Linear and nonlinear theory of eigenfunction scars. Ann. Phys. (NY) 264, 171–206 (1998) [KM] Keating, J.P., Mezzadri, F.: Pseudo-symmetries of Anosov maps and spectral statistics. Nonlinearity 13, 747–775 (2000) [KR1] Kurlberg, P., Rudnick, Z.: Hecke theory and equidistribution for the quantization of linear maps of the torus. Duke Math. J. 103, 47–77 (2001) [KR2] Kurlberg, P., Rudnick, Z.: On quantum ergodicity for linear maps of the torus. Commun. Math. Phys. 222, 201–227 (2001) [KR3] Kurlberg, P., Rudnick, Z.: On the distribution of matrix elements for the quantum cat map. Ann. Math. 161, 489–507 (2005) [Lak] Lakshminarayan, A.: On the quantum baker’s map and its unusual traces. Ann. Phys. (NY) 239, 272–295 (1995) [LV] Lebœuf, P., Voros, A.: Chaos revealing multiplicative representation of quantum eigenstates. J. Phys. A 23, 1765–1774 (1990) [Lin] Lindenstrauss, E.: Invariant measures and arithmetic quantum unique ergodicity. Ann. Math. 163, 165–219 (2006) [LS] Luo, W., Sarnak, P.: Quantum variance for Hecke eigenforms Ann. Sci. Ecole Norm. Sup. 37, 769–799 (2004) [MO’K] Marklof, J., O’Keefe, S.: Weyl’s law and quantum ergodicity for maps with divided phase space; Appendix by Zelditch, S.: Converse quantum ergodicity. Nonlinearity 18, 277–304 (2005) [MR] Marklof, J., Rudnick, Z.: Quantum unique ergodicity for parabolic maps. Geom. Func. Anal. 10, 1554–1578 (2000) [Ma] Martinez, A.: An introduction to semiclassical and microlocal analysis. Berlin-HeidelbergNew York: Springer-Verlag, 2002 [O’CTH] O’Connor, P.W., Tomsovic, S., Heller, E.J.: Accuracy of semiclassical dynamics in the presence of chaos J. Stat. Phys. 68, 131–152 (1992) [Per] Perelomov, A.M.: Generalized coherent states and their applications. Heidelberg: Springer Verlag, 1986 [Rob] Robert, D.: Remarks on time dependent Schrödinger equation, bound states and coherent states. In: Multiscale methods in quantum mechanics, Trends Maths, Boston: Birkhäuser 2004, pp. 139–158 [Ros] Rosenzweig, L.: Quantum unique ergodicity for maps on T2 . M.Sc. Thesis, Tel Aviv University, 2004 [RubSal] Rubin, R., Salwen, N.: A canonical quantization of the Baker’s Map. Ann. Phys. (NY) 269, 159–181 (1998) [RudSar] Rudnick, Z., Sarnak, P.: The behaviour of eigenstates of arithmetic hyperbolic manifolds. Commun. Math. Phys. 161, 195–213 (1994) [RuSo] Rudnick, Z., Soundararajan, K.: In preparation, 2004 [Sa] Saraceno, M.: Classical structures in the quantized baker transformation. Ann. Phys. (NY) 199, 37–60 (1990) [SaVo] Saraceno, M., Voros, A.: Towards a semiclassical theory of the quantum baker’s map. Physica D 79, 206–268 (1994) [Sar1] Sarnak, P.: Spectra of hyperbolic surfaces. Bull. Amer. Math. Soc. 40, 441–478 (2003) [Sar2] Sarnak, P.: Quantum vesus classical fluctuations on the modular surface. Talk given at the meeting: “Random Matrix Theory and Arithmetic Aspects of Quantum Chaos” at the Isaac Newton Institute, Cambridge, June 2004. Audio file available at http://www.newton.cam.ac.uk/webseminars/ [Schu1] Schubert, R.: Semiclassical localization in phase space. Ph.D. Thesis, Universität Ulm, 2001. Available at http://vts.uni-ulm.de

352 [Schu2] ˇ [Sni] [Wil] [Zel1] [Zel2] [ZZ]

M. Degli Esposti, S. Nonnenmacher, B. Winn Schubert, R.: Upper bounds on the rate of quantum ergodicity. Preprint 2005, http://arXiv.org/list/math-ph/0503045, 2005 ˇ Snirel’man, A.I.: Ergodic properties of eigenfunctions. Usp. Mat. Nauk. 29, 181–182 (1974) Wilkinson, M.: A semiclassical sum rule for matrix elements of classically chaotic systems. J. Phys. A 9, 2415–2423 (1987) Zelditch, S.: Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55, 919–941 (1987) Zelditch, S.: On the rate of quantum ergodicity. I. Upper bounds. Commun. Math. Phys. 160, 81–92 (1994) Zelditch, S., Zworski, M.: Ergodicity of eigenfunctions for ergodic billiards. Commun. Math. Phys. 175, 673–682 (1996)

Communicated by P. Sarnak


Communications in


A Canonical Ensemble Approach to the Fermion/Boson Random Point Processes and Its Applications H. Tamura1 , K.R. Ito2 1 2

Department of Mathematics, Kanazawa University, Kanazawa 920-1192, Japan. E-mail: [email protected] Department of Mathematics and Physics, Setsunan University, Neyagawa, Osaka 572-8508, Japan. E-mail: [email protected]; [email protected]

Received: 1 February 2005 / Accepted: 28 July 2005 Published online: 3 Febraury 2006 – © Springer-Verlag 2006

Abstract: We introduce the boson and the fermion point processes from the elementary quantum mechanical point of view. That is, we consider quantum statistical mechanics of the canonical ensemble for a fixed number of particles which obey Bose-Einstein, Fermi-Dirac statistics, respectively, in a finite volume. Focusing on the distribution of positions of the particles, we have point processes of the fixed number of points in a bounded domain. By taking the thermodynamic limit such that the particle density converges to a finite value, the boson/fermion processes are obtained. This argument is a realization of the equivalence of ensembles, since resulting processes are considered to describe a grand canonical ensemble of points. Random point processes corresponding to para-particles of order two are discussed as an application of the formulation. Statistics of a system of composite particles at zero temperature are also considered as a model of determinantal random point processes.

1. Introduction As special classes of random point processes, fermion point processes and boson point processes have been studied by many authors since [1, 16, 17]. Among them, [8, 6, 9, 7] made a correspondence between boson processes and locally normal states on the C ∗ -algebra of operators on the boson Fock space. A functional integral method is used in [15] to obtain these processes from quantum field theories of finite temperatures. On the other hand, [23] formulated both the fermion and boson processes in a unified way in terms of the Laplace transformation and generalized them. Let Q(R) be the space of all the locally finite configurations over a Polish space R and K a locally trace class integral operator on L2 (R) with a Radon function f measure λ on R . For any nonnegative having bounded support and ξ = j δxj ∈ Q(R), we set < ξ, f >= j f (xj ). Shirai and Takahashi [23] have formulated and studied the random processes µα,K which have

354

H. Tamura, K.R. Ito

Laplace transformations E[e

−

µα,K (dξ ) e−

]≡ Q(R)

−1/α = Det I + α 1 − e−f K 1 − e−f

(1.1)

for the parameters α ∈ {2/m; m ∈ N} ∪ {−1/m; m ∈ N }. Here the cases α = ±1 correspond to boson/fermion processes, respectively. In their argument, the generalized Vere-Jones’ formula [29] 1 Det(1 − αJ )−1/α = det α (J (xi , xj ))ni,j =1 λ⊗n (dx1 · · · dxn ) (1.2) n! R n has played an essential role. Here J is a trace class integral operator, for which we need the condition ||αJ || < 1 unless −1/α ∈ N, Det( · ) the Fredholm determinant and detα A the α-determinant defined by α n−ν(σ ) Aiσ (i) (1.3) det α A = σ ∈Sn

i

for a matrix A of size n × n, where ν(σ ) is the number of cycles in σ . The formula (1.2) is Fredholm’s original definition of his functional determinant in the case α = −1. The purpose of the paper is to construct both the fermion and boson processes from a viewpoint of elementary quantum mechanics in order to get a simple, clear and straightforward understanding of them in connection with physics. Let us consider the system of N free fermions/bosons in a box of finite volume V in Rd and the quantum statistical mechanical state of the system with a finite temperature. Giving the distribution function of the positions of all particles in terms of the square of the absolute value of the wave functions, we obtain a point process of N points in the box. As the thermodynamic limit, N, V → ∞ and N/V → ρ, of these processes of finite points, fermion and boson processes in Rd with density ρ are obtained. In the argument, we will use the generalized Vere-Jones’ formula in the form: 1 ⊗N (dx1 · · · dxN ) det α (J (xi , xj ))N i,j =1 λ N! dz = Det(1 − zαJ )−1/α , (1.4) N+1 2πiz Sr (0) where r > 0 is arbitrary for −1/α ∈ N, otherwise r should satisfy ||rαJ || < 1. Here and hereafter, Sr (ζ ) denotes the integration contour defined by the map θ → ζ + r exp(iθ ), where θ ranges from −π to π , r > 0 and ζ ∈ C. In the terminology of statistical mechanics, we start from the canonical ensemble and end up with formulae like (1.1) and (1.2) of a grand canonical nature. In this sense, the argument is related to the equivalence of ensembles. The use of (1.4) makes our approach simple. The thermodynamic limit has been discussed in [11] and [18] in the contexts of local current algebras for boson and fermion gases respectively at zero temperature. In our approach, we need neither quantum field theories nor the theory of states on the operator algebras to derive the boson/fermion processes. It is interesting to apply the method to the problems which have not been formulated in statistical mechanics on quantum field theories yet. Here, we study the system of para-fermions and para-bosons

Canonical Ensemble Approach to the Fermion/Boson Random Point Processes

355

of order 2. Para statistics was first introduced by Green [10] in the context of quantum field theories. For its review, see [21]. References [19] and [12, 27] formulated it within the framework of quantum mechanics of a finite number of particles. See also [20]. Recently statistical mechanics of para-particles are formulated in [28, 2, 3]. However, it does not seem to be fully developed so far. We formulate here point processes as the distributions of positions of para-particles of order 2 with finite temperature and positive density through the thermodynamic limit. It turns out that the resulting processes correspond to the cases α = ±1/2 in [23]. We also try to derive point processes from ensembles of composite particles at zero temperature and positive density in this formalism. The resulting processes also have their Laplace transforms expressed by Fredholm determinants. This paper is organized as follows. In Sect. 2, the random point processes of fixed numbers of fermions as well as bosons are formulated on the base of quantum mechanics in a bounded box. Then, the theorems on thermodynamic limits are stated. The proofs of the theorems are presented in Sect. 3 as applications of a theorem of rather abstract form. In Sects. 4 and 5, we consider the systems of para-particles and composite particles, respectively. In the Appendix, we calculate complex integrals needed for the thermodynamic limits. 2. Fermion and Boson Processes Consider L2 (L ) on L = [−L/2, L/2]d ⊂ Rd with the Lebesgue measure on L . Let

L be the Laplacian on HL = L2 (L ) satisfying periodic boundary conditions at ∂L . We deal with periodic boundary conditions in this paper, however, all the arguments except that in Sect. 5 may be applied to other boundary conditions. Hereafter we regard − L as the quantum mechanical Hamiltonian of a single free particle. The usual factor (L) 2 /2m is set at unity. For k ∈ Zd , ϕk (x) = L−d/2 exp(i2π k · x/L) is an eigenfunction (L) of L , and {ϕk }k∈Zd forms an complete orthonormal system [CONS] of HL . In the following, we use the operator GL = exp(β L ) whose kernel is given by GL (x, y) =

e−β|2πk/L| ϕk (x)ϕk (y) 2

(L)

(L)

(2.1)

k∈Zd (L)

for β > 0. We put gk = exp(−β|2πk/L|2 ), the eigenvalue of GL for the eigenfunction (L) ϕk . We also need G = exp(β ) on L2 (Rd ) and its kernel G(x, y) =

dp −β|p|2 +ip·(x−y) exp(−|x − y|2 /4β) e = . d (4πβ)d/2 Rd (2π)

Note that GL (x, y) and G(x, y) are real symmetric and GL (x, y) = G(x, y + kL).

(2.2)

k∈Zd

Let f : Rd → [0, ∞) be an arbitrary continuous function whose support is compact. In the course of the thermodynamic limit, f is fixed and we assume that L is so large that L contains the support, and regard f as a function on L .

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H. Tamura, K.R. Ito

2.1. Fermion processes. In this subsection, we construct the fermion process in Rd as a limit of the process of N points in L . Suppose there are N identical particles which obey the Fermi-Dirac statistics in a finite box L . The space of the quantum mechanical states of the system is given by F = {AN f | f ∈ ⊗N HL }, HL,N

where AN f (x1 , . . . , xN ) =

1 sgn(σ )f (xσ (1) , . . . , xσ (N) ) N!

( x1 , . . . , xN ∈ L )

σ ∈SN

(L)

is anti-symmetrization in the N indices. Using the CONS {ϕk }k∈Zd of HL = L2 (L ), we make the element 1 sgn(σ )ϕk1 (xσ (1) ) · · · · · ϕkN (xσ (N) ) (2.3) k (x1 , . . . , xN ) = √ N ! σ ∈S N

F of HL,N for k = (k1 , . . . , kN ) ∈ (Zd )N . Let us introduce the lexicographic order ≺ in Zd and put (Zd )N = {(k1 , . . . , kN ) ∈ (Zd )N | k1 · · · kN }. Then {k }k∈(Zd )N

F . forms a CONS of HL,N According to the idea of the canonical ensemble in quantum statistical mechanics, the probability density distribution of the positions of the N free fermions in the periodic box L at the inverse temperature β is given by N

F pL,N (x1 , . . . , xN ) = ZF−1

k∈(Zd )N

= ZF−1

j =1

(L)

gkj

|k (x1 , . . . , xN )|2

k (x1 , . . . , xN )

k∈(Zd )N

× (⊗N GL )k (x1 , . . . , xN ),

(2.4)

where ZF is the normalization constant. We can define the point process of N points N in L from the density (2.4). I.e., consider a map N j =1 δxj ∈ L (x1 , . . . , xN ) → Q(Rd ). Let µFL,N be the probability measure on Q(Rd ) induced by the map from the F probability measure on N L which has the density (2.4). By EL,N , we denote expectation with respect to the measure µFL,N . The Laplace transform of the point process is given by − F EL,N dµFL,N (ξ ) e− e = =

=

Q(Rd )

N L

exp(−

N

F f (xj ))pL,N (x1 , . . . , xN ) dx1 . . . dxN

j =1

Tr HF [(⊗N e−f )(⊗N GL )] L,N

Tr HF [⊗N GL ] L,N


˜ L )AN ] Tr ⊗N HL [(⊗N G Tr ⊗N HL [(⊗N GL )AN ]

˜ N det −1 GL (xi , xj ) dx1 . . . dxN = L , N det −1 GL (xi , xj ) dx1 · · · dxN

=

357

(2.5)

L

˜ L is defined by where G ˜ L = G1/2 e−f G1/2 , G L L

(2.6)

where e−f represents the operator of multiplication by the function e−f . 1/2 The fifth expression follows from [⊗N GL , AN ] = 0, cyclicity of the trace and 1/2 1/2 ˜ L and so on. The last expression can be obtained (⊗N GL )(⊗N e−f )(⊗N GL ) = ⊗N G N by calculating the trace on ⊗ HL using its CONS {ϕk1 ⊗ · · · ⊗ ϕkN | k1 , . . . , kN ∈ Zd }, where det−1 is the usual determinant, see Eq. (1.3). Now, let us consider the thermodynamic limit, where the volume of the box L and the number of points N in the box L tend to infinity in such a way that the densities tend to a positive finite value ρ, i.e., L, N → ∞,

N/Ld → ρ > 0.

(2.7)

Theorem 2.1. The finite fermion processes {µFL,N } defined above converge weakly to the fermion process µFρ whose Laplace transform is given by Q(Rd )

e− dµFρ (ξ ) = Det 1 − 1 − e−f z∗ G(1 + z∗ G)−1 1 − e−f (2.8)

in the thermodynamic limit (2.7), where z∗ is the positive number uniquely determined by ρ=

dp z∗ e−β|p| = (z∗ G(1 + z∗ G)−1 )(x, x). d (2π) 1 + z∗ e−β|p|2 2

Remark. The existence of µFρ which has the above Laplace transform is a consequence of the result of [23] we have mentioned in the introduction.

2.2. Boson processes. Suppose there are N identical particles which obey Bose-Einstein statistics in a finite box L . The space of the quantum mechanical states of the system is given by B HL,N = {SN f | f ∈ ⊗N HL },

where SN f (x1 , . . . , xN ) =

1 f (xσ (1) , . . . xσ (N) ) N! σ ∈SN

( x1 , . . . , xN ∈ L )

358

H. Tamura, K.R. Ito (L)

is symmetrization in the N indices. Using the CONS {ϕk }k∈Zd of L2 (L ), we make the element 1 ϕk1 (xσ (1) ) · · · · · ϕkN (xσ (N) ) (2.9) k (x1 , . . . , xN ) = √ N!n(k) σ ∈SN

B for k = (k , . . . , k ) ∈ Zd , where n(k) = of HL,N 1 N l∈Zd ({n ∈ {1, . . . , N} | kn = l}!). d N Let us introduce the subset (Z )≺ = {(k1 , . . . , kN ) ∈ (Zd )N | k1 ≺ · · · ≺ kN } of (Zd )N . B . Then {k }k∈(Zd )N≺ forms a CONS of HL,N As in the fermion’s case, the probability density distribution of the positions of the N free bosons in the periodic box L at the inverse temperature β is given by B (x1 , . . . , xN ) = ZB−1 pL,N

N k∈(Zd )N ≺

j =1

(L)

gkj

|k (x1 , . . . , xN )|2 ,

(2.10)

where ZB is the normalization constant. We can define a point process of N points µB L,N from the density (2.10) as in the previous section. The Laplace transform of the point process is given by ˜ L )SN ] − Tr ⊗N HL [(⊗N G B EL,N e = N Tr ⊗N HL [(⊗ GL )SN ]

˜ N det 1 GL (xi , xj ) dx1 · · · dxN = L , N det 1 GL (xi , xj ) dx1 · · · dxN

(2.11)

L

where det1 denotes permanent, see Eq. (1.3). We set 2 dp e−β|p| ρc = , 2 d Rd (2π) 1 − e−β|p| which is finite for d > 2. Now, we have

(2.12)

Theorem 2.2. The finite boson processes {µB L,N } defined above converge weakly to the boson process µB whose Laplace transform is given by ρ e− dµB ρ (ξ ) Q(Rd )

= Det[1 +

1 − e−f z∗ G(1 − z∗ G)−1 1 − e−f ]−1

(2.13)

in the thermodynamic limit (2.7) if 2 dp z∗ e−β|p| ρ= = (z∗ G(1 − z∗ G)−1 )(x, x) < ρc . 2 d Rd (2π) 1 − z∗ e−β|p| Remark 1. For the existence of µB ρ , we refer to [23]. Remark 2. In this paper, we only consider the boson processes with low densities : ρ < ρc . The high density cases ρ ρc are related to the Bose-Einstein condensation. ˜ L to deal with these cases. It We need the detailed knowledge about the spectrum of G will be reported in another publication.


359

3. Thermodynamic Limits 3.1. A general framework. It is convenient to consider the problem in a general framework on a Hilbert space H over C. The proofs of the theorems of Sect. 2 are given in the next subsection. We denote the operator norm by || · ||, the trace norm by || · ||1 and the Hilbert-Schmidt norm by || · ||2 . Let {VL }L>0 be a one-parameter family of Hilbert-Schmidt operators on H which satisfy the conditions ∀L > 0 : || VL || = 1,

lim || VL ||2 = ∞,

L→∞

and A a bounded self-adjoint operator on H satisfying 0 A 1. Then GL = VL∗ VL , ˜ L = V ∗ AVL are self-adjoint trace class operators satisfying G L ˜ L GL 1, ||GL || = 1 and ∀L > 0 : 0 G

lim Tr GL = ∞.

L→∞

We define I−1/n = [0, ∞) for n ∈ N and Iα = [0, 1/|α|) for α ∈ [−1, 1] − {0, −1, −1/2, . . . }. Then the function (α)

hL (z) =

Tr [zGL (1 − zαGL )−1 ] Tr GL

is well defined on Iα for each L > 0 and α ∈ [−1, 1] − {0}. Theorem 3.1. Let α ∈ [−1, 1]−{0} be arbitrary but fixed. Suppose that for every z ∈ Iα , (α) there exist a limit h(α) (z) = limL→∞ hL (z) and a trace class operator Kz satisfying lim || Kz − (1 − A)1/2 VL (1 − zαVL∗ VL )−1 VL∗ (1 − A)1/2 ||1 = 0.

L→∞

(3.1)

Then, for every ρˆ ∈ [0, supz∈Iα h(α) (z)), there exists a unique solution z = z∗ ∈ Iα of h(α) (z) = ρ. ˆ Moreover suppose that a sequence L1 < L2 < · · · < LN < · · · satisfies lim N/Tr GLN = ρ. ˆ

(3.2)

N→∞

Then

σ ∈SN

˜ LN U (σ )] α N−ν(σ ) Tr ⊗N H [⊗N G

σ ∈SN

α N−ν(σ ) Tr ⊗N H [⊗N GLN U (σ )]

lim

N→∞

= Det[1 + z∗ αKz∗ ]−1/α

(3.3)

holds. Here the operator U (σ ) on ⊗N H is defined by U (σ )ϕ1 ⊗ · · · ⊗ ϕN = ϕσ −1 (1) ⊗ · · · ⊗ ϕσ −1 (N) for σ ∈ SN and ϕ1 , . . . , ϕN ∈ H. In order to prove the theorem, we prepare several lemmas under the same assumptions of the theorem. Lemma 3.2. h(α) is a strictly increasing continuous function on Iα and there exists a unique z∗ ∈ Iα which satisfies h(α) (z∗ ) = ρ. ˆ

360

H. Tamura, K.R. Ito (α)

(α)

Proof. From hL (z) = Tr [GL (1−zαGL )−2 ]/Tr GL , we have 1 hL (z) (1−zα)−2 (α)

for α > 0 and 1 hL (z) (1−zα)−2 for α < 0, i.e., {hL }{L>0} is equi-continuous on (α) Iα . By Ascoli-Arzelà’s theorem, the convergence hL → h(α) is locally uniform and (α) hence h is continuous on Iα . It also follows that h(α) is strictly increasing. Together (α) with h(α) (0) = 0, which comes from hL (0) = 0, we get that h(α) (z) = ρˆ has a unique solution in Iα . (α)

Lemma 3.3. There exists a constant c0 > 0 such that ˜ L ||1 = Tr [VL∗ (1 − A)VL ] c0 ||GL − G uniformly in L > 0. Proof. Since 1 − zαGL is invertible for z ∈ Iα and VL is Hilbert-Schmidt, we have Tr [VL∗ (1 − A)VL ] = Tr [(1 − zαGL )1/2 (1 − zαGL )−1/2 ×VL∗ (1 − A)VL (1 − zαGL )−1/2 (1 − zαGL )1/2 ] ||1 − zαGL ||Tr [(1 − zαGL )−1/2 VL∗ (1 − A)VL (1 − zαGL )−1/2 ] = ||1 − zαGL ||Tr [(1 − A)1/2 VL (1 − zαGL )−1 VL∗ (1 − A)1/2 ] = (1 − (α ∧ 0)z)(Tr Kz + o(1)).

(3.4)

Here we have used |Tr B1 CB2 | ||B1 || ||B2 || ||C||1 = ||B1 || ||B2 ||Tr C for bounded operators B1 , B2 and a positive trace class operator C and Tr W V = Tr V W for Hilbert-Schmidt operators W, V . ˜ L in decreasing order Let us denote all the eigenvalues of GL and G g0 (L) = 1 g1 (L) · · · gj (L) · · · and g˜ 0 (L) g˜ 1 (L) · · · g˜ j (L) · · · , respectively. Then we have Lemma 3.4. For each j = 0, 1, 2, . . . ,

gj (L) g˜ j (L)

holds.

Proof. By the min-max principle, we have g˜ j (L) =

min

max

min

max

ψ0 ,...,ψj −1 ∈HL ψ∈{ψ0 ,...,ψj −1 }⊥ ψ0 ,...,ψj −1 ∈HL ψ∈{ψ0 ,...,ψj −1

}⊥

˜ L ψ) (ψ, G ||ψ||2 (ψ, GL ψ) = gj (L). ||ψ||2


361

Lemma 3.5. For N large enough, the conditions ˜ LN (1 − α z˜ N G ˜ LN )−1 ] = N Tr [zN GLN (1 − αzN GLN )−1 ] = Tr [˜zN G

(3.5)

determine zN , z˜ N ∈ Iα uniquely. zN and z˜ N satisfy zN z˜ N ,

|˜zN − zN | = O(1/N ) and

lim zN = lim z˜ N = z∗ .

N→∞

N→∞

Proof. From the proof of Lemma 3.2, HN (z) = Tr [zGLN (1−zαGLN )−1 ] = hLN (z)Tr GLN is a strictly increasing continuous function on Iα and HN (0) = 0. Let us pick z0 ∈ Iα such that z0 > z∗ . Since h(α) is strictly increasing, h(α) (z0 ) − h(α) (z∗ ) = > 0. We have (α)

h(α) (z0 ) HN (z0 ) Tr GLN = hLN (z0 ) → =1+ , (3.6) N N ρˆ ρˆ which shows HN (sup Iα )−0 HN (z0 ) > N for large enough N . Thus zN ∈ [0, z0 ) ⊂ Iα is uniquely determined by HN (zN ) = N . ˜ LN (1 − zα G ˜ LN )−1 ]. Then by Lemma 3.4, H˜ N is well-defined Put H˜ N (z) = Tr [zG on Iα and H˜ N HN there. Moreover ˜ LN )(1 − αzG ˜ LN )−1 ] HN (z) − H˜ N (z) = Tr [(1 − αzGLN )−1 z(GLN − G ˜ LN )−1 ||zTr [GLN − G ˜ LN ] ||(1 − αzGLN )−1 ||||(1 − αzG zc0 Cz = (1 − (α ∨ 0)z)2 holds. Together with (3.6), we have HN (z0 ) − Cz0 Cz H˜ N (z0 ) >1+ − 0, N N 2ρˆ N hence H˜ N (z0 ) > N , if N is large enough. It is also obvious that H˜ N is strictly increasing and continuous on Iα and H˜ N (0) = 0. Thus z˜ N ∈ [0, z0 ) ⊂ Iα is uniquely determined by H˜ N (˜zN ) = N . (α) The convergence zN → z∗ is a consequence of hLN (zN ) = N/Tr GLN → ρˆ = (α)

(α)

h(α) (z∗ ), the strict increasingness of h(α) , hL and the pointwise convergence hL → h(α) . We get zN z˜ N from HN H˜ N and the increasingness of HN , H˜ N . Now, let us show |˜zN − zN | = O(N −1 ), which together with zN → z∗ , yields z˜ N → z∗ . From 0 = N − N = HN (zN ) − H˜ N (˜zN ) ˜ LN )(1 − α z˜ N G ˜ LN )−1 ] = Tr [(1 − αzN GLN )−1 (zN GLN − z˜ N G ˜ LN )(1 − α z˜ N G ˜ LN )−1 ] = zN Tr [(1 − αzN GLN )−1 (GLN − G ˜ LN (1 − α z˜ N G ˜ LN )−1 ], −(˜zN − zN )Tr [(1 − αzN GLN )−1 G

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H. Tamura, K.R. Ito

we get z˜ N − zN ˜ LN (1 − α z˜ N G ˜ LN )−1 (1 − αzN GLN )−1/2 ] Tr [(1 − αzN GLN )−1/2 z˜ N G z˜ N ˜ LN )−1 ]. ˜ LN )(1 − α z˜ N G = zN Tr [(1 − αzN GLN )−1 (GLN − G It follows that z˜ N − zN ˜ z˜ N − zN HN (˜zN ) N= z˜ N z˜ N z˜ N − zN = Tr [(1 − αzN GLN ) z˜ N ˜ LN (1 − α z˜ N G ˜ LN )−1 (1 − αzN GLN )−1/2 ] ×(1 − αzN GLN )−1/2 z˜ N G z˜ N − zN ˜ LN ||1 − αzN GLN ||Tr [(1 − αzN GLN )−1/2 z˜ N G z˜ N ˜ LN )−1 (1 − αzN GLN )−1/2 ] ×(1 − α z˜ N G ˜ LN )−1 ] ˜ LN )(1 − α z˜ N G = ||1 − αzN GLN ||zN Tr [(1 − αzN GLN )−1 (GLN − G ˜ LN )−1 || ˜ LN ] ||(1 − α z˜ N G zN ||1 − αzN GLN || ||(1 − αzN GLN )−1 || Tr [GLN − G c0 z0 (1 − (α ∧ 0)z0 )/(1 − (α ∨ 0)z0 )2 for N large enough, because zN , z˜ N < z0 . Thus, we have obtained z˜ N −zN = O(N −1 ). We put v (N) = Tr [zN GLN (1 − αzN GLN )−2 ]

˜ LN (1 − α z˜ N G ˜ LN )−2 ]. and v˜ (N) = Tr [˜zN G

Then we have : Lemma 3.6. (i) v (N) , v˜ (N) → ∞, Proof.

(ii)

v (N) → 1. v˜ (N)

(i) follows from the lower bound v (N) = Tr [zN GLN (1 − αzN GLN )−2 ] Tr [zN GLN (1 − αzN GLN )−1 ] ||1 − αzN GLN ||−1 N (1 + o(1))/(1 − (α ∧ 0)z∗ ),

since zN → z∗ . The same bound is also true for v˜ (N) . (ii) Using v (N) = Tr [−zN GLN (1 − αzN GLN )−1 + α −1 (1 − αzN GLN )−2 − α −1 ] = −N + α −1 Tr [(1 − αzN GLN )−2 − 1]


363

and the same for v˜ (N) , we get ˜ LN )−2 − (1 − αzN GLN )−2 ]| |v˜ (N) − v (N) | = |α −1 Tr [(1 − α z˜ N G ˜ LN )−1 |α −1 Tr [ (1 − α z˜ N G ˜ LN )−1 (1 − α z˜ N G ˜ LN )−1 ]| −(1 − αzN G ˜ LN )−1 +|α −1 Tr [ (1 − αzN G ˜ LN )−1 ]| −(1 − αzN GLN )−1 (1 − α z˜ N G ˜ LN )−1 +|α −1 Tr [(1 − αzN GLN )−1 (1 − α z˜ N G ˜ LN )−1 ]| −(1 − αzN G +|α −1 Tr [(1 − αzN GLN )−1 ˜ LN )−1 − (1 − αzN GLN )−1 ]| × (1 − αzN G ˜ LN )−1 || ||(1 − α z˜ N G ˜ LN )−1 (˜zN − zN ) ||(1 − α z˜ N G ˜ LN (1 − αzN G ˜ LN )−1 ||1 ×G ˜ LN ) ˜ LN )−1 || ||(1 − αzN G ˜ LN )−1 zN (GLN − G +||(1 − α z˜ N G ×(1 − αzN GLN )−1 ||1 ˜ LN )−1 (˜zN − zN ) +||(1 − αzN GLN )−1 || ||(1 − α z˜ N G ˜ LN (1 − αzN G ˜ LN )−1 ||1 ×G ˜ LN )−1 zN (GLN − G ˜ LN ) +||(1 − αzN GLN )−1 || ||(1 − αzN G ×(1 − αzN GLN )−1 ||1 ˜ LN )−1 || + ||(1 − αzN GLN )−1 ||) (||(1 − α z˜ N G z˜ N − zN ˜ LN × ||˜zN G z˜ N ˜ LN )−1 ||1 ||(1 − αzN G ˜ LN )−1 || ×(1 − α z˜ N G ˜ LN ||1 ˜ LN )−1 || ||GLN − G +zN ||(1 − αzN G ×||(1 − αzN GLN )−1 || = O(1). In the last step, we have used Lemmas 3.3 and 3.5. This, together with (i), implies (ii). Lemma 3.7. lim

N →∞

S1 (0)

lim

N →∞

2π v (N) 2π v˜ (N) S1 (0)

−1/α dη Det 1 − αzN (η − 1)GLN (1 − αzN GLN )−1 = 1, N+1 2πiη dη ˜ LN )−1 −1/α = 1. ˜ LN (1 − α z˜ N G Det 1 − α z˜ N (η − 1)G N+1 2πiη

364

H. Tamura, K.R. Ito

Proof. Put s = 1/|α| and (N)

pj

=

|α|zN gj (LN ) . 1 − αzN gj (LN )

Then the first equality is nothing but Proposition A.2(i) for α < 0 and Proposition A.2(ii) for α > 0. The same is true for the second equality. Proof of Theorem 3.1. Since the uniqueness of z∗ has already been shown, it is enough to prove (3.3). The main apparatus of the proof is Vere-Jones’ formula in the following form: Let α = −1/n for n ∈ N. Then Det(1 − αJ )−1/α =

∞ 1 n−ν(σ ) α Tr ⊗n H [(⊗n J )U (σ )] n! n=0

σ ∈Sn

holds for any trace class operator J . For α ∈ [−1, 1] − {0, −1, −1/2, . . . , 1/n, . . . }, this holds under an additional condition ||αJ || < 1. This has actually been proved in Theorem 2.4 of [23]. We use the formula in the form 1 N−ν(σ ) α Tr ⊗N H [(⊗N GLN )U (σ )] N! σ ∈SN dz Det(1 − zαGLN )−1/α = N+1 SzN (0) 2πiz

(3.7)

˜ LN . Here, recall that zN , z˜ N ∈ Iα . We and in the form in which GLN is replaced by G calculate the right-hand side by the saddle point method. Using the above integral representation and the property of the products of the Fredholm determinants followed by the change of integral variables z = zN η, z = z˜ N η, we get N−ν(σ ) Tr N ˜ ⊗N H [⊗ GLN U (σ )] σ ∈S α N N−ν(σ ) Tr ⊗N H [⊗N GLN U (σ )] σ ∈SN α −1/α dz/2πizN+1 ˜ Sz˜ N (0) Det(1 − zα GLN ) = −1/α dz/2πizN+1 Sz (0) Det(1 − zαGLN ) N

N ˜ LN ]−1/α zN Det[1 − z˜ N αGLN ]−1/α Det[1 − z˜ N α G = N Det[1 − zN αGLN ]−1/α Det[1 − z˜ N αGLN ]−1/α z˜ N −1 −1/α dη/2π iηN+1 ˜ ˜ S (0) Det[1 − z˜ N (η − 1)α GLN (1 − z˜ N α GLN ) ] × 1 . −1 −1/α dη/2π iηN+1 S1 (0) Det[1 − zN (η − 1)αGLN (1 − zN αGLN ) ]

Thus the theorem is proved if the following behaviors in N → ∞ are valid: (a) N zN N z˜ N

z˜ N − zN = exp − N + o(1) , zN


365

(b) z˜ N − zN Det[1 − z˜ N αGLN ]−1/α = exp N + o(1) , −1/α Det[1 − zN αGLN ] zN (c) ˜ LN ]−1/α Det[1 − z˜ N α G → Det[1 + z∗ αKz∗ ]−1/α , Det[1 − z˜ N αGLN ]−1/α (d) S1 (0)

˜ LN (1 − z˜ N α G ˜ LN )−1 ]−1/α dη/2π iηN+1 Det[1 − z˜ N (η − 1)α G

S1 (0) Det[1 − zN (η

− 1)αGLN (1 − zN αGLN )−1 ]−1/α dη/2π iηN+1

→ 1.

In fact, (a) is a consequence of Lemma 3.5. For (b), let us define a function k(z) = log Det[1 − zαGL ]−1/α = −α −1 ∞ log(1 − zαgj (L)). Then by Taylor’s formula j =0 and (3.5), we get (˜zN − zN )2 2 ∞ ∞ αgj2 gj (˜zN − zN )2 = (˜zN − zN ) + 1 − zN αgj (1 − z¯ αgj )2 2

k(˜zN ) − k(zN ) = k (zN )(˜zN − zN ) + k (¯z)

j =0

=N

j =0

z˜ N − zN + δ, zN

where z¯ is a mean value of zN and z˜ N and |δ| = O(1/N ) by Lemma 3.5. From the property of the product and the cyclic nature of the Fredholm determinants, we have ˜ LN ] Det[1 − z˜ N α G Det[1 − z˜ N αGLN ] = Det[1 + z∗ α(1 − A)1/2 VLN (1 − z∗ αGLN )−1 VL∗N (1 − A)1/2 ] ˜ LN )(1 − z˜ N αGLN )−1 ] + Det[1 + z˜ N α(GLN − G ˜ LN )(1 − z∗ αGLN )−1 ] . −Det[1 + z∗ α(GLN − G The first term converges to Det[1 + z∗ αKz∗ ] by the assumption (3.1) and the continuity of the Fredholm determinants with respect to the trace norm. The brace in the above equation tends to 0, because of the continuity and ˜ LN )(1 − z˜ N αGLN )−1 − z∗ α(GLN − G ˜ LN )(1 − z∗ αGLN )−1 ||1 ||˜zN α(GLN − G ˜ LN ||1 ||(1 − z˜ N αGLN )−1 || |˜zN − z∗ | |α| ||GLN − G ˜ LN ||1 ||(1 − z˜ N αGLN )−1 − (1 − z∗ αGLN )−1 || → 0, +z∗ |α| ||GLN − G where we have used Lemmas 3.3 and 3.5. Thus, we get (c). (d) is a consequence of Lemma 3.6 and Lemma 3.7.

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H. Tamura, K.R. Ito

3.2. Proofs of the theorems. To prove Theorem 2.1[2.2], it is enough to show that (2.5)[(2.11)] converges to the right-hand side of (2.8) [(2.13), respectively] for every f ∈ Co (Rd )[5]. We regard HL = L2 (L ) as a closed subspace of L2 (Rd ). Corresponding to the orthogonal decomposition L2 (Rd ) = L2 (L ) ⊕ L2 (cL ), we set VL = eβ L /2 ⊕ 0. Let A = e−f be the multiplication operator on L2 (Rd ), which can be decomposed as A = e−f χL ⊕ χcL for large L since supp f is compact. Then GL = VL∗ VL = eβ L ⊕ 0

˜ L = VL∗ AVL = eβ L /2 e−f eβ L /2 ⊕ 0 and G

can be identified with those in Sect. 2. We begin with the following fact, where we denote (L)

k

=

1 1 d

2π k+ − , L 2 2

for

k ∈ Zd .

Lemma 3.8. Let b : [0, ∞) → [0, ∞) be a monotone decreasing continuous function such that b(|p|)dp < ∞. Rd

Define the function bL : Rd → [0, ∞) by bL (p) = b(|2πk/L|)

if

(L)

p ∈ k

for k ∈ Zd .

Then bL (p) → b(|p|) in L1 (Rd ) as L → ∞ . Proof. There exist positive constants c1 and c2 such 2 |p|) holds for √ that bL (p) c1 b(c√ all L 1 and p ∈ Rd . Indeed, c1 = b(0)/b(2π d/(d + 8)), c2 = 2/ d + 8 satisfy (L) the condition, since inf{c1 b(c2 |p|) | p ∈ 0 } b(0) for ∀L > 1 and sup{c2 |p| | p ∈ (L) k } 2π |k|/L for k ∈ Zd − {0}. Obviously c1 b(c2 |p|) is an integrable function of p ∈ Rd . The lemma follows by the dominated convergence theorem. Finally we confirm the assumptions of Theorem 3.1. Proposition 3.9. (i) ∀L > 0 : ||VL || = 1,

lim Tr GL /Ld = (4πβ)−d/2 .

L→∞

(3.8)

(ii) The following convergences hold as L → ∞ for each z ∈ Iα :

(α)

Tr [zGL (1 − zαGL )−1 ] Tr GL 2 dp ze−β|p| d/2 = h(α) (z), → (4πβ) 2 d Rd (2π) 1 − zαe−β|p| || 1 − e−f GL (1 − zαGL )−1 −G(1 − zαG)−1 1 − e−f ||1 → 0.

hL (z) =

(3.9)

(3.10)


367

2 2 ˜ Proof. By applying the above lemma to b(|p|) = e−β|p| and b(|p|) = ze−β|p| / 2 (1 − zαe−β|p| ), we have (3.8) and (3.9). By Grüm’s convergence theorem, it is enough to show 1 − e−f GL (1 − zαGL )−1 1 − e−f → 1 − e−f G(1 − zαG)−1 1 − e−f

strongly and Tr [ 1 − e−f GL (1 − zαGL )−1 1 − e−f ] = (1 − e−f (x) ) GL (1 − zαGL )−1 (x, x)dx d R → (1 − e−f (x) ) G(1 − zαG)−1 (x, x)dx Rd = Tr [ 1 − e−f G(1 − zαG)−1 1 − e−f ] for (3.10). These are direct consequences of |zGL (1 − zαGL )−1 (x, y) − zG(1 − zαG)−1 (x, y)| dp ˜ |eL (p, x − y)b˜L (p) − e(p, x − y)b(|p|)| = (2π )d dp ˜ ˜ ˜ + |eL (p, x − y) − e(p, x − y)|b(|p|) →0 |bL (p) − b(|p|)| d (2π ) ˜ uniformly in x, y ∈ supp f . Here we have used the above lemma for b(|p|) and we put e(p, x) = eip·x and eL (p; x) = e(2πk/L; x)

(L)

if p ∈ k

for

k ∈ Zd .

Thanks to (3.8), we can take a sequence {LN }N∈N which satisfies (3.2). On the relation between ρ in Theorems 2.1, 2.2 and ρˆ in Theorem 3.1, ρˆ = (4πβ)d/2 ρ is derived from (2.7). We have the ranges of ρ in Theorem 2.2 and Theorem 2.1, since sup h(1) (z) = (4πβ)d/2 z∈I1

dp e−β|p| = (4πβ)d/2 ρc 2 d Rd (2π) 1 − e−β|p| 2

and supz∈I−1 h(−1) (z) = ∞ from (3.9). Thus we get Theorem 2.1 and Theorem 2.2 using Theorem 3.1. 4. Para-Particles The purpose of this section is to apply the method which we have developed in the preceding sections to statistical mechanics of gases which consist of identical particles obeying para-statistics. Here, we restrict our attention to para-fermions and para-bosons of order 2. We will see that the point processes obtained after the thermodynamic limit are the point processes corresponding to the cases of α = ±1/2 given in [23].

368

H. Tamura, K.R. Ito

In this section, we use the representation theory of the symmetric group ( cf. e.g. [13, 22, 25]). We say that (λ1 , λ2 , . . . , λn ) ∈ Nn is a Young frame of length n for the symmetric group SN if n

λj = N,

λ1 λ2 · · · λn > 0.

j =1

We associate the Young frame (λ1 , λ2 , . . . , λn ) with the diagram of λ1 -boxes in the first row, λ2 -boxes in the second row,..., and λn -boxes in the nth row. A Young tableau on a Young frame is a bijection from the numbers 1, 2, . . . , N to the N boxes of the frame. 4.1. Para-bosons of order 2. Let us select one Young tableau, arbitrary but fixed, on each Young frame of length less than or equal to 2, say the tableau Tj on the frame (N − j, j ) for j = 1, 2, . . . , [N/2] and the tableau T0 on the frame (N ). We denote by R(Tj ) the row stabilizer of Tj , i.e., the subgroup of SN consists of those elements that keep all rows of Tj invariant, and by C(Tj ) the column stabilizer whose elements preserve all columns of Tj . Let us introduce the three elements 1 1 a(Tj ) = σ, b(Tj ) = sgn(σ )σ #R(Tj ) #C(Tj ) σ ∈R(Tj )

σ ∈C (Tj )

and e(Tj ) =

dTj N!

sgn(τ )σ τ = cj a(Tj )b(Tj )

σ ∈R(Tj ) τ ∈C (Tj )

of the group algebra C[SN ] for each j = 0, 1, . . . , [N/2], where dTj is the dimension of the irreducible representation of SN corresponding to Tj and cj = dTj #R(Tj )#C(Tj )/N !. As is known, a(Tj )σ b(Tk ) = b(Tk )σ a(Tj ) = 0

(4.1)

hold for any σ ∈ SN and 0 j < k [N/2]. The relations a(Tj )2 = a(Tj ),

b(Tj )2 = b(Tj ),

e(Tj )e(Tk ) = δj k e(Tj )

(4.2)

also hold. For later use, let us introduce d(Tj ) = e(Tj )a(Tj ) = cj a(Tj )b(Tj )a(Tj )

(j = 0, 1, . . . , [N/2]).

(4.3)

They satisfy d(Tj )d(Tk ) = δj k d(Tj )

for

0 j, k [N/2],

(4.4)

as is shown readily from (4.1) and (4.2). The inner product < ·, · > of C[SN ] is defined by < σ, τ >= δσ τ

for σ, τ ∈ SN

and extended to all elements of C[SN ] by sesqui-linearity.


369

The left representation L and the right representation R of SN on C[SN ] are defined by

L(σ )g = L(σ )

g(τ )τ =

τ ∈SN

and

R(σ )g = R(σ )

τ ∈SN

g(τ )σ τ =

τ ∈SN

g(τ )τ =

g(σ −1 τ )τ

τ ∈SN

g(τ )τ σ −1 =

τ ∈SN

g(τ σ )τ,

τ ∈SN

respectively. Here and hereafter we identify g : SN → C and τ ∈SN g(τ )τ ∈ C[SN ]. They are extended to the representation of C[SN ] on C[SN ] as L(f )g = f g = f (σ )g(τ )σ τ = f (σ τ −1 )g(τ ) σ σ,τ

and R(f )g = g fˆ =

σ

g(σ )f (τ )σ τ −1 =

σ,τ

τ

σ

g(σ τ )f (τ ) σ,

τ

where fˆ = τ fˆ(τ )τ = τ f (τ −1 )τ = τ f (τ )τ −1 . The character of the irreducible representation of SN corresponding to the tableau Tj is obtained by χTj (σ ) = (τ, σ R(e(Tj ))τ ) = (τ, σ τ e(T j )). τ ∈SN

τ ∈SN

We introduce a tentative notation χg (σ ) ≡ (τ, σ R(g)τ ) = (τ, σ τ γ −1 )g(γ ) = g(τ −1 σ τ ) τ ∈SN

τ,γ ∈SN

(4.5)

τ ∈SN

for g = τ g(τ )τ ∈ C[SN ]. Let U be the representation of SN ( and its extension to C[SN ]) on ⊗N HL defined by U (σ )ϕ1 ⊗ · · · ⊗ ϕN = ϕσ −1 (1) ⊗ · · · ⊗ ϕσ −1 (N)

for ϕ1 , . . . , ϕN ∈ HL ,

or equivalently by (U (σ )f )(x1 , . . . , xN ) = f (xσ (1) , . . . , xσ (N) )

for f ∈ ⊗N HL .

Obviously, U is unitary: U (σ )∗ = U (σ −1 ) = U (σ )−1 . Hence U (a(Tj )) is an orthogonal projection because of U (a(Tj ))∗ = U (a(T j )) = U (a(Tj )) and (4.2). So are U (b(Tj ))’s, [N/2] U (d(Tj ))’s and P2B = j =0 U (d(Tj )). Note that Ran U (d(Tj )) = Ran U (e(Tj )) because of d(Tj )e(Tj ) = e(Tj ), e(Tj )d(Tj ) = d(Tj ). We refer to the literature [19, 12, 27] for quantum mechanics of para-particles. (See also [20].) The arguments of the literatures indicate that the state space of N para-bosons 2B = P ⊗N H . It is obvious that there is of order 2 in the finite box L is given by HL,N 2B L (L)

(L)

2B which consists of the vectors of the form U (d(T ))ϕ a CONS of HL,N j k1 ⊗ · · · ⊗ ϕkN ,

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which are the eigenfunctions of ⊗N GL . Then, we define a point process of N free para-bosons of order 2 as in Sect. 2 and its generating functional is given by ˜ L )P2B ] − Tr ⊗N HL [(⊗N G 2B e = . EL,N N Tr ⊗N HL [(⊗ GL )P2B ] Let us give expressions, which have a clear correspondence with (2.11). Lemma 4.1. 2B EL,N

e

−

[N/2] j =0

σ ∈SN

= [N/2] =

˜ L )U (σ )] χTj (σ )Tr ⊗N HL [(⊗N G

N σ ∈SN χTj (σ )Tr ⊗N HL [(⊗ GL )U (σ )] j =0 [N/2] ˜ L (xi , xj )}dx1 · · · dxN det Tj {G j =0 N L . [N/2] det Tj {GL (xi , xj )}dx1 · · · dxN j =0 N L

(4.6)

(4.7)

2B = P N Remark 1. HL,N 2B ⊗ HL is determined by the choice of the tableaux Tj ’s. The spaces corresponding to different choices of tableaux are different subspaces of ⊗N HL . However, they are unitarily equivalent and the generating functional given above is not affected by the choice. In fact, χTj (σ ) depends only on the frame on which the tableau Tj is defined.

N Remark 2. detT A = σ ∈SN χT (σ ) i=1 Aiσ (i) in (4.7) is called immanant, another generalization of determinant than det α . Proof. Since ⊗N G commutes with U (σ ) and a(Tj )e(Tj ) = e(Tj ), we have Tr ⊗N HL (⊗N GL )U (d(Tj )) = Tr ⊗N HL (⊗N GL )U (e(Tj ))U (a(Tj )) = Tr ⊗N HL U (a(Tj ))(⊗N GL )U (e(Tj )) = Tr ⊗N HL (⊗N GL )U (e(Tj )) . (4.8) On the other hand, we get from (4.5) that χg (σ )Tr ⊗N HL (⊗N G)U (σ ) = g(τ −1 σ τ )Tr ⊗N HL (⊗N G)U (σ ) σ ∈SN

=

τ,σ

=

τ,σ

= N!

τ,σ ∈SN

g(σ )Tr ⊗N HL (⊗N G)U (τ σ τ −1 )

g(σ )Tr ⊗N HL (⊗N G)U (τ )U (σ )U (τ −1 )

g(σ )Tr ⊗N HL (⊗N G)U (σ ) = N !Tr ⊗N HL (⊗N G)U (g) ,

(4.9)

σ

where we have used the cyclicity of the trace and the commutativity of U (τ ) with ⊗N G. Putting g = e(Tj ) and using(4.8), the first equation is derived. The second one is obvious.


371

N Let ψTj be the character of the induced representation IndS R(Tj ) [1], where 1 is the representation R(Tj ) σ → 1, i.e.,

ψTj (σ ) =

< τ, σ R(a(Tj ))τ >= χa(Tj ) (σ ).

τ ∈SN

Then the determinantal form [13] χTj = ψTj − ψTj −1 χT0 = ψT0

(j = 1, . . . , [N/2])

(4.10)

yields the following result: Theorem 4.2. The finite para-boson processes defined above converge weakly to the point process whose Laplace transform is given by −2 Eρ2B e− = Det 1 + 1 − e−f z∗ G(1 − z∗ G)−1 1 − e−f in the thermodynamic limit, where z∗ ∈ (0, 1) is determined by ρ = 2

z∗ e−β|p| dp = (z∗ G(1 − z∗ G)−1 )(x, x) < ρc , (2π)d 1 − z∗ e−β|p|2 2

and ρc is given by (2.12). Proof. Using (4.10) in the expression in Lemma 4.1 and (4.9) for g = a(T[N/2] ), we have 2B − EL,N e

˜ L )U (σ ) ψT[N/2] (σ )Tr H⊗N (⊗N G L = N σ ∈SN ψT[N/2] (σ )Tr H⊗N (⊗ GL )U (σ ) L ˜ L )U (a(T[N/2] ) Tr ⊗N H (⊗N G L = Tr ⊗N H (⊗N GL )U (a(T[N/2] ) L [N/2] ˜ L )S[(N +1)/2] Tr [N/2] ˜ L )S[N/2] Tr ⊗[(N +1)/2] H (⊗[(N +1)/2] G G ⊗ HL (⊗ L = . Tr ⊗[(N +1)/2] H (⊗[(N +1)/2] GL )S[(N +1)/2] Tr ⊗[N/2] H (⊗[N/2] GL )S[N/2]

σ ∈S N

L

L

In the last equality, we have used a(T[N/2] ) =

σ ∈R1

#R1

σ

τ ∈R2

#R2

τ

,

where R1 is the symmetric group of [(N + 1)/2] numbers which are on the first row of the tableau T[N/2] and R2 that of [N/2] numbers on the second row. Then, Theorem 2.2 yields the theorem.

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4.2. Para-fermions of order 2. For a Young tableau T , we denote by T the tableau obtained by interchanging the rows and the columns of T . In another word, T is the transpose of T . The tableau Tj is on the frame (2, . . . , 2, 1, . . . , 1) and satisfies j

R(Tj ) = C(Tj ),

N−2j

C(Tj ) = R(Tj ).

The generating functional of the point process for N para-fermions of order 2 in the finite box L is given by N [N/2] ˜ − j =0 Tr ⊗N HL (⊗ G)U (d(Tj )) 2F EL,N e = [N/2] N j =0 Tr ⊗N HL (⊗ G)U (d(Tj )) as in the case of para-bosons of order 2. Let us recall the relations χTj (σ ) = sgn(σ )χTj (σ ),

ϕTj (σ ) = sgn(σ )ψTj (σ ),

where we have denoted by ϕTj (σ ) =

< τ, σ R(b(Tj ))τ >

τ N [ sgn ], where sgn is the representathe character of the induced representation IndS C (T ) j

tion C(Tj ) = R(Tj ) σ → sgn(σ ). Thanks to these relations, we can easily translate the argument of para-bosons to that of para-fermions and get the following theorem.

Theorem 4.3. The finite para-fermion processes defined above converge weakly to the point process whose Laplace transform is given by 2 Eρ2B e− = Det 1 − 1 − e−f z∗ G(1 + z∗ G)−1 1 − e−f in the thermodynamic limit, where z∗ ∈ (0, ∞) is determined by 2 dp z∗ e−β|p| ρ = = (z∗ G(1 + z∗ G)−1 )(x, x). 2 (2π)d 1 + z∗ e−β|p|2 5. Gas of Composite Particles Most gases are composed of composite particles. In this section, we formulate point processes which yield the position distributions of constituents of such gases. Each composite particle is called a “molecule", and molecules consist of “atoms". Suppose that there are two kinds of atoms, say A and B, such that both of them obey Fermi-Dirac or Bose-Einstein statistics simultaneously, that N atoms of kind A and N atoms of kind B are in the same box L and that one A-atom and one B-atom are bounded to form a molecule by the non-relativistic interaction described by the Hamiltonian HL = − x − y + U (x − y) with periodic boundary conditions in L2 (L × L ). Hence there are totally N such molecules in L . We assume that the interaction between atoms in different molecules


373

can be neglected. We only consider such systems of zero temperature, where N molecules are in the ground state and (anti-)symmetrizations of the wave functions of the N atoms of type A and the N atoms of type B are considered. In order to avoid difficulties due to boundary conditions, we have set the masses of two atoms A and B equal. We also assume that the potential U is infinitely deep so that the wave function of the ground state has a compact support. We put 1 (R) (r) HL = − R − 2 r + U (r) = HL + HL , 2 where R = (x + y)/2, r = x − y. The normalized wave function of the ground state (R) (r) of HL is the constant function L−d/2 . Let ϕL (r) be that of the ground state of HL . −d/2 Then, the ground state of HL is ψL (x, y) = L ϕL (x − y). The ground state of the N-particle system in L is, by taking the (anti-)symmetrizations, −1 L,N (x1 , . . . , xN ; y1 , . . . , yN ) = Zcα

α N−ν(σ ) α N−ν(τ )

σ,τ ∈SN

=

N

ψL (xσ (j ) , yτ (j ) )

j =1

N N! N−ν(σ ) α ϕL (xj − yσ (j ) ), (5.1) Zcα LdN/2 σ j =1

where Zcα is the normalization constant and α = ±1. Recall that α N−ν(σ ) = sgn(σ ) for α = −1. The distribution function of positions of 2N -atoms of the system with zero temperature is given by the square of magnitude of (5.1), cα pL,N (x1 , . . . , xN ; y1 , . . . , yN ) =

×

α N−ν(σ )

σ,τ ∈SN

N

(N !)2 2 LdN Zcα

ϕL (xj − yτ (j ) )ϕL (xσ (j ) − yτ (j ) ).

(5.2)

j =1

Suppose that we are interested in one kind of atoms, say of type A. We introduce the operator ϕL on HL = L2 (L ) which has the integral kernel ϕL (x −y). Then the Laplace transform of the distribution of the positions of N A-atoms can be written as N − cα cα e = EL,N e− j =1 f (xj ) pL,N (x1 , . . . , xN ; y1 , . . . , yN ) 2N

×dx1 · · · dxN dy1 · · · dyN N−ν(σ ) Tr N ∗ −f ϕ )U (σ )] L ⊗N H [(⊗ ϕL e σ ∈S α = N . ∗ N−ν(σ ) N Tr ⊗N H [(⊗ ϕL ϕL )U (σ )] σ ∈SN α In order to take the thermodynamic limit N, L → ∞, V /Ld → ρ, we consider a Schrödinger operator in the whole space. Let ϕ be the normalized wave function of the ground state of Hr = −2 r + U (r) in L2 (Rd ). Then ϕ(r) = ϕL (r) (∀r ∈ L ) holds for large L by the assumption on U . The Fourier series expansion of ϕL is given by ϕL (r) =

2π d/2 2πk ei2πk·r/L ϕˆ , L L Ld/2 d

k∈Z

374

H. Tamura, K.R. Ito

where ϕˆ is the Fourier transform of ϕ: ϕ(p) ˆ = ϕ(r)e−ip·r Rd

dr . (2π)d/2

By ϕ, we denote the integral operator on H = L2 (Rd ) having kernel ϕ(x − y). Now we have the following theorem on the thermodynamic limit, where the density ρ > 0 is arbitrary for α = −1, ρ ∈ (0, ρcc ) for α = 1 and ρcc =

2 |ϕ(p)| ˆ dp . 2 − |ϕ(p)| 2 (2π)d |ϕ(0)| ˆ ˆ

Theorem 5.1. The finite point processes defined above for α = ±1 converge weakly to the process whose Laplace transform is given by −1/α Eρcα e− = Det 1 + z∗ α 1 − e−f ϕ(||ϕ||2L1 − z∗ αϕ ∗ ϕ)−1 ϕ ∗ 1 − e−f in the thermodynamic limit (2.7), where the parameter z∗ is the positive constant uniquely determined by 2 dp z∗ |ϕ(p)| ˆ ρ= = (z∗ ϕ(||ϕ||2L1 − z∗ αϕ ∗ ϕ)−1 ϕ ∗ )(x, x). 2 − z α|ϕ(p)| 2 (2π )d |ϕ(0)| ˆ ˆ ∗ ˆ k/L)}k∈Zd . Since ϕ Proof. The eigenvalues of the integral operator ϕL is {(2π )d/2 ϕ(2π is the ground state of the Schrödinger operator, we can assume ϕ 0. Hence the largest eigenvalue is (2π )d/2 ϕ(0) ˆ = ||ϕ||L1 . We also have 2π d 2π k 2 2 1 = ||ϕ||2L2 (Rd ) = |ϕ(p)| ˆ dp = ||ϕL ||2L2 ( ) = ϕˆ . (5.3) L L L Rd d k∈Z

Set VL = ϕL /||ϕ||L1 so that ||VL || = 1,

||VL ||22 = Ld /||ϕ||2L1 .

Then Theorem 3.1 applies as follows: For z ∈ Iα , let us define functions d, dL on Rd by d(p) =

2 z|ϕ(p)| ˆ 2 − zα|ϕ(p)| 2 |ϕ(0)| ˆ ˆ

and dL (p) = d(2π k/L) Then

Rd

(L)

if p ∈ k

for

k ∈ Zd .

dp dL (p) = L−d ||zVL (1 − zαVL∗ VL )−1 VL∗ ||1 (2π)d

and the following lemma holds:

(5.4)


375

Lemma 5.2. lim ||dL − d||L1 = 0.

L→∞

Proof. Put ϕˆ[L] (p) = ϕ(2π ˆ k/L)

(L)

if p ∈ k

for

k ∈ Zd

and note that compactness of supp ϕ implies ϕ ∈ L1 (Rd ) and uniform continuity of ϕ. ˆ ˆ 2 ||L∞ → 0 and ||dL − d||L∞ → 0. On the other hand, we Then we have || |ϕˆ[L] |2 − |ϕ| ˆ 2 ||L1 from (5.3). It is obvious that get || |ϕˆ[L] |2 ||L1 = || |ϕ| | ||dL ||L1 − ||d||L1 |

|| |ϕˆ[L] |2 − |ϕ| ˆ 2 ||L1 z . 2 2 (1 − z(α ∨ 0)) |ϕ(0)|

Hence the lemma is derived by using the following fact twice: If f, f1 , f2 , · · · ∈ L1 (Rd ) satisfy ||fn − f ||L∞ → 0 and || fn ||L1 → || f ||L1 , then ||fn − f ||L1 → 0 holds. In fact, using |fn (x)| dx = |f (x)| dx |x|>R |x|>R + (|f (x)| − |fn (x)|) dx + || fn ||L1 − || f ||L1 , |x|R

we have

||fn − f ||L1

|x|R

|fn (x) − f (x)| dx +

2

|x|R

|x|>R

(|fn (x)| + |f (x)|) dx

|fn (x) − f (x)| dx

+2

|x|>R

|f (x)| dx + || fn ||L1 − || f ||L1 .

For any > 0, we can choose R large enough to make the second term of the right hand side smaller than . For this choice of R, we set n so large that the first term and the remainder are smaller than and then ||fn − f ||L1 < 3. (Continuation of the proof of Theorem 5.1). Using this lemma, we can show (α)

Tr [zVL (1 − zαVL∗ VL )−1 VL∗ ] Tr VL∗ VL 2 z|ϕ(p)| ˆ 2 → |ϕ(0)| ˆ dp = h(α) (z), 2 2 d | ϕ(0)| ˆ − zα| ϕ(p)| ˆ R || 1 − e−f VL (1 − zαVL∗ VL )−1 VL∗ −ϕ(||ϕ||2L1 − zαϕ ∗ ϕ)−1 ϕ 1 − e−f ||1 → 0,

hL (z) =

as in the proof of (3.9) and (3.10). We have the conversion ρˆ = ||ϕ||2L1 ρ and hence ρcc = supz∈I1 h(1) (z)/||ϕ||2L1 . Hence the proof is completed by Theorem 3.1.

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Acknowledgements. We would like to thank Professor Y. Takahashi and Professor T. Shirai for many useful discussions. K. R. I. thanks the Grant-in-Aid for Science Research (C)15540222 from JSPS.

A. Complex Integrals Lemma A.1.

(i) For 0 p 1 and −π θ π, 2p(1 − p)θ 2

|1 + p(eiθ − 1)| exp − π2

holds. For 0 p 1 and −π/3 θ π/3, p(1 − p) 2 4p(1 − p)|θ |3 θ | | log 1 + p(eiθ − 1) − ipθ + √ 2 9 3 holds. (ii) For p 0 and −π θ π, the following inequalities hold: 2p(1 + p) θ 2

, 1 + 4p(1 + p) π 2 p(1 + p) 2 p(1 + p)(1 + 2p)|θ |3 | log 1 − p(eiθ − 1) + ipθ − θ | . 2 6 |1 − p(eiθ − 1)| exp

Proof. (i) The first inequality follows from |1 + p(eiθ − 1)|2 = 1 − 2p(1 − p)(1 − cos θ) exp(−2p(1 − p)(1 − cos θ)) exp(−4p(1 − p)θ 2 /π 2 ),

(A.1)

where 1 − cos θ 2θ 2 /π 2 for θ ∈ [−π, π] is used in the second inequality. Put f (θ) = log(1 + p(eiθ − 1)). Then we have f (0) = 0, i(1 − p) , f (0) = ip, 1 − p + peiθ p(1 − p)eiθ f (θ) = − , f (0) = −p(1 − p), (1 − p + peiθ )2 f (θ) = i −

and f (3) (θ ) = −

ip(1 − p)eiθ (1 − p − peiθ ) . (1 − p + peiθ )3

By (A.1) and θ ∈ [−π/3, √ π/3], we have |1+p(eiθ −1)|2 1−p(1−p) 3/4. Hence, (3) |f (θ )| 8p(1 − p)/3 3 holds. Taylor’s theorem yields the second inequality.


377

(ii) The first inequality follows from |1 − p(eiθ − 1)|2 = 1 + 2p(1 + p)(1 − cos θ) 2p(1 + p)(1 − cos θ)

exp 1 + 4p(1 + p) 4p(1 + p) θ 2

exp . 1 + 4p(1 + p) π 2 x/(1+a) for x ∈ [0, a] in the first inequality, which is derived Here we have used 1+x

x e from log(1 + x) = 0 dt/(1 + t) x/(1 + a). Put f (θ) = log(1 − p(eiθ − 1)). Then we have f (0) = 0,

i(1 + p) , f (0) = −ip, 1 + p − peiθ p(1 + p)eiθ f (θ ) = , f (0) = p(1 + p), (1 + p − peiθ )2 f (θ ) = i −

and ip(1 + p)eiθ (1 + p + peiθ ) . (1 + p − peiθ )3

f (3) (θ ) =

Hence, we have |f (3) (θ )| p(1 + p)(1 + 2p). Thus we get the second inequality. (N)

Proposition A.2. Let s > 0 and a collection of numbers {pj }j,N satisfy (N)

p0

(N)

p1

(N)

p2

(N)

· · · pj

· · · 0,

∞

(N)

spj

= N.

j =0 (N)

(i) Moreover, if p0

1 and

v (N) ≡

∞

(N)

(N)

spj (1 − pj ) → ∞ (N → ∞),

j =0

then lim

N→∞

v (N) S1 (0)

holds. (N) (ii) If {p0 } is bounded, then (N) lim w N→∞

S1 (0)

∞ dη 1 (N) (1 + pj (η − 1))s = √ N+1 2πiη 2π j =0

dη 1 1 =√ 2πiηN+1 ∞ (1 − p (N) (η − 1))s 2π j =0 j

holds, where w

(N)

≡

∞ j =0

(N)

(N)

spj (1 + pj ).

378

H. Tamura, K.R. Ito

√

∞ Proof. (i) Set η = exp(ix/ v (N) ). Then the integral is written as −∞ hN (x) dx/2π , where √

hN (x) = χ[−π √v (N ) ,π √v (N ) ] (x)e−iNx/

v (N )

∞

(N)

1 + pj (eix/

√ v (N )

s − 1) .

j =0

By Lemma A.1(i), we have |hN (x)|

∞

(N )

e−2spj

(N )

(1−pj

)x 2 /π 2 v (N )

= e−2x

2 /π 2

∈ L1 (R).

j =0

√ If N is so large that |x/ v (N) | π/3, we also get ∞ √ Nx (N ) (N) +s log 1 + pj (eix/ v − 1) hN (x) = χ[−π √v (N ) ,π √v (N ) ] (x) exp − i √ v (N) j =0 Nx = χ[−π √v (N ) ,π √v (N ) ] (x) exp − i √ v (N) (N) (N) (N) 2 ∞ pj x pj (1 − pj )x (N) +s − + δ i√ j 2v (N) v (N) j =0

x2 2 = χ[−π √v (N ) ,π √v (N ) ] (x) exp − + δ (N) −→ e−x /2 , N→∞ 2 where |δ

(N)

|=|

∞

(N) sδj |

j =0

∞ 4sp (N) (1 − p (N) )x 3 4|x 3 | j j = . √ √ √ 3 9 3v (N) 9 3 v (N ) j =0

The dominated convergence theorem yields ∞ ∞ dx −x 2 /2 dx 1 −→ e hN (x) =√ . 2π N→∞ −∞ 2π 2π −∞ √ (N) → ∞ as N → ∞. Set η = exp(ix/ w (N) ). Then the integral is (ii) Note that

∞w written as −∞ kN (x) dx/2π, where kN (x) =

√ (N ) e−iNx/ w √ √ √ χ[−π w(N ) ,π w(N ) ] (x) (N) ix/ w(N ) ∞ j =0 1 − pj (e

s .

− 1)

(N)

By LemmaA.1(ii) and the boundedness of {p0 }, we have, with some positive constant c, |kN (x)|

∞ j =0

exp −

x2

2 e−cx ∈ L1 (R) (N) (N) π 2 w (N) 1 + 4p (1 + p ) (N)

(N)

0

0

2spj (1 + pj )


379

and Nx kN (x) = χ[−π √w(N ) ,π √w(N ) ] (x) exp − i √ w (N) ∞ √ (N ) (N) −s log 1 − pj (e−ix/ w − 1) j =0

Nx = χ[−π √w(N ) ,π √w(N ) ] (x) exp − i √ w (N) (N) (N) (N) ∞ pj x pj (1 + pj )x 2 (N) −s −i√ + + δ j 2w (N) w (N) j =0 x2 2 = χ[−π √w(N ) ,π √w(N ) ] (x) exp − + δ (N) −→ e−x /2 , N→∞ 2 where |δ

(N)

|=|

∞ j =0

(N) sδj |

∞ p (N) (1 + p (N) )(1 + 2p (N) )|x 3 | (N) (1 + 2p0 ) 3 j j j |x |. √ √ 3 6 w (N) 6 w (N) j =0

The result is obtained by the dominated convergence theorem.

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17. Macchi, O.: The fermion process–a model of stochastic point process with repulsive points. In: Transactions of the Seventh Prague Conference on Information Theory, Statistical Decision Functions, Random Processes and of the Eighth European Meeting of Statisticians Vol. A, Prague 1974. Dordrecht: Reidel Publishing, 1977, pp. 391–398 18. Menikoff, R.: The Hamiltonian and generating functional for a nonrelativistic local current algebra. J. Math. Phys. 15, 1138–1152 (1974) 19. Messiah,A.M.L., Greenberg, O.W.: Symmetrization postulate and its experimental foundation. Phys. Rev. 136, B248–B267 (1964) 20. Ohnuki, Y., Kamefuchi, S.: Wavefunctions of identical particles. Ann. Phys. 51, 337–358 (1969) 21. Ohnuki, Y., Kamefuchi, S.: Quantum field theory and parastatistics. Berlin: Springer-Verlag 1982 22. Sagan, B.E.: The Symmetric Group. New York: Springer-Verlag, 1991 23. Shirai, T., Takahashi, Y.: Random point fields associated with certain Fredholm determinants I: fermion, Poisson and boson point processes. J. Funct. Anal. 205, 414–463 (2003) 24. Simon, B.: Trace ideals and their applications. London Mathematical Society Lecture Note Series, Vol. 35, Cambridge: Cambridge University Press, 1979 25. Simon, B.: Representations of Finite and Compact Groups. Providence, RI: A.M.S, 1996 26. Soshnikov, A.: Determinantal random point fields. Russ. Math. Surv. 55, 923–975 (2000) 27. Stolt, R.H., Taylor, J.R.: Classification of paraparticles. Phys. Rev. D1, 2226–2228 (1970) 28. Suranyi, P.: Thermodynamics of parabosonic and parafermionic systems of order two. Phys. Rev. Lett. 65, 2329–2330 (1990) 29. Vere-Jones, D.: A generalization of permanents and determinants. Linear Algebra Appl. 111, 119–124 (1988) Communicated by J.L. Lebowitz


Communications in


The Pearcey Process Craig A. Tracy1 , Harold Widom2 1 2

Department of Mathematics, University of California, Davis, CA 95616, USA Department of Mathematics, University of California, Santa Cruz, CA 95064, USA

Received: 1 February 2005 / Accepted: 19 September 2005 Published online: 10 February 2006 – © Springer-Verlag 2006

Abstract: The extended Airy kernel describes the space-time correlation functions for the Airy process, which is the limiting process for a polynuclear growth model. The Airy functions themselves are given by integrals in which the exponents have a cubic singularity, arising from the coalescence of two saddle points in an asymptotic analysis. Pearcey functions are given by integrals in which the exponents have a quartic singularity, arising from the coalescence of three saddle points. A corresponding Pearcey kernel appears in a random matrix model and a Brownian motion model for a fixed time. This paper derives an extended Pearcey kernel by scaling the Brownian motion model at several times, and a system of partial differential equations whose solution determines associated distribution functions. We expect there to be a limiting nonstationary process consisting of infinitely many paths, which we call the Pearcey process, whose space-time correlation functions are expressible in terms of this extended kernel. 1. Introduction Determinantal processes are at the center of some recent remarkable developments in probability theory. These processes describe the mathematical structure underpinning random matrix theory, shape fluctuations of random Young tableaux, and certain 1 + 1 dimensional random growth models. (See [2, 9, 10, 18, 20] for recent reviews.) Each such process has an associated kernel K(x, y), and certain distribution functions for the process are expressed in terms of determinants involving this kernel. (They can be ordinary determinants or operator determinants associated with the corresponding operator K on an L2 space.) Typically these models have a parameter n which might measure the size of the system and one is usually interested in the existence of limiting distributions as n → ∞. Limit laws then come down to proving that the operator Kn , where we now make the n dependence explicit, converges in trace class norm to a limiting operator K. In this context universality theorems become statements that certain canonical operators K are the limits for a wide variety of Kn . What canonical K can we expect to encounter?

382

C. A. Tracy, H. Widom

In various examples the kernel Kn (x, y) (or, in the case of matrix kernels, the matrix entries Kn,ij (x, y)) can be expressed as an integral

f (s, t) eφn (s,t; x,y) ds dt. C1

C2

To study the asymptotics of such integrals one turns to a saddle point analysis. Typically one finds a nontrivial limit law when there is a coalescence of saddle points. The simplest example is the coalescence of two saddle points. This leads to the fold singularity φ2 (z) = 13 z3 + λz in the theory of Thom and Arnold and a limiting kernel, the Airy kernel [19] or the more general matrix-valued extended Airy kernel [17, 11]. After the fold singularity comes the cusp singularity φ3 (z) = 41 z4 + λ2 z2 + λ1 z. The diffraction integrals, which are Airy functions in the case of a fold singularity, now become Pearcey functions [16]. What may be called the Pearcey kernel, since it is expressed in terms of Pearcey functions, arose in the work of Brézin and Hikami [6, 7] on the level spacing distribution for random Hermitian matrices in an external field. More precisely, let H be an n × n GUE matrix (with n even), suitably scaled, and H0 a fixed Hermitian matrix with eigenvalues ±a each of multiplicity n/2. Let n → ∞. If a is small the density of eigenvalues is supported in the limit on a single interval. If a is large then it is supported on two intervals. At the “closing of the gap” the limiting eigenvalue distribution is described by the Pearcey kernel. Bleher, Kuijlaars and Aptekarev [4, 5, 3] have shown that the same kernel arises in a Brownian motion model. Okounkov and Reshetikhin [15] have encountered the same kernel in a certain growth model. Our starting point is with the work of Aptekarev, Bleher and Kuijlaars [3]. With n even again, consider n nonintersecting Brownian paths starting at position 0 at time τ = 0, with half the paths conditioned to end at b > 0 at time τ = 1 and the other half conditioned to end at −b. At any fixed time this model is equivalent to the random matrix model of Brézin and Hikami since they are described by the same distribution function. If b is of the order n1/2 there is a critical time τc such that the limiting distribution of the Brownian paths as n → ∞ is supported by one interval for τ < τc and by two intervals when τ > τc . The limiting distribution at the critical time is described by the Pearcey kernel. It is in searching for the limiting joint distribution at several times that an extended Pearcey kernel arises.1 Consider times 0 < τ1 ≤ · · · ≤ τm < 1 and ask for the probability that for each k no path passes through a set Xk at time τk . We show that this probability is given by the operatordeterminant det(I − K χ ) with an m × m matrix kernel K(x, y), where χ (y) = diag χ Xk (y) . We then take b = n1/2 and scale all the times near the critical time by the substitutions τk → 1/2 + n−1/2 τk and scale the kernel by x → n−1/4 x, y → n−1/4 y. (Actually there are some awkward coefficients involving 21/4 which we need not write down exactly.) The resulting limiting kernel, the extended Pearcey kernel, has i, j entry −

1 4π 2

i∞

C

−i∞

e−s

4 /4+τ s 2 /2−ys+t 4 /4−τ t 2 /2+xt j i

ds dt s−t

(1.1)

1 It was in this context that the extended Airy kernel, and other extended kernels considered in [21], arose.

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plus a Gaussian when i < j . The t contour C consists of the rays from ±∞eiπ/4 to 0 and the rays from 0 to ±e−iπ/4 . For m = 1 and τ1 = 0 this reduces to the Pearcey kernel of Brézin and Hikami.2 These authors also asked the question whether modifications of their matrix model could lead to kernels involving higher-order singularities. They found that this was so, but that the eigenvalues of the deterministic matrix H0 had to be complex. Of course there are no such matrices, but the kernels describing the distribution of eigenvalues of H0 + H make perfectly good sense. So in a way this was a fictitious random matrix model. In Sect. V we shall show how to derive analogous extended kernels and limiting processes from fictitious Brownian motion models, in which the end-points of the paths are complex numbers. For the extended Airy kernel the authors in [21] derived a system of partial differential equations, with the end-points of the intervals of the Xk as independent variables, whose solution determines det(I − K χ ).3 Here it is assumed that each Xk is a finite union of intervals. For m = 1 and X1 = (ξ, ∞) these partial differential equations reduce to ordinary differential equations which in turn can be reduced to the familiar Painlevé II equation. In Sect. IV of this paper we find the analogous system of partial differential equations where now the underlying kernel is the extended Pearcey kernel.4 Unlike the case of the extended Airy kernel, here it is not until a computation at the very end that one sees that the equations close. It is fair to say that we do not really understand, from this point of view, why there should be such a system of equations. The observant reader will have noticed that so far there has been no mention of the Pearcey process, supposedly the subject of the paper. The reason is that the existence of an actual limiting process consisting of infinitely many paths, with correlation functions and spacing distributions described by the extended Pearcey kernel, is a subtle probabilistic question which we do not now address. That for each fixed time there is a limiting random point field follows from a theorem of Lenard [13, 14] (see also [18]), since that requires only a family of inequalities for the correlation functions which are preserved in the limit. But the construction of a process, a time-dependent random point field, is another matter. Of course we expect there to be one. 2. Extended Kernel for the Brownian Motion Model Suppose we have n nonintersecting Brownian paths. It follows from a theorem of Karlin and McGregor [12] that the probability density that at times τ0 , . . . , τm+1 their positions are in infinitesimal neighborhoods of x0i , . . . , xm+1,i is equal to m

det (P (xm,i , xm+1,j , σm )),

(2.1)

k=0

where σk = τk+1 − τk 2

In the external source random matrix model, an interpretation is also given for the coefficients of s 2 and t 2 in the exponential. It is not related to time as it is here. 3 Equations of a different kind in the case m = 2 were found by Adler and van Moerbeke [1]. 4 In the case m = 1 the kernel is integrable, i.e., it is a finite-rank kernel divided by x −y. (See footnote 7 for the exact formula.) A system of associated PDEs in this case was found in [7], in the spirit of [19], when X1 is an interval. This method does not work when m > 1, and our equations are completely different.

384


and P (x, y, σ ) = (π σ )−1/2 e−(x−y)

2 /σ

.

The indices i and j run from 0 to n − 1, and we take τ0 = 0, τm+1 = 1. We set all the x0i = ai and xm+1,j = bj , thus requiring our paths to start at ai and end at bj . (Later we will let all ai → 0.) By the method of [8] (modified and somewhat simplified in [21]) we shall derive an “extended kernel” K(x, y), which is a matrix kernel (Kk (x, y))m k,=1 such that for general functions f1 , . . . , fm the expected value of m n−1 (1 + fk (xki )) k=1 i=0

is equal to det (I − K f ), where f (y) = diag (fk (y)). In particular the probability that for each k no path passes through the set Xk at time τk is equal to det (I − K χ ), where χ (y) = diag (χ Xk (y)). (The same kernel gives the correlation functions [8]. In particular the probability density (2.1) is equal to (n!)−m det(Kk (xki , xj ))k,=1,...,m; i,j =0,...,n−1 .) The extended kernel K will be a difference H − E, where E is the strictly uppertriangular matrix with k, entry P (x, y, τ − τk ) when k < , and where Hk (x, y) is given at first by the rather abstract formula (2.5) below and then by the more concrete formula (2.6). Then we let all ai → 0 and find the integral representation (2.11) for the case when all the Brownian paths start at zero. This representation will enable us to take the scaling limit in the next section. We now present the derivation of K. Although in the cited references the determinants at either end (corresponding to k = 0 and m in (2.1)) were Vandermonde determinants, it is straightforward to apply the method to the present case. Therefore, rather than go through the full derivation again we just describe how one finds the extended kernel. For i, j = 0, . . . , n − 1 we find Pi (x), which are a linear combination of the P (x, ak , σ0 ) and Qj (y), which are a linear combination of the P (y, bk , σm ) such that

···

Pi (x1 )

m−1 k=1

P (xk , xk+1 , σk ) Qj (xm ) dx1 · · · dxm = δij .

The Pearcey Process

385

Because of the semi-group property of the P (x, y, τ ) this is the same as Pi (x) P (x, y, τm − τ1 ) Qj (y) dx dy = δij .

(2.2)

We next define for k < , Ek (xk , x ) =

···

−1

P (xr , xr+1 , σr ) dxk+1 · · · dx−1 = P (xk , x , τ − τk ).

r=k

Set Pi = P1i ,

Qmj = Qj ,

and for k > 1 define Pki (y) = E1k (y, u) Pi (u) du = P (y, u, τk − τ1 ) Pi (u) du,

(2.3)

and for k < m define Qkj (x) = Ekm (x, v) Qj (v) dv = P (x, v, τm − τk ) Qj (v) dv.

(2.4)

(These hold also for P1i and Qmj if we set Ekk (x, y) = δ(x − y).) The extended kernel is given by K = H − E where Hk (x, y) =

n−1

(2.5)

Qki (x) Pi (y),

i=0

and Ek (x, y) is as given above for k < and equal to zero otherwise. This is essentially the derivation in [8] applied to this special case. We now determine Hk (x, y) explicitly. Suppose Pi (x) = pik P (x, ak , σ0 ), k

Qj (y) =

qj P (y, b , σm ).

If we substitute these into (2.2) and use the fact that σ0 + τm − τ1 + σm = 1 we see that it becomes 1 2 pik qj e−(ak −b ) = δij . √ 2π k, Thus, if we define matrices P , Q and A by P = (pij ), Q = (qij ), A = (e−(ai −bj ) ), √ then we require P AQt = 2π I . 2

386


Next we compute Pri (y) =

P (y, u, τr − τ1 ) Pi (u) du =

pik P (y, ak , τr ),

k

Qsj (x) =

P (x, v, τm − τs ) Qj (v) dv =

qj P (x, b , 1 − τs ).

Hence Hrs (x, y) =

Qri (x) Psi (y) =

i

P (x, b , 1 − τr ) qi pik P (y, ak , τs ).

i,k,

√ The internal sum over i is equal to the , k entry of Qt P = 2π A−1 . So the above can be written (changing indices) √ Hk (x, y) = 2π P (x, bi , 1 − τk ) (A−1 )ij P (y, aj , τ ). i,j

If we set B = (e2 ai bj ) then this becomes Hk (x, y) =

√ 2 2 2π P (x, bi , 1 − τk ) ebi (B −1 )ij eaj P (y, aj , τ ).

(2.6)

i,j

This gives the extended kernel when the Brownian paths start at the aj . Now we are going to let all aj → 0. There is a matrix function D = D(a0 , . . . , an−1 ) such that for any smooth fuction f,    f (0)  f (a0 )   f (a1 )    f (0)   lim D(a0 , . . . , an−1 )  = .. .  ..   ai →0   . . (n−1) f (an−1 ) (0) f Here limai →0 is short for a certain sequence of limiting operations. Now B −1 applied to the column vector 2

(eaj P (y, aj , τ )) equals (DB)−1 applied to the vector 2

D (eaj P (y, aj , τ )). When we apply limai →0 this vector becomes j

(∂a ea

2 /2

P (y, a, τ )|a=0 ),

The Pearcey Process

387

while DB becomes the matrix ((2 bj )i ), which is invertible when all the bj are distinct. If we set V = (bj i ) then the limiting (DB)−1 is equal to V −1 diag (2−j ). Thus we have shown that when all ai = 0, √ 2 2 j P (x, bi , 1 − τk ) ebi (V −1 )ij 2−j ∂a ea P (y, a, τ )|a=0 . (2.7) Hk (x.y) = 2π i,j

The next step is to write down an integral representation for the last factor. We have

i∞ τ 2 y y2 1 a2 1−τ s +2s a− 1−τ 1−τ e P (y, a, τ ) = √ e ds. e πi 2(1 − τ ) −i∞ Hence 2−j ∂a ea P (y, a, τ )|a=0 = j

2

y2 1 e 1−τ πi 2(1 − τ )

√

i∞ −i∞

τ

s j e 1−τ

2sy s 2 − 1−τ

ds. (2.8)

Next we are to multiply this by (V −1 )ij and sum over j . The index j appears in (2.8) only in the factor s j in the integrand, so what we want to compute is n−1

(V −1 )ij s j .

(2.9)

j =0

Cramer’s rule in this case tells us that the above is equal to i /, where denotes the Vandermonde determinant of b = {b0 , . . . , bn−1 } and i the Vandermonde determinant of b with bi replaced by s. This is equal to s − br . bi − b r r=i

Observe that this is the same as the residue s − br 1 res , t = bi . t − br s − t r

(2.10)

This allows us to replace the sum over i in (2.7) by an integral over t. In fact, using (2.8) and the identification of (2.9) with (2.10) we see that the right side of (2.7) is equal to i∞ τ y2 2sy s − b ds dt 1 1 2 s 2 − 1−τ r − , P (x, t, 1 − τk ) et e 1−τ e 1−τ √ 2π π(1 − τ ) t − b s−t r C −i∞ r where the contour of integration C surrounds all the bi and lies to one side (it doesn’t matter which) of the s contour. Thus y2 x2 1 1 1−τ − 1−τk e √ 2π 2 (1 − τk ) (1 − τ ) i∞ τ τ 2sy s − b 2xt − k t 2 + 1−τ + 1−τ s 2 − 1−τ r ds dt k × e 1−τk . (2.11) t − b r s−t C −i∞ r

Hk (x, y) = −

388


In this representation the s contour (which passes to one side of the closed t contour) may be replaced by the imaginary axis and C by the contour consisting of the rays from ±∞eiπ/4 to 0 and the rays from 0 to ±∞e−iπ/4 . (We temporarily call this the “new” contour C.) To see this let CR denote the new contour C, but with R replacing ∞ and the ends joined by two vertical lines (where t 2 has positive real part). The t contour may be replaced by CR if the s contour passes to the left of it. To show that the s contour may be replaced by the imaginary axis it is enough to show that we get 0 when the s contour is the interval [−iR, iR] plus a curve from iR to −iR passing around to the left of CR . If we integrate first with respect to s we get a pole at s = t, and the resulting t integral is zero because the integrand is analytic inside CR . So we can replace the s contour by the imaginary axis. We then let R → ∞ to see that CR may be replaced by the new contour C. 3. The Extended Pearcey Kernel The case of interest here is where half the br equal b and half equal −b. It is convenient to replace n by 2n, so that the product in the integrand in (2.11) is equal to 2 n s − b2 . t 2 − b2 We take the case b = n1/2 . We know from [3] that the critical time (the time when the support of the limiting density changes from one interval to two) is 1/2, and the place (where the intervals separate) is x = 0. We make the replacements τk → 1/2 + n−1/2 τk and the scaling x → n−1/4 x, y → n−1/4 y. More exactly, we define Kn,ij (x, y) = n−1/4 Kij (n−1/4 x, n−1/4 y), with the new definition of the τk . (Notice the change of indices from k and to i and j . This is for later convenience.) The kernel En,ij (x, y) is exactly the same as Eij (x, y). As for Hn,ij (x, y), its integral representation is obtained from (2.11) by the scaling replacements and then by the substitutions s → n1/4 s, t → n1/4 t in the integral itself. The result is (we apologize for its length) 1 1 2x 2 2y 2 Hn,ij (x, y) = − 2 − 1/2 exp 1/2 π n − 2τj n − 2τi (1 − 2n−1/2 τi ) (1 − 2n−1/2 τj ) i∞ 1 + 2n−1/2 τi 2 4xt × exp −n1/2 t + 1 − 2n−1/2 τi 1 − 2n−1/2 τi C −i∞ −1/2 τ 4ys j 2 1/2 1 + 2n × exp n s − 1 − 2n−1/2 τj 1 − 2n−1/2 τj i n 1/2 2 ds dt 1 − s /n . (3.1) × 2 1/2 1 − t /n s−t

The Pearcey Process

389

We shall show that this has the limiting form i∞ ds dt 1 4 2 4 2 Pearcey Hij (x, y) = − 2 e−s /2+4 τj s −4ys+t /2−4 τi t +4xt , π C −i∞ s−t

(3.2)

where, as in (2.11) and (3.1), the s integration is along the imaginary axis and the contour C consists of the rays from ±∞eiπ/4 to 0 and the rays from 0 to ±∞e−iπ/4 . Precisely, we shall show that Pearcey

lim Hn,ij (x, y) = Hij

n→∞

(x, y)

(3.3)

uniformly for x and y in an arbitrary bounded set, and similarly for all partial derivatives.5 The factor outside the integral in (3.1) converges to −1/π 2 . The first step in proving the convergence of the integral in (3.1) to that in (3.2) is to establish pointwise convergence of the integrand. The first exponential factor in the integrand in (3.1) is exp − (n1/2 + 4τi + O(n−1/2 )) t 2 + (4 + O(n−1/2 )) xt , while the second exponential factor is exp (n1/2 + 4τj + O(n−1/2 )) s 2 − (4 + O(n−1/2 )) ys .

(3.4)

When s = o(n1/4 ), t = o(n1/4 ) the last factor in the integrand is equal to exp n1/2 t 2 + t 4 /2 + o(t 4 /n) − n1/2 s 2 − s 4 /2 + o(s 4 /n) . Thus the entire integrand (aside from the factor 1/(s − t)) is exp − (1 + o(1)) s 4 /2 + (4τj + O(n−1/2 )) s 2 − (4 + O(n−1/2 )) ys} × exp (1 + o(1)) t 4 /4 − (4τi + O(n−1/2 )) t 2 + (4 + O(n−1/2 )) xt .

(3.5)

In particular this establishes pointwise convergence of the integrands in (3.1) to that in (3.2). For the claimed uniform convergence of the integrals and their derivatives it is enough to show that they all converge pointwise and boundedly. To do this we change the t contour C by rotating its rays slightly toward the real line. (How much we rotate we say below. We can revert to the original contour after taking the limit.) This is so that on the modified contour, which we denote by C , we have t 2 > 0 as well as t 4 < 0. The function 1/(s − t) belongs to Lq for any q < 2 in the neighborhood of s = t = 0 on the contours of integration and to Lq for any q > 2 outside this neighborhood. To establish pointwise bounded convergence of the integral it therefore suffices to show that for any p ∈ (1, ∞) the rest of the integrand (which we know converges pointwise) has Lp norm which is uniformly bounded in x and y.6 The rest of the integrand is the 5 The constants in (3.2) are different from those in (1.1), a matter of no importance. In the next section we shall make the appropriate change so that they agree. 6 That this suffices follows from the fact, an exercise, that if {f } is a bounded sequence in Lp conn verging pointwise to f then (fn , g) → (f, g) for all g ∈ Lq , where p = q/(q − 1). We take fn to be the integrand in (3.1) except for the factor 1/(s − t) and g to be 1/(s − t), and apply this twice, with q < 2 in a neighborhood of s = t = 0 and with q > 2 outside the neighborhood.

390


product of a function of s and a function of t and we show that both of these functions have uniformly bounded Lp norms. 4 2 From (3.5) it follows that for some small ε > 0 the function of t is O(e t /2+O(|t| ) ) 1/4 uniformly in x if |t| ≤ ε n . Given this ε we choose C to consist of the rays of C rotated slightly toward the real axis so that if θ = arg t 2 when t ∈ C then cos θ = ε2 /2. 4 2 4 When t ∈ C and |t| ≤ ε n1/4 the function of t is O(e t /2+O(|t| ) ) = O(ecos 2θ |t| /2 ). p Since cos 2θ < 0 the L norm on this part of C is O(1). When t ∈ C and |t| ≥ ε n1/4 we have |1 − t 2 /n1/2 |2 = 1 + n−1 |t|4 − 2n−1/2 cos θ |t|2 . But n−1 |t|4 − 2n−1/2 cos θ |t|2 = n−1/2 |t|2 (n−1/2 |t|2 − ε 2 ) ≥ 0 when |t| ≥ εn1/4 . Thus |1−t 2 /n1/2 | ≥ 1 and the function of t is O(e− cos θ n |t| +O(|t| ) ) 1/2 2 = O(e− cos θ n |t| /2 ), and the Lp norm on this part of C is o(1). We have shown that on C the function of t has uniformly bounded Lp norm. For the Lp norm of the function of s we see from (3.4) that the integral of its p th power is at most a constant independent of y times ∞ 1/2 2 2 e−pn s +τ s (1 + n−1/2 s 2 )pn ds. 1/2

2

2

0

(We replaced s by is, used evenness, and took any τ > −4p τj .) The variable change s → n1/4 s replaces this by ∞ 2 2 1/2 2 n1/4 e−pn (s −log (1+s ))+n τ s ds. 0

The integral over (1, ∞) is exponentially small. Since s 2 − log (1 + s 2 ) ≥ s 4 /2 when s ≤ 1, what remains is at most

1

n1/4 0

e−pn s

4 /2+O(n1/2 s 2 )

ds =

n1/4

e−p s

4 /2+O(s 2 )

ds,

0

which is O(1). This completes the demonstration of the bounded pointwise convergence of Hn,ij Pearcey (x, y) to Hij . Taking any partial derivative just inserts in the integrand a polynomial in x, y, s and t, and the argument for the modified integral is virtually the same. This completes the proof of (3.3). 4. Differential Equations for the Pearcey Process We expect the extended Pearcey kernel to characterize the Pearcey process, a point process which can be thought of as infinitely many nonintersecting paths. Given sets Xk , the probability that for each k no path passes through the set Xk at time τk is equal to det (I − K χ ),

The Pearcey Process

391

where χ (y) = diag (χ Xk (y)). The following discussion follows closely that in [21]. We take the case where each Xk is a finite union of intervals with end-points ξkw , w = 1, 2, . . ., in increasing order. If we set R = (I − K χ )−1 K, with kernel R(x, y), then ∂kw det (I − K χ ) = (−1)w+1 Rkk (ξkw , ξkw ). (We use the notation ∂kw for ∂ξkw .) We shall find a system of PDEs in the variables ξkw with the right sides above among the unknown functions. In order to have the simplest coefficients later we make the further variable changes s → s/21/4 , t → t/21/4 and substitutions x → 21/4 x/4, y → 21/4 y/4, τk → 21/2 τk /8. The resulting rescaled kernels are (we omit the superscripts “Pearcey”) i∞ 1 ds dt 4 2 4 2 Hij (x, y) = − 2 e−s /4+τj s /2−ys+t /4−τi t /2+xt , 4π C −i∞ s−t which is (1.1), and 2

Eij (x, y) =

1 − (x−y) e 2(τj −τi ) . 2π (τj − τi )

Define the vector functions i∞ 1 1 t 4 /4−τk t 2 /2+xt −s 4 /4+τk s 2 /2−ys e dt , ψ(y) = e ds . ϕ(x) = 2π i C 2π i −i∞ We think of ϕ as a column m-vector and ψ as a row m-vector. Their components are Pearcey functions.7 The vector functions satisfy the differential equations ϕ (x) − τ ϕ (x) + x ϕ(x) = 0, ψ (y) − ψ (y) τ − y ψ(y) = 0, where τ = diag (τk ). 7

In case m = 1 the kernel has the explicit representation K(x, y) =

ϕ (x) ψ(y) − ϕ (x) ψ (y) + ϕ(x) ψ (y) − τ ϕ(x)ψ(y) x−y

in terms of the Pearcey functions. (Here we set τ = τ1 . This is the same as the i, i entry of the matrix kernel if we set τ = τi .) This was shown in [6]. Another derivation will be given in footnote 9.

392


Define the column vector function Q and row vector function P by Q = (I − K χ )−1 ϕ, P = ψ (I − χ K)−1 .

(4.1)

The unknowns in our equations will be six vector functions indexed by the end-points kw of the Xk and three matrix functions with the same indices. The vector functions are denoted by q, q , q , p, p , p . The first three are column vectors and the second three are row vectors. They are defined by = Qi (ξiu ), qiu = Qi (ξiu ), qiu = Qi (ξiu ), qiu

and analogously for p, p , p . The matrix function unknowns are r, rx , ry defined by riu,j v = Rij (ξiu , ξj v ), rx,iu,j v = Rxij (ξiu , ξj v ), ry,iu,j v = Ryij (ξiu , ξj v ). (Here Rxij , for example, means ∂x Rij (x, y).) The equations themselves will contain the matrix functions rxx , rxy , ryy defined analogously, but we shall see that the combinations of them that appear can be expressed in terms of the unknown functions. The equations will be stated in differential form. We use the notation ξ = diag (ξkw ), dξ = diag (dξkw ), s = diag ((−1)w+1 ). Recall that q is a column vector and p a row vector. Our equations are dr drx dry dq dp dq dp dq dp

= −r s dξ r + dξ rx + ry dξ, (4.2) = −rx s dξ r + dξ rxx + rxy dξ, (4.3) = −r s dξ ry + dξ rxy + ryy dξ, (4.4) = dξ q − r s dξ q, (4.5) = p dξ − p dξ s r, (4.6) = dξ q − rx s dξ q, (4.7) = p dξ − p dξ s ry , (4.8) = dξ (τ q − ξ q + r s q − ry s q + ryy s q − r τ s q) − rxx s dξ q, (4.9) = (p τ + p ξ + p s r − p s rx + p s rxx − p s τ r) dξ − p dξ s ryy .(4.10)

One remark about the matrix τ in Eqs. (4.9) and (4.10). Earlier τ was the m × m diagonal matrix with k diagonal entry τk . In the equations here it is the diagonal matrix with kw diagonal entry τk . The exact meaning of τ when it appears will be clear from the context. As in [21], what makes the equations possible is that the operator K has some nice commutators. In this case we also need a miracle at the end.8 8

That it seems a miracle to us shows that we do not really understand why the equations should close.

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Denote by ϕ ⊗ ψ the operator with matrix kernel (ϕi (x) ψj (y)), where ϕi and ψj are the components of ϕ and ψ, respectively. If we apply the operator ∂x + ∂y to the integral defining Hij (x, y) we obtain the commutator relation [D, H ] = −ϕ ⊗ ψ, where D = d/dx. Since also [D, E] = 0 we have [D, K] = −ϕ ⊗ ψ.

(4.11)

This is the first commutator. From it follows [D, K χ ] = −ϕ ⊗ ψ χ + K δ. Here we have used the following notation: δkw is the diagonal matrix operator whose k th diagonal entry equals multiplication by δ(y − ξkw ), and δ= (−1)w+1 δkw . kw

It appears above because D χ = δ. Set ρ = (I − K χ )−1 , R = ρ K = (I − K χ K)−1 − I. It follows from the last commutator upon left- and right-multiplication by ρ that [D, ρ] = −ρ ϕ ⊗ ψ χ ρ + R δ ρ. From the commutators of D with K and ρ we compute [D, R] = [D, ρ K] = ρ [D, K] + [D, ρ] K = −ρ ϕ ⊗ ψ (I + χ R) + R δ R. Notice that I + χ R = (I − χ K)−1 . If we recall (4.1) we see that we have shown [D, R] = −Q ⊗ P + R δ R.

(4.12)

To obtain our second commutator we observe that if we apply ∂t + ∂s to the integrand in the formula for Hij (x, y) we get zero for the resulting integral. If we apply it to (s − t)−1 we also get zero. Therefore we get zero if we apply it to the numerator, and this operation brings down the factor t 3 − τi t + x − s 3 + τj s − y. The same factor results if we apply to Hij (x, y) the operator ∂x3 + ∂y3 − (τi ∂x + τj ∂y ) + (x − y). We deduce [D 3 − τ D + M, H ] = 0.

394


One verifies that also [D 3 − τ D + M, E] = 0. Hence [D 3 − τ D + M, K] = 0.

(4.13)

This is the second commutator.9 From it we obtain [D 3 − τ D + M, K χ ] = K [D 3 − τ D + M, χ ] = K (DδD + D 2 δ + δD 2 − τ δ), and this gives [D 3 − τ D + M, ρ] = R (DδD + D 2 δ + δD 2 ) ρ − R τ δ ρ,

(4.14)

which in turn gives [D 3 −τ D + M, R] = [D 3 −τ D + M, ρ K] = R (DδD+D 2 δ+δD 2 ) R − R τ δ R.

(4.15)

Of our nine equations the first seven are universal — they do not depend on the particulars of the kernel K or vector functions ϕ or ψ. (The same was observed in [21].) What are not universal are Eqs. (4.9) and (4.10). For the equations to close we shall also have to show that the combinations of the entries of rxx , rxy and ryy which actually appear in the equations are all expressible in terms of the unknown functions. The reader can check that these are the diagonal entries of rxx + rxy and rxy + ryy (which also give the diagonal entries of rxx − ryy ) and the off-diagonal entries of rxx , rxy and ryy . What we do at the beginning of our derivation will be a repetition of what was done in [21]. First, we have ∂kw ρ = ρ (K ∂kw χ ) ρ = (−1)w R δkw ρ.

(4.16)

From this we obtain ∂kw R = (−1)w R δkw R, and so ∂kw riu,j v = (∂kw Rij )(ξiu , ξj v ) + Rxij (ξiu , ξj v ) δiu,kw + Ryij (ξiu , ξj v ) δj v,kw . = (−1)w riu,kw rkw,j v + rx,iu,j v δiu,kw + ry,iu,j v δj v,kw . Multipliying by dξkw and summing over kw give (4.2). Equations (4.3) and (4.4) are derived analogously. Next we derive (4.5) and (4.7). Using (4.16) applied to ϕ we obtain ∂kw qiu = Qi (ξiu ) δiu,kw + (−1)w (R δkw Q)i (ξiu ) = qiu δiu,kw + (−1)w riu,kw qkw .

Multiplying by dξkw and summing over kw give (4.5). If we multiply (4.16) on the left by D we obtain ∂kw ρx = (−1)w Rx δkw ρ and applying the result to ϕ we obtain (4.7) similarly. For (4.9) we begin as above, now applying D 2 to (4.16) on the left and ϕ on the right to obtain w ∂kw qiu = Q i (ξiu ) δiu,kw + (−1) rxx,iu,kw qkw .

(4.17)

9 From (4.11) we obtain also [D 3 , K] = −ϕ ⊗ ψ + ϕ ⊗ ψ − ϕ ⊗ ψ . Combining this with (4.11) itself and (4.13) for m = 1 with τ = τ1 we obtain [M, K] = ϕ ⊗ ψ − ϕ ⊗ ψ + ϕ ⊗ ψ − τ ϕ ⊗ ψ. This is equivalent to the formula stated in footnote 7.

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395

Now, though, we have to compute the first term on the right. To do it we apply (4.14) to ϕ and use the differential equation satisfied by ϕ to obtain Q (x) − τ Q (x) + x Q(x) = −Ry δ Q + Ryy δ Q + R δ Q − R τ δ Q. (4.18) This gives Q i (ξiu ) = τi qiu − ξiu qiu + (−ry s q + ryy s q + r s q − r τ s q)iu .

If we substitute this into (4.17), multiply by dξkw and sum over kw we obtain Eq. (4.9). This completes the derivation of the equations for the differentials of q, q and q . We could say that the derivation of the equations for the differentials of p, p and p is analogous, which is true. But here is a better way. Observe that the P for the operator K is the transpose of the Q for the transpose of K, and similarly with P and Q interchanged. It follows from this that for any equation involving Q there is another one for P obtained by replacing K by its transpose (and so interchanging ∂x and ∂y ) and taking transposes. The upshot is that Eqs. (4.6), (4.8) and (4.10) are consequences of (4.5), (4.7) and (4.9). The reason for the difference in signs in the appearance of ξ on the right sides of (4.9) and (4.10) is the difference in signs in the last terms in the differential equations for ϕ and ψ. Finally we have to show that the diagonal entries of rxx + rxy and rxy + ryy and the off-diagonal entries of rxx , rxy and ryy are all known, in the sense that they are expressible in terms of the unknown functions. This is really the heart of the matter. We use ≡ between expressions involving R, Q and P and their derivatives to indicate that the difference involves at most two derivatives of Q or P and at most one derivative of R. The reason is that if we take the appropriate entries evaluated at the appropriate points we obtain a known quantity, i.e., one expressible in terms of the unknown functions. If we multiply (4.12) on the left or right by D we obtain Rxx + Rxy = −Q ⊗ P + Rx δ R,

Rxy + Ryy = −Q ⊗ P + R δ Ry .

In particular Rxx +Rxy ≡ 0,

Rxy +Rxy ≡ 0,

so in fact all entries of rxx + rxy and rxy + ryy are known. From (4.12) we obtain consecutively [D 2 , R] = −Q ⊗ P + Q ⊗ P + DR δ R + R δ RD,

(4.19)

[D 3 , R] = −Q ⊗ P +Q ⊗ P −Q ⊗ P +D 2 R δ R+DR δRD+R δ RD 2 . (4.20) If we subtract (4.15) from (4.20) we find [τ D − M, R] = −Q ⊗ P + Q ⊗ P − Q ⊗ P +Rxx δ R − Rx δRy + R δ Ryy + Ry δRx − Ryy δ R − R δRxx + R τ δ R. We use Rxx − Ryy = Q ⊗ P − Q ⊗ P + Rx δ R − R δ Ry

396


and Ry = −Rx − Q ⊗ P + R δ R to see that this equals −Q ⊗ P + Q ⊗ P − Q ⊗ P + (Q ⊗ P − Q ⊗ P − R δ Ry ) δ R −R δ (Q ⊗ P − Q ⊗ P + Rx δ R − R δ Ry ) + Rx δ Q ⊗ P

(4.21)

−(Q ⊗ P − R δ R) δ Rx + R τ δ R. We first apply D acting on the left to this, and deduce that D [τ D, R] ≡ −Q ⊗ P + Rxx δ Q ⊗ P . If we use (4.18) and the fact that Ryy ≡ Rxx we see that this is ≡ 0. This means that τi Rxx,i,j + τj Rxy,i,j ≡ 0. Since Rxx,i,j + Rxy,i,j ≡ 0 we deduce that Rxx,i,j and Rxy,i,j are individually ≡ 0 when i = j . Therefore rxx,iu,j v and rxy,iu,j v are known then. But we still have to show that riu,iv is known when u = v, and for this we apply D 2 to (4.21) rather than D. We get this time D 2 [τ D − M, R] ≡ −Q ⊗ P + Q ⊗ (P − P δ R) − Rxx δ Ry δ R −Rxx δ (Q ⊗ P − Q ⊗ P + Rx δ R − R δ Ry )

(4.22)

+Rxxx δ Q ⊗ P + Rxx δ R δ Rx + Rxx τ δ R. We first compare the diagonal entries of D 2 [τ D, R] on the left with those of Rxx τ δ R on the right. The diagonal entries of the former are the same as those of τ (Rxxx + Rxxy ) ≡ τ Rxx δ R. (Notice that applying D 2 to (4.12) on the left gives Rxxx + Rxxy ≡ Rxx δ R.) The difference between this and Rxx τ δ R is [τ, Rxx ] δ R. Only the off-diagonal entries of Rxx occur here, so this is ≡ 0. If we remove these terms from (4.22) the left side becomes −D 2 [M, R] and the resulting right side we write, after using the fact Rx + Ry = −Q ⊗ P + R δ R twice, as −Q ⊗ P + Q ⊗ (P − P δ R) −Rxx δ (Q ⊗ P − Q ⊗ P − Q ⊗ P δ R + R δ Q ⊗ P ) + Rxxx δ Q ⊗ P . (4.23) Now we use (4.18) and the facts Ryy ≡ Rxx , Rxy ≡ −Rxx , Rxxx − Rxyy ≡ Rxx δ R (the last comes from applying D to (4.19) on the left) to obtain Q ≡ Rxx δ Q, Q ≡ Rxx δ Q + (Rxxx − Rxx δ R) δ Q. Substituting these expressions for Q and Q into (4.23) shows that it is ≡ 0. This was the miracle. We have shown that D 2 [M, R] ≡ 0, in other words (x − y) Rxx (x, y) ≡ 0. If we set x = ξiu , y = ξiv we deduce that Rxx (ξiu , ξiv ) = rxx,iu,iv is known when u = v.

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397

5. Higher-Order Singularities We begin with the fictitious Brownian motion model, in which the end-points of the paths are complex numbers. The model consists of 2Rn nonintersecting Brownian paths starting at zero, with n of them ending at each of the points ±n1/2 br (r = 1, . . . , R). The product in the integrand in (2.11) becomes n R 2 b − s 2 /n r

r=1

br2 − t 2 /n

,

(5.1)

and we use the same contours as before. We shall first make the substitutions τk → 1/2 + n−δ τk with δ to be determined. The first exponential in (2.11) becomes exp − (1 + 4 n−δ τk + O(n−2δ )) t 2 + (4 + O(n−δ )) xt + O(x 2 ) , (5.2) and the second exponential becomes exp (1 + 4 n−δ τ + O(n−2δ )) s 2 − (4 + O(n−δ )) ys + O(y 2 ) . If we set ar = 1/br2 the product (5.1) is the exponential of

n−1 2 4 n−R R+1 2R+2 2R+2 (t ar (t − s 4 ) + · · · + ar −s ) ar (t 2 − s 2 ) + 2 R+1 +O(n−R−1 (|t|2R+4 + |s|2R+4 )).

If R > 1 we choose the ar such that ar2 = · · · = arR = 0. ar = 1, The ar are the roots of the equation a R − a R−1 +

1 R−2 (−1)R − ··· + a = 0, 2! R!

(5.3)

from which it follows that

arR+1 =

(−1)R+1 . R!

In general the ar will be complex, and so the same will be true of the end-points of our Brownian paths. In the integrals defining Hij we make the substitutions t → nδ/2 t, s → nδ/2 s, where δ=

R , R+1

398


and in the kernel we make the scaling x → n−δ/2 x, y → n−δ/2 y. This gives us the kernel Hn,ij (x, y). As before Eij is unchanged, and the limiting form of Hn,ij (x, y) is now 1 − 2 π

C

i∞ −i∞

R s 2R+2 /(R+1)!+4 τ s 2 −4ys+(−1)R+1 t 2R+2 /(R+1)!−4 τ t 2 +4xt j i

e(−1)

ds dt . s−t

This is formal and it is not at first clear what the C contour should be, although one might guess that it consists of four rays, one in each quadrant, on which (−1)R+1 t 2R+2 is negative and real. We shall see that this is so, and that the rays are the most vertical R ones, those between 0 and ±∞e± 2R+2 πi . The orientation of the rays is as in the case R = 1. (The s integration should cause no new problems.) After the variable changes the product of the two functions of t in the integrand in (2.11) is of the form e−n t 2 −δ 2 e(−4 τk t +4xt+O(n (|t|+|t| )) . 2 δ−1 2 n (br − n t ) δ 2

(5.4)

The main part of this is the quotient. Upon the substitution t → n(δ−1)/2 t the quotient becomes e−nt 2 . (br − t 2 )n 2

(5.5)

Suppose we want to do a steepest descent analysis of the integral of this over a nearly vertical ray from 0 in the right half-plane. (This nearly vertical ray would be the part of the t contour in (2.11) in the first quadrant.) No pole ±br is purely imaginary, as is clear from the equation the ar satisfy. So there are R poles in the right half plane and R in the left. There are 2R + 2 steepest descent curves emanating from the origin, half starting out in the right half-plane. These remain there since, as one can show, the integrand is positive and increasing on the imaginary axis. We claim that there is at least one pole between any two of these curves. The reason is that otherwise the integrals over these two curves would be equal, and so have equal asymptotics. That means, after computing the asymptotics, that the integrals

∞eikπ/(R+1)

R+1 t 2R+2

e(−1)

dt

0

would be the same for two different integers k ∈ (−(R + 1)/2, (R + 1)/2). But the integrals are all different. Therefore there is a pole between any two of the curves. Let be the curve which R starts out most steeply, in the direction arg t = 2R+2 π . It follows from what we have just shown that there is no pole between and the positive imaginary axis. This is what we wanted to show. The curve we take for the t integral in (2.11) is n = n(1−δ)/2 . The original contour for the t-integral in the representation of Hn,ij can be deformed to this one. (We are speaking now, of course, of one quarter of the full contour.)

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We can now take care of the annoying part of the argument establishing the claimed asymptotics. The curve is asymptotic to the positive real axis at +∞. Therefore for A 2 δ 2 sufficiently large (5.5) is O(e−n|t| /2 ) when t ∈ , |t| > A. Hence (5.4) is O(e−n |t| /4 ) (1−δ)/2 2 when t ∈ n , |t| > n A. It follows that its L norm over this portion of n is exponentially small. When t ∈ , |t| > ε then (5.5) is O(e−nη ) for some η > 0, and it follows that (5.4) is O(e−nη/2 ) when t ∈ n , |t| > n(1−δ)/2 ε and also |t| < n(1−δ)/2 A. Therefore the norm of (5.4) over this portion of n is also exponentially small. So we need consider only the portion of n on which |t| < n(1−δ)/2 ε, and for this we get the limit ∞e2iπ R/(R+1) R+1 2R+2 −4 τ t 2 +x t k e(−1) t dt 0

with appropriate uniformity, in the usual way. Just as with the Pearcey kernel we can search for a system of PDEs associated with det (I − K χ ). Again we obtain two commutators, which when combined show that K is an integrable kernel when m = 1. In this case we define ϕ and ψ by 1 R+1 2R+2 /(R+1)!−4τ t 2 /2+4xt k ϕ(x) = e(−1) t dt , πi C i∞ 1 R 2R+2 /(R+1)!+4τ t 2 /2−4yt k ψ(y) = e(−1) s ds . πi −i∞ They satisfy the differential equations cR ϕ (2R+1) (x) − 2τ ϕ (x) + 4xϕ(x) = 0, cR ψ (2R+1) (y) − 2ψ (y)τ − 4yψ(y) = 0, where (−1)R+1 . 42R+1 R! The first commutator is [D, K] = −4 ϕ ⊗ ψ, which also gives cR = 2

[D n , K] = −4

n−1

(−1)k ϕ (n−k−1) ⊗ ψ (k) .

(5.6)

k=0

The second commutator is [cR D 2R+1 − 2 τ D + 4M, K] = 0. In case m = 1 (or for a general m and a diagonal entry of K), combining this with the commutator [D 2R+1 , K] obtained from (5.6) and the differential equations for ϕ and ψ we get an expression for [M, K] in terms of derivatives of ϕ and ψ up to order 2R. This gives the analogue of the expression for K(x, y) in footnote 7. For a system of PDEs in this case we would have many more unknowns, and the industrious reader could write them down. However there will remain the problem of showing that certain quantities involving 2R th derivatives of the resolvent kernel R (too many Rs!) evaluated at endpoints of the Xk are expressible in terms of the unknowns. For the case R = 1 a miracle took place. Even to determine what miracle has to take place for general R would be a nontrivial computational task. Acknowledgement. This work was supported by the National Science Foundation under grants DMS0304414 (first author) and DMS-0243982 (second author).

400


References 1. Adler, M., van Moerbeke, P.: A PDE for the joint distributions of the Airy process. http:// arxiv.org/list/math.PR/0302329, 2003; PDEs for the joint distribution of the Dyson, Airy and Sine processes. Ann. Prob. 33, 1326–1361 (2005) 2. Aldous, D., Diaconis, P.: Longest increasing subsequences: from patience sorting to the Baik-DeiftJohansson theorem. Bull. Amer. Math. Soc. 36, 413–432 (1999) 3. Aptekarev, A.I., Bleher, P.M., Kuijlaars, A.B.J.: Large n limit of Gaussian random matrices with external source, Part II. Commun. Math. Phys. 259, 367–389 (2005) 4. Bleher, P.M., Kuijlaars, A.B.J.: Random matrices with an external source and multiple orthogonal polynomials. Int. Math. Res. Not. 2004, No. 3, 109–129 5. Bleher, P.M., Kuijlaars, A.B.J.: Large n limit of Gaussian random matrices with external source, Part I. Commun. Math. Phys. 252, 43–76 (2004) 6. Brézin, E., Hikami, S.: Universal singularity at the closure of a gap in a random matrix theory. Phys. Rev. E 57, 4140–4149 (1998) 7. Brézin, E., Hikami, S.: Level spacing of random matrices in an external source. Phys. Rev. E 58, 7176–7185 (1998) 8. Eynard, B., Mehta, M.L.: Matrices coupled in a chain. I. Eigenvalue correlations. J. Phys. A: Math. Gen. 31, 4449–4456 (1998) 9. Ferrari, P.L., Prähofer, M., Spohn, H.: Stochastic growth in one dimension and Gaussian multi-matrix models. http://arxiv.org./list/math-ph/0310053, 2003 10. Johansson, K.: Toeplitz determinants, random growth and determinantal processes. Proc. Inter. Congress of Math., Vol. III (Beijing, 2002), Beijing: Higher Ed. Press, 2002, pp 53–62 11. Johansson, K.: Discrete polynuclear growth and determinantal processes. Commun. Math. Phys. 242, 277–329 (2003) 12. Karlin, S., McGregor, J.: Coincidence probabilities. Pacific J. Math. 9, 1141–1164 (1959) 13. Lenard, A.: States of classical statistical mechanical systems of infinitely many particles, I. Arch. Rat. Mech. Anal. 59, 219–239 (1975) 14. Lenard, A.: States of classical statistical mechanical systems of infinitely many particles, II. Arch. Rat. Mech. Anal. 59, 241–256 (1975) 15. Okounkov, A., Reshetikhin, N.: Private communication with authors, June, 2003; Poitiers lecture, June 2004; Random skew plane partitions and the Pearcey process, http://front.math.ucdavis.edu/ math.CO/0503508, 2005 16. Pearcey, T.: The structure of an electromagnetic field in the neighborhood of a cusp of a caustic. Philos. Mag. 37, 311–317 (1946) 17. Prähofer, M., Spohn, H.: Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108, 1071–1106 (2002) 18. Soshnikov, A.: Determinantal random point fields. Russ. Math. Surv 55, 923–975 (2000) 19. Tracy, C.A. Widom, H.: Level-spacing distributions and the Airy kernel. Commun. Math. Phys. 159, 151–174 (1994) 20. Tracy, C.A., Widom, H.: Distribution functions for largest eigenvalues and their applications. Proc. Inter. Congress of Math., Vol.I (Beijing, 2002), Beijing: Higher Ed. Press, 2002, pp 587–596 21. Tracy, C.A., Widom, H.: Differential equations for Dyson processes. Commun. Math. Phys. 252, 7–41 (2004) Communicated by H. Spohn


Communications in


Semiclassical Orthogonal Polynomials, Matrix Models and Isomonodromic Tau Functions M. Bertola1,2 , B. Eynard1,3 , J. Harnad1,2 1

Centre de recherches mathématiques, Université de Montréal C. P. 6128, succ. centre ville, Montréal, Québec, Canada H3C 3J7. E-mail: [email protected], [email protected] 2 Department of Mathematics and Statistics, Concordia University 7141 Sherbrooke W., Montréal, Québec, Canada H4B 1R6 3 Service de Physique Théorique, CEA/Saclay Orme des Merisiers, 91191 Gif-sur-Yvette Cedex, France. E-mail: [email protected] Received: 2 February 2005 / Accepted: 16 August 2005 Published online: 31 January 2006 – © Springer-Verlag 2006

Abstract: The differential systems satisfied by orthogonal polynomials with arbitrary semiclassical measures supported on contours in the complex plane are derived, as well as the compatible systems of deformation equations obtained from varying such measures. These are shown to preserve the generalized monodromy of the associated rank-2 rational covariant derivative operators. The corresponding matrix models, consisting of unitarily diagonalizable matrices with spectra supported on these contours are analyzed, and it is shown that all coefficients of the associated spectral curves are given by logarithmic derivatives of the partition function or, more generally, the gap probabilities. The associated isomonodromic tau functions are shown to coincide, within an explicitly computed factor, with these partition functions. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Generalized Orthogonal Polynomials and Partition Functions . . 2.1 Orthogonality measures and integration contours . . . . . 2.1.1 Definition of the boundary-free contours. . . . . . 2.1.2 Definition of the hard-edge contours. . . . . . . . 2.2 Recursion relations, derivatives and deformations equations 2.2.1 Existence of orthogonal polynomials and relation to random matrices. . . . . . . . . . . . . . . . . . 2.2.2 Wave vector equations. . . . . . . . . . . . . . . . 2.2.3 Wave vector of the second kind. . . . . . . . . . . 3. Folding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 n-windows and Christoffel–Darboux formula . . . . . . .

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Research supported in part by the Natural Sciences and Engineering Research Council of Canada, the Fonds FCAR du Québec and EC ITH Network HPRN-CT-1999-000161.

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M. Bertola, B. Eynard, J. Harnad

3.2

Folded version of the deformation equations for changes in the potential . . . . . . . . . . . . . . . . . . . . . . . 3.3 Folding of the endpoint deformations . . . . . . . . . . . . 3.4 Folded version of the recursion relations and ∂x relations 4. Spectral Curve and Spectral Invariants . . . . . . . . . . . . . . 4.1 Virasoro generators and the spectral curve . . . . . . . . . 4.2 Spectral residue formulæ . . . . . . . . . . . . . . . . . . 5. Isomonodromic Tau Function . . . . . . . . . . . . . . . . . . . 5.1 Isomonodromic deformations and residue formula . . . . . 5.2 Traceless gauge . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . .

. . . . . . . . .

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416 418 419 422 422 426 429 429 430

1. Introduction The partition function for Hermitian random matrix models with measures that are exponentials of a polynomial potential was shown in [4] to be equal, within a multiplicative factor independent of the deformation parameters, to the Jimbo-Miwa–Ueno isomonodromic tau function [11] for the rank 2 linear differential system satisfied by the corresponding set of orthogonal polynomials. The results of [4] were in fact more general, in that polynomials orthogonal with respect to complex measures supported along certain contours in the complex plane were considered. These may be viewed as corresponding to unitarily diagonalizable matrix models in which the spectrum is constrained to lie on these contours. The purpose of the present work is to extend these considerations to the more general setting of complex measures whose logarithmic derivatives are arbitrary rational functions, the associated semiclassical orthogonal polynomials and generalized matrix models. By also including contours with endpoints, the latter viewed as further deformation parameters, the gap probability densities are included as special cases of partition functions. To place the results in context, we first briefly recall the main points of [4], restricting to the more standard case of Hermitian matrices and real measures. Consider orthogonal −1 polynomials πn (x) ∈ L2 (R, e− V (x) dx) supported on the real line, with the measure defined by exponentiating a real polynomial potential V (x) =

d tJ J x . J

(1-1)

J =1

(Here we assume V (x) is of even degree and with positive leading coefficient, although these restrictions are unnecessary in the more general setting of [4].) The small parameter is usually taken as O(N −1 ) when considering the limit N → ∞. Any two consecutive polynomials satisfy a first order system of ODE’s, d πn−1 (x) π (x) = Dn (x) n−1 , (1-2) πn (x) πn (x) dx where Dn (x) is a 2 × 2 matrix with polynomial coefficients of degree at most d − 1 = deg(V (x)). The infinitesimal deformations corresponding to changes in the coefficients {tJ } result in a sequence of Frobenius compatible, overdetermined systems of PDE’s, ∂ πn−1 (x) π (x) = Tn,J (x) n−1 J = 1, . . . , d, (1-3) πn (x) πn (x) ∂tJ

Orthogonal Polynomials, Matrix Models and Isomonodromic Tau Functions

403

where the matrices Tn,J (x) are polynomials in x of degree J which satisfy the compatibility conditions ∂ d − Dn (x) = 0 . − Tn,J (x), (1-4) ∂tJ dx It follows that the generalized monodromy data of the sequence of rational covariant d derivative operators dx − Dn (x) are invariant under these deformations, and independent of the integer n. This is a particular case of the general problem of rational isomonodromic deformation systems [11]. An important rôle is played in this theory by the isomonodromic tau function τnI M associated with any solution of an isomonodromic deformation system. This function on the space of deformation parameters is obtained by integrating a closed differential whose coefficients are given by residues involving the fundamental solutions of the system. The main results of [4] were the following. First, the coefficients of the associated spectral curve, given by the characteristic equation of the matrix Dn (x), can be obtained by applying certain first order differential operators with respect to the deformation parameters (Virasoro generators) to ln(Zn (V )), where the partition function Zn (V ) of the associated n × n matrix model is 1 Zn (V ) := dM exp − TrV (M) . (1-5) Hn Second, this partition function is equal to the isomonodromic tau function up to a multiplicative factor that does not depend on the deformation parameters τnI M = Zn (V ) Fn .

(1-6)

The present work generalizes these results to the case of measures whose logarithmic derivatives are arbitrary rational functions, including those supported on curve segments in which the endpoints may play the rôle of further deformation parameters. The latter are of importance in the calculation of gap probabilities in matrix models [17] since these may, in this way, be put on the same footing as partition functions using measures supported on such segments [6]. A Frobenius compatible system of first order differential and deformation equations satisfied by the corresponding orthogonal polynomials is derived (Propositions 3.2, 3.4) and the coefficients of the associated spectral curve are again shown to be obtained by applying suitable Virasoro generators to ln(ZN (V )) (Theorems 4.1–4.2). A formula that generalizes (1-6) is also derived (Theorem 5.1): τnI M = Zn (V )Fn (V ) ,

(1-7)

where the factor Fn (V ) is an explicitly computed function of the deformation parameters determining V , which can in fact be eliminated by making a suitable scalar gauge transformation. The results of Theorems 4.1, 4.2 give a precise meaning, for finite n, to formulæ that are usually derived in the asymptotic limit n → ∞ (n ∼ O(1)) through saddle point computations, relating the free energy to the asymptotic spectral curve. It is well known [9] that the free energy in the large n limit is given by solving a minimization problem (in the Hermitian matrix model) F0 := − lim 2 ln Zn = min V (x)ρ(x)dx − ρ(x)ρ(x ) ln |x − x | , n→∞

ρ(x)≥0

(1-8)

404


giving the equilibrium density ρeq for the eigenvalue distribution. If, for instance, the potential is a real polynomial bounded from below, it is known [8] that the support of the equilibrium density is a union of finite segments I ⊂ R. The density ρeq is obtained from the variational equation ρeq (x)dx = V (x) (1-9) 2P x − x (for x ∈ I ), and is related to the resolvent by

ρeq (x) 1 = dx , z∈C\I . ω(z) := lim Tr n→∞ M −z x−z I

(1-10)

In terms of this, the spectral density may be recovered as the jump-discontinuity of ω(z) across I , and all its moments are given by zJ 1 xJ J lim TrM = dx ρeq (x) = − res ω(z)dz . (1-11) ∂tJ F0 = z=∞ J J n→∞ J I The function y = −ω(x) satisfies an algebraic relation given by y 2 = yV (x) + R(x) ,

(1-12)

where R(x) is a polynomial of degree less than V (x) that is uniquely determined by the consistency of (1-9) and (1-12). The point to be stressed here is that this asymptotic spectral curve should be compared with the spectral curve of Theorem 4.1, given by the characteristic equation (4-28), which also contains all the relevant information about the finite n case. In the n → ∞ limit, logarithmic derivatives of the partition function are expressed in (1-11) as residues of the meromorphic differentials zk ydz on the curve. The same formulæ are shown in Theorem 4.2 to hold as exact relations for the finite n case if we replace the “asymptotic” spectral curve by the spectral curve given by the characteristic equation of the matrix Dn (x). The paper is organized as follows. In Sect. 2 the problem is defined in terms of polynomials orthogonal with respect to an arbitrary semiclassical measure supported on complex contours, and the corresponding generalized matrix model partition functions. In Sect. 2.2 the recursion relations, differential systems and deformation equations which these satisfy are expressed in terms of the semi-infinite “wave vector” formed from the orthogonal polynomials. In Sect. 3 the notion of “folding” is introduced and used (Propositions 3.2–3.4) to express the preceding equations as an infinite sequence of compatible overdetermined 2 × 2 systems of linear differential equations and recursion relations satisfied by pairs of consecutive orthogonal polynomials. In Sect. 4 the results of folding are used to express the spectral curve in terms of logarithmic derivatives of the partition function and it is shown that the n → ∞ relation between the free energy and the spectral curve is also valid as an exact result for finite n. In Sect. 5 the definition of the isomonodromic tau function [11] is recalled and it is computed by relating it to the spectral invariants of the rational matrix generalizing Dn (x) in (1-2). These invariants are shown to give the logarithmic derivatives of the tau functions in terms of residues of meromorphic differentials on the spectral curves through formulæ that are nearly identical to those for the partition function, This leads to the main result, Theorem 5.1, which gives the explicit relation between Zn and τnI M .


405

2. Generalized Orthogonal Polynomials and Partition Functions 2.1. Orthogonality measures and integration contours. Given a measure on the real line, the associated orthogonal polynomials are those that diagonalize the quadratic form associated to the corresponding (complex) moment functional; i.e., the linear form over the space of polynomials obtained by integration with respect to the measure L : C[x] → C , p(x) → L(p(x)) = R p(x)dµ(x) .

(2-1)

A natural generalization consists of including moment functionals that are expressed by integration along more general contours in the complex x plan, with respect to a complex measure defined by locally analytic weight functions that may have isolated essential singular points and complex power-like branch points. We thus consider linear forms on polynomials given by integrals of the form L(p(x)) = p(x)µ(x)dx, κ

1

µ(x) = e− V (x) , K V (x) := Tr (x) ,

(2-2)

r=0

where T0 (x) := t0,0 + Tr (x) :=

dr J =1

d0 t0,J J x , J

J =1

tr,J − tr,0 ln(x − cr ) J (x − cr )J

−∂x ln µ(x) = V (x) =

d 0 −1

t0,J x J −1 −

J =1

K d r +1 r=1 J =1

tr,J −1 , (x − cr )J

(2-3)

and the symbol κ denotes integration over linear combinations of contours on which the integrals are convergent, as explained below. This class of linear functionals is sometimes referred to as semiclassical moment functionals [3, 14, 15]. We consider the corresponding monic generalized orthogonal polynomials pn (x), which satisfy pn (x)pm (x)µ(x)dx = hn δnm . , hn ∈ C \ {0} . (2-4) κ

If all the contours are contained in the real axis and the weight is real and positive, we reduce to the usual notion of semiclassical orthogonal polynomials. The small parameter introduced in (2-2) is not of essential importance here; it is only retained in the formulæ below to recall that, when taking the large n limit, it plays the rôle of small parameter for which n remains finite as n → ∞. (j ) To describe the contours of integration, we first define sectors Sr , r = 0, . . . K, k = 1, . . . dr around the points cr for which dr > 1 (c0 := ∞) in such a way that (V (x)) −→ +∞ . x → cr , x∈

(j ) Sr

(2-5)

406


The number of sectors for each pole in V is equal to the degree of that pole; that is, d0 for the pole at infinity and dr for the pole at cr . Explicitly

2kπ −arg(t0,d0 )− π2 2kπ −arg(t0,d0 )+ π2 (0) Sk := x ∈ C; , (2-6) < arg(x) < d0 d0 k = 0 . . . d0 − 1 ;

2kπ +arg(tr,dr )− π2 2kπ +arg(tr,dr )+ π2 (r) , (2-7) < arg(x − cr ) < Sk := x ∈ C; dr dr k = 0, . . . , dr − 1, r = 1, . . . , K . These sectors are defined precisely so that approaching within them any of the essential singularities of µ(x) (i.e., a cr such that dr > 0), the function µ(x) tends to zero faster than any power of the local parameter. 2.1.1. Definition of the boundary-free contours. The definition of the contours follows [16] (see Fig. 1). 1. For any cr for which there is no essential singularity in the measure (i.e., dr = 0), there are two subcases: (a) For the cr ’s that are branch points or poles in µ (i.e., tr,0 ∈ / N), we take a loop (0) starting at infinity in some fixed sector Sk encircling the singularity and going back to infinity in the same sector. (Note that if cr is just a pole; i.e., tr,0 ∈ −N+ , the contour could equivalently be taken as a circle around cr .) (b) For the cr ’s that are regular points (tr,0 ∈ N ), we take a line joining cr to infinity, (0) approaching ∞ in a sector Sk as before. 2. For any cr for which there is an essential singularity in µ (i.e., dr > 0) we define dr (r) (r) contours starting from cr in the sector Sk and returning to it in the next sector Sk+1 . Also, if tr,0 ∈ / Z, we join the singularity cr to ∞ by a path approaching ∞ within one (0) fixed sector Sk . (0) 3. For c0 := ∞, we take d0 − 1 contours starting at c0 in the sector Sk and returning (0) at c0 in the next sector Sk+1 . Note that, with these definitions, the integrals involved are convergent and we can perform integration by parts. Moreover, any contour in the complex plane for which the integral of µ(x)p(x)dx is convergent for all polynomials p(x) is equivalent to a linear combination of the contours defined above, no two of which are, in this sense, equivalent. 2.1.2. Definition of the hard-edge contours. We also include some additional contours in the complex plane {mj }j =1,...,L , starting at some points aj , j = 1 . . . L and going to (0) ∞ within one of the sectors Sk . These could be viewed as corresponding to additional points in 1(b) for which both dr = 0 and tr,0 = 0, but we prefer to deal with them separately since integration by parts on these contours does give a contribution. In total there are S := d0 + K r=1 (dr + 1) boundary-free contours σ , = 1, . . . , S and L hard-edge contours mh , h = 1, . . . , L. The moment functional is an arbitrary linear combination of integrals taken along these contours L S := κj + κL+j . (2-8) κ

j =1

mj

j =1

σj


Sk

407

L

*c2

a3

a2

a1

c1

c3

Fig. 1. The types of contours considered in the x Riemann sphere P1 . Here we have c1 with d1 = 3 and c2 with d2 = 0, t2,0 Z (logarithmic singularity in the potential), c3 with d3 = 0, t3,0 ∈ N and the degree of the potential at infinity c0 = ∞ is d0 = 5. The essential singularity in µ at c1 is of the form exp (x − c1 )−3 and there is also a cut extending from c1 to ∞ if t1,0 ∈ Z. The point c2 is a branch point of µ(x) since t2,0 ∈ Z, and the cut extends to infinity “inside” the contour (as shown here). If it were a pole (t2,0 ∈ −N+ ), the contour would be replaced by a circle around it. The point c3 is a regular point with t3,0 ∈ N× , and the contour extending from it to infinity is no different from the ones starting at the regular points a1 , a2 , a3 . The latter are the “hard-edge” segments joining the points a1 , a2 and a3 to ∞ (0) within one of the sectors Sk

Note that, by taking appropriate linear combinations of the contours, we could alternatively have had contours consisting of finite segments joining the points aj . 2.2. Recursion relations, derivatives and deformations equations. 2.2.1. Existence of orthogonal polynomials and relation to random matrices. Recall [7] that orthogonal polynomials satisfying (2-4) exist provided all the Hankel determinants formed from the moments are nonzero: i+j

n (κ) := det x µ(x)dx = 0 , ∀n ∈ N. (2-9) κ

0≤i,j ≤n−1

Since the n (κ)’s are homogeneous polynomials in the coefficients κj , the zero locus excluded by (2-9) is of zero measure (in the space of κj ’s), and hence “generically” the conditions (2-9) are fulfilled. The development to follow will in fact only involve orthogonal polynomials up to some arbitrarily large fixed degree, say N , and hence the conditions n (κ) = 0, n ≤ N − 1 determine a Zariski closed set in {κj }, (and a closed set of measure zero in the space of coefficients of V ).

408


The orthogonal polynomials considered here are related to models of unitarily diagonalizable random matrices M ∈ gl(n, C) with spectra supported on the contours defined above. More specifically we have the partition function 1 Zn := Cn dMe− TrV (M) spec(M)∈ κ 1 n = dx1 · · · dxn (x)2 e− j =1 V (xj ) (2-10) κ

κ

= n! n (κ, V ) = n!

n−1

hj ,

(2-11)

(2-12)

j =0

where CN :=

1 U (n) dU

is the inverse of the U (n) group volume, and

(x) := (xi − xj )

(2-13)

i<j

is the usual Vandermonde determinant. The notation spec(M) ∈ κ in the first integral just means that M is unitarily diagonalizable M = U D U† ,

D := diag(x1 , . . . xn ) ,

U ∈ U (n) ,

(2-14)

and the eigenvalues {x1 , . . . xn } of M are constrained to lie on the contours entering in κ . In particular, as in the standard case, the orthogonal polynomials may be shown to be equal to the expectation values of the characteristic polynomials in such models pn (x) = det(xI − M) n 1 n 1 = dx1 · · · dxn (x − xi ) (x)2 e− j =1 V (xj ) , Zn κ κ

(2-15)

i=1

and all correlation functions between the eigenvalues may be expressed as determinants in terms of the standard Christoffel-Darboux kernel formed from them Kn (x, y) :=

n−1 1 1 pj (x)pj (y)e− 2 (V (x)+V (y)) . hj

(2-16)

j =0

More precisely, this is valid when there are no “hard-edge” contours present. Inclusion of the latter however allows one to interpret these determinants as certain conditional correlators, known as “Janossy distribution” correlators [6], giving the probability densities for a certain number of eigenvalues to lie at given locations within the complementary part of the support, while the remaining ones lie within it. The partition function Zn in this case can be reinterpreted as the corresponding gap probability [6, 17].


409

2.2.2. Wave vector equations. We now define the normalized orthogonal polynomials 1 πn (x) := √ pn (x) hn

(2-17)

and what will be referred to as the “orthonormal quasi-polynomials” 1

ψn (x) := πn (x)e− 2 V (x) , satisfying

(2-18)

κ

ψn (x)ψm (x)dx = δmn .

(2-19)

We now form the semi-infinite “wave vectors” (x) := [π0 (x), π1 (x), . . . , πn (x), . . .]t .

(2-20)

As in the theory of ordinary orthogonal polynomials, we have x(x) = Q(x) ,

(2-21)

where Q is a symmetric tridiagonal semi-infinite matrix with components Qij = γj δi,j −1 + βi δij + γi δi,j +1 ,

i, j ∈ N ,

(2-22)

defining a three term recursion relation of the form xπj (x) = γj +1 πj +1 (x) + βj πj (x) + γj πj −1 (x) .

(2-23)

Now introduce semi-infinite matrices P , Ai , Cr , Tr,J such that ∂x (x) = ∂ai (x) = ∂cr (x) = ∂tr,J (x) =

P (x), Ai (x) , Cr (x) , Tr,J (x) ,

i = 1, . . . , L, r = 1, . . . , K, r = 0, . . . , K, J = 0, . . . , dr .

Their matrix elements are determined simply by integration Xnm = (∂πn (x)) πm (x)µ(x)dx , κ

(2-24) (2-25) (2-26) (2-27)

(2-28)

where ∂ denotes any of the derivatives ∂x , ∂ai , ∂cr , ∂tr,J above for which X becomes the corresponding matrices P , Ai , Cr or Tr,J on the RHS of (2-24) – (2-27). Remark 2.1. Such wave vectors and associated deformation equations have been studied in many previous works relating orthogonal polynomials, matrix models and integrable systems (see, e.g. [2, 18]). However, considerations of the deformation theory have mainly been within the formal setting, with the potential V (x) replaced by some initial value, V0 (x), plus a perturbation consisting of an infinite power series with arbitrary coefficients, without regard to domains of convergence. Results obtained in this formal setting cannot be directly applied to the study of isomonodromic deformations, where the local analytic structure in the neighborhood of a number of isolated singular points is of primary interest.

410


For any such semi-infinite square matrix X, let X0 , X+ , X− denote the diagonal, upper and lower triangular parts, respectively, and let 1 (2-29) X0 + X − . 2 Proposition 2.1. The matrices P , Ai , Cr and Tr,J are all lower semi-triangular (with P strictly lower triangular), and are given by X−0 :=

P = V (Q)−0 −

L

Ai = V (Q)− −

i=1 t

L

(Ai )− ,

Ai = κi ((ai ) (ai ))−0 µ(ai ), Cr =

dr J =0

(2-30)

i=1

−1 tr,J (Q − cr )−J , −0

r = 1, . . . , K,

1 1, 2 1 = QJ−0 , J = 1, . . . , d0 , J 1 = (Q − cr )−J −0 , r = 1, . . . , K, J = 1, . . . , dr , J = − ln(Q − cr )−0 , r = 1, . . . , K ,

(2-31) (2-32)

T0,0 =

(2-33)

T0,J

(2-34)

Tr,J Tr,0 where (Q − cr

(2-35) (2-36)

)−J

and ln(Q − cr ) are defined by the formulæ πn (z)πm (z) −J µ(z)dz, (Q − cr )nm := (z − cr )J κ ln(Q − cr )nm := ln(z − cr )πn (x)πm (z)µ(z)dz . κ

(2-37) (2-38)

The diagonal matrix elements for each of the above is given by the formula Xjj = − ∂(ln hj ) , (2-39) 2 where ∂ = ∂x , ∂ai , ∂cr and ∂tr,J , respectively, for X = P , Ai , Cr and Tr,J . In particular, they vanish for P , which is strictly lower triangular, and hence

V (Q)jj =

L

κi ψj (ai )2 .

(2-40)

Proof. We make use of the orthogonality relations (x)t (x)µ(x)dx = 1.

(2-41)

i=1

κ

Equations (2-32) – (2-36) are obtained as follows. Consider a deformation ∂ with respect to any of the above cr ’s or tr,J ’s and denote by X the corresponding matrix; then pn (x) 1 1 = − (∂ ln(hn )) πn (x) + √ ∂pn (x) ∂πn (x) = ∂ √ 2 hn hn 1 = − (∂ ln(hn )) πn (x) + lower degree polynomials , 2

(2-42) (2-43)


411

since the polynomials πn are monic. It follows that the deformation matrix X is lower semi-triangular. On the other hand, differentiating Eq. (2-41) gives 0= t ∂µ(x)dx ∂ t + ∂t µ(x)dx + κ κ t t = X+X + ∂µ(x)dx. (2-44) κ

Applying the operators for each case to µ as defined in (2-2) and using Eq. (2-21) then gives the result. Now consider the deformations of the endpoints ai of the “hard-edge” contours. Differentiating (2-41) gives 0 = ∂ai t µ(x)dx κ t = −κi (ai ) (ai )µ(ai ) + (∂ai )t + ∂ai t µ(x)dx κ

= −κi (ai )t (ai )µ(ai ) + Ai + Ati , where

(2-45)

Ai :=

It follows that

κ

∂ai t µ(x)dx.

(2-46)

Ai = κi (t )−0 x=a µ(ai ) ,

(2-47)

i

proving Eq. (2-31), and also that (Ai )nn = − ∂ai ln(hn ) = κi ψn2 (ai ) . 2 2

(2-48)

To determine the matrix P , note that it is strictly lower triangular and µ(ai )t

∂κ

= −

L

Ai + Ati

i=1

t = + t + t ∂x ln µ(x) µ(x)dx κ t + t − V (x) t µ(x)dx = κ

= P + P t − V (Q).

(2-49)

This implies that P = V (Q)−0 −

L i=1

Ai = V (Q)− −

L

(Ai )− .

(2-50)

i=1

This last equality follows from (2-40), which, in turn, follows from integration by parts in the definition of V (Q)nn . It may be seen as a consequence of the invariance of the

412


partition function under an infinitesimal change in the integration variables xj → xj + in (2-11); i.e., translational invariance. From (2-39) and (2-11) follows a relation between the diagonal elements of the deformation matrices and the logarithmic derivatives of the partition function that will be very important in what follows. Define the truncated trace of a semi-infinite matrix X to be Tr n X :=

n−1

(2-51)

Xjj .

j =0

Corollary 2.1. For ∂ = ∂aj , ∂cr , and ∂tr,J , ∂ ln Zn = −2Tr n X ,

(2-52)

with X = Aj , Cr and Tr,J , respectively. For the cases ∂cr and ∂tr,J , ∂ ln Zn = Tr n ∂V (Q) ,

(2-53)

while for the ∂ai ’s we have

L

∂ai ln Zn = −V (Q)nn .

(2-54)

i=1

Proof. The first of these relations follows from (2-39) and (2-11) directly, the second from the explicit expressions for the deformation matrices (2-32)–(2-36) and of the potential V (x), and the third is a restatement of the (2-40) (translational invariance). Corollary 2.2. The compatibility conditions [G, H] = 0 ,

(2-55)

are satisfied, where G, H are any of the following operators: ∂ai − Ai , ∂tr,J − Tr,J , ∂cr − Cr , ∂x − P , x − Q

(2-56)

and r = 0, . . . K, J = 0, . . . dr . Proof. This follows immediately from the fact that the orthogonal polynomials entering in Eqs. (2-24)–(2-27) are linearly independent. Remark 2.2. Note that [∂x − P , x − Q] = 0

(2-57)

is just the string equation, while the other compatibility conditions involving x − Q imply the Lax equations: ∂ai Q = [Ai , Q],

∂tr,J Q = [Tr,J , Q],

∂cr Q = [Cr , Q] ,

(2-58)

showing that the spectrum of the matrix Q is invariant under these deformations.


413

2.2.3. Wave vector of the second kind. We now consider solutions of the second kind,

1

φn (x) := e V (x)

1

e− V (z) π(z) dz , x−z κ

(2-59)

which may be combined to form the components of a wave vector of the second kind (x) := [φ0 (x), φ1 (x), . . . , φn (x), . . .]t .

(2-60)

Denote by ∇Q V (x) :=

V (x) − V (Q) x−Q

(2-61)

the semi-infinite square matrix with elements 1 V (x) − V (Q) V (x) − V (z) = dze− V (z) πn (z)πm (z) , x−Q x−z κ nm

(2-62)

and define U (x) to be the semi-infinite column vector (with only its zeroth component nonvanishing) given by 1 (U (x))n := h0 e V (x) δn,0 . (2-63) The following lemma gives the effect of multiplication of (x) by x and of application of ∂x to it. It may be deduced immediately from Eqs. (2-21) and (2-24), applied inside the integral, together with integration by parts. Lemma 2.1. x(x) = Q(x) + U (x),

(2-64)

∂x (x) = P (x) + ∇Q V (x)U (x) +

L i=1

κi

e

1 (V (x)−V (ai ))

x − ai

(ai ). (2-65)

The next proposition, which is similarly verified, gives the effects of the above deformations on the wave vector of the second kind. Proposition 2.2. ∂ai (x) = Ai (x) 1

e (V (x)−V (ai )) −κi (ai ), x − ai ∂cr (x) = Cr (x) +

dr

tr,J

J =0

i = 1, . . . , L,

(Q−cr )−J −1 −(x −cr )−J −1 U (x), Q−x

(2-66)

r = 1, . . . , K, (2-67)

∂t0,J (x) = T0,J (x) +

QJ − x J U (x), Q−x

J = 1, . . . , d0 ,

(2-68)

414


∂tr,J (x) = Tr,J (x) +

(Q−cr )−J −(x −cr )−J U (x) , Q−x

J = 1, . . . , dr ,

r = 1, . . . , K, (2-69)

∂tr,0 (x) = Tr,0 (x) ln(Q − cr ) − ln(x − cr ) + U (x), Q−x

r = 1, . . . , K.

(2-70)

The content of Eqs. (2-26)–(2-27) and (2-67)–(2-70) may be summarized uniformly as follows. Let v(x) be any function that is analytic at each point of the contours except, possibly, the points cr , and for which the following integrals are convergent:

1 v(z)πn (z)πm (z)e− V (z) dz = v(z)ψn (z)ψm (z)dz, κ κ v(x) − v(z) v(x) − v(Q) := := ψn (z)ψm (z)dz. (2-71) x−Q x−z κ nm

v(Q)nm := (∇Q v(x))nm

Define the deformation matrix under the infinitesimal variation of the potential V (x) → V (x) + v(x) to be Xv := v(Q)−0 .

(2-72)

Then the two infinite systems δv (x) := Xv (x), δv (x) := Xv (x) + ∇Q v(x)U (x),

(2-73) (2-74)

describe the infinitesimal deformation of the orthogonal polynomials and the secondkind solutions under such infinitesimal variations of the potential. Equivalently, define the 2 × ∞ matrix (x) := [(x), (x)] .

(2-75)

In terms of (x), all the recursion, differential and deformation equations (2-21), (2-24)–(2-27) and (2-64)–(2-70) may be expressed simultaneously as x = Q + (0, U ) , ∂x = P + 0, ∇Q V U +

K i=1

e

1 (V (x)−V (ai ))

x − ai

δv = Xv + (0, ∇Q vU ), 1 e (V (x)−V (ai )) ∂ai = Ai − 0, κi (ai ) , x − ai

(ai ) ,

(2-76) (2-77) (2-78) (2-79)

where v signifies any of the infinitesimal deformations of the potential ∂ci , ∂tr,J V (i = 1, . . . L, r = 0, . . . K, J = 1, . . . , dr ).


415

3. Folding 3.1. n-windows and Christoffel–Darboux formula. Let in be the ∞ × 2 matrix that represents the injection of the 2-dimensional subspace spanned by the (n − 1, n) basis elements into the (semi-)infinite space corresponding to the components of or . Its matrix elements are thus: (in )j k = δk,1 δj,n−1 + δk,2 δj,n ,

j = 0, 1, 2, . . . ,

k = 1, 2.

(3-1)

Let inT denote its transpose, which is the corresponding projection operator. The nth 2×2 block (or “window”) of is then given by:

n (x) :=

inT

πn−1 (x) = πn (x)

φn−1 (x) φn (x)

(3-2)

.

By “folding” the infinite recursion and differential-deformation equations (2-21), (2-24)–(2-27), (2-64)–(2-70), (2-76)–(2-79), we mean the corresponding sequence of recursion relations, ODEs and PDEs satisfied by the n (x)’s. To derive these, a form of the Christoffel–Darboux identity for orthogonal polynomials will repeatedly be used. Let 0n denote the semi-infinite square matrix whose only nonvanishing entries are 1’s on the diagonal in positions 0 to n (i.e., the projection onto the first n + 1 components) (0n )ij

:=

if

δij 0

0 ≤ i, j ≤ n otherwise .

(3-3)

Let σ :=

0 1

−1 0

(3-4)

be the standard 2 × 2 symplectic matrix, and let n := in σ inT

(3-5)

denote its projection onto the 2 × 2 subspace in position (n − 1, n). Proposition 3.1. The following extended Christoffel-Darboux formulæ are satisfied: (x − x ) T (x)0n−1 (x ) =

(3-6)

γn Tn (x)σ n (x )

+

0 1

e V (x)

1

−e V (x ) 1 1 1 e (V (x)+V (x )) κ e− 2 V (z) x−z −

= γn T (x) n (x ) 1 0 −e V (x ) + 1 1 1 1 e V (x) e (V (x)+V (x )) κ e− 2 V (z) x−z −

1 x −z

(3-7)

dz

1 x −z

dz

.

(3-8)

416


Equivalently, in components, (x − x )

n−1

πj (x)πj (x ) = γn πn (x)πn−1 (x ) − πn−1 (x)πn (x )

(3-9)

j =0

(x − x )

n−1

1 πj (x)φj (x ) = γn πn (x)φn−1 (x )−πn−1 (x)φn (x ) −e V (x )

(3-10)

j =0

(x − x )

n−1

φj (x)φj (x ) = γn φn (x)φn−1 (x ) − φn−1 (x)φn (x )

j =0

+e

1 (V (x)+V (x ))

κ

e

− 21 V (z)

1 1 − dz, x − z x − z (3-11)

and evaluating at x = x gives det n (x) = πn−1 (x)φn (x) − φn−1 (x)πn (x) = −

1 1 V (x) e . γn

(3-12)

Proof. Equation (3-9) is the standard Christoffel-Darboux relation for orthogonal polynomials. The extended system may be derived as follows. Multiplying the expression (x)0n−1 (x ) by (x − x ), and applying the relation (2-76) with respect to both x and x gives (x − x ) T (x)0n−1 (x ) 0 = T (x) Q0n−1 − πn−1 Q (x ) 1 0 −e V (x ) + 1 1 1 1 e V (x) e (V (x)+V (x )) κ e− 2 V (z) x−z −

(3-13) 1 x −z

dz

.

(3-14)

The result (3-8) is obtained by substituting the following identity, which holds for any tridiagonal symmetric matrix of the form Q Q0n−1 − 0n−1 Q = γn in σ inT .

(3-15)

3.2. Folded version of the deformation equations for changes in the potential. Under infinitesimal changes of the parameters in the potential V and the end-points of the “hardedge” contours, the wave vectors (x) and (x) and the combined system (x) undergo changes determined by Eqs. (2-25)–(2-27), (2-66)–(2-70)) and (2-79)–(2-78). Besides the deformations induced by infinitesimal changes of the endpoints {aj }, all these deformations have the same general form, depending only on the function v(x) = δV (x) that gives the infinitesimal deformation of the potential. We deal with them all on the same footing in the following proposition, which expresses the explicit form they take on the window n (x).


417

Proposition 3.2. The deformation equations (2-73), (2-74) (2-78) are equivalent to the infinite sequence of 2 × 2 equations, δv n (x) = Vn (x) n (x),

(3-16)

where the folded matrix of the deformation is defined by v(x)− 21 v(Q)n−1,n−1 0 Vn (x) = 1 0 2 v(Q)nn ∇Q v(x)n−1,n−1 ∇Q v(x)n−1,n +γn σ. ∇Q v(x)n,n−1 ∇Q v(x)nn

(3-17)

For the deformations in (2-24)–(2-27) and (2-66)–(2-70), this gives the following equations corresponding to changes in the potential, ∂cr n (x) = Cr;n (x) n (x), ∂tr,J n (x) = Tr,J ;n (x) n (x),

(3-18) (3-19)

where the sequence of 2 × 2 matrices Cr;n and Tr,J ;n (x) are rational in x, with poles at the points {cr }, obtained by making the following substitutions in Eq. (3-17): Cr : v(x) →

dr

tr,J (x − cr )−J −1 ,

J =0

1 (x − cr )−J , J 1 : v(x) → x J , J : v(x) → − ln(x − cr ) .

Tr,J : v(x) → T0,J Tr,0

(3-20)

Proof. Using the definition (2-71) of ∇Q v(x) and the extended Christoffel-Darboux relation (3-8), we have 1 T (y) Tn (x) γn ∇Q v(x) n = γn (3-21) dy e− V (y) (v(y) − v(x)) (y) y−x κ 1 = dye− V (y) (v(y) − v(x)))(y)T (y)0n−1 (x) (3-22) κ 1 V (x) − 1 V (y) v(y) − v(x) + 0, −e (y) (3-23) dye y−x κ = v(Q)0n−1 (x)−v(x)0n−1 (x)− 0, ∇Q v(x)U (x) . (3-24) Applying the projector inT and noting that inT v(Q)0n−1 = inT v(Q)−0 +

1 v(Q)n−1,n−1 0 2

0 n, −v(Q)n,n

(3-25)

418


we obtain δv n (x) = inT δv (x) = inT Xv (x) + (0, ∇Q vU (x) = =

=

(3-26)

inT v(Q)−0 (x) + (0, inT ∇Q v(x)U (x)) inT v(Q)0n−1 (x) 1 − 2 v(Q)n−1,n−1 0 n + 1 0 2 v(Q)n,n +(0, inT ∇Q v(x)U (x)) γn inT ∇Q v(x)inT σ n (x) v(x) − 21 v(Q)n−1,n−1 0 + n (x), 1 0 2 v(Q)n,n

(3-27)

(3-28)

(3-29)

proving the relation (3-17). Remark 3.1. Note that formula (3-17) for the deformation of the measure in Proposition 3.2, as well as those below, (3-36), (3-37), which are obtained through folding of the ∂x operator, could also be derived for arbitrary locally analytic potentials V (x), provided all the integrals involved are convergent [13]. However applicability of the subsequent isomonodromic analysis would be lost if the derivatives were not rational, since the resulting deformation equations would then have essential singularities. 3.3. Folding of the endpoint deformations. The case (2-25) and (2-66) involving deformations of the locations of the “hard edge” end–points must be considered separately. Proposition 3.3. The following gives a closed system for the nth window of Eqs. (2-25) and (2-66): ∂ai n (x) = Ai,n (x) n (x), where

(3-30)

2 (a ) κi γn ψn−1 (ai )ψn (ai ) −ψn−1 i ψn2 (ai ) −ψn−1 (ai )ψn (ai ) ai − x 2 (a ) κi −ψn−1 0 i + 0 ψn2 (ai ) 2 2 κi γn ψn−1 (ai ) ψn−1 (ai )ψn (ai ) = σ ψn2 (ai ) ai − x ψn−1 (ai )ψn (ai ) 2 (a ) κi −ψn−1 0 i + . 0 ψn2 (ai ) 2

Ai,n :=

(3-31)

Proof. This is very similar to the proof of Prop. 3.2. Using the definition (2-31) of the matrices Ai and the extended Christoffel–Darboux relation (3-8) we have ∂ai n (x) = inT ∂ai (x) T = in κi (ai ) T (ai )

1

e (V (x)−V (ai )) (x) − 0, κi (ai ) −0 x − ai


κi (ai ) (ai ) 2 (a ) κi −ψn−1 0 i + n (x) 0 ψn2 (ai ) 2 γn κi = inT (ai ) T (ai ) n (x) ai − x 2 (a ) κi −ψn−1 0 i + n (x) . 0 ψn2 (ai ) 2

=

inT

T

0n−1 (x) −

419

1

e (V (x)−V (ai )) 0, κi (ai ) x − ai

(3-32)

Recalling the definition (3-5) of n and computing the matrix product yields the result in the statement. Q.E.D. 3.4. Folded version of the recursion relations and ∂x relations. We now consider the recursion relations (2-21), (2-64) and (2-76) and the action of the ∂x operator in (2-24), (2-65) and (2-77) which, in their folded form are given by the following. Proposition 3.4. The folded forms of the relations (2-76) and (2-77) are n+1 (x) = Rn (x) n (x) , ∂x n (x) = Dn (x) n (x),

n ≥ 1,

(3-33) (3-34)

,

(3-35)

where Rn :=

0

1

γn − γn+1

x−βn γn+1

and Dn (x) = Dn(0) (x) +

L 2 (a ) κi γn ψn−1 (ai )ψn (ai ) −ψn−1 i ψn2 (ai ) −ψn−1 (ai )ψn (ai ) x − ai

(3-36)

i=1

with Dn(0) (x)

V (x) = 0 V (x) = 0

∇Q V (x) ∇Q V (x) n−1,n 0 −γn 0 n−1,n−1 + 0 γn 0 ∇ V (x) ∇ Q V (x) Q n,n−1 (x)nn V (x) − ∇ V ∇ 0 n−1,n−1 Q Q n−1,n . (3-37) + γn 0 − ∇Q V (x) n,n−1 ∇Q V (x) nn

Remark 3.2. Note that formula (3-36) implies that Tr(Dn (x)) = V (x) .

(3-38)

Proof. The folded form (3-33) of the recursion relations follows directly from Eqs. (2-21) and (2-64)), xπn (x) = γn+1 πn+1 (x) + βn πn (x) + γn πn−1 (x),

xφn (x) = γn+1 φn+1 (x) + βn φn (x) + γn φn−1 (x) + δn0 h0 e

(3-39) 1 V (x)

.

(3-40)

420


To prove (3-34), note that the folding relations (3-16) may be expressed inT δv n = Vn n

(3-41)

for any infinitesimal variation v = δV in the potential. Choosing δ := −

K

∂cr +

d 0 −1

J t0,J +1 ∂t0,J + t0,1 ∂t0,0 ,

(3-42)

J =1

r=1

we have V (x) ≡ δV . Using (2-30) and (2-65), we have ∂x n =

P + 0 , ∇Q V (x)U −

inT

= inT (V (Q)−0 −

L

1

e (V (x)−V (ai )) κi (ai ) x − ai

Ai )

i=1

+ 0 , ∇Q (δV )(x)U −

L i=1

= inT ((δV )(Q)−0 −

L

1

1

e (V (x)−V (ai )) κi (ai ) x − ai

Ai )

i=1

+ 0 , ∇Q (δV )(x)U − = inT δ −

L i=1

(3-43)

L

L i=1

e (V (x)−V (ai )) κi (ai ) x − ai

∂ai ,

(3-44)

i=1

where we have used the deformation equations (2-25)–(2-27), (2-66)–(2-70). Applying the folded relations (3-16), (3-17) and (3-30), this gives K K ˆ ¯ ∂x n = Vn − Ai,n − Ai,n n , (3-45) i=1

where

and

i=1

2 (a ) κi γn ψn−1 (ai )ψn (ai ) −ψn−1 i ˆ Ai,n := , ψn2 (ai ) −ψn−1 (ai )ψn (ai ) ai − x 2 (a ) κi −ψn−1 0 i A¯ i,n := 2 (a ) , 0 ψ 2 n i

V (x)− 21 V (Q)n−1,n−1 0 Vn (x) = 1 0 2 V (Q)nn

(3-46) (3-47)


+

∇Q V (x)n−1,n−1 ∇Q V (x)n−1,n ∇Q V (x)n,n−1 ∇Q V (x)nn

0 −γn γn 0

421

.

(3-48)

It follows from (2-40) that the diagonal V (Q) terms in Vn (x) are canceled by the sum in the last term of (3-45), giving the stated result (3-36), (3-37). Combining the differential, recursion and deformations relations (3-34), (3-33), (318), (3-19) and (3-30), the fact that the invertible matrices n are simultaneous fundamental systems for all these equation implies the compatibility of the cross-derivatives; i.e., the corresponding set of zero-curvature equations. Corollary 3.1. For n ≥ 0 the set of PDE’s and recursion equations, ∂x n (x) = Dn (x) n (x), ∂cr n (x) = Cr;n (x) n (x), n+1 (x) = Rn (x) n (x),

∂ai n (x) = Ai;n (x) n (x), ∂tr,J n (x) = Tr,J ;n (x) n (x), (3-49)

are simultaneously satisfied by the invertible matrices n (x), and hence the zero-curvature equations, [∂x − Dn , ∂ai − Ai;n ] = 0, [∂x − Dn , ∂ai − Cr;n ] = 0, [∂x − Dn , ∂ai − Tr,J ;n ] = 0 , [∂ai = Ai;n , ∂ai − Cr;n ] = 0, [∂ai − Ai;n , ∂ai − Tr,J ;n ] = 0 , [∂ai − Cr;n , ∂ai − Tr,J ;n ] = 0, ∂ai Rn = Ai;n+1 Rn − Rn Ai;n , ∂cr Rn = Cr;n+1 Rn − Rn Cr;n , ∂tr,J Rn = Tr,J ;n+1 Rn − Rn Tr,J ;n (3-50) are satisfied. Remark 3.3. (The Riemann–Hilbert method.) The Riemann-Hilbert method for characterizing orthogonal polynomials [10, 8] provides an alternative approach to deriving the results of this section. This is a well-established approach, and will not be developed in detail here, except to indicate briefly how it could be applied to deducing the differential and deformation equations satisfied by the fundamental systems. The fundamental system n (x) has, by construction, a jump-discontinuity across any of the contours defining the orthogonality measure. Denoting the limiting values when approaching any of these contours from the left or the right by n,± we have the jump discontinuity conditions 1 2iπκj (3-51) n,+ (x) = n,− (x) , x ∈ γj . 0 1 Furthermore, the local asymptotic behavior near the singularities at ∞ are specified as in Sect. 5.2. To be more precise the function n (x) has local formal asymptotic form, within any of the Stokes sectors,  1   1 + O(x − c ) e− 2 Tr (x)σ3 x → cr C  r r     1 e− 2 V (x) n (x) ∼ x → aj (3-52) Aj 1+O(x −aj ) e−κj ln(x−aj )σ+ .     1 1   e− 2 T0 (x)σ3 +(n− r t0,r )σ3 ln(x) x → ∞  C0 1 + O x

422


It follows from the usual argument based on Liouville’s theorem that any two fundamental solutions (with the same Stokes matrices, given in fact by the same matrices (3-51) ) satisfying the above Riemann–Hilbert conditions are equal, within a constant scalar multiple. Also, from Liouville’s theorem it follows that the first column of (x) consists of polynomials (the orthogonal polynomials). Using similar arguments one can show that the following matrix is rational with poles, of the correct order, at the singular points cr , aj , ∞: 1 Dn (x) := ∂x n (x) n (x)−1 + V (x)1. 2

(3-53)

By comparing the local singular behavior of the logarithmic (matrix) derivatives of any two solutions and applying Liouville’s theorem, it follows again that these globally combine to define rational matrix functions which give the deformation matrices with respect to the various parameters at the poles. 4. Spectral Curve and Spectral Invariants The aim of this section is to express the spectral curve of the ODE (3-34) (i.e., the characteristic equation of Dn (x)) in terms of the partition function. In fact we will prove an exact finite n analog (Thm. 4.2) of the formulæ that are obtained by variational methods in the n → ∞ limit [9]. We start by expressing the explicit relation between the partition function and the spectral curve of the isomonodromic system.

4.1. Virasoro generators and the spectral curve. To express the result in a compact form, introduce the following local Virasoro generators: (r)

V−J :=

d r −J

Mtr,M+J

M=1

∂ , ∂tr,M

J = 0, . . . , dr − 1 , r = 0, . . . , K

(4-1)

in terms of which we define the following differential operator with coefficients that are rational functions of x: D(x) : =

L i=1

−

d 0 −3 1 ∂ (0) − x J V−J −2 x − ai ∂ai J =0

K d r +1 r=1 J =2

1 1 ∂ (r) V − . 2−J (x − cr )J x − cr ∂cr K

(4-2)

r=1

Theorem 4.1. The characteristic polynomial of the matrix Dn (x) in the differential system (3-34) is given by

V (M) − V (x) det y − Dn (x) = y 2 − yV (x) + Tr M −x −

L i=1

2 ∂a ln(Zn ), x − ai i

(4-3)


= y 2 − yV (x) + Tr n −

L i=1

V (Q) − V (x) Q−x

423

2 ∂a ln(Zn ), x − ai i

= y 2 − yV (x) + n

d 0 −1

(4-4)

t0,J +1 x J −1 − 2 D(x) ln Zn ,

(4-5)

J =1

and the quadratic trace invariant is TrDn (x)2 = V (x)2 − 2n

d 0 −1

t0,J +1 x J −1 + 22 D(x) ln Zn .

(4-6)

J =1

Proof. The equivalence of (4-3) and (4-4) follows from the well–known relation Tr(f (M)) = Tr n (f (Q))

(4-7)

for any scalar function f (x) for which the Tr(f (M)) is a convergent integral. The equivalence of (4-5) and (4-6) follows from (3-38). To prove (4-4) we use the recursion relation (3-33) and the explicit expression of Dn , we obtain Dn+1 = Rn Dn Rn −1 + Rn Rn −1 , 0 0 0 − γ1n Rn Rn −1 = , Rn −1 Rn = . 1 0 0 γn+1 0 Therefore,

(4-8) (4-9)

Tr Dn+1 (x)2 = Tr Dn (x)2 + 2Tr Dn (x)Rn −1 Rn 2 +2 Tr Rn Rn−1 V (Q) − V (x) = Tr Dn (x)2 − 2 Q−x nn −22

L κi ψ 2 (ai ) n

x − ai i=1 V (Q) − V (x) = Tr Dn (x)2 − 2 Q−x nn +22

L i=1

1 ∂a ln(hn ), x − ai i

(4-10)

(4-11)

(4-12)

where we have used Eqs. (3-36), (3-37) in (4-11) and (2-48) in (4-12). These equations imply that Tr(Dn (x) ) = Tr(D1 (x) ) − 2 2

2

n−1 V (Q) − V (x) j =1

Q−x

jj

424


+ 22

n−1 L

1 ∂a ln(hj ). x − ai i

j =1 i=1

(4-13)

From the definition of D1 , we have D1 =

π0 φ0 π1 φ1

π0 φ0 π1 φ 1

−1 (4-14)

.

Using (3-12), this gives 1

det(D1 (x)) = 2 γ1 e− V (x) φ0 π1

− 1 V (z)

1 h1 − 1 V (x) 1 d e e e+ V (x) dz h0 h0 h1 dx κ x−z − 1 V (z) − 1 V (z) 2 1 e e = V (x) dz − dz 2 h0 κ x−z κ (x − z) − 1 V (z) 2 1 e = V (x) dz h0 κ x−z 1 − 1 V (z) ∂ − e dz ∂z x − z κ K 2 (a ) (V (x) − V (z))ψ02 (z) κ ψ i i 0 = , dz + 2 x−z x − ai κ

= 2

(4-15) (4-16)

(4-17)

(4-18) (4-19)

i=1

and hence Tr(D12 (x)) = −2 det(D1 (x)) + Tr(D1 (x))2 (V (x) − V (z))ψ02 (z) 2 = ((V (x)) − 2 dz x−z κ −22

K κi ψ 2 (ai ) 0

x − ai i=1 V (x) − V (Q) 2 = (V (x)) − 2 x−Q 00 +22

L i=1

1 ∂a ln(h0 ). x − ai i

(4-20)

(4-21)

(4-22)

Combining this with (4-13) gives Tr(Dn (x)2 ) = (V (x))2 −2

n−1 V (Q) − V (x) j =0

Q−x

+22 jj

L n−1 j =0 i=1

1 ∂a ln(hj ), x − ai i

(4-23) which, taking the expression (2-11) for the partition function into account, completes the proof of Eq. (4-4).


425

We now proceed to the proof of Eq. (4-5). By expanding the third term on the right of (4-4), we obtain d J −1 0 −1 V (x) − V (Q) = t0,J +1 x M QJ −M−1 x−Q J =1

M=0

K d r +1

+

tr,J −1

r=1 J =1

=

d 0 −1

J −1 M=0

t0,J +1 x J −1 1 +

J =1

tr,J −1

r=1 J =1

=

d 0 −1

J −1 M=0

t0,J +1 x

J −1

1+

J =1

x M t0,J +1 QJ −M−1

1 (Q − cr )−M−1 (x − cr )J −M

d 0 −3

x

r=1 M=1

1 (x − cr )M

dr

M

t0,J QJ −M−1

J =M+2

M=0

K d r +1

+

d −2 0 −1 J J =2 M=0

K d r +1

+

1 (Q − cr )−M−1 (x − cr )J −M

d r +1

tr,J −1 (Q − cr )M−J −1 . (4-24)

J =M

Now recall that for any deformation matrix X corresponding to an infinitesimal variation ∂ we have n−1 j =0

Xjj = − ∂ ln(Zn ). 2

(4-25)

Summing the diagonal terms of (4-24) up to n−1 and substituting into (4-4) we therefore obtain Tr(Dn (x)2 ) = (V (x))2 − 2n − 22 − 2

2

d 0 −2

t0,J +1 x J −1

J =1

xM

d0

(J − M − 1)t0,J +1

M=1 J =M+1 K d r +1 r=1 M=2

− 22

d 0 −1

K r=1

∂ ln(Zn ) ∂t0,J −M−1

d r +1 1 ∂ ln(Zn ) (J − M + 1)tr,J −1 (x − cr )M ∂tr,J −M+1 J =M

1 ∂ ln(Zn ) 1 ∂ ln(Zn ) + 22 x − cr ∂cr x − ai ∂ai L

i=1

Using Eq. (4-25) we finally get d 0 −1 t0,J +1 x J −1 det y − Dn (x) = y 2 − yV (x) + n J =1

(4-26)

426


+2 +

2

d 0 −3

xM

(J − M − 1)t0,J +1

∂ ln(Zn ) ∂t0,J −M−1

M=0

J =M+2

K d r +1

d r +1 1 ∂ ln(Zn ) (J − M + 1)tr,J −1 (x − cr )M ∂tr,J −M+1

r=1 M=2

+2

d 0 −1

K r=1

J =M

1 ∂ ln(Zn ) 1 ∂ ln(Zn ) − 2 , x − cr ∂cr x − ai ∂ai L

(4-27)

i=1

which completes the proof of Eq. (4-5).

4.2. Spectral residue formulæ. Theorem 4.1, which determines all the coefficients of the spectral curve as logarithmic derivatives of the partition function, may be expressed in another form, in which the individual deformation parameters, as well as the logarithmic derivatives with respect to them, may be directly expressed as spectral invariants. The characteristic equation of Dn (x) det (y(x)I − Dn (x)) = 0,

(4-28)

defines a hyperelliptic curve Cn as a 2–sheeted branched cover of the Riemann sphere, on which y is a meromorphic function. It follows from (3-36) and Theorem 4.1 that y, viewed as a double valued function of x, has the same pole structure and degree as Dn (x) at the points {c0 = ∞, cr , ai }, but that the points {ai } are branch points. Let Y± (x) denote the two values of y(x). Defining W (x) := Tr n

L V (x) − V (Q) 2 ∂aj ln Zn − , x−Q x − aj

(4-29)

j =1

it follows from the explicit expression (4-4) for the spectral curve that, near any of the poles c0 = ∞, c1 , . . . , cK , the two branches have the asymptotic form ! 1 1 Y± (x) = V (x) ± (V (x))2 − W 2 4 1 W2 1 O(x −2d0 −1 ) x→∞ ∼ V (x) ∓ W+ +. . . + . 2 O((x − cr )2dr +2 ) x → cr 0 V (x) (V (x) ) (4-30) Before proceeding we have to point out that here we have assumed that d0 ≥ 1, for otherwise ∞ is a branch-point of the spectral curve. Theorem 4.2. The following residue formulæ express the deformation parameters and the logarithmic derivatives of Zn as spectral invariants of the matrix Dn (x): Y+ (x) dx, J = 1 . . . d0 , xJ = − res (x −cr )J Y+ (x)dx, r = 1, . . . , K, J = 1, . . . , dr ,

t0,J = − res

(4-31)

tr,J

(4-32)

x=∞ x=cr


2 ∂t0,0 ln Zn = res Y− (x)dx = −n,

427

( for d0 ≥ 1),

x=∞

(4-33)

1 res Y− (x)x J dx, J = 1, . . . , d0 , (4-34) J x=∞ 1 1 2 ∂tr,J ln Zn = dx, r = 1, . . . , K, J = 1, . . . , dr , (4-35) res Y− (x) J x=cr (x −cr )J 2 ∂cr ln Zn = − res Y− (x)Tr (x)dx, r = 1, . . . , K, (4-36)

2 ∂t0,J ln Zn =

x=cr

1 ∂aj ln Zn = res Tr(Dn2 (x))dx . 2 x=aj 2

(4-37)

Note that identity (4-33) can be established only for d0 ≥ 1 for otherwise the residue would not make sense since ∞ becomes a branch-point. Proof. Considering the various deformations associated to the poles and end-points we have: At infinity: 1 x J /J V (x) − V (Q) 2 J res x Y− dx = res Tr n − ∂a ln Zn x=∞ V (x) J x=∞ x−Q x − aj j x J /J V (x) − V (Q) Tr = − Tr n QJ = 2 ∂t0,J ln Zn , n x=∞ V (x) x−Q J J = 1, . . . d0 − 2. (4-38)

= res

Note that this computation does not provide the derivatives with respect to the two highest coefficients t0,d0 and t0,d0 −1 , which will be computed below. Moreover we should remark that the last equality follows from the following interchange of order of integrals: n−1 1 x J /J V (x) − V (Q) x J /J V (x) − V (z) 2 res Tr = πj (z)e− V (z) dz n x=∞ V (x) x=∞ V (x) κ x−Q x−z res

j =0

=

n−1 j =0

=−

1 x J /J V (x) − V (z) 2 πj (z)e− V (z) dz (x) x=∞ V x − z κ

res

n−1 1 zJ 2 1 πj (z)e− V (z) dz = − Tr n QJ . (4-39) J κ J j =0

The exchange is justified by the usual arguments observing that the expression V (x)−V x−z has no singularities at coinciding points x = z (away from the singularities of V ). At the poles cr :

(z)

(x − cr )−J /J V (x) − V (Q) 1 res (x − cr )−J Y− dx = res Tr n x=cr J x=cr V (x) x−Q = − Tr n (Q − cr )−J = 2 ∂tr,J ln Zn , J = 1, . . . , dr ; J T (x) V (x) − V (Q) − res Y− (x)Tr (x)dx = − res r Tr n = Tr n Tr (Q) x=cr x=cr V (x) x−Q = 2 ∂cr ln Zn . (4-40)

428


The last equalities in (4-40) are obtained by a similar argument used for the deformations at c0 = ∞ here above. At the endpoints aj : V (x) − V (Q) 1 1 2 res Tr(Dn (x)) = res (V (x))2 − 2Tr n 2 x=aj 2 x=aj x−Q  L 2 +2 ∂a ln Zn  x − aj j j =1

= ∂aj ln Zn . 2

(4-41)

The determination Y− has the asymptotic behavior   L V (x) − V (Q) 2 1  Y− (x) ∼ ∂a ln Zn  Tr n − V (x) x−Q x − aj j j =1 & 2 2 n − x2 + O(x −d0 −2 ) x → ∞ + . O((x − cr )dr +1 ) x → cr

(4-42)

Identities (4-35) and (4-36) follow immediately from the expressions in (4-38), (4-40) and the asymptotic forms (4-42), as do the identities (4-34) for J ≤ d0 − 2. For the remaining two values of J (d0 − 1, d0 ), we compute 1 − res T0 (x)Y− (x)dx = − res (V (x) + O(1/x)) x=∞ x=∞ V (x) n2 2 −3 W − 2 + O x ) dx x =−

K

Tr n Tr (Q) +

L

2 ∂aj ln Zn

j =1

r=1

= Tr n T0 (Q),

(4-43)

where the last equality follows from Eq. 2-40 (translational invariance). This identity, together with Eqs. (4-34) for j ≤ d0 − 2 implies res x d0 −1 Y− (x)dx = 2

x=∞

∂ ∂t0,d0 −1

ln Zn ,

(4-44)

which is the case J = d0 − 1 of (4-34). Similarly − res xT0 (x)Y− (x)dx = − res (xV (x) + x=∞

x=∞

+O(1/x)) = n2 2 − n

K

tr,0

r=1

1 n2 2 −3 + O x ) dx W − V (x) x2 K r=1

= Tr n QT0 (Q),

tr,0 −

K r=1

Tr n QTr (Q) +

L

2 aj ∂aj ln Zn

j =1

(4-45)


429

where the last equality holds because of dilation invariance. This, together with the above proves (4-34) for J = d0 . The formulæ (4-37) follow from equating the residues at the poles x = aj in Eq. (4-6). Finally we examine (4-33): the second equality follows from the fact that t0,0 appears t0,0

in the integral (2-11) defining Zn only in the overall normalization factor e− . The first equality in (4-33) for the cases d0 ≥ 2 follows from the asymptotic expression of Y− . For the case d0 = 1 instead one has to use (4-43) and the fact that T0 (x) = t0,1 =constant. Remark 4.1. In the formulæ (4-33), (4-34), (4-36), (4-37) we may replace Y− by −(Y+ − V (x)); this corresponds to the fact that, in the large n limit, the behavior of Y (x) on the physical sheet (i.e. Y+ ) is related to the resolvent of the model by ' ( Y+ = V (x) + Tr(x − M)−1 . (4-46) 5. Isomonodromic Tau Function 5.1. Isomonodromic deformations and residue formula. In this section we briefly recall the definition of the isomonodromic tau-function given in [11] and compute its logarithmic derivatives in the present case in order to compare it with the partition function. This will lead to the main result of this section, Theorem 5.1, which explicitly gives this relation. Consider a rational covariant derivative operator on a rank p vector bundle over CP 1 , Dx = ∂x − A(x) ,

(5-1)

where the connection component A(x) is a p × p matrix, rational in x. Deformations of such an operator that preserve its (generalized) monodromy (i.e. including the Stokes’ data) are determined infinitesimally by requiring compatibility of the equations ∂x (x) = A(x), ∂ui (x) = Ui (x)(x) ,

i = 1, . . . ,

(5-2) (5-3)

where in the second set of equations Ui (x) are also p × p matrices, rational in x, viewed as components of a connection over the extended space consisting of the product of CP 1 with the space of deformation parameters {u1 , . . .}. The invariance of the generalized monodromy of Dx follows [11] from the compatibility of this overdetermined system, which is equivalent to the zero-curvature equations [∂x − A(x), ∂ui − Ui (x)] = 0 ,

[∂ui − Ui (x), ∂uj − Uj (x)] = 0.

(5-4)

Near a pole x = cν of A(x) a fundamental solution can be found that has the formal asymptotic behavior, in a suitable sector: (x) ∼ Cν Yν (x)eTν (x) ,

(5-5)

Yν (x) = 1 + O(x − cν )

(5-6)

where Cν is a constant matrix,

is a formal power series in the local parameter (x −cν ) (or 1/x for the pole at infinity) and Tν (x) is a Laurent-polynomial matrix in the local parameter, plus a possible logarithmic

430


term t0 ln(x − c). In the generic case Tν (x) is a diagonal matrix, and, more generally, may be an element of a maximal Abelian subalgebra containing an element with no multiple eigenvalues. The locations of the poles cν and the coefficients of the nonlogarithmic part of Tν (x) are the independent deformation parameters. The deformation of the connection matrix A(x) is determined by the requirement that the (generalized) monodromy data be independent of all these isomonodromic deformation parameters. Given a solution of such an isomonodromic deformation problem, one is led to consider the associated isomonodromic τ -function [11], determined by integrating the following closed differential on the space of deformation parameters:

ω :=

ν

res Tr Yν−1 Yν · dTν (x) = d ln τ I M ,

x=cν

(5-7)

where the sum is over all poles of A(x) (including possibly one at x = ∞), and the differential is over all the independent isomonodromic deformation parameters. In the present situation A(x) is our 2 × 2 matrix Dn (x) and the (generalized) monodromy of the operator ∂x − Dn (x) is invariant under changes in the parameters cr , tr,J , aj and n.

5.2. Traceless gauge. For convenience in the computations we perform a scalar gauge transformation of the ODE by choosing quasipolynomials rather than polynomials. Explicitly we set 1

n (x) := e− 2 V (x) n (x) =

ψn−1 (x) ψn (x)

)n−1 (x) ψ )n (x) , ψ

n (x) = An (x)n (x), 1 An (x) = Dn (x) − V (x)1, 2

(5-8)

where )n := e− 2 V (x) φn . ψ 1

(5-9)

In this gauge the matrix of the ODE is traceless and the infinitesimal deformation matrices are transformed correspondingly by addition of the identity element multiplied by the derivatives of − 21 V (x) with respect to the parameters {cr , trJ , aj }. This choice gives a consistent reduction of the general gl(p, C) isomonodromic deformation problem to sl(p, C). (To be precise, this would require a further, x–independent diagonal −1

1

2 gauge transformation of the form diag(hn−1 , hn2 ) to render the infinitesimal deformation matrices also traceless.) At each of the poles c0 := ∞, c1 , . . . we then have the following asymptotic expansions. (To simplify notation, the index n is omitted in labeling the fundamental system and its local asymptotic form.)

    1 1 (x) ∼ Cr Yr (x) exp − Tr (x) + δr0 n + tr,0  ln(x) σ3  . (5-10) 2 2 

r≥1


Here we have set Y0 (x) := 1 + Yr (x) := 1 +  Cr =

∞ Y0;k k=1 ∞

xk

C0 =

,

0

√

√1 hn

431

hn−1 , 0

Yr;k (x − cr )k ,

k=1



Vr (cr ) −1 √ 2 , ,  πn−1 (cr )e ˇ (cr − Q)n−1,0 h0 eˇ √ Vr (cr ) Vr (cr ) 2 πn (cr )e− 2 (cr − Q)−1 h e 0 n,0 ˇ

ˇ

(cr ) − Vr2

(5-11)

where Vˇr (x) = V (x) − Tr (x) is the holomorphic part of the potential at cr . The asymptotic forms given by (5-10)–(5-11) follow from the fact that, in any Stokes’ sector near cr , the second-kind solutions behave like ∞

)n (x) ∼ e 2 V (x) ψ 1

x→∞

x −k

k=n+1

1

κ

πn (z)e− V zk−1

1 1 = e 2 V (x) x −n−1 hn 1 + O , x 1 1 e− V (z) πn (z) V (x) ) ψn (x) ∼ e 2 dz (1 + O(x − cr )) x→cr cr − z κ 1 = e V (x) (cr − Q)−1 n,0 h0 (1 + O(x − cr )).

(5-12)

(5-13)

Near the endpoints aj we have 0 1 , σ+ := (x) ∼ Aj · Yj (x) · exp −κj ln(x − aj )σ+ , 0 0   1 − V (z) V (aj ) V (a ) e (πn−1 (z)−πn−1 (aj )) − j   πn−1 (aj )e 2 e 2 κ dz aj −z (5-14) Aj =  , − 1 V (z) V (aj ) V (aj ) e (π (z)−π (a )) n n j − 2 πn (aj )e e 2 κ dz aj −z since the matrix

(x) · exp −σ+  =

−  πn−1 (x)e

πn (x)e−

1

e− V (z) dz x−z κ

V (x) 2

V (x) 2

e

 − 1 V (z) e (πn−1 (z)−πn−1 (x)) dz κ x−z  − 1 (z) V (x) (πn (z)−πn (x)) e e 2 κ dz x−z

V (x) 2

(5-15) is analytic in a neighborhood of aj and has the limiting value indicated in (5-14). The − 1 V (z) function − κ dz e x−z in the exponential of the second matrix in this formula has the same singularity as κj ln(x − aj ). (The signs in (5-14) follow from the orientation of the contour originating at aj ).

432


The differential (5-7) can now be written κj daj −1 1 d ln τnI M = res dTr (x)Tr Yr−1 Yr σ3 + res Tr Yj Yj σ+ , x=cr x=aj x − aj 2 r=0

j

(5-16) where the differential involves the isomonodromic parameters only d K r ∂ ∂ ∂ ∂ d := + dtr,J + dcr daj = d(r) + daj . ∂tr,J ∂cr ∂aj ∂aj r=0

J =1

j

r=0

j

(5-17) We now derive residue formulæ for the deformation parameters and the logarithmic derivatives of the tau function for our rational 2 × 2 isomonodromic deformation problem. These essentially are the same as the formulæ of Thm. 4.2 giving the latter quantities in terms of logarithmic derivatives of the partition function of the matrix model1 . Consider the quadratic spectral invariant near any of the singularities: by virtue of the asymptotics (5-10, 5-14) we have, near cr and aj respectively (setting S := 2n + K r=1 tr,0 ) 2 2 2 −1 2 2 −1 Tr(A (x)) = Tr(( ) ) = Tr Yr Yr δr0 S −1 −Tr Yr Yr σ3 Tr − x 2 δr0 S 1 Tr − , (5-18) + 2 x Tr(A2 (x)) = 2 Tr(( −1 )2 ) 2 2κj + Tr Yj−1 Yj σ+ . = 2 Tr Yj−1 Yj x − aj

(5-19)

Taking the principal part at each singularity and using Liouville’s theorem (since TrA2 is a priori a rational function) we find K 1 δr0 S 2 1 δr0 S 2 −1 Tr(A (x)) = + Tr − Tr − Tr(Yr Yr σ3 ) 2 x x r=0 r,+ L 2 κj Tr(Yj (aj )σ+ ) , (5-20) + x − aj j =1

where the subscripts r,+ mean the singular part at the pole x = cr (including the constant for x = c0 = ∞). Consider now the spectral curve of the connection ∂x − A(x), w2 =

1 TrA2 (x), 2

(5-21)

1 The case of an arbitrary rank rational, nonresonant isomonodromic deformation problem will be developed elsewhere [5], together with further properties that allow us to view these as nonautonomous Hamiltonian systems, in which the logarithmic derivatives of the τ -function computed below are interpreted as the Hamiltonians generating the deformation dynamics.


433

One immediately finds w± (x) 0 1 11 δr0 S 2 1 δr0 S −1 2 Tr(Yr Yr σ3 ) Tr − Tr − − =± 4 r x 2 x r,+

r,+

+

1 κj Tr(Yj (aj )σ+ ) . x − aj j (5-22)

Near any of the poles one has the asymptotic behavior  1 δr0 S 1 −1   Tr − − T Tr(Yr Yr σ3 )   2Tr (x) r 2  r,+ 2x   dr +1    O((x − c ) ) near x = c r r   + ±w± =  O(x −d0 −2 ) near x = ∞ .         κj Tr(Yj (aj )σ+ )    (1 + O(x − aj )) near x = aj (x − aj )

(5-23)

This immediately implies the following identities: 1 n + tr,0 = ∓ res w± dx. x=∞ 2 r 1 1 t0,J = ∓ res J w± dx , J ≥ 1, x=∞ x 2 1 tr,J = ∓ res (x − cr )J w± dx, x=cr 2

(5-24)

and 2

∂ (x − cr )−J w± dx, ln τnI M = ∓ res x=cr ∂tr,J J

∂ xJ w± dx, ln τnI M = ∓ res x=∞ J ∂t0,J ∂ 2 ln τnI M = ± res Tr (x)w± dx, x=cr ∂cr ∂ 2 ln τnI M = res (w± )2 dx . x=aj ∂aj

2

(5-25) (5-26) (5-27) (5-28)

In order to compare with the formulæ given in Thm. (4.2) we note that the eigenvalues w of A(x) and Y of Dn (x) are related as follows due to the change of gauge (5-8): 1 V (x) + w± . 2 Comparing Eqs. (4-34)–(4-37) with Eqs. (5-26)–(5-28) we obtain Y± =

Zn 1 res x J V , = I M τn 2J x=∞ 1 Zn 1 res ln I M = V (x)dx, τn 2J x=cr (x − cr )j

(5-29)

2 ∂t0,J ln 2 ∂tr,J

r = 1, . . . , K, J = 1, . . . , dr ,

434


2 ∂cr ln

Zn 1 = res Tr (x)V (x)dx, I M τn 2 x=cr

r = 1, . . . , K,

2 ∂aj ln Zn = 2 ∂aj ln τnI M .

(5-30)

These relations define a closed differential Zn d ln =: d ln(Fn ) , τnI M

(5-31)

where the quantity Fn is determined up to a multiplicative factor independent of the a ,a } isomonodromic deformation parameters {cr , trJ j J ≥1 , but which may depend on n. This may be explicitly integrated to give Fn 1 ln res T (x)Tq (x) , (5-32) =− 2 x=cr r n! 2 0≤qm

ζl ζm

kl,m ,

where µ(α; k) =

1 1 (α; q)k (q 2 t − 2 )k . −1 (αqt ; q)k

448

J. Shiraishi

Thus it is explicitly seen that I (α) is lower-triangular with respect to the dominance ordering on the monomial basis. It is easy to see that the diagonal elements are given by Eq. (35), and that all of them are distinct since the parameters are generic. Hence there is no obstruction for the construction of an eigenfunction of the form Eq. (34) corresponding to the eigenvalue Eq. (35). The uniqueness follows from the normalization cj1 ,j2 ,...,jn−1 = 1. In view of the equality µ(α; k + l) = µ(α; k)µ(αq k ; l), it immediately follows that all the eigenfunctions are related with each other. Proposition 4.2. We have fj1 ,j2 ,...,jn−1 (ζ1 , . . . , ζn ) =

n

j

ζi i−1

−ji

(Tq,si )−ji−1 +ji · f0,0,...,0 (ζ1 , . . . , ζn ). (36)

i=1

Remark. The family fj1 ,j2 ,...,jn−1 (ζ1 , . . . , ζn ) form a basis of Fn . At this moment, Conjecture 1.2 remains open when n ≥ 3, since properties of the eigenfunctions are not well known. For n = 3, one may guess an explicit formula of them by a brute force calculation. Conjecture 4.3. The first eigenfunction of I (α) for n = 3 is given by f0,0 (ζ1 , ζ2 , ζ3 ) =

∞

(qt −1 , qt −1 , t, t; q)k (qs1 /s3 )k (ζ3 /ζ1 )k (37) (q, qs1 /s2 , qs2 /s3 , qs1 /s3 ; q)k k=0 k+1 −1 −1 q t , qt si /sj , × (1 − ζj /ζi ) · 2 φ1 ; q, tζj /ζi . q k+1 si /sj 1≤i<j ≤3

Remark. No dependence on the parameter α is observed in f0,0 (ζ1 , ζ2 , ζ3 ), nor in the complete set of eigenfunctions (from Proposition 4.2), from which Conjecture 1.2 follows immediately. 5. Modified Macdonald Difference Operator D 1 In this section, we study some properties of the modified Macdonald difference operator D 1 given by Definition 1.3. We will see that these properties for D 1 are just the same for I (α), though we do not have any a priori reason for such coincidence. Proposition 5.1. For any set of nonnegative integers j1 , j2 , . . . , jn−1 , there exist a unique solution to the equation D 1 fj1 ,j2 ,...,jn−1 (ζ1 , . . . , ζn ) = λj1 ,j2 ,...,jn−1 fj1 ,j2 ,...,jn−1 (ζ1 , . . . , ζn ),

(38)

with the expansion Eq. (34) and the normalization cj1 ,j2 ,...,jn−1 = 1, if and only if λj1 ,j2 ,...,jn−1 =

n

si q −ji−1 +ji .

i=1

The eigenfunctions of D 1 satisfy the relation Eq. (36).

(39)

Family of Integral Transformations and Basic Hypergeometric Series

449

Remark. One can regard the eigenfunctions as a basic analogue of the Heckman-Opdam hypergeometric function [6]. Proof. The proof parallels the case of I (α). We have j1 j2 j1 j2 ζ3 ζ3 ζn jn−1 ζ2 ζn jn−1 1 ζ2 D ··· = ··· ζ1 ζ2 ζn−1 ζ1 ζ2 ζn−1 n 1 − q −1 tζi /ζj 1 − qt −1 ζk /ζi × si q −ji−1 +ji , (40) 1 − q −1 ζi /ζj 1 − qζk /ζi i=1

j

k>i

where the rational factors should be understood as the series in Eq. (8). Hence it is explicit in Eq. (40) that D is lower-triangular in the monomial basis with respect to the dominance order, with distinct diagonals given by Eq. (39). In view of Eq. (40), it follows that the relation Eq. (36) holds. We study the simplest case n = 2. Proposition 5.2. For n = 2, all the eigenfunctions of I (α) and D 1 are the same. Hence Conjecture 1.4 is true for n = 2. n Proof. Set f0 (ζ1 , ζ2 ) = (1 − ζ )g(ζ ) and g(ζ ) = ∞ n=0 gn ζ , where ζ = ζ2 /ζ1 . The difference equation for g reads s1 (1 − qt −1 ζ )g(qζ ) + s2 (1 − q −1 tζ )g(q −1 ζ ) = (s1 + s2 )(1 − ζ )g(ζ ). Solving this with the condition g0 = 1, we have gn =

(qt −1 , qt −1 s1 /s2 ; q)n n t . (q, qs1 /s2 ; q)n

Hence the first eigenfunction of D 1 is

qt −1 , qt −1 s1 /s2 (41) ; q, tζ2 /ζ1 . qs1 /s2 From Lemma 5.3 below, we find that f0 (ζ1 , ζ2 ) in Eq. (41) agrees with Eq. (24) written for j = 0. Since we have the same relations Eq. (36) for I (α) and D 1 , it follows that all the eigenfunctions of I (α) and D 1 must be the same. Hence we have the commutativity I (α)D 1 = D 1 I (α) for n = 2. f0 (ζ1 , ζ2 ) = (1 − ζ2 /ζ1 ) 2 φ1

Lemma 5.3. We have (1 − z) 2 φ1

a, b ; q, zq/b = 4 W3 (a/q; b/q; q, zq/b) . aq/b

Proof. ∞

∞

(a, b; q)n−1 (a, b; q)n (q/b)n zn − (q/b)n−1 zn (aq/b, q; q)n (aq/b, q; q)n−1 n=0 n=1 ∞ (a, b; q)n (1 − aq n /b)(1 − q n ) 1 − (b/q) = (zqb−1 )n (aq/b, q; q)n (1 − aq n−1 )(1 − bq n−1 )

LHS =

n=0

=

∞ n=0

(a, b; q)n (1 − bq −1 )(1 − aq 2n−1 ) (zqb−1 )n = RHS. (aq/b, q; q)n (1 − aq n−1 )(1 − bq n−1 )

(42)

450

J. Shiraishi

At present, it remains open how to construct an explicit formula for the eigenfunctions of D 1 for general n. For n = 3, a brute force calculation suggests the following. Conjecture 5.4. For n = 3, Eq. (37) also gives the first eigenfunction of the difference operator D 1 , from which Conjecture 1.4 follows. 6. Weyl Group Symmetry In this section, we study a hidden symmetry of the eigenfunctions of I (α) or D 1 in terms of the Weyl group of type An−1 . This Weyl group symmetry appears when the series for the eigenfunctions become truncated under the specialization t = q m (m = 1, 2, 3, . . .). Let W (An−1 ) be the Weyl group of type An−1 generated by σ1 , σ2 , · · · , σn−1 with the braid relations σi2 = id and σi σi+1 σi = σi+1 σi σi+1 . Let Pn be the space of polynomials in ζ1 , ζ2 , . . . , ζn with coefficients which are rational expressions in si ’s and q. Definition 6.1. For a positive integer m, we define the action πm of W (An−1 ) on Pn by

=

πm (σi )f (ζ1 , . . . , ζi , ζi+1 , . . . , ζn , s1 , . . . , si , si+1 , . . . , sn ) m−1 si − q k si+1 f (ζ1 , . . . , ζi+1 , ζi , . . . , ζn , s1 , . . . , si+1 , si , . . . , sn ). s − q k si k=1 i+1

(43)

We make a conjecture. Conjecture 6.2. If t = q m (m = 1, 2, 3, . . .), the eigenfunctions of I (α) or D 1 become truncated. We have n

(n−k)m

ζk

f0,0,...,0 (ζ1 , ζ2 , . . . , ζn ) ∈ Pn ,

(44)

k=1

and the antisymmetry with respect to the action πm (σ ) (σ ∈ W (An−1 )) πm (σ ) ·

n k=1

(n−k)m

ζk

f0,0,...,0 (ζ1 , ζ2 , . . . , ζn ) = −

n

(n−k)m

ζk

f0,0,...,0 (ζ1 , ζ2 , . . . , ζn ).

k=1

(45) Consider the case n = 2. An easy calculation gives us the following. Proposition 6.3. For m = 1, 2, 3, . . . , we have the W (A1 )-symmetric polynomial in P2 : −1 −1 qt , qt s1 /s2 m−1 πm (σ1 ) · ζ1 2 φ1 ; q, tζ2 /ζ1 m qs1 /s2 t=q −1 −1 qt , qt s1 /s2 = ζ1m−1 2 φ1 ; q, tζ2 /ζ1 . (46) m qs1 /s2 t=q

Hence from Eq. (41) Conjecture 6.2 is true for n = 2. Next, we look at the case n = 3. One may find a family of W (A2 )-symmetric polynomials.


451

Lemma 6.4. Let m be a positive integer and k be a nonnegative integer. Set (qt −1 , qt −1 , t, t; q)k (q, qs1 /s2 , qs2 /s3 , qs1 /s3 ; q)k k+1 −1 −1 q t , qt si /sj ; q, tζj /ζi 2 φ1 q k+1 si /sj

ϕk = (ζ3 /ζ1 )k (qs1 /s3 )k ×

1≤i<j ≤3

(47) . t=q m

Then we have ζ12m−2 ζ2m−1 ϕk ∈ P3 and the W (A2 )-symmetry πm (σ ) · ζ12m−2 ζ2m−1 ϕk = ζ12m−2 ζ2m−1 ϕk . (48) From Conjecture 4.3 and the lemma above, we have f0,0 = i<j (1 − ζj /ζi ) k ϕk , from which Conjecture 6.2 follows for n = 3. 7. Product Formula for the Eigenfunction In this section, we study the special case (s1 , s2 , . . . , sn ) = (1, t, . . . , t n−1 ),

(49)

and consider an infinite product formula for the first eigenfunction. Proposition 7.1. We have the infinite product formula for the first eigenfunction of D 1 as follows: (qt −1 ζj /ζi ; q)∞ f0,0,...,0 (ζ1 , ζ2 , . . . , ζn ) = (1−ζj /ζi ) . (tζj /ζi ; q)∞ n−1 (s1 ,s2 ,···,sn )=(1,t,···,t

)

1≤i<j ≤n

(50) Remark. Conjecture 1.4 suggests that the same is true also for I (α). Proof. We have D(1, t, . . . , t n−1 ; q, t)

(1 − ζj /ζi )

1≤i<j ≤n

= t n−1

(1 − ζj /ζi )

1≤i<j ≤n

= (1 + t + · · · + t n−1 )

(qt −1 ζj /ζi ; q)∞ (tζj /ζi ; q)∞

n (qt −1 ζj /ζi ; q)∞ t −1 ζi − ζj (tζj /ζi ; q)∞ ζi − ζ j i=1 j =i

(1 − ζj /ζi )

1≤i<j ≤n

(qt −1 ζj /ζi ; q)∞ . (tζj /ζi ; q)∞

Here we have used the identity n −1 t ζi − ζ j = (1 + t −1 + · · · + t −n+1 ), ζi − ζ j i=1 j =i

which can be proved by decomposition in partial fractions.

452

J. Shiraishi

It is of some interest to check this factorization by restricting the general formulas for the eigenfunctions. For n = 2, setting s1 = 1, s2 = t, ζ = ζ2 /ζ1 in Eq. (41), and using the q-binomial theorem (Eq. (1.3.2) in [5]), we have −1 −2 qt , qt f0 (ζ1 , ζ2 ) = (1 − ζ ) × 2 φ1 ; q, tζ (s1 ,s2 )=(1,t) qt −1 (qt −1 ζ ; q)∞ . (tζ ; q)∞ Next we look at the case n = 3. Using the q-Pfaff-Saalschütz formula (Eq. (1.7.2) in [5]), we have the equality, k+1 −1 −3 ∞ q t , qt (t; q)k (t; q)k −2 k (qt ζ ) φ ; q, tζ 2 1 (q; q)k (qt −2 ; q)k q k+1 t −2 k=0 ∞ (qt −1 ; q)m (qt −3 ; q)m m m t, t, q −m = t ζ φ ; q, q 3 2 (qt −2 ; q)m (q; q)m qt −1 , q −m t 3 = (1 − ζ )

=

m=0 (qt −1 ζ ; q)∞

. (tζ ; q)∞ Hence from Conjecture 4.3, it follows that f0,0 (ζ1 , ζ2 , ζ3 ) 2 =

(s1 ,s2 ,s3 )=(1,t,t ) (qt −1 ζ2 /ζ1 ; q)∞ (1 − ζj /ζi ) × (tζ2 /ζ1 ; q)∞ 1≤i<j ≤3

×

∞ k=0

=

(qt −1 ζ3 /ζ2 ; q)∞ (tζ3 /ζ2 ; q)∞

k+1 −1 −3 (qt −1 , qt −1 , t, t; q)k q t , qt , −2 k k (qt ) (ζ3 /ζ1 ) · 2 φ1 ; q, tζ3 /ζ1 (q, qt −1 , qt −1 , qt −2 ; q)k q k+1 t −2 (1 − ζj /ζi )

1≤i<j ≤3

(qt −1 ζj /ζi ; q)∞ . (tζj /ζi ; q)∞

8. Raising Operators for Macdonald Polynomials In this section, we connect the modified Macdonald operator D 1 with raising operators for the Macdonald symmetric polynomials. We briefly recall the notion of the Macdonald polynomials [3]. Let x1 , x2 , . . . , xn be a set of indeterminates, and n = Z[x1 , . . . , xn ]Sn denotes the ring of symmetric polynomials. The ring of symmetric functions is defined as the inverse limit of the n in the category of graded rings. In this section, we regard q and t as independent indeterminates. Let F = Q(q, t) be the field of rational functions in q and t, and set F = ⊗Z F . Let pn = i xin be the power sum symmetric functions, and denote pλ = pλ1 pλ2 · · · for any partition λ = (λ1 , λ2 , . . .). The scalar product is introduced by 1−q λj pλ , pµ q,t = δλ,µ i mi mi ! , (51) 1−t λj i≥1 j≥1 where mi = mi (λ) is the multiplicity of the part i in the partition λ.


453

The Macdonald symmetric functions Pλ (x; q, t) ∈ F are uniquely characterized by the following two conditions [3]: (a) Pλ = mλ + uλµ mµ , (52) µn2p f¯0 − E(f¯0 | U¯ n B0 ) 2 = O(1/n2 ), 0 L which is summable. This proves (50). Let h¯ = E(f¯0 | B0 ). This function is constant along the stable leaves, and has zero integral (since f¯0 also has zero integral). Hence, it induces a function h on the quotient 0 . Since f¯0 ∈ L2 , it satisfies h ∈ L2 ( 0 ). The following lemma is an easy consequence of the Hölder properties of the invariant measure and (52), see [You98, Sublemma, p. 612] for details. Lemma 5.4. There exist constants C > 0 and τ < 1 such that, for all x, y in the same ¯ 0,i , unstable leaf of a set ¯ ¯ |h(x) − h(y)| CA(x)τ s(x.y) . The function A is integrable. Hence, by [Gou04, Lemma 3.4], this implies that the 0 h is Hölder continuous on 0 . By [Gou04, Corollary 3.3], we get: function U 0n h tends exponentially fast to 0 U in the space of Hölder continuous functions on 0 . A computation gives

E(f¯0 | U¯ −n B0 ) 2 2 = 0 L

(55)

n

0n h) ◦ U0n hL2 (U 0 h) ◦ U0n 2 h · (U L

n = hL2 U0 h L2 .

Hence, this term is exponentially small. This proves (51) and concludes the proof of Lemma 5.3. The return time ϕ also satisfies a central limit theorem, by the same argument. Hence, by Theorem 5.1 (applied with b = 1), there exists σ12 0 such that n−1 ¯ ¯k k=0 f ◦ U → N (0, σ12 ). √ n ¯ to X, it implies that Going from n−1 k k=0 f ◦ T → N (0, σ12 ). √ n

(56)

Moreover, the return √ √ time√ϕ+ : X → N satisfies a limit theorem with normalization n log n. Since n = o( n log n), we can unfortunately not apply Theorem 5.1 with b = 1. However, if we can prove the following lemma, then this theorem applies with b < 1. Lemma 5.5. For all b > 1/2, n−1 1 f ◦Tk → 0 |n|b k=0

almost everywhere in X when n → ±∞.

(57)

Limit Theorems in the Stadium Billiard

507

Proof. We first estimate the decay of correlations of f¯0 for U¯ 0 . We will use the notations of the proof of Lemma 5.3. We have f¯0 · f¯0 ◦ U¯ 02n = f¯0 · E(f¯0 ◦ U¯ 0n | B0 ) ◦ U¯ 0n + f¯0 · f¯0 ◦ U¯ 02n − E(f¯0 ◦ U¯ 0n | B0 ) ◦ U¯ 0n . (58) The contraction properties of U¯ 0 along stable manifolds and (53) give |f¯0 ◦ U¯ 0n (x) − E(f¯0 ◦ U¯ 0n | B0 )(x)| CA(U¯ 0n x)λαn . Hence, the second integral in (58) is at most

|f¯0 | · A ◦ U¯ 02n λαn f¯0 L2+ε2 ALp λαn , 1 + p1 = 1. Hence, this term decays exponentially fast. where p < 2 is chosen so that 2+ε 2 In the first integral of (58), the function E(f¯0 ◦ U¯ 0n | B0 ) ◦ U¯ 0n is B0 -measurable (i.e., constant along stable leaves). Hence, this integral is equal to h¯ · E(f¯0 ◦ U¯ 0n | B0 ) ◦ U¯ 0n . (59)

Let h¯ n = E(f¯0 ◦ U¯ 0n | B0 ), it is B0 -measurable and defines a function hn on the quotient 0 . The integral (59) is then equal to 0n h · hn . h · hn ◦ U0n = U (60) 0

The L2 -norm of hn

is bounded independently of n. By (55), (60) is exponentially small. This proves that f¯0 · f¯0 ◦ U¯ 02n decays exponentially. In the same way, f¯0 · f¯0 ◦ U¯ 02n+1 decays exponentially. Since the correlations of f¯0 decay exponentially fast and f¯0 ∈ L2 , [Kac96, Theorem ¯ ¯k 16] implies that n1b n−1 k=0 f0 ◦ U0 tends to zero almost everywhere when n → +∞, for all b > 1/2. ¯ ¯k ¯ Now to see that n1b n−1 k=0 f ◦ U tends to zero almost everywhere in when n → +∞, for all b > 1/2, we use [MT04, Lemma 2.1 (a)] which gives this convergence on ¯ 0 . However, by the ergodicity of U¯ , the set on which this convergence holds must have ¯ 0 has positive measure, we get this convergence almost either full or zero measure. As ¯ Finally, this implies the same for f in X. We have proved (57) for everywhere on . any b > 1/2 when n → +∞. To deal with n → −∞, we go to the natural extension. It is sufficient to prove the ¯ 0 , since the previous reasoning still applies (using the fact that the result for f¯0 in natural extension is functorial, i.e., the natural extension and commutes with induction ¯ of ¯ 0 , we have f¯ · f¯ ◦ U¯ −n = f¯0 ◦ U¯ n · f¯0 , projections). In the natural extension 0 0 0 0 0 which is exponentially small. Hence, [Kac96, Theorem 16] still applies and gives the desired result. Remark 5.6. As µ0 (X) > 0, we may apply [MT04, Lemma 2.1 (a)] just as we did in the proof above to see that Lemma 5.5 implies n−1 1 f0 ◦ T0k → 0 |n|b k=0

almost everywhere when n → ±∞, for any b > 1/2.

508

P. Bálint, S. Gouëzel

Proof of Theorem 1.4. The convergence (56), together with Lemma 5.5 and Theorem 5.1, implies (2). We still have to prove the zero √ variance statement. If√ f0 = χ − χ ◦ T0 for some measurable function χ , then Sn f0 / n = (χ − χ ◦ T0n )/ n tends in probability to 0, which implies σ = 0. Conversely, assume that σ = 0. The function f¯0 on the basis ¯ 0 of the Young tower satisfies a central limit theorem with zero variance. The proof of Gordin’s theorem then ensures the existence of a measurable function χ¯ 0 such that ¯ 0 . This implies that f¯ is a coboundary f¯0 = χ¯ 0 − χ¯ 0 ◦ U¯ 0 , i.e., f¯0 is a coboundary on ¯ as follows: let π¯ 0 : ¯ → ¯ 0 be the projection on the basis of the tower. Defining on , ¯ → R by χ¯ : χ¯ (x) = χ¯ 0 (π¯ 0 x) −

ω(x)−1

f¯(U¯ k π¯ 0 x),

(61)

k=0

we have f¯ = χ¯ − χ¯ ◦ U¯ . Since the function f¯ = f ◦ πX is a coboundary, general results on coboundaries (see e.g. [Gou05a, Theorem 1.4]) ensure that f also is a coboundary on X for T . Finally, this implies that f0 is a coboundary on X0 for T0 , using a formula similar to (61). 5.3. Proof of Proposition 1.5. We work in the stadium billiard with = ∗ , for which the free flight τ0 satisfies a usual central limit theorem. Define a function τ : X → R ϕ+ (x)−1 ∗ k by τ (x) = k=0 τ0 (T0 x). Since the function τ0∗ does not satisfy (P 1), Lemma 2.5 does a priori not apply. Nevertheless, due to the geometric properties of the free flight, the function τ satisfies the following inequality: if x, y ∈ X are two points sliding n times along the semicircles, then |τ (x) − τ (y)| C(d(x, y) + d(T x, T y)). This estimate is sufficient to carry out the proofs of Lemmas 2.5 and 2.6. Hence, there exist two ¯ → R and g : → R such that τ ◦ πX = g ◦ π + u¯ − u¯ ◦ U¯ on , ¯ functions u¯ : ¯ and g is Hölder continuous on . Let 0 be the basis of the tower u¯ is bounded on ϕ0 (x)−1 g(U k x), , and let g0 be the function induced by g on 0 (given by g0 (x) = k=0 where ϕ0 is the return time from 0 to itself). Assume that σ = 0. By Theorem 1.4, this implies that τ0∗ is a coboundary. In turn, arguments similar to the end of the proof of Theorem 1.4 show that g0 itself is a coboundary. Since g is Hölder continuous, g0 satisfies the assumptions of [Gou05b, Theorem 1.1]. This theorem implies that the function g0 is essentially bounded. Let us show that this is not the case. The function τ is bounded from above: since E(τ0 ) = 2, the function τ is O(1) on the set of points bouncing n times between the segments. Moreover, on the set of points sliding n times along the circles, the function τ is equal to −2n + O(1). Since u¯ is bounded, this implies that there exists a constant C1 such that g C1 , and that g = −2n + O(1) on a set of measure at least C/n4 . Let An ⊂ be the set of points in the tower where ϕ0 (π0 x) n/(2C1 ) (where π0 : → 0 is the projection on the basis) and g −n. Since µ {ϕ0 (π0 x) > n/(2C1 )} is exponentially small in n, while µ {g −n} C/n4 , the set An has nonzero measure for n large enough. ϕ0 (y)−1 If y ∈ π0 (An ), then g0 (y) −n + k=0 C1 −n/2. Since π0 (An ) has positive measure, this shows that the function g0 is not bounded from below. This contradiction concludes the proof.


509

Acknowledgement. We are very grateful to D. Szász and T. Varjú for useful discussions and for their valuable remarks on earlier versions of the manuscript. This paper has grown out of discussions we had at the CIRM conference on multi-dimensional non-uniformly hyperbolic systems in Marseille in May 2004, and while S. G. visited the Institute of Mathematics of the BUTE in October 2004. The hospitality of both institutions, along with the financial support of Hungarian National Foundation for Scientific Research (OTKA), grants TS040719, T046187 and TS049835 is thankfully acknowledged.

A. Proof of Lemma 3.6 Let U0 be the map induced by U on the basis 0 of the tower. Denote by ϕ the first return time on the basis, so that U0 (x) = U ϕ(x) (x). Note that ϕ(x) can also be defined for x ∈ \ 0 as the first hitting time of the basis. Let F be a finite subset of N. Let (ni )i∈F be positive integers. Let K(F, ni ) = {x ∈ 0 | ∀i ∈ F, ϕ(U0i x) = ni }. Lemma A.1. There exists a constant C such that, for all F and ni as above, $ (Cρ ni ). µ (K(F, ni )) i∈F

Proof. The proof is by induction on max F , and the result is trivial when F = ∅. Write F = {i − 1 | i ∈ F, i 1} and, for i ∈ F , set ni = ni+1 . If 0 ∈ F , K(F, ni ) = U0−1 (K(F , ni )). Since U0 preserves µ and max F < max F , we get the result. Otherwise, 0 ∈ F . Then K(F, ni ) = U0−1 (K(F , ni )) ∩ {x ∈ 0 , ϕ(x) = n0 }. By bounded distortion, we get µ (K(F, ni )) Cµ (K(F , ni ))µ {x ∈ 0 , ϕ(x) = n0 } Cµ (K(F , ni ))ρ n0 . Lemma A.2. There exist C > 0 and θ < 1 such that, for all n ∈ N, τ n Cθ n . U −n 0

Proof. Let κ > 0 be very small (how small will be specified later in the proof). Then U −n 0 ⊂ {x ∈ | n (x) κn} ∪ {x ∈ | ϕ(x) n/2} ∪ {x ∈ | ϕ(x) < n/2, n (x) < κn}. On the first of these sets, τ n τ κn , whence the integral of τ n is exponentially small. The second of these sets has exponentially small measure. Finally, the last of these sets n/2 is contained in i=0 U −i n , where n = {x ∈ 0 |

ϕ(U0i x) n/2}.

0i κn

To conclude the proof of the lemma, it is sufficient to prove that the measure of n is exponentially small.

510

P. Bálint, S. Gouëzel

Take L ∈ N such that ∀n L, (Cρ)n ρ n/2 , where C is the constant given by Lemma A.1. For x ∈ n , let F (x) := {0 i κn | ϕ(U0i x) L}. Then n ϕ(U0i x) − L (1/2 − Lκ)n. 2 i∈F (x)

i∈F (x)

This implies that

n ⊂

F ⊂[0,κn] i∈F

By Lemma A.1, we get µ (n )

F ⊂[0,κn]

i∈F

κn k=0

2κn

κn k

K(F, ni ).

ni L ni (1/2−Lκ)n

$

(Cρ ni )

i∈F ni L ni (1/2−Lκ)n

(Cρ n0 ) . . . (Cρ nk−1 )

n0 ,...,nk−1 L ni (1/2−Lκ)n

ρ

ni /2

2κn

0k κn n0 ,...,nk−1 L ni (1/2−Lκ)n

For r ∈ N,

ρ

ni /2

n0 +···+nk−1 =r

ni = r} = ρ

r/2

ρ

ni /2

.

0k κn n0 ,...,nk−1 ∈N ni (1/2−Lκ)n

= ρ r/2 Card{n0 , . . . , nk−1 | (r + k)k r +k ρ r/2 . k k!

Hence,

µ (n ) 2κn

ρ r/2

0k κn r (1/2−Lκ)n

(r + k)k . k! κ

k

The sequence ur = ρ r/2 (r+k) satisfies uur+1 ρ := ρ 1/2 e 1/2−Lκ for all r (1/2 − k! r Lκ)n and k κn. If κ is small enough, ρ < 1, and we get " #k 1 κn (1/2−Lκ)n/2 (1/2 − Lκ)n + κn µ (n ) 2 ρ k! 1 − ρ 0k κn

The sequence

nk k!

2κn 1−ρ

ρ (1/2−Lκ)n/2

0k κn

nk . k!

is increasing for k n. Hence, we finally get µ (n )

2κn (1/2−Lκ)n/2 nκn . ρ (κn + 1) 1−ρ κn!


511

Using Stirling’s Formula, it is easy to check that this expression is exponentially small if κ is small enough. This concludes the proof. Proof of Lemma 3.6. Let θ be given by Lemma A.2. Choose α > 0 so that eεα θ < 1. Then

U −n 0 ⊂ {x ∈ | ω(x) αn} ∪ {x ∈ | ω(x) < αn} ∩ U −n 0 . Hence,

U −n 0

eεω τ n

ωαn

eεω + eεαn

U −n 0

τ n .

The first term is exponentially small since eε ρ < 1. Lemma A.2 and the definition of α also imply that the second term is exponentially small. References [Aar97]

Aaronson, J.: An introduction to infinite ergodic theory. Volume 50 of Mathematical Surveys and Monographs. Providence RI: American Mathematical Society, 1997 [AD01] Aaronson, J., Denker, M.: A local limit theorem for stationary processes in the domain of attraction of a normal distribution. In N. Balakrishnan, I.A. Ibragimov, V.B. Nevzorov, eds., Asymptotic methods in probability and statistics with applications. Papers from the international conference, St. Petersburg, Russia, 1998, Basel: Birkhäuser, 2001, pp. 215–224 [Bun79] Bunimovich, L.: On the ergodic properties of nowhere dispersing billiards. Commun. Math. Phys. 65, 295–312 (1979) [BY93] Baladi, V., Young, L.-S.: On the spectra of randomly perturbed expanding maps. Commun. Math. Phys. 156, 355–385 (1993) [Che97] Chernov, N.: Entropy, Lyapunov exponents and mean free path for billiards. J. Stat. Phys. 88, 1–29 (1997) [Che99] Chernov, N.: Decay of correlations and dispersing billiards. J. Stat. Phys. 94, 513–556 (1999) [CZ05] Chernov, N., Zhang, H.: Billiards with polynomial mixing rates. Nonlinearity 18, 1527–1554 (2005) [Eag76] Eagleson, G.K.: Some simple conditions for limit theorems to be mixing. Teor. Verojatnost. i Primenen. 21(3), 53–660 (1976) [Gor69] Gordin, M.: The central limit theorem for stationary processes. Dokl. Akad. Nauk SSSR 188, 739–741 (1969) [Gou03] Gouëzel, S.: Statistical properties of a skew-product with a curve of neutral points. http://name.math.Univ-rennes1.fr/Sebastien.gouezel/articles/skewproduct. pdf, 2004 [Gou04] Gouëzel, S.: Central limit theorem and stable laws for intermittent maps. Probab. Theory and Rel. Fields 128, 82–122 (2004) [Gou05a] Gouëzel, S.: Berry-Esseen theorem and local limit theorem for non uniformly expanding maps. http://name.math.Univ-rennes1.fr/Sebastien.gouezel/articles/vitesse-TCL.pdf, Annales de l’IHP Probabilités et Statistiques, 41, 997–1024 [Gou05b] Gouëzel, S.: Regularity of coboundaries for non uniformly expanding Markov maps. Proc. Am. Math. Soc. 134(2), 391–401 (2005) [Hen93] Hennion, H.: Sur un théorème spectral et son application aux noyaux lipschitziens. Proc. Amer. Math. Soc. 118, 627–634 (1993) [Kac96] Kachurovski˘ı, A.G.: Rates of convergence in ergodic theorems. Russian Math. Surveys 51, 653–703 (1996) [KL99] Keller, G., Liverani, C.: Stability of the spectrum for transfer operators. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28(1), 141–152 (1999) [Mac83] Machta, J.: Power law decay of correlations in a billiard problem. J. Statist. Phys. 32, 555–564 (1983) [Mar04] Markarian, R.: Billiards with polynomial decay of correlations. Ergodic Theory Dynam. Systems 24, 177–197 (2004) [MT04] Melbourne, I., Török, A.: Statistical limit theorems for suspension flows. Israel J. Math. 144, 191–210, 2004

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P. Bálint, S. Gouëzel Szász, D., Varjú, T.: Local limit theorem for the Lorentz process and its recurrence on the plane. Ergodic Theory Dynam. Systems 24, 257–278 (2004) Szász, D., Varjú, T.: Markov towers and stochastic properties of billiards. In: Modern dynamical systems and applications, Cambridge: Cambridge University Press, 2004, pp. 433–445 Szász, D., Varjú, T.: Limit laws and recurrence for the planar Lorentz process with infinite horizon. Preprint, 2005 Young, L.-S.: Statistical properties of dynamical systems with some hyperbolicity. Ann. Math. (2) 147, 585–650 (1998) Young, L.-S.: Recurrence times and rates of mixing. Israel J. Math. 110, 153–188 (1999)

Communicated by G.Gallavotti


Communications in


Local Energy Statistics in Disordered Systems: A Proof of the Local REM Conjecture Anton Bovier1,2 , Irina Kurkova3 1

Weierstraß–Institut für Angewandte Analysis und Stochastik, Mohrenstrasse 39, 10117 Berlin, Germany. E-mail: [email protected] 2 Institut f¨ ur Mathematik, Technische Universität Berlin, Strasse des 17. Juni 136, 12623 Berlin, Germany 3 Laboratoire de Probabilités et Modèles Aléatoires, Université Paris 6, 4, place Jussieu, B.C. 188, 75252 Paris, Cedex 5, France. E-mail: [email protected] Received: 29 March 2005 / Accepted: 13 September 2005 Published online: 23 February 2006 – © Springer-Verlag 2006

Abstract: Recently, Bauke and Mertens conjectured that the local statistics of energies in random spin systems with discrete spin space should, in most circumstances, be the same as in the random energy model. Here we give necessary conditions for this hypothesis to be true, which we show to be satisfied in wide classes of examples: short range spin glasses and mean field spin glasses of the SK type. We also show that, under certain conditions, the conjecture holds even if energy levels that grow moderately with the volume of the system are considered. 1. Introduction In a recent paper [BaMe], Bauke and Mertens have formulated an interesting conjecture regarding the behaviour of local energy level statistics in disordered systems. Roughly speaking, their conjecture can be formulated as follows. Consider a random Hamiltonian, HN (σ ), i.e., a real-valued random function on some product space, S N , where S is a finite set, typically S = {−1, 1}, of the form HN (σ ) =

A (σ ),

(1.1)

A⊂N

where N are finite subsets of Zd of cardinality, say, N . The sum runs over subsets, A, of N and A are random local functions, typically of the form A (σ ) = JA

σx ,

(1.2)

x∈A Research supported in part by the DFG in the Dutch-German Bilateral Research Group “Mathematics of Random Spatial Models from Physics and Biology” and by the European Science Foundation in the Programme RDSES.

514

A. Bovier, I. Kurkova

where JA , A ⊂ Zd , is a family of (typically independent) random variables, defined on some probability space, (, F, P), √ whose distribution is not too singular. In such a situation, for typical σ , HN (σ ) ∼ N , while supσ HN (σ ) ∼ N . Bauke and Mertens then ask the following question: Given a fixed number, E, what is the statistics of the values N −1/2 HN (σ ) that are closest to this number, and how are configurations, σ , for which these good approximants of E are realised, distributed on S N ? Their conjectured answer, which at first glance seems rather surprising, is quite simple: find δN,E such that P(|N −1/2 HN (σ ) − E| ≤ bδN,E ) ∼ |S|−N b; then, the collection of points, −1 δN,E |N −1/2 HN (σ ) − E|, over all σ ∈ S N , converges to a Poisson point process on R+ . Furthermore, for any finite k, the k-tuple of configurations, σ 1 , σ 2 , . . . , σ k , where the k-best approximations are realised, is such that all of its elements have maximal Hamming distance between each other. In other words, the asymptotic behavior of these best approximants of E is the same, as if the random variables HN (σ ) were all independent Gaussian random variables with variance N , i.e., as if we were dealing with the random energy model (REM) [Der1]. Bauke and Mertens call this “universal REM like behaviour”. Mertens previously proposed such a conjecture in the special case of the number partitioning problem[Mer1]. In that case, the function HN is simply given by HN (σ ) =

N

Xi σ i ,

(1.3)

i=1

with Xi i.i.d. random variables uniformly distributed on [0, 1], σi ∈ {−1, 1}, and one is interested in the distribution of energies near the value zero (which corresponds to an optimal partitioning of the N random variables, Xi , into two groups such that their sum in each group is as similar as possible). In this case, his conjecture was proven to hold by Borgs, Chayes, and Pittel [BCP] and the same authors with Mertens [BCMP]. It should be noted that in this problem, one needs, of course, take care of the symmetry of the Hamiltonian under the transformation σ → −σ . An extension of this result in the spirit of the REM conjecture was proven recently in [BCMN], i.e., when the value zero is replaced by an arbitrary value, E. In [BK2] we generalised this result to the case of the k-partitioning problem, where the random function to be considered is vector-valued (consisting of the vector of differences between the sums of the random variables in each of the k subsets of the partition). To be precise, we considered the special case where the subsets of the partition are required to have the same cardinality, N/k (restricted k-partitioning problem). The general approach to the proof we developed in that paper sets the path towards the proof of the conjecture by Bauke and Mertens that we will present here. The universality conjecture suggests that correlations are irrelevant for the properties of the local energy statistics of disordered systems for energies near “typical energies”. On the other hand, we know that correlations must play a rôle for the extremal energies near the maximum of HN (σ ). Thus, there are two questions beyond the original conjecture that naturally pose themselves: (i) assume we consider instead of fixed E, N -dependent energy levels, say, EN = N α C. How fast can we allow EN to grow for the REM-like behaviour to hold? and (ii) what type of behaviour can we expect once EN grows faster than this value? We will see that the answer to the first question depends on the properties of HN , and we will give an answer in models with Gaussian couplings. The answer to question (ii) requires a detailed understanding of HN (σ ) as a random process, and we will be able to give a complete answer only in the case of the GREM,

Local REM Conjecture

515

when HN is a hierarchically correlated Gaussian process. This will be discussed in a companion paper [BK3]. Our paper will be organized as follows. In Chapter 2, we prove an abstract theorem, that implies the REM-like-conjecture under three hypothesis. This will give us some heuristic understanding why and when such a conjecture should be true. In Chapter 3 we then show that the hypothesis of the theorem are fulfilled in two classes of examples: p-spin Sherrington-Kirkpatrick like models and short range Ising models on the lattice. In both cases we establish conditions on how fast EN can be allowed to grow, in the case when the couplings are Gaussian. Note added. After this paper was submitted, Borgs et al. published an interesting preprint [BCMN2] where the following results were obtained: (i) In the p-spin SK models, with p = 1, 2, our growth conditions on EN are optimal, i.e. for EN ∼ CN 1/4 (p = 1), resp. EN ∼ CN 1/2 (p = 2), the REM conjecture cannot hold, at least if c is small enough. They also extended our results in these examples to the case of non-Gaussian interactions, provided they have some finite exponential moments. 2. Abstract Theorems In this section we will formulate a general result that implies the REM property under some concise conditions, that can be verified in concrete examples. This will also allow us to present the broad outline of the structure of the proof without having to bother with technical details. Our approach to the proof of the Mertens conjecture is based on the following theorem, which provides a criterion for Poisson convergence in a rather general setting. Theorem 2.1. Let Vi,M ≥ 0, i ∈ N, be a family of non-negative random variables satisfying the following assumptions: for any ∈ N and all sets of constants bj > 0, j = 1, . . . , , lim

M↑∞

P(∀j =1 Vij ,M < bj ) →

(i1 ,...,i )⊂{1,...,M}

bj ,

(2.1)

j =1

where the sum is taken over all possible sequences of different indices (i1 , . . . , i ). Then the point process PM =

M

δVi,M ,

(2.2)

i=1

on R+ , converges weakly in distribution, as M ↑ ∞, to the standard Poisson point process, P on R+ (i.e., the Poisson point process whose intensity measure is the Lebesgue measure). Remark. Theorem 2.1 was proven (in a more general form, involving vector valued random variables) in [BK2]. It is very similar in its spirit to an analogous theorem for the case of exchangeable variables proven in [BM] in an application to the Hopfield model. The rather simple proof in the scalar setting can be found in Chapter 13 of [Bo].

516


Naturally, we want to apply this theorem with Vi,M given by |N −1/2 HN (σ ) − EN |, properly normalised. We will now introduce a setting in which the assumptions of Theorem 2.1 are verified. Consider a product space S N where S is a finite set. We define on S N a real-valued random process, YN (σ ). Assume for simplicity that

Define on

EYN (σ ) = 0, E(YN (σ ))2 = 1.

(2.3)

bN (σ, σ ) ≡ cov(YN (σ ), YN (σ )).

(2.4)

SN , SN

Let us also introduce on the Gaussian process, ZN , that has the same mean and the same covariance matrix as YN . Let G be the group of automorphisms on SN , such that, for g ∈ G, YN (gσ ) = YN (σ ), and let F be the larger group, such that, for g ∈ F , |YN (gσ )| = |YN (σ )|. Let EN = cN α , c, α ∈ R, 0 ≤ α < 1/2,

(2.5)

be a sequence of real numbers, that is either a constant, c ∈ R, if α = 0, or converges to plus or minus infinity, if α > 0. We will define sets N as follows: If c = 0, we denote by N the set of residual classes of S N modulo G; if c = 0, we let N be the set of residual classes modulo F . We will assume throughout that | N | > κ N , for some κ > 1. Set 2 δN = π2 eEN /2 | N |−1 . (2.6) Note that for α < 1/2, δN is exponentially small in N . δN is chosen such that, for any b ≥ 0, lim | N |P(|ZN (σ ) − EN | < bδN ) = b.

N↑∞

(2.7)

⊗ For ∈ N, and any collection, σ 1 , . . . , σ ∈ N , we denote by BN (σ 1 , . . . , σ ) the covariance matrix of YN (σ ), with elements

bi,j (σ 1 , . . . , σ ) ≡ bN (σ i , σ j ). Assumption A.

(2.8)

η

(i) Let RN, denote the set

⊗ : ∀1≤i<j ≤ |bN (σ i , σ j )| ≤ N −η }. RN, ≡ {(σ 1 , . . . , σ ) ∈ N η

(2.9)

Then there exists a continuous decreasing function, ρ(η) > 0, on ]η0 , η˜ 0 [ (for some η˜ 0 ≥ η0 > 0), and µ > 0, such that η |RN, | ≥ 1 − exp −µ(η)N ρ(η) | N | . (2.10) (ii) Let ≥ 2, r = 1, . . . , − 1. Let ⊗ LN,r = (σ 1 , . . . , σ ) ∈ N : ∀1≤i<j ≤ |YN (σ i )| = |YN (σ j )|, rank(BN (σ 1 , . . . , σ )) = r .

(2.11)

Then there exists dr, > 0, such that, for all N large enough, |LN,r | ≤ | N |r e−dr, N .

(2.12)


517

(iii) For any ≥ 1, any r = 1, 2, . . . , , and any b1 , . . . , b ≥ 0, there exist con⊗ stants, pr, ≥ 0 and Q < ∞, such that, for any σ 1 , . . . , σ ∈ N for which 1 rank(BN (σ , . . . , σ )) = r, r N pr, . (2.13) P ∀i=1 : |YN (σ i ) − EN | < δN bi ≤ QδN Theorem 2.2. Assume Assumptions A hold. Assume that α ∈ [0, 1/2[ is such that, for some η1 ≤ η2 ∈]η0 , η˜ 0 [, we have: α < η2 /2,

(2.14)

α < η/2 + ρ(η)/2, ∀η ∈]η1 , η2 [,

(2.15)

α < ρ(η1 )/2.

(2.16)

and

η

1 , Furthermore, assume that, for any ≥ 1, any b1 , . . . , b > 0, and (σ 1 , . . . , σ ) ∈ RN, P ∀i=1 : |YN (σ i ) − EN | < δN bi = P ∀i=1 : |ZN (σ i ) − EN | < δN bi

+o(| N |− ).

(2.17)

Then, the point process, PN ≡

σ ∈ N

δ{δ −1 |YN (σ )−EN |} → P N

(2.18)

converges weakly, as N ↑ ∞, to the standard Poisson point process P on R+ . Moreover, for any > 0 and any b ∈ R+ , P ∀N0 ∃N≥N0 : ∃σ,σ :|bN (σ,σ )|> : |YN (σ ) − EN | ≤ |YN (σ ) − EN | ≤ δN b = 0. (2.19) Remark. Before giving the proof of the theorem, let us comment on the various assumptions. (i) Assumption A (i) holds with some η in any reasonable model, but the function ρ(η) is model dependent. (ii) Assumptions A (ii) and (iii) are also apparently valid in most cases, but can be tricky sometimes. An example where (ii) proved difficult is the k-partitioning problem, with k > 2 [BK2]. (iii) Condition (2.19) is essentially a local central limit theorem. In the case α = 0 it holds, if the Hamiltonian is a sum over independent random interactions, under mild decay assumptions on the characteristic function of the distributions of the interactions. Note that some such assumptions are obviously necessary, since if the random interactions take on only finitely many values, then also the Hamiltonian will take values on a lattice, whose spacings are not exponentially small, as would be necessary for the theorem to hold. Otherwise, if α > 0, this will require further assumptions on the interactions. We will leave this problem open in the present paper. It is of course trivially verified, if the interactions are Gaussian.

518


−1 Proof. We just have to verify the hypothesis of Theorem 2.1, for Vi,M given by δN |YN (σ ) − EN |, i.e., we must show that P ∀i=1 : |YN (σ i ) − EN | < bi δN → b1 · · · b . (2.20) ⊗l (σ 1 ,...,σ )∈ N

η

1 We split this sum into the sums over the set RN, and its complement. First, by the assumption (2.17) P ∀i=1 : |YN (σ i ) − EN | < bi δN η

1 (σ 1 ,...,σ )∈RN,

=

P ∀i=1 : |ZN (σ i ) − EN | < bi δN + o(1).

(2.21)

η1 (σ 1 ,...,σ )∈RN,

But, with C(EN ) = { x = (x1 , . . . , x ) ∈ R : ∀i+1 |EN − xi | ≤ δN bi },

P ∀i=1 : |ZN (σ i ) − EN | < bi δN = C (EN )

−1

e−(z,BN (σ ,...,σ )z)/2 dz, (2π)/2 det(BN (σ 1 , . . . , σ )) 1

(2.22) where BN (σ 1 , . . . , σ ) is the covariance matrix defined in (2.8). Since δN is exponenη1 tially small in N, we see that, uniformly for (σ 1 , . . . , σ ) ∈ RN, , the integral (2.22) equals √ −1 1 (2.23) (2δN / 2π) (b1 · · · b )e−(EN ,B (σ ,...,σ )EN )/2 (1 + o(1)), where we denote by EN the vector (EN , . . . , EN ). η2 η1 We treat separately the sum (2.21) taken over the smaller set, RN, ⊂ RN, , and the η1 η2 one over RN, \ RN, . 2 N −η2 → 0, by (2.17), (2.22), and (2.23), Since, by (2.14), η2 is chosen such that EN η2 each term in the sum over RN, equals √ 1 2 −η2 (2δN / 2π) (b1 · · · b )e− 2 EN (1+O(N )) (1 + o(1)) = (b1 · · · b )| N |− (1 + o(1)), uniformly for (σ 1 , . . . , σ ) ∈

(2.24)

η2 RN,l .

Hence by Assumption A (i) P ∀i=1 : |ZN (σ i ) − EN | < bi δN

η

2 (σ 1 ,...,σ )∈RN, 2 = |RN,l || N |− (b1 · · · b )(1 + o(1)) → b1 · · · bl .

η

η

(2.25)

η

2 1 (if it is non-empty, i.e., if η1 < η2 ), Now let us consider the remaining set RN, \ RN, 0 1 n and let us find η1 = η < η < · · · < η = η2 , such that

α < ηi /2 + ρ(ηi+1 )/2 ∀i = 0, 1, . . . , n − 1,

(2.26)


519 η

η

1 2 which is possible due to the assumption (2.15). Then let us split the sum over RN,l \RN, ηi+1 ηi into n sums, each over RN, \ RN, , i = 0, 1, . . . , n − 1. By (2.17), (2.22), and (2.23),

ηi

we have, uniformly for (σ 1 , . . . , σ ) ∈ RN, , √ 1 2 −ηi P ∀i=1 : |ZN (σ i ) − EN | < bi δN = (2δN / 2π ) (b1 · · · b )e− 2 EN (1+O(N )) (1 + o(1)) ≤ C| N |− eN

2α−ηi

,

(2.27)

for some constant C < ∞. Thus by Assumption A (i), P(∀i=1 : |ZN (σ i ) − EN | < bi δN ) η

η

i \R i+1 RN,l N,l 2α−ηi

⊗l ≤ C| N \ R i+1 || N |− eN N,l i+1 i ≤ C exp −µ(ηi+1 )N ρ(η ) + N 2α−η , η

(2.28) that, by (2.26), converges to zero, as N → ∞, for any i = 0, 1, . . . , n − 1. So the sum η1 η2 (2.21) over RN,l \ RN,l vanishes. η1 Now we turn to the sum over collections, (σ 1 , . . . , σ ) ∈ RN,l . We distinguish the cases when det(BN (σ 1 , . . . , σ )) = 0 and det(BN (σ 1 , . . . , σ )) = 0. For the contributions from the latter case, using Assumptions A (i) and (iii), we get readily that, ρ(η1 ) P ∀i=1 |YN (σ i ) − EN | < δN bi ≤ | N | e−µ(η1 )N Q|δN | N p η (σ 1 ,...,σ ) ∈RN1

rank(BN (σ 1 ,...,σ ))=

2 ≤ CN p exp −µ(η1 )N ρ(η1 ) + EN /2 .

(2.29)

The right-hand side of (2.29) tends to zero exponentially fast, if condition (2.16) is verified. Finally, we must deal with the contributions from the cases when the covariance matrix is degenerate, namely P(∀i=1 : |YN (σ i ) − EN | < bi δN ), (2.30) ⊗l (σ 1 ,...,σ )∈ N 1 rank(BN (σ ,...,σ ))=r

for r = 1, . . . , − 1. In the case c = 0, this sum is taken over the set LrN, , since σ and σ in N are different, iff |YN (σ )| = |YN (σ )|, by definition of N . In the case c = 0, this sum is taken over -tuples (σ 1 , . . . , σ ) of different elements of N , i.e., such that YN (σ i ) = YN (σ j ), for any 1 ≤ i < j ≤ . But for all N large enough, all terms in this sum over -tuples, (σ 1 , . . . , σ ), such that YN (σ i ) = −YN (σ j ), for some 1 ≤ i < j ≤ , equal zero, since the events {|YN (σ i ) − EN | < bi δN } and {| − YN (σ i ) − EN | < bj δN }, with EN = cN α , c = 0, are disjoint. Therefore (2.30) is

520


reduced to the sum over LrN, in the case c = 0 as well. Then, by Assumptions A (ii) and (iii), it is bounded from above by |LrN, |Q(δN )r N pr, ≤ | N |r e−dr, N Q(δN )r N pr, ≤ Ce−dr, N eEN /2 N pr, . (2.31) 2

2 = c2 N 2α , with α < 1/2. This bound converges to zero exponentially fast, since EN This concludes the proof of the first part of the theorem. The second assertion (2.19) is elementary: by (2.29) and (2.31), the sum of terms η1 ⊗2 P(∀2i=1 : |YN (σ i ) − EN | < δN b) over all pairs, (σ 1 , σ 2 ) ∈ N \ RN,2 , such that σ 1 = 2 σ , converges to zero exponentially fast. Thus (2.19) follows from the Borel-Cantelli lemma.

Finally, we remark that the results of Theorem 2.2 can be extended to the case when EYN (σ ) = 0, if α = 0, i.e., EN = c. Note that, e.g. the unrestricted number partitioning problem falls into this class. Let now ZN (σ ) be the Gaussian process with the same mean and covariances as YN (σ ). Let us consider both the covariance matrix, BN , and the mean of YN , EYN (σ ), as random variables on the probability space ( N , BN , Eσ ), where Eσ is the uniform law on N . Assume that, for any ≥ 1, D

BN (σ 1 , . . . , σ ) → Id , N ↑ ∞,

(2.32)

where Id denotes the identity matrix, and D

EYN (σ ) → D, N ↑ ∞, where D is some random variable D. Let δN = π2 K −1 | N |−1 ,

(2.33)

(2.34)

where K ≡ Ee−(c−D)

2 /2

.

(2.35)

Theorem 2.3. Assume that, for some R > 0, |EYN (σ )| ≤ N R , for all σ ∈ N . Assume that (2.10) holds for some η > 0 and that (ii) and (iii) of Assumptions A are valid. Assume η that there exists a set, QN ⊂ RN, , such that (2.17) is valid for any (σ 1 , . . . , σ ) ∈ QN , γ η and that |RN, \ QN | ≤ | N | e−N , with some γ > 0. Then, the point process δ δ −1 |YN (σ )−EN | → P (2.36) PN ≡ σ ∈ N

N

converges weakly to the standard Poisson point process P on R+ . Proof. We must prove again the convergence of the sum (2.20), that we split into three η sums: the first over QN , the second over RN, \ QN , and the third over the complement η of the set RN, . By assumption, (2.17) is valid on QN , and thus the terms of the first sum are reduced to −1 1

e−((z−EYN (σ ))BN (σ ,...,σ )(z−EYN (σ )))/2 dz (2π)/2 det(BN (σ 1 , . . . , σ )) ∀i=1,...,:|zi −c|< δN bi

√ −1 1 = (2δÑ / 2π ) (b1 · · · b )e−(c−EYN (σ ))B (σ ,...,σ )(c−EYN (σ ))/2 (1 + o(1)), (2.37)


521

with c ≡ (c, . . . , c), and E YN (σ ) ≡ (EYN (σ 1 ), . . . , EYN (σ )), since δN is exponentially small and |EYN (σ )| ≤ N R . By definition of δÑ , the quantities (2.37) are at most O(| N |− ), while, by the estimate (2.10) and by the assumption on the cardinality of η η η ⊗l \ RN, and in RN, \ QN RN, \ QN , the number of -tuples of configurations in N is exponentially smaller than | N | . Hence P(∀i=1 : |YN (σ i ) − EN | < bi δN ) (σ 1 ,...,σ )∈QN

=

√ −1 1 (2δÑ / 2π) (b1 · · · b )e−(c−EYN (σ ))B (σ ,...,σ )(c−EYN (σ ))/2

(σ 1 ,...,σ )∈QN

×(1 + o(1)) + o(1) √ −1 1 = (2δÑ / 2π) (b1 · · · b )e−(c−EYN (σ ))B (σ ,...,σ )(c−EYN (σ ))/2 ⊗ (σ 1 ,...,σ )∈ N

×(1 + o(1)) + o(1) b 1 · · · b = | N | K

e−(c−EYN (σ ))B

−1 (σ 1 ,...,σ )( c−EYN (σ ))/2

⊗ (σ 1 ,...,σ )∈ N

×(1 + o(1)) + o(1).

(2.38)

The last quantity converges to b1 · · · b , by the assumptions (2.32), (2.33) and (2.35). The sum of the probabilities, P(∀i=1 : |YN (σ ) − EN | < δN bi ), over all -tuples of −γ η RN, \ QN , contains at most | N | e−N terms, while, by Assumption A (iii), (and η since, for any (σ 1 , . . . , σ ) ∈ RN, , the rank of BN (σ 1 , . . . , σ ) equals ) each term is at most of order | N |− , up to a polynomial factor. Thus this sum converges to zero. ⊗l \ Finally, the sum of the same probabilities over the collections (σ 1 , . . . , σ ) ∈ N η RN, converges to zero, exponentially fast, by the same arguments as those leading to (2.29) and (2.31), with η1 = η. 3. Examples We will now show that the assumptions of our theorem are verified in a wide class of physically relevant models: 1) the Gaussian p-spin SK models, 2) SK-models with non-Gaussian couplings, and 3) short-range spin-glasses. In the last two examples we consider only the case α = 0. 3.1. p-spin Sherrington-Kirkpatrick models, 0 ≤ α < 1/2. In this subsection we illustrate our general theorem in the class of Sherrington-Kirkpatrick models. Consider S = {−1, 1}: √ N Ji1 ,...,ip σi1 σi2 · · · σip (3.1) HN (σ ) = N p

1≤i1 δ. Changing variables s = Np |IN4 (σ 1 , . . . , σ )| ≤ Q2−N N p/2

−(−1) N p (1+o(1)) φ(sm )2 dsm .

(3.36)

s >ζ m=1

Assumption B made on φ(s) implies that φ(s) is aperiodic, and thus |φ(s)| < 1, for any s = 0. Moreover, for any ζ > 0, there exists h(ζ ) > 0, such that |φ(s)| < 1 − h(ζ ), for all s with |s| > ζ /. Therefore, the right-hand side of (3.36) does not exceed Q2

−N

N

p/2

(1 − h(ζ ))

2−(−1) N p (1+o(1))−2

φ(sm )2 dsm ,

(3.37)

s >η m=1

where the integral is finite again due to Assumption B. Therefore, IN4 (σ 1 , . . . , σ ) is exponentially smaller than 2−N . This proves (2.17) on QN and hence the theorem. 3.3. Short range spin glasses. As a final example, we consider short-range spin glass models. To avoid unnecessary complications, we will look at models on the d-dimensional torus, N , of length N . We consider Hamiltonians of the form rA J A σ A , (3.38) HN (σ ) ≡ −N −d/2 A⊂N

where e σA ≡ x∈A σx , rA are given constants, and JA are random variables. We will introduce some notation: (a) Let AN denote the set of all A ⊂ N , such that rA = 0. (b) For any two subsets, A, B ⊂ N , we say that A ∼ B, iff there exists x ∈ N such that B = A + x. We denote by A the set of equivalence classes of AN under this relation. We will assume that the constants, rA , and the random variables, JA , satisfy the following conditions: (i) rA = rA+x , for any x ∈ N ; (ii) there exists k ∈ N, such that any equivalence class in A has a representative A ⊂ k ; we will identify the set A with a uniquely chosen set of representatives contained in k . 2 d (iii) A⊂N : rA = N . (iv) JA , A ∈ Zd , are a family of independent random variables, such that (v) JA and JA+x are identically distributed for any x ∈ Zd ; (vi) EJA = 0 and EJA2 = 1, and EJA3 < ∞;

530


(vii) For any A ∈ A, the Fourier transform φA (s) ≡ E exp (isJA ), of JA satisfies |φA (s)| = O(|s|−1 ) as |s| → ∞. Observe that EHN (σ ) = 0, b(σ, σ ) ≡ N −d EHN (σ )HN (σ ) = N −d

rA2 σA σA ≤ 1,

(3.39)

A⊂N

where equality holds, if σ = σ . Note that YN (σ ) = YN (σ ) (resp. YN (σ ) = −YN (σ ) ), if and only if, for all A ∈ AN , σA = σA (resp. σA = −σA ). E.g., in the standard Edwards-Anderson model, with nearest neighbor pair interaction, if σx differs from σx on every second site, x, then YN (σ ) = −YN (σ ), and if σ = −σ , YN (σ ) = YN (σ ). In general, we will consider two configurations, σ, σ ∈ S N , as equivalent, iff for all A ∈ AN , σA = σA . We denote the set of these equivalence classes by N . We will assume in the sequel that there is d a finite constant, ≥ 1, such that | N | ≥ 2N −1 . In the special case of c = 0, the equivalence relation will be extended to include the case σA = −σA , for all A ∈ AN . In most reasonable examples (e.g. whenever nearest neighbor pair interactions are included in the set A), the constant ≤ 2 (resp. ≤ 4, if c = 0)). Theorem 3.7. Let c ∈ R, and N be the space of equivalence classes defined before. 2 /2 π −1 c Let δN ≡ | N | e 2 . Then the point process PN ≡

σ ∈ N

δ{δ −1 |HN (σ )−c|} , N

(3.40)

converges weakly to the standard Poisson point process on R+ . If, moreover, the random variables J A are Gaussian, then, for any c ∈ R, and 2α c2 /2 π −1 N 0 ≤ α < 1/4, with δN ≡ | N | e 2 , the point process PN ≡

σ ∈ N

δ{δ −1 |HN (σ )−cN α |} , N

(3.41)

converges weakly to the standard Poisson point process on R+ . Proof. We will now show that Assumptions A of Theorem 2.3 hold. First, the point (i) of Assumption A is verified due to the following proposition. η

Proposition 3.8. Let RN, be defined as in (2.9). Then, in the setting above, for all 0 ≤ η < 21 , d(1−2η) η , (3.42) |RN, | ≥ | N | 1 − e−hN with some constant h > 0. Proof. Let Eσ denote the expectation under the uniform probability measure on {−1, 1}N . We will show that there exists a constant, K > 0, such that, for any σ , and any 0 ≤ δN ≤ 1, 2 d N ). Pσ (σ : b(σ, σ ) > δN ) ≤ exp(−KδN

(3.43)


531

Note that without loss, we can take σ ≡ 1. We want to use the exponential Chebyshev inequality and thus need to estimate the Laplace transform   −d 2 (3.44) rA σA  . Eσ exp tN A∈N

Let us assume for simplicity that N = nk is a multiple of k, and introduce the sub-lattice, N,k ≡ {0, k, . . . , (n − 1)k, nk}d . Write

rA2 σA =

2 rA+y+x σA+y+x ≡

A∈A y∈N,k x∈k

A∈N

(3.45)

Zx (σ ),

x∈k

where

Zx (σ ) =

(3.46)

Yy,x (σ )

y∈N,k

has the nice feature that, for fixed x, the summands 2 rA+y+x σA+y+x Yx,y (σ ) ≡ A∈A

are independent for different y, y ∈ n,k (since the sets A + y + x and A + y + x are disjoint for any A, A ∈ k ). Using the Hölder inequality repeatedly,   k −d d Zx (σ ) ≤ Eσ etk Zx (σ ) Eσ exp t x∈k

x∈k

=

Eσ etk

dY x,y (σ )

k −d

x∈k y∈N,k

N d k −d d = Eσ etk Y0,0 (σ ) .

(3.47)

It remains to estimate the Laplace transform of Y0,0 (σ ),   Eσ exp tk d Y0,0 (σ ) = Eσ tk d rA2 σA  ,

(3.48)

A∈k

and, since Eσ σA = 0, using that ex ≤ 1 + x +  Eσ exp tk d



x 2 |x| 2 e ,





rA2 σA  ≤ Eσ exp 

A∈k

≡ Eσ exp

t2 2

k 2d 

 rA2  e

tk d

2 A∈k rA

 

A∈k

t 2 tD , Ce 2

(3.49)

532


so that

 Eσ exp tN −d

x∈k



2 −d t N −d tD  , Ce Zx (σ ) ≤ exp N 2

(3.50)

with constants, C, C , D, that do not depend on N . To conclude the proof of the lemma, the exponential Chebyshev inequality gives,

t2 −d (3.51) Pσ b(σ, σ ) > δN ≤ exp −δN t + N −d C etN D . 2 Choosing t = N d δN , this gives 2 d Pσ b(σ, σ ) > δN ≤ exp −δN N 1 − C eδN D /2 .

(3.52)

Choosing small enough, but independent of N , we obtain the assertion of the lemma. To verifyAssumptionsA (ii) and (iii), we need to introduce the matrix C = C(σ 1 , . . . , with columns and |AN | rows, indexed by the subsets A ∈ AN : the elements of each of its column are rA σA1 , rA σA2 , . . . , rA σA , so that C T C is the covariance matrix, BN (σ 1 , . . . , σ ), up to a multiplicative factor N d . Assumption (ii) is verified due to Proposition 3.3. In fact, let us reduce C to the matrix ˜ ˜ 1 , . . . , σ ) with columns σ 1 , σ 2 , . . . , σ , without the constants rA . Then, C = C(σ A A A exactly as in the case of p-spin SK models, by Proposition (3.3), for any (σ 1 , . . . , σ ) ∈ ˜ 1 , . . . , σ ) can contain at most 2r − 1 different columns. Hence, LN d ,r the matrix C(σ σ )

d

d

|LN d ,r | = O((2r − 1)N ) while | N |r ≥ (2N / )r . Assumption (iii) is verified as well, and its proof is completely analogous to that of Proposition 3.5. The key observation is that, again, the number of possible non-degenerate matrices C¯ r×r that can be obtained from Cp (σ 1 , . . . , σ ) is independent of N . But this is true since, by assumption, the number of different constants rA is N -independent. Finally, we define QN as follows. For any A ∈ A, let j η,A QN, = (σ 1 , . . . , σ ) : ∀1≤i<j ≤ rA2 σAi σA < |A|−1 N −η . (3.53) x∈Zd :x+A⊂N

Let us define QN =

η,A A∈A QN,

η

⊂ RN, . By Proposition (3.8), applied to a model

where |A| = 1, for any A ∈ A, we have |SN⊗ \ QN, | ≤ 2N exp(−hA N d(1−2η) ), with η some hA > 0. Hence, |RN, \QN | has cardinality smaller than | N | exp(−hN d(1−2η) ), with some h > 0. The verification of (2.17) on QN is analogous to the one in Theorem 3.4, using the analogue of Proposition 3.6. We only note a small difference in the analysis of the term IN4 , where we use the explicit construction of QN . We represent the corresponding generating function as the product of |A| terms over different equivalence classes of A, with representatives A ⊂ k , each term being x∈Zd :x+A∈N φ(N −d/2 rA )). Next, we use the fact that for any (σ 1 , . . . , σ ) ∈ Q each 1 × (t1 σx+A + · · · + t σx+A N of these |A| terms is a product of at least 2 − 1 (and of course at most 2 ) different η,A

d


533

terms, each is taken to the power |A|−1 N d 2− (1 + o(1)). This proves the first assertion of the theorem. The proof of the second assertion, i.e., the case α > 0 with Gaussian variables JA is immediate from the estimates above and the abstract Theorem 2.2, in view of the fact that the condition (2.17) is trivially verified. Acknowledgement. We would like to thank Stephan Mertens for interesting discussions.

References [BFM]

Bauke, H., Franz, S., Mertens, S.: Number partitioning as random energy model. J. Stat. Mech.: Theory and Experiment, page P04003 (2004) [BaMe] Bauke, H., Mertens, S.: Universality in the level statistics of disordered systems. Phys. Rev. E 70, 025102(R) (2004) [BCP] Borgs, C., Chayes, J., Pittel, B.: Phase transition and finite-size scaling for the integer partitioning problem. Random Structures Algorithms 19(3–4), 247–288 (2001) [BCMP] Borgs, C., Chayes, J.T., Mertens, S., Pittel, B.: Phase diagram for the constrained integer partitioning problem. Random Structures Algorithms 24(3), 315–380 (2004) [BCMN] Borgs, C., Chayes, J.T., Mertens, S., Nair, C.: Proof of the local REM conjecture for number partitioning. Preprint 2005, available at http://research.microsoft.com/∼chayes/ [BCMN2] Borgs, C., Chayes, J.T., Mertens, S., Nair, C.: Proof of the local REM conjecture for number partitioning II: Growing energy scales. http://arvix.org/list/ cond-mat/0508600, 2005 [Bo] Bovier, A.: Statistical mechanics of disordered systems. In: Cambridge Series in Statistical and Probabilisitc mathematics, Cambridge University Press, to appear May 2006 [BK1] Bovier, A., Kurkova, I.: Derrida’s generalised random energy models. I. Models with finitely many hierarchies. Ann. Inst. H. Poincaré Probab. Statist. 40(4), 439–480 (2004) [BK2] Bovier, A., Kurkova, I.: Poisson convergence in the restricted k-partioning problem. Preprint 964, WIAS, 2004, available at http://www.wias-berlin.de/people/files/publications.html, to appear in Random Structures Algorithms (2006) [BK3] Bovier, A., Kurkova, I.: A tomography of the GREM: beyond the REM conjecture. Commun. Math. Phys. 263(2), 535–552 (2006) [BKL] Bovier, A., Kurkov, I., Löwe, M.: Fluctuations of the free energy in the REM and the p-spin SK models. Ann. Probab. 30, 605–651 (2002) [BM] Bovier, A., Mason, D.: Extreme value behaviour in the Hopfield model. Ann. Appl. Probab. 11, 91–120 (2001) [Der1] Derrida, B.: Random-energy model: an exactly solvable model of disordered systems. Phys. Rev. B (3) 24(5), 2613–2626 (1981) [Der2] Derrida, B.: A generalisation of the random energy model that includes correlations between the energies. J. Phys. Lett. 46, 401–407 (1985) [Mer1] Mertens, S.: Phase transition in the number partitioning problem. Phys. Rev. Lett. 81(20), 4281–4284 (1998) [Mer2] Mertens, S.: A physicist’s approach to number partitioning. Theoret. Comput. Sci. 265(1–2), 79–108, (2001) Communicated by M. Aizenman


Communications in


A Tomography of the GREM: Beyond the REM Conjecture Anton Bovier1,2 , Irina Kurkova3 1 2 3

Weierstraß–Institut für Angewandte Analysis und Stochastik, Mohrenstrasse 39, 10117 Berlin, Germany. E-mail: [email protected] Institut für Mathematik, Technische Universität Berlin, Strasse des 17. Juni 136, 12623 Berlin, Germany Laboratoire de Probabilités et Modèles Aléatoires, Université Paris 6, 4, place Jussieu, B.C. 188, 75252 Paris, Cedex 5, France. E-mail: [email protected]

Received: 29 March 2005 / Accepted: 13 September 2005 Published online: 23 February 2006 – © Springer-Verlag 2006

Abstract: In a companion paper we proved that in a large class of Gaussian disordered spin systems the local statistics of energy values near levels N 1/2+α with α < 1/2 are described by a simple Poisson process. In this paper we address the issue as to whether this is optimal, and what will happen if α = 1/2. We do this by analysing completely the Gaussian Generalised Random Energy Models (GREM). We show that the REM behaviour persists up to the level βc N, where βc denotes the critical temperature. We show that, beyond this value, the simple Poisson process must be replaced by more and more complex mixed Poisson point processes. 1. Introduction In a companion paper [4] we proved (for a large class of models) a conjecture by Bauke and Mertens [1] on the universality of the local energy level statistics in disordered systems, the so-called local REM conjecture. While in the original form this conjecture concerns the distribution of the values, HN (σ ), of the (random) Hamiltonian, HN , near √ energies EN = E N (where N is the volume of the system), we could show that, at least in the case of Gaussian couplings, the result still holds for energies EN = EN 1/2+α , with α > 0. In the case of short range spin glasses, it is true for all α < 1/4, and in the case of the p-spin SK models it holds for α < 1/4 if p = 1, and even for α < 1/2, if p ≥ 2. It is natural to ask whether these are true thresholds, and also what should happen for larger values of the energy. As we have noted in [4], the thresholds cannot be surpassed with the method of proof that was used there.1 In this paper we hope to provide some Research supported in part by the DFG in the Dutch-German Bilateral Research Group “Mathematics of Random Spatial Models from Physics and Biology” and by the European Science Foundation in the Programme RDSES. 1 After this paper had been submitted, Borgs et al [2] published a preprint that recovers our results on the p-spin model with p = 1, 2, and extends them to non-Gaussian couplings. They also prove via the estimates on moments, that if EN = cN 3/4 and p = 1, [resp. EN = cN and p = 2] with c small, convergence to a Poisson process cannot hold. No explicit analysis of what happens then is, however, given.

536


insight into these questions by studying a model that allows more explicit computations and hence lets us provide the full picture for all energies, the Generalised Random Energy Model (GREM) of Derrida. The result we obtain gives a somewhat extreme microcanonical picture of the GREM, exhibiting in a somewhat tomographic way the distribution of states in a tiny vicinity of any value of the energy. Let us briefly recall the definition of the GREM. We consider parameters α0 = 1 < α1 , . . . , αn < 2 with ni=1 αi = 2, a0 = 0 < a1 , . . . , an < 1, ni=1 ai = 1. Let N = {−1, 1}N be the space of 2N spin configurations σ . Let Xσ1 ···σl , l = 1, . . . , n, be independent standard Gaussian random variables indexed by configurations σ1 √ . . . σl ∈ {−1, 1}N ln(α1 ···αl )/ ln 2 . We define the Hamiltonian of the GREM as HN (σ ) ≡ N Xσ , with Xσ ≡

√ √ a1 Xσ1 + · · · + an Xσ1 ···σn .

(1.1)

Then cov (Xσ , Xσ ) = A(dN (σ, σ )), where dN (σ, σ ) = N −1 [min{i : σi = σi } − 1], and A(x) is a right-continuous step function on [0, 1], such that, for any i = 0, 1, . . . , n, A(x) = a0 + · · · + ai , for x ∈ [ln(α0 α1 , · · · αi )/ ln 2 , ln(α0 α1 , · · · αi+1 )/ ln 2). Set J0 ≡ 0, and, define, for l > 0, ln(αJl−1+1 · · · αJ ) ln(αJ +1 · · · αm ) Jl = min n ≥ J > Jl−1 : < ∀m ≥ J +1 , (1.2) aJl−1+1 +· · ·+aJ aJ +1 +· · · + am up to Jk = n. Then, the k segments connecting the points (a0 + · · · + aJl , ln(α0 α1 · · · αJl )/ ln 2), for l = 0, 1, . . . , k form the concave hull of the graph of the function A(x). Let a¯ l = aJl−1 +1 + aJl−1 +2 + · · · + aJl , α¯ l = αJl−1 +1 αJl−1 +2 · · · αJl ..

(1.3)

ln α¯ 1 ln α¯ 2 ln α¯ k < < ··· < . a¯ 1 a¯ 2 a¯ k

(1.4)

Then

Moreover, as it is shown in Proposition 1.4 of [3], for any l = 1, . . . , k, and for any Jl−1 + 1 ≤ i < Jl , we have ln(αJl−1 +1 · · · αi )/(aJl−1 +1 + · · · + ai ) ≥ ln(α¯ l )/a¯ l . Hence ln(αJl−1 +1 · · · αj ) ln α¯ l = min . j =Jl−1 +1,Jl−1 +2,...,n aJl−1 +1 + · · · + aj a¯ l

(1.5)

To formulate our results, we also need to recall from [3] (Lemma 1.2) the point process of Poisson cascades P l on Rl . It is best understood in terms of the following iterative construction. If l = 1, P 1 is the Poisson point process on R1 with the intensity measure K1 e−x dx. To construct P l , we place the process P l−1 on the plane of the first l − 1 coordinates and through each of its points draw a straight line orthogonal to this plane. Then we put on each of these lines independently a Poisson point process with intensity measure Kl e−x dx. These points on Rl form the process P l . The constants K1 , . . . , Kl > 0 (that are different from 1 only in some degenerate cases) are defined in the formula (1.14) of [3].

Beyond the REM Conjecture

537

We will also need the following facts concerning P l from Theorem 1.5 of [3]. Let γ1 > γ2 > · · · > γl > 0. There exists a constant h > 0, such that, for all y > 0, P ∃(x1 , . . . , xl ) ∈ P l , ∃j = 1, . . . , l : γ1 x1 + γ2 x2 + · · · + γj xj > (γ1 + · · · + γj )y ≤ exp(−hy). (1.6) Here and below we identify the measure P l with its support, when suitable. Furthermore, for any y ∈ R, #{(x1 , . . . , xl ) ∈ P l : x1 γ1 + · · · + xl γl > y} < ∞ a.s. Moreover, let β > 0 be such that βγ1 > · · · > βγl > 1. The integral l = eβ(γ1 x1 +···γl xl ) P l (dx1 , . . . , dxl )

(1.7)

(1.8)

Rl

is understood as limy→−∞ Il (y) with eβ(γ1 x1 +···+γl xl ) P l (dx1 , . . . , dxl ) Il (y) = (x1 ,...,xl )∈Rl : ∃i,1≤i≤l:γ1 x1 +···+γi xi >(γ1 +···+γi )y

=

l

eβ(γ1 x1 +···+γl xl ) P l (dx1 , . . . , dxl , ). (1.9)

j =1

(x1 ,...,xl )∈Rl : ∀i=1,...,j −1:γ1 x1 +···+γi xi ≤(γ1 +···+γi )y γ1 x1 +···+γj xj >(γ1 +···+γj )y

It is finite, a.s., by Proposition 1.8 of [3]. To keep the paper self-contained, let us recall how this fact can be established by induction starting from l = 1. The integral (1.8), ∞ in the case l = 1, is understood as limy→−∞ I1 (y). Here I1 (y) = y eβγ1 x1 P1 (dx) is finite, a.s., since P1 contains a finite number of points on [y, ∞[, a.s. Furthermore, by [5] or Proposition 1.8 of [3], limy→−∞ I1 (y) is finite, a.s., since E supy ≤y (I1 (y ) − I1 (y)) converges to zero exponentially fast, as y → −∞, provided that βγ1 > 1. If l ≥ 1, each term in the representation (1.9) is determined and finite, a.s., by induction. In fact, to see this for the j th term, given any realization of P l in Rl , take its projection on the plane of the first j coordinates. Then by (1.7), there exists only a finite number of points (x1 , . . . , xj ) of P j , such that γ1 x1 + · · · + γj xj > (γ1 + · · · + γj )y, a.s. Whenever the first j coordinates of a point of P l in Rl are fixed, the remaining l − j coordinates are distributed as P l−j on Rl−j . Then the integral over the function eβ(γj +1 xj +1 +···+γl xl ) over these coordinates is defined by induction and is finite, a.s., provided that βγj +1 > · · · > βγl > 1. Thus the j th term in (1.9) is the sum of an a.s. finite number of terms and each of them is a.s. finite. Finally, again by Proposition 1.8 of [3], limy→−∞ Il (y) is finite, a.s., since E supy ≤y (Il (y ) − Il (y)) → 0 as y → −∞ exponentially fast provided that βγ1 > · · · > βγl > 1. Let us define the constants dl , l = 0, 1, . . . , k, where d0 = 0 and dl ≡

l

i=1

a¯ i 2 ln α¯ i .

(1.10)

538


Finally, we define the domains Dl , for l = 0, . . . , k − 1, as   k   2 ln α¯ l+1 a¯ j . Dl ≡ |y| < dl +   a¯ l+1

(1.11)

j =l+1

It is not difficult to verify that D0 ⊂ D1 ⊂ · · · ⊂ Dk−1 . We are now ready to formulate the main result of this paper. Theorem 1.1. Let a sequence cN ∈ R be such that lim sup cN ∈ D0 and lim inf cN ∈ D0 . Then, the point process M0N =

N→∞

σ ∈N

δ

2N +1 (2π)−1/2 e

2 N/2 −cN

Xσ −cN √N

N→∞

(1.12)

converges to the Poisson point process with intensity measure the Lebesgue measure. Let, for l = 1, . . . , k − 1, c ∈ Dl \ Dl−1 (where Dl−1 is the closure of Dl−1 ). Define c˜l = |c| − dl , βl =

c˜l , a¯ l+1 + · · · + a¯ k

γi =

(1.13)

a¯ i /(2 ln α¯ i ), i = 1, . . . , l,

(1.14)

and Rl (N ) =

l 2(α¯ l+1 · · · α¯ k )N exp(−N c˜l βl /2)

(4N π ln α¯ j )−βl γj /2 . 2π(a¯ l+1 + · · · + a¯ k ) j =1

(1.15)

Then, the point process MlN =

σ ∈N

δR (N)√a X 1 σ l

√ √ an Xσ1 ...σn −c N

1 +···+

(1.16)

converges to mixed Poisson point process on [0, ∞[: given a realization of the random variable l , its intensity measure is l dx. The random variables l are defined in terms of the Poisson cascades Pl via (1.17) l = eβl (γ1 x1 +···γl xl ) P l (dx1 , . . . , dxl ).

Rl

Remark. Note that d0 + 2 lna¯ 1α¯ 1 kj =1 a¯ j = 2 lna¯ 1α¯ 1 = βc (see [3]), the inverse critical temperature in the GREM. Thus the first part of the theorem asserts that the REM conjecture holds for exactly those energies that satisfy EN < Nβc . It is tempting to conjecture that this model independent formulation of the result might be true more generally. The next section will be devoted to the proof of this result. Before doing this, we conclude the present section with a heuristic interpretation of the main result. Let us first look at (1.12). This statement corresponds to the REM-conjecture of Bauke and Mertens [1]. It is quite remarkable that this conjecture holds in the case of the GREM for energies of the form cN (namely for c ∈ D0 ).


539

In the REM [7], Xσ are 2N independent standard √Gaussian random variables and a statement (1.12) would hold for all c with |c| < 2 ln 2: it is a well√known result from the theory of independent random variables [9]. The value c = 2 ln 2 corresponds to the maximum √ of 2N independent standard Gaussian random variables, i.e., √ −1/2 maxσ ∈N N Xσ → 2 ln 2 a.s. Therefore, at the level c = 2 ln 2, one has the emergence of the extremal process. More precisely, the point process δ√2N ln 2X −√2N ln 2+ln(4πN ln 2)/√8N ln 2 , (1.18) σ

σ ∈N

that is commonly written as uN (x) =

σ ∈N δu−1 N (Xσ )

with

√ ln(4πN ln 2) x 2N ln 2 − √ +√ , 2 2N ln 2 2N ln 2

(1.19)

√ converges to the Poisson point process P 1 defined above (see e.g. [9]). For√c > 2 ln 2, the probability that one of the Xσ will be outside of the domain {|x| < c N }, goes to zero, and thus it makes no sense to consider such levels. In the GREM, N −1/2 maxσ ∈N Xσ converges to√the value dk ∈ ∂Dk−1 (1.10) (see Theorem 1.5 of [3]) that is generally smaller than 2 ln 2. Thus it makes no sense to consider levels with c ∈ D k−1 . However, the REM-conjecture is not true for all levels in Dk−1 , but only in the smaller domain D0 . To understand the statement of the theorem outside D0 , we need to recall how the extremal process in the GREM is related to the Poisson cascades introduced above. Let us set Nwl ≡ {−1, 1}Nwl , where wl = ln(α¯ 1 · · · α¯ l )/ ln 2

(1.20)

with the notation (1.3). Let us also define the functions Ul,N (x) ≡ N 1/2 dl − N −1/2

l

γi ln(4π N ln α¯ i )/2 + N −1/2 x

(1.21)

i=1

with the notations (1.3), (1.10), (1.14), and set σj ≡ X

j √

a i Xσ1 ...σi ,

Xˇ σj ≡

i=1

n √ a i Xσ1 ...σi .

From what was shown in [3], for any l = 1, . . . , k, the point process, δ −1 Jl El,N ≡ σˆ ∈N wl

(1.22)

i=j +1

Ul,N (Xσˆ )

(1.23)

converges in law to the Poisson cluster process, El , given in terms of the Poisson cascade, P l , as (1.24) El ≡ P (l) (dx1 , . . . , dxl )δl γi xi . i=1

Rl

540


In view of this observation, we can re-write the definition of the process MlN as follows: J √ δ MlN = Jl )) , (1.25) Rl (N)Xˇ l − N |c|−dl +N −1 (l,N −U −1 (X σˆ σˇ

σˆ ∈wl N σˇ ∈(1−wl )N

l,N

σˆ

with the abbreviation l,N ≡

l

γi ln(4πN ln α¯ i )/2

(1.26)

i=1

(c is replaced by |c| due to the symmetry of the standard Gaussian distribution). The normalizing constant, Rl (N ), is chosen such that, for any finite value, U , the point process J √ δ (1.27) Rl (N)Xˇ l − N |c|−dl +N −1 (l,N −U ) , σˇ ∈(1−wl )N

σˆ σˇ

converges to the Poisson point processes on R+ , with intensity measure given by eU times Lebesgue measure, which is possible precisely because c ∈ Dl \ Dl−1 , that is |c| − dl is smaller than the a.s. limit of N −1/2 maxσˇ ∈(1−w )N Xˇ σJˆ lσˇ . This is completely l analogous to the analysis in the domain D0 . Thus each term in the sum over σˆ in (1.25) −1 l l , (Xσˆ ), i.e., to an element of the extremal process of X that gives rise to a “finite” Ul,N σˆ gives rise to one Poisson process with a random intensity measure in the limit of MlN . This explains how the statement of the theorem can be understood, and also shows what the geometry of the configurations realizing these mixed Poisson point processes will be. Let us add that, if c ∈ ∂Dk−1 , i.e. |c| = dk , then one has the emergence of the extremal point process (1.23) with l = k, i.e. σ ∈N δ{√N(Xσ −dk √N+N −1/2 k,N )} converges to (1.24) with l = k, see [3]. 2. Proof of Theorem 1.1 Note that (1.17) is finite a.s. since γ1 > · · · > γl by (1.4) and βl γl > 1 by the definition of βl . Note also that c can be replaced by |c| in (1.12) and (1.16) due to the symmetry of the standard Gaussian distribution. Let MlN (A) be the number of points of MlN in a Borel subset A ⊂ R+ . We will p show that for any finite disjoint union of intervals, A = ∪q=1 [aq , bq ), the avoidance function converges P(MlN (A) = 0) → E exp(−|A|l ),

(2.1)

where of course 0 = 1 in the case l = 0. Note that in that case, the right-hand side is the avoidance function of a Poisson point process with intensity 1, while in all other cases, this is the avoidance function of a mixed Poisson point process. To conclude the proof in the case l = 0, it is enough to show that for any segment A = [a, b), EM0N (A) → (b − a), N → ∞.

(2.2)


541

Then the result (1.12) would follow from Kallenberg’s theorem, see [10] or [9]. In the cases l = 1, . . . , k − 1 we will prove that the family {MlN }∞ N=1 is uniformly tight: by Proposition 9.1V of [6], this is equivalent to the fact that, for any compact segment, A = [a, b], and for any given > 0, one can find a large enough integer, R, such that P(MlN (A) > R) < , ∀N ≥ 1.

(2.3)

Finally, we will show that the limit of any weakly convergent subsequence of MlN is a simple point process, that is without double points (see Definition 7.1IV in [6]). Theorem 7.3II of [6] asserts that a simple point process is uniquely characterized by its avoidance function, which then implies the result (1.16) claimed in Theorem (1.1). To prove (2.1), we need the following lemma. p

Lemma 2.1. Let A = ∪q=1 [aq , bq ), 0 ≤ a1 < b1 < a2 < b2 < · · · < aq < bq , with p |A| = q=1 (bq − aq ). Let 0 < f < 1, K(N) > 0 be a polynomial in N . We write p K(N)f N A ≡ ∪q=1 [K(N)f N aq , K(N )f N bq ). For any i = 1, 2, . . . , n, any > 0, δ > 0 small enough, and M > 0, there exists N0 , such that, for all N ≥ N0 and for all y, such that +···+2 ln αn +2 ln f + ) max max (ai +···+an )(2 lnaαmm+···+a , n m=i+1,...,n (2 ln αi+1 + · · · + 2 ln αn + 2 ln f + ) ≤ y 2 ≤ M, (2.4) the probability, P ∀σˇ ∈ {−1, 1}N(ln(αi ···αn )/ ln 2) : √

√ − y N ∈ K(N )f N A ,

(2.5)

with Xˇ σi−1 defined by (1.22), is bounded from above and below, respectively, by ˇ 2 N · · · αnN e−y N/2 . exp − (1 ± δ)|A|(2π)−1/2 2K(N)f N αiN αi+1

(2.6)

Xˇ σi−1 ˇ

ai + · · · + a n

Proof. Let us define the quantity PN (i, y, f, K(N ))≡P ∃σˇ ∈{−1, 1}(lnαi+1+···αn )/ln2: √

Xˇ σiˇ

ai +· · ·+an

√ −y N∈K(N )f N A . (2.7)

We will show that, for any > 0 small enough and M > 0 large enough, we have N PN (i, y, f, K(N)) ∼ (2π)−1/2 2K(N)f N |A|αi+1 · · · αnN e−y

2 N/2

, as N → ∞, (2.8)

uniformly for the parameter y in the domain max

m=i+1,...,n

(ai + · · · + an )(2 ln αm + · · · + 2 ln αn + 2 ln f + ) ≤ y 2 ≤ M. (2.9) am + · · · + a n

542


α N Then, the probability (2.5) equals 1 − PN (i, y, f, K(N )) i , where the asymptotics of the quantity PN (i, y, f, K(N)) is established in (2.8). Moreover, by the assumption (2.4), PN (i, y, f, K(N )) ≤ (2π)−1/2 2K(N )|A| exp(− N/2) → 0.

(2.10)

Then the elementary inequality, −x − ≤ ln(1 − x) ≤ −x, that holds for |x| < 1/2, leads to (2.6). Therefore we concentrate on the proof of the asymptotics (2.8). Let X be a standard Gaussian random variable. Then x2

PN (n, y, f, K(N )) √ 2 = P(|X − y N | ∈ K(N)f N A) ∼ (2π)−1/2 2K(N )f N |A|e−y N/2 , N → ∞, (2.11) uniformly for y 2 ≤ M. This implies (2.8) for i = n. Note also that √ N PN (i, y, f, K(N )) ≤ αi+1 · · · αnN P(|X − y N | ∈ K(N )f N A),

(2.12)

so that the upper bound for (2.8) is immediate. We will establish the lower bound by induction downwards from i = n to i = 1, using the identity √ √ √ ∞ 2 ai +· · · + an y N − ai t dt e−t /2

PN (i, y, f, K(N )) = , f, 1− 1−PN i +1, √ N (ai+1 +· · ·+an ) 2π −∞ √ α N ai + · · · + a n i+1 . (2.13) K(N ) √ ai+1 + · · · + an By the induction hypothesis for i + 1, √ √ √ ai + · · · + an y − ai t ai + · · · + a n PN i + 1,

K(N ) , f, √ ai+1 + · · · + an N (ai+1 + · · · + an ) √ √ √ √ ( ai +···+an y N− ai t)2 a +· · ·+a − i n N N N N −1/2 2(ai+1 +···+an ) 2K(N)f |A|αi+2 αi+3 · · · αn e , ∼ (2π ) √ ai+1 + · · · +an (2.14) uniformly for all y, t that satisfying (ai+1 + · · · + an )(2 ln αm + · · · + 2 ln αn + 2 ln f + i+1 ) m=i+2,...,n am + · · · + a n √a + · · · + a y √N − √a t 2 i n i

≤ ≤ Mi+1 , (2.15) N (ai+1 + · · · + an ) max

for any i+1 > 0 small enough and Mi+1 > 0 large enough. The right-hand side of this inequality reads √ √ √ √ ai +· · · + an y − ai+1 +· · · + an Mi+1 − NT1 (y) ≡ N ≤t √ ai √ √ √ ai + · · ·+an y + ai+1 +· · ·+an Mi+1 √ ≤ N = N T1+ (y). (2.16) √ ai


543

Obviously, the left-hand side of (2.15) holds for all t ∈ (−∞, ∞), if ln αn + · · · + ln αi+2 + 2 ln f < 0 and i+1 is small enough. Otherwise, it holds, if either √ √ ai+1 + · · · + an N t≥ √ max ai + · · · + a n y + √ m=i+2,...,n: ai ln α +···+ln α +2 ln f ≥0 am + · · · + a n n m

√ 2 ln αm + · · · + 2 ln αn + 2 ln f + i+1 ≡ N T2+ (y), (2.17) or

√ √ N ai+1 + · · · + an t≤ √ min ai + · · · + a n y − √ m=i+2,...,n: ai ln α +···+ln α +2 ln f ≥0 am + · · · + a n n m √

2 ln αm + · · · + 2 ln αn + 2 ln f + i+1 ≡ N T2− (y).

(2.18)

Let us put for convenience T2+ (y) = −∞ and T2− (y) = ∞, if 2 ln αn + · · · + 2 ln αi+2 + 2 ln f < 0. Finally, √ √ √ ai + · · · + a n y − ai t ai + · · · + a n N αi+1 PN i + 1,

K → 0, (2.19) , f, √ ai+1 + · · · + an N (ai+1 + · · · + an ) uniformly in the domain where √a +· · ·+a y √N − √a t 2 i n i

≥ 2 ln αi+1 + · · · + 2 ln αn +2 ln f + i+1 . (2.20) N(ai+1 + · · · + an ) This domain is equivalent to −∞ < t < +∞, if 2 ln αn + · · · + 2 ln αi+1 + 2 ln f < 0 and i+1 > 0 is small enough. Otherwise, it is reduced to the union of the domains √

N √ t ≥ √ ai +· · · an y + (ai+1 +· · ·+an )(2 ln αi+1 +· · ·+2 ln αn + 2 ln f + i+1 ) ai √ + ≡ T3 (y) N (2.21) and

√

N √ t ≤ √ ai +· · · an y − (ai+1 +· · ·+an )(2 ln αi+1 +· · ·+2 ln αn + 2 ln f + i+1 ) ai √ ≡ T3+ (y) N. (2.22)

Then, using the elementary inequalities −x − x 2 ≤ ln(1 − x) ≤ −x, 1 + x ≤ ex ≤ 1 + x + x 2 for |x| < 1/2, (2.23) it is easy to deduce from (2.13), (2.14), and (2.19) the following asymptotic lower bound, if 2 ln αn + · · · + 2 ln αi+1 + 2 ln f ≥ 0: √ ai + · · · + a n −1 N N N P (i, y, f, K(N )) ≥ (2π) √ 2K(N )f N αi+1 αi+2 αi+3 · · · αnN ai+1 +· · ·+an   √ √ − min(T2− (y),T T1+ (y) N 3 (y)) N    × +   (

−

×e

√

√ T1− (y) N

√ √ ai +···+an y N− ai t)2 2(ai+1 +···+an )

√ max(T2+ (y),T3+ (y)) N

e−t

2 /2

dt.

(2.24)

544


If 2 ln αi+1 + · · · + 2 ln αn + 2 ln f < 0, then from the same assertions we deduce the same √ bound, but √ with the domain of integration ranging over the entire interval [T1− (y) N, T1+ (y) N ]. By the change of variables, √ √ ai + · · · + an t − ai y s= , (2.25) √ ai+1 + · · · + an the right-hand side of (2.24) equals 2K(N ) N N N N 2 f αi+1 αi+2 αi+3 · · · αnN e−y N/2 2π  ×  where

√ − min(S2− (y),S 3 (y)) N

√ (y) N S1+

+ √ S1− (y) N

√ max(S2+ (y),S3+ (y)) N

  −s 2 /2 e ds, 

√ √ ai+1 + · · · + an y ± ai + · · · + an Mi+1 , √ ai

S2− (y) = min (ai+1 + · · · + an )/ai

S1− (y), S1+ (y) =

(2.26)

(2.27)

m=i+1,...,n: ln αn +···+ln αl +2 ln f≥0

√ ai + · · · + a n

× y− √ ln αm +· · ·+ln αn +ln f + i+1 , (2.28) am + · · · + a n

+ max (ai+1 + · · · + an )/ai S2 (y) = m=i+1,...,n: lnαn +···+ln αm +2 ln f ≥0

√ ai +· · ·+an

× y+ √ ln αl +· · ·+ln αn +ln f + i+1 , am +· · ·+an

(2.29)

if T2± (y) are finite, and, of course, S2+ (y) = −∞, if T2+ (y) = −∞, S2− (y) = +∞, if T2− (y) = +∞, and finally √ √ √ ai+1 +· · ·+an y ± ai +· · ·+an ln αi+1 +· · ·+ln αn +ln f + i+1 ± S3 (y) = . √ ai (2.30) Now let us take any > i+1 and M = Mi+1 . Then, there exist δ > 0 and Q > 0, such that, for all y ≥ 0 satisfying (2.9), we have S1− (y) ≤ −Q and min(S2− (y), S3− (y)) ≥ δ; and for all y < 0 satisfying (2.9), we have S1+ (y) ≥ Q and max(S2+ (y), S3+ (y)) ≤ −δ. Hence   √ √ − (y) N min(S2− (y),S S1+ 3 (y)) N   −s 2 /2 e (2π )−1/2  + ds  

≥ (2π)−1/2

√ S1− (y) N √ δ N

√ −Q N

e−s

√ max(S2− (y),S3− (y)) N

2 /2

ds → 1,

(2.31)


545

as N → ∞. In the case when 2 lnn + · · ·√ + 2 ln αi+1 + √2 ln f < 0, we have the analogue of (2.24) with the integral over [T1− (y) N , T1+ (y) N ], and by the same change we get the bound (2π )−1/2

√ (y) N S1+

e−s

2 /2

ds ≥ (2π)−1/2

√ S1− (y) N

√ Q N

e−s

2 /2

ds → 1, N → ∞. (2.32)

√ −Q N

Since i+1 [resp. Mi+1 ] could be chosen arbitrarily small [resp. large], by the induction hypothesis, the estimates (2.24), (2.26), and (2.31), (2.32) show that, for any > 0 small enough, and M > 0 large enough, the assertion (2.8) holds uniformly in the domain (2.9). This finishes the proof of the lemma. Lemma (2.1) implies the next lemma.

Lemma 2.2. Let l ∈ {0, . . . , k − 1}, c be with |c| < 2 ln α¯ l+1 (a¯ l+1 + · · · + a¯ k )/a¯ l+1 . For any , δ > 0 small enough, and M > 0, there exists N0 = N0 ( , δ, M), such that, for all N ≥ N0 , the probability P ∀σˇ ∈ {−1, 1}(1−wl )N : √ ∈ K(N )e

c2 N/2

Xˇ σJˇ l

a¯ l+1 + · · · + a¯ k (α¯ l+1 · · · α¯ k )−N A

√ − (|c| + z) N

is bounded from above and below, respectively, by 2 exp − (1 ± δ)(2π)−1/2 2K(N)|A|e−(2|c|z+z )N/2

(2.33)

(2.34)

for any − < z < M.

2 Proof. If |c| < 2 ln α¯ l+1 (a¯ l+1 + · · · + a¯ k )/a¯ l+1 , then by (1.5) we have ec /2 (α¯ l+1 · · · α¯ k )−1 < 1 and with some 0 > 0 small enough: max

(aJl+1 +· · ·+an )(2ln αm +· · ·+2ln αn +2(c2 /2−ln(α¯ Jl+1 · · ·α¯ Jk ))+ 0 ) , m=Jl +2,...,n am +· · ·+an (2 ln αJl +2 +· · ·+2 ln αn + 2(c2 /2−ln(α¯ Jl+1 · · · α¯ Jk ))+ 0 < c2 . max

(2.35) This last inequality remains true with c2 replaced in the left-hand side by (|c| + z)2 if z > − with > 0 small enough. Then Lemma (2.1) applies with i = Jl + 1 and 2 f = ec /2 (α¯ l+1 · · · α¯ k )−1 and gives the asymptotics (2.34). √ Lemma (2.2) with l = 0, z = 0, K(N) = 2π/2 implies immediately the convergence of the avoidance function (2.1) in the case l = 0. To conclude the proof of (1.12), let us note that √ 2 EM0N (A) = P |Xσ − cN N | ∈ 2−N−1 (2π )ecN N/2 A (2.36) σ ∈N

546


is the sum of 2N identical terms, each of them being 2−N |A|(1 + o(1)) by the trivial estimate for standard Gaussian random variables (2.11). Then (2.36) converges to |A| and the proof of (1.12) is finished. To prove the convergence of the avoidance function (2.1) in the case l ≥ 1, let us write the event {MlN (A) = 0} in terms of the functions Ul,N defined in (1.21) as {MlN (A) = 0} = ∀σˆ ∈ wl N , σˇ ∈ (1−wl )N √ −1 Jl : Xˇ σJˆ lσˇ − N c˜l +N −1 l,N −Ul,N (Xσˆ ) ∈ Rl (N )−1 A (2.37) with the abbreviations (1.20), (1.22), (1.26). Let us introduce the following event with a parameter y > 0: l (y) = ∀j = 1, . . . , l, ∀σˆ ∈ wl N : 2j,N − 2N dj BN

−1 Jj −(γ1 +· · ·+γj )y < Uj,N (Xσˆ ) < y(γ1 +· · ·+γj ) .

(2.38)

By the convergence (1.23) to (1.24), the property (1.6) and the symmetry of the standard Gaussian distribution, the probability of the complementary event satisfies the following bound: l (y)) ≤ 2 exp(−hy), lim sup P(B¯ N

(2.39)

N→∞

with some constant h > 0. Now, let us fix any arbitrarily large y > 0 and consider Jj , ∀lj =1 , ∀ σ ∈ wl N ) P(MlN (A) = 0) = E 1I{B l (y)} E(1I{Ml (A)=0} | X σˆ N N Jj , ∀lj =1 , ∀ σ ∈ wl N ) . +E 1I{B¯ l (y)} E(1I{Ml (A)=0} | X σˆ N

(2.40)

N

Due to the representation (2.37), the conditional expectation E(1I{Ml

N (A)=0}

Jj , | X σˆ

∀lj =1 , ∀ σ ∈ wl N ) can be viewed as the product over σ ∈ wl N of the quantities (2.33) with c˜l |c| = √ , a¯ l+1 + · · · + a¯ k

√ l 2π K(N ) = (4N π ln α¯ j )βl γj /2 , 2

(2.41)

j =1

and −1 Jl (Xσˆ ) , σˆ ∈ wl N . (2.42) z = z(σˆ ) = (a¯ l+1 + · · · + a¯ k )−1/2 N −1 l,N − Ul,N l (y), we have z( l σ ) ∈ (− , √a¯ 2d + )∀σˆ ∈ wl N (with Furthermore, on BN l+1 +···+a¯ k some small enough > 0), so that Lemma (2.2) applies to 1I{B l (y)} E(1I{Ml (A)=0} N

N

j , ∀l , ∀ |X j =1 σ ∈ wl N ). Hence, by (2.40) and by Lemma (2.2), for any δ > 0 small σˆ J


547

enough, there exists N0 (δ, y) such that for all N ≥ N0 , 2 l exp − (1 − δ)(2π)−1/2 2K(N)|A|e− 2|c|z(σˆ )+z (σˆ ) N/2 +P(B¯ N (y)) E σˆ ∈wl N

≥ E 1I{B l

N (y)}

≥

2 exp −(1−δ)(2π)−1/2 2K(N )|A|e− 2|c|z(σˆ )+z (σˆ ) N/2

σ∈ ˆ wl N

l +P(B¯ N (y)) l P(MN (A) = 0)

≥ E 1I{B l

N (y)}

exp − (1 + δ)(2π)

−1/2

2K(N )|A|e

− 2|c|z(σˆ )+z2 (σˆ ) N/2

σˆ ∈wl N

2 ≥E exp − (1 + δ)(2π)−1/2 2K(N )|A|e− 2|c|z(σˆ )+z (σˆ ) N/2 σˆ ∈wl N l −P(B¯ N (y)).

(2.43)

Using the convergence (1.23) to (1.24), we derive that for any y > 0 large enough and δ > 0 small enough, E

(x1 ,...,xl )∈Pl

l exp(−(1 − δ)|A|eβl (γ1 x1 +···+γl xl ) ) + lim sup P(B¯ N (y)) N→∞

≥ lim sup P(MN (A) = 0) ≥ lim inf P(MN (A) = 0) N→∞

≥E

(x1 ,...,xl )∈Pl

N→∞

l exp(−(1 + δ)|A|eβl (γ1 x1 +···+γl xl ) ) − lim sup P(B¯ N (y)). (2.44) N→∞

Thus (2.44) and (2.39) imply the following bounds: E exp(−(1 − δ)|A|l ) + 2 exp(−hy) ≥ lim sup P(MlN (A) = 0) N→∞

≥ lim inf P(MlN (A) = 0) ≥ E exp(−(1 + δ)|A|l )) − 2 exp(−hy). (2.45) N→∞

Since y > 0 can be chosen arbitrarily large and δ > 0 fixed arbitrarily small, this finishes the proof of the convergence of the avoidance function (2.1) in the case of l = 1, 2, . . . , k − 1. To proceed with the proof of tightness (2.3), we need the following lemma.

Lemma 2.3. Let l ∈ {0, . . . , k − 1}, |c| < 2 ln α¯ l+1 (a¯ l+1 + · · · + a¯ k )/a¯ l+1 , K(N ) > 0 is polynomial in N, z ∈ R. For any segment B ⊂ R+ , let us define an integer-valued random variable

548

A. Bovier, I. Kurkova c,z,K(N)

Tl,N

(B)

√ Xˇ σJˇ l − N (|c| + z) = # σˇ ∈ (1−wl )N : √ a¯ l+1 + · · · + a¯ k c2 N/2 ∈ K(N )e (α¯ l+1 · · · α¯ k )−N B .

(2.46)

(i) For any bounded segment A ⊂ R+ , any , δ > 0 small enough and M > 0 there exists N0 = N0 (δ, M, ) such that for all N ≥ N0 , for any B ⊂ A and any z ∈] − , M[ we have: √ c,z,K(N) 2 P Tl,N (B) ≥ 1 ≤ (1 + δ)|B|K(N)(2/ 2π )e−(2|c|z+z )N/2 . (2.47) (ii) For any bounded segment A ⊂ R+ , any δ > 0 small enough, K > 0 large enough and M > 0 there exists N0 = N0 (δ, M, K) such that for all N ≥ N0 , for any segment B ⊂ A with |B| < K −1 and for any ln(2K(N )/√2π) − ln K z = zN ∈ , M (2.48) |c|N we have: c,z,K(N) (B) ≥ 2 P Tl,N 2 √ √ 2 2 ≤ δ|B|K(N )(2/ 2π)e−(2|c|z+z )N/2 + |B|K(N )(2/ 2π )e−(2|c|z+z )N/2 /2. (2.49) Remark. The bound (2.49) is far from being the optimal one, but it is enough for our purpose. Therefore, we do not prove a precise bound that requires much more tedious computations. Proof. The right-hand side of (2.47) is bounded from above by √ 2 (2.50) (α¯ l+1 · · · α¯ k )N P |X − N (|c| + z)| ∈ K(N )ec N/2 (α¯ l+1 · · · α¯ k )−N B with X a standard Gaussian random variable. Since by the assumption of the lemma 2 and by (1.5) we have ec /2 (α¯ l+1 · · · α¯ k )−1 < 1, then (2.47) is obvious from the trivial estimate (2.11). c,z,K(N) (B) just equals (2.50), whence To prove (ii), note that ETl,N c,z,K(N)

ETl,N

√ 2 (B) ≤ (1 + δ)|B|K(N)(2/ 2π )e−(2|c|z+z )N/2 .

(2.51)

Finally

c,z,K(N) c,z,K(N) c,z,K(N) (B) ≥ 2 ≤ ETl,N (B) − 1 − P Tl,N (B) = 0 , (2.52) P Tl,N

c,z,K(N) where by Lemma 2.2 P Tl,N (B) = 0 is bounded from above by the exponent (2.34). The assumption (2.48) and the fact that |B| < 1/K assure that the argument of this exponent is smaller than 1 by absolute value, i.e. √ 2 0 < (1 − δ)|B|K(N )(2/ 2π)e−(2|c|z+z )N/2 < 1 − δ. (2.53)


549

Then (2.52), (2.51), the bound (2.34) with (2.53) and the elementary fact that e−x ≤ 1 − x + x 2 /2 for 0 < x < 1 yield the estimate (2.49). We are now ready to prove the tightness (2.3) of the family {MlN }∞ N=1 for l = 1, . . . , k − 1. For a given > 0, let us first fix y large enough and N1 (y) such that l P(B¯ N (y)) < /4 ∀N ≥ N1 = N1 (y),

(2.54)

which is possible due to (2.39). Now let us split the segment A = [a, b] into R disjoint segments A1 , . . . , AR of size (b − a)/R, R > 1. Then P({MlN (A)

>

l R} ∩ BN (y))

≤

R

l P({MlN (Ai ) ≥ 2} ∩ BN (y))

i=1

≤

R

l l P(CN (Ai , σ ) ∩ BN (y, σ ))

i=1 σˆ ∈wl N

+

R

l l P(DN (Ai , τ ) ∩ DN (Ai , η)

i=1 τˆ ,η∈ ˆ wl N ,τˆ =ηˆ l l ∩BN (y, τ ) ∩ BN (y, η)),

(2.55)

where

√ −1 Jl l CN (Ai , σ ) = ∃η, ˇ τˇ ∈ (1−wl )N , ηˇ = τˇ : Xˇ σJˆ lσˇ − N & cl + N −1 (l,N − Ul,N (Xσˆ )) ∈ Rl (N )−1 Ai for σˇ = η, ˇ σˇ = τˇ , √ −1 Jl l DN (Ai , σ ) = ∃σˇ ∈ (1−wl )N :Xˇ σJˆ lσˇ − N & cl + N −1 (l,N − Ul,N (Xσˆ )) (2.56) ∈ Rl (N )−1 Ai ,

and −1 Jj l BN (y, σ ) = ∀j = 1, . . . , l : 2j,N − 2N dj − (γ1 + · · · + γj )y < Uj,N (Xσˆ ) < y(γ1 + · · · + γj ) . (2.57) Each term in the first sum of (2.55) equals E 1IB l

E1I

N (y,σˆ )

= E 1IB l

Jj l X ,∀

l (A ,σˆ ) CN i

E1I

N (y,σˆ )

c,z,K(N)

σˆ

c,z(σˆ ),K(N )

Tl,N

j =1

(Ai )≥2

|X Jj , ∀lj =1 σˆ

(2.58)

defined in Lemma (2.3) and with parameters with the random variables Tl,N l (y, c, K(N ), z( σ ) defined by (2.41) and (2.42). Furthermore, on BN σ ), the parameter z( σ ) satisfies the condition (2.48) with the constant K = eβl (γ1 +···+γl )y and M = 2dl (a¯ l+1 + · · · + a¯ k )−1/2 + with some small > 0. Therefore, if |Ai | = (a − b)/R
0. Then the assertion (i) of Lemma 2.3 applies to the conditional expectations in (2.59). Thus by Lemma 2.3, for any δ > 0, there exists N2 (y, δ) such that for all N ≥ N2 , R

l P({M0N (Ai ) ≥ 2} ∩ BN (y))

i=1

≤

R i=1

+

√ δ(2/ 2π)K(N )(b − a)R −1 E 1IB l

N (y,σˆ )

σˆ ∈wl N

R

e− 2|c|z(σˆ )+z2 (σˆ ) N/2

(4/2π )K(N )2 (b − a)2 R −2

i=1

1 2 ×E 1IB l (y,σˆ ) e− 2|c|z(σˆ )+z (σˆ ) N 2 N σˆ ∈wl N 2 (η) ˆ ˆ N/2 e− 2|c|z(τˆ )+z2 (τˆ )+2|c|z(η)+z + 1IB l (y,τˆ ),B l (y,η) ˆ N

τˆ ,η∈ ˆ wl N :τˆ =ηˆ

N

= δ(b − a)IN (y) + R −1 (b − a)2 JN (y)/2, where

e− 2|c|z(σˆ )+z2 (σˆ ) N/2 ,

√ 1IB l IN (y) = (2/ 2π)K(N )E σˆ ∈wl N

N (y,σˆ )

JN (y) = (4/(2π ))K(N ) E 1IB l 2

σˆ ∈wl N

2 e− 2|c|z(σˆ )+z2 (σˆ ) N/2 .

N (y,σˆ )

Here, the quantity IN (y) converges to eβ(γ1 x1 +···γl xl ) Pl (dx1 . . . , dxl ) I (y) = E ∀1≤j ≤l: γ1 x1 +···+γj xj 0 there exists r0 such that ˜ l (Ar,i ) ≥ 2) < ∀r ≥ r0 . P(∃i = 1, . . . , 2r : M

(2.66)

˜ l can have double points within A with probability smaller than . Since > 0 Then M ˜ l is simple. Thus the proof of the theorem is complete. is arbitrary, it follows that M

552


References 1. Bauke, H., Mertens, St.: Universality in the level statistics of disordered systems. Phys. Rev. E, 70, 025102(R) (2004) 2. Borgs, C., Chayes, J.T., Mertens, S., Nair, C.: Proof of the local REM conjecture for number partitioning II: Growing energy scales. http://arxiv.org/list/cond-mat/0508600, 2005 3. Bovier, A., Kurkova, I.: Derrida’s generalized random energy models. I. Models with finitely many hierarchies. Ann. Inst. H. Poincaré Probab. Statist. 40(4), 439–480 (2004) 4. Bovier, A., Kurkova, I.: Local energy statistics in disordered systems: a proof of the local REM conjecture. Commun. Math. Phys. 263(2), 513–533 (2006) 5. Bovier, A., Kurkova, I., Lowe, M.: Fluctuations of the free energy in the REM and the p-spin SK models. Ann. Probab. 30, 605–651 (2002) 6. Daley, D.J., Vere-Jones, D.: An introduction to the theory of point processes. Springer Series in Statistics, Berlin-Heidelberg New York: Springer-Verlag, 1988 7. Derrida, B.: Random-energy model: an exactly solvable model of disordered systems. Phys. Rev. B (3), 24(5), 2613–2626 (1981) 8. Derrida, B.: A generalization of the random energy model that includes correlations between the energies. J. Phys. Lett. 46, 401–407 (1985) 9. Leadbetter, M.R., Lindgren, G., Rootzén, H.: Extremes and related properties of random sequences and processes. Springer Series in Statistics. New York: Springer-Verlag, 1983 10. Kallenberg, O.: Random Measures. Fourth ed., Berlin: Akademie Verlag, 1986 Communicated by M. Aizenman


Communications in


Homotopy Algebras Inspired by Classical Open-Closed String Field Theory Hiroshige Kajiura1 , Jim Stasheff2 1 2

Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan. E-mail: [email protected] Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, USA. E-mail: [email protected]

Received: 18 October 2004 / Accepted: 10 November 2005 Published online: 3 March 2006 – © Springer-Verlag 2006

Abstract: We define a homotopy algebra associated to classical open-closed strings. We call it an open-closed homotopy algebra (OCHA). It is inspired by Zwiebach’s openclosed string field theory and also is related to the situation of Kontsevich’s deformation quantization. We show that it is actually a homotopy invariant notion; for instance, the minimal model theorem holds. Also, we show that our open-closed homotopy algebra gives us a general scheme for deformation of open string structures (A∞ -algebras) by closed strings (L∞ -algebras). Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Strong Homotopy Algebra . . . . . . . . . . . . . . . . . . 2.1 Strong homotopy associative algebras . . . . . . . . . 2.2 The coalgebra description and the Gerstenhaber bracket 2.3 The tree description . . . . . . . . . . . . . . . . . . . 2.4 L∞ -algebras and morphisms . . . . . . . . . . . . . . 2.5 The symmetric coalgebra description . . . . . . . . . . 2.6 The tree description . . . . . . . . . . . . . . . . . . . 2.7 Open-closed homotopy algebra (OCHA) . . . . . . . . 2.8 The coalgebra description . . . . . . . . . . . . . . . . 2.9 The tree description . . . . . . . . . . . . . . . . . . . 3. Cyclic Structures . . . . . . . . . . . . . . . . . . . . . . . 4. Minimal Model Theorem and Decomposition Theorem . . . 5. Deformations and Moduli Spaces of A∞ -Structures . . . . . 5.1 Deformations and Maurer-Cartan equations . . . . . . 5.2 Gauge equivalence and moduli spaces . . . . . . . . .

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554 555 556 557 558 559 561 561 562 567 567 569 571 573 573 577

H. K is supported by JSPS Research Fellowships for Young Scientists. J. S. is supported in part by NSF grant FRG DMS-0139799 and US-Czech Republic grant INT0203119.

554

H. Kajiura, J. Stasheff

1. Introduction In this paper we define a strong homotopy algebra inspired by Zwiebach’s classical open-closed string field theory [61] and examine its homotopy algebraic structures. It is known that classical closed string field theory has an L∞ -structure [60, 52, 34] and classical open string field theory has an A∞ -structure [12, 61, 46, 30]. As described by Zwiebach [60, 61] and others, string field theory is presented in terms of decompositions of moduli spaces of the corresponding Riemann surfaces into cells. The associated Riemann surfaces are (respectively) spheres with (closed string) punctures and disks with (open string) punctures on the boundaries. That is, classical closed string field theory is related to the conformal plane C with punctures and classical open string field theory is related to the upper half plane H with punctures on the boundary from the viewpoint of conformal field theory. The algebraic structure that the classical open-closed string field theory has is similarly interesting since it is related to the upper half plane H with punctures both in the bulk and on the boundary, which also appeared recently in the context of deformation quantization [36, 5]. In operad theory (see [44]), the relevance of the little disk operad to closed string theory is known, where a (little) disk is related to a closed string puncture on a sphere in the Riemann surface picture above. The homology of the little disk operad defines a Gerstenhaber algebra [6, 16], in particular, a suitably compatible graded commutative algebra structure and graded Lie algebra structure. The framed little disk operad is in addition equipped with a BV-operator which rotates the disk boundary S 1 . The algebraic structure on the homology is then a BV-algebra [14], where the graded commutative product and the graded Lie bracket are related by the BV-operator. Physically, closed string states associating to each disk boundary S 1 are constrained to be the S 1 -invariant parts, the kernel of the BV-operator. This in turn leads to concentrating on the Lie algebra structure, where two disk boundaries are identified by twist-sewing as Zwiebach did [60]. On the other hand, he worked at the chain level (‘off shell’), discovering an L∞ -structure. This was important since the multi-variable operations of the L∞ -structure provided correlators of closed string field theory. Similarly for open string theory, the little interval operad and associahedra are relevant, the homology corresponding to a graded associative algebra, but the chain level reveals an A∞ -structure giving the higher order correlators of open string field theory. The corresponding operad for the open-closed string theory is the Swiss-cheese operad [58] that combines the little disk operad with the little interval operad; it was inspired also by Kontsevich’s approach to deformation quantization. The algebraic structure at the homology level has been analyzed thoroughly by Harrelson [24]. In contrast, our work in the open-closed case is at the level of strong homotopy algebra, combining the known but separate L∞ - and A∞ -structures. There are interesting relations (not yet fully explored) between an algebra over the Swiss-cheese operad and the homotopy algebra we define in the present paper. In particular, we leave for later work the inclusion of the appropriate homotopy algebra corresponding to the graded commutative product and the BV-operator. For possible structures to be added to our structure, see [54]. We call our structure an open-closed homotopy algebra (OCHA) (since it captures a lot of the operations in existing open-closed string field theory algebra structure [32]). We show that this description is a homotopy invariant algebraic structure, i.e. that it transfers well under homotopy equivalences or quasi-isomorphisms. Also, we show that an open-closed homotopy algebra gives us a general scheme for deformation of open string structures (A∞ -algebras) by closed strings (L∞ -algebras).

Homotopy Algebras Inspired by Classical Open-Closed String Field Theory

555

We first present our notion of open-closed homotopy algebras in Sect. 2. An openclosed homotopy algebra consists of a direct sum of graded vector spaces H = Hc ⊕Ho . It has an L∞ -structure on Hc and reduces to an A∞ -algebra if we set Hc = 0. Moreover, the operations that intertwine the two are a generalization of the strong homotopy analog of H. Cartan’s notion of a Lie algebra g acting on a differential graded algebra E [4, 9]. We present the basics of these notions of homotopy algebra from three points of view: multi-variable operations, coderivation differentials and tree diagrams. In a more physically oriented paper [32], we give an alternate interpretation in the language of homological vector fields on a supermanifold. The motivating physics of string interactions suggests that the homotopy algebra should be appropriately cyclic [44, 17]. In Sect. 3, we make it precise in terms of an odd symplectic/cyclic structure which is strictly invariant with respect to the OCHA structure. It would be worth investigating a strongly homotopy invariant analog in the sense of [55, 56], which we do not discuss in this paper. One of the key theorems in homotopy algebra is the minimal model theorem which was first proved for A∞ -algebras by Kadeishvili [29]. The minimal model theorem states the existence of minimal models for homotopy algebras analogous to Sullivan’s minimal models [53] for differential graded commutative algebras introduced in the context of rational homotopy theory. For an A∞ - or L∞ -algebra, the minimal model theorem is now combined with various stronger results; those employing the techniques of homological perturbation theory (HPT) (for instance see [19, 27, 20–23]), what is called the decomposition theorem in [31, 33], Lefèvre’s approach [40], etc. These theorems are very powerful and make clear the homotopy invariant nature of the algebraic properties (for instance [42, 28, 33]). In Sect. 4 we describe these theorems for our open-closed homotopy algebras, pointing out subtleties of the open-closed case in addition to those for L∞ -algebras in comparison to the existing versions for A∞ -algebras. In Sect. 5, we show that an open-closed homotopy algebra gives a general scheme of deformation of the A∞ -algebra Ho as controlled by Hc . A particular example of the deformation point of view applied in an open-closed setting occurs in analyzing Kontsevich’s deformation quantization theorem, which we shall explain explicitly in the sequel to this paper [32]. We discuss this deformation theory also from the viewpoint of generalized Maurer-Cartan equations for an open-closed homotopy algebra and the moduli space of their solution space. We include an appendix by M. Markl, where A∞ -algebras over L∞ -algebras are interpreted as a colored version of strongly homotopy algebras in the sense in [44]. We have taken care to provide the detailed signs which are crucial in calculations, but which are conceptually unimportant and can be ignored at first reading. The majority of this paper is entirely mathematics, and in the sequel [32] we show how our structures are related to those in Zwiebach [61], deformation quantization by Kontsevich [36], as well as those discussed in [25, 26], where an open-closed homotopy algebra is applied to topological open-closed strings. It should be very interesting to investigate the application to homological mirror symmetry [59, 35, 3, 26].

2. Strong Homotopy Algebra An open-closed homotopy algebra, as we propose in this paper, is a strong homotopy algebra (or ∞-algebra) which combines two typical strong homotopy algebras, an A∞ algebra and an L∞ -algebra. Let us begin by recalling those definitions. We restrict our

556


arguments to the case that the characteristic of the field k is zero. We further let k = C for simplicity. There are various equivalent ways of defining/describing strong homotopy algebras: in terms of multi-variable operations and relations among them, in terms of a coderivation differential of square zero on an associated coalgebra or as a representation of a particular operad of trees. We will treat all three of these in turn. The reader who is familiar with these approaches to the ‘classical’ A∞ -algebras and L∞ -algebras can move ahead to Subsect. 2.7, being warned that the definitions we give are different from the original ones [49, 39, 44] in the degrees of the multi-linear maps and hence of the relevant signs. Both are in fact equivalent and related by suspension [44], as we explain further below.

2.1. Strong homotopy associative algebras. Definition 1 (A∞ -algebra (strong homotopy associative algebra)[49]). Let A be a Zgraded vector space A = ⊕r∈Z Ar and suppose that there exists a collection of degree one multi-linear maps m := {mk : A⊗k → A}k≥1 . (A, m) is called an A∞ -algebra when the multi-linear maps mk satisfy the following relations:

k

(−1)o1 +···+oi−1 mk (o1 , · · · , oi−1 , ml (oi , · · · , oi+l−1 ), oi+l , · · · , on ) = 0

k+l=n+1 i=1

(2.1) for n ≥ 1, where oj on (−1) denotes the degree of oj . A weak A∞ -algebra consists of a collection of degree one multi-linear maps m := {mk : A⊗k → A}k≥0 satisfying the above relations, but for n ≥ 0 and in particular with k, l ≥ 0. Remark 1. The relation (2.1) is different from the original one [49] in the definition of the degrees of the multi-linear maps mk and hence of the signs. Both are in fact equivalent and related by desuspension [44]. In [49], the mk are multi-linear maps on ↓A where (↓A)r+1 = Ar ; we denote desuspension by ↓. (The algebraic geometry tradition would use [−1].) Note that, in that notation [49], a differential graded (dg) algebra is an A∞ -algebra with a differential m1 , a product m2 , and m3 = m4 = · · · = 0. The ‘weak’ version is fairly new, inspired by physics, where m0 : C → A, regarded as an element m0 (1) ∈ A, is related to what physicists refer to as a ‘background’. The augmented relation then implies that m0 (1) is a cycle, but m1 m1 need no longer be 0, rather m1 m1 = ±m2 (m0 ⊗ 1) ± m2 (1 ⊗ m0 ). Definition 2 (A∞ -morphism). For two A∞ -algebras (A, m) and (A , m ), suppose that there exists a collection of degree zero (degree preserving) multi-linear maps fk : A⊗k → A ,

k≥1.


557

The collection {fk }k≥1 : (A, m) → (A , m ) is called an A∞ -morphism iff it satisfies the following relations: 1≤k1 0 map L into ↑Coder(T c A) and the rest extend that to an L∞ -map. This is just the higher homotopy structure a mathematician would construct by the usual procedures of strong homotopy algebra (see the Appendix by M. Markl). Here the sign exponent µp,i (σ ) is given explicitly by µp,i (σ ) = (σ ) + (cσ (1) + · · · + cσ (p) ) + (o1 + · · · + oi ) +(o1 + · · · + oi )(cσ (p+1) + · · · + cσ (n) ) ,

(2.20)

corresponding to the signs effected by the interchanges. The sign can be seen easily in the coalgebra and tree expressions. We can also write the defining equation (2.19) in the following shorthand expression,

⊗m 0= (−1)(σ ) n1+r,m (lp ⊗ 1⊗r c ⊗ 1o )(cσ (I ) ; o1 , . . . , om ) σ ∈Sp+r=n

+

⊗p ⊗j (−1)(σ ) np,i+1+j (1c ⊗ 1⊗i ⊗ n ⊗ 1 )(c ; o , . . . , o ) , r,s m σ (I ) 1 o o

σ ∈Sp+r=n i+s+j =m

where the complicated sign is absorbed into this expression. Note that the rule for the action of tensor products of graded multi-linear maps on (Hc )⊗n ⊗(Ho )⊗m is determined in a canonical way; for instance for (f ⊗ . . . )(c1 , . . . , cn ; o1 , . . . , om ) with the first multi-linear map f : (Hc )⊗k ⊗ (Ho )⊗l → H, we may bring (c1 , . . . , ck ; o1 , . . . , ol ) to the first f with the associated sign, do the same thing for the next multi-linear map in . . . and repeat this in order. String field theory suggests that an open-closed homotopy algebra includes the addition of the maps np,0 : L⊗p → A and in particular n1,0 : L → A corresponding to the opening of a closed string to an open one. Definition 11 (Open-Closed Homotopy Algebra (OCHA)). An open-closed homotopy algebra (OCHA) 1 (H = Hc ⊕ Ho , l, n) consists of an L∞ -algebra (Hc , l) and ⊗p ⊗q a family of maps n = {np,q : Hc ⊗ Ho → Ho } for p, q ≥ 0 with the exception of (p, q) = (0, 0) satisfying the compatability conditions (2.19): 0= (−1)(σ ) n1+r,m (lp (cσ (1) , . . . , cσ (p) ), cσ (p+1) . . . , cσ (n) ; o1 , . . . , om ) σ ∈Sp+r=n

+

(−1)µp,i (σ ) np,i+1+j (cσ (1) , . . . , cσ (p) ; o1 , . . . , oi ,


nr,s (cσ (p+1) , . . . , cσ (n) ; oi+1 , . . . , oi+s ), oi+s+1 , . . . , om ),

(2.21)

for the full range n, m ≥ 0, (n, m) = (0, 0). A weak OCHA consists of a weak L∞ -algebra (Hc , l) with a family of maps n = ⊗p ⊗q {np,q : Hc ⊗ Ho → Ho } now for p, q ≥ 0 satisfying the analog of the above relation. 1

The authors worked with the acronym for several weeks before realizing it is Japanese for ‘tea’.

566


For an OCHA (H, l, n), the multi-linear maps {np,q }p≥1,q≥0 still correspond to an adjoint L∞ -map Hc → Coder(T c Ho ), as in the case of an A∞ -algebra over an L∞ -algebra. This has a particular importance in terms of deformation theory, cf. Subsect. 5.1, where the addition of maps np,0 leads in turn to the deformation of the A∞ -structure m to a weak A∞ -structure. Definition 12 (Open-closed homotopy algebra (OCHA) morphism). For two weak OCHAs (H, l, n) and (H , l , n ), consider a collection f of degree zero (degree preserving) multi-linear maps fk : (Hc )⊗k → Hc , fk,l : (Hc )

⊗k

⊗ (Ho )

⊗l

for k ≥ 0, → Ho ,

for k, l ≥ 0 ,

where fk and fk,l are graded symmetric with respect to (Hc )⊗k . We call f : (H, l, n) → (H , l , n ) a weak OCHA-morphism when {fk }k≥0 : (Hc , l) → (Hc , l ) is a weak L∞ -morphism and {fk,l }k,l≥0 further satisfies the following relations:

⊗m (−1)(σ ) f1+r,m (lp ⊗ 1⊗r c ⊗ 1o )(cσ (I ) ; o1 , . . . , om ) σ ∈Sp+r=n

+

⊗p ⊗j (−1)(σ ) fp,i+1+j (1c ⊗ 1⊗i ⊗ n ⊗ 1 )(c ; o , . . . , o ) r,s 1 m σ (I ) o o


=

σ ∈S(r1 +···+ri )+(p1 +···+pj )=n (q1 +···+qj )=m

(−1)(σ ) i!

ni,j (fr1 ⊗ · · ⊗fri ⊗ fp1 ,q1 ⊗ · · ⊗fpj ,qj )(cσ (I ) ; o1 , . . . , om ) .

(2.22)

The right-hand side is written explicitly as

ni,j (fr1 ⊗ · · · ⊗ fri ⊗ fp1 ,q1 ⊗ · · · ⊗ fpj ,qj )(cσ (I ) ; o1 , . . . , om ) → p ,− q (σ ) n (f (c = (−1)τ−→ i,j r1 σ (1) , ··, cσ (r1 ) ), . . . , fri (cσ (¯ri−1 +1) , . . . , cσ (¯ri ) ); fp1 ,q1 (cσ (¯ri +1) , . . . , cσ (p¯1 ) ; o1 , . . . , oq1 ), . . . , fpj ,qj (cσ (p¯j −1 +1) , . . . , cσ (p¯j ) ; oq¯j −1 +1 , . . . , oq¯j )),

→ → where r¯k := r1 + · · · + rk , p¯ k := r¯i + p1 + · · · + pk , q¯k := q1 + · · · + qk and τ− p ,− q (σ ) is given by

→ → τ− p ,− q (σ )

=

j −1

(cσ (p¯k +1) + · · · + cσ (p¯k+1 ) )(o1 + · · · + oq¯k ).

k=1

In particular, if (H, l, n) and (H , l , n ) are OCHAs and if f0 = f0,0 = 0, we call it an OCHA-morphism. Definition 13 (OCHA-quasi-isomorphism). Given two OCHAs (H, l, n), (H , l , n ) and an OCHA-morphism f : (H, l, n) → (H , l , n ), f is called an OCHA-quasi-isomorphism if f1 + f0,1 : H → H induces an isomorphism between the cohomology spaces of the complexes (H, d := l1 + n0,1 ) and (H , d ). In particular, if f1 + f0,1 is an isomorphism, we call f an OCHA-isomorphism.


567

2.8. The coalgebra description. Consider an OCHA H = Hc ⊕ Ho . Recall that the separate L∞ - and A∞ -structures are described by coderivation differentials on, respectively, C(Hc ) and T c (Ho ). The defining multi-linear maps for H are to be extended to coderivations of C(Hc ) ⊗ T c (Ho ). The coproduct on T c (Hc ) ⊗ T c (Ho ) is the standard tensor product coproduct defined by ((c1 ⊗ · · · ⊗ cm ) ⊗ (o1 ⊗ · · · ⊗ on )) n m = (−1)η(p,q) (c1 ⊗ · · · ⊗ cp ⊗ o1 ⊗ · · · ⊗ oq ) p=0 q=0

⊗(cp+1 ⊗ · · · ⊗ cm ⊗ oq+1 ⊗ · · · ⊗ on ),

(2.23)

where η(p, q) = (cp+1 + · · · + cm )(o1 + · · · + oq ). The relevant subcoalgebra is C(Hc ) ⊗ T c (Ho ). Now we define the total coderivation l + n by lifting (lk + mk ) +

(2.24)

np,q ,

p≥1,q≥0

k≥1

with mk = n0,k . Thus we have an OCHA iff l + n is a codifferential: (l + n)2 = 0.

(2.25)

If this is true with the addition of l0 and m0 , we have a weak OCHA. Also, given two OCHAs (H, l, n) and (H , l , n ), an OCHA-morphism f : (H, l, n) → (H , l , n ) can be lifted to the coalgebra homomorphism f : C(Hc ) ⊗ T c (Ho ) → C(Hc ) ⊗ T c (Ho ) and the condition for an OCHA-morphism is written as f ◦ (l + n) = (l + n ) ◦ f.

2.9. The tree description. We associated the k-corolla of planar rooted trees to the multilinear map mk of an A∞ -algebra, and the k-corolla of non-planar rooted trees to the graded symmetric multi-linear map lk of an L∞ -algebra. For an OCHA (H, l, n), the corolla corresponding to nk,l should be expressed as the following mixed corolla,

1

··· ···

nk,l ←→

k

1

···

l

··· ,

(2.26)

which is partially symmetric (non-planar), that is, only symmetric with respect to the k leaves. Let us consider such corollas for 2k +l +1 ≥ 3 together with non-planar corollas {lk }k≥2 . Since we have two kinds of edges, we have two kinds of grafting; grafting of edges associated to Hc (closed string edges) and those for Ho (open string edges). We denote them by ◦i and •i , respectively. For these corollas, we have three types of the

568


composite; in addition to the composite l1+k ◦i ll in L∞ , there is a composite nk,m ◦i lp described by 1

···

k

1

···

···

m

···

1

2

p

3 ···

◦i

[

··· ] [ ··· ] ··· ···

( ··· ···

)

=

,

(2.27) where in the right, hand side the labels are given by [i, . . . , i + p − 1][1, . . . , i − 1, i + p, . . . , p + k − 1](1, . . . , m), and the composite np,q •i nr,s 1

··· ···

p

1

··· ···

q

1

•i

···

r

···

1

···

[ · · · ] ( · · · ) [ · · · ] (· · · ) ( · · · ) ·· ·· ·· ·· ··

s

···

=

(2.28) with labels [1, . . . , p](1, . . . , i − 1)[p + 1, . . . , p + r](i, . . . , i + s − 1)(i + s, . . . , q + s − 1). To these resulting trees, grafting of a corolla lk or nk,l can be defined in a natural way, and we can repeat this procedure. Let us consider tree graphs obtained in this way, that is, by grafting the corollas lk and nk,l recursively, together with the action of permutations of the labels for closed string leaves. Each of them has a closed string root edge or an open string root edge. The tree graphs with closed string root edge, with the addition of the identity ec ∈ L∞ (1), generate L∞ as stated in Subsect. 2.6. On the other hand, the tree graphs with open string root edge are new; the graded vector space generated by them with k closed string leaves and l open string leaves we denote by N∞ (k; l). In particular, we formally add the identity eo generating N∞ (0; 1), and N∞ (1; 0) is generated by a corolla n1,0 . For N∞ := ⊕k,l N∞ (k; l), the tree operad relevant here is then OC ∞ := L∞ ⊕ N∞ . For each tree T ∈ OC ∞ , its grading is given by the number of the vertices v(T ). For trees in OC ∞ , let T → T indicate that T is obtained from T by contracting a closed or an open internal edge. A degree one differential d : OC ∞ → OC ∞ is given by d(T ) =

±T ,

T →T

so that the following compatibility holds: d(T ◦i T ) = d(T ) ◦i T + (−1)v(T ) T ◦i d(T ), d(T •i T ) = d(T ) •i T + (−1)v(T ) T •i d(T ).


569

Thus, OC ∞ forms a dg operad. In particular, d(lk ) is given by Eq. (2.12), and d(nn,m ) is as follows: [

−

··· ] [ ··· ] ··· ···

( ··· ···

)

−

σ ∈Sp+r=n

[ · · · ] ( · · · ) [ · · · ] (· · · ) ( · · · ) ·· ·· ·· ·· ··

,


(2.29) where the labels for the first and the second terms are [σ (1), . . . , σ (p)][σ (p + 1), . . . , σ (n)](1, . . . , m) and [σ (1), . . . , σ (p)](1, . . . , i)[σ (p + 1), . . . , σ (n)](i + 1, . . . , i + s)(i + s + 1, . . . , m), respectively. An algebra H := Hc ⊕ Ho over OC ∞ is obtained by a representation φ : L∞ (k) → Hom(Hc⊗k , Hc ) ,

φ : N∞ (k; l) → Hom((Hc )⊗k ⊗ (Ho )⊗l , Ho )

which is compatible with respect to the grafting ◦i , •i and the differential d. Here, regarding elements in both Hom(Hc⊗k , Hc ) and Hom((Hc )⊗k ⊗ (Ho )⊗l , Ho ) as those in Coder(C(Hc ) ⊗ T c (Ho )), the differential in the algebra side is given by [l1 + n0,1 , ]. By combining it with Eq. (2.29) one can recover the condition of an OCHA (2.19). If we adjust the notation for grading as ↓↓Hc and ↓Ho , the degree of the multilinear map lk is 3 − 2k as stated previously and the degree of nk,l turns out to be 1 + (1 − l) − 2k = 2 − (2k + l). The grading of a tree T ∈ N∞ (k; l) is then replaced by int (T ) + (2 − 2k − l), which is equal to minus the dimension of the corresponding boundary piece of the compactified moduli space of a disk with k points interior and l points on the boundary (see [61]). 3. Cyclic Structures Now we consider an additional structure, cyclicity, on open-closed homotopy algebras. Algebras with invariant inner products (ab, c = a, bc or [a, b], c = a, [b, c]) are very important in mathematical physics; the analogous definition for strong homotopy algebras is straightforward (cf. [44], Sects. II.5.1 and II.5.2). The string theory motivation for this additional structure is that punctures on the boundary of the disk inherit a cyclic order from the orientation of the disk and the operations are to respect this cyclic structure, just as the L∞ -structure reflects the symmetry of the punctures in the interior of the disk or on the sphere. In our context, cyclicity is defined in terms of constant symplectic inner products. (The terminology is that used for symplectic structures on supermanifolds [1]; see also [31] and references therein. These inner products are also essential to the description of the Lagrangians appearing in string field theory.) Definition 14 (Constant symplectic structure). Bilinear maps, ωc : Hc ⊗ Hc → C and ωo : Ho ⊗ Ho → C, are called constant symplectic structures when they have fixed integer degrees |ωc |, |ωo | ∈ Z and are non-degenerate and skew-symmetric. Here ‘skew-symmetric’ indicates that ωc (c2 , c1 ) = −(−1)c1 c2 ωc (c1 , c2 ) ,

ωo (o2 , o1 ) = −(−1)o1 o2 ωo (o1 , o2 )

570


for any c1 , c2 ∈ Hc , o1 , o2 ∈ Ho , and degree |ωc | and |ωo | implies that ωc (c1 , c2 ) = 0 except for deg(c1 ) + deg(c2 ) + |ωc | = 0 and ωo (o1 , o2 ) = 0 except for deg(o1 ) + deg(o2 )+|ωo | = 0. We further denote the constant symplectic structure on H = Hc ⊕Ho by ω := ωc ⊕ ωo . Suppose that an OCHA (H, l, n) is equipped with constant symplectic structures ωc : Hc ⊗Hc → C and ωo : Ho ⊗Ho → C as in Definition 14. For {lk }k≥1 and {np,q }p+q≥1 , let us define two kinds of multi-linear maps by Vk+1 = ωc (lk ⊗ 1c ) : (Hc )⊗(k+1) → C, Vp,q+1 = ωo (np,q ⊗ 1o ) : (Hc )⊗p ⊗ (Ho )⊗(q+1) → C or more explicitly Vk+1 (c1 , . . . , ck+1 ) = ωc (lk (c1 , . . . , ck ), ck+1 ) and Vp,q+1 (c1 , . . . , cp ; o1 , . . . , oq+1 ) = ωo (np,q (c1 , . . . , cp ; o1 , . . . , oq ), oq+1 ). The degree of Vk+1 and Vp,q+1 are |ωc | + 1 and |ωo | + 1. Definition 15 (Cyclic open-closed homotopy algebra (COCHA)). An OCHA (H, ω, l, n) is a cyclic open-closed homotopy algebra (COCHA) when Vk+1 is graded symmetric with respect to any permutation of (Hc )⊗(k+1) and Vp,q+1 has cyclic symmetry with respect to cyclic permutations of (Ho )⊗(q+1) , that is, if Vk+1 (c1 , . . . , ck+1 ) = (−1)(σ ) Vk+1 (cσ (1) , . . . , cσ (k+1) ) ,

σ ∈ Sk+1

and Vp,q+1 (c1 , . . . , cp ; o1 , . . . , oq+1 ) = (−1)o1 (o2 +...oq+1 ) Vp,q+1 (c1 , . . . , cp ; o2 , . . . , oq+1 , o1 ). The graded commutativity of Vp,q+1 with respect to permutations of (Hc )⊗p , that is, Vp,q+1 (c1 , . . . , cp ; o1 , . . . , oq+1 ) = (−1)(σ ) Vp,q+1 (cσ (1) , . . . , cσ (p) ; o1 , . . . , oq+1 ), σ ∈ Sp automatically holds by the definition of n. Since we have non-degenerate inner products ωc and ωo , we can identify H with its linear dual, then reverse the process and define further maps rp−1,q+1 : (Hc )⊗(p−1) ⊗ (Ho )⊗(q+1) → Hc with relations amongst themselves and with the operations already defined, which can easily be deduced from their definition. In particular, for n1,0 : Hc → Ho we have r0,1 : Ho → Hc . Namely, for the cyclic case the fundamental object is the multi-linear map Vp,q+1 , where np,q and rp−1,q+1 are equivalent under the relation above. However, we get a codifferential (2.25) since we took np,q instead of rp−1,q+1 for defining an OCHA. Physically, for the multi-linear map Vp,q+1 , choosing Ho as a root edge instead of Hc as in Eq. (2.26) is related to a standard compactification of the corresponding Riemann surface (a disk with p points interior and (q + 1) points on the boundary).


571

4. Minimal Model Theorem and Decomposition Theorem Homotopy algebras are designed to have homotopy invariant properties. A key and useful theorem in homotopy algebras is then the minimal model theorem. For A∞ -algebras, it was proved by Kadeishvili [29]. For the construction of minimal models of A∞ structures, in particular on the homology of a differential graded algebra, homological perturbation theory (HPT) is developed by [19, 27, 20–23], for instance, and the form of a minimal model is also given explicitly and more recently in [45, 37]. There are various results referred to as minimal model theorems: the weakest form asserts the existence of a quasi-isomorphism as A∞ -algebras H (A) → A for an A∞ structure on H (A), by noticing that all the relevant obstructions vanish because the homology of A and H (A) agree. A stronger result constructs an A∞ -structure on H (A) and the quasi-isomorphism, then a decomposition theorem is proved from which the inverse quasi-isomorphism follows [31, 40, 33, 29]. Alternatively, the full strength of homological perturbation theory gives the maps in both directions and the homotopy for the composition A → H (A) → A all together. The corresponding theorems for L∞ -algebras are more recent: [48] for the two step procedure, [28] for the full HPT treatment. The latter points out that, although L∞ -algebras can be constructed by symmetrization of A∞ -algebras, the corresponding constructions of the maps and homotopy are more subtle. It is not surprising that the minimal models and decompositions exist also for our OCHAs. These theorems imply that, for an OCHA (H, l, n), the higher multi-linear structures lk , k ≥ 2 and np,q , (p, q) = (0, 1) have been transformed to those on H (H), where H (H) is the cohomology of the complex (H, d = l1 + n0,1 ). Even though some of those higher structures may have been zero on the original OCHA H, those on H (H) need not be. We present these statements more precisely below, leaving detailed proofs to the industrious reader. Definition 16 (Minimal open-closed homotopy algebra). An OCHA (H = Hc ⊕ Ho , l, n) is called minimal if l1 = 0 on Hc and n0,1 = 0 on Ho . Definition 17 (Linear contractible open-closed homotopy algebra). A linear contractible OCHA (H, l, n) is a complex (H, d = l1 +n0,1 ) which has trivial cohomology, that is, an OCHA (H = Hc ⊕ Ho , l, n) such that ll = 0 for l ≥ 2, np,q = 0 except for (p, q) = (0, 1), and the complexes (Hc , l1 ), (Ho , n0,1 ) having trivial cohomologies. Theorem 3 (Decomposition theorem for open-closed homotopy algebras). Any OCHA is isomorphic to the direct sum of a minimal OCHA and a linear contractible OCHA. A weak version of the minimal model theorem follows from the decomposition theorem above: Theorem 4 (Minimal model theorem for open-closed homotopy algebras). For a given OCHA (H, l, n), there exists a minimal OCHA (H (H), l , n ) and an OCHAquasi-isomorphism f : (H (H), l , n ) → (H, l, n). In particular, the minimal model can be taken so that l2 = H (l2 ), n0,2 = H (n0,2 ) and n1,0 = H (n1,0 ). To obtain a homotopy equivalence from an initial quasi-isomorphism f above, one way is to employ the decomposition theorem (Theorem 3). Alternatively, it can be obtained

572


directly by the methods of HPT (see Theorem 5 below). In either approach, for a given OCHA (H, l, n), one first considers a Hodge decomposition of the complex (H, d = l1 + n0,1 ). Namely, decompose H into a direct sum isomorphic to H = H (H) ⊕ C, C := Y ⊕ dY with a contracting homotopy h : dY → Y of degree minus one. Together with the inclusion ι and the projection π, let us express these data as (H (H) o

/

ι

H , h) .

π

Then the decomposition theorem (Theorem 3) states that there exists an OCHA-isomorphism fisom : (H (H), l , n )⊕(C, d) → (H, l, n), where (C, d) is the linear contractible OCHA. The OCHA-isomorphism is obtained by first decomposing the L∞ -algebra (Hc , l) into the direct sum of a minimal part H (Hc ) and a linear contractible part Cc , and then decomposing the OCHA (Cc , dc ) ⊕ (H (Hc ) ⊕ Ho , l , n) in a similar way as in the A∞ case. Because we have the OCHA-isomorphism, we may consider a homotopy equivalence between (H (H), l , n ) and (H (H), l , n ) ⊕ (C, d). In fact, the maps ι and π naturally extend to OCHA-quasi-isomorphisms between them, and the corresponding homotopy is obtained as in the sense in Theorem 5 below (see [31] for the A∞ case) or as a path between them with some appropriate compatibility ([33] for the A∞ case). Alternatively, one can refine the standard HPT machinery to function in the category of OCHAs and their morphisms or apply the known results to the L∞ -algebra Hc and then extend to H, regarding Ho as an analog of an sh-algebra over Hc . The extra detail of the HPT form of the minimal model theorem is then: Theorem 5 (HPT minimal model theorem for open-closed homotopy algebras). Given an OCHA (H, l, n) and a Hodge decomposition with a contraction (H (H) o

/

ι π

H , h),

the linear maps π and ι can be extended to coalgebra maps and perturbed so that there exists a corresponding contraction of coalgebras (C(H (Hc )) ⊗ T c (H (Ho )) o

ι¯ π¯

/ C(H ) ⊗ T c (H ) , h) ¯ , o c

(4.1)

where h¯ is a degree minus one linear homotopy on C(Hc ) ⊗ T c (Ho ), not necessarily a coalgebra map. In the same way as in the case of A∞ -algebras, the minimal model theorem together with these additional theorems implies various corollaries. For instance, Corollary 1 (Uniqueness of minimal open-closed homotopy algebras). For an OCHA (H, l, n), its minimal OCHA H (H) is unique up to an isomorphism on H (H). Corollary 2 (Existence of an inverse quasi-isomorphism). For two OCHAs (H, l, n) and (H , l , n ), suppose there exists an OCHA quasi-isomorphism f : (H, l, n) → (H , l , n ). Then, there exists an inverse OCHA quasi-isomorphism f−1 : (H , l , n ) → (H, l, n). In particular, Corollary 2 guarantees that quasi-isomorphisms do in fact define a (homotopy) equivalence relation and in addition give bijective maps between the moduli spaces of the solution space of the corresponding Maurer-Cartan equations for quasi-isomorphic sh-algebras (see Theorem 9). The same facts should hold also for cyclic OCHAs.


573

5. Deformations and Moduli Spaces of A∞ -Structures 5.1. Deformations and Maurer-Cartan equations. Consider an OCHA (H = Hc ⊕ Ho , l, n). We will show how the combined structure implies the L∞ -algebra (Hc , l) controls some deformations of the A∞ -algebra (Ho , {mk }k≥1 ). We will further investigate the deformations of this control as H is deformed. We first review some of the basics of deformation theory from a homotopy algebra point of view. The philosophy of deformation theory which we follow (due originally, we believe, to Grothendieck 2 cf. [47, 18, 7]) regards any deformation theory as ‘controlled’ by a dg Lie algebra g (unique up to homotopy type as an L∞ -algebra). For the deformation theory of an (ungraded) associative algebra A, the standard controlling dg Lie algebra is Coder(T c A) with the graded commutator as the graded Lie bracket [51]. Under the identification (including a shift in grading) of Coder(T c A) with Hom(T c A, A) (which is the Hochschild cochain complex), this bracket is identified with the Gerstenhaber bracket and the differential with the Hochschild differential, which can be written as [m, ] [13]. The generalization to a differential graded associative algebra is straightforward; the differential is now: [dA + m2 , ]. For an A∞ -algebra, the differential similarly generalizes to [m, ]. Deformations of A correspond to certain elements of Coder(T c A), namely those that are solutions of an integrability equation, now known more commonly as a Maurer-Cartan equation. Definition 18 (The classical Maurer-Cartan equation). In a dg Lie algebra (g, d, [ , ]), the classical Maurer-Cartan equation is 1 dθ + [θ, θ] = 0 2

(5.1)

for θ ∈ g1 = (↓ L)1 . For an A∞ -algebra (A, m) and θ ∈ Coder 1 (T c A), a deformed A∞ -structure is given by m + θ iff (m + θ )2 = 0. Teasing this apart, since we start with m2 = 0, we have equivalently Dθ + 1/2[θ, θ] = 0,

(5.2)

hence the Maurer-Cartan name. (Here D is the natural differential on Coder(T c A) ⊂ End(T c A), i.e. Dθ = [m, θ]. ) Notice that we call this the Maurer-Cartan equation for the dg Lie algebra (Coder(T c A), D, [ , ]) but not for (A, m). For L∞ -algebras, the analogous remarks hold, substituting the Chevalley-Eilenberg complex for that of Hochschild, i.e. using Coder C(L) Hom(C(L), L). In any case, formal deformation theory controlled by a dg Lie algebra (g, d, [ , ]) proceeds as follows. Consider a formal solution θ of the Maurer-Cartan equation (5.1) in θ ∈ g1 ⊗C[[]], where is a formal parameter. We express it as θ = θ(1) +θ(2) 2 +· · · , where θ(i) ∈ g1 . The Maurer-Cartan equation holds separately in different powers of , so we have 2

See [8] for an extensive annotated bibliography of deformation theory.

574


dθ(1) = 0 , 1 dθ(2) + [θ(1) , θ(1) ] = 0 , 2 dθ(3) + [θ(1) , θ(2) ] = 0 , ······ .

()1 : ()2 : ()3 : ···

(5.3) (5.4) (5.5)

The first order solution θ(1) is defined by the first equation (5.3), that is, θ(1) is a cocycle. This is also known as an infinitesimal deformation. We may proceed to second order if there is some θ(2) satisfying the second equation (5.4). Similarly, we can ask for θ(3) satisfying the third equation (5.5), etc. Since deformation theory is controlled by a dg Lie algebra up to homotopy (see Theorem 8), the Maurer-Cartan equation should be extended to that for an L∞ -algebra. We present the definition in the suspended (L = ↑g) notation. In addition to the convergence problem which would occur in the dg Lie algebra case, for an L∞ -algebra on L the Maurer-Cartan equation itself does not make sense in general since it consists of an infinite sum (see below). One way to avoid these problems is again to consider formal deformation theory; one usually considers a homotopy algebra on a graded vector space V over C[[]], or more generally a finite dimensional nilpotent commutative associative algebra. In particular, for an Artin algebra A and its maximal ideal mA , the standard way is to consider V ⊗ mA , where the degree of A is set to be zero. The multi-linear operations on V are extended to those on V ⊗ mA trivially. From now on, we shall assume but not mention explicitly that any homotopy algebra V we consider has been tensored with mA for some fixed mA and denote the result also by V . Definition 19 (The strong homotopy Maurer-Cartan equation). In an L∞ -algebra (L, l), the (generalized) Maurer-Cartan equation is 1 lk (c, ¯ · · · , c) ¯ =0 k! k≥1

for c¯ ∈ L0 . Note that the degree of c¯ is zero since g1 = L0 . We denote the set of solutions of the Maurer-Cartan equation as MC(L, l). In the same sense as in the dg Lie algebra case, a cocycle c¯ ∈ L0 , l1 c¯ = 0 play the role of a first order solution. Recall that for an OCHA (H, l, n), the adjoints of the maps np,q constitute an L∞ map ρ : Hc → ↑Coder(T c Ho ).

(5.6)

Since it is known that an L∞ -morphism preserves the solutions of the Maurer-Cartan equations, we obtain the following: Theorem 6. If c¯ ∈ Hc is a Maurer-Cartan element, then ρ(c) ¯ ∈ ↑Coder(T c Ho ) gives a deformation of Ho as a weak A∞ -algebra. In particular, a first order solution for c¯ ∈ Hc is preserved to be a first order solution in ↑Coder(T c Ho ) by an L∞ -morphism. The corresponding situation is the chain map (2.18) considered previously: ρ(dg (↑X)) = [m, ρ(↑X)],


575

where ↑X ∈ Hc and ρ(↑X) ∈ ↑Coder(T c Ho ). If ↑X is a first order solution, the chain map gives us [m, ρ(↑X)] = 0. However, notice that this ρ(↑X), a first order solution in ↑Coder(T c Ho ), in general includes a constant term C → Ho coming from n1,0 . Namely, the first order deformation of m turns out to be a ‘weak’ homotopy derivation, a natural extension of a strong homotopy derivation in Definition 9 by including a map θ0 : C → A. Rather than treat this result in isolation, we look at more general deformations of H as an OCHA. In order to do it, let us first explain another aspect of the Maurer-Cartan equation for a dg Lie algebra or an L∞ -algebra more generally. Lemma 1. For an L∞ -algebra (L, l) and a graded vector space L , consider a coalgebra isomorphism f : C(L ) → C(L), that is, a collection of degree zero graded symmetric maps {f0 , f1 , . . . } such that f1 : L → L is an isomorphism. Then, the inverse of f exists, and a unique weak L∞ -structure l is induced by l = (f)−1 ◦ l ◦ f so that f : (L , l ) → (L, l) is a weak L∞ -isomorphism. It is clear by definition that l is a degree one coderivation and (l )2 = 0. Moreover, if we take {f0 = c¯ ∈ L0 , f1 = 1, f2 = · · · = 0} for f, the explicit form of l is given as follows (see Getzler [15] and Schuhmacher [48]): 1 ll (c1 , . . . , cl ) := l≥0. (5.7) ln+l (c¯⊗n , c1 , . . . , cl ) , n! n≥0

Here recall that f0 : C → Hc so we identify f0 with its image c. ¯ Notice that l0 = 1 ⊗k k≥1 k! lk (c¯ ). Thus, l gives a (strict) L∞ -structure iff c¯ ∈ MC(L, l). In this argument, we can also begin with a weak L∞ -algebra (L, l) together with a straightforward modification of the Maurer-Cartan equation for a weak L∞ -algebra. The same fact holds true also for (weak) A∞ -algebras, as explained in Subsect. 2.4 in [31] (the explicit form of the deformed A∞ -structures can be found in [10, 11]). Let us consider the same story for an OCHA. Lemma 2. For an OCHA (H, l, n) and a graded vector space H , consider a coalgebra map f : C(Hc ) ⊗ T c (Ho ) → C(Hc ) ⊗ T c (Ho ) such that f1 + f0,1 : H → H is an isomorphism. Then, a unique weak OCHA-structure l + n is induced by l + n = (f)−1 ◦ (l + n) ◦ f so that f : (H , l , n ) → (H, l, n) is a weak OCHA-isomorphism. Again, the fact that f is a coalgebra map and l+n is a degree one coderivation implies that l + n is in fact a degree one coderivation, and (l + n )2 = 0 follows from (l + n)2 = 0. The reason the structure is weak is the presence of the operations l0 and n0,0 . In particular, when we take a weak OCHA-isomorphism f given by f0 = c¯ ∈ Hc0 ,

f0,0 = o¯ ∈ Ho0 ,

f1 = 1c ,

f0,1 = 1o

and other higher multi-linear maps set to be zero, the deformed weak OCHA structure is given by l in Eq. (5.7) and 1 np,q (c1 , . . . , cp ; o1 , . . . , oq ) := nn+p,m0 +...+mk +q (c¯⊗n , c1 , . . . , cp ; n! n,m0 ,... ,mk ≥0 ⊗m0

o¯

, o1 , o¯ ⊗m1 , . . . , o¯ ⊗mq−1 , oq , o¯ ⊗mq ) (5.8)

for p ≥ 0 and q ≥ 0. Now, we can spell out generalized Maurer-Cartan equations for OCHAs.

576


Definition 20 (Maurer-Cartan equations for (H, l, n)). For an OCHA (H, l, n) and degree zero elements c¯ ∈ Hc and o¯ ∈ Ho , we define ¯ := l∗ (c)

1 ¯ . . . , c) ¯ , lk (c, k!

n∗ (c; ¯ o) ¯ :=

k

1 ¯ . . . , c; ¯ o, ¯ . . . , o) ¯ . (5.9) nk,l (c, k! k,l

We call the following pair of equations ¯ , 0 = l∗ (c)

0 = n∗ (c; ¯ o) ¯

(5.10)

the Maurer-Cartan equations for the OCHA (H, l, n). The solution space of the Maurer-Cartan equations is denoted by ¯ = 0, n∗ (c; ¯ o) ¯ =0}. MC(H, l, n) = {(c, ¯ o) ¯ ∈ (Hc0 , Ho0 ) | l∗ (c) The Maurer-Cartan equations (5.10) are nothing but the condition that l0 = 0 and n0 = 0, since l0 = l∗ (c) ¯ and n0 = n∗ (c; ¯ o). ¯ In particular, the first equation is just the Maurer-Cartan equation for the L∞ -algebra (Hc , l). Now, one gets the following. Theorem 7 (Maurer-Cartan elements as deformations). (c, ¯ o) ¯ ∈ MC(H, l, n) gives a deformation of (Ho , m) as a (strict) A∞ -algebra. The explanations are as follows. First of all, for a weak OCHA (H = Hc ⊕ Ho , l , n ) given in Eq. (5.7) and Eq. (5.8), let us consider its restriction to Hc = 0, that is, consider the defining equation for a (weak) OCHA (2.19) and set c1 = · · · = cn = 0. Then, only the equations for n = 0 survive, which are given by 0 = n1,m (l0 ; o1 , . . . , om ) + n0,i+1+j (∅; o1 , . . .

(−1)β(s,i)

i+s+j =m , oi , n0,s (∅; oi+1 , . . .

, oi+s ), oi+s+1 , . . . , om ) .

Here l = 0 iff c¯ ∈ MC(Hc , l), then the first term on the right hand side drops out and the second term turns out to be the defining equation for a weak A∞ -algebra. This is just the situation of Theorem 6, where ρ(c) ¯ is given explicitly by ↓ρ(c) ¯ =

p≥1,q≥0

1 np,q (c¯⊗p ; , . . . , ) ∈ Hom(T c Ho , Ho ). p!

Note that (c, ¯ 0) ∈ (Hc0 , Ho0 ) need not belong to MC(H, l, n) even if c¯ ∈ MC(H, l) because of the existence of nk,0 (c, ¯ . . . , c) ¯ terms in the second equation in Eq. (5.9).Alternatively, for c¯ ∈ MC(Hc , l), if we can find an element o¯ such that (c, ¯ o) ¯ ∈ MC(H, l, n), n0,0 also vanishes and one gets a deformed (strict) A∞ -algebra. Thus we obtain Theorem 7 above.


577

5.2. Gauge equivalence and moduli spaces. Continuing with the general philosophy of deformation theory, we regard two deformations as equivalent if they are related by gauge equivalence, that is, if they differ by the action of the group obtained by exponentiating the action of g0 of the controlling dg Lie algebra g. For the case of L∞ -algebras instead of dg Lie algebras, it is more subtle to show that the gauge equivalence given in a similar way in fact defines an equivalence relation, that is, the composition of gauge transformations is a gauge transformation. In order to avoid such conceptually irrelevant subtlety, we give a definition of gauge equivalence in a more formal way in terms of piecewise smooth paths, though these definitions should be equivalent under some appropriate assumptions (see [11]). Definition 21 (Gauge equivalence). Given an L∞ -algebra (Hc , l), two elements c¯0 ∈ MC(Hc , l) and c¯1 ∈ MC(Hc , l) are called gauge equivalent iff there exists a piecewise smooth path c¯t ∈ MC(Hc , l), t ∈ [0, 1] such that 1 d c¯t = (5.11) l1+k (α(t), c¯t⊗k ) dt k! k≥0

for a degree minus one element α(t) ∈ Hc−1 . By this definition, it is clear that the gauge equivalence actually defines an equivalence relation. One can also express this gauge transformation in terms of a path ordered integral as c1 = c1 ({lk }, c0 , α(t)) = c0 + · · · [31, 33]. Definition 22 (Moduli space). For an L∞ -algebra (Hc , l) and the solution space of its Maurer-Cartan equation MC(Hc , l), the corresponding moduli space M(Hc , l) is defined as M(Hc , l) := MC(Hc , l)/ ∼ , where ∼ is the gauge equivalence in Definition 21. The moduli space for an A∞ -algebra (Ho , m) is also defined in a similar way and denoted by M(Ho , m) := MC(Ho , m)/ ∼. The following classical fact is known (for instance see [36, 10, 11, 31, 33]; some of these include the case of A∞ -algebras, for which a similar fact holds). Theorem 8. For two L∞ -algebras (Hc , l) and (Hc , l ), suppose there exists an L∞ morphism f : (Hc , l) → (Hc , l ). Then there exists a well-defined map f∼ : M(Hc , l) → M(Hc , l ) and in particular f∼ gives an isomorphism if f is an L∞ -quasi-isomorphism. Then, as a corollary of Theorem 6 we have the following: Corollary 3 (A∞ -structure parameterized by the moduli space of L∞ -structures). For an L∞ -algebra (Hc , l) and an A∞ -algebra (Ho , m), suppose there exists an OCHA (H = Hc ⊕ Ho , l, n) such that (Ho , {n0,k }) = (Ho , m). Also, let (Hc , l ) be an L∞ algebra obtained by the suspension of the dg Lie algebra Coder(T c A) with D = [m, ] and Lie bracket [ , ]. The OCHA (H, l, n) then gives a map from M(Hc , l) to M(Hc , l ) and it is in particular an isomorphism if the L∞ -morphism (Hc , l) → (Hc , l ) is an L∞ -quasi-isomorphism.

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In a similar way as in the A∞ or L∞ case, we can define the moduli space of the solution space of the Maurer-Cartan equations for an OCHA. Definition 23 (Open-closed gauge equivalence). Given an OCHA (H, l, n), we call two elements (c¯0 , o¯ 0 ) ∈ MC(H, l, n) and (c¯1 , o¯ 1 ) ∈ MC(H, l, n) gauge equivalent iff there exists a piecewise smooth path (c¯t , o¯ t ) ∈ MC(H, l, n), t ∈ [0, 1] such that c¯t satisfies differential equation (5.11) and o¯ t satisfies 1 d ⊗p ⊗q o¯ t = n1+p,q (α(t), c¯t ; o¯ t ) dt p! p,q≥0

+

p,q,q ≥0

1 ⊗p ⊗q ⊗q np,q+1+q (c¯t ; o¯ t , β(t), o¯ t ) p!

for degree minus one elements (α(t), β(t)) ∈ (Hc−1 , Ho−1 ). By definition, when (c¯0 , o¯ 0 ) and (c¯1 , o¯ 1 ) are gauge equivalent in the sense of an OCHA, c¯0 and c¯1 are gauge equivalent in the sense of the L∞ -algebra. Definition 24 (Moduli space for an OCHA). For an OCHA (H, l, n) and the solution space of its Maurer-Cartan equations MC(H, l, n), the moduli space for the OCHA (H, l, n) is defined by M(H, l, n) := MC(H, l, n)/ ∼ , where ∼ is the gauge equivalence in Definition 23. Then, due to the theorems in Sect. 4 and in particular Corollary 2, the following theorem is obtained in a similar way as in the A∞ and L∞ -cases. Theorem 9. Suppose we have an OCHA homomorphism f : (H, l, n) → (H , l , n ) between two OCHAs. Then, f induces a well-defined map between two moduli spaces f∼ : M(H, l, n) → M(H , l , n ). Furthermore, if f is an OCHA quasi-isomorphism, it induces an isomorphism between the two moduli spaces. Thus, the moduli space M(H, l, n) is also a homotopy invariant notion and in particular the equivalence class of deformations given by Theorem 7 is described by M(H, l, n). Acknowledgements. H. K would like to thank A. Kato, T. Kimura, H. Ohta, A. Voronov for valuable discussions. Also, he is very grateful to the Department of Mathematics of the University of Pennsylvania for hospitality, where this work was almost completed. J. S. would like to thank E. Harrelson, T. Lada, M. Markl and B. Zwiebach for valuable discussions.

Appendix (by M. Markl): Operadic Interpretation of A∞ -Algebras over L∞ Algebras This part of the paper assumes some knowledge of the language of operads and related notions, see the book [44], namely Sect. II.3.7 of this book. We explain here how A∞ algebras over L∞ -algebras can be interpreted using a ‘colored’ version of the standard theory of strong homotopy algebras in the form formulated in [44, Prop. II.3.88]. Let us briefly recall some necessary background material.


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Assume that P is a quadratic Koszul operad governing the algebraic structure we have in mind (such as associative algebra, Lie algebra, etc.) and let P ! denote the quadratic dual of P. Proposition II.3.88 of [44] then says that a strongly homotopy P-algebra on a graded vector space V is the same as a degree +1 differential on the cofree nilpotent P ! -coalgebra TP ! (↓ V ) on the desuspension of V . Using a colored version of this proposition, we show that A∞ -algebras over L∞ -algebras are in fact strongly homotopy Leibniz pairs. Let ρ : g → Der A be a Leibniz pair as in Definition 8. These Leibniz pairs are algebras over a two-colored operad Leib, with the white color denoting inputs/output in g and the black color inputs/output in A. The operad Leib is a quadratic {◦, •}-colored operad generated by one antisymmetric binary operation l of type (◦, ◦) → ◦ for the Lie multiplication in g, one binary operation m of type (•, •) → • for the associative multiplication in A, and one binary operation ρ of type (◦, •) → • for the action of g on A. The relations defining Leib as a quadratic colored operad can be easily read off from Eq. (2.14) and Eq. (2.15). We may safely leave as an exercise to verify that the quadratic dual Leib! of the operad Leib describes objects (C, A) consisting of a commutative associative algebra C, an associative algebra A and an action of C on A that satisfies X(ab) = X(a)b = aX(b), for X ∈ C and a, b ∈ A, and (XY)(a) = X(Y(a)) = Y(X(a)), for X, Y ∈ C and a ∈ A. It is equally simple to prove that the cofree nilpotent Leib! -algebra cogenerated by a colored space V◦ ⊕ V• equals C(V◦ ) ⊗ T c (V• ), where C(V◦ ) is the cofree nilpotent cocommutative coassociative coalgebra cogenerated by V◦ and T c (V• ) is the cofree nilpotent coassociative coalgebra cogenerated by X• (the tensor coalgebra). It immediately follows from these calculations that the obvious colored version of the above mentioned [44, Prop. II.3.88] identifies A∞ -algebras over L∞ -algebras in the coalgebra description as in Subsect. 2.8 (where V◦ and V• correspond to Hc and Ho ) with strongly homotopy Leibniz pairs. The only nontrivial thing which we left aside was to prove that Leib is a Koszul quadratic colored operad, in the sense of [57]. This is, according to [42], necessary for the homotopy invariance of these strongly homotopy Leibniz pairs, though the above constructions make sense even without the Koszulity. We believe that Koszulness of Leib would follow from a spectral sequence argument similar to that used in the proof of [41, Theorem 4.5]. References 1. Alexandrov, M., Kontsevich, M., Schwartz, A., Zaboronsky, O.: The Geometry of the master equation and topological quantum field theory. Int. J. Mod. Phys. A 12, 1405 (1997) 2. Alekseev, A., Meinrenken, E.: Equivariant cohomology and the Maurer-Cartan equation. Duke Mathematical Journal, 130(3), 479–522 (2005) 3. Barannikov, S., Kontsevich, M.: Frobenius manifolds and formality of Lie algebras of polyvector fields. Internat. Math. Res. Notices 1998, no. 4, 201–215 4. Cartan, H.: Notions d’algébre difféntielle; application aux groupes de Lie et aux variétés oú opére un groupe de Lie (French). In: Colloque de topologie (espaces fibrés), Bruxelles, 1950, Liége: Georges Thone, Paris: Masson et Cie., 1951, pp 15–27 5. Cattaneo, A.S., Felder, G.: A path integral approach to the Kontsevich quantization formula. Commun. Math. Phys. 212, 591–611 (2000)

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6. Cohen, F.R.: The homology of Cn+1 -spaces, n ≥ 0. In: The homology of iterated loop spaces, Lecture Notes in Mathematics, Vol. 533, Berlin-New York: Springer-Verlag, 1976 7. Deligne, D.: Letter to W. M. Goldman, J. J. Millson 8. Doran, C.F., Wong, S.: Deformation Theory: An Historical Annotated Bibliography. In: Deformation of Galois Representations, Chap. 2, to appear in the AMS-IP Studies in Advanced Mathematics Series 9. Flato, M., Gerstenhaber, M., Voronov, A.A.: Cohomology and deformation of Leibniz pairs. Lett. Math. Phys. 34, no. 1, 77–90 (1995) 10. Fukaya, K.: Deformation theory, Homological Algebra, and Mirror symmetry. In: Geometry and physics of branes, U. Bruzzo, V. Gorinu, U. Moschelli, eds., Bristol-Philadelphia: Inst. of Phys. Publishing, 2002, pp. 121–210 11. Fukaya, K., Oh,Y.-G., Ohta, H., Ono, K.: Lagrangian intersection Floer theory-anomaly and obstruction. To appear, International Press 12. Gaberdiel, M.R., Zwiebach, B.: Tensor constructions of open string theories I: Foundations. Nucl. Phys. B 505, 569 (1997) 13. Gerstenhaber, M.: The cohomology structure of an associative ring. Ann. Math. 78, 267–288 (1963); On the deformation of rings and algebras. Ann. Math. 79, 59–103 (1964) 14. Getzler, E.: Batalin-Vilkovisky algebras and two-dimensional topological field theories. Commun. Math. Phys. 159, 265 (1994) 15. Getzler, E.: Lie theory for nilpotent L-infinity algebras. http://arxiv.org/list/math.AT/0404003, 2004 16. Getzler, E., Jones, J.D.S.: Operads, homotopy algebra and iterated integrals for double loop spaces. Preprint, Department of Mathematics, MIT, March 1994, http://arxiv.org/list/hep-th/9403055, 1994 17. Getzler, E., Kapranov, M.M.: Cyclic operads and cyclic homology. In: Geometry, topology, & physics, Conf. Proc. Lecture Notes Geom. Topology, IV, Cambridge, MA: Internat. Press, 1995, pp. 167–201 18. Goldman, W.M., Millson, J.J.: The deformation theory of representations of fundamental groups ´ of compact Kähler manifolds. Inst. Hautes Etudes Sci. Publ. Math. No. 67, 43–96 (1988); The homotopy invariance of the Kuranishi space. Illinois J. Math. 34, no. 2, 337–367 (1990) 19. Gugenheim, V.K.A.M.: On a perturbation theory for the homology of the loop-space. J. Pure Appl. Algebra 25, 197–205 (1982) 20. Gugenheim, V.K.A.M., Stasheff, J.D.: On perturbations and A∞ -structures. Bull. Soc. Math. Belg. Sér. A 38, 237–246 (1986) 21. Gugenheim, V.K.A.M., Lambe, L.A.: Perturbation theory in differential homological algebra. I. Illinois J. Math. 33, no. 4, 566–582 (1989) 22. Gugenheim, V.K.A.M., Lambe, L.A., Stasheff, J.D.: Algebraic aspects of Chen’s twisting cochain. Illinois J. Math. 34, no. 2, 485–502 (1990) 23. Gugenheim, V.K.A.M., Lambe, L.A., Stasheff, J.D.: Perturbation theory in differential homological algebra II. Illinois J. Math. 35, no. 3, 357–373 (1991) 24. Harrelson, E.: On the homology of open/closed string theory. http://arxiv.org/list/math.AT/0412249, 2004 25. Hofman, C., Ma, W.K.: Deformations of topological open strings. JHEP 0101, 035 (2001) 26. Hofman, C.: On the open-closed B-model. JHEP 0311, 069 (2003) 27. Huebschmann, J., Kadeishvili, T.: Small models for chain algebras. Math. Z. 207, 245–280 (1991) 28. Huebschmann, J., Stasheff, J.: Formal solution of the Master Equation via HPT and deformation theory. Forum Mathematicum, 14, 847–868 (2002) 29. Kadeishvili, T.V.: The algebraic structure in the homology of an A(∞)-algebra. (Russian) Soobshch. Akad. Nauk Gruzin. SSR 108, no. 2, 249–252 (1982) 30. Kajiura, H.: Homotopy algebra morphism and geometry of classical string field theories. Nucl. Phys. B 630, 361 (2002) 31. Kajiura, H.: Noncommutative homotopy algebras associated with open strings. Doctoral thesis, Graduate School of Mathematical Sciences, Univ. of Tokyo, http://arxiv.org/list/math.QA/0306332, 2003 32. Kajiura, H., Stasheff, J.: Open-closed homotopy algebra in mathematical physics. J. Math. Phys. 47, 023506 (2006) 33. Kajiura, H., Terashima, Y.: Homotopy equivalence of A∞ -morphisms and gauge transformations. Preprint, 2003 34. Kimura, T., Stasheff, J., Voronov, A.A.: On operad structures of moduli spaces and string theory. Commun. Math. Phys. 171, 1 (1995) 35. Kontsevich, M.: Homological algebra of mirror symmetry. In: Proceedings of the International Congress of Mathematicians, Vol. 1 (Zürich, 1994), Basel, Birkhäuser: 1995, pp. 120–139 36. Kontsevich, M.: Deformation quantization of Poisson manifolds, I. Lett. Math. Phys. 66, 157–216 (2003)


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Communications in


Higher-Dimensional Generalizations of Affine Kac-Moody and Virasoro Conformal Lie Algebras M. Golenishcheva-Kutuzova Department of Mathematics, University of Florida, PO Box 118105, Gainesville, FL 32611-8105, USA. E-mail: [email protected] Received: 18 December 2004 / Accepted: 27 August 2005 Published online: 28 February 2006 – © Springer-Verlag 2006

Abstract: We discuss the generalizations of the notion of Conformal Algebra and Local Distribution Lie algebras for multi-dimensional bases. We replace the algebra of Laurent polynomials on C∗ by an infinite-dimensional representation (with some additional structures) of a simple finite-dimensional Lie algebra g in the space of regular functions on the corresponding Grassmann variety M g that can be described as a “right” higherdimensional generalization of C∗ from the point of view of a corresponding group action. For g = sl2 it gives us the usual Vertex Algebra notion. We construct the higher dimensional generalizations of the Virasoro and the Affine Kac-Moody Conformal Lie algebras explicitly and in terms of the Operator Product Expansion. Contents 1. 2. 3. 4. 5. 6. 7.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generalized Delta-Function . . . . . . . . . . . . . . . . . . . . . . . . . Local Distributions on V g . . . . . . . . . . . . . . . . . . . . . . . . . The Space V g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Explicit Realization of the Space V sl3 . . . . . . . . . . . . . . . . . . . Generalized Affine Kac-Moody Algebras Associated with the Space V sl3 Generalized Virasoro Algebra Associated with the Space V sl3 . . . . . .

. . . . . . .

583 587 590 593 595 602 608

1. Introduction The basic object of d-dimensional Quantum Field Theory (QFT) are d-dimensional fields (operator-valued distributions on M d ) and the representation of the Poincaré group in the Hilbert Space H that behave according to the Wightman axioms of QFT. In 1984 Belavin, Polyakov and Zamolodchikov ([BPZ]) initiated the study of two-dimensional Conformal Field Theory (CFT). In CFT we have a nice split of variables that leads to the notion

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of the “chiral part” of a conformal field theory, where fields are just operator-valued distributions on C∗ . This arises a huge interest in the theory of infinite-dimensional Lie algebras and their representations. It turns out that the symmetries of most models of CFT are classified along the representations of Affine Kac-Moody Lie algebras and the Virasoro algebra. A mathematical definition of the “chiral part” of a conformal field theory, called a Vertex algebra, was proposed by Borcherds ([B]). The axioms of Vertex algebras ([K2]) are mathematical description of the operator product expansion in CFT. The Vertex algebra approach simplifies the problem of classification of infinite dimensional symmetries in CFT. However, until now a classification of Vertex algebras seems to be far away. There is the solution to the classification problem of simple conformal algebras only when the chiral algebra is generated by a finite number of quantum fields, closed under the operator product expansions (in a sense that only derivatives of the generating field may occur). Roughly speaking, in the finitely generated case there is nothing but the Affine Kac-Moody algebras and the Virasoro Vertex algebra (Heisenberg algebra can be treated as a trivial Affine algebra). This fact was proven in ([DK]) using the notion of a conformal algebra, which is related to a chiral algebra in the same way as a Lie algebra is related to its universal enveloping algebra. On the space of fields (mutually local formal distributions in complex variable z) there is the action of the operator ∂, given by ∂a(z) = ∂z a(z) and the fields that appear in the right part of the commutation relations of two fields a(z) and b(w) can be viewed as n-products of these two fields. The definition of conformal algebra is: Definition 1.1 (V.Kac). A (Lie) conformal algebra is a C[∂]-module R, endowed with a family of C-bilinear products a(n) b, n ∈ Z+ , satisfying axioms (C1)–(C4): (C1) a(n) b = 0 f or n >> 0, (C2) (∂a)(n) b = −na(n−1) b, a(n) ∂b = ∂(a(n) b) − (∂a)(n) b, n+j ∂ (j ) (b a), (C3) a(n) b = − ∞ j =0 (−1) m(n+j)m (C4) a(m) (b(n) c) − b(n) (a(m) c) = j =0 j (a(j ) b)(m+n−j ) c, j

where ∂ (j ) = ∂n! . A conformal algebra R is called f inite if R is a finitely generated C[∂]-module. The rank of conformal algebra R is its rank as a C[∂]-module. Definition 1.2. Let g be an arbitrary Lie algebra. A formal distribution Lie algebra (g, F ) is the space F of all mutually local g-valued distributions in complex variable z. It is very important that we have a functor from the category of formal distribution Lie algebras to the category of conformal algebras as well as a functor in the opposite direction that canonically associates to a conformal algebra R a formal distribution Lie algebra ([K2]). It means that when we use a very formal language of conformal algebras we do not lose the information about the physical origin. It is especially important to have it in mind when we would like to construct higher dimensional generalizations. The approach to higher dimensional chiral algebras suggested by Beilinson and Drinfeld is based more on algebraic geometry than representation theory. In ([BD]) they introduced the notion of “Chiral algebra” as a quantization of what they call the “coisson algebra” (a Poisson algebra on X in the compound setting). A really challenging problem is to find out what a chiral algebra on higher dimensional X is (the coisson algebras live in any dimension). A more algebraic approach to the higher dimensional vertex algebras was suggested by R. Borcherds in ([B2]). He introduced the notion of G-vertex algebra. The main

Higher-Dimensional Generalizations of Conformal Lie Algebras

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component of G vertex algebras is the notion of a vertex group G (cocommutative Hopf algebra ) with a space of functions with some allowable singularities on it. The similar idea to replace the C[∂] in the definition of conformal algebra R by a Hopf algebra H = U (g), where g is a finite-dimensional Lie algebra and U (g) is its universal enveloping algebra was explored by Bakalov, D’Andrea, and Kac in ([BDK, BDK1]). They introduced the notion of Lie pseudoalgebra. A Lie pseudoalgebra is defined as an H -module L endowed with an H -bilinear map L ⊗ L → (H ⊗ H )⊗H L

(1.1)

subject to a certain skewsymmetry and Jacobi identity axioms. Unfortunately, for Lie pseudoalgebras the functor to the higher dimensional distribution Lie algebras was not explicitly constructed. In particular it means that there is no sensible way to construct the corresponding underlying space (of functions with singularities on G) for the associated higher dimensional quantum field theories. We suggest an approach to the higher dimensional CFT that preserves the connection between the higher dimensional distribution Lie algebras and a higher dimensional conformal algebra. We replace the C[∂] in the definition of the conformal algebra R by a universal enveloping algebra of some noncommutative Lie algebra n. In the case when n is a nilpotent subalgebra of a simple Lie algebra g we have a natural geometrical realization of the algebra of functions with singularities on n as the algebra V g of all regular functions on some complex manifold M g that is an infinite dimensional representation of g. The corresponding local distribution Lie algebra has most of the properties of two dimensional CFT. It means that we have the consistent definition of the OPE and the normal product of two fields in the dimension higher than two. The corresponding conformal algebra is a U (n)-module with the eγ -products subject to a certain skewsymmetry and Jacobi identity axioms. Here {eγ } is a basis in some representation of g. When g = sl2 our construction leads to the standard two dimensional situation, where M g = C∗ , and the n-product of two fields corresponds to the (en )-product, where {en } is a basis in some representation of sl2 . As it was mentioned above, in the case of a simple finite conformal algebra, there are only two principal solutions to the system of axioms (C1)–(C4) (in this paper we do not discuss conformal superalgebras, where we have more possibilities): 1. Current conformal algebras Cur g = C[∂] ⊗ g associated to the Lie algebra g. The only non-trivial n-product is the 0-product: a(0) b = [a, b], a, b ∈ g. We identify g with the subspace of Cur g spanned by elements 1 ⊗ g, g ∈ g. The corresponding −1 formal distribution Lie algebra is (˜g, R) where g˜ = g ⊗ C[t, t ]. Formal distributions g(z) = g ⊗ δ(t − z) = n∈Z g ⊗ t n ⊗ z−n−1 , defined for every g ∈ g, satisfy the commutation relations: [g1 (z), g2 (w)] = [g1 , g2 ](w)δ(z − w). 2.

(1.2)

Virasoro conformal algebra Conf (Vect C∗ ). The centerless Virasoro algebra of algebraic vector fields on C∗ is spanned by the vector fields Ln = −t n+1 ∂t . The VectC∗ valued formal distribution L(z) = δ(t − z) ∂t satisfies [L(z), L(w)] = ∂w L(w) δ(z − w) + 2L(w) ∂w δ(z − w).

(1.3)

The conformal algebra Conf (Vect C∗ ) = C[∂]L associated to (Vect C∗ , {L(z)}) is defined by the n-products: L(0) L = ∂L,

L(1) L = 2L,

L(n) L = 0 if

n ≥ 2.

(1.4)

586


The conformal algebra associated to the semidirect sum (Vect C∗ )+g˜ is the semidirect sum Conf(Vect C∗ ) + Cur g, defined by (a ∈ g) : L(0) a = ∂a, L(1) a = a, L(n) a = 0 for n > 1.

(1.5)

What makes these two cases so special? The answer is that they both are related to the action of the Lie algebra sl2 on the space V sl2 of regular functions on a subset M C∗ of the flag manifold SL(2, C)/B− CP 1 and the corresponding fields are expressed in terms of the formal delta function δ(t − w) = n∈Z t n w −n−1 , associated with this space. Our approach is based on the construction of a complex manifold M g that can be described as a “right” higher-dimensional generalization of C∗ from the point of view of a corresponding group action. In Sect. 2 we define the space V g for any simple Lie algebra g and the formal delta function associated with this space. In Sect. 3 we define the Lie algebra of formal distributions on V g and the higher dimensional analogues of the n-products and λ-product for the conformal algebra associated with the Lie algebra of formal distributions on V g . In Sect. 4 we discuss the basic ideas about the geometrical realization of the space V g in the general case for any simple Lie algebra g, and in Sect. 5 we present an explicit realization of V g for g = sl3 . In Sect. 6 we define the 3-dimensional analogues of the Affine Kac-Moody algebras associated with the action of the Lie algebra sl3 . We call these algebras the Generalized Affine Kac-Moody algebras gV associated with the space V sl3 . We think that it is especially important to consider in more detail the sl3 -case, because it is a well known fact that it is difficult to make a step from sl2 to sl3 and it is almost straightforward from sl3 to go to the general case of any simple Lie algebra (at least to sln ). The constructed algebras have many of the good properties of “the generalized Affine Kac-Moody algebras”. The basic idea of Borcherds is to think of generalized Affine Kac-Moody algebras as infinite dimensional Lie algebras which have most of the good properties of finite dimensional reductive Lie algebras ([B]). For instance, we construct the normalized invariant form ( , ) and the Cartan involution of gV . In Sect. 7 we discuss the higher dimensional version of the Virasoro conformal algebra associated with the space V sl3 . The Virasoro algebra appears in many different contexts related to the Lie algebra sl2 and contains sl2 as a subalgebra. We will discuss one more context related to the conformal Virasoro (1.4) algebras and similarly we define the generalized Virasoro conformal algebra associated with the space V sl3 . This algebra contains sl3 as a subalgebra and it is a rank one module over U (n+ ), where n+ is the upper nilpotent subalgebra of sl3 . Also we have the semidirect sum of the generalized Virasoro conformal algebra and conformal algebra gV . The generalized Virasoro conformal algebra (the root lattice Virasoro conformal algebra) associated to any simple Lie algebra g will be constructed in future publications of the author. We would like note that our definition of the generalized Virasoro conformal algebras is different from the Virasoro pseudoalgebras defined in the ([BDK, BDK1]). In future publications we shall present: 1. The explicit geometrical realization of the space V g for any simple Lie algebra g; 2. The representations of the Generalized Affine Kac-Moody algebras gV associated with the space V g and the generalized Virasoro conformal algebra associated with this space;


587

3. The axioms for the higher dimensional Vertex Algebras associated with the space V g; 4. The higher dimensional generalizations of the N = 1 and N = 2 superconformal algebras. 2. Generalized Delta-Function In this section we introduce the notion of the generalized delta-function associated with a vector space V g , where g is a simple Lie algebra. First we recall the basic definitions of formal distributions and in particular the formal delta-function. A formal distribution in the indeterminates z, w, . . . ∈ (C∗ )N with values in a vector space W is a formal expression of the form am,n,... zm w n . . . , m,n... ,∈Z

where am,n,... are elements of a vector space W . They form a vector space denoted by W [[z, z−1 , w, w−1 ]]. Given a formal distribution a(z) = m∈Z am zm , we define the trace (integral) by the usual formula Resz a(z) = a−1 .

(2.1)

For any test function b(z) on C∗ and a formal distribution a(z) we can define a pairing by < a(z), b(z) >= Resz a(z)b(z).

(2.2)

Since Resz ∂a(z) = 0, we have the usual integration by parts: < ∂a(z), b(z) >= − < a(z), ∂b(z) >,

(2.3)

where ∂a(z) = a(z). We would like to remark that the bilinear form (2.2) is invariant under the action of the Lie algebra sl2 given by: ∂ ∂z

X = ∂,

H = −2z∂ − 1,

Y = −z2 ∂ − z,

(2.4)

where X, H, Y is the standard basis in sl2 . This bilinear form defines the pairing between the Verma module of sl2 with the highest weight − α2 in the space V + C[z] and the Verma module with the lowest weight α2 in the space V − 1z C[ 1z ]. The action (2.4) results from the natural action of the group G = SL(2, C) on the Flag manifold G/B− , where B− is the lower Borel subgroup of G. The space V + is isomorphic to the space of regular functions on the big cell U = N+ · [1] ∈ G/B− and the space V − is isomorphic to the space of regular functions on the dual cell U ∗ = N− ·sα [1] ∈ G/B− factorized by constants. Here sα denote the action of the generator of the Weyl group of sl2 on the flag manifold. The space V + is the maximum isotropic subspace with respect to the bilinear form (2.2). The weight basis in V + is en = zn , n ∈ Z+ and the dual basis in V − is en ∗ = z−n−1 , n ∈ Z+ . The complex torus C∗ is the intersection of U and U ∗ and the space of all regular functions on C∗ is C[z, z− ] = V + ⊕ V − . The subspaces V + and V − are invariant under the multiplication and the action of g sl2 , given by (2.4). We will denote the space C[z, z−1 ] by Vz for g = sl2 .

588


Recall that the formal delta function is the following formal distribution in z and w with values in C δ(z − w) = zn · w −n−1 . (2.5) n∈Z

g g We can think about the formal delta function as an element of the space Vz ⊗ Vw of the form δ(z − w) = en ⊗ en∗ + en∗ ⊗ en = δ(z − w)− + δ(z − w)+ , (2.6) n∈Z+

n∈Z+

where + δ(z − w)− ∈ Vz+ ⊗ Vw− and δ(z − w)+ ∈ Vz− ⊗ Vw .

(2.7)

The well known properties of the formal delta-function ([K1]) result from (2.6). Let us mention some of them: (a) For any formal distribution f (z) ∈ U [[z, z−1 ]] one has: Resz f (z)δ(z−w) = f (w), j (b) ∂z δ(z − w) = (−∂w )j δ(z − w). As we have mentioned before, the formal delta function is connected with the action of the Lie algebra sl2 on functions on the Flag manifold and it is an element of the space g g Vz ⊗ Vw (2.4). The construction of the generalized formal delta functions is based on the same idea. First we will give a formal definition of the space V g for a simple Lie algebra g and the generalized formal delta functions connected with this space. Then in the next sections we will give the explicit construction for g = sl3 and discuss the general construction for any simple Lie algebra g. Let g be a simple Lie algebra of rank l. As a vector space, it has the triangular decomposition g = n+ ⊕ h ⊕ n− ,

(2.8)

where h is a Cartan subalgebra and n± are the upper and lower nilpotent subalgebras. Let b± = h ⊕ n±

(2.9)

be the upper and lower Borel subalgebras. Definition 2.1. A space V g is a vector space endowed with a C-valued non-degenerated symmetric bilinear form < ·, · > : V g ⊗ V g −→ C (2.10) such that the following axioms hold (V1) V g is a commutative associative algebra with unitary element 1 with respect to the multiplication. (We can think of V g as a space of complex-valued functions on a complex manifold M or orbifold). (V2) V g is a g-module, such that the action of n+ is a derivation of V g ; this means that x(f · g) = x(f ) · g + f · x(g) for any x ∈ n+ and any f, g ∈ V g . In particular, the unitary element of V g is annihilated by all elements from n+ .


589

(V3) The bilinear form < ·, · > is invariant under the multiplication in V g and the action of the Lie algebra n+ : < f · g, h >=< g, f · h >< x(f ), g > + < f, x(g) >= 0,

(2.11)

for any x ∈ n+ and any f, g, h ∈ V g . (V4) There is a maximal isotropic with respect to the bilinear form subspace V+ in V g , such that V+ is invariant under the multiplication and the action of n+ . Then V g = V+ ⊕ V− ,

(2.12)

where V− V+∗ is the dual to the subspace V+ in V g with respect to the given bilinear form. In the next section we will construct the explicit realization of the space V g as the ring of complex-valued regular functions on a complex manifold M. Because the vector space V g is self-dual, we can identify V g with (V g )∗ . Let {eγ }, γ ∈ be a basis in V+ and {eγ∗ } is the dual basis in V − , where is a discrete set that numerates the basis. Remark 2.1. In the previous definition instead of a bilinear form we can postulate that the space V g has a trace Res invariant with respect to the action of the nilpotent subalgebra n+ . If such a trace exists, the bilinear form on V g can be defined as < f , g >= Res (f · g).

(2.13)

In the other direction, if we have a bilinear form with the given properties on V g , and eγ0 = 1 for some γ0 ∈ , then the invariant trace can be defined as Res (f ) =< f , 1 >= coefficient of eγ∗0

(2.14)

in the decomposition with respect to the basis {eγ } ∪ {eγ∗ }, γ ∈ . Definition 2.2. The generalized formal delta functions associated with the space V g are defined as an element of the space V g ⊗ (V g )∗ = V+ ⊗ V− ⊕ V− ⊗ V+ of the form δV g = eγ ⊗ eγ∗ ⊕ eγ∗ ⊗ eγ . (2.15) γ ∈

Now suppose that the space V g is realized as the space Fun(M) of formal distributions on M and z = (z1 , z2 , . . . , zn ) are coordinates on M, then the space V g ⊗ (V g )∗ can be identified with the space of distributions in two sets of coordinates z = (z1 , z2 , . . . , zn ) and w = (w1 , w2 , . . . , wn ). In this case we will use the notation δV g (z − w),

(2.16)

or for short δV (z − w). The action of the nilpotent subalgebra n+ on Fun(M) is given by vector fields. Thus, we have a Lie algebra homomorphism n+ → VectM. So defined generalized formal delta functions have most of the properties of the standard delta function with respect to the trace (2.2) and differentiations from n+ :

590


(a) for any formal distribution f (z) ∈ V g one has: Resz f (z) δV (z − w) = f (w),

(2.17)

(b) for any element a ∈ n+ one has: (ξaz )j δ(z − w) = (−ξaw )j δ(z − w),

(2.18)

where ξaz is the image of a in Vect M. 3. Local Distributions on V g By definition the space V g = V+ ⊕ V− is a U (n+ )-module. We assume that V g is realized as a space of regular functions on some complex manifold M g . Let {eγ }, γ ∈ be a basis in V+ and let {eγ∗ } be the dual basis in V− constructed in the previous section. Fix a triangular decomposition of g: g = n+ ⊕ h ⊕ n− ,

(3.1)

where h is the Cartan subalgebra and n± are the upper and the lower nilpotent subalgebras. Let b± = h ⊕ n±

(3.2)

be the upper and lower Borel subalgebras. Let hi = αi , i = 1, . . . , l, be the i th coroot of g and let l be the rank of g. The set {hi }i=1,... ,l is a basis of h. We choose a root basis of n± , {eα }α∈± , where + (− ) is the set of positive (negative) roots of g, so that [h, eα ] = α(h)eα for all h ∈ h. Let ξα1 be the image of eαi , α ∈ + in Der (V g ). Fix some ordering of + . Then we can write ∂i instead of ξαi and (V g ) is a C[∂1 , . . . , ∂p ]-module, where p = |+ |. k

The C[∂1 , . . . , ∂p ] has a Poincaré-Birkhoff-Witt basis of the form {∂1k1 ∂2k2 . . . ∂pp }. Let W be a vector space. Consider the space End W ⊗ V g of End W -valued formal distributions associated with the space V g . Any element a V ∈ End W ⊗ V g is an expression of the form V V aV = aeγ∗ ⊗ eγ + aeγ ⊗ eγ∗ = a+ + a− , (3.3) γ ∈

γ ∈

where aeγ∗ , aeγ ∈ End W. The coefficients in (3.3) are defined via the trace (2.14). For any element a V ∈ End W ⊗ V g and any f ∈ V g we can define afV ∈ End W as afV = ResV g ((1 ⊗ f ) · a V ).

(3.4)

We have a natural action of C[∂1 , . . . , ∂p ] on End W ⊗ V g , induced by the action on V g , so that (∂i a)Vf = −a∂Vi f ,

(3.5)

or, more generally: (∂1k1 ∂2k2 . . . ∂pp a)Vf = (−1)k1 +k2 +...kp a Vkp k

kp−1

k

∂p ∂p−1 ...∂1 1 f

.

(3.6)


591

We will say that two formal distributions a V and bV are mutually local if the commutator can be expressed as a finite linear combination of the δV and its derivatives: kp k1 k2 V ]= cV k1 kp ∗ ∂1,w ∂2,w . . . ∂p,w δV . (3.7) [azV , bw k1 ,...kp

( ∂1 ...∂p vµ ) ; w

g g Since δV ∈ Vz ⊗ (Vw )∗ in the previous formula, we need to specify to which factor we apply the operator ∂k . ∂k,w means that we apply it to the second factor in the tensor product. g The coefficient cV k1 kp ∗ ∈ End W ⊗ Vw in this decomposition can be viewed ( ∂1 ...∂p vµ ) ; w

as the vk1 ,...kp -product of fields a V and bV . The element vk1 ,...kp is in the space Vµ∗ , dual to some representation of g with a lowest weight µ and the lowest vector vµ ∈ Vµ , and k

is dual to ∂1k1 ∂2k2 . . . ∂pp in the sense that p−1 ∂pp ∂p−1 . . . ∂1k1 vs1 ,...sp = (−1)k1 +k2 +...kp vµ∗ .

k

k

(3.8)

In particular, ∂i vµ∗ = 0.

g g For any f and g ∈ V g we can define [afV , bgV ] ∈ End W ⊗ Vz ⊗ Vw as: [afV , bgV ] = Rezw Rezz ([a V , bV ] · (1 ⊗ f ⊗ 1) · (1 ⊗ 1 ⊗ g)).

(3.9)

To consider the space of formal distributions with the values in a Lie algebra a we V bV -products need to impose the skewsymmetry and the Jacobi identity axioms to the a(v) g ∗ of two fields, v ∈ Vµ . Denote by R the set of all local fields (distributions) on V g with the values in a. The v-products define a C-linear map A : Vµ∗ ⊗ R g ⊗ R g −→ R g .

(3.10)

This map satisfies the following axioms that are analogues of axioms (C1)–(C2) in Def.1.1: (H1) For any a, b ∈ R g the map A : V ∗ ⊗ a ⊗ b −→ R g is non zero only on a finite µ

∗ ⊂ V ∗, dimensional subspace Vab µ (H2) A◦(1⊗∂i ⊗1) = A◦(∂i ⊗1⊗1) and A◦(1⊗1⊗∂i ) = ∂i ◦A+A◦(1⊗∂i ⊗1).

Definition 3.1. A(Lie) generalized conformal algebra R g , associated with the space Vλ∗ is a C[∂1 , . . . , ∂p ]-module, endowed with the map (3.10), satisfying axioms (H1)-(H2) as well as the skewsymmetry and the Jacobi identity axioms. A conformal algebra R g is called f inite if R g is a finitely generated C[∂1 , . . . , ∂p ]-module. The rank of the conformal algebra R g is its rank as a C[∂1 , . . . , ∂p ]-module. Since n+ is a non commutative algebra, it is difficult to write explicitly in terms of A the skewsymmetry and the Jacobi identity axioms. As for Conformal algebras these axioms have a more simple form in terms of the so-called λ − bracket (see for reference [K2]) defined as: λ(n) a(n) b. (3.11) [aλ b] = n∈Z

592


We define a non-commutative analogue of the λ − bracket as follows. Consider a k k Poincaré-Birkhoff-Witt basis B = {eαk11 . . . eαpp } ∈ U (n+ ). Let λ¯ = λ1k1 . . . λpp be the k k symbol of eαk11 . . . eαpp in C[λ1 , . . . , λp ] and ∂¯ = ∂1k1 . . . ∂pp its image in C[∂1 , . . . , ∂p ]. Denote by t λ¯ = (−λp )kp . . . (−λ1 )k1 the image of λ¯ under the transposition map. Define ¯ the λ-bracket of two elements a, b ∈ R g as: (k ) (k ) [a λ¯ b] = λ1 1 . . . λp p a k1 kp ∗ b. (3.12) k

( ∂1 ...∂p vµ )

kp

∂1 1 ...∂p ∈B

The axioms (H1)-(H2) are rephrased as follows: (H1) [a λ¯ b] ∈ C[λ1 , . . . , λp ] ⊗ R g , (H2) [∂i a λ¯ b] = −λi [a λ¯ b], [a λ¯ ∂i b] = (∂i + λi ) [a λ¯ b]. The skewsymmetry axiom: (H3) [a λ¯ b] = −[b

t (∂+ ¯ λ¯ )

a],

the Jacobi identity axiom: (H4) [a λ¯ [b µ¯ c] = [[a λ¯ b] λ¯ +µ¯ c] + [b µ¯ [a λ¯ c]]. In this setting the axioms (C1)–(C4) of conformal algebra R in Def.1.1, applied to the affine Kac-Mody and Virasoro cases say that there are only two opposite cases: (i) R/∂R is a Lie algebra and Vµ is a one dimensional module. Then for a, b ∈ R/∂R we define a(vµ∗ ) b = [a, b]. This is the current conformal algebra case. (ii) R/∂R is a one dimensional (rank one conformal) algebra and Vµ is a self-dual C[∂]-module that has the structure of a Lie algebra. More precisely in the second case we take µ = −α , where α is the root of sl2 . Then, Vα sl2 with vα = Y and the action ∂ given by ∂ = ad (X). Here X, H, Y is the standard basis of sl2 . Normalize this basis as vα = Y, H = ∂Y, −2X = ∂ 2 Y and ∂X = 0. The dual basis is X, H /2, −Y /2. Take L = Y and define the products, corresponding to the elements of the dual basis as: L(0) L = L(X) L = [X, L] = ∂L, L(1) L = L(H /2) L = [−H, L] = 2L. (3.13) The axiom (H2) reads as (∂L)(e) L = L(∂e) L, in particular (∂L)(H /2) L = L(∂H /2) L = −L(X) L. These relations between the commutation relations of the sl2 Lie algebra and n-products of the Virasoro conformal algebra is one more manifestation of the link between Vir and sl2 Lie algebras. Given a generalized conformal algebra R g , a Lie algebra of V g -local formal distributions associated to it is defined as follows. Fix a basis {eγ , eγ∗ }, γ ∈ in V g . Consider a vector space over C with the basis aeVγ , aeV∗ , where a V ∈ R g . Then the Lie γ algebra of V g -local formal distributions is a quotient of this space by the C-span of all elements of the form (λa V + µbV )eγ − (λa V )eγ − µ(bV )eγ , (λa V + µbV )eγ∗ − (λa V )eγ∗ − µ(bV )eγ∗ , (∂i a)Veγ + a∂Vi eγ ,

(∂i a)Veγ∗ + a∂Vi eγ∗ ,

λ, µ ∈ C.

(3.14) (3.15)


593

4. The Space V g In this section we discuss the basic ideas about the realization of the space V g in the general case for any simple Lie algebra g. We identify the space V g with the space of a regular function on an open subset (manifold) of the flag manifold B− \G, where G is the simply-connected Lie group corresponding to the Lie algebra g. As a vector space, the Lie algebra g has a triangular decomposition g = n+ ⊕ h ⊕ n− ,

(4.1)

where h is the Cartan subalgebra and n± are the upper and lower nilpotent subalgebras. Let b± = h ⊕ n±

(4.2)

be the upper and lower Borel subalgebras. Let N± (respectively, B± be the upper and lower unipotent subgroups (respectively, Borel subgroups) of G corresponding to n± (respectively, b± ). Let {αi } i = 1, . . . , l be the root basis of g and {hi }i=1,..,l , the coroot basis of h. We choose a root basis of n± , {eα }α∈± , where + (− ) is the set of positive (negative) roots of g, so that [h, eα ] = α(h)eα for all h ∈ h. Consider the flag manifold B− \G. It has a unique open N+ -orbit, the so-called big cell U = [1] · N+ ⊂ B− \G, isomorphic to N+ . Since N+ is a unipotent Lie group, the exponential map n+ → N+ is an isomorphism and we have U C3 . From the action of N+ on U we can introduce a system {yα }α∈+ of homogeneous coordinates on U . Homogeneous means that h · yα = −α(h)yα

(4.3)

for all h ∈ h. The action of G on B− \G gives us a map from g to the Lie algebra of vector fields on B− \G, and hence on its open subset U N+ . Thus we obtain a Lie algebra homomorphism g → Vect N+ . With respect to this action the space Fun N+ of regular functions on U has the structure of the contragradient Verma module M0∗ with lowest weight 0 (for more details se [F1]). We have a natural pairing U (n+ ) × Fun N+ → C, which maps (P , A) to the value of the function P · A at the identity element of N+ , for any A ∈ Fun N+ and P ∈ U (n+ ). The vector 1 ∈ Fun N+ is annihilated by n+ and has weight 0 with respect to h. Hence there is a non-zero homomorphism Fun N+ → M0∗ sending 1 ∈ Fun N+ to a non-zero vector v0∗ ∈ M0∗ of weight 0. Since both Fun N+ and M0∗ are isomorphic to U (n+ )∨ as n+ -modules, this homomorphism is an isomorphism. It is known ([F1]) that we can identify the Module Mχ∗ with an arbitrary weight χ with Fun N+ , where the latter is equipped with a modified action of g. Recall that we have a canonical lifting of g to D≤1 (N+ ) of differential operators on U of order one, in a way that a → ξa . The modified action is obtained by adding to each ξa a function φa ∈ Fun N+ . The modified differential operators ξa + φa satisfy the commutation relations of g if and only if the linear map g → Fun N+ given by a → φa is a one-cocycle of g with coefficients in Fun N+ . If we impose the extra condition that the modified action of h on Fun N+ remains diagonalizable, we get that our cocycle should be h-invariant: φ[h,a] = ξh · φa , for all h ∈ h, a ∈ g. The space of the h-invariant one-cocycle of g with coefficients in Fun N+ is canonically isomorphic to the first cohomology of g with coefficients in Fun N+ (see [F1]). By Shapiro’s lemma we have H 1 (g, Fun N+ .) = H1 (g, M0∗ )

594


H1 (b− , C0 ) = (b− /[b− , b− ])∗ h∗ . Thus, for each χ ∈ h we have an embedding ρχ : g → D≤1 (N+ ) and the structure of the h∗ -graded g-module on Fun N+ . More detailed analysis shows that the action of n+ on Fun N+ is not modified and the action of h ∈ h is modified by h → h + χ (h). This comes from the reason that the weight of any monomial in Fun N+ is equal to the sum of negative roots (4.6) and the h-invariance of the one-cocycle ξh · φeα = φ[h,eα ] = α(h)φeα , α ∈ + . Therefore φeα = 0 for all α ∈ + . The vector 1 ∈ Fun N+ is still annihilated by n+ , but now it has weight χ with respect to h. Hence there is a non-zero homomorphism Fun N+ → Mχ∗ sending 1 ∈ Fun N+ to a non-zero vector vχ∗ ∈ Mχ∗ of weight χ . Since both Fun N+ and Mχ∗ are isomorphic to U (n+ )∨ as n+ -modules, this homomorphism is an isomorphism. For reasons explained below we will consider the modified action of g on Fun N+ with highest weight 1 χ = −ρ, where ρ = α. (4.4) 2 α∈+

Proposition 4.1. With respect to the modified action of the Lie algebra g the space Fun N+ has the structure of the Verma module M−ρ with highest weight −ρ. The space Fun N+ is closed under the multiplication and n+ acts on this space by derivations. We will construct the bigger space with the same action of g that has all the properties of Definition 3.1. We have the natural action of the Weyl group W of g on the flag manifold B− \G. From the definition W N (T l )/T l , where T l is maximum torus in G and N (T l ) is its normalizer. We have a particular element w0 ∈ W called the longest element (see [[W]]), satisfying the following three conditions: (i) w0 (+ ) = − , (ii) w0 · ρ = −ρ, (iii) l(w0 ) = |+ |, 2 where ρ = 21 α∈+ α. This element is uniquely defined, and satisfies w0 = e. For example, for type Al and any simple positive root αi , w0 αi = −αl+1−i , 1 ≤ i ≤ l. It means that w0 N+ w0−1 = N− . Consider the big cell U ∗ ∈ B− \G dual to U = [1] · N+ ⊂ B− \G. We have U ∗ = [1]w0 · N− = [1]w0 w0 N+ w0 = U w0 . Fun U∗

U ∗.

(4.5)

U∗

Let be the ring of regular function on We have N− . From the action of N− on U ∗ we can introduce a system {xα }α∈+ of homogeneous coordinates on U ∗ . Homogeneous means that h · xα = α(h)xα

(4.6)

for all h ∈ h. On the intersection U ∩ U ∗ we can consider the change of variables on U ∩ U ∗ from {yα } to {xα }. There is an ideal in Fun U ∗ that has a structure of the Verma module Mρ∨ with the lowest weight ρ with respect to the modified action of the Lie algebra g. Proposition 4.2. There is an element f ∈ Fun U that has the following properties: (i) f has weight −2ρ with respect to the action of h, (ii) φ = (f )−1 is an element of Fun U∗ of weight ρ,


595

(iii) φ is annihilated by n− , (iv) the ideal Vρ = φ · FunU∗ ∈ FunU∗ has the structure of the Verma module Mρ with the lowest weight ρ with respect to the modified action of the Lie algebra g. We have a natural pairing < ·, · > between the Verma module M−ρ Fun U with highest weight −ρ and the Verma module Mρ Vρ ∈ Fun U∗ with lowest weight ρ. Consider the space V0 = Fun U ⊕ φ · Fun U∗ . The element φ ∈ V0 defines an invariant trace on V0 in a way: Res f = < f, 1 >,

f ∈ V0 .

(4.7)

This space has almost all properties of the space from Definition 2.1, with the exception that it is not closed under the multiplication. Now we will construct the bigger space V g , that contains V0 as the subspace and closed under the multiplication. Consider a complex submanifold M ∈ B− \G that is the intersection of U and U ∗ , M = U ∩ U ∗.

(4.8)

As a set M is isomorphic to the complex space Cn \ Dφ n = |+ |, where Dφ is the divisor defined by φ = φ1 (yα1 , . . . , yαn ) · . . . · φl (yα1 , . . . , yαn ) = 0 and each divisor Dφi : φi (yα1 , . . . , yαn ) = 0, i = 1, . . . , l is a simple divisor. Denote by V g the space Fun M of regular functions on M. Theorem 4.1. The space V g has all the properties listed in Definition 2.1. The proof of this theorem, Proposition 4.2, and the explicit construction for the manifold M = U ∩ U ∗ for the Lie algebra g of one of the types An , Bn , Cn , Dn will be given in future publications of the author. The case g = A2 is discussed in the next section. 5. Explicit Realization of the Space V sl3 In this section we give an explicit construction of the space V g for g = sl3 as the space of a regular function on an open subset (manifold) of the flag manifold B− \G, where G = SL(2, C) is the simply- connected Lie group corresponding to g = sl3 . For this section we assume that g = sl3 . Fix a triangular decomposition g = n+ ⊕ h ⊕ n− ,

(5.1)

where h is a Cartan subalgebra and n± are the upper and lower nilpotent subalgebras. Let b± = h ⊕ n±

(5.2)

be the upper and lower Borel subalgebras. Let N± (respectively, B± be the upper and lower unipotent subgroups (respectively, Borel subgroups) of G corresponding to n± (respectively, b± ). Let hi = αi∨ , i = 1, 2 be the i th coroot of g. The set {hi }i=1,2 is a basis of h. We choose the root basis of n± , {eα }α∈± , where + (− ) is the set of positive ( negative) roots of sl3 , so that [h, eα ] = α(h)eα for all h ∈ h.

596


Consider the flag manifold B− \G. It has a unique open N+ -orbit, the so-called big cell U = N+ · [1] ⊂ B− \G, isomorphic to N+ , it means that U C3 . From the action of N+ on U we can introduce a system {yα }α∈+ of homogeneous coordinates on U in the following way. Let   1ac n = 0 1 b 001 be an element of N+ and a, b, c ∈ C, then (a, b, c) are homogeneous coordinates on U . We have a Lie algebra homomorphism sl3 → Vect N+ . With respect to this action the space Fun N+ of regular functions on U has the structure of the contragradient Verma module M0∗ with lowest weight 0. We have a natural pairing U (n+ ) × Fun N+ → C, which maps (P , A) to the value of the function at the identity element of N+ , for any A ∈ Fun N+ and P ∈ U (n+ ). This pairing defines an isomorphism between the Fun N+ and M0∗ . Consider the modified action of sl3 on Fun N+ described in the previous section with the highest weight χ = −ρ = −(α1 + α2 ) = −α3 ,

(5.3)

where {α1 , α2 , α3 = α1 +α2 } are positive roots for g = sl3 . The corresponding basis in sl3 is {e1 , e2 , e3 , h1 , h2 , f1 , f2 , f3 }. The explicit formulas for the modified sl3 -action in terms of homogeneous coordinates a, b, c on U follow: ξα1 = ξ−ρ (e1 ) = ξα2 = ξ−ρ (e2 ) =

∂ ∂ +a = ∂2 , ∂b ∂c

ξα3 = ξ−ρ (e3 ) = ξh1 = ξ−ρ (h1 ) = −2a ξh2 = ξ−ρ (h2 ) = a ξ−α1 = ξ−ρ (f1 ) = −a 2

∂ = ∂3 , ∂c

∂ ∂ ∂ +b − c − 1, ∂a ∂b ∂c

∂ ∂ ∂ − 2b − c − 1, ∂a ∂b ∂c

∂ ∂ ∂ − (c − ab) − ac − a, ∂a ∂b ∂c

ξ−α2 = ξ−ρ (f2 ) = c ξ−α3 = ξ−ρ (f3 ) = −ac

∂ = ∂1 , ∂a

∂ ∂ − b2 − b, ∂a ∂b

∂ ∂ ∂ − b(c − ab) − c2 − (2c − ab). ∂a ∂b ∂c

(5.4)


597

Proposition 5.1. With respect to the action of the Lie algebra sl3 given by (5.4), the space Fun N+ has the structure of the Verma module M−ρ with highest weight −ρ. The space Fun N+ is closed under the multiplication and n+ acts on this space by derivations. We will construct the bigger space with the same action of sl3 that has all the properties of Definition 3.1. We have the natural action of the Weyl group W of sl3 on the flag manifold B− \G. From the definition W N (T l )/T l , where T l is a maximum torus in G = SL(3, C) and N(T l ) is its normalizer. In the case of sl3 the Weyl group W S3 , where S3 is a group of permutations. The longest element w0 ∈ W corresponds to the permutation (1, 3) via this identification. Consider the big cell U ∗ ∈ B− \G dual to U = [1] · N+ ⊂ B− \G. We have U ∗ = [1]w0 · N− = U w0 .

(5.5)

Let Fun U∗ be the ring of regular function on U ∗ . We have U ∗ N− . From the action of N− on U ∗ we can introduce a system {yα }α∈− of homogeneous coordinates on U ∗ . Let   100 n = s 1 0 t r 1 be an element of N− and s, r, t ∈ C. Then s, r, t are homogeneous coordinates on U ∗ . With respect to the action of the Cartan subalgebra (5.4) any monomial of the form s n r m t k has weight equal to the −ρ + nα1 + mα2 + kα3 . The intersection U ∩U ∗ is an open subset in B− \G. On U ∩U ∗ the relations between the homogeneous coordinates a, b, c and s, r, t are given by: r s 1 b a 1 a= , b=− , c = ; or s = − , r= , t= . t t − rs t c − ab c c

(5.6)

We will show that there is an ideal in Fun U ∗ that has the structure of the Verma module Mρ∨ with the lowest weight ρ with respect to the action of the Lie algebra g given by (5.4). Proposition 5.2. There is an element f (a, b, c) ∈ Fun U that has the following properties: (i) f has weight −2ρ with respect to h, (ii) φ = (f )−1 is an element of Fun U∗ of weight ρ, (iii) φ is annihilated by n− , (iv) the ideal Vρ = φ · FunU∗ ∈ FunU∗ has the structure of the Verma module Mρ with lowest weight ρ with respect to the action of the Lie algebra g given by (5.4), da∧db∧dc (v) the differential form da∧db∧dc c(c−ab) is invariant under the transformation (5.6): c(c−ab) −→

dr∧ds∧dt t (t−sr)

.

Proof. Consider a function f (a, b, c) = c(c − ab) = (t (t − rs))−1 . Straightforward calculations show that this element has all the properties listed above. We have a natural pairing < ·, · > between the Verma module M−ρ Fun U with highest weight −ρ and the Verma module Mρ Vρ ∈ Fun U∗ with lowest weight ρ.

598


Consider the space V0 = Fun U ⊕ φ · Fun U∗ . This space has almost all properties of the space from Definition 2.1, with the exception that it is not closed under multiplication. Now we will construct the bigger space V g , that is the algebraic closer (with respect to multiplication) of the space V0 and we will show that V g Fun U ∩ U ∗ . This explains the choice of the highest weight χ = −ρ for the modified action of g on Fun N+ . Consider a complex submanifold M ∈ B− \G that is the intersection of U and U ∗ , M = U ∩ U ∗.

(5.7)

Denote by V g the space Fun M of regular functions on M. In order to obtain a nice description of M and of the space V g we identify a point (a flag) in the flag manifold B− \G with two orthogonal projective vectors: {(z1 : z2 : z3 ), (w1 : w2 : w3 ) |

3

zi wi = 0}.

(5.8)

i=1

We have [1] ∈ B− \G = {(0, 0, 1); (1, 0, 0)} and U = {(z1 : z2 : z3 ), (w1 : w2 : w3 )| z3 = 0, w1 = 0}.

(5.9)

In terms of homogeneous coordinates (a, b, c) we have: U = {(−c + ab, −b, 1), (1, a, c), a, b, c ∈ C}.

(5.10)

The natural action of the Weyl group W is just the corresponding permutation of coordinates. If s ∈ W S3 , then Us = {(z1 : z2 : z3 ), (w1 : w2 : w3 )| zs(3) = 0, ws(1) = 0},

(5.11)

where Us is the big cell in B− \G, that is the image of U under the action of the element s ∈ W . In particular, for the dual (opposite) cell we have U ∗ = Uw0 . Thus, M = {(z1 : z2 : z3 ), (w1 : w2 : w3 ) |

3

zi wi = 0,

i=1

and zi = 0, wi = 0, i = 1, 3} is the intersection of C∗ × C × C∗ × C with the hypersurface CP 2 × CP 2 . If we fix homogeneous coordinates (a, b, c), a =

w2 w3 z2 , c= , b=− w1 w1 z3

(5.12) 3

i=1 zi wi

= 0 in

(5.13)

on U C3 , then M is the total space of a non trivial bundle with the base (C)2a,b - the two-dimensional complex space and the fiber C∗c \ {c = ab} : M

C∗c \{c=ab}

−→

(C)2a,b .

(5.14)


599

The action (5.4) of the n+ on Fun M in terms of projective coordinates (z1 : z2 : z3 ) and (w1 : w2 : w3 ) is: ∂ ∂ ∂ ∂ + w1 , ξα2 = −z3 + w2 , ξα1 = −z2 ∂z1 ∂w2 ∂z2 ∂w3 ∂ ∂ ξα3 = −z3 + w1 , (5.15) ∂z1 ∂w3 and the element φ from Proposition 5.2 is z3 w1 · . (5.16) φ= z1 w 3 This element defines the n+ -invariant trace on V g Fun M in the following sense. The space V g is the space of all regular functions on M or all functions on C3 with poles only at c = 0 and c = ab. We can choose a basis {enml ; fnml } in V g as: enml = a n cm (c − ab)l , n ≥ 0, m, l ∈ Z; fnml = bn cm (c − ab)l , n > 0, m, l ∈ Z.

(5.17)

Proposition 5.3. Every regular function on M is a unique linear combination of the monomials of the basis (5.17). The proof follows immediately from the description of M in terms of projective coordinates (5.12). The element z3 w1 φ= · (5.18) = e0,−1,−1 z1 w 3 is from this basis and for any regular function f on M we can define a trace: Resf = coefficient at φ in the decomposition with respect to this basis. (5.19) This trace is invariant with respect to the action of n+ in the sense that Res (ξαi (f )) = 0, for i = 1, 2, 3.

(5.20)

This invariant trace defines a bilinear invariant form on V g Fun M as: < f, g = Res f · g

for

f, g ∈ Fun M.

(5.21)

Remark 5.1. This invariant form, being restricted to the subspace V0 = F un U ⊕ φ · F un U ∗ ⊂ V g is the natural pairing between the Verma module M−ρ Fun U with highest weight −ρ and the Verma module Mρ Vρ ∈ Fun U∗ with lowest weight ρ. Consider the subspace V+ in Fun M that consists of all regular functions on M+ = (C)3a,b,c \D{c=0} . As a linear space M+ is spanned by the elements {enml , n, l ≥ 0, m ∈ Z; fnml , n > 0, l ≥ 0, m ∈ Z} of the basis (5.17). For simplicity, we will denote the basis in V+ by {eγ }, γ ∈ , where is the set of all indexes of {enml , }, {fnml }. We assume that e0 = 1. We have < eγ , φ >= 0 for

γ = 0

(5.22)

and < e0 , φ >= 1. The above can be summarized in the following proposition.

(5.23)

600


Proposition 5.4. (i) The space V+ contains the unit function 1 and it is closed under the multiplication. (ii) V+ is invariant under the action (5.15) of n+ . (iii) The bilinear form (5.21) is non-degenerated on V g . (iiii) V+ is the maximal isotropic subspace in V g with respect to the bilinear form (5.21). Let V− ∈ Fun M be the subspace dual to V+ in V g with respect to the bilinear form (5.21) and let {eγ∗ }, γ ∈ be the basis in V− dual to {eγ }, γ ∈ . We have e0∗ = φ. The space V sl3 of all regular functions on M is the direct sum of two mutually dual subspaces V sl3 = V+ ⊕ V− . The space V sl3 satisfies all properties (V1)–(V2) of the Definition 2.1. So we can introduce the generalized formal delta function associated with the space V g as an element of the space V g ⊗ (V g )∗ = V+ ⊗ V− ⊕ V− ⊗ V+

(5.24)

in a way described in Sect. 2: δV g =

eγ ⊗ eγ∗ ⊕ eγ∗ ⊗ eγ .

(5.25)

γ ∈

The space V g is realized as the space of the function Fun M or, more generally, as the space of formal distributions on M and z = (z1 , z2 , z3 ) are global coordinates on M. Here z1 = a, z2 = b, z3 = c, where (a, b, c) are homogeneous coordinates on M. The space V g ⊗ (V g )∗ can be identified with the space of distributions in two sets of coordinates z = (z1 , z2 , z3 ) and w = (w1 , w2 , w3 ). In this case we will use the notation δV g (z − w), or for short, δV (z − w). We also can define δV (z − w)± as eγ∗ ⊗ eγ ∈ V−z ⊗ V+w δV (z − w)+ =

(5.26)

(5.27)

eγ ∈V+

and δV (z − w)− =

eγ ∈V+

eγ ⊗ eγ∗ ∈ V+z ⊗ V−w .

(5.28)

Thus defined, the generalized formal delta function has most of the properties of the standard delta function with respect to the trace (2.14) and differentiations from n+ : (a) For any formal distribution f (z) ∈ V g one has: Resz f (z) δ(z − w) = f (w),

(5.29)

(zi − wi ) δ(z − w) = 0;

(5.30)

(b) for any element a ∈ n+ one has: (ξaz )j δ(z − w) = (−ξaw )j δ(z − w), where ξaz is the image of a in Vect M.

(5.31)


601

We would like to remark that for the Lie algebra sl2 the complex manifold M U U ∗ C∗ is contactable to the one dimensional real manifold S 1 . The trace (2.1) is the integral over the S 1 :| z |= 1:

a(z) dz (5.32) Resz a(z) = |z|=1

and the formal Cauchy formula can be written in the form Resz a(z)∂z(k) Resz a(z)∂z(k)

1 = (−1)k ∂ (k) a(w)+ , z−w

1 = (−1)k+1 ∂ (k) a(w)− , z−w

| z |>| w |,

for

for

| z | 0 if g ∈ Gn , g = 0 and n = 0. Proposition 6.3. The Generalized Affine Kac-Moody algebras gV associated with the space V sl3 are defined by the same conditions with one small change: we replace Condition 3 by 3*. G is graded as G = ⊕n∈Zk Gn with Gn not necessarily finite dimensional and with ω acting as −1 on the “Cartan subalgebra” G0 . Here k = 2 is the rank of sl3 . In order to prove this statement we need to construct an invariant symmetric bilinear form ( , ) and a (Cartan) involution ω on gV . The normalized invariant form ( , ) on gV can be described as follows. Take the normalized invariant form ( , ) on g and extend ( , ) to the whole gV by (A ⊗ f (t), B ⊗ g(t)) = (A, B)(Rest φ · f · g); (Ki , Kj ) = 0; (Lsl3 g, CK ⊕ CK ⊕ CK ) = 0. 1

2

3

(6.18)

To define a Cartan involution of gV consider the Cartan involution ω˜ of g and the transformation w0 on M given by a→−

a 1 b , b→ , c→ , c − ab c c

(6.19)

where t = (a, b, c) are homogeneous coordinates on U. This transformation is defined by the change from the homogeneous coordinates on U to the homogeneous coordinates on U ∗ , given by (5.6). The Cartan involution of gV can be expressed as: ˜ ⊗ f (w0 (t)) − λ1 K1 − λ2 K2 . ω(A ⊗ f (t) + λ1 K1 + λ2 K2 ) = ω(A)

(6.20)

It is not difficult to see that for any element g = A⊗f (t) ∈ Lsl3 g we have (g, ω(g)) > 0. In QFT it is important to consider the correlation functions of two or more fields. In 2-dimensional conformal field theory the correlator of two fields < v ∗ |A(z)B(w)|v >

(6.21)

is a rational function of z and w in the domain |z| > |w| with poles only at hyperplanes z = 0, w = 0, and z = w. This important property in quantum field theory in terms of OPE (operator product expansion) means that the product of two fields at nearby points can be expanded in terms of other fields and the small parameter z − w. In terms of the commutator, the OPE can be expressed as A(z)B(w) = [A− (z), B(w)]+ : A(z)B(w) :,

(6.22)

where [A− (z), B(w)] is a singular part of the OPE and : A(z)B(w) := A(z)+ B(w) + B(w)A(z)−

(6.23)

606


is the normal ordered product of two fields. The singular part of the OPE of two local fields A(z) and B(w), given by N

[A− (z), B(w)] =

A(n) B(w) ∂w(n) δ(z − w)+ ,

(6.24)

n=0 1 n is a finite linear combination of other fields A(n) B(w) from the theory. Here ∂ (n) = n! ∂ . The δ(z − w)+ is a regular function with respect to w in the domain |z| > |w| and has the properties: ∂w δ(z − w)+ = −∂z δ(z − w)+ and δ(z − w)+ |w=0 = 1/z. From these properties we have

δ(z − w)+ =

1 in the domain |z| > |w|. z−w

(6.25)

For the regular delta function we don’t need these properties to find the sum δ(z − w)+ =

∞

z−n−1 w n =

n=0

1 , z−w

(6.26)

but for the generalized delta function this can give us some idea. In order to have a consistent definition of the correlator of two fields in dimension higher than 2, we need to introduce the OPE and the normal product of two fields on M. Consider two fields a sl3 (z), bsl3 (w) ∈ gV . Define a sl3 (z)± as a sl3 (z)± = a ⊗ δV (z − w)± ,

(6.27)

where δV (z − w)± are defined by (5.27). Then the OPE can de defined as:

2 sl sl w 3 3 a (z) b (w) = (a, b) K · L δ (z − w) i

i

V

+

i=1

+[a, b]sl3 (w) δV (z − w)+ + : a sl3 (z) bsl3 (w) :,

(6.28)

where : a sl3 (z) bsl3 (w) := a sl3 (z)+ bsl3 (w) + bsl3 (w)a sl3 (z)− .

(6.29)

It would be nice to have a formula similar to (6.25) for the δV (z − w)+ , defined as (5.27). The space V sl3 is a sum of two dual subspaces V g = V+ ⊕ V− . Here g = sl3 . The positive part of the delta function is defined as eγ∗ ⊗ eγ ∈ V−z ⊗ V+w . (6.30) δV (z − w)+ = eγ ∈V+

The space V+ of all regular functions on M+ is the direct sum of two subspaces V+ = V++ ⊕V+− , where V++ is a space of all regular functions on U C3 . For the dual space V− we have the dual decomposition V− = V−+ ⊕V−− . According to this decomposition we can write δV (z − w)+ = δV (z − w)++ + δV (z − w)+− .

(6.31)


607

The distribution z w ⊗ V++ δV (z − w)++ ∈ V−−

(6.32)

is a regular function with respect to w = (w1 , w2 , w3 ) in the domain U C3 . Thus, this function is well defined, when wi = 0. From the construction of the basis (5.17), we have δV (z − w)++ |w1 =0, w2 =0, w3 =0 = φ(z1 , z2 , z3 ) =

1 . z3 (z3 − z1 z2 )

(6.33)

Additionally, we have ∂iw δV (z − w)++ = −∂iz δV (z − w)++ , i = 1, 2, 3.

(6.34)

This equality we understand at the level of distributions. Consider the function of two sets of variables z = (z1 , z2 , z3 ) and w = (w1 , w2 , w3 ) given by z1 z2 − w1 w2 (z1 − w1 )(z2 − w2 )(z3 − w3 )(z3 − z1 z2 − (w3 − w1 w2 )) z1 = (z1 − w1 )(z2 − w2 )(z3 − z1 z2 − (w3 − w1 w2 )) w2 + . (6.35) (z2 − w2 )(z3 − w3 )(z3 − z1 z2 − (w3 − w1 w2 ))

F (z, w) =

Proposition 6.4. In the domain D1 : |z1 | > |w1 | > 0, |z2 | > |w2 | > 0, |z3 ||w3 |, |z3 − z1 z2 ||w3 − w1 w2 | the function F (z, w) = δV (z − w)++ in the sense that Resz F (z, w) · f (z) = f (z)++ ,

(6.36)

and in the domain D2 : |z1 | > |w1 | > 0, |z2 | > |w2 | > 0, |z3 | < |w3 |, |z3 − z1 z2 ||w3 − w1 w2 | the function −F (z, w) = δV (z − w)+− in the sense that Resz F (z, w) · f (z) = −f (z)+− .

(6.37)

The proof is based on a realization of the trace as an integral over the 3-dimensional cycle S and the corresponding formal power series expansion for F (z, w) on S. Then, the OPE expansion of two fields a(z), b(w) ∈ gV can be written in terms of the function F (z, w) if we replace in the OPE (6.28 ) the δV (z − w)+ by the sum F (z, w) − F (z, w) |z1 | > |w1 | > 0, |z2 | > |w2 | > 0, |z1 | > |w1 | > 0, |z2 | > |w2 | > 0, |z3 | > |w3 |, |z3 − z1 z2 | > |w3 − w1 w2 |; |z3 | < |w3 |, |z3 − z1 z2 ||w3 − w1 w2 |. This form of OPE is very similar to what we have in 2-dimensional conformal field theory.

608


7. Generalized Virasoro Algebra Associated with the Space V sl3 In Sect. 3 we constructed the n-products of the Virasoro conformal algebra from the commutation relations of sl2 (3.13). Now we will apply the same ideas to construct a generalized Virasoro conformal algebra associated to the Lie algebra sl3 . For g = sl3 positive roots are {α1 , α2 , α3 = α1 + α2 } and the corresponding weight basis in sl3 is {e1 , e2 , e3 , h1 , h2 , f1 , f2 , f3 }. Denote ∂i = ad (ei ). The Lie algebra sl3 has the structure of the U (n+ )-module with the lowest weight −α3 . Via the PoincaréBirkhoff-Witt basis we can identify U (n+ ) C[∂1 , ∂2 , ∂3 ]. Denote by L = f3 the lowest vector with respect to this action, then sl3 C[∂1 , ∂2 , ∂3 ] L is the rank one C[∂1 , ∂2 , ∂3 ]-module. Consider a basis in sl3 of the form L = f3 , ∂1 L = −f2 ,

∂2 L = f1 ;

∂12 ∂2 L = −2e1 ,

∂1 ∂2 L = h1 . ∂3 L = h1 + h2 ;

∂2 ∂3 L = −e2 ,

∂1 ∂2 ∂3 L = −e3 .

(7.1)

The dual basis (with respect to the normalized bilinear form ) is 1 1 1 2 1 h1 − h2 , h1 + h2 ; − f1 , −f2 , −f3 . (7.2) 3 3 3 3 2 For any element e of the dual basis we define a bilinear operation L(e) L product as: e3 , −e2 ,

e1 ;

L(f3∗ ) L = [e3 , L] = ∂3 L,

L((∂1 f3 )∗ ) L = [−e2 , L] = −∂2 L,

L((∂2 f3 )∗ ) L = [e1 , L] = ∂1 L,

L((∂3 f3 )∗ ) L = [−2h1 − h2 , L] = 3L.

(7.3)

The other products are 0 because [fi , f3 ] = 0 for i = 1, 2, 3. Let δV = δ sl3 (t − z) V

(7.4)

be a delta function associated with the space V sl3 , defined in (5.25). Denote by Lg (z) the field associated with the space V sl3 Fun M with the commutation relation defined by products (7.3) : [Lg (z), Lg (w)] = ∂ Lg (w) δ (z − w) − ∂ Lg (w) ∂ w δ (z − w) 3

2

V

1

V

+∂1 Lg (w) ∂2w δV (z − w) + 3Lg (w) ∂3w δV (z − w).

(7.5)

Theorem 7.1. The defined above commutation relation satisfies to the skewsymmetry and Jacobi identity axioms. Proof. We have [Lg (w), Lg (z)] = ∂3 Lg (z) δV (z − w) − ∂2 Lg (z) ∂1z δV (z − w) +∂ Lg (z) ∂ z δ (z − w) + 3Lg (z) ∂ z δ (z − w) 1

2 V

3 V

= ∂3 Lg (w) δV (z − w) + ∂2 Lg (w) ∂1w δV (z − w) +∂ ∂ Lg (w) δ (z − w) − ∂ Lg (w) ∂ w δ (z − w) 1 2

V

1

2

V

−∂2 ∂1 Lg (w) δV (z − w) − 3Lg (w) ∂3w δV (z − w) −3 ∂3 L(w) δV (z − w) = −[Lg (z), Lg (w)].

(7.6)


609

Define a field on V sl3 by Lg (z) = −δV (t − z) ∂3t + ∂1t δV (t − z)∂2t − ∂2t δV (t − z)∂1t .

(7.7)

Straightforward calculations show that this field satisfies the commutation relation (7.5). g For any function f ∈ V sl3 defines an operator Lf as Lf = Resz (f (z)Lg (z)) so that g Lf = −f ∂3t + ∂1 f ∂2t − ∂2 f ∂1t . (7.8) From direct calculations we have g g g [Lf , Lg ] = L{f, g} ,

(7.9)

where {f, g} defines a Lie bracket on the space V sl3 given by {f, g} = g ∂3 f − f ∂3 g + ∂1 f ∂2 g − ∂1 g ∂2 f.

(7.10)

This bracket is skewsymmetric and satisfies the Jacobi identity {h, {f, g}} + {f, {g, h}} + {g, {h, f }} = 0

(7.11)

for any h, f, g ∈ V g , so it is a Lie bracket. This completes the proof.

Proposition 7.1. The Lie bracket defined in (7.10) corresponds to the contact Lie bracket on the real space MR = V gR ( we now suppose that (a, b, c) are real coordinates). The g operators Lf are contact vector fields on MR with the contact Hamiltonian f. The corresponding contact structure is defined by a 1-form β, such that β(∂3 ) = −1, β(∂1 ) = 0, β(∂2 ) = 0, dβ(∂3 , ·) = 0 and β ∧ d β = 0. Proof. Let β be a contact 1-form on a 3-dimensional real manifold M : β ∧ d β = 0. The vector field ξf on M is contact with the contact hamiltonian f ∈ Fun M, if Lξf preserves the foliation β = 0 and β(ξf ) = f . Then ξf = f · α + θ(f ) where α is the vector field defined by β(α) = 1 and dβ(α, ·) = 0 and θ is the bivector field on M, inverse to the 2-form dβ. The contact Lie bracket on Fun M is defined by [ξf , ξg ] = ξ{f, g} or more explicitly: {f, g} = f α(g) − gα(f ) + θ (f, g).

(7.12)

Take α = −∂3 , θ = ∂1 ∧ ∂2 and β = −dc + adb. This completes the proof.

Proposition 7.2. The Lie algebra V sl3 with the bracket (7.10) contains the Lie subalgebra isomorphic to sl3 . Proof. For simplicity we will use the homogeneous coordinates (a, b, c) on M g . The action of sl3 in terms of these coordinates is given by (5.4). Operators L1 , La ,

Lb ,

Lc ,

Lc−ab ,

Lac ,

L−b(c−ab) ,

Lc(c−ab)

(7.13)

are closed with respect to the Lie bracket (7.9) and form the Lie algebra isomorphic to the sl3 . The Cartan subalgebra generators Lc = −c ∂3 −a ∂1 , Lc−ab = −(c−ab) ∂3 −b ∂2 are grading operators from the previous section and they are analogues of the energy operator L0 in two dimensional CFT.

610


We constructed natural higher dimensional analogues of the Virasoro algebra and the Virasoro conformal algebra associated with the space V sl3 . We suggest to denote this algebra by V ir sl3 . We also define a semidirect sum of V ir sl3 and the generalized affine Kac-Moody algebra gV associated with the space V sl3 . The V ir sl3 acts on gV via its action as derivations of V sl3 . In terms of fields this action is given by [Lg (z), a sl3 (w)] = ∂3 a sl3 (w) δV (z − w) − ∂2 a sl3 (w) ∂1w δV (z − w) +∂ a sl3 (w) ∂ w δ (z − w) + 2a sl3 (w) ∂ w δ (z − w), (7.14) 1

where a sl3 (w) ∈ gV .

2

V

3

V

Acknowledgements. The author wants to thank A. Gerasimov for very useful discussions and M. Olshanesky for reading the manuscript and suggesting improvements.

References [B]

Borcherds, R.: Vertex algebras, Kac-Moody algebras and the monster. Proc. Nat. Acad. Sci. USA 83, 3068–3071 (1986) [B1] Borcherds, R.: What is Moonshine? In: Proceedings of the International Congress of of Mathematicians, Berlin 1998, Vol. I, Berlin: Documenta Mathematica Mathematicians, 1998, pp. 607–615 [B2] Borcherds, R.: Vertex algebras. In: Topological field theory, primitive formes and related topics, (Kyoto, 1996), Progr. Math., 160, Boston, MA: Birkhauser, 1998, pp. 35–77 [BPZ] Belavin,A., Polykov,A., Zamolodchikov,A.: Infinite conformal symmetries in two-dimensional quantum field theory. Nucl. Phys. B241, 333–380 (1984) [BD] Beilinson, A., Drinfeld, V.: Chiral Algebras. AMS Colloquium Publications 51, Providence, RI: Amer Math. Soc., 2004 [BDK] Bakalov, B., D’Andrea, A., Kac, V.G.: Theory of finite pseudoalgebras. Adv. Math. 162, 1–140 (2001) [BDK1] Bakalov, B., D’Andrea, A., Kac, V.G.: Irreducible modules over finit simple Lie pseudoalgebras I. Primitive Pseudoalgebra of type W and S. http://arXiv.org/list/math.QA/0410213 vl, 2004 [DK] D’Andrea, A., Kac, V.G.: Structure Theory of Finite Conformal Algebras. Selecta Math. (N.S.) 4, no. 3, 337–418 (1998) [BE] Ben-Zvi, D., Frenkel, E.: Geometric realization of the Segal-Sugavara construction. http://arxiv.org/list/math.AG/0301206 v2, 2003 [F1] Frenkel, E.: Wakimoto modules, opers and the center at the critical level. Adv. Math. 195, no. 2, 297–404 (2005) [K1] Kac, V.G.: Infinite dimensional Lie Algebras. Third edition, Cambridge: Cambridge Univ. Press, 1990 [K2] Kac, V.G.: Vertex algebras for beginners. University lecture series 10, Providence, RI: Amer. Math. Soc. 2nd edition, 1998 [W] Wakimoto, M.: Infinite-Dimensional Lie Algebras. Translations of Math. Monographs, 195, Providence, RI: Amer. Math. Soc., 2000 Communicated by L. Takhtajan

Commun. Math. Phys. 263, 611–657 (2006) Digital Object Identifier (DOI) 10.1007/s00220-005-1528-x

Communications in


Algebraic Curve for the SO(6) Sector of AdS/CFT N. Beisert1,2 , V.A. Kazakov3, , K. Sakai3 1

Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Am Mühlenberg 1, 14476 Potsdam, Germany 2 Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USA. E-mail: [email protected] 3 Laboratoire de Physique Théorique de l’Ecole Normale Supérieure et l’Université Paris-VI, 75231 Paris, France. E-mails: [email protected]; [email protected] Received: 31 January 2005 / Accepted: 2 October 2005 Published online: 4 March 2006 – © Springer-Verlag 2006

Abstract: We construct the general algebraic curve of degree four solving the classical sigma model on R × S 5 . Up to two loops it coincides with the algebraic curve for the dual sector of scalar operators in N = 4 SYM, also constructed here. 1. Introduction Since ’t Hooft’s discovery of the planar limit in field theories [1], the idea that the planar non-abelian gauge theory could be exactly solvable, or integrable, always fascinated string and field theorists. The analogy between planar graphs of the 4D YM theory and the dynamics of string world sheets of a fixed low genus (described by some unidentified 2D CFT) already pronounced in [1] lead to numerous attempts aimed at the precise formulation of the YM string. This circle of ideas lead to a dual, matrix model formulation [2, 3] of completely integrable toy models of the string theory and 2D quantum gravity, having also their dual description in the usual world sheet formalism [4–6]. Big planar graphs of the matrix models find their description in terms of the Liouville string theory proposed in [7]. However the problem of such dual description is still open in the original bosonic 4D YM theory. Fortunately, these ideas are beginning to work in the 4D world thanks to supersymmetry. The work of Maldacena [8], inspired by some earlier ideas and observations [9, 10], lead to a precise formulation of the string/gauge duality at least in the case of IIB superstrings on AdS5 × S 5 and the conformal N = 4 SYM theory. A lot of work has been done since then to find the AdS/CFT dictionary identifying the string analogs of physical operators and correlators in the N = 4 SYM theory, see [11, 12] for reviews. The two most recent important advances in this field are the BMN correspondence [13] (see [14–18] for reviews) and the semiclassical spinning strings duality [19] (see

Membre de l’Institut Universitaire de France

612

N. Beisert, V.A. Kazakov, K. Sakai

[20–22] for reviews). These promise to enable quantitative comparisons between both theories even though the duality is of a strong/weak type. Let us only comment briefly on classical spinning strings on AdS5 × S 5 which were investigated in [23–28]. These were argued to be dual to long SYM operators first investigated in [29, 30] where a remarkable agreement was found up to two-loops. Agreement for other particular examples [31– 34] as well as at the level of the Hamiltonian [35–42] was obtained until discrepancies surfaced at three-loops, first in the (near) BMN correspondence [43–46], later also for spinning strings [47]. This problem, which might be the order-of-limits issue explained in [48], is not resolved at the moment. We will observe further evidence of two-loop agreement/three-loop discrepancy in this work. Luckily, in the last few years the first signs of integrability were observed on both sides of the duality. The first, striking observation of integrability in N = 4 SYM was made by Minahan and Zarembo [49]. They investigated the sector of single-trace local operators of N = 4 supersymmetric gauge theory composed from scalars Tr m1 m2 . . . mL .

(1.1)

It was found that the planar one-loop dilatation operator, which measures their anomalous scaling dimensions, is the Hamiltonian of an integrable spin chain. This chain has so(6) symmetry and the spins transform in the vector representation. A basis of the spin chain Hilbert space is thus given by the states |m1 , m2 , . . . , mL

(1.2)

which correspond, up to cyclic permutations, to single-trace local operators. It was subsequently shown that integrability not only extends to all local single-trace operators of N = 4 SYM [50, 51] (extending earlier findings of integrability in gauge theory, cf. [52] for a review), but more surprisingly also to higher loops (at least in some subsectors) [53, 54] (see also [55] for integrability in a related theory). The hypothesis of all-loop integrability together with input from the BMN conjecture [13] has allowed to make precise predictions of higher-loop scaling dimensions [53, 56, 47, 48], which have just recently been verified by explicit computations [57], see also [58, 59]. Integrability and the Bethe ansatz was also an essential tool in obtaining scaling dimensions for states dual to spinning strings. For reviews of gauge theory results and integrability, see [60, 61]. Integrability in string theory on AdS5 × S 5 , whose sigma model was explicitly formulated in [62, 63], is based on a so-called Lax pair, a family of flat connections of the two-dimensional world sheet theory. Its existence is a common feature of sigma models on coset spaces and it can be used to construct Pohlmeyer charges [64, 65]. These multilocal charges have been discussed in the context of classical bosonic string theory on AdS5 × S 5 in [66], while a family of flat connections for the corresponding superstring was identified in [67]. The Lax pair of the string sigma model was first put to use in [41] in the case of the restricted target space R × S 3 , where R represents the time coordinate of AdS5 . There the analytic properties of the monodromy of the flat connections around the closed string were investigated and translated to integral equations similar to the ones encountered in the algebraic Bethe ansatz of gauge theory. This lead to the first rigorous proof of two-loop agreement of scaling dimensions for an entire sector of states. The possibility to quantize the sigma model by discretizing the continuous equations, by analogy with the finite chain Bethe ansatz for gauge theory, was suggested in [41]. A concrete proposal for the quantization of these equations was given in [68]. √ It reproduces the near BMN results of [43–46] and, even more remarkably, a generic 4 λ behavior at large coupling in agreement predicted in [69]. Curiously, this proposal appears to have

Algebraic Curve for the SO(6) Sector of AdS/CFT

613

a spin chain correspondence in the weak-coupling extrapolation [70], which however does not agree with gauge theory. Integrability is usually closely related with the theory of algebraic curves. In most cases, integrable models of 2D field theory or integrable matrix models are completely characterized by their algebraic curves. Often the algebraic curve unambiguously defines also the quantum version of the model. We also know many examples of algebraic curves characterizing the massive 4D N = 2 SYM theories, starting from the famous SeibergWitten curve [71], as well as for the N = 1 SYM theories [72]. However, the curves describe in that cases only particular BPS sectors of the gauge theories characterized by massive moduli. The N = 4 SYM CFT gives us the first hope for an entirely integrable 4D gauge theory, including the non-BPS states. There is an increasing evidence that the full integrability on both sides of AdS/CFT duality might be governed by similar algebraic curves. On the SYM side, such a curve for the quasi-momentum can be built so far only for small ’t Hooft coupling. Its mere existence is due to the perturbative integrability, confirmed up to three-loops [54, 47] and hopefully existing for all-loops [48] and even non-perturbatively. On the string sigma model side, we already know the entire classical curve, also for the quasi-momentum, for the R × S 3 [41] and AdS3 × S 1 [42] sectors of the theory, dual to su(2) and sl(2) sectors of the gauge theory, respectively. In this paper we will make a new step in the direction of construction of the full algebraic curve of the classical AdS5 ×S 5 sigma model and of its perturbative counterpart on the SYM side following from the one-loop integrability of the full SYM theory found in [50, 51]. We will accomplish this program for the so(6) sector of SYM, which is closed at one-loop as well as in the thermodynamic limit [73], and its dual, the sigma model for the string on Rt × S 5 . We will show in both theories that the projection of the algebraic curve onto the complex plane is a Riemann surface with four sheets corresponding to the four-dimensional chiral spinor representation of SU(4) ∼ SO(6). In the SYM case such a curve solves the classical limit of the corresponding Bethe equations. We will identify and fix all the parameters of this curve on both sides of AdS/CFT. On the gauge side, the main tool of our analysis will be transfer matrices in the algebraic Bethe ansatz framework (see e.g. [74] for an introduction). These can be derived from a Lax-type formulation of the Bethe equations proposed in [75] for the su(m) algebras. On the string side, we will construct the so-called finite-gap solution [76, 77] of the R × S 5 sigma model, also based on the Lax method [78]. This paper is organized as follows. In Sect. 2 we review the vector so(6) spin chain which is dual to the one-loop planar dilatation operator of N = 4 SYM in the sector of local operators composed from scalar fields. Special attention is paid to various transfer matrices and their analytic properties. In the thermodynamic limit we will then construct the generic solution in terms of an algebraic curve and illustrate by means of two examples. All this is meant to serve as an introduction to the treatment of the string sigma model in the sections to follow. We start in Sect. 3 by investigating the properties of the monodromy of the Lax pair around the string. They are similar to the ones encountered for the spin chain, but there is an additional symmetry for the spectral parameter. These are used in Sect. 4 to reconstruct an algebraic curve associated to each solution of the equations of motion. We will show that the algebraic curve is uniquely defined by the analytic properties of the monodromy matrix. It thus turns out that solutions are completely characterized by their B-periods (mode numbers) and filling fractions (excitation numbers). In Sect. 5 we construct equations similar to the Bethe equations of the spin chain. These are equivalent to the algebraic curve, but allow for a particle/scattering

614


interpretation.1 We conclude in Sect. 6. In the appendix we present results which are not immediately important for the AdS/CFT correspondence. Let us mention that the electronic arxiv version of this article [80] contains several examples and further details concerning the spin chain operators. 2. The so(6) Spin Chain In this section we will review the integrable so(6) spin chain with spins transforming in the vector representation of the symmetry algebra. Due to the isomorphism of the algebras so(6) and su(4) we can rely on a vast collection of results on the integrable spin chain with unitary symmetry algebra. We will use these firm facts to gain a better understanding of the Bethe ansatz in the thermodynamic limit for higher-rank symmetry groups. We identify some key properties of the resolvents which describe the distribution of Bethe roots. In the following chapters we will derive similar properties for the sigma model which later on will be used to (re)construct a similar Bethe ansatz for classical string theory.

2.1. Spin Chains Operators. Dilatation Operator. Let us consider single-trace local operators of N = 4 SYM composed from L scalars without derivatives. These are isomorphic to states of a quantum so(6) spin chain with spins transforming in the vector (6) representation. The planar one-loop dilatation generator of N = 4 SYM closes on these local operators; it can thus be written in terms of a spin chain Hamiltonian H as follows: D = L + g2 H + · · · ,

g2 =

2 N gYM . 8π 2

(2.1)

The Hamiltonian was derived in [49], it is given by the nearest-neighbor interaction H2 H=

L

Hp,p+1 ,

H = 2P 15 + 3P 1 = I − S + 21 K6,6 .

(2.2)

p=1

The spin chain operators P 20 , P 15 , P 1 project to the modules 1, 15, 20 which appear in the tensor product of two spins, 6 × 6. These can be written using the operators I, S, K6,6 which are the identity, the permutation of two spins and the trace (K6,6 )ij kl = δ ij δkl of two so(6) vectors, respectively

P 20 = 21 I + 21 S − 16 K6,6 ,

P 15 = 21 I − 21 S,

P 1 = 16 K6,6 .

(2.3)

1 These equations were proposed independently by M. Staudacher and confirmed by comparing to explicit solutions of the string equations of motion [79]. 2 We shall distinguish between global and local spin chain operators by boldface and curly letters, respectively. For their eigenvalues we shall use regular letters.


615

R-matrices. Minahan and Zarembo have found out that the Hamiltonian (2.2) obtained from N = 4 SYM is integrable [49]: It coincides with the Hamiltonian of a standard so(m) spin chain investigated by Reshetikhin [81, 82]. In this article we focus on the case m = 6. Integrability for a standard quantum spin chain means that the (nearest-neighbor) Hamiltonian density H can be obtained via R(u) = S 1 + iuH + O(u2 )

(2.4)

from the expansion of an R-matrix R(u) (see e.g. [60] for an introduction in the context of gauge theory) which satisfies the Yang-Baxter relation R12 (u1 − u2 )R13 (u1 − u3 )R23 (u2 − u3 ) = R23 (u2 − u3 )R13 (u1 − u3 )R12 (u1 − u2 ).

(2.5)

In addition to the YBE, we would like to demand the inversion formula R12 (u1 − u2 )R21 (u2 − u1 ) = I.

(2.6)

The R-matrix for two vectors of so(6) is given by [81, 82]3 u − i 15 (u − i)(u − 2i) 1 P + P u+i (u + i)(u + 2i) i u iu = S+ I− K6,6 . u+i u+i (u + i)(u + 2i)

R6,6 (u) = P 20 +

(2.7)

It yields, via (2.4), the spin chain Hamiltonian (2.2). For completeness, we shall also state the R-matrices between a vector and a (anti)chiral spinor R4,6 (u) = P 20 + ¯

R4,6 (u) = P 20 +

u− u+ u− u+

3i 2 3i 2 3i 2 3i 2

¯

P4 = I − P4 = I −

i 2

u+

3i 2

i 2

u+

K4,6 , ¯

3i 2

K4,6 .

(2.8)

These also satisfy the Yang-Baxter equation (2.5) when we assign any of the three representations 6, 4, 4¯ to the three spaces labeled by 1, 2, 3. Here we can again express ¯ the projectors in terms of identity I and spin-trace K4,6 , K4,6 defined with Clifford ¯ ˙ j β˙ gamma matrices by (K4,6 )βj αi = (γ j γi )β α and (K4,6 )βj αi ˙ = (γ γi ) α˙ , P 20 = I − 16 K4,6 ,

¯

P 4 = 16 K4,6 ,

¯

P 20 = I − 16 K4,6 ,

¯

P 4 = 16 K4,6 .

(2.9)

3 This R-matrix coincides with the one given in [49, 81, 82] up to an overall factor and a redefinition of u. The redefinition is needed to comply with (2.6).

616


Transfer matrices. An R-matrix describes elastic scattering of two spins, it gives the phase shift for both spins at the same time. For a spin chain, it can also be viewed as a (quantum) SO(6) lattice link variable. If we chain up the link variables around the closed chain, we obtain a Wilson loop. The open Wilson loop is also known as the monodromy matrix a (u), where a labels the auxiliary space of the Wilson line. The complex number u of the Wilson loop is the spectral parameter. In the 6 representation it is convenient to use the combination 6a (u) =

(u + i)L 6,6 6,6 Ra,1 (u − i)R6,6 a,2 (u − i) · · · Ra,L (u − i). uL

(2.10)

The closed Wilson loop is also known as the transfer matrix T6 (u) = Tr a 6a (u).

(2.11)

We can also write the monodromy and transfer matrices in the spinor representations (u + 2i )L 4,6 Ra,1 (u − i) · · · R4,6 a,L (u − i), uL (u + 2i )L 4,6 ¯ ¯ ¯ 4a (u) = Ra,1 (u − i) · · · R4,6 a,L (u − i), uL

4a (u) =

T4 (u) = Tr a 4a (u), ¯

T4¯ (u) = Tr a 4a (u).

(2.12)

The transfer matrices commute due to the Yang-Baxter relation (2.5), as one can easily convince oneself by inserting the inversion relation (2.6) between both Wilson loops. Local Charges. The transfer matrices give rise to commuting charges when expanded in powers of u. Local charges Qr are obtained from T6 (u), (the representation of the Wilson loop coincides with the spin representation) when expanded around u = i, i.e. ∞

(u + i)L T6 (u + i) = U exp i ur−1 Qr . L (u + 2i)

(2.13)

r=2

The operator U is a global shift operator, it shifts all spins by one site. For gauge theory we are interested in the subspace of states with zero momentum, i.e. with eigenvalue U =1

(2.14)

of U. This is the physical state condition. The second charge Q2 is the Hamiltonian Q2 = H;

(2.15)

this fact can be derived from (2.4). Therefore all transfer matrices and charges are also conserved quantities. The third charge Q3 leads to pairing of states, a peculiar property of integrable spin chains [83, 53]. Global Charges. An interesting value of the spectral parameter is u = ∞ where one finds the generators of the symmetry algebra, in our case of so(6) = su(4). Let us first note the symmetry generators for different representations


J 6,6 = S − K6,6 ,

617

J 4,6 = 21 I − 21 K4,6 ,

¯

¯

J 4,6 = 21 I − 21 K4,6 .

(2.16)

The two vector spaces on which these operators act are interpreted as follows: The first ¯ specifies the parameters for the rotation which acts on the second space (6 in (6, 4, 4) all cases). To be more precise, consider the operator J αi βj , where α, β belong to 6, 4, 4¯ while i, j belong to the second 6. In all three cases, the indices α, β can be combined into an index of the adjoint representation which determines the parameters of the rotation. Note that the expressions in (2.16) respect the symmetry properties of the adjoint representation in the tensor products 6 × 6, 4 × 4¯ and 4¯ × 4. We now express the R-matrices in terms of these symmetry generators and find iu u 2 I+ J 6,6 − S, u+i (u + i)(u + 2i) (u + i)(u + 2i) u+i i R4,6 (u) = I+ J 4,6 , u + 3i2 u + 3i2 u+i i ¯ ¯ R4,6 (u) = I+ J 4,6 . 3i 3i u+ 2 u+ 2 R6,6 (u) =

(2.17)

One then finds that the expansion at infinity i u + i 6,6 R (u − i) = I + J 6,6 + O(1/u2 ), u u u + 2i 4,6 i R (u − i) = I + J 4,6 + O(1/u2 ), u u u + 2i 4,6 i ¯ ¯ R (u − i) = I + J 4,6 + O(1/u2 ). u u

(2.18)

Here we have used the same shifts and prefactors as in the construction of the monodromy matrices (2.10,2.12). In all three cases, the monodromy matrix therefore has the expansion Ra (u) = Ia +

i R J + O(1/u2 ) u a

(2.19)

at u = ∞ with the global rotation operators JaR =

L

R,6 Ja,p .

(2.20)

p=1

If we expand further around u = ∞ we will find multi-local operators along the spin chain. These are the generators of the Yangian, see e.g. [84, 85] and [86–88] in the context of N = 4 SYM. 2.2. Bethe ansatz. States. Consider a spin chain state

  Kj 3 {uj,k }, L ∼  Bj (uj,k ) |0, L. j =1 k=1

(2.21)

618


The vacuum state |0, L is the tensor product of L spins in a highest weight configuration of the 6. In other words, |0, L is the ferromagnetic vacuum with all spins aligned to give a maximum total spin. The operator Bj (u), u ∈ C, j = 1, 2, 3, creates an excitation with rapidity u and quantum numbers of the j th simple root of su(4). A state with a given weight [r1 , r2 , r3 ] (Dynkin labels) of su(4) has excitation numbers Kj given by (cf. [51]): K1 = 21 L − 43 r1 − 21 r2 − 41 r3 , K2 = L − 21 r1 − r2 − 21 r3 , K3 = 21 L − 41 r1 − 21 r2 − 43 r3 .

(2.22)

Transfer matrices. Now let us assume that the state |{uj,k }, L is an eigenstate of all transfer matrices TR (u) for all values of the spectral parameter u. Then it can be shown that the eigenvalue of the transfer matrix in the 6 representation is given by (see App. A for a derivation) R2 (u −

T6 (u) =

3i 2) R2 (u − 2i ) R2 (u + 2i ) R3 (u − i) R2 (u − 2i ) R3 (u)

V (u + i) V (u)

+

R1 (u − i) R1 (u)

+

R1 (u + i) R1 (u)

R3 (u − i) V (u − i) V (u + i) R3 (u) V (u) V (u)

+

R1 (u − i) R1 (u)

R3 (u + i) V (u − i) V (u + i) R3 (u) V (u) V (u)

+

R1 (u + i) R2 (u − 2i ) R3 (u + i) V (u − i) V (u + i) R1 (u) R2 (u + 2i ) R3 (u) V (u) V (u)

+

R2 (u +

3i 2) R2 (u + 2i )

V (u − i) V (u + i) V (u) V (u)

V (u − i) . V (u)

(2.23)

Note that the monodromy matrix 6a (u) is a 6 × 6 matrix in the auxiliary space labelled by a. This explains why the transfer matrix as its trace consists of six terms. For convenience, we have defined the functions Rj (u), V (u)

Rj (u) =

Kj

(u − uj,k ),

V (u) = uL ,

(2.24)

k=1

which describe two and one-particle scattering, respectively. Let us also state the eigenvalues of the transfer matrices in the spinor representations


T4 (u) =

619

V (u + 2i ) V (u)

R1 (u −

3i 2) R1 (u − 2i ) R1 (u + 2i ) R2 (u − i) + R1 (u − 2i ) R2 (u)

V (u + 2i ) V (u)

R2 (u + i) R3 (u − 2i ) V (u − 2i ) R2 (u) R3 (u + 2i ) V (u)

+

R3 (u +

3i 2) R3 (u + 2i )

+

V (u − 2i ) V (u)

(2.25)

and its conjugate 3i i 2 ) V (u + 2 ) R3 (u − 2i ) V (u) R3 (u + 2i ) V (u + 2i ) R3 (u − 2i ) V (u) V (u − 2i )

R3 (u −

T4¯ (u) = +

R2 (u − i) R2 (u)

+

R1 (u − 2i ) R2 (u + i) R1 (u + 2i ) R2 (u)

+

R1 (u +

3i 2) R1 (u + 2i )

V (u) V (u − 2i ) . V (u)

(2.26)

Bethe Equations. As they stand, the above expressions for TR (u) are rational functions of u. From the definition of TR (u) in (2.10,2.11,2.12) and RR,6 (u) in (2.7) it follows that TR (u) is a polynomial in 1/u4 of degree at most 2L (for R = 6; for R = 4 or R = 4¯ the maximum degree is L). This means that a state |{uj,k }, L cannot be an eigenstate of the transfer matrices if TR (u) has poles anywhere in the complex plane except the obvious singularity at u = 0 from the definition of Ra (u). From the cancellation of poles for all 1/u ∈ C one can derive a set of equations which in effect allowed rapidities {uj,k } to make up an eigenstate. These are precisely the Bethe equations [81, 82, 89] R1 (u1,k + i) R2 (u1,k − 2i ) R1 (u1,k − i) R2 (u1,k + 2i )

= −1,

R1 (u2,k − 2i ) R2 (u2,k + i) R3 (u2,k − 2i ) V (u2,k + 2i ) = − , R1 (u2,k + 2i ) R2 (u2,k − i) R3 (u2,k + 2i ) V (u2,k − 2i ) R2 (u3,k − 2i ) R3 (u3,k + i) = −1. R2 (u3,k + 2i ) R3 (u3,k − i) 4

The expansion in 1/u instead of the more common one in u is due to our definitions.

(2.27)

620


Effectively, they ensure that T6 (u), T4 (u) and T4¯ (u) are all analytic for 1/u ∈ C. Using the identity K

j

K

Rj (uj ,k + a) =

k =1

Kj j

(uj ,k − uj,k + a) = (−1)

Kj Kj

k=1 k =1

Kj

Rj (uj,k − a)

k=1

(2.28) it is easy to see that the product of all Bethe equations yields the constraint 1=

K2 V (u2,k + 2i ) k=1

V (u2,k − 2i )

=

R2 (− 2i )L R2 (+ 2i )L

.

(2.29)

Local Charges. In T6 (u) all terms but one are proportional to V (u − i) = (u − i)L . Thus the first L terms in the expansion in u around i are determined by this one term alone (unless there are singular roots at u = + 2i which would lower the bound) R2 (u − 2i ) V (u + i) + O(uL ). T6 (u + i) = V (u + 2i) R2 (u + 2i )

(2.30)

According to (2.13) this is precisely the combination for the expansion in terms of local charges. Comparing (2.30) to (2.13) we obtain for the global shift and local charge eigenvalues U, Qr , U =

K2 u2,k −

i 2 i 2

, u2,k + K2 1 i 1 . − Qr = i r−1 i r−1 r −1 (u (u + ) − ) 2,k 2,k 2 2 k=1 k=1

(2.31)

Note that the eigenvalue of the second charge Q2 is the energy E = Q2 , eigenvalue of the Hamiltonian, see (2.15). The momentum U must satisfy U L = 1 due to (2.29) in agreement with the fact that the shift operator obeys UL = 1. The expansion in terms of local charges is a distinctive feature of T6 (u), which is in the same representation as the spins. For T6 (u) we can expand around u = ±i and only one of the six terms does contribute in the leading few powers as in (2.30). In contradistinction, at least two terms contribute to the expansion of T4 (u) and T4¯ (u) at every point u. Therefore, neither T4 (u) nor T4¯ (u) can be used to yield local charges, which are the sums of the magnon charges as in (2.31).

2.3. Thermodynamic Limit. In the thermodynamic limit the length L of the spin chain as well as the number of excitations Kj approach infinity while focusing on the low-energy spectrum [90, 29]. Let us now rescale the parameters {Kj , u, rj , D, g} → L{Kj , u, rj , D, g},

(2.32)


621

while E → E/L. The Bethe roots uj,k condense on (not necessarily connected) curves Cj in the complex plane with a density function ρj (u), i.e. Kj

... → L

k=1

Cj

du ρj (u) . . . .

This fixes the normalization of the densities to

du ρj (u) = Kj . Cj

It is useful to note the following limits of fractions involving R and V : Rj (u + a) V (u + a) a−b → exp (b − a)Gj (u) , → exp Rj (u + b) V (u + b) u with the resolvent

(2.33)

(2.34)

(2.35)

Gj (u) =

dv ρj (v) . Cj v − u

(2.36)

We can now determine the limit of the transfer matrices. Let us start with the fundamental representation (2.25); we obtain T4 (u) → exp ip1 (u) + exp ip2 (u) + exp ip3 (u) + exp ip4 (u) . (2.37) The four exponents p1,2,3,4 (u) read ˜ 1 (u), p1 (u) = G ˜ 2 (u) − G ˜ 1 (u), p2 (u) = G ˜ 3 (u) − G ˜ 2 (u), p3 (u) = G ˜ 3 (u), p4 (u) = −G

(2.38)

˜ j (u) as where we have defined the singular resolvents G ˜ 1 (u) = G1 (u) + 1/2u, G ˜ 2 (u) = G2 (u) + 1/u, G ˜ 3 (u) = G3 (u) + 1/2u. G

(2.39)

Note that the exponents add up to zero p1 (u) + p2 (u) + p3 (u) + p4 (u) = 0.

(2.40)

The limit of a transfer matrix in an arbitrary representation R now reads simply TR (u) →

R

exp ipkR (u) .

(2.41)

k=1

The functions p4 (u) = p(u) are related to the transfer matrix in the fundamental rep¯ resentation. From (2.23,2.26) we can derive up similar functions p 6 (u), p 4 (u) for the

622


Fig. 1. Transfer matrix in 4 and 4¯ representation

vector and conjugate fundamental representation. For each component of the multiplet there is an exponent pkR (u): p 4 = (p1 , p2 , p3 , p4 ), p6 = (p1 + p2 , p1 + p3 , p1 + p4 , p2 + p3 , p2 + p4 , p3 + p4 ), = (p1 + p2 , p1 + p3 , p1 + p4 , −p1 − p4 , −p1 − p3 , −p1 − p2 ), ¯

p4 = (p1 + p2 + p3 , p1 + p2 + p4 , p1 + p3 + p4 , p2 + p3 + p4 ) = (−p4 , −p3 , −p2 , −p1 ).

(2.42)

2.4. Properties of the Resolvents. Bethe equations and sheets. We know that T4 (u) is a polynomial in 1/u. It therefore has no singularities except at u = 0 and it should remain analytic in the thermodynamic limit. This is ensured by the Bethe equations (2.27) whose limit reads ˜ 2 (u) = /p1 (u) − /p2 (u) = 2π n1,a , ˜ 1 (u) − G 2/ G ˜ 2 (u) − G ˜ 1 (u) − G ˜ 3 (u) = /p2 (u) − /p3 (u) = 2π n2,a , 2/ G ˜ 3 (u) − G ˜ 2 (u) = /p3 (u) − /p4 (u) = 2π n3,a , 2/ G

u ∈ C1,a , u ∈ C2,a , u ∈ C3,a .

(2.43)

Here we have split up the curves Cj into their connected components Cj,a with Cj = Cj,1 ∪ · · · ∪ Cj,Aj

(2.44)

and introduced a mode number nj,a for each curve to select the branch of the logarithm that was used to bring Eqs. (2.27) into the form (2.43). Furthermore G /˜ and /p are the ˜ principal values of G and p, respectively, at a cut, e.g. ˜ j (u − ) + 1 G ˜ G /˜ j (u) = 21 G 2 j (u + ).

(2.45)

Let us explain the meaning of the Bethe equations in words. The first one implies that a cut in p1 (u) or p2 (u) at C1,a can be analytically continued by the function p2 (u) and p1 (u), respectively (up to a shift by ±2πn1,a ). For the transfer matrices in (2.41) neither the interchange between p1 and p2 nor a shift by an integer multiple of 2π has any effect. Similarly, p2 and p3 or p3 and p4 are connected by cuts along C2 or C3 as depicted in Fig. 1. Therefore the transfer matrices are analytic except at u = 0. In total, the functions p1,2,3,4 (u) (modulo 2π ) make up four sheets of a Riemann surface, an algebraic curve of degree four. The function p = (p1 , p2 , p3 , p4 ) is not single valued


623

Fig. 2. Transfer matrix in 6 representation

due to the ambiguities by multiples of 2π. In dp the (constant) ambiguities drop out. The differential dp therefore is a holomorphic function on the algebraic curve except at the singular points u = 0 on each sheet, see (2.39,2.42). The configurations of sheets and their connections are displayed in Figs. 1 ,2.5 The ˜ j (u) sheet function pkR (u) is obtained by summing up the outgoing singular resolvents G and subtracting the incoming ones, cf. (2.38,2.42). Let us note here that Eqs. (2.43) are reminiscent of the saddle point equations of [91] for the RSOS type multi-matrix models. However the potential part is different and, most importantly, the right hand sides of (2.43) would be zero for RSOS models. Local Charges. The expansion of T6 (u) at u = i gives the local charges. In the thermodynamic limit, this point is scaled to u = 0 and from (2.30,2.35) we find [92] ∞

˜ 2 (u) = p1 (u) + p2 (u) = 1 + G ur−1 Qr , u

(2.46)

r=1

where Qr has been rescaled by Lr−1 . The first charge Q1 is the total momentum around the spin chain which should equal Q1 = 2πn0

(2.47)

for gauge theory states. The second charge Q2 = E = (D − 1)/g 2

(2.48)

is the energy eigenvalue of the Hamiltonian. The other two resolvents are non-singular ˜ 3 (u) = O(u0 ) ˜ 1 (u), G G

(2.49)

and their expansion (thus) does not correspond to local quantities. For convenience, we also display the expansion of the sheet functions at zero +p1 (u), +p2 (u), −p3 (u), −p4 (u) =

1 + O(u0 ). 2u

(2.50)

5 The cuts are not necessarily along the real axis as might be suggested by the figures. In fact, for compact spin representations, they usually cross the real axis at right angles.

624


Global Charges. The charges of the symmetry algebra ae obtained from the monodromy matrix at u = ∞, see Sect. 2.1. When we expand the resolvents Gj (u) at infinity

Kj 1 dv ρj (v) + O(1/u2 ) = − (2.51) + O(1/u2 ), Gj (u) = − u Cj u we find the fillings of the cuts (2.34), which are related to the representation [r1 , r2 , r3 ] of the state via (2.22). The singular resolvents directly relate to the Dynkin labels as follows (L = 1 after rescaling) ˜ 1 (u) = 1 1 − K1 + O(1/u2 ) = G 2 u 1 ˜ G2 (u) = ( 1 − K2 ) + O(1/u2 ) = u ˜ 3 (u) = 1 1 − K3 + O(1/u2 ) = G 2 u

1 3 r1 + 21 r2 + 41 r3 + O(1/u2 ), 4 u 1 1 r1 + r2 + 21 r3 + O(1/u2 ), (2.52) 2 u 1 1 r1 + 21 r2 + 43 r3 + O(1/u2 ). 4 u

For convenience, we also display the expansion of the sheet functions at infinity 1 3 + 4 r1 + u 1 1 p2 (u) = − 4 r1 + u 1 1 p3 (u) = − 4 r1 − u 1 1 p4 (u) = − 4 r1 − u p1 (u) =

1 2 r2

+ 41 r3 + O(1/u2 ),

1 2 r2

+ 41 r3 + O(1/u2 ),

1 2 r2

+ 41 r3 + O(1/u2 ),

1 2 r2

− 43 r3 + O(1/u2 ).

(2.53)

2.5. Algebraic curve. Let us now try to restore the function p(x) from the information derived in the previous subsection,6 namely, from the Riemann-Hilbert equations (2.43)

/pk (x) − /pk+1 (x) = 2πna

for x ∈ Ca ,

(2.54)

where Ca connects sheets k and k + 1 and from the behavior at the various sheets (2.38) at x → ∞ (2.53) pk ∼

1 + O(1/u2 ) u

(2.55)

as well as x → 0 (2.50) +p1 , +p2 , −p3 , −p4 =

1 + 0 log u + O(u0 ) 2u

(2.56)

From the discussion in Sect. 2.4 we know that exp(ip) is a single valued holomorphic function on the Riemann surface with four sheets except at the points 0 and ∞. It is however not an algebraic curve because it has an essential singularity of the type exp(i/u) at u = 0. While p only has pole-singularities, it is defined only modulo 2π . 6 In the su(2) case the corresponding hyperelliptic curve was constructed in [41] using the method proposed in [93].


625

This problem is overcome in the derivative p which is has a double pole 1/u2 at u = 0, but no single pole 1/u, neither at u = 0 nor at u = ∞. All this suggests that there exists a function p(u), the quasi-momentum, such that its derivative y(u) = u2

dp (u) du

(2.57)

satisfies a quartic algebraic equation F (y, u) = P4 (u) y 4 + P2 (u) y 2 + P1 (u) y + P0 (u) = P4 (u)

4

y − yk (u) = 0.

k=1

(2.58) For a solution with finitely many cuts we may assume the coefficients Pk (u) to be polynomials in u. The term y 3 is absent because p1 + p2 + p3 + p4 = 0. We have adjusted y to approach a constant limiting value at x = 0 as well as at x = ∞. It follows that all the polynomials Pk (u) have the same order 2A and a non-vanishing constant coefficient. Altogether the function F (y, u) which determines the curve is parameterized by 8A + 4 coefficients minus one overall normalization. Let us now investigate the analytic structure of the solution of F (y, u) √ = 0 and compare it to the structure of p. In general we can expect that p behaves like u − u∗ √ ∗ ∗ at a branch point u , consequently y ∼ 1/ u − u . To satisfy the equation F (y, u) = 0 at y = ∞ we should look for zeros of P4 (u)/P2 (u). Incidentally, we find precisely the correct behavior for y due to the missing of the y 3 term.7 For a generic P2 (u), the branch points are thus the roots of P4 (u), P4 (u) =

A

(u − aa )(u − ba ).

(2.59)

a=1

Therefore, A is the number of cuts and aa , ba are the branch points. The algebraic equation (2.58) potentially has further cuts. The associated singularities are of the undesired form y ∼ (u − u∗ )r+1/2 or p ∼ (u − u∗ )r+3/2 and we have to ensure their absence. Their positions can be obtained as roots of the discriminant of the quartic equation R = −4P12 P23 + 16P0 P24 − 27P14 P4 + 144P0 P12 P2 P4 − 128P02 P22 P4 + 256P03 P42 = P45 (y1 − y2 )2 (y1 − y3 )2 (y1 − y4 )2 (y2 − y3 )2 (y2 − y4 )2 (y3 − y4 )2 .

(2.60)

All solutions of R(u) = 0 with odd multiplicity give rise to undesired branch cuts, in other words we have to demand that the discriminant is a perfect square R(u) = Q(u)2 with a polynomial free coefficients.

Q(u).8

(2.61)

This fixes 5A coefficients and we remain with only 3A + 3

7 A pole on a single sheet could never be cancelled in p + p + p + p = 0. In contrast, a branch 1 2 3 4 √ √ singularity +α/ u − u∗ will be cancelled by an accompanying singularity −α/ u − u∗ on the sheet which is connected along the branch cut. 8 An equivalent condition is: All solutions to the equations dF (y, u) = 0 and P (u) = 0 lie on the 4 curve F (y, u) = 0. The condition dF (y, u) = 0 ⇒ F (y, u) = 0 eliminates branch points and P4 (u) = 0 preserves the desired ones.

626


First of all we can fix the coefficients of the double pole in p at u = 0 according to (2.56). This fixes three coefficients, P4 (0) = −8P2 (0) = 16P0 (0) and P1 (0) = 0. The function p(u) has to be single-valued (modulo 2π ) on the curve. We can put the A-cycles to zero9 dp = 0. (2.62) Aa

The cycle Aa surrounds the cut Ca . Note that there are only A − 3 independent A-cycles in agreement with the genus of the algebraic curve, A − 3. The sum of all A-cycles on each of the three independent sheets can be joined to a cycle around the punctures at u = 0. Here we expect a double pole, but not a single pole, (2.56) dpk = 0. (2.63) 0

The A-cycles together with the absence of single poles at u = 0 yield A constraints. Next we consider the B-periods. The cycle Ba connects the points u = ∞ of two sheets k, k + 1 going through a cut Ca which connects these sheets.10 We now rewrite the Bethe equations (2.54) as A integer B-periods

dp = 2πna , (2.64) Ba

where na is the mode number associated to the cut Ca . We can now integrate p (u) and obtain p(u). The integration constants are determined by the value at u = ∞, (2.55). At this point we are left with precisely A undefined coefficients. These can be identified with the filling fractions 1 Ka = − p(u) du. (2.65) 2πi Aa In the integral representation these correspond to the quantities

Ka = ρ(u) du. Ca

(2.66)

When all filling fractions Ka and integer mode numbers na are fixed, we can calculate ˜ 2 = p1 + p2 = −p3 − p4 in principle any function of physical interest. In particular, G gives an infinite set of local charges (2.46), including the anomalous dimension. Note that so far we have not considered the momentum constraint (2.47) which serves as a physicality condition for gauge theory states.11 This reduces the number of independent continuous parameters by one since n0 is discrete. We can even express the constraint fully in terms of Ka and na , n0 =

A

na K a ∈ Z

(2.67)

a=1

by integrating the Bethe equations (2.43) over all cuts. 9 Even though we should assume multiples of 2π as the periods, the cuts can be chosen in such a way as to yield single-valued functions pk [41]. 10 Here the property p ∼ 1/x 2 , (2.55), is useful to mark the points u = ∞. 11 Spin chain states which do not obey (2.47) are perfectly well-defined, they merely have no correspondence in gauge theory.


627

3. Classical Sigma-Model on R × S m−1 In this section we will investigate the analytic properties of the monodromy of the Lax pair around the closed string. The string is the two-dimensional non-linear sigma model on R × S m−1 supplemented by the Virasoro constraints. This is an interesting model, because (classically) it is a consistent truncation of the superstring on AdS5 × S 5 . 3.1. The sigma-model. Consider the two-dimensional sigma model on R × S m−1 . Let {Xi }m i=0 denote the target space coordinates. While X0 can take any value on R, the other coordinates {Xi }m i=1 satisfy a constraint X1 2 + · · · + Xm 2 = 1. Let us also introduce the following vector notation:   X1 . 

=

2 = 1 X  ..  , X Xm

(3.1)

(3.2)

by and a matrix hV associated with X

X

T hV = 1 − 2X

(i.e.

hV ij = δij − 2Xi Xj ).

(3.3)

det hV = −1; it satisfies hV hTV = 1, The matrix describes a reflection along the vector X, i.e. it is orthogonal, hV ∈ O(m). Furthermore it is symmetric, hV = hTV , and therefore it equals its own inverse h−1 V = hV .

(3.4)

The action of a bosonic string rotating on the S 5 sphere and restricted to a time-like geodesic R of AdS5 is given by √ 2 λ

· ∂aX

− ∂ a X0 ∂ a X 0 + X

−1 , Sσ = − dσ dτ ∂a X (3.5) 4π

to the unit sphere (3.1). Here it is where is a Lagrange multiplier that constrains X useful to introduce light-cone coordinates σ± = 21 (τ ± σ ),

∂± = ∂τ ± ∂σ .

(3.6)

Then the equations of motion read

+ (∂+ X

· ∂− X)

X

= 0, ∂ + ∂− X

∂+ ∂− X0 = 0.

(3.7)

A solution for the time coordinate which we use to fix the residual gauge of the string is Dτ D(σ+ + σ− ) X0 (τ, σ ) = √ = , √ λ λ

(3.8)

where D is the dimension in the AdS/CFT interpretation. In addition to the action, the string must satisfy the Virasoro constraints

2 = (∂± X0 )2 , (∂± X)

(3.9)

628


which read

2= (∂± X)

D2 λ

(3.10)

in the gauge (3.8).

3.2. Flat and conserved currents. Let us next define the right current jV and the left current V by jV := h−1 V dhV ,

V := −dhV h−1 V .

(3.11)

Due to the special property (3.4) these currents coincide. In this case they are simply

as expressed in terms of X

X

T − dX

X

T ), jV = V = 2(Xd

(3.12)

or in terms of their components, (jV )a,ij = (V )a,ij = 2 Xi ∂a Xj − Xj ∂a Xi .

(3.13)

From (3.11) it is clear that jV satisfies the flatness condition djV + jV ∧ jV = 0.

(3.14)

The kinetic term of the Lagrangian of the sigma model on S m−1 is given by Tr jV ∧∗jV . It is invariant under global right (and left) multiplication to hV and jV is the associated conserved current d(∗jV ) = 0

(i.e. ∂a jVa = 0).

(3.15)

This relation can be verified using the equations of motion (3.7), in fact it is equivalent to them. The current jV is an element of so(m) in the vector representation. Similarly, we can write a current in the spinor representation. Let {γi } form the basis of the Clifford algebra of SO(m), i.e. they satisfy {γi , γj } = 2δij .

(3.16)

hS = γ · X

(3.17)

Now we introduce the matrix

in the spinor representation. It satisfies hS = h−1 S ,

(3.18)

therefore it can be regarded as a spinor equivalent of hV . As above this gives rise to equal right and left currents jS = S . These are in fact equivalent to jV through jS ∼ γi γj (jV )ij .


629

3.3. Lax pair and monodromy matrix. Having a flat and conserved current j , one can construct a family of flat currents 1 x j+ ∗j 1 − x2 1 − x2

a(x) =

(3.19)

parameterized by the spectral parameter x. These give rise to a pair of Lax operators (M, L) = d + a, 1 j− j+ − , L(x) = ∂σ + aσ (x) = ∂σ + 2 1−x 1+x 1 j− j+ M(x) = ∂τ + aτ (x) = ∂τ + + , 2 1−x 1+x

(3.20)

where we make use of ∗(jτ , jσ ) = (jσ , jτ ) and define j± = jτ ± jσ . The conservation and flatness conditions for j are interpreted as the flatness condition for a(x) for all values of x, da(x) + a(x) ∧ a(x) = 0

L(x), M(x) = 0.

or

(3.21)

Here we have made use of the relations ∗b ∧ c = −b ∧ ∗c and ∗b ∧ ∗c = −b ∧ c for one-forms b, c in two dimensions. We can now compute the monodromy of the operator d + a(x) around the closed string. This is the Wilson line along the curve γ (σ, τ ) which winds once around the string and which is starting and ending at the point (τ, σ ),

−a(x) .

(x, τ, σ ) = P exp

(3.22)

γ (τ,σ )

As the current a(x) is flat, the actual shape of the curve γ is irrelevant. The monodromy (x) depends on the starting point (τ, σ ) through the defining equations of the Wilson line, d(x) + a(x), (x) = 0,

(3.23)

which generates a similarity transformation. Physical information should be invariant under the choice of specific points on the world sheet. Therefore, the monodromy (x) is not physical, but only its conjugacy class, i.e. the set of its eigenvalues. For our purposes this means that neither the curve nor its starting point is relevant. We can thus choose the curve γ to be given by τ = 0, σ ∈ [0, 2π ]. The monodromy matrix becomes

(x) = P exp 0

2π

1 dσ 2

j+ j− + , x−1 x+1

(3.24)

where the path ordering symbol P puts the values of σ in decreasing order from left to right.

630


3.4. Eigenvalues of the monodromy matrix. Let us now choose the current in the vector representation j = jV . Since jVT = −jV and x ∈ C, V is a complex orthogonal matrix12 V VT = 1.

(3.25)

Only the conjugacy class of V (x), characterized by its eigenvalues, corresponds to physical observables. V ∈ SO(m, C) is diagonalized into the following general form V (x) diag eiq1 (x) , e−iq1 (x) , eiq2 (x) , e−iq2 (x) , . . . , eiq[m/2] (x) , e−iq[m/2] (x) for m even, diag eiq1 (x) , e−iq1 (x) , eiq2 (x) , e−iq2 (x) , . . . , eiq[m/2] (x) , e−iq[m/2] (x) , 1 for m odd, (3.26) [m/2]

where we express the eigenvalues in terms of quasi-momenta {qk (x)}k=1 . However, there still remains the freedom of permutation of the eigenvalues, switching the sign and adding integer multiples of 2πi. A more convenient quantity is the characteristic polynomial V (α) = det(α − V ) [m/2] − eiqk )(α − e−iqk ), for m even k=1 (α = [m/2] iq −iq k k (α − 1) k=1 (α − e )(α − e ), for m odd ≡

m

(−1)m−k α k TV[k] .

(3.27)

k=0

It is an mth order polynomial and each coefficient TV[k] is a symmetric polynomial of the eigenvalues. TV[k] is the trace of the monodromy matrix in the k th antisymmetric tensor product of the vector representation. Note that TV[m] = TV[0] = 1,

TV[m−k] = TV[k]

(3.28)

in agreement with representation theory. For the monodromy matrix in the spinor representation we find S diag exp(± 2i q1 ± 2i q2 · · · ± 2i q[m/2] )

(3.29)

with all 2[m/2] choices of signs. When n is even, we can reduce S further into its chiral and antichiral parts S± , = S

S+ 0 0 S−

.

(3.30)

For S+ and S− we should only take those eigenvalues with an even or odd number of plus signs in (3.29), respectively. 12 In fact it satisfies the reality condition (V (x ∗ ))∗ = V (x), i.e. the complex values in V (x) are introduced only through a complex x.


631

3.5. Analyticity. The monodromy matrix (x) depends analytically on x except at the two singular points x = ±1, see Sect. 3.7. This, however, does not directly imply the same for its eigenvalues or the qk ’s. Most importantly, at those points {xb∗ } where two eigenvalues eiqk , eiql degenerate we should expect the generic behavior qk,l (x) = qk,l (xb∗ ) ± αb x − xb∗ + O(x − xb∗ ) (3.31) with some coefficients αb . This square-root singularity not only violates analyticity locally at the point xb∗ , but also requires a square-root branch cut originating from it. At the branch cut, the eigenvalues are permuted. Furthermore, qk is defined only modulo 2π. Finally, the labelling of qk ’s is defined at our will. Although the qk ’s could fluctuate randomly from one point to the next according to these two ambiguities, we shall assume qk to be analytic except at two singular points and at a (finite) number of branch cuts Ca . At the cuts, the qk ’s can be permuted and shifted by multiples of 2π . Such a transformation is captured by the equations /qk (x) ∓ /ql (x) = 2πna

for x ∈ Ca .

(3.32)

Here /q (x) means the principal part of qk (x) across the cut /qk (x) = 21 qk (x + i ) + 21 qk (x − i ).

(3.33)

The integer na is called the mode number of Ca and it is assumed to be constant along the cut. Without loss of generality we can restrict ourselves to a subset of allowed permutations, qk with qk+1 , i.e. /qk (x) − /qk+1 (x) = 2πnk,a

for x ∈ Ck,a .

(3.34)

These must be supplemented by (for m even) /q[m/2]−1 (x) + /q[m/2] (x) = 2πn[m/2],a

for x ∈ C[m/2],a

(3.35)

or (for m odd) 2/q[m/2] (x) = 2πn[m/2],a

for x ∈ C[m/2],a .

(3.36)

3.6. Asymptotics. Let us investigate the expansion at x = ∞. In leading order, the family of flat connections 1 a(x) = − ∗j + O(1/x 2 ) x

(3.37)

yields the dual of the conserved current j . The expansion of the monodromy matrix (x) at x = ∞ thus yields

1 2π 1 4π J (x) = I + dσ jτ + O(1/x 2 ) = I + (3.38) √ + O(1/x 2 ). x 0 x λ Here, the conserved charges J of the sigma-model √ 2π λ J = dσ jτ , 4π 0

(3.39)

632


appear as the first order in the expansion in 1/x. The eigenvalues of J V in the vector representation are given by diag iJ1 , −iJ1 , iJ2 , −iJ2 , . . . , iJ[m/2] , −iJ[m/2] for m even, JV (3.40) diag iJ1 , −iJ1 , iJ2 , −iJ2 , . . . , iJ[m/2] , −iJ[m/2] , 0 for m odd. The charge eigenvalues Jk are related to the Dynkin labels [s1 , s2 , . . . ] for even m by Jk =

[m/2] j =k

sj − 21 s[m/2]−1 − 21 s[m/2] ,

(3.41)

and for odd m by Jk =

[m/2] j =k

sj − 21 s[m/2] .

(3.42)

When we compare (3.38,3.26,3.40) we find the expansion of the qk ’s at x = ∞. Since qk ’s are defined as in (3.26), there is a freedom of choosing signs and the ordering of qk ’s as well as the branch of the logarithm. We fix them so that qk has asymptotic behavior qk (x) =

1 4πJk √ + O(1/x 2 ). x λ

(3.43)

In particular we fix the branch of all the logarithms such that all qk (x) vanish at x = ∞. 3.7. Singularities. To study the asymptotics of the monodromy matrix (3.24) at x → ±1 we assume that at all values of σ the unitary matrix u± (σ ) diagonalizes j± (σ ),13 u± (σ ) j± (σ ) u−1 ± (σ ) = j±

diag

(σ ).

(3.44)

We furthermore assume that the function u± (x, σ ) is some analytic continuation of u± (σ ) at x = ±1, i.e. u± (x, σ ) = u± (σ ) + O(x ∓ 1).

(3.45)

When we now do a gauge transformation using u± (x, σ ) we find that u± (x) L(x) u± (x)−1 = ∂σ −

diag

1 j± + O (x ∓ 1)0 , 2 x∓1

(3.46)

0 where the gauge term ∂σ u± u−1 ± is of order O((x ∓ 1) ). The higher-order off-diagonal terms in L(x) can be removed order by order by adding the appropriate terms to u± (x). Therefore u± (x) can be used to completely diagonalize L(x) for all σ . Thus we can drop the path ordering and write diag j± 1 2π −1 V 0 u± (x, 2π ) (x) u± (x, 0) = exp . dσ + O (x ∓ 1) 2 0 x∓1

(3.47) 13

We thank Gleb Arutyunov for explaining to us the expansion.


633

In other words, to compute the leading singular behavior of the eigenvalues of V (x), it suffices to integrate the eigenvalues of j± . In App. B we will compute the next few terms in this series and thus find some local commuting charges of the sigma model. From (3.12) we infer that the current jV in the vector representation has only two nonzero eigenvalues. They are imaginary, have equal absolute value but opposite signs. The

2 = (∂± X0 )2 . absolute value is determined through the Virasoro constraint (3.9), (∂± X) We then find using (3.12), 2 2 2

2 = ∂± X0 2 = D (∂± τ )2 = D . − 18 Tr jV,± = ∂± X λ λ

(3.48)

diag

The diagonalized matrix jV ± thus takes the form 2iD diag jV,± = √ diag(+1, −1, 0, 0, 0, 0, . . . ) λ

(3.49)

and hence does not depend on σ . This leads to the following asymptotic formula for the eigenvalues of the monodromy matrix: qk = δk1 √

2πD λ (x ∓ 1)

+ O (x ∓ 1)0

for x → ±1.

(3.50)

This formula allows to relate the asymptotic data at x → ±1 with the energy of a classical state of the string and hence identify the anomalous dimension of an operator due to the AdS/CFT correspondence. 3.8. Inversion symmetry. For a sigma model with right and left currents j = h−1 dh and = −dh h−1 , the families of associated flat currents a(x) and a (x) are related by an inversion of the spectral parameter x: h d + a(x) h−1 = h dh−1 +

1 x hj h−1 + h ∗j h−1 2 1−x 1 − x2 1 x =d +− − ∗ 2 1−x 1 − x2 x2 x =d− − ∗ 2 1−x 1 − x2 1 1/x =d+ + ∗ 1 − 1/x 2 1 − 1/x 2 = d + a (1/x).

(3.51)

In our S m−1 model where j = this is particularly interesting since it implies the symmetry hV LV (x) h−1 V = LV (1/x)

and

hS LS (x) h−1 = LS (1/x) S

and

hS (2π ) S (x) hS (0)−1 = S (1/x). (3.53)

(3.52)

and consequently hV (2π ) V (x) hV (0)−1 = V (1/x)

634


For a closed string we have h(0) = h(2π), therefore (x) and (1/x) are related by a similarity transformation and thus have the same set of eigenvalues. For the vector representation it means that each of the quasi-momenta {q1 , −q1 , q2 , −q2 , . . . } transforms into one of {q1 , −q1 , q2 , −q2 , . . . } under the x ↔ 1/x symmetry. When m is even, the spinor representation can be reduced into its chiral and antichiral components. We know that a gamma matrix γ interchanges both chiralities. Therefore the matrix hS inverts chirality while jS = h−1 S dhS preserves it. It follows that the inversion symmetry relates monodromy matrices of opposite chiralities hS (2π) S± (x) hS (0)−1 = S∓ (1/x).

(3.54)

To be consistent with the singular behavior (3.50), q1 has to transform as q1 (1/x) = 4πn0 − q1 (x).

(3.55)

The integer constant n0 reflects the difference of branches of the logarithm at x = 0 and x = ∞. We need a factor of 4π, because for spinor representations (3.29) we find the exponentials exp(± 2i q1 ), which must not change sign. Furthermore, for even m we require that chiral and antichiral representations are interchanged. This is achieved by an odd number of sign flips for all the qk . A possible transformation rule for the quasi-momenta that works for all values of m is14 qk (1/x) = (1 − 2δk1 ) qk (x) + 4π n0 δk1 ,

(3.56)

i.e. q1 flips sign while all other qk are invariant. We shall assume that it is the correct rule although there might be other consistent choices for m > 4. Note that there are no additional constant shifts for qk , k = 1, because these would be in conflict with the even transformation rule. 3.9. The sigma-model on R × S 3 . To test out results, we will consider the case R × S 3 which was extensively studied in [41]. The isometry group SO(4) of S 3 is locally isomorphic to SU(2)L × SU(2)R , which played a key role in the preceding discussion. Now we need not to make use of the isomorphism, nevertheless it is useful to interpret our formulation also in terms of SU(2)L × SU(2)R . The spinor representations 2L , 2R of SO(4) can be viewed as the fundamental representations of SU(2)L , SU(2)R , respectively. Monodromy matrices in these representations are then viewed as the SU(2) monodromy matrices. They are diagonalized as S+ ∼ diag(eipL , e−ipL ),

S− ∼ diag(eipR , e−ipR ).

(3.57)

Now we have two independent quasi-momenta pL , pR , due to the reducibility of SO(4). They can be related to the quasi-momenta q1 , q2 by pL = 21 q1 + 21 q2 ,

pR = 21 q1 − 21 q2 .

(3.58)

The inversion symmetry now gives rise to p(x) := pR (x) = −pL (1/x) + 2π n0 . 14

(3.59)

We cannot exclude different transformation rules at this point. It would be interesting to see whether there exist such solutions (which are not merely obtained by a relabelling of the eigenvalues).


635

This means one can assemble the two quasi-momenta to a single quasi-momentum p(x) without inversion symmetry. This special fact played a crucial role in the previous analysis in [41]. In fact this p(x) is the quasi-momentum discussed there. Let us deduce properties of p(x) from our general results. The pole structure reads πD p(x) = √ (3.60) + O (x ∓ 1)0 . λ (x ∓ 1) It exhibits the asymptotic behavior 2πrR 1 p(x) = √ for x → ∞ + O(1/x 2 ) λ x while the asymptotic behavior of pL (x) for x → ∞ is interpreted as

(3.61)

2πrL p(x) = 2πn0 − √ x + O(x 2 ) for x → 0. (3.62) λ The Dynkin labels rL = J1 + J2 and rR = J1 − J2 specify the quantum numbers of SU(2)L and SU(2)R ; they equal twice the invariant SU(2) spin. Both analyticity conditions (3.34,3.35) are now simply given by 2/p (x) = 2πna ,

x ∈ Ca .

(3.63)

All of these agree exactly with the previous results [41]. 4. Algebraic Curve for the Sigma-Model on R × S 5 In this section we will show that the generic solution to the string sigma model on R×S 5 is uniquely characterized by a set of mode numbers and fillings. These are related to certain cycles of the derivative of the quasi-momentum, qk (x), which is an algebraic curve of degree four. 4.1. SO(6) vs. SU(4). The isometry group SO(6) of S 5 is locally isomorphic to SU(4). This enables us to formulate the model in terms of the su(4) algebra and the spinor representation which turns out to simplify the structure of the algebraic curve. Here we will translate the properties obtained in the previous section in terms of the quasi-momentum p corresponding to the spinor representation instead of q which corresponds to the vector representation. The chiral spinor representation 4 of SO(6) is equivalent to the fundamental representation of SU(4). Therefore S+ can be regarded as the SU(4) monodromy matrix, which is diagonalized as S+ ∼ diag(eip1 , eip2 , eip3 , eip4 )

(4.1)

with p1 + p2 + p3 + p4 = 0. The quasi-momenta pk are identified as p1 = 21 ( q1 + q2 − q3 ), p2 = 21 ( q1 − q2 + q3 ), p3 = 21 (−q1 + q2 + q3 ), p4 = 21 (−q1 − q2 − q3 ) in our general notation.

(4.2)

636


Fig. 3. Structure of sheets and branch cuts in the 4 representation for the sigma model on S 5 . There are three types of cuts, C1,2,3 , corresponding to the simple roots of SU(4). For each cut C , there is a mirror cut C −1 . Whether or not it connects the same two sheets depends on the type of cut. The total number of cuts including the mirror images is denoted by A1 , A2 , A3

The inversion symmetry (3.56) in terms of pk is now written as15 p1,2 (1/x) = 2πn0 − p2,1 (x),

p3,4 (1/x) = −2π n0 − p4,3 (x).

(4.3)

This leads to the structure of branch cuts as depicted in Fig. 3. The cuts of type C1 , C2 , C3 correspond to the three simple roots of SU(4). While cuts C1 and C3 connect sheets 1, 2 and 3, 4, respectively, cuts C2 may connect either the sheets 2, 3 or the sheets 1, 4 due to the symmetry (4.3). The total number of cuts of either type will be denoted by A1 , A2 , A3 , mirror cuts are assumed to be explicitly included. The pole structure (3.50) reads p1,2 (x) = −p3,4 (x) = √

πD λ (x ∓ 1)

+ O (x ∓ 1)0

for x → ±1,

(4.4)

while the asymptotic behavior at x = ∞ (3.43) now reads 1 4π 3 √ 4 r1 + x λ 1 4π p2 (x) = √ − 41 r1 + x λ 1 4π p3 (x) = √ − 41 r1 − x λ 1 4π p4 (x) = √ − 41 r1 − x λ p1 (x) =

1 2π + 41 r3 + · · · = √ ( J1 + J2 − J3 ) + · · · x λ 1 2π 1 1 2 r2 + 4 r3 + · · · = x √ ( J1 − J2 + J3 ) + · · · λ 1 2π 1 1 2 r2 + 4 r3 + · · · = x √ (−J1 + J2 + J3 ) + · · · λ 1 2π 1 3 2 r2 − 4 r3 + · · · = x √ (−J1 − J2 − J3 ) + · · · λ

1 2 r2

, , , . (4.5)

Here the Dynkin labels [r1 , r2 , r3 ] of SU(4) are related to the Dynkin labels [s1 ; s2 , s3 ] and charges (J1 , J2 , J3 ) of SO(6) by r1 = s2 = J2 − J3 , r2 = s1 = J1 − J2 , r3 = s3 = J2 + J3 .

(4.6)

This is due to the difference in the labelling of simple roots between the Lie algebras of SU(4) and SO(6): The labels 1 and 2 are interchanged (see Fig. 4). 15 This is based on the assumption (3.56). Other possibilities for m = 6 are p (1/x) ∈ 2π Z −p (x) 1,2 1,2 and/or p3,4 (1/x) ∈ 2π Z −p3,4 (x). This will slightly change the counting of individual constraints below, but the overall number of moduli will remain the same.


637

Fig. 4. Dynkin diagrams of SO(6) and SU(4)

4.2. Branch cuts. The monodromy matrix S+ (x) has similar analytic properties as the one for the spin chain of Sect. 2. Therefore, as discussed in Sect. 2.5, the derivative of the quasi-momentum, p = (p1 , p2 , p3 , p4 ), is again an algebraic curve of degree four. First of all, let us define yk (x) = (x − 1/x)2 x pk (x).

(4.7)

This removes the poles at x = ±1 (4.4) and leads to a simple transformation rule under the symmetry (4.3). We can now write y as the solution to an algebraic equation of the same type as (2.58), F (y, x) = P4 (x) y 4 + P2 (x) y 2 + P1 (x) y + P0 (x) = 0.

(4.8)

As explained in Sect. 2.5 we know that the branch points are given by the roots of P4 (x); let us assume there are A cuts P4 (x) ∼

A

x − aa x − ba .

(4.9)

a=1

√ √ Together with P3 (x) = 0 this can easily be seen to yield a 1/ x − aa and 1/ x − ba behavior at aa , ba as expected from √ (3.31). We need to remove all further branch points, which are generically of the type x − x ∗ . These would lead to unexpected (x − x ∗ )3/2 behavior in p(x), cf. Sect. 3.5. Their positions x ∗ can be obtained as roots of the discriminant R of the quartic equation R = −4P12 P23 + 16P0 P24 − 27P14 P4 + 144P0 P12 P2 P4 − 128P02 P22 P4 + 256P03 P42 . (4.10) This means that the discriminant must be a perfect square R(x) = Q(x)2 .

(4.11)

4.3. Asymptotics. The asymptotics p(x) ∼ 1/x at x = ∞, (4.5) translate to y(x) ∼ x

at x = ∞

(4.12)

for y as defined in (4.7). This requires that Pk (x) ∼ x −k P0 (x) for the highest-order terms. Similarly, the asymptotics p(x) ∼ const. + x at x = 0 obtained through the symmetry (4.3) translate to y(x) ∼ 1/x

at x = 0.

(4.13)

638


This requires that Pk (x) ∼ x k P0 (x) for the lowest-order terms. In total this leads to polynomials of the form Pk (x) = ∗x k + · · · + ∗x 2A+8−k .

(4.14)

Consequently, the discriminant (4.10) takes the form R(x) = ∗x 8 + · · · + ∗x 10A+32 .

(4.15)

4.4. Symmetry. The symmetry of the quasi-momentum p in (4.3) translates to y1,2 (1/x) = y2,1 (x),

y3,4 (1/x) = y4,3 (x).

(4.16)

In order for the solution to the algebraic equation (4.8) to have this symmetry, the polynomials must transform according to16 Pk (1/x) = x −2A−8 Pk (x).

(4.17)

Similarly, the resolvent satisfies R(1/x) = x −10A−40 R(x).

(4.18)

In other words, the coefficients of the polynomials are the same when read backwards and forwards. Note that in (4.9) we have not made the symmetry for P4 (x) manifest. It requires that aa = 1/ab and ba = 1/bb for a pair of cuts Ca,b which interchange under the symmetry.17 The symmetry of F (y, x), however, merely guarantees that yk (1/x) = yπ(k) (x) with some permutation π(k). In the most general case, there can only be the trivial permutation π(k) = k. This can be seen by looking at the fixed points x = ±1 of the map x → 1/x. If y1 (±1) = y2 (±1) there is no chance that the permutation in (4.16) is realized. To permit (4.16) we need to make sure that y1 (x) = y2 (x) and y3 (x) = y4 (x)

for

x = +1

and

x = −1.

(4.19)

This yields four constraints on the coefficients of F (y, x). At this point the trivial permutation π(k) = k is still an option. However, now the choice between π(1) = 1 and π(1) = 2 is merely a discrete one; there are no further constraints which remove a continuous degree of freedom. In fact, as the solution to F (y, x) = 0 degenerates into two pairs at x = ±1, the discriminant must have a quadruple pole at these points, i.e. we can write R(x) = x 8 (x 2 − 1)4 ∗x 0 + · · · + ∗x 10A+16 .

(4.20)

We could also assume Pk (1/x) = −x −2A−8 Pk (x), but it turns out to be too restrictive. In principle we should also allow symmetric cuts with ba = 1/aa . Apparently these do not occur for solutions which correspond to gauge theory states at weak coupling. At weak coupling one cut should grow to infinity while the other shrinks to zero. This is not compatible with symmetric cuts. 16 17


639

Fig. 5. Branch cut Ca between sheets k and k + 1 with associated A-cycles and B-period

4.5. Singularities. Let us now consider the poles at x = ±1. The expansion of a generic solution for y yields pk (x) =

αk± βk± + + O (x ∓ 1)0 . 2 (x ∓ 1) x∓1

(4.21)

We have already demanded that α1± = α2± and α3± = α4± . The symmetry furthermore requires β1± = −β2± and β3± = −β4± . As the sum of all sheets must be zero, ± ± ± p1 + p2 + p3 + p4 = 0, it moreover follows that α1,2 = −α3,4 whereas β1,2 and ± β3,4 are independent. This means there are three independent coefficients each for the singular behavior at x = ±1. Now the residue of p at x = ±1 is proportional to the energy or dimension D. This we cannot fix as it will be the (hopefully) unique result of the calculation. However, we know that the residues at both x = ±1 are equal, (4.4), which gives one constraint on the α’s. Furthermore, there is no logarithmic behavior in p, (4.4), therefore all β’s must be zero which gives four constraints. In total there are five constraints from the poles at x = ±1. 4.6. A-cycles. The eigenvalues exp(ipk (x)) of S+ (x) are holomorphic functions of x. This however does not exclude the possibility of cuts where the argument pk (x) jumps by multiples of 2π but is otherwise smooth. Such cuts originate from logarithmic or branch-cut singularities; they are required when the closed integral around the singularity does not vanish. We know that there are no logarithmic singularities, therefore we merely need to ensure that dp ∈ 2π Z, (4.22) Aa

where the cycles Aa surround a cut Ca , see Fig. 5. As was shown in [41] we can even demand that all A-cycles are zero, which conveniently reduces the number of cuts. Assume first Ca connects sheets 1, 2 or sheets 3, 4.18 Then there is another cut Cb as the image of Ca under (4.3) between the same two sheets. The values of the A-cycles are related dp1 = − dp2 = dp1 . (4.23) Aa

Ab =1/Aa

Ab

The two signs flips are explained as follows: The first is related to the symmetry (4.3) and the second to changing the sheet back to 1. For a cut which connects sheets 2, 3, the mirror image will connect sheets 1, 4. In this case we have 18

In Fig. 3 we have illustrated how the sheets are connected by the cuts.

640


Aa

dp2 = −

1/Aa

dp1 = −

Ab

(4.24)

dp1 .

In particular this means that there is only one constraint dp = 0

(4.25)

Aa

for each pair of cuts. Moreover, the cycle around all cuts on sheet 1 is just the negative of the corresponding one on sheet 2, equivalently for sheets 3 and 4. As there are no further single poles on any sheet, the cycle around all cuts can be contracted and must be zero. The total number of constraints from A-cycles is thus 21 A − 2. 4.7. B-periods. We know that the set of eigenvalues exp(ipk (x)) of S+ (x) depends analytically on x. Their labeling k = 1, 2, 3, 4, however, is artificial. This allows for the presence of cuts Ca where the pk permute, see Sect. 3.5. In addition they can also shift by multiples of 2π without effect on exp(ipk (x)). This shift can be expressed through the integral of dp along the curve Ba which connects the points x = ∞ on the involved sheets through the cut Ca . We know that p(x) is analytic along Ba except at the intersection of Ba with Ca . Moreover we assume that p(∞) = 0 on both sheets, therefore the period

dp ∈ 2πZ (4.26) Ba

describes the shift in p(x) at Ca and must be a multiple of 2π . Note that the symmetry x → 1/x does not map B-periods directly to B-periods due to the explicit reference to the point x = ∞. First, we should therefore consider the integral

∞

dpk = pk (∞) − pk (0) = −pk (0),

(4.27)

0

where we have made use of our choice pk (∞) = 0. From (4.3) it follows that p1,2 (0) = −p3,4 (0) = 2π n0

(4.28)

which is the momentum constraint. It reduces the number of degrees of freedom by one, because n0 must be integer. Now consider a B-period between sheets 1, 2 or sheets 3, 4. Due to the symmetry

Ba

dp =

xa

∞

∞

dp1 +

dp2 = −

xa

∞

=− 0

dp2 −

xb ∞

= p2 (0) − p1 (0) +

dp2 −

1/xa

0

∞ xb

xb ∞

dp1 +

dp2 −

dp1 − ∞ xb

0

dp1 1/xa 0

∞

dp2 =

(4.29)

dp1

Bb

dp,


641

we see that the cycles Ba and Bb have the same value. Equivalently, for Ba between sheets 2, 3 which is related to Ba between sheets 1, 4,

xa

∞

1/xa

0

dp = dp2 + dp3 = − dp1 − dp4 Ba

∞

0

xa

∞

=−

dp1 −

0

xb ∞

= p1 (0) − p4 (0) −

dp1 −

∞ xb

xb ∞

1/xa

dp1 −

dp4 − ∞

0 ∞

(4.30)

dp4

dp4 = 4π n0 −

xb

Bb

dp.

Note that by demanding that both values of cycles Ba and Bb are multiples of 2π , it follows that n0 is integer.19 So by fixing

dp = 2πna , (4.31) Ba

we automatically determine the value of mirror period Bb . Consequently, the B-periods together with the momentum constraint fix 21 A + 1 coefficients. 4.8. Fillings. The polynomial F (y, x) has 8A + 22 coefficients in total, see (4.14). Of them 4A + 9 are incompatible with the symmetry (4.17) and another 4 are constrained by enabling non-trivial permutations of the yk , see (4.19). The overall normalization of F (y, x) is irrelevant for the F (y, x) = 0, this removes one degree of freedom. The discriminant R, (4.20), has 5A + 8 non-trivial pairs of roots related by the symmetry (4.18). These should all have even multiplicity, (4.11), which fixes 25 A + 4 coefficients. The residues of the poles and absence of logarithmic singularities at x = ±1 leads to 5 constraints. The A-cycles and B-periods yield 21 A − 2 and 21 A + 1 constraints, respectively. In total there are 21 A continuous degrees of freedom remaining. These can be used to assign one filling to each pair of cuts. We define the filling of a cut Ca as √ √ 1 λ λ 1 x + dx 1 − 2 p(x) = dp. (4.32) Ka = − 2 8π i Aa x 8π 2 i Aa x The second form which directly relates to dp is obtained by partial integration. In addition to the fillings we define one further similar quantity which we call the “length” L=D+

√ A/2 λ dx p1 (x) + p2 (x) 2 2 4π i x a=1 Aa

√ A/2 λ 1 dp1 (x) + dp2 (x) . =D− 2 4π i x a=1 Aa

(4.33)

19 This is different from spin chain for which there are states with non-integer total momentum which are perfectly well-defined.

642


A/2 Note that the sum a=1 extends only over one cut from each pair of cuts related by the inversion symmetry.20 The length is related to the fillings by the constraint

n0 L =

A 1 /2

n1,a K1,a +

A 2 /2

a=1

n2,a K2,a +

a=1

A 3 /2

n3,a K3,a =

a=1

A/2

na Ka

(4.34)

a=1

which means that among {L, Ka } there are only A independent continuous parameters: A − 1 independent fillings Ka and one expansion parameter λ/L2 . This matches the counting for one-loop gauge theory since the loop counting parameter λ/L2 is absent. Note that the case n0 = 0 forces us to view the length as fundamental rather than depending on A independent fillings. To derive the constraint, consider the integral √ 4 λ 1 2 dx 1 − 2 pk (x). 64π 3 i ∞ x

(4.35)

k=1

On the one hand it is immediately zero due to pk (x) ∼ 1/x at x → ∞. On the other hand we can split up the contour of integration around the singularities and cuts and obtain the constraint √ √ 4 A λ λ 1 2 0=− dx 1 − 2 pk (x) p (0) + p2 (0) n0 + 4π 1 64π 3 i Aa x a=1 k=1 = n0 L − n0

A 2 /2

K2,a −

a=1

1 2

A a=1

na Ka = n0 L −

A/2

na K a .

(4.36)

a=1

We have made use of the identity √ 4 λ 1 2 dx 1 − 2 pk (x) = − 21 na Ka , 64π 3 i Aa x k=1

(4.37)

which one gets after pulling the contour Aa tightly around the cut Ca . Furthermore, the value of p1 (0) + p2 (0) follows from the residue at x = ∞, see the following subsection. Finally, the filling and mode number of the inverse cut C2,a = 1/C2,a are given by K2,a = −K2,a , n2,a = 2n0 − n2,a , cf. Sect. 4.6, 4.7. Similarly, the fillings and mode numbers for the other types of cuts are invariant under inversion.

4.9. Global charges. Now let us compute the global charges at x = ∞, see (4.5). These are obtained as the cycles of pk (x) dx around x = ∞ which we can also write as the sum of cycles around all singularities on the same sheet (cf. Fig. 3 for the structure of cuts) 20 This definition of length is ambiguous but in the comparison to gauge theory it becomes clear which cut to select from each pair. A potential self-symmetric cut should be counted with weight 1/2.


√ λ dx p − p = 1 2 8π 2 i ∞ √ λ = dx p2 − p3 = 2 8π i ∞

r1 = r2

= L+

A 1 /2

K1,a − 2

A 2 /2

a=1

r3

√ A A 2 /2 1 /2 λ K2,a − 2 K1,a , p1 − p2 (0) = 4π a=1 a=1 √ λ p − p1 (0) 4π 4

K2,a +

a=1

643

A 3 /2

K3,a ,

a=1

√ √ A A 2 /2 3 /2 λ λ = dx p − p − p K − 2 K3,a . = (0) = p 3 4 2,a 4 8π 2 i ∞ 4π 3 a=1

a=1

(4.38) Here we have made use of the symmetry to write our findings in terms of the fillings Kk,a . As in gauge theory, the fillings are directly related to the Dynkin labels rk and the length L. Let us also note the particularly useful combination √

1 2 r1

+ r2 +

1 2 r3

λ dx p1 + p2 2 8π i ∞ √ A 2 /2 λ =− p1 (0) + p2 (0) = L − K2,a . 4π

=

(4.39)

a=1

4.10. Comparison to gauge theory. Here we will show that the algebraic curve of the SYM theory in the so(6) sector coincides with the algebraic curve of the string sigma model on R × S 5 at one loop, in accordance with the proposal of [73]. Let us compare the analytical data defining the curves in the Frolov-Tseytlin limit √ λ/L2 → 0. We will define for convenience a rescaled variable u = ( λ/4π L) x, which makes it similar to the spectral parameter u of Sect. 2 (in Sect. 5.6 we refine the relationship for higher loops). In this limit, for each pair of mutually symmetric branch points, (ua , λ/16π 2 L2 ua ), one goes to zero and one remains finite. This means half of the cuts approach x = ∞ and half of them approach x = 0. We will use this distinction to A/2 select half of the cuts: The sums a=1 introduced in Sect. 4.8 refer to the long cuts with x → ∞ which remain finite in the u-plane. The other half of the cuts becomes infinitely short in the limit and needs to be handled separately. We are thus left with half of the cuts having no symmetry with respect to inversion, as in the case of the SYM curve. Both curves enjoy the following common properties: • It is easy to see that Eq. (4.8) becomes (2.58) in this limit, for similar definitions of y(u). • Four sheets for the quasi-momentum (which we call p(u) even after rescaling) in the u-projection, connected by finite cuts, as discussed in Sects. 2, 4. • The same condition of zero A-cycles. • The same set of Eqs. (2.43) and (5.31) defining the symmetric part of p(u) on the cuts and hence the same B-periods (2.64) of the cuts. • The same asymptotics for p(u) at u → ∞ for all the sheets, given through the SU(4) charges by the formulas (2.53) and (4.5), when we rescale rk → Lrk in (4.5).

644


To understand the expansion of p(u) at u = 0 we need to take the inverse cuts into account which approach u = 0. For this purpose, we shall define a contour C in the x-plane which encircles the poles at x = ±1 and all the short cuts 1/Ca . Equivalently, this may be considered a contour which excludes x = ∞ and all the long/finite cuts Ca . After rescaling C merely encircles the point u = 0 in the u-plane which can be used to obtain the expansion of p(u) as follows: ∂ r−1 pk 4πL r−1 1 dx (0) = √ pk (x) r . (4.40) ∂ur−1 2πi x λ C Using the identities and definitions in Sect. 4.8,4.9 we find the useful relations √ λ 1 L= dx 1 − 2 p1 (x) + p2 (x) , 2 8π i C x √ λ 1 D= p1 (x) + p2 (x) , dx 1 + 2 2 8π i C x dx 1 2π n0 = p1 (x) + p2 (x) . 2πi C x

(4.41)

This determines the expansion of p1 (u) + p2 (u) as follows: D+L 1 8π 2 L2 D − L + 2πn0 + u + O(u2 ). 2L u λ L The residues of the poles at u = 0 are obtained in a similar manner 1 p1,2 (u) = −p3,4 (u) = 21 + O(λ/L2 ) + ··· , u p1 (u) + p2 (u) =

(4.42)

(4.43)

where O(λ/L2 ) represents various integrals which are suppressed in the one-loop approximation. The two curves therefore have • The same poles at u = 0, p1,2 (u) = −p3,4 = 1/2u + O(u0 ), cf. (2.50) in gauge theory. The extra poles at zero for the sigma model come from the poles at √ u = ± λ/4π L when λ/L2 → 0. The small cuts contribute to the residue only at higher loop orders. • The same expansion p1 (u) + p2 (u) = 1/u + 2πn0 + uE + · · · at u = 0. We make use of D = L + O(λ/L) to match the residue of 1/u. Integrality of n0 corresponds to cyclicity of the trace U = 1, or (2.46,2.47) in SYM. The anomalous dimension E = (D − L)/Lg 2 with g 2 = λ/8π 2 L2 extracted from both curves also coincides. These properties define the one-loop algebraic curves and their relation to the physical data unambiguously and consequently they coincide. At two loops the full proof of the equivalence of two curves was only for the su(2) sector [41], the only one where the two-loop dilatation operator is actually calculated [53]. But we can borrow the idea of [73] where the closure of the so(6) sector was demonstrated in higher loops in the classical limit and the Bethe equation in the second loop was guessed. We can do the comparison of the curves at two loops along the same guidelines. In the next section it will be done using the Bethe equations. As we also know [47], at three loops the curves do not match already in the su(2) sector, most probably, due to yet unidentified non-perturbative corrections arising on the way from the weak to strong coupling.


645

Fig. 6. Dynkin diagram of SO(m) for even and odd m

5. Bethe Ansatz for the Sigma-Model on R × S m−1 Having constructed the algebraic curve for the classical string on R × S 5 and having convinced ourselves that we have identified all relevant parameters, we proceed by constructing an integral representation of the curve (for all R×S m−1, m 5). The obtained equations are similar to the Bethe equations for integrable spin chains in the thermodynamic limit which in fact form a Riemann-Hilbert problem. We finally compare the obtained equations to the one derived for gauge theory and find agreement up to two loops. 5.1. Simple roots. To reveal the group theory structure of the equations, we will now introduce singular resolvents H˜ k (x) which can be associated to the simple roots of so(m). See Fig. 6 for the Dynkin diagram of the algebra and the labelling of the simple roots. They are related to the quasi-momenta qk (x) by H˜ k =

k

(5.1)

qj ,

j =1

with the exceptions for the simple roots associated to spinors for even m, H˜ [m/2]−1 =

[m/2]−1 j =1

1 2 qj

− 21 q[m/2] ,

H˜ [m/2] =

[m/2]−1 j =1

1 2 qj

+ 21 q[m/2] ,

(5.2)

and for odd m H˜ [m/2] =

[m/2] j =1

1 2 qj .

(5.3)

Let us now collect the facts about the analytic properties of H˜ k (x). First of all we know that the expansion at x = ∞ is related to the representation [s1 , s2 , . . . ] of the state. Using the Cartan matrix Mkj of SO(m) (see App. C) it can be summarized as [m/2] 1 −1 4πsj ˜ Hk (x) = Mkj √ + O(1/x 2 ). x λ

(5.4)

j =1

Secondly, we have derived the singular behavior at x = ±1,21 H˜ k (x) = −1 We could also write Mk1 as vector representation. 21

−1 D 1 2πMk1 + O 1/(x ∓ 1)0 . √ x∓1 λ

j

(5.5)

−1 V Mkj Vj , where VjV = (1, 0, 0, . . . ) are the Dynkin labels of the

646


Finally, we will use the assumption (3.56) for the symmetry of H˜ k (x) under the map x → 1/x,22 −1 ˜ −1 H˜ k (1/x) = H˜ k (x) − 2Mk1 H1 (x) + 4π n0 Mk1 .

(5.6)

5.2. Integral representation. We make properties (5.5) and (5.6) manifest by defining −1 H˜ k (x) = Hk (x) + Hk (1/x) − 2Mk1 H1 (1/x) 4πD −1 1 −1 −1 + + 2π n0 Mk1 , √ Mk1 + ck − c1 Mk1 x − 1/x λ

(5.7)

where the ck ’s are a set of constants. The resolvents Hk (x) are assumed to be analytic except at a collection of branch cuts Ck and approach zero at x = ∞. Note that this representation of H˜ k (x) is ambiguous. We can add to Hk an antisymmetric function −1 Hk (x) → Hk (x) + fk (x) − fk (1/x) + 2Mk1 f1 (1/x),

(5.8)

this has no effect on the physical function H˜ k (x). Let us introduce the density ρk (x) which describes the discontinuity across a cut ρk (x) =

1 − 1/x 2 Hk (x − i ) − Hk (x + i ) 2πi

for x ∈ Ck .

(5.9)

The factor of 1 − 1/x 2 was introduced for later convenience and will allow the interpretation of ρk (x) as a density. The apparent pole at x = 0 is irrelevant as long as the cuts do not cross this point. We could also demand positivity of the density, dx ρk (x) > 0. This would fix the position of the cuts Ck in the complex plane, but will not be essential for the treatment of the classical sigma model. From ρk (x) we can reconstruct the function Hk (x),

dy ρk (y) 1 Hk (x) = . (5.10) 2 1 Ck − 1/y y − x 5.3. Asymptotic behavior. We should now relate the SO(m) representation of a state to the cuts and densities. For that purpose, we note the expansion of the resolvents Hk (x) at x = ∞, 1 4π Kk Hk (x) = − (5.11) √ + Hk (0) + O(1/x 2 ), x λ and x = 0 Hk (x) = Hk (0) + x Hk (0) + O(x 2 ), where we have defined the normalizations or fillings of the densities, √ λ Kk = dy ρk (y). 4π Ck 22

Again, H˜ 1 =

j

H˜ j VjV could be written in a more ‘covariant’ way.

(5.12)

(5.13)


647

For H˜ k (x) we find the asymptotic behavior at x → ∞, −1 −1 −1 H˜ k (x) = ck − c1 Mk1 + 2πn0 Mk1 + Hk (0) − 2Mk1 H1 (0) 4π Kk 1 4πD −1 −1 − √ + + O(1/x 2 ) . √ Mk1 − 2H1 (0)Mk1 x λ λ

(5.14)

We compare this to (5.4) and find the relation between fillings and Dynkin labels

[m/2] 4πD −1 4π sj Mkj √ − 2H1 (0) − √ , λ λ j =1

(5.15)

−1 −1 −1 − 2πn0 Mk1 − Hk (0) + 2Mk1 H1 (0). ck = c1 Mk1

(5.16)

−1 Kk = Mk1

as well as the constants

In fact, this equation for k = 1 cannot be solved for c1 , it drops out. This leads to an additional condition for the resolvents, the momentum constraint H1 (0) = 2πn0 ,

(5.17)

whereas c1 is not fixed. When substituting the constants into (5.7) we obtain H˜ k (x) = Hk (x) + Hk (1/x) − Hk (0) −1 + Mk1 −2H1 (1/x) + 2H1 (0) +

1 4π D √ x − 1/x λ

.

(5.18)

Now the expansion of the functions H˜ k is fixed at the points x = ∞ and x = 0 and it turns out that the ambiguity (5.8) must not modify Hk (x) at x = ∞.23 5.4. Bethe equations. The singular resolvents H˜ k now satisfy the desired symmetries and expansions at specific points. They however have branch cuts along the curves Ck . These must not be seen in the transfer matrices, which are analytic except at the special points. This leads us to the Bethe equations, which are manifestations of the analyticity conditions for the monodromy matrix, see Sect. 3.5, [m/2]

Mkj H /˜ j (x) = 2πnk,a ,

for x ∈ Ck,a .

(5.19)

j =1

As explained in Sects. 2.4, 3.5, they ensure that across a cut only the labelling of sheets and the branch of the logarithm changes. The slash through a resolvent implies a principal part prescription, H / k (x) = 21 Hk (x + i ) + 21 Hk (x − i ). 23

Hk (x) was assumed to be zero at x = ∞ anyway.

(5.20)

648


Here we have split up the curves Ck into their connected components Ck,a . For each connected curve we have introduced a mode number nk,a due to the allowed shift by multiples of 2π i in the exponent. When we substitute (5.18) the Bethe equations read 2π nk,a =

[m/2]

Mkj H / j (x) + Hj (1/x) − Hj (0)

j =1

+ δk1 −2H1 (1/x) + 2H1 (0) +

1 4π D √ x − 1/x λ

,

for x ∈ Ck,a . (5.21)

Note the explicit appearance of the dimension/energy D which constitutes the physical quantity of main interest. For a given set of mode numbers nk,a and fillings, √ λ Kk,a = dy ρk (y), (5.22) 4π Ck,a Eqs. (5.21) should only have a solution if D has the appropriate value. The Bethe equations of the spin chain in Sect. 2.4 are qualitatively different: They should always be soluble and the dimension is subsequently read off from (2.46, 2.48). It is useful to go to the u-plane which is related to the x-plane by [48] x(u) = 21 u + 21 u2 − 4 , u(x) = x + 1/x. (5.23) We can introduce a resolvent in the u-plane by

dy ρk (y) dv ρk (v) H¯ k (u) = = . y + 1/y − u v−u

(5.24)

Note that ρk (x) transforms as a density, i.e. dx ρk (x) = du ρk (u). It is related to a symmetric combination of the resolvents in the x-space Hk (x) + Hk (1/x) = H¯ k (x + 1/x) + Hk (0). (5.25) This allows us to write the Bethe equations in the u-plane [m/2]

/¯ j (u) + δk1 Fstring (u) = 2πnk,a Mkj H

for u ∈ C¯k,a

(5.26)

j =1

with 4πD (5.27) √ + 2H1 (0) − 2H1 1/x(u) . λ u2 − 4 It might be favorable to replace the dimension D, which is intended to be the final result of the computation, by some known quantities. We can rewrite the definition of the length (4.33) as an energy formula √ √ λ dy ρ1 (y) 1 λ D =L+ H (0) . =L+ (5.28) 2 2 2π C1 1 − 1/y y 2π 1 Fstring (u) = √

1

When we substitute this in (5.27) we obtain Fstring (u) = √

2H (0) 4πL − 2H1 1/x(u) . √ + 2H1 (0) + √ 1 2 2 λ u −4 u −4 1

(5.29)


649

5.5. The sigma-model on R×S 5 . Let us now apply our results to the case m = 6, i.e. the sigma model on R × S 5 . Here we shall adopt a SU(4) notation instead of the one for SO(6). The benefit of SU(4) is that it is manifestly a subgroup of SU(2, 2|4), the full supergroup of the superstring on AdS5 × S 5 . The change merely amounts to swapping the labels of the first two simple roots, cf. Fig. 4. We introduce the singular resolvents ˜ k (x) corresponding to the simple roots of SU(4) by G ˜ 1 (x) = H˜ 2 (x), G ˜ 2 (x) = H˜ 1 (x), G ˜ 3 (x) = H˜ 3 (x). G

(5.30)

We also interchange labels 1, 2 for the densities ρk and fillings Kk . The Bethe equations are written as ˜ 1 (x) − G ˜ 2 (x) = /p1 (x) − /p2 (x) = 2π n1,a , 2/ G ˜ 2 (x) − G ˜ 1 (x) − G ˜ 3 (x) = /p2 (x) − /p3 (x) = 2π n2,a , 2/ G ˜ 3 (x) − G ˜ 2 (x) = /p3 (x) − /p4 (x) = 2π n3,a , 2/ G

x ∈ C1,a , x ∈ C2,a , (5.31) x ∈ C3,a .

Now one can draw the Riemann sheets picture as in Sect. 2.4. The Riemann surface ˜ k describe consists of four sheets each of which corresponds to pk while resolvents G how to connect the sheets with cuts. The main difference is that due to the inversion symmetry all the cuts appear in pairs as depicted in Fig. 3. 5.6. Comparison to gauge theory. Let us now consider √ the limit where the Dynkin labels rk and the dimension D are large with respect to λ. For this purpose we rescale according to 4πL {x, u} → √ {x, u}, λ

{D, rk , Kk } → L{D, rk , Kk }

(5.32)

while keeping ρk (x), Gk (x) fixed. Here L is defined to be the limiting value of D (before rescaling) at λ = 0 corresponding to the classical dimension in gauge theory.24 For convenience, we define the effective coupling constant g as g2 =

λ g2 N = YM2 2 . 2 2 8π L 8π L

(5.33)

The Bethe equations (5.26) are left invariant, but the function Fstring (u) changes to25 2 g 2 G2 (0) 1 Fstring (u) = + 2G2 (0) + − 2G2 g /2x(u) , (5.34) u2 − 2g 2 u2 − 2g 2 24 The ‘length’ L was conjectured to be an action variable in [94]. If true, it would be interesting to relate it to our definition. See also [40] on the definition of ‘length’ in the coherent approach. 25 This equation along with the generic form of the Bethe equations (5.26) was proposed independently by M. Staudacher [79]. He also showed that the solutions discussed in [31] for this deformation of the equations yield precisely the energies computed from the string equations of motion [26, 19, 73, 40]. We would like to thank him for insightful discussions.

650

where now x(u) = 21 u +

N. Beisert, V.A. Kazakov, K. Sakai 1 2

u2 − 2g 2 . When we expand in g we obtain

Fstring (u) = Fgauge (u) + O(g 4 ).

(5.35)

This means that the functions Fstring (u) and Fgauge (u) agree up to and including order g 2 corresponding to two loops for the scaling dimension D. We have thus demonstrated the generic two-loop matching of scaling dimensions in gauge theory26 and energies of spinning strings in the SO(6) sector.27 This complies with the one-loop results of [40] in the coherent state approach to spinning strings [35–39] and also with the matching of integrable charges in special cases [95].

6. Discussion In this work we continued the investigation of integrability of the multi-color N = 4 SYM theory and its close relation (and hopefully equivalence) to the AdS5 × S 5 string sigma model. The general solutions of the one-loop SYM theory and of the classical sigma model, constructed here in the R × S 5 or so(6) sector, give, as expected, the same result in the weakly coupled region of the classical, BMN limit. Elsewhere, the algebraic curves of the two models appear to have a very similar structure and differ only in the details. We hope that a quantized version of the sigma model will reproduce the known SYM perturbative data precisely and give in addition the complete non-perturbative information on the gauge theory, compatible with these perturbative data. We see the most natural way to prove this complete AdS/CFT duality in construction of the full algebraic curve of the model, with the following quantization based on this curve. Often the quantization means an appropriate discretization of the model and the curve gives a hint on the right procedure. For example, the matrix model with a finite size N of a matrix, is often completely defined by its large-N algebraic curve and can be considered as its quantum counterpart. The other example is the discrete Bethe ansatz equations, as those considered here, which provide the right quantization of the classical algebraic curve. This procedure of quantization is carried out in one-loop here. There were recently some interesting attempts to find the quantum version of the AdS5 × S 5 sigma model [68], though it is too early to claim that we are close to the whole resolution of this formidable problem. As a next important step in this program we would consider the generalization of the present construction, to the algebraic curve of the one-loop SYM theory for the full dilatation Hamiltonian of [50], on the one hand, and of the full AdS5 × S 5 classical sigma model, on the other hand. The present paper, together with [42, 41], provides most of the necessary technique for the completion of this task. Acknowledgements. We would like to thank Gleb Arutyunov, Andrei Mikhailov, Arkady Tseytlin, Kostya Zarembo and especially Matthias Staudacher for helpful discussions and remarks. N. B. would like to thank the Ecole Normale Supérieure and the Kavli Institute for Theoretical Physics for kind hospitality during parts of this project. The work of N. B. is supported in part by the 26 By higher-loop gauge theory we mean the higher-loop Bethe ansatz for the so(6) sector in the thermodynamic limit [73]. It has not yet been shown that this ansatz indeed matches gauge theory at two or higher loops. 27 In both theories we have focussed on the low-lying modes of the spectrum. States which have no expansion in the effective coupling g are disregarded.


651

U.S. National Science Foundation Grants No. PHY99-07949 and PHY02-43680. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. V. K. would like to thank the Princeton Institute for Advanced Study for the kind hospitality during a part of this work. The work of V. K. was partially supported by European Union under the RTN contracts HPRN-CT-2000-00122 and 00131 and by NATO grant PST.CLG.978817. The work of K. S. is supported by the Nishina Memorial Foundation.

A. Antisymmetric Transfer Matrices of su(4) There is a nice formula to obtain expressions for the transfer matrices for the Bethe ansatz in all totally antisymmetric products of the fundamental representation of su(m), see [75]. In this appendix we shall present it for our case of interest, su(4). It allows us ¯ It is based on a differential operator u4 , to obtain the transfer matrices in 4, 6, 4.

u4

R1 (u) V (u + 2i) = exp(i∂u ) − R1 (u + i) V (u + 3i2 ) R2 (u − 2i ) R1 (u + i) V (u + i) · exp(i∂u ) − R2 (u + 2i ) R1 (u) V (u + 2i )

R3 (u − i) R2 (u + 2i ) V (u − i) · exp(i∂u ) − R3 (u) R2 (u − 2i ) V (u − 2i ) R3 (u) V (u − 2i) · exp(i∂u ) − . R3 (u − i) V (u − 3i2 )

(A.1)

The operator is slightly modified from [75] to accommodate for a non-fundamental spin representation and a different normalization of rapidities. When this operator is expanded in powers of exp(i∂u ), which shifts u by i, it should yield

u4 =

V (u + 2i)V (u + i)V (u − i)V (u − 2i) V (u +

i i 3i 3i 2 )V (u + 2 )V (u − 2 )V (u − 2 )

− exp( 2i ∂u ) + exp(i∂u )

V (u + 3i2 ) V (u − 3i2 ) T ¯ (u) exp( 2i ∂u ) V (u + i) V (u − i) 4 V (u) V (u +

V (u) i 2)

V (u − 2i )

T6 (u) exp(i∂u )

− exp( 3i2 ∂u ) T4 (u) exp( 3i2 ∂u ) + exp(4i∂u ).

(A.2)

652


Here we can read off the expressions for TR (u), they agree with (2.23, 2.25, 2.26). Alternatively one can use the conjugate operator R3 (u) V (u + 2i) 4¯ u = exp(i∂u ) − R3 (u + i) V (u + 3i2 ) R2 (u − 2i ) R3 (u + i) V (u + i) · exp(i∂u ) − R2 (u + 2i ) R3 (u) V (u + 2i ) R1 (u − i) R2 (u + 2i ) V (u − i) · exp(i∂u ) − R1 (u) R2 (u − 2i ) V (u − 2i ) R1 (u) V (u − 2i) · exp(i∂u ) − (A.3) R1 (u − i) V (u − 3i2 ) which expands as follows: V (u + 2i)V (u + i)V (u − i)V (u − 2i)

¯

u4 =

V (u +

3i i i 3i 2 )V (u + 2 )V (u − 2 )V (u − 2 ) V (u + 3i2 ) V (u − 3i2 ) T4 (u) − exp( 2i ∂u ) V (u + i) V (u − i)

+ exp(i∂u )

V (u)

V (u)

V (u + 2i ) V (u − 2i )

exp( 2i ∂u )

T6 (u) exp(i∂u )

− exp( 3i2 ∂u ) T4¯ (u) exp( 3i2 ∂u ) + exp(4i∂u ).

(A.4)

In the thermodynamic limit, see Sect. 2.3, we find the following limits for the operators: u4 → exp(i∂u /L) − exp(ip1 ) · exp(i∂u /L) − exp(ip2 ) · exp(i∂u /L) − exp(ip3 ) · exp(i∂u /L) − exp(ip4 ) (A.5) and the conjugate one ¯ u4 → exp(i∂u /L) − exp(−ip4 ) · exp(i∂u /L) − exp(−ip3 ) · exp(i∂u /L) − exp(−ip2 ) · exp(i∂u /L) − exp(−ip1 ) .

(A.6)

B. Higher Charges of the Sigma Model Here we shall continue the expansion of qk at the singular points x = ±1 to higher orders. Note, first of all, that the straight perturbative diagonalization fails if there are degenerate eigenvalues. In the case at hand the eigenvalue zero is indeed degenerate. However, the zero subspace can be decoupled completely from the non-zero eigenvectors in perturbation theory. This is not a problem, because the zero subspace does not


653

display singular behavior by definition; there the monodromy matrix behaves as for any other point x and we cannot expect to be able to find a simple expression for qk (x), k = 1 at x = ±1. For q1 (x) the situation is different; it starts with a pole whose residue is non-degenerate. We would now like to find a solution of the equation L(x, σ )V (x, σ ) = if (x, σ )V (x, σ )

(B.1)

such that the eigenvalue f (x, σ ) is singular at x = +1. As explained around (3.46), the leading-order eigenvector V (x, σ ) √ is an eigenvector of j+ (σ ). It therefore must be

we find X

− iX

a linear combination of X and X+ := ( λ/D)∂+ X; 2 + with eigenvalue √ f (x, σ ) = D/ λ(x − 1) + · · · . When we substitute this in the above equation for L we can solve for the subleading terms. In the first few orders we find the eigenstate 1

− iX

+ + − i X

V (x) = X 2 + − 2 X++ (x − 1) 2 1 i

+ + i X

+ + 8i X 4 3+ − 8 (X·X4+ )X + 4 (X·X4+ )X+ (x − 1) i 3 1 3 1

+ − i X

+ − 16 X 8 3+ + 8 X4+ + 16 (X·X4+ )X++ + 20 (X·X5+ )X+ (x − 1) + O (x − 1)4 , (B.2) √ n X.

n+ is defined as X

The value q1 (x) is now given as the

n+ := ( λ/D)n ∂+ where X integral of f (x, σ ) over the closed string

q1 (x) =

2π

dσ f (x, σ )

2π 1 D dσ + =√ x−1 λ 0 0

+(x −1)

1 2

2

X

+− + 1 X·

X

4+ + (x − 1) − 18 + 41 X· 8 1 1 16 − 8 X·X+−

−

1 16 X·X4+

+O (x − 1)

3

. (B.3)

We make use of the following two identities: Tr jV,+ jV,− =

8D 2 X·X+− , λ

Tr ∂± jV,± ∂± jV,± =

8D 4

X

4± 1 − X· 2 λ

(B.4)

to express the expansion of q1 (x) in terms of the currents jV , q1 (x) = √

3/2 Q πD λ3/2 Q± ± 2λ ∓ (x ∓ 1) + O (x ∓ 1)3 , ± √ + (x ∓ 1) 3 3 64D 128D λ (x ∓ 1) λ (B.5)

2π D

with

Q± =

2π

dσ 0

2D 2 Tr jV,+ jV,− − Tr ∂± jV,± ∂± jV,± . λ

(B.6)

Here we have also included the expansion around x = −1. The charges Q± are the first two non-trivial local commuting charges of the sigma model.

654


C. Cartan Matrices The Cartan metric for so(m) are given by   +2 −1 . .    −1 . . . .    .. ..   Mj k =  . . −1 ,    −1 +2 −1 −1    −1 +2 −1 +2

for m even

(C.1)

and by 

Mj k

+2 −1 .   −1 . .  .. = .  

 ..

.

..

. −1 −1 +2 −2 −2 +4

   ,  

for m odd.

The inverse metric is given by   1 1 1 1 1 ··· 1 2 2  1 2 2 ··· 2 1 1   3 3  1 2 3 ··· 3 2 2     . . . . . . . −1 . . . . . . . ,  . . . . Mj k =  . . .   1 2 3 · · · 1 (m − 4) 1 (m − 4) 1 (m − 4)    2 4 4  1 3 1 1 1  2 1 2 · · · 4 (m − 4) 8 (m − 0) 8 (m − 4)  1 2

and by

1

···

3 2



Mj−1 k

1 1  1  =  .. .  1 1 2

1 2 2 .. .

(C.2)

for m even

(C.3)

1 1 1 4 (m − 4) 8 (m − 4) 8 (m − 0)

1 2 3 .. .

··· ··· ··· .. .

1 2 3 .. .

1 2



    , ..  .  1 1 2 3 · · · 2 (m − 3) 4 (m − 3)  1 23 · · · 41 (m − 3) 18 (m − 1)

For so(6) this reduces to   +2 −1 −1 , Mj k =  −1 +2 −1 +2

1 3 2



for m odd.

1

1 Mj−1 k = 2 1 2

1 2 3 4 1 4

1  2 1  4 3 4,

(C.4)

(C.5)

while for the su(4) notation we need to permute the first two rows and columns 3 1 1   4 2 4 +2 −1 1 1 Mj k =  −1 +2 −1  , (C.6) Mj−1 k =  2 1 2 . −1 +2 1 1 3 4 2 4


655

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Communications in


The Algebraic Curve of Classical Superstrings on AdS5 × S 5 N. Beisert1 , V.A. Kazakov2, , K. Sakai2 , K. Zarembo3, 1

Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USA. E-mail: [email protected] 2 Laboratoire de Physique Théorique de l’Ecole Normale Supérieure et l’Université Paris-VI, 75231 Paris, France. E-mail: [email protected]; [email protected] 3 Department of Theoretical Physics, Uppsala University, 751 08 Uppsala, Sweden. E-mail: [email protected] Received: 6 May 2005 / Accepted: 2 October 2005 Published online: 4 March 2006 – © Springer-Verlag 2006

Abstract: We investigate the monodromy of the Lax connection for classical IIB superstrings on AdS5 × S 5 . For any solution of the equations of motion we derive a spectral curve of degree 4 + 4. The curve consists purely of conserved quantities, all gauge degrees of freedom have been eliminated in this form. The most relevant quantities of the solution, such as its energy, can be expressed through certain holomorphic integrals on the curve. This allows for a classification of finite gap solutions analogous to the general solution of strings in flat space. The role of fermions in the context of the algebraic curve is clarified. Finally, we derive a set of integral equations which reformulates the algebraic curve as a Riemann-Hilbert problem. They agree with the planar, one-loop N = 4 supersymmetric gauge theory proving the complete agreement of spectra in this approximation. 1. Introduction and Overview Strings in flat space have been solved a long time ago. The solution of the classical equations of motion is straight forward and obtained by a Fourier transformation, or mode decomposition, of the world sheet. The string is then represented by a collection of independent harmonic oscillators, one for each mode and orientation in target space. The oscillators are merely coupled by the Virasoro and level-matching constraints. The conserved, physical quantities of the string are the absolute values of oscillator amplitudes. Quantization of this system essentially poses no problem. The harmonic oscillators are excited in quanta and the amplitudes turn into integer-valued excitation numbers. Maldacena’s conjecture [1–3] however brought about special attention on strings in curved target spaces with ‘RR-flux’, in particular IIB superstrings on AdS5 × S 5 . There, a solution and quantization is much more involved due to the highly non-linear

Membre de l’Institut Universitaire de France Also at ITEP, Moscow, Russia

660

N. Beisert, V.A. Kazakov, K. Sakai, K. Zarembo

nature of the string action [4]. A direct quantization of the world sheet theory is furthermore obstructed by conformal and kappa symmetry which require gauge fixing. This introduces a number of additional terms and usually makes the problem intractable. One path to quantization is related to the maximally supersymmetric plane-wave background [5, 6] and the correspondence to gauge theory [7]. In this background the solution and quantization closely resembles its flat space counterpart [8, 9]. The full AdS5 × S 5 background may be regarded as a deformation of plane waves. Following this idea, one can obtain a quantum string on AdS5 × S 5 in a perturbation series around plane waves [10]. This approach has yielded several important insights into the quantum nature of the string, but there are drawbacks: The perturbative expansion is very involved, the first order is feasible [11–14], but beyond there are no definite answers available yet. Even if this problem might be overcome, still we would be limited to a certain region of the parameter space of full AdS5 × S 5 which is insensitive to global aspects. Another approach to strings in curved space is to consider classical solutions, see e.g. [15–19]. For these solutions with large spins one can show that quantum effects are suppressed and already the classical solution yields a good approximation for the full energy. Even more excitingly, Frolov and Tseytlin discovered that many of these spinning string solutions have an expansion which is in qualitative agreement with the loop expansion of gauge theory [20]. Their conjecture of a quantitative agreement has been confirmed in several cases in [21, 22] and many more works since,1 see [24–28] for reviews of the subject. Finding exact solutions is not trivial, the complexity of the functions increases with the complexity of the solution. The functions that occur are of algebraic, elliptic or hyperelliptic type and many of those which can be expressed using conventional functions have been found. While in principle each and every solution can be found using suitable (unconventional) functions, it is impossible to catalog infinitely many of them in order to understand their generic structure. Finding the energy spectrum of superstrings on AdS5 × S 5 therefore appears a too difficult problem to be solved explicitly. Instead one can ask a more moderate question: How is the spectrum of string solutions organized? In other words, can we classify string solutions even though we cannot write them explicitly? Understanding the classification at the classical level might be an essential step towards understanding the quantum string. The classification was started in [29] for bosonic strings on R × S 3 which is a subspace of the full AdS5 × S 5 background. It was shown that for each solution of the equations of motion there exists a corresponding hyperelliptic curve. The key physical data of the solution, such as the energy and Noether charges, were identified in the algebraic curve.2 At this point one can turn the logic around and investigate the moduli space of admissible curves, i.e. those curves which correspond to some classical solution. This leads to a solution of the spectral problem in terms of algebraic curves, which is probably as close to an explicit solution as it can be. However, one would have to ensure that all relevant constraints on the structure of admissible curves have been correctly identified. A survey of the moduli space of admissible curves suggests that this is indeed the case: There turns out to be one continuous modulus per genus and each handle of the curve 1 Here, as well as in the case of near plane-wave strings there are discrepancies starting at three gauge theory loops [10, 23]. This puzzle can also be reformulated as the question why it works at one and two loops in the first place. We have little to add on this issue. 2 This explains, among other things, why the classical energy, one of these charges, is typically expressed through hyperelliptic functions. The various integration constants of the classical solution turn into moduli of the algebraic curve which appear as parameters to the hyperelliptic functions. See also [30, 31] for a discussion of the moduli of some particular curves.

Algebraic Curve of Classical Superstrings on AdS5 × S 5

661

can be interpreted as a particular string mode. This count matches with strings in flat space, which has one amplitude per string oscillator. Although two distinct theories are compared here, one can expect that the number of local degrees of freedom of the string should be independent of the background. We furthermore believe that the (conserved) moduli of a curve represent a complete set of action variables for the string. The moduli space of admissible curves would thus represent half the phase space of the string model. Another interesting option is to reformulate the problem of finding admissible curves as a Riemann-Hilbert problem. This is achieved by representing the curve as a collection of Riemann sheets connected by branch cuts. The branch cuts are represented by integrals over contours and densities in the complex plane. The admissibility conditions turn into integral equations on these contours and densities. This representation reveals an underlying scattering problem and the branch cuts represent the fundamental particles. The integral equations select equilibrium states of the scattering problem. This can be compared to a direct Fourier transformation of the string: The Fourier transformation transforms the equations of motion into equations among the different Fourier modes. Conceptually, the resulting equations are very similar to the integral equations. The main difference between the two approaches is that there are interactions between arbitrarily many Fourier modes due to the highly non-linear nature of the strings, while the interactions for the integral equations are only pairwise! In some sense, the algebraic curve can thus be interpreted as a clever mode decomposition specifically tailored for the particular curved background. The pairwise, i.e. factorized, nature of the scattering problem leads us to integrability. Indeed, the algebraic curve was constructed using the Lax connection, a family of flat connections on the two-dimensional string world sheet. For sigma models on group manifolds and symmetric coset spaces, such as SU(2) = S 3 , this connection is well-known [32] and related to integrability as well as an infinite set of conserved charges [33–36] of the two-dimensional theory. Integrable structures were also found in the AdS/CFT dual N = 4 gauge theory: The dual of the world-sheet Hamiltonian, the planar dilatation operator (see [26] for a review), was shown to be integrable at leading loop order [37, 38]. Moreover, there are indications that integrability is not broken by higher-loop effects [39, 40]. In gauge theory, integrability enables one to construct a Bethe ansatz [41] to diagonalize local operators, which are isomorphic to quantum spin chains. This leads to a set of algebraic equations [37, 38, 23, 42] whose solutions are in one-to-one correspondence to eigenstates of the dilatation operator. In the limit of states with a large number of partons, which is at the heart of the spinning-strings correspondence, the discrete Bethe equations turn into integral equations [43, 21]. These are very similar to the integral equations from the string sigma model. In fact, it was shown that the higherloop Bethe equations in the su(2) sector [23, 42] match with the equations from classical string theory on R × S 3 up to two gauge-theory loops [29]. This proves the equality of energy spectra in this limit and sector. Alternatively, one can also derive an algebraic curve for gauge theory and compare it to the one for the sigma model [29]. An altogether different approach to showing the agreement of spectra uses coherent states [44, 45]. The solution of the spectral problem in terms of algebraic curves has since been extended to three other subsectors of the full superstring: Bosonic strings on AdS3 × S 1 [46], on R × S 5 [47] and on AdS5 × S 1 [48]. Also some features of the assembly of full AdS5 and full S 5 are known [49]. In all previous analyses, however, fermions have been excluded.3 While this is justified (for almost all practical purposes) at the classical level, 3 Within the Frolov-Tseytlin correspondence fermions have been treated in [50, 51] using the coherent state approach.

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they are certainly required to give a consistent quantum theory. It is therefore essential to include them, even at the classical level. This can indeed be done, even though it is a classical setting.4 In the present article we shall derive the solution of the spectrum of IIB superstrings on AdS5 × S 5 in terms of algebraic curves. The starting point will be the family of flat connections found by Bena, Polchinski and Roiban [52]. Using its open Wilson loop around the closed-string world sheet, the so-called monodromy, we can derive an algebraic curve. As was demonstrated in [52] the Lax connection exists prior to gauge fixing. We therefore do not fix any gauge, neither of conformal nor of kappa symmetry, in contrast to [29, 46, 47] and especially [49]. The emergent curve is neither a regular algebraic curve, nor an algebraic supercurve, i.e. not a supermanifold. It almost splits in two parts, but it is held together by the fermions. Each part has degree four and corresponds to one of the S 5 and AdS5 coset models. The bosonic degrees of freedom give rise to square-root branch points and cuts connecting them. These appear only within each set of four Riemann sheets. We shall show that, conversely, the fermions give rise to poles. Poles come in pairs, one of them is on the S 5 -part of the curve, the other on the AdS5 -part while their residues are the same.5 Their position within the algebraic curve is determined by the bosonic background. The two parts of the curve are furthermore linked by the Virasoro constraint: It relates a set of fixed poles between the two parts of the curve. These poles are an important general characteristic of the model and do not correspond to fermions. The precise structure of the algebraic curve and its representation in the form of integral equations constitute the key information from string theory for a comparison with gauge theory [53, 38, 26, 48] via the Frolov-Tseytlin proposal. Using an integral representation for the curve, we are able to show agreement of the spectra at leading order in the effective coupling constant. A more detailed comparison will be performed in the follow-up article [54]. The structure of this article is as follows: In Sect. 2 we will investigate the monodromy of the Lax connection and derive an algebraic curve from it. The remainder of the section is devoted to finding the analytic properties of the curve and relating them to data of the associated string solution. Then we decouple from the underlying string solution in Sect. 3 and consider the set of admissible curves. After counting the number of moduli, we shall identify them with certain integrals on the curve. Their relationship to the global charges is established. In the final Sect. 4 we shall represent the algebraic curve by means of its branch cuts between the Riemann sheets. The resulting equations are closely related to the equations one obtains from spin chains in the thermodynamic limit. We show that they agree with one-loop gauge theory. We conclude and give an outlook in Sect. 5. The appendices contain a review of supermatrices (App. A), the relation between coset and vector models (App. B) and explicit but lengthy expressions related to the full supersymmetric sigma model (App. C and D). 2. Supersymmetric Sigma Model We start by investigating the AdS5 × S 5 supersymmetric sigma model on a closed string worldsheet. First of all we present the sigma model in terms of its fields, currents and 4 Having fermions in classical equations is not a problem, but we run into difficulties when we try to find explicit solutions, which would require the introduction of actual Grassmann numbers. 5 The residues are products of two Grassmann-odd numbers. Therefore, they are Grassmann-even, but not of zeroth degree, i.e. they cannot be represented by common numbers. This is why fermions can be neglected for almost all practical purposes. The derivations are however simplified by ignoring this fact.


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constraints. Then we review the Lax connection and its monodromy and show that the essential physical information (action variables) is described by an algebraic curve. The remainder of this section is devoted to special properties of the curve and relating them to physical quantities. The AdS5 × S 5 superspace can be represented as the coset space of the supergroup PSU(2, 2|4) over Sp(1, 1) × Sp(2). Up to global issues, but preserving the algebraic structure, we can change the signature of the target spacetime. Here we will consider the coset PSL(4|4, R)/Sp(4, R) × Sp(4, R). This choice is convenient as we can completely avoid complex conjugation which may be somewhat confusing, especially in a supersymmetric setting. See e.g. [55, 56] for an explicit treatment of the PSU(2, 2|4) coset model. The global issues that we should keep in mind are whether the string can wind around the manifold. For S 5 this is certainly the case, while for AdS5 there should be no windings. Note that the physical AdS5 is a universal cover and there cannot be windings around the unfolded time circle. Likewise, the involved group manifolds are considered to be universal coverings.

2.1. The coset model. The Metsaev-Tseytlin string is a coset space sigma model. To represent the coset, we consider a group element g of PSL(4|4, R) and two constant (4|4) × (4|4) matrices6 E0 0 0 E1 = , E2 = , (2.1) 0 0 0E which break PSL(4|4, R) to Sp(4, R) × Sp(4, R). Here, E is an antisymmetric 4 × 4 matrix7 0 +I E= , (2.2) −I 0 where each entry corresponds to a 2 × 2 block and I is the identity matrix. We shall denote the pseudo-inverses of E1 , E2 by −1 0 0 E 0 ¯ ¯ , E2 = E1 = . (2.3) 0 0 0 E −1 These are defined such that a product of Ea and E¯ b is a projector to the even/odd subspace if a = b or zero if a = b. Finally, let us introduce a grading matrix +I 0 η= (2.4) 0 −I which will be useful at various places. The breaking of PSL(4|4, R) to Sp(4, R) × Sp(4, R) is achieved as follows: The matrix E1 is invariant under E1 → hE1 hST for elements h of a subgroup Sp(4, R) × SL(4, R) of PSL(4|4, R). Similarly, E2 → hE2 hST is invariant under a SL(4, R) × 6

For a short review of the algebra of supermatrices, cf. App. A. In fact, any E = −E T with εαβγ δ E αβ E γ δ = 0 would suffice and one could as well pick distinct matrices E for E1 and E2 . 7

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Sp(4, R). The combined map (E1 , E2 ) → (hE1 hST , hE2 hST ) leaves (E1 , E2 ) invariant precisely for h ∈ Sp(4, R) × Sp(4, R). Thus the element (gE1 g ST , gE2 g ST ) with g ∈ PSL(4|4, R) parametrizes the AdS5 × S 5 superspace.8 We now introduce the supermatrix-valued field g(τ, σ ) ∈ PSL(4|4, R) on the worldsheet. It satisfies sdet g = 1 and we identify group elements which are related by an abelian rescaling, g = ξg.9 The field g is not necessarily strictly periodic but g(τ, σ + 2π) = g(τ, σ ) h(τ, σ )

(2.5)

with h(τ, σ ) an element of Sp(4, R) × Sp(4, R). We define the standard g-connection J as J = −g −1 dg.

(2.6)

It is flat and supertraceless dJ = J ∧ J

and

str J = 0

(2.7)

by means of the usual identities and sdet g = 1. The algebra psl(4|4, R) can be decomposed into four parts obeying a Z4 -grading [4, 57–59].10 The connection decomposes as follows: J = H + Q1 + P + Q2 .

(2.8)

We use the constant supermatrices E1,2 , E¯ 1,2 to project to the various components H = 21 E1 E¯ 1 J E1 E¯ 1 − 21 E1 J ST E¯ 1 + Q1 = 21 E1 E¯ 1 J E2 E¯ 2 + 21 E1 J ST E¯ 2 + P = 21 E1 E¯ 1 J E1 E¯ 1 + 21 E1 J ST E¯ 1 + Q2 = 1 E1 E¯ 1 J E2 E¯ 2 − 1 E1 J ST E¯ 2 + 2

2

1 ¯ 2 E 2 E2 J 1 ¯ 2 E 2 E2 J 1 ¯ 2 E 2 E2 J 1 ¯ 2 E 2 E2 J

E2 E¯ 2 − 21 E2 J ST E¯ 2 , E1 E¯ 1 − 21 E2 J ST E¯ 1 , E2 E¯ 2 + 1 E2 J ST E¯ 2 , 2

E1 E¯ 1 + 21 E2 J ST E¯ 1 .

(2.9)

They satisfy the Z4 -graded Bianchi identities in [52] dH dQ1 dP dQ2

=H =H =H =H

∧ H + Q 1 ∧ Q2 + P ∧ P + Q 2 ∧ Q1 , ∧ Q1 + Q1 ∧ H + P ∧ Q 2 + Q2 ∧ P , ∧ P + Q1 ∧ Q1 + P ∧ H + Q 2 ∧ Q 2 , ∧ Q2 + Q1 ∧ P + P ∧ Q1 + Q2 ∧ H,

(2.10)

and their supertraces vanish str H = str Q1 = str P = str Q2 = 0.

(2.11)

Note that str H = str Q1 = str Q2 = 0 is satisfied by means of the projections (2.9) while str P = str J = 0 holds due to (2.7). Note that E1 is an antisymmetric supermatrix, E1ST = −ηE1 , while E2 is symmetric, E2ST = +ηE2 . Therefore also g(E1 ± E2 )g ST or g(E1 ± iE2 )g ST parametrize the coset as we can disentangle the contributions from E1 and E2 by projecting to the symmetric and antisymmetric parts. 9 For (4|4) × (4|4) supermatrices, sdet ξ I = 1 for any number ξ . 10 The Z -grading is directly related to supertransposing, cf. App. A. 4 8


665

The action of the IIB superstring on AdS5 ×S 5 given in [4] in terms of the connections P , Q1,2 reads [59] √ λ 1 Sσ = str P ∧ ∗P − 21 str Q1 ∧ Q2 + ∧ str P . (2.12) 2 2π We have introduced the Lagrange multiplier to enforce str P = 0. In fact, we cannot remove the part proportional to the identity matrix because of the identity str I = 0. The equations of motion read 0 = P ∧ Q2 − ∗P ∧ Q2 + Q2 ∧ P − Q2 ∧ ∗P , d∗P = H ∧ ∗P + Q1 ∧ Q1 + ∗P ∧ H − Q2 ∧ Q2 + d , 0 = P ∧ Q1 + ∗P ∧ Q1 + Q1 ∧ P + Q1 ∧ ∗P .

(2.13)

The appearance of in the equations of motion is related to the projective identification g= ξg. The equations of motion can also be written as the g-covariant conservation of the global psl(4|4, R) symmetry current K, d∗K − J ∧ ∗K − ∗K ∧ J = 0,

K = P + 21 ∗Q1 − 21 ∗Q2 − ∗ .

(2.14)

The above equations of motion follow after decomposition into the Z4 -graded components. The dependence of K on the unphysical Lagrange multiplier reflects the ambiguity in the definition of the abelian part of K in psl(4|4, R). In the fixed frame,11 which is related to the moving one by k = gKg −1 , the equations for the current are even shorter d∗k = 0.

(2.15)

The global symmetry charges are consequently given by √ 2π √ λ λ ∗k = dσ kτ . s= 2π γ 2π 0

(2.16)

These do not depend on the form of the path γ around the closed loop and are thus conserved physical quantities. For later convenience, we rewrite s = g(0)Sg −1 (0) in terms of the moving-frame current K as follows: √ 2π λ S= dσ g −1 (0)g(σ )Kτ (σ )g −1 (σ )g(0) 2π 0 √ 2π −1 σ σ λ dσ P exp dσ Jσ (σ ) Kτ (σ ) P exp dσ Jσ (σ ) . (2.17) = 2π 0 0 0 Here, as for the remainder of the article, the path ordering symbol P puts the values of σ in decreasing order from left to right. In addition to the equation of motion, the Virasoro constraints following from variation of the world-sheet metric (which appears only within the dualization ∗) are given by str P±2 = 0.

(2.18)

Here we have introduced the light-cone coordinates σ± = 21 (τ ± σ ),

∂± = ∂τ ± ∂σ ,

P± = Pτ ± Pσ .

(2.19)

11 We shall distinguish between a moving frame and a fixed frame. In the moving frame E is a constant matrix and the fundamental field is g. The gauge connection is D = d − J . In the fixed frame the matrix corresponding to E is gEg ST . It is not constant but rather the fundamental field. The gauge connection is trivial, D = d. See App. B for a comparison of both formalisms. We use uppercase and lowercase letters for the moving and fixed frames, respectively.

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Ω(z) γ

Fig. 1. The monodromy (z) is the open Wilson loop of the Lax connection A(z) around the string

2.2. Lax connection and monodromy. A family of flat connections a(κ) for the superstring on AdS5 × S 5 was derived in [52].12 This was expressed in the fixed frame, which is related to moving one by j = gJ g −1 and similarly for H, Q1 , P , Q2 . The Lax connection is given by a(κ) = α(κ) p + β(κ) (∗p − ) + γ (κ) (q1 + q2 ) + δ(κ) (q1 − q2 )

(2.20)

with α(κ) = −2 sinh2 κ, γ (κ) = 1 − cosh κ, β(κ) = 2 sinh κ cosh κ, δ(κ) = sinh κ.

(2.21)

We will employ a more convenient parametrization by setting z = exp κ. The coefficient functions become α(z) = 1 − 21 z2 − 21 z−2 , γ (z) = 1 − 21 z − 21 z−1 , β(z) = 21 z2 − 21 z−2 , δ(z) = 21 z − 21 z−1 .

(2.22)

We would now like to transform the connection to the moving frame using J = g −1 jg and compute d − A(z) = g −1 d + a(z) g = d − J + g −1 a(z)g = d − H + (α − 1) P +β (∗P − )+(γ − 1) (Q1 + Q2 )+δ (Q1 − Q2 ), (2.23) where the Lax connection reads A(z) = H + 21 z2 + 21 z−2 P + − 21 z2 + 21 z−2 (∗P − ) + z−1 Q1 + z Q2 . (2.24) As was shown in [52], it satisfies the flatness condition 2 d − A(z) = 0

(2.25)

by means of the equations of motion. It is also traceless for obvious reasons, str A(z) = 0. As emphasized in [29, 47], an important object for the solution of the spectral problem is the open Wilson loop of the Lax connection around the closed string. It is given by 12 For complex values of κ, there is only one family of flat connections. The other family mentioned in [52] is trivially obtained by replacing κ with iπ − κ.


2π

0 (z) = P exp

667

dσ Aσ (z) P exp

A(z).

(2.26)

0

The monodromy which is defined as13 (z) = −1 0 (1) 0 (z)

(2.27)

is independent of the path γ around the closed string; it merely depends on the point γ (2π ) = γ (0) where the path is cut open. More explicitly, a shift of γ (0) leads to a similarity transformation ( ), see e.g. [47]. Therefore, the eigenvalues of (z) are invariant, physical quantities. Note that we did not specify any particular gauge of conformal or kappa symmetry. Under kappa symmetry the Lax connection transforms by conjugation [60] and consequently leaves the eigenvalues invariant as well. For definiteness we define (z) through the path σ ∈ [0, 2π] at τ = 0. Also note that str A(z) = 0 leads to sdet (z) = 1. In the Hamiltonian formulation, the eigenvalues of the monodromy represent action variables of the sigma model.14 We have a one-parameter family of them and it is not inconceivable that they form a complete set. So we might have a sufficient amount of information to fully characterize the class of solution. The time-dependent angle variables and all gauge degrees of freedom are completely projected out in the eigenvalues of (z). This is a very good starting point for a quantum theory: For quantum eigenstates we can measure all the action variables exactly, but information of the angle variables is obscured by the uncertainty principle. 2.3. The Algebraic Curve. The physical information of the monodromy matrix is its conjugation class. Let u(z) diagonalize (z) as follows: u(z)(z)u−1 (z)

= diag ei p˜1 (z) , ei p˜2 (z) , ei p˜3 (z) , ei p˜4 (z) ei pˆ1 (z) , ei pˆ2 (z) , ei pˆ3 (z) , ei pˆ4 (z) . (2.28) ei p˜k

ei pˆl

Note that the eigenvalues and corresponding to the two gradings are distinguishable, they cannot be interchanged by a (bosonic) similarity transformation. We can associate p˜ k to S 5 while pˆ k corresponds to AdS 5 . In contrast, we may freely interchange eigenvalues within each set of four. Unimodularity, sdet (z) = 1, translates to the condition p˜ 1 + p˜ 2 + p˜ 3 + p˜ 4 − pˆ 1 − pˆ 2 − pˆ 3 − pˆ 4 ∈ 2π Z.

(2.29)

The monodromy (2.27) depends analytically on the spectral parameter z by definition except at the singular points z = 0 and z = ∞. This however does not imply that also the eigenvalues {ei p˜k ||ei pˆk } enjoy the same property. Let us first consider a point z˜ a where two eigenvalues ei p˜k , ei p˜l corresponding to the 5 S -part of the sigma model degenerate. The restriction of (z) to the subspace of the two corresponding eigenvalues then takes the general form ab = (2.30) cd For z = 1 the Lax connection A(z) = J is the gauge connection. The additional factor −1 0 (1) = g(0)−1 g(2π ) = h(0) therefore transforms the monodromy back to the tangent space at σ = 0. 14 See [56] for an investigation of the Poisson brackets of the monodromy. 13

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with some coefficients a, b, c, d depending analytically on z. Its eigenvalues are given by the general formula

1 γ1,2 = a + d ± (a − d)2 + 4bc . (2.31) 2 At z = z˜ a the combination f = (γ1 − γ2 )2 = (a − d)2 + 4bc = (Tr )2 − 2 Tr 2 under the square root vanishes, f (˜za ) = 0. In the generic case, one can expect f (˜za ) = 0. This implies the well-known fact that crossing of eigenvalues usually gives rise to a square-root singularity:

ei p˜k,l (z) = ei p˜k (˜za ) 1 ± α˜ a z − z˜ a + O(z − z˜ a ) . (2.32) Similarly, coincident AdS5 -eigenvalues ei pˆk and ei pˆl at zˆ a lead to square-root singularities

ei pˆk,l (z) = ei pˆk (ˆza ) 1 ± αˆ a z − zˆ a + O(z − zˆ a ) . (2.33) The behavior around a point za∗ where eigenvalues of opposite gradings, ei p˜k and coincide is quite different: Consider the submatrix of (z) on the subspace of the two associated eigenvectors ab = . (2.34) cd ei pˆl ,

The eigenvalues of this supermatrix are given by γ1 =

bc +a, d −a

γ2 =

bc +d, d −a

(2.35)

where again a, b, c, d are given by analytic functions in z. At z = za∗ , the combination f = a − d = γ1 − γ2 = str in the denominator is zero by definition, f (za∗ ) = 0. Generically, we cannot however expect that also the numerator bc vanishes and therefore we find a pole singularity at za∗ , ∗ αa ∗ ∗ ei p˜k (z) = ei p/˜k (za ) + 1 + O(z − z ) = ei pˆl (z) . (2.36) a z − za∗ ∗

/ a ) of eip(z) at z = z∗ are the same Note that the residue αa∗ as well as the regular part ei p(z a 15 for both eigenvalues. We thus learn that the set of eigenvalues of (z) depends analytically on z except at a set of points {0, ∞, z˜ a , zˆ a , za∗ }. Let us assume that there are only finitely many singularities of this kind. The cases of an infinite number of singularities can hopefully be viewed as limits of this finite setting. A unique labelling of eigenvalues cannot be achieved globally, because a full circle around one of the square-root singularities z˜ a , zˆ a will result in an interchange of the two eigenvalues associated to the singularity. Therefore we need to introduce several branch cuts Cã and Câ in ei p˜k (z) and

It might be worthwhile to point out that αa∗ = −bc is the product of two Grassmann-odd quantities and thus, in principle, cannot be an ordinary number. It satisfies a nilpotency condition (αa∗ )2 = 0 which, however, does not quite make it trivial. When quantizing the string, these factors give rise to fermionic excitations due to quantum ∼ 1/L effects. This effect can already be seen in the one-loop spin chain for the N = 4 gauge theory [38] where there are no nilpotent objects. 15


669

ei pˆk (z) , respectively, which connect the square-root singularities. The functions p˜ k (z) and pˆ k (z) are therefore analytic except at {0, ∞, Cã , Câ , za∗ }. Alternatively, we could ˜ ˆ ˜ and M ˆ of C. ¯ In and ei p(z) as one function on suitable four-fold coverings M view ei p(z) i p(z) ˜ i p(z) ˆ ∗ that case, the functions e and e are analytic except at {0, ∞, z˜ a , zˆ a , za }. At za∗ i p(z) ˜ i p(z) ˆ both functions e and e have poles with equal residues and regular parts. Finally, 2 2 at 0 and ∞ there are essential singularities of the type eiα0 /z , eiα∞ z . ˆ , ei p(z) ˜ Except for the last two singularities, the functions ei p(z) would satisfy all requirements for algebraic curves. In order to turn the essential singularities at {0, ∞} into regular singularities, we take the logarithmic derivative of the eigenvalues. Let us define the matrix Y (z) according to ∂ u(z)Y (z)u−1 (z) = −iz (2.37) log u(z)(z)u−1 (z) , ∂z where u(z) diagonalizes (z). In other words, the eigenvalues of Y (z) are the logarithmic derivatives of the eigenvalues of (z). The corresponding eigenvectors are the same. We can now reduce Y (z) to the following expression: Y (z) = −1 (z) −iz (z) + [U (z), (z)] , U (z) = −izu−1 (z)u (z). (2.38) As (z) is non-zero and its only singularities are at {0, ∞}, any further singularities can only originate from U (z). The diagonalization matrix u(z) has square roots and branch cuts. It appears that all the branch points of u(z) are turned into single poles in U (z).16 Consequently, U (z) has poles at {˜za , zˆ a , za∗ }, but all the branch cuts are removed. Therefore Y (z) is single-valued and analytic on the complex plane except at the singularities ¯ C\{0, ∞, z˜ a , zˆ a , za∗ }. Now we can read off the eigenvalues y(z), ˜ y(z) ˆ of Y (z) from its characteristic function F (y, z), F (y(z), ˜ z) = 0,

F (y(z), ˆ z) = ∞

(2.39)

with F (y, z) =

F˜ (y, z) F˜4 (z) . sdet y − Y (z) = Fˆ (y, z) Fˆ4 (z)

(2.40)

We have included polynomial prefactors F˜4 (z), Fˆ4 (z) in the definition of F = F˜ /Fˆ which clearly do not change the algebraic curve. The purpose of the prefactors is to remove the poles originating from U (z). The roots of these prefactors are thus given by the singularities {˜za , za∗ } or {ˆza , za∗ }, respectively. They enable us to write both F˜ and Fˆ as polynomials, not only in y (obvious), but also in z. As Y (z) has only pole-type singularities at {0, ∞, z˜ a , zˆ a , za∗ }, the above equation de˜ and M, ˆ respectively. fines two algebraic curves y(z) ˜ and y(z) ˆ on the Riemann surfaces M We can even unite the two curves into one curve y(z) = {y(z)|| ˆ y(z)} ˜ on ˜ ∪ M. ˆ At the points {˜za }, {ˆza }, the functions y(z), ˜ y(z) ˆ have inverse square-root M=M singularities. At {za∗ } both functions y(z), ˜ y(z) ˆ have double poles with equal coefficients. Similarly, at {0, ∞}, there are singularities of the type −2α0 /z2 , 2α∞ z2 . Finally, there are no single poles anywhere, because they would lead to a singular matrix , which cannot happen. 16 This may require a special matrix u(z). The point is that one can redefine u(z) → a(z)u(z) with any diagonal matrix a(z). This is a possible source of non-analyticity in U (z), which however drops out in [U (z), (z)].

670


2.4. The central element. Consider the local transformation g(τ, σ ) → ξ(τ, σ ) g(τ, σ )

(2.41)

with ξ a number-valued field which is nowhere zero. Here we would like to demonstrate that this transformation does not have any physical effect. First of all, it changes the current J by J → J − ξ −1 dξ , but note that str J = 0 remains true due to str I = 0. The transformation can now be easily seen to affect only the P -component of J , P → P − ξ −1 dξ.

(2.42)

In the equations of motion (2.13) the variation drops out when the Lagrange multiplier shifts accordingly,

→ − ξ −1 ∗dξ − i dζ − iυ dσ.

(2.43)

The additional transformation parameters are the field ζ (τ, σ ) and the constant υ. We cannot include υ in ζ as ζ → ζ + υ σ as ζ would not be periodic. The action is also invariant except for the term proportional to υ. This actually leads to a change of the global charges (2.17), √ √ √ λ λ S → S − dζ − υ dσ = S − λ υ. (2.44) 2π 2π This change of the central element of S is unphysical because the global symmetry is merely PSU(2, 2|4), not SU(2, 2|4). The family of flat connections changes up to a central gauge transformation A(z) → A(z) − ( 21 z2 + 21 z−2 ) ξ −1 dξ − ( 21 z2 − 21 z−2 )(i dζ + iυ dσ ).

(2.45)

As this is an abelian shift, it completely factorizes from the monodromy and we get (z) → (z) exp 0

dσ (1− 21 z2 − 21 z−2 ) ξ −1 ∂σ ξ −( 21 z2 − 21 z−2 )(i∂σ ζ + iυ) .

2π

(2.46) The first term measures the winding number of ξ around 0 when going once around the string. This winding affects both the AdS5 and S 5 parts of g. However, in the physical setting, the background is a universal cover and windings around the time-circle of AdS5 are not permitted. Therefore the term involving ξ does not contribute. Also the term involving ζ vanishes because ζ is periodic. We end up with (2.47) (z) → (z) exp −iπ υ(z2 − z−2 ) . The factor is abelian and does not change the eigenvectors. We thus find Y (z) → Y (z) − 2πυ(z2 + z−2 ).

(2.48)

This means that we can shift the curve y(z) by a term proportional to (z2 + z−2 ) as long as the factor of proportionality is the same for all sheets.


671

2.5. Symmetry. Let us introduce the (antisymmetric) supermatrix C = E1 − iE2 .

(2.49)

Then there is another useful way of expressing (2.9), H = 41 J − 41 C J ST C −1 + 41 η J η − 41 η C J ST C −1 η, Q1 = 41 J − 4i C J ST C −1 − 41 η J η + 4i η C J ST C −1 η, P = 41 J + 41 C J ST C −1 + 41 η J η + 41 η C J ST C −1 η, Q2 = 41 J + 4i C J ST C −1 − 41 η J η − 4i η C J ST C −1 η,

(2.50)

where η is the grading matrix ( 2.4). This form reveals that a conjugation of the four components H, Q1 , P , Q2 of J with C is equivalent to their supertranspose up to a sign determined by their grading under Z4 , C −1 H C = −H ST , C −1 Q1 C = −iQST 1 , C −1 P C = +P ST , C −1 Q2 C = +iQST 2 .

(2.51)

When we apply this conjugation to the flat connections we obtain ST C −1 A(z)C = −H ST + 21 z2 + 21 z−2 P ST + − 21 z2 + 21 z−2 ∗P ST − iz−1 QST 1 +iz Q2 = −AST (−iz).

(2.52)

This, in turn, implies a symmetry relation for the monodromy C −1 (z) C = −ST (−iz). The inverse is due to the overall sign in (2.52) and the transpose puts the Wilson loop in the original path ordering. In other words17 (iz) = C −ST (z) C −1

(2.53)

is related to (z) by conjugation, inversion and supertranspose. This translates to the following symmetry of Y (z) and U (z): Y (iz) = −C Y ST (z) C −1 ,

U (iz) = −C U ST (z) C −1 .

(2.54)

In particular, the characteristic function has the symmetry F (y, iz) =

Fˆ4 (z) F˜4 (iz) sdet y − Y (iz) = sdet y + Y (z) = F (−y, z). ˜ ˆ F4 (z) F4 (iz)

(2.55)

It therefore depends analytically only on the combinations z4 , yz2 , y 2 . In other words, y(iz) = −y(z) and consequently p(iz) = −p(z) + 2π Z with some permutation of the sheets. To determine the permutation, let us consider the action on the diagonalized matrix −1 ST Ydiag (iz) = −Cdiag (z)Ydiag (z) Cdiag (z) with 17

Cdiag (z) = u(iz) C uST (z). (2.56)

The contribution from −1 0 (1) = h(0) can be seen to cancel, because h ∈ Sp(4, R) × Sp(4, R).

672


ST As both Ydiag (iz) and Ydiag (iz) = Ydiag (iz) are diagonal, Cdiag (z) must be a permutation matrix and thus constant (up to branch cuts). In particular, we should investigate the fixed points of z → iz; these are the singular points {0, ∞}. At these points, Cdiag (z) as defined in (2.56) must approach an antisymmetric matrix related to C. As it is constant, it must always be an antisymmetric permutation matrix which acts non-trivially with period 2. We therefore find that the eigenvalues obey the symmetry

y˜k (iz) = −y˜k (z),

yˆk (iz) = −yˆk (z),

(2.57)

where we are free to choose the following permutation of sheets: k = (2, 1, 4, 3)

for

k = (1, 2, 3, 4).

(2.58)

pˆ k (iz) = −pˆ k (z).

(2.59)

For the quasi-momentum we find p˜ k (iz) = 2πmεk − p˜ k (z), Here we have introduced εk = (+1, +1, −1, −1)

for

k = (1, 2, 3, 4).

(2.60)

The constant shift 2π m in p˜ k (iz) is related to winding around S 5 . It must be absent for the AdS5 counterpart pˆ k (iz) because there cannot be windings in the time direction. Finally, we see that y must depend analytically on z2 . We can thus introduce the variable x defined by x=

1 + z2 , 1 − z2

z2 =

x−1 , x+1

(2.61)

which is precisely the variable commonly used for bosonic sigma models as in [29, 47]. The points associated to local and global charges, discussed in the following subsections, and the symmetry are related as follows, see also Fig. 2: x x x x x

= ∞ = 0 = +1 = −1 → 1/x

⇔ z = ±1, ⇔ z = ±i, ⇔ z = 0, ⇔ z = ∞, ⇔ z → iz.

(2.62)

Note the relation of differentials dx dz = dκ , = 1 − 1/x 2 z where κ = log z is the spectral parameter used in [52].

(2.63)


673

multi-local Z4

z = +i x=0 multi-local z = −1 x=∞

local+

local− z=∞ x = −1

multi-local z = +1 x=∞

z=0 x = +1 multi-local z = −i x=0

Fig. 2. Special points of the quasi-momenta. The expansion around z = 0, ∞ yields one sequence of local charges each, see Sect. 2.6. At z = ±1, ±i one finds the Noether charges, discussed in Sect. 2.8, and multi-local charges. All other points are related to non-local charges

2.6. Local charges. At the points z = 0, ∞ the expansion of the Lax connection is singular A( ±1 ) = 21 −2 (P ± ∗P ∓ ) + −1 Q1,2 + H + Q2,1 + 21 2 (P ∓ ∗P ± ). (2.64) The expansion of the quasi-momentum p(z) at these points is thus related to local charges. As was shown in, e.g., [47], in the absence of the fermionic contributions Q1,2 , the leading coefficient of p(z) in is directly related to eigenvalues of the leading contribution to Aσ . Let us repeat the argument for the point z = 0. Consider the transformed ¯ connection A(z) in the σ -direction given by ¯ ∂σ − A(z) = T (z) ∂σ − Aσ (z) T −1 (z). (2.65) Here T (z) and A(z) are given by their expansion in z, T (z) =

∞

z r Tr ,

¯ A(z) =

r=0

∞

zr A¯ r .

(2.66)

r=−2

We demand that T0 diagonalizes the leading term A¯ −2 = 21 T0 P+ T0−1 + 21 σ = diag(α˜ 1 , α˜ 2 , α˜ 3 , α˜ 4 ||αˆ 1 , αˆ 2 , αˆ 3 , αˆ 4 ).

(2.67)

Since P satisfies CP ST C −1 = P , cf. (2.51), its eigenvalues must be doubly degenerate, α˜ 1 = α˜ 2 , α˜ 3 = α˜ 4 , αˆ 1 = αˆ 2 , αˆ 3 = αˆ 4 . Furthermore, P+ satisfies the Virasoro constraint str P+2 = 0. This requires α˜ 1 = αˆ 1 , α˜ 3 = αˆ 3 . The abelian shift by σ is compatible with this construction and we find [49] αI 0 . (2.68) A¯ −2 = diag(α, α, β, β||α, α, β, β) = 0 βI

674


Here we have introduced a (2|2) × (2|2) block decomposition of the (4|4) × (4|4) supermatrix, i.e. each block is a supermatrix. As the eigenvalues α and β are (generically) distinct, we can use the Tr (z) to bring A¯ r−2 (z) to a block-diagonal form a(z) 0 ar 0 ¯ ¯ or A(z) = . (2.69) Ar = 0 b(z) 0 br When this is done order by order in perturbation theory, the resulting ar and br are local combinations of the fields. However, the diagonalization of the Lax connection is not yet complete and a complete diagonalization will lead to non-local results. Still we can obtain local charges: Although the open Wilson loop is in general non-local, its superdeterminant is the exponential of a local charge. Here sdet = 1 is trivial, but we can consider only one block of T (2π )T (0)−1 , ω(z) = P exp

−1

2π

a(1)

0

2π

P exp

a(z) .

(2.70)

0

Then sdet ω(z) = exp iq(z) with q(z) = −i

2π

dσ str a(z) − str a(1) .

(2.71)

0

The expression for the other block involving b(z) is in fact equivalent due to str a+str b = str A¯ = 0. The expansion of q(z) into qr gives a sequence of local charges. The term q−2 vanishes because a−2 is proportional to the identity. We can also perform a similar construction around z = ∞ leading to similar charges and thus we have found two infinite sequences of local charges. Let us express q(z) through the quasi-momentum p(z). As exp iq is the superdeterminant of the block ω of T (2π )T (0)−1 we can also write q(z) as a sum over a half of the quasi-momenta, q(z) = p˜ 1 (z) + p˜ 2 (z) − pˆ 1 (z) − pˆ 2 (z).

(2.72)

Using (2.60) we write the generator of local charges in the concise form q(z) =

4

εk

1

1 2 p˜ k (z) − 2 pˆ k (z)

.

(2.73)

k=1

Expanded around z = 0, ∞ it yields the conserved local charges. In App. C we will construct the first of these charges. Note that besides the local charges there is a larger set of conserved non-local charges.

2.7. Singularities. We would like to understand the singular behavior of the quasimomentum p(z) at z = 0 better. In the bosonic case we would be finished after the semi-diagonalization of the previous section because all singular terms have been diagonalized and can be integrated up. In the supersymmetric case, the remaining singular term a−1 is not diagonal and might lead to further singularities at z = 0. Here we will show that this does not happen. The difficulty of the proof is that any attempt to diagonalize further would lead to non-local terms.


675

¯ We shall start with one block a(z) of the semi-diagonalized connection A(z). Let us investigate the logarithm of the monodromy ω(z) and expand near z = 0,18 log ω(z) = 0

2π

2π

dσ a(σ, z) +

σ

dσ 0

0

dσ

1 2

a(σ, z), a(σ , z) + . . . .

(2.74)

The further terms involve nested commutators of a(z) at various points σ . The term a−2 = αI is abelian and thus contributes only to the first term. This is not necessarily the most singular term, as a−1 may appear many times within the nested commutators. To resolve this problem we note that the full Lax connection obeys the Z4 -symmetry relation (2.51). This reduces to a similar relation for the block a(z), c a ST (z) c−1 = −a(iz)

or

c arST c−1 = i 2+r ar ,

(2.75)

where c = e1 − ie2 with e1,2 as in (2.1), but with e being a 2 × 2 instead of a 4 × 4 antisymmetric matrix. This means that ar has Z4 -grading r. Note that the grading is obeyed by commutators, i.e. when x and y have gradings r and s, respectively, the commutator [x, y] has grading r + s. Now consider two (2|2) × (2|2) supermatrices x, y of grading −1. Then it can be shown (explicitly) that their commutator [x, y] is proportional to the identity matrix I . It therefore drops out of any further commutators and nested commutators can never produce terms of grading less than −2. Furthermore, all terms of grading −2 are proportional to the identity. The grading coincides with the power of z and we find log ω(z) = d−2 z−2 I + d−1 z−1 + O(z0 )

(2.76)

with d−2 a number and d−1 a matrix of grading −1. To finally diagonalize log ω(z) we first use a matrix exp(t−1 z−1 ) which, using the Baker-Campbell-Hausdorff identity and for the same reasons as above, removes the term d−1 without lifting the degeneracy of double poles or creating even higher poles. Afterwards ω(z) can be diagonalized perturbatively. Of course, all of the above holds true for the other block. We assemble the two blocks and find for the quasi-momentum p˜ k (z) ∼ pˆ k (z) ∼ (α0 + εk β0 ) z−2 + O(z−1 )

(2.77)

with some coefficients α0 , β0 not directly related to α, β. We have thus proved that the structure of residues found in [49] is not affected by the fermions. Note that this distribution on the pk is compatible with the permutation of sheets in Sect. 2.5. Similarly, at z = ∞ the expansion of the quasi-momenta is given by p˜ k (z) ∼ pˆ k (z) ∼ (α∞ + εk β∞ ) z2 + O(z).

(2.78)

2.8. Global charges. At z = 1 the expansion of the Lax connection A(1 + ) = J − 2 ∗K + O( 2 )

(2.79)

18 For convenience we omit contributions from the second term in ω; they do not change the principal result.

676


is related to the psu(2, 2|4) Noether current. The expansion of the monodromy yields (1 + )

2π

= I −2 0

dσ P exp

σ

−1 dσ Jσ (σ ) Kτ (σ ) P exp

0

σ

dσ Jσ (σ ) + O( 2 )

0

(2.80) which equals 4π S (1 + ) = I − √ + O( 2 ) λ

(2.81)

by means of (2.17). Not only z = +1, but also z = −1 and z = ±i are related to the global charges, as can be seen from the symmetry discussed in Sect. 2.5. The higher orders in the expansion yield multi-local charges. These are the Yangian generators discussed in [52, 55, 61, 56]. The expansion of the quasi-momenta p˜ k (z) associated to S 5 at z = 1 is [47] 4π p˜ 1 (1 + ) = − √ + 43 r˜1 + λ 4π 1 p˜ 2 (1 + ) = − √ − 4 r˜1 + λ 4π 1 p˜ 3 (1 + ) = − √ − 4 r˜1 − λ 4π 1 p˜ 4 (1 + ) = − √ − 4 r˜1 − λ

1 2 r˜2

+ 41 r˜3 + 41 r ∗ + . . . ,

1 2 r˜2

+ 41 r˜3 + 41 r ∗ + . . . ,

1 2 r˜2

+ 41 r˜3 + 41 r ∗ + . . . ,

1 2 r˜2

− 43 r˜3 + 41 r ∗ + . . . .

(2.82)

Here, [˜r1 , r˜2 , r˜3 ] are the Dynkin labels of SU(4) related to the spins of SO(6) by r˜1 = J2 − J3 , r˜2 = J1 − J2 , r˜3 = J2 + J3 . The label r ∗ is an unphysical label related to the U(1) hypercharge. It transforms under the transformation described in Sect. 2.4 as √ ∗ ∗ r → r + υ λ . Similarly, the expansion for pˆ k (z) associated to AdS5 reads [49] 4π pˆ 1 (1 + ) = √ + 43 rˆ1 + λ 4π 1 pˆ 2 (1 + ) = √ − 4 rˆ1 + λ 4π 1 pˆ 3 (1 + ) = √ − 4 rˆ1 − λ 4π 1 pˆ 4 (1 + ) = √ − 4 rˆ1 − λ

1 2 rˆ2

+ 41 rˆ3 − 41 r ∗ + . . . ,

1 2 rˆ2

+ 41 rˆ3 − 41 r ∗ + . . . ,

1 2 rˆ2

+ 41 rˆ3 − 41 r ∗ + . . . ,

1 2 rˆ2

− 43 rˆ3 − 41 r ∗ + . . . .

(2.83)

The Dynkin labels [ˆr1 , rˆ2 , rˆ3 ] of SU(2, 2) are related to the spins of SO(2, 4) by rˆ1 = S1 − S2 , rˆ2 = −E − S1 , rˆ3 = S1 + S2 .


677

2.9. Bosonic AdS5 × S 5 , R × S 5 and AdS5 × S 1 sectors. The restriction to the classical bosonic string on AdS5 × S 5 [49], R × S 5 [47] and AdS5 × S 1 [48] is straight forward: First of all we remove all possible fermionic poles. This implies K ∗ = 0 but we can also set r ∗ = B = 0 and obtain the bosonic string on AdS5 × S 5 . Then the expansion at z = 1 (2.82, 2.83) as well as the structure of poles at z = 0, ∞, cf. Sect. 2.7, agrees with [49] under the change of spectral parameter (2.61). In the next step we either reduce AdS5 to R or S 5 to S 1 . The isometry groups of both factors R and S 1 are abelian. For the monodromy corresponding to this factor we can therefore remove the path ordering

−1

(z) = P exp

A(1)

P exp

A(z) = exp

A(z) − A(1) .

(2.84)

We now substitute A(z) from (2.24) with H = Q1 = Q2 = 0, P = −g −1 dg and solve (z) = eip(z) for the quasi-momentum −1 1 2 1 −2 1 2 1 −2 p(z) = i(1 − 2 z + 2 z ) g dg − i(− 2 z + 2 z ) g −1 ∗dg. (2.85) The first integral represents the winding number m; it must vanish for R and can be non-trivial for S 1 . The second integral represents the global charge; it is proportional to the energy E for R and to the spin J for S 1 . By comparing to (2.83) we find that in the case of R × S 5 the full quasi-momentum for AdS5 is given by πE pˆ k (z) = εk √ − 21 z2 + 21 z−2 . λ

(2.86)

When the residues at z = 0, ∞ are matched between p˜ k and pˆ k we find perfect agreement with [47]. Equivalently in the case of AdS5 × S 1 the full quasi-momentum for S 5 is obtained by comparing to (2.82),19 πJ p˜ k (z) = εk √ − 21 z2 + 21 z−2 + εk π m 1 − 21 z2 − 21 z−2 . λ

(2.87)

Again, after matching the residues, this is in agreement with [48]. 3. Moduli of the Curve In this section we investigate the moduli space of admissible curves. Admissible curves are algebraic curves which satisfy all the properties derived in the previous section and which can thus arise from a classical string configuration on AdS5 × S 5 . For a fixed degree of complexity of the solution, which manifests as the genus of the curve, we count the number of degrees of freedom for admissible curves. Although it is not obvious that all admissible curves indeed represent string solutions (in other words that we have identified all relevant properties of admissible curves) we see that this number agrees with strings in flat space. We take this as evidence that our classification of string solutions in terms of admissible curves is complete. We finally identify the discrete parameters and continuous moduli with certain cycles on the curve and interpret them. For the 19

The integral of g −1 dg yields odd multiples of iπ when g(2π ) = −g(0), which is an allowed case.

678


pˆ1 pˆ2

1/x∗2 x∗2

−1 +1

x∗1

C˜1

1/C˜1

1/x∗1

1/Cˆ2

p˜1 Cˆ2

1/Cˆ1

Cˆ1

pˆ4

p˜3 p˜4

−1 +1 pˆ3

p˜2

Fig. 3. Some configuration of cuts and poles for the sigma model. Cuts Cã between the sheets p˜ k correspond to S 5 excitations and likewise cuts Câ between the sheets pˆ k correspond to AdS 5 excitations. Poles xa∗ on sheets p˜ k and pˆ l correspond to fermionic excitations. The dashed line in the middle is related to physical excitations, cuts and poles which cross it contribute to the total momentum, energy shift and local charges

comparison to gauge theory we investigate the Frolov-Tseytlin limit of the algebraic curve corresponding to a loop expansion in gauge theory. 3.1. Properties. Let us collect the analytic properties of the quasi-momentum

p(x) = p˜ 1 (x), p˜ 2 (x), p˜ 3 (x), p˜ 4 (x) pˆ 1 (x), pˆ 2 (x), pˆ 3 (x), pˆ 4 (x) ,

(3.1)

see Fig. 3 for an illustration. All sheet functions p˜ k (x) and pˆ l (x) are analytic almost everywhere. The singularities are as follows: • At x = ±1 there are single poles, cf. Sect. 2.6. The four sheets p˜ 1,2 (x), pˆ 1,2 (x) all have equal residues; the same holds for the remaining four sheets p˜ 3,4 (x), pˆ 3,4 (x). • Bosonic degrees of freedom are represented by branch cuts {Cã }, a = 1, . . . , 2A˜ and ˆ The cut Cã connects the sheets kã and lã of p˜ (x). Equivalently, {Câ }, a = 1, . . . , 2A. Câ connects the sheets kâ and lâ of pˆ (x). At both ends of the branch cut, xã± or xâ± , there is a square-root singularity on both sheets. • Fermionic degrees of freedom are represented by poles at {xa∗ }, a = 1, . . . , 2A∗ . The pole xa∗ exists on the sheets ka∗ of p(x) ˜ and la∗ of p(x) ˆ with equal residue. Further properties are: • For definiteness, we assume the quasi-momentum to approach zero at x = ∞ on all sheets, cf. Sect. 2.8 p(x) ˜ = O(1/x),

p(x) ˆ = O(1/x).

(3.2)

• The quasi-momentum obeys the symmetry x → 1/x, see Sect. 2.5, as follows p˜ k (1/x) = −p˜ k (x) + 2πmεk , We use the permutation k

pˆ k (1/x) = −pˆ k (x).

(3.3)

of k and a sign εk for each sheet k = (1, 2, 3, 4) as defined

in (2.58, 2.60) k = (2, 1, 4, 3),

εk = (+1, +1, −1, −1).

(3.4)


679

∞

p˜kã

∞

p˜˜la

Ba∗

∞

p˜ka∗

Ba∗

∞

Bã

Aã Cã Bã

A∗a x∗a

pˆla∗

A∗a

pˆkâ

Bâ

∞

Câ

pˆˆla

Bâ

Aâ

∞

Fig. 4. Cycles for S 5 -cuts (top), fermionic poles (middle) and AdS5 -cuts (bottom). Generically, S 5 -cuts are along aligned in the imaginary direction while AdS5 -cuts are along the real axis

The branch cuts and poles must respect the symmetry. We therefore consider the cut CÃ+a = 1/Cã to be the image of Cã . The independent cuts are thus labelled by ˜ ˜ Similarly for AdS5 -cuts Câ and fermionic poles xa∗ ,20 a = 1, . . . , A. CÃ+a = 1/Cã , ˜

CÂ+a = 1/Câ , ˆ

xA∗ ∗ +a = 1/xa∗ .

(3.5)

Note that there is an arbitrariness of which cuts are considered fundamental and which are their images under the symmetry. E.g. we might replace Cã by 1/Cã which effectively interchanges Cã and CÃ+a without changing the curve. ˜ • The unimodularity condition (2.29) together with (3.2) translates to p˜ 1 + p˜ 2 + p˜ 3 + p˜ 4 = pˆ 1 + pˆ 2 + pˆ 3 + pˆ 4 .

(3.6)

• A common shift of all sheets p(x) ˜ → p(x) ˜ +

4πυ , x − 1/x

p(x) ˆ → p(x) ˆ +

4π υ x − 1/x

(3.7)

is considered unphysical, cf. Sect. 2.4. For the cuts and poles we define several cycles and periods, cf. Fig. 4: • We define the cycles A˜ a , Aˆ a which surround the cuts Cã , Câ , respectively. The cuts, which connect the branch points {xã± }, {xâ± }, have been arranged in such a way that d p˜ = 0, d pˆ = 0. (3.8) A˜ a

20

Aˆ a

Within sums a self-symmetric cut will be counted with weight 1/2.

680


˜ Z) This can be achieved by a reorganization of cuts which corresponds to a Sp(2A, ˆ Z) transformation, respectively [29]. or Sp(2A, • We define the cycle A∗a which surrounds the fermionic pole xa∗ . There are no logarithmic singularities at xa∗ , d p˜ = d pˆ = 0. (3.9) A∗a

A∗a

At the singular points x = ±1 there are no logarithmic singularities either, d p˜ k = d pˆ k = 0. ±1

±1

(3.10)

• We define periods Bã , Bâ which connect x = ∞ on sheet kã , kâ to x = ∞ on sheet lã , lâ through the cuts Cã , Câ , respectively, see Fig. 4. These must be integral d p˜ = 2π n˜ a , d pˆ = 2π nˆ a , (3.11) Bã

Bâ

because the monodromy at both ends of the B-period is trivial, (∞) = I . Together with the asymptotic behavior (3.2) and single-valuedness (3.8, 3.9, 3.10) this implies that p(x), ˜ p(x) ˆ must jump by 2π n˜ a , 2π nˆ a when passing through the cut Cã , Câ , respectively. This is written as the equivalent condition p/˜ lã (x) − p/˜ kã (x) = 2π n˜ a

for

x ∈ Cã ,

p/ˆ lâ (x) − p/ˆ kâ (x) = 2π nˆ a

for

x ∈ Câ .

(3.12)

˜ It • The period Ba∗ for a fermionic pole connects x = ∞ to x = xa∗ on sheet ka∗ of p(x). then continues from x = xa∗ to x = ∞ on sheet la∗ of p(x). ˆ As fermionic singularities arise for coinciding eigenvalues, the regular parts of p(x) ˜ and p(x) ˆ must be equal modulo a shift by 2πn∗a , p/ˆ lâ (xa∗ ) − p/˜ kã (xa∗ ) = 2π n∗a .

(3.13)

Expressed as a B-period this yields − dp = 2πn∗a .

(3.14)

Ba∗

• In addition to (∞) = I we also have (0) = I . This means that a period connecting x = 0 with x = ∞ must be a multiple of 2π . In fact, the symmetry (3.3) enforces 0 0 0 d p˜ 1,2 = − d p˜ 3,4 = 2π m, pˆ k (0) = d pˆ k = 0. p˜ 1,2 (0) = −p˜ 3,4 (0) = ∞

∞

∞

(3.15) The integral for the AdS5 -part must vanish, because there cannot be windings on the time circle of AdS5 [46]. In fact, for physical applications one needs to consider the universal covering of AdS5 where time circle has been decompactified.


681

• When no confusion arises, we may use a unified notation Aa and Ba with a = 1, . . . , 2A for cuts and poles, A˜ a , Aˆ a , A∗a and Bã , Bâ , Ba∗ . The total number of cuts and poles is A = A˜ + Aˆ + A∗ . In this case we label the sheets pk by k = 1, . . . , 8 according to p1,2 = pˆ 1,2 ,

p3,4,5,6 = p˜ 1,2,3,4 ,

p7,8 = pˆ 3,4 .

(3.16)

This ordering leads to the configuration of sheets as depicted in Fig. 3. Some details of this representation are discussed in App. D. It makes physical excitations and the comparison to gauge theory more transparent. 3.2. Ansatz. The characteristic function of our algebraic curve is rational F (y, x) =

F˜ (y, x) F˜4 (x)y 4 + F˜3 (x)y 3 + F˜2 (x)y 2 + F˜1 (x)y + F˜0 (x) = , Fˆ (y, x) Fˆ4 (x)y 4 + Fˆ3 (x)y 3 + Fˆ2 (x)y 2 + Fˆ1 (x)y + Fˆ0 (x)

(3.17)

˜ y(x)} ˆ obeys the algebraic with F˜k (x), Fˆk (x) polynomials in x. The curve y(x) = {y(x)|| equation F˜ (y(x), ˜ x) = 0,

Fˆ (y(x), ˆ x) = 0.

(3.18)

We define the curve y(x) with a different prefactor as compared to the previous section as y(x) = (x − 1/x)2 x p (x).

(3.19)

This definition removes the poles at x = ±1 [47]. Branch points and fermionic poles. Bosonic branch points xã± of the S 5 part manifest themselves as inverse square roots in y(x). ˜ An asymptotic analysis shows that they are obtained when while F˜4 (xã± ) = 0 = F˜3 (xã± ). (3.20) F˜4 (xã± ) = F˜3 (xã± ) = 0 Similarly for branch points xâ± in the AdS5 part Fˆ4 (xâ± ) = Fˆ3 (xâ± ) = 0 Fermionic singularities is achieved by

xa∗

Fˆ4 (xâ± ) = 0 = Fˆ3 (xâ± ).

while

(3.21)

manifest themselves as double poles in y(x). A double pole

F˜4 (xa∗ ) = F˜4 (xa∗ ) = Fˆ4 (xa∗ ) = Fˆ4 (xa∗ ) = 0

F˜3 (xa∗ ) = 0 = Fˆ3 (xa∗ ). (3.22)

while

The behavior of F2,1,0 is generic at these points. Here we see that a non-zero F3 , unlike in [47], is required due to fermions. All these singularities are encoded in F4 (x) as F˜4 (x) = x 4 Fˆ4 (x) = x

4

2A˜

(x − xã+ )

2A˜

a=1

a=1

2Aˆ

2Aˆ

a=1

(x

− xâ+ )

a=1

(x − xã− )

∗ 2A

(x − xa∗ )2 ,

a=1

(x

− xâ− )

∗ 2A

(x − xa∗ )2 .

(3.23)

a=1

The factor x 4 is introduced for convenience as we shall see below. For F˜4 (x), Fˆ4 (x) there are in total 4A˜ + 4Aˆ + 2A∗ degrees of freedom.

682


Asymptotics. At x = ∞ the curve behaves as y(x) ∼ x and at x = 0 as y(x) ∼ 1/x. This is achieved by the following range of exponents in the polynomials: ∗ ˜ F˜k (x) = ∗x 4A+4A +8−k + . . . + ∗x k , ∗ ˆ Fˆk (x) = ∗x 4A+4A +8−k + . . . + ∗x k .

(3.24)

We can now count the remaining number of free coefficients. In F˜k (x), Fˆk (x), k < 4, there are 4A˜ + 4A∗ + 9 − 2k and 4Aˆ + 4A∗ + 9 − 2k degrees of freedom, respectively. This leaves 20A˜ + 20Aˆ + 34A∗ + 48 relevant coefficients in total. Unimodularity. The unimodularity condition yˆ1 + yˆ2 + yˆ3 + yˆ4 = y˜1 + y˜2 + y˜3 + y˜4 is imposed as a relation of the two leading coefficients of the algebraic equation, Fˆ3 (x) F˜3 (x) . = F˜4 (x) Fˆ4 (x)

(3.25)

This requires F˜3 (x) = F3∗ (x)

2A˜

(x − xã+ )

a=1

Fˆ3 (x) = F3∗ (x)

2Aˆ

2A˜

(x − xã− ),

a=1

(x

− xâ+ )

a=1

2Aˆ

(x − xâ− ),

(3.26)

a=1

with some polynomial ∗ +5

F3∗ (x) = ∗x 4A

+ . . . + ∗x 3 .

(3.27)

It reduces the number of degrees of freedom by 4A˜ + 4Aˆ + 4A∗ + 3 to 16A˜ + 16Aˆ + 30A∗ + 45. Symmetry. The symmetry y(1/x) = y(x) is realized by the conditions ∗ ˜ F˜k (1/x) = x −4A−4A −8 F˜k (x), ∗ ˆ Fˆk (1/x) = x −4A−4A −8 Fˆk (x), ∗ −8

F3∗ (1/x) = x −4A

F3∗ (1/x).

(3.28)

∗ +19 constraints and leaves 8A+8 ∗ +26 coefficients. ˜ A+15A ˆ ˜ A+15A ˆ This yields 8A+8

Singularities. We have to group up the residues at x = ±1 according to Sect. 2.6: Out of the 16 residues, there should only be 4 independent ones. This gives 12 constraints, but two of them have already been imposed by the unimodularity condition. As the singularities are at the fixed points x = ±1 of the symmetry x → 1/x, all 10 constraints can be imposed independently. This leaves 8A˜ + 8Aˆ + 15A∗ + 16 degrees of freedom.


683

Unphysical branch points. In addition to the physical branch points at xã± , xâ± the algebraic curve might have further ones. Generically, these singularities are square roots in contrast to the physical one which are inverse square roots. We can remove them using a condition of the discriminants21 R˜ = −4F˜12 F˜23 F˜4 + 16F˜0 F˜24 F˜4 − 27F˜14 F˜42 + 144F˜0 F˜12 F˜2 F˜42 − 128F˜02 F˜22 F˜42 +256F˜03 F˜43 +18F˜13 F˜2 F˜3 F˜4 −80F˜0 F˜1 F˜22 F˜3 F˜4 −192F˜02 F˜1 F˜3 F˜42 −6F˜0 F˜12 F˜32 F˜4 +144F˜02 F˜2 F˜32 F˜4 + F˜12 F˜22 F˜32 −4F˜0 F˜23 F˜32 −4F˜13 F˜33 +18F˜0 F˜1 F˜2 F˜33 −27F˜02 F˜34 , (3.29) ˆ The discriminants measure the product of squared distances of soluand similarly for R. ˜ ˆ tions y˜k (x) or yˆk (x). A single root of R(x) = 0 or R(x) = 0 thus implies a square root behavior which can only occur at x = xã± or x = xâ± . The discriminants must therefore have the form ˜ R(x) = x 12 (x 2 − 1)4

2A˜

(x − xã+ )

a=1

ˆ R(x) = x 12 (x 2 − 1)4

2Aˆ a=1

2A˜

˜ 2, (x − xã− ) Q(x)

a=1

(x

− xâ+ )

2Aˆ

ˆ 2. (x − xâ− ) Q(x)

(3.30)

a=1

It is clear that xã± and xâ± are roots, because all terms in (3.29) contain F˜4 or F˜3 . Noting the generic form of the discriminants ∗ ˜ ˆ R(x) = ∗x 24A+24A +36 + . . . + ∗x 12 , ∗ ˆ ˜ R(x) = ∗x 24A+24A +36 + . . . + ∗x 12 .

together with the inversion symmetry we find 5A˜ + 5A˜ + 3A˜ + 3Aˆ + 3A∗ + 8 remaining degrees of freedom.

12A∗

(3.31) + 8 constraints and

Single poles and A-cycles. We need to remove all the single poles and A-cycles from the curve y(x) which would otherwise give rise to undesired logarithmic behavior in the quasi-momentum when restoring the quasi-momentum from its derivative. The symmetry x → 1/x allows for 8 independent single poles in y(x) at x = ±1. There are A˜ + Aˆ independent A-cycles around bosonic cuts. Fermionic singularities contribute 2A∗ independent single poles: one for y˜ and one for yˆ at each x = xa∗ modulo inversion symmetry. Among all these single poles and A-cycles, there are 4 relations from the sum over all residues, one for each pair of sheets related by the symmetry. In total this yields A˜ + Aˆ + 2A∗ + 4 constraints and leaves 2A˜ + 2Aˆ + A∗ + 4 coefficients. B-periods. For each bosonic cut and for each fermionic singularity there is a B-period which must be integral. Furthermore, for each pair of sheets related by the symmetry, the B-period connecting 0 and ∞ must also be integral. Due to the unimodularity condition, ∗ +3 constraints ˜ A+A ˆ only three of these periods are independent. In total we obtain A+ ˜ ˆ and are left with A + A + 1 degrees of freedom. 21 We could also use the equivalent condition: All solutions to dF = 0 are on the curve unless there is a physical singularity at this value of x. However, it is not quite clear how to count the number of constraints from this condition.

684


Hypercharge. One degree of freedom corresponds to an irrelevant shift of the Lagrange ˆ multiplier, cf. Sect. 2.4. The final number of moduli for admissible curves is A˜ + A. 3.3. Mode Numbers and Fillings. We will now associate each of the A˜ + Aˆ moduli of the curve to one parameter per pair of bosonic cuts. We define the filling of an S 5 -cut Cã connecting sheets kã and lã as √ √ 1 λ λ 1 K˜ a = − 2 x + dx 1 − 2 p˜ kã (x) = d p˜ kã . (3.32) 8π i A˜ a x 8π 2 i A˜ a x Our definition uses the sheet kã , alternatively we might use lã and invert the sign. Equivalently, we define the filling for an AdS5 -cut Câ , but now using the sheet lâ , √ √ λ λ 1 1 ˆ x+ dx 1 − 2 pˆ lâ (x) = d pˆ lâ . (3.33) Ka = − 2 8π i Aˆ a x 8π 2 i Aˆ a x The corresponding definition using the sheet kâ would require an opposite sign. For completeness, we also define a filling for fermionic singularities xa∗ , √ √ λ λ 1 1 Ka∗ = − 2 x + dx 1 − 2 p˜ ka∗ (x) = d p˜ ka∗ , (3.34) 8π i A∗a x 8π 2 i A∗a x which we could also write using pˆ la∗ . It is not an independent modulus and it measures the residue at xa∗ . In addition to the fillings, a curve is specified by the mode numbers 1 1 1 n˜ a = − dp. dp, nˆ a = dp, n∗a = (3.35) 2π Bã 2π Bâ 2π Ba∗ These are discrete parameters and therefore do not count as moduli. Note that the Bperiods all start at x = ∞ on sheet kã , kâ , ka∗ and end at x = ∞ on sheet lã , lâ , la∗ , respectively. Furthermore, there is one overall winding number defined as 0 0 1 1 p˜ 1,2 = − p˜ 3,4 . (3.36) m= 2π ∞ 2π ∞ It is defined through the S 5 -part of the curve and there is no corresponding quantity for the AdS5 -part, because there cannot be windings in the non-compact time direction of the universal covering of AdS5 [46]. In most cases, the fillings give the right number of moduli, but for m = 0 there is a constraint among the fillings as we shall see below. Therefore, let us introduce one further modulus which we call the length,22 √

λ L= 16π 2 i

√ 4 A √ 4 λ λ dx dx εk p˜ k + dx ε p ˜ + εk p˜ k . k k 16π 2 i −1 8π 2 i Aa x 2 +1 a=1 k=1 k=1 k=1 4

(3.37) 22

The term ‘length’ is due to analogy with spin chains. For an alternative approach to identifying this conserved charge in the sigma model, see [62, 63].


685

Note that we use only half of the 2A cuts for the definition of length, one from each pair related by inversion symmetry. This definition depends on which of the two cuts we select from each pair and is therefore ambiguous. In a particular limit, however, this choice is obvious as we shall see in Sect. 3.7. The length is related to the fillings by the constraint23 mL =

A

(3.38)

na K a ,

a=1

which means that among {L, Ka } there are only A˜ + Aˆ independent continuous parame˜ A−1 ˆ ters: A+ independent fillings Ka and the length L. To derive it, consider the integral √ 4 2 λ dx p˜ k (x) − pˆ k2 (x) 3 32π i ∞ k=1 √ √ 4 4 2 2 λ λ 2 = p ˜ p˜ k (x) − pˆ k2 (x) dx (x) − p ˆ (x) + dx k k 3 3 32π i +1 32π i −1 k=1 k=1 √ 2A 4 λ 2 2 + dx (x) − p ˆ (x) p ˜ k k 32π 3 i Aa

0=

a=1

= mL −

k=1

A

(3.39)

na K a .

a=1

The first integral is zero due to p(x) ∼ 1/x at x = ∞. We then split up the contour of integration around the singularities and cuts. To obtain the last line, we split up the integrals around x = ±1 evenly in two and also split up the sum 2A into A a=1 a=1 and 2A . Then we transform half of the integrals to 1/x, a=A+1 dx dx f (x) = − f (1/x), (3.40) 2 AA+a Aa x and use the inversion symmetry 4

4 4 2 εk p˜ k (x) + 16π 2 m2 p˜ k2 (1/x) − pˆ k2 (1/x) = p˜ k (x) − pˆ k2 (x) − 4π m

k=1

k=1

k=1

(3.41) to transform them back. The terms proportional to m2 drop out from the integrals, they contain no residue, while the terms multiplying m sum up to L. The remaining integrals around x = ±1, √ 4 λ 1 2 dx 1 − (3.42) p˜ k (x) − pˆ k2 (x) = 0 3 2 64π i ±1 x k=1

23

This constraint reveals the ambiguity of L: For some cuts the mode numbers and fillings of the mirror cut are related by nA+a = 2m − na , KA+a = −Ka . If we interchange the cut Ca with its mirror image CA+a , L changes by 2Ka .

686


sum up to zero as discussed in Sect. 2.7. In the final step we have employed the identity √

4 λ 1 2 p˜ k (x) − pˆ k2 (x) = −na Ka dx 1 − 2 3 32π i Aa x k=1

(3.43)

which one obtains after pulling the contour Aa tightly around the cut Ca . Then the integrand p2 (x + ) − p2 (x − ) can be split into symmetric and antisymmetric parts. The antisymmetric part is equal on two sheets up to a sign. The symmetric parts then combine using (3.12, 3.13) and yield 2πna . The remaining integral is the filling. A more direct way to derive the constraint uses the Riemann bilinear identity, 1 1 dp dq − dp dq = Resa (p dq), (3.44) 2π i a 2π i a Aa Ba Ba Aa valid for any curve with a set of independent cycles Aa , Ba and two arbitrary holomorphic differentials dp, dq. Let us briefly sketch the proof: We take as p the quasi-momentum and dq = p dx and count the S 5 -part and AdS5 -parts with opposite signs. The first product of integrals will be zero due to (3.8). According to (3.11, 3.32, 3.33) the second product of integrals leads to the sum a na Ka over the bosonic cuts when the symmetry is taken into account as explained above. The sum of residues of p2 yields the contributions from the fermions using (3.14, 3.34). The residues from x = ±1 cancel and the term mL appears during symmetrization as above. 3.4. Moduli of string solutions. At this point we briefly summarize our results on the number of moduli and compare it to the general solution of strings in flat space or on plane waves. We have found one continuous modulus, the filling Ka , and one discrete parameter, na , per pair of cuts (related by inversion symmetry). Furthermore we need to specify which of the 4|4 sheets are connected by the cut through ka , la . The situation for fermionic poles is similar, only that their filling is not an independent parameter. In addition, there is one continuous global modulus, the length L, and one discrete global parameter, m, but also one global constraint which relates Ka , na , L, m. Note that we have discarded λ which can be considered as an external parameter. The classification for (classical) strings in flat space or on plane waves is similar: Consider a solution with only a finite number of active string modes. Let us furthermore assume a light-cone gauge to focus on the physical excitations. Then each mode is described by its mode number (na ), amplitude (Ka ) and orientation (ka , la ), where we have indicated in brackets the corresponding quantities in our sigma model. The amplitudes of fermions cannot be specified by regular numbers and thus should not be counted as continuous moduli. One overall level matching constraint relates the amplitudes and mode numbers (Ka , na ). The string tension (λ) will again be considered external. The only difference between strings in flat space and our model is the lack of a modulus describing the effective curvature (L) and a parameter describing winding (m). While the relation between amplitudes and fillings as well as integers n and mode numbers is obvious, the relation between sheets and orientation of the string needs further explanations. For cuts related to S 5 we see that there are 6 pairs of sheets and thus 6 choices for (kã , lã ). Similarly for AdS 5 . Fermions have to connect one sheet of each type and thus there are 16 choices. It thus seems that there are (6 + 6)|16 orientations. There


687

is however a further criterion which we use to distinguish cuts and poles. We denote the cuts/poles with εk = εl as physical. The cuts/poles with εk = εl are considered auxiliary. The explanation for this classification is that precisely the physical cuts/poles appear within the combination q(x) in (2.73) which is used to define the local charges (and also the energy shift, cf. the following subsection). Among the 6 types of bosonic cuts each, there are 4 physical and 2 auxiliary ones. The 16 types of fermionic poles split up evenly into 8 physical and 8 auxiliary ones. Thus the counting of orientations for physical modes, (4 + 4)|8, is as expected for a superstring. In conclusion we see that the moduli of admissible curves are in one to one correspondence to the moduli describing closed superstrings in flat space. We expect that the number of moduli and their types should be mostly independent of the background. The only relevant properties for the enumeration of moduli (open/closed, bosonic/supersymmetric, number of spacetime dimensions, smoothness of the target space, . . .) are the same in both theories. We take this as compelling evidence that all admissible curves, as discussed in this section, indeed correspond to at least one string solution. We thus believe that we have not missed a relevant characteristic feature in Sect. 2 for the construction of admissible curves and that our classification is complete.24 3.5. Global charges. Here we shall relate the global charges of PSU(2, 2|4) to the fillings. Let us concentrate on S 5 at first and define global fillings A √ λ 1 3 1 1 1 ˜ K1 = − dx 1 − 2 4 p˜ 1 − 4 p˜ 2 − 4 p˜ 3 − 4 p˜ 4 , 8π 2 i Aa x a=1 A √ λ 1 1 ˜ K2 = − dx 1 − 2 p˜ 1 + 21 p˜ 2 − 21 p˜ 3 − 21 p˜ 4 , 2 2 8π i Aa x a=1 √ A λ 1 1 ˜ K3 = − dx 1 − 2 p˜ 1 + 41 p˜ 2 + 41 p˜ 3 − 43 p˜ 4 . (3.45) 4 2 8π i Aa x a=1

These can also be represented as a sum of fillings K˜ a of the individual cuts and residues Ka∗ of fermionic poles. We will not do this explicitly, as there are too many pairs of sheets and thus too many types of cuts. The Dynkin labels [˜r1 , r˜2 , r˜3 ] of SU(4) are given by the following combinations: √ λ dx p˜ j (x) − p˜ j +1 (x) . (3.46) r˜j = 8π 2 i ∞ Their relation to the global fillings is as follows: r˜1 = K˜ 2 − 2K˜ 1 , K˜ 1 = 21 L − 43 r˜1 − 21 r˜2 − 41 r˜3 , r˜2 = L − 2K˜ 2 + K˜ 1 + K˜ 3 , K˜ 2 = L − 21 r˜1 − r˜2 − 21 r˜3 , r˜3 = K˜ 2 − 2K˜ 3 , K˜ 3 = 21 L − 41 r˜1 − 21 r˜2 − 43 r˜3 .

(3.47)

To derive these, it is convenient to make use of the inversion symmetry, cf. the previous subsection. 24 We only refer to the action variables of string solutions. Of course, the (time-dependent) angle variables are not described by the algebraic curve. According to standard lore, they correspond to a set of marked point on the Jacobian of the curve.

688


For AdS5 the results are very similar. Again we define the global fillings A

√

λ 8π 2 i a=1 A √ λ Kˆ 2 = 8π 2 i a=1 A √ λ Kˆ 3 = 8π 2 i Kˆ 1 =

a=1

1 3 dx 1 − 2 pˆ 1 − 41 pˆ 2 − 41 pˆ 3 − 41 pˆ 4 , 4 x Aa

1 1 dx 1 − 2 pˆ 1 + 21 pˆ 2 − 21 pˆ 3 − 21 pˆ 4 , 2 x Aa

1 1 dx 1 − 2 pˆ 1 + 41 pˆ 2 + 41 pˆ 3 − 43 pˆ 4 , 4 x Aa

(3.48)

which we might write as sums of the individual fillings. Then the Dynkin labels are given by √ λ rˆj = dx pˆ j +1 (x) − pˆ j (x) 2 8π i ∞

(3.49)

and related to the global fillings by rˆ1 = Kˆ 2 − 2Kˆ 1 , rˆ2 = −L − δE − 2Kˆ 2 + Kˆ 1 + Kˆ 3 , rˆ3 = Kˆ 2 − 2Kˆ 3 ,

Kˆ 1 = − 21 L − 21 δE − 43 rˆ1 − 21 rˆ2 − 41 rˆ3 , Kˆ 2 = − L − δE − 21 rˆ1 − rˆ2 − 21 rˆ3 , (3.50) Kˆ 3 = − 21 L − 21 δE − 41 rˆ1 − 21 rˆ2 − 43 rˆ3 .

Here we have introduced a new quantity δE, the energy shift δE =

A a=1

√

4 A √ λ dx λ dx q(x) (3.51) −εk p˜ k + εk pˆ k = − 2 2 2 8π i Aa x 4π i Aa x 2 a=1 k=1

with q(x) defined in (2.73). When we write rˆ2 in terms of the AdS5 energy E, E = −r2 − 21 r1 − 21 r3 = L + Kˆ 2 + δE,

(3.52)

we see that δE is indeed the energy shift when L + Kˆ 2 is interpreted as the bare energy. Finally, we introduce the global fermionic filling ∗

K =−

A a=1

√

4 λ 1 1 1 dx 1 − 2 2 p˜ k + 2 pˆ k . 8π 2 i Aa x k=1

(3.53)

It is related to the hypercharge eigenvalue r ∗ , √ 4 1 λ r = dx p˜ k (x) + 21 pˆ k (x) = 2B − K ∗ . 2 2 8π i ∞ ∗

(3.54)

k=1

We have introduced a charge B which is related to the √Lagrange multiplier, see Sect. 2.4. Under the symmetry it transforms as B → B + 21 υ λ.


689

3.6. Superstrings on AdS3 × S 3 . Let us consider solutions of the supersymmetric AdS5 × S 5 sigma model which extend only over a supersymmetric AdS3 × S 3 subspace, which in fact is given by the group manifold PSU(1, 1|2). For this class of solutions, the algebraic curve will split into two disconnected parts. The first component consists of p˜ 2 , p˜ 3 and pˆ 1 , pˆ 4 and the other component consists of the remaining four sheets. There are no branch cuts or fermionic poles connecting the two parts. Both components are isomorphic to algebraic curves obtained from the PSU(1, 1|2) sigma model [64– 66]. One of them corresponds to the monodromy in the fundamental representation, the other one to the monodromy in the antifundamental representation. These two curves are not unrelated, for a sigma model on a group manifold they should map into each other under inversion x → 1/x. Indeed, this is precisely what the AdS5 × S 5 sigma model implies, see Sect. 2.5. There are several conceptual differences which make it interesting to consider the AdS3 × S 3 model separately. First of all, the AdS3 × S 3 model leads to one algebraic curve without inversion symmetry (or, equivalently, two related algebraic curves) whereas the full AdS5 × S 5 model has only one self-symmetric curve. This also means that we can distinguish between a cut and its image under inversion: They reside on different components of the algebraic curve and we shall consider only one component. The definitions of length (3.37) and energy shift (3.51) thus become natural and unambiguous. In fact they become two of the global charges. Together with the spin S on AdS3 and spin J2 on S 3 they are the four charge eigenvalues of the isometry group PSU(1, 1|2) × PSU(1, 1|2). Again we can express the global charges through the fillings A √ λ 1 1 ˜ J2 = K = − dx 1 − 2 p˜ 2 − 21 p˜ 3 , 2 2 8π i Aa x a=1 √ A λ 1 1 S = Kˆ = (3.55) dx 1 − pˆ 1 − 21 pˆ 4 . 2 2 2 8π i Aa x a=1

Here we have defined J2 and S to match the fillings. The expansion of the quasi-momentum at x = ∞ is related to the charges of one of the global PSU(1, 1|2) factors √ λ dx p˜ 2 − p˜ 3 = L − 2K˜ = J1 − J2 , 2 8π i ∞ √ λ (3.56) dx pˆ 4 − pˆ 1 = −L − δE − 2Kˆ = −E − S. 2 8π i ∞ For convenience, we have defined the spin J1 and energy E to replace the length and energy shift as follow: L = J1 + J2 ,

δE = E − L − S.

(3.57)

The expansion at x = 0 relates to the charges of the other global PSU(1, 1|2) factor √ λ dx p˜ 2 − p˜ 3 = L = J1 + J2 , 2 2 8π i 0 x √ λ dx pˆ 4 − pˆ 1 = −L − δE = −E + S. (3.58) 2 2 8π i 0 x These expressions agree precisely with the rank-one subsectors of R×S 3 and AdS3 ×S 1 considered in [29, 46].

690


pˆ1 pˆ2 0

u∗1

C˜1

u∗2

p˜1 Cˆ2

p˜2 p˜3 p˜4

0 pˆ3 pˆ4

Cˆ1

Fig. 5. The Frolov-Tseytlin limit of the configuration of cuts and poles in Fig. 3. All inverse cuts and poles as well as the poles at x = ±1 have been scaled to u = 0 and absorbed into an effective pole

3.7.√The Frolov-Tseytlin limit. In this section we shall discuss the Frolov-Tseytlin limit L/ λ → ∞ of the curve.25 In this limit, half of the cuts and poles approach x = ∞ and half of them approach x = 0.26 Let us label those cuts and poles which escape to x = ∞ by a = 1, . . . , A, those which approach x = 0 will be labelled by a = A + 1, . . . , 2A. Therefore, there is a natural choice for those cuts which contribute to the definition of the length in (3.37). We will define for convenience a rescaled variable27 √ λ x. u= 4π

(3.59)

In other words, in the u-plane all cuts and poles with a = 1, . . . , A approach finite values of u while their images approach u = 0. Similarly, the poles at x = ±1 approach u = 0, see Fig. 5. This means that the point u = 0 is special and p(u) near u = 0 is not directly related to p(x) near x = 0, but there are contributions from the cuts and poles. To understand the expansion of p(u) at u = 0 we shall define a contour C in the x-plane which encircles the poles at x = ±1 and all the cuts and poles with a = A + 1, . . . , 2A. Equivalently, this may be considered a contour which excludes x = ∞ and all the cuts and poles with a = 1, . . . , A. After rescaling C merely encircles the point u = 0 in the u-plane which can be used to obtain the expansion of p(u) according to the formula28 ∂ r−1 pk (0) = ∂ur−1 25

4πL √ λ

r−1

1 dx pk (x). 2πi C x r

(3.60)

At the level of the action and the Hamiltonian, this limit was studied in [44, 45]. Solutions with self-symmetric cuts do not have a proper√ Frolov-Tseytlin limit. 27 This relationship needs to be refined at higher orders in λ/L [29, 42]. 28 This formula explains why there are two different transfer matrices T (x), T¯ (u) in [42]. The transfer matrix T¯ (u) is the suitable one for finite g, while T (x) is the effective one according to this formula. It should also be useful to understand the relationship between conserved local charges in string theory and gauge theory in [67, 68]. 26


691

Using the identities and definitions in Sect. 3.3, 3.5 we find the singular behavior of all sheets at u = 0, A 1 1 dx 2π p˜ k (x) dx = √ B + εk L − p˜ (x), 2 k 2π i C 2π i λ Aa x a=1 A 1 2π 1 dx pˆ k (x) dx = √ B + εk L + εk δE − pˆ k (x), (3.61) 2π i C 2π i Aa x 2 λ a=1 and the first few moments of the generator of local charges (2.73), 1 2π δE dx q(x) = − √ , 2πi C λ dx 1 q(x) = 2πm, 2πi C x 1 dx 2π δE q(x) = √ . 2πi C x 2 λ

(3.62)

Now we note that the energy shift (3.51) δE = −

A a=1

√

A λ dx du λ 1 q(x) = − q(u) 4π 2 i Aa x 2 8π 2 L 2π i Aa u2 a=1

(3.63)

is of order O(λ/L). After rescaling we obtain the singular behavior of p(u) at u = 0 from (3.60, 3.61) 1 p˜ k (u) ∼ pˆ k (u) ∼ u

εk B + 2 2L

(3.64)

and the first few local charges from (3.60, 3.62) q(u) = 2πm +

8π 2 L δE u + O(u2 ). λ

(3.65)

In particular, the momentum constraint is q(0) = 2π Z while the individual sheets pk (u) no longer have a fixed finite value at u = 0. The above curve apparently is the spectral curve of the supersymmetric Landau-Lifshitz model in [50].29 This model is related to the coherent state approach to gauge theory [44, 45]. Unlike the curve of the full superstring, this curve has only one singular point at u = 0 and thus seems to be similar to the one of the classical Heisenberg magnet discussed in [29]. We expect the expansion of the function q(u) around u = 0 to yield the local charges of the model, while the point u = ∞ should be related to Noether and multi-local charges. 29

We thank A. Mikhailov for discussions on this point.

692


4. Integral Representation of the Sigma Model We can reformulate the algebraic curve in terms of a Riemann-Hilbert problem, i.e. as integral equations on some density functions. This formulation is similar to the thermodynamic limit of the Bethe equations for the gauge theory counterpart. It is thus suited well for a comparison of both theories, especially at higher loops (once the gauge theory results become available). We start by representing the various discontinuities of cuts of the algebraic curve by integrals over densities. We then match this representation to the properties derived earlier and thus fix several of the parameters. The remaining properties lead to equations which are of the same nature as the Bethe equations in the thermodynamic limit. In order to be more explicit, we specify the equations for a number of subsectors, while full equations are written out only in App. D. Finally, we compare to one-loop gauge theory and find complete agreement.

4.1. Parametrization of the quasi-momentum. The quasi-momentum p(x) is a function with two sets of four sheets p˜ k (x) and pˆ k (x), k = 1, 2, 3, 4. These are analytic functions of x except at the singular points x = ±1, a set of branch cuts and additional fermionic poles. Ansatz. Let us construct a generic ansatz for p(x): It is straightforward to incorporate the poles at x = ±1 with undetermined residues a˜ k± and aˆ k± . The branch cuts and fermionic poles will be contained in several resolvents G(x). All the resolvents G(x) will be defined to vanish at x = ∞, consequently we should add an undetermined constant b˜k , bˆk for each sheet. The branch cuts connect two sheets of either p(x) ˜ or p(x). ˆ The discontinuity of the ˜ kl (x) or G ˆ kl (x). As branch cuts between sheets k and l is contained in the resolvent G the graded sum of all sheets pk should be zero, the sum of discontinuities must cancel. ˜ kl and G ˆ kl must be antisymmetric in k, l. A fermionic pole appears In other words, G on a sheet p˜ k and a sheet pˆ l with the same residue for both sheets. They are contained in the resolvent G∗kl (x). We shall include only the cuts/poles with a = 1, . . . , A in the resolvent G(x). Their images with a = A + 1, . . . , 2A under the inversion symmetry (3.3) will be incorporated by G(1/x). This leaves some discrete arbitrariness of which cuts/poles belong to G(x) and which to G(1/x). Note that although the algebraic curve is invariant under such permutations, our interpretation of some quantities will have to change. Taking the above constraints into account, we arrive at the following ansatz: p˜ k (x) = pˆ k (x) =

4

a˜ + a˜ − ˜ kl (x) − G ˜ k l (1/x) + G∗kl (x) − G∗ (1/x) + k + k + b˜k , G kl x−1 x+1

l=1 4 l=1

aˆ + aˆ − ˆ lk (1/x) + G∗lk (x) − G∗ (1/x) + k + k + bˆk , ˆ lk (x) − G G lk x−1 x+1 (4.1)

where the permutation k of sheets is defined in (2.58). We will now determine the constants using the known properties of p(x).


693

˜ ˆ Resolvents. The bosonic resolvents G(x), G(x) are defined in terms of the densities ρ(x), ˜ ρ(x) ˆ as follows: ˜ kl (x) = G

dy ρ˜kl (y) 1 , 2 ˜ Ckl 1 − 1/y y − x

ˆ kl (x) = G

Cˆkl

dy ρˆkl (y) 1 . 1 − 1/y 2 y − x

(4.2)

.

(4.3)

The fermionic resolvent G∗kl (x) is given by a set of poles30 ∗

G∗kl (x)

=

Akl

∗ αkl,a

∗2 1 − 1/xkl,a a=1

1 ∗ xkl,a

−x

All the resolvents vanish at x = ∞ and are analytic functions except on the curves Ca or at the poles xa∗ . They are obviously single-valued, i.e. the cycles of dG around cuts/poles vanish. The filling of a cut (3.32, 3.33, 3.34) is given by K˜ kl,a =

√ λ dy ρ˜kl (y), 4π C˜kl,a

Kˆ kl,a =

√ λ dy ρˆkl (y), 4π Cˆkl,a

∗ Kkl,a

√ λ ∗ α . = 4π kl,a (4.4)

Singularities. From Sect. 2.7 we know the general structure of the residues at x = ±1. The residues a˜ k± for S 5 are linked to the residues aˆ k± for AdS5 . Furthermore, all residues are paired. We can thus write them in terms of four independent constants c1,2 , d1,2 using (2.60), a˜ k± = aˆ k± =: 21 c1 ± 21 c2 + 21 d1 εk ± 21 d2 εk .

(4.5)

Asymptotics. The asymptotics p(x) ∼ 1/x at x = ∞ fixes all the constants b, b˜k =

˜ k l (0) + G∗ (0) , G kl

4 l=1

bˆk =

ˆ lk (0) + G∗ (0) . G lk

4

(4.6)

l=1

Unimodularity. The graded sum 4

p˜ k (x) − pˆ k (x) = 0

(4.7)

k=1

˜ kl , G ˆ kl in k, l. of all sheets indeed vanishes trivially by antisymmetry of G 30 Strictly speaking the residues must be nilpotent numbers because they represent a product of two Grassmann odd numbers, but we can mostly ignore this fact.

694


Symmetry. The inversion symmetry leads to the following expressions: p˜ k (1/x) = −p˜ k (x) − a˜ k+ + a˜ k− + b˜k + b˜k = −p˜ k (x) + 2π εk m, !

! p˜ k (1/x) = −pˆ k (x) − aˆ k+ + aˆ k− + bˆk + bˆk = −pˆ k (x)

(4.8)

with the permutation k of sheets is defined in ( 2.58). When we substitute the above expressions for a˜ k± , aˆ k± , we find c2 =

4

1 2

G∗kl (0),

d2 =

k,l=1

1 2

4

ˆ lk (0) + G∗lk (0) εk G

(4.9)

k,l=1

as well as the (momentum) constraint 1 2

4

ˆ kl (0) + G∗kl (0) − G∗lk (0) = 2π m. ˜ kl (0) + G εk G

(4.10)

k,l=1

Length. We substitute p˜ k in the definition of length and obtain √ √ 4 4 λ λ ˜ + − εk (a˜ k + a˜ k ) − εk Gkl (0) + G∗ L= kl (0) . 8π 4π k=1

(4.11)

k,l=1

This leads to 2π L d1 = √ + λ

1 2

4

˜ kl (0) + G∗ εk G kl (0) .

(4.12)

k,l=1

Note that we have assumed that all cuts a = 1, . . . , A˜ are captured by G(x) while the cuts a = A˜ + 1, . . . , 2A˜ are captured by G(1/x). Hypercharge. The remaining constant c1 corresponds to a shift of the Lagrange multiplier and is irrelevant. We shall express it by means of the hypercharge B defined in (3.54), c1 =

1 2

4

2π B G∗ . kl (0) + √ λ k,l=1

(4.13)

Energy Shift. We substitute pk in the definition of energy shift and obtain √ 4 λ ˜ ∗ ˆ εk Gkl (0) + G∗ δE = kl (0) − Glk (0) − Glk (0) . 4π k,l=1

(4.14)


695

4.2. Integral equations. Let us now assemble and simplify the various findings of the previous section. As a first step, we write G(x), but not G(1/x), G(0) or G (0), in terms of the inversion-symmetric function Hkl (x) := Gkl (x) + Gkl (1/x) − Gkl (0). The terms in the integrand of H combine as follows: 1 1 1 1 1 + − = , 2 1 − 1/y y−x y − 1/x y (y + 1/y) − (x + 1/x) which means that we can write H as

H (x) :=

du ρ(u) , C u − (x + 1/x)

where ρ transforms as a density, dx ρ(x) = du ρ(u), under the map u(x) = x + 1/x, x(u) = 21 u + 21 u 1 − 4/u2 .

(4.15)

(4.16)

(4.17)

(4.18)

The conversion of G(x) to H (x) creates a few more instances of Gkl (1/x) which turn out to pair up with existing instances of Gk l (1/x) in all cases. These can be rewritten by applying f k + fk =

1 2

4

fl + 21 εk

l=1

4

εl f l .

(4.19)

l=1

This identity for any fk can easily be verified by evaluating it for all possible values k = 1, 2, 3, 4. It is convenient to introduce the following combinations of resolvents: ˜ sum (x) = G

1 2

4

˜ kl (x) + G∗kl (x) , εk G

k,l=1

ˆ sum (x) = G

1 2

4

ˆ lk (x) + G∗lk (x) , εk G

k,l=1

G∗sum (x) =

1 2

4

G∗kl (x),

k,l=1

˜ sum (x) − G ˆ sum (x). Gmom (x) = G

(4.20)

We can now write down the simplified quasi-momentum p˜ k (x) = pˆ k (x) =

4

H˜ kl (x) + Hkl∗ (x) + εk F˜ (x) + F ∗ (x),

l=1 4 l=1

Hˆ lk (x) + Hlk∗ (x) + εk Fˆ (x) + F ∗ (x).

(4.21)

696


All the terms which do not follow the regular pattern of resolvents H could be absorbed into three potentials F ,

ˆ sum (0) 2π L 1/x G ˜ sum (1/x) + Gmom (0), ˜ + −G √ + Gsum (0) 2 1 − 1/x 1 − 1/x 2 λ ˆ sum (0) 1/x G 2π L ˆ sum (1/x), ˜ sum (0) + −G Fˆ (x) = √ + G 2 1 − 1/x 1 − 1/x 2 λ 2π B 1/x G∗ (0) F ∗ (x) = √ + G∗ (0) + sum 2 − G∗sum (1/x) . (4.22) sum 2 1 − 1/x 1 − 1/x λ F˜ (x) =

It might be useful to note the transformation of the potentials F under the symmetry31 F˜ (1/x) = −F˜ (x) − H˜ sum (x) + Gmom (0), Fˆ (1/x) = −Fˆ (x) − Hˆ sum (x),

∗ F ∗ (1/x) = −F ∗ (x) − Hsum (x).

(4.23)

The integral equations (3.12, 3.13) enforcing integrality of the B-periods (3.11, 3.14) read p/˜ l (x) − p/˜ k (x) = 2π n˜ kl,a p/ˆ l (x) − p/ˆ k (x) = 2π nˆ kl,a p/ˆ l (x) − p/˜ k (x) = 2πn∗kl,a

for x ∈ C˜kl,a , for x ∈ Cˆkl,a ,

∗ for x = xkl,a .

(4.24)

These equations must be supplemented by the momentum constraint (4.10) Gmom (0) = 2πm.

(4.25)

Note that the potential can appear in various combinations depending on the type of cut/pole. Let us denote those cuts/poles with εk = εl as physical, the others are considered as auxiliary. The physical cuts are precisely the ones that contribute to Gmom which in turn contains the total momentum for the momentum constraint (4.10) and the energy shift (4.14). They connect sheets 1, 2 to sheet 3, 4 of either type. A physical cut is subject to the potential 2F˜ or 2Fˆ depending on whether it is of S 5 -type or AdS5 -type. For an auxiliary bosonic cut there is no effective potential. For physical fermionic poles we get the potential F˜ + Fˆ and F˜ − Fˆ for auxiliary fermions. The global charges are found at x = ∞, they are determined through the fillings K, the length L and the energy shift δE: The expansion of H gives the total filling K of the cuts in H , 31

The summed resolvents Hsum are defined in analogy to (4.20).


H (x) = −

697

A 1 4π Ka 1 4π K + O(1/x 2 ) = − √ + O(1/x 2 ). √ x x λ λ a=1

(4.26)

The expansion of F provides the length and the energy shift (4.14), 1 2π L F˜ (x) = √ + O(1/x 2 ), x λ

1 2π (L + δE) Fˆ (x) = + O(1/x 2 ). (4.27) √ x λ

The energy shift is given by √ λ (0). G δE = 2π mom

(4.28)

4.3. Rank-one sectors. We will now investigate the cases when only one of the physical ˜ G ˆ or G∗ . resolvents is non-zero. The final result depends on the type of resolvent, G, ˜ 23 and consider one quasi-momentum Bosonic, Compact. We turn on only G = G p = p˜ 2 = −p˜ 3 . This reduces to the case of strings on R × S 3 investigated in [29], 2G (0)/x 4π L 1/x 2p(x) / = +2H / (x) + 2F˜ (x) = 2/ G(x) + + √ 2 1 − 1/x λ 1 − 1/x 2 = −2πna for x ∈ Ca .

(4.29)

Note that the term G(1/x) has precisely cancelled out and p(x) is no longer symmetric under x → 1/x. This is related to the fact that spacetime is now a group manifold. In this case, the image under inversion is given by a different quasi-momentum, here p˜ 1 . Also the length L now becomes a true global charge next to J . This is related to the left and right symmetry of group manifolds. ˆ 14 and consider the single quasi-momenBosonic, non-compact. We turn on only G = G tum p = pˆ 1 = −pˆ 4 . This reduces to the case of strings on AdS3 × S 1 investigated in [46], 2G(0)/x 2 4π L 1/x + √ 2 1 − 1/x λ 1 − 1/x 2 for x ∈ Ca . (4.30)

2p(x) / = −2H / (x) + 2Fˆ (x) = −2/ G(x) − = −2πna

Fermionic. We turn on only G = G∗24 and consider the quasi-momenta p˜ = p˜ 2 , pˆ = pˆ 4 . The two quasi-momenta are given by p(x) ˜ = H (x) + F˜ (x) + F ∗ (x) and p(x) ˆ = ∗ ˆ H (x) − F (x) + F (x), 2π (B + L) 1/x + G (0) p(x) ˜ = G(x) + , (4.31) √ 1 − 1/x 2 λ p(x) ˆ = G(x) +

2π (B − L) 1/x G(0)/x 2 + . √ 1 − 1/x 2 1 − 1/x 2 λ

The relevant combination for the integral equation is the difference of sheets p(x) ˜ − p(x) ˆ = F˜ (x) + Fˆ (x),

698


p(x) /˜ − p(x) /ˆ =−

G(0)/x 2 G (0)/x 4πL 1/x + + √ = −2π na 1 − 1/x 2 1 − 1/x 2 λ 1 − 1/x 2

for x = xa∗ . (4.32)

This agrees precisely with the expression derived from the near-plane-wave limit in [41]. 4.4. Superstrings on AdS3 × S 3 . The above three subsectors can be combined into one larger sector. Let us consider only the following four sheets p1 = pˆ 1 , p2 = p˜ 2 , p3 = p˜ 3 , ˆ 41 , G ˜ 23 , G∗ , G∗ , G∗ , G∗ so that again there p4 = pˆ 4 and corresponding resolvents G 21 31 24 34 is no apparent inversion symmetry. By inspection of the quasi-momenta there are only three independent combinations of resolvents appearing:

Gmom

ˆ 41 , −G G1 = −G∗21 − G∗31 ˜ 23 − G∗31 + G∗24 − G ˆ 41 , = G2 = +G G3 = +G∗34

ˆ 41 . + G∗24 − G

(4.33)

The quasi-momenta then read

2π(B + L) G1 (0)/x 2 , + G2 (0) − G1 (0) − √ 1 − 1/x 2 λ 2π(B + L) 1/x G1 (0)/x 2 p2 (x) = G2 (x) − G1 (x)+ (0) − G (0) − , + G √ 2 1 2 1 − 1/x 1 − 1/x 2 λ 2π(B − L) 1/x G3 (0)/x 2 p3 (x) = G3 (x) − G2 (x)+ (0) − G (0) + , + G √ 3 2 1 − 1/x 2 1 − 1/x 2 λ 2π(B − L) 1/x G3 (0)/x 2 p4 (x) = G3 (x) + (0) − G (0) + . + G √ 3 2 1 − 1/x 2 1 − 1/x 2 λ (4.34) p1 (x) =

− G1 (x) +

1/x 1 − 1/x 2

The differences of adjacent sheets which appear in the integral equations are given by p1 (x) − p2 (x) = −G2 (x),

2G2 (0)/x 4π L 1/x + √ 2 1 − 1/x λ 1 − 1/x 2 G (0)/x G1 (0)/x 2 ˜ 1 (x) − 1 −G − 2 1 − 1/x 1 − 1/x 2 G (0)/x G3 (0)/x 2 − G3 (x) − 3 − , 1 − 1/x 2 1 − 1/x 2 p3 (x) − p4 (x) = −G2 (x).

p2 (x) − p3 (x) = +2G2 (x) +

(4.35)

Differences of non-adjacent sheets are obtained by summing up the equations. 4.5. Comparison to gauge theory. Let us briefly comment on the comparison to N = 4 gauge theory. An in-depth comparison of the spectral curves can be found in [54]. The complete one-loop dilatation generator has been derived in [53, 26]. It is integrable and one can use a Bethe ansatz to find its energy (scaling dimension) eigenvalues [53]. In the thermodynamic limit [43, 21], which should be related to string theory [20], the


699

Bethe equations have been written in [26]. Their form does not resemble the equations (4.21, 4.22, 4.24) very much, but rather the one in the previous Sect. 4.4. We shall refrain from transforming the equations here and refer the reader to App. D. The resulting equations (D.11) 7

Mj,j H / j (x) + Fj (x) = −2πnj,a

for x ∈ Cj,a

(4.36)

j =1

can be seen to agree with the equations in [26]. Also the expressions for the momentum constraint and energy shift as well as the local and global charges agree. Note that in the Frolov-Tseytlin limit, see Sect. 3.7, the potential Fj reduces to a term proportional to Vj L/u, cf. [29, 47] for similar results. We have thus proven the agreement of the spectra of one-loop planar gauge theory and classical string theory. 5. Conclusions and Outlook We solve the problem of describing all classical solutions of the superstring sigma-model in AdS5 ×S 5 in terms of their spectral curves. Let us underline the importance of dealing with the whole supersymmetric string theory on the AdS5 ×S 5 space, including the fermionic degrees of freedom, for its quantization. For the classical string we can drop the fermions and the two bosonic sectors, AdS5 and S 5 , appear to be completely factorized (up to the constraints on the fixed poles √ and total momentum). Conversely, the quantum corrections at higher powers of ∼ 1/ λ , make the two sectors interact nontrivially, an effect which is already seen in the super spin chain for the one-loop approximation to gauge theory. It is also clear that the direct quantization of sigma models in closed subsectors, like R × S 5 , does not make much sense since those models are even not conformal. It seems that it is better to attack directly the full supersymmetric quantum theory on AdS5 × S 5 . Our paper shows that, at least at the classical level, the full string theory has no principal difficulties comparing to the simpler subset sigma models. The curves are solutions of a Riemann-Hilbert problem and as such can be encoded in the set of singular integral equations which we have derived. We hope that this classical result will be a useful starting point for a quantization. Some indications that the integrable structures persist in the quantum regime are found in [69–71, 49, 61, 72, 73]. There are several benefits of the formulation in terms of algebraic curves which might facilitate quantization: For one, the formulation is completely gauge independent, at no point one is required to fix a gauge; especially we can preserve full kappa symmetry [52]. Moreover, the curve consists only of action variables. Due to the Heisenberg principle this is all we can ask for to know in the quantum theory. Finally, there are no unphysical degrees of freedom associated to the curve. These would usually contain spurious infinities and their absence should make the curve completely finite. Our integral equations can be interpreted as the classical limit of the yet to be found discrete Bethe equations, which describe the exact quantum spectrum of the string, cf. the ansatz by Arutyunov, Frolov and Staudacher [69] and a corresponding ‘string chain’ [70] for some initial steps in this direction. The existence of such equations is an assumption, but since many quantum integrable systems are solvable by a Bethe ansatz, in particular some sigma models [74–77], this assumption does not look inconceivable. We believe that in any event integrability will be an important ingredient in solving string theory in AdS5 × S 5 , be it a Bethe ansatz or some other method, and hope that our findings will be helpful in attacking this challenging problem.

700


We should mention that many classical string solutions (those without self-symmetric cuts) admit a regular expansion in the ’t Hooft coupling.32 We have compared the energy spectrum of the classical string with the spectrum of anomalous dimensions in N = 4 SYM which at one loop is given by a Bethe ansatz. Our comparison is based on Bethe equations but it can also be performed at the level of spectral curves. Although we should not expect agreement beyond third order in the effective coupling [23], we might use the present result as a source of inspiration for higher-loop gauge theory. The method of solving the spectral problem in terms of algebraic curves which we employ for this sigma model is usually called the finite gap method (see the book [78] for a good introduction). This means that we look for a finite genus algebraic curve characterizing some solution. It was first proposed in [79, 80] for the KdV system and later generalized to KP equations in [81]. In principle, for any given algebraic curve one can construct explicitly the corresponding solution of the equations of motion in terms of Riemann theta functions. The dependence on time enters linearly in the argument of the Riemann theta function and the frequencies are given by the periods of certain Abelian differentials. Note that in our investigation we have the angle variables; these enter as the initial phase for the arguments of the theta function. Moreover, to construct the solution for the AdS5 × S 5 coset model, one would first have to fix a gauge for the local symmetries. Finally, only up to genus one the theta functions can be expressed in terms of conventional algebraic and elliptic functions. Beyond that they are known only as integrals or series and therefore less efficient. Acknowledgement. It is a pleasure to thank A. Agarwal, G. Arutyunov, C. Callan, L. Dolan, A. Gorsky, A. Mikhailov, J. Minahan, R. Plesser, S. G. Rajeev, L. Rastelli, R. Roiban, A. Ryzhov, F. Smirnov, D. Serban, A. Sorin, M. Staudacher A. Tseytlin, H. Verlinde and M. Wijnholt for discussions. N. B. would like to thank the Kavli Institute for Theoretical Physics, the Ecole Normale Supérieure and the Albert-EinsteinInstitut for the kind hospitality during parts of this work. K. Z. would like to thank the Kavli Institute for Theoretical Physics for hospitality. The work of N. B. is supported in part by the U.S. National Science Foundation Grants No. PHY99-07949 and PHY02-43680. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. V. K. would like to thank the Princeton Institute for Advanced Study for the kind hospitality during a part of this work. The work of V. K. was partially supported by European Union under the RTN contracts HPRN-CT-2000-00122 and 00131 and by NATO grant PST.CLG.978817. The work of K. S. is supported by the Nishina Memorial Foundation. The work of K.Z. was supported in part by the Swedish Research Council under contracts 621-2002-3920 and 621-2004-3178, by the Göran Gustafsson Foundation, and by the RFBR grant 02-02-17260.

A. Supermatrices We shall write supermatrices as matrices where horizontal/vertical bars separate between rows/columns with even and odd grading. We shall only consider bosonic supermatrices. The grading matrix η consequently is given by +I 0 η= . (A.1) 0 −I For example, it can be used to define the supertrace of a supermatrix as a regular trace of a supermatrix, str A = tr η A.

(A.2)

32 Typically, this is an expansion in inverse powers of some large conserved charge. In our framework a useful conserved charge is the length L, see also [62, 63]. The length is related to angular momentum J on S 5 in many cases and thus gives a natural generalization of the Frolov-Tseytlin proposal [20, 24].


701

The supertrace is cyclic str A B = str B A.

(A.3)

The superdeterminant is defined as det A det(A − BD −1 C) AB sdet = , = CD det D det(D − CA−1 B)

(A.4)

it obeys sdet(AB) = sdet A sdet B

(A.5)

and is compatible with the identity sdet exp A = exp str A.

(A.6)

Supertranspose. The supertranspose is defined as ST T AB A CT . = CD −B T D T

(A.7)

Like the common transpose, it inverts the order within a product of matrices (A B)ST = B ST AST

(A.8)

and does not change supertraces and superdeterminants str AST = str A,

sdet AST = sdet A.

(A.9)

Unlike the common transpose, it is not an involution, but a Z4 -operation due to the identity (AST )ST = η A η.

(A.10)

Furthermore, a supersymmetric or a superantisymmetric matrix require a slightly modified definition A = +η AST ,

A = −η AST .

(A.11)

(1|1) × (1|1) Supermatrices. Let us collect some formulas for (1|1) × (1|1) supermatrices ab A= . (A.12) cd The inverse is given by −1

A

=

1 a

+ abc 2d c − ad

b − ad 1 bc d − ad 2

.

(A.13)

a bc − 2. d d

(A.14)

The supertrace and superdeterminant read str A = a − d,

sdet A =

702


It can be diagonalized by the matrices bc b 1 − 2(a−d) + a−d 2 T = , c bc − a−d 1 + 2(a−d) 2

T −1 =

1−

bc 2(a−d)2 c + a−d

b − a−d

1+

bc 2(a−d)2

, (A.15)

such that T AT

−1

=

α1 0 0 α2

,

α1 = a +

bc , a−d

α2 = d +

bc . a−d

(A.16)

The eigenvalues satisfy the sum and product rules α1 − α2 = str A,

α1 = sdet A α2

(A.17)

sdet(A − α2 ) = ∞.

(A.18)

as well as the characteristic equation sdet(A − α1 ) = 0,

Clearly α1 and α2 are associated to different gradings. For str A = 0, i.e. a = d, the eigenvalues degenerate and some problems arise as in the case of bosonic matrices. B. Cosets and Vectors In this appendix we shall explain the relationship between the vector model used in [47] and the coset model used in this paper. We will start with the coset model. The physically relevant cosets S 5 = SU(4)/Sp(2) and AdS5 = SU(2, 2)/Sp(1, 1) require several i’s at various places. These can be avoided by considering the coset SL(4, R)/Sp(4, R). Its algebraic structure is precisely the same but the formulas are slightly easier to handle. For the breaking of SL(4, R) we can use a fixed 4 × 4 antisymmetric matrix E, say 0 +I E= , (B.1) −I 0 where each block corresponds to a 2 × 2 matrix and I is the identity. The currents of the coset model in the moving frame are J = −g −1 dg, H = 21 J − 21 EJ T E −1 , K = 21 J + 21 EJ T E −1 .

(B.2)

In the fixed frame, which is related to the moving frame by j = gJ g −1 , etc., they are given by j = −dg g −1 , h = − 21 dg g −1 + 21 gEdg T g −T E −1 g −1 , k = − 21 dg g −1 − 21 gEdg T g −T E −1 g −1 .

(B.3)


703

We can now rewrite k as k = − 21 dg g −1 − 21 gEdg T (gEg T )−1 = − 21 d(gEg T ) (gEg T )−1

(B.4)

and see that it can be rewritten as k = − 21 dX X−1

with

X = gEg T .

(B.5)

We would now like to interpret X as the fundamental field of the theory. For all g ∈ Sp(4, R) we find X = E, thus X parametrizes the coset SL(4, R)/Sp(4, R). Note that we can define a norm for X by εαβγ δ X αβ X γ δ = εαβγ δ E αβ E γ δ det g = −8.

(B.6)

Starting from a generic matrix X, the conditions X = −X T and εαβγ δ X αβ X γ δ = −8 leave 5 degrees of freedom for X and therefore such an X indeed parametrizes the coset SL(4, R)/Sp(4, R) which has 15 − 10 = 5 dimensions. We now parametrize X as X = σ · X,

2 = −1, X

(B.7)

is a vector of SO(3, 3) and σ is a chiral component of the Clifford algebra. where X This reveals the connection between the sigma model (B.5) and the vector model (B.7), they are merely reparametrizations of the same model. More explicitly the matrix X is by given through the components of the vector X 

0  −X1 − X4 X= −X2 − X5 −X3 − X6

+X1 + X4 0 −X3 + X6 +X2 − X5

+X2 + X5 +X3 − X6 0 −X1 + X4

 +X3 + X6 −X2 + X5  +X1 − X4  0

(B.8)

such that αβ γ δ 1 8 εαβγ δ X X

2 = X1 2 + X2 2 + X3 2 − X4 2 − X5 2 − X6 2 . =X

(B.9)

The corresponding expressions for SO(6) are 

0  −X1 − iX2 X= −X3 − iX4 −X5 − iX6 where again 18 εαβγ δ X αβ X γ δ SO(2, 4) we find  0  −X5 − iX0 X= −X1 − iX2 −X3 − iX4

+X1 + iX2 0 −X5 + iX6 +X3 − iX4

+X3 + iX4 +X5 − iX6 0 −X1 + iX2

 +X5 + iX6 −X3 + iX4  , +X1 − iX2  0

(B.10)

2 , but with a positive signature of the norm. For = X +X5 + iX0 0 −X3 + iX4 +X1 − iX2

+X1 + iX2 +X3 − iX4 0 +X5 − iX0

 X3 + iX4 −X1 + iX2  , −X5 + iX0  0

(B.11)

2 has signature +−−−−+. These expressions only differ from the expressions where X for SO(3, 3) by relabelling the Xk and multiplying some of them by i.

704


C. A Local Charge Here we compute the first local charge as outlined in Sect. 2.6. Our starting point is (2.68) where we have already diagonalized the leading order A¯ −2 of the Lax connection using some matrix T0 , αI 0 A¯ −2 = 21 T0 (P+ − σ )T0−1 = . (C.1) 0 βI Let us see how this matrix can be used to block-diagonalize A¯ −1 = X¯ + T1 T0−1 , A¯ −2

with

X¯ = T0−1 Q1,σ T0 =

uv xy

.

(C.2)

The key insight is that the double commutator

¯ = (α − β)2 A¯ −2 , [A¯ −2 , X]

0v x0

(C.3)

can be used to extract the off-diagonal elements. We can thus cancel them in A¯ −1 by setting T1 =

1 ¯ 0 [A¯ −2 , X]T (α − β)2

and obtain A¯ −1 = X¯ −

1 ¯ = A¯ −2 , [A¯ −2 , X] 2 (α − β)

(C.4)

u0 0y

.

(C.5)

We can continue and block-diagonalize A¯ r order by order in this fashion. Note that Ar is block-diagonal if and only if A¯ −2 , A¯ r = 0. (C.6) Together with the identity for any matrix Y¯ A¯ −2 , A¯ −2 , [A¯ −2 , Y¯ ] = (α − β)2 [A¯ −2 , Y¯ ]

(C.7)

one can construct the higher order transformation matrices quite conveniently. The local charges are defined via the trace of only one block ar of A¯ r . Again this can be achieved using the matrix A¯ −2 as follows: 1 1 str A¯ −2 A¯ r = α str ar + β str br = str ar . α−β α−β

(C.8)

Here it is important that str A¯ r = str ar − str br = 0. Finally, we would like to express the local charges in terms of the physical currents P , Q1,2 . Note that all the expressions occurring in the conjugated T0−1 A¯ r T0 are commutators of the currents, e.g. T0−1 A¯ −1 T0 = Q1,σ − −2 (C.9) + P+ , [P+ , Q1,σ ] ,


705

where + = 2(α − β) is the difference of eigenvalues of P+ . The only exception is a term related to the diagonalization using T0 , i.e. T0−1 ∂σ T0 . Within the trace they can be eliminated by making use of (C.10) A¯ −2 , ∂σ A¯ −2 = 0 which is equivalent to the statement that ∂σ A¯ −2 is block-diagonal. This leads to (C.11) P+ , [P+ , T0−1 ∂σ T0 ] = P+ , ∂σ P+ and is sufficient to write every instance of T0−1 ∂σ T0 within str ar in terms of ∂σ P+ . Putting everything together we find −5 str a2 = 21 −1 + str P+ P− + + str[P+ , Dσ P+ ][P+ , Dσ P+ ] − 6−5 + str [P+ , Q1,σ ], Q1,σ [P+ , Dσ P+ ]

+ 2−3 + str[P+ , Q1,σ ]Dσ Q1,σ

− 2−3 + str[P+ , Q1,σ ][P+ , Q2,σ ] −5 − + str [P+ , Q1,σ ], Q1,σ [P+ , Q1,σ ], Q1,σ [P+ , Q1,σ ], P+ , Q1,σ . − 5−7 + str [P+ , Q1,σ ], P+ , Q1,σ

(C.12)

Here Dσ X = ∂σ X − [Hσ , X] is the world-sheet covariant derivative. The conserved charge corresponding to str a2 is the integral q2+ = −i

2π

dσ str a2 .

(C.13)

0

Furthermore there exists a world-sheet parity conjugate charge q2− from the expansion around z = ∞ instead of z = 0. It is obtained from q2+ with the replacements P± → −P∓ and Q1,2 → Q2,1 . D. Sleeping Beauty This appendix contains lengthy expressions related to the complete superalgebra using the ‘Beauty’ form of psu(2, 2|4) [38], cf. Fig. 6. In this form, the grading of the sheets corresponding to the fundamental representation reads ηk = (−1, −1, +1, +1, +1, +1, −1, −1).

(D.1)

The sheets of the quasi-momentum are arranged as follows p1,2,7,8 = pˆ 1,2,3,4 ,

p3,4,5,6 = p˜ 1,2,3,4 .

Fig. 6. ‘Beauty’ Dynkin diagram of su(2, 2|4) [38]

(D.2)

706


D.1. Global charges. The global fillings are defined as Kj =

A a=1

√

j λ 1 dx 1 − ηk pk (x). 8π 2 i Ca x2 k=1

(D.3)

The global filling Kj essentially measures the total filling of all (k, l)-cuts with k ≤ j < l. The global fillings are directly related to the Dynkin labels [r1 ; r2 ; r3 , r4 , r5 ; r6 ; r7 ] of a solution. The Dynkin labels are obtained through the residues at infinity √ λ η¯ j rj = dx pj (x) − pj +1 (x) , 2 8π i ∞

(D.4)

where η¯ j = [−1; +1; +1, +1, +1; +1; −1] are conventional factors for the definition of the Dynkin labels. These are given by r1 r2 r3 r4 r5 r6 r7

= K2 − 2K1 , = K3 − K1 + 21 δE, = K2 + K4 − 2K3 , = L − 2K4 + K3 + K5 , = K4 + K6 − 2K5 , = K5 − K7 + 21 δE, = K6 − 2K7

(D.5)

or for short η¯ j rj = Vj L + V¯j δE − Mj,j Kj .

(D.6)

The labels V¯j = [0; 21 ; 0, 0, 0; 21 ; 0] indicate the change of (fermionic) Dynkin labels induced by the energy shift. Note that the Dynkin labels obey the central charge constraint −r1 + 2r2 + r3 = r5 + 2r6 − r7 .

(D.7)

The inverse relation is given by K1 = − K2 = − K3 = − K4 = K5 = − K6 = − K7 = −

+ 21 B − 41 r ∗ − L + B − 21 r ∗ − 1 1 1 ∗ 2L + 2B − 4r − − 1 1 1 ∗ L − B + r − 2 2 4 1 ∗ L − B + 2r − 1 1 1 ∗ 2L − 2B + 4r − 1 2L

3 4 r1 1 2 r1 1 2 r1 1 2 r1 1 2 r1 1 2 r1 1 4 r1

+ + + + + + +

+ + r2 + r2 + r2 + r2 + 1 r 2 2+ 1 2 r2 r2

+ 21 r4 + 21 r5 + + r4 + r5 + + 21 r4 + 43 r5 + + 21 r5 + 1 + 2 r4 + 41 r5 + r3 + r4 + r5 + 1 1 1 2 r3 + 2 r4 + 2 r5 + 1 2 r3 r3 1 4 r3 1 2 r3 3 4 r3

The constant B represents the hypercharge of the vacuum.

− − r6 − r6 − r6 − r6 − 1 2 r6 − 1 2 r6 r6

1 4 r7 1 2 r7 1 2 r7 1 2 r7 1 2 r7 1 2 r7 3 4 r7

− − − − − − −

1 2 δE,

δE, δE, δE, δE, δE, 1 2 δE. (D.8)


707

D.2. Integral representation. We present the reduction of a full set of resolvents into seven simple ones Gj for the AdS5 × S 5 superstring. The easiest way to reduce the expressions is to drop all but the resolvents between adjacent sheets. When the remaining resolvents are replaced by the suitably defined simple resolvents Gj ,

Gmom

ˆ 21 + . . . G1 = −G G2 = −G∗12 + . . . ˜ 12 + . . . G3 = +G ˜ 23 + . . . = G4 = +G ˜ 34 + . . . G5 = +G

, ,

, , , ∗ G6 = +G43 + . . . , ˆ 43 + . . . , G7 = −G

(D.9)

the original expressions are recovered. The quasi-momenta in terms of simple resolvents read G2 (0) (c1 + d1 )/x p1 (x) = − H1 (x) + G2 (1/x) − + , 2 1 − 1/x 1 − 1/x 2 G2 (0) (c1 + d1 )/x p2 (x) = H1 (x) − H2 (x) + G2 (1/x) − + , 1 − 1/x 2 1 − 1/x 2 G2 (0) (c1 + d1 )/x p3 (x) = H3 (x) − H2 (x) + G2 (1/x) − + −G4 (1/x)+G4 (0), 1 − 1/x 2 1 − 1/x 2 G2 (0) (c1 + d1 )/x p4 (x) = H4 (x) − H3 (x) + G2 (1/x) − + −G4 (1/x)+G4 (0), 2 1 − 1/x 1 − 1/x 2 G6 (0) (c1 − d1 )/x p5 (x) = H5 (x) − H4 (x) − G6 (1/x) + + +G4 (1/x)−G4 (0), 2 1 − 1/x 1 − 1/x 2 G6 (0) (c1 − d1 )/x p6 (x) = H6 (x) − H5 (x) − G6 (1/x) + + +G4 (1/x)−G4 (0), 2 1 − 1/x 1 − 1/x 2 G6 (0) (c1 − d1 )/x + , p7 (x) = H6 (x) − H7 (x) − G6 (1/x) + 1 − 1/x 2 1 − 1/x 2 G6 (0) (c1 − d1 )/x p8 (x) = H7 (x) − G6 (1/x) + + , (D.10) 2 1 − 1/x 1 − 1/x 2 √ √ with c1 = 2π B/ λ + 21 G6 (0) − 21 G2 (0) and d1 = 2π L/ λ + G4 (0) − 21 G2 (0) − 1 2 G6 (0). The integral equations are given by p/ j +1 (x) − p/ j (x) = −

7

Mj,j H / j (x) − Fj (x) = 2π nj,a

for x ∈ Cj,a , (D.11)

j =1

with Mj,j the Cartan matrix. Here, the non-zero potentials Fj (x) read F2 (x) = F6 (x) = G4 (1/x) − G4 (0), 2G4 (0)/x G2 (0)/x G2 (0) + G (1/x) − − F4 (x) = −2G4 (1/x) + 2G4 (0) + 2 1 − 1/x 2 1 − 1/x 2 1 − 1/x 2 G (0)/x G6 (0) 4π L 1/x +G6 (1/x) − − 6 + √ . (D.12) 2 2 1 − 1/x 1 − 1/x λ 1 − 1/x 2

708


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Communications in


Hamiltonian Perspective on Generalized Complex Structure Maxim Zabzine School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, UK. E-mail: [email protected] Received: 25 February 2005 / Accepted: 20 September 2005 Published online: 31 January 2006 – © Springer-Verlag 2006

Abstract: In this note we clarify the relation between extended world-sheet supersymmetry and generalized complex structure. The analysis is based on the phase space description of a wide class of sigma models. We point out the natural isomorphism between the group of orthogonal automorphisms of the Courant bracket and the group of local canonical transformations of the cotangent bundle of the loop space. Indeed this fact explains the natural relation between the world-sheet and the geometry of T ⊕ T ∗ . We discuss D-branes in this perspective.

1. Introduction The concept of generalized complex structure was introduced by Hitchin [5] and studied by Gualtieri in his thesis [4]. The generalized complex structure and related constructions such as generalized Kähler and generalized Calabi-Yau structures appear naturally in the context of geometry of the sum of the tangent and cotangent bundles, T ⊕ T ∗ . At the same time there are indications that the geometry of T ⊕ T ∗ plays a profound role within modern string theory. Actually before Hitchin’s work [5] some of the relevant mathematical notions were anticipated in the string literature (e.g., the algebraic definition of a generalized complex (Kähler) geometry is discussed in [7]). This note intends to further examine the relation between the geometry of T ⊕ T ∗ and string theory. In particular we want to explore the relation between the generalized complex geometry and extended supersymmetry of world-sheet theories. The objective of this note is to clarify and extend some of the results from [12]. Typically the extended supersymmetry for the low dimensional sigma models is related to complex geometry and this is a model independent statement. Let us recall the simple algebraic argument for this fact. We start from the ansatz δ()µ = Dν J µν ,

(1.1)

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where is a superfield corresponding to the map X : → M. A simple calculation of the algebra gives the following expression: [δ(1 ), δ(2 )]µ = −21 2 ∂λ (J µν J νλ ) µ

−21 2 Dλ Dρ (J νλ,ρ J µν − J νρ J λ,ν ).

(1.2)

To reproduce the supersymmetry algebra [δ(1 ), δ(2 )]µ = 21 2 ∂µ .

(1.3)

J is thus a complex structure. The main idea of this note is try to repeat this simple algebraic argument in phase space (, S), where S is the momentum conjugated to . In the phase space writing down the ansatz for the transformation is equivalent to the choice of symplectic structure and the generator for the transformation. Using the most general form for the generator for second supersymmetry and the standard (twisted) symplectic structure we arrive at the main result of the paper that the phase space realization of extended supersymmetry is related to generalized complex structure. Unlike [12] all results presented in this note are obtained in a model independent way. Indeed the phase space picture offers a natural explanation of the appearance of the geometry of T ⊕ T ∗ and it agrees with the recent work [1] where the role of the Courant bracket has been discussed in this context. The note is organized as follows. In Sect. 2 we start by reviewing the standard description of the string phase space in terms of cotangent bundle T ∗ LM of the loop space LM. Then we introduce the N = 1 version of T ∗ LM and explain the notation. We point out the natural isomorphism between the group of orthogonal automorphism of the Courant bracket and the group of local canonical transformations of T ∗ LM (or its supersymmetric version). In Sect. 3 we explain the relation between extended suspersymmetry and generalized complex geometry. We also explain how the real Dirac structures may arise in this context. In the following Sect. 4 we deal with D-branes in the present context. We replace the loop space LM by the interval space P M. We define the N = 1 version of T ∗ P M and explore the relation to generalized complex submanifolds. Finally, in Sect. 5 we give a summary of the paper with a discussion of the open problems and the relation of our result to previous results in the literature. There are two appendices at the end of the paper. In the first appendix we establish our conventions for N = 1 superspace. In the second appendix the basic facts about T ⊕ T ∗ geometry are stated. 2. Hamiltonian Formalism A wide class of sigma models share the following phase space description (e.g., see [1]). For the world-sheet = S 1 × R the phase space can be identified with a cotangent bundle T ∗ LM of the loop space LM = {X : S 1 → M}. Using local coordinates X µ (σ ) and their conjugate momenta pµ (σ ) the standard symplectic form on T ∗ LM is given by ω = dσ δX µ ∧ δpµ , (2.4) S1 where δ is de Rham differential on T ∗ LM. The symplectic form (2.4) can be twisted by a closed three form H ∈ 3 (M), dH = 0 as follows:

ω= S1

1 dσ (δX µ ∧ δpµ + Hµνρ ∂X µ δX ν ∧ δX ρ ), 2

(2.5)

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where ∂ ≡ ∂σ is derivative with respect to σ . For both symplectic structures the following transformation is canonical: Xµ → X µ , pµ → pµ + bµν ∂X ν

(2.6)

associated with a closed two form, b ∈ 2 (M), db = 0. There are also canonical transformations which correspond to Diff (M) when X transforms as a coordinate and p as a section of cotangent bundle T ∗ M. In fact the group of local canonical transformations1 for T ∗ LM is a semidirect product of Diff (M) and 2closed (M). Therefore we come to the following proposition Proposition 1. The group of local canonical transformations on T ∗ LM is isomorphic to the group of orthogonal automorphisms of the Courant bracket. For the description of orthogonal automorphisms of the Courant bracket see Appendix B. This construction is supersymmetrized in rather straightforward fashion (see Appendix A for superspace conventions). Let S 1,1 be a “supercircle” with coordinates (σ, θ ). Then the corresponding superloop space is LM = { : S 1,1 → M}. The phase space is given by the cotangent bundle T ∗ LM of LM, however with reversed parity on the fibers. In local coordinates we have a scalar superfield (σ, θ) and a conjugate momenta, spinorial superfield Sµ (σ, θ ) with the following expansion: µ (σ, θ ) = Xµ (σ ) + θ λµ (σ ),

Sµ (σ, θ ) = ρµ (σ ) + θpµ (σ ),

(2.7)

where λ and ρ are fermions (their linear combinations can be related to the standard world-sheet fermions ψ+ and ψ− ). S is a section of the pullback X∗ ( T ∗ M) of the cotangent bundle of M, considered as an odd bundle. The corresponding symplectic structure on T ∗ LM is 1 dσ dθ (δµ ∧ δSµ − Hµνρ Dµ δν ∧ δρ ), (2.8) ω= 2 S 1,1

such that the bosonic part of (2.8) coincides with (2.5). Therefore C ∞ ( T ∗ LM) carries the structure of super-Poisson algebra. Again as in the purely bosonic case the group of local canonical transformations of T ∗ LM is a semidirect product of Diff (M) and 2closed (M). The b-transform now is given by µ → µ ,

Sµ → Sµ − bµν Dν ,

(2.9)

or in components Xµ → X µ , pµ → pµ + bµν ∂X ν + bµν,ρ λν λρ , λµ → λµ , ρµ → ρµ − bµν λν .

(2.10) (2.11)

1 By local canonical transformation we mean those canonical transformations when the new pair ˜ (X, p) ˜ is given as a local expression in terms of the old one (X, p). For example, in the discussion of T-duality one uses non-local canonical transformations, i.e. X˜ is non-local expression in terms of X.

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3. Supersymmetry in Phase Space In this section we describe the conditions under which extended supersymmetry can be introduced on T ∗ LM. We start from the case H = 0. By construction of T ∗ LM the generator of manifest supersymmetry is given by Q1 () = − dσ dθ Sµ Qµ , (3.12) S 1,1

where Q is the operator introduced in (A.1) and is an odd parameter. Using (2.8) we can calculate the Poisson brackets for supersymmetry generators {Q1 (), Q1 (˜ )} = P(2 ˜ ),

(3.13)

where P is generator of translations along σ , P(a) = dσ dθ aSµ ∂µ

(3.14)

S 1,1

with a being an even parameter. Next we study when there exists a second supersymmetry. The second supersymmetry should be generated by some Q2 () such that it satisfies the following brackets: {Q1 (), Q2 (˜ )} = 0,

{Q2 (), Q2 (˜ )} = P(2 ˜ ).

(3.15)

By dimensional arguments there is a unique ansatz for the generator Q2 () on T ∗ LM which does not involve any dimensionful parameters 1 Q2 () = − dσ dθ (2Dρ Sν J νρ + Dν Dρ Lνρ + Sν Sρ P νρ ). (3.16) 2 S 1,1

We can combine D and S into a single object D = S

(3.17)

which can be thought of as a section of pullback of X ∗ ( (T ⊕ T ∗ )). The tensors in (3.16) can be combined into a single object2 −J P , (3.18) J = L Jt which is understood now as J : T ⊕ T ∗ → T ⊕ T ∗ . With this new notation we can rewrite (3.16) as follows: 1 Q2 () = − dσ dθ , J . (3.19) 2 S 1,1

If the generators Q1 () and Q2 () satisfy the algebra (3.13) and (3.15) then we say that there is N = 2 supersymmetry. The following proposition tells us when there exists N = 2 supersymmetry. 2

To relate to other notation (e.g., in [12]), the whole problem is invariant under the change J → −J .


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Proposition 2. T ∗ LM admits N = 2 supersymmetry if and only if M is a generalized complex manifold. Proof. We have to impose the algebra (3.15) on Q2 (). The calculation of the second bracket is lengthy but straightforward. The coordinate expressions coincide with those given in [12]. Therefore we give only the final result of the calculation. Thus the algebra (3.15) is satisfied if and only if J 2 = −12d ,

∓ [ ± (X + η), ± (Y + η)]c = 0,

(3.20)

where ± = 21 (12d ± iJ ). Thus (3.20) together with the fact that J (see (3.18)) respects the natural pairing (J t I = −IJ ) implies that J is a generalized complex structure. ± project to two maximally isotropic involutive subbundles L and L¯ such ¯ Thus we have shown that T ∗ LM admits N = 2 superthat (T ⊕ T ∗ ) ⊗ C = L ⊕ L. symmetry if and only if M is a generalized complex manifold. Our derivation is algebraic in nature and does not depend on the details of the model. The canonical transformations of T ∗ LM cannot change any brackets. Thus the canonical transformation corresponding to a b-transform (2.9),

D S

→

1 0 −b 1

D S

(3.21)

induces the following transformation of the generalized complex structure: Jb =

10 1 0 J , b1 −b 1

(3.22)

and thus gives rise to a new extended supersymmetry generator. Therefore Jb is again the generalized complex structure. This is a physical explanation of the behavior of generalized complex structure under b-transform. Using δi ()• = {Qi (), •} we can write down the explicit form for the second supersymmetry transformations as follows: δ2 ()µ = Dν J µν − Sν P µν ,

(3.23)

1 δ2 ()Sµ = D(Sν J νµ ) − Sν Sρ P νρ,µ + D(Dν Lµν ) + Sν Dρ J νρ,µ 2 1 − Dν Dρ Lνρ,µ . (3.24) 2 Indeed it coincides with the supersymmetry transformation for the topological model analyzed in [12]3 . Alternatively we can relate the generalized complex structure with an odd differential δ on C ∞ ( T ∗ LM). Indeed the supersymmetry transformations (A.3) and (3.23)-(3.24) 3 Namely, in [12] the transformations (4.2)-(4.3) subject to (4.5)-(4.6) coincide with (3.23)-(3.24) in the present paper, modulo obvious identifications.

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can be thought of as odd transformations (by putting formally = 1) which squares to the translations, ∂. Thus we can define the odd generator q = Q1 (1) + iQ2 (1) i =− dσ dθ (Sµ Qµ + iDρ Sν J νρ + Dν Dρ Lνρ 2 S 1,1

i + Sν Sρ P νρ ), 2

(3.25)

which gives rise to the following transformations: δµ = Qµ + iDν J µν − iSν P µν ,

(3.26)

i δSµ = QSµ + iD(Sν J νµ ) − Sν Sρ P νρ,µ + iD(Dν Lµν ) + iSν Dρ J νρ,µ 2 i ν ρ − D D Lνρ,µ . (3.27) 2 Thus δ 2 = 0 if and only if J defined in (3.18) is a generalized complex structure. In doing the calculations one should remember that now δ is an odd operation and whenever it passes through an odd object (e.g., D, Q and S) there is extra minus. The existence of the odd nilpotent operation (3.26)-(3.27) is related to the topological twist of N = 2 algebra (for the related discussion see [8] and [9]). The odd generator (3.25) is reminiscent of the solution of master equations proposed in [14] (see also [15]). However there are principal differences related to the setup and to the definitions of basic operations (e.g., D). It is straightforward to generalize all results to the case when H = 0. In all formulas we can generate H by non-canonical transformations µ → µ , Sµ → Sµ + Bµν Dν

(3.28)

with Hµνρ = Bµν,ρ + Bνρ,µ + Bρµ,ν with Bµν,ρ ≡ ∂ρ Bµν . The transformation (3.28) is just a technical trick and all final formulas contain only H . Thus the generator of manifest supersymmetry is Q1 () = − dσ dθ (Sµ + Bµν Dν )Qµ =

S 1,1

1 dσ (pµ λµ − ρµ ∂X µ − Hµνρ λµ λν λρ ), 3

(3.29)

S1

and the generator of translations is 1 P(a) = dσ dθ a(Sµ ∂µ − Hµνρ Dµ Dν Dρ ). 6

(3.30)

S 1,1

Assuming the full symplectic structure (2.8), Q1 () and P(a) obey the same algebra (3.13) as before. The ansatz for the generator Q2 () of the second supersymmetry is the


717

same as before (3.16) and the algebra (3.15) should be imposed. In its turn the algebra implies the following conditions: J 2 = −12d , ∓ [ ± (X + η), ± (Y + η)]H = 0,

(3.31)

where [ . ]H is a twisted Courant bracket. Therefore now J is a twisted generalized complex structure. There is a possibility to modify the supersymmetry algebra slightly [6]. Namely for the “pseudo-supersymmetry” algebra the last condition in (3.15) is replaced by the following one: {Q2 (), Q2 (˜ )} = −P(2 ˜ ).

(3.32)

Geometrically it implies that J 2 = 12d , ∓ [ ± (X + η), ± (Y + η)]H = 0,

(3.33)

where ± = 21 (12d ± J ). Using the fact J respects the natural pairing we conclude that ± project to maximally isotropic subbundles which are involutive with respect to a (twisted) Courant bracket. Therefore we get two complementary (twisted) Dirac structures L+ and L− such that T ⊕ T ∗ = L+ ⊕ L− . For any M there is always a trivial “pseudo-supersymmetry” δ2 ()µ = Dµ ,

δ2 ()Sµ = −DSµ ,

(3.34)

which corresponds to the choice J = 12d . Another interesting example of “pseudosupersymmetry” is given by the following choice: J =

1d P 0 −1d

,

(3.35)

where P is a Poisson structure on M. In this case the transformations are δ2 ()µ = Dµ − Sν P µν ,

1 δ2 ()Sµ = −DSµ − Sν Sρ P νρ,µ . 2

(3.36)

In analogy with the discussion of the standard N = 2 supersymmetry we can consider the topological twist of “pseudo-supersymmetry”. Namely we can introduce an odd nilpotent operation on T ∗ LM as follows δ = δ1 (1) + δ2 (1). Thus for the example (3.35) the corresponding nilpotent operation δ is reminiscent of the BV-transformations of the Poisson sigma model [2] (however we are working in a Hamiltonian setup). The above discussion about the odd transformations can be summarized as follows: Proposition 3. The super-Poisson algebra C ∞ ( T ∗ LM) admits an odd differrential δ if either M is a generalized complex manifold or M is a Dirac manifold with two complementary Dirac structures.

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4. D-branes We can generalize the previous discussion to the world-sheets with the boundary. For the world-sheet = P 1 × R with P 1 being the interval [0, 1] the phase space can be identified with the cotangent bundle T ∗ P M of the path space P M = {X : [0, 1] → M, X(0) ∈ D0 , X(1) ∈ D1 }, where D0 and D1 are submanifolds of M. To write down the symplectic structure on T ∗ P M we have to require that D0 and D1 are generalized submanifolds of M, i.e. there exists B i ∈ 2 (D i ) such that dB i = H |Di . Hence the symplectic structure is given by 1 ω=

1 dσ (δX µ ∧ δpµ + Hµνρ ∂X µ δX ν ∧ δX ρ ) + 2

0

1 0 1 1 + Bµν (X(0))δX µ (0) ∧ δX ν (0) − Bµν (X(1))δX µ (1) ∧ δX ν (1), 2 2

(4.37)

where the boundary contributions are needed in order ω to be closed. Next we proceed with the supersymmetrization of T ∗ P M. In analogy with the previous discussion we introduce the “superinterval” P 1,1 with coordinates (σ, θ ) and the superinterval space PM = { : P 1,1 → M}. The phase space is given by the cotangent bundle T ∗ PM of PM, with reversed parity on the fibers. As before we introduce two superfields and S, see (2.7). Let us start to discuss the situation when H = 0. The canonical symplectic structure on T ∗ PM is given by dσ dθ δµ ∧ δSµ . (4.38) ω= P 1,1

However this symplectic form is not supersymmetric unless the extra data is specified. In presence of boundaries the superfield expressions are not automatically supersymmetric due to possible boundary terms. Namely the transformation of symplectic form (4.38) under manifest supersymmetry (A.3) gives rise to a boundary term δ1 ()ω = (δX µ (1) ∧ δρµ (1) − δX µ (0) ∧ δρµ (0)).

(4.39)

Moreover the supersymmetry algebra has boundary term {Q1 (), Q1 (˜ )} = P (2 ˜ ) + 2(ρµ (1)λµ (1) − ρµ (0)λµ (0))

(4.40)

and Q1 () is not translation-invariant {P(a), Q1 ()} = a(pµ (1)λµ (1) − pµ (0)λµ (0) + ρµ (0)∂X µ (0) − ρµ (1)∂X µ (1)). (4.41) These boundary terms spoil wanted properties. The problem can be cured by imposing the appropriate boundary conditions on the fields such that the boundary terms vanish. In fact the required boundary conditions have a simple geometrical form D(1) (1) = ∈ X∗ ( (T D1 ⊕ N ∗ D1 )), S(1) D(0) (0) = ∈ X∗ ( (T D0 ⊕ N ∗ D0 )), (4.42) S(0)


719

where T Di is tangent and N ∗ Di is conormal bundles of Di correspondingly. The conditions (4.42) are understood as conditions on each component of superfields. Next we consider the case when H = 0. We should apply the same logic as before. Namely we have to impose such boundary conditions on the fields that there are no unwanted boundary terms which spoil supersymmetry. Consequently we arrive at the following symplectic form for T ∗ PM, 1 ω= dσ dθ (δµ ∧ δSµ − Hµνρ Dµ δν ∧ δρ ) + 2 P 1,1

1 0 1 1 + Bµν (X(0))δX µ (0) ∧ δX ν (0) − Bµν (X(1))δX µ (1) ∧ δX ν (1), 2 2

(4.43)

with the fields satisfying the following boundary conditions: (1) ∈ X∗ ( τDB1 ), 1

(0) ∈ X∗ ( τDB0 ), 0

(4.44)

i

where τDBi is a generalized tangent bundle (B.8) of generalized submanifold (Di , B i ). With these boundary conditions the symplectic form (4.43) is supersymmetric and there are no boundary terms in the supersymmetry algebra. Actually the boundary conditions i (4.44) can be thought of as a B-transform of the conditions (4.42). The spaces τDBi are maximally isotropic with respect to the natural pairing , on T ⊕ T ∗ , i.e. (0), (0) = 0,

(1), (1) = 0.

(4.45)

Finally we have constructed the supersymmetric version of T ∗ PM, where the boundary conditions (4.44) play the crucial role. Now we turn to the discussion of extended supersymmetry. As in the previous section we should write down the generator for the second supersymmetry (3.16) and check the algebra (3.15). The only difference with the discussion from Sect. 3 is that we should keep track of the boundary terms. For example, we can check if Q2 () is translation-invariant, i.e. 1 1 {P(a), Q2 ()} = a dθ , J (0) − a dθ , J (1). (4.46) 2 2 Thus translation-invariance is spoiled by the boundary terms. We can restore it by imposing the additional property , J (0) = 0,

, J (1) = 0.

(4.47)

Indeed this property (together with (4.44)) is sufficient to cancel all other unwanted boundary terms, e.g. in (3.15) or in δ2 ()ω. Thus the property (4.47) together with i (4.44) implies that the subbundles τDBi are stable under J , i.e. (Di , B i ) are the generalized complex manifolds introduced in [4]. We summarize the above discussion in proposition Proposition 4. The cotangent bundle T ∗ PM admits N = 2 supersymmetry if and only if M is a (twisted) generalized complex manifold and (Di , Bi ) are generalized complex submanifolds.

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This result agrees with the previous considerations in [10] and [13]. However now it is applicable to a wide class of sigma models. If instead we consider the “pseudo-supersymmetry” algebra (3.32) then appropriate i boundary conditions would require that τDBi is invariant under J which defines two transversal Dirac structures. This is a real analog of the generalized complex submanifold. In the example (3.35), τD0 is stable under J if a submanifold D is coisotropic with respect to P , see [3]. 5. Concluding Remarks In this short note we clarified and extended results from [12]. The first order actions discussed in [11] and [12] can be thought of as phase space actions and therefore the Hamiltonian formalism should naturally arise in the problem. Indeed the Hamiltonian formalism offers a deep insight on the relation between the world-sheet and the geometry of T ⊕ T ∗ . The main result of the paper is that the phase space realization of extended supersymmetry is related to generalized complex structure. This result is model independent and it is applicable to the wide range of sigma models, e.g. the standard sigma model, the Poisson sigma model, the twisted Poisson sigma model, etc. The next step would be to specify the Hamiltonian H (i.e., choose the concrete model) and check that Q2 is in fact the symmetry of the Hamiltonian. At this stage the compatibility between the geometrical data used in H (e.g., a metric g, a Poisson structure π , etc.) and J will arise. We hope to come back to this elsewhere. Acknowledgements. I am grateful to Anton Alekseev, Alberto Cattaneo, Giovanni Felder, Matthias Gaberdiel and Ulf Lindström for discussions on this and related subjects. I thank Ulf Lindström and Pierre Vanhove for reading and commenting on the manuscript. The research is supported by EU-grant MEIFCT-2004-500267.

A. Appendix: Superspace Conventions We use the superspace conventions. The odd coordinate is labeled by θ and the covariant derivative D and supersymmetry generator Q are defined as follows: D = ∂ θ − θ ∂σ ,

Q = ∂θ + θ∂σ

(A.1)

such that D 2 = −∂σ ,

Q2 = ∂σ ,

QD + DQ = 0.

(A.2)

In terms of covariant derivatives, a supersymmetry transformation4 of a superfields is then given by δ1 ()µ ≡ Qµ ,

δ1 ()Sµ ≡ QSµ .

(A.3)

The components of superfields can be found via projection as follows, | ≡ Xµ ,

D| ≡ λµ ,

Sµ | ≡ ρµ ,

DSµ | ≡ pµ ,

(A.4)

We give the expressions for the case H = 0. Analogously using the generator (3.29) one can write down the expressions for the case H = 0. 4


721

where a vertical bar denotes “the θ = 0 part of”. Thus, in components, the N=1 supersymmetry transformations are given by δ1 ()X µ = λµ ,

δ1 ()λµ = −∂X µ ,

δ1 ()ρµ = pµ ,

δ1 ()pµ = −∂ρµ . (A.5)

The N=1 spinorial measure in terms of covariant derivatives dθ L = DL|.

(A.6)

B. Appendix: Basics on T ⊕ T ∗ Consider the vector bundle T ⊕T ∗ which is the sum of the tangent and cotangent bundles of an d-dimensional manifold M. T ⊕ T ∗ has a natural pairing t X Y I . (B.1) X + ξ, Y + η ≡ (iY ξ + iX η) ≡ ξ η The smooth sections of T ⊕ T ∗ have a natural bracket operation called the Courant bracket and defined as follows: 1 (B.2) [X + ξ, Y + η]c = [X, Y ] + LX η − LY ξ − d(iX η − iY ξ ), 2 where [ , ] is a Lie bracket on T . Given a closed three form H we can define a twisted Courant bracket [X + ξ, Y + η]H = [X + ξ, Y + η]c + iX iY H.

(B.3)

The orthogonal automorphism (i.e., such which preserves , ) F : T ⊕ T ∗ → T ⊕ T ∗ of (twisted) Courant bracket F ([X + ξ, Y + η]H ) = [F (X + ξ ), F (Y + η)]H is semidirect product of Diff (M) and form is given as follows:

2closed (M),

(B.4)

where the action of the closed two

eb (X + ξ ) ≡ X + ξ + iX b

(B.5)

The transformation (B.5) is called b-transform. The maximally for b ∈ isotropic subbundle L of T ⊕ T ∗ , which is involutive with respect to (twisted) Courant bracket is called (twisted) Dirac structure. We can consider two complementary (twisted) Dirac structures L+ and L− such that T ⊕ T ∗ = L+ ⊕ L− . Alternatively we can define L± by proving a map J : T ⊕ T ∗ → T ⊕ T ∗ with the following properties: 2closed (M).

J t I = −IJ ,

J 2 = 12d ,

∓ [ ± (X + ξ ), ± (Y + η)]H = 0,

(B.6)

± J ) are projectors on L± . where ± = The (twisted) generalized complex structure is the complex version of two comple¯ We can define mentary (twisted) Dirac subbundles such that (T ⊕ T ∗ ) ⊕ C = L ⊕ L. ∗ the generalized complex structure as a map J : (T ⊕ T ) ⊗ C → (T ⊕ T ∗ ) ⊗ C with the following properties: 1 2 (12d

J t I = −IJ , where ± =

1 2 (12d

J 2 = −12d ,

∓ [ ± (X + ξ ), ± (Y + η)]H = 0,

± iJ ) are the projectors on L and L¯ correspondingly.

(B.7)

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The generalized submanifold is a submanifold D with a two form B ∈ 2 (M) such that dB = H |D . For generalized submanifold (D, B) we can define the generalized tangent bundle τDB = {X + ξ ∈ T D ⊕ T ∗ M|D , ξ |D = iX B}.

(B.8)

The submanifold (D, B) is (twisted) generalized complex submanifold if τDB is stable under the action of map J defined in (B.7). For further details the reader may consult the Gualtieri’s thesis [4]. References 1. Alekseev, A., Strobl, T.: Current algebra and differential geometry. JHEP 0503, 035 (2005) 2. Cattaneo, A.S., Felder, G.: A path integral approach to the Kontsevich quantization formula. Commun. Math. Phys. 212, 591 (2000) 3. Cattaneo, A.S., Felder, G.: Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model. Lett. Math. Phys. 69, 157 (2004) 4. Gualtieri, M.: Generalized complex geometry. Oxford University DPhil thesis, http://arxiv.org/list/math.DG/0401221, 2004 5. Hitchin, N.: Generalized Calabi-Yau manifolds. Q. J. Math. 54, no. 3, 281–308, (2003) 6. Hull, C.M.: Actions for (2,1) sigma models and strings. Nucl. Phys. B 509, 252 (1998) 7. Kapustin, A., Orlov, D.: Vertex algebras, mirror symmetry, and D-branes: The case of complex tori. Commun. Math. Phys. 233, 79 (2003) 8. Kapustin, A.: Topological strings on noncommutative manifolds. Int. J. Geom. Meth. Mod. Phys. 1, 49 (2004) 9. Kapustin, A., Li, Y.: Topological sigma-models with H-flux and twisted generalized complex. http://arxiv.org/list/hep-th/0407249, 2004 10. Lindström, U., Zabzine, M.: N = 2 boundary conditions for non-linear sigma models and LandauGinzburg models. JHEP 0302, 006 (2003) 11. Lindström, U.: Generalized N = (2,2) supersymmetric non-linear sigma models. Phys. Lett. B 587, 216 (2004) 12. Lindström, U., Minasian, R., Tomasiello, A., Zabzine, M.: Generalized complex manifolds and supersymmetry. Commun. Math. Phys. 257, 235 (2005) 13. Zabzine, M.: Geometry of D-branes for general N = (2,2) sigma models. Lett. Math. Phys. 70, 211 (2004) 14. Zucchini, R.: A sigma model field theoretic realization of Hitchin’s generalized complex geometry. JHEP 0411, 045 (2004) 15. Zucchini, R.: Generalized complex geometry, generalized branes and the Hitchin sigma model. JHEP 0503, 022 (2005) Communicated by M.R. Douglas


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Area-Preserving Surface Diffeomorphisms Zhihong Xia Department of Mathematics, Northwestern University, Evanston, Il 60208, USA. E-mail: [email protected] Received: 4 March 2005 / Accepted: 2 October 2005 Published online: 26 January 2006 – © Springer-Verlag 2006

Abstract: We prove some generic properties for C r , r = 1, 2, . . . , ∞, area-preserving diffeomorphism on compact surfaces. The main result is that the union of the stable (or unstable) manifolds of hyperbolic periodic points are dense in the surface. This extends the result of Franks and Le Calvez [10] on S 2 to general surfaces. The proof uses the theory of prime ends and Lefschetz fixed point theorem.

1. Introduction and Statement of Main Results Let Diffrµ (M) be the set of all C r , r = 1, 2, . . . , ∞ diffeomorphisms of compact orientable surface M that preserves a smooth area element µ.A property for C r area-preserving diffeomorphisms of M is said to be generic, if there is a residual subset R ⊂ Diffrµ (M) such that the property holds for all f ∈ R. The following are some examples of known generic properties: G1 All the periodic points are either elliptic or hyperbolic. This is proved by Robinson [22]. G2 All elliptic periodic points are Moser stable. Moser stable means that the normal form at the elliptic periodic points are non-degenerate in the sense of KAM theory (cf. Siegel & Moser [24]). This implies that there are invariant curves surrounding each elliptic periodic point. This condition requires that the map is sufficiently smooth, say r ≥ 16 (cf. Douady [7]). G3 For any two hyperbolic periodic points p, q, the intersections of the stable manifold W s (p) and the unstable manifold W u (q) are transversal. This is again proved by Robinson [22]. G1 and G3 together are often referred to as the Kupka-Smale condition for area-preserving diffeomorphisms.

Research supported in part by National Science Foundation.

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G4 For any hyperbolic periodic point p, let 1 and 2 be any two branches of the stable manifold or unstable manifold of p, then 1 = 2 . For S 2 , this is proved by Mather [15]. The general case is proved by Oliveira [16]. G5 If M is a two-sphere S 2 or a two torus T 2 , let 1 be a branch of the stable manifold of a hyperbolic periodic point of p and 2 be a branch of the unstable manifold of p, then 1 ∩ 2 = ∅. For S 2 , this is proved by Robinson [23] and Pixton [18]. For T2 , it is proved by Oliveira [16]. For general surfaces, it is proved by Oliveira [17] for most homotopy classes of maps. For the C 1 case and any compact manifold of arbitrary dimension, the result is proved by Takens [26] (see also Xia [29] for a stronger result). Franks & Le Calvez [10] recently showed another remarkable C r generic property for area-preserving diffeomorphisms on S 2 . Their results state that the stable and unstable manifolds of hyperbolic periodic points are dense in S 2 generically. In this paper, we extend this result to arbitrary orientable surfaces and to arbitrary homotopy classes. More precisely, we state our main result. Theorem 1.1. Let M be a compact orientable surface. There exists a residual subset R ⊂ Diffrµ (M) such that if f ∈ R and P is the set of all hyperbolic periodic points of f , then both the sets ∪p∈P W s (p) and ∪p∈P W u (p) are dense in M. Furthermore, let U ⊂ M be an open connected subset such that it contains no periodic point for f and suppose that, for some hyperbolic periodic point p, U ∩W s (p) = ∅, then W s (p) is dense in U . The main tool of our proof is the theory of prime ends [4, 15] and Lefschetz fixed point theorem. Also, Arnold’s conjecture for symplectic fixed points, as proved by Conley & Zehnder [5] for the case of torus, turned out to be essential for our results on T2 . Our work was motivated in our effort to prove the so-called C r closing lemma, as conjectured by Poincaré [19]. Our result provides strong evidence supporting the C r closing lemma. We will discuss this and other related problems in the end of the paper. 2. Prime Ends and a Theorem of Mather Let U be a simply connected domain of S 2 whose complement contains more than one point. One can define the prime end compactification of U , introduced by Caratheodory [4], by adding the circle S 1 . Each point in S 1 is defined by a sequence of nested arcs in U . The circle S 1 inherits a topology from U . We refer to Mather [15] for definitions and general discussions. Here we only summarize what we will need in this paper. The compactification Uˆ = U S 1 of U , with inherited topology on S 1 , is homeomorphic to a closed disk D. If U ⊂ S 2 is an invariant subset for a homeomorphism on S 2 , then the function f extends continuously to a homeomorphism fˆ : Uˆ → Uˆ . Moreover, fˆ restricted to the prime ends S 1 is a circle homeomorphism. When the rotation number ρ of fˆ|S 1 is rational, then fˆ has a periodic points on the prime ends. But it may happen in general that f does not have any periodic points on ∂U . However, if f is area-preserving, then this will never happen. Lemma 2.1. Let f be an area-preserving homeomorphism on S 2 and let U be a simply connected, invariant set whose complement contains more than one point. Let fˆ: Uˆ → Uˆ be the extension of f to the prime end compactification of U . If fˆ has a periodic point in its prime ends then f has a periodic point in ∂U .

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The proof is simple, we refer to Mather [15] and Franks & Le Calvez [10]. The next theorem, which is due to Mather [15] states that in generic situations, there are no periodic points on the prime ends. Theorem 2.2. Let f be an area-preserving diffeomorphism on S 2 and let U be a simply connected, invariant set whose complement contains more than one point. Further assume that f satisfies the generic conditions G1, G2 and G3. Then there is no periodic point for f on the boundary of U and as a consequence, there is no periodic point for fˆ on the prime ends and the rotation number of fˆ on the prime ends is irrational. The proof is based on the following ideas: by generic conditions G1 and G2, f can not have any elliptic periodic points on ∂U . If there are any hyperbolic periodic points on ∂U , then the boundary of U has to contain some branches of the stable and unstable manifolds of the hyperbolic periodic points and this will lead to a contradiction to the generic property G3. For details of the proof, see Mather [15] and Franks & Le Calvez [10]. The theory of prime ends can be easily generalized to other orientable compact surfaces M, possibly with boundaries. Let U ⊂ M be a connected open domain in M. Assume that the boundary of U contains a finite number of connected pieces and each piece contains more than one point. Then U can be compactified by adding prime ends. In this case, the prime ends are homeomorphic to a union of finitely many circles S 1 . The number of circles S 1 added is the number of boundary pieces seen from the inside of U . It may be more than the number of connected components of ∂U . The compactification of U , denoted as Uˆ , is a compact surface with boundaries, each S 1 will be a hole for Uˆ . If f : M → M is a homeomorphism and U is invariant for f , then f extends continuously to a homeomorphism on Uˆ . We denote this extension by fˆ. Lemma 2.1 and Theorem 2.2, with obvious modifications, are both true for Uˆ . In the case where U has infinitely many connected boundary components, or where some of the boundary components are single points, one can do a prime end compactification to finitely many isolated connected boundary pieces that contain more than one point. Again, the number of boundary pieces are defined to be the number seen from inside U . 3. Periodic-point-Free Regions Let R be a residual subset of Diffrµ (M) such that for any map in R the generic properties G1, G2, G3 and G4 hold. Whenever G4 is assumed, we require the map to have sufficient smoothness, say r ≥ 16. Our final result will be true for any r = 1, 2, . . . ,. Fix a map f ∈ R. Let U ⊂ M be a connected open set such that f has no periodic point in U . Our goal is to show that either U is filled by stable and unstable manifolds of some hyperbolic periodic points, or by an arbitrarily small C r perturbation of f ; we can create a periodic point in U . Let AU be the set of all points whose orbit under f passes through U , i.e., i AU = ∪∞ i=−∞ f (U ).

Since f is area-preserving, almost every point is recurrent. Since M has finite area, AU has finite number of connected components. Each of these components are open and periodic. Without loss of generality, we may assume that AU has only one component

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and this component certainly contains U . The case with more than one component can be reduced to this case simply by considering powers of f . The set AU is invariant under f and it is periodic point free. We are interested in whether the closure of AU contains any periodic point. The set AU typically has infinitely many connected boundary components, so we can not apply Lemma 2.1 and Theorem 2.2, or other results obtained by the prime ends directly. We have the following simple lemma. Lemma 3.1. The closure of AU , AU , is a compact invariant set which contains no elliptic periodic point for f . Here we assume f has sufficient smoothness so that the generic condition G2 holds. Proof. Recall that the map f ∈ R satisfies the generic property G2, thus all elliptic periodic points are Moser stable. This implies that surrounding each elliptic periodic point there are infinitely many invariant curves. If AU contains an elliptic periodic point, then AU has to intersect an invariant curve with irrational rotation number close to the elliptic point. The set AU is invariant, thus it completely covers that curve. By the Birkhoff fixed point theorem, or more generally the Aubry-Mather theory, any neighborhood of the invariant curve contains infinitely many periodic orbits. This contradicts the assumption that AU is periodic point free. This proves the lemma. The closure of AU may still contain hyperbolic periodic points. We show that this happens only if AU contains stable or unstable manifolds of hyperbolic periodic points. Lemma 3.2. If the closure of AU , AU , contains a hyperbolic periodic point p, then U ∩ W s (p) = ∅ and U ∩ W u (p) = ∅. Proof. Recall that for f ∈ R,the generic property G4 implies that W s (p) = W u (p), and it suffices to show that either AU ∩ W s (p) = ∅ or AU ∩ W u (p) = ∅. First, we remark that if M = S 2 or M = T2 and the generic property G5 is assumed, then the lemma can be easily proved. In this case, each branch of the stable and unstable manifolds of p intersect and the intersection is transversal, there is a homoclinic tangle in the neighborhood of p and this homoclinic tangle divides a small neighborhood of p into infinitely many small rectangles. Choose a small neighborhood p such that all of these rectangles have areas smaller than that of U . If p is in the closure of AU , then, for some integer i, f i (U ) intersects one of these small rectangles. By the area preserving property of f , f i (U ) can not be totally contained in any single rectangle and thus it has to intersect the boundary of the rectangle. This implies that f i (U ) contains a point on either the stable manifold or the unstable manifold of p. For the general case, suppose that there is a hyperbolic periodic point p ∈ M such that AU ∩ W s (p) = ∅; we want to show that p is not in the closure of AU . Let V be the connected component of M\W s (p) containing AU . Since the complement of V is connected, the boundary of V consists of finitely many connected pieces and each piece contains more than one point. Therefore V has a prime end extension, Vˆ with the extended map fˆ on Vˆ . By Mather’s theorem [15], which is proved in this general setting, Vˆ has no periodic points on the boundary of Vˆ . This implies that p ∈ / V and since AU ⊂ V , this implies that the closure of AU contains no periodic point. This proves the lemma.


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To get rid of isolated points in the boundary of AU , we let A = int(A¯U ). Then any isolated connected boundary piece contains more than one point. If the closure of AU is periodic point free, then so is the closure of A. 4. Isotopic To Identity Cases In this section we prove our main lemma in the case where the map is isotopic to identity. Lemma 4.1. Let f ∈ R be a generic diffeomorphism of M and f is isotopic to identity. Let A ⊂ M be an open, connected, periodic point free, f -invariant set whose closure ¯ Then either A = M = T2 contains no periodic point. We further assume that A = intA. or A is homeomorphic to an open annulus and its prime end extension is a closed annulus. Proof. We first assume that M = T2 . Since the Euler characteristic of M is non-zero, by the Lefschetz fixed point theorem, f contains at least one fixed point. This implies that A = M. Let Fix(f ) be the set of all fixed points for f . It contains a finite number of points and it is contained in the interior of M\A. Let B ⊂ M be the union of the connected components of M\A intersecting Fix(f). The set B is closed and has a finite number of connected components and each contains at least one fixed point of f . Let C = M\B, obviously C is open, A ⊂ C and the closure of C contains no fixed point. Let Cˆ be the prime end extension of C and let fˆ: Cˆ → Cˆ be the extension of f . Then fˆ has no fixed point. The set Cˆ is a compact surface with boundary, its topology is uniquely determined by the number of handles and the number of holes. Let the number of handles of Cˆ be k and the number of holes be l. The map fˆ keeps invariant all the boundary pieces. The Euler characteristic number of Cˆ is 2 − 2k − l. Even though fˆ may not neccessarily be homotopic to identity, its induced map on homology is identity. This is because all the generators of the homology are mapped to themselves. Hence its Lefschetz number is the same as its Euler characteristic number. Since fˆ has no fixed point, this implies that 2 − 2k − l = 0. This implies that either k = 0, l = 2 or k = 1, l = 0. The latter implies that C is a torus and thus M is a torus, which we assumed was not the case. The first case implies that Cˆ is a closed annulus. ¯ i.e., x is in the interior of We claim that A = C. If not, there is a point x ∈ C\A. ¯ But C is open, and this implies that C\A. For if no such point exists, then C ⊂ A. C ⊂ intA¯ = A, by the assumption on A. Let Bx be the connected component in C\A¯ containing x. Then Bx is closed, simply connected and Bx ∩∂C = ∅. For if Bx ∩∂C = ∅, then Bx ⊂ B, where B is the set of connected components containing fixed points in the complement of A, as defined previously. Since Bx has positive area, it must be periodic under f . Let k be the period of Bx . Let Fix(f k ) be the set of all periodic points of f with period k. Again, Fix(f k ) contains only a finite number of points. Let Bk be the union of all connected components of C\A and f i (Bx ), i = 0, 1, 2, . . . , k − 1. The set Bk has a finite number of, say l > 0, connected components and all components are periodic with period k. Moreover, Bk ∩ ∂C = ∅. Let Cx = C\Bk , then Cx is an open set and Cx = intC¯ x . The prime end extension of Cx is a closed annulus with l interior disks removed and there are no periodic points of period k in Cx . However, the Lefschetz number for f k on Cx is l = 0; this implies that f k has to have at least one fixed point in Cx . This contradiction proves that A = C, i.e., A is an annulus. If M = T2 , then either A = T 2 or there is a point x ∈ int(M\A). Let B be the connected component containing x in the complement of A. Again, B must be periodic

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and let this period be k. Let B be the union of all the connected components of M\A that are fixed under f k . Then B ⊂ B and B has finitely many connected components. Let C = A\B. Then this case follows from The above arguments by considering f k on C. This proves our lemma. 5. A Lemma on Periodic Points Now we consider the case where f is not necessarily isotopic to identity. To prove the same result, we need to prove a simple theorem on the existence of periodic points for maps of compact surfaces with boundaries. Lemma 5.1. Let M be a compact, connected surface, possibly with boundary. Assume that M has nonzero Euler characteristic. Suppose that f: M → M is a homeomorphism, then f has a periodic point. Moreover, for any positive integer n, there exists infinitely many positive integers i such that the Lefschetz number L(f ni ) is negative. Proof. Let L(f ) be the Lefschetz number of the map f . By the Lefschetz fixed point theorem, it suffices to show that L(f n ) is nonzero for some large integer n. We consider homology groups of M with real coefficients. If H2 (M) = 0, by taking an iterate of f , we may assume that the induced map on H2 (M) is the identity. Let tr1 (f∗ ) be the trace of the induced map f∗: H1 (M) → H1 (M), then L(f n ) = 2 − tr1 (f∗n ) if M is orientable and without boundary, and L(f n ) = 1 − tr1 (f∗n ) if M is non-orientable or has boundary. The theorem follows easily for the cases where the dimension of H1 (M) is zero (the sphere and the projective plane). The torus, the Klein bottle, the Möbius strip and the annulus all have Euler characteristic zero and hence are excluded. The only cases we need to consider are where the dimension of the first homology H (M) is greater than or equal to 3. Let λ1 , λ2 , . . . , λl be the eigenvalues of the induced isomorphism f∗ : H1 (M) → H1 (M), where l is the dimension of H1 (M). We can write λi = ri αi , where ri is a positive real number and αi is a complex number on the unit circle, for nk i = 1, 2, . . . , l. There exists a sequence of integers {nk }∞ k=1 such that αi → 1 as k → ∞, for all i = 1, 2, . . . , l. If ri > 1 for some i = 1, 2, . . . , l, then tr1 (f∗nk ) = li=1 λni k → ∞ as k → ∞. This n implies that for large k, L(f k ) = 0, the lemma follows. If there exists i such that ri < 1 for some i = 1, 2, . . . , l, in the area preserving cases we consider here, det(f∗ ) = 1, there exists i such that ri > 1, the lemma again follows. In general cases where f is not area-preserving, we can consider f −1 , the inverse of f . The same argument shows that L(f n ) = 0 for some negative integer n. Since L(f n ) = L(f −n ), the lemma again follows. Finally, if ri = 1 for all i = 1, 2, . . . , l, then tr1 (f∗nk ) = li=1 λni k → l as k → ∞. As j ≥ 3, 2 − tr1 (f∗nk ) < 0 and 1 − tr1 (f∗nk ) < 0 for large k. This shows that for infinitely many positive integers k, the Lefschetz number L(f nk ) is negative.The Lefschetz fixed point theorem concludes that there is at least one fixed point for f nk for such k. For any positive integer n, replacing f with f n , the above arguments show that there are infinitely many positve integer i such that L(f ni ) is negative. This proves the lemma. For orientable compact surfaces, only T2 and annulus have zero Euler characteristics. Lemma 4.1 would follow from the above lemma if the periodic point free set A in Lemma 4.1 has a prime end extension that makes it into a compact manifold with boundary. We have to show that A can’t have infinitely many boundary pieces. The proof of the lemma is basically a verification of this fact.


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6. General Case Let f ∈ R be a generic diffeomorphism on the compact surface M. Let A ⊂ M be an open, connected, periodic point free, f -invariant set whose closure contains no periodic ¯ For any integer i, we let Fix(f i ) be the set of all periodic point. Assume A = int(A). points of f with period i. If A = M, M\A is nonempty and contains at least one connected component. The number of the connected components in M\A may be infinite. Let Bi be the union of all connected components of M\A containing a periodic point of period i. The set Bi is closed and Bi ∩ Fix(f i ) = ∅, if Fix(f i ) = ∅. The set Bi has a finite number of connected components. For any positive integers i, j , each component of Bj is either a component of Bi or disjoint from Bi . If j = ki for some positive integer k, then Bi ⊂ Bj . Let Ci = M\Bi . Then Ci has a finite number of boundary components. Let Cˆ i be the prime end extension of Ci , it is a compact surface with boundary. Let the number of handles of Cˆ i be m and the number of holes be n. For any integer k, the number of handles of Cˆ ki is smaller than or equal to m. Therefore, there exists an integer i ∗ such that the number of handles of Cˆ ki ∗ is a constant for all positive integers k. If Fix(f i ) is empty for all positive integer i, then either A = M, then we set Ci = M for all i, or A = M, then we pick a point x ∈ M\A and let Bi be the connected component of M\A and its iterates under f . Since f is area preserving, Bi has a finite number of components. By our construction, fî : Cˆ i → Cˆ i has no fixed points for all i ≥ 1. We are ready to prove Lemma 4.1 for the general case. Lemma 6.1. Let f ∈ R be a generic diffeomorphism of M. Let A ⊂ M be an open, connected, periodic point free, f -invariant set whose closure contains no periodic point. ¯ Then either A = M = T2 or A is homeomorphic to We further assume that A = intA. an open annulus and its prime end extension is a closed annulus. Proof. First if there exists a positive integer k such that f k is isotopic to identity, the lemma is proved in the same way as Lemma 4.1. One just needs to consider f k . We first suppose that M = T2 . As described before, there exists an integer i ∗ such that the number of handles of ∗ Cˆ ki ∗ is a constant for all positive integers k. We now consider fî : Cˆ i ∗ → Cˆ i ∗ . It is a homeomorphism on a compact surface, keeping invariant all boundary pieces. Suppose ∗ fî : Cˆ i ∗ → Cˆ i ∗ has no periodic points, then by Lemma 5.1, Ci ∗ is an open annulus and Cˆ i ∗ is a closed annulus. We claim that A = Ci ∗ . Suppose not, then there is a point x ∈ Ci ∗ , but x ∈ / A. Since A = intA¯ and by the definition of Ci ∗ , x ∈ / ∂A. Let Cx be the connected component of Ci ∗ \A¯ containing x. Then Cx is an open disk in the annulus Ci ∗ . Since f is area preserving, Cx must be periodic. Let Cˆ x be the prime end compactification of Cx , then Cˆ x is a closed disk, periodic under fˆ. Mather’s Theorem 2.2 implies that f contains a periodic point in Cx . This contradicts the assumption that Ci ∗ has no periodic point. ∗ Now suppose that fî : Cˆ i ∗ → Cˆ i ∗ has a periodic point x with period p. By the ¯ By Lemma 5.1, there is a positive integer j such that the assumption on A, x ∈ / A. ∗ ∗ Lefschetz number of the pj ’s iterate of fî on Ci ∗ is negative., i.e., L((fî )pj |Ci ∗ ) < 0, where p is the period of x. Now consider the prime end extension of Ci ∗ pj . By our choice of i ∗ , Cˆ i ∗ pj is topologically Cˆ i ∗ with finitely many, say k, k > 0, open disks removed and

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these disks are periodic with period pj under the extended map fî . We have the follow∗ ∗ ing relations on the Lefschetz numbers L(fî pj |Ci ∗ pj ) = L((fî )pj |Ci ∗ ) − k < 0.The ∗ Lefschetz fixed point theorem implies that fî pj has a fixed point on Ci ∗ pj , which is ∗ impossible by the definition of Ci ∗ pj . This contradiction shows that fî : Cˆ i ∗ → Cˆ i ∗ has no periodic point and hence A = Cˆ i ∗ is an annulus. We are left with one case: M = T2 and A = M. The same argument works in this case too. This proves the lemma. 7. Maps on Annulus i Let U be a connected open subset of M and let AU = ∪∞ i=−∞ f (U ). If the closure of AU contains no periodic point, then A = int(A¯ U ) is open, containing no periodic points in its closure. By the above lemma, if M = T2 , then A is a union of finite disjoint open annuli, periodic under f . The dynamics on Annulus have been well studied (cf. Franks [8], Le Calvez & Yoccoz [3]). The following lemma shows that, if A is an annulus, we can perturb f with an arbitrarily small C r perturbation to create a periodic point in U . The same result was also used in [10].

Lemma 7.1. Fix f ∈ R and assume M = T2 . Let U be a connected open subset of i S 2 and AU = ∪∞ i=−∞ f (U ). Assume that U does not intersect any stable or unstable manifolds of hyperbolic periodic points of f . Then for any C r neighborhood V of f , there exists g ∈ V such that the support of g ◦ f −1 is contained in the interior of the closure of AU and g has a periodic point in U . Proof. Since U does not intersect any stable or unstable manifolds of hyperbolic periodic points of f , by the above lemmas, f has no periodic point in AU and no periodic point in the closure of AU and therefore, A = int(A¯ U ) is a union of finite disjoint periodic annuli. Without loss of generality, we may assume that AU itself is an open annulus. Using prime end extension, we obtain an area-preserving continuous map on the prime end closure of AU , still denoted by AU . Let (x, y), x ∈ R (mod 1), y ∈ [0, 1] be a coordinate on AU . Since f preserves invariant measure µ, by Birkhoff Ergodic Theorem, for µ-almost every point z = (x, y) ∈ AU , the rotation number πx f˜i (z) i→∞ i

ρ(z, f ) = lim

is well defined. Here πx is the projection on AU into its first coordinate and f˜ is a lift of f to its universal cover R × [0, 1]. A different lift of f yields a different rotation number that differs by an integer. Since there are no periodic points in AU , by Franks theorem [8], there exists an irrational number α ∈ R such that for almost all z ∈ AU , the rotation number ρ(z, f ) exists and ρ(z, f ) = α. In particular, if z is in the boundaries of AU then ρ(z, f ) = α. Let γ be a simple closed curve in the interior of AU . We may assume that γ is homotopically non-trivial in AU . Take a small tubular neighborhood γδ of γ in the interior of AU and parametrize this tubular neighborhood by γδ : S 1 × [−δ, δ] →: AU for some small δ. In fact, for convenience we may even assume that γδ is area-preserving. Let β : [−δ, δ] → R be a C ∞ function such that β(t) > 0 for all −δ < t < δ and β(−δ) = β(δ) = 0 and β is C ∞ flat at ±δ. i.e., all the derivatives of β(t) at ±δ are zero.


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Let h : AU → AU be a C ∞ diffeomorphism such that if z ∈ / γδ , h (z) = z and if z ∈ γδ , h = γδ ◦ T ◦ (γδ )−1 where T (θ, t) = (θ + β(t), t) for all θ ∈ S 1 and t ∈ [−δ, δ]. We remark that h → Id in C ∞ topology as → 0 and the mean rotation number for h with respect to the area µ is δ 1 ρµ (AU , h ) = β(t)dt. µ(AU ) −δ Therefore for any > 0, ρµ (AU , h ◦ f ) = ρµ (AU , h ) + ρµ (AU , f ) > α, this implies that there exists a point y ∗ ∈ AU , such that ρ(y ∗ , h ◦ f ) > α. Since ρ(z, h ◦ f ) = ρ(z, f ) = α for all z ∈ ∂AU , we conclude, from Franks’ theorem [8], that for any rational number p/q, such that α < p/q < ρ(y ∗ , h ◦ f ), there exists a periodic point of period p/q for the map h ◦ f . Thus, for any > 0, there are infinitely many periodic points for h ◦ f in the interior of AU . In fact, all of these periodic points have to pass through the strip γδ . However, these periodic points may be far away from U . To find periodic points in U , we need to do some estimates on these orbits. Since AU = ∪i∈Z f i (U ), for any point z ∈ γδ ⊂ AU , there exists an integer nz ∈ Z and a neighborhood of z, Wz ⊂ AU , such that f nz (Wz ) ⊂ U . The collection {Wz , z ∈ γδ } forms an open cover for the compact set γδ . Let Wz1 , Wz2 , . . . , Wzk be a finite subcover of γδ and let N = max{|nz1 |, |nz2 |, . . . , |nzk |}. The integer N chosen above has a very important property: for any z ∈ γδ , the orbit segment {f −N (z), f −N +1 (z), . . . , f N (z)} intersects U at least once. Or equivalently, i the set ∪N i=−N f (U ) covers γδ . Since U is open, this same property holds for all g sufficiently close to f in C 0 topology., i.e., the orbit segment {g −N (z), g −N+1 (z), . . . , g N (z)} intersects U for all z ∈ γδ , provided that g is sufficiently close to f . The above arguments show that if > 0 is small enough, h ◦ f has infinitely many periodic orbits and all of these periodic orbits intersect U . This proves the lemma. 8. Maps on Torus and Arnold’s Conjecture The final case is where M = T2 and f has no periodic point. We will show that such f is not generic and it can be perturbed to create a periodic point. Let f∗1 : H1 (T2 , R) = R2 → R2 be the induced map on the first homology of 2 T . Let λ1 , λ2 be the eigenvalues of f∗1 , λ1 = λ¯ 2 . Since f has no periodic point, the Lefschetz number L(f k ) = 0 for all k. This implies that λk1 + λk2 = 2, for all k. We must have λ1 = λ2 = 1. Since f∗1 and its inverse are both integer matrices, we have only two choices: f is isotopic to identity, where f∗1 = I or, modulo a conjugation by an element of SL(n, Z), f is isotopic to a Dehn twist, i.e., for some integer k = 0, 1k f∗1 = . 01 . We first consider the case where f is isotopic to identity. The proof is basically an application of the Arnold conjecture [1] as proved by Conley and Zehnder [5]. Lemma 8.1. Let f : T2 → T2 be an area-preserving diffeomorphism such that f is isotopic to identity. Then for any C r neighborhood V of f , there exists g ∈ V such that g : T2 → T2 has a periodic point.

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Proof. Let f˜ : R2 → R2 be a lift of the map f on T2 = R2 /Z2 to its universal cover R2 . Let πi , i = 1, 2 be the projection of R2 to its first and second coordinates respectively. The average rotation numbers for the map f˜, are defined to be ρi (f˜) = (πi (f˜(p)) − πi (p))dµ. The combination ρ(f ) = (ρ1 (f˜), ρ2 (f˜)), mod Z2 is called the average (or mean) rotation vector for f . The rotation vectors are well defined for maps isotopic to identity. We want to do a small perturbation to f so that each component of the average rotation vector is a rational number. This is easy: one composes the map f with T(1 ,2 ) (x, y) = (x1 +1 , x2 +2 ), then ρ(f ·T(1 ,2 ) ) = ρ(f )+(1 , 2 ), mod Z2 . By properly choosing small 1 and 2 , we obtain a rational rotation vector for f · T(1 ,2 ) . There is a positive number i such that the mean rotation vector for (f · T(1 ,2 ) )ik is an integer vector, which is equivalent to zero on the torus. The map (f · T(1 ,2 ) )ik is isotopic to identity and preserves a smooth area element. The Arnold conjecture, as proved by Conley and Zehnder [5] in the case of torus, implies that (f · T(1 ,2 ) )ik has at least three fixed points, four if all are non-degenerate. This implies that f · T(1 ,2 ) has periodic points of period ki. This proves the lemma. We use the Poincaré-Birkhoff Theorem for the case where f is isotopic to a Dehn twist (cf. Doeff [6]; the author thanks the referee for pointing out the reference). Lemma 8.2. Let f : T2 → T2 be an area-preserving diffeomorphism such that f is isotopic to a Dehn twist. Then for any C r neighborhood V of f , there exists g ∈ V such that g : T2 → T2 has infinitely many periodic points. Let ((x, y), mod Z2 ) be a coordinate system on T2 such that f is isotopic to the map (x, y) → ((x + ky, y), mod Z2 ) for some non-zero integer k. We may assume, without loss of generality, that k > 0. In this homotopy class, there are maps without any periodic point. For example, the map (x, y) → (x + ky, y + α) has no periodic point if α is irrational. However, there are periodic points when α is rational. Our first step is to perturb the map so that it has a rational vertical rotation number. Lift the map f in the y direction; we obtain a map on the infinite cylinder f˜y : 1 S × R → S 1 × R. Define the mean vertical rotation number y ˜ ρ2 (f ) = (π2 (f˜y (p)) − π2 (p))dµ, where µ is the area element. π2 (f˜y (p)) − π2 (p) is independent of choices of the covering points. We define the mean vertical rotation number of f to be ρ2 (f ) = ρ2 (f˜y ), mod 1. This is independent of the lift. By composing f with the map (x, y) → (x, y + ) for some small , we obtain a map g on the torus such that its mean vertical rotation number is rational. This implies that there is a positive integer l such that the vertical rotation number of g l is zero. Now, choose a lift of g l , Gy , in the y direction such that its mean vertical rotation number is zero (instead of being a non-zero integer). Then Gy is an area preserving map on the infinite cylinder S 1 × R which is also exact, i.e., an integral of the 1-form ydx over any closed curve on the cylinder is invariant under the map. Moreover, if we let G be the lift of Gy to R2 , we have that π1 (G(p)) − x − kly is uniformly bounded for any p = (x, y) ∈ R2 .


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Poincaré-Birkhoff twist map theorem implies that there are at least two fixed points and infinitely many periodic points for Gy . This implies that there are infinitely many periodic points for g on T2 . This proves the lemma. The Poincaré-Birkhoff twist map theorem (or Poincaré’s last geometric theorem) shows the existence of fixed points for area-preserving maps on the annulus with twist condition. Here we have an infinite cylinder with infinite twists on two ends. There are two ways to work around this. One is to directly apply Birkhoff’s proof [2] to the annulus {|y| ≤ M} with large M > 0. The annulus is not invariant but the proof works in the same way, as long as one has exactness in the area preserving property. Another way to prove the result is to modify the map so that all horizontal lines are fixed for large values of |y|. Again, the exactness is necessary. Since these techniques are well known, we will not give details here. 9. Proof of the Main Theorem Let M be a compact surface and let R ⊂ Diffrµ (M) be the set of area-preserving diffeomorphisms on M satisfying G1-G4. We first assume that r ≥ 16. In addition, we assume that R satisfies the following two conditions: for any f ∈ R, G6 Every periodic open annulus contains a periodic point; G7 If M = T2 , then there is a periodic point. By Lemma 8.1, G7 is an open and dense condition, hence generic. We claim that G6 is also a generic condition. By the proof of Lemma 7.1, every periodic open annulus can be perturbed to create a periodic point. Let {Ui }∞ i=1 be a collection of countable open sets that forms a basis of the topology M. For each Ui , if it is contained in an invariant open annulus, then we can perturb the map to create a periodic point inside Ui . We can further make the periodic point nondegenrate (hyperbolic or elliptic). Since having a nondegenerate periodic point in Ui is an open condition, the following property is an open and dense property: every periodic open annulus containing Ui contains a periodic point in Ui . Every open set contains an element of {Ui }∞ i=1 ; this implies that, for a residual set of area-preserving diffeomorphisms, every periodic open annulus contains a periodic point. Therefore, the set R, where G1-G7 are satisfied, is a residual set. We claim that for any f ∈ R, the stable and unstable manifolds of hyperbolic periodic points are dense in M. Suppose this is not true and there exists an open set U ⊂ M such that U does not intersect the stable manifold and unstable manifold of any hyperbolic periodic point. Then neither does the invariant set AU = ∪i∈Z f i (U ). Then by Lemma 3.1 and Lemma 3.2, the closure of AU contains no periodic points. Let A = int(AU ), then the closure of A contains no periodic point. Lemma 6.1 shows that A must be an open annulus or M must be a torus. This contradicts Conditions G6 and G7. This proves the first part of our theorem for r ≥ 16. For lower smoothness, i.e., for r = 1, 2, . . . , 15, we first note that the residual set R ⊂ Diffrµ (M), r ≥ 16 constructed above is dense in Diffrµ (M) with r = 1, 2, . . . , 15. Moreover, for each open set U ⊂ M, if U intersects a piece of stable (or unstable) manifold of a hyperbolic fixed point for some f ∈ Diffrµ (M), then the same is true for any map close to f . Since M has a countable basis of open sets, there is a residual subset R r ⊂ Diffrµ (M) for all r = 1, 2, . . . , such that every open set intersects a piece of stable (and unstable) manifold of some hyperbolic periodic point. This proves the first part of our main theorem. For the second part of the theorem, we need the following lemma.

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Lemma 9.1. Every prime end is an accumulation point of periodic points for a generic C r surface diffeomorphism. More precisely, there is a residual subset R ⊂ Diffrµ (M) (this set can be chosen to be the same as above) such that for any f ∈ R, we have the following property: Let V be an open, f invariant set V with finite number of connected boundary pieces and each boundary piece containing more than one point, and let Vˆ be the prime end extension of U , let z ∈ Vˆ be a prime end, then there is a sequence of periodic points of f , {pn }∞ i=1 such that pn → z as n → ∞. We will not give a detailed proof of this lemma. The proof follows from Corollary 8.9 in Franks & Le Calvez [10], which uses the Conley index in a small neigborhood of the prime end circle to obtain periodic points (cf. Franks [9] and Le Calvez & Yoccoz [3]). We remark that even though their results are on S 2 , there is no difference nearby one piece of prime ends. We also remark that each prime end circle has irrational rotation, if a sequence of periodic points approaches the prime end circle, then the sequence of periodic orbits approaches every point in the prime end circle. Now let f ∈ R and U be an open connected set that contains no periodic point for f . Assume that W s (p) ∩ U = ∅ for some hyperbolic periodic point p. Suppose that U is not contained in the closure of W s (p); we will derive a contradiction. Let V be a connected component of M\W s¯(p) whose intersection with U is non-empty. Such a V exists by our assumption. Let Vˆ be the prime end extension of V , then U , as a subset of Vˆ , is an open neighborhood of an arc in the prime ends. Since the rotation number on the prime end circle is irrational, this implies, by the above lemma, U contains infinitely many priodic points. But U is periodic point free; by our assumption, this contraction shows that U ⊂ W s¯(p). This proves our main theorem. 10. Other Problems and Conjectures of Poincaré Poincaré already noted the importance of the generic properties of area-preserving diffeomorphisms in his study of the three-body problem. The following two fundamental conjectures are due to Poincaré [19]. Conjecture 10.1. For generic C r area-preserving diffeomorphisms on a compact surface M, the set of all periodic points are dense. Conjecture 10.2. There exist a residual set R ∈ Diffrµ (M) such that if f ∈ R and p is a hyperbolic periodic point of f then the homoclinic points of p are dense in both stable and unstable manifolds of p. In other words, let J be a segment in W s (p) (or W u (p)), then W s (p) ∩ W u (p) ∩ J = J . In C 1 topology, both of the above conjectures are proved to be true. The first one is a consequence of the so-called closing lemma. It is proved by Pugh [20] and later improved to various cases by Pugh & Robinson [21]. A different proof was given by Liao [13] and Mai [14]. The second conjecture in C 1 topology is a result of Takens [26]. The high dimensional analog was proved by Xia [29]. It can also be regarded as a so-called C 1 connection lemma, first proved by Hayashi [12] and later simplified and generalized by Xia [29], Wen & Xia [27, 28]. In C r topology with r > 1, little progress has been made for these two conjectures and it’s known to be an extremely difficult problem (cf. Smale [25]). The local perturbation methods used in the C 1 case no longer seem to work and examples suggest that a more global approach has to be developed (Gutierrez [11]).


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While unable to prove these two conjectures, our main result offers very strong evidence supporting the conjectures. Moreover, as an easy consequence, our result implies that Conjecture 2 implies Conjecture 1. Acknowledgement. I am grateful to Kamlesh Parwani for useful comments and corrections to the original manuscript. I would also like to thank the referees of this paper for their careful reading, comments and suggestions.

References 1. Arnold,V.I.: Mathematical Methods of Classical Mechanics. Berlin-Heidelberg-NewYork: SpringerVerlag, 1978 2. Birkhoff, G.D.: Dynamical Systems. Volume 9. American Math. Soc. Colloquium Publications, Providence, RI:AMS, 1966 3. Calvez, P.Le., Yoccoz, J.C.: Un théorème d’indice pour les homéomorphismes du plan au voisinage d’un point fixe. Ann. of Math. (2) 146(2), 241–293 (1997) ¨ 4. Caratheodory, C.: Uber die begrenzung einfach zusammenhangender gebiete. Math. Ann. 73, 323– 370 (1913) 5. Conley, C., Zehnder, E.: The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnold. Invent. Math. 73(1), 33–49 (1983) 6. Doeff, E.: Rotation measures for homeomorphisms of the torus homotopic to a Dehn twist. Ergod. Theor. Dyn. Syst. 17, 575–591 (1997) 7. Douady, R.: Applications du théorème des tores invariants. Thèse de troisiéme cycle, Université de Paris 7, 1992 8. Franks, J., Rotation vectors and fixed points of area preserving surface diffeomorphisms. Trans. Amer. Math. Soc. 348(7), 2637–2662 (1996) 9. Franks, J.: The Conley index and non-existence of minimal homeomorphisms. Illinois J. Math. 43 457–464 (1999) 10. Franks, J., Le Calvez, P.: Regions of instability for non-twist maps. Ergod. Theor. Dynam. Sys. 23(1),111–141 (2003) 11. Gutierrez, C.: A counter-example to a C 2 closing lemma. Ergod. Theor. Dyn. Syst. 7(4), 509–530 (1987) 12. Hayashi, S.: Connecting invariant manifolds and the solution of the C 1 stability and -stability conjectures for flows. Ann. of Math. 145(1), 81–137 (1997) 13. Liao, S.T.: An extension of the C 1 closing lemma. Acta Sci. Natur. Univ. Pekinensis. 2, 1–41 (1979) 14. Mai, J.: A simpler proof of C 1 closing lemma. Scientia Sinica 10, 1021–1031 (1986) 15. Mather, J.: Topological proofs of some purely topological consequences of caratheodory’s theory of prime ends. In: Selected Studies. Eds. Th. M. Rassias, G. M. Rassias, 1982, pp. 225–255 16. Oliveira, F.: On the generic existence of homoclinic points. Ergodic Theory Dynamical Systems 7, 567–595 (1987) 17. Oliveira, F.: On C ∞ genericity of homoclinic orbits. Nonlinearity 13, 653–662 (2000) 18. Pixton, D.: Planar homoclinic points. J. Differ. Eq. 44, 1365–382 (1982) 19. Poincaré, H.: Les méthodes nouvelles de la mécanique céleste. Paris, 1892 20. Pugh, C.: The closing lemma. Amer. J. Math. 89, 956–1021 (1967) 21. Pugh, C., Robinson, C.; The C 1 closing lemma, including hamiltonians. Ergod. Theor. Dyn. Sys. 3, 261–313 (1983) 22. Robinson, C.: Generic properties of conservative systems, i, ii. Amer. J. Math. 92, 562–603, 897–906 (1970) 23. Robinson, C.: Closing stable and unstable manifolds on the two-sphere. Proc. Amer. Math. Soc. 41, 299–303 (1973) 24. Siegel, C.L., Moser, J.K.: Lectures on Celestial Mechanics. Berlin-Heidelberg-New York: Springer, 1971 25. Smale, S.: Mathematical problems for the next century. Math. Intelligencer 20(2), 7–15 (1998) 26. Takens, F.: Homoclinic points in conservative systems. Invent. Math. 18, 267–292 (1972) 27. Wen, L., Xia, Z.: A basic C 1 perturbation theorem. J. Differ. Eqs. 154(2), 267–283 (1999) 28. Wen, L., Xia, Z.: On C 1 connecting lemmas. Trans. Amer. Math. Soc. 352,(10) (2000) 29. Xia, Z.: Homoclinic points in symplectic and volume-preserving diffeomorphism. Commun. Math. Phys. 177, 435–449 (1996) Communicated by G. Gallavotti


Communications in


A Pairing Between Super Lie-Rinehart and Periodic Cyclic Homology Tomasz Maszczyk1,2, 1 2

Institute of Mathematics, Polish Academy of Sciences, Sniadeckich 8, 00–956 Warszawa, Poland Institute of Mathematics, University of Warsaw, Banacha 2, 02–097 Warszawa, Poland. E-mail: [email protected]

Received: 14 March 2005 / Accepted: 26 July 2005 Published online: 29 November 2005 – © Springer-Verlag 2005

Abstract: We consider a pairing producing various cyclic Hochschild cocycles, which led Alain Connes to cyclic cohomology. We are interested in geometrical meaning and homological properties of this pairing. We define a non-trivial pairing between the homology of a Lie-Rinehart (super-)algebra with coefficients in some partial traces and relative periodic cyclic homology. This pairing generalizes the index formula for summable Fredholm modules, the Connes-Kubo formula for the Hall conductivity and the formula computing the K 0 -group of a smooth noncommutative torus. It also produces new homological invariants of proper maps contracting each orbit contained in a closed invariant subset in a manifold acted on smoothly by a connected Lie group. Finally we compare it with the characteristic map for the Hopf-cyclic cohomology. 1. Introduction Let G be a simply-connected Lie group acting smoothly on a smooth manifold N and Z be a closed invariant submanifold. Let a smooth map N → M contract these orbits. In the dual language of algebras of smooth functions the situation can be described as follows. We have a Lie algebra g acting by derivations on an algebra B, fixing an ideal J . We have also a homomorphism of algebras π ∗ : A → B such that g(π ∗ A) ⊂ J , or equivalently, we have a homomorphism of algebras A → B ×B/J (B/J )g .

(1)

The last homomorphism of algebras of smooth functions describes a continuous map of topological spaces N P P /G → M.

The author was partially supported by the KBN grant 1P03A 036 26.

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We generalize this construction to the case of families, parameterized by commutative super-spaces, of noncommutative super-spaces, with the total space acted by superLie-Rinehart algebras over the base algebra, as follows. Let (L, R) be a Z/2-graded Lie-Rinehart algebra over a Z/2-graded-commutative ring R containing rational numbers, with a subring of constants k := H 0 (L, R; R) = R L , acting (from the left) by super-derivations on a Z/2-graded associative R-algebra B. Provided a homomorphism of Z/2-graded associative k-algebras A → B ×B/J (B/J )L is given, we prove the following theorem. Theorem 1. There exists a nontrivial canonical k-bilinear pairing Hp (L, R; H 0 (B, (J p )∗ )) ⊗k S(H Cp+2 (A/k)) → k.

(3)

Here Hp (L, R; −) denotes the pth super-Lie-Rinehart homology [14, 18], H 0 (B, −) denotes the 0th Hochschild cohomology, (−)∗ = H omk (−, k) and S : H Cp+2 (A/k) → H Cp (A/k)) is the periodicity map of Connes on the relative cyclic homology [5]. The above theorem implies, after passing to the inverse limit with respect to S, the existence of a canonical bilinear pairing with the relative periodic cyclic homology of A. Corollary 1. There exists a nontrivial canonical bilinear pairing Hp (L, R; H 0 (B, (J p )∗ )) ⊗k H Pp (A/k) → k.

(4)

The latter pairing induces (is equivalent to, if k = R and k is a field) the following k-linear map: Hp (L, R; H 0 (B, (J p )∗ )) → H P p (A/k),

(5)

which can be regarded as a kind of characteristic map. We will compare it with the Connes-Moscovici characteristic map [7, 8] and with the cup-product of the second kind of Khalkhali-Rangipour [13] in Hopf-cyclic cohomology. We will show four classes of examples for which our pairing (or the characteristic map) is known to be, in general, non-trivial and its values have important geometric interpretations. The first one is the creation of nontrivial homology classes by contracting orbits making sense in classical differential geometry, the second is the index formula for summable Fredholm modules [5, 6], the third is the Connes-Kubo formula for the Hall conductivity in the quantum Hall effect [3, 1, 5, 2, 15, 19], and the fourth computes K 0 -group of a noncommutative torus in terms of characteristic numbers of smooth Powers-Rieffel projections [4, 16, 17]. Analogous considerations give us the following “dual” variant, seemingly more fundamental, of our construction for ((L, R), J ⊂ B) and A → B as above. Theorem 2. There exists a nontrivial canonical k-linear “dual characteristic map” H Pp (A/k) → H p (L, R; H0 (B, J p )).

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2. Construction We consider the bilinear pairing Cp (L, R; H 0 (B, (J p )∗ )) ⊗k

p+1 k

A → k,

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A Pairing Between Super Lie-Rinehart and Periodic Cyclic Homology

(τ ⊗ X1 ∧ · · · ∧ Xp ) ⊗ (a0 ⊗ . . . ⊗ ap ) →

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(±)τ (a0 Xσ (1) (a1 ) · · · Xσ (p) (ap )),

σ ∈p

where the sign is determined uniquely by the convention of transposition of homogeneous symbols from the left hand side to the position on the right hand side. In the sequel we will use homogeneous elements, p the above sign convention and the abbreviated notation X = X1 ∧ · · · ∧ Xp ∈ R L. Let τ ∈ H 0 (B, (J p )∗ ) = (J p /[B, J p ])∗ . The latter space is a right (L, R)-module. The super-Lie-Rinehart boundary operator p ∂ : Cp (L, R; −) → Cp−1 (L, R; −), where Cp (L, R; −) = (−) ⊗R R , computing homology with values in (L, R)-modules is an obvious minimal common generalization of the super-Lie boundary operator from [14] and the Lie-Rinehart boundary operator from [18]. By Zp (L, R; −) (resp. Bp (L, R; −)) we denote cycles (resp. boundaries) in this complex. By b (resp. t, B) we denote the Hochschild boundary (resp. cyclic operator, Connes B-operator) used in cyclic homology [4]. In the lemmas below we apply the above pairing to various pairs of submodules of super-Lie-Rinehart and Hochschild chains. Lemma 1. Cp (L, R; H 0 (B, (J p )∗ )) · im(b) = 0.

(8)

(τ ⊗ X) · b(a0 ⊗ · · · ⊗ ap ) = 0.

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Zp (L, R; H 0 (B, (J p )∗ )) · im(1 − t) = 0.

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Proof.

Lemma 2.

Proof. (τ ⊗ X) · (1 − t)(a0 ⊗ · · · ⊗ ap ) = ±∂(τ ⊗ X) · (ap a0 ⊗ a1 ⊗ · · · ⊗ ap−1 ). (11) From the last two lemmas we get Corollary 2. There exists a canonical bilinear pairing Zp (L, R; H 0 (B, (J p )∗ )) ⊗ H Cp (A/k) → k.

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One could expect that the above pairing descends to Lie algebra homology. But it is not true without an appropriate replacement on the level of cyclic homology. Lemma 3. Bp (L, R; H 0 (B, (J p )∗ )) · ker(B : H Cp (A/k) → H Hp+1 (A/k)) = 0.

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Proof. The following formula is an analog of the Stokes formula (τ ⊗ X) · B(a0 ⊗ · · · ⊗ ap−1 ) = ±p ∂(τ ⊗ X) · (a0 ⊗ . . . ⊗ ap−1 ).

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By the long exact sequence of Connes I

S

B

· · · → H Hp+2 (A/k) → H Cp+2 (A/k) → H Cp (A/k) → H Hp+1 (A/k) → · · · , (15) we have ker(B : H Cp (A/k) → H Hp+1 (A/k)) = im(S : H Cp+2 (A/k) → H Cp (A/k)). Together with Lemma 3 and Corollary 2 this gives the pairing Hp (L, R; H 0 (B, (J p )∗ )) ⊗k im(S : H Cp+2 (A/k) → H Cp (A/k)) → k

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desired in Theorem 1. In order to show that it is non-trivial and interesting we consider the following classes of examples. 3. Example: Contracting Orbits Before we present non-classical examples, we want to explain the classical case in differential topology. Let N be a compact manifold (resp. singular with boundary) acted on by a connected Lie group G with Lie algebra g, P be a closed invariant subset (resp. containing the singular locus or boundary) and J ⊂ B := C ∞ (N ) be a g-invariant ideal of smooth functions on N vanishing along P . The action of g on differential forms on N is a representation of a Z-graded super-Lie algebra linearly spanned by symbols (d, ιX , LX ), where X ∈ g, of degrees (1, −1, 0), subject to the relations [d, d] = 0, [ιX , ιY ] = 0, [d, LX ] = 0, [d, ιX ] = LX , [LX , ιY ] = ι[X,Y ] , [LX , LY ] = L[X,Y ] .

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We will use the following consequence of these relations [d, ιX1 . . . ιXp ] = (−1)i−1 ιX1 . . . ι (−1)i+j −1 ι[Xi ,Xi ] ιX1 . . . ι Xi . . . ιXp LXi + Xi . . . ι Xj . . . ιXp . i

i<j

(18) Every smooth measure µ on N \ P (i.e. a differential top degree form with values in the orientation bundle), such that for every element f ∈ J p the product f µ extends to a smooth measure on the whole N , defines an element (−)µ ∈ H 0 (B, (J p )∗ ) = (J p )∗ . (19) Y

The right g-action on such element reads as (−)µ · X = − (−)LX µ. Y

Y

(20)


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Proposition 1. If a chain (−)µ ⊗ X1 ∧ · · · ∧ Xp ∈ Cp (g, H 0 (B, (J p )∗ )),

(21)

Y

where µ’s are smooth measures on Y as above, is a cycle (resp. a boundary) then the differential form ιX1 . . . ιXp µ is closed (resp. exact). Proof. The cycle condition for our chain (−)µ ⊗ X1 ∧ · · · ∧ Xp = 0 ∂

(22)

Y

is equivalent to i . . . ∧ Xp ⊗ LXi µ (−1)i X1 ∧ . . . X i

i . . . X j . . . ∧ Xp ⊗ µ = 0, (23) (−1)i+j −1 [Xi , Xi ] ∧ X1 ∧ . . . X + i<j

which implies that (−1)i ιX1 . . . ι Xi . . . ιXp LXi µ i

(−1)i+j −1 ι[Xi ,Xi ] ιX1 . . . ι + Xi . . . ι Xj . . . ιXp µ = 0,

(24)

i<j

which is equivalent to

[d, ιX1 . . . ιXp ]µ = 0.

(25)

Since µ is a top degree form dµ = 0, which gives finally d ιX1 . . . ιXp µ = 0.

(26)

The proof of the implication “boundary ⇒ exact” is similar.

Let us consider now a smooth map π : N → M into a compact manifold (resp. singular variety, with boundary) M, contracting each orbit contained in P to a point. Let A := C ∞ (M) be an algebra of smooth functions on M. Then g(π ∗ A) ⊂ J ⊂ B. Comparing with the canonical map from De Rham homology of currents to periodic cyclic cohomology we see that our characteristic map associates with the Lie homology class of the above cycle a homology class of the closed current j , where (π ∗ ω)(X˜ 1 , . . . , X˜ p )µ = ± (π ∗ ω) ∧ ιX1 . . . ιXp µ. (27) j (ω) := N

N

Here by X˜ we mean the vector field corresponding to an element X ∈ g. Note that if ιX1 . . . ιXp µ extends to the whole N then we get the push-forward of the homology class of the closed current

(−)µ ⊗ X1 ∧ . . . ∧ Xp → ±π! (−)ιX1 . . . ιXp µ (28) N

in periodic cyclic cohomology.

N

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T. Maszczyk

Take for example N = M = G = S 1 and π = id : S 1 → S 1 , P = ∅, µ the Haar measure and X ∈ g normalized so that ιX µ = 1. Then S 1 (−)µ ⊗ X is a cycle and the characteristic map gives

, (29) (−)µ ⊗ X → S1

S1

i.e. the fundamental class of S 1 . Though it is a nontrivial example, it is not very enlightening. Therefore we need a more complicated example to show the point. This time ιX1 . . . ιXp µ will not extend to the whole N , so the push-forward of the respective current will not be defined. However, our characteristic map still will define a nontrivial homology class on M. To see this, let us take a cylinder N = S 1 × [−π, π ] with coordinates (ϕ, ψ) (ϕ - circular coordinate, ψ ∈ [−π, π ]). Consider the following smooth action of the additive Lie group G = R on N , (ϕ + t, 2 arctan(tan ψ2 + t)) ifψ = ±π, t · (ϕ, ψ) := (ϕ + t, ±π)) ifψ = ±π. It is generated by a vector field X=

∂ ∂ + (1 + cos ψ) . ∂ϕ ∂ψ

Let us take P = ∂N, which is the union of compact orbits of the above action and the ideal J = (1 + cos ψ) in the algebra B = C ∞ (N ), vanishing along P . The form µ=

1 dψ ∧ dϕ 2π 1 + cos ψ

is defined on N \ P and invariant. We have ιX µ =

1 1 dψ dϕ − , 2π 2π 1 + cos ψ

which does not extend onto the whole N. Take now a subvariety M ⊂ R3

2 x 2 2 z =ε + 1 − ( x 2 + y 2 − 1)2 , x2 + y2

(x, y) = (0, 0),

where a parameter ε ∈ (0, 1/2), homeomorphic to a torus with one basic cycle contracted to the unique singular point (−1, 0, 0), and a smooth map π : N → M of the form π ∗ x = cos ψ(1 + ε(1 + cos ψ) cos ϕ), π ∗ y = sin ψ(1 + ε(1 + cos ψ) cos ϕ), π ∗ z = ε(1 + cos ψ) sin ϕ. One can check that X(π ∗ x), X(π ∗ y), X(π ∗ z) ∈ J which implies that for A = C ∞ (M), X(π ∗ A) ⊂ J. We have H1 (N, Z) = Z generated by the homology class of one boundary circle (ψ = π ), H1 (M, Z) = Z generated by the homology class of the ellipse (x 2 + y 2 = 1, z = ε(x + 1)). The topology of the map π : N → M follows. It contracts the boundary


743

circles of the cylinder N to the singular point of M. Therefore it kills the generator of H1 (N, Z). But it also creates the generator of H1 (M, Z). The killing property of π is described by the nullity of the induced map H1 (N, Z) → H1 (M, Z). We will show that the creating property of π is described by our characteristic map. Let ω=

1 xdy − ydx 2π x 2 + y 2

be a closed 1-form on M, whose period over the generator of H1 (M, Z) is equal to 1. We have π ∗ ω = dψ. Let us compute our pairing of the Lie homology class with the De Rham cohomology class of ω,

1 (−)µ ⊗ X · [ω] = − (π ∗ ω) ∧ ιX µ = − 2 dψ ∧ dϕ = 1. 4π N N N Therefore our characteristic map applied to N (−)µ ⊗ X gives the homology class of the current homological to the period over the generator of H1 (M, Z). 4. Example: Index Formula Let us assume that k = C and we have a p-summable even involutive Fredholm module (A, H, F ) [5, 6], i.e. A is a Z/2-graded ∗-algebra, H is a Z/2-graded Hilbert space with a grading preserving ∗-representation A → B(H ), and F is an odd self-adjoint involution on H such that [F, A] ⊂ Lp (H ),

(30)

where Lp (H ) denotes the p th Schatten ideal in B(H ). Let us define now a Z/2-graded abelian super-Lie algebra generated by one odd element d, g := C · d,

(31)

B := B(H )

(32)

J := Lp (H ).

(33)

db := [F, b]

(34)

a Z/2-graded associative algebra

and an ideal

The formula

defines the left action of g on B by derivations and obviously J is a g-ideal. The projection into the first cartesian factor defines an isomorphism of Z/2-graded associative algebras ∼ =

B ×B/J (B/J )g → {b ∈ B(H ) | [F, b] ∈ Lp (H )}

(35)

and a ∗-homomorphism of Z/2-graded associative C ∗ -algebras A → B ×B/J (B/J )g is equivalent to a structure of a p-summable even involutive Fredholm module (A, H, F ).

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T. Maszczyk

By functoriality of Connes’ long exact sequence it is enough to consider our pairing for A := B ×B/J (B/J )g . Since the super-trace is a linear functional on the p th power of the ideal J := Lp (H ) which vanishes on super-commutators and the super-Lie algebra g = C · d is abelian, the element str ⊗ d ∧ . . . ∧ d ∈ H 0 (A, (J p )∗ ) ⊗ p g is a cycle. On the other hand, for any even self-adjoint idempotent e ∈ A the element e ⊗ · · · ⊗ e ∈ p+1 A is a cyclic cycle for even p, which is in the image of the periodicity operator S. We can compute our pairing of homology classes of these two cycles which gives the index of a Fredholm operator [str ⊗ d ∧ · · · ∧ d] · [e ⊗ · · · ⊗ e] = cp Index(e11 F01 e00 ),

(36)

in general a non-zero number. Here F01 : H0 → H1 (resp. e00 : H0 → H0 , e11 : H1 → H1 ) is a unitary block of F (resp. self-adjoint idempotent block of e) under the orthogonal decomposition H = H0 ⊕ H1 into an even and odd part. 5. Example: Connes-Kubo Formula Let g be an abelian Lie algebra and A = B = J . If τ is a g-invariant trace on A then this is obvious that for all X1 , . . . , Xp ∈ g the chain τ ⊗ X 1 ∧ · · · ∧ Xp

(37)

is a cycle, hence defines a homology class. This construction is next adapted to the geometry of the Brillouin zone. Its pairing with an appropriately normalized even dimensional class [e ⊗ e ⊗ e] in H P2 (A) computes the Hall conductivity σ in noncommutative geometric models of quantum Hall effect, [τ ⊗

g

Xi ∧ Xi+g ] · [e ⊗ e ⊗ e] = σ,

(38)

i=1

in general a non-zero integer [3, 1, 5, 2, 19] or rational number [15], depending on the model. 6. Example: K0 of a Noncommutative Torus Formally it is the same construction as in the previous example adapted to the context of non-commutative geometry of the noncommutative 2-torus [17]. Let A be the dense subalgebra of “smooth functions on the noncommutative 2-torus" [4] of the C ∗ -algebra generated by two unitaries U, V subject to the relation U V = e2πiθ V U

(39)

with an irrational real θ . The Lie group S 1 × S 1 acts on A by automorphisms and its Lie algebra g spanned by commuting elements X, Y acts by derivations such that X(U ) = 2πiU, X(V ) = 0,

(40)

Y (U ) = 0, Y (V ) = 2πiV .

(41)


745

Every element a ∈ A can be uniquely expanded as a = amn U m V n . Then the functional τ (a) = a00 is a g-invariant trace. Again, we have a homology class [τ ⊗ X ∧ Y ]. It is known that K0 (A) = Z + Z · θ ⊂ C, where the identification is done by this trace [16, 17]. Any selfadjoint idempotent e ∈ A is determined by its trace τ (e) = p − q · θ uniquely up to unitary equivalence. This is our pairing in dimension zero, [τ ] · [e] = p − q · θ.

(42)

Our pairing in dimension two computes the number q, [τ ⊗ X ∧ Y ] · [e ⊗ e ⊗ e] = q · 2π i.

(43)

This means that our pairings, defined a priori over C, detect fully the K0 -group isomorphic to Z ⊕ Z. 7. Comparison with Other Constructions In [7, 8] the following pairing (Connes-Moscovici characteristic map) is considered: p

H P(δ,σ ) (H ) ⊗ T r(δ,σ ) (A) → H P p (A)

(44)

for any Hopf algebra H with (δ, σ ) a modular pair in involution, acting on an algebra A, where by T r(δ,σ ) (A) we denote the space of (δ, σ )-traces on A. Taking H = U (g), δ = , σ = 1 one has [8], p Hi (g), (45) H P(,1) (U (g)) = i≡p (mod 2)

T r(,1) (A) = H 0 (A, A∗ )g .

(46)

Then we have the following commuting diagram → Hp (g, H 0 (A, A∗ )) Hp (g) ⊗ H 0 (A, A∗ )g ↓ ↓ p H P(,1) (U (g)) ⊗ T r(,1) (A) → H P p (A),

(47)

where left vertical and upper horizontal arrows are canonical, the bottom horizontal arrow is the Connes-Moscovici characteristic map and the right vertical arrow is our characteristic map for A = B = J . The main difference between these two characteristic maps is the position of traces: in the Connes-Moscovici map traces are paired with cyclic periodic cohomology while in our map they are coefficients of Lie algebra homology. Recently [13] a new pairing with values in cyclic cohomology (the cup product of the second kind) p

q

H CH (C, M) ⊗ H CH (A, M) → H C p+q (A)

(48)

has been presented, which allows to consider in this pairing cyclic cohomology with nontrivial coefficients in the sense of [9]. It is defined for a Hopf algebra H , an H -module algebra A, an H -comodule algebra B, an H -module coalgebra C acting on A in a suitable sense and any stable anti-Yetter-Drinfeld (SAYD) module M

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T. Maszczyk

over H . For C = H = U (g), q = 0 and M = k(,1) trivial one dimensional SAYDmodule one gets again the Connes-Moscovici characteristic map. Since U (g) is a cocommutative Hopf algebra, any U (g)-module M with a trivial U (g)-comodule structure is a SAYD-module. Taking C = H = U (g), q = 0 and M = H 0 (A, A∗ ) one has the Khalkhali-Rangipour cup product of the second kind H CU (g) (U (g), H 0 (A, A∗ )) ⊗ H CU0 (g) (A, H 0 (A, A∗ )) → H C p (A). p

(49)

The trace evaluation map H 0 (A, A∗ ) ⊗ A → k defines a distinguished element in H CU0 (g) (A, H 0 (A, A∗ )) and consequently the following characteristic map: H CU (g) (U (g), H 0 (A, A∗ )) → H C p (A), p

(50)

by taking the above cup product with this distinguished element. In fact this map comes from the morphism of cyclic objects, so it can be pushed to the periodic cyclic cohomology, hence we get a map H PU (g) (U (g), H 0 (A, A∗ )) → H P p (A). p

(51)

By Theorem 5.2 of [10] (note that in [10] authors use cyclic objects related to cyclic objects from [9] by Connes’s cyclic duality [12], transforming homology into cohomology) one has H PU (g) (U (g), H 0 (A, A∗ )) = p

Hi (g, H 0 (A, A∗ )).

(52)

i≡p (mod 2)

In this case, our characteristic map factorizes through (51),

p H PU (g) (U (g), H 0 (A, A∗ )) →

Hp (g, H 0 (A, A∗ )) ↓ H P p (A),

(53)

where the south-west arrow is an embedding onto a direct summand in the decomposition (52). However, in general, there is no way to extend a partial trace from the ideal J p to the trace defined on the whole algebra, hence there is no canonical element to pair with as in (50). Therefore the characteristic map a´ la Khalkhali-Rangipour ìs not defined in general. In particular, the index pairing discussed in paragraph 4 cannot be obtained in this way. In spite of this discrepancy we expect that, after appropriate modifications of Hopfcyclic cohomology (or its extended version [11] working in the case of enveloping algebras of Lie-Rinehart algebras) with appropriate coefficients, our construction could be generalized to (super, extended) Hopf-cyclic cohomology. The crucial property this generalization should satisfy is the above index pairing.


747

References 1. Bellissard, J., van Elst, A., Schulz-Baldes, H.: The non-commutative geometry of the quantum Hall effect. J. Math. Phys. 35, 5373–5451 (1994) 2. McCann, P.J.: Geometry and the integer quantum Hall effect. In: Geometric Analysis and Lie Theory in Mathematics and Physics. Edited by A.L. Carey, M.K. Murray, Cambridge: Cambridge Univ. Press, 1998, pp. 132–208 3. Carey, A., Hannabus, K., Mathai, V., McCann, P.: Quantum Hall Effect on the hyperbolic plane. Commun. Math. Phys. 190, No. 3, 629–673 (1976) 4. Connes, A.: C ∗ -algébres et géométrie différentielle. C. R. Acad. Sci. Paris Ser. A-B, 290 (1980) 5. Connes, A.: Noncommutative differential geometry. Publ. Math. I.H.E.S. 62, 257–360 (1986) 6. Connes, A.: Noncommutative geometry. San Diego, CA: Acad. Press, Inc., 1994 7. Connes, A., Moscovici, H.: Cyclic cohomology and Hopf algebras. Lett. Math. Phys. 48, 97–108 (1999) 8. Connes, A., Moscovici, H.: Hopf algebras, cyclic cohomology and the transverse index theorem. Commun. Math. Phys. 198, 199–246 (1998) 9. Hajac, P.M., Khalkhali, M., Rangipour, B., Sommerhäuser, Y.: Hopf-cyclic homology and cohomology with coefficients. C. R. Math. Acad. Sci. Paris 338, No. 9, 667–672 (2004) 10. Jara, P., Stefan, D.: Cyclic homology of Hopf Galois extensions and Hopf algebras. http:arxiv.org/list/math.KT/0307099, 2003 11. Khalkhali, M., Rangipour, B.: Cyclic cohomology of (extended) Hopf algebras. In: Noncommutative geometry and quantum groups (Warsaw, 2001), Banach Center Publ. 61, 2003, pp. 59–89 12. Khalkhali, M., Rangipour, B.: A note on cyclic duality and Hopf algebras. Commun. Alg. 33, No. 3, 763–773 (2005) 13. Khalkhali, M., Rangipour, B.: Cup Products in Hopf-Cyclic Cohomology. C. R. Acad. Sci. Paris, Ser. I. 340, 9–14 (2005) 14. Leites, D.A., Fuks, D.B.: Cohomology of Lie superalgebras. Dokl. Bolg. Akad. Nauk 37, No. 10, 1294–1296 (1984) 15. Marcolli, M.; Mathai, V.: Twisted Higher Index Theory on Good Orbifolds, II: Fractional Quantum Numbers. Commun. Math. Phys. 201, No. 1, 55–87 (2001) 16. Pimsner, M., Voiculescu, D.: Exact sequences for K groups and Ext groups of certain cross products C ∗ -algebras. J. Operator Theory 4, 93–118 (1980) 17. Rieffel, M.: C ∗ -algebras associated with irrational rotations. Pac. J. Math. 93, 415–429 (1981) 18. Rinehart, G.: Differential forms on general commutative algebras. Trans. Amer. Math. Soc. 108, 195–222 (1963) 19. Xia, J.: Geometric invariants of the quantum Hall effect. Commun. Math. Phys. 119, 29–50 (1988) Communicated by A. Connes


Communications in


Construction of Perfect Crystals Conjecturally Corresponding to Kirillov-Reshetikhin Modules over Twisted Quantum Affine Algebras Satoshi Naito, Daisuke Sagaki Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan. E-mail: [email protected], [email protected] Received: 16 March 2005 / Accepted: 16 August 2005 Published online: 1 March 2006 – © Springer-Verlag 2006

Abstract: Assuming the existence of the perfect crystal bases of Kirillov-Reshetikhin modules over simply-laced quantum affine algebras, we construct certain perfect crystals for twisted quantum affine algebras, and also provide compelling evidence that the constructed crystals are isomorphic to the conjectural crystal bases of Kirillov-Reshetikhin modules over twisted quantum affine algebras.

Introduction The finite-dimensional irreducible modules over quantum affine algebras Uq (g) with (classical) weight lattice Pcl have been extensively studied from various points of view, but the study of these modules from the point of view of the crystal base theory still seems to be insufficient. This is mainly because unlike (infinite-dimensional) integrable highest weight modules over quantum affine algebras Uq (g) with (affine) weight lattice P , finite-dimensional irreducible Uq (g)-modules do not have crystal bases in general. (It is not even known which finite-dimensional irreducible Uq (g)-modules have crystal bases.) However, it is conjectured (see Conjecture 1.5.1 below, and also [HKOTY, §2.3], [HKOTT, §2.3]) that a certain important class of finite-dimensional irreducible Uq (g)-modules, called Kirillov-Reshetikhin modules (KR modules for short), do have crystal bases. In this paper, assuming the existence of the perfect crystal bases B i,s of the KR modules over the simply-laced quantum affine algebras Uq (g), we construct ceri,s for twisted quantum affine algebras Uq ( tain perfect crystals B g ). Furthermore, we i,s decompose, when regarded describe explicitly, in almost all cases, how the crystals B as a Uq ( gI0 )-crystal by restriction, into a disjoint union of the crystal bases of irreducible highest weight Uq ( gI0 )-modules, where Uq ( gI0 ) is the quantized universal enveloping algebra of the (canonical) finite-dimensional, reductive Lie subalgebra gI0 of g. i,s are isomorphic to the crystal These results motivate a conjecture that the crystals B

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S. Naito, D. Sagaki

bases of certain KR modules with specified Drinfeld polynomials (see §1.5 below) over the twisted quantum affine algebras Uq ( g ), since they coincide with the conjectural “branching rules” (with q = 1) in [HKOTT, Appendix A]. We now describe our results more precisely. Let g be a simply-laced affine Lie algebra over C, that is, let g be the Kac-Moody algebra g(A) over C associated to the generalized (1) (1) Cartan matrix (GCM for short) A = (aij )i,j ∈I of type A2n−1 (n ≥ 2), A2n (n ≥ 1), (1)

(1)

(1)

Dn+1 (n ≥ 3), D4 , or E6 , where I is an index set for the simple roots (numbered as in §2.1 below). We denote by Uq (g) (resp., Uq (g)) the associated quantum affine algebra over C(q) with weight lattice P (resp., Pcl ). For each i ∈ I0 := I \ {0}, s ∈ Z≥1 , and (i) ζ ∈ C(q)× := C(q) \ {0}, let Ws (ζ ) denote the finite-dimensional irreducible module over the quantum affine algebra Uq (g) whose Drinfeld polynomials Pj (u) ∈ C(q)[u] for j ∈ I0 are given by:

Pj (u) =

 s   (1 − ζ q s+2−2k u)  

if j = i,

k=1

   

1

otherwise.

(Here we are using the Drinfeld realization of Uq (g), and the classification of its finitedimensional irreducible modules of “type 1” by Drinfeld polynomials; see [CP1] for (i) details.) We call this Uq (g)-module Ws (ζ ) a Kirillov-Reshetikhin module (KR module for short) over Uq (g). It is conjectured (see Conjecture 1.5.1 below, and also [HKOTY,

§2.3]) that for every i ∈ I and s ∈ Z≥1 , there exists some ζs ∈ C(q)× such that the (i) (i) KR module Ws (ζs ) has a crystal base. Let ω : I → I be a nontrivial diagram automorphism such that ω(0) = 0, and g the corresponding orbit Lie algebra of g (see §2.2 below for the definition). Note that the orbit Lie algebra g is a twisted affine Lie algebra, i.e., that g is the Kac-Moody algebra (2) (2) g(A) over C associated to the GCM A = ( aij )i,j ∈I of type Dn+1 (n ≥ 2), A2n (n ≥ 1), (2) (3) (2) A2n−1 (n ≥ 3), D4 , or E6 , where I ⊂ I is a certain complete set (containing 0 ∈ I ) of representatives of the ω-orbits in I , and also an index set for the simple roots of g. In this paper, we assume that for (arbitrarily) fixed i ∈ I0 := I\ {0} and s ∈ Z≥1 , there exists (i)

(ωk (i))

(ζs ) some ζs ∈ C(q)× such that for every 0 ≤ k ≤ Ni − 1, the KR module Ws k over Uq (g) has a crystal base, denoted by B ω (i),s , where Ni ∈ Z≥1 is the number of (i) elements of the ω-orbit of i ∈ I0 in I ; here we note that the ζs ∈ C(q)× is assumed to (i)

(i)

k

be independent of 0 ≤ k ≤ Ni − 1. We further assume that the B ω (i),s , 0 ≤ k ≤ Ni − 1, are all perfect Uq (g)-crystals of level s (in the sense of Definition 1.4.12). Now, for the i,s equipped (fixed) i ∈ I0 and s ∈ Z≥1 , we define the tensor product Uq (g)-crystal B with the Kashiwara operators ej and fj , j ∈ I , by: i,s = B i,s ⊗ B ω(i),s ⊗ · · · ⊗ B ωNi −1 (i),s , B

on which the diagram automorphism ω : I → I acts in a canonical way (see §2.3 i,s for j ∈ I by (see below). Also, we define ω-Kashiwara operators ej and fj on B

Construction of Perfect Crystals

751

(2.4.1), and Remark 2.2.1):

xj =

 2 xj xω(j ) xj

if Nj = 2 and aj,ω(j ) = aω(j ),j = −1,

x x j ω(j ) · · · xωNj −1 (j )

if aj,ωk (j ) = 0 for all 1 ≤ k ≤ Nj − 1,

where x is either e or f . Then, the ω-Kashiwara operators ej and fj , j ∈ I, stabilize i,s i,s the fixed point subset B of B under the action of the diagram automorphism ω, i,s with a structure of Uq ( and hence equip the B g )-crystal, where Uq ( g ) denotes the i,s is a perquantum affine algebra associated to g. Furthermore, we prove that the B fect Uq ( g )-crystal of level s (in the sense of Definition 1.4.12), thereby establishing Theorem 2.4.1. i,s for the (fixed) i ∈ I0 = I\ {0} and s ∈ Z≥1 is perfect Because the Uq ( g )-crystal B (and hence regular), it decomposes, under restriction, into a direct sum of the crystal bases of integrable highest weight modules over the quantized universal enveloping algebra Uq ( gI0 ) of the finite-dimensional, reductive Lie subalgebra gI0 of g corresponding to I0 ⊂ I. In fact, we can give, in almost all cases, an explicit description (see §5.2 – i,s §5.6) of the branching rule with respect to the restriction to Uq ( gI0 ), i.e., how the B decomposes into a disjoint union of connected components as a Uq ( gI0 )-crystal. This i,s is isomorphic as a Uq ( result deserves to be supporting evidence that the B g )-crystal to the conjectural crystal base of a certain KR module with specified Drinfeld polynomi(i) s(i) ( ζs ), over Uq ( g ), since our branching rule for the als (see §1.5 below), denoted by W i,s B indeed coincides with the “branching rule” (with q = 1) conjectured in [HKOTT, (i) s(i) ( ζs ) over Uq ( g ), regarded as a Uq ( gI0 )-module Appendix A] for the KR module W by restriction. i,s , i ∈ I0 , Finally, we should mention the relation between the Uq ( g )-crystals B i,s , s ∈ Z≥1 , and “virtual” crystals defined in [OSS1, OSS2]. Since the Uq ( g )-crystals B i ∈ I0 , s ∈ Z≥1 , are perfect (and hence simple), their crystal graphs are connected. i,s coincide with virtual crystals at least in the case that g )-crystals B Therefore, the Uq ( (1)

(2)

(1)

g is of type Dn+1 for n ≥ 2, the case that g is of type Dn+1 and g is g is of type A2n−1 and (2)

(1)

(3)

g is of type D4 , and the case of type A2n−1 for n ≥ 3, the case that g is of type D4 and (1) (2) g is of type E6 . This clarifies the representation-theoretical that g is of type E6 and meaning of virtual crystals. The organization of this paper is as follows. In §1, we briefly review some basic notions in the theory of crystals for quantum affine algebras, and recall a conjecture on the existence of the crystal bases of KR modules. In §2, we first fix the notation for diagram automorphisms of simply-laced affine Lie algebras g, and also the notation for i,s , i ∈ I0 , corresponding orbit Lie algebras g. Then we define the Uq ( g )-crystals B s ∈ Z≥1 , and state our main result (Theorem 2.4.1). In §3, we show some technical propositions and lemmas concerning the fixed point subsets of regular crystals, or those of tensor products of regular crystals under the action of the diagram automorphism ω, i,s , which will be needed later. In §4, we prove the perfectness of the Uq ( g )-crystals B i ∈ I0 , s ∈ Z≥1 , thereby establishing Theorem 2.4.1. In §5, we give explicit descripi,s , i ∈ I0 , s ∈ Z≥1 , with a few g )-crystals B tions of the branching rules for the Uq ( i,s is exceptions, and then propose a conjecture that for each i ∈ I0 and s ∈ Z≥1 , the B

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isomorphic as a Uq ( g )-crystal to the conjectural crystal base of a certain KR module g ). over the twisted quantum affine algebra Uq ( 1. Crystals for Quantum Affine Algebras 1.1. Cartan data. Let A = (aij )i,j ∈I be a symmetrizable generalized Cartan matrix (GCM for short). A Cartan datum for the GCM A is, by definition, a quintuplet (A, P , P ∨ , , ∨ ) consisting of the GCM A, a free Z-module P ∨ of finite rank, its dual P := HomZ (P ∨ , Z), a subset ∨ := hj j ∈I of P ∨ , and a subset := αj j ∈I of P satisfying αk (hj ) = aj k for j, k ∈ I . Further, we assume that the elements hj , j ∈ I , of ∨ ⊂ P ∨ are linearly independent; however, we do not assume that the elements αj , j ∈ I , of ⊂ P are linearly independent. 1.2. Crystals. Let us briefly recall some basic notions in the theory of crystals from [HK, Chap. 4, §4.5] (see also [Kas2, §7]). Let A = (aij )i,j ∈I be a symmetrizable GCM, and let (A, P , P ∨ , , ∨ ) be a Cartan datum for the GCM A. A crystal associated to the Cartan datum (A, P , P ∨ , , ∨ ) is a set B equipped with maps wt : B → P , ej , fj : B ∪ {θ } → B ∪ {θ}, j ∈ I , and εj , ϕj : B → Z ∪ {−∞}, j ∈ I , satisfying Conditions (1) – (7) of [HK, Definition 4.5.1] (with ej , fj , 0 replaced by ej , fj , θ , respectively). We call the map ej (resp., fj ) the raising (resp., lowering) Kashiwara operator with respect to αj ∈ , and understand that ej θ = fj θ = θ for all j ∈ I . A crystal B is said to be semiregular if εj (b) = max m ≥ 0 | ejm b = θ ,

ϕj (b) = max m ≥ 0 | fjm b = θ , (1.2.1)

for all b ∈ B and j ∈ I ; every crystal treated in this paper is semiregular. Let B1 , B2 be crystals associated to the Cartan datum (A, P , P ∨ , , ∨ ) above. We define the tensor product crystal B1 ⊗ B2 of B1 and B2 as in [HK, Definition 4.5.3] (see also [Kas2, §7.3]). Note that the convention used for tensor product crystals in [OSS1] and [OSS2] is different from that in [HK, Definition 4.5.3], and hence from ours (see, for example, [OSS1, (2.10) and (2.11)]). 1.3. Quantum affine algebras. For the remainder of this paper, we assume that a GCM A = (aij )i,j ∈I is of affine type. Take a special vertex 0 ∈ I as in [Kac, §4.8, Tables Aff 1 – Aff 3], and set I0 := I \ {0}. Let g = g(A) be the affine Lie algebra over the field C of complex numbers associated to the GCM A of affine type. Then 

h=

 Chj  ⊕ Cd

(1.3.1)

j ∈I

is a Cartan subalgebra of g, with hj , j ∈ I , the simple coroots, and d the scaling element. The simple roots αj ∈ h∗ := HomC (h, C), j ∈ I , and fundamental weights j ∈ h∗ , j ∈ I , are defined by (see [HK, Chap. 10, §10.1]): αk (hj ) = aj k ,

αk (d) = δk,0 ,

k (hj ) = δk,j ,

k (d) = 0,

(1.3.2)


753

for j, k ∈ I . Let Ej , Fj , j ∈ I , be the Chevalley generators of g, where Ej (resp., Fj ) corresponds to the simple root αj (resp., −αj ), and let δ=

a j αj ∈ h ∗

and

c=

j ∈I

j ∈I

aj∨ hj ∈ h

(1.3.3)

be the null root and the canonical central element of g, respectively. We take a dual weight lattice P ∨ and a weight lattice P as follows:     1 Zhj  ⊕Zd ⊂ h and P =  Zj  ⊕Z δ ⊂ h∗ . (1.3.4) P ∨ = a0 j ∈I

j ∈I

Clearly, we have P ∼ = HomZ (P ∨ , Z). Here we should note that a0 = 1 except the case (2) (2) in which g is of type A2n , and a0 = 2 in the case that g is of type A2n . It is easily ∨ ∨ seen that the quintuplet (A, P , P , , ) is a Cartan datum for the affine type GCM A = (aij )i,j ∈I . Let Uq (g) = Ej , Fj , q h | j ∈ I, h ∈ P ∨ be the quantized universal enveloping algebra of g over the field C(q) of rational functions in q (with complex coefficients) with weight lattice P , and Chevalley generators Ej , Fj , j ∈ I . Now, we set Chj ⊂ h and Pcl∨ := Zhj ⊂ P ∨ . (1.3.5) hcl := j ∈I

j ∈I

For each λ ∈ h∗ , we define cl(λ) ∈ h∗cl := (hcl )∗ to be the restriction λ|hcl of λ ∈ h∗ to hcl (we simply write λ for cl(λ) when this abbreviation causes no confusion). It is clear that h∗cl = cl(h∗ ) ∼ = h∗ /Cδ as C-vector spaces, and h∗cl =

C cl(j ).

(1.3.6)

j ∈I

We then the classical weight lattice Pcl to be cl(P ) ⊂ h∗cl . We have Pcl ∼ = define HomZ Pcl∨ , Z ∼ P /(Cδ ∩ P ) as (free) Z-modules, and = Pcl =

Zj ,

(1.3.7)

j ∈I

where cl(j ) ∈ h∗cl is simply denoted by j for j ∈ I . Further, we set Pcl+ := j ∈I Z≥0 j , (Pcl )0 := µ ∈ Pcl | µ(c) = 0 , for each s ∈ Z≥0 . (Pcl+ )s := µ ∈ Pcl+ | µ(c) = s

(1.3.8)

It is easily seen that the quintuplet (A, Pcl , Pcl∨ , , ∨ ) is also a Cartan datum for the affine type GCM A = (aij )i,j ∈I . For simplicity, a crystal associated to the Cartan datum (A, Pcl , Pcl∨ , , ∨ ) is called a Uq (g)-crystal, where Uq (g) denotes the C(q)-subalgebra of Uq (g) generated by Ej , Fj , j ∈ I , and q h , h ∈ Pcl∨ (which is the quantized universal enveloping algebra of g over C(q) with weight lattice Pcl ).

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1.4. Perfect crystals for quantum affine algebras. We keep the of §1.3. Let notation us fix a proper subset J of I . We set AJ := (aij )i,j ∈J , J := αj j ∈J ⊂ Pcl , J∨ := hj j ∈J ⊂ Pcl∨ , and denote by gJ the Lie subalgebra of the affine Lie algebra g generated by Ej , Fj , j ∈ J , and hcl . Then the quintuplet (AJ , Pcl , Pcl∨ , J , J∨ ) is a Cartan datum for the GCM AJ . For simplicity, a crystal associated to this Cartan datum (AJ , Pcl , Pcl∨ , J , J∨ ) is called a Uq (gJ )-crystal, where Uq (gJ ) denotes the C(q)-subalgebra of Uq (g) generated by Ej , Fj , j ∈ J , and q h , h ∈ Pcl∨ (which is the quantized universal enveloping algebra of gJ over C(q) with weight lattice Pcl ). If B is a Uq (g)crystal and J is a (proper) subset of I , then the set B equipped with the Kashiwara operators ej , fj , j ∈ J , and the maps wt : B → Pcl , εj , ϕj : B → Z ∪ {−∞}, j ∈ J , is a Uq (gJ )-crystal. Definition 1.4.1 (see [AK, §1.4]). A Uq (g)-crystal B is said to be regular if for every proper subset J I , the B, regarded as a Uq (gJ )-crystal in the way above, is isomorphic to the crystal base of an integrable Uq (gJ )-module (for details about crystal bases, see, for example, [HK, Chap. 4, §4.2] and [Kas2, §4]). Let W := rj | j ∈ I ⊂ GL(h∗ ) be the Weyl group of g, where rj ∈ GL(h∗ ) is the simple reflection with respect to αj ∈ h∗ . Note that the weight lattice P ⊂ h∗ is stable under the action of the Weyl group W , and that there exists an action of W on Pcl induced from that on P , since W δ = δ. We can define an action of the Weyl group W on a regular Uq (g)-crystal B as follows (see [Kas1, §7]). For each j ∈ I , we define Sj : B → B by: m fj b if m := (wt b)(hj ) ≥ 0 for b ∈ B. (1.4.1) Sj b = −m ej b if m := (wt b)(hj ) < 0 Proposition 1.4.2. Let B be a regular Uq (g)-crystal. Then, there exists a unique action S : W → Bij(B), w → Sw , of the Weyl group W on the set B such that Srj = Sj for all j ∈ I , where Bij(B) denote the group of all bijections from the set B to itself. In addition, wt(Sw b) = w(wt b) holds for w ∈ W and b ∈ B. Definition 1.4.3 (see [AK, §1.4]). Let B be a regular Uq (g)-crystal. An element b ∈ B is said to be extremal (or more accurately, W -extremal) if for every w ∈ W , either ej Sw b = θ or fj Sw b = θ holds for each j ∈ I . Remark 1.4.4. It follows immediately from the definition above that if b ∈ B is an extremal element, then Sw b ∈ B is an extremal element of weight w( wt b) for each w ∈ W. We know the following lemma from [AK, Lemma 1.6 (1)] and its proof (see also Lemma 4.2.4 below). Lemma 1.4.5. Let B1 , B2 be regular Uq (g)-crystals of finite cardinality such that the weights of their elements are all contained in (Pcl )0 . Let b1 ∈ B1 and b2 ∈ B2 be extremal elements whose weights are contained in the same Weyl chamber with respect to the simple coroots hj , j ∈ I0 = I \ {0}. Then, b1 ⊗ b2 ∈ B1 ⊗ B2 is an extremal element. Also, Sw (b1 ⊗ b2 ) = Sw b1 ⊗ Sw b2 holds for all w ∈ W . Definition 1.4.6. A regular Uq (g)-crystal B is said to be simple if it satisfies the following conditions:


755

(S1) The set B is of finite cardinality, and the weights of elements of B are all contained in (Pcl )0 . (S2) The set of all extremal elements of B coincides with a Weyl group orbit in B. (S3) Let b ∈ B be an extremal element, and set µ := wt b ∈ (Pcl )0 . Then the subset Bµ ⊂ B of all elements of weight µ consists only of the element b, i.e., Bµ = {b}. Remark 1.4.7. Let B be a regular Uq (g)-crystal satisfying condition (S1) of Definition 1.4.6. Then we see that there exists at least one extremal element in B (see the comment following the proof of [Kas1, Proposition 9.3.2]). Lemma 1.4.8. Let B be a simple Uq (g)-crystal. Then there exists a unique extremal element u ∈ B such that (wt u)(hj ) ≥ 0 for all j ∈ I0 . Proof. The existence of an extremal element with the desired property follows immediately from Remark 1.4.4. Therefore, it remains to show the uniqueness. Let u1 , u2 be extremal elements whose weights are both dominant with respect to the simple coroots hj , j ∈ I0 , and set µ1 := wt u1 and µ2 := wt u2 . By condition (S2) of Definition 1.4.6, there exists some w ∈ W such that Sw u2 = u1 . It follows that µ1 = wµ2 ∈ W µ2 . We recall from [Kac, Proposition 6.5] that the Weyl group W decomposes into the semidirect product WI0 T of the Weyl group WI0 := rj | j ∈ I0 (of finite type) and the abelian group T of translations. Since µ2 ∈ Pcl is of level zero by condition (S1) of Definition 1.4.6, it follows from [Kac, Chap. 6, formula (6.5.5)] that W µ2 = WI0 µ2 , and hence µ1 ∈ WI0 µ2 . Since both µ1 and µ2 are dominant with respect to the simple coroots hj , j ∈ I0 , we deduce that µ1 = µ2 . Therefore, by condition (S3) of Definition 1.4.6, we obtain u1 = u2 , which shows the uniqueness. Remark 1.4.9. Let B be a simple Uq (g)-crystal, and let u ∈ B be the unique extremal element such that (wt u)(hj ) ≥ 0 for all j ∈ I0 . We can show by an argument similar to that in the proof of [Kas3, Corollary 5.2], using [AK, Lemma 1.5], that the weights of elements of B are all contained in the convex hull of the W -orbit of wt u. Hence, by Lemma 1.4.8, we see that the weights of elements of B are all contained in the set wt u − j ∈I0 Z≥0 αj . We know from [NS3, Remark 2.5.7] that the definition of simple Uq (g)-crystals above is equivalent to [AK, Definition 1.7], and also to [Kas3, Definition 4.9]. Thus we know the following from [AK, Lemmas 1.9 and 1.10]. Proposition 1.4.10. (1) The crystal graph of a simple Uq (g)-crystal is connected. (2) Let B1 , B2 be simple Uq (g)-crystals. Then, the tensor product B1 ⊗ B2 is also a simple Uq (g)-crystal. In particular, the crystal graph of B1 ⊗ B2 is connected. Let B be a simple Uq (g)-crystal. We define maps ε, ϕ : B → Pcl+ by: ε(b) = εj (b)j and ϕ(b) = ϕj (b)j for b ∈ B. j ∈I

(1.4.2)

j ∈I

Further, we define a positive integer lev B (called the level of B) and a subset Bmin of B by: lev B = min (ε(b))(c) | b ∈ B ∈ Z>0 , (1.4.3) Bmin = b ∈ B | (ε(b))(c) = lev B ⊂ B. (1.4.4)

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Remark 1.4.11. It can easily be seen from the definition of crystals that ϕ(b)−ε(b) = wt b for every b ∈ B. Since wt b ∈ (Pcl )0 by condition (S1) ofDefinition 1.4.6, we have (ε(b))(c) = (ϕ(b))(c) for all b ∈ B, and hence lev B = min (ϕ(b))(c) | b ∈ B . Definition 1.4.12. A simple Uq (g)-crystal B is said to be perfect if the restrictions of the maps ε, ϕ : B → Pcl+ to Bmin induce bijections Bmin → (Pcl+ )s , where s := lev B. Remark 1.4.13. (1) In the definition of perfect Uq (g)-crystals B, it is often required that B is isomorphic to the crystal base of a finite-dimensional Uq (g)-module as a Uq (g)crystal (see, for example, Condition (1) of [HK, Definition 10.5.1]); but we do not require it in this paper. (2) Our definition of perfect Uq (g)-crystals seems to be slightly different from the one in [HK, Definition 10.5.1], and also from the one in [HKOTT, §2.2]. We can deduce from Remark 1.4.9, condition (S3) of Definition 1.4.6, and Proposition 1.4.10 that if B is a perfect Uq (g)-crystal in the sense of Definition 1.4.12, then B satisfies Conditions (2) – (5) of [HK, Definition 10.5.1]. However, in [HKOTT, §2.2], it is required that the perfect Uq (g)-crystal B is “finite” (in the sense of [HKKOT, Definition 2.5]); we do not require this “finiteness” condition in our definition of perfect Uq (g)-crystals, since it does not seem to be essential for our purposes. (3) If B is a perfect Uq (g)-crystal in the sense of Definition 1.4.12, and isomorphic to the crystal base of a finite-dimensional Uq (g)-module as a Uq (g)-crystal, then B is perfect in the sense of [OSS1, §2.10]. Lemma 1.4.14. Let B1 , B2 be perfect Uq (g)-crystals of the same level s. Then, the tensor product B1 ⊗ B2 is also a perfect Uq (g)-crystal of level s. Proof. We know from Proposition 1.4.10 (2) that the tensor product B1 ⊗ B2 is a simple Uq (g)-crystal. In addition, by elementary arguments using the tensor product rule for crystals, we can easily show that the B1 ⊗ B2 is of level s, and the maps ε, ϕ : (B1 ⊗ B2 )min → (Pcl+ )s are bijective. We know the following proposition from [OSS1, Theorem 2.4]. Proposition 1.4.15. (1) Let B1 , B2 be perfect Uq (g)-crystals isomorphic to the crystal bases of finite-dimensional Uq (g)-modules as a Uq (g)-crystal. Then, there exists a ∼

unique isomorphism (called a combinatorial R-matrix ) R : B1 ⊗ B2 → B2 ⊗ B1 of Uq (g)-crystals. (2) Let B be a perfect Uq (g)-crystal isomorphic to the crystal base of a finitedimensional Uq (g)-module as a Uq (g)-crystal. Then, there exists a Z-valued function H : B ⊗ B → Z (called an energy function ) satisfying  H (b1 ⊗b2 ) + 1 if j = 0 and ϕ0 (b1 ) ≥ ε0 (b2 ),    H (ej (b1 ⊗ b2 )) = H (b1 ⊗ b2 ) − 1 if j = 0 and ϕ0 (b1 ) < ε0 (b2 ),    H (b1 ⊗b2 ) if j = 0, for all j ∈ I and b1 ⊗ b2 ∈ B⊗B such that ej (b1 ⊗ b2 ) = θ .

(1.4.5)


757

Remark 1.4.16. With the notation and assumption of Proposition 1.4.15 (2), we have  H (b1 ⊗ b2 ) − 1 if j = 0 and ϕ0 (b1 ) > ε0 (b2 ),    H (fj (b1 ⊗ b2 )) = H (b1 ⊗ b2 ) + 1 if j = 0 and ϕ0 (b1 ) ≤ ε0 (b2 ),    H (b1 ⊗ b2 ) if j = 0,

(1.4.6)

for all j ∈ I and b1 ⊗ b2 ∈ B ⊗ B such that fj (b1 ⊗ b2 ) = θ. 1.5. A conjectural family of perfect crystals. In this subsection, let g be either a simply-laced affine Lie algebra, or a twisted affine Lie algebra. More specifically, let g be (1) (1) (1) (2) the affine Lie algebra of type An (n ≥ 2), Dn (n ≥ 4), E6 , or of type Dn+1 (n ≥ 2), (2)

(2)

(3)

(2)

A2n (n ≥ 1), A2n−1 (n ≥ 3), D4 , E6 , with the index set I for the simple roots numbered as in [Kac, §4.8, Tables Aff 1 – Aff 3]. For each i ∈ I0 = I \ {0} = {1, 2, . . . , n}, (i) s ∈ Z≥1 , and ζ ∈ C(q)× := C(q) \ {0}, we denote by Ws (ζ ) the finite-dimensional irreducible module over the quantum affine algebra Uq (g) whose Drinfeld polynomials Pj (u) ∈ C(q)[u] are specified as follows (see [KNT, Definition 5.3]): (1)

(1)

(1)

Case 1. The cases of type An (n ≥ 2), Dn (n ≥ 4), E6 . In these cases, the Drinfeld polynomials Pj (u), j ∈ I0 = {1, 2, . . . , n}, are given by:

Pj (u) =

 s   (1 − ζ q s+2−2k u)  

if j = i,

k=1

   

1

otherwise.

(2)

(2)

Case 2. The case of type Dn+1 (n ≥ 2) (resp., A2n−1 (n ≥ 3)). In this case, the Drinfeld polynomials Pj (u), j ∈ I0 = {1, 2, . . . , n − 1, n}, are given by:

Pj (u) =

 s     (1 − ζ q di (s+2−2k) u)

if j = i,

k=1

   

1

otherwise,

where di = 1 if i = n (resp., i = n), and di = 2 otherwise. (3)

(2)

Case 3. The case of type D4 (resp., E6 ). In this case, the Drinfeld polynomials Pj (u), j ∈ I0 = {1, 2} (resp., j ∈ I0 = {1, 2, 3, 4}), are given by:

Pj (u) =

 s     (1 − ζ q di (s+2−2k) u)

if j = i,

k=1

   

1

otherwise,

where di = 1 if i = 1 (resp., i = 1, 2), and di = 3 (resp., di = 2) otherwise.

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Case 4. The case of type A2n (n ≥ 1). In this case, we should remark that the index set for the Drinfeld polynomials Pj (u) is not I0 = {1, 2, . . . , n}, but {0, 1, . . . , n − 1}; the Drinfeld polynomials Pj (u), j ∈ I0 = {0, 1, . . . , n − 1}, are given by:

Pj (u) =

 s     (1 − ζ q 2(s+2−2k) u)

if j = n − i,

k=1

   

1

otherwise.

(In all the cases above, we used the Drinfeld realization of Uq (g), and the classification of its finite-dimensional irreducible modules of type 1 by Drinfeld polynomials; see [CP1, (i) CP2] for details.) We call this Uq (g)-module Ws (ζ ) a Kirillov-Reshetikhin module (KR module for short) over Uq (g). Now, let us fix (arbitrarily) i ∈ I0 and s ∈ Z≥1 . Conjecture 1.5.1 (cf. [HKOTT, Conjecture 2.1 (1)]). For some ζs ∈ C(q)× = (i) (i) C(q) \ {0}, the KR module Ws (ζs ) over Uq (g) has a crystal base B i,s that is a perfect Uq (g)-crystal of level s. (i)

Remark 1.5.2. Conjecture 1.5.1 has already been proved in some cases (see [HKOTY, Remark 2.3] and [HKOTT, Remark 2.6]; see also Remark 2.3.3 below). Lemma 1.5.3. Assume that Conjecture 1.5.1 holds for the fixed i ∈ I0 and s ∈ Z≥1 . Let (i) (i) Li,s ⊂ Ws (ζs ) be the crystal lattice corresponding to the crystal base B i,s . Suppose (i) (i) that (L, B) is another pair of crystal lattice and crystal base for Ws (ζs ) such that the i,s crystal graph of B is connected. Then, L = f (q)L holds for some f (q) ∈ C(q) \ {0}, (i) (i) and B is isomorphic to B i,s as a Uq (g)-crystal. Namely, the crystal base of Ws (ζs ) is unique, up to a nonzero constant multiple. Proof. Let b ∈ B i,s be an extremal element, and set µ := wt b ∈ (Pcl )0 . Note that the set (B i,s )µ consists only of the element b by condition (S3) of Definition 1.4.6, and hence (i) (i) that the µ-weight space of Ws (ζs ) is one-dimensional. Let v ∈ Li,s be an element of weight µ corresponding to the b under the canonical projection Li,s Li,s /qLi,s . Here we recall that the crystal graph of B i,s is connected by Proposition 1.4.10 (1). By using Nakayama’s lemma, we can show that Li,s is identical to the A-module generated by all elements of the form xj1 xj2 · · · xj k v, j1 , j2 , . . . , jk ∈ I , k ≥ 0, where A := f (q) ∈ C(q) | f (q) is regular at q = 0 , and xj is either the raising Kashiwara (i) (i) operator ej or the lowering Kashiwara operator fj on Ws (ζs ) for each j ∈ I : Li,s =

A xj1 xj2 · · · xjk v.

(1.5.1)

j1 , j2 , ..., jk ∈I ; k≥0

Similarly, take an element v ∈ L of weight µ corresponding to the unique element b ∈ B of weight µ. Because the crystal graph of B is connected by the assumption, we obtain


759

L=

A xj1 xj2 · · · xjk v

(1.5.2)

j1 , j2 , ..., jk ∈I ; k≥0

in the same way as above. (i) (i) Since the µ-weight space of Ws (ζs ) is one-dimensional as mentioned above, it follows that v = f (q)v for some f (q) ∈ C(q) \ {0}. Combining this fact with (1.5.1) and (1.5.2), we have L = f (q)Li,s . Thus we have a C-linear isomorphism ∼ (i) (i) : Li,s /qLi,s → L/qL induced from the transformation v → f (q)v on Ws (ζs ). i,s Furthermore, we can deduce from the connectedness of the crystal bases B and B that the restriction |Bi,s of to B i,s gives an isomorphism of Uq (g)-crystals from B i,s onto B. This proves the lemma.

2. Construction of Perfect Crystals for Twisted Quantum Affine Algebras (1)

For the remainder of this paper, let g = g(A) be the affine Lie algebra of type An (n ≥ (1) (1) 2), Dn (n ≥ 4), or E6 , and let ω : I → I be a nontrivial diagram automorphism satisfying the (additional) condition that ω(0) = 0. 2.1. Diagram automorphisms of simply-laced affine Lie algebras. Here we give all pairs (g, ω) of an affine Lie algebra g and a nontrivial diagram automorphism ω : I → I satisfying the condition that ω(0) = 0, after introducing our numbering of the index set = ( I . (For the definition of the matrix A aij )i,j ∈I, see §2.2 below.) (1)

Case (a). The affine Cartan matrix A = (aij )i,j ∈I is of type A2n−1 (n ≥ 2), and the diagram automorphism ω : I → I is given by: ω(0) = 0 and ω(j ) = 2n − j for = ( j ∈ I0 = I \ {0} (note that the order of ω is 2). Then the matrix A aij )i,j ∈I is the (2)

affine Cartan matrix of type Dn+1 : 1

A:

A:

2

n−1

n

0

0

2n − 1

2n − 2

n+1

1

2

n−1 (1)

n

Case (b). The affine Cartan matrix A = (aij )i,j ∈I is of type A2n (n ≥ 1), and the diagram automorphism ω : I → I is given by: ω(0) = 0 and ω(j ) = 2n + 1 − j for = ( j ∈ I0 = I \ {0} (note that the order of ω is 2). Then the matrix A aij )i,j ∈I is the (2)

affine Cartan matrix of type A2n :

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S. Naito, D. Sagaki

If n ≥ 2, then

If n = 1, then 1

A:

A:

n−1

2

n

1

0

0

0

2n

2n − 1

n+2

n+1

1

2

n−1

n

2

0

1

(1)

Case (c). The affine Cartan matrix A = (aij )i,j ∈I is of type Dn+1 (n ≥ 3), and the diagram automorphism ω : I → I is given by: ω(j ) = j for j ∈ I \ {n, n + 1}, and ω(n) = n + 1, ω(n + 1) = n (note that the order of ω is 2). Then the matrix (2) = ( A aij )i,j ∈I is the affine Cartan matrix of type A2n−1 : n

0

A: 1

2

3

n−1

n+1 0

A:

1

2

3

n−1

n (1)

Case (d). The affine Cartan matrix A = (aij )i,j ∈I is of type D4 , and the diagram automorphism ω : I → I is given by: ω(0) = 0, ω(1) = 1, ω(2) = 3, ω(3) = 4, and = ( ω(4) = 2 (note that the order of ω is 3). Then the matrix A aij )i,j ∈I is the affine (3)

Cartan matrix of type D4 :


761 2

0

1

3

A: 4

A: 0

1

2 (1)

Case (e). The affine Cartan matrix A = (aij )i,j ∈I is of type E6 , and the diagram automorphism ω : I → I is given by: ω(0) = 0, ω(1) = 1, ω(2) = 2, ω(3) = 5, ω(4) = 6, ω(5) = 3, and ω(6) = 4 (note that the order of ω is 2). Then the matrix (2) = ( A aij )i,j ∈I is the affine Cartan matrix of type E6 :

0

1

3

4

5

6

3

4

2

A:

A: 0

1

2

We define C-linear isomorphisms ω : h → h and ω∗ : h∗ → h∗ by: ω(hj ) = hω(j )

for j ∈ I ,

ω(d) = d,

(ω∗ (λ))(h) = λ(ω−1 (h)) for λ ∈ h∗ and h ∈ h.

(2.1.1)

It is clear that the subsets P ∨ , hcl , and Pcl∨ of h are all stable under ω ∈ GL(h), and that P ⊂ h∗ is stable under ω∗ ∈ GL(h∗ ). In addition, ω(c) = c,

ω∗ (δ) = δ,

ω∗ (αj ) = αω(j ) ,

ω∗ (j ) = ω(j )

for j ∈ I .

(2.1.2)

Also, we have a C(q)-algebra automorphism ω ∈ Aut(Uq (g)) such that ω(Ej ) = Eω(j ) , ω(Fj ) = Fω(j ) for j ∈ I , and ω(q h ) = q ω(h) for h ∈ P ∨ . Since Pcl∨ is stable under ω ∈ GL(h), we see that the C(q)-subalgebra Uq (g) is stable under ω ∈ Aut(Uq (g)), thus obtaining a C(q)-algebra automorphism ω of Uq (g). Further, we define a C-linear automorphism ω∗ : h∗cl → h∗cl by: ω∗ (j ) = ω(j ) . Note that this C-linear automorphism of h∗cl = (hcl )∗ can be thought of as the one induced from ω∗ ∈ GL(h∗ ) since

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h∗cl ∼ = h∗ /Cδ, as well as the contragredient map of the restriction of ω ∈ GL(h) to hcl . We set (hcl )0 := h ∈ hcl | ω(h) = h , (h∗cl )0 := λ ∈ h∗cl | ω∗ (λ) = λ . (2.1.3) 2.2. Orbit Lie algebras. We choose (and fix) a complete set I (containing 0 ∈ I ) of representatives of the ω-orbits in I in such a way that if j ∈ I, then j ≤ ωk (j ) for all k ∈ Z≥0 (see the figures in §2.1). Now we set Nj −1

cij :=

ai, ωk (j )

for i, j ∈ I,

and cj := cjj

for j ∈ I,

(2.2.1)

k=0

where Nj is the number of elements of the ω-orbit of j ∈ Iin I . (In fact, Nj is equal to 1, 2, or 3.) Remark 2.2.1 (cf. [FSS, §2.2]). We see that cj = 2 except the case in which the pair (g, ω) is in Case (b) and j = n; if cj = 2, then the subdiagram of the Dynkin diagram of A corresponding to the ω-orbit of the j is of type A1 ×· · ·×A1 (Nj times). Contrastingly, if the pair (g, ω) is in Case (b) and j = n, then cj = 1; in this case, the subdiagram of the Dynkin diagram of A corresponding to the ω-orbit of the j is of type A2 . Further, we set aij := 2cij /cj for i, j ∈ I. := ( Lemma 2.2.2 (see [FSS, §2]). The matrix A aij )i,j ∈I is a generalized Cartan matrix is as in §2.1. of (twisted ) affine type. Moreover, the explicit type of the GCM A be the (affine) Kac-Moody algebra over C associated to the GCM A Let g := g(A) above, which iscalled the orbit Lie algebra (corresponding to the diagram automorphism is a Cartan subalgebra of ∨ := { ω). Then, h= C h hj }j ∈I the g, with j ⊕C d j ∈I ∗ ∗ := set of simple coroots, and dthe scaling element. Denote by αj ⊂ h := (h) j ∈I

j ∈ the set of simple roots, and h∗ , j ∈ I, the fundamental weights for the orbit Lie = δj,0 and = 0 for j ∈ I. Let j (d) algebra g (of affine type); note that αj (d) δ :=

aj αj ∈ h∗

and

c :=

j ∈I

aj∨ h hj ∈

(2.2.2)

j ∈I

be the null root and the canonical central element of g, respectively. We take a dual ∨ ⊂ ⊂ weight lattice P h and a weight lattice P h∗ as follows:     1 ∨     P = P= δ ⊂ h∗ . Z hj ⊕Z d ⊂ h, Z j ⊕Z (2.2.3) a0 j∈I

j∈I

∨ ⊂ cl ⊂ Define hcl , P h, and P h∗cl := ( hcl )∗ for the orbit Lie algebra g, and also define cl + + cl as in §1.3. Note that (A, P cl , P ∨ , , ∨) subsets (Pcl )0 , Pcl , and (Pcl )s , s ∈ Z≥0 , of P cl Let Uq ( is a Cartan datum for the GCM A. g ) be the quantized universal envelop, and define its ing algebra of the orbit Lie algebra g over C(q) with weight lattice P


763

C(q)-subalgebra Uq ( g ) as in §1.3 (which is the quantized universal enveloping algecl ). We call a crystal associated to the Carbra of g over C(q) with weight lattice P P cl , P ∨ , , ∨ ) a Uq ( tan datum (A, g )-crystal. Further, for a proper subset J of I, cl g, and the C(q)-subalgebra Uq ( gJ) of Uq ( g) let us define the Lie subalgebra gJ of corresponding to the subset J as in §1.4. We call a crystal associated to the Cartan , P cl , P ∨ , J := ( J, ∨ ) for the GCM A datum (A aij )i,j ∈J a Uq ( gJ)-crystal, where cl J J ∨ ∨ αj j ∈J ⊂ Pcl and J := hj j ∈J ⊂ Pcl . J := 0 Now, let us define C-linear hcl from the fixed point isomorphisms ∗Pω :∗ (hcl ) → 0 j subspace (hcl ) onto hcl = j ∈I C hj , and Pω : h → (h∗ )0 from h∗ = j ∈I C cl

cl

onto the fixed point subspace (h∗cl )0 by:   Nj −1 Nj −1 1 j ) = hj and Pω∗ ( hωk (j )  = ωk (j ) Pω  Nj k=0

cl

for j ∈ I

(2.2.4)

k=0

(note that (h0cl )∗ can be identified with (h∗cl )0 in a natural way). Then it is easily seen that (Pω∗ ( λ))(h) = λ(Pω (h)) for λ ∈ h∗cl and h ∈ (hcl )0 , and that Pω∗ ( αj ) =

c, Pω (c) =

Nj −1 2 αωk (j ) cj

for j ∈ I.

(2.2.5)

k=0

Furthermore, we can identify rj | j ∈ I of the orbit Lie algebra the Weyl ∗group W := ∗ := w ∈ W | ω w = wω of the Weyl group W = rj | j ∈ I g with the subgroup W of g as follows. Define wj ∈ W by:  if cj = 1, rj rω(j ) rj (2.2.6) wj = r r j ω(j ) · · · rωNj −1 (j ) if cj = 2, for all j ∈ I. Also, for each j ∈ I (see Remark 2.2.1). Then it is clear that wj ∈ W →W such that we see from [FRS, §3] that there exists a group isomorphism : W , and ( ( w )|(h∗ )0 = Pω∗ ◦ w ◦ (Pω∗ )−1 for each w ∈W rj ) = wj for all j ∈ I. cl

2.3. Fixed point subsets. Let I ⊂ I be the index set (chosen as in §2.2) for the orbit Lie algebra g corresponding to the ω. Let us fix (arbitrarily) i ∈ I0 := I\ {0} and s ∈ Z≥1 . For the rest of this section, we make the following assumption (cf. Conjecture 1.5.1): Assumption 2.3.1. There exists some ζs ∈ C(q)× (independent of 0 ≤ k ≤ Ni − 1) (ωk (i)) (i) such that for every 0 ≤ k ≤ Ni − 1, the KR module Ws (ζs ) over Uq (g) has a (i)

crystal base, denoted by B ω Uq (g)-crystals of level s.

k (i),s

. Further, the B ω

k (i),s

, 0 ≤ k ≤ Ni − 1, are all perfect

Remark 2.3.2. Let 0 ≤ k ≤ Ni − 1. Then, since the Uq (g)-crystal B ω (i),s is perfect (and hence simple) by Assumption 2.3.1, it follows from Lemma 1.4.8 that there exists a k unique extremal element of B ω (i),s , denoted by uωk (i),s , such that (wt uωk (i),s )(hj ) ≥ 0 for all j ∈ I0 . In addition, we can show that wt uωk (i),s = s ωk (i) , where ωk (i) := ωk (i) − aω∨k (i) 0 ∈ Pcl . k

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S. Naito, D. Sagaki

Remark 2.3.3 (see [HKOTY, Remark 2.3]). For Cases (a) and (b), we know from [KMN] that Assumption 2.3.1 is satisfied for all i ∈ I0 and s ∈ Z≥1 . For Case (c) (resp., Case (d)), we know from [KMN] that Assumption 2.3.1 is satisfied if i = 1, n (resp., i = 2), and from [Ko] that Assumption 2.3.1 is satisfied if i = n and s = 1 (resp., if s = 1). First, we define a bijection τω : B i,s → B ω(i),s such that τω ◦ ej = eω(j ) ◦ τω and τω ◦fj = fω(j ) ◦τω for all j ∈ I (τω (θ ) is understood to be θ ), and such that wt(τω (b)) = (i) (i) ω∗ (wt b) for each b ∈ B i,s as follows. Let ρ : Uq (g) → EndC(q) (Ws (ζs )) be the representation map affording the KR module Ws (ζs ) over Uq (g). It is clear that the (i)

(i)

representation of Uq (g) on the (same) C(q)-vector space Ws (ζs ) given by ρ ◦ ω−1 , (i)

(i)

denoted by (ρ ◦ ω−1 , Ws (ζs )), is finite-dimensional and irreducible. In addition, we can easily check that if Pj (u) ∈ C(q)[u], j ∈ I0 , are the Drinfeld polynomi(i) (i) als of the KR module Ws (ζs ) over Uq (g) (see §1.5), then the Drinfeld polynomi(i)

(i)

als Pjω (u) ∈ C(q)[u], j ∈ I0 , of the representation (ρ ◦ ω−1 , Ws (ζs )) of Uq (g) are given by: Pjω (u) = Pω−1 (j ) (u) for each j ∈ I0 . (Here we have used Assump(i)

(i)

tion 2.3.1 that the ζs ∈ C(q)× is independent of 0 ≤ k ≤ Ni − 1.) Because the finite-dimensional irreducible Uq (g)-modules (of type 1) are parametrized by their Drinfeld polynomials up to Uq (g)-module isomorphism, it follows that the representation (i)

(ρ ◦ ω−1 , Ws (ζs )) of Uq (g) is equivalent to the KR module Ws (i)

(i)

(ω(i))

(i)

(i)

(ω(i))

(i)

(ζs ) over Uq (g). (i)

(ζs ) a nonzero intertwining map between We denote by τω : Ws (ζs ) → Ws (i) (i) (ω(i)) (i) (ζs ) denotes a these two representations of Uq (g). Namely, τω : Ws (ζs ) → Ws C(q)-linear isomorphism such that τω (xv) = ω(x)τω (v)

for x ∈ Uq (g) and v ∈ Ws(i) (ζs(i) ).

(2.3.1)

It follows immediately from (2.3.1) that for each µ ∈ Pcl , the µ-weight space of (i) (i) (ω(i)) (i) (ζs ) by τω . Also, we can easily Ws (ζs ) is sent to the ω∗ (µ)-weight space of Ws deduce that τω ◦ ej = eω(j ) ◦ τω ,

and

τω ◦ fj = fω(j ) ◦ τω

for all j ∈ I ,

(2.3.2)

where ej (resp., fj ), j ∈ I , denote the raising (resp., lowering) Kashiwara operators on (i) (i) (ω(i)) (i) (ζs ). Let us denote by Li,s the crystal lattice of Ws (ζs ), and also those on Ws (i) (i) Ws (ζs ). By (2.3.2), we see that the image τω (Li,s ) of the Li,s is stable under the action (ω(i)) (i) (ζs ). Therefore, these Kashof the Kashiwara operators ej and fj , j ∈ I , on Ws (ω(i)) (i) (ζs ) induce operators, also denoted by ej and fj , j ∈ I , iwara operators on Ws on the C-vector space τω (Li,s )/qτω (Li,s ). If τω : Li,s /qLi,s → τω (Li,s )/qτω (Li,s ) denotes the induced C-linear map, then it follows from (2.3.2) that the set τω (B i,s ) ∪ {θ} is stable under the Kashiwara operators ej and fj , j ∈ I , on τω (Li,s )/qτω (Li,s ), which (ω(i)) (i) (ζs ). Hence it follows from implies that (τω (Li,s ), τω (B i,s )) is a crystal base of Ws i,s ω(i),s ∼ Lemma 1.5.3 that τω (B ) = B as Uq (g)-crystals. Thus we have obtained a bijection τω : B i,s → B ω(i),s such that τω ◦ ej = eω(j ) ◦ τω ,

and

τω ◦ fj = fω(j ) ◦ τω

for all j ∈ I ,

wt(τω (b)) = ω∗ (wt b) for each b ∈ B i,s .

(2.3.3)


765

Here (and below) we understand that τω (θ ) = θ . Similarly, for each 1 ≤ k ≤ Ni − 1, k k+1 we obtain a bijection τω : B ω (i),s → B ω (i),s such that τω ◦ ej = eω(j ) ◦ τω and τω ◦ fj = fω(j ) ◦ τω for all j ∈ I , and such that wt(τω (b)) = ω∗ (wt b) for each k b ∈ B ω (i),s . Next, we set  i,s if Ni = 1, B    i,s i,s ω(i),s := B ⊗ B if Ni = 2, B (2.3.4)   2  i,s B ⊗ B ω(i),s ⊗ B ω (i),s if Ni = 3. Since the B ω (i),s , 0 ≤ k ≤ Ni − 1, are perfect Uq (g)-crystals of (the same) level s i,s is a perfect Uq (g)-crystal by Assumption 2.3.1, it follows from Lemma 1.4.14 that B i,s as follows. If of level s. We define an action of the diagram automorphism ω on B i,s → B i,s is defined to be τω . If Ni = 2, then we first define a Ni = 1, then ω : B bijection from B i,s ⊗ B ω(i),s onto B ω(i),s ⊗ B i,s by: b1 ⊗ b2 → τω (b1 ) ⊗ τω (b2 ) for b1 ⊗ b2 ∈ B i,s ⊗ B ω(i),s . By Proposition 1.4.15 (1), we have an isomorphism (a com∼ binatorial R-matrix) B ω(i),s ⊗ B i,s → B i,s ⊗ B ω(i),s of Uq (g)-crystals. We now define i,s → B i,s to be the composite of these maps: ω:B k

τω ⊗τω

∼

i,s = B i,s ⊗ B ω(i),s −→ B ω(i),s ⊗ B i,s → B i,s ⊗ B ω(i),s = B i,s . (2.3.5) ω:B i,s to be the composite of the Similarly, if Ni = 3, then we define an action of ω on B map τω ⊗ τω ⊗ τω with combinatorial R-matrices: i,s = B i,s ⊗ B ω(i),s ⊗ B ω2 (i),s ω:B

τω ⊗τω ⊗τω ∼

−→

B ω(i),s ⊗ B ω

→ B i,s ⊗ B ω(i),s ⊗ B

2 (i),s

ω2 (i),s

⊗ B i,s i,s . (2.3.6) =B

In all the cases above, we can deduce from the tensor product rule for crystals, (2.3.3), and the comment following (2.3.3) that ω ◦ ej = eω(j ) ◦ ω

and

ω ◦ fj = fω(j ) ◦ ω

i,s for all j ∈ I , on B

i,s , wt(ω(b)) = ω∗ (wt b) for each b ∈ B where ω(θ) is understood to be θ . Finally, we set i,s := b ∈ B i,s | ω(b) = b . B

(2.3.7)

(2.3.8)

i,s are all contained in (Pcl )0 ∩ (h∗ )0 by condition Note that the weights of elements of B cl (S1) of Definition 1.4.6 and the second equation of (2.3.7). 2.4. Main result. i,s ∪ {θ } by: B

For each j ∈ I, we define ω-Kashiwara operators ej and fj on

xj =


if cj = 1,

x x j ω(j ) · · · xωNj −1 (j )

if cj = 2,

where x is either e or f . The main result of this paper is the following theorem.

(2.4.1)

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S. Naito, D. Sagaki

Theorem 2.4.1. Let i ∈ I0 = I\ {0} and s ∈ Z≥1 (fixed as in §2.3 ). We keep Assumpi,s ∪ {θ } is stable under the ω-Kashiwara i,s ∪ {θ } of B tion 2.3.1. Then, the subset B i,s operators ej and fj on B ∪ {θ} for all j ∈ I. Moreover, if we set  i,s , cl b := (Pω∗ )−1 (wt b) ∈ P wt for b ∈ B    i,s and j ∈ I, ej )m b = θ for b ∈ B εj (b) := max m ≥ 0 | (    i,s and j ∈ I, ϕj (b) := max m ≥ 0 | (fj )m b = θ for b ∈ B

(2.4.2)

i,s equipped with the ω-Kashiwara operators ej and fj , j ∈ I, the maps then the set B i,s i,s →P → Z≥0 , j ∈ I, becomes a perfect Uq ( cl , and :B wt εj , ϕj : B g )-crystal of level s. We will establish Theorem 2.4.1 under the following plan. First, in §3.2, we show i,s is a regular Uq ( i,s that B g )-crystal. Next, in §4.2, we prove that the Uq ( g )-crystal B i,s is equal to s, and that the is simple. Finally, in §4.3, we show that the level of B i,s )min induce bijections (B i,s )min → (P + )s , where restrictions of the maps ε, ϕ to (B cl i,s → P i,s )min is defined as + are defined as in (1.4.2), and the set (B the maps ε, ϕ:B cl in (1.4.4). 3. Fixed Point Subsets of Crystals under the Action of ω (1)

(1)

(1)

Let g = g(A) be the affine Lie algebra of type An (n ≥ 2), Dn (n ≥ 4), or E6 , and let ω : I → I be a nontrivial diagram automorphism satisfying the condition that ω(0) = 0. 3.1. Fixed point subsets of crystal bases. Let us fix a proper subset J of I such that ω(J ) = J . For an integral weight λ ∈ Pcl that is dominant with respect to the simple coroots hj , j ∈ J , which we call a J -dominant integral weight, we denote by VJ (λ) the integrable highest weight Uq (gJ )-module of highest weight λ. Further, let us denote by BJ (λ) the crystal base of VJ (λ) with raising Kashiwara operators ej , j ∈ J , and lowering Kashiwara operators fj , j ∈ J . Let us take (and fix) a J -dominant integral weight λ ∈ Pcl such that ω∗ (λ) = λ. Then, as in [NS1, §3.2], we obtain an action ω : BJ (λ) → BJ (λ) of the diagram automorphism ω on the crystal base BJ (λ) satisfying the condition: ω ◦ ej = eω(j ) ◦ ω

and

ω ◦ fj = fω(j ) ◦ ω

for all j ∈ I ,

wt(ω(b)) = ω∗ (wt b) for each b ∈ BJ (λ), where ω(θ) is understood to be θ . We set BJω (λ) := b ∈ BJ (λ) | ω(b) = b .

(3.1.1)

(3.1.2)

It follows immediately from (3.1.1) that wt b ∈ Pcl ∩ (h∗cl )0 for all b ∈ BJω (λ). Set cl that is dominant with respect to the J := J ∩ I I. For an integral weight λ∈P simple coroots hj , j ∈ J, which we call a J-dominant integral weight, we denote by ( J( V λ) the integrable highest weight Uq ( gJ)-module of highest weight λ. Also, B J λ)


767

J( denotes the crystal base of V λ). Further, for each j ∈ J, we define ω-Kashiwara operators ej and fj on BJ (λ) ∪ {θ} by: xj =


if cj = 1,

x x j ω(j ) · · · xωNj −1 (j )

if cj = 2,

(3.1.3)

where x is either e or f . We know the following theorem from [NS2, Theorem 2.2.1 (1) – (3)]; note that the restriction ω|J of ω to the subset J of I is a diagram automorphism for the finite-dimensional, reductive Lie subalgebra gJ of g, and the Lie subalgebra gJ of g can be thought of as the orbit Lie algebra of gJ corresponding to the ω|J . Theorem 3.1.1. Let λ ∈ Pcl be a J -dominant integral weight such that ω∗ (λ) = λ, and set λ := (Pω∗ )−1 (λ). Then, the subset BJω (λ) ∪ {θ } of BJ (λ) ∪ {θ} is stable under the ω-Kashiwara operators ej and fj on BJ (λ) ∪ {θ } for all j ∈ J. Moreover, the set BJω (λ) equipped with the ω-Kashiwara operators ej , fj , j ∈ J, and the maps  cl b := (Pω∗ )−1 (wt b) ∈ P wt for b ∈ BJω (λ),    ej )m b = θ ∈ Z≥0 for b ∈ BJω (λ) and j ∈ J, (3.1.4) εj (b) := max m ≥ 0 | (    ϕj (b) := max m ≥ 0 | (fj )m b = θ ∈ Z≥0 for b ∈ BJω (λ) and j ∈ J, ( gJ)-crystal isomorphic to the crystal base B becomes a Uq ( J λ) of the integrable highest J( weight Uq ( gJ)-module V λ) of highest weight λ. For each m ∈ Z≥1 and j ∈ I, we define operators e(m)j and f(m)j on B(λ) ∪ {θ} by:

x (m)j =

 m 2m m xj xω(j ) xj

if cj = 1,

x m x m

if cj = 2,

j

m ω(j ) · · · xωNj −1 (j )

(3.1.5)

where x is either e or f . We know the following from [NS2, Theorem 2.2.1 (4)]. Proposition 3.1.2. Let λ ∈ Pcl be a J -dominant integral weight such that ω∗ (λ) = λ. Then, for every m ∈ Z≥1 and j ∈ J, we have e(m)j = ( ej )m and f(m)j = (fj )m on ω BJ (λ) ∪ {θ }. Proposition 3.1.3. Let λ ∈ Pcl be a J -dominant integral weight such that ω∗ (λ) = λ. Then, for each b ∈ BJω (λ), we have εj (b) = εωk (j ) (b) and ϕj (b) = ϕωk (j ) (b) for all j ∈ J and 0 ≤ k ≤ Nj − 1. Proof. Let b ∈ BJω (λ). Then we see from (3.1.1) that εωk (j ) (b) = εj (b) and ϕωk (j ) (b) = εj (b) = ϕj (b) for all j ∈ J and 0 ≤ k ≤ Nj − 1. Therefore, we need only show that εj (b) and ϕj (b) = ϕj (b). But, these equations follow from [NS2, Lemma 2.1.3 and Theorem 2.2.1 (2)].

768

S. Naito, D. Sagaki

3.2. Fixed point subsets of regular crystals. Let B be a regular Uq (g)-crystal with an action ω : B → B of the diagram automorphism ω satisfying the condition: ω ◦ ej = eω(j ) ◦ ω

and

ω ◦ fj = fω(j ) ◦ ω

for all j ∈ I ,

wt(ω(b)) = ω∗ (wt b) for each b ∈ B. Here (and below) we understand that ω(θ ) = θ . We set B ω := b ∈ B | ω(b) = b ,

(3.2.1)

(3.2.2)

and assume that B ω = ∅. Note that wt b ∈ Pcl ∩ (h∗cl )0 for all b ∈ B ω by the second equation of (3.2.1). For each j ∈ I, we define ω-Kashiwara operators ej and fj on B ∪ {θ} by:  2 if cj = 1, xj xω(j ) xj (3.2.3) xj = x x j ω(j ) · · · xωNj −1 (j ) if cj = 2, : Bω where x is either e or f . Further, we define maps wt j ∈ I , by:  cl b := (Pω∗ )−1 (wt b) ∈ P wt    ej )m b = θ ∈ Z≥0 εj (b) := max m ≥ 0 | (    ϕj (b) := max m ≥ 0 | (fj )m b = θ ∈ Z≥0

→ Pcl and εj , ϕj : B ω → Z≥0 , for b ∈ B ω , for b ∈ B ω and j ∈ I, (3.2.4) for b ∈ B ω and j ∈ I.

Proposition 3.2.1. Let B be a regular Uq (g)-crystal with an action ω : B → B of the diagram automorphism ω satisfying (3.2.1). Then, the subset B ω ∪ {θ } of B ∪ {θ } is stable under the ω-Kashiwara operators ej and fj on B ∪ {θ } for all j ∈ I. Moreover, ω the fixed point subset B equipped with the maps wt, ej , fj , j ∈ I, and εj , ϕj , j ∈ I, becomes a regular Uq ( g )-crystal. Proof. Step 1. For a proper subset J of I, we set J := ωk (j ) | 0 ≤ k ≤ Nj − 1, j ∈ J I . Since B is a regular Uq (g)-crystal, it follows that B is isomorphic as a Uq (gJ )crystal to the crystal base of an integrable Uq (gJ )-module, and hence to a direct sum of the crystal bases of integrable highest weight Uq (gJ )-modules. Namely, there exists an isomorphism of Uq (gJ )-crystals: ∼

J : B → BJ (λ1 ) BJ (λ2 ) · · · BJ (λp ),

(3.2.5)

for some J -dominant integral weights λ1 , λ2 , . . . , λp ∈ Pcl . Set Bt := J−1 (BJ (λt )), and bt := J−1 (vλt ) for 1 ≤ t ≤ p, where vλt is the highest weight element of BJ (λt ). Assume that B ω ∩ Bt = ∅ for any 1 ≤ t ≤ p , and B ω ∩ Bt = ∅ for all p + 1 ≤ t ≤ p (1 ≤ p ≤ p). Then the highest weight elements bt ∈ Bt , 1 ≤ t ≤ p , are all fixed by ω : B → B, i.e., ω(bt ) = bt for all 1 ≤ t ≤ p . Indeed, if 1 ≤ t ≤ p and b ∈ B ω ∩Bt = ∅, then there exist j1 , j2 , . . . , jl ∈ J such that bt = ej1 ej2 · · · ejl b. Hence it follows from (3.2.1) that ω(bt ) = eω(j1 ) eω(j2 ) · · · eω(jl ) b, since ω(b) = b. Here we note that ω(j1 ), ω(j2 ), . . . , ω(jl ) ∈ J , since J is stable under ω. Because Bt is a connected component of B regarded as a Uq (gJ )-crystal, it follows that ω(bt ) = eω(j1 ) eω(j2 ) · · · eω(jl ) b


769

is also contained in Bt . In addition, we see from (3.2.1) that ω(bt ) ∈ Bt is the highest weight element with respect to ej , j ∈ J . Therefore, we conclude that ω(bt ) = bt by the uniqueness of the highest weight element in Bt , which is isomorphic to BJ (λt ) as a Uq (gJ )-crystal, and hence that ω∗ (λt ) = λt . Note also that ω(Bt ) = Bt since Bt is a connected Uq (gJ )-crystal. Moreover, since Bt is connected as a Uq (gJ )-crystal, it follows from (3.1.1) and (3.2.1) that the following diagram commutes: ∼

Bt −−−−→ BJ (λt ) J |Bt    ω ω ∼

Bt −−−−→ BJ (λt ). J |Bt

Hence we deduce that J (B ω ) = BJω (λ1 ) BJω (λ2 ) · · · BJω (λp ).

(3.2.6)

Note that by Theorem 3.1.1, the set on the right-hand side of (3.2.6), equipped with ω( Kashiwara operators, becomes a Uq ( gJ)-crystal isomorphic to the crystal base B J λ1 ) ( B λ )· · · B ( λ ) of the integrable U ( g )-module V ( λ )⊕ V ( λ )⊕· · ·⊕ V q J J 2 J p J 1 J 2 J(λp ), where λt := (Pω∗ )−1 (λt ) for 1 ≤ t ≤ p . Step 2. First, we show that the set B ω ∪ {θ } is stable under the ω-Kashiwara operators ej , j ∈ I. Let us fix (arbitrarily) j ∈ I, and let b ∈ B ω be such that ej b = θ . Set k J := j I , J := ω (j ) | 0 ≤ k ≤ Nj − 1 I , and let J be the isomorphism of Uq (gJ)-crystals in (3.2.5). Then, from the definition (3.1.3) of the ω-Kashiwara operator ej on BJ (λ1 )BJ (λ2 )· · ·BJ (λp ) ∪{θ } and the definition (3.2.3) of the ω-Kashiwara ej ◦ J = J ◦ ej , i.e., that ej b = J−1 ( ej (J (b))). operator ej on B ∪ {θ}, we see that ω Also, we know from (3.2.6) that the image J (B ) is a disjoint union of the fixed point subsets BJω (λt ) under the action of ω of the crystal bases BJ (λt ) of integrable highest weight Uq (gJ )-modules. Hence we see by Theorem 3.1.1 that ej (J (b)) is contained −1 ω ω in J (B ), i.e., that ej b = J ( ej (J (b))) is contained in B . Similarly, we can show that the set B ω ∪ {θ} is stable under the ω-Kashiwara operators fj , j ∈ I. This proves the first assertion. Next, let us prove the second assertion. We show only the equation ϕj (b) = εj (b) + b)( (wt hj ) for each b ∈ B ω and j ∈ I (i.e., Condition (1) of [HK, Definition 4.5.1]); the other conditions follow immediately from the definition (3.2.3) of the ω-Kashiwara εj , operators ej , fj , j ∈ I, the definitions (3.2.4) of the maps wt, ϕj , j ∈ I, for B ω , and k Eq. (2.2.5). Let us fix (arbitrarily) j ∈ I , and set J := {j } I , J := ω (j ) | 0 ≤ k ≤ Nj − 1 I as above. Then we deduce from (3.2.6) and Theorem 3.1.1 that for each b ∈ Bω , J (b)) ( ϕj (J (b)) = εj (J (b)) + wt( hj ). (3.2.7) In addition, since J is an isomorphism of Uq (gJ )-crystals (see (3.2.5)), we see from the definition (3.1.4) of the maps wt, εj , ϕj for BJω (λ1 ) BJω (λ2 ) · · · BJω (λp ) and the corresponding definition (3.2.4) for B ω that (J (b)) = wt b, wt

εj (J (b)) = εj (b),

ϕj (J (b)) = ϕj (b)

(3.2.8)

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for each b ∈ B ω . Consequently, by combining (3.2.7) and (3.2.8), we obtain ϕj (b) = b)( εj (b) + (wt hj ), as desired. Finally, the regularity of the Uq ( g )-crystal B ω follows from (3.2.6) and the comment following it. This proves the second assertion. By an argument similar to that in the proof of Proposition 3.2.1, we also obtain the following Lemmas 3.2.2 and 3.2.3, using Propositions 3.1.2 and 3.1.3, respectively. Lemma 3.2.2. Let B be a regular Uq (g)-crystal with an action ω : B → B of the diagram automorphism ω satisfying (3.2.1). Then, we have e(m)j = ( ej )m and f(m)j = m ω (fj ) on the fixed point subset B for every m ≥ 0 and j ∈ I , where e(m)j and f(m)j are defined by :  m 2m m if cj = 1, xj xω(j ) xj (3.2.9) x (m)j = x m x m · · · x m if cj = 2, Nj −1 j ω(j ) ω

(j )

where x is either e or f . Lemma 3.2.3. Let B be a regular Uq (g)-crystal with an action ω : B → B of the diagram automorphism ω satisfying (3.2.1). Then, for every element b of the fixed point subset B ω , we have εj (b) = εωk (j ) (b) and ϕj (b) = ϕωk (j ) (b) for all j ∈ I and k ≥ 0. Let B be a regular Uq (g)-crystal with an action ω : B → B of the diagram automorphism ω satisfying (3.2.1). Then it follows from Proposition 3.2.1 that the fixed point subset B ω becomes a regular Uq ( g )-crystal. Therefore, there exists a unique action ω of the orbit Lie algebra S : W → Bij(B ), w → Sw g on , of the Weyl group W the set B ω such that Srj = Sj for all j ∈ I, where Sj is defined as in (1.4.1) (see Proposition 1.4.2). ω , where : W → W is Lemma 3.2.4. We have Sw ∈ W = S ( w) on B for all w the isomorphism of groups introduced at the end of §2.2. In particular, the equation Sj = Swj holds on B ω for all j ∈ I.

Proof. We need only show that the equation Sj = Swj holds on B ω for all j ∈ I. Fix ω b)( an arbitrary j ∈ I. Let b ∈ B , and set m := (wt hj ). Here we note that Nj −1 1 ∗ k (wt b)(hj ) = (ω ) (wt b) (hj ) since wt b ∈ (h∗cl )0 Nj k=0

Nj −1 Nj −1 1 1 −k = (wt b)(ω (hj )) = (wt b)(hω−k (j ) ) Nj Nj k=0

=

k=0

(wt b)(Pω−1 ( hj ))

by the definition (2.2.4) of Pω b)( hj ) = (wt hj ). = (Pω∗ )−1 (wt b) (

(3.2.10)

Therefore, we have (wt b)(hj ) = m. In addition, since wt b ∈ (h∗cl )0 , it follows that (wt b)(hωk (j ) ) = m

for all 0 ≤ k ≤ Nj − 1.

(3.2.11)


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Now, we assume that m ≥ 0. Then we obtain  m 2m m fj fω(j ) fj b Sj b = (fj )m b = f(m)j b = f m f m · · · f m Nj −1 j ω(j ) ω

if cj = 1, (j )

b

if cj = 2,

from the definition of Sj and Lemma 3.2.2. Also, we obtain  m 2m m if cj = 1, fj fω(j ) fj b Swj b = f m f m · · · f m b if cj = 2. Nj −1 j ω(j ) ω

(j )

Indeed, if cj = 1, then we have wj = rj rω(j ) rj by Remark 2.2.1 and the definition (2.2.6) of wj , and hence Swj b = Sj Sω(j ) Sj b = Sj Sω(j ) fjm b by definition. Since cj = 1, we deduce from Remark 2.2.1 and (3.2.11) that (wt(fjm b))(hω(j ) ) = (wt b − mαj )(hω(j ) ) = (wt b)(hω(j ) ) − mαj (hω(j ) ) = m − m × (−1) = 2m. 2m f m b. Similarly, we obtain S f 2m Therefore, we obtain Sj Sω(j ) fjm b = Sj fω(j j ω(j ) ) j 2m f m b. Hence we conclude that S b = f m f 2m f m b. The proof for fjm b = fjm fω(j wj j ω(j ) j ) j the case in which cj = 2 is easier. Thus we have shown that Sj b = Swj b if m ≥ 0. The proof for the case in which m ≤ 0 is similar. This proves the lemma.

3.3. Fixed point subsets of tensor products of regular crystals. Let B1 (resp., B2 ) be a regular Uq (g)-crystal with an action of the diagram automorphism ω satisfying (3.2.1), with B = B1 (resp., B = B2 ). Define an action ω : B1 ⊗ B2 → B1 ⊗ B2 of the diagram automorphism ω by: ω(b1 ⊗ b2 ) = ω(b1 ) ⊗ ω(b2 ) for b1 ⊗ b2 ∈ B1 ⊗ B2 . Then, we can easily check by the tensor product rule for Uq (g)-crystals that this action ω : B1 ⊗ B2 → B1 ⊗ B2 satisfies condition (3.2.1), with B = B1 ⊗ B2 . Hence it follows from Proposition 3.2.1 that the fixed point subset (B1 ⊗B2 )ω of B1 ⊗B2 under this action g )-crystal, when equipped with the ω-Kashiwara operators ej becomes a regular Uq ( and fj , j ∈ I. But, set-theoretically, we have (B1 ⊗ B2 )ω = b1 ⊗ b2 | b1 ∈ B1ω , b2 ∈ B2ω , (3.3.1) where B1ω and B2ω are the fixed point subsets of B1 and B2 under the action of ω, respectively. We know from Proposition 3.2.1 that both of the fixed point subsets B1ω and B2ω become regular Uq ( g )-crystals, when equipped with the ω-Kashiwara operators ej and B2ω the tensor product Uq ( fj , j ∈ I. Denote by B1ω ⊗ g )-crystal of B1ω and B2ω ; we use instead of ⊗ to emphasize that it denotes the tensor product of Uq ( the symbol ⊗ g ) crystals. Also, we denote by ej , j ∈ I , and fj , j ∈ I , the raising Kashiwara operators and lowering Kashiwara operators, respectively, on the tensor product Uq ( g )-crystal B2ω . B1ω ⊗ B2ω → (B1 ⊗ B2 )ω be the map defined by : (b1 ⊗ b2 ) = Lemma 3.3.1. Let : B1ω ⊗ ω b2 ∈ B1 ⊗ B2ω . Then, is an isomorphism of Uq ( b1 ⊗ b2 for b1 ⊗ g )-crystals from B2ω onto (B1 ⊗ B2 )ω . B1ω ⊗

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Proof. It is obvious by (3.3.1) that is bijective. In addition, it is clear that preserves weights. Therefore, it remains to show that ◦ ej = ej ◦ and ◦ fj = fj ◦ for ω ω ω B2 and (B1 ⊗ B2 ) are semiregular Uq ( all j ∈ I, since both B1 ⊗ g )-crystals. We show the equation ◦ ej = ej ◦ only for the case in which cj = 1 (see Remark 2.2.1), since the proof of the equation ◦ fj = fj ◦ is similar, and the proof for the case in which cj = 2 is easier (cf. [OSS1, Proposition 6.4] for the case in which b2 ∈ B1ω ⊗ B2ω . We deduce from Lemma 3.2.3 and the tensor product cj = 2). Let b1 ⊗ B2ω , together with rule for the Uq (g)-crystal B1 ⊗B2 and that for the Uq ( g )-crystal B1ω ⊗ a computation similar to (3.2.10), that εj (b1 ⊗ b2 ) = εj (b1 ⊗ b2 ) (note that b1 ⊗ b2 = b2 ) by definition), where b2 ) := max m ≥ 0 | b2 ) = θ . Also, (b1 ⊗ εj (b1 ⊗ ejm (b1 ⊗ we see from Lemma 3.2.3 that εj (b1 ⊗ b2 ) = εj (b1 ⊗ b2 ). Combining these equations, b2 ∈ B1ω ⊗ B2ω , we deduce that for b1 ⊗ b2 ) = εj (b1 ⊗ εj (b1 ⊗ b2 ).

(3.3.2)

b2 ) = θ if and only if b2 )) = Therefore, we conclude that ej (b1 ⊗ ej ((b1 ⊗ ej (b1 ⊗ b2 ) = θ. b2 ) = θ ; note that ej (b1 ⊗ b2 ) = θ , as seen above. Now, we assume that ej (b1 ⊗ We may further assume that ϕj (b1 ) ≥ εj (b2 ), since the proof for the case in which b2 ) = b2 by the defini ϕj (b1 ) < εj (b2 ) is similar. As a result, we have ej (b1 ⊗ e j b1 ⊗ tion of ej . Set b1 ⊗ b2 := ej (b1 ⊗ b2 ). Because ϕj (b1 ) ≥ εj (b2 ) by the assumption, it follows from Lemma 3.2.3 that ϕj (b1 ) ≥ εj (b2 ). Hence we deduce that 2 2 ej (b1 ⊗ b2 ) = (ej eω(j b1 ⊗ b2 = ) ej )(b1 ⊗ b2 ) = ej eω(j ) (ej b1 ⊗ b2 ).

b2 ∈ B1ω ⊗ B2ω . In addition, since Here, obviously, we have b1 ∈ B1ω , b2 ∈ B2ω since b1 ⊗ ω b1 ⊗ b2 = ej (b1 ⊗ b2 ) ∈ (B1 ⊗ B2 ) by Proposition 3.2.1, it follows from (3.3.1) that b1 ∈ B1ω , b2 ∈ B2ω . Thus, we have wt b1 , wt b1 , wt b2 , wt b2 ∈ (h∗cl )0 . Therefore, by the tensor product rule for crystals, we obtain 2 b1 ⊗ b2 = ej eω(j ) (ej b1 ⊗ b2 ) 2 = (ej eω(j ) ej b 1 ) ⊗ b 2

or

(eω(j ) ej b1 ) ⊗ (ej eω(j ) b2 ).

2 Indeed, the b1 ⊗b2 cannot be equal, for example, to (eω(j ) ej b1 )⊗(ej b2 ) since wt(ej b2 ) = ∗ 0 wt b2 + αj is not contained in (hcl ) . Moreover, we have the following claim. Claim. The b1 ⊗ b2 cannot be equal to eω(j ) ej b1 ⊗ ej eω(j ) b2 .

Proof of Claim. Suppose that b1 ⊗ b2 = eω(j ) ej b1 ⊗ ej eω(j ) b2 . Then we have b1 = eω(j ) ej b1 . Let J := j, ω(j ) I , and denote by B1 (b1 ) the connected component of B1 , regarded as a Uq (gJ )-crystal, containing the b1 ∈ B1 . Note that b1 is also contained in B1 (b1 ). We see that B1 (b1 ) is isomorphic as a Uq (gJ )-crystal to the crystal base BJ (λ) for some J -dominant integral weight λ ∈ Pcl (see (3.2.5)). Because B1ω ∩ B1 (b1 ) contains b1 and b1 (and hence is nonempty), we deduce as in Step 1 of the proof of Proposition 3.2.1 that ω∗ (λ) = λ, and that B1ω ∩ B1 (b1 ) equipped with ( ω-Kashiwaraoperators becomes a Uq ( gJ)-crystal isomorphic to the crystal base B J λ), ( where J := j and λ := (Pω∗ )−1 (λ). Since B λ) is connected, it follows that both J


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b1 and wt b1 are contained in the set b1 − wt b1 ∈ Z wt λ + Z αj , and hence that wt αj . But, we obtain by (2.2.5) that 1 b1 = wt b1 + eω(j ) ej b1 = wt b1 + Z wt αj ∈ wt αj , 2 which is a contradiction. This shows the claim. 2 ej b1 ⊗b2 , from which Thus we have proved that ej (b1 ⊗b2 ) = (ej eω(j ) ej b1 )⊗b2 = the equation ( ej (b1 ⊗ b2 )) = ej ((b1 ⊗ b2 )) follows immediately. This completes the proof of the lemma.

Because the fixed point subsets B1ω , B2ω , and (B1 ⊗ B2 )ω , equipped with the ω-Kashiwara operators ej and fj , j ∈ I , are all regular Uq ( g )-crystals by Proposition 3.2.1, there exist actions of the Weyl group W of g on them by Proposition 1.4.2, all of which . With this notation, we have the following lemma. are denoted by Sw ∈W , w -extremal elements whose weights are Lemma 3.3.2. Let b1 ∈ B1ω and b2 ∈ B2ω be W contained in the same Weyl chamber with respect to the simple coroots hj , j ∈ I0 . Then, -extremal element, and the equation b1 ⊗ b2 ∈ (B1 ⊗ B2 )ω is also a W Sw (b1 ⊗ b2 ) = . Sw Sw ∈W b1 ⊗ b2 holds for all w B2ω is regular, there exists an action of Proof. Since the tensor product Uq (g)-crystal B1ω ⊗ ω ω on B ⊗ B2 (see Proposition 1.4.2). In addition, since : B1ω ⊗ B2ω → the Weyl group W 1 ω (B1 ⊗ B2 ) is an isomorphism of Uq ( g )-crystals by Lemma 3.3.1, it follows from the on B ω ⊗ B2ω and the one on (B1 ⊗B2 )ω that definitions of the action of the Weyl group W 1 -extre◦ Sw ∈ W . Therefore, we see that b1 ⊗b2 ∈ (B1 ⊗B2 )ω is a W = Sw ◦ for all w −1 mal element since (b1 ⊗b2 ) = b1 ⊗ b2 is a W -extremal element of the tensor product B2ω by Lemma 1.4.5. Also, the equation Uq ( g )-crystal B1ω ⊗ Sw Sw Sw (b1 ⊗b2 ) = b1 ⊗ b2 follows immediately from Lemma 1.4.5 and the equation ◦ Sw = Sw ◦ . 4. Proof of the Main Result In this section, we use the notation of §2.3 and keep Assumption 2.3.1. Recall that we i,s is a regular Uq (g)fixed (arbitrarily) i ∈ I0 = I \ {0} and s ∈ Z≥1 . By (2.3.7), B i,s i,s → B of the diagram automorphism ω satisfying crystal with an action ω : B i,s . Hence we can apply the results in §3.2 to the case in condition (3.2.1), with B = B i,s , B ω = B i,s , and those in §3.3 to the case in which B1 = B2 = B i,s , which B = B ω ω i,s B1 = B2 = B . 4.1. Proof of regularity. The following proposition follows immediately from Propoi,s (note that B ω = B i,s ). sition 3.2.1 applied to the case B = B i,s ∪ {θ } is stable under the ω-Kashiwara i,s ∪ {θ } of B Proposition 4.1.1. The subset B i,s i,s equipped with the operators ej and fj on B ∪ {θ } for all j ∈ I. Furthermore, the B maps wt, ej , fj , j ∈ I, and εj , ϕj , j ∈ I, becomes a regular Uq ( g )-crystal.

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i,s is 4.2. Proof of simplicity. In this subsection, we prove that the Uq ( g )-crystal B simple. First we show the following proposition. i,s i,s is of finite cardinality, and the weights of elements of B Proposition 4.2.1. The set B i,s are all contained in (Pcl )0 . In other words, the Uq ( g )-crystal B satisfies condition (S1) of Definition 1.4.6. i,s is the tensor product of the crystal bases B ωk (i),s , 0 ≤ k ≤ Ni − 1, Proof. Since B i,s is a finite set, and hence so is B i,s . Therefore, it of finite cardinality, it follows that B i,s . Let b ∈ B i,s . Then we deduce that cl )0 for all b ∈ B b ∈ (P suffices to show that wt b)( c) (wt c ) = ((Pω∗ )−1 (wt b))(

= (wt b)(Pω−1 ( c )) (see the comment following (2.2.4)) = (wt b)(c) by (2.2.5).

(4.2.1)

i,s is perfect (and hence simple), we have (wt b)(c) = 0, Because the Uq (g)-crystal B and hence (wt b)( c ) = 0. This proves the proposition. i,s satisfies The rest of this subsection is devoted to proving that the Uq ( g )-crystal B i,s condition (S2), and also condition (S3), of Definition 1.4.6. Since B is a perfect (and hence simple) Uq (g)-crystal, it follows from Lemma 1.4.8 that there exists a unique W i,s , denoted by u, such that wt u is I0 -dominant, i.e., (wt u)(hj ) ≥ 0 extremal element of B i,s for all j ∈ I0 = I \ {0} (this u is identical to the element ui,s ∈ B in Remark 5.1.3 below). i,s is contained in the fixed point subLemma 4.2.2. The W -extremal element u ∈ B i,s i,s . set B . Moreover, the element u is a W -extremal element of the Uq ( g )-crystal B , either Namely, for every w ∈W ej Sw Sw u = θ or fj u = θ holds for each j ∈ I. Proof. We first show that ω( u) = u. Note that wt(ω( u)) is I0 -dominant. In addition, for each w ∈ W , we have u)) = ω(Sw u) Sw (ω(

(4.2.2)

i,s and (2.3.7) for some w ∈ W . Indeed, we see from the definition of the action of W on B i,s . Therefore, if w = rj1 rj2 · · · rjp that Sj (ω(b)) = ω(Sω−1 (j ) b) for all j ∈ I and b ∈ B is a reduced expression of w ∈ W , then we have u)) = Sj1 Sj2 · · · Sjp (ω( u)) Sw (ω( = ω(Sω−1 (j1 ) Sω−1 (j2 ) · · · Sω−1 (jp ) u) = ω(Sw u), i,s is a W -extremal where w := rω−1 (j1 ) rω−1 (j2 ) · · · rω−1 (jp ) ∈ W . Since Sw u ∈ B element, we deduce from (2.3.7) and (4.2.2) that ω( u) is also a W -extremal element, whose weight wt(ω( u)) is I0 -dominant. Hence it follows from the uniqueness of u (see i,s . Lemma 1.4.8) that ω( u) = u. Thus we conclude that u∈B . We know from Lemma 3.2.4 that Now, let w ∈ W Sw u = S ( u. Since u is w) i,s , either ej Sw a W -extremal element of the Uq (g)-crystal B u = ej S ( u = θ or w) fj u = fj S ( u = θ holds for each j ∈ I . Hence it follows from the definition of Sw w) the ω-Kashiwara operators ej , fj , j ∈ I, that ej u = θ or fj u = θ for each j ∈ I. Sw Sw This proves the lemma.


775

i,s be a W i,s . Then there -extremal element of the regular Uq ( Let b ∈ B g )-crystal B such that wt 0 -dominant, i.e., wt ( ( exists some w ∈W Sw Sw hj ) ≥ 0 for b ) is I b ) ( i,s satisfies all j ∈ I0 = I \ {0}. Therefore, in order to prove that the Uq ( g )-crystal B condition (S2) of Definition 1.4.6, it suffices to show the next proposition. i,s be a W -extremal element of the regular Uq ( Proposition 4.2.3. Let b ∈ B g )-crystal i,s such that wt b is I0 -dominant. Then we have b = B u. In the proof below of Proposition 4.2.3, we need some lemmas. cl )0 that are contained in the same Weyl chamLemma 4.2.4. Let µ, ν be elements of (P , w ber with respect to the simple coroots hj , j ∈ I0 . Then, for each w ∈ W ( µ) and w ( ν ) are contained in the same Weyl chamber with respect to the simple coroots hj , , we have either of the following : (i) ( j ∈ I0 . Moreover, for each w ∈W w ( µ))( h0 ) ≤ 0 and ( w ( ν ))( h0 ) ≤ 0, (ii) ( w ( µ))( h0 ) ≥ 0 and ( w ( ν ))( h0 ) ≥ 0. of Proof. Recall that the Weyl group W g decomposes into the semidirect product I T of the Weyl group W I := W rj | j ∈ I0 (of finite type) and the abelian group 0 0 , there exists I and T of translations. Therefore, for each w ∈W z∈W t ∈ T such that 0 cl )0 by the assumption, it follows from [Kac, Chap. 6, formula w = z t. Since µ, ν ∈ (P (6.5.5)] that t( µ) = µ, t( ν) = ν, and hence that w ( µ) = z( µ), w ( ν) = z( ν ). Thus, , w for each w ∈W ( µ) and w ( ν ) are contained in the same Weyl chamber with respect to the simple coroots hj , j ∈ I0 . . Note that Now, we fix an arbitrary w ∈W h0 = c− γ ∨ , where γ ∨ is a root of the dual root system of gI0 . Since w ( µ) and w ( ν ) are contained in (Pcl )0 , we have ( w ( µ))( h0 ) = ( w ( µ))( c− γ ∨ ) = −( w ( µ))( γ ∨ ), w ( ν ))( c− γ ∨ ) = −( w ( ν ))( γ ∨ ). ( w ( ν ))( h0 ) = (

(4.2.3)

Because w ( µ) and w ( ν ) are contained in the same Weyl chamber as shown above, we conclude from (4.2.3) that either ( w ( µ))( h0 ) ≤ 0 and ( w ( ν ))( h0 ) ≤ 0, or ( w ( µ))( h0 ) ≥ 0 and ( w ( ν ))(h0 ) ≥ 0 holds. This proves the lemma. cl )0 that are I0 -dominant, and assume that Lemma 4.2.5. Let µ, ν be elements of (P I such that ( µ = ν. Then, there exists some z∈W z( µ))( h0 ) = ( z( ν ))( h0 ), and such 0 that ( z( µ))(h0 ) ≤ 0, ( z( ν ))(h0 ) ≤ 0. such that ( Proof. First, let us show that there exists some w ∈ W w ( µ))( h0 ) = ( w ( ν ))( h0 ), and such that ( w ( µ))( h0 ) ≤ 0, ( w ( ν ))( h0 ) ≤ 0. Suppose that ( w ( µ))( h0 ) = , or equivalently ( w ( ν ))( h0 ) for all w ∈W µ( h) = ν( h) for all h∈W h0 .

(4.2.4)

It is easy to show (cf. the argument in the hint for [Kac, Exercise 6.9]) that hcl is gener ated by W h0 as a C-vector space: hcl = h∈W h0 Ch. Therefore, it follows immediately from (4.2.4) that µ( h) = ν( h) for all h ∈ hcl , and hence that µ = ν, which contra. w ( ν ))( h0 ) for some w ∈ W dicts the assumption. This shows that ( w ( µ))( h0 ) = ( Furthermore, for this w ∈ W , we see from Lemma 4.2.4 that either ( w ( µ))(h0 ) ≤ 0 and ( w ( ν ))( h0 ) ≤ 0, or ( w ( µ))( h0 ) ≥ 0 and ( w ( ν ))( h0 ) ≥ 0 holds. If ( w ( µ))( h0 ) ≥ 0 and ( w ( ν ))( h0 ) ≥ 0, then ( r0 w ( µ))( h0 ) ≤ 0 and ( r0 w ( ν ))( h0 ) ≤ 0, with ( r0 w ( µ))( h0 ) =

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such that ( ( r0 w ( ν ))( h0 ). Thus, we have obtained an element w ∈ W w ( µ))( h0 ) = ( w ( ν ))( h0 ), and such that ( w ( µ))( h0 ) ≤ 0, ( w ( ν ))( h0 ) ≤ 0. in the form w I and We write the w ∈ W = z t, with z ∈ W t ∈ T. Then we 0 see that w ( µ) = z( µ) and w ( ν) = z( µ) (see the proof of Lemma 4.2.4), and hence that ( z( µ))( h0 ) = ( z( ν ))( h0 ), and ( z( µ))( h0 ) ≤ 0, ( z( ν ))( h0 ) ≤ 0. This proves the lemma. i,s is a perfect Uq (g)-crystal isomorProof of Proposition 4.2.3. Recall from §2.3 that B phic to the crystal base of a tensor product of finite-dimensional irreducible Uq (g)-modules. By Proposition 1.4.15 (2), we have an energy function H on the tensor product i,s ⊗ B i,s . We show that if b were not equal to Uq (g)-crystal B u, then the energy funci,s ⊗ B i,s , which tion H would take infinitely many different values on the finite set B proves the proposition by contradiction. i,s by Lemma 4.2.2, and b ∈ B i,s by assumpNow, suppose that b = u. Since u∈B tion, it follows that µ := wt u ∈ (P cl )0 ∩ (h∗cl )0 and ν := wt b ∈ (Pcl )0 ∩ (h∗cl )0 . i,s )µ = Note that µ = ν since (B u by condition (S3) of Definition 1.4.6. If we set ∗ −1 cl )0 . ν := (Pω∗ )−1 (ν), then by Proposition 4.2.1, we have µ, ν ∈ (P µ := (Pω ) (µ) and z( µ))(h0 ) = ( z( ν ))(h0 ), and such that ( z( µ))(h0 ) ≤ 0, Let us take z ∈ WI0 such that ( ( z( ν ))( h0 ) ≤ 0 (see Lemma 4.2.5). i,s → B i,s ⊗ B i,s of the diagram automorphism i,s ⊗ B Step 1. Define an action of ω : B i,s ⊗ B i,s , and let (B i,s ⊗ B i,s )ω ω by: ω(b1 ⊗ b2 ) = ω(b1 ) ⊗ ω(b2 ) for b1 ⊗ b2 ∈ B i,s ⊗ B i,s under this action of ω as in §3.3. Obviously, be the fixed point subset of B i,s ⊗ B i,s )ω . Furthermore, we deduce u ⊗ b is contained in the fixed point subset (B from Lemma 3.3.2 that u ⊗ b is a W -extremal element of the regular Uq ( g )-crystal i,s i,s ω ⊗B ) equipped with ω-Kashiwara operators, and that (B Sz ( u ⊗ b) = Sz u⊗ Sz b. We set p := −( z( µ))( h0 ), and q := −( z( ν ))( h0 ); note that p = q, and p ≥ 0, q ≥ 0. Then, because wt Sz u⊗ Sz b ( h0 ) = z( µ) + z( ν ) ( h0 ) = −(p + q) ≤ 0, we have by the definition of S0 , S0 u ⊗ b) = ( e0 )p+q ( Sz u⊗ Sz b) Sz ( p+q =e (Sz u ⊗ Sz b) 0

-extremal elements of the (note that e0 = e0 ). In addition, since Sz u and Sz b are W i,s , and since (wt Uq ( g )-crystal B Sz u)( h0 ) = ( z( µ))( h0 ) = −p ≤ 0, (wt Sz b)( h0 ) = ( z( ν ))(h0 ) = −q ≤ 0, we obtain ε0 ( Sz u) = ε0 ( Sz u) = 0 and ε0 ( Sz b) = ε0 ( Sz b) = 0, ϕ0 (Sz u) = ϕ0 (Sz u) = p and ϕ0 (Sz b) = ϕ0 ( Sz b) = q (note that e0 = e0 , f0 = f0 ). Using these equations, we can easily show by the tensor i,s ⊗ B i,s that product rule for the Uq (g)-crystal B  u ⊗ el ( for 0 ≤ l ≤ q, z b) 0 S Sz l u ⊗ Sz b = (4.2.5) e0 Sz q  l−q u) ⊗ e0 ( Sz b) for q ≤ l ≤ p + q. e0 (Sz


777

i,s ) successively, we obtain Therefore, by using (1.4.5) (applied to the case B = B q H S0 Sz ( u ⊗ b) = H Sz u ⊗ e0 ( Sz b) + p. (4.2.6) Indeed, we deduce that p+q H S0 Sz ( u ⊗ b) = H e0 ( Sz u⊗ Sz b) p−1 q Sz u) ⊗ e0 ( Sz b) = H e0 e0 (

by (4.2.5)

p−1 q Sz u) ⊗ e0 ( Sz b) + 1 = H e0 (

q since ε0 e0 ( Sz b) = 0

p−2 q Sz u) ⊗ e0 ( Sz b) + 1 = H e0 e0 ( p−2 q Sz u) ⊗ e0 ( Sz b) + 2 = H e0 (

by (4.2.5)

q since ε0 e0 ( Sz b) = 0.

Continuing in this way, we finally obtain (4.2.6). Furthermore, again by using (1.4.5) successively, we obtain q H Sz u ⊗ e0 ( Sz b) = H Sz u⊗ Sz b − q. (4.2.7) Indeed, we deduce that q q−1 H Sz u ⊗ e0 ( Sz b) = H e0 u ⊗ e0 ( Sz b) by (4.2.5) Sz q−1 =H Sz u ⊗ e0 ( Sz b) − 1

q−1 Sz since ϕ0 u = 0 and ε0 e0 ( Sz b) > 0

q−2 Sz u ⊗ e0 ( Sz b) = H e0

by (4.2.5)

q−2 u ⊗ e0 ( Sz b) − 2 =H Sz

q−2 since ϕ0 u = 0 and ε0 e0 ( Sz b) > 0. Sz

Continuing in this way, we finally obtain (4.2.7). Consequently, by combining (4.2.6) and (4.2.7), we have H S0 u ⊗ b) = H Sz u⊗ Sz b + p − q. Sz ( i,s ) that Also, we deduce from (1.4.5) and (1.4.6) (applied to the case B = B H Sz u⊗ Sz b = H Sz ( u ⊗ b) = H ( u ⊗ b), since Sz is defined by using only ej and fj , j ∈ I0 , and hence by using only ej and fj , j ∈ I0 . Thus we conclude that H S0 u ⊗ b) = H ( u ⊗ b) + p − q. (4.2.8) Sz (

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in the form I and Step 2. We write r0 ∈ W r0 = z t ∈ T. Then we see t, with z ∈ W 0 that cl )0 z ( t ( r0 r0 z( µ)) = z z( µ)) since r0 z( µ) ∈ (P = r0 ( r0 z( µ)) = z( µ),

(4.2.9)

and similarly that z r0 z( ν) = z( ν ). Set b1 ⊗ b2 := Sz u ⊗ b). It follows from S0 Sz ( Lemma 3.3.2 that b1 = Sz u) and b2 = Sz S0 Sz ( S0 Sz (b). Hence, by (4.2.9), we have b1 = b2 = wt z( µ) and wt z( ν ). Now, by an argument similar to that in Step 1 above, we can deduce that H ( S0 (b1 ⊗ b2 )) = H (b1 ⊗ b2 ) + p − q. I , the Note that since z ∈ W Sz is defined by using only ej , fj , j ∈ I0 , and hence by 0 using only ej , fj , j ∈ I0 . Therefore, by (1.4.5) and (1.4.6), we obtain H ( S0 (b1 ⊗ b2 )) = H (b1 ⊗ b2 ) + p − q = H Sz u ⊗ b) + p − q S0 Sz ( u ⊗ b) + p − q Sz ( =H S0 = H ( u ⊗ b) + 2(p − q) by (4.2.8). Repeating this argument, we can show that for every k ∈ Z≥0 , there exists some x ∈ i,s )ω ⊂ B i,s ⊗ B i,s such that H (x) = H ( i,s ⊗ B u ⊗ b) + 2k(p − q). This contradicts (B i,s i,s the fact that B ⊗ B is a finite set. Thus we have proved the proposition. i,s satisfies condition (S3) of Definition 1.4.6. Finally, we show that the B i,s , and i,s be a W -extremal element of the Uq ( g )-crystal B Proposition 4.2.6. Let b ∈ B i,s i,s µ⊂B of all elements of weight b. Then, the subset (B set µ := wt µ consists only ) i,s ) = b . of the element b, i.e., (B µ Proof. By Proposition 4.2.3, together with the comment preceding it, we see that b is i,s : b = . Furthermore, it follows -orbit of Sw u for some w ∈W contained in the W u∈B from Lemma 3.2.4 that b = S ( u , and hence that b is contained also in the W -orbit w) i,s of u. Note that since u ∈ B is a W -extremal element of the simple Uq (g)-crystal i,s i,s B , so is the (S3) of Definition 1.4.6, we have b ∈ B . In particular, by condition i,s )wt b = b . Therefore, we conclude that (B i,s ) (B µ = b , as desired. g )-crystal By combining Propositions 4.2.1, 4.2.3, and 4.2.6, the simplicity of the Uq ( i,s B is established. 4.3. Proof of bijectivity. i,s is equal to s. Moreover, Proposition 4.3.1. The level of the simple Uq ( g )-crystal B + i,s the maps ε, ϕ : (B )min → (Pcl )s are bijective. i,s and B ω = B i,s Proof. We see from Lemma 3.2.3 applied to the case in which B = B ∗ i,s that Pω ( ε(b)) = ε(b) for all b ∈ B . Therefore, we deduce that ( ε(b))( c ) = (Pω∗ )−1 (ε(b)) ( c) = (ε(b))(Pω−1 ( c )) (see the comment following (2.2.4)) = (ε(b))(c) by (2.2.5),


779

i,s ⊂ B i,s since B i,s is a perfect and hence that ( ε(b))( c ) = (ε(b))(c) ≥ s for all b ∈ B i,s Uq (g)-crystal of level s. This implies that the level of B is greater than or equal to s. i,s is equal to s, it suffices to show that there Therefore, to prove that the level of B i,s such that ( + )s ; note that (P + )s = ∅ exists some b ∈ B ε(b))( c ) = s. Let µ ∈ (P cl cl + )s . Then we deduce that (Pω∗ ( 0 ∈ (P for all s ∈ Z≥1 since s µ))(c) = µ(Pω (c)) = cl i,s is a perfect Uq (g)-crystal of µ( c ) = s, and hence that Pω∗ ( µ) ∈ (Pcl+ )s . Because B i,s )min such that ε(b) = Pω∗ ( µ) ∈ (Pcl+ )s . It is level s, there exists a (unique) b ∈ (B i,s , i.e., that ω(b) = b. Indeed, we see from easy to check that this b is contained in B (2.3.7) that εj (ω(b)) = εω−1 (j ) (b) for all j ∈ I , and hence that ε(ω(b)) = ω∗ (ε(b)). µ) ∈ (Pcl+ )s ∩ (h∗cl )0 , we have ω∗ (ε(b)) = ε(b). As a result, we But, since ε(b) = Pω∗ ( i,s )min → (P + )s is bijective, we conclude that have ε(ω(b)) = ε(b). Because ε : (B cl ε(b)) = ε(b) and Pω∗ ( µ) = ε(b) that ω(b) = b. Also, it follows from the formulas Pω∗ ( + )s . Thus, we have ε(b) = µ. Therefore, we have ( ε(b))( c) = µ( c) = s since µ ∈ (P cl i,s = s. shown that lev B i,s )min → (P + )s is bijective. The argument above Next, we prove that the map ε : (B cl + i,s )min such that )s , there exists some b ∈ (B shows that for each µ ∈ (P ε(b) = µ, cl + i,s )min → (P )s is surjective. Let us show that which means that the map ε : (B cl i,s )min → (P i,s )min . + )s is injective. Assume that ε : (B ε(b) = ε(b ) for b, b ∈ (B cl Note that, by Lemma 3.2.3, ε(b) = Pω∗ ( ε(b)) = Pω∗ ( ε(b )) = ε(b ), and hence that i,s )min . (ε(b))(c) = ( ε(b))( c ) = s, (ε(b ))(c) = ( ε(b ))( c ) = s, i.e., that b, b ∈ (B i,s )min → Therefore, we conclude from the equation ε(b) = ε(b ) that b = b since ε : (B i,s )min → (P + )s is injec(Pcl+ )s is bijective. Thus, we have shown that the map ε : (B cl i,s )min → (P + )s is tive, and hence bijective. Similarly, we can show that the map ϕ : (B cl bijective. This proves the proposition. i,s is a perfect Uq (g)-crystal of level s, By Proposition 4.3.1, we can conclude that B thereby completing the proof of Theorem 2.4.1. 4.4. Relation to virtual crystals. Let u be the unique W -extremal element of the simple i,s whose weight is I0 -dominant. Then, by Lemma 4.2.2, the element Uq (g)-crystal B u i,s i,s is contained in the fixed point subset B . In addition, since B is a perfect (and hence i,s is simple) Uq ( g )-crystal by Theorem 2.4.1, we see by Proposition 1.4.10 (1) that B i,s identical to the set of elements of B obtained by applying the ω-Kashiwara operators ej , fj , j ∈ I, successively to u. Therefore, if the pair (g, ω) is in Case (a) (resp., (c), i,s ) coincides with the virtual crystal for (D (2) , A(1) ) i,s , B (d), (e)) in §2.1, then (B n+1 2n−1 (2)

(1)

(3)

(1)

(2)

(1)

defined in [OSS1, §6.7] (resp., (A2n−1 , Dn+1 ), (D4 , D4 ), (E6 , E6 ) defined in [OSS2, Definition 2.6]). With regard to combinatorial R-matrices, we have the following proposition (see also [OSS2, Definition–Conjecture 3.4] and the comment following it). Proposition 4.4.1. Let us fix i1 ∈ I0 , s1 ∈ Z≥1 and i2 ∈ I0 , s2 ∈ Z≥1 for which i2 ,s2 → i1 ,s1 ⊗ : B B Assumption 2.3.1 is satisfied. Then, there exists an isomorphism R i ,s i ,s 2 2 ⊗ 1 1 of Uq ( B B g )-crystals. i2 ,s2 → B i1 ,s1 ⊗ B i2 ,s2 of the diagram i1 ,s1 ⊗ B Proof. As in §3.3, we define an action ω : B i1 ,s1 ⊗ B i2 ,s2 , and let automorphism ω by: ω(b1 ⊗ b2 ) = ω(b1 ) ⊗ ω(b2 ) for b1 ⊗ b2 ∈ B i1 ,s1 ⊗ B i2 ,s2 )ω be the fixed point subset under this action of ω. Similarly, we define (B

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i2 ,s2 ⊗ B i1 ,s1 → B i2 ,s2 ⊗ B i1 ,s1 of ω as above, and let (B i2 ,s2 ⊗ B i1 ,s1 )ω an action ω : B be the fixed point subset under this action of ω. i2 ,s2 → B i2 ,s2 ⊗ B i1 ,s1 be the combinatorial R-matrix in i1 ,s1 ⊗ B Claim. Let R : B i ,s 1 1 ⊗ B i2 ,s2 )ω ⊂ (B i1 ,s1 )ω . i2 ,s2 ⊗ B Proposition 1.4.15 (1 ). Then we have R (B u2 ) be the unique W -extremal element of the simple Proof of Claim. Let u1 (resp., i1 ,s1 (resp., B i2 ,s2 ) whose weight is I0 -dominant (see Lemma 1.4.8). Uq (g)-crystal B i1 ,s1 ⊗ B i2 ,s2 )ω and Then we see from Lemma 4.2.2 and (3.3.1) that u1 ⊗ u2 ∈ (B i ,s i ,s ω i ,s i ,s ω i ,s 2 2 1 1 1 1 2 2 1 i2 ,s2 )ω u2 ⊗ u1 ∈ (B ⊗B ) . Now, let b ∈ (B ⊗B ) . Because (B 1 ⊗ B i ,s i ,s 1 1 2 2 is isomorphic to B as a Uq ( g )-crystal by Lemma 3.3.1, it follows from Theo⊗B i1 ,s1 ⊗ B i2 ,s2 )ω is simple. g )-crystal (B rem 2.4.1 and Proposition 1.4.10 (2) that the Uq ( i2 ,s2 )ω can be written as: b = i1 ,s1 ⊗ B xj1 xj2 · · · xjl ( u1 ⊗ u2 ), Therefore, the b ∈ (B where xj is either ej or fj for each j ∈ I. Since R is an isomorphism of Uq (g)crystals, we see from the definition of the ω-Kashiwara operators ej and fj , j ∈ I, that R(b) = xj1 xj2 · · · xjl R( u1 ⊗ u2 ). Furthermore, it follows from Lemma 1.4.5 that i1 ,s1 ⊗B i2 ,s2 u1 ⊗ u2 (resp., u2 ⊗ u1 ) is a W -extremal element of the simple Uq (g)-crystal B i2 ,s2 ⊗ B i1 ,s1 ) whose weight is I0 -dominant. Because such an element of a sim(resp., B u1 ⊗ u2 ) = u2 ⊗ u1 , ple Uq (g)-crystal is unique by Lemma 1.4.8, we conclude that R( and hence that R(b) = xj1 xj2 · · · xjl ( u2 ⊗ u1 ). In addition, we deduce from Proposii2 ,s2 ⊗ B i1 ,s1 that R(b) ∈ (B i2 ,s2 ⊗ B i1 ,s1 )ω since tion 3.2.1 applied to the case B = B i2 ,s2 ⊗ B i1 ,s1 )ω . This proves the claim. u2 ⊗ u1 ∈ (B i1 ,s1 ⊗ i2 ,s2 ⊗ i2 ,s2 → B i1 ,s1 so that the following : B B B Now we define a map R diagram commutes: ∼ i1 ,s1 ⊗ i1 ,s1 i2 ,s2 −−− B B −→ (B 1   R

i2 ,s2 )ω ⊗B   R

(4.4.1)

∼ i2 ,s2 ⊗ i1 ,s1 −−− i2 ,s2 ⊗ B i1 ,s1 )ω , B B −→ (B 2

g )-crystals on the top (resp., on the botwhere the isomorphism 1 (resp., 2 ) of Uq ( tom) is given by Lemma 3.3.1. It follows immediately from this commutative diagram i2 ,s2 → B i1 ,s1 is an embedding of Uq ( i1 ,s1 ⊗ i2 ,s2 ⊗ : B B B that R g )-crystals. In addi i ,s 1 1 and B i2 ,s2 are perfect (and hence simple) by Theotion, since the Uq ( g )-crystals B i2 ,s2 ⊗ i1 ,s1 is a simple Uq ( B rem 2.4.1, we see from Proposition 1.4.10 (2) that B g )-crysi ,s i ,s 2 2 ⊗ i1 ,s1 ⊗ 1 1 is connected. Therefore, we deduce that R i2 ,s2 → : B B B tal, and that B i2 ,s2 ⊗ i1 ,s1 is surjective, and hence bijective. Thus we have proved the proposition. B B ∼

i,s ⊗ B i,s )ω ⊂ B i,s ⊗ B i,s of i,s → (B i,s ⊗ B Similarly, using the isomorphism : B in Lemma 3.3.1, we can prove the following proposition.

Uq ( g )-crystals

i,s ⊗ i,s → Z by: :B B Proposition 4.4.2. We define a Z-valued function H i,s , i,s ⊗ (b1 ⊗ b2 ) = H ((b1 ⊗ b2 )) = H (b1 ⊗ b2 ) for b1 ⊗ b2 ∈ B B H

(4.4.2)


781

i,s ⊗ B i,s → Z is the energy function in Proposition 1.4.15 (2 ). Then, the where H : B function H enjoys the following property :  (b1 ⊗ b2 ) + 1 if j = 0 and ϕ0 (b1 ) ≥ ε0 (b2 ), H    ( (b1 ⊗ b2 )) = H b2 ) − 1 if j = 0 and H ej (b1 ⊗ ϕ0 (b1 ) < ε0 (b2 ), (4.4.3)    H (b1 ⊗ b2 ) if j = 0, i,s ⊗ i,s such that b2 ∈ B B b2 ) = θ. for all j ∈ I and b1 ⊗ ej (b1 ⊗ gI0 ) 5. Branching Rules with Respect to Uq ( In this section, we use the notation of §2.3 and keep Assumption 2.3.1 for the (arbii,s is a perfect (and hence regular) trarily) fixed i ∈ I0 = I\ {0} and s ∈ Z≥1 . Since B Uq ( g )-crystal for i ∈ I0 = I \ {0} and s ∈ Z≥1 by Theorem 2.4.1, it decomposes, as a Uq ( gI0 )-crystal, into a direct sum of the crystal bases of integrable highest weight Uq ( gI0 )-modules. In this section, we describe explicitly the branching rule, i.e., how the i,s decomposes, with respect to the restriction to Uq ( B gI0 ), for almost all i ∈ I0 and s ∈ Z≥1 . 5.1. Preliminary results.

Let us set i,s | i,s )h.w. := b ∈ B ej b = θ for all j ∈ I0 (B

(5.1.1)

i,s , we have for i ∈ I0 and s ∈ Z≥1 . Then, by the regularity of the Uq ( g )-crystal B i,s ∼ (wt b) as Uq ( B B gI0 )-crystals, (5.1.2) = I0 i,s )h.w. b∈(B

( where B gI0 )-module I0 λ) denotes the crystal base of the integrable highest weight Uq ( VI0 (λ) of (I0 -dominant) highest weight λ ∈ Pcl . Similarly, we set i,s )h.w. := b ∈ B i,s | ej b = θ for all j ∈ I0 (B (5.1.3) i,s , we have for i ∈ I0 and s ∈ Z≥1 . Then, by the regularity of the Uq (g)-crystal B i,s ∼ B BI0 (wt b) as Uq (gI0 )-crystals, (5.1.4) = i,s )h.w. b∈(B

where BI0 (λ) denotes the crystal base of the integrable highest weight Uq (gI0 )-module VI0 (λ) of (I0 -dominant) highest weight λ ∈ Pcl . Lemma 5.1.1. We have

i,s )h.w. = (B i,s )h.w. ∩ B i,s = b ∈ (B i,s )h.w. | ω(b) = b . (B

(5.1.5)

i,s )h.w. ∩ B i,s is obvious from the definition of the i,s )h.w. ⊃ (B Proof. The inclusion (B i,s )h.w. raising ω-Kashiwara operators ej , j ∈ I. For the reverse inclusion, let b ∈ (B (note that ω(b) = b by definition). Then we see that εj (b) = 0 for all j ∈ I0 , and hence by Lemma 3.2.3 that εj (b) = 0 for all j ∈ I0 . This implies that ej b = θ for all j ∈ I0 . i,s )h.w. ∩ B i,s . This proves the lemma. Therefore, we have b ∈ (B

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i,s as a Uq (gI0 )-crystal is Lemma 5.1.2. Assume that the decomposition (5.1.4) of B multiplicity-free. Then we have i,s )h.w. = b ∈ (B i,s )h.w. | ω∗ (wt b) = wt b . (B

(5.1.6)

Proof. The inclusion ⊂ is obvious by Lemma 5.1.1. For the reverse inclusion, let b ∈ i,s )h.w. be such that ω∗ (wt b) = wt b. It follows immediately from (2.3.7) that ω(b) (B i,s )h.w. (note that ω(I0 ) = I0 ). In addition, we have wt(ω(b)) = is also contained in (B ∗ ω (wt b) = wt b. Therefore, we deduce that ω(b) = b, since the decomposition (5.1.4) i,s )h.w. i,s as a Uq (gI0 )-crystal is multiplicity-free by the assumption. Thus, the b ∈ (B of B i,s i,s i,s is contained in the set (B )h.w. ∩ B , and hence in the set (B )h.w. by Lemma 5.1.1. This proves the lemma. In §5.2 – §5.6 below, by using Lemma 5.1.2, we give an explicit description of the i,s for i ∈ I0 and s ∈ Z≥1 with respect to the restriction to Uq ( branching rule for B gI0 ), i,s as a Uq (gI0 )-crystal is not except a few cases in which the decomposition (5.1.4) of B multiplicity-free. In the following, we use the notation:

i : = i − ai∨ 0 ∈ Pcl N i −1

for i ∈ I0 ,

(5.1.7)

ωk (i) ∈ Pcl

for i ∈ I0 ,

and

0 := 0,

(5.1.8)

cl 0 ∈ P i − ai∨

i : =

for i ∈ I0 ,

and

0 := 0.

(5.1.9)

i : =

k=0

Note that Pω∗ ( i ) = i for all i ∈ I0 , and also for i = 0. Remark 5.1.3. For i ∈ I0 and s ∈ Z≥1 , we set ui,s := ui,s ⊗ uω(i),s ⊗ · · · ⊗ uωNi −1 (i),s , k

where the uωk (i),s ∈ B ω (i),s , 0 ≤ k ≤ Ni − 1, are the W -extremal elements in Remark 2.3.2. Then, by Lemma 1.4.5, this ui,s is the W -extremal element of the simi,s such that (wt ui,s )(hj ) ≥ 0 for all j ∈ I0 . In fact, we have ple Uq (g)-crystal B wt ui,s = s i . Therefore, it follows from Remark 1.4.9 that the weights of elements of i,s are all contained in the set s B i − j ∈I0 Z≥0 αj . Also, the weights of elements of i,s are all contained in the set s B i − j ∈I0 Z≥0 αj . 5.2. Branching rule for Case (a).

We know from [KMN] that

B i,s ∼ = BI0 (s i ) as Uq (gI0 )-crystals

(5.2.1)

for i ∈ I0 and s ∈ Z≥1 . Here we recall the (well-known) fact that the integrable highest weight Uq (gI0 )-module VI0 (λ) of I0 -dominant highest weight λ ∈ Pcl has the same character as the integrable highest weight gI0 -module of the same highest weight λ ∈ Pcl , and this character is equal to b∈BI (λ) ewt b . By using this fact, we deduce from 0


783

Decomposition Rule on p. 145 of [L] that if 1 ≤ i ≤ n − 1, then as Uq (gI0 )-crystals, i,s = B i,s ⊗ B ω(i),s ∼ B = BI0 (s i ) ⊗ BI0 (s ω(i) ) ∼ =

by (5.2.1)

BI0 (s0 0 + s1 1 + · · · + si i )

s0 +s1 +···+si =s s0 , s1 , ..., si ∈Z≥0

=

BI0 (s1 1 + · · · + si i )

s1 +···+si ≤s s1 , ..., si ∈Z≥0

n,s = B n,s ∼ for s ∈ Z≥1 . If i = n, then B = BI0 (s n ) as Uq (gI0 )-crystals for s ∈ Z≥1 . i,s above is multiplicity Note that for every i ∈ I0 and s ∈ Z≥1 , the decomposition of B free, and that the highest weights appearing in the decomposition are all fixed by ω∗ . Consequently, by using Lemma 5.1.2, we obtain the branching rule as follows. Proposition 5.2.1. For i ∈ I0 and s ∈ Z≥1 , we have  (s1  B 1 + · · · + si i ) if 1 ≤ i ≤ n − 1,  I0    s1 +···+si ≤s i,s ∼ B = s1 , ..., si ∈Z≥0      (s B if i = n, I0 n )

(5.2.2)

as Uq ( gI0 )-crystals. 5.3. Branching rule for Case (b). following proposition.

In exactly the same way as in Case (a), we have the

Proposition 5.3.1. For i ∈ I0 and s ∈ Z≥1 , we have i,s ∼ (s1 B B 1 + · · · + si i ), = I0

(5.3.1)

s1 +···+si ≤s s1 , ..., si ∈Z≥0

as Uq ( gI0 )-crystals. (i)

(i)

5.4. Branching rule for Case (c). We know from [C] that the KR module Ws (ζs ) (i) over Uq (g) for i ∈ I0 , s ∈ Z≥1 , and ζs ∈ C(q)× , decomposes under the restriction to Uq (gI0 ) as follows: (i) (i) Ws (ζs ) ∼ =   VI0 (spi pi + spi +2 pi +2 + · · · + si i ) if 1 ≤ i ≤ n − 1,     spi +spi +2 +···+si =s spi , spi +2 , ..., si ∈Z≥0 (5.4.1)      VI0 (s i ) if i = n, n + 1,

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as Uq (gI0 )-modules, where the pi ∈ 0, 1 for 1 ≤ i ≤ n is defined to be 0 (resp., 1) if i is even (resp., odd), and VI0 (λ) is the integrable highest weight Uq (gI0 )-module of (I0 -dominant) highest weight λ ∈ Pcl . Accordingly, from (5.4.1), we obtain the follow(i) (i) ing decomposition of the crystal base B i,s of Ws (ζs ), regarded as a Uq (gI0 )-crystal by restriction: B i,s ∼ =   BI0 (spi pi + spi +2 pi +2 + · · · + si i ) if 1 ≤ i ≤ n − 1,     spi +spi +2 +···+si =s spi , spi +2 , ..., si ∈Z≥0 (5.4.2)      BI0 (s i ) if i = n, n + 1, as Uq (gI0 )-crystals for s ∈ Z≥1 . Therefore, as in Case (a), we deduce from Decomposition Rule on p. 145 of [L] that as Uq (gI0 )-crystals, n,s = B n,s ⊗ B n+1,s ∼ B = BI0 (s n ) ⊗ BI0 (s n+1 )

∼ =

spn +spn +2 +···+sn =s spn , spn +2 , ..., sn ∈Z≥0

by (5.4.2)

BI0 (spn pn + spn +2 pn +2 + · · · + sn n )

for s ∈ Z≥1 . Also, if 1 ≤ i ≤ n − 1, then as Uq (gI0 )-crystals,

i,s = B i,s ∼ B =

spi +spi +2 +···+si =s spi , spi +2 , ..., si ∈Z≥0

=


BI0 (spi pi + spi +2 pi +2 + · · · + si i )

by (5.4.2)

BI0 (spi pi + spi +2 pi +2 + · · · + si i ).

i,s above is multiplicity-free, and the Because in all the cases, the decomposition of B highest weights appearing in the decomposition are all fixed by ω∗ , we obtain the following proposition by using Lemma 5.1.2. Proposition 5.4.1. For i ∈ I0 and s ∈ Z≥1 , we have

i,s ∼ B =


as Uq ( gI0 )-crystals.

(spi B pi + spi +2 pi +2 + · · · + si i ) I0

(5.4.3)


785

5.5. Branching rule for Case (d). First of all, we should warn the reader that our numbering of the index set I is different from the one in [HKOTY] and [L]. As in Case (c), we obtain that as Uq (gI0 )-crystals, 1,s ∼ B BI0 (s1 1 ), (5.5.1) = 0≤s1 ≤s

2,s ∼ B = BI0 (s 2 ) ⊗ BI0 (s 3 ) ⊗ BI0 (s 4 ),

(5.5.2)

for s ∈ Z≥1 . Note that in general, the irreducible decomposition of the tensor product VI0 (s 2 )⊗VI0 (s 3 )⊗VI0 (s 4 ) as a Uq (gI0 )-module is not multiplicity-free, and hence 2,s , regarded as a Uq (gI0 )-crystal by restriction, is that the decomposition (5.1.4) of B not multiplicity-free. Hence we exclude this case, i.e., the case in which i = 2. Then, by using Lemma 5.1.2 as above, we obtain the following proposition. Proposition 5.5.1. For s ∈ Z≥1 , we have (s1 1,s ∼ B B 1 ) = I0

(5.5.3)

0≤s1 ≤s

gI0 )-crystals. as Uq ( 5.6. Branching rule for Case (e). First of all, we should warn the reader that our numbering of the index set I is different from the one in [HKOTY], and also from the one in [L]. Assume that i = 2 (note that the decomposition of B 2,s is known not to (3) be multiplicity-free in general; cf. the formula for Ws in the case that Xn = E6 on p. 278 of [HKOTY, Appendix A]). Then, as in Case (c), we obtain from [C] the following decomposition of B i,s , regarded as a Uq (gI0 )-crystal by restriction:   if i = 4, 6, BI0 (s i )        BI0 (s1 1 ) if i = 1,      0≤s ≤s 1     i,s ∼ BI0 (s3 3 + s6 6 ) if i = 3, B =   s +s =s   s33 ,s66≥0         BI0 (s4 4 + s5 5 ) if i = 5,     s4 +s5 =s s4 ,s5 ≥0

as Uq (gI0 )-crystals for s ∈ Z≥1 . Therefore, as in Cases (a), (c), we deduce from Decomposition Rule on p. 145 of [L] that as Uq (gI0 )-crystals, 1,s = B 1,s ∼ B BI0 (s1 1 ), (5.6.1) = 0≤s1 ≤s

4,s = B 4,s ⊗ B 6,s ∼ B = BI0 (s 4 ) ⊗ BI0 (s 6 ) ∼ =

s1 +s4 ≤s s1 , s4 ∈Z≥0

BI0 (s1 1 + s4 4 ),

(5.6.2)

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for s ∈ Z≥1 . Here we have excluded the case in which i = 3, since the decomposition 3,s = B 3,s ⊗ B 5,s , regarded as a Uq (gI0 )-crystal by restriction, is known not to of B be multiplicity-free in general. Then, by using Lemma 5.1.2 as above, we obtain the following proposition. Proposition 5.6.1. For s ∈ Z≥1 , we have (s1 1,s ∼ 4,s ∼ B B 1 ), B = = I0 0≤s1 ≤s

s1 +s4 ≤s s1 , s4 ∈Z≥0

(s1 B 1 + s4 4 ), I0

(5.6.3)

as Uq ( gI0 )-crystals. 5.7. Comments. In addition to Assumption 2.3.1 for g, let us assume that Conjecture 1.5.1 with g replaced by g holds for the (fixed) i ∈ I0 and s ∈ Z≥1 . Denote the KR (i) (i) s(i) ( module over Uq ( g ) having a crystal base by W ζs ), where ζs ∈ C(q)× . Then we make the following observation. (a)

Observation. By comparing (5.2.2) (resp., (5.3.1), (5.4.3)) with the formula for Ws in (2) (2) (2) the case that g = Dn+1 (resp., A2n , A2n−1 ) on p. 247 (resp., p. 247, p. 246) of [HKOTT, i ”, and “q” equated with Appendix A] (with “a” replaced by “i”, “i ” replaced by “ “1”), we can check that they indeed coincide. Similarly, by comparing (5.5.3) (resp., (1) (1) (4) (3) (5.6.3)) with the formula for Ws (resp., Ws and Ws ) in the case that g = D4 (2) (resp., E6 ) on p. 249 (resp., p. 248) of [HKOTT, Appendix A] (with “i ” replaced by “ i ”, and “q” equated with “1”), we can check that they also coincide. Here the formula (a) for Ws (with q = 1) describes the conjectural “branching rule” for the KR module (a) (a) s ( ζs ) over Uq ( g ) with respect to the restriction to Uq ( gI0 ). W Motivated by this observation, we propose the following conjecture. Conjecture 5.7.1. Let us fix (arbitrarily) i ∈ I0 and s ∈ Z≥1 , and keep Assumption 2.3.1. i,s is isomorphic to the (conjectural ) crystal base of g )-crystal B Then, the perfect Uq ( (i) (i) s ( ζs ) over the twisted quantum affine algebra Uq ( g ). the KR module W References [AK] [C] [CP1] [CP2] [FRS] [FSS] [HKKOT]

Akasaka, T., Kashiwara, M.: Finite-dimensional representations of quantum affine algebras. Publ. Res. Inst. Math. Sci. 33, 839–867 (1997) Chari, V.: On the fermionic formula and the Kirillov–Reshetikhin conjecture. Int. Math. Res. Not. 2001, no. 12, 629–654 (2001) Chari, V., Pressley, A.: Quantum affine algebras and their representations. In: Allison, B.N., Cliff, G.H., (eds.), “Representations of Groups”, CMS Conf. Proc. Vol. 16, Providence, RI: Amer. Math. Soc., 1995, pp. 59–78 Chari, V., Pressley, A.: Twisted quantum affine algebras. Commun. Math. Phys. 196, 461– 476 (1998) Fuchs, J., Ray, U., Schweigert, C.: Some automorphisms of generalized Kac–Moody algebras. J. Algebra 191, 518–540 (1997) Fuchs, J. Schellekens, B., Schweigert, C.: From Dynkin diagram symmetries to fixed point structures. Commun. Math. Phys. 180, 39–97 (1996) Hatayama, G., Koga, Y., Kuniba, A., Okado, M., Takagi, T.: Finite crystals and paths. In: Koike, K., Kashiwara, M., et al., (eds.), “Combinatorial Methods in Representation Theory”, Adv. Stud. Pure Math. Vol. 28, Tokyo: Kinokuniya, 2000, pp. 113–132

Construction of Perfect Crystals [HKOTT]

[HKOTY]

[HK] [Kac] [KMN] [Kas1] [Kas2] [Kas3] [Ko] [KNT] [L] [NS1] [NS2] [NS3] [OSS1] [OSS2]

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Hatayama, G., Kuniba, A., Okado, M., Takagi, T., Tsuboi, Z.: Paths, crystals and fermionic formulae. In: Kashiwara, M., Miwa, T., (eds.), “Math Phys Odyssey 2001, Integrable Models and Beyond”. Progress in Mathematical Physics, Vol. 23, Boston: Birkhäuser, 2002, pp. 205–272 Hatayama, G., Kuniba, A., Okado, M., Takagi, T., Yamada, Y.: Remarks on fermionic formula. In: Jing, N., Misra, K.C, (eds.), “Recent Developments in Quantum Affine Algebras and Related Topics”, Contemp. Math. Vol. 248, Providence, RI: Amer. Math. Soc., 1999, pp. 243–291 Hong, J., Kang, S.-J.: “Introduction to Quantum Groups and Crystal Bases”. Graduate Studies in Mathematics Vol. 42, Providence, RI: Amer. Math. Soc., 2002 Kac, V.G.: “Infinite Dimensional Lie Algebras”. 3rd Edition. Cambridge: Cambridge University Press, 1990 Kang, S.-J., Kashiwara, M., Misra, K.C., Miwa, T., Nakashima, T., Nakayashiki, A.: Perfect crystals of quantum affine Lie algebras. Duke Math. J. 68, 499–607 (1992) Kashiwara, M.: Crystal bases of modified quantized enveloping algebra. Duke Math. J. 73, 383–413 (1994) Kashiwara, M: On crystal bases. In: Allison, B.N., Cliff, G.H., (eds.), “Representations of Groups”, CMS Conf. Proc. Vol. 16, Providence, RI: Amer. Math. Soc., 1995, pp. 155–197 Kashiwara, M.: On level-zero representations of quantized affine algebras., Duke Math. J. 112, 117–175 (2002) (1) (1) (1) Koga, Y.: Level one perfect crystals for Bn , Cn and Dn . J. Algebra 217, 312–334 (1999) Kuniba, A., Nakanishi, T., Tsuboi, Z.: The canonical solutions of the Q-systems and the Kirillov–Reshetikhin conjecture. Commun. Math. Phys. 227, 155–190 (2002) Littelmann, P.: On spherical double cones. J. Algebra 166, 142–157 (1994) Naito, S., Sagaki, D.: Crystal bases and diagram automorphisms. In: Shoji, T., Kashiwara, M. et al., (eds.), “Representation Theory of Algebraic Groups and Quantum Groups”, Adv. Stud. Pure Math. Vol 40, Tokyo: Math. Soc. Japan, 2004, pp. 321–341 Naito, S., Sagaki, D.: Crystal base elements of an extremal weight module fixed by a diagram automorphism. Algebr. Represent. Theory 8, 689–707 (2005) Naito, S., Sagaki, D.: Crystal of Lakshmibai-Seshadri paths associated to an integral weight of level zero for an affine Lie algebra. Int. Math. Res. Not. 2005, no. 14, 815–840 Okado, M., Schilling, A., Shimozono, M.: Virtual crystals and fermionic formulas of type (2) (2) (1) Dn+1 , A2n , and Cn . Represent. Theory 7, 101–163 (2003) Okado, M., Schilling, A., Shimozono, M.: Virtual crystals and Kleber’s algorithm. Commun. Math. Phys. 238, 187–209 (2003)

Communicated by L. Takhtajan


Communications in


On the Spectral Dynamics of the Deformation Tensor and New A Priori Estimates for the 3D Euler Equations Dongho Chae Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Korea. E-mail: [email protected] Received: 22 March 2005 / Accepted: 3 May 2005 Published online: 24 November 2005 – © Springer-Verlag 2005

Abstract: In this paper we study the dynamics of eigenvalues of the deformation tensor for solutions of the 3D incompressible Euler equations. Using the evolution equation of the L2 norm of spectra, we deduce new a priori estimates of the L2 norm of vorticity. As an immediate corollary of the estimate we obtain a new sufficient condition of L2 norm control of vorticity. We also obtain decay in time estimates of the ratios of the eigenvalues. In the remarks we discuss what these estimates suggest in the study of searching initial data leading to possible finite time singularities. We find that the dynamical behaviors of L2 norm of vorticity are controlled completely by the second largest eigenvalue of the deformation tensor.

1. Introduction We are concerned with the following Euler equations for the homogeneous incompressible fluid flows in ⊂ R3 . ∂v + (v · ∇)v = −∇p, ∂t div v = 0, v(x, 0) = v0 (x),

(1.1) (1.2) (1.3)

where v = (v1 , v2 , v3 ), vj = vj (x, t), j = 1, 2, 3, is the velocity of the flow, p = p(x, t) is the scalar pressure, and v0 is the given initial velocity, satisfying div v0 = 0. For simplicity of presentation we assume = T3 , the 3D periodic box. Most of the results in this paper, however, are valid also in the whole of R3 , or bounded domain with smooth Part of this work was done while the author was visiting CSCAMM, University of Maryland, USA. The author would like to thank to Professor E. Tadmor for his hospitality.

790

D. Chae

boundary, at least after obvious modifications. Given m ∈ N ∪ {0}, let H m (T3 ) be the standard Sobolev space on T3 , m 3 2 3 2 H (T ) = {f ∈ L (T ) | f H m = |D α f (x)|2 dx < ∞}, 3 |α|≤m T

where α = (α1 , α2 , α3 ) with |α| = α1 + α2 + α3 is the usual multi-index notation. We introduce the space of solenoidal vector fields, m 3 3 Hm σ = {u ∈ [H (T )] | div u = 0}.

Then, for v0 ∈ Hm σ with m > 5/2, the local in time unique existence of solution to (1.1)– (1.3), which belongs to C([0, T ]; Hm σ ) for some T = T (v0 H m ), was established in [17, 22]. This was later extended to the local existence in various other function spaces by many authors([18, 8, 9, 24, 25, 3–6]). The question of finite time blow-up/global regularity of such a locally constructed solution is one of the most outstanding open problems in mathematical fluid mechanics. For physical meaning and other significance of this problem as well as many instructive examples of solutions we refer to [10, 21]. For a mathematical or numerical test of the actual finite time blow-up of a given solution, it is important to have a good blow-up criterion. In this direction there is a celebrated result by Beale-Kato-Majda([2]), now called the BKM criterion. This criterion is later refined in [19, 6, 9], using refined versions of logarithmic Sobolev inequalities. As for another approach to the blow-up criterion, there is a pioneering work on the geometric type of blow-up criterion due to Constantin-Fefferman-Majda([13])(see also [10]), which was initiated in [12], and the idea of which was used in the recent works in [7, 15]. For a different type of geometric approach to the blow-up problem (considering vortex tube initial data) we refer to [14]. We also mention a recent interesting result in [1] for rotating flows, where they discovered that rotation has some sense of regularization effect. In this paper we study the regularity/blow-up problem, using the spectral dynamics of the deformation tensor for the solution of the Euler equations. Previous spectral approaches to the singularity problems in the nonlinear partial differential equations are studied in [20], however their study is not for the real Euler equations, but for its model equations, avoiding the difficulty of the nonlocality caused by the Riesz transform appearing in the equations when the pressure is eliminated. Moreover, their spectrum is for the matrices of the velocity gradient, not for the deformation tensor, which is the symmetric part of the velocity gradient. In the next section we derive an evolution equation, in the L2 sense, of the eigenvalues of the deformation tensor. From this equation we derive new a priori estimates for the L2 norm of vorticity. The inequality itself already tells us an interesting dynamical mechanism of compression and stretching of infinitesimal fluid volume elements leading to possible blow-up. The inequality immediately leads to a very simple and elegant sufficient condition of L2 norm control of vorticity of the 3D Euler equations. In Sect. 3 we consider special classes of initial data. For such initial data we can have better estimates of exponential growth/decay of the L2 norm of vorticity. We also deduce decay estimates in time of a ratio of eigenvalues of the deformation tensor. 2. Dynamics of Eigenvalues of the Deformation Tensor We use the following notations for matrix components: Vij =

∂vj , ∂xi

Sij =

Vij + Vj i , 2

Aij =

Vij − Vj i , 2

Spectral Dynamics of Deformation Tensor and Estimates for 3D Euler Equations

791

where i, j = 1, 2, 3. Then, obviously we have Vij = Sij + Aij . For a given 3D velocity field v(x, t), describing fluid motions, Sij is called the deformation tensor, while Aij is related to the vorticity field ω = curl v by 1 εij k ωk , 2 3

Aij =

k=1

where εij k , the skewsymmetric tensor with the normalization ε123 = 1. For an incom pressible fluid we have T r(S) = 3i=1 Sii = div v = 0. We now state the theorem on the evolutions of the eigenvalues of the deformation tensor associated with the solution of the Euler system (1.1)–(1.3). Theorem 2.1. Let λ1 (x, t), λ2 (x, t), λ3 (x, t) be the eigenvalues of the deformation tensor S = (Sij )3i,j =1 associated to the classical solution v(x, t) of (1.1)–(1.3). Then, the following equation holds: d (λ21 + λ22 + λ23 )dx = −4 λ1 λ2 λ3 dx. (2.1) dt T3 T3 Proof. We take the L2 inner product (1.1) with u, and integrate by part to derive 1 d 2 (v · ∇)v · vdx ∇vL2 = 2 dt T3 3 ∂vj ∂vk ∂vj =− dx 3 T ∂xk ∂xi ∂xi i,j,k=1

=− =− =−

3 3 i,j,k=1 T

3 3 i,j,k=1 T

3 3 i,j,k=1 T

Skj Vik Vij dx

Skj (Sik + Aik )(Sij + Aij )dx (Skj Aik Aij + Skj Sik Sij )dx

3 3 1 =− Skj δkj ωm ωm − ωj ωk dx 3 4 m=1 i,j,k=1 T − =

3 3 i,j,k=1 T

Skj Sik Sij dx

3 3 1 Sj k ωj ωk dx − Skj Sik Sij dx. 3 3 4 j,k=1 T i,j,k=1 T

(2.2)

Next, we consider the vorticity equation for the 3D Euler equations, ∂ω + (v · ∇)ω = (ω · ∇)v, ∂t

(2.3)

792

D. Chae

which is obtained from (1.1) by taking the curl(·) operation. Taking the L2 inner product (2.3) with ω, we obtain, after integration by part, 3 1 d ω2L2 = (ω · ∇)v · ωdx = Sj k ωj ωk dx. 3 2 dt T3 j,k=1 T Since we have the equality,

(2.4)

T3

|∇v|2 dx =

T3

|ω|2 dx,

(2.5)

from (2.2) and (2.4) we obtain 3 3 3 1 Sj k ωj ωk dx − Skj Sik Sij dx = Sj k ωj ωk dx. 3 3 3 4 j,k=1 T i,j,k=1 T j,k=1 T Hence, 3 j,k=1

3 4 Sj k ωj ωk dx = − Skj Sik Sij dx. 3 T3 T3

(2.6)

i,j,k=1

We also have the following pointwise equality, |∇v|2 =

3 j,k=1

=

3

Vj k V j k =

3 j,k=1

(Sj k + Aj k )(Sj k + Aj k )

j,k=1

1 + εj km εj kn ωm ωn 4 m,n 3

Sj k S j k

j,k=1

=

3

1 Sj k Sj k + |ω|2 . 2

(2.7)

Integrating (2.7) over T3 , and using (2.5), we obtain |ω| dx = 2 2

T3

3 3 j,k=1 T

Sj k Sj k dx = 2

T3

(λ21 + λ22 + λ23 )dx.

(2.8)

We also observe, 3

Skj Sik Sij = λ31 + λ32 + λ33 = 3λ1 λ2 λ3 ,

(2.9)

i,j,k=1

which follows from the following algebra, using λ1 + λ2 + λ3 = 0: 0 = (λ1 + λ2 + λ3 )3 = λ31 + λ32 + λ33 + 3λ21 (λ2 + λ3 ) + 3λ22 (λ1 + λ3 ) + 3λ3 (λ1 + λ2 ) + 6λ1 λ2 λ3 = λ31 + λ32 + λ33 − 3(λ31 + λ32 + λ33 ) + 6λ1 λ2 λ3 . Substituting (2.8) and (2.6), combined with (2.9), into (2.4), we have (2.1).


793

The following is a new a priori estimate for the L2 norm of vorticity for the 3D incompressible Euler equations. Theorem 2.2. Let v(t) ∈ C([0, T ); Hm σ ), m > 5/2 be the local classical solution of the 3D Euler equations with initial data v0 ∈ Hm σ . Let λ1 (x, t) ≥ λ2 (x, t) ≥ λ3 (x, t) be the ∂v ∂vi eigenvalues of the deformation tensor Sij (v) = 21 ( ∂xji + ∂x ). We denote λ+ 2 (x, t) = j

max{λ2 (x, t), 0}, and λ− 2 (x, t) = min{λ2 (x, t), 0}. Then, the following (a priori) estimates hold: t 1 + − ω0 L2 exp inf λ (x, t) − sup |λ2 (x, t)| dt ≤ ω(t)L2 2 x∈T3 2 0 x∈T3 t 1 + − sup λ2 (x, t) − inf |λ2 (x, t)| dt ≤ ω0 L2 exp 2 x∈T3 0 x∈T3 (2.10) for all t ∈ (0, T ). Remark 2.1. The above estimate says, for example, that if we have the following comparability conditions: − sup λ+ 2 (x, t) inf |λ2 (x, t)| g(t) x∈T3

x∈T3

for some time interval [0, T ], then t

ω(t)L2 O exp C g(s)ds

∀t ∈ [0, T ]

0

for some constant C. On the other hand, we note the following connection of the above result to the previous one. In [13] the authors obtained, D|ω| = α|ω|, Dt

α(x, t) =

ω · Sω , |ω|2

which implies immediately ω(t)L2 ≤ ω0 L2 exp

sup α(x, s)ds 0 x∈T3

≤ ω0 L2 exp

t

t

sup λ1 (x, s)dτ ,

0 x∈T3

where we used the fact λ3 ≤ α ≤ λ1 , the well-known estimate for the Rayleigh quotient. Remark 2.2. We note that λ+ 2 (x, t) > 0 implies we have stretching of infinitesimal fluid volume in two directions and compression in the other one direction (planar stretching) at (x, t), while |λ− 2 (x, t)| > 0 implies stretching in one direction and compressions in two directions (linear stretching). The above estimate says that the dominance competition between planar stretching and linear stretching is an important mechanism controlling the growth/decay in time of the L2 norm of vorticity.

794

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Proof of Theorem 2.2. Since λ1 + λ2 + λ3 = 0, and λ1 ≥ λ2 ≥ λ3 , we have λ1 ≥ 0, λ3 ≤ 0, and |λ2 | ≤ min{λ1 , |λ3 |}.

(2.11)

We first observe that from (2.8),

T3

|ω|2 dx = 2 =4 =4

3 T 3 T

T3

(λ21 + λ22 + λ23 )dx (λ21 + λ1 λ2 + λ22 )dx

(λ3 = −λ1 − λ2 )

(λ22 + λ2 λ3 + λ23 )dx

(λ1 = −λ2 − λ3 ).

(2.12)

We estimate the ‘vortex stretching term’ as λ1 λ2 λ3 dx = −4 λ+ λ λ dx − 4 λ− 2 1 3 2 λ1 λ3 dx 3 3 3 T T T =4 λ+ λ (λ + λ )dx − 4 |λ− 1 1 2 2 2 |(λ2 + λ3 )λ3 dx 3 3 T T 2 2 =4 λ+ |λ− 2 (λ1 + λ1 λ2 )dx − 2 2 |(2λ2 λ3 + 2λ3 )dx 3 3 T T ≤ 4 sup λ+ (x, t) (λ21 + λ1 λ2 + λ22 )dx 2

−4

x∈T3

T3

x∈T3

T3

(x, t)| (λ22 + λ2 λ3 + λ23 )dx −2 inf |λ− 2 3 x∈T3 T + (λ21 + λ1 λ2 + λ22 )dx = 4 sup λ2 (x, t) −2 inf |λ− 2 (x, t)| x∈T3

T3

(λ21 + λ1 λ2 + λ22 )dx,

(2.13)

where we used (2.11) and (2.12). This, combined with (2.1) and (2.12), yields d + − 2 |ω(x, t)| dx ≤ 2 sup λ2 (x, t) − inf |λ2 (x, t)| |ω(x, t)|2 dx. 3 3 dt T3 3 x∈ T T x∈T (2.14) Applying the Gronwall lemma, we have the second inequality of (2.10). In order to prove the first inequality of (2.10) we estimate from below starting from the equality part of (2.13):


−4

T3

λ1 λ2 λ3 dx = 4

3 T

λ+ 2 λ1 (λ1 + λ2 )dx − 4

T3

|λ− 2 |(λ2 + λ3 )λ3 dx

2 |λ− 2 |(λ2 λ3 + λ3 )dx 2 2 2 2 ≥2 λ+ (λ + λ λ + λ )dx − 4 |λ− 1 2 1 2 2 2 |(λ2 + λ2 λ3 + λ3 )dx 3 3 T T + 2 2 ≥ 2 inf λ2 (x, t) (λ1 + λ1 λ2 + λ2 )dx x∈T3 T3 (λ22 + λ2 λ3 + λ23 )dx, (2.15) −4 sup |λ− 2 (x, t)|

=2

3 T

2 λ+ 2 (2λ1

795

+ 2λ1 λ2 )dx − 4

T3

T3

x∈T3

where we used (2.11) again. Similarly to the above, combining this with (2.1) and (2.12), yields d − |ω(x, t)|2 dx ≥ inf λ+ |ω(x, t)|2 dx, 2 (x, t) − 2 sup |λ2 (x, t)| dt T3 x∈T3 T3 x∈T3 (2.16) and, applying the Gronwall lemma we finish the first inequality of (2.10).

Corollary 2.1. Let v0 ∈ Hm σ be given, and λ1 (x, t), λ2 (x, t), λ3 (x, t) are as in Theorem 2.2. Suppose lim sup ω(t)L2 = ∞.

(2.17)

t→T∗

Then, necessarily 0

T∗

∞ λ+ 2 (t)L dt = ∞.

(2.18)

Proof. We just observe that from (2.10), we have immediately t ∞ ω(t)L2 ≤ ω0 L2 exp λ+ (s) ds . L 2 0

This implies the corollary.

Remark 2.3. The above corollary says that if singularity happens in the L2 norm of vorticity, then it should be caused by the uncontrollable intensification of planar stretching. Remark 2.4. In the 3D Navier-Stokes equations the L2 norm singularity of vorticity is equivalent to that of any high norms in Sobolev space (see e.g. [23, 11]). Hence, the above corollary says that the regularity/singularity of the 3D Navier-Stokes equations t ∞ are controlled by the integral, 0 λ+ (s) ds. L 2 Remark 2.5. In [16] the author also investigated another sufficient condition for the singularity of the L2 norm of vorticity of the 3D Euler equations, using simultaneously the eigenvector and eigenvalues of the deformation tensor and the hessian of the pressure. Our condition is completely different from it, and has direct physical interpretation.

796

D. Chae

3. Applications for Some Classes of Initial Data In order to state our theorem in this section we introduce some definitions. Given a differentiable vector field f = (f1 , f2 , f3 ) on T3 , we denote by the scalar field λi (f ), i=1,2,3, the eigenvalues of the deformation tensor associated with f . Below we always assume the ordering, λ1 (f ) ≥ λ2 (f ) ≥ λ3 (f ). We also fix m > 5/2 below. We recall that if f ∈ Hm σ , then λ1 (f ) + λ2 (f ) + λ3 (f ) = 0, which is another representation of div f = 0. Let us begin with introduction of admissible classes A± defined by 3 A+ = {f ∈ Hm σ (T ) | inf λ2 (f )(x) > 0 }, x∈T3

and 3 A− = {f ∈ Hm σ (T ) | sup λ2 (f )(x) < 0 }. x∈T3

Physically A+ consists of solenoidal vector fields with planar stretching (see Remark 2.2) everywhere, while A− consists of everywhere linear stretching vector fields. Although they do not represent real physical flows, they might be useful in the study of searching initial data leading to finite time singularity for the 3D Euler equations. Given v0 ∈ Hm σ , let T∗ (v0 ) be the maximal time of unique existence of solution in m m Hσ for the system (1.1)–(1.3). Let St : Hm σ → Hσ be the solution operator, mapping from initial data to the solution v(t). Given f ∈ A+ , we define the first zero touching time of λ2 (f ) as T (f ) = inf{t ∈ (0, T∗ (v0 )) | ∃x ∈ T3 such that λ2 (St f )(x) < 0}. Similarly for f ∈ A− , we define T (f ) = inf{t ∈ (0, T∗ (v0 )) | ∃x ∈ T3 such that λ2 (St f )(x) > 0}. The following theorem is actually an immediate corollary of Theorem 2.2, combined with the above definition of A± and T (f ). We just observe that for v0 ∈ A+ (resp. A− ) + + − 3 we have λ− 2 = 0, λ2 = λ2 (resp. λ2 = 0, λ2 = λ2 ) on T × (0, T (v0 )). Theorem 3.1. Let v0 ∈ A± be given. We set λ1 (x, t) ≥ λ2 (x, t) ≥ λ3 (x, t) as the eigenvalues of the deformation tensor associated with v(x, t) = (St v0 )(x) defined t ∈ (0, T (v0 )). Then, for all t ∈ (0, T (v0 )) we have the following estimates: (i) If v0 ∈ A+ , then t 1 ω(t)L2 exp inf |λ2 (x, s)|ds ≤ 3 2 0 x∈T ω0 L2 t ≤ exp sup |λ2 (x, s)|ds .

(3.1)

0 x∈T3

(ii) If v0 ∈ A− , then exp −

t

sup |λ2 (x, s)|ds

0 x∈T3

ω(t)L2 ω0 L2 1 t ≤ exp − inf |λ2 (x, s)|ds . (3.2) 2 0 x∈T3

≤


797

Remark 3.1. If we have the comparability conditions, inf |λ2 (x, t)| sup |λ2 (x, t)| g(t)

x∈T3

∀t ∈ (0, T (v0 )),

x∈T3

which is the case for the sufficiently small box T3 , then we have   exp 0t g(s)ds ω(t)L2  exp − t g(s)ds ω0 L2 0

if

v0 ∈ A +

if

v0 ∈ A −

for t ∈ (0, T (v0 )). In particular, if we could find v0 ∈ A+ such that inf |λ2 (x, t)| O

x∈T3

1 t∗ − t

(3.3)

for time interval near t∗ , then such data would lead to singularity at t∗ . Below we have some decay in time estimates for some ratio of eigenvalues. Theorem 3.2. Let v0 ∈ A± be given, and we set λ1 (x, t) ≥ λ2 (x, t) ≥ λ3 (x, t) as in Theorem 3.1. We define ε(x, t) =

|λ2 (x, t)| λ(x, t)

∀(x, t) ∈ T3 × (0, T (v0 )),

(3.4)

where we set λ(x, t) =

λ1 (x, t) −λ3 (x, t)

if v0 ∈ A+ if v0 ∈ A− .

Then, there exists a constant C = C(v0 , ||), with || denoting the volume of the box = T3 , such that C ε(x, s) < √ t (x,s)∈T3 ×(0,t) inf

∀t ∈ (0, T (v0 )).

(3.5)

Remark 3.2. Regarding the problem of searching a finite time blowing up solution, again, the proof of the above theorem, in particular, the estimate (3.10) below, combined with Remark 2.3, suggests the following, : Given δ > 0, let us suppose we could find v0 ∈ A+ such that for the associated solution v(x, t) = (St v0 )(x) the estimate 1 inf , (3.6) ε(x, s) O 1 (x,s)∈T3 ×(0,t) t 2 +δ holds true, for sufficiently large time t. Then such v0 will lead to the finite time singularity. In order to check the behavior (3.6) for a given solution we need a sharper and/or localized version of Eq. (2.1) for the dynamics of eigenvalues of the deformation tensor.

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D. Chae

Proof of Theorem 3.2. We divide the proof into two separate cases. (i) The case v0 ∈ A+ . We parameterize the eigenvalues of the deformation tensor of the solution v(x, t) of (1.1)–(1.3) by λ1 (x, t) = λ(x, t) > 0, λ2 = ε(x, t)λ(x, t) > 0, λ3 (x, t) = −(1 + ε(x, t))λ(x, t) < 0 for all (x, t) ∈ T3 × (0, T ). We observe that 0 < ε(x, t) ≤ 1 ∀(x, t) ∈ T3 × [0, T (v0 )). Equation (2.1) can be written as d λ2 (ε 2 + ε + 1)dx = 2 λ3 (ε 2 + ε)dx dt T3 T3 From the estimate λ2 (ε 2 + ε + 1)dx =

2

λ2 (ε 2 + ε) 3

(3.7)

∀t ∈ (0, T (v0 )).

(3.8)

(ε 2 + ε + 1)

dx 2 (ε 2 + ε) 3

2

13 3 (ε 2 + ε + 1)3 3 2 ≤ λ (ε + ε)dx dx (ε 2 + ε)2 T3 T3

1

2 3 3 1 3 3 2 ≤ √ dx λ (ε + ε)dx , 3 4 4 T3 ε T3

T3

T3

where we used (3.7), we have

2 λ (ε + ε)dx ≥ √ 3 27 T 3

1 dx 4 3 ε T

2

− 1

3

2

λ (ε + ε + 1)dx 2

T3

2

2

dx,

which, combined with (3.8), yields

− 1

3 2 2 1 4 d 2 2 2 2 λ (ε + ε + 1)dx ≥ √ dx λ (ε + ε + 1)dx . (3.9) 4 dt T3 27 T3 ε T3 Setting

1

y(t) =

T3

2

λ (ε + ε + 1)dx 2

2

,

we have 2 dy ≥√ dt 27

1 dx 4 3 ε T

− 1 2

y2.

Solving the differential inequality, we have y(t) ≥ 1−

y0 t

2y0 √ 27 0

1 T3 ε4 dx

− 1 2

. ds


Since y 2 (t) = 21 ω(t)2L2 , we have just derived √ 2ω0 L2 ω(t)L2 ≥ − 1 √ 2ω t 2 2 − √0 L2 0 T3 ε14 dx ds

799

∀t ∈ [0, T (v0 )).

27

Since the denominator should be positive for all t ∈ [0, T (v0 )], we obtain that 2ω0 L2 √ 27

t 0

1 dx 4 T3 ε

− 1 2

ds
0,

λ2 = −ε(x, t)λ(x, t) > 0,

λ3 (x, t) = −λ(x, t) > 0, where as previously we have 0 < ε(x, t) ≤ 1 for all (x, t) ∈ T3 × (0, T (v0 )). Equation (2.1) can now be written as d λ2 (ε 2 + ε + 1)dx = −2 λ3 (ε 2 + ε)dx. (3.11) 3 dt T3 T Similarly to the above, we obtain

− 1

3 2 2 1 d 2 2 2 2 2 λ (ε + ε + 1)dx ≤ − √ dx λ (ε + ε + 1)dx . 4 dt T3 27 T3 ε T3

(3.12)

Hence, by a similar procedure to the previous case, we have √ 2ω0 L2 ω(t)L2 ≤ − 1 . √ 2ω0 L2 t 2 1 √ 2+ ds 0 T3 ε4 dx

(3.13)

27

Now we recall the helicity conservation (see e.g. [21]), H (t) = v(x, t) · ω(x, t)dx = v0 (x) · ω0 (x)dx = H0 , T3

T3

which implies H0 ≤ v(t)L2 ω(t)L2 = v0 L2 ω(t)L2 =

2E0 ω(t)L2 ,

where we used the energy conservation E(t) =

1 1 v(t)2L2 = v0 2L2 = E0 . 2 2

(3.14)

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D. Chae

Combining (3.13) with (3.14), we have H0 ≤ √ √ 2E0 from which we derive t 0

1 dx 4 T3 ε

√

2+

− 1 2

2ω0 L2 2ω0 L2 t 1 √

27

√ ds ≤ 27

0

T3 ε4 dx

− 1 2

, ds

√ E0 1 . −√ H0 2ω0 L2

Estimating from below the left hand side of (3.15), we deduce √ √ 1 E 1 0 t inf ε 2 (x, s) ≤ 27|| 2 −√ H0 (x,s)∈T3 ×[0,s] 2ω0 L2 for all t ∈ (0, T (v0 )). This finishes the proof of (3.5) for v0 ∈ A− .

(3.15)

(3.16)

Acknowledgement. This work was supported by Korea Research Foundation Grant KRF-2002-015CS0003.

References 1. Babin, A., Mahalov, A., Nicolaenko, B.: 3D Navier-Stokes and Euler equations with Initial Data Characterized by Uniformly Large Vorticity. Indiana Univ. Math. J. 50(1), 1–35 (2001) 2. Beale, J.T., Kato, T., Majda, A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94, 61–66 (1984) 3. Chae, D.: On the Well-Posedness of the Euler equations in the Besov and the Triebel-Lizorkin spaces. In: Proceedings of the Conference, “Tosio Kato’s Method and Principle for Evolution Equations in Mathematical Physics”, June 27–29, 2001 4. Chae, D.: On the Well-Posedness of the Euler equations in the Triebel-Lizorkin spaces. Commun. Pure Appl. Math. 55, 654–678 (2002) 5. Chae, D.: On the Euler equations in the critical Triebel-Lizorkin spaces. Arch. Rat. Mech. Anal. 170(3), 185–210 (2003) 6. Chae, D.: Local existence and blow-up criterion for the Euler equations in the Besov spaces. Asymp. Anal. 38(3–4), 339–358 (2004) 7. Chae, D.: Remarks on the blow-up criterion of the 3D Euler equations. Nonlinearity, 18, 1021–1029 (2005) 8. Chemin, J.-Y.: Régularité des trajectoires des particules d’un fluide incompressible remplissant l’espace. J. Math. Pures Appl. 71(5), 407–417 (1992) 9. Chemin, J.-Y.: Perfect incompressible fluids. Oxford: Clarendon Press, 1998 10. Constantin, P.: Geometric statistics in turbulence. SIAM Rev. 36, 73–98 (1994) 11. Constantin, P., Foias, C.: Navier-Stokes equations. Chicago: Univ. Chicago Press, 1988 12. Constantin, P., Fefferman, C.: Direction of vorticity and the problem of global regularity for the Navier-Stokes equations. Indiana Univ. Math. J. 42, 775–789 (1993) 13. Constantin, P., Fefferman, C., Majda, A.: Geometric constraints on potential singularity solutions for the 3D Euler equations. Commun. Partial Diff. Eqs. 21, 559–571 (1996) 14. Cordoba, D., Fefferman, C.: On the collapse of tubes carried by 3D incompressible flows. Commun. Math. Phys. 222(2), 293–298 (2001) 15. Deng, J., Hou, T.Y., Yu, X.: Geometric and nonblowup of 3D incompressible Euler flow. Commun. P.D.E 30, 225–243 (2005) 16. He, X.: A sufficient condition for a finite-time L2 singularity of the 3d Euler Equations. To appear in Math. Proc. Camb. Phil. Soc. (2005) 17. Kato, T.: Nonstationary flows of viscous and ideal fluids in R3 . J. Funct. Anal. 9, 296–305 (1972) 18. Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier-Stokes equations. Commun. Pure Appl. Math. 41, 891–907 (1988)


801

19. Kozono, H., Taniuchi, Y.: Limiting case of the Sobolev inequality in BMO, with applications to the Euler equations. Commun. Math. Phys. 214, 191–200 (2000) 20. Liu, H., Tadmor, E.: Spectral dyanamics of the velocity gradient field in restricted flows. Commun. Math. Phys. 228, 435–466 (2002) 21. Majda, A., Bertozzi, A.: Vorticity and incompressible flow. Cambridge: Cambridge Univ. Press, 2002 22. Temam, R.: On the Euler equations of incompressible flows. J. Funct. Anal. 20, 32–43 (1975) 23. Temam, R.: Navier-Stokes equations. 2nd ed., Amsterdam: North-Holland, 1986 24. Vishik, M.: Hydrodynamics in Besov spaces. Arch. Rat. Mech. Anal, 145, 197–214 (1998). 25. Vishik, M.: Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type. ´ Norm. Sup., 4e série, t. 32, 769–812 (1999) Ann. Scient. Ec. Communicated by P. Constantin


Communications in


Lower Bounds for an Integral Involving Fractional Laplacians and the Generalized Navier-Stokes Equations in Besov Spaces Jiahong Wu Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA. E-mail: [email protected] Received: 24 March 2005 / Accepted: 25 July 2005 Published online: 29 November 2005 – © Springer-Verlag 2005

Abstract: When estimating solutions of dissipative partial differential equations in Lp -related spaces, we often need lower bounds for an integral involving the dissipative term. If the dissipative term is given by the usual Laplacian −, lower bounds can be derived through integration by parts and embedding inequalities. However, when the Laplacian is replaced by the fractional Laplacian (−)α , the approach of integration by parts no longer applies. In this paper, we obtain lower bounds for the integral involving (−)α by combining pointwise inequalities for (−)α with Bernstein’s inequalities for fractional derivatives. As an application of these lower bounds, we establish the existence and uniqueness of solutions to the generalized Navier-Stokes equations in Besov spaces. The generalized Navier-Stokes equations are the equations resulting from replacing − in the Navier-Stokes equations by (−)α . 1. Introduction This paper is concerned with the generalized incompressible Navier-Stokes (GNS) equations ∂t u + u · ∇u + ∇P = −ν(−)α u,

∇ · u = 0,

(1.1)

where ν > 0 and α > 0 are real parameters, and the fractional Laplacian (−)α is defined in terms of the Fourier transform α u(ξ ) = (2π|ξ |)2α (−) u(ξ ).

We accomplish two major goals. First, we obtain lower bounds for the integral |f |p−2 f · (−)α f dx, D(f ) ≡ Rd

(1.2)

804

J. Wu

where p ≥ 2 and α ≥ 0. Second, we apply these lower bounds to establish the existence and uniqueness of solutions to the GNS equations in homogeneous Besov spaces. We shall now explain in some detail our major results together with background information necessary for understanding these results. When α = 1, the GNS equations (1.1) reduce to the usual Navier-Stokes equations. One advantage of working with the GNS equations is that they allow simultaneous consideration of their solutions corresponding to a range of α’s. For example, the 3-D GNS equations with any α ≥ 45 always possess global classical solutions [15]. For the general d-D GNS equations, we have shown in [17] that α ≥ 21 + d4 guarantees global regularity. In this paper, we consider the GNS equations with a general fraction α ≥ 0 and one of the difficulties is how to obtain a lower bound for D(f ) defined in (1.2). The quantity D(f ) arises very naturally in the process of bounding solutions of the GNS equations in Lp -related spaces. In the special case when α = 1, lower bounds for D(f ) are often derived through integrating by parts ([1, 2, 16]). However, for a general fraction α ≥ 0, (−)α is a nonlocal operator and this approach fails. In this paper, we establish lower bounds for D(f ) with a general fraction α ≥ 0 by combining the pointwise inequalities for (−)α and Bernstein’s inequalities for fractional derivatives. In [10] and [11], A. Córdoba and D. Córdoba showed that for any 0 ≤ α ≤ 1 and any f ∈ C 2 (Rd ) that decays sufficiently fast at infinity, the pointwise inequality 2 f (x) (−)α f (x) ≥ (−)α f 2 (x),

x ∈ Rd

(1.3)

holds. By modifying the proof in [10], N. Ju proved in [14] that if p ≥ 0 and f is, in addition, nonnegative, then (p + 1) f p (x) (−)α f (x) ≥ (−)α f p+1 (x). We obtain here the inequality in the general form (p1 + p2 ) f p1 (x) (−)α f p2 (x) ≥ p2 (−)α f p1 +p2 (x),

(1.4)

where p1 = kl11 and p2 = kl22 with l1 and l2 being odd and k1 l2 + k2 l1 being even (see Proposition 3.2 for more details). Another type of generalization of (1.3) was considered by P. Constantin, who established an identity for (−)α acting on the product of two functions. This identity allowed him to obtain a calculus inequality involving fractional derivatives [7]. In this paper, we combine suitable pointwise inequalities with Bernstein’s inequalities for fractional derivatives to derive several lower bounds for D(f ). In particular, we have p

D(f ) ≥ C 22αj f Lp ,

(1.5)

which is valid for any f ∈ C 2 (Rd ) satisfying supp f ⊂ {ξ ∈ Rd : K1 2j ≤ |ξ | ≤ K2 2j }. The precise statement of this result is provided in Theorem 3.4. As an application of these lower bounds, we study the solutions of the GNS equations r (Rd ) with general indices and establish several in the homogeneous Besov space B˚ p,q existence and uniqueness results. In particular, it is shown that the GNS equations posr (Rd ) with r = 1 − 2α + d for any initial datum sess a unique global solution in B˚ p,q p r (Rd ). This result holds for any 1 ≤ q ≤ ∞ and for u0 that is comparable to ν in B˚ p,q

Fractional Laplacians and Generalized Navier-Stokes Equations in Besov Spaces

805

either 21 < α and p = 2 or 21 < α ≤ 1 and 2 < p < ∞. We defer the exact statement to Theorem 6.1. The proof of this theorem is based on the contraction mapping principle and two major a priori inequalities. The first one states that any solution u of the GNS equations satisfies d q q q u ˚ r + C qνu r+ 2α ≤ C qu 1−2α+ pd u r+ 2α , Bp,q q q dt ˚ B ˚ p,q Bp,q B˚ p,q

(1.6)

where 0 < α ≤ 1, r ∈ R and 1 ≤ q < ∞. The inequality for the case q = ∞ is slightly different (see Theorem 4.1 for details). The second inequality d q q q q q q F ˚ r +C qνF r+ 2α ≤ C q v 1−2α+ d w r+ 2α +w 1−2α+ d v r+ 2α Bp,q q q q p p dt B˚ p,q B˚ p,q B˚ p,q B˚ p,q B˚ p,q (1.7) bounds solutions of the equation ∂t F + ν(−)α F = P(v · ∇w),

(1.8)

where P is the matrix operator projecting onto the divergence free vector fields. For r (Rd ), the local existence and uniqueness of arbitrarily large initial datum u0 ∈ B˚ p,q r (Rd ) is established. The proof of the local existence result requires solutions in B˚ p,q different a priori bounds and they are provided in Theorems 5.2 and 5.4 of Sect. 5. These results for the GNS equations together with those in [4, 8, 9, 17–20] contribute significantly to understanding how the general fractional dissipation effects the regularity of solutions to dissipative partial differential equations. The results of this paper have two important special consequences. First, the 3-D Navier-Stokes equations −1+ 3

have a unique global solution for any initial datum u0 comparable to ν in B˚ p,q p (R3 ), and a unique local solution for any large datum in this space, where 1 ≤ p < ∞ and −1+ 3

1 ≤ q ≤ ∞. Solutions of the 3-D Navier-Stokes equations in B˚ p,q p (R3 ) have previously been studied in [5] and [13]. Second, these existence and uniqueness results also hold for solutions of the GNS equations in the usual Sobolev spaces W˚ r,p (Rd ) with r reduces to W ˚ r,p when p = q. r = 1 − 2α + pd . This is because B˚ p,q The rest of this paper is divided into five sections. Section 2 provides the definition of the homogeneous Besov spaces and states Bernstein’s inequalities for both integer and fractional derivatives. Section 3 presents the general form of the pointwise inequality (1.4) and the lower bound (1.5). Section 4 derives the a priori bounds (1.6) and (1.7). Section 5 establishes a priori bounds for the GNS equations and for Eq. (1.8) in r ) and L r ). Section 6 proves the existence and uniqueness q ((0, T ); B˚ p,q Lq ((0, T ); B˚ p,q results. 2. Besov Spaces In this section, we provide the definition of the homogeneous Besov space and state the Bernstein inequalities for integer and fractional derivatives. We first fix some notation. Let S be the usual Schwarz class and S the space of tempered distributions. The Fourier transform f of a L1 -function f is given by

806

J. Wu

f(ξ ) =

Rd

f (x) e−2π i x·ξ dx.

(2.1)

For f ∈ S , the Fourier transform of f is obtained by (f, g) = (f, g) for any g ∈ S. The Fourier transform is a bounded linear bijection from S to S whose inverse is also bounded. For this reason, the fractional Laplacian (−)α with α ∈ R can be defined through its Fourier transform, namely, α f (ξ ) = (2π |ξ |)2α f(ξ ). (−) 1

For notational convenience, we sometimes write for (−) 2 . We use S0 to denote the following subset of S, γ φ(x)x dx = 0, |γ | = 0, 1, 2, · · · . S0 = φ ∈ S, Rd

Its dual S0 is given by S0 = S /S0⊥ = S /P, where P is the space of multinomials. In other words, two distributions in S are identified as the same in S0 if their difference is a multinomial. We now introduce a dyadic partition of Rd . For each j ∈ Z, we define Aj = {ξ ∈ Rd : 2j −1 < |ξ | < 2j +1 }. Now, we choose φ0 ∈ S(Rd ) such that supp φ0 = {ξ : 2−1 ≤ |ξ | ≤ 2}

and φ0 > 0 on A0 .

Then, we set φj (ξ ) = φ0 (2−j ξ ) and define j ∈ S by φ (ξ ) j (ξ ) = j .

j φj (ξ ) j and j satisfy It is clear that

0 (2−j ξ ), j (ξ ) =

j ⊂ Aj , supp

j (x) = 2j d 0 (2j x).

Furthermore, ∞ k=−∞

k (ξ ) =

1 if ξ ∈ Rd \ {0}, 0 if ξ = 0.


807

Thus, for a general function ψ ∈ S, we have ∞

k (ξ )ψ (ξ ) = ψ (ξ ) for ξ ∈ Rd \ {0}.

k=−∞

But, if ψ ∈ S0 , then ∞

k (ξ )ψ (ξ ) = ψ (ξ )

for all ξ ∈ Rd .

k=−∞

That is, for ψ ∈ S0 , ∞

k ∗ ψ = ψ

k=−∞

and hence ∞

k ∗ f = f

(2.2)

k=−∞

in the weak* topology of S0 for any f ∈ S0 . To define the homogeneous Besov space, we set j f = j ∗ f,

j = 0, ±1, ±2, · · · .

(2.3)

s if f ∈ S and For s ∈ R and p, q ∈ [1, ∞], we say that f ∈ B˚ p,q 0 ∞

2j s j f Lp

q

0 and integer j , then α f Lq ≤ C 2

j α+j d( p1 − q1 )

f Lp .

2) If supp f ⊂ {ξ ∈ Rd : K1 2j ≤ |ξ | ≤ K2 2j } for some K1 , K2 > 0 and integer j , then C2

j α+j d( p1 − q1 )

1

1

j α+j d( p − q ) f Lp ≤ α f Lq ≤ C˜ 2 f Lp .

Proposition 2.3 is a simple extension of Proposition 2.2. I am indebted to David Ullrich who communicated Proposition 2.3 to me. We remark that a lemma in Danchin’ work [12, p. 632] implies a special case of 2) of Proposition 2.3, which states that for s > 0 and for any even integer p, f satisfying supp f ⊂ {ξ ∈ Rd : K1 2j ≤ |ξ | ≤ K2 2j } implies p

p

s (f 2 )L2 ≥ C 2sj f 2 L2 . We now point out several simple facts concerning the operators j : j k = 0, if |j − k| ≥ 2;

(2.4)


Sj ≡

j −1

k → I, as j → ∞;

809

(2.5)

k=−∞

j (Sk−1 f k f ) = 0, if |j − k| ≥ 3.

(2.6)

I in (2.5) denotes the identity operator and (2.5) is simply another way of writing (2.2). Finally, we provide here two elementary inequalities involving summations over infinite terms: Hölder’ inequality and Minkowski’s inequality. Lemma 2.4. Let p ∈ [1, ∞] and

1 p

+

1 q

= 1. If {ak } ∈ l p and {bk } ∈ l q , then

∞ ∞ 1 ∞ 1 p q p q ≤ a b |a | |b | . k k k k k=1

k=1

(2.7)

k=1

Lemma 2.5. Let p ∈ [1, ∞]. If ak ≥ 0 and bk ≥ 0 for k = 1, 2, 3, . . . , then ∞ k=1

1

p

(ak + bk )p

≤

∞ k=1

1

p

p ak

+

∞

1

p

p bk

.

k=1

3. Lower Bounds In this section, we extend the pointwise inequality of Córdoba and Córdoba to a more general form and prove the lower bound (1.5) for D(f ). The extended pointwise inequality is given in Proposition 3.2 and the lower bound in Theorem 3.4. In [10] and [11], Córdoba and Córdoba proved the following pointwise inequality. Proposition 3.1. Let 0 ≤ α ≤ 1 and assume f ∈ C 2 (Rd ) decays sufficiently fast at infinity. Then, for any x ∈ Rd , 2 f (x) (−)α f (x) ≥ (−)α f 2 (x). As they pointed out, the condition that f ∈ C 2 (Rd ) can be weakened. This proposition actually holds for any f such that f , (−)α f and (−)α f 2 are defined, and are, respectively, the limits of fm , (−)α fm and (−)α fm2 for fm ∈ C 2 (Rd ) that decays sufficiently fast at infinity. For later applications, we extend the inequality in Proposition 3.1 to a general form stated in the following proposition. Proposition 3.2. Let 0 ≤ α ≤ 1. Let p1 = kl11 ≥ 0 and p2 = kl22 ≥ 1 be rational numbers with l1 and l2 being odd, and with k1 l2 + k2 l1 being even. Then, for any x ∈ Rd and any function f ∈ C 2 (Rd ) that decays sufficiently fast at infinity, (p1 + p2 ) f p1 (x) (−)α f p2 (x) ≥ p2 (−)α f p1 +p2 (x).

(3.1)

We make several remarks. First, this theorem also applies to functions that are not necessarily in C 2 (Rd ). The estimate still holds if f , (−)α f p2 and (−)α f p1 +p2 are p p +p defined, and are respective limits of fm , (−)α fm 2 and (−)α fm 1 2 for a sequence fm ∈ C 2 (Rd ) that decays sufficiently fast at infinity. Secondly, the condition that k1 l2 + k2 l1 is even can not be removed.

810

J. Wu

Proof. When α = 0 or α = 1, (3.1) can be directly verified. When p1 and p2 are integers, (3.1) can be proven by slightly modifying the proof of Córdoba and Córdoba for Proposition 3.1. As shown in [11], (−)α f can be represented as the integral f (x) − f (y) dy, x ∈ Rd , (−)α f (x) = Cα P.V. d+2α Rd |x − y| where Cα is a constant depending on α only and P.V. means the principal value. Therefore, f p1 (x) (−)α f p2 (x) f (x)p1 +p2 − f (x)p1 f (y)p2 = Cα P.V. dy |x − y|d+2α Rd p1 f (x)p1 +p2 − (p1 + p2 )f (x)p1 f (y)p2 + p2 f (y)p1 +p2 = Cα P.V. dy (p1 + p2 )|x − y|d+2α Rd p2 f (x)p1 +p2 − f (y)p1 +p2 + Cα P.V. dy. (3.2) p1 + p 2 |x − y|d+2α Rd When k1 l2 + k2 l1 is even, p1 + p2 is even and we have Young’s inequality p1 f (x)p1 +p2 − (p1 + p2 )f (x)p1 f (y)p2 + p2 f (y)p1 +p2 ≥ 0. Consequently, the first integral on the right of (3.2) is nonnegative. The second integral is 2 (−)α f p1 +p2 (x). Therefore, (3.2) implies simply the integral representation of p1p+p 2 (3.1). Now, we consider the case when p1 = kl11 ≥ 0 and p2 = kl22 ≥ 1 are rational num1

bers. For notational convenience, we write F (x) = f (x) l1 l2 . Since l1 and l2 are odd, f ∈ C02 (Rd ) implies F ∈ C02 (Rd ). Because (3.1) has been shown for integers, we have f p1 (x) (−)α f p2 (x) = F k1 l2 (x) (−)α F k2 l1 (x) k2 l1 (−)α F k1 l2 +k2 l1 (x) ≥ k1 l2 + k2 l1 p2 = (−)α f p1 +p2 (x). p1 + p 2 That is, (3.1) holds in this case. This completes the proof of Proposition 3.2. If we are willing to assume the function f ≥ 0, then the assumption on the indices p1 and p2 can be reduced. The following proposition is due to N. Ju [14]. Proposition 3.3. Let 0 ≤ α ≤ 1 and let p ≥ 0. Then, for any f ∈ C 2 (Rd ) that decays sufficiently fast at infinity, f ≥ 0, and any x ∈ Rd , (p + 1) f p (x) (−)α f (x) ≥ (−)α f p+1 (x). We now derive the lower bound (1.5) for D(f ). Theorem 3.4. Assume either 0 ≤ α and p = 2 or 0 ≤ α ≤ 1 and 2 0 and some integer j , then p D(f ) ≡ |f |p−2 f · (−)α f dx ≥ C 22αj f Lp (Rd ) , Rd

811

(3.4)

where C is a constant depending on d, p, K1 and K2 only. The restriction that 0 ≤ α ≤ 1 comes from applying the pointwise inequalities. When p = 2, (3.4) is a direct consequence of Plancherel’s theorem and thus α is not required to satisfy α ≤ 1. We also remark that the requirement f ∈ C 2 (Rd ) can be reduced. In fact, this theorem applied to any function f with the property that f and (−)α f j (j = 1, 2, . . . , p2 ) j are defined, and are, respectively, limits of fm and (−)α fm with each fm ∈ C 2 (Rd ). This also explains why we do not assume the functions are in C 2 (Rd ) when we apply this theorem in the subsequent sections. Proof. When p = 2, the lower bound in (3.4) is a direct consequence of Plancherel’s theorem. When p > 2, Proposition 3.3 implies p α p2 2 |f |p−2 f · (−)α f dx ≥ C D(f ) ≡ (f ) dx = C α (f 2 )2L2 . Rd

Rd

It then follows from 2) of Proposition 2.3 that p

p

D(f ) ≥ C 22αj f 2 2L2 ≥ C 22αj f Lp . Other useful lower bounds for D(f ) can also be established using the pointwise inequality in Proposition 3.3. These lower bounds which do not require the support of f satisfy the condition (3.3). Theorem 3.5. Assume either 0 ≤ α and p = 2 or 0 ≤ α ≤ 1 and 2 < p < ∞. Then D(f ) can be bounded as follows. (a) If p = 2 and 2α = d, then D(f ) ≥ C f 2Lq (Rd )

(3.5)

for any q ∈ [2, ∞) and some constant depending on q only. (b) If p = 2 and 2α < d, then D(f ) ≥ C f 2

2d

L d−2α (Rd )

for some constant C depending on α and d only. (c) If p > 2 and 2α = d, then, for any f ∈ L2 (Rd ), (1+β)p

−β p

D(f ) ≥ C f Lp (Rd ) f L2 (Rd ) , 2q−4 where β = pq−2q for any q ∈ [p, ∞), and C is a constant depending on p and q only. (d) If p > 2 and 2α < d, then for any f ∈ L2 (Rd ), (1+γ )p

−γ p

D(f ) ≥ C f Lp (Rd ) f L2 (Rd ) , where γ =

4α d(p−2)

and C is a constant depending on d, α and p only.

The proof of this theorem is left to the appendix.

812

J. Wu

r 4. A Priori Estimates in ˚ Bp,q r : one for solutions of the GNS equations This section derives two a priori bounds in B˚ p,q

∂t u + u · ∇u + ∇P = −ν(−)α u, ∇ · u = 0,

x ∈ Rd ,

x ∈ Rd ,

t > 0,

t > 0,

(4.2)

x ∈ Rd ,

u(x, 0) = u0 (x),

(4.1)

(4.3)

and one for solutions of the equation ∂t F + ν(−)α F = P(v · ∇w),

x ∈ Rd ,

t > 0,

x ∈ Rd ,

F (x, 0) = F0 (x),

(4.4) (4.5)

where P = I − ∇−1 ∇· is the matrix operator projecting onto divergence free vector fields and I is the identity matrix. The bounds are given in Theorems 4.1 and 4.3. We work with the following form of the GNS equations: ∂t u + ν(−)α u = −P(u · ∇u),

(4.6)

which is equivalent to (4.1) and (4.2). This form of the GNS equations can be seen as a special case of (4.4). Before stating and proving the theorems, we first briefly introduce s ) and L s ). ρ ((a, b); B˚ p,q the spaces Lρ ((a, b); B˚ p,q s ) is defined in the standard For ρ, p, q ∈ [1, ∞] and s, a, b ∈ R, Lρ ((a, b); B˚ p,q s ) denotes the space of Lρ -integrable functions from fashion. That is, Lρ ((a, b); B˚ p,q s s ) was introduced in [6] and later used in [5] ρ ((a, b); B˚ p,q (a, b) to B˚ p,q . The space L s ) if ρ ((a, b); B˚ p,q and [13]. We say that f ∈ L {µj } ∈ l q , where µj is defined by µj = 2

b

js a

ρ j f (·, t)Lp dt

µj = 2j s sup j f (·, t)Lp

ρ1

for ρ < ∞,

for ρ = ∞.

t∈(a,b) s ) is given by ρ ((a, b); B˚ p,q The norm in L

f Lρ ((a,b);B˚ s

p,q )

= µj l q .

ρ (B˚ s ) for L s ). ρ ((0, t); B˚ p,q For notational convenience, we sometimes write L t p,q We now state the first major theorem of this section.


813

Theorem 4.1. Let r ∈ R and 1 ≤ q ≤ ∞. Assume either 0 < α and p = 2 or 0 < α ≤ 1 and 2 < p < ∞. Then any solution u of the GNS equations (4.1) and (4.2) or of (4.6) satisfies d q q q u ˚ r + C qνu r+ 2α ≤ C qu 1−2α+ pd u r+ 2α for q ∈ [1, ∞) Bp,q q q dt ˚ Bp,q B˚ p,q B˚ p,q

(4.7)

and, for q = ∞, u(·, t)B˚ r

p,∞

+CνuL1 (B˚ r+2α ) ≤ u0 B˚ r t

p,∞

p,∞

+C sup u(·, τ ) 0≤τ ≤t

d 1−2α+ p

B˚ p,∞

uL1 (B˚ r+2α ) , t

p,∞

(4.8) where C’s are constants depending on d, α, r and p only. The inequality in (4.7) contains rich information about solutions of the GNS equa1−2α+ d

tions. For example, if we know u is comparable to ν in B˚ p,q p , then u is bounded r for any r ∈ R. In particular, when r = 1 − 2α + d , (4.7) becomes a “closed” in B˚ p,q p inequality. We state the result in this special case as a corollary. Corollary 4.2. Let 1 ≤ q ≤ ∞. Assume either 0 < α and p = 2 or 0 < α ≤ 1 and 2 < p < ∞. Then any solution u of (4.1) and (4.2) or of (4.6) satisfies d q q q u 1−2α+ d +C qνu 1−2α+ d + 2α ≤ C qu 1−2α+ pd u 1−2α+ d + 2α for q ∈ [1, ∞) p p q p q dt ˚ B ˚ ˚ p,q Bp,q Bp,q B˚ p,q and, for q = ∞, u(·, t)

d 1−2α+ p

B˚ p,∞

+ Cνu

d

1 (B˚ 1+ p ) L p,∞ t

≤ u0

d 1−2α+ p

B˚ p,∞

+C sup u(·, τ ) 0≤τ ≤t

d 1−2α+ p

B˚ p,∞

u

d

1 (B˚ 1+ p ) L p,∞ t

,

where C’s are constants depending on d, α and p only. A special indication of this inequality is that any solution of the GNS equations will remain in the Besov space 1−2α+ d

B˚ p,q p for all time if the corresponding initial datum u0 is in this Besov space and is comparable to ν. Proof of Theorem 4.1. Let j ∈ Z. Applying j to (4.6) yields ∂t j u + (u · ∇)j u + ν(−)α j u = −[Pj , u · ∇]u, where the brackets represent the commutator operator, namely [Pj , u · ∇]u ≡ Pj (u · ∇u) − u · ∇ Pj u. We then dot both sides of (4.9) by p|j u|p−2 j u and integrate over Rd . Since (u · ∇)j u · |j u|p−2 j u dx = 0, Rd

(4.9)

814

we obtain

J. Wu

d p |j u|p−2 j u · (−)α j udx j uLp + p ν d dt R = −p |j u|p−2 j u · [Pj , u · ∇]u dx. Rd

(4.10)

According to Theorem 3.4, the dissipative part admits the following lower bound: p |j u|p−2 j u · (−)α j udx ≥ C ν22αj j uLp . (4.11) pν Rd

We now estimate the nonlinear part |j u|p−2 j u · [Pj , u · ∇]u dx. I ≡ −p Rd

By Hölder’s inequality, p−1

|I | ≤ C j uLp [Pj , u · ∇]uLp . To estimate [Pj , u · ∇]uLp , we use Bony’s notion of paraproduct to write [Pj , u · ∇]u = I1 + I2 + I3 + I4 + I5 , where I1 =

(4.12)

Pj ((Sk−1 u · ∇)k u) − (Sk−1 u · ∇)Pj k u,

k∈Z

I2 =

Pj ((k u · ∇)Sk−1 u),

k∈Z

I3 = −

(k u · ∇)j Sk−1 u, k∈Z

I4 =

Pj ((k u · ∇)l u),

|k−l|≤1

I5 = −

(k u · ∇)j l u.

|k−l|≤1

We now estimate the Lp -norms of these terms. According to (2.6), the summation ˜ j be the convolution kernel in I1 is only over those k satisfying |k − j | ≤ 2. Let

associated with the operator Pj . Since each entry in P is the difference between 1 and ˜ j is smooth. Using

˜ j , we write a product of two Riesz transforms,

˜ j (x − y) (Sk−1 u(y) − Sk−1 u(x)) · ∇k u(y) dy. I1 =

d |k−j |≤2 R

We integrate by parts and use the fact that ∇ · u = 0 to obtain ˜ j (x − y) · (Sk−1 u(y) − Sk−1 u(x)) k u dy. ∇

I1 = − d |k−j |≤2 R


By Young’s inequality,

I1 Lp ≤ C

∇Sk−1 uL∞ k uLp

|k−j |≤2

=C

Rd

815

˜ j (x)| dx |x||∇

∇Sk−1 uL∞ k uLp .

(4.13)

|k−j |≤2

Similarly, the summations in I2 and I3 are also only over k satisfying |k − j | ≤ 2. Applying Young’s inequality and using the fact that the norm of the operator P in Lp with 1 < p < ∞ is 1 yield I2 ||Lp ≤ C ∇Sk−1 uL∞ k uLp , (4.14) |k−j |≤2

I3 Lp ≤ C

k uLp j ∇Sk−1 uL∞

|k−j |≤2

≤C

k uLp ∇Sk−1 uL∞ .

(4.15)

|k−j |≤2

The summation in I4 is over k with |k − j | ≤ 3. The Lp norm of I4 is bounded by I4 Lp ≤ C k uLp ∇l uL∞ . (4.16) |k−j |≤3,|k−l|≤1

The estimate of I5 is similar, namely, I5 Lp ≤ C

k uLp ∇l uL∞ .

(4.17)

|k−j |≤2,|k−l|≤1

Collecting the estimates (4.13), (4.14), (4.15), (4.16) and (4.17), we find that [j , u · ∇]uLp ≤ C ∇Sk−1 uL∞ k uLp |k−j |≤2

+C

k uLp ∇l uL∞ .

(4.18)

|k−j |≤3,|k−l|≤1

We emphasize that the summations in (4.18) are only over a finite number of k s. It suffices to consider the term with k = j in the first summation and the term with k = l = j in the second summation. By Bernstein’s inequality, ∇j uL∞ ≤ C 2 ∇Sj −1 uL∞ ≤

(1+ pd )j

j uLp ,

∇m uL∞ ≤ C

m<j −1

2

(1+ pd )m

m uLp .

m<j −1

Therefore, [j , u · ∇]uLp ≤ C j uLp

m<j −1

2

(1+ pd )m

m uLp + C 2

(1+ pd )j

j u2Lp .

816

J. Wu

Consequently, we obtain the following bound for the nonlinear term in (4.10):

p

|I | ≤ Cj uLp

2

(1+ pd )m

m uLp + C 2

(1+ pd )j

p+1

j uLp .

(4.19)

m<j −1

Combining (4.10), (4.11) and (4.19), we find d p p j uLp + C ν22αj j uLp dt (1+ d )m p p+1 (1+ d )j ≤ Cj uLp 2 p m uLp + C 2 p j uLp .

(4.20)

m<j −1

Equivalently, d j uLp + C ν22αj j uLp dt (1+ d )m (1+ d )j ≤ Cj uLp 2 p m uLp + C 2 p j u2Lp .

(4.21)

m<j −1 q−1

For 1 ≤ q < ∞, we multiply both sides by q2rj q j uLp and sum over j ∈ Z to get d q q u ˚ r + C qνu r+ 2α ≤ J1 + J2 , Bp,q q dt B˚ p,q

(4.22)

where J1 and J2 are given by J1 ≡ C q

q

2rj q j uLp

2

(1+ pd )m

m uLp ,

m<j −1

j

J2 ≡ C q

2

(1+ pd )j

q

j uLp 2rj q j uLp .

j

To estimate J1 , we write J1 = C q

2

(r+ 2α q )j q

q

j uLp

2

(1+ pd −2α)m

m uLp 22α(m−j ) .

m<j −1

j

Applying the elementary inequality (2.7), we have m<j −1

2

(1+ pd −2α)m



≤

m uLp 22α(m−j ) 1 

q

2

(1+ pd −2α)mq

m<j −1

= C u

d 1−2α+ p

B˚ p,q

q m uLp 



1 q¯

2

2α(m−j )q¯ 

m<j −1

,

(4.23)


where q¯ is the conjugate of q, namely

+

1 q

J1 ≤ C qu

1 q¯

817

= 1. Therefore, J1 is bounded by

d 1−2α+ p

B˚ p,q

u

q r+ 2α

.

q B˚ p,q

To bound J2 , we first write (1−2α+ d )j (r+ 2α p q )j q uq . J2 = C q 2 j uLp 2 j Lp j

It is clear that 1

 2

(1−2α+ pd )j

j uLp ≤ 

q

2

(1−2α+ pd )j q

q j uLp 

= u

j

d 1−2α+ p

B˚ p,q

. (4.24)

Therefore, J2 is bounded by J2 ≤ C qu

d 1−2α+ p

B˚ p,q

u

q r+ 2α

.

q B˚ p,q

Finally, we insert the estimates for J1 and J2 in (4.22) to obtain (4.7). We now deal with the case when q = ∞. We multiply (4.21) by 2rj , integrate over (0, t) and take the supremum over j ∈ Z to get u(·, t)B˚ r

p,∞

+ C νuL1 (B˚ r+2α ) ≤ u0 B˚ r t

p,∞

p,∞

+ J1 + J2 ,

(4.25)

where J1 and J2 are given by t (1+ d )m 2j r j uLp 2 p m uLp dτ, J1 = C sup j

0

m<j −1

J2 = C sup j

t

2j r 2

(1+ pd )j

0

j u2Lp dτ.

To estimate J1 , we rearrange its terms as t (1−2α+ pd )m 2(r+2α)j j uLp 22α(m−j ) 2 m uLp dτ. J1 = C sup 0

j

m<j −1

It is then clear that t 2(r+2α)j j uLp dτ J1 ≤ C sup j

0

× sup sup

j 0≤τ ≤t m<j −1 t (r+2α)j

≤ C sup j

2

22α(m−j ) 2

(1−2α+ pd )m

j uLp dτ sup sup 2 0≤τ ≤t j

0

= C uL1 (B˚ r+2α ) sup u(·, τ ) t

p,∞

m u(·, τ )Lp

0≤τ ≤t

d 1−2α+ p

B˚ p,∞

.

(1−2α+ pd )j

j u(·, τ )Lp (4.26)

818

J. Wu

J2 can be bounded as follows. t (1−2α+ pd )j J2 = C sup 2 j uLp 2(r+2α)j j uLp dτ j

0

≤ C sup sup 2

(1−2α+ pd )j

j 0≤τ ≤t

0≤τ ≤t

d 1−2α+ p

B˚ p,∞

2(r+2α)j j uLp dτ

0

j

≤ C sup u(·, τ )

t

j uLp sup uL1 (B˚ r+2α ) . t

(4.27)

p,∞

Combining the estimates in (4.25), (4.26) and (4.27) yields (4.8). This completes the proof of Theorem 4.1. We now present the priori bound for solutions of the general equation (4.4). This type of estimates will be needed in Sect. 6 when we study the existence and uniqueness of the solutions to the GNS equations. To establish the estimate, we need to restrict to α > 21 . Theorem 4.3. Let r ∈ R and q ∈ [1, ∞]. Assume either 21 < α and p = 2 or 21 < α ≤ 1 and 2 < p < ∞. Assume that v and w are in the class r+ 2α

r ) ∩ Lq ([0, T ); B˚ p,q q ) L∞ ([0, T ); B˚ p,q

for 0 < T ≤ ∞. Then any solution F of (4.4) satisfies d q q F ˚ r + C qνF r+ 2α B q dt p,q B˚ p,q ≤ C q v

q d 1−2α+ p

B˚ p,q

w

q r+ 2α

q B˚ p,q

+ w

q d 1−2α+ p

B˚ p,q

v

q r+ 2α

,

(4.28)

q B˚ p,q

where C is a constant depending on d and p only. For the sake of conciseness, we did not include in this theorem the inequality for the case when q = ∞. It can be stated and derived analogously as (4.28). Proof. We apply j to (4.4) and then multiply by p|j F |p−2 j F . Bounding the dissipative part by the lower bound as in (4.11) and estimating the right-hand side by Hölder’s inequality, we obtain d p p p−1 j F Lp + C pν22αj j F Lp ≤ C pj (v · ∇)wLp j F Lp . dt That is, d j F Lp + C ν22αj j F Lp ≤ C j (v · ∇)wLp . dt To estimate the term on the right, we write j (v · ∇)w = K1 + K2 + K3 ,

(4.29)


819

where K1 , K2 and K3 are given by K1 = j ((Sk−1 v · ∇)k w) , k

K2 =

j ((k v · ∇)Sk−1 w) ,

k

K3 =

j ((k v · ∇)l w) .

|k−l|≤1

To estimate these terms, we first notice that the summations in K1 , K2 and K3 are only over k satisfying |k − j | ≤ 2. Therefore, it suffices to estimate the representative term with k = j . Applying Young’s inequality and Bernstein’s inequality, we obtain K1 Lp ≤ CSj −1 vL∞ ∇j wLp ≤ C m vL∞ 2j j wLp m<j −1

≤C

2

d pm

m vLp 2j j wLp ,

m<j −1

K2 Lp ≤ C ∇Sj −1 wL∞ j vLp ≤ C

2

(1+ pd )m

m wLp j vLp ,

m<j −1

K3 Lp ≤ C j vLp ∇j wL∞ ≤ C 2

(1+ pd )j

j vLp j wLp . q−1

Inserting these estimates in (4.29), then multiplying by q 2rj q j F Lp and summing over all j , we obtain d q q F ˚ r + C qν F r+ 2α ≤ L1 + L2 + L3 , Bp,q q dt B˚ p,q where L1 , L2 and L3 are given by dm q−1 2rj q j F Lp 2j j wLp 2 p m vLp , L1 = C q m<j −1

j

L2 = C q

q−1 2rj q j F Lp j vLp

L3 = C q

2

(1+ pd )m

m wLp ,

m<j −1

j

2

rj q

q−1 (1+ d )j j F Lp 2 p j vLp j wLp .

j

To bound L1 , we write L1 = C q

2

(r+ 2α q )j (q−1)

q−1

j F Lp 2

(r+ 2α q )j

j wLp

j

×

m<j −1

2

(1+ pd −2α)m

m vLp 2(2α−1)(m−j ) .

(4.30)

820

J. Wu

When α > 21 , (2α − 1)(m − j ) < 0 and we obtain as in (4.23)

2

(1+ pd −2α)m

m vLp 2(2α−1)(m−j ) ≤ C v

d 1−2α+ p

B˚ p,q

m<j −1

.

After applying the elementary inequality (2.7), we find L1 ≤ C q v

d 1−2α+ p

B˚ p,q

w

r+ 2α

q B˚ p,q

F

q−1 r+ 2α

.

q B˚ p,q

L2 is bounded similarly, but we do not need α > 21 , L2 ≤ C qw

d 1−2α+ p

B˚ p,q

v

r+ 2α

q B˚ p,q

F

q−1 r+ 2α

.

q B˚ p,q

To bound L3 , we write (r+ 2α )j (q−1) d q−1 (r+ 2α q q )j w p 2(1+ p −2α)j v p . L3 = C q 2 j F Lp 2 j L j L j

Estimating 2

(1+ pd −2α)j

j vLp as in (4.24), we obtain

L3 ≤ C qv

w

d 1−2α+ p

B˚ p,q

r+ 2α

q B˚ p,q

F

q−1 r+ 2α

.

q B˚ p,q

Combining these estimates for L1 , L2 and L3 with (4.30), we obtain d q q F ˚ r + C qν F r+ 2α Bp,q q dt B˚ p,q ≤ C q v

d 1−2α+ p

B˚ p,q

w

+ w

r+ 2α

q B˚ p,q

d 1−2α+ p

B˚ p,q

v

r+ 2α

q B˚ p,q

F

q−1 r+ 2α

.

q B˚ p,q

Applying Young’s inequality then yields (4.28). s ) and in L s ) q ((0, T ); ˚ 5. A Priori Estimates in Lq ((0, T ); ˚ Bp,q Bp,q

In this section, we establish a priori estimates for solutions of the GNS equations and for s ) and L s ). The q ((0, T ); B˚ p,q those of (4.4) in two different type of spaces: Lq ((0, T ); B˚ p,q major results are stated in Theorems 5.1, 5.2, 5.3 and 5.4. We will need these estimates when we study solutions of the GNS equations in the next section. The first theorem provides an a priori estimate for solutions of the GNS equations in r+ 2α

Lq ([0, T ); B˚ p,q q ). The derivation of this bound requires q > 2. Theorem 5.1. Let 2 < q ≤ ∞. Assume either 0 < α and p = 2 or 0 < α ≤ 1 and ˚r 2 0, A(T ) ≤ C ν −1

(1 − Ej (qT )) 2rj q j u0 Lp + C ν −(q−2) q

j

T

A2 (t) dt, (5.1)

0

where Ej (t) ≡ exp(−C ν22αj t) and C’s are constants depending on d, p and q. In particular, A(T ) ≤ C ν −1 u0 ˚ r + C ν −(q−2) q Bp,q

T

A2 (t) dt.

0

Proof. As derived in the proof of Theorem 4.1, u satisfies (4.21), namely, d j uLp + C ν22αj j uLp dt (1+ d )m (1+ d )j ≤ Cj uLp 2 p m uLp + C 2 p j u2Lp . m≤j −1

We first convert it into the following integral form: t (1+ d )j Ej (t − s) 2 p j u(·, s)2Lp ds j u(·, t)Lp ≤ CEj (t)j u0 Lp + C 0 t (1+ d )m +C Ej (t − s) j u(·, s)Lp 2 p m u(·, s)Lp ds, 0

m<j −1 (r+ 2α )j

q where Ej (t) ≡ exp(−C ν22αj t). Multiplying both sides by 2 , raising them to th the q power, summing over all j and integrating over (0, T ), we obtain

u

q

r+ 2α q

≤ M1 + M 2 + M 3 ,

(5.2)

Lq (0,T );B˚ p,q

where M1 , M2 and M3 are given by

T

M1 ≡ C

0

0

q

j u0 Lp dt,

2

(1+ pd +r+ 2α q )j q

2

(r+ 2α q )j q

M21 (t)dt,

j T

M3 ≡ C

(r+ 2α q )j q

j T

M2 ≡ C

q

Ej (t) 2

0

M31 (t)dt

j

with

t

M21 = 0

q Ej (t − s)j u(·, s)2Lp ds

(5.3)

822

J. Wu

and



t

M31 = 

Ej (t − s) j u(·, s)Lp

0

q

2

(1+ pd )m

m u(·, s)Lp ds  . (5.4)

m<j −1

We now estimate M1 , M2 and M3 . Inserting T q Ej (t) dt = C ν −1 2−2αj (1 − Ej (qT )) 0

in M1 , we have M1 = Cν −1

q

(1 − Ej (qT )) 2rj q j u0 Lp .

(5.5)

j

In particular, M1 ≤ C ν −1 u0 B r . To bound M2 , we start with an estimate for M21 . p,q For q > 2, Hölder’s inequality implies t q t q−2 t 2 q q q−2 2 Ej (t − s)j u(·, s)Lp ds ≤ Ej (t − s) ds j uLp ds q

0

≤ C ν −(q−2) 2−2αj (q−2)

1−E

0

q t q −2

0

q−2

t 0

q

2

j uLp ds

.

(5.6)

Therefore, M2 is bounded by t 2 T q (1+ pd −2α+ 4α +r+ 2α )j q −(q−2) q q 2 j uLp ds dt. M2 ≤ C ν 0

For r = 1 +

d p

− 2α +

2α q ,

0

j

we have

M2 ≤ C ν −(q−2)

T

0

≤ C ν −(q−2) = Cν

2

(r+ 2α q )j q

0

j

q

T

t



0 T

u

0

2

j uLp ds



0

−(q−2)

t

dt 2

2

(r+ 2α q )j q

q j uLp ds 

dt

j

2q

r+ 2α q

dt.

(5.7)

Lq (0,t);Bp,q

We now bound M3 . First, we estimate M31 . As in the estimate of (5.6), we obtain  q t t (1+ d )m q  M31 ≤ C ν −(q−2) 2−2αj (q−2) j uLp ds 2 p m uLp  ds 0

t

= Cν −(q−2)

0

q j uLp ds

0



t

 0

2

m<j −1

m<j −1

(1+ pd −2α+ 4α q )m

m uLp 2

q 2α(1− q2 )(m−j )

ds.


For q > 2 and r = 1 +

2

m<j −1

d p

(1+ pd −2α+ 4α q )m



≤

− 2α +

2α q ,

m uLp 2

we obtain by applying (2.7),

2α(1− q2 )(m−j )

1 

q

2

(r+ 2α q )mq

823

q m uLp  

m<j −1

(5.8) 1 q¯

2

2α(1− q2 )(m−j )q¯

 = C u

r+ 2α q

,

Bp,q

m<j −1

where q¯ is the conjugate of q. Therefore, M31 ≤ C ν

−(q−2)

t

u

0

= Cν

−(q−2)

q r+ 2α q Bp,q

q u Lq

t

q

j uLp ds

ds

r+ 2α q (0,t);Bp,q

0

t

0

q

j uLp ds.

(5.9)

Inserting this bound in M3 , we obtain M3 ≤ C ν −(q−2)

T 0

= C ν −(q−2)

2

(r+ 2α q )j q

0

j T

0

u

t

q

j uLp ds u

2q r+ 2α q

Lq ((0,t);Bp,q

q r+ 2α q

Lq ((0,t);Bp,q

dt )

(5.10)

dt. )

Combining (5.5), (5.7), (5.10) with (5.2) yields (5.1). This completes the proof of Theorem 5.1. r+ 2α

q ([0, t); B˚ p,q q ). We now provide the estimate for solutions of the GNS equations in L Theorem 5.2. Let 1 < q ≤ ∞. Assume either 0 < α and p = 2 or 0 < α ≤ 1 and r 2 0, B(T ) ≤ C ν −1

(1 − Ej (qT )) 2rj q j u0 Lp + C ν −(q−1) B 2 (T ). q

j

In particular, B(T ) ≤ C ν −1 u0 ˚ r + C ν −(q−1) B 2 (T ). q Bp,q

(5.11)

824

J. Wu

Proof. Following a similar procedure as in the proof of Theorem 5.1, we obtain u

q r+ 2α q (0,T );B˚ p,q q L

≤ N1 + N2 + N3 ,

(5.12)

where N1 , N2 and N3 are given by N1 ≡ C

2

(r+ 2α q )j q

2

(1+ pd +r+ 2α q )j q

q

T

M21 (t)dt, 0

j

N3 ≡ C

q

Ej (t) j u0 Lp dt,

0

j

N2 ≡ C

T

2

(r+ 2α q )j q

T

M31 (t)dt, 0

j

with M21 defined in (5.3) and M31 in (5.4). The bound for N1 is the same as that for M1 , which is given in (5.5). To estimate N2 , we apply Young’s inequality to obtain

T

T

M21 (t) dt ≤

0

q q−1

Ej

0

q−1

0 T 0

d p

q j uLp dt

(t) dt

≤ C ν −(q−1) 2−2αj (q−1) If r = 1 +

2

T

2 q

j uLp dt

.

− 2α, then N2 is bounded by

N2 ≤ C ν

−(q−1)

2

2α (1+ pd −2α+ 2α q +r+ q )j q

= Cν

T 0

j

−(q−1)

2

(r+ 2α q )j q

T 0

j



≤ C ν −(q−1) 

2

(r+ 2α q )j q

= C ν −(q−1) u

2q

T 0

r+ 2α

2 q j uLp dt

2

j

2 q j uLp dt

j uLp dt  q

.

(5.13)

q (0,T );B˚ p,q q L

To bound N3 , we again apply Young’s inequality to get q T d q q (1+ p )m p M31 dt ≤ C Ej q j uLq (0,T ) 2 u m L L q−1 (0,T ) 0 m<j −1 q L (0,T )  q T T (1+ d )m q  = C ν −(q−1) 2−2αj (q−1) j uLp dt 2 p m uLp  dt. 0

0

m<j −1


Since r = 1 +

d p

T

C ν −(q−1)

0

825

− 2α, the right-hand side can be written as  q T 2α 1 q (r+ q )m 2α(1− q )(m−j )   j uLp dt 2 m uLp 2 dt. 0

m<j −1

For q > 1, we have, according to (2.7), 

2

(r+ 2α q )m

m uLp 2

2α(1− q1 )(m−j )

≤C 

m<j −1

1 q

2

(r+ 2α q )mq

q m uLp 

.

m<j −1

Therefore, T

M31 dt ≤ C ν

−(q−1)

0

= C ν −(q−1)

T 0

N3 ≤ C ν

2

q

(r+ 2α q )j q

T 0

j

= C ν −(q−1) u

2

(r+ 2α q )mq

2q

r+ 2α

T 0

j uLp dt u

Therefore, −(q−1)

m

T 0

q j uLp dt

q

m uLp dt

q . r+ 2α q q ˚ L (0,T );Bp,q

q

j uLp dt u

q r+ 2α q q ˚ L (0,T );Bp,q

.

(5.14)

q (0,T );B˚ p,q q L

Inserting the estimates (5.5), (5.13) and (5.14) in (5.12), we obtain (5.11). The a priori estimates of Theorems 5.1 and 5.2 can be extended to solutions of the r+ 2α

equations in (4.4). The bound in Lq (0, T ); B˚ p,q q is stated in Theorem 5.3, while r+ 2α q (0, T ); B˚ p,q q is provided in Theorem 5.4. These theorems can be the bound in L shown by combining the arguments in the proofs of Theorems 4.3, 5.1 and 5.2, so we omit the details. Theorem 5.3. Let 2 < q ≤ ∞. Assume either 21 < α and p = 2 or ˚r 2 0. Then any solution of (4.4) and (4.5) satisfies q q ≤ C ν −1 F (1 − Ej (qT )) 2rj q j F0 Lp 2α r+ q

Lq (0,T );B˚ p,q

j

+C ν −(q−2)

T 0

v

q Lq

r+ 2α q (0,t);B˚ p,q

wq

r+ 2α q

dt.

Lq (0,t);B˚ p,q

(5.15)

826

J. Wu

In particular, (5.15) holds when the first term on the right of (5.15) is replaced by C ν −1 F0 B˚ r . p,q

Theorem 5.4. Let 1 < q ≤ ∞. Assume either 21 < α and p = 2 or r and 2 0. Then any solution of (4.4) and (4.5) satisfies q r+ 2α q (0,T );B˚ p,q q L

F

≤ C ν −1

q

(1 − Ej (qT )) 2rj q j F0 Lp

j

+C ν −(q−1) v

q wq . r+ 2α r+ 2α q q q q ˚ ˚ L (0,T );Bp,q L (0,T );Bp,q

(5.16)

In particular, (5.16) holds when the first term on the right of (5.16) is replaced by C ν −1 F0 B˚ r . p,q

6. Existence and Uniqueness This section presents three existence and uniqueness results for the GNS equations. r (Rd ) with The first one asserts the global existence and uniqueness of solutions in B˚ p,q r (Rd ). The second one r = 1 − 2α + pd for initial data that are comparable to ν in B˚ p,q r (Rd ) for arbitrarily establishes the local existence and uniqueness of solutions in B˚ p,q s+ 2α

r (Rd ). The third one concerns solutions in Lq ((0, T ); B ˚ p,q q (Rd )) with large data in B˚ p,q d 2α s = 1 − 2α + p + q . The precise statements are given in Theorems 6.1, 6.2 and 6.3.

Theorem 6.1. Let 1 ≤ q ≤ ∞. Assume either 21 < α and p = 2 or 21 < α ≤ 1 and r (Rd ) satisfies 2 < p < ∞. Let r = 1 − 2α + pd . Assume that u0 ∈ B˚ p,q u0 B˚ r

p,q

≤ C0 ν

for some suitable constant C0 depending on d and p only. Then the GNS equations (4.1),(4.2) and (4.3) have a unique global solution u satisfying r+ 2α

r u ∈ C([0, ∞); B˚ p,q ) ∩ Lq ((0, ∞); B˚ p,q q ) for 1 ≤ q < ∞, d

r 1 ((0, ∞); B˚ 1+ p ) for q = ∞. )∩L u ∈ C([0, ∞); B˚ p,∞ p,∞

In addition, for any t > 0, u(·, t)B˚ r + C νu p,q

u(·, t)B˚ r

p,∞

r+ 2α

q Lq ((0,t);B˚ p,q )

+ C νu

d

1 ((0,t);B˚ 1+ p ) L p,∞

≤ C1 ν for 1 ≤ q < ∞, ≤ C1 ν for q = ∞,

for some constants C and C1 depending on d and p only.


827

Although this theorem does not explicitly mention the case when 1 ≤ p < 2, we can r easily extend it to cover any initial datum u0 ∈ B˚ p,q with 1 ≤ p < 2. In fact, by the Besov embedding (see Part 2) of Proposition 2.1), B˚ pr11 ,q (Rd ) ⊂ B˚ pr22 ,q (Rd ) for any 1 ≤ q ≤ ∞, 1 ≤ p1 ≤ p2 ≤ ∞ and r1 = r2 + d( p11 − p12 ). Thus, if r u0 ∈ B˚ p,q with 1 ≤ p < 2 and r = 1 − 2α + pd , then u0 ∈ B˚ pr22 ,q for any p2 > 2 and r2 = 1 − 2α +

d p2 . Theorem

6.1 then implies that the GNS equations have a unique r with 1 ≤ p < 2 and r = 1 − 2α + d . global solution associated with any u0 ∈ B˚ p,q p

Proof of Theorem 6.1. To apply the contraction mapping principle, we write the GNS equations in the integral form t u(t) = Gu(t) ≡ exp(−ν(−)α t)u0 − exp(−ν(−)α (t − s)) P(u · ∇u)(s)ds, 0

where exp(−ν(−)α t) for each t ≥ 0 is a convolution operator with α t)(ξ ) = exp(−ν(2π |ξ |)2α t). exp(−ν(−) We shall only provide the proof for the case when 1 ≤ q < ∞ since the proof for q = ∞ is analogous. To set up, we write r+ 2α r X = C([0, ∞); B˚ p,q ), Y = Lq (0, ∞); B˚ p,q q , Z = X ∩ Y. For the norm in Z, we choose uZ = max{uX + C ν uY }, where C is a suitable constant depending on d, p and q only. In addition, we use D to denote the subset D = {u ∈ Z : uZ ≤ C1 ν}. We aim to show that G is a contractive map from D to D. For notational convenience, we write Gu as Gu = u0 − F (u, u), where u0 and F (v, w) are given by u0 = exp(−ν(−)α t)u0 , t F (v, w) = exp(−ν(−)α (t − s))P(v · ∇w)(s)ds. 0

Obviously, u0 satisfies the equation ∂t u0 + ν(−)α u0 = 0,

u0 (x, 0) = u0 (x).

828

J. Wu

As shown in the proof of Theorem 4.1, we have u

0

q (·, t) ˚ r Bp,q

t

+ C qν

u(·, s)

0

q r+ 2α q B˚ p,q

q Bp,q

ds ≤ u0 ˚ r .

Consequently, u0 Z ≤ C0 ν. To bound F in D, we first notice that F satisfies ∂t F + ν(−)α F = P(u · ∇u),

F (u, u)(x, 0) = 0.

Applying the result of Theorem 4.3, we have d q q q q F ˚ r + C νF r+ 2α ≤ C u ˚ r u r+ 2α . Bp,q Bp,q q q dt ˚ Bp,q B˚ p,q Therefore, for u ∈ D and suitable C1 , F (u, u)Z ≤ C1 ν. Thus, G maps D to D. To see that G is contractive, we write the difference Gu − Gv into two parts, namely, Gu − Gv = −(F (u, u − v) + F (u − v, v)). Since F (u, u − v) satisfies ∂t F + ν(−)α F = P(u · ∇(u − v)),

F (u, u)(x, 0) = 0.

Again the result of Theorem 4.3 shows F (u, u − v)Z ≤ CuZ u − vZ . F (u − v, v) admits a similar bound. Therefore, Gu − GvZ ≤ C (uZ + vZ )u − vZ . For suitable C1 , C (uZ + vZ ) < 1 and G is contractive. The result of Theorem 6.1 then follows from the contraction mapping principle. We now state and prove the local existence and uniqueness result. For the sake of conciseness, the statement for the case when q = ∞ is omitted from the theorem. Theorem 6.2. Let 1 < q ≤ ∞. Assume either 21 < α and p = 2 or 21 < α ≤ 1 and r (Rd ). Then there exists a T > 0 2 < p < ∞. Let r = 1 − 2α + pd . Assume u0 ∈ B˚ p,q such that the GNS equations (4.1),(4.2) and (4.3) have a unique solution u on [0, T ) in the class r+ 2α

r q ((0, T ); B˚ p,q q ). C([0, T ); B˚ p,q )∩L


829

Proof. The proof combines the contraction mapping principle with the a priori bounds in Theorems 5.2 and 5.4. The operator G is the same as in the proof of the previous theorem, but the functional setting is different. For T > 0 and R > 0 to be selected, we define r+ 2α

q ((0, T ); B˚ p,q q ), Z1 = L

D1 = u ∈ Z1 : uZ1 ≤ R .

The goal is to show that G is a contractive map from D1 to D1 . We again write Gu = u0 − F (u, u). Since u0 satisfies ∂t u0 + ν(−)α u0 = 0,

u0 (x, 0) = u0 (x),

we can show as in the proof of Theorem 5.2, q q u0 Z1 ≤ C ν −1 (1 − Ej (qT )) 2rj q j u0 Lp , j r , this inequality especially implies where Ej (t) ≡ exp(−C ν22αj t). Since u0 ∈ B˚ p,q 0 that u Z1 is finite. In addition, by the Dominated Convergence Theorem, q (1 − Ej (qT )) 2rj q j u0 Lp → 0 as T → 0. j

Therefore, u0 Z1 is small for small T > 0. Since F (u, u) satisfies ∂t F + ν(−)α F = P(u · ∇u),

F (u, u)(x, 0) = 0,

we have, according to Theorem 5.4, F (u, u)Z1 ≤ Cu2Z1 ≤ C R 2 for any u ∈ D1 . Thus for small T > 0 and suitable R > 0, Gu ∈ D1 . As in the proof of Theorem 6.1, we can show by applying Theorem 5.4 again that Gu − GvZ1 ≤ C (uZ1 + vZ1 )u − vZ1 ≤ 2C R u − vZ1 , which implies G is a contraction for suitable selected R. By applying the contraction mapping principle, we find that the GNS equations have a solution in Z1 , namely, in r+ 2α

q ((0, T ); B˚ p,q q ). L r , we can derive similarly as in the proof of Theorem 5.2 the bound To show u ∈ B˚ p,q u(·, t)B˚ r

p,q

≤ u0 B˚ r + C u2Z1 p,q

r . This for any solution u of the GNS equations. Thus, for t ∈ [0, T ), u(·, t) is in B˚ p,q completes the proof of Theorem 6.2. r+ 2α

q ((0, T ); B˚ p,q q ) was used in the proof of the previous theorem. If we The space L s+ 2α

use Lq ((0, T ); B˚ p,q q ) instead, we have the following local existence and uniqueness theorem. This result can be proven by the contraction mapping principle combined with the a priori estimates in Theorems 5.1 and 5.3. We omit details of the proof.

830

J. Wu

Theorem 6.3. Let 2 < q ≤ ∞. Assume either 21 < α and p = 2 or 21 < α ≤ 1 and d ˚s 2 0 such that the GNS equations have a unique solution u on [0, T ) satisfying s+ 2α

s C([0, T ); B˚ p,q ) ∩ Lq ((0, T ); B˚ p,q q ).

Appendix We provide the proof of Proposition 3.5 in Sect. 3. Proof of Proposition 3.5. (a) When p = 2, we have α f (−) f dx = |α f |2 dx. D(f ) = Rd

Rd

(A.1)

Since α = 1 and d = 2, we obtain (3.5) by applying the Sobolev embedding d

W r ,r (Rd ) ⊂ Lq (Rd )

(A.2)

valid for any r ∈ (1, ∞) and q ∈ [r, ∞). The inequality in (b) is a consequence of (A.1) and the Sobolev embedding rd

Ws,r (Rd ) ⊂ L d−rs (Rd )

(A.3)

for any sr < d. To prove (c), we first apply the pointwise inequality in Proposition 3.2 to obtain

α p/2 2 p/2 α p/2 D(f ) ≥ C dx = C f (−) f dx. f Rd

Rd

By (A.2), we have for any q ∈ [p, ∞), p

D(f ) ≥ C f

L

Since 2 < p

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Communications in Mathematical Physics - Volume 184

Communications in Mathematical Physics - Volume 261

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Communications In Mathematical Physics - Volume 293

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Communications In Mathematical Physics - Volume 298

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Communications In Mathematical Physics - Volume 288

Communications in Mathematical Physics - Volume 304

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