Communications in Mathematical Physics - Volume 259

Commun. Math. Phys. 259, 1–44 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1387-5 Communications in Mathe...

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Commun. Math. Phys. 259, 1–44 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1387-5

Communications in

Mathematical Physics

Integrable Structure of the Dirichlet Boundary Problem in Multiply-Connected Domains I. Krichever1,2,3 , A. Marshakov3,4,5 , A. Zabrodin3,6 1 2 3 4 5 6

Department of Mathematics, Columbia University, New York, USA. E-mail: [email protected] Landau Institute, Moscow, Russia ITEP, Moscow, Russia Max Planck Institute of Mathematics, Bonn, Germany Lebedev Physics Institute, Moscow, Russia Institute of Biochemical Physics, Moscow, Russia

Received: 21 September 2003 / Accepted: 3 February 2004 Published online: 8 July 2005 – © Springer-Verlag 2005

Abstract: We study the integrable structure of the Dirichlet boundary problem in two dimensions and extend the approach to the case of planar multiply-connected domains. The solution to the Dirichlet boundary problem in the multiply-connected case is given through a quasiclassical tau-function, which generalizes the tau-function of the dispersionless Toda hierarchy. It is shown to obey an infinite hierarchy of Hirota-like equations which directly follow from properties of the Dirichlet Green function and from the Fay identities. The relation to multi-support solutions of matrix models is briefly discussed. 1. Introduction The Dirichlet boundary problem [1] is to reconstruct a harmonic function in a bounded domain from its values on the boundary. Remarkably, this standard problem of complex analysis, related however to string theory and matrix models, possesses a hidden integrable structure [2], which we clarify further in this paper. It turns out that variation of a solution to the Dirichlet problem under variation of the domain is described by an infinite hierarchy of non-linear partial differential equations known (in the simply-connected case) as dispersionless Toda hierarchy. It is a particular example of the universal hierarchy of Whitham equations introduced in [3, 4]. The quasiclassical tau-function or, more precisely, its logarithm F , is the main new object associated with a family of domains in the plane. Any domain in the complex plane with sufficiently smooth boundary can be parameterized by its moments with respect to a basis of harmonic functions. The F -function is a function of the full infinite set of the moments. The first order derivatives of F are then moments of the complementary domain. This gives a formal solution to the inverse potential problem, considered for the simply-connected case in [5, 6]. The second order derivatives are coefficients of the Taylor expansion of the Dirichlet Green function and therefore they solve the Dirichlet boundary problem. These coefficients are constrained by an infinite number of universal (i.e. domain-independent) relations which, unified in a generating form, just constitute

2

I. Krichever, A. Marshakov, A. Zabrodin

the dispersionless Hirota equations. For the third order derivatives (their role in problems of complex analysis is not yet quite clear) there is a nice “residue formula” which allows one to prove [7] that F obeys the WDVV equations. Below we are going to demonstrate that for planar multiply-connected domains the solution to the Dirichlet boundary problem can be performed in a similar way. Specifically, we consider domains which are obtained by cutting several “holes” in the complex plane. Boundaries of the holes are assumed to be smooth simple non-intersecting curves. In this case, the complete set of independent variables can be again identified with the set of harmonic moments. However, a choice of the proper basis of harmonic functions in a multiply-connected domain becomes crucial for our approach. It turns out that the Laurent polynomials which were used in the simply-connected case should be replaced by the basis analogous to the one introduced in [8] – a “global” generalization of the Laurent basis for algebraic curves of arbitrary genus. The basis has to be also enlarged to include harmonic functions with multi-valued analytic part. This results in an additional finite set of extra variables. We construct the F -function and prove that its second derivatives satisfy non-linear relations, which generalize the Hirota equations of the dispersionless Toda hierarchy. These relations are derived from the Fay identities [9] for the Riemann theta functions on the Jacobian of Riemann surface obtained as the Schottky double of the plane with given holes. We note that extra variables, specific for the multiply-connected case, can be chosen in different ways and possess different geometric interpretations, depending on the choice of basis of homologically non-trivial cycles on the Schottky double. The corresponding F -functions are shown to be connected by a duality transformation – a (partial) Legendre transform, with the generalized Hirota relations being the same. Now let us give a bit more expanded description of the Dirichlet problem in planar domains. Let Dc be a domain in the complex plane bounded by one or several non-intersecting smooth curves. It will be convenient to realize Dc as a complement to another domain D, having one or more connected components, and to consider the Dirichlet problem in Dc : to find a harmonic function u(z) in Dc such that it is continuous up to the boundary, ∂Dc , and equals a given function u0 (ξ ) on the boundary. The problem has a unique solution written in terms of the Dirichlet Green function G(z, ξ ): 1 u(z) = − u0 (ξ )∂n G(z, ξ )|dξ | , (1.1) 2π ∂Dc where ∂n is the normal derivative on the boundary with respect to the second variable, the normal vector n is directed inward Dc , and |dξ | := dl(ξ ) is an infinitesimal element of the length of the boundary ∂Dc . The main object to study is, therefore, the Dirichlet Green function. It is uniquely determined by the following properties [1]: (G1) The function G(z, z ) is symmetric and harmonic everywhere in Dc (including ∞ if Dc ∞) in both arguments except z = z , where G(z, z ) = log |z − z | + · · · as z → z ; (G2) G(z, z ) = 0 if any one of the variables z, z belongs to the boundary ∂Dc . Note that the definition implies that G(z, z ) < 0 inside Dc . In particular, ∂n G(z, ξ ) is strictly negative for all ξ ∈ ∂Dc . If Dc is simply-connected (note that we assume ∞ ∈ Dc ), i.e., the boundary has only one component, the Dirichlet problem is equivalent to finding a bijective conformal map from Dc onto the complement to unit disk or any other reference domain for which the

Integrable Structure of the Dirichlet Boundary Problem

3

Green function is known explicitly. Such a bijective conformal map w(z) exists by virtue of the Riemann mapping theorem, then w(z) − w(z ) , G(z, z ) = log (1.2) w(z)w(z ) − 1 where bar means complex conjugation. It connects the Green function at two points with the conformal map normalized at some third point (say at z = ∞: w(∞) = ∞). It is this formula which allows one to derive the Hirota equations for the tau-function of the Dirichlet problem in the most economic and transparent way [2] (see also Sect. 2 below). For multiply-connected domains, formulas of this type based on conformal maps do not really exist. In general, there is no canonical choice of the reference domain, moreover, the shape of a reference domain depends on Dc itself. In fact, as we demonstrate in the paper, the correct extension of (1.2) needed for derivation of the generalized Hirota equations follows from a different direction which is no longer explicitly related to bijective conformal maps. Namely, logarithm of the conformal map log w(z) should be replaced now by the Abel map from the Schottky double of Dc to the Jacobi variety of this Riemann surface, and the rational function under the logarithm in (1.2) is substituted by ratio of the prime forms or Riemann theta-functions. We show that the Green function of multiply-connected domains admits a representation through the logarithm of the tau-function of the form 1 1 1 (1.3) G(z, z ) = log − + ∇(z)∇(z )F . z z 2 Here ∇(z) is a certain vector field on the moduli space of boundary curves, therefore it can be represented as a (first-order) differential operator w.r.t. harmonic moments with constant (in moduli) coefficients depending, however, on the point z as a parameter. In this paper we also obtain similar formulas for the harmonic measures of the boundary components and for the Abel map. A combination of these formulas with the Fay identities yields the generalized Hirota-like equations for the tau-function F . Our main tool is the Hadamard variational formula [10] which gives the variation of the Dirichlet Green function under small deformations of the domain in terms of the Green function itself: 1 δG(z, z ) = ∂n G(z, ξ )∂n G(z , ξ )δn(ξ )|dξ |. (1.4) 2π ∂Dc Here δn(ξ ) is the normal displacement (with sign!) of the boundary under the deformation, counted along the normal vector at the boundary point ξ . It was shown in [2] that this remarkable formula is a key to all integrable properties of the Dirichlet problem. An extremely simple “pictorial” derivation of the formula (1.4) is presented in Fig. 1. We start with a brief recollection of the results for the simply-connected case in Sect. 2. However, instead of “bump” deformations used in [2] we work here with their rigorously defined versions – a family of infinitesimal deformations which we call elementary ones. This approach is basically motivated by the theory of interface dynamics in viscous fluids, which is known to be closely connected with the formalism developed in [2] and in the present paper (see [5] for details). In Sect. 3 we introduce local coordinates in the space of planar multiply-connected domains and express the elementary deformations in these coordinates. Using the Hadamard formula, we then observe remarkable symmetry or “zero-curvature” relations

4

I. Krichever, A. Marshakov, A. Zabrodin z δn ξ

Fig. 1. A “pictorial” derivation of the Hadamard formula. We consider a small deformation of the domain, with the new boundary being depicted by the dashed line. According to (G2) the Dirichlet Green function vanishes, G(z, ξ ) = 0, if ξ belongs to the old boundary. Then the variation δG(z, ξ ) is simply equal to the new value, i.e. in the leading order δG(z, ξ ) = −δn(ξ )∂n G(z, ξ ). Now notice that δG(z, ξ ) is a harmonic function (the logarithmic singularity cancels since it is the same for both old and new functions) with the boundary value −δn(ξ )∂n G(z, ξ ). Applying (1.1) one obtains (1.4). The argument is the same for both simply-connected and multiply-connected domains

which connect elementary deformations of the Green function and harmonic measures. The existence of the tau-function and the formula (1.3) for the Green function directly follow from these relations. In Sect. 4 we make a Legendre transform to another set of local coordinates in the space of algebraic multiply-connected domains, which is in a sense dual to the original one. In these coordinates, Eq. (1.3) gives another version of the Green function which solves the so-called modified Dirichlet problem. We also discuss the relation to multi-support solutions of matrix models in the planar large N limit. In Sect. 5 we combine the results outlined above with the representation of the Green function in terms of the prime form on the Schottky double. This allows us to obtain an infinite system of partial differential equations on the tau-function which generalize the dispersionless Hirota equations. 2. The Dirichlet Problem for Simply-Connected Domains and Dispersionless Hirota Equations In this section we rederive the results from [2] for the simply-connected case in a slightly different manner, more suitable for further generalizations. At the same time we show that the results of [2] obtained for analytic curves can be easily extended to the smooth case. Let D be a connected domain in the complex plane bounded by a simple smooth curve. We consider the exterior Dirichlet problem in Dc = C \ D which is the complement of D in the whole (extended) complex plane. Without loss of generality, we assume that D is compact and contains the point z = 0. Then Dc is a simply-connected domain on the Riemann sphere containing ∞. 2.1. Harmonic moments and deformations of the boundary. Let tk be moments of the domain Dc = C \ D defined with respect to the harmonic functions {z−k /k}, k > 0: 1 tk = − z−k d 2 z , k = 1, 2, . . . , (2.1) πk Dc


5

1 and {t¯k } be the complex conjugate moments, i.e. t¯k = − πk formula represents the harmonic moments as contour integrals

tk =

1 2πik

Dc

z−k z¯ dz

d 2 z¯z−k . The Stokes

(2.2)

∂D

providing, in particular, a regularization of possibly divergent integrals (2.1). Besides, we denote by t0 the area (divided by π) of the domain D: 1 t0 = π

d 2z .

(2.3)

D

The harmonic moments of Dc are coefficients of the Taylor expansion of the potential (z) = −

2 π

log |z − z |d 2 z

(2.4)

D

induced by the domain D filled by two-dimensional Coulomb charges with the uniform density ρ = −1. Clearly, ∂z ∂z¯ (z) = −1 if z ∈ D and vanishes otherwise, so around the origin (recall that D 0) the potential equals to −|z|2 plus a harmonic function, i.e. (z) − (0) = −|z|2 +

tk zk + t¯k z¯ k ,

(2.5)

k≥1

and one can verify that tk are just given by (2.1). For analytic boundary curves, one may introduce the Schwarz function associated with the curve. The function ∂z (z) = −

1 π

D

d 2 z z − z

is continuous across the boundary and holomorphic for z ∈ Dc while for z ∈ D the function ∂z + z¯ is holomorphic. If the boundary is an analytic curve, both these functions can be analytically continued outside the regions where they were originally defined, and, therefore, there exists a function, S(z), analytic in some strip-like neighborhood of the boundary contour, such that S(z) = z¯ on the contour. In other words, S(z) is the analytic continuation of z¯ away from the boundary contour, this function completely determines the shape of the boundary and is called the Schwarz function [11]. In general we are going to work with smooth curves, not necessarily analytic, when the Schwarz function does not exist as an analytic function. Nevertheless, it appears to be useful below to define the class of boundary contours with nice algebro-geometric properties. The basic fact of the theory of deformations of closed smooth curves is that the (in general complex) moments {tk , t¯k } ≡ {t±k } supplemented by the real variable t0 form a set of local coordinates in the “moduli space” of smooth closed curves [12] (see also [13]).

6


Important remark . This means that: (a) under any small deformation of the domain the set t = {t0 , t±k } is subject to a small change; (b) on the space of smooth closed curves there exist vector fields ∂tk such that ∂tk tn = δkn , which are represented in terms of infinitesimal normal displacements of the boundary that change xk = Re tk or yk = Im tk keeping all the other moments fixed; (c) the corresponding infinitesimal displacements can be locally integrated. The latter means that for each domain Dc with moments {t0 , t±k } and for an arbitrary integer N there exist constants m , |m| ≤ N , such that for } with |t − t | < , m ≤ N, t = t , |m| > N , in the neighborhood any set {t0 , t±k m m m m m }. We adopt this restricted c of D there is a unique domain with the moments {t0 , t±k notion of the local coordinates throughout the paper. It would be very interesting to find conditions on the infinite sets k for the corresponding rectangles to form an open set in an infinite-dimensional variety of smooth curves. We plan to address this problem elsewhere. Let us present a proof of this statement which later will be easily adjusted to the case of multiply-connected domains. At the same time this proof allows one to derive a deformation of the domain with respect to the variables tk . Suppose there is a one-parametric deformation D(t) (with some real parameter t) of D = D(0) such that all tk are preserved: ∂t tk = 0, k ≥ 0. Let us prove that such a deformation is trivial. The proof is based on two key observations: • The difference of the boundary values ∂t C ± (ζ )dζ of the derivative of the Cauchy integral C(z)dz =

dz 2πi

∂D

ζ¯ dζ ζ −z

(2.6)

is a purely imaginary differential on the boundary of D. Indeed, let ζ (σ, t) be a parameterization of the curve ∂D(t). Denote the value of the differential (2.6) by C − (z)dz for z ∈ Dc and by C + (z)dz for z ∈ D. Taking the t-derivative of (2.6) and integrating by parts one gets

ζ¯t ζσ + ζ¯ ζt, σ ζ¯ ζσ ζt − ζ −z (ζ − z)2 ∂D ¯ dz ζt ζσ − ζ¯σ ζt = dσ . 2πi ∂D ζ −z

dz ∂t C(z)dz = 2πi

dσ (2.7)

Hence,

∂t C + (ζ ) − ∂t C − (ζ ) dζ = ∂t ζ¯ dζ − ∂t ζ d ζ¯ = 2iIm ∂t ζ¯ dζ

is indeed purely imaginary. • If a t-deformation preserves all the moments tk , k ≥ 0, the differential ∂t ζ¯ dζ − ∂t ζ d ζ¯ extends to a holomorphic differential in Dc . If |z| < |ζ | for all ζ ∈ ∂D, then we can expand: ∂ ∂t C (z)dz = ∂t +

∞ dz k z ζ −k−1 ζ¯ dζ 2πi ∂D k=0

=

∞ k=1

k (∂t tk ) zk−1 dz = 0

(2.8)


7

and, since C + is analytic in D, we conclude that ∂t C + ≡ 0. The expression ∂t ζ¯ dζ − ∂t ζ d ζ¯ is the boundary value of the differential −∂t C − (z)dz which has at most simple pole at the infinity and holomorphic everywhere else in Dc . The equality 1 (∂t ζ¯ dζ − ∂t ζ d ζ¯ ) = 0 ∂t t 0 = 2πi ∂D then implies that the residue at z = ∞ vanishes, therefore ∂t C − (z)dz is holomorphic. Any holomorphic differential which is purely imaginary along the boundary of a simply-connected domain must be zero in this domain. Indeed, the real part of the harmonic continuation of the integral of this differential is a harmonic function with a constant boundary value. Such a function must be constant by virtue of the uniqueness of the solution to the Dirichlet problem. Another proof relies on the Schwarz symmetry principle and the standard Schottky double construction (see the next section for details). Consider the compact Riemann surface obtained by attaching to Dc its complex conjugated copy along the boundary. Since ∂t C − dz is imaginary along the boundary, we conclude, from the Schwarz symmetry principle, that ∂t C − dz extends to a globally defined holomorphic differential on this compact Riemann surface, which has genus zero. Therefore, such a differential is equal to zero. Hence we conclude that ∂t ζ¯ dζ − ∂t ζ d ζ¯ = 0. This means that the vector ∂t ζ is tangent to the boundary. Without loss of generality we can always assume that a parameterization of ∂D(t) is chosen so that ∂t ζ (σ, t) is normal to the boundary. Thus, the t-deformation of the boundary preserving all harmonic moments is trivial. The fact that the set of harmonic moments is not overcomplete follows from the explicit construction of vector fields in the space of domains that changes any harmonic moment keeping all the others fixed (see below). 2.2. Elementary deformations and the operator ∇(z). Fix a point z ∈ Dc and consider a special infinitesimal deformation of the domain such that the normal displacement of the boundary is proportional to the gradient of the Green function G(z, ξ ) at the boundary point (Fig. 2): δn(ξ ) = − ∂n G(z, ξ ) . 2

(2.9)

For any sufficiently smooth initial boundary this deformation is well-defined as → 0. We call infinitesimal deformations from this family, parametrized by z ∈ Dc , the elementary deformations. The point z is referred to as the base point of the deformation. Note that since ∂n G < 0 (see the remark after the definition of the Green function in the Introduction), δn for the elementary deformations is either strictly positive or strictly negative depending of the sign of . Let δz be a variation of any quantity under the elementary deformation with the base point z. It is easy to see that δz t0 = , δz tk = z−k /k. Indeed, 1 δn(ξ )|dξ | = − ∂n G(z, ξ )|dξ | = , δz t0 = π 2π (2.10) 1 −k −k −k δz tk = ξ δn(ξ )|dξ | = − ξ ∂n G(z, ξ )|dξ | = z πk 2πk k

8


z

Fig. 2. The elementary deformation with the base point z

by virtue of the Dirichlet formula (1.1). Note that the elementary deformation with the base point at ∞ keeps all moments except t0 fixed. Therefore, the deformation which changes only t0 is given by δn(ξ ) = − 2 ∂n G(∞, ξ ). Now we can explicitly define the deformations that change only either xk = Re tk or yk = Im tk keeping all other moments fixed. As is clear from (2.10), the corresponding δn(ξ ) is given by the real or imaginary part of normal derivative of the function 1 Hk (ξ ) = zk ∂z G(z, ξ )dz (2.11) 2πi ∞ at the boundary. Here the contour integral goes around infinity. Namely, the normal displacements δn(ξ ) = Re (∂n Hk (ξ )) and δn(ξ ) = Im (∂n Hk (ξ )) change the real and imaginary part of tk by ± respectively keeping all other moments fixed. These deformations allow one to introduce the vector fields ∂ , ∂t0

∂ , ∂xk

∂ ∂yk

in the space of domains which are locally well-defined. Existence of such vector fields means that the variables tk are independent. For k > 0 it is more convenient to use their linear combinations ∂ ∂ ∂ ∂ 1 ∂ 1 ∂ , = −i = +i ∂tk 2 ∂xk ∂yk ∂ t¯k 2 ∂xk ∂yk which span the complexified tangent space to the space of simply-connected domains (with fixed area t0 ). If X is any functional of our domain locally representable as a function of harmonic moments, X = X(t), the vector fields ∂t0 , ∂tk , ∂t¯k can be understood as partial derivatives acting to the function X(t). Consider the variation δz X of a functional X = X(t) under the elementary deformation with the base point z. In the leading order in we have: δz X =

∂X k

∂tk

δz tk = ∇(z)X

(2.12)


where the differential operator ∇(z) is given by z−k z¯ −k ∇(z) = ∂t0 + ∂tk + ∂t¯k . k k

9

(2.13)

k≥1

The right-hand side suggests that for functionals X such that the series ∇(z)X converges everywhere in Dc up to the boundary, δz X is a harmonic function of the base point z. Note that in [2] we have used the “bump” deformation and continued it harmonically into Dc . In fact, it was the elementary deformation (2.10) δz ∝ |dξ |∂n G(z, ξ )δ bump (ξ ) that was really used. The “bump” deformation should be understood as a (carefully taken) limit of δz when the point z tends to the boundary ∂Dc . 2.3. The Hadamard formula as integrability condition. Variation of the Green function under small deformations of the domain is known due to Hadamard, see Eq. (1.4). To find how the Green function changes under small variations of the harmonic moments, we fix three points a, b, c ∈ C \ D and compute δc G(a, b) by means of the Hadamard formula (1.4). Using (2.12), one can identify the result with the action of the vector field ∇(c) on the Green function: 1 ∇(c)G(a, b) = − ∂n G(a, ξ )∂n G(b, ξ )∂n G(c, ξ )|dξ | . (2.14) 4π ∂D Remarkably, the r.h.s. of (2.14) is symmetric in all three arguments, i.e. ∇(a)G(b, c) = ∇(b)G(c, a) = ∇(c)G(a, b) .

(2.15)

This is the key relation which allows one to represent the Dirichlet problem as an integrable hierarchy of non-linear differential equations [2], (2.15) being the integrability condition of the hierarchy. It follows from (2.15) (see [2] for details) that there exists a function F = F (t) such that 1 1 1 G(z, z ) = log − + ∇(z)∇(z )F . (2.16) z z 2 We note that existence of such a representation of the Green function was first conjectured by Takhtajan. For the simply-connected case, this formula was obtained in [14] (see also [13] for a detailed proof and discussion). The function F is (logarithm of) the tau-function of the integrable hierarchy. In [14] it was called the tau-function of the (real analytic) curves – the boundary contours ∂D or ∂Dc . 2.4. Dispersionless Hirota equations. Combining (2.16) and (1.2), we obtain the relation w(z) − w(z ) 2 1 1 2 log = log − + ∇(z)∇(z )F (2.17) z z w(z)w(z ) − 1 which implies an infinite hierarchy of differential equations on the function F . It is convenient to normalize the conformal map w(z) by the conditions that w(∞) = ∞ and ∂z w(∞) is real, so that z (2.18) w(z) = + O(1) as z → ∞ , r

10


where the real number r = limz→∞ dz/dw(z) is called the (external) conformal radius of the domain D (equivalently, it can be defined through the Green function as log r = limz→∞ (G(z, ∞) + log |z|), see [15]). Then, tending z → ∞ in (2.17), one gets log |w(z)|2 = log |z|2 − ∂t0 ∇(z)F .

(2.19)

The limit z → ∞ of this equality yields a simple formula for the conformal radius: log r 2 = ∂t20 F .

(2.20)

Let us now separate holomorphic and antiholomorphic parts of these equations, introducing the holomorphic and antiholomorphic parts of the operator ∇(z) (2.13): D(z) =

z−k k≥1

k

∂tk ,

¯ z) = D(¯

z¯ −k k≥1

k

∂t¯k .

(2.21)

Rewrite (2.17) in the form w(z) − w(z ) 1 1 1 log − − ∂t + D(z) ∇(z )F − log z z 2 0 w(z)w(z ) − 1

1 w(z) − w(z ) 1 1 ¯ = − log + log − + ∂t + D(¯z) ∇(z )F . z¯ z¯ 2 0 w(z )w(z) − 1 The l.h.s. is a holomorphic function of z while the r.h.s. is antiholomorphic. Therefore, both are equal to a z-independent term which can be found from the limit z → ∞. As a result, we obtain the equation w(z) − w(z ) z log = log 1 − (2.22) + D(z)∇(z )F z w(z) − (w(z ))−1 which, as z → ∞, turns into the formula for the conformal map w(z): 1 log w(z) = log z − ∂t20 F − ∂t0 D(z)F 2

(2.23)

(here we also used (2.20)). Proceeding in a similar way, one can rearrange (2.22) in order to write it separately for holomorphic and antiholomorphic parts in z : log

w(z) − w(z ) 1 = − ∂t20 F + D(z)D(z )F , z − z 2

− log 1 −

1 w(z)w(z )

¯ z )F . = D(z)D(¯

(2.24)

(2.25)

Writing down Eqs. (2.24) for the pairs of points (a, b), (b, c) and (c, a) and summing up the exponentials of both sides of each equation one arrives at the relation (a − b)eD(a)D(b)F + (b − c)eD(b)D(c)F + (c − a)eD(c)D(a)F = 0

(2.26)

which is the dispersionless Hirota equation (for the KP part of the two-dimensional Toda lattice hierarchy) written in the symmetric form. This equation can be regarded as


11

a very degenerate case of the trisecant Fay identity [9]. It encodes the algebraic relations between the second order derivatives of the function F . As c → ∞, we get these relations in a more explicit but less symmetric form: 1 − eD(a)D(b)F =

D(a) − D(b) ∂t1 F a−b

(2.27)

which makes it clear that the totality of second derivatives Fij := ∂ti ∂tj F are expressed through the derivatives with one of the indices put equal to unity. More general equations of the dispersionless Toda hierarchy obtained in a similar way by combining Eqs. (2.23), (2.24) and (2.25) include derivatives w.r.t. t0 and t¯k : (a − b)eD(a)D(b)F = ae−∂t0 D(a)F − be−∂t0 D(b)F , ¯

1 − e−D(z)D(¯z)F =

1 ∂t ∇(z)F . e 0 z¯z

(2.28) (2.29)

These equations allow one to express the second derivatives ∂tm ∂tn F , ∂tm ∂t¯n F with m, n ≥ 1 through the derivatives ∂t0 ∂tk F , ∂t0 ∂t¯k F . In particular, the dispersionless Toda equation, 2

∂t1 ∂t¯1 F = e∂t0 F

(2.30)

which follows from (2.29) as z → ∞, expresses ∂t1 ∂t¯1 F through ∂t20 F . For a comprehensive exposition of Hirota equations for dispersionless KP and Toda hierarchies we refer the reader to [16, 17]. 2.5. Integral representation of the tau-function. Equation (2.16) allows one to obtain a ˜ representation of the tau-function as a double integral over the domain D. Set (z) := ∇(z)F . One is able to determine this function via its variation under the elementary deformation: ˜ δa (z) (2.31) = −2 log a −1 − z−1 + 2G(a, z) ˜ with the which is read from Eq. (2.16) by virtue of (2.12). This allows one to identify ˜ “modified potential” (z) = (z) − (0) + t0 log |z|2 , where is given by (2.4). Thus we can write vk 2 ˜ ∇(z)F = (z) = − z−k . (2.32) log |z−1 − ζ −1 |d 2 ζ = v0 + 2Re π D k k>0

The last equality is to be understood as the Taylor expansion around infinity. The coefficients vk are moments of the interior domain (the “dual” harmonic moments) defined as 1 2 vk = zk d 2 z (k > 0) , v0 = −(0) = log |z|d 2 z . (2.33) π D π D From (2.32) it is clear that vk = ∂tk F ,

k ≥ 0,

(2.34)

12


i.e., the moments of the complementary domain D (the “dual” moments) are completely determined by the function F of harmonic moments of Dc . In a similar manner, one arrives at the integral representation of the tau-function. Comparing (2.32) with (2.31) one can easily notice that the meaning of the elementary deformation δξ or the operator ∇(ξ ) formally applied at the boundary point ξ ∈ ∂D (where G(z, ξ ) = 0) is attaching a “small piece” to the integral over the domain D (the “bump” operator from [2]). Using this fact and interpreting (2.32) as a variation δz F we arrive at the double-integral representation of the tau-function 1 F =− 2 log |z−1 − ζ −1 |d 2 zd 2 ζ (2.35) π D D or F =

1 2π

D

2 ˜ (z)d z=

1 2π

((z) − 2(0)) d 2 z .

(2.36)

D

As we see below, the main formulas from this paragraph remain intact in the multiplyconnected case. 3. The Dirichlet Problem and the Tau-Function in the Multiply-Connected Case Let now Dα , α = 0, 1, . . . , g, be a collection of g + 1 non-intersecting bounded cong nected domains in the complex plane with smooth boundaries ∂Dα . Set D = ∪α=0 Dα , c so that the complement D = C \ D becomes a multiply-connected unbounded domain in the complex plane (see Fig. 3). Let bα be the boundary curves. They are assumed to g be positively oriented as boundaries of Dc , so that ∪α=0 bα = ∂Dc while bα = −∂Dα has the clockwise orientation. Comparing to the simply-connected case, nothing is changed in posing the standard Dirichlet problem. The definition of the Green function and the formula (1.1) for the solution of the Dirichlet problem through the Green function remain to be the same. A difference is in the nature of harmonic functions. Any harmonic function is still the real part of an analytic function but in the multiply-connected case these analytic functions are not necessarily single-valued (only their real parts have to be single-valued). In other words, the harmonic functions may have non-zero “periods” over non-trivial cycles1 . In our case, the non-trivial cycles are the boundary curves bα . In general, the Green function has non-zero “periods” over all boundary contours. Hence it is natural to introduce new objects, specific to the multiply-connected case, which are defined as “periods” of the Green function. First, the harmonic measure ωα (z) of the boundary component bα is the harmonic function in Dc such that it is equal to unity on bα and vanishes on the other boundary curves. Thus the harmonic measure is the solution to the particular Dirichlet problem. From the general formula (1.1) we conclude that 1 ωα (z) = − ∂n G(z, ζ )|dζ |, α = 1, . . . , g (3.1) 2π bα so the harmonic measure is the period of the Green function w.r.t. one of its arguments. From the maximum principle for harmonic functions it follows that 0 < ωα (z) < 1 1

Here and below by “periods” of a harmonic function f we mean the integrals trivial cycles.

∂n f dl over non-


13

D2 b2

z2

D0 z1 b1

z =0 0

b0

D1

z

3

C\D b3

D3

Fig. 3. A multiply-connected domain Dc = C \ D for g = 3. The domain D = 3α=0 Dα consists of g + 1 = 4 disconnected parts Dα with the boundaries bα . To define the complete set of harmonic moments, we also need the auxiliary points zα ∈ Dα which should be always located inside the corresponding domains

g in internal points. Obviously, α=0 ωα (z) = 1. In what follows we consider the linear independent functions ωα (z) with α = 1, . . . , g. Further, taking “periods” in the remaining variable, we define 1 αβ = − ∂n ωα (ζ )|dζ |, α, β = 1, . . . , g . (3.2) 2π bβ The matrix αβ is known to be symmetric, non-degenerate and positively-definite. It will be clear below that the matrix Tαβ = iπ αβ can be identified with the matrix of periods of holomorphic differentials on the Schottky double of the domain Dc (see formula (3.5)). For brevity, we refer to both Tαβ and αβ as period matrices. For the harmonic measure and the period matrix there are variational formulas similar to the Hadamard formula (1.4). They can be derived either by a direct variation of (3.1) and (3.2) using the Hadamard formula or (much easier) by a “pictorial” argument like in Fig. 1. The formulas are: 1 δωα (z) = ∂n G(z, ξ ) ∂n ωα (ξ ) δn(ξ ) |dξ | , (3.3) 2π ∂D δ αβ

1 = 2π

∂n ωα (ξ ) ∂n ωβ (ξ ) δn(ξ ) |dξ | .

(3.4)

∂D

3.1. The Schottky double. It is customary to associate with a planar multiply-connected domain its Schottky double (see, e.g., [18], Ch. 2.2), a compact Riemann surface without

14


D0

ξ0 aα ξ

b

α

α

c

D =C\D

Dα

Fig. 4. The domain Dc with the aα -cycle, going one way along the “upper sheet” and back along the “lower sheet” of the Schottky double of Dc . For such a choice one clearly gets the intersection form aα ◦ bβ = δαβ for α, β = 1, . . . , g.

boundary endowed with antiholomorpic involution, the boundary of the initial domain being the set of the fixed points of the involution. The Schottky double of the multiply-connected domain Dc can be thought of as two copies of Dc (“upper” and “lower” g sheets of the double) glued along the boundaries ∪α=0 bα = ∂Dc , with points at infin¯ In this set-up the holomorphic coordinate on the upper sheet is ity added (∞ and ∞). z inherited from Dc , while the holomorphic coordinate2 on the other sheet is z¯ . The Schottky double of Dc with two infinities added is a compact Riemann surface of genus g = #{Dα } − 1. A meromorphic function on the double is a pair of meromorphic functions f, f˜ on Dc such that f (z) = f˜(¯z) on the boundary. Similarly, a meromorphic differential on the double is a pair of meromorphic differentials f (z)dz and f˜(¯z)d z¯ such that f (z)dz = f˜(¯z)d z¯ along the boundary curves. On the double, one may choose a canonical basis of cycles. We fix the b-cycles to be just the boundaries of the holes bα for α = 1, . . . , g. Note that regarded as the oriented boundaries of Dc (not D) they have the clockwise orientation. The aα -cycle connects the α th hole with the 0th one. To be more precise, fix points ξα on the boundaries, then the aα cycle starts from ξ0 , goes to ξα on the “upper” (holomorphic) sheet of the double and goes back the same way on the “lower” sheet, where the holomorphic coordinate is z¯ , see Fig. 4. Being harmonic, ωα can be represented as the real part of a holomorphic function: ωα (z) = Wα (z) + Wα (z) , where Wα (z) are holomorphic multivalued functions in Dc . The differentials dWα are holomorphic in Dc and purely imaginary on all boundary contours. So they can be 2 More precisely, the proper coordinates should be 1/z (and 1/¯z), which have first order zeros instead ¯ of poles at z = ∞ (and z¯ = ∞).


15

extended holomorphically to the lower sheet as −dWα (z). In fact this is the canonically normalized basis of holomorphic differentials on the double: according to the definitions,

aα

ξα

dWβ =

dWβ (z) +

ξ0

−dWβ (z) = 2Re

ξ0 ξα

ξα

dWβ (z)

ξ0

= ωβ (ξα )−ωβ (ξ0 ) = δαβ . Then the matrix of b-periods of these differentials reads

Tαβ

i = dWβ = − 2 bα

∂n ωβ dl = iπ αβ ,

(3.5)

bα

i.e. the period matrix Tαβ of the Schottky double is a purely imaginary non-degenerate matrix with positively definite imaginary part π αβ (3.2).

3.2. Harmonic moments of multiply-connected domains. One may still use harmonic moments to characterize the shape of a multiply-connected domain. However, the set of harmonic functions should be extended by adding functions with poles in any hole (not only in D0 as before) and functions whose holomorphic parts are not single-valued. To specify them, let us mark points zα ∈ Dα , one in each hole (see Fig. 3). Without loss of generality, it is convenient to put z0 = 0. Then one may consider single-valued 2 analytic functions in Dc of the form (z − zα )−k and harmonic functions log 1 − zzα with multi-valued analytic part. The arguments almost identical to the ones used in the simply-connected case show that the parameters t0 , Mn,α , φα , where as in (2.3) t0 = Area(D)/π , Mn, α = −

1 π

Dc

(z − zα )−n d 2 z,

α = 0, 1, . . . , g, n ≥ 1

(3.6)

together with their complex conjugate, and 1 φα = − π

zα 2 2 log 1 − d z, z Dc

α = 1, . . . , g

(3.7)

uniquely define Dc , i.e. any deformation preserving these parameters is trivial. Note that the extra moments φα are essentially the values of the potential (2.4) at the points zα , φα = (0) − (zα ) − |zα |2 .

(3.8)

A crucial step for what follows is the change of variables from Mn,α to the variables τk which are finite linear combinations of the Mn,α ’s. They can be directly defined as moments with respect to new basis of functions: τ0 = t0 , τk =

1 2πi

Ak (z)¯zdz = − ∂D

1 π

Dc

d 2 zAk (z), k > 0 .

(3.9)

16


The functions Ak (z) are analogous to the Laurent-Fourier type basis on Riemann surfaces introduced in [8]. They are explicitly defined by the following formulas (here the indices α and β are understood modulo g + 1): Am(g+1)+α = R −m (z)

α−1

(z − zβ )−1 ,

β=0

R(z) =

g

(z − zβ ) .

(3.10)

β=0

In a neighbourhood of infinity Ak (z) = z−k + O(z−k−1 ). Any analytic function in Dc vanishing at infinity can be represented as a linear combination of Ak which is convergent in domains such that |R(z)| > const. In the case of one hole (g = 0) the formulas (3.10) give the basis used in the previous section: Ak = z−k . Note that A0 = 1, A1 = 1/z, therefore τ0 = t0 and τ1 = M1,0 = t1 . 3.3. Local coordinates in the space of multiply-connected domains. Now we are going to prove that the parameters τk , φα can be treated as local coordinates in the space of multiply-connected domains. (Here we use the same restricted notion of the local coordinates, as in the simply connected case (see the remark in Sect.2)). It is instructive and simpler first to prove this for another choice of parameters. Instead of φα one may use the areas of the holes 1 Area(Dα ) 1 = d 2z = z¯ dz , α = 1, . . . , g . (3.11) sα = π π Dα 2πi ∂Dα In order to prove that any deformation that preserves τk and sα is trivial, we introduce the basis of differentials dBk which satisfy the defining “orthonormality” relations 1 Ak dBk = δk,k (3.12) 2πi ∂D for all integer k, k . It is easy to see that explicitly they are given by: dBm(g+1)+α =

α−1 dzR m (z) (z − zβ−1 ) , z − zg

(3.13)

β=0

where we identify z−1 ≡ zg . The existence of a well-defined “dual” basis of differentials obeying the orthonormality relation is the key feature of the basis functions Ak , which makes τk good local coordinates comparing to the Mn,α . For the functions (z − zα )−n one cannot define the dual basis. The summation formulas ∞

dzdζ = dζ An (ζ )dBn (z), |R(z)| < |R(ζ )| , ζ −z n=1 ∞

dzdζ dζ A−n (ζ )dB−n (z), |R(z)| > |R(ζ )| , =− ζ −z n=0

(3.14)


17

which can be checked directly, allow us to repeat arguments of Sect. 2. Indeed, the Cauchy integral (2.6), dz C(z)dz = 2πi

ζ¯ dζ , ζ −z

∂D

(3.15)

where the integration now goes along all boundary components, defines in each of the holes Dα analytic differentials C α (z)dz (analogs of C + (z)dz in the simply-connected case). In the complementary domain Dc the Cauchy integral still defines the differential C − (z)dz holomorphic everywhere in Dc except for infinity where it has a simple pole. The difference of the boundary values of the Cauchy integral is equal to z¯ : C α (z) − C − (z) = z¯ ,

z ∈ ∂Dα .

From Eq. (2.7), which can be written separately for each contour, it follows that • The difference of the boundary values

∂t C α (ζ ) − ∂t C − (ζ ) dζ of the derivative of the Cauchy integral (3.15) is, for all α, a purely imaginary differential on the boundary bα . The expansion (3.14) of the Cauchy kernel implies that • If a t-deformation preserves all the moments τk , k ≥ 0, then ∂t ζ¯ dζ − ∂t ζ d ζ¯ extends to a holomorphic differential in Dc . Indeed, since |R(z)| is small for z close enough to any of the points zα , one can expand ∂t C α (z) for any α as ∂t C α (z)dz =

∞ ∞ 1 dBk ∂t Ak (ζ )ζ¯ dζ = ∂t τk dBk (z) , 2πi ∂D k=1

(3.16)

k=1

and conclude that it is identically zero provided ∂t τk = 0. Hence −∂t C − (z)dz is the desired extension of ∂t ζ¯ dζ −∂t ζ d ζ¯ . It has no pole at infinity due to the equation ∂t τ0 = 0. Using the Schwarz symmetry principle we obtain that ∂t C − (z)dz extends to a holomorphic differential on the Schottky double. If the variables sα are also preserved under the t-deformation, then this holomorphic differential has zero periods along all the cycles bα . Therefore, it is identically zero. This completes the proof of the statement that any deformation of the domain preserving τk and sα is trivial. In this proof the variables sα were used only at the last moment in order to show that the extension of ∂t C − (z)dz as a holomorphic differential on the Schottky double is trivial. The variables φα can be used in a similar way. Namely, let us show that if they are preserved under t-deformation then aα -periods of the extension of ∂t C − (z)dz are trivial, and therefore this extension is identically zero. Indeed, the variable φα (3.7) can be represented in the form 2 φα = − Re π

zα

dz 0

Dc

d 2ζ . z−ζ

(3.17)

18


d2ζ α The differential dz z + C − (z))dz for π Dc z−ζ is equal to C (z)dz for z ∈ Dα and (¯ c z ∈ D . Let ξ0 , ξα be the points where the integration path from 0 to zα intersects the boundary contours b0 , bα . Then ξ0 zα ξα C 0 (z)dz + C α (z)dz + (¯z + C − (z))dz . (3.18) φα = −2 Re 0

ξα

ξ0

It is shown above that if a t-deformation preserves the variables τk then all ∂t C α (z)dz = 0. Thus vanishing of the t-derivative ∂t φα = 0 implies ξα ∂t C − (z)dz . (3.19) 0 = −∂t φα = 2 Re ξ0

The r.h.s. of this equation is just the aα -period of the holomorphic extension of the differential ∂t C − (z)dz. φ φ Let us construct the deformations ∂xk and ∂yk of the boundary that change the real or imaginary parts of the variable τk = xk + iyk , k ≥ 1, keeping all the other moments φ φ φ and the variables φα fixed. It is convenient to set ∂τk = 21 (∂xk − i∂yk ). The argument is similar to the proof of the fact that any deformation that preserves all the variables is trivial. φ φ φ φ • Suppose that the deformations ∂xk and ∂yk exist. Then the differential ∂τk ζ¯ dζ −∂τk ζ d ζ¯ c extends from ∂D to the Schottky double . Its extension is a meromorphic differential φ d k with the only pole at the infinity point ∞ on the upper sheet. In a neighborhood of ∞ it has the form

d k (z) = dBk (z) + O(z−2 )dz . φ

(3.20)

φ

The a-periods of d k are equal to φ d k = aα

zα

(3.21)

dBk .

0 φ

First of all, it is clear that the meromorphic differential d k on is uniquely defined by its asymptotics at ∞ and by the normalization (3.21) of its a-periods. To deduce these properties, we notice, using (3.16), that ∂xk C α (z)dz = dBk (z). Therefore, the φ φ differential ∂xk ζ¯ dζ − ∂xk ζ d ζ¯ extends to Dc as d φxk = −∂xk C − (z)dz + dBk .

(3.22)

Using the Schwarz symmetry principle we conclude that it extends to the Schottky douφ ble as a meromorphic differential. Around the two infinities it has the form d xk =

z→∞

dBk + O(z−2 )dz and d xk φ

−d B¯ k + O(¯z−2 )d z¯ . In the same way one gets that

=

¯ z¯ →∞

φ φ the differential ∂yn ζ¯ dζ − ∂yn ζ d ζ¯ extends to the double as a meromorphic differenφ φ tial d yk , which at the two infinities has the form d yk = idBk + O(z−2 )dz and φ d yk

=

¯ z¯ →∞

id B¯ k + O(¯z−2 )d z¯ respectively. Since φ

2d k = d φxk − id φyk ,

z→∞


19

φ

the first statement is proven. From ∂xk φα = 0 and (3.17), (3.22) it follows that ξ0 zα ξα

φ 0 = Re −d xk + dBk + dBk + dBk . 0

Hence

ξ0

aα

In the same way one gets

ξα

d φxk = 2Re

aα

zα

dBk . 0

d φyk

= −2Im

zα

dBk . 0

The last two equations are equivalent to (3.21). Normal displacement of the boundary that accomplishes the deformations can be explicitly found using the following elementary proposition: • Let D(t) be a deformation with real parameter t such that the differential d = ∂t ζ¯ dζ − ∂t ζ d ζ¯ extends to a meromorphic differential d globally defined on the Schottky double . Then the corresponding z normal displacement of the boundary is proportional to normal derivative of Re d at the boundary point ξ : ξ 1 δn(ξ ) = δt ∂n Re d . (3.23) 2 Conversely, if δn(ξ ) = 21 δt ∂n H (ξ ), where H is a real-valued function such that dH = 0 along the boundary contours and ∂z H is meromorphic in Dc then the differential ∂t ζ¯ dζ − ∂t ζ d ζ¯ is meromorphically extendable to the Schottky double as 2∂z H dz on the upper sheet and −2∂z¯ H d z¯ on the lower sheet. In our case normal displacements of the boundary that change xk or yk keeping all the other moments and the variables φα fixed are thus given by ξ ξ 1 1 δn(ξ ) = δxk ∂n Re (3.24) d φxk , δn(ξ ) = δyk ∂n Re d φyk 2 2 respectively. Note that since the differentials d xk (z), d yk (z) (but not d k (z)!) are purely imaginary on the boundaries, d Re xk (z) = d Re yk (z) = 0 along each component of the boundary. With formulas (3.24) at hand, one can directly verify that these deformations indeed change xk or yk only and keep fixed all other moments. We leave this to the reader. φ In terms of the differential d k formulas (3.24) acquire the form ξ ξ φ φ δn(ξ ) = δxk ∂n Re (3.25) d k , δn(ξ ) = −δyk ∂n Im d k (cf. (2.11) for the simply-connected case). Indeed, taking the real part of 2 k (ξ ) = xk (ξ ) − i yk (ξ ), we get 2 Re k (ξ ) = Re xk (ξ ) + Im yk (ξ ). But the normal derivative of Im yk (ξ ) vanishes since, by virtue of the Cauchy-Riemann identities, it is equal

20


to the tangential derivative of the conjugate harmonic function Re yk (ξ ). This proves the first formula in (3.25). The second one is proven in a similar way by taking imaginary part of 2 k (ξ ). φ φ The construction of the vector fields ∂τ0 (which changes τ0 only) and ∂α (which changes φα only) is quite similar and even simpler since the derivative (3.16) vanishes. So, we present the results without going into details. φ

• The deformation ∂τ0 corresponds to the normal displacement 1 δn(ξ ) = − δτ0 ∂n G(∞, ξ ) . 2 φ φ The differential −(∂τ0 ζ¯ dζ − ∂τ0 ζ d ζ¯ ) extends from ∂Dc to the Schottky double . Its extension is a meromorphic third-kind Abelian differential d 0 which has simple poles at the infinities on the two sheets of the Schottky double (with residues ±1) and vanishing a-periods. φ • The deformation ∂α corresponds to the normal displacement

δn(ξ ) =

1 δφα ∂n ωα (ξ ) , 4

where ωα is the harmonic measure of the boundary component bα (see (3.1)). The φ φ differential ∂α ζ¯ dζ − ∂α ζ d ζ¯ holomorphically extends from ∂Dc to the Schottky double . Its extension is the canonically normalized holomorphic differential dWα = ∂z ωα (z)dz on the upper sheet (and dWα = −∂z¯ ωα (z)d z¯ on the lower sheet). φ

φ

φ

φ

So we see that ∂xk , ∂yk , ∂τ0 and ∂α are well-defined vector fields on the space of multiply-connected domains. This fact allows us to treat φα , τk as local coordinates on this space. At this stage it becomes clear why we prefer to use the moments τk rather than Mk,α . Although the latter are finite linear combinations of the former, they can not be treated as local coordinates because the vector fields ∂/∂Mk,α , being in general infinite φ linear combinations of the ∂τk , are not well-defined. 3.4. -variables. Up to now the roles of the variables sα and φα have been in some sense dual to each other. It is necessary to emphasize that this duality does not go beyond the framework of our proof of the statement that the first or the second sets together with the variables τk are local coordinates in the space of multiply-connected domains. For analytic boundary curves one can define the Schwarz function, which is a unique function analytic in some strip-like neighborhoods of all boundaries such that S(z) = z¯

on the boundary curves .

(3.26)

Then the variables sα are b-periods of the differential S(z)dz. At the same time, the variables φα in general can not be identified with periods of this differential (or its extension) over any cycles on the Schottky double. Now we are going to introduce new variables, α , which can be called virtual a-periods of the differential S(z)dz on the Schottky double, since in all the cases when the Schwarz function has a meromorphic extension to the double they indeed coincide with the a-periods of the corresponding differential (see below in this section).


Let us consider the differential φ

d k = d k −

21

Bk (zα )dWα

(k ≥ 1) ,

(3.27)

α

where

z

Bk (z) =

dBk

(3.28)

0

is a polynomial of degree k. It is a meromorphic differential on with the only pole at ∞ on the upper sheet, where it has the form d k (z) = dBk (z) + O(z−2 )dz. From (3.21) it is clear that the differential d k has vanishing a-periods d k = 0 ,

(3.29)

(3.30)

aα

i.e. it is a canonically normalized meromorphic differential. The normal displacements of the boundary given by real and imaginary parts of the normal derivative ∂n k define a complex tangent vector field Bk (zα )∂αφ (3.31) ∂τk = ∂τφk − α

to the space of multiply-connected domains. These vector fields keep fixed the formal variable Bk (zα )τk . (3.32) α = φα + 2Re k

In a general situation this variable is only a formal one because the sum generally does not converge. Thus, we call α the virtual a-period of the Schwarz differential S(z)dz, since in the case when the Schwarz function has a meromorphic extension to the double , the sum does converge and the corresponding quantity does coincide with the aα -period of the extension of the Schwarz differential. 3.5. Elementary deformations and the operator ∇(z). Like in Sect. 2, we introduce the elementary deformations δa δ (α)

with δn(ξ ) = − ∂n G(a, ξ ) , a ∈ Dc , 2 with δn(ξ ) = − ∂n ωα (ξ ) α = 1, . . . , g , 2

(3.33)

where ωα (z) is the harmonic measure of the boundary component bα (see (3.1)). The deformations δ (α) were considered in [19] in connection with the so-called quadrature domains [19, 20]. In complete analogy with Sect. 2 one can derive the following formulas for variations of the local coordinates under elementary deformations: zα 2 δa τk = Ak (a), δa φα = log 1− , δ (α) τk = 0, δ (α) φβ = −2δαβ . (3.34) a

22


The first two formulas are particular cases of 2 h(ζ )d ζ = h(ζ )∂n G(a, ζ )|dζ | = −π h(a) δa 2 ∂Dc Dc which is valid for any harmonic function h in Dc (the last equality is just the formula for solution of the Dirichlet problem). Similarly, (α) 2 δ h(ζ )d ζ = h∂n ωα |dζ | = ∂n h |dζ | = −i ∂ζ h dζ 2 ∂Dc 2 bα Dc ∂Dα (the Green formula was used), and the last two formulas in (3.34) correspond to the particular choices of h(z). Variations of the variables α (in the case when they are well-defined) then read: δz α = 0,

δ (α) β = −2δαβ .

(3.35)

Therefore, for any functional X on the space of the multiply-connected domains the following equations hold: δz X = ∇(z)X ,

(3.36)

δ (α) X = −2∂αφ X = −2∂α X .

(3.37)

The differential operator ∇(z) in the multiply-connected case is defined by the formula Ak (z)∂τk + Ak (z)∂τ (3.38) ∇(z) = ∂τ0 + ¯k . k≥1

The functional X can be regarded as a function X = Xφ (φα , τk ) on the space of the local coordinates φα , τk , or as a function X = X (α , τk ) on the space of the local coordinates α , τk . We would like to stress once again, that although in the latter case the variables α are formal their variations under elementary deformations and the vector-fields ∂τk , which keep them fixed, are well-defined. For completeness, let us characterize elementary deformations δa in terms of meromorphic differentials on the Schottky double (as we have already seen, deformations δ (α) correspond to holomorphic differentials). • Let ∂t (a) be the vector field in the space of multiply-connected domains corresponding to the elementary deformation δa . Then the differential −(∂t (a) ζ¯ dζ −∂t (a) ζ d ζ¯ ) extends from ∂Dc to the Schottky double . Its extension is a meromorphic third-kind Abelian ¯ which has simple poles at the points a and a differential d (a,a) ¯ on the two sheets of the Schottky double (with residues ±1) and vanishing a-periods. In terms of the Green function we have: 2∂z G(a, z)dz on the upper sheet (a,a) ¯ d = −2∂z¯ G(a, z)dz on the lower sheet (cf. (3.23) and (3.33)). Note that the differential d 0 introduced before coincides with ¯ . d (∞,∞)


23

Let K(z, ζ )dζ be a unique meromorphic Abelian differential of the third kind on with simple poles at z and ∞ on the upper sheet with residues ±1 normalized to zero a-periods. (Note that as a function of the variable z it is multi-valued on .) Then 2∂ζ G(z, ζ )dζ − 2∂ζ G(∞, ζ )dζ = K(z, ζ )dζ + K(¯z, ζ )dζ and the differential d k (ζ ) can be represented in the form dζ d k (ζ ) = K(u, ζ )dBk (u) , 2πi ∞

(3.39)

(3.40)

where the u-integration goes along a big circle around infinity. Using (3.14) we obtain that K(u, ζ )du dζ Ak (z)d k (ζ ) = − = K(z, ζ )dζ . (3.41) 2πi ∞ u − z k≥1

Therefore, the following expansion of the derivative of the Green function holds: ¯ k (ζ ) . 2∂ζ G(z, ζ )dζ = d 0 (ζ ) − (3.42) Ak (z)d k (ζ ) + Ak (z)d k≥1

¯ k is a unique meromorphic differential on with the only pole at infinity on Here d the lower sheet with the principal part −dBk (z) and vanishing a-periods. This formula generalizes Eq. (3.8) from [2] to the multiply-connected case.

3.6. The F -function. Applying the variational formulas (1.4), (3.3), (3.4), we can find variations of the Green function, harmonic measure and period matrix under the elementary deformations. In this way we obtain a number of important relations which connect elementary deformations of these objects: δa G(b, c) = δb G(c, a) = δc G(a, b) , δa ωα (b) = δ (α) G(a, b) = δb ωα (a) , δ (α) ωβ (z) = δ (β) ωα (z) , δz αβ = δ (α) ωβ (z) , δ (α) βγ = δ (β) γ α = δ (γ ) αβ .

(3.43)

From (3.36), (3.37) it follows that the formulas (3.43) can be rewritten in terms of the differential operators ∇(z) and ∂α := ∂/∂φα = ∂/∂α : ∇(a)G(b, c) = ∇(b)G(c, a) = ∇(c)G(a, b) , ∇(a)ωα (b) = −2∂α G(a, b) , ∂α ωβ (z) = ∂β ωα (z) , ∇(z) αβ = −2∂α ωβ (z) , ∂α βγ = ∂β γ α = ∂γ αβ .

(3.44)

These integrability relations generalize formulas (2.15) to the multiply-connected case. The first line just coincides with (2.15) while the other ones extend the symmetry of the derivatives to the harmonic measure and the period matrix.

24


Again, (3.44) can be regarded as a set of compatibility conditions of an infinite hierarchy of differential equations. They imply that there exists a function F = F (α , τ ) such that 1 G(a, b) = log a −1 − b−1 + ∇(a)∇(b)F , (3.45) 2 ωα (z) = − ∂α ∇(z)F ,

(3.46)

Tαβ = iπ αβ = 2πi ∂α ∂β F .

(3.47)

The function F is the (logarithm of the) tau-function of multiply-connected domains.

3.7. Dual moments and integral representation of the tau-function. To obtain the integral representation of the function F , we proceed exactly in the same manner as in Sect. 2 (see also [2] for more details). ˜ ˜ Again, set (z) = ∇(z)F . Equations (3.45) and (3.46) determine the function (z) for z ∈ Dc via its variations under the elementary deformations: ˜ δa (z) = −2 log a −1 − z−1 + 2G(a, z) , (3.48) ˜ δ (α) (z) = 2ωα (z) . It is easy to verify that the function 2 ˜ (z) =− log |z−1 − ζ −1 |d 2 ζ = (z) − (0) + τ0 log |z|2 π D

(3.49)

satisfies (3.48). Indeed, using (3.33), variation of (3.49) reads 2 −1 −1 2 δa − log |z − ζ |d ζ = |dξ |∂n G(a, ξ ) log |z−1 − ξ −1 | π D π ∂Dc = |dξ |∂n G(a, ξ ) log |z−1 − ξ −1 | − G(z, ξ ) π ∂Dc = −2 log a −1 − z−1 + 2G(a, z) , where we have used properties of the Dirichlet Green function and the fact that the Dirichlet formula restores harmonic function from its value at the boundary. Similarly, for z ∈ Dc we obtain: 2 (α) −1 −1 2 δ − log |z − ζ |d ζ = |dξ |∂n ωα (ξ ) log |z−1 − ξ −1 | π D π ∂Dc = |dξ |∂n ωα (ξ ) log |z−1 − ξ −1 | − G(z, ξ ) π ∂Dc = |dξ |∂n log |z−1 − ξ −1 | − G(z, ξ ) π bα = 2ωα (z) .


The same calculation for z ∈ D yields 0 (α) ˜ δ (z) = 2δαβ

25

if z ∈ D0 . if z ∈ Dβ , β = 1, . . . , g

(3.50)

˜ given by (2.32), where D is We see that the expression in (3.49) coincides with ˜ at infinity now understood as the union of all Dα ’s. The coefficients of an expansion of define the dual moments νk : 2 ˜ ∇(z)F = (z) =− log |z−1 − ζ −1 |d 2 ζ = v0 + 2Re νk Ak (z) . (3.51) π D k>0

The coefficients in the r.h.s. of (3.51) are moments of the union of the interior domains with respect to the dual basis 1 νk = Bk (z)d 2 z . (3.52) π D From Eq. (3.51) it follows that νk = ∂τk F .

(3.53)

The same arguments show that the derivatives sα := − ∂α F

(3.54)

are just areas of the holes (3.11). Indeed, Eqs. (3.46), (3.47) determine these quantities via their variations: δa sα = ωα (a), δ (β) sα = αβ . A direct check, using (3.33), shows that 1 1 2 2 Area(Dα ) ∇(z)F = − log − d ζ , ∂α F = − . (3.55) π D z ζ π For example, 1 (α) 2 ˜ (z)d δ z 2π D 1 ˜ ˜ ) d 2ζ . =− |dξ |∂n ωα (ξ )(ξ ) + δ (α) (ζ 4π ∂Dc 2π D

δ (α) F =

In the last term we use (3.50) and obtain the result: ˜ )+ ˜ d 2 ζ + sα = 2sα δ (α) F = − |dξ |∂n (ξ d 2ζ = − 4π bα π Dα 4π Dα (here = 4∂z ∂z¯ is the Laplace operator). The integral representation of F is found in the same way through its variations which are read from (3.55). The result is given by the same formulas (2.35) and (2.36) as in g the simply-connected case with the understanding that D = ∪α=0 Dα is now the union of all Dα ’s.

26


3.8. Algebraic domains. In what follows we restrict our analysis by the class of algebraic domains. In the simply-connected case dealt with in the previous section the algebraic domains are simply images of the exterior of the unit disk under one-to-one conformal maps given by rational functions whose singularities are all in the other “half” of the plane, i.e. inside the unit circle. Note that the boundary of the unit circle is the set of fixed points of the inversion w → 1/w¯ which is the antiholomorphic involution of the w-plane compactified by a point at infinity (the Riemann sphere). Planar multiply-connected algebraic domains can be defined as the domains for which the Schwartz function has a meromorphic extension to a higher genus Riemann surface (a complex algebraic curve) with antiholomorphic involution. More precisely, let be a real Riemann surface by which we mean a complex algebraic curve of genus g with an antiholomorphic involution such that the set of fixed points consists of exactly g + 1 closed contours (such curves are sometimes called M-curves). Then can be naturally divided in two “halves” (say upper and lower sheets) which are interchanged by the involution. Algebraic domains with g holes in the plane can be defined as images of the upper half of the real Riemann surface under bijective conformal maps given by rational (meromorphic) functions on (see, e.g. [21]). For the purpose of this paper, it is convenient to use another, more direct characterization of algebraic domains. The domain Dc is algebraic if and only if the Cauchy integrals (3.15) 1 ζ¯ dζ α C (z) = for z ∈ Dα 2πi ∂D ζ − z are extendable to a rational (meromorphic) function J (z) on the whole complex plane with a marked point at infinity (see [21]). It is important to stress that this function is required to be the same for all α. The equality S(z) = J (z) − C − (z) valid by definition for z ∈ ∂Dc can be used for analytic extension of the Schwarz function. The function C − (z) is analytic in Dc . Therefore, J (z) and S(z) have the same singular parts at their poles in Dc . One may treat S(z) as a function on the Schottky double extending it to the lower sheet as z¯ . It is also convenient to introduce z J (z)dz (3.56) V (z) = 0

which is multi-valued if J (z) has simple poles (to fix a single-valued branch, we make cuts from ∞ to all simple poles of J (z)). In fact we need only the real part of V (z). In neighborhoods of the points zα one has J (z)dz = τk dBk (z), V (z) = τk Bk (z) . (3.57) k≥1

k≥1

The formula (3.32) α = φα + 2 Re V (zα ),

zα ∈ Dα

(3.58)

shows that for the algebraic domains the variables α , introduced in the general case as formal quantities, are well-defined. It is easy to show that they are equal to the a-periods of the differential S(z)dz on the Schottky double . Indeed, using the fact that C 0 (z)


27

and C α (z) represent restrictions of the same function J (z), one can rewrite (3.18) in the form zα ξα φα = −2 Re J (z)dz + (¯z + C − (z) − J (z))dz . 0

ξ0

Under the second integral we recognize the Schwarz function. Combining this equality with the definition of α (3.32), we obtain: ξα ξα α = 2 Re (S(z) − z¯ )dz = S(z)dz . (3.59) S(z)dz − z¯ dS(z) = ξ0

aα

ξ0

As an example of algebraic domains, it is instructive to consider the case when only a finite-number of the moments τk are non-zero. Let AN be the space of multiply-connected domains such that τk = 0,

(3.60)

k>N.

Then the arguments similar to the ones used above show that • S(z) extends to a meromorphic function on with a pole of order N − 1 at ∞ and a ¯ simple pole at ∞. The function z extended to the lower sheet of the Schottky double as S(z) has a simple ¯ For a domain Dc ∈ AN the moments with pole at ∞ and a pole of order N − 1 at ∞. respect to the Laurent basis (cf. (2.1)) 1 tk = − z−k d 2 z (3.61) πk Dc coincide with the coefficients of the expansion of the Schwarz function near ∞: S(z) =

N

ktk zk−1 + O(z−1 ),

z → ∞.

(3.62)

k=1

The normal displacement of the bondary of an algebraic domain, which changes the variable tk keeping all zthe other moments (and α ) fixed is defined by normal derivative ˜ k . Here d ˜ k is a unique normalized meromorphic differential d of the function 2 Re on with the only pole at ∞ of the form k = d(zk + O(z−1 )), k = 0 . d d (3.63) aα

k is well-defined for a generic, not necessarily algebraic Note that the differential d z ˜ k defines a tangent d domain. Therefore, the normal derivative of the function 2 Re vector field ∂t on the whole space of multiply-connected domains. k The space AN is a particular case of algebraic orbits of the universal Whitham hierarchy. In this case the general formula (7.42) from [4] for the τ -function of the Whitham hierarchy, after proper change of the notation, acquires the form 1 1 2F = − τ02 + τ0 v0 + (2 − k)(τk νk + τ¯k ν¯ k ) − α sα 2 2 g

k≥1

α=1

(3.64)

28


which is a quasi-homogeneity condition obeyed by F (compare with formula (5.11) from [2]). Let d 0 be a unique normalized meromorphic differential on with simple poles ¯ Its Abelian integral at the infinities ∞ and ∞. z log w(z) = d 0 (3.65) ξ0

defines in the neighborhood of ∞ a function w(z) which has a simple pole at infinity. The dependence of the inverse function z(w) on the variables tk is described by the Whitham equations for the two-dimensional Toda lattice hierarchy. These equations have the form k (w), z(w)} := ∂t z(w) = { k

k (w) d dz k (w) ∂t0 z(w) − ∂t0 . d log w d log w

(3.66)

Algebraic domains of a more general form correspond to the universal Whitham hierarchy. Let AN1 ,... ,Nl be the space of domains such that the extension of the Schwarz function S(z) to Dc has poles of orders Nj − 1 at some points zj (which possibly ¯ Then, according to [4], the variables include ∞ and ∞). zj 1 t0,j = S(z)dz, tk,j = reszj (z − zj )k−1 S(z)dz, k = 1, . . . , Nj − 1 (3.67) k ξ0 together with the variables sα (or α ) provide a set of local Whitham coordinates on the space AN1 ,... ,Nl . Note that the definition of the algebraic orbits of the universal Whitham hierarchy is a bit more general than the definition of algebraic domains given above. It corresponds to the case when the differential dS of the Schwarz function is extendable to Dc as a meromorphic differential (in [21] such domains are called Abelian domains). (1) For example, let AN be the space of multiply-connected domains such that (1) Tk

=

Ak dS =

∂D

S (z)Ak dz = − ∂D

∂D

Ak S(z)dz = 0,

k > N . (3.68)

This space is characterized by the following property: there are constants Kα such that S(z) + Kα extends to a meromorphic function on Dc with a pole of order N at ∞. The variables 1 Kα , sα , tk = z−k z¯ dz, k ≥ 1, (3.69) 2πik ∂D (1)

are local coordinates on AN . The two cases when S(z) or its derivative S (z) have a meromorphic extension to c D are particular examples of the whole hierarchy of integrable domains, which can be defined in a similar way by the condition that the mth order derivative of the Schwarz function S(z) admits a meromorphic extension to the Schottky double . For example, (2) let AN be the space of multiply-connected domains such that (2)

Tk

= ∂D

S (z)Ak dz = −

∂D

Ak S(z)dz = 0, k > N .

(3.70)


29

This space is characterized by the following property: there are linear functions kα (z) = Kα0 + Kα1 z such that S(z) + kα (z) extends to a meromorphic function on Dc with a pole of order N + 1 at ∞. The variables 1 Kα (z), sα , tk = z−k z¯ dz, k ≥ 1, (3.71) 2πik ∂D (2)

(m)

are local coordinates on AN . The other spaces AN with m > 2 can be defined in a similar way. 4. The Duality Transformation The independent variables α (3.59) or φα used in the previous section are not as transparent as the dual variables sα (3.11), which are simply areas of the holes Dα . In this section we show how to pass to the set of independent variables s1 , . . . , sg (3.11) (together with the infinite set of τk ’s). This transformation is similar to the passing from “external” to “internal” moments in the simply-connected case (see Sect. 5 of [2]). The difference is that only a finite number of times are subject to the transformation while the infinite set of τk ’s remains the same. The change to the variables sα can be done in the general case of domains with smooth boundaries. However, it is the change α → sα rather than φα → sα that leads to a transparent duality. Since α ’s are only defined as formal (“virtual”) variables for domains with smooth boundaries, we shall restrict our consideration to the class of algebraic domains discussed at the end of the previous section. In this case the variables α are well defined. 4.1. The Legendre transform. Passing from α to sα is a particular duality transformation which is equivalent to the interchanging of the a and b cycles on the Schottky double . This is achieved by the (partial) Legendre transform F (α , τ ) −→ F˜ (sα , τ ), where F˜ = F +

g

(4.1)

α sα .

α=1

The function F˜ is the “dual” tau-function. Below in this section, it is shown that F˜ solves the modified Dirichlet problem and can be identified with the free energy of a matrix model in the planar large N limit in the case when the support of eigenvalues consists of a few disconnected domains (a so-called multi-support solution, see [22] and references therein). The main properties of F˜follow from those of F . According to (2.34), (3.55) we have dF = − α sα dα + k νk dτk (for brevity, k is assumed to run over all integer values, τ−k ≡ τ¯k , etc.), so d F˜ = α α dsα + k νk dτk . This gives the first order derivatives: ∂ F˜ ∂ F˜ α = , νk = . (4.2) ∂sα ∂τk The second order derivatives are transformed as follows (see e.g. [23]). Set Fαβ =

∂ 2F , ∂α ∂β

Fαk =

∂ 2F , ∂α ∂τk

Fik =

∂ 2F ∂τi ∂τk

30


and similarly for F˜ . Then Fαβ = −(F˜ −1 )αβ , g (F˜ −1 )αγ F˜γ k , Fαk = γ =1

Fik = F˜ik −

(4.3) g

F˜iγ (F˜ −1 )γ γ F˜γ k .

γ ,γ =1

Here (F˜ −1 )αβ means the matrix element of the matrix inverse to the g × g matrix F˜αβ . Using these formulas, it is easy to see that the main properties (3.45), (3.46) and (3.47) of the tau-function are translated to the dual tau-function as follows: 1 ˜ G(a, b) = log |a −1 − b−1 | + ∇(a)∇(b)F˜ , 2

(4.4)

2πi ω˜ α (z) = − ∂sα∇(z)F˜ ,

(4.5)

∂ 2 F˜ 2πi T˜αβ = , ∂sα ∂sβ

(4.6)

where τk -derivatives in ∇(z) are taken at fixed sα . The objects in the left-hand sides of these relations are: ˜ G(a, b) = G(a, b) + iπ

g

ωα (a)T˜αβ ωβ (b) ,

(4.7)

α,β=1

ω˜ α (z) =

g

T˜αβ ωβ (z) ,

(4.8)

β=1

i T˜ = −T −1 = −1 . π

(4.9)

˜ is the Green function of the modified Dirichlet problem to be discussed The function G below. The matrix T˜ is the matrix of a-periods of the holomorphic differentials d W˜ α on the double (so that ω˜ α (z) = W˜ α (z) + W˜ α (z)), normalized with respect to the b-cycles − d W˜ β = δαβ , d W˜ β = T˜αβ , bα

aα

i.e. more precisely, the change of cycles is aα → bα , bα → −aα . An important remark is in order. By a simple rescaling of the independent variables one is able to write the group of relations (4.4)–(4.6) for the function F˜ in exactly the same form as the ones for the function F (3.45)–(3.47), so that they differ merely by the notation. We use this fact in Sect. 5.


31

4.2. The modified Dirichlet problem. The modified Green function (4.7) solves the modified Dirichlet problem which can be formulated in the following way. One may eliminate all except for one of the periods of the Green function G, thus making it similar, in this respect, to the Green function of a simply-connected domain (recall that the latter has the non-zero period 2π over the only boundary curve b0 ). This leads to the following modified Dirichlet problem (see e.g. [24]): given a function u0 (z) on the boundary, to find a harmonic function u(z) in Dc such that it is continuous up to the boundary and equals u0 (z) + Cα on the α’s boundary component. Here, Cα ’s are some constants. It is important to stress that they are not given a priori but have to be determined from the condition that the solution u(z) has vanishing periods over the boundaries b1 , . . . , bg . One of these constants can be put equal to zero. We set C0 = 0. This problem also has a unique solution. It is given by the same formula (1.1) in terms ˜ of the modified Green function G(z, ζ ). The definition of the latter is similar to that of the G(z, ζ ) but differs in two respects: ˜ ˜ (G1) G(z, ζ ) is required to have zero periods over the boundaries b1 , . . . , bg ; ˜ ˜ ˜ (G2) The derivative of G(z, ζ ) along the boundary (not G(z, ζ ) itself!) vanishes on the boundary. ˜ Under the condition that G(z, ζ ) = 0 on b0 such a function is unique. The function given by (4.7) just meets these requirements. We conclude that the modified Green function is expressed through the dual tau-function F˜ . Note that variations of the modified Green function under small deformations of the domain are described by the same Hadamard formula (1.4), where each Green function ˜ This follows, after some algebra, from the formula for G ˜ in terms of is replaced by G. G, ωα and αβ . Therefore, all the arguments of Sect. 3 could be repeated in a completely parallel way starting from the modified Dirichlet problem. One may also say that ˜ differ merely by a preferred basis of cycles on the double: the the functions G and G ˜ has vanishing differential ∂z Gdz has vanishing periods over the a-cycles while ∂z Gdz periods over the b-cycles. ˜ The function wa (z) such that G(z, a) = − log |wa (z)| maps Dc onto the exterior of the unit circle which is slit along g concentric circular arcs (see Fig. 5). Since the periods

˜ Fig. 5. The image of a triply-connected domain under the conformal map wa (z) such that G(z, a) = − log |wa (z)|. The function wa maps the domain onto the exterior of the unit (dashed) circle with g = 3 concentric circular cuts. Positions and lengths of the arcs depend on the shape of Dc in Fig. 3 and depend also on the normalization point z = a which is mapped to ∞

32


˜ vanish, the function wa is single-valued. Positions of the arcs depend of the function G on the shape of Dc as well as on the point a which is mapped to ∞. The radii of the arcs, Rα , are expressed through the dual tau-function as log Rα2 = ∂sα∇(a)F˜ .

(4.10)

In particular, for a = ∞ we have log Rα2 = ∂sα∂τ0 F˜ (cf. Eq. (2.20) for the conformal radius).

4.3. Relation to multi-support solutions of matrix models. The partition function of the model3 , written as an integral over eigenvalues, reads:   N N N 1 1 ZN = exp  log |zi − zj |2 + U (zi ) d 2 zj . (4.11) N! i<j

i=1

j =1

The matrix model potential U is usually chosen to be of the form U (z) = −z¯z + V (z) + V (z) ,

(4.12)

where V (z) is at the moment some polynomial; however, we will see immediately that the coincidence of the notation with (3.57) is not accidental. The parameter , in the large N limit, tends to zero simultaneously with N → ∞ in such a way that t0 = N is kept finite and fixed. In the leading order, one can apply the saddle point method. The saddle point condition is

N j =1,=i

1 + ∂zi U (zi ) = 0 . zi − z j

The density of eigenvalues is a sum of two-dimensional delta-functions: ρ(z) = π δ (2) (z − zi ) . i

In the large N limit, one treats it as a continuous function normalized as t0 . In terms of the continuous density the saddle point equation reads 1 π

ρ(ζ )d 2 ζ + ∂z U (z) = 0 . z−ζ

1 π

ρ(z)d 2 z =

(4.13)

The solution is well known and easy to obtain: the extremal density ρ0 (z) is constant in some domain D (the support of eigenvalues which can be a union of disconnected domains Dα ) and zero otherwise. More precisely, 1 z∈D ρ0 (z) = . 0 z ∈ C \ D ≡ Dc 3

One may have in mind the model of all complex or mutually hermitian conjugated or normal matrices.


33

Note that the saddle point condition is imposed only for z inside the support of eigenvalues. Writing (cf. (3.57)) V (z) =

p

τk Bk (z) ,

k=1

multiplying Eq. (4.13) by Ak (z) and integrating it over the boundary of the support of eigenvalues, one finds that the domain D is such that the coefficients τk are moments of its complement with respect to the basis functions Ak and the higher moments (with numbers greater than p) vanish. As is proven in Sect. 3, these conditions, together with the normalization condition for the density, locally determine the shape of the support of p eigenvalues. Equivalently, one might parametrize the polynomial as V (z) = k=1 tk zk , −k then tk are moments of the C \ D with respect to the functions z . An interesting problem is to obtain, for a given value of t0 , necessary and sufficient conditions on the polynomial V for the support of eigenvalues to be a union of g + 1 disconnected domains (”droplets”) with non-zero filling. One may approach this problem from a “classical” limit of very small (point-like) droplets. Clearly, for our choice of the signs in (4.11) and (4.12), the stable point-like droplets are located at minima of −U (z), or equivalently at maxima of U (z). As soon as we use the basis which explicitly depends on the marked points zα ∈ Dα , it is natural to consider the germ configuration with point-like eigenvalue droplets at the points zα . It is easy to see that the sufficient conditions for the potential (4.12) to have maxima at the points zα are z¯ α = V (zα ),

|V (zα )| < 1

for all α .

(4.14)

The first one means that there is an extremum of the potential U (z) at the point zα . The second one ensures that eigenvalues of the matrix of second derivatives of the potential at the extremum are both negative, i.e. the extremum is actually a maximum of U (z). The first condition literally coincides with the one used in [22] for the completely degenerate curve (point-like droplets) while the second one requires that not all extrema are filled, so that the “smooth” genus g is always less than the maximal possible genus (p−1)2 −1. Let now V0 (z) be a minimal degree polynomial that obeys these conditions. Then perturbed polynomials of the form V (z) = V0 (z) + τk Bk (z) k≥g+2

obey the same conditions for sufficiently small τk , and this is the advantage of using the basis (3.10), (3.13) from the perspective of matrix models. The saddle point equation (4.13) just means that the “effective potential” π1 D log |z− ζ |2 d 2 ζ + U (z) is constant in each Dα , i.e. for z ∈ Dα it holds: 2 log |z − ζ |d 2 ζ + U (z) = v0 + α , (4.15) π D where v0 = π2 D log |z|d 2 z, and so in our normalization 0 = 0. Let Nα be the number of eigenvalues in Dα , then limN→∞ Nα = sα . First we find the free energy with some fixed sα : 1 lim 2 log ZN ρ0 (z) log |z − ζ |ρ0 (ζ )d 2 zd 2 ζ = 2 N→∞ π sα fixed 1 + ρ0 (z)U (z)d 2 z, π

34


where one should substitute (4.15). The result is g 1 lim 2 log ZN =− 2 log |z−1 − ζ −1 |d 2 zd 2 ζ + α sα N→∞ π D D sα fixed α=1

= F˜ (sα , τ ) .

(4.16)

This quantity depends on sα ’s. It is given by the value of the integrand in (4.11) at the saddle point with fixed sα . If one wants to take into account the “tunneling” of eigenvalues between different components of the support, sα are no longer free parameters, and the free energy F (0) of the “planar limit” of the matrix model should be obtained by extremizing F˜ with respect to sα (with fixed t0 ). It follows from the above that ∂sα F˜ = α and so the extremum is at α = 0. Therefore, F (0) = F , (4.17) α =0

where F is given by (2.35). The results of Sect. 3 imply that the growth of the support of eigenvalues, when N → N + δN and the tunneling is taken into account, is given by the normal displacement of the boundary proportional to the normal derivative of the Green function with the minus sign, like in (2.9). Since this quantity is always non-negative, we conclude that if a point belongs to the support of eigenvalues, it does so as t0 increases. In particular, if one starts with point-like droplets at the points zα , as is discussed above, then these points always remain inside the droplets. Let us note that different aspects of multi-support solutions of the 2-matrix model and matrix models with complex eigenvalues were discussed in [22, 26, 25]. For matrix models one usually restricts oneself to the algebraic case of a finite amount of nonvanishing moments playing the role of the coefficients of the matrix model potential (4.12). In such a case the corresponding complex curve can be described by an algebraic equation [22], which can be thought of as an auxiliary constraint to the second derivatives of F . These auxiliary constraints look similar to the reduction conditions in the case of Landau-Ginzburg topological theories (see, for example, discussion of such conditions in the context of dispersionless Hirota equations and WDVV equations in [7]). Their meaning is that the derivatives w.r.t. the times {τk } for k > g + 1 (where g is the genus of the corresponding algebraic complex curve) can be expressed through the derivatives restricted to k ≤ g + 1. 5. Green Function on the Schottky Double and Generalized Hirota Equations Let us now turn to the generalization of the dispersionless Hirota equations to the multiply-connected case. For a unified treatment of the two “dual” representations of the Dirichlet problem discussed in the two previous sections, we make a simple change of variables. Namely, let us introduce the generic “period” variables Xα which are identified either with α or 2πi sα depending on the choice of the set of cycles, and the function F(Xα , τ ) Xα equal to F (Xα , τ ) or F˜ ( 2πi , τ ) respectively. Then the main relations (3.45)–(3.47) and (4.4)–(4.6) acquire the form 1 G(a, b) = log |a −1 − b−1 | + ∇(a)∇(b)F , 2

(5.1)


35

ωα (z) = − ∂α ∇(z)F ,

(5.2)

Tαβ = 2πi ∂α ∂β F ,

(5.3)

where ∂α := ∂Xα and G, ω and T stand for the corresponding objects with or without tilde, depending on the chosen basis of cycles. We will see below that, in analogy to the simply-conected case, any second order derivative of the function F w.r.t. τk (and τ¯k ), Fik , will be expressed through the derivatives {Fαβ }, where α, β = 0, . . . , g together with {Fατi } and their complex conjugated. To be more precise, one can consider all second derivatives as functions of {Fαβ , Fαk } modulo certain relations on the latter, like the relation (5.14) to be discussed below. Sometimes on this “small phase space” more extra constraints arise, which can be written in the form similar to the Hirota or WDVV equations [27]; we are not going to discuss this issue here, restricting ourselves to the generic situation. 5.1. The Abel map. To derive equations for the function F = F in Sect. 2, we used the representation (2.23) for the conformal map w(z) in terms of F and Eq. (1.2) relating the conformal map to the Green function which, in its turn, is expressed through the second order derivatives of F. In the multiply-connected case, our strategy is basically the same, with the suitable analog of the conformal map w(z) (or rather of log w(z)) being the embedding of Dc into the g-dimensional complex torus Jac, the Jacobi variety of the Schottky double. This embedding is given, up to an overall shift in Jac, by the Abel map z → W(z) := (W1 (z), . . . , Wg (z)), where z dWα (5.4) Wα (z) = ξ0

is the holomorphic part of the harmonic measure ωα . By virtue of (5.2), the Abel map is represented through the second order derivatives of the function F: z dWα = −∂α D(z)F , (5.5) Wα (z) − Wα (∞) = ∞

2 Re Wα (∞) = ωα (∞) = −∂τ0 ∂α F .

(5.6)

The last formula immediately follows from (5.2).

5.2. The Green function and the prime form. The Green function of the Dirichlet boundary problem, appearing in (5.1), can be written in terms of the prime form (A.4) on the Schottky double (cf. (1.2)): E(z, ζ ) . G(z, ζ ) = log (5.7) E(z, ζ¯ ) Here by ζ¯ we mean the (holomorphic) coordinate of the “mirror” point on the Schottky double, i.e. the “mirror” of ζ under the antiholomorphic involution. The pairs of such mirror points satisfy the condition Wα (ζ ) + Wα (ζ¯ ) = 0 in the Jacobian (i.e., the sum

36


should be zero modulo the lattice of periods). The prime form4 is written through the Riemann theta functions and the Abel map as follows: E(z, ζ ) =

θ∗ (W(z) − W(ζ )) h(z) h(ζ )

(5.8)

when the both points are on the upper sheet and E(z, ζ¯ ) =

θ∗ (W(z) + W(ζ ))

(5.9)

ih(z) h(ζ )

when z is on the upper sheet and ζ¯ is on the lower one (for other cases we define E(¯z, ζ¯ ) = E(z, ζ ), E(¯z, ζ ) = E(z, ζ¯ )). Here θ∗ (W) ≡ θδ ∗ (W|T ) is the Riemann theta function (A.2) with the period matrix Tαβ = 2πi ∂α ∂β F and any odd characteristics δ ∗ , and h2 (z) = −z2

g

θ∗,α (0)∂z Wα (z) = z2

α=1

g

θ∗,α (0)

α=1

Ak (z)∂α ∂τk F .

(5.10)

k≥1

Note that in the l.h.s. of (5.9) the bar means the reflection in the double while in the r.h.s. the bar means complex conjugation. The notation is consistent since the local coordinate in the lower sheet is just the complex conjugate one. However, one should remember that E(z, ζ¯ ) is not obtained from (5.8) by a simple substitution of the complex conjugated argument. On different sheets so defined prime “form” E is represented by different functions. In our normalization (5.9) iE(z, z¯ ) is real (see also Appendix B) and lim

E(z, ζ ) = 1. − ζ −1

ζ →z z−1

In particular, limz→∞ zE(z, ∞) = 1. 5.3. The prime form and the tau-function. In (5.7), the h-functions in the prime forms cancel, so the analog of (2.17) reads 2 θ∗ (W(z) − W(ζ )) 2 = log 1 − 1 + ∇(z)∇(ζ )F . log (5.11) z ζ θ∗ (W(z) + W(ζ )) This equation already explains the claim made in the beginning of this section. Indeed, the r.h.s. is the generating function for the derivatives Fik while the l.h.s. is expressed through derivatives of the form Fαk and Fαβ only. The expansion in powers of z, ζ allows one to express the former through the latter. The analogs of Eqs. (2.19), (2.20) are, respectively: θ∗ (W(z) − W(∞)) 2 = − log |z|2 + ∂τ ∇(z)F , log (5.12) 0 θ∗ (W(z) + W(∞)) Given a Riemann surface with local coordinates 1/z and 1/¯z we trivialize the bundle of − 21 -differentials and “redefine” the prime form E(z, ζ ) → E(z, ζ )(dz)1/2 (dζ )1/2 so that it becomes a function. However for different coordinate patches (the “upper” and “lower” sheets of the Schottky double) one gets different functions, see, for example, formulas (5.8) and (5.9) below. 4


37

2 h (∞) 2 = ∂2 F . log τ0 θ (ω(∞)) ∗

(5.13)

Here ω(z) ≡ 2 Re W(z) = (ω1 (z), . . . , ωg (z)) and h2 (∞) = lim z θ∗ z→∞

z ∞

g dW = − θ∗,α (0)∂α ∂τ1 F . α=1

¯ A simple check shows that the l.h.s. of (5.13) can be written as −2 log(iE(∞, ∞)). ¯ + O(z−1 ) as As is seen from the expansion G(z, ∞) = − log |z| − log(iE(∞, ∞)) ¯ −1 is a natural analog of the conformal radius, and (5.13) indeed z → ∞, (iE(∞, ∞)) turns to (2.20) in the simply-connected case (see Appendix B for an explicit illustrative example). However, now it provides a nontrivial relation on Fαβ ’s and Fαi ’s: 

 2 θ∗,α ∂α ∂τ1 F  θ∗,β ∂β ∂τ¯1 F  = θ∗2 (ω(∞))e∂τ0 F (5.14) α

β

so that the “small phase space” contains the derivatives modulo this relation. The next steps are exactly the same as in Sect. 2: we are going to decompose these equalities into holomorphic and antiholomorphic parts. The results are conveniently written in terms of the prime form. The counterpart of (2.22) is E(ζ, z)E(∞, ζ¯ ) ζ log + D(z)∇(ζ )F . (5.15) = log 1 − z E(ζ, ∞)E(z, ζ¯ ) Tending ζ → ∞, we get: log

¯ E(z, ∞) ¯ − ∂τ0 D(z)F . = log z + log E(∞, ∞) E(z, ∞)

(5.16)

Separating holomorphic and antiholomorphic parts of (5.15) in ζ , we get analogs of (2.24) and (2.25): log

E(z, ζ ) = log(z − ζ ) + D(z)D(ζ )F E(z, ∞)E(∞, ζ ) − log

¯ E(z, ζ¯ )E(∞, ∞) ¯ ζ¯ )F . = D(z)D( ¯ E(z, ∞)E(∞, ζ¯ )

(5.17)

(5.18)

Combining these equalities (with merging points z → ζ in particular), one is able to obtain the following representations of the prime form itself: 1 2 E(z, ζ ) = (z−1 − ζ −1 )e− 2 (D(z)−D(ζ )) F ,

(5.19)

1 ¯ ¯ 2 iE(z, ζ¯ ) = e− 2 (∂τ0 +D(z)+D(ζ )) F .

(5.20)

Note also the nice formula 1

iE(z, z¯ ) = e− 2 ∇

2 (z)F

.

(5.21)

38


5.4. Generalized Hirota relations. For higher genus Riemann surfaces there are no simple universal relations connecting values of prime forms at different points, which, via (5.19), (5.20), could be used to generate equations on F. The best available relation [9] is the celebrated Fay identity (A.5). Although it contains not only prime forms but Riemann theta functions themselves, it is really a source of closed equations on F, since all the ingredients are in fact representable in terms of second order derivatives of F in different variables. An analog of the KP version of the Hirota equation (2.26) for the function F can be obtained by plugging Eqs. (5.5) and (5.19) into the Fay identity (A.5). As a result, one obtains a closed equation which contains second order derivatives of the F only (recall that the period matrix in the theta-functions is essentially the matrix of the derivatives Fαβ ). A few equivalent forms of this equation are available. First, shifting Z → Z − W3 + W4 in (A.5) and putting z4 = ∞, one gets the relation a c b D(a)D(b)F (a − b)e θ dW + dW − Z θ dW − Z ∞

+ (b − c)eD(b)D(c)F θ

+ (c − a)eD(c)D(a)F θ

∞

b ∞ c

∞

dW +

∞ a

dW +

∞

c

∞

dW − Z θ dW − Z θ

a ∞ b

∞

dW − Z

dW − Z

= 0 . (5.22)

The vector Z is arbitrary (in particular, zero). We see that (2.26) gets “dressed” by the theta-factors. Each theta-factor is expressed through F only. For example,   z 2 dW = exp −2π 2 nα nβ ∂αβ F − 2π i nα ∂α D(z)F  . θ ∞

nα ∈Z

α

αβ

Another form of this equation, obtained from (5.22) for a particular choice of Z, reads a −1 21 (D(a)+D(b))2 F h(c)θ∗ dW (a − b)c e +

b

∞

∞

dW + [cyclic per-s of a, b, c] = 0 .

(5.23)

Taking the limit c → ∞ in (5.22), one gets an analog of (2.27): a b θ ( ∞ dW + ∞ dW − Z) θ (Z) D(a)D(b)F 1− e a b θ ( ∞ dW − Z) θ( ∞ dW − Z) a g θ ( ∞ dW − Z) D(a) − D(b) 1 ∂ ∂τ1 F + = log b ∂α ∂τ1 F , (5.24) a−b a−b ∂Zα θ( W − Z) α=1

∞

which also follows from another Fay identity (A.6). Equations on F with τ¯k -derivatives follow from the general Fay identity (A.5) with some points on the lower sheet. Besides, many other equations can be derived as various combinations and specializations of the ones mentioned above. Altogether, they form an infinite hierarchy of consistent differential equations of a very complicated structure which deserves further investigation. The functions F corresponding to different choices of independent variables (i.e., to different bases in homology cycles on the Schottky double) provide different solutions to this hierarchy.


39

5.5. Higher genus analogs of the dispersionless Toda equation. Let us show how the simplest equation of the hierarchy, the dispersionless Toda equation (2.30), is modified in the multiply-connected case. Applying ∂z ∂ζ¯ to both sides of (5.18) and setting ζ = z, we get: ¯ z))F = −∂z ∂z¯ log E(z, z¯ ) . (∂D(z))(∂¯ D(¯

Here ∂D(z) is the z-derivative of the operator D(z): ∂D(z) = k Ak (z)∂τk . To transform the r.h.s., we use the identity (A.8) (Appendix A) and specialize it to the particular local parameters on the two sheets: θ (ω(z) + Z)θ (ω(z) − Z) θ 2 (Z)E 2 (z, z¯ ) +|z|4 (log θ (Z)), αβ ∂z Wα (z)∂z¯ Wβ (z) .

|z|4 ∂z ∂z¯ log E(z, z¯ ) =

α,β

Tending z to ∞, we obtain a family of equations (parametrized by an arbitrary vector Z) which generalize the dispersionless Toda equation for the tau-function: ∂τ1 ∂τ¯1 F =

θ (ω(∞)+Z)θ (ω(∞)−Z) ∂τ2 F e 0 θ 2 (Z) g − (log θ (Z)), αβ (∂α ∂τ1 F)(∂β ∂τ¯1 F).

(5.25)

α,β=1

(Here we used the z → ∞ limits of (5.5) and (5.21).) The following two equations correspond to special choices of the vector Z: ∂τ1 ∂τ¯1 F +

g

(log θ (0)), αβ (∂α ∂τ1 F)(∂β ∂τ¯1 F) =

α,β=1

∂τ1 ∂τ¯1 F = −

g

log θ∗ (ω(∞))

,αβ

θ 2 (ω(∞)) ∂τ2 F e 0 , θ 2 (0)

(∂α ∂τ1 F)(∂β ∂τ¯1 F) .

(5.26)

(5.27)

α,β=1

Finally, let us specify Eq. (5.25) for the genus g = 1 case. In this case there is only one extra variable X := X1 (1 or 2πis1 ). The is

Riemann theta-function

θ(ω(∞) + Z) then replaced by the Jacobi theta-function ϑ ∂X ∂τ0 F − Z T ≡ ϑ3 ∂X ∂τ0 F − Z T , where the elliptic modular parameter is T = 2πi ∂X2 F, and the vector Z ≡ Z has only one component. The equation has the form:

ϑ3 ∂X ∂τ0 F +Z 2πi ∂X2 F ϑ3 ∂X ∂τ0 F −Z 2π i ∂X2 F ∂τ2 F

∂τ1 ∂τ¯1 F = e 0 ϑ32 Z| 2πi ∂X2 F (5.28) − ∂Z2 log ϑ3 Z| 2πi ∂X2 F (∂X ∂τ1 F)(∂X ∂τ¯1 F) . Note also that Eq. (5.14) acquires the form

2 2 ϑ1 ∂X ∂τ0 F 2π i ∂X2 F

(∂X ∂τ1 F)(∂X ∂τ¯1 F) = e ∂τ 0 F , 2 ϑ1 0| 2πi ∂X F

(5.29)

40


where ϑ∗ ≡ ϑ1 is the only odd Jacobi theta-function. Combining (5.28) and (5.29) one may also write the equation

ϑ3 ∂X ∂τ0 F +Z 2πi ∂X2 F ϑ3 ∂X ∂τ0 F −Z 2π i ∂X2 F

∂τ1 ∂τ¯1 F = ϑ32 Z| 2πi ∂X2 F 

2 2 ϑ1 ∂X ∂τ0 F 2πi ∂X2 F

∂Z2 log ϑ3 Z| 2π i ∂X2 F  e∂τ0 F (5.30) − 2 ϑ1 0| 2πi ∂X F whose form is close to (2.30) but differs by the nontrivial “coefficient” in the square brackets. In the limit T → i∞ the theta-function ϑ3 tends to unity, and we obtain the dispersionless Toda equation (2.30). 6. Conclusions In this paper we have considered the Dirichlet boundary problem in planar multiplyconnected domains. A planar multiply-connected domain Dc is the complex plane with several holes. We study how the solution of the Dirichlet problem depends on small deformations of boundaries of the holes. General properties of such deformations allow us to introduce the quasiclassical tau-function associated to the variety of planar multiply-connected domains. By the tau-function, we actually mean its logarithm, which only makes sense for the quasiclassical or Whitham-type integrable hierarchies. Namely, the key properties are the specific “exchange” relations (3.43) which follow from the Hadamard variational formula for the Green function and the harmonic measure. They have the form of integrability conditions and thus ensure the existence of the tau-function. The tau-function corresponds to a particular solution of the universal Whitham hierarchy [4] and generalizes the dispersionless tau-function which describes deformations of simply-connected domains. The algebro-geometric data associated with the multiply-connected geometry include a Riemann surface with antiholomorphic involution, the Schottky double of the domain Dc = C\D endowed with particular holomorphic coordinates z and z¯ on the two sheets of the double, respecting the involution. This Riemann surface has genus g = #{holes} − 1. The (logarithm of) tau-function, F, describes small deformations of these data as functions of an infinite set of independent deformation parameters which are basically harmonic moments of the domain. These variables can be equivalently redefined as periods of the generating one-form S(z)dz over non-trivial cycles on the double and the residues of the one-forms Ak (z)S(z)dz, where Ak (z) z−k is some proper global basis (3.10) z→∞

of harmonic functions. We have obtained simple expressions for the period matrix, the Abel map and the prime form on the Schottky double in terms of the function F. Specifically, all these objects are expressed through second order derivatives of the F in its independent variables. The generalized dispersionless Hirota equations on F for the multiply-connected case (equivalent to the Whitham hierarchy) are obtained by incorporating the above mentioned expressions into the Fay identities. As a result, one comes to a series of quite non-trivial equations for (second derivatives of) the function F , which have not been written before (except for certain relations for the second derivatives of the Seiberg-Witten prepotential [28]). When the Riemann surface degenerates to the Riemann


41

sphere with two marked points, they turn into Hirota equations of the dispersionless Toda hierarchy. Algebraic orbits of the universal Whitham hierarchy describe the class of domains which can be obtained as conformal images of a “half” of a complex algebraic curve with the antiholomorphic involution under conformal maps given by rational functions on the curve. In particular, all domains having only a finite amount of non-vanishing harmonic moments, are in this class. In this case one can define the curve by a polynomial equation written explicitly in [22]. This situation is an analog of the (Laurent) polynomial conformal maps in the simply-connected case and literally corresponds to multi-support solutions of matrix models with polynomial potentials. The definition of the tau-function for multiply-connected domains proposed above holds in a broader set-up of general algebraic domains. It does not rely on the finiteness of the amount of non-vanishing moments. In general any effective way to describe the complex curve associated to a multiply-connected domain by a system of polynomial or algebraic equations is not known. The curve may be thought of as a spectral curve corresponding to a generic finite-gap solution to the Toda lattice hierarchy. Acknowledgement. We are indebted to V.Kazakov, M.Mineev-Weinstein, S.Natanzon, L.Takhtajan and P.Wiegmann for illuminating discussions and especially to A.Levin for very important comments on Sect. 5. We are also grateful to the referee for a very careful reading of the manuscript and valuable remarks. The work was also partially supported by NSF grant DMS-01-04621 (I.K.), RFBR under the grant 01-01-00539 (A.M. and A.Z.), by INTAS under the grants 00-00561 (A.M.), 99-0590 (A.Z.) and by the Program of support of scientific schools under the grants 1578.2003.2 (A.M.), 1999.2003.2 (A.Z.). The work of A.Z. was also partially supported by the NATO grant PST.CLG.978817. A.M. is grateful for the hospitality to the Max Planck Institute of Mathematics in Bonn, where an essential part of this work has been done.

Appendix A. Theta Functions and Fay Identities Here we present some definitions and useful formulas from [9]. The Riemann theta function θ (W) ≡ θ (W|T ) is defined as θ (W) = eiπn·T ·n + 2πin·W . (A.1) n∈Zg

The theta function with (half-integer) characteristics δ = (δ 1 , δ 2 ), where δα = Tαβ δ1,β + δ2,α and δ 1 , δ 2 ∈ 21 Zg is θδ (W) = eiπ δ 1 ·T ·δ 1 +2πi δ 1 ·(W+δ 2 ) θ(W + δ) = eiπ(n+δ 1 )·T ·(n+δ 1 )+2πi(n+δ 1 )·(W+δ 2 ) .

(A.2)

n∈Zg

Under shifts by a period of the lattice, it transforms according to θδ (W + eα ) = e2πiδ1,α θδ (W) ,

θδ (W + Tαβ eβ ) = e−2πiδ2,α −iπTαα −2πiWα θδ (W) .

(A.3)

The prime form E(z, ζ ) is defined as θ∗ (W(z) − W(ζ )) , α θ∗,α dWα (z) β θ∗,β dWβ (ζ )

E(z, ζ ) =

(A.4)

42


where θ∗ is any odd theta function, i.e., the theta function with any odd characteristic δ ∗ (the characteristics is odd if 4δ ∗1 · δ ∗2 = odd). The prime form does not depend on the particular choice of the odd characteristics. In the denominator, ∂θ∗ (W) θ∗,α = θ∗,α (0) = ∂Wα W=0 is the set of θ-constants. The data we use in the main text contain also distinguished coordinates on a Riemann surface: the holomorphic co-ordinates z and z¯ on two different sheets, and we do not distinguish, unless it is necessary between the prime form (A.4) and a function E(z, ζ ) ≡ E(z, ζ )(dz)1/2 (dζ )1/2 “normalized” onto the differentials of a distinguished co-ordinate. Let us now list the Fay identities [9] used in the paper. The basic one is Fay’s trisecant formula (Eq. (45) from p. 34 of [9]) θ (W1 − W3 − Z) θ(W2 − W4 − Z) E(z1 , z4 )E(z3 , z2 ) + θ (W1 − W4 − Z) θ(W2 − W3 − Z) E(z1 , z3 )E(z2 , z4 ) = θ (W1 + W2 − W3 − W4 − Z) θ(Z) E(z1 , z2 )E(z3 , z4 ) .

(A.5)

Here Wi ≡ W(zi ). This identity holds for any four points z1 , . . . , z4 on a Riemann surface and any vector Z ∈ Jac. In the limit z3 → z4 ≡ ∞ one gets (formula (38) from p. 25 of [9]) z z θ ( ∞1 dW + ∞2 dW − Z) θ(Z) E(z1 , z2 ) z1 z2 θ ( ∞ dW − Z) θ( ∞ dW − Z) E(z1 , ∞)E(z2 , ∞) z g θ( ∞1 dW − Z) (z1 ,z2 ) = d (∞) + dWα (∞) ∂Zα log z2 , (A.6) θ( ∞ dW − Z) α=1 where d (z1 ,z2 ) (∞) = dz log

E(z, z1 ) E(z, z2 )

(A.7)

is the normalized Abelian differential of the third kind with simple poles at z1 and z2 and residues ±1. Another relation from [9] we use (see e.g. (29) on p.20 and (39) on p.26) is θ (W1 − W2 − Z)θ (W1 − W2 + Z) θ 2 (Z)E 2 (z1 , z2 ) g = ω(z1 , z2 ) + (log θ (Z)),αβ dWα (z1 )dWβ (z2 ),

(A.8)

α,β=1

where (log θ (Z)), αβ =

∂ 2 log θ (Z) ∂Zα ∂Zβ

and ω(z1 , z2 ) = dz1 dz2 log E(z1 , z2 )

(A.9)

is the canonical bi-differential of the second kind with the double pole at z1 = z2 .


43

Appendix B. Degenerate Schottky Double For an illustrative purpose we would like to adopt some of the above formulas to the simplest possible case, which is the Riemann sphere realized as the Schottky double of the complement to the disk of radius r. In this case dW =

dz d z¯ = idϕ = − z z¯

(B.1)

is purely imaginary on the circle and obviously satisfies the condition dW (z)+dW (¯z) = 0. Further, (cf. (3.65)) z z z W (z) = dW = dW = log (B.2) r ξ0 r and z (B.3) r which is nothing but the conformal map of the exterior of the circle |z| ≥ r onto the exterior of the unit circle |w| ≥ 1. Note that on the “lower” sheet of the double z¯ r dW = log , (B.4) W (¯z) = z¯ r w(z) = eW (z) =

and instead of (B.3) one gets w(¯z) =

r z¯

(B.5)

which is the conformal map of the exterior of the disk |¯z| ≥ r on the lower sheet onto the interior of the unit circle |w| ≤ 1. The prime form on the genus zero Riemann surface is (cf. (A.4)) w1 − w2 E(z1 , z2 ) = √ , √ dw1 dw2

(B.6)

¯ which is understood in the main text, where wi ≡ w(zi ). Let us compute E(∞, ∞), as “normalized” on the values of the local coordinates z∞ = 1/z and z¯ ∞ = 1/¯z in the ¯ on two sheets of the double. One gets (cf. (5.9)) points ∞ and ∞ ¯ =− E 2 (∞, ∞)

¯ 2 (w(∞) − w(∞)) . ¯ (dw(∞)/dz∞ )(dw(∞)/dz ¯ ) ∞

(B.7)

Substituting into (B.7) the formulas (B.3), (B.5) and dz z2 = − lim dz∞ , z→∞ r z→∞ r ¯ = r dz∞ dw(∞) ¯ ,

dw(∞) = lim

(B.8)

one finally gets 1 z2 /r 2 = 2, z→∞ (z2 /r)r r

¯ 2 = lim E(∞, ∞)

and this demonstrates that (5.14) indeed turns into (2.20) in the limit.

(B.9)

44


References 1. Hurwitz, A., Courant, R.: Vorlesungen u¨ ber allgemeine Funktionentheorie und elliptische Funktionen. Herausgegeben und ergänzt durch einen Abschnitt u¨ ber geometrische Funktionentheorie. Berlin-Heidelberg-New York: Springer-Verlag, 1964 (Russian translation, adapted by M.A. Evgrafov: Theory of functions, Moscow: Nauka, 1968) 2. Marshakov, A., Wiegmann, P., Zabrodin, A.: Commun. Math. Phys. 227, 131 (2002) 3. Krichever, I.M.: Funct. Anal. Appl. 22, 200–213 (1989) 4. Krichever, I.M.: Commun. Pure. Appl. Math. 47, 437 (1994) 5. Mineev-Weinstein, M., Wiegmann, P.B., Zabrodin, A.: Phys. Rev. Lett. 84, 5106 (2000) 6. Wiegmann, P.B., Zabrodin, A.: Commun. Math. Phys. 213, 523 (2000) 7. Boyarsky, A., Marshakov, A., Ruchayskiy, O., Wiegmann, P., Zabrodin, A.: Phys. Lett. B515, 483– 492 (2001) 8. Krichever, I., Novikov, S.: Funct. Anal. Appl. 21, No 2, 46–63 (1987) 9. Fay, J.D.:“Theta Functions on Riemann Surfaces”. Lect. Notes in Mathematics 352, Berlin-Heidelberg-New York: Springer-Verlag, 1973 10. Hadamard, J.: Mém. présentés par divers savants a` l’Acad. sci., 33, (1908) 11. Davis, P.J.: The Schwarz function and its applications. The Carus Mathematical Monographs, No. 17, Buffalo, N.Y.: The Math. Association of America, 1974 12. Krichever, I.: 2000, unpublished 13. Takhtajan, L.: Lett. Math. Phys. 56, 181–228 (2001) 14. Kostov, I.K., Krichever, I.M., Mineev-Weinstein, M., Wiegmann, P.B., Zabrodin, A.: τ -function for analytic curves. In: Random matrices and their applications, MSRI publications, 40, Cambridge: Cambridge Academic Press, 2001 15. Hille, E.: Analytic function theory. V.II, Oxford: Ginn and Company, 1962 16. Gibbons, J., Kodama, Y.: Proceedings of NATO ASI “Singular Limits of Dispersive Waves”. ed. N. Ercolani, London – New York: Plenum, 1994; Carroll, R., Kodama, Y.: J. Phys. A: Math. Gen. A28, 6373 (1995) 17. Takasaki, K., Takebe, T.: Rev. Math. Phys. 7, 743–808 (1995) 18. Schiffer, M., Spencer, D.C.: Functionals of finite Riemann surfaces. Princeton, NJ: Princeton University Press, 1954 19. Gustafsson, B.: Acta Applicandae Mathematicae 1, 209–240 (1983) 20. Aharonov, D., Shapiro, H.: J. Anal. Math. 30, 39–73 (1976); Shapiro, H.: The Schwarz function and its generalization to higher dimensions. University of Arkansas Lecture Notes in the Mathematical Sciences, Volume 9, W.H. Summers, Series Editor, New York: A Wiley-Interscience Publication, John Wiley and Sons, 1992 21. Etingof, P., Varchenko, A.: Why does the boundary of a round drop becomes a curve of order four? University Lecture Series. 3, Providence, RI: American Mathematical Society, 1992 22. Kazakov, V., Marshakov, A.: J. Phys. A: Math. Gen. 36, 3107–3136 (2003) 23. De Wit, B., Marshakov, A.: Theor. Math. Phys. 129, 1504 (2001) [Teor. Mat. Fiz. 129, 230 (2001)] 24. Gakhov, F.: The boundary value problems. Moscow: Nauka, 1977 (in Russian) 25. Eynard, B.: Large N expansion of the 2-matrix model, multicut case. http://arxiv:org/list/mathph/0307052, 2003 26. Bertola, M.: Free energy of the two-matrix model/dToda tau-function. Nucl. Phys. B669, 435–461 (2003) 27. Marshakov, A., Mironov, A., Morozov, A.: Phys. Lett. B389, 43–52 (1996); Braden, H., Marshakov, A.: Phys. Lett. B541, 376–383 (2002) 28. Gorsky, A., Marshakov, A., Mironov, A., Morozov, A.: Nucl. Phys. B527, 690–716 (1998) Communicated by L. Takhtajan


Communications in


Massless D-Branes on Calabi–Yau Threefolds and Monodromy Paul S. Aspinwall1 , R. Paul Horja2 , Robert L. Karp1 1

Center for Geometry and Theoretical Physics, Box 90318, Duke University, Durham, NC 27708-0318, USA 2 Department of Mathematics, University of Michigan, East Hall, 525 E University Avenue, Ann Arbor, MI 48109-1109, USA Received: 15 December 2003 / Accepted: 17 February 2004 Published online: 14 June 2005 – © Springer-Verlag 2005

Abstract: We analyze the link between the occurrence of massless B-type D-branes for specific values of moduli and monodromy around such points in the moduli space. This allows us to propose a classification of all massless B-type D-branes at any point in the moduli space of Calabi–Yau’s. This classification then justifies a previous conjecture due to Horja for the general form of monodromy. Our analysis is based on using monodromies around points in moduli space where a single D-brane becomes massless to generate monodromies around points where an infinite number become massless. We discuss the various possibilities within the classification.

1. Introduction The derived category approach to B-type D-Branes [1–5] appears to be extremely powerful. It allows one to go beyond the picture of D-branes as vector bundles over submanifolds so that α -corrections can be correctly understood. For example, the fact that B-type D-branes must undergo monodromy as one moves about the moduli space of complexified Kähler forms can be expressed in the derived category language [6–8]. The main purpose of this paper is to try to classify which D-branes can become massless at a given point in the moduli space. Again the language of derived categories will be invaluable. In order for an object in the bounded derived category of coherent sheaves to represent a D-brane it must be “-stable”. Criteria for -stability have been discussed in [9–12] although it is not clear that we yet have a mathematically rigorous algorithm for determining stability. Despite this, in simple examples such as in the above references and [13] one can compute stability with a fair degree of confidence. In particular if you have reason to believe that a certain set of a D-branes is stable at a given point in the moduli space then one can move along a path in moduli space and see how the spectrum of stable states changes. There is considerable evidence [12] that such changes

46

P.S. Aspinwall, R.P. Horja, R.L. Karp

in -stability depend only on the homotopy class of the path in the moduli space of conformal field theories. The fact that changes in -stability do depend on the homotopy class of such paths was used in [12] to “derive” Kontsevich’s picture of monodromy at least in the case of the quintic Calabi–Yau threefold. The moduli space of conformal field theories may be compactified by including the “discriminant locus” consisting of badly-behaved worldsheet theories. Typically one expects such theories to be bad because some D-brane has become massless [14]. Indeed, the monodromy seen in [2, 13] around parts of this discriminant locus was intimately associated to massless D-branes. It is this link between massless D-branes and monodromy that we wish to study more deeply in this paper. In simple cases as one approaches a point in the discriminant locus, a single D-brane becomes massless. Of more interest to us is the case where an infinite number become massless. In [7,8] one of the authors studied components of the discriminant locus corresponding to what was called “EZ-transformations”. Namely if one has a Calabi–Yau threefold X with some complex subspace E, there may be a point in Kähler moduli space where E collapses to a complex subspace Z of lower dimension than E. We will see that it is then the derived category of Z that describes the massless D-branes associated to this transformation. A particular autoequivalence was naturally associated to a particular EZ-transformation and it was conjectured in [7,8] that such an autoequivalence resulted from the associated monodromy. We will call this conjecture the “EZ-monodromy conjecture”. One purpose of this paper is to justify this conjecture. Because of the nature of our understanding of D-branes and string theory it will not be possible to rigorously prove any hard theorems about D-branes. Instead we will have to play with a number of conjectures whose interdependence leads to considerable evidence of the validity of the overall story. In particular, on the one hand we have the EZ-monodromy conjecture and, on the other hand, we have our conjecture concerning which D-branes become massless. These two conjectures are interlinked by -stability as we discuss in Sect. 2. In particular, in Sect. 2.1 we discuss an older conjecture concerning single massless D-branes. In Sect. 2.2 we then review a framework for the more general case which is linked to the simpler case in Sect. 2.3 for a particular example. The physical interpretation of the general case is then given in Sect. 2.4. The link discussed in Sect. 2.3 between the simple case of a single D-brane becoming massless and an infinite number becoming massless depends upon a mathematical result which is derived in Sect. 3. This section is more technical than the other sections and may be omitted by the reader if need be. That said, it shows how the sophisticated methods of derived categories are directly relevant to the physics of D-branes. In Sect. 4 we discuss a natural hierarchy of cases. The familiar “conifold”-like situation arises where Z is a point and only one soliton becomes massless. If Z has dimension one then the derived category of Z has more structure. This case corresponds to the Seiberg–Witten theory of some nonabelian gauge group. We study an explicit example of this elsewhere [15]. The case where Z has complex dimension two is more complicated as the derived category now has a rich structure. We show that it appears to be similar to the spectrum of massless D-branes one gets from a decompactification. We also see that it demonstrates how 2-branes wrapped around a 2-torus can become massless. At first sight this appears to contradict T-duality but we will see that this is not actually the case. Finally, for completeness, in Sect. 4.4 we discuss the case of an exoflop which is awkward to fit into our general classification but still yields a simple result.

Massless D-Branes on Calabi–Yau Threefolds and Monodromy

47

2. Monodromy and Massless D-Branes 2.1. A single massless D-brane. B-type D-branes on X correspond to objects in the bounded derived category of coherent sheaves on X [1–5]. A given object A is represented by a complex. We may then construct another object A[n] by shifting this complex n places to the left. Such a shift or “translation” is a global symmetry of physics if it is applied simultaneously to all objects [2]. Relative shifts are significant — an open string stretched between A and B is not equivalent to an open string stretched between A[n] and B if n = 0. We would like to consider the case of moving to a point in moduli space where a single physical D-brane A becomes massless. Because of the global shift symmetry all of its translates A[n] are equally massless. Thus an infinite number of objects in D(X) are becoming massless even though only one D-brane counts towards any physical effects of this masslessness as it would be computed by Strominger [14] for example. The analysis of -stability in [12, 13] showed that monodromy is intimately associated to massless D-branes. This should not be surprising since monodromy can only occur around the discriminant and the discriminant is associated with singularities in the conformal field theory associated with massless solitons [14]. Consider an oriented open string f stretched between two D-branes in a Calabi–Yau threefold X. In the derived category language this is written as a morphism between two objects in D(X), f : A → B.

(1)

These two objects may or may not form a bound state according to the mass of the open string f . If f is tachyonic then we have a bound state a` la Sen [16]. (As we emphasize shortly A is really an anti-brane in such a bound state.) A real number1 (dubbed a “grade” in [2]) ϕ is associated to each stable D-brane. We assume ϕ varies continuously over the moduli space and is defined mod 2 by the central charge Z: ϕ=−

1 arg(Z) π

(mod 2).

(2)

The precise definition of ϕ is discussed at length in [12]. In [2] it was argued that the mass squared of the open string in (1) is then proportional to ϕ(B) − ϕ(A) − 1 allowing the stability of this bound state to be determined. One of the key features of the derived category which makes it so useful for the study of solitons is the way that bound states are described using distinguished triangles. The open string f between A and B is best represented in the context of a distinguished triangle

A

[1]

C _@ @@ @@ @@ f / B.

(3)

1 It has been suggested that ϕ is defined modulo some integer such as 6 [2, 17]. Periodicity can also appear, if desired, in Floer cohomology (see [18, 19] for example) which is supposedly mirror to the structure we are considering. For simplicity we ignore such a possibility. To take such an effect into account one should probably quotient the derived category by such translations.

48


The “[1]” represents the fact that one must shift one place left when performing the corresponding map. The object C, which is equivalent to the “mapping cone” Cone(f : A → B), is then potentially a bound state of A[1] and B. As explained in [2], A[odd] should be thought of as an anti-A. The triangle also tells us that B is potentially a bound state of A and C. Equally A is a bound state of B and C[−1]. The “[1]” could be interpreted as keeping track of which brane should be treated as an anti-brane. The fact that D(X) copes so well with anti-branes demonstrates its power to analyze D-branes. The other approach, namely K-theory, should be considered the derived category’s weaker cousin since it only knows about D-brane charge! Now suppose that A is stable and becomes massless at a particular point P in the moduli space. Furthermore, let us assume that the only massless D-branes at P are of the form A[m] for any m. Let us take a generic complex plane with polar coordinates (r, θ ) passing through P at the origin and assume that Z(A) behaves as cr exp(−iθ ) near P for c some real and positive constant. That is, we assume that Z(A) has a simple zero at P . Suppose B does not have vanishing mass. It follows that Z(B) and Z(C) are equal at P and nonzero. In particular if we circle the point P by varying θ , these central charges will be constant close to P . Furthermore, if C can be a marginally bound state of anti-A and B near P , then, according to the rules of [12], we have ϕ(B) = ϕ(C) near P . This allows us to rewrite (3) including the differences in the ϕ’s for the open strings (i.e., sides of the triangle) to give 1+a−b+ πθ [1]

A

C _@ @@ @@0 @@ f / B,

(4)

b−a− πθ

where ϕ(B) = b and ϕ(A) = a at θ = 0. The stability of a given vertex of this triangle depends upon the number on the opposite side being less than 1. By “stability” we mean relative to this triangle only. A given D-brane may decay by other channels. It follows that C becomes stable for θ > π(b − a − 1) while B becomes unstable for θ > π(b − a). Note that A is always stable near P consistent with our assumptions. Based on this idea that we “gain” C and “lose” B as θ increases, we can try to formulate a picture for monodromy around P . The meaning of monodromy is that after traversing this loop in the moduli space we should be able to relabel the D-branes in such a way as to restore the physics we had before we traversed the loop. It is important to note that monodromy is not really the statement that a certain D-brane manifestly “becomes” another D-brane explicitly as we move through the moduli space. It is much more accurately described as a relabeling process. Since stability is a physical quality, we are forced to relabel B since it has decayed. The obvious candidate in the above case is to call it C. Thus monodromy would transform B into C. Life can be more complicated than this however. If we have an open string f : A → C, then, since ϕ(B) = ϕ(C) when B decays to C + A[1], C will immediately decay further to D = Cone(f : A → C) plus another A[1]. Suppose A is “spherical” in the sense of [20] which means Hom(A, A[m]) = C for m = 0 or 3, and Hom(A, A[m]) = 0 otherwise. This condition is always satisfied in the context of this subsection — i.e., only A and its translates become massless. A long exact sequence associated to (3) then implies dim Hom(A, C) = dim Hom(A, B) − 1.

(5)


49

It follows that this second decay will occur if dim Hom(A, B) > 1. Iterating this process one sees that B will decay splitting off an A[1] a total of dim Hom(A, B) times. Finally we should also worry about homomorphisms between B and A[m] for other values of m. We refer to the example in Sect. 4 of [13] for a detailed example of exactly how this happens in a fairly nontrivial example. All said, allowing for all these decays, C becomes a number of A’s (probably shifted) together with Cone . . . ⊕ Ab0 ⊕ A[−1]b1 ⊕ A[−2]b2 ⊕ . . . → B , (6) where bn = dim Hom(A[−n], B) = dim Extn (A, B).

(7)

The cone (6) may be written more compactly as KA (B) = Cone(hom(A, B) ⊗ A → B),

(8)

where hom(A, B) is the complex of C-vector spaces 0

0

0

0

. . . → Ext0 (A, B) → Ext1 (A, B) → Ext2 (A, B) → . . . .

(9)

We refer to [20] for further explanation of the notation.2 We can also write more heuristically   KA (B) = Cone 

As much massless stuff that can bind to B as possible.

→ B .

(10)

Interpreted na¨ıvely, we have shown that, upon increasing θ from −∞ to +∞, an object B will decay and a canonically associated object KA (B) will become stable and appear as one of the decay products of B. What is desired however is monodromy once around P , i.e., θ should only increase by 2π. We will indeed claim that monodromy once around P replaces B by KA (B). For some objects, increasing θ by only 2π (at the appropriate starting point) will cause the complete decay of B into KA (B). Thanks to its rather simple cohomology, this always happens for Ox , the structure sheaf of a point x ∈ X. Therefore the relabeling process under monodromy should replace B by KA (B). There are undoubtedly many other objects B under which this increase in θ by only 2π would not induce the entire decay to KA (B ). This doesn’t matter however, we can still leave physics invariant by relabeling B by KA (B ). For example in an extreme case, both B and KA (B ) may be stable with respect to the above triangle both before and after increasing θ by 2π . It is therefore harmless to relabel one of these states as the other. Well, it is fine saying that it is harmless to relabel B by KA (B ), but why are we forced to relabel like this? The reason is that we know that physics must be completely invariant under monodromy which implies that the relabeling must amount to an autoequivalence 2 Note that since “left-derived” L’s or “right-derived” R’s should be added to every functor in this paper, we may consistently omit them without introducing any ambiguities!

50


of D(X). One can indeed show that KA defines an autoequivalence3 of D(X) so long as A is spherical [8, 20]. What’s more, as argued in [12, 21], it is pretty well the only autoequivalence that works. To be more precise, once we have argued that the specific objects Ox undergo monodromy given by KA then all other objects must undergo the same monodromy up to some possible multiplication by some fixed line bundle L. Note that the central charge Z is also a physical quantity. Insisting that monodromy acts correctly in this case amounts to insisting that the D-brane “charges” ch(B) transform under monodromy. This is precisely the same monodromy on H even (X, Z) that one deduces from mirror symmetry as in [22]. This determines that the above line bundle L is trivial (in a considerably overdetermined way!). It is known (see [7] for example) that KA then induces the correct transformation on these charges — indeed this was the reason why KA was conjectured as the monodromy action in the first place [6]! It is worth noting that in some special cases the transformation KA has nothing to do with decay. Consider how the spherical object A itself transforms: Cone(hom(A, A) ⊗ A → A) = Cone((C → 0 → 0 → C) ⊗ A → A) = Cone((A → 0 → 0 → A) → A) ∼ =0→0→A = A[−2],

(11)

where we use the convention of [4] by underlining the zero position when necessary. Such a transformation cannot be argued from -stability however. Clearly an open string between A and itself (perhaps translated) cannot have a mass that depends upon some angle as we orbit the conifold point as clearly the mass is constant. Instead one could argue that the transform (11) occurs simply because Z(A) has a simple zero at the conifold point and thus ϕ(A) shifts by −2 as we loop around the conifold point. Then we can apply the rule ϕ(A[n]) = ϕ(A) + n from [12].4 We must therefore view (8) as being motivated by -stability for most but not all of the objects in D(X). Note that the fact that the obvious physical requirement that monodromy be an autoequivalence of D(X) can force (8) to be the required transform for all the objects in D(X) once -stability has established it for a few elements. This was the basis of the proof in the case of the quintic in [12]. Anyway, all said we have motivated the following conjecture (which, in perhaps a slightly different form, is due to Kontsevich [6], Horja [7] and Morrison [23]): Conjecture 1. If we loop around a component of the discriminant locus associated with a single D-brane A (and thus its translates) becoming massless then this results in a relabeling of D-branes given by an autoequivalence of the derived category in which B becomes Cone(hom(A, B) ⊗ A → B). This transformation was also motivated by its relation to mirror symmetry and studied at length by Seidel and Thomas [20]. 3 Pedants will object that the cone construction is only defined up to a non-canonical isomorphism making the transformation on morphisms badly-defined. Fortunately, as is well-known and we discuss at length in Sect. 3, this transformation can be written as a Fourier–Mukai transform removing this objection. 4 In [17] it was suggested that the monodromy action on the derived category should be translated by 2 to undo this action on A. Since monodromy is a relabeling process, one is free to do this, but it looks unnatural from the perspective of associating monodromy with -stability.


51

2.2. General monodromies. It is then natural to ask what happens more generally, i.e., if more than just a single D-brane A becomes massless. In order to answer this we need to set up a general description of how one might analyze monodromy in a multi-dimensional moduli space. There are two paradigms for monodromy — both of which are useful: 1. The discriminant locus decomposes into a sum of irreducible divisors. Pick some base point in the moduli space and loop around a component of the discriminant “close” to the base point. 2. Restrict attention to a special rational curve C in the moduli space. This rational curve contains two “phase limit points”, in a sense to be described below, and a single point in the discriminant. The loop in question is around this unique discriminant point. In the case of the one parameter models, such as the quintic, these two paradigms coincide. The moduli space is C ∼ = P1 and the discriminant locus is a single point. If a component of the discriminant intersects C transversely then we can again have agreement between these two pictures of monodromy. In general the discriminant need not intersect C transversely — a fact we use to our advantage in Sect. 2.3. We now recall the relationship between the discriminant locus and phases as analyzed in [24, 25]. The following is a very rapid review. Please refer to the references for more details. To make the discussion easier we suffer a little loss of generality and assume we are in the “Batyrev-like” [26] case X being a hypersurface in a toric variety. The data for X is then presented in the form of a point set A which is the intersection of some convex polytope with some lattice N . See [27], for example, for more details of this standard construction. The conformal field theory associated to this data then has a phase structure where each “phase” is associated to a regular triangulation of A [28, 29]. The real vector space in which the Kähler form lives is naturally divided into a “secondary fan” of all possible phases. One cone of this fan is the Kähler cone for X where we have the “Calabi–Yau” phase. Mirror to X, Y is described as the zero-set of a polynomial W in many variables. The points in A are associated one-to-one with each monomial in W . Thus the data A is associated to deformations of complex structure of Y via the monomial-divisor mirror map [30]. If we model the moduli space of complex structures on Y by the space of coefficients in W , then the discriminant locus can be computed by the failure of W to be transversal. This can be mapped back to the space of complexified Kähler forms on X. The result is that part of the discriminant asymptotically lives in each wall dividing adjacent phases in the space of Kähler forms. That is to say, if we tune the B-field suitably we can always hit a bad conformal field theory as we pass from one phase to another. Thus we may associate singular conformal field theories with phase transitions. The discriminant itself is generically reducible. The combinatorial structure of this reduction has been studied in detail in [31]. In particular, any time an m-dimensional face of the convex hull of the set A contains more than m + 1 points, the resulting linear relationship between these points yields a component of . One may then follow an algorithm presented in [25] to compute the explicit form of each component. The general picture then is of a discriminant with many components with each component having “fingers” which separate the phases from each other. Each phase transition is associated with fingers from one or more component of . Torically each maximal cone in the secondary fan is associated to a point in the moduli space which gives the limit point in the “deep interior” of the associated phase.

52


The real codimension-one wall between two maximal cones corresponds to a rational curve C passing through two such limit points. The rational curve C will intersect the discriminant locus in one point as promised earlier in this section. One component of is distinguished — it corresponds to the case of viewing the full convex hull as a face of itself. This is called the “primary” component of . Closely tied in with Conjecture 1 (and at least partially attributed to the same authors) is the following conjecture: Conjecture 2. At any point on the primary component of (reached by a suitable path from a suitable basepoint) the 6-brane associated with the structure sheaf OX and its translates become massless. At a generic point no other D-branes become massless. This idea was perhaps first discussed in [32]. It is certainly a very natural conjecture — the primary component of the discriminant is a universal feature for any Calabi–Yau manifold and so must be associated with the masslessness of a very basic D-brane. The fact that it works for the quintic was explicitly computed in [12], and presumably it is possible to verify the conjecture in a much larger class of examples. We will assume this conjecture to be true. The Kähler cone is a particular maximal cone in the secondary fan corresponding to the “Calabi–Yau” phase. Let us concentrate on the walls of the Kähler cone. A typical situation as we approach the wall of the Kähler cone is that an exceptional set E collapses to some space Z. We depict this as

E

i

/X

(12)

q

Z

where i is an inclusion (which may well be the identity) and q is a fibration with a strict inequality dim(E) > dim(Z). Associated with such a wall in the secondary fan we have a rational curve C in the moduli space connecting the large radius limit point with some other limit point. We wish to consider the monodromy associated to circling the point in the discriminant in C. The resulting autoequivalence on the derived category has been studied in [8] where it was dubbed an “EZ-transformation”. The simplest example would be the case of the quintic Calabi–Yau threefold which has only one deformation of the Kähler form. This single component of the Kähler form gives the overall size of the manifold. Thus the “wall” (i.e., the origin) corresponds to X collapsing to a point. In this case i is the identity map and Z is a point. This is the case discussed above in Sect. 2.1. Indeed, it appears that in all cases where Z is a point, the resulting monodromy amounts to a transform of the type studied in Sect. 2.1. It is precisely when Z is more than just a point the case of interest to us. 2.3. New monodromies from old. The precise form of the “EZ-monodromy conjecture” which associates an autoequivalence of D(X) with a given EZ-transform was given in [8]. Rather than appealing this conjecture, let us derive the simplest example of a more general case, by assuming the conjectures above, dealing with the case of a single massless D-brane.


53

We will look at the well-known example [33], where X is a degree 8 hypersurface in the resolution of a weighted projective space P4{2,2,2,1,1} . The mirror Y is then a quotient of the same hypersurface with defining equation a0 z1 z2 z3 z4 z5 + a1 z14 + a2 z24 + a3 z34 + a4 z48 + a5 z58 + a6 z44 z44 .

(13)

The “algebraic” coordinates on the moduli space are then given by a 4 a5 . a62

(14)

0 = (1 − 28 x)2 − 218 x 2 y.

(15)

x=

a1 a2 a3 a6 , a04

y=

The primary component of can be computed as

The edge of the convex hull containing the points labeled by a4 , a5 , a6 leads to another component 1 = 1 − 4y,

(16)

with = 0 1 . X can be viewed as a K3-fibration π : X → P1 . In this case the component of the 1 Kähler form given asymptotically (for x, y 1) by 2πi log(x) controls the size of the 1 K3 fibre. The component of the Kähler form given asymptotically by 2πi log(y) gives the size of the P1 base. The base P1 is made very large by setting y → 0. In this case, we hit the primary component 0 of the discriminant when x = 2−8 . Let us refer to this point as P1 . Increasing x beyond this value moves one out of the Calabi–Yau phase into the hybrid “P1 -phase” where the model is best viewed as a fibration with base P1 and a Landau– Ginzburg orbifold as fibre [29]. Fixing y = 0 and varying x spans a rational curve C in the moduli space shown in Fig. 1. We would like to analyze the monodromy around the singularity P1 in Fig. 1. Clearly the transition associated with this monodromy consists of collapsing X onto the P1 base. In the language of (12), E = X, i.e., the inclusion map i is the identity, and Z ∼ = P1 . The map q is given by the fibration map π. In a way, we have constructed the simplest possible example where Z is more than just a point.

Large Radius Limit ( x = 0)

Calabi-Yau Singularity P1 ( x = 1/256) Hybrid P

1

H x=∞

Fig. 1. Moduli Space for y = 0

54

Hybrid Limit


L

P1

C

Large Radius Limit

H ∆0

K

Fig. 2. Full Moduli Space around P1

Now the useful trick is that the monodromy around the singularity in Fig. 1 can be written in terms of other monodromies that we already understand. This was originally described in [33], while this feature was also exploited in [7, 34]. The full moduli space near P1 is shown in Fig. 2 (with complex dimensions shown as real). The rational curve C given by y = 0 corresponds to an infinite radius limit and as such we understand the monodromy around it (see for example [7, 34]). (We will follow closely the notation and analysis of [34]). Let L refer to the autoequivalence of D(X) we apply upon looping this curve. It follows that L(B) = B ⊗ OX (S),

(17)

where S is the divisor class of a K3 fibre in X. Meanwhile let K refer to the autoequivalence of D(X) we apply upon looping the primary component 0 = 0 (i.e., denote KOX of Sect. 2.1 by K). Then from Conjectures 1 and 2 we know that K(B) = Cone(hom(OX , B) ⊗ OX → B).

(18)

It follows (see, for example Sect. 5.1 of [34] for an essentially identical computation) that the autoequivalence for the desired loop shown in Fig. 1 around P1 is given by L−1 KLK.

(19)

The desired goal therefore is to find the autoequivalence of D(X) obtained by combining the transforms in the above form. The result is that L−1 KLK = H,

(20)

where H is an autoequivalence that acts on D(X) by H(B) = Cone(π ∗ π∗ B → B).

(21)

Section 3 is devoted to the proof of this statement. Let us review briefly what is exactly meant by the rather concise notation of (21). Given the map π : X → Z and a sheaf E on X we may construct the “push-forward” sheaf π∗ E on Z by associating π∗ E (U ) with E (π −1 U ) for any open set U ⊂ Z. The π∗ appearing in (21) is the right-derived functor of this push-forward map. This π∗ “knows” about the cohomology of the fibre of π (see, for example, Chapter III of [35]). The pull-back map π ∗ is defined for sheaves of OZ -modules, and in particular for locally free sheaves, and thus for vector bundles.


55

The map π ∗ appearing in (21) is the corresponding left-derived functor. It is a central result of the theory of derived categories [36] that π ∗ is the left-adjoint of π∗ : HomX (π ∗ E, F) ∼ = HomZ (E, π∗ F),

(22)

for any E ∈ D(Z) and F ∈ D(X). It follows that HomX (π ∗ π∗ B, B) ∼ = HomZ (π∗ B, π∗ B).

(23)

Thus the most natural morphism that would appear in (21) is the image of the identity on the right-hand side of Eq. (23) under this natural isomorphism. One can show that this is indeed the case. 2.4. Interpretation of monodromy. Let us interpret (21) in light of our discussion of monodromy from -stability in Sect. 2.1. To aid our discussion consider how one might rewrite the monodromy result (18) for the primary component of the discriminant. Let c : X → x be the constant map of X to a single point. One can then show, using the fact that sheaf cohomology is equivalent to c∗ , which, in turn, is also given by the global section functor Hom(OX , −) [35], that (18) is equivalent to K(B) = Cone(c∗ c∗ B → B).

(24)

Now, the only D-brane (up to translation in D(X)) which becomes massless in this case is OX , which is equal to c∗ C, where we denote the trivial (very trivial!) line bundle on the point x as C. That is, massless D-branes for the primary component of the discriminant are given by c∗ (something). The c∗ in (24) then gives a natural map to form a cone as required. The expression (10) immediately dictates that we may interpret Cone(π ∗ π∗ B → B) in a similar way. The D-branes becoming massless at the point P1 in the moduli space correspond to π ∗ z for some z ∈ D(Z). The push-forward map π∗ can be viewed as the natural ingredient required to form the cone from (23). There is one technical subtlety here which needs to be mentioned. The set of objects of the form π ∗ z for any z ∈ D(Z) is not closed under composition by the cone construction. If we write C = Cone(f : π ∗ a → π ∗ b),

(25)

for two objects a, b ∈ D(Z), then we may only write C = π ∗ Cone(f : a → b),

(26)

when there is a relationship between the morphisms f = π ∗ f . Unfortunately such an f need not exist for arbitrary f . Thus we might more properly say that the set of massless D-branes are generated by objects of the form π ∗ z, where we allow for composing such states. Flushed with success at interpreting the autoequivalence given by this example we now write down the most obvious generalization for the more general EZ-transform (12). First of all, we expect q ∗ (something) to be a massless D-brane on E. E is mapped into X by the inclusion map i. The push-forward map i∗ is then “extension by zero” of a sheaf which is the obvious way of mapping D-branes on E into D-branes on X. Our next conjecture is then

56


Conjecture 3. Any D-Brane which becomes massless at a point on a component of the discriminant associated with an EZ-transform is generated by objects of the form i∗ q ∗ z for z ∈ D(Z). This implies a corresponding autoequivalence for the monodromy from -stability: B → Cone(i∗ q ∗ ζ B → B),

(27)

for some “natural” map ζ : D(X) → D(Z). To compute ζ we use the same trick as (23). Introduce the functor “i ! ” as the right-adjoint of i∗ : HomX (i∗ E, F) ∼ = HomE (E, i ! F),

(28)

for any E ∈ D(E) and F ∈ D(X). The existence of i ! is one of the most important features of the derived category in algebraic geometry [36] and (28) may be regarded as a generalization of Serre Duality. Now we have HomX (i∗ i ! B, B) ∼ = HomE (i ! B, i ! B).

(29)

This leads to a natural map i∗ q ∗ q∗ i ! B → i∗ i ! B → B. This implies Conjecture 4. The monodromy around the discriminant in the wall associated to a phase transition given by an EZ-transform leads to the following autoequivalence on D(X): B → Cone(i∗ q ∗ q∗ i ! B → B).

(30)

This is equivalent to the EZ-monodromy conjecture of [8]. In particular, it was proven there that this is indeed an autoequivalence of D(X). We should perhaps emphasize that we have not proven Conjectures 3 and 4. Rather we have used the known connection between -stability and monodromy and generalized the example we considered in the simplest and most obvious way. Note that we may derive Conjecture 1 from Conjectures 3 and 4 as follows. Suppose Z is a point, then i∗ q ∗ z can only be one thing, so we have a single massless D-brane. Furthermore, q∗ now becomes the cohomology functor giving i∗ q ∗ q∗ i ! B = i∗ (homE (OE , i ! B) ⊗ OE ) = homX (i∗ OE , B) ⊗ i∗ OE .

(31)

This also shows that the massless D-brane is i∗ OE — i.e., the D-brane wrapping around E as observed in [2, 10, 37]. For completeness we should also obtain the monodromy as one circles the discriminant in the opposite direction. Going back to the triangle (4) we see that decreasing θ would result in C being replaced by B = Cone(C → A[1])[−1]. This implies we modify the above monodromy arguments to consider the transformation under which C → Cone(C → i∗ q ∗ ηC)[−1],

(32)

for some “natural” map η : D(X) → D(Z). That is, the massless objects bind “to the right” of C in the mapping cone rather than to the left. The “[−1]” is needed because of the asymmetrical definition of the mapping cone — we need to keep C in its original position.


57

The only nontrivial step in copying the above argument is that to construct η we need a left-adjoint functor for q ∗ . Given that q ! F = q ∗ F ⊗ q ! OZ we can construct such a functor from HomE (E, q ∗ F) = HomE (E ⊗ q ! OZ , q ! F) = HomZ (q∗ (E ⊗ q ! OZ ), F). Thus our desired transform is given by C → Cone C → i∗ q ∗ q∗ (i ∗ C ⊗ q ! OZ ) [−1].

(33)

(34)

It was shown in [8] that this is indeed the inverse of the transformation given in Conjecture 4. 3. Composing Transforms In this section we will prove (20). For completeness, we choose to adopt a more general point of view and work with the class of Calabi–Yau fibrations over projective spaces. Thus we cover examples such as elliptic fibrations over P2 as discussed in Sect. 4.3, as well as the case of K3 fibrations over P1 as desired in Sect. 2.3. Readers not familiar with manipulations in the derived category may well wish to accept the result and skip this section. Having said that, some of the methods used in this section are very powerful and may have many other applications to D-brane physics. For the sake of brevity we use the formalism of kernels to describe the Fourier-Mukai transforms. For the convenience of the reader we review some of the key notions involved. The notations follow those of [8]. For X a non-singular projective variety, an object G ∈ D(X × X) determines an exact functor of triangulated categories G : D(X) → D(X) by the formula G (−) := p2∗ (G ⊗ p1∗ (−)),

(35)

where p1 : X × X → X is projection on the first factor, while p2 is projection on the second factor. The object G ∈ D(X × X) is called the kernel. The convenience in using kernels comes about because of the following natural isomorphism of functors: G G ∼ = G ◦ G . The composition of the kernels G , G ∈ D(X × X) is defined as ∗ ∗ G G := p13∗ p23 (G ) ⊗ p12 (G) ,

(36)

(37)

where pij is the obvious projection from X×X×X to the relevant two factors. There is an identity element for the composition of kernels: (X )∗ (OX ), where X : X → X × X is the diagonal morphism. In this section X is assumed to be a smooth Calabi–Yau fibration of dimension n over Z ∼ = Pd , with π : X → Z the fibration map. For us, the Calabi–Yau fibration structure simply means that π : X → Z is a flat morphism (see Sect. III.9 of [35]) with the generic fibre a Calabi–Yau variety of dimension n − d. Further assumptions on the Calabi–Yau fibration will be added shortly.

58


In order to set the functors L, K and H of the previous section on firm mathematical footing and to define them in the more general context of this section, we need to describe the kernels that induce them as exact functors according to formula (35). The following commutative diagram contains most of the maps that we use in the sequel: X _

(38)

j

p2 /X X×X E E z EE z EE π zz π1 zz EE 2 z π π π×π EE z z EE z z E z E" |zz / Z. Z × Z Zo s1 s2 Xo

p1

The maps are mostly projections, that are obvious from the context, except for j := X : X → X × X the diagonal of X and π : X → Z the fibration map. We now define the Fourier–Mukai functor L to be the autoequivalence of D(X) induced by the kernel L = j∗ (π ∗ OZ (1)). This functor acts on D(X) by (compare to (17)) L(B) = B ⊗ π ∗ OZ (1).

(39)

Note that the use of the notation L for this functor is consistent with the one used in the previous section, since in the case when X is a K3 fibration over Z ∼ = P1 we have ∗ OX (S) = π OZ (1). We also define the exact functor K induced by the kernel K = Cone(OX×X → O ), with O := j∗ OX and OX×X → O the natural restriction map. We can quote for example Lemma 3.2 of [20] to conclude that the action of this functor on D(X) is indeed given by (18). We make the assumption that the sheaf OX is spherical (as defined in Sect. 2.1). This ensures that the functor K is an autoequivalence of D(X). Finally, we define the exact functor H to be the so-called fibrewise Fourier– Mukai transform associated to the Calabi–Yau fibration π : X → Z (see, for example, [20, 38, 39]). The functor H is induced by the kernel H = Cone(OX×Z X → O ), with OX×Z X viewed as a sheaf on X × X (extension by zero), and OX×Z X → O the restriction map. To ensure that the functor H is indeed an autoequivalence (Fourier–Mukai functor), we assume that the sheaf OX is EZ–spherical. In the language of [8], this means that there exists a distinguished triangle in D(Z) (Z ∼ = Pd ) of the form5 OZ → π∗ OX → OZ (−d − 1)[−n + d] → OZ [1].

(40)

Note that the sphericity and EZ-sphericity conditions are both satisfied in the specific example of a K3 fibration over P1 analyzed in the previous section of this paper. We now justify the use of the notation H to denote the fibrewise Fourier–Mukai functor by showing that its action on D(X) is indeed, even in the higher dimensional situation, given by the formula (21) of the previous section. 5

Lemma 3.12 in [20], as well as Example 3.3 of [8] provide sufficient conditions for (40) to hold.


59

The fibre product X ×Z X fits in the fibre square diagram X×Z X

q1

q2

X

/X

(41)

π

π

/ Z,

and let k : X ×Z X → X × X denote the canonical embedding. Since π is of finite type and flat, we can apply “cohomology commutes with the base change” (Prop. III 9.3 of [35] or for a more general form Prop. II 5.12 of [36]): π ∗ π∗ B ∼ = q2∗ q1∗ B,

(42)

for some B in D(X). On the other hand, q1 = p1 ◦ k, and q2 = p2 ◦ k, so we can write π ∗ π∗ B ∼ = q2∗ q1∗ B ∼ = p2∗ k∗ k ∗ p1∗ B ∼ = p2∗ (k∗ OX×Z X ⊗ p1∗ B).

(43)

Note that the last line in the previous formula represents the action on D(X) of the exact functor OX×Z X induced as in (35) by the kernel OX×Z X (shorthand for k∗ OX×Z X ). This shows that H(B) = Cone( OX×Z X (B) → B) = Cone(π ∗ π∗ B → B) as desired. We are now ready to start discussing the main goal of this section which is to prove the following relation between the defined Fourier–Mukai functors: (L−d KLd ) . . . (L−1 KL)K ∼ = H.

(44)

Equivalently, the same formula can be expressed using kernels as (L−d K Ld ) . . . (L−1 K L) K ∼ = H,

(45)

where, for any integer i, Li = j∗ (π ∗ OZ (i)). Of course, the case d = 1 (Z ∼ = P1 ) of (44) is precisely formula (20) of the previous section. Before we move on with the technicalities of the proof, a few remarks are in order. Note that the parentheses in the two formulae are simply decorative: the composition of functors, as well as the composition of kernels are associative (but, of course, not commutative!). For a fixed object G in D(X × X), the functors G − and − G from D(X × X) to D(X × X) are exact functors between triangulated categories (i.e. they preserve the distinguished triangles). Therefore, for any integer i, we can start with the distinguished triangle defining the kernel K, OX×X → O → K → OX×X [1],

(46)

and apply to it the operations L−i − and − Li from the left, and right, respectively. But L−i OX×X Li ∼ = (π × π )∗ (OZ (−i) OZ (i)), = π2∗ OZ (−i) ⊗ π1∗ OZ (i) ∼

(47)

and L−i O Li ∼ = O .

(48)

60


The notation OZ (−i) OZ (i) (the exterior tensor product) will be used quite often in what follows and simply designates s2∗ OZ (−i) ⊗ s1∗ OZ (i). As a shorthand for later convenience we introduce the following objects in D(X×X) : Ti := (π × π)∗ (OZ (−i) OZ (i)) ∼ = π2∗ OZ (−i) ⊗ π1∗ OZ (i)

(49)

Ci := L−i K Li ∼ = Cone(Ti → O ),

(50)

and

for i ∈ Z. To justify the definition of the kernel Ci , we need to explain how to define the morphism Ti → O . We start with the canonical pairing map on Z × Z, OZ (−i) OZ (i) → OZ ,

(51)

(π × π)∗ (OZ (−i) OZ (i)) → (π × π )∗ (OZ ).

(52)

(π × π)∗ (OZ ) ∼ = OX×Z X .

(53)

and lift it to X × X ,

We claim that

Indeed, for the fibre square X ×Z X _

t

/Z

π×π

/ Z×Z

Z

k

X×X

(54)

with t : X ×Z X → Z the “diagonal map” of the fibre square (41), and π × π flat, we can apply again “cohomology commutes with the base change” to obtain (π × π )∗ (Z )∗ OZ = k∗ t ∗ OZ . Since t ∗ OZ = OX×Z X , the previous formula can be written as (π × π )∗ (Z )∗ OZ = OX×Z X , which is exactly (53). The morphism Ti → O is then defined as the composition Ti ∼ = OX×Z X → O . = (π × π )∗ (OZ (−i) OZ (i)) → (π × π)∗ (OZ ) ∼

(55)

Note that C0 ∼ = K. Therefore we have to show that Cd . . . C1 C0 ∼ = H.

(56)

Note that the case d = 0 is immediate since in that case Z reduces to a point, and the Fourier–Mukai transforms H and K coincide. An important rôle in what follows will be played by Beilinson’s resolution of the diagonal in Z × Z ∼ = Pd × Pd [40], 0 → OZ (−d) dZ (d) → . . . → OZ (−1) 1Z (1) → OZ×Z → OZ → 0, (57) where iZ is the sheaf of holomorphic i-forms on Pd .


61

For any integer i, 0 ≤ i ≤ d, we define the complexes Si on Z ×Z to be the following truncated versions of Beilinson’s resolution 0 → OZ (−i) iZ (i) → . . . → OZ (−1) 1Z (1) → OZ×Z → 0,

(58)

arranged such that the sheaf OZ×Z is located at the 0th position. Define Si := (π × π)∗ (Si ).

(59)

We claim that there exists a natural map 6 Si → O ,

(60)

that can be defined at the level of complexes, where, as usual, O denotes the complex on X × X with the only non-zero component located at the 0th position. To see this, we first make the remark that there exists a map of complexes Si → OZ . Such a map of complexes is well defined, since the complex Si is a piece of Beilinson’s resolution. We can now proceed as in (55) and define the desired morphism in D(X × X) as the composition Si = (π × π)∗ (Si ) → (π × π)∗ (OZ ) ∼ = OX×Z X → O .

(61)

The key result to be proved in this section is the following: Claim. For any integer i, 0 ≤ i ≤ d, Ci . . . C1 C0 ∼ = Cone(Si → O ).

(62)

Before proving it, let us convince ourselves that the claim implies (56). The complex Sd is quasi-isomorphic to the sheaf OZ (more precisely, to the complex on Z × Z having the sheaf OZ at the 0th position). Therefore, the i = d case of the claim states that Cd . . . C1 C0 ∼ = Cone((π × π)∗ (OZ ) → O ).

(63)

Since by (53) (π × π)∗ (OZ ) ∼ = OX×Z X , we see that the kernel Cd . . . C1 C0 is indeed isomorphic to H. We now proceed with the inductive proof of the claim. The induction is performed with respect to i. The case i = 0 is clear, since S0 = OZ×Z and S0 = (π ×π )∗ (OZ×Z ) = OX×X . To prove the inductive step i ⇒ (i + 1), we start with the natural maps Si → O and Ti+1 → O . Applying the functors − Si and Ti+1 − we get two more maps, and we can form the commutative square Ti+1 Si

/ Si

Ti+1

/ O .

(64)

6 In fact, by assuming that the sheaf O is EZ–spherical, it can be shown that Hom ∼ X X×X (Si , O ) = C. Therefore, the described nonzero morphism from Si to O is essentially unique.

62


There is a nice result due to Verdier, guaranteeing that a commuting square X

/ Y

X

/Y

(65)

extends to a “9–diagram” of the form X

/ Y

/ Z

/ X [1]

X

/Y

/Z

/ X[1]

X

/ Y

/ Z

/ X [1]

X [1]

/ Y [1]

/ Z [1]

/ X [2] ,

(66)

where all the rows and columns are distinguished triangles, every square commutes, except for the last one (containing the shift operator [2]), which anticommutes (for more details see p. 24 in [41]). Applying Verdier’s “9–diagram” construction to (64) yields Ti+1 Si

/ Si

/ Ci+1 Si

/ Ti+1 Si [1]

/ O

/ Ci+1

/ Ti+1 [1]

/ Ci . . . C0

/ Ci+1 (Ci . . . C0 )

/ Ti+1 (Ci . . . C0 )[1]

/ Si [1]

/ Ci+1 Si [1]

/ Ti+1 Si [2] .

Ti+1 Ti+1 (Ci . . . C0 ) Ti+1 Si [1]

(67)

We are interested in the term Ci+1 Ci . . . C0 . To compute it, return for a moment to the commutative diagram (66), and consider the “diagonal” map Y → Z . The axioms of a triangulated category guarantee that the morphism Y → Z can be included in a distinguished triangle of the form A → Y → Z → A[1].

(68)

A crucial piece of the proof of Verdier’s “9–diagram” (p. 24 in [41]) provides another distinguished triangle that involves A, namely A → X → Y [1] → A[1] .

(69)

Returning to diagram (67), we obtain the distinguished triangles X → O → Ci+1 Ci . . . C0 → X [1] ,

(70)


63

and X → Ti+1 Ci . . . C0 → Si [1] → X [1] ,

(71)

for some element X in D(X × X). We plan on using the latter triangle to compute the object X , but for that we need to first understand the term Ti+1 Ci . . . C0 . The leftmost column of diagram (67) shows that Ti+1 Ci . . . C0 ∼ = Cone(Ti+1 Si → Ti+1 ).

(72)

∗ T ∗ By definition Ti+1 Si = p13∗ (p23 i+1 ⊗ p12 Si ). After inspecting the definitions of the kernels Ti+1 and Si , it is not hard to see that computing Ti+1 Si requires the calculation of kernels of the type j ∗ ∗ (π3∗ OZ (−i − 1) ⊗ π2∗ OZ (i + 1)) ⊗ p12 (π2∗ OZ (−j ) ⊗ π1∗ Z (j ))) ∼ p13∗ (p23 = j ∗ ∗ ∼ (73) = (π × π ) OZ (−i − 1) (j ) ⊗ ( X )∗ (π OZ (i + 1 − j )) , Z

with 0 ≤ j ≤ i, and X : X → {pt} the projection to a point. But ( X )∗ (π ∗ OZ (i + 1 − j )) ∼ = ( Z )∗ (π∗ OX ⊗ OZ (i + 1 − j )). Since OX is EZ– spherical, the long exact cohomology sequence induced by the distinguished triangle (40) implies that ( Z )∗ (π∗ OX ⊗ OZ (i + 1 − j )) ∼ = HomZ (OZ , OZ (i + 1 − j )).

(74)

Summing up our work, we can conclude that Cone(Ti+1 Si → Ti+1 ) ∼ = (π × π)∗ (OZ (−i − 1) Ui ),

(75)

where Ui is the following complex in D(Z) : 0 → iZ (i) ⊗ HomZ (OZ , OZ (i + 1 − i)) j

→ . . . → Z (j ) ⊗ HomZ (OZ , OZ (i + 1 − j )) → . . . → 1Z (1) ⊗ HomZ (OZ , OZ (i + 1 − 1)) → OZ ⊗ HomZ (OZ , OZ (i + 1)) → OZ (i + 1) → 0,

(76)

arranged such that the sheaf OZ (i + 1) is located at the 0th position. But what is the complex Ui after all? Again the answer can be obtained by employing Beilinson’s resolution. Consider the following two quasi-isomorphic complexes obtained by truncating (57) at the appropriate place: 0 → dZ (d) OZ (−d) → . . . → i+1 Z (i + 1) OZ (−i − 1) → 0, 0 → iZ (i) OZ (−i) → . . . → OZ×Z → OZ → 0,

(77)

arranged such that the sheaf OZ in the second complex is located at the 0th position th (and, as a result, the sheaf i+1 Z (i + 1) OZ (−i − 1) is located at the (−i − 1) position in the first complex). Call these complexes Ai and Bi , respectively.

64


On one hand, the well known properties of the cohomology groups of the projective space Z ∼ = Pd , give that s2∗ (Bi ⊗ s1∗ OZ (i + 1)) ∼ = Ui .

(78)

On the other hand, the same cohomology properties give that s2∗ (Ai ⊗ s1∗ OZ (i + 1)) ∼ = i+1 Z (i + 1)[i + 1].

(79)

But Bi and Ai are quasi-isomorphic (i.e. isomorphic in D(Z × Z)), hence Ui ∼ = i+1 Z (i + 1)[i + 1],

(80)

and Ti+1 Ci . . . C0 ∼ = Cone(Ti+1 Si → Ti+1 ) ∗ ∼ = (π × π) (OZ (−i − 1) i+1 (i + 1))[i + 1] . Z

(81)

The distinguished triangle (71) and the definition (59) of the complexes Si show that in fact our unknown complex X is nothing else but Si+1 . The proof by induction of the main claim of this section is then finished by invoking the distinguished triangle (70). 4. Applications 4.1. Z is a point. It was discussed above that the case of Z being a point amounts to a single D-brane becoming massless. This is the case originally studied by Strominger in [14] and yielding monodromy of the form studied in detail by Seidel and Thomas [20]. There are three possibilities: 1. E = X in which case we are looking at the primary component of the discriminant. The quintic was studied at length in [12]. 2. E is a complex surface of codimension one in X. This could arise from the blow-up of an isolated quotient singularity. This was studied for example in [37]. 3. E is a rational curve. This is the flop case and was studied in [13]. 4.2. Z is a curve. Now we have an infinite number of massless D-branes arising from the derived category of an algebraic curve. There are two possibilities: 1. E = X in which case X is a K3-fibration and Z ∼ = P1 . This is the case we studied in Sect. 2. 2. E is a ruled surface arising from blowing up a curve of quotient singularities in X. In either case we are essentially looking at nonperturbatively enhanced gauge symmetry [42, 43]. Putting z = Op for some point p ∈ Z we obtain a soliton q ∗ z = OF which is the structure sheaf of a single fibre F of the map q. These correspond to the charged vector bosons responsible for the enhanced gauge symmetry. Clearly these bosons classically have a moduli space given by Z since p may vary in Z. Upon including quantum effects this leads to a number of massless hypermultiplets in the theory given by the genus of Z [43, 44].


65

Putting z = OZ we obtain q ∗ z = OX which corresponds to one of the massless monopoles. In fact we may analyze the complete spectrum of solitons in Seiberg–Witten theory [45] by using the “geometric engineering” approach of [46] to “zoom in” on the point in the moduli space where the nonabelian gauge symmetry appears. We will explain exactly what happens in detail in [15]. It is worth speculating that such analysis of Seiberg–Witten theory may shed some light on the “local mirror symmetry” story of papers such as [47]. The massless B-type D-branes associated with the derived category of Z may be related to the A-type D-brane story of [48] by some kind of local homological mirror symmetry. 4.3. Z is a surface. There is only one possibility, namely X = E is an elliptic fibration over Z. We will denote this fibration π : X → S. Let z ∈ D(S) correspond to the skyscraper sheaf of a point s ∈ S. Then π ∗ z ∈ D(X) corresponds to the structure sheaf of an elliptic fibre e ⊂ X over s. Let us consider the case where the size of S becomes infinite. According to our rules then, the 2-brane wrapping e should become massless when we hit the discriminant moving from the large radius phase to the phase where the elliptic fibres such as e have collapsed. At first sight this looks peculiar. One does not usually expect a 2-brane wrapped around a 2-torus to become massless for a particular radius of the torus! Actually we will argue that this indeed happens and that it is when the 2-torus is zero sized that the 2-brane becomes massless. Why T-duality doesn’t interfere with this will become apparent.7 As a specific example let X be given by the following equation in P4{9,6,1,1,1} : x12 + x23 + x318 + x418 + x518 .

(82)

This has a quotient singularity which may be resolved with an exceptional divisor P2 . One may regard [x3 , x4 , x5 ] as homogeneous coordinates on this S ∼ = P2 . Fixing a point 2 on S one then has an elliptic fibre e given by a sextic in P{3,2,1} . See, for example, [49] for more details about such fibrations. The moduli space of interest to us regards the area of e. This area is infinite for the Calabi–Yau limit point and shrinks down as we approach the other limit point. For a more precise statement let us consider the mirror e˜ of e given by the following equation in P2{3,2,1} : 1

x12 + x23 + x36 + 432 6 ψx1 x2 x3 .

(83)

Varying the size of e is then mirror to varying the complex structure of e˜ by varying ψ. Going through the usual story of solving the Picard–Fuchs equation and using the mirror map to map back to the B + iJ plane for e we obtain the shaded region in Fig. 3. The result is that for the Calabi–Yau limit point we √ have ψ → ∞ and thus J → ∞ as expected. For the other limit point ψ = 0 and J = 3/2. The discriminant is given by ψ = 1 and corresponds to J = 0, i.e., e has zero area. This is where the 2-branes wrapping e (and all the other D-branes given by π ∗ z for z ∈ D(S)) become massless. So what about T-duality for e? Note that the moduli space in Fig. 3 corresponds to two fundamental regions for the action of SL(2, Z) on the upper-half plane. T-duality 7

We are grateful to R. Plesser for an invaluable discussion on this point.

66


ψ

∞

ψ =0 ψ =1 Fig. 3. Moduli Space for Elliptic Curve

should map the point at ψ = 1 to the point at ψ = ∞. It is important to remember however that T-duality does not act only upon e — one must also shift the string dilaton by an amount related to the resulting change in the area of the torus. As such, the point at ψ = 1 is related to ψ = ∞ only with the string coupling shifted off to infinity, implying again that the D-brane mass is zero. Therefore if we fix the dilaton to be a finite value we cannot use T-duality to relate ψ = 1 to any large radius torus. This explains why we really do have a massless 2-brane appearing wrapped around a zero-sized torus. Note also that if we allow the base S ∼ = P2 to have finite size then the T-duality group ceases to exist anyway as in [50]. The fact that there is a T-duality relating ψ = ∞ to ψ = 1 shows that their physics must be similar. In particular ψ = 1 must be an infinite distance away in the moduli space. Indeed, in many respects the spectrum of stable D-branes at ψ = 1, coming from the derived category of P2 , must be similar to the spectrum one would see upon going to a large radius limit. It would be interesting to investigate this in more detail and generality. 4.4. The Exoflop. Finally let us note that not all the walls of the Kähler cone correspond directly to some subspace E collapsing to Z. Even so, it appears that we can still fit many, if not all examples into the general EZ language. We illustrate this with an “exoflop”. Let X be the degree 12 hypersurface in P4{3,3,3,2,1} . Its mirror, Y , then has defining equation a0 z1 z2 z3 z4 z5 + a1 z14 + a2 z24 + a3 z34 + a4 z46 + a5 z512 + a6 z42 z58 + a7 z44 z54 ,

(84)

and we may use the following algebraic coordinates on the moduli space of complex structures of Y : x=

a1 a2 a3 a62 a04 a5

,

y=

a 4 a6 , a72

z=

a 5 a7 . a62

(85)

We have chosen coordinates so that the Kähler cone of X appears naturally as a positive octant in the secondary fan. In particular this means that the 3 rational curves


67

in the moduli space connecting the large radius limit to each of the three neighbouring phase limit points are given by setting x = y = 0, x = z = 0 or y = z = 0 respectively. The discriminant has two components given by 0 = −1 − 64x + 768xz + 32768x 2 z − 196608x 2 z2 − 4194304x 3 z2 + 294912x 2 yz2 +16777216x 3 (yz2 + z3 ) − 75497472x 3 yz3 + 113246208x 3 y 2 z4 (86) and 1 = −1 + 4y + 4z − 18yz + 27y 2 z2 .

(87)

The phase transition of interest occurs for the rational curve C for which y = z = 0. For small x we are in the Calabi–Yau phase. For large x the Calabi–Yau X undergoes an “exoflop” [51]. That is X becomes reducible with one component consisting of a threefold with a singularity. The other component consists of a fibration of a Landau– Ginzburg orbifold theory over a P1 . These components intersect at a point which is the singularity in the threefold component. This then is not an EZ transformation. Note however that the discriminant 0 inter1 sects C transversely at (x, y, z) = (− 64 , 0, 0) and so the monodromy within C is exactly given by monodromy around the primary component of the discriminant. Therefore exactly one D-brane becomes massless at the transition point — the D6-brane wrapping X. This exoflop transition is equivalent, as far as monodromy is concerned, to an EZ transformation with X = E and Z given by a point. This is not entirely surprising given the following. Classically one would describe the exoflop wall of the Kähler cone as a wall where X J 3 = 0. Thus the classical volume of X is going to zero, even though the volume of some surfaces and curves within X, as measured by the Kähler form, do not vanish. Since the volume of X vanishes, one should expect the D6-brane to have vanishing mass. The fact that no other D-branes become massless is not obvious. It would be interesting to show that all phase transitions give rise to monodromies that can be associated with EZ transformations. Acknowledgements. It is a pleasure to thank J. Distler, D. Morrison and R. Plesser for useful conversations. P.S.A. is supported in part by NSF grant DMS-0074072 and by a research fellowship from the Alfred P. Sloan Foundation. R.L.K. was partly supported by NSF grants DMS-9983320 and DMS-0074072.

References 1. Kontsevich, M.: Homological Algebra of Mirror Symmetry. In: “Proceedings of the International Congress of Mathematicians”, Basel-Boston: Birkhäuser, 1995, pp. 120–139 2. Douglas, M. R.: D-Branes, Categories and N=1 Supersymmetry. J. Math. Phys. 42, 2818–2843 (2001) 3. Lazaroiu, C. I.: Unitarity, D-Brane Dynamics and D-brane Categories. JHEP 12, 031 (2001) 4. Aspinwall, P. S., Lawrence, A. E.: Derived Categories and Zero-Brane Stability. JHEP 08, 004 (2001) 5. Diaconescu, D.-E.: Enhanced D-brane Categories from String Field Theory. JHEP 06, 016 (2001) 6. Kontsevich, M.: 1996, Rutgers Lecture, unpublished 7. Horja, R. P.: Hypergeometric Functions and Mirror Symmetry in Toric Varieties. http://arxic.org/list/math.AG/9912109, 1999 8. Horja, R. P.: Derived Category Automorphisms from Mirror Symmetry. Duke Math. J. 127, 1–34 (2005) 9. Douglas, M. R., Fiol, B., Römelsberger, C.: Stability and BPS Branes. http://arxiv.org/list/hepth/0002037, 2000 10. Douglas, M. R., Fiol, B., Romelsberger, C.: The Spectrum of BPS Branes on a Noncompact CalabiYau. http://arxiv.org/list/hep-th/0003263, 2000

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Douglas, M. R.: Topics in D-geometry. Class. Quant. Grav. 17, 1057–1070 (2000) Aspinwall, P. S., Douglas, M. R.: D-Brane Stability and Monodromy. JHEP 05, 031 (2002) Aspinwall, P. S.: A Point’s Point of View of Stringy Geometry. JHEP 01, 002 (2003) Strominger, A.: Massless Black Holes and Conifolds in String Theory. Nucl. Phys. B451, 96–108 (1995) Aspinwall, P. S., Karp, R. L.: Solitons in Seiberg–Witten Theory and D-Branes in the Derived Category. JHEP 04, 049 (2003) Sen, A.: Tachyon Condensation on the Brane Antibrane System. JHEP 08, 012 (1998) Distler, J., Jockers, H., Park, H.: D-Brane Monodromies, Derived Categories and Boundary Linear Sigma Models. http://arxiv.org/list/hep-th/0206242, 2002 Fukaya, K.: Floer Homology and Mirror Symmetry I. AMS/IP Stud. in Adv. Math. 23, Providence, RI: Amer. Math. Soc., 2001, pp. 15–43 Seidel, P.: Graded Lagrangian Submanifolds. Bull. Soc. Math. France 128, 103–149 (2000) Seidel, P., Thomas, R. P.: Braid Groups Actions on Derived Categories of Coherent Sheaves. Duke Math. J. 108, 37–108 (2001) Bridgeland, T., Maciocia, A.: Fourier-Mukai transforms for Quotient Varieties. http://arxiv.org/list/math.AG/9811101, 1998 Candelas, P., de la Ossa, X. C., Green, P. S., Parkes, L.: A Pair of Calabi–Yau Manifolds as an Exactly Soluble Superconformal Theory. Nucl. Phys. B359, 21–74 (1991) Morrison, D. R.: Geometric Aspects of Mirror Symmetry. In: Enquist, B., Schmid, W. (eds.), “Mathematics Unlimited – 2001 and Beyond”, Berlin-Heidelberg-NewYork: Springer-Verlag, 2001, pp. 899–918 Aspinwall, P. S., Greene, B. R., Morrison, D. R.: Measuring Small Distances in N = 2 Sigma Models. Nucl. Phys. B420, 184–242 (1994) Morrison, D. R., Plesser, M. R.: Summing the Instantons: Quantum Cohomology and Mirror Symmetry in Toric Varieties. Nucl. Phys. B440, 279–354 (1995) Batyrev, V. V.: Dual Polyhedra and Mirror Symmetry for Calabi–Yau Hypersurfaces in Toric Varieties. J. Alg. Geom. 3, 493–535 (1994) Aspinwall, P. S., Greene, B. R.: On the Geometric Interpretation of N = 2 Superconformal Theories. Nucl. Phys. B437, 205–230 (1995) Witten, E.: Phases of N = 2 Theories in Two Dimensions. Nucl. Phys. B403, 159–222 (1993) Aspinwall, P. S., Greene, B. R., Morrison, D. R.: Multiple Mirror Manifolds and Topology Change in String Theory. Phys. Lett. 303B, 249–259 (1993) Aspinwall, P. S., Greene, B. R., Morrison, D. R.: The Monomial-Divisor Mirror Map. Internat. Math. Res. Notices, 1993, pp. 319–338 Gelfand, I. M., Kapranov, M. M., Zelevinski, A. V.: Discriminants, Resultants and Multidimensional Determinants. Basel-Boston: Birkhäuser, 1994 Greene, B. R., Kanter,Y.: Small Volumes in Compactified String Theory. Nucl. Phys. B497, 127–145 (1997) Candelas, P. et al.: Mirror Symmetry for Two Parameter Models — I. Nucl. Phys. B416, 481–562 (1994) Aspinwall, P. S.: Some Navigation Rules for D-brane Monodromy. J. Math. Phys. 42, 5534–5552 (2001) Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics 52, Berlin-Heidelberg-New York: Springer-Verlag, 1977 Hartshorne, R.: Residues and Duality. Lecture Notes in Math. 20, Berlin-Heidelberg-New York: Spinger-Verlag, 1966 Diaconescu, D.-E., Gomis, J.: Fractional Branes and Boundary States in Orbifold Theories. JHEP 10, 001 (2000) Bridgeland, T., Maciocia, A.: Fourier-Mukai Transforms for K3 and Elliptic Fibrations. http://arxiv.org/list/math.AG/9908022, 1999 Andreas, B., Curio, G., Hernandez Ruiperez, D., Yau, S.-T.: Fourier–Mukai Transforms and Mirror Symmetry for D-Branes on Elliptic Calabi–Yau. http://arxiv.org/list/math.AG/0012196, 2000 Beilinson, A. A.: Coherent Sheaves on Pn and Problems in Linear Algebra. Funct. Anal. Appl. 12, 214–216 (1978) Beilinson, A. A., Bernstein, J. N., Deligne, P.: Faisceaux pervers. Astérisque 100, (1982) Aspinwall, P. S.: Enhanced Gauge Symmetries and Calabi–Yau Threefolds. Phys. Lett. B371, 231– 237 (1996) Katz, S., Morrison, D. R., Plesser, M. R.: Enhanced Gauge Symmetry in Type II String Theory. Nucl. Phys. B477, 105–140 (1996) Witten, E.: Phase Transitions in M-Theory and F-Theory. Nucl. Phys. B471, 195–216 (1996)

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45. Seiberg, N., Witten, E.: Electric - Magnetic Duality, Monopole Condensation, and Confinement in N=2 Supersymmetric Yang-Mills Theory. Nucl. Phys. B426, 19–52 (1994) (erratum-ibid. B430, 485–486 (1994)) 46. Kachru, S. et al.: Nonperturbative Results on the Point Particle Limit of N=2 Heterotic String Compactifications. Nucl. Phys. B459, 537–558 (1996) 47. Katz, S., Mayr, P., Vafa, C.: Mirror Symmetry and Exact Solution of 4D N = 2 Gauge Theories. I. Adv. Theor. Math. Phys. 1, 53–114 (1998) 48. Klemm, A. et al.: Self-Dual Strings and N=2 Supersymmetric Field Theory. Nucl. Phys. B477, 746–766 (1996) 49. Vafa, C., Witten, E.: Dual String Pairs With N = 1 and N = 2 Supersymmetry in Four Dimensions. In: “S-Duality and Mirror Symmetry”, Nucl. Phys. (Proc. Suppl.) B46, 225–247 (1996) 50. Aspinwall, P. S., Plesser, M. R.: T-Duality Can Fail. J. High Energy Phys. 08, 001 (1999) 51. Aspinwall, P. S., Greene, B. R., Morrison, D. R.: Calabi–Yau Moduli Space, Mirror Manifolds and Spacetime Topology Change in String Theory. Nucl. Phys. B416, 414–480 (1994) Communicated by N.A. Nekrasov


Communications in


The First Eigenvalue of the Dirac Operator on Compact Spin Symmetric Spaces Jean-Louis Milhorat Laboratoire Jean Leray, UMR CNRS 6629, Département de Mathématiques, Université de Nantes, 2, rue de la Houssinière, BP 92208, 44322 Nantes Cedex 03, France. E-mail: [email protected] Received: 12 February 2004 / Accepted: 21 March 2005 Published online: 8 July 2005 – © Springer-Verlag 2005

Abstract: We give a formula for the first eigenvalue of the Dirac operator acting on spinor fields of a spin compact irreducible symmetric space G/K. 1. Introduction It is well-known that symmetric spaces provide examples where detailed information on the spectrum of Laplace or Dirac operators can be obtained. Indeed, for those manifolds, the computation of the spectrum can be (theoretically) done using group theoretical methods. However the explicit computation is far from being simple in general and only a few examples are known. On the other hand, many results require some information about the first (nonzero) eigenvalue, so it seems interesting to get this eigenvalue without computing all the spectrum. In that direction, the aim of this paper is to prove the following formula for the first eigenvalue of the Dirac operator: Theorem 1. Let G/K be a compact, simply-connected, n-dimensional irreducible symmetric space with G compact and simply-connected, endowed with the metric induced by the Killing form of G sign-changed. Assume that G and K have the same rank and that G/K has a spin structure. Let βk , k = 1, . . . , p, be the K-dominant weights occurring in the decomposition into irreducible components of the spin representation under the action of K. Then the square of the first eigenvalue of the Dirac operator is 2 min βk 2 + n/8 , 1≤k≤p

(1)

where · is the norm associated to the scalar product < , > induced by the Killing form of G sign-changed. Remark 1. The proof uses a lemma of R. Parthasarathy in [Par71], which allows to express (1) in the following way. Let T be a fixed common maximal torus of G and K. Let be the set of non-zero roots of G with respect to T . Let δG , (resp. δK ) be

72

J.-L. Milhorat

the half-sum of the positive roots of G, (resp. K), with respect to a fixed lexicographic ordering in . Then the square of the first eigenvalue of the Dirac operator is given by 2 δG 2 + 2 δK 2 − 4 max < w · δG , δK > +n/8 , w∈W

(2)

where W is a certain (well-defined) subset of the Weyl group of G.

2. The Dirac Operator on a Spin Compact Symmetric Space We first review some results about the Dirac operator on a spin symmetric space, cf. for instance [CFG89] or [Bär91]. A detailed survey on the subject may be found, among other topics, in the reference [BHMM]. Let G/K be a spin compact symmetric space. We assume that G/K is simply connected, so G may be chosen to be compact and simply connected and K is the connected subgroup formed by the fixed elements of an involution σ of G, cf. [Hel78]. This involution induces the Cartan decomposition of the Lie algebra G of G into G = K ⊕ P, where K is the Lie algebra of K and P is the vector space {X ∈ G ; σ∗ · X = −X}. This space P is canonically identified with the tangent space to G/K at the point o, o being the class of the neutral element of G. We also assume that the symmetric space G/K is irreducible, so all the G-invariant scalar products on P, hence all the G-invariant Riemannian metrics on G/K are proportional. We consider the metric induced by the Killing form of G sign-changed. With this metric, G/K is an Einstein space with scalar curvature Scal = n/2, (cf. for instance Theorem 7.73 in [Bes87]). The spin condition implies that the homomorphism α : K → SO(P) SOn , k → AdG (k)|P lifts to a homomorphism α : K → Spinn , cf. [CG88]. Let ρ : Spinn → HomC (, ) be the spin representation. The composition ρ ◦ α defines a “spin” representation of K which is denoted ρK . The spinor bundle is then isomorphic to the vector bundle := G ×ρK . Spinor fields on G/K are then viewed as K-equivariant functions G → , i.e. functions: :G→

s.t.

∀g ∈ G , ∀k ∈ K , (gk) = ρK (k −1 ) · (g) .

Let L2K (G, ) be the Hilbert space of L2 K-equivariant functions G → . The Dirac operator D extends to a self-adjoint operator on L2K (G, ). Since it is an elliptic operator, it has a (real) discrete spectrum. Now if the spinor field is an eigenvector of D for the eigenvalue λ, then the spinor field σ ∗ · is an eigenvector for the eigenvalue −λ, hence the spectrum of the Dirac operator is symmetric with respect to the origin. Thus the spectrum of D may be deduced from the spectrum of its square D2 . By the PeterWeyl theorem, the natural unitary representation of G on the Hilbert space L2K (G, ) decomposes into the Hilbert sum ⊕ Vγ ⊗ HomK (Vγ , ) ,

γ ∈G

The First Eigenvalue of the Dirac Operator on Compact Spin Symmetric Spaces

73

is the set of equivalence classes of irreducible unitary complex representations where G and HomK (Vγ , ) is the vector space of of G, (ργ , Vγ ) represents an element γ ∈ G K-equivariant homomorphisms Vγ → , i.e. HomK (Vγ , ) = {A ∈ Hom(Vγ , ) s.t. ∀k ∈ K , A ◦ ργ (k) = ρK (k) ◦ A} . The injection Vγ ⊗ HomK (Vγ , ) → L2K (G, ) is given by v ⊗ A → g → (A ◦ ργ (g −1 ) ) · v . Note that Vγ ⊗ HomK (Vγ , ) consists of C ∞ spinor fields to which the Dirac operator can be applied. The restriction of D2 to the space Vγ ⊗ HomK (Vγ , ) is given by the Parthasaraty formula, [Par71]: Scal v ⊗ A, (3) 8 where Cγ is the Casimir operator of the representation (ργ , Vγ ). Now since the representation is irreducible, the Casimir operator is a scalar multiple of identity, Cγ = cγ id, Hence if HomK (Vγ , ) = {0}, where the eigenvalue cγ only depends on γ ∈ G. 2 cγ + n/16 belongs to the spectrum of D . Let ρK = ⊕ρK,k be the decomposition of the spin representation K → into irreducible components. Denote by m(ργ |K , ρK,k ) the multiplicity of the irreducible K-representation ρK,k in the representation ργ restricted to K. Then dim HomK (Vγ , ) = m(ργ |K , ρK,k ) . D2 (v ⊗ A) = v ⊗ (A ◦ Cγ ) +

k

So the spectrum of the square of the Dirac operator is s.t. ∃k s.t. m(ργ , ρK,k ) = 0} . Spec(D2 ) = {cγ + n/16 ; γ ∈ G |K

(4)

3. Proof of the Result We assume that G and K have the same rank. Let T be a fixed common maximal torus. Let be the set of non-zero roots of the group G with respect to T . According to a classical terminology, a root θ is called compact if the corresponding root space is contained in KC (that is, θ is a root of K with respect to T ) and noncompact if the root + space is contained in PC . Let + G be the set of positive roots of G, K be the set of + positive roots of K, and n be the set of positive noncompact roots with respect to a fixed lexicographic ordering in . The half-sums of the positive roots of G and K are respectively denoted δG and δK and the half-sum of noncompact positive roots is denoted by δn . The Weyl group of G is denoted WG . The space of weights is endowed with the WG -invariant scalar product < , > induced by the Killing form of G sign-changed. Let + W := {w ∈ WG ; w · + G ⊃ K } .

(5)

By a result of R. Parthasaraty, cf. Lemma 2.2 in [Par71], the spin representation ρK of K decomposes into the irreducible sum ρK,w , (6) ρK = w∈W

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where ρK,w has for dominant weight βw := w · δG − δK .

(7)

βw0 2 = min βw 2 ,

(8)

if ∃w1 = w0 ∈ W such that βw1 2 = min βw 2 , then βw1 ≺ βw0 ,

(9)

Now define w0 ∈ W such that w∈W

and w∈W

where ≺ is the usual ordering on weights. Lemma 1. The weight βwG0 := w0−1 · βw0 = δG − w0−1 · δK , is G-dominant. Proof. Let G = {θ1 , . . . , θr } ⊂ + G be the set of simple roots. It is sufficient to prove

that 2 is a non-negative integer for any simple root θi . Since T is a maximal common torus of G and K, βw0 , which is an integral weight for K is also an integral weight for G. Now since the Weyl group WG permutes the weights, βwG0 = w0−1 · βw0

is also a integral weight for G, hence 2 is an integer for any simple root θi . So we only have to prove that this integer is non-negative. G ,θi > Let θi be a simple root. Since 2 is WG -invariant, one gets 2

< βwG0 , θi > < θi , θi >

=1−2

< δK , w0 · θi > . < θi , θi >

(10)

Suppose first that w0 · θi ∈ K . If w0 · θi is positive then w0 · θi is necessarily a Ksimple root. Indeed let K = {θ1 , . . . , θl } ⊂ + K be the set of K-simple roots. One has w0 · θi = lj =1 bij θj , where the bij are non-negative integers. But since w0 ∈ W , there are l positive roots α1 , . . . , αl in + G such that w0 · αj = θj , j = 1, . . . , l. So l r θi = j =1 bij αj . Now each αj is a sum of simple roots k=1 aj k θk , where the aj k are non-negative integers. So θi = j,k bij aj k θk . By the linear independence of simple roots, one gets j bij aj k = 0 if k = i, and j bij aj i = 1. Hence there exists a j0 such that bij0 = aj0 i = 1, the other coefficients being zero. So w0 · θi = θj0 is a K-simple K ,w0 ·θi > root. Now since 2

2

G ,θ > K ,w0 ·θi > 2 1. < θi , θi >

Since δK can be expressed as δK = a K-simple root

θj

such that
< 0, which is equivalent to 2

l

i=1 ci θi , where the ci

θj , w0

· θi > 0, and since 2

(11) are nonnegative, there exists

is an integer, this

implies that 2

< θj , w0 · θi > < θj , θj >

≥ 1.

(12)

So θj − w0 · θi is a root (cf. for instance § 9.4 in [Hum72]). Moreover, from the bracket relation [K, P] ⊂ P, it is a noncompact root. Now ±(θj − w0 · θi ) is a positive noncompact root, so by the description of the weights of the spin representation ρK , (they are of the form: δn −(a sum of distinct positive noncompact roots), cf. §2 in [Par71]), (w0 · δG − δK ) ± (θj − w0 · θi ) is a weight of ρK . Now, (w0 · δG − δK ) + (θj − w0 · θi ) can not be a weight of ρK . Otherwise since σi · δG = δG − θi , (w0 σi · δG − δK ) + θj is a weight of ρK . But since w0 σi ∈ W , µ := w0 σi · δG − δK is a dominant weight of ρK . So µ is a dominant weight but not the highest weight of an irreducible component of ρK . Hence there exists an irreducible representation of ρK with dominant weight λ = w · δG − δK , w ∈ W , whose set of weights

contains µ. Furthermore µ ≺ λ. Now since µ ∈ , µ + δK 2 ≤ λ + δK 2 , with equality only if µ = λ, (cf. for instance Lemma C, §13.4 in [Hum72]). But µ + δK 2 = δG 2 = λ + δK 2 , so µ = λ, contradicting the fact that µ ≺ λ. Thus only µ0 := (w0 · δG − δK ) − (θj − w0 · θi ) ,

(13)

can be a weight of ρK . Now one has µ0 2 = w0 · δG − δK + w0 · θi 2 −2 < w0 · δG − δK + w0 · θi , θj > +θj 2 . Since w0 · δG − δK is a dominant weight, < w0 · δG − δK , θj >≥ 0, and from (12), 2 < w0 · θi , θj > −θj 2 ≥ 0, so µ0 2 ≤ (w0 · δG − δK ) + w0 · θi 2 .

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Now (w0 · δG − δK ) + w0 · θi 2 = w0 · δG − δK 2 +2 < δG − w0−1 · δK , θi > +θi 2 . But, as we supposed 2 2
+θi

< 0, one has 2

≤ 0, hence

2 θi 2

≤ −1, so

(w0 · δG − δK ) + w0 · θi 2 ≤ w0 · δG − δK 2 , so µ0 2 ≤ w0 · δG − δK 2 . Now, being a weight of ρK , µ0 is conjugate under the Weyl group of K to a dominant weight of ρK , say w1 · δG − δK , with w1 ∈ W . Note that w1 = w0 , otherwise since µ0 ≺ w1 · δG − δK , (cf. Lemma A, § 13.2 in [Hum72]), the noncompact root θj − w0 · θi should be a linear combination with integral coefficients of compact simple roots. But, by the bracket relation [K, K] ⊂ K, that is impossible. Thus, by the definition of w0 , cf. (8), w0 · δG − δK 2 ≤ w1 · δG − δK 2 = µ0 2 , so µ0 2 = w1 · δG − δK 2 = w0 · δG − δK 2 . But by the condition (9), the last equality is impossible, otherwise since µ0 ≺ w1 ·δG −δK and w1 · δG − δK ≺ w0 · δG − δK , the noncompact root θj − w0 · θi should be a linear combination with integral coefficients of compact simple roots. Hence 2 also if w0 · θi ∈ / K .

G ,θ > . On the other hand βw0 2 = δG 2 + δK 2 − 2 < w0 · δG , δK > . Hence cγ0 = 2 βw0 2 + δG 2 − δK 2 . Now, the Casimir operator of K acts on the spin representation ρK as scalar multiplication by δG 2 − δK 2 , (cf. Lemma 2.2 in [Par71]). Indeed, each dominant weight of ρK being of the form w · δG − δK , w ∈ W , the eigenvalue of the Casimir operator on each irreducible component is given by: (w · δG − δK ) + δK 2 − δK 2 = w · δG 2 − δK 2 = δG 2 − δK 2 . On the other hand, the proof of the formula (3) shows that the Casimir operator of K acts on the spin representation ρK as scalar multiplication by Scal 8 = n/16 (cf. [Sul79]), hence δG 2 − δK 2 = n/16 . So cγ0 + n/16 = 2 βw0 2 + n/8 . Hence the result.

(14)

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In order to conclude, we have to prove that Lemma 4. 2 βw0 2 + n/8 , is the lowest eigenvalue of the square of the Dirac operator. be such that there exists w ∈ W such that m(ργ , ρK,w ) ≥ 1. Let Proof. Let γ ∈ G |K βγ be the dominant weight of ργ . First, since the Weyl group permutes the weights of ργ , w −1 · βw = δG − w −1 · δK is a weight of ργ . Hence βγ + δG 2 ≥ w −1 · βw + δG 2 , (cf. for instance Lemma C, §13.4 in [Hum72]). So, from the Freudenthal formula, cγ = βγ + δG 2 − δG 2 ≥ w −1 · βw + δG 2 − δG 2 . But, using (14) w−1 · βw + δG 2 − δG 2 = 2 βw 2 + δG 2 − δK 2 = 2 βw 2 + n/16 . Hence by the definition of βw0 , cγ ≥ 2 βw 2 + n/16 ≥ 2 βw0 2 + n/16 . Hence the result.

References [Bär91]

Bär, C.: Das Spektrum von Dirac-Operatoren. Dissertation, Universität Bonn, 1991, Bonner Mathematische Schriften 217. [Bes87] Besse, A.: Einstein Manifolds. Berlin: Springer-Verlag, 1987 [BHMM] Bourguignon, J.P., Hijazi, O., Milhorat, J.-L., Moroianu, A.: A Spinorial Approach to Riemannian and Conformal Geometry. Monograph (in preparation) [CFG89] Cahen, M., Franc, A., Gutt, S.: Spectrum of the Dirac Operator on Complex Projective Space P2q−1 (C). Lett. Math. Phys. 18, 165–176 (1989) [CG88] Cahen, M., Gutt, S.: Spin Structures on Compact Simply Connected Riemannian Symmetric Spaces. Simon Stevin 62, 209–242 (1988) [Hel78] Helgason, S.: Differential Geometry, Lie Groups and Symmetric Spaces, Pure and Applied mathematics, Vol. 80. San Diego: Academic Press, 1978 [Hum72] Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory. Berlin-Heidelberg New York: Springer-Verlag, 1972 [Par71] Parthasarathy, R.: Dirac operator and the discrete series. Ann. Math. 96, 1–30 (1971) [Sul79] Sulanke, S.: Die Berechnung des Spektrums des Quadrates des Dirac-Operators auf der Sphäre. Doktorarbeit, Humboldt-Universität, Berlin, 1979 Communicated by P. Sarnak


Communications in


SU (3)-Instantons and G2 , Spin(7)-Heterotic String Solitons Petar Ivanov, Stefan Ivanov University of Sofia “St. Kl. Ohridski”, Faculty of Mathematics and Informatics, Blvd. James Bourchier 5, 1164 Sofia, Bulgaria. E-mail: [email protected] (P. Ivanov); [email protected] (S. Ivanov) Received: 4 May 2004 / Accepted: 25 February 2005 Published online: 8 July 2005 – © Springer-Verlag 2005

Abstract: Necessary and sufficient conditions to the existence of a hermitian connection with totally skew-symmetric torsion and holonomy contained in SU (3) are given. A formula for the Riemannian scalar curvature is obtained. Non-compact solution to the supergravity-type I equations of motion with non-zero flux and non-constant dilaton is found in dimension 6. Non-conformally flat non-compact solutions to the supergravity-type I equations of motion with non-zero flux and non-constant dilaton are found in dimensions 7 and 8. A Riemannian metric with holonomy contained in G2 arises from our considerations and Hitchin’s flow equations, which seems to be new. Compact examples of SU (3), G2 and Spin(7) instanton satisfying the anomaly cancellation conditions are presented. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. General Properties of SU (3), G2 and Spin(7)-Structures . . . . . . . 2.1 SU(3)-structures in d = 6 . . . . . . . . . . . . . . . . . . . . . 2.2 G2 -structures in d = 7 . . . . . . . . . . . . . . . . . . . . . . 2.3 Spin(7)-structures in d = 8 . . . . . . . . . . . . . . . . . . . 3. The Supersymmetry Equations in Dimensions 6, 7 and 8 . . . . . . . 4. Non-Compact G2 -Solution Induced from a SU (3)-Instanton . . . . . 5. Non-Compact Spin(7)-Solution Induced from a G2 -Instanton . . . . 6. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 (SU (3), G2 , Spin(7))-instanton and conformally flat non-compact solution . . . . . . . . . . . . . . . . . . . . . . . 6.2 (SU (3), G2 , Spin(7))-instanton and non-conformally flat non-compact solution . . . . . . . . . . . . 7. Almost Contact Metric Structures and Non-Compact SU (3)-Solutions in Dimension 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Non-conformally flat local SU (3)-solutions on S 5 × S 1 . . . . .

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P. Ivanov, S. Ivanov

1. Introduction Supersymmetric backgrounds of string/M theory with non-vanishing fluxes are currently an active area of study for at least two reasons. Firstly they provide a framework of searching for new models with realistic phenomenology and secondly, they appear in generalizations of the AdS/CFT correspondence. The supersymmetric geometries of the common NS-NS sector of type IIA, IIB and heterotic/type I supergravity are analyzed in [42]. The bosonic geometry is of the form R1,9−p × Mp , where the Riemannian metric g, the dilaton function φ and the three form H are non-trivial only on Mp but all R-R fields and fermions are set to zero in type II theories. The type I/heterotic geometries, which is the main object of interest in the present note, allow in addition non-trivial gauge field A with field strength F A . We recall the basic notations [73, 54, 52, 41, 42]. We search for solutions to lowest nontrivial order in α of the equations of motion that follow from the bosonic action 2 1 2 1 10 √ −2φ g g 2 Scal + 4(∇ φ) − H − α T r F A S = 2 d x −ge 2k 12 which also preserves at least one supersymmetry. The three form H satisfies a modified Bianchi identity dH = 2α T r(F A ∧ F A ). The so-called Bianchi identity reads ˜ ˜ dH = 2α T r F A ∧ F A − T r R ∇ ∧ R ∇ ,

(1.1)

(1.2)

˜ ˜ where R ∇ is the curvature of the metric connection ∇˜ with torsion T ∇ = −H related 1 g g to the Levi-Civita connection ∇ by ∇˜ = ∇ − 2 H . The second term on the right-hand side of (1.2) is the leading string correction to the supergravity expression arising from the anomaly cancellation but for the consistency of the theory, a modification to the action should be included [10] (see also [22, 45]). In terms of characteristic classes (1.2) means that dH is proportional to the difference ˜ of the connections A, ∇, ˜ respectively. of the first Pontrjagin 4-forms (p1 (A) − p1 (∇)) A heterotic/type I geometry will preserve supersymmetry if and only if, in 10 dimensions, there exists at least one Majorana-Weyl spinor such that the supersymmetry variations of the fermionic fields vanish, i.e. 1 g δλ = ∇m = ∇m + Hmnp np = 0, 8 1 δ = m ∂m φ + Hmnp mnp = 0, (1.3) 12 A mn δξ = Fmn = 0,

where λ, , ξ are the gravitino, the dilatino and the gaugino, fields, respectively. The equations of motion corresponding to the action S are presented explicitly in [42]. It is known [25, 41] that the equation of motions of type I supergravity are automatically satisfied if one imposes, in addition to the preserving supersymmetry equations (1.3), the modified Bianchi identity (1.1).

SU (3)-Instantons and G2 , Spin(7)-Heterotic String Solitons

81

According to no-go (vanishing) theorems (a consequence of the equations of motion [31, 25]; a consequence of the supersymmetry [58] for SU(n)-case and [42] for the general case) there are no compact solutions with non-zero flux and non-constant dilaton satisfying the supersymmetry equations (1.3) and the modified Bianchi identity (1.1) simultaneously. In dimensions 7 and 8 the only known heterotic/type I solutions to the equations of motion preserving at least one supersymmetry, i.e. satisfying (1.3) and (1.1), are those constructed in [27, 37, 54] in dimension 8 and those presented in [52] in dimension 7. All these solutions are conformal to a flat space. In dimension 6, the possibility of the existence of a non-conformally flat solution on the complex Iwasawa manifold was discussed in [73, 21, 42, 63]. In the present note we concentrate our attention to find non-compact solutions to the supergravity equations (1.3) including the modified Bianchi identity (1.1) as well as the anomaly cancellation condition (1.2). In dimensions 7 and 8 we find non-locally-conformally flat non-compact solutions to the gravitino, gaugino and dilatino equations with non-zero flux and non-constant dilaton which obey the Bianchi identities (1.1) and (1.2) and therefore satisfy the equations of motion, due to the result in [41]. In dimension 6, we present a (non-conformally-flat) non-compact solution to the equations of motion showing that it obeys (1.3),(1.1),(1.2). All these non-compact solutions seem to be new. We present a non conformally flat (resp. conformally flat) SU (3), G2 and Spin(7)instantons which satisfy the anomaly cancellation condition (1.2) as well as the modified Bianchi identity (1.1). We obtain compact 6,7 and 8-manifolds which solve the gravitino and gaugino equations and satisfy the compatibility conditions (1.2) and (1.1) but do not solve the dilatino equation which is consistent with the no-go theorems. Geometrically, the vanishing of the gravitino variation is equivalent to the existence of a non-trivial spinor parallel with respect to a metric connection ∇ with totally skew symmetric torsion T = H which is related to the Levi-Civita connection ∇ g by 1 ∇ = ∇ g + H. 2 The presence of a ∇-parallel spinor leads to restriction of the holonomy group H ol(∇) of the torsion connection ∇. Namely, H ol(∇) has to be contained in SU (3), d = 6 [73, 59, 58, 51, 21, 6, 7], the exceptional group G2 , d = 7 [32, 40, 34], the Lie group Spin(7), d = 8 [40, 57]. A detailed analysis of the possible geometries is carried out in [42]. Complex Non-Kähler geometries appear in string compactifications and are studied intensively [73, 46, 42, 40, 41, 46, 45, 6, 7]. Some types of non-complex 6-manifold have been also invented recently in the string theory due to the mirror symmetry and T-duality [63, 50, 49, 11, 21, 22]. Another special dimension turns out to be dimension 5. The existence of a ∇-parallel spinor in dimension 5 determines an almost contact metric structure whose properties as well as solutions to gravitino and dilatino equations are investigated in [32, 33]. We use these considerations in our construction in Sect. 6 of a new SU (3)-instanton and non-compact solution to the equations of motion in dimension 6. Almost Hermitian manifolds with totally skew-symmetric Nijenhuis tensor arise as target spaces of a class of (2,0)-supersymmetric two-dimensional sigma models [69]. For the consistency of the theory, the Nijenhuis tensor has to be parallel with respect to the torsion connection with holonomy contained in SU (n). The known models are those on group manifolds. We present a 6-dimensional nil-manifold as an example which is not a group manifold.

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Starting with a SU (3)-structure in dimension 6 we analyze the five classes discovered recently by Chiossi and Salamon [23] from the point of the existence of a SU (3)connection having totally skew-symmetric torsion. We obtain necessary and sufficient conditions for the existence of a connection solving the gravitino equation in dimension 6, i.e. the existence of a linear connection preserving the almost hermitian structure with torsion 3-form and holonomy contained in SU (3), in terms of the given SU (3)structure and present a formula for the Riemannian scalar curvature (Theorem 4.1). It turns out that the corresponding almost complex structure may not be integrable. In case that the almost complex structure is integrable, we derive that the SU (3)-structure is holomorphic if and only if the corresponding hermitian structure is balanced, i.e. has co-closed fundamental form (Corollary 4.3). On the other hand, any SU (3)-weak holonomy manifold (Nearly Kähler manifold) automatically solves both the gravitino and gaugino equations. It turns out that the Nearly Kähler 6-sphere S 6 satisfies in addition the compatibility conditions (1.2), (1.1). We present a six dimensional non-conformally flat nil-manifold Nil 6 = G/ , i.e. a compact quotient of a nilpotent Lie group G with a discrete subgroup , which solves both the gravitino and gaugino equations and satisfies the compatibility conditions (1.2), (1.1) but it is neither complex nor Nearly Kähler. Consequently, it does not solve the dilatino equation [73]. However, the SU (3)-structure on G is half-flat and therefore it determines a Riemannian metric with holonomy contained in G2 on G × R ∼ = R7 according to the procedure discovered by Hitchin [55], which seems to be a new one. We propose a simple way to lift a 6-dimensional solution to the gravitino and gaugino equations, i.e. a SU (3)-instanton solving the gravitino equation and satisfying the conditions (1.1) (resp. (1.2)), to a G2 -instanton on the product with the real line which solves the three supersymmetry equations (1.3) as well as the compatibility condition (1.1) (resp. (1.2)). We show that N il 6 ×R, (resp. S 6 ×R) is a non-compact solution to the equations of motion in dimension 7 with non-zero flux and non-constant dilaton which preserves at least one supersymmetry and is not locally conformally flat (resp. locally conformally flat). Consequently, the compact spaces N il 6 × S 1 , S 6 × S 1 admit a G2 instanton structure satisfying all the Eqs. (1.3), (1.2), (1.1) except the dilatino equation. It turns out that any G2 -weak holonomy manifold (Nearly-parallel manifold) automatically solves both the gravitino and gaugino equations. We show that the Nearly parallel 7-sphere satisfies in addition the compatibility conditions (1.2), (1.1). The same lifting procedure is applicable to the Spin(7) case. Namely, any G2 -instanton solving the gravitino equation and satisfying the conditions (1.1) (resp. (1.2)) can be lifted to a Spin(7)-instanton on the product with the real line which solves the three supersymmetry equations (1.3) as well as the compatibility condition (1.1) (resp. (1.2)). We show that N il 6 × R × R, (resp. S 6 × R × R, S 7 × R) is a non-compact solution to the equations of motion in dimension 8 with non-zero flux and non-constant dilaton which preserves at least one supersymmetry and is not locally conformally flat (resp. locally conformally flat). Consequently, the compact spaces Nil 6 ×S 1 ×S 1 , S 6 ×S 1 ×S 1 , S 7 ×S 1 admit a Spin(7)-instanton structure satisfying all Eqs. (1.3), (1.2), (1.1) except the dilatino equation. Starting with a Tanno deformed Einstein Sasaki structure in dimension 5, we lift it to a SU (3)-instanton on the product with the real line which satisfies all the supersymmetry equations (1.3) in dimension 6. In this way we obtain local solutions to the equations of motion in dimension 6. Consider S 5 as a Sasakian space form, i.e. a Tanno deformation of the standard Einstein-Sasaki structure, we show that S 5 ×R is a non-compact solution to the equations of motion in dimension 6 with non-zero flux and non-constant dilaton


83

which preserves at least one supersymmetry. Consequently, the compact space S 5 × S 1 admits a SU (3)-instanton structure satisfying all the Eqs. (1.3), (1.2), (1.1). 2. General Properties of SU (3), G2 and Spin(7)-Structures In this section we recall necessary properties of SU (3), G2 and Spin(7) structures. 2.1. SU(3)-structures in d = 6. Let (M 6 , g, J ) be an almost Hermitian 6-manifold with Riemannian metric g and an almost complex structure J , i.e. (g, J ) define an U (3)-structure. The Nijenhuis tensor N , the Kähler form F and the Lee form θ 6 are defined by N = [J., J.] − [., .] − J [J., .] − [., J.],

F = g(., J.),

θ 6 (.) = δF (J.), (2.4)

respectively. A SU √(3)-structure is determined by an additional non-degenerate (3,0)-form = + + −1 − , or equivalently by a non-trivial spinor. To be more explicit, we may choose a local orthonormal frame e1 , . . . , e6 , identifying it with the dual basis via the metric. Write ei1 i2 ...ip for the monomial ei1 ∧ei2 ∧· · ·∧eip . A SU (3)-structure is described locally by √ √ √ = −(e1 + −1e2 ) ∧ (e3 + −1e4 ) ∧ (e5 + −1e6 ), F = −e12 − e34 − e56 , (2.5) + = −e135 + e236 + e146 + e245 , − = −e136 − e145 − e235 + e246 . The subgroup of SO(6) fixing the forms F and simultaneously is SU (3). The two forms F and determine the metric completely. The Lie algebra of SU (3) is denoted su(3). The failure of the holonomy group of the Levi-Civita connection to reduce to SU (3) can be measured by the intrinsic torsion τ , which is identified with ∇ g F or ∇ g J and can be decomposed into five classes [23], τ ∈ W1 ⊕ · · · ⊕ W5 . The intrinsic torsion of an U (n)- structure belongs to the first four components described by Gray-Hervella [48]. The five components of a SU (3)-structure are first described by Chiossi-Salamon [23] (for interpretation in physics see [21, 50, 49]) and are determined by dF, d + , d − as well as by dF and N. We describe those of them which we will use later. τ ∈ W1 : The class of Nearly Kähler (weak holonomy) manifold defined by dF to be (3,0)+(0,3)-form. τ ∈ W2 : The class of almost Kähler manifolds, dF = 0. τ ∈ W3 : The class of balanced hermitian manifold determined by the conditions N = θ 6 = 0, i.e these are complex manifolds with vanishing Lee form. These spaces are investigated in [39, 68, 2]. τ ∈ W4 : The class of locally conformally Kähler spaces characterized by dF = θ 6 ∧ F . τ ∈ W1 ⊕ W3 ⊕ W4 : The class called by Gray-Hervella G1 -manifolds determined by the condition that the Nijenhuis tensor is totally skew-symmetric. This is the precise class which we are interested in. The class of a half-flat SU(3)-manifold [23] may be characterized by the conditions d + = 0, θ 6 = 0. The half-flat structures can be lifted to a G2 -holonomy metric on

84


the product with the real line and vice versa due to the Hitchin theorem [55]. In fact, many new G2 -holonomy metrics are obtained in this way [43, 14]. We recall [23] that the fifth component W5 and the two scalar components of W1 are determined by the expressions ∗d + ∧ + , d + ∧ F = W1+ vol., d − ∧ F = W1− vol., respectively. If all five components are zero then we have a Ricci-flat Kähler (Calabi-Yau) 3-fold. 2.2. G2 -structures in d = 7. Endow R7 with its standard orientation and inner product. Let e1 , . . . , e7 be an oriented orthonormal basis. Consider the three-form ω on R7 given by ω = e127 − e236 + e347 + e567 − e146 − e245 + e135 .

(2.6)

The subgroup of GL(7) fixing ω is the exceptional Lie group G2 . It is a compact, connected, simply-connected, simple Lie subgroup of SO(7) of dimension 14 [16]. The Lie algebra is denoted by g2 and it is isomorphic to the two forms satisfying 7 linear equations, namely g2 ∼ = {α ∈ 2 (M)|∗(α ∧ ω) = −α}. The 3-form ω corresponds to a real spinor and therefore, G2 can be identified as the isotropy group of a non-trivial real spinor. The Hodge star operator supplies the 4-form ∗ω given by ∗ω = e3456 + e1457 + e1256 + e1234 + e2357 + e1367 − e2467 .

(2.7)

A 7-dimensional Riemannian manifold is called a G2 -manifold if its structure group reduces to the exceptional Lie group G2 . The existence of a G2 -structure is equivalent to the existence of a global non-degenerate three-form which can be locally written as (2.6). The 3-form ω is called the fundamental form of the G2 -manifold [15]. From the purely topological point of view, a 7-dimensional paracompact manifold is a G2 -manifold if and only if it is an oriented spin manifold [66]. We will say that the pair (M, ω) is a G2 -manifold with G2 -structure (determined by) ω. The fundamental form of a G2 -manifold determines a Riemannian metric implicitly through gij = 16 kl ωikl ωj kl [47]. This is referred to as the metric induced by ω. We write ∇ g for the associated Levi-Civita connection. In [29], Fernandez and Gray divide G2 -manifolds into 16 classes according to how the covariant derivative of the fundamental three-form behaves with respect to its decomposition into G2 irreducible components (see also [23, 40]). If the fundamental form is parallel with respect to the Levi-Civita connection, ∇ g ω = 0, then the Riemannian holonomy group is contained in G2 . In this case the induced metric on the G2 -manifold is Ricci-flat, a fact first observed by Bonan [15]. It was shown by Gray [47] (see also [16, 71]) that a G2 -manifold is parallel precisely when the fundamental form is harmonic, i.e. dω = d ∗ ω = 0. The first examples of complete parallel G2 -manifolds were constructed by Bryant and Salamon [17, 44]. Compact examples of parallel G2 -manifolds were obtained first by Joyce [60–62] and recently by Kovalev [65]. The Lee form θ 7 is defined by [19] 1 θ 7 = − ∗(∗dω ∧ ω). 3

(2.8)

If the Lee form vanishes, θ 7 = 0, then the G2 -structure is said to be balanced. If the Lee form is closed, dθ 7 = 0, then the G2 -structure is locally conformally equivalent to a balanced one [34]. If the G2 -structure satisfies the condition d ∗ω = θ 7 ∧ ω then it is called integrable and an analog of the Dolbeault cohomology is investigated in [30].


85

2.3. Spin(7)-structures in d = 8. Now, let us consider R8 endowed with an orientation and its standard inner product. Let {e0 , ..., e7 } be an oriented orthonormal basis. Consider the 4-form on R8 given by = e0127 − e0236 + e0347 + e0567 − e0146 − e0245 + e0135 +e3456 + e1457 + e1256 + e1234 + e2357 + e1367 − e2467 .

(2.9)

The 4-form is self-dual ∗ = and the 8-form ∧ coincides with the volume form of R8 . The subgroup of GL(8, R) which fixes is isomorphic to the double covering Spin(7) of SO(7) [53]. Moreover, Spin(7) is a compact simply-connected Lie group of dimension 21 [16]. The Lie algebra of Spin(7) is denoted by spin(7) and it is isomorphic to the two forms satisfying 7 linear equations, namely spin(7) ∼ = {α ∈

2 (M)| ∗ (α ∧ ) = −α}. The 4-form corresponds to a real spinor φ and therefore, Spin(7) can be identified as the isotropy group of a non-trivial real spinor. A Spin(7)-structure on an 8-manifold M is by definition a reduction of the structure group of the tangent bundle to Spin(7); we shall also say that M is a Spin(7) manifold. This can be described geometrically by saying that there exists a nowhere vanishing global differential 4-form on M which can be locally written as (2.9). The 4-form is called the fundamental form of the Spin(7) manifold M [15]. The fundamental form of a Spin(7)-manifold determines a Riemannian metric implic1 itly through gij = 24 klm iklm j klm [47]. This is referred to as the metric induced by . In general, not every 8-dimensional Riemannian spin manifold M 8 admits a Spin(7)structure. We explain the precise condition [66]. Denote by p1 (M), p2 (M), X(M), X(S± ) the first and the second Pontrjagin classes, the Euler characteristic of M and the Euler characteristic of the positive and the negative spinor bundles, respectively. It is well known [66] that a spin 8-manifold admits a Spin(7) structure if and only if X(S+ ) = 0 or X(S− ) = 0. The latter conditions are equivalent to p12 (M) − 4p2 (M) + 8X(M) = 0, for an appropriate choice of the orientation [66]. Let us recall that a Spin(7) manifold (M, g, ) is said to be parallel (torsion-free [61]) if the holonomy of the metric H ol(g) is a subgroup of Spin(7). This is equivalent to saying that the fundamental form is parallel with respect to the Levi-Civita connection ∇ g of the metric g. Moreover, H ol(g) ⊂ Spin(7) if and only if d = 0 [16] (see also [71]) and any parallel Spin(7) manifold is Ricci flat [15]. The first known explicit example of complete parallel Spin(7) manifold with H ol(g) = Spin(7) was constructed by Bryant and Salamon [17, 44]. The first compact examples of parallel Spin(7) manifolds with H ol(g) = Spin(7) were constructed by Joyce [60, 61]. There are 4-classes of Spin(7) manifolds according to the Fernandez classification [28] obtained as irreducible representations of Spin(7) of the space ∇ g . The Lee form θ 8 is defined by [18] 1 θ 8 = − ∗ (∗d ∧ ) = 1/7 ∗ (δ ∧ ). 7

(2.10)

The 4 classes of Fernandez classification can be described in terms of the Lee form as follows [18]: W0 : d = 0; W1 : θ 8 = 0; W2 : d = θ 8 ∧ ; W : W = W1 ⊕ W2 . A Spin(7)-structure of the class W1 (i.e. Spin(7)-structure with zero Lee form) is called a balanced Spin(7)-structure. If the Lee form is closed, dθ 8 = 0 then the Spin(7)-structure is locally conformally equivalent to a balanced one [57]. It is shown in [18] that the Lee form of a Spin(7) structure in the class W2 is closed and therefore

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such a manifold is locally conformally equivalent to a parallel Spin(7) manifold. The compact spaces with closed but not exact Lee form (i.e. the structure is not globally conformally parallel) have very different topology than the parallel ones [57]. Coeffective cohomology and coeffective numbers of Riemannian manifolds with Spin(7)-structure are studied in [74]. 3. The Supersymmetry Equations in Dimensions 6, 7 and 8 Dimension d=6 . Necessary conditions to have a solution to the system of dilatino and gravitino equations in dimension 6 were derived by Strominger in [73] and then studied by many authors [40–42, 21, 22, 6–8, 45] Necessary conditions to solve the gravitino equation are given in [32]. The presence of a parallel spinor in dimension 6 leads firstly to the reduction to U (3), i.e. the existence of an almost hermitian structure, secondly to the existence of a linear connection preserving the almost hermitian structure with torsion 3-form, and thirdly to the reduction of the holonomy group of the torsion connection to SU(3). It is shown in [32] that there exists a unique linear connection preserving an almost hermitian structure having totally skew-symmetric torsion if and only if the Nijenhuis tensor is a 3-form, i.e. the intrinsic torsion τ ∈ W1 ⊕ W3 ⊕ W4 . The torsion connection ∇ is determined by 1 ∇ = ∇g + T , 2 1 T = J dF + N = −dF (J., J., J.) + N = −dF + (J., J., J.) + N, 4

(3.11)

where dF + denotes the (1,2)+(2,1)-part of dF . The (3,0)+(0,3)-part dF − is determined completely by the Nijenhuis tensor [38]. If N is a three form then (see e.g. [32]) 3 dF − = − J N. 4

(3.12)

In addition, the dilatino equation forces the almost complex structure to be integrable and the Lee form to be closed (for applications in physics the Lee form has to be exact) determined by the dilaton due to θ 6 = 2dφ [73]. When the almost complex structure is integrable, N = 0, the torsion connection is also known as the Bismut connection and was used by Bismut to prove the local index theorem for the Dolbeault operator on the Hermitian non-Kähler manifold [12]. This formula was recently applied in string theory [7]. Vanishing theorems for the Dolbeault cohomology on the compact Hermitian non-Kähler manifold were found in terms of the Bismut connection [4, 58, 59]. Dimension d=7 . The precise conditions to have a solution to the gravitino equation in dimension 7 are found in [32]. Namely, there exists a non-trivial parallel spinor with respect to a G2 -connection with torsion 3-form T if and only if there exists a G2 -structure (ω, g) satisfying the equations d ∗ω = θ 7 ∧ ∗ω.

(3.13)

In this case the torsion connection ∇ is unique, the torsion 3-form T is given by 1 ∇ = ∇g + T , 2

H =T =

1 (dω, ∗ω)ω − ∗dω + ∗(θ 7 ∧ ω), 6

(3.14)


87

and the Riemannian scalar curvature has the following expression [34] sg =

1 1 (dω, ∗ω) + 2||θ 7 ||2 − ||T ||2 + 3δθ 7 . 18 12

(3.15)

The necessary conditions to have a solution to the system of dilatino and gravitino equations were derived in [40, 32, 34] and sufficiency was proved in [32, 34]. The general existence result [32, 34] states that there exists a (local) non-trivial solution to both dilatino and gravitino equations in dimension 7 if and only if there exists a G2 -structure (ω, g) satisfying the equations d ∗ω = θ 7 ∧ ∗ω,

dω ∧ ω = 0,

θ 7 = 2dφ.

(3.16)

The torsion 3-form (the flux H ) is given by 1 ∇ = ∇g + T , 2

H = T = −∗dω + 2∗(dφ ∧ ω).

The Riemannian scalar curvature satisfies s g = 8||dφ||2 −

1 2 12 ||T ||

(3.17)

+ 6δdφ.

Dimension d=8. It is shown in [57] that the gravitino equation always has a solution in dimension 8. Namely, any Spin(7)-structure admits a unique Spin(7)-connection with totally skew-symmetric torsion T = ∗d − ∗(θ 8 ∧ ). The necessary conditions to have a solution to the system of dilatino and gravitino equations were derived in [40, 57] and sufficiency was proved in [57]. The general existence result [57] states that there exists a (local) non-trivial solution to both dilatino and gravitino equations in dimension 8 if and only if there there exists a Spin(7)-structure (, g) with closed Lee form, dθ 8 = 0, which is equivalent to the statement that the Spin(7)-structure is locally conformally balanced. The torsion 3-form (the flux H ) and the Lee form are given by 1 ∇ = ∇g + T , 2

H = T = ∗d − 2∗(dφ ∧ ),

θ8 =

12 dφ. 7

(3.18)

1 ||T ||2 + 6δdφ. The Riemannian scalar curvature satisfies s g = 8||dφ||2 − 12 In addition to these equations, the vanishing of the gaugino variation requires the 2-form F A to be of instanton type: ([24, 73, 54, 70, 26, 42])

Case d=6 A Donaldson-Uhlenbeck-Yau SU (3)-instanton, i.e. the gauge field A is a SU (3)-connection with curvature 2-form F A ∈ su(3). The SU(3)-instanton condition can be written in local holomorphic coordinates in the form [24, 73] ¯

A = Fα¯Aβ¯ = 0, FαAβ¯ F α β = 0. Fαβ

(3.19)

Case d=7 A G2 -instanton, i.e. the gauge field A is a G2 -connection and its curvature 2-form F A ∈ g2 . The latter can be expressed in any of the following two equivalent ways A mn Fmn ω p=0

A ⇔ Fmn =

1 A F (∗ω)pq mn . 2 pq

(3.20)

88


Case d=8 A Spin(7)-instanton, i.e. the gauge field A is a Spin(7)-connection and its curvature 2-form F A ∈ spin(7). The latter is equivalent to A = Fmn

1 A pq F mn . 2 pq

(3.21)

4. Non-Compact G2 -Solution Induced from a SU (3)-Instanton In this section we show how to construct a local solution to the equations of motion in dimension 7 if we have a solution to the gravitino and gaugino equations satisfying the modified Bianchi identity (1.1) in dimension 6. We first investigate a necessary and sufficient condition to have a solution to the gravitino equation in dimension 6, i.e. to have a ∇-parallel spinors. We prove the following Theorem 4.1. Let (M 6 , g, J, ) be a 6-dimensional smooth manifold with a SU (3)structure (g, J, ) or equivalently, the almost hermitian manifold (M 6 , g, J ) has topologically trivial canonical bundle trivialized by a (3,0)-form . The next two conditions are equivalent a) There exists a unique SU (3)-connection with torsion 3-form, i.e. a linear connection with torsion 3-form which preserves the almost hermitian structure whose holonomy is contained in SU(3). b) The Nijenhuis tensor N is totally-skew symmetric and the following conditions hold 1 d + = θ 6 ∧ + − (N, + )∗F, 4

1 d − = θ 6 ∧ − − (N, − )∗F. (4.22) 4

The torsion is given by 1 1 T = −∗dF + ∗(θ 6 ∧ F ) + (N, + ) + + (N, − ) − . 4 4

(4.23)

The Riemannian scalar curvature is expressed in the following way sg =

1 1 1 (N, + )2 + (N, − )2 + 2||θ 6 ||2 − ||T ||2 + 3δθ 6 . 8 8 12

(4.24)

In particular, if the structure is complex and balanced then the Riemannian scalar curvature is non-positive. Proof. Suppose condition a) holds. Then the Nijenhuis tensor N is a three form due to Theorem 10.1 in [32]. The conditions ∇ = ∇ + = ∇ − = 0 imply the constraints (4.22) on the exterior derivative of the form . This can be checked directly using (3.11) and (3.12). To prove the converse we consider the product M 7 = M 6 × R with the G2 -structure ω defined by [55, 23] ω = −F ∧ e7 − + ,

(4.25)

where e7 is the standard 1-form on R. We adopt the convention to indicate the object on the product by a superscript 7, 7 i.e. ∗7 , T 7 , θ 7 , ∇ 7 , R ∇ are the Hodge star operator, the torsion 3-form, the Lee form,


89

the torsion connection and its curvature, respectively. The same objects on M 6 have superscript 6. Our idea is to check that the G2 -structure on the product M 7 = M 6 × R satisfies the conditions (3.13) and to apply the Friedrich-Ivanov result from [32] assuring the existence of G2 -connection ∇ 7 with torsion 3-form T 7 . We show that the torsion satisfies the condition T (e7 , ., .) = 0 and therefore ∇ 7 e7 = 0. Hence, the connection ∇ 7 descends on M 6 to a connection ∇ 6 which preserves the SU (3)-structure and has totally skew-symmetric torsion. We get from (4.25) applying (3.12) and (4.22) the following sequence of equalities 1 θ 7 = − ∗7 (∗7 dω ∧ ω) 3 1 6 6 = −∗ (∗ dF ∧ F ) + ∗6 (∗6 dF ∧ + )e7 − ∗6 ∗6 d + ∧ + 3 1 = θ 6 + (N, − )e7 , (4.26) 4 where we used the identities ∗6 (∗6 d + ∧ + ) = ∗6 (∗6 (θ ∧ + ) ∧ + ) = −2θ, ∗7 dω = ∗6 dF − ∗6 d + ∧ e7 , θ = −∗6 dF ∧ dF, 3 ∗6 (∗6 dF ∧ + ) = −( + , dF ) = − ( − , N ). 4

(4.27)

Applying the equalities d ∗6 F = −∗6 J θ 6 = θ 6 ∧ ∗6 F and the conditions (4.22) we obtain (3.13). Hence, there exists a G2 -connection ∇ 7 with torsion 3-form T 7 given by (3.14) on the product M 7 = M 6 × R. To compute T 7 we use (3.12) and (4.26). We have 3 (dω, ∗7 ω) = ∗7 (dω ∧ ω) = 2∗7 (dF ∧ + ) = − (N, + ), 2 (4.28) 1 6 6 6 6 + 7 7 ∗ (θ ∧ ω) = ∗ (θ ∧ F ) − ∗ (θ ∧ )e7 + (N, − ) − . 4 Now, the formula (3.14) and (4.22) give the desired expression (4.23) which implies that the torsion T 7 does not depend on e7 and therefore the connection ∇ 7 descends to M 6 . Substituting (4.28) and (4.26) into (3.15) we get (4.24) for the Riemannian scalar curvature on the product which clearly coincides with the scalar curvature on (M 6 , g 6 ). Corollary 4.2. In dimension 6 the following conditions are equivalent: a) There exists a non-trivial solution to the system of gravitino and dilatino equations with non-zero flux H and non-constant dilaton φ. b) There exists a SU (3)-structure (F, ) satisfying the conditions d + = 2dφ ∧ + ,

d − = 2dφ ∧ − .

The flux H is given by H = T = −∗dF + 2∗(dφ ∧ F ).

(4.29)

The Riemannian scalar curvature of the solution has the expression s g = 8||dφ||2 −

1 ||T ||2 + 6δdφ. 12

(4.30)

90


A part of the necessary conditions we presented are known [73, 42, 40, 41, 21]. The formula (4.29) was discovered in [40], the first formula in (4.27) has already appeared in [21]. Corollary 4.3. A closed SU (3)-structure (d + = d − =0) admits a SU (3)-connection with torsion 3-form if and only if the corresponding almost Hermitian structure is a balanced Hermitian structure, (N = θ 6 = 0). In particular, a holomorphic SU (3)-structure on a complex manifold supports a linear connection with torsion 3-form and holonomy contained in SU (3) if and only if it is balanced. In the latter case the Riemannian scalar curvature is non-positive. Consequently, such a structure is half-flat and therefore it determines a Riemannian metric with holonomy contained in G2 on the product with the real line . Remark 4.4. We note the coincidence of the formulas for the Riemannian scalar curvature of solutions to the gravitino and dilatino equations in dimensions 6, 7, and 8. Actually, it is proved in [59] (see also [45]) that the SU(n)-geometry arising from any solution to the gravitino and dilatino equations satisfies the identity Ricg (X, Y ) −

1 H (X, ei , ej )H (Y, ei , ej ) + 2∇ g X ∇ g Y φ 4 i,j

1 + dH (X, J Y, ei , J ei ) = 0, 4

(4.31)

i

which is consistent with the first equation of motion. The trace in (4.31) gives (4.30) due to the identity 1 1 dH (X, J Y, ei , J ei ) = 8||dφ||2 + 4δdφ − ||H ||2 4 3 i

shown in [4] (see also (3.24) in [59]). Remark 4.5. We note that the Riemannian scalar curvature for a half-flat SU (3)-structure is computed in [50]. On the other hand, not every half-flat structure admits an SU (3)connection with torsion 3-form since it may have a nonzero W2 component. For example, the SU (3)-structure on the nilpotent Lie algebras described in [23], Example 2 and 3, are half-flat but do not admit an SU (3) connection with torsion 3-form since d − = f F ∧F . Another consequence of Theorem 4.1 is the following Theorem 4.6. Let (M 6 , g, J, F, ) be a smooth almost complex 6-manifold with totally skew-symmetric Nijenhuis tensor equipped with a SU (3)-structure solving the gravitino equation, i.e. the conditions (4.22) hold. Assume also the conditions dθ 6 = 0,

(N, + ) = 0,

(N, − ) = const. = 0.

(4.32)

i) Then the G2 -structure ω = −F ∧ e7 − + defined on the product M 7 = M 6 × R solves both the gravitino and dilatino equations with non-constant dilaton. ii) If in addition the torsion connection ∇ 6 is a SU (3)-instanton then the corresponding torsion connection ∇ 7 is a G2 -instanton.


91

iii) Suppose moreover that the torsion connection ∇ 6 satisfies the modified Bianchi 6 identity (1.1), (resp. (1.2)) with F A = R ∇ . Then ∇ 7 also obeys (1.1), (resp. (1.2)) 7 with F A = R ∇ and therefore solves the equations of motion with non zero flux and non-constant dilaton provided θ 6 is exact. Proof. Equations (4.26) and (4.28) imply (3.16) due to the conditions of the theorem. 7 6 ˜7 ˜6 We know from Theorem 4.1 that T 7 = T 6 , R ∇ = R ∇ , R ∇ = R ∇ and conse7 7 6 6 7 7 ˜ ˜ ˜6 ˜6 quently, T r(R ∇ ∧ R ∇ ) = T r(R ∇ ∧ R ∇ ), T r(R ∇ ∧ R ∇ ) = T r(R ∇ ∧ R ∇ ). A 7 glance at the structure of Lie algebras su(3) and g2 implies that R ∇ satisfies the G2 6 instanton equations (3.20) provided R ∇ obeys the SU (3)-instanton equations (3.19) 5. Non-Compact Spin(7)-Solution Induced from a G2 -Instanton In this section we shall show that a G2 -instanton on (N 7 , ω) induces a Spin(7)-instanton on the product N 8 = N 7 × R. We denote the Hodge star operator on N 7 by ∗7 . On the product N 8 = N 7 × R there exists a Spin(7)-structure defined by = e0 ∧ ω + ∗7 ω,

(5.33)

where e0 = dt is the standard 1-form on R. We indicate the object on the product by a 8 superscript 8, i.e. ∗8 , T 8 , θ 8 , ∇ 8 , R ∇ are the Hodge star operator, the torsion 3-form, the Lee form, the torsion connection and its curvature, respectively. The same objects on N 7 have superscript 7. Theorem 5.1. Suppose (N 7 , ω7 , g 7 , ∇ 7 , T 7 ) is a smooth G2 -manifold which solves the gravitino equation, i.e. (3.13) holds. Then the Spin(7)-structure on the product N 7 ×R determined with (5.33) has the properties 6 7 1 θ + (dω, ∗7 ω)e0 , T 8 = T 7 , 7 7 8 8 7 7 T r(R ∇ ∧ R ∇ ) = T r(R ∇ ∧ R ∇ ).

θ8 =

R∇ = R∇ , 8

7

Assume in addition the conditions dθ 7 = 0,

(dω, ∗7 ω) = const. = 0.

i) Then the Spin(7)-structure defined on the product N 8 = N 7 × R solves both the gravitino and dilatino equations with non-zero flux and non-constant dilaton. ii) If in addition the torsion connection ∇ 7 is a G2 -instanton then the corresponding torsion connection ∇ 8 is a Spin(7)-instanton. iii) Suppose moreover that the torsion connection ∇ 7 satisfies the modified Bianchi 7 identity (1.1), (resp. (1.2)) with F A = R ∇ . Then ∇ 8 also obeys (1.1), (resp. (1.2)) 8 with F A = R ∇ and therefore solves the equations of motion with non-zero flux and non-constant dilaton provided θ 7 is exact.

92


Proof. Take the exterior derivative in (5.33) and use (3.16) to get the identity d = −e0 ∧ dω + θ 7 ∧ ∗7 ω. The latter yields (5.34) ∗8 d = −∗7 dω − e0 ∧ ∗7 (θ 7 ∧ ∗7 ω), 8 7 7 7 7 7 7 ∗ (θ ∧ ) = −∗ (θ ∧ ω) − e0 ∧ ∗ (θ ∧ ∗ ω), (5.35) 1 θ 8 = − ∗8 ∗8 d ∧ = (5.36) 7 1 − ∗8 e0 ∧ ∗7 dω ∧ ω − e0 ∧ ∗7 θ 7 ∧ ∗7 ω ∧ ∗7 ω 7

−∗8 (∗7 dω ∧ ∗7 ω) = 1 − ∗7 ∗7 dω ∧ ω − ∗7 ∗7 θ 7 ∧ ∗7 ω ∧ ∗7 ω 7 6 1 −(dω, ∗7 ω)e0 = θ 7 + (dω, ∗7 ω)e0 , 7 7

where we used the conditions (3.16), (2.8) and the general identity ∗7 (∗7 γ ∧ ∗7 ω ∧ ∗7 ω) = 3γ valid for any 1-form γ on (M 7 , ω). Substitute (5.34), (5.35) and (5.36) into the formula (3.18) and compare the result with (3.17) to get T 8 = T 7 . The vector field e0 is parallel with respect to the Levi-Civita connection of g 8 and satis8 7 ˜8 fies T 8 (e0 , ., .) = T 7 (e0 , ., .) = 0. Therefore ∇ 8 e0 = 0 yielding R ∇ = R ∇ , R ∇ = ˜7 R ∇ . Consequently, 7 0

∇ Rmnkl mn ij = 8

7

∇ Rmnkl (∗7 ω)mn ij = 2Rij∇ kl = 2Rij∇ kl , 7

7

8

1

since R ∇ is a G2 -instanton. Clearly, the modified Bianchi identity (1.1), (resp. (1.2)) is 8 satisfied for F A = R ∇ . 7

The reverse procedure to find types of SU (3)-structures on a 6-manifold induced by different types of G2 -structures on 7-manifold is discussed recently in [8, 11, 50, 49]. 6. Examples Theorem 4.6 and Theorem 5.1 allow us to produce a number of examples of G2 and Spin(7)-instantons and solutions to the equations of motion with gauge connection A = ∇ starting from certain types of almost complex 6-manifolds or certain types of G2 manifolds. We recall the well known curvature identity 1 ˜ R ∇ (X, Y, Z, V ) = R ∇ (Z, V , X, Y ) + dT (X, Y, Z, V ). 2

(6.37)

It helps us to handle the Bianchi identity (1.2) with F A = R ∇ provided the next equality holds R ∇ (X, Y, Z, V ) = R ∇ (Z, V , X, Y ).

(6.38)


93

Combine (6.37) and (6.38) to get 1 ˜ R ∇ = R ∇ + dT . 2 In view of (6.39), the Bianchi identity (1.2) with A = ∇ takes the form 1 dT = α T r(R ∇ ∧ dT ) − T r(dT ∧ dT ) . 2

(6.39)

Clearly, the condition (6.38) is a sufficient SU (3) (resp. G2 , Spin(7)) -connection to satisfy the SU (3) (resp. G2 , Spin(7)) -instanton condition (3.19) (resp. (3.20),(3.21)). The symmetry (6.38) of the curvature of a metric connection ∇ with torsion 3-form T holds exactly when ∇T is a 4-form which is equivalent to the condition ∇ g T = 41 dT [56]. In particular, if the torsion is ∇-parallel, ∇T = 0, then we have the additional relations 1 1 1 g Rij∇ kl = Rij∇ kl − Tij m Tkl m − Tj km Til m − Tkim Tj l m ; (6.40) 2 4 4

m m m dTij kl = 2 Tij m Tkl + Tj km Til + Tkim Tj l . 6.1. (SU (3), G2 , Spin(7))-instanton and conformally flat non-compact solution. Any Nearly-Kähler 6-manifold is an SU (3)-instanton since the torsion T = 41 N = J ∇ g J is ∇- parallel [64], (see also [9, 32]) and therefore the curvature R ∇ satisfies (6.38). Take + = dF ; we obtain a SU (3)-instanton solving the gravitino and gaugino equations according to Theorem 4.6 which, however, does not solve the dilatino equation since the almost complex structure is not integrable [73]. There are known only four compact Nearly Kähler 6-manifolds, namely S 6 , S 3 × S 3 , CP 3 and the flag F = U (3)/(U (1) × U (1) × U (1)) [55, 72]. We consider the six-sphere (S 6 , J, g) endowed with the standard nearly Kähler structure (g, J ) inherited from the imaginary octonions in R7 [47]. We claim that (S 6 , g, J, ∇, A = ∇) satisfies both the modified Bianchi identity (1.1) and the anomaly cancellation condition (1.2). It is well known that any 6-dimensional Nearly Kähler manifold is Einstein and of constant type. Consequently, the following identities hold [32]: 1 Tij m Tkl m = a 2 (gik gj l − gj k gil − Fik Fj l + Fj k Fil ), dT = a 2 F ∧ F = −2a 2 ∗F, 2 where a 2 is a non-zero constant which can be identified with the Riemannian scalar curvature s g , 15a 2 = s g . Applying the fact that (S 6 , g) is a space of constant sectional g curvature, i.e. Rij∇ kl = 21 a 2 (gj k gil − gik gj l ) and (6.39), we calculate the Pontrjagin forms 3a 2 ∇ ab 16π 2 p1 (∇) = T r(R ∇ ∧ R ∇ ) = Rij∇ ab Rkl dT ; dx i ∧ dx j ∧ dx k ∧ dx l = − 4 2 ˜ = T r(R ∇˜ ∧ R ∇˜ ) = 9a dT . 16π 2 p1 (∇) 4 Remark 6.1. Observe that if we rescale the metric homothetically by a constant c, g¯ = e2c g, then the new torsion T¯ = e2c T in the case of SU (3)-structure and T¯ = e4c T in the case of G2 or Spin(7)-structure while the Pontrjagin 4-forms remain unchanged, ¯

¯

T r(R ∇ ∧ R ∇ ) = T r(R ∇ ∧ R ∇ ),

¯˜

¯˜

˜

˜

T r(R ∇ ∧ R ∇ ) = T r(R ∇ ∧ R ∇ ) (see [22] for more

94


precise discussion of this phenomena). Hence, if dT is proportional to the difference of the Pontrjagin 4-forms with a constant then we can always rescale the structure by a suitable constant in order to get the formulas (1.1) and (1.2). Keeping Remark 6.1 in mind we obtain Theorem 6.2. The Nearly Kähler 6-sphere solves the gravitino equation, the gaugino equation with F A = R ∇ and satisfies the modified Bianchi identity (1.1) and (1.2) with negative α . Consequently: a) The product (S 6 × R, ω, A = ∇ 7 ) with the G2 -structure described in Sect. 4 solves all the supersymmetry equations (1.3) with non-zero flux, non-constant dilaton and satisfies the Bianchi identity (1.1), (1.2). Therefore it solves the equations of motion in dimension 7. The product (S 6 × S 1 , ω, A = ∇ 7 ) is a compact space solving locally the supersymmetry equations (1.3) which satisfies the Bianchi identity (1.1), (1.2) in dimension 7. b) The product (S 6 ×R×R, , A = ∇ 8 ) with the Spin(7)-structure described in Sect. 5 solves the supersymmetry equations (1.3) with non-zero flux, non-constant dilaton and satisfies the Bianchi identity (1.1), (1.2). Therefore it solves the equations of motion in dimension 8. The product (S 6 × S 1 × S 1 , , A = ∇ 8 ) is a compact space solving locally the supersymmetry equations (1.3) which satisfies the Bianchi identity (1.1), (1.2) in dimension 8. Similarly to the SU(3)-case, any G2 -weak holonomy manifold (nearly-parallel G2 manifold) is a G2 -instanton. Indeed, it is well known that any 7-dimensional nearly-parallel G2 manifold is Einstein and the following identities hold dω = −λ∗ω,

(dω, ∗ω) = −λ,

(6.41)

8 g where λ2 = 21 s is a non-zero constant. The torsion T = − 16 λω is ∇-parallel [32]. Hence, any nearly-parallel G2 -manifold is a G2 -instanton which solves the gravitino equation and the gaugino equation with A = ∇ 7 but does not solve the dilatino equation according to the result in [34]. There are many known examples of compact nearly parallel G2 -manifolds: S 7 , SO(5)/SO(3) [17, 71], the Aloff-Wallach spaces N (g, l) = SU (3)/U (1)gl [20], any Einstein-Sasakian and any 3-Sasakian 7-manifold [35, 36]. We consider the seven sphere (S 7 , ω, g) endowed with the standard nearly-parallel G2 -structure induced by the octonions in R8 , namely, consider the seven sphere as a totally umbilical hypersurface in R8 [29]. Clearly (S 7 , ω, g, ∇, A = ∇) is a G2 -instanton. We claim that it satisfies the modified Bianchi identity (1.1) and the anomaly cancellation condition (1.2). Indeed, we easily calculate from (6.40) applying (6.41), (6.39), the fact that (S 7 , g) is a space of constant sectional curvature and some G2 -algebra, that ∇ ab 16π 2 p1 (∇) = T r(R ∇ ∧ R ∇ ) = Rij∇ ab Rkl dx i ∧ dx j ∧ dx k ∧ dx l =

−

λ4 λ2 ∗ω = − dT ; 8.27 36

˜ = T r(R ∇˜ ∧ R ∇˜ ) = 1 λ4 ∗ω = 1 λ2 dT . 16π 2 p1 (∇) 54 9 We obtain using Remark 6.1 the following


95

Theorem 6.3. The nearly-parallel 7-sphere solves the gravitino equation, the gaugino equation with F A = R ∇ and satisfies the modified Bianchi identity (1.1) and (1.2) with negative α . Consequently, the product (S 7 ×R, , A = ∇ 8 ) with the Spin(7)-structure described in Sect. 5 solves the supersymmetry equations (1.3) with non-zero flux, non-constant dilaton and satisfies the Bianchi identity (1.1), (1.2). Therefore it solves the equations of motion in dimension 8. The product (S 7 × S 1 , , A = ∇ 8 ) is a compact space solving locally the supersymmetry equations (1.3) which satisfies the Bianchi identity (1.1), (1.2) in dimension 8. We note that these sphere-solutions are (locally) conformally flat. 6.2. (SU (3), G2 , Spin(7))-instanton and non-conformally flat non-compact solution. In this section we present a non-locally-conformally flat solution starting with a nilpotent 6-dimensional Lie group. Let G be the six-dimensional connected simply connected and nilpotent Lie group, determined by the left invariant 1-forms {e1 , . . . , e6 } such that de2 = de3 = de6 = 0, de1 = e3 ∧ e6 , de4 = e2 ∧ e6 ,

(6.42) de5 = e2 ∧ e3 .

In terms of the standard coordinates x1 , . . . , x6 on R6 the left invariant forms {e1 , . . . , e6 } are described by the expressions e2 = dx2 , e3 = dx3 , e6 = dx6 , e1 = dx1 − x6 dx3 , e4 = dx4 − x6 dx2 , e5 = dx5 + x2 dx3 . 6 Consider the metric on G ∼ = R6 defined by g = i=1 ei2 , or equivalently

(6.43)

ds 2 = dx12 + (1 + x62 )dx22 + (1 + x22 + x62 )dx32 + dx42 + dx52 + dx62 −x6 (dx1 dx3 + dx2 dx4 ) + x2 dx3 dx5 . (6.44) Let (F, ) be the SU (3)-structure on G given by (2.5). Then(G, F, ) is an almost complex manifold with a SU (3)-structure. We show below that this space is a new non-conformally flat SU (3)-instanton solving both the gravitino and gaugino equations satisfying the modified Bianchi identity (1.1) as well as the anomaly cancellation condition (1.2) but not solving the dilatino equation. We compute the Riemannian curvature R g . The general Koszul formula 2g(∇ g X Y, Z) = Xg(Y, Z) + Y g(X, Z) − Zg(X, Y ) +g([XY ], Z) − g([Y, Z], X) − g([X, Z], Y )

(6.45)

gives the following essential non-zero terms 2∇ g e6 e3 = e1 , 2∇ g e2 e3 = −e5 , 2∇ g e3 e6 = −e1 , 2∇ g e3 e2 = e5 , 2∇ g e1 e6 = −e3 , 2∇ g e5 e2 = e3 ,

2∇ g e6 e2 = e4 , 2∇ g e2 e6 = −e4 , 2∇ g e4 e6 = −e2 .

(6.46)

Then we obtain R g (e5 , e6 , e2 , e1 ) = − 41 = 0. Hence, the metric is not locally conformally flat since the Weyl tensor W g (e5 , e6 , e2 , e1 ) = R g (e5 , e6 , e2 , e1 ) = − 41 = 0.

96


It is easy to verify using (2.4) and (6.42) that dF = −3e236 , N = − − , θ 6 = d + = (N, + ) = 0.

d − = ∗F,

(N, − ) = −4, (6.47)

Hence, (G, , g, J ) is neither complex nor Nearly Kähler manifold but it fulfills the conditions (4.22) of Theorem 4.1 and therefore there exists a SU (3)-holonomy connection with torsion 3-form on (G, , g, J ). The expression (4.23) and (6.47) give T = −2e145 + e136 + e235 − e246 ,

dT = −2(e1256 + e3456 + e1234 ) = 2∗F. (6.48)

Plug (6.46) and (6.48) into (3.11) to get that the nonzero essential terms of the torsion connection are ∇e1 e6 = −e3 ,

∇e5 e2 = e3 ,

∇e4 e5 = −e1 ,

∇e5 e1 = −e4 ,

∇e4 e6 = −e2 , ∇e1 e4 = −e5 .

(6.49)

It follows from (6.49) and (6.48) that the torsion tensor T as well as the Nijenhuis tensor N are parallel with respect to the connection ∇. Hence, ∇ defines an SU (3)-instanton. To verify the Bianchi identities for H we calculate the curvature R ∇ by means of (6.49). We obtain the following non-zero terms R ∇ (e6 , e2 , e6 , e2 ) = R ∇ (e6 , e3 , e6 , e3 ) = R ∇ (e2 , e3 , e2 , e3 ) = 1, R ∇ (e4 , e5 , e4 , e5 ) = R ∇ (e4 , e1 , e4 , e1 ) = R ∇ (e5 , e1 , e5 , e1 ) = 1, R ∇ (e2 , e6 , e5 , e1 ) = R ∇ (e3 , e6 , e4 , e5 ) = R ∇ (e2 , e3 , e1 , e4 ) = −1.

(6.50)

Applying (6.50), (6.48) and (6.39) it is straightforward to compute the first Pontrjagin ˜ Compare the result with the second equality in (6.48) to get 4-forms p1 (∇) and p1 (∇). dT =

1 ˜ ˜ T r(R ∇ ∧ R ∇ ) = −T r(R ∇ ∧ R ∇ ). 2

(6.51)

The coefficient of the structure equations of the Lie algebra given by (6.42) are integers. Therefore, the well-known theorem of Malcev [67] states that the group G has a uniform discrete subgroup such that N il 6 = G/ is a compact 6-dimensional nilmanifold. The SU (3)-structure, described above, descends to N il 6 and therefore we obtain a compact SU (3)-instanton. With the help of Remark 6.1 and (6.51) we derive from Theorem 4.6 and Theorem 5.1 the following Theorem 6.4. The non conformally flat almost hermitian 6-manifold (G, g, F, , A = ∇) solves the gravitino and gaugino equations and satisfies the Bianchi identity (1.1), (1.2) with positive α . Consequently, a) The product (G × R, ω, A = ∇ 7 ) with the G2 -structure described in Sect. 4 solves all the supersymmetry equations (1.3) with non-zero flux, non-constant dilaton and satisfies the Bianchi identity (1.1), (1.2). Therefore it solves the equations of motion in dimension 7. The product (N il 6 × S 1 , ω, A = ∇ 7 ) is a compact space solving locally the supersymmetry equations (1.3) which satisfies the Bianchi identity (1.1) and (1.2) in dimension 7;


97

b) The product (N il 6 × R × R, , A = ∇ 8 ) with the Spin(7)-structure described in Sect. 5 solves the supersymmetry equations (1.3) with non-zero flux, non-constant dilaton and satisfies the Bianchi identity (1.1), (1.2). Therefore it solves the equations of motion in dimension 8. The product (N il 6 × S 1 × S 1 , , A = ∇ 8 ) is a compact space solving locally the supersymmetry equations (1.3) which satisfies the Bianchi identity (1.1), (1.2) in dimension 8. The G2 -analog of the Dolbeault cohomology on G2 -manifold was studied on N il 6 × S 1 in [30]. Remark 6.5. The space (G, g, J ) is an example of an almost complex 6-manifold with totally skew-symmetric Nijenhuis tensor N and zero Lee form θ 6 which is neither complex nor Nearly Kähler but it is half-flat and therefore it determines a Riemannian metric with holonomy contained in G2 on G × R ∼ = R7 [55] which seems to be new. The explicit expression of this metric can be found solving the Hitchin flow equations dF =

∂( + ) , ∂t

d − = −F ∧

∂(F ) , ∂t

where the SU (3)-structure depends on a real parameter t ∈ R [55]. G2 -holonomy metrics arising from types of Hermitian 6-manifolds are studied recently in [5]. Remark 6.6. The torsion tensor T as well as the Nijenhuis tensor N of (N il 6 , g, J ) are parallel with respect to the torsion connection ∇ but ∇R ∇ = 0 since the space is not naturally reductive due to the inequality −1 = g([e3 , e6 ], e1 ) = −g([e3 , e1 ], e6 ) = 0. Hence, (N il 6 , g, J ) is an example of a compact non-naturally reductive almost Hermitian 6-manifold with totally skew-symmetric Nijenhuis tensor which is neither complex nor Nearly Kähler. The torsion as well as the Nijenhuis tensor are parallel with respect to the torsion connection. Remark 6.7. Spaces for which the covariant derivative of the torsion is a four form (in particular zero) become automatically of ‘instanton type’. Spaces with parallel torsion are studied in [1] in connection with the string model; almost Hermitian 6-manifolds with parallel torsion are investigated very recently (after the first version of the present article was posted to the arXiv) in [3]. Remark 6.8. In general, on any almost hermitian manifold the Nijenhuis tensor N is the (3,0)+(0,3)-part of the torsion of any linear connection ∇ compatible with the almost hermitian structure (see e.g. [38]). Therefore, the condition ∇T = 0 always implies ∇N = 0 because ∇ preserves the type decomposition induced from the almost complex structure.

7. Almost Contact Metric Structures and Non-Compact SU (3)-Solutions in Dimension 6 We construct in this section a new non-compact solution to the type I-supergravity equations of motion in dimension 6. We derive our solution from Sasakian structures in dimension 5. Solutions to the gravitino and dilatino equations in dimension 5 are investigated in [32, 33]. In dimension five any solution to the gravitino equation, i.e. any parallel spinor

98


with respect to a metric connection with torsion 3-form defines an almost contact metric structure (g, ξ, η, ψ), which is preserved by the torsion connection [32, 33]. It is shown in [33] that solutions to both gravitino and dilatino equations are connected with a special type ‘conformal’ transformations of the almost contact structure introduced in [33]. We recall that an almost contact metric structure consists of an odd dimensional manifold M 2k+1 equipped with a Riemannian metric g, vector field ξ of length one, its dual 1-form η as well as an endomorphism ψ of the tangent bundle such that ψ(ξ ) = 0,

ψ 2 = −id + η ⊗ ξ,

g(ψ., ψ.) = g(., .) − η ⊗ η.

The Nijenhuis tensor N and the fundamental form F of an almost contact metric structure are defined by F (., .) = g(., ψ.),

N = [ψ, ψ] + dη ⊗ ξ.

There are many special types of almost contact metric structures. We introduce those which are relevant to our considerations: – – – –

normal almost contact structures determined by the condition N = 0; contact metric structures characterized by dη = 2F ; quasi-Sasaki structures, N = 0, dF = 0. Consequently, ξ is a Killing vector [13]; Sasaki structures, N = 0, dη = 2F . Consequently, ξ is a Killing vector [13].

An almost contact metric structure admits a linear connection ∇ with torsion 3-form preserving the structure, i.e. ∇g = ∇ξ = ∇ψ = 0, if and only if the Nijenhuis tensor is totally skew-symmetric and the vector field ξ is a Killing vector field [32]. In this case the torsion connection is unique. The torsion T of ∇ on a Sasakian manifold is expressed by T = η ∧ dη = 2η ∧ F and the torsion T is ∇-parallel, ∇T = 0 [32]. We restrict our attention to the Sasakian manifold in dimension five. The spinor bundle of a 5-dimensional contact metric spin manifold decomposes under the action of the fundamental 2-form F 5 into the sum = 0 ⊕ 1 ⊕ 2 , dim 0 = dim 2 = 1, dim 1 = 2. Spinors of type 1 parallel with respect to the torsion connection on quasi-Sasakian 5-manifold are studied in [33]. We are interested in ∇ 5 -parallel spinors of type 0 or 2 on Sasakian 5-manifold. We recall ([32], Theorem 9.2) that a 5-dimensional simply connected Sasakian manifold admits a ∇ 5 -parallel spinor of type 0 or 2 if and only if the Riemannian Ricci tensor Ricg has the form Ricg = 6g − 2η ⊗ η.

(7.52)

The Tanno deformation of a Sasakian structure satisfying (7.52), defined by the formulas ψ = ψ,

ξ =

3 ξ, 4

η =

4 η, 3

g =

4 4 g + η ⊗ η, 3 9

yields an Einstein-Sasakian structure with Ricci tensor Ricg = 4g and vice versa. We may choose locally an orthonormal basis e1 , e2 , e3 , e4 , e5 = ξ such that F 5 = e1 ∧ e 2 + e 3 ∧ e 4 ,

dη = de5 = 2F 5 ,

T 5 = 2η ∧ F 5 = 2e5 ∧ (e1 ∧ e2 + e3 ∧ e4 ).

(7.53)


99

Let M 6 = M 5 × R. We indicate the objects on M 6 with a superscript 6. Consider the product Riemannian manifold M 6 = M 5 ×R with the product metric and the compatible almost complex structure J determined by the fundamental form F 6 = F 5 + e 5 ∧ e6 ,

(7.54)

where e6 = dt is the standard 1-form on R. The identity (7.54) yields dF 6 = 2F 5 ∧ e6 = 2e6 ∧ F 6 . This equality tell us that the almost hermitian manifold (M 6 , g 6 , F 6 ) is locally conformally Kähler. In particular, the almost complex structure is integrable and the Lee form θ 6 = 2e6 = 2dt is a closed 1-form on R. It is easy to check that the torsion T 6 of the corresponding Bismut connection ∇ 6 is determined by the equality T 6 = T 5 . Consequently, ∇ 6 T 6 = 0. The Bismut connection defines an SU (3)-instanton on M 5 × R which solves the three supersymmetry equations (1.3) provided M 5 is a Sasakian 5-manifold whose Riemannian Ricci tensor satisfies (7.52). 7.1. Non-conformally flat local SU (3)-solutions on S 5 × S 1 . Let S 5 be the five-sphere with the standard Einstein Sasakian structure induced on S 5 by the usual complex structure on C3 considering S 5 as a totally umbilic hypersurface in the complex space C3 . We consider (S 5 , g, ψ, η, ξ ) as a Sasakian space form, i.e. a Tanno deformation of the standard Einstein-Sasakian structure on S 5 . We may assume that the Riemannian curvature is given by (see e.g. [13]) 1

4

g Rij kl = gj k gil − gik gj l + Fkj Fli − Fki Flj + 2Fij Flk 3 3 1

+ ηi ηk gj l − ηj ηk gil + ηj ηl gik − ηi ηl gj k . (7.55) 3 Consider the locally conformally Kähler structure on S 5 × R determined by (7.54). We claim that the corresponding Bismut connection satisfies the modified Bianchi identity (1.1) as well as the anomaly cancellation condition (1.2). Indeed, Eq. (7.53) yields dT 6 = dT 5 = dη ∧ dη = 4F 5 ∧ F 5 . Use (6.40), apply (7.53) and (7.55), to get 1 4

6 Rij∇ kl = gj k gil − gik gj l + ηi ηk gj l − ηj ηk gil + ηj ηl gik − ηi ηl gj k + dTij6 kl . 3 6 The latter equality as well as (6.39) help to calculate the Pontrjagin forms 8 6 6 16π 2 p1 (∇ 6 ) = T r(R ∇ ∧ R ∇ ) = − dT 6 , 3 Keeping Remark 6.1 in mind we obtain

16π 2 p1 (∇˜ 6 ) =

16 6 dT . 3

Theorem 7.1. The Sasakian space form (S 5 , g 5 , ψ, η, ξ ) solves the gravitino equation 5 and satisfies the modified Bianchi identity (1.1) and (1.2) for F A = R ∇ with negative α . Consequently: a) The product (S 5 × R, g 6 , F 6 , ∇ 6 , A = ∇ 6 ) solves all the supersymmetry equations (1.3) with non-zero flux, non-constant dilaton and satisfies the Bianchi identity (1.1), (1.2). Therefore it solves the equations of motion in dimension 6. The product (S 5 × S 1 , g 6 , F 6 , A = ∇ 6 ) is a compact space solving locally the supersymmetry equations (1.3) which satisfies the Bianchi identity (1.1), (1.2) in dimension 6.

100


b) The product (S 5 × R × R, ω, A = ∇ 7 ) with the G2 -structure described in Sect. 4 solves all the supersymmetry equations (1.3) with non-zero flux, non-constant dilaton and satisfies the Bianchi identity (1.1), (1.2). Therefore it solves the equations of motion in dimension 7. The product (S 5 × S 1 × S 1 , ω, A = ∇ 7 ) is a compact space solving locally the supersymmetry equations (1.3) which satisfies the Bianchi identity (1.1), (1.2) in dimension 7. c) The product (S 5 × R × R × R, , A = ∇ 8 ) with the Spin(7)-structure described in Sect. 5 solves the supersymmetry equations (1.3) with non-zero flux, non-constant dilaton and satisfies the Bianchi identity (1.1), (1.2). Therefore it solves the equations of motion in dimension 8. The product (S 5 × S 1 × S 1 × S 1 , , A = ∇ 8 ) is a compact space solving locally the supersymmetry equations (1.3) which satisfies the Bianchi identity (1.1), (1.2) in dimension 8. Remark 7.2. Note that all compact examples we have presented in Sects. 6 and 6 solve the supersymmetry equations only locally since the closed Lee form θ is actually a closed 1-form on a circle and therefore it can not be exact. This is consistent with the vanishing results claiming that there are no compact solutions with globally defined non-constant dilaton and non-zero flux in type II and type I supergravities. Acknowledgments. The research was done during the visit of S.I. at the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. S.I. thanks the Abdus Salam ICTP for providing support and an excellent research environment. S.I. is a member of the EDGE, Research Training Network HPRN-CT2000-00101, supported by the European Human Potential Programme. The research is partially supported by Contract MM 809/1998 with the Ministry of Science and Education of Bulgaria, Contract 586/2002 with the University of Sofia “St. Kl. Ohridski”. We thank Tony Pantev for his interest in this work, for the useful suggestions and stimulating discussions. We are also grateful to Jerome Gauntlett for his helpful comments and remarks. We would like to thank the referee for his valuable comments and suggestions on clarifying the phenomenological background, especially the form of the Bianchi identity including string corrections.

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Communications in


Lagrangian Supersymmetries Depending on Derivatives. Global Analysis and Cohomology 1

Giovanni Giachetta , Luigi Mangiarotti1 , Gennadi Sardanashvily2 1 2

Department of Mathematics and Informatics, University of Camerino, 62032 Camerino (MC), Italy Department of Theoretical Physics, Physics Faculty, Moscow State University, 117234 Moscow, Russia

Received: 28 May 2004 / Accepted: 20 July 2004 Published online: 18 February 2005 – © Springer-Verlag 2005

Abstract: Lagrangian contact supersymmetries (depending on derivatives of arbitrary order) are treated in a very general setting. The cohomology of the variational bicomplex on an arbitrary graded manifold and the iterated cohomology of a generic nilpotent contact supersymmetry are computed. In particular, the first variational formula and conservation laws for Lagrangian systems on graded manifolds using contact supersymmetries are obtained. 1. Introduction At present, BRST transformations in the BV formalism [7, 24] provide the most interesting example of Lagrangian contact supersymmetries, depending on derivatives and preserving the contact ideal of graded exterior forms. Much that is already known regarding Lagrangian BRST theory (including the short variational complex, BRST cohomology [4, 5, 8], Noether’s conservation laws [5, 16, 19]) has been formulated in terms of jet manifolds of vector bundles (see [5] for a survey) since the jet manifold formalism provides the algebraic description of Lagrangian and Hamiltonian systems of both even and odd variables. In spite of this formulation, most authors however assume the base manifold X of these bundles to be contractible because, e.g., the relative (local in the terminology of [4, 5]) cohomology are not trivial even when X = Rn . Stimulated by the BRST theory, we consider Lagrangian systems of odd variables and contact supersymmetries in a very general setting. For this purpose, one usually calls into play fiber bundles over supermanifolds [12, 13, 17, 34]. We describe odd variables and their jets on an arbitrary smooth manifold X as generating elements of the structure ring of a graded manifold whose body is X [32, 38, 39]. This definition differs from that of jets of a graded fiber bundle [27], but reproduces the heuristic notion of jets of ghosts in the field-antifield BRST theory on Rn [5, 9]. Our goal is the following. Firstly, we construct the Z2 -graded variational bicomplex on a graded manifold with an arbitrary body X, and obtain the cohomology of its short

104

G. Giachetta, L. Mangiarotti, G. Sardanashvily

variational subcomplex and the complex of one-contact graded forms (Theorem 4.1). In particular, the first variational formula and conservation laws for Lagrangian systems on graded manifolds using contact supersymmetries are obtained (formulae (5.4) – (5.5)). Secondly, the iterated cohomology of a generic nilpotent contact supersymmetry is computed (Theorems 6.2, 6.4 and 6.5). In the most interesting case of the form degree n = dim X, it coincides with the above mentioned relative cohomology. Therefore, we extend the results of [5] and our recent work [21] to an arbitrary nilpotent contact supersymmetry. As is well-known, generalized (depending on derivatives) symmetries of differential equations have been intensively investigated [3, 11, 29, 30, 35]. Generalized symmetries of Lagrangian systems on a local coordinate domain have been described in detail [11, 35]. The variational bicomplex constructed in the framework of the infinite order jet formalism enables one to provide the global analysis of Lagrangian systems on a fiber bundle and their symmetries [2, 22, 32, 42]. Sketched in Sect. 2 of our work, this analysis is extended to Lagrangian systems on graded manifolds (Sect. 3). Recall that an r-order Lagrangian on a fiber bundle Y → X is defined as a horizontal n

density L : J r Y → ∧ T ∗ X, n = dim X, on the r-order jet manifold J r Y of sections of Y → X. With the inverse system of finite order jet manifolds r πr−1

π01

π

X ←− Y ←− J 1 Y ←− · · · J r−1 Y ←− J r Y ←− · · · ,

(1.1)

we have the direct system π∗

r ∗ πr−1

π01 ∗

∗ O∗ (X) −→ O∗ (Y ) −→ O1∗ −→ · · · Or−1 −→ Or∗ −→ · · ·

(1.2)

of graded differential algebras (henceforth GDAs) of exterior forms on jet manifolds r ∗ . Its direct limit is the GDA O ∗ with respect to the pull-back monomorphisms πr−1 ∞ consisting of all the exterior forms on finite order jet manifolds modulo the pull-back ∗ is decomposed into the sum d = d + d identification. The exterior differential on O∞ H V ∗ into a bicomplex. of the total and the vertical differentials. These differentials split O∞ Introducing the projector (2.3) and the variational operator δ, one obtains the var∗ . Its d - and δ-cohomology (Theorem 2.1) has been iational bicomplex (2.4) of O∞ H obtained in several steps [1, 2, 22, 35, 41–43]. In order to define the variational bicomplex on graded manifolds (Sect. 3), let us recall that, by virtue of Batchelor’s theorem [6], any graded manifold (A, X) with a body X is isomorphic to the one whose structure sheaf AQ is formed by germs of sections of the exterior product 2

∧Q∗ = R ⊕ Q∗ ⊕ ∧ Q∗ ⊕ · · · , X

X

(1.3)

X

where Q∗ is the dual of some real vector bundle Q → X. In field models, a vector bundle Q is usually given from the beginning. Therefore, we consider graded manifolds (X, AQ ) where Batchelor’s isomorphism holds. We agree to call (X, AQ ) the simple graded manifold constructed from Q. Accordingly, r-order jets of odd fields are defined as generating elements of the structure ring of the simple graded manifold (X, AJ r Q ) constructed from the jet bundle J r Q → X of Q which is also a vector bundle [32, 39]. Let CJ∗ r Q be the bigraded differential algebra (henceforth BGDA) of Z2 -graded (or, simply, graded) exterior forms on the graded manifold (X, AJ r Q ). A linear bundle r : J r Q → J r−1 Q yields the corresponding monomorphism of BGDAs morphism πr−1 ∗ ∗ CJ r−1 Q → CJ r Q [6, 32]. Hence, there is the direct system of BGDAs

Lagrangian Supersymmetries Depending on Derivatives π01∗

105

r ∗ πr−1

∗ CQ −→ CJ∗ 1 Q · · · −→ CJ∗ r Q −→ · · · ,

(1.4)

∗ consists of graded exterior forms on graded manifolds (X, A r ), whose direct limit C∞ J Q r ∈ N, modulo the pull-back identification. This definition of odd jets enables one to describe odd and even variables (e.g., fields, ghosts and antifields in BRST theory) on the same footing. Namely, let Y → X be an ∗ ⊂ O ∗ the C ∞ (X)-subalgebra of exterior forms whose coeffiaffine bundle and P∞ ∞ cients are polynomial in the fiber coordinates on jet bundles J r Y → X. This notion is ∗ is an exterior form on some finite order jet manifold intrinsic since any element of O∞ ∗ of graded and all jet bundles J r Y → X are affine. Let us consider the product S∞ ∗ ∗ ∗ algebras C∞ and P∞ over their common subalgebra O (X) of exterior forms on X. It is a BGDA which is split into the Z2 -graded variational bicomplex, analogous to that of ∗ . O∞ In Sect. 4, we obtain cohomology of some subcomplexes of the variational bicom∗ when X is an arbitrary manifold (Theorem 4.1). They are the short variational plex S∞ complex (4.1) of horizontal (local in the terminology of [5, 8]) graded exterior forms and the complex (4.2) of one-contact graded forms. For this purpose, one however must: (i) ∗ to the BGDA (S∗ ) of graded exterior forms of locally finite jet enlarge the BGDA S∞ ∞ order, (ii) compute the cohomology of the corresponding complexes of (S∗∞ ), and (iii) ∗ . Following this proceprove that this cohomology of (S∗∞ ) coincides with that of S∞ dure, we show that cohomology of the complex (4.1) equals the de Rham cohomology of X, while the complex (4.2) is globally exact. Note that the exactness of the short variational complex (4.1) on X = Rn has been repeatedly proved [5, 8, 14]. One has also considered its subcomplex of graded exterior forms whose coefficients are constant on Rn . Its dH -cohomology is not trivial [5]. The exactness of the complex (4.2) enables us to generalize the first variational formula and Lagrangian conservation laws in the calculus of variations on fiber bundles to graded Lagrangians and contact supersymmetries of arbitrary order (Sect. 5). Cohomology of the short variational complex (4.1) and its modification (6.2) is the main ingredient in a computation of the iterated cohomology of nilpotent contact supersymmetries. By analogy with a contact symmetry (Proposition 2.3), an infinitesimal contact supertransformation or, simply, a contact supersymmetry υ is defined as a graded derivation 0 such that the Lie derivative L preserves the contact ideal of the BGDA of the R-ring S∞ υ ∗ . The BRST transformation υ (5.7) in gauge theory on principal bundles exemplifies S∞ a first order contact supersymmetry such that the Lie derivative Lυ of horizontal graded exterior forms is nilpotent. This fact motivates us to study nilpotent contact supersymmetries in a general setting. The key point is that the Lie derivative Lυ along a contact supersymmetry and the total differential dH mutually commute. When Lυ is nilpotent (Lemma 5.3), we suppose 0,∗ that the dH -complex S∞ of horizontal graded exterior forms is split into the bicomplex k,m {S } with respect to the nilpotent operator

sυ φ = (−1)|φ| Lυ φ,

0,∗ φ ∈ S∞ ,

(1.5)

and the total differential dH . In the case of the above mentioned BRST transformation υ (5.7), sυ (1.5) is the BRST operator. One usually studies the relative cohomology H ∗,∗ (sυ /dH ) of sυ with respect to the total differential dH (see [15] for the BRST cohomology modulo the exterior differential d). This cohomology is not trivial even when X = Rn , but it can be related to the total (sυ + dH )-cohomology only in the form

106


degree n = dim X. We consider the iterated cohomology H ∗,∗ (sυ |dH ) of the bicomplex {S k,m } (Sect. 6). In the most interesting case of form degree n = dim X, relative and iterated cohomology groups coincide. They naturally characterize graded Lagrangians 0,∗ L ∈ S ∗,n , for which υ is a variational symmetry, modulo Lie derivatives Lυ ξ , ξ ∈ S∞ , and dH -exact graded exterior forms. Using the fact that dH -cocycles are represented by exterior forms on X and that any exterior form on X is sυ -closed, we obtain the iterated cohomology H ∗,m0

k

◦ hk ◦ hn ,

(φ) =

| |≥0

(−1)| | θ i ∧ [d (∂i φ)],

>0,n φ ∈ O∞ ,

(2.3)

Lagrangian Supersymmetries Depending on Derivatives

107

∗ such that ◦ d = 0 and the nilpotent variational operator δ = ◦ d on O ∗,n . of O∞ H V ∞ k,n ∗ is split into the variational bicomplex Put Ek = (O∞ ). Then the GDA O∞

dV

0 →

1,0 O∞ dV

0 → R →

−→ dH

0,1 O∞

π ∞∗ d

dV dH

−→ · · ·

6

−→

6 0

1,1 O∞

dV

O 0 (X) −→

6

6

dV dH

6

0 O∞ π ∞∗

0 → R →

6

dV dH

−→ · · ·

6

6 1,m O∞

π ∞∗ d

6

dV dH

−→ · · ·

6

0,m O∞

1,n O∞

dH

−→ · · ·

6

0,n O∞ π ∞∗

6

6

−δ

6

−→

E1 → 0

≡

6

0,n O∞

(2.4)

d

O m (X) −→ · · ·

O n (X) −→ 0

6

0

−δ

6

dV

d

O 1 (X) −→ · · ·

. . .

. . .

. . .

. . .

. . .

6

0

0

Its cohomology has been obtained in several steps (see the outline of proof of Theorem 2.1 in Appendix A). Theorem 2.1. (i) The second row from the bottom and the last column of this bicomplex make up the variational complex dH

dH

δ

δ

0 0,1 0,n 0 → R → O∞ −→ O∞ · · · −→ O∞ −→ E1 −→ E2 −→ · · · .

(2.5)

Its cohomology is isomorphic to the de Rham cohomology of the fiber bundle Y . (ii) The rows of contact forms of the bicomplex (2.4) are exact sequences. One can think of the elements 0,n L = Lω ∈ O∞ , δL = (−1)| | d (∂i L)θ i ∧ ω ∈ E1 ,

ω = dx 1 ∧ · · · ∧ dx n ,

| |≥0

of the variational complex (2.5) as being a finite order Lagrangian and its Euler–Lagrange operator, respectively. Corollary 2.2. The exactness of the row of one-contact forms of the variational bicom1,n plex (2.4) at the term O∞ relative to the projector provides the R-module decomposition 1,n 1,n−1 O∞ = E1 ⊕ dH (O∞ ) 0,n , the corresponding decomposition and, given a Lagrangian L ∈ O∞

dL = δL − dH .

(2.6)

The form in the decomposition (2.6) is not uniquely defined. It reads λν ...ν

= Fi s 1 θνis ...ν1 ∧ ωλ , Fiνk ...ν1 = ∂iνk ...ν1 L − dλ Fiλνk ...ν1 +hνi k ...ν1 , ωλ = ∂λ ω, s=0 (ν ν

)...ν

1 0 obey the relations hν = 0, h k k−1 = 0. It follows where local functions h ∈ O∞ i i that L = + L is a Lepagean equivalent of a finite order Lagrangian L [25]. The decomposition (2.6) leads to the first variational formula (2.15) and the Lagrangian conservation law (2.16) as follows.

108


0 of the R-ring O 0 such that the Lie derivative L preserves A derivation υ ∈ dO∞ υ ∞ ∗ (i.e., the Lie derivative L of a contact form is a contact the contact ideal of the GDA O∞ υ form) is called an infinitesimal contact transformation or, simply, a contact symmetry (by analogy with C-transformations in [30] though υ need not come from a morphism of J ∞ Y ). Proposition 2.3 below shows that, restricted to a coordinate chart (2.1) and a GDA Or∗ of finite jet order, any contact symmetry υ is the jet prolongation of a generalized vector field in [35]. 0 is isomorphic to the O 0 -dual (O 1 )∗ Proposition 2.3. (i) The derivation module dO∞ ∞ ∞ 1 0 is of the module of one-forms O∞ . (ii) Relative to an atlas (2.1), a derivation υ ∈ dO∞ given by the expression i υ = υ λ ∂λ + υ i ∂i + υ ∂i , (2.7) | |>0

i are local smooth functions of finite jet order obeying the transformation where υ λ , υ i , υ law

υ = λ

∂x λ µ υ , ∂x µ

υ = i

∂y i j ∂y i µ υ + µυ , ∂y j ∂x

υ = i

∂y i ||≤| |

j

∂y

j

υ +

∂y i µ υ . ∂x µ (2.8)

(iii) A derivation υ (2.7) is a contact symmetry iff i i υ = d (υ i − yµi υ µ ) + yµ+ υ µ,

0 < | |.

(2.9)

∗ is generated by elements df , f ∈ O 0 . It suffices Proof. (i) At first, let us show that O∞ ∞ 1 0 -linear combination of elements df , to justify that any element of O∞ is a finite O∞ 0 . Indeed, every φ ∈ O 1 is an exterior form on some finite order jet manifold f ∈ O∞ ∞ r J Y . By virtue of the Serre–Swan theorem extended to non-compact manifolds [36, 40], the C ∞ (J r Y )-module Or∗ of one-forms on J r Y is a projective module of finite rank, i.e., φ is represented by a finite C ∞ (J r Y )-linear combination of elements df , 0 . Any element ∈ (O 1 )∗ yields a derivation υ (f ) = (df ) f ∈ C ∞ (J r Y ) ⊂ O∞ ∞ 0 1 is generated by elements df , f ∈ O 0 , different of the R-ring O∞ . Since the module O∞ ∞ 1 )∗ provide different derivations of O 0 , i.e., there is a monomorphism elements of (O∞ ∞ 1 )∗ → dO 0 . By the same formula, any derivation υ ∈ dO 0 sends df → υ(f ) (O∞ ∞ ∞ 0 is generated by elements df , it defines a morphism : O 1 → O 0 . and, since O∞ υ ∞ ∞ Moreover, different derivations υ provide different morphisms υ . Thus, we have a 0 → (O 1 )∗ . monomorphism and, consequently, an isomorphism dO∞ ∞ 1 0 -module generated by the (ii) Restricted to a coordinate chart (2.1), O∞ is a free O∞ i . Then dO 0 = (O 1 )∗ restricted to this chart consists of eleexterior forms dx λ , θ ∞ ∞ i . The transformation rule (2.8) results ments (2.7), where ∂λ , ∂i are the duals of dx λ , θ from the transition functions (2.1). The interior product υ φ and the Lie derivative Lυ φ, ∗ , obey the standard formulae. Restricted to a coordinate chart, the Lie derivative φ ∈ O∞ Lυ sends each finite jet order GDA Or∗ to another finite jet order GDA Os∗ . Since the ∗ . atlas (2.1) is finite, Lυ φ preserves O∞ (iii) The expression (2.9) results from a direct computation similar to that of the first part of Bäcklund’s theorem [29]. One can then justify that local functions (2.9) satisfy the transformation law (2.8).


109

Any contact symmetry admits the horizontal splitting υ = υH + υV = υ λ dλ + (ϑ i ∂i +

| |>0

ϑ i = υ i − yµi υ µ , (2.10)

d ϑ i ∂i ),

0 [32]. relative to the canonical connection ∇ = dx λ ⊗ dλ on the C ∞ (X)-ring O∞

Lemma 2.4. Any vertical contact symmetry υ = υV obeys the relations υ dH φ = −dH (υ φ), Lυ (dH φ) = dH (Lυ φ),

∗ φ ∈ O∞ .

(2.11) (2.12)

Proof. It is easily justified that, if φ and φ satisfy the relation (2.11), then φ ∧ φ does i . The well. Then it suffices to prove the relation (2.11) when φ is a function and φ = θ result follows from the equalities i i υ θ = υ ,

i i dH (υ ) = υλ+ dx λ ,

i dH θλi = dx λ ∧ θλ+ , (2.13)

i i ∂i = v ∂i ◦ dλ . dλ ◦ v

The relation (2.12) is a corollary of the equality (2.11).

(2.14)

0,n Proposition 2.5. Given a Lagrangian L = Lω ∈ O∞ , its Lie derivative Lυ L along a contact symmetry υ (2.10) fulfills the first variational formula

Lυ L = υV δL + dH (h0 (υ

L )) + LdV (υH ω),

(2.15)

where L is a Lepagean equivalent, e.g., a Poincaré–Cartan form of L. Proof. The formula (2.15) comes from the splitting (2.6) and the relation (2.11) as follows: Lυ L = υ dL + d(υ L) = [υV dL − dV L ∧ υH ω] + [dH (υH L) + dV (LυH ω)] = υV dL + dH (υH L) + LdV (υH ω) = υV δL − υV dH + dH (υH L) + LdV (υH ω) = υV δL + dH (υV

+ υH L) + LdV (υH ω), where υV

= h0 (υ

) since is a one-contact form, υH L = h0 (υ L), and L =

+ L. Let υ be a variational symmetry of L (in the terminology of [35]), i.e., Lυ L = dH σ , 0,n−1 σ ∈ O∞ . By virtue of the expression (2.15), this condition implies that υ is projected onto X. Then the first variational formula (2.15) restricted to Ker δL leads to the weak conservation law 0 ≈ dH (h0 (υ

L ) − σ ).

(2.16)

110


3. Z2 -Graded Variational Bicomplex Let (X, AQ ) be the simple graded manifold constructed from a vector bundle Q → X of fiber dimension m. Its structure ring AQ of sections of AQ consists of sections of the exterior bundle (1.3) called graded functions. Given bundle coordinates (x λ , q a ) on Q with transition functions q a = ρba q b , let {ca } be the corresponding fiber bases for Q∗ → X, together with transition functions c a = ρba cb . Then (x λ , ca ) is called the local basis for the graded manifold (X, AQ ) [6, 32]. With respect to this basis, graded functions read f =

m 1 fa ...a ca1 · · · cak , k! 1 k k=0

where fa1 ···ak are local smooth real functions on X. Given a graded manifold (X, AQ ), by the sheaf dAQ of graded derivations of AQ is meant a subsheaf of endomorphisms of the structure sheaf AQ such that any section u of dAQ over an open subset U ⊂ X is a Z2 -graded derivation of the Z2 -graded ring AQ (U ) of graded functions on U , i.e., u(ff ) = u(f )f + (−1)[u][f ] f u(f ),

f, f ∈ AQ (U ),

where [.] denotes the Grassmann parity. One can show that sections of dAQ over U exhaust all Z2 -graded derivations of the ring AQ (U ) [6]. Let dAQ be the Lie superalgebra of Z2 -graded derivations of the R-ring AQ . Its elements are called Z2 -graded (or, simply, graded) vector fields on (X, AQ ). Due to the canonical splitting V Q = Q × Q, the vertical tangent bundle V Q → Q of Q → X can be provided with the fiber bases {∂a } which is the dual of {ca }. Then a graded vector field takes the local form u = uλ ∂λ + ua ∂a , where uλ , ua are local graded functions. It acts on AQ by the rule u(fa...b ca · · · cb ) = uλ ∂λ (fa...b )ca · · · cb + ud fa...b ∂d (ca · · · cb ).

(3.1)

This rule implies the corresponding transformation law u = uλ , λ

u = ρja uj + uλ ∂λ (ρja )cj . a

Then one can show [32, 38] that graded vector fields on a simple graded manifold can be represented by sections of the vector bundle VQ → X which is locally isomorphic to the vector bundle ∧Q∗ ⊗X (Q ⊕X T X), and is equipped with the bundle coordinates (x˙aλ1 ...ak , vbi 1 ...bk ), k = 0, . . . , m, together with the transition functions x˙ i1 ...ik = ρ −1 ia11 · · · ρ −1 iakk x˙aλ1 ...ak , j i −1 b1 −1 bk v j1 ...jk = ρ j1 · · · ρ jk ρji vb1 ...bk + λ

k! λ i ∂λ ρbk . x˙ (k − 1)! b1 ...bk−1

Using this fact, we can introduce graded exterior forms on the simple graded mani∗ , where V ∗ → X is the pointwise fold (X, AQ ) as sections of the exterior bundle ∧ VQ Q ∧Q∗ -dual of VQ . Relative to the dual bases {dx λ } for T ∗ X and {dcb } for Q∗ , graded one-forms read φ = φλ dx λ + φa dca ,

φa = ρ −1ba φb ,

φλ = φλ + ρ −1ba ∂λ (ρja )φb cj .


111

The duality morphism is given by the interior product u φ = uλ φλ + (−1)[φa ] ua φa . ∗ with respect to the bigraded exterior Graded exterior forms constitute the BGDA CQ product ∧ and the exterior differential d. The standard formulae of a BGDA hold. Since the jet bundle J r Q → X of a vector bundle Q → X is a vector bundle, let us consider the simple graded manifold (X, AJ r Q ) constructed from J r Q → X. Its local a }, 0 ≤ | | ≤ r, together with the transition functions basis is {x λ , c a j a c λ+ = dλ (ρja c ), dλ = ∂λ + cλ+ ∂a , (3.2) | |0

>0,n φ ∈ S∞ ,

similar to (2.3). The graded variational operator δ = ◦d is introduced. Then the BGDA ∗ is split into the Z -graded variational bicomplex S∞ 2 ∗,∗ k,n (O∗ (X), S∞ , Ek = (S∞ ); d, dH , dV , , δ),

(3.6)

analogous to the variational bicomplex (2.4). 4. Cohomology of Z2 -Graded Complexes We aim to study the cohomology of the short variational complex dH

dH

δ

0 0,1 0,n 0 −→ R −→ S∞ −→ S∞ · · · −→ S∞ −→ E1

(4.1)

and the complex of one-contact graded forms dH

dH

1,0 1,1 1,n 0 → S∞ −→ S∞ · · · −→ S∞ −→ E1 → 0

(4.2)

∗ . One can think of the elements of the BGDA S∞ 0,n L = Lω ∈ S∞ , δ(L) = (−1)| | θ A ∧ d (∂A L) ∈ E1 | |≥0

of the complexes (4.1) – (4.2) as being a graded Lagrangian and its Euler–Lagrange operator, respectively.


113

Theorem 4.1. The cohomology of the complex (4.1) equals the de Rham cohomology H ∗ (X) of X. The complex (4.2) is exact. The proof of Theorem 4.1 follows the scheme of the proof of Theorem 2.1, but all sheaves are sheaves over X. The proof falls into the three steps. (i) We start by showing that the complexes (4.1) – (4.2) are locally exact. Lemma 4.2. The complex (4.1) on X = Rn is exact. Referring to [5], Theorems 4.1 – 4.2, for the proof, we summarize a few formulae 0,∗ quoted in the sequel. Any horizontal graded form φ ∈ S∞ admits the decomposition , φ = φ0 + φ

= φ

1 0

dλ A s ∂A φ, λ

(4.3)

| |≥0

0,m0

A , but all s A are odd. where all υ A are smooth real functions on X, ∂A are the duals of ds

6. Cohomology of Nilpotent Contact Supersymmetries Let υ be a nilpotent contact supersymmetry. Since the Lie derivative Lυ obeys the rela0,∗ tion (5.3), let us assume that the R-module S∞ of graded horizontal forms is split into a bicomplex {S k,m } with respect to the nilpotent operator sυ (1.5) and the total differential dH which obey the relation

120


sυ ◦ dH + dH ◦ sυ = 0.

(6.1)

This bicomplex dH : S k,m → S k,m+1 ,

sυ : S k,m → S k+1,m

is graded by the form degree 0 ≤ m ≤ n and an integer k ∈ Z, though it may happen that S k,∗ = 0 starting from some number k. For the sake of brevity, let us call k the charge number. 0,∗ For instance, the BRST bicomplex S∞ (C, VG P ) is graded by the charge number r . In this case, s k which is the polynomial degree of its elements in odd variables C υ r (1.5) is the BRST operator. Since the ghosts C are characterized by the ghost number 1, k ∈ N, is the ghost number. The bicomplex defined by the contact supersymmetry (5.10) has the similar gradation, but taken with the sign minus (i.e., k = 0, −1, . . . ) because the nilpotent operator sυ decreases the odd polynomial degree. Let us consider the relative and iterated cohomology of the nilpotent operator sυ (1.5) with respect to the total differential dH . Recall that a horizontal graded form φ ∈ S ∗,∗ is said to be a relative closed form, i.e., (sυ /dH )-closed form if sυ φ is a dH -exact form. This form is called exact if it is a sum of an sυ -exact form and a dH -exact form. Accordingly, we have the relative cohomology H ∗,∗ (sυ /dH ). If a (sυ /dH )-closed form φ is also dH -closed, it is called an iterated (sυ |dH )-closed form. This form φ is said to be exact if φ = sυ ξ + dH σ , where ξ is a dH -closed form. Thus, we obtain the iterated cohomology H ∗,∗ (sυ |dH ) of the (sυ , dH )-bicomplex S ∗,∗ . It is the term E2∗,∗ of the spectral sequence of this bicomplex [31]. There is an obvious isomorphism H ∗,n (sυ /dH ) = H ∗,n (sυ |dH ) of relative and iterated cohomology groups on horizontal graded densities. Forthcoming Theorems 6.2 and 6.5 extend our results on iterated cohomology in [21] to an arbitrary nilpotent contact supersymmetry. Proposition 6.1. Let us consider the complex dH

dH

dH

0 0,1 0,n 0 −→ R −→ S∞ −→ S∞ · · · −→ S∞ −→ 0.

(6.2)

Its cohomology groups H m 0 :

γn ≤ C exp(−an) ∀n ≥ 0.

In terms of the weighted sequence spaces 2 = {(xn ) : |xn |2 2 (n) < ∞}, Sobolev or analytic functions v, v(x) =

vk exp(2πikx),

can be characterized as having their Fourier-coefficient sequences in 2 , where = (1 + n2 )a/2 , or = exp(an), a > 0, respectively. T. Kappeler and B. Mityagin [19, 20] raised the general question about the relationship between the two conditions v ∈ H () and (γn ) ∈ 2 , where

1D Dirac Operators

141

H () = {v : (vk ) ∈ 2 },

(1.3)

for general (submultiplicative) weights. They showed that v ∈ H () ⇒ (γn ) ∈ 2 .

(1.4)

(γn ) ∈ 2 ⇒ v ∈ H ()

(1.5)

The opposite implication

required a delicate analysis of special non-linear equations in sequence spaces and a priori estimates of the Sobolev norms of their solutions. This has been done in [3–5] for, roughly speaking, each submultiplicative weight sequence of subexponential growth, i.e., lim (log n ) /n = 0. This is not just a technical restriction. For with superexponential growth like exp(|n|b ), b > 1, the implications (1.4) and (1.5) are not valid, but the proper adjustment can be made, and this is presented in [6]. Analysis of non-self-adjoint Hill operators, i.e., the case of complex-valued potentials, is done in [7]; see further references there. 3. Let us return to Dirac operators. Surprisingly enough, we could not find in the literature even a Hochstadt–McKean–Trubowitz [17, 18, 24] type statement in this case. Still, after [19, 20] the approach developed there for the Schrödinger-Hill case has been used in the Dirac case in [15, 16] to get claims about the decay rate of spectral gaps: γn2 2n < ∞ (1.6) P ∈ H () ⇒ n∈Z

under some rigid and (as we will see) unnecessary restrictions on . The main goal of the present paper is to show that for subexponential weights the H ()-smoothness of a potential P , i.e., the condition P ∈ H (), follows from 2 -decay of the two-sided sequence (γn ), i.e., (1.7) γn2 2n < ∞ ⇒ P ∈ H () (see Theorem 11, Sect. 4, for accurate formulation). This result has been announced in [8], Thm 2(i). Maybe, it’s worth mentioning that there is an analogue of this implication (and equivalence) in the non-self-adjoint case (see [8], Thm 2(ii); this result will be given with detailed proofs in [9]). In particular, (1.6) and (1.7) tell us the following. (A) (γn ) decays faster than any power of 1/n if and only if P ∈ C ∞ (compare to [17, 24]). (B) (γn ) decays faster than exp(−an) for some a > 0 if and only if P is analytic in a strip around the real axis (compare to [30]). (C) (γn ) decays faster than exp(−anβ ), β ∈ (0, 1), for some a > 0 if and only if the Fourier coefficients (pk ) of P decay faster than exp(−A|k|β ), for some A > 0 (compare to [5]). The statements (A), (B) and (C) are given in a precise form in Sect. 5.1. In the case of Schrödinger - Hill operators we have proven similar statements in [5] and [7]. The general scheme of the present paper is close to the scheme of our paper [5]. However, the technical details and difficulties are quite different, because

142

P. Djakov, B. Mityagin

(i) the Dirac operator is not semibounded; (ii) its resolvent is not a trace class operator. We are going to make this point explicit and specific in our proofs and comments below. The structure of our paper is as follows. Abstract Introduction Basic equation and formulae for gaps Weights; Carleman sequences Basic results: estimates on the smoothness of the potential in terms of the decay rate of spectral gaps 5. Conclusions and comments References 1. 2. 3. 4.

2. Basic Equation and Formulae for Spectral Gaps 1. The Dirac operator

L0 y = i

1 0 dy , 0 −1 dx

y=

y1 , y2

(2.1)

considered on the interval [0, 1] with periodic (y(0) = y(1)) or antiperiodic (y(0) = −y(1)) boundary conditions, has a discrete spectrum {2kπ, k ∈ Z} or {(2k+1)π, k ∈ Z}, respectively. Each eigenvalue nπ, both for periodic (if n is even), or antiperiodic (if n is odd) boundary conditions has multiplicity 2, and 1 −inπx 0 inπx 1 2 en (x) = e , en (x) = e (2.2) 0 1 are eigenfunctions corresponding to the eigenvalue nπ. Moreover, if the Hilbert space H = L2 [0, 1] × L2 [0, 1] is equipped with the scalar product 1 g f1 , 1 = (2.3) f1 (x)g1 (x) + f2 (x)g2 (x) dx, f2 g2 0 1 , e2 , k ∈ Z} and {e1 2 then each of the systems {e2k 2k 2k+1 , e2k+1 , k ∈ Z} is an orthonormal basis in H. The operator 0 P (x) 0 L = L + V, V = , (2.4) Q(x) 0

where P and Q are 1-periodic functions, may be considered as a perturbation of L0 . Further we always assume that P , Q ∈ L2 [0, 1]; then the operator L, considered with periodic or antiperiodic boundary conditions, has also a discrete spectrum. The following statement is known (see, for example [22, 23, 25, 26], in particular, [27], Thm. 4.1 and Prop. 4.3). Lemma 1. There exists N0 = N0 (P , Q) such that for each |n| ≥ N0 the open disc with center π n and radius π/2 contains exactly two (counted with multiplicity) periodic (if + n is even), or antiperiodic (if n is odd) eigenvalues {λ− n , λn } of L, i.e., |λ± n − πn| < π/2,

|n| ≥ N0 .

(2.5)

1D Dirac Operators

143

2. Suppose that λ = nπ + z, |n| ≥ N0 , is a periodic (or antiperiodic) eigenvalue of L with |z| < π/2 and y = 0 is a corresponding eigenvector. Let En0 = [en1 , en2 ] be the eigenspace of L0 that corresponds to nπ, and let H(n) be its orthogonal complement. We denote by Pn0 and Q0n , respectively, the orthogonal projectors on En0 and H(n). Then the equation (nπ + z − L)y = 0 is equivalent to the following system of two equations: Q0n (nπ + z − L0 − V )Q0n y + Q0n (nπ + z − L0 − V )Pn0 y = 0,

(2.6)

Pn0 (nπ + z − L0 − V )Q0n y + Pn0 (nπ + z − L0 − V )Pn0 y = 0.

(2.7)

Taking into account that Pn0 Q0n = Q0n Pn0 = 0 and Pn0 L0 Q0n = Q0n L0 Pn0 = 0 we obtain that (2.6) and (2.7) can be written as Q0n (nπ + z − L0 − V )Q0n y − Q0n V Pn0 y = 0,

(2.8)

−Pn0 V Q0n y − Pn0 V Pn0 y + zPn0 y = 0.

(2.9)

A = A(n, z) := Q0n (nπ + z − L0 − V )Q0n : H(n) → H(n)

(2.10)

The operator

is invertible for large |n| (see below (2.18), (2.26) and (2.27)). Thus, solving (2.8) for Q0n y, we obtain Q0n y = A−1 Q0n V Pn0 y, where Pn0 y = 0 (otherwise Q0n y = 0 which implies y = Pn0 y + Q0n y = 0). Now (2.9) implies (after plugging the above expression for Q0n y into (2.9)) that (S − z)Pn0 y = 0, where the operator S is given by S := Pn0 V A−1 Q0n V Pn0 + Pn0 V Pn0 : En0 → En0 .

(2.11)

12

S 11 S be the matrix representation of the two-dimensional operator S with S 21 S 22 respect to the basis en1 , en2 ; then Let

S 11 = en1 , Sen1 ,

S 22 = en2 , Sen2 ,

S 12 = en1 , Sen2 ,

S 21 = en2 , Sen1 . (2.12)

Hence we obtain (since Pn0 y = 0) 11 S − z S 12 = 0. det S 21 S 22 − z

(2.13)

In the self-adjoint case (Q(x) = P (x)), if λ is a double eigenvalue, then there exists another eigenvector y˜ (corresponding to λ), such that y and y˜ are linearly independent. Then Pn0 y and Pn0 y˜ are linearly independent also. Indeed, if Pn0 y = cPn0 y˜ then ˜ Q0n y = A−1 Q0n V Pn0 y = cA−1 Q0n V Pn0 y˜ = cQ0n y, which leads to a contradiction:

y = Pn0 y + Q0n y = c Pn0 y˜ + Q0n y˜ = cy. ˜

144


Thus S ≡ 0, i.e., if λ = πn + z is a double eigenvalue of a self-adjoint Dirac operator L, then (for large enough n) S 11 − z = 0,

S 12 = 0,

S 21 = 0,

S 22 − z = 0.

(2.14)

1 , m ∈ Z} and 3. Let H1 and H2 be the subspaces of H generated, respectively, by {em 2 1 2 {em , m ∈ Z}, and let H (n) and H (n) be, respectively, the intersections of these spaces with H(n). Then H = H1 ⊕ H2 , so each operator B : H → H may be identified with a 2 × 2 operator matrix (B ij ), where B ij : Hj → Hi , i, j = 1, 2. If we consider the 1 , e2 , k ∈ Z} (or {e1 2 matrix representation of B in the basis {e2k 2k 2k+1 , e2k+1 , k ∈ Z}) ij then this matrix itself combines the matrix representations of B . Of course, a similar remark holds for operators acting in H(n). Further we always work with one of the bases (2.2) (respectively, using the first basis in the case of periodic boundary conditions, and the second one in the case of antiperiodic boundary conditions). However, we don’t specify below which basis is used because the formulas for the matrix representations in these bases are formally the same (with running indices even in the first case and odd in the second case). Let P (x) = p(m)eimπx and Q(x) = q(m)eimπx , (2.15) m∈Z

m∈Z

where p(m) = q(m) = 0 for odd m, be the Fourier expansions of the functions P and Q. It is easy to see that the operator V has the following matrix representation 0 V 12 12 21 = p(−k − m), Vkm = q(k + m). (2.16) V = , Vkm V 21 0 The diagonal operator Q0n (nπ + z − L0 )Q0n : H(n) → H(n) is invertible in H(n) for any z with |z| ≤ π/2. Let Dn denote its inverse operator; then the matrix representation of Dn is 11 δkm Dn 0 11 22 , D . (2.17) = D = Dn = 22 n n 0 Dn km km π(n − k) + z The operator A defined in (2.10) can be written as A = Q0n (nπ + z − L0 )Q0n (1 − Tn Q0n ),

(2.18)

Tn = Dn Q0n V : H → H(n).

(2.19)

where

Thus A = A(n, z) is invertible if and only if 1 − Tn Q0n is invertible in H(n). By (2.16) and (2.17) one can easily see that the operator (2.19) has a matrix representation 0 Tn12 , (2.20) Tn = Tn21 0 where (Tn12 )km =

p(−k − m) , π(n − k) + z

(Tn21 )km =

q(k + m) , π(n − k) + z

k, m ∈ Z, k = n. (2.21)

1D Dirac Operators

145

We need also to know the matrix representation of its square Tn2 . From (2.20) and (2.21) it follows that 12 21 T T 0 Tn2 = n n , (2.22) 0 Tn21 Tn12 where (Tn12 Tn21 )km =

p(−k − j )q(j + m) , [π(n − k) + z][π(n − j ) + z]

j =n

(Tn21 Tn12 )km =

(k, m ∈ Z, k = n) q(k + j )p(−j − m) . [π(n − k) + z][π(n − j ) + z]

j =n

(2.23)

Lemma 2. The norm of the operator Tn2 : H → H(n) tends to 0 as |n| → ∞. More precisely, if |z| < π/2, then 1/2 1/2   P Q Tn2 ≤ C √ |q(k)|2  + CQ  |p(k)|2  , + CP  |n| |k|≥|n| |k|≥|n| (2.24) where C is an absolute constant. Proof. The norm of Tn2 does not exceed its Hilbert-Schmidt norm, so, by (2.22), it is less than the sum of the Hilbert-Schmidt norms of the operators Tn12 Tn21 and Tn21 Tn12 . We estimate in detail only the Hilbert-Schmidt norm Tn12 Tn21 H S because 21 Tn Tn12 H S could be estimated in the same way. One can easily see that 1 1 ≤ |π(n − k) + z| |n − k|

for

|z| < π/2, k = n,

(2.25)

so by (2.23) we have Tn12 Tn21 2H S

 2 |p(−k − j )||q(j + m)|   ≤ 1 + 2 + 3 , ≤ |n − k||n − j | m k =n

j =n

where

1 =



 ... ,

m |k−n|≥ |n| 2



2

2 =

j =n

3 =

  k =n m

  

m |k−n|< |n| 2

|j −n|< |n| 2

2  ... .

|j −n|≥ |n| 2

2  ...

146


Now we estimate each of these sums separately. By the Cauchy inequality we obtain    1  

1 ≤ |p(−k − j )|2 |q(j + m)|2  2 (n − j )2 (n − k) |n| m |k−n|≥

≤

π2 3

j =n

2

|k−n|≥|n|/2

j =n

1 C1 |p(−k − j )|2 |q(j + m)|2 ≤ P 2 Q2 . 2 (n − k) n m j =m

The sum 2 can be estimated in an analogous way, so

2 ≤

C1 P 2 Q2 . n

Finally, the sum 3 does not exceed     |p(−k − j )|2 |q(j + m)|2  1    2 2 (n − k) (n − j ) |n| m |n| |k−n|
|n|

1/2  |p(i)|2 



|i|>|n|

1/2 |q(i)|2 

→0

as |n| → ∞.

|i|>|n|

(2.53) The Cauchy inequality yields 

2 ≤ P Q 

|n−k|>|n|/2

1/2 1  (n − k)2

= O(1/ |n|).

(2.54)

On the other hand, (2.16), (2.20) and (2.21) imply that

V Tn Dn Tn en1 , en1 = 0,

V Dn Tn en1 , en2 = 0.

(2.55)

Next we estimate V Tn Dn Tn en1 , en2 . Set Un = Tn Dn Tn ; then, by (2.16) and (2.20)– (2.23) the absolute value of each term in the matrix representation of Un does not exceed the absolute value of the corresponding term in the matrix representation of (Tn )2 , and therefore, by the proof of Lemma 2, Un = Tn Dn Tn → 0

as |n| → ∞.

(2.56)

Of course, (2.56) implies that

V Tn Dn Tn en1 , en2 → 0

as |n| → ∞.

(2.57)

Now, in view of (2.48) and (2.49), the formulae (2.50)–(2.57) show that (2.43) holds. Theorem 5. Let L be a self-adjoint Dirac operator given by (1.1), and let (γn ) be the sequence of its spectral gaps. Then there exist N2 > 0 and a sequence of positive numbers (εn ), εn → 0, such that 2|βn (z)|(1 − εn ) ≤ γn ≤ 2|βn (z)|(1 + εn ),

|n| ≥ N2 ,

(2.58)

where z = zn , |zn | ≤ π/2.

(2.59)

2 ± Proof. By Lemma 1, if |n| ≥ N0 , then there are exactly two eigenvalues λ± n = n + zn ± of L (periodic for even n and antiperiodic for odd n) such that |zn | < π/2. Moreover, we know (see (2.26) and (2.27)) that there exists N1 > N0 such that, for |n| ≥ N1 , zn− and zn+ are roots of the quasi-quadratic equation (2.13). Since the operator L is self-adjoint, zn− and zn+ are real numbers, zn− ≤ zn+ , and

γn = zn+ − zn− .

(2.60)

By (2.40) and (2.41) in Lemma 3, the quasi-quadratic equation (2.13) becomes the equation (z − αn (z))2 − |βn (z)|2 = 0,

(2.61)

1D Dirac Operators

151

which splits into two equations z − αn (z) − |βn (z)| = 0,

(2.62)

z − αn (z) + |βn (z)| = 0.

(2.63)

δn = sup |α (n, z)| + sup |β (n, z)|.

(2.64)

Set |z|≤π/2

|z|≤π/2

By Lemma 4, δn → 0 as |n| → ∞. Choose N2 > N1 so that for |n| ≥ N2 .

δn < 1/8

(2.65)

+ Fix an n such that |n| ≥ N2 . If γn = 0, then λ− n = λn is a double eigenvalue of L, so (2.14) and (2.42) yield (2.58). If zn− < zn+ , set

ζn+ = zn+ − αn (zn+ ),

ζn− = zn− − αn (zn− ).

(2.66)

|ζn− | = |βn (zn− )|.

(2.67)

Then, by (2.62) and (2.63), |ζn+ | = |βn (zn+ )|, By (2.66), ζn+

− ζn+

=

zn+

zn−

1 − αn (z) dz.

Thus, in view of Lemma 4, (zn+ − zn− )(1 − δn ) ≤ |ζn+ − ζn− | ≤ (zn+ − zn− )(1 + δn ),

(2.68)

which yields (since δn < 1/8 by (2.65)) the inequalities |ζn+ − ζn− | (1 − δn ) ≤ zn+ − zn− ≤ |ζn+ − ζn− | (1 + 2δn ) ≤ 2|ζn+ − ζn− |.

(2.69)

Since zn+ and zn− are roots of (2.61), each of these numbers is a root of either (2.62) or (2.63). Hypothetically, there are two cases: (i) zn+ and zn− are roots of different equations; (ii) zn+ and zn− are roots of one and the same equation. In Case (i) we have, by (2.62), (2.63) and (2.67), that |ζn+ − ζn− | = |βn (zn+ )| + |βn (zn− )| = |ζn+ | + |ζn− |. On the other hand, since βn (zn+ ) − βn (zn− ) =

zn+ zn−

βn (t)dt, (2.64) and (2.66) imply that

|βn (zn+ ) − βn (zn− )| ≤ (zn+ − zn− )δn ≤ |ζn+ − ζn− | · 2δn . Thus, (2.67) and (2.70) yield + |ζ | − |ζ − | = |βn (z+ )| − |βn (z− )| ≤ |ζ + | + |ζ − | · 2δn n

n

n

n

(2.70)

n

n

(2.71)

152


so, since 2|ζn+ | = |ζn+ | + |ζn− | + |ζn+ | − |ζn− | , + |ζn | + |ζn− | (1 − 2δn ) ≤ 2|ζn+ | ≤ |ζn+ | + |ζn− | (1 + 2δn ) , and therefore, since δn < 1/8, 2|ζn+ | (1 − 2δn ) ≤ |ζn+ | + |ζn− | ≤ 2|ζn+ | (1 + 4δn ) .

(2.72)

Finally, using again that δn < 1/8, we obtain by (2.69), (2.70) and (2.72) that (2.58) holds with z = zn+ and εn = 8δn . Case (ii), where zn+ and zn− are simultaneously roots of one of Eqs. (2.62) and (2.63), is impossible. Indeed, by (2.71), we would have, since δn < 1/8, 1 |ζn+ − ζn− | = |βn (zn+ )| − |βn (zn− )| ≤ |ζn+ − ζn− | · 2δn ≤ |ζn+ − ζn− |, 4 which implies ζn+ = ζn− . But then (2.69) yields zn+ = zn− , which contradicts our assump tion that zn+ = zn− . 3. Weights and Carleman sequences 1. A sequence of positive numbers (n), n ∈ Z, is called a weight, or a weight sequence, if (−n) = (n),

(n) ∞

as

n ∞, n ≥ n0 > 0.

(3.1)

Each weight generates the weighted 2 -space, 2 (, Z) = {x = (xn )n∈Z : x2 =

|xn |2 ((n))2 < ∞}.

n∈Z

We say that two weights 1 and 2 are equivalent if ∃C > 0 :

C −1 1 (n) ≤ 2 (n) ≤ C1 (n),

n ∈ Z.

(3.2)

Obviously equivalent weights yield equivalent norms, so they generate one and the same weighted 2 -space. A weight is called submultiplicative if ∃C > 0 :

(n + m) ≤ C(n)(m),

n, m ∈ Z.

(3.3)

Of course, if 1 and 2 are equivalent weights, then whenever one of them is submultiplicative, the other one is submultiplicative also. Obviously, if satisfies (3.3), then ˜ = C satisfies (3.3) with C = 1. Therefore, we may assume that (3.3) holds with C = 1 by passing to an equivalent weight. Moreover, it is easy to see that if (3.3) holds for |n|, |m| ≥ n0 , then it holds for all n, m ∈ Z, maybe with another constant C. A weight is said to be slowly increasing if sup (2n)/ (n) < ∞.

(3.4)

n

It is easy to see that (3.4) implies ∃ m > 0, C > 0 :

(n) ≤ C|n|m ,

for |n| ≥ 1.

(3.5)

1D Dirac Operators

153

Indeed, if M = supn≥1 (2n)/ (n), then (3.4) implies that (2k ) ≤ (1)M k = (1)(2k )m ,

m = log2 (M).

Now (3.5) follows (since is monotone increasing for n ≥ n0 ) : if n0 ≤ 2k ≤ n < 2k+1 then (n) ≤ (2k+1 ) ≤ M(2k ) ≤ M(1)(2k )m ≤ M(1)nm . Further we consider weights of the form (n) = exp(h(|n|)),

|n| ≥ n0 > 0,

(3.6)

or (n) = exp(ϕ(log |n|)),

|n| ≥ n0 > 0,

(3.7)

and characterize some properties of in terms of the functions h and ϕ. Remark. Observe that in (3.6) or (3.7) we don’t care to define for all n because our main object is the corresponding weighted 2 -space. Therefore, weights are important only “up to equivalence” and the values of (n) for |n| < n0 may be chosen in an arbitrary way since the corresponding 2 -spaces will coincide. Of course, with the formulae ϕ(t) = h(et ),

h(n) = ϕ(log(n)),

one can easily pass from representation (3.6) to (3.7), and back. It is more convenient to give concrete weights in the form (3.6). For example, m (n) = |n|m ,

m > 0,

(3.8)

are known as the Sobolev weights, and a,b (n) = exp(a|n|b ),

a > 0, b ∈ (0, 1),

(3.9)

are the Gevrey weights. Lemma 6. A weight of the form (3.6) is submultiplicative if h is an increasing concave function. Proof. Indeed, one can easily see that if h : [n0 , ∞) → R is an increasing concave function, then there exists an increasing concave function h1 : [0, ∞) → [0, ∞) such that h1 (n) = h(n) + C for n ≥ n0 , n ∈ N. Then the weight is equivalent to the weight 1 (·) = exp(h1 (·)), so it is enough to show that 1 is submultiplicative. On the other hand, since h1 is concave we have for m, n > 0, h1 (0) + h1 (m + n) ≤ h1 (m) + h1 (n), which implies (in view of (3.3) and (3.6)) that the weight 1 is submultiplicative.

2. The next lemma brings our attention to a class of rapidly increasing submultiplicative weights of the form (3.7). In particular, this class contains the Gevrey weights (3.9).

154


Lemma 7. Suppose ϕ : [0, ∞) → [0, ∞), ϕ(0) = 0, is a twice differentiable function such that the following conditions hold: ϕ (t) ∞ as t ∞;

(3.10)

et /ϕ (t) ∞ as t ∞.

(3.11)

(a) Let ψ(s) be the Young dual function of ϕ, i.e. ψ(s) = sup[st − ϕ(t)],

s ≥ 0.

(3.12)

t≥0

Then ek :=

1 exp(ψ (k)) ∞ as k ∞. k

(3.13)

ϕ (t) − ϕ (t) > 1, log ϕ (t)

(3.14)

(b) In addition, if lim inf t→∞

then ∃p ∈ N, τ > 1 :

k

τ

ek epk

k ≤ 1 for k ≥ k0 .

(3.15)

Proof. (a) Since (st −ϕ(t))t = s −ϕ (t) one can easily see, by (3.10), that the expression st − ϕ(t) attains its maximum at the point t (s) = (ϕ )−1 (s),

(3.16)

ψ(s) = st (s) − ϕ(t (s)).

(3.17)

thus

The function s → t (s) is increasing because ϕ is increasing. From the identity ϕ (t (s)) = s and (3.17) it follows that ψ (s) = t (s) + st (s) − ϕ (t (s))t (s) = t (s),

(3.18)

ψ (s) = t (s) = 1/ϕ (t (s)).

(3.19)

Therefore, (3.11) implies that

ek = eψ (k) /k = et (k) /ϕ (t (k)) ∞. (b) One can easily see that (3.15) is equivalent to epk k ∃p ∈ N : lim inf > 1. log k log k ek

(3.20)

1D Dirac Operators

155

By (3.13), we have log ek = ψ (k) − log k, and therefore, (3.16) and (3.19) imply that epk log = [ψ (pk) − log(pk)] − [ψ (k) − log k] ek p p 1 1 1 =k ψ (uk) − du = k − du. uk ϕ [t (uk)] ϕ [t (uk)] 1 1 For large enough k it follows from (3.14) and (3.16) that uk = ϕ [t (uk)] > ϕ [t (uk)], so 1 1 ϕ [t (uk)] − ϕ [t (uk)] − > . ϕ [t (uk)] ϕ [t (uk)] u2 k 2 Thus (again, by (3.14)) we obtain that p epk k ϕ [t (uk)] − ϕ [t (uk)] 1 > log · 2 du log k ek log ϕ [t (uk)] u 1

(3.21)

for large enough k. Let > 1 be the liminf in (3.14). Choose p ∈ N so that +1 2 (1 − 1/p) > 1. Since ( + 1)/2 < there exists k0 such that for k ≥ k0 , (3.21) holds and there the integral is greater than p +1 1 +1 · 2 du = (1 − 1/p) > 1. 2 u 2 1 This completes the proof of the lemma.

Remark. Obviously, if ϕ satisfies the condition ϕ (t) − ϕ (t) =∞ t→∞ log ϕ (t) lim

(3.22)

then (3.14) holds. One can easily see that the Gevrey weights (3.9) satisfy (3.22). Now we present a family of weights that satisfy (3.14) but don’t satisfy (3.22). Consider the weights (3.7) generated by t ϕ(t) = eω(u) du, 0

where ω(u) = βu − (1 − β) cos u + αue−βu ,

α > 1, β ∈ (0, 1).

Then ϕ (t) = eω(t) ,

ϕ (t) = eω(t) β + (1 − β) sin t + α(1 − βt)e−βt ,

so ϕ (t) − ϕ (t) eω(t) = (1 − β)(1 − sin t) + α(βt − 1)e−βt log ϕ (t) ω(t)

(3.23)

156


which is greater than α

eω(t) βt − 1 (βt − 1)e−βt = α exp[(β − 1) cos t + αte−βt ]. ω(t) ω(t)

(3.24)

Let (tk ) be a sequence of positive numbers such that tk → ∞. Observe that if lim inf k (1− sin(tk )) > 0 then the expression (3.23) with t = tk goes to ∞ as k → ∞, while whenever limk (1 − sin(tk )) = 0 the expression (3.24) with t = tk tends to α. On the other hand for t = tk = (4k + 1)π/2, k = 1, 2, . . . the expressions (3.23) and (3.24) coincide. By these observations it is easy to see that lim inf [ϕ (t) − ϕ (t)]/ log ϕ (t) = α. t→∞

Since α > 1, the inequality (3.14) holds, while (3.22) fails. 3. We say that a sequence of positive numbers (Mk )∞ k=0 is a Carleman sequence if M0 = 1,

Mk / (kMk−1 ) ∞.

(3.25)

We attach to any Carleman sequence (Mk ) the following sequences: m0 = 1,

mk = Mk /Mk−1 ,

e0 = 1,

ek = mk /k, k ≥ 1.

(3.26)

We set also E0 := e0 ,

Ek = e1 . . . ek = Mk /k!,

k ≥ 1.

(3.27)

Observe that if a sequence (ek )∞ k=0 satisfies the condition ek ∞, then it generates a corresponding Carleman sequence Mk = k!Ek with Ek defined by (3.27). Suppose ϕ ∈ (3.7) is a weight that grows faster than any power of n. For a technical reason we need to characterize the relation x = (xn ) ∈ 2 () by the sequence of 1 -norms xk = x0 + |xn ||n|k , k = 1, 2, . . . . (3.28) It turns out that this can be done in terms of an appropriate Carleman sequence generated by the function ϕ. For every function ϕ such that (3.10), (3.11) and (3.14) hold we denote by (Mk (ϕ)) the Carleman sequence generated by the formula mk (ϕ) = exp(ψ (k)),

k = 1, 2, . . . ,

that is Mk (ϕ) = exp(ψ (1) + · · · + ψ (k)),

k = 1, 2, . . . ,

where ψ is the Young dual function of ϕ. We may assume, without loss of generality, that the function ϕ is defined on [0, ∞), and moreover, that the condition ϕ (t) − ϕ (t) > 0 holds for t ≥ 0 (since otherwise one can consider an equivalent weight generated by a suitable function ϕ). ˜ Moreover, the condition (3.11) implies that the weight ϕ is submultiplicative. Indeed, since ϕ (n) = exp[ϕ(log |n|)] = eψ(|n|)

with

ψ(s) = ϕ(log s),

1D Dirac Operators

157

we obtain, in view of (3.11), that the derivative ϕ(log s) ϕ(log s) = s elog s is decreasing, so ψ(s) is a concave function. Thus, by Lemma 6, the weight ϕ is submultiplicative. ψ (s) =

Lemma 8. If ϕ satisfies (3.10), (3.11), (3.14) and ϕ (|n|) = exp(ϕ(log |n|)) then we have (a)

x = (xn ) ∈ 2 (ϕ ) ⇒ xk ≤ CMk (ϕ1 ),

ϕ1 (t) = ϕ(t) − t;

(3.29)

(b)

xk ≤ CMk (ϕ) ⇒ x = (xn ) ∈ 2 (ϕ2 ),

ϕ2 (t) = ϕ(t) − 4t.

(3.30)

Proof. (a) Observe that (with ψ(0) = 0 ) ψ(k) ≤ ψ (1) + · · · + ψ (k) ≤ ψ(k + 1). Therefore sup n

|n|k = exp sup k log |n| − ϕ(log |n|) ϕ (n) n ≤ exp(ψ(k)) ≤ Mk (ϕ) = exp(ψ (1) + · · · + ψ (k)).

If x = (xn ) ∈ 2 (ϕ ) then, with = ϕ , by the Cauchy inequality we obtain xk = |xn ||n|k = |xn |ϕ (n) |n|k / ϕ (n) 1/2 1 |n|k+1 2 |n|k ≤ x2 (ϕ ) · ≤ C sup ≤ CMk (ϕ1 ), n2 ϕ (n) ϕ (n)/n n where ϕ1 (t) = ϕ(t) − t. (b) Suppose that xk ≤ CMk (ϕ); then for large |n| we have xk |n|k ≥ |xn | sup Mk k k Mk ≥ |xn | exp sup k log |n| − (ψ (1) + · · · + ψ (k)) k ≥ |xn | exp sup k log |n| − ψ(k + 1) k −2 ≥ |xn ||n| exp sup s log |n| − ψ(s) = |xn ||n|−2 exp(ϕ(log |n|)),

C ≥ sup

s>0

that is |xn

||n|−2

ϕ (n)

≤ C. Therefore 1 ϕ (n) |xn | 4 ≤ C < ∞, n n2 n n

which implies that x = (xn ) ∈ 1 (ϕ2 ) ⊂ 2 (ϕ2 ) with ϕ2 (t) = ϕ(t) − 4t.

158


Lemma 9. Suppose (ek )∞ k=1 is a sequence of positive numbers such that ek ↑ ∞ and let E0 = 1,

Ek =

k

k ≥ 1.

ej ,

1

Then the following implications hold: sup k τ (ek /epk )k < ∞

∃ p ∈ N, τ > 0 :

⇒

sup k τ (Ek )2 /E2k < ∞;

k ∞

(ek /epk )k < ∞

⇒

k=1 ∞

(3.31)

k ∞

(Ek )2 /E2k < ∞;

(3.32)

k=1

(Ek )2 /E2k < ∞

⇒

m Ej Em−j

Q := sup m

k=1

j =0

Em

< ∞,

(3.33)

and moreover, sup

m s +···+s µ 0

Es0 . . . Esµ < Qµ , E m =m

µ = 1, 2, . . . .

(3.34)

Remark. This lemma is a “multidimensional” version of the statements on p. 164 in [3]. It improves Lemma 5 on p. 251 in [4], where we can now omit the factor k p−2 in the hypothesis (5.7). Proof. If k = pν + r with 0 ≤ r < p, then we have (Ek )2 e1 . . . eν eν+1 . . . ek e1 . . . eν = · ≤ ≤ E2k ek+1 . . . ek+ν ek+ν+1 . . . e2k ek+1 . . . ek+ν

eν epν

ν

(because ei < ej for i < j ). Thus (3.31) and (3.32) hold. To prove (3.33) and (3.34) let us consider the sums Tm =

m Ej Em−j j =0

Em

.

Then Ej Em−j Ej Em+1−j ≤ , Em+1 Em

0 ≤ j ≤ m,

because (3.35) is equivalent to em+1−j ≤ em+1 , which holds since the sequence (ek ) is increasing. By symmetry Tm = 2

0≤j ≤m/2

Ej Em−j − δm , Em

(3.35)

1D Dirac Operators

159

where

δm =

0 , m = 2n + 1 En2 /E2n , m = 2n.

The next sum is

Tm+1 = 2

0≤j ≤m/2

Ej Em+1−j + δm+1 , Em+1

and (3.35) implies that Tm+1 ≤ Tm + δm + δm+1 . Therefore, we have Tm ≤ 2 +

∞

(δk + δk+1 ) = 2 + 2

∞

En2 /E2n < ∞,

n=1

k=1

thus (3.33) holds. Now we prove (3.34) by induction in µ. Let us denote by Sµ (m) the set of all (µ+1)tuples of integers s = (s0 , . . . , sµ ) such that 0 ≤ si ≤ m and |s| = s0 + · · · + sµ = m, i.e., Sµ (m) = {s = (s0 , . . . , sµ ) :

0 ≤ si ≤ m, |s| = m}.

(3.36)

By (3.33), the inequality (3.34) holds for µ = 1. Assume that (3.34) holds for some µ ≥ 1. Then we have   m Es0 . . . Esµ+1 Es0 . . . Esµ Esµ+1 Em−sµ+1   = Em Em−sµ+1 Em sµ+1 =0

s∈Sµ+1 (m)

≤ Qµ

s∈Sµ (m−sµ+1 )

m Esµ+1 Em−sµ+1

Em

sµ+1 =0

This proves (3.34).

≤ Qµ+1 .

4. The next statement (Lemma 10) has as its prototypes Lemma 6 in [3] and Theorem 3 in [4] (see also the proof of Prop. 4 there). But, influenced by the proof of Lemma 1.1 in [2], now we use “maxima” instead of “sums” in the statement, which makes the lemma more convenient for applications. The proof of Lemma 10 uses the same idea that was used to prove its prototypes, but it is simpler. Lemma 10. Let (fk )∞ k=1 be a sequence of positive numbers such that fk ∞,

(3.37)

and let F0 = 1,

Fk =

k j =1

fj ,

k = 1, 2, . . . .

(3.38)

160


If T > 0 and (Xk )∞ k=0 is a sequence of positive numbers such that X0 = 1 and Fs0 . . . Fsµ Xk ≤ max T , sup max (3.39) Xs0 . . . Xsµ , k ≥ 2, Fk µ si T k+1 /Fk+1 }.

(3.41)

It is easy to see, by (3.37) and (3.38), that T k /Fk → 0 as k → ∞; thus, k0 is well defined, and moreover, in view of (3.40) and (3.41), we have T ≤

Tk Fk

2 ≤ k ≤ k0

for

and T >

Tk Fk

for

k ≥ k0 + 1,

(3.42)

and for k > k0 .

fk > T

(3.43)

Step 2. Claim. The following inequalities hold: Xk ≤ T k /Fk ,

k = 0, 1, . . . , k0 .

(3.44)

We prove (3.44) by induction. In view of (3.38) and (3.40) our claim holds for k = 0, 1. Let Ps =

Fs0 . . . Fsµ Xs0 . . . Xsµ , F|s|

s = (s0 , . . . , sµ ).

(3.45)

Assume that (3.44) holds for k = 1, . . . , m for some m with 1 ≤ m < k0 . Then, for each µ and for each (µ + 1)-tuple s = (s0 , . . . , sµ ) ∈ Sµ (m + 1), we have by (3.44), Ps ≤

Fs0 . . . Fsµ T s0 T sµ T m+1 · ··· = . Fm+1 Fs0 Fsµ Fm+1

By (3.42) T m+1 /Fm+1 ≥ T ; thus, (3.39) implies that Xm+1 ≤ T m+1 /Fm+1 , i.e., (3.44) holds for k = m + 1. The claim is proven. Step 3. Here we show that Xk ≤ T

for k ≥ k0 + 1.

(3.46)

For technical convenience we prove also that Ps < T

for s = (s0 , . . . , sµ ) with

sj < |s| = k,

k ≥ k0 + 1.

(3.47)

1D Dirac Operators

161

Observe that in view of (3.39), if the inequalities (3.47) hold for some k, then (3.46) holds for the same k also. We are proving (3.46) and (3.47) by induction for k ≥ k0 + 1. Let k = k0 + 1. For each (µ + 1)-tuple s = (s0 , . . . , sµ ) ∈ Sµ (k0 + 1), with sj < k0 + 1, we obtain, by (3.44) and (3.42), that Ps ≤

Fs0 . . . Fsµ T s0 T sµ T k0 +1 · ... = < T. Fk+1 Fs0 Fsµ Fk0 +1

Thus, (3.47), and of course (3.46), hold for k = k0 + 1. Let m ≥ k0 + 1; assume that (3.46) and (3.47) hold for every k, k0 + 1 ≤ k ≤ m. Then, we claim that (3.46) and (3.47) hold for k = m + 1. Indeed, fix any (µ + 1)-tuple s = (s0 , . . . , sµ ), with |s| = m + 1 and sj < m + 1. There are several cases: (a) If sj ≤ k0 for every j = 0, . . . , µ, then the numbers Xsj satisfy the estimates (3.44). Thus, one can easily see (as in the proof for k = k0 + 1 ) that Ps ≤ T m+1 /Fm+1 < T , so in this case (3.47) holds. (b) Suppose that there exists j with sj > k0 , say j = 0. (Since a transposition of s0 , . . . , sµ does not change Ps , one may assume without loss of generality that j = 0.) Then we have two subcases: (b1) where m+1−s0 ≤ k0 , and (b2) where m+1−s0 > k0 . In the subcase (b1) we estimate Ps by using (3.46) for Xs0 and (3.44) for Xs1 , . . . , Xsµ . Since T < fk for k > k0 by (3.43), we obtain that Ps ≤

Fs0 Fs1 . . . Fsµ T s1 T sµ T m+1−s0 ·T · ··· =T · < T; Fm+1 Fs1 Fsµ fs0 +1 . . . fm+1

thus, (3.47) holds for k = m + 1. In the case (b2) we have Xs Fs Fm+1−s0 Ps = 0 0 Fm+1

Fs1 . . . Fsµ Xs1 . . . Xsµ . Fm+1−s0

The expression in the brackets equals Ps˜ with s˜ = (s1 , . . . , sµ ), |˜s | = m + 1 − s0 . Thus, by the inductive assumption Ps˜ < T . Since Xs0 < T (by (3.46) with k = s0 ) we have T Fs0 Fm+1−s0 Ps < · T, Fm+1 so it remains to show that the expression in the square brackets does not exceed 1. By (3.42) Fk0 ≤ T k0 −1 , and therefore, Fs0 = Fk0 fk0 +1 . . . fs0 ≤ T k0 −1 fk0 +1 . . . fs0 . Thus, T Fs0 Fm+1−s0 T k0 fk0 +1 . . . fs0 ≤ < 1, Fm+1 fm+2−s0 . . . fm+1 because, due to (3.37) and (3.43), each factor in the numerator of the latter fraction is strictly less than the corresponding factor in the denominator.

162


4. Basic Results; Estimates on the Smoothness of the Potential in Terms of the Decay Rate of Spectral Gaps 1. Our main result is the following statement. Theorem 11. Let L be a self-adjoint Dirac operator given by (1.1), with a potential function P ∈ L2 ([0, 1])), P (x) = p(2n)e2πinx . Let be a submultiplicative weight (see (3.1) and (3.3)) such that either is slowly increasing (i.e., (3.4) holds), or is a rapidly increasing weight of the form (n) = exp(ϕ(log |n|)), where ϕ has the properties (3.10), (3.11) and (3.14). Then − 2 2 |λ+ ⇒ |p(2n)(2n)|2 < ∞. (4.1) n − λn | ((2n)) < ∞ n∈Z

n∈Z

An implication in the opposite direction is given by Theorem 12 below; see further comments in Sect. 5.1. Theorem 12. Let L be a self-adjoint given by (1.1), with a potential Dirac operator function P ∈ L2 ([0, 1])), P (x) = p(2n)e2πinx . If is a submultiplicative weight, then − 2 2 |p(2n)|2 ((2n))2 < ∞ ⇒ |λ+ (4.2) n − λn | ((2n)) < ∞. Proof. By Theorem 5, for large enough |n|, − γn = λ+ n − λn 2|βn (zn )| with

|zn | ≤ π/2,

(4.3)

where, in view of (2.42) and (2.36)–(2.38), βn (n, zn ) = p(−2n) +

∞

21 S2ν (n, zn ).

(4.4)

ν=1

Therefore, by (2.25), we have |βn (zn )| ≤ |p(−2n)| +

∞ ∞ 21 σν (n, r), S2ν (n, zn ) ≤ |r(2n)| + ν=1

(4.5)

ν=1

where r = (r(m))m∈Z , and σν (n, r) =

j1 ,... ,j2ν =n

r(m) = max(|p(m)|, |p(−m)|),

(4.6)

r(n + j1 )r(−j1 − j2 )r(j2 + j3 ) . . . r(−j2ν−1 − j2ν )r(j2ν + n) . |n − j1 ||n − j2 | . . . |n − j2ν | (4.7)

Consider the operator σ :

r = (r(m)) ∈ 2 (Z) → (σ (n, r)) ∈ 2 (Z),

(4.8)

where σ (n, r) =

∞

σν (n, r).

(4.9)

ν=1

Thus, in view of (4.3)–(4.9), the following statement completes the proof of Theorem 12.

1D Dirac Operators

163

Proposition 13. If is a submultiplicative weight, then for each sequence of nonnegative numbers, |r(2n)|2 ((2n))2 < ∞ ⇒ |σ (n, r)|2 ((2n))2 < ∞. (4.10) Proposition 13 is proven in Sect. 4.2 as a corollary of some basic properties of the operator σ. Proof of Theorem 11. The proof of Theorem 11 follows from the properties of the operator σ also, but it is much more complicated. Set ζ (n) = |β(n, zn )|; then Theorem 11 will be proven if we show that |p(2n)|2 ((2n))2 < ∞. |ζ (n)|2 ((2n))2 < ∞ ⇒

(4.11)

Under the above notations we have, by (4.4), that |p(−2n)| ≤ |βn (zn )| +

∞ ∞ 21 σν (n, r). S2ν (n, zn ) ≤ |ζ (n)| + ν=1

(4.12)

ν=1

In the same way, changing n to −n one can see that ∞ ∞ 21 |p(2n)| ≤ |β−n (z−n )| + σν (n, r). S2ν (−n, z−n ) ≤ |ζ (−n)| + ν=1

(4.13)

ν=1

Thus, by (4.12) and (4.13), we obtain, with ξ(n) = max(ζ (n), ζ (−n)), r(2n) ≤ ξ(n) +

∞

σν (n, r) = ξ(n) + σ (n, r).

ν=1

Thus, in view of the above discussion, Theorem 11 would be proven if we prove the following statement. Theorem 14. Let be a submultiplicative weight (see (3.1) and (3.3)) such that either slowly increasing (i.e., (3.4) holds), or is a rapidly increasing weight of the form (n) = exp(ϕ(log n)), where ϕ has the properties (3.10), (3.11) and (3.14). If ξ = (ξ(m))m∈Z and r = (r(m))m∈Z are two sequences of non-negative numbers such that r ∈ 2 (Z),

r(m) = 0 for odd m,

r(2n) ≤ ξ(n) + σ (n, r),

|n| ≥ n∗ ,

(4.14) (4.15)

then n∈Z

|ξ(n)(2n)|2 < ∞ ⇒

n∈Z

|r(2n)(2n)|2 < ∞.

(4.16)

164


The remaining part of this section is devoted to the proof of Theorem 14. Some of the steps of this proof are interesting by themselves (e.g., Lemma 15 and Proposition 16 give a proof of Proposition 13). Therefore, the claims that follow below are formulated and proven as independent statements, although they are steps in the proof of Theorem 14. 2. Throughout the paper we assume that the weights are submultiplicative. The following property of the operator σ (n, r) reveals why this assumption is so important. Lemma 15. If is a submultiplicative weight such that (k + m) ≤ (k)(m) ∀k, m ∈ Z, (i.e., (3.3) holds with C = 1) then, for each sequence of non-negative numbers r = (r(m))m∈Z , σ (n, r)(2n) ≤ σ (n, r˜ ) where r˜ = (r(m)(m))m∈Z .

(4.17)

Proof. Since the weight is submultiplicative, we have, for each 2ν-tuple (j1 , . . . , j2ν ), that (2n) ≤ (n + j1 )(−j1 − j2 )(j2 + j3 ) · · · (−j2ν+1 − j2ν )(j2ν + n), and therefore, r(n + j1 )r(−j1 − j2 ) · · · r( j2ν + n)(2n) ≤ r˜ (n + j1 )˜r (−j1 − j2 ) · · · r˜( j2ν + n). Thus, in view of (4.7), we obtain σν (n, r)(2n) ≤ σν (n, r˜ )

ν = 1, 2, . . . ,

so, by (4.9), σ (n, r)(2n) =

∞

σν (n, r)(2n) ≤

ν=1

∞

σν (n, r˜ ) = σ (n, r˜ ).

ν=1

Next, we use the properties of the operator σ to prove the following crucial estimate. Proposition 16. Under the above notations |n|≥N

|σ (n, r)|2 ≤

2 + (R(N ))2 , N

N > N ∗,

(4.18)

where R(N) :=

|n|≥N

|r(n)|2 .

(4.19)

1D Dirac Operators

165

Proof. By (4.7), the sequence (σ (n, r)) is the sum of the sequences (σν (n, r)), and therefore, by the triangle inequality for 2 -norms, we have 1/2  1/2  ∞ 2 2   |σ (n, r)|  ≤ |σν (n, r)|  . (4.20) |n|≥N

To estimate in (4.7)

|n|≥N

|n|≥N

ν=1

|σν (n, r)|2 , for fixed ν ∈ N, we divide the set of summation indices j1 , . . . , j2ν = n}

J (n) = {j = (j1 , . . . , j2ν ) : into several subsets by setting

a = {α = (α1 , . . . , α2ν ) : αs ∈ {0, 1}}, and

J (n) = (j1 , . . . , j2ν ) ∈ J (n) : α

|α| = α1 + · · · + α2ν ,

|n − js | ≤ |n|/2 if αs = 0 |n − js | > |n|/2 if αs = 1 .

Notice that card (a) = 22ν . Then J (n) =

J α (n), α∈a

so

··· =

··· ,

α∈a j ∈J α (n)

J (n)

and therefore, the triangle inequality implies that  1/2    |σν (n)|2  ≤  |n|≥N

α∈a

2 1/2  · · ·  . |n|≥N j ∈J α (n)

(4.21)

By the Cauchy inequality, |n|≥N

 

2

··· ≤

Aα (n)Bα (n),

(4.22)

|n|≥N

j ∈J α (n)

where Aα (n) =

j ∈J α (n)

 ≤

|n−k|≤|n|/2

2ν−|α|  1   (n − k)2

1 (n − j1 )2 . . . (n − j2ν )2

|n−k|>|n|/2

(4.23)

|α| 2 2ν−|α| |α| 1 4  ≤ π , (n − k)2 3 N

166


and

Bα (n) =

|r(n + j1 )|2 |r(−j1 − j2 )|2 · · · |r(j2ν + n)|2 .

(4.24)

j ∈J α (n)

In order to estimate

|n|≥N

Bα (n) we change the indices of summation to

i1 = n + j1 , i2 = −j1 − j2 , . . . , i2ν = −j2ν−1 − j2ν , i2ν+1 = j2ν + n. Then

Bα (n) ≤

|n|≥N

|r(i1 )|2 · · · |r(i2ν+1 )|2 ,

(4.25)

i∈I α

where I α = I α (N ) is the set of indices i = (i1 , . . . , i2ν+1 ) given by I α = I1 (α) × · · · × I2ν+1 (α), where Is (α) =

Z {is : |is | ≥ N }

if αs = 1 if αs = 0

for

s = 1, 2ν + 1

and Is (α) =

Z {is : |is | ≥ N}

if αs−1 = 1 or αs = 1 , if αs−1 = 0 and αs = 0

2 ≤ s ≤ 2ν.

Indeed, α1 = 0 (or α2ν+1 = 0) means that |n − j1 | ≤ |n|/2 (respectively, |n − j2ν | ≤ |n|/2). Thus, |i1 | = |n + j1 | = |2n − (n − j1 )| ≥ |2n| − |n|/2 > |n| ≥ N, and the same argument shows that |i2ν+1 | ≥ N. Fix an s such that 2 ≤ s ≤ 2ν. If αs−1 = αs = 0 then |n − js−1 | ≤ |n|/2,

|n − js | ≤ |n|/2;

thus, |is | = |js−1 + js | = |2n − (n − js−1 ) − (n − js )| ≥ |2n| − 2(|n|/2) ≥ |n| ≥ N. Now we have |n|≥N

Bα (n) ≤

2ν+1

|r(is )|2 ≤ (R(N ))γ (α) (r2 )2ν+1−γ (α) ,

(4.26)

s=1 is ∈Is (α)

where γ (α) := card{s : Is (α) = Z} ≥ 2ν + 1 − 2|α|.

(4.27)

1D Dirac Operators

167

Indeed, one can easily see, by the definition of Is (α), that γ (α) = (1 − α1 ) + (1 − α2ν ) +

2ν

(1 − αs )(1 − αs−1 )

s=2 2ν

≥ (1 − α1 ) + (1 − α2ν ) +

(1 − αs − αs−1 ) = 2ν + 1 − 2|α|.

s=2

Taking into account (4.22), (4.23), (4.26) and (4.27), we obtain  2 2 2ν 2ν+1−γ (α) 2 2|α| π   ··· ≤ (4.28) √ (R(N ))γ (α) r2 3 N α |n|≥N

j ∈J (n)

≤ K 2ν+1 (ρ(N ))2|α|+γ (α)) ≤ K 2ν+1 (ρ(N ))2ν+1 , where 2 ρ(N ) = √ + R(N), N

K=

π2 (r2 + 1). 3

Obviously ρ(N ) → 0 as N → ∞, so there is N ∗ such that −1 ρ(N ) < 256K 3 for N ≥ N ∗ .

(4.29)

Since card(a) = 22ν , the inequalities (4.21), (4.28) and (4.29) imply, for N ≥ N ∗ , that  1/2 ∞ ∞  |σν (n, r)|2  ≤ 4ν (Kρ(N ))ν+1/2 ν=1

|n|≥N

ν=1

≤ 4(Kρ(N ))3/2

∞

2−ν ≤ 8(Kρ(N ))3/2 ≤

ν=0

1 ρ(N ). 2

Thus, by (4.20),

|σ (n.r)|2 ≤

|n|≥N

which completes the proof.

1 2 (ρ(N ))2 ≤ + (R(N ))2 , 4 N

Proof of Proposition 13. Suppose that is a submultiplicative weight (we may assume that (3.3) holds with C = 1) and r = (r(n))n∈Z is a sequence of non-negative numbers such that r(m) = 0 for odd m and (4.30) (r(2n)(2n))2 < ∞. Lemma 15 implies that σ (n, r)(2n) ≤ σ (n, r˜ ),

where

r˜ = (r(m)(m)) .

(4.31)

168


Therefore, in view of (4.30), we have r˜ ∈ 2 (Z); thus, by Proposition 16, there exists N∗ > 0 such that 2  2 + |˜r (n)|2  < ∞. (σ (n, r˜ ))2 ≤ N |n|≥N∗

|n|≥N∗

Thus, by (4.31), we obtain that (σ (n, r)(2n))2 ≤ (σ (n, r˜ ))2 < ∞, which proves Proposition 13. 3. Two elementary lemmas. ∞ Lemma 17. If (B(n))∞ 1 and (R(n))1 are decreasing sequences of positive real numbers such that

B(n) 0,

R(n) 0,

R(2n) ≤ C1 B(n) + C1 (R(n))2 ,

(4.32)

C1 > 0, n = 1, 2, . . . ,

(4.33)

and B(n) ≤ C2 B(2n),

C2 > 0, n = 1, 2, . . . ,

(4.34)

then there exists a constant C > 0 such that R(2n) ≤ CB(n),

n = 1, 2, . . . .

(4.35)

Proof. By (4.32) there exists n1 such that R(n)
0 :

(m) ≤ |m|a

for |m| > 1.

(4.43)

For convenience the proof is divided into two steps. Step 1. Proof of the claim in the case where a < 1/4. By (4.41), |r(2n)|2 ≤ 2 |ξ(n)|2 + 2 |σ (n, r)|2 , |n|≥N

|n|≥N

(4.44)

|n|≥N

and therefore, by Proposition 16, we have for N ≥ 4 (since ((N ))2 ≤ N 1/2 ≤ N/2), R(2N ) ≤ 2X(N) + where X(n) =

2 + 2(R(N ))2 , ((N ))2

|ξ(n)|2 .

|n|≥N

On the other hand, we have εN := X(N )((N ))2 ≤

|n|≥N

|ξ(n)|2 ((n))2 → 0,

(4.45)

(4.46)

1D Dirac Operators

171

and therefore, X(N) = εN /((N ))2

with

εN → 0.

(4.47)

Consider the sequence (B(N )) given by B(N) := X(N) +

1 . ((N ))2

(4.48)

Since the weight is slowly increasing, (4.47) and (4.48) imply that sup B(N )/B(2N ) < ∞,

˜ B(N ) ≤ C/((N ))2 .

(4.49)

N

By (4.45), we have R(2N ) ≤ B(N) + 2(R(N ))2 ; thus, in view of (4.49), Lemma 17 gives us that R(2N ) ≤ C1 B(N) ≤ C1

C˜ . ((N ))2

(4.50)

On the other hand, by (4.44) and Proposition 16, we obtain R(2N ) ≤ 2X(N) + 4/N + 2(R(N ))2 .

(4.51)

Notice that (4.43), with a < 1/4, implies ((N ))4 /N → 0. Thus, since is slowly increasing weight, (4.50) and (4.51) yield R(2N ) ≤ 2X(N) +

C2 . ((N ))4

(4.52)

Now (4.47) and (4.52) imply that R(2N )(N )2 → 0

as

N → ∞.

(4.53)

Moreover, (4.52) implies R(2N ) ((N ))2 − ((N − 1))2 < ∞.

(4.54)

N

Indeed, since (ξn ) ∈ 2 (), by Lemma 18 we have that X(N) ((N ))2 − ((N − 1))2 < ∞. N

On the other hand, ((N ))2 − ((N − 1))2 1 1 < ∞; ≤ − ((N ))4 ((N − 1))2 ((N ))2 N

N

thus, (4.54) holds. By (4.53), (4.54) and Lemma 18, we have i.e., (4.42) holds if a < 1/4.

|r(m)|2 ((m))2 < ∞,

172


Step 2. Proof of the claim in the case where a ≥ 1/4. If (m) ≤ |m|a with a ≥ 1/4, then we choose k0 so that a/k0 < 1/4, set k (m) = ((m))k/k0 ,

k = 1, . . . , k0 ,

(4.55)

and prove that the claim holds for k by induction in k. Since 1 (m) ≤ |m|1/4 , by Step 1, the claim holds for k = 1. Assume that r = (r(m)) ∈ 2 (k ) for some k, 1 ≤ k < k0 . Multiplying both sides of (4.41) by k (2n) and using that k is submultiplicative, we obtain r˜ (2n) ≤ ξ˜ (n) + σ (n, r˜ ),

(4.56)

where r˜ (m) = r(m)k (m),

ξ˜ (m) = ξ(m)k (2m),

m ∈ Z.

Since (r(m)) ∈ 2 (k ) and (ξm ) ∈ 2 () we have r˜ = (˜r (m)) ∈ 2 ,

(ξ˜ (m)) ∈ 2 (k0 −k ) ⊂ 2 (1 ).

By Step 1, it follows that (˜r (m)) ∈ 2 (1 ); thus, r = (r(m)) ∈ 2 (k+1 ). Hence, r = (r(m)) ∈ 2 (k ) for k = 1, . . . , k0 . By (4.55), k0 = . This proves Proposition 19. 5. Finally, we prove Theorem 14 for rapidly increasing weights of the form ϕ (|n|)) = exp(ϕ(log |n|)). Proposition 20. Suppose (Mk )∞ k=0 is a Carleman sequence (see (3.25) - (3.27)) such that Mk = k!Ek with √ k(Ek )2 /E2k → 0, (4.57)

∃τ ∈ (0, 1) :

∞

(Ek )2 /E2k

τ

→ 0.

(4.58)

k=1

If r = (r(n))n∈Z and ξ = (ξn )n∈Z are sequences of non-negative numbers such that r(n) = 0 for odd n,

r ∈ 2

and r(2n) ≤ ξ(n) + σ (n, r),

|n| ≥ n∗ ,

(4.59)

where σ is the operator defined by (4.7) - (4.9), then ˜ k ∀k. |ξ |k := |ξn ||2n|k ≤ CMk ∀k ⇒ rk = |r(2n)||2n|k ≤ CM n

n

(4.60)

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173

Proof. By Proposition 19, |ξ |k < ∞

⇒

rk < ∞,

∀k ∈ N.

Set rk = Xk Mk r0 ,

k = 0, 1, 2, . . . .

(4.61)

The lemma will be proven if we show that the sequence (Xk ) is bounded. Let us multiply both sides of (4.59) by |2n|k+1 and sum up over n, |n| ≥ N∗ > n∗ . (Later N∗ will be chosen large enough.) This leads us to the inequality |r(2n)||2n|k+1 + |ξ(n)||2n|k+1 + σ (n, r)|2n|k+1 rk+1 ≤ |n| |n|/2 if αs = 1 By the definition of J α (n) we have |n − j1 | . . . |n − j2ν | ≥ (|n|/2)|α| ≥ (N∗ /2)|α|

for

j ∈ J α (n).

With this estimate for the denominator in (4.63), we have (N∗ /2)−|α| |2n|k+1 r(n + j1 . . . r(j2ν + n). Sν ≤ α∈a

(4.64)

|n|≥N∗ J α (n)

Set a = {α ∈ a : |α| = |α1 + · · · α2ν | ≥ ν},

a = a \ a ,

(4.65)

174


and split the sum in (4.64) into two subsums: ··· = ··· + ··· . α∈a

α∈a

(4.66)

α∈a

First, we estimate α∈a . Taking into account that card (a) = 22ν we obtain · · · ≤ (8/N∗ )ν |2n|k+1 r(n + j1 ) . . . r(j2ν + n). (4.67) α∈a

n j1 ,... ,j2ν

By the multinomial formula, |2n|k+1 = |(n + j1 ) + (−j1 − j2 ) + (j2 + j3 ) + · · · + (j2ν + n)|k+1 k + 1 |n + j1 |s0 · | − j1 − j2 |s1 · · · |j2ν + n|s2ν ; ≤ s |s|=k+1

thus, (4.67) implies ν k+1 8 rs0 rs1 . . . rs2ν ··· ≤ s N∗ |s|=k+1 α∈a ν ν k+1 8 8 2ν rs0 rs1 . . . rs2ν . = (2ν + 1)r0 rk+1 + s N∗ N∗ |s| = k + 1 si < k + 1 Obviously, there exists N1 > 0 such that ∞

(8/N∗ )ν (2ν + 1)r2ν 0 < 1/2

for N∗ > N1 .

ν=1

Thus, we have ∞ ν=1 α∈aν

∞ 8 ν 1 ≤ rk+1 + 2 N∗ ν=1

|s| = k + 1 si < k + 1

k+1 rs0 rs1 . . . rs2ν s (4.68)

aν

a

to show the dependence of on ν). (where the notation is used Next, we estimate the sum α∈a . Consider the new indices i1 = n + j1 , i2 = −j1 − j2 , . . . , i2ν = −j2ν−1 − j2ν , i2ν+1 = j2ν + n.

(4.69)

Such change of the summation indices has been used in the proof of Proposition 16. It is easy to check, by the definition of J α (n), that if j ∈ J α (n) then α1 = 0 ⇒ |i1 | = |n + j1 | > |n|,

α2ν = 0 ⇒ |i2ν+1 | = |j2ν + n| > |n|

and αs−1 = αs = 0 ⇒ |is | = |js−1 + js | > |n|, (see the proof of Proposition 16 for details).

2 ≤ s ≤ 2ν

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175

Let γ (α) denote the number of expressions is in (4.69) such that |is | > n for j ∈ J α (n). Of course, γ (α) is the same function that has been used in the proof of Proposition 16; thus, by (4.27), we have γ (α) ≥ 2ν + 1 − 2|α|. In particular, since |α| ≤ ν − 1 for α ∈ a , we obtain for α ∈ a .

γ (α) ≥ 3

Choose indices s1 , s2 so that the corresponding is1 and is2 in (4.69) satisfy the inequalities |is1 | > |n|,

|is2 | > |n| for j = (j1 , . . . , j2ν ) ∈ J α (n).

Set k1 = k2 =

k+1 for odd k, 2

k1 =

k k , k2 = 1 + for even k. 2 2

(4.70)

Then |n|k+1 ≤ |is1 |k1 |is2 |k2

for j ∈ J α (n).

Thus, by changing the indices of summation according to the formulae (4.69), we have by (4.64), 2 |α| ··· ≤ 2k+1 |is1 |k1 |is2 |k2 r(i1 ) . . . r(i2ν+1 ), N ∗ |n|≥N∗ J α (n) α∈a α∈a ! 2|α| 2 ≤ 2k+1 |is1 |k1 |is2 |k2 r(i1 ) . . . r(i2ν+1 ), (4.71) N ∗ α I (N∗ )

α∈a

α where I α (N∗ ) = I1α × · · · × I2ν+1 with

Isα = {m ∈ Z : and Isα = Z otherwise. Let R(N∗ ) =

|m| > N∗ }

|r(n)|,

if |is | > n ∀j ∈ J α (n),

ρ(N∗ ) =

2/N∗ + R(N∗ ).

(4.72)

|n|>N∗

With these notations, (4.71) implies the inequality ! 2|α| 2 2ν+1−γ (α) ··· ≤ 2k+1 rk1 rk2 | (R(N∗ ))γ (α)−2 r0 N ∗ I α (N∗ ) α∈a α∈a ≤ (ρ(N∗ ))2|α|+γ (α)−2 2k+1 rk1 rk2 |K 2ν−1 , α∈a

where K = max(1, r0 ). Since card a ≤ 2ν ,

2|α| + γ (α) ≥ 2ν + 1,

176


by (4.27), we have

· · · ≤ (2Kρ(N∗ ))2ν−1 2k+1 rk1 rk2 |.

(4.73)

α∈a

From (4.72) it follows that ρ(N∗ ) → 0 as N∗ → ∞, so there exists N2 > 0 such that 2Kρ(N∗ ) < 1/2

for N∗ ≥ N2 .

Thus, (4.73) implies that ∞ ν=1

· · · ≤ 2k+1 rk1 rk2

for N∗ ≥ N2 .

(4.74)

α∈a

Now we sum up the above inequalities. From (4.62), (4.64), (4.68) and (4.74) it follows that for N∗ > max(N1 , N2 ), rk+1 ≤ C1 (2N∗ )k+1 + ξ k+1 + 2k+1 rk1 rk2 ∞ 8 ν k+1 +2 rs0 rs1 . . . rs2ν . N∗ s ν=1 |s| = k + 1 si < k + 1 By substituting the norms of r from (4.61), and estimating from above the norm of ξ by (4.60), we obtain (with Mk = k!Ek , and after dividing with (k + 1)!Ek+1 r0 ): 2C1 k1 !k2 ! Ek1 Ek2 (2N∗ )k+1 · · + 2C + 2k+2 r0 (k + 1)!Ek+1 (k + 1)! Ek+1 ν ∞ 8r20 Es0 . . . Es2ν +2 Xs0 . . . Xs2ν , N∗ Ek+1 ν=1 |s| = k + 1 si < k + 1

Xk+1 ≤

(4.75)

where k1 and k2 are given in (4.70). Obviously, the first term in the above estimate of Xk+1 goes to 0 as k → ∞, so it is bounded. The same holds for the third term. Indeed, if k + 1 is even, say k + 1 = 2m, then k1 = k2 = m, and by the Stirling formula we have, in view of (4.57), that 22m

√ m!m! Em Em · m(Em )2 /E2m → 0. (2m)! E2m

If k + 1 is odd, say k + 1 = 2m + 1, then k1 = m, k2 = m + 1, and we obtain em+1 2m+1 m!(m + 1)! Em Em+1 2m m!m! Em Em 2m + 2 2 → 0, · · · =2 (2m + 1)! E2m+1 (2m)! E2m 2m + 1 e2m+1 because the expression in the square brackets is bounded. Thus, we have 2C1 (2N∗ )k+1 Ek1 Ek2 k+2 k1 !k2 ! D := sup < ∞. · + 2C + 2 · r0 (k + 1)!Ek+1 (k + 1)! Ek+1

(4.76)

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177

By Lemma 9, the assumption (4.58) implies that Es . . . Es τ 0 2ν ∃Q > 0 : sup < Q2ν . E k+1 k |s|=k+1

Therefore, the double sum in (4.75) does not exceed the expression ∞ ν 8r2 Q2 Es0 . . . Es2ν 1−τ 0 2 sup max Xs0 . . . Xs2ν · . Ek+1 N∗ ν≥1 |s| = k + 1 ν=1 si < k + 1 If N∗ > N3 := 40r20 Q2 , then the sum in the above expression is less than 1/4. By (4.76), we have     Es0 . . . Es2ν 1−τ   max Xs0 . . . Xs2ν  . Xk+1 ≤ max 2D, sup   E k+1 ν≥1 |s| = k + 1 si < k + 1 Hence, by Lemma 10 (with T = 2D, Fk = (Ek )1−τ ) we obtain that the sequence (Xk ) is bounded. This completes the proof of Proposition 20. Now we complete the proof of Theorem 14 for weights of the form ϕ (n) = exp(ϕ(|n|)), where ϕ has the properties (3.10), (3.11) and (3.14). Let ξ = (ξ(m))m∈Z and r = (r(m))m∈Z be sequences with non-negative terms such that r(2n) ≤ ξ(n) + σ (n, r), We have to prove that 2 ξ(n)ϕ (2n) < ∞

⇒

n ≥ n∗ .

(4.77) 2

r(m)ϕ (m)

< ∞.

(4.78)

Let ξ = (ξ (m)), By part (a) of Lemma 9, 2 ξ (m)ϕ (m) < ∞

ξ (2m) = ξ(m),

⇒

∃C > 0 :

ξ 2m+1 = 0.

ξ = |ξ |k ≤ CMk (ϕ1 ),

where ϕ1 (t) = ϕ(t) − t and (Mk (ϕ1 )) is the Carleman sequence generated by ϕ1 (see the text after (3.4) prior to Lemma 8). By Proposition 20 there exists C˜ > 0 such that ξ k = |ξ |k ≤ CMk (ϕ1 )

⇒

˜ k (ϕ1 ) rk ≤ CM

k = 0, 1, 2, . . . .

On the other hand, the part (b) of Lemma 8 yields 2 ˜ k (ϕ1 ) ∀k ⇒ rk ≤ CM r(m)ϕˆ (m) < ∞, with ϕ(t) ˆ = ϕ1 (t) − 4t = ϕ(t) − 5t.

(4.79)

178


Consider the sequences rˆ = r(k)ϕˆ (k) ,

ξˆ = ξ(k)ϕˆ (2k) .

By multiplying (4.77) by ϕˆ (2n), we obtain, by Lemma 15, that rˆ (2n) ≤ ξˆ (n) + σ (n, rˆ ).

(4.80)

By (4.79), we have rˆ ∈ 2 (Z). Since ϕ (m) = ϕˆ (m) · |m|5 , the left side of (4.16) yields 2 2 ξˆ (k)|2m|5 = ξ(m)ϕ (2m) < ∞.

(4.81)

In view of (4.80) and (4.81), Proposition 19 can be applied to the sequences rˆ , ξˆ and the 2 weight (n) = |n|5 , so we have rˆ (m)|m|5 < ∞. Thus,

2

r(m)ϕ (m)

=

rˆ (m)|m|5

2

< ∞.

This completes the proof of Theorem 14. Therefore, Theorem 11 has been proven as well.

5. Conclusions and Comments 1. Theorems 12, 14 and our main result, Theorem 11, are formulated and proven in terms of weight sequences . This could be a slight obstruction in understanding these statements as results about classes of smooth (or infinitely differentiable) functions. Since these classes are basic for many analytic problems, let us write a few examples of weight sequences [satisfying the hypotheses of Theorem 11], and the corresponding corollaries of Theorem 11. Observe, that the weights in Examples 1-3 below are subexponential, i.e., lim (log (n)) /n = 0 (compare to (110) in [5]). Example 1. Consider the slowly increasing weight (n) = (1 + |n|)α , α > 0. Then H () = H α is a Sobolev space. Corollary 21. Let L be a self-adjoint Dirac operator given by (1.1). Then λ+ − λ− 2 (1 + n2 )α < ∞ n n Z

if and only if P ∈ H α . Of course, Corollary 21 explains Statement (A) of Sect. 1.

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179

Example 2. Let (n) = (1 + |n|)α exp a|n|b ,

α ∈ R, b ∈ (0, 1), a > 0.

(5.1)

These weights are more general than (3.9). Let us denote by G(b, a; α) the corresponding Gevrey space H (). We rewrite as (n) = exp h(|n|),

h(t) = at b + α log(1 + t), t > 0,

or (n) = exp (ϕ(log(|n|)) ,

where ϕ(τ ) = h(eτ ) = aebτ + α log 1 + eτ , τ > 0.

Then −1 ϕ (τ ) = abebτ + α 1 + e−τ , and the conditions (3.10) and (3.11) hold. In addition, ϕ (τ ) − ϕ (τ ) ab(1 − b)ebτ ∞, log ϕ (τ ) bτ so (3.14) holds also. These elementary calculus exercises show that ∈ (5.1) is a rapidly increasing weight with the properties (3.10), (3.11) and (3.14). Therefore, Theorem 11 gives the following. Corollary 22. Let L be as in Corollary 21, and let α ∈ R, a > 0, b ∈ (0, 1). Then λ+ − λ− 2 (1 + n2 )α exp 2a|n|b < ∞ n

n

Z

if and only if P is a function in the Gevrey class G(b, a; α). Corollary 22 explains Statement (C) of Sect. 1. Example 3. Let (n) = 1 + log(e + |n|)γ ,

γ > 0,

(5.2)

or (n) = (1 + |n|)α 1 + log(e + |n|)β ,

α > 0, β ∈ R.

(5.3)

Like in Example 1, this is a slowly increasing weight. We do not write explicitly the corresponding corollary of Theorem 11. But it is worth to mention that in the case (5.2) the space H (γ ) is N OT a subspace of any Sobolev space H α , α > 0, although, vice versa, H α ⊂ H (γ ) for any α, γ > 0.

180


Example 4. Let (n) = a (n) = exp(a|n|),

a > 0.

(5.4)

In this case the functions P ∈ H () are analytic in the strip {z = x+iy | |y| < a/(2π )}. If P (x), P (x + 1) = P (x) is analytic in the strip |y| < b, then P ∈ H (α ) for any a < 2π b. We could be more specific and talk about the boundary values of H (α )functions being in L2 (I ±),

I ± = {x + iy | 0 ≤ x ≤ 1, y = ±a/(2π )}.

But (n) = ea|n| is not a subexponential weight: lim (log (n)) /n = a > 0.

n→∞

(5.5)

Still, the hypotheses of Theorem 12 hold and Theorem 12 implies: Corollary 23. If P ∈ H (a ) then λ+ − λ− 2 e2a|n| < ∞. n n Z

In a weaker form, if P (x), P (x + 1) = P (x) is real-valued on R and analytic in the strip |y| < b/(2π ) then λ+ − λ− 2 e2a|n| < ∞ (5.6) n n Z

for any a, ; 0 < a < b. However, with (5.5), a does not satisfy the hypotheses of Theorem 11 and we cannot use it immediately. Still, we can modify our constructions from Sects. 3 and 4 to prove an analogue of Theorem 11 in the case of weights (5.4). Proposition 24. If P (x) = p(k)e2πikx is an L2 -potential function, and the sequence − of spectral gaps (γn ) = (λ+ n − λn ) of Dirac operator (1.1) satisfies (5.6), then P (x) can be extended as an analytic function in the strip {x + iy | |y| < A} with some A > 0. The constant A cannot be chosen as a function of a; it depends on the norm P 2 as well. We’ll give all technical details and necessary adjustments of the constructions of Sects. 3 and 4 in [9]. We have done such analysis and adjustments of our constructions in Sect. 5.4, Prop. 15, in [5], in the case of Schrödinger operators to give an alternative proof of E. Trubowitz’ result [30]. Now, in the case of Dirac operators, we have Statement (B) of Sect. 1, as a corollary of Corollary 23 and Proposition 24. 2. If L is a Dirac operator of the form (2.4), not necessarily self-adjoint, then the left − side inequality in (2.58), Theorem 5, does not hold. But in any case, |λ+ n − λn | could be estimated from above if we use the basic equation (2.13) and Lemma 4. More precisely, the following is true.

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181

Lemma 25. If L is a Dirac operator of the form (2.4), then there exists n∗ > 0 such that 12 21 − |λ+ − λ | ≤ 2 max (n, z) + 2 max (n, z) (5.7) S S , |n| ≥ n∗ . n n |z|≤π/2

|z|≤π/2

Proof. Indeed, with αn (z) = S 11 (n, z) = S 22 (n, z) and ζ = z − αn (z),

(5.8)

ζ 2 = S 11 (n, z)S 22 (n, z).

(5.9)

Eqn. (2.13) becomes

By Lemma 4, there exists n∗ > N0 , where N0 is the constant from Lemma 1, such that dαn (z) 1 dz ≤ 2 for |z| ≤ π/2, |n| ≥ n∗ . Thus, (5.8) defines a holomorphic mapping ζ (z) = z − αn (z) in the disc |z| < π/2 such that 1/2 ≤ |dζ /dz| ≤ 3/2. From here, it follows that 1 + |z − zn− | ≤ |ζ (zn+ ) − ζ (zn− )| ≤ 2|zn+ − zn− |, 2 n where, in view of Lemma 1, |zn± | < π/2 for |n| ≥ n∗ . So, by taking into account that − + − |λ+ n − λn | = |zn − zn |, we have 1 + + − + − |λ − λ− n | ≤ |ζn − ζn | ≤ 2|λn − λn |, 2 n where ζn+ = ζ (zn+ ) and ζn− = ζ (zn− ). On the other hand, (5.9) implies that 1/2 ± 12 ζ = S (n, z± )S 21 (n, z± ) ≤ 1 S 12 (n, z± ) + 1 S 21 (n, z± ) . n n n n n 2 2 Therefore, |ζn+ − ζn− | ≤ |ζn+ | + |ζn− | ≤ max S 12 (n, z) + max S 21 (n, z) ; |z|≤π/2

hence, (5.7) holds.

|z|≤π/2

Theorem 26. Let L be a Dirac operator of the form (2.4) with potential functions P (x) = p(2n)e2π inx and Q(x) = q(2n)e2πinx . If is a submultiplicative weight, then − 2 2 |p(2n)|2 + |q(2n)|2 ((2n))2 < ∞ ⇒ |λ+ n − λn | ((2n)) < ∞. (5.10)

182


Proof. Set r(m) = max(|p(−m)|, |p(m)|, |q(−m)|, |q(m)|).

(5.11)

In view of (2.36)–(2.39), we obtain, by (2.25), that max |S 12 (n, z)| ≤ σ (n, r),

|z|≤π/2

max |S 21 (n, z)| ≤ σ (n, r),

|z|≤π/2

(5.12)

where r = (r(m)) and σ (n, r) is defined by (4.7)–(4.9). Now, the claim follows from Proposition 13. Under some rigid assumptions on , which, a for example, exclude such weights as (m) = exp(a|m|), or (k) = log(e + |k|) , a > 0, the claim (5.10) can be found in [15] or [16]. 3. The present paper deals only with the case of subexponential growth of the weight , i.e., (m) ≤ ea|m| , a > 0. The case of superexponential weights could be analyzed as well. We will present this analysis elsewhere. For Hill–Schrödinger operators see such analysis in [6]. References 1. Ablowitz, M. A., Segur, H.: Solitons and the inverse scattering transform. Philadelphia: SIAM, 1981 2. Costin, O., Kruskal, M.: Optimal uniform estimates and rigorous asymptotics beyond all orders for a class of ordinary differential equations. Proc. Roy. Soc. London Ser. A 452, 1057–1085 (1996) 3. Djakov, P., Mityagin, B.: Smoothness of solutions of a nonlinear ODE. Integral Equations and Operator Theory 44, 149–171 (2002) 4. Djakov, P., Mityagin, B.: Smoothness of solutions of nonlinear ODE’s. Math. Ann. 324, 225–254 (2002) 5. Djakov, P., Mityagin, B.: Smoothness of Schrödinger operator potential in the case of Gevrey type asymptotics of the gaps. J. Funct. Anal. 195, 89–128 (2002) 6. Djakov, P., Mityagin, B.: Spectral gaps of the periodic Schrödinger operator when its potential is an entire function, Adv. in Appl. Math. 31(3), 562–596 (2003) 7. Djakov, P., Mityagin, B.: Spectral triangles of Schrödinger operators with complex potentials. Selecta Mathematica 9, 495–528 (2003) 8. Djakov, P., Mityagin, B.: Spectra of 1D periodic Dirac operators and smoothness of potentials. Math. Reports Acad. Sci. Royal Soc. Canada 25, 121–125 (2003) 9. Djakov, P., Mityagin, B.: Uspehi Mat. Nauk, in preparation 10. Dubrovin, B. A.: The inverse problem of scattering theory for periodic finite-zone potentials. Funktsional. Anal. i Prilozhen. 9, 65–66 (1975) 11. Dubrovin, B. A., Krichever, I.M., Novikov, S.P.: Integrable systems I. In: Encycl. of Math. Sci., Dynamical systems IV, Arnold, V. I., Novikov, S.P. (eds.), Berlin-Heidelberg-NewYork: Springer, 1990, pp. 173–283 12. Faddeev, L. D., Takhtajan, L. A.: Hamiltonian methods in the theory of solitons. Berlin, New York: Springer-Verlag, 1987 13. Gelfand, I. M., Levitan, B. M.: On a simple identity for the eigenvalues of a second order differential operator. Dokl. Akad. Nauk SSSR 88, 593–596 (1953) (Russian) 14. Goldberg, W.: On the determination of a Hill’s equation from its spectrum. Bull. Amer. Math. Soc. 80, 1111–1112 (1974) 15. Grébert, B., Kappeler, T., Mityagin, B.: Gap estimates of the spectrum of the Zakharov-Shabat system. Appl. Math. Lett. 11, 95–97 (1998) 16. Grébert, B., Kappeler, T.: Estimates on periodic and Dirichlet eigenvalues for the Zakharov-Shabat system. Asymptotic Analysis 25, 201–237 (2001) 17. Hochstadt, H.: Estimates on the stability intervals for the Hill’s equation. Proc. Amer. Math. Soc. 14, 930–932 (1963) 18. Hochstadt, H.: On the determination of a Hill’s equation from its spectrum. Arch. Ration. Mech. Anal. 19, 353–362 (1965)

1D Dirac Operators

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19. Kappeler, T., Mityagin, B.: Gap estimates of the spectrum of Hill’s Equation and Action Variables for KdV. Trans. AMS 351, 619–646 (1999) 20. Kappeler, T., Mityagin, B.: Estimates for periodic and Dirichlet eigenvalues of the Schrödinger operator. SIAM J. Math. Anal. 33, 113–152 (2001) 21. Levitan, B. M., Sargsian.: “Introduction to spectral theory; Self-adjoint ordinary differential operators”. Translation of Mathematics Monographs, Vol. 39, Providence, RI: AMS, 1975 22. Li, Y., McLaughlin, D.: Morse and Melnikov functions for NLS PDEs. Commun. Math. Phys. 162, 175–214 (1994) 23. Marchenko, V. A.: Sturm-Liouville operators and applications. Oper. Theory Adv. Appl., Vol. 22, Basel-Boston: Birkhäuser, 1986 24. McKean, H., Trubowitz, E.: Hill’s operator and hyperelliptic function theory in the presence of infinitely many branch points. Comm. Pure Appl. Math. 29, 143–226 (1976) 25. Misyura, T.: Properties of the spectra of periodic and antiperiodic boundary value problems generated Dirac operators I, II (in Russian). Teor. Funktsii Funktsional. Anal. i Prilozhen. 30, 90–101 (1978); 31, 102–109 (1979) 26. Mityagin, B.: Convergence of expansions in the eigenfunctions of the Dirac operator, (Russian), Dokl. Acad. Nauk 393, 456–459 (2003). [English transl.: Doklady Math. 68, 388–391 (2003)] 27. Mityagin, B.: Spectral expansions of one-dimensional periodic Dirac operator. Dynamics of PDE 1, 125–191 (2004) 28. Novikov, S.P.: The periodic problem for Korteweg-De Vries equation. Funktsional. Anal. i Prilozhen. 8:3 , 54–66 (1974). English transl., Functional Analysis and its applications, 8, 236–246 January 1975 29. Tkachenko, V.: Non-selfadjoint periodic Dirac operators, Operator Theory; Advances and Applications, Vol. 123, Basel: Birkhäuser Verlag, 2001, pp. 485–512 30. Trubowitz, E.: The inverse problem for periodic potentials. CPAM 30, 321–342 (1977) 31. Zakharov, V., Manakov, S., Novikov, S., Pitayevskii, L. : Theory of solitons; the inverse scattering method. New York: Consultants Bureau, 1984 32. Zakharov, V., Shabat, A.: Exact theory of two-dimensional self-focusing and one-dimensional selfmodulation of waves in non-linear media. Sov. Phys. JETP 34, 62–69 (1971) 33. Zakharov, V., Shabat, A.: A scheme of integrating the non-linear equations of mathematical physics by the method of inverse scattering problem I. J. Funct. Anal. Appl. 8, 226–235 (1974) 34. Zakharov, V., Shabat, A.: Integration of the non-linear equations of mathematical physics by the method of inverse scattering problem II. J. Funct. Anal. Appl. 13, 166–174 (1979) Communicated by B. Simon


Communications in


Equality of the Bulk and Edge Hall Conductances in a Mobility Gap A. Elgart1 , G.M. Graf 2 , J.H. Schenker2 1 2

Department of Mathematics, Stanford University, Stanford, CA 94305-2125, USA Theoretische Physik, ETH Zürich, 8093 Zürich, Switzerland

Received: 8 September 2004 / Accepted: 7 February 2005 Published online: 21 June 2005 – © Springer-Verlag 2005

Abstract: We consider the edge and bulk conductances for 2D quantum Hall systems in which the Fermi energy falls in a band where bulk states are localized. We show that the resulting quantities are equal, when appropriately defined. An appropriate definition of the edge conductance may be obtained through a suitable time averaging procedure or by including a contribution from states in the localized band. In a further result on the Harper Hamiltonian, we show that this contribution is essential. In an appendix we establish quantized plateaus for the conductance of systems which need not be translation ergodic. 1. Introduction Two conductances, σB and σE , are associated to the Quantum Hall Effect (QHE), depending on whether the currents are ascribed to the bulk or to the edge. The equality σB = σE , suggested by Halperin’s analysis [17] of the Laughlin argument [21], has been established in the context of an effective field theory description [14]. It was later derived in a microscopic treatment of the integral QHE [32, 12, 24] for the case that the Fermi energy lies in a spectral gap of the single-particle Hamiltonian HB . We prove this equality, by quite different means, in the more general setting that HB exhibits Anderson localization in – more precisely, dynamical localization (see (1.2) below). The result applies to Schrödinger operators which are random, but does not depend on that property. We therefore formulate the result for deterministic operators. The relation to recent work [7] will be discussed below. The bulk is represented by the lattice Z2 x = (x1 , x2 ) with Hamiltonian HB = HB∗ on 2 (Z2 ). We assume its matrix elements HB (x, x ), x, x ∈ Z2 , to be of short range in the sense that HB (x, x ) (eµ|x−x | − 1) =: C1 < ∞ (1.1) sup x∈Z2 x ∈Z2

186

A. Elgart, G.M. Graf, J.H. Schenker

for some µ > 0, where |x| = |x1 | + |x2 |. Our hypothesis on the bounded open interval ⊂ R is that for some ν ≥ 0, g(HB )(x, x ) (1 + |x|)−ν eµ|x−x | =: C2 < ∞, sup (1.2) g∈B1 ()

x,x ∈Z2

where B1 () denotes the set of Borel measurable functions g which are constant in {λ|λ < } and in {λ|λ > } with |g(x)| ≤ 1 for every x. In particular C2 is a bound when g is of the form gt (λ) = e−itλ E (λ) and the supremum is over t ∈ R, which is a statement of dynamical localization. By the RAGE theorem this implies that the spectrum of HB is pure point in (see [20] or [10, Theorem 9.21] for details). We denote the corresponding eigen-projections by E{λ} (HB ) for λ ∈ E , the set of eigenvalues λ ∈ . We assume that no eigenvalue in E is infinitely degenerate, dim E{λ} (HB ) < ∞ ,

λ ∈ E .

(1.3)

The validity of these assumptions is discussed below (but see also [2, 4]). The zero temperature bulk Hall conductance at Fermi energy λ is defined by the Kubo-Stˇreda formula [5] σB (λ) = −i tr Pλ [ [Pλ , 1 ] , [Pλ , 2 ] ] ,

(1.4)

where Pλ = E(−∞,λ) (HB ) and i (x) is the characteristic function of x = (x1 , x2 ) ∈ Z2 | xi < 0 . Under the above assumptions σB (λ) is well-defined for λ ∈ , but independent thereof, i.e., it shows a plateau. (This result, first proved in [6], is strengthened here in an appendix, since we do not assume translation covariance or ergodicity of the Schrödinger operator. We also show the integrality of 2πσB therein, though it is not needed in the sequel.) We remark that (1.3) is essential for a plateau: for the Landau Hamiltonian (though defined on the continuum rather than on the lattice) Eqs. (1.1, 1.2) hold if properly interpreted, but (1.3) fails in an interval containing a Landau level, where indeed σB (λ) jumps. The sample with an edge is modeled as a half-plane Z × Za , where Za = {n ∈ Z | n ≥ −a}, with the height −a of the edge eventually tending to −∞. The Hamiltonian Ha = Ha∗ on 2 (Z × Za ) is obtained by restriction of HB under some largely arbitrary boundary condition. More precisely, we assume that Ea = Ja Ha − HB Ja : 2 (Z × Za ) → 2 (Z2 ) satisfies sup

x∈Z2 x ∈Z×Z

Ea (x, x ) eµ(|x2 +a|+|x1 −x1 |) ≤ C3 < ∞ ,

(1.5)

(1.6)

a

where Ja : 2 (Z × Za ) → 2 (Z2 ) denotes extension by 0. For instance with Dirichlet boundary conditions, Ha = Ja∗ HB Ja , we have Ea = (Ja Ja∗ − 1)HB Ja , i.e., −HB (x, x ) , x2 < −a , Ea (x, x ) = 0, x2 ≥ −a ,

Equality of the Bulk and Edge Hall Conductances in a Mobility Gap

187

whence (1.6) follows from (1.1). We remark that Eq. (1.1) is inherited by Ha with a constant C1 that is uniform in a, but not so for Eq. (1.2) as a rule. The definition of the edge Hall conductance requires some preparation. The current operator across the line x1 = 0 is −i [Ha , 1 ]. Matters are simpler if we temporarily assume that is a gap for HB , i.e., if σ (HB ) ∩ = ∅, in which case one may set [32] σE := −i tr ρ (Ha ) [Ha , 1 ] , where ρ ∈ C ∞ (R) satisfies

(1.7)

ρ(λ) =

1, λ < , 0, λ > .

(1.8)

The heuristic motivation for (1.7) is as follows. We interpret ρ(Ha ) as the 1-particle density matrix of a stationary quantum state. Though some current is flowing near the edge we should discard it, as it is supposed to be canceled by current flowing at an opposite edge located at x2 = +∞. If the chemical potential is now lowered by δ at the first edge, but not at the second, a net current δ I = −i tr ((ρ(Ha + δ) − ρ(Ha )) [Ha , 1 ]) = −i dt tr ρ (Ha − t) [Ha , 1 ] 0

is flowing. Since σE is independent of ρ as long as it conforms with (1.8), see [32] and Theorem 1 below, it is indeed the conductance σE = I /δ for sufficiently small δ. The operator in (1.7) is trace class essentially because i [H, 1 ] is relevant only on (single-particle) states near x1 = 0, and ρ (Ha ) only near the edge x2 = −a, so that the intersection of the two strips is compact. In the situation (1.2) considered in this paper the operator appearing in (1.7) is not trace class, since the bulk operator may have spectrum in , which can cause the above stated property to fail for ρ (Ha ). In search of a proper definition of σE , we consider only the current flowing across the line x1 = 0 within a finite window −a ≤ x2 < 0 next to the edge. This amounts to modifying the current operator to be i i − (2 [Ha , 1 ] + [Ha , 1 ] 2 ) = − { [Ha , 1 ] , 2 } , 2 2

(1.9)

with which one may be tempted to use i lim − tr ρ (Ha ) { [Ha , 1 ] , 2 } 2

a→∞

(1.10)

as a definition for σE . Though we show that this limit exists, it is not the physically correct choice. We may in fact expect that the dynamics of e−itHa acting on states supported far away from the edge resembles for quite some time the dynamics generated by HB . Being bound states or, more likely, resonances, such states may carry persistent currents (whence the operator in (1.7) is not trace class), but no or little net current across the line x1 = 0. This cancelation is the rationale for ignoring the part x2 ≥ 0 of the line x1 = 0 by means of the cutoff 2 in (1.9), however the cancelation is not achieved on states located near the end point x = (0, 0). In the limit a → ∞ we pretend these states are bound, which yields the contribution missed by (1.10): i − (ψλ , { [HB , 1 ] , 1 − 2 } ψλ ) = Im (ψλ , 1 HB 2 ψλ ) , 2

(1.11)

188


from each bound state ψλ of HB , with corresponding energy λ ∈ E . We incorporate them with weight ρ (λ) in our definition of the edge conductance: i (1) σE := lim − tr ρ (Ha ) { [Ha , 1 ] , 2 } a→∞ 2 + ρ (λ) Im tr E{λ} 1 HB 2 E{λ} .

(1.12)

λ∈E

We will show that the sum on the r.h.s. is absolutely convergent, and its physical meaning will be further discussed at the end of the introduction. We will also show it to be nonzero on average for the Harper Hamiltonian with an i.i.d. random potential in Theorem 3. The terms of this sum involve HB , though the few states for which they are sizeable are supported near x = (0, 0) and hence far from the edge x2 = −a. Since the mere appearance of HB in the definition of an edge property may be objectionable, we present an alternative. The basic fact that the net current of a bound state is zero, −i (ψλ , [HB , 1 ] ψλ ) = 0 ,

(1.13)

can be preserved by the regularization provided the spatial cutoff 2 is time averaged. In fact, let 1 T iHa t −iHa t AT ,a (X) = e Xe dt (1.14) T 0 be the time average over [0, T ] of a (bounded) operator X with respect to the Heisenberg evolution generated by Ha , with a finite or a = B. If a limit ∞ 2 = lim T →∞ AT ,B (2 ) were to exist, it would commute with HB so that −

i ψλ , [HB , 1 ] , ∞ 2 ψλ = 0 . 2

This motivates our second definition, (2)

σE

i := lim lim − tr ρ (Ha ) [Ha , 1 ] , AT ,a (2 ) . T →∞ a→∞ 2

(1.15)

The two definitions allow for the following result. Theorem 1. Under the assumptions (1.1, 1.2, 1.3, 1.6, 1.8) the sum in (1.12) is absolutely convergent, the limits there and in (1.15) exist, and (1)

σE

(2)

= σE

= σB .

In particular (1.12, 1.15) depend neither on the choice of ρ nor on that of Ea . Remark 1. i) The hypotheses (1.1, 1.2) hold almost surely for ergodic Schrödinger operators whose Green’s function G(x, x ; z) = (HB − z)−1 (x, x ) satisfies a moment condition [3] of the form s

sup lim sup E G(x, x ; E + iη) ≤ Ce−µ|x−x | (1.16) E∈

η→0


for some s < 1. The implication is through the dynamical localization bound E sup g(HB )(x, x ) ≤ Ce−µ|x−x | ,

189

(1.17)

g∈B1 ()

although (1.2) has also been obtained by different means, e.g., [16]. The implication (1.16) ⇒ (1.17) was proved in [1] (see also [2, 11, 4]). The bound (1.17) may be better known for supp g ⊂ , but is true as stated since it also holds [6, 2] for the projections g(HB ) = Pλ , Pλ⊥ = 1 − Pλ , (λ ∈ ). ii) Condition (1.3), in fact simple spectrum, follows from the arguments in [34], at least for operators with nearest neighbor hopping, HB (x, y) = 0 if |x − y| > 1. iii) When σ (HB ) ∩ = ∅, the operator appearing in (1.7) is known to be trace class. (1) (2) In this case, the conductance σE = σE defined here coincides with σE defined in (1.7). This statement follows from Theorem 1 and the known equality σE = σB [32, 12], but can also be seen directly. For completeness, we include a proof of this fact in Sect. 2 below. A point of view which combines both definitions of the edge conductance is expressed by the following result. Theorem 2. Under the assumptions of Theorem 1, i lim − tr ρ (Ha ) [Ha , 1 ] , 2;a (t) a→∞ 2 = σB + ρ (λ) Im tr E{λ} [HB , 1 ] eiHB t 2 e−iHB t E{λ} ,

(1.18)

λ∈E

with 2;a (t) = eiHa t 2 e−iHa t . (1)

In particular, this reduces to σE = σB for t = 0 by (1.11, 1.12). On the other hand, (2) σE = σB results, as we will show, from the time average of (1.18). A recent preprint [7] contains results which are topically related, to but substantially different from, those presented here. In that work, two contiguous media are modeled by positing a potential of the form U (x1 , x2 ) = V0 (x2 )χ (x2 < 0) + V (x1 , x2 )χ (x2 ≥ 0) (in our notation), where V0 is independent of x1 . The role of V is that of a bulk potential, and that of V0 as of a wall, provided it is large. The kinetic term is given by the Landau Hamiltonian on the continuum L2 (R2 ), whose unperturbed spectrum is the familiar set (2N+1)B, with B the magnitude of the constant magnetic field. A result is the following: if model (a), with V0 = 0, exhibits localization in ⊂ [(2N − 1)B, (2N + 1)B] for some positive integer N, and hence σE = 0, then model (b), with V0 (x2 ) ≥ (2N + 1)B, has 2π σE = N. The result is established by showing that the difference between 2π σE in cases (b) and (a) is independent of V , and equals N if V = 0, the two models then being solvable thanks to the translation invariance w.r.t. x1 . In comparison to our work, the following features may be noted: i) The localization assumption on the reference model (a) is made for a system which has itself an interface. (Our Eq. (1.2) concerns a bulk model serving as reference.) ii) The validity of that assumption is limited to small V , because the interface of (a) will otherwise produce extended edge states with energies in . The result σE = σB thus applies to perturbations of the free Landau Hamiltonian of size B. (Our comparison σE = σB does not require either side to be explicitly computable.)

190

A. Elgart, G.M. Graf, J.H. Schenker (1)

iii) The definition of σE for (b) depends on eigenstates in of (a), like our σE , but (2) not σE . A model without bulk potential, but allowing interactions between diluted particles, was studied from a related perspective in [25]. In (1.11, 1.12) we argued that the limit (1.10) is not identical to σB . To indeed prove this, we show that the sum on the right-hand side of (1.12) does not vanish for the Harper Hamiltonian with i.i.d. Cauchy randomness on the diagonal. The Harper Hamiltonian models the hopping of a tightly bound charged particle in a uniform magnetic field. The hopping terms H (x, x ) are zero except for nearest neighbor pairs, for which they are of modulus one, H (x, x ) = 0, |x − x | = 1 , (1.19) 1, |x − x | = 1 , where the non-zero matrix elements are interpreted as H (x, x ) = ei

x

x

A(y)· d1 y

,

with A the magnetic vector potential and the line integral computed along the bond connecting x, x . The magnetic flux through any region D ⊂ R2 is B(x)d2 x = A(y) · d1 y , D

∂D

so, for a uniform field, the flux is proportional to the area A(y) · d1 y = φ|D| . ∂D

Thus, we require that H (x (1) , x (4) )H (x (4) , x (3) )H (x (3) , x (2) )H (x (2) , x (1) ) = ei

∂P

A(y)·d1 y

= eiφ , (1.20)

where x (1) , x (2) , x (3) , x (4) are the vertices of a plaquette P , listed in counter-clockwise order, and φ is the flux through any plaquette. There are many choices of nearest neighbor hopping terms which satisfy (1.19) and (1.20), all interrelated by gauge transformations. For our purposes, it suffices to fix a gauge and take   x = x ± e1 , 1 , iφx 1 Hφ (x, x ) := e (1.21) , x = x + e2 ,  e−iφx1 , x = x − e , 2 with e1 = (1, 0) and e2 = (0, 1) the lattice generators. This choice of Hφ comes from representing the constant field B = φ via the vector potential A = φ(0, x1 ). We note that the bulk and edge Hall conductances are gauge invariant quantities, so Theorem 3 stated below holds for any other choice of Hφ . We refer the reader to ref. [26] and references therein for further discussion of the Harper Hamiltonian.


191

To guarantee localized spectrum, we consider a bulk Hamiltonian which consists of Hφ plus a diagonal random potential, HB = Hφ + αV , where V ψ(x) = V (x)ψ(x) and V (x), x ∈ Z2 are independent identically distributed Cauchy random variables. Here α is a coupling parameter (the “disorder strength”) and “Cauchy” signifies that the distribution of v = V (x) is 1 1 dv . π 1 + v2 We use Cauchy variables because it is possible to calculate certain quantities explicitly for such variables: E (f (v)) = f (i) for a function f having a bounded analytic continuation to the upper half plane. It is clear that HB is short range, i.e., (1.1) holds. For simplicity we consider Ha which are defined via non-random boundary conditions, i.e., the operators Ea appearing in (1.5) do not depend on the random couplings V (x). We then have the following result. Theorem 3. For HB , Ha as above, there is jB ∈ C ∞ such that i E − lim tr ρ (Ha ) { [Ha , 1 ] , 2 } = − ρ (λ)jB (λ)dλ , 2 a→∞

(1.22)

whenever ρ ∈ C0∞ (R). The expectation is well defined and may be interchanged with the limit. Furthermore, jB (λ) has the following asymptotic behavior jB (λ) = −

4|α| sin(φ)(cos(φ) + 1)λ−5 + O(λ−6 ) , π

|λ| → ∞ .

(1.23)

The result is relevant in relation to (1.12) since it has in fact been shown that (1.2) holds for HB at large energies: Theorem ([1]). There is E0 (α) such that (1.17) holds for HB and = ± with − = (−∞, −E0 (α)] and + = [E0 (α), ∞). Hence (1.2) holds almost surely. Remark 2. i) For any α = 0 the spectrum of HB is (almost surely) the entire real line, so the eigenvalues of HB in ± make up a (random) dense subset which we denote E± . In fact, this pure point spectrum is almost surely simple, as can be shown using the methods in [34]. ii) For sufficiently large α we have E0 (α) = 0, i.e., the spectrum is completely localized. iii) Localization also holds inside the spectral gaps of Hφ , for small α, via the methods in [1, 4]. The mentioned result implies σB (λ) = 0 for λ ∈ ± , because σB is insensitive to λ in that range and Pλ → 1 or 0 as λ → ∞ or −∞, respectively. Thus for ρ as in (1.8) (1) with supp ρ ⊂ ± we have σE = 0 by Theorem 1. On the other hand, for the first term on the r.h.s. of (1.12), JB (ρ), we have by Theorem 3, 4|α| (1.24) sin(φ)(cos(φ) + 1) ρ (λ)λ−5 dλ + O λ−6 E (JB (ρ)) = 0 π |λ|≥E0 (α) as λ0 = inf |λ||λ ∈ supp ρ → ∞. Clearly the right-hand side can be non-zero for appropriately chosen ρ, and the same then holds for the expectation of the last term in (1.12).

192


The definitions (1.12, 1.15) may be related, heuristically, to concepts from classical electro-magnetism of material media [31]. There the macroscopic (or average) current is split as jf + ∂ P /∂t + rot M into free, polarization, and magnetization currents. (The magnetization M is a scalar in two dimensions.) The distinction depends on the existence of units (free electrons, atoms, molecules, ...) each with conserved charge, whose current densities are effectively of the form ∂ j( x , t) = q r˙ (t)δ( x − r(t)) + δ( x − r(t))p(t) + rot (δ( x − r(t))m(t)) , (1.25) ∂t where q, p(t), m(t) are the unit’s charge and electric/magnetic moments respectively. The macroscopic quantities emerge as a weak limit of the microscopic ones     x , t)   qk r˙ k (t)   jf ( δ( x − rk (t)) , p (t) P ( x , t)  k    m (t) k M( x , t) k or more precisely after integration against compactly supported test functions which vary slowly over the interatomic distance. The microscopic current across the portion x2 ≤ 0 of the line x1 = 0 is then d2 x (x1 )(x2 )jk,1 ( x , t) I =− k

=−

+

∂ P1 x , t) + d x (x1 )(x2 ) jf,1 ( ( x , t) ∂t 2

d2 x (x1 ) (x2 )M( x , t) .

(1.26)

The derivation assumes that is smooth over interatomic distances. The last term in x − rk (t))mk (t). It (1.26) comes from the corresponding term in (1.25), which is ∂2 δ( cannot be replaced by adding (rotM)1 = ∂2 M within the square brackets, which would correspond to the macroscopic current. In fact, it differs from that by a boundary term, which would vanish if (x2 ) were compactly supported. Let now the macroscopic fields be stationary and slowly varying on the scale of . In the QHE we expect that the (free) edge currents are located near the edge, so that (1.26) becomes dx2 jf,1 ( x ) + M(0) . I = x1 =0

When M(0) is subtracted from the l.h.s., we obtain an expression for the edge current, which is the role of the second term in (1.12). In this analogy the definition (1.15) corresponds to replacing (x2 ) in the first line of (1.26) by (e2 · rk,T ), where rk,T is the time average of rk (t). Then the last term no longer arises. The above discussion neglects the weighting ρ (λ) of energies in (1.12). This will be remedied in the following heuristic argument in support of σB = σE . In a finite sample of volume V the Stˇreda relation [35] asserts ∂N ∼ = σB V , ∂φ

(1.27)


193

where N is the total charge of carriers, i.e., N = tr ρ(HV ) in the situation considered here. For the total magnetization M we have −

∂HV ∂M = tr ρ (HV ) , ∂µ ∂φ

(1.28)

where µ is the chemical potential, as can be seen from the Maxwell relation [15] −

∂N ∂M = . ∂µ ∂φ

(1.29)

To compute ∂HV /∂φ we use a gauge equivalent to (1.21), with trivial phases along bonds in direction e2 , and obtain for (1.28), −

1 i 1 ∂M = tr ρ (HV ) { [HV , X1 ] , X2 } . V 2 V ∂µ (1)

By (1.27, 1.29) this quantity is formally σB . To relate it to σE it should be noted that the total magnetization is not the integral of the bulk magnetization, even in the thermodynamic limit. For instance, for classical, spinless particles M vanishes [22], but consists [27] of a diamagnetic, bulk contribution and an opposite contribution from states close to the edge. These two contributions (in reverse order) may be identified in the quantum mechanical context with the two terms of (1.12). In this example, the expected edge term is negative for φ > 0. This should also emerge from (1.24) when sup supp ρ → −∞, and it does if one also takes into account that −H is the counterpart to the continuum Hamiltonian. In Sect. 2 we will present the main steps in the proof of Theorems 1 and 2, with details supplied in Sect. 3. The proof of Theorem 3 will be given in Sect. 4. The appendix is about properties of σB . 2. Outline of the Proof A reasonable first step is to make sure that the traces in (1.12, 1.15) are well-defined. We will show this for σE (a, t) := −i tr ρ (Ha ) [Ha , 1 ] 2;a (t) ,

(2.1)

with 2;a (t) = eitHa 2 e−itHa , by proving that i [Ha , 1 ] 2;a (t) ∈ I1 in Lemma 5. Here, I1 denotes the ideal of trace class operators, and we denote the trace norm by ·1 . Then σE (a, t) = −i tr 2;a (t) [Ha , 1 ] ρ (Ha ) = −i tr ρ (Ha )2;a (t) [Ha , 1 ] , where we used that tr AB = tr BA if AB , BA ∈ I1 , e.g., [33, Corollary 3.8]. The definition (1.15) then reads 1 T (2) σE = lim lim dt Re σE (a, t) . T →∞ a→∞ T 0

(2.2)

(2.3)

194


By the argument given in the Introduction, the trace norm of the operator in (2.1) diverges as a → ∞. To see that its trace nevertheless converges we subtract from it an operator Z(a, t) ∈ I1 , to be specified below, with tr Z(a, t) = 0, implying

σE (a, t) = −i tr ρ (Ha ) [Ha , 1 ] 2;a (t) − Z(a, t) . (2.4) The idea, of course, is to choose Z(a, t) so that sup ρ (Ha ) [Ha , 1 ] 2;a (t) − Z(a, t)1 < ∞ .

(2.5)

a

An operator of zero trace is [ρ(Ha ), 1 ] 2 ; it is trace class (see Lemma 5) and its trace, computed in the position basis, is seen to vanish. Though it does not quite suffice for (2.5), we consider it since [ρ(Ha ), 1 ] and ρ (Ha ) [Ha , 1 ] are closely related: From the Helffer-Sjöstrand representations (see Sect. 3 for details) 1 ρ(Ha ) = (2.6a) dm(z)∂z¯ ρ(z)R(z), 2π 1 ρ (Ha ) = − dm(z)∂z¯ ρ(z)R(z)2 , (2.6b) 2π with R(z) = (Ha − z)−1 , we obtain

1 dm(z)∂z¯ ρ(z)R(z) [Ha , 1 ] R(z), 2π 1 ρ (Ha ) [Ha , 1 ] = − dm(z)∂z¯ ρ(z)R(z)2 [Ha , 1 ] . 2π [ρ(Ha ), 1 ] = −

(2.7a) (2.7b)

The two expressions, multiplied from the right by 2 , respectively by 2;a (t) as in (2.1), would have an even more similar structure if in the second a resolvent could be moved to the right. This can be achieved under the trace by setting Z(a, t) = [ρ(Ha ), 1 ] 2

1 − dm(z)∂z¯ ρ(z)R(z) R(z) [Ha , 1 ] 2;a (t)−[Ha , 1 ] 2;a (t)R(z) , 2π (2.8) for which tr Z(a, t) = 0. Then (2.4) reads σE (a, t) = tr a (t) with a i

ia (t) := − [ρ(Ha ), 1 ] 2 1 + − dm(z)∂z¯ ρ(z)R(z) [Ha , 1 ] 2;a (t)R(z) 2π

(2.9)

a (t) i

ia (t)

(2.10) = [ρ(Ha ), 1 ] 2;a (t) − 2 " ! 1 + − dm(z)∂z¯ ρ(z)R(z) [Ha , 1 ] R(z) Ha , 2;a (t) R(z), 2π ia (t)


195

where, to obtain! the last expression, (2.7a) " ! multiplied "by 2;a (t) has been added and subtracted, and R(z), 2;a (t) = −R(z) Ha , 2;a (t) R(z) has been used. We remark that equality of (2.9) and (2.10) also holds for HB , i.e., if we replace Ha by HB and 2;a (t) by 2;B (t) = eiHB t 2 e−iHB t , and set R(z) = (HB − z)−1 . We will show σE (a, t) = tr a (t) −−−→ tr B (t) a→∞

(2.11)

and, incidentally, (2.5) by establishing: Lemma 1. Under assumptions (1.1, 1.6), but without making use of (1.2, 1.3, 1.8), we have for ρ ∈ C0∞ (R), Ja (t)J ∗ − (t) −−−→ 0 , (2.12) a a B 1 a→∞ Ja (t)J ∗ − (t) −−−→ 0, (2.13) a a B 1 a→∞

uniformly for t in a compact interval. ∗ on 2 (Z × Note that the replacement A → Ja AJ extends by zero an operator a simply 2 2 ∗ ∗ Za ) to one on (Z ). In particular Ja AJa 1 = A1 and tr Ja AJa = tr A. For the rest of this section we shall only be concerned with bulk quantities like tr B (t). By (2.1, 2.3, 2.11), the statements to be proven are ρ (λ) Im tr E{λ} [HB , 1 ] eiHB t 2 e−iHB t E{λ} Re tr B (t) = σB + λ∈E

for Theorem 2 and part of Theorem 1, and 1 T lim tr B (t)dt = σB T →∞ T 0

(2.14)

for the other part, where actually the real part of the l.h.s. would suffice. It may be noted (1) (2) that the ρ’s allowed by (1.8) form an affine space and that B (t), like σE , σE , is affine in ρ. The relation to σB will be made through the following decomposition, which exhibits the same property for this quantity. Lemma 2. Let ⊂ R be as in Theorem 1 and let E− , E+ be the spectral projections for HB onto {λ | λ < }, resp. {λ | λ > }. Then, for λ0 ∈ , ! " ! " σB (λ0 ) = i tr E− Pλ0 , 1 2 E− + i tr E+ Pλ0 , 1 2 E+ + i tr E Tλ0 E , (2.15) where Pλ0 = E(−∞,λ0 ) , Tλ0 = Pλ0 1 Pλ⊥0 2 Pλ0 − Pλ⊥0 1 Pλ0 2 Pλ⊥0 ,

(2.16)

and the traces are well defined. Moreover, the last term in (2.15) can be further decomposed as ! " i tr E{λ} Pλ0 , 1 2 E{λ} , (2.17) i tr E Tλ0 E = λ∈E

with absolutely convergent sum.

196


Since σB is independent of λ0 ∈ , (2.15) with the last term replaced by the r.h.s. of (2.17) also holds if Pλ0 is replaced by ρ satisfying (1.8), since ρ(HB ) = − dλ0 ρ (λ0 )Pλ0 . The proof of Lemma 2, which is given in Sect. 3, makes use of E{λ} , (2.18) 1 = E− + E+ + E , E = λ∈E

where the sum is strongly convergent. Using this decomposition on B (t) ∈ I1 we obtain

B (t) E− + tr E+ B (t) E+ B + B + tr B (t) = tr E− + tr E B (t)E .

(2.19)

Though the two contributions (2.9) to B (t) are not separately trace class, they become E± also appear in (2.15), and E± (t)E± vanish so in (2.19). In fact, those of E± B B by integration by parts since E± R(z) and R(z)E± are analytic on the support of ρ(z) or of ρ(z) − 1. We thus find that tr B (t) = σB + i dλ0 ρ (λ0 ) tr E Tλ0 E + tr E B (t)E . (2.20) At this point the analysis of the last term splits into two tracks with the purpose of (1) (2) showing σE = σB , resp. σE = σB . 2.1. Track 1. We decompose the projection E into its atoms as in (2.18), which by s Xn − → 0 , Y ∈ I1 ⇒ Xn Y 1 → 0 , Y Xn∗ 1 → 0 (2.21) yields a trace class norm convergent sum for E B (t)E . Thus

B (t) E{λ} . B + tr E{λ} tr E B (t)E = λ∈E

E{λ} are themselves trace class as they match those of Again, the contributions E{λ} B (2.17), canceling the second term of (2.20). We conclude that tr B (t) = σB = σB

i dm(z)∂z¯ ρ(z) tr E{λ} R(z) [HB , 1 ] 2;B (t)R(z)E{λ} + 2π λ∈E − i ρ (λ) tr E{λ} e−iHB t [HB , 1 ] eiHB t 2 E{λ} , (2.22) λ∈E

where we used that f (HB )E{λ} = f (λ)E{λ} . By its derivation this sum is absolutely (1) convergent for each t. This proves Thm. 2 and hence σE = σB . 2.2. Track 2. Here we do not decompose E , but use (2.10) whose two terms are separately trace class, tr E B (t)E = tr E B (t)E + tr E B (t)E .


197

Lemma 3. For ⊂ R as in Theorem 1 we have 1 T

T

0

tr E B (t)E dt −−−→ 0 , T →∞

(2.23)

and ! "

−i tr E Pλ0 , 1 AT ,B (2 ) − 2 E −−−→ i tr E Tλ0 E T →∞

(2.24)

for λ0 ∈ , the expression on the l.h.s. being uniformly bounded in λ0 ∈ , T > 0. By dominated convergence (2.24) implies 1 T

0

T

tr E B (t)E dt −−−→ −i T →∞

dλ0 ρ (λ0 ) tr E Tλ0 E . (2)

Together with (2.20, 2.23), this proves (2.14) and hence σE = σB . 2.3. Alternate Track 2. We now show that the last result can also be inferred from (2.22), at least if assumption (1.3) is strengthened to a uniform upper bound on the degeneracies: dim E{λ} (HB ) ≤ C4 < ∞ ,

λ ∈ E .

(2.25)

Then, the sum (2.22) is uniformly convergent in t ∈ R, as stated in Lemma 4. Assuming (1.1, 1.2, 2.25), we have sup tr E{λ} e−iHB t [HB , 1 ] eiHB t 2 E{λ} < ∞ . λ∈E t∈R

(2.26)

In order to prove (2.14), it suffices in view of (2.26) to show 1 lim T →∞ T

0

T

tr E{λ} e−iHB t [HB , 1 ] eiHB t 2 E{λ} = 0

(2.27)

for each λ ∈ E . Because tr E{λ} e−iHB t [HB , 1 ] eiHB t 2 E{λ} = i

d tr E{λ} e−iHB t 1 eiHB t 2 E{λ} , dt

(2.28)

the expression under the limit is just $ i # tr E{λ} e−iHB t 1 eiHB t 2 − tr E{λ} 1 2 . T

(2.29)

Since each term inside the square brackets is bounded by C4 < ∞, Eq. (2.27) follows. (2) This concludes the alternate proof of σE = σB .

198


2.4. Edge conductance in a spectral gap. We conclude this section by showing as mentioned above in the remark following Theorem 1 that −i tr ρ (Ha ) [Ha , 1 ] = σB if σ (HB ) ∩ = ∅. By translation invariance of σB , see Lemma 7 below, it suffices to show this for a = 0, in which case we drop the subscript a of the edge Hamiltonian. It s has been shown in (A.8) of [12] that ρ (H ) [H, 1 ] ∈ I1 . Since 2,a := 2 (· − a) − →1 as a → ∞, we have by (2.21), −i tr ρ (H ) [H, 1 ] = −i lim tr ρ (H ) [H, 1 ] 2,a a→∞ " ! = −i lim tr ρ (H a ) H a , 1 2 . a→∞

(2.30)

Here H a is the operator on 2 (Z × Za ) obtained from H by a shift (0, −a); it is not the restriction to Z × Za of a fixed Bulk Hamiltonian HB , as Ha was, but instead of an equally shifted one, HBa . The estimates (1.1, 1.6) therefore still apply, which is all that matters for (2.12, 2.13). The r.h.s. of (2.30) thus equals lima→∞ tr Ba (0), where Ba (t) pertains to HBa . Since the sum in (2.22) vanishes, tr Ba (t) = σBa , which is independent of a. 3. Details of the Proof We give some details about the Helffer-Sjöstrand representations (2.6). The integral is over z = x + iy ∈ C with measure dm(z) = dxdy, ∂z¯ = ∂x + i∂y , and ρ(z) is a quasianalytic extension of ρ(x) which, see [18], for given n can be chosen so that

dm(z) |∂z¯ ρ(z)| |y|−p−1 ≤ C

n+2 (k) ρ k=0

k−p−1

(3.1)

k for p = 1, ..., n, provided the appearing norms f k = dx(1 + x 2 ) 2 |f (x)| are finite. This is the case for ρ with ρ ∈ C0∞ (R). For p = 1 this shows that (2.6b) is norm convergent. The integral (2.6a), which would correspond to the case p = 0, is nevertheless a strongly convergent improper integral, see e.g., (A.12) of [12]. A further preliminary is the Combes-Thomas bound [8] δ(x) Ra (z)e−δ(x) ≤ e where δ can be chosen as

C , | Im z|

δ −1 = C 1 + | Im z|−1

(3.2)

(3.3)

for some (large) C > 0 and (x) is any Lipschitz function on Z2 with |(x) − (y)| ≤ |x − y| (see e.g. [2, Appendix D] for details).

(3.4)


199

Lemma 5. We have [Ha , 1 ] 2;a (t) ∈ I1 ,

(3.5)

and for ρ ∈ C ∞ (R) with supp ρ compact also [ρ(Ha ), 1 ] 2;a (t) ∈ I1 .

(3.6)

In particular, Z(a, t) as given in (2.8) is trace class. Proof. We first prove the finite propagation speed estimate (see [13] and [23]): Let µ > 0 be as in (1.1). Then, for 0 ≤ δ ≤ µ and as (3.4), δ(x) iHa t −δ(x) e e (3.7) ≤ eC|t| e for some C < ∞. Indeed, let A(t) = eδ(x) eiHa t e−δ(x) , d −δ(x) −iHa t 2δ(x) iHa t −δ(x) d e e e e = A(t)∗ BA(t) , A(t)∗ A(t) = e dt dt ! " where B = −ie−δ(x) Ha , e2δ(x) e−δ(x) has matrix elements

iB(x, x ) = Ha (x, x )(eδ((x )−(x)) − eδ((x)−(x )) ). By (1.1) which, as remarked in the Introduction, is inherited by Ha , and by Holmgren’s bound B ≤ max sup |B(x, x )| , sup |B(x, x )| , (3.8) x

x

x

x

we have 2C := B < ∞ and hence A(t)2 = A(t)∗ A(t) ≤ e2C|t| . We factorize [Ha , 1 ] 2;a (t) = [Ha , 1 ] eδ|x1 | · e−δ|x1 | e−δ|x2 | · eδ|x2 | 2;a (t) , and note that

−δ|x1 | −δ|x2 | e ≤ Cδ −2 , e

(3.9)

1

since this is a summable function of (x1 , x2 ) ∈ Z2 . It is therefore enough for (3.5) to show (3.10) [Ha , 1 ] eδ|x1 | ≤ C , δ|x2 | 2;a (t) ≤ Ceδa (3.11) e for small δ, where the first estimate also holds for a = B. Indeed, the first operator has matrix elements

T (x, x ) = Ha (x, x )(1 (x ) − 1 (x))eδ|x1 | .

200


They vanish if |x1 − x1 | ≤ |x1 | since x1 ≥ 0 (resp. x1 < 0) then implies the same for x1 . Therefore T (x, x ) ≤ 2 Ha (x, x ) eδ|x1 −x1 | ≤ 2 Ha (x, x ) eδ|x−x | ,

which together with T (x, x) = 0 yields |T (x, x )| ≤ C|Ha (x, x )|(eδ|x−x | − 1). Now (3.10) follows from (1.1) and (3.8). The estimate for eδ|x2 | 2;a (t) = eδ|x2 | eiHa t e−δ|x2 | · eδ|x2 | 2 e−iHa t follows from (3.7) and from eδ|x2 | 2 = eδa < ∞. The proof of (3.6) is similar: Using (2.6) we write 1 dm(z)∂z¯ ρ(z) [Ra (z), 1 ] [ρ(Ha ), 1 ] = 2π

(3.12)

and claim that [Ra (z), 1 ] eδ|x1 | ≤

C | Im z|2

(3.13)

for δ = δ(z) as in (3.3). Together with (3.1, 3.9, 3.11) this implies (3.6). To derive (3.13), note that the operator to be bounded is −Ra (z)[Ha , 1 ]eδ|x1 | · e−δ|x1 | Ra (z)eδ|x1 | and the bound follows from (3.2, 3.10). The conclusion about Z(a, t) follows from (3.6) at t = 0 and (3.1, 3.5). 3.1. Proof of Lemma 1. It follows from (3.6) that a (t) is trace class. While (3.13) holds uniformly in a, including the bulk case, (3.11) fails in this respect. Nevertheless B (t) ∈ I1 , since (3.14) sup eδ|x2 | (2;a (t) − 2 ) ≤ C a,B

for t in a compact interval. In fact e

δ|x2 |

2;a (t) − 2 = e

δ|x2 |

t

eiHa s i [Ha , 2 ] e−iHa s ds

0

with

sup eδ|x2 | eiHa t [Ha , 2 ] e−iHa t ≤ C ,

(3.15)

a,B

because of (3.7) and of eδ|x2 | [Ha , 2 ] ≤ C, cf. (3.10). To prove (2.12) we use (3.12) and Ja∗ Ja = 1 to write 1 ∗ dm(z)∂z¯ ρ(z) Ja a (t)Ja = − 2π

×Ja [Ra (z), 1 ] eδ|x1 | Ja∗ · e−δ|x1 | e−δ|x2 | · Ja eδ|x2 | 2;a (t) − 2 Ja∗ .


201

It is enough to establish convergence to the bulk expression pointwise in z, since domination is provided by (3.13, 3.9, 3.14, 3.1). We thus may show s

Ja [Ra (z), 1 ] eδ|x1 | Ja∗ −−−→ [RB (z), 1 ] eδ|x1 | , a→∞

s Ja 2;a (t) − 2 Ja∗ eδ|x2 | −−−→ (2B (t) − 2 ) eδ|x2 | .

(3.16) (3.17)

a→∞

Since the l.h.s.’s are uniformly bounded in a by (3.13, 3.14) it suffices to prove convergence on the dense subspace of compactly supported states in 2 (Z×Z), which amounts to dropping eδ|xi | in (3.16, 3.17). Equation (1.5) implies the geometric resolvent identity Ja Ra (z) − RB (z)Ja = −RB (z)Ea Ra (z), and by taking the adjoint

s Ja Ra (z)Ja∗ − RB (z) = − Ja Ra (z)Ea∗ + 1 − Ja Ja∗ RB (z) −−−→ 0, a→∞

s

s

because Ea∗ −−−→ 0 by (1.6) and because 1 − Ja Ja∗ −−−→ 0 is the projection onto a→∞ a→∞ states supported in {x2 < −a}. This implies [30, Thm. VIII.20] s-lim Ja f (Ha )Ja∗ = f (HB )

(3.18)

a→∞

for any bounded continuous function f , and in particular the modified limits (3.16, 3.17). The proof of (2.13) is similar. We write the integrand of Ja a (t)Ja∗ as " ! Ja [Ra (z), 1 ] eδ|x1 | Ja∗ · e−δ|x1 | e−δ|x2 | · Ja eδ|x2 | Ha , 2;a (t) Ra (z)Ja∗ . Since the estimates for the first two factors have already been given, all we need are ! " sup eδ|x2 | Ha , 2;a (t) ≤ C , !

a,B

" s Ja Ha , 2;a (t) Ja∗ −−−→ [HB , 2B (t)] . a→∞

The first estimate is just (3.15) and the second is again implied by (3.18).

3.2. Proof of Lemma 2. Let Pλ⊥0 = 1 − Pλ0 . By the definition (1.4) we have σB (λ0 ) = i tr Pλ0 1 Pλ⊥0 2 Pλ0 − Pλ0 2 Pλ⊥0 1 Pλ0 . Since the two terms are separately trace class by (A.2), we also have −iσB (λ0 ) = tr Tλ0 with Tλ0 as in (2.16); see (2.2). Now (2.18) yields   −iσB (λ0 ) = tr E− Tλ0 E− + E+ Tλ0 E+ + E{λ} Tλ0 E{λ}  , λ∈E

and the claim follows from " ! tr P Tλ0 P = tr P Pλ0 , 1 2 P

202


for P = P ∗ with P Pλ⊥0 = 0 or P Pλ0 = 0, since one or the other holds true for P = E± , E{λ} . Indeed, in the first case, which also entails Pλ⊥0 P = 0, we have

P Tλ0 P = P Pλ0 1 Pλ⊥0 2 Pλ0 P = P Pλ0 1 2 − 1 Pλ0 2 P ! " = P Pλ0 , 1 2 P . The other case is similar:

P Tλ0 P = −P Pλ⊥0 1 Pλ0 2 Pλ⊥0 P = −P Pλ⊥0 1 2 − 1 Pλ⊥0 2 P # $ " ! = −P Pλ⊥0 , 1 2 P = P Pλ0 , 1 2 P .

3.3. Consequences of localization. We now discuss the technical consequences of assumption (1.2). In fact, all that we say in this section is a consequence of the following (weaker) estimate: g(HB )(x, x ) e−ε|x| eµ|x−x | =: Dε < ∞ , sup (3.19) g∈B1 ()

x,x ∈Z2

for every ε > 0, where the factor (1+|x|)−ν of (1.2) has been replaced by an exponential. Note that (3.19) follows from (1.2) since e−ε|x| ≤ Cε,ν (1 + |x|)−ν . (We require (1.2) to prove integrality of 2πσB (Prop. 3 below), otherwise (3.19) would suffice for the results described here.) In terms of operators, rather than of matrix elements, (3.19) implies that for some µ > 0 and all ε > 0, sup eµ(x) e−ε|x| g(HB )e−µ(x) ≤ Dε < ∞ , (3.20) g,

where the supremum with g ∈ B1 () is also taken over Lipschitz functions as in (3.4). In fact, the norm in (3.20) is estimated by Holmgren’s bound (3.8) as the larger of sup eµ((x)−(x )) e−ε|x| g(HB )(x, x ) (3.21) x

x

and a similar quantity with x, x under the supremum and summation interchanged. After bounding the supremum by a sum, both quantities are estimated by (3.19). Conversely, we take (x) = |x − x | and consider the (x, x ) matrix element of the operator in (3.20), eµ|x−x | e−ε|x| g(HB )(x, x ) ≤ Dε . (3.22) The sum in (3.19) is finite if µ is replaced there by µ/2 and ε by 2ε. We say that a bounded operator X is confined in direction i (i = 1, 2) if for some δ > 0 and all (small) ε > 0, −ε|x| δ|xi | X(i) (3.23) := e 0. In fact, Xg(HB )e−ε|x| eδ|x2 | ε ε ε ε ≤ Xe− 2 |x| eδ|x2 | · e−(δ|x2 |− 2 |x|) g(HB )e− 2 |x| e(δ|x2 |− 2 |x|) , and for sufficiently small ε, δ > 0 the Lipschitz norm of δ|x2 | − 2ε |x| is smaller than µ, whence (3.20) applies. Lemma 6. Let S ⊂ R be a Borel set that either contains or is disjoint from {λ|λ < } and similarly for {λ|λ > }, i.e., ES ∈ B1 (). Let X be a confined operator in direction i (i = 1, 2). i) The following operators are also confined in direction i, as indicated by the estimates: (i) [X, g(HB )](i) ε,δ ≤ C X 2ε ,δ , (g ∈ B1 ()) , ⊥ E XES (i) ≤ C X(i) . ε S ,δ ε,δ

(3.26) (3.27)

2

ii) If in addition S ⊂ , then the following operators are also confined: ! " HB , AT ,B (X) ES (i) ≤ C X(i) , ε ε,δ 2 ,δ T

(i) (i) AT ,B (X) − X ES ≤ C X ε ,δ , ε,δ

(3.28) (3.29)

2

and given S ⊂ R with d = dist(S, S ) > 0, ES AT ,B (X)ES (i) ≤ C X(i) . ε ε,δ 2 ,δ T

(3.30)

(i)

iii) Properties (i, ii) also hold for X = i , with Xε,δ replaced by 1. The constants C depend on ε, δ, but not on the remaining quantities, except for (3.30) which depends on d. The main use of confined operators will be through the following remark: If Xi , (i = 1, 2), is confined in direction i, then X2 X1∗ ∈ I1 with X2 X ∗ ≤ C X2 (2) X1 (1) 1 1 ε,δ ε,δ

(3.31)

for 2ε < δ. In particular, if also X1∗ X2 ∈ I1 , (3.31) is a bound for tr X1∗ X2 = tr X2 X1∗ . Indeed, (3.31) follows from e−δ|x2 | e2ε|x| e−δ|x1 | = e−(δ−2ε)|x| ∈ I1 .

204


3.4. Proof of Lemma 6. For X confined, (3.26) is implied by (3.24, 3.25). We thus consider X = i , where it is enough to estimate [i , g(HB )] e−ε|x| e±δxi = i g(HB )(1 − i )e−ε|x| e±δxi +(1 − i )g(HB )i e−ε|x| e±δxi . In the + case, for instance, the second term is bounded because i eδxi is. By (3.20) this holds for the first one too. From now on the switch functions and the confined operators will be treated simultaneously. Equation (3.27) follows from (3.26) and ES⊥ XES = ES⊥ [X, ES ]. To prove (3.28) we consider ! " T · i HB , AT ,B (X) ES = (eiHB T Xe−iHB T − X)ES = eiHB T Xe−iHB T ES − e−iHB T ES X ES − ES⊥ XES . (3.32) The term in parentheses is bounded by (3.26) for g(λ) = e−iλT ES (λ). The norm (3.23) of (3.32) is uniformly bounded in T ∈ R by (3.24, 3.25, 3.27). The same bound applies to 1 T dt (eiHB t Xe−iHB t − X)ES . (AT ,B (X) − X)ES = T 0 We now turn to (3.30), which is related to an integration by parts lemma of [19]. Since S ⊂ and d > 0, there is a contour γ in the complex plane (of length ≤ 4|| + 2d) encircling S once, but not S , at a distance ≥ d/2 from both. Then 1 X = dzR(z)ES XES R(z) 2π γ is convergent in the norm (3.23) because of (3.24, 3.25, 3.27) (note that (2/d)·ES (λ)(z− λ)−1 ∈ B1 ()). Its commutator with HB is ! " =− 1 i HB , X dz [HB − z, R(z)ES XES R(z)] 2πi γ 1 dz(ES XES R(z) − R(z)ES XES ) = ES XES . =− 2πi γ ! " ES and the claim Therefore, ES AT ,B (X)ES = AT ,B (ES XES ) = ES i HB , AT ,B (X) follows from (3.28). 3.5. Proof of Lemma 3. We first prove (2.23) and begin by recalling, see (2.10, 1.14), that 1 T i tr E B (t)E = dm(z)∂z¯ ρ(z) tr E R(z) [HB , 1 ] · T 0 2π ! " ·R(z) HB , AT ,B (2 ) R(z)E . (3.33) By (3.24, 3.25, 3.28) we have for small δ > 0, ! " R(z) HB , AT ,B (2 ) R(z)E (2) ≤ C | Im z|−2 , ε,δ T


205

and, together with (3.10), −1 [HB , 1 ] R(z)E (1) . ε,δ ≤ C| Im z|

By (3.31) the trace in (3.33) is bounded by a constant times T −1 | Im z|−3 . As the constant is independent of z, (2.23) now follows by means of (3.1). The operator under the trace in (2.24) is E Pλ0 1 (AT ,B (2 ) − 2 )E − E 1 Pλ0 (AT ,B (2 ) − 2 )E = E Pλ0 1 Pλ⊥0 · (AT ,B (2 ) − 2 )E −E Pλ⊥0 1 Pλ0 · (AT ,B (2 ) − 2 )E .

(3.34)

We claim that the two terms on the r.h.s. are separately trace class. In fact (3.27) implies Pλ0 1 Pλ⊥0 e−ε|x| eδ|x1 | ≤ C, and similarly with Pλ0 , Pλ⊥0 interchanged, and the bound (3.14) also applies with AT ,B (2 ) in place of 2,B (t). (Note however that the bound so obtained is not uniform in T .) A factor Pλ0 , resp. Pλ⊥0 , may now be cycled around the traces of the two terms on the r.h.s. of (3.34). The trace (2.24) thus equals tr E Pλ0 1 Pλ⊥0 · Pλ⊥0 AT ,B (2 )Pλ0 E − tr E Pλ⊥0 1 Pλ0 · Pλ0 AT ,B (2 )Pλ⊥0 E − tr E Tλ0 E ,

(3.35)

where we used that the two terms of Tλ0 , see (2.16), are separately trace class. We next show that the first two terms of (3.35) are uniformly bounded in λ0 ∈ , T > 0. Indeed, X1 = Pλ⊥0 1 Pλ0 E and X2 = Pλ⊥0 AT ,B (2 )Pλ0 E = Pλ⊥0 (AT ,B (2 )− 2 )Pλ0 E + Pλ⊥0 2 Pλ0 E are uniformly confined by (3.27, 3.29) and the conclusion is by (3.31). Finally, we will show that these two terms vanish as T → ∞, pointwise in λ0 ∈ . The first one is split according to Pλ0 = Pλ + (Pλ0 − Pλ ) for any λ < λ0 , λ ∈ : tr Pλ⊥0 AT ,B (2 )Pλ0 E · E Pλ0 1 Pλ⊥0 = tr Pλ⊥0 AT ,B (2 )Pλ E · E Pλ0 1 Pλ⊥0 + tr Pλ⊥0 AT ,B (2 )Pλ0 E · (Pλ0 − Pλ ) · E Pλ0 1 Pλ⊥0 ≡ I + II . In II, we extract the weights of the confined operators, so that the middle factor becomes e2ε|x| e− 2 (|x1 |+|x2 |) · e−ε|x| e 2 (|x1 |−|x2 |) (Pλ0 − Pλ )e 2 (|x2 |−|x1 |) e−ε|x| · δ

δ

δ

· e− 2 (|x1 |+|x2 |) e2ε|x| . δ

For δ/2 > 2ε the operators on the sides are trace class, and the middle one is uniformly bounded in λ ∈ by (3.20). Moreover, it converges weakly to zero as λ ↑ λ0 , as this s holds true by Pλ0 − Pλ − → 0 for matrix elements between states from the dense subspace of compactly supported states in 2 (Z2 ). Using w

Xn − →0,

Y1 , Y2 ∈ I1

⇒

Y1 Xn Y2 1 → 0 ,

206


we conclude that II can be made uniformly small in T by picking λ close to λ0 . The term I is then seen to be O(T −1 ) by (3.30) with S = (−∞, λ) ∩ and S = [λ0 , ∞). The second trace in (3.35) is dealt with slightly differently. We insert Pλ0 = Pλ + E (Pλ0 − Pλ )E for λ < λ0 , λ ∈ , which yields two well-defined traces. The second can be made uniformly small in T , as was the case for II above. The first one, which by (2.2) equals tr Pλ AT ,B (2 )Pλ⊥0 E · E Pλ⊥0 1 Pλ , is O(T −1 ) by (3.30), this time with S = [λ0 , ∞) ∩ , S = (−∞, λ). 3.6. Proof of Lemma 4. We shall need a particular choice of basis ψλ;j for ran E{λ} , which is related to a SULE basis [11]. (The issue is only of relevance if λ ∈ E is degenerate, since otherwise ψλ is unique up to a phase.) a basis

can be chosen ) We claim so that (3.20) applies not only to g(Hλ ) = E{λ} = ψλ;j ψλ;j , · , but also to the rank one projections into which it is decomposed (upon changing µ, Dε , depending on C4 ). Since φ (ψ, · ) = φ ψ, this amounts to (3.36) sup eµ(x) e−ε|x| ψλ;j e−µ(x) ψλ;j ≤ Dε . )

In fact, since x E{λ} (x, x)=tr E{λ}≤C4 , we may pick x0 ∈ Z2 such that E{λ} (x0 , x0 ) = maxx E{λ} (x, x). Let ψλ;0 (x) = E{λ} (x, x0 )/E{λ} (x0 , x0 )1/2 . This normalized eigenfunction satisfies the bounds ε|x0 | e−µ|x−x0 | /E (x , x )1/2 , {λ} 0 0 ψλ;0 (x) ≤ Dε e E{λ} (x0 , x0 )1/2 . The first one follows from (3.22) for g(HB ) = E{λ} , and the second from E{λ} (x, x0 ) ≤ E{λ} (x, x)1/2 E{λ} (x0 , x0 )1/2 ≤ E{λ} (x0 , x0 ) . 1

µ

Combining them into a geometric mean yields |ψ(x)| ≤ Dε2 e 2 |x0 | e− 2 |x−x0 | and, by the triangle inequality, µ µ ψλ;0 (x)ψ λ;0 (x ) ≤ Dε eε|x0 | e− 2 (|x−x0 |+|x −x0 |) ≤ Dε eε|x| e−( 2 −ε)|x−x | .

For small ε the bound (3.22) is reproduced for ψλ;0 ψλ;0 , · in place of E{λ} , with a smaller value of µ. Since the rank of E{λ} − ψλ;0 ψλ;0 , · is one less than the rank of E{λ} , the task is completed by induction. After these preliminaries, we turn to the proof of Lemma 4 proper. We denote by E the eigenvalues in E listed according to multiplicity. More precisely, we let E be the set of pairs ζ = (λ; n) with λ ∈ E and n a non-negative integer less than the multiplicity of λ. The eigenvectors {ψζ , ζ ∈ E } constructed above are an ortho-normal basis for ranE . Let, for ζ ∈ E ,

Mζ = min 1 ψζ , (1 − 1 )ψζ , 2 ψζ , (1 − 2 )ψζ . We claim that

ζ ∈E

Mζ < ∞ .

ε

(3.37)


207

This states that almost all eigenfunctions are localized in at least one among the left, right, upper, and lower half planes, and hence in at most two (intersecting) ones. In particular almost no eigenfunction encircles the origin, which makes them insensitive to a flux tube applied there – a fact

used in some explanations [17, 28] of the QHE. We apply (3.36) to ψζ ψζ , · and use that for rank one operators φ (ψ, ·) = φ ψ to obtain eµ(x) e−ε|x| ψζ e−µ(x) ψζ ≤ Dε . For (x) = x1 we have 1 (x) ≤ e−µ(x) , implying −1 Dε 1 ψζ ≥ eµx1 e−ε|x| ψζ , similar estimates for 1 − 1 , 2 , and 1 − 2 have x1 on the r.h.s. replaced by −x1 , x2 , and −x2 respectively. Therefore, −2 −2 −2 −2 Mζ−2 = max 1 ψζ , (1 − 1 )ψζ , 2 ψζ , (1 − 2 )ψζ 1 1 ψζ −2 + (1 − 1 )ψζ −2 + 2 ψζ −2 + (1 − 2 )ψζ −2 ≥ 4 1 −2ε|x| 2µx1 −2µx1 2µx2 −2µx2 ≥ ψ e ψζ , e + e + e + e ζ 4Dε2 1 (µ−2ε)|x| , e ψ ψ ≥ ζ ζ 4Dε2 −1 1 −(µ−2ε)|x| ψ , e ψ , ≥ ζ ζ 4Dε2 where we use e2µ|x1 | + e2µ|x2 | ≥ eµ(|x1 |+|x2 |) and, in the last step, the Cauchy-Schwarz inequality 2

2 δ δ 1 = ψζ , ψζ = e 2 |x| ψζ , e− 2 |x| ψζ ≤ ψζ , eδ|x| ψζ ψζ , e−δ|x| ψζ . Now let ε > 0 be small enough that δ := µ − 2ε > 0. We can then estimate the distribution function N(t) of the Mζ : + * t2 N(t) := # ζ ∈ E | Mζ > t ≤ # ζ ∈ E | ψζ , e−δ|x| ψζ > 4Dε2 * + t2 ≤ # x ∈ Z2 | e−δ|x| > 4Dε2 ≤ C ln t , where in the step before last we used the min-max principle, see e.g., [29, Theorem XIII.1]. Together with N (t) = 0 for t ≥ 1, we have ζ ∈E

proving (3.37).

Mζ = −

1

0+

t dN (t) =

1 0+

dt N (t) < ∞ ,

208


We can now estimate the traces in (2.26): tr E{λ} e−iHB t [HB , 1 ] eiHB t 2 E{λ} ≤ ψζ , [HB , 1 ] eiHB t 2 ψζ . ζ =(λ;·)

(3.38) By inserting 2 = 1 − (1 − 2 ), the terms on the right-hand side may also be expressed as ψζ , [HB , 1 ] eiHB t (1 − 2 )ψζ . Using

ψζ , [HB , 1 ] φ = 1 ψζ , (λ − HB )φ = − (1 − 1 )ψζ , (λ − HB )φ , ) one sees that (3.38) is bounded by a constant times ζ =(λ;·) Mζ , so the right-hand side ) of (2.26) is bounded by ζ Mζ . 4. Analysis of the Harper Hamiltonian In this section we prove Theorem 3 which shows that the contribution from bulk states in (1.12) can be non-zero. We begin with the following proposition: Proposition 1. Let f ({Vx }x∈Zd ) be a function which is bounded and continuous in the Zd product topology on {Vx }x∈Zd | Im Vx ≤ 0 = C− . If f is separately analytic in each Vx , then E (f ) = f ({−i}x∈Zd ) ,

(4.1)

where E (·) represents the average with respect to the product measure dP({Vx }x∈Zd ) :=

, x∈Zd

dVx , π(1 + Vx2 )

d supported on {Vx }x∈Zd |Vx ∈ R = RZ . The same statement holds for C+ , +i in place of C− , −i. Proof. Let Sj be an increasing sequence of finite sets with limj Sj = ∪j Sj = Zd , and let Fjc denote the σ -algebra generated by {Vx }x∈Sjc . So conditional expectation with respect to Fjc is given by “averaging out” the variables {Vx }x∈Sj . Thus fj ({Vx }x∈Sjc ) := E

f |Fjc

=

, x∈Sj

dVx f ({Vx }x∈Sj × {Vx }x∈Sjc ) . π(1 + Vx2 )

Because f is bounded and separately analytic in each Vx , we may evaluate the integrals on the right-hand side by residues to obtain fj ({Vx }x∈Sjc ) = f ({−i}x∈Sj × {Vx }x∈Sjc ) .


209

Because f is continuous and limj →∞ {−i}x∈Sj × {Vx }x∈Sjc = {−i}x∈Zd in the product topology on C−

Zd

, we have lim fj ({Vx }x∈Sjc ) = f ({−i}x∈Zd )

j →∞

d for any {Vx }x∈Zd ∈ RZ . Since fj are uniformly bounded and E fj = E (f ) for every j , we conclude by dominated convergence that (4.1) holds. Turning now to the proof of Theorem 3, we first recall that, by Lemma 1, −

i lim tr ρ (Ha ) {[Ha , 1 ] , 2 } = Re tr B (0) , 2 a→∞

where iB (0)

1 =− 2π

(4.2)

! ! " " dm(z)∂z¯ ρ(z) tr RB (z) Hφ , 1 RB (z) Hφ , 2 RB (z) . TB (z)

In going from (2.10) to the above expression for B (0) we have replaced HB by Hφ in the commutators [HB , i ] since the random potential commutes with each switch function i . By Lemma 1, we have supa tr ρ (Ha ) { [Ha , 1 ] , 2 } ≤ C < ∞, with a constant C that depends on ρ and on the bounds C1 , C3 in (1.1, 1.6), but not on the random constant C2 in (1.2). Since the constants C1 , C3 are non-random in our setup, the expectation in (1.22) is well defined, and furthermore can be exchanged with the limit. We claim that for Im z = 0, E (tr TB (z)) = tr Tφ (z + iασ (z)) , (4.3) ! ! " " where Tφ (z) = Rφ (z) Hφ , 1 Rφ (z) Hφ , 2 Rφ (z), with Rφ (z) = (Hφ − z)−1 , and σ (z) = Im z/| Im z| denotes the sign of the imaginary part of z. Indeed, for Im z > 0, it suffices to verify that fz ({Vx }) = tr TB (z) obeys the hypotheses of Proposition 1. For that purpose, it is useful to note that Gz ({Vx }x∈Zd ) := (Hφ + αV − z)−1 is a continuous map from {Vx }x∈Zd | Im Vx ≤ 0 to the bounded operators on 2 (Z2 ) endowed with the strong operator topology. Indeed, z is in the resolvent set of Hφ + αV since the numerical range of this operator is contained in the closed lower half plane. Thus Gz is well defined, SOT-continuous (since {Vx }x → Hφ + αV and A → A−1 are SOT-continuous), and Gz ({Vx }

≤

x∈Zd )

1 1 . ≤ dist(z, num. range(Hφ + αV )) | Im z|

Furthermore, the Combes-Thomas bound (3.2) extends to Gz , i.e., C δ(x) Gz ({Vx }x∈Zd )e−δ(x) ≤ , δ −1 = C 1 + | Im z|−1 , e | Im z|

(4.4)

(4.5)

210


with (x) as in (3.4). The resolvent of e±δ(x) (Hφ + αV )e∓δ(x) , considered as a perturbation of Hφ + αV , is in fact as stable as in (3.2), where Hφ was self-adjoint, since the same bound (4.4) still holds for Im z > 0. Furthermore, we see in this way that {Vx }x∈Zd → eδ(x) Gz ({Vx }x∈Zd )e−δ(x) is SOT-continuous. Thus, for Im z > 0,

" ! tr TB (z) = tr Gz ({Vx }x∈Zd ) Hφ , 1 eδ|x1 | · e−δ|x1 | Gz ({Vx }x∈Zd )eδ|x1 | · ! " · e−δ(|x1 |+|x2 |) · eδ|x2 | Hφ , 2 Gz ({Vx }x∈Zd ) ,

is a continuous function, which is bounded by |tr TB (z)| ≤

! " δ|x | C Gz ({Vx } d )2 H e 1 · , φ 1 x∈Z δ2 ! " · e−δ|x1 | Gz ({Vx }x∈Zd )eδ|x1 | eδ|x2 | Hφ , 2

≤C

(4.6)

(1 + | Im z|−1 )2 , | Im z|3

with the factor of 1/δ 2 coming from the estimate (3.9) on the trace of e−δ|x| . A similar argument is used for Im z < 0. Since the separate analyticity of fz (·) = tr TB (z) is clear, Proposition 1 applies. We see that

1 (4.7) Im dm(z)∂z¯ ρ(z) tr Tφ (z + iασ (z)) , E Re tr B (0) = − 2π

where the interchange of dm(z) and E is justified by Fubini’s theorem and (4.6) since we may arrange for ∂z¯ ρ(z) to vanish faster than | Im z|5 as z approaches the real axis. We note that Cα tr Tφ (z + iασ (z)) ≤ . (4.8) [x 2 + (|y| + α)2 ]3/2 In fact, now that V = 0, | Im z|−1 in (4.4) may be replaced by dist(z, σ (Hφ ))−1 ≤ dist(z, [−2, 2])−1 and the same replacement carries over to the denominator in the estimate (4.6) for tr Tφ (z). The only singularities in the integrand on the right - hand side of (4.7) are jump discontinuities at Im z = 0. Integrating by parts, on the upper and lower half planes separately, we find ∞

1 E Re tr B (0) = dxρ(x) tr Tφ (x + αi) − Tφ (x − αi) , (4.9) Re 2π −∞ since by (4.8)

∞ there are no contributions from the boundary at infinity. Upon writing ρ(x) = − x ρ (λ)dλ, and interchanging λ and x integration we obtain ∞ λ

1 E Re tr B (0) = − dλρ (λ) Re tr Tφ (x + αi) − Tφ (x − αi) dx . 2π −∞ −∞ (4.10)


211

This proves (1.22) with jB (λ) =

1 Re 2π

λ

−∞

tr Tφ (x + αi) − Tφ (x − αi) dx .

To obtain the asymptotic expression (1.23), note that for |λ| > 2, α 1 idη tr Tφ (λ + iη) , Re jB (λ) = 2π −α

(4.11)

(4.12)

because the difference of the right-hand sides of (4.11, 4.12) is the real part of an integral around a closed contour, which may be deformed to infinity, of the analytic function tr Tφ (z), which vanishes like 1/|z|2 as z → ∞. (It is of interest to note that for λ in an internal gap of the spectrum of Hφ , the corresponding contour integral gives (φ) the bulk conductance σB (λ) for the Hamiltonian Hφ at Fermi energy λ, so jB (λ) =

(φ) α 1 σB (λ) + 2π Re i −α dη tr Tφ (λ + iη).) It is useful to rewrite (4.12) as α 1 idη tr Tφ (λ + iη) − tr Tφ (λ − iη) , (4.13) Re jB (λ) = 2π 0 which follows by considering the contributions from η < 0 and η > 0 separately, and using Re i w = − Re i w. We obtain (1.23) from the series for TB (λ+iη)−tr TB (λ − iη) produced by expanding each resolvent in a Neumann series. For sufficiently large |λ|, Rφ (λ + iη) = −

. ∞ 1 Hφ − iη n λ λ

(4.14)

n=0

is absolutely convergent, and ∞ "

n ! 1 1 tr Hφ − iη 1 Hφ , 1 · λ3 λN N=0 n1 +n2 +n3 =N

n2 !

n " · Hφ − iη Hφ , 2 Hφ − iη 3 .

tr Tφ (λ + iη) = −

To prove convergence here, it is useful to note that in addition to (4.14), the series e

δ|x|

Rφ (λ + iη)e

−δ|x|

.n ∞ 1 eδ|x| Hφ e−δ|x| − iη = − λ λ n=0

is also absolutely convergent, in light of (1.1). By cyclicity of the trace tr Tφ (λ + iη) = −

∞ N=0

1 λN+3

N n=0

n ! " (n + 1) tr Hφ − iη Hφ , 1 ·

"

N−n ! · Hφ − iη Hφ , 2 ,

212


and, making use of the identity tr T = tr T ∗ , ∞

N

"

n ! (N − n + 1) tr Hφ − iη Hφ , 1 · λN+3 n=0 N=0 "

N−n ! · Hφ − iη Hφ , 2 .

tr Tφ (λ − iη) = −

1

Thus tr Tφ (λ + iη) − tr Tφ (λ − iη) = −

∞ N=0

N

1 λN+3

n !

N−n ! " " (2n − N ) tr Hφ − iη Hφ , 1 · Hφ − iη Hφ , 2 ,

n=0

which is the desired expansion. The first term (N = 0) of this series vanishes trivially. The second (N = 1) also vanishes, because ! " " "! "

!

! tr Hφ , 1 Hφ − iη Hφ , 2 − tr Hφ − iη Hφ , 1 Hφ , 2 ! ! "" ! " = − tr Hφ , Hφ , 1 Hφ , 2 ! ! "" ! " =− Hφ , Hφ , 1 (x, y) Hφ , 2 (y, x) = 0 , (4.15) x

y

" ! "" ! ! since Hφ , 2 (y, x) = 0 only for |x − y| = 1 and Hφ , Hφ , 1 (x, y) = 0 only for |x − y| = 0, 2, as only nearest neighbor hopping terms are present in Hφ . However the coefficient of λ−5 (N = 2) is non-zero, and given by ! " " "! "

2 !

2 ! 2 tr Hφ , 1 Hφ − iη Hφ , 2 − 2 tr Hφ − iη Hφ , 1 Hφ , 2 ! " ! " ! "! " = 2 tr Hφ , 1 Hφ2 Hφ , 2 − 2 tr Hφ2 Hφ , 1 Hφ , 2 !! " ! "" = −2 tr Hφ2 Hφ , 1 , Hφ , 2 , 2 since the term proportional " ! by (4.15) "" and the term proportional to η is the !! to η vanishes trace of a commutator, tr Hφ , 1 , Hφ , 2 = 0. To calculate this term explicitly, recall that i = I [xi < 0] so, by (1.21), " ! Hφ , 1 (x, x ) = (1 (x ) − 1 (x))Hφ (x, x )   x = (0, x2 ) , x = (−1, x2 ) , 1 , = −1 , x = (−1, x2 ) , x = (0, x2 ) ,  0 , all other x, x ,

which is more succinctly expressed in Dirac notation: " ! |0, a −1, a| − |−1, a 0, a| . Hφ , 1 = a∈Z

Similarly, " eiφa |a, 0 a, −1| − e−iφa |a, −1 a, 0| . Hφ , 2 =

!

a∈Z


Thus

213

!! " ! "" Hφ , 1 , Hφ , 2 = (e−iφ − 1) |0, 0 −1, −1| + |−1, 0 0, −1| −(eiφ − 1) |0, −1 −1, 0| + |−1, −1 0, 0| ,

and " ! "" Hφ , 1 , Hφ , 2 = (e−iφ − 1) −1, −1| Hφ2 |0, 0 + 0, −1| Hφ2 |−1, 0 − c.c.

tr Hφ2

!!

Finally, since −1, −1| Hφ2 |0, 0 = 1 + e−iφ ,

0, −1| Hφ2 |−1, 0 = 1 + eiφ ,

we have 2 tr Hφ2

!!

" ! "" Hφ , 1 , Hφ , 2 = 4(e−iφ − 1) (cos(φ) + 1) − c.c. = −8i sin(φ)(cos(φ) + 1) .

Therefore tr Tφ (λ + iη) − tr Tφ (λ − iη) = 8i sin(φ)(cos(φ) + 1)λ−5 + O(λ−6 ) , and jB (λ) = −

4α sin(φ)(cos(φ) + 1)λ−5 + O(λ−6 ) , π

which gives (1.23). This completes the proof of Theorem 3.

Appendix: Conductance Plateaus Localization is an essential prerequisite for the QHE. Some localization condition, valid at energies in an interval , is proven and used in [6, 2]. It ensures that σB (λ) is 1. well defined as given by (1.4), 2. constant in λ ∈ , and 3. 2π σB (λ) ∈ Z. These results also rest on a homogeneity assumption for the Hamiltonian HB , or on its Fermi projections Pλ , namely that they be invariant or ergodic under magnetic translations. The purpose of the Appendix is to establish (1.–3.) under assumptions (1.1–1.3), which do not entail translation invariance. Proposition 2. Assume (1.1) and (1.2). Then σB (λ) is well-defined. If in addition (1.3) holds, then σB (λ) is constant in λ ∈ . Proposition 3. Assume (1.1) and (1.2). Then 2πσB (λ) ∈ Z for λ ∈ . We remark that here constancy is proven without combining integrality and continuity.

214


A.1. Proof of Prop. 2. We consider Borel sets S ⊂ R that either contain or are disjoint from {λ|λ < } and similarly for {λ|λ > }. The class of such sets S is closed under unions and complements. We associate a bulk Hall conductance to S by setting σB (S) = −i tr ES [[ES , 1 ] , [ES , 2 ]] = i tr ES 1 ES⊥ 2 ES − ES 2 ES⊥ 1 ES ,

(A.1)

where ES⊥ = 1 − ES and the second line follows from ES [ES , 1 ] = ES [ES , 1 ] ES⊥ = ES 1 ES⊥ . Note that σB (λ0 ) = σB ((−∞, λ0 )). We claim that, if S1 ∩ S2 = ∅, then ES1 1 ES2 2 ES1 ∈ I1 , σB (S1 ∪ S2 ) = σB (S1 ) + σB (S2 ) ,

(A.2) (A.3)

lim σB (Sn ) = 0 if Sn ↓ ∅ .

(A.4)

and moreover n→∞

In particular, (A.2) and its adjoint for S1 = S, S2 = R \ S imply that the two terms in the final expression of (A.1) are separately trace class. (A.2): In the factorization ES1 1 ES2 2 ES1 = ES1 1 ES2 e3δ|x1 | e−δ|x| · e−δ|x| · e−δ|x| e3δ|x2 | ES2 2 ES1 ,

(A.5)

the middle e−δ|x| = e−δ|x1 | e−δ|x2 | is trace class by (3.9), so that we need to show (A.6) ES1 i ES2 e3δ|xi | e−δ|x| < ∞ , (i = 1, 2) . This follows from (3.25, 3.27) and part (iii) of Lemma 6, with a bound which is uniform in S1 , S2 . (A.4): By (A.1, A.5) and (2.21) it suffices to show s

ESn i ES⊥n e3δ|xi | e−δ|x| −−−→ 0 . n→∞

Since the l.h.s. is uniformly bounded in norm by the remark just made, we may drop the exponentials as explained in connection with (3.16, 3.17). Then the claim becomes obvious. (A.3): From ES1 ∪S2 = ES1 + ES2 and (2.2) we have σB (S1 ∪ S2 ) =

2

tr ESi i ES⊥1 ∪S2 2 ESi − tr ES⊥1 ∪S2 1 ESi 2 ES⊥1 ∪S2

i=1

We use ES⊥1 ∪S2 = ES⊥i − ESi+1 (with i + 1 defined mod 2) and obtain σB (S1 ∪ S2 ) =

2

σB (Si ) −

i=1

+

2

2

tr ESi 1 ESi+1 2 ESi

i=1

tr ESi+1 1 ESi 2 ESi+1

i=1

= σB (S1 ) + σB (S2 ) .

.


215

We finally prove constancy by showing that σB ([a, b]) = 0 for any [a, b] ⊂ . Since σ (HB ) is pure point in we have En :=

n i=1

s

E{λi } −−−→ EE[a,b] = E[a,b] , n→∞

where λi is any labeling of the eigenvalues λ ∈ E[a,b] . Now En is a finite dimensional projection by (1.3), whence the two terms in

σB ∪ni=1 {λi } = −i tr (En 1 En 2 En − En 2 En 1 En ) = 0 are separately trace class. They cancel by (2.2). We conclude by (A.3, A.4) that

σB ([a, b]) = σB ∪ni=1 {λi } + σB E[a,b] \ ∪ni=1 {λi } −−−→ 0 . n→∞

A.2. Proof of Prop. 3. As in [5] we are going to establish that 2π σB (λ) is an integer by relating it to the index of a pair of projections. We first allow the functions i in (1.4) to switch values at points other than the origin. Let p = (p1 , p2 ) ∈ Z2∗ = Z2 + ( 21 , 21 ) be the center of a plaquette and set !! " ! "" σp = −i tr Pλ Pλ , 1,p , Pλ , 2,p ! " ! " ! " ! " = i tr Pλ , 1,p Pλ⊥ Pλ , 2,p − Pλ , 2,p Pλ⊥ Pλ , 1,p , (A.7) where i,p = (xi − pi ), (i = 1, 2). (Since (n) = (n + 21 ) for n ∈ Z, σB (λ) is just σp for p = −( 21 , 21 ).) To define the index, let θp (x) = arg(x −p) be the angle of sight of x ∈ Z2 from p, and set Up (x) = eiθp (x) . The relevant index is Np = Ind(Up Pλ Up∗ , Pλ ), where Ind(P , Q) denotes the index of a pair of projections introduced in ref. [5]: Ind(P , Q) := dim ran P ∩ ker Q − dim ran Q ∩ ker P .

(A.8)

We recall the following basic properties of Ind(·, ·): 1. If P − Q is compact, Ind(P , Q) is well defined and finite. 2. If (P − Q)2n+1 is trace class for some integer n ≥ 0, then tr(P − Q)2n+1 = Ind(P , Q) .

(A.9)

Since Np is an integer by (A.8), Prop. 3 is a consequence of the identity 2πσB (λ) = Np , to be proved below. Indeed, this is the same strategy employed in refs. [5, 2]. The starting point for our proof is the observation that σp and Np are independent of p even without ergodicity for the underlying projection. Lemma 7. The index Np is well defined for any p ∈ Z2∗ , and for any a ∈ Z2 , i) Np+a = Np , ii) σp+a = σp .

216


Proof. Part (i) follows from [5, Prop. 3.8] once we verify that Np is well defined. For this we follow [2] and show that (Pλ − Up Pλ Up∗ )3 is trace class, using Lemma ([2, Lemma 1]). For an operator with the matrix elements Tx,y , 1/3 3 1/3 3 T 3 ≡ (tr |T | ) ≤ |Tx+b,x | . x

b

In our case, with T =

Pλ − Up Pλ Up∗ ,

we have (see [2, Eq. (4.13)])

|T (x + b, x)| = |1 − ei(θp (x+b)−θp (x)) ||Pλ (x + b, x)| |b| |b| ≤ C |Pλ (x + b, x)| ≤ C(1 + |p|) |Pλ (x + b, x)| . 1 + |x − p| 1 + |x| (Here and in the sequel, C denotes a generic constant, whose value is independent of any lattice sites in the given inequality, though that value may change from line to line.) Since (1.2) holds for g(HB ) = Pλ , we have |Pλ (x + b, x)| ≤ C2 (1 + |x|)ν e−µ|b| , but we also have |Pλ (x + b, x)| ≤ 1, because Pλ ≤ 1. Combing these two estimates gives 1 |b| ≤ 2ν µ ln(|x| + 1) , µ (A.10) |Pλ (x + b, x)| ≤ 2ν − 2 |b| |b| > µ ln(|x| + 1) . C2 e Thus

x

1/3 |T (x + b, x)|

3



 ≤ C(1 + |p|)|b| 

µ

#

|x|<e 2ν

≤ C(1 + |p|)|b| e

− µ2 |b|

|b|

µ

C2 e− 2 |b|

−1

+ e

(1 + |x|)3 µ − 6ν |b|

1/3

$3 +

µ

|x|≥e 2ν

|b|

−1

1   (1 + |x|)3

.

Since the last line is clearly summable over b, we see that (Up Pλ Up∗ − P )3 is trace class, and therefore the index Np is well defined. Turning now to part (ii), we note that we may just treat the case p = −( 21 , 21 ), a = (a1 , 0), the case of translation in the 2-direction being similar. By (A.1, A.2, 2.2) we need to show that tr(Pλ (1 )Pλ⊥ 2 Pλ ) − tr(Pλ 2 Pλ⊥ (1 )Pλ ) = tr(Pλ (1 )Pλ⊥ 2 Pλ ) − tr(Pλ⊥ (1 )Pλ 2 Pλ⊥ )

(A.11)

vanishes, where 1 (x) = (x1 ) − (x1 − a1 ) is compactly supported in x1 . We claim that (1 )Pλ⊥ 2 Pλ ∈ I1 . This follows like (A.2) through the factorization (1 )Pλ⊥ 2 Pλ = (1 )e3δ|x1 | e−δ|x| · e−δ|x| · e−δ|x| e3δ|x2 | Pλ⊥ 2 Pλ ,


217

by noticing that the first factor, which is new, is bounded. Likewise (1 )Pλ 2 Pλ⊥ ∈ I1 . Therefore (A.11) equals tr(1 )Pλ⊥ 2 Pλ − tr(1 )Pλ 2 Pλ⊥ = tr(1 ) [2 , Pλ ] = 0 , by evaluating the trace in the position basis.

The proof of Prop. 3 is now completed by the following result, with the translation invariance required in the argument of [5] now provided by Lemma 7. Lemma 8. Let L = {−L, . . . , L}2 ⊂ Z2 . Then + −2i N/2π Pλ (x, y)Pλ (y, z)Pλ (z, x) Area(x, y, z) , = lim 2 σB (λ) L→∞ (2L + 1) 2 y,z∈Z x∈L

(A.12) where N, resp. σB (λ) are the translation invariant values of Np , resp. σp , and Area (x, y, z) is the triangle’s oriented area, namely 21 (x − y) ∧ (y − z). Remark 3. The r.h.s. of (A.12) is the trace per unit volume of −iPλ [[Pλ , X1 ] , [Pλ , X2 ]] , which may be interpreted as the macroscopic version of (1.4). Proof. The first statement makes use of Connes’ area formula [9] in the version [5] adapted to the lattice [2]: For a fixed triplet u(1) , u(2) , u(3) ∈ Z2 , let αi (p) ∈ (−π, π ) be the angle of view from p ∈ Z2∗ of u(i+2) relative to u(i+1) (with αi (p) = 0 if p lies between them). Then 3

sin αi (p) = 2π Area(u(1) , u(2) , u(3) ) .

(A.13)

p∈Z2∗ i=1

By the computation of [5], Np = tr(Up Pλ Up − Pλ )3 = −2i

Pλ (x, y)Pλ (y, z)Pλ (z, x)S(p, x, y, z),

x,y,z∈Z2

with S(p, x, y, z) = sin ∠(x, p, y) + sin ∠(y, p, z) + sin ∠(z, p, x). Letting ∗L = 2 −L + 21 , . . . , L + 21 ⊂ Z2∗ we have that N (2L + 1)2 is the sum of the r.h.s. over ∗ p ∈ L . We would like to replace the sum over x ∈ Z2 , p ∈ ∗L by that over x ∈ L , p ∈ Z2∗ . The error is estimated by |f (p, x)| + |f (p, x)| , (A.14) x∈Z2 \L p∈∗L

x∈L p∈Z2∗ \∗L

218


where f (p, x) := −2i

Pλ (x, y)Pλ (y, z)Pλ (z, x)S(p, x, y, z) .

y,z∈Z2

By (1.2) for g(HB ) = Pλ the points y, z are exponentially clustered around x, so we have |f (p, x)| ≤ Cx (1 + |p − x|)−3 . However because of the pre-factor (1 + |x|)ν in (1.2), the constant Cx carries some dependence on x (as indicated), which must be controlled in order to bound (A.14). In fact, the following estimate for |f (p, x)| is true: |f (p, x)| ≤ C

[1 + ln(1 + |x|)]5 . 1 + |x − p|3

(A.15)

Before proving (A.15), let us see how it allows us to complete the proof. Indeed, since 1 = O (L ln L) , L → ∞ , (1 + |x − p|)3 x∈L p∈Z2∗ \∗L

as far as the second term of (A.14) is concerned, we have 1 |f (p, x)| ≤ C[ln L]5 = O(L[ln L]6 ) . (1 + |x − p|)3 x∈L p∈Z2∗ \∗L

x∈L p∈Z2∗ \∗L

For the first term we note that [1 + ln(1 + |x|)]5 ≤ C(ln L)5 [1 + ln(1 + |x − p|)]5 , for x, p in the indicated range and large L, resulting in

|f (p, x)| ≤ C[ln L]5

p∈∗L x∈Z2 \L

x∈Z2 \L p∈∗L

Therefore,

N(2L + 1)2 =

= −2i

[1 + ln(1 + |x − p|)]5 = O(L[ln L]11 ) . (1 + |x − p|)3

f (p, x) + O(L[ln L]11 )

x∈L p∈Z2∗

Pλ (x, y)Pλ (y, z)Pλ (z, x)

S(p, x, y, z) + O(L[ln L]11 ) ,

p∈Z2∗

x∈L y,z∈Z2

which gives (A.12) for N/2π after applying Connes’ area formula and taking the limit L → ∞. As for the proof of (A.15), we consider separately the cases (i) |p−x| < 2ν µ ln(|x|+1) and (ii) |p − x| ≥ conclude

2ν µ

|f (p, x)| ≤ 6

ln(|x| + 1). In case (i), we use the bound |S(p, x, y, z)| ≤ 3 to y,z∈Z2

|Pλ (x, y)Pλ (y, z)Pλ (z, x)| ≤ 6

y∈Z2

|Pλ (x, y)| ,


219

since

 |Pλ (y, z)Pλ (z, x)| ≤ 

z∈Z2

|Pλ (y, z)|2

z∈Z2

z∈Z2 1/2

≤ [Pλ (y, y)Pλ (x, x)]

1/2 |Pλ (z, x)|2  ≤ 1.

Now by (A.10), 2 ν |Pλ (x, y)| ≤ 4 ln(|x| + 1) + 1 + C2 µ 2

y∈Z

|b|> 2ν µ ln(|x|+1)

≤ C [1 + ln(|x| + 1)]2 ≤ C

µ

e− 2 |b|

[1 + ln(|x| + 1)]5 , (1 + |x − p|)3

where in the last step we have used that |x − p| ≤ 2ν µ ln(|x| + 1). This implies (A.15) in case (i). To prove (A.15) in case (ii), consider separately the contributions to f (p, x) coming when both y and z fall inside the ball of radius |p − x| around x and when one of y or z falls outside the ball. The latter contribution is exponentially small in |x − p|, since it is bounded by     6 

|y−x|≥|p−x| z∈Z2

≤ 12

+

|z−x|≥|p−x| y∈Z2

  |Pλ (x, y)Pλ (y, z)Pλ (z, x)|  µ

|Pλ (x, y)| ≤ Ce− 2 |x−p| ,

|y−x|≥|p−x|

where in the last step we have used (A.10) and the fact that |x − p| > 2ν µ ln(|x| + 1). To bound the former contribution note that in this case both |∠(y, p, x)| and |∠(z, p, x)| are smaller than π2 , and make use of the following estimates: (1) given α, β ∈ (− π2 , π2 ), |sin α + sin β − sin(α + β)| ≤ |sin α|3 + |sin β|3 , and (2) given y with |y − x| < |p − x|, |sin ∠(y, p, x)| ≤

|y − x| . 1 + |p − x|

Putting these two estimates together gives the following bound for the contribution with y, z in the ball of radius |x − p| around x, C (1 + |p − x|)3 ≤

|Pλ (x, y)Pλ (y, z)Pλ (z, x)| |y − x|3 + |z − x|3

|y−x|,|z−x| 0 is a small parameter which will tend to zero and > 0, which is the Elsasser number, is considered to be fixed O(1). We add to this system the boundary conditions ε uε\∂ = 0, j\∂ · n = 0,

(4)

where n is the normal to the boundary of the domain and the initial condition uε (0, x) = u0 (x),

(5)

224

F. Rousset

where u0 is independent of ε and divergence free. This system with such a scaling is relevant to modelize the motion of the Earth liquid core, ε is known to be very small (ε ∼ 10−7 ) and hence it is physically relevant to investigate the limit of the model when ε goes to zero. The possible instabilities in the boundary layers could have some effect in the geodynamo process. We refer to [11] for more details on this model and the underlying physics. For ε > 0 fixed, the mathematical status of the system (1), (2), (3), (4), (5) is very similar to the one of the incompressible Navier-Stokes equation: there is a classical theory of local in time existence and uniqueness of strong solutions (for example in H s with s sufficiently large) that we shall use in this article and a theory of global weak solutions (without uniqueness in 3-d) analogous to Leray solutions of the incompressible Navier-Stokes equation, we refer to [24] for a survey on this topic. In this paper we shall focus on the study of the asymptotic behavior of the solution when ε goes to zero. In a formal way, when ε tends to zero, because of the stiff terms in (1), uε tends to be in the kernel of the operator Lu = P e × u + e × P (e × u) , where P is the Leray projection on divergence free vector fields. The kernel of this operator is described by the classical Taylor-Proudman theorem which gives that u is a two-dimensional vector field, u = (u1 (t, x1 , x2 , 0), u2 (t, x1 , x2 , 0), 0) and j = 0. Since a two-dimensional vector field cannot match the boundary conditions (4), a boundary layer which is of size ε (the Ekman-Hartmann boundary layer) appears in the vicinity of the boundary. This boundary layer which is well-known in physics was mathematically studied in [8, 22]: a matched asymptotic expansion using boundary layers was computed in [8] and its justification was performed in [8] under a smallness assumption and in [22] under a more general spectral assumption. Actually, the model considered in [8, 22] is slightly more general: there is an evolution equation for the magnetic field which replaces (2). Nevertheless, as described in [11], the model that we consider here is already physically relevant and moreover the stability problems only involve part (2) of the equation as it was proved in [22]. It seems possible to generalize the results of this paper to the more general model, but computations should be more complicated. The main restriction in the two works [8, 22] is that the problem was studied for well-prepared data; this means that it was assumed that at t = 0 the initial datum of (1), (2), (3) tends sufficiently quickly to a two-dimensional vector field when ε goes to zero. Here, we shall study (1), (2), (3) for ill-prepared data, i.e. with a data u0 in (5) which is an arbitrary three-dimensional vector field which is divergence free and verifies the boundary condition (4). The aim of this paper is to prove that the main result of [22] that is the convergence under the assumption of linear stability of the boundary layers of uε to a two-dimensional vector field solution of an Euler equation is still true in this general setting. More precisely, we assume that u0 is a three dimensional vector field which matches the boundary condition (4) and we define 1 1 0,wp 0,wp u1 = u01 (t, x1 , x2 , x3 ) dx3 , u2 = u02 (t, x1 , x2 , x3 ) dx3 . 0

0

int Note that u0,wp is divergence free. Let uint = (uint 1 (t, x1 , x2 ), u2 (t, x1 , x2 )) be the solution of the two-dimensional Euler equation with damping

Stability of Large Amplitude Ekman-Hartmann Boundary Layers in MHD

∂t uint + uint · ∇uint + ∇p + β uint = 0, where

β=

2 , tan τ2

tan

225

∇ · uint = 0, t > 0, (x1 , x2 ) ∈ R2 , (6)

τ 1 . = √ 2 + 1 + 2

We recall that the damping term in this equation which was computed in [8] is due to energy dissipation in the boundary layers. We add to this equation the initial condition uint (0, x1 , x2 ) = u0,wp (x1 , x2 ).

(7)

Next, for each uint (t, x1 , x2 ) = q, we can define near each boundary x3 = 0, x3 = 1, an Ekman-Hartmann boundary layer ub (q, Z) = (ub1 , ub2 , 0), j b (q, Z) = (j1b , j2b , 0), where Z = x3 /ε or (1 − x3 )/ε . The profile ub is a solution of the ordinary differential equation −1 b b ∂ZZ u = Bu , B = 1 such that ub (q, 0) = −q, ub (q, +∞) = 0. We easily find that the solution of this problem is given by √ +iZ

ub1 + iub2 = −e−

(q1 + iq2 ).

Moreover, the current in the boundary layer is given by j1b + ij2b = −i(ub1 + iub2 ). To characterize the stability of the lower boundary layer, we can linearize (1), (2), (3) about (q + ub , j b ) and set (t , x1 , x2 , Z) = (t/ε, x1 /ε, x2 /ε, x3 /ε). Dropping the ’, and keeping only the 1/ε terms in the equations yield the evolution problem in R2 ×(0, +∞): ∂t v + (q + ub ) · ∇v + v3 ∂Z ub + ∇p + e × u + e × j = v, j = ∇ϕ − e × v, ∇ · v = 0, ∇ · j = 0. Note that, if we set j = curl b, we recover exactly the same system as in [22], Sect. 2. We add to this system the boundary condition u = 0, j3 = 0 for Z = 0, and we shall say that the boundary layer is linearly stable if this system does not have any solution which grows exponentially in time. By a normal mode analysis, we can look for solutions under the form (u, j ) = e(γ +iτ )t+iξ ·(x1 ,x2 ) (U (Z), J (Z)), γ ≥ 0, τ ∈ R, ξ ∈ R2 . This yields an ordinary differential equation with ζ = (γ , τ, ξ ) as parameters and by setting V = U3 , ∂Z2 U3 , iξ1 U2 − iξ2 U1 , J3 , ∂Z U3 , ∂Z3 U3 , ∂Z (iξ1 U2 − iξ2 U1 ), ∂Z J3 , we reduce the eigenvalue problem to the search of solutions of ∂Z V = A(Z, ζ, q)V ,

Z>0

such that V (0) = 0, V (+∞) = 0, where V = (V1 , V3 , V4 , V5 ) and

(8)

226

F. Rousset

A(Z, ζ, q) = 

0 4 I4 M A

0  −|ξ |2 P1 − iξ · ∂ 2 ub θ M=  i∂θ ub · ξ ⊥ 0

1 P2 0 0

, 0 0 P1 0

 0 0  , 0  |ξ |2

P1 = γ + iτ + i(q + ub ) · ξ + |ξ |2 , P2 = γ + iτ + i(q + ub ) · ξ + 2|ξ |2 + ,   0 0 0 0  0 0 1 0  A= . −1 0 0 −  0 0 −1 0 We recall from [22], that the space of solutions of ∂Z Z = AV which tend to zero as Z tends to +∞ has dimension four and is smooth for ξ = 0, γ ≥ 0. Hence by choosing a basis (V 1 , V 2 , V 3 , V 4 ) of this stable subspace, we can define an Evans function for this problem by D(ζ, q) = det V 1 (0, ζ, q), V 2 (0, ζ, q), V 3 (0, ζ, q), V 4 (0, ζ, q) . By definition, there is an unstable mode if and only if the Evans function vanishes for γ > 0. This kind of construction was very much used to characterize the stability of travelling waves in reaction-diffusion equations [1, 17] or viscous conservation laws [12]. It was also used in the setting of hydrodynamic stability to perform numerical computations [3]. Finally, we point out that as in [22], the same Evans function characterizes the stability of the upper boundary layer. With the help of this Evans function, we can state our main theorem. It just remains to define the anisotropic Sobolev space H m,0 () = {u ∈ L2 (), ∇xm1 ,x2 u ∈ L2 ()}. Theorem 1. Let u0 be a divergence free vector field such that u0/∂ = 0 and that u0 , ∇u0 ∈ H m+s,0 (), and ∇ 2 u0 ∈ H m,0 () for m ≥ 2 and s > 6. Let uint be the solution of (6), (7), assuming that D(ζ, uint (t, x1 , x2 )) = 0, ∀ζ = (γ , τ, ξ ), γ ≥ 0, ξ = 0

(H) for every t ∈

[0, T ∗ ], (x1 , x2 ) ||u − u ε

+ε

int

−1 2

∈

(9)

R2 , then the solution of (1), (2), (3), (4), (5) is such that

||L∞ ((0,T ∗ ),L2 ()) + ||uε − uint ||L2 ((0,T ∗ )×)) loc

||j ε ||L2 ((0,T ∗ )×) → 0

(10)

when ε goes to zero. This theorem says that we have a nonlinear stability result as long as the boundary layer remains linearly stable which is the meaning of the Assumption (H). The method that we use needs a lot of regularity for u0 . Nevertheless, the regularity that we require here is not too surprising when we compare it to other results (see [21] for example). As it was proved in [8] by direct L2 energy estimates, this assumption is matched when a Reynolds number R = sup ||uint ||L∞ t

is sufficiently small (R 5). More generally, it can be checked numerically (see [9]) that Assumption (H) is still matched for larger Reynolds number R 40. When the


227

Reynolds number becomes larger than a critical value Rc ∼ 40 the Ekman-Hartmann boundary layer becomes linearly unstable [9] and it was proved in [10] that it also implies nonlinear instability. Consequently, the combination of Theorem 1 and the result of [10] gives a complete description of the problem: there is linear and nonlinear stability for general data when R < Rc and linear and nonlinear instability for R > Rc . There are basically two parts in the proof of Theorem 1. At first, though it does not appear in the statement of the theorem, we need to generalize the construction of an approximate solution to absorb the part of the initial data which depends on the vertical variable: this new approximate solution will be made of various waves, initial layers and boundary layers. In the second part, we have to prove that the interaction of these new waves with the main Ekman-Hartmann boundary layer does not affect its stability, and hence that we can still get from our Assumption (H) a good energy estimate as in [22]. In the case of rotating fluid (that we recover by taking formally = 0), the vertical part of the initial data creates waves, called inertial waves, the situation was mathematically investigated in [6] for an anisotropic viscosity where ε is replaced by ε∂x23 x3 + H u. The crucial part in the convergence proof in [6] was the study of the dispersion of the inertial waves. Here, we shall see that the inertial waves are damped by magnetic effect; hence the situation is less subtle, nevertheless, we have to take into account the possible instabilities in the boundary layer. This last phenomenon is not present in [6] since the Ekman layer is always stable in the case of anisotropic viscosities. In our setting, for boundary layers of moderate amplitude which verify our Assumption (H) but not the smallness assumption of [8], the L2 norm is not decreasing any more and hence for the stability part, the method of [8] does not apply. To get a result under the sharp Assumption (H), we shall use the microlocal analysis of [22, 23] which was inspired from [21]. In this part of the proof, the fact that the inertial waves are damped is crucial; our analysis does not readily extend to the Ekman case ( = 0). For related works in the periodic case, we refer to [5, 4, 20]. The first part of the paper is devoted to the construction of an approximate solution U app for (1), (2), (3), (5), (4). This means that U app will match exactly the divergence free condition (3) and the boundary condition (4) and that U app makes an error which is O(εM ) with M sufficiently large when we plug it into Eqs. (1), (2). Our approximate solution is basically made of three pieces. The first part U wp is just the approximate solution which corresponds to the two-dimensional part of the initial data and that was already used in [8, 13]. Next, we add a second part U ip in order to lift the remaining part of the initial data. In this part, we shall study precisely the damping of the inertial waves. Each wave will create a boundary layer of size ε. At this stage, we have an approximate solution such that √ ||u0 − (uwp + uip )/t=0 ||L2 ε. Actually, the right-hand side comes from the L2 norm of the boundary layers which do not vanish at t = 0. This estimate is not sufficient to close a nonlinear stability argument. Actually, when we deal with large linearly stable boundary layers as in Theorem 1, we cannot obtain an L2 energy estimate by the standard energy method. We still get an L2 energy estimate on the linearized problem thanks to (H) through a microlocal analysis. Due to this method we cannot work in the class of Leray’s weak solutions of (1), the existence and uniqueness of a solution of the nonlinear problem in an interval of time independent of ε come from a fixed point argument as in [18]. To realize this program, app we need that ||u0 − u/t=0 || = O(εk ) with k > 21 . Consequently, we need to add a third part in the approximate solution to cancel the boundary layers at t = 0; this part U tl will

228

F. Rousset

be made of initial layers. The crucial point is that the special structure of our system (1), (2), (3) gives for utl a linear equation which is just a heat equation with damping. The second part of the paper is devoted to the study of the linearization of (1), (2), (3) about the approximate solution U app . The main result of this part is that despite the presence of the new waves and boundary layers, Assumption (H) on the main boundary layer still allows to get the same type of energy estimate as in [22]. Actually, we need to derive a better energy estimate to deal with the nonlinear problem: here the use of time derivatives is not appropriate in this setting for non-zero initial data because of the initial layers. In the last part, we give the proof of Theorem 1 by a fixed point argument. Notations. At first, we set U = (u, j, p, ϕ) and we define the operators corresponding to (1), (2): e×u e×j ∇p + + − εu, ε ε ε M(U ) = j − (∇ϕ − e × u).

N S(U ) = ∂t u + u · ∇u +

Throughout this paper, we use the notation ˜ for horizontal operators and quantities: ˜ = ∂1 f , u ˜ = ∂12 u + ∂22 u, u˜ = u1 . ∇f ∂2 f u2 Moreover, we shall often use for the variables the notations z = x3 and y = (x1 , x2 ). Next, we need to define some norms. For u, v ∈ R3 , |u|2 and u · v are the standard euclidean norm and scalar product. We also set

|∂yα u(t, y, x3 )|2 dy, |u|2m = |u(t, ·, x3 )|2H m (R2 ) = 2 |α|≤m R

||u||2m = ||u||H m,0 ()2 =

|α|≤m

|∂yα u(t, x)|2 dx.

The norm || · ||0 will be denoted by || · || and the associated scalar product will be denoted by (·, ·). Note that the norm || · ||m is not the usual norm of H m (), we do not require any regularity with respect to the z variable. Moreover, we deal throughout the paper with m such that m > 1 so that H m (R2 ) is an algebra. In a similar way, we define

|u||m,p = ||u||W m,p,0 () = ||∂yα u||Lp () , for p ∈ [1, +∞]. |α|≤m

As previously, for p = 2, we omit the index p. Actually, we only need the cases p = 1 and p = +∞. We shall also use weighted norms; we set for a parameter γ ≥ 1 which will be chosen sufficiently large (and independent of ε)

α ||u||2m,γ = ||Zαm u||2 , Zm u = γ m−|α| ∂yα u |α|≤m


and |||u|||2m,γ =

|α|≤m

|||Zαm u|||2 , |||u|||2 =

+∞

229

||u(t)||2 dt.

0

The notation || · ||m,p , || · ||m,γ may seem a little confusing, nevertheless, we will always use the letter p or p = 1, ∞ for the W m,p,0 norm and the letter γ for the weighted norm. Throughout the paper , C, O(1) stand for harmless numbers which are independent of ε and γ ≥ 1. We shall sometimes need to make precise the dependence of these numbers with respect to the regularity of the initial value u0 : we will use the notations Cm for a number which only depends on ||u0 ||m and Cm,β for a number which only depends on k≤β ||∂zk u0 ||m := ||u0 ||H m,β . Finally a(τ ) stands for a nonnegative function such that +∞ a(τ ) dτ < +∞ 0

which may change from line to line. 1. Construction of an Approximate Solution The aim of this section is to construct an approximate solution U a = (ua , pa , j a , ϕ a ) of (1, 2, 3, 4, 5). This means that U a matches exactly the boundary condition (4) and the divergence free condition (3) and is an approximate solution of (1), (2), in the sense that N S(U a ) = R1ε , M(U a ) = R2ε , where R1ε and R2ε goes to zero when ε goes to zero (we will give a precise statement later). As in [6], we can use a special basis of L2 (0, 1) to decompose the initial value under the form 0 0,k u1 u1 = cos(kπ z), (11) u02 u0,k 2 k≥0

and since

u0

must be divergence free, we have the corresponding decomposition

1 ∇˜ · u˜ 0,k sin(kπ z). (12) u03 = − kπ k≥1

Thanks to (11), (12), we can set 0,k  0,0    u1 u1 cos kπ z .  and u0,ip =  k≥1 u0,k u0,wp =  u0,0 2 2 1 0,k 0 − k≥1 kπ ∇˜ · u˜ sin kπ z We consider three different parts in the construction of the approximate solution. The first part corresponds to u0,wp in the decomposition of u0 , hence this part is associated to a two dimensional initial value, this is the well prepared part of the solution since the high rotation forces the fluid to be invariant in the direction of the rotation axis and hence to be two-dimensional. This part of the approximate solution was already built in [8, 13]: it is made of an interior part which is two dimensional and a boundary layer part which is used to match the boundary conditions.

230

F. Rousset

Next we consider a second part in the approximate solution, this part is associated to the remaining components in the decomposition of the initial data; this part is made of waves and boundary layers. The waves are oscillating at high frequency in time and are damped by magnetic effect. This is the main difference with the pure Ekman case studied in [6] where the waves are not damped; they tend to zero due to dispersion effects. Finally the third part of the approximate solution is made of an initial layer: this part is used to make the approximate solution sufficiently close to u0 at t = 0 to apply our nonlinear argument as explained in the introdution. 1.1. The well-prepared part. In this section, we recall the construction of [8, 13] Proposition 2 ([8, 13]). There exists U wp under the form U wp (t, x) =

M

k=0

z 1 − z ε k U wp,k t, x, , ε ε

(13)

such that U wp matches exactly the boundary condition (4) and the divergence free condition (3). Moreover, we have N S(U wp ) = R11 , M(U wp ) = R21 , where for every m ≥ 0, T ε ||R11 ||2m dt + 0

T 0

||R11 ||m dt

2

+ ε −1

T 0

||R21 ||2m dt ≤ Cm+2 ε 2 .

Moreover, at t = 0, we have wp u (0, x) − u0,wp ≤ C e−αx/ε + e−α(1−x)/ε + r 1 ,

(14)

(15)

||r 1 ||2m + ε 2 ||∇r 1 ||2m + ε 4 ||∇ 2 r 1 ||2m ≤ Cm+2 ε 2 . Proof. We shall not give the complete proof of this lemma, but for the sake of clarity and since it will be useful later, we describe more precisely the approximate solution U wp . In (87) each term is under the form z 1 − z z 1−z U wp,k t, x, , = U int,k (t, x)+ Uˇ wp,k (t, x1 , x2 , )+ Uˆ wp,k (t, x1 , x2 , ), ε ε ε ε where the boundary layer terms Uˆ 1,k , Uˇ 1,k are exponentially decreasing with respect to Z = z/ε or (1 − z)/ε: there exists α, C > 0 such that |∂Zα Uˇ wp,k (t, x1 , x2 , Z)| + |∂Zα Uˆ wp,k (t, x1 , x2 , Z)| ≤ Ce−αZ , ∀t, x, Z.

(16)

Moreover, uint,0 = (uint,0 (t, x1 , x2 ), uint,0 (t, x1 , x2 ), 0), j int,0 = 0 and uint,0 = uint 1 2 is the solution of the two dimensional Euler equation (6) with the initial condition (7). The other important term in (87) is the leading term of the boundary layer terms. We have pˇ 1,0 = ϕˇ 1,0 = ˇ31,0 = uˇ 1,0 3 = 0.

(17)

1,0 ˇ 1,0 ˜ = (uˇ 1,0 , uˇ 2,0 ) The remaining components are such that ˇ11,0 = uˇ 1,0 2 , ˇ2 = −u 1 and u is a solution of the linear ordinary differential equation


∂θ2 u˜ = B u, ˜ B=

−1 1

231

(18)

with the constraints u(+∞) ˜ = 0, u˜ /θ=0 = −uint,0 . It is easy to show that this ordinary differential equation has a unique solution which moreover decreases at an exponential rate. We have just given the form of the leading term of the approximate solutions U wp , as usual in WKB type expansions the next terms U wp,k are solutions of linearized equations about the leading order solution U wp,0 with a source term which depends only on (U wp,l )l≤k−1 . In particular, in our case, we have the same equations for the boundary layer terms because the leading order boundary layer equations are linear and we have a linearized Euler equation for the interior part uint,k which has the form ∂t uint,k + uint,0 · ∇uint,k + uint,k · ∇uint,0 + βuint,k + ∇p = Fk , wp,0

= uˆ 3

wp,0

εe

−αZ

A useful remark is that since uˇ 3 |ub3 |m

∇ · uint,k = 0.

= 0, we have the improved estimates

, |∂z u3,b |, 1.

(19)

Finally, (15) just comes from the fact that the difference between the approximate solution and u0,wp is made of the leading boundary layer term plus correction terms which are at least of order ε. As explained in the introduction, the property (15) is not good enough for our nonlinear argument; we have to add an additional corrector which is an initial layer to cancel the boundary layers in the approximate solution at the initial time t = 0. 1.2. Additional initial layer. In this section, we consider an initial layer under the form t z t 1−z U tl = Uˇ tl,0 ( , x1 , x2 , ) + Uˆ tl,0 ( , x1 , x2 , ) + ε··· , ε ε ε ε and we look for functions which are fastly decreasing with respect to the variables τ = t/ε and Z = z/ε or (1 − z)/ε. Moreover, we want that the initial layer corrects the term created by the boundary layer at t = 0: wp,0

wp,0

uˇ tl,0 ˇ /t=0 , uˆ tl,0 ˆ /t=0 . /t=0 = −u /t=0 = −u We will also require that the initial layer matches exactly the boundary condition (4). By plugging this ansatz in the equation, we first find as usual for boundary layer terms that tl,0 ptl,0 = ϕ tl,0 = utl,0 =0 3 = j3

and then, we get that

(uˇ tl,0 ˇ tl,0 1 ,u 2 )

and

(uˆ tl,0 ˆ tl,0 1 ,u 2 )

(20)

are solutions of

∂τ v − ∂ZZ v + Bv = 0

(21)

v(τ, 0) = 0,

(22)

v(0, Z) = v0 (Z).

(23)

with the boundary condition and the initial condition Here, B is the two by two matrix defined in (18) and v0 (Z) stands for −uˇ wp,0 (0, x1 , x2 , Z) or −uˆ wp,0 (0, x1 , x2 , Z) which are both exponentially decreasing with respect to Z.

232

F. Rousset

Moreover, the current in the initial layer is given by tl,0 = utl,0 j2tl,0 = −utl,0 1 , j1 2 .

(24)

Finally, note that the variables (x1 , x2 ) are parameters in this problem they just appear through the dependence of the initial condition with respect to them. We also point out that the initial condition for (21) does not verify the boundary condition (22), hence the solution of this parabolic equation will have a singularity at t = 0, z = 0 that we shall need to consider carefully in the future estimates. The aim of the next lemma is to study the properties of the solutions of the system (21), (22), (23). Lemma 3. Let v be a solution of ∂τ v − ∂ZZ v + Bv = F, Z > 0, τ > 0

(25)

with the boundary condition (22) and the initial value f ; we assume that the source term and the initial value are such that |F (τ, ·, Z)|m + |∂Z F (τ, ·, Z)|m ≤ Km e−αZ a(τ ), |f (·, Z)|m + |∂Z f (·, Z)|m ≤ Km e−αZ ,

(26) (27)

for some α > 0. Then we have for every p ∈ [1, +∞], the estimates ||v(τ )||m,p Km ,

(28)

||v(τ )||m,p + ||v(τ )||2m,p + ||∂Z v(τ )||m,p + ||∂Z v(τ )||2m Km a(τ ), ε−1 |∂Zk v(τ, ·, ε−1 )|m + ε −2 |∂Zk v(τ, ·, ε−1 )|2m Km a(τ ), k ≤ 2,

(29)

1 2

||Z ∂Z v||m Km .

(30) (31)

The proof of this lemma which only relies on elementary convolution estimates on the explicit formula for the solution of (25) which is just a heat equation with damping is postponed to Appendix A. Thanks to this lemma, we are now able to give the precise construction of our initial layer corrector; we take U tl = U tl,0 + ε(U tl,1 + U tl,2 ), where U tl is given by (20), (21), (22), (23), (24). Since the initial condition for (21) is a boundary layer, the assumption (26), (27) of Lemma 3 is verified (note that in this case, there is no source term) and hence the estimates of Lemma 3 hold for U tl,0 . As usual, the two correctors, U tl,1 , U tl,2 are needed to recover exactly the divergence free condition (3), and the boundary condition (4). To recover the divergence free condition, we first choose +∞ uˇ tl,1 (τ, x , x , Z) = (∂1 uˇ tl,0 ˇ tl,0 1 2 3 1 + ∂2 u 2 )(τ, x1 , x2 , θ ), dθ, Z +∞ (∂1 ˇ1tl,0 + ∂2 ˇ2tl,0 )(τ, x1 , x2 , θ ), dθ, ˇ3tl,1 (τ, x1 , x2 , Z) = Z +∞ tl,1 (∂1 uˆ tl,0 ˆ tl,0 uˆ 3 (τ, x1 , x2 , Z) = − 1 + ∂2 u 2 )(τ, x1 , x2 , θ ), dθ, Z +∞ (∂1 ˆ1tl,0 + ∂2 ˆ2tl,0 )(τ, x1 , x2 , θ ), dθ. ˆ3tl,1 (τ, x1 , x2 , Z) = − Z


233

Note that thanks to Lemma 3, the corrector U tl,1 enjoys the estimates ||U tl,1 ||m,∞ + ||∂Z U tl,1 ||m,∞ ≤ Cm+1 ,

(32)

||U ||m,∞ + ||U tl,1 ||2m,∞ ≤ Cm+1 a(τ ), ||∂Z U tl,1 ||m,∞ + ||∂Z U tl,1 ||2m,∞ ≤ Cm+1 a(τ ), ||∂Zk U tl,1 ||m + ||∂Zk U tl,1 ||2m ≤ Cm+1 a(τ ), ∀k ≤

(33)

tl,1

(34) 2.

(35)

Moreover, note that thanks to (21), we have for example +∞ tl,1 ∂1 (−B uˇ tl,0 )1 + ∂2 (−B uˇ tl,0 )2 dθ − ∂1Z uˇ tl,0 ˇ tl,0 ∂τ uˇ 3 = 1 − ∂2Z u 2 , Z

hence thanks to Lemma 3, we get the estimate ||∂τ U tl,1 ||m + ||∂τ U tl,1 ||2m ≤ Cm+1 a(τ ).

(36)

Finally, the second corrector U tl,2 (τ, x1 , x2 , z) will be used to recover the exact boundary condition. Consequently, we choose it as in [13] such that in : ∇ · utl,2 = 0, ∇ · j tl,2 = 0, and with the boundary conditions utl,2 (τ, x1 , x2 , 0) = −ε−1 uˆ tl,0 (τ, x1 , x2 , ε−1 )− uˇ tl,1 (τ, x1 , x2 , 0)− uˆ tl,1 (τ, x1 , x2 , ε−1 ), utl,2 (τ, x1 , x2 , 1) = −ε−1 uˇ tl,0 (τ, x1 , x2 , ε−1 )− uˇ tl,1 (τ, x1 , x2 , ε−1 )− uˆ tl,1 (τ, x1 , x2 , 0), j3tl,2 (τ, x1 , x2 , 0) = −ε −1 ˆ3tl,0 (τ, x1 , x2 , ε−1 )−ˇ3tl,1 (τ, x1 , x2 , 0)−ˆ3tl,1 (τ, x1 , x2 , ε−1 ), j3tl,2 (τ, x1 , x2 , 1) = −ε −1 ˆ3tl,0 (τ, x1 , x2 , ε−1 )−ˇ3tl,1 (τ, x1 , x2 , ε−1 )−ˆ3tl,1 (τ, x1 , x2 , 0). Note that thanks to (31), the terms like ε−1 uˆ tl,0 (τ, x1 , x2 , ε−1 ) are uniformly bounded in ε and integrable with respect to τ . By using the same technique as in [13] and thanks to the estimates (29), (30) and (33), we find that the solution of the previous system enjoys the estimates: ||U tl,2 ||m,∞ + ||∂z U tl,2 ||m,∞ ≤ Cm+1 , ||U ||m,∞ + ||U tl,2 ||2m,∞ + ||∂z U tl,2 ||m,∞ + ||∂z U tl,2 ||2m,∞ ||∂zk U tl,2 ||m + ||∂zk U tl,2 ||2m ≤ Cm+1 a(τ ), ∀k ≤ 2. tl,2

(37) ≤ Cm+1 a(τ ),

(38) (39)

Moreover, note that thanks to (30), Eq. (25) also gives that ε−1 |∂τ v(τ, ·, ε−1 )|m + ε −2 |∂τ v(τ, ·, ε−1 )|2m Km a(τ ), where v stands for uˆ tl,0 or uˇ tl,0 and hence this implies that ||∂τ utl,2 ||m + ||∂τ utl,2 ||2m ≤ Cm+1 a(τ ).

(40)

We are now able to improve the statement of Proposition 2. Lemma 4. Let U = U wp + U tl , then U verifies exactly the boundary condition (4), and the divergence free condition (3); moreover, we have

234

F. Rousset

N S(U ) = R 1 , M(U ) = R 2 , where ||R 1 ||m + ||R 1 ||2m + ε −2 ||R 2 ||2m Cm+2 (ε + a(τ )), and at t = 0, let

r = U /t=0 −

u˜ 00 0

(41)

,

then ||r||2m + ε 2 ||∇r||2m + ε 4 ||∇ 2 r||2m ≤ Cm+2 ε 2 .

(42)

The main improvement in the estimates of Lemma 4 with respect to Lemma 2 is that (15) is changed in (42) which will be sufficient for our nonlinear argument. Proof. We begin with the estimate of ||R 2 ||m which is the easiest one. By definition, we have R 2 = R21 + ε(j tl,1 + j tl,2 − e × (utl,1 + utl,2 )), and hence the estimate (41) for R 2 follows from (14) and (35). Next, we note that R 1 = E1 + E2 + E3 , where 1

E1 = R 1 + (∂τ utl,1 (t/ε) + ∂τ utl,2 (t/ε)) + e × utl,2 ˜ tl , −ε2 (utl,1 + utl,2 ) − ε u

(43)

E2 = uwp · ∇utl + utl · ∇uwp , E3 = utl · ∇utl .

(44) (45)

The estimate (41) for E1 follows from (14), Lemma 3 and (32–36), (37–40). The fact ˜ tl . that this estimate depends on Cm+2 comes from the term u ∞ To estimate the second line, we use L estimates for uwp , (19) and the fact that wp utl3 = 0, u3 = 0 to get ˜ tl ||m + ε||∂z utl ||m , ||E2 ||m ||utl ||m + ||∇u and hence, thanks to (29), (35), (39), we get the desired estimate. Finally, for E3 , we use (28), (32) and (37), to get ˜ tl ||m + ε||∂z utl ||m , ||E3 ||m ∇u which gives the wanted estimate thanks to (29), (35), (39). 1.3. The ill-prepared part. We construct in this section an approximate solution U ip for the linear problem e×j ∇p e×u ∂t u + + + − εu = 0, (46) ε ε ε j = ∇ϕ − e × u, (47) ∇ · u = 0, ∇ · j = 0, (48) in with the boundary condition (4) such that at t = 0, uip is sufficiently close to the part of the initial condition that is not absorbed in the construction of the previous


235

paragraph: this means that we shall require: ip

||u/t=0 − u0,ip ||2m ε2 . For this construction, the first step is to get a good spectral representation of the operator Lu = P e × u + e × (P (e × u)) , where P is the Leray projection on divergence free vector fields or equivalently, to understand the behaviour of the solutions of the linear equation ∂t u +

Lu =0 ε

in R2 × (0, 1) with the boundary condition u3/∂ = 0. As in [6], the first step is to reformulate (46), (47), (48). Let us denote by F the Fourier transform with respect to horizontal variables (x1 , x2 ) and by ξ the variable dual to (x1 , x2 ), we define ω = F(∂1 u2 − ∂2 u1 ), d = F(∂1 u1 + ∂2 u2 ), w = F(u3 ), η = F(∂1 j2 − ∂2 j1 ), c = F(∂1 j1 + ∂2 j2 ), J = F(j3 ). And we rewrite the system with new unknowns (ω, d, w, η, c, J ) (in this section with an abuse of notation we still denote by p and ϕ the Fourier transform of p and ϕ). The new system becomes ω d + = |ξ |2 ϕ, ε ε ε d ω |ξ |2 p ∂t d + − = , ε ε ε −∂z p , ∂t w = ε c = −|ξ |2 ϕ + ω, η = −d, J = ∂z ϕ, ∂z w = −d, ∂z J = −c.

∂t ω +

(49) (50) (51) (52) (53)

Following the method of [6], we can use the basis (cos kπ z)k≥0 of L2 (0, 1) to decompose v = (ω, d) as

v= vk (t) cos kπ z, k≥1

where vk = (ωk , dk ). Thanks to the divergence free condition, w must be under the form

w=− (kπ )−1 dk (t) sin kπ z. k≥1

As usual, the divergence free conditions (53) determine the pressure and the magnetic potential ((kπ)2 + |ξ |2 )ϕk = ωk , ((kπ )2 + |ξ |2 )pk = dk − ωk . Next we get thanks to (49), (50), ∂ t vk =

Dk vk , ε

236

F. Rousset

where D = k

−αk2 −1 αk2 −αk2

, αk =

(kπ )2 (kπ )2 + |ξ |2

21 .

The resolution of this system gives vk (t) = Lk (t/ε)v k (0), where 2 cos(αk τ ) sin(αk τ )/αk Lk (τ ) = exp(τ D k ) = e−αk τ . −αk sin(αk τ ) cos(αk τ ) Note that when = 0, we have the same expression as in [6]. Here, when is positive, the oscillatory waves created by the high rotation are damped by the high magnetic field, hence the situation is very different and less subtle. The aim of the following lemma is to study precisely this damping effect: Lemma 5. Let (u, j ) be a solution of the linear equation ∂τ u + e × u + e × j + ∇p = 0, j = ∇ϕ − e × u, ∇ · u = 0, ∇ · j = 0,

(54) (55) (56)

in with the boundary condition u3/∂ = j3/∂ = 0

(57)

and an initial condition u0 such that ∇ · u0 = 0 and u0/∂ = 0. Then, we have for s > 4, the following estimates: ||∂zβ u(τ )||m ≤ C(||u0 ||m+s+β + ||u0 ||H m,β ), ||∂zβ u(τ )||m

≤ C(||∂z u ||m+s+β + ||u ||H m,β )a(τ ), 0

0

||u(τ )||m,∞ ≤ C||∂z u ||m+s a(τ ). 0

(58) (59) (60)

Remark 6. Note that by obvious interpolation between (58) and (59), we also have ∞ ||∂zβ u(τ )||2m dτ ≤ C(||∂z u0 ||2m+s+β + ||u0 ||2H m,β ). (61) 0

Proof. At first, we prove (58), (59) for β = 0. In this case, we use the explicit formula for the solution of (54), (55), (56). We have, thanks to the same notations as previously

v(τ, ξ, z) = Lk (τ )vk0 cos(kπ z), w(τ, ξ, z) = − (kπ )−1 (Lk (τ )vk0 )2 sin(kπ z). k≥1

k≥1

(62) Thanks to the Bessel identity, we shall get ||Fξ−1 v(τ, ·, ·)||2m + ||Fξ−1 w(τ, ·, ·)||2m + ||Fξ−1 ∂z w(τ, ·, ·)||2m ≤ C||u0 ||2m+1 . (63) Indeed, since ∀τ ≥ 0, ||Lk (τ )|| ≤ Ce−αk τ (1 + |ξ |) ≤ C(1 + |ξ |), 2

we have


||v(τ, ·, ·)||2m =

k≥1 ||v 0 ||2m+1

≤

|Lk (τ )vk0 |2m ≤ ≤

2 k≥1 R C||u0 ||2m+2 .

237

(1 + |ξ |2 )(1 + |ξ |2 )m |vk0 |2 dξ

In a similar way, we get the estimates for w and ∂z w. We deduce (58) from (63) and classical estimates for the inversion of the Laplacian in R2 since ˜ 1 = F −1 (−iξ2 ω + iξ1 ∂z u3 ). ˜ 2 = F −1 (iξ1 ω − iξ2 ∂z u3 ), u u ξ ξ

(64)

Next, we prove (59) for β = 0. Using (62), we rewrite

vk (τ, ξ ) cos(kπ z) v(τ, ξ, z) =

(65)

k≥1

and we note that vk (τ, ξ, z) is a linear combination of terms under the form 0 (ξ ), Ik (τ, ξ ) = e−αk τ ak (ξ )vk,i 2

(66)

0 stands for a component of v 0 (ξ ). where |ak (ξ )| ≤ C(1 + |ξ |) and vk,i k +∞ We want to estimate 0 |Ik (τ, ·)|L2 dτ . Towards this, we use that ξ

+∞ 0

|Ik (τ, ·)|L2 dτ = ξ

+∞

sup

(Ik (τ, ·), ϕ(τ, ·))L2 dτ

(67)

ξ

|ϕ(τ,·)|L2 ≤1 0 ξ

and we perform the decomposition +∞

|(Ik (τ, ·), ϕ(τ, ·))L2 | dτ ≤ ξ

0

+∞

l≥−1 0

|(Ikl (τ, ·), ϕ(τ, ·))L2 | dτ,

(68)

ξ

where Ikl (τ, ξ ) = ϕ l (ξ )Ik (ξ ) and ϕ l = ϕ(ξ/2l ) is the usual dyadic partition of unity of the Littlewood-Paley decomposition (see [2] or [7] for example). Now, by the CauchySchwarz inequality, we get that +∞ +∞ 2 l 0 |(Ik (τ, ·), ϕ(τ, ·))L2 | dτ |e−αk τ ak (ξ )ϕ l (ξ )vk,i (ξ )ϕ(τ, ξ )| dξ dτ 0

0

+∞ 0 |vk,i ϕ l |L2 ξ

ξ

0

ξ ∈supp(ϕ l )

ξ

1 2 2 e−2αk τ |ak (ξ )|2 |ϕ(τ, ξ )|2 dξ dτ.

Now, since on the support of ϕ l , we have |ξ | 2l , we get that e−αk τ e−αk (2 )τ , |ak (ξ )|2 22l , 2

2

l

and we find +∞ 0 |(Ikl (τ, ·), ϕ(τ, ·))L2 | dτ 2l |vk,i ϕ l |L2 sup(|ϕ(τ, ·)|L2 ) 0

ξ

2

3l

ξ τ 0 l |vk,i ϕ |L2 . ξ

ξ

0

+∞

e−αk (2 )τ dτ 2

l

238

F. Rousset

Thanks to (68), this yields for s > 3, +∞

0 (Ik (τ, ·), ϕ(τ, ·))L2 dτ 23l |vk,i ϕ l |L2 ξ

0

ξ

l≥−1

1 2

0 22sl |vk,i ϕ l |2L2

l≥−1 0 |vk,i |s .

ξ

2(6−2s)l

1 2

l≥−1

In the last step, we have used the classical characterization of Sobolev spaces by dyadic decomposition (again see [2] or [7] for example. Consequently, going back to (67), we have proven that +∞ 0 |Ik (τ, ·)|L2 dτ |vk,i |s . ξ

0

Consequently, we easily get thanks to (65) by a combination of the last estimate and the Sobolev embedding that for s > 4, +∞ +∞

−1 ||Fξ v(τ )||m,∞ + ||Fξ−1 v(τ )||m ≤ C ||Fξ−1 vk0 ||m+s . 0

0

Moreover, since

u0/∂

k≥1

= 0, we have thanks to an integration by parts

||Fξ−1 vk0 ||m+s ≤ C(kπ )−1 ||Fξ−1 (∂z v 0 )k ||m+s and hence the Cauchy Schwarz inequality gives +∞ +∞ −1 ||Fξ v(τ )||m,∞ + ||Fξ−1 v(τ )||m ≤ C||Fξ−1 ∂z v 0 ||m+s Cm+s,1 . (69) 0

0

By the same technique, we can prove thanks to (62) that the same inequality as (69) holds for w and ∂z w with Cm+s,1 replaced by Cm+s+1,1 . At last, we get (59) for β = 0 by using again (64). It remains to prove (58), (59) for β > 0. Note that it seems difficult to use directly the representation (62) since the series obtained by taking the derivative with respect to z of each term of (62) are not convergent any more. To overcome this difficulty, we shall make direct energy estimates on the partial differential equations (54), (55), (56). At first, we notice that the divergence free relations (56) determine the pressure and the magnetic potential as ϕ = ∇ · (e × u) = −(∂1 u2 − ∂2 u1 ), p = −∇ · e × u + e × j = ∂1 u2 − ∂2 u1 + (∂1 j2 − ∂2 j1 ) .

(70) (71)

Consequently, by classical elliptic regularity, we get for β ≥ 1 the estimates ||∇∂zβ ϕ||m ||∂zβ−1 u||m+1 , ||∂zβ−1 ϕ||m+2 ||∂zβ−1 u||m+1 ,

(72)

||∇∂zβ p||m ||∂zβ−1 u||m+1 + ||∂zβ−1 j ||m+1 .

(73)

Note that thanks to (55) we also have ||∂zβ j || ||∇∂zβ ϕ||m + ||∂zβ u||m , hence the combination of (73) and (74), (72) gives

(74)


239

||∇∂zβ p||m ||∂zβ−1 u||m+1 .

(75)

Next, by a standard energy estimate on (54), we have 1 d ||∂ β u(τ )||2m + (e × ∂zβ j, ∂zβ u)m + (∇∂zβ p, ∂zβ u)m = 0, 2 dτ z and hence thanks to (55), we get 1 d ˜ 2m (||∇∂zβ p||m + ||∇∂zβ ϕ||m )||∂zβ u||m . ||∂ β u(τ )||2m + ||∂zβ u|| 2 dτ z Since by the divergence free condition, we also have ||∂zβ u3 ||2m ||∂zβ−1 u|| ˜ m+1 ||∂zβ u3 ||m , we finally get 1 d ||∂ β u(τ )||2m + ||∂zβ u||2m (||∂zβ−1 u|| ˜ m+1 + ||∇∂zβ p||m + ||∇∂zβ ϕ||m )||∂zβ u||m . 2 dτ z Consequently, the use of (72) and (75) and an integration in time yield τ τ β β ||∂z u(τ )||m + ||∂z u(s)||m ds ≤ C ||∂zβ−1 u(s)||m+1 ds + ||∂zβ u(0)||m , (76) 0

0

where C is independent of τ . Since we have already proven (58), (59) for β = 0, we get the estimates in the general case thanks to (76) by induction. Thanks to Lemma 5, we can now give the construction of the approximate solution which corresponds to the ill-prepared part of the initial data. Lemma 7. There exists an approximate solution U ip of (1), (2), (3) under the form U ip = U int,ip (t/ε, x) + Uˇ ip (t/ε, y, z/ε) + Uˆ ip (t/ε, y, (1 − z)/ε) +εU c (t, t/ε, x, z/ε, (1 − z)/ε), where U int,ip is the solution of the linear equation (54), (55), (56) given by Lemma 5, Uˇ ip , and Uˆ ip are boundary layer terms and U c is an higher order correction term. This approximate solution matches exactly the divergence free conditions (3) and the boundary condition (4), and we have ip

ip

N S(U ip ) = R1 , M(uip ) = R2 , with the estimates ||R1 ||m + ||R1 (τ )||2m + ε −2 ||R2 ||2m ≤ (Cm+s+2,1 + Cm,2 )(ε + a(τ )). (77) ip

ip

ip

Moreover, at t = 0, we have ip

ip

ip

||u/t=0 − u0,ip ||2m + ε 2 ||∇(u/t=0 − u0,ip )||2m + ε 4 ||∇ 2 (u/t=0 − u0,ip )||2m ≤ (Cm+s+2,1 + Cm,2 )ε 2 .

(78)

Finally, the boundary terms Uˇ ip and Uˆ ip satisfy the estimates (28), (29), (30), (31) of Lemma 3 and the corrector term U c verifies the estimates (33), (34), (35).

240

F. Rousset

Proof. We first choose U int,ip as the solution of (54), (55), (56) with the boundary condition (57) given by Lemma 5. Since it does not match the full boundary condition (4), we must add the boundary layers Uˇ ip , Uˆ ip . As usual for boundary layers, we find p = ϕ = u3 = j3 = 0, the remaining components (u1 , u2 ) = v are solutions of ∂τ v − ∂ZZ v + Bv = 0, τ > 0, Z > 0 with the prescribed boundary condition int,ip

int,ip

vˇ/Z=0 = −(u1/z=0 , u2/z=0 ) for the boundary layer corresponding to the lower boundary and int,ip

int,ip

vˆ/Z=0 = −(u1/z=1 , u2/z=1 ) for the boundary layer corresponding to the upper one. To get an approximate solution sufficiently accurate in order to get the estimate (78) at t = 0, we choose the initial condition v /t=0 = 0. Again, note that the initial condition does not match the boundary condition, hence the boundary layers are not smooth in the close domain (τ ≥ 0, Z ≥ 0). Since we are studying an initial boundary value problem with an inhomogeneous boundary condition, it is more convenient to change it into an homogeneous one. We set v = v − v(τ, y, 0)e−Z . After this change of unknown, we find for v the inhomogeneous equation with homogeneous boundary condition ∂τ v − ∂ZZ v + Bv = F, v(τ, y, 0) = 0 with the source term F = e−Z (∂τ + B + Id)v(τ, y, 0). The initial condition is now v(0, y, Z) = −v(0, y, 0)e−Z . Next, we note that thanks to Lemma 5, the assumptions (26), (27) of Lemma 3 is matched with Km replaced by Cm+s,1 . Consequently, we get the existence of Uˇ ip and Uˆ ip which satisfy the estimates (28), (29), (30), (31) thanks to Lemma 5. Next, the construction of the corrector εU c = ε(U c,1 + U c,2 ) to recover the divergence free conditions and the exact boundary condition is similar as in Sect. 1.2. Moreover, the estimates on these terms are the same with Cm+1 replaced by Cm+s+1,1 since they only depend on the estimates on Lemma 3. ip ip Finally, we have to compute the error terms R1 , R2 . Again since the estimates on the terms Uˆ ip , Uˇ ip and U c are the same as in Sect. 1.2, most of the terms except one type of terms can be estimated as in this section, and we shall not give the details again. The only difference occurs for the estimates of the terms involving quadratic quantities in the Navier-Stokes equation and more precisely for the terms under the form (there are similar terms involving the other boundary layer that we can estimate in the same way): N = u3 ∂z uˇ ip = ε−1 u3 ip

int,ip

∂Z Uˇ ip + uc3 ∂Z Uˇ ip .

ip ip The last equality comes from the fact that uˇ 3 = uˆ 3 term, we use the same technique as for the term E3 in the

(79)

= 0. To estimate the second proof of Lemma 4. It remains to estimate the first term. Note that this term did not appear in Sect. 1.2 since for the


241

well-prepared part, we have uint 3 = 0. For the estimate of this term, we use the classical Hardy - like estimate which is classical in boundary layer stability problems [14–16]: int,ip since ∂ym u3 (τ, y, 0) = 0, we have int,ip

|u3

int,ip

(τ, z)|2m ≤ Cz||∂z u3

hence we write int,ip

||u3

∂Z uˇ ip ||2m ε −2 ||∂z u3

int,ip

int,ip

||∂z u3

(τ )||2m

(τ )||2m ,

+∞ 0

z|∂Z uˇ ip (τ, ·, z/ε)|2m dz

(80)

1

(τ )||2m ||Z 2 ∂Z uˇ ip (τ, ·, ·)||2m int,ip

Cm+s+1 ||∂z u3

(τ )||2m .

(81)

Here, we have used the uniform bound (31). Finally, thanks to (59), (61), we get int,ip

||u3

int,ip

∂Z uˇ ip ||2m + ||u3

∂Z uˇ ip ||m (Cm+s+1 + Cm,1 )a(τ )

which is a part of (77). 1.4. Final approximate solution. By collecting the results of Lemma 4–7, we can finally get our final approximate solution Theorem 8. Let U app = U + U ip , then U app verifies the boundary condition (4), the divergence free condition (3) and app

app

N S(U app ) = R1 , M(U app ) = R2 , with the estimates ||R1 (τ )||m + ||R2 (τ )||2m + ε −2 ||R2 ||2m ≤ (Cm+s+2,1 + Cm,2 )(ε + a(τ )), app

app

app

(82)

where a ∈ + ). Moreover, at t = 0, we have L1 (R

app

app

app

||u/t=0 − u0 ||2m + ε 2 ||∇(u/t=0 − u0 )||2m + ε 4 ||(u/t=0 − u0 )||2m ≤ (Cm+s+2,1 + Cm,2 )ε 2 .

(83)

We do not detail the proof of this theorem since the estimates (82), (83) are obtained by collecting the results of Lemma 4–7 and by estimating the new interaction terms uip · ∇u + u · ∇uip . Nevertheless there is no new difficulty for the estimate of these terms: as previously, we get a bound on the most difficult term (which appears only in the first term of the above int,wp sum since u3 = 0) thanks to (80). 2. Linear Stability In this section, we set V = e−γ t (U ε − U app ), where γ is a parameter which will be chosen sufficiently large (but independent of ε) later and the aim is to derive an energy estimate for V which is the solution of the system

242

F. Rousset e×j ∂t v + γ v + uwp · ∇v + v · ∇uwp + e×v ε + ε + j = ∇ϕ − e × v + F2 , ∇ · v = 0, ∇ · j = 0,

∇p ε

= εv + F1 ,

(84) (85) (86)

where app

F1 = R1 F2 =

− (uapp − uwp ) · ∇v − v · ∇(uapp − uwp ) − eγ t v · ∇v,

app R2

(87) (88)

with the boundary condition (4) and the initial condition v(0, x) = v0 (x).

(89)

Note that in the Navier-Stokes equation, we have incorporated both the nonlinear term and some part of the linear terms in the source term. In this section, we shall make an estimate of the solution of (84), (85), (86) with a general source term F = (F1 , F2 ). In the next section, we shall prove Theorem 1 by using this estimate in a fixed point argument which will take into account the form of the source term given by (87), (88). We have chosen to put the terms (uapp − uwp ) · ∇v + v · ∇(uapp − uwp ) which are linear in v in the source term and to postone their estimate to the next section because they need to be estimated in a different way: the fast decay in time of the ill-prepared part and time layer part of the approximate solution makes them easier to handle, nevertheless, we have to be careful with the singularity that they carry at the space time corner t = 0, z = 0. The main theorem of this section is Theorem 9. Assume (H), then there exists γ0 > 0 and ε0 > 0 such that for every γ ≥ γ0 , ε ≤ ε0 and m ∈ N, there exists C > 0 such that ∀T ≥ 0, T ||v(T )||m,γ + ε 2 ||∇v(T )||2m,γ + ε||∇v(s)||2m,γ + ε −1 ||j (s)||2m,γ ds 0 ≤ C ||v0 ||2m,γ + ε 2 ||∇v0 ||2m,γ + ε 4 ||v0 ||2m,γ T T 2 2 −1 2 . (90) + ε||F1 (s)||m,γ + ε ||F2 (s)||m,γ ds + ||F1 (s)||m,γ ds 0

0

Note that the estimate (90) can be seen as an improvement of the estimate in Theorem 1 3 of [23]: the main amelioration is that we have an estimate L∞ T (H ()) for v which is 2 1 better than the LT (H ) given in [23]. Moreover, there are no derivatives with respect to z of the source term in the estimate (90). These two points are crucial for the nonlinear stability argument since we cannot use time and normal derivative estimates as in [23] because of the singularities in the initial layers. The proof of this theorem is quite long and hence, we split it in various lemmas. The aim of the two first lemmas is to show that we get (90) as soon as we have a good estimate T of ε 0 ||∇v||2m,γ . Hence the situation will be reduced to the one in [23, 22], and the aim of the next three lemmas will be to sharpen the estimates in [23, 22] to control the singularity in the source term.


243

Lemma 10. There exists C1 > 0 such that for γ ≥ γ0 and ε ≤ ε0 , we have T 2 ||v(T )||m,γ + ε −1 ||j (s)||2m,γ + ε||∇v||2m,γ ds ≤ C1 ||v0 ||2m,γ

T

+C1 0

0

ε||∇v(s)||2m,γ +||F1 (s)||m,γ ||v(s)||m,γ +ε−1 ||F2 (s)||2m,γ ds. (91)

Note that in this lemma, we do not use the spectral Assumption (H). Nevertheless, for the moment this estimate does not allow to get (90) by the Gronwall inequality because of the term C1 ε||∇v||2m,γ , which cannot be absorbed by the left-hand side when C1 is large. To control this term, we need other estimates which use the spectral Assumption (H) as in [23]. α to (84), (85), (86), we get Proof. After the application of the operator Zm α α α α v + γ Zm v + uwp · Zm ∇v + Zm v · ∇uwp ∂t Zm αv αj αp e × Zm e × Zm ∇Zm + + + ε ε ε α α v + Zm F1 − C α , = εZm α α α j = ∇Zm ϕ − e × Zm v + Zm F2 , α α ∇ · Zm v = 0, ∇ · Zm j = 0,

(92) (93) (94)

where the commutator C α is defined as α α , uwp · ∇]v + [Zm , ∇uwp ]v. C α = [Zm

A standard energy estimate for this system gives d 1 α 2 α α α ||Z v|| + γ ||Zm v||2 + ε −1 ||Zm j ||2 + ε||∇Zm v||2 dt 2 m α α α α α ||Zαm F1 || ||Zm v|| + ε −1 ||Zm F2 ||2 + |(C α , Zm v)| + |(Zm v · ∇uwp , Zm v)|. (95) α v, Z α v) cancels since ∇ · uwp = 0 and that We just recall that the term (uwp · ∇Zm m thanks to (93), (94), we have α α α α α α α j, Zm u) = −(Zm j, e × Zm u) = ||Zm j ||2 − (Zm j, ∇Zm ϕ) (e × Zm α α +O(1)||Zm j || ||Zm F2 || 1 α 2 α α α α = ||Zm j ||2 + O(1)||Zm j || ||Zm F2 || ≥ ||Zm j || − C||Zm F2 ||2 . 2

Next, we estimate the last term in (95) in a classical way by the same method as in (80): α α (96) v · ∇uwp , Zm v)| ≤ C ||v||2m,γ + ε||∇v||2m,γ . |(Zm We recall that C is large when the boundary layer has large amplitude. The commutator C α was already estimated in [23](estimates (40) and (41)), we have

α |(C α,wp , Zm v)| ||v||2m + (ε 2 + εγ −1 )||∇v||2m . (97) |α|≤m

Finally, (91) follows by integrating (95) in time and by using (97) and (96).

244

F. Rousset

The next step towards the proof of Theorem 9 is to estimate ε 2 ||∇v(T )||2m,γ . Lemma 11. There exists C2 > 0 such that for γ ≥ γ0 and ε ≤ ε0 we have T 2 2 ||∂t v(s)||2m,γ ds ε ||∇v(T )||m,γ + ε 0

≤ C2 ε 2 ||∇v0 ||2m,γ + +ε−1 ||j ||2m,γ

(98)

T

||v||2m,γ + ε||∇v||2m,γ + ε||F1 ||2m,γ + ε −1 ||F2 ||2m,γ ds .

(99)

0

(100)

Note that this lemma will give control of ε2 ||∇v(T )||2m,γ . This estimate will be crucial in the nonlinear stability argument and is better than in the pure rotating fluid case where only an estimate of ε3 ||∇v(T )||2 was derived in [8]. α v, a standard energy estimate, thanks to the boundary Proof. We multiply (84) by ε∂t Zm condition (4) and the divergence free condition (94) gives

d α α α α α α v||2 + ε||∂t Zm v||2 + (e × Zm v, ∂t Zm v) + (e × Zm j, ∂t Zm v) (101) ||∇Zm dt α wp α α (u · ∇v)||2 + ||Zm (v · ∇uwp )||2 + ||Zm F1 ||2 . (102) ε ||Zm

ε2

Next, we write thanks to the Young inequality, α α α α |(e × Zm j, ∂t Zm v)| ≤ ε η||∂t Zm v||2 + c(η)ε −1 ||Zm j ||2 ,

(103)

where η > 0 independent of ε and γ will be chosen sufficiently small later. Moreover, by a new use of (93) and (94), we have α α α α α α α α (e × Zm v, ∂t Zm u) = (∇Zm ϕ, ∂t Zm v) − (Zm j, ∂t Zm v) + (Zm F2 , ∂t Zm v) α α α α = −(Zm j, ∂t Zm v) + (Zm F2 , ∂t Zm v),

hence we get α α α α α |(e × Zm v, ∂t Zm v)| ≤ ηε||∂t Zm v||2 + ε −1 c(η)(||Zm j ||2 + ||Zm F2 ||2 ).

(104)

By choosing η sufficiently small, we get from (101), (103), (104) that T ||∂t v(s)||2m,γ ds ε2 ||∇v(T )||2m,γ + ε 0

T

α α ε −1 ||j ||2m,γ + ε −1 ||Zm F2 ||2 + ε||Zm F1 ||2 0 α α app +ε||Zm (u · ∇v)||2m,γ + ε||Zm (v · ∇uapp )||2m,γ ds . (105)

ε

2

||∇v(0)||2m,γ

+

To estimate the two last terms in the last inequality, we write α wp ε||Zm (u · ∇v)||2 ε||uwp ||2m,∞ ||∇v||2m,γ ε||∇v||2m,γ

and α α ˜ wp ||2m,∞ ||v||2m,γ + ε||Zm ε||Zm (v · ∇uwp )||2m ε||∇u (v3 ∂z uwp )||2m α (v3 ∂z uapp )||2m . ||v||2m,γ + ε||Zm

(106)


245

To estimate the second term, we use that α α ε||Zm (v3 ∂z uwp )||2 ε||v||2m,γ + ε −1 ||Zm (v3 ∂Z uwp,b )||2 .

Again, thanks to (80), we have α ε−1 ||Zm (v3 ∂Z uwp,b )||2 ε||∇v||2m,γ .

Consequently, we have proved that α ε||Zm (v · ∇uwp )||2m ||v||2m,γ + ε||∇v||2m,γ .

We end the proof by collecting (105) and (106), (107).

(107)

For the moment, the estimates in Lemmas 10–11 are not sufficient to conclude since T the term ε 0 ||∇v||2m,γ cannot be absorbed by the left-hand side when the boundary layer has large amplitude. As in [23, 22], we must use other estimates which come from the spectral Assumption (H). These estimates rely on a cut-off of the frequency domain following the idea of [21]. To use the Fourier transform in time, it is more convenient to deal with a vanishing initial value, hence we set v = v − e−t/ε v0 for t ≥ 0. Since v0 is such that ∇ · v0 = 0 and v0/∂ = 0, v is now a solution of ∂t v + γ v + uwp · ∇v + v · ∇uwp +

e×v e×j ∇p + + ε ε ε

(108)

= εv + F1 + F10 , j = ∇ϕ − e × v + F2 + F20 , ∇ · v = 0, ∇ · j = 0,

(109) (110)

where e × v0 F10 = e−t/ε (γ − ε −1 )v0 + uwp · ∇v0 + v0 · ∇uwp + − εv0 , ε F20 = −e−t/ε e × v0 , which still satisfies the boundary condition (4). Next we continue the source terms F , F0 by 0 for t < 0 and t > T , and we choose a smooth continuation of uwp for t ∈ R. By standard arguments for parabolic equations (see [19, 21]), the solution of the new equation will coincide with v on [0, T ], that’s why we do not use different letters for the functions and their continuation. Finally, since v /t=0 = 0, we set v = 0 for t < 0 and v will be a solution of (108) for t ∈ R. To get better estimates than in [23], we use a slightly different frequency partition. The first step here will be to elimitate low spatial frequencies. Let χ l (ξ ) ∈ Cc∞ (R2 ) such that χ l = 1 for |ξ | ≤ r and χ l = 0 for |ξ | ≥ 2r. We define v l as v l (t, y, z) = χ l (ε∂y )v(t, y, z), where χ l (ε∂y ) is the Fourier multiplier defined as Fy (χ l (ε∂y )v)(t, ξ, z) = χ l (εξ )Fy (v)(t, ξ, z). Note that the variables t and z are considered as parameters in this definition. By choosing r sufficiently small, we can get a better estimate for v l than the classical one which was obtained in Lemma 10. This is the aim of the following lemma.

246

F. Rousset

Lemma 12. There exists r > 0, C3 > 0, such that for γ ≥ γ0 , and ε ≤ ε0 , we have γ |||v l (s)|||2m,γ + ε|||∇v l (s)|||2m,γ + ε −1 |||j l |||2m,γ ≤ C3 |||v|||2m,γ + ε(γ −1 + ε)|||∇v|||2m,γ T (||F1 (s)||m,γ + ||F10 (s)||m,γ )||v(s)||m,γ + 0 +ε −1 (||F2 (s)||2m,γ + ||F20 (s)||2m,γ ) ds .

(111)

α to (108), (109), (110), this yields Proof. We apply the operator χ (ε∂y )Zm α l α l α α l ∂t Zm v + γ Zm v + uwp · Zm ∇v l + Zm v · ∇uwp +

α vl e × Zm ε

α jl α pl e × Zm ∇Zm α l α l α 0,l = εZm v + Zm F1 + Z m F1 − C α,l , + ε ε α l α l α l ϕ − e × Zm v + Zm F2 , j l = ∇Zm

+

∇

α l · Zm v

= 0, ∇

α l · Zm j

= 0,

(112) (113) (114)

where the commutator C α,l is defined by C α,l = C1α,l + C2α,l + C3α,l , where α α C1α,l = [Zm , uwp · ∇]v l + [Zm , ∇uwp ]v l , α C2α,l = Zm [χ l , uwp · ∇]v , α C3α,l = Zm [χ l , ∇uwp ]v .

We perform the same kind of energy estimate as in the proof of Lemma 10. Nevetheless, to get the improved bounds (111), we have to estimate in a different way the crucial term α v l · ∇uwp , Z α v l ). As in [23] Sect. 2.3 and [22] Sect. 3.3, we have (Zm m α l α l |(Zm v · ∇uwp , Zm v )| ≤ Cεr||∇v l ||2m,γ .

(115)

The factor r in this estimate comes from the spectral localization of χ l . Moreover, thanks to the proof of [23] Lemma 1, we also have α l |(C α,l , Zm v )| ||v||2m,γ + ε 2 ||∇v||2m,γ + εγ −1 ||∇v||2m,γ .

(116)

Note that the estimate of these commutators rely on [21] Appendix B and in particular on the fact that t and x3 playing the part of parameters: |[χ (ε∂y ), m]v(t, x3 , ·)| ≤ Cε|v(t, x3 , ·)| for m ∈ W 1,∞ .

(117)


247

Consequently, the usual energy estimate gives that γ |||v l |||2m,γ + ε|||∇v l |||2m,γ + ε −1 |||j l |||2m,γ ≤ C εr|||∇v l |||2m,γ + |||v|||2m,γ + ε(γ −1 + ε)|||∇v|||2m,γ T (||F1 (s)||m,γ + ||F10 (s)||m,γ )||v(s)||m,γ + 0 +ε −1 (||F2 (s)||2m,γ + ||F20 ||2m,γ ) ds, and hence we find (111) by choosing r sufficiently small (Cr < 1).

The previous lemma gives a good estimate of γ |||v l |||2m,γ + ε|||∇v l |||2m,γ . The aim of the two following lemmas is to estimate the remaining part, i.e. γ |||(1 − χ l (ε∂y )v|||2m,γ + ε|||∇(1 − χ l (ε∂y ))v|||2m,γ . Towards this, we need to consider differently large and bounded space-time frequencies. We choose a smooth bounded function ψ(s) on R such that ψ = 1 for |s| ≤ R and ψ = 0 for |s| ≥ 2R and we define χ (ζ ) = ψ(< ζ >), where ζ = (γ , τ, ξ ) and 1 < ζ >= (γ 2 + τ 2 + |ξ |4 ) 4 . Further, we write (1 − χ l (ε ∂y ))v = (1 − χ l )χ (ε ∂)v + (1 − χ l )(1 − χ (ε ∂))v = χ s (ε ∂) v + χ L (ε ∂)v = vs + vL, where χ (ε ∂) is the Fourier multiplier defined by Ft,y (χ (ε ∂)f ) = χ (ε ζ )Ft,y f. Note that the spectrum of v s is supported in {< εζ >≤ 2R} ∩ {ε|ξ | ≥ r} and that of v L is supported in {< εζ >≥ R} ∩ {ε|ξ | ≥ r}. We first give an estimate for v L : Lemma 13. There exists R > 0, C4 > 0 such that for γ ≥ γ0 and ε ≤ ε0 , we have γ |||v L |||2m,γ + ε|||∇v L |||2m,γ + ε −1 |||j L |||2m,γ ≤ C4 |||v|||2m,γ + ε(γ −1 + ε)|||∇v|||2m,γ

+ε(|||F1 |||2m,γ + |||F10 |||2m,γ ) + ε −1 (|||F2 ||2m,γ + |||F20 |||2m,γ ) .

(118)

α to (108), Proof. We use the same technique as in [22] Lemma 5, we apply χ L (ε∂)Zm α v L , where κ(ζ ) = i/sgn τ for |τ | ≥ 1. (109), (110) and we multiply it by (1 + κ(ε∂))Zm A standard energy estimate together with the fact that the spectrum of v L is contained in < εζ >≥ R gives

γ |||v L |||2m,γ + ε|||∇v L |||2m,γ + R 2 ε −1 |||v L |||2m,γ + ε −1 |||j L |||2m,γ (119) −1 L 2 −1 2 ≤ C ε |||v |||m,γ + ε(ε + γ )|||∇v|||m,γ + |||F1 |||m,γ + |||F10 |||m,γ |||v|||m,γ + ε −1 (|||F2 |||2m,γ + |||F20 |||2m,γ ) , (120) where C is independent of R, ε and γ . Consequently, we find (118) for R sufficiently large.

248

F. Rousset

It remains to estimate v s ; this is the crucial part, where the spectral Assumption (H) is needed. Lemma 14. Assuming (H), there exists C5 > 0, such that for γ ≥ γ0 and ε ≤ ε0 , we have ε|||∇v s |||2m,γ ≤ C5 ε|||v|||2m,γ + (ε 2 + εγ −1 )|||∇v|||2m,γ +ε(|||F1 |||2m,γ + |||F10 |||2m,γ ) + ε −1 (|||F2 |||2m,γ + |||F20 |||2m,γ ) . (121) Proof. Again we apply the operator χ s (ε∂) to (108), (109), (110), and we rewrite the equation obtained from (108) as ∂t v s + uwp,0 · ∇v s + v s · ∇uwp,0 +

e × vs e × js + = εv s + H, ε ε

(122)

where H = F1s + F10,s − (uwp − uwp,0 ) · ∇v s − v s · ∇(uwp − uwp,0 ) + C, where C is the commutator defined by C = [χ s , uwp · ∇]v + [χ s , ∇uwp ]v. Moreover, we recall that uwp,0 stands for the leading term in the asymptotic expansion (87). Note that thanks to a new use of (117), we have the estimate |||H |||2m,γ |||F1 |||2m,γ + |||F10 |||2m,γ + ε 2 |||∇v|||2m,γ + |||v|||2m,γ .

(123)

Now, we introduce the operators ˜ + , T1 (ε∂) = ε∂t + εγ + u˜ wp,0 · ∇˜ − ε wp,0 ˜ ˜ T2 (ε∂) = ε∂t + εγ + u˜ · ∇ − ε , and we rewrite (122), (109), (110) in spatial dynamics. We set t ps ϕ s V = v s1 , v s2 , ε∂z v s1 , ε∂z v s2 , v 3 , , , j3s ε ε and we get the system 1 G(ε∂, t, x1 , x2 , z/ε)V + H, ε where the symbol of the operator is given by  0 0 1 0 0 0 0  0 0 0 1 0 0 0   T1 (ζ ) −1 0 0 ∂Z uwp,b,0 ξ1 −ξ2  1  1 T (ζ ) 0 0 ∂ uwp,b,0 ξ ξ1  1 Z 2 2 G(ζ, q, z/ε) =  0 0 0  −ξ1 −ξ2 0 0  0 0 −ξ1 −ξ2 −T2 (ζ ) 0 0   0 0 0 0 0 0 0 ξ2 −ξ1 0 0 0 0 −ξ12 − ξ22 ∂z V =

where q is a placeholder for uint,0 (t, x1 , x2 ). Moreover, H is defined as

(124)  0 0  0  0 , 0  0  1 0


0,s s − F2,3 , −∇˜ · (F˜2s + F˜20,s ) . 0, 0, −H1 , −H2 , 0, H3 , −F2,3

249

(125)

The boundary condition for (124) becomes V1 = V2 = V5 = V8 = 0 on ∂.

(126)

Next, we notice that for ξ = 0, the matrix G∞ (ζ, q) = limZ→+∞ G(ζ, q, Z) is similar to A∞ (ζ, q) = limZ→+∞ A(ζ, q, Z), hence the dimension of the stable subspace of G∞ is also 4. Since we have four boundary conditions, we can construct an Evans function E by the same argument as in the introduction. Moreover, the existence of a nontrivial solution to (8) is equivalent to the existence of a nontrivial solution for (124), (126). Hence the nonvanishing of D is equivalent to the nonvanishing of E for ξ = 0. Consequently, since we have a problem under the abstract form (124), (126), we can use the fact that the Evans function E does not vanish as in [21, 23, 22] to get an energy estimate for such an elliptic boundary value problem. We find |||V |||2m,γ ε2 |||H|||2m,γ . This yields, in particular, thanks to (125) and (123), ε −1 |||v s |||2m,γ + ε|||∂z v s1 |||2m,γ + ε|||∂z v s2 |||2m,γ ε(|||F1 |||2m,γ + |||F10 |||2m,γ ) + ε|||∇˜ · (F˜2s + F˜20,s )|||2m,γ +(ε 2 + εγ −1 )|||∇v|||2m,γ + ε|||v|||2m,γ . Finally, we note that |ξ |2 ≤ C(R)ε −2 on the support of χ s , hence we get an estimate for ε|||∂z v s1 |||2m,γ + ε|||∂z v s2 |||2m,γ as in (121) by using that ε|||∇˜ · (F˜2s + F˜20,s )|||2m,γ ≤ C(R)ε −1 (|||F2 |||2m,γ + |||F20 |||2m,γ ). Next, we use that thanks to (110), we also have ε|||∂z v s3 |||2m,γ ≤ C(R)ε −1 |||v s |||2m,γ and that ˜ s |||2m,γ ≤ C(R)ε −1 |||v s |||2m,γ ε|||∇v to conclude the proof of Lemma 14.

2.1. Proof of Theorem 9. To prove (90), it suffices to collect carefully the estimates of the previous lemmas. At first, by summing (111), (118) and (121), we get γ |||v|||2m,γ + ε|||∇v|||2m,γ ≤ C6 |||v|||2m,γ + ε(γ −1 + ε)|||∇v|||2m,γ T ||F1 (s)||m,γ + ||F10 (s)||m,γ ||v(s)||m,γ ds + 0

+ε(|||F1 |||2m,γ + ||||F10 |||2m,γ ) + ε −1 (|||F2 |||2m,γ + |||F20 |||2m,γ ,

250

F. Rousset

where C6 is independent of γ ≥ γ0 and ε ≤ ε0 . Consequently, we can enlarge γ0 to get γ |||v|||2m,γ + ε|||∇v|||2m,γ T ||F1 (s)||m,γ + ||F10 (s)||m,γ ||v(s)||m,γ ds 0

+ε(|||F1 |||2m,γ + ||||F10 |||2m,γ ) + ε −1 (|||F2 |||2m,γ + |||F20 |||2m,γ .

(127)

Now, thanks to the expressions of F10 and F20 , we have that ||F1 (t)||m,γ e−t/ε (γ + ε −1 )||v0 ||m,γ + ||∇v0 ||m,γ + ε||v0 ||m,γ , (128) ||F2 (t)||m,γ e−t/ε ||v0 ||m,γ .

(129)

Consequently, we get from (127) and (128), (129) the estimate T 1 2 2 γ |||v|||m,γ + ε|||∇v|||m,γ Nγ ,ε (v0 ) + ε −1 a(t/ε)Nγ ,ε (v0 ) 2 ||v(s)||m,γ ds + 0

0

T

||F1 (s)||m,γ ||v(s)||m,γ ds + ε|||F1 |||2m,γ + ε −1 |||F2 |||2m,γ ,

where we have set Nε,γ (v0 ) = ||v0 ||2m,γ + ε 2 ||∇v0 ||2m,γ + ε 4 ||v0 ||2m,γ . Finally since v = v − e−t/ε v0 , we find that γ |||v|||2m,γ + ε|||∇v|||2m,γ T 1 Nε,γ (v0 ) + ε −1 a(t/ε)Nε,γ (v0 ) 2 ||v(s)||m,γ ds +

0

T

||F1 (s)||m,γ ||v(s)||m,γ ds + ε|||F1 |||2m,γ + ε −1 |||F2 |||2m,γ . (130)

0

Now, consider A(130) + (91) + η(98), where A and η are chosen independent of ε and γ and such that A > C1 + ηC2 and 1 − C2 η > 0. This yields for every T > 0, ||v(T )||2m,γ + ε 2 ||∇v(T )||2m,γ T + ε −1 ||j (s)||2m,γ + ε||∇v||2m,γ + ε||∂t v(s)||2m,γ ds 0

Nε,γ (v0 ) + +

T

T

ε −1 ||F2 ||2m,γ + ε||F1 ||2m,γ ds

0

1 ||F1 (s)||m,γ + ε −1 a(s/ε)Nε,γ (v0 ) 2 ||v(s)||m,γ ds.

0

We conclude by a Gronwall type inequality, we define for T ∈ [0, t]; t z(T ) = Nε,γ (v0 ) + ε −1 ||F2 ||2m,γ + ε||F1 ||2m,γ ds + 0

T

0

1 ||F1 (s)||m,γ + ε −1 a(s/ε)Nε,γ (v0 ) 2 ||v(s)||m,γ ds.

(131)


251

Note that the second member of (131) is majorated by z(T ) for T ∈ [0, t], hence we find 1 z (T ) ||F1 (T )||m,γ + ε −1 a(T /ε)Nε,γ (v0 ) 2 z(T ), and the integration of this inequality gives z(T ) z(0) + Nε,γ (v0 ) + Since z(0) = Nε,γ (v0 ) +

T 0

||F1 (s)||m,γ ds

2 .

(132)

t 0

ε −1 ||F2 ||2m,γ + ε||F1 ||2m,γ ds,

we get (90) by combining (131) and (132) for T = t. 3. Nonlinear Stability In this section, we prove a refined version of Theorem 1. We will deduce (10) in Theorem 1 as a consequence. Let U app be the approximate solution given by Theorem 8. Theorem 15. Under the same assumptions as in Theorem 1, there exists C > 0 such that the solution of (1), (2), (3), (4), (5) is defined on [0, T ∗ ] and such that ∀t ∈ [0, T ∗ ], t ε app 2 ||u (t) − u (t)||m + ε −1 ||j ε − j app ||2m + ε||∇(uε − uapp )||2m ds ≤ Cε2 . 0

Proof. As in Sect. 2, we set V = e−γ t (U ε − U app ) and hence, we study the system (84), (85), (86), but now, we take into account that app

F1 = R1

+ L + Q,

L = −v ip · ∇v − v · ∇v ip ,

Q = −eγ t v · ∇v,

app

F2 = R2 , where v ip = uapp − uwp . Note that by Theorem 8, we have Nε,γ (v0 ) ≤ C0 ε 2 . The existence of classical solutions for (84), (85), (86) is classical and we can define T ε ≤ T ∗ , the maximum of times T such that there exists a solution of (84), (85), (86) such that u ∈ C([0, T ), H m,1 ) and t Ym,γ ,ε (v(t)) + ε||∇v(s)||2m,γ ds ≤ Rε2 , ∀t ∈ [0, T ), (133) 0

where R is chosen such that R > C0 and Ym,γ ,ε (v(t)) = ||v(t)||2m,γ + ε 2 ||∇v(t)||2m,γ . As usual, to prove that T ε = T ∗ , it suffices to prove that we can never reach equality in (133) when ε is sufficiently small. Towards this, we shall use the a priori estimate of Theorem 9. To evaluate the right-hand side of (90), we note thanks to Theorem 8 that T app app ε ||R1 ||2m,γ + ε −1 ||R2 ||2m,γ ds Nm,ε,γ (v0 ) + +

0

0

T

1 ||Rapp ||m,γ ds

2

≤ Cε 2 ,

(134)

252

F. Rousset

where C is independent of ε and R. In this section, γ is fixed and hence in this section C and stand for a number which is independent of ε and R. Next, we have, thanks to Theorem 8 and Lemma 4, 5, 7, that ||v ip · ∇v||m,γ ||v ip ||m,∞ ||∇v||m,γ a(t/ε)||∇v||m,γ , ||v ip · ∇v||2m,γ ||v ip ||2m,∞ ||∇v||2m,γ a(t/ε)||∇v||2m,γ , which gives

t

0

||v ip · ∇v||m,γ ds

t

ε 0

||v ip · ∇v||2m,γ

2

2 a(s/ε)||∇v||m,γ ds 0 t t a(s/ε) a(s/ε)||∇v||2m,γ ds 0 0 t ε a(s/ε)||∇v||2m,γ ds, 0 t ds ε a(s/ε)||∇v||2m,γ ds.

t

(135) (136)

0

In a similar way, by classical Sobolev embeddings for m ≥ 2, we have that 1 1 1 1 1 2 2 2 2 ||v · ∇v ip (t)||m,γ ||v||m,γ ||∇v||m,γ ||∇uip ||m,γ √ ||v||m,γ ||∇v||m,γ a(t/ε), ε

hence we find by a successive use of the Cauchy-Schwarz and Young inequalities that T 2 ||v · ∇v ip ||m,γ ds 0

2 T 1 1 1 T 2 2 ||v||m,γ ||∇v||m,γ a(t/ε) dt ||v||m,γ ||∇v||m,γ a(t/ε) dt ε 0 0 T ε −1 a(t/ε)||v||2m,γ + εa(t/ε)||∇v||2mγ ds (137)

0

and

T

ε 0

||v · ∇v ip ||2m,γ

T 0

0

T

||v||m,γ ||∇v||m,γ a(t/ε) dt ε −1 a(t/ε)||v||2m,γ + εa(t/ε)||∇v||2m,γ dt.

(138)

Finally, we evaluate the nonlinear term Q. For m ≥ 3, we have for every T ∈ [0, T ε ) that T eγ t ||v · ∇v||2m,γ dt ε 0

T

≤C ε T∗

0

≤ CT ∗ Rε 2

||v||m,γ ||∇v||3m,γ

T 0

||∇v||2m,γ dt

dt ≤ C ε sup (||v||m,γ ||∇v||m,γ ) T∗

[0,T ]

0

T

||∇v||2m,γ dt (139)


253

and that 2 T eγ t ||v · ∇v||m,γ dt 0

≤ CT ∗

T 0 3

≤ CT ∗ R ε 2

1

3

2 2 ||v||m,γ ||∇v||m,γ dt

T 0

2

≤ CT ∗

T 0

||v||2m,γ dt

1

T

2

0

||∇v||2m,γ dt.

||∇v||2m,γ dt

3 2

(140)

Finally, we collect (90) and (134), (135), (136), (137), (138), (139), (140) and we note 3 that for ε sufficiently small, we have that CT ∗ R(ε 2 + ε 2 ) < 1/2; hence, we find ε t Ym,γ ,ε (v(t)) + ||∇v||2m,γ 2 0 t ≤ C1 ε 2 + C2 ε −1 a(s/ε)Ym,γ ,ε (v(s)) ds ∀t ∈ [0, T ε ) 0

and the Gronwall inequality gives that ∀t ∈ [0, T ε ), t ε t Ym,γ ,ε (v(t)) + ||∇v||2m,γ ≤ C1 ε 2 exp C2 ε −1 a(s/ε) ds ≤ C1 eC3 ε 2 . 2 0 0 To conclude, we choose R such that R > C1 eC3 ; this shows that inequality is impossible in (133). By classical arguments we get that T ε = T ∗ . This ends the proof of Theorem 15. 3.1. Proof of Theorem 1. We recall from Theorem 8 that T ||U ip (s)||2 + ||U tl ||2 ds ε. 0

The crucial part in the obtention of this estimate is Lemma 5. We get at once from Theorem 15 that ||uε − uint ||2L2 ((0,T ∗ )×) + ε −1 ||j ε ||2L2 ((0,T ∗ )×) ε. ∗ 2 It remains to prove that uε − uint goes to zero in L∞ loc ((0, T ), L ()). Again, we easily get from Theorem 15 that √ ||uε − uint ||L∞ ((0,T ∗ ),L2 ()) ||uip ||L∞ ((0,T ∗ ),L2 ()) + ε. loc

loc

||v ip ||

Consequently, it suffices to prove that tends to zero. From (65) ∗ 2 L∞ loc ((0,T ),L ()) −1 and (66), it suffices to prove that ||F Ik (·/ε)||L∞ ((0,T ∗ ),L2 ()) tends to zero. Let η > 0, loc we choose R such that 0 ||1|ξ |≥R ak (ξ )vk,i || ≤ η.

Then, we get ||Ik (t/ε)|| ≤ ||1|ξ |≤R Ik || + η ≤ Ce and hence, we easily get (10).

−

π 2 t ε(π 2 +R 2 )

+ η,

254

F. Rousset

4. Proof of Lemma 3 We set v = v1 + iv2 , F = F1 + iF2 and f = f1 + if2 , then v is a solution of ∂τ v + ( + i)v − ∂ZZ v = F,

(141)

in the domain Z > 0 with the initial condition v(0, y, Z) = f(y, Z) and the boundary condition v(τ, y, Z) = 0. As usual for boundary value problems for the heat equation, we can choose odd continuations of F and f and then it is equivalent to solve (141) for Z ∈ R with F and f replaced by their continuation (by an abuse of notations we will use the same letters for the functions and their continuations). Note that f is not continuous since we did not assume that f(0) = 0. The solution of this problem is then given by τ v(τ, y, z) = e−(+i)τ kτ f + e−(+i)(τ −s) kτ −s F(s) ds, (142) 0

where

Z2 exp − 4τ 4πτ is the standard one dimensional heat kernel and stands for the convolution with respect to the Z variable. We recall that in this problem the variable y only plays the part of a parameter. By standard convolution estimates, we have τ ||v(τ )||m,p e−τ ||f||m,p + e−(τ −s) ||F(s)||m,p ds kτ (Z) = √

1

0

which gives (28). Moreover, since by (26), (27), we also have +∞ τ +∞ e−(τ −s) ||F(s)||m,p dsdτ Km a(τ ) dτ, 0

and

0

+∞ τ

0

0

0

e−(τ −s) ||F(s)||m,p ds

2

2 dτ Km

2

+∞

a(τ ) dτ

,

0

we also get (29). To prove the estimates involving Z derivatives, we note, thanks to an integration by parts, that ∂Z (kτ f) = ∂Z kτ f = − ∂µ kτ (Z − µ)f(µ) dµ R

= 2kτ (Z)f(y, 0) + kτ ∂Z f. Consequently, we get

τ

kτ (Z)f(y, 0) + 2 e−(+i)(τ −s) kτ −s (Z)F(s)Z=0 ds 0 τ +e−(+i)τ kτ ∂Z f + e−(+i)(τ −s) kτ −s ∂Z F(s). (143)

∂Z v = 2e

−(+i)τ

0

We deduce from this that τ 1 1 ||∂Z v||2m Kn e−τ (1 + √ ) + e−(τ −s) a(s)(1 + √ ) ds , τ τ −s 0 +∞ e−s √ ds, < +∞, we get (29). and hence, since 0 s


255

We now turn to the proof of (30). We only prove the case k = 2, the others being easier. Thanks to (143), we get τ e−(+i)(τ −s) ∂Z kτ −s (Z)F(s)Z=0 ds ∂Z2 v = 2e−(+i)τ ∂Z kτ (Z)f(y, 0) + 2 0 τ −(+i)τ +e ∂ Z kτ ∂ Z f + e−(+i)(τ −s) ∂Z kτ −s ∂Z F(s). 0

Again by standard convolution estimates, this yields τ 1 1 − 2 −1 −1 −τ − 4ε2 τ e + e−(τ −s) e 4ε2 (τ −s) a(s) ds . |∂Z v(τ, ·, ε )|m Km τ e 0

We deduce from this that +∞ |∂Z2 v(τ, ·, ε−1 )|m dτ Km 1 + 0

and since τ −1 e

−

1 4ε 2 τ

+∞

+∞

a(s) ds 0

e−τ τ −1 e

−

1 4ε 2 τ

dτ,

0

ε2 for every τ ≥ 0, we get +∞ |∂Z2 v(τ, ·, ε−1 )|m dτ Km ε 2 . 0

This gives the first part of (30). In a similar way, we have +∞ |∂Z2 v(τ, ·, ε−1 )|2m dτ 0 +∞ 2 +∞ − 1 a(s) ds e−τ τ −2 e 2ε2 τ Km ε 4 , Km 1 + 0

0

and hence the second part of (30) is proved. It remains to prove (31). Again, we will use (143). At first, we notice that 2 1 2 2 ||z kτ ∂Z f||m |Z| kτ (Z − µ)|∂Z f(µ)|m dµ dZ R R 2 1 |z − µ| 2 kτ (Z − µ)|∂Z f(µ)|m dµ dZ R R 2 1 + kτ (Z − µ)|µ| 2 |∂Z f(µ)|m dµ dZ R R 1 1 |Z 2 kτ |2L1 (Z) + |kτ |2L1 (Z) ||∂z f||2m + ||Z 2 ∂Z f||2m 1

Km (τ 2 + 1), thanks to (26), (27). Next, thanks to the previous computation and (143), we get that 1 1 1 ||Z 2 ∂Z v||m Km e−τ (||Z 2 kτ ||m + 1 + τ 4 ) τ 1 1 + e−(τ −s) (||Z 2 kτ −s ||m + 1 + (τ − s) 4 )a(s) ds , 0

1

1

and hence we find (31) since τ 4 e−τ is uniformly bounded and ||Z 2 kτ −s ||m = O(1).

256

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References 1. Alexander, J., Gardner, R., Jones, C.: A topological invariant arising in the stability analysis of travelling waves. J. Reine Angew. Math. 410, 167–212 (1990) 2. Alinhac, S., Gérard, P.: Opérateurs pseudo-différentiels et théorème de Nash-Moser. Savoirs Actuels. [Current Scholarship]. Paris: InterEditions, 1991 3. Allen, L., Bridges, T.J.: Hydrodynamic stability of the ekman boundary layer including interaction with a compliant surface: a numerical framework. European J. Mech. B Fluids 22, 239–258 (2003) 4. Babin, A., Mahalov, A., Nicolaenko, B.: Global splitting, integrability and regularity of 3D Euler and Navier-Stokes equations for uniformly rotating fluids. European J. Mech. B Fluids 15(3), 291–300 (1996) 5. Babin, A., Mahalov, A., Nicolaenko, B.: Global regularity of 3D rotating Navier-Stokes equations for resonant domains. Indiana Univ. Math. J. 48(3), 1133–1176 (1999) 6. Chemin, J.-Y., Desjardins, B., Gallagher, I., Grenier, E.: Ekman boundary layers in rotating fluids. ESAIM Control Optim. Calc. Var. 8, 441–466 (electronic), 2002. A tribute to J. L. Lions 7. Chemin, J.-Y.: Perfect incompressible fluids. In: Volume 14 of Oxford Lecture Series in Mathematics and its Applications. New York: The Clarendon Press Oxford University Press, 1998. Translated from the 1995 French original by Isabelle Gallagher and Dragos Iftimie 8. Desjardins, B., Dormy, E., Grenier, E.: Stability of mixed Ekman-Hartmann boundary layers. Nonlinearity 12(2), 181–199 (1999) 9. Desjardins, B., Dormy. E., Grenier, E.: Instability of Ekman-Hartmann boundary layers, with application to the fluid flow near the core mantle boundary. Physics of the Earth and Planetary Interior 124, 283–294 (2001) 10. Desjardins, B., Grenier, E.: Linear instability implies nonlinear instability for various boundary layers. Ann. Inst. H. Poincaré Anal. Non-Lineaire 20(1), 87–106 (2000) 11. Dormy, E.: Modélisation numérique de la dynamo terrestre. PhD thesis, Institut de Physique du Globe, 1997 12. Gardner, R.A., Zumbrun, K.: The gap lemma and geometric criteria for instability of viscous shock profiles. Comm. Pure Appl. Math. 51(7), 797–855 (1998) 13. Gerard-Varet, D.: A geometric optics type approach to fluid boundary layers. Comm. Partial Differ. Eqs. 28(9–10), 1605–1626 (2003) ´ 14. Gisclon, M., Serre, D.: Etude des conditions aux limites pour un système strictement hyberbolique via l’approximation parabolique. C. R. Acad. Sci. Paris Sér. I Math. 319(4), 377–382 (1994) 15. Grenier, E., Guès, O.: Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems. J. Differ. Eqs. 143(1), 110–146 (1998) 16. Grenier, E., Masmoudi, N.: Ekman layers of rotating fluids, the case of well prepared initial data. Comm. Partial Differ. Eqs. 22(5–6), 953–975 (1997) 17. Kapitula, T., Sandstede, B.: Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations. Phys. D 124(1–3), 58–103 (1998) 18. Kato, T., Fujita, H.: On the nonstationary Navier-Stokes system. Rend. Sem. Mat. Univ. Padova 32, 243–260 (1962) 19. Kreiss, H.-O., Lorenz, J.: Initial-boundary value problems and the Navier-Stokes equations. Boston, MA: Academic Press Inc., 1989 20. Masmoudi, N.: Ekman layers of rotating fluids: the case of general initial data. Comm. Pure Appl. Math. 53(4), 432–483 (2000) 21. Métivier, G., Zumbrun, K.: Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems. To appear in Mem. of the Amer. Math. Soc. available at http://www.ufr-mi.u-bordeaux.fr/˜ metivier/preprints. html, Preprint, 2002 22. Rousset, F.: Large mixed Ekman-Hartmann boundary layers in magnetohydrodynamics. Nonlinearity 17(2), 503–518 (2004) 23. Rousset, F.: Stability of large Ekman boundary layers in rotating fluids. Arch. Rat. Mech. Anal. 172(2), 213–245 (2004) 24. Sermange, M., Temam, R.: Some mathematical questions related to the MHD equations. Comm. Pure Appl. Math. 36(5), 635–664 (1983) Communicated by P. Constantin


Communications in


Loop-Erased Random Walk on a Torus in Dimensions 4 and Above Itai Benjamini, Gady Kozma Department of Mathematics, Weizmann Institute of Science, Rehorot 76100, Israel Received: 7 October 2003 / Accepted: 28 February 2005 Published online: 15 July 2005 – © Springer-Verlag 2005

Abstract: We show that the statistics of loop erased random walks above the upper critical dimension, 4, are different between the torus and the full space. The typical length of the path connecting a pair of sites at distance L, which scales as L2 in the full space, changes under the periodic boundary conditions to Ld/2 . The results are precise for dimensions ≥ 5; for the dimension d = 4 we prove an upper bound, conjecturally sharp up to subpolyonmial factors. 1. Introduction A well known phenomenon in probabilistic constructions in Rd or Zd is that usually some critical dimension d exists, above which the geometry of Rd ceases to play any significant role, and the process behaves like a similar non-geometric object, usually a tree. At the critical dimension itself, similar behavior is also expected, but when compared to the non-geometric object one gets a logarithmic correction. The phenomenon was discovered in statistical mechanics and field theory, (see e.g. [W71]) for which it is known that various critical exponents stabilize beyond suitable, model dependent, upper critical dimensions. Many of these physical models can be translated to questions in probability, some of which were solved rigorously. Notable examples include the Ising model [A82], the self-avoiding walk [BS85, HS92], percolation [HS90], directed percolation [NY95], and lattice trees [DS98]. In particular, the problem of loop-erased random walk on Zd is well studied. Loop-erased random walk is a process that starts from a random walk on some graph and then removes all loops in chronological order, or in other words, whenever the random walk hits the partial path, the loop just created is erased and the process continues. The result is a random simple path. Originally [L80] suggested as a model for the selfavoiding walk (a random walk conditioned not to hit itself), better understanding of its structure has situated it as an important object in combinatorics and mathematical physics. In particular, loop-erased random walk can be used to describe the uniform spanning

258

I. Benjamini, G. Kozma

tree, which is the limit of the q-Potts model as q → 0. See [S00] for a survey, and the complementary [LSW04]. For a survey with a different focus, see [L99]. Note also the recent [BKPS04] for the rich structure of the phase transitions of the uniform spanning tree in various dimensions. For other recent results of interest, see [BLPS01, K, LPS03]. It is well known that the critical dimension of loop-erased random walk on Zd is 4, since above this dimension a random walk does not intersect itself enough and the process of loop-erasure is local and uninteresting. See [L96, Chap. 7]. Further, loop-erased walk is one of the few models where the logarithmic correction is known precisely, with a correction of log−1/3 loop-erased random walk on Z4 is similar to the regular random walk on Z4 , see [L95]. With so much known, it seems strange that a small change in settings could provoke significant difficulties. To understand why, let us examine the question we are interested in precisely. Let T be a discrete torus, Zd /(N Z)d for some large N . Let b and e be two points on a torus, and let R be a random walk starting from b and stopped on e. We wish to say something about the loop-erasure of R. The results for Zd all use the fact that the random walk does not intersect itself enough. However, in our settings the random walk does a very long walk — of the order of N d — in a relatively small space, and intersects itself over and over again. Thus it is definitely not true that the random walk and its loop erasure are similar! The random walk is essentially a random set that covers a large portion of the torus. Its loop-erasure is much thinner — as we will see, the expected size is N d/2 . The estimates discussed here may be related to a phenomenon which is of broader interest, that for certain problems, periodic boundary conditions change the answer in a fundamental way. Very roughly, if the problem features long paths with high winding number, then the correct geometry less model to consider is actually the complete graph. A few conjectures of similar nature pertaining to critical Ising model and percolation were communicated to us by Michael Aizenman together with convincing heuristic arguments. To the best of our knowledge, the results of this paper are the first rigorous demonstration of this phenomenon. Let us therefore analyze the complete graph. There are a number of ways it can be done, but our favorite is using the notion of the Laplacian random walk. A Laplacian random walk from b to e, two points on an arbitrary graph G, is constructed inductively by solving, at each step, the discrete Dirichlet problem f (e) = 1, f |γ ≡ 0, f |G\(γ ∪{e}) ≡ 0,

(1)

where γ is the partially constructed path and is the discrete Laplacian. The walk then continues to the next point using f as weights. This model was suggested in [LEP86] and was shown to be equivalent to loop-erased random walk in [L87] (though the core Markov property is already in [L80]). The case of the complete graph is very easy to analyze, since if the partially constructed curve γ has length i then   v∈γ 0 , f (v) = 1 v=e   1 otherwise i+1 and then the probability of the walk to terminate in the next step is closed formula P(# LE(R) = k) = k/n

k−1 i=2

1−

i+1 . n

i+1 n .

This gives a

Loop-Erased Random Walk on a Torus in Dimensions 4 and Above

259

√ √ In particular, we see that the correct scaling is N and that # LE(R)/ N converges 2 to a limiting distribution with density te−t /2 . Unfortunately, we do not know how to analyze more interesting graphs using the Laplacian random walk, nor can we show the existence of a limiting distribution for # LE(R) on, say, the torus 1 . have a good basis to claim that mean field behavior in our case should be √ Thus wed/2 |T | = N . For d < 4 this does not happen — indeed known results for d = 1 (trivial) and d = 2 ([K00a, K00b], see also [LSW04]) and computer simulations for d = 3 [GB90] show that even a single branch of the loop-erased walk is too big2 (the sizes of a loop-erased random walk reaching to distance N are, respectively, N , N 5/4 and N 1.62±0.01 ). We shall show that mean field behavior does occur for d > 4. In the critical dimension itself, we can only show an upper bound, and we do not calculate the precise logarithmic correction (we do have some good evidence for a conjecture on the precise logarithmic correction needed — log1/6 N — see the end of this section). Namely, our results are Theorem 1. If d > 4 then a loop-erased random walk L on the (N, d)-torus starting from a point b and stopped when hitting a point e has the estimate P(#L > λN d/2 ) ≤ Ce−cλ . If d = 4 then P(#L > λN 2+ ) ≤ Ce−cλ ∀ > 0. Where the constants C and c may depend on d and on . Theorem 2. Let d ≥ 5. Let b be a point in T = TNd and let e be a random, uniform point in T . Let R be a random walk on T starting from b and stopped at e. Let λ ≥ N −1/2 . Then P(# LE(R) ≤ λN d/2 ) ≤ Cλ log λ−1 . Returning to the cases of d ≤ 3, we see that the reason for non-mean-field behavior is strong local intersections and these increase the size of the loop-erased walk. Therefore we are tempted to conjecture Conjecture. Let G be a vertex transitive finite graph, and let b and e be two random points in G. Let R be a random walk starting from b and stopped when hitting e. Then E# LE(R) ≥ c |G|. A graph G is vertex transitive when, for every two vertices v and w there exists a graph automorphism of G carrying v to w. The requirement that G is vertex transitive is supported by the standard “extreme non-transitive” example of a tree of size N , where the loop-erased random walk between b and e is of course the only path between b and e and its length is bounded by C log N . We wish to end this introduction with one last conjecture. Returning to the analysis of the complete graph using the Laplacian random walk, we note that this analysis does 1 As this paper was being prepared to print, this was proved (in dimension 5 and above) by Peres and Revelle, see [PR]. 2 We believe that the growth exponents in d = 2, 3 are the same on Zd and T d , but this is beyond the N scope of this paper.

260


not change by much if one considers α-power Laplacian random walk, which is a walk one gets if one takes as weights for any step the function f α , where f is defined by (1)— this generalization was also discussed in [LEP86], and has since been investigated non-rigorously by physicists, see e.g. [H02]. For the complete graph we get that the size of a typical path is N 1/(1+α) . We ask: is this behavior replicated in a d-dimensional torus for d > dαcrit ? 2(1+α) Conjecture. For α ≤ 1, dαcrit = 2(1+α) the typical path of a α , i.e. for any d > α α-weighted Laplacian random walk on a d-dimensional torus is of size N d/(1+α) while for smaller d’s this does not hold.

We have no good conjecture on the value of the critical dimension for α > 1, though it does seem (again, we have no proof of that) that for α = ∞ (which corresponds to a deterministic process which simply proceeds to the point where f attains its maximum) the process gives a straight line from b to e in all dimensions, so one might say the critical dimension is 1. Also note that we do not believe this also describes the critical dimension in Zd . 1.1. About the proof. The basic question behind the solution is “what is the probability of a random walk of length L will hit a loop-erased walk of length L?” (in dimension 4 we need to differentiate between these two lengths, but only by a sub-polynomial factor). When the probability is larger than some constant c > 0, then this is the L we seek, as this means that the probability of a loop-erased random walk to go further than λL is exponentially small in λ. Since a loop-erased walk is a complicated object, let us first ask “what is the probability of a random walk of length L will hit some set of size L?” This probability is largest when is rather spread out. Take as an example to be a random collection of points on the torus. It is easy to calculate the expected number of intersections of a random walk with and the second moment and to derive from both the estimate that the probability is ≈L2 N −d , so this gives that the L we look for is 4 two or at most three steps are necessary to get the true estimate, N d/2 . This argument is done in Lemma 1. 1.2. Reading recommendations. Section 2 is probably the one deserving most attention. While the main ideas are sketched above, the devil is in the details and the interested reader might want to read through the proof and do √ the “exercise” — not so designated explicitly — of simplifying the proof with a cost of log in the final result. Section 3 is technical and most readers would probably agree that the conclusion (Theorem 3) is not surprising. The proof of Lemma 5 is the core — as for Lemma 4, you might opt to read its statement but skip its proof. And again, verify that the claim is trivial if one is willing to lose a factor of log (the argument is contained in the first half-page of the proof of


261

Lemma 5). Section 4 contains the proof of Theorem 2 and is quite short. While there are alternative, more complicated approaches that might prove a little more we have not included them. There are some comments and hints at the end of Sect. 4 — we hope they make at least some sense. We have collected some well known and unsurprising facts we use (and their proofs) in the Appendix. We hope this makes the paper more accessible to non-experts and students. Lemmas with numbers like “A.7” are to be found in the appendix.

1.3. Standard notations. In the sequel we denote by C and c positive constants which may depend on the dimension but on nothing else. C will usually pertain to constants which are “large enough” and c to constants which are “small enough”. The notation x ≈ y is a short-hand for cx ≤ y ≤ Cx. In dimension 4 we shall prove only imprecise estimates, namely that the length of the loop-erased walk is < N 2+ . All constants C and c may depend on this as well. Similarly, all constants implicit in notations such as O and ≈ might depend on d and . Occasionally we shall number constants for clarity. When we write log x we always mean max{log x, 1} and log 0 = 1. The (N, d)-torus, denoted by TNd is the set Zd /(N Z)d endowed with the graph structure derived from Zd and the distance derived from the l2 norm on Zd . The distance of v and w will be denoted by |v − w|, while distance of sets will be denoted by d(·, ·). A ball of radius r and center v in either Zd or TNd will be denoted by B(v, r) and its inner boundary (namely, all points in B with an edge leading outside of B) by ∂B(v, r). 2. The Upper Bound We will need to examine the effect of adding a section to a path and how it might increase the length of its loop-erasure. We shall always assume that the section we add starts at 0, so that we are looking at a path γ : {−m, . . . , n} → T and define, in addition to the usual loop-erasure of γ , which we will denote by LE(γ ), the continued loop-erasure, which we shall denote by LE+ (γ ). Here are both definitions: Definition. For a finite path γ : {−m, . . . , n} → T in a graph T we define its loop erasure, LE(γ ), which is a simple path in T , by the consecutive removal of loops from γ . Formally, LE(γ )0 := γ (−m), LE(γ )i+1 := γ (ji + 1) ji := max{j : γ (j ) = LE(γ )i }. Naturally, this is defined for all i such that ji < n. The continued loop-erasure is a subset of LE(γ ) defined by LE+ (γ )i := LE(γ )I +i I := min{i : ji ≥ 0}. The notations LE(γ [A, B]) and LE+ (γ [A, B]) stand for the loop-erasure and continued loop-erasure of the segment of γ going from A to B. When we write −∞ in place of A we just mean the beginning of the path, nothing more. Definition. Let d ≥ 4 be the dimension and let N ∈ N. Let R be a path in T = TNd such that the negative part is fixed and the positive part is a random walk on T . Let

262


b = R(0). Let v ∈ T and 0 < r < 18 N , and assume for simplicity that b ∈ B(v, 2r). Let ti be stopping times defined by t0 = 0 and then inductively t2i+1 := min{t ≥ t2i : R(t) ∈ ∂B(v, 2r)}, t2i := min{t ≥ t2i−1 : R(t) ∈ ∂B(v, 4r)} .

(2)

Let f : R → R be an increasing function. Then we say that the (d-dimensional) random walk has the f -property if one has P(#(LE+ (R[−∞, ti ]) ∩ B(v, r)) > λf (r) | R[t2j , t2j +1 ]∀j ) ≤ Ce−cλ

(3)

which should hold for every such v and r, every λ > 0, every i ∈ N and any path we put in the negative portion of R. The conditioning here, in words, is on any arbitrary set of paths between t2j and t2j +1 , and in particular on the points R(ti ) themselves. Notice that we do not condition on the value of the ti ’s. Let us remark that for the proof of the upper bound it is enough to consider the case where R has no negative part, and then LE+ ≡ LE. Lemma 1. Let d ≥ 4. Then 1. If the d-dimensional random walk satisfies the r α logβ r-property for r α logβ r r d−2 log−3 r, then it also satisfies the r α/2+1 log(β+3)/2 r-property. d−2 log−3 r-property, then it also 2. If the d-dimensional √ random walk satisfies the r d/2 satisfies the r log log r-property. 3. If it satisfies the r α logβ -property for r α logβ r r d−2 log−3 r, then it satisfies the r d/2 -property. Case 2 is not really necessary for the proof of the theorem, we include it here mainly for completeness. Proof. Denote the function given to us (e.g. r α logβ r) by f (r) and the result (e.g. r α/2+1 log(β+3)/2 r) by g(r). Let ti be the stopping times from the definition of the f -property. The main part of the lemma will consider the events in R[ti , ti+1 ] for some particular odd i. Therefore let us fix i > 0. Denote Li,v,r := #(LE+ (R[−∞, ti ]) ∩ B(v, r)). Clearly L2i+1,v,r ≤ L2i,v,r so if we prove the lemma for all i odd it will also hold fori even. To fix notations, we consider the time span [−∞, ti ] as the “past” and ti , ti+1 is the “present”. We start by examining the past. Let w ∈ B(v, r) and s ≤ 18 r. The first step is to show that (3) holds if we replace the ball but keep the stopping times, i.e P(#(LE+ (R[−∞, ti ]) ∩ B(w, s)) > λf (s) | R[t2j , t2j +1 ]∀j ) ≤ Ce−cλ .

(4)

We generalize the notation Li,v,r to Li,w,s := #(LE+ (R[−∞, ti ]) ∩ B(w, s)), that is, again, the loop-erased random walk inside a smaller ball measured at the stopping times pertaining to the larger ball. Here our conditioning by everything outside the ball is crucial. Let Kj ∈ N be some arbitrary numbers, and let γj,k be paths (1 ≤ k ≤ Kj ) in B(v, 4r) \ B(w, 2s) such that γj,1 is a path going from R(t2j −1 ) ∈ ∂B(v, 2r) to ∂B(w, 2s), γj,k for 1 < k < Kj is a path from ∂B(w, 4s) to ∂B(w, 2s) and γj,Kj is a path from ∂B(w, 4s) to R(t2j ) ∈ ∂B(v, 4r). If Kj = 1 then let γj,1 be a path from R(t2j −1 ) to R(t2j ). Then we can sum


263

over all such combinations of K and γ as follows. Denote by X the event Li,w,s > λf (s). Let YK,γ be the event that for all j , the random walk on [t2j −1 , t2j ] follows γj,1 until ∂B(w, 2s), then stays within B(w, 4s), then follows γj,2 , etc. until finally exiting from B(v, 4r). Then P(X | R[t2j , t2j +1 ]∀j ∩ YK,γ ) · P(YK,γ | R[t2j , t2j +1 ]∀j ) P(X | R[t2j , t2j +1 ]∀j ) = K,γ

≤ Ce−cλ

P(YK,γ | R[t2j , t2j +1 ]∀j ) = Ce−cλ .

(5)

K,γ

(i−1)/2 Of course, we used the f -property for w, s and the index j =1 Kj ; and the fact that B(w, 4s) ⊂ B(v, 2r). The inequality (4) is not as useful as it should be since most balls of radius s (for s r) are empty anyway. However, another consequence of the conditioning is the fact that (4) is independent from the event LE+ (R[−∞, ti ]) ∩ B(w, 4s) = ∅. The reason is that Li,w,4s = 0 if and only if the segment inside B(w, 4s) is cut “from the root”, i.e. for some u1 < u2 < · · · < u2n , n ∈ {1, 2, . . . } we must have R[u2i−1 , u2i ] ∩ B(w, 4s) = ∅ and R(u2i ) = R(u2i+1 ). Whether this happens in the positive or negative part of R is immaterial — in both cases this is an event that happens outside B(w, 4s), therefore it is an event we condition on. We get P(Li,w,s > λf (s) | R[t2j , t2j +1 ]∀j ) ≤ Ce−cλ P(Li,w,4s = 0 | R[t2j , t2j +1 ]∀j ) . (6) Let γ = γi be the (chronologically) first G elements of LE+ (R[−∞, ti ]) ∩ B(v, r), where G is some number. If LE+ (R[−∞, ti ]) ∩ B(v, r) contains less than G elements, take γ = LE+ (R[−∞, ti ]) ∩ B(v, r). Equations (4) and (6) allow us to get a “secondorder estimate for γ ”. By this we mean the quantity Vs := #{w1 , w2 ∈ γ : |w1 − w2 | ≤ s} which has the estimate P(#γ > δELi,v,r and Vs > λ log(s/δ)f (s)#γ ) ≤ Ce−cλ

(7)

for any parameters λ > 0 and 0 < δ < 1. Before starting the proof of (7) let us just remark that the first condition and the variable δ are unfortunate technicalities. The “essentials” of (7) are really the stronger claim P(Vs > λ(log s)f (s)#γ ) ≤ Ce−cλ , but we don’t know how to prove it. Also note that it is rather easy to show P(Vs > λ(log r)f (s)#γ√) ≤ Ce−cλ , saving us all the mucking with δ later on, but this inequality will cost us a log r in the final result of Theorem 1. Proof of (7). Cover B(v, r) by balls {Bj } of radius 2s such that any two points of distance ≤ s are inside at least one Bj , and such that each point is covered ≤ C times. Examine one Bj = B(wj , 2s). We have (not writing the “| R[t2j , t2j +1 ]∀j ” for brevity) ELi,v,r > c

j

(6)

P(Li,wj ,8s > 0) ≥ cecλ

P(Li,wj ,2s > λf (s)) ∀λ.

j

Denote by Xµ the total volume of the balls Bj , where Li,wj ,2s > µf (s) and get EXµ ≤ Ce−cµ s d ELi,v,r . This gives, using P(Xµ > ecµ EXµ ) ≤ e−cµ , P(Xµ > Cs d e−cµ ELi,v,r ) ≤ e−cµ

∀µ,

264


and shoving in #γ in a way that might look, for now, a little artificial, we get P(#γ > δELi,v,r and Xµ > Cs d δ −1 e−cµ #γ ) ≤ e−cµ

∀µ.

Taking µk = λ log(s/δ) + Ck and assuming that λ > C for some C sufficiently large (as we may, without loss of generality), we get P(#γ > δELi,v,r and for some k, Xµk > c−k #γ ) ≤ Ce−cλ Now, since Vs ≤ j #(γ ∩ Bj ) · Li,wj ,2s , then Vs ≤ #γ (λ log(s/δ) + C)f (s) +

∞

.

(8)

Xµk (λ log(s/δ) + Ck)f (s).

k=1

If it happens that Xµk ≤ c−k #γ for all k, i.e. the opposite of the second half of the event in (8), then Vs ≤ λ log(s/δ)f (s)#γ +

∞

(c−k #γ )f (s)(λ log(s/δ) + Ck)

k=1

≤ Cλ log(s/δ)f (s)#γ , 1 r but (7) holds for larger s too and we get (7). This argument works for any s ≤ 16 (there’s not much point in s > 2r of course) — we only have to pay in the constant C.

We want (7) to hold not for one particular s but for all s and the simplest version of such an inequality is P(#γ > δELi,v,r and ∃s s.t. Vs > λ log2 (s/δ)f (s)#γ ) ≤ Ce−c1 λ

(9)

which follows from using (7) with λs := λ log(s/δ) and summing over s. Continuing the proof of the lemma, it is now time to examine the present. We keep the notations of G, γ and Vs . For an odd i we want to estimate the probability pi := P(R[ti , ti+1 ] ∩ γ = ∅)

.

Lemma A.5 allows us to consider a unconditioned random walk starting from R(ti ) and stopped on ∂B(v, 4r) instead of R. Denote it by R . Denote by Xi the number of intersections of R with γ , so pi ≈ P(Xi > 0). We have E(Xi | past) =

ti+1

P(R (t) = w | past)

.

t=ti w∈γ

For r 2 ≤ t − ti ≤ 2r 2 we have for half of the w ∈ B(v, r) that P({R (t) = w} ∩ {t < ti+1 }) > cr −d (“half of the w’s” means that we need t − ti + ||w − R(ti )||1 to be even, otherwise the probability is zero). Therefore E(Xi | past) > cr 2−d #γ

.

(10)


265

Next estimate E(Xi2 | past). Assume until further notice that Vs ≤ λ log2 (s/δ)f (s)#γ for some δ and λ and for all s. Then E(Xi2 | past) = P(R (ti ) = wi ) ≤ t1 ,t2 ,w1 ,w2 ∞ ∞

P R(t) = w1 , R(t + ) = w2 ,

≤2

=0 k=0 t,w1 ,w2

√ √ k ≤ |w1 − w2 | < (k + 1) .

(11)

√ Examine one couple of w1 , w2 ∈ γ with k ≤ |w1 − w2 |. Remembering the independence of the past from the present we can estimate the probability of one summand with a standard estimate on the end point of a random walk of length starting from w1 . We get P(R(t) = w1 , R(t + ) = w2 ) ≤ Cr −d −d/2 e−k

2 /2

.

We sum over all t. Since, easily, P(ti+1 − ti > ) ≤ Ce−c/r and since E(ti+1 − ti | ti+1 − ti > ) ≤ C max{r 2 , }, we get 2 2 P(R(t) = w1 , R(t + ) = w2 ) ≤ Ce−c/r max{r 2 , }r −d −d/2 e−k /2 . 2

t

Plugging this into (11) we get E(Xi2 | past) ≤ C

∞ ∞

e−c/r max{r 2 , }r −d −d/2 e−k 2

2 /2

V(k+1)√ .

(12)

=0 k=0

For all our functions f (that is, all the specific functions we named in the statement of the lemma) we have ∞

e−k

2 /2

V(k+1)√ ≤ λ#γ

∞

e−(k−1)

2 /2

√ √ f (k ) log2 (k /δ) ≤

k=1

k=0

√ ≤ Cλ#γf ( ) log2 (/δ)

.

Similarly, for all our functions f we have ∞

√ 2 e−c/r max{r 2 , }−d/2 f ( ) log2 (/δ) ≤

=0 2

≤ Cr

2

r

√ −d/2 f ( ) log2 (/δ)

.

=0

Equations (12) and (13) give 2

E(Xi2 | past)

≤ Cλr

2−d

#γ

r =0

√ −d/2 f ( ) log2 (/δ)

(13)

266


and then with (10) and the standard inequality P(X > 0) ≥ (EX)2 /EX2 we get r 2−d #γ P(Xi > 0 | past) > c √ λ −d/2 f ( ) log2 (/δ)

(14)

.

This inequality is the heart of the proof. We recall that we assumed Vs ≤ λ log2 (s/δ) · f (s)#γ to get it. Fix G = µg(r), where µ > 1 is some variable which we will fix

later and where g is as defined in the beginning of the lemma. Let H = 2 g(r)r −2 , where · is the integer value. Let X1 = X1 (µ) be the event that #γi = G, let X2 = X2 (λ, δ, µ) be the event that Vs ≤ λ log2 (s/δ)f (s)G for all s (λ and δ are two additional variables) and let X3 = X3 (µ) be the event that R[tj +1 − tj ] ∩ γ = ∅ for all odd i ≤ j ≤ i + H . The events comprising X3 are (conditioning on the R(tj )) independent, therefore we may use (14) 21 H times to get

r 2−d µg(r) P(X3 | X1 ∩ X2 ) ≤ 1 − c 2 √ λ r=1 −d/2 f ( ) log2 (/δ) cµ ≤1− . λ log2 δ −1

1H 2

(15)

To see the rightmost inequality in (15), for each of the cases in the formulation of the lemma, apply the corresponding f and g and estimate the sum. Indeed, (15) is the inequality that governs the connection between f and g. Note that the formulation of the lemma is a little lax: if f (r) = r α logβ r with √ α > d − 2 then we can actually prove the lemma with g = r α/2 log(β+2)/2 , i.e. one log r factor better than the formulation √ of the lemma. This additional log r factor is here only for the case α = d − 2 and β < −3. Have no fear — this factor will disappear in the conclusion of Theorem 1. The proof of the lemma will now follow by induction over i. We use a “jumping induction” that assumes that for some k and K we have the inequality P(Li,v,r > νg(r)) ≤ Ke−kν for all ν > 0 and then proves the same for Li+H,v,r (the case i = 0 needs no explanation). Therefore we need first to calculate how much Li,v,r can change in between. Clearly, if R([tj , tj +1 ]) does not intersect LE([R[0, tj ]) then Lj +1,v,r − Lj,v,r ≤ tj +1 − tj

.

These variables have the simple estimate P(tj +1 − tj > νr 2 ) ≤ Ce−cν

(16)

irrespectively of R(tj +1 ) and R(tj ) for all j odd. Denote by Ai the sum of 21 H of those, and get a similar estimate (see Lemma A.9): P (Ai > νg(r)) ≤ Ce−c2 ν

Ai :=

i+H

tj +1 − tj

.

(17)

j =i j odd

Next we make the following important assumption: G > δELi,v,r

∀i.

(18)


267

Actually, we want it to be true independently of the value of µ, so we really need g(r) > δELi,v,r . This holds for δ sufficiently small, but it is inconvenient to fix the value of δ at this point, as it depends on some constants (depending on d only) which are determined only later. Therefore we shall perform the necessary calculations with δ a variable and finally fix its value as some constant when we have all the information at hand, see (20). With a value of δ satisfying (20), or smaller, (18) will hold. It is time to compare Li,v,r with Li+H,v,r . Li+H,v,r might be larger than νg(r) for the simple reason that Ai is very large. Let τ ≤ ν be yet another variable describing what “very large” means and we may estimate this phenomenon simply by ν

P({Li,v,r > (ν − n − 1)g(r)} ∩ {Ai > ng(r)})

.

n=τ

“Simply” because we ignore any effect of intersections. If, however, Ai is not as large we need both Li,v,r to be rather large, and X3 , i.e. to have no intersections with a path of length G = µg(r) during the last H “moves”. We need to assume µ + τ < ν for this to make sense, and this assumption holds until (19) below and we will not repeat it. All in all we get P(Li+H,v,r > νg(r)) ≤ P({Li,v,r > (ν − τ )g(r)} ∩ X3 ) ν + P({Li,v,r > (ν − n − 1)g(r)} ∩ {Ai > ng(r)}) ∀i, ν, τ, µ n=τ

(the parameter µ hides in the definition of X3 ). For the first summand we have by (9), (18), (15) and the induction hypothesis that P({Li,v,r > (ν − τ )g(r)} ∩ X3 ) ≤ ≤ P({Li,v,r > (ν − τ )g(r)} \ X2 ) + P({Li,v,r > (ν − τ )g(r)} ∩ X3 ∩ X2 ) cµ ≤ Ce−c1 λ + Ke−k(ν−τ ) 1 − ∀i, ν, τ, λ, µ, δ, λ log2 δ −1 and estimating the other summands using (17) we get cµ P(Li+H,v,r > νg(r)) ≤ Ke−k(ν−τ ) 1 − + Ce−c1 λ + λ log2 δ −1 ν Ke−k(ν−n−1) · Ce−c2 n ∀i, ν, τ, λ, µ, δ. +

(19)

n=τ

Having arrived at this closed formula, we only need to pick our variables carefully. First pick τ = C log δ −1 for some C sufficiently large. This will give, if k < c2 /2, that ν n=τ

Ke−k(ν−n−1) · Ce−c2 n ≤ C

−1

e−C log δ Ke−k(ν−τ ) ≤ CδKe−k(ν−τ ) 1 − e−c2 /2

.

Next we pick λ = Cν and µ = 21 ν, and the requirement µ + τ < ν translates to ν > C log δ −1 . We get from everything that

−1 1 − log2cδ −1 + Cδ + Ce−cν . P(Li+H,v,r > νg(r)) ≤ Ke−kv ekC log δ

268


Pick k = c log−3 δ −1 and get, for δ sufficiently small and ν > C log δ −1 that c . P(Li+H,v,r > νg(r)) ≤ Ke−kν 1 − log2 δ −1 Pick K sufficiently large so that the inequality P(Li,v,r > νg(r)) ≤ Ke−kν will hold trivially for ν ≤ C log δ −1 — notice that because k = c log−3 δ −1 we have that K does not depend on δ — and our induction is complete. With these k and K, the inequality P(Li,v,r > νg(r)) ≤ Ke−kν is preserved from i to i + H and since it clearly holds for i ≤ H then it holds for all i. Is this the end of the lemma? Almost. We still need to justify the assumption (18). The estimate P(Li,v,r > νg(r)) ≤ Ke−kν gives ELi,v,r ≤ g(r) Kk ≤ Cg(r) log3 δ −1 . Therefore (remember that G > g(r)) the assumption reduces to the inequality g(r) > g(r) · (Cδ log3 δ −1 )

(20)

.

Taking δ sufficiently small this will hold, and the lemma is proved.

Lemma 2. The d-dimensional random walk has the f -property for r d/2 d > 4 . fd (r) := 2+ r d=4

(21)

Proof. Trivially, the d-dimensional random walk has the r d -property. Therefore we may apply Lemma 1 twice for d > 6, thrice for d = 6 or 5 and log −1 times for d = 4. Proof of Theorem 1. Lemma 2 gives P(Li,v,r > λf (r)) ≤ Ce−cλ . where Li,v,r = #(LE(R[0, ti ]) ∩ B(v, r)) for any v and r satisfying b ∈ B(v, 2r), where f is defined by (21). Note that at this point we do not need the formulation in terms of continued process, and we may set the negative part of R to empty. If in addition e ∈ B(v, 4r), then the event that R is stopped between tI and tI +1 is external to the ball, therefore we get that (21) holds for I . Since the section of the walk from tI until the time when R hits e can only decrease LE(R) ∩ B(v, r) we get P(#(L ∩ B(v, r)) > λf (r)) ≤ Ce−cλ . However, we can cover our torus by balls B(vi,j , N 2−i ) with the property b, e ∈ B(vi,j , 4N 2−i ) and with the number of j ’s corresponding to each i bounded by a constant. Therefore for some constant c3 sufficiently small we have P(#L > λf (N )) ≤ P(∃i, j s.t. L ∩ B(vi,j , r) > c3 λ2i/4 f (r))

c log N

≤

Ce−cλ2

i/4

≤ Ce−cλ

.

(22)

i=0


269

Remark. The same techniques can be improved to show that P(#L > λf (N )) ≤ Ce−cλ , 2

where f is given by (21). The basic phenomenon behind this estimate is that to get a path of length λf (N ), we need to have that each of the λ sections of the random walk, which are essentially independent, would not intersect any other. Since there are cλ2 couples, the true estimate of the probability is square-exponential, as above. The analysis required to get this estimate is not inherently more difficult than that of the exponential estimate, but is more technical and we decided to represent the simpler exponential estimate. On the other hand, we are not aware of a simpler version of the proof that gives an estimate of the decay of the probability worse than exponential. This follows from the recursive character of the proof. Thus, Lemma 1 may be simplified by removing the requirement that the probability decays exponentially, but it then cannot be used recursively to get a reasonable final result. Similarly, the very strong independence condition in Lemma 1, that the probability estimate inside every ball is independent of everything that happens outside the ball, cannot be relaxed without destroying the ability of the lemma to be used recursively. We wish to reiterate that the only major simplification we are aware of of this proof is the one discussed after (7). It saves the discussion after (5), i.e. the one leading √ to (6), as well as each and every appearance of the parameter δ. The cost is an added log factor in the formulation of the theorem. Conjecture. The accurate upper bound in dimension 4 is N 2 log1/6 N . The method above may be refined in many points and an estimate of the type N 2 logα N may be achieved for rather small α’s. However, a fundamental difficulty is the fact that the sum in the denominator of (14) truly depends on N , which means that the second moment methods used here alone cannot give a precise result. 3. Absolute Times The proof of the lower bound is, as will be seen in Sect. 4, quite simple once a good estimate of the upper bound is available. Actually, one might think about the recursive nature of the proof of the upper bound in the following terms: “the proof of the upper bound was only possible once a good estimate of the upper bound was available”. Unfortunately, we were not able to get a reasonable proof of the lower bound using only Lemma 1. The problem is that we need to know what happens at absolute times, i.e. to fix some t and get an estimate for LE(R[0, t]). Calculations true for ti do not hold automatically for a fixed t. Apriori, one cannot rule out behavior such as “the loop-erased random walk is much denser if t is divisible by 1024”, since the ti ’s might avoid those “bad absolute times”. The purpose of this section is to show that this ridiculous behavior does not occur. The first step is to learn something about the distribution of the ti ’s. Since ti is a sum of the return times to some sphere, and these return times are more-or-less independent, we would expect a central limit theorem. We don’t need something so precise — we shall prove below (Lemma 4) a large deviation estimate of the sort one would expect from a Gaussian variable, and this will be enough. We start with

270


Lemma 3. Let X1 , . . . , Xn be variables with the properties P(|Xi | > λ | X1 , . . . , Xi−1 , Xi+1 , . . . , Xn ) ≤ Ce−cλ , E(Xi1 · · · Xik | Xik+1 , . . . , Xil ) ≤

k

C exp(−c min |ij − im |), 1≤m≤l m=j

j =1

(23) (24)

where (24) needs to hold only for i1 , . . . , il all different. Then for all λ < cn1/4 ,

√ 2 P Xi > λ n ≤ Ce−cλ . We interpret the condition (24) in the case k = l = 1 as saying EXi = 0 for all i. In the case k > 1, we call (24) a “pseudo independence” relation, because, rather than claiming that E Xi = 0, as we would have for independent variables, we get that it is exponentially small in the distance, so that if the ik ’s are relatively sparse, it will be extremely small. Actually, it is possible to replace exp(−ck) with any sequence ak with ak < C. The proof is a pretty standard exercise: a calculation (which can be done either directly or√by comparing to the case of independent exponential variables) can show that for k < c n, E

2k Xi

≤ (Ckn)k

.

Taking k = cλ2 and using Markov’s inequality will give the lemma. We skip the gory details. Lemma 4. Let b ∈ TNd and let R be a random walk on T starting from b. Let C < r < 1 d 8 N, v ∈ TN and let ti be the stopping times defined by (2). Then there exists numbers E = E(r) ≈ N d r 2−d and σ = σ (r) ≈ E such that √ 2 P(|tn − nE| > λσ n) ≤ Ce−cλ

(25)

for all n ∈ N and λ < cn1/4 . Proof. The point is of course to show that the variables ti+1 − ti are pseudo independent and apply Lemma 3. The first thing to note is that the distributions of R(ti ) converge exponentially. Let q1 and q2 be two distributions on ∂B(v, 2r), and denote :=

|q1 (x) − q2 (x)|

.

x∈B(v,2r)

Let Rµ , µ = 1, 2 be random walks starting from a point on ∂B(v, 2r) chosen with the distribution qµ and stopped when hitting ∂B(v, 4r). Let pµ be the distributions on the hit points of Rµ . Then p1 (w) − p2 (w) =

x∈∂B(v,2r)

(q1 (x) − q2 (x))π(x, w),

(26)


271

where π(x, w) is the probability of a random walk starting from x to hit w. Let A+ ⊂ ∂B(v, 2r) be the set where q1 (x) ≥ q2 (x), and define D + (w) = |q1 (x) − q2 (x)|π(x, w) . x∈A+

Clearly w∈∂B(v,4r)

D + (w) =

|q1 (x) − q2 (x)|π(x, w) =

x∈A+ w

1 2

and similarly for D − defined equivalently using A− := ∂B(v, 2r)\A+ . Furthermore, the inequality π(x, w) ≈ r 1−d (see Lemma A.4) gives that D ± (w) ≈ r 1−d and therefore |D + (w) − D − (w)| ≤ (1 − c)(D + (w) + D − (w)) for some constant c > 0. This gives |p1 (w) − p2 (w)| = |D + (w) − D − (w)| w

w∈∂B(v,4r)

≤ (1 − c)

D + (w) + D − (w) = (1 − c)

(27)

w

and we see that the L1 distance between the distributions has contracted. An identical calculation works when the random walk starts from ∂B(v, 4r) and stops at ∂B(v, 2r) (see the remark following Lemma A.4) therefore we see that there is only one limiting distribution as i increases, and that the L1 distance to this distribution decreases expoµ nentially with i. In other words, if ti are stopping times defined by (2) for the walks Rµ then we get |P(R1 (ti1 ) = w) − P(R2 (ti2 ) = w)| ≤ e−ci . (28) w

This

L1

estimate allows to get a uniform estimate for every w and i > 0: |P(R1 (ti1 ) = w) − P(R2 (ti2 ) = w)| ≤ Ce−ci min P(Rµ (ti ) = w). µ

µ=1,2

(29)

µ

Indeed, take the distributions of Ri−1 as the qµ ’s in (26) and together with (28) and π(x, w) ≤ Cr 1−d get that |P(R1 (ti1 ) = w) − P(R2 (ti2 ) = w)| ≤ Cr 1−d e−ci

.

µ P(Rµ (ti )

In the other direction, π(x, w) ≥ cr 1−d gives = w) ≥ cr 1−d and we get (29). To make notations simpler, let Bi be ∂B(v, 2r) if i is odd and ∂B(v, 4r) if i is even. Now, each ti+1 − ti has an exponential distribution3 , with its expectation being less than or equal to Cr 2 i is odd Ui := (30) CN d r 2−d i is even 3 For i even, t 2 i+1 − ti has a rather large (>c) probability to be very small, of the order of r . However, since there is also a probability >c to escape B(v, 21 N), this fact has negligible impact on the moments of ti+1 − ti .

272


even after conditioning on the entry and exit points. In a formula, P(ti+1 − ti > λUi | R(ui ) = y1 and R(ui+1 ) = y2 ) ≤ Ce−cλ

(31)

for every y1 ∈ Bi and y2 ∈ Bi+1 (see Lemmas A.8 and A.11). Define the variables Xi := (ti+1 − ti − E(ti+1 − ti ))/U0

.

We wish to use Lemma 3 for the Xi ’s. To get (23) we use (31) to see that E(ti+1 −ti )/U0 ≤ CUi /U0 ≤ C and then use (31) again to get P(Xi > λ | R(ti ), R(ti+1 )) ≤ Ce−cλ

(32)

.

Denote by X the event X1 , . . . , Xi−1 , Xi+1 , . . . , Xn and then P(Xi > λ | X ) = EP(Xi > λ | R(ti ), R(ti+1 ), X ) = EP(Xi > λ | R(ti ), R(ti+1 )) ≤ ECe−cλ = Ce−cλ , where the expectation above is with respect to R(ti ) and R(ti+1 ). This gives (23). The argument for (24) requires the convergence of the distributions. Start with the case of one i. Denote by Y the event R(ti ), R(ti+1 ) and by Z the event R(ti− ), R(ti+1+ ) for some ∈ {0, 1, . . . }. Then P(Y = y | Z) · E(Xi | Y = y) E(Xi | Z) = y∈Bi ×Bi+1

=

P(Y = y | Z) − P(Y = y) · E(Xi | Y = y)

y∈Bi ×Bi+1

≤C

P(Y = y | Z) − P(Y = y),

(33) (34)

y∈Bi ×Bi+1

where the equality (33) is due to EXi = 0. Denote by πk (w, x) the probability to start from w and hit x after k moves of going from Bj to Bj +1 . In a formula πk (w, x) := P(R(uj +k ) = x | R(uj ) = w)

.

Of course, we mean that if w ∈ ∂B(v, 2r) then we take j odd and in the opposite case we take j > 0 even. Other than that the value of πk is independent of j . With these notations we get P(Y = (y1 , y2 )) = P(R(ti ) = y1 )π1 (y1 , y2 ), π (z1 , y1 )π1 (y1 , y2 )π (y2 , z2 ) P(Y = (y1 , y2 ) | Z = (z1 , z2 )) = , π2+1 (z1 , z2 ) so

|P(Y = y | Z = z) − P(Y = y)| ≤ π1 (y1 , y2 ) |P(R(ti ) = y1 ) − π (z1 , y1 )| + π (y2 , z2 ) + . (35) − 1 π (z1 , y1 ) π2+1 (z1 , z2 )


273

Summing over y the first half of (35) we get π1 (y1 , y2 )|P(R(ti ) = y1 ) − π (z1 , y1 )| y1 ,y2

=

|P(R(ti ) = y1 ) − π (z1 , y1 )| ≤ 2e−c ,

(36)

y1

where the last inequality is due to the exponential convergence of the distributions in the form (28) — take q1 to be the distribution of R(ti− ) and q2 = δ{z1 } (the distance between any two distributions is always ≤ 2). For the second half of (35), we use the form (29) for and get, under the assumption > 0, π (y2 , z2 ) π (z1 , y1 )π1 (y1 , y2 ) − 1 π (z , z ) y1 ,y2

≤ Ce−c

2+1

1

2

π (z1 , y1 )π1 (y1 , y2 ) = Ce−c

.

(37)

y1 ,y2

We used here (29) with q1 = δ{y2 } and q2 the distribution of R(uj ++1 ) | R(uj ) = z1 for the point z2 . Using (36), (37) and (35) in (34) gives E(Xi | R(ti− ), R(ti+1+ )) ≤ Ce−c

(38)

(the case = 0 doesn’t follow from the argumentation above, but can be deduced, say, from (32)). With (38), proving (24) is easy. Let i1 , . . . , il be some integers, all different, and let 1 , min |ij − im | − 1 j = 2 1≤m≤l m=j

so that the intervals ij − j , ij + 1 + j are disjoint. Let X be the event R(ti1 −1 ), R(ti1 +1+1 ), . . . , R(tik −k ), R(tik +1+k )

.

Then conditioning by X the events Xi are independent so we get E(Xi1 · · · Xik | X ) =

k

E(Xj | X ) =

j =1 (38)

≤

k

k

E(Xj | R(tij −j ), R(tij +1+j ))

j =1

Ce−cj ≤

j =1

k

C exp(−c min |ij − im |)

j =1

1≤m≤l m=j

which immediately gives (24) since E(Xi1 · · · Xik | Xik+1 , . . . , Xil ) = E E(Xi1 · · · Xik | X ) Xik+1 , . . . , Xil k ≤E C exp(−c min |ij − im |) Xik+1 , . . . , Xil j =1

=

k j =1

1≤m≤l m=j

C exp(−c min |ij − im |) 1≤m≤l m=j

274


with (23) and (24) established we can invoke Lemma 3 and get P(|tn − Etn | > λU0 ) ≤ Ce−cλ

2

.

Lemma 4 now follows since (29) shows that Et2i+1 −t2i converge exponentially to some Eeven and Et2i+2 − t2i+1 converge exponentially to some Eodd so Etn − n 1 (Eeven + Eodd ) ≤ CU0 2 √ and for λ < C n this translation affects only the multiplicative constant. Therefore taking E = 21 (Eeven + Eodd ) ≈ N d r 2−d and σ = U0 ≈ E we are done. Lemma 5. Let b ∈ TNd and let R be a random walk on T starting from b. Let r < 18 N , v ∈ TNd . Let t ∈ N be some time. Then P(LE(R[0, t]) ∩ B(v, r) > λf (r)) ≤ Ce−cλ ∀λ > 0,

(39)

where f is defined by (21). Proof. Let λ > 0 be some number. We note that we may assume t < λN d since in time λN d the probability to hit b is > 1 − Ce−cλ and in this case the process starts afresh, memoryless. Let ti be stopping times defined by (2). Let E and σ be defined by Lemma 4 so that (25) holds. The first case is λ > C1 log r for some C1 sufficiently large. This case is uninteresting for the following reason: Lemma 4 gives that for some C2 sufficiently large, if

n = C2 λr d−2 then P(tn ≤ t) ≤ Ce−cλ . Let k := max{l : tl ≤ t}. If k is even then LE(R[0, t]) ∩ B(v, r) ⊂ LE(R[0, tk ]) ∩ B(v, r). If k is odd then #((LE(R[0, t]) \ LE(R[0, tk ])) ∩ B(v, r)) ≤ tk+1 − tk and this variable has the estimate (16) so it is uninteresting. Therefore it is enough to calculate the loop-erased at the times tk . We get P(#(LE(R[0, t]) ∩ B(v, r)) > λf (r)) n 1 ≤ Ce−cλ + P(#(LE(R[0, tk ]) ∩ B(v, r)) > λf (r)) 2 k=1

≤ Cne−cλ ≤ Cr d−2 λe−cλ ≤ Cr d−2 e−cλ ≤ Cr d−2−cC1 e−cλ (of course, all c’s in the last line are different). This shows that for C1 sufficiently large— namely, (d − 2)/c, where c is the last c on the last line above, (39) holds. Thus this case is proved. Therefore we shall assume that λ < C log r. Let n± 1 be defined by √ n− 1 := max{n even : t − nE > λσ n}, √ n+ 1 := min{n : t − nE < −λσ n}. Note that

− 3/2 n+ r d−2 . 1 − n1 ≤ Cλ t/E ≤ Cλ


275

− + + − n E| > λσ Let E1 be the event |tn− − n− E| > λσ n or |t n+ n 1 1 1 1 . Lemma 4 gives 1

1

us that P(E1 ) < Ce−cλ . We note that under ¬E1 we can “locate” t, tn− < t < tn+ 1 1 √ and the interval is not very large, tn+ − tn− < CN d λ3/2 r 2−d . Let E2 be the event 1

1

# LE(R[0, tn1 ]) ∩ B(v, r)) > λf (r). Lemma 2 gives us that P(E2 ) < Ce−cλ . We continue to define a short sequence of n± j inductively: √ n− j := max{n even : t − tNj−−1 − nE > λσ n}, √ n+ j := min{n : t − tN − − nE < −λσ n}, j −1

Nj± := n± j +

j −1

n− k.

k=1 ± Unlike n± 1 which are just numbers, nj , j

E2i−1 to be the event |tN ± i

> 1 are events depending on R[0, tN − ]. Define i−1 − t| > λσ n± (as before, we mean that either happens). i

Again, we get P(E2i−1 ) < Ce−cλ . Under ¬(E1 ∪ E3 ∪ · · · ∪ E2i−3 ) we have n± i λf (r)}, Li := LE+ (Ri [−tN − , tN − − tN − ]), i−1

i

i−1

(42) (43)

Note that we have now defined all the exceptional events Ei : the even ones are (42) and the odd ones have been defined slightly above. When we said that the series n± i is short, we meant that we shall take it until I defined by   d≥7 2 I= 3 d = 5, 6 ,  C log −1 d = 4

276


where is from (21), which we consider as a constant, so I ≤ C. In particular P(E1 ∪ · · · ∪ E2I ) ≤ CI e−cλ ≤ Ce−cλ . The reason for this selection of I is that with this I it is possible to do a simple estimate of the path between tN − and t. For any i we have I

Ni+

< Cλ (t − tN − )/E ≤ Cλ (tN + − tN − )/E

− Ni−

i−1

(41)

2−2−i

≤ Cλ

r

2−i (d−2)

i−1

≤ Cr

2−i (d−2)

i−1

log2 r

(remember that λ < C log r) and for I this gives NI+ − NI− ≤ Cf (r)r −2

.

Therefore we may use (16) NI+ − NI− times, to get P

NI+

tj +1 − tj > λf (r) ≤ Ce−cλ ,

(44)

j =NI− j odd

which of course bounds also #(LE(R[0, t]) ∩ B(v, r)) − #(LE(R[0, tN − ]) ∩ B(v, r)). I Finally, the definitions of Ri , LE, LE+ and Li (43) give LE(R[0, tN − ]) ⊂ L1 ∪ L2 ∪ · · · ∪ LI , I

and assuming ¬(E2 ∪ E4 ∪ · · · ∪ E2i ) we have from (42) that #(LE(R[0, tN − ]) ∩ B(v, r)) ≤ I λf (r) ≤ Cλf (r) I

and with (44) we finally get P(#(LE(R[0, t]) ∩ B(v, r)) > λf (r)) ≤ Ce−cλ and the lemma is proved.

,

Remark. By now the reader would not be surprised to learn that here too, if one is willing to let go of a log factor then the proof gets much simpler. Indeed, the arguments used for the case λ > C log r can be used for any λ to get this result, and for this case one does not need the precise estimates of Lemma 4 either, and the entire section may be reduced to half a page. Theorem 3. Let b ∈ TNd and let R be a random walk on T starting from b. Let t ∈ N be some time. Then P(LE(R[0, t]) > λf (N )) ≤ Ce−cλ ∀λ > 0, where f is defined by (21). The theorem follows from Lemma 5 like Theorem 1 follows from Lemma 2 (cover T by balls, etc.) and we shall omit the proof.


277

4. The Lower Bound We will use the concept of a cut time Definition. Let R be a random walk on a graph, possibly with a stopping condition. A time t is called a cut time for R if R[0, t] ∩ R ]t, ∞[ = ∅. Clearly, if t is a cut time then R(t) ∈ LE(R). Further, all R(ti )’s for different cut times ti are different. Therefore it is possible to estimate the length of a loop-erased random walk by counting cut times. Lemma 6. Let d ≥ 5. Let R be a random walk on TNd of length L for some L = N d/2 , sufficiently small and N > N0 (). Let X be the number of cut times of R. Then EX > cL VX < C 2 L2 .

(45)

As usual V denotes the variance, i.e. VX := EX 2 − (EX)2 . Proof. Denote by Et the event that t is a cut time. Easily, 1 − P(Et ) ≤

L t

P(R(s1 ) = R(s2 ))

.

s1 =0 s2 =t+1

Now for |si − t| ≤ N this is identical to the equivalent problem on Zd which is well known (see [L96]) so we get P(R[max{0, t − N}, t] ∩ R ]t, min{L, t + N }] = ∅) < 1 − c

.

For other si we use the easy P(R(s1 ) = R(s2 )) ≤ C min{N 2 , |s1 − s2 |}−d/2

(46)

to get 1 − P(Et ) < 1 − c + C 2 + CN 2−d/2 , therefore for sufficiently small and N sufficiently large we get P(Et ) > c which gives the first part of (45) — EX > cL. For the second part,

we examine the covariance of Et1 and Et2 for some t1 < t2 . Denote t = 21 (t1 + t2 ) and E1 = P(R[0, t1 ] ∩ ]t1 , t] = ∅) E2 = P(R[t, t2 ] ∩ R ]t2 , L] = ∅)

.

We note that E1 and E2 are independent and therefore cov E1 , E2 = 0. On the other hand, summing (46) we get |P(E1 ) − P(Et1 )|

≤ P(R[0, t1 ] ∩ R[t, L] = ∅) ≤

t1 L

C min{N 2 , |s2 − s1 |}−d/2

s1 =0 s2 =t

≤C

t1

|t − s|1−d/2 + N −d/2 ≤ C(|t2 − t1 |2−d/2 + 2 ),

s1 =0

so we get the same for the covariance of Eti , cov Et1 , Et2 ≤ C|t2 − t1 |2−d/2 + C 2

.

Summing these for all ti ’s we get the second half of the lemma.

278


Lemma 7. Let d ≥ 5. Let b ∈ TNd and let R be a random walk on T starting from b. Let t ∈ N, t > N d/2 and λ > N −1/2 . Then P(# LE(R[0, t]) ≤ λN d/2 ) ≤ Cλ. Proof. We may assume without loss of generality that λ ≤ c for some constant. Let C1 be some constant which will be fixed later. Define u := t − C1 λN d/2 (we assume here λ < 1/C1 , as we may). Denote by X the number of cut times in the segment [u, t]. Lemma 6 shows that EX > c(t − u) = cC1 λN d/2 . Pick C1 sufficiently large such that EX > 3λN d/2 . Lemma 6 also gives VX ≤ Cλ4 N d/2 and then P(X ≤ 2λN d/2 ) ≤ Cλ2 . Next we want to estimate P(LE(R[0, u]) ∩ R[u + N 2 , t] = ∅). Define Y = #{LE(R[0, u]) ∩ R[u + N 2 , t]}. If we assume # LE(R[0, u]) ≤ µN d/2 , then because R(u + N 2 ) is distributed ≈ uniformly on T we get E(Y | # LE(R[0, u]) ≤ µN d/2 ) ≈ N −d (# LE(R[0, u]))(t − u − N 2 ) ≈ µλ (this is the only place we use the assumption λ > N −1/2 ). Without the assumption # LE(R[0, u]) ≤ µN d/2 we get EY ≤ ≤

∞ µ=0 ∞

P(# LE(R[0, u]) ≤ µN d/2 ) · E(Y | # LE(R[0, u]) ≤ (µ + 1)N d/2 ) Ce−cµ µλ ≤ Cλ,

µ=0

and hence P(Y > 0) ≤ Cλ. Under the assumption Y = 0 every cut point of R[u, t] above u + N 2 is in LE(R[0, t]) and the lemma follows. Proof of Theorem 2. Let R be a random walk starting from b with no stopping condition. Define events X (v, t) = {R (t) = v ∧ v ∈ R [0, t[}, Y(v, t) = {R (t) = v ∧ # LE(R[0, t]) ≤ λN d/2 }.

Now, v P(X (v, t)) is simply the probability that a random walk reaches its end point for the first time, or equivalently by symmetry, the probability that it never returned to its starting point, therefore it is easy to calculate −d P(X (v, t)) ≤ Ce−ctN ∀t. v∈T

Next, for t > N d/2 , Lemma 7 gives P(Y(v, t)) ≤ Cλ v∈T

∀t > N d/2 .


279

Finally, note that ∞

P(X (v, t)) = 1

∀v ∈ T .

t=0

With these three facts we get, for any parameter µ > 0, P(X (v, t) \ Y(v, t)) t,v



≥N − d

d/2 N

t=0

+

∞ t=µN d

 

d µN

t=N d/2

v∈T

P(X (v, t)) −

v∈T

P(Y(v, t))

≥ N d (1 − Ce−cµ − N −d/2 − Cµλ). Picking µ = C log λ for some C sufficiently large will prove the theorem.

4.1. Remarks on alternative approaches. The first alternative approach to the proof of the lower bound is as follows: prove a conditioned version of Lemma 6, namely Lemma. Let b and e be two points on TNd with |b − e| > cN . Let R be a random walk on TNd of length L for some L = N d/2 , sufficiently small starting from b and conditioned to end at e. Let X be the number of cut times of R. Then EX > cL VX < C 2 L2 . This lemma allows to prove a version of Theorem 2 for any points far enough, not just two random points. Further, it allows to avoid the need to use absolute times, and just work directly with the times ti for some arbitrary ball. In other words, to show that the loop-erased random walk from b to e is long with high probability, define an arbitrary ball B, show that at the stopping times ti corresponding to B the entire loop-erased random walk is quite small (this is quite simple) and then show that the random walk from the last ti to e has many cut points using the lemma above. The proof of this lemma requires no new ideas when compared with Lemma 6. However, it is very technical, and quite long, which is the main reason we chose the approach above. In some sense we do not consider the length of Sect. 3 as an indication that the approach we chose is more complicated because the result (Theorem 3) is trivial if one can afford to lose a log factor (and also because the result is quite natural). Another approach is the use of the uniform spanning tree and Wilson’s algorithm (see [W96]). Roughly, one might hope to show that the loop-erased random walk is long by constructing an appropriate partial UST, and then showing that the random walk R starting from some point b and stopped on the partial UST is not too long (therefore no complicated self interactions, as in Lemma 6) and not too short, so LE(R) can be proved to be long. Since the loop-erased random walk from b to some other point e (say inside the partial UST) contains LE(R), this will be enough. Alternatively, one can take two random walks R and R starting from b and e respectively and stopped on the partially constructed UST, and calculate the probabilities that at least one is long and that they do not intersect. Both approaches allow to generalize Theorem 2 from a random end point to any end point (naturally, if b and e are very close then with positive probability the

280


loop-erased random walk from b to e is short. However, one can show that there is a positive probability for the loop-erased random walk to be long, i.e. ≈N d/2 ). A third strategy using the UST is as follows. Notice that the harmonic measure on a partially constructed UST is roughly uniform — this follows since the escape probabilities from a typical small ball are positive. If one wants to estimate the probability that the loop-erased random walk between b and e is ≤ λN d/2 , construct a partial UST containing b up until its size is ≈ (1/λ)N d/2 , and then estimate that the number of vertices in the tree with distant ≤ λN d/2 from b is ≈ λN d/2 , so the harmonic measure is ≈ λ2 . This approach gives (in addition to the fact that e may be arbitrary) stronger estimates than the λ log λ−1 of Theorem 2 — formalizing these arguments we were able to show P(# LE(R) ≤ λN d/2 ) ≤ Cλ2 log λ−1 , and we believe that the true value is, as in the case of the complete graph, λ2 . The difference between λ log λ−1 and λ2 log λ−1 is significant in the following sense: the weaker estimate does not prove that the UST has a true branching nature: even points that are distributed linearly along a path of length N d/2 satisfy the requirement that P(LE ≤ λN d/2 ) ≤ Cλ. However, the estimate λ2 log λ−1 allows to deduce non-trivial facts about the branching structure of the UST. None of these methods work in dimension 4, and the culprit is always the same: in dimension 4 our methods do not show that within the mixing time the probability of hitting the loop-erased random walk is small. In other words, to get a lower bound for dimension 4 one must either show a very precise upper estimate (not much different from the conjectured precise value) or alternatively show indirectly that the mixing time is smaller than the hitting time of the loop-erased random walk. Appendix A. Proofs of Known and Unsurprising Facts The harmonic potential on Zd , d > 2, is the unique bounded function a satisfying 1 z = 0 a(z) = 0 otherwise and a(∞) = 0, where stands for the discrete Laplacian. It is well known (see e.g. [L96, Theorem 1.5.4] 4 or [KS04, Theorem 5]) that a(v) = α|v|2−d + O(|v|−d ). Lemma A.1. Let B1 = B(x1 , r1 ) ⊂ B2 = B(x2 , r2 ) ⊂ TNd , r2 ≤ C1 r1 . Let v ∈ B2 \ B1 satisfy d(v, ∂B2 ) ≥ c1 r1 . Let R be a random walk starting from v and stopped on ∂B1 ∪ ∂B2 . Let p be the probability that R hits ∂B1 . Then p ≥ c(c1 , C1 ). We assume here that a ball (e.g. B2 ) satisfies r2 < 21 N i.e. it does not wrap itself because we are on a torus. This assumption holds for all balls in this appendix, and we will not repeat it. Proof. Clearly, we may assume r1 is sufficiently large in the sense that r1 > C(c1 , C1 ). Since we are dealing with a process completely inside B2 , we may assume we are in Zd . Assume first that |x1 − v| < 21 d(x1 , ∂B2 ). Since a1 (v) := a(v − x1 ) is harmonic on B2 \ B1 , a1 (R) is a martingale, and if we define τ to be the stopping time on ∂B1 ∪ ∂B2 then we get a1 (v) = Ea1 (R(τ )), so a1 (v) = pE(a1 (R(τ )) | R(τ ) ∈ ∂B1 ) + (1 − p)E(a1 (R(τ )) | R(τ ) ∈ ∂B2 ) ≤ pαr12−d (1 + o(1)) + (1 − p)αd(x1 , ∂B2 )2−d (1 + o(1)) 4

(47)

[L96] only shows a(v) = α|v|2−d + O(|v|−d ), but this is completely sufficient for our purposes.


281

(the o(1) notations are as r1 → ∞ and may depend on c1 and C1 ) and from a1 (v) = α|x1 − v|2−d (1 + o(1)) we get p≥

|x1 − v|2−d − d(x1 , ∂B2 )2−d r12−d − d(x1 , ∂B2 )2−d

(1 + o(1)) ≥ c

(48)

for r1 sufficiently large. In the case |x1 − v| ≥ 21 d(x1 , ∂B2 ), we can find a sequence of balls B(yn , sn ) of length ≤ C(c1 , C1 ) and each sn ≥ c(c1 , C1 )r1 such that |y1 − v| ≤ 21 d(y1 , ∂B2 ) and ∀w ∈ ∂B(yi , si ), |yi+1 − w| ≤ 21 d(yi+1 , ∂B2 ). Notice that this is possible because d(v, ∂B2 ) ≥ c1 r1 . The previous case now gives that the probability that the random walk, after hitting B(yi , si ) will continue to B(yi+1 , si+1 ) is ≥ c2 (c1 , C1 ). Since it needs to perform only C(c1 , C1 ) such steps in order to hit B1 , we get p ≥ c2C = c3 (c1 , C1 ). Lemma A.2. Let B(x1 , r1 ), B(x2 , r2 ) be two disjoint balls, r2 ≤ C1 r1 and |x1 − x2 | ≤ C1 r1 ; and let v ∈ B2 ∪ B1 satisfy d(v, B2 ) ≥ c1 r2 . Then p ≥ c(c1 , C1 ), where p is as above. Proof. Assume first that d(v, B1 ) ≤ 21 d(B2 , B1 ) where d(B1 , B2 ) stands for the distance between the two balls in the usual sense. Let Bi be the sets Bi considered as subsets of Zd and let Si = Bi + N Zd , i.e. Si is the preimage of Bi by the quotient map Zd → TNd . Let R be a simple random walk on Zd starting from v (we consider v and the Bi ’s as subsets of Zd as well, say by locating them in [0, N]d ). Then p = P(R hits S1 before S2 ) ≥ P(R hits B1 before S2 ) (∗)

≥ P(R hits B1 before ∂B(x1 , r1 + d(B1 , B2 ))) ≥ c, where (∗) comes from the same harmonic potential arguments as (47)-(48). If d(v, B1 ) > 21 d(B2 , B1 ) but we have both d(v, B1 ) ≤ (2C1 + 2)r1 , d(v, B2 ) ≥ c1 r1 , then the same ball-sequence argument as in the previous lemma gives p ≥ c; If d(v, B1 ) > (2C1 + 2)r1 , let τ be the hitting time of B3 := B(x1 , (2C1 + 2)r1 ), then p = EP(R starting from R(τ ) hits B1 before B2 ) ≥ Ec = c,

(49)

where here R is a simple random walk on TNd (differing from R only by the starting point), the expectation E is over the distribution of R(τ ) and the inequality comes from the previous two cases. Finally, if d(v, B2 ) < c1 r1 define τ the hitting time of B4 := B(x2 , c1 r1 ). The harmonic potential at x2 with a calculation similar to (47)-(48) shows that the probability to hit B4 before B2 is ≥ c. After hitting B4 a calculation similar to (49) gives that p ≥ c. Lemma A.3. Let d(B(x1 , r1 ), B(x2 , r2 )) ≥ c1 r1 , r2 ≥ c1 r1 and |x1 − x2 | ≤ C1 r2 ; and let v ∈ ∂B1 . Let R be a random walk starting from v and stopped on ∂B2 ∪ {x1 }. Let p be the probability that R hits x1 . Then p ≈ r12−d .

282


In the formulation of the lemma, and in its proof, all constants implicit in the ≈ signs might depend on c1 and C1 . Proof. Let B3 = B(x1 , 21 r1 ). Define stopping times ti similarly to (2), as follows: t0 := 0 and t2i+1 := {t > t2i : R(t) ∈ ∂B3 ), t2i := {t > t2i−1 : R(t) ∈ ∂B1 ∪ {x1 }}. Define also τ the hitting time of ∂B2 . The usual harmonic potential calculations (use the harmonic potential around x1 ) show that the probability of a random walk starting from any v ∈ ∂B3 to hit x1 before exiting B1 is ≈ r12−d . Hence, P(R(t2i ) = x1 |R(t2i−1 )) ≤ Cr12−d . Lemma A.2 shows that a random walk starting from any point in ∂B1 has probability ≥ c to hit B2 before hitting B3 . Therefore, the probability to get ti > τ decreases exponentially in i, i.e. P(t2i < τ ) ≤ Ce−ci and hence P(R(t2i ) = x1 and t2i < τ ) ≤ Ce−ci r12−d so p≤

∞

P(R(t2i+1 ) = x1 and t2i+1 < τ ) ≤ Cr12−d .

i=0

The inequality p ≥ cr12−d follows easily from (47)–(48) for the harmonic potential at x1 and the requirement d(B1 , B2 ) ≥ c1 r1 . Lemma A.4. Let |v| < (1−c1 )r and w ∈ ∂B(0, r). Then the probability p that a random walk starting from v will exit B in w is ≈ r 1−d . Without the restriction |v| < (1 − c1 )r one has p ≤ C(r − |v|)1−d In the formulation of the lemma, and in its proof, all constants implicit in the ≈ signs might depend on c1 . Proof. In Zd , d > 2, the probability of a walk starting from v to never return is > c. Hence the probability to hit ∂B(0, r) before returning to v is > c, and this event is identical on Zd and on the torus. The symmetry of the random walk shows that p is ≈ to the probability that a random walk starting from w will hit ∂B ∪ {v} in v (the quotient is exactly the probability of a random walk starting from v to return to v before exiting ∂B, which, as we just discussed, is ≈ 1) 5 . This probability can be calculated in three steps as follows. First, the probability of a random walk starting from w to hit ∂B(0, 23 r + 13 |v|) is ≈ (r − |v|)−1 : this uses an argument similar to (47)-(48) using the harmonic potential a at 0, but here we need the precise estimate a(x) = |x|2−d + O(|x|−d ) or at least a(x) = |x|2−d + O(|x|1−d ) (see, e.g. [K87, Lemma 3] for a detailed version of this calculation). Next, if |v| < r(1−c1 ), use Lemma A.1 to show that continuing from any point on ∂B(0, 23 r + 13 |v|) the probability to hit B(v, 13 (r − |v|)) is ≈ 1 — if |v| ≥ r(1 − c1 ), we only estimate that this probability is ≤ 1. Finally, the same (47)-(48) argument with the harmonic potential at v shows that starting from any point on ∂B(v, 13 (r − |v|)), the probability to hit v before hitting ∂B is ≈ (r − |v|)2−d . 5

When we say “hit” we mean at time > 0, so that these probabilities are not simply 1.


283

A similar calculation works when (1 + c1 )r ≤ ||v|| and p is the probability the random walk will hit B in w, using Lemma A.2 instead of Lemma A.1 and Lemma A.3 in the third step. Lemma A.5. Let v ∈ B(0, r) ⊂ B(0, 2r) ⊂ TNd . Let R be a random walk starting from v and stopped on ∂B(0, 2r). Let Rx be a random walk starting from v and conditioned to hit ∂B(0, 2r) at a specific point x. Then R ∩ B(0, r) ≈ Rx ∩ B(0, r), where ≈ means that the probabilities of any event are equal up to a constant. Proof. Let t be the last time when R(t) ∈ B(0, r). Let w = R(t). For any w, the probability of an (unconditioned) random walk starting from w to hit B(0, r) ∪ ∂B(0, 2r) in ∂B(0, 2r) is ≈ r −1 . The probability to hit x is ≈ r −d . This independence from w finishes the lemma. Both estimates are easily proved as in the previous lemma. Lemma A.6. Let R be a random walk starting from v ∈ B(0, r) and let w ∈ B(0, r). Let t > r 2 . Let p be the probability that R[0, t] ⊂ B(0, r) and R(t) = w. Then p ≤ Cr −d e−ctr

−2

.

Proof. Starting from any v ∈ B(0, r), after r 2 steps the random walk has probability > c to exit B(0, r). This shows, clearly, that the probability that R[0, t − r 2 ] ⊂ B(0, r) is −2 ≤ Ce−ctr . For any x ∈ B(0, r), the probability that a random walk R starting from x satisfies R (r 2 ) = w is ≤ Cr −d . Lemma A.7. Let v ∈ B(0, r) and let R be a random walk starting from v and stopped on ∂B(0, r). Let w ∈ ∂B(0, r). Then the probability p that R hits w satisfies p ≤ C1 |v − w|1−d . Proof. Denote s = |v − w|. We shall prove the lemma using an induction process that assumes the lemma holds for 1, . . . , 21 s and proves it for 21 s + 1, . . . , s. Denote d(v, ∂B(0, r)) = s. The first thing to note is that the lemma holds if > c with no need for induction (in the sense that p ≤ C2 (c)|v − w|1−d ), due to the second part of Lemma A.4. It is for the case of small that we need the induction process. Let δ = c1 log−1 −1 for some c1 which will be fixed later. Let D := B(0, r) \ B(0, r − δs), 1 E : = D ∩ B(v, s) . 2 Examine the exit probabilities of R from E. Let p2 be the probability that R exits E at ∂B(0, r − δs) ∩ ∂E. Then p2 ≤ P(R exits D at ∂B(0, r − δs)) ≤ C/δ, where the second inequality comes from the harmonic potential at zero. Let p3 be the probability that R exits E at ∂B(v, 21 s) ∩ ∂E. It is easy to see that for any x ∈ D, the probability that R exits B(x, 2δs) without hitting ∂D is λr 2 ) ≤ Ce−cλ . Proof. We may assume λ > 1. Let R be an unconditioned walk starting from v. Let An = (B(w, 2n ) \ B(w, 2n−1 )) ∩ B(0, r). Lemma A.6 shows that P(R [0, λr 2 ] ⊂ B(0, r) and R (λr 2 ) ∈ An ) ≤ C#An r −d e−cλ . Lemma A.7 shows that for every x ∈ An , the probability of a random walk starting at x to exit B(0, r) at w is ≤ C2n(1−d) . Therefore P(R [0, λr 2 ] ⊂ B(0, r) and R (λr 2 ) ∈ An and R hits w) ≤ C(#An )2n(1−d) r −d e−cλ ≤ Cr −d 2n e−cλ , and we get

P(R [0, λr ] ⊂ B(0, r) and R hits w) ≤ Cr 2

1−d −cλ

e

log r

r −1 2n ≤ Cr 1−d e−cλ .

n=1

Since the probability of R to hit w is > cr 1−d (by Lemma A.4), we are done.

Lemma A.9. Let Xi be events with a past-independent exponential estimate, namely P(Xi > λE | Xi−1 , . . . , X1 ) ≤ C1 e−c1 λ . Then P

n

Xi > λnE

≤ Ce−cλ .

i=1

As usual, C, c and all constants in the proof might depend on C1 and c1 .


285

Proof. Clearly we may assume E = 1. Let Yi be i.i.d. variables with Yi ∼ C2 (1 + G), where G is a standard exponential variable (namely with density e−t ) and C2 is some constant sufficiently large such that P(Yi > λ) ≥ min{1, C1 e−c1 λ }. A simple induction now shows that

P Xi > λn ≤ P Yi > λn , n−1

t .A and the sum of the Yi has the distribution C2 (n + ), where has density e−t (n−1)! −cλ simple calculation shows that P( > λn) ≤ Ce .

Lemma A.10. Let v ∈ TNd , and let R be a random walk starting from v going to a length of C1 N d r 2−d for some C1 sufficiently large. Then P(R hits B(0, r)) ≥

1 2

∀v ∈ TNd .

Proof. Define S to be the preimage of B in Zd (namely B +N Zd ) and let R be a random walk on Zd starting from some preimage of v. Then P(R hits B(0, r)) = P(R [0, N d r 2−d ] ∩ S = ∅). Define stopping times ti as follows: t0 = 0 and for every i let zi be the element of N Zd closest to R(ti ). Define inductively ti+1 = min{t > ti : R(t) ∈ ∂B(zi , r) ∪ ∂B(zi , 2N )}. Since d(R(ti ), zi ) ≤ N then using the harmonic potential at zi shows that P(R(ti+1 ) ∈ S) ≥ P(R(ti+1 ) ∈ ∂B(zi , r)) ≥ c(r/N )d−2 independently of the value of R(ti ). This immediately gives that, for C2 sufficiently large,

r d−2 C2 (N/r)d−2 1 P(R[0, tC2 (N/r)d−2 ] ∩ S = ∅) ≤ 1 − c ≤ . (50) N 4 On the other hand, it is easy to see that P(ti+1 − ti > λN 2 ) ≤ Ce−cλ independently of the past, and using Lemma A.9 we get that P(tn > λnN 2 ) ≤ Ce−cλ . Using this for n = C2 (N/r)d−2 and λ sufficiently large we get that 1 . 4 Equations (50) and (51) together show that the C in (51) may serve as our C1 . P(tn > CN d r 2−d ) ≤

(51)

Lemma A.11. Let v ∈ ∂B(0, 2r) and w ∈ ∂B(0, r). Let R be a random walk started from v and conditioned to hit B(0, r) in w. Let t be the hitting time. Then P(t > λN d r 2−d ) ≤ Ce−cλ . The proof is identical to that of Lemma A.8 with the use of Lemmas A.4 and A.7 replaced by the comments following them, respectively, and using Lemma A.10 to show that the probability to not hit a ball of radius r after λN d r 2−d steps is ≤ Ce−cλ and hence the equivalent of Lemma A.6. We omit the details. Acknowledgement. We wish to thank Chris Hoffman, Dan Romik and Oded Schramm for useful discussions.

286


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Communicated by M. Aizenman


Communications in


Topological Calculation of the Phase of the Determinant of a Non Self-Adjoint Elliptic Operator Alexander G. Abanov1, , Maxim Braverman2, 1 2

Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794, USA. E-mail: [email protected] Department of Mathematics, Northeastern University, Boston, MA 02115, USA. E-mail: [email protected]

Received: 30 January 2004 / Accepted: 6 April 2005 Published online: 15 July 2005 – © Springer-Verlag 2005

Abstract: We study the zeta-regularized determinant of a non self-adjoint elliptic operator on a closed odd-dimensional manifold. We show that, if the spectrum of the operator is symmetric with respect to the imaginary axis, then the determinant is real and its sign is determined by the parity of the number of the eigenvalues of the operator, which lie on the positive part of the imaginary axis. It follows that, for many geometrically defined operators, the phase of the determinant is a topological invariant. In numerous examples, coming from geometry and physics, we calculate the phase of the determinants in purely topological terms. Some of those examples were known in physical literature, but no mathematically rigorous proofs and no general theory were available until now. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Preliminaries on Determinants of Elliptic Operators . . . . . 3. Operators Whose Spectrum is Symmetric with Respect to the Imaginary Axis . . . . . . . . . . . . . . . . . . . . . . . . 4. First Examples . . . . . . . . . . . . . . . . . . . . . . . . 5. A Dirac-Type Operator on a Circle . . . . . . . . . . . . . . 6. The Phase of the Determinant and the Degree of the Map . .

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1. Introduction In the recent years several examples appeared in the physical literature when the phase of the determinant of a geometrically defined non-self-adjoint Dirac-type operator is a topological invariant (see e.g., [11, 12, 2, 1]). Many of those examples appear in the study of the non-linear σ -model for Dirac fermions coupled to chiral bosonic fields

The first author was partially supported by the Alfred P. Sloan foundation. The second author was partially supported by the NSF grant DMS-0204421.

288

A.G. Abanov, M. Braverman

[2, 1]. The topologically invariant phase is called the θ-term. It has a dramatic effect on the dynamics of the Goldstone bosons but also has a great interest for geometers. Unfortunately, no mathematically rigorous proofs of the topological invariance of the phase of the determinant were available until now. This paper is an attempt to better understand the above phenomenon. In particular, we find a large class of operators whose determinants have a topologically invariant phase. We also develop a technique for calculation of this phase. In particular, we get a first mathematically rigorous derivation of several examples which appeared in the physical literature. In many cases, we also improve and generalize those examples. Our first result is Theorem 3.1 which states that the determinant of an elliptic operator D with a self-adjoint leading symbol, which acts on an odd-dimensional manifold and whose spectrum is symmetric with respect to the imaginary axis, is real. Moreover, the sign of this determinant is equal to (−1)m+ , where m+ is the number of the eigenvalues of D (counted with multiplicities) which lie on the positive part of the imaginary axis. Note that this result is somewhat surprising. Indeed, if one calculates the determinant of a finite matrix D with the spectrum symmetric with respect to an imaginary axis, then one comes to a different result. E.g., the determinant is not necessarily real. Suppose now that we are given a family D(t) of operators as above. Assuming that the eigenvalues of D(t) depend continuously on t one easily concludes (cf. Theorem 3.2) that the sign of the determinant of D(t) is independent of t. In particular, it follows that, if the definition of the operator D depends on some geometric data (Riemannian metric on a manifold, Hermitian metric on a vector bundle, etc.), then (provided the spectrum of D is symmetric) the sign of the determinant is independent of these data, i.e., is a topological invariant. We present numerous examples of this phenomenon. In all those examples we calculate the signs of the determinants in terms of the standard topological invariants, such as the Betti numbers or the degree of a map. The paper is organized as follows: In Sect. 2, we briefly recall the basic facts about the ζ -regularized determinants of elliptic operators. In Sect. 3, we formulate and prove our main result (Theorem 3.1) and discuss its main implications. In Sect. 4, we present the simplest (but still interesting) geometric examples of applications of Theorem 3.1. In Sect. 5, we consider an operator D on a circle, which appeared in the study of a quantum spin in the presence of a planar, time-dependent magnetic field. This operator depends on a map from a circle to itself. We calculate the phase of the determinant of D in terms of the winding number of this map. In Sect. 6, we extend some of the examples considered by P. Wiegmann and the first author in [2]. The operator in question is a Dirac type operator D on an odd dimensional manifold M, whose potential depends on a section n of the bundle of spheres in R ⊕ T M. In particular, if the manifold M is parallelizable, n is a map from M to a dim M-dimensional sphere. We show that the sign of the determinant of D is equal to (−1)deg n , where deg n is the topological degree of n. 2. Preliminaries on Determinants of Elliptic Operators Let E be a vector bundle over a smooth compact manifold M and let D : C ∞ (M, E) → C ∞ (M, E) be an elliptic differential operator of order m ≥ 1. Let σL (D) denote the leading symbol of D.

Topological Calculation of the Phase of the Determinant

289

2.1. The choice of an angle. Our aim is to define the ζ -function and the determinant of D. For this we will need to define the complex powers of D. As usual, to define complex powers we need to choose a spectral cut in the complex plane. We will restrict ourselves to the simplest spectral cuts given by a ray Rθ = ρeiθ : 0 ≤ ρ < ∞ , 0 ≤ θ ≤ 2π. (2.1) Consequently, we have to choose an angle θ ∈ [0, 2π ). Definition 2.1. The angle θ is a principal angle for an elliptic operator D if spec σL (D)(x, ξ ) ∩ Rθ = ∅, for all x ∈ M, ξ ∈ Tx∗ M\{0}. If I ⊂ R we denote by LI the solid angle LI = ρeiθ : 0 < ρ < ∞, θ ∈ I . Definition 2.2. The angle θ is an Agmon angle for an elliptic operator D if it is principal angle for D and there exists ε > 0 such that spec (D) ∩ L[θ−ε,θ+ε] = ∅. 2.2. The ζ -function and the determinant. Let θ be an Agmon angle for D. Assume, in addition, that D is injective. The ζ -function ζθ (s, D) of D is defined as follows. Let ρ0 > 0 be a small number such that spec (D) ∩ z ∈ C; |z| < 2ρ0 = ∅. Define the contour = θ,ρ0 ⊂ C consisting of three curves = 1 ∪ 2 ∪ 3 , where 1 = ρeiθ : ρ0 ≤ ρ < ∞ , 2 = ρ0 eiα : θ < α < θ + 2π , 3 = ρei(θ+2π) : ρ0 ≤ ρ < ∞ . (2.2) Assume that θ = 0. For Re s > Dθ−s =

dim M m ,

i 2π

the operator

θ,ρ0

−1 λ−s dλ θ (D − λ)

(2.3)

is a pseudo-differential operator with smooth kernel Dθ−s (x, y), cf. [14, 15]. Here λ−s θ := e−s logθ λ , where logθ λ denotes the branch of the logarithm in C\Rθ which takes real values on the positive real axis. We define dim M . (2.4) ζθ (s, D) = Tr Dθ−s = tr Dθ−s (x, x) dx, Re s > m M It was shown by Seeley [14] (see also [15]) that ζθ (s, D) has a meromorphic extension to the whole complex plane and that 0 is a regular value of ζθ (s, D). More generally, let Q be a pseudo-differential operator of order q. We set ζθ (s, Q, D) = Tr Q Dθ−s ,

Re s > (q + dim M)/m.

(2.5)

290


If Q is a projection, i.e., Q2 = Q then [18, §6], [19] (see also [10] for a shorter proof), the function ζθ (s, D; Q) also has a meromorphic extension to the whole complex plane which is regular at 0. Finally, we define the ζ -regularized determinant of D by the formula d (2.6) Det θ (D) = exp − s=0 ζθ (s, D) . ds 2.3. The case of an operator close to self-adjoint. Let us assume now that σL (D)∗ = σL (D),

(2.7)

where σL (D)∗ denotes the dual of σL (D) with respect to some fixed scalar product on the fibers on E. This assumption implies that D can be written as a sum D = D + A, where D is self-adjoint and A is a differential operator of a smaller order. In this situation we say that D is an operator close to self-adjoint, cf. [3, §6.2], [9, §I.10]. Though the operator D is not self-adjoint in general, the assumption 2.7 guarantees that it has nice spectral properties. More precisely, cf. [9, §I.6], the space L2 (M, E) of square integrable sections of E is the closure of the algebraic direct sum of finite dimensional D-invariant subspaces L2 (M, E) =

k

(2.8)

such that the restriction of D to k has a unique eigenvalue λk and limk→∞ |λk | = ∞. In general, the sum 2.8 is not a sum of mutually orthogonal subspaces. The spaces k are called the space of root vectors of D with eigenvalue λk . We call the dimension of the space k the multiplicity of the eigenvalue λk and we denote it by mk . By Lidskii’s theorem [8], [13, Ch. XI], the ζ -function 2.4 is equal to the sum (including the multiplicities) of the eigenvalues of Dθ−s . Hence, ζθ (s, D) =

∞

k=1

mk λ−s k

=

∞

mk e−s logθ λk ,

(2.9)

k=1

where logθ (λk ) denotes the branch of the logarithm in C\Rθ which take the real values on the positive real axis. 2.4. Dependence of the determinant on the angle. Assume now that θ is only a principal angle for D. Then, cf. [14, 15], there exists ε > 0 such that spec (D)∩L[θ−ε,θ+ε] is finite and spec (σL (D)) ∩ L[θ−ε,θ+ε] = ∅. Thus we can choose an Agmon angle θ ∈ (θ − ε, θ + ε) for D. In this subsection we show that Detθ (D) is independent of the choice of this angle θ . For simplicity, we will restrict ourselves with the case when D is an operator close to self-adjoint, cf. Subsect. 2.3. Let θ

> θ be another Agmon angle for D in (θ − ε, θ + ε). Then there are only finitely many eigenvalues λr1 , . . . , λrk of D in the solid angle L[θ ,θ

] . We have logθ λk , if k ∈ {r1 , . . . , rk }; logθ

λk = (2.10) logθ λk + 2πi, if k ∈ {r1 , . . . , rk }.


291

Hence, ζθ (0, D) − ζθ

(0, D) =

k

d mk e−s logθ (λri ) (1 − e−2πis ) ds s=0 i=1

= 2π i

k

(2.11)

mri ,

i=1

and Det θ

D = Detθ D.

(2.12)

Note that the equality 2.12 holds only because both angles θ and θ

are close to a given principal angle θ so that the intersection spec (D) ∩ L[θ ,θ

] is finite. If there are infinitely many eigenvalues of D in the solid angle L[θ ,θ

] then Det θ

(D) and Det θ (D) might be quite different. 3. Operators Whose Spectrum is Symmetric with Respect to the Imaginary Axis In this section M is an odd-dimensional closed manifold, E → M is a complex vector bundle over M, and D is a differential operator of order m ≥ 1 which is close to self-adjoint (cf. Subsect. 2.3) and invertible. 3.1. The phase of the determinant and the imaginary eigenvalues. Suppose that the spectrum of D is symmetric with respect to the imaginary axis. More precisely, we assume that, if λ = ρeiα is an eigenvalue of D with multiplicity m, then ρe−i(π+α) is also an eigenvalue of D with the same multiplicity. Since the leading symbol of D is self-adjoint, ± π2 are principal angles of D, cf. Definition 2.1. Hence, cf. Subsect. 2.4, we can choose an Agmon angle θ ∈ ( π2 , π) such that there are no eigenvalues of D in the solid angles L(π/2,θ] and L(−π/2,θ −π ] . Let m+ denote the number of eigenvalues of D (counted with multiplicities) on the positive part of the imaginary axis, i.e., on the ray Rπ/2 (cf. Definition 2.1). Our first result is the following Theorem 3.1. In the situation described above Im ζθ (0, D) = − π m+ . (3.1) In particular, Detθ D = exp − ζθ (0, D) is a real number, whose sign is equal to (−1)m+ . Remark 3.1. a. For 3.1 to hold we need the precise assumption on θ which we specified above. However, if we are only interested in the sign of the determinant of D, the result remains true for all θ ∈ (−π, π). This follows from 2.12. b. Note that only the eigenvalues on the positive part of the imaginary axis contribute to the sign of the determinant. This asymmetry between the positive and the negative part of the imaginary axis is coursed by our choice of the spectral cut Rθ in the upper half plane. If we have chosen the spectral cut in the lower half plane the sign of the determinant would be determined by the eigenvalues on the negative imaginary axis.

292


Proof. Let π , j = 1, 2, . . . 2 be all the eigenvalues of D which lie in the solid angle L(θ−π,π/2) (here and below all the eigenvalues appear in the list the number of times equal to their multiplicities). Since the spectrum of D is symmetric with respect to the imaginary axis, ρj e−i(π+αj ) (j = 1, 2, . . . ) are all the eigenvalues of D in the solid angle L(θ−2π,−π/2) . 1 Finally, let θ − π < αj
0,

and

Detθ Da > 0 if a < 0.

(4.1)


295

Remark 4.1. The determinant of the operator Da can be calculated explicitly. In fact [6] provides a formula for this determinant in terms of the monodromy operator associated to Da . Using this formula, one easily gets Det θ Da = e−aβ − 1, where β is the length of the circle. Clearly, that agrees with 4.1. 4.2. The deformed DeRham-Dirac operator. Suppose M is a closed manifold of odd dimension N = 2l + 1. Let d : ∗ (M) → ∗+1 (M) be the DeRham differential and let d ∗ : ∗ (M) → ∗−1 (M) be the adjoint of d with respect to a fixed Riemannian metric on M. Let βj (M) = dim H j (M) (j = 0, . . . , N) denote the Betti numbers of M. The operator Da := d + d ∗ + ia,

a ∈ R, a = 0

has exactly one imaginary eigenvalue λ = ia and its multiplicity is equal to the sum N e duality, this sum is j =0 βj (M) of the Betti numbers of M. Because of the Poincar´ an even number. Thus Theorem 3.1 implies that Det θ Da > 0,

a ∈ R, 0 < θ < π.

for all

To construct a more interesting example let us fix non-zero real numbers a0 , . . . , aN and consider the operator A : ∗ (M) → ∗ (M) defined by the formula A : ω → aj ω,

if

ω ∈ j (M).

Then one easily concludes from Theorem 3.1 that βj (M) Det θ d + d ∗ + iA . Detθ d + d ∗ + iA = (−1) {j :aj >0} Another interesting example can be constructed as follows. Let ∗ : ∗ (M) → N−∗ (M) denote the Hodge-star operator. Consider the operator : ∗ (M) → N−∗ (M), defined by the formula : α → i

N (N +1) 2

(−1)

j (j +1) 2

∗ α = i l+1 (−1)

j (j +1) 2

∗ α,

α ∈ j (M). (4.2)

Since N = 2l + 1 is odd, is self-adjoint, satisfies 2 = 1, and commutes with d + d ∗ . In particular, acts on Ker(d + d ∗ ) and this action has exactly 2 eigenvalues ±1, which have equal multiplicities 21 N j =0 βj (M). Hence, the operator D := d + d ∗ + i , has exactly 2 imaginary eigenvalues ±i, and multiplicities of these eigenvalues are equal to 21 N j =0 βj (M). Theorem 3.1 implies now that 1

Det θ D = (−1) 2

N

j =0

βj (M)

Det θ D .

Remark 4.2. All the results of this subsection can be easily extended to operators acting on the space of differential forms with values in a flat vector bundle F → M. (The DeRham differential should be replaced by the covariant differential and the Betti numbers should be replaced by the dimensions of the cohomology of M with coefficients in F ). We leave the details to the interested reader.

296


5. A Dirac-Type Operator on a Circle In this section we consider the operator D on the circle, which appears, e.g., in the study of a quantum spin in the presence of a planar, time-dependent magnetic field. In the case when the magnetic field is changing adiabatically in time the wave function of spin 1/2 acquires the phase π k, where k is an integer number of rotations that the direction n of magnetic field makes around the origin during the time evolution. This adiabatic phase is called the Berry phase [5]. For adiabatic evolution of magnetic field the Berry phase is equal up to a trivial dynamic factor to the determinant of the operator D defined in 5.2 below, cf., e.g., [2]. The main result of this section is Theorem 5.1 which calculates the phase of this determinant. This theorem is known in physical literature (see e.g., [2]), but no mathematically rigorous proofs were available until now. 5.1. The setting. Let S 1 be the circle, which we view as the interval [0, β] (β > 0) with identified ends. Let n : S 1 −→ z ∈ C : |z| = 1 be a smooth map. Then there exists a smooth function φ : R → R, satisfying the periodicity conditions φ(t + β) = 2πk + φ(t),

k ∈ Z,

(5.1)

such that n = eiφ . The number k above is called the topological degree (or the winding number) of the map n. 0 eiφ and consider the family of operators depending on a real paramSet n = −iφ e 0 eter m d d 0 eiφ D = i + im n = i + im −iφ , (5.2) e 0 dt dt acting on the space of vector-functions ξ : [0, β] → C2 with boundary conditions ξ(β) = eiπν ξ(0),

ξ˙ (β) = eiπν ξ˙ (0),

ν = 0, 1.

(5.3)

We shall study the determinant of D. The following lemma shows that this determinant is non-zero for m sufficiently large. ˙ Lemma 5.1. For m > maxt∈[0,β] |φ(t)|, zero is not in the spectrum of D. Proof. Consider the following scalar product on the vector valued functions on [0, β]: β (ξ, η) = ξ(t), η(t) dt, 0

where ·, · stands for the standard scalar product on C2 . Let ξ = (ξ, ξ )1/2 denote the norm of the vector function ξ . Integrating by parts the expression for Dξ 2 we obtain, for ξ satisfying the boundary condition (5.3), Dξ 2 = (ξ˙ , ξ˙ ) + m( nξ, ξ˙ ) + m(ξ˙ , nξ ) + m2 ξ 2 ≥ −m( n˙ ξ, ξ ) − m( nξ˙ , ξ ) + m(ξ˙ , nξ ) + m2 ξ 2 ˙ = −m( n˙ ξ, ξ ) + m2 ξ 2 ≥ m m − max |φ(t)| ξ 2 . t∈[0,β]


297

˙ Theorem 5.1. Let m > maxt∈[0,β] |φ(t)|. For every θ ∈ (0, π ) such that there are no eigenvalues of D on the ray Rθ the following equality holds Det θ D = −(−1)k+ν Det θ D , (5.4) where k is defined in 5.1 and ν is defined in 5.3. Remark 5.1. Theorem 5.1 relates the sign of Det θ D with the topological invariant of the map eiφ . This realizes the program outlined in Subsect. 3.3. We precede the proof of the theorem with some discussion of the spectral properties of D. 5.2. The spectral properties of D. In order to study the spectrum of D it is convenient to replace it by a conjugate operator as follows. The operator iφ/2 e 0 Uφ := 0 e−iφ/2 maps the space of vector-functions with boundary conditions 5.3 to the space of functions ξ : [0, β] → C2 with new boundary conditions ξ(β) = eiπ(ν+k) ξ(0), Thus the operator

ξ˙ (β) = eiπ(ν+k) ξ˙ (0).

(5.5)

˙ := U −1 ◦ D ◦ Uφ = i d + −φ/2 im D φ ˙ im φ/2 dt

(5.6)

acting on the space of vector functions with boundary conditions 5.5 is isospectral to D. We now consider the following deformation of D: ˙ a := i d + −a φ/2 im , a ∈ [0, 1]. (5.7) D ˙ im a φ/2 dt The same arguments which were used in the proof of Lemma 5.1 show that a is invertible for all a ∈ [0, 1], and all sufficiently large Lemma 5.2. The operator D m > 0. Let

˙ −im d −a φ/2 Da = −i + , ˙ −im a φ/2 dt

a . be the complex conjugate of the operator D a (and, hence, of D) is symmetric The following lemma shows that the spectrum of D with respect to both the real and the imaginary axis. a , −D a∗ are conjugate to each other. Therefore, a , and D Lemma 5.3. The operators D they have the same spectral decomposition (2.8). In particular, the operators D, −D, and D ∗ are conjugate to each other. Proof. An easy calculation shows that 1 0 1 0 a∗ , · Da · = D 0 −1 0 −1

01 01 a . · Da · = −D 10 10

298


are conjugate to each other 5.3. Proof of Theorem 5.1. Since the operators D and D =D a is their determinants are equal. By Theorem 3.2, the sign of the determinant of D equal to the sign of the determinant of the operator d 0 im + . D0 = i im 0 dt 0 are given by the formula It is easy to see that all the eigenvalues of D λ± n = ±im +

π (2n − k − ν), β

n ∈ Z.

0 does not have any eigenvalues on the ray Rπ/2 if k + ν is odd and has exactly Hence, D one eigenvalue λ+ (k+ν)/2 = im on this ray if k + ν is even. Theorem 5.1 follows now from Theorem 3.1. 6. The Phase of the Determinant and the Degree of the Map This section essentially generalizes the previous section to manifolds of higher dimensions. For the case of a sphere of dimension N = 4l + 1 the results of this section have been obtained in [2] as topological terms in non-linear σ -models emerging as effective models for Dirac fermions coupled to chiral bosonic fields. However, no mathematically rigorous proofs were available until now. Note also that our result is more precise, since the equality 6.6 was obtained in [2] from the gradient expansion, i.e., only asymptotically for m → ∞. The section is organized as follows: first we formulate the problem in purely geometric terms as a question about the determinant of the DeRham-Dirac operator with potential. We state our main result as Theorem 6.6. Then, in Subsect. 6.3, we reformulate the result in terms of an operator acting on the tensor product of the two spaces of spinors. This formulation is closer to the one considered in physical literature. Finally, we present the proof of Theorem 6.6 based on the application of Theorems 3.1 and 3.2. 6.1. The setting. Let M be a closed oriented manifold of odd dimension N = 2r + 1. We fix a Riemannian metric on M and use it to identify the tangent and the cotangent j bundles, T M T ∗ M. Let ∗ T M = N j =0 T M denote the exterior algebra of T M viewed as a vector bundle over M. The space ∗ (M) of complex-valued differential forms on M coincides with the space of sections of the complexification ∗ T M ⊗ C of this bundle. The bundle ∗ T M ⊗ C (and, hence, the space ∗ (M)) carries 2 anti-commuting actions of the Clifford algebra of T M (the “left” and the “right” action) defined as follows cL (v) ω = v ∧ ω − ιv ω, cR (v) ω = v ∧ ω + ιv ω,

v ∈ T M, ω ∈ ∗ (M),

where ιv denotes the interior multiplication by v.

(6.1)


299

The DeRham-Dirac operator ∂ can be written now (cf. [4, Prop. 3.53]) as ∂ = d + d∗ =

N

j =1

cL (ej ) ∇eLC , j

(6.2)

where ∇ LC denotes the Levi-Civita covariant derivative and e1 , . . . , eN is an orthonormal frame of T M. We view the direct sum R ⊕ T M as a vector bundle over M. Consider the corresponding sphere bundle S := (t, a) ∈ R ⊕ T M : t 2 + |a|2 = 1 . (6.3) Let n be a smooth section of the bundle S. In other words, n = (n0 , n), where n0 ∈ C ∞ (M), n ∈ C ∞ (M, T M) and n20 + |n|2 = 1. Remark 6.1. Suppose M is a parallelizable manifold, i.e., there given an identification between T M and the product M × RN . Then n can be considered as a map (6.4) M −→ S N := y ∈ RN+1 : |y|2 = 1 . Also, if M ⊂ RN+1 is a hypersurface, then, for every x ∈ M, the space R ⊕ Tx M is naturally identified with RN+1 . Hence, n again can be considered as a map M → S N . Note, however, that, even if M is parallelizable, this map is different from 6.4. Consider the map : R ⊕ T M → End ∗ T M ⊗ C,

: n = (n0 , n) → in0 + cR (n),

and define the family of deformed DeRham-Dirac operators Dmn = ∂ + m (n) : ∗ (M) −→ ∗ (M).

(6.5)

We are interested in the phase of Detθ Dmn for sufficiently large m. The following lemma shows that this determinant is well defined. Lemma 6.1. Fix an orthonormal frame e1 , . . . , eN of T M and set N

LC LC ∇ n(x) = ∇ n(x) , ej

For m > maxx∈M

j =1

|(∇ LC n(x)| + |∇n

N

∇n0 (x) = ∇e n0 (x) . j j =1

0 (x)| , zero is not in the spectrum of D.

The lemma is a particular case of a more general Lemma 6.2, cf. below. 6.2. The degree of a section. Note that the bundle S → M has a natural section σ : M → S, σ (x) = (1, 0). Definition 6.1. The topological degree deg(n) of the map n is the intersection number of the manifolds σ (M) and n(M) inside S. Remark 6.2. Suppose M is parallelizable and consider n and σ as maps M → S N , cf. Remark 6.1. Then σ is the constant map σ (x) = (1, 0). Hence, deg(n) is the usual topological degree of the map n : M → S N . Theorem 6.1. Let m > maxx∈M |(∇ LC n(x)| + |∇n0 (x)| . For every θ ∈ (0, π ) such that there are no eigenvalues of Dmn on the ray Rθ the following equality holds: (6.6) Detθ Dmn = (−1)deg n Det θ Dmn .

300


6.3. Reformulation in terms of spinors. Consider the (left) chirality operator L := i

N +1 2

cL (e1 ) cL (e2 ) · · · cL (eN ),

where e1 , . . . , eN is an orthonormal frame of T M. This operator is independent of the choice of the frame [4, Lemma 3.17] (in fact, it coincides with the operator defined in 4.2). Moreover, L2 = 1 and L commutes with cL (v) and anti-commutes with cR (v) for all v ∈ T M. Consider the map : R ⊕ T M → End ∗ T M,

n = (n0 , n) → n := i L n0 + L cR (n).

Then n2 = − n20 + |n|2 . Hence, the map n → n defines a Clifford action of R ⊕ T M on ∗ T M. Assume now that M is a spin-manifold (without this assumption the construction of this subsection is true only locally, in any coordinate neighborhood). In particular, there exists a bundle S → M whose fibers are isomorphic to the space of spinors over R⊕T M. Then (cf. [4, Prop. 3.35]) there exists a bundle S → M, such that ∗ T M ⊗ C → M can be decomposed as the tensor product S ⊗ S, and the operators n (n ∈ R ⊕ T M) act only on the second factor. More precisely, if we denote by cS : R ⊕ T M → End S the Clifford action of R ⊕ T M on S, then n = 1 ⊗ cS (n). We introduce now a new Clifford action c : T M → End ∗ T M ⊗ C of T M on ∗ T M ⊗ C, defined by the formula c(v) = L cL (v),

v ∈ T M.

(6.7)

One readily sees that n and c(v) commute for all v ∈ T M, n ∈ R ⊕ T M. It follows (cf. [4, Prop. 3.27]) that there is a Clifford action cS : T M → End S such that c(v) = cS (v) ⊗ 1. Comparing dimensions we conclude that S is a spinor bundle over M. It follows from (6.2), that ∂ = L ∂ S ⊗ 1, where ∂ S is the Dirac operator on S. Hence, the operator (6.5) takes the form Dmn = ∂ + m L n = L ∂ S ⊗ 1 + m · 1 ⊗ cS (n) : C ∞ (S ⊗ S) −→ C ∞ (S ⊗ S). In this form this and similar operators have appeared in physical literature. In particular, for the case when M is a (4l + 1)-dimensional sphere this operator 3 was considered in [2]. Also a result similar to our Theorem 6.6 was obtained in [2] for the operator L · Dmn = ∂ S ⊗ 1 + m 1 ⊗ cS (n) on a (4l + 3)-dimensional sphere. 3

Note, however, that there is a sign discrepancy between our notation and the notation accepted in physical literature. Our operators cS (v), cS (n) are skew-adjoint and satisfy the equalities cS (v)2 = −|v|2 , cS (n) = −|n|2 . Consequently, the operator ∂ S is self-adjoint.


301

6.4. The idea of the proof. The rest of this section is devoted to the proof of Theorem 6.6, which is based on an application of Theorems 3.1 and 3.2. More pre whose determinant has the same cisely, we will deform operator Dmn to an operator D sign (in view of Theorem 3.2). We then calculate the number of imaginary eigenvalues which, in view of Theorem 3.1, will give us the sign of the determinants of D and of D, Dmn . First, we need to define the class of operators in which we will perform our deformation. This is done in the next subsection. 6.5. Extension of the class of operators. Let a : M → R and v : M → T M be a smooth function and a smooth vector field on M respectively. Set D(a, v) := ∂ + ia + cR (v). Clearly, Dmn = D(mn0 , mn). Also the following analogue of Lemma 6.1 holds: Lemma 6.2. Suppose a(x)2 + |v(x)|2 > 0 for all x ∈ M. Fix m0 > max |(∇ LC v(x)| + |∇a(x)| . x∈M

Then, for all m ≥ m0 , zero is not in the spectrum of D(m0 a, mv). Proof. Set ∂ m = ∂ + (m − m0 )cR (v). Then D(m0 a, mv) = ∂ m + i m0 a + m0 cR (v). ∗ (M).

Let α ∈ Using 6.2, we obtain, D(m0 a, mv) α 2 2 = ∂ m α + m20 α2 + m0 ∂ m (cR (v) + ia) + (cR (v) − ia) ∂ m α, α ≥ m20 α2 + m0

N

j =1

cL (ej ) cR (∇eLC v) + ∇ej a α, α j

+ 2 m0 (m − m0 ) |v|2 α, α ≥ m0 m0 − max |(∇ LC v(x)| + |∇a(x)| α2 . x∈M

The following lemma shows that we can apply Theorem 3.1 to the study of Detθ (a, v) (and, hence, of Detθ Dmn ). Lemma 6.3. The operators D(a, v) and −D(a, v)∗ are conjugate to each other. Consequently, they have the same spectral decomposition (2.8). ∗ are conjugate to each other. In particular, the operators Dmn and −Dmn Proof. Let N : ∗ (M) → ∗ (M) be the grading operator defined by the formula N ω = (−1)j ω,

ω ∈ j (M).

(6.8)

Then N ◦ D(a, v) ◦ N = − ∂ + ia − cR (v) = −D(a, v)∗ .

(6.9)

302


6.6. Deformation of Dmn . Let n = (n0 , n) be as in Theorem 6.6. Suppose that deg(n) = ±k, where k is a non-negative integer. Then there exists a section n = (n 0 , n ) of S, which is homotopic to n and has the following properties: • There exist k distinct points x1 , . . . , xk ∈ M such that n 0 (xj ) = 1,

n (xj ) = 0,

j = 1, . . . , k.

• There exists a Morse function f : M → R and a neighborhood U of the set {x1 , . . . , xk } such that n (x) = ∇f (x),

for all

x ∈ U,

and n (x) = 0 for all x ∈ U \{x1 , . . . , xk }. • If n (x) = 0 and x ∈ {x1 , . . . , xk }, then n 0 (x) = −1 and ∇f (x) = 0. Let xk+1 , . . . , xl be the rest of the critical points of f . Then n (x) = 0 for all x ∈ {x1 , . . . , xl }. Fix open neighborhoods Vj (j = 1, . . . , l) of xj whose closures are mutually disjoint and such that Vj ⊂ U for all j = 1, . . . , k. We will assume that Vj are small enough so that n 0 (x) = 0 and n (x) = 0 for all x ∈ Vj \{xj }. For each j = 1, . . . , l fix a neighborhood Wj of xj , whose closure lies inside Vj . Let a : M → [−1, 1] be a smooth function such that 1, if x ∈ kj =1 Wj ; a(x) = (6.10) −1, if x ∈ kj =1 Vj . Consider the deformation (n0 (t), n(t)) of the section (n 0 , n ) ∈ S given by the formulas ta + (1 − t)n 0 , 0≤t ≤1 n0 (t) = , a, 1≤t ≤2 n , 0≤t ≤1 n(t) = .

(t − 1)∇f + (2 − t)n , 1≤t ≤2 Clearly, (n0 (t), n(t)) = 0 for all t ∈ [0, 2]. Hence, by Lemma 6.2, for large m0 and every m > m0 , t ∈ [0, 2], zero is not in the spectrum of the operator D m0 n0 (t), mn(t) . Theorem 3.2 implies now that the determinant of Dmn has the same sign as the determinant of D m0 n0 (2), mn(2) = D(m0 a, m∇f ). 6.7. The spectrum of the operator D(0, m∇f ). Before investigating the operator D(m0 a, m∇f ) we consider a simpler operator ∗ D(0, m∇f ) = ∂ + m cR (∇f ) = e−mf d emf + e−mf d emf . This is a self-adjoint operator whose spectrum was studied by Witten [17] (see, for example, [16] for a mathematically rigorous exposition of the subject). In particular, D(0, m∇f ) has the following properties:


303

• There exist a constant C > 0 and a function r(m) > 0 such that limt→0 r(m) = 0 and, for all sufficiently large m > 0, the spectrum of D(0, m∇f ) lies inside the set √ √ − ∞, −C m ∪ − r(m), r(m) ∪ C m, ∞). • Let Em denote the span of the eigenvectors of D(0, m∇f ) with eigenvalues in the interval − r(m), r(m) . Then, for all sufficiently large m, the space Em has a basis α1,m , . . . , αl,m (αj,m = 1) such that each αj,m is concentrated in Wj in the following sense: αj,m ∧ ∗αj,m = 1 − o(1), as m → ∞. (6.11) Wj

(Here o(1) stands for a vector whose norm tends to 0 as m → ∞.) In particular, dim Em = l. Note that (6.11) and (6.10) imply that αj,m (x) + o(1), a(x) αj,m (x) = −αj,m (x) + o(1),

for j = 1, . . . , k; for j = k + 1, . . . , l.

(6.12)

6.8. The spectrum of the operator D(m0 a, m∇f ). Let m0 be as in Subsect. 6.6 and let C be as in Subsect. 6.7. Choose m large enough so that 4(m0 + 1) 2 m > , C and r(m) < 1. We view the operator D(m0 a, m∇f ) = D(0, m∇f ) + i m0 a(x) as a perturbation of D(0, m∇f ). Lemma 6.4. The number of eigenvalues λ (counting with multiplicities) of D(m0 a, m∇f ) which satisfy |λ| < 2(m0 + 1),

Im λ > 0,

is equal to k = deg n. Proof. The spectral projection of the operator D(0, m∇f ) onto the space Em (cf. Subsect. 6.7) is given by the Cauchy integral −1 1 λ − D(0, m∇f ) dλ, (6.13) Pm = 2πi γ where γ is the boundary of the disk B = z ∈ C : |z| < 2(m0 + 1) . Note that, for all λ ∈ γ , we have −1 1 1 = . (6.14) ≤ λ − D(0, m∇f ) 2m0 + 1 dist λ, spec D(0, m∇f )

304


Hence, for all λ ∈ γ , we obtain −1 λ − D(m0 a, m∇f ) −1 −1 ≤ λ − D(0, m∇f ) · 1 − (λ − D(0, m∇f ))−1 m0 a ≤

1 1 1 = · . 2m0 + 1 1 − 2mm00+1 m0 + 1

(6.15)

In particular, γ is contained in the resolvent set of D(m0 a, m∇f ).

denote the span of the root vectors of D(m a, m∇f ) with eigenvalues in B. Let Em 0

is given by the formula The spectral projection of D(m0 a, m∇f ) onto Em Pm =

1 2πi

λ − D(m0 a, m∇f )

−1

dλ.

γ

Using (6.14) and (6.15), we obtain −1 −1 1 λ − D(0, m∇f ) m a λ − D(m a, m∇f ) dλ 0 0 2π γ 1 1 2m0 · m0 · = < 1. (6.16) ≤ 2(m0 + 1) · 2m0 + 1 m0 + 1 2m0 + 1

Pm − P = m

In particular,

dim Em = dim Em = l,

. Recall that the basis α and the projection Pm maps Em isomorphically onto Em 1,m

, . . . , αl,m of Em was defined in Subsect. 6.7. Then Pm α1,m , . . . , Pm αl,m is a basis of

. Em From (6.12), we get

D(m0 a, m∇f ) αj,m =

i m0 αj,m + o(1), −i m0 αj,m + o(1),

for j = 1, . . . , k for j = k + 1, . . . , l.

Since the operators D(m0 a, m∇f ) and Pm commute we obtain D(m0 a, m∇f ) Pm αj,m =

i m0 Pm αj,m + o(1), −i m0 Pm αj,m + o(1),

for j = 1, . . . , k for j = k + 1, . . . , l.

has exactly k eigenvalues (counting with Hence, the restriction of D(m0 a, m∇f ) to Em multiplicities) with positive imaginary part.


305

6.9. Proof of Theorem 6.1. Clearly, all the eigenvalues of D(m0 a, m∇f ) satisfy Im λ ≤ m0 . (6.17) In particular, all the eigenvalues of D(m0 a,m∇f ) which lie on the ray Rπ/2 belong to the disc B = z ∈ C : |z| < 2(m0 + 1) . Since the spectrum of D(m0 a, m∇f ) is symmetric with respect to the imaginary axis the number of these eigenvalues (counting with multiplicities) has the same parity as the number of all eigenvalues, which lie in B and have positive imaginary part. Theorem 6.6 follows now from Theorem 3.1 and Lemma 6.4. Acknowledgements. The first author would like to thank the Theory Institute of Strongly Correlated and Complex Systems at Brookhaven for hospitality and support. The second author would like to thank the Max-Planck-Institut für Mathematik, where most of this work was completed, for hospitality and providing the excellent working conditions. He would also like to thank Mikhail Shubin and Raphael Ponge for valuable discussions.

References 1. Abanov, A.G.: Hopf term induced by fermions. Phys.Lett. B492, 321–323 (2000) 2. Abanov, A.G., Wiegmann, P.B.: Theta-terms in nonlinear sigma-models. Nucl.Phys. B570, 685–698 (2000) 3. Agranovich, M.S.: Elliptic operators on closed manifolds. Current problems in mathematics. Fundamental directions, Vol. 63 (Russian), Itogi Nauki i Tekhniki, Moscow: Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., 1990, pp. 5–129 4. Berline, N., Getzler, E., Vergne, M.: Heat kernels and Dirac operators. Berlin-Heidelberg-New York: Springer-Verlag, 1992 5. Berry, M.: Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A 392, 45–57 (1984) 6. Burghelea, D., Friedlander, L., Kappeler, T.: On the determinant of elliptic differential and finite difference operators in vector bundles over S 1 . Comm. Math. Phys. 138(1), 1–18 (1991) 7. Kato, T.: Perturbation theory for linear operators. Berlin-Heidelberg-New York: Springer-Verlag, 1966 8. Lidski˘ı, V.B.: Non-selfadjoint operators with a trace. Dokl. Akad. Nauk SSSR 125, 485–487 (1959) 9. Markus, A.S.: Introduction to the spectral theory of polynomial operator pencils. Translations of Mathematical Monographs, Vol. 71, Providences RI: Amer. Math, Soc. 1998 10. Ponge, R.: Spectral asymetry, zeta function and the noncommutative residue. Preprint, to appear in J. Funct. Anal. 11. Redlich, A.N.: Gauge Noninvariance and Parity Nonconservation of Three-Dimensional Fermions. Phys. Rev. Lett. 52, 18–21 (1984) 12. Redlich, A.N.: Parity violation and gauge noninvariance of the effective gauge field action in three dimensions. Phys. Rev. D 29, 2366–2374 (1984) 13. Retherford, J.R.: Hilbert space: compact operators and the trace theorem. London Mathematical Society Student Texts, Vol. 27, Cambridge: Cambridge University Press, 1993 14. Seeley, R.: Complex powers of elliptic operators. Proc. Symp. Pure and Appl. Math. AMS 10, 288–307 (1967) 15. Shubin, M.A.: Pseudodifferential operators and spectral theory. Berlin, New York: Springer Verlag, 1980 16. Shubin, M.A.: Semiclassical asymptotics on covering manifolds and Morse inequalities. Geom. Funct. Anal. 6, 370–409 (1996) 17. Witten, E.: Supersymmetry and Morse theory. J. of Diff. Geom. 17, 661–692 (1982) 18. Wodzicki, M.: Local invariants of spectral asymmetry. Invent. Math. 75(1), 143–177 (1984) 19. Wodzicki, M.: Noncommutative residue. I. Fundamentals. K-theory, arithmetic and geometry (Moscow, 1984–1986), Lecture Notes in Math., Vol. 1289, Berlin: Springer, 1987 pp. 320–399 20. Wojciechowski, K.P.: Heat equation and spectral geometry. Introduction for beginners. Geometric methods for quantum field theory (Villa de Leyva, 1999), River Edge, NJ: World Sci. Publishing, 2001, pp. 238–292 Communicated by P. Sarnak


Communications in


Birkhoff’s Polytope and Unistochastic Matrices, N = 3 and N = 4 Ingemar Bengtsson1 , Åsa Ericsson1 , Marek Ku´s2 , Wojciech Tadej3 , 2,4 ˙ Karol Zyczkowski 1

Stockholm University, AlbaNova, Fysikum, 106 91 Stockholm, Sweden. E-mail: [email protected]; [email protected] 2 Centrum Fizyki Teoretycznej, Polska Akademia Nauk, Al. Lotników 32/44, 02–668 Warszawa, Poland. E-mail: [email protected] 3 Cardinal Stefan Wyszynski University, Warszawa, Poland. E-mail: [email protected] 4 Instytut Fizyki im. Smoluchowskiego, Uniwersytet Jagiello´ nski, ul. Reymonta 4, 30–059 Kraków, Poland. E-mail: [email protected] Received: 8 March 2004 / Accepted: 24 March 2005 Published online: 15 July 2005 – © Springer-Verlag 2005

Abstract: The set of bistochastic or doubly stochastic N × N matrices is a convex set called Birkhoff’s polytope, which we describe in some detail. Our problem is to characterize the set of unistochastic matrices as a subset of Birkhoff’s polytope. For N = 3 we present fairly complete results. For N = 4 partial results are obtained. An interesting difference between the two cases is that there is a ball of unistochastic matrices around the van der Waerden matrix for N = 3, while this is not the case for N = 4. 1. Introduction Unistochastic matrices arise in many different contexts including error correcting codes, quantum information theory and particle physics. To define them, we first recall that an N × N matrix B is said to be bistochastic if its matrix elements satisfy i: Bij ≥ 0, ii: Bij = 1, iii: Bij = 1 . (1) i

j

The set of bistochastic matrices is a convex polytope known as Birkhoff’s polytope. One way of constructing a bistochastic matrix is to begin with a unitary matrix U and let Bij = |Uij |2 .

(2)

However, it is well-known [1] that not all bistochastic matrices arise in this way. If there is such a U , then we will call B unistochastic. If U is also real, that is orthogonal, then we call B orthostochastic. (Much of the mathematics literature uses the term orthostochastic to mean any matrix satisfying (2) and does not distinguish the subclass for which U is real. We will see later that the distinction is important.) In this paper, we consider the problem of characterizing the unistochastic subset of Birkhoff’s polytope. Before summarizing our results, we mention some physical applications. In quantum mechanics, the transition probabilities associated with a finite basis form bistochastic

˙ I. Bengtsson, Å. Ericsson, M. Ku´s, W. Tadej, K. Zyczkowski

308

matrices. In studies of the foundations of quantum theory, the attempt to build some group structure into these transition probabilities leads to the requirement that they form unistochastic matrices. A sample of the literature includes Landé [2], Rovelli [3] and Khrennikov [4]. In the attempt to formulate quantum mechanics on graphs (in the laboratory on thin strips of, say, gold film) the question of what Markov processes have quantum counterparts in the given setting again leads to unistochastic matrices [5–7]. In this connection studies of the spectra and entropies of unistochastic matrices chosen at random have been made [8]. In particle physics, a related question arises. In the theory of weak interactions one encounters the unitary Kobayashi-Maskawa matrices (one for quarks and one for neutrinos), and Jarlskog raised the question to what extent such a matrix can be parametrized by the easily measured moduli of its matrix elements. The physically interesting case here is N = 3 [9], and possibly also N = 4, should a fourth generation of quarks be discovered [10]. The question of determining U from B also arises in scattering theory, with no restriction on N [11]. Our main result involves the van der Waerden matrix JN , whose matrix elements satisfy (JN )ij = N1 . This matrix is unistochastic, and any corresponding unitary matrix is known as a complex Hadamard matrix. An example is the Fourier matrix, whose matrix elements are 1 Uj k = √ q j k , N

0 ≤ j, k ≤ N − 1 .

(3)

Here q = e2π i/N is a root of unity. Complex Hadamard matrices have a long history in mathematics [12–14], and have recently arisen in quantum information theory [15–17]. In this paper we study the set of unistochastic matrices, and the precise way in which it forms a subset of Birkhoff’s polytope. Our main result is that for N = 4 every neighborhood of the van der Waerden matrix contains matrices that are not unistochastic. This is in striking contrast with the N = 3 case for which J3 is at the center of a ball of unistochastic matrices inside a star-shaped region bounded by the set of orthostochastic matrices. This paper is organized as follows. In Sect. 2 we consider the set of all bistochastic matrices, and describe the cases N = 3 and N = 4 in detail (N = 2 is trivial). In Sect. 3 we discuss some generalities concerning unistochastic matrices, and then characterize the unistochastic subset in the case N = 3. Most of our results can be found elsewhere but, we believe, not in this coherent form. In Sect. 4 we consider N = 4, prove our main result, and relate some already known facts [10] to our explicit description of Birkhoff’s polytope. Section 5 summarises our conclusions. Some technical matters are found in three appendices. 2. Birkhoff’s Polytope The set BN of bistochastic N × N matrices has (N − 1)2 dimensions. To see this, note that the last row and the last column are fixed by the conditions that the row and column sums should equal one. The remaining (N − 1)2 entries can be chosen freely, within limits. Birkhoff proved that BN is a convex polytope whose extreme points, or corners, are the N ! permutation matrices [18]. It is called Birkhoff’s polytope. All its corners are equivalent in the sense that they can be transformed into each other by means of orthogonal transformations. A bistochastic matrix belongs to the boundary of BN if and only if at least one of its entries is zero. The boundary consists of corners, edges, faces,

Birkhoff’s Polytope and Unistochastic Matrices, N = 3 and N = 4

309

3-faces and so on; the highest dimensional faces are called facets and consist of matrices with only one zero entry. For a detailed account of BN , especially its face structure, see Brualdi et al. [19]. We will be even more detailed concerning B3 and B4 . We will use a quite explicit notation for the 24 permutation matrices in B4 ; see Appendix A for the details. It is convenient to regard the convex polytope BN as a subset of a vector space, with the van der Waerden matrix JN as its origin. The distance squared between two matrices is chosen to be D 2 (A, B) = Tr(A − B)(A† − B † ) ,

(4)

where the dagger denotes Hermitian conjugation. The distance squared between an arbitrary bistochastic matrix B and the van der Waerden matrix JN is then given by D 2 (B, JN ) = Bij2 − 1 . (5) i,j

In particular, the distance between JN and a corner of the polytope becomes √ D = N − 1. Permutations of rows or columns are orthogonal transformations of the polytope, since they preserve distance and leave the van der Waerden matrix invariant. They also take permutation matrices (corners) into permutation matrices, hence they are symmetry operations of Birkhoff’s polytope as well. The (Shannon) entropy of a bistochastic matrix is defined as the entropy of the rows averaged over the columns, 1 S=− Bij ln Bij . (6) N i

j

Its maximum value ln N is attained at JN . For some of its properties consult Słomczyński [20] et al. [8]. When N = 2 there are just two permutation matrices and B2 is a line segment between these two points. A general bistochastic matrix can be parametrized as 2 2 π c s B= 2 2 , c ≡ cos θ , s ≡ sin θ , 0≤θ ≤ . (7) s c 2 When N = 3 we have six permutation matrices forming the vertices of a four dimensional polytope. It admits a simple description: Theorem 1. The 6 corners of B3 are the corners of two equilateral triangles placed in two totally orthogonal 2-planes and centered at J3 . To prove this we form two triangles as convex combinations of permutation matrices. Using a notation that is consistent with Appendix A they are   p0 p3 p4 1 = p0 P0 + p3 P3 + p4 P4 =  p4 p0 p3  , p0 + p3 + p4 = 1 (8) p 3 p4 p 0 and

 p1 p2 p5 2 = p1 P1 + p2 P2 + p5 P5 =  p2 p5 p1  , p5 p 1 p 2 

p1 + p2 + p5 = 1 .

(9)


310

The calculation we have to do is to check that D 2 (P0 , P3 ) = D 2 (P0 , P4 ) = D 2 (P3 , P4 ) = 6 and similarly for the other triangle, and also that Tr(1 − J3 )(†2 − J3 ) = 0

(10)

for all values of pi . This is so. There are thus 6 corners and 6 · 5/2 = 15 edges, all of which are extremal. The last is a rather exceptional property; in 3 dimensions only the simplex has it. There are 9 short edges of length squared D 2 = 4 and 6 long edges of length squared D 2 = 6, namely the sides of the two equilateral triangles. A useful overview of B3 is given by its graph, where we exhibit all corners and all edges (see Fig. 1). All the 2-faces are triangles with one long and two short edges. The 3-faces in a 4 dimensional polytope are facets and here they are made of matrices with a single zero. They are irregular tetrahedra with two long edges, one from each equilateral triangle (see Fig. 4). The volume of B3 is readily computed because it can be triangulated using only three simplices. The total volume is 9/8. As N grows the total volume of BN becomes increasingly hard to compute; mathematicians know it for N ≤ 10 [21]. The next case is the 9 dimensional polytope B4 . It has 24 corners and 276 edges. The latter come in four types and we give the classification including the angle they subtend at J4 and whether they consist of unistochastic matrices or not (see Sects. 3 and 4): 4U 6 8 8U

Length squared Unistochastic Angle at origin Number of edges 4 Yes Acute 72 6 No 90 degrees 96 8 No Obtuse 72 8 Yes Obtuse 36

All edges except the 8U ones are extremal. The 2-faces consist of triangles and squares. (Interestingly, for all N it is true that the 2-faces of Birkhoff’s polytope BN are either triangles or rectangles [19].) There are 18 squares bounded by edges of type 4U and their diagonals are of type 8U . Three squares meet at each corner. If we pick four permutation matrices we obtain a 3-face, with six exceptions. The exceptions form 6 regular tetrahedra centered at J4 , whose edges are non-extremal 8U edges. They are denoted Ti

Fig. 1. Left: Birkhoff’s polytope for N = 2 (centered at J2 ). Right: The graph of Birkhoff’s polytope for N = 3; single lines have D 2 = 4 and double D 2 = 6. The double edges form the triangles mentioned in Theorem 1


311

Fig. 2. How to begin to draw the surface of B4 . Two tetrahedra whose edges are the non-extremal diagonals of squares are shown. The dashed line goes through the polytope; it connects the midpoints of two opposing 8U edges of two tetrahedra that are otherwise disjoint

and explicitly listed in Appendix A. When regular tetrahedra are mentioned below it is understood that we refer to one of these six. In a sense the structure can now be drawn; see Fig. 2. The facets consist of matrices with one zero, so there are 16 facets. A subset of B4 that has no counterpart for B3 is the set of matrices that are tensor products of two by two bistochastic matrices. This subset splits naturally into several two dimensional components, and it turns out that they sit in B4 as doubly ruled surfaces inside the regular tetrahedra. Thus the following matrix, parametrised with two angles, is a tensor product of two matrices of the form (7): 

c12 c22 c12 s22 s12 c22 s12 s22



 c2 s 2 c2 c2 s 2 s 2 s 2 c2   1 2 1 2 1 2 1 2  2 2 2 2 2 2 2 2 ,  s 1 c 2 s 1 s 2 c1 c2 c 1 s 2 

c1 ≡ cos θ1 , etc.

(11)

s12 s22 s12 c22 c12 s22 c12 c22 These matrices form a doubly ruled surface inside the regular tetrahedron T1 , analogous to that depicted in Fig. 4. An interesting way to view B4 , and one that will recur in Sect. 4, stems from the following observation: Theorem 2. The 24 corners of B4 belong to a set of nine orthogonal hyperplanes through J4 . Each regular tetrahedron belongs to six hyperplanes and contains the normal vectors of the remaining three hyperplanes. Each hyperplane contains four regular tetrahedra and its normal vector is the intersection of the remaining two regular tetrahedra. Again the proof is a simple calculation, once the explicit form of the hyperplanes is known. They are denoted i and listed in Appendix A. From now on, hyperplane always refers to one of these nine. Figure 3 in a sense illustrates the theorem.


312

Fig. 3. A regular tetrahedron centered at J4 . It contains the normal vectors of three orthogonal hyperplanes and belongs entirely to another six. There are six such regular tetrahedra and pairs of them intersect along the normal vectors they contain. (Note that the dashed line in Fig. 2 represents such a normal vector.)

It is quite helpful to have an incidence table for tetrahedra and hyperplanes available. It is T1 T2 T3 T4 T5 T6

1 2 X X X X X X X X

3 4 5 6 X X X X X X X X X X X X X X X X

7 8 9 X X X X X X X X X X X X

(12)

where the tetrahedra Ti and the hyperplanes i are listed in Appendix A. For later purposes we will need some information about exactly how the hyperplanes divide the space into 29 hyperoctants. For this reason we look at the rays Bi (t) = J4 + tVi ,

(13)

where Vi is a vector constructed in terms of the normal vectors n1 , . . . , n9 of the hyperplanes (see Appendix A), namely   9 −3 −3 −3 1  −3 1 1 1  V1 ≡ n1 + n2 + n3 + n4 + n5 + n6 + n7 + n8 + n9 =   , (14) 4 −3 1 1 1 −3 1 1 1 

 7 −1 −1 −5 1  −1 −1 −1 3  V2 ≡ n1 + n2 + n3 + n4 + n5 + n6 + n7 + n8 − n9 =   , (15) 4 −1 −1 −1 3 −5 3 3 −1


313



 5 1 −3 −3 1  1 −3 1 1  V3 ≡ n1 + n2 + n3 + n4 − n5 + n6 + n7 + n8 − n9 =   . (16) 4 −3 1 −3 5 −3 1 5 −3 All other cases can be obtained from one of these three by permutations of rows and columns. The various hyperoctants are convex cones centered on these rays. This gives a classification of the hyperoctants into six different types (since the parameter t can be positive or negative) called respectively type I± , II± and III± . Type I has 16 representatives and is especially noteworthy. For type I− the centered ray hits the boundary in the center of one of the 16 facets, at the matrix B1 (− 19 ). In the other direction we also hit a quite distinguished point. There are 16 ways of setting one entry of a bistochastic matrix equal to one, and this gives rise to 16 copies of B3 sitting in the boundary of B4 . For the octants I+ the centered ray hits the boundary precisely at the center of such a B3 , at the matrix B1 ( 13 ). In Sect. 4 we will see how the structure of the unistochastic subset is related to the structure of Birkhoff’s polytope, and in particular to those of its features that we have stressed. 3. The Unistochastic Subset, N = 3 Let us begin with some generalities concerning the unistochastic subset UN of BN . The dimension of BN is (N −1)2 , and the dimension of the group of unitary N by N matrices, U (N ), is N 2 . Therefore the map U (N ) → BN cannot be one-to-one. Now it is clear that multiplying a row or a column by a phase factor—an operation that we refer to as rephasing—will result in the same bistochastic matrix via Eq. (2). Therefore the map is naturally defined as a map from a double coset space to BN . The double coset space is U (1) × · · · × U (1) \ U (N) / U (1) × · · · × U (1) ,

(17)

with N U (1) factors acting from the right and N − 1 factors from the left, say. The dimension of this set is (N − 1)2 , so now the dimensions match. There is a complication because the double coset space is not a smooth manifold. The action from the left of the U (1) factors on the right coset space (in itself a well behaved flag manifold) has fixed points. These fixed points are easy to locate however (and always map to the boundary of BN ), so that for most practical purposes we can think of our map as a map between smooth manifolds. In general we will see that the image of our map is a proper subset of BN , and the map is many-to-one. There is not much we can usefully say about the general case, except for two remarks: The unistochastic subset UN has the full dimension (N − 1)2 while the unistochastic subset of the boundary of BN has dimension (N − 1)2 − 2; why this is so will presently become clear. For N = 2 every bistochastic matrix is orthostochastic. A unitary matrix that maps to the matrix in Eq. (7) is π c s U= , c ≡ cos θ , s ≡ sin θ , 0≤θ ≤ . (18) s −c 2 The matrix is given in dephased form. This means that the first row and the first column are real and positive. This fixes the U (1) factors mentioned above (unless there is a zero entry in one of these places) and from now on we shall present all unitary matrices


314

in this form. For any N it is straightforward to check whether a given edge of BN is unistochastic. For N = 3 the edges of length squared equal to 4 are unistochastic, and for N = 4 we have the results given in table (11). Given a 3 × 3 bistochastic matrix

it is easy to check whether it is unistochastic or not [22, 9]. We form the moduli rij = Bij and write down the matrix 

 r00 r01 • U =  r10 r11 eiφ11 •  . r20 r21 eiφ21 •

(19)

This matrix is given in dephased form. If it is unitary, the original matrix B is unistochastic. The unitarity conditions simply say that the first two columns are orthogonal. The last column by construction has the right moduli and does not impose any further restrictions, hence it is not written explicitly. The problem is whether phases φ11 and φ21 can be found so that the matrix is unitary. This problem can be translated into the problem of forming a triangle from three line segments of given lengths L0 = r00 r01 ,

L1 = r10 r11 ,

L2 = r20 r21 .

(20)

This is possible if and only if the “chain–links” conditions are fulfilled, i.e. |L1 − L2 | ≤ L0 ≤ L1 + L2 .

(21)

The bistochastic matrix B corresponding to U sits at the boundary of U3 if and only if one of these inequalities is saturated. When the inequalities (21) hold the solution for the phases is cos φ11 =

L22 − L20 − L21 , 2L0 L1

cos (φ11 − φ21 ) =

cos φ21 =

L21 − L22 − L20 , 2L0 L2

L20 − L21 − L22 . 2L1 L2

(22)

(23)

There is a two-fold ambiguity (corresponding to taking the complex conjugate of the matrix, U → U ∗ ). The area A of the triangle is easily computed and the chain–links conditions are equivalent to the single inequality A ≥ 0. As a matter of fact we can form six so called unitarity triangles in this way, depending on what pair of columns or rows that we choose. Although their shapes differ their area is the same, by unitarity [9]. Because we can easily decide if a given matrix is unistochastic, it is easy to characterize the unistochastic set U3 . We single out the following facts (some of which are known [22, 23]) for attention: Theorem 3. The unistochastic subset U3 of B3 is a non-convex star shaped four dimensional set whose boundary consists of√the set of orthostochastic matrices. It contains a unistochastic ball of maximal radius 2/3, centered at J3 . The set meets the boundary of B3 in a doubly ruled surface in each facet.


315

Fig. 4. Birkhoff’s polytope for N = 3. Left: One of the two orthogonal equilateral triangles centered at J3 , with its unistochastic subset (the boundary is the famous hypocycloid). Right: A facet, an irregular tetrahedron, with its doubly ruled surface of unistochastic matrices

The relative volume of the unistochastic subset is, according to our numerics, vol(U3 ) ≈ 0.7520 ± 0.0005 . vol(B3 )

(24)

We did not attempt an analytical calculation; details of our numerics are in Appendix B. Theorem 3 is easy to prove. To see that U3 is non-convex we just draw its intersection with one of the equilateral triangles that went into the definition of the polytope, and look at it (see Fig. 4). An amusing side remark is that the boundary of the unistochastic set in this picture is a 3-hypocycloid [8]. It can be obtained by rolling a circle of radius 1/3 inside the unit circle. The maximal unistochastic ball is centered at J3 and touches the boundary at the hypocycloid, as one might guess from the picture; its radius was deduced from results presented in ref. [24]. To see that the boundary consists of orthostochastic matrices, observe that when the chain–links conditions are saturated the phases in U will equal ±1. That the set is star shaped then follows from an explicit check that there is only one orthostochastic matrix on any ray from J3 . Finally Fig. 4 includes a picture of the unistochastic subset of a facet. The reason why it has codimension one is that a matrix on the boundary of BN has a zero entry, which means that the number of phases available in the dephased unitary matrix drops with one, and then the dimension of the unistochastic set also drops with one; the argument goes through for any N . Finally let us make some remarks on entropy. First we compare the Shannon entropy averaged over B3 to the Shannon entropy averaged over U3 , using the flat measure in both cases. Numerically we find that S B3 ≈ 0.883

and

S U3 ≈ 0.908 ,

(25)

with all digits significant. Observe that the latter average is larger since some matrices of small entropy close to the boundary of B3 are not unistochastic and do not contribute to the average over U3 . The above data may be compared with the maximal possible entropy Smax = ln 3 ≈ 1.099, attained at J3 , and also with S Haar =

1 1 + ≈ 0.833 , 2 3

(26)

which is the average taken over U3 with respect to the measure induced by the Haar measure on U (3). This analytical result follows from the observation that S Haar coincides

316


with the average entropy of squared components of complex random vectors, which was computed by Jones [25]. 4. The Unistochastic Subset, N = 4 The case N = 4 is more difficult. It is also clear from the outset that it will be qualitatively different—thus the dimension of the orthogonal group is too small for the boundary of the unistochastic set U4 to be formed by orthostochastic matrices alone. There are other differences too, as we will see.

Given a bistochastic matrix we can again define rij = Bij and consider   r00 r01 r02 •  r10 r11 eiφ11 r12 eiφ12 •   U = (27)  r20 r21 eiφ21 r22 eiφ22 •  . r30 r31 eiφ31 r32 eiφ32 • Phases must now be chosen so that this matrix is unitary, and more especially so that the three columns we focus on are orthogonal. Geometrically this is the problem of forming three quadrilaterals with their sides given and six free angles. This is not a simple problem, and in practice we have to resort to numerics to see whether a given bistochastic matrix is unistochastic (see Appendix B for details). There are some easy special cases though. One easy case is that of a matrix belonging to the boundary of BN . Then the matrix U must contain one zero entry and when we check the orthogonality of our three columns two of the equations reduce to the problem of forming triangles. This fixes four of the angles, and the final orthogonality relation is easily dealt with. Another easy case concerns the regular tetrahedra. They turn out to consist of orthostochastic matrices; for the tetrahedron T1 (see Appendix A) a corresponding orthonormal matrix is √ √ √   √ √p0 √p7 √p16 √p23  p − p − √p23 √p16  O1 =  √ 7 √ 0 (28)  . √p16 √p23 −√p0 −√p7 p23 − p16 p 7 − p0 This saturates a bound saying that the maximum number of N × N permutation matriN

ces whose convex hull is unistochastic is not larger than 2 2 , where [N/2] denotes the integer part of N/2 [26]. Let us now turn our attention to J4 . Hadamard [27] observed that up to permutations of rows and columns the most general form of the complex Hadamard matrix is   1 1 1 1 1  1 eiφ −1 −eiφ  . H (φ) =  (29) 2  1 −1 1 −1  1 −eiφ −1 eiφ One can show that this is a geodesic in U (N ). What is new, compared to N = 3, is that the van der Waerden matrix is orthostochastic because H (0) is real. Moreover, there is a continuous set of dephased unitaries mapping to the same B. In a calculational tour de force, Auberson et al. [10] were able to determine all bistochastic matrices whose dephased unitary preimages contain a continuous ambiguity (and they found that the


317

ambiguity is given by one parameter in all cases). There are three such families. Using the notation of ref. [10] they consist of matrices of the following form:  Type A:

a b e f

b a f e

c d g h

 d c , h g 

Type B:

 Type C:

a a

b b   c c d d

s12 s22 c12 s22 c32 c22 s32 c22

1 2 1 2 1 2 1 2

−a −b −c −d

1 2 1 2 1 2 1 2

−a



−b  , −c

(30)

−d



 s 2 c2 c2 c2 c2 s 2 s 2 s 2   1 2 1 2 3 2 3 2  2 2 2 2 2 2 2 2 .  c1 c4 s 1 c4 s 3 s 4 c3 s 4 

(31)

c12 s42 s12 s42 s32 c42 c32 c42 Here c1 = cos θ1 , s1 ≡ sin θ1 , and so on. Type A consists of nine five dimensional sets, type B of nine four dimensional sets, and type C of six three dimensional sets. In trying to understand their location in B4 the observation in Sect. 2 concerning the nine orthogonal hyperplanes begins to pay dividends. (In particular, consult the incidence table (12).) Type A consists of the linear subspaces obtained by taking all intersections of four hyperplanes that contain exactly two regular tetrahedra. Type C consists of the linear subspaces obtained by taking all intersections of six hyperplanes that contain no permutation matrices at all. Type B finally consists of curved manifolds confined to one hyperplane. Auberson’s families are not exclusive. In particular tensor product matrices belong to families A and B, which means that there are two genuinely different ways of introducing a free phase in the corresponding unitary matrix. Outside the three sets A, B and C Auberson et al. find a 12-fold discrete ambiguity in the dephased unitaries, dropping to 4-fold for symmetric matrices [10]. Tensor product matrices B4 = B2 ⊗ B2 appear because 4 = 2 × 2 is a composite number. That they are always unistochastic follows from a more general result: Lemma 1. Let BK and BM be unistochastic matrices of size K and M, respectively. Then the matrix BN = BK ⊗ BM of size KM is unistochastic. The corresponding dephased unitary matrices contain at least (K − 1)(M − 1) free phases. That BN is unistochastic follows from properties of the Hadamard and the tensor products. By definition, the Hadamard product A ◦ B of two matrices is the matrix whose matrix elements are the products of the corresponding matrix elements of A and B. Then ∗ implies that B = (U ◦ U ∗ ) ⊗ (U ◦ U ∗ ) = BK = UK ◦ UK∗ and BM = UM ◦ UM N K M K M ∗ ∗ (UK ⊗ UM ) ◦ (UK ⊗ UM ), so it is unistochastic. The existence of free phases is an easy generalization of Proposition 2.9 in Haagerup [28]. The hyperplane structure of B4 reverberates in the structure of the unistochastic set in several ways. Let us consider how the tangent space of U (N ) behaves under the map to BN . In equations, this means that we fix a unitary matrix U0 and expand 1 U (t) = eiht U0 = (1 + iht − h2 t 2 + . . . )U0 , 2

(32)

where h is an Hermitian matrix. Then we study bistochastic matrices with elements Bij (t) = |Uij (t)|2 to first order in t. The following features are true for all N :


318

• Generically the tangent space of U (N ) maps onto the tangent space of BN . We checked this statement by generating unitary matrices at random using the Haar measure on the group. It implies that the dimension of the unistochastic set is equal to that of BN . • A matrix element in B receives a first order contribution only if it is non-vanishing. Hence the map of the tangent space of U (N ) to the tangent space of BN is degenerate at the boundary of the polytope. In general such behaviour is to be expected at the boundary of the unistochastic set UN . • If U0 is real the map is degenerate in the sense that the tangent space maps to an N(N − 1)/2 dimensional subspace of the tangent space of BN . • If U0 maps to a corner of the polytope then the first order contributions vanish. To second order we pick up the tip of a convex cone whose extreme rays are the N (N −1)/2 edges of type 4U , emanating from that corner. For N = 4 the story becomes interesting when we choose U0 equal to the Hadamard matrix H (φ). Then we find that the tangent space at U0 maps into one of the nine hyperplanes; which particular one depends on how we permute rows and columns in Eq. (29). The question therefore arises whether the orthostochastic van der Waerden matrix belongs to the boundary of the unistochastic set—or not since a priori such degeneracies can occur also in the interior of the set. We know that we can form curves of unistochastic matrices starting from J4 and moving out into the nine hyperplanes. Can we form such curves that go directly out into one of the 29 hyperoctants? Here the division of the 29 hyperoctants into six different types becomes relevant. We have investigated whether their central rays given in Eqs. (13–16) consist of unistochastic matrices, or not. Let us begin with the 16 hyperoctants of type I, where the central ray B1 (t) = J4 + tV1 hits the boundary in the center of one of the 16 B3 sitting in the boundary (at t = 1/3), and in the center of one of the 16 facets (at t = −1/9). Of these two points, the first is unistochastic, the second is not. A one parameter family of candidate unitary matrices that maps to the central ray is √ √  √ • √1 − 3t iφ √1 − 3t √1 − 3t φ 11 12 1  1+t •  , √1 + te √1 + te √ (33) U (t) =  2  √1 + t √1 + teiφ21 √1 + teφ22 •  φ iφ 1+t 1 + te 31 1 + te 32 • where t > 0 and we permuted the columns relative to Eq. (14) in order to get the unitarity equations in a pleasant form. (We do not need to give the phases for the last column.) The conditions that the first three columns be orthogonal read eiφ11 + eiφ21 + eiφ31 + L = 0, eiφ12 + eiφ22 + eiφ32 + L = 0, i(φ11 −φ12 ) i(φ21 −φ22 ) e +e + ei(φ31 −φ32 ) + L = 0,

(34) (35) (36)

where L=

1 − 3t . 1+t

In Appendix C we prove that the system of equations (34–36) 1. has no real solutions for L > 1,

(37)


319

2. for 0 < L < 1 has the solution φ11 = 0, φ21 = φ, φ31 = −φ , φ12 = φ, φ22 = 0, φ32 = −φ

cos φ =

L+1 t −1 =− . t +1 2

(38)

It follows that the central ray is unistochastic for the hyperoctants of type I+ (and the unitary matrices on the central ray tend to the real Hadamard matrix at t = 0). In the other direction the central ray is not unistochastic for type I− . Thus we have proved Theorem 4. For N = 4 there are non-unistochastic matrices in every neighbourhood of the van der Waerden matrix J4 . At J4 the map U (4) → B4 aligns the tangent space of U (4) with one of the nine orthogonal hyperplanes. The structure of the unistochastic set is dramatically different depending on whether N = 3 or N = 4. It is only in the former case that there is a ball of unistochastic matrices surrounding the van der Waerden matrix. On the other hand, the hyperoctants are not empty—some of them do contain unistochastic matrices all the way down to J4 . Concerning the other hyperoctants, for types II− , III+ , and III− the central rays hit the boundary of the polytope in points that are not unistochastic, but numerically we find that a part of the ray close to J4 is unistochastic. For type II+ we hit the boundary in a unistochastic point and numerically we find the entire ray to be unistochastic. There is still much that we do not know. We do not know if the hyperoctants of type I− are entirely free of unistochastic matrices, nor do we know if U4 is star shaped, or what its relative volume may be. What is clear from the results that we do have is that the global structure of Birkhoff’s polytope reverberates in the structure of the unistochastic subset in an interesting way—it is a little bit like a nine dimensional snowflake, because the nine hyperplanes in B4 can be found through an analysis of the behaviour of U4 in the neighbourhood of J4 . 5. Conclusions Our reasons for studying the unistochastic subset of Birkhoff’s polytope have been summarized in the introduction. Because the problem is a difficult one we concentrated on the cases N = 3 and N = 4. Our descriptions of Birkhoff’s polytope for these two cases are given in Theorems 1 and 2, respectively, and a characterization sufficient for our purposes of the unistochastic set for N = 3 is given in Theorem 3. For N = 4 the dimension of the unistochastic set is again equal to that of the polytope itself, but its structure differs dramatically from the N = 3 case. In particular Theorem 4 states that for N = 4 there are non-unistochastic matrices in every neighbourhood of the van der Waerden matrix. Hence there does not exist a unistochastic ball surrounding the van der Waerden matrix. We observed that the structure of the unistochastic set at the center of the polytope reflects the global structure of the latter in an interesting way. It is natural to ask to what extent the difference between the two cases is due to the fact that 3 is prime while 4 is not. Although this is not the place to discuss the cases N > 4, let us mention that we have reasons to believe that the dimension of the unistochastic set is equal to that of BN for all values of N [29]. On the other hand it is only when N is a prime number that we have been able to show that there is a unistochastic ball surrounding the van der Waerden matrix.


320

Acknowledgements. We thank Göran Björck, Prot Pakoński, Wojciech Słomczyński, and Gregor Tanner for discussions, Petre D˘i¸ta for email correspondence, Uffe Haagerup for supplying us with a copy of Petrescu’s thesis, and an anonymous referee for comments. Financial support from the Swedish Research CouncilVR, and from the Polish Ministry of Scientific Research under grant No PBZ-MIN-008/P03/2003, is gratefully acknowledged.

Appendix A: Notation For N = 4 we have defined the 24 permutation P0 , . . . , P23 matrices in a lexicographical order. They can be regarded as the corners of 6 regular tetrahedra, that can be written in the form   p0 p7 p16 p23 p p p p  T1 = p0 P0 + p7 P7 + p16 P16 + p23 P23 =  7 0 23 16  , p16 p23 p0 p7 p23 p16 p7 p0 

T2 = p1 P1 + p6 P6 + p17 P17 + p22 P22

p1  p6 = p22 p17

p6 p1 p17 p22



T3 = p2 P2 + p10 P10 + p13 P13 + p21 P21

p2  p13 = p10 p21 

T4 = p3 P3 + p11 P11 + p12 P12 + p20 P20

p3  p12 = p20 p11 

T5 = p4 P4 + p8 P8 + p15 P15 + p19 P19

p4  p19 = p8 p15 

T6 = p5 P5 + p9 P9 + p14 P14 + p18 P18

p5  p18 = p14 p9

p17 p22 p6 p1

 p22 p17  , p1  p6

p10 p21 p2 p13

p13 p2 p21 p10

 p21 p10  , p13  p2

p11 p20 p12 p3

p12 p3 p11 p20

 p20 p11  , p3  p12

p8 p15 p4 p19

p15 p8 p19 p4

 p19 p4  , p15  p8

p9 p14 p18 p5

p14 p9 p5 p18

 p18 p5  . p9  p14

These expressions also implicitly define our numbering convention for the permutation matrices. The nine hyperplanes mentioned in Theorem 2 consist of matrices of the form     B00 B01 • • B00 B01 • • B B • •  • • • • 1 =  10 11 , 2 =  , B20 B21 • •  • • • • • • •• • • ••




B00  • 3 =  • B30 

B00  • 5 =  B20 • 

B00  B10 7 =  • •

• • • •

 • • , • •

B02 • B22 •

 • • , • •

B01 • • B31 • • • •

•• •• •• ••

 B03 B13  , •  • 

B00  • 9 =  • B30



B00  B10 4 =  • • 

B00  • 6 =  • B30 

B00  • 8 =  B20 • •• •• •• ••

321

• • • •

B02 B12 • •

 • • , • •

• • • •

B02 • • B32

 • • , • •

•• •• •• ••

 B03 •  , B23  •

 B03 •  , •  B33

where the matrix elements that are explicitly written are assumed to sum to one (hence this holds also for the remaining three blocks taken separately). The normal vectors of these hyperplanes are the matrices   1 1 −1 −1 1  1 1 −1 −1  n1 =   4 −1 −1 1 1 −1 −1 1 1 and so on. Appendix B: Numerics I. Entropy averaged over B3 . To generate a random bistochastic matrix from the flat measure on B3 ⊂ R4 , we have drawn at random a point (x, y, z, t) in the 4-dimensional hypercube. It determines a minor of a N = 3 matrix B, and the remaining five elements of B may be determined by the unit sum conditions in Eq. (1). Condition i is fulfilled if the sums in both rows and both columns of the minor does not exceed unity, and the sum of all four elements is not smaller than one. If this was the case, the random matrix B was accepted to the ensemble of random bistochastic matrices. If additionally, the chain links condition (21) were satisfied, the matrix was accepted to the ensemble of unistochastic matrices, generated with respect to the flat measure on U3 . The mean entropies, (25), were computed by taking an average over both ensembles consisting of 107 random matrices, respectively. II. Numerical verification, whether a given bistochastic matrix B is unistochastic. We have performed a random walk in the space of unitary matrices. Starting from an arbitrary random initial point U0 we computed B0 = U0 ◦ U0∗ and its distance to the analyzed matrix, D0 = D(B0 , B), as defined in (4). We fixed a small parameter α ≈ 0.1, generated a random Hermitian matrix H from the Gaussian unitary


322

ensemble [30], and found a unitary perturbation V = exp(−iαH ). The matrix Un+1 = V Un was accepted as a next point of the random trajectory if the distance Dn+1 was smaller than the previous one, Dn . If a certain number (say 100) of random matrices V did not allow us to decrease the distance, we reduced the angle α by half, to start a finer search. A single run was stopped if the distance D was smaller than = 10−6 (numerical solution found), or α got smaller than a fixed cut off value (say αmin = 10−4 ). In the latter case, the entire procedure was repeated a hundred times, starting from various unitary random matrices U0 , generated from the Haar measure on U (4) [31]. The smallest distance Dmin and the closest unistochastic matrix Bmin = Un ◦ U¯ n were recorded. To check the accuracy of the algorithm we constructed several random unistochastic matrices, B = U ◦ U ∗ , and verified that the random walk procedure gave their approximations with Dmin < . Appendix C: A System of Equations In order to curtail a plethora of indices in Eqs. (34–36) and ease the subsequent notation, let us introduce shorthand: ϕj = φj 1 , ψj = −φj 2 , j = 1, 2, 3. With that the system reads eiϕ1 + eiϕ2 + eiϕ3 = −L, eiψ1 + eiψ2 + eiψ3 = −L, ei(ϕ1 +ψ1 ) + ei(ϕ2 +ψ2 ) + ei(ϕ3 +ψ3 ) = −L.

(39) (40) (41)

We shall prove the following: Lemma 2. The system of Eqs. (39–41) 1. has no real solutions for L > 1, 2. for 0 < L < 1 has the solution ϕ1 = 0, ϕ2 = φ, ϕ3 = −φ , ψ1 = −φ, ψ2 = 0, ψ3 = φ

cos φ =

L+1 t −1 =− , t +1 2

(42)

unique up to obvious permutations, 3. has continuous families of solutions for L = 0, 1. Indeed, each of the unimodal numbers eiϕk , k = 1, 2, 3 is a root of: P (λ) = (λ − eiϕ1 )(λ − eiϕ2 )(λ − eiϕ3 ) = λ3 − (eiϕ1 + eiϕ2 + eiϕ3 )λ2 + (ei(ϕ1 +ϕ2 ) + ei(ϕ1 +ϕ3 ) + ei(ϕ2 +ϕ3 ) )λ −e(iϕ1 +ϕ2 +ϕ3 ) (43) = λ3 − (eiϕ1 + eiϕ2 + eiϕ3 )λ2 + (e−iϕ3 + e−iϕ2 + e−iϕ1 )e(iϕ1 +ϕ2 +ϕ3 ) λ −e(iϕ1 +ϕ2 +ϕ3 ) = λ3 + λ2 L − λLei − ei = λ2 (λ + L) − (1 + λL)ei , where = ϕ1 + ϕ2 + ϕ3 , and we used (39) and the reality of L. Thus each λ = eiϕk , (k = 1, 2, 3), fulfills: λ2 (λ + L) = (1 + λL)ei .

(44)


323

Analogously, µ = eiψk , (k = 1, 2, 3), fulfills µ2 (µ + L) = (1 + µL)ei ,

(45)

with = ψ1 + ψ2 + ψ3 . Observe now, that if λ = eiϕk and µ = eiψk are solutions of (39–41) with the same number k (k = 1, 2, 3) then, upon the same reasoning applied to (41), λµ fulfills λ2 µ2 (λµ + L) = (1 + λµL)ei(+ ) .

(46)

Multiplying (44) by (45) and finally by (46) after exchanging its sides, we obtain, after division by λ2 µ2 ei(+ ) = 0, (L + λ)(L + µ)(Lλµ + 1) = (Lλ + 1)(Lµ + 1)(L + λµ),

(47)

which, upon substitution λ = eiϕk , µ = eiψk and putting everything on one side factorizes to L(L − 1)(eiϕk − 1)(eiψk − 1)(ei(ϕk +ψk ) − 1) = 0,

(48)

(any computer symbolic manipulation program can be helpful in revealing (48) from (47)). Hence, if L = 0, 1, then for each pair (ϕk , ψk ), k = 1, 2, 3, either: a) one of the angles is zero or b) they are opposite. The latter case can not occur for all three pairs since then ei(ϕ1 +ψ1 ) + ei(ϕ2 +ψ2 ) + ei(ϕ3 +ψ3 ) = 3 = −L, hence at least one of ϕk or ψk equals zero. Up to unimportant permutations we can assume ϕ3 = 0, but then, since eiϕ1 + eiϕ2 + eiϕ3 = −L ∈ R, we immediately get ϕ1 = −ϕ2 . This determines also all other angles (also up to some unimportant permutation), and we end up with the solution announced in point 2 above as the only possibility, but such a solution exists only if L ≤ 1. To prove point 3, observe that 1. for L = 0, ϕ1 = ϕ, ψ1 = ψ,

ϕ2 = ϕ + 2π/3, ϕ3 = ϕ + 4π/3, ψ2 = ψ + 2π/3, ψ3 = ψ + 4π/3,

(49) (50)

is a legitimate solution of (39–41) for arbitrary ϕ and ψ, 2. for L = 1, ϕ1 = ϕ, ϕ2 = π, ϕ3 = ϕ + π, ψ1 = −ϕ + π, ψ2 = π, ψ3 = −ϕ, is a solution for an arbitrary ϕ.

(51) (52)

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References 1. Marshall, A.W., Olkin, I.: Inequalities: Theory of Majorization and its Applications. New York: Academic Press 1979 2. Landé, A.: From Dualism to Unity in Quantum Physics. Cambridge: Cambridge U. P. 1960 3. Rovelli, C.: Relational Quantum Mechanics. Int. J. of Theor. Phys. 35, 1637 (1996) 4. Khrennikov, A.: Linear representations of probabilistic transformations induced by context transition. J. Phys. A34, 9965 (2001) 5. Tanner, G.: Unitary-stochastic matrix ensembles, and spectral statistics. J. Phys. A34, 8485 (2001) ˙ 6. Pakoński, P., Zyczkowski, K., Ku´s, M.: Classical 1D maps, quantum graphs and ensembles of unitary matrices. J. Phys. A34, 9303 (2001) ˙ 7. Pakoński, P., Tanner, G., Zyczkowski, K.: Families of Line-Graphs and Their Quantization. J. Stat. Phys. 111, 1331 (2003) ˙ 8. Zyczkowski, K., Ku´s, M., Słomczyński, W., Sommers, H.-J.: Random unistochastic matrices. J. Phys. A36, 3425 (2003) 9. Jarlskog, C., Stora, R.: Unitarity polygons and CP violation areas and phases in the standard electroweak model. Phys. Lett. B208, 268 (1988) 10. Auberson, G., Martin,A., Mennessier, G.: On the Reconstruction of a Unitary Matrix from its Moduli. Commun. Math. Phys. 140, 523 (1991) 11. Mennessier, G., Nyuts, J.: Some unitarity bounds for finite matrices. J. Math. Phys. 15, 1525 (1974) 12. Sylvester, J.J.: Thoughts on inverse orthogonal matrices, simultaneous sign-successions, and tessellated pavements in two or more colours, with applications to Newton’s rule, ornamental tile-work, and the theory of numbers. Phil. Mag. 34, 461 (1867) 13. Petrescu, M.: Existence of continuous families of complex Hadamard matrices of certain prime dimensions and related results. UCLA thesis, Los Angeles 1997 14. D˘i¸ta, P.: Some results on the parametrization of complex Hadamard matrices. J. Phys. A37, 5355 (2004) ˙ 15. Zeilinger, A., Zukowski, M., Horne, M.A., Bernstein, H.J., Greenberger, D.M.: Einstein-PodolskyRosen correlations in higher dimensions. In: J. Anandan, J. L. Safko, eds. Fundamental Aspects of Quantum Theory, Singapore: World Scientific, 1993 16. Törmä, P., Stenholm, S., Jex, I.: Hamiltonian theory of symmetric optical network transforms. Phys. Rev. A52, 4853 (1995) 17. Werner, R.F.: All Teleportation and Dense Coding Schemes. J. Phys. A34, 7081 (2001) 18. Birkhoff, G.: Tres observaciones sobre el algebra lineal. Univ. Nac. Tucumán Rev. A5, 147 (1946) 19. Brualdi, R.A., Gibson, P.M.: Convex Polyhedra of Doubly Stochastic Matrices. I. Applications of the Permanent Function. J. Comb. Theory A22, 194 (1977) 20. Słomczyński, W.: Subadditivity of Entropy for Stochastic Matrices. Open Sys. Inf. Dyn. 9, 201 (2002) 21. Beck, M., Pixton, D.: The volume of the 10th Birkhoff polytope. arXiv: math.CO/0305322 22. Au-Yeung, Y.-H., Poon, Y.-T.: 3 × 3 Orthostochastic Matrices and the Convexity of Generalized Numerical Ranges. Lin. Alg. Appl. 27, 69 (1979) 23. Nakazato, H.: Sets of 3 × 3 Orthostochastic Matrices. Nikonkai Math. J. 7, 83 (1996) 24. Gadiyar, H.G., Maini, K.M.S., Padma, R., Sharatchandra, H.S.: Entropy and Hadamard matrices. J. Phys. A36, L109 (2003) 25. Jones, K.R.W.: Entropy of random quantum states. J. Phys. A23, L1247 (1990) 26. Au-Yeung, Y.-H., Cheng, C.-M.: Permutation Matrices Whose Convex Combinations Are Orthostochastic. Lin. Alg. Appl. 150, 243 (1991) 27. Hadamard, M.J.: Résolutions d’une question relative aux déterminants. Bull. Sci. Math. 17, 240 (1893) 28. Haagerup, U.: Orthogonal maximal Abelian ∗-subalgebras of the n × n matrices and cyclic n-roots. In: Operator Algebras and Quantum Field Theory, Rome (1996), Cambridge, MA: Internat. Press, 1997 29. Tadej, W.: Unpublished 30. Mehta, M.L.: Random Matrices. II ed., New York: Academic, 1991 ˙ 31. Po´zniak, M., Zyczkowski, K., Ku´s, M.: Composed ensembles of random unitary matrices. J. Phys. A31, 1059 (1998) Communicated by M.B. Ruskai


Communications in


A New Quantum Deformation of ‘ax + b’ Group W. Pusz, S. L. Woronowicz Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, Ho˙za 74, 00-682 Warszawa, Poland. E-mail: [email protected]; [email protected] Received: 19 April 2004 / Accepted: 5 April 2005 Published online: 15 July 2005 – © Springer-Verlag 2005

Abstract: The paper is devoted to locally compact quantum groups that are related to the classical ‘ax + b’ group. We discuss in detail the quantization of the deformation parameter assumed with no justification in the previous paper. Next we construct (on the C∗ -level) a larger family of quantum deformations of the ‘ax + b’ group corresponding to the deformation parameter q 2 running over an interval in the unit circle. To this end, beside the reflection operator β known from the previous paper we use a new unitary generator w. It commutes with a, b and βwβ = s sgn b w, where s ∈ S 1 is a new deformation parameter related to q 2 . At the end we discuss the groups at roots of unity.

0. Introduction In the last years a lot of effort was devoted to constructing explicit examples of (noncompact) locally compact quantum groups. The present paper inscribes into this line of research. It is devoted to quantum deformations of the group ‘ax + b’ of affine transformations of the real line. Such quantum ‘ax + b’ groups were presented first in [19]. We shall use the adjective old to distinguish them from the new ‘ax + b’ groups constructed in Sect. 4 of present paper. We go back to the subject for the following reasons. At first the quantizations of the deformation parameter introduced in the previous paper were not discussed in detail. π Now we give strong arguments that the values of = 2k+3 are the only ones allowed within the setting considered in [19]. Secondly one of the important formulae in [19] was not proven. We fill this gap. Third, the old quantum ‘ax + b’ admits a large set of automorphisms. In [19] we identified only four of them. Now we show that the group of automorphisms is as large as S 1 . These automorphisms play an important role in constructing the new quantum ‘ax + b’ groups. The new groups do exist for running over an interval in R (no more quantization of deformation parameter).

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W. Pusz, S.L. Woronowicz

The new quantum groups constructed in the present paper seem to be very important. They will serve as building blocks in the construction of the quantum SL(2, R) group. This is our next target. Let G be the ‘ax + b’ group. On the classical level G consists of all transformations of the form R x −→ ax + b ∈ R,

(0.1)

where a and b are real parameters labeling the elements of the group. We shall assume that a > 0. Assigning to each element of the group the values of the parameters we define two unbounded continuous real functions on G. To denote the functions we shall use the same letters: a, b ∈ C(G). Then the C∗ -algebra C∞ (G) of all continuous functions vanishing at infinity on G is generated by log a and b: uniformly closed linear envelope C∞ (G) = f (log a)g(b) : f, g ∈ C∞ (R) . Functions a and b may be considered as elements affiliated with C∞ (G). Composing two transformations of the form (0.1) with parameters (a1 , b1 ) and (a2 , b2 ) one obtains the transformation with parameters (a1 a2 , a1 b2 + b1 ). This result leads to the following formulae describing the comultiplication: (a) = a ⊗ a, (b) = a ⊗ b + b ⊗ I.

(0.2)

At the moment the elements of G are considered as affine transformations of R. However one may realize them as unitary operators acting on a Hilbert space. To this end, to any transformation of the form (0.1) we assign the unitary operator V(a,b) ∈ B(L2 (R)) introduced by the formula: V(a,b) f (x) = a −1/2 f a −1 (x − b) for any f ∈ L2 (R). Then G may be identified with the set of unitary operators: G = V(a,b) : a, b ∈ R; a > 0 .

(0.3)

This identification preserves the group structure and the topology. More precisely V(1,0) = I and V(a1 ,b1 ) V(a2 ,b2 ) = V(a1 a2 ,a1 b2 +b1 )

(0.4)

for any a1 , a2 ∈ ]0, ∞[ and b1 , b2 ∈ R. Moreover a sequence V(an ,bn ) converges to V(a∞ ,b∞ ) in strong topology if and only if an → a∞ > 0 and bn → b∞ . In particular (0.3) with the strong operator topology is a locally compact space. One can also show that (0.3) is a closed subset of B(L2 (R)) (in strong operator topology). For any Hilbert space H we denote by K(H ) the C∗ -algebra of all compact operators acting on H . According to the general theory [13] the strongly continuous family of unitaries (0.3) is described by a single unitary V ∈ M(K(L2 (R)) ⊗ C∞ (G)). The C∗ -algebra C∞ (G) is generated (in the sense of [13]) by V . Formula (0.4) means that (id ⊗ )V = V12 V13 .

A New Quantum Deformation of ‘ax + b’ Group

327

This way we arrive at the notion of a (quantum) group of unitary operators. Let H be a Hilbert space. We shall consider pairs (A, V ), where A is a C∗ -algebra and V is a unitary element of the multiplier algebra M(K(H ) ⊗ A). If A is generated by V ∈ M(K(H ) ⊗ A) then (A, V ) is called a quantum family of unitary operators. We say that the family is closed with respect to operator multiplication if there exists a morphism ∈ Mor(A, A ⊗ A) such that (id ⊗ )V = V12 V13 .

(0.5)

Then is unique (because A is generated by V ). Finally (A, V ) is said to be a quantum group of unitary operators, if it is closed with respect to the operator multiplication and if (A, ) is a locally quantum group in the sense of Kustermans and Vaes [3]. It should be possible to formulate the last condition directly in terms of (A, V ). However this is not the subject of the present paper. Let us go back to the ‘ax + b’ group. In the quantum setting functions a and b are replaced by selfadjoint elements a = a ∗ > 0 and b = b∗ that no longer commute. Instead they satisfy the relation ab = q 2 ba,

(0.6)

where the deformation parameter q 2 is a number of modulus 1. Unfortunately in our case elements a and b are represented by unbounded operators and the products ab and ba may not be well defined because of the domain problem. For this reason we replace (0.6) by the so-called Zakrzewski relation. It says that for any τ ∈ R: a iτ ba −iτ = eτ b. In this formula is a real constant such that q 2 = e−i . For technical reasons we shall assume that 0 < < π2 . The reader should notice that for τ = −i the above relation reduces to (0.6). The second problem is related to the comultiplication. We would like to keep formulae (0.2). However in general a ⊗ b + b ⊗ I is not selfadjoint and in the best case we may expect that (b) is a selfadjoint extension of a ⊗ b + b ⊗ I : a ⊗ b + b ⊗ I ⊂ (b). To choose the extension in a well defined way we have to use additional operators independent of a and b. For old quantum ‘ax + b’ groups we use a selfadjoint unitary β commuting with a and anticommuting with b. For new groups the situation is even more complicated. It means that the algebra A is no longer generated by log a and b. It is not obvious how to present the quantum ‘ax + b’ group as a quantum group of unitary operators (A, V ). The crucial point is the formula V = V (a, b, . . . ) expressing V in terms of a, b and perhaps some other elements related to A. Equation (0.5) takes the form V (a ⊗ a, [a ⊗ b + b ⊗ I ], . . . ) = V (a ⊗ I, b ⊗ I, . . . )V (I ⊗ a, I ⊗ b, . . . ), where [a ⊗ b + b ⊗ I ] is a suitable selfadjoint extension of a ⊗ b + b ⊗ I . To find solutions of this equation we spent a lot of time making use of our experience in the area of quantum exponential functions and quantum groups (cf. [15, 14, 11, 18, 17, 19, 7, 9]). As a result we got formulae (3.8) and (4.8) that are starting points in our presentation.

328


Let us briefly discuss the content of the paper. Sections 1 and 2 are devoted to mathematical tools used in the paper. In the first one we recall the Zakrzewski commutation relation and related quantum exponential function (with a slightly modified notation). Most of the results presented in that section come from [17]; the essentially new result is contained in Proposition 1.4. The second section deals with the notion of a C∗ -algebra generated by affiliated elements. We prove a number of results used in the main part of the paper. Section 3 is devoted to the quantum ‘ax + b’ groups introduced in [19]. These groups π exist only for special values of deformation parameter q 2 = e−i with = 2k+3 , where k = 0, 1, 2, . . . . This fact was not really shown in [19]. The special values of the deformation parameter were chosen to proceed with some computations. It was not clear that (at the expense of some complications) one is not able to construct the quantum ‘ax + b’ group for a larger set of values of the deformation parameter. Now, presenting the ‘ax + b’ group as a quantum group of unitary operators we obtain the quantization of the deformation parameter as a precise mathematical statement (cf. Theorem 3.3). More precisely for q 2 = e−i we shall construct a C∗ -algebra A with distinguished selfadjoint elements a, b and iβb affiliated with it (the so called reflection operator β is a unitary involution which is not affiliated with A). These elements satisfy (in a well defined sense) the relations ab = q 2 ba, aβ = βa and bβ = −βb. The algebra A is generated by a unitary element V ∈ M(K(L2 (R))⊗A). The pair (A, V ) is defined for all 0 < < π/2. However, the existence of satisfying the condition (0.5) selects a much smaller subset of admissible ’s. We shall prove that exists if and only if is of the form indicated above. Next we derive formulae showing how acts on generators of A. In particular we prove an elegant formula describing the action of the comultiplication on the reflection operator. This formula appeared (with no proof) in the previous paper (cf. formula (4.16) of [19]). At the end of Sect. 3 we find an interesting action of S 1 on the algebra A: for any s ∈ S 1 we have an automorphism φs of A and φss = φs oφs . If π exists (i.e. if = 2k+3 , where k = 0, 1, 2, . . . ) then oφs = (φs ⊗ φs )o. New quantum groups related to the classical ‘ax +b’ group are constructed in Sect. 4. Using one of the automorphism φs described at the end of Sect. 3 we consider the corresponding crossed product. In other words we extend the algebra A by adding a new unitary generator w implementing φs . This enlargement of the algebra opens new possibilities. In particular we obtain new admissible values of the deformation parameter. Now = πp , where p is a number larger than 2 such that −eiπp = s. The latter relation distinguishes a discrete set of possible p. However the quantization of disappears because changing s we may cover the whole interval ∈]0, π2 [. For s = 1 we obtain p = 2k + 3, where k = 0, 1, 2, . . . . In this case the new quantum ‘ax + b’ group reduces to a semidirect product of the old one by S 1 . For s = 1 we get essentially new examples of locally compact quantum groups. In the next section (Sect. 5) we investigate the multiplicative unitaries for the quantum groups constructed in Sect. 4. We prove their modularity and find the unitary antipode and scaling group. In particular the objects constructed in Sect. 4 satisfy all the axioms of Kustermans and Vaes [3] and the ones of Masuda, Nakagami and Woronowicz [4]. At the end of the section we briefly discuss the duals of the new quantum ‘ax + b’ groups. The last section is devoted to new quantum ‘ax + b’ groups with q 2 being a root of unity. In this case we may pass to groups with smaller size. To this end we have to assume that the unitary generator w satisfies the additional relation of the form w N = I . The word size used in the previous paragraph has a precise meaning. It is based on the Stone - von Neumann theorem. Let (A, ) be one of the quantum ‘ax + b’ group


329

considered in this paper and π be a representation of A acting on a Hilbert space Hπ such that ker(b) = {0}. Then operators log(π(a)) and log(|π(b)|) satisfy the same commutation relations as position xQM and momentum pQM in quantum mechanics. By the Stone - von Neumann theorem the pair (log(π(a)), log(|π(b)|) is unitarily equivalent to a direct sum of k copies of (xQM , pQM ). The number k will be called the multiplicity of π . We say that the size of the group (A, ) is equal to k if there exists a faithful representation of A with multiplicity k and if k is the smallest number with this property. In the classical situation the algebra of functions A is generated by log a and b. This is not the case when we consider quantum ‘ax + b’ groups: we use additional generators such as β and w. It means that together with the quantum deformation we pass to a sort of extension of the group. The size tells us how large the extension is. The old quantum ‘ax + b’ groups are of size 2. This is the minimal value. One can show that the old quantum ‘ax + b’ groups are the only ones with size 2. The new groups introduced in Sect. 4 are of infinite size. On the other hand the groups at roots of unity considered in the last section are of size 2N, where N is the number appearing in the relation w N = I . Our approach extensively uses the C∗ -algebra language and the theory of selfadjoint operators on Hilbert space. For the basic facts concerning the general C∗ -algebra theory we refer to [1, 6]. The notation used in the paper follows the one explained in [13, 12]. In particular M(A) is the multiplier algebra of a C∗ -algebra A. The affiliation relation in the sense of C∗ -algebra theory is denoted by “η” and Aη is the set of all affiliated elements (“unbounded multipliers”). It is known that M(A) ⊂ Aη . A morphism from A to a C∗ -algebra B is by definition any ∗ -homomorphism π : A −→ M(B) such that π(A)B is dense in B. Let us recall that any such π has the unique extension to a unital ∗-homomorphism π : M(A) −→ M(B) and to a ∗ -preserving map π : Aη −→ B η respectively (both denoted by the same symbol). The set of all morphisms from A to B is denoted by Mor(A, B). With some abuse of notation, the symbol Rep(A) will stay for the “set” of all nondegenerate representations of a C∗ -algebra A. For any π ∈ Rep(A), we denote by Hπ the carrier Hilbert space of π. Then π ∈ Mor(A, K(Hπ )). In the paper we mostly deal with concrete C∗ -algebras. By definition they are norm closed ∗ -subalgebras of the algebra B(H ) of all bounded operators acting on some (separable) Hilbert space H . As a rule, C∗ -algebras we deal with are separable. Non separable ones will appear only as multiplier algebras. In particular B(H ) = M(K(H )). We shall denote by C ∗ (H ) the set of all non-degenerate separable C∗ -algebras of operators acting on a Hilbert space H . We recall that an algebra A ⊂ B(H ) is non-degenerate if AH is dense in H . We shall use functional calculus for strongly commuting selfadjoint operators. If T and β are selfadjoint operators acting on a Hilbert space H and T and β strongly commute then T =

⊕

β=

r dE(r, ),

⊕

dE(r, ),

where dE(r, ) is the common spectral measure supported by the joint spectrum ⊂ R2 of (T , β). Moreover for any measurable complex valued function on we have f (T , β) =

⊕

f (r, )dE(r, ).

330


In this context the characteristic function χ will appear quite often. By definition for any sentence R, we have

0 if R is false, χ (R) = 1 if R is true. Typically R is a formula involving an (in)equality sign. For example χ (r ≤ 0) is equal to 0 for positive r and 1 for r = 0 or negative. Consequently χ (T ≤ 0) is the spectral projection assigned to the negative part of the spectrum of a selfadjoint operator T . The corresponding spectral subspace will be denoted by H (T ≤ 0): H (T ≤ 0) = χ (T ≤ 0)H . Similarly χ(T = λ) is the orthogonal projection on the eigenspace H (T = λ) of T corresponding to the eigenvalue λ ∈ R. We refer to [17] for a more detailed explanation of this notation. Let Np(r) = rχ (r < 0). Then for any selfadjoint T , Np(T ) = T χ (T < 0)

(0.7)

is a selfadjoint operator acting on H . This is the negative part of the operator T . Another function frequently used in the paper is the one that returns the sign of the argument: sgn r = χ (r > 0) − χ (r < 0). Then sgn T is the partial isometry that appears in the polar decomposition: T =(sgn T ) |T |. 1. A Special Function and Selfadjoint Extensions In this section we recall (in a slightly modified version with a certain loss of generality) the basic definitions and statements of [17]. The only essentially new result is contained in formula (1.11). Later on it will help us to prove the formula announced in [19, formula (4.16)]. We start with a modified version of the quantum exponential function introduced in [17]. Let ∈ R and 0 < < π2 . Instead of function F defined on the set R− × {−1, 1} ∪ R+ × {0} we shall use function G defined on R × {−1, 1}. It is related to the function F by the formula G (r, ) = F (r, χ (r < 0))

(1.1)

for any r ∈ R and = ±1. Taking into account the definition [17, formula (1.19)] we obtain  Vθ (log r) for r > 0   G (r, ) = (1.2) π   1 + i|r| Vθ log |r| − π i for r < 0, where θ = 2π and Vθ is the meromorphic function on C such that

∞ 1 dt log(1 + t −θ ) Vθ (x) = exp 2πi 0 t + e−x for all x ∈ C such that |x| < π . In addition G (0, ±1) = 1. Then G (r, ) is a continuous function on R × {−1, 1} and G (r, ) = G (r, ) ⇐⇒ χ (r < 0) = χ (r < 0) . (1.3)


331

The asymptotic behavior of G (r, ) for large r is described by the formula

(log |r|)2 G (r, ) ≈ C exp , 2i

(1.4)

where C is a phase factor depending only on sgn r and ρ and ‘≈’means that the difference goes to 0 when r → ±∞ (see Statements 9 and 10 of [17, Theorem 1.1]). It is known that the quantum exponential function assumes values of modulus 1. Therefore if T and β are operators acting on a Hilbert space H , T is selfadjoint and β is unitary selfadjoint commuting with T , then G (T , β) is unitary. Now we recall the concept of selfadjoint extension of a symmetric operator defined by a reflection operator. Let Q be a symmetric operator acting on a Hilbert space H and ρ be a unitary selfadjoint operator (ρ ∗ = ρ, ρ 2 = I ) anticommuting with Q. Then we denote by [Q]ρ the restriction of Q∗ to the domain {x ∈ D(Q∗ ) : (ρ − I )x ∈ D(Q)}. It is known (cf. [17, Prop. 5.1]) that [Q]ρ is a selfadjoint extension of Q. We shall use the following simple Proposition 1.1. Let Q, X and ρ be operators acting on a Hilbert space H such that Q is symmetric, X is selfadjoint, ρ is unitary selfadjoint, ρQ = −Qρ and ρX = −Xρ. Assume that the restrictions of Q and X to H (ρ = −1) coincide: Q|H (ρ=−1) = X|H (ρ=−1) .

(1.5)

Then X = [Q]ρ . Proof. Let H1 = H (ρ = −1) and H2 = H (ρ = 1). Then H = H1 ⊕ H2 and (all) bounded and (some) unbounded operators may be represented by 2 × 2 matrices. In particular −I , 0 ρ= . 0 ,I Remembering that Q and X anticommute with ρ we obtain: 0 , X− 0 , Q− and X = , Q= Q+ , 0 X+ , 0 where Q+ and X+ are operators acting from H1 to H2 and Q− and X− are operators ∗ (X is acting from H2 to H1 . Clearly Q+ ⊂ Q∗− (Q is symmetric) and X− = X+ selfadjoint). Assumption (1.5) means that Q+ = X+ . Therefore 0 , Q∗+ . X= Q+ , 0 On the other hand Q∗ =

0 , Q∗+ Q∗− , 0

.

It shows that X ⊂ Q∗ and D(X) = {x ∈ D(Q∗ ) : (ρ − I )x ∈ D(Q)}.

332


Let ∈ R. We shall use the Zakrzewski relation o (cf. [17]). Let R and S be selfadjoint operators acting on a Hilbert space H with the polar decompositions R = sgn R |R| and S = sgn S |S|. For simplicity we shall assume that one of the operators R and S has trivial kernel. If ker S = {0}, then sgn S is unitary selfadjoint and   sgn S commutes with R   R o S ⇐⇒  and |S|−iλ R |S|iλ = eλ R  . for any λ ∈ R. If ker R = {0}, then sgn R is unitary selfadjoint and   sgn R commutes with S   R o S ⇐⇒  and |R|iλ S |R|−iλ = eλ S  . for any λ ∈ R. If ker R = ker S = {0}, then the two above conditions are equivalent. One can easily show that antiunitary operators reverse the direction of the Zakrzewski relation: R o S and J is an (1.6) ⇒ J SJ o J RJ . antiunitary involution

Let R and S be selfadjoint operators with trivial kernels and R o S. It is known [17, Example 3.1] that in this case, the operators ei /2 S −1 R and ei /2 SR −1 are selfadjoint and sgn ei /2 S −1 R = sgn ei /2 SR −1 = (sgn R)(sgn S). We shall use the following result (cf. [17, Theorem 5.2]): Proposition 1.2. Let R, S and τ be operators acting on a Hilbert space H . Assume that

R and S are selfadjoint with trivial kernels, R o S, and that τ is unitary, selfadjoint anticommuting with R and S. We set T = ei /2 S −1 R. Then T is a selfadjoint operator with trivial kernel, T commutes with τ , R + S is a closed symmetric operator and the selfadjoint extension [R + S]τ = G (T , τ )∗ SG (T , τ ) = G (T −1 , τ )RG (T −1 , τ )∗ .

(1.7)

Remark 1.3. If τ is another unitary, selfadjoint operator anticommuting with R and S and if in addition there exists a unitary selfadjoint operator ρ that commutes with τ, τ and S and anticommutes with R then (1.8) [R + S]τ = [R + S]τ ⇒ τ = τ . Indeed if [R + S]τ = [R + S]τ , then (cf. (1.7)) G (T , τ )∗ SG (T , τ ) = G (T , τ )∗ SG (T , τ ).


333

It shows that the unitary operator U = G (T , τ )G (T , τ )∗ commutes with S and hence with |S|. Clearly G (T , τ )∗ = G (T , τ )∗ U.

(1.9)

Moreover T o S due to the Zakrzewski relation R o S and ρ anticommutes with T . As we know τ and τ anticommute with S, hence they commute with |S|. We shall use Proposition 2.4 (see the next section). Setting R1 = R2 = T , ρ1 = τ , ρ2 = τ , U1 = I , U2 = U and replacing S by |S| we have all the assumptions of that proposition satisfied. Therefore (1.9) implies the equality τ Np(T ) = τ Np(T ). It means that τ and τ coincide on H (T < 0). Then τ and τ coincide on ρH (T < 0) for any operator ρ commuting with τ and τ . If ρ commutes with S and anticommutes with R then it anticommutes with T and ρH (T < 0) = H (T > 0). In this case τ and τ coincide on H (T < 0) ⊕ H (T > 0) = H (this is because ker T is trivial). Hence τ = τ . We shall prove a result of the same flavor as (1.7): Proposition 1.4. Let R and S be strictly positive selfadjoint operators acting on a Hil

bert space H such that R o S and let τ , ρ, σ and ξ be unitary selfadjoint operators commuting with R and S. Assume that τ commutes with ξ and anticommutes with ρ and σ and ξ χ (τ = −1) = αρσ χ (τ = −1),

(1.10)

iπ 2 2

where α = i e . We set: T = ei /2 S −1 R. Then T is a positive selfadjoint operator π π with trivial kernel, σ S + ρR is a closed symmetric operator anticommuting with τ and the selfadjoint extension π π π = G (τ T , ξ )∗ σ S G (τ T , ξ ) σ S + ρR −τ (1.11) π −1 −1 ∗ = G (τ T , ξ )ρR G (τ T , ξ ) . Proof. At first we shall prove the first equality of (1.11). Inserting S −1 instead of R and R instead of S in [17, Example 3.1] we see that T is a positive selfadjoint operator with trivial kernel and T ik = e− 2 k S −ik R ik = e i 2

i 2 2 k

R ik S −ik

(1.12)

for any k ∈ R. Denote by X the right-hand side of the first equality in (1.11). We know that G (τ T , ξ ) π is unitary (in what follows we write G (τ T , ξ )−1 instead of G (τ T , ξ )∗ ). Operator S π commutes with σ and τ whereas σ and τ anticommute. Therefore σ S is a selfadjoint operator anticommuting with τ . So is X. π π Let Q = σ S + ρR . Clearly Q is a symmetric operator anticommuting with τ . By virtue of Proposition 1.1 it is sufficient to show that Q|H (τ =1) = X|H (τ =1) . π

(1.13)

Restricting G (τ T , ξ )∗ σ S G (τ T , ξ ) to H (τ = 1) we may replace the second τ by 1 and the first τ by −1 (this is because σ maps H (τ = 1) onto H (τ = −1)): X|H (τ =1) = G (−T , ξ )−1 σ S G (T , ξ )|H (τ =1) π

334


and using (1.2) we obtain −1 π −1 π X|H (τ =1) = 1 + iξ T Vθ log T − πi σ S Vθ (log T )|H (τ =1) . (1.14) π

Now we shall move σ S to the right end of (1.14). It is known (cf. [17, relation (1.30)]) that the function Vθ (x) has no poles and no zeroes in strip = {x ∈ C : 0 ≤ x ≤ π }. Therefore functions Vθ (x) and Vθ (x)−1 are continuous on and holomorphic inside . Moreover (cf. [17, the asymptotic formula (1.37)]), Vθ (x) −→ 1 when x −→ −∞ whereas x stays bounded and using formula (1.32) of [17] one can easily show that for 2 2 any λ > 0, functions e−λx Vθ (x) and e−λx Vθ (x)−1 are bounded on . Furthermore T is a strictly positive selfadjoint operator and T Statement (3) of Theorem 3.1 of [17] we obtain π

o

S. Therefore T

π

π

o

S and using

π

S Vθ (log T ) = Vθ (log T + iπ ) S . Inserting this formula into (1.14) and using in the second step formula (1.28) of [17] we get: π −1 π X|H (τ =1) = 1 + iξ T Vθ (log T − πi)−1 Vθ (log T + π i)σ S |H (τ =1) π −1 2π π = 1 + iξ T 1 + T σ S |H (τ =1) π π = 1 − iξ T S σ |H (τ =1) . On the other hand multiplying both sides of (1.10) by σ from the right we obtain ξ σ χ(τ = 1) = αρχ (τ = 1). Therefore ρ|H (τ =1) = αξ σ |H (τ =1) and π π Q|H (τ =1) = S + αξ R σ |H (τ =1) . To end this part of the proof it is sufficient to show that π π π π S + αξ R = 1 − iξ T S .

(1.15)

We shall use (1.12). It shows that for any x, y ∈ H and any k ∈ R we have i 2 y S ik x − i e 2 k y ξ R ik x = I + iξ T −ik y S ik x . π π π Let x ∈ D S ∩ D R . If y ∈ D(T ) then both sides of the above formula have continuous holomorphic continuation to the strip − π ≤ k ≤ 0. Inserting k = −i π we obtain π π π π y S x + α y ξ R x = I + iξ T y S x . π π π This formula holds for any y in the domain of I +iξ T . Therefore S x ∈ D I − iξ T π π π π and S x + αξ R x = I − iξ T S x. This way we showed that π π π π S + αξ R ⊂ I − iξ T S .

(1.16)


335

π π To prove the converse inclusion we use again (1.12). Let x ∈ D S and S x ∈ π D T . Then for any y ∈ H and k ∈ R: e− 2 k

i 2

π π R −ik y S −ik x = y T ik S x .

π

If y ∈ D(R ) then both sides of the above formula have continuous holomorphic continuation to the strip − π ≤ k ≤ 0. Inserting k = −i π we obtain π π π iα R y x = y T S x . π

π

This formula holds for any y ∈ D(R ). Therefore x ∈ D(Rπ). This wayweπ showed the π π π π π inclusion D(T S ) ⊂ D(R ). Consequently D I −iξ T S ⊂D S + αξ R . Combining this result with (1.16) we get (1.15) and (1.13). This way the first equality of (1.11) is shown. The second equality may be shown in the same manner. However it is simpler to use the following trick based on (1.6). Let J be an antiunitary involutive operator acting on H and Rn = J SJ, ρn = J σ J,

Sn = J RJ, σn = JρJ,

τn = J τ J, ξn = J ξ J.

The subscript ‘n’ stands for ‘new’. One can easily show that the new operators satisfy all the assumptions of our theorem. In particular ξn χ (τn = −1) = J ξ χ (τ = −1)J = J αρσ χ(τ = −1)J = ασn ρn χ (τn = −1) = (ασn ρn χ (τn = −1))∗ = αρn σn χ(τn = −1). In the present case Tn = ei /2 Sn−1 Rn = J e−i /2 R −1 SJ = J T −1 J and the first equality of (1.11) takes the form:

π

π

σn Sn + ρn Rn

−τn

π

= G (J τ T −1 J, J ξ J )∗ σn Sn G (J τ T −1 J, J ξ J ) π

= J G (τ T −1 , ξ )ρR G (τ T −1 , ξ )∗ J. A moment of reflection shows that the left-hand side of this formula equals π π J and the second equality of (1.11) follows immediately. J σ S + ρR −τ

π

π

To end the proof we have to show that the operator Q = σ S + ρR is closed. π π Operator ξ T is selfadjoint. Therefore operator I − iξ T is invertible with the inverse π −1 ∈ B(H ). Using this fact one can easily show that the composition I − iξ T π π I − iξ T S σ is a closed operator. Restricting this operator to H (τ = 1) we obtain Q|H (τ =1) . Hence Q|H (τ =1) is closed. Remembering that Q anticommute with τ we conclude that Q is closed.

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Remark 1.5. According to (1.10) operator αρσ χ (τ =−1) is selfadjoint. Using this fact and remembering that ρ and σ anticommute with τ one can show that ρσ = α 2τ σρ. Conversely let τ, ρ, σ be unitary selfadjoint operators commuting with R, S and let τ anticommute with ρ and σ . If the above relation is satisfied, then using (1.10) to define ξ on H (τ = −1) and extending it in an arbitrary way to a unitary selfadjoint operator defined on the whole space we obtain the quadruple (ρ, σ, τ, ξ ) of operators satisfying the assumptions of Proposition 1.4. We end this section with the reformulation of Theorem 6.1 of [17]. Theorem 1.6. Let (R, S) be a pair of selfadjoint operators acting on a Hilbert space H

such that ker R = ker S = {0} and R o S and let ρ, σ be unitary selfadjoint operators on H . Assume that ρ commutes with R, ρ anticommutes with S, σ commutes with S and σ anticommutes with R. We set: T = ei /2 S −1 R, τ = αρσ χ (S < 0) + ασρχ (S > 0), iπ 2

where α = i e 2 . Then

1. T is selfadjoint, sgn T = (sgn R) (sgn S), T o R and T o S. 2. τ is unitary selfadjoint, τ commutes with T and τ anticommutes with R and S. 3. G satisfies the following exponential function equality: G (R, ρ)G (S, σ ) = G (T , τ )∗ G (S, σ )G (T , τ ) = G ([R + S]τ , σ),

(1.17)

where [R + S]τ is the selfadjoint extension of R + S corresponding to the reflection operator τ and σ = G (T , τ )∗ σ G (T , τ ). Proof. By direct computation one can easily show that τ 2 = I , τ ∗ = τ and τ χ (T < 0) = αρχ (R < 0)σ χ (S < 0) + ασ χ (S < 0)ρχ (R < 0). Now, our theorem follows immediately from [17, Theorem 6.1].

Remark 1.7. In Theorem 1.6, operator τ may be replaced by τ = αρσ χ (R > 0) + ασρχ (R < 0). Operator σ is not affected by this change. Indeed, using the formula sgn T = sgn R sgn S, one can verify that τ χ (T < 0) = τ χ(T < 0). It shows that G (T , τ ) = G (T , τ ).


337

2. The Special Functions and Affiliation Relation In this section we shall use the concept of a C∗ -algebra generated by a set of affiliated elements [13, Def. 4.1, p. 501]. Let C, A be C∗ -algebras and V be an element affiliated with C ⊗ A. We say that A is generated by an element V η (C ⊗ A) if and only if for any π ∈ Rep(A) and any B ∈ C ∗ (Hπ ) we have: (id⊗π)V η (C⊗B) ⇒ π ∈ Mor(A, B) . (2.1) In general the above condition is not easy to verify. We shall use the following criterion (cf. [13, Example 10, p. 507]): Proposition 2.1. Let C, A be C∗ -algebras and V be a unitary element of M(C ⊗ A). Assume that there exists a faithful representation φ of C such that: 1. For any φ-normal linear functional ω on C we have (ω ⊗ id)V ∈ A. 2. The smallest ∗ -subalgebra of A containing {(ω ⊗ id)V : ω is φ-normal} is dense in A. Then A is generated by V ∈ M(C ⊗ A). We recall that a linear functional ω on C is said to be φ-normal if there exists a trace-class operator ρ acting on Hφ such that ω(c) = Tr(ρφ(c)) for all c ∈ C. Let be the locally compact space obtained from R × {−1, 1} by gluing points (r, −1) and (r, 1) for all r ≥ 0. Then:

f (r, −1) = f (r, 1) C∞ () = f ∈ C∞ R × {−1, 1} : . for all r ≥ 0 If R, ρ are operators acting on a Hilbert space H , R is selfadjoint, ρ is unitary selfadjoint and ρ commutes with R, then the mapping C∞ () f −→ π(f ) = f (R, ρ) ∈ B(H )

(2.2)

is a representation of C∞ () acting on H . Operators R and ρ Np(R) are determined by π. Indeed R = π(f1 ) and ρ Np(R) = π(f2 ), where f1 , f2 are elements of C∞ ()η = C() introduced by the formulae f1 (r, ) = r,

f2 (r, ) = Np(r)

(2.3)

for any r ∈ R and = ±1. Using [13, Example 2, p. 497] we see that f1 , f2 generate C∞ (). Therefore for any π ∈ Rep (C∞ ()) and any B ∈ C ∗ (Hπ ) we have: π(f1 ), π(f2 ) η B ⇒ π ∈ Mor(C∞ (), B) ⇒ π(f ) η B for any f ∈ C() . In particular for π introduced by (2.2) we obtain the following result: R, ρ Np(R) η B ⇒ f (R, ρ) η B . f ∈ C()

(2.4)

Our special function G is continuous and satisfies the relation G (r, −1) = G (r, 1) for all r ≥ 0. In other words G ∈ C(). For any r ∈ R, = ±1 and t > 0 we set: F (t; r, ) = G (r, )G (tr, ).

(2.5)

Let R+ = {t ∈ R : t > 0}. Then F is a continuous function on R+ × with values of modulus 1 and we may treat F as a unitary element of M (C∞ (R+ ) ⊗ C∞ ()). We shall prove the following

338


Proposition 2.2. The C∗ -algebra C∞ () is generated by F ∈ M (C∞ (R+ ) ⊗ C∞ ()). Proof. We shall use Proposition 2.1 with C = C∞ (R+ ), A = C∞ () and V = F . Let φ be the natural representation of C∞ (R+ ) acting on L2 (R+ ). For any g ∈ C∞ (R+ ), φ(g) is the multiplication by g. Then φ is faithful and a linear functional ω on C∞ (R+ ) is φ-normal if and only if it is of the form ω(g) =

R+

g(t)ϕ(t) dt,

where ϕ ∈ L1 (R+ ). Applying ω ⊗ id to F ∈ M (C∞ (R+ ) ⊗ C∞ ()) we obtain an element of M (C∞ ()), i.e. a bounded continuous function on . Clearly for any r ∈ R and = ±1 we have (ω ⊗ id)F (r, ) =

F (t; r, )ϕ(t) dt G (tr, )ϕ(t) dt. = G (r, ) R+

(2.6)

R+

Taking into account the asymptotic behavior (1.4) and using the Riemann–Lebesgue lemma one can verify that the integral on the right-hand side tends to 0 when r → ±∞. In other words, (ω ⊗ id)F ∈ C∞ (). Using Statement 7 of Theorem 1.1 of [17] one can easily show that lim

t→0+

1 r G (tr, ) − 1 = t 2i sin (/2)

(2.7)

for all r ∈ R and = ±1. Let r, r ∈ R and , = ±1. Assume for the moment that (ω ⊗ id)F (r, ) = (ω ⊗ id)F (r , ) for all φ-normal functionals ω. Then G (r, )G (tr, ) = G (r , ) G (tr , ) for all t > 0. Going to the limit when t → +0 we get G (r, ) = G (r , ). Comparing this formula with the previous one we see that G (tr, ) = G (tr , ) for all t > 0. Formula (2.7) shows now that r = r and by (1.3) χ (r < 0) = χ (r < 0). This way we have shown that the functions (2.6) separate points of . Now, using the Stone - Weierstrass theorem (applied to the one point compactification of ) we conclude that the smallest ∗ -algebra containing all functions (2.6) is dense in C∞ (). The following proposition will be very useful in proving many technical details important in future considerations. Proposition 2.3. Let R, ρ, U , S be operators acting on a Hilbert space H and C ∈ C ∗ (H ). Assume that: 1. 2. 3. 4.

R is selfadjoint and ρ is unitary selfadjoint commuting with R, U is unitary, S is positive selfadjoint, ker S = {0}, S commutes with ρ and U and R o S, Operators R, ρ Np(R), U and log S are affiliated with C.

Then G (R, ρ) ∈ M(C) and


339

1. For any φ ∈ Rep(C) and any B ∈ C ∗ (Hφ ) we have: φ(R), φ(ρ Np(R)), φ(U ) φ(log S), φ G (R, ρ)∗ U ⇒ . are affiliated with B are affiliated with B 2. For any φ1 , φ2 ∈ Rep(C) such that Hφ1 = Hφ2 we have:   φ1 (R) = φ2 (R), φ1 (S) = φ2 (S),   ⇒  φ1 (ρ Np(R)) = φ2 (ρ Np(R)),  . φ1 G (R, ρ)∗ U = φ2 G (R, ρ)∗ U φ1 (U ) = φ2 (U ) Proof. Relation G (R, ρ) ∈ M(C) follows immediately from (2.4). Ad 1. Let λ ∈ R. Using the commutation relations satisfied by operators R, ρ, U, S we have: S −iλ G (R, ρ)∗ U S iλ = G (tR, ρ)∗ U, where t = eλ > 0. Applying a representation φ of C to both sides of the above relation we get φ(S)−iλ φ G (R, ρ)∗ U φ(S)iλ = φ G (tR, ρ)∗ U . If φ(log S), φ G (R, ρ)∗ U η B, then all factors on the left-hand side of the above equation belong to M(B) and depend continuously on λ (we use strict topology on M(B)). Therefore φ G (tR, ρ)∗ U ∈ M(B) for any t ∈ R+ and the mapping R+ t −→ φ G (tR, ρ)∗ U ∈ M(B) is strictly continuous. Applying the hermitian conjugation and multiplying from the left by φ G (R, ρ)∗ U ∈ M(B) we see that φ G (R, ρ)∗ G (tR, ρ) = φ F (t; R, ρ) ∈ M(B) and the mapping R+ t −→ φ F (t; R, ρ) ∈ M(B) (2.8) is strictly continuous. In the above relations F is the function introduced by (2.5).According to the general theory [13], strictly continuous bounded mappings from R+ into M(B) correspond to elements of M(C∞ (R+ )⊗B). A moment of reflection shows that the mapping (2.8) corresponds to the element (id ⊗ φ oπ)F , where π is the representation of C∞ () introduced by (2.2). This way we have shown that (id ⊗ φ oπ)F ∈ M(C∞ (R+ ) ⊗ B). Using now Proposition 2.2 we conclude that φ oπ ∈ Mor(C∞ (), B). Therefore φ oπ maps continuous functions on into elements affiliated with B. Applying this rule to functions f1 , f2 (cf. (2.3)) and G we obtain: φ(R), φ(ρ Np(R)) η B and φ(G (R, ρ)) ∈ M(B). Comparing the last relation with the assumed one φ G (R, ρ)∗ U ∈ M(B) we see that φ(U ) ∈ M(B). Statement 1 is shown. Ad 2. Let φ = φ1 ⊕ φ2 . Then Hφ = Hφ1 ⊕ Hφ2 and φ(c) = φ1 (c) ⊕ φ2 (c). In our case Hφ1 = Hφ2 . We set: B = m ⊕ m : m ∈ K(Hφ1 ) . Then B ∈ C ∗ (Hφ ). One can easily verify that for any c η C we have: φ(c) η B ⇐⇒ φ1 (c) = φ2 (c) . Now Statement 2 follows immediately from Statement 1.

340


We shall use a slightly different version of Statement 2 of the above proposition. Proposition 2.4. Let R1 , ρ1 , U1 , R2 , ρ2 , U2 , S be operators acting on a Hilbert space H . Assume that for each k = 1, 2 the operators Rk , ρk , Uk , S satisfy Assumptions 1-3 of the previous proposition. Then   R1 = R2 ,   G (R1 , ρ1 )∗ U1 = G (R2 , ρ2 )∗ U2 ⇒  ρ1 Np(R1 ) = ρ2 Np(R2 ),  . (2.9) U1 = U2 . Proof. Let C = K(H ) ⊕ K(H ) and for any m1 , m2 ∈ K(H ) we set φk (m1 ⊕ m2 ) = mk (k = 1, 2). We use Proposition 2.3 with R, ρ, U and S replaced by R1 ⊕ R2 , ρ1 ⊕ ρ2 , U1 ⊕ U2 and S ⊕ S. Now (2.9) follows immediately from Statement 2 of Proposition 2.3. Proposition 2.5. Let X and Y be selfadjoint operators acting on Hilbert spaces K and H respectively. Assume that the spectral measure of X is absolutely continuous with respect to the Lebesgue measure. Then for any A ∈ C ∗ (H ) we have: iX⊗Y e is affiliated ⇒ Y is affiliated with A . with K(K) ⊗ A Proof. For any normal linear functional ω on B(K) and t ∈ R we set fω (t) = ω eitX . Then fω is a continuous function on R. Remembering that the spectral measure of X is absolutely continuous with respect to the Lebesgue measure and using the RiemannLebesgue lemma one can easily show that fω (t) → 0 when t → ±∞. Therefore fω ∈ C∞ (R). Let t, t ∈ R, t = t . Assume for the moment that fω (t) = fω (t ) for all ω. Then itX = eit X and ei(t−t )X = I . It shows that the spectral measure of X is supported e 2π by the set t−t Z, which is in contradiction with the assumption saying that the spectral measure of X is absolutely continuous with respect to the Lebesgue measure. This way we showed that functions fω separate points of R. By the Stone – Weierstrass theorem, the smallest ∗ -subalgebra of C∞ (R) containing all fω is dense in C∞ (R). By the general theory strongly continuous mappings from R into the set of unitary operators acting on K correspond to unitary multipliers of K(K) ⊗ C∞ (R). Let X ∈ M(K(K) ⊗ C∞ (R)) be the unitary corresponding to the mapping R t −→ eitX ∈ B(K). Then for any normal linear functional ω on B(K) we have (ω ⊗ id)X = fω . Using Proposition 2.1 we see that C∞ (R) is generated by X ∈ M(K(K) ⊗ C∞ (R)). For any f ∈ C∞ (R) we set: π(f ) = f (Y ). Then π is a representation of C∞ (R) acting on the Hilbert space Hπ = H . A moment of reflection shows that (id ⊗ π)X = eiX⊗Y . If eiX⊗Y is affiliated with K(K) ⊗ A then π ∈ Mor(C∞ (R), A) and π maps continuous functions on R into elements affiliated with A. Applying this rule to the coordinate function f (t) = t we obtain Y = π(f ) η A.


341

3. Constructions Related to Old Quantum ‘ax + b’ Groups In this section we recall the main results of [19]. The quantum ‘ax +b’ group will be presented as a quantum group of unitary operators. We shall construct a pair (A, V ), where A is a C∗ -algebra and V is a unitary element of M(K(K) ⊗ A), where K is a Hilbert space endowed with a certain structure and K(K) denotes the algebra of all compact operators acting on K. (A, V ) may be treated as a quantum family of unitary operators acting on K ‘labeled by elements’ of quantum space related to the C∗ -algebra A. Our construction will depend on a real parameter . We shall assume that 0 < < π/2. Negative value of leads to the C∗ -algebra anti-isomorphic to that with positive . On the other hand the restriction < π/2 is related to the technical assumption used in the theory of the quantum exponential function [17]. The main result of this section is contained in Theorem 3.2. It states that (A, V ) is a π quantum group if and only if = 2k+3 with k = 0, 1, 2, . . . . To define A we consider three operators a, b and β acting on the Hilbert space L2 (R). Operator a is strictly positive selfadjoint and such that for any τ ∈ R and any x ∈ L2 (R) we have a iτ x (t) = eτ/2 x(eτ t).

In other words a is the analytic generator of the one-parameter group of unitaries corresponding to the homotheties of R. Operator b is the multiplication operator: (bx)(t) = tx(t). By definition domain D(b) consists of all x ∈ L2 (R) such that the right-hand side of the above equation is square integrable. Finally, β is the reflection: for any x ∈ L2 (R) we have: (βx)(t) = x(−t). Clearly β is unitary selfadjoint. One can easily verify that aβ = βa and bβ = −βb. By the last relation ibβ is selfadjoint. Moreover a iτ ba −iτ = eτ b for any τ ∈ R. This relation means that a

o

(3.1)

b.

Theorem 3.1. Let

A=

norm closed

f , f , g ∈ C∞ (R) linear envelope . f1 (b) + βf2 (b) g(log a) : 1 2 f2 (0) = 0

Then: 1. A is a nondegenerate C∗ -algebra of operators acting on L2 (R), 2. log a, b and ibβ are affiliated with A: log a, b, ibβ η A, 3. log a, b and ibβ generate A.

(3.2)

342


Proof. Ad 1. Using the relation bβ = −βb one can easily show that norm closed

f , f ∈ C∞ (R) linear envelope B = f1 (b) + βf2 (b) : 1 2 f2 (0) = 0

(3.3)

is a non-degenerate C∗ -algebra of operators acting on L2 (R). Let C0 (R, B) denote the set of all continuous mappings from R into B with compact support. Then

norm closure it A= f (t)a dt : f ∈ C0 (R, B) . (3.4) R

To prove this formula it is sufficient to notice that for f (t) = f1 (b) + βf2 (b) ϕ(t), where t ∈ R and ϕ ∈ C0 (R) we have f (t)a it dt = f1 (b) + βf2 (b) g(log a),

R

where g(λ) = R ϕ(t)eiλt dt (λ ∈ R) and by the Riemann-Lebesque Lemma, g ∈ C∞ (R). On the other hand (3.1) shows that the unitaries a it (t ∈ R) implement a one parameter group of automorphisms of B. Using now the standard technique of the theory of crossed products (cf. [6, Sect. 7.6]) one can easily show that (3.4) is a non-degenerate C∗ -algebra of operators acting on L2 (R). Statement 1 is proven. Ad 2. We recall (cf. [5, 13]) that a closed operator T is affiliated with a C∗ -algebra A 1 1 if the z-transform zT = T (I + T ∗ T )− 2 ∈ M(A) and if (I + T ∗ T )− 2 A is dense in A. − 1 Inspecting definition (3.2) one can easily show that zlog a = (log a) I + (log a)2 2 − 1 is a right multiplier of A and that A I + (log a)2 2 is dense in A. Passing to adjoint ∗ operators we see that zlog a = zlog a is a left multiplier (hence zlog a ∈ M(A)) and that 1 − I + (log a)2 2 A is dense in A. It shows that log a is affiliated with A. 1

1

For T = b and T = iβb we have zT = b(I + b2 )− 2 and zT = iβb(I + b2 )− 2 1 1 respectively. In both cases (I + T ∗ T )− 2 = (I + b2 )− 2 . Taking into account definition 1 (3.2) one can easily show that (I +T ∗ T )− 2 A is dense in A and that zT is a left multiplier of A. However in both cases zT is selfadjoint. Therefore zT is also a right multiplier and zT ∈ M(A). It shows that b and iβb are affiliated with A. −1 Ad 3. We shall use Theorem 3.3 of [13]. By definition (3.2), (I +b2 )−1 I + (log a)2 ∈ A. To end the proof it is sufficient to show that a, b, iβb separate representations of A. If c ∈ A is of the form c = f1 (b) + βf2 (b) g(log a), (3.5) where f1 , f2 , g ∈ C∞ (R), f2 (0) = 0 and f2 is differentiable at point 0 ∈ R, then f2 (t) = ith(t), where t ∈ R and h ∈ C∞ (R) and (3.6) π(c) = f1 (π(b)) + π(iβb)h(π(b)) g(π(log a)) for any representation π of A. One can easily see that elements of the form (3.5) form a dense subset of A. Formula (3.6) shows now that π is determined uniquely by π(log a), π(b) and π(iβb).


343

Now we pass to the description of the Hilbert space K (cf. the first paragraph of this section). The structure of K is determined by a triple of selfadjoint operators (a, b, β) acting on K and having the following properties:

1. a > 0, ker a = ker b = {0} and a o b, 2. β is a unitary involution, β commutes with a and anticommutes with b. One of the possible choices is: K = L2 (R) and (a, b, β) = (a, b, β). However there is another possibility that is even more interesting: (a, b, β) = (|b|−1 , ei /2 b−1 a, αβ),

(3.7)

where α = ±1. The reader easily verifies that these operators possess the required properties.

The Zakrzewski relation a o b implies that the spectral measures of a and b are absolutely continuous with respect to the Lebesgue measure. Moreover Sp(a) = R+ and Sp(b) = R. The latter fact follows from the relation β b = −bβ. Let V = G (b ⊗ b, β ⊗ β)∗ e log a⊗log a . i

(3.8)

This is the basic object considered in this section. We shall prove Theorem 3.2. 1. V is a unitary operator and V ∈ M(K(K) ⊗ A), 2. A is generated by V ∈ M(K(K) ⊗ A). i

Proof. Let R = b ⊗ b, ρ = β ⊗ β, U = e log a⊗log a , S = a −1 ⊗ I and C = K(K) ⊗ A. Then all the assumptions of Proposition 2.3 are satisfied. Clearly V = G (R, ρ)∗ U ∈ M(C) and Statement 1 is proved. Let π ∈ Rep(A) and B ∈ C ∗ (Hπ ). Then id ⊗ π is a representation of C acting on K ⊗ Hπ . The reader should notice that (id ⊗ π)S = a −1 ⊗ I is affiliated with K(K) ⊗ B. Assume that (id ⊗ π)V ∈ M(K(K) ⊗ B). By Statement 1 of Proposition 2.3, operators: i (id⊗π )R = b⊗π(b), (id⊗π )(ρ Np(R)) and (id⊗π )U = e log a⊗π(log a) are affiliated with K(K) ⊗ B. Using now Proposition A.1 of [19] we see that π(b) is affiliated with B. One can easily verify that β ⊗ I commutes with ρ and anticommutes with R. Therefore ρ Np(R) − (β ⊗ I )ρ Np(R)(β ⊗ I ) = ρ(Np(R) − Np(−R)) = ρR, and applying id ⊗ π to both sides we get (id ⊗ π )(ρ Np(R)) − (β ⊗ I )(id ⊗ π )(ρ Np(R))(β ⊗ I ) = (id ⊗ π )(ρR) = −i β b ⊗ π(iβb). The operators β ⊗ I and (id ⊗ π )(ρ Np(R)) appearing on the left-hand side are affiliated with K(K) ⊗ B. Therefore i β b ⊗ π(iβb) η K(K) ⊗ B and using again Proposition A.1 of [19] we see that π(iβb) is affiliated with B. Moreover, remembering that i e log a⊗π (log a) η K(K) ⊗ B and using Proposition 2.5 we see that π(log a) is affiliated with B. According to Statement 3 of Theorem 3.1, b, iβb and log a generate A. Therefore π ∈ Mor(A, B). We showed that (id ⊗ π)V ∈ M(K(K) ⊗ B) implies π ∈ Mor(A, B). It means that A is generated by V ∈ M(K(K) ⊗ A).

344


Now we are able to formulate the main result of this section: Theorem 3.3.   There exists ∈ Mor(A, A ⊗ A) π   such that , k = 0, 1, 2, . . . .   ⇐⇒ = 2k + 3 (id ⊗ )V = V12 V13 Proof. Let iπ 2

α = ie 2 , T = I ⊗ ei /2 b−1 a ⊗ b, τ = (I ⊗ β ⊗ β) αχ (b ⊗ b ⊗ I < 0) + αχ (b ⊗ b ⊗ I > 0) and W = G (T , τ )∗ e

− i I ⊗log|b|⊗log a

.

(3.9)

(3.10)

Clearly W is a unitary operator acting on K ⊗ L2 (R) ⊗ L2 (R). We shall prove that ∗

V12 V13 = W V12 W .

(3.11)

To make our formulae shorter we set Z = e− log|b|⊗log a .

i

i

U = e log a⊗log a , Using the relations a

o

b, a β = β a and a

o

b one can easily verify that

U (b ⊗ I )U ∗ = b ⊗ a, U (β ⊗ I )U ∗ = β ⊗ I, Z(a ⊗ I )Z ∗ = a ⊗ a.

(3.12) (3.13)

With the above notation V = G (b ⊗ b, β ⊗ β)∗ U and V12 V13 = G (b ⊗ b ⊗ I, β ⊗ β ⊗ I )∗ U12 G (b ⊗ I ⊗ b, β ⊗ I ⊗ β)∗ U13 . Using (3.12) we get U12 G (b ⊗ I ⊗ b, β ⊗ I ⊗ β)∗ = G (b ⊗ a ⊗ b, β ⊗ I ⊗ β)∗ U12 and

∗ V12 V13 = G (b ⊗ a ⊗ b, β ⊗ I ⊗ β) G (b ⊗ b ⊗ I, β ⊗ β ⊗ I ) U12 U13 . (3.14)

Let us consider the first factor in (3.14). We apply Theorem 1.6 with R = b ⊗ a ⊗ b, ρ = β ⊗ I ⊗ β, S = b ⊗ b ⊗ I, σ = β ⊗ β ⊗ I. Then T and τ are given by (3.9) and G (b ⊗ a ⊗ b, β ⊗ I ⊗ β) G (b ⊗ b ⊗ I, β ⊗ β ⊗ I ) = G (T , τ )∗ G (b ⊗ b ⊗ I, β ⊗ β ⊗ I ) G (T , τ ).

(3.15)


345

Now (3.14) takes the form V12 V13 = G (T , τ )∗ G (b ⊗ b ⊗ I, β ⊗ β ⊗ I )∗ G (T , τ ) U12 U13 .

(3.16)

We shall move G (T , τ ) to the end of the right-hand side of this formula. Performing simple computations and using (3.13) we obtain: i

U12 U13 = e log a⊗log(a⊗a) ∗ . = Z23 U12 Z23

It turns out that log a ⊗ log(a ⊗ a) commutes with T , log a ⊗ log(a ⊗ a) commutes with τ.

(3.17) (3.18)

Indeed the Zakrzewski relation a o b implies b−1 o a. Using both relations we see that a ⊗ a commutes with ei /2 b−1 a ⊗ b. Therefore log(a ⊗ a) commutes with ei /2 b−1 a ⊗ b and log a ⊗ log(a ⊗ a) commutes with T = I ⊗ ei /2 b−1 a ⊗ b. Relation (3.17) is shown.

To prove (3.18) we use Zakrzewski relations a o b and a o b. They show that a commutes with sgn b and a commutes with sgn b. Therefore log a⊗log(a⊗a) commutes with sgn(b ⊗ b ⊗ I ) = sgn b ⊗ sgn b ⊗ I and (3.18) follows. Taking into account (3.17) and (3.18) we see that G (T , τ ) commutes with U12 U13 . Now relation (3.16) takes the form: ∗ V12 V13 = G (T , τ )∗ G (b ⊗ b ⊗ I, β ⊗ β ⊗ I )∗ Z23 U12 Z23 G (T , τ ). (3.19)

Finally b ⊗ I and β ⊗ I commute with log |b| ⊗ log a. Therefore G (b ⊗ b ⊗ I, β ⊗ β ⊗ I ) commutes with Z23 . Clearly G (b ⊗ b ⊗ I, β ⊗ β ⊗ I )∗ U12 = V12 and W = G (T , τ )∗ Z23 . Now (3.11) follows immediately from (3.19). By the Zakrzewski relation a iλ ba −iλ = eλ b for all λ ∈ R. Multiplication by a strictly positive number does not change the sign of an operator. Using this fact one can easily show that τ commutes with a iλ ⊗ I ⊗ I . Consequently τ commutes with a ⊗ I ⊗ I . Since T = I ⊗ ei /2 b−1 a ⊗ b and I ⊗ log |b| ⊗ log a obviously commute with a ⊗ I ⊗ I , we conclude that W commutes with a ⊗ I ⊗ I . Now we are ready to prove the main statement. ⇒ . Let ∈ Mor(A, A ⊗ A) and (id ⊗ )V = V12 V13 . We go back to the notation used in the proof of Theorem 3.2. In particular C = K(K) ⊗ A. For any c ∈ C we set: φ1 (c) = (id ⊗ )(c), φ2 (c) = W (c ⊗ I )W ∗ . Then φ1 and φ2 are representations of C acting on the same Hilbert space K ⊗ L2 (R) ⊗ L2 (R). One can easily verify that φ1 (a ⊗ I ) = a ⊗ I ⊗ I = φ2 (a ⊗ I ). Formula (3.11) shows that φ1 (V ) = φ2 (V ). In our notation (cf. the beginning of the proof of Theorem 3.2), a ⊗ I = S and V = G (R, ρ)∗ U , where in particular R = b ⊗ b. Statement 2 of Theorem 2.3 shows now that φ1 (R) = φ2 (R). It means that ∗

b ⊗ (b) = W (b ⊗ b ⊗ I ) W .

346


Taking into account (3.10) and using Proposition 1.2 we get: b ⊗ (b) = G (T , τ )(b ⊗ b ⊗ I )G (T , τ )∗ = b⊗a⊗b+b⊗b⊗I τ .

(3.20)

We recall that

τ = (I ⊗ β ⊗ β) αχ (b ⊗ b ⊗ I < 0) + αχ (b ⊗ b ⊗ I > 0) .

Inspecting the last two formulae we observe that b is the only operator appearing in the first leg position. We know that b is selfadjoint. Therefore replacing in both sides of (3.20) operator b by a real number λ we obtain a formula that should hold for almost all λ ∈ Sp b. For positive λ we get (b) = a ⊗ b + b ⊗ I , (3.21) τ+

where

τ+ = (β ⊗ β) αχ (b ⊗ I < 0) + αχ (b ⊗ I > 0) .

On the other hand for negative λ we have (b) = a ⊗ b + b ⊗ I where

τ−

,

τ− = (β ⊗ β) αχ (b ⊗ I > 0) + αχ (b ⊗ I < 0) .

(3.22)

(3.23)

(3.24)

Clearly the two expressions for (b) must coincide. Let us notice that the operator I ⊗ β commutes with τ+ , τ− and b ⊗ I and anticommutes with a ⊗ b. Therefore τ+ = τ− by iπ 2

Remark 1.3. Comparing (3.22) and (3.24) we get α = α. Remembering that α = i e 2 π and 0 < < π2 we conclude that = 2k+3 (k = 0, 1, 2, . . . ). π ⇐. Assume that = 2k+3 for some k = 0, 1, 2, . . . . Then formula (3.10) essentially simplifies. In this case α = (−1)k , τ = (−1)k (I ⊗ β ⊗ β) and W = W23 = I ⊗ W , where ∗ i (3.25) W = G ei /2 b−1 a ⊗ b, (−1)k β ⊗ β e− log|b|⊗log a . Formula (3.11) takes the form ∗ V12 V13 = W23 V12 W23 .

(3.26)

(c) = W (c ⊗ I )W ∗ .

(3.27)

For any c ∈ A we set Then is a representation of A acting on L2 (R)⊗L2 (R). We know that V

∈ M(K(K)⊗

A). Formula (3.26) shows that (id ⊗ )V = V12 V13 . Clearly V12 , V13 ∈ M(K(K) ⊗ A ⊗ A). Therefore (id ⊗ )V = V12 V13 ∈ M(K(K) ⊗ A ⊗ A). Remembering that A is generated by V we conclude that ∈ Mor(A, A ⊗ A).


347

π Let = 2k+3 (k = 0, 1, 2, . . . ). Then formula (3.27) makes it possible to calculate (c) for any c ∈ A. The same holds for any c affiliated with A. We shall show that

(a) = a ⊗ a, (b) = a ⊗ b + b ⊗ I (−1)k β⊗β , (ib2k+3 β) = a 2k+3 ⊗ ib2k+3 β + ib2k+3 β ⊗ I − sgn(b⊗b) .

(3.28)

Formula for (a) follows immediately from (3.13); the reader should notice that operators ei /2 b−1 a ⊗ b and β ⊗ β commute with a ⊗ a. The formula for (b) was in fact shown in the proof of Theorem 3.3; in the present case τ+ = τ− = (−1)k β ⊗ β and the second formula of (3.28) coincides with (3.21) (and with (3.23) as well). It remains to prove the third formula. We know that |b| commutes with ib2k+3 β. Taking into account (3.25) we obtain (ib2k+3 β) = W (ib2k+3 β ⊗ I )W ∗ ∗ ib2k+3 β⊗I = G ei /2 b−1 a⊗b, (−1)k β⊗β ×G ei /2 b−1 a⊗b, (−1)k β⊗β .

(3.29)

To compute the right-hand side we use Proposition 1.4 with R = a ⊗ |b| ,

S = |b| ⊗ I,

τ = sgn(b ⊗ b),

ξ = (−1)k β ⊗ β,

ρ = I ⊗ i(sgn b)β, σ = i(sgn b)β ⊗ I. Remembering that β 2 = I and β anticommutes with b and hence commutes with |b| one can easily check that these operators fulfill all assumption of Proposition 1.4. In this case we have T = (ei /2 |b|−1 a) ⊗ |b| and τ T = ei /2 b−1 a ⊗ b. According to our assumption π = 2k + 3 is an odd positive integer. Therefore π σ S = i(sgn b)β ⊗ I |b|2k+3 ⊗ I = ib2k+3 β ⊗ I, π ρR = I ⊗ i(sgn b)β a 2k+3 ⊗ |b|2k+3 = a 2k+3 ⊗ ib2k+3 β, and formula (1.11) takes the form ib2k+3 β ⊗ I + a 2k+3 ⊗ ib2k+3 β

− sgn(b⊗b) ∗ 2k+3

ib = G ei /2 b−1 a⊗b, (−1)k β⊗β ×G ei /2 b−1 a⊗b, (−1)k β⊗β .

β⊗I

(3.30)

Comparing (3.29) with (3.30) we get the last formula of (3.28). This formula appeared without proof in [19].

348


Remark 3.4. Let s ∈ S 1 be a number of modulus 1. Replacing in the above computations σ = i(sgn b)β ⊗ I and ρ = i(sgn b)β ⊗ I by σ = s sgn b β ⊗ I and ρ = s sgn b β ⊗ I respectively, one can prove that (s sgn b |b|2k+3 β) = a 2k+3 ⊗ s sgn b |b|2k+3 β + s sgn b |b|2k+3 β ⊗ I (3.31) . − sgn(b⊗b)

If s = i then s sgn b = i sgn b and (3.31) reduces to the previous formula. For s = 1 we get . (3.32) (|b|2k+3 β) = a 2k+3 ⊗ |b|2k+3 β + |b|2k+3 β ⊗ I − sgn(b⊗b)

Assume now that K = L2 (R) and that the operators a, b, β are given by (3.7). Then operator (3.8) coincides with (3.25): V = W . Relation (3.26) takes the form: W23 W12 = W12 W13 W23 . This is the famous pentagon equation of Baaj and Skandalis [2]. It means that W is a multiplicative unitary. It is known that W is modular [8]. This property enables us to introduce the unitary antipode, scaling group and Haar weight (see [8, 16, 19, 10, 20] for details). In [19] we discussed the cyclic group of four elements acting on a quantum ‘ax + b’ group. In fact this action may be extended to an action of S 1 . At the beginning we set no condition for ∈ R. For any s ∈ S 1 and any closed operator c we set φs (c) = ws∗ c ws , where ws is the unitary operator introduced by ws = s χ(b 0) and W = G (T , τ )∗ e− I ⊗log|b|⊗log a (I ⊗ I ⊗ w)I ⊗L⊗I . i

(4.10)

Clearly W is a unitary operator acting on K ⊗ L2 (R × S 1 ) ⊗ L2 (R × S 1 ). We shall prove that ∗

V12 V13 = W V12 W .

(4.11)

In order to make our formulae shorter we set i

U = e log a⊗log a (I ⊗ w)L⊗I , Z = e− log|b|⊗log a (I ⊗ w)L⊗I . i

Using the commutation relations, one can easily verify that U (b ⊗ I )U ∗ = b ⊗ a, Z(a ⊗ I )Z ∗ = a ⊗ a,

U (β ⊗ I )U ∗ = (β ⊗ I )(I ⊗ w)− sgn b⊗I , (4.12) Z(w ⊗ I )Z ∗ = w ⊗ w. (4.13)

The last formula follows from (4.4). With the above notation V = G (b ⊗ b, β ⊗ β)∗ U and V12 V13 = G (b ⊗ b ⊗ I, β ⊗ β ⊗ I )∗ U12 G (b ⊗ I ⊗ b, β ⊗ I ⊗ β)∗ U13 . Taking into account (4.12) we get U12 G (b ⊗ I ⊗ b, β ⊗ I ⊗ β)∗ = G (b ⊗ a ⊗ b, (β ⊗ I ⊗ β)(I ⊗ w ⊗ I )− sgn b⊗I ⊗I )∗ U12 ∗ = G b ⊗ a ⊗ b, β ⊗ (I ⊗ β)(w ⊗ I )I ⊗sgn b U12 . The second equality follows from (1.3). Therefore ∗ V12 V13 = G (R, ρ) G (S, σ ) U12 U13 ,

(4.14)

(4.15)

where R = b ⊗ a ⊗ b, ρ = β ⊗ (I ⊗ β)(w ⊗ I )I ⊗sgn b , S = b ⊗ b ⊗ I, σ = β ⊗ β ⊗ I.

(4.16)

One can easily verify that R, S are selfadjoint, ρ, σ are unitary selfadjoint, R commutes with ρ and anticommutes with σ and S anticommutes with ρ and commutes

352


with σ . Operator T = ei /2 S −1 R = I ⊗ ei /2 b−1 a ⊗ b coincides with the operator T introduced by (4.9). Moreover σρ = I ⊗ (β ⊗ β)(w ⊗ I )I ⊗sgn b ≡ I ⊗ βw − sgn b ⊗ β, ρσ = I ⊗ (I ⊗ β)(w ⊗ I )I ⊗sgn b (β ⊗ I ) = I ⊗ (β ⊗ β)(βwβ ⊗ I )I ⊗sgn b = I ⊗ (β ⊗ β)(s sgn b w ⊗ I )I ⊗sgn b ≡ s −1 I ⊗ βw − sgn b ⊗ β, where ‘≡’ denotes the equivalence relation: x ≡ y if and only if xχ (T < 0) = yχ(T < 0). Consequently αρσ χ (S < 0) + ασρχ (S > 0) ≡ τ, where τ is given by (4.9). Theorem 1.6 shows now that G (R, ρ)G (S, σ ) = G (T , τ )∗ G (S, σ ) G (T , τ ) and (4.15) takes the form V12 V13 = G (T , τ )∗ G (b ⊗ b ⊗ I, β ⊗ β ⊗ I )∗ G (T , τ ) U12 U13 .

(4.17)

Performing simple computations and using (4.13) we obtain i

U12 U13 = e log a⊗log(a⊗a) (I ⊗ w ⊗ w)L⊗I ⊗I ∗ . = Z23 U12 Z23

Repeating the arguments used in the proof of Theorem 3.3 we see that G (T , τ ) i commutes with e log a⊗log(a⊗a) . One can easily check that T commute with L ⊗ I ⊗ I , I ⊗ w ⊗ w and τ commutes with L ⊗ I ⊗ I . Moreover (I ⊗ w ⊗ w)∗ τ (I ⊗ w ⊗ w) = τ (I ⊗ s sgn b ⊗ s sgn b ) ≡ τ. Therefore G (T , τ ) commutes with (I ⊗ w ⊗ w)L⊗I ⊗I and in (4.17) we may move G (T , τ ) to the most right position: ∗ G (T , τ ) (4.18) V12 V13 = G (T , τ )∗ G (b ⊗ b ⊗ I, β ⊗ β ⊗ I )∗ Z23 U12 Z23

Finally one easily verifies that b ⊗ I and β ⊗ I commute with log |b| ⊗ log a, L ⊗ I and I ⊗ w. Therefore G (b ⊗ b ⊗ I, β ⊗ β ⊗ I ) commutes with Z23 . Clearly G (b ⊗ b ⊗ I, β ⊗ β ⊗ I )∗ U12 = V12 and W = G (T , τ )∗ Z23 . Now (4.11) follows immediately from (4.18). Also in the present case W commutes with a ⊗ I ⊗ I . The same proof applies. Now we are ready to prove the main statement. ⇒ . Let ∈ Mor(A, A ⊗ A) and (id ⊗ )V = V12 V13 . Repeating the reasoning used in the proof of Theorem 3.3 we easily arrive at the formula (4.19) (b) = a ⊗ b + b ⊗ I τ = a ⊗ b + b ⊗ I τ , +

where

−

τ+ = (βw− sgn b ⊗ β) αs −1 χ (b ⊗ I < 0) + αχ (b ⊗ I > 0) , τ− = (βw− sgn b ⊗ β) αs −1 χ (b ⊗ I > 0) + αχ (b ⊗ I < 0) .

(4.20)


353

Clearly the two expressions for (b) must coincide. Let us notice that the operator I ⊗ β commutes with τ+ , τ− and b ⊗ I and anticommutes with a ⊗ b. Therefore τ+ = τ− by iπ 2

Remark 1.3. Using (4.20) we get s = α 2 . Remembering that α = i e 2 and 0 < < π2 we conclude that = πp , where p ∈ R, p > 2 and eiπp = −s. ⇐. Assume that = πp , for some p ∈ R such that p > 2 and eiπp = −s. Then formula (4.10) essentially simplifies. In this case αs −1 = α, τ = I ⊗ αβw − sgn b ⊗ β = I ⊗ αwsgn b β ⊗ β and W = W23 = I ⊗ W , where ∗ i W = G ei /2 b−1 a ⊗ b, αw sgn b β ⊗ β e− log|b|⊗log a (I ⊗ w)L⊗I . (4.21) Formula (4.11) takes the form ∗ V12 V13 = W23 V12 W23 .

(4.22)

(c) = W (c ⊗ I )W ∗ .

(4.23)

For any c ∈ A we set

Then is a representation of A acting on L2 (R × S 1 ) ⊗ L2 (R × S 1 ). We know that V ∈ M(K(K) ⊗ A). Formula (4.22) shows that (id ⊗ )V = V12 V13 . Clearly V12 , V13 ∈ M(K(K) ⊗ A ⊗ A). Therefore (id ⊗ )V = V12 V13 ∈ M(K(K) ⊗ A ⊗ A). Remembering that A is generated by V we conclude that ∈ Mor(A, A ⊗ A). Let s = −eiπp and = πp for some p > 2. Formula (4.23) enables us to calculate (c) for any c ∈ A. The same holds for any c affiliated with A. We shall show that (a) = a ⊗ a, (b) = a ⊗ b + b ⊗ I , αwsgn b β⊗β β |b|p = (w ⊗ I )−I ⊗sgn b (a p ⊗ β |b|p ) + β |b|p ⊗ I

− sgn(b⊗b)

,

(4.24)

(w) = w ⊗ w. Repeating the reasoning preceding the formula (4.18) one can show that a ⊗ a and w ⊗ w commute with G ei /2 b−1 a ⊗ b, αw sgn b β ⊗ β and formulae for (a) and (w) follow immediately from (4.13). The formula for (b) coincides with (4.19). It remains to prove the third formula. According to (4.21) operator W is the com ∗ i position of two unitaries: G ei /2 b−1 a ⊗ b, αw sgn b β ⊗ β and e− log|b|⊗log a (I ⊗ w)L⊗I . Formula (4.23) shows now that = ψ oϕ, where ϕ(c) = e− log|b|⊗log a (I ⊗ w)L⊗I (c ⊗ I ) (I ⊗ w)−L⊗I e log|b|⊗log a , ∗ ψ(d) = G ei /2 b−1 a ⊗ b, αw sgn b β ⊗ β d G ei /2 b−1 a ⊗ b, αw sgn b β ⊗ β . i

i

One can easily verify that ϕ(b) = b ⊗ I and ϕ(β) = β ⊗ I . Therefore ϕ(β |b|p ) = β |b|p ⊗ I

354


and β |b|p = ψ β |b|p ⊗ I .

(4.25)

To compute the right-hand side we use Proposition 1.4 with R = a ⊗ |b| ,

S = |b| ⊗ I,

τ = sgn(b ⊗ b),

ξ = αw sgn b β ⊗ β,

ρ = (w ⊗ I )−I ⊗sgn b (I ⊗ β),

σ = β ⊗ I.

One can easily check that these operators fulfill all assumptions of Proposition 1.4. In this case we have T = (ei /2 |b|−1 a) ⊗ |b| and τ T = ei /2 b−1 a ⊗ b. According to our assumption π = p. Therefore π

σ S = β |b|p ⊗ I,

π ρR = (w ⊗ I )−I ⊗sgn b a p ⊗ β |b|p , and formula (1.11) takes the form

β |b|p ⊗ I + (w ⊗ I )−I ⊗sgn b a p ⊗ β |b|p

− sgn(b⊗b)

= ψ β|b|p ⊗I . (4.26)

Comparing (4.25) with (4.26) we get the third formula of (4.24).

5. Modularity and All That Now we shall investigate the unitary W introduced by (4.21). We shall prove that W is a modular multiplicative unitary. Throughout this section s = −eiπp and = πp , where p > 2. Let K be the Hilbert space complex conjugate to K. The structure of K is established by an antiunitary mapping K x ←→ x ∈ K. For any closed operator c acting on K, we denote by c the transpose of c. By definition c is an operator acting on K with domain D(c ) = {x : x ∈ D(c∗ )} such that c x = c∗ x 1

for any x ∈ D(c∗ ). In what follows Q = a 2 . Proposition 5.1. Let V be the unitary operator introduced by (4.8) and i = G −b⊗ei /2 ba −1 , −β ⊗αw sgn b β e log a ⊗log a (I ⊗ w)L ⊗I . V

(5.1)

is unitary and for any x, z ∈ K, y ∈ D(Q−1 ), u ∈ D(Q) we have: Then V x ⊗ Q−1 y . (x ⊗ u V z ⊗ y) = z ⊗ Qu V

(5.2)


355

Proof. Let us notice that χ (−b⊗ei /2 ba −1 < 0) = χ (b ⊗ b > 0). By virtue of (1.3), we may replace β ⊗ αw sgn b β = β ⊗ αβw − sgn b by α(I ⊗ β) τ , where τ = (β ⊗ I )(I ⊗ w)− sgn b

⊗I

(5.3)

.

Therefore i = G −b⊗ei /2 ba −1 , −α(I ⊗ β) τ e log a ⊗log a (I ⊗ w)L ⊗I . V

(5.4)

We know that β L = (L − sgn b)β. Therefore L β = β (L − sgn b ) and ⊗I

(I ⊗ w)L

⊗I −sgn b ⊗I

(β ⊗ I ) = (β ⊗ I )(I ⊗ w)L

.

It shows that ⊗I

τ (I ⊗ w)L

⊗I

= (I ⊗ w)L

(β ⊗ I ).

(5.5)

We shall follow the proof of Proposition 2.3 of [19]. The reader should notice that in large part that proof is independent of the particular value of . To make our formulae shorter we set: i

U = e log a⊗log a ,

= e i log a ⊗log a , U

=U (I ⊗ w)L ⊗I , U = U (I ⊗ w)L⊗I , U " " " " = "b ⊗ ei /2 ba −1 " . B = "b ⊗ b" , B

(5.6)

We know that sgn b and Q commute. Therefore we may assume that u and y are eigenvectors of sgn b. Similarly we may assume that x and z are common eigenvectors of sgn b. Proceeding in the same way as in [19] we reduce (5.2) to the following three equations (cf. [19, formula (2.23) and the next two]): − π i)U x ⊗ Q−1 y , (5.7) x ⊗ u Vθ (log B)∗ U z ⊗ y = z ⊗ Qu Vθ (log B U x ⊗ Q−1 y , x ⊗ u Vθ (log B − πi)∗ U z ⊗ y = z ⊗ Qu Vθ (log B) (5.8)

x⊗u

∗ π i β ⊗ β B Vθ (log B − πi) U z ⊗ y

π Vθ (log B − π i)U x ⊗ Q−1 y . = z ⊗ Qu − iα(I ⊗ β) τB In these formulae θ = 2π = 2p. The left-hand side of the last formula π LHS of (5.9) = −i βx ⊗ βu B Vθ (log B − π i)∗ U z ⊗ y . Similarly remembering that β commutes with Q we have: π Vθ (log B − π i)U x ⊗ Q−1 y . τB RHS of (5.9) = −iα z ⊗ Qβu

(5.9)

356


We shall move τ to the most right position. Clearly (cf. (5.3)) this operator commutes i with B and U = e log a ⊗log a . Taking into account (5.5) we obtain π Vθ (log B − π i)U βx ⊗ Q−1 y . RHS of (5.9) = −iα z ⊗ Qβu B Replacing βx and βu by x and u respectively we see that (5.9) is equivalent to the equation π x ⊗ u αB Vθ (log B − πi)∗ U z ⊗ y π − πi)U x ⊗ Q−1 y . Vθ (log B = z ⊗ Qu B (5.10) Let us notice that our crucial formulae (5.7), (5.8) and (5.10) fit the same pattern: U x ⊗ Q−1 y , (5.11) (x ⊗ u fi (B)U z ⊗ y) = z ⊗ Qu gi (B) where fi and gi (i = 1, 2, 3) are functions on positive reals: f1 (t) = Vθ (log t),

g1 (t) = Vθ (log t − π i),

f2 (t) = Vθ (log t − π i),

g2 (t) = Vθ (log t),

π

π

f3 (t) = αt Vθ (log t − π i), g3 (t) = t Vθ (log t − π i) for all t > 0. (cf. (5.6)) we obtain a simplified version by U and U Replacing operators U and U of (5.11): U x ⊗ Q−1 y . (5.12) x ⊗ u fi (B)U z ⊗ y = z ⊗ Qu gi (B) It is known that the last equality holds in all three cases i = 1, 2, 3 (cf. [19, proof of Proposition 2.3]). We shall show that (5.11) follows from (5.12). We shall use the expansion (I ⊗ w)L⊗I =

∞ #

χ (L = m) ⊗ w m .

(5.13)

m=−∞

Inserting in (5.12), χ (L = m)x and wm y instead of x and y we obtain: x⊗u fk (B)U χ (L=m)⊗w m z⊗y U χ (L =m)⊗w m x⊗Q−1 y . = z⊗Qu gk (B) Summing over m and using (5.13) we obtain (5.11). The proof is complete.

We recall the basic definitions [2, 16, 8]. Let H be a Hilbert space and W be a unitary operator acting on H ⊗ H . We say that W is multiplicative unitary if it satisfies the pentagonal equation W23 W12 = W12 W13 W23 .


357

A multiplicative unitary W is said to be modular if there exist strictly positive selfadjoint acting on H ⊗ H such that operators Q and Q acting on H and a unitary operator W Q ⊗ Q commutes with W and x ⊗ Q−1 y (5.14) (x ⊗ u W z ⊗ y) = z ⊗ Qu W for any x, z ∈ H , u ∈ D(Q) and y ∈ D(Q−1 ). In this definition H is the complex conjugate Hilbert space related to H by an antiunitary mapping H x ←→ x ∈ H . The main result of this section is contained in the following Theorem 5.2. The operator W introduced by (4.21) is a modular multiplicative unitary acting on L2 (R × S 1 ) ⊗ L2 (R × S 1 ). Proof. Assume that K = L2 (R × S 1 ). One can easily verify that operators a = |b|−1 ,

β = αw sgn b β,

b = ei /2 b−1 a, L = L

(5.15)

obey the properties listed in (4.6). In particular β 2 = I , β ∗ = β and β Lβ = wsgn b βLβw − sgn b = w sgn b Lw − sgn b = L − sgn b = L − sgn b. With this choice, the right-hand side of (4.8) coincides with that of (4.21): V = W and relation (4.22) takes the form: W23 W12 = W12 W13 W23 . Hence W is a multiplicative unitary operator. Let Q = a 1/2 and Q = |b|1/2 . Inserting in (5.1) operators (5.15) we obtain a uni satisfying formula (5.14). To end the proof we have to show that W tary operator W commutes with Q ⊗ Q. We know that a commutes with β and w. One can easily check that |b| commutes with αw sgn b β and L. Therefore Q ⊗ Q = |b|1/2 ⊗ a 1/2 commutes with αw sgn b β ⊗ β, L ⊗ I and I ⊗ w. Clearly it commutes with log |b| ⊗ log a. Moreover due to the Zakrzewski

relation a o b, Q ⊗ Q commutes with ei /2 b−1 a ⊗ b. Inspecting formula (4.21) we see that Q ⊗ Q commutes with W . Now we can use the full power of the theory of multiplicative unitaries [2, 16, 8]. Denoting by B(L2 (R × S 1 ))∗ the set of all normal functionals on B(L2 (R × S 1 )) we have: norm closure A = (ω ⊗ id)W : ω ∈ B(L2 (R × S 1 ))∗ . Indeed according to Theorem 1.5 of [16], the set on the right-hand side is a C∗ -algebra generated by W and the above equality follows immediately from Theorem 4.2 (in the present setting V = W ). Formula (4.22) shows that (4.8) is an adapted operator in the sense of [16, Definition 1.3]. Comparing (5.1) with Statement 5 of Theorem 1.6 of [16] one can easily find the unitary antipode R of our quantum group. It acts on a, b, β, w as follows: a R = a −1 ,

β R = −αwsgn b β,

bR = −ei /2 ba −1 , wR = w∗ .

358


The action of the scaling group is described by the formulae: τt (a) = a,

τt (β) = β,

τt (b) = et b, τt (w) = w. In the following, Tr denotes the trace of operators acting on L2 (R × S 1 ) and E0 denotes the orthogonal projection onto kernel of L − χ (b > 0): E0 = χ L − χ (b > 0) = 0 = χ L = 0 and b < 0 + χ L = 1 and b > 0 . The reader should notice that E0 commutes with all operators (5.15). Therefore E0 ⊗ I commutes with the multiplicative unitary W . For any positive c ∈ A we set h(c) = Tr E0 QcQE0 = Tr E0 |b|1/2 c |b|1/2 E0 . Let c = g(log a)f (b), where f, g ∈ C∞ (R). Then c ∈ A. In what follows, dµ(z) denotes the normalized Haar measure on S 1 . One can verify that the operator c QE0 = 1 c |b| 2 E0 is an integral operator: 1 Kc (t , z ; t, z) x(t, z) dt dµ(z) (c |b| 2 E0 x)(t , z ) = R×S 1

with the kernel " "−1/2 g (t /t)f (t) (z /z)χ(t>0) , Kc (t , z ; t, z) = "t " where g () =

1 g(τ )iτ/ dτ 2π R

for > 0 and g () = 0 for < 0. Therefore " " ∗ "Kc (t , z ; t, z)"2 dt dµ(z ) dt dµ(z) h(c c) = R×S 1 ×R×S 1 ∞ 2 d

=

| g ()|

0

=

R

|f (t)|2 dt

1 |g(τ )|2 dτ |f (t)|2 dt < ∞ π R R

for g, f ∈ L2 (R). Let > 0 and c = (I + log2 a)−1 (I + b2 )−1 . Then h(c∗ c ) < ∞ for any > 0. Clearly c → I in strict topology, when → +0. Therefore the left ideal {c ∈ A : h(c∗ c) < ∞} is dense in A. According to the theory developed by Van Daele [10], h is a right Haar weight on our quantum group. See also [20], where the right invariance of h is verified by a straightforward computation. One can easily construct the reduced dual of our quantum group. By definition (see [2, 16]) this is a quantum group (A, ) related to the multiplicative unitary W = W ∗


359

( denotes the flip operator acting on the tensor product of a Hilbert space by itself: (x ⊗ y) = y ⊗ x). In particular norm closure . A = (id ⊗ ω)W ∗ : ω ∈ B(L2 (R × S 1 ))∗ Let a, b, β and L be operators introduced by (5.15). One can show that log a, b, β |b| and L are affiliated with A. Furthermore A is generated by these operators. The action of is described by the formula: (c) = W (c ⊗ I )W ∗ = W ∗ (I ⊗ c)W . In particular (a) = a ⊗ a, (b) = b ⊗ a + I ⊗ b , β⊗α wsgn b β β |b|p = (I ⊗ w)− sgn b⊗I β|b|p ⊗ a p + I ⊗ β|b|p

− sgn(b⊗b)

,

(5.16)

(L) = L ⊗ I + I ⊗ L, where w = α −2L = s −L . To derive the second and third formulae one has to use the second versions of formulae (1.7) and (1.11). The details are left to the reader. The last relation in (5.16) shows that (w) = w ⊗ w. It is easy to verify that operators a, b, β and w obey the same commutation relations as a, b, β and w. Using this fact one can show that there exists ψ ∈ Mor(A, A) such that ψ(a) = a, ψ(b) = b, ψ(ibβ) = i bβ and ψ(w) = s −L . Let opp be the comultiplication opposite to : opp = flipo. Comparing formulae (4.24) with (5.16) we see that opp (ψ(c)) = (ψ ⊗ ψ)(c)

(5.17)

for c = a, b, β |b|p , w. Functions of these operators generate A, so (5.17) holds for any c ∈ A. Let us notice that the operator (id ⊗ ψ)W = G (b ⊗ b, β ⊗ β) e log a⊗log a s −L⊗L i

commutes with . There exists an independent proof of (5.17) based on this observation. 6. ‘ax + b’-Groups at Roots of Unity iπ 2

In this section we shall assume that q 2 = e−i is a root of unity. Then s = α 2 = −e is a root of unity. Let N be the smallest natural number such that s N = 1. Formula (4.1) shows now that wN commutes with β. Consequently wN commutes with all elements of A and the set norm closure CN = (w N − I )c : c ∈ A

360


is a two-sided ideal in A. We know that (w) = w ⊗ w. Therefore (w N − I ) = w N ⊗ w N − I ⊗ I = w N ⊗ (w N − I ) + (w N − I ) ⊗ I. It shows that (CN ) ⊂ A ⊗ CN + CN ⊗ A and the comultiplication goes down to the quotient algebra AN = A/CN . More precisely there exists N ∈ Mor(AN , AN ⊗ AN ) such that N (π(c)) = (π ⊗ π )(c). In this formula π ∈ Mor(A, AN ) denotes the canonical epimorphism from A onto AN = A/CN . From now until the end of this section we shall work with the quantum group (AN , N ). To simplify notation we shall omit π and write a, b, β |b| and w instead of π(a), π(b), π(β |b|) and π(w). These operators are affiliated with AN and we have the following commutation relation: a ∗ = a, a > 0, b∗ = b, a

o

b,

β ∗ = β, β 2 = I, βaβ = a, βbβ = −b, w∗ w = ww∗ = I, w∗ aw = a, w∗ bw = b,

(6.1)

w∗ βw = s sgn b β, wN = I . The action of N is described by the formulae identical with (4.24). It is not difficult to describe AN as a concrete C∗ -algebra and find the multiplicative unitary corresponding to (AN , N ). To this end one has to repeat the considerations of Sect. 4 replacing S 1 by the cyclic group of N elements: ZN = s : = 0, 1, . . . , N − 1 = z ∈ S 1 : zN = 1 . In particular elements of AN will be operators acting on L2 (R × ZN ). To define a, b, β and w we shall use the same formulae (4.2) with necessary reinterpretation: now x ∈ L2 (R × ZN ) and z runs over ZN . One can easily verify that a, b, β and w satisfy the relations (6.1). Now the formula (4.5) defines a C∗ -algebra acting on L2 (R × ZN ). This algebra is isomorphic to AN . It is not possible to find a selfadjoint operator L acting on L2 (R × ZN ) such that Sp L ⊂ Z and w∗ Lw = L + I . However there is a replacement for (I ⊗ w)L⊗I . Instead of L we shall use an operator u acting on L2 (R × ZN ) according to the formula: (ux)(t, z) = x(t, s −1 z). One can easily verify that u is a unitary operator commuting with a, b, β such that wuw∗ = su and uN = I . By the last relation Sp u = ZN . Similarly Sp w ⊂ ZN . We shall use the bicharacter describing the selfduality of the group ZN : Ch : ZN × ZN −→ S 1 . By definition Ch(s , s k ) = s −k for any k, ∈ Z. The reader should notice that Ch(s , z) = z− and Ch(sz , z) = z−1Ch(z , z) for any ∈ Z and z, z ∈ ZN . Using the last formula and remembering that wuw∗ = su we obtain (w ⊗ I ) Ch(u ⊗ I, I ⊗ w)(w ∗ ⊗ I ) = Ch(su ⊗ I, I ⊗ w) = (I ⊗ w ∗ ) Ch(u ⊗ I, I ⊗ w). Therefore Ch(u ⊗ I, I ⊗ w)(w ⊗ I ) Ch(u ⊗ I, I ⊗ w)∗ = w ⊗ w.

(6.2)


361

One should compare this formula with (4.4). It shows that Ch(u ⊗ I, I ⊗ w) is the right replacement for (I ⊗ w)L⊗I . For the moment we shall use the Hilbert space K and operators a, b, β and L the same as in Sect. 4. Remembering that Ch(s , z) = z− we obtain Ch(s −L ⊗ I, I ⊗ w) = (I ⊗ w)L⊗I and formula (4.8) takes the form: V = G (b ⊗ b, β ⊗ β)∗ e log a⊗log a Ch(w ⊗ I, I ⊗ w), i

(6.3)

where w = s −L . Taking into account (4.6) we see that w is a unitary operator with Sp w ⊂ ZN , it commutes with a and b and β wβ = s −L+sgn b = s sgn b w. We should note that in order to define V ∈ M(K(K) ⊗ AN ) we need only the operators a, b, β, w acting on K. These operators must have the following properties: 1. a, b are selfadjoint, a > 0, ker a = ker b = {0} and a

o

b,

2. β is a unitary involution, β commutes with a and anticommutes with b, 3. w is unitary with Sp w ⊂ ZN , w commutes with a and b,

(6.4)

4. β w β = s sgn b w. In this way the operator L disappears from our setup. Let us notice that operators a, b, β, w satisfy the same commutation relations as a, b, β, w. From the beginning of this section we assumed that the deformation parameters s 2 and are related by the formula s = α 2 = −eiπ / . Repeating (with the necessary modifications indicated above) the considerations of Sect. 4 we obtain the following formulae: ∗ i W = G ei /2 b−1 a ⊗ b, αw sgn b β ⊗ β e− log|b|⊗log a Ch(u ⊗ I, I ⊗ w), (6.5) ∗ V12 V13 = W23 V12 W23 , (6.6) N (c) = W (c ⊗ I )W ∗ (6.7) for any c ∈ AN . Assume that K = L2 (R × ZN ), a = |b|−1 , β = αw sgn b β, b = ei /2 b−1 a, and w = u. One can easily verify that these operators obey the properties listed in (6.4). In particular β 2 = I , β ∗ = β, β w β = wsgn b uw − sgn b = s sgn b u = s sgn b w, where in the second step we used the relation wuw∗ = su. With this choice, operators (6.3) and (6.5) coincide: V = W and (6.6) shows that W is a multiplicative unitary. Using the method described in Sect. 5 one can show that W is modular with Q = a 1/2 and Q = |b|1/2 . This is the multiplicative unitary corresponding to the quantum group (AN , N ). It can be used to determine the action of the unitary antipode and scaling group, the Haar weight and the reduced dual (AN , N ). We know that a, b, β and w satisfy the same commutation relations as a, b, β and w. Furthermore formula (6.3) is symmetric: replacing operators without ‘hats’ by corresponding operators with ‘hats’ we obtain an element of M(A ⊗sym A). Using these facts one can show that the quantum group (AN , N ) is opp isomorphic to (AN , N ). Acknowledgement. The authors acknowledge the financial support of the Polish Committee for Scientific Research (KBN, grants 115/E-343/SPB/6.PRUE/DIE50/2005-2008 and 2 P03A 040 22) and of the Foundation for Polish Science.

362


References 1. Arveson, W.: An invitation to C ∗ -Algebra. New York-Heidelberg-Berlin: Springer-Verlag, 1976 2. Baaj, S., Skandalis, G.: Unitaries multiplicatifs et dualité pour les produits croisé de C ∗ -algèbres. Ann. Sci. Ec. Norm. Sup. 4e série 26, 425–488 (1993) 3. Kustermans, J., Vaes, S.: Locally compact quantum groups. Ann. Sci. de l’Ecole Normale Supérieure 33(6), 837–934 (2000) 4. Masuda, T., Nakagami, Y., Woronowicz, S.L.: A C ∗ -algebraic framework for the quantum groups. Int. J. Math. 14(9), 903–1001 (2003) 5. Napiórkowski, K., Woronowicz, S.L.: Operator theory in the C ∗ -algebra framework. Rep. Math. Phys. 31(3), 353–371 (1992) 6. Pedersen, G.K.: C∗ -algebras and their Automorphism Groups. London-New York-San Francisco: Academic Press, 1979 7. Rowicka, M.: Exponential Equations Related to the Quantum ‘ax +b’ Group. Commun. Math. Phys. 244, 419–453 (2004) 8. Sołtan, P.M., Woronowicz, S.L.: A remark on manageable multiplicative unitaries. Lett. Math. Phys. 57, 239–252 (2001) 9. Sołtan, P.M.: New deformations of the group of affine transformations of the plane. Doctor dissertation (in Polish), University of Warsaw (2003) 10. Van Daele, A.: The Haar measure on some locally compact quantum groups. http://arxiv.org/list/math.OA/0109004v1, 2001 11. Van Daele, A., Woronowicz, S.L.: Duality for the quantum E(2) group. Pa. J. Math. 173, 375–385 (1996) 12. Woronowicz, S.L.: Unbounded elements affiliated with C ∗ -algebras and non-compact quantum groups. Commun. Math. Phys. 136, 399–432 (1991) 13. Woronowicz, S.L.: C∗ -algebras generated by unbounded elements. Rev. Math. Phys. 7(3), 481–521 (1995) 14. Woronowicz, S.L.: Quantum E(2) group and its Pontryagin dual. Lett. Math. Phys. 23, 251–263 (1991) 15. Woronowicz, S.L.: Operator equalities related to quantum E(2) group. Commun. Math. Phys. 144, 417–428 (1992) 16. Woronowicz, S.L.: From multiplicative unitaries to quantum groups. Int. J. Math. 7, 127–149 (1996) 17. Woronowicz, S.L.: Quantum exponential function. Rev. Math. Phys. 12(6), 873–920 (2000) 18. Woronowicz, S.L.: Quantum ‘az + b’ group on complex plane. Int. J. Math. 12, 461–503 (2001) 19. Woronowicz, S.L., Zakrzewski, S.: Quantum ‘ax + b’ group. Rev. Math. Phys. 14(7 & 8), 797–828 (2002) 20. Woronowicz, S.L.: Haar weight on some quantum groups. In: Group 24: Physical and mathematical aspects of symmetries, Proceedings of the 24th International Colloquium on Group Theoretical Methods in Physics Paris, 15 - 20 July 2002, Inst. of Physics, Conference Series Number 173, pp. 763–772 Communicated by A. Connes


Communications in


Orthomodular Lattices Generated by Graphs of Functions W. Cegła1 , J. Florek2 1 2

Institute of Theoretical Physics, University of Wrocław, pl. Maxa Borna 9, 50–204 Wrocław, Poland. E-mail: [email protected] Institute of Mathematics, University of Economics, ul. Komandorska 118/120, 53–345 Wrocław, Poland. E-mail: [email protected]

Received: 12 July 2004 / Accepted: 8 December 2004 Published online: 19 May 2005 – © Springer-Verlag 2005

Abstract: In a subset Z ⊆ R × M, where R is the real line and M is an arbitrary topological space, an orthogonality relation is constructed from a family of graphs of continuous functions from connected subsets of R to M. It is shown that under two conditions on this family a complete lattice of double orthoclosed sets is orthomodular. 1. Introduction An orthomodularity condition is important in the quantum logic approach to quantum theory [4]. Any complete orthomodular lattice can be constructed using the orthogonality space (Z, ⊥), where Z is a nonempty set and ⊥, called the orthogonality relation, is a symmetric and irreflexive relation on Z, i.e. x ⊥ y ⇒ y ⊥ x and x ⊥ x for all x, y ∈ Z. Having (Z, ⊥) one defines A⊥ := {x ∈ Z; x ⊥ a ∀ a ∈ A}, A⊥⊥ := (A⊥ )⊥ for all A ⊆ Z and one considers the family of double orthoclosed sets ζ (Z, ⊥) := {A ⊆ Z; A = A⊥⊥ }. This family ζ (Z, ⊥) partially ordered by set-theoretical inclusion and equipped with the orthocomplementation A → A⊥ , with l.u.b. and g.l.b. given respectively by the formulas Aj = (∪Aj )⊥⊥ Aj = ∩Aj , forms a complete ortholattice [1], which in general is not orthomodular. To verify the orthomodularity of ζ (Z, ⊥) it is enough to prove one of the equivalent conditions discussed in [3], namely, if A is an orthogonal subset of Z (which means that x ⊥ y for all x, y ∈ A with x = y), then (OM)

x ∈ Z, x ∈ / A⊥ , and x ∈ / A⊥⊥ ⇒ A⊥ ∩ (x ⊥ ∩ A⊥ )⊥ = ∅.

Let R × M be the topological product of the real line R and arbitrary topological space M and p be a canonical projection of R × M on R. Let G be a set of continuous functions of the form g : Dg → M, where the domain Dg of g is a connected subset of R. We shall identify the function with its graph.

364

W. Cegła, J. Florek

We define a space (Z, ≤)G and a space (Z, ⊥)G generated by the family G as follows: Z= g, g∈G

x≤y x⊥y

iff there is g ∈ G such that {x, y} ⊆ g and p(x) ≤ p(y), iff x ≤ y and y ≤ x iff there is no g ∈ G such that {x, y} ⊆ g.

Of course (Z, ≤)G is antisymmetric and reflexive so (Z, ⊥)G is an orthogonality space. Two conditions for the set G listed in Sect. 3 ensure that the resulting lattice ζ (Z, ⊥)G is orthomodular. The first one says the (Z, ≤)G is a partial order space, the second one is purely topological. Section 3 contains the proof of orthomodularity of ζ (Z, ⊥). Three examples are discussed in Sect. 4. 2. Definitions and Symbols Let the space (Z, ≤)G and the orthogonality space (Z, ⊥)G be generated by the family G and p be a canonical projection of R × M on R. For a ∈ Z and A ⊆ Z we define: + a : = {z ∈ Z : a ≤ z} = z ∈ Z : p(z) ≥ p(a) ∧ ∃ (z ∈ g ∧ a ∈ g) , g∈G − a : = {z ∈ Z : z ≤ a} = z ∈ Z : p(z) ≤ p(a) ∧ ∃ (z ∈ g ∧ a ∈ g) , g∈G a ⊥ : = {z ∈ Z : z ⊥ a} = z ∈ Z : ∀ (z ∈ g ⇒ a ∈ / g) , g∈G + + A := a , a∈A

A− : =

a− ,

a∈A ⊥

A :=

⊥

a = z ∈ Z : ∀ (z ∈ g ⇒ g ∩ A = ∅) . g∈G

a∈A

One can check that: (I) a ≤ z iff z ∈ a + iff a ∈ z− , (II) a ⊥ = (a + ∪ a − ) and A⊥ = (A+ ∪ A− ) , where denotes the set complement in Z. For A ⊆ Z and g ∈ G we denote p(g ∩ A) = {p(a) ∈ Dg : a ∈ g ∩ A} . Then it is easy to see that: (III) p(g ∩ ( Ai )) = p(g ∩ Ai ) , i∈I i∈I (IV) p(g ∩ ( Ai )) = p(g ∩ Ai ) . i∈I

i∈I

Because g ⊆ Z so p(g ∩ Z) = Dg . Hence we have (V) p(g ∩ A ) = [p(g ∩ A)] , where A and [p(g ∩ A)] denote the set complement in Z and in Dg respectively.

Orthomodular Lattices Generated by Graphs of Functions

365

3. Basic Results From now on we assume that G satisfies the following conditions: (*)

∀

x,y,z∈Z

(**) ∀

z∈Z

(x ≤ y

∧

y ≤ z ⇒ x ≤ z),

z+ \ {z} and z− \ {z} are open sets in R × M.

Remark 1. The condition (*) is equivalent to the following one:

∀ ∀ ∀ p(y) ∈ p(g ∩ x + ) ∧ p(z) > p(y) => p(z) ∈ p(g ∩ x + ) .

x∈Z g∈G y,z∈g

Lemma 1. Let f ∈ G, A ⊆ Z, f ∩ A = ∅. (i) f ∩ A+ = ∅ ∧ f ⊆ A+ ⇒ f, A+ = (t, ∞) ∩ Df , where t ∈ Df , (ii) f ∩ A− = ∅ ∧ f ⊆ A− ⇒ f, A− = (−∞, s) ∩ Df , where s ∈ Df , (iii) f ∩ A⊥ = ∅ ∧  f ⊆ A⊥ ⇒ for f ∩ A+ = ∅ and f ∩ A− = ∅,  [s, t] ⊥ ⇒ p(f ∩ A ) = (−∞, t] ∩ Df for f ∩ A+ = ∅ and f ∩ A− = ∅,  [s, ∞) ∩ D for f ∩ A+ = ∅ and f ∩ A− = ∅, f + − where t = inff, A and s = supf, A . Proof of Lemma 1. We shall prove (i). Let A = {a}, f ∩ a + = ∅ and f ⊆ a + . Because f is continuous on Df and by (∗∗) a + \ {a} is an open set in Z so p(f ∩ a + ) = p(f ∩ (a + \ {a})) is an open set in Df . Hence using Remark 1, there exists ta ∈ Df such that p(f ∩ a + ) = (ta , ∞) ∩ Df . It is enough to see that by (*) and (III), p(f ∩ A+ ) = p(f ∩ a + ) = (ta , ∞) ∩ Df = (t, ∞) ∩ Df , where t ∈ Df . a∈A

a∈A

The proof in the case (ii) goes in a similar way. We shall prove (iii). If f ∩ A⊥ = ∅ then f ⊆ A+ and f ⊆ A− . If f ⊆ A⊥ then f ∩ A+ = ∅ or f ∩ A− = ∅. By (II), (III), (V) we obtain p(f ∩ A⊥ ) = [p(f ∩ A+ )] ∩ [p(f ∩ A− )] . Hence by (i), (ii) we get (iii). Theorem 1. If (Z, ⊥)G is the orthogonality space generated by the family G satisfying conditions (*) and (**) then ζ (Z, ⊥)G = {A ⊆ Z : A = A⊥⊥ } is an orthomodular lattice. Proof of Theorem 1. We shall prove that (Z, ⊥)G satisfies the (OM) condition. Assume that A ⊆ Z, x ∈ Z, x ∈ / A⊥ and x ∈ / A⊥⊥ . From the assumption x ∈ / A⊥⊥ = {z ∈ ⊥ Z : ∀ (z ∈ g ⇒ g ∩ A = ∅)} it follows that there exists f ∈ G such that x ∈ f g∈G

and f ∩ A⊥ = ∅. From the assumption x ∈ / A⊥ it follows, by (II), that x ∈ A+ or − + x ∈ A . We consider only the case x ∈ A . The proof in the second case proceeds in a similar way. Because f ∩ A⊥ = ∅ and x ∈ f ∩ A+ , so by Lemma 1(i), (iii), there exists y ∈ f ∩ A⊥ such that p(f ∩ A+ ) = (p(y), ∞) ∩ Df and p(y) < p(x). For every g ∈ G such that y ∈ g we have the following: (j) z ∈ g ∧ p(z) ≤ p(y) ⇒ z ∈ x − , (jj) z ∈ g

∧

p(z) > p(y) ⇒ z ∈ A+ ,

(jjj) g ∩ (x ∪ A)⊥ = ∅ .

366

W. Cegła, J. Florek

Case (j). Because z ∈ g, y ∈ g and p(z) ≤ p(y) therefore z ∈ y − . Because y ∈ f , x ∈ f and p(y) < p(x) so y ∈ x − . Hence, by Remark 1, we obtain z ∈ x − . Case (jj). Because y ∈ g, z ∈ g and p(z) > p(y) therefore y ∈ z− . Because y ∈ f ∩z− so p(y) ∈ p(f ∩ z− ). Hence, by (IV) and by Lemma 1(ii), p(f ∩ A+ ∩ z− ) = p(f ∩ A+ ) ∩ p(f ∩ z− ) = (p(y), ∞) ∩ p(f ∩ z− ) is an open interval. Then there exists v ∈ A+ ∩ z− . Hence, by (I), z ∈ v + , v ∈ A+ and, by (*), z ∈ A+ . Case (jjj). Observe that, by (j), (jj) and (II), g ⊆ x − ∪ A+ ⊆ (x ∪ A)− ∪ (x ∪ A)+ = ((x ∪ A)⊥ ) . By (jjj), y ∈ (x∪A)⊥⊥ = {z ∈ Z : ∀ (z ∈ g ⇒ g∩(x∪A)⊥ = ∅)}. But y ∈ A⊥ , so by g∈G

a property of an orthogonality relation, we have y ∈ A⊥ ∩(x∪A)⊥⊥ = A⊥ ∩(x ⊥ ∩A⊥ )⊥ . 4. The Examples Example 1. Let M be a normed vector space and Z be an open set in R × M. Let G be the family of all functions g : Dg ⊆ R → M on connected sets Dg of R with values in M satisfying the sharp Lipschitz condition with constant α > 0, ∀

such that Z =

t,s∈Dg

g(t) − g(s) < α|t − s| ,

g.

g∈G

The family G satisfies the condition (*), which is obvious, and condition (**) follows because Z is an open set in R×M and the balls in the normed vector space M are convex sets. The following example is a special case of Example 1. Example 2. Let Z = R × Rn and G be the family of all translations of graphs of linear functions g : R → Rn satisfying sharp Lipschitz condition with constant α = 1. The orthogonality relation in this case means a space-like or light-like separation. It was shown in [2] that ζ (Z, ⊥)G forms the orthomodular lattice. Example 3. If Z = {0} × M, G = {{(0, m)}; m ∈ M} and g = {(0, m)} ∈ G is a function with domain Dg = {0} such that g(0) = m. Two points (0, m) and (0, n) are orthogonal iff m = n. Of course ζ (Z, ⊥) is isomorphic to the complete, atomic Boolean lattice of all subsets of the set M. Acknowledgements. The authors are very grateful to an anonymous referee for the very detailed remarks and suggestions which improved the paper.

References 1. Birkhoff, G.: Lattice Theory. Amer. Math. Soc. Colloq. Publ. XXV, Providence, RI: Amer. Math. Soc., 1967 2. Cegła, W., Jadczyk, A.Z.: Commun. Math. Phys. 57, 213–217 (1977) 3. Foulis, D.J., Randall, C.H.: Lexicographic Orthogonality. J. Combin. Theory 11, 157–162 (1971) 4. Pták, P., Pulmannová, S.: Orthomodular Structures as Quantum Logics. Number 55 in “Fundamental Theories of Physics”, Dordrecht: Kluwer, 1991 Communicated by M.B. Ruskai


Communications in


Large n Limit of Gaussian Random Matrices with External Source, Part II Alexander I. Aptekarev1 , Pavel M. Bleher2 , Arno B.J. Kuijlaars3 1

Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya Square 4, Moscow 125047, Russia. E-mail: [email protected] 2 Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 N. Blackford St., Indianapolis, IN 46202, U.S.A. E-mail: [email protected] 3 Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, 3001 Leuven, Belgium. E-mail: [email protected] Received: 24 August 2004 / Accepted: 8 February 2005 Published online: 2 June 2005 – © Springer-Verlag 2005

Abstract: We continue the study of the Hermitian random matrix ensemble with external source 1 −nTr( 1 M 2 −AM) 2 e dM, Zn where A has two distinct eigenvalues ±a of equal multiplicity. This model exhibits a phase transition for the value a = 1, since the eigenvalues of M accumulate on two intervals for a > 1, and on one interval for 0 < a < 1. The case a > 1 was treated in Part I, where it was proved that local eigenvalue correlations have the universal limiting behavior which is known for unitarily invariant random matrices, that is, limiting eigenvalue correlations are expressed in terms of the sine kernel in the bulk of the spectrum, and in terms of the Airy kernel at the edge. In this paper we establish the same results for the case 0 < a < 1. As in Part I we apply the Deift/Zhou steepest descent analysis to a 3 × 3-matrix Riemann-Hilbert problem. Due to the different structure of an underlying Riemann surface, the analysis includes an additional step involving a global opening of lenses, which is a new phenomenon in the steepest descent analysis of Riemann-Hilbert problems. 1. Introduction This paper is a continuation of [6] to which we will frequently refer in this paper. It will be followed by a third part [8], which deals with the critical case. In these papers, we The first and third author are supported in part by INTAS Research Network NeCCA 03-51-6637 and by NATO Collaborative Linkage Grant PST.CLG.979738. The first author is supported in part by RFBR 05-01-00522 and the program “Modern problems of theoretical mathematics” RAS(DMS). The second author is supported in part by the National Science Foundation (NSF) Grant DMS-0354962. The third author is supported in part by FWO-Flanders projects G.0176.02 and G.0455.04 and by K.U.Leuven research grant OT/04/24 and by the European Science Foundation Program Methods of Integrable Systems, Geometry, Applied Mathematics (MISGAM) and the European Network in Geometry, Mathematical Physics and Applications (ENIGMA)

368

A.I. Aptekarev, P.M. Bleher, A.B.J. Kuijlaars

study the random matrix ensemble with external source A, µn (dM) =

1 −nTr(V (M)−AM) e dM, Zn

(1.1)

defined on n × n Hermitian matrices M, with Gaussian potential V (M) =

1 2 M , 2

(1.2)

and with external source A = diag(a, . . . , a , −a, . . . , −a ). n/2

(1.3)

n/2

In the physics literature, the ensemble (1.1) was studied in a series of papers of Brézin and Hikami [9–13], and P. Zinn-Justin [33, 34]. The Gaussian ensemble, V (M) = 21 M 2 , has been solved, in the large n limit, in the papers of Pastur [29] and Brézin-Hikami [9– 13], by using spectral methods and a contour integration formula for the determinantal kernel. The contour integration technique has been extended in the recent work of Tracy and Widom [30] to the large n double scaling asymptotics of the determinantal kernel at the critical point. Our aim is to develop a completely different approach to the large n asymptotics of the Gaussian ensemble with external source. Our approach is based on the Riemann-Hilbert problem and it is applicable, in principle, to a general V . We develop the Deift-Zhou nonlinear steepest descent method for Riemann-Hilbert (RH) problems [18], thereby extending the works [3, 4, 16, 17, 25, 26] who treated the unitary invariant case (i.e., A = 0) with RH techniques. While the unitary invariant case is connected with orthogonal polynomials [15, 27], the ensemble (1.1) is connected with multiple orthogonal polynomials [5]. These are characterized by a matrix RH problem [32], and the eigenvalue correlation kernel of (1.1) has a direct expression in terms of the solution of this RH problem, see [5, 14] and also formula (1.14) below. The RH problem for (1.1) has size (r + 1) × (r + 1) if r is the number of distinct eigenvalues of A. So with the choice (1.3), the RH problem is 3 × 3-matrix valued. The asymptotic analysis of RH problems has been mostly restricted to the 2 × 2 case. The analysis of larger size RH problems presents some novel technical features as already demonstrated in [6, 24]. In the present paper another new feature appears, namely at a critical stage in the analysis we perform a global opening of lenses. This global opening of lenses requires a global understanding of an associated Riemann surface, which is explicitly known for the Gaussian case (1.2). This is why we restrict ourselves to (1.2) although in principle our methods are applicable to a more general polynomial V . The Gaussian case has some special relevance in its own right as well. Indeed, first of all we note that for (1.2) we can complete the square in (1.1), and then it follows that M = M0 + A,

(1.4)

where M0 is a GUE matrix. So in the Gaussian case the ensemble (1.1) is an example of a random + deterministic model, see also [9–13]. A second interpretation of the Gaussian model comes from non-intersecting Brownian paths. This can be seen from the joint probability density for the eigenvalues

Large n Limit of Gaussian Random Matrices with External Source, Part II

369

of M, which by the HarishChandra/Itzykson-Zuber formula [19, 27], takes the form 1 ˜ Zn

n 1 n 2 (λj − λk ) det enλj ak j,k=1 e− 2 nλj

(1.5)

j =1

1≤j 0. It satisfies ρ(x) =

1 | Im ξ(x)| , π

(1.16)

where ξ = ξ(x) is a solution of the cubic equation, ξ 3 − xξ 2 − (a 2 − 1)ξ + xa 2 = 0.

(1.17)

The support of ρ consists of those x ∈ R for which (1.17) has a non-real solution.


371

(a) For 0 < a < 1, the support of ρ consists of one interval [−z1 , z1 ], and ρ is real analytic and positive on (−z1 , z1 ), and it vanishes like a square root at the edge points ±z1 , i.e., there exists a constant ρ1 > 0 such that ρ1 ρ(x) = |x ∓ z1 |1/2 (1 + o(1)) as x → ±z1 , x ∈ (−z1 , z1 ). (1.18) π (b) For a = 1, the support of ρ consists of one interval [−z1 , z1 ], and ρ is real analytic and positive on (−z1 , 0) ∪ (0, z1 ), it vanishes like a square root at the edge points ±z1 , and it vanishes like a third root at 0, i.e., there exists a constant c > 0 such that ρ(x) = c|x|1/3 (1 + o(1)) ,

as x → 0.

(1.19)

(c) For a > 1, the support of ρ consists of two disjoint intervals [−z1 , −z2 ] ∪ [z2 , z1 ] with 0 < z2 < z1 , ρ is real analytic and positive on (−z1 , −z2 ) ∪ (z2 , z1 ), and it vanishes like a square root at the edge points ±z1 , ±z2 . Remark. Theorem 1.1 is a very special case of a theorem of Pastur [29] on the eigenvalues of a matrix M = M0 + A, where M0 is random and A is deterministic as in (1.4). Since in this paper our interest is in the case 0 < a < 1, we show in Sect. 9 how Theorem 1.1 follows from our methods for this case. See [6] for the case a > 1. Remark. Theorem 1.1 has the following interpretation in terms of non-intersecting Brownian motions starting at 0 and ending at some specified points bj . We suppose n is even, and we let half of the bj ’s coincide with b > 0 and the other half with −b. Then as explained before, at time t ∈ (0, 1) the (rescaled) positions of the Brownian paths coincide with the eigenvalues of the Gaussian random matrix with external source (1.3) where t a=b . (1.20) 1−t The phase transition at a = 1 corresponds to t = tc ≡

1 . 1 + b2

So, by Theorem 1.1, the limiting distribution of the Brownian paths as n → ∞ is supported by one interval when t < tc and by two intervals when t > tc . At the critical time tc the two groups of Brownian paths split, with one group ending at t = 1 at b and the other at −b. As in [6] we formulate our main result in terms of a rescaled version of the kernel Kn , Kˆ n (x, y) = en(h(x)−h(y)) Kn (x, y)

(1.21)

for some function h. The rescaling (1.21) does not affect the correlation functions (1.10). Theorem 1.2. Let 0 < a < 1 and let z1 and ρ be as in Theorem 1.1 (a). Then there is a function h such that the following hold for the rescaled kernel (1.21): (a) For every x0 ∈ (−z1 , z1 ) and u, v ∈ R, we have

1 u v sin π(u − v) ˆ lim Kn x0 + , x0 + = . n→∞ nρ(x0 ) nρ(x0 ) nρ(x0 ) π(u − v)

(1.22)

372


(b) For every u, v ∈ R we have

1 u v ˆ n z1 + K , z + 1 n→∞ (ρ1 n)2/3 (ρ1 n)2/3 (ρ1 n)2/3 Ai(u) Ai (v) − Ai (u) Ai(v) , = u−v lim

(1.23)

where Ai is the usual Airy function, and ρ1 is the constant from (1.18). Theorem 1.2 is similar to the main theorems Theorem 1.2 and Theorem 1.3 of [6]. It expresses that the local eigenvalue correlations show the universal behavior as n → ∞, both in the bulk and at the edge, that is well-known from unitary random matrix models. So the result itself is not that surprising. To obtain Theorem 1.2 we use the Deift/Zhou steepest descent method for RH problems and a main tool is the three-sheeted Riemann surface associated with Eq. (1.17) as in [6]. There is however an important technical difference with [6]. For a > 1, the branch points of the Riemann surface are all real, and they correspond to the four edge points ±z1 , ±z2 of the support as described in Theorem 1.1 (c). For a < 1, two branch points are purely imaginary and they have no direct meaning for the problem at hand. The other two branch points are real and they correspond to the edge points ±z1 as in Theorem 1.1 (a). See Fig. 1 for the sheet structure of the Riemann surface. The branch points on the non-physical sheets result in a non-trivial modification of the steepest descent method. As already mentioned before, one of the steps involves a global opening of lenses, and this is the main new technical contribution of this paper. The rest of the paper is devoted to the proof of the theorems with the Deift/Zhou steepest descent method for RH problems. It consists of a sequence of transformations which reduce the original RH problem to a RH problem which is normalized at infinity, and whose jump matrices are uniformly close to the identity as n → ∞. In this paper there are four transformations Y → U → T → S → R. A main role is played by the Riemann surface (1.17) and certain λ-functions defined on it. These are introduced in the next section, and they are used in Sect. 3 to define the first transformation Y → U . This transformation has the effect of normalizing the RH problem at infinity, and in addition, of producing “good” jump matrices that are amenable to subsequent analysis. However, contrary to earlier works, some of the jump matrices for U have entries that are exponentially growing as n → ∞. These exponentially growing entries disappear after the second transformation U → T in Sect. 4 which involves the global opening of lenses. The remaining transformation follows the pattern of [6, 16, 17] and other works. The transformation T → S in Sect. 5 involves a local opening of lenses which turns the remaining oscillating entries into exponentially decaying ones. Then a parametrix for S is built in Sects. 6 and 7. In Sect. 6 a model RH problem is solved which provides the parametrix for S away from the branch points, and in Sect. 7 local parametrices are built around each of the branch points with the aid of Airy functions. Using this parametrix we define the final transformation S → R in Sect. 8. It leads to a RH problem for R which is of the desired type: normalized at infinity and jump matrices tending to the identity as n → ∞. Then R itself tends to the identity matrix as n → ∞, which is then used in the final Sect. 9 to prove Theorems 1.1 and 1.2.


373

2. The Riemann Surface and λ-Functions We start from the cubic equation (1.17) which we write now with the variable z instead of x ξ 3 − zξ 2 + (1 − a 2 )ξ + za 2 = 0.

(2.1)

It defines a Riemann surface that will play a central role in the proof. The inverse mapping is given by the rational function z=

ξ 3 − (a 2 − 1)ξ . ξ 2 − a2

(2.2)

There are four branch points ±z1 , ±iz2 with z1 > z2 > 0, which can be found as the images of the critical points under the inverse mapping. The mapping (2.2) has three inverses, ξj (z), j = 1, 2, 3, that behave near infinity as 1 + O(1/z2 ), z 1 ξ2 (z) = a + + O(1/z2 ), 2z 1 ξ3 (z) = −a + + O(1/z2 ). 2z ξ1 (z) = z −

(2.3)

The sheet structure of the Riemann surface is determined by the way we choose the analytical continuations of the ξj ’s. It may be checked that ξ1 has an analytic continuation to C \ [−z1 , z1 ], which we take as the first sheet. The functions ξ2 and ξ3 have analytic continuations to C \ ([0, z1 ] ∪ [−iz2 , iz2 ]) and C \ ([−z1 , 0] ∪ [−iz2 , iz2 ]), respectively, which we take to be the second and third sheets, respectively. So the second and third sheet are connected along [−iz2 , iz2 ], the first sheet is connected with the second sheet along [0, z1 ], and the first sheet is connected with the third sheet along [−z1 , 0], see Fig. 1.

ξ1

ξ2

ξ3 Fig. 1. The Riemann surface ξ 3 − zξ 2 + (1 − a 2 )ξ + za 2 = 0

374


We note the jump relations ξ1∓ = ξ2± ξ1∓ = ξ3± ξ2∓ = ξ3±

on (0, z1 ), on (−z1 , 0), on (−iz2 , iz2 ).

(2.4)

z The λ-functions are primitives of the ξ -functions λj (z) = ξj (s)ds, more precisely z λ1 (z) = ξ1 (s)ds, z1 z ξ2 (s)ds, λ2 (z) = (2.5) z1 z ξ3 (s)ds + λ1− (−z1 ). λ3 (z) = −z1+

The path of integration for λ3 lies in C\((−∞, 0]∪[−iz2 , iz2 ]), and it starts at the point −z1 on the upper side of the cut. All three λ-functions are defined on their respective sheets of the Riemann surface with an additional cut along the negative real axis. Thus λ1 , λ2 , λ3 are defined and analytic on C \ (−∞, z1 ], C \ ((−∞, z1 ] ∪ [−iz2 , iz2 ]), and C \ ((−∞, 0] ∪ [−iz2 , iz2 ]), respectively. Their behavior at infinity is 1 2 z − log z + 1 + O(1/z), 2 1 (2.6) λ2 (z) = az + log z + 2 + O(1/z), 2 1 λ3 (z) = −az + log z + 3 + O(1/z) 2 for certain constants j , j = 1, 2, 3. The λj ’s satisfy the following jump relations: λ1 (z) =

λ1∓ λ1− λ1+ λ2∓ λ2∓ λ1+ λ2+ λ3+

= λ2± = λ3+ = λ3− − πi = λ3± = λ3± − πi = λ1− − 2πi = λ2− + πi = λ3− + πi

on (0, z1 ), on (−z1 , 0), on (−z1 , 0), on (0, iz2 ), on (−iz2 , 0), on (−∞, −z1 ), on (−∞, 0), on (−∞, −z1 ),

(2.7)

where the segment (−iz2 , iz2 ) is oriented upwards. We obtain (2.7) from (2.4), (2.5), and the values of the contour integrals around the cuts in the positive direction ξ2 (s)ds = πi, ξ3 (s)ds = π i, ξ1 (s)ds = −2πi, which follow from (2.3). Remark. We have chosen the segment [−iz2 , iz2 ] as the cut that connects the branch points ±iz2 . We made this choice because of symmetry and ease of notation, but it is not essential. Instead we could have taken an arbitrary smooth curve lying in the region bounded by the four smooth curves in Fig. 2 (see the next section) that connect the points x0 , iz2 , −x0 , and −iz2 . For any such curve, the subsequent analysis would go through without any additional difficulty.


375

3. First Transformation Y → U We define for z ∈ C \ (R ∪ [−iz2 , iz2 ]), U (z) = diag e−n1 , e−n2 , e−n3 Y (z) 1 2 × diag en(λ1 (z)− 2 z ) , en(λ2 (z)−az) , en(λ3 (z)+az) .

(3.1)

This coincides with the first transformation in [6]. Then U solves the following RH problem: • U : C \ (R ∪ [−iz2 , iz2 ]) → C3×3 is analytic. • U satisfies the jumps  n(λ −λ ) n(λ −λ ) n(λ −λ )  e 1+ 1− e 2+ 1− e 3+ 1−  U+ = U−  0 en(λ2+ −λ2− ) 0 0 0 en(λ3+ −λ3− ) and

 1 0 0  0 U+ = U− 0 en(λ2+ −λ2− ) n(λ −λ ) 3+ 3− 0 0 e

• U (z) = I + O(1/z)

on R,

(3.2)



on [−iz2 , iz2 ].

(3.3)

as z → ∞.

The asymptotic condition follows from (1.13), (2.6) and the definition of U . The jump on the real line (3.2) takes on a different form on the four intervals (−∞, −z1 ], [−z1 , 0), (0, z1 ], and [z1 , ∞). Indeed we get from (2.7), (3.2), and the fact that n is even,   1 en(λ2+ −λ1− ) en(λ3+ −λ1− )  on (−∞, −z1 ], U+ = U− 0 (3.4) 1 0 0 0 1  n(λ −λ ) n(λ −λ )  e 1+ 1− e 2+ 1− 1  0 1 0 U+ = U−  on (−z1 , 0), (3.5) n(λ −λ ) 3+ 3− 0 0 e  n(λ −λ )  e 1+ 1− 1 en(λ3 −λ1− )  U+ = U−  (3.6) on (0, z1 ), 0 en(λ2+ −λ2− ) 0 0 0 1   1 en(λ2 −λ1 ) en(λ3 −λ1 )  on [z1 , ∞). U+ = U− 0 (3.7) 1 0 0 0 1 Now to see what has happened it is important to know the sign of Re (λj − λk ) for j = k. Figure 2 shows the curves where Re λj = Re λk . From each of the branch points ±z1 , ±iz2 there are three curves emanating at equal angle of 2π/3. We have Re λ1 = Re λ2 on the interval [0, z1 ] and on two unbounded curves from z1 . Similarly, Re λ1 = Re λ3 on the interval [−z1 , 0] and on two unbounded curves from −z1 . We have Re λ2 = Re λ3 on the curves that emanate from ±iz2 . That is, on the vertical half-lines [iz2 , +i∞) and (−i∞, −iz2 ] and on four other curves, before

376

A.I. Aptekarev, P.M. Bleher, A.B.J. Kuijlaars 3 Re λ = Re λ 1 2 Re λ1 = Re λ3 Re λ2 = Re λ3 2

iz2

Im z

1

0

−z

−x

1

z

x

0

1

0

−iz

−1

2

−2

−3 −3

−2

−1

0 Re z

1

2

3

Fig. 2. Curves where Re λ1 = Re λ2 (dashed lines), Re λ1 = Re λ3 (dashed-dotted lines), and Re λ2 = Re λ3 (solid lines). This particular figure is for the value a = 0.4

they intersect the real axis. The points where they intersect the real axis are ±x0 for some x0 ∈ (0, z1 ). After that point we have Re λ1 = Re λ3 for the curves in the right half-plane and Re λ1 = Re λ2 for the curves in the left half-plane. Figure 2 was produced with Matlab for the value a = 0.4. The picture is similar for other values of a ∈ (0, 1). As a → 0+ or a → 1−, the imaginary branch points ±iz2 tend to the origin. Using Fig. 2 and the asymptotic behavior (2.6) we can determine the ordering of Re λj , j = 1, 2, 3 in every domain in the plane. Indeed, in the domain on the right, bounded by the two unbounded curves emanating from z1 , we have Re λ1 > Re λ2 > Re λ3 because of (2.6). Then if we go to a neighboring domain, we pass a curve where Re λ1 = Re λ2 , and so the ordering changes to Re λ2 > Re λ1 > Re λ3 . Continuing in this way, and also taking into account the cuts that we have for the λj ’s, we find the ordering in any domain. Inspecting the jump matrices for U in (3.3)–(3.7), we then find the following: (a) The non-zero off-diagonal entries in the jump matrices in (3.4) and (3.7) are exponentially small, and the jump matrices tend to the identity matrix as n → ∞. (b) The non-constant diagonal entries in the jump matrices in (3.5) and (3.6) have modulus one, and they are rapidly oscillating for large n. (c) The (1, 2)-entry in the jump matrix in (3.5) is exponentially decreasing on (−z1 , −x0 ), but exponentially increasing on (−x0 , 0) as n → ∞. Similarly, the (1, 3)-entry in the jump matrix in (3.6) is exponentially decreasing on (x0 , z1 ), and exponentially increasing on (0, x0 ).


377

(d) The entries in the jump matrix in (3.3) are real. The (2, 2)-entry is exponentially increasing as n → ∞, and the (3, 3)-entry is exponentially decreasing. The exponentially increasing entries observed in items (c) and (d) are undesirable, and this might lead to the impression that the first transformation Y → U was not the right thing to do. However, after the second transformation which we do in the next section, all exponentially increasing entries miraculously disappear. 4. Second Transformation U → T The second transformation involves the global opening of lenses already mentioned in the introduction. It is needed to turn the exponentially increasing entries in the jump matrices into exponentially decreasing ones. Let be a closed curve, consisting of a part in the left half-plane from −iz2 to iz2 , symmetric with respect to the real axis, plus its mirror image in the right half-plane. The part in the left half-plane lies entirely in the region where Re λ2 < Re λ3 and it intersects the negative real axis in a point −x ∗ with x ∗ > z1 , see Fig. 3. So avoids the region bounded by the curves from ±iz2 to ±x0 . In a neighborhood of iz2 we take to be the analytic continuation of the curves where Re λ2 = Re λ3 . As a result, this means that λ2 − λ3

is real on in a neighborhood of iz2 .

(4.1)

3

2

Σ 1

iz

Im z

2

*

0

−x

−z1

z1

*

x

−iz

2

−1

−2

−3

−3

−2

−1

0 Re z

1

2

3

Fig. 3. Contour which is such that Re λ2 < Re λ3 on the part of in the left half-plane and Re λ2 > Re λ3 on the part of in the right half-plane

378


This will be convenient for the construction of the local parametrix in Sect. 7. The contour encloses a bounded domain and we make the second transformation in that domain only. So we put T = U outside and inside we put   1 0 0 1 0 for Re z < 0 inside , T = U 0 n(λ −λ ) 2 3 1 0 −e   (4.2) 10 0 T = U 0 1 −en(λ3 −λ2 )  for Re z > 0 inside . 00 1 Then T is defined and analytic outside the contours shown in Fig. 4. Using the jumps for U and the definition (4.2) we calculate the jumps for T on any part of the contour. We get different expressions for six real intervals, for the vertical segment [−iz2 , iz2 ], and for (oriented clockwise) in the left and right half-planes. The result is that T satisfies the following RH problem: • T : C \ (R ∪ [−iz2 , iz2 ] ∪ ) → C3×3 is analytic. • T satisfies the following jump relations on the real line   1 en(λ2+ −λ1− ) en(λ3+ −λ1− )  on (−∞, −x ∗ ], T+ = T− 0 1 0 0 0 1   n(λ −λ ) 1 0 e 3+ 1−  on (−x ∗ , −z1 ], T+ = T− 0 1 0 00 1  n(λ −λ )  e 1+ 1− 0 1  0 1 0 on (−z1 , 0), T+ = T−  0 0 en(λ3+ −λ3− )  n(λ −λ )  e 1+ 1− 1 0 T+ = T−  on (0, z1 ), 0 en(λ2+ −λ2− ) 0 0 0 1

(4.3)

(4.4)

(4.5)

(4.6)

2

Σ

1.5

iz

2

1

•

Im z

0.5

•

0 *

−x

•

•

−z1

z1

• *

x

−0.5 •

−1

−iz

2

−1.5

−2 −4

−3

−2

−1

0 Re z

1

2

Fig. 4. T has jumps on the real line, the interval [−iz2 , iz2 ] and on

3

4




1 en(λ2 −λ1 )  T+ = T− 0 1 0 0  1 en(λ2 −λ1 )  T+ = T− 0 1 0 0

 0 0 1

on [z1 , x ∗ ),

 en(λ3 −λ1 )  0 1

The jump on the vertical segment is   1 0 0  1 T+ = T− 0 0 n(λ −λ ) 3+ 3− 0 −1 e

379

(4.7)

on [z1 , ∞).

(4.8)

on [−iz2 , iz2 ].

(4.9)

The jumps on are 

1 T+ = T− 0 0  1 T+ = T− 0 0

0 1

en(λ2 −λ3 )

 0 0 1 

0 0 1 en(λ3 −λ2 )  0 1

on {z ∈ | Re z < 0},

(4.10)

on {z ∈ | Re z > 0}.

(4.11)

• T (z) = I + O(1/z) as z → ∞. Now the jump matrices are nice. Because of our choice of we have that the jump matrices in (4.10) and (4.11) converge to the identity matrix as n → ∞. Also the jump matrices in (4.3), (4.4), (4.7) and (4.8) converge to the identity matrix as n → ∞. The (3,3)-entry  in the jump matrix in (4.9) is exponentially small, so that this matrix tends 1 0 0 to 0 0 1. 0 −1 0 The jump matrices in (4.6) and (4.7) have oscillatory entries on the diagonal, and they are turned into exponential decaying off-diagonal entries by opening a (local) lens around (−z1 , z1 ). This is the next transformation. 5. Third Transformation T → S We are now going to open up a lens around (z1 , z1 ) as in Fig. 5. There is no need to treat 0 as a special point. The jump matrix on (−z1 , 0), see (4.5), has factorization   n(λ −λ ) e 1 3+ 0 1  0 1 0 T−−1 T+ =  n(λ −λ ) − 1 3 0 0e     1 00 1 00 0 01 0 1 0 , 0 1 0  0 1 0  (5.1) = −λ ) −λ ) n(λ n(λ − + 3 3 1 1 −1 0 0 01 e 01 e

380

A.I. Aptekarev, P.M. Bleher, A.B.J. Kuijlaars 2

Σ

1.5

iz2

1

•

Im z

0.5

• •

0

iy*

•

−x*

•

−z

z

1

•

−0.5

1

•

x*

*

−iy

•

−1

−iz2

−1.5

−2 −4

−3

−2

−1

0 Re z

1

2

3

4

Fig. 5. Opening of lens around [−z1 , z1 ]. The new matrix-valued function S has jumps on the real line, the interval [−iz2 , iz2 ], on , and on the upper and lower lips of the lens around [−z1 , z1 ]

and the jump matrix on (0, z1 ), see (4.6), has factorization 

en(λ1 −λ2 )+ T−−1 T+ =  0 0  1 = en(λ1 −λ2 )− 0

1

 0 0 1

en(λ1 −λ2 )− 0    1 00 00 0 10 1 0 −1 0 0 en(λ1 −λ2 )+ 1 0 . 0 01 01 0 01

(5.2)

We open up the lens on [−z1 , z1 ] and we make sure that it stays inside . We assume that the lens is symmetric with respect to the real and imaginary axis. The point where the upper lip intersects the imaginary axis is called iy ∗ . Then we define S = T outside the lens and   1 00 0 1 0 S=T in upper part of the lens in left half-plane, −λ ) n(λ 1 3 −e 01   1 00 0 1 0 S=T in lower part of the lens in left half-plane, n(λ −λ ) 1 3 e 01   (5.3) 1 00 S = T −en(λ1 −λ2 ) 1 0 in upper part of the lens in right half-plane, 0 01   1 00 S = T en(λ1 −λ2 ) 1 0 in lower part of the lens in right half-plane. 0 01 Outside the lens, the jumps for S are as those for T , while on [−z1 , z1 ] and on the upper and lower lips of the lens, the jumps are according to the factorizations (5.1) and (5.2). The result is that S satisfies the following RH problem:


381

• S is analytic outside the real line, the vertical segment [−iz2 , iz2 ], the curve , and the upper and lower lips of the lens around [−z1 , z1 ]. • S satisfies the following jumps on the real line:   1 en(λ2+ −λ1− ) en(λ3+ −λ1− )  S+ = S− 0 on (−∞, −x ∗ ], (5.4) 1 0 0 0 1   1 0 en(λ3+ −λ1− )  on (−x ∗ , −z1 ], S+ = S− 0 1 (5.5) 0 00 1   0 01 S+ = S−  0 1 0 on (−z1 , 0), (5.6) −1 0 0   0 10 S+ = S− −1 0 0 on (0, z1 ), (5.7) 0 01   1 en(λ2 −λ1 ) 0 on [z1 , x ∗ ), S+ = S− 0 (5.8) 1 0 0 0 1   1 en(λ2 −λ1 ) en(λ3 −λ1 )  on [x ∗ , ∞). S+ = S− 0 (5.9) 1 0 0 0 1 S has the following jumps on the segment [−iz2 , iz2 ]:   1 0 0  1 S+ = S− 0 0 on (−iz2 , −iy ∗ ), −λ ) n(λ 0 −1 e 3+ 3−   1 0 0  0 0 1 S+ = S−  on (−iy ∗ , 0), n(λ −λ ) n(λ −λ ) e 1 3− −1 e 3+ 3−   1 0 0  0 0 1 S+ = S−  on (0, iy ∗ ), n(λ −λ ) n(λ −λ ) −e 1 3− −1 e 3+ 3−   1 0 0  1 S+ = S− 0 0 on (iy ∗ , iz2 ). n(λ −λ ) 0 −1 e 3+ 3− The jumps on are



1 S+ = S− 0 0  1 S+ = S− 0 0

0 1

en(λ2 −λ3 )

 0 0 1 

0 0 1 en(λ3 −λ2 )  0 1

(5.10)

(5.11)

(5.12)

(5.13)

on {z ∈ | Re z < 0},

(5.14)

on {z ∈ | Re z > 0}.

(5.15)

382


Finally, on the upper and lower lips of the lens, we find jumps   1 00 0 1 0 on the lips of the lens in the left half-plane. S + = S−  en(λ1 −λ3 ) 0 1 (5.16)   1 00 S+ = S− en(λ1 −λ2 ) 1 0 on the lips of the lens in the right half-plane. 0 01 (5.17) • S(z) = I + O(1/z)

as z → ∞.

So now we have 14 different jump matrices (5.4)–(5.17). As n → ∞, all these jumps have limits. Most of the limits are the identity matrix, except for the jumps on (−z1 , z1 ), see (5.6) and (5.7), and on (−iz2 , iz2 ), see (5.10)–(5.13). In the next section we will solve explicitly the limiting model RH problem. The solution to the model problem will be further used in the construction of parametrix away from the branch points. 6. Parametrix Away from Branch Points The model RH problem is the following. Find N such that • N : C \ ([−z1 , z1 ] ∪ [−iz2 , iz2 ]) → C3×3 is analytic. • N satisfies the jumps   0 01 N+ = N−  0 1 0 on [−z1 , 0), −1 0 0   0 10 N+ = N− −1 0 0 on (0, z1 ], 0 01   1 0 0 N+ = N− 0 0 1 on [−iz2 , iz2 ]. 0 −1 0 • N(z) = I + O(1/z)

(6.1)

(6.2)

(6.3)

as z → ∞.

To solve the model RH problem we lift it to the Riemann surface (2.1) with the sheet structure as in Fig. 1, see also [6, 24], where the same technique was used. Consider to that end the range of the functions ξk on the complex plane, k = ξk (C) for k = 1, 2, 3. Then 1 , 2 , 3 give a partition of the complex plane into three regions, see Fig. 6. In this figure q, p and p0 are such that q = ξ1 (z1 ) = ξ2 (z1 ) = −ξ1 (−z1 ) = −ξ3 (−z1 ), ip = −ξ2 (iz2 ) = −ξ3 (iz2 ) = ξ2 (−iz2 ) = ξ3 (−iz2 ), ip0 = ξ1+ (0) = −ξ1− (0).

(6.4)


383

Ω1

ip 0 Γ ip Ω3 −q

Ω2 a

−a

q

−ip

−ip 0 Fig. 6. Partition of the complex ξ -plane

Let be the boundary of 1 . Then we have ξ1± ([−z1 , z1 ]) = ∩ {±Im z ≥ 0}, ξ2− ([−iz2 , 0]) = [ip, ip0 ], ξ2− ([0, iz2 ]) = [−ip0 , −ip], ξ3− ([−iz2 , 0]) = [ip, 0], ξ3− ([0, iz2 ]) = [0, −ip],

(6.5)

and ξ2± (iy) = ξ3∓ (iy) for −z2 ≤ y ≤ z2 . According to our agreement, on the interval −iz2 ≤ y ≤ iz2 the minus side is on the right. We are looking for a solution N in the following form:  N1 (ξ1 (z)) N1 (ξ2 (z)) N1 (ξ3 (z)) N (z) = N2 (ξ1 (z)) N2 (ξ2 (z)) N2 (ξ3 (z)) , N3 (ξ1 (z)) N3 (ξ2 (z)) N3 (ξ3 (z)) 

(6.6)

where N1 (ξ ), N2 (ξ ), N3 (ξ ) are three scalar analytic functions on C \ ( ∪ [−ip0 , ip0 ]). To satisfy the jump conditions on N (z) we need the following jump relations for Nj (ξ ), j = 1, 2, 3: Nj + (ξ ) = Nj − (ξ ), ξ ∈ ( ∩ {Im z ≤ 0}) ∪ [−ip0 , −ip] ∪ [ip, ip0 ], Nj + (ξ ) = −Nj − (ξ ), ξ ∈ ( ∩ {Im z ≥ 0}) ∪ [−ip, ip].

(6.7)

So the Nj ’s are actually analytic across the curve in the lower half-plane and on the segments [ip, ip0 ] and [−ip0 , −ip]. What remains are the curve in the upper halfplane and the segment [−ip, ip], where the functions change sign. Since ξ1 (∞) = ∞, ξ2 (∞) = a, ξ3 (∞) = −a, then to satisfy N (∞) = I we require N1 (∞) = 1, N2 (∞) = 0, N3 (∞) = 0,

N1 (a) = 0, N2 (a) = 1, N3 (a) = 0,

N1 (−a) = 0; N2 (−a) = 0; N3 (−a) = 1.

(6.8)

384


Thus, we obtain three scalar RH problems on N1 , N2 , N3 . Equations (6.7)–(6.8) have the following solution: N1 (ξ ) =

ξ 2 − a2 (ξ 2

+ p 2 )(ξ 2

− q 2)

,

N2,3 (ξ ) = c2,3

ξ ±a (ξ 2

+ p 2 )(ξ 2 − q 2 )

, (6.9)

with cuts at ∩ {Im ξ ≥ 0} and [−ip, ip]. The constants c2,3 are determined by the equations N2,3 (±a) = 1. We have that (ξ 2 + p 2 )(ξ 2 − q 2 ) = ξ 4 − (1 + 2a 2 )ξ 2 + (a 2 − 1)a 2 ≡ R(ξ ; a),

(6.10)

− √i

. Thus, the solution to the model RH and as in Sect. 6 of [6], we obtain c2 = c3 = 2 problem is given by   ξ12 (z) − a 2 ξ22 (z)−a 2 ξ32 (z)−a 2 √ √  √ R(ξ2 (z);a) R(ξ3 (z);a)  R(ξ1 (z); a)       ξ1 (z) + a √ ξ2 (z)+a √ ξ3 (z)+a  , (6.11) N(z) =  −i −i −i √   2R(ξ (z);a) 2R(ξ (z);a) 2 3 2R(ξ1 (z); a)      ξ1 (z) − a ξ2 (z)−a ξ3 (z)−a  √ √ −i √ −i 2R(ξ (z);a) −i 2R(ξ (z);a) 2 3 2R(ξ1 (z); a) with cuts on [−z1 , z1 ] and [−iz2 , iz2 ]. 7. Local Parametrices Near the branch points N will not be a good approximation to S. We need a local analysis near each of the branch points. In a small circle around each of the branch points, the parametrix P should have the same jumps as S, and on the boundary of the circle P should match with N in the sense that P (z) = N (z) (I + O(1/n))

(7.1)

uniformly for z on the boundary of the circle. The construction of P near the real branch points ±z1 makes use of Airy functions and it is the same as the one given in [6, Sect. 7] for the case a > 1. The parametrix near the imaginary branch points ±iz2 is also constructed with Airy functions. We give the construction near iz2 . We want an analytic P in a neigborhood of iz2 with jumps   1 0 0 1 0 on left contour, P+ = P− 0 0 en(λ2 −λ3 ) 1   10 0 P+ = P− 0 1 en(λ3 −λ2 )  on right contour, (7.2) 00 1   1 0 0  1 P+ = P− 0 0 on vertical part. −λ ) n(λ 3+ 3− 0 −1 e In addition we need the matching condition (7.1). Except for the matching condition (7.1), the problem is a 2 × 2 problem.


385

Let us consider λ2 − λ3 near the branch point iz2 . We know that (λ2 − λ3 )(iz2 ) = 0, see (2.7) and since ξ2 − ξ3 has square root behavior at iz2 it follows that

z

(λ2 − λ3 )(z) =

(ξ2 (s) − ξ3 (s))ds = (z − iz2 )3/2 h(z)

iz2

with an analytic function h with h(iz2 ) = 0. So we can take a 2/3-power and obtain a conformal map. To be precise, we note that arg((λ2 − λ3 )(iy)) = π/2,

for y > z2 ,

and so we define

2/3 3 f (z) = (λ2 − λ3 )(z) 4

(7.3)

such that arg f (z) = π/3,

for z = iy, y > z2 .

Then s = f (z) is a conformal map, which maps [0, iz2 ] into the ray arg s = − 2π 3 , and which maps the parts of near iz2 in the right and left half-planes into the rays arg s = 0 and arg s = 2π 3 , respectively. [Recall that λ2 − λ3 is real on these contours, see (4.1).] We choose P of the form   0 1 1 0 , 0 P (z) = E(z) n2/3 f (z) 0 e 2 n(λ2 −λ3 ) − 21 n(λ2 −λ3 ) 0 0 e

(7.4)

where E is analytic. In order to satisfy the jump conditions (7.2) we want that is defined and analytic in the complex s-plane cut along the three rays arg s = k 2πi 3 , k = −1, 0, 1, and there it has jumps 

1

+ = − 0 0  1

+ = − 0 0  1

+ = − 0 0

 0 0 1  0 0 0 1 −1 1  00 1 1 01 0 1 1

for arg s = 2π/3,

for arg s = −2π/3,

(7.5)

for arg s = 0.

Put y0 (s) = Ai(s), y1 (s) = ω Ai(ωs), y2 (s) = ω2 Ai(ω2 s) with ω = 2π/3 and Ai the standard Airy function. Then we take as

386




1

= 0 0  1

= 0 0  1

= 0 0

 0 0 y0 −y2  for 0 < arg s < 2π/3, y0 −y2  0 0 y0 y1  for − 2π/3 < arg s < 0, y0 y1  0 0 −y1 −y2  for 2π/3 < arg s < 4π/3. −y1 −y2

(7.6)

This satisfies the jumps (7.5). In order to achieve the matching (7.1) we define the prefactor E as

with

E = N L−1

(7.7)

   1 0 0 1 0 0 1  −1/6 −1/4  0 1 i  , f 0 0n L= √ 2 π 0 1/6 1/4 0 −1 i 0 n f

(7.8)

where f 1/4 has a branch cut along the vertical segment [0, iz2 ] and it is real and positive where f is real and positive. The matching condition (7.1) now follows from the asymptotics of the Airy function and its derivative 2 3/2 1 1 + O s −3/2 , Ai(s) = √ s −1/4 e− 3 s 2 π 2 3/2 1 Ai (s) = − √ s 1/4 e− 3 s 1 + O s −3/2 , 2 π 1/4

as s → ∞, | arg s| < π. On the cut we have f+  1 0 L+ = L− 0 0 0 −1

1/4

= if− . Then (7.8) gives  0 1 , 0

which is the same jump as satisfied by N , see (6.3). This implies that E = N L−1 is analytic in a punctured neighborhood of iz2 . Since the entries of N and L have at most fourth-root singularities, the isolated singularity is removable, and E is analytic. It follows that P defined by (7.4) does indeed satisfy the jumps (7.2) and the matching condition (7.1). A similar construction gives the parametrix in the neighborhood of −iz2 . 8. Fourth Transformation S → R Having constructed N and P , we define the final transformation by R(z) = S(z)N (z)−1 −1

R(z) = S(z)P (z)

away from the branch points, near the branch points.

(8.1)


387

2

Σ

1.5

iz2

1

•

Im z

0.5

• •

0

−x*

iy*

•

•

−z1

z1 •

−0.5

•

x*

*

−iy

•

−1

−iz2

−1.5

−2 −4

−3

−2

−1

0 Re z

1

2

3

4

Fig. 7. R has jumps on this system of contours

Since jumps of S and N coincide on the interval (−z1 , z1 ) and the jumps of S and P coincide inside the disks around the branch points, we obtain that R is analytic outside a system of contours as shown in Fig. 7. On the circles around the branch points there is a jump R+ = R− (I + O(1/n)),

(8.2)

which follows from the matching condition (7.1). On the remaining contours, the jump is R+ = R− (I + O(e−cn ))

(8.3)

for some c > 0. Since we also have the asymptotic condition R(z) = I + O(1/z) as z → ∞, we may conclude as in [5, Sect. 8] that

1 as n → ∞, (8.4) R(z) = I + O n(|z| + 1) uniformly for z ∈ C, see also [15–17, 23]. 9. Proof of Theorems 1.1 and 1.2 We follow the expression for the kernel Kn as we make the transformations Y → U → T → S. From (1.14) and the transformation (3.1) it follows that Kn has the following expression in terms of U , for any x, y ∈ R,   1 2 2 e−nλ1+ (x) e 4 n(x −y ) nλ2+ (y) nλ3+ (y) −1  . (9.1) U+ (y)U+ (x)  Kn (x, y) = e 0 0e 2πi(x − y) 0 Then from (4.2) we obtain for y ≥ 0 inside the contour , and for any x ∈ R,   1 2 2 e−nλ1+ (x) e 4 n(x −y ) nλ2+ (y) −1 , Kn (x, y) = 0 T+ (y)T+ (x)  0 0e 2πi(x − y) 0

(9.2)

388


and from (5.3), we have when x, y ∈ [0, z1 ),

 −nλ (x)  1 2 2 e 1+ e 4 n(x −y ) nλ1+ (y) nλ2+ (y) −1 Kn (x, y) = e 0 S+ (y)S+ (x) e−nλ2+ (x)  . (9.3) −e 2π i(x − y) 0

Since λ1+ and λ2+ are each others complex conjugates on [0, z1 ), we can rewrite (9.3) for x, y ∈ [0, z1 ) as Kn (x, y) =

en(h(y)−h(x)) ni Im λ1+ (y) −ni Im λ1+ (y) e 0 −e 2πi(x − y)  −ni Im λ (x)  1+ e −1 (x)  , ni Im λ  1+ ×S+ (y)S+ (x) e 0

(9.4)

where 1 h(x) = Re λ1+ (x) − x 2 . 4

(9.5)

Note that (9.4) is exactly the same as Eq. (5.14) in [6]. Therefore we can almost literally follow the proofs in Sect. 9 of [6] to complete the proof of Theorem 1.1 and 1.2. Indeed as in [6] the limiting mean density (1.15) follows from (9.4) and (8.4) in case x > 0, where ρ(x) =

1 Im ξ1+ (x), π

x ∈ R.

(9.6)

The case x < 0 follows in the same way and also by symmetry. Recalling that the choice of the cut [−iz2 , iz2 ] was arbitrary as remarked at the end of Sect. 2, we note that we might as well have done the asymptotic analysis on a contour that does not pass through 0, so that we obtain (1.15) for x = 0 as well. The statement in part (a) on the behavior of ρ follows immediately from (9.6) and the properties of ξ1 as the inverse mapping of (2.2). This completes the proof of Theorem 1.1. The proof of part (a) of Theorem 1.2 for the case x0 > 0 follows from (9.4) and (8.4) exactly as in Sect. 9 of [6]. The case x0 < 0 follows by symmetry, and the case x0 = 0 follows as well, since we might have done the asymptotic analysis on a cut different from [−iz2 , iz2 ], as just noted above. The proof of part (b) follows as in [6] as well. Note however that the proof of part (b) relies on the local parametrix at the branch point z1 , which we have not specified explicitly in Sect. 7. However, the formulas are the same as those in [6] and the proof can be copied. This completes the proof of Theorem 1.2. References 1. Aptekarev, A.I., Branquinho, A., Van Assche, W.: Multiple orthogonal polynomials for classical weights. Trans. Amer. Math. Soc. 355, 3887–3914 (2003) 2. Baik, J.: Random vicious walks and random matrices. Commun. Pure Appl. Math. 53, 1385–1410 (2000) 3. Bleher, P., Its, A.: Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and the universality in the matrix model. Ann. Math. 150, 185–266 (1999) 4. Bleher, P., Its, A.: Double scaling limit in the random matrix model. The Riemann-Hilbert approach. Commun. Pure Appl. Math. 56, 433–516 (2003)


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5. Bleher, P.M., Kuijlaars, A.B.J.: Random matrices with external source and multiple orthogonal polynomials. Internat. Math. Research Notices 2004, 3, 109–129 (2004) 6. 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) 7. Bleher, P.M., Kuijlaars, A.B.J.: Integral representations for multiple Hermite and multiple Laguerre polynomials. http://arxiv.org/abs/math.CA/0406616, 2004 to appear in Annales de l’Institut Fourier 8. Bleher, P.M., Kuijlaars, A.B.J.: Large n limit of Gaussian random matrices with external source, Part III: double scaling limit in the critical case. In preparation 9. Brézin, E., Hikami, S.: Spectral form factor in a random matrix theory. Phys. Rev. E 55, 4067–4083 (1997) 10. Brézin, E., Hikami, S.: Correlations of nearby levels induced by a random potential. Nucl. Phys. B 479, 697–706 (1996) 11. Brézin, E., Hikami, S.: Extension of level spacing universality. Phys. Rev. E 56, 264–269 (1997) 12. Brézin, E., Hikami, S.: Universal singularity at the closure of a gap in a random matrix theory. Phys. Rev. E 57, 4140–4149 (1998) 13. Brézin, E., Hikami, S.: Level spacing of random matrices in an external source. Phys. Rev. E 58, 7176–7185 (1998) 14. Daems, E., Kuijlaars, A.B.J.: A Christoffel-Darboux formula for multiple orthogonal polynomials. J. Approx. Theory 130, 190–202 (2004) 15. Deift, P.: Orthogonal Polynomials and Random Matrices: a Riemann-Hilbert approach. Courant Lecture Notes in Mathematics, Vol. 3, Providence R.I.: Amer. Math. Soc., 1999 16. Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R., Venakides, S., Zhou, X.: Uniform asymptotics of polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Commun. Pure Appl. Math. 52, 1335–1425 (1999) 17. Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R., Venakides, S., Zhou, X.: Strong asymptotics of orthogonal polynomials with respect to exponential weights. Commun. Pure Appl. Math 52, 1491– 1552 (1999) 18. Deift, P., Zhou, X.: A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation. Ann. Math. 137, 295–368 (1993) 19. Itzykson, C., Zuber, J.B.: The planar approximation II. J. Math. Phys. 21, 411–421 (1980) 20. Johansson, K.: Non-intersecting paths, random tilings and random matrices. Probab. Theo Related Fields 123, 225–280 (2002) 21. Karlin, S., McGregor, J.: Coincidence probabilities. Pacific J. Math. 9, 1141–1164 (1959) 22. Katori, M., Tanemura, H.: Scaling limit of vicious walks and two-matrix model. Phys. Rev. E 66, Art. No. 011105 (2002) 23. Kuijlaars, A.B.J.: Riemann-Hilbert analysis for orthogonal polynomials. In: Orthogonal Polynomials and Special Functions E. Koelink, W. Van Assche, (eds), Lecture Notes in Mathematics, Vol. 1817, Berlin-Heiderberg-New York, Springer-Verlag, 2003, pp. 167–210 24. Kuijlaars, A.B.J., Van Assche, W., Wielonsky, F.: Quadratic Hermite-Padápproximation to the exponential function: a Riemann-Hilbert approach. http://Constr.org/list/math.CA/0302357, 2003 Approx. 21, 351–412 (2005) 25. Kuijlaars, A.B.J., Vanlessen, M.: Universality for eigenvalue correlations from the modified Jacobi unitary ensemble. Internat. Math. Research Notices 2002, 1575–1600 (2002) 26. Kuijlaars, A.B.J., Vanlessen, M.: Universality for eigenvalue correlations at the origin of the spectrum. Commun. Math. Phys. 243, 163–191 (2003) 27. Mehta, M.L.: Random Matrices. 2nd edition, Boston: Academic Press, 1991 28. Nagao, T., Forrester, P.J.: Vicious random walkers and a discretization of Gaussian random matrix ensembles. Nuclear Phys. B 620, 551–565 (2002) 29. Pastur, L.A.: The spectrum of random matrices (Russian). Teoret. Mat. Fiz. 10, 102–112 (1972) 30. Tracy, C.A., Widom, H.: The Pearcey process. http://arxiv.org/abs/math.PR/0412005, 2004 31. Van Assche, W., Coussement, E.: Some classical multiple orthogonal polynomials. J. Comput. Appl. Math. 127, 317–347 (2001) 32. Van Assche, W., Geronimo, J.S., Kuijlaars, A.B.J.: Riemann-Hilbert problems for multiple orthogonal polynomials. In: Special Functions 2000: Current Perspectives and Future Directions, J. Bustoz et al., (eds), Dordrecht: Kluwer, 2001, pp. 23–59 33. Zinn-Justin, P.: Random Hermitian matrices in an external field. Nucl. Phys. B 497, 725–732 (1997) 34. Zinn-Justin, P.: Universality of correlation functions of Hermitian random matrices in an external field. Commun. Math. Phys. 194, 631–650 (1998) Communicated by J.L. Lebowitz

Commun. Math. Phys. 259, 391–411 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1348-z

Communications in


Abelianizing Vertex Algebras Haisheng Li1,2, 1 2

Department of Mathematical Sciences, Rutgers University, Camden, NJ 08102, USA Department of Mathematics, Harbin Normal University, Harbin, China

Received: 10 September 2004 / Accepted: 25 November 2004 Published online: 12 April 2005 – © Springer-Verlag 2005

Abstract: To every vertex algebra V we associate a canonical decreasing sequence of subspaces and prove that the associated graded vector space gr(V ) is naturally a vertex Poisson algebra, in particular a commutative vertex algebra. We establish a relation between this decreasing sequence and the sequence Cn introduced by Zhu. By using the (classical) algebra gr(V ), we prove that for any vertex algebra V , C2 -cofiniteness implies Cn -cofiniteness for all n ≥ 2. We further use gr(V ) to study generating subspaces of certain types for lower truncated Z-graded vertex algebras. 1. Introduction Just as with classical (associative or Lie) algebras, abelian or commutative vertex algebras (should be) are the simplest objects in the category of vertex algebras. It was known (see [B]) that commutative vertex algebras exactly amount to differential algebras, namely unital commutative associative algebras equipped with a derivation. Related to the notion of a commutative vertex algebra, is the notion of a vertex Poisson algebra (see [FB]), where a vertex Poisson algebra structure combines a commutative vertex algebra structure, or equivalently, a differential algebra structure, with a vertex Lie algebra structure (see [K, P]). As it was shown in [FB], vertex Poisson algebras can be considered as classical limits of vertex algebras. In the classical theory, a well known method to abelianize an associative algebra is to use a good increasing filtration and then consider the associated graded vector space. A typical example is the universal enveloping algebra U (g) of a Lie algebra g with the filtration {Un }, where for n ≥ 0, Un is linearly spanned by the vectors a1 · · · am for m ≤ n, a1 , . . . , am ∈ g. In this case, the associated graded algebra grU (g) is naturally a Poisson algebra and the well known Poincaré-Birkhoff-Witt theorem says that the associated graded Poisson algebra grU (g) is canonically isomorphic to the symmetric

Partially supported by an NSA grant

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algebra S(g) which is also a Poisson algebra. This result and the canonical isomorphism have played a very important role in Lie theory. Motivated by this classical result, in [Li2] we introduced and studied a notion of what we called good increasing filtration for a vertex algebra V and we proved that the associated graded vector space grV of V with respect to a good increasing filtration isnaturally a vertex Poisson algebra. Furthermore, for any N-graded vertex algebra V = n∈N V(n) with V(0) = C1, we constructed a canonical good increasing filtration of V . This increasing filtration was essentially used in [KL, GN, Bu1, 2, ABD and NT] in the study on generating subspaces of V with a certain property analogous to the well known Poincaré-Birkhoff-Witt spanning property. In this paper, we introduce and study “good” decreasing filtrations for vertex algebras. To any vertex algebra V we associate a canonical decreasing sequence E of subspaces En for n ≥ 0 and we prove that the associated graded vector space gr E (V ) is naturally an N-graded vertex Poisson algebra, where for n ∈ Z, En is linearly spanned by the vectors (1)

(r)

u−1−k1 · · · u−1−kr v for r ≥ 1, u(i) , v ∈ V , ki ≥ 0 with k1 + · · · + kr ≥ n. Notice that unlike the increasing filtration which uses the weight grading, this decreasing sequence uses only the vertex algebra structure. For any vertex algebra V , there has been a fairly well known decreasing sequence C = {Cn }n≥2 introduced by Zhu [Z1, 2], where for n ≥ 2, Cn is linearly spanned by the vectors u−n v for u, v ∈ V . The notion of C2 was introduced and used in the fundamental study of Zhu on modular invariance, where the finiteness of dim V /C2 played a crucial role. It was shown in [Z2] that V /C2 has a natural Poisson algebra structure. In this paper, we relate our decreasing sequence E with Zhu’s sequence C. In particular, we show that C2 = E1 and C3 = E2 . We then show that the degree zero subspace E0 /E1 of gr E (V ), which is naturally a Poisson algebra, is exactly Zhu’s Poisson algebra V /C2 . We further show that gr E (V ) as a differential algebra is generated by the degree zero subalgebra V /C2 . As an application, we show that for any vertex algebra V , if V is C2 -cofinite, then V is En -cofinite and Cn+2 -cofinite for all n ≥ 0. Similarly we show that if V is a C2 -cofinite vertex algebra and if W is a C2 -cofinite V -module, then W is Cn -cofinite for all n ≥ 2. Under the assumption that V is an N-graded vertex algebra with dim V(0) = 1, it has been proved before by [GN] (see also [NT, Bu1, 2]) that C2 -cofiniteness implies Cn+2 -cofinite for all n ≥ 0. On the other hand, the original method of [GN] and [KL] used this assumption in an essential way. As we show in this paper, for certain vertex algebras, both sequences E and C are trivial in the sense that En = Cn+2 = V for all n ≥ 0. On the other hand,by using the connection between the two decreasing sequences we prove that if V = n≥t V(n) is a lower truncated Z-graded vertex algebra such as a vertex operator algebra in the sense of [FLM] and [FHL], then for any k, Cn , En ⊂ m≥k V(m) for n sufficiently large. Consequently, ∩n≥0 En = ∩n≥0 Cn+2 = 0. (In this case, both sequences are filtrations.) Furthermore, using this result and gr E (V ) we show that if a graded subspace U of V gives rises to a generating subspace of V /C2 as an algebra, then U generates V with a certain spanning property. Similar results have been obtained before in [KL, GN, Bu1, 2 and NT] under a stronger condition. This paper is organized as follows: In Sect. 2, we define the sequence E and show that the associated graded vector space is an N-graded vertex Poisson algebra. In Sect. 3, we

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relate the sequences E and C. In Sect. 4, we study generating subspaces of certain types for lower truncated Z-graded vertex algebras. 2. Decreasing Sequence E and the Vertex Poisson Algebra grE (V ) In this section we first recall the definition of a vertex Poisson algebra from [FB] and we then construct a canonical decreasing sequence E for each vertex algebra V and show that the associated graded vector space gr E V is naturally a vertex Poisson algebra. We also show that if V is an N-graded vertex algebra, then the sequence E is indeed a filtration of V . Let V be a vertex algebra. We have Borcherds’ commutator formula and iterate formula: m [um , vn ] = (ui v)m+n−i , (2.1) i i≥0 m (um v)n w = um−i vn+i w − (−1)m vm+n−i ui w (−1)i (2.2) i i≥0

for u, v, w ∈ V , m, n ∈ Z. Define a (canonical) linear operator D on V by D(v) = v−2 1

for v ∈ V .

(2.3)

Then Y (v, x)1 = ex D v

for v ∈ V .

(2.4)

Furthermore, [D, vn ] = (Dv)n = −nvn−1

(2.5)

for v ∈ V , n ∈ Z. If V is a vertex operator algebra in the sense of [FLM] and [FHL], then D = L(−1), a component of the vertex operator Y (ω, x) = n∈Z L(n)x −n−2 associated to the conformal (or Virasoro) vector ω. (See for example [LL] for an exposition of such facts.) A vertex algebra V is called a commutative vertex algebra if [um , vn ] = 0

for u, v ∈ V , m, n ∈ Z.

(2.6)

It is well known (see [B, FHL]) that (2.6) is equivalent to un = 0

for u ∈ V , n ≥ 0.

(2.7)

Remark 2.1. Let A be any unital commutative associative algebra with a derivation d. Then one has a commutative vertex algebra structure on A with Y (a, x)b = (exd a)b for a, b ∈ A and with the identity 1 as the vacuum vector (see [B]). On the other hand, let V be any commutative vertex algebra. Then V is naturally a commutative associative algebra with u · v = u−1 v for u, v ∈ V and with 1 as the identity and with D as a derivation. Furthermore, Y (u, x)v = (ex D u)v for u, v ∈ V . Therefore, a commutative vertex algebra exactly amounts to a unital commutative associative algebra equipped with a derivation, which is often called a differential algebra.

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A vertex algebra V equipped with a Z-grading V = n∈Z V(n) is called a Z-graded vertex algebra if 1 ∈ V(0) and if for u ∈ V(k) with k ∈ Z and for m, n ∈ Z, um V(n) ⊂ V(n+k−m−1) . (2.8) We say that a Z-graded vertex algebra V = n∈Z V(n) is lower truncated if V(n) = 0 for n sufficiently small. In particular, every vertex operator algebra in the sense of [FLM] and [FHL] is a lower truncated Z-graded vertex algebra. An N-graded vertex algebra is defined in the obvious way. We say that a vertex algebra V is Z-gradable (N-gradable) if there exists a Z-grading (N-grading) such that V becomes Z-graded (N-graded) vertex algebra. We see that a commutative Z-graded vertex algebra is naturally a Z-graded differential algebra. The following definition of the notion of vertex Lie algebra is due to [K and P]: Definition 2.2. A vertex Lie algebra is a vector space V equipped with a linear operator D and a linear map Y− : V → Hom (V , x −1 V [x −1 ]), vn x −n−1 v → Y− (v, x) =

(2.9)

n≥0

such that for u, v ∈ V , m, n ∈ N, (Dv)n = −nvn−1 , m 1 um v = (−1)m+i+1 D i vm+i u, i! i=0 m m [um , vn ] = (ui v)m+n−i . i

(2.10) (2.11)

(2.12)

i=0

A module (see [K]) for a vertex Lie algebra V is a vector space W equipped with a linear map Y−W : V → Hom (W, x −1 W [x −1 ]), vn x −n−1 v → Y−W (v, x) =

(2.13)

n≥0

such that (2.10) and (2.12) hold. Recall the following notion of vertex Poisson algebra from [FB] (cf. [DLM]): Definition 2.3. A vertex Poisson algebra is a commutative vertex algebra A, or equivalently, a (unital) commutative associative algebra equipped with a derivation ∂, equipped with a vertex Lie algebra structure (Y− , ∂) such that Y− (a, x) ∈ x −1 (Der A)[[x −1 ]]

for a ∈ A.

(2.14)

A module for a vertex Poisson algebra A is a vector space W equipped with a module structure for A as an associative algebra and a module structure for A as a vertex Lie algebra such that Y−W (u, x)(vw) = (Y−W (u, x)v)w + vY−W (u, x)w for u, v ∈ V , w ∈ W .

(2.15)


395

The following result obtained in [Li2] gives a construction of vertex Poisson algebras from vertex algebras through certain increasing filtrations: Proposition 2.4. Let V be a vertex algebra and let F = {Fn }n∈Z be a good increasing filtration of V in the sense that 1 ∈ F0 , un Fs ⊂ Fr+s

(2.16)

for u ∈ Fr , r, s, n ∈ Z and un Fs ⊂ Fr+s−1

for n ≥ 0. (2.17) Then the associated graded vector space gr F V = n∈Z Fn+1 /Fn is naturally a vertex Poisson algebra with (u + Fm−1 )(v + Fn−1 ) = u−1 v + Fm+n−1 , ∂(u + Fm−1 ) = Du + Fm−1 , (ur v + Fm+n−2 )x −r−1 Y− (u + Fm−1 )(v + Fn−1 ) =

(2.18) (2.19) (2.20)

r≥0

for u ∈ Fm , v ∈ Fn with m, n ∈ Z. Furthermore, the following construction of good increasing filtrations was also given in [Li2]: an N-graded vertex algebra such that V(0) = C1. Theorem 2.5. Let V = n∈N V(n) be Let U be a graded subspace of V+ = n≥1 V(n) such that (1)

(r)

V = span{u−k1 · · · u−kr 1 | r ≥ 0, u(i) ∈ U, ki ≥ 1}. In particular, we can take U = V+ . For any n ≥ 0, denote by FnU the subspace of V linearly spanned by the vectors (1)

(r)

u−k1 · · · u−kr 1 for r ≥ 0, for homogeneous vectors u(1) , . . . , u(r) ∈ U and for k1 , . . . , kr ≥ 1 with wt u(1) + · · · + wt u(r) ≤ n. Then the sequence FU = {FnU } is a good increasing filtration1 of V . Furthermore, FU does not depend on U . Next, we give a construction of vertex Poisson algebras from vertex algebras using decreasing filtrations. First, we formulate the following general result, which is similar to Proposition 2.4 and which is classical in nature: Proposition 2.6. Let V be any vertex algebra and let E = {En }n≥0 be a decreasing sequence of subspaces of V such that 1 ∈ E0 and un v ∈ Er+s−n−1

for u ∈ Er , v ∈ Es , r, s ∈ N, n ∈ Z,

(2.21)

where byconvention Em = V for m < 0. Then the associated graded vector space gr E V = n≥0 En /En+1 is naturally an N-graded vertex algebra with (u + Er+1 )n (v + Es+1 ) = un v + Er+s−n 1

This good increasing filtration was denoted by EU = {EnU } in [Li2].

(2.22)

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for u ∈ Er , v ∈ Es , r, s ∈ N, n ∈ Z and with 1 + E1 ∈ E0 /E1 as the vacuum vector. Furthermore, gr E V is commutative if and only if un v ∈ Er+s−n

for u ∈ Er , v ∈ Es , r, s, n ∈ N.

(2.23)

Assume (2.21) and (2.23). Then the commutative vertex algebra gr E V is a vertex Poisson algebra where ∂(u + Er+1 ) = Du + Er+2 ,

(un v + Er+s−n+1 )x −n−1 Y− (u + Er+1 , x)(v + Es+1 ) =

(2.24) (2.25)

n≥0

for u ∈ Er , v ∈ Es with r, s ∈ N. Proof. Notice that the condition (2.21) guarantees that the operations given in (2.22) are well defined. Just as with any classical algebras, it is straightforward to check that gr E V is an N-graded vertex algebra and it is also clear that gr E V is commutative if and only if (2.23) holds. Assuming (2.21) and (2.23) we have a commutative associative N-graded algebra gr E (V ) with derivation ∂ defined by ∂(u + En+1 ) = (u + En+1 )−2 (1 + E1 ) = u−2 1 + En+2 = Du + En+2 , noticing that by (2.21) we have Du = u−2 1 ∈ En+1 . The condition (2.23) guarantees that the linear map Y− in (2.24) is well defined. It is straightforward to check that gr E (V ) equipped with Y− and ∂ is a vertex Lie algebra. Now, we check the compatibility condition (2.14). Let u ∈ Er , v ∈ Es , w ∈ Ek with r, s, k ∈ N. For m ≥ 0, using the Borcherds’ commutator formula for V we have m um (v−1 w) = v−1 (um w) + (ui v)m−1−i w i i≥0

= v−1 (um w) + (um v)−1 w +

m−1 i=0

m (ui v)m−1−i w. i

(2.26)

For 0 ≤ i ≤ m − 1, using (2.23) (twice) we have (ui v)m−1−i w ∈ Er+s+k−m+1 . Thus um (v−1 w) + Er+s+k−m+1 = v−1 (um w) + (um v)−1 w + Er+s+k−m+1 . This proves Y− (u, x) ∈ algebra.

x −1 (Der(gr

E

(V )))[x −1 ]. Therefore, gr

(2.27)

E (V ) is a vertex Poisson

In the following, for each vertex algebra we construct a canonical decreasing sequence E = {En }n≥0 which satisfies all the conditions assumed in Proposition 2.6. Definition 2.7. Let V be a vertex algebra and let W be a V -module. Define a sequence EW = {En (W )}n∈Z of subspaces of W , where for n ∈ Z, En (W ) is linearly spanned by the vectors (1)

(r)

u−1−k1 · · · u−1−kr w for r ≥ 1, u(1) , . . . , u(r) ∈ V , w ∈ W, k1 , . . . , kr ≥ 0 with k1 + · · · + kr ≥ n.

(2.28)


397

Our main task is to establish the properties (2.21) and (2.23) for the sequence E. The following are some immediate consequences: Lemma 2.8. For any V -module W we have En (W ) ⊃ En+1 (W ) for any n ∈ Z, En (W ) = W for any n ≤ 0, u−1−k En (W ) ⊂ En+k (W ) for u ∈ V , k ≥ 0, n ∈ Z.

(2.29) (2.30) (2.31)

The following gives a stronger spanning property for En (W ): Lemma 2.9. Let W be a V -module. For any n ≥ 1, we have En (W ) = span{u−1−i w | u ∈ V , i ≥ 1, w ∈ En−i (W )}.

(2.32)

Furthermore, for n ≥ 1, En (W ) is linearly spanned by the vectors (1)

(r)

u−k1 −1 · · · u−kr −1 w

(2.33)

for r ≥ 1, u(1) , . . . , u(r) ∈ V , w ∈ W, k1 , . . . , kr ≥ 1 with k1 + · · · + kr ≥ n. Proof. Notice that (2.33) follows from (2.32) and induction. Denote by En (W ) the space on the right-hand side of (2.32). To prove (2.32), we need to prove that each spanning vector of En (W ) in (2.28) lies in En (W ). Now we use induction on r. If r = 1, we have (1) k1 ≥ n ≥ 1 and w ∈ W = En−k1 (W ), so that u−1−k1 w ∈ En (W ). Assume r ≥ 2. If k1 ≥ 1, we have u−1−k1 u−1−k2 · · · u−1−kr w ∈ En (W ) (1)

(2)

(2)

(r)

(r)

because u−1−k2 · · · u−1−kr w ∈ En−k1 (W ) with k2 + · · · + kr ≥ n − k1 . If k1 = 0, (2)

(r)

we have k2 + · · · + kr ≥ n, so that u−1−k2 · · · u−1−kr w ∈ En (W ). By the inductive (2) u−1−k2

hypothesis, we have 1, w ∈ En−k (W ), we have

(r) · · · u−1−kr w

∈ En (W ). Furthermore, for any b ∈ V , k ≥

u−1 b−1−k w = b−1−k u−1 w + (1)

(1)

−1 i≥0

i

(ui b)−2−k−i w . (1)

From the definition we have u−1 w ∈ En−k (W ), so that b−1−k u−1 w ∈ En (W ). On the other hand, for i ≥ 0, we have w ∈ En−k (W ) ⊂ En−k−i−1 (W ), so that (1)

(1)

(ui b)−2−k−i w ∈ En (W ). (1)

Therefore, u−1 b−1−k w ∈ En (W ). This proves that u−1−k1 u−1−k2 · · ·u−1−kr w ∈ En (W ), completing the induction. (1)

(1)

(2)

We have the following special case of (2.21) and (2.23) for E:

(r)

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Lemma 2.10. Let W be any V -module. For a ∈ V , m, n ∈ Z, we have am En (W ) ⊂ En−m−1 (W ).

(2.34)

Furthermore, am En (W ) ⊂ En−m (W )

for m ≥ 0.

(2.35)

Proof. By (2.31), (2.34) holds for m ≤ −1. Assume m ≥ 0. Since En−m (W ) ⊂ En−m−1 (W ) it suffices to prove (2.35). We now prove the assertion by induction on n. If n ≤ 0, we have En−m (W ) = W (because n − m ≤ 0), so that am En (W ) ⊂ W = En−m (W ). Assume n ≥ 1. From (2.32), En (W ) is spanned by the vectors u−1−k w for u ∈ V , k ≥ 1, w ∈ En−k (W ). Let u ∈ V , k ≥ 1, w ∈ En−k (W ). In view of Borcherds’ commutator formula we have m am u−1−k w = u−1−k am w + (ai u)m−k−i−1 w. i i≥0

Since w ∈ En−k (W ) with n − k < n, from the inductive hypothesis we have am w ∈ En−k−m (W ), (ai u)m−k−i−1 w ∈ En−m+i (W ) ⊂ En−m (W )

for i ≥ 0.

Furthermore, using the inductive hypothesis and Lemma 2.8 we have u−1−k am w ∈ u−1−k En−k−m (W ) ⊂ En−m (W ). Therefore, am u−1−k w ∈ En−m (W ). This proves am En (W ) ⊂ En−m (W ), completing the induction and the whole proof. Now we have the following general case: Proposition 2.11. Let W be a V -module and let u ∈ Er (V ), w ∈ Es (W ) with r, s ∈ Z. Then un w ∈ Er+s−n−1 (W )

for n ∈ Z.

(2.36)

un w ∈ Er+s−n (W )

for n ≥ 0.

(2.37)

Furthermore, we have

Proof. We are going to use induction on r. By Lemma 2.10, we have un w ∈ Es−n−1 (W ). If r ≤ 0, we have r +s −n−1 ≤ s −n−1, so that un w ∈ Es−n−1 (W ) ⊂ Er+s−n−1 (W ). Assume r ≥ 0 and u ∈ Er+1 (V ). In view of (2.32) it suffices to consider u = a−2−i b for some a ∈ V , 0 ≤ i ≤ r, b ∈ Er−i (V ). By the iterate formula (2.2) we have j −2 − i a−2−i−j bn+j w − (−1)i bn−2−i−j aj w . (−1) (a−2−i b)n w = j j ≥0

(2.38) If n ≥ 0, using the inductive hypothesis (with b ∈ Er−i (V )) and Lemma 2.10 we have a−2−i−j bn+j w ∈ a−2−i−j Er−i+s−n−j (W ) ⊂ Er+1+s−n (W ), bn−2−i−j aj w ∈ bn−2−i−j Es−j (W ) ⊂ Er+1+s−n (W ),


399

from which we have that (a−2−i b)n w ∈ Er+1+s−n (W ). If n ≤ −1, we have a−2−i−j bn+j w ∈ a−2−i−j Er−i+s−n−j −1 (W ) ⊂ Er+s−n (W ), bn−2−i−j aj w ∈ bn−2−i−j Es−j (W ) ⊂ Er+1+s−n (W ) ⊂ Er+s−n (W ), so that (a−2−i b)n w ∈ Er+s−n (W ) = E(r+1+s)−n−1 (W ). This concludes the proof.

Combining Propositions 2.6 and 2.11 we immediately have: Theorem 2.12. Let V be any vertex algebra and let E = {En (V )} be the decreasing sequence defined in Definition 2.7 for V . Set

En /En+1 . (2.39) gr E (V ) = n≥0

Then gr E (V ) equipped with the multiplication defined by (a + Er+1 )(b + Es+1 ) = a−1 b + Er+s+1

(2.40)

is a commutative and associative N-graded algebra with 1 + E1 ∈ E0 /E1 as identity and with a derivation ∂ defined by ∂(u + En+1 ) = D(u) + En+2

for u ∈ En , n ∈ N.

Furthermore, gr E (V ) is a vertex Poisson algebra where (un v + Er+s−n+1 )x −n−1 Y− (a + Er+1 , x)(b + Es+1 ) =

(2.41)

(2.42)

n≥0

for a ∈ Er , b ∈ Es with r, s ∈ N. Proposition 2.13. Let W be any V -module and EW the decreasing sequence defined in Definition 2.7 for W . Then the associated graded vector space gr E (W ) = n≥0 En (W )/En+1 (W ) is naturally a module for the vertex Poisson algebra gr E (V ) with (v + Er+1 (V )) · (w + Es+1 (W )) = v−1 w + Er+s+1 (W ), (vn w + Er+s−n+1 )x −n−1 Y−W (v + Er+1 )(w + Es+1 (W )) =

(2.43) (2.44)

n≥0

for v ∈ Er (V ), w ∈ Es (W ). Proof. With the properties (2.36) and (2.37) the actions given by (2.43) and (2.44) are well defined. Clearly, 1 + E1 acts on gr E (W ) as identity and we have (u + Er+1 (V )) · ((v + Es+1 (V )) · (w + Ek+1 (W ))) = u−1 v−1 w + Er+s+k+1 (W ). By the iterate formula (2.2) we have (u−1 v)−1 w = (u−1−i v−1+i w + v−2−i ui w) , i≥0

where for i ≥ 1, using (2.37) and (2.36) we have u−1−i v−1+i w ∈ u−i−1 Es+k+1−i (W ) ⊂ Er+s+k+1 (W )

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H. Li

and for i ≥ 0, similarly we have v−2−i ui w ∈ v−2−i Es+k−i (W ) ⊂ Er+s+k+1 (W ). Thus (u−1 v)−1 w ∈ u−1 v−1 w + Er+s+k+1 (W ). This proves that gr E (W ) is a module for gr E (V ) as an associative algebra. It is straightforward to check that it is a module for the vertex Lie algebra. Other properties are clear from the proof of Proposition 2.6. Notice that so far we have not excluded the possibility that the associated sequence EV is trivial in the sense that En (V ) = V for all n ≥ 0. Indeed, as we shall see in the next section, for some vertex algebras the associated sequence E is trivial. Nevertheless, we have: Lemma 2.14. Let V = n≥0 V(n) be an N-graded vertex algebra and E = {En } be the decreasing sequence defined in Definition 2.7 for V . Then En (V ) ⊂

V(m)

for n ≥ 0.

(2.45)

m≥n

Furthermore, the associated decreasing sequence E = {En } for V is a filtration, i.e., ∩n≥0 En (V ) = 0. Proof. By definition we have E0 = V = spanned by the vectors (1)

n≥0 V(n) . For n

(2.46) ≥ 1, recall that En is linearly

(r)

u−1−k1 · · · u−1−kr v for r ≥ 1, u(1) , . . . , u(r) , v ∈ V , k1 , . . . , kr ≥ 1 with k1 + · · · + kr ≥ n. If the vectors u(1) , . . . , u(r) , v are homogeneous, we have wt

(1)

(r)

u−1−k1 · · · u−1−kr v

= wt u(1) + k1 + · · · + wt u(r) + kr + wt v ≥ k1 + · · · + kr ≥ n. This proves (2.45) for n ≥ 1. Clearly, each subspace En of V is graded. From (2.45) we immediately have (2.46). In the next section we shall generalize Lemma 2.14 from an N-graded vertex algebra to a lower truncated Z-graded vertex algebra by using a relation between the decreasing sequence E and a sequence introduced by Zhu.


401

3. The Relation Between the Sequences E and C In this section we first recall the sequence C introduced by Zhu and we then give a relation between the two decreasing sequences E and C. We show that if V is a lower truncated Z-graded vertex algebra, then both sequences are decreasing filtrations of V . The following definition is (essentially) due to Zhu ([Z1,2]): Definition 3.1. Let V be a vertex algebra and W a V -module. For any n ≥ 2 we define Cn (W ) to be the subspace of W , linearly spanned by the vectors v−n w for v ∈ V , w ∈ W . A V -module W is said to be C n -cofinite if W/Cn (W ) is finite-dimensional. In particular, if V /Cn (V ) is finite-dimensional, we say that the vertex algebra V is C n -cofinite. The following are easy consequences: Lemma 3.2. Let V be any vertex algebra, let W be a V -module and let n ≥ 2. Then Cm (W ) ⊂ Cn (W ) for m ≥ n, (3.1) u−k Cn (W ) ⊂ Cn (W ) for u ∈ V , k ≥ 0, (3.2) u−n v−k w ≡ v−k u−n w mod Cn+k (W ) for u, v ∈ V , w ∈ W, k ≥ 0. (3.3) Proof. For v ∈ V , r ≥ 2 we have v−r−1 = 1r (Dv)−r . From this we immediately have Cr+1 (W ) ⊂ Cr (W ) for r ≥ 2, which implies (3.1). Let u, v ∈ V , w ∈ W, k ≥ 0. Using the commutator formula (2.1) and (3.1) we have −k (ui v)−k−n−i w ∈ Cn (W ), u−k v−n w = v−n u−k w + i i≥0

proving that u−k Cn (W ) ⊂ Cn (W ). We also have −n (ui v)−n−k−i w ∈ Cn+k (W ), u−n v−k w − v−k u−n w = i i≥0

proving (3.3).

We also have the following more technical results: Lemma 3.3. Let V be any vertex algebra, let W be a V -module and let k ≥ 2. Then u−k Ck (W ) ⊂ Ck+1 (W )

for u ∈ V .

Proof. For u, v ∈ V , w ∈ W , in view of the iterate formula (2.2) we have (u−1 v)−2k+1 w = (u−1−i v−2k+1+i w + v−2k−i ui w) .

(3.4)

(3.5)

i≥0

Now we examine each term in (3.5). Notice that (u−1 v)−2k+1 w ∈ Ck+1 (W ) as −2k + 1 ≤ −k −1 and that v−2k−i ui w ∈ Ck+1 (W ) for i ≥ 0 as −2k −i ≤ −k −1. If i ≥ k, we have −1−i ≤ −k −1, so that u−1−i v−2k+1+i w ∈ Ck+1 (W ). For 0 ≤ i ≤ k −2, we have −2k + 1 + i ≤ −k − 1, so that v−2k+1+i w ∈ Ck+1 (W ). Then by Lemma 3.2 we have u−1−i v−2k+1+i w ∈ Ck+1 (W ) for 0 ≤ i ≤ k − 2. Therefore, the only remaining term u−k v−k w in (3.5) must also lie in Ck+1 (W ). This proves u−k Ck (W ) ⊂ Ck+1 (W ).

402

H. Li

Proposition 3.4. Let V be any vertex algebra, let W be a V -module and let n be any nonnegative integer. Then (1)

(r)

u−k1 · · · u−kr w ∈ Cn+2 (W )

(3.6)

for r ≥ 2n , u(1) , . . . , u(r) ∈ V , w ∈ W, k1 , . . . , kr ≥ 2. Proof. Since u−i Cn+2 (W ) ⊂ Cn+2 (W ) for u ∈ V , i ≥ 0 (by Lemma 3.2), it suffices to 1 prove the assertion for r = 2n . Also, since u−k = (k−1)! (Dk−2 u)−2 for u ∈ V , k ≥ 2, it suffices to prove the assertion for k1 = · · · = kr = 2. We are going to use induction on n. If n = 0, by definition we have v−2 w ∈ C2 (W ) for v ∈ V , w ∈ W . Assume the assertion holds for n = p, some nonnegative integer. Assume that r = 2p+1 and set s = 2p . Let u(1) , . . . , u(r) ∈ V , w ∈ W . By inductive hypothesis we have (s+1)

u−2

(r)

· · · u−2 w ∈ Cp+2 (W ),

so that (1)

(r)

(1)

(s)

u−2 · · · u−2 w ∈ u−2 · · · u−2 Cp+2 (W ).

(3.7)

Consider a typical spanning vector a−p−2 w of Cp+2 (W ) for a ∈ V , w ∈ W . Using (3.3) and (3.2) we have u−2 · · · u−2 a−p−2 w ≡ a−p−2 u−2 · · · u−2 w (1)

(s)

(1)

(s)

mod Cp+4 (W ).

(3.8)

Furthermore, by inductive hypothesis, we have u−2 · · · u−2 w ∈ Cp+2 (W ), (1)

(s)

which together with Lemma 3.3 gives a−p−2 u−2 · · · u−2 w ∈ a−p−2 Cp+2 (W ) ⊂ Cp+3 (W ). (1)

(s)

(3.9)

Thus by (3.8) we have u−2 · · · u−2 a−p−2 w ∈ Cp+3 (W ), (1)

(s)

(1)

(s)

proving that u−2 · · · u−2 Cp+2 (W ) ⊂ Cp+3 (W ).

(3.10)

Therefore, by (3.7) we have (1)

(r)

u−2 · · · u−2 w ∈ Cp+3 (W ). This finishes the induction steps and completes the proof.

The relation between the two decreasing sequences {En (W )} and {Cn (W )} is described as follows:


403

Theorem 3.5. Let W be any module for vertex algebra V and let EW = {En (W )} be the associated decreasing sequence. Then for any n ≥ 2, Cn (W ) ⊂ En−1 (W ), Em (W ) ⊂ Cn (W )

(3.11)

whenever m ≥ max{1, (n − 2)2

n−2

}.

(3.12)

Furthermore, ∩n≥0 En (W ) = ∩n≥0 Cn+2 (W ).

(3.13)

Proof. From the definitions of Cn (W ) and En−1 (W ) we immediately have Cn (W ) ⊂ En−1 (W ). Consider a generic spanning element of Em (W ) (with m ≥ 1): (1)

(r)

X = u−1−k1 · · · u−1−kr w, where r ≥ 1, u(1) , . . . , u(r) ∈ V , w ∈ W, k1 , . . . , kr ≥ 1 with k1 + · · · + kr ≥ m. (i) If ki ≥ n − 1 for some i, by (3.1) we have u−1−ki W ⊂ C−1−ki (W ) ⊂ Cn (W ) and then by (3.2) we have X ∈ Cn (W ). If r ≥ 2n−2 , by Proposition 3.4 X ∈ Cn (W ). Since k1 + · · · + kr ≥ m ≥ (n − 2)2n−2 , we have either ki ≥ n − 1 for some i or r ≥ 2n−2 . Therefore, X ∈ Cn (W ) whenever m ≥ (n − 2)2n−2 . This proves (3.12). Combining (3.12) and (3.11) we have (3.13). Corollary 3.6. For any vertex algebra V and any V -module W , we have E1 (W ) = C2 (W ), E2 (W ) = C3 (W ).

(3.14)

Proof. By (3.11) we have C2 (W ) ⊂ E1 (W ) and C3 (W ) ⊂ E2 (W ). On the other hand, by (3.12) with m = 1, n = 2 we have E1 (W ) ⊂ C2 (W ) and by (3.12) with m = 2, n = 3 we have E2 (W ) ⊂ C3 (W ). Recall the following result of Zhu [Z1, 2]: Proposition 3.7. Let V be any vertex algebra. Then V /C2 (V ) is a Poisson algebra with u¯ · v¯ = u−1 v,

[u, ¯ v] ¯ = u0 v

for u, v ∈ V ,

(3.15)

where u¯ = u + C2 (V ), and with 1 + C2 (V ) as the identity element. It is clear that the degree zero subspace E0 /E1 of gr E (V ) is a Poisson algebra where (u + E1 )(v + E1 ) = u−1 v + E1 ,

[u + E1 , v + E1 ] = u0 v + E1

for u, v ∈ V . With E0 (V ) = V and E1 (V ) = C2 (V ), we see that this Poisson algebra is nothing but Zhu’s Poisson algebra V /C2 (V ). Thus we have: Proposition 3.8. Let V be any vertex algebra. The degree zero subspace gr E (V )(0) = E0 (V )/E1 (V ) of the N-graded vertex Poisson algebra gr E (V ) is naturally a Poisson algebra which coincides with Zhu’s Poisson algebra V /C2 (V ) = E0 /E1 . The following result generalizes the result of Lemma 2.14:

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H. Li

Proposition 3.9. Let V = Then

n≥t

V(n) be a lower truncated Z-graded vertex algebra.

Cn (V ) ⊂

(3.16)

V(k)

k≥2t+n−1

for n ≥ 2. Furthermore, Em (V ) ⊂

V(k)

whenever m ≥ (n − 2)2n−2 ,

(3.17)

k≥2t+n−1

∩n≥0 En (V ) = ∩n≥2 Cn (V ) = 0.

(3.18)

Proof. For homogeneous vectors u, v ∈ V and for any n ≥ 2 we have wt (u−n v) = wt u + wt v + n − 1 ≥ 2t + n − 1. In view of this we have Cn (V ) ⊂

V(k)

k≥2t+n−1

for n ≥ 2. This proves (3.16), from which we immediately have that ∩n≥0 Cn+2 (V ) = 0. Using Theorem 3.5, we obtain (3.17) and (3.18). For the rest of this section, we consider vertex algebras whose associated decreasing sequence E is trivial. First we have: Lemma 3.10. Let V be a vertex algebra and let W be a V -module. If W = C2 (W ), then En (W ) = Cn+2 (W ) = W

for all n ≥ 0.

(3.19)

Proof. Since W = C2 (W ), we have E1 (W ) = C2 (W ) = W . Assume that Ek (W ) = W for some k ≥ 1. Then v−2 W = v−2 Ek (W ) ⊂ Ek+1 (W )

for v ∈ V .

From this we have W = C2 (W ) ⊂ Ek+1 (W ), proving Ek+1 (W ) = W . By induction, we have En (W ) = W for all n ≥ 0. In view of Theorem 3.5 we have Cn (W ) = W for all n ≥ 2. Suppose that V is a vertex algebra such that C2 (V ) = V . By Lemma 3.10 we have Cn+2 (V ) = V for n ≥ 0, so that V = ∩n≥0 Cn+2 (V ). Furthermore, if there exists a lower truncated Z-grading V = n∈Z V(n) with which V becomes an Z-graded vertex algebra, by (3.18) (Proposition 3.9) we have ∩n≥0 Cn+2 (V ) = 0, so that V = ∩n≥0 Cn+2 (V ) = 0. Therefore we have proved: Proposition 3.11. Let V be a nonzero vertex algebra such that C2 (V ) = V . Then there does not exist a lower truncated Z-grading V = n∈Z V(n) with which V becomes an Z-graded vertex algebra.


405

From [B and FLM], associated to any nondegenerate even lattice L of finite rank, we have a vertex algebra VL . Furthermore, VL is a vertex operator algebra if and only if L is positive-definite in the sense that α, α > 0 for 0 = α ∈ L. In this case, VL is N-graded by L(0)-weight (with 1-dimensional weight-zero subspace), so that Lemma 2.14 (and Proposition 3.9) applies to VL . On the other hand, we have: Proposition 3.12. Let L be a finite rank nondegenerate even lattice that is not positivedefinite and let VL be the associated vertex algebra. Then Cn+2 (VL ) = En (VL ) = VL for n ≥ 0. Furthermore, there does not exist a lower truncated Z-grading on VL with which VL becomes a lower truncated Z-graded vertex algebra. Proof. First we show that there exists α ∈ L such that α, α < 0. Since L is not positivedefinite, there exists 0 = β ∈ L such that β, β ≤ 0. If β, β = 0, that is, β, β < 0, then we can simply take α = β. Suppose β, β = 0. Since L is nondegenerate, there exists γ ∈ L such that γ , β = 0. For m ∈ Z, we have

γ + mβ, γ + mβ = γ , γ + 2m γ , β. We see that γ + mβ, γ + mβ < 0 for some m. Then we can take α = γ + mβ with the desired property. Let α ∈ L be such that α, α < 0 and set α, α = −2k with k ≥ 1. Using the explicit expression of the vertex operators in [FLM], we have (eα )−2k−1 e−α = 1, so that 1 ∈ C2k+1 (VL ) ⊂ C2 (VL ). Then v = v−1 1 ∈ C2 (VL ) for v ∈ VL . Thus C2 (VL ) = VL . By Lemma 3.10 we have Cn+2 (VL ) = En (VL ) = VL for n ≥ 0. The last assertion follows immediately from Proposition 3.11. 4. Generating Subspaces of Vertex Algebras In this section we shall use the differential algebra structure on gr E (V ) to study certain kinds of generating subspaces of lower truncated Z-graded vertex algebras. First we prove the following results for classical algebras: Lemma 4.1. Let (A, ∂) be an N-graded (unital) differential algebra such that (∂A)A = A+ , where

A(n) . (4.1) A+ = n≥1

Let S be a generating subspace of A(0) as an algebra. Then A is linearly spanned by the vectors ∂ n1 (a1 ) · · · ∂ nr (ar )

(4.2)

for r ≥ 0, n1 ≥ n2 ≥ · · · ≥ nr ≥ 0, a1 , . . . , ar ∈ S, or equivalently, S generates A as a differential algebra. In particular, A(0) generates A as a differential algebra. Furthermore, A is linearly spanned by the vectors ∂ n1 (a1 ) · · · ∂ nr (ar ) for r ≥ 1, n1 > n2 > · · · > nr ≥ 0, a1 , . . . , ar ∈ A(0) .

(4.3)

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H. Li

Proof. First, we show that A as a differential algebra is generated by A(0) . Let A be the differential subalgebra of A, generated by A(0) . We are going to show (by induction) that kn=0 A(n) ⊂ A for all k ≥ 0. From the definition, we have A(0) ⊂ A . Assume that kn=0 A(n) ⊂ A for some k ≥ 0. Consider the subspace A(k+1) of A. From our assumption, we have A(k+1) ⊂ A+ = A∂A, so A(k+1) is linearly spanned by the vectors a∂b for a ∈ A(r) , b ∈ A(s) with r + s + 1 = k + 1. For any a ∈ A(r) , b ∈ A(s) with r + s + 1 = k + 1, since r, s ≤ k (with r, s ≥ 0), by the inductive hypothesis, we have a, b ∈ A . Consequently, a∂b ∈ A . Thus A(k+1) ⊂ A . This proves that kn=0 A(n) ⊂ A for all k ≥ 0. Therefore, we have A = A , proving that A as a differential algebra is generated by A(0) . It follows that if S generates A(0) as an algebra, then S generates A as a differential algebra. For a positive integer n, let A(n) be the subspace of A(n) spanned by the vectors ∂ k1 (a1 ) · · · ∂ kr (ar )b

(4.4)

for r ≥ 1, k1 > k2 > · · · > kr ≥ 1, a1 , . . . , ar , b ∈ A(0) with k1 + · · · + kr = n. We must prove A(n) = A(n) for all n ≥ 1. For a positive integer n, denote by Pn the set of partitions of n. We now endow Pn with the reverse order of the lexicographic order on Pn . Set P = ∪n≥1 Pn . For α ∈ Pm , β ∈ Pn , combining α and β together we get a partition of m + n, which we denote by α ∗ β. Clearly, this defines an abelian semigroup structure on P . Furthermore, for α, β ∈ Pn , γ ∈ P , if α > β, then α ∗ γ > β ∗ γ . That is, the order is compatible with the multiplication. For α ∈ Pn , define Aα(n) to be the linear span of the vectors ∂ k1 (a1 ) · · · ∂ kr (ar )b for r ≥ 1, k1 ≥ k2 ≥ · · · ≥ kr ≥ 1, a1 , . . . , ar , b ∈ A(0) with k1 + · · · + kr = n and (k1 , . . . , kr ) ≤ α. Since A(0) generates A as a differential algebra, {Aα(n) } is a (finite) increasing filtration of A(n) . For a, b ∈ A(0) and k ≥ 1, we have ∂ (ab) = 2k

2k 2k i=0

which can be rewritten as 2k k ∂ (a)∂ k (b) k = ∂ (ab) − ∂ (a)b − ∂ (b)a − 2k

2k

2k

i

∂ 2k−i (a)∂ i (b),

k−1 2k i=1

i

∂ 2k−i (a)∂ i (b) + ∂ i (a)∂ 2k−i (b) . (4.5)

We see that (k, k) > (2k), (2k − i, i) for 1 ≤ i ≤ k − 1.


407

Now consider a typical element of Aα(n) , X = ∂ k1 (a1 ) · · · ∂ kr (ar )b for (k1 , . . . , kr ) ∈ Pn , a1 , . . . , ar , b ∈ A(0) with (k1 , . . . , kr ) ≤ α. If all k1 > k2 > · · · > kr , then X ∈ A(n) . Otherwise, using (4.5) we see that X∈

β

A(n) .

β · · · > kr ≥ 0. In particular, gr E (V ) as a differential algebra is generated by the subspace E0 /E1 (= V /C2 (V )). The following result generalizes a theorem of [GN] (see also [NT]): Proposition 4.4. Let V be any vertex algebra. If V is C2 -cofinite, then V is En -cofinite and Cn+2 -cofinite for any n ≥ 0. Proof. Since dim V /C2 < ∞, it follows from Corollary 4.3 that for each n ≥ 0, the degree n subspace En /En+1 of gr E (V ) is finite dimensional. Consequently, dim V /En = dim E0 /En < ∞ for all n ≥ 0. For any n ≥ 2, by (3.17) we have Em ⊂ Cn for m = (n − 2)2n−2 . Then dim V /Cn ≤ dim V /Em < ∞. Furthermore we have (cf. [Bu1, 2]): Proposition 4.5. Let V be any vertex algebra and W any V -module. If V and W are C2 -cofinite, then W is Cn -cofinite for all n ≥ 2. Proof. In the proof of Proposition 4.4, we showed that gr E (V ) is an N-graded differential algebra with finite-dimensional homogeneous subspaces. Since dim W/C2 (W ) < ∞, it follows from (4.7) that all the homogeneous subspaces of gr E (W ) are finite-dimensional. The same argument of Proposition 4.4 shows that W is Cn -cofinite for all n ≥ 2. Remark 4.6. It has been proved in [Bu1 and NT] that if V is a vertex operator algebra with nonnegative weights and with V(0) = C1 and if V is C2 -cofinite, then any irreducible V -module W is Cn -cofinite for all n ≥ 2. The following result generalizes a theorem of [GN] (cf. [Bu1-2, ABD]): Theorem 4.7. Let V = n≥t V(n) be any lower truncated Z-graded vertex algebra such as a vertex operator algebra in the sense of [FLM and FHL]. Then for any graded subspace U of V , V = U + C2 (V ) if and only if V is linearly spanned by the vectors (1)

(r)

u−n1 · · · u−nr 1

(4.9)

for r ≥ 0, n1 > · · · > nr ≥ 1, u(1) , . . . , u(r) ∈ U . Proof. Assume that V = U + C2 (V ). Denote by A the vertex Poisson algebra gr E (V ) obtained in Theorem 2.12. In particular, A is an N-graded (unital) differential algebra. Recall that A = n∈N A(n) , where A(n) = En /En+1 for n ∈ N. Let K be the subspace of V , spanned by those vectors in (4.9). Clearly, K is a graded subspace. For m ≥ 0, set Km = K ∩ Em . For any linear operator F on a vector space and for any nonnegative integer n, we set F (n) = F n /n!. From Corollary 4.3, for any m ≥ 0, Em /Em+1 is linearly spanned by the vectors ∂ (k1 ) (u(1) + E1 ) · · · ∂ (kr ) (u(r) + E1 )


409

for r ≥ 1, u(i) ∈ U, k1 > k2 > · · · > kr ≥ 0 with k1 + · · · + kr = m. By definition we have ∂ (k1 ) (u(1) + E1 ) · · · ∂ (kr ) (u(r) + E1 ) = (D(k1 ) u(1) + Ek1 +1 ) · · · (D(kr ) u(r) + Ekr +1 ) (1)

(r)

= u−1−k1 · · · u−1−kr 1 + Em+1 . It follows that Em = Km + Em+1 . Then V = E0 = K0 + K1 + · · · + Kn + En+1 ⊂ K + En+1 for any n ≥ 0. Since K and En+1 are graded subspaces and since Em ⊂ for m ≥ (n − 2)2n by (3.17), we must have

k≥2t+n−1 V(k)

V = K = K0 + K1 + K2 + · · · , proving the desired spanning property. Conversely, assume the spanning property. Notice that if r ≥ 2, we have n1 ≥ 2, so (1) (r) (1) (r) that u−n1 · · · u−nr 1 ∈ C2 (V ). If nr ≥ 2, we also have u−n1 · · · u−nr 1 ∈ C2 (V ). Then we get V ⊂ U + C2 (V ), proving V = U + C2 (V ). By slightly modifying the proof of Theorem 4.7 we immediately obtain the following result (cf. [KL]): Theorem 4.8. Let V = n≥t V(n) be a lower truncated Z-graded vertex algebra such as a vertex operator algebra in the sense of [FLM and FHL] and let S be a graded subspace of V such that {u + C2 (V ) | u ∈ S} generates V /C2 (V ) as an algebra. Then V is linearly spanned by the vectors (1)

(r)

u−n1 · · · u−nr 1 for r ≥ 0, u(1) , . . . , u(r) ∈ S, n1 ≥ · · · ≥ nr ≥ 1. Furthermore, if S is linearly ordered, V is linearly spanned by the above vectors with u(i) > u(i+1) when ni = ni+1 . Definition 4.9. Let S be a subset of a vertex algebra V . We say that S is a type 0 generating subset of V if V is the smallest vertex subalgebra containing S, S is a type 1 generating subset of V if V is linearly spanned by the vectors (1)

(r)

u−k1 · · · u−kr 1

(4.10)

for r ≥ 0, u(i) ∈ S, ki ≥ 1. S is called a type 2 generating subset of V if for any linear order on S (if S is a vector space, replace S with a basis), V is linearly spanned by the above vectors with u(i) > u(i+1) when ni = ni+1 . Remark 4.10. A type 0 generating subset is just a generating subset in the usual sense and a type 1 generating subset of V is also called a strong generating subset V in [K]. Theorem 4.11. Let V be a lower truncated Z-graded vertex algebra and let U be a graded subspace. Then the following three statements are equivalent: (a) U is a type 1 generating subspace of V . (b) U is a type 2 generating subspace of V . (c) U/C2 (V ) = {u + C2 (V ) | u ∈ U } generates V /C2 (V ) as an algebra.

410

H. Li

Proof. By definition, (b) implies (a) and by Theorem 4.8, (c) implies both (a) and (b). Now it suffices to prove that (a) implies (c). Assuming (a) we have that V /C2 (V ) is (1) (r) linearly spanned by the vectors u−k1 · · · u−kr 1 + C2 (V ) for r ≥ 0, u(i) ∈ U, ki ≥ 1. If (1)

(r)

ki ≥ 2 for some i, we have u−k1 · · · u−kr 1 ∈ C2 (V ). Then V /C2 (V ) is linearly spanned (1)

(r)

by the vectors u−1 · · · u−1 1 + C2 (V ) for r ≥ 0, u(i) ∈ U . That is, U/C2 (V ) generates V /C2 (V ) as an algebra. With Lemma 4.2, from the proof of Theorem 4.7 we immediately have: Proposition 4.12. Let V be a lower truncated Z-graded vertex algebra and let U be a graded subspace of V such that U generates V /C2 (V ) as an algebra. Let W be a lower truncated Z-graded V -module and let W 0 be a graded subspace of W such that W = W 0 + C2 (W ). Then W is spanned by the vectors (1)

(r)

u−1−k1 · · · u−1−kr w for r ≥ 1, u(1) , . . . , u(r) ∈ U, w ∈ W 0 , k1 > · · · > kr ≥ 0. References [ABD] Abe, T., Buhl, G., Dong, C.: Rationality, regularity and C2 -cofiniteness. http:// arxiv.org/list/math.QA/0204021, 2002 [AN] Abe, T., Nagatomo, K.: Finiteness of conformal blocks over the projective line. In: Vertex Operator Algebras in Mathematics and Physics, Proc. of Workshop at Fields Institute for Research in Mathematical Sciences, 2000, Berman, S., Billig, Y., Huang, Y.-Z., Lepowsky, J. (eds.), Fields Institute Communications 39, Providence, RI: Amer. Math. Soc., 2003 [B] Borcherds, R. E.: Vertex algebras, Kac-Moody algebras, and the Monster. Proc. Natl. Acad. Sci. USA 83, 3068–3071 (1986) [Bu1] Buhl, G.: A spanning set for VOA modules. J. Algebra 254, 125–151 (2002) [Bu2] Buhl, G.: Rationality and C2 -cofiniteness is regularity. Ph.D. thesis, University of California, Santa Cruz, 2003 [DLM] Dong, C., Li, H.-S., Mason, G.: Vertex Lie algebra, vertex Poisson algebras and vertex algebras. In: Recent Developments in Infinite-Dimensional Lie Algebras and Conformal Field Theory, Proceedings of an International Conference at University of Virginia, May 2000, Contemp Math. 297, 69–96 (2002) [FB] Frenkel, E., Ben-Zvi, D.: Vertex Algebras and Algebraic Curves. Mathematical Surveys and Monographs, Vol. 88, Providence, RI: Amer. Math. Soc., 2001 [FHL] Frenkel, I., Huang, Y.-Z., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. Memoirs Amer. Math. Soc. 104, 1993 [FLM] Frenkel, I., Lepowsky, J., Meurman, A.: Vertex Operator Algebras and the Monster. Pure and Appl. Math. Vol. 134, Boston: Academic Press, 1988 [FZ] Frenkel, I., Zhu,Y.: Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J. 66, 123–168 (1992) [GN] Gaberdiel, M., Neitzke, A.: Rationality, quasirationality and finite W-algebras. Commun. Math. Phys. 238, 305–331 (2003) [K] Kac, V. G.: Vertex Algebras for Beginners. University Lecture Series 10, Providence, RI: Amer. Math. Soc., 1997 [KL] Karel, M., Li, H.-S.: Certain generating subspaces for vertex operator algebras. J. Alg. 217, 393–421 (1999) [LL] Lepowsky, J., Li, H.-S.: Introduction to Vertex Operator Algebras and Their Representations. Progress in Math. 227, Boston, MA: Birkhäuser, 2003 [Li1] Li, H.-S.: Some finiteness properties of regular vertex operator algebras. J. Algebra 212, 495– 514 (1999) [Li2] Li, H.-S.: Vertex algebras and vertex Poisson algebras. Commun. Contemp. Math. 6, 61–110 (2004) [M] Miyamoto, M.: Modular invariance of vertex operator algebras satisfying C2 -cofiniteness. Duke Math. J. 122, 51–91 (2004)

Abelianizing Vertex Algebras [NT] [P] [Z1] [Z2]

411

Nagatomo, K., Tsuchiya, A.: Conformal field theories associated to regular chiral vertex operator algebras I: theories over the projective line. http://arxiv.org/list/math.QA/0206223, 2002 Primc, M.: Vertex algebras generated by Lie algebras. J. Pure Appl. Alg. 135, 253–293 (1999) Zhu, Y.-C.: Vertex operator algebras, elliptic functions and modular forms. Ph.D. thesis, Yale University, 1990 Zhu, Y.: Modular invariance of characters of vertex operator algebras. J. Amer. Math. Soc. 9, 237–302 (1996)

Communicated by Y. Kawahigashi


Communications in


The Structure of the Ladder Insertion-Elimination Lie Algebra Igor Mencattini1 , Dirk Kreimer2 1

Boston University, Department of Mathematics and Statistics, Boston University, 111 Cummington Street, Boston, MA 02215, USA. E-mail: igorre@@math.bu.edu 2 CNRS at IHES, 35, route de Chartres, 91440 Bures-sur-Yvette, France. E-mail: [email protected] Received: 17 September 2004 / Accepted: 18 November 2004 Published online: 15 April 2005 – © Springer-Verlag 2005

Abstract: We continue our investigation into the insertion-elimination Lie algebra LL of Feynman graphs in the ladder case, emphasizing the structure of this Lie algebra relevant for future applications in the study of Dyson–Schwinger equations. We work out the relation to the classical infinite dimensional Lie algebra gl + (∞) and we determine the cohomology of LL . 1. Introduction In the last few years perturbative QFT has been shown to have a rich algebraic structure [9] leading to relations with apparently unrelated sectors of mathematics like noncommutative geometry and Riemann-Hilbert like problems [4, 5]. Such extraordinary relations can be summarized, to some extent, by the existence of a commutative, non co-commutative Hopf algebra H defined on the set of Feynman diagrams. We will continue the investigation started in [15] where we discussed first relations of perturbative QFT with the representation theory of Lie algebras. In that paper we introduced the ladder Insertion-Elimination Lie algebra LL and we discussed relations of this Lie algebra with some more classical (infinite dimensonal) Lie algebras. In what follows we describe in greater detail the structure of this Insertion-Elimination Lie algebra. The plan of the paper is as follows: in Section Two we give some motivations for the relevance of the ladder insertion elimination Lie algebra LL for full QFT. In particular we stress the relation of LL to the quantum equations of motion or Dyson-Schwinger equations (DSEs). This section is meant to give the motivations from physics for the mathematical endeavor undertaken in the following sections. In Section Three we recollect some basic fact about the Lie algebra LL taken from [15]. Sections Four and Five are the core of this paper: in Sect. Four we give a structure D.K. supported by CNRS; both authors supported in parts by NSF grant DMS-0401262, Ctr. Math. Phys. at Boston Univ.; BUCMP/04-06.

414

I. Mencattini, D. Kreimer

theorem that stresses the relation of the Lie algebra LL with the classical infinite dimensional Lie algebra gl + (∞). Finally, in Sect. Five we collect some basic results about the cohomology of the Lie algebra LL . 2. The Significance of Zn,m The Lie algebra LL on generators Zn,m is an insertion elimination Lie algebra [15] obtained from these operations applied to a cocommutative and commutative Hopf algebra Hcomm built on generators (ladders) tn , n ≥ 0, (tn ) = nj=0 tj ⊗ tn−j , on which it acts as a derivation Zi,j (tn ) = (n − j )tn−j +i , where (n − j ) is defined as (n − j ) = 1 for n − j ≥ 0, and 0 otherwise. This seems to give just a glimpse of the full insertion-elimination Lie algebra of [6], which acts as a derivation on the full Hopf algebra of Feynman graphs in a renormalizable quantum field theory. Nevertheless, a full understanding of Zn,m goes a long way in understanding the full insertion elimination Lie algebra [13], using the fact that LL acts on elements in the full Hopf algebra which are homogenous in the appropriate grading resulting from the Hochschild cohomology of that very Hopf algebra [10]. There are two strong reasons for that: i) quantum field theory sums over all skeleton graphs in a symmetric fashion, ii) non-linear Dyson–Schwinger equations (DSEs) modify linear DSEs precisely by the anomalies generated by a non-vanishing β-function. The first fact ensures that we can work on homogeneous elements in the Hopf algebra; the second one ensures that there are effective methods available to deal with the operadic aspects of graph insertions. Here, DSEs are introduced combinatorially via a fixpoint equation in the Hochschild cohomology of a connected graded commutative Hopf algebra. Let us summarize the main features which emerged in recent work [11, 10, 12, 13]. Under the Feynman rules the Hochschild 1-cocycles, provided by the Hopf algebra of graphs, map to integral operators provided by the underlying skeletons of the theory. Renormalization conditions are determined by suitable boundary conditions for the integral equations so generated. The DSEs determine the Green functions from this Hochschild cohomology of the Hopf algebra of Feynman graphs, which is itself derived from free quantum field theory and the choice of renormalizable interactions. Indeed, following [12, 10], the identification of these 1-cocycles leads to a combinatorial Dyson–Schwinger equation: r = 1 +

[1] p∈HL res(p)=r

= 1+

∈HL res()=r

α |p| p B (Xp ) Sym(p) + α || . Sym()

(1)

The first sum in (1) is over a finite (or countable) set of Hopf algebra primitives Feynman graphs p, such that: (p) = p ⊗ e + e ⊗ p, p B+

(2)

each p indexing the closed Hochschild 1-cocycles above. The second sum in (1) is over all one-particle irreducible graphs contributing to the desired Green function, all

Insertion-Elimination Lie Algebra

415

weighted by their symmetry factors. Here, Xp is a polynomial in all r , and the superscript r ranges over the finite set (in a renormalizable theory) of superficially divergent Green functions. It indicates the number and type of external legs reflecting the monomials in the underlying Lagrangian. We use res(p) = r to indicate that the external legs of the graph p are of type r. The structure of these equations allows for a proof of locality using Hochschild cohomology [10], which is also evident using a coordinate space approach [2]. These fixpoint equations are solved by the following Ansatz: r = 1 +

∞

r

α k ck .

(3)

k=1

We grant ourselves the freedom to call such an equation a DSE or a combinatorial equation of motion for a simple reason: the DSEs of any renormalizable quantum field theory can be cast into this form. Crucially, in the above it can be shown (see [13], which we follow here) that |p|

Xp = res(p) Xcoupl ,

(4)

where Xcoupl is a connected Green function which maps to an invariant charge under the Feynman rules. This is rather obvious: consider, as an example, the vertex function in quantum electrodynamics. A n loop primitive graph p, contributing to it, provides 2n+1 internal vertices, 2n internal fermion propagators and n internal photon propagators. An invariant charge [8] is provided by a vertex function multiplied by the squareroot of the photon propagator and the fermion propagator. Thus the integral kernel corresponding to p is dressed by 2n invariant charges, and one vertex function. This is a general fact: each integral kernel corresponding to a Green function with external legs r in a renormalizable quantum field theory is dressed by a suitable power of invariant charges proportional to the grading of that kernel, and one additional appearance of r itself. This immediately shows that for a vanishing β-function the DSEs are reduced to a linear set of equations, and that the general case can be most efficiently handled by an expansion in the breaking of conformal symmetry induced by a non-vanishing β-function. Thus, a complete understanding of the linear case goes a long way in understanding the full solution. This emphasizes the crucial role which the insertion-elimination Lie algebra [6] in the ladder case [15] plays in the full theory: it defines an algebra of graphs which provide an underlying field of residues, which is then extended by the contributions resulting from a non-trivial β-function. The resulting scaling anomalies extend the Hopf algebra of graphs to a non-cocommutative one. Moreover, they force the appearance of new transcendental numbers and they result in the appearance of non-trivial representations of the symmetric group in the operad of graph insertions [13]. Here, we study the underlying linear DSEs which would suffice for a vanishing β-function. Indeed, we now define the linear DSE associated to the system above: r

lin = 1 +

[1] p∈HL res(p)=r

p

α |p| r p B ( ). Sym(p) + lin

(5)

The Hochschild closedness of B+ then ensures that we obtain a Hopf algebra isomorphic to the word Hopf algebra based on letters p, which we obtain as the underlying Dyson skeletons in the expansion of r .

416


The solutions of the linear DSE above are graded by the order in α and by the augmentation degree r

lin = 1 +

∞

∞

α j cj = 1 +

j =1

(6)

dj .

j =1

Here, cj is the sum of all words of order α j , where the degree | w | of a word w is the sum of the degree of its letters, and the degree of a letter is the loop number of the accompanying skeleton graph. These words uniquely correspond to Feynman graphs obtained by inserting primitive graphs into each other, where insertion now happens at a single vertex or edge in accordance with that linear DSE. On the other hand, dj is the sum α |w| w, (7) dj = j

w∈Haug

of all words made out of j letters, and we set | w |aug = j , the augmentation degree. Having defined the associated linear system, the propagator-coupling dualities [3] provide the general solution once the representation theory of the symmetric group has been established, which reflects the operadic nature of graph insertions [13]. But a complete understanding of linear Dyson–Schwinger equations comes first. To this end, it is profitable to study how the insertion elimination Lie algebra acts on the Hopf algebra of graphs in that case. In this paper, we start some groundwork by clarifying the structure of the insertion elimination Lie algebra which relates to the Hopf algebra structure of a linear DSE. The crucial point is always the identification of the Hochschild closed 1-cocycles in p the Hopf algebra of graphs B+ , typically parametrized by primitive elements p of the Hopf algebra of graphs [10]. We first mention that the Hopf algebra of graphs contains, as a corollary of the results in [6], a sub-Hopf algebra Hw of graphs generated by the linear DSE. It is naturally based on graphs which can be regarded as words, with the corresponding insertion-elimination Lie algebra Lw . It acts on the Hopf algebra Hw as w1 v if w = w2 v for some v Zw1 ,w2 (w) = (8) 0, if w has not this form. The Lie bracket in Lw is then [Zw1 ,w2 , Zw3 ,w4 ] = ZZw

1 ,w2 (w3 ),w4

−ZZw

− Zw3 ,Zw

3 ,w4 (w1 ),w2

K Z −δw 2 ,w3 w1 ,w4

2 ,w1 (w4 )

+ Zw1 ,Zw

4 ,w3 (w2 )

K + δw Z . 1 ,w4 w3 ,w2

(9)

See [6] for notation. of the Lie algebra Zn,m comes from the fact that the map B+ = The|p|significance p p α B+ maps the linear DSE to the fundamental DSE (which also underlies the polylog [12]) X = 1 + B+ (X),

(10)

where B+ is of order α. Note that B+ is not homogenous in α (there are primitive graphs of any degree in the coupling), but it is homogenous in the augmentation degree: all terms in its defining sum enhance this degree by one.


417

There is a natural inclusion ιH from Hcomm to Hw which sends tn → dn . This induces a map Zw1 ,w2 ιL : L → Lw , Zn,m → , (11) #(m) |w1 |aug =n,|w2 |aug =m

where #(m) is the number of words of degree m, such that ιH (Zn,m (tk ))=ιL (Zn,m )(ι(tk )). It is compatible with the Lie bracket: [ιL (Zn1 ,m1 ), ιL (Zn2 ,m2 )] (ιH (tn )) = ιH [Zn1 ,m1 , Zn2 ,m2 ](tn ) . (12) As long as we study linear DSEs, the ladder insertion elimination Lie algebra on generators Zn,m suffices, where it now acts by increasing and decreasing the augmentation degree. In [13] the reader can find a discussion of the Galois theory, which is missing, to handle the general case. The study of such questions in QFT is a beautiful mathematical problem in its own right. It gives mathematical justification to early ideas [14] of the use of anomalous dimensions and bootstrap equations in QFT to absorb short-distance singularities. Progress along these lines following [12, 13] will be reported in future work. We now continue to treat LL . 3. Generalities about the Ladder Insertion-Elimination Lie Algebra Let us recall some basic definition from [15] to which we refer for the details which are omitted in what follows. Let us introduce the Lie algebra LL via generators and relations: Definition 3.1. LL = spanC Zn,m | n, m ∈ Z≥0 , with: Zn,m , Zl,s = (l − m)Zl−m+n,s − (s − n)Zl,s−n+m −(n − s)Zn−s+l,m + (m − l)Zn,m−l+s −δm,l Zn,s + δn,s Zl,m , where

(13)

(l − m) = 0 if l < m, (l − m) = 1 if l ≥ m,

and where δn,m is the usual Kronecker delta: δn,m = 1 if m = n, δn,m = 0 if n = m. Then we have Corollary 3.2. [15] 1) LL is Z-graded Lie algebra: LL = ⊕i∈Z li , where for each Zn,m ∈ li , deg(Zn,m ) = i = n − m and dimC li = +∞; 2) LL has the following decomposition: LL = L+ ⊕ L0 ⊕ L− , where L+ = ⊕n>0 ln , L− = ⊕n m, n < m and n = m. Remark 3.4. The previous proposition is equivalent to the following (vector space) decomposition of the Lie algebra LL : LL = [D, D] ⊕ D, where we define D = a+ ⊕ a− ⊕ C, and a+ = spanC Zn,0 : n > 0, a− = spanC Z0,n : n > 0 and C is the trivial Lie algebra generated by Z0,0 . In fact a+ and a− are commutative sub algebras of LL and Z0,0 is a central element. 4. Structure of the Lie Algebra LL Let us start this section with two statements whose proofs are collected at the end of the this section. Theorem 4.1. The center of the Lie algebra LL has dimension one and it is generated by the element Z0,0 . Theorem 4.2. l 0 is a maximal abelian sub-algebra of LL . In what follows we will show that the Lie algebra LL is not simple. Let us introduce the following: Definition 4.3. [15] gl + (∞) = spanC Ei,j : Zi,j − Zi+1,j +1 | i, j ∈ Z≥0 . We have: Proposition 4.4. 1) [Ei,j , Er,k ] = Ei,k δj,r − Er,j δk,i ; 2) gl+ (∞) is an ideal in LL .


419

Proof. The proof of 1) and 2) is a simple but tedious application of the commutator formula (13). We can now define the quotient Lie algebra: C = LL /gl + (∞) and consider the exact sequence: π

0 → gl + (∞) → LL → C → 0.

(17)

It is now clear that to have a better understanding of the Lie algebra LL we need to study carefully the structure of the Lie algebra C. The crucial ingredient will be the following proposition: Proposition 4.5. gl+ (∞) = [LL , LL ]. Proof. Let us prove the two inclusions. gl + (∞) ⊂ [LL , LL ] since from the definition of gl+ (∞): Ei,j = Zi,j − Zi+1,j +1 = [Zi,0 , Z0,j ] − [Zi+1,0 , Z0,j +1 ], where the second equality follows from the formula (16). To show the other inclusion, i.e [LL , LL ] ⊂ gl + (∞), it suffices to observe that for any two generators, say Zh,p and Zr,q , of LL we have that their commutator is given by the difference between two elements having same degree (see formula (13)), say Zn,m and Zl,s , such that n − m = l − s. Under the hypothesis that k = n − m = l − s > 0 and that s > m (the other cases are completely analogous), we can write their difference as follows: Zn,m − Zl,s = Zm+k,m − Zs+k,s = Zm+k,m − Zm+k+1,m+1 + +Zm+k+1,m+1 − .... − Zs+k−1,s−1 + Zs+k−1,s−1 − Zs+k,s , which expresses the difference between Zn,m and Zl,s as finite linear combination of elements in gl + (∞). In particular we can rephrase the previous proposition as follows: Lemma 4.6. Two generators Zn,m and Zl,s are gl+ (∞)-equivalent if and only if they have the same degree, i.e: Zn,m ∼ Zl,s ⇐⇒ deg(Zm,n ) = deg(Zl,s ). Proof. Using the same argument we used to prove Proposition 4.5, we can conclude that if Zn,m and Zl,s have the same degree then they are equivalent. Suppose now that the difference between Zn,m and Zl,s can be written as a (finite) linear combination of elements in gl+ (∞) and also that n − m = l − s (w.l.o.g. we can assume that n − m > 0 and that l − s > 0). Under these assumptions, and from formula (16), it follows also that: Zn−m,0 − Zl−s,0 = f inite ai Epi ,qi . But this has as a consequence that each of these two elements are finite linear combinations of (homogeneous) elements in gl+ (∞). Accordingly we can write: Zn−m,0 = f inite ci Eri ,ki and Zl−s,0 = f inite ci Eti ,vi .

420


Rewriting the right-hand side of each of those two equalities in terms of the generators Zn,m , it follows that such equations can not hold. From Proposition 4.5 it follows that C is a (maximal) commutative Lie algebra coming from a quotient of LL . Let us now introduce a set of (natural) generators for C. Since the set Zn,m | n, m ∈ Z≥0 is a basis for LL and since π : LL −→ C is a surjection, it follows that Z n,m = π(Zn,m )| n, m ∈ Z≥0 is a set of generators for C. Moreover from Lemma 4.6 it follows that when n > m, then Zn,m ∼ Zn−m,0 , when m > n, then Zn,m ∼ Z0,m−n and, finally, when n = m, then Zn,m ∼ Z0,0 . So defining Zn = Z n,0 , Z−n = Z 0,n for n > 0 and Z0 = Z 0,0 , we get C = spanC Zn |n ∈ Z. The fact that such elements are also linearly independent (i.e. they form a basis for C) follows easily from Lemma 4.6. We now want to look more closely at the exact sequence (17). In particular we will prove the following result: Theorem 4.7. The exact sequence (17) does not split, i.e the Lie algebra LL is not the semi-direct product of the Lie algebra gl+ (∞) with the (commutative) Lie algebra C. Before addressing the proof of Theorem 4.7, we will introduce some preliminaries to make the paper as self-contained as possible. From the exact sequence (17) and from what we explained above, we can conclude that the Lie algebra LL is a non-abelian extension of the commutative Lie algebra C by the Lie algebra gl+ (∞). Let us explain with some care the meaning of such a statement (for more details we refer to the paper [1]). In what follows, g, h and e will be Lie algebras. Definition 4.8. [1] We will say that the Lie algebra e is an extension of the Lie algebra g by the Lie algebra h, if g, h and e fit into the following exact sequence: π

0 → h → e → g → 0.

(18)

Moreover we will say that two such extensions, e and e , are equivalent, if and only if e and e are isomorphic as Lie algebras. Let Der(h) be the Lie algebra of derivations of h, α , α ∈ HomC (g, Der(h)) and ρ , ρ ∈ HomC (2 g, h). On the set of the couples (α, ρ) introduced above, we define the following equivalence relation: (α, ρ) ∼ (α , ρ ) ⇐⇒ ∃ b ∈ H omC (g, h) such that: α (x).ξ = α(x).ξ + [b(x), ξ ]h , ρ (x, y) = ρ(x, y) + α(x).b(y) − α(y).b(x) − b([x, y]g ) + [b(x), b(y)]h .


421

Then we have that Theorem 4.9. [1] 1) The classes of isomorphism of the extensions of the Lie algebra g by the Lie algebra h given by the exact sequence (18), are in one-to-one correspondence with the equivalence classes [(α, ρ)], such that: [α(x), α(y)]Der(h) .ξ − α([x, y]g ).ξ = [ρ(x ∧ y), ξ ]h , α(x).ρ(y, z) − ρ([x, y]g , z) = 0, cyclic

for every x, y, z ∈ g and ξ ∈ h. 2) The Lie algebra structure induced by the datum (α, ρ), on the vector space e = h ⊕ g, is given by [(ξ1 , x1 ), (ξ2 , x2 )]e = ([ξ1 , ξ2 ]h + α(x1 ).ξ2 − α(x2 ).ξ1 + ρ(x1 , x2 ), [x1 , x2 ]g ). (19) We apply this result to our setting, where g = C and h = gl + (∞). The exact sequence (17) tells us that we have: LL gl + (∞) ⊕ C, where such a splitting holds in the category of vector spaces. We first prove that Proposition 4.10. The Lie algebra structure on LL , given by the bracket (13), corresponds to the couple (α, ρ) defined by (En+k,j δi,k − Ei,k δn+k,j ) α(Zn ).(Ei,j ) = (n) k≥0

+(−n)

(Ek,j δk+n,i − Ei,k+n δj,k ) k≥0

for n = 0 and α(Z0 ) ≡ 0, while ρ(Zn , Zm ) = 0 if n, m ≥ 0 or n, m ≤ 0 and ρ(Zn , Z−m ) =

m−1

En−m+k,k

k=0

if n > m, and ρ(Zn , Z−m ) =

m−1

Ek,m−n+k ,

k=0

if n < m. Proof. The proof follows comparing formula (13) with formula (19).

422


We now remark that Lemma 4.11. [1] Given π

0 → h → e → g → 0,

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as in (18), any splitting s : g −→ e (at the vector space level) of the previous exact sequence, induces a map αs ∈ H omC g, Der(h) , via the following: αs (X).ξ = [s(X), ξ ], for each X ∈ g and each ξ ∈ h.

Proposition 4.12. The map α ∈ HomC C, Der(gl + (∞)) , defined in Proposition 4.10, is induced by the linear map s : C −→ LL , which is defined as follows: s(Zn ) = (n)Zn,0 + (−n)Z0,n − δn,0 Z0,0 .

(21)

Proof. The map s defined in formula (21) is a section of the projection π : LL −→ C defined by the exact sequence (17). In other words, s ∈ H omC (C, LL ) such that s ◦ π = I dC . From Lemma (4.11) we know that such a section s induces a linear map: αs : C −→ Der(gl+ (∞)), defined by: αs (x).ξ = [s(x), ξ ]LL . It is now easy to check that this map is the same defined in Proposition 4.10.

We are now almost ready to prove Theorem 4.7. We only need to remark the following. From Theorem 4.9 we have that a given extension (α, ρ) of the Lie algebra C by the Lie algebra gl+ (∞) will split, i.e. will be equivalent to a semi-direct product of these two Lie algebras, if and only if (α, ρ) ∼ (α , 0), or, in other words, if and only if α is a morphism of Lie algebras. Theorem 4.9 tells us that this is equivalent to ask for the existence of a linear map b : C −→ gl+ (∞), such that s + b : C −→ LL is a morphism of Lie algebras. Moreover, since we are working with the category of graded Lie algebras, the map b has to be grade preserving. In conclusion, to prove Theorem 4.7, we are left to show that such a map b does not exist. Proof of Theorem 4.7. Suppose we can define a linear map b : C −→ gl+ (∞) such that s + b : C −→ LL is a morphism of (graded) Lie algebras. That means that we can N find elements M i=1 ahi Ehi +1,hi ∈ gl+ (∞) and i=1 bkj Ekj ,kj +1 ∈ gl+ (∞) such that M N b(Z1 ) = i=1 ahi Ehi +1,hi , b(Z−1 ) = i=1 bkj Ekj ,kj +1 and furthermore 0 = [(s + b)(Z1 ), (s + b)Z−1 ] = [Z1,0 +

M i=1

ahi Ehi +1,hi , Z0,1 +

N

bkj Ekj ,kj +1 ].

j =1

We can calculate such a commutator by re-writing each of the terms Ei,j in the sums in terms of the generators Zn,m , and applying to such terms the brackets given in formula (13). The result, written in terms of the generators Ei,j , takes the form: −E0,0 +

N j =1

bkj (Ekj +1,kj +1 − Ekj ,kj ) +

M i=1

ahi (1 + bhi )(Ehi +1,hi +1 − Ehi ,hi ) = 0.


423

The right-hand side of the previous sum can be reorganized in terms of the summands Ej +1,j +1 − Ej,j as follows: −

L

φj (Ej +1,j +1 − Ej,j ),

i≥0

where L is the biggest between N and M and the φj ’s are coefficients. Then we have that E0,0 =

L

φj (Ej +1,j +1 − Ej,j ) = −φ0 E0,0 +

(φj +1 − φj )Ej,j + φL EL+1,L+1 ,

j ≥0

i≥0

that clearly give us a contradiction.

Proof of Theorems 4.1 and 4.2. We now conclude this section by giving the proofs for Theorems 4.1 and 4.2. We recall from [15] that the Lie algebra LL has an obvious module: Definition 4.13. [15] S=

Ctn = C[t0 , t1 , t2 , t3 .....].

n≥0

We will assign a degree equal to k to the generator tk for each k ≥ 0. LL acts on S via the following: Zn,m tk = 0 if m > k, Zn,m tk = tk−m+n if m ≤ k.

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In what follows we will indicate by Z(LL ) the center of the Lie algebra LL . Proof of Theorem 4.1. It is obvious that CZ0,0 ⊂ Z(LL ). Let us prove the other inclusion. Let us suppose that there is some element α ∈ LL , not proportional to Z0,0 and that belongs to the center of LL . W.l.o.g. we assume α=

k i=1

ai Zni ,mi =

i: ni ,mi =0

bi Zni ,mi +

i: n˜ i =0

ci Zn˜ i ,0 +

di Z0,m˜ i ,

(23)

i: m ˜ i =0

where all the n˜ i ’s (m ˜ i ’s) are different from 0 and n˜ i = n˜ j (m ˜ i = m ˜ j ), if i = j , and (ni , mi ) = (nj , mj ), if i = j . We will prove that α, defined above, is equal to zero by showing that the coefficients bi , ci and di are all equal to zero. We will split the proof of this assertion into two lemmas. Lemma 4.14. If α ∈ Z(LL ), where α is defined as above, then bi = di = 0 for each i.

424


Proof. Let us consider some element Zn,0 ∈ LL such that 0 < n ≤ mini {mi , m ˜ i }. Then using formula (13), we get: [Zn,0 , α] = bi [Zn,0 , Zni ,mi ] + di [Zn,0 , Z0,m˜ i ] i

=

i

bi (Zni +n,mi − Zni ,mi −n ) +

i

di (Zn,m˜ i − Z0,m˜ i −n ).

i

Note that all the m ˜ i ’s are different (and different from 0), while in the set of the mi ’s (which are also all different from 0) we can have repetitions. . Let us now define the set M = {m1 , ...., mk , m ˜ 1 , ..., m ˜ r }, and let us consider the disjoint union: M = M1 ∪ · · · ∪ Ms . Each Mi corresponds to the set of all indices in M which are equal to some given index li , say. We remark once more that for each i, Mi ∩ {m ˜ 1, · · · , m ˜ r } contains at most one element, since in the set {m ˜ 1 , ..., m ˜ r } we do not have repetitions. Let us now consider p1 = l1 − n (which is ≥ 0, as a consequence of the condition we imposed on n), and let us also consider the corresponding element tp1 ∈ S. Since α belongs to Z(LL ), and since n > 0, we have:

0 = [Zn,0 , α](tp1 ) = − bi tp1 −mi +n+ni + di tp1 −m˜ i +n . (24) i: mi ∈M1

i: m ˜ i ∈M1

Remark 4.15. We observe that all the indices in M1 are equal to l1 and that p1 = l1 − n. Moreover the ni ’s in the first sum of the right-hand side in formula (24) are all different (since by assumption we have that (ni , mi ) = (nj , mj ) unless i = j and in our case all the mi belong to the class M1 ). Finally, we notice that the last sum, if not equal to zero, contains only one term. ˜ 1, · · · , m ˜ r } are both not Let us now suppose that M1 ∩ {m1 , · · · , mk } and M1 ∩ {m empty (the cases where one, or both, of those intersections are empty, are completely analogous). From the previous remark it follows that   0 = [Zn,0 , α](tp1 ) = −  bi tl1 −n−l1 +n+ni + di tl1 −n−l1 +n  =−

i: mi ∈M1

i: m ˜ i ∈M1

bi tni + d1 t0 .

i

Since all the ni in the first sum are different, we have that d1 = 0 and bi = 0 for each i. We can apply the same argument to the sets M2 ,...,Ms , to show that each of the coefficients bi and ci are equal to 0. From Lemma 4.14 we conclude that if α ∈ Z(LL ), α defined as in Eq. (23), then α= ci Zni ,0 . i


425

To conclude the proof of Theorem 4.1, we need to show that Lemma 4.16. If α ∈ Z(LL ) and α = i ci Zni ,0 , then ci = 0 for each i. Proof. We first notice that we can suppose all ni = 0 and n1 < n2 .... Let us now consider some element Z0,n , such that n ≥ maxi {ni }. Since we suppose α = i ci Zni ,0 to be in the center of LL , we can write: ci [Zni ,0 , Z0,n ] = ci (Zni ,n − Z0,n−ni ). 0 = [α, Z0,n ] = i

i

By the hypothesis on n and the one on the ni ’s, we conclude that all the ci ’s are equal to zero. Proof of Theorem 4.2. Let us suppose that l 0 is not a maximal abelian sub-algebra of 0 LL , i.e. that there exists LL α ∈ / l , α = ni=1 ai Zni ,mi , such that [α, Zk,k ] = 0, ∀ k > 0. Without loss of generality we can suppose that in each of (ni , mi )’s, ni = mi (if no, α = β + i fi Zni ,ni and [β, Zk,k ] = [α, Zk,k ]). Such an element can be written as bi Zni ,mi + ci Zn˜ i ,0 + di Z0,m˜ i . (25) α= i: mi =0, ni =0

i: n˜ i =0

i: m ˜ i =0

Remark 4.17. We note that in formula (25) all the ni ’s and the mi ’s are different from 0 and also that n˜ i = n˜ j and m ˜ i = m ˜ j for each i = j . We will prove that such an element is identically equal to zero, showing that each of the coefficients in Eq. (25) is equal to zero. We will divide the proof of this statement in two lemmas. Lemma 4.18. Given α ∈ l 0 , defined as in formula (25), we have that ci = di = 0 for all i. Proof. Let us fix integer k, 0 < k ≤ mini {ni , mi , n˜ i , m ˜ i }. Then we get bi [Zni ,mi , Zk,k ] + ci [Zn˜ i ,0 , Zk,k ] + di [Z0,m˜ i , Zk,k ] [α, Zk,k ] = i: mi =0,ni =0

=

ci (Zk+n˜ i ,k − Zn˜ i ,0 ) +

i: n˜ i =0

i: n˜ i =0

i: m ˜ i =0

di (Z0,m˜ i − Zk,k+m˜ i ),

i: m ˜ i =0

since [Zni ,mi , Zk,k ] = 0, ∀ {ni , mi } such that ni ≥ k, mi ≥ k, [Zn˜ i ,0 , Zk,k ] = Zk+n˜ i ,k − Zn˜ i ,0 if 0 < k ≤ n˜ i , and [Z0,m˜ i , Zk,k ] = −Zk,m˜ i +k + Z0,m˜ i if 0 < k ≤ m ˜ i. Since α commutes with all the elements of the sub-algebra l 0 , we have ci (Zk+n˜ i ,k − Zn˜ i ,0 ) + 0= di (Z0,m˜ i − Zk,k+m˜ i ). i: n˜ i =0

i: m ˜ i =0

426


But in the right-hand side of the previous formula the first sum contains only elements of positive degree while the second sum contains only those of negative degree; thus the sum is equal to zero if and only if separately ci (Zk+n˜ i ,k − Zn˜ i ,0 ) = 0 and di (Z0,m˜ i − Zk,k+m˜ i ) = 0. i: n˜ i =0

i: m ˜ i =0

From this it follows that all ci ’s and di ’s are equal to zero. Indeed, consider the sum containing the ci ’s (the one containing the di ’s can be treated in the same way): ci (Zk+n˜ i ,k − Zn˜ i ,0 ) = 0. i: n˜ i =0

Since k = 0 and since n˜ i = n˜ j , if i = j , all the elements Zk+n˜ i ,k − Zn˜ i ,0 are linearly independent. Summarizing, so far we have proved that if a given element α commutes with each of the elements in l 0 , then: α= bi Zni ,mi . (26)

Lemma 4.19. If α, l

0

i: ni =0,mi =0

= 0, with α defined as in (26), then all the bi ’s are equal to 0.

Proof. Let us decompose the element α in term of elements of positive and negative degree, i.e: α= ai Zni ,mi = bi Zri +sj ,ri + ci Zpi ,pi +tj . i

j

i≥0

j

i≥0

Remark 4.20. We remark that in α elements of the same (negative or positive) degree could be present; as an example of such an element (of positive degree) we can consider: βj = bi Zri +sj ,ri , for a given j i

or the element (of negative degree): γj = ci Zpi ,pi +tj , for a given j . i

From the previous remark let us re-write α as: α= βj + γj , j

j

each βj ∈ L+ and each γj ∈ L− . Let us now consider some element Zk,k ∈ l 0 and let us take the commutator of such an element with α, [α, Zk,k ] = [βj , Zk,k ] + [γj , Zk,k ]. j

j

Since LL is a graded Lie algebra and since deg Zk,k = 0, we have that deg [βj , Zk,k ] = sj , ∀j


427

and similarly deg [γj , Zk,k ] = −tj , ∀j. Hence [α, Zk,k ] = 0 ⇐⇒ [βj , Zk,k ] = 0 and [γj , Zk,k ] = 0 , ∀j. We are left to prove that any homogeneous element commuting with all the elements in l 0 can not exist. So, to fix ideas, let us now consider some element of positive degree s, say, β = l i=1 ai Zni +s,ni and let us suppose that [β, Zk,k ] = 0 ∀ k ≥ 1.

(27)

Without loss of generality we can further assume that 0 < n1 < n2 < ... < nk (that β fulfills the hypothesis is constrained by the assumptions given for the element α defined in formula (26), which translates for β into the condition ni = 0). To conclude, it suffices to show that each of the ai ’s of β = li=1 ai Zni +s,ni is equal to 0. So let us consider k = n2 in formula (27). Applying formula (13) to this case, we get: ai [Zni +s,ni , Zn2 ,n2 ] [β, Zk,k ] = a1 [Zn1 +s,n1 , Zn2 ,n2 ] +

i≥2

= a1 Zn2 +s,n2 − (n2 − n1 − s)Zn2 ,n2 −s

−(n1 + s − n2 )Zn1 +s,n1 + δn1 +s,n2 Zn2 ,n1 ,

since i≥2 ai [Zni +s,ni , Zn2 ,n2 ] = 0. By the previous formula and the hypothesis for the ni ’s, we conclude that [β, Zn2 ,n2 ] = 0 ⇐⇒ a1 = 0. Taking k = n3 , n4 , ...., and using the same argument, we can conclude that each of the ai ’s is equal to zero. 5. Cohomology of the Lie Algebra LL In what follows we will describe in some detail the cohomology of the Lie algebra LL . We will start with an explicit calculation for the dimension of the first cohomology group (with trivial coefficients) and we will continue using the general machinery to calculate the higher cohomology groups. Let us first introduce the derivation Y , acting on LL as follows: Y.Zn,m ≡ [Y, Zn,m ] = (n − m)Zn,m .

(28)

Let us now consider the following extension of the Lie algebra LL : Definition 5.1. Lˇ L = spanC Zn,m , Y |n, m ∈ Z≥0 , where the commutator [Zn,m , Zl,s ] is given by formula (13) and [Y, Zn,m ] = (n − m)Zn,m . We have the following: Theorem 5.2. 1. dimC H 1 (Lˇ L , C) = 1; 2. dimC H 1 (LL , C) = +∞. Proof. The elements of H 1 (LL , C) are in one-to-one correspondence with the elements φ ∈ HomC (LL , C) such that

428


φ([Zn,m , Zl,s ]) = 0, for each Zn,m , Zl,s ∈ LL . As a consequence of Proposition 3.3, Eq. (16), we have that the value of φ on a given element Zn,m , depends only on the degree of such an element. In fact, given: Zn,m = [Zn,0 , Z0,m ] + (n − m)Zn−m,0 + (m − n)Z0,m−n − δn−m,0 Z0,0 , φ(Zn,m ) = (n − m)φ(Zn−m,0 ) + (m − n)φ(Z0,m−n ) − δn−m,0 φ(Z0,0 ). From this remark, it follows that dimC H 1 (LL , C) = +∞. This proves the second assertion. To show that also the first holds, let us observe that, since: 0 = φ([Y, Zn,m ]) = (n − m)φ(Zn,m ), for each Zn,m ∈ LL , then φ(Zn,m ) = 0 for any element Zn,m with degree different from zero, so that we can write: φ(Zn,n ) = cφ (n) ∈ C. On the other hand, since the value of φ on a given element depends only on the degree of such an element, we have that: φ(Zn,m ) = cφ δn−m,0 , or, in other words: H 1 (Lˇ L , C) C. To go to the higher cohomology groups we need to introduce some notation and some (classical) results about the cohomology of Lie algebras. Let us start with the following lemma: Lemma 5.3. Let gl(n) the (Lie) algebra of n × n matrices (with entries in C). Let us define the direct system of (Lie) algebras: ... → gl(n − 1) → gl(n) → gl(n + 1) → ...,

(29)

where the arrows are given by the standard inclusions, i.e A ∈ gl(n) is mapped into A˜ ∈ gl(n + 1) such that a˜ i,j = ai,j , ∀i ≤ n, j ≤ n and a˜ i,j = 0 if i > n or j > n. Then the direct limit of such a direct system is isomorphic to the Lie algebra gl+ (∞) introduced in Definition 4.3. Proof. The proof follows immediately from Definition 4.3.

Let us next quote a result about the cohomology of the Lie algebra of the general linear group gl(n) [7]: Theorem 5.4. 1) The cohomology ring of the Lie algebra gl(n) is an exterior algebra in n generators of degree 1, 3, ..., 2n − 1: H • (gl(n)) = [c1 , c3 , ....., c2n−1 ];


429

2) for any given n, the (inclusion) map defined in formula (29): i : gl(n − 1) −→ gl(n) induces a map i ∗ in cohomology: i ∗ : H • (gl(n)) −→ H • (gl(n − 1)), such that i ∗ : H p (gl(n)) −→ H p (gl(n − 1)) is an isomorphism for p < n, and it maps to zero the top degree generator when p = 2n − 1; 3) from 1), 2) and the previous Lemma 5.3, it follows that the cohomology ring of the Lie algebra gl+ (∞) is a (non finitely generated) exterior algebra having generators only in odd degree: H • (gl + (∞)) = [c1 , c3 , .....]. Proof. We refer to the book [7] for the proof of 1) and 2). Part 3) follows from Lemma 5.3, where we have identified the direct limit of the Lie algebras gl(n) with the Lie algebra gl+ (∞). Now let us go back to the exact sequence (17). This induces the following exact sequence in cohomology ([7, 17]): f1

δ

0 → H 1 (C) → H 1 (LL ) → H 1 (gl + (∞))C → H 2 (C) f2

δ

→ H 2 (LL ) → H 2 (gl + (∞))C → · · ·.

(30)

Here, by H i (gl + (∞))C we understand the (sub)-vector space of C-invariant elements in H i (gl + (∞)), i.e the space {a ∈ H i (gl + (∞))|x.a = 0, ∀ x ∈ C}. Using Theorem 5.4 and the previous exact sequence we have one of our main results. Theorem 5.5. The cohomology groups H i (LL ), for i = 1, 2, are infinite dimensional. In particular the Lie algebra LL has infinite many non equivalent central extensions. Proof. Since C is an (infinite dimensional) abelian Lie algebra the groups H n (C) are infinite dimensional. Let us now consider the following segment of the exact sequence (30): δ

f p+1

· · · → H p (gl + (∞))C → H p+1 (C) −→ H p+1 (LL ) → H p+1 (gl + (∞))C → · · ·. (31) Let us suppose first that p is an odd number. From Theorem 5.4, it follows that H p+1 (gl + (∞)) = 0 and H p (gl + (∞)) < ∞, so that f p+1 is surjective and ker( f p+1 ) is finite dimensional. We can argue in an analogous way for p an even number; in this case the map f p+1 is injective as H p (gl + (∞)) = 0. The statement about central extensions follows now from the fact that those are in one-to-one correspondence with the elements of the group H 2 (LL ).

430


We conclude this section with the following proposition: Proposition 5.6. The map f 1 , defined in the exact sequence (30), is not an isomorphism. The proof of this statement follows from: Claim 5.7. H 1 (gl + (∞))C C. Proof. Let us start by observing that H 1 (gl + (∞)) gl + (∞)/[gl + (∞), gl + (∞)] C, and identifying [gl + (∞), gl + (∞)] with sl+ (∞), i.e. with the Lie algebra of infinite matrices of finite rank, having trace equal to zero. In particular, this implies that the only non trivial class [φ] ∈ H 1 (gl + (∞)) corresponds to a (closed) cochain φ ∈ C 1 (gl + (∞)) whose kernel is sl+ (∞). Let us now define the action of the (abelian) Lie algebra C on H 1 (gl + (∞)): for any φ ∈ C 1 (gl + (∞)) and [Z] ∈ C LL /gl + (∞), define ([Z].φ)(α) = φ([Z + β, α]),

(32)

where Z ∈ LL and β ∈ gl + (∞)). On the other hand, since φ is a cocycle, we have that φ([Z + β, α]) = φ([Z, α]). It is a simple calculation to show that [LL , gl + (∞)] ⊂ sl+ (∞) so that, from the hypothesis on φ, we conclude that φ([Z, α]) = 0, i.e. [Z].φ = 0, or that φ is C-invariant.

Remark 5.8. In this remark we want to compare the Lie algebra gl + (∞) with its finite dimensional analogue, e.g. gl(n). These two Lie algebras are not simple; in fact the Lie algebra of the matrices having trace equal to zero is a non trivial ideal in both cases (sl + (∞) in the infinite dimensional case and sl(n) in the finite dimensional case). Moreover in both cases the quotient is the trivial Lie algebra, e.g. C. While in the finite dimensional case the quotient gl(n)/sl(n) C Z(gl(n)), where Z(gl(n)) is the center of gl(n), in the infinite dimensional case the quotient gl + (∞)/sl + (∞) does not correspond to any ideal in gl + (∞). In particular Z(gl + (∞)) = {0}. Remark 5.9. Because of the (vector space) isomorphism LL gl + (∞)⊕C, and because of the definition of the Lie algebra: gl + (∞) lim gl(n), −→

it is reasonable to ask if we can describe the Lie algebra LL as a direct limit of (finite) dimensional Lie algebras. Such a question becomes even more interesting, if we compare the Lie algebra LL with a Lie algebra that plays an important role in some recent developments of CFT [16]. This Lie algebra, LV in the notation of [16] is actually the central extension of the (infinite) symplectic Lie algebra sp(∞, R), it contains as an ideal the Lie algebra gl + (∞) (a property which is shared with LL ), and it allows for a finite dimensional approximation.


431

Even if it is not clear to us if there is any reason to believe that these two Lie algebras are related, the question if it is possible or not to describe LL as direct limit of finite dimensional Lie algebra stands, in our opinion, is an interesting problem per se. At the moment we do not have any definitive answer to this question. Anyhow we can note the following. Let us define An = gl + (∞) ⊕ V (2n − 1), n ≥ 1, where V (2n − 1) is a vector space of dimension 2n − 1 whosegenerators, Z±k with k = {0, 1, . . . , n − 1}, are defined by the following: Zk = j ≥0 Ek+j,j , Z−k = E and Z = E . Then it is easy to show that: j,j +k 0 j,j j ≥0 j ≥0 Theorem 5.10. {An }n≥1 defines a filtration of Lie algebras, and: LL = lim An −→

in the category of Lie algebras. 6. Conclusion and Outlook In this paper we gave results about the structure of the Lie algebra LL . We first discussed its relevance for the structure of quantum field theory. Having motivated its study, we showed that LL is the (non-abelian) extension via gl+ (∞) of a commutative Lie algebra. We also showed that this extension does not split. Furthermore, we described the cohomology of LL and proved that the second cohomology group of this Lie algebra is infinite dimensional, allowing for infinitely many non-equivalent central extensions. It should be very interesting to understand the physical meaning of the central extensions of this Lie algebra in the future, in particular their relations with those DSEs. In future work we will study more closely the representation theory of this Lie algebra as well as the representation and the cohomology of the Lie algebra Lˇ L defined in 5.1, with the hope to shed some light on these problems. Acknowledgements. I.M. thanks the IHES for hospitality during a stay in May of 2004. We thank Takashi Kimura for several useful conversations. I.M wants to thank Pavel Etingof and Victor Kac and Zoran Skoda for stimulating discussions and valuable advice. We thank Claudio Bartocci and Ivan Todorov for carefully reading a preliminary version of the manuscript and for suggesting several improvements.

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Commun. Math. Phys. 259, 433–450 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1372-z

Communications in


Density of Eigenvalues of Random Normal Matrices Peter Elbau, Giovanni Felder Department of Mathematics, ETH-Zentrum, 8092 Zurich, Switzerland. E-mail: [email protected]; [email protected] Received: 27 September 2004 / Accepted: 7 February 2005 Published online: 14 June 2005 – © Springer-Verlag 2005

Abstract: The relation between random normal matrices and conformal mappings discovered by Wiegmann and Zabrodin is made rigorous by restricting normal matrices to have spectrum in a bounded set. It is shown that for a suitable class of potentials the asymptotic density of eigenvalues is uniform with support in the interior domain of a simple smooth curve. 1. Introduction In recent work initiated by P. Wiegmann and A. Zabrodin [1–4], a connection between the normal matrix model and conformal mappings was discovered. In this model one considers random normal N × N complex matrices with probability measure −1 PN (M) dM = ZN exp{−(N/t0 )tr(M ∗ M − p(M) − p(M)∗ )} dM,

(1)

where dM is a natural measure on the variety of normal matrices and ZN is the normalization factor. The result is that, as N → ∞, the density of eigenvalues is 1/π t0 times the characteristic function of a bounded domain in the complex plane. This domain is characterized by the fact that its exterior harmonic moments are the coefficients tj of the polynomial p appearing in the measure. Moreover, the Riemann mapping of the exterior of the unit disk onto the exterior of the domain obeys, as a function of the tj , the equations of the integrable dispersionless Toda hierarchy. These fascinating results remain however at the level of formal manipulations of undefined objects, as the integrals diverge, except in the simplest case of a polynomial p of degree 2, where the domain is bounded by an ellipse. The purpose of this note is to give a setting in which the above statements make mathematical sense and to give a proof of these statements. The problem of divergence of the integral over normal matrices is solved in a naive way, by restricting the integral to normal matrices whose eigenvalues lie in a compact

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domain D of the complex plane. Then for small t0 the results can be formulated in terms of polynomial curves, i.e., curves in the complex plane admitting a parametrization of the form w → h(w) = rw+ nj=0 aj w −j , |w| = 1. For simple polynomial curves the prob 1 −j dz lem of determining the curve out of its exterior harmonic moments tj = 2πi γ z¯ z and the area π t0 of the interior domain (the interior domain is the bounded connected component of the complement of the curve) has a unique solution for small t0 , as we show in Sect. 5. Our main result is then: Theorem 1.1. Let D ⊂ C be the closure of a bounded open set containing the origin. Let p(z) = t2 z2 + · · · + tn+1 zn+1 be a polynomial such that |z|2 − p(z) − p(z) has a non-degenerate absolute minimum in D at z = 0. Then there exists a δ > 0 so that for all 0 < t0 < δ, (i) there exists a unique simple polynomial curve γ with exterior harmonic moments t1 = 0, t2 , . . . , tn+1 , 0, 0, . . . and area of interior domain π t0 ; (ii) the expectation value of the density of eigenvalues of random normal matrices with spectrum in D and distribution (1) converges as N → ∞ to a uniform distribution with support in the interior domain of γ . The condition on p implies the Hessian condition |t2 | < 21 and it is fulfilled if the Hessian condition holds and t3 , . . . , tn are sufficiently small. It then follows from results of [5, 1] that the curve γ as a function of the tj in this range provides a solution of the integrable dispersionless 2D Toda hierarchy obeying the string equation. The paper is organized as follows: the basic definitions of the random normal matrix model are recalled in Sect. 2. We then introduce the “equilibrium measure” as the unique solution of a variational problem in Sect. 3 and show in Sect. 4 that the density of eigenvalues converges to it. These are either known results or adaptations of results known for hermitian matrices to our case. In Sect. 5 we introduce the notion of polynomial curve and prove Theorem 5.3, which is a stronger form of part (i) of Theorem 1.1. In Sect. 6 we prove part (ii) of Theorem 1.1, see Theorem 6.1, and discuss our results in Sect. 7. 2. Eigenvalues of Random Normal Matrices We consider the probability measure PN (M) dM =

1 −N tr V (M) e dM, ZN

ZN =

NN (D)

PN (M) dM,

defined by a potential V , on the set NN (D) = {M ∈ MatC (N ) | [M, M ∗ ] = 0, σ (M) ⊂ D} of normal N × N complex matrices with spectrum in some compact domain D ⊂ C. The measure dM is the Riemannian volume form on (the smooth part of) NN (D) 2 with respect to the metric induced from the standard metric on the vector space CN of all N × N matrices. In a parametrization by eigenvalues and unitary matrices it is given by [6] dM = dU

|zi − zj |2

1≤i<j ≤N

N i=1

d 2 zi ,

Density of Eigenvalues of Random Normal Matrices

435

∗ where M = U diag (zi )N i=1 U and dU denotes the normalized U(N ) invariant measure on U(N )/U(1)N . This leads to the probability measure N N 1 −N N V (zi ) i=1 d 2 zi = e |zi − zj |2 d 2 zi , PN (zi )N i=1 ZN i=1

1≤i<j ≤N

ZN =

DN

e−N

N

i=1 V (zi )

i=1

|zi − zj |2

1≤i<j ≤N

N

d 2 zi

i=1

on the space of eigenvalues zi ∈ D, 1 ≤ i ≤ N . 3. The Equilibrium Measure We are interested in the behavior of the function PN (z) as N → ∞. Because the probability that two eigenvalues are equal is always zero, we may consider PN as a function on the set D0N = {z ∈ D N | zi = zj ∀ i = j }. Introducing the probability measure δz (A) =

N 1 χA (zi ), N i=1

z = (zi )N i=1 ,

on D (χA shall denote the characteristic function of the set A), we write for z ∈ D0N , 1 2 −1 V (ζ ) dδz (ζ ) + exp −N log |ζ − ξ | dδz (ξ ) dδz (ζ ) . PN (z) = ZN ζ = ξ Letting N → ∞, only the infimum of the coefficient of −N 2 , I0 := inf I (µ), µ∈M(D) I (µ) := V (z) dµ(z) +

z = ζ

log |z − ζ |−1 dµ(ζ ) dµ(z),

will be relevant. Here M(D) denotes the set of all Borel probability measures on D without point masses (a measure with point masses could only arise from measures δz with some zi = zj , but then PN (z) = 0). Therefore, we safely can neglect the restriction on the double integral. The precise sense in which the infimum of I controls the large N behavior of PN will be discussed in the next section. Here we consider this reasoning only as a motivation for introducing the variational problem. χD Because of I ( |D| λ) < ∞ (λ the Lebesgue measure on C and |D| = λ(D) the area of D), I0 is finite. Definition. An equilibrium measure for V on D ⊂ C is a Borel probability measure µ on D without point masses so that I (µ) = I0 . Theorem 3.1. Every continuous function V on a compact subset D has a unique equilibrium measure.

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In the remaining part of this section we prove this theorem, which is a known fact from potential theory, see e.g. [7], and give necessary and sufficient conditions for µ to be an equilibrium measure. The constructions are adapted from the corresponding results for Hermitian matrices, see [9]. 3.1. Existence. To show the infimum is achieved, we choose a sequence (µn )∞ n=1 in M(D) with I (µn ) → I0 . Lemma 3.2. The space of all Borel probability measures on D is sequentially compact. Proof. By the theorem of Riesz-Markov each Borel measure µ on D corresponds to exactly one positive, linear functional φµ ∈ C(D)∗ and by the theorem of Alaoglu, the closed unit-sphere in C(D)∗ is weak-*-compact. Therefore, for each sequence (µn )∞ n=1 or Borel probability measures on D, the sequence (φµn )∞ n=1 contains a weak-*-convergent subsequence (φµn(k) )∞ k=1 , i.e. ∃ φ ∈ C(D)∗ : φµn(k) (f ) → φ(f )

∀f ∈ C(D).

Now we find a measure µ on D with φ = φµ . This measure fulfills f dµn(k) → f dµ (k → ∞) ∀f ∈ C(D), and hence is again a Borel probability measure. Because of this lemma there exists a convergent subsequence (µn(k) ) of (µn ) and a Borel probability measure µ with µn(k) → µ. To prove that I (µ) = I0 , we estimate with an arbitrary real constant L, lim I (µn(k) ) = lim V (z) dµn(k) (z) + lim log |z − ζ |−1 dµn(k) (ζ ) dµn(k) (z) k→∞ k→∞ k→∞ ≥ V (z) dµ(z)+ lim min{log |z − ζ |−1 , L} dµn(k) (ζ ) dµn(k) (z). k→∞

Approximating uniformly the second integrand according to the theorem of StoneWeierstraß up to ε > 0 with a polynomial in z, z¯ , ζ and ζ¯ and using Fubini’s theorem, we get a lower bound for the limit: lim I (µn(k) ) ≥ V (z) dµ(z) + min{log |z − ζ |−1 , L} dµ(ζ ) dµ(z) − 2ε. k→∞

Letting first ε → 0 and then L → ∞ shows that µ has no point masses (otherwise the right-hand side would diverge) and I0 = I (µ). 3.2. Uniqueness. Next, we want to show that there is exactly one measure µ ∈ M(D) with I (µ) = I0 . So suppose µ˜ ∈ M(D) also fulfills I (µ) ˜ = I0 . Then we consider the family µt = t µ˜ + (1 − t)µ = µ + t (µ˜ − µ), t ∈ [0, 1], in M(D).


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Lemma 3.3. Let µ and µ˜ be probability measures such that the function log |z − ζ |−1 is integrable with respect to µ ⊗ µ and to µ˜ ⊗ µ. ˜ Then log |z − ζ |−1 is also integrable with respect to µ ⊗ µ. ˜ Additionally, we have the inequality (2) log |z − ζ |−1 d(µ˜ − µ)(ζ ) d(µ˜ − µ)(z) ≥ 0, with equality if and only if µ = µ. ˜ Proof. We start with the distributional identity log |z|−1 ϕ(z) d2 z = −2π ϕ(0) for any Schwarz function ϕ. Introducing the Fourier transform ϕˆ of ϕ, we find 1 (|k|2 ϕ(k)) ˆ d2 k log |z|−1 ϕ(z) d2 z = − |k|2

i 1 1 ¯ 2 (kz+k z¯ ) − f (k) d 2 z d 2 k = ϕ(z) e 2π |k|2 1 1 i (kz+k¯ z¯ ) 2 = − f (k) d2 k ϕ(z) d2 z, e 2π |k|2 where f denotes a real, continuous function which is one in the vicinity of zero and becomes zero at infinity. So we have for all z ∈ C \ {0} the equation 1 1 i (kz+k¯ z¯ ) −1 2 − f (k) d2 k + C(f ) e log |z| = 2π |k|2 with some real constant C(f ). So, with Tonelli’s theorem, we see that log |z − ζ |−1 d(µ˜ − µ)(ζ ) d(µ˜ − µ)(z) 2 i 1 1 ¯ z¯ ) (kz+ k e2 d(µ˜ − µ)(z) d2 k = 2 2π |k| is non-negative and finite, which immediately implies the integrability of log |z − w|−1 with respect to µ ⊗ µ. ˜ To achieve equality in (2), we need i i ¯ z¯ ) ¯ (kz+ k dµ(z) = e 2 (kz+k z¯ ) dµ(z) ˜ e2 for all k ∈ C, which reads µ = µ. ˜

So we can expand I (µt ) and obtain −1 V (z) + 2 log |z − ζ | dµ(ζ ) d(µ˜ − µ)(z) I (µt ) = I (µ) + t +t 2 log |z − ζ |−1 d(µ˜ − µ)(ζ ) d(µ˜ − µ)(z).

(3)

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Lemma 3.3 now states that the coefficient of t 2 is non-negative and so the function t → I (µt ) is convex on [0, 1]. In particular, ˜ + (1 − t)I (µ) = I0 , I (µt ) ≤ tI (µ) which implies I (µt ) ≡ I0 . This requires the last summand in (3) to vanish and so, again with Lemma 3.3, we see that µ = µ. ˜ 3.3. A variational form. To determine if a given measure µ is the equilibrium measure for the potential V on the domain D, we may check a variational principle. Proposition 3.4. The probability measure µ is the equilibrium measure for the potential V on the domain D if and only if the function E(z) = V (z) + 2 log |z − ζ |−1 dµ(ζ ) (4) fulfills the relation E(z) dµ(z) ˜ ≥ E(z) dµ(z) =: E0 for all µ˜ ∈ M(D).

(5)

Additionally, we have the property E(z) ≡ E0 µ-almost everywhere.

(6)

Proof. Let us first assume µ is the equilibrium measure. Then, for an arbitrary measure µ˜ ∈ M(D), the condition d I (µt ) t=0 ≥ 0, µt = t µ˜ + (1 − t)µ, t ∈ [0, 1], dt has to hold. This means V (z) + 2 log |z − ζ |−1 dµ(ζ ) d(µ˜ − µ)(z) ≥ 0, which immediately implies Eq. (5). Assume on the other side µ fulfills condition (5). Then we obtain with the equilibrium measure µ0 , I (µ0 ) = I (µ) + V (z) + 2 log |z − ζ |−1 dµ(ζ ) d(µ0 − µ)(z) + log |z − ζ |−1 d(µ0 − µ)(ζ ) d(µ0 − µ)(z) ≥ I (µ), and so µ = µ0 . To show the additional statement, we consider for the probability measure µ the set B = {z ∈ D | E(z) < E0 }. If µ(B) > 0, the variational principle (5) for the measure µ˜ = χB (z) E0 ≤ E(z) dµ(z) < E0 , µ(B) and so µ(B) has to vanish. Thus, we get condition (6).

χB µ(B) µ

would yield

Instead of verifying condition (5), we prefer to use the following statement:


439

Corollary 3.5. If for a measure µ ∈ M(D) the function E, defined by Eq. (4), fulfills, for some real constant E0 , E(z) ≡ E0 on the support of µ and E(z) ≥ E0 everywhere, then µ is the equilibrium measure. 4. The Eigenvalue Density Definition. The k-point correlation function R (k) is given by (k) RN (zi )ki=1 =

N! (N − k)!

D N −k

N PN (zi )N d 2 zi . i=1 i=k+1

So, the one-point correlation function is up to normalization nothing but the density of the eigenvalues. As indicated in the previous section, all the correlation functions can be calculated in the limit N → ∞ out of the equilibrium measure µ, as in the case of hermitian matrix models, see [8, 9]. Theorem 4.1. For all φ ∈ C(D k ) we have the equality lim

N→∞ D k

k

(k) 2 1 φ (zi )ki=1 RN (zi )ki=1 d zi = k N

I.e. the measure

k (k) 1 R (zi )ki=1 i=1 Nk N

i=1

k φ (zi )ki=1 dµ(zi ). i=1

d2 zi on D k converges weakly to

k

i=1

(7)

dµ(zi ).

Proof. Substituting in the left-hand-side of Eq. (7) the definition of the correlation functions and turning our attention to the highest order in N , we obtain (because PN is invariant under the symmetric group) k (k) 1 (k) 1 φ, k RN = k φ (zi )ki=1 RN (zi )ki=1 d 2 zi N N Dk =

1 Nk

N

k i1 ,... ,ik =1 D

i=1

N φ (zij )kj =1 PN (zi )N d2 zi + o(1). i=1 i=1

Since for large values of N the probability distribution localizes at values z ∈ D0N with I (δz ) ≈ I0 , let us consider the sets AN,η = {z ∈ D0N | I (δz ) ≤ I0 + η},

η > 0.

Lemma 4.2. The probability PN (D N \ AN,η ) drops for N → ∞ exponentially to zero. 1 2 Proof. For an absolutely continuous

equilibrium measure dµ(z) = ψ(z) d z, we get with Jensen’s theorem and I (δz ) dµ(zi ) = I0 + o(1),

ZN ≥

e

N

−N 2 I (δz )−

{z∈D N

i=1 log ψ(zi )

| ψ(zi ) = 0 ∀i}

N

dµ(zi ) ≥ e−N

2 I +o(N 2 ) 0

,

i=1

1 This case suffices for our needs, but the restriction is in fact not necessary. Indeed, we could perform an analogous argument for the measures dµε (z) = ψε (z) d2 z, ψε (z) = 1 2 Bε (z) dµ. In the limit πε ε → 0, where I (µε ) → I0 , this would yield the desired statement.

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and therefore, PN (D N \ AN,η ) ≤

DN

eN

2 I +o(N 2 )−N 2 (I +η) 0 0

N

d2 zi = o(e−N

2 η/2

).

i=1

Let the continuous function N1k φ (zij )kj =1 take its maximum on the compact set AN,η at ζ , and set νN,η = δζ . Then, 1 (k) 1 φ, k RN ≤ k N N

N i1 ,... ,ik =1

φ (ζij )kj =1 =

k φ (zi )ki=1 dνN,η (zi ). i=1

Because of Lemma 3.2, we find a convergent subsequence νN(n),η → νη (n → ∞) with lim

N→∞

φ,

k 1 (k) k ≤ φ (z R ) dνη (zi ). i i=1 Nk N i=1

Lemma 4.3. We have νη ∈ M(D) and, in the limit η → 0, I (νη ) → I0 . Proof. Using ζ ∈ AN(n),η , we obtain with the cut-off L ∈ R: N(n) 1 1 I0 + η ≥ V (ζi ) + min{log |ζi − ζj |−1 , L} N (n) N (n)2 i=1 1≤i=j ≤N(n) = V (z) dνN(n),η (z) L + min{log |z − ζ |−1 , L} dνN(n),η (ζ ) dνN(n),η (z) − . N (n)

Sending first n and then L to infinity brings us to νη ∈ M(D) and therefore I0 ≤ I (νη ) ≤ I0 + η. So, letting η → 0, a subsequence of νη converges to the equilibrium measure µ, and thus, k 1 (k) k lim φ, k RN ≤ φ (zi )i=1 dµ(zi ). N→∞ N i=1

Arguing in the same way for the limes inferior concludes the proof.

5. Polynomial Curves Definition. A polynomial curve of degree n is a smooth simple closed curve in the complex plane with a parametrization h : S 1 ⊂ C → C of the form h(w) = rw + a0 + a1 w −1 + · · · + an w −n ,

|w| = 1,

(8)

with r > 0 and an = 0. The standard (counterclockwise) orientation of the circle induces an orientation on the curve. We say that a polynomial curve is positively oriented if this orientation is counterclockwise, i.e., if the tangent vector to the curve makes one full turn in the counterclockwise direction as we go around the unit circle.


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Proposition 5.1. Let γ be a positively oriented polynomial curve with parametrization h of the form (8). Then h, viewed as a homolorphic map on C× , restricts to a biholomorphic map from the exterior of the unit disk onto the exterior of γ . Proof. We have to show that h (w) = 0 for all w in the complement of the unit disk. Let t denote the tangent vector map w → t (w) = h (w)iw = i(rw − j aj w −j ). Since γ is a simple closed curve, the map w → t (w)/|t (w)| is a map of degree 1 from the unit circle to itself. Therefore we have 1 1= d arg(t (w)) 2π |w|=1 1 t (w) = dw 2πi |w|=1 t (w) 1 t (w) =N+ dw. 2πi |w|=R t (w) Here N ≥ 0 denotes the number of zeros of t (w), counted with multiplicity, in the complement of the unit disk and R is so large that it contains them all. The latter integral is 1 as can be seen by sending R to infinity. Thus N = 0, and h has no zeros in the complement of the unit disk. A simple consequence of this proposition is that a polynomial curve is uniquely parametrized by a map of the form (8) with r > 0. Indeed, any other conformal mapping of the complement differs by an automorphism of the complement of the unit disk. But non-trivial automorphisms are given by fractional linear transformations which do not preserve the conditions. From now on, we will only consider polynomial curves encircling the origin, i.e., such that the origin is contained in their interior domain. This can always be achieved by a translation, i.e., a shift of a0 . Definition. The harmonic moments (tj )∞ j =1 of the exterior domain D− of a polynomial curve (or more generally of an analytic curve) encircling the origin are defined by 1 1 −j 2 tj = − z d z= z¯ z−j dz, πj D− 2πij γ where only the right integral should be taken as a definition for j ≤ 2. Proposition 5.2. Let γ be a positively oriented polynomial curve of degree n encircling the origin. (i) The exterior harmonic moments tj of γ vanish for all j > n + 1. (ii) There exist universal polynomials Pj,k ∈ Z[r, a0 , . . . , ak−j ], 1 ≤ j ≤ k, so that for j = 1, . . . , n + 1, j tj = a¯ j −1 r −j +1 +

n

a¯ k r −k Pj,k (r, a0 , . . . , ak−j ).

(9)

k=j

Moreover, Pj,k is a homogeneous polynomial of degree k − j + 1 and it is also weighted homogeneous of degree k − j + 1 for the assignment deg(aj ) = j + 1, deg(r) = 0.

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(iii) The area of the domain enclosed by γ is πt0 , where t0 = r 2 −

n

j |aj |2 .

j =1

Proof. Since γ encircles the origin, h(w) never vanishes for |w| ≥ 1. Hence the contour in the formula for tj may be computed by taking residues at infinity. For j ≥ 1, 1 ¯ −1 )h (w)h(w)−j dw h(w j tj = 2π i |w|=1 −j n n n a¯ k −j k−j −l−1 −l−1 =r r− 1+ lal w al w /r dw. w 2π i k=0

l=1

l=0

The integrals in this sum vanish if k ≤ j − 2. The formula for tj in terms of ak , r is obtained by expanding the geometric series and picking the coefficient of w−1 in the integrand. This proves (i) and the first part of (ii). The homogeneity property is clear. The weighted homogeneity follows by rescaling w in the integral. The same formula ¯ −1 ), which does not contribute can be used to compute t0 , but the first term rw −1 in h(w to the integral and was omitted for j ≥ 1, must be added here. Examples. The terms in tj involving polynomials Pj,k with k ≤ 3 are t1 = a¯ 0 − r −1 a¯ 1 a0 − r −2 a¯ 2 (2 a1 r − a0 2 ) + r −3 a¯ 3 (3 a0 a1 r − 3 a2 r 2 − a0 3 ) + · · · , 2 t2 = r −1 a¯ 1 − 2 r −2 a¯ 2 a0 − 3 r −3 a¯ 3 (a1 r − a0 2 ) + · · · , 3 t3 = r −2 a¯ 2 − 3 r −3 a¯ 3 a0 + · · · . Theorem 5.3. Let t2 , . . . , tn+1 be complex numbers so that |t2 | < 1/2. Then there exists an A0 = A0 (t2 , . . . , tn+1 ) > 0 so that for all A, t1 with 0 < A < A0 and |t1 |2 < A(1/2 − |t2 |), there exists a unique positively oriented polynomial curve of degree ≤ n encircling the origin, with area A and exterior harmonic moments t1 , . . . , tn+1 , 0, 0, . . . . Proof. The idea is to invert the map (r, a0 , . . . , an ) → (t0 , . . . , tn+1 ) for small r and a0 . Set αj = r −j aj , ρ = r 2 and consider instead the polynomial map F : (ρ, α0 , . . . , αn ) → (t0 , . . . , tn+1 ), as a map from R × Cn+1 to itself. The first claim is that this map has a smooth inverse in some neighborhood of any point t ∈ R × Cn+1 such that t0 = t1 = 0 and |t2 | = 1/2. By Prop. 5.2, this map is given by t0 = ρ −

n

ρ j j |αj |2 ,

j =1

j tj = α¯ j −1 +

n

α¯ k Pj,k (r, α0 , rα1 , . . . , r k−j αk−j )

k=j

= α¯ j −1 +

n k=j

α¯ k Pj,k (ρ, α0 , α1 , . . . , αk−j ).


443

From the contour integral representation of the harmonic moments we get the integral Pj,k (ρ, α0 , . . . , αk−j ) −j n n 1 k−j l −l−1 l −l−1 1− 1+ w lαl ρ w αl ρ w dw = 2π i |w|=R l=1

l=0

for any sufficiently large R. By computing the residue at infinity, we can calculate Pj,k and thus tj up to terms of at least second order in α0 , ρ, t0 = ρ (1 − |α1 |2 ) + · · · , j tj = α¯ j −1 − j α¯ j α0 − (j + 1)ρ α¯ j +1 α1 + · · · ,

j ≥ 1.

Hence F (0, 0, 2 t¯2 , . . . , (n + 1) t¯n+1 ) = (0, 0, t2 , . . . , tn+1 ) and the tangent map at this point sends (ρ, ˙ α˙ 0 , . . . , α˙ n ) to (t˙0 , . . . , t˙n ) with t˙0 = (1 − 4|t2 |2 )ρ, ˙ ¯ j t˙j = α˙ j −1 − j (j + 1) tj +1 α˙ 0 − 2j (j + 1)(j + 2) tj +2 t2 ρ, ˙

j ≥ 1.

The tangent map is invertible if |t2 | = 1/2. By the inverse function theorem, F has a smooth inverse on some neighborhood of (0, 0, t2 , . . . , tn+1 ). If |t2 | < 1/2, F preserves the positivity of the first coordinate. In terms of the original variables, this means that given any t = (t0 , t1 , t2 , . . . , tn+1 ) with small t0 > 0, t1 and such that |t2 | = 1/2, there is a curve w → h(w) with h(w) = rw + α0 + rα1 w −1 + · · · + r n αn w −n and αj (j + 1) t¯j +1 . It remains to show that if r > 0 is small enough, h parametrizes a positively oriented simple closed curve containing the origin. We first show that h is an immersion. Since h (w) = r − rα1 w −2 + O(r 2 ) and limr→0 α1 = 2 t¯2 , we see that as long as |t2 | = 1/2, h (w) does not vanish on the unit circle. Similarly, we show that h : S 1 → C is injective: we have

|h(w) − h(w )|2 = r|w − w + 2 t¯2 (w −1 − w −1 )| + O(r 2 ) = r|w − w + 2 t¯2 (w¯ − w¯ )| + O(r 2 ). But the expression in the absolute value can only vanish for w = w if |2 t¯2 | = 1 which is excluded by the hypothesis. Moreover h(w) = rw + t¯1 + 2r t¯2 w −1 + O(r 2 ). Therefore h parametrizes a perturbation of an ellipse centered at t¯1 . The condition on t1 is a sufficient condition for this ellipse to contain the origin. Example. Let k ≥ 3 and let us consider curves with tj = 0 for all j = k. Then aj = 0 for all j = k − 1, so that h(w) = rw + ak−1 w −k+1 . The relation between (t0 , tk ) and (r, ak−1 ) is t0 = r 2 − (k − 1)|ak−1 |2 , k tk = a¯ k−1 r −k+1 . This map is a diffeomorphism from the region 0 < r < (k − 1) |ak−1 |, which is the condition for h(w) to be an embedding of the unit circle, onto the region 0 < t0 < (k(k − 1)|tk |)−2/(k−2) (k − 2)/(k − 1). As t0 approaches the upper bound for given tk , the curve develops cusp singularities.

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6. The Equilibrium Measure for a Polynomial Curve In this section we evaluate the equilibrium measure corresponding to potentials 1 V (z) = t0

|z| − 2 Re 2

n+1

tk z

k

.

(10)

k=1

We anticipate the result: 1 Theorem 6.1. For any set (tk )∞ k=1 ⊂ C with t1 = 0, |t2 | < 2 and tk = 0 for k > n + 1 and any compact domain D ⊂ C containing the origin as an interior point and such that t0 V , V given by Eq. (10), is positive on D \ {0}, there exists a δ > 0 so that for all 0 < t0 < δ the equilibrium measure µ for V on D is given by

µ=

1 χD λ, πt0 +

where D+ denotes the interior domain of the polynomial curve γ defined by the harmonic moments (tk )∞ k=0 , and λ is the Lebesgue measure on C. The rest of the section is dedicated to the proof of this theorem.

6.1. The Schwarz reflection. We first need the notion of a reflection on an analytic curve (see e.g. [10]). Definition. The Schwarz function of an analytic curve γ is defined as the analytic continuation (in a neighbourhood of the curve) of the function S(z) = z¯ on γ . The Schwarz reflection ρ for the analytic curve in this domain is the anti-holomorphic map ρ(z) = S(z). Definition. Under the critical radius R of the polynomial curve defined by the parametrization h we understand the value R = max{|w| | h (w) = 0, w ∈ C}, which, by definition of the map h, is less than 1. Lemma 6.2. Let γ be a polynomial curve parametrized by h and R its critical radius. Then the Schwarz function S and the Schwarz reflection ρ of the curve γ restricted to h(B1/R \ B¯ R ), where BR denotes the open disk with radius R around zero, are biholomorphic respectively anti-biholomorphic maps. They are given by S(z) = h¯

1 −1 h (z)

1 and ρ(z) = h . h¯ −1 (¯z)

(11)

Therefore, ρ maps γ identically on itself, h(B1 \ B¯ R ) to h(B1/R \ B¯ 1 ) and vice versa. Also, we have ρ 2 = id.


445

Proof. By definition of the Schwarz function, in a neighborhood of |w| = 1, ¯ −1 ). S(h(w)) = h(w Because for w ∈ C \ B¯ R the function h is biholomorphic, we may write 1 1 ¯ S(z) = h . , ρ(z) = S(z) = h h−1 (z) h¯ −1 (¯z) Taking the derivatives, we find that they do not vanish for R < |h−1 (z)|
1 and |t2 | < 21 . The last bracket we can estimate by ¯ −1 ) ≥ |t2 | − |t2 ww ¯ −1 | = 0. |t2 | − Re(t2 ww It therefore remains to show (|w|2 − 1)(1 + 2|t2 ||w|−2 ) + (1 + 2|t2 |2 ) log(|w|−2 ) ≥ 0, which follows immediately out of the following lemma. Lemma 6.5. For 0 ≤ α ≤ 1 the function f (x) = (x − 1)(1 + αx −1 ) − (1 + α) log x is non-negative on the interval [1, ∞). Proof. For the function f and its derivatives f and f we have 2α 1 ≥ 0. f (1) = 0, f (1) = 0 and f (x) = 2 1 + α − x x

So we showed that E(z) ≥ 0 for all z ∈ C \ D+ . Because of Lemma 6.3, we also know that in the interior domain D+ , E is zero and we therefore can apply Corollary 3.5 to see that µ is indeed the equilibrium measure for the potential V on C. 6.3. The proof of Theorem 6.1. As in the Gaussian case we are going to show that the measure µ given in the theorem fulfills the conditions of Corollary 3.5 and is therefore the uniquely defined equilibrium measure. Because Theorem 6.1 is only valid for interior domains with small area, we are going to consider the asymptotical behaviour t0 → 0, where the harmonic moments (tk )∞ k=1 are kept fixed. To catch the asymptotical behavior of the corresponding polynomial curve, let us parametrize it as in the proof of Theorem 5.3: h(w) = rw +

n

r j αj w −j .

j =0

Then, for r → 0, we have r 2 t0 , αj (j + 1)t¯j +1 , j ≥ 1, and, because we set t1 = 0, α0 r 2 .

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Lemma 6.6. The critical radius of h is asymptotically constant for r → 0: R = |α1 | + O(r). Proof. The√roots of the function h (w) = r −rα1 w −2 −· · ·−nr n αn w −n−1 are in zeroth order at ± α1 and (n − 1-times degenerated) at zero. We consider now for z ∈ D respectively for w ∈ h−1 (D \ D+ ) the functions −1 z log − 1 d2 ζ and (15) E(z) = V (z) + ζ D+ w 1 ¯ w˜ −1 )h (w) E(w) = E(h(w)) = |h(w)|2 − |h(1)|2 + 2 Re h( ˜ dw˜ . (16) t0 1 We already showed in Lemma 6.3 that E(z) ≡ 0 in D+ , so the first condition of the corollary is satisfied (this is essentially the way we have chosen our potential V ). Now, also with Lemma 6.3, we see that E ≥ 0 in the vicinity of the curve γ , strictly speaking in the domain h(B1/R \ B¯ 1 ). Indeed, if we look at the connected components of the contour lines of the function E (which are smooth curves in the considered domain because there ∂z¯ E = 0), we see that the gradient vector ∂z¯ E always points outwards, i.e. in the exterior domain of the contour line. Therefore, the value of E on the contour lines is increasing outwards as desired. A bit farther from the curve, i.e. for 1/R ≤ |w| < r −α , 0 < α < 13 , the function E equals asymptotically the one of the corresponding ellipse h(0) (w) = rw + α0 + rα1 w −1 , we denote it by E (0) . Indeed, remarking that for the area of this ellipse we have (0) t0 = t0 + O(r 4 ) and that E (0) (w) = O(r −2α ), we obtain, uniformly in w, 1 (|h(w)|2 − |h(0) (w)|2 − |h(1)|2 + |h(0) (1)|2 ) t0 w 2 + Re (h(w˜ −1 ) − h(0) (w˜ −1 ))h (w) ˜ dw˜ t0 1 w 2 + Re (h (w) ˜ − h(0) (w))h ˜ (0) (w˜ −1 ) dw˜ t0 1 1 (0) t0 − t0 E (0) (w) + t0 = O(r 1−α ) + O(r 1−3α ) + O(r 1−2α ) + O(r 2−2α ) → 0 (r → 0).

E(w) − E (0) (w) =

Because E (0) (w) ≥ C > 0 for all |w| ≥ R1 and r > 0, we may choose r so small that |E(w) − E (0) (w)| < E (0) (w) and so E(w) > 0 for all w ∈ Br −α \ B1/R . The domain remains, where |w| ≥ r −α . For k ≥ 2, we obtain  l k n k (rw)k−l  r j αj w −j  h(w)k − (rw)k = l l=1

j =0

l=1

j =0

 l k n k k k−2l  j −1 = r w r αj w 1−j  = O(r 2 ). l


449

And since t0 V (z) = |z|2 − t2 z2 − t¯2 z¯ 2 + o(|z|2 ) 1

= |z − 2t2 z¯ |2 + (1 − 4|t2 |2 )|z|2 + o(|z|2 ) 2 for z → 0 and V > 0 on D \{0}, we may find a constant C > 0 such that t0 V (z) ≥ C|z|2 for all z in the compact domain D. Therefore, for r → 0, C V (h(w)) = V (rw) + O(1) ≥ |rw|2 + O(1), t0 which tends to infinity at least as r −2α . On the other side, the integral over the logarithm in (15) diverges for r → 0 only as log r. So we have for r small enough E(h(w)) > 0 for all w ∈ h−1 (D \ D+ ) \ Br −α . This proves now E(z) ≥ 0 for all z ∈ D and therefore, with Corollary 3.5, that µ is the equilibrium measure. 6.4. Shifting the origin. As a little generalization, we consider the case where t1 = 0. This corresponds to a shift of the origin. Therefore, we like to define the harmonic moments also for a curve which does not encircle the origin. Definition. Let h(w) = rw +

n

aj w −j

j =0

parametrize a polynomial curve of degree n. Then the harmonic moments (tk )n+1 k=1 are given by the equation system (9). All other harmonic moments are set to zero. Proposition 5.2 tells us that this definition coincides with the previous one if the origin is in the interior domain of the curve. 1 Corollary 6.7. For any set (tk )∞ tk = 0 for k > n + 1 and k=1 ⊂ C with |t2 | < 2 and 2 k any compact domain D ⊂ C such that U (z) = |z| − 2 Re n+1 k=1 tk z has exactly one absolute minimum in the interior of D, there exists a δ > 0 so that for all 0 < t0 < δ the equilibrium measure µ for V = t10 U on D is given by

1 χD λ, πt0 + where D+ denotes the interior domain of the polynomial curve γ defined by the harmonic moments (tk )∞ k=0 . µ=

Proof. Let us first shift the origin by a0 , such that V (z + a0 ) has its absolute minimum in 0. Thereby, the potential gets the form n+1 1 |z|2 − 2 Re V (z + a0 ) = tk zk + V (a0 ), t0 k=2

where the tk are the harmonic moments of the shifted curve γ − a0 . Indeed, the coefficients of V and the harmonic moments depend polynomially on the shift a0 . Because we know them to coincide as long as the origin is inside D+ , they do so for all a0 . Applying now Theorem 6.1 for the shifted potential gives the desired result.

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7. Discussion We have shown that under suitable assumptions on the polynomial p appearing in the potential and on the integration range D of the eigenvalues, the asymptotic density is uniform with support on a domain uniquely determined by the coefficients of the polynomial p. It would be interesting to understand what happens at the range of validity of our assumptions. If the potential V has more than one minimum in D then one should expect for small t0 to have an equilibrium measure with disconnected support, so that a description by a polynomial curve cannot be valid. Also as t0 gets bigger, polynomial curves develop singularities and become non-simple. The question is then what happens to the eigenvalues. Finally we note that polynomial curves are (real sections of complex) rational curves. Curves of higher genus should arise by replacing p by more general holomorphic functions. References 1. Wiegmann, P. B., Zabrodin, A.: Conformal Maps and integrable hierarchies. Commun. Math. Phys. 213(3), 523–538 (2000) 2. Kostov, I. K., Krichever, I., Mineev-Weinstein, M., Wiegmann, P. B., Zabrodin, A.: τ -function for analytic curves. In: Random matrix models and their applications, Math. Sci. Res. Inst. Publ. 40, Cambridge: Cambridge Univ. Press, 2001, pp. 285–299 3. Marshakov, A., Wiegmann, P., Zabrodin, A.: Integrable Structure of the Dirichlet Boundary Problem in Two Dimensions. Commun. Math. Phys. 227(1), 131–153 (2002) 4. Krichever, I., Marshakov, A., Zabrodin, A.: Integrable Structure of the Dirichlet Boundary Problem in Multiply-Connected Domains. http://arxiv.org/list/hep-th/0309010, 2003 5. Takasaki, K., Takebe, T.: Integrable Hierarchies and Dispersionless Limit. Rev. Math. Phys. 7(5), 743–808 (1995) 6. Chau, L.-L., Zaboronsky, O.: On the Structure of Correlation Functions in the Normal Matrix Model. Commun. Math. Phys. 196(1), 203–247 (1998) 7. Saff, E. B., Totik, V.: Logarithmic Potentials with External Fields. Grundlehren der mathematischen Wissenschaften 316, Berlin-Heidelberg-New York: Springer, 1997 8. Johansson, K.: On Fluctuations of Eigenvalues of Random Hermitian Matrices. Duke Math J. 91, 151–204 (1998) 9. Deift, P. A.: Orthogonal polynomials and random matrices: a Riemann-Hilbert Approach. Courant Lecture Notes in Mathematics, Vol 3, New York: Courant Institute of Mathematical Sciences, 1999 10. Davis, P. J.: The Schwarz function and its applications. The Carus Mathematical Monographs, No. 17, Washington, DC: The Mathematical Association of America, 1974 Communicated by L. Takhtajan

Commun. Math. Phys. 259, 451–474 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1373-y

Communications in


Traveling Fronts in a Reactive Boussinesq System: Bounds and Stability Brandy Winn Department of Mathematics, University of Chicago, Chicago, IL 60637, USA. E-mail: [email protected] Received: 28 September 2004 / Accepted: 4 January 2005 Published online: 14 June 2005 – © Springer-Verlag 2005

Abstract: This paper considers a simplified model of active combustion in a fluid flow, with the reaction influencing the flow. The model consists of a reaction-diffusion-advection equation coupled with an incompressible Navier-Stokes system under the Boussinesq approximation in an infinite vertical strip. We prove that for certain ignition nonlinearities, including all that are C 2 , and for any domain width, planar traveling front solutions are nonlinearly and exponentially stable within certain weighted H 2 spaces, provided that the Rayleigh number ρ is small enough. The same result holds for bistable nonlinearities in unweighted H 2 spaces. We also obtain uniform bounds on the Nusselt number, the bulk burning rate, and the average maximum vertical velocity for chemistries that include bistable and ignition nonlinearities.

1. Introduction Transition processes in nature, such as the transition from one equilibrium state to another, are often manifested as traveling waves. Traveling waves are observed in combustion [17], chemical kinetics [4], propagation of dominant genes [3, 7], and propagation of nerve impulses [6], to name a few applications. The use of parabolic equations to reproduce wave propagation dates back to the work of Kolmogorov, Petrovski˘ı, and Piskunov [7] and Fisher [3] in 1937. Since the 1970’s, parabolic systems of reaction diffusion equations have been actively used and studied to model traveling wave phenomena in a wide variety of physical, chemical, and biological problems [14]. Many of these problems of interest take place in fluids and are strongly influenced by the fluid dynamics [9, 17]. Appropriately, numerous studies have investigated hydrodynamical effects on reaction processes. Most of these studies consider passive advection, in which the reactants are carried by the flow, but the flow is unaffected by the reaction. Only a few studies rigorously examine the effects of active advection, in which the fluid flow is influenced by the reaction [2, 8, 10, 11, 13]. The above references are merely a small

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sampling of the literature; additional references can be found within the recent reviews [1, 16] as well as the above sources. This paper considers a reactive Boussinesq system in an infinite vertical strip = [0, ]x × Rz : ∂t T + u · ∇T = T + f (T ), ∂t u + u · ∇u + ∇P = σ u + σρT eˆz , ∇ · u = 0.

(1.1) (1.2) (1.3)

In this model a reaction-diffusion-advection equation is coupled with an incompressible Navier-Stokes system under the Boussinesq approximation. We regard the system as a simplified model of combustion in a fluid flow, with the reaction influencing the flow. The reaction chemistry is given by f ; of particular interest are ignition, bistable, and KPP nonlinearities. The boundary conditions in the x-direction are periodic, and in the z-direction are given by lim T = 1,

z→−∞

lim T = 0,

z→∞

lim u = 0.

z→±∞

(1.4)

The corresponding one-dimensional problem without x variation on a domain R and with an ignition, bistable, or KPP nonlinearity is well known to have traveling front solutions connecting the lower regions of hot fluid with the higher regions of cold fluid. These fronts have the form T = T (z−ct), u = 0 and are easily seen to solve the full system (1.1)–(1.4) as well. One of the central questions concerning this system is whether or not these planar fronts are stable with respect to small perturbations in the initial data. This issue was analyzed in [2] for ignition, bistable, and KPP nonlinearities. The authors proved that for small aspect ratio < C1 and small Rayleigh number ρ < C2 /3 for some constants C1 and C2 , the only traveling front solutions of (1.1)–(1.4) are the planar fronts corresponding to the one-dimensional problem, and any other solution of the system becomes planar in the sense that ||∇ ⊥ · u(t)||L2 () + ||∂x T (t)||L2 () → 0 as t → ∞. The authors also found that planar traveling fronts of this system with speed two, T = T (z − 2t), u = 0, are linearly unstable if ρ > 2C/σ and is large enough. In this case the planar fronts lose stability due to longwave perturbations that grow exponentially. Note that in general a Boussinesq system might have nonplanar traveling front solutions. This is certainly the case for reactive Boussinesq systems with bistable nonlinearities and large Rayleigh numbers; in [10, 11] such systems are proven to admit nonplanar fronts as solutions. This paper examines the nonlinear stability of planar fronts for ignition and bistable nonlinearities. We prove that for certain ignition nonlinearities, including all that are C 2 , and for any domain width , these planar traveling fronts are nonlinearly and exponentially stable within certain weighted H 2 spaces, provided that the Rayleigh number ρ is small enough to satisfy ρ < δ(, σ ) for some δ with δ(, σ ) → 0 as → ∞ or σ → ∞. The same result holds for bistable nonlinearities in unweighted H 2 spaces provided that the initial data for T decay exponentially at infinity. Our other main result concerns uniform bounds. We obtain bounds on the Nusselt number, the bulk burning rate, and the average maximum vertical fluid velocity, provided that the initial data for the temperature satisfy front-like exponential decay conditions and provided that the chemistry f satisfies certain conditions which include bistable and ignition nonlinearities. Similar bounds were obtained in [2] for the case of concave KPP nonlinearities.

Traveling Fronts in a Reactive Boussinesq System

453

The organization of this paper is as follows. Secttion 2 takes a closer look at the reactive Boussinesq system, noting some basic properties and symmetries of the equations, as well as setting up notation and assumptions to be used throughout this paper. Bounds for the temperature are derived in Sect. 3.1. The temperature bounds are then used to derive bounds for the Nusselt number, the bulk burning rate, and the average maximum vertical velocity in Sect. 3.2. The next section examines the nonlinear stability of the planar waves. First, Sect. 4.1 studies the spectrum of a relevant linear operator L and gives some bounds on the resolvent operator. Then the exponential of L and fractional powers of L are defined and a few needed bounds involving these are stated. Finally, Sect. 4.2 proves a stability result for ignition and bistable nonlinearities. 2. The Reactive Boussinesq System We consider the system ∂t T + u · ∇T = T + f (T ), ∂t u + u · ∇u + ∇P = σ u + σρT eˆz , ∇ ·u = 0

(2.1) (2.2) (2.3)

along with its vorticity ω = ∇ ⊥ · u = ∂x u2 − ∂ξ u1 , ∂t ω + u · ∇ω = σ ω + σρTx

(2.4)

on the domain = [0, ]x × Rz . We interpret T as the temperature, u as the fluid velocity, P as the pressure, and eˆz = (0, 1) as the unit vector in the direction opposite gravity. The positive constants σ and ρ are nondimensional, with σ representing the Prandtl number or normalized fluid viscosity, and ρ representing the Rayleigh number. The chemistry f (T ) is Lipschitz continuous, nontrivial, and satisfies f (0) = f (1) = 0. Additional conditions on f will be introduced as needed; of particular interest are ignition and bistable nonlinearities. For ignition f (T ) = 0 in [0, q∗ ] ∪ {1} and f (T ) > 0 in (q∗ , 1) for some q∗ ∈ (0, 1). For bistable nonlinearities f (T ) < 0 in (0, q∗ ) and f (T ) > 0 in (q∗ , 1) for some q∗ ∈ (0, 1). The boundary conditions are periodic in the x-direction, T (t, x, z) = T (t, x + , z), u(t, x, z) = u(t, x + , z), ω(t, x, z) = ω(t, x + , z), (2.5) and are given by lim T = 1,

z→−∞

lim T = 0,

z→∞

lim u = 0, and

z→±∞

lim ω = 0

z→±∞

(2.6)

in the z-direction. We assume the initial condition T0 (x, z) satisfies 0 ≤ T0 (x, z) ≤ 1. Note that the incompressibility of the system and the boundary conditions at z = ±∞ force the mean of u2 to be independent of z and hence zero: 0= (∂x u1 + ∂z u2 ) dx = ∂z u2 dx, 0

and so

0

0

u2 dx = 0.

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This system is symmetric with respect to reflections so that if u, T , ω is a solution, then so is ˜˜ x, z) = (−u1 (t, − x, z), u2 (t, − x, z)) , u(t, T˜˜ (t, x, z) = T (t, − x, z), ˜˜ x, z) = −ω(t, − x, z). ω(t,

We assume that the initial data u0 , T0 is invariant with respect to this symmetry. With ˜˜ x, z) are solutions with the same initial data this assumption, T , u and T˜˜ (t, x, z), u(t, and boundary conditions. So this invariance is preserved in time and u1 (t, x, z) = −u1 (t, − x, z), T (t, x, z) = T (t, − x, z),

u2 (t, x, z) = u2 (t, − x, z), ω(t, x, z) = −ω(t, − x, z).

Under these circumstances, u1 and ω are odd, and hence have mean zero: u1 dx = 0, ω dx = 0. 0

0

This symmetry assumption on the initial data will be used only in Sect. 4.2 to obtain Poincaré inequalities for u and ω, thereby completing the proof of stability. This assumption can be replaced by any other condition that ensures Poincaré inequalities for u and ω. 3. Some Bounds 3.1. Bounds for T . First note that by the maximum principle, our assumption that 0 ≤ T0 ≤ 1 forces 0 ≤ T ≤ 1 for all t ≥ 0. Proposition 1. Suppose T0 satisfies T0 (x, z) ≤ ke−az for some positive constants k and a and suppose f (T ) ≤ 0 for T ∈ [0, q∗ ] for some q∗ ∈ (0, 1). Choose b ≥ a 2 + f ∞ . Then t u2 (s)∞ ds T (x, z, t) ≤ k exp −az + bt + a (3.1) 0

for all t ≥ 0. Proof. Define t u2 (s)∞ ds . θ+ (z, t) = k exp −az + bt + a 0

Then

So

∂t θ+ + u · ∇θ+ − θ+ = θ+ b + au2 (t)∞ − au2 − a 2 ≥ θ+ f ∞ . ∂t (θ+ − T ) + u · ∇(θ+ − T ) − (θ+ − T ) ≥ f ∞ θ+ − f (T ).


455

Note that f (T ) < f ∞ T for all T > 0. Define η := min f ∞ T − f (T ) > 0. T ∈[q∗ ,1]

We claim that θ+ − T ≥ 0 for all t ≥ 0. Suppose not. Then we may choose a number µ such that 0 > µ > inf(θ+ − T ) and 0 > µ > − 2fη . Moreover, there exists a point ∞ pµ = (xµ , zµ , tµ ) with (θ+ − T )(pµ ) = µ and inf (θ+ − T ) = µ. Then at pµ we ×[0,tµ ]

have

0 ≥ ∂t (θ+ − T ) + u · ∇(θ+ − T ) − (θ+ − T ) ≥ f ∞ θ+ − f (T ). If T ≤ q∗ , then f ∞ θ+ − f (T ) > 0 automatically. If T > q∗ , then at pµ f θ+ − f (T ) = f (θ+ − T ) + f T − f (T ) ∞ ∞ ∞ ≥ f ∞ µ + η η > . 2

Thus f θ+ − f (T ) > 0 for all T at pµ , a contradiction. ∞

Proposition 2. Suppose T0 satisfies 1 − T0 (x, z) ≤ keaz for some positive constants k and a and suppose f (T ) ≥ 0 for T ∈ [q∗ , 1] for some q∗ ∈ (0, 1). Choose b ≥ a 2 + f ∞ . Then t u2 (s)∞ ds (3.2) 1 − T (x, z, t) ≤ k exp az + bt + a 0

for all t ≥ 0. Proof. Note that ∂t (1 − T ) + u · ∇(1 − T ) = (1 − T ) − f (1 − (1 − T )) . Define T˜ = 1 − T and f˜(T˜ ) = −f (1 − T˜ ). Then ∂t T˜ + u · ∇ T˜ = T˜ + f˜(T˜ ) and f˜(T˜ ) ≤ 0 for T˜ ∈ [0, 1 − q∗ ]. Applying the proof of Proposition 1 to T˜ yields the result.

3.2. Some General Bounds. We assume in this section that f (T ) ≤ 0 for T ∈ [0, q∗1 ] and f (T ) ≥ 0 for T ∈ [q∗2 , 1] for some 0 < q∗1 ≤ q∗2 < 1 so that Propositions 1 and 2 hold. In particular f could be an ignition or bistable nonlinearity. We would like to obtain bounds on the Nusselt number N u, the bulk burning rate V∞ , and the average maximum vertical velocity W (t). These are given by the relations

1 2 ∇T (t)L2 , N u = lim N (t), where N (t) = t→∞ 1 V∞ = lim V (t) , where V (t) = ∂t T (x, z, t) dxdz, t→∞

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and W (t) = u2 (t)∞ . Here φ(t) =

1 t

t

φ(t) dt 0

denotes the time average. At this point it will be useful to introduce a weighted L2 norm that is independent of the domain width : 1 φ2L2 = |φ|2 dxdz. (3.3) Lemma 1. Suppose the initial data T0 of system (2.1)–(2.3) satisfies (3.1) so that T ≤ k1 e−a1 z+B1 (t) ,

(3.4)

where

t

B1 (t) = b1 t + a1 0

u2 (s)∞ ds = b1 t + a1 tW (t).

Then V (t) ≤ W (t) +

b1

+ a1 t

(3.5)

for all t ≥ 0, where is a constant depending on the initial data. Proof. V (t) = = = ≤ = =

1 t 1 ∂t T (x, z, t) dxdz dt t 0 11 [T (x, z, t) − T0 (x, z)] dxdz t B1 (t)/a1 ∞ 0 11 (T − T0 )dz+ (T − T0 )dz+ (T − T0 )dz dx t 0 0 −∞ B1 (t)/a1 B1 (t)/a1 ∞ 0 11 −a1 z+B1 (t) (1 − T0 ) dz + 1 dz+ k1 e dz dx t 0 0 −∞ B1 (t)/a1

1 B1 (t) k1

0 + + t a1 a1

0 + k1 /a1 b1 + + W (t), t a1

where 0 is a constant depending only on the initial data T0 .


457

Lemma 2. Suppose the initial data T0 of system (2.1)-(2.3) satisfies (3.2) so that 1 − T ≤ k2 ea2 z+B2 (t) , where

t

B2 (t) = b2 t + a2 0

(3.6)

u2 (s)∞ ds = b2 t + a2 tW (t).

Then 1 b2 + , a2 t where is a constant depending on the initial data. N (t) ≤ V (t) + W (t) +

(3.7)

Proof. First note that by Eq. (2.1),

1 f (T ) dxdz. V (t) = Let ϕ(T ) = T (1 − T ). Making use of Eq. (2.1) we find (∂t + u · ∇)ϕ(T ) = ϕ (T ) [∂t T + u · ∇T ] = ϕ (T ) [T + f (T )] .

(3.8)

Integrating over and using integration by parts gives d

2 ϕ(T )dxdz = − ϕ (T )|∇T | dxdz + ϕ (T )f (T ) dxdz dt ≥2 |∇T |2 dxdz − f (T ) dxdz

= 2∇T 2L2 − V (t). Taking the time average and dividing by 2 yields 1 1 N(t) ≤ V (t) + [ϕ(T ) − ϕ(T0 )] dxdz 2 2t 1 1 1 = V (t) + (T − T0 )dxdz + (T0 − T )(T0 + T )dxdz 2 2t 2t 1 = V (t) + (T0 − T )(T0 + T )dxdz. (3.9) 2t The last integral in (3.9) is equivalent to 0 −B2 (t)/a2 (T0 + T ) (T0 + T ) 1 dz + (T0 − T ) dz (T0 − T ) t 0 2 2 −∞ −B2 (t)/a2 ∞ (T0 + T ) + dz dx (T0 − T ) 2 0 −B2 (t)/a2 0 ∞ 1 ≤ (1 − T ) dz + 1 dz + T0 dz dx t 0 −∞ −B2 (t)/a2 0 ∞ −B2 (t)/a2 1 B2 (t) ≤ k2 exp{a2 z + B2 (t)} dz + + T0 dz dx t 0 a2 −∞ 0 b2 11 ∞ k2 + + W (t) + T0 dzdx. = a2 t a2 t 0 0

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Inserting this last relation into (3.9) reveals that b2 1 k2 1 ∞ N(t) ≤ V (t) + W (t) + + + T0 dzdx . a2 t a2 0 0

Lemma 3. Any solution of system (2.1)–(2.3) satisfies 1 W (t) ≤ C max{2 , } ρ N (t) + √ ω0 L2 σt

(3.10)

for some constant C. Here ω0 is the initial data of the vorticity ω. Proof. We will use bar notation to denote averages in x: φ(z, t) := 1 0 φ(x, z, t)dx. Recall that u2 = 0. Now any function φ satisfying φ = 0 also satisfies 2 |φ|2 ≤ 0 |∂x φ|dx ≤ 0 |∂x φ|2 dx. In particular

|u2 |2 ≤

|∂x u2 |2 dx,

(3.11)

0

|∂x u2 | ≤ 2

0

and

|∂z u2 |2 ≤

|∂x2 u2 |2 dx,

(3.12)

|∂x ∂z u2 |2 dx.

(3.13)

0

Treating u2 as a function in z alone, the Sobolev inequalities tell us that ∞ ∞ |u2 |2 dz + |∂z u2 |2 dz . |u2 |2 ≤ C −∞

−∞

(3.14)

Inserting the inequalities (3.11)–(3.13) into the right-hand side of (3.14) yields ∞ ∞ |u2 |2 ≤ C 3 |∂x2 u2 |2 dxdz + |∂x ∂z u2 |2 dxdz . (3.15) −∞ 0

−∞ 0

We now apply integration by parts to continue transforming the inequality: ∞ ∞ |u2 |2 ≤ C 3 |∂x2 u2 |2 dxdz + (∂x2 u2 )(∂z2 u2 )dxdz −∞ 0 −∞ 0 |u2 |2 dxdz ≤ C max{3 , } 3 |∂x ω|2 dxdz = C max{ , }

≤ C max{4 , 2 }∇ω2L2 .

It now follows that

W (t) = u2 (t)∞ ≤ C max{2 , } ∇ωL2 .


459

Applying Cauchy-Schwartz in time, we obtain 1/2 W (t) ≤ C max{2 , } ∇ω2L2 .

(3.16)

Multiplying the vorticity equation (2.4) by ω, integrating over , and applying integration by parts yields 1 d ω(t)2L2 + σ ∇ω(t)2L2 2 dt = σρ

ω∂x T dxdz

= −σρ (∂x ω)(T − T ) dxdz σ ≤ |∂x ω|2 + ρ 2 |T − T |2 dxdz 2 σ σρ 2 2 ≤ ∇ω(t)2L2 + ∇T 2L2 , by Sobolev’s inequality. 2 2 Taking the time average and dividing by , transforms the above inequality into 1 1 ω(t)2L2 + σ ∇ω(t)2L2 ≤ σρ 2 2 N (t) + ω(0)2L2 . (3.17) t t Combining (3.16) and (3.17) gives the desired bound.

Theorem 1. If a solution of system (2.1)–(2.3) has front-like initial data obeying (3.4) and (3.6) then 1 b1 b2 1+ + + , W (t) ≤ 2C 2 max{6 , 4 }ρ 2 + 2C max{2 , } √ ω0 L2 + 2a1 2a2 2 t σt (3.18) 1 2b 2b +

1 2 N(t) ≤ 4C 2 max{6 , 4 }ρ 2 + 4C max{2 , } √ ω0 L2 + + +2 a1 a2 t σt (3.19) and 1 3b1 b2 1 + 3 V (t) ≤ 2C 2 max{6 , 4 }ρ 2 +2C max{2 , } √ ω0 L2 + , + + 2a1 2a2 2 t σt (3.20) where and are constants depending on the initial data T0 , and C is the Sobolev constant from (3.14). Proof. Applying Young’s inequality to (3.10) yields W (t) ≤

1 1 N (t) + C 2 max{6 , 4 }ρ 2 + C max{2 , } √ ω0 L2 . 4 σt

(3.21)

Combining the relations (3.7), (3.5), and (3.21) in a straightforward way yields the desired results.

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Corollary 1. If a solution of system (2.1)-(2.3) has front-like initial data obeying (3.4) and (3.6) then 2b1 2b2 2 7 5 2 N u ≤ 4C max{ , }ρ + (3.22) + a1 a2 and V∞ ≤ 2C 2 max{6 , 4 }ρ 2 +

3b1 b2 + . 2a1 2a2

(3.23)

4. Stability of Planar Traveling Waves We wish to investigate the stability of the special family of traveling wave solutions u = 0, T = U (z − c0 t + γ ), γ ∈ R, where U solves Uzz + c0 Uz + f (U ) = 0,

(4.1)

lim U = 1, and lim U = 0.

(4.2)

z→∞

z→−∞

The constant c0 is unique for ignition and bistable nonlinearities and can assume all values in the interval [2 f (0), ∞) for KPP nonlinearities. In all three cases U < 0. To obtain stability, we will switch to a set of moving coordinates ξ = z − c0 t + εc1 (t, ε), ε > 0, that move approximately with the traveling waves. The extra degree of freedom from allowing the shift γ = εc1 (t, ε) to vary in time will prove helpful. After performing the switch ∂t T − (c0 + ε∂t c1 )∂ξ T + u · ∇T ∂t u − (c0 + ε∂t c1 )∂ξ u + u · ∇u + ∇p ∇ ·u ∂t ω − (c0 + ε∂t c1 )∂ξ ω + u · ∇ω lim T = 0,

ξ →∞

lim T = 1

ξ →−∞

= T + f (T ), = σ u + σρT eˆξ , = 0, = σ ω + σρTx ,

lim u = 0, and

ξ →±∞

(4.3) (4.4) (4.5) (4.6)

lim ω = 0.

ξ →±∞

We now perturb about the special solution 0, U (ξ ) by writing T (t, x, ξ ) = U (ξ ) + ετ (t, x, ξ ), u(t, x, ξ ) = 0 + ερv(t, x, ξ ), ω(t, x, ξ ) = 0 + ερ ω(t, ˜ x, ξ ). Then τ , v, and ω˜ satisfy ∂t τ = −Lτ + F, 1 ∂t v = σ v + (c0 + ε∂t c1 )∂ξ v − ερv · ∇v − ∇ p˜ + σ τ eˆξ + σ U eˆξ , ε ∇ · v = 0, ∂t ω˜ = σ ω˜ + (c0 + ε∂t c1 )∂ξ ω˜ − ερv · ∇ ω˜ + σ ∂x τ,

(4.7) (4.8) (4.9) (4.10)


lim τ = 0,

ξ →±∞

461

lim v = 0, and

ξ →±∞

lim ω˜ = 0,

ξ →±∞

where −L = + c0 ∂ξ + f (U ), F = ε∂t c1 ∂ξ τ − ερv · ∇τ + (∂t c1 − ρv2 )Uξ + εN, 1 N = 2 [f (U + ετ ) − f (U ) − f (U )ετ ]. ε

(4.11) (4.12) (4.13)

The decay of τ and hence the stability of the system is determined to a great degree by the spectrum of L. By differentiating Eq. (4.1), we see that LUξ = 0 so that zero may be an eigenvalue of L. Clearly Uξ does not decay in time; so the zero-eigenspace could potentially be a problem. In the case of a zero eigenvalue, the pursuit of stability requires one to restrict to the orthogonal complement of the zero-eigenspace. To obtain stability, it will then suffice for the remaining spectrum of −L to lie strictly in the left half-plane with a gap from the imaginary axis. For some nonlinearities, such as the bistable nonlinearity, a spectral gap occurs automatically in H 2 (); however for other nonlinearities such as ignition and KPP, it is necessary to introduce weights to create a gap. Ultimately, the existence of a gap and the need for weights is determined by the behavior of f at the zeros of f , since these points are equilibria of the system.

4.1. Properties of the Linear Operator L. Consider the linear operator L given by −L = + c0 ∂ξ + f (U ),

(4.14)

where f is either an ignition chemistry with the properties f (T ) = 0 for T ∈ [0, q∗ ] ∪ {1}, 0 < q∗ < 1, f (T ) > 0 for T ∈ (q∗ , 1), f (T ) is Lipschitz continuous,

(4.15)

or a bistable chemistry with the properties f (0) = f (q∗ ) = f (1) = 0, 0 < q∗ < 1, f (T ) < 0 for T ∈ (0, q∗ ), f (T ) > 0 for T ∈ (q∗ , 1), f (T ) is Lipschitz continuous,

(4.16)

and c0 > 0 is the constant uniquely determined by (4.1)–(4.2). Consider for a moment the case of ignition. Note that f (U (−∞)) = f (1) < 0 and f (U (∞)) = f (0) = 0, so that 1 is a stable solution of the problem ∂t T = f (T ), but 0 is not. To obtain a spectral gap it is necessary to reweight L near ξ = ∞. We take the domain of L to be a weighted Sobolev space H 2 (w, ) on = [0, ]x × Rξ with weight w(ξ ) = (1 + eξ c0 /2 )2 . The corresponding weighted L2 space will be denoted by L2 (w, ) or L2 (w) when is clear, and its norm will be denoted by −L2 (w,) =

1/2 | − |2 w(ξ ) d

.

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B. Winn

The corresponding inner product will be written as g, hw, =

1/2 ghw d

.

On the other hand, for bistable nonlinearities f (0) < 0 and f (1) < 0. So 0 and 1 are both stable solutions of the problem ∂t T = f (T ), and it is not necessary to work in a weighted space to obtain a spectral gap. To investigate the properties of the operator L, it is helpful to first consider the one dimensional operator −Lξ = ∂ξ2 + c0 ∂ξ + f (U )

(4.17)

on H 2 (w, ˜ R), where w˜ = w for ignition and w˜ = 1 for bistable. In Sattinger [12] the spectrum of −Lξ as an operator on weighted L∞ spaces is analyzed for a wide variety of nonlinearities, including ignition, bistable, and KPP. The methods and proofs in [12] can be modified in a straightforward way to apply to −Lξ as an operator on weighted H 2 spaces as well; although some of the exact details are different, the main ideas are identical [15]. The results can then be extended to the full operator −L by using a Fourier representation of τ . In this way, we obtain a theorem trapping the nonzero spectrum of −L in a parabola that lies strictly in the left half-plane and explicitly bounding the resolvent on regions exterior to this parabola, [15]. Before stating the theorem, we make a few definitions. First define λ1 by c02 2 2

. λ1 := inf λ ∈ [−c0 /4, 0) : (λ, 0) contains no eigenvalues of M = ∂ξ + f (U ) − 4 Next define the regions P ± by     2 c c 0 , P ± = λ : Re λ − f (U (±∞)) + 0 >  4 2 and define the region P by P=

P − , if f is ignition, P − ∩ P + , if f is bistable.

Note that P ± is the region exterior to a parabola that lies strictly in the left half-plane and intersects the real axis at f (U (±∞)). We will also be interested in shifts of these regions by an amount . Define P± = λ + : λ ∈ P ± . Theorem 2. Assume f is either an ignition nonlinearity with the properties (4.15) or a bistable nonlinearity with the properties (4.16) and f ∈ C 2 on neighborhoods of 0 and 1. Define the weight w˜ to be w(ξ ) = (1 + eξ c0 /2 )2 if f is ignition and 1 if f is bistable.


463

Then zero is a simple isolated eigenvalue of −L as an operator on H 2 (w, ˜ ) with eigenfunction Uξ , and the rest of the spectrum lies in a parabolic region strictly in the left half plane. More specifically the set {λ = 0} ∩ {| arg(λ − λ1 )| < π} ∩ P ∩ {| arg(λ + (2π/)2 )| < π }

(4.18)

is in the resolvent of −L. Furthermore, for all > 0, φ ∈ ( π2 , π), 0 < β < 1, and λ ∈ {λ = 0} ∩ {| arg(λ − λ1 )| < φ} ∩ P ∩ {Re λ ≥ −β(2π/)2 } ∪ {| arg(λ)| ≤ φ} we have the bounds (λ + L)−1 ψ

L2 (w,) ˜

(λ + L)−1 ψ

−1

(λ + L)

ψ

H 2 (w,) ˜

≤

C(, φ, β) ψL2 (w,) , ˜ |λ|

(4.19)

≤

C(, φ, β) ψL2 (w,) , ˜ F(|λ|)

(4.20)

H 1 (w,) ˜

≤ C(, φ, β)(1 +

(λ + L)−1 ∂ξ ψ

L2 (w,) ˜

≤

−2

1 ψL2 (w,) ) 1+ , (4.21) ˜ F(|λ|)

C(, φ, β) ψL2 (w,) , ˜ F(|λ|)

(4.22)

and (λ + L)−1 ψ

H 1 (w,) ˜

≤

where F(|λ|) = {|λ| if |λ| ≤ 1;

C(, φ, β) ||ψ||H 1 (w,) , if f

∞ < ∞, ˜ 2 F(|λ|)

(4.23)

√ |λ| if |λ| ≥ 1}.

Note that for all n ∈ Z, −(2π n/)2 is an eigenvalue of −L with eigenfunction ξ . So the spectral gap of −L depends on , and for large the spectral gap shrinks. In fact ≤ (2π/)2 , and for large , = (2π/)2 . Consider now the adjoint operator −L∗ of −L in H 2 (w, ˜ ), given by ei(2πn/)x U

1 [ − c0 ∂ξ + f (U (ξ ))]w˜ w ˜    + 1 − √2 c ∂ + f (U (ξ )) − 0 ξ w =   − c0 ∂ξ + f (U (ξ )) for bistable.

−L∗ =

c0

c02 e 2 2w

ξ

for ignition,

Zero is a simple eigenvalue of −L∗ with eigenfunction Uξ∗ = C w1˜ ec0 ξ Uξ , where the constant C is chosen so that Uξ , Uξ∗ = 1. w, ˜

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B. Winn

Corollary 2. Assume f is either an ignition nonlinearity with the properties (4.15) or a bistable nonlinearity with the properties (4.16) and f ∈ C 2 on neighborhoods of 0 and 1. Define the weight w˜ to be w(ξ ) = (1 + eξ c0 /2 )2 if f is ignition and 1 if f is bistable. Then zero is a simple isolated eigenvalue of −L∗ as an operator on H 2 (w, ˜ ), and the rest of the spectrum lies in a parabolic region strictly in the left half plane. More specifically the set {λ = 0} ∩ {| arg(λ − λ1 )| < π} ∩ P ∩ {| arg(λ + (2π/)2 )| < π } is in the resolvent of −L∗ . Furthermore, for all > 0, φ ∈ ( π2 , π), 0 < β < 1, and λ ∈ {λ = 0} ∩ {| arg(λ − λ1 )| < φ} ∩ P ∩ {Re λ ≥ −β(2π/)2 } ∪ {λ : | arg(λ)| ≤ φ} we have the bounds C(, φ, β) ψL2 (w,) ≤ , (λ + L∗ )−1 ψ 2 ˜ L (w,) ˜ |λ| (λ + L∗ )−1 ψ ∗ −1

(λ + L ) and

ψ

H 2 (w,) ˜

H 1 (w,) ˜

≤

C(, φ, β) ψL2 (w,) , ˜ F(|λ|)

≤ C(, φ, β)(1 +

(λ + L∗ )−1 ∂ξ ψ

L2 (w,) ˜

where F(|λ|) = {|λ| if |λ| ≤ 1;

≤

−2

1 ψL2 (w,) ) 1+ , ˜ F(|λ|)

C(, φ, β) ψL2 (w,) , ˜ F(|λ|)

√ |λ| if |λ| ≥ 1}.

Assume from here on that f is an ignition or bistable nonlinearity as in (4.15) or (4.16) and f ∈ C 2 on neighborhoods of 0 and 1. We now take a moment to consider the restriction of −L to the orthogonal complement of its zero-eigenspace. Define P1 to be the projection operator onto the zero-eigenspace of −L, and let ∗ ∗ P2 = I −P1 . Then P1 ψ = ψ, Uξ Uξ . Note that LP1 ψ = ψ, Uξ LUξ = 0 and w, ˜ w, ˜ P1 Lψ = Lψ, Uξ∗ Uξ = ψ, L∗ Uξ∗ Uξ = 0, so that L commutes with P1 and w, ˜

w, ˜

hence also with P2 . Finally, define ϒ to be the orthogonal complement of the zero-eigenspace of −L in H 2 (w, ˜ ): 2 2 ∗ ϒ = ϒH 2 (w,) = P2 H (w, ˜ ) = ψ ∈ H (w, ˜ ) : ψ, Uξ = 0 . (4.24) ˜ w, ˜

Denote by L|ϒ the restriction of L to ϒ. Since zero is an isolated point of the spectrum of −L, it is in the resolvent of L|ϒ . So (λ + L|ϒ )−1 not only satisfies the bounds (4.19)–(4.23), but is also bounded near λ = 0. Thus L|ϒ (and L|∗ϒ ) is a sectorial operator [5], and −L|ϒ is the infinitesimal generator of an analytic semigroup {e−tL|ϒ }t≥0 , where 1 e−tL|ϒ = (λ + L|ϒ )−1 eλt dλ, 2πi C and C is a contour in the resolvent of −L|ϒ that


465

1. lies strictly in the left half-plane 2. stays outside a sector containing the spectrum of −L|ϒ 3. tends to infinity along rays with arguments ±γ for some γ ∈ (π/2, π ). Since L|ϒ is sectorial and Re σ (L|ϒ ) > 0, fractional powers of L|ϒ may be defined for any µ > 0 as ∞ 1 −µ L|ϒ = t µ−1 e−tL|ϒ dt, (µ) 0 µ −µ −1 , L|ϒ = L|ϒ where

(µ) =

∞

t µ−1 e−t dt.

0

For 0 < µ < 1, this definition is equivalent to sin(π µ) ∞ −µ −µ L|ϒ = λ (λ + L|ϒ )−1 dλ. π 0 The bounds (4.19)–(4.23) on the resolvent operator and the definitions of the exponential of −L|ϒ and fractional powers of L|ϒ combine in a straightforward way to yield the following bounds which will be needed in our proof of stability. Similar bounds hold for the adjoint operator L|∗ϒ . Proposition 3. Assume f is either an ignition nonlinearity with the properties (4.15) or a bistable nonlinearity with the properties (4.16) and f ∈ C 2 on neighborhoods of 0 and 1. Define the weight w˜ to be w(ξ ) = (1 + eξ c0 /2 )2 if f is ignition and 1 if f is bistable. Fix α with 0 < α < min{(2π/)2 , −λ1 , k(f )} ≤ , where k(f ) = −f (1) for ignition, and k(f ) = min{−f (0), −f (1)} for bistable. Then the exponential satisfies the bounds e−tL|ϒ ψ e−tL|ϒ ψ

H 2 (w,) ˜ H 1 (w,) ˜

e−αt ψL2 (w,) , ˜ t e−αt ≤ C(α, ) √ ψL2 (w,) , ˜ t ≤ C(α, )

e−tL|ϒ ψ 1 ≤ C(α, )e−αt ||ψ||H 1 (w,) , if f

∞ < ∞, ˜ H (w,) ˜ −tL|ϒ ψ 2 ≤ C(α, )e−αt ψL2 (w,) . e ˜ ˜ L (w,)

For

1 2

< µ < 1 the fractional powers of L|ϒ satisfy the bound −µ

L|ϒ ψ

H 1 (w,) ˜

Moreover, for µ ≥ 0 we have the bound µ −tL|ϒ L|ϒ e 2 2

≤ C(µ) ψL2 (w,) . ˜

L (w,)→L ˜ (w,) ˜

≤ C(µ, α, )t −µ e−αt .

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B. Winn

4.2. Stability of the Traveling Waves. We return now to investigating the stability of the family of planar traveling wave solutions. Recall that τ , v, and ω˜ satisfy ∂t τ = −Lτ + F,

(4.25)

1 ∂t v = σ v + (c0 + ε∂t c1 )∂ξ v − ερv · ∇v − ∇ p˜ + σ τ eˆξ + σ U eˆξ , ε ∇ · v = 0, ∂t ω˜ = σ ω˜ + (c0 + ε∂t c1 )∂ξ ω˜ − ερv · ∇ ω˜ + σ ∂x τ, lim τ = 0,

ξ →±∞

lim v = 0, and

ξ →±∞

(4.26) (4.27) (4.28)

lim ω˜ = 0,

ξ →±∞

where −L = + c0 ∂ξ + f (U ), F = ε∂t c1 ∂ξ τ − ερv · ∇τ + (∂t c1 − ρv2 )Uξ + εN, 1 N = 2 [f (U + ετ ) − f (U ) − f (U )ετ ]. ε

(4.29) (4.30) (4.31)

Again we assume f is an ignition or bistable nonlinearity as in (4.15) or (4.16) and f ∈ C 2 on neighborhoods of 0 and 1. Our method follows closely many of the ideas of Xin and Malham in [8], which examines a slightly different coupled Boussinesq system and establishes neutral stability of nonplanar fronts generated by a shear flow. Let us attempt to simplify the situation and to deal with any zero-eigenspace by applying the projections P1 , P2 to τ . We begin by examining the projection P1 τ which has the equation ∂t P1 τ = P1 F = ε∂t c1 ∂ξ τ − ερv · ∇τ + (∂t c1 − ρv2 )Uξ + εN, Uξ∗ Uξ w, ˜

+ −ερv · ∇τ − ρv2 Uξ + εN, Uξ∗ = ∂t c1 1 + ε ∂ξ τ, Uξ∗ w, ˜

w, ˜

Let us simplify this equation by choosing ∂t c1 so that P1 F = 0: − −ερv · ∇τ − ρv2 Uξ + εN, Uξ∗ w, ˜ ∂t c1 = . ∗ 1 + ε ∂ξ τ, Uξ

Uξ .

(4.32)

w, ˜

This choice of ∂t c1 is clearly possible for small ε and small time, so that the denominator stays nonzero. We will later show that for ε sufficiently small, our choice of ∂t c1 will stay bounded so that it is valid for all time. Then P1 τ (t) = P1 τ0 = τ0 , Uξ∗ Uξ w, ˜ 1 [T0 (x, ξ ) − U (ξ )]Cec0 ξ Uξ (ξ ) dξ dx = Uξ ε # " C = Uξ T 0 (ξ ) − U (ξ ) ec0 ξ Uξ (ξ ) dξ, ε R


467

where T 0 (ξ ) = 1 0 T0 (x, ξ ) dx. Assume T0 decays exponentially at infinity, so that ˜ , ) for some 0 < c˜ < c . For ignition, this is a reasonable assumption, T0 ∈ L2 (ecξ 0 2 ˜ , ). For bistable nonlinearities, this is an additional restricsince L (w, ) ⊂ L2 (ecξ tion. Since U is monotonic and T 0 −U → 0 as ξ → ±∞, there must be a γ ∈ R such that # " T 0 (ξ ) − U (ξ + γ ) ec0 ξ Uξ (ξ + γ ) dξ = 0. R

To see this, define

# U (ξ + γ ) − T 0 (ξ ) ec0 ξ |Uξ (ξ + γ )| dξ R " # −c0 γ =e U (y) − T 0 (y − γ ) ec0 y |Uξ (y)| dy. "

G(γ ) =

R

We begin by showing that G is negative for large γ . Let p1 = U −1 (1/4) so that U (y) < 1/4 for y > p1 . Choose b ≥ p∗ large enough so that ∞ 1. q∗ ec0 p∗ b [U (y) − T (y)]dy ≤ 1. ec0 p1 1 2. Uz ∞ + 1 < inf |ec0 y Uz (y)| · (b − p1 ). c0 2 y≥p1 Since T 0 (z) → 1 as z → −∞, there exists a γ1 ≥ 0 such that T 0 (z) ≥ 3/4 for z ≤ b−γ and γ ≥ γ1 . Then for γ ≥ γ1 we have p1 b ∞ # " ec0 γ G(γ ) = + + U (y) − T 0 (y − γ ) ec0 y |Uξ (y)| dy −∞ p1 b

c p 0 1 e 1 3 ≤ Uz ∞ + − inf ec0 y Uz (y) · (b − p1 ) c0 4 4 y≥p1 ∞ [U (y) − T (y − γ )]dy +q∗ ec0 p∗ b

ec0 p1 1 = Uz ∞ − inf ec0 y Uz (y) · (b − p1 ) − q∗ ec0 p∗ c0 2 y≥p1 ∞ c0 p∗ +q∗ e [U (y) − T (y)]dy

b

T (y)dy b−γ

b

ec0 p1 1 ≤ Uz ∞ − inf ec0 y Uz (y) · (b − p1 ) + 1 c0 2 y≥p1 < 0. We now show that G is positive for γ negative enough. −c0 γ c0 y G(γ ) = e U (y)e |Uξ (y)| dy − T 0 (ξ )ec0 ξ |Uξ (ξ + γ )| dξ ≥ e−c0 γ

R R

R

U (y)ec0 y |Uξ (y)| dy − 1/2

R

˜ |T 0 (ξ )|2 ecξ dξ

1/2

˜ |Uξ (ξ + γ )|2 e(2c0 −c)ξ dξ R c˜ = e−c0 γ U (y)ec0 y |Uξ (y)| dy − e−(c0 − 2 )γ T 0 L2 (ecξ˜ ,R) Uξ L2 (e(2c0 −c)ξ ˜ ,R) ,

R

468

B. Winn

which is positive for γ negative enough. So the continuity of G implies that it has a zero somewhere. Without loss of generality we may assume that U is chosen so that γ = 0. Then P1 τ = 0. So the zero-eigenspace is not a problem. Let us now examine the projection P2 τ . Since P1 τ = 0, we have τ = P2 τ and τ0 = P2 τ0 . We also know P1 F = 0, and hence P2 F = F . So the solution τ satisfies ∂t τ = −L|ϒ τ + P2 F and τ (t, x, ξ ) = e

−tL|ϒ

t

τ0 (x, ξ ) +

e−(t−s)L|ϒ P2 F (s) ds.

(4.33)

0

Theorem 3. Assume f is an ignition nonlinearity with the (4.15) or a bistable properties nonlinearity with the properties (4.16), f ∈ C 1 [0, 1], f

∞ < ∞, and f ∈ C 2 on neighborhoods of 0 and 1. In addition, assume T0 decays exponentially at infinity and assume the initial data u0 and T0 are invariant with respect to the symmetries of the equations, as mentioned in Sect. 2. Define the weight w˜ to be w(ξ ) = (1 + eξ c0 /2 )2 if f is ignition and 1 if f is bistable. Choose c1 (t, ε) to satisfy (4.32). Write M(t) = ||τ (t)||H 1 (w,) + ˜

√

ρ ||v(t)||H 1 () + |∂t c1 (t, ε)|.

Then for any domain width , there exist constants K = K() > 0 and δ = δ(, σ, M(0)) > 0, so that if ε, ρ < δ(, σ ), then M(t) ≤ KM(0)e−βt ,

(4.34)

where β = β() is any constant satisfying 0 < β < 41 min{4π 2 −2 , σ −2 , −f (1), −λ1 }. Moreover, δ = O((1 + σ )−2 ) and δ → 0 as → ∞. Proof. Aside from the weights, the proofs for ignition and bistable nonlinearities are identical; so we only provide details here for the slightly more complicated ignition case. The plan is to use a supersolution argument to obtain stability. First, we need to derive an inequality for M. Let us begin by examining |∂t c1 |. Recall that ∂t c1 was chosen so that ∂t c1 = −ε∂t c1 ∂ξ τ + ερv · ∇τ + ρv2 Uξ − εN, Uξ∗

w,

.

Using Taylor’s Theorem we have 1 |f (U + ετ ) − f (U ) − f (U )ετ | ε2 1 1 |f (U + µετ ) − f (U )| dµ = |τ | ε 0 1

1 f |µετ | dµ ≤ |τ | ∞ ε 0 1 = f

∞ |τ |2 . 2

|N | =

(4.35)


469

Consequently

|∂t c1 | ≤ ε|∂t c1 | ∂ξ τ L2 (w,) Uξ∗

√ ∗ ∇τ + ερ||v|| 2 2 L () L (w,) Uξ w L2 (w,) ∞

f ∞ τ 2L2 (w,) Uξ∗ +ρ||v2 ||L2 () ||Uξ Uξ∗ w||L2 () + ε ∞ 2 " ≤ C ε|∂t c1 | ||τ ||H 1 (w,) + ερ ||v||H 1 () ||τ ||H 1 (w,) + ρ ||v||H 1 () +ε ||τ ||2H 1 (w,) . (4.36)

We now examine ||v||H 1 () . Since v has mean zero in x, it satisfies the Poincaré inequality ||v||L2 () ≤ ||∂x v||L2 () . So ||v||H 1 () ≤

1 + 2 ||∇v||L2 () = 1 + 2 ||ω|| ˜ L2 () .

Multiplying the vorticity equation (4.28) by ω˜ and integrating over yields 1 d ˜ 2L2 () + σ ||ω|| ˜ L2 () ||τ ||H 1 () . ||ω|| ˜ 2L2 () ≤ −σ ||∇ ω|| 2 dt Since ω˜ has mean zero in x, it also satisfies a Poincaré inequality: ||ω|| ˜ L2 () ≤ ||∂x ω|| ˜ L2 () ≤ ||∇ ω|| ˜ L2 () . Hence d σ ||ω|| ˜ L2 () ≤ − 2 ||ω|| ˜ L2 () + σ ||τ ||H 1 (w,) . dt Applying Gronwall’s inequality, we obtain t 2 −(σ/2 )t ||ω(t)|| ˜ ≤ || ω(0)|| ˜ e + σ e−(σ/ )(t−s) ||τ (s)||H 1 (w,) ds. 2 2 L () L () 0

Thus ||v(t)||H 1 () t 2 −(σ/2 )t 2 ≤ 1 + 2 ||ω(0)|| ˜ e + σ 1 + e−(σ/ )(t−s) ||τ (s)||H 1 (w,) ds. L2 () 0

(4.37) Finally, let’s examine the term ||τ ||H 1 (w,) . Recalling the equality (4.33), we have ||τ (t)||H 1 (w,)

t

e−(t−s)L|ϒ P2 F (s) 1 ds H (w,) t ≤ C()e−αt ||τ0 ||H 1 (w,) + ερ e−(t−s)L|ϒ P2 (v · ∇τ )(s)

≤ e

−tL|ϒ

τ0

H 1 (w,)

+

0

0

t e−α(t−s) +C() {ε|∂t c1 | ∂ξ τ (s)L2 (w,) √ t −s √0 +ρ Uξ w ∞ ||v2 (s)||L2 () + ε N (s)L2 (w,) } ds,

H 1 (w,)

ds

470

B. Winn

where 0 < α < min{(2π/)2 , −f (1), −λ1 } ≤ . By the equality (4.35) 2 1 f

∞ τ L2 (w,) 2 √ 2 1 ≤ f

∞ τ w L4 () 2 √ √ ≤ C||τ w||L2 () ||∇(τ w)||L2 () , by Gagliardo-Nirenberg $ c % 0 τ L2 (w,) ||τ ||H 1 (w,) . ≤ C max 1, 2

N L2 (w,) ≤

We proceed now to bound the term e−(t−s)L|ϒ P2 (v · ∇τ )(s) 1 2

with < µ1 < 1, and then choose µ2 with 0 < µ2 < and µ2 < 21 . So e−(t−s)L|ϒ P2 (v · ∇τ )(s) −µ

. Choose µ1

H 1 (w,) 1−µ1 . Note that 21 < µ1 +µ2

H 1 (w,)

= L|ϒ 1 L|ϒ1 e−(t−s)L|ϒ P2 (v · ∇τ )(s) 1 H (w,) µ1 −(t−s)L|ϒ P2 (v · ∇τ )(s) 2 ≤ C(µ1 ) L|ϒ e L (w,) µ1 +µ2 −(t−s)L|ϒ −µ2 = C(µ1 ) L|ϒ e L|ϒ P2 (v · ∇τ )(s) 2 L (w,) −µ2 µ1 +µ2 −(t−s)L|ϒ ≤ C(µ1 ) L|ϒ e P (v · ∇τ )(s) 2 L| 2 ϒ 2 µ

L (w,)→L (w,)

≤ C(µ1 , µ2 , )

e−α(t−s) (t − s)µ1 +µ2

−µ2 L|ϒ P2 (v · ∇τ )(s)

2 , −f (1), −λ } ≤ . where 0 < α < min{(2π/) 1 −µ2 Examining the term L|ϒ P2 (v · ∇τ )(s) 2

L (w,)

−µ

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Communications in Mathematical Physics - Volume 253

Communications in Mathematical Physics - Volume 222

Communications in Mathematical Physics - Volume 259

Communications in Mathematical Physics - Volume 221

Communications in Mathematical Physics - Volume 220

Communications in Mathematical Physics - Volume 235

Communications in Mathematical Physics - Volume 223

Communications In Mathematical Physics - Volume 283

Communications In Mathematical Physics - Volume 270

Communications in Mathematical Physics - Volume 208

Communications in Mathematical Physics - Volume 186

Communications In Mathematical Physics - Volume 294

Communications in Mathematical Physics - Volume 217

Communications In Mathematical Physics - Volume 274

Communications in Mathematical Physics - Volume 239

Communications in Mathematical Physics - Volume 306

Communications in Mathematical Physics - Volume 264

Communications in Mathematical Physics - Volume 227

Communications in Mathematical Physics - Volume 184

Communications in Mathematical Physics - Volume 261

Communications in Mathematical Physics - Volume 225

Communications In Mathematical Physics - Volume 263

Communications in Mathematical Physics - Volume 211

Communications In Mathematical Physics - Volume 293

Communications in Mathematical Physics - Volume 246

Communications In Mathematical Physics - Volume 298

Communications in Mathematical Physics - Volume 234

Communications In Mathematical Physics - Volume 288

Communications in Mathematical Physics - Volume 304

Communications In Mathematical Physics - Volume 292

Communications in Mathematical Physics - Volume 233

Communications in Mathematical Physics - Volume 253

Communications in Mathematical Physics - Volume 222

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