Communications in Mathematical Physics - Volume 217

Commun. Math. Phys. 217, 1 – 31 (2001) Communications in Mathematical Physics © Springer-Verlag 2001 Integrable Stru...

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Commun. Math. Phys. 217, 1 – 31 (2001)

Communications in

Mathematical Physics

© Springer-Verlag 2001

Integrable Structures in Classical Off-Shell 10D Supersymmetric Yang–Mills Theory J.-L. Gervais, H. Samtleben Laboratoire de Physique Théorique de l’École Normale Supérieure , 24 rue Lhomond, 75231 Paris Cedex 05, France Received: 20 April 2000 / Accepted: 10 September 2000

Abstract: The field equations of supersymmetric Yang–Mills theory in ten dimensions may be formulated as vanishing curvature conditions on light-like rays in superspace. In this article, we investigate the physical content of the modified SO(7) covariant superspace constraints put forward earlier [11]. To this end, group-algebraic methods are developed which allow to derive the set of physical fields and their equations of motion from the superfield expansion of the supercurl, systematically. A set of integrable superspace constraints is identified which drastically reduces the field content of the unconstrained superfield but leaves the spectrum including the original Yang–Mills vector field completely off-shell. A weaker set of constraints gives rise to additional fields obeying first order differential equations. Geometrically, the SO(7) covariant superspace constraints descend from a truncation of Witten’s original linear system to particular one-parameter families of light-like rays. 1. Introduction Recently, progress was made in applying exact integration methods to supersymmetric Yang–Mills theory in ten dimensions. The starting point was the flatness conditions in superspace which have been known for some time to be equivalent to the field equations [15, 19]. It was shown in [11], that there exists an on-shell light cone gauge, where the superfields may be entirely expressed in terms of a scalar superfield satisfying two sets of equations. The first is linear and a general solution was derived; the second is similar to Yang’s equations and has been handled by methods similar to earlier studies of self-dual Yang–Mills in four dimensions. A general class of exact solutions1 has been obtained [11] and a Bäcklund transformation put forward [8]. Work supported in part by EU contract ERBFMRX-CT96-0012.

UMR 8549: Unité Mixte du Centre National de la Recherche Sientifique, et de l’Ecole Normale Suérieure. 1 Keeping, however, only the dependence upon one time and one space coordinates in contrast with the

dimensional reduction which will be discussed below. This is probably not essential.

2

J.-L. Gervais, H. Samtleben

So far, however, it has not been possible to simultaneously solve the two sets of equations. Only a particular class of solutions of theYang type equations has been found, which is not general enough to solve the other (linear) set. Returning to a general gauge, one may see that deriving the scalar superfield satisfying the linear subset of equations is equivalent [8] to solving a particular symmetrized form of the flatness conditions. This symmetrized form was shown to be explicitly integrable directly, since it arises [9, 10] as a compatibility condition of a Lax representation, similar to the one of Belavin and Zakharov [2], which may be handled by the same powerful techniques as in the case of self-dual Yang–Mills in four dimensions. The goal of this article is a systematic study of the symmetrized form of the original flatness conditions in superspace. For the above mentioned reasons we refer to these modified conditions as the integrable superspace constraint – as opposed to the strong superspace constraint which describes vanishing of the full super field strength and is equivalent to the Yang–Mills system. Since the proof of this equivalence is recursive and rather tedious [15, 19, 13, 1], a priori it is not clear which modification of the field content and dynamics is implied by a modification of the strong superspace constraint. An important property of the integrable superspace constraint is that it explicitly breaks the original SO(9, 1) Lorentz symmetry down to SO(2, 1) × SO(7). For WeylMajorana spinors and in particular the Grassmann coordinates in superspace, this leads to the separation 16 → 8 + 8, (where the r.h.s. denotes a pair of SO(7) spinor representations which form a doublet under SO(2, 1)). This is instrumental in defining the involution which is the key to applying methods modeled over the bosonic self-duality requirement in four dimensions; here, the analogous construction involves the exchange of the two spinor representations. In the rest of the paper we concentrate upon deriving the field content and dynamical equations induced by the integrable superspace constraint. Since this constraint is weaker than the original (strong) superspace constraint, we effectively go partially off-shell. In particular, this gives rise to the appearance of more physical fields appearing in the superfield components and a modification of the dynamics. At this point, it is worth recalling that, in the standard treatment of Refs. [13, 1], the method used to eliminate the unphysical components of the superfields makes essential use of the equations of motion. It is thus not applicable to our case. One of the aims of the present work is to devise a more direct and general method, which is also applicable to our modified equations. After reviewing superspace notations, in Sect. 2 we introduce the original (strong) and the modified (integrable) superspace constraints. We explain the geometrical origin of the integrable superspace constraint as particular truncation of the Lax representation in superspace [19]. This original Lax representation, associated with the strong superspace constraint, is formulated for light-like rays in ten dimensions, which play the role of spectral parameters. Restricting the connection to certain one-parameter families of light rays (spanning a three-manifold) makes the system accessible to the techniques developed in [2] for four-dimensional self-dual Yang–Mills theory. With a particular choice of three-manifold, this gives back the integrable superspace constraint. For the subsequent analysis it is helpful to further introduce a slightly stronger version of this constraint corresponding to an effective reduction of the Lax representation to seven dimensions and referred to as the reduced integrable constraint. Due to the fact that this reduced constraint commutes with the action of the symmetric group on spinor indices, it eventually turns out to be completely soluble, which essentially simplifies the analysis of the integrable constraint.

Integrable Structures in Yang–Mills Theory

3

The rest of the paper is devoted to studying field content and dynamics induced by the integrable and the reduced integrable superspace constraints. Section 3 presents a systematic general study of the expansion of the superfield equations in powers of the odd variables θ . We derive recursion relations with an interesting structure. The elimination of unphysical components of superfields is done recursively and involves two operators noted S and T . The first satisfies a simple quadratic equation while the second is nilpotent. Thus our equations bear some analogy with the descent equations [20]. The field equations are enforced by further applying a projector K, and we thus study the interplay between S, T , and K on general ground. Applying this method to the integrable superspace constraints, in Sect. 4, we explicitly identify the induced physical field content. As a result, we find a spectrum which is essentially larger than the original Yang–Mills system. For the reduced integrable constraint it consists of 384 + 384 fields which correspond to three copies of the ten-dimensional off-shell multiplet [3]; the integrable constraint then gives rise to additional 31 + 16 fields. Section 5 is devoted to deriving explicit recurrent relations which determine the higher order superfield components in terms of these fields. Finally, in Sect. 6 we derive the field equations which are induced by the superspace constraints. The spectrum associated with the reduced integrable constraint remains completely off-shell, whereas the additional fields appearing with the integrable constraint satisfy a set of first order differential equations. A discussion of this dynamics and some concluding remarks are given at the end.

2. Superspace Constraints and Lax Representation 2.1. Superfield conventions. The notations are essentially the same as in the previous references. The physical fields are noted as follows: Xa (x) is the vector potential, φ α (x) is the Majorana-Weyl spinor. Both are matrices in the adjoint representation of the gauge group G. Latin indices a = 0, . . . , 9 describe Minkowski components, Greek indices α = 1, . . . , 16 denote spinor components. We use the Dirac matrices 016×16 (σ a )αβ 116×16 0 a 11 , = . (1) = 0 −116×16 016×16 (σ a )αβ α We will use the superspace formulation with 16 odd coordinates θ . The general superfield expansions of a superfield x, θ may be written as

(x, θ) =

16

[p] (x, θ ) ≡

p=0

16 p=0 α1 ,... ,αp

θ α1 · · · θ αp [p] α1 ...αp (x). p!

(2)

The grading is given by the operator R = θ α ∂α ,

R , [p] = p [p] .

(3)

Superderivatives are defined by Dα = ∂α − θ β σ a αβ ∂a ,

such that

Dα , Dβ

+

= −2 (σ a )αβ ∂a .

(4)

4


The odd super vector valued in the adjoint representation of the gauge group, potential, is denoted by Aα x, θ . We define its supercurl Mαβ as

Mαβ ≡ Dα Aβ + Dβ Aα + Aα , Aβ + ≡ Fαβ − 2 σ a αβ Aa , (5) Aa ≡ −

1 (σa )αβ Mαβ . 32

(6)

This gives the decomposition of the supercurl into the super field strength Fαβ and the even vector potential Aa x, θ as functions of Aα .2 The superfield formalism is invariant under gauge transformations Aα → g −1 Aα g + g −1 Dα g, Fαβ → g −1 Fαβ g,

(7)

Aa → g −1 Aa g + g −1 ∂a g, with an even superfield g x, θ as gauge parameter. Imposing the so-called recursion gauge condition θ α Aα = 0

(8)

restricts the freedom (7) to ordinary gauge freedom, i.e. to gauge parameters g with [R, g] = 0. 2.2. Superspace constraints. It is known that the equations of motion of super Yang Mills theory in ten dimensions may equivalently be expressed as vanishing of the super curvature Fαβ = 0.

(9)

More precisely, it has been shown in [15,19,13,1] that the recursion gauge condition (8) together with the flatness conditions (9) implies the Yang–Mills equations of motion

(σ a )αβ

1 ∂ a Fab + Aa , Fab = (σb )αβ χ α , χ β , 2

β β ∂a χ + A a , χ = 0,

(10)

for the superfields Aa and χ α , the latter being defined as χ α ≡ (σ a )αβ Faβ with the curvature

Faβ ≡ Dβ Aa − ∂a Aβ + Aβ , Aa . (11) Moreover, (8) and (9) yield a unique recurrent prescription of the higher order superfield components in Aa , χ α and Aα as functions of the zero order contributions Xa ≡ A[0] a ,

φ α ≡ χ α [0] .

(12)

These zero order components in particular satisfy the usual supersymmetric Yang–Mills equations of motion. 2 In the terminology of Ref. [7], we have hence resolved the “conventional constraint” (σ )αβ F a αβ = 0.


5

For the purpose of this paper we rewrite the vanishing super curvature condition (9) in terms of the supercurl Mαβ . It is convenient to introduce the general space of superfields symmetric in two additional spinor indices (αβ), which we denote by M. This space carries the grading (2): M =

16

M[p] ≡

p=0

16

v(αβ),[γ1 ...γp ] θ γ1 . . . θ γp ,

and its elements have the general decomposition according to Mαβ = −2 σ a αβ Aa + 5!1 σ a1 ...a5 αβ Ba1 ...a5 , (16×16)s

(13)

p=0

10

(14)

126

with selfdual Ba1 ...a5 = − 5!1 !a1 ...a10 B a6 ...a10 . In terms of this decomposition, the superspace constraint (9) corresponds to setting Ba1 ...a5 = 0 and may be written as a projection condition KYM on the supercurl Mαβ = (KYM M)αβ , with

α β

(KYM )αβ =

(15) 1 16

σa

αβ

(σa )α β .

This paper is devoted to a study of other (weaker) superfield constraints which replace (15) and have appeared as completely integrable superfield equations in [9, 10]. Obviously, (15) is the only SO(9, 1) covariant constraint that can be imposed on the supercurl. We hence break the original SO(9, 1) Lorentz invariance of the system down to SO(2, 1) × SO(7); the reason for this particular choice will become clear the following. It is in this setting that the modified constraints have appeared in [11, 9, 10]. For Weyl-Majorana spinors, this symmetry breaking leads to the separation 16 → (2, 8) :

χ α → (χ µ , χ µ¯ ),

µ, µ¯ = 1, . . . , 8,

(16)

where the r.h.s. denotes a pair of SO(7) spinor representations which form a doublet under SO(2, 1) such that µ ↔ µ¯ denotes the SO(7) invariant involution. In Appendix A we have collected the conventions about decomposing the SO(9, 1) σ -matrices into SO(8) γ -matrices. The supercurl Mαβ correspondingly decomposes into Mαβ = 10+126 → (3, 1) + (1, 7) + (1, 21) + (3, 35).

(17)

The superspace constraints which we are going to study in this paper are the following projections: KYM : Mαβ → (3, 1) + (1, 7), KI : Mαβ → (3, 1) + (1, 7) + (1, 21), KIR : Mαβ → (1, 7) + (1, 21).

(18) (19) (20)

The first constraint is the original vanishing super curvature condition (15) which corresponds to the Yang–Mills system. The latter two constraints have appeared in [9, 10] as compatibility equations of completely integrable Lax representations. We will refer to

6


them as to the integrable and the reduced integrable constraint, respectively. Note that the truncation (20) is gauge covariant only after dimensional reduction of the system to seven dimensions. Before analyzing the field content and dynamics implied by these superfield constraints we first recall how they may be obtained as compatibility equations of Lax representations in superspace. 2.3. Lax representations. In this section we recall the original linear system [19] associated with the vanishing curvature condition (18), and show how the integrable constraints (19), (20) may be obtained as certain truncations thereof. The flatness conditions (9), (18) possess a Lax type representation in superspace [19]. They imply the existence of a G-valued superfield #[$] for any light-like ten-dimensional vector $, which is defined by the linear system (21) $a (σa )αβ Dβ + Aβ #[$] = 0, $a {∂a − Aa } #[$] = 0. In turn, the compatibility conditions of (21) imply (9). Clearly, these equations are invariant under multiplication by an overall constant, so that # only depends upon the light-like ray considered. The system of equations (21) may be considered as a Lax representation of the field equations (10) where the light-like vector $ plays the role of the spectral parameter. As they stand, however, they have not been very useful in practice, i.e. with regard to the powerful solution generating methods applicable in lower dimensional systems. The constraints (19) and (20), in contrast, are derivable from a Lax representation where the spectral parameter is a complex number λ instead of a light ray, such that methods inspired from Ref. [2] become applicable [10, 8]. Starting from Eqs. (21), the integrable constraints geometrically correspond to keeping only the flatness conditions associated with a particular one parameter subset $(λ) of light-like rays. Breaking the original SO(9, 1) invariance we introduce the following parametrization for light-like rays $ = $(λ, v): $± = ±i

1∓λ , 1±λ

$i = v i ,

for

i = 1, . . . , 8,

$± ≡ $0 ± $9 ,

(22)

with a complex number λ and an eight-dimensional unit vector v i . The vector v i prov = v·γ vides a mapping between the two spinor representations of SO(8) via γµ¯ µ¯ν ν (cf. Appendix A). The linear system (21) then takes the form v v (vµ + Bµv + λ((µ + B µ ) #[λ, v] = 0, (23)

1−λ 1+λ (∂+ +A+ ) − (∂− +A− ) − i v·(∂ +A) #[λ, v] = 0, 1+λ 1−λ

with v Dρ , (vµ = Dµ + iγµρ

Bµv

=

v Aµ + iγµρ Aρ ,

v

v (µ = Dµ − iγµρ Dρ ,

v Bµ

=

v Aµ − iγµρ Aρ .

(24)


7

We are now going to show that the superspace constraints (18), (19), and (20) arise as compatibility equations of certain truncations of (23): KYM : impose (23) for all vectors v,

(25)

KI : impose (23) for a fixed vector v, KIR : impose (23) for a fixed vector v, and reduce the linear system to seven dimensions. The first relation is the result of [19] and follows from computing the commutator of the Lax connection (23) with itself, thereby implying v v (1+λ)2 Fµν − (1−λ)2 γµvµ¯ γνvν¯ Fµν + i(1−λ2 ) (γµ¯ ν Fνν + γν µ¯ Fµµ ) = 0

for all values of λ and v and hence vanishing of the supercurvature (9). For a fixed choice of the vector v on the other hand, these conditions imply Mµν = 4 δµν A+ =

µ ν 1 8 δµν δ

Mµ ν ,

Mµν = 4 δµν A− =

µ ν

Mµ ν ,

1 8 δµν

v v (γµ¯ ν Mνν + γν µ¯ Mµµ ) = −2 δµν v·A =

δ

1 4 δµν

γv

µ ν¯

(26)

Mµ ν¯ .

This precisely corresponds to the projection (19). If furthermore we assume independence of the solution # of the Lax pair (23) of the three coordinates ∂± # = 0 = v·∂ #,

(27)

the second equation of (23) shows that

A± = # ∂± # −1

λ=∓1

= 0,

v·A = # (v·∂ + i∂9 )# −1

(28)

λ=0

− iA9 = 0.

Together with (26), this implies the following stronger truncation on the supercurl: Mµν = 0,

(29)

Mµν = 0, v (γµ¯ ν Mνν

+ γνvµ¯ Mµµ )

= 0,

corresponding to the projection (20). This finishes the proof of (25). We have hence shown that the superspace constraints (19) and (20) arise as integrability conditions of the Lax pair (23) upon truncating the spectral parameter to a particular one parameter family of light-like rays. The advantage of the reduced Lax connection comes from the fact that for fixed choice of v, the linear system (23) is similar to the one proposed by Belavin and Zakharov for the four-dimensional self-dual Yang–Mills theory [2] and similar techniques may successfully be applied. The role of the involution which in that case describes selfduality is played by exchanging the two SO(8) spinor representations by means of v·γ µρ , here. This explains the particular choice of breaking the original Lorentz symmetry

8


down to SO(9, 1) → SO(2, 1) × SO(7). Upon this reduction, the system (23) may be solved starting from an ansatz which is meromorphic in λ. This leads to purely algebraic equations – coming from the fact that the bracket in Eq. (23) is linear in λ – which may be solved in essentially the same way as was done for the self-dual Yang–Mills theory. In this sense, the constraints (19) and (20) arise as completely integrable superfield equations. In contrast, it seems impossible to carry out the next step and solve the system for all v, which would really give a solution of the full Yang–Mills equations in ten dimensions. Indeed, Eq. (24) implies that the bracket in (23) should be linear in v · γ , a very strong requirement, which to satisfy there seems to exist no systematic method. The situation further simplifies upon dropping the coordinate dependence according to (27). It should be noted that this is a natural but stronger requirement than solely restricting the coordinate dependence of the superfield components of Mαβ (cf. [4] for a discussion of the reduction to four dimensions). With the original linear system (21) for example, dimensionally reduced physical configurations generically induce functions # which still depend on the compactified coordinates. Restricting to dimensionally reduced functions # corresponds to imposing further superfield constraints (29). Relaxing the original superspace constraints however corresponds to going partially off-shell and gives rise to additional fields appearing in the higher order superfield components. Our goal in the following is to extract the field and the dynamical content associated with the integrable subset of superspace constraints. 3. Systematics of the Supercurl Expansion In the following, we study the purely algebraic problem to determine the field and dynamical content, induced by imposing the constraints (18)–(20) on the superfields. To this end, we first review the level structure of the superfields and show how to systematically extract field content and equations. For the original set of constraints (26) this structure has been discussed in detail in [13, 1] (and likewise in [12] for the reduction to N = 3 supersymmetric Yang–Mills theory in four dimensions). However, this discussion makes an essential use of the Yang–Mills field equations, and thus does not apply to our case. The purpose of this paper is to present an alternative method. As the central object we consider the supercurl Mαβ . 3.1. Algebraic structure of the recursion gauge condition. If we do not impose any constraint on the superfields, we simply have to take into account the fact that the gauge freedom (7) has been fixed by the recursion gauge condition (8) to gauge parameters which do not have higher order superfield components. For the components of the super vector potential this implies θ α Aα = 0

⇐⇒

[p]

A[α,γ1 ...γp ] = 0,

(30)

which is still invariant under ordinary gauge transformations. This shows that the independent components in the superfield Aα are given by the following sum of Young diagrams3 for the spinor representation 16 of SO(9, 1), 3 Here and in the rest of the paper, the notion of Young diagrams always refers to the (anti-)symmetrization of the factors of a tensor product V ⊗N of a given representation V , i.e. always to the Young diagrams of the corresponding permutation group SN .


A[1] α

+

A[2] α

+

9

A[3] α

[p]

+

...

+ p

      

Aα

+

...

..    .   

Fig. 1. Independent components in the superfield Aα

Recurrence relations. Since later on we are going to study the field equations implied by further constraining the supercurl Mαβ , we first identify the remaining independent field components in Mαβ after imposing the recursion gauge condition. Equation (30) yields (1 + R) Aα = θ β Mαβ

A[p+1] = α

⇐⇒

1 [p] θ γ Mαγ . p+2

(31)

At order p we get from (5), [p]

[p+1]

Mαβ = ∂α Aβ

+ ∂β A[p+1] α

[p−1] [p−q] − σ m αγ θ γ ∂m Aβ − σ m βγ θ γ ∂m A[p−1] + . A[q] α α , Aβ p−1

+

q=1

Using (31) we may then re-express this relation entirely in terms of Mαβ . It is convenient to write it in the form (S + R) M +

R+2 T M = C, R

(32)

where we have introduced two linear operators on the set of symmetric superfields, by α β α β (SM)αβ = Sαβ Mα β , and (T M)αβ = Tαβ Mα β , with α β

β

Sαβ = δαα θ β ∂β + δβ θ α ∂α , α β β Tαβ = θ γ θ β σ a βγ δαα + θ α σ a αγ δβ ∂a .

(33) (34)

Moreover, the non-linear term C is given by [p]

Cαβ = (p + 2)

p−1 q=1

[q−1]

θ γ Mαγ

[p−q−1]

, θ δ Mβδ

(q + 1) (p − q + 1)

+

.

(35)

Note that the operator S commutes with R whereas T raises the level by 2. Thus, (32) indeed builds a recursive system, relating the higher levels of Mαβ to the image of the lower ones under T .

10


Algebraic properties of S and T . By explicit computation one verifies that the operator S satisfies the equation (S − 2) (S + R) = 0.

(36)

Thus, at a given level R = p, the operator S has only two different eigenvalues. We may hence decompose M into the eigenspaces of S: M = M+ + M− ,

(S + R) M+ = 0,

(S − 2) M− = 0.

(37)

With (33) one finds that (S −2) and (S +R) are proportional to the projectors onto the Young diagrams given in Fig. 2.

p+1

       

p

..    .  

    

..  .   

 

M+ [p]

M− [p]

Fig. 2. Eigenspace decomposition of the supercurl M[p] under S

Moreover, one may verify the algebraic relations T 2 = 0,

(38)

(S − 2) T = 0 = T (S + R) ;

(39)

i.e. the level raising operator T is nilpotent and acts nontrivially only between M+ and M− : T : M+ [p] → M− [p+2] .

(40)

The non-linear terms of the field equations (32) are lumped into C. In the weak field approximation the right-hand side of this equation is negligible. Since T is nilpotent, there is then an interesting analogy between Eq. (32) and the descent equations [20]. However, these involve in general two nilpotent operators, whereas in our case S satisfies Eq. (36) instead of being nilpotent. General solution. Let us separate the two eigenvalues of S in Eq. (32) according to (37). It is easy to verify that C + = 0. Thus one gets T M − = 0,

RM − + T M + = C.

The first relation is automatically satisfied because of (39). In conclusion, M + is arbitrary, and M − = R−1 C − T M + . (41)


11

Thus, M + contains the independent components in Mαβ left over by the gauge fixing (30). Comparing the Young diagrams from Figs. 3.1 and 2 we hence recover the independent components identified in the vector potential Aα after imposing the recursion gauge. The total and the independent number of components of M [p] are respectively given by [p]

dim M

16 = 136 , p

+ [p]

dim M

17 = (p + 1) . p+2

(42)

We give a computation of these numbers in Appendix B. Altogether, Mαβ contains 983041 independent components. Since the gauge fixing (30) is defined by covariant constraints on the superfield, these components are necessarily expressible in terms of representations of the supersymmetry algebra (4). Of course, supersymmetry is not realized level by level. Decomposing M + into SO(9) multiplets, we find the following structure:

dim M+[p] = 983041

(43)

p

= 1 + (44 + 84 + 128) × (9 + 16 + 36 + 126 + 128 + 231 + + 432 + 576 + 594 + 768 + 924). The 256 = (44 + 84 + 128) corresponds precisely to the smallest irreducible off-shell multiplet of the 10d supersymmetry algebra (4) [3]. Consistently, M + forms a multiple of this multiplet. The additional singlet in (43) corresponds to the fact that we have not fixed the ordinary gauge invariance. Dual space. For future use, let us recall that we can introduce the dual space Mdual of superfields by means of the bilinear form F |G =

dθ

αβ x, θ Gαβ x, θ , F

(44)

αβ

where |G ∈ M, [p] αβ α1 ···αp = F

F | ∈ Mdual ,

1 !α1 ···α16 F [16−p] αβ , αp+1 ···α16 . (16 − p)! α ,··· ,α p+1

16

0 , Breaking the O(9, 1) invariance, one may identify M and Mdual by means of σαβ for example. With respect to the decomposition SO(9, 1) → SO(2, 1) × SO(7), the bilinear form (44) then yields an SO(7) invariant scalar product. We are going to use this scalar product in the subsequent analysis of the superfield constraints (25). Note finally that with respect to this scalar product the operator S from (33) is self-adjoint

S ad = S.

(45)

12


3.2. Extracting dynamics from the superspace constraints. So far in this section, we have restricted the supercurl Mαβ only by the recursion gauge condition (30), thereby restricting the gauge freedom (7). Further restrictions and in particular dynamical equations arise from imposing further constraints (18), (19), and (20), respectively, on the supercurl. These constraints have been casted into the form of projections under an operator K, K 2 = K, K ≡ I − K, such that Mαβ is subject to α β

K αβ Mα β = 0.

(46)

This defines a decomposition of the superfields in M into M = KM + KM ≡ M + M⊥ .

(47)

The role of K is twofold. First, it further restricts the field content in the superfield Mαβ by certain algebraic relations; secondly, it implies field equations for the remaining independent superfield components. The explicit projectors for the dynamical constraints (18), (19), and (20) are given by a α β 1 (KYM )αβ = 16 (48) σ αβ (σa )α β ,

ν (KI )µ µν = µ ν (KI )µν µ ν (KI )µν

= =

µ ν 1 , 8 δµν δ 1 8 δµν 1 8 δµν

δ

µ ν

,

δ

µ ν

+

(49)

1 8

γ i µν γi µ ν +

1 16

ν (KIR )µ µν = 0, µ ν (KIR )µν µ ν

γ ij µν γij µ ν , (50)

= 0,

(KIR )µν =

1 8

γ i µν γi µ ν +

1 16

γ ij µν γij µ ν ,

as one extracts from (15), (26) and (29) (putting for simplicity v i = δ i8 , cf. Appendix A). ad , etc. These projectors are self-adjoint w.r.t. to the scalar product (44), i.e. KYM = KYM Note that KI is the weakest of these constraints in the sense that KYM M ⊂ KI M,

KIR M ⊂ KI M.

(51)

In the following, we are going to analyze the content of these sets of superspace constraints. To this end, we first give the general recipe how to obtain field content and field equations implied by a constraint of the type (46) and subsequently apply this formalism to the constraints (48), (49), and (50). Field content. To identify the physical field content among the components of the supercurl Mαβ , we collect the constraints that have been imposed on Mαβ . These are given by the recursion gauge condition (32) and the constraint (46): (S + R) M = − K M = 0.

R+2 T M + C, R

(52)


13

This obviously leaves + M+ ≡ M ∩ M = ker K ∩ ker (S + R),

(53)

undetermined. The independent (or physical) superfield components in Mαβ are hence given by M+ , the space of eigenvectors of the operator KSK with eigenvalue −p. The remaining part of Mαβ is consequently determined by the system (52) in terms of derivatives and nonlinear combinations of the physical fields. The fact that this part is in fact overdetermined by (52) then in turn implies the field equations as we shall discuss now. Field equations. The dynamical equations arise from combining the two equations of (52) into [p]

(S + R) M +

R+2 [p−2] = C [p] . T M R

(54)

[p]

This defines M in terms of the lower levels unless we project out onto vectors z| such that [p]

z| (S + R) M

= z| (S + R) K M [p] = 0,

in which case (54) implies a restriction on the image of T . The relevant vectors z| are hence simultaneous eigenvectors of K ad with zero eigenvalue and of S ad with eigenvalue 2, z| (S + R) K = 0, − |. These are eigenvectors of K S ad K with eigenvalue 2. (As we denote them as z⊥ discussed above, for the superspace constraint we find that S, KI , and KIR are self-adjoint − w.r.t. the scalar product induced by (44).) For any such eigenvector z⊥ |, (52) yields the dynamical equation ad

ad

p+2 − [p−2] − = z⊥ | C [p] . z ⊥ T M p

(55)

− Vice versa, if (55) is satisfied for all vectors of the form z⊥ |, the system (52) has a [p] solution for Mαβ in terms of the lower levels. Thus, the basic information about the content of the dynamical constraint (46) con− cerns the set of simultaneous eigenvectors z⊥ | and |z+ , respectively. We denote the − + corresponding spaces by M⊥ and M , respectively. Counting of dimensions yields the identity + dim M⊥ − dim M+ = dim M− ⊥ − dim M ,

(56)

where the numbers on the l.h.s. can simply be extracted from the representation tables of SO(9, 1) and SO(2, 1) × SO(7), respectively. For the lowest levels, these tables are collected in Appendix C.

14


4. The Physical Field Content In this chapter, we will determine the field content which is induced by the different dynamical constraints KYM , KI , and KIR . The result for the latter is summarized in Tables 1 and 2. We recall that with the strong superspace constraint (15), the arbitrariness in the supercurl Mαβ is restricted to the levels p = 0 and p = 1, i.e. all higher levels are determined. By analyzing the Bianchi identities for the supercurvature one verifies that in this case the following superfield relation holds [1]: R(R+1) Mαβ = where

1 2

(σ a )αβ (σa bc )γ1 γ2 θ γ1 θ γ2 Fbc ,

(57)

Fab = ∂a Ab − ∂b Ab + [Aa , Ab ]− ,

is now the curvature of the superfield Aa . Together with (31), one hence obtains recurrence relations which completely determine Mαβ in terms of its lowest components – the physical fields Xa and φ α . The field content associated with KYM hence precisely coincides with the ten-dimensional Yang–Mills multiplet. With the integrable (19) and the reduced integrable constraint (20) the situation becomes essentially more complex. In particular, there will be more superfield components left undetermined by the recurrent relations, i.e. the spectrum turns out to be considerably larger. In this section we analyze the physical field content associated with these integrable superfield constraints KIR and KI . According to the general discussion above, the independent components in the supercurl Mαβ are given by the space M+ , i.e. by the intersection of the kernels of (S+R) and K. We start from (56) − + dim M+ = dim M − dim M⊥ + dim M⊥ ,

(58)

and will in the following determine the r.h.s. of this equation for KIR and KI . To this end we first describe the decomposition of superfields into irreducible representations of SO(2, 1). 4.1. Decomposition into SO(2, 1) representations. The integrable constraints KI and KIR from (49), (50) are still invariant under the action of SO(2, 1) corresponding to the second factor in SO(9, 1) → SO(2, 1) × SO(7). This provides a convenient way to organize the spectrum. Explicitly, this group acts on the supercurl as given in (99) below. The generators are pairwise adjoint with respect to the scalar product defined in (44), δ0ad = δ0 ,

ad δ±1 = δ∓1 .

The supercurl Mαβ may hence be decomposed according to its SO(2, 1) spin. We label the total SO(2, 1) spin by $ and its z-component (i.e. the eigenvalue of δ0 which is raised resp. lowered by δ± ) by $0 . According to the action of δ0 , the value of $0 is given by the difference of barred and unbarred indices in a superfield Mαβ,γ1 ...γp . Specifically, [p]

M$0 is spanned by vectors    Mµν,µ1 ...µk−1 ,ρ 1 ...ρ l+1 M = Mµν,µ1 ...µk ,ρ 1 ...ρ l ,   Mµν,µ1 ...µk+1 ,ρ 1 ...ρ l−1

with

p = k + l,

$0 = 21 (l − k),

(59)


15

and the spin $ states are generated by highest weight states at $0 = $ obtained from (59) by dividing out the action of δ+ , i.e. satisfying    Mµν,µ1 ...µk−2 [ρ 1 ,ρ 2 ...ρ l+2 ] = 0 kMµν,µ1 ...µk−1 [ρ 1 ,ρ 2 ...ρ l+1 ] = Mµν,µ1 ...µk−1 ,ρ 1 ...ρ l+1 . (60)   (k+1) Mµν,µ1 ...µk [ρ 1 ,ρ 2 ...ρ l ] = Mµν,µ1 ...µk ,ρ 1 ...ρ l + Mµν,µ1 ...µk ,ρ 1 ...ρ l Furthermore, the space M[p] may be decomposed according to the action of the symmetric group on the p + 2 spinor indices. This is most conveniently described in terms of Young diagrams, where we use the standard notation [a1 , . . . , an ] to describe the Young diagram with n rows of length a1 , . . . , an . By [a1 , . . . , an ] we denote the conjugated Young diagram consisting of n columns of length a1 , . . . , an . Each box of the Young diagrams now represents a 8 of SO(7). The relations (59), (60) then imply [p] M$=0 = [2] × [l −1, k+1] + [l, k] + [l +1, k−1] + [1, 1] × [l, k] , [p]

M$=0 = [2] × [k+1, k−1] + [1, 1] × [k, k] .

(61)

M[p]

Since the decomposition (37) of has been defined purely in terms of permuting the spinor indices, it commutes with the action of the symmetric group. The Young diagram decomposition of the eigenspaces M+ and M− may be obtained from Fig. 2 by analyzing the decomposition of the Young diagrams under 16 → 8+8. Specifically, we find M+ $=0 = [l +1, k+1] + [l, k+2] + [l, k+1, 1] + [l +2, k] + [l +1, k, 1] , M+ $=0 = [k+2, k] + [k+1, k, 1] .

(62)

In particular, this gives the dimension 8 9 17k + 7 + [p] . dim M$=0 = k+1 k (k + 1)(k + 2) 4.2. The reduced integrable constraint. Here, we analyze the field content associated with the reduced integrable constraint KIR , given by (50). This constraint may be equivalently rewritten as 1 (63) Mµν − Mνµ , KIR Mµν = 0 = KIR Mµν , KIR Mµν = 2 and hence commutes with the action of the symmetric group on the 8 spinor indices (in contrast to KYM and KI ). This fact allows to completely resolve this case without any explicit reference to the decomposition of the superfield into irreducible representations of SO(7) or SO(9, 1), respectively. According to (58) we have to determine the spaces M⊥ and M− ⊥ . We start with M⊥ = K IR M. According to (63) and (60), the spin $ sector of M⊥ is given by the vectors satisfying    Mµν,µ1 ...µk−2 [ρ 1 ,ρ 2 ...ρ l+2 ] = 0 kMµν,µ1 ...µk−1 [ρ 1 ,ρ 2 ...ρ l+1 ] = Mµν,µ1 ...µk−1 ,ρ 1 ...ρ l+1 . (64)   (k+1) Mµν,µ1 ...µk [ρ 1 ,ρ 2 ...ρ l ] = 2Mµν,µ1 ...µk ,ρ 1 ...ρ l

16


In other words, each vector is given by its part Mµν , satisfying the constraint Mµν,µ1 ...µk−2 [ρ 1 ρ 2 ρ 3 ,ρ 4 ...ρ l+2 ] = 0, the other parts of M are determined from this by (64). This gives the Young diagram decomposition of K IR M: K IR M$=0 = 2 · [l +1, k, 1] + 2 · [l, k+1, 1] + [l +2, k] + [l, k+2]

(65)

+ [l −1, k+1, 1, 1] + [l +1, k−1, 1, 1] + [l +2, k−1, 1] + [l −1, k+2, 1] + [l +1, k+1] + [l, k, 1, 1] ,

K IR M$=0 = [k+1, k, 1] + [k+2, k] + [k+1, k−1, 1, 1] + [k+2, k−1, 2] . It remains to determine M− ⊥ , the space of common eigenvectors of S and K IR . For this, we consider the operator SIR ≡ K IR SK IR whose action on Mµν is found from (33) and (64) to be: [p]

(SIR M [p] )µν,µ1 ...µk+1 ,ρ 1 ...ρ l−1 = 2(l −1) Mµρ 1 ,µ1 ...µk+1 ,νρ 2 ...ρ l−1

(66)

[p]

− (k+1) Mµµ1 ,νµ2 ...µk+1 ,ρ 1 ...ρ l−1 [p]

+ (k+1)(l −1) Mµµ1 ,ρ 1 µ2 ...µk+1 ,νρ 2 ...ρ l−1 . The operator SIR obviously does not commute with its ancestor S and correspondingly has eigenvalues which do not necessarily coincide with those of S. However, since K IR is an orthogonal projector, it follows that the eigenvalues of SIR lie in the interval [−p, 2]. Moreover, eigenvectors of SIR with eigenvalues −p and 2, respectively, are necessarily also eigenvectors of S. Diagonalizing the action (66), SIR finally may be decomposed into projectors P[... ] onto the Young diagrams from (65), respectively: (67) = 2 P[l−1,k+1,1,1] + P[l+1,k−1,1,1] + P[l,k,1,1] + P[l+2,k−1,1] SIR l>k + 2 P[l−1,k+2,1] + P[l+1,k,1] + P[l,k+1,1] + (2−k) P[l+2,k] −

2k+l−1 2

P[l+1,k,1] −

k+2l 2

P[l,k+1,1] +

2−k−l 2

P[l+1,k+1]

+ (1−l) P[l,k+2] , SIR

l=k

= 2 P[k+1,k−1,1,1] + P[k+2,k−1,1] + (2−k) P[k+2,k] +

1−3k 2

P[k+1,k,1] .

The eigenspaces with eigenvalue 2 in this decomposition span the space M− ⊥ . Putting (62), (65), and (67) together, we find from (58),  8  dim([l +2] ) = l+2 for k = 0 + . (68) dimM =  0 otherwise The exceptional role of k = 0 stems from the fact that for this value the eigenvalue of the corresponding Young diagram [l +2, k] in (67) takes the extremal value 2 such that at k = 0 this eigenspace becomes part of M− ⊥.


17

With (58) we hence have obtained the entire physical field content in the superfield Mαβ induced by the reduced integrable superspace constraint KIR . We collect the result in Table 1, organized by level p and SO(2, 1) spin $. The total number of states is 769 = 384 + 384 + 1, where 384 + 384 corresponds to 3 copies of the irreducible offshell multiplet (128+128) of [3] and the singlet captures the remaining bosonic gauge freedom. This counting is the first hint, that the field content of the reduced integrable constraint remains completely off-shell, a fact that we shall show in the next chapters. Table 1. Spectrum induced by the reduced integrable constraint KIR 1 2

$

0

p=0 p=1 p=2 p=3 p=4 p=5 p=6

7+21

3 2

1

5 2

2

3

8+48 1+7+27+35 8+48 7+21 8 1

4.3. The integrable constraint. Here, we analyze the field content associated with the integrable constraint KI , given by (49). Comparing to the result from Table 1 for the stronger constraint KIR , even more fields must appear in this case. Note that in this section M and M⊥ refer to the decomposition (47) with respect to KI . Nevertheless, it is K I M = K IR M. To make use of the result of the previous section, we rewrite (58) as − + dim M+ = dim M − dim M⊥ + dim M⊥ = dim M+ − dim (K IR M)+ + dim (K IR M)+ ,

(69)

where the term in the brackets on the r.h.s. has been determined in (68) above and (K IR M)+ is defined to be the intersection of (K IR M)+ ≡ (SIR −2) K IR M, and

(K IR M) ≡ K IR KI M.

This space hence contains the fields that enlarge the spectrum with respect to Table 1. Its dimension remains to be computed. We first consider the case l = k, i.e. the SO(2, 1) singlets. The space (K IR M)+ then is generated by vectors v = v1 + v2 such that v1 and v2 are eigenvectors of SIR with eigenvalues (2−k) and 21 (1−3k), respectively – cf. (67) –, which in particular satisfy KI SIR v = KI SIR KI v =

1 4

(k−2) v.

(70)

The second equality is obtained from contracting (66) over µν. Since KI is an orthogonal projector, comparing (70) to the eigenvalues of SIR shows that it can be satisfied only if 1 2

(1−3k) ≤

1 4

(k−2) ≤ (2−k)

"⇒

p = 2k ≤ 4.

(71)

18


For p = 4 one may show by a similar but slightly more complicated analysis of the operator (KI SIR KI SIR KI ), that (K IR M)+ is empty also at this level. We leave details to the reader. At p = 2, in contrast, it is K IR M = (K IR M)+ and hence dim (K IR M)+ = dim (K IR KI M) = 7 + 21.

(72)

For states of arbitrary SO(2, 1) spin $ = 21 (l −k) one shows by similar reasoning, that dim (K IR M)+

=

8 k+1

iff l = 1.

(73)

The complete result is summarized in Table 2. The total number of fields in this case is 816 = 416 + 400. Table 2. Spectrum induced by the integrable constraint KI $ p=0 p=1 p=2 p=3 p=4 p=5 p=6

0

1 2

7+21

1

3 2

2

5 2

3

1 8+8+48

7+21

1+7+27+35 8+48 7+21 8 1

5. Recurrent Relations Having determined the field content, we will derive the recurrent relations which explicitly determine the higher level superfield components in terms of the lower level components. For simplicity, we restrict for the rest of the paper to purely bosonic configurations, e.g. we set all fermionic fields to zero. This is just for the sake of clarity, the techniques may likewise be applied to determine the structure of the fermionic fields and field equations. In particular, supersymmetry is unbroken up to this point. For the strong superfield constraint, the superfield Mαβ is entirely determined by its zero level components which are the physical Yang–Mills fields. The recurrent relations which determine the higher superfield levels have been given in (57). With the integrable constraints (19), (20) the picture becomes more complicated. In view of the field content given in Tables 1 and 2, a huge amount of additional fields has to be introduced to eventually obtain closed recurrence relations which replace (57). To keep things tractable, we will most of the time restrict the analysis to those fields which transform as singlets under the SO(2, 1) symmetry underlying (23). Despite this technical restriction, the method described in the following allows straightforward although more tedious generalization to the higher spin fields.


19

5.1. The reduced integrable constraint. In the spin zero sector the field content associated with KIR according to Table 1 contains an antisymmetric tensor field in addition to the seven dimensional Yang–Mills vector field. We have shown that in principle all higher levels p > 0 of Mαβ are uniquely determined in terms of these fields. To make this dependence explicit, we start from the system (52) [p−2] (S + p) M [p] = − p+2 + C [p] . p TM

(74)

Contracting this equation with KIR , we obtain [p−2] + KIR C [p] . (KIR SKIR + pKIR ) M [p] = − p+2 p KIR T M

(75)

Recall that the space M$=0 decomposes into K IR M = [k+1, k, 1] + [k+2, k] + [k+1, k−1, 1, 1] + [k+2, k−1, 2] , KIR M = [k+1, k, 1] + [k+2, k] + [k, k, 2] . Together with the eigenvalue decomposition (62) of S this gives the eigenvalues of the operator (KIR SKIR ) at each level p, which are 2, − p2 , and 41 (6 − p). This allows to effectively invert the system (75) to obtain M [p] = 9[p] T M [p−2] −

p p+2

9[p] C [p] ,

8 9[p] ≡ − 3p2 (p+2) (KIR SKIR )2 +

2(p+14) 3p2 (p+2)

(76) (KIR SKIR ) −

3p2 +10p+24 3p2 (p+2)

KIR .

Thus, we have completely resolved this sector with an explicit recurrent definition of all higher level components of the superfield Mαβ . In a similar way, one may obtain the defining relations for the higher SO(2, 1) spin sectors. As an illustration, we evaluate the lowest two bosonic levels of the supercurl Mαβ . They are determined by the level zero fields: the seven dimensional Yang–Mills vector field Xi and an antisymmetric tensor field Bij : [0] Mµν = γ i µν Xi + γ ij µν Bij ,

[2] Mµν =

1 2

ij

γµν δρ1 ρ 2 −

1 2

(77) ij m

ij mn

1 mn m γ γµν ρ1 ρ 2 + 4 γµν γρ1 ρ 2

θ ρ1 θ ρ 2 Yij

i mn imn + 2 γµν δρ1 ρ 2 + γµν γρ1 ρ 2 θ ρ1 θ ρ 2 Dj Bij m ij km km ij m km ij m + γµν γρ1 ρ 2 − γµν γρ1 ρ 2 + 2 γµν γρ1 ρ 2 θ ρ1 θ ρ 2 Dk Bij

ij m ij m mn ij mn + 2 γµν γρ1 ρ 2 + 2 γµν δρ1 ρ 2 + γµν γρ1 ρ 2 θ ρ1 θ ρ 2 Bik , Bj k

ij k l lm ij km + 2 γµν γρ1 ρ 2 + γµν γρ1 ρ 2 θ ρ1 θ ρ 2 Bij , Bkl .

Here, Yij is the Yang–Mills field strength

Yij = ∂i Xj − ∂j Xi + Xi , Xj − ,

(78)

20


and Dk denotes the gauge covariant derivative Dk Bij = ∂k Bij + [Xk , Bij ].

(79)

5.2. The integrable constraint. For the reduced integrable KIR constraint, we have given the complete recurrent solution (76). Since the integrable constraint KI is weaker in the sense of (51), its field content is larger and the recurrent relations will involve more fields. From Table 2 we know already that the spectrum associated with KI in its spin 0 sector contains another copy of the vector and tensor fields. The system (74) in this case gives rise to the recurrent relations p p M [p] = 9[p] T M [p−2] − p+2 C [p] + p+2 9[p] S + I KI K IR N [p] , (80) where 9[p] has been defined in (76) above, and N is a superfield which satisfies:   Nµν = δµν θ ρ ∂ρ n N = KI K IR N "⇒ N = Nµν = δµν n (81)  N = δ θρ∂ n µν µν ρ with a scalar superfield n, further constrained by (60). Taking different projections of (74) one may obtain the remaining recurrent relations which determine the higher levels of these scalar superfields in terms of the lower levels in Mαβ and n. These however become more tedious due to the fact that the integrable constraint can no longer be expressed entirely in terms of permuting spinor indices. Here, we restrict to giving the first two levels of n which have the particularly simple form n[0] = 0,

(82)

n[2] = γ i ρ1 ρ 2 θ ρ1 θ ρ 2 Zi + γ ij ρ1 ρ 2 θ ρ1 θ ρ 2 Cij . The level p = 2 is completely undetermined and hence contains the additional physical fields denoted by Zi and Cij whose existence has been anticipated in Table 2. Evaluating (80) we find for the supercurl [0] Mµν = γ i µν Xi + γ ij µν Bij ,

(83) ij

[2] Mµν = δµν γρi 1 ρ 2 θ ρ 1 θ ρ 2 Zi + δµν γρ 1 ρ 2 θ ρ 1 θ ρ 2 Cij ,

[2] Mµν = δµν γρi1 ρ 2 +

1 2

i γµν δρ1 ρ 2 +

ij + δµν γρ1 ρ 2 + +

1 2

ij

1 2

γµν δρ1 ρ 2 −

1 4

ij

mn imn γµν γρ1 ρ 2 θ ρ1 θ ρ 2 Zi

γµν δρ1 ρ 2 + 1 2

ij m

1 2

ij m

m γµν γρ1 ρ 2 + ij mn

1 mn m γ γµν ρ1 ρ 2 + 4 γµν γρ1 ρ 2

i mn imn + 2 γµν δρ1 ρ 2 + γµν γρ1 ρ 2 θ ρ1 θ ρ 2 Dj Bij

1 4

ij mn

mn γµν γρ1 ρ 2

θ ρ1 θ ρ 2 Yij

θ ρ1 θ ρ 2 Cij


21

m ij km km ij m km ij m + γµν γρ1 ρ 2 − γµν γρ1 ρ 2 + 2 γµν γρ1 ρ 2 θ ρ1 θ ρ 2 Dk Bij

ij m ij m mn ij mn + 2 γµν γρ1 ρ 2 + 2 γµν δρ1 ρ 2 + γµν γρ1 ρ 2 θ ρ1 θ ρ 2 Bik , Bj k

ij k l lm ij km + 2 γµν γρ1 ρ 2 + γµν γρ1 ρ 2 θ ρ1 θ ρ 2 Bij , Bkl ,

[2] = δµν γρi1 ρ2 θ ρ1 θ ρ2 Zi + δµν γρij1 ρ2 θ ρ1 θ ρ2 Cij . Mµν

Summarizing, we have shown that in the sector of SO(2, 1) singlets, the supercurl Mαβ in recursion gauge and with the integrable superspace constraint (49) imposed, is determined in all orders by the set of physical fields Xi , Bij , Zi , Cij ,

(84)

which enter as components at the levels p = 0 and p = 2 of the superfield expansion of Mαβ as given in (83). In the following, we will study what kind of dynamical relations we may in addition extract for these fields. Since KI is the weakest of the three constraints we are studying, the other two cases may be embedded as particular truncations of (83). It is easy to see that they correspond to KYM : Cij = Yij , Zi = 0 , Bij = 0,

(85)

KIR : Cij = 0 , Zi = 0. 6. Field Equations In this section we will determine the field equations implied by the integrable constraints KI , KIR for the physical fields. As in the previous chapter we restrict to the dimensionally reduced situation where all fields depend only on the coordinates x i , i = 1, . . . , 7, thereby consistently truncating the system to singlets under SO(2, 1). The field content in this sector is given by (84) for KI , KIR implies the further truncation (85). Following the general discussion of Sect. 3, the dynamical content arises from projecting the image of the operator T according to (55) onto the space M− ⊥ . Applying this to the integrable constraints, we find that KIR in fact does not imply any field equation on the fields Xi , Bij , such that the corresponding spectrum remains completely off-shell. The weaker constraint KI which has a larger spectrum, will give rise to first order dynamical equations for the additional fields Zi and Cij , coupled to the off-shell fields Xi and Bij . Let us first recapitulate the case of the strong constraint. At level p = 2 the supercurl is uniquely determined by the Yang–Mills fields according to the lowest order component of Eq. (57), [2] Mαβ = (σ a )αβ (σa bc )γ1 γ2 θ γ1 θ γ2 Ybc .

(86)

Since there are no new fields arising on this level, relation (56) yields M+ =∅

"⇒

+ dim M− ⊥ = dim M⊥ − dim M ,

(87)

22


where the numbers on the r.h.s. may be extracted from the representation tables collected in Appendix C. In particular, this shows that no field equations arise on this level. At level p = 4, Eq. (87) with Table 4 shows that M− ⊥ is nonempty but contains e.g. the vector representation 10 of SO(9, 1). According to (55) the dynamical equation is given by the scalar product [2] [4] 2 [4] T M , M− − C ⊥ 10 3

with

T M[2] ∼ ∂ b Yab .

(88)

Since this is the nondegenerate scalar product on a space of multiplicity one, it suffices to show that T M[2] = 0 (with M[2] given by (86)) to indeed obtain the bosonic part of the Yang–Mills field equations Db Yab = 0.

(89)

One might expect to find further relations in the SO(9, 1) representations 120 and 126, respectively, in which according to Table 4 the space M− ⊥ also has nonvanishing contributions. However, the first one contains precisely the Bianchi identities of Yab which are automatically satisfied, whereas there is no nontrivial image of T into the 126 as one may easily verify. Thus, in agreement with [1], there arise no further restrictions than the Yang–Mills equations of motion (89), here. In the rest of this section, we will repeat this analysis for the integrable constraints KI and KIR .

6.1. The reduced integrable constraint. For the reduced integrable constraint KIR , the entire constraint is encoded in the system (74) of which we have already solved the projection (75) by imposing the recurrent relations (76). It remains to study the complementary projection: [p−2] K IR (S + p) M [p] = − p+2 + K IR C [p] . p K IR T M

(90)

Plugging in the explicit solution (76), one obtains after some calculation K IR (2S + (p−4)) K IR (4S + (3p−2)) K IR

T M [p−2] −

p p+2

C [p] = 0.

(91)

This encodes the entire dynamics of this constraint. Comparing (91) with (67), one recognizes (55); the operator on the l.h.s. of this equation is precisely the projector onto M− ⊥. However, it turns out (as we have explicitly checked for p ≤ 6 and are confident that it holds on all levels) that Eq. (91) becomes an identity when M [p−2] is expressed by the recurrent relation (76). Hence, we conclude that for the reduced integrable constraint, the system (74) is solved by (76) without imposing any further relations on the physical fields. The system remains completely off-shell.


23

6.2. The integrable constraint. Let us turn to the integrable constraint KI . Recall that this is a weaker constraint than KIR and hence gives rise to a larger spectrum. Likewise, the dynamical equations implied by this constraint must be compatible with the truncation (85) since the system associated with KIR was completely off-shell. In other words, setting Zi = 0 = Cij must solve all dynamical equations without imposing further dynamics on Xi , Bij . As we have shown above, at the level p = 2 of the supercurl Mαβ we find new fields − arising, the explicit formulae have been given in (83). From Table 5 we find that M⊥ is empty at this level, i.e. there are no field equations arising at p = 2, the projection (55) turns out to be satisfied without imposing any restrictions on the level zero fields. At level p = 4, we expect some dynamical equations to appear in analogy with (88) for the strong constraint. We discuss the different irreducible representations of SO(7), starting with the singlet 1. According to Table 6, this appears in a particularly simple way, namely with multiplicity one. Moreover, this table shows that [4] mult M− 1 = 1, ⊥

i.e. according to (55), a dynamical equation arises from the scalar product [4] [2] 2 [4] M− T M , with M [2] given by (83). − C ⊥1 3 Similarly to (88), this scalar product is particularly simple to compute because it lives on a space M[4] 1 of multiplicity one. With the explicit expression from (83) we arrive at the first field equation for the enlarged system

Di Zi = 43 Cij , B ij . (92) Note that this equation has no analogue in the original Yang–Mills system since in that system there is no combination of fields and derivatives transforming as a singlet of SO(9, 1) at this order. Let us continue with the vector part 7 which should contain the analogue of theYang– Mills equations of motion. For illustration, we will describe this sector in some detail. According to the general proceeding outlined above, we first determine the subspace M− ⊥ which by projection gives rise to the dynamical equations of the system. It follows [4] is nonempty with multiplicity one. To determine this from Table 6 and (87) that M− 7 ⊥ space explicitly, it suffices to diagonalize the operator S from (33) on the space M⊥ [4] 7 . A basis of the latter is e.g. given by mnk ki ki mn (w1 )iµν , µ1 µ2 ρ 1 ρ 2 = γµν γ − γ γ γµmn (93) µ1 µ2 ρ 1 ρ 2 , 1 µ2 ρ 1 ρ 2 imnk γµmn , (w2 )iµν , µ1 µ2 ρ 1 ρ 2 = γµν γ k − γµk1 µ2 γρmn 1 µ2 ρ 1 ρ 2 1ρ2 imn m (w3 )iµν , µ1 µ2 ρ 1 ρ 2 = γµν γµ1 µ2 γρn1 ρ 2 , imn mk (w4 )iµν , µ1 µ2 ρ 1 ρ 2 = γµν γµ1 µ2 γρnk1 ρ 2 ,

where the other components of these vectors are obtained from the conditions (60), [4] is discussed above. Computing the action of S on this basis (93) one finds that M− 7 ⊥ spanned by [4] i i i M− (94) = w − w + 4w 1 2 3 . ⊥7

24


The dynamical equations are finally obtained according to (55) by projecting the image of M [2] – the latter being entirely given by (83) – under T onto the constraint vector (94) [4] [2] 2 [4] M− . T M − C ⊥7 3 Explicit computation gives the following result:

Dm Cmi = 21 Z m , Bim .

(95)

This gives the analogue of the Yang–Mills equations for the enlarged system associated to the integrable dynamical constraint. For the strong constraint KYM this equation according to (85) consistently reproduces theYang–Mills equations of motion. Moreover, it is compatible with the absence of dynamics in the truncation to the reduced integrable constraint KIR . Similarly, one may continue with all the other SO(7) subrepresentations contained in Mαβ . As is clear from the above proceeding, the existence of a dynamical equation first requires the corresponding subspace M− ⊥ to be nonempty and in addition a nontrivial projection (55) of the image of T . Whereas validity of the first criterion may simply be extracted from the tables collected in Appendix C, the second condition requires a more careful calculation and has been done on the computer using Mathematica. We give the result for this level in linearized order, where the complete set of dynamical equations is given by ∂(m Zn) = 0,

(96)

∂ m Cmi = 0, ∂[i Cj k] = 0. The full nonlinear extensions of the first two equations have been given in (92), (95) above; likewise, the third equation acquires nonlinear contributions, such that the field Cij does not satisfy the pure Bianchi identities of a covariant field strength. The linearized equations however are sufficient to extract the propagating degrees of freedom contained in Zi and Cij . E.g. one of the main results of (96) is the absence of an equation in the antisymmetric 21 for ∂[m Zn] (although the space M− ⊥ 21 is nonempty, the projection (55) has trivial image). Instead, we find an equation for ∂(m Zn) in the symmetric 27 + 1. Whereas Cij hence carries the dynamics of a propagating vector field, the role of Zi remains somewhat unclear. We close this section with a remark on the dynamical equations for the higher SO(2, 1) spin fields at this level. Since the fields appearing in the spectrum of KIR in Table 1 remain off-shell, the only dynamical equations of higher spin can appear for the (3, 1) fields of level p = 0 from Table 2. These fields are part of the original Yang–Mills vector field, as such their dynamical equation is expected in the (3, 1) at [4] level p = 4. However, according to (87) and Table 6 the space M − (3,1) is empty, ⊥ such that there is no analogue of this part of the originalYang–Mills equations of motion. Equations (96) hence contain the complete dynamical content at this level. The task of studying the higher superfield levels which might induce higher order equations for the fields Zi and Cij is left for future work. A strong consistency check for the arising equations is provided by their compatibility with the different truncations (85) to the Yang–Mills and the off-shell system, respectively.


25

7. Summary In this paper we have analyzed the field content and the dynamical equations induced by certain modifications of the constraint of vanishing super curvature which is equivalent to ten dimensional supersymmetric Yang–Mills theory. The geometric origin of the modified constraints is a truncation of the ten-dimensional linear system, breaking the original Lorentz symmetry SO(9, 1) down to SO(2, 1) × SO(7). The Lax representation thereby reduces to a system with scalar spectral parameter (23) which bears strong similarity with the Lax connection for selfdual four-dimensional Yang–Mills theory. Applying the general formalism of Sect. 3, two different scenarios have been revealed for the integrable and the reduced integrable constraint KI and KIR , respectively. The latter induces a spectrum of (384+384) fields given in Table 1 thereby drastically reducing the field content (43) of the unconstrained supercurl but still remaining completely off-shell. The integrable constraint KI gives rise to additional (31+16) fields which are strongly restricted by first order differential equations which in linearized form have been given in (96). The complete spectrum of SO(2, 1) singlets thus involves two pairs of fields (Xi , Zi and Bij , Cij ) with the same tensorial structure, but with different dimensions, since Xi , Bij , and Zi , Cij have dimensions one and two, respectively. The fields Xi and Bij do not obey any equations of motion whereas Cij and Zi appear with dynamical equations coupled to the off-shell fields. According to (96), Cij contains the degrees of freedom of a propagating vector field X˜ i , whereas Zi apparently is associated with an off-shell two-form B˜ ij . Hence, we find an intriguing duality with the original fields Xi , Bij which remains to be explored in more detail. If we reduce to four dimensions, there is a striking analogy with the case of electromagnetism in the presence of magnetic charge (see e.g. Ref. [21]), where the field strength is build from two pieces, a homogeneous one such that the Bianchi identity gives zero, and another piece for which the Bianchi identity gives the magnetic current. Thus, we conclude that our dynamics in general involves magnetic charges. On the other hand, the field Bij has the features of a two-form vector potential. An intriguing question is the role of the corresponding gauge transformations Bij → Bij + D[i >j ] + . . . . However, the form of possible interactions with higher form gauge potentials appears to be highly restricted on general grounds (see [14] for a recent discussion). We leave all these questions to future studies. Another point we have omitted so far is the dynamics of the fermionic fields of the theory which may be analyzed with exactly the same methods that have been presented here for the bosonic sector. In particular, the (possibly broken) supersymmetry should help to better understand the nature of the underlying physical system. In view of the geometric origin of the integrable constraints, a natural problem is the generalization of the present approach by studying more general reductions of the original linear system (21). Here, we have analyzed the dynamical systems associated with the particular truncation (22) to a one-parameter set of light-like rays whose spatial part spans a two-dimensional plane. Truncation to more general subvarieties may refine the dynamics. Allowing for a more general dependence of the linear system on the vector v i bears some striking similarities with the harmonic superspace approach to four-dimensional N = 2 supersymmetric Yang–Mills theory [6]. We finally mention the possibility to recover in this framework and upon dimensional reduction some of the classical higher spin gauge theories, which have been constructed by Vasiliev (see e.g. [18] for a review) and recently [16] been brought into the context of a possible M-theoretic origin.

26


Acknowledgement. This work was done in part while one of us (J.-L. G.) was visiting the Physics Department of the University of California at Los Angeles. He is grateful for the warm hospitality, and generous financial support extended to him. The work of H. S. was supported by EU contract ERBFMRX-CT96-0012. Discussions with E. Cremmer, E. D’Hoker, P. Forgacs and K. Stelle have been very useful.

A. Reduction of σ -Matrices In this appendix we collect our conventions of SO(9, 1) σ -matrices and their decomposition into SO(8) γ matrices. We use the following particular realization: αβ −18×8 08×8 σ9 = σ9 = , (97) 08×8 18×8 αβ αβ 18×8 08×8 0 0 σ =− σ = , 08×8 18×8 αβ i αβ 0 γµν i i σ = σ = iT , i = 1, . . . 8, γ 0 αβ νµ i denote the SO(8) γ -matrices obeying where γµν

γ i γ j T + γ j γ i T = 2δ ij ,

i, j = 1, . . . , 8.

(98)

Our index convention here is as follows: Greek letters from the beginning of the alphabet run from 1 to 16, letters from the middle of the alphabet from 1 to 8, denoting the two spinor representations of SO(8). Choosing a particular eight-dimensional vector v i breaks SO(8) down to SO(7) and provides a mapping between the two spinor represenv = v i γ i . For notational convenience we put v i = δ i and γ 8 = δ , such tations by γµν µν 8 µν µν that µ ↔ µ¯ denotes an SO(7) covariant involution. It then follows from the Dirac algebra (98) that the matrices γ i , i = 1, . . . , 7 are antisymmetric. In the main text, unless otherwise stated, Roman indices i, j, . . . from the middle of the alphabet exclusively denote the coordinates 1, . . . , 7. To make the SO(2, 1) covariance of the decomposition (97) manifest, we define the action of the generators δk , k = −1, 0, 1 on the supercurl Mαβ as:

δk Mαβ = (Jk M + M JkT )αβ + θ α (Jk )α β ∂β Mαβ , with J0 = σ 0 σ 9 ,

J±1 =

1 2

(99)

σ 8σ ±.

B. The O(9, 1) Characters In this appendix, we compute the characters of the reducible representations which appear in Sect. 3. The path followed is similar to the calculation of string characters of Ref. [5], and we shall refer to paper for details. In general, the O(9, 1) characters that ! v H i i , where vi are arbitrary parameters, where the are defined as χ (v) = Tr e i trace is taken in the representation considered, and Hi , i = 1, . . . , 5 are a set of five commuting elements of the Lie algebra. Using a parametrization analogous to the one


27

used in Ref. [5] for O(8) spinors, one easily sees for instance that the character of the 16 representation4 is given by " 1 χ16 (v) = e 2 vi !i . (100) !1 ,··· ,!5 =±1 odd # =1

i

B.1. The unconstrained character. In this subsection, we first compute the character associated with the representation span by the full Mαβ , by considering the trace over the full space M. The αβ indices then contribute a factor " 1 1 " 1 vi !i 2 2v ! i i χ16⊗16 (v) = . e2 + e2 2 ! ,··· ,! =±1 ! ,··· ,! =±1 s

1

i

5 odd # =1

1

i

5 odd # =1

Concerning the θ part, one works in an occupation number basis where Nα = θ α ∂α is simultaneously diagonal for α = 1, · · · , 16. Since the Lie group generators commute with the grading operator R, it isconvenient to introduce in general characters of the ! R type χ (v|q) = Tr q e i vi Hi . Then the calculation becomes identical to a part of the string calculation, where the role of R is played by the Virasoro generator L0 . Altogether, one finds that the character without any constraint denoted χu is given by " 1 1 " 1 vi !i 2 × e2 + e 2 2vi !i χu (v|q) = 2 ! ,··· ,! =±1 ! ,··· ,! =±1 1

×

i

5 odd # =1

1

"

1+q

"

i

5 odd # =1

1 e 2 vi !i .

(101)

i

!1 ,··· ,!5 =±1 odd # =1

B.2. The character corresponding to M± . To determine them, we first compute the character with S introduced. This is straightforward since S is a group invariant. Using again the occupation number operators Nα = θ α ∂α , one may verify that !5 (102) χS (v|q) ≡ Tr e i=1 vi Hi S =

! " Tr θ q γ Nγ evi Hi

α,β;α,β

i

α,β

evi

!

ρ ρ (Hi ) ρ Nρ

Nα + N β

,

s

where the trace over αβ only involves the symmetric states. After some straightforward computation one finds that " 1 " " 1 1+q e 2 vi (!i +!i ) e 2 vi ηi 2χu (v|q) − χS (v|q) = {!}{! } i

+

{η}={!},

" {!}

i

evi (!i )

"

1+q

{η}={!}

4 In this appendix boldface dimensions refer to O(9, 1) representations

i

" i

1 e 2 vi ηi .

(103)

28


Clearly, " " 2χu − χS ≡ Tr (S − 2) q R evi Hi = Tr M+ (R + 2) q R evi Hi . i

i

An easy computation then gives " R vi Hi −2 + =q e χ (v|q) ≡ Tr M+ q i

q 0

dxx {2χu (v|x) − χS (v|x)} .

(104)

This gives χ + (v|q)) =

14

15

1

e2

!

v.

p+2 r=1 η r

{η1 } 0 2

2d

2d

2d

2

2

M= { S ,R ,H }

M= H

reff < 0, K(v) < ξ

2

M= H 2

Fig. 3

S2

R2

H2

β>0

β>0

β > | r|

β = | r|

S2

S2

S2

R2

H2

β =0, reff > 0

β =0, reff> 0

β =0, reff> 0

β =0, reff= 0

β =0, reff < 0

H2

2

H2

2

2

β < | r|

Fig. 4

Then we have

  in the case (P) d(v) −1 K (v) = ξ −1 tan(ξ d(v)) in the case (E)  ξ −1 tanh(ξ d(v)) in the case (H).

(14)

As one can see from Eqs. (9)–(14), the geometric optics (described in terms of parameters d, s) depend only on the value reff . This fact allows to study the linearized dynamics problems on M(r, β) using the corresponding results for the non-magnetic surfaces of the constant Gaussian curvature reff = r + β 2 . The corresponding transition M(r, β) → M(reff , 0) is illustrated by Fig. 4. 3. Stability of Generalized Two-Periodic Trajectories Let Q be a billiard table on M(r, β). For each v ∈ V let t (v) be the corresponding past semitrajectory in Q. Consider the curvature evolution of an infinitesimal beam along t (v). Starting with B(φ −k · v, χ ) for arbitrary χ , we obtain after k steps forward the infinitesimal beam B(v, χ (k) ), χ (k) = φ k · χ . Eqs. (9)–(11) and (12) describe the action of the billiard map on the curvature of infinitesimal beams. Assuming k to be infinity, we obtain a formal continued fraction corresponding to the semitrajectory t (v): c(v) = χ (∞) = a0 +

b0 a−1 +

b−1 a−2 · · ·

.

(15)

Hyperbolic Magnetic Billiards on Surfaces of Constant Curvature

41

The coefficients of the continued fraction are determined by di = d(φ i · v), and by the lengths si of consecutive billiard segments as follows: (P) (E) (H)

ai = −2si−1 + 2di−1 , bi = −si−2 ; ai = −2 cot(ξ si ) + 2 cot(ξ di ), bi = − sin−2 (ξ si ); ai = −2 coth(ξ si ) + 2 coth(ξ di ), bi = − sinh−2 (ξ si ).

The continued fractions (15) determines the stability type of the trajectory: t (v) is unstable if c(v) is convergent (see e.g., [Si]). Since for a given sequence of di and si , c(v) is completely determined by reff , one can reduce the problem of stability trajectories on M(r, β) to the corresponding “non-magnetic” problem on M(reff , 0). As it has been mentioned in the introduction, we are interested in the stability properties of generalized two periodic trajectories. A trajectory is a generalized two periodic trajectory (g.t.p.t.) if its parameters di are periodic: d2i+1 = d1 , d2i = d2 and si = s are the same along the trajectory (see Fig. 2). Obviously, a g.t.p.t. yields a periodic continued fraction. We denote by T (d1 , d2 , s) the g.t.p.t. with parameters (d1 , d2 , s) and by c(d1 , d2 , s) the associated continued fraction. The stability of T (d1 , d2 , s), or equivalently, the convergence of the two periodic continued fraction c(d1 , d2 , s) has been studied in [GSG] for non-magnetic surfaces of constant curvature. On the basis of the equivalence between the magnetic and nonmagnetic problems we can immediately generalize the results of [GSG] to the case β = 0. Proposition 1. The continued fraction c(d1 , d2 , s) converges if and only if the following inequalities are satisfied. (P) (E) (H)

(s − d1 )(s − d2 )(s − d1 − d2 )s ≥ 0; sin(ξ(s − d1 )) sin(ξ(s − d2 )) sin(ξ(s − d1 − d2 )) sin(ξ s) ≥ 0; sinh(ξ(s − d1 )) sinh(ξ(s − d2 )) sinh(ξ(s − d1 − d2 )) sinh(ξ s) ≥ 0.

Below we reformulate Proposition 1 explicitly as conditions for the instability of the corresponding g.t.p.t. (P) T (d1 , d2 , s) is unstable if and only if   [d1 , d2 ] ∪ [d1 + d2 , ∞) if d1 , d2 ≥ 0 s ∈ [0, ∞) (16) if d1 , d2 ≤ 0  [0, d + d ] ∪ [d , ∞) if d ≥ 0, d ≤ 0. 2 1 1 2 1 (E) In this case 0 ≥ ξ s ≥ 2π , and we set π¯ = π · ξ −1 , s if s ≤ π¯ s mod π¯ = s − π¯ if s > π¯ . Then T (d1 , d2 , s) is unstable if and only if   ¯ ∪ [d1 , d2 ] [d1 + d2 , π]   [0, d + d + π¯ ] ∪ [π¯ − d , π¯ − d ] 1 2 1 2 s mod π¯ ∈  [d , π ¯ + d ] ∪ [0, d + d ] 2 1 1 2   [d , π¯ + d ] ∪ [π¯ + d + d , π¯ ] 2 1 2 1

if d1 , d2 ≥ 0 if d1 , d2 ≤ 0 if d1 ≤ 0, d2 ≥ 0, |d2 | ≥ |d1 | if d1 ≤ 0, d2 ≥ 0, |d2 | ≤ |d1 |. (17)

42

B. Gutkin

(H) It matters whether vi ∈ V A or vi ∈ V B for i = 1, 2. We say that T (d1 , d2 , s) is of type (A − A) if v1 ∈ V A and v2 ∈ V A . The other types: (A − B), (B − A), and (B − B) are defined analogously. We formulate the explicit criteria of instability for T (d1 , d2 , s) type-by-type. Type (A − A):  A A A A A A  [d1 , d2 ] ∪ [d1 + d2 , ∞) if d1 , d2 ≥ 0 s ∈ [0, ∞) (18) if d1A , d2A ≤ 0  [0, d A + d A ] ∪ [d A , ∞) A A if d1 ≥ 0, d2 ≤ 0, 1 2 1 Type (B − B):

s∈

[d1B + d2B , ∞) if d1B + d2B ≥ 0 [0, ∞) if d1B + d2B ≤ 0,

Types (A − B) or (B − A):

s∈

[d1A , ∞) [0, ∞)

if d1A ≥ 0 if d1A ≤ 0.

(19)

(20)

It is worth mentioning that in Proposition 1 (resp. Eqs. (16)–(20)) the hyperbolicity of T (d1 , d2 , s) corresponds to strict inequalities (resp. inclusions in the interior). The equality case (resp. boundary case) corresponds to the parabolicity of T (d1 , d2 , s). There are also two special cases when T (d1 , d2 , s) is parabolic independently of the value of s: (P), d1 = d2 = −∞ and (H), |d1A | = |d2A | = ∞. We call the right-hand side of Eqs. (16)–(20) the instability set of T (d1 , d2 , s). In general, it is a union of two intervals, where one of them degenerates when |d1 | = |d2 |, while the other is always nontrivial. Following the terminology of our previous work [GSG], we will say that the interval which persists is a “big interval”, while the other one is a “small interval”. We will say that T (d1 , d2 , s) is (strictly) B-unstable if s belongs to the (interior of the) big interval of instability. The proposition below makes this terminology explicit. Proposition 2. The g.t.p.t. T (d1 , d2 , s) is B-unstable if (and only if) the triple (d1 , d2 , s) satisfies the following conditions: (P)   [d1 + d2 , ∞) if d1 , d2 ≥ 0 s ∈ [0, ∞) (21) if d1 , d2 ≤ 0  [d , ∞) if d1 ≥ 0, d2 ≤ 0, 1 (E)   if d1 , d2 ≥ 0 [d1 + d2 , π¯ ] s mod π¯ ∈ [0, d1 + d2 + π¯ ] if d1 , d2 ≤ 0  [d , π¯ + d ] if d1 ≤ 0, d2 ≥ 0, 2 1

(22)


43

(H) The case (A − A)  A A A A  [d1 + d2 , ∞) if d1 , d2 ≥ 0 s ∈ [0, ∞) if d1A , d2A ≤ 0  [d A , ∞) if d1A ≥ 0, d2A ≤ 0, 1 or |d1A | = |d2A | = ∞ and arbitrary s. (H) The case (B − B) [d1B + d2B , ∞) if d1B + d2B ≥ 0 s∈ [0, ∞) if d1B + d2B ≤ 0. (H) The cases (A − B) or (B − A) [d1A , ∞) if d1A ≥ 0 s∈ [0, ∞) if d1A ≤ 0.

(23)

(24)

(25)

Obviously, the conditions (21)–(25) for B-unstable g.t.p.t.s are the same as those which appeared in [GSG] for the corresponding non-magnetic cases. 4. The Main Theorem Let Q be a billiard table and v ∈ V be an arbitrary point in the phase space of the billiard map. Set v1 = v, v2 = φ(v), di = d(vi ), i = 1, 2, and let s = s(v) be the length of the particle trajectory between the origin points of v1 and v2 respectively. We will associate with v a formal g.t.p.t. T (v) = T (d1 , d2 , s), whose parameters are defined by the triplet ¯ (d1 , d2 , s). We denote by λ(v) the Lyapunov exponent of the billiard Q and by λ(v) the Lyapunov exponent of T (v) (see Sect. 1), which are defined for µ-almost all v ∈ V . Using Proposition 2 we introduce the following special class of points of the phase space of the billiard map. Definition 1. A point v ∈ V of the billiard phase space is a) B-hyperbolic (or strictly B-unstable) if the corresponding g.t.p.t. T (v) is strictly Bunstable; b) B-parabolic if the corresponding g.t.p.t. T (v) is B-unstable and parabolic (i.e., s belongs to the boundary of the appropriate interval (21–25)); c) B-unstable if the corresponding g.t.p.t. T (v) is B-unstable (i.e., B-parabolic or Bhyperbolic); d) eventually strictly B-unstable if there is some integer n such that T (φ i (v)) is Bunstable for 0 ≤ i < n and T (φ n (v)) is strictly B-unstable. Below we formulate the main theorem of the present work. Theorem 1. Let Q be a billiard table on M(r, β). If µ-almost every point of the billiard phase space is eventually strictly B-unstable, then the Lyapunov exponent λ is positive µ-almost everywhere.

44

B. Gutkin

Proof. The proof of the theorem is based on the cone field method which has been initially applied to the planar billiards in [Wo1,Wo2]. A cone in Tv V corresponds to an interval in the projectivization Bv . Therefore, a cone field, W, is determined by a function, W (·), on V , where each W (v) is an interval in the projective coordinate χ. We define the function W (v) as in [GSG]. For completeness, we repeat this definition below. [K(v), +∞] if K(v) ≥ 0 (P) and (E) W (v) = , [−∞, K(v)] if K(v) ≤ 0 [K(v), +∞] if K(v) ≥ ξ (H) W (v) = . [−∞, K(v)] if K(v) ≤ ξ As it follows from Lemma 2 in [GSG], this cone field is eventually strictly preserved by the billiard map if the conditions of Theorem 1 are satisfied. By this fact the proof of the theorem follows immediately from Wojtkowski’s theorem (Theorem 1 in [Wo2]). Applying the method developed in [Wo2], one can actually estimate from below the Lyapunov exponent using the cone field defined above. The result is given by the next theorem. Theorem 2. Let Q be a billiard table satisfying the assumptions of Theorem 1, then ¯ h(φ) = λ(v) dµ ≥ λ(v) dµ. V

V

Proof. The proof follows immediately by the repetition of calculations given in the proof of the analogous theorem for the non-magnetic case (see Theorem 2 in [GSG]). 5. Applications and Examples Theorem 1 together with Proposition 2 lead to a simple geometric criterion for billiard tables with hyperbolic dynamics. In this section we apply this criterion to construct various classes of hyperbolic billiards on M(r, β). 5.1. Elementary billiard tables. There is a class of billiard tables, where the application of Theorem 1 gives an especially simple criterion for hyperbolicity. This class consists of billiard tables Q, whose boundary is a finite union of arcs, .i , of constant geodesic curvature, κ(.i ) = κi . We call these tables elementary. We will use the notation .i+ (resp. .i− ) if κ(.i ) > 0 (resp. κ(.i ) ≤ 0). Let Ci be the curve of constant geodesic curvature such that .i ⊆ Ci and Di ⊂ M be the corresponding disk (Ci = ∂Di ). Since the representation ∂Q = ∪N i=1 .i is unique, we call .i the components of ∂Q. In the following, we consider elementary billiard tables for which |κi | ≥ β. One may easily see that the fulfillment of this inequality is necessary for billiards satisfying the conditions of Theorem 1 (see discussion in the Sect. 5.2 for billiards with boundaries of general type). (E) Elliptic case (reff > 0). Let D ⊂ M be a disc such that ∂D is the circle whose geodesic curvature κ satisfies κ ≥ β. We define the component −D ⊂ M as the set of


β

π

45

+D

−1

π

π

+D

+D

−D

−D

−D

Fig. 5 a

Fig. 5 b

Fig. 5 c

the points m ∈ M satisfying the condition m m = π¯ for some point m ∈ D, where m m is the length of the particle trajectory between the points m, m . We will refer to −D as the dual component of D ≡ +D. Straightforward analysis shows that −D is the ring whose width equals the diameter of D and its radius is defined by ξ (for M = R2 its radius is β −1 ), see Figs. 5 a, b, c. When M = S2 and β = 0, −D is the disk obtained from D by reflection about the center of S2 , as it has been defined in [GSG]. Let us also introduce the terminology: If R ⊂ S ⊂ M are regions with piecewise C 1 boundaries, we call an inclusion R ⊂ S proper if ∂R ∩ int S = ∅. The application of Theorem 1 to the elementary billiard tables in the case reff > 0 leads to the following criterion for hyperbolicity. Corollary E. Let Q ⊂ M be an elementary billiard table whose boundary consists of N > 1 components of type plus or minus. Suppose Q satisfies the following conditions: Condition E1. For every component .i+ of ∂Q we have Di ⊂ Q. Besides, either −Di ⊂ Q, or −Di ⊂ M \ Q, where the inclusions are proper; Condition E2. For every component .j− we have Dj ⊂ M \ Q, and the inclusions −Dj ⊂ M \ Q, or −Dj ⊂ Q are proper. Then the billiard in Q is hyperbolic. Outline of proof. The assumptions of Corollary E imply those of Theorem 1. Remark. Suppose Q = M \ Q is connected. If Q satisfies Conditions E1 and E2, then Q also does, and hence the billiard in Q is hyperbolic. Examples. “Lorenz gas” billiards. Such billiards are obtained by removing from M a number of disjoint discs Di , so that Q = M \∪Di . If all the intersections Di ∩±Dj i = j , are empty, then the billiard in Q is hyperbolic by Corollary E. The simplest example of such hyperbolic billiard is obtained by removing two disks from the magnetic plane, see Fig. 6 a. The intersections Di ∩ ±Dj i = j , are always empty, if all the discs are contained inside of a free-flight particle trajectory (i.e., if all the discs lie inside a circle of geodesic curvature β). Such billiards are the “magnetic” analogs of the non-magnetic hyperbolic billiard tables on the sphere, obtained by removing a finite number of disjoint disks from one hemisphere [GSG]. The examples of hyperbolic billiards of this type on S2 , R2 and H2 are shown in Figs. 6 b, c, d. One can consider also unbounded billiard tables Q obtained by removing an infinite number of disjoint disks from R2 , H2 . The simplest example of this type is obtained by

46

B. Gutkin particle trajectory

β

1

−1

β

−D 1 +D 2

+D1

Q

Q

Fig. 6 a

Fig. 6 b

particle trajectory

Q

Q particle trajectory

Fig. 6 c

Fig. 6 d

...

removing a chain of equal disks from M = R2 , as shown in Fig. 7 a (this billiard can be also seen as cylinder with one hole). Because of the translation symmetry, one needs to check the non-intersection condition only for one disk. The non-intersection condition is also necessary for hyperbolicity of such billiards. If it is not satisfied, then Q has at least two stable g.t.p.t.s (see Fig. 7 a).

Q

β

...

Q ...

−1

−D

−D

...

β

Fig. 7 a

...

+D

...

+D

−1

Fig. 7 b


47

Q

Q’

β

−1

Q

Q’

Q

Q’

Fig. 8 a

Fig. 8 b

Fig. 8 c

Another type of unbounded hyperbolic billiard tables can be obtained by removing a lattice of the disks from M = R2 , H2 . The example of such a billiard shown in Fig. 7 b, is equivalent to the torus with one hole. Here, again, because of the translation symmetry, one has to check the non-intersection condition only for a single disk. “Flowers” like billiards. Consider a simply connected billiard table Q, whose boundary consists of several circular arcs of positive and negative curvature satisfying the condition |κi | ≥ β. Such billiards were originally introduced by Bunimovich [Bu1, Bu2] as examples of planar (non-magnetic) hyperbolic billiards with convex boundary. It has been demonstrated that for r = 0, β = 0 such billiards are hyperbolic if the conditions Di ⊆ Q are satisfied for each convex component of the boundary. For reff > 0 we have by Corollary E the additional requirement: ∂Q ∩ −Di = ∅ for each component of the boundary (compare with the analogous conditions in [GSG] for the case β = 0, M = S2 ). The examples of hyperbolic billiards Q on S2 , R2 , H2 of “flower” type satisfying the conditions of Corollary E are shown in Figs. 8 a, b, c. It follows from the remark above that billiards in the domain Q = M \ Q are also hyperbolic. (H+P) Hyperbolic and parabolic cases (reff ≤ 0). The criterion for hyperbolicity in this case is given by the following corollary. Corollary H. Let Q ⊂ M be an elementary billiard table, and let ∂Q consist of N > 1 components. If Q satisfies conditions: Condition H1. For every convex component .i+ of ∂Q, we have Di ⊂ Q; Condition H2. For every concave component of ∂Q, we have κ(.i− ) ≤ −β and for every convex component κ(.i+ ) ≥ ξ . Then the billiard in Q is hyperbolic. Outline of proof. The assumptions of Corollary H imply those of Theorem 1. Remark. When β → 0 and r → 0, condition H2 is automatically fulfilled and Corollary H turns to be the classical criterion of Bunimovich [Bu2] for hyperbolicity of planar, non-magnetic billiard tables. Examples. Analogs of Sinai billiards. The boundary of these billiards consists of concave arcs .i− of constant curvature (see Fig. 9). If the condition κ(.i− ) ≤ −β is satisfied for each component of the boundary, then the billiard is hyperbolic by Corollary H. Analogs of Bunimovich billiards. The example of hyperbolic billiard table with convex components satisfying the conditions H1, H2 is shown in Fig. 10.

48

B. Gutkin

Q

Fig. 9

Q

Fig. 10

Remark. The assumptions in Corollaries E and H that N > 1 and that the inclusions be proper are needed only to exclude certain degenerate situations, where each v ∈ V is B-parabolic. This is the case, for instance, if Q is a disc, or the annulus between concentric circles. 5.2. Hyperbolic billiard tables with boundary of general type. Let us consider billiard tables on M(r, β) with piecewise smooth boundary, ∂Q = ∪i γi of general type. The components γi are C 2 smooth curves parameterized by the arclength l, whose curvature κi (l) has the same sign along each γi . We will refer to γi as convex component if κi (l) > 0, or as concave component if κi (l) ≤ 0. Let us denote κ(γi ) = max{κi (l), l ∈ γi } for the convex components, and κ(γi ) = min{κi (l), l ∈ γi } for the concave components. Following the terminology in [Wo2], we introduce the class of convex scattering curves on M(r, β). Definition 2. A smooth convex curve γ ⊂ M is (strictly) convex scattering if for any v ∈ V , such that the origin points of v and φ(v) belong to γ , the corresponding g.t.p.t. T (v) is (strictly) B-unstable. A curve γ is convex scattering if one of the relevant conditions (21-25) is satisfied for each pair of points on γ . Regarding the planar non-magnetic case, this leads to the definition of Wojtkowski [Wo2] for convex scattering curve. Let us introduce the parameter R(l) = (κ(l) − β)−1 . Considering the infinitesimally close points on γ we show in the Appendix that the condition R (l) ≤ 0 is necessary for γ to be convex scattering. It should be noted, that this condition is also sufficient in the planar, nonmagnetic case (see [Wo2]), but not for generic parameters r, β (see [GSG] for β = 0 case). In what follows, we formulate the principles for design of hyperbolic billiards satisfying the conditions of Theorem 1. Let Q be a billiard table satisfying the conditions of Theorem 1. Then each convex component of ∂Q has to be convex scattering and consequently, the condition R ≤ 0 holds along each convex component of the boundary. There is an additional restriction on the curves γi which compose the boundary of Q. It follows from Proposition 2 that for billiards satisfying the conditions of Theorem 1 the sign of K(v) (d(v)) depends only on the origin point of v (there is no dependence on θ) for any v ∈ V , i.e., K(v) (d(v)) has the same sign along γi as κ(γi ). This happens if for each component γi , |κ(γi )| ≥ β (the magnetic field is sufficiently weak). Thus, in what follows we particularly exclude from our consideration the magnetic billiards with flat boundaries. Such billiards do not satisfy the conditions of Theorem 1.


β

−1

49

β

−1

Q

Q

Fig. 11 a

Fig. 11 b

Design of hyperbolic billiard tables in the (E) case. By Definition 2 a curve γ is convex scattering if it is convex and the condition d1 + d2 ≤ s ≤ π¯ ,

(26)

holds for any pair of points on γ . For simplicity of exposition, we will restrict our attention for M = R2 , H2 to the bounded billiard tables and for M = S2 to the billiard tables which can be placed in a hemisphere. Theorem 1 yields the following principles for the design of piecewise billiard tables with hyperbolic dynamics in (E) case: P1. |κ(γi )| ≥ β for all components. P2. All convex components of ∂Q are convex scattering. P3. Any convex component of ∂Q has to be “sufficiently far”, but not “too far”, from any other component. Any concave component has to be not “too far”, from any other concave component. The precise meaning of P3 is that the parameters of any two consecutive bouncing points, which belong to different components of the boundary, satisfy the condition (22). In particular it implies the set of restrictions on the angles between consecutive components of the boundary. It can be formulated as an additional principle. P4. Let γi , γi+1 ⊂ ∂Q be two adjacent components, meeting at a vertex. If both γi and γi+1 are convex, then the interior angle at the vertex is greater than π. If γi and γi+1 have different sign of curvature, then the angle in question is greater or equal to π . Another restriction which arises from P3 is that the length (equivalently the time) of free-flight between any two consequent bouncing points on the boundary of the billiard has to be not greater than π. ¯ In other words, the billiard table has to be “smaller” than the circle drawn by a free-flight particle on M(r, β). Examples. The examples of the hyperbolic billiards on R2 satisfying the above principles are shown in Fig. 11 a, b. A bounded Sinai-like billiard, whose boundary consists of (strictly) concave components (Fig. 11 a) always satisfies the principles P1–P4 for sufficiently weak magnetic field. The example of a convex billiard is shown in Fig. 11 b. It is a cardioid, whose boundary is strictly convex scattering curve for β = 0 (see [Wo2]). For β = 0 this billiard is hyperbolic, as it follows from Theorem 1. Since a strictly convex scattering curve remains to be such under small perturbations of β, the billiard in Fig. 11 b is hyperbolic for sufficiently weak magnetic field.

50

B. Gutkin

Design of hyperbolic billiard tables in the (P+H) case. Definition 2 leads to the following geometric conditions on the convex scattering curve in the (P+H) case. A convex curve γ is convex scattering if κ(γ ) ≥ ξ and for each pair of points on γ , d1 + d2 ≤ s.

(27)

Theorem 1 yields the following principles for the design of billiard tables with hyperbolic dynamics in the (P+H) case: P1. κ(γi ) ≥ ξ for any convex component of ∂Q and κ(γi ) ≤ −β for any concave component of ∂Q. P2. All convex components of ∂Q are convex scattering. P3. Any convex component of ∂Q is “sufficiently far” from any other component. More precisely, condition P3 means that any two consecutive bouncing points of the billiard ball, which belong to different components, satisfy Eqs. (23)–(25). In particular, this yields, the same inequalities (P4) as in (E) case, for the interior angles between consecutive components of ∂Q. Examples. In the (P+H) case, any concave billiard is hyperbolic if the condition κ(γi ) ≤ −β is fulfilled for each component of the boundary. As in the case (E), the examples of the convex hyperbolic billiards can be obtained from their non-magnetic counterparts satisfying the conditions of Theorem 1. Finally, it should be noted that the principles formulated above for design of hyperbolic billiards on M(r, β) are robust under small perturbations of β, r and the billiard wall. Generally, one can construct hyperbolic billiards on magnetic surfaces of constant curvature on the basis of the corresponding non-magnetic planar billiards satisfying Wojtkowski’s criterion. 6. Conclusions In the present paper we have formulated the criterion for hyperbolic dynamics in billiards on surfaces of constant Gaussian curvature r in the presence of a homogeneous magnetic field β perpendicular to the surface. The criterion is valid for all values of r, β and its geometric realization depends only on the type of linearized dynamics (elliptic, parabolic or hyperbolic). In this way we extend our recent results in [GSG] to the case of magnetic surfaces of constant curvature. The basic property, which allows unification of the hyperbolicity criteria for the magnetic and non-magnetic billiards on surfaces of constant curvature, is the equivalence between the geometric optics in both cases. In fact, in terms of special parameters di , si the geometric optics depend only on the effective curvature reff = r + β 2 of the surface. It is important to stress, that the dynamics in magnetic and non-magnetic billiards are very different (e.g., the magnetic field breaks time reve rsal symmetry). It is the only linearized dynamics, which are the same for the considered systems. Applying the hyperbolicity criterion, we were able to construct the different classes of hyperbolic billiards for each type of the linearized dynamics (equivalently for each of the signs of reff ). There are two types of necessary conditions which arise for hyperbolic billiards satisfying our criterion. The first one is a requirement for the convex components of the boundary to be convex scattering. As a consequence, the inequality R (l) ≤ 0 has


51

to be satisfied along each convex component. This inequality is a generalization of the well-known Wojtkowski condition [Wo2] for a convex component of the planar (nonmagnetic) hyperbolic billiard. It has been demonstrated for planar non-magnetic billiards in [Bu3, Bu4, Do] that Wojtkowski’s criterion can be considerably strengthened. This suggests, in particular, that condition R (l) ≤ 0 can be relaxed for general parameters r, β by employing invariant cone fields, different from the one used in the present paper (see the discussion in [GSG]). The second type of conditions is specific for magnetic billiards. This is a requirement of “weakness” for the magnetic field compared to the curvature of the billiard boundary. For generic systems, such a condition is expected, in order to prevent stable skipping orbits close to the boundary. It has been shown in [BR], (see also [BK]) that a billiard with sufficiently smooth boundary possesses invariant tori corresponding to skipping trajectories. It seems that in the strong field regime a part of stable periodic orbits has to survive even if the smoothness of the boundary is broken. It remains, however an open question, whether the condition |κi | ≥ β can be relaxed for generic billiard. The positive Lyapunov exponent for a billiard implies strong mixing properties: a countable number of ergodic components, positive entropy, Bernoulli property, etc. It should be pointed out, however, that ergodicity does not automatically follow from the positivity of Lyapunov exponent. Nevertheless, one can expect that billiards satisfying the conditions of Theorem 1 will be typically ergodic. It seems that the methods developed for the proof of ergodicity of planar hyperbolic billiards can be extended to the class of billiards considered in the present paper. Acknowledgement. The author is indebted to Professor U. Smilansky for proposing this investigation and for critically reading the manuscript. The author would like to thank Andrey Shapiro de Brosh for interesting and inspiring discussions, and various valuable remarks. This work was supported by the Minerva Center for Nonlinear Physics of Complex Systems.

7. Appendix We will investigate the conditions under which a convex arc on the surface of constant curvature M in the presence of magnetic field β is convex scattering. For simplicity of exposition, we consider the case, when M is a magnetic plane. Let γ (l) ⊂ M be any smooth curve parameterized by arclength l, and let κ(l) be the geodesic curvature of γ . Let now γ (l0 ) and γ (l1 ) be two points on γ , such that the arc of γ between γ (l0 ) and γ (l1 ) lies entirely on one side of a straight line passing through γ (l0 ) and γ (l1 ). We choose a cartezian coordinate system (x, y) in such a way that y(l0 ) = y(l1 ) = 0, x(l0 ) = −x(l1 ) and the arc of γ between γ (l0 ) and γ (l1 ) lies above x-axis, see Fig. 12. Let α(l) be the angle, which dγ dl makes with x-axis, then dx = cos α; dl

dy = sin α; dl

dα = −κ. dl

(28)

We introduce also an auxiliary variable δ, such that βx = sin δ. For β > 0 there are two different particle trajectories connecting the points γ (l0 ) and γ (l1 ) (resp. two different g.t.p.t.s corresponding to these points), see Fig. 12. Below, we consider the trajectory which lies in the lower halfplane. Then, the results for trajectory in the upper halfplane are obtained by the change of the sign of β to the opposite. Let θ = α + δ. Then, at the points l0,1 , θ(l0,1 ) are the angles between γ and the particle trajectory connecting γ (l0 ) and γ (l1 ). Set 7 = s − d1 − d2 . By Eq. (14) we get

52

B. Gutkin y

γ(l 0 ) θ

γ( l1 ) x

0 δ β−1

Fig. 12

β sin θ d arctan + dδ κ − β cos θ

cos α α −κ sin θ + κ κ + β sin sin δ cos δ − cos θ = dl κ 2 + β 2 − 2βκ cos θ

7 = β −1

(29)

We separate the last integral into the sum of two parts. The first one is

R −κ sin θ , = dy I = dl κ 2 + β 2 − 2βκ cos θ 1 + 4R 2 κβ sin2 θ/2 where R −1 (l, β) = κ(l) − β. Since y(l0 ) = y(l1 ) = 0, we obtain yR (4R 2 κβ sin2 θ2 ) yR − I = − dl 1 + 4R 2 κβ sin2 θ2 (1 + 4R 2 κβ sin2 θ2 )2 R L3 κ =− + O(L4 ), 12

(30)

where L = l1 −l0 is the length of the curve between the points γ (l0 ), γ (l1 ). Analogously, for the second part we have cos α α − cos θ κ κ + β sin sin δ cos δ II = dl κ 2 + β 2 − 2βκ cos θ

sin δ sin θ κ cos δ κ sin α + β = dy = O(L4 ). κ 2 + β 2 − 2βκ cos θ Adding both parts we obtain finally 7 = I + II = −

R L3 κ + O(L4 ). 12

(31)

Thus, if the curve γ is convex scattering, then the condition R (l, β) ≤ 0 holds everywhere on γ . Considering trajectories of the second type (i.e., trajectories which lay in the upper halfplane), we obtain the condition R (l, −β) ≤ 0 for convex scattering curves. However, it is easy to see that R (l, β) ≤ 0 actually implies R (l, −β) ≤ 0.


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Repeating the same analysis for general M(r, β) we have found (see also [GSG] for β = 0 case) that Eq. (31) holds for all surfaces of constant curvature. As a consequence, R ≤ 0 is a necessary condition for convex scattering on M(r, β). On the contrary, if the strict inequality R < 0 holds along γ , then by Eq. (31), any sufficiently small piece of γ is convex scattering. References [BK] [BR] [Bu1] [Bu2] [Bu3] [Bu4] [Do] [GSG] [K] [KSS] [Si] [Ta1] [Ta2] [Ta3] [Tab] [Vet1] [Vet2] [Vi] [Wo1] [Wo2]

Berglund, N., Kunz, H.: J. Stat. Phys. 83, 81–126 (1996) Berry, M.V., Robnik, M.: J. Phys. A Math. Gen. 18, 1361–1378 (1985) Bunimowich, L.A.: Mathem. Sbornik 95, 49–73 (1974) Bunimowich, L.A.: Commun. Math. Phys. 65, 295–312 (1979) Bunimowich, L.A.: Chaos 1 (2), 187 (1991) Bunimowich, L.A.: On absolutely focusing mirrors. Lecture Notes in Math. Vol. 1514, 1991, pp. 62–82 Donnay, V.J.: Commun. Math. Phys. 141, 225–257 (1991) Gutkin, B., Smilansky, U., Gutkin, E.: Commun. Math. Phys. 208, 65–90 (1999) Kovàcs, Z.: Phys. Rep. 290, 49–66 (1997) Kramli, A., Simanyi, N., Szasz, D.: Commun. Math. Phys. 125, 439–457 (1989) Sinai, Ya.G.: Russian Mathem. Surveys 25, 137–189 (1970) Tasnadi, T.: Commun. Math. Phys. 187, 597–621 (1997) Tasnadi, T.: J. Math. Phys. 39, 3783–3804 (1998) Tasnadi, T.: J. Math. Phys. 37, 5577–5598 (1996) Tabachnikov, S.: Billiards, Societe Mathematique de France, 1995 Vetier, A.: Sinai billiard in potential field (constraction of stable and unstable fibers). Coll. Math. Soc. J. Bolyai 36, 1079–1146 (1982) Vetier,A.: Sinai billiard in potential field (absolute continuity). In Proc. 3rd Pann. Symp. J. Mogyorody, I. Vincze, W. Wertz (eds.). Budapest: Hungarian Academy of Sciences, pp, 341–345, 1982 Vinberg, A.P.: Geometry 2, Encycl. of Math. Sc. Vol. 29. Berlin–Heidelberg–New York: Springer, 1993 Wojtkovski, M.: Erg. Theor. Dyn. Sys. 5, 145–161 (1985) Wojtkovski, M.: Commun. Math. Phys. 105, 391–414 (1986)

Communicated by Ya. G. Sinai

Commun. Math. Phys. 217, 55 – 87 (2001)

Communications in



Twisted Index Theory on Good Orbifolds, II: Fractional Quantum Numbers Matilde Marcolli1 , Varghese Mathai1,2 1 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA.

E-mail: [email protected]

2 Department of Mathematics, University of Adelaide, Adelaide 5005, Australia.

E-mail: [email protected] Received: 4 November 1999 / Accepted: 22 September 2000

Abstract: This paper uses techniques in noncommutative geometry as developed by Alain Connes [Co2], in order to study the twisted higher index theory of elliptic operators on orbifold covering spaces of compact good orbifolds, which are invariant under a projective action of the orbifold fundamental group, continuing our earlier work [MM]. We also compute the range of the higher cyclic traces on K-theory for cocompact Fuchsian groups, which is then applied to determine the range of values of the Connes–Kubo Hall conductance in the discrete model of the quantum Hall effect on the hyperbolic plane, generalizing earlier results in [Bel+E+S], [CHMM]. The new phenomenon that we observe in our case is that the Connes–Kubo Hall conductance has plateaux at integral multiples of a fractional valued topological invariant, namely the orbifold Euler characteristic. Moreover the set of possible fractions has been determined, and is compared with recently available experimental data. It is plausible that this might shed some light on the mathematical mechanism responsible for fractional quantum numbers. Introduction This paper uses techniques in noncommutative geometry as developed by Alain Connes [Co2] in order to prove a twisted higher index theorem for elliptic operators on orbifold covering spaces of compact good orbifolds, which are invariant under a projective action of the orbifold fundamental group. These higher indices are basically the evaluation of pairings of higher traces (which are cyclic cocycles arising from the orbifold fundamental group and the multiplier defining the projective action) with the index of the elliptic operator, considered as an element in the K-theory of some completion of the twisted group algebra of the orbifold fundamental group. This paper is the continuation of [MM] and generalizes the results there. The main purpose for studying the twisted higher index theorem on orbifolds is to highlight the fact that when the orbifold is not smooth, then the twisted higher index can be a fraction. In particular, we determine the range of the higher cyclic traces on K-theory for general cocompact Fuchsian groups. We adapt and

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M. Marcolli, V. Mathai

generalize the discrete model of the quantum Hall effect of Bellissard and his collaborators [Bel+E+S] and also [CHMM], to the case of general cocompact Fuchsian groups and orbifolds, which can be viewed equivalently as the generalization to the equivariant context. The new phenomenon that we observe in our case is that the Connes–Kubo Hall conductance has plateaux at integral multiples of a fractional valued topological invariant, namely the orbifold Euler characteristic. The presence of denominators is caused by the presence of cone points singularities and by the hyperbolic geometry on the complement of these cone points. The negative curvature of the hyperbolic structure replaces interaction and simulates, in our single electron model, the presence of Coulomb interactions. We also have a geometric term in the Hamiltonian (arising from the cone point singularities) which accounts partly for the effect of Coulomb interactions. This geometric model of interaction is fairly simple, hence the agreement of our fractions with the experimental values is only partial. Among the observed fractions, for instance, we can derive 5/3, 4/3, 4/5, 2/3, and 5/2 from genus one orbifolds, and 2/5, 1/3, 4/9, 4/7, 3/5, 5/7, 7/5 from genus zero orbifolds, see Sect. 5. However, fractions like 3/7 and 5/9, seem unobtainable in this model, even including higher genus orbifolds. Their explanation probably requires a more sophisticated term describing the electron interaction. It is not unreasonable to expect that this term may also be geometric in nature, but we leave it to future studies. There are currently several different models which describe the occurrence of fractional quantum numbers in the quantum Hall effect. Usually quantum field theoretic techniques are involved. Most notably, there is a sophisticated Chern–Simons theory model for the fractional quantum Hall effect developed by Frohlich and his collaborators, cf. [Froh]. Also within the quantum field theoretic formalism it can be noticed that possibly different models are needed in order to explain the occurrence of different sets of fractions. For example, the fraction 5/2 requires by itself a separate model. After reviewing some preliminary material in Sect. 1, we establish in Sect. 2 a twisted higher index theorem which adapts the proofs of the index theorems of Atiyah [At], Singer [Si], Connes and Moscovici [CM], and Gromov [Gr2], [Ma1], to the case of good orbifolds, that is, orbifolds whose orbifold universal cover is a smooth manifold. This theorem generalizes the twisted index theorem for 0-traces of [MM] to the case of higher degree cyclic traces. The result can be summarized as follows. Let R be the algebra of rapidly decreasing sequences, i.e. R = (ai )i∈N : sup i k |ai | < ∞ ∀ k ∈ N . i∈N

Let be a discrete group and σ be a multiplier on . Let C(, σ ) denote the twisted group →M algebra. We denote the tensor product C(, σ ) ⊗ R by R(, σ ). Let → M is a smooth denote the universal orbifold cover of a compact good orbifold M, so that M manifold. Suppose given a multiplier σ on and assume that there is a projective By considering (, σ¯ )-action on L2 sections of -invariant vector bundles over M. (, σ¯ )-invariant elliptic operators D acting on L2 sections of these bundles, we will define a (, σ )-index element in K-theory Indσ (D) ∈ K0 (R(, σ )). We will compute the pairing of Indσ (D) with higher traces. More precisely, given a normalized group cocycle c ∈ Z k (, C), we define a cyclic cocycle tr c ∈ ZC k (C(, σ ))

Twisted Higher Index Theory on Good Orbifolds

57

of dimension k on the twisted group algebra C(, σ ), which extends continuously to a k-dimensional cyclic cocycle on R(, σ ). This induces a map on K-theory, [tr c ] : K0 (R(, σ )) → C. A main theorem established in this paper is a cohomological formula for Ind(c,,σ ) (D) = [tr c ] (Indσ (D)) . Our method consists of applying the Connes–Moscovici local higher index theorem to all of which represent a family of idempotents constructed from the heat operator on M, the (, σ )-index. Let be a Fuchsian group of signature (g; ν1 , . . . , νn ), that is, is the orbifold fundamental group of the 2 dimensional hyperbolic orbifold (g; ν1 , . . . , νn ) of signature (g; ν1 , . . . , νn ), where g ∈ Z, g ≥ 0 denotes the genus and 2π/νj , νj ∈ N denotes the cone angles at the cone points of the orbifold. In [MM] we computed the K-theory of the twisted group C ∗ algebra. Under the assumption that the Dixmier–Douady invariant of the multiplier σ is trivial, we obtained  n Z2−n+ i=1 νi if j = 0; ∗ Kj (C (, σ )) ∼ = Z2g if j = 1. Here we use a result of [Ji], which is a twisted analogue of a result of Jollissant and which says in particular that, when is a cocompact Fuchsian group, then the natural inclusion map j : R(, σ ) → C ∗ (, σ ) induces an isomorphism in K-theory K• (R(, σ )) ∼ = K• (C ∗ (, σ )). Using this, together with our twisted higher index theorem for good orbifolds and some results in [MM], and under the same assumptions as before, we determine, in Sect. 3, the range of the higher trace on K-theory [tr c ](K0 (C ∗ (, σ ))) = φ Z, where −φ = 2(1 − g) + (ν − n) ∈ Q is the orbifold Euler characteristic of n (g; ν1 , . . . , νn ). Here we have ν = j =1 1/νj and c is the area 2-cocycle on , i.e. c is the restriction to of the area 2-cocycle on P SL(2, R). In Sect. 4 we study the hyperbolic Chern–Simons formula for the Hall conductance in the discrete model of the quantum Hall effect on the hyperbolic plane, where we consider Cayley graphs of Fuchsian groups which may have torsion subgroups. This generalizes the results in [CHMM] where only torsion-free Fuchsian groups were considered. We recall that the results in [CHMM] generalized to hyperbolic space the noncommutative geometry approach to the Euclidean quantum Hall effect that was pioneered by Bellissard and collaborators [Bel+E+S], Connes [Co] and Xia [Xia]. We first relate the hyperbolic Connes–Kubo Hall conductance cyclic 2-cocycle and the area cyclic 2-cocycle on the algebra R(, σ ), and show that they define the same class in cyclic cohomology. Then we use our theorem on the range of the higher trace on K-theory to determine the range of values of the Connes–Kubo Hall conductance cocycle in the Quantum Hall Effect. The new phenomenon that we observe in this case is that the Hall conductance has plateaux at all energy levels belonging to any gap in the spectrum of the Hamiltonian (known as the generalized Harper operator), where it is now shown to be equal to an integral multiple

58


of a fractional valued topological invariant φ, which is the negative of the orbifold Euler characteristic of the good orbifold (g; ν1 , . . . , νn ). If we fix the genus g , then the set of possible denominators is finite by the Hurwitz theorem [Sc], and has been explicitly determined in the low genus cases [Bro]. This provides a topological explanation of the appearance of fractional quantum numbers. In the last section we compare our results with some observed values. In Sect. 5, we provide lists of specific examples of good 2-dimensional orbifolds for which φ is not an integer. First we observe how the presence of both the hyperbolic structure and the cone points is essential in order to have fractional quantum numbers. In fact, φ is an integer whenever the hyperbolic orbifold is smooth, i.e. whenever 1 = ν1 = . . . = νn , which is the case considered in [CHMM]. Similarly, by direct inspection, it is possible to see that all Euclidean orbifolds also produce only integer values of φ. We use the class of orbifolds which are spheres or tori with cone points, having a (singular) hyperbolic structure, to represent in our physical model some of the fractions observed in the FQHE. We also list the examples arising from quotients of low genus surfaces [Bro], and we discuss some phenomenology on the role of the orbifold points and of the minimal genus g of the covering surface. Summarizing, one key advantage of our model is that the fractions we get are obtained from an equivariant index theorem and are thus topological in nature. Consequently, as pointed out in [Bel+E+S], the Hall conductance is seen to be stable under small deformations of the Hamiltonian. Thus, this model can be easily generalized to systems with disorder as in [CHM]. This is a necessary step in order to establish the presence of plateaux [Bel+E+S]. The main limitation of our model is that there is a small number of experimental fractions that we do not obtain in our model, and we also derive other fractions which do not seem to correspond to experimentally observed values. To our knowledge, however, this is also a limitation occuring in the other models available in the literature. 1. Preliminaries Recall that, if H denotes the hyperbolic plane and is a Fuchsian group of signature (g; ν1 , . . . , νn ), that is, is a discrete cocompact subgroup of P SL(2, R) of genus g and with n elliptic elements of order ν1 , . . . , νn respectively, then the corresponding compact oriented hyperbolic 2-orbifold of signature (g; ν1 , . . . , νn ) is defined as the quotient space (g; ν1 , . . . , νn ) = \H, where g denotes the genus and 2π/νj , νj ∈ N denotes the cone angles at the cone points of the orbifold. A compact oriented 2-dimensional Euclidean orbifold is obtained in a similar manner, but with H replaced by R2 . All Euclidean and hyperbolic 2-dimensional orbifolds (g; ν1 , . . . , νn ) are good, being in fact orbifold covered by a smooth surface g cf. [Sc], i.e. there is a finite group G acting on g with quotient (g; ν1 , . . . , νn ), where g = 1+ #(G) 2 (2(g−1)+(n−ν)) and where ν = nj=1 1/νj . For fundamental material on orbifolds, see [Sc], [FuSt] and [Bro]. See also [MM], Sect. 1. Let M be a good, compact orbifold, and E → M be an orbifold vector bundle over be its lift to the universal orbifold covering space → M → M, M, and E → M which is by assumption a simply-connected smooth manifold. We have a (, σ¯ )-action


59

E),where (where σ is a multiplier on and σ¯ denotes its complex conjugate) on L2 (M, such that ω is also -invariant, although η is we choose ω = dη an exact 2-form on M not assumed to be -invariant, and the Hermitian connection ∇ = d + iη with curvature ∇ 2 = iω. The projective action is on the trivial line bundle over M, defined as follows: ω− ω = d(γ ∗ η − η) ∀ γ ∈ . Firstly, observe that since ω is -invariant, 0 = γ ∗ ∗ therefore So γ η − η is a closed 1-form on the simply connected manifold M, γ ∗ η − η = dφγ

∀ γ ∈ ,

satisfying in addition, where φγ is a smooth function on M ∀ γ , γ ∈ ; • φγ (x) + φγ (γ x) − φγ γ (x) is independent of x ∈ M • φγ (x0 ) = 0 for some x0 ∈ M ∀ γ ∈ . Then σ¯ (γ , γ ) = exp(−iφγ (γ · x0 )) defines a multiplier on i.e. σ¯ : × → U (1) satisfies the following identity for all γ , γ , γ ∈ σ¯ (γ , γ )σ¯ (γ , γ γ ) = σ¯ (γ γ , γ )σ¯ (γ , γ ). let Sγ u = eiφγ u and Uγ u = (γ −1 )∗ u and Tγ = Uγ ◦ Sγ be the E), For u ∈ L2 (M, composition. Then T defines a projective (, σ¯ )-action on L2 -spinors, i.e. Tγ Tγ = σ¯ (γ , γ )Tγ γ . This defines a (, σ¯ )-action, provided that the Dixmier–Douady invariant δ(σ ) = 0, see [MM]. As in [MM], we shall consider the twisted group von Neumann algebra W ∗ (, σ ), the commutant of the left σ¯ -regular representation on (2 () and W ∗ (σ ) as the commutant E). of the (, σ¯ )-action on L2 (M, We have an identification (see [MM]) F )), W ∗ (σ ) ∼ = W ∗ (, σ ) ⊗ B(L2 (F, E| F )) denotes the algebra of all bounded operators on the Hilbert space where B(L2 (F, E| F ), and F is a relatively compact fundamental domain in M for the action of L2 (F, E| . We have a semifinite trace tr : W ∗ (σ ) → C defined as in the untwisted case due to Atiyah [At], Q→ tr(kQ (x, x))dx, M

where kQ denotes the Schwartz kernel of Q. Note that this trace is finite whenever kQ × M. is continuous in a neighborhood of the diagonal in M We also consider, as in [MM], the subalgebra C ∗ (σ ) of W ∗ (σ ), whose elements have the additional property of some off-diagonal decay, and one also has the identification (cf. [MM]) F )). C ∗ (σ ) ∼ = C ∗ (, σ ) ⊗ K(L2 (F, E|

60


In [MM] we considered the C ∗ algebra C ∗ (M) = C(P ) SO(m), where P is the bundle of oriented frames on the orbifold tangent bundle. The relevent K-theory is the orbifold K-theory 0 0 Korb (M) ≡ K0 (C ∗ (M)) = K0 (C(P ) SO(m)) ∼ (P ). = KSO(m)

In the case when M is a good orbifold, one can show that the C ∗ algebras C ∗ (M) and C0 (X) G are strongly Morita equivalent, where X is smooth and G → X → M is an orbifold cover. In particular, 0 0 (M) ∼ (X). Korb = K 0 (C0 (X) G) = KG j

The relevant cohomology is the orbifold cohomology Horb (M) = H j (X, G), for j = 0, 1, which is the delocalized equivariant cohomology for a finite group action on a smooth manifold [BC]. The Baum–Connes equivariant Chern character is a homomorphism 0 (X) → H 0 (X, G), chG : KG that is, a homomorphism 0 0 (M) → Horb (M). ch : Korb

Let B = \E be the classifying space of proper actions, as defined in [BCH]. In our case, the orbifold (g; ν1 , . . . , νn ), viewed as the quotient space \H, is B(g; ν1 , . . . , νn ). Equivalently, B(g; ν1 , . . . , νn ) can be viewed as the classifying space of the orbifold fundamental group (g; ν1 , . . . , νn ). Let S denote the set of all elements of which are of finite order. Then S is not empty, since 1 ∈ S. acts on S by conjugation, and let F denote the associated permutation module over C, i.e.

F = λα [α] λα ∈ C and λα = 0 except for a finite number of α . α∈S

Let C k (, F ) denote the space of all antisymmetric F -valued -maps on k+1 , where acts on k+1 via the diagonal action. The coboundary map is ∂c(g0 , . . . , gk+1 ) =

k+1

(−1)i c(g0 , . . . , gˆ i . . . gk+1 )

i=0

for all c ∈ C k (, F ) and where gˆ i means that gi is omitted. The cohomology of this complex is the group cohomology of with coefficients in F , H k (, F ), cf. [BCH]. They also show that H k (, F ) ∼ = H j (, C) ⊕m H k (Z(Cm ), C), where S = {1, Cm |m = 1, . . . } and the isomorphism is canonical. Also, for any Borel measurable -map µ : E → , there is an induced map on cochains µ∗ : C k (, F ) → C k (E, ) which induces an isomorphism on cohomology, µ∗ : H k (, F ) ∼ = H k (E, ) [BCH]. Here H j (E, ) denotes the Z-graded (delocalised) equivariant cohomology of E,


61

which is a refinement of what was discussed earlier, and which is defined in [BCH] using sheaves (and cosheaves), but we will not recall the definition here. Let M be a good orbifold with orbifold fundamental group . We have seen that is classified by a continuous map f : M → B, or the universal orbifold cover M → E. The induced map is f ∗ : H j (B, C) ≡ equivalently by a -map f : M orb ) ≡ H k (M, C) and therefore in particular one has f ∗ ([c]) ∈ H k (E, ) → H k (M, orb k (M, C) for all [c] ∈ H k (, C). This can be expressed on the level of cochains by Horb easily modifying the procedure in [CM], and we refer to [CM] for further details. Finally, we add here a brief comment on the assumption used throughout [MM] on the vanishing of the Dixmier–Douady invariant of the multiplier σ . We show here that the condition is indeed necessary, since we can always find examples where δ(σ ) = 0. Let be the Fuchsian group of signature (g; ν1 , . . . , νn ), as before. Consider the long exact sequence of the change of coefficient groups, as in [MM], e∗2π

i∗

δ

√ −1

· · · H 1 (, U (1)) → H 2 (, Z) → H 2 (, R) → δ

(1)

H 2 (, U (1)) → H 3 (, Z) → H 3 (, R). The argument of [MM] shows that H 3 (, R) = 0 and H 2 (, R) = R. Moreover, we observe that H 1 (, Z) = Hom(, Z) ∼ = Z2g , H 1 (, R) = Hom(, R) ∼ = R2g and n 1 2g 2 H (, U (1)) = Hom(, U (1)) ∼ = U (1) ×j =1 Zνj . Now H (, Z) = Z ⊕j Zνj , see [Patt], which is consistent with the result in [MM] that the group of the orbifold line bundles over the orbifold \H has 1 − n + nj=1 νj generators. It is also proved in [Patt] that H 2 (, U (1)) = U (1) ×nj=1 Zνj . Using the long exact sequence and the remarks above, we see that H 3 (, Z) = Tor(H 2 (, U (1))) = ×nj=1 Zνj . Thus, in the sequence √

√

we have Ker(i∗ ) = ⊕j Zνj , Im(i∗ ) = Z = Ker(e∗2π −1 ), Im(e∗2π −1 ) = U (1). So we can identify all the classes of multipliers with trivial Dixmier–Douady invariant with U (1) = Ker(δ). Finally, we have Im(δ) = H 3 (, Z) = H 2 (, U (1))/Ker(δ) = ⊕j Zνj . The calculations of the cohomology of the Fuchsian group = (g; ν1 , . . . , νn ) are summarized in Table 1.

Table 1. j

H j (, Z)

H j (, R)

H j (, U(1))

0

Z

R

U(1)

1

Z2g

R2g

U(1)2g ⊕j Zνj

2

Z ⊕j Zνj

R

U(1) ⊕j Zνj

3

⊕j Zνj

0

62


2. Twisted Higher Index Theorem In this section, we will define the higher twisted index of an elliptic operator on a good orbifold, and establish a cohomological formula for any cyclic trace arising from a group cocycle, and which is applied to the twisted higher index. We adapt the strategy and proof in [CM] to our context. 2.1. Construction of the parametrix and the index map. Let M be a compact, good → M is a smooth manifold and we will orbifold, that is, the universal cover → M given by Tγ = Uγ ◦Sγ ∀ γ ∈ . assume, as before, that there is a (, σ¯ )-action on L2 (M) F be Hermitian vector bundles on M and let E, F be the corresponding lifts to Let E, Then there are induced (, σ¯ )-actions on -invariants Hermitian vector bundles on M. and L2 (M, which are also given by Tγ = Uγ ◦ Sγ ∀ γ ∈ . E) F) L2 (M, → L2 (M, be a first order (, σ¯ )-invariant elliptic operator. E) F) Now let D : L2 (M, be an open subset that contains the closure of a fundamental domain for the Let U ⊂ M Let ψ ∈ Cc∞ (M) be a compactly supported smooth function such that -action on M. supp(ψ) ⊂ U , and

γ ∗ ψ = 1. γ ∈

be a compactly supported smooth function such that φ = 1 on supp(ψ). Let φ ∈ Cc∞ (M) Since D is elliptic, we can construct a parametrix J for it on the open set U by standard methods, J Du = u − H u

U ), ∀ u ∈ Cc∞ (U, E|

where H has a smooth Schwartz kernel. Define the pseudodifferential operator Q as

Tγ φJ ψTγ∗ . (2) Q= γ ∈

We compute, QDw =

γ ∈

since Tγ D = DTγ

Tγ φJ ψDTγ∗ w

E), ∀ w ∈ Cc∞ (M,

∀ γ ∈ . Since D is a first order operator, one has D(ψw) = ψDw + (Dψ)w

so that (3) becomes =

γ ∈

Tγ φJ DψTγ∗ w −

γ ∈

Tγ φJ (Dψ)Tγ∗ w.

Using (2), the expression above becomes

= Tγ ψTγ∗ w − Tγ φH ψTγ∗ w − Tγ φJ (Dψ)Tγ∗ w. γ ∈

Therefore (3) becomes

γ ∈

QD = I − R0 ,

γ ∈

(3)


where

R0 =

γ ∈

63

Tγ (φH ψ + J (Dψ)) Tγ∗

has a smooth Schwartz kernel. It is clear from the definition that one has Tγ Q = QTγ and Tγ R0 = R0 Tγ ∀ γ ∈ . Define R1 =t R0 + DR0t Q − DQ(t R0 ). Then Tγ R1 = R1 Tγ

∀ γ ∈ , R1 has a smooth Schwartz kernel and satisfies DQ = I − R1 .

Summarizing, we have obtained the following → M be the universal Proposition 2.1. Let M be a compact, good orbifold and → M F be orbifold covering space. Let E, F be Hermitian vector bundles on M and let E, We will assume the corresponding lifts to -invariants Hermitian vector bundles on M. given by Tγ = Uγ ◦ Sγ ∀ γ ∈ , as before that there is a (, σ¯ )-action on L2 (M) and L2 (M, which are also given by Tγ = E) F) and induced (, σ¯ )-actions on L2 (M, Uγ ◦ Sγ ∀ γ ∈ . → L2 (M, be a first order (, σ¯ )-invariant elliptic oper E) F) Now let D : L2 (M, ator. Then there is an almost local, (, σ¯ )-invariant elliptic pseudodifferential operator Q and (, σ¯ )-invariant smoothing operators R0 , R1 which satisfy QD = I − R0

Define the idempotent

e(D) =

and

DQ = I − R1 .

R02 (R0 + R02 )Q . R1 D 1 − R12

Then e(D) ∈ M2 (R(, σ )), where R(, σ ) = C(, σ ) ⊗ R is as defined in Sect. 1. The R(, σ )-index is by fiat Indσ (D) = [e(D)] − [E0 ] ∈ K0 (R(, σ )), where E0 is the idempotent

E0 =

00 . 01

It is not difficult to see that Indσ (D) is independent of the choice of (, σ¯ )-invariant parametrix Q that is needed in its definition. Let j : R(, σ ) → Cr∗ (, σ ) be the canonical inclusion, which induces the morphism in K-theory j∗ : K• (R(, σ )) → K• (Cr∗ (, σ )). Then we have E) Definition. The Cr∗ (, σ )-index of a (, σ¯ )-invariant elliptic operator D : L2 (M, 2 F) is defined as → L (M, Ind(,σ ) (D) = j∗ (Indσ (D)) ∈ K0 (Cr∗ (, σ )).

64


2.2. Heat kernels and the index map. Given D as before, for t > 0, we use the stan∗ dard off-diagonal estimates for the heat kernel. Recall that the heat kernels e−tD D ∗ −tDD and e are elements in the R(, σ ) (see the appendix). Define the idempotent et (D) ∈ M2 (R(, σ )) (see the Appendix) as follows −tD ∗ D ∗ ∗ e−tD D e−t/2D D (1−eD ∗ D ) D ∗ . et (D) = ∗ ∗ e−t/2DD D 1 − e−tDD It is sometimes known as the Wasserman idempotent. The relationship with the idempotent e(D) constructed earlier can be explained as follows. Define for t > 0, ∗ 1 − e−t/2D D ∗ D . Qt = D∗D ∗

Then one easily verifies that Qt D = 1 − e−t/2D D = 1 − R0 (t) and DQt = 1 − ∗ e−t/2DD = 1 − R1 (t). That is, Qt is a parametrix for D for all t > 0. Therefore one can write

R0 (t)2 (R0 (t) + R0 (t)2 )Qt et (D) = . R1 (t)D 1 − R1 (t)2 In particular, one has for t > 0 Indσ (D) = [et (D)] − [E0 ] ∈ K0 (R(, σ )). M,

We use the same notation as in [MM]. A first order elliptic differential operator D on D : L2 (M, E) → L2 (M, F)

on the smooth manifold is by fiat a -equivariant first order elliptic differential operator D M, → L2 (M, : L2 (M, E) F). D which is compatible with the action and the Hermitian Given any connection ∇ W on W to act on sections of E ⊗ W , metric, we define an extension of the elliptic operator D, F ⊗ W, ⊗ W ⊗ ∇W ) → (M, F ) D : (M, E ⊗ W

as in [MM]. 2.3. Group cocycles and cyclic cocycles. Using the pairing theory of cyclic cohomology and K-theory, due to [Co], we will pair the (, σ )-index of a (, σ¯ )-invariant elliptic with certain cyclic cocycles on R(, σ ). The cyclic cocycles that we operator D on M consider come from normalised group cocycles on . More precisely, given a normalized group cocycle c ∈ Z k (, C), for k = 0, . . . , dim M, we define a cyclic cocycle tr c of dimension k on the twisted group ring C(, σ ), which is given by a0 . . . ak c(g1 , . . . , gk ) tr(δg0 δg1 . . . δgk ) if g0 . . . gk = 1, tr c (a0 δg0 , . . . , ak δgk ) = 0 otherwise,


65

where aj ∈ C for j = 0, 1, . . . , k. To see that this is a cyclic cocycle on C(, σ ), we first define, as done in [Ji], the twisted differential graded algebra =• (, σ ) as the differential graded algebra of finite linear combinations of symbols g0 dg1 . . . dgn

gi ∈

with module structure and differential given by (g0 dg1 . . . dgn )g =

n

(−1)n−1 σ (gj , gj +1 )g0 dg1 . . . d(gj gj +1 ) . . . dgn dg

j =1

+ (−1)n σ (gn , g)g0 dg1 . . . d(gn g) d(g0 dg1 . . . dgn ) = dg0 dg1 . . . dgn . We now recall normalised group cocycles. A group k-cocycle is a map h : k+1 → C satisfying the identities h(gg0 , . . . ggk ) = h(g0 , . . . gk ) 0=

k+1

(−1)i h(g0 , . . . , gi−1 , gi+1 . . . , gk+1 ).

i=0

Then a normalised group k-cocycle c that is associated to such an h is given by c(g1 , . . . , gk ) = h(1, g1 , g1 g2 , . . . , g1 . . . gk ) and it is defined to be zero if either gi = 1 or if g1 . . . gk = 1. Any normalised group cocycle c ∈ Z k (, C) determines a k-dimensional cycle via the following closed graded trace on =• (, σ ) c(g1 , . . . , gk ) tr(δg0 δg1 . . . δgk ) if n = k and g0 . . . gk = 1, g0 dg1 . . . dgn = 0 otherwise. Of particular interest is the case when k = 2, when the formula above reduces to c(g1 , g2 )σ (g1 , g2 ) if g0 g1 g2 = 1; g0 dg1 dg2 = 0 otherwise. The higher cyclic trace tr c is by fiat this closed graded trace. 2.4. Twisted higher index theorem – the cyclic cohomology version. Let M be a compact p → orbifold of dimension n = 4(. Let → M M be the universal cover of M and the orbifold fundamental group is . Let D be an elliptic first order operator on M and D be the lift of D to M, → L2 (M, E) F). : L2 (M, D commutes with the -action on M. Note that D Define Now let ω be a closed 2-form on M such that ω = p∗ ω = dη is exact on M. and the ∇ = d + iη. Then ∇ is a Hermitian connection on the trivial line bundle over M,

66


curvature of ∇, (∇)2 = i ω. Then ∇ defines projective (, σ¯ )-actions on L2 sections as in Sect. 1. over M Consider the twisted elliptic operator on M, → L2 (M, E) F). ⊗ ∇ : L2 (M, D ⊗ ∇ no longer commutes with , but it does commute with the projective (, σ¯ ) Then D action. In Sect. 2.1, we have defined the higher index of such an operator, ⊗ ∇) ∈ K0 (R(, σ )). Indσ (D Given a group cocycle c ∈ Z 2q (), one can define the associated cyclic cocycle τc on R(, σ ) as in Sect. 2.3. Then τc induces a homomorphism on K-theory [τc ] : K0 (R(, σ )) → R . The real valued higher index is the image of the higher index under this homomorphism, i.e. ⊗ ∇) = [τc ](Indσ (D ⊗ ∇)). Ind(c,,σ ) (D To introduce the next theorem, we will briefly review some material on characteristic classes for orbifold vector bundles. Let M be a good orbifold, that is the universal orbifold → M of M is a smooth manifold. Then the orbifold tangent bundle T M cover → M on M. Similar comments apply to of M can be viewed as the -equivariant bundle T M ∗ the orbifold cotangent bundle T M and, more generally, to any orbifold vector bundle on M. It is then clear that, choosing -invariant connections on the -invariant vector one can define the Chern–Weil representatives of the characteristic classes bundles on M, These characteristic classes are -invariant and of the -invariant vector bundles on M. so define cohomology classes on M. For further details, see [Kaw]. Theorem 2.2. Let M be a compact, even dimensional, good orbifold, and let be be a first order, -invariant elliptic differential its orbifold fundamental group. Let D 2 where → M → operator acting on L sections of -invariant vector bundles on M, M is the universal orbifold cover of M. Let ω be a closed 2-form on M such that Define ∇ = d + iη, which is a Hermitian connection ω = p ∗ ω = dη is exact on M. whose curvature is (∇)2 = i ω. Recall that ∇ defines on the trivial line bundle over M 2 as in Sect. 1. Then, for any group cocycle projective (, σ¯ )-actions on L sections over M c ∈ Z 2q (), one has ⊗ ∇) = Ind(c,,σ ) (D

q! (2πi)q (2q!) T d(M) ∪ ch(symb (D)) ∪ p ∗ f ∗ (φc ) ∪ p ∗ eω , [T ∗ M] ,

(4)

where T d(M) denotes the Todd characteristic class of the complexified orbifold tangent bundle of M which is pulled back to the orbifold cotangent bundle T ∗ M, ch(symb(D)) is the Chern character of the symbol of the operator D, φc is the Alexander-Spanier cocycle on B that corresponds to the group cocycle c, f : M → B is the map that → M, cf. Sect. 1 and p : T ∗ M → M is the classifies the orbifold universal cover M projection.


67

Proof. Choose a bounded, almost everywhere smooth Borel cross-section β : M → M, which can then be used to define the Alexander–Spanier cocycle φc corresponding to ), cf. Sect. 1. As in Sect. 2.2, for c ∈ Z 2q (), and such that [φc ] = f ∗ [c] ∈ H 2q (M, t > 0, there is an index idempotent,

R0 (t)2 (R0 (t) + R0 (t)2 )Qt et (D) = ∈ M2 (R(, σ )), R1 (t)D 1 − R1 (t)2 where for t > 0, ∗ 1 − e−t/2D D Qt = D∗ D∗D

, R0 (t) = e−t/2D

∗D

,

∗

R1 (t) = e−t/2DD .

Then as in Sect. 2.2, one sees that R0 (t), R1 (t) are smoothing operators and Qt is a parametrix for D for all t > 0. The R(, σ )-index map is then ⊗ ∇) = [et (D ⊗ ∇)] − [E0 ] ∈ K0 (R(, σ )), Indσ (D where E0 is the idempotent

0 0 . E0 = 0 1

⊗ ∇) − E0 . Then one has Let Rt = et (D ⊗ ∇) = lim tr c (Rt , Rt , . . . Rt ). Ind(c,,σ ) (D t→0

(5)

One can directly adapt the strategy and proof in [CM] to our situation to deduce that ⊗ ∇) = lim Ind(c,,σ ) (D φc (x0 , x1 , . . . x2q )tr(Rt (x0 , x1 ) . . . t→0 M 2q+1

Rt (x2q , x0 ))dx0 dx1 . . . dx2q ,

The where we have identified M with a fundamental domain for the action on M. proof is completed by applying the local higher index Theorems 3.7 and 3.9 in [CM], ⊗ ∇). # to obtain the desired cohomological formula (4) for Ind(c,,σ ) (D $ 3. Twisted Kasparov Map and Range of the Higher Trace on K-Theory In this section, we compute the range of the 2-trace tr c on K-theory of the twisted group C ∗ algebra, where c is a 2-cocycle on the group, generalising the work of [CHMM]. Suppose as before that is a discrete, cocompact subgroup of P SL(2, R) of signature (g; ν1 , . . . , νn ). That is, is the orbifold fundamental group of a compact hyperbolic orbifold (g; ν1 , . . . , νn ) of signature (g; ν1 , . . . , νn ). Then for any multiplier σ on such that δ(σ ) = 0, one has the twisted Kasparov isomorphism, • µσ : Korb ((g; ν1 , . . . , νn )) → K• (Cr∗ (, σ )),

Proposition 2.14 in [MM]. Its construction is recalled in this section, as we need to refine it by factoring it through the K-theory of the dense subalgebra R(, σ ) of Cr∗ (, σ ). This is necessary in order to be able to use the pairing theory of Connes [Co], [CM] between higher cyclic traces and K-theory. We note that using a result of [Ji], that R(, σ ) is

68


indeed a dense subalgebra of Cr∗ (, σ ) in our case. In particular, given any projection P in Cr∗ (, σ ) there is both a projection P˜ in the same K0 class but lying in the dense subalgebra R(, σ ). This fact will also be utilized in the next section. On the other hand, by the results of the current section, given any such projection P there is a higher topological index that we can associate to it cf. Theorem 3.3. The main result we prove here is that the range of the 2-trace tr c on K-theory of the twisted group C ∗ algebra is always an integer multiple of a rational number. This will enable us to compute the range of values of the Hall conductance in the quantum Hall effect on hyperbolic space, generalizing the results in [CHMM]. 3.1. Twisted Kasparov map. Let be as before, that is, is the orbifold fundamental group of the hyperbolic orbifold (g; ν1 , . . . , νn ). Then for any multiplier σ on , we will factor the twisted Kasparov isomorphism, • ((g; ν1 , . . . , νn )) → K• (Cr∗ (, σ )) µσ : Korb

(6)

in [MM] through the K-theory of the dense subalgebra R(, σ ) of Cr∗ (, σ ). Let E → (g; ν1 , . . . , νn ) be an orbifold vector bundle over (g; ν1 , . . . , νn ) defining an element [E] in K 0 ((g; ν1 , . . . , νn )). As in [Kaw], one can form the twisted Dirac operator ∂E+ : L2 ((g; ν1 , . . . , νn ), S + ⊗ E) → L2 ((g; ν1 , . . . , νn ), S − ⊗ E), where S ± denote the 21 spinor bundles over (g; ν1 , . . . , νn ). One can lift the twisted (g; ν1 , . . . , νn ), ∂E+ on H = Dirac operator ∂E+ as above, to a -invariant operator which is the universal orbifold cover of (g; ν1 , . . . , νn ), + ⊗ E) → L2 (H, S − ⊗ E). ∂E+ : L2 (H, S

For any multiplier σ of with √δ([σ ]) = 0, there is a R-valued 2-cocycle ζ on with [ζ ] ∈ H 2 (, R) such that [e2π −1ζ ] = [σ ]. By the argument of [MM], Sect. 2.2, we know that we have an isomorphism H 2 (, R) ∼ = H 2 (g , R), and therefore there is a √ 2π −1ω 2-form ω on g such that [e ] = [σ ]. Of course, the choice of ω is not unique, but this will not affect the results that we are concerned with. Let ω denote the lift of ω to the universal cover H. Since the hyperbolic plane H is contractible, it follows that ω = dη, where η is a 1-form on H which is not in general invariant. Now ∇ = d +iη is a Hermitian connection on the trivial complex line bundle on H. Note that the curvature + ˜ Consider now the twisted Dirac operator ∂ E which is twisted again of ∇ is ∇ 2 = i ω. by the connection ∇, + ⊗ E) → L2 (H, S − ⊗ E). ∂E+ ⊗ ∇ : L2 (H, S

It does not commute with the action, but it does commute with the projective (, σ¯ )action which is defined by the connection ∇ as in Sect. 1. In Sect. 2.1, we have defined the higher index of such an operator ∂E+ ⊗ ∇) ∈ K0 (R(, σ )), Indσ ( where as before, R denotes the algebra of rapidly decreasing sequences on Z2 . Then the twisted Kasparov map (6) is ∂E+ ⊗ ∇)) = Ind(,σ ) ( ∂E+ ⊗ ∇) ∈ K0 (C ∗ (, σ )), µσ ([E]) = j∗ (Indσ (


69

where j : R(, σ ) = C(, σ ) ⊗ R → Cr∗ (, σ ) ⊗ K is the natural inclusion map, and as before, K denotes the algebra of compact operators. Then j∗ : K0 (R(, σ )) → K0 (Cr∗ (, σ )) is the induced map on K0 . The twisted Kasparov map was defined for certain torsionfree groups in [CHMM] and the general case in [Ma1]. It is related to the Baum–Connes assembly map [BC], [BCH], as is discussed in [Ma1]. 3.2. Range of the higher trace on K-theory. The first step in the proof is to show that given a bounded group cocycle c ∈ Z 2 () we may define canonical pairings with K 0 ((g; ν1 , . . . , νn )) and K0 (Cr∗ (, σ )) which are related by the twisted Kasparov isomorphism, by adapting some of the results of Connes and Connes–Moscovici to the twisted case. As (g; ν1 , . . . , νn ) = B is a negatively curved orbifold, we know (by [Mos] and [Gr]) that degree 2 cohomology classes in H 2 () have bounded representatives i.e. bounded 2-cocycles on . The bounded group 2-cocycle c may be regarded as a skew symmetrised function on × × , so that we can use the results in Sect. 2 to obtain a cyclic 2-cocycle tr c on C(, σ ) ⊗ R by defining: tr c (f 0 ⊗ r 0 , f 1 ⊗ r 1 , f 2 ⊗ r 2 )

= Tr(r 0 r 1 r 2 )

f 0 (g0 )f 1 (g1 )f 2 (g2 )c(1, g1 , g1 g2 )σ (g1 , g2 ).

g0 g1 g2 =1

Since the only difference with the expression obtained in [CM] is σ (g1 , g2 ), and since |σ (g1 , g2 )| = 1, we can use Lemma 6.4, part (ii) in [CM] and the assumption that c is bounded, to obtain the necessary estimates which show that in fact tr c extends continuously to the bigger algebra R(, σ ). By the pairing of cyclic theory and K-theory in [Co], one obtains an additive map [tr c ] : K0 (R(, σ )) → R. tr c (e, · · · , e) − tr c (f, · · · , f ), where e, f are idempotent Explicitly, [tr c ]([e] − [f ]) = matrices with entries in (R(, σ ))∼ = unital algebra obtained by adding the identity to R(, σ ) and tr c denotes the canonical extension of tr c to (R(, σ ))∼ . Let ∂E+ ⊗ ∇ be the Dirac operator defined in the previous section, which is invariant under the projective action of the fundamental group defined by σ . Recall that by definition, the (c, , σ )index of ∂E+ ⊗ ∇ is Ind(c,,σ ) ( ∂E+ ⊗ ∇) = [tr c ](Indσ ( ∂E+ ⊗ ∇)) = '[tr c ], µσ ([E])( ∈ R. It only depends on the cohomology class [c] ∈ H 2 (), and it is linear with respect to [c]. We assemble this to give the following theorem. Theorem 3.1. Given [c] ∈ H 2 () and σ ∈ H 2 (, U (1)) a multiplier on , there is a canonical additive map 0 ((g; ν1 , . . . , νn )) → R, '[c], ( : Korb

which is defined as ∂E+ ⊗ ∇) = [tr c ](Indσ ( ∂E+ ⊗ ∇)) = '[tr c ], µσ ([E])( ∈ R. '[c], [E]( = Ind(c,,σ ) ( Moreover, it is linear with respect to [c].

70


The area cocycle c of the Fuchsian group is a canonically defined 2-cocycle on that is defined as follows. Firstly, recall that there is a well known area 2-cocycle on P SL(2, R), cf. [Co2], defined as follows: P SL(2, R) acts on H such that H ∼ = P SL(2, R)/SO(2). Then c(g1 , g2 ) = Area(H(o, g1 .o, g2 −1 .o)) ∈ R, where o denotes an origin in H and Area(H(a, b, c)) denotes the hyperbolic area of the geodesic triangle in H with vertices at a, b, c ∈ H. Then the restriction of c to the subgroup is the area cocycle c of . Corollary 3.2. Let c, [c] ∈ H 2 (), be the area cocycle, and E → (g; ν1 , . . . , νn ) be an orbifold vector bundle over the orbifold (g; ν1 , . . . , νn ). Then in the notation above, one has '[c], [E]( = φ rank E ∈ φZ, where −φ = 2(1 − g) + (ν − n) ∈ Q is the orbifold Euler characteristic of (g; ν1 , . . . , νn ) and ν = nj=1 1/νj . Proof. By Theorem 2.2, one has [tr c ](Indσ ( ∂E+ ⊗ ∇)) =

1 2π #(G)

g

E ˆ ˜ A(=) tr(eR )eω ψ ∗ (c),

(7)

where g is smooth and G → g → (g; ν1 , . . . , νn ) is a finite orbifold cover. Here ψ : g → g is the lift of the map f : (g; ν1 , . . . νn ) → (g; ν1 , . . . νn ) (since B = (g; ν1 , . . . νn ) in this case) which is the classifying map of the orbifold universal cover (and which in this case is the identity map) and [c] ˜ degree 2 cohomology class on g that is the lift of c to g . We next simplify the right hand side of (7) using the fact ˆ that A(=) = 1 and that E

tr(eR ) = rank E + tr(R E ), ψ ∗ (c) ˜ = c, ˜ eω = 1 + ω. We obtain rank E [tr c ](Indσ ( '[c], ˜ [g ](. ∂E+ ⊗ ∇)) = 2π #(G) When c, [c] ∈ H 2 (), is the area 2-cocycle, then c˜ is merely the restriction of the area cocycle on P SL(2, R) to the subgroup g . Then one has '[c], ˜ [g ]( = −2π χ (g ) = 4π(g − 1). The corollary now follows from Theorem 3.1 above together with the fact that g = n $ 1 + #(G) j =1 1/νj . # 2 (2(g − 1) + (n − ν)), and ν = We next describe the canonical pairing of K0 (Cr∗ (, σ )), given [c] ∈ H 2 (). Since (g; ν1 , . . . , νn ) is negatively curved, we know from [Ji] that

2 k R(, σ ) = f : → C | |f (γ )| (1 + l(γ )) < ∞ for all k ≥ 0 , γ ∈


71

where l : → R+ denotes the length function, is a dense and spectral invariant subalgebra of Cr∗ (, σ ). In particular it is closed under the smooth functional calculus, and is known as the algebra of rapidly decreasing L2 functions on . By a theorem of [Bost], the inclusion map R(, σ ) ⊂ Cr∗ (, σ ) induces an isomorphism Kj (R(, σ )) ∼ = Kj (Cr∗ (, σ )),

j = 0, 1.

(8)

The desired pairing is the one obtained from the canonical pairing of K0 (R(, σ )) with [c] ∈ H 2 () using the canonical isomorphism. Therefore one has the equality '[c], µ−1 σ [P ]( = '[tr c ], [P ]( for any [P ] ∈ K0 (R(, σ )) ∼ = K0 (Cr∗ (, σ )). Using the previous corollary, one has Theorem 3.3 (Range of the higher trace on K-theory). Let c be the area 2-cocycle on . Then c is known to be a bounded 2-cocycle, and one has '[tr c ], [P ]( = φ(rank E 0 − rank E 1 ) ∈ φZ, where −φ = 2(1 − g) of (g; ν1 , + (ν − n) ∈ Q is the orbifold Euler characteristic . . . , νn ) and ν = nj=1 1/νj . Here [P ] ∈ K0 (R(, σ )) ∼ = K0 (Cr∗ (, σ )), and E 0 , E 1 are orbifold vector bundles over (g; ν1 , . . . , νn ) such that 0 1 0 µ−1 σ ([P ]) = [E ] − [E ] ∈ Korb ((g; ν1 , . . . , νn )).

In particular, the range of the the higher trace on K-theory is [tr c ](K0 (C ∗ (, σ ))) = φ Z .

Note that φ is in general only a rational number and we will give examples to show that this is the case; however it is an integer whenever the orbifold is smooth, i.e. whenever 1 = ν1 = . . . = νn , which is the case considered in [CHMM]. We will apply this result in the next section to compute the range of values the Hall conductance in the quantum Hall effect on the hyperbolic plane, for orbifold fundamental groups, extending the results in [CHMM]. In the last section we provide a list of specific examples where fractional values are achieved, and discuss the physical significance of our model. 4. The Area Cocycle, the Hyperbolic Connes–Kubo Formula and the Quantum Hall Effect In this section, we adapt and generalize the discrete model of the quantum Hall effect of Bellissard and his collaborators [Bel+E+S] and also [CHMM], to the case of general cocompact Fuchsian groups and orbifolds, which can be viewed equivalently as the generalization to the equivariant context. We will first derive the discrete analogue of the hyperbolic Connes–Kubo formula for the Hall conductance 2-cocycle, which was derived in the continuous case in [CHMM]. We then relate it to the Area 2-cocycle on the twisted group algebra of the discrete Fuchsian group, and we show that these define the same cyclic cohomology class. This enables us to use the results of the previous section to show that the Hall conductance has plateaux at all energy levels belonging to

72


any gap in the spectrum of the Hamiltonian, where it is now shown to be equal to an integral multiple of a fractional valued topological invariant, namely the orbifold Euler characteristic. The presence of denominators is caused by the presence of cone points singularities and by the hyperbolic geometry on the complement of these cone points. Moreover the set of possible denominators is finite and has been explicitly determined in the next section, and the results compared to the experimental data. It is plausible that this might shed light on the mathematical mechanism responsible for fractional quantum numbers in the quantum Hall effect. The arguments of this section are formulated in the case of orbifolds with positive genus of the underlying topological surface. In order to include the examples that appear in Sect. 5 of hyperbolic orbifolds with underlying topological surface of genus zero, one can suitably modify the arguments of this section, by working equivariantly on a finite orbifold-cover g of positive genus. We consider the Cayley graph of the Fuchsian group of signature (g; ν1 , . . . , νn ), which acts freely on the complement of a countable set of points in the hyperbolic plane. The Cayley graph embeds in the hyperbolic plane as follows. Fix a base point u ∈ H such that the stabilizer (or isotropy subgroup) at u is trivial and consider the orbit of the action through u. This gives the vertices of the graph. The edges of the graph are geodesics constructed as follows. Each element of the group may be written as a word of minimal length in the generators of and their inverses. Each generator and its inverse determine a unique geodesic emanating from a vertex x and these geodesics form the edges of the graph. Thus each word x in the generators determines a piecewise geodesic path from u to x. Recall that the area cocycle c of the Fuchsian group is a canonically defined 2cocycle on that is defined as follows. Firstly, recall that there is a well known area 2-cocycle on P SL(2, R), cf. [Co2], defined as follows: P SL(2, R) acts on H such that H ∼ = P SL(2, R)/SO(2). Then c(γ1 , γ2 ) = Area(H(o, γ1 .o, γ2 −1 .o)) ∈ R, where o denotes an origin in H and Area(H(a, b, c)) denotes the hyperbolic area of the geodesic triangle in H with vertices at a, b, c ∈ H. Then the restriction of c to the subgroup is the area cocycle c of . This area cocycle defines in a canonical way a cyclic 2-cocycle tr c on the group algebra C(, σ ) as follows;

a0 (γ0 )a1 (γ1 )a2 (γ2 )c(γ1 , γ2 )σ (γ1 , γ2 ). tr c (a0 , a1 , a2 ) = γ0 γ1 γ2 =1

We will now describe the hyperbolic Connes–Kubo formula for the Hall conductance in the Quantum Hall Effect. Let =j denote the (diagonal) operator on (2 () defined by =j f (γ ) = =j (γ )f (γ )

where =j (γ ) =

γ .o o

αj

∀ f ∈ (2 ()

∀ γ ∈ ,

j = 1, . . . , 2g,

and where {αj }j =1,... ,2g = {aj }j =1,... ,g ∪ {bj }j =1,... ,g

(9)

is a collection of harmonic V -forms on the orbifold (g; ν1 , . . . , νn ), generating H 1 (g , R) = R2g , cf. [Kaw2, pp.78–83]. These correspond to harmonic G-invariant forms on g and to harmonic -invariant forms on H.


73

Notice that we can write equivalently =j (γ ) = cj (γ ), where the group cocycles cj form a symplectic basis for H 1 (, Z) = Z2g , with generators {αj }j =1,... ,2g , as in (9) and can be defined as the integration on loops on the Riemann surface of genus g underlying the orbifold (g; ν1 , . . . , νn ), cj (γ ) = αj . γ

For j = 1, . . . , 2g, define the derivations δj on R(, σ ) as being the commutators δj a = [=j , a]. A simple calculation shows that δj a(γ ) = =j (γ )a(γ )

∀ a ∈ R(, σ ) ∀ γ ∈ .

Thus, we can view this as the following general construction. Given a 1-cocycle a on the discrete group , i.e. a(γ1 γ2 ) = a(γ1 ) + a(γ2 )

∀ γ1 , γ2 ∈

one can define a derivation δa on the twisted group algebra C(, σ ) δa (f )(γ ) = a(γ )f (γ ). Then we verify that δa (f g)(γ ) = a(γ )f g(γ )

f (γ1 )g(γ2 )σ (γ1 , γ2 ) = a(γ ) γ =γ1 γ2

=

a(γ1 ) + a(γ2 ) f (γ1 )g(γ2 )σ (γ1 , γ2 )

γ =γ1 γ2

=

δa (f )(γ1 )g(γ2 )σ (γ1 , γ2 ) + f (γ1 )δa (g)(γ2 )σ (γ1 , γ2 )

γ =γ1 γ2

= (δa (f )g)(γ ) + (f δa g)(γ ). As determined in Sect. 1, the first cohomology of the group = (g; ν1 , . . . , νn ) is a free Abelian group of rank 2g. It is in fact a symplectic vector space over Z, and assume that aj , bj , j = 1, . . . g is a symplectic basis of H 1 (, Z), as in (9). We denote δaj by δj and δbj by δj +g . Then these derivations give rise to cyclic 2-cocycle on the twisted group algebra C(, σ ), tr K (f0 , f1 , f2 ) =

g

tr(f0 (δj (f1 )δj +g (f2 ) − δj +g (f1 )δj (f2 ))).

j =1

tr K is called the Connes–Kubo Hall conductance cyclic 2-cocycle. In terms of the =j , note that we have the simple estimate |=j (γ )| ≤ ||aj ||(∞) d(γ .o, o),

74


where d(γ .o, o) and the distance d (γ , 1) in the word metric on the group are equivalent. This then yields the estimate |δj a(γ )| ≤ CN d (γ , 1)−N

∀ N ∈ N,

i.e. δj a ∈ R(, σ ) ∀ a ∈ R(, σ ). Note that since ∀ γ , γ ∈ , the difference =j (γ γ ) − =j (γ ) is a constant independent of γ , we see that -equivariance is preserved. For j = 1, . . . , 2g, define the cyclic 2-cocycles tr K j (a0 , a1 , a2 ) = tr(a0 (δj a1 δj +g a2 − δj +g a1 δj a2 )). These compute the Hall conductance for currents in the (j + g)th direction which are induced by electric fields in the j th direction, as can be shown using the quantum adiabatic theorem of Avron–Seiler–Yaffe [Av+S+Y ] just as in Sect. 6 of [CHMM], in the continuous model. Then the hyperbolic Connes–Kubo formula for the Hall conductance is the cyclic 2-cocycle given by the sum K

tr (a0 , a1 , a2 ) =

g

j =1

tr K j (a0 , a1 , a2 ).

Theorem 4.1 (The Comparison Theorem). [tr K ] = [tr c ] ∈ H C 2 (R(, σ ))

Proof. Our aim is now to compare the two cyclic 2-cocycles and to prove that they differ by a coboundary i.e. tr K (a0 , a1 , a2 ) − tr c (a0 , a1 , a2 ) = bλ(a0 , a1 , a2 ) for some cyclic 1-cochain λ and where b is the cyclic coboundary operator. The key to this theorem is a geometric interpretation of the hyperbolic Connes–Kubo formula. We begin with some calculations, to enable us to make this comparison of the cyclic 2-cocycles. tr K (a0 , a1 , a2 ) =

g

a0 (γ0 ) δj a1 (γ1 )δj +g a2 (γ2 )

j =1 γ0 γ1 γ2 =1

− δj +g a1 (γ1 )δj a2 (γ2 ) σ (γ0 , γ1 )σ (γ0 γ1 , γ2 )

=

g

a0 (γ0 )a1 (γ1 )a2 (γ2 ) =j (γ1 )=j +g (γ2 )

j =1 γ0 γ1 γ2 =1

− =j +g (γ1 )=j (γ2 ) σ (γ1 , γ2 )

since by the cocycle identity for multipliers, one has σ (γ0 , γ1 )σ (γ0 γ1 , γ2 ) = σ (γ0 , γ1 γ2 )σ (γ1 , γ2 ) = σ (γ0 , γ0−1 )σ (γ1 , γ2 ) = σ (γ1 , γ2 )

since

since

γ 0 γ1 γ2 = 1

σ (γ0 , γ0−1 ) = 1.


75

So we are now in a position to compare the two cyclic 2-cocycles. Define Oj (γ1 , γ2 ) = =j (γ1 )=j +g (γ2 ) − =j +g (γ1 )=j (γ2 ). Let P : H → R2g denote the Abel–Jacobi map

x x x x a1 , b1 , . . . , ag , bg , P : x *→ o

o

o

o

x

where o means integration along the unique geodesic in H connecting o to x. The origin o is chosen so that it satisfies .o ∼ = . The map P is a symplectic map, that is, if ω and ωJ are the respective symplectic 2-forms, then one has P∗ (ωJ ) = ω. One then has the following geometric lemma. Lemma 4.2.

g

Oj (γ1 , γ2 ) =

j =1

HE (γ1 ,γ2 )

ωJ ,

where HE (γ1 , γ2 ) denotes the Euclidean triangle with vertices at P(o), P(γ1 .o) and P(γ2 .o), and ωJ denotes the flat Kähler 2-form on the Jacobi variety. That is, g j =1 Oj (γ1 , γ2 ) is equal to the Euclidean area of the Euclidean triangle HE (γ1 , γ2 ). Proof. We need to consider the expression g

j =1

Oj (γ1 , γ2 ) =

g

=j (γ1 )=j +g (γ2 ) − =j +g (γ1 )=j (γ2 ).

j =1

Let s denote the symplectic form on R2g given by: s(u, v) =

g

(uj vj +g − uj +g vj ).

j =1

The so-called ‘symplectic area’ of a triangle with vertices P(o) = 0, P(γ1 .o), P(γ2 .o) may be seen to be s(P(γ1 .o), P(γ2 .o)). To appreciate this, however, we need to use an argument from [GH, pp. 333–336]. In terms of the standard basis of R2g (given in this case by vertices in the integer period lattice arising from our choice of basis of harmonic one forms) and corresponding coordinates u1 , u2 , . . . u2g the form s is the two form on R2g given by g

duj ∧ duj +g . ωJ = j =1

Now the “symplectic area” of a triangle in R2g with vertices P(o) = 0, P(γ1 .o), P(γ2 .o) is given by integrating ωJ over the triangle and a brief calculation reveals that this yields s(P(γ1 .o), P(γ2 .o))/2, proving the lemma. # $ We also observe that since ω = P∗ ωJ , one has ω= c(γ1 , γ2 ) = H(γ1 ,γ2 )

P(H(γ1 ,γ2 ))

ωJ .

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Therefore the difference g

Oj (γ1 , γ2 ) − c(γ1 , γ2 ) =

HE (γ1 ,γ2 )

j =1

ωJ −

=

∂HE (γ1 ,γ2 )

P(H(γ1 ,γ2 ))

ωJ

RJ −

∂P(H(γ1 ,γ2 ))

RJ ,

where RJ is a 1-form on the universal cover of the Jacobi variety such that dRJ = ωJ . Therefore one has g

Oj (γ1 , γ2 ) − c(γ1 , γ2 ) = h(1, γ1 ) − h(γ1−1 , γ2 ) + h(γ2−1 , 1), j =1

where h(γ1−1 , γ2 ) = P(((γ1 ,γ2 )) RJ − m(γ1 ,γ2 ) RJ , where ((γ1 , γ2 ) denotes the unique geodesic in H joining γ1 .o and γ2 .o and m(γ1 , γ2 ) is the straight line in the Jacobi variety joining the points P(γ1 .o) and P(γ2 .o). Since we can also write h(γ1−1 , γ2 ) = D(γ1 ,γ2 ) ωJ , where D(γ1 , γ2 ) is a disk in the Jacobi variety with boundary P(((γ1 , γ2 ))∪ m(γ1 , γ2 ), we see that h is -invariant. We now define the cyclic 1-cochain λ on R(, σ ) as

h(1, γ1 )a0 (γ0 )a1 (γ1 )σ (γ0 , σ1 ), λ(a0 , a1 ) = tr((a0 )h a1 ) = γ0 γ1 =1

where (a0 )h is the operator on (2 () whose matrix in the canonical basis is h(γ1 , γ2 ) a0 (γ1 γ2−1 ). Firstly, one has by definition bλ(a0 , a1 , a2 ) = λ(a0 a1 , a2 ) − λ(a0 , a1 a2 ) + λ(a2 a0 , a1 ). We compute each of the terms seperately

h(1, γ2 )a0 (γ0 )a1 (γ1 )a2 (γ2 )σ (γ1 , γ2 ), λ(a0 a1 , a2 ) = γ0 γ1 γ2 =1

λ(a0 , a1 a2 ) =

h(1, γ1 γ2 )a0 (γ0 )a1 (γ1 )a2 (γ2 )σ (γ1 , γ2 ),

γ0 γ1 γ2 =1

λ(a2 a0 , a1 ) =

h(1, γ1 )a0 (γ0 )a1 (γ1 )a2 (γ2 )σ (γ1 , γ2 ).

γ0 γ1 γ2 =1

Now by -equivariance, h(1, γ1 γ2 ) = h(γ1−1 , γ2 ) and h(1, γ2 ) = h(γ2−1 , 1). Therefore one has bλ(a0 , a1 , a2 )

a0 (γ0 )a1 (γ1 )a2 (γ2 ) h(γ2−1 , 1) − h(γ1−1 , γ2 ) + h(1, γ1 ) σ (γ1 , γ2 ). = γ0 γ1 γ2 =1

Using the formula above, we see that bλ(a0 , a1 , a2 ) = tr K (a0 , a1 , a2 ) − tr c (a0 , a1 , a2 ). It follows from Connes pairing theory of cyclic cohomology and K-theory [Co2], by the range of the higher trace Theorem 3.3 and by the Comparison Theorem 4.1 above that


77

Corollary 4.3 (Rationality of conductance). The Connes–Kubo Hall conductance cocycle tr K is rational. More precisely, one has tr K (P , P , P ) = tr c (P , P , P ) ∈ φZ for all projections P ∈ R(, σ ), where −φ = 2(1 − g) + (ν − n) ∈ Q is the orbifold Euler characteristic of (g; ν1 , . . . , νn ). Finally, suppose that we are given a very thin sample of pure metal, with electrons situated along the Cayley graph of , and a very strong magnetic field which is uniform and normal in direction to the sample. Then at very low temperatures, close to absolute zero, quantum mechanics dominates and the discrete model that is considered here is a model of electrons moving on the Cayley graph of which is embedded in the sample. The associated discrete Hamiltonian Hσ for the electron in the magnetic field is given by the Random Walk operator in the projective (, σ ) regular representation on the Cayley graph of the group . It is also known as the generalized Harper operator and was first studied in this generalized context in [Sun], see also [CHMM]. We will see that the Hamiltonian that we consider is in a natural way the sum of a free Hamiltonian and a term that models the Coulomb interaction. We also add a restricted class of potential terms to the Hamiltonian in our model. # $ Because the charge carriers are Fermions, two different charge carriers must occupy different quantum eigenstates of the Hamiltonian. In the limit of zero temperature they minimize the energy and occupy eigenstates with energy lower that a given one, called the Fermi level and denoted E. Let PE denote denote the corresponding spectral projection of the Hamiltonian. If E is not in the spectrum of the Hamiltonian, then then PE ∈ R(, σ ) and the hyperbolic Connes–Kubo formula for the Hall conductance σE at the energy level E is defined as follows; σE = tr K (PE , PE , PE ). As mentioned earlier, it measures the sum of the contributions to the Hall conductance at the energy level E for currents in the (j + g)th direction which are induced by electric fields in the j th direction, cf. Sect. 6 [CHMM]. By Corollary 4.3, one knows that the Hall conductance takes on values in φZ whenever the energy level E lies in a gap in the spectrum of the Hamiltonian Hσ . In fact we notice that the Hall conductance is a constant function of the energy level E for all values of E in the same gap in the spectrum of the Hamiltonian. That is, the Hall conductance has plateaux which are integer multiples of the fraction φ on the gap in the spectrum of the Hamiltonian. We now give some details. Recall the left σ -regular representation (U (γ )f )(γ ) = f (γ −1 γ )σ (γ , γ −1 γ ). For all f ∈ (2 () and for all γ , γ ∈ . It has the property that U (γ )U (γ ) = σ (γ , γ )U (γ γ ). −1 −1 Let S = {Aj , Bj , A−1 i = 1, . . . , n} be a symmetric j , Bj , Ci , Ci : j = 1, . . . , g, set of generators for . Then the Hamiltonian is explicitly given as

Hσ : (2 () → (2 (),

U (γ ) Hσ = γ ∈S

78


and is clearly by definition a bounded self adjoint operator. Notice that the Hamiltonian can be decomposed as a sum of a free Hamiltonian containing the torsionfree generators and a term simulating Coulomb interactions, that contains the torsion generators. Hσ = Hσfree + Hσinteraction , where Hσfree =

g

∗ U (Aj ) + U (Bj ) + U (Aj ) + U (Bj )

j =0

and Hσinteraction =

n

U (Ci ) + U (Ci )∗ .

i=1

Let V ∈ C(, σ ) be any “potential”, and Hσ,V = Hσ + V . Lemma 4.4. If E ∈ spec(Hσ,V ), then PE ∈ R(, σ ), where PE = χ[0,E] (Hσ,V ) is the spectral projection of the Hamiltonian to energy levels less than or equal to E. Proof. Since E ∈ spec(Hσ,V ), then PE = χ[0,E] (Hσ,V ) = ϕ(Hσ,V ) for some smooth, compactly supported function ϕ. Now by definition, Hσ ∈ C(, σ ) ⊂ R(, σ ), and since R(, σ ) is closed under the smooth functional calculus by the result of [Ji], it follows that PE ∈ R(, σ ). # $ Therefore by Corollary 4.3 and the discussion following it, we have, Theorem 4.5 (Fractional Quantum Hall Effect). Suppose that the Fermi energy level E lies in a gap of the spectrum of the Hamiltonian Hσ,V , then the Hall conductance σE = tr K (PE , PE , PE ) = tr c (PE , PE , PE ) ∈ φZ. That is, the Hall conductance has plateaux which are integer multiples of φ on any gap in the spectrum of the Hamiltonian, where −φ = 2(1 − g) + (ν − n) ∈ Q is the orbifold Euler characteristic of (g; ν1 , . . . , νn ). Remarks 4.6. The set of possible denominators φ for low genus coverings can be derived easily from the results of [Bro] and is reproduced in the second table in the next section. It is plausible that this Theorem might shed light on the mathematical mechanism responsible for fractional quantum numbers that occur in the quantum Hall effect, as we attempt to explain in the following section. 5. Fractional Quantum Numbers: Phenomenology We first discuss the characteristics of our model explaining the appearance of fractional quantum numbers in the quantum Hall effect. In particular, we point out the main advantages and limitations of the model. Our model is a single electron model. It is well known that the FQHE is a consequence of the Coulomb interaction between electrons, hence it should not be seen by a single particle model. However, in our setting, the negative curvature of the hyperbolic structure provides a geometric replacement for interaction. The equivalence between negative


79

curvature and interaction is well known from the case of classical mechanics where the Jacobi equation for a single particle moving on a negatively curved manifold can be interpreted as the Newton equation for a particle moving in the presence of a negative potential energy [Arn]. The main advantage of this setting is that the fractions derived in this way are topological. In fact, they are obtained from an equivariant index theorem. Moreover, they are completely determined by the geometry of the orbifold. In fact, we have φ = −χorb ((g; ν1 , . . . , νn )). Let us recall that the orbifold Euler characteristic χorb () of an orbifold , is a rational valued invariant that is completely specified by the following properties, cf. [Tan]: 1. it is multiplicative under orbifold covers; 2. it coincides with the topological Euler characteristic in the case of a smooth surface; 3. it satisfies the volume formula, χorb (1 ∪ · · · ∪ k ) =

k

j =1

χorb (j ) −

χorb (i ∩ j )

i,j

+ · · · (−1)k+1 χorb (1 ∩ · · · ∩ k ), whenever all the intersections on the right hand side are suborbifolds of k , and all the j are orbifolds of the same dimension.

1 ∪ · · · ∪

This characterization allows for ease of computation and prediction of expected fractions. Most notably, as pointed out in [Bel+E+S], the topological nature of the Hall conductance makes it stable under small deformations of the Hamiltonian. Thus, this model can be easily generalized to systems with disorder, cf. [CHM]. This is a necessary step in order to establish the presence of plateaux [Bel+E+S]. The identifications of fractions with integer multiples of the orbifold Euler characteristic imposes some restrictions on the range of possible fractions from the geometry of the orbifolds. For instance, it is known from the Hurwitz theorem that the maximal order of a finite group acting by isometries on a smooth Riemann surface g is #(G) = 84(g − 1). Moreover, this maximal order is always attained. Thus, the smallest 2(g −1) possible fraction that appears in our model is φ = 84(g −1) = 1/42. This is, in some respects, an advantage of the model, in as it gives very clear prediction on which fractions can occur, and at the same time its main limitation, in as we do not get a complete agreement between the set of fractions we obtain and the fractions that are actually observed in experiments on the FQHE. In order to compare our predictions with experimental data, we restrict our attention to orbifolds with a torus or a sphere as underlying topological surface. Recall that, as explained above, we think of the hyperbolic structure induced by the presence of cone points on these surfaces as a geometric way of introducting interaction in this single electron model, hence we would consider equivalently the underlying surface with many interacting electrons (fractions observed in FQHE experiments) or as a hyperbolic surface with one electron. We report a table of comparison between the values obtained experimentally and our prediction (Table 2). Notice how the fraction 5/2 which appears in the experimental values and caused major problems of interpretation in the many-particle models appears here naturally as the orbifold Euler characteristic of (1; 6, 6, 6) (which we may as

80


Table 2. Experimental

g = 1 or g = 0

5/3 4/3 7/5 4/5 5/7 2/3 3/5 4/7 5/9 4/9 3/7 2/5 1/3 5/2

(1; 6, 6) (1; 3, 3) (0; 5, 5, 10, 10) (1; 5) (0; 7, 14, 14) (1; 3) (0; 5, 10, 10) (0; 7, 7, 7) ??? (0; 3, 9, 9) ??? (0; 5, 5, 5) (0; 3, 6, 6) (1; 6, 6, 6)

well refer to as the Devil’s orbifold). Despite the small number of discrepancies in the table above, the agreement between values of orbifold Euler characteristics and experimentally observed fractions in the quantum Hall effect is far from being satisfactory. In particular, not only there is a small number of observed values which are not orbifold Euler characteristics, but there are also many rational numbers that are realized as orbifold Euler characteristics, which do not seem to appear among the experimental data. For instance, by looking at the values of Table 3, reported also in Fig. 1, we see clearly that we have some fractions with even denominator, such as 1/4, 1/2, and 1/6, which do not correspond to experimental values. As pointed out in the introduction, the reason for this discrepancy is that a more sophisticated model for the Coulomb interaction is needed in general. In the remaining of this section, we discuss some phenomenology, with particular emphasis on the nature of the cone points and the role of the minimal genus of the covering surface g . We hope to return to these topics in some future work. Every orbifold (g; ν1 , . . . , νn ) is obtained as a quotient of a surface g with respect to the action of a finite group G, cf. [Sc]. In general both g and G are not unique. For instance, the orbifold (1; 2, 2) is obtained as the quotient of 2 by the action of Z2 , or as the quotient of 3 by the action of Z4 , or by the action of Z2 × Z2 , cf. [Bro]. For every (g; ν1 , . . . , νn ) there is a minimal g such that the orbifold is obtained as a quotient of g by a finite group action. In [Bro], Broughton has derived a complete list of all the good two dimensional orbifolds which are quotients of Riemann surfaces g with genus g = 2 or 3. In a physical model one can distinguish between two types of disorder: a mobility disorder and a sample disorder, cf. [Bel+E+S]. We can argue phenomenologically that, if an orbifold can be realized by a covering of low genus, this corresponds to a lower density of atoms in the sample, as opposed to the case of a surface of high genus, as one can see by looking at the Cayley graph of g . Thus, we can consider the minimal genus of the smooth coverings as a measure of mobility. This means that, in an experiment, the fractions derived from orbifolds with low genus coverings will be easier to observe (have more clearly marked plateaux) than fractions which are only realized by quotients of surfaces of higher genus. Thus, we can consider the list of examples given in [Bro] and compute the corresponding fractions. We list the result in Table 3.


81

Table 3. φ 4/3 2/3 4/7 1/2 4/9 2/5 1/3 1/4 1/5 4/21 1/6 1/8 1/12 1/24 1/42

g = 2 (0; 3, 3, 3, 3) (0; 2, 2, 4, 4)∗ (0; 2, 2, 2, 2, 2)∗ (0; 5, 5, 5) (0; 3, 6, 6) (0; 2, 2, 3, 3)∗ (0; 2, 8, 8)∗ (0; 4, 4, 4)∗ (0; 2, 2, 2, 4)∗ (0; 2, 5, 10) (0; 3, 4, 4)∗ (0; 2, 6, 6)∗ (0; 2, 2, 2, 3)∗ (0; 2, 4, 8)∗ (0; 2, 4, 6)∗ (0; 3, 3, 4)∗ (0; 2, 3, 8)∗

g = 3 (0; 3, 3, 3, 3, 3) (1; 3, 3) (0; 2, 2, 6, 6) (0; 2, 3, 3, 6) (0; 2, 2, 2, 2, 3) (1; 3) (0; 7, 7, 7) (0; 4, 8, 8) (1; 2) (0; 3, 9, 9) (0; 2, 12, 12) (0; 3, 4, 12) (0; 4, 4, 6) (0; 2, 2, 2, 6)

(0; 3, 7, 7) (0; 2, 4, 12) (0; 3, 3, 6)

(0; 2, 3, 7)

In the table the orbifolds that are marked with a ∗ can be realized both as quotient of 2 and of 3 . It seems also reasonable to think that if the same fraction is realized by several different orbifolds, for fixed g , then the corresponding plateau will be more clearly marked in the experiment. This would make φ = 1/3 the most clearly pronounced plateau, which is in agreement with the experimental data. However, higher genus corrections are not always negligible. In fact, by only considering genus g = 2 and g = 3 contributions, we would expect a more marked plateau for the fraction φ = 2/3 than for the fraction φ = 2/5, and the experimental results show that this is not the case. It seems important to observe that this model produces equally easily examples of fractions with odd or even denominators (e.g. φ = 1/4 appears in the table above). It is interesting to compare this datum with the difficulty encountered within other models in explaining the appearance of the fraction 5/2 in the experiments. Its presence is only justified by introducing a different physical model (the so called non-abelian statistics). In figure 1 we sketch the plateaux as they would appear in the result of an experiment, using only the low genus g = 2 and g = 3 approximation. As we already mentioned in the introduction, both the hyperbolic structure and the cone points are essential in order to have fractional quantum numbers. In fact, φ is an integer whenever the hyperbolic orbifold is smooth, i.e. whenever 1 = ν1 = . . . = νn , which is the case considered in [CHMM]. On the other hand, by direct inspection, it is possible to see that all euclidean orbifolds also produce only integer values of φ. (Notice that sometimes hyperbolic orbifolds with cone points may still produce integers: the orbifold (1; 2, 2) has φ = 1, cf. [Bro].) Models of FQHE on euclidean orbifolds have been considered, in a different, string-theoretic context, e.g. [Sk-Th]. We can argue that the cone points can also be thought of as a form of “disorder”. In fact, we may identify the preimage of the cone points in the universal covering H with sample disorder (with respect to the points in the Cayley graph of g ). The same fraction can often be obtained by orbifolds with a varying number of cone points (for fixed g ), as illustrated in the previous table. This can be rephrased by saying that the

82


1/42 1/12 1/6 1/5 1/4 1/24 1/8 4/21

1/3

4/9 1/2 4/7 2/5

2/3

4/3

Fig. 1. Phenomenology of fractions in the low genus approximation

system allows for more or less sample disorder, and in some cases this can be achieved without affecting the mobility measured by g . Appendix The main purpose of this appendix is to establish Lemma F, which is used in the paper. We follow closely the approach in [BrSu]. We use the notation of the previous sections. S ⊗ E) with Schwartz kernel kA and also commuting Let A be an operator on L2 (M, with the given (, σ¯ )-action. Then one has eiφγ (x) kA (γ x, γ y) e−iφγ (y) = kA (x, y)

∀ γ ∈ ,

(10)

with the fibre at γ x ∈ M. If kA is smooth, where we have identified the fibre at x ∈ M then one can define the von Neumann trace just as Atiyah did in the untwisted case, tr (kA ) = tr (kA (x, x)) dx, F

and where tr denotes the where F denotes a fundamental domain for the action of on M pointwise or local trace. The von Neumann trace is well defined, since as a consequence The following lemma establishes of (10), tr(kA (x, x)) is a -invariant function on M. that it is a trace.


83

Lemma (A). Let A, B be operators on L2 (M, S ⊗ E) with smooth Schwartz kernels and also commuting with the given (, σ¯ )-action. Then one has tr (AB) = tr (BA) . Proof. Let kA , kB denote the smooth Schwartz kernels of A, B respectively, and kAB , kBA denote the smooth Schwartz kernels of AB, BA respectively. Then one has tr (kAB (x, y) − kBA (x, y)) tr (AB − BA) = x∈F = tr (kA (x, y)kB (y, x) − kB (x, y)kA (y, x)) x∈F y∈M

= tr (kA (x, γ y)kB (γ y, x) − kB (x, γ y)kA (γ y, x)) γ ∈ x∈F

y∈F

=0 since each term in the summand vanishes by symmetry, and we have used the fact that the fundamental domain F is compact in order to interchange the order of the summation and integral. # $ S ⊗ E|F ). We will also adopt a more operator theoretic approach. Let H = L2 (F, ∼ = 2 2 Then T : L (M, S ⊗ E) → ( (, H) is given by (Ts)(γ ) = RF (Tγ s) ∀ γ ∈ , where S ⊗ E) → H denotes the restriction map to the fundamental domain F. RF : L2 (M, As in Sect. 1, let W ∗ (σ ) denote the commutant, i.e. W ∗ (σ ) = A ∈ (2 (, H) : [Tγ , A] = 0 ∀ γ ∈ . Then one has the following simple lemma, Lemma (B). W ∗ (σ ) is a semifinite von Neumann algebra. Proof. We need to show that W ∗ (σ ) is a ∗-algebra which is weakly closed. We will establish that it is has a semifinite trace a bit later on. Let A, B ∈ W ∗ (σ ). Since [Tγ , AB] = [Tγ , A]B + A[Tγ , B], it follows that AB ∈ ∗ W (σ ). Since [Tγ , A] = −[Tγ∗ , A∗ ] = −[Tγ −1 , A∗ ] it follows A∗ ∈ W ∗ (σ ). Clearly the identity operator is in W ∗ (σ ). Finally, if An ∈ W ∗ (σ ) ∀ n ∈ N and An converges weakly to A, it follows that for all γ ∈ , Tγ An converges weakly to Tγ A and also to ATγ . By uniqueness of weak limits, we deduce that A ∈ W ∗ (σ ). # $ ) ∈ B(H) as For A ∈ W ∗ (σ ), define its generalized Fourier coefficients A(γ )v = Tγ (Aδ1v )(1), A(γ where δ1v ∈ (2 (, H) is defined for all v ∈ H as δ1v (γ )

=

v

if γ = 1;

0

otherwise.

84


Since Tγ δ1v (γ ) = δ1v (γ γ )σ (γ , γ ), one has Tγ δ1v (γ )

=

v

if γ = γ −1 ;

0

otherwise,

since σ (γ −1 , γ ) = 1 ∀ γ ∈ . In particular, it follows that for all f ∈ (2 (, H), one has

f (γ ) f (γ ) = Tγ1 δ1 2 γ1 γ2 =γ

so that one has the following Fourier expansion Af (γ ) =

γ1 γ2 =γ

=

f (γ2 )

ATγ1 δ1

=

γ1 γ2 =γ

f (γ2 )

Tγ1 Aδ1

1 )(f (γ2 )). A(γ

γ1 γ2 =γ

The following elementary properties are satisfied by the Fourier coefficients. Lemma (C). For A, B ∈ W ∗ (σ ) and for all γ ∈ , for all f ∈ (2 (, H), one has

1 )(f (γ2 )); (1) Af (γ ) = A(γ γ1 γ2 =γ

−1 ))∗ ; ∗ (γ ) = (A(γ (2) A

)= 2 ); 1 )B(γ (3) AB(γ A(γ γ1 γ2 =γ

∗ (1) = (4) AA (5) ||A|| ≤

)A(γ ); A(γ

γ

)||; ||A(γ

γ

) − B(γ ).

(6) A − B(γ ) = A(γ Proof. The proof follows by straightforward calculations as done above. The reader is warned that the righthand side of the inequality in part (5) is not necessarily finite. # $ ) ∈ K ∀ γ ∈ , Define Co (, K) to be the set of all A ∈ W ∗ (σ ) such that A(γ ) = 0 for all but finitely many γ ∈ . Then the completion of Co (, K) with and A(γ respect to the operator norm is denoted, as in Sect. 1 of [MM], by Cr∗ (, σ ) ⊗ K, and called the twisted crossed product algebra associated to the twisted action (α, σ ). Then one has the following useful containment criterion,

)|| < ∞, then A ∈ Cr∗ (, σ ) ⊗ Lemma (D). If A ∈ W ∗ (σ ) and also satisfies ||A(γ K.

If A ∈ W ∗ (σ ) and also satisfies

k, then A ∈ R(, σ ).

γ

γ

)|| < ∞, for all positive integers d(γ , 1) ||A(γ k


85

Proof. Let K1 ⊂ K2 ⊂ · · · be a sequence of finite subsets of which is an exhaustion of , i.e. j ≥1 Kj = . For all j ∈ N, define Aj ∈ W ∗ (σ ) by ) if γ ∈ Kj ; A(γ Aj (γ ) = 0 otherwise. Then in fact Aj ∈ Co (, K) by definition, and using the previous lemma, we have

||A

− Aj (γ )|| ||A − Aj || ≤ γ

=

)−A j (γ )|| ||A(γ

γ

=

)||. ||A(γ

γ ∈\Kj

By hypothesis,

)|| < ∞, therefore ||A(γ

γ

)|| → 0 as j → ∞, since Kj ||A(γ

γ ∈\Kj

is an increasing exhaustion of . This proves that A ∈ Cr∗ (, σ ) ⊗ K. The second part is clear from the definition, once we identify R with the algebra of sequences aγ γ ∈ sup d(γ , 1)k |aγ | < ∞ ∀ k ∈ N . $ # γ ∈

The following off-diagonal estimate is well known, cf. [BrSu]. + Lemma (E). Let D = ∂ E ⊗ ∇ s be a twisted Dirac operator. Then the Schwartz kernel ∗ k(t, x, y) of the heat operator e−tD D is smooth ∀ t > 0. It also satisfies the following off-diagonal estimate 2 |k(t, x, y)| ≤ C1 t −n/2 e−C2 d(x,y) /t ×M for any T > 0, where d denotes the Riemannian distance uniformly in (0, T ] × M function on M. The same result is true for the the Schwartz kernel of the heat operator ∗ e−tDD . ∗ ∗ + Lemma (F). Let D = ∂ E ⊗ ∇ s be a twisted Dirac operator. Then e−tD D , e−tDD ∈ ∗ R(, σ ) ⊂ Cr (, σ ) ⊗ K ∀ t > 0. ∗

∗

Proof. By the Lemma above, it follows that e−tD D , e−tDD are bounded operators ∗ ∗ commuting with the given twisted action, i.e. e−tD D , e−tDD ∈ W ∗ (, H). Since the −tD ∗ D (γ ), e

−tDD ∗ (γ ) are smooth ∀ γ ∈ by the Lemma above, Schwartz kernels of e

−tD ∗ D (γ ), e

−tDD ∗ (γ ) ∈ K ∀ γ ∈ . Let d denote the word metric with it follows that e

Then it is respect to a given finite set of generators, and d the Riemannian metric on M. well known that d (γ1 , γ2 ) ≤ C3 ( inf d(γ1 x, γ2 y) + 1) x,y∈M

for some positive constant C3 . By the Lemma 5 above, one has, −tD D (γ )|| ≤ C e−C5 d (γ ,1) ||e

4 ∗

2

86


−tDD ∗ (γ ). Setting for some positive constants C4 , C5 , and a similar estimate holds for e

r(γ ) = d (γ , 1) observe that one has the estimate

# {γ ∈ | r(γ ) ≤ R} ≤ C6 eC7 R for some positive constants C6 , C7 , since the volume growth rate of is at most exponential. Therefore one has

−tD ∗ D (γ )|| < ∞ and −tDD 8 (γ )|| < ∞ d(γ , 1)k ||e

d(γ , 1)k ||e

γ

γ

for all positive integers k. By the Lemma above, it follows that e−tD R(, σ ) ⊂ Cr∗ (, σ ) ⊗ K ∀ t > 0. # $

∗D

∗

, e−tDD ∈

Acknowledgements. We thank J. Bellissard for his encouragement and for some useful comments. The second author thanks A. Carey and K. Hannabuss for some helpful comments concerning the Sect. 4. The first author is partially supported by NSF grant DMS-9802480. Research by the second author is supported by the Australian Research Council. The second author acknowledges that this work was completed in part for the Clay Mathematical Institute.

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Communicated by A. Connes

Commun. Math. Phys. 217, 89 – 106 (2001)

Communications in



Classification of Subsystems for Local Nets with Trivial Superselection Structure Sebastiano Carpi1 , Roberto Conti2, 1 Dipartimento di Matematica, Università di Roma “La Sapienza”, Piazzale A. Moro, 00185 Roma, Italy.


2 Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma,

Italy. E-mail: [email protected] Received: 26 January 2000 / Accepted: 28 September 2000

Dedicated to S. Doplicher and J. E. Roberts on the occasion of their 60th birthdays Abstract: Let F be a local net of von Neumann algebras in four spacetime dimensions satisfying certain natural structural assumptions. We prove that if F has trivial superselection structure then every covariant, Haag-dual subsystem B is of the form F1G ⊗ I for a suitable decomposition F = F1 ⊗ F2 and a compact group action. Then we discuss some application of our result, including free field models and certain theories with at most countably many sectors. 1. Introduction In the algebraic approach to QFT [31] the main objects under investigation are (isotonous) nets of von Neumann algebras over bounded regions in Minkowski spacetime, satisfying pertinent additional requirements. Any such correspondence is usually denoted by O → F(O). Internal symmetries of a net F can be defined as those automorphisms of the C ∗ inductive limit (∪O∈K F(O))−· (the quasi-local C ∗ -algebra; it is customary to denote it in the same way as the net), that leave every local algebra F(O) globally invariant; unbroken internal symmetries leave the vacuum state invariant. Given a certain (compact) group G of (unbroken) internal symmetries of F, the fixpoint net F G defined by F G (O) = F(O)G is an example of subsystem (sometimes also called subnet or subtheory in the literature), i.e. a net of (von Neumann) subalgebras of F. This is the typical situation allowing one to recover an observable net from a field net via a principle of gauge invariance. However, in certain situations one can easily produce examples of subsystems that can hardly be seen to arise in this way. See e.g. the discussion in [46, 1, 12]. In this work we address the problem of classifying subsystems of a given net F. Some related work has been already done in [37, 38, 17, 15, 11, 9]. Our main result states that if Supported by EU.

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F satisfies certain structural properties then all the reasonably well-behaved subsystems essentially arise in the way explained above, namely they are fixpoints for a compact group action on F or on one component F1 in a tensor product decomposition F = F1 ⊗ F 2 . We confine our discussion to nets F satisfying the usual postulates such as Poincaré covariance, Bisognano–Wichmann and the split property, plus an additional condition, the absence of nontrivial sectors, whose meaning has been recently clarified in [15]. At first sight the latter condition might appear very strong but it is generic in a sense: under reasonable conditions for the observable net it is verified by the canonical field net attainable by performing the Doplicher–Roberts reconstruction procedure [15]. Our assumptions are sufficiently general to cover many interesting situations, including the well-known Bosonic free field models (massive or massless). In particular in the case of (finitely many) multiplets of the massive scalar free fields we (re)obtain a classification result of Davidson [17], but with a different method of proof. Moreover our discussion applies to the massless case as well. In a different direction, we also provide a first solution to a long-standing open problem, proposed by S. Doplicher, concerning the relationship between an observable net A and the subsystem C generated by the canonical local implementations of spacetime translations whose generators are the abstract analogue for the local energy-momentum tensor [21, 12]. As to the main ingredients, now A is required to have the split property and at most countably many superselection sectors, all with finite statistical dimension1 (and Bosonic), in order to guarantee that the canonical field net F has trivial superselection structure. Still our assumptions are restrictive enough to rule out the occurrence of models with undesirable features. This allows us to overcome certain technical difficulties that cannot be handled in too general (perhaps pathological) situations. This paper is organized in the following way. In the next section we describe our setup and collect some preliminaries. The third section contains the stated classification result. In the fourth section we present some applications. Some of the assumptions can be relaxed to some extent, at the price of much more complicated proofs and no sensible improvement. We comment on this in the fifth section. We end the article with some brief comments and suggestions for future work. An appendix is included to provide some technical results about scalar free field theories. 2. Preliminaries Throughout this article we denote P the connected component of the identity of the Poincaré group in four spacetime dimensions and K the set of open double cones of R4 . We will denote the elements of P by pairs (, x), where is an element of the restricted Lorentz group and x ∈ R4 is a spacetime translation, or alternatively by a single letter L. Double cones and wedges will be denoted O and W respectively, with subscripts if necessary. We consider a net F over K, i.e. a map O → F(O) from double cones to von Neumann algebras acting on a separable Hilbert space H, satisfying the following assumptions: (i)

Isotony. If O1 ⊂ O2 , O1 , O2 ∈ K, then F(O1 ) ⊂ F(O2 ).

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1 If one can rule out the occurrence of sectors with infinite statistics for A, the other two facts are easily implied by the split property for the canonical field net F, that is anyhow needed from the start to define the subsystem C.

Classification of Subsystems for Local Nets

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Locality. If O1 , O2 ∈ K and O1 is spacelike separated from O2 then F(O1 ) ⊂ F(O2 ) .

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(iii) Covariance. There is a strongly continuous unitary representation U of P such that, for every L ∈ P and every O ∈ K, there holds U (L)F(O)U (L)∗ = F(LO).

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(iv)

Existence and uniqueness of the vacuum. There exists a unique (up to a phase) unit vector which is invariant under the restriction of U to the subgroup of spacetime translations. (v) Positivity of the energy. The joint spectrum of the generators of the spacetime translations is contained in the closure V + of the open forward light cone V+ . (vi) Reeh–Schlieder property. The vacuum vector is cyclic for F(O) for every O ∈ K. (vii) Haag duality. For every double cone O ∈ K there holds F(O ) = F(O) ,

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where O is the interior of the spacelike complement of O and, for every open set S ⊂ R4 , F(S) denote the algebra defined by F(S) = ∨O⊂S F(O).

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(viii) TCP covariance. There exists an antiunitary involution (the TCP operator) such that: U (, x) = U (, −x) ∀(, x) ∈ P; F(O) = F(−O). (ix)

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Bisognano–Wichmann property. Let WR = x ∈ R4 : x 1 > |x 0 | be the standard wedge and let and J be the modular operator and the modular conjugation of the algebra F(WR ) with respect to , respectively. Then there hold: it = U ((−2π t), 0); J = U (R1 (π ), 0),

(x)

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where (t) and R1 (θ ) are the one-parameter groups of Lorentz boosts in the x 1 -direction and of spatial rotations around the first axis, respectively. Split property. Let O1 , O2 ∈ K be open double cones such that the closure of O1 is contained in O2 (as usual we write O1 ⊂⊂ O2 ). Then there is a type I factor N(O1 , O2 ) such that F(O1 ) ⊂ N(O1 , O2 ) ⊂ F(O2 ).

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Using standard arguments (cf. [16]) it can be shown that the previous assumptions imply the irreducibility of the net F, namely the algebra F(R4 ) coincides with the algebra B(H) of all bounded operators on H. Another easy consequence of the assumptions is that is U -invariant. Moreover the algebra F(W) is a factor (in fact a type III1 factor), for every wedge W, see e.g. [5, Theorem 5.2.1]. Strictly speaking, it is also possible to deduce (viii) from the other assumptions [29, Theorem 2.10] and the separability of H from (vi) and (x) [23]. From Haag duality it follows that the algebra associated with a double cone coincides with intersection of the algebras associated to the wedges containing it, i.e. F(O) = ∩O⊂W F(W),

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for every O ∈ K. Thus our net F corresponds to a particular case of the AB-systems described in [46], see also [45]. Moreover the Bisognano–Wichmann property implies wedge duality, i.e. F(W) = F(W ),

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for every wedge W, where W denotes the interior of the causal complement of W. Another important fact is that, due to the split property, the net F satisfies Property ˜ O, O ˜ ∈ K, for each nonzero selfadjoint projection B for double cones: given O ⊂⊂ O, ˜ with E = W W ∗ . Moreover, for every E ∈ F(O) there exists an isometry W ∈ F(O) 4 nonempty open set S ⊂ R , the algebra F(S) is properly infinite. Definition 2.1. A covariant subsystem B of F is an isotonous nontrivial net of von Neumann algebras over K, such that: B(O) ⊂ F(O); U (L)B(O)U (L)∗ = B(LO),

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for every O ∈ K and every L ∈ P. We use the notation B ⊂ F to indicate that B is a covariant subsystem of F. As in the case of F, for every open set S ⊂ R4 we define B(S) by B(S) = ∨O⊂S B(O).

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Definition 2.2. We say that a covariant subsystem B of F is Haag-dual if B(O) = ∩O⊂W B(W) ∀O ∈ K.

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If a covariant subsystem B is not Haag-dual, one can associate to it an Haag-dual covariant subsystem Bd (the dual subsystem) defined by Bd (O) = ∩O⊂W B(W),

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cf. [45, 46]. Note that B(W) = Bd (W) for every wedge W. Given a covariant subsystem B of F we denote HB the closure of B(R4 ) and EB the corresponding orthogonal projection. It is trivial that the algebras B(O), O ∈ K ˆ leave HB stable. Hence we can consider the reduced von Neumann algebras B(O) :=


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ˆ the corresponding net. B(O)EB , O ∈ K acting on the Hilbert space HB and denote B It is straightforward to verify that ˆ B(S)EB = ∨O⊂S B(O),

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ˆ is unambiguous. Moreover, for every open set S ⊂ R4 . Therefore the notation B(S) ˆ due to the Reeh–Schlieder property (for F), the map B(S) B → Bˆ := BEB ∈ B(S) is an isomorphism of von Neumann algebras, whenever the interior S of the causal complement of S is nonempty. The following result is due in large part to Wichmann [46] and Thomas and Wichmann [45]. Proposition 2.3. Let B be a Haag-dual subsystem of F. Then the following properties hold: (a) and U commute with EB . Accordingly we can consider the reduced operators ˆ := EB and Uˆ := UEB on HB ; (b) All the properties from (i) to (x) listed in the beginning of this section hold with F, ˆ HB , Uˆ , , ˆ respectively. H, U , , replaced by B, Proof. For (a) and (b), properties from (i) to (ix), we refer the reader to [46] and [45, Sect. 5]. More precisely, since by assumption B(O) = ∩W⊃O B(W) we are in the position to apply [46, Theorem 4] (cf. [45, Theorem 5.3]). Accordingly, there hold (a), ˆ ˆ (b), properties (i)–(vi) and (viii)–(ix), and furthermore B(O) = ∩W⊃O B(W) for every O ∈ K. Thus we have ˆ ˆ ˆ ˆ ), B(O) = ∨W⊃O B(W) = ∨W ⊂O B(W ) = B(O

i.e. property (vii). ˆ corresponds to show that the split property is hereditary. This fact Proving (x) for B is well known (cf. e.g. [20, Sect. 5]) but we include here a proof for convenience of the reader. Let O1 , O2 ∈ K be such that O1 ⊂⊂ O2 . It is sufficient to show that there is a faithful ˆ 1 ) ∨ B(O ˆ 2 ) , i.e. a faithful normal state φ satisfying normal product state on B(O ˆ 1 ), ∀B ∈ B(O ˆ 2 ) , φ(BB ) = φ(B)φ(B ) ∀B ∈ B(O

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ˆ satisfies Haag duality and see e.g. [23]. B ˆ 1 ) ∨ B(O ˆ ) = [B(O1 ) ∨ B(O )]E B(O 2 2 B is isomorphic to B(O1 )∨B(O2 ), HB being separating for the latter algebra. Therefore it remains to show the existence of a faithful normal product state on B(O1 ) ∨ B(O2 ). This trivially follows from the existence of a faithful normal product state for F(O1 )∨F(O2 ), which is a consequence of the split property for F.2 2 A similar argument shows that split for wedges (cf. [41]) is inherited by subsystems satisfying wedge duality; here the space-time dimension is not important.

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ˆ satisfies Haag From the previous proposition it follows that if B is Haag-dual then B duality.3 It is quite easy to show that also the converse is true. This remark should make it clear that considering only Haag-dual subsystems is not too serious a restriction in our framework; these subsystems are exactly those satisfying Haag duality on their own vacuum Hilbert space. If B is a covariant subsystem of F, we can consider the net Bc defined by Bc (O) = B(R4 ) ∩ F(O), Bc

(20) Bc

is trivial, then we say that B is full (in F). If is nontrivial, then it is cf. [17, 5]. If easy to check that it is a Haag-dual covariant subsystem of F (the coset subsystem). It follows from the definition that B ⊂ Bcc , and Bc = Bccc . For later use it is convenient to introduce the notions of tensor product and of unitary equivalence of two nets. Let F1 and F2 be two nets acting on H1 and H2 respectively, and let U1 , U2 and 1 , 2 be the corresponding representations of the Poincaré group and the vacuum vectors. By tensor product of nets F1 ⊗ F2 we mean the net K O → F1 (O) ⊗ F2 (O) acting on H1 ⊗ H2 together with the representation U1 ⊗ U2 of P and the vacuum 1 ⊗ 2 . It follows that F1 ⊗ F2 satisfies properties (i)–(x) if F1 and F2 do so. We say that F1 and F2 are unitarily equivalent if there exists a unitary operator W : H1 → H2 with W F1 (O)W ∗ = F2 (O) (O ∈ K), W U1 (L)W ∗ = U2 (L). Note that since the vacuum is unique up to a phase, one can always choose W so that W 1 = 2 . 3. General Classification Results We recall that a representation of (the quasi-local C ∗ -algebra) F satisfies the DHR selection criterion, or is localizable, if it is unitarily equivalent to the vacuum representation of F in restriction to the C ∗ -subalgebra associated with the spacelike complement of any double cone O (the C ∗ -subalgebra of F generated by ∪O1 ⊂O F(O1 )). Unitary equivalence classes of such irreducible representations are called DHR superselection sectors or simply sectors.4 In this section we consider a net F satisfying all the properties (i)–(x) described in the previous section. Moreover we will assume the following condition (cf. [15]): (A) Every representation of F satisfying the DHR selection criterion is a multiple of the vacuum representation. Let us observe that condition (A) is equivalent to the seemingly weaker condition that all the irreducible representations satisfying the selection criterion are equivalent to the vacuum representation. This is a consequence of the fact that almost all the irreducible representations occurring in the direct integral decomposition of a localizable5 representation are localizable (see [34, Appendix B]). Now let B be a Haag-dual, covariant subsystem of F and let π be the correspondˆ on H, i.e. the representation defined by π(B) ˆ = B for B ∈ ing representation of B 0 ∪O∈K B(O). We denote π the identical (vacuum) representation of F on H and π0 the ˆ i.e. its identical representation on HB . Note that by the asvacuum representation of B, sumptions H and HB are both (infinite dimensional and) separable. The following result is already known (see e.g. [15]) but we include a proof for the sake of completeness. 3 This is not true in two spacetime dimensions. 4 For the basic notions concerning the DHR theory of superselection sectors we refer the reader to [31] and

references therein. 5 In this article the word localizable refering to representations means localizable in double cones.


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Lemma 3.1. π satisfies the DHR criterion. ˆ ) are isomorphic. Proof. For every O ∈ K the von Neumann algebras B(O ) and B(O Moreover, as noted in the previous section, these von Neumann algebras are properly infinite with properly infinite commutants. By [32, Theorem 7.2.9.] and [32, Proposition 9.1.6.] we can find a unitary operator UO : HB → H such that ˆ O ∗ = B ∀B ∈ B(O ). UO BU Hence if O1 ∈ K is contained in O there holds ˆ 1 ). ˆ = UO ∗ π(B)U ˆ O ∀Bˆ ∈ B(O π0 (B) Actually, this is the DHR criterion.

ˆ is As usual we say that an endomorphism σ of a quasi-local C ∗ -algebra, say B, ˆ ˆ ˆ ˆ localized in a given double cone O if σ (B) = B for all B ∈ ∪O1 ⊂O B(O1 ); furthermore, a localized endomorphism (i.e. localized in a double cone) is transportable if it is inner equivalent to an endomorphism localized in any other double cone. It is a very important ˆ satisfies the DHR selection criterion if and only if it is fact that a representation of B unitarily equivalent to a representation of the form π0 ◦ σ , where σ is a localized and transportable endomorphism. ˆ π0 ◦ σ Proposition 3.2. For every irreducible localized transportable morphism σ of B, is equivalent to a subrepresentation of π. Moreover σ is covariant with positive energy and it has finite statistical dimension. Proof. Since π satisfies the DHR criterion we can find a transportable localized morˆ such that there holds the unitary equivalence phism ρ of B π π0 ◦ ρ,

(21)

cf. [40, Proposition 3.4.]. Let us consider the extension σˆ of σ to F [15], cf. [40]. Then the assumption (A) for F implies that π 0 ◦ σˆ ⊕i π 0 ,

(22)

where the index i in the direct sum on the r.h.s. runs over a set whose cardinality is at most countable. Restricting these representations to B we find π ◦ σ ⊕i π,

(23)

ρσ ⊕i ρ.

(24)

and therefore using Eq. (21)

Since ρ contains the identity sector we have σ ≺ ρσ and hence σ ≺ ⊕i ρ.

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ˆ satisfying the DHR Thus, σ being arbitrary, every irreducible representation of B criterion is contained in a countable multiple of ρ. The latter multiple is a representation ˆ on a separable Hilbert space. Hence there are at most countably many sectors of B. π being a direct integral of irreducible DHR representations [34, Appendix B] and appealing to some standard arguments (see e.g. [18, 19]) one gets that π is in fact a direct sum. From Eq. (25) it is not difficult to show that, σ being irreducible, we have σ ≺ ρ i.e. π0 ◦ σ is unitarily equivalent to a subrepresentation of π . Since B is covariant π is covariant with positive energy. We have to show that every irreducible subrepresentation has the same property, cf. [4]. Since the action induced by the representation U of the Poincaré group leaves B(R4 ) globally invariant it leaves globally invariant also its centre. The latter being purely atomic (due to the decomposition of π into irreducibles) and P connected, it follows that the orthogonal projection E[σ ] ∈ B(R4 ) ∩ B(R4 ) onto the isotypic subspace corresponding to σ must commute with U . Let U[σ ] and π[σ ] be the restrictions to E[σ ] H of U and π respectively. Then we have the unitary equivalence π[σ ] (π0 ◦ σ ) ⊗ I.

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ˆ [σ ] (L)∗ = π[σ ] (Uˆ (L)Bˆ Uˆ (L)∗ ), U[σ ] (L)π[σ ] (B)U

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Moreover, using the relation

where B ∈ ∪O∈K B(O), L ∈ P, and a classical result by Wigner on projective unitary representations of P [47, 2], it is quite easy to show that U[σ ] (L) Uσ (L) ⊗ Xσ (L),

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where Uσ and Xσ are unitary continuous representations of (the covering group of) P and Uσ is such that ˆ σ (L)∗ = σ (Uˆ (L)Bˆ Uˆ (L)∗ ). Uσ (L)σ (B)U

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Since U[σ ] satisfies the spectrum condition, both Uσ and Xσ have to satisfy it.6 Hence σ is covariant with positive energy. Finally, from ρσ σρ and Eq. (24) it follows that id ≺ σρ. Therefore, σ being covariant with positive energy, it has finite statistical dimension because of [22, Prop. A.2]. A related result has been independently obtained by R. Longo, in the context of nets of subfactors [39]. ˆ as defined in [25, Sect. 3]. In a natural way Let FB be the canonical field net of B FB can be considered as a Haag-dual subsystem of F containing B [15, Theorem 3.5]. In fact one finds that FB (O) coincides with the von Neumann algebra generated by the family of Hilbert spaces Hσˆ in F, where σ runs over all the transportable morphisms of B which are localized in O and σˆ denotes the functorial extension of σ to F. From the fact that the latter extension commutes with spacetime symmetries, namely (σL )ˆ = (σˆ )L for every L ∈ P it is also easy to show that FB is a covariant subsystem. (Besides, by [13, Prop. 2.1] FB coincides with its covariant companion, cf. [25].) 6 This follows from the fact that if S and S are two orbits of the restricted Lorentz group such that 1 2 S1 + S2 ⊂ V + then S1 ⊂ V + and S2 ⊂ V + .


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Theorem 3.3. F B has no DHR sectors other than the vacuum. Proof. By the previous proposition it is enough to consider sectors with finite statistical dimension. Let R be the canonical field net of F B . Then R is a Haag-dual covariant subsystem of F, and as such it inherits the split property. By the results discussed in ˜ of the (unbroken) [8] this is sufficient7 to deduce that FB = R.8 In fact the group G symmetries of R extending the gauge automorphisms of FB is compact in the strong ˜ operator topology by (the proof of) [23, Theorem 10.4], and obviously RG = B. The conclusion follows by the uniqueness of the canonical field net [25]. c Theorem 3.4. There exists a unitary isomorphism of F with F B ⊗ B which maps F B c ˆ ˆ into F ⊗ B for every O ∈ K, F ∈ FB (O) and B ∈ B (O). In particular FB = Bcc , and if B is full 9 in F then FB = F. Proof. Let π˜ be the representation of F B on H (the vacuum Hilbert space of F) arising from the embedding FB ⊂ F and π˜ 0 the vacuum representation of F B on HFB ⊂ H. By the previous theorem F has no nontrivial sectors. Moreover Lemma 3.1 applied to B FB instead of B implies that π˜ is (spatially) equivalent to a multiple of π˜ 0 and therefore to π˜ 0 ⊗I, on HFB ⊗H1 , where H1 is a suitable Hilbert space. Let W : H → HFB ⊗H1 be a unitary operator implementing this equivalence. Thus we have W F W ∗ = Fˆ ⊗ I for all F ∈ FB . For every double cone O it holds ˜ F B (O ) ⊗ I ⊂ F(O ),

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˜ where F(O) = W F(O)W ∗ . Therefore, using Haag duality for F B, ˜ F B (O) ⊗ I ⊂ F(O) ⊂ FB (O) ⊗ B(H1 ).

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˜ F B (W) ⊗ I ⊂ F(W) ⊂ FB (W) ⊗ B(H1 )

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It follows that

for every wedge W. The algebras of wedges are factors. By the results in [27] (cf. also [44]) there exists a von Neumann algebra M(W) ⊂ B(H1 ) such that ˜ F(W) = F B (W) ⊗ M(W).

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Taking on both sides of this equality the intersection over all the wedges containing a given O ∈ K we find ˜ F(O) = F B (O) ⊗ M(O),

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M(O) = ∩O⊂W M(W).

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where

7 This idea is not new, see e.g. [42, Sect. 2], however some technical difficulties are circumvented when the assumptions made in this paper are used. 8 Alternatively, the same result may be deduced combining Proposition 3.2 with [15]. 9 Irreducible subsystems, namely those satisfying B ∩ F = C, are full.

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Now, using the commutation theorem for von Neumann tensor products, it is straightforward to show that I ⊗ M(O) = W Bc (O)W ∗ for every O ∈ K. The previous equation implies the existence of a representation τ of c on H1 such that W BW ∗ = I ⊗ τ (B), ˆ B ∈ Bc (O) for every O ∈ K. Moreover, since B c is contained in I ⊗ τ , M acts irreducibly on H1 and the vacuum representation π c of B c τ is spatially equivalent to π and thus the mapping O → M(O) gives a net unitarily c . Therefore without loss of generality we can assume that H1 = HBc equivalent to B c and that W F(O)W ∗ = F B (O) ⊗ B (O), O ∈ K. Furthermore we can also assume that ∗ ˆ ˆ W F BW = F ⊗ B, where F ∈ FB (O) and B ∈ Bc (O). The conclusion follows by noticing that W U W ∗ = UEFB ⊗ UEBc . Here we omit the easy details. Applying the previous theorem to Bc in place of B we get that Bc has no nontrivial sectors, since FBc = Bccc = Bc . Corollary 3.5. Let B be a Haag-dual covariant subsystem of F, then the net of inclusions G K O → B(O) ⊂ F(O) is (spatially) isomorphic to O → F B (O) ⊗ I ⊂ FB (O) ⊗ ˆ c (O), where G is the canonical gauge group of B. B Proof. From Theorem 3.4 the net of inclusions K O → B(O) ⊂ F(O) is spatially c isomorphic to O → B(O)EFB ⊗ I ⊂ F B (O) ⊗ B (O). From [15, Theorem 3.5] O → ˆ B(O)E ⊂ F B (O) is isomorphic to the canonical embedding of B as a fixpoint net of FB

FB [25], hence the conclusion follows.

Corollary 3.6. If B is a Haag-dual covariant subsystem of F and if FB is full (in particular if B is full) then there exists a compact group G of unbroken internal symmetries of F such that B = F G . Now let C be the (local) net generated by the canonical implementations of the translations on F [12]. It is a covariant subsystem of F. Since C is (irreducible thus) full in F and Cd ⊂ F Gmax , where Gmax is the (compact) group of all the unbroken internal symmetries of F, we have Corollary 3.7. In the situation described above it holds Cd = F Gmax .

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4. Applications 4.1. Free fields. Our standing assumptions are satisfied in the case where F is generated by a finite set of free scalar fields [26, 7] and also by suitable infinite sets of such fields [24]. They are also satisfied in other Bosonic theories, e.g. when F is generated by the free electromagnetic field, see [7]. Therefore from our Corollary 3.6 one can obtain all the results in [17] in the case of full subsystems, even without assuming the existence of a mass gap. Concerning subsystems that are not full, one has to study the possible decompositions c F B (O) ⊗ B (O) = F(O)

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(up to unitary equivalence). In the case where F is generated by a finite set of free scalar fields, it turns out that FB and Bc are always free scalar theories generated by two suitable disjoint subsets of the generating fields of F. We present a detailed proof of this fact in the appendix.10 In particular, if F is generated by a single scalar free field ϕ(x) of mass m ≥ 0, no such nontrivial decomposition is possible and hence all the subsystems of F are full. Accordingly, in this case, the unique Haag-dual covariant proper subsystem of F is the fixed point net F Z2 under the action of the group of (unbroken) internal symmetries. Note that when m = 0 there are covariant subsystems which are not Haag-dual. For instance the subsystem A ⊂ F generated by the derivatives ∂µ ϕ(x) is Poincaré covariant but not Haag-dual and in fact one has F = Ad [7]. However it is not hard to show that conformally covariant subsystems of F are always Haag-dual (actually the latter fact still holds in the more general context where the conformally covariant net F is not necessarily generated by free fields). 4.2. Theories with countably many sectors. In this subsection we discuss the classification problem for subsystems of a canonical field net obtained through the DR recipe, with a special emphasis on the subsystem generated by the canonical local implementations of the spacetime translations. Of course, a closely related problem is to look for the structural hypotheses on A ensuring that F = FA will have the required properties. It has been known for some time that if A has only a finite number of DHR sectors with finite statistical dimension (i.e. A is rational), all of which are Bosonic, then F (is local and) has no nontrivial DHR sectors with finite statistical dimension [13, 42]. This result is not sufficient for our purposes, because it does not rule out the possible presence of irreducible DHR representations of F with infinite statistical dimension. However, a solution to this problem can be reached by using the stronger results given in [15]. Theorem 4.1. Let A be a local net satisfying the split property and Haag duality in its (irreducible) vacuum representation. If A has at most countably many (DHR) sectors, all of which are Bosonic and with finite statistical dimension, then any DHR representation of A is (equivalent to) a direct sum of irreducible ones. Moreover, the canonical field net F of A has no nontrivial sectors with any (finite or infinite) statistical dimension. Proof. In view of [15, Theorem 4.7] it is enough to show the first statement. But using the split property and the results in [34, Appendix B] one finds that every DHR representation of A is a direct integral of irreducible DHR representations. Taking into account the bound on the number of sectors, it follows that every DHR representation of A is (equivalent to) a direct sum of irreducible ones, cf. the proof of Proposition 3.2. This result11 shows that F satisfies the condition (A) of section 3. Moreover if A satisfies all of the conditions (i)–(vii) then the same is true for F [25]. In order to apply the above result about classification of subsystems and solve the problem concerning the abstract analogue for the energy-momentum tensor, we need to know conditions on A implying the validity of properties (viii)–(x). Concerning (x), it would be a consequence of the split property for A if G were finite and abelian [20]. In other cases one can invoke some version of nuclearity for A, implying that F is split [6]. But it is also necessary 10 Davidson obtained this result in the purely massive case [17]. 11 As in [13], in the case of rational theories a different argument could be given when the local algebras

are factors, based on a restriction-extension argument (cf. [34, Lemma 27]).

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to know if the existence of a TCP symmetry and the special condition of duality for A imply the same for its canonical field system F. The relationship between the validity of conditions (viii)–(ix) for A and its canonical field system F has been discussed in [35, 36] (the TCP symmetry has been also treated in [14] under milder hypotheses). The conclusion is that if A satisfies the usual axioms (and all its sectors are covariant with positive energy, a condition which in the present situation can be omitted in view of the results in [28, Sect. 7]), moreover it is purely Bosonic and satisfies a suitable version of nuclearity (implying, among other things, the existence of at most countably many sectors), TCP covariance and the Bisognano–Wichmann property, then we know how to classify all the subsystems of F satisfying Haag duality. Corollary 4.2. Let A be an observable net satisfying the properties (i)–(ix) above, without DHR sectors with infinite statistical dimension or para-Fermi statistics of any finite order, whose (Bosonic) canonical field net F has the split property. Then, if C is the net generated by the canonical local implementation of the spacetime translations, one has Cd = F Gmax . Moreover A = Cd if and only if A has no proper full Haag-dual subsystem (in which case A has no unbroken internal symmetries). Proof. Since A satisfies the split property and has at most countably many sectors, all with finite statistics, the first statement follows by the previous result and Corollary 3.3. If G denotes the canonical gauge group of A, so that A = F G , the equality A = Cd is equivalent to the equality G = Gmax , which, due to Corollary 3.6, means that there is no proper subsystem of A full (or irreducible) in F. To complete the proof we only need to show that every full subsystem of A is full in F, when G = Gmax . Let B be a (Haag-dual) subsystem of A. Due to the results in the previous section, for every wedge W the inclusions B(W) ⊂ A(W) ⊂ F(W) are spatially isomorphic to ˆ c ˜ B(W) ⊗ I ⊂ A(W) ⊂ F B (W) ⊗ B (W), with A˜ isomorphic to A. Moreover, from G = Gmax it follows that ˆ c (W). ˜ A(W) ⊂ B(W) ⊗B Arguing as in the proof of Theorem 3.4 we find that if B is not full in F then for every O ∈ K, the algebra B(R4 ) ∩ A(O) is nontrivial. It follows that B is not full in A. 5. Comments on the Assumptions Some of the results of the previous sections are in fact still true even after relaxing some conditions. We will briefly discuss some aspects here. The hypothesis (x) is useful to derive property B (also for the subsystems), to apply the results in [34] and also to define the local charges. If we renounce to (x), and possibly (A), taking F as the DHR field net of A ⊃ B in its vacuum representation on H (here it is not even essential to require the condition of covariance, nor the additional assumptions


101

of the main theorem in [15]), it is still possible to deduce that π˜ π˜ 0 ⊗ I as in the proof of Theorem 3.4. For this purpose one needs to know that A and B both satisfy property B, ˆ is quasi-contained in and that π˜ in restriction to B (thought of as a representation of B) ˆ the canonical embedding of B into its field net. By the results in [15], the latter property holds if it is possible to rule out the occurrence of representations with infinite statistics ˆ acting on H (e.g. if [A : B] < ∞ in the case of nets of subfactors). In fact we for B do not even need to know a priori that π satisfies the DHR selection criterion. Relaxing covariance is necessary to discuss QFT on (globally hyperbolic) curved spacetimes. Possibly results resembling those presented here should hold also in that context (cf. [30]). The Bisognano–Wichmann property for F and TCP covariance may also be relaxed, but, for the time being, F and the considered subsystems always have to satisfy Haag duality in order to deduce some nice classification result. However, let us discuss the inheritance of the split property in a slightly more general situation. We start with a subsystem B ⊂ F, but now both F and B are only assumed to satisfy essential duality (cf. [31]) in their respective vacuum representation, namely ˆ d = (B) ˆ dd (this is consistent with the notation adopted in the previous F d = F dd and (B) sections). Moreover we require the split property for F d . In the situation where one has ˆ d inside F d ,12 we may deduce the split property for (B) ˆ d by our an embedding of (B) previous argument. For instance if F satisfies the Bisognano–Wichmann property (thus ˆ satisfies the same in particular wedge duality, which implies essential duality), then B property as well [46] and moreover there exists the embedding alluded above, therefore ˆ d .13 the split property for F d entails the split property for (B) 6. Conclusion and Outlooks Summing up, we have shown a classification result for Haag-dual subsystems of a purely Bosonic net F with trivial superselection structure (including infinite statistics) and with the split property. Moreover we have exhibited an interesting class of examples, namely (multiplets of) the free fields, to which our results apply. Finally we have considered the much more general situation where F is the canonical field net of an observable net A. In this article we have not discussed graded local (Fermionic) nets. As far as we can see, it should be possible to obtain classification results also in this case, once the natural changes in the assumptions, the statements and the proofs are carried out. We hope to return on this subject in the future. A. Appendix In this appendix we study the possible tensor product decompositions of a net generated by a finite number of scalar free fields. We consider a net O → F(O), acting irreducibly on its vacuum Hilbert space H, generated by a finite family of Hermitian scalar free fields ϕ1 (x), ϕ2 (x) . . . , ϕn (x), where n = n1 + n2 + . . . + nk and ϕ1 (x), . . . , ϕn1 (x) have mass m1 , ϕn1 +1 (x), . . . , ϕn1 +n2 (x) have mass m2 , and so forth, and 0 ≤ m1 < . . . < mk . 12 This may be true or not and is related to the validity of the equality (B) ˆ d = (Bd )ˆ. 13 As a matter of fact, the same argument goes through when we just have essential duality for F and wedge

ˆ see e.g. [15, Sect. 3]. duality for B,

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Accordingly, for each O ∈ K, F(O) is the von Neumann algebra generated by the Weyl unitaries eiϕj (f ) for j = 1, . . . , n and real-valued f ∈ S(R4 ) with support in O. We denote U, , the corresponding representation of P, TCP operator and vacuum vector respectively. For every i we let Ki be the closed subspace of H generated by the vectors ϕi (f ) with f ∈ S(R4 ). Each Ki is U -invariant, and the restriction Vi of U to Hi is the irreducible representation of P with spin 0 and corresponding mass. Moreover the generating fields are chosen so that Ki is orthogonal to Kj for i # = j . If K = ⊕ni=1 Ki and V = ⊕ni=1 Vi , then H can be identified with the (symmetric) Fock space .(K) and U with the second quantization representation .(V ), see e.g. [43]. If Fi is the covariant subsystem of F generated by ϕi (x), then HFi can be identified with .(Ki ) and from the relation F(O) = ∨i Fi (O) and the properties of the second quantization functor it follows that the net F is isomorphic to Fˆ1 ⊗ . . . ⊗ Fˆn on ⊗i .(Ki ). Note that there is some freedom in the choice of the generating fields, corresponding to the internal symmetry group G = O(n1 ) × . . . × O(nk ). nh−1 +nh K , where Let Emh be the orthogonal projection from H onto Kmh := ⊕i=n h−1 +1 i by convention n0 = 0. For each m ≥ 0, let Pm be the orthogonal projection onto Ker(P 2 −m2 ), where P 2 denotes the mass operator corresponding to U . It is not difficult to see that Pm (K + C )⊥ = 0 by a direct calculation on the k-particle subspaces of H (note that Pm = 0 whenever m ∈ / {0} ∪ {m1 , . . . , mk }). It follows that Pmh = Emh if mh > 0, while for mh = 0 we have Pmh = Emh + P , where P ∈ U (P) ∩ U (P) is the orthogonal projection onto C . In particular, for any h ∈ {1, . . . , k} we have Emh ∈ U (P) ∩ U (P) . The following simple lemma will be used to study the tensor product decomposition of F. Lemma A.1. Let U1 and U2 be subrepresentations of U on subspaces H1 and H2 of H both orthogonal to C . Then there are no eigenvectors for the mass operator corresponding to the representation U1 ⊗ U2 . Proof. We consider the net F˜ = F ⊗F and the corresponding representation U˜ = U ⊗U of P. Obviously the net F˜ is of the same type as F, with the same masses but different multiplicities. U1 ⊗ U2 is a subrepresentation of U˜ on H1 ⊗ H2 . If P˜ 2 is the mass operator corresponding to U˜ and P˜m is the orthogonal projection onto Ker(P˜ 2 − m2 ), we only have to show that for every m ≥ 0 we have P˜m H1 ⊗ H2 = 0. But this follows by the discussion in the last paragraph before the statement, since H1 ⊗H2 is orthogonal ˜ where K ˜ = K ⊗ + ⊗ K is the one-particle subspace of H ⊗ H. to C( ⊗ ) + K, We are now ready to study the possible tensor product decompositions FA ⊗FB of F. In the sequel we assume to have such a decomposition, and deduce some consequences. Then H is given by HA ⊗ HB so that = A ⊗ B and U = UA ⊗ UB . ˜ A , and analogously for B, so that H = HA ⊗ HB = We set HA = C A ⊕ H ˜ B )⊕(H ˜ A ⊗ B )⊕(H ˜ A ⊗H ˜ B ). We also set F0 = P , FA = [ A ⊗ H ˜ B ], C ⊕( A ⊗ H ˜ A ⊗ B ], FAB = [H Ã ⊗H ˜ B ]. Notice that these orthogonal projections FB = [H commute not only with U but also with . Lemma A.2. For each h = 1, . . . , k it holds Emh FAB = 0.


103

Proof. It is an immediate consequence of Lemma A.1.

Since Emh F0 = 0, the previous lemma implies that Emh (FA + FB ) = Emh , for ˜ A ⊗ B ⊕ A ⊗ H ˜ B . As a consequence, h = 1, . . . , k. This amounts to say that K ⊂ H with the aid of some linear algebra and the fact that FA and FB commute with , it is not difficult to show that there is a partition in two disjoint sets {1, . . . , n} = αA ∪ αB along with a suitable choice of the generating fields such that, for every f ∈ S(R4 ), ˜ A ⊗ B for i ∈ αA , ϕi (f ) ∈ H

˜ B for i ∈ αB . ϕi (f ) ∈ A ⊗ H

(38)

Because of Eqs. (38), for every f ∈ S(R4 ) and i ∈ αA one can define a vector ˜ A by Ti (f ) ∈ H ϕi (f )( A ⊗ B ) =: Ti (f ) ⊗ B .

(39)

It follows that if supp(f ) ⊂ O, f real, and XA ∈ FA (O ), XB ∈ FB (O ), we get that ϕi (f )(XA A ⊗ XB B ) = ϕi (f )(XA ⊗ XB )( A ⊗ B ) = (XA ⊗ XB )ϕi (f )( A ⊗ B ) = XA Ti (f ) ⊗ XB B , i ∈ αA .

(40)

By a continuity argument (we are assuming ϕi (f ) to be closed), ϕi (f )(XA A ⊗ ξ ) = XA Ti (f ) ⊗ ξ

∀ ξ ∈ HB .

Therefore, for every T ∈ B(HB ), (I ⊗ T )(XA A ⊗ XB B ) belongs to the domain of ϕi (f ) and (I ⊗ T )ϕi (f )(XA A ⊗ XB B ) = ϕi (f )(I ⊗ T )(XA A ⊗ XB B ).

(41)

Hence again by continuity we find that, for every X ∈ F(O ), (I ⊗ T )ϕi (f )X = ϕi (f )(I ⊗ T )X , i ∈ αA .

(42)

Similarly, for each T ∈ B(HA ), (T ⊗ I )ϕi (f )X = ϕi (f )(T ⊗ I )X , i ∈ αB .

(43)

Our next goal is to show that F(O ) is a core for ϕi (f ) for any real f as above and i = 1, . . . , n. This will entail that eiϕi (f ) ∈ (I ⊗ B(HB )) = B(HA ) ⊗ I for every real-valued test function f with compact support (by arbitrariness of O in the argument above) and i ∈ αA , and similarly eiϕi (f ) ∈ I ⊗ B(HB ) for i ∈ αB , from which it is easy to see that ∨i∈αA Fi (O) = FA (O) ⊗ I and ∨i∈αB Fi (O) = I ⊗ FB (O), O ∈ K. Proposition A.3. For any f ∈ S(R4 ) real, O ∈ K and i = 1, . . . , n, F(O) contains a core for ϕi (f ). In particular if supp(f ) ⊂ O then F(O ) is a core for ϕi (f ).

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Proof. We use some techniques concerning energy-bounds, cf. [3, Sect. 13.1.3]. Let N be the total number operator acting on H = .(K). Then N is the closure of i Ni with Ni the number operator on .(Ki ). Using well known estimates about free fields (see [43, Sect. X.7]) for every real f and ψ in the domain of N we have √ ϕi (f )ψ ≤ c(f ) N + I ψ ≤ c(f )(N + I )ψ (44) for some constant c(f ) depending only on f . Moreover ϕi (f ) is essentially self-adjoint on any core for N . We define a self-adjoint operator H as (the closure of) the sum of the Hi , where Hi on .(Ki ) is the conformal Hamiltonian if ϕi (x) has vanishing mass and the generator of time translations otherwise. Note that Ni2 ≤ ci2 Hi2 , where ci is the inverse of the mass corresponding to ϕi (x) if that is different from 0, and equal to 1 otherwise. It follows that, for ψ in the domain of H , ϕi (f )ψ ≤ b(f )(H + I )ψ

(45)

for some constant b(f ). Thus, since N is essentially self-adjoint on the domain of H , ϕi (f ) is essentially self-adjoint on any core for H . To complete the proof we only need to show that, for each O ∈ K, F(O) contains a core for H . But this follows from [10, Appendix], after noticing that given O1 ⊂⊂ O then eitH F(O1 )e−itH ⊂ F(O) for |t| small enough. Summing up, we have thus proved the following result. Theorem A.4. Let F be the net generated by a finite family of free Hermitian scalar fields and let F = FA ⊗ FB be a tensor product decomposition, then, for a suitable choice ϕ1 (x), . . . , ϕn (x) of the generating fields for F and a k ∈ {1, . . . , n}, FA ⊗ I is generated by ϕ1 (x), . . . ,ϕk (x) and I ⊗ FB by ϕk+1 (x), . . . ,ϕn (x). Acknowledgements. We thank D. R. Davidson, S. Doplicher, and J. E. Roberts for some useful comments and discussions at different stages of this research. Part of this work has been done while R. C. was visiting the Department of Mathematics at the University of Oslo. He thanks the members of the operator algebras group in Oslo for their warm hospitality and the EU TMR network “Non-commutative geometry” for financial support.

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Commun. Math. Phys. 217, 107 – 126 (2001)

Communications in



A Change of Coordinates on the Large Phase Space of Quantum Cohomology Alexandre Kabanov1, , Takashi Kimura2, 1 Mathematik Departement, ETH-Zentrum, Rämistrasse 101, 8092 Zurich, Switzerland.


2 Department of Mathematics, 111 Cummington Street, Boston University, Boston, MA 02215, USA.

E-mail: [email protected] Received: 2 August 1999 / Accepted: 30 September 2000

Abstract: The Gromov–Witten invariants of a smooth, projective variety V , when twisted by the tautological classes on the moduli space of stable maps, give rise to a family of cohomological field theories and endow the base of the family with coordinates. We prove that the potential functions associated to the tautological ψ classes (the large phase space) and the κ classes are related by a change of coordinates which generalizes a change of basis on the ring of symmetric functions. Our result is a generalization of the work of Manin–Zograf who studied the case where V is a point. We utilize this change of variables to derive the topological recursion relations associated to the κ classes from those associated to the ψ classes. 0. Introduction Notation. All (co)homologies are understood to have Q coefficients unless otherwise stated. Summation over repeated upper and lower indices is assumed. The theory of Gromov–Witten invariants of a smooth projective variety V has developed at a rapid pace cf. [4, 6, 32, 44]. These are multilinear operations on the cohomology H • (V ) which can be constructed from intersection numbers on the moduli space of stable maps into V , Mg,n (V ). In particular, the genus zero Gromov–Witten invariants endow H • (V ) with the structure of the quantum cohomology ring of V . The existence of these invariants was foreseen by physicists who encountered these operations as correlators of a topological sigma model coupled to topological gravity [47]. These invariants are of great mathematical interest, for example, because they are symplectic invariants of V [36] and because of their close relationship to problems in enumerative geometry [32]. Gromov–Witten invariants satisfy relations (factorization identities) parametrized by the relations between cycles on the moduli space of stable curves Mg,n . These Research of the first author was partially supported by NSF grant number DMS-9803553.

Research of the second author was partially supported by NSF grant number DMS-9803427.

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A. Kabanov, T. Kimura

relations can be formalized by stating that the space (H • (V ), η) (where η is the Poincaré pairing) is endowed with the structure of a cohomological field theory (CohFT) in the sense of Kontsevich–Manin [32]. The Gromov–Witten invariants are characterized by its generating function (the small phase space potential) (x), where x := { x α } are coordinates associated to a basis of H • (V ). Restricting to genus zero, (x) essentially endows (H • (V ), η) with the structure of a (formal) Frobenius manifold [11, 22, 38]. It is precisely the structure of a CohFT which was used by Kontsevich–Manin to compute the number of rational curves on CP2 [32] and the number of elliptic curves by Getzler [17] (where the number is counted with suitable multiplicities). Furthermore, there are tautological cohomology classes (denoted by ψi ) associated to the universal curve on Mg,n (V ) for all i = 1, . . . , n which are the first Chern class of tautological line bundles over Mg,n (V ). These classes are a generalization of the ψ classes on Mg,n due to Mumford. What is remarkable is that by twisting the Gromov– Witten invariants by these ψ classes to obtain the so-called gravitational descendents, one endows (H • (V ), η) with the structure of a formal family of CohFT structures whose base is equipped with coordinates t := { taα }, where the integer a ≥ 1 and α is as above. The associated generating function F(x; t) (the large phase space potential) reduces to (x) when t vanishes. The large phase space potential F is itself a remarkable object as its exponential is conjectured to satisfy a highest weight condition for the Virasoro algebra [12], a conjecture which has nontrivial consequences [21]. Indeed, when V is a point, this condition is equivalent to the Witten conjecture [47] proven by Kontsevich [31]. There are other tautological cohomology classes on Mg,n (V ) associated to its universal curve. In this paper, we define generalizations to Mg,n (V ) of the “modified” κ classes (due to Arbarello–Cornalba [1]) on Mg,n . We define Gromov–Witten invariants twisted by the κ classes and prove that we obtain a formal family of CohFTs on (H • (V ), η) whose base is endowed with coordinates s := { saα }, where a ≥ 0. We then prove that the generating function G(x; s) associated to this family can be identified with the large phase space potential F(x; s) through an explicit change of variables. This change of variables can be interpreted as a change of basis in the space of symmetric functions whose variables take values in H • (V ). The variables s can be interpreted as another canonical set of coordinates on the large phase space. We also utilize this change of variables to derive topological recursion relations for G in terms of those of F. When V is convex, the κ classes on M0,n (V ) had already been introduced in [25] and the genus zero topological recursion relations were proven. This paper generalizes those results to situations where Mg,n (V ) need not have the expected dimension (and, hence, the technicalities of the virtual fundamental class cannot be avoided) as well as deriving the change of variables on the large phase space. When V is a point, our formula reduces to the work of Kaufmann–Manin–Zagier [27] and [39] who noted (see also [34]) that, in addition, the coordinates s are additive with respect to the tensor product in the category of CohFTs. Manin–Zograf [39] used this formula to compute asymptotic Weil–Peterson volumes of the moduli spaces Mg,n as n → ∞ (this was done for g = 1 in [24]). However, this additivity property need not hold for a general variety V . It is worth pointing out several generalizations. First of all, when V is a point, Manin– Zograf use the Witten conjecture to show that their change of variables can be directly interpreted as arising from an analogous change of the cohomology classes appearing in the potential functions. It would be interesting to obtain an analogous result for a general

Change of Coordinates on Large Phase Space of Quantum Cohomology

109

V . Secondly, the above construction should be feasible for any CohFT and there should be coordinates which are additive under tensor product – such a construction would be useful in studying the ring of CohFTs. Work towards this direction is in progress [23]. The third is the fact that there are yet another set of tautological classes (called λ) on Mg,n associated to the Hodge bundles. Twisting Gromov–Witten invariants by both the κ and λ classes, one obtains the very large space [24, 39] (see also [13]). A subspace of the latter yields coordinates on the moduli space of nondegenerate rank one CohFTs in genus 1 [24] which are additive under tensor product. It would be interesting to understand the role of these additional coordinates for general V . The first section of the paper is a review of the technicalities necessary to push forward and pull back cohomology classes on the moduli space of stable maps. This includes Gysin morphisms and the flat push-forward. In the second section, we review the basic properties of the moduli space of stable maps, the structure of the boundary classes, and properties of the virtual fundamental classes. In the third section we introduce the tautological κ and ψ classes, and prove their restriction properties on the boundary classes. In the fourth section, we define the notion of a CohFT and its potential function. We review the large phase space potential F. We prove that by introducing the κ classes, (H • (V ), η) is endowed with a formal family of CohFT structures together with coordinates on the base of the family. In the fifth section, we prove that after an explicit change of coordinates, the potential G can be identified with the large phase space potential F. In the final section, we derive the topological recursion relations for G in genus 0 and 1 and derive the usual topological recursion relations for F through the change of variables.

1. Technical Preliminaries In this section we present several technical points needed in the sequel. They are concerned with the Gysin morphisms in homology and cohomology. You may skip this section provided you are willing to accept that everything works at a “naive” level. An article of Fulton and MacPherson [15] may serve as a general reference to this section. All references mentioned in this section deal with schemes rather than stacks, but the sheaf-theoretic approach allows one to work in the category of stacks. If F is a functor, then RF denotes the corresponding derived functor. Let π : Y → X be a flat representable morphism of Deligne–Mumford stacks with fibers of pure dimension d. As explained in [8], π defines the natural morphism of T rπ : R 2d Q → Q, which induces the corresponding flat push-forward in cohomology with compact supports π∗ : Hck (Y ) → Hck−2d (X). (The axioms uniquely defining the morphism T r are also given in [45] and [35].) One of the axioms defining the T r morphism states that it commutes with base change, that is, with pull-back on cohomology in a fibered square. However, in this paper we will need to consider commutative squares which are a little more general than the fibered squares. That is the reason for giving the following definition.

110


Definition 1.1. Let X1 , Y1 , X, Y be Deligne–Mumford stacks. A commutative square f1

Y1 −−−−→  π 1

Y  π

f

X1 −−−−→ X is called close to a fibered square if the induced morphism g : Y1 → X1 ×X Y is a proper birational morphism, and there is an open subset U of X1 ×X Y whose intersection with each fiber of π1 is a dense subset of the fiber such that g|g −1 U is an isomorphism. Lemma 1.2. If the commutative square f1

Y1 −−−−→  π 1

Y  π

f

X1 −−−−→ X is close to a fibered one, and π and π1 are representable flat morphisms with fibers of pure dimension d, then π∗ f ∗ = f1∗ π1∗ : H • (Y ) → H • (X1 ). Proof. Consider the following diagram g

pr2

Y1 −−−−→ X1 ×X Y −−−−→     pr1 π1 X1

X1

Y   π

f

−−−−→ X,

where pr2 g = f1 . Since the right square is a fibered square it follows from the properties of the T r morphism that π∗ f ∗ = pr2∗ pr1∗ . Therefore it remains to show that pr1∗ = g ∗ π1∗ . It follows from the construction in [8, Sect. 2] that the T r morphism is determined by a Zariski open subset whose intersection with each fiber is dense. In other words, T r : R pr1! Q → Q coincides with R pr1! Q → R pr1! Rg∗ Q = Rπ1! Q → Q. We have used the fact that R g∗ = R g! since g is proper.

Dually, a flat morphism π : Y → X with fibers of pure dimension d determines a flat pull-back π ∗ : Hk (X) → Hk+2d (Y ). It is shown in [35, Sect. 6] that π ∗ agrees with the flat pull-back π ∗ : Ak (X) → Ak+d (Y ) via the cycle map. (In the set up of bivariant intersection theory [15, 2.3], each flat morphism π determines a canonical element in T −2d (Y → X).) We also need to define the Gysin morphisms associated to regular imbeddings. A closed imbedding i : X1 → X is called a regular imbedding of codimension d if the conormal sheaf of X1 in X is a locally free sheaf on X1 of rank d [14, B.7.1]. Let i : X1 → X be a regular embedding of codimension d. The corresponding canonical


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element θi ∈ H 2d (X, X − X1 ) is constructed in [3, IV.4] and [46, Sect. 5]. (In bivariant intersection theory H 2d (X, X − X1 ) = T 2d (X1 → X).) If i1

Y1 −−−−→ Y     f1 f i

X1 −−−−→ X is a fibered square, then the pull-back f ∗ θi determines an element in H 2d (Y, Y − Y1 ). Accordingly, it defines Gysin homomorphisms: i ! : Hk (Y ) → Hk−2d (Y1 ) k

i! : H (Y1 ) → H

k+2d

and

(Y )

by the cap-product (or cup-product) with f ∗ θi . However, we will denote i ! : Hk (X) → Hk−2d (X1 ) by i ∗ , and i! : H k (X1 ) → H k+2d (X) by i∗ . This agrees with the notation from [14]. If E → X is a rank d vector bundle, and X1 is the zero scheme of a section i : X → E, then i∗ 1 = cd E [14, Sect. 19.2]. The Gysin morphism i ! defined above agrees with the Gysin morphism i ! on the level of Chow groups via the cycle map [46]. Remark. More generally, one can define the Gysin morphisms for local complete intersection morphisms. If a flat morphism is at the same time a local complete interesection morphism, then the two definitions agree. The morphisms π∗ , π ∗ , i! , i ! satisfy the expected projection type formulae and commute with the standard pull-backs and push-forwards [15, 2.5]. We will use these properties without explicitly mentioning them. 2. The Moduli Spaces of Stable Maps We adopt the notation from [18]. Let Mg,n be the moduli space of stable curves. The stability implies that 2g − 2 + n > 0. Let & be a stable graph of genus g with n tails. We denote by M(&) ⊂ Mg,n the closure in Mg,n of the locus of stable curves with dual graph &, and by i& the corresponding inclusion. Let M(&) := Mg(v),n(v) . v∈V (&)

Then Aut(&) acts on M(&). The natural morphism µ& : M(&) → M(&) identifies M(&) with M(&)/ Aut(&). We denote by ρ& the composition µ&

i&

ρ& : M(&) −→ M(&) −→ Mg,n . The previous considerations apply word for word to the moduli spaces of prestable curves Mg,n , g ≥ 0, n ≥ 0, their subspaces M(&), and the products M(&) [18, Sect. 2].

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Note that Mg,n is an open dense substack of Mg,n when 2g − 2 + n > 0, and, more generally, M(&) is an open dense substack of M(&) when & is a stable graph. We adopt a similar notation for the substacks of Mg,n (V , β) determined by decorated stable graphs. Let H2+ (V , Z) denote the semigroup generated by those homology classes represented by the image of a morphism from a curve into V . Let G be a stable graph of genus g with n tails whose vertices are decorated by elements of H2+ (V , Z). (Henceforth, such decorated graphs will be denoted by G.) Then we denote by M(G, V ) the closure in Mg,n (V , β) of those points in the moduli space of stable maps whose dual graph is G. Let M(G, V ) be determined by the following fibered square (cf. [18, Sect. 6]): , M(G, V ) −−−−→ Mg(v),n(v) (V , β(v)) v∈V (G)

 ev V E(G)

,1

 ev

V E(G) × V E(G) ,

−−−−→

where ,1 is the diagonal morphism. In the sequence µ(G) i(G) , Mg(v),n(v) (V , β(v)) ←− M(G, V ) −→ M(G, V ) −→ Mg,n (V , β), v∈V (G)

the morphism µ(G) is the quotient by Aut(G) identifying M(G, V ) with the quotient, and i(G) is the inclusion of a substack. We denote the composition of two morphisms on the right by ρ(G). We will also need to introduce some other notation to describe the pull back of the virtual fundamental classes with respect to the inclusions of the strata (cf. [18, Sect. 6]. Let & be a graph of genus g with n tails, not necessarily stable. We define M(&, V , β) := Mg,n (V , β) ×Mg,n M(&). It is the closure of the subset of Mg,n (V , β) whose points correspond to the graph & after forgetting the decoration. If & is a stable graph, then M(&, V , β) = Mg,n (V , β) ×Mg,n M(&) since M(&) is dense in M(&). If G is a decorated graph we denote by G0 the underlying non-decorated graph. Let V , β) := M(&, M(G, V ).

G:G0 =&

V , β) → As before, , : M(&, G0 =& v∈V (G) Mg(v),n(v) (V , β(v)) is determined by the diagonal morphism. One has the natural morphism: µ(&) i(&) V , β) −→ M(&, V , β) −→ Mg,n (V , β). ρ(&) : M(&,

Here i(&) is an inclusion of a substack, and µ(&) factors as V , β) → M(&, V , β)/ Aut(&) → M(&, V , β), M(&, where the second morphism is a proper, surjective, birational morphism. The difference with the previous situation is explained by the fact that two substacks M(G, β) and M(G , β) of Mg,n (V , β) whose underlying undecorated graphs are the same may have a nonempty intersection.


113

3. Tautological Classes In this section we introduce the tautological κ classes on the moduli spaces of stable maps which generalize the corresponding tautological classes on the moduli spaces of stable curves. We will also show how these classes restrict to the boundary strata. Let π : Mg,n+1 (V , β) → Mg,n (V , β) be the universal curve. We assume that π “forgets” the (n + 1)st marked point. The morphism π has n canonical sections σ1 , . . . , σn . Each of these sections determines a regular embedding. We denote by ω the relative dualizing sheaf of π . Definition 3.1. For each i = 1, . . . , n the tautological line bundle Li on Mg,n (V , β) is σi∗ ω. The tautological class ψi ∈ H 2 (Mg,n (V , β)) is the first Chern class c1 (Li ). Remark. It is shown in [18, Sect. 5] that ψi = p∗ 0i , where p : Mg,n (V , β) → Mg,n , where 0i is the tautological class in H 2 (Mg,n ). One can also pull back cohomology classes from V to Mg,n (V , β) using the evaluation maps to obtain the Gromov–Witten classes. The definition of the κ classes involves both, powers of the ψ classes and these pull backs. Definition 3.2. The tautological class κa in H • (Mg,n (V , β)) ⊗ H • (V )∗ for a ≥ −1 is defined as follows. For each γ ∈ H • (V ), the cohomology class κa (γ ) is a+1 ∗ π∗ (ψn+1 evn+1 (γ )), where π is the universal curve defined above. In particular, if γ has definite degree |γ | then κa (γ ) has degree 2a + |γ |. If { eα }α∈A , is a homogeneous basis for H • (V ), then κa,α denotes the cohomology class κa (eα ). Remark. The class κ−1 (γ ) vanishes due to dimensional reasons if |γ | < 2. In addition, all classes κ−1 (γ ) vanish on Mg,n (V , 0). The classes κ−1 (γ ) are not needed in the change of coordinates formula in Sect. 5. Our definition corresponds to the “modified” κ classes defined by Arbarello and Cornalba [1] rather than the “classical” κ classes defined by Mumford [40]. The following lemma shows how the κ classes restrict to the boundary substacks of Mg,n (V , β). Lemma 3.3. Let G be a stable H2+ (V , Z) decorated genus g, degree β graph with n tails. Denote the class κa (γ ) on Mg,n (V , β) (resp. M(v), where v ∈ V (G)) by κ (resp. κv ). Then ρ(G)∗ (κ) = ,∗

κv .

v∈V (G)

Proof. Let v ∈ V (G). Denote by G(v) the graph obtained from G by attaching a tail labeled n + 1 to the vertex v of G. For each v ∈ V (G) the graph G(v) determines a substack of Mg,n+1 (V , β), and there are natural morphisms π:

w∈V (G(v))

M(w) →

w∈V (G)

M(w), and π : M(G(v), V ) → M(G, V ).

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Consider the following commutative diagram:

v∈V (G) w∈G(v)   π

w∈G M(w)

,

M(w) ←−−−−

v∈V (G)

,

←−−−−

ρ(G ) M(G(v), V ) −−−−→ Mg,n+1 (V , β)

  π

M(G, V)

 π ρ(G)

−−−−→

Mg,n (V , β).

Note that the left square is a fibered square, the right square is close to a fibered square in the sense of Definition 1.1, and all morphisms π, π are representable and flat. Therefore, one can apply Lemma 1.2. Also note that for each v ∈ V (G) one has ρ(G(v))∗ ψn+1 = ,∗ ψn+1

∗ ∗ ρ(G(v))∗ evn+1 γ = ,∗ evn+1 γ.

and

Now a+1 ∗ ρ(G)∗ (κ) = ρ(G)∗ π∗ (ψn+1 evn+1 γ ) =

=

v∈V (G)

π∗ ,

∗

a+1 ∗ (ψn+1 evn+1 γ )

=

v∈V (G)

v∈V (G)

a+1 ∗ π∗ ρ(G(v))∗ (ψn+1 evn+1 γ )

a+1 ∗ ,∗ π∗ (ψn+1 evn+1 γ ) = ,∗

κv .

v∈V (G)

The above lemma shows that the class κa (γ ) restricts to the sum of the κa (γ ) classes. It follows that exp(κa (γ )) restricts to the product of exp(κa (γ )). More gen

α erally, exp( ∞ a=−1 κa,α sa ), where si ’s are formal variables, restricts to the product of

∞ α exp( a=−1 κa,α sa ). This will be used in Sect. 4. 4. Cohomological Field Theories In this section, we define a cohomological field theory in the sense of Kontsevich–Manin [32]. We prove that the Gromov–Witten invariants twisted by the κ classes endows H • (V ) together with its Poincaré pairing with the family of CohFT structures. In genus zero, this reduces to endowing H • (V ) with a family of formal Frobenius manifold structures arising from the Poincaré pairing and deformations of the cup product on H • (V ). These deformations contain quantum cohomology as a special case. Definition 4.1. Let (H, η) be an r-dimensional vector space H with an even, symmetric nondegenerate, bilinear form η. A (complete) rank r cohomological field theory (or CohFT) with state space (H, η) is a collection 4 := { 4g,n }, where 4g,n is an even element in Rg,n := H • (Mg,n ) ⊗ T n H∗ (where T n H∗ denotes the n-fold tensor product) defined for stable pairs (g, n) satisfying (i) to (iii) below (where the summation convention has been used): (i) 4g,n is invariant under the diagonal action of the symmetric group Sn on T n H and Mg,n .


115

(ii) For each partition of [n] = J1 J2 such that |J1 | = n1 and |J2 | = n2 and nonnegative g1 , g2 such that g = g1 + g2 and 2gi − 2 + ni + 1 > 0 for all i, consider the inclusion map ρ : Mg1 ,J1 ∗ × Mg2 ,J2 ∗ → Mg1 +g2 ,n , where ∗ denotes the two marked points that are attached under the inclusion map. The forms satisfy the restriction property ρ ∗ 4g,n (γ1 , γ2 . . . , γn ) γα ) ⊗ eµ ) ηµν ⊗ 4g2 ,n2 (eν ⊗ γα ), = ± 4g1 ,n1 (( α∈J1

α∈J2

where the sign ± is the usual one obtained by applying the permutation induced by the partition to (γ1 , γ2 , . . . γn ) taking into account the grading of { γi } and where { eα } is a homogeneous basis for H. (iii) Let ρ0 : Mg−1,n+2 → Mg,n be the canonical map corresponding to attaching the last two marked points together, then ρ0∗ 4g,n (γ1 , γ2 , . . . , γn ) = 4g−1,n+2 (γ1 , γ2 , . . . , γn , eµ , eν ) ηµν . (iv) If, in addition, there exists an even element e0 in H such that π ∗ 4g,n (γ1 , . . . , γn ) = 4g,n+1 (γ1 , . . . , γn , e0 ) and

M0,3

40,3 (e0 , γ1 , γ2 ) = η(γ1 , γ2 )

for all γi in H then 4 endows (H, η) with the structure of a CohFT with flat identity e0 . A cohomological field theory of genus g consists of only those 4g ,n , where g ≤ g which satisfy the subset of axioms of a cohomological field theory which includes only objects of genus g ≤ g. The strata maps ρ and ρ0 in the above definition can be extended to arbitrary boundary strata on Mg,n . Let & be a stable graph, then there is a canonical map ρ& obtained by composition of the canonical maps Mg(v),n(v) → M& → Mg,n . v∈V (&)

Since the map ρ& can be constructed from morphisms in (ii) and (iii) above, 4g,n satisfies a restriction property of the form

ρ&∗ 4g,n = η&−1 4g(v),n(v) , (1) v∈V (&)

where η&−1 :

v∈V (&)

Rg(v),n(v) → Rg,n

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is the linear map contracting tensor factors of H using the metric η induced from successive application of Eqs. (ii) and (iii) above. Notice that the definition of a cohomological field theory is valid even when the ground ring is enlarged from C to another ring K. Finally, axioms (i) to (iii) in the definition of a CohFT are equivalent to endowing (H, η) with the structure of an algebra over the modular operad H• (M) := { H• (Mg,n ) } [20]. Definition 4.2. Let 8 consist of formal symbols q β for all β ∈ H2+ (V , Z) together with the multiplication (q β q β ) → q β+β . Let C[[8]] consist of formal sums

β β β ∈ H2+ (V ,Z) aβ q , where aβ are elements in C. Assign to each q , the degree of −2c1 (V ) ∩ β. The product is well-defined according to [30, Prop. II.4.8]. This endows 8 with the structure of a semigroup with unit. Furthermore, let C[[8, s]] := C[[8]][[s]], formal power series in the variables s with coefficients in C[[8]]. Notation. Let V be a topological space and let H • (V , C) be given a homogeneous basis e := { eα }α∈A and let e0 denote the identity element. Let s := { saα | a ≥ −1, α ∈ A } be a collection of formal variables with grading |saα | = 2a + |eα |. All formal power series and polynomials in a collection of variables (e.g. s) are in the Z2 -graded sense. It will be useful to associate a generating function (called the potential) to each CohFT. Definition 4.3. Let 4 be a rank r CohFT with state space (H, η). Its potential function in λ−2 C[[H, λ]] is defined by (x) := g (x) λ2g−2 , g≥0

where (x) :=

∞ 1 4g,n (x, x, . . . , x) n! Mg,n n=3

α and x = r−1 α=0 x eα for a given homogeneous basis { e0 , . . . , er−1 } for H. The formal parameter λ is even. In genus zero, the potential function yields yet another formulation of a CohFT which is essentially the definition of a formal Frobenius manifold structure on its state space. Theorem 4.4. Let (H, η) be an r dimensional vector space with metric. An element 0 (x) in C[[H]] is the potential of a rank r, genus zero CohFT (H, η) if and only if [32, 38] it contains only terms which are cubic and higher order in the variables x 0 , . . . , x r and it satisfies the WDVV equation (∂a ∂b ∂e 0 ) ηef (∂f ∂c ∂d 0 ) = (−1)|xa |(|xb |+|xc |) (∂b ∂c ∂e 0 ) ηef (∂f ∂a ∂d 0 ), where ηab := η(ea , eb ), ηab is in inverse matrix to ηab , ∂a is derivative with respect to x a , and the summation convention has been used. Furthermore, any genus zero CohFT is completely characterized by its genus zero potential 0 (x).


117

The theorem follows from the work of Keel [28] who proved that all relations between boundary divisors on M0,n arise from lifting the basic codimension one relation on M0,4 . As before, one can extend the ground ring C above to C[[8, s]] in the definition of the potential of a genus zero CohFT and the above theorem extends, as well. In our setting, the potential is a formal function on H := H • (V , C[[8, s]]) and η is the Poincaré pairing extended linearly to C[[8, s]]. belongs to λ−2 C[[8, s, λ]][[x 0 , . . . , x r ]]. Again, if H • (V ) consists entirely of even dimensional classes, then plugging in numbers (almost all of which are zero) for all saα , where a = −1, 0, 1, . . . and α = 0, 1, . . . , r − 1 and setting λ = 1, one obtains families of CohFT structures on H • (V , C[[8]]). Notation. We define sκ to be

∞ a=−1

κa,α saα . Note that each term has even parity.

Theorem 4.5. Let V be a smooth projective variety. For each pair (g, n) such that 2g − 2 + n > 0, let 4g,n be the element of Rg,n (V )[[8, s]] defined by 4g,n (γ1 , . . . , γn ) :=

β∈H2+ (V ,Z)

st∗ ( ev1∗ γ1 · · · evn∗ γn exp(κs) ∩ [Mg,n (V , β)]virt ) q β ,

where γ1 , γ2 , . . . , γn are elements in H • (V , C). Then 4 := { 4g,n } endows (H • (V , C[[8, s]]), η) with the structure of a CohFT, where η is the Poincaré pairing extended C[[8, s]]-linearly. Proof. It is clear that the morphisms 4g,n are Sn -equivariant. In order to prove the restriction properties fix β ∈ H2+ (V , Z), (g, n) such that 2g − 2 + n > 0, and a stable graph & of genus g with n tails. Let G be the set of all H2+ (V , Z) decorated graphs such that the underlying graph without decoration is &. Let XG , where XG := M(v), X := G∈G

v∈G

and let [XG ]virt ∈ H• (XG ) be the product of the corresponding virtual fundamental classes. Consider the following commutative diagram: X   st M(&)

i(&) , V , β) −−µ(&) ←−−−− M(&, −−→ M(&, V , β) −−−−→ Mg,n (V , β)       st st st

M(&)

µ&

−−−−→

M(&)

i&

−−−−→

Mg,n .

We want to see how the β summand of 4g,n restricts to H • (M(&)). In the sequence of equations below we will use the following properties. The right square of the above diagram is a fibered square. All vertical morphisms st are proper. If x ∈ H • (M(&)) is invariant under the action of Aut(&), then µ∗& µ&∗ x = N x, where N := | Aut(&)|. In addition, we use the following result of Getzler [18, Thm. 13]: 1 [XG ]virt , i&! [Mg,n (V , β)]virt = µ(&)∗ ,!1 N G∈G

118


where ,1 : V E(G) → V E(G) × V E(G) is the diagonal morphism. If G ∈ G, and v ∈ V (G), then we denote by γ v the tensor product of the corresponding γi ’s on M(v), and by κ v s the formal sum on M(v). For the sake of brevity we will write µ for µ& , i for i& , and γ for ⊗γi . The sums below are always taken over G ∈ G.

µ∗ i ∗ st∗ ev ∗ γ exp(κs) ∩ [Mg,n (V , β)]virt

= µ∗ st ∗ i ! ev ∗ γ exp(κs) ∩ [Mg,n (V , β)]virt

= µ∗ st ∗ (i(&)∗ ev ∗ γ exp(κs)) ∩ i ! [Mg,n (V , β)]virt 1 = µ∗ st ∗ (i(&)∗ ev ∗ γ exp(κs)) ∩ µ(&)∗ ,!1 [XG ]virt N

1 ∗ = µ st∗ µ(&)∗ (µ(&)∗ i(&)∗ ev ∗ γ exp(κs)) ∩ ,!1 [XG ]virt N = st∗ ,∗ (⊗v∈G ev ∗ γ v exp(κ v s)) ∩ ,!1 [XG ]virt

= ⊗v∈G st ∗ ev ∗ γ v exp(κ v s) ∩ ,∗ ,!1 [XG ]virt . Summing over all β gives the statement of the theorem taking into account that ,∗ ,!1 is the cap-product with the Poincaré dual of the diagonal in V E(&) × V E(&) . Theorem 4.5 provides a CohFT determined by the κ classes. One can similarly construct a CohFT determined by the ψ classes. Its potential is the usual potential. A more general construction will appear in [23]. Remark. The potential of the CohFT defined in the previous theorem coincides with the usual notion of potential of Gromov–Witten invariants up to terms quadratic in the variables x which correspond to contributions from the moduli spaces Mg,n (V ), where 2g − 2 + n ≤ 0. 5. The Change of Coordinates In this section we prove the change of coordinate formula on the large phase space. Throughout the rest of this section, we fix a homogeneous basis {eα }, where α ∈ A of H • (V ) such that e0 is the identity element. We also fix a total ordering on A. Remark. In this section we will not use the tautological classes κ−1 (γ ). Definition 5.1. Let β ∈ H2+ (V , Z), and eα , α = 0, . . . , r − 1 be a basis of H • (V ). Assume that all di > 0 and all ai ≥ 0. We define σ ν1 . . . σνn τd1 ,µ1 . . . τdk ,µk κa1 ,α1 . . . κal ,αl g,β := ev1∗ (eν1 ) . . . evn∗ (eνn ) [Mg,n (V ,β)]virt

dk d1 ∗ ∗ π∗ (ψn+1 evn+1 (eµ1 ) . . . ψn+k evn+k (eµk ))κa1 ,α1 . . . κal ,αl ,

where π : Mg,n+k (V , β) → Mg,n (V , β) “forgets” the last k marked points.


119

Remark. This definition differs from the standard one. However, if no κ classes are present, then the intersection number above is the standard intersection number of the ψ and the pull-back classes with [Mg,n+k (V , β)]virt . This definition is motivated by the representation of the large phase space on the level of cohomology classes in Sect. 4. Also, it will be easier to work with this definition to derive the coordinate change below. Let the sequence ν1 , . . . , νn contain rν elements ν, ν ∈ A, the sequence (d1 , µ1 ), (d2 , µ2 ), . . . , (dk , µk ) contain md,µ pairs (d, µ), where d > 0, µ ∈ A, and the sequence (a1 , α1 ), (a2 , α2 ), . . . , (al , αl ) contain pa,α pairs (a, α), where a ≥ 0, α ∈ A. Then we also denote the intersection number above by σ r τ m κ p g,β . One has to be careful if H • (V ) has elements of odd degree. In this case σ r τ m κ p g,β denotes the intersection number above with the following ordering. If i < j then, using the chosen order on A, a) νi ≤ νj ; b) di < dj , or di = dj and µi ≤ µj ; c) ai < aj , or ai = aj and αi ≤ αj . Definition 5.2. We define σ r τ m κ p g :=

σ r τ m κ p g,β q β ,

β∈H2+ (V ,Z)

where q is a formal variable. In the sequel we will consider the following collection of formal variables: x = (x ν ), µ t = (ti ), s = (saα ), i > 0, a ≥ 0, ν, µ, α ∈ A. These variables have the following µ degrees: |x ν | = |ν| − 2, |ti | = 2(i − 1) + |µ|, and |saα | = 2a + |α|. Note that the Z/2Z-degree is determined by the upper index. Let µ xr := (x ν )rν , tm := (td )md,µ , sp := (saα )pa,α . ν

d,µ

a,α

Again one has to exercise care in case there are variables of odd degree. In this case we order the products above so that x ν1 precedes x ν2 if ν1 ≥ ν2 ; µ µ td11 precedes td22 if d1 > d2 , or d1 = d2 and µ1 ≥ µ2 ; saα11 precedes saα22 if a1 > a2 , or a1 = a2 and α1 ≥ α2 . That is, we require the order on the products to be the opposite to the order on the intersection numbers. Definition 5.3. We define Kg ∈ C[[8, x, t, s]] by Kg (x, t, s) :=

r,m,p

where p :=

ν

pν and p! :=

ν

σ r τ m κ p g

sp t m x r , p! m! r!

pν ! (and similarly for m and r).

Remark. In the above definition one could have chosen an arbitrary ordering for the intersection numbers, and then chosen the opposite ordering on the corresponding variables.

120


The various degrees chosen for the variables together with the dimensions of the cohomology classes and the virtual fundamental class insures the Kg has degree 2(3 − d)(1 − g). Note that K(x, t, 0) = F(x, t), the standard large phase space potential if one sets x ν = t0ν . Similarly, K(x, 0, s) = G(x, s), the potential of the family of CohFTs determined by the κ classes including the terms with 2g − 2 + n ≤ 0. Theorem 5.4. Let t(s) be determined by the following equation in H • (V ): µ e0 − θ d−1 td eµ = exp − θ a saα eα ,

(2)

a≥0

d≥1

where θ is an even formal parameter. Then Fg (x, t(s)) = Gg (x, s) for every g ≥ 0. Remark. In the case when V = pt, Thm. 5.4 reduces to Thm. 4.1 from [39]. The polynomials ta (s) are the Schur polynomials. We will prove the above theorem in a sequence of lemmata. Lemma 5.5. Let I and J be two sets such that I ∩ J = {1, . . . , n} and I ∪ J = {1, . . . , n + N }. Let I := I − {1, . . . , n} and J := J − {1, . . . , n}. Consider the following commutative diagram: ρ

Mg,I (V , β) ←−−−− Mg,n+N (V , β)   π π ρ

Mg,n (V , β) ←−−−− Mg,J (V , β), where the horizontal morphisms ρ “forget” the marked points from J , and the vertical morphisms π “forget” the marked points from I . The morphisms π and ρ are flat, and ρ ∗ π∗ = π∗ ρ ∗ and ρ∗ π ∗ = π ∗ ρ∗ . Proof. The morphisms π and ρ are flat as compositions of flat morphisms. Let I be {1, . . . , n + 1}, and J be {1, . . . , n, n + 2}. Then the commutative diagram above is close to a fibered square in the sense of Def. 1.1 (cf. [1]). Therefore one has ρ ∗ π∗ = π∗ ρ ∗ and ρ∗ π ∗ = π ∗ ρ∗ . Iterating, one obtains the statement of the lemma. Consider the universal curve π : Mg,n+1 (V , β) → Mg,n (V , β). It has n canonical sections σi , and each of these sections is a regular embedding of codimension one. Therefore, the image of each of these sections determines a Cartier divisor on Mg,n+1 (V , β). We denote the corresponding Chern classes by Di,n+1 ∈ H 2 (Mg,n+1 (V , β)). Equivalently, Di,n+1 = σi∗ 1. The equalities below hold in H • (Mg,n+1 (V , β)): Di,n+1 Dj,n+1 = 0 if i ! = j, ψi Di,n+1 = ψn+1 Di,n+1 = 0. In addition, σi∗ Di,n+1 = −ψi . Let π : Mg,n+1 (V , β) → Mg,n (V , β) be the universal curve. In the next two lemmas we will use the following properties. Firstly, π ∗ ψi = ψi − Di,n+1 proved in a [18, Prop. 11]. It follows that π ∗ ψia = ψia + (−1)a Di,n+1 . Secondly, π ∗ evi∗ = evi∗ . The following lemma and its proof are similar to those in [1, Sect. 1].


121

a ev ∗ γ . Lemma 5.6. If γ ∈ H • (V ), then π ∗ κa (γ ) = κa (γ ) − ψn+1 n+1

Proof. Consider the following commutative diagram close to a fibered square: ρ

Mg,n+1 (V , β) ←−−−− Mg,n+2 (V , β)   π π ρ

Mg,n (V , β) ←−−−− Mg,J (V , β), where J = {1, . . . , n, n + 2}. Let σ : Mg,n+1 (V , β) → Mg,n+2 (V , β) associated to the (n + 1)st marked point. One has a+1 ∗ a+1 ∗ π ∗ κa (γ ) = π ∗ ρ∗ (ψn+2 evn+2 γ ) = ρ∗ π ∗ (ψn+2 evn+2 γ ) a+1 ∗ a+1 ∗ evn+2 γ ) + (−1)a+1 ρ∗ (Dn+1,n+2, evn+2 γ) = ρ∗ (ψn+2 a ∗ a ∗ evn+2 γ ) = κa (γ ) − ψn+1 evn+1 γ. = κa (γ ) + (−1)a+1 ρ∗ σ∗ σ ∗ (Dn+1,n+2,

Definition 5.7. Let γ ∈ H • (V ). Define the homomorphism Da (γ ) : H • (Mg,n+1 (V , β)) → H • (Mg,n (V , β)) by a+1 ∗ Da (γ )(x) := π∗ (ψn+1 evn+1 γ x).

Note that Da (γ )(1) = κa (γ ). Lemma 5.8. Assume that all ai > 0 for i = 1, . . . , N. Then aN a1 ∗ ∗ Da1 −1 (γ1 ) . . . DaN −1 (γN )(x) = πN∗ (ψn+1 evn+1 γ1 . . . ψn+N evn+N γN x),

where πN : Mg,n+N (V , β) → Mg,n (V , β) “forgets” the last N marked points. Proof. We proceed by induction. When N = 1 the statement of the lemma is trivial. Assume that the statement is true for N and prove it for N + 1. We denote by π the universal curve Mg,n+N+1 (V , β) → Mg,n+N (V , β). One has Da1 −1 (γ1 ) . . . DaN +1 −1 (γN+1 )(x) a

N +1 ∗ = Da1 −1 (γ1 ) . . . DaN −1 (γN )(π∗ (ψn+N+1 evn+N+1 γN+1 x))

a

aN a1 N +1 ∗ ∗ ∗ evn+1 γ1 . . . ψn+N evn+N γN π∗ (ψn+N+1 evn+N+1 γN+1 x)) = πN∗ (ψn+1 ∗ = πN+1∗ ((ψn+1 − Dn+1,n+N+1 )a1 evn+1 γ1 . . . a

N +1 ∗ ∗ (ψn+N − Dn+N,n+N+1 )aN evn+N γN ψn+N+1 evn+N+1 γN+1 x))

a

aN a1 N +1 ∗ ∗ ∗ = πN+1∗ (ψn+1 evn+1 γ1 . . . ψn+N evn+N γN ψn+N+1 evn+N+1 γN+1 x).

It follows from Lemma 5.8 that the operators Da (γ ) super-commute.

122


Remark. If some of the numbers ai are equal to zero, then Lem. 5.8 does not necessarily hold. Lemma 5.9. Assume the conditions of Lem. 5.8. Then aN a1 ∗ ∗ πN∗ (ψn+1 evn+1 γ1 . . . ψn+N evn+N γN )κa (γ ) aN a1 a+1 ∗ ∗ ∗ γN ψn+1+N evn+N+1 γ) = πN+1∗ (ψn+1 evn+1 γ1 . . . ψn+N evn+N k ai +a ∗ a1 ∗ − (−1)|γi+1 ...γN ||γ | πN∗ (ψn+1 evn+1 γ1 . . . ψn+i evn+i (γi γ ) i=1

aN ∗ . . . ψn+N evn+N γN ).

Proof. We proceed by induction. Let N = 1, and πN = π . Then, using Lemma 5.6 and Lemma 5.8, one gets a1 a1 +a ∗ ∗ evn+1 γ1 π ∗ κa (γ )) = Da1 −1 (γ1 )(κa (γ )) − π∗ (ψn+1 evn+1 (γ1 γ )). π∗ (ψn+1

This proves the statement of the lemma when N = 1. Now assume that the statement is true for N = N − 1 and prove it for N . Denote by πN the natural morphism Mg,n+N (V , β) → Mg,n+1 (V , β), aN a1 ∗ ∗ evn+1 γ1 . . . ψn+N evn+N γN )κa (γ ) πN∗ (ψn+1 aN a1 a2 ∗ ∗ ∗ evn+1 γ1 πN∗ (ψn+2 evn+2 γ2 . . . ψn+N evn+N γN ) = π∗ (ψn+1 a ∗ (κa (γ ) − ψn+1 evn+1 γ )).

The rest follows applying the induction hypothesis to the product aN a2 ∗ ∗ πN ∗ (ψn+2 evn+2 γ2 . . . ψn+N evn+N γN )κa (γ ),

and using Lemma 5.8.

Lemma 5.9 provides a recursion relation for the intersection numbers of the ψ and µ the κ classes. Let {eα }, α = 0, . . . , r, be the chosen basis of H • (V ). Define cα1 ,... ,αj by the formula eα1 . . . eαj = cαµ1 ,... ,αj eµ . µ

µ

(We assume summation over the repeating indices.) In particular, cα = δα . The following recursion relation follows from Lemma 5.9, σ r τd1 ,µ1 . . . τdk ,µk κa,α κ p g,β = σ r τd1 ,µ1 . . . τdk ,µk τa+1,α κ p g,β k µ − (−1)|eµi+1 ...eµk ||eα | cµ σ r τd1 ,µ1 . . . τdi +a,µ . . . τdk ,µk κ p g,β . i ,α i=1

Note that the above relations also holds if one replaces . . .g,β with . . .g . It turns out that the equation above implies that for each g ≥ 0, ∞

∂Kg ∂Kg µ ν ∂Kg = α − cν,α ti µ . α ∂sa ∂ta+1 ∂ti+a i=1

(3)


123

We leave it to the reader to check that all signs agree. Let us introduce the following standard notation: tν unless i = 1 and ν = 0, ν t˜i := i0 t1 − 1 if i = 1 and ν = 0. Then one can rewrite (3) as ∞

∂Kg µ ˜ν ∂Kg = − cν,α ti µ . α ∂sa ∂ti+a

(4)

i=1

Proof of Theorem 5.4. We assume that t(s) is determined by (2). It follows that t(0) = 0, and   µ ∂t θ d−1 dα eµ = −θ a eα exp − θ a1 saα11 eα1  − ∂sa a1 ≥0 d≥1 µ = θ d−1 t˜d−a eµ eα . d≥a+1

It follows that for each d, a, and α such that d ≥ a + 1 one has µ

∂td ν eµ = −t˜d−a eν eα ∂saα

µ

or, equivalently,

∂td ν µ = −t˜d−a cν,α , ∂saα

(5)

µ

and ∂td /∂saα = 0 if d ≤ a. Consider the function Kg (x, t(s0 + s), −s), where s0 is a constant. Differentiating it with respect to saα provides using (5) and (4), ∂ Kg (x, t(s0 + s), −s) α ∂sa =−

µ

∂t ∂Kg ∂Kg d (x, t(s0 + s), −s) + (s0 + s) µ (x, t(s0 + s), −s) α α ∂sa ∂sa ∂td d≥1

=−

∂Kg (x, t(s0 + s), −s) − ∂saα

d≥1

µ ˜ν cν,α td (s0 + s)

∂Kg µ (x, t(s0 + s), −s) ∂td+a

= 0. Therefore Kg (x, t(s0 + s), −s) does not depend on s. It follows that for all values s1 and s2 one has Kg (x, t(s1 + s2 ), 0) = Kg (x, t(s1 ), s2 ) = Kg (x, 0, s1 + s2 ). In particular, Fg (x, t(s)) = Gg (x, s) for every g ≥ 0.

Remark. Note that the condition t(s) = 0 and (5) are equivalent to (2), and determine t(s) completely. Note also that the coordinate change given by (2) is invertible. Remark. The function t(s) has the following Taylor coefficients: µ

∂ k td |s=0 = (−1)k+1 cαµ1 ,... ,αk δd,a1 +...+ak +1 . ∂saα11 . . . ∂saαkk

124


6. Topological Recursion Relations In this section we will derive the topological recursion relations for G0 and G1 using the change of coordinates formula (2). In [24] we represented the cohomology classes by graphs to obtain the topological recursion relations when V is a convex variety and genus g = 0. However, we were not able to extend this technique to the general case since it is not clear that one can pull back homology classes w.r.t. µ(&) from Sect. 2. We will use the fact that Gg (x, s) = Fg (x, t(s)), where t(s) is determined by (2). Notice that ∂Gg /∂x α = ∂Fg /∂x α since the coordinate change t(s) does not depend on x. We will raise and lower indices in the usual manner. Proposition 6.1. Let a ≥ 1. Then ∂ 3 G0 ∂ 3 G0 ∂ 2 G0 = α ∂x ρ ∂x ∂x µ ∂x ν . ∂saα ∂x µ ∂x ν ∂sa−1 ρ Proof. Applying the chain rule one gets: ∂t ξ ∂ 3 G0 ∂ 3 F0 d = ξ α µ ν α ∂sa ∂x ∂x ∂sa ∂t ∂x µ ∂x ν d≥a+1 d =

∂t ξ ∂ 2 F0 ∂ 3 F0 d . ξ α ∂sa ∂t ∂x ρ ∂xρ ∂x µ ∂x ν d≥a+1 d−1

The second equation uses that F0 satisfies the topological recursion relations. Similarly, ξ

∂t ∂ 2 G0 ∂ 3 G0 ∂ 2 F0 ∂ 3 F0 d = . α α ξ ρ µ ν ∂sa−1 ∂x ∂xρ ∂x ∂x ∂sa−1 ∂t ∂x ρ ∂xρ ∂x µ ∂x ν d≥a d ξ

ξ

α = ∂td+1 /∂saα , and the proposition follows. Equation (5) implies that ∂td /∂sa−1

Proposition 6.2. Let |α| ≤ 2. Then

∂G ∂ 3 G0 ∂ 3 G0 ∂ 3 G0 0 ξ = Dα + cρ,α xξ , α µ ν ρ µ ν ∂s0 ∂x ∂x ∂x ∂xρ ∂x ∂x ∂xρ ∂x µ ∂x ν where the differential operator Dα is the C[[x, t, s]]-linear operator defined by Dα q β := q β eα β

for all α. Proof. We use the chain rule, (5), and the topological recursion relations for F0 : ζ ξ ∂ 2 F0 ∂ 3 G0 ∂ 3 F0 ˜ = − c t ξ,α d ζ µ ν ∂s0α ∂x µ ∂x ν ∂t ∂x ρ ∂xρ ∂x ∂x d≥1

= Dα

∂F 0 ∂x ρ

d−1 ∂ 3 F0 ∂xρ ∂x µ ∂x ν

ξ + cρ,α xξ

∂ 3 F0 . ∂xρ ∂x µ ∂x ν

In the second equation we used the divisor equation for F [19, 2.6].

(6)

(7)


125

Remark. The first term of the right-hand side in (6) contributes only when |eα | = 2. When α = 0 one can get (6) from ∂G0 ∂G0 = x ρ ρ − 2G0 . 0 ∂x ∂s0 ρ This equation can be derived using the dilaton equation for F0 [19, 2.7]. Similarly, one can derive the topological recursion relations for G in genus 1 using known topological recursion relations for F. We state the results without proofs. Proposition 6.3. Let a ≥ 1. Then ∂ 3 G0 ∂ 2 G0 ∂G1 1 ∂G1 = α + . α α ρ ∂sa ∂sa−1 ∂x ∂xρ 24 ∂sa−1 ∂x ρ ∂xρ Acknowledgement. We would like to thank D. Abramovich for useful conversations and J. Stasheff for his comments on an earlier version of the manuscript.

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Communications in


Ground State Energy of the One-Component Charged Bose Gas Elliott H. Lieb1, , Jan Philip Solovej2, 1 Departments of Physics and Mathematics, Jadwin Hall, Princeton University, PO Box 708, Princeton,

NJ 08544-0708, USA. E-mail: [email protected]

2 Department of Mathematics, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark.

E-mail: [email protected] Received: 23 August 2000 / Accepted: 5 October 2000

Dedicated to Leslie L. Foldy on the occasion of his 80th birthday Abstract: The model considered here is the “jellium” model in which there is a uniform, fixed background with charge density −eρ in a large volume V and in which N = ρV particles of electric charge +e and mass m move – the whole system being neutral. In 1961 Foldy used Bogolubov’s 1947 method to investigate the ground state energy of this system for bosonic particles in the large ρ limit. He found that the energy per particle −3/4 is −0.402 rs me4 /h¯ 2 in this limit, where rs = (3/4πρ)1/3 e2 m/h¯ 2 . Here we prove that this formula is correct, thereby validating, for the first time, at least one aspect of Bogolubov’s pairing theory of the Bose gas. 1. Introduction Bogolubov’s 1947 pairing theory [B] for a Bose fluid was used by Foldy [F] in 1961 to calculate the ground state energy of the one-component plasma (also known as “jellium”) in the high density regime – which is the regime where the Bogolubov method was thought to be exact for this problem. Foldy’s result will be verified rigorously in this paper; to our knowledge, this is the first example of such a verification of Bogolubov’s theory in a three-dimensional system of bosonic particles. Bogolubov proposed his approximate theory of the Bose fluid [B] in an attempt to explain the properties of liquid Helium. His main contribution was the concept of pairing of particles with momenta k and −k; these pairs are supposed to be the basic constituents of the ground state (apart from the macroscopic fraction of particles in the “condensate”, or k = 0 state) and they are the basic unit of the elementary excitations of the system. The pairing concept was later generalized to fermions, in which case the pairing was between © 2000 by the authors. This article may be reproduced in its entirety for non-commercial purposes.

Work partially supported by U.S. National Science Foundation grant PHY98 20650-A01. Work partially supported by EU TMR grant, by the Danish Research Foundation Center MaPhySto, and

by a grant from the Danish Research Council.

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E. H. Lieb, J. P. Solovej

particles having opposite momenta and, at the same time, opposite spin. Unfortunately, this appealing concept about the boson ground state has neither been verified rigorously in a 3-dimensional example, nor has it been conclusively verified experimentally (but pairing has been verified experimentally for superconducing electrons). The simplest question that can be asked is the correctness of the prediction for the ground state energy (GSE). This, of course, can only be exact in a certain limit – the “weak coupling” limit. In the case of the charged Bose gas, interacting via Coulomb forces, this corresponds to the high density limit. In gases with short range forces the weak coupling limit corresponds to low density instead. Our system has N bosonic particles with unit positive charge and coordinates xj , and a uniformly negatively charged “background” in a large domain of volume V . We are interested in the thermodynamic limit. A physical realization of this model is supposed to be a uniform electron sea in a solid, which forms the background, while the moveable “particles” are bosonic atomic nuclei. The particle number density is then ρ = N/V and this number is also the charge density of the background, thus ensuring charge neutrality. The Hamiltonian of the one-component plasma is N

H =

1 2 pj + Upp + Upb + Ubb , 2

(1)

j =1

where p = −i∇ is the momentum operator, p2 = −, and the three potential energies, particle-particle, particle-background and background-background, are given by |xi − xj |−1 , (2) Upp = 1≤i<j ≤N

Upb = −ρ

N j =1

Ubb = 21 ρ 2

|xj − y|−1 d 3 y,

(3)

|x − y|−1 d 3 xd 3 y.

(4)

In our units h¯ 2 /m = 1 and the charge is e = 1. The “natural” energy unit we use is two Rydbergs, 2Ry = me4 /h¯ 2 . It is customary to introduce the dimensionless quantity rs = (3/4πρ)1/3 e2 m/h¯ 2 . High density is small rs . The Coulomb potential is infinitely long-ranged and great care has to be taken because the finiteness of the energy per particle in the thermodynamic limit depends, ultimately, on delicate cancellations. The existence of the thermodynamic limit for a system of positive and negative particles, with the negative ones being fermions, was shown only in 1972 [LLe] (for the free energy, but the same proof works for the ground state energy). Oddly, the jellium case is technically a bit harder, and this was done in 1976 [LN] (for both bosons and fermions). One conclusion from this work is that neutrality (in the thermodynamic limit) will come about automatically – even if one does not assume it – provided one allows any excess charge to escape to infinity. In other words, given the background charge, the choice of a neutral number of particles has the lowest energy in the thermodynamic limit. A second point, as shown in [LN], is that e0 is independent of the shape of the domain provided the boundary is not too wild. For Coulomb systems this is not trivial and for real magnetic systems it is not even generally true. We take

Ground State Energy of One-Component Charged Bose Gas

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advantage of this liberty and assume that our domain is a cube [0, L] × [0, L] × [0, L] with L3 = V . We note the well-known fact that the lowest energy of H in (1) without any restriction about “statistics” (i.e., on the whole of ⊗N L2 (R3 )) is the same as for bosons, i.e., on the symmetric subspace of ⊗N L2 (R3 ). The fact that bosons have the lowest energy comes from the Perron–Frobenius Theorem applied to −. Foldy’s calculation leads to the following theorem about the asymptotics of the energy for small rs , which we call Foldy’s law. Theorem 1.1 (Foldy’s Law). Let E0 denote the ground state energy, i.e., the bottom of the spectrum, of the Hamiltonian H acting in the Hilbert space ⊗N L2 (R3 ). We assume that = [0, L] × [0, L] × [0, L]. The ground state energy per particle, e0 = E0 /N , in the thermodynamic limit N, L → ∞ with N/V = ρ fixed, in units of me4 /h¯ 2 , is −3/4

+ o(ρ 1/4 ) lim E0 /N = e0 = −0.40154 rs 1/4 4π = −0.40154 ρ 1/4 + o(ρ 1/4 ), 3

V →∞

(5)

where the number −0.40154 is, in fact, the integral ∞ 31/4 4 "(3/4) 1 p 2 (p 4 + 2)1/2 − p 4 − 1 dp = − √ A = 61/4 ≈ −0.40154. (6) π 5 π "(5/4) 0 Actually, our proof gives a result that is more general than Theorem 1.1. We allow the particle number N to be totally arbitrary, i.e., we do not require N = ρV . Our lower bound is still given by (5), where now ρ refers to the background charge density. In [F] 0.40154 is replaced by 0.80307 since the energy unit there is 1 Ry. The main result of our paper is to prove (5) by obtaining a lower bound on E0 that agrees with the right side of (5) An upper bound to E0 that agrees with (5) (to leading order) was given in 1962 by Girardeau [GM], using the variational method of himself and Arnowitt [GA]. Therefore, to verify (5) to leading order it is only necessary to construct a rigorous lower bound of this form and this will be done here. It has to be admitted, as explained below, that the problem that Foldy and Girardeau treat is slightly different from ours because of different boundary conditions and a concommitant different treatment of the background. We regard this difference as a technicality that should be cleared up one day, and do not hesitate to refer to the statement of 1.1 as a theorem. Before giving our proof, let us remark on a few historical and conceptual points. Some of the early history about the Bose gas, can be found in the lecture notes [L]. Bogolubov’s analysis starts by assuming periodic boundary condition on the big box and writing everything in momentum (i.e., Fourier) space. The values of the momentum, k are then discrete: k = (2π/L)(m1 , m2 , m3 ) with mi an integer. A convenient tool for taking care of various n! factors is to introduce second quantized operators ak# (where a # denotes a or a ∗ ), but it has to be understood that this is only a bookkeeping device. Almost all authors worked in momentum space, but this is neither necessary nor necessarily the most convenient representation (given that the calculations are not rigorous). Indeed, Foldy’s result was reproduced by a calculation entirely in x-space [LS]. Periodic boundary conditions are not physical, but that was always chosen for convenience in momentum space. We shall instead let the particle move in the whole space, i.e., the operator H acts in the Hilbert space L2 (R3N ), or rather, since we consider bosons, in the the subspace

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consisting of the N -fold fully symmetric tensor product of L2 (R3 ). The background potential defined in (2) is however still localized in the cube . We could also have confined the particles to with Dirichlet boundary conditions. This would only raise the ground state energy and thus, for the lower bound, our setup is more general. There is, however, a technical point that has to be considered when dealing with Coulomb forces. The background never appears in Foldy’s calculation; he simply removes the k = 0 mode from the Fourier transform, ν of the Coulomb potential (which is ν(k) = 4π|k|−2 , but with k taking the discrete values mentioned above, so that we are thus dealing with a “periodized” Coulomb potential). The k = 0 elimination means that we set ν(0) = 0, and this amounts to a subtraction of the average value of the potential – which is supposed to be a substitute for the effect of a neutralizing background. It does not seem to be a trivial matter to prove that this is equivalent to having a background, but it surely can be done. Since we do not wish to overload this paper, we leave this demonstration to another day. In any case the answers agree (in the sense that our rigorous lower bound agrees with Foldy’s answer), as we prove here. If one accepts the idea that setting ν(0) = 0 is equivalent to having a neutralizing background, then the ground state energy problem is finished because Girardeau shows [GM] that Foldy’s result is a true upper bound within the context of the ν(0) = 0 problem. The potential energy is quartic in the operators ak# . In Bogolubov’s analysis only terms in which there are four or two a0# operators are retained. The operator a0∗ creates, and a0 destroys particles with momentum 0 and such particles are the constituents of the “condensate”. In general there are no terms with three a0# operators (by momentum conservation) and in Foldy’s case there is also no four a0# term (because of the subtraction just mentioned). For the usual short range potential there is a four a0# term and this is supposed to give the leading term in the energy, namely e0 = 4πρa, where a is the “scattering length” of the two-body potential. Contrary to what would seem reasonable, this number, 4πρa is not the coefficient of the four a0# term, and to to prove that 4πρa is, indeed, correct took some time. It was done in 1998 [LY] and the method employed in [LY] will play an essential role here. But it is important to be clear about the fact that the four a0# , or “mean field” term is absent in the jellium case by virtue of charge neutrality. The leading term in this case presumably comes from the two a0# terms, and this is what we have to prove. For the short range case, on the other hand, it is already difficult enough to obtain the 4πρa energy that going beyond this to the two a0# terms is beyond the reach of rigorous analysis at the moment. The Bogolubov ansatz presupposes the existence of Bose–Einstein condensation (BEC). That is, most of the particles are in the k = 0 mode and the few that are not come in pairs with momenta k and −k. Two things must be said about this. One is that the only case (known to us) in which one can verify the correctness of the Bogolubov picture at weak coupling is the one-dimensional delta-function gas [LLi] – in which case there is presumably no BEC (because of the low dimensionality). Nevertheless the Bogolubov picture remains correct at low density and the explanation of this seeming contradiction lies in the fact that BEC is not needed; what is really needed is a kind of condensation on a length scale that is long compared to relevant parameters, but which is fixed and need not be as large as the box length L. This was realized in [LY] and the main idea there was to decompose into fixed-size boxes of appropriate length and use Neumann boundary conditions on these boxes (which can only lower the energy, and which is fine since we want a lower bound). We shall make a similar decomposition here, but, unlike the case


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in [LY] where the potential is purely repulsive, we must deal here with the Coulomb potential and work hard to achieve the necessary cancellation. The only case in which BEC has been proved to exist is in the hard core lattice gas at half-filling (equivalent to the spin-1/2 XY model) [KLS]. Weak coupling is sometimes said to be a “perturbation theory” regime, but this is not really so. In the one-dimensional case [LLi] the asymptotics near ρ = 0 is extremely difficult to deduce from the exact solution because the “perturbation” is singular. Nevertheless, the Bogolubov calculation gives it effortlessly, and this remains a mystery. One way to get an excessively negative lower bound to e0 for jellium is to ignore the kinetic energy. One can then show easily (by an argument due to Onsager) that the potential energy alone is bounded below by e0 ∼ −ρ 1/3 . See [LN]. Thus, our goal is to show that the kinetic energy raises the energy to −ρ 1/4 . This was done, in fact, in [CLY], but without achieving the correct coefficient −0.803(4π/3)1/4 . Oddly, the −ρ 1/4 law was proved in [CLY] by first showing that the non-thermodynamic N 7/5 law for a two-component bosonic plasma, as conjectured by Dyson [D], is correct. The [CLY] paper contains an important innovation that will play a key role here. There, too, it was necessary to decompose R3 into boxes, but a way had to be found to eliminate the Coulomb interaction between different boxes. This was accomplished by not fixing the location of the boxes but rather averaging over all possible locations of the boxes. This “sliding localization” will play a key role here, too. This idea was expanded upon in [GG]. Thus, we shall have to consider only one finite box with the particles and the background charge in it independent of the rest of the system. However, a price will have to be paid for this luxury, namely it will not be entirely obvious that the number of particles we want to place in each box is the same for all boxes, i.e., ρ(3 , where ( is the length of box. Local neutrality, in other words, cannot be taken for granted. The analogous problem in [LY] is easier because no attractive potentials are present there. We solve this problem by choosing the number, n, in each box to be the number that gives the lowest energy in the box. This turns out to be close to n = ρ(3 , as we show and as we know from [LN] must be the case as ( → ∞. Finally, let us remark on one bit of dimensional analysis that the reader should keep in mind. One should not conclude from (5) that a typical particle has energy ρ 1/4 and hence momentum ρ 1/8 or de Broglie wavelength ρ −1/8 . This is not the correct picture. Rather, a glance at the Bogolubov–Foldy calculation shows that the momenta of importance are of order ρ −1/4 , and the seeming paradox is resolved by noting that the number of excited particles (i.e., those not in the k = 0 condensate) is of order Nρ −1/4 . This means that we can, hopefully, localize particles to lengths as small as ρ −1/4+) , and cut off the Coulomb potential at similar lengths, without damage, provided we do not disturb the condensate particles. It is this clear separation of scales that enables our asymptotic analysis to succeed. 2. Outline of the Proof The proof of our Main Theorem 1.1 is rather complicated and somewhat hard to penetrate, so we present the following outline to guide the reader. 2.1. Section 3. Here we localize the system whose size is L into small boxes of size ( independent of L, but dependent on the intensive quantity ρ. Neumann boundary conditions for the Laplacian are used in order to ensure a lower bound to the energy. We

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always think of operators in terms of quadratic forms and the Neumann Laplacian in a box Q is defined for all functions in ψ ∈ L2 (Q) by the quadratic form |∇ψ(x)|2 dx. (ψ, −Neumann ψ) = Q

The lowest eigenfunction of the Neumann Laplacian is the constant function and this plays the role of the condensate state. This state not only minimizes the, kinetic energy, but it is also consistent with neutralizing the background and thereby minimizing the Coulomb energy. The particles not in the condensate will be called “excited” particles. To avoid localization errors we take ( ρ −1/4 , which is the relevant scale as we mentioned in the Introduction. The interaction among the boxes is controlled by using the sliding method of [CLY]. The result is that we have to consider only interactions among the particles and the background in each little box separately. The N particles have to be distributed among the boxes in a way that minimizes the total energy. We can therefore not assume that each box is neutral. Instead of dealing with this distribution problem we do a simpler thing which is to choose the particle number in each little box so as to achieve the absolute minimum of the energy in that box. Since all boxes are equivalent this means that we take a common value n as the particle number in each box. The total particle number which is n times the number of boxes will not necessarily equal N , but this is of no consequence for a lower bound. We shall show later, however, that it equality is nearly achieved, i.e., the the energy minimizing number n in each box is close to the value needed for neutrality. 2.2. Section 4. It will be important for us to replace the Coulomb potential by a cutoff Coulomb potential. There will be a short distance cutoff of the singularity at a distance r and a large distance cutoff of the tail at a distance R, with r ≤ R (. One of the unusual features of our proof is that r are R are not fixed once and for all, but are readjusted each time new information is gained about the error bounds. In fact, already in Sect. 4 we give a simple preliminary bound on n by choosing R ∼ ρ −1/3 , which is much smaller than the relevant scale ρ −1/4 , although the choice of R that we shall use at the end of the proof is of course much larger than ρ −1/4 , but less than (. 2.3. Section 5. There are several terms in the Hamiltonian. There is the kinetic energy, which is non-zero only for the excited particles. The potential energy, which is a quartic term in the language of second quantization, has various terms according to the number of times the constant function appears. Since we do not have periodic boundary conditions we will not have the usual simplification caused by conservation of momentum, and the potential energy will be correspondingly more complicated than the usual expression found in textbooks. In this section we give bounds on the different terms in the Hamiltonian and use these to get a first control on the condensation, i.e., a control on the number of particles n+ in each little box that are not in the condensate state. The difficult point is that n+ is an operator that does not commute with the Hamiltonian and so it does not have a sharp value in the ground state. We give a simple preliminary bound on its average n+ in the ground state by again choosing R ∼ ρ −1/3 . In order to control the condensation to an appropriate accuracy we shall eventually need not only a


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bound on the average, n+ , but also on the fluctuation, i.e, on n2+ . This will be done in Sect. 8 using a novel method developed in Appendix A for localizing off-diagonal matrices. 2.4. Section 6. The part of the potential energy that is most important is the part that is quadratic in the condensate operators a0# and quadratic in the excited variables ap# with p = 0. This, together with the kinetic energy, which is also quadratic in the ap# , is the part of the Hamiltonian that leads to Foldy’s law. Although we have not yet managed to eliminate the non-quadratic part up to this point we study the main “quadratic” part of the Hamiltonian. It is in this section that we essentially do Foldy’s calculation. It is not trivial to diagonalize the quadratic form and thereby reproduce Foldy’s answer because there is no momentum conservation. In particular there is no simple relation between the resolvent of the Neumann Laplacian and the Coulomb kernel. The former is defined relative to the box and the latter is defined relative to the whole of R3 . It is therefore necessary for us to localize the wavefunction in the little box away from the boundary. On such functions the boundary condition is of no importance and we can identify the kinetic energy with the Laplacian in all of R3 . This allows us to have a simple relation between the Coulomb term and the kinetic energy term since the Coulomb kernel is in fact the resolvent of the Laplacian in all of R3 . When we cut off the wavefunction near the boundary we have to be very careful because we must not cut off the part corresponding to the particles in the condensate. To do so would give too large a localization energy. Rather, we cut off only functions with sufficiently large kinetic energy so that the localization energy is relatively small compared to the kinetic energy. The technical lemma needed for this is a double commutator inequality given in Appendix B. 2.5. Section 7. At this point we have bounds available for the quadratic part (from Sect. 6) and the annoying non-quadratic part (from Sect. 5) of the Hamiltonian. These depend on r, R, n, n+ , and n2+ . We avail ourselves of the bounds previously obtained for n and n+ and now use our freedom to choose different values for r and R to bootstrap to the desired bounds on n and n+ , i.e., we prove that there is almost neutrality and almost condensation in each little box. 2.6. Section 8. In order to control n2+ we utilize, for the first time, the new method for localizing large matrices given in Appendix A. This method allows us to restrict to states with small fluctuations in n+ , and thereby bound n2+ , provided we know that the terms that do not commute with n+ have suffciently small expectation values. We then give bounds on these n+ “off-diagonal” terms. Unfortunately, these bounds are in terms of positive quantities coming from the Coulomb repulsion, but for which we actually do not have independent a-priori bounds. Normally, when proving a lower bound to a Hamiltonian, we can sometimes control error terms by absorbing them into positive terms in the Hamiltonian, which are then ignored. This may be done even when we do not have an a-priori bound on these positive terms. If we want to use Theorem A.1 in Appendix A, we will need an absolute bound on the “off-diagonal” terms and we can therefore not use the technique of absorbing them into the positive terms. The decision when to use the theorem in Appendix A or use the technique of absorption into positive terms is resolved in Sect. 9.

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2.7. Section 9. Since we do not have an a-priori bound on the positive Coulomb terms as described above we are faced with a dichotomy. If the positive terms are, indeed, so large that enough terms can be controlled by them we do not need to use the localization technique of Appendix A to finish the proof of Foldy’s law. The second possibility is that the positive terms are bounded in which case we can use this fact to control the terms that do commute with n+ and this allows us to use the localization technique in Appendix A to finish the proof of Foldy’s law. Thus, the actual magnitude of the positive repulsion terms is unimportant for the derivation of Foldy’s law. 3. Reduction to a Small Box As described in the previous sections we shall localize the problem into smaller cubes of size ( L. We shall in fact choose ( as a function of ρ in such a way that ρ 1/4 ( → ∞ as ρ → ∞. We shall localize the kinetic energy by using Neumann boundary conditions on the smaller boxes. We shall first, however, describe how we may control the electostatic interaction between the smaller boxes using the sliding technique of [CLY]. Let t, with 0 < t < 1/2, be a parameter which we shall choose later to depend on ρ in such a way that t → 0 as ρ → ∞. The choice of ( and t as functions of ρ will be made at the end of Sect. 9 when we complete the proof of Foldy’s law. Let χ ∈ C0∞ (R3 ) satisfy supp χ ⊂ [(−1 + t)/2, (1 − t)/2]3 , 0 ≤ χ ≤ 1, χ (x) = 1 for x in the smaller box [(−1 + 2t)/2, (1 − 2t)/2]3 , and χ (x) = χ (−x). Assume that all m-th order derivatives of χ are bounded by Cm t −m , where the constants Cm√depend only on m and are, in particular, independent of t. Let χ ( (x) = χ (x/(). Let η = 1 − χ . 1 χ We shall assume that χ is defined such that2 η is also C . Let η( (x) = η(x/(). Using −1 we define the constant γ by γ = χ (y) dy, and note that 1 ≤ γ ≤ (1 − 2t)−3 . We also introduce the Yukawa potential Yν (x) = |x|−1 e−ν|x| for ν > 0. As a preliminary to the following Lemma 3.1 we quote Lemma 2.1 in [CLY]. Lemma. Let K : R3 → R be given by

K(z) = r −1 e−νr − e−ωr h(z)

with r = |z| and ω > ν ≥ 0. Let h satisfy (i) h is a C 4 function of compact support; (ii) h(z) = 1 + ar 2 + O(r 3 ) near z = 0. Let h(z) = h(−z), so that K has a real Fourier transform. Then there is a constant, C3 (depending on h) such that if ω − ν ≥ C3 then K has a positive Fourier transform and, moreover,

ei ej K(xi − xj ) ≥

1≤i<j ≤N

1 (ν − ω)N 2

for all x1 , . . . xN ∈ R3 and all ei = ±1. Lemma 3.1 (Electrostatic decoupling of boxes using sliding). There exists a function of the form ω(t) = Ct −4 (we assume that ω(t) ≥ 1 for t < 1/2) and a constant γ with 1 ≤ γ ≤ (1 − 2t)−3 such that if we set w(x, y) = χ ( (x)Yω(t)/( (x − y)χ ( (y)

(7)


135

then the potential energy satisfies Upp + Upb + Ubb ≥γ

dµ N j =1

+

w xi + (µ + λ)(, xj + (µ + λ)(

1≤i<j ≤N

λ∈Z3 µ∈[− 21 , 21 ]3

−ρ

1 2 2ρ ×

w xj + (µ + λ)(, y + (µ + λ)( dy ω(t)N . w (x + (µ + λ)(, y + (µ + λ)() dx dy − 2(

Proof. We calculate dµ γ χ (x + (µ + λ))Yω (x − y)χ (y + (µ + λ)) λ∈Z3µ∈[−1/2,1/2]3

=

γ χ (x + z)Yω (x − y)χ (y + z) dz = h(x − y)Yω (x − y),

where we have set h = γ χ ∗ χ . Note that h(0) = 1 and that h satisfies all the assumptions in Lemma 2.1 in [CLY]. We then conclude from Lemma 2.1 in [CLY] that the Fourier transform of the function F (x) = |x|−1 − h(x)Yω(t) (x) is non-negative, where ω is a function such that ω(t) → ∞ as t → 0. [The detailed bounds from [CLY] show that we may in fact choose ω(t) = Ct −4 , since ω(t) has to control the 4th derivative of h.] Note, moreover, that limx→0 F (x) = ω(t). Hence 1≤i<j ≤N

F (yi − yj ) − ρ

N −1 j =1 (

F (yj − y) dy + 21 ρ 2

F (x − y) dx dy ≥ −

(−1 ×(−1

N ω(t) . 2

The lemma follows by writing |x|−1 = F (x) + h(y)Yω(t) (x) and by rescaling from boxes of size 1 to boxes of size (. As explained above we shall choose the parameters t and ( as functions of ρ at the very end of the proof. We shall choose them in such a way that t → 0 and ρ 1/4 ( → ∞ as ρ → ∞. Moreover, we will have conditions of the form ρ −τ (ρ 1/4 () → 0,

and t ν (ρ 1/4 () → ∞

as ρ → ∞, where τ, ν are universal constants. Consider now the n-particle Hamiltonian n Hµ,λ = − 21

n j =1

(j )

Qµ,λ + γ Wµ,λ ,

(8)

136

E. H. Lieb, J. P. Solovej (j )

where we have introduced the Neumann Laplacian Qµ,λ of the cube Qµ,λ = (µ + 3

λ)( + − 21 (, 21 ( and the potential Wµ,λ (x1 , . . . , xn ) = w xi + (µ + λ)(, xj + (µ + λ)( 1≤i<j ≤n n

−ρ

j =1

w xj + (µ + λ)(, y + (µ + λ)( dy

+

1 2 2ρ

w (x + (µ + λ)(, y + (µ + λ)() dx dy.

× n be the ground state energy of the HamilLemma 3.2 (Decoupling of boxes). Let Eµ,λ n given in (8) considered as a bosonic Hamiltonian. The ground state energy tonian Hµ,λ E0 of the Hamiltonian H in (1) is then bounded below as ω(t)N n E0 ≥ inf Eµ,λ dµ − . 1≤n≤N 2( 3 λ∈Z

µ∈[− 21 , 21 ]3

Proof. If ;(x1 , . . . , xN ) ∈ L2 (R3N ) is a symmetric function. Then µ,λ ;) dµ − ω(t)N , (;, H (;, H ;) ≥ 2( 3 λ∈Z

µ∈[− 21 , 21 ]3

where µ,λ ;) = (;, H

N j =1 xj ∈Qµ,λ

|∇j ;(x1 , . . . , xN )|2 dx1 . . . dxN

Wµ,λ (x1 , . . . , xN )|;(x1 , . . . , xN )|2 dx1 . . . dxN .

+γ

µ,λ ;) ≥ inf 1≤n≤N E n . The lemma follows since it is clear that (;, H µ,λ

n fall in three groups depending on λ. The first kind For given µ the Hamiltonians Hµ,λ for which Qλ,µ ∩ = ∅. They describe boxes with no background. The optimal energy for these boxes are clearly achieved for n = 0. The second kind for which Qλ,µ ⊂ . These Hamiltonians are all unitarily equivalent to γ H(n , where

H(n =

n j =1

+

− 21 γ −1 (,j − ρ

1≤i<j ≤n

w(xj , y) dy

w(xi , xj ) + 21 ρ 2

(9) w(x, y) dx dy,

where −( is the Neumann Laplacian for the cube [−(/2, (/2]3 . Finally, there are operators of the third kind for which Qµ,λ intersects both and its complement. In


137

this case the particles only see part of the background. If we artificially add the missing background only the last term in the potential Wµ,λ increases. (The first term does not change and the second can only decrease.) In fact it will increase by no more than 1 2 1 2 w(x, y) dx dy ≤ 2 ρ |x − y|−1 dx dy ≤ Cρ 2 (5 . 2ρ x∈[−(/2,(/2]3 y∈[−(/2,(/2]3

n of the third kind are bounded below by an operator which is Thus the operator Hµ,λ unitarily equivalent to γ H(n − Cρ 2 (5 . We now note that the number of boxes of the third kind is bounded above by C(L/()2 . The total number of boxes of the second or third kind is bounded above by (L+()3 /(3 = (1 + L/()3 . We have therefore proved the following result.

Lemma 3.3 (Reduction to one small box). The ground state energy E0 of the Hamiltonian H in (1) is bounded below as E0 ≥ (1 + L/()3 γ

inf inf Spec H(n − C(L/()2 ρ 2 (5 −

1≤n≤N

ω(t)N , 2(

where H(n is the Hamiltonian defined in (9). In the rest of the paper we shall study the Hamiltonian (9). 4. Long and Short Distance Cutoffs in the Potential The potential in the Hamiltonian (9) is w given in (7). Our aim in this section to replace w by a function that has long and short distance cutoffs. We shall replace the function w by wr,R (x, y) = χ ( (x)Vr,R (x − y)χ ( (y),

(10)

where Vr,R (x) = YR −1 (x) − Yr −1 (x) =

e−|x|/R − e−|x|/r . |x|

(11)

Here 0 < r ≤ R ≤ ω(t)−1 (. Note that for x r then Vr,R (x) ≈ r −1 − R −1 and for |x| R then Vr,R (x) ≈ |x|−1 e−|x|/R . In this section we shall bound the effect of replacing w by wr,R . We shall not fix the cutoffs r and R, but rather choose them differently at different stages in the later arguments. We first introduce the cutoff R alone, i.e., we bound the effect of replacing w by wR (x, y) = χ ( (x)VR (x − y)χ ( (y), where VR (x) = |x|−1 e−|x|/R = YR −1 (x). Thus, since R ≤ ω(t)−1 (, the Fourier transforms satisfy 1 1 ω/( (k) − V R (k) = 4π Y ≥ 0. − k 2 + (ω(t)/()2 k 2 + R −2

138


(We use the convention that fˆ(k) = f (x)e−ikx dx.) Hence w(x, y) − wR (x, y) = χ ( (x) Yω/( − VR (x − y)χ ( (y) defines a positive semi-definite kernel. Note, more over, that Yω/( − VR (0) = R −1 − ω/( ≤ R −1 Thus,

w(xi , xj ) − ρ

1≤i<j ≤n



−

j =1

wR (xi , xj ) − ρ

1≤i<j ≤n

=

n 1 2 n

−

n

w(xj , y) dy + 21 ρ 2 n j =1

w(x, y) dx dy

wR (xj , y) dy + 21 ρ 2

δ(x − xi ) − ρ (w − wr )(x, y)

n

i

 wR (x, y) dx dy 

δ(y − xi ) − ρ

dx dy

i

1 χ ( (xi )2 Yω/( − VR (0) ≥ − 21 n Yω/( − VR (0) = − 21 nR −1 . 2

(12)

i

We now bound the effect of replacing wR by wr,R . I.e., we are replacing VR (x) = |x|−1 e−|x|/R by |x|−1 e−|x|/R − e−|x|/r . This will lower the repulsive terms and for the attractive term we get n n wR (xj , y) dy ≥ − ρ wr,R (xj , y) dy −ρ j =1

j =1

− nρ sup x

≥ −ρ

n

χ ( (x)

e−|x−y|/r χ ( (y) dy |x − y|

(13)

wr,R (xj , y) dy − Cnρr 2 .

j =1

If we combine the bounds (12) and (13) we have the following result. Lemma 4.1 (Long and short distance potential cutoffs). Consider the Hamiltonian n n 1 −1 − 2 γ (,j − ρ wr,R (xj , y) dy + wr,R (xi , xj ) H(,r,R = j =1 1≤i<j ≤n (14) + 21 ρ 2

wr,R (x, y) dx dy,

where wr,R is given in (10) and (11) with 0 < r ≤ R ≤ ω(t)−1 ( and −( as before is the Neumann Laplacian for the cube [−(/2, (/2]3 . Then the Hamiltonian H(n defined in (9) obeys the lower bound n − 21 nR −1 − C1 nρr 2 . H(n ≥ H(,r,R

A similar argument gives the following result. Lemma 4.2. With the same notation as above we have for 0 < r " ≤ r ≤ R ≤ R " ≤ ω(t)−1 ( that n n −1 1 H(,r − C1 nρr 2 . " ,R " ≥ H(,r,R − 2 nR


139

Proof. Simply note that Vr " ,R " (x)−Vr,R (x) = YR " −1 (x)−YR −1 (x)+Yr −1 (x)−Yr " −1 (x) and now use the same arguments as before. Corollary 4.3 (The particle number n cannot be too small). There exists a constant C > 0 such that if ω(t)−1 ρ 1/3 ( > C then H(n ≥ 0 if n ≤ Cρ(3 . Proof. Choose R = ρ −1/3 and r = 21 R. Then we may assume that R ≤ ω(t)−1 ( since ω(t)−1 ρ 1/3 ( is large. From Lemma 4.1 we see immediately that n n 1 2 H( ≥ − wr,R (x, y) dx dy − CnρR 2 ρ wr,R (xj , y) dy + 2 ρ j =1

≥ −n sup ρ x

wr,R (x, y) dy +

wr,R (x, y) dx dy − CnρR 2 .

1 2 2ρ

The corollary follows since supx wr,R (x, y) dy ≤ 4π R 2 and with the given choice of R and r it is easy to see that 21 wr,R (x, y) dx dy ≥ cR 2 (3 . 5. Bound on the Unimportant Part of the Hamiltonian n In this section we shall bound the Hamiltonian H(,r,R given in (14). We emphasize that we do not necessarily have neutrality in the cube, i.e., n and ρ(3 may be different. We n are simply looking for a lower bound to H(,r,R , that holds for all n. The goal is to find a lower bound that will allow us to conclude that the optimal n, i.e., the value for which the energy of the Hamiltonian is smallest, is indeed close to the neutral value. We shall express the Hamiltonian in second quantized language. This is purely for convenience. We stress that we are not in any way changing the model by doing this and the treatment is entirely rigorous and could have been done without the use of second quantization. Let up , (p/π ∈ (N ∪ {0})3 be an orthonormal basis of eigenfunctions of the Neumann Laplacian −( such that −( up = |p|2 up . I.e.,

up (x1 , x2 , x3 ) = cp (−3/2

3

cos

j =1

pj π(xj + (/2) , (

√ where the normalization satisfies c0 = 1 and in general 1 ≤ cp ≤ 8. The function u0 = (−3/2 is the constant eigenfunction with eigenvalue 0. We note that for p = 0 we have (up , −( up ) ≥ π 2 (−2 . n We now express the Hamiltonian H(,r,R ∗ ∗ operators ap = a(up ) and ap = a(up ) .

in terms of the creation and annihilation

Define

w pq,µν =

wr,R (x, y)up (x)uq (y)uµ (x)uν (y) dx dy.

We may then express the two-body repulsive potential as wr,R (xi , xj ) = 21 w pq,µν ap∗ aq∗ aν aµ , 1≤i<j ≤n

(15)

pq,µν

140


where the right-hand side is considered restricted to the n-particle subspace. Likewise the background potential can be written −ρ

n

wr,R (xj , y) dy = −ρ(3

pq

j =1

w 0p,0q ap∗ aq

and the background-background energy 1 2 wr,R (x, y) dx dy = 21 ρ 2 (6 w ρ 00,00 . 2 We may therefore write the Hamiltonian as n = 21 γ −1 |p|2 ap∗ ap + H(,r,R p

− ρ(

3

pq

w 0p,0q ap∗ aq

1 2

pq,µν

w pq,µν ap∗ aq∗ aν aµ (16)

+ 21 ρ 2 (6 w 00,00 .

We also introduce the operators n0 = a0∗ a0 and n+ = p=0 . These operators represent the number of particles in the condensate state created by a0∗ and the number of particle not in the condensate. Note that on the subspace where the total particle number is n, both of these operators are non-negative and n+ = n − n0 . Using the bounds on the long and short distance cutoffs in Lemma 4.1 we may immediately prove a simple bound on the expectation value of n+ . Lemma 5.1 (Simple bound on the number of excited particles). There is a constant C > 0 such that if ω(t)−1 ρ 1/3 ( > C then for any state such that the expectation H(n ≤ 2 0, the expectation of the number of excited particles satisfies n+ ≤ Cnρ −1/6 ρ 1/4 ( . Proof. We simply choose r = R = ρ −1/3 in Lemma 4.1. This is allowed since R ≤ ω(t)−1 ( is ensured from the assumption that ω(t)−1 ρ 1/3 ( is large. We then obtain H(n

≥

n j =1

− 21 γ −1 (,j

−

−1 1 2 nR

− Cnρr ≥ 2

n j =1

− 21 γ −1 (,j − Cnρ 1/3 . n

The bound on n+ follows since the bound on the gap (15) implies that n+ π 2 (−2 .

j =1 −(,j

≥

Motivated by Foldy’s use of the Bogolubov approximation it is our goal to reduce the n so that it has only what we call quadratic terms, i.e., terms which Hamiltonian H(,r,R contain precisely two ap# with p = 0. More precisely, we want to be able to ignore all terms containing the coefficients • w 00,00 . • w p0,q0 = w 0p,0q , where p, q = 0. These terms are in fact quadratic, but do not appear in the Foldy Hamiltonian. We shall prove that they can also be ignored. • w p0,00 = w 0p,00 = w 00,p0 = w 00,0p , where p = 0. • w pq,µ0 = w µ0,pq = w qp,0µ = w 0µ,qp , where p, q, µ = 0.


141

• w pq,µν , where p, q, µ, ν = 0. The sum of all these terms form a non-negative contribution to the Hamiltonian and can, when proving a lower bound, either be ignored or used to control error terms. We shall consider these cases one at a time. n Lemma 5.2 (Control of terms with w 00,00 ). The sum of the terms in H(,r,R containing w 00,00 is equal to 2 3 1 n0 − ρ( 00,00 − n0 2w 2 3 2 3 1 = 2w 00,00 n − ρ( + ( n+ ) − 2 n − ρ( n0 . n+ −

Proof. The terms containing w 00,00 are 2 1 00,00 a0∗ a0∗ a0 a0 − 2ρ(3 a0∗ a0 + ρ 2 (6 = 21 w 00,00 a0∗ a0 − ρ(3 − 21 w 00,00 a0∗ a0 2w using the 0commutation relation [ap , aq∗ ] = δp,q .

n containing Lemma 5.3 (Control of terms with w p0,q0 ). The sum of the terms in H(,r,R 0p,0q with p, q = 0 is bounded below by w p0,q0 or w

−4π[ρ − n(−3 ]+ n+ R 2 − 4π n2+ (−3 R 2 , where [t]+ = max{t, 0}. 0p,0q are Proof. The terms containing w p0,q0 or w ∗ ∗ ∗ ∗ 3 ∗ 1 1 w a a a a + w a a a a − ρ( w a a 0p,0q p q 2 p0,q0 p 0 0 q 2 0p,0q 0 p q 0 p =0 q=0

= ( n0 − ρ(3 ) Note that n0 commutes with

p=0 q=0

We have that w p0,q0 = (−3 Hence

w p0,q0 ap∗ aq = (−3

w p0,q0 ap∗ aq . wr,R (x, y) dyup (x)uq (x) dx. 

wr,R (x, y) dy 

p=0

≤ (−3 sup x"

= (−3 sup x"

w p0,q0 ap∗ aq .

p=0 q=0

p =0 q=0

wr,R (x " , y) dy wr,R (x " , y) dy

 up (x)ap∗  

 

p=0

p=0

p=0

∗ up (x)ap∗  dx.

 up (x)ap∗  

ap∗ ap = (−3 sup x"

p=0

∗ up (x)ap∗  dx.

wr,R (x " , y) dy n+ .

142


Since

wr,R (x, y) dy ≤

sup x

Vr,R (y) dy ≤ 4π R 2

we obtain the operator inequality 0≤

w p0,q0 ap∗ aq ≤ 4π (−3 R 2 n+ ,

p =0 q=0

and the lemma follows. Before treating the last two types of terms we shall need the following result on the structure of the coefficients w pq,µν . Lemma 5.4. For all p" , q " ∈ (π/() (N ∪ {0})3 and α ∈ N there exists Jpα" q " ∈ R with Jpα" q " = Jqα" p" such that for all p, q, µ, ν ∈ (π/() (N ∪ {0})3 we have w pq,µν =

α

α α Jpµ Jqν .

(17)

Moreover we have the operator inequalities 0≤

p,p" =0

w pp" ,00 ap∗ ap" =

and

0≤

p,p" ,m=0

p,p" =0

w p0,0p" ap∗ ap" ≤ 4π (−3 R 2 n+

(18)

w pm,mp" ap∗ ap" ≤ r −1 n+ .

Proof. The operator A with integral kernel wr,R (x, y) is a non-negative Hilbert–Schmidt operator on L2 (R3 ) with norm less than supk Vˆr,R (k) ≤ 4π R 2 . Denote the eigenvalues of A by λα , α = 1, 2, . . . and corresponding orthonormal eigenfunctions by ϕα . We may assume that these functions are real. The eigenvalues satisfy 0 ≤ λα ≤ 4π R 2 . We then have λα up (x)uµ (x)ϕα (x) dx uq (y)uν (y)ϕα (y) dy. w pq,µν = α

α = λ1/2 u (x)u (x)ϕ (x) dx. The identity (17) thus follows with Jpµ p µ α α If P denotes the projection onto the constant functions we may also consider the operator (I − P )A(I − P ). Denote its eigenvalues and eigenfunctions by λ"α and ϕα" . Then again 0 ≤ λ"α ≤ 4πR 2 . Hence we may write −3

w p0,0p" = (

α

λ"α

up (x)ϕα" (x) dx

up" (y)ϕα" (y) dy.


143

Thus, since all ϕα" are orthogonal to constants we have p,p" =0

w p0,0p" ap∗ ap" = (−3

α

= (−3

α

 λ"α 



up (x)ϕα" (x) dx ap∗  

p=0

λ"α a

p=0

∗

ϕα" a ϕα" .

∗ up (x)ϕα" (x) dx ap∗ 

The inequalities (18) from this. follow immediately The fact that p,p" ,m=0 w pm,mp" ap∗ ap" ≥ 0 follows from the representation (17). Moreover, since the kernel wR,r (x, y) is a continuous function we have that wr,R (x, x) = 2 for almost all x and hence λ ϕ (x) α α α m=0

w pm,mp" =

up (x)up" (x)wr,R (x, x) dx − w p0,0p" .

We therefore have p,p" ,m=0

w pm,mp" ap∗ ap" ≤ =

p,p" =0

 wr,R (x, x) 

p=0

≤ sup wr,R (x " , x " )

x"

up (x)up" (x)Wr,R (x, x) dx ap∗ ap" 

up (x)ap∗  

 

p=0 "

p=0

∗ up (x)ap∗  dx



up (x)ap∗  

p=0

∗ up (x)ap∗  dx

"

= sup wr,R (x , x ) n+ x"

and the lemma follows since supx " wr,R (x " , x " ) ≤ r −1 .

n containing Lemma 5.5 (Control of terms with w p0,00 ). The sum of the terms in H(,r,R 0p,00 , w00,p0 , or w 00,0p , with p = 0 is, for all ε > 0, bounded below by w p0,00 , w

n0 n+ − ε w00,00 ( n0 + 1 − ρ(3 )2 , −ε−1 4π(−3 R 2

(19)

and by p=0

w p0,00 (n − ρ(3 )ap∗ a0 + a0∗ ap (n − ρ(3 ) − ε −1 4π (−3 R 2 n0 n+ − ε w00,00 ( n+ − 1)2 .

(20)

144


Proof. The terms containing w p0,00 , w 0p,00 , w00,p0 , or w 00,0p are p=0

1 p0,00 2w

=

p=0

=

2ap∗ a0∗ a0 a0 + 2a0∗ a0∗ a0 ap − 2ρ(3 a0∗ ap − 2ρ(3 ap∗ a0

w p0,00 ( n0 − ρ(3 ) n0 − ρ(3 )ap∗ a0 + a0∗ ap (

α p=0

α α Jp0 J00 ap∗ a0 ( n0 + 1 − ρ(3 ) + ( n0 + 1 − ρ(3 )a0∗ ap .

In the last term we have used the representation (17) and the commutation relation [ n0 , a0 ] = a0 . For all ε > 0 we get that the above expression is bounded below by

ε−1

α p,p" =0

α α Jp0 Jp" 0 n0 ap∗ ap" − ε

= −ε

α

−1

p,p" =0

α J00

2

( n0 + 1 − ρ(3 )2

w p0,0p" n0 ap∗ ap" − ε w00,00 ( n0 + 1 − ρ(3 )2 .

The bound (19) follows from (18). The second bound (20) follows in the same way if we notice that the terms containing 0p,00 , w00,p0 , or w 00,0p may be written as w p0,00 , w p=0

w p0,00 (n − ρ(3 )ap∗ a0 + a0∗ ap (n − ρ(3 ) +

α p=0

α α Jp0 J00 ap∗ a0 (1 − n+ ) + (1 − n+ )a0∗ ap .

n containing Lemma 5.6 (Control of terms with w pq,m0 ). The sum of the terms in H(,r,R pq,0m , wp0,qm , or w 0p,qm , with p, q, m = 0 is bounded below by w pq,m0 , w

−ε −1 4π(−3 R 2 n0 n+ − ε n+ r −1 − ε

p,m,p" ,m" =0

for all ε > 0.

∗ ∗ w mp" ,pm" am ap" am" ap ,


145

Proof. The terms containing w pq,m0 , w pq,0m , wp0,qm , or w 0p,qm are ∗ w pqm0 ap∗ aq∗ am a0 + a0∗ am aq a p pqm=0    α ∗  α ∗  = Jq0 aq a0 Jpm a p am 

α



+

q=0

pm=0

≥ −



pm=0

∗ 

α ∗ Jpm a p am  



ε −1 

α

 +ε 

q=0

q=0

pm=0

∗ 

α ∗   Jq0 a q a0



α ∗  Jq0 a q a0

q=0

 α ∗ Jpm a m ap  

 α ∗  Jq0 a 0 aq

pm=0

 α ∗ Jpm ap am  .

α = J α we may write this as Using that Jpm mp

−ε −1

qq " =0

w q0,0q " aq∗ aq " a0 a0∗ − ε

= − ε −1

qq " =0

−ε

p,m,p" ,m" =0

w q0,0q " aq∗ aq " a0 a0∗ − ε

p,m,m" =0

∗ w mp" ,pm" am ap ap∗ " am"

p,m,p" ,m" =0

∗ ∗ w mp" ,pm" am ap" am" ap

∗ w mp,pm" am am" .

The lemma now follows from Lemma 5.4.

6. Analyzing the Quadratic Hamiltonian In this section we consider the main part of the Hamiltonian. This is the “quadratic” Hamiltonian considered by Foldy. It consists of the kinetic energy and all the terms with the coefficients w pq,00 , w 00,pq w p0,0q , and w 0p,q0 with p, q = 0, i.e., HFoldy = 21 γ −1 +

1 2

p

pq=0

= 21 γ −1

p

|p|2 ap∗ ap w pq,00 ap∗ a0∗ a0 aq + a0∗ ap∗ aq a0 + ap∗ aq∗ a0 a0 + a0∗ a0∗ ap aq (21) |p|2 ap∗ ap +

w pq,00 ap∗ aq a0∗ a0 + 21 ap∗ aq∗ a0 a0 + 21 a0∗ a0∗ ap aq .

pq=0

146


In order to compute all the bounds we found it necessary to include the first term in (20) into the “quadratic” Hamiltonian. We therefore define |p|2 ap∗ ap + w p0,00 (n − ρ(3 )ap∗ a0 + a0∗ ap (n − ρ(3 ) HQ = 21 γ −1 +

p

pq=0

w pq,00

p=0

ap∗ aq a0∗ a0

+ 21 ap∗ aq∗ a0 a0 + 21 a0∗ a0∗ ap aq .

(22)

Note that HFoldy = HQ in the neutral case n = ρ(3 . Our goal is to give a lower bound on the ground state energy of the Hamiltonian HQ . For the sake of convenience we first enlarge the one-particle Hilbert space L2 [−(/2, (/2]3 . In fact, instead of considering the symmetric Fock space over L2 [−(/2, (/2]3 we now consider the symmetric Fock space over the one-particle Hilbert space L2 [−(/2, (/2]3 ⊕ C. Note that the larger Fock space of course contains the original Fock space as a subspace. On the larger space we have a new pair of creation and annihilation operators that we denote a0∗ and a0 . These operators merely 2 3 create vectors in the C component of L [−(/2, (/2] ⊕ C, and so commute with all other operators. We shall now write ∗ ap , if p = 0 ap , if p = 0 ∗ ap = and ap = . (23) a0 , if p = 0 a0∗ , if p = 0 We now define the Hamiltonian Q = 1 γ −1 |p|2 ap∗ ap + w p0,00 (n − ρ(3 ) ap∗ a0 + a0∗ ap (n − ρ(3 ) H 2 +

pq

p

p

w pq,00 aq a0∗ a0 + 21 ap∗ aq∗ a0 a0 + 21 a0∗ a0∗ ap aq , ap∗

(24)

where we no longer restrict p, q to be different from 0. Note that for all states on the Q = HQ . larger Fock space for which a0∗ a0 = 0 we have H 2 3 For any function ϕ ∈ L [−(/2, (/2] we introduce the creation operator (up , ϕ) ap∗ . a ∗ (ϕ) = p

Note that sum includes p = 0. the difference from a ∗ (ϕ) is given by a ∗ (ϕ)−a ∗ (ϕ) = the (u0 , ϕ) a0∗ − a0∗ . Then [ a (ϕ), a ∗ (ψ)] = (ϕ, ψ). We have introduced the “dummy” operator a0∗ in order for this relation to hold. One could just as well have stayed in the old space, but then the relation above would hold only for functions orthogonal to constants. For any k ∈ R3 denote χ (,k (x) = eikx χ ( (x) and define the operators a ∗ (χ (,k )a0 bk∗ =

and

bk = a (χ (,k )a0∗

They satisfy the commutation relations a (χ (,k ) a ∗ (χ (,k " ) [bk , bk∗" ] = a0∗ a0 χ (,k , χ (,k " − 2 (k " − k) − = a0∗ a0 χ a (χ (,k ) a ∗ (χ (,k " ) (

(25)


147

We first consider the kinetic energy part of the Hamiltonian. We shall bound it using the double commutator bound in Appendix B. First we need a well known comparisson between the Neumann Laplacian and the Laplacian in the whole space. Lemma 6.1 (Neumann resolvent is bigger than free resolvent). Let P( denote the projection in L2 (R3 ) that projects onto L2 ([−(/2, (/2]3 ) (identified as a subspace). Then if − denotes the Laplacian on all of R3 and −( is the Neumann Laplacian on [−(/2, (/2]3 we have the operator inequality (−( + a)−1 ≥ P( (− + a)−1 P( , for all a > 0. Proof. It is clear that for all f ∈ L2 (R3 ) (P( (−( + a)1/2 P( (− + a)−1/2 f (2 ≤ (f (2 , and hence ((− + a)−1/2 P( (−( + a)1/2 P( f (2 ≤ (f (2 . Now simply use this with f = (−( + a)−1/2 u. Lemma 6.2 (The kinetic energy bound). There exists a constant C " > 0 such that if C " t < 1, where t is the parameter used in the definition of χ ( in Sect. 3, we have p

|p|2 ap∗ ap

−3

≥ (2π )

"

2 −1

(1 − C t) n

R3

|k|4 b∗ b dk |k|2 + ((t 3 )−2 k k

for all states with a0∗ a0 = 0 and particle number equal to n, i.e., # " ∗ 2 = n2 . p a p ap

∗ p a p ap

!2

=

Proof. Let s, with 0 < s ≤ t, be a parameter to be chosen below. Recall that t is the parameter used in the definition of χ ( in Section 3. Then since χ 2( + η(2 = 1 we have (−( )2 (−( )2 (−( )2 = 21 (χ 2( + η(2 ) + 21 (χ 2 + η(2 ) −2 −2 −( + (((s) −( + ((s) −( + ((s)−2 ( (−( )2 (−( )2 χ ( + η( = χ( η( −2 −( + ((s) −( + ((s)−2 (−( )2 (−( )2 χ( , χ( + , η + , , η ( ( −( + ((s)−2 −( + ((s)−2

−( ≥

(−( )2 (−( )2 χ ( + η( η( −( + ((s)−2 −( + ((s)−2 −( − C((t)−2 − C(−2 s 2 t −4 , −( + ((s)−2

≥ χ(

148


where the last inequality follows from Lemma B.1 in Appendix B. We can now repeat this calculation to get −( (−( )2 −2 χ( −( ≥ χ ( − C((t) −( + ((s)−2 −( + ((s)−2 −( (−( )2 −2 η( − C(−2 s 2 t −4 + η( − C((t) −( + ((s)−2 −( + ((s)−2 −( −( −2 χ χ − C((t) , ( , ( + , η( , η( . −( + ((s)−2 −( + ((s)−2 If we therefore use (53) in Lemma B.1 and recall that s ≤ t we arrive at −( (−( )2 −2 χ( −( ≥ χ ( − C((t) −( + ((s)−2 −( + ((s)−2 −( (−( )2 −2 η( − C(−2 s 2 t −4 . + η( − C((t) −( + ((s)−2 −( + ((s)−2 Note that for α > 0 we have α

(−( )2 −( − C((t)−2 ≥ −Cα −1 s 2 t −4 (−2 . −2 −( + ((s) −( + ((s)−2

Thus if we also assume that α < 1 we have −( ≥ (1 − α)χ (

(−( )2 χ ( − Cα −1 s 2 t −4 (−2 . −( + ((s)−2

Thus if u is a normalized function on L2 (R3 ) which is orthogonal to constants we have according to the bound on the gap (15) that for all 0 < δ < 1 (−( )2 χ (u (u, −( u) ≥ (1 − δ)(1 − α) u, χ ( −( + ((s)−2 − C(1 − δ)α −1 s 2 t −4 (−2 + δπ 2 (−2 . We choose α = δ = C " st −2 for an appropriately large constant C " > 0 and assume that s and t are such that δ is less than 1. Then (−( )2 " −2 2 χ (u . (u, −( u) ≥ (1 − C st ) u, χ ( −( + ((s)−2 If we now use Lemma 6.1 we may write this as (u, −( u) ≥ (1 − C " st −2 )2 u, χ ( (

1 ( χ ( u − + ((s)−2 (−)2 " −2 2 χ χ = (1 − C st ) u, ( (u , − + ((s)−2

where in the last inequality we have used that χ = ( χ and χ = χ ( .


149

We now choose s = t 3 and we may then write this inequality in second quantized form as $ ∗ % |k|4 2 ∗ −3 " 2 a (χ (,k ) |p| ap ap ≥ (2π) (1 − C t) a (χ (,k ) dk 2 + ((t 3 )−2 3 |k| R p $ ∗ % using that a0 a0 = 0. Since we consider only states with particle number n the inequality still holds if we insert n−1 a0 a0∗ as in the statement of the lemma. With the same notation as in the above lemma we may write wr,R (x, y) = (2π )−3 Vˆr,R (k)χ (,k (x)χ (,k (y) dk. The last two sums in the Hamiltonian (24) can therefore be written as & χ ( (k)bk∗ + χ ( (k)bk (2π()−3 Vˆr,R (k) (n − ρ(3 )(−3/2 ' ∗ ∗ + 21 bk∗ bk + b−k b−k + bk∗ b−k + bk b−k dk − w pq,00 ap∗ aq . pq

Note that it is important here that the potential wr,R contains the localization function χ (. ˆ ˆ χ ( (k) = χ ( (−k) we have for states with $ ∗ Thus, % since Vr,R (k) = Vr,R (−k) and a0 a0 = 0 that $

% Q ≥ H

R3

! % w pq,00 aq , ap∗ hQ (k) dk −

$

(26)

pq

where hQ (k) =

∗ (1 − C " t)2 |k|4 ∗ bk bk + b−k b−k 3 2 3 −2 4(2π) γ n |k| + ((t ) Vˆr,R (k) & 3 −3/2 ∗ (27) χ ( (k)(bk + b−k χ ( (k)(bk∗ + b−k ) + + )( ) (n − ρ( 2(2π()3 ' ∗ ∗ + bk∗ bk + b−k b−k + bk∗ b−k + bk b−k .

Theorem 6.3 (Simple case of Bogolubov’s method). For arbitrary constants A ≥ B > 0 and κ ∈ C we have the inequality ∗ ∗ ∗ A(bk∗ bk + b−k b−k ) + B(bk∗ b−k + bk b−k ) + κ(bk∗ + b−k ) + κ(bk + b−k ) 2 ( 2|κ| ∗ . ≥ − 21 (A − A2 − B 2 )([bk , bk∗ ] + [b−k , b−k ]) − A+B

150


Proof. We may complete the square ∗ ∗ ∗ b−k ) + B(bk∗ b−k + bk b−k ) + κ(bk∗ + b−k ) + κ(bk + b−k ) A(bk∗ bk + b−k

∗ ∗ ∗ + a) + D(b−k + αbk + a)(b−k + αb−k + a) = D(bk∗ + αb−k + a)(bk + αb−k

∗ − Dα 2 ([bk , bk∗ ] + [b−k , b−k ]) − 2D|a|2 ,

if D(1 + α 2 ) = A,

2Dα = B, aD(1 + α) = κ. ( We choose the solution α = A/B − A2 /B 2 − 1. Hence Dα 2 = Bα/2 = 21 (A −

(

A2 − B 2 ),

D|a|2 =

|κ|2 |κ|2 = . D(1 + α 2 + 2α) A+B

Usually when applying Bogolubov’s method the commutator [bk , bk∗ ] is a positive constant. In this case the lower bound in the theorem is actually the bottom of the spectrum of the operator. If moreover, A > B the bottom is actually an eigenvalue. In our case the commutator [bk , bk∗ ] is not a constant, but according to (25) we have ∗ (28) [bk , bk ] ≤ χ ( (x)2 dxa0∗ a0 ≤ (3 a0∗ a0 . From this and the above theorem we easily conclude the following bound. Lemma 6.4 (Lower bound on quadratic Hamiltonian). On the subspace with n particles we have 2 HQ ≥ −I n5/4 (−3/4 − 21 n − ρ(3 w 00,00 − 4π n5/4 (−3/4 (n()−1/4 , where I = 21 (2π)−3

R3

g(k) = 4π

f (k) − (f (k)2 − g(k)2 )1/2 dk with 1 1 − 4π 2 k 2 + (n1/4 (−3/4 R)−2 k + (n1/4 (−3/4 r)−2

and f (k) = g(k) + 21 γ −1 (1 − C " t)2

|k|4 . |k|2 + (n1/4 (1/4 t 3 )−2

Q . We shall use (26). Proof. We consider a state with a0∗ a0 = 0. Then HQ = H Note first that w pq,00 ap∗ aq = w p0,0q ap∗ aq ≤ 4π (−3 R 2 n+ ≤ 4π (−1 n pq

p,q=0

by (18) and the fact that R ≤ (. We may of course rewrite (−1 n = n5/4 (−3/4 (n()−1/4 . By Theorem 6.3, (27) and (28) we have hQ (k) ≥ −(Ak −

)

A2k − Bk2 )n(3 −

* Vˆr,R (k)2 (n − ρ(3 )2 ** χ ( (k)*2 , 2(2π )6 (9 (Ak + Bk )


151

where Bk =

Vˆr,R (k) , 2(2π()3

Ak = Bk +

|k|4 (1 − C " t)2 . 4(2π )3 γ n |k|2 + ((t 3 )−2

Since Ak > Bk we have that hQ (k) ≥ −(Ak −

)

A2k − Bk2 )n(3 −

* Vˆr,R (k)(n − ρ(3 )2 ** χ ( (k)*2 . 3 6 2(2π ) (

Note that

* Vˆr,R (k)(n − ρ(3 )2 ** χ ( (k)*2 dk 2(2π)3 (6 n 2 χ ( (x)Vr,R (x − y)χ ( (y) dx dy = = 21 3 − ρ (

1 2

n − ρ(3

2

w 00,00 .

The lemma now follows from (26) by a simple change of variables in the k integral.

As a consequence we get the following bound for the Foldy Hamiltonian. Corollary 6.5 (Lower bound on the Foldy Hamiltonian). The Foldy Hamiltonian in (21) satisfies HFoldy ≥ −I n5/4 (−3/4 − 4π n5/4 (−3/4 (n()−1/4 .

(29)

There is constant C > 0 such that if ρ 1/4 R > C, ρ 1/4 (t 3 > C, and t < C −1 then the Foldy Hamiltonian satisfies the bound HFoldy ≥

1 4

p

|p|2 ap∗ ap − Cn5/4 (−3/4 .

(30)

Proof. Lemma 6.4 holds for all ρ hence also if we had replaced ρ by n/(3 in this case we get (29). The integral I satisfies the bound I ≤ 21 (2π)−3

R3

max g(k), 21 g(k)2 (f (k) − g(k))−1 dk.

By Corollary 4.3 we may assume that n ≥ cρ(3 . Hence I is bounded by a constant as long as ρ 1/4 R and ρ 1/4 (t 3 are sufficiently large and t is sufficiently small (which also ensures that γ is close to 1). Note that we do not have to make any assumptions on r. Moreover, if this is true we also have that n( ≥ cρ(4 is large and hence (n()−1 is small. This would give the bound in the corollary except for the first positive term. The above argument, however, also holds (with if we replace the kinetic energy different constants) 2 ∗ in the Foldy Hamiltonian by 21 γ −1 − 21 p |p| ap ap (assuming that γ < 2). This proves the corollary.

152


Note that if n1/4 (−3/4 R → ∞, n1/4 (−3/4 r → 0, n1/4 (1/4 t 3 → ∞, and t → 0

(31)

it follows by dominated convergence that I converges to −3 1 2 (2π)

R3

1/2 4π |k|−2 + 21 |k|2 − (4π |k|−2 + 21 |k|2 )2 − (4π |k|−2 )2 dk

∞

= (2/π )

3/4

1+x −x 4

2

x +2 4

1/2

0

4π dx = − 3

1/4 A,

where A was given in (6). Thus if we can show that n ∼ ρ(3 we see that the term −I n5/4 (−3/4 ∼ −Iρ 1/4 n agrees with Foldy’s calculation (5) for the little box of size (. Our task is now to show that indeed n ∼ ρ(3 , i.e., that we have approximate neutrality in each little box and that the term above containing the integral I is indeed the leading term.

7. Simple Bounds on n and n+ The Lemmas 4.1, 5.2, 5.3, 5.5, and 5.6 together with Lemma 6.4 or Corollary 6.5 control all terms in the Hamiltonian H(n except the positive term 1 2

p,m,p" ,m" =0

∗ ∗ w mp" ,pm" am ap" am" ap .

If we use (30) in Corollary 6.5 together with the other bounds we obtain the following bound if ρ 1/4 R and ρ 1/4 (t 3 are sufficiently large and t is sufficiently small H(n ≥

1 4

+

p

|p|2 ap∗ ap − Cn5/4 (−3/4 − 21 nR −1 − Cnρr 2

1 00,00 2w

2 3 n0 − ρ( − n0

− 4π[ρ − n(−3 ]+ n+ R 2 − 4π n2+ (−3 R 2

− ε −1 8π(−3 R 2 n0 n+ − ε w00,00 ( n0 + 1 − ρ(3 )2 ∗ ∗ − ε n+ r −1 + ( 21 − ε) w mp" ,pm" am ap" am" ap . p,m,p" ,m" =0

The assumptions on ρ 1/4 R, ρ 1/4 (t 3 , and t are needed in order to bound the integral I above by a constant. If we choose ε = 1/4, use w 00,00 ≤ 4π R 2 (−3 and ignore the last


153

positive term in the bound above we arrive at H(n ≥

1 4

p

2 n0 − ρ(3 |p|2 ap∗ ap − Cn5/4 (−3/4 − 21 nR −1 − Cnρr 2 + 41 w 00,00

− 4π[ρ − n(−3 ]+ n+ R 2 − 4π n2+ (−3 R 2 − 32π(−3 R 2 n0 n+ − 4π R 2 (−3 n+ r −1 n0 − 21 ρ(3 + 41 − 41 2 n0 − ρ(3 ≥ 41 |p|2 ap∗ ap − Cn5/4 (−3/4 − 21 nR −1 − Cnρr 2 + 41 w 00,00 p

− 48π(−3 R 2 n n+ − 4π R 2 (−3 n+ r −1 , n0 + 41 − 41 (32) where in the last inequality we have used that ρ(3 ≤ 2n, n0 ≤ n and n+ ≤ n. Lemma 7.1 (Simple bound on n). Let ω(t) be the function described in Lemma 3.1. There is a constant C > 0 such that if (ρ 1/4 ()t 3 > C and (ρ 1/4 ()ρ −1/12 , t, and ω(t)(ρ 1/4 ()−1 are smaller than C −1 then for any state with H(n ≤ 0 we have C −1 ρ(3 ≤ n ≤ Cρ(3 . Proof. The lower bound follows from Corollary 4.3. To prove the upper bound on n we choose R = ω(t)−1 ( (the maximally allowed value) and r = bω(t)−1 (, where we shall choose b sufficiently small, in particular b < 1/2. We then have that ρ 1/4 R = ω(t)−1 ρ 1/4 ( is large. Moreover w 00,00 ≥ CR 2 (−3 = Cω(t)−2 (−1 for some constant C > 0 and we get from (32) and Lemma 5.1 that 2 H(n ≥ (−1 [−Cn5/4 (1/4 − 21 nω(t) − Cb2 ω(t)−2 n2 + Cω(t)−2 n0 − ρ(3 − 48π ω(t)−2 ρ −1/6 ((ρ 1/4 )2 n2 − 4π ω(t)−2 n + 41 − 41 nb−1 ω(t)], where we have again used that cρ(3 ≤ n, n0 ≤ n and n+ ≤ n. Note that n5/4 (1/4 ≤ Cω(t)−2 n2 (ρ 1/4 ()−2 ρ −1/4 ω(t)2 and nω(t) ≤ Cω(t)−2 n2 ρ −1 ω(t)3 . From Lemma 5.1 we know that n0 ≥ n(1 − Cρ −1/6 ((ρ 1/4 )2 ). By choosing b small enough we see immediately that n ≤ Cρ(3 . Using this result as an input in (32) we can get a better bound on n than above and a better bound on n+ than given in Lemma 5.1. In particular, the next lemma in fact implies that we have near neutrality, i.e., that n is nearly ρ(3 . Lemma 7.2 (Improved bounds on n and n+ ). There exists a constant C > 0 such that if (ρ 1/4 ()t 3 > C and (ρ 1/4 ()ρ −1/12 , t,and ω(t)(ρ 1/4 ()−1 are smaller than C −1 then for any state with H(n ≤ 0 we have p |p|2 ap∗ ap ≤ Cρ 5/4 (3 (ρ 1/4 () and n+ ≤ Cnρ

−1/4

(ρ

1/4

()

3

and

n − ρ(3 ρ(3

2

≤ Cρ −1/4 (ρ 1/4 ()3 .

n " For any other state with H(,r n+ " if r " ≤ " ,R " ≤ 0 we have the same bound on ρ −3/8 (ρ 1/4 ()1/2 and R " ≥ a(ρ 1/4 ()−2 ( where a > 0 is an appropriate constant.

154


Proof. Inserting the bound n ≤ Cρ(3 into (32) gives H(n ≥

1 4

p

2 |p|2 ap∗ ap − Cρ 5/4 (3 − 21 ρ(3 R −1 − Cρ 2 (3 r 2 + 41 w 00,00 n0 − ρ(3

− CR 2 ρ n+ − CR 2 ρ + 41 (−3 − 41 n+ r −1 . We now choose r = ρ −3/8 (ρ 1/4 ()1/2 and R = a(ρ 1/4 ()−2 (, where we shall choose a below, independently of ρ, ρ 1/4 (, and t. Note that since ω(t)(ρ 1/4 ()−2 is small we may assume that R ≤ ω(t)−1 ( as required and since (ρ 1/4 ()ρ −1/12 is small we may assume −1 −1/8 (ρ 1/4 ()3/2 (−2 and R 2 ρ = a 2 (ρ 1/4 ()−4 (2 ρ = that r ≤ R. Moreover r 2 =∗ ρ 2 −2 a ( . Hence, since p |p| ap ap ≥ π 2 (−2 n+ (see 15), we have H(n ≥

1 8

p

|p|2 ap∗ ap +

π2 8

− a 2 − 41 ρ −1/8 (ρ 1/4 ()3/2 (−2 n+

2 n0 − ρ(3 + 41 w 00,00 1 − ( 2a + C)ρ 5/4 (3 (ρ 1/4 () − Ca 2 ρ 5/4 (3 (ρ 1/4 ()−5 (1 + (ρ 1/4 ()−3 ρ −1/4 ). 1/4 and t) we immediately get the By choosing a appropriately (independently of ρ, ρ (,5/4 bound on p |p|2 ap∗ ap and the bound (−2 n+ ≤ Cρ (3 (ρ 1/4 (), which implies the stated bound on n . The bound on (n − ρ(3 )2 (ρ(3 )−2 follows since we also have 2+ 3 n0 − ρ( ≤ Cρ 5/4 (3 (ρ 1/4 () and w 00,00

2 2 w 00,00 n0 − ρ(3 ≥ CR 2 (−3 n0 − ρ(3 2 ≥ Ca 2 (ρ 1/4 ()−4 (2 n − ρ(3 − nCρ −1/4 ((ρ 1/4 )3 , where we have used the bound on n+ which we have just proved. n " The case when H(,r " ,R " ≤ 0 follows in the same way because we may everywhere n replace H(n by H(,r " ,R " and use Lemma 4.2 instead of Lemma 4.1. Note that in this case we already know the bound on n since we still assume the existence of the state such that H(n ≤ 0. 8. Localization of n+ Note that Lemma 7.2 may be interpreted as saying that we have neutrality and condensation, in the sense that n+ is a small fraction of n, in each little box. Although this bound on n+ is sufficient for our purposes we still need to know that n2+ ∼ n+ 2 . We shall however not prove this for a general state with negative energy. Instead we shall show that we may change the ground state, without changing its energy expectation significantly, in such a way that the possible n+ values are bounded by Cnρ −1/4 (ρ 1/4 ()3 . To do this we shall use the method of localizing large matrices in Lemma A.1 of Appendix A. n We begin with any normalized n-particle wavefunction n ; of the operator H( . Since ; is an n-particle wave function we may write ; = m=0 cm ;m , where for all m = 1, 2, . . . , n, ;m , is a normalized eigenfunctions of n+ with eigenvalue m. We may now


155

consider the (n + 1) × (n + 1) Hermitean matrix A with matrix elements Amm" = n " . ;m , H(,r,R ψm

We shall use Lemma A.1 for this matrix and the vector ψ = (c0 , . . . , cn ). We shall choose M in Lemma A.1 to be of the order of the upper bound on n+ derived in Lemma 7.2, e.g., M is the integer part of nρ −1/4 (ρ 1/4 ()3 . Recall that with the assumption in Lemma 7.2 we have M 1. With the notation in LemmaA.1 we have λ = (ψ, Aψ) = n n (;, H(,r,R ;). Note also that because of the structure of H(,r,R we have, again with the notation in Lemma A.1, that dk = 0 if k > 3. We conclude from Lemma A.1 that there with the property that the corresponding exists a normalized wavefunction ; n+ values belong to an interval of length M and such that n n − CM −2 (|d1 | + |d2 |). , H(,r,R ; ; ≥ ; ;, H(,r,R

We shall discuss d1 , d2 , which depend on ;, in detail below, but first we give the result on the localization of n+ that we shall use. Lemma 8.1 (Localization of n+ ). There is a constant C > 0 with the following property. If (ρ 1/4 ()t 3 > C and (ρ 1/4 ()ρ −1/12 , t, and ω(t)(ρ 1/4 ()−1 are less than C −1 and r ≤ ρ 3/8 (ρ 1/4 ()1/2 , R ≥ C(ρ 1/4 ()−2 ( , and ; is a normalized wavefunction such that

n ;, H(,r,R ; ≤ 0 and

n ;, H(,r,R ; ≤ −C(nρ −1/4 (ρ 1/4 ()3 )−2 (|d1 | + |d2 |) (33)

, which is a linear combination of eigenthen there exists a normalized wave function ; functions of n+ with eigenvalues less than Cnρ −1/4 (ρ 1/4 ()3 only, such that

n n − C(nρ −1/4 (ρ 1/4 ()3 )−2 (|d1 | + |d2 |). , H(,r,R ;, H(,r,R ; ; ≥ ;

(34)

Here d1 and d2 , depending on ;, are given as explained in Lemma A.1. Proof. As explained above we choose M to be of order nρ −1/4 (ρ 1/4 ()3 . We then choose as explained above. Then (34) holds. We also know that the possible ; n+ values of ; range in an interval of length M. We do not know however, where this interval is located. The assumption (33) will allow us to say more about the location of the interval. n In fact, it follows from (33), (34) that ; , H(,r,R ; ≤ 0. It is then a consequence , of Lemma 7.2 that ; n+ ; ≤ Cnρ −1/4 (ρ 1/4 ()3 . This of course establishes that the allowed n+ values are less than C " nρ −1/4 (ρ 1/4 ()3 for some constant C " > 0. n Our final task in this section is to bound d1 and d2 . We have that d1 = (;, H(,r,R (1)ψ), n n where H(,r,R (1) is the part of the Hamiltonian H(,r,R containing all the terms with the coefficents w pq,µν for which precisely one or three indices are 0. These are the terms bounded in Lemmas 5.5 and 5.6. These lemmas are stated as one-sided bounds. It is clear from the proof that they could have been stated as two sided bounds. Alternatively n n we may observe that H(,r,R (1) is unitarily equivalent to −H(,r,R (1). This follows by ∗ applying the unitary transform which maps all operators ap and ap with p = 0 to −ap∗ and −ap . From Lemmas 5.5 and 5.6 we therefore immediately get the following bound on d1 .

156


Lemma 8.2 (Control of d1 ). With the notation above we have for all ε > 0 n0 n+ ;) + ε ;, n+ r −1 + w 00,00 ( n0 + 1 − ρ(3 )2 ; |d1 | ≤ ε −1 8π(−3 R 2 (;, ∗ ∗ + ε ;, w mp" ,pm" am ap" am" ap ; . p,m,p" ,m" =0

n n Likewise, we have that d2 = (;, H(,r,R (2)ψ), where H(,r,R (2) is the part of the Hamiln tonian H(,r,R containing all the terms with precisely two a0 or two a0∗ . i.e., these are the terms in the Foldy Hamiltonian, which do not commute with n+ .

Lemma 8.3 (Control of d2 ). There exists a constant C > 0 such that if (ρ 1/4 ()t 3 > C and (ρ 1/4 ()ρ −1/12 , t, and ω(t)(ρ 1/4 ()−1 are less than C −1 and ; is a wave function with (;, H(n ;) ≤ 0 then with the notation above we have n+ n0 ;) . |d2 | ≤ Cρ 5/4 (3 (ρ 1/4 () + 4π (−3 R 2 (;, Proof. If we replace all the operators ap∗ and ap with p = 0 in the Foldy Hamiltonian by −iap∗ and iap we get a unitarily equivalent operator. This operator however differs from n the Hamiltonian HFoldy only by a change of sign on the part that we denoted H(,r,R (2). Since both operators satisfy the bound in Corollary 6.5 we conclude that 2 ∗ ∗ ∗ ∗ ∗ 1 −1 1 |p| ap ap + 2 w pq,00 ap a0 a0 aq + a0 ap aq a0 ; |d2 | ≤ ;, 2 γ p

5/4 −3/4

+ Cn

(

pq=0

.

Note that both sums above define positive operators. This is trivial for the first sum. For the second it follows from (18) in Lemma 5.4 since a0∗ a0 commutes with all ap∗ and ap with p = 0. The lemma now follows from (18) and from Lemma 7.2. 9. Proof of Foldy’s Law We first prove Foldy’s law in a small cube. Let ; be a normalized n-particle wave function. We shall prove that with an appropriate choice of ( 1/3 ;, H(n ; ≥ 4π Aρ(3 ρ 1/4 + o ρ 1/4 , (35) 3 where A is given in (6). Note that A < 0. It then follows from Lemma 3.3 that 1/3 ω(t)N . Aρ(3 ρ 1/4 + o ρ 1/4 − C(L/()2 ρ 2 (5 − E0 ≥ (1 + L/()3 γ 4π 3 2( Thus, since N = ρL3 we have −1 1/3 1/4 E0 1/4 1/4 1/4 − Cρ ≥ γ 4π A ρ + o ρ ω(t) ρ ( . 3 L→∞ N lim

Foldy’s law (5) follows since we shall choose (see below) t and ( in such a way that as ρ → ∞ we have t → 0 and hence γ → 1 and ω(t)(ρ 1/4 ()−1 → 0 (see condition (41) below).


157

It remains to prove (35). First we fix the long and short distance potential cutoffs (36) R = ω(t)−1 (, and r = ρ −3/8 (ρ 1/4 ()−1/2 . We may of course assume that ;, H(n ; ≤ 0. Thus n satisfies the bound in Lemma 7.2. We proceed in two steps. In Lemma 9.1 Foldy’s law in the small boxes is proved under the restrictive assumption given in (37) below. Finally, in Theorem 9.2 Foldy’s law in the small boxes is proved by considering the alternative case that (37) fails. Let us note that, logically speaking, this could have been done in the reverse order. I.e., we could, instead, have begun with the case that (37) fails. At the end of the section we combine Theorem 9.2 with Lemma 3.3 to show that Foldy’s law in the small box implies Foldy’s law Theorem 1.1. At the end of this section we show how to choose ( and t so that Theorem 9.2 implies (35) and hence Theorem 1.1, as explained above. Lemma 9.1 (Foldy’s law for H(n : restricted version). Let R and r be given by (36). There exists a constant C > 0 such that if (ρ 1/4 ()t 3 > C and (ρ 1/4 ()ρ −1/12 , t, and ω(t)(ρ 1/4 ()−1 are less than C −1 then, whenever n+ ;) n(−3 R 2 (;, 00,00 ( n0 − ρ(3 )2 + ≤ C −1 ;, w

p,m,p" ,m" =0

∗ ∗ w mp" ,pm" am ap" am" ap ; ,

(37)

we have that ;, H(n ; ≥ −I n5/4 (−3/4 −Cρ 5/4 (3 ω(t)(ρ 1/4 ()−1 + ω(t)−2 ρ −1/8 (ρ 1/4 ()13/2 + +ρ −1/8 (ρ 1/4 ()7/2 , with I as in Lemma 6.4. Proof. We assume ;, H(n ; ≤ 0. We proceed as in the beginning of Sect. 7, but we now use (29) of Corollary 6.5 instead of (30). We then get H(n ≥ − I n5/4 (−3/4 − 4π n5/4 (−3/4 (n()−1/4 − 21 nR −1 − Cnρr 2 2 3 1 n0 − ρ( + 2w 00,00 − n0 − 4π[ρ − n(−3 ]+ n+ R 2 − 4π n2+ (−3 R 2

− ε −1 8π(−3 R 2 n0 n+ − ε w00,00 ( n0 + 1 − ρ(3 )2 ∗ ∗ − ε n+ r −1 + ( 21 − ε) w mp" ,pm" am ap" am" ap . p,m,p" ,m" =0

n0 ≤ n, and w 00,00 ≤ If we now use the assumption (37) and the facts that n+ ≤ n, 4πR 2 (−3 we see with appropriate choices of ε and C that H(n ≥ − I n5/4 (−3/4 − 4πn5/4 (−3/4 (n()−1/4 − 21 nR −1 − Cnρr 2 − CR 2 (−3 (n + 1) − CR 2 (−3 |n − ρ(3 |( n+ + 1) − C n+ r −1 .

If we finally insert the choices of R and r and use Lemma 7.2 we arrive at the bound in the lemma.

158


Theorem 9.2 (Foldy’s law for H(n ). There exists a C > 0 such that if (ρ 1/4 ()t 3 > C and (ρ 1/4 ()ρ −1/12 , t, and ω(t)(ρ 1/4 ()−1 are less than C −1 then for any normalized n-particle wave function ; we have ;, H(n ; ≥ −I n5/4 (−3/4 − Cρ 5/4 (3 ω(t)(ρ 1/4 ()−1 + ω(t)−1 ρ −1/16 (ρ 1/4 ()29/4 + ρ −1/8 (ρ 1/4 ()7/2 , (38) where I is defined in Lemma 6.4 with r and R as in (36). Proof. According to Lemma 9.1 we may assume that n+ ;) n(−3 R 2 (;, 00,00 ( n0 − ρ(3 )2 + ≥ C −1 ;, w

p,m,p" ,m" =0

∗ ∗ w mp" ,pm" am ap" am" ap ; ,

(39)

where C is at least as big as the constant in Lemma 9.1. We still assume that ;, H(n ; ≤ 0. We begin by bounding d1 and d2 using Lemmas 8.2 and 8.3. We have from Lemmas 7.2 and 8.3 that |d2 | ≤ Cρ 5/4 (3 (ρ 1/4 () + C(−1 ω(t)−2 n2 ρ −1/4 (ρ 1/4 ()3 ≤ C[nρ −1/4 (ρ 1/4 ()3 ]2 ρ 5/4 (3 (ρ 1/4 ()−11 + ω(t)−2 (ρ 1/4 ()−7 ≤ C[nρ −1/4 (ρ 1/4 ()3 ]2 ρ 5/4 (3 ω(t)−2 (ρ 1/4 ()−7 . In order to bound d1 we shall use (39). Together with Lemma 8.2 this gives (choosing ε = 1/2 say) n+ ;) + 21 ;, 00,00 (n − ρ(3 + 1) ; . n+ r −1 + w |d1 | ≤ C(−3 R 2 n (;, Inserting the choices for r and R and using Lemma 7.2 gives |d1 | ≤ C[nρ −1/4 (ρ 1/4 ()3 ]2 ρ 5/4 (3 ω(t)−2 (ρ 1/4 ()−7 + ρ −1/8 (ρ 1/4 ()−17/2 , where we have also used that we may assume that ρ −1/8 (ρ 1/4 ()−9/2 is small. The assumption (33) now reads n ; ≤ −Cρ 5/4 (3 ω(t)−2 (ρ 1/4 ()−7 + ρ −1/8 (ρ 1/4 ()−17/2 . ;, H(,r,R If this is not satisfied we see immediately that the bound (38) holds. Thus from Lemma 8.1 it follows that we can find a normalized n-particle wavefunction with ; ≤ Cnρ −1/4 (ρ 1/4 ()3 and ≤ Cn2 ρ −1/2 (ρ 1/4 ()6 , , (40) ; n2+ ; ; n+ ; such that n n − Cρ 5/4 (3 ω(t)−2 (ρ 1/4 ()−7 + ρ −1/8 (ρ 1/4 ()−17/2 . , H(,r,R ; ; ≥ ; ;, H(,r,R


159

we proceed as in the beginning of Sect. 7. This , H n ; In order to analyze ; (,r,R time we use Lemmas 4.1, 5.2, 5.3, 5.5, and 5.6 together with Lemma 6.4 instead of Corollary 6.5. We obtain n H(,r,R

≥

1 00,00 2w

n − ρ(

3

2

+ ( n+ ) − 2 n − ρ( 2

3

n0 n+ −

− 4π[ρ − n(−3 ]+ n+ R 2 − 4π n2+ (−3 R 2 − ε n+ r −1 − ε −1 8π (−3 R 2 n0 n+ 2 ∗ ∗ 1 − ε w00,00 ( n+ − 1) + ( 2 − ε) w mp" ,pm" am ap" am" ap −

1 2

n − ρ(3

2

p,m,p" ,m" =0

w 00,00 − 4π n5/4 (−3/4 (n()−1/4 − I n5/4 (−3/4 .

This time we shall however not but rather big. Note that since choose ε small, ∗ a ∗ a a ≤ r −1 wr,R (x, y) ≤ r −1 we have w mp" ,pm" am n+ ( n+ − 1), which " p m" p p,m,p" ,m" =0

follows immediately from p,m,p" ,m" =0

∗ ∗ w mp" ,pm" am ap" am" ap

=

wr,R (x, y)

p,m=0

um (x)up (y)am ap

∗ p,m=0

um (x)up (y)am ap dx dy.

We therefore have n ≥ − I n5/4 (−3/4 − 4π n5/4 (−3/4 (n()−1/4 − CR 2 (−3 n0 H(,r,R

− C(−3 R 2 |ρ(3 − n| n+ − 4π n2+ (−3 R 2 − ε n+ r −1 − ε −1 8π (−3 R 2 n0 n+ − εCR 2 (−3 n2+ − ε n2+ r −1 .

, If we now insert the choices of r and R, take the expectation in the state given by ; and use (40) and the bound on n from Lemma 7.2 we arrive at & n ≥ − I n5/4 (−3/4 − Cρ 5/4 (3 (ρ 1/4 ()−1 + ω(t)−2 (ρ 1/4 ()−1 , H(,r,R ; ;

+ ω(t)−2 ρ −1/8 (ρ 1/4 ()11/2 + ω(t)−2 ρ −1/4 (ρ 1/4 ()8 + ερ −1/8 (ρ 1/4 ()7/2 ' + ε −1 ω(t)−2 (ρ 1/4 ()5 + εω(t)−2 ρ −1/4 (ρ 1/4 ()8 + ερ −1/8 (ρ 1/4 ()19/2 . If we now choose ε = ω(t)−1 ρ 1/16 (ρ 1/4 ()−9/4 we arrive at (38).

Completion of the proof of Foldy’s law, Theorem 1.1. We have accumulated various errors and we want to show that they can all be made small. There are basically two parameters that can be adjusted, ( and t. Instead of ( it is convenient to use X = ρ 1/4 (. We shall choose X as a function of ρ such that X → ∞ as ρ → ∞. From Lemma 7.1 we know that for some fixed C > 0 C −1 ρ(3 ≤ n ≤ Cρ(3 . Hence according to (31) with r and R

160


given in (36) we have that I → −

4π 1/3 3

X → ∞,

(41)

1/4

X → ∞,

(42)

t X → ∞, t → 0.

(43) (44)

ω(t) ρ

A as ρ → ∞ if

−1

3

The hypotheses of Theorem 9.2 are valid if (41), (43), (44), and ρ −1/12 X → 0

(45)

hold. From Lemma 7.2, for which the hypotheses are now automatically satisfied, we have that n = ρ(3 (1 + O(ρ −1/8 X 3/2 ) and from (45) we see that n is ρ(3 to leading order. With these conditions we find that the first term on the right side of (38) is, in the limit ρ → ∞, exactly Foldy’s law. The conditions that the other terms in (38) are of lower order are (X/ω(t))4/25 ρ −1/100 X → 0, ρ −1/28 X → 0

(46) (47)

together with (41). It remains to show that we can satisfy the conditions (41–47). Condition (42) is trivially satisfied since both ρ and X tend to infinity. Since ω(t) ∼ t −4 for small t we see that (43) is implied by (41). Condition (45) is implied by (47), which is in turn implied by (41) and (46). The remaining two conditions (41) and (46) are easily satisfied by an approriate choice of X and t as functions for ρ with X → ∞ and t → 0 as ρ → ∞. In fact, we simply need ρ 1/116 t −16/29 X t −4 . The bound (35) has now been established. Hence Foldy’s law Theorem 1.1 follows as discussed in the beginning of the section.

Appendix A. Localization of Large Matrices The following theorem allows us to reduce a big Hermitean matrix, A, to a smaller principal submatrix without changing the lowest eigenvalue very much. ( The k th supra(resp. infra-) diagonal of a matrix A is the submatrix consisting of all elements ai,i+k (resp. ai+k,i ). ) Theorem A.1 (Localization of large matrices). Suppose that A is an N ×N Hermitean matrix and let Ak , with k = 0, 1, ..., N −1, denote the matrix consisting of the k th supraand infra-diagonal of A. Let ψ ∈ CN be a normalized vector and set dk = (ψ, Ak ψ) and λ = (ψ, Aψ) = N−1 k=0 dk . (ψ need not be an eigenvector of A.) Choose some positive integer M ≤ N . Then, with M fixed, there is some n ∈ [0, N − M] and some normalized vector φ ∈ CN with the property that φj = 0 unless n + 1 ≤ j ≤ n + M (i.e., φ has length M) and such that (φ, Aφ) ≤ λ +

N−1 M−1 C 2 k |d | + C |dk |, k 2 M k=1

k=M

where C > 0 is a universal constant. (Note that the first sum starts with k = 1.)

(48)


161

Proof. It is convenient to extend the matrix Ai,j to all −∞ < i, j < +∞ by defining Ai,j = 0 unless 1 ≤ i, j ≤ N . Similarly, we extend the vector ψ and we define the numbers dk and the matrix Ak to be zero when k ∈ [0, N − 1]. We shall give the construction for M odd, the M even case being similar. For s ∈ Z set f (s) = AM [M + 1 − 2|s|] if 2|s| < M and f (s) = 0 otherwise. Thus, = 0 for precisely M values of s. Also, f (s) = f (−s). AM is chosen so that f (s) 2 s f (s) = 1. (m) For each m ∈ Z define the vector φ (m) by φj = f (j − m)ψj . We then define K (m) = (φ (m) , Aφ (m) ) − (λ + σ )(φ (m) , φ (m) ). (The number σ will be chosen later.) After this, we define K = m K (m) . Using the fact that s f (s)2 = 1, we have that (φ (m) , Aφ (m) ) = (φ (m) , Ak φ (m) ) = f (s)f (k + s)(ψ, Ak ψ) m

m k=0

=

s

and λ=λ

s

k

f (s)f (k + s)dk

k=0

(φ (m) , φ (m) ) = f (s)2 (ψ, Ak ψ) = f (s)2 dk m

s

s

k=0

(49)

k

Hence K=

m

K (m) = −σ −

N−1

d k γk

(50)

k=1

with

1 (51) [f (s) − f (s + k)]2 . 2 s (m) = 0. Recalling that not all of the Let us choose σ = − N−1 mK k=1 dk γk . Then, φ (m) equal zero, we conclude that there is at least one value of m such that (i) φ (m) = 0 and (ii) (φ (m) , Aφ (m) ) ≤ (λ + σ )(φ (m) , φ (m) ). k2 This concludes the proof of (48) except for showing that γk ≤ C k 2 +M 2 for all M and k. This is evident from the easily computable large M asymptotics in (51). γk =

B. A Double Commutator Bound Lemma B.1. Let −N be the Neumann Laplacian of some bounded open set O. Given θ ∈ C ∞ (O) with supp |∇θ | ⊂ O satisfying (∂i θ( ≤ Ct −1 , (∂i ∂j θ ( ≤ Ct −2 , (∂i ∂j ∂k θ( ≤ Ct −3 , for some 0 < t and all i, j, k = 1, 2, 3. Then for all s > 0 we have the operator inequality −N (−N )2 , θ , θ ≥ −Ct −2 − Cs 2 t −4 . (52) −2 −N + s −N + s −2 We also have the norm bound + + + + −N 2 −2 4 −4 + , θ ,θ + + − + s −2 + ≤ C(s t + s t ). N

(53)

162


Proof. We calculate the commutator 1 1 (−N )2 , θ = s −2 [−N , θ ] (−N ) −2 −2 −N + s −N + s −N + s −2 +

−N [−N , θ ] . −N + s −2

Likewise we calculate the double commutator −N −N (−N )2 ,θ ,θ = − [[−N , θ ] θ] −N + s −2 −N + s −2 −N + s −2 −N −N + [[−N , θ ] θ] −2 −N + s −N + s −2 1 1 1 − 2s −4 . [−N , θ ] [θ, −N ] −N + s −2 −N + s −2 −N + s −2

(54)

+ [[−N , θ ] θ]

Note that [[−N , θ ] θ] = −2 (∇θ )2 and thus the first term above is positive. We claim that [−N , θ ] [θ, −N ] ≤ −Ct −2 N + Ct −4 .

(55)

To see this we simply calculate [−N , θ ] [θ, −N ] = −

3

4∂i (∂i θ )(∂j θ)∂j + (∂i2 θ )(∂j2 θ) + 2(∂i θ )(∂i ∂j2 θ)

i,j

The last two terms are bounded by Ct −4 . For the first term we have by the CauchySchwarz inequality for operators, BA∗ + AB ∗ ≤ ε −1 AA∗ + εBB ∗ , for all ε > 0, that −

3 i,j

∂i (∂i θ )(∂j θ )∂j =

3

3 ∗ ∂i (∂i θ )(∂i θ)∂i (∂i (∂i θ)) ∂j (∂j θ) ≤ −3

i,j

i

and this is bounded above by −3t −2 N and we get (55). Inserting (55) into (54), recalling that the first term is positive, we obtain −N −N (−N )2 , θ , θ ≥ − 2(∇θ)2 −2 (∇θ)2 −N + s −2 −N + s −2 −N + s −2 − Ct −2

−N − Cs 2 t −4 . −N + s −2

Again using the Cauchy–Schwarz inequality, we have −N −N +2 (∇θ)2 −2 −N + s −N + s −2 1/2 1/2 −N −N −N −2 4 −2 ≤ 2t + 2t (∇θ ) −N + s −2 −N + s −2 −N + s −2 −N ≤ Ct −2 , −N + s −2

2(∇θ )2


163

and (52) follows. The bound (53) is proved in the same way. Indeed,

1 1 −N , θ , θ = −s −2 [[−N , θ ], θ ] −N + s −2 −N + s −2 −N + s −2 1 1 1 + 2s −2 , [−N , θ ] [θ − N ] −2 −2 −N + s −N + s −N + s −2

and (53) follows from [[−N , θ ] θ] = −2 (∇θ )2 and (55).

References [B]

Bogolubov, N.N.: J. Phys. (U.S.S.R.) 11, 23 (1947); Bogolubov, N.N. and Zubarev, D.N.: Sov. Phys. JETP 1, 83 (1955) [CLY] Conlon, J.G.. Lieb, E.H. and Yau, H-T.: The N 7/5 law for charged bosons. Commun. Math. Phys. 116, 417–448 (1988) [D] Dyson, F.J.: Ground-state energy of a finite system of charged particles. J. Math. Phys. 8, 1538–1545 (1967) [F] Foldy, L.L.: Charged boson gas. Phys. Rev. 124, 649–651 (1961); Errata. ibid 125, 2208 (1962) [GM] Girardeau, M.: Ground state of the charged Bose gas. Phys. Rev. 127, 1809–1818 (1962) [GA] Girardeau, M. and Arnowitt, R.: Theory of many-boson systems: Pair theory. Phys.Rev. 113, 755–761 (1959) [GG] Graf, G.M.: Stability of matter through an electrostatic inequality. Helv. Phys. Acta 70, 72–79 (1997) [KLS] Kennedy, T., Lieb, E.H. and Shastry, S.: The XY model has long-range order for all spins and all dimensions greater than one. Phys. Rev. Lett. 61, 2582–2585 (1988) [L] Lieb, E.H.: The Bose fluid. In: Lecture Notes in Theoretical Physics VIIC, edited by W.E. Brittin. Univ. of Colorado Press, 1964, pp. 175–224 [LS] Lieb, E.H. and Sakakura, A.Y.: Simplified approach to the ground state energy of an imperfect Bose gas II. Charged Bose gas at high density. Phys. Rev. A 133, 899–906 (1964) [LLe] Lieb, E.H. and Lebowitz, J.L.: The constitution of matter: Existence of thermodynamics for systems composed of electrons and nuclei. Adv. in Math. 9, 316–398 (1972) [LLi] Lieb, E.H. and Liniger, W.: Exact analysis of an interacting Bose gas I. The general solution and the ground state. Phys. Rev. 130, 1605–1616 (1963). See Fig. 3 [LN] Lieb, E.H. and Narnhofer, H.: The thermodynamic limit for jellium. J. Stat. Phys. 12, 295–310 (1975); Errata. 14, 465 (1976) [LY] Lieb, E.H. and Yngvason, J.: Ground state energy of the low density Bose gas. Phys. Rev. Lett. 80, 2504–2507 (1998) Communicated by M. Aizenman

Commun. Math. Phys. 217, 165 – 180 (2001)

Communications in



Self-Similarity of Volume Measures for Laplacians on P. C. F. Self-Similar Fractals Jun Kigami1 , Michel L. Lapidus2, 1 Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan.


2 Department of Mathematics, University of California, Riverside, CA 92521-0135, USA.

E-mail: [email protected] Received: 6 September 1999 / Accepted: 7 October 2000

Abstract: Our main goal in this paper is to obtain a precise analogue of Weyl’s asymptotic formula for the eigenvalue distribution of Laplacians on a certain class of “finitely ramified” (or p.c.f.) self-similar fractals, building, in particular, on the work of [7, 9, 22, 24]. Our main result consists in precisely identifying (for the class of “decimable fractals”) the volume measures constructed by the second author in [24] for general p.c.f. fractals and showing that they are self-similar. From a physical point of view, our results should be relevant to the study of the density of states for diffusions and wave propagation in fractal media. 1. Introduction In this paper, we will obtain a refined version of Weyl’s formula for the eigenvalue distribution of Laplacians on certain self-similar fractals. There is now a well-developed theory of Laplacians and diffusions on “finitely ramified” self-similar sets. (See, for example, Kusuoka [23], Goldstein [14], Barlow and Perkins [4].) Before discussing our results, we first recall Weyl’s classical formula for Laplacians on Riemannian manifolds. (See, for example, Hörmander [17] and in the Euclidean case, Reed and Simon [31].) Let − be the positive Laplacian (or Laplace–Beltrami operator) on a closed, compact d-dimensional smooth (connected) Riemannian manifold M. Then it is well-known that − has a discrete spectrum {λj }∞ j =1 which can be written in non-decreasing order according to multiplicity as follows: 0 ≤ λ1 ≤ λ2 ≤ · · · ≤ λj ≤ · · · → ∞. For x > 0, let ρ(x) = #{j ≥ 1 : λj ≤ x} denote the eigenvalue counting function of −. Then Weyl’s asymptotic formula in this context states that ρ(x) = cd Vol(M)x d/2 (1 + o(1))

(1.1)

Work supported by the US National Science Foundation under Grants DMS-9623002 and DMS-0070497.

166

J. Kigami, M. L. Lapidus

as x → ∞, where cd is a positive constant depending only on d and where Vol(M) denotes the Riemannian volume of M. Henceforth, o(1) stands for a function that tends to zero as x → ∞. We note that if M is a compact manifold with smooth boundary, an entirely analogous formula holds for the Dirichlet Laplacian on M. (See also, for example, [2 and 31] for various physical applications of Weyl’s formula in the case where M is a bounded smooth domain in Euclidean space.) If, in addition, M is a (closed) spin manifold, Connes ([7, 8, §VI.1]) has used the notion of Dixmier trace (a suitable scale-invariant trace which is well-suited for dealing with logarithmic divergences) to reconstruct the Riemannian volume measure of M and hence to reinterpret Weyl’s formula within the framework of noncommutative geometry. In the case of a “finitely ramified” (that is, p. c. f. ) self-similar fractal K instead of a smooth manifold, Kigami and Lapidus [22] have obtained a partial analogue of Weyl’s formula for the Dirichlet Laplacian on K. When K is in “general position” (the “non-lattice case”), the counterpart of (1.1) is then given by ρ(x) = Cx dS /2 (1 + o(1))

(1.2)

as x → ∞, where C is a positive constant depending on K and dS > 0 is a suitable “spectral exponent” defined in Theorem 3.2 below. On the other hand, in the “lattice case” (also called the “arithmetic case” in probability theory), the analogue of (1.1) is given by ρ(x) = (G(log x/2) + o(1))x dS /2

(1.3)

as x → ∞, where G is a positive periodic function that is bounded away from zero and infinity; so that ρ(x) x dS /2 . (See Theorem 3.2 below.) Motivated by the above mentioned work of Connes [7] and using, in particular, the results of [22], Lapidus [24] has constructed a “volume measure” ν on the p. c. f. selfsimilar set K, associated with the Dirichlet Laplacian on K. (See Theorem 4.1 below.) Moreover, he has shown that the total mass of ν, namely, ν(K), is given by the constant C appearing in (1.2) in the non-lattice case, and by the mean-value of the periodic function G occurring in (1.3) in the lattice case. In part by analogy with the work of Connes and Sullivan on the “quantized calculus” on limit sets of quasi-Fuchsian groups ([9, 8, §IV.3]), such as certain hyperbolic Julia sets, the second author has also conjectured that this volume measure (or rather, the associated probability measure ν/ν(K)) is “approximately self-similar”. (See [24, §5.1 and 25, §6.) In the present paper, under a certain hypothesis, we will identify the volume measure ν constructed in [24] and show that it is equal to a constant multiple of a self-similar measure on K. (See Theorem 4.7 in conjunction with Hypothesis 4.6.) Moreover, we will verify that this hypothesis holds for the class of p. c. f. self-similar sets satisfying the eigenvalue decimation property, which was first introduced by the physicists Rammal and Toulouse [30] and Rammal [29] for the case of the Sierpinski gasket. Several examples of such “decimable fractals” are provided in Sect. 5 below. A sample of physics papers studying finitely ramified fractals includes Dhar [11], Alexander and Orbach [1], Berry [5, 6], Hattori et al. [15], along with the survey articles by Liu [26], Havlin and Bunde [16] and by Nakayama et al. [28]. We note that in the mathematics literature, the eigenvalue decimation method – which provides an explicit algorithm to compute the eigenvalues and the eigenfunctions of the Laplacian – has been justified rigorously by Fukushima and Shima 13] for the Sierpinski gasket and

Laplacians on P. C. F. Self-Similar Fractals

167

later on, by Shima [33] for the more general class of p.c.f. self-similar sets considered here. (See also the recent work by Teplyaev [34].) We believe that Hypothesis 4.6 under which our main result is established should hold more generally than for “decimable fractals”, but unfortunately, we cannot prove it at this point. We also remark that under our hypothesis, the normalized volume measure ν/ν(K) coincides with the original selfsimilar measure defining the mass distribution of K if (and only if) dS coincides with the spectral dimension of K, as defined in [22]. In that case, ν was proposed in [24, 25] to be thought of as an analogue of Riemannian volume on K. As an immediate consequence of our results (combined with the earlier works in [22] and [24]), one obtains a more precise version of Weyl’s classical formula in the present context of Laplacians on (certain) self-similar fractals. Part of our present joint results was announced in Sect. 6 of [25]. The interested reader can find in [24, 25] further discussion of the possible connections between aspects of noncommutative geometry [8] and of spectral and fractal geometry. The rest of this paper is organized as follows. In Sect. 2, we briefly review the analytic definition of Laplacians on p. c. f. self-similar fractals. In Sect. 3, we recall the main result of [22] concerning the eigenvalue distribution of Laplacians on p.c.f. fractals and provide some preparatory lemmas and definitions. In Sect. 4, we recall the main result of [24] concerning the construction of volume measures on fractals. We also briefly discuss the notion of Dixmier trace and introduce Hypothesis 4.6 as well as derive its main consequence, Theorem 4.7, which proves the self-similarity of ν/ν(K). Finally, in Sect. 5, we establish a sufficient condition for the self-similarity of volume measures (that is, for Hypothesis 4.6 to be satisfied); see Theorem 5.2. We also provide several examples illustrating our results.

2. Laplacians on P. C. F. Self-Similar Sets In this section, we will define post critically finite self-similar sets and construct Laplacians on them. See [18, 19] for details. Definition 2.1. Let K be a compact metrizable topological space and let S be a finite set. In this paper, S = {1, 2, · · · , N }. Also, let Fi , for i ∈ S, be a continuous injection from K to itself. Then, (K, S, {Fi }i∈S ) is called a self-similar structure if there exists a continuous surjection π : → K such that Fi ◦ π = π ◦ i for every i ∈ S, where = S N is the one-sided shift space and i : → is defined by i(w1 w2 w3 · · · ) = iw1 w2 w3 · · · for each w1 w2 w3 · · · ∈ . Note that if (K, S, {Fi }i∈S ) is a self-similar structure, then K is self-similar in the following sense: K=

Fi (K).

(2.1)

i∈S

Notation. Wm = S m is the collection of words with length m. For w = w1 · · · wm ∈ Wm , we define Fw : K → K by Fw = Fw1 ◦ · · · ◦ Fwm and Kw = Fw (K). In particular, W0 = {∅} and F∅ is the identity map. Also we define W∗ = ∪m≥0 Wm .

168


Definition 2.2. Let (K, S, {Fi }i∈S ) be a self-similar structure. We define the critical set C ⊂ and the post critical set P ⊂ by C = π −1 ( (Ki ∩ Kj )) and P = σ n (C), n≥1

i=j

where σ is the shift map from to itself defined by σ (ω1 ω2 · · · ) = ω2 ω3 · · · . A selfsimilar structure is called post critically finite (p. c. f. for short) if and only if #(P) is finite. Now, we fix a p. c. f. self-similar structure (K, S, {Fi }i∈S ). Definition 2.3. Let V0 = π(P). For m ≥ 1. Also set Vm = Fw (π(P)) and V∗ = Vm . m≥0

w∈Wm

It is easy to see that Vm ⊂ Vm+1 and that K is the closure of V∗ . In particular, V0 is thought of as the “boundary” of K. Next we explain how to construct Laplacians on a p. c. f. self-similar set. First we define a Laplacian on a finite set. Definition 2.4. Let V be a finite set. We denote the collection of real-valued functions on V by "(V ). The space "(V ) is equipped with the standard inner product (u, v) = p∈V u(p)v(p) for u, v ∈ "(V ). A symmetric linear operator H : "(V ) → "(V ) is called a Laplacian on V if it satisfies (L1) H is non-positive definite, (L2) H u = 0 if and only if u is a constant on V , and (L3) Hpq ≥ 0 for all p = q ∈ V . We use L(V ) to denote the collection of Laplacians on V . For H ∈ L(V ), EH (·, ·) is a non-negative symmetric bilinear form defined by EH (u, v) = −(H u, v) for u, v ∈ "(V ). Proposition 2.5. Let D ∈ L(V0 ) and let r = (r1 , · · · , rN ), where ri > 0 for i ∈ S. Define a symmetric bilinear form E (m) on "(Vm ) by E (m) (u, v) = w∈Wm rw −1 ED (u ◦ Fw , v ◦ Fw ), where rw = rw1 · · · rwm for w = w1 · · · wm ∈ Wm . Then there exists Hm ∈ L(Vm ) that satisfies E (m) = EHm . Definition 2.6. (D, r) is said to be a harmonic structure if and only if E (m) (u, u) = min{E (m+1) (v, v) : v ∈ "(Vm+1 ), v|Vm = u}

(2.2)

for all m ≥ 0 and for any u ∈ "(Vm ). It is known that (2.2) holds for all m ≥ 0 if and only if it holds for m = 0. Definition 2.7. If (D, r) is a harmonic structure, then we define F = {u : u ∈ "(V∗ ), lim E (m) (u|Vm , u|Vm ) < ∞} m→∞

and E(u, v) = limm→∞ E (m) (u|Vm , v|Vm ) for u, v ∈ F. Also F0 = {u ∈ F : u|V0 = 0}. Since E (m) is defined in a self-similar fashion, E naturally satisfies the following selfsimilarity property.


169

Proposition 2.8. u ∈ F if and only if u ◦ Fi ∈ F for all i ∈ S. Also E(u, v) = ri −1 E(u ◦ Fi , v ◦ Fi ) i∈S

for any u, v ∈ F.

Proposition 2.9 (Self-similar measure). If µi > 0 for each i ∈ S and i∈S µi = 1, then there exists a unique Borel regular probability measure µ on K such that f dµ = µi f ◦ Fi dµ K

i∈S

K

for any continuous function on K. µ is called a self-similar measure on K with weight (µ1 , · · · , µN ). If µ is a self-similar measure, then µ(Kw ) = µw , where µw = µw1 · · · µwm for w = w1 · · · wm ∈ Wm . Now we give a direct definition of the Laplacian associated with (E, F) and a measure µ. Let C(K) be the collection of all real-valued continuous functions on K. Definition 2.10. For p ∈ Vm , let ψpm be the unique function in F that attains the following minimum: min{E(u, u) : u ∈ F, u(p) = 1, u(q) = 0 for q ∈ Vm \{p}}. For u ∈ C(K), if there exists f ∈ C(K) such that lim

max |µ−1 m,p (Hm u)(p) − f (p)| = 0,

m→∞ p∈Vm \V0

where µm,p = K ψpm dµ, then we define the µ-Laplacian µ by µ u = f . The domain of µ is denoted by Dµ . Proposition 2.11. For u ∈ Dµ and p ∈ V0 ,

− lim (Hm u)(p) = −(Du)(p) + m→∞

K

ψp0 µ udµ.

The above limit is denoted by (du)p and is called the Neumann derivative at p. There is a natural relation between µ , (E, F) and Neumann derivatives. Proposition 2.12 (Gauss-Green’s formula). For u ∈ F and v ∈ Dµ , u(p)(dv)p − uµ vdµ. E(u, v) = p∈V0

K

Theorem 2.13. Let (D, r) be a harmonic structure on a p. c. f. self-similar structure (K, S, {Fi }i∈S ). Also let µ be a self-similar measure on K with weight (µ1 , · · · , µN ). If µi ri < 1 for all i ∈ S, then F is naturally embedded in L2 (K, µ). (E, F) and (E, F0 ) are local regular Dirichlet forms on L2 (K, µ). Moreover, let HN and HD be non-negative self-adjoint operators on L2 (K, µ) associated with (E, F) and (E, F0 ) respectively, then both HN and HD have compact resolvent.

170


The operators HN and HD are defined through the abstract theory of closed quadratic forms on a Hilbert space. See [10, 32] for the general theory. For example, let u and f be in L2 (K, µ), u ∈ Dom(HN ) and HN u = f if and only if u ∈ F and E(v, u) = (v, f )µ for all v ∈ F, where (u, v)µ is the inner product of L2 (K, µ). The operator −HN is thought to be a Laplacian on K with Neumann boundary conditions while −HD is thought to be a Laplacian on K with Dirichlet boundary conditions. In fact, if DN = {u ∈ Dµ : (du)p = 0

for any p ∈ V0 },

then the above characterization of HN along with Proposition 2.12 implies that DN ⊂ Dom(HN ) and µ = −HN on DN . Similarly, if DD = {u ∈ Dµ : u|V0 = 0}, then DD ⊂ Dom(HD ) and µ = −HD on DD . Moreover we can verify the following theorem. Theorem 2.14. The operators −HN and −HD are the Friedrichs extensions of µ |DN and µ |DD , respectively. 3. Eigenvalue Distribution of Laplacians In this section, we will discuss results concerning the eigenvalue distributions of Laplacians on p. c. f. self-similar sets. Throughout the rest of this paper, (K, S, {Fi }i∈S ) is a p. c. f. self-similar structure with S = {1, 2, · · · , N } and (D, r) is a harmonic structure, where r = (r1 , · · · , rN ). Further, µ is a self-similar measure on K with weight (µ1 , · · · , µN ) that satisfies ri µi < 1 for all i ∈ S. In the following, the symbol ∗ always represents D or N. Definition 3.1 (Eigenvalues and Eigenfunctions). For k ∈ R, we define E∗ (k) = {u : u ∈ Dom(H∗ ), H∗ u = ku}. If dim E∗ (k) ≥ 1, then k is called a ∗-eigenvalue and u ∈ E∗ (k) is said to be a ∗-eigenfunction belonging to the ∗-eigenvalue k. It is known that if u ∈ E∗ (k), then u ∈ D∗ and µ u = −ku. See [22, 28]. Since H∗ has compact resolvent, the ∗-eigenvalues are non-negative, of finite multiplicity and the only accumulation point is ∞. Precisely, there exist a complete orthonormal system of L2 (K, µ), {ϕj∗ }j ≥1 ⊂ D∗ and {kj∗ }j ≥1 such that H∗ ϕj∗ = kj∗ ϕj∗ and kj∗ ≤ kj∗+1 for all j ≥ 1. Hence if we let ρ∗ (x, µ) = dim E∗ (k) = #{j : kj∗ ≤ x}, k≤x

ρ∗ (x, µ) is well-defined and ρ∗ (x, µ) → ∞ as x → ∞. We call ρ∗ (x, µ) the eigenvalue counting function. The following theorem gives an analogue of Weyl’s asymptotic formula for the eigenvalue counting functions.


171

Theorem 3.2 ([22]). Let dS be the unique positive number d that satisfies γid = 1, √ where γi = ri µi for i ∈ S. Then

i∈S

0 < lim inf ρ∗ (x, µ)/x dS /2 ≤ lim sup ρ∗ (x, µ)/x dS /2 < ∞ x→∞

x→∞

for ∗ = D, N . The positive number dS is called the spectral exponent of (E, F, µ). Moreover, we have the following dichotomy: (1) Non-lattice case : If i∈S Z log γi is a dense subgroup of R, then the limit d /2 S limx→∞ ρ∗ (x, µ)/x exists. (2) Lattice case : If i∈S Z log γi is a discrete subgroup of R, let T > 0 be its generator. Then ρ∗ (x, µ) = (G(log x/2) + o(1))x dS /2 , where G is a right-continuous T -periodic function such that 0 < inf G(x) ≤ sup G(x) < ∞ and o(1) denotes a term which vanishes as x → ∞. It is known that 0 ≤ ρN (x, µ) − ρD (x, µ) ≤ #(V0 ). See [22, 18]. Hence the limit limx→∞ ρ∗ (x, µ)/x dS /2 (or the periodic function G) is independent of the boundary conditions. In fact, if R(x) = ρD (x, µ) − i∈S ρD (γi2 x, µ), then −1 ∞ dS /2 lim ρ∗ (x, µ)/x = − νi log νi dS U (t)dt (3.1) x→∞

−∞

i∈S

in the non-lattice case and ∞ −1 νi log νi dS T U (t + j T ) G(t) = − i∈S

(3.2)

j =−∞

in the lattice case, where νi = γidS for i ∈ S and U (t) = e−dS t R(e2t ). In light of (3.2), we immediately deduce the following lemma. Lemma 3.3. In the lattice case, we have −1 ∞ 1 T G(t)dt = − νi log νi dS U (t)dt. T 0 −∞

(3.3)

i∈S

By analogy with Weyl’s classical theorem (see (1.1) or [22, Theorem 0.1] for example), the limit (3.1) may represent a kind of volume of the space in the non-lattice case. Even in the lattice case, we may use the integral average (3.3) as a substitute for the value of the limit. Definition 3.4 (Spectral Volume). The spectral volume vol(K, µ) is defined by −1 ∞ vol(K, µ) = − νi log νi dS U (t)dt. (3.4) i∈S

−∞

172


Note that 0 < vol(K, µ) < ∞ by (3.1) and (3.3). To justify this analogy, we need some kind of natural measure ν defined on K that satisfies ν(K) = vol(K, µ). Such a measure was in fact defined by Lapidus in [24]. We will introduce it in the next section. In the meantime, we derive a formula for the spectral volume. Let kj denote the j th Dirichlet eigenvalue kjD for j ≥ 1. Proposition 3.5. −1 νi log νi lim (q(x) − νi q(γi2 x)) vol(K, µ) = − x→∞

i∈S i∈S −1 = − νi log νi lim (q(t) ˜ − νi q(t/ν ˜ i )),

where q(x) =

t→0

i∈S

−dS /2 kj ≤x kj

and q(t) ˜ =

i∈S

−dS /2

t≤kj

−d /2 kj S .

Proof. We need to show that ∞ e−dS t R(e2t )dt = lim (q(x) − νi q(γi2 x)). dS x→∞

−∞

i∈S

Although R(x) = ρD (x, µ) − formula of integration by parts. Then ∞ −dS t 2t dS e R(e )dt =

2 i∈S ρD (γi x, µ)

−∞

Now ρD (e2t , µ) = Hence we have

−∞

j δtj ,

t

−∞

∞

is a step function, we can still use the e−dS t (R(e2t )) dt.

where tj = log kj /2 and δx is the Dirac point mass at x.

e−dS t (ρD (e2t , µ)) dt =

tj ≤t

−dS /2

kj

.

Therefore it follows that t e−dS t (R(e2t )) dt = q(e2t ) − νi q(γi2 e2t ). −∞

i∈S

By letting t → ∞, we deduce the proposition.

4. Volume Measures First we will recall the notion of volume measures introduced by Lapidus in [24]. Combining [24, Theorem 4.41] and [24, Corollary 4.45], we obtain the following result. Theorem 4.1. There exists a unique positive Borel regular measure ν on K such that −d /2 f dν = Trw (Mf ◦ HD S ) K

for any f ∈ C(K), where Trw (·) is the Dixmier trace of operators (as explained just below) and Mf is the multiplication operator on L2 (K, µ) defined by Mf (u) = f u. Moreover, the total mass of K with respect to ν is equal to the spectral volume. In other words, vol(K, µ) = ν(K).


173

The Borel regular measure ν in the above theorem is called the volume measure associated with (E, F, µ) and is denoted by νµ . Next, we briefly recall the notion of Dixmier trace ([12, 8, §IV.2]), which is a very useful tool in Connes’ noncommutative geometry and quantized calculus. (See, for example, [8, Chapters IV and VI].) Given a compact (nonnegative and self-adjoint) operator 1+ (the R on a Hilbert space H, with eigenvalues {κj (R)}∞ j =1 ↓ 0, we say that R ∈ L “Matsaev ideal” [8]) if the sequence (ln J )−1 Jj=1 κj (R) is bounded. (In Theorem 4.1, the Hilbert space H is equal to L2 (K, µ).) Then, roughly speaking, the Dixmier trace of R is defined by −1

Trw (R) = Lim (ln J ) w

J

κj (R),

(4.1)

j =1

where “Limw ” is a suitable notion of limit of (bounded sequences) with nice scaleinvariance (i. e., renormalization) properties. See, e.g., [7, 8, §IV.2] and [24, §4.1] for more details and additional relevant references. (Intuitively, Trw (R) captures the “semiclassical information” contained in R.) Further, Tr w extends to a finite, positive (nonnormal and unitary) trace on L1+ . The following proposition summarizes some of the basic properties of Tr w . Proposition 4.2. Let A and B belong to L1+ . (1) Trw (A ◦ B) = Trw (B ◦ A). (2) If A belongs to the trace class, then Trw (A) = 0. (3) If A is non-negative, then Trw (A) ≥ 0. Our main interest in this paper is to determine the nature of the volume measure. In particular, we conjecture that the normalized volume measure νµ /νµ (K) is the selfsimilar measure with weight (ν1 , · · · , νN ). Recall that νi = γi dS for i ∈ S. In the next section, we will prove this conjecture for a class including the standard Laplacians on the Sierpinski gaskets. 0 = {u ∈ F : u|V1 = 0}. It is easy to see that (E, F 0 ) becomes a local regular Set F 2 Dirichlet form on L (K, µ). Let HD be a non-negative self-adjoint operator associated 0 ). Note that E(u, v) = (u, H Then Proposition 2.8 implies D v)µ for all v ∈ F. with (E, F the following lemma. Lemma 4.3. Let ϕj denote the j th Dirichlet eigenfunction ϕjD for all j ≥ 1. Set ϕj,i = (µi )−1/2 Si ϕj , where f (Fi−1 (x)) if x ∈ Kw , Si (f )(x) = 0 otherwise. D ϕj,i = Then {ϕj,i }j ≥1,i∈S is a complete orthonormal system of L2 (K, µ). Moreover, H kj ri µi ϕj,i .

Lemma 4.4. For all f ∈ C(K), −dS /2

Mf ◦ H D where Ri (u) = u ◦ Fi .

=

i∈S

νi Si ◦ Mf ◦Fi ◦ HD −dS /2 ◦ Ri ,

174


Remark. For all i ∈ S, Ri ◦ Si is the identity and Si ◦ Ri u = χKi u, where χKi is the characteristic function of Ki . Proof. Let u = j,i αj,i ϕj,i , then D −dS /2 u = H

i∈S

D −dS /2 = This implies H proposed equality.

i∈S νi Si

νi

j ≥1

−dS /2

αj,i kj

ϕj,i .

◦ HD −dS /2 ◦ Ri . Now we can easily obtain the

Proposition 4.5. For all f ∈ C(K), D −dS /2 )). νµ (f ) − νi νµ (f ◦ Fi ) = Trw (Mf ◦ (HD −dS /2 − H i∈S

Proof. By Lemma 4.4, −dS /2

Trw (Mf ◦ H D

)= =

i∈S

νi Trw (Si ◦ Mf ◦Fi ◦ HD −dS /2 ◦ Ri ) νi Trw (Mf ◦Fi ◦ HD −dS /2 ),

i∈S

where we also use Proposition 4.2 (1). This immediately implies the proposition.

The following hypothesis is a key to show self-similarity of volume measures in the present approach. We believe that it is always satisfied but unfortunately, so far, we do not know how to verify it in general. D −dS /2 belongs to the trace class and Hypothesis 4.6. The operator HD −dS /2 − H −1 D −dS /2 ). vol(K, µ) = − νi log νi tr(HD −dS /2 − H

(4.2)

i∈S

In the next section, we will show that the above hypothesis holds for the Laplacians associated with strong harmonic structures in the sense of Shima [33], where the eigenvalue decimation method can be applied. This class includes the standard Laplacians on the Sierpinski gaskets. We give several examples in the next section. Theorem 4.7. Define the normalized volume measure ν˜ µ by ν˜ µ = νµ /νµ (K). If Hypothesis 4.6 is true, then the normalized volume measure ν˜ µ is the self-similar measure with weight (ν1 , · · · , νN ). D −dS /2 belongs to the trace class. Then, since the trace Proof. Assume HD −dS /2 − H class is an ideal in the algebra of all bounded linear operators (see Reed & Simon [32] D −dS /2 ) also belongs to the trace class. Hence, by (2) for example), Mf ◦ (HD −dS /2 − H −dS /2 − H D −dS /2 )) = 0. So Proposition 4.5 implies of Proposition 4.2, Trw (Mf ◦ (HD νµ (f ) = i∈S νi νµ (f ◦ Fi ) for any f ∈ C(K). Using Proposition 2.9, we see that ν˜ µ is the self-similar measure with weight (ν1 , · · · , νN ).


175

Remark. If Hypothesis 4.6 is true, then vol(K, µ) = νµ (K) = Trw (HD −dS /2 ) −1 D −dS /2 ) = − νi log νi tr(HD −dS /2 − H i∈S −1 = − νi log νi lim (q(x) − νi q(γi2 x)). i∈S

x→∞

i∈S

In the rest of this section, we discuss properties of volume measures assuming Hypothesis 4.6. Note that in general the self-similar measure ν˜ µ has a different weight from that of the original self-similar measure µ. More precisely, µ = ν˜ µ if and only if the harmonic structure (D, r) is regular (i. e., 0 < ri < 1 for all i ∈ S) and µi = ridH for all i ∈ S, where dH is defined as the unique d > 0 that satisfies i∈S rid = 1. Assume that the harmonic structure (D, r) is regular. Let µ∗ be the self-similar measure which satisfies µ∗ = ν˜ µ∗ . Then by the appendix of Kigami–Lapidus [22], µ∗ is the unique self-similar measure that attains the following maximum max{dS : µ is a self-similar measure on K} H and dS = d2d .Also, Kigami [20] has shown that dH is equal to the Hausdorff dimension H +1 of K with respect to the effective resistance metric. If µ = µ∗ , νµ and µ are mutually singular. In [24], the measure νµ∗ = vol(K, µ∗ )µ∗ is called the “natural volume measure” on K (associated with the harmonic structure (D, r)) and is suggested to be a counterpart of the usual Riemannian volume measure for this class of self-similar fractals, by analogy with the work of Connes in [7] for smooth Riemannian (spin) manifolds. In general, the value of the Dixmier trace may depend on the choice of the mean w used to define Trw in (4.1); see [8, §IV.2.β]. It follows from [24] that the total mass of ν, namely, ν(K) = vol(K, µ), is always independent of w. (See Theorem 4.1 above.) Moreover, Theorem 4.7 implies that the measure ν itself is independent of the choice of w under Hypothesis 4.6.

5. A Sufficient Condition for Self-Similarity and Examples In this section, we will give a sufficient condition related to localized eigenfunctions for Hypothesis 4.6 to be satisfied. To state our sufficient condition, we need to recall some notions about localized (and non-localized) eigenfunctions and corresponding eigenvalue counting functions. ⊥

Definition 5.1. We define E W (k) = ED (k) ∩ EN (k) and E F (k) = ED (k) ∩ E W (k) . We also define corresponding eigenvalue counting functions as follows: ρ W (x, µ) = dim E W (k) and ρ F (x, µ) = dim E F (k). k≤x

k≤x

Obviously, ρD (x, µ) = ρ W (x, µ) + ρ F (x, µ). If u ∈ E W (k) for some k > 0, then u is called a pre-localized eigenfunction.

176


Theorem 5.2. Suppose that there exists a pre-localized eigenfunction. If κF = lim sup x→∞

log ρ F (x, µ) dS < , log x 2

(5.1)

then Hypothesis 4.6 is satisfied. Recall Theorem 3.2, where we obtain that ρD (x, µ) x dS /2 as x → ∞. Hence the above condition requires that the counting function of non-localized eigenfunctions ρ F (x, µ) is asymptotically much smaller than that of localized eigenfunctions ρ W (x, µ). In [21], (5.1) is conjectured to be true whenever there exists a pre-localized eigenfunction. In particular, it was shown in [21, Theorem 4.5] that (5.1) is true if the harmonic structure is a strong harmonic structure in the sense of Shima [33]. In this paper, we will not go into the details. Instead, we will give examples where (5.1) has been verified in [21]. Example 5.3 (Sierpinski gasket). Let {p1 , p2 , p3 } ⊂ C satisfy |pi − pj | = 1 for any i = j . Define Fi : C → C by Fi (z) = (z − pi )/2 + pi for i ∈ S, where S = {1, 2, 3}. The Sierpinski gasket is the unique non-empty compact set K that satisfies (2.1). Clearly (K, S, {Fi }i∈S ) is a p. c. f. self-similar structure and V0 = {p1 , p2 , p3 }. Now if   −2 1 1 3 3 3 D =  1 −2 1  and r = ( , , ), 5 5 5 1 1 −2 then (D, r) is a harmonic structure. Also let µ be the self-similar measure on K with weight (1/3, 1/3, 1/3). The Laplacian associated with (D, r) and µ is called the standard Laplacian on the Sierpinski gasket K. By Theorem 4.4 of [21], we can verify (5.1). In fact, κF = log 2/ log 5 < dS /2 = log 3/ log 5. Hence Hypothesis 4.6 is true. So the normalized volume measure ν˜ µ is a self-similar measure. Since µi ri = 1/5 for all i ∈ S, it follows that ν˜ µ is the self-similar measure with weight (1/3, 1/3, 1/3) and hence it coincides with µ. Analogous results are also valid for the higher-dimensional Sierpinski gaskets. We have discussed only the above case for simplicity. Example 5.4 (Vicsek set, [21, Example 4.6]). For 1 ≤ √ j ≤ 5, define Fj : C → √ C by Fj = (z − pj )/3 + pj , where p1 = 1, p2 = −1, p3 = −1, p4 = − −1 and p5 = 0. The Vicsek set K is the unique non-empty compact set that satisfies (2.2), where S = {1, 2, 3, 4, 5}. (K, S, {Fi }i∈S ) is a p. c. f. self-similar structure and V0 = {p1 , p2 , p3 , p4 }. Define D ∈ L(V0 ) by Dpj pk = 1 for 1 ≤ j = k ≤ 4 and Dpj pj = −3 for all j and let r = (s, s, s, s, t), where t > 0, s > 0 and 2s + t = 1. t Then (D, r) is a regular harmonic structure. Moreover, set µ1 = µ2 = µ3 = µ4 = 4t+s and µ5 =

s 4t+s .

Then in [21], it was shown that dS /2 =

log 5 log n0

and κF =

log 3 log n0 ,

where

n0 = 4t+s st . So by Theorem 5.2 and Theorem 4.1, the normalized volume measure ν˜ µ is a self-similar measure. As µi ri = n−1 0 for all i ∈ S, νi = 1/5 for all i ∈ S. Therefore, µ = ν˜ µ if and only if s = t = 1/3. Example 5.5 (modified Koch curve, [2], [21, Example 4.7]). Let fp,q (z) = (q − p)z + p for p, q ∈ C. Define F1 = f√0,1/3 , F2 = f2/3,1 , F3 = f1/3,2/3 , F4 = f1/3,c and F5 = fc,2/3 , where c = 21 + √−1 . The modified Koch curve is the unique compact 2 3 set K that satisfies (2.1), where S = {1, 2, 3, 4, 5}. Obviously, (K, S, {Fi }i∈S ) is a


177

1 p. c. f. self-similar structure and V0 = {0, 1}. Set D = −1 1 −1 and r = (s, s, t, h, h) 2ht with 2s + t+2h = 1 for s, t, h > 0. Then (D, r) is a harmonic structure. Note that one of the numbers t or h can be arbitrarily large. In such a case, (D, r) is not a regular harmonic structure. Now set µ1 = µ2 = (n0 s)−1 , µ3 = (n0 t)−1 and µ4 = µ5 = (n0 h)−1 , where log 5 log 4 n0 = 2s −1 + t −1 + 2h−1 . Then it was shown in [21] that dS /2 = log n0 and κF = log n0 . So by Theorem 5.2 and Theorem 4.1, the normalized volume measure ν˜ µ is a self-similar measure. As µi ri = n−1 0 for all i ∈ S, νi = 1/5 for all i ∈ S. Hence µ = ν˜ µ if and only if s = t = h = 3/8. In the rest of this section, we will prove Theorem 5.2. First we will introduce some properties of pre-localized eigenfunctions. A pre-localized eigenfunction can generate a sequence of infinitely many pre-localized eigenfunctions as follows. Proposition 5.6 ([3, Lemma 4.2]). Let u be a pre-localized eigenfunction with u ∈ E W (k). Define uw = Sw1 ◦ · · · ◦ Swm (u) for any w = w1 · · · wm ∈ W∗ . Then uw is also a pre-localized eigenfunction belonging to the eigenvalue rwkµw . Note that Sj (E W (µj rj k)) ⊂ E W (k). Naturally, the eigenfunctions in Sj (E W (µj rj k)) are thought to be offsprings of the preceding eigenfunctions in E W (µj rj k). From such an observation, we can divide E W (k) into offsprings E2W (k) and generators E1W (k). Definition 5.7. E2W (k) =

Si (E W (kµi ri ))

and

E1W (k) = (E2W (k))⊥ ∩ E W (k).

i∈S

Now we can choose kjW and φj ∈ E1W (kjW ) for j ≥ 1 so that kjW ≤ kjW+1 and {φj }∞ j =1 is a complete orthonormal system of E1W = ⊕k E1W (k). Then {φj,w |j ≥ 1, w ∈ W∗ } is a complete orthonormal system of E W = ⊕k E W (k), where φj,w = (µw )−1/2 Sw1 ◦ · · · ◦ Swm (φj ) for w = w1 · · · wm ∈ W∗ . Note that φj,w ∈ E2W (kiW /(µw rw )) if w ∈ / W0 and {φj,w }j ≥1,w∈W∗ \W0 is a complete orthonormal system of E2W = ⊕k E2W (k). The following proposition was obtained in [21]. Proposition 5.8 ([21, Theorem 3.5]). Suppose that there exists a pre-localized eigenfunction. (1) In the lattice case, ρ W (x, µ) = (GW (log x/2) + o(1))x dS /2 as x → ∞, where GW is a discontinuous T -periodic function with 0 < inf GW ≤ sup GW < ∞. W dS /2 exists and is positive. (2) In the non-lattice case, the limit limx→∞ ρ (x, µ)/x (3) j ≥1 (kjW )−dS /2 < ∞ and −1 νi log νi (kjW )−dS /2 , cW = − i∈S

where

cW =

T

j ≥1

GW (t)dt in the lattice case, limx→∞ ρ W (x, µ)/x dS /2 in the non-lattice case. 1 T

0

(5.2)

178


By the above proposition, we have the following lemma. Lemma 5.9. If (5.1) is satisfied, then −1 νi log νi (kjW )−dS /2 . vol(K, µ) = cW = − i∈S

j ≥1

Proof. If (5.1) is satisfied, then we see that G = GW in the lattice case and limx→∞ ρ F (x, µ)/x dS /2 = 0 in the non-lattice case. Hence comparing the definitions of vol(K, µ) and cW , we obtain vol(K, µ) = cW . Next we choose kjF > 0 and ξj ∈ E F (kjF ) for j ≥ 1 so that kjF ≤ kjF+1 and

{ξj }j ≥1 is a complete orthonormal system of E F = ⊕k E F (k). It follows immediately that L2 (K, µ) = E F ⊕ E1W ⊕ E2W and {ξj , φj,w }j ≥1,w∈W∗ is a complete orthonormal system of L2 (K, µ). Lemma 5.10. If (5.1) is satisfied, then j ≥1 (kjF )−dS /2 < ∞. Proof. Choose α so that κF < α < dS /2. Note that ρ F (x, µ) = #{j : kjF ≤ x}. So by (5.1), we obtain that there exists c > 0 such that cj 1/α ≤ kjF for any j ≥ 1. Therefore (kjF )−dS /2 ≤ cj −dS /(2α) . Now as 1 < dS /(2α), j ≥1 j −dS /(2α) < ∞. Lemma 5.11. Let ξj,i = Si (ξj ) for any j ≥ 1 and i ∈ S. Then {ξj,i }j ≥1,i∈S is a complete orthonormal system of E F ⊕ E1W . Proof. Applying the same argument as in Lemma 4.3 to {ξj , φj,w }j ≥1,w∈W∗ , we see that {ξj,i , φj,w }j ≥1,i∈S ,w∈W∗ \W0 is a complete orthonormal system of L2 (K, µ). Recall that {φj,w }j ≥1,w∈W∗ \W0 is a complete orthonormal system of E2W . Hence {ξj,i }j ≥1,i∈S is a complete orthonormal system of the orthogonal complement of E2W , which is E F ⊕E1W . Proof of Theorem 5.2. Let PF , P1 and P2 be the orthogonal projection of L2 (K, µ) D −dS /2 . By onto E F , E1W and E2W , respectively. Also let A = HD −dS /2 and B = H W d −d /2 S Proposition 5.6 and Lemma 4.3, Aφj,w = Bφj,w = (µw rw ) (kj ) S /2 φj,w for j ≥ 1 and w ∈ W∗ \W0 . Hence A ◦ P2 = B ◦ P2 . Therefore, A − B = A1 + AF − BF 1 where AF = A ◦ PF , A1 = A ◦ P1 and BF 1 = B ◦ (PF + P1 ). Note that AF ξj = (kjF )−dS /2 ξj , A1 φj = (kjW )−dS /2 φj and BF 1 ξj,i = νi (kjF )−dS /2 ξj,i . So it is easy to see that AF , A1 and BF 1 are bounded non-negative self-adjoint operators. Now by Lemma 5.9 and Lemma 5.10, it follows that tr(AF ) = j ≥1 (kjF )−dS /2 < ∞, tr(A1 ) = W −dS /2 < ∞ and tr(B ) = F −dS /2 = F −dS /2 < F1 j ≥1 (kj ) j ≥1,i∈S νi (kj ) j ≥1 (kj ) ∞. Hence AF , A1 and BF 1 belong to the trace class. Therefore A − B belongs to the trace class. Moreover, tr(A − B) = tr(AF ) + tr(A1 ) − tr(BF 1 ) = tr(A1 ) (kjW )−dS /2 . = j ≥1

This along with Lemma 5.9 implies (4.2).


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Acknowledgements. The authors wish to thank the University of California, Riverside, the Isaac Newton Institute for Mathematical Sciences, the University of Cambridge and Kyoto University, where this research was carried out and completed.

References 1. Alexander, S. and Orbach, R.: Densities of states on fractals: Fractons. J. Physique Lettres 43, L625–L631 (1982) 2. Baltes, H.B. and Hilf, E.R.: Spectra of Finite Systems. Vienna: B.I. Wissenschaftsverlag, 1976 3. Barlow, M.T. and Kigami, J.: Localized eigenfunctions on p.c.f. self-similar sets. London Math. Soc. (2) 56, 320–332 (1997) 4. Barlow, M.T. and Perkins, E.A.: Brownian motion on the Sierpinski gasket. Probab. Theory Related Fields 79, 542–624 (1988) 5. Berry, M.V.: Distribution of modes in fractal resonators. In: Structural Stability in Physics, W. Güttinger and H. Eikemeier, eds., Berlin–Heidelberg–New York: Springer, 1979, pp. 51–53 6. Berry, M.V.: Some geometric aspects of wave motion: Wavefront dislocations, diffraction catastrophes, diffractals. In: Geometry of the Laplace Operator, Proc. Symp. Pure Math. vol 36, Providence, RI: Amer. Math. Soc., 1980, pp. 13–38 7. Connes, A.: The action functional in non-commutative geometry. Commun. Math. Phys. 117, 673–683 (1988) 8. Connes, A.: Noncommutative Geometry. New York–London: Academic Press, 1994 9. Connes, A. and Sullivan, D.: Quantized calculus on S 1 and quasi-fuchsian groups. In preparation 10. Davies, E.B.: Spectral Theory and Differential Operators. Cambridge Studies in Advanced Math. vol. 42, Cambridge: Cambridge University Press, 1995 11. Dahr, D.: Lattices of effectively nonintegral dimensionality. J. Math. Phys. 18, 577–585 (1977) 12. Dixmier, J.: Existence de traces non normales. C. R. Acad. Sci. Paris 262, 1107–1108 (1966) 13. Fukushima, M. and Shima, T.: On a spectral analysis for the Sierpinski gasket. Potential Analysis 1, 1–35 (1992) 14. Goldstein, S.: Random walks and diffusions on fractals. In: Percolation Theory and Ergodic Theory of Infinite Particle Systems H. Kersten, ed., IMA Math Appl., vol. 8, Berlin–Heidelberg–NewYork: Springer, 1987, pp. 121–129 15. Hattori, K., Hattori, T. and Watanabe, H.: Gaussian field theories on general networks and the spectral dimensions. Progr. Theoret. Phys. Suppl. 29, 108–143 (1987) 16. Havlin, P. and Bunde, A.: Percolation II. In: Fractals and Disordered Systems, Berlin–Heidelberg–New York: Springer, 1991, pp. 97–149 17. Hörmander, L.: The Analysis of Linear Partial Differential Operators III & IV. Berlin–Heidelberg–New York: Springer, 1985 18. Kigami, J.: Analysis on Fractals. Cambridge: Cambridge University Press, to appear 19. Kigami, J.: Harmonic calculus on p.c.f. self-similar sets. Trans. Amer. Math. Soc. 335, 721–755 (1993) 20. Kigami, J.: Effective resistances for harmonic structures on p.c.f. self-similar sets. Proc. Cambridge Phil. Soc. 115, 291–303 (1994) 21. Kigami, J.: Distributions of localized eigenvalues of Laplacians on p.c.f. self-similar sets. J. Funct. Anal. 156, 170–198 (1998) 22. Kigami, J. and Lapidus, M.L.: Weyl’s problems for the spectral distribution of Laplacians on p.c.f. selfsimilar fractals. Commun. Math. Phys. 158, 93–125 (1993) 23. Kusuoka, S.: A diffusion process on a fractal. In: Proc. of Taniguchi International Symp. (Katata & Kyoto, 1985) K. Ito and N. Ikeda, eds., Tokyo: Kinokuniya, 1987, pp. 251–274 24. Lapidus, M.L.: Analysis on fractals, Laplacians on self-similar sets, noncommutative geometry and spectral dimensions. Topological Methods in Nonlinear Analysis 4, 137–195 (1994) 25. Lapidus, M.L.: Towards a noncommutative fractal geometry? – Laplacians and volume measures on fractals. Contemp. Math. 208, 211–252 (1997) 26. Liu, S.H.: Fractals and their applications in condensed matter physics. Solid State Phys. 39, 207–273 (1986) 27. Malozemov, L.: The integrated density of states for the difference Laplacian on the modified Koch curve. Commun. Math. Phys. 156, 387–397 (1993) 28. Nakayama, T.,Yakubo, K. and Orbach, R.L.: Dynamical properties of fractal networks: Scaling, numerical simulation, and physical realization. Rev. Modern Phys. 66, 381–443 (1994) 29. Rammal, R.: Spectrum of harmonic excitations on fractals. J. Physique 45, 191–206 (1984) 30. Rammal, R. and Toulouse, G.: Random walks on fractal structures and percolation clusters. J. Physique Lettres 44, L13–L22 (1983)

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Commun. Math. Phys. 217, 181 – 201 (2001)

Communications in



From Large N Matrices to the Noncommutative Torus G. Landi1,3 , F. Lizzi2,3 , R. J. Szabo4 1 Dipartimento di Scienze Matematiche, Università di Trieste, P.le Europa 1, 34127 Trieste, Italy.


2 Dipartimento di Scienze Fisiche, Università di Napoli Federico II, Mostra d’Oltremare Pad. 20,

80125 Napoli, Italy. E-mail: [email protected]

3 INFN, Sezione di Napoli, Napoli, Italy 4 The Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen Ø, Denmark. E-mail: [email protected]

Received: 24 December 1999 / Accepted: 7 October 2000

Abstract: We describe how and to what extent the noncommutative two-torus can be approximated by a tower of finite-dimensional matrix geometries. The approximation is carried out for both irrational and rational deformation parameters by embedding the C ∗ -algebra of the noncommutative torus into an approximately finite algebra. The construction is a rigorous derivation of the recent discretizations of noncommutative gauge theories using finite dimensional matrix models, and it shows precisely how the continuum limits of these models must be taken. We clarify various aspects of Morita equivalence using this formalism and describe some applications to noncommutative Yang–Mills theory. 1. Introduction The relationship between large N matrix models and noncommutative geometry in string theory was suggested early on in studies of the low energy dynamics of D-branes, where it was observed [1] that a system of N coincident D-branes has collective coordinates which are described by mutually noncommuting N × N matrices. Various aspects of the large N limit of such systems have been important to the Matrix theory conjecture [2] and the representation of branes in terms of large N matrices [3]. The connection between finite dimensional matrix algebras and noncommutative Riemann surfaces is the basis for the fact that large N Matrix theory contains M2-branes. A more precise connection to noncommutative geometry came with the observation [4] that the most general solutions to the quotient conditions for toroidal compactification of the IKKT matrix model [5] are given by connections of vector bundles over a noncommutative torus. The resulting large N matrix model is noncommutative Yang–Mills theory which is dual to the low-energy dynamics of open strings ending on D-branes in the background of a constant Neveu–Schwarz two-form field [6]. The description of noncommutative tori and their gauge bundles as the large N limit of some sort of tower of finite-dimensional matrix geometries is therefore an important,

182

G. Landi, F. Lizzi, R. J. Szabo

yet elusive, problem. This correspondence was described at a very heuristic level in [7], while a definition of noncommutative gauge theory as the large N limit of a matrix model has been made more precise recently in [8, 9]. In particular, in [9] it was shown how the standard projective modules [10, 11] over the noncommutative two-torus can be discretized in terms of finite-dimensional matrix algebras. This immediately raises an apparent paradox. A standard result asserts that the noncommutative torus cannot be described by any approximately finite dimensional algebra. This means that it cannot be written explicitly as the large N limit of some sequence of finite dimensional matrix algebras. One way to understand this is in terms of K-theory. K-theory groups are stable under deformations of algebras, and those of the ordinary torus T2 are non-trivial. The deformation of the algebra of functions on T2 to the noncommutative torus therefore preserves this non-trivial K-theory structure. On the other hand, the K 1 group of any approximately finite dimensional algebra is trivial (see for instance [12]). In fact, it is precisely this K-theoretic stability which immediately implies that there is a canonical map between gauge bundles on ordinary T2 and gauge bundles on the noncommutative torus. This canonical map is constructed explicitly in [6]. However, this mathematical reasoning would seem to put very stringent restrictions on the allowed observables of field theories defined on the noncommutative torus. The generators of a noncommutative torus with a deformation parameter θ that is a rational number can be represented by finite dimensional (clock and shift) matrices. There is no such matrix description in the case that θ is irrational. However, an irrational (or rational) θ can always be represented as the limit of a sequence θn of rational numbers. From a physical standpoint, we would expect any correlation function C of a field theory on such noncommutative tori to be a continuous function of θ, so that C(θ) = limn C(θn ). This means that there must be some sense in which observables of noncommutative Yang– Mills theory can be approximated as the large N limit of a sequence of those for finite dimensional matrix models. Such an approximation scheme is reminescent of fuzzy spaces [13], whereby the multiplication law of the algebra of functions is approximated by a particular matrix multiplication. Although the space of functions on a manifold is not an approximately finite dimensional algebra, its product is approximated arbitrarily well as N → ∞. However, the algebras which are deformations of function algebras are somewhat distinct from fuzzy spaces which are typically finite dimensional [14], and the algebraic approximation in the case of the noncommutative torus must come about in a different way. In this paper we will show precisely how to do this. The main point is that although the algebra of the noncommutative torus is not approximately finite, it can be realized as a subalgebra of an algebra which is built from a certain tower of finite dimensional matrix algebras [15]. As an important byproduct we solve what has been a problem for the physical interpretation of the deformation parameter of the algebra of the torus. The mathematical properties of the noncommutative torus depend crucially on whether or not the parameter θ is a rational number. Certain distinct values of θ are connected by Morita equivalence, and the set of equivalent θ’s is dense on the real line. This is similar (and in some cases equivalent) to the phenomenon of T-duality in string theory [6, 16]. Nevertheless, with a particular choice of background fields, θ is in principle an observable variable, and it would be wrong to expect that the fact that θ is rational or not could have measurable physical consequences. In what follows we will see how it is possible to approximate the algebra with irrational or rational θ by a sequence of finite dimensional matrix algebras. As an immediate corollary, the physical quantities that one calculates as the limit (which we show exists) are continuous functions of θ . In fact, we will show that

From Large N Matrices to Noncommutative Torus

183

all Morita equivalent noncommutative tori can be embedded into the same approximately finite algebra, so that the present construction shows that all noncommutative gauge theories can be approximated within a unifying framework. This description is therefore useful for analysing the phase structure of noncommutative Yang–Mills theory, as a function of θ, using matrix models. The results presented in the following give a very precise meaning to the definition of noncommutative Yang–Mills theory as the large N limit of a matrix model, and at the same time clarify in a rigorous manner the way that the field content, observables and correlators of the matrix model must be mapped to the continuum gauge theory. This is particularly important for numerical computations in which the interest is in determining quantities in noncommutative Yang–Mills theory in terms of those of large matrices at finite N . Such large N limits are also important for describing the dynamics of Matrix theory, whereby the N × N matrix geometries coincide with the parameter spaces of systems of N D0-branes. This paper is organized as follows. In Sect. 2 we shall describe this construction, and discuss exactly in what sense the generators of any noncommutative torus can be approximated by large N matrices. In Sect. 3 we will then show that this procedure can be used to approximate correlation functions for field theories on the noncommutative torus in terms of expectation values constructed from matrices acting on a finite dimensional vector space. In Sect. 4 we show how to express geometries on the noncommutative torus, including gauge bundles, in terms of a tower of matrix geometries. Section 5 contains some concluding remarks.

2. AF-Algebras and the Noncommutative Torus The algebra Aθ of smooth functions on the “noncommutative two-torus” T2θ is the unital ∗-algebra generated by two unitary elements U1 , U2 with the relation U1 U2 = e2πiθ U2 U1 .

(2.1)

A generic element a ∈ Aθ is written as a convergent series of the form a=

amn (U1 )m (U2 )n ,

(2.2)

(m,n)∈Z2

where amn is a complex-valued Schwarz function on Z2 , i.e. a sequence of complex numbers {amn ∈ C | (m, n) ∈ Z2 } which decreases rapidly at “infinity”. When the deformation parameter θ = M/N is a rational number, with M and N positive integers which we take to be relatively prime, the algebra AM/N is intimately related to the algebra C ∞ (T2 ) of smooth functions on the ordinary torus T2 . Precisely, AM/N is Morita equivalent to C ∞ (T2 ), i.e., AM/N is a twisted matrix bundle over C ∞ (T2 ) of topological charge M whose fibers are N × N complex matrix algebras. Physically, this implies that noncommutative U (1) Yang–Mills theory with rational deformation parameter θ = M/N is dual to a conventional U (N ) Yang–Mills theory with M units of ’t Hooft flux. The algebra AM/N has a “huge” center C(AM/N ) which is generated by the elements (U1 )N and (U2 )N . One identifies C(AM/N ) with the algebra C ∞ (T2 ), while the appearence of finite dimensional matrix algebras can be seen as follows. With

184


ω = e2πiM/N , one introduces the N  1  ω   ω2 1 =  U  ..  .

× N clock and shift matrices    0 1 0 0 1       .. ..    2 =  . . U (2.3) , .    .   .. 1  ωN−1 1 0 N−1 k These matrices are traceless (since k=0 ω = 0), they obey the relation (2.1), and they satisfy

N N 2 = IN . 1 = U (2.4) U Since M and N are relatively prime, the matrices (2.3) generate the finite dimensional algebra MN (C) of N × N complex matrices [17].1 Furthermore, there is a surjective algebra morphism π : AM/N → MN (C)

(2.5)

given by  π



amn (U1 )m (U2 )n  =

(m,n)∈Z2

m n 1 2 , U amn U

(2.6)

(m,n)∈Z2

under which the whole center C(AM/N ) is mapped to C. When MN (C) is thought of as the Lie algebra gl(N, C), a basis is provided by the N × N matrices Tp(N) =

i N p1 p2 /2 p1 p2 U1 U2 , ω 2π M

(2.7)

N−3 N−1 where pa ∈ {− N−1 2 , − 2 , . . . , 2 }. These matrices obey the commutation relations

N πM (N) Tp(N) , Tq(N) = (2.8) sin (p1 q2 − p2 q1 ) Tp+q (mod N) πM N

which in the limit N → ∞ with M/N → 0 become

(∞) Tp(∞) , Tq(∞) = p1 q2 − p2 q1 Tp+q .

(2.9)

Equation (2.9) is recognized as the Poisson-Lie algebra of functions on T2 with respect to the usual Poisson bracket. In a unitary representation of the algebra (2.8), anti-Hermitian (N) combinations of the traceless matrices Tp span the Lie algebra su(N ). This identifies the symplectomorphism algebra (2.9) of the torus with su(∞) [18] which is an example of a universal gauge symmetry algebra [7]. This identification has been exploited recently in [19] to study the perturbative renormalizability properties of noncommutative Yang– Mills theory. For finite N , su(N ) may be regarded as the Lie algebra of infinitesimal reparametrizations of the algebra described by (2.7) and (2.8). Given these connections, 1 If M and N are not coprime then the generated algebra would be a proper subalgebra of M (C). N


185

it follows that the noncommutative two-torus coincides with the parameter space of Matrix theory. In what follows we shall be interested in taking the limit where both N, M → ∞ with the ratio M/N approaching a fixed irrational or rational number. This is the type of limit considered in [9], and it yields the appropriate embeddings of matrix algebras into the infinite dimensional C ∗ -algebra which describes the noncommutative spacetime of D0-branes in Matrix theory [2]. For finite N , the matrix model consists of maps of a quantum Riemann surface (the noncommutative toroidal M2-brane) into a noncommutative transverse space. In the case where θ is an irrational number, the algebra (2.1) cannot be mapped to any subalgebra of su(∞). We would like to investigate how and to what extent the geometries for Aθ can be approximated by towers of matrix geometries. Naively, one could think of considering the algebra Aθ as the inductive limit of a sequence of finite dimensional ∗-algebras. This would be tantamount to (the closure of) Aθ being an approximately finite dimensional C ∗ -algebra. As we mentioned in the previous section, this is not the case, as can be easily seen for any value of θ using cohomological arguments. The K-theory groups of T2θ are Kn (T2θ ) = Z ⊕ Z, n = 0, 1, just as for the ordinary torus T2 . On the other hand, the group K1 of any approximately finite algebra is necessarily trivial [12]. 2.1. AF-algebras. In [15], Pimsner and Voiculescu have shown that there is the possibility to realize the C ∗ -algebra Aθ , which is the norm closure of the algebra of smooth functions Aθ , as a subalgebra of a larger, approximately finite dimensional C ∗ -algebra. In a classical sense, this would mean that an embedded submanifold of T2θ is induced by the parameter space geometries. This is analogous to what happens in Matrix theory, whereby the noncommutative target space is realized as a “submanifold” of the matrix parameter space of N D0-branes. Before describing this embedding, we shall in this subsection briefly describe some general properties of the class of approximately finite algebras [20]. A unital C ∗ -algebra A is said to be approximately finite dimensional (AF for short) if there exists an increasing sequence ρ1

ρ2

ρ3

ρn

ρn+1

A0 → A1 → A2 → · · · → An → · · ·

(2.10)

of finite dimensional C ∗ -subalgebras of A such that A is the norm closure of the union n An , A = n An . The maps ρn are injective ∗-morphisms. Without loss of generality one may assume that each An contains the unit I of A and that the maps ρn are unital. The algebra A is the inductive limit of the inductive system of algebras {An , ρn }n∈Z+ [12]. As a set, n An is made of coherent sequences, ∞

An = a = (an )n∈Z+ , an ∈ An ∃N0 , an = ρn (an−1 ) ∀ n > N0 .

(2.11)

n=0

The sequence (an An )n∈Z+ is eventually decreasing since an+1 ≤ an (the maps ρn are norm decreasing) and is therefore convergent. The norm on A is given by (an )n∈Z+ = lim an . (2.12) n→∞

An

Since the maps ρn are injective, the expression (2.12) gives a true norm directly and not merely a semi-norm, and there is no need to quotient out the zero norm elements.

186


Since each subalgebra An is finite dimensional, it is a direct sum of matrix algebras, An =

kn

Md (n) (C),

(2.13)

k

k=1

where Md (C) is the algebra of d × d matrices with complex entries and endowed with its usual Hermitian conjugation and operator On the other hand, a unital norm. given 1 2 embedding A1 → A2 of the algebras A1 = nj =1 Md (1) (C) and A2 = nk=1 Md (2) (C), j

k

one can always choose suitable bases in A1 and A2 in such a way as to identify A1 with a subalgebra of A2 having the form A1 ∼ =

n2 n1 k=1 j =1

Nkj Md (1) (C).

(2.14)

j

Here, for any two non-negative integers p, q, the symbol p Mq (C) denotes the algebra p Mq (C) ∼ = Mq (C) ⊗C Ip , and one identifies

n1

j =1 Nkj

(2.15)

Md (1) (C) with a subalgebra of Md (2) (C). The non-negative j

k

integers Nkj satisfy the condition

n1 j =1

(1)

Nkj dj

(2)

= dk .

(2.16)

One says that the algebra Md (1) (C) is partially embedded in Md (2) (C) with multiplicity j

k

Nkj . A useful way of representing the algebras A1 , A2 and the embedding A1 → A2 is by means of a diagram, the so-called Bratteli diagram [20], which can be constructed out (1) (2) of the dimensions dj , j = 1, . . . , n1 and dk , k = 1, . . . , n2 of the diagonal blocks of the two algebras, and out of the numbers Nkj that describe the partial embeddings. One draws two horizontal rows of vertices, the top (bottom resp.) one representing A1 (A2 resp.) and consisting of n1 (n2 resp.) vertices, one for each block which are labeled (1) (1) (2) (2) by the corresponding dimensions d1 , . . . , dn1 (d1 , . . . , dn2 resp.). Then, for each (1) (2) j = 1, . . . , n1 and k = 1, . . . , n2 , one has a relation dj Nkj dk to denote the fact that Md (1) (C) is partially embedded in Md (2) (C) with multiplicity Nkj . j

k

For any AF-algebra A one repeats this procedure for each level, and in this way one obtains a semi-infinite diagram which completely defines A up to isomorphism. This diagram depends not only on the collection of An ’s but also on the particular sequence {An , ρn }n∈Z+ which generates A. However, one can obtain an algorithm which allows one to construct from a given diagram all diagrams which define AF-algebras that are isomorphic to the original one [20]. The problem of identifying the limit algebra or of determining whether or not two such limits are isomorphic can be very subtle. In [21] an invariant for AF-algebras has been devised in terms of the corresponding K-theory which completely distinguishes among them. Note that the isomorphism class of an AF-algebra n An depends not only on the collection of algebras An but also on the way that they are embedded into one another.


187

2.2. Embedding the noncommutative torus in an AF-algebra: Irrational case. We are now ready to describe the realization [15] of the algebra Aθ as a subalgebra of a larger, AF algebra A∞ which is determined by the K-theory of Aθ (to be precise K0 (Aθ )). While in [15] the values of θ are taken to be irrational and to lie in the interval (0, 1), we shall repeat the construction for an arbitrary real-valued deformation parameter. In this subsection we shall take θ to be irrational. The case of rational θ will be described in the next subsection. It is known [22] that any θ ∈ R−Q has a unique representation as a simple continued fraction expansion θ = lim θn

(2.17)

n→∞

in terms of positive integers ck > 0 (k ≥ 1) and c0 ∈ Z. The nth convergents θn of the expansion are given by θn ≡

pn = c0 + qn

1

.

1

c1 + c2 +

(2.18)

1 ..

. cn−1 +

1 cn

One also writes this as θ = [c0 , c1 , c2 , . . . ].

(2.19)

The relatively prime integers pn and qn may be computed recursively using the formulae pn = cn pn−1 + pn−2 , qn = cn qn−1 + qn−2 ,

p0 = c0 , q0 = 1,

p1 = c0 c1 + 1, q1 = c1

(2.20)

for n ≥ 2. Note that all qn ’s are strictly positive, qn > 0, while pn ∈ Z, and that both qn and |pn | are strictly increasing sequences which therefore diverge as n → ∞. For each positive integer n, we let Mqn (C) denote the finite dimensional C ∗ -algebra of qn × qn complex matrices acting on the finite dimensional Hilbert space Cqn which (n) is endowed with its usual inner product and its canonical orthonormal basis ej , 1 ≤ j ≤ qn . Then, for any integer n, consider the semi-simple algebra An = Mqn (C) ⊕ Mqn−1 (C)

(2.21)

ρn

and introduce the embeddings An−1 → An defined by2   M   ..  cn .  ρn M   −→  M  N 



N

  ,  

(2.22)

M

2 In [15], in order to explicitly construct the embedding of the noncommutative torus algebra in the limit AF-algebra, the embeddings (2.22) are conjugated with suitable (and rather involved) unitary operators

Wn : Cqn−1 ⊕ · · · ⊕ Cqn−1 −→ Cqn . ! cn times

Since the two embeddings are the same up to an inner automorphism, the limit algebra is the same [20].

188


where M and N are qn−1 × qn−1 and qn−2 × qn−2 matrices, respectively, and we have used (2.20). The norm closure of the inductive limit ∞

A∞ =

An

(2.23)

n=0

is the AF-algebra that we are looking for. As mentioned in the previous subsection, the elements of A∞ are coherent sequences {Gn }n∈Z+ , Gn ∈ An , with Gn = ρn (Gn−1 ) for n sufficiently large, or limits of coherent sequences. It is useful to visualize them as infinite matrices and we shall also loosely write A∞ ∼ = M∞ (C). From the discussion of the previous subsection it follows that the embeddings ρn An−1 → An are completely determined by the collection of partial embeddings {cn }. The corresponding Bratteli diagram is shown in Fig. 1. Associated with them we have positive maps ϕn : Z2 → Z2 defined by qn qn−1 cn 1 = ϕn , ϕn = . (2.24) 1 0 qn−1 qn−2 As a consequence, the group K0 (A∞ ) can be obtained as the inductive limit of the inductive system {ϕn : K0 (An−1 ) → K0 (An )}n∈Z+ of ordered groups. Since K0 (An ) = Z ⊕ Z (with the canonical ordering Z+ ⊕ Z+ ) it follows that [23] K0 (A∞ ) = Z + θ Z with ordering defined by taking the cone of non-negative elements to be K0+ (A∞ ) = (z, w) ∈ Z2 z + θw ≥ 0 .

(2.25)

(2.26)

This is a total ordering since for all pairs of integers (z, w), one has either z + θw ≥ 0 or z+θ w < 0. We shall comment more on the K-theory group (2.25) later on. Furthermore, these K-theoretic properties will enable us in Sect. 4 to map a gauge bundle over a matrix algebra to a gauge bundle over the noncommutative torus. .. . qn−1

s ❅

❅ ❅

cn qn

s qn−2

❅

❅

s

❅s qn−1

.. . Fig. 1. Bratteli diagram for the algebra A∞ in the case of irrational θ . The labels of the vertices denote the dimensions of the corresponding matrix algebras. The labels of the links denote the partial embeddings (not written when equal to unity)


189

At each finite level labelled by the integer n, let Aθn be the algebra of the noncommutative two-torus with rational deformation parameter θn = pn /qn given in (2.18), and (n) generators Ua , a = 1, 2 obeying the relation (n)

(n)

U 1 U2

(n)

(n)

= e2πipn /qn U2 U1 .

(2.27)

From (2.5) and (2.6) it follows that there exists a surjective algebra homomorphism a(n) , a = 1, 2 π : Aθn → Mqn (C), π Ua(n) ≡ U (2.28) (n) we may take the qn × qn clock and cyclic shift matrices, respec(n) and U and for U 1 2 tively,

2πi(j −1)pn /qn (n) (n) U = e δ , U = δk,j −1 , k, j = 1, . . . , qn (mod qn ), kj 1 2 kj

kj

(2.29) which also obey a relation like (2.27), (n) U (n) = e2πipn /qn U (n) U (n) . U 1 2 2 1

(2.30)

Thus, within each finite dimensional matrix algebra An there is the subalgebra π(Aθn ) ⊕ π(Aθn−1 ) which is represented by clock and shift matrices. The main result of Ref. [15] is the statement that the algebra π(Aθn ) ⊕ π(Aθn−1 ) can be taken to be a finite dimensional approximation of the algebra Aθ of the noncommutative torus in the following sense. a(n−1) ⊕ U a(n−2) ) = U a(n) ⊕ U a(n−1) . Then, we have First of all, notice that ρn (U Proposition 1 (Pimsner–Voiculescu). a(n−1) ⊕ U a(n) ⊕ U a(n−2) − U a(n−1) lim ρn U n→∞

An

= 0,

a = 1, 2.

Proposition 1 can be proven similarly to Proposition 3 below, and will therefore be omitted. It implies that there exist unitary operators Ua ∈ A∞ , a = 1, 2, which are not themselves coherent sequences, but which can be written as a limit of such a sequence with respect to the operator norm of A∞ . Because of (2.17), (2.18) and (2.30), the operators Ua so defined satisfy (2.1) and therefore generate the subalgebra Aθ ⊂ A∞ . Thus, there exists a unital injective ∗-morphism ρ : Aθ → A∞ .3 This also means that at sufficiently large level n in the AF-algebra A∞ , the generators of the algebra (2.30) may be well approximated by the images under the injection ρn of the corresponding matrices generating Aθn−1 . It is in this sense that the elements of the algebra Aθ may be approximated by sufficiently large finite dimensional matrices. In what follows we shall show how to use this approximation to describe aspects of field theories over the noncommutative torus T2θ . An important consequence of these results is the fact that Morita equivalent noncommutative tori can be embedded in the same AF-algebra A∞ . From (2.25) and (2.26) we know that K0 (A∞ ) = Z + θ Z as an ordered group. On the other hand, it is known 3 The canonical representation of A is on the Hilbert space L2 (T2 ), which by Fourier expansion coincides θ with &2 (Z2 ).

190


[23]

a b that Z + θ Z and Z + θ Z are order isomorphic if and only if there is an element c d ∈ GL(2, Z) such that

θ =

aθ + b . cθ + d

(2.31)

From the point of view of continued fraction expansions, if θ = [c0 , c1 , c2 , . . . ] and θ = [c0 , c1 , c2 , . . . ], the relation (2.31) is the statement that the two expansions have the same tails, i.e. that cn = cn+m for some integer m and for n sufficiently large [22]. But (2.31) is just the Morita equivalence relation between Aθ and Aθ [24]. Thus, on the one hand we rediscover the known fact that Morita equivalent tori have the same K0 group,4 but we can also infer that Morita equivalent algebras can be embedded in the same (up to isomorphism) AF-algebra A∞ . Morita equivalent algebras can be embedded in the same A∞ because their sequences of embeddings are the same up to a finite number of terms. In Sect. 4 this will be the key property which allows the construction of projective modules within the same approximation, and the physical consequences will be that dual noncommutative Yang–Mills theories all lie within the same AF-algebra A∞ . Let us now describe the infinite dimensional Hilbert space H∞ on which A∞ is represented as (bounded) operators. It is similarly defined by an inductive limit determined by the Bratteli diagram of Fig. 1. For any integer n, consider the finite dimensional Hilbert space Hn = Cqn ⊕ Cqn−1

(2.32) ρñ

on which the algebra An in (2.21) naturally acts. Next, consider the embeddings Hn−1 → Hn defined by 

√ v 1+cn

    ρñ v  −→  w    

where v =

qn

j =1 v

j e (n) j

∈ Cqn and w =

.. .

√ v 1+cn

 

 cn

w √ v 1+cn

qn−1 j =1

H∞ =

  

∞

(n−1)

w j ej

     ,    

(2.33)

∈ Cqn−1 . Then

Hn .

(2.34)

n=0

The normalization factors (1 + cn )−1/2 in (2.33) are inserted so that the linear transformations ρñ are isometries, # " # " = v ⊕ w, v ⊕ w . (2.35) ρñ (v ⊕ w), ρñ (v ⊕ w ) Hn

4 It is a general fact that Morita equivalent algebras have the same K-theory.

Hn−1


191

This ensures that the vectors of H∞ , which are built from the coherent sequences of n Hn , are indeed convergent. Note that the elements of a coherent sequence are related inductively at each level by v n ⊕ w n = ρñ (v n−1 ⊕ w n−1 ) for n sufficiently large, or v n−1 v n−1 ⊕ ··· ⊕ √ ⊕w n−1 , vn = √ 1 + cn 1 + cn !

v n−1 wn = √ . 1 + cn

(2.36)

cn times

The inner product in H∞ is given by # " # " (ψn )n∈Z+ , (ψm )m∈Z+ = lim ψn , ψn n→∞

Hn

.

(2.37)

In the same spirit by which we think of elements of A∞ as infinite matrices, we also visualize elements of H∞ as square summable complex sequences and write H∞ ∼ = &2 Z+ .

2.3. Embedding the noncommutative torus in an AF-algebra: Rational case. Everything we have said in the previous subsection is true for irrational θ , but in many instances one is still interested in the case of rational deformation parameters. Even though Morita equivalence implies that the algebra Aθ is then equivalent in a certain sense to the algebra of functions on the ordinary torus T2 , the physical theories built on the two algebras can have different characteristics (analogously to the case of T-duality between different brane worldvolume field theories). Indeed, physical correlation functions should not have a discontinuous behaviour between rational and irrational deformation parameters. Furthermore, as shown in [25], the noncommutative Yang–Mills description is the physically significant one in the infrared regime as a local field theory of the light degrees of freedom, even though this theory is equivalent by duality to ordinary Yang–Mills theory. When θ is rational one can repeat, to some extent, the constructions of the previous subsection, but one needs to excercise some care due to the occurence of continued fraction expansions which are not simple, i.e. some cn ’s in the expansion vanish. In this case, although the second equality in (2.18) does not make sense if cn = 0, one can nonetheless define the n-convergent θn by the first equality in (2.18), i.e. θn = pn /qn , with pn and qn defined recursively by the formulae (2.20) (recall that qn > 0 always). Thus, we let θ = p/q with p, q relatively prime. The simple continued fraction expansion of θ, which is unique, will terminate at some level n0 , so that

p (2.38) θ = = c0 , c1 , . . . , cn0 . q However, we may still approximate θ by an infinite but not simple continued fraction expansion in the following manner. First, above the level n0 , we take all even c’s to vanish, cn0 +2n = 0,

n ≥ 0.

(2.39)

Consequently, from (2.20) we get pn0 +2n = p,

qn0 +2n = q;

n≥0

(2.40)

192


so that θn0 +2n =

p , q

n ≥ 0.

(2.41)

As for the odd c’s (above the level n0 ), we shall not specify cn0 +1 at the moment, while we take cn0 +2n+1 = 1,

n > 0.

(2.42)

From (2.20) we get pn0 +2n+1 = np + pn0 +1 ,

qn0 +2n+1 = nq + qn0 +1 ;

n≥0

(2.43)

so that θn0 +2n+1 =

np + pn0 +1 nq + qn0 +1

p . q

n→∞

−→

(2.44)

Thus, we can write the rational number p/q as the infinite but not simple continued fraction expansion

p (2.45) = c0 , c1 , . . . , cn0 , cn0 +1 , 0, 1, 0, 1, . . . . q If necessary, we shall use the arbitrariness in cn0 +1 to fix pn0 +1 and qn0 +1 in such a way that pn0 +2n+1 and qn0 +2n+1 are relatively prime integers. In this way we obtain infinite, strictly increasing sequences of relatively prime integers qn0 +2n+1 and |pn0 +2n+1 |, and the constructions and proofs of the previous subsection can be adapted to the present situation. We are now ready to construct the AF-algebra A∞ in which to embed the noncommutative torus with rational deformation parameter. Note that, generally, the isomorphism class of an AF-algebra is completely characterized by the infinite tail of its Bratteli diagram, which for the present case is depicted in Fig. 2a. A comparison with Fig. 1 for the irrational case shows that the algebra for rational θ is of the same kind, with the additional rule that for vanishing c’s in the fractional expansion there is no link in the Bratteli diagram. From Fig. 2a we see that by going from an odd level to the next even one, one simply exchanges the factors in the decomposition, and thus it is better to ‘glue’ an odd level to the next even one. This produces the Bratteli diagram in Fig. 2b, which we stress describes the very same AF-algebra A∞ . There we have defined qñ = qn0 +2n+1 ,

n ≥ 0.

(2.46)

The finite dimensional algebras at level n are then Bn = Mqñ (C) ⊕ Mq (C)

(2.47)

ρn

with embeddings Bn−1 → Bn given by

M

N

ρn



−→ 

M

 N

N

,

(2.48)

where M and N are qñ−1 × qñ−1 and q × q matrices, respectively. The norm closure of the inductive limit (2.47,2.48) is the desired AF-algebra A∞ . Note that, aside from


193

.. .

s ❅

qn0 +2n+1

s q ❅

.. .

❅ ❅

q

❅ ❅s qn0 +2n+1

s ❅

qñ

s

s q

qñ+1

s

s q

❅

❅ ❅

❅ ❅s q

s

qn0 +2n+3

.. .

(a)

.. .

(b)

Fig. 2a,b. Equivalent Brattelli diagrams for the algebra A∞ in the case of rational θ . The labels of the vertices denote the dimensions of the corresponding matrix algebras. All partial embeddings are equal to unity

the fact that it contributes to the increase of dimension in the first factor of Bn , the constant part Mq (C) is required at each level for K-theoretic reasons. The positive maps ϕn : Z2 → Z2 associated with the embeddings (2.48) are now given by qñ−1 qñ 1 1 . (2.49) = ϕn , ϕn ≡ ϕ = 0 1 q q As a consequence one finds K0 (A∞ ) = Z ⊕ Z

(2.50)

with the cone of non-negative elements, which defines the ordering, given by K0+ (A∞ ) =

∞

ϕ −r Z+ ⊕ Z+

r=1

= (a, b) ∈ Z b > 0 ∪ (a, 0) ∈ Z2 a ≥ 0 .

(2.51)

2

In analogy with (2.46) we also define p˜ n = pn0 +2n+1 ,

n≥0

(2.52)

and θñ = θn0 +2n+1 =

pn0 +2n+1 , qn0 +2n+1

n ≥ 0.

(2.53)

Then, exactly as it happens for the irrational situation, within each finite dimensional matrix algebra Bn there is the subalgebra π(Aθñ ) ⊕ π(Ap/q ) with Aθñ and Ap/q rational noncommutative tori and π the representation in finite dimensional matrices as given in (2.5), (2.6) and (2.28), (2.29), i.e., in terms of clock and shift matrices. In contrast to

194


the irrational case, however, it now

follows from the form of the second factor in the 0 finite dimensional algebras that ρ ⊕ π(A ) = 0qñ ⊕ π(Ap/q ), while it is still n qñ−1 p/q

true that ρn π(Aθñ−1 ⊕ 0q ) = π Aθñ−1 ⊕ 0q . Consequently we have an analogue of Proposition 1 and the statement that the algebra π(Aθñ ) ⊕ π(Ap/q ) can be taken to be a finite dimensional approximation of the algebra Aθ of the noncommutative torus with rational deformation parameter θ = p/q. Finally, the infinite dimensional Hilbert space H∞ on which A∞ is represented is given at level n by the finite dimensional vector space Hn = Cqñ ⊕ Cq

(2.54) ρñ

on which the algebra Bn in (2.47) naturally acts. The embeddings Hn−1 → Hn can be read off from the Bratteli diagram in Fig. 2b and are given by ρñ (v n−1 ⊕ w) = v n ⊕ w,

1

v n = √ v n−1 ⊕ w . 2

(2.55)

3. Approximating Correlation Functions Consider an operator G ∈ Aθ and states ψ , ψ ∈ H∞ . The element G is a particular combination of the generators Ua , a = 1, 2, of the noncommutative torus and the vectors ψ , ψ may be represented by particular coherent sequences {ψn }n∈Z+ , {ψm }m∈Z+ with ψn , ψn ∈ Hn . We are interested in evaluating the correlation function C(θ ) = "ψ , Gψ#,

(3.1)

where, for simplicity, we indicate only the dependence of the correlator on the deformation parameter of the algebra. According to Proposition 1 (and its counterpart for the rational case), there is a corresponding sequence of operators Gn ∈ π(Aθn ) ⊕ π(Aθn−1 ), a(n) ⊕ U a(n−1) everywhere, which approximate G obtained by replacing the Ua ’s by U in the sense that limn Gn − G = 0. Using this sequence we can also consider the correlation functions Cn (θn ) = "ψn , Gn ψn #Hn .

(3.2)

We wish to show that the correlators (3.2) for sufficiently large n give a “good” approximation to the correlation function (3.1), i.e., C(θ ) = limn Cn (θn ). This will be true if, as one moves from one level to the next in the coherent sequence, the corresponding expectation values of the operator Gn+1 are approximated by the functions (3.2). This property will follow immediately from the following Proposition 2. Given any two sequences of vectors ψn−1 , ψn−1 ∈ Hn−1 , define

" # a(n−1) ⊕ U a(n−2) ψn−1 Ua(n) ≡ ψn−1 , U Hn−1 " # a(n) ⊕ U a(n−1) ◦ ρñ (ψn−1 ) − ρñ (ψn−1 ), U

Hn

for a = 1, 2. Then

lim Ua(n) = 0.

n→∞

(3.3)


195

Proof. We will give the proof for the case of irrational θ . The proof for the rational case is a straightforward modification of the normalizations of the immersions. Let , with v qn−1 and ψn−1 = v n−1 ⊕ wn−1 and ψn−1 = v n−1 ⊕ wn−1 n−1 , v n−1 ∈ C q n−2 wn−1 , wn−1 ∈ C . The quantity (3.3) for a = 1 can be calculated to be (n)

U1

qn−2

=

j =1

j j w n−1 wn−1 e2πiθn−2 (j −1) − e2πiθn (j −1+cn qn−1 )

cn −1 q n−1 1 j j + v n−1 vn−1 e2πiθn−1 (j −1) − e2πiθn (j −1+kqn−1 ) . 1 + cn

(3.4)

k=0 j =1

In the first sum in (3.4), we add and subtract e2πiθn−1 (j −1) to each of the differences of exponentials there. From (2.17) it follows that the differences 2πiθn−2 (j −1) − e2πiθn−1 (j −1) e

(3.5)

each vanish in the limit n → ∞. For the remaining differences 2πiθn−1 (j −1) − e2πiθn (j −1+cn qn−1 ) , e

(3.6)

we use the inequality [15] 2πiθn−1 l − e2πiθn (l+mqn−1 ) = e2πiθn−1 (l+mqn−1 ) − e2πiθn (l+mqn−1 ) e 2π ≤ 2π qn θn−1 − θn = qn−1

(3.7)

which holds for every pair of integers l, m with |l+mqn−1 | ≤ qn . From (2.20) it therefore follows that qn−2 j j w n−1 wn−1 e2πiθn−2 (j −1) − e2πiθn (j −1+cn qn−1 ) j =1 % 2π $ ≤ εn + w n−1 , w n−1 Cqn−2 , qn−1

(3.8)

where εn → 0 and we have assumed that n is sufficiently large. Because the vectors ψn−1 and ψn−1 are Cauchy sequences in H∞ , the sequence of inner products in (3.8) converges. Since qn → ∞, this shows that the first sum in (3.4) vanishes as n → ∞. In a similar way one proves that the second sum in (3.4) vanishes as n → ∞. For a = 2 the expression (3.3) can be written as (n)

U2

q

=

n−1 1 v n−1 vn−1

1 + cn

q

n−2 1 + w n−1 wn−1 −

q

q

n−1 1 + w n−2 v 1 v n−1 wn−1 n−1 n−1 . √ 1 + cn

(3.9)

196

G. Landi, F. Lizzi, R. J. Szabo (n)

Using Eq. (2.36) we may deduce how the U2 (n+1) U2

q

change with n, and we find

q

n−2 1 w n−1 vn−1 v n−1 v 1 = + n−1 n−1 √ 1 + cn (1 + cn+1 ) 1 + cn ' qn−1 1 ( v n−1 vn−1 1 qn−2 1 . −& w n−1 vn−1 + √ 1 + cn (1 + cn )(1 + cn+1 )

(3.10)

(n+m)

By using (3.9), (3.10) and an induction argument, we find that in general U2 can be bounded by the product of a convergent constant Mm , determined by the uniform bounds on the vectors ψn−1 and ψn−1 , and a product of normalization factors (1+cn )−1/2 . Since each cn ≥ 1, we then find 1 m (n+m) (3.11) U2 ≤ Mm √ 2 which establishes the Proposition for a = 2.

& %

Proposition 2 can be generalized straightforwardly to arbitrary powers of the Ua ’s, and also to products U1 U2 by inserting a complete set of states of H∞ in between U1 and U2 . It represents the appropriate limiting procedure that one could use in a numerical simulation of the correlation functions. Namely, one starts with sufficiently large vectors and matrices which approximate a correlation function (3.1) and then iterates the vectors to the next level according to the embedding (2.33) (or (2.55) for the rational case). From this procedure one may in fact estimate the rate of convergence of the approximation to the desired correlator. As a simple example, we have checked numerically the convergence of the quantities " a(n+m) ⊕ U a(n+m−1) ρñ+m ◦ ρñ+m−1 ◦ · · · ◦ ρñ (ψ ), U # ◦ ρñ+m ◦ ρñ+m−1 ◦ · · · ◦ ρñ (ψ) (3.12) Hn+m

for various cases. For the deformation parameter we have taken the Golden Ratio θ = √ 5+1 which is characterized by cn = 1, ∀ n ≥ 0, and which is known to be the 2 slowest converging continued fraction. In this case pn = qn−1 is the n-th element of the Fibonacci sequence. Nevertheless, the convergence of the θn to θ is quite rapid: for n = 15 the accuracy is of one part in 106 and the matrices are of size 610 × 610. Starting with various choices of ψ , ψ and n, the expression (3.12) converges to definite values quite fast in m, with the difference between successive evaluations steadily decreasing. For example, for random vectors ψ and ψ with a starting value n = 5 and for m = 13 immersions, the difference between successive evaluations is less than a part in 103 at the end of the iterations. For other irrational θ ’s the convergence will be faster, and so will be the growth in dimension of the matrices. 4. Approximating Geometries Thus far the approximating schemes we have discussed have been at the level of C ∗ algebras. In the context of noncommutative geometry, this means that all of our equivalences hold only at the level of topology (this is actually the geometrical meaning of


197

Morita equivalence). The algebra Aθ on its own does not specify the geometry of the underlying noncommutative space, and the latter is determined by the specification of a K-cycle [10, 26]. The algebra AM/N is essentially just a matrix algebra, and for it there exists choices of K-cycles corresponding to the deformed torus, the fuzzy two-sphere, and even the fuzzy three-sphere [13]. In this section we will describe how to obtain the K-cycle appropriate to the noncommutative torus T2θ from the embedding of Aθ into the AF-algebra A∞ . In a more physical language, this will tell us how to approximate derivative terms for field theories on the noncommutative torus and also how to approximate gauge theories, as in [9]. As far as large N Matrix theory is concerned, this choice of K-cycle will be just one possible D0-brane parameter space geometry in the noncommutative spacetime. On T2θ , there are natural linear derivations δa defined by δa (Ub ) = 2π i δab Ub ,

a, b = 1, 2.

(4.1)

These derivations can be used to construct the canonical Dirac operator on T2θ , and hence the K-cycle appropriate to the (noncommutative) Riemannian geometry of the two-torus. With the canonical derivations (4.1), a connection ∇a on a vector bundle H over the noncommutative torus may be defined as a Hermitian operator acting on H and satisfying the property [∇a , Ub ] = 2π δab Ub ,

a, b = 1, 2.

(4.2)

Here the bundle H is taken to be a finitely-generated, left projective module over the noncommutative torus and (4.2) is a statement about operators acting on the left on H. Indeed, it is nothing but the usual Leibniz rule. In general, it is not possible to approximate the defining property (4.2) by finite dimensional matrices. It is, however, straightforward to construct an exponentiated version of this constraint in each algebra An . For this, it is convenient to use a different representation for the generators of the algebra (2.30), namely

(n) U = e2πi(j −1)/qn δkj , 1 kj

(4.3) (n) U = δ k, j = 1, . . . , q (mod q ). k,j −pn +1 n n 2 kj

(n)

(n)

(n)

We seek unitary matrices ei∇a ∈ An , (∇a )† = ∇a ,5 which conjugate elements of π(Aθn ) in the sense (n)

(n)

(n)

(n) ei∇a = e2πiδab ra e−i∇a U b

/qn

(n) , U b

a, b = 1, 2,

(4.4)

(n)

where ra are sequences of integers such that (n)

ra = Ra , n→∞ qn lim

a = 1, 2

(4.5)

5 The construction given below, as well those of [15] and in the preceeding sections of this paper, are strictly speaking only true in the continuous category, i.e. at the level of the Lie group of unitary matrices. Once we have the required approximation at hand, however, we may pass to the corresponding Lie algebra of Hermitian matrices and hence to the smooth category wherein the connections lie.

198


are fixed, finite real numbers whose interpretation will be given below. A set of operators obeying the conditions (4.4) is given by (n) ei∇1 = δk−r (n) +1,j , 1 kj (4.6) (n) (n) ei∇2 = e2πi(j −1)r2 /pn qn δkj , k, j = 1, . . . , qn (mod qn ). kj

Note that e

(n) i∇a

∈ / π(Aθn ), and that the matrices (4.6) obey the commutation relation (n)

(n)

(n) (n) r2 /pn qn

ei∇1 ei∇2 = e−2πir1

(n)

(n)

ei∇2 ei∇1 .

(4.7)

We are interested in the behaviour of these matrices as n → ∞. Proposition 3. (n−1) (n−2) (n) (n−1) ⊕ ei∇a − ei∇a ⊕ ei∇a lim ρn ei∇a n→∞

An

= 0,

a = 1, 2.

Proof. Again we will explicitly demonstrate this in the case of irrational θ, the rational case being a straightforward modification. For a = 1 the eigenvalues of the matrix (n−1)

i∇a e

(n−1)

(n−2)

⊕ ·· · ⊕ ei∇a ! ⊕ ei∇a

(n−1)

⊕ ei∇a

(n)

(n−1)

− ei∇a ⊕ ei∇a

(4.8)

cn times

are readily found to be all equal to 0 (for any n). For a = 2, the eigenvalues of (4.8) (n−1)

are of the generic form eicj

/pn−1 qn−1

(n) dj /pn qn

& %

→ 0 as n → ∞.

(n)

− eidj

/pn qn

(n)

(n−1)

, where cj

/pn−1 qn−1 → 0 and

(n−1)

∈ An are norm convergent Proposition 3 implies that the operators ei∇a ⊕ ei∇a to unitary operators ei∇a ∈ A∞ − Aθ . It follows from (4.4) and (4.5) that these operators conjugate elements of the algebra Aθ according to e−i∇a Ub ei∇a = e2πiRa δab Ub ,

a, b = 1, 2.

(4.9)

Iterating (4.9) and continuing to s ∈ R, this property is seen to be the s = 1 limit of the equation e−is∇a Ub eis∇a = e2πisRa δab Ub .

(4.10)

Differentiating (4.10) with respect to s and then setting s = 0 yields [∇a , Ub ] = 2π Ra δab Ub .

(4.11)

From this commutator we infer that the operators ∇a satisfy the appropriate Leibniz rule and therefore define a connection on a bundle over the noncommutative torus T2θ . The matrices (4.6) thereby give a finite dimensional approximation, in the spirit of the present paper, to the connection ∇a . From (4.11) we see that the numbers Ra defined by (4.5) represent the lengths of the two sides of T2 . Moreover, from (4.7) we find that the connection ∇a has constant curvature

2π iR R 1 2 . (4.12) ∇ 1 , ∇2 = θ


199

The objects presented here thereby define connections of the modules H0,1 over the noncommutative torus which have rank |p − qθ | = θ and topological charge q = 1 [11]. Gauge fields may be introduced in the usual way now by constructing functions of (n) elements in the commutants of the algebras generated by Ua and Ua . The more general class of constant curvature modules Hp,q [11] can likewise be constructed using the tensor product decomposition described in [9]. We will omit the details of this somewhat tedious generalization. Notice that at the finite dimensional level, all of the operators we have defined live in the same algebra A∞ . In the inductive limit however, while (n) the Ua go to the algebra of the noncommutative torus, the unitary operators giving the connection ∇a go to a Morita equivalent one. Thus in the large N limit here we reproduce the known fact [11] that the endomorphism algebra of Hp,q is a noncommutative torus which is Morita equivalent to the original one. The reason for this correct reproduction of gauge theories in the limit is K-theoretic and was discussed in Sect. 2.

5. Conclusions The constructions presented in this paper show that it is indeed possible to represent both geometrical and physical quantities defined over the noncommutative torus as a certain limit of finite dimensional matrices. These results give a systematic and definitive way to realize the spectral geometry, and also the noncommutative gauge theory, of T2θ for any θ ∈ R by an infinite tower of finite dimensional matrix geometries. It should be stressed though that the types of large N limits described in this paper are somewhat different in spirit than those used for brane constructions from matrix models [2, 3, 5], which are rooted in the fuzzy space approximations to function algebras [13]. The present matrix approximations are more suited to the definition of noncommutative Yang–Mills theory in terms of Type IIB superstrings in D-brane backgrounds [8]. It would be interesting to carry out the constructions of string theoretical degrees of freedom in terms of the above decompositions of the noncommutative torus into finite dimensional matrices, and thus test the correspondence between noncommutative gauge theoretic predictions with those of the matrix models. The constructions of this paper also shed some light on the precise meaning of Morita equivalence in such physical models. Although Morita equivalence does imply a certain duality between (noncommutative) Yang–Mills theories, within the matrix approximations there is essentially no distinction between rational and irrational deformation parameters and hence no reason for a model with rational θ to be regarded as completely equivalent to an ordinary (commutative) gauge theory. This is in agreement with the recent hierarchical classification of noncommutative Yang–Mills theories given in [25]. It should always be understood that Morita equivalence is a duality between C ∗ -algebras, and as such it is topological. The equivalence at the level of geometry typically goes away upon the introduction of appropriate K-cycles (as is the usual case for T-duality equivalences as well). On the other hand, we have shown that dual Yang–Mills theories all originate from the same AF-algebra A∞ . We close with some remarks about how these results may be generalized to higher dimensional noncommutative tori and hence to more physically relevant noncommutative Yang–Mills theories. The algebra of functions on a d-dimensional noncommutative torus Tdθ is generated by d unitary operators satisfying the relations Ua Ub = e2πiθab Ub Ua ,

a, b = 1, . . . , d,

(5.1)

200


where θ = [θab ] is an antisymmetric, real-valued d × d matrix. It is always possible to rotate θ into a canonical skew-diagonal form with skew-eigenvalues ϑa ,   0 ϑ1   −ϑ1 0   ..   . ,  (5.2) θ =  0 ϑr     −ϑ 0 r

0d−2r where 2r is the rank of θ . Thus one may embed the algebra of a higher dimensional noncommutative torus into a d-fold tensor product of algebras corresponding to r noncommutative two-tori T2ϑa and an ordinary (d − 2r)-torus Td−2r . This embedding preserves the appropriate K-theory groups K0 (Td ) = Z ⊕ ·· · ⊕ Z! .

(5.3)

2d−1 times

However, the issue of generalizing the constructions of the present paper to higher dimensions in this manner is still a delicate issue. It turns out [27] that for almost all noncommutative tori (precisely, for a set of deformation parameters of Lebesgue measure 1) one may can construct an AF algebra in which to embed the algebra of functions on Tdθ . Acknowledgements. We thank L. Dabrowski, G. Elliott, R. Nest, J. Madore, J. Nishimura, M. Rieffel, M. Sheikh-Jabbari, A. Sitarz and J. Várilly for interesting discussions. This work was supported in part by the Danish Natural Science Research Council.

References 1. 2. 3. 4. 5. 6. 7.

8.

9. 10. 11. 12. 13. 14.

Witten, E.: Nucl. Phys. B 460, 335 (1996)[hep-th/9510135] Banks, T., Fischler, W., Shenker, S.H. and Susskind, L.: Phys. Rev. D 55, 5112 (1997) [hep-th/9610043] Banks, T., Seiberg, N. and Shenker, S.H.: Nucl. Phys. B 490, 91 (1997) [hep-th/9612157] Connes, A., Douglas, M.R. and Schwarz, A.: J. High Energy Phys. 9802, 003 (1998)[hep-th/9711162] Ishibashi, N., Kawai, H., Kitazawa, Y. and Tsuchiya, A.: Nucl. Phys. B 498, 467 (1997) [hep-th/9612115] Seiberg, N. and Witten, E.: J. High Energy Phys. 9909, 032 (1999) [hep-th/9908142] Rajeev, S.G.: Phys. Rev. D 42, 2779 (1990); 44, 1836 (1991); Lee, C.-W.H. and Rajeev, S.G.: Nucl. Phys.B 529, 656 (1998) [hep-th/9712090]; Lizzi, F. and Szabo, R.J.: Chaos Solitons Fractals 10, 445 (1999) [hep-th/9712206]; J. High Energy Phys. Proc. corfu98/073 [hep-th/9904064] Li, M.: Nucl. Phys. B 499, 149 (1997) [hep-th/9612222]; Aoki, H., Ishibashi, N., Iso, S., Kawai, H., Kitazawa, Y. and Tada, T.: Nucl. Phys. B 565, 176 (2000) [hep-th/9908141]; Ishibashi, N., Iso, S., Kawai, H. and Kitazawa, Y.: Nucl. Phys. B 573, 573 (2000) [hep-th/9910004]; Bars, I. and Minic, D.: Phys. Rev. D 62, 105018 (2000) [hep-th/9910091] Ambjørn, J., Makeenko, Y.M., Nishimura, J. and Szabo R.J.: J. High Energy Phys. 9911, 029 (1999) [hep-th/9911041] Connes, A.: Noncommutative Geometry. Academic Press, 1994 Connes, A.: C.R. Acad. Sci. Paris Sér. A 290, 599 (1980); Connes, A. and Rieffel, M.A.: Contemp. Math. 62, 237 (1987) Wegge-Olsen, N.E.: K-Theory and C ∗ -Algebras. Oxford: Oxford Science Publications, 1993 de Wit, B., Hoppe, J. and Nicolai, H.: Nucl. Phys. B 305[FS23], 545 (1988); Madore, J.: An Introduction to Noncommutative Geometry and its Physical Applications, Second Edition. Cambridge: Cambridge University Press, 1999 Rieffel, M.A.: Questions on Quantization. In: Proc. Int. Conf. Operator Algebras and Operator Theory, July 4–9 1997, Shanghai [quant-ph/9712009]


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15. Pimsner, M. and Voiculescu, D.: J. Oper. Theory 4, 201 (1980) 16. Schwarz, A.: Nucl. Phys. B 534, 720 (1998) [hep-th/9805034]; Landi, G., Lizzi, F. and Szabo, R.J.: Commun. Math. Phys. 206, 603 (1999) [hep-th/9806099]; Brace, D., Morariu, B. and Zumino, B.: Nucl. Phys. B 545, 192 (1999) [hep-th/9810099]; Pioline, B. and Schwarz, A.: J. High. Energy Phys. 9908, 021 (1999) [hep-th/9908019] 17. Weyl, H.: The Theory of Groups and Quantum Mechanics. Dover, 1931 18. Fairlie, D.B., Fletcher, P. and Zachos, C.K.: Phys. Lett. B 218, 203 (1989); J. Math. Phys. 31, 1088 (1990); Fairlie, D.B. and Zachos, C.K.: Phys. Lett. B 224, 101 (1989) 19. Sheikh-Jabbari, M.M.: J. High Energy Phys. 9906, 015 (1999) [hep-th/9903107] 20. Bratteli, O.: Trans. Am. Math. Soc. 171, 195 (1972); Effros, E.G.: CBMS Reg. Conf. Ser. Math., no. 46 (Am. Math. Soc., 1981) 21. Elliott, G.A.: J. Algebra 38, 29 (1976) 22. Hardy, G.H. and Wright, E.M.: An Introduction to the Theory of Numbers. Oxford, 1954 23. Effros, E.G. and Shen, C.L.: Indiana J. Math. 29, 191 (1980) 24. Rieffel, M.A.: Pacific J. Math. 93, 415 (1981); Contemp. Math. 105, 191 (1990) 25. Hashimoto, A. and Itzhaki, N.: J. High Energy Phys. 9912, 007 (1999) [hep-th/9911057] 26. Landi, G.: An Introduction to Noncommutative Spaces and their Geometries. Springer-Verlag, 1997; Várilly, J.C.: An Introduction to Noncommutative Geometry, Lectures at the EMS Summer School on Noncommutative Geometry and Applications, September 1997, Portugal [physics/9709045]; Fröhlich, J., Grandjean, O. and Recknagel, A.: In: Quantum Symmetries, Connes, A., Gaw¸edzki, K. and Zinn-Justin, J. (eds.). Les Houches Session 64. Amsterdam: Elsevier, p. 221 [hep-th/9706132] 27. Boca, F.P.: J. Reine Angew. Math. 492, 179 (1997) Communicated by R. H. Dijkgraaf

Commun. Math. Phys. 217, 203 – 228 (2001)

Communications in



The Formulae of Kontsevich and Verlinde from the Perspective of the Drinfeld Double C. Klimˇcík Institute de Mathématiques de Luminy, 163, Avenue de Luminy, 13288 Marseille, France Received: 10 December 1999 / Accepted: 8 October 2000

Abstract: A two dimensional gauge theory is canonically associated to every Drinfeld double. For particular doubles, the theory turns out to be e.g. the ordinary Yang–Mills theory, the G/G gauged WZNW model or the Poisson σ -model that underlies the Kontsevich quantization formula. We calculate the arbitrary genus partition function of the latter. The result is the q-deformation of the ordinary Yang–Mills partition function in the sense that the series over the weights is replaced by the same series over the qweights. For q equal to a root of unity the series acquires the affine Weyl symmetry and its truncation to the alcove coincides with the Verlinde formula. 1. Introduction The original motivation of this article was to elucidate a relation between theYang–Mills theory in two dimensions and the G/G gauged WZNW model. It is known [1,2] that the latter can be understood as a sort of a nonlinear deformation of the former. The first main result of this work shows that it exists a whole moduli space of two dimensional gauge theories which contains the both theories mentioned above as special points. Throughout this paper, G will denote a simple compact connected and simply connected Lie group. We shall argue that the moduli space of Yang–Mills-like theories in two dimensions based on the group G coincides with the space of doubles D(G) of G and we shall refer to the points in this moduli space as to Poisson–Lie Yang–Mills theories. We define a double D(G) of an n-dimensional real Lie group G to be any 2ndimensional real Lie group D (containing G as its subgroup) such that its Lie algebra D is equipped with an symmetric invariant non-degenerate R-bilinear form . , . with respect to which the Lie algebra G of G is isotropic (i.e. G, G = 0). Our second main result is the observation that the Poisson σ -models, corresponding to the Poisson–Lie structures on group manifolds, are special points in our moduli space. More precisely, they are the Poisson–Lie Yang–Mills theories with vanishing coupling constant. A particular example of the Poisson σ -model is the BF theory which indeed

204

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can be obtained from the ordinary Yang–Mills theory by setting to zero the coupling constant of the latter. The Poisson σ -models were introduced by Ikeda [3] and Schaller and Strobl [4] for manifolds without boundary. Their actions read S=

1 (Ci ∧ dX i + α ij (X)Ci ∧ Cj ). 2

(1)

Here Xi is the set of coordinates on the manifold M viewed as functions on the worldsheet , α ij (X) denotes a bivector field (i.e. a section of ∧2 T M) which defines the Poisson structure on M and Ci is a set of 1-form fields on which can be interpreted as sections of the bundle X ∗ (T ∗ M) ⊗ T ∗ . Recently, Cattaneo and Felder [5] have shown that certain correlators of the Poisson σ -models, corresponding to insertions at the boundary of the disc, are computed by the Kontsevich formula [6]. This is in the sense of the perturbation expansion in the field theoretic Planck constant h. ¯ In our picture, this Planck constant h¯ turns out to be the parameter which multiplies the Poisson–Lie bracket on G and therefore it can be interpreted as the Planck constant also from this point of view. The third main result of this paper is the computation of an arbitrary genus partition function of the Poisson–Lie Yang–Mills theory corresponding to the Lu–Weinstein– Soibelman (LWS) Drinfeld double D(G). This model possesses the gauge symmetry based on the group G. For the special case of the vanishing coupling constant, we thus obtain the partition function of the Poisson σ -model for the LWS Poisson–Lie structure ˜ If we set q = e2π h¯ B(ψ,ψ) , where B(. , .) is the Killing–Cartan form on the dual group G. on G and ψ is the longest root, this partition function Z(q) has an interesting behaviour in the complex plane q. In fact, for q = 1 it gives the ordinary BF partition function and for q = 1 but equal to a root of unity it gives the standard Verlinde formula [7]. It therefore appears natural to refer to the partition function of the Poisson σ -model for an arbitrary double D(G) as to a generalized D(G) Verlinde formula. All those results suggest that the Kontsevich and the Verlinde formulae are in fact cousins; a general correlator of the Poisson σ -model with bulk and boundary insertions, on arbitraty genus and for arbitrary double D(G) is the object that appears to generalize both of them. In Sect. 2, we shall define the Poisson–Lie Yang–Mills theory as the gauge theory canonically associated to every double D(G) and we shall indicate the doubles which give respectively the ordinary Yang–Mills theory and the G/G gauged WZNW model. In Sect. 3, we review the definition of the LWS double, we identify its Poisson–Lie Yang–Mills theory and show how the Poisson σ -model (1) emerges if the coupling constant vanishes. We calculate the corresponding partition function in Sect. 4. We use an appropriate generalization of the method of Blau and Thompson worked out for the ordinary Yang–Mills theory and for the G/G gauged WZNW model in [2, 8]. We shall finish with a short outlook.

2. The Poisson–Lie Yang–Mills Theory The Poisson–LieYang–Mills theory, that we shall associate to every double of a Lie group G, is simply obtained by an isotropic gauging of the WZNW model on the double. Its

Formulae of Kontsevich and Verlinde from Drinfeld Double Perspective

205

action reads S(l, η10 , η01 ) 1 −1 10 01 −1 01 −1 10 [∂+ ll , η − η , l ∂− l − lη , l , η ] + εi ωOi (l), = I (l) + 2π (2) where l is a map from the world-sheet into the double D, and η10 and η01 are respectively (1, 0) and (0, 1) forms on the world-sheet with values in the isotropic subalgebra G of the Lie algebra D of D. εi are coupling constants, ω is a volume form on the world-sheet and Oi (l) are functions on the group manifold D which are separately invariant with respect to the left and right action of G on D. ∂− and ∂+ denote the Minkowski version of the Dolbeault coboundary operators, in particular, acting on functions they are given by the standard light-cone derivatives ∂± = ∂τ ± ∂σ .

(3)

Note that the (2) is written entirely in the language of the differential forms, though we have suppressed the symbol of the wedge product. The pure WZNW model action is given in the standard way 1 1 −1 −1 I (l) = (4) ∂+ ll , ∂− ll + d −1 dll −1 , [dll −1 , dll −1 ]. 4π 24π Of course, it is obvious how to make sense of (4) also when D is not a matrix group. In this section and in the following one the Poisson–Lie Yang–Mills theory (2) will be considered only at the classical level; the world-sheet will be always a cylinder equipped with the Minkowski metric. This means, that the variation problem with the fixed boundary conditions at the initial and final time is well-defined (see [9] for more details about the WZNW model on the cylinder). When we fix a gauge, we shall always tacitly assume that those initial and final boundary conditions are compatible with this fixing. The model (2) is gauge invariant with respect to two mutually commuting gauge symmetries: l → glk −1 ,

η10 → gη10 g −1 − ∂− gg −1 ,

η01 → kη01 k −1 − ∂− kk −1 ,

(5)

where g, k ∈ G are mappings from the world-sheet into the maximally isotropic subgroup G. The crucial property of the action (4), which is needed for verifying the gauge invariance (5), is the validity of the Polyakov–Wiegmann formula [10]: 1 I (l1 l2 ) = I (l1 ) + I (l2 ) + ∂+ l2 l2−1 , l1−1 ∂− l1 . (6) 2π Let us consider a compact group G and for its double D(G) we take its cotangent bundle T ∗ G. This bundle is of course trivializable, hence we can represent every point in its total space as a pair (g, X), where g ∈ G and X ∈ G ∗ . The group law is then (g1 , X1 )(g2 , X2 ) = (g1 g2 , Coadg1 X2 + X1 )

(7)

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and the Lie algebra of T ∗ G is the semidirect sum of G and G ∗ , where G acts on G ∗ in the coadjoint way. Finally, the invariant nondegenerate bilinear form . , . on D is given by (α, X), (β, Y ) ≡ Y (α) + X(β),

α, β ∈ G,

X, Y ∈ G ∗ .

(8)

If we now partially fix the gauge by setting l = (1, X), we obtain from (2) the following theory 1 X(dη + η ∧ η) + εi ωOi (X), (9) S=− 2π where η = η10 + η01 . If we set all εi , but ε1 , to zero and choose O1 (X) = B(X, X)

(10)

(we denote by the same symbol the dual of the Killing–Cartan form B(. , .)), we find that (9) is nothing but the action of the standard two dimensional Yang–Mills theory. Moreover, if ε1 vanishes, we obtain the BF theory. In both cases, the theory possesses the standard gauge symmetry with respect to the group G and in our picture it is just the residual gauge symmetry (5) that respects the gauge condition l = (1, X). It is given by X → Coadk X,

η10 → kη10 k −1 − ∂− kk −1 ,

η01 → kη01 k −1 − ∂+ kk −1 .

(11)

An important example is given by another double of the same group G. We simply take D(G) = G × G and the invariant form . , . on its Lie algebra D ≡ G ⊕ G is given by (α1 , α2 ), (β1 , β2 ) ≡ B(α1 , β1 ) − B(α2 , β2 ),

(12)

where B(. , .) is the standard Killing–Cartan form on G. Clearly, the diagonal embedding of G into G × G is isotropic, i.e. (α, α), (β, β) = 0.

(13)

Now we fix the gauge l = (g, 1) and evaluate the action of our Poisson–Lie Yang–Mills theory (2). We obtain 1 B(∂+ gg −1 , η10 ) − B(η01 , g −1 ∂− g) S = IB (g) + 2π (14) − B(gη01 g −1 , η10 ) + B(η01 , η10 ) , where IB (g) =

1 4π

B(∂+ gg −1 , ∂− gg −1 ) +

1 24π

d −1 B(dgg −1 , [dgg −1 , dgg −1 ]) (15)

and we have set all coupling constants εi to zero. Needless to say, the model thus obtained is the gauged G/G WZNW model with respect to the form B(. , .). The residual gauge symmetry (5) which preserves the gauge l = (g, 1) is now g → kgk −1 ,

η10 → kη10 k −1 − ∂− kk −1 ,

η01 → kη01 k −1 − ∂+ kk −1 .

Of course, this is the standard gauge symmetry of the G/G WZNW model.

(16)


207

3. Lu–Weinstein–Soibelman Doubles The previous two choices of the doubles of G have lead to the well-known gauge theories in two dimensions. We shall now consider another important choice of D(G), where the double of the simple, compact, connected and simply connected group G is the so called Lu–Weinstein–Soibelman Drinfeld double DLW S (G) [11]. We shall see that this choice will lead to the theories known as the Poisson-σ -models [3–5]. Recall, that a general Drinfeld double [12] is a 2n-dimensional real Lie group D whose Lie algebra D is equipped with an symmetric invariant non-degenerate R-bilinear form ˜ is required, such . , .. Moreover, an existence of two n-dimensional subgroups G and G that their Lie algebras G and G˜ are isotropic with respect to . , . and D is the direct sum ˜ of the vector spaces G and G. Clearly, any Drinfeld double D containing G as one of its isotropic subgroups is the double D(G) in the sense described in the introduction. The converse need not be true, for we may have a double of G which is not the Drinfeld double. Inspite of this fact, we refer to the points in our moduli space as to the Poisson–Lie Yang–Mills theories. The reason is the following: if the double is indeed the Drinfeld double then Poisson– Lie brackets are simultaneously induced respectively on the group manifolds G and ˜ In particular, to every such Poisson–Lie structure we associate the corresponding G. deformation of the ordinary Yang–Mills theory. The LWS double is simply the complexification (viewed as the real group) GC of G. So, for example, the LWS double of SU (2) is SL(2, C). The invariant non-degenerate form . , .h¯ on the Lie algebra Dlws of DLWS is given by x, yh¯ =

1 ImB(x, y), h¯

(17)

or, in other words, it is just the imaginary part of the Killing–Cartan form divided by a real parameter h. ¯ Since G is the real form of GC , clearly the imaginary part of B(x, y) vanishes if x, y ∈ G. Hence, G is indeed isotropically embedded in GC . Note the presence of the parameter h¯ which indicates that we have actually in mind a oneparameter family of doubles. It turns out that GC is in fact the Drinfeld double, because GC is at the same time ˜ which coincides with the so the double of its another dim G-dimensional subgroup G called AN group in the Iwasawa decomposition of GC : GC = GAN.

(18)

For the groups SL(n, C) the group AN can be identified with upper triangular matrices of determinant 1 and with positive real numbers on the diagonal. In general, the elements of AN can be uniquely represented by means of the exponential map as follows g˜ = eφ exp[ α>0 vα Eα ] ≡ eφ n.

(19)

Here α’s denote the roots of GC , vα are complex numbers and φ is an Hermitian element1 of the Cartan subalgebra of G C . Loosely said, A is the “noncompact part” of the complex ˜ = AN follows from (19); maximal torus of GC . The isotropy of the Lie algebra G˜ of G 1 Recall that the Hermitian element of any complex simple Lie algebra G C is an eigenvector of the involution which defines the compact real form G; the corresponding eigenvalue is (−1) . The anti-Hermitian elements that span the compact real form are eigenvectors of the same involution with the eigenvalue equal to 1. For elements of sl(n, C) Lie algebra, the Hermitian element is indeed a Hermitian matrix in the standard sense.

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the fact that G and G˜ generate together the Lie algebra D of the whole double is evident from (18). In general, a Poisson bracket α on a manifold M is a smooth section of the bivector bundle on M with vanishing Schouten bracket [α, α]S = 0.

(20)

Moreover, the Poisson–Lie bracket on a group manifold G has to be compatible with the group multiplication, i.e. {F1 , F2 }G×G = {F1 , F2 }G .

(21)

Here F (g1 , g2 ) = F (g1 g2 ) is the standard coproduct on the algebra of functions on the group manifold and {. , .}G×G is the product Poisson structure on G × G: {F1 (x)G1 (y), F2 (x)G2 (y)}G×G = {F1 (x), F2 (x)}G G1 (y)G2 (y) + F1 (x)F2 (x){G1 (y), G2 (y)}G ,

(22)

where x and y are coordinates on the first and second copy of G respectively. Of course, we have by definition {F1 , F2 }G ≡ α(dF1 , dF2 ).

(23)

Since the bivector bundle on the group manifold is trivializable by the left invariant vector fields, we loose no information about the Poisson–Lie structure α if we trade it for another object, namely a map ; : G → ∧2 G defined as follows ;(g) ≡ ;ij (g) T i ⊗ T j ≡ Lg −1 ∗ αg ,

(24)

where T i is some basis of G, αg is the value of the Poisson bivector α at the point g of the group manifold and Lg −1 ∗ is the push-forward map with respect to the left translation by the element g −1 . The conditions (20) and (21) for the Poisson–Lie structure α translate under (24) to the following conditions for ;(g): ;ij (g) = −;j i (g), 1 1 ;kl (∇ k ;ij + f kmi ;mj − f kmj ;mi ) + cycl(l, i, j ) = 0 2 2

(25)

;(gh) = ;(h) + Adh−1 ;(g).

(27)

(26)

and

Here f km j are the structure constants of G defined as [T k , T m ] = f kmj T j

(28)

and ∇ k is a differential operator acting on functions on G as follows ∇ k F (g) ≡

d k F (getT )|t=0 . dt

(29)

Note that the condition (27) simply says that ;(g) is a 1-cocycle in the group cohomology of G with values in ∧2 G.


209

Now let us introduce an h-dependent family of the Poisson–Lie brackets on the ¯ ˜ = AN , which are called the LWS Poisson–Lie structures. group manifolds G and G They are completely determined by the adjoint representation of GC . To describe them, ˜ The duality means it is convenient to introduce a basis T i in G and its dual basis T˜i in G. the following relation j

T˜i , T j h¯ = δi ,

(30)

[T˜i , T˜j ] = f˜ij k T˜k .

(31)

moreover we have

A convenient choice of T i ’s and of T˜i ’s is given, respectively, by the set (Eα + E−α ), i(Eα −E−α ), iHi and its dual −i hE ¯ α , hE ¯ α , hH ¯ i . Here Hi is an (Hermitian) orthonormal basis of the Cartan subalgebra T with respect to the Killing–Cartan form and Eα , E−α are eigenvectors of T corresponding to roots α. Of course, any other basis performs equally well. In fact, we could choose also a basis independent description. It seems to us, however, that in these particular cirmumstances the work with some chosen basis will positively influence the clarity of the exposition. Now for each h, ¯ define the following matrices (cf. [13]) j ˜ = g˜ −1 T˜i g, ˜ T j h¯ , A˜ i (g)

˜ B˜ ij (g) ˜ = g˜ −1 T i g, ˜ T j h¯ , g˜ ∈ G

(32)

and Aij (g) = g −1 T i g, T˜j h¯ , Bij (g) = g −1 T˜i g, T˜j h¯ ,

g ∈ G.

(33)

It is then a simple matter to check, that the objects j ˜ g) ˜ ij (g) ˜ T˜i ⊗ T˜j = B˜ ki (g) ˜ A˜ k (g) ˜ T˜i ⊗ T˜j ;( ˜ =;

(34)

;(g) = ;ij (g)T i ⊗ T j = Bki (g)Akj (g)T i ⊗ T j

(35)

and ˜ and G [14]. This means, define respectively the Poisson–Lie structures on the groups G that the conditions (25),(26) and (27) and their dual analogues are verified. The bivectors (34) and (35) are called the LWS Poisson–Lie structures. The existence of the global decomposition (18) enables us to define a natural left ˜ = AN which is called the dressing action ([12, 14, action of the group G on its dual G ˜ as follows 15, 11]). An element g ∈ G acts on g˜ ∈ G g

g˜ = P˜ (gg ˜ −1 ),

(36)

˜ induced by the Iwasawa decomposition (18). It is where P˜ is the map from GC onto G easy to verify that, indeed, (36) defines an action of G, i.e. (g1 g2 )

g˜ =

g1 g2

(

g). ˜

(37)

There is a useful formula which clarifies the relation between the dressing transformation ˜ Indeed, the infinitesimal action of an element ˜ on G. (36) and the Poisson–Lie structure ; β = βi T i ∈ G on a function F (g) ˜ is given by ˜ ˜ ij (g)β δβ F (g) ˜ =; ˜ j ∇˜ i F (g) ˜ ≡ (β Ag˜ )i ∇˜ i F (g). ˜

(38)

210

C. Klimˇcík

The fact that this is really an action, i.e. [δβ , δγ ]F (g) ˜ = −δ[β,γ ] F (g) ˜

(39)

follows from the (dual of the) Jacobi identity (26) and from (the infinitesimal version of) the cocycle condition (27): j ˜ li ˜ ij (g) = f ij − f˜kl i ; ˜ lj (g) ˜ + f˜kl ; (g). ˜ ∇˜ k ; k

(40)

Before proceeding further, let us study a limit h¯ → 0. We fix a basis T i in G. Then it is clear that in the limit h¯ → 0 the commutators of the dual generators T˜i ∈ G˜ tend to zero ˜ becomes an Abelian group isomorphic to G ∗ . In the same sense, the Lie algebra and G C ˜ where G acts on G˜ in the G becomes isomorphic to the semidirect sum of G and G, coadjoint way. The dressing action (36) becomes the standard coadjoint action of G on G ∗ in this limit, the Poisson–Lie structure ;(g) vanishes (clearly, ;(g) is proportional ˜ g) to h). ˜ becomes nothing but the standard linear ¯ Moreover, the Poisson–Lie structure ;( Kirillov Poisson bracket on G ∗ . We conclude, that the limit h¯ → 0 corresponds to the previously considered case of the cotangent bundle T ∗ G as the double D(G). Hence our Poisson–LieYang–Mills theory (2) on the LWS double GC is a 1-parameter deformation of the standard Yang–Mills theory (9). Its action reads 1 S(l, η10 , η01 ) = I (l) + [∂+ ll −1 , η10 h¯ − η01 , l −1 ∂− lh¯ − lη01 l −1 , η10 h¯ ] 2π ε + 2 ω tr(l † l − 1). 2h¯ (41) Note that we have set all but one εi in (2) to zero, and we have chosen canonically the biinvariant term Oi (l) where tr is the trace in the adjoint representation. With an isotropic gauge choice ˜ l = P˜ (l) = g˜ ∈ G,

(42)

the action of the Poisson–Lie Yang–Mills theory (41) for GC becomes 1 S(g, ˜ η10 , η01 ) = ˜ h¯ − η01 , g˜ −1 η10 g ˜ h¯ ] [∂+ g˜ g˜ −1 , η10 h¯ − η01 , g˜ −1 ∂− g 2π (43) ε + 2 ω tr(g˜ † g˜ − 1). 2h¯ Moreover, if we set ε = 0 and define j

i −1 10 i ˜ A10 i T ≡ Ai (g˜ )ηj T ,

i 01 i A01 i T ≡ ηi T .

(44)

then we have (43) as S(g, ˜ A10 , A01 ) =

1 2π

01 ˜ g)(A ˜ A10 h¯ − A01 , g˜ −1 ∂− g ˜ h¯ − ;( ˜ , A10 )], [g˜ −1 ∂+ g,

(45)


211

˜ g) where ;( ˜ is the LWS Poisson–Lie structure (34). Introduce a differential form A, A ≡ A10 + A01 , then we can rewrite (45) as 1 S(g, ˜ A) = − 2π

(46)

1 ∧ ˜ g)(A ˜ h¯ + ;( , A) . A ∧, g˜ −1 d g ˜ 2

(47)

This is precisely the Poisson σ -model (1) (written in the left-invariant frame) for the ˜ Of course, we could introduce some coordinates X i and Poisson–Lie group manifold G. write (47) directly in the form (1). We conclude that for the vanishing coupling constant ε = 0, the Poisson–Lie Yang–Mills theory gives the Poisson σ -model. The gauge fixing (42) is only partial, the residual group of gauge symmetry (5) consists of the dressing gauge transformations by elements k(ξ+ , ξ− ) ∈ G: g˜ → k g, ˜ A

01 −1

→ kA k

(48) − ∂+ kk

−1

,

(49)

˜ Ak ]k −1 − ∂− kk −1 . A10 → k[A10 − ((A10 )Ag˜ + g˜ −1 ∂− g)

(50)

01

˜

Of course, Ak is defined in the dual way to (38), namely for W = W i T˜i ∈ G˜ we have W Ak = (W Ak )i T i ≡ ;ij (k)W j T i .

(51)

In the limit h¯ → 0 the Poisson–Lie structure ;(k) vanishes and (50) becomes the standard gauge transformation law like (11). The reader may convince himself, that the prescription (48)–(50) defines indeed the action of the gauge group on the triple of the fields (g, ˜ A10 , A01 ). She or he may also directly check the gauge invariance of the action (43) or (47) with respect to the gauge transformation (48)–(50). Infinitesimal version of the transformations (48)–(50) is given by (38) and by j k 01 i Aj βk ; jk ˜ l −1 ˜ k −∂− βi − f i A10 j βk − fik [(g˜ ∂− g)

δβ A10 i = −∂+ βi − f δβ A10 i =

(52) ˜ g) + ;( ˜ km A10 m ]βl .

(53)

The transformation (38) is the same as in [5] for the case of the Poisson–Lie groups, but (52) and (53) are different. This is actually an interesting issue. Ikeda[3], Schaller and Strobl [4], and Cattaneo and Felder [5] have remarked a gauge symmetry of the Poisson σ -models that closes only on shell if the Poisson structure is not linearly dependent on the coordinates Xi . The absence of the off shell closure then requires to use the Batalin– Vilkovisky quantization. Our gauge symmetry (38), which together with (49) and (50) closes even off-shell, acts in the same way on the Poisson manifold as the one in [5]. It is due to this fact that we find plausible to conjecture that the Kontsevich formula can be derived by the standard Faddeev–Popov procedure in the special case of the Poisson–Lie structures. As an example, consider the double SL(2, C) of SU (2). We choose the basis of su(2) as Tj =

i j σ , 2

(54)

212

C. Klimˇcík

where σ j are the Pauli matrices, and the dual basis of G˜ as 1 1 1 01 0 −i 1 0 T˜1 = h¯ , T˜2 = h¯ , T˜3 = h¯ . 00 0 0 0 −1 2 2 4 ˜ is as follows The coordinate parametrization of the group G 1 e 4 h¯ X3 0 1 21 h(X ¯ 1 − iX 2 ) . g˜ = 1 3 0 1 0 e− 4 h¯ X

(55)

(56)

With these data, the Poisson–Lie Yang–Mills theory (43) becomes 1 3 S= Ci ∧ dX i i=1 2π

1 1 1 ¯ 1 2 −h¯ X3 + ) + h¯ XX C1 ∧ C2 (1 − e X C2 ∧ C 3 + X C3 ∧ C 1 + h¯ 2π 4

1 ε 8 1 3 ¯X , ¯ 2h + (57) d 2 ξ 2 (cosh h¯ X 3 − 1) + XXe 2 2 h¯ where we have set X ≡ X1 + iX 2 ,

X¯ ≡ X1 − iX 2 .

(58)

The parameter ε is the coupling constant. Clearly, for h¯ → 0 we recover the ordinary SU (2) Yang–Mills theory, because the terms in the square brackets in the second and ¯ in this limit. If the coupling third lines of (57) become respectively X 3 and (X 3 )2 + XX constant ε vanishes, (57) gives the Poisson σ -model which is the h¯ -deformation of the BF theory. ˜ is generated by the following vector The infinitesimal dressing transformation on G fields 1 1 1 1 2 3 ˜ i1 ∇˜ i = v1 = ; (1 − e−h¯ X ) − h¯ Re(XX) ∂X2 − X 2 ∂X3 + hX ¯ X ∂X1 , (59) h¯ 4 2 1 1 1 1 2 ˜ i2 ∇˜ i = − (1 − e−h¯ X3 ) − h¯ Re(XX) ∂X1 + X 1 ∂X3 − hX v2 = ; ¯ X ∂X 2 , h¯ 4 2 (60) 3 i3 2 1 ˜ ∇˜ i = X ∂X1 − X ∂X2 . v =; (61) One can check that those vector fields leave invariant the term

1 8 1 3 ¯ X3 . ¯ 2h hX (cosh − 1) + XXe ¯ 2 h¯ 2 We do not write the gauge transformations (49) and (50) explicitely, because the corresponding formulas are cumbersome and not too illuminating anyway. Their basic ingredients are given, however, by components of the Poisson–Lie bivectors. They read 1 1 ¯ ˜ i )12 = − (1 − e−h¯ X3 ) + h¯ XX, ;(X h¯ 4 ;(u, v)12 = hv ¯ ¯ v,

;(u, v)23 =

˜ i )23 = −X 1 , ;(X

1 h(uv + u¯ v), ¯ ¯ 2

˜ i )31 = −X 2 ; ;(X (62)

;(u, v)31 =

1 i h( ¯ u¯ v¯ − uv). 2

(63)


213

Here an element g of SU (2) is parametrized by two complex coordinates u, v fulfilling uu ¯ + vv ¯ = 1; u −v¯ g= . (64) v u¯ Much of what we said in this section about the LWS doubles remains true in a more general situation. Actually, if the double of a Lie group G is a Drinfeld double D, it ˜ which is also the isotropic subgroup of D. If, follows that it exists the dual group G moreover, every element l ∈ D can be unambiguously represented as ˜ l = k k,

˜ k ∈ G, k˜ ∈ G

(65)

˜ is a diffeomorphism, then the Poisson–Lie structures and the induced map D → G × G ˜ are again given by the expressions (34) and (35). Of course, one on the groups G and G uses the invariant bilinear form that corresponds to the double in question. The Poisson– Lie Yang–Mills theory, with the gauge group based on G, corresponding to the double D and with the vanishing coupling constant, is then again given by the Poisson σ -model ˜ ˜ is the Poisson–Lie structure on G. (47) where ; 4. The Partition Function The partition function of the ordinary Euclidean Yang–Mills theory has been computed by many methods [1, 2, 16–19]. Here we shall calculate this quantity for the LWS deformation of theYang–Mills theory introduced in the previous section. We use an appropriate generalization of the method of Blau and Thompson [2, 8]. 4.1. The Wick rotation. The definition of an Euclidean version of the Poisson–LieYang– Mills theory (2) requires some care. The reason is the chiral gauge symmetry (5). If we naively replace ∂− by ∂z and ∂+ by ∂z¯ we cannot view the elements ∂z gg −1 and ∂z¯ kk −1 as elements of G because they are actually the elements of G C . This suggests that the fields η10 and η01 are also independent elements of G C . This would change, however, the number of degrees of freedom of our theory. We may try to use the standard prescription in gauge theory, namely, take η10 and η01 in G C and declare η01 to be an anti-Hermitian conjugate of η10 . This would balance the correct number of degrees of freedom but the independent chiral gauge transformations (5) of η10 and η01 would not respect such a constraint. The way out of the trouble is a partial gauge fixing. We see in the examples (9) and (14) that we can partially fix the gauge (11) or (16) in such a way that the residual gauge symmetry acts in the same way on η10 and η01 . This makes possible to take η10 as the anti-Hermitian conjugate of η10 and indeed this is the standard way how the ordinary Yang–Mills theory and the gauged G/G model are put on the Riemann surface. Unfortunately, the gauge fixing (42) in the LWS case still leads to the residual gauge symmetry (49) and (50) which acts differently on η10 and η01 (or, rather, on A10 and A01 ). We can consider, however, another gauge fixing which does make possible to define the Euclidean version of the theory2 . It uses the Cartan decomposition [20] of the 2 The reason why we did not consider immediately this new gauge fixing is simple: we wanted to make link to the Poisson σ -model (1) that underlies the Kontsevich formula and this link was explicit in the gauge (42).

214

C. Klimˇcík

group GC which says that every element l ∈ GC can be represented as l = pg,

(66)

where g ∈ G and p ∈ P . This decomposition is unique. Here our notation is standard; if we consider the set of the Hermitian elements of G C (cf. footnote 1), we have P ≡ iG and P = exp P.

(67)

The exponential mapping in (67) is one-to-one. All this makes possible to choose conveniently the new gauge fixing as l = Pˆ (l) = p ∈ P ,

(68)

where the map Pˆ : GC → P is induced by the Cartan decomposition (xy). The residual gauge symmetry (5) in this gauge becomes p → kpk −1 ,

η10 → kη10 k −1 − ∂− kk −1 , η01 → kη01 k −1 − ∂+ kk −1 ,

k ∈ G.

(69)

Now it is straightforward to write the Euclidean version of the Poisson–Lie Yang–Mills theory (41) in the gauge (68): SE (p, η10 , η01 , ε, h) ¯ = IE (p) i ¯ −1 , η10 h¯ − η01 , p−1 ∂ph¯ − pη01 p −1 , η10 h¯ ∂pp + 2π g ε + 2 ω tr(p2 − 1), 2h¯ g where IE (p) =

i 4π +

g

i 24π

(70)

¯ −1 , ∂pp−1 h¯ ∂pp g

d −1 dpp −1 , [dpp−1 , dpp−1 ]h¯ .

(71)

The gauge symmetry is given by the following transformations p → kpk −1 ,

η10 → kη10 k −1 − ∂kk −1 ,

¯ −1 . η01 → kη01 k −1 − ∂kk

(72)

Remark that we use in (70) and (71) the language of differential forms though we do not indicate explicitly the wedge products between the forms. The operators ∂ and ∂¯ are the Dolbeault coboundary operators with respect to the chosen complex structure on the Riemann surface g (g indicates the genus of the surface). The forms η10 and η01 are respectively the (1, 0) and (0, 1) forms in the Dolbeault complex and the form η10 + η01 is in G ⊗ T ∗ g and is interpreted as a connection on the (for the simply connected G necessarily) trivial G bundle over g . In particular, it means that η01 = −(η10 )† .

(73)


215

In other words, η10 is the anti-Hermitian conjugate of η01 , where the operation † is the Hermition conjugation on G C tensored with the complex conjugation on T ∗C g . Note that only the term proportional to the coupling constant ε depends on the measure on the Riemann surface, which itself is normalized as g

ω = 1.

(74)

The reader should avoid a pitfall in understanding the formula (70). It has to do with the fact that the LWS double DLWS of the compact simple connected and simply connected group G is isomorphic to the complexification GC of G. For the purpose of defining the Euclidean version of the Poisson–Lie Yang–Mills theory, we have declared the 1-forms η10 and η01 to be the elements of G C , hence seemingly to be the elements of the Lie algebra of the double. In fact, it is indeed correct to say that in the Euclidean version η10 and η01 are the elements of a complexification of the Lie algebra G, but it is not correct to interpret η10 and η01 as the elements of the Lie algebra Dlws of the double. The solution of this apparent paradox is that two different (though mathematically isomorphic) complexifications of G play role here. In order to disentangle the two different complexifications, let us work from the very beginning with the real group DLWS as if we did not know that it can be identified C with the complexification of G. Then consider the complexified group DLWS and its Lie C C algebra Dlws . Upon the complexification of Dlws to Dlws , the subalgebra G ⊂ Dlws gets complexified to (G C ) . We indicate by that this complexification is not the same as the complexification G C = Dlws . In fact, the forms η10 and η01 are to be understood as the elements of (G C ) in full agreement with the Euclidean treatment of the coset models C (cf. [21]). Of course, the invariant bilinear form . , .h¯ on Dlws gets extended onto Dlws by bilinearity (not sesquilinearity!).

4.2. The measure of the path integral. We now wish to quantize the theory (70). It actually resembles (the gauging of) the WZNW model on the symmetric space P as defined in [21]. The difference is, however, that there the Killing–Cartan form B(. , .) was used while we are using the invariant bilinear form . , .h¯ 3 . The two models have nevertheless some common features. They live both on the symmetric space P which is a contractible manifold diffeomorphic to the Euclidean space Rdim G . This fact means that the d −1 of the WZNW 3-form (based on whatever invariant nondegenerate bilinear form) does exist globally. As the consequence, we do not have to extend the map g → P to a 3-manifold, whose boundary is g , if we want to determine the contribution of the WZNW-term. Hence the level of the WZNW model (which is equal to 1/h¯ in our case) does not get quantized and it can be an arbitrary positive real number. We also note that the gauge symmetry (72) is diagonal and hence it is not anomalous. By the way, also the chiral symmetries (5) of the original Poisson–LieYang–Mills theory (2) are not anomalous since the Lie algebra G is isotropic. We may interpret it by saying that also at the quantum level the theory (70) is the gauge fixed version of the Poisson–Lie Yang–Mills theory (41). 3 For example, our WZNW action I (a) vanishes for a ∈ A which is not the case in [21]. E

216

C. Klimˇcík

The partition function of the model (70) on the genus g Riemann surface is given by the following path integral. 1 Z(ε, h, (DpDη10 Dη01 )g exp −SE (p, η10 , η01 , ε, h), (75) ¯ g) = ¯ Vol(G g ) where the action SE is explicitly written in (70) and Vol(G g ) is the volume of the gauge group. It is natural to expect that the G-invariant measures (Dη10 Dη01 )g and (Dp)g should be based on the bilinear form . , .h¯ that underlies the model (41) or (70). On the other hand, there appears an immediate trouble in using . , .h¯ for defining the measure on the fields η10 and η01 ; indeed, these fields are isotropic with respect to . , .h¯ , hence it is not clear how to build up a non-zero norm on the field space. We can circumvent the trouble by borrowing some inspiration from the Lie group theory. The standard measure on a simple complex group GC is only indirectly defined by the Killing–Cartan form B(. , .) on G C . Actually, people define another G-invariant bilinear form [22] as follows K(X, Y ) = B(X† , Y ),

X, Y ∈ G C ,

(76)

where we remind that † means the Hermitian conjugation in G C . Equipped with the form K(. , .), the Lie algebra G C becomes an Euclidean space. By left transport of this Euclidean form from the origin of the group manifold GC everywhere, the Riemannian metric and, hence, the Riemannian measure on GC is canonically defined. The reason4 for such a construction is simple. With the choice of the positive definite bilinear form K(. , .), a standardly K-normalized measure is defined at the same time also for all Lie subgroups of GC . In our case, there also exists the G-invariant way of turning the bilinear form . , .h¯ into a positive definite bilinear form on G C . For this and also for further purposes it is convenient to fix canonically a real basis of the Lie algebra Dlws : Dlws = SpanR (R α , J α , K j , rα , jα , kj ),

(77)

where i2Hαj 1 −i R α = √ (Eα + E−α ), J α = √ (Eα − E−α ), K j = ; B(Hαj , Hαj ) 2 2 −i h¯ −h¯ rα = √ (Eα + E−α ), jα = √ (Eα − E−α ), kj = hH ¯ λj . 2 2

(78) (79)

Here our conventions and normalizations are the same as in [23]. This means, in particular, B(Eα , E−α ) = −1, Eα† = −E−α , [Eα , E−α ] = −Hα ; [H, Eα ] = α(H )Eα , α(H ) = B(Hα , H ),

(80) (81)

where H is an arbitrary element of the Cartan subalgebra T ,αj ’s are the simple roots and λj ’s the fundamental weights. We recognize in −iK j ’s the simple coroots to which the fundamental weights λj ’s are dual. 4 We are indebted to P. Delorme for this explanation.


217

Note that the (anti-Hermitian) capital generators R α , J α , K j generate the compact real form G of Dlws . The small generators rα , jα , kj do not generate a Lie subalgebra of Dlws but they span the vector subspace in Dlws that coincides with P (cf. the discussion between Eqs. (66), and (67)). The commutation relations of the elements of the basis (xy) give rise to real structure constants thus they define the real Lie algebra Dlws . In this basis, the invariant bilinear form . , .h¯ is given as follows R α , rβ h¯ = δβα ,

J α , jβ h¯ = δβα ,

K i , kj h¯ = δji ,

(82)

and all other inner products vanish. Thus we see that small generators are in some sense dual to the capital ones, but this decomposition does not give rise to a Manin triple because P is not a Lie algebra. Consider an R-linear flip map θ, θ 2 = 1, defined by changing the capital character into the small one in the canonical basis (77) (e.g. θ(R α ) = rα ). Then we define the G-invariant positive definite bilinear form Kh¯ (. , .) as follows Kh¯ (X, Y ) ≡ θ(X), Y h¯ .

(83)

Note the similarity between (76) and (83); θ is the analog of † in (76). A 2n-dimensional Euclidean volume form dDlws on the Lie algebra Dlws , originating from Kh¯ (. , .), is given by (dR α ∧ dJ α )) ∧ ( dK j ) ∧ ( (drα ∧ djα )) ∧ ( dkj ) dDlws = ( α∈K+ α∈K+ j j (84) ≡ dT ⊥ ∧ dT ∧ dA⊥ ∧ dA. Here dR α , dJ α , dK j , drα , djα , dkj is by definition the dual basis (of the dual space of the Lie algebra Dlws ) with respect to the Kh¯ -orthonormal basis R α , J α , K j , rα , jα , kj . We use the symbol Kh¯ , because the volume form on Dlws computed from Kh¯ (. , .) differs the volume form coming from K(. , .) by a factor c(G)h¯ dim G , where c(G) is an h-independent constant. ¯ A measure on the gauge fields η = η10 + η01 coming from Kh¯ (. , .) is defined by an inner product on the tangent space at each point of the connection space η: 1 Kh (δη1 ∧, δη2 ). (85) (δη1 , δη2 ) = 4π g ¯ It is independent on the point η in the connection space and is gauge invariant by virtue of the G-invariance of the form Kh¯ . It can be easily checked that a measure on η defined by K(. , .) differs from our measure (85) by a constant independent on h. ¯ This fact will play an important role in what follows. By using Kh¯ (. , .), we have landed (up to a normalization) on the same volume form as the one standardly used in group theory. It is also clear that our measure on P will differ only by the normalization factor from the standard measure on the symmetric space P [24, 25]. Indeed, let us define this measure in the way useful also for further applications. First of all we define a volume form dDLWS on the 2n-dimensional group manifold DLWS . We do that by the left transport of the Lie algebra volume form dDlws . Thus (dDLWS )l at a point l ∈ DLWS is now defined as (dDLWS )l = L∗l −1 dDlws ,

(86)

218

C. Klimˇcík

where L∗l −1 is the pull-back of the form dDlws (defined in the unit element of the group) by the left translation diffeomorphism. Note that the invariant volume form dDLWS is thus canonically normalized by the bilinear form Kh¯ . The measure dP on the symmetric space P = DLWS /G is the most simply defined in the following way: consider the “projection” map Pˆ : DLWS → P defined in (68).Then the integral of an arbitrary function with compact support f (p) on P is defined by the prescription 1 f (p)dP ≡ (87) (Pˆ ∗ f )(l)dDLWS , Vol(G) where Pˆ ∗ f is the pull-back of the function f by the “projection” map Pˆ and V ol(G) is the volume of the compact group G. Of course, the measure on the subgroup G ⊂ DLWS is also standardly Kh¯ -normalized hence it makes sense to speak about the volume of G. In fact, we can readily write the volume form dG on G, it is given by (dG)g = L∗g −1 [dT ⊥ ∧ dT ].

(88)

We finish this subsection by noting that the measure (Dp)g of the path integral (75) is given by the Riemann surface point-wise product of the measures (87). 4.3. The generalized Weyl integral formula. Our strategy for computing (75) will be similar as in [2, 8]. It means that we shall first Abelianize the theory by finding a generalized version of the Weyl integral formula and then we shall compute the Abelian partition function in the standard way [2]. The non-Abelian origin of the Abelianized theory will be remembered in the determinants produced by the Abelianization procedure. It turns out that the generalization of the Weyl integral formula, which would work in our setting, indeed exists. It is given in [24, p. 186] and, in more general setting and including the normalization, in [25]. This formula is based on another form of the Cartan decomposition which says that any element l of GC can be (non uniquely) written as l = gak −1 ,

g, k ∈ G,

a ∈ A.

(89)

In particular, it follows from the Cartan decomposition (66) that the elements of P can be represented as p = kak −1 ,

k ∈ G,

a ∈ A.

(90)

The ambiguity of this representation of p is clearly parametrized by the elements of the normalizer of A in G; we denote this group as NG (A). Evidently, there is a normal subgroup ZG (A) ⊂ NG (A) containing the elements of G which commute with A. This subgroup is called the centralizer of A in G and in our case it coincides with the maximal torus T of G. From the fact that exp T = T and exp iT = A, we conclude that the quotient group NG (A)/ZG (A) is nothing but the Weyl group of G C . Thus the decomposition (90) is unique if we view k as a class in G/T and a as an element of A+ . Here A+ = exp A+ and A+ is the fundamental domain (=the Weyl chambre) of the action of the Weyl group on A = iT . For this unique parametrization of P , we can infer the generalized Weyl integral formula [24, 25] which holds for the functions satisfying f (p) = f (kpk −1 ),

k ∈ G.

(91)


It reads

P

f (p)dP =

Vol(G) Vol(T )

219

A+

J (a)f (a)dA,

(92)

where J (a) = ;α∈K

1 α |a − a −α |. 2h¯

(93)

With a parametrization j a = exp φ j kj = exp hφ ¯ Hλj ≡ exp hφ, ¯

φ ∈ A+ ,

(94)

we have J (φ) = ;α∈K+

sinh2 (hα(φ)) ¯ . h¯ 2

(95)

The set of all roots of G C is denoted as K, the set of positive roots as K+ . The measure dA is the standard measure on A ⊂ DLWS in the sense discussed above; it is given by the Kh¯ -normalized volume form (dA)a = L∗a −1 dA.

(96)

The volume Vol(G) is computed with respect to the standard measure dG, defined in (88), and Vol(T ) with respect to (dT )t = L∗t −1 dT ,

t ∈ T.

(97)

The formula (95) gives the Jacobian J (φ); for h¯ = 1 it coincides with the Jacobian in [24, 25]. Note that the limit h¯ → 0 makes sense and it produces the Jacobian which arises in the Weyl integral formula for the Lie algebra G [2]. In what follows, we shall use a notation often used in the world of quantum groups; i.e. [x]h¯ =

sinhhx ¯ h¯

(98)

for an arbitrary number x. With this notation, the Jacobian J (φ) becomes the product of the q-numbers: J (φ) = ;α∈K+ [α(φ)]2h¯ .

(99)

Let us now give the proof of (95). It is clear that the integral f (p)dP in the l.h.s. of (92) reduces to some integral over A+ since both the function f (p) and the measure dP are invariant with respect to the conjugation by elements of G (the latter fact follows from the simultaneous left and right G-invariance of dDLWS ). It is not difficult to find the volume form corresponding to this integration. For this, define first a map Aˆ + : P → A+ that associates to every p ∈ P the element a ∈ A+ under the Cartan decomposition (90). Clearly, the function f (p) on P satisfying (91) is the pull-back of some function f˜(a) on A+ by the map Aˆ + . We are looking for a function J (a) such that 1 f˜(a)J (a)dA+ = (Pˆ ∗ Aˆ ∗+ f˜)(l)dDLWS . (100) Vol(G)Vol(G/T )

220

C. Klimˇcík

Here Vol(G/T ) is calculated from the measure on the homogeneous space G/T defined in a similar way as the measure on DLWS /G (cf. (87)). It then follows Vol(G/T ) =

Vol(G) . Vol(T )

(101)

We see from (100) that J (a)(dA+ )a = i{ L Ta⊥ } i{ R Ga } (dDLWS )a ,

(102)

where the multivector { R Ga } is defined as { R Ga } ≡

α∈K+

and the multivector { L Ta⊥ } as { L Ta⊥ } ≡

α∈K+

( R Raα ∧

R α Ja ) ∧ (

( L Raα ∧

L α Ja ).

R

j

Ka )

(103)

j

(104)

Here e.g. R Jaα realizes the right action of the generator J α of G on the group manifold DLWS at the point a and the multivector { L Ta⊥ } corresponds to the left action of T ⊥ on DLWS at the same point a. Clearly, iV ω denotes the insertion of the multivector V into the form ω. Every such a generator, say R Jaα , can be written as R α Ja

= La∗ J α ,

J α ∈ G,

(105)

where La∗ is the push-forward map corresponding to the left transport. In a similar way, we have { L Ta⊥ } = Ra∗ {T ⊥ } = La∗ (Ada −1 {T ⊥ }).

(106)

Thus we immediately arrive at J (a)(dA+ )a = iRa∗ {T ⊥ } iLa∗ {G } L∗a −1 (dT ⊥ ∧ dT ∧ dA⊥ ∧ dA) = L∗a −1 (iAd

a −1 {T

⊥}

(dA⊥ ∧ dA)).

(107)

We calculate sinhh¯ α(φ) ; h¯ sinhh¯ α(φ) Ada −1 J α = J α coshhα(φ) − rα . ¯ h¯

+ jα Ada −1 R α = R α coshhα(φ) ¯

(108) (109)

Inserting (108) and (109) into (107) and taking into account (84) and (107), it follows J (φ) = ;α∈K+

sinh2 (hα(φ)) ¯ = ;α∈K+ [α(φ)]2h¯ . h¯ 2

(110)


221

4.4. The maximal torus bundles. Consider the gauge φ = φ j Hλj

p = a = exp hφ, ¯

φ ∈ A+

(111)

j

and introduce a pair of real ghosts cαr , cα for each positive root α. By taking into account (70), (92), (110) and (111), the formula (75) for the partition functions becomes 1 10 01 r j (DφDη Dη Dc Dc )g exp −ε ω [α(φ)]2h¯ Z(ε, h, ¯ g) = Vol(T g ) α∈K i {dφ j ∧ Aj + exp − ([α(φ)]h¯ Bα ∧ Cα + ω[α(φ)]2h¯ cαr cαj )}. (112) 2π α∈K+

Here η10 = A10 m K m + B 10 α R α + C 10 α J α ; η

01

=A

01

mK

m

+B

01

αR

α

+C

01

αJ

α

;

(113) (114)

10 10 01 01 10 and the component forms A10 m , Bα , Cα and Am , Bα , Cα are pairwise complex con∗C jugated forms in T g . Note that the term IE (p = a) in (70) gives no contribution to (112). This seems to be a trivial fact because A is commutative and isotropic with respect to . , .h¯ . However, the isotropy and commutativity together with the cohomological triviality of the WZNW term explains the vanishing of IE (a). In the case of the G/G gauged WZNW model the WZNW term is not cohomologically trivial and it may and does contribute [2, 27]. This can be easily understood by realizing that in the cohomologically nontrivial situation one has to consider the mappings extended to three-dimensional domain of which g is the boundary. This extension need not respect the (isotropic and/or commutative) gauge choice on g . Our formula (112) almost coincides with the result of the Abelianization of the ordinaryYang–Mills theory (formula (2.58) of [2]). Up to a trivial overall 2π normalization5 , the only difference in the action consists in replacing the “ordinary” numbers α(φ) of Blau and Thompson by our quantum numbers [α(φ)]h¯ . Moreover, the measure of our path integral differs from that of the ordinary Yang–Mills case only by an h-independent ¯ constant. This constant originates from the difference between our measure-defining form Kh¯ (. , .) and the ordinary Yang–Mills measure-defining form K(. , .), where the h-dependent part of this difference is already taken into account in the Jacobian J (φ) or, ¯ in other words, in the ghost part of the action. Moreover, also the 2π renormalization of the measure of the gauge fields play role (cf. our Eq. (85) and Eq. (2.11) of [2]). We shall eventually use a freedom to renormalize our measure-defining bilinear form by all these h-independent constants in such a way that the limit h¯ → 0 in (112) gives a correctly ¯ normalized ordinary Yang–Mills partition function. There is a very important aspect of exploiting the (generalized) Weyl integral formula in the ordinary Yang–Mills case [2] and also in our Poisson–Lie Yang–Mills setting. It has to do with the following fact: the validity of the Cartan decomposition (90) of an 5 Blau and Thompson have chosen the overall normalization to be in accord with the fixed point theorems while we are using the standard WZNW normalization.

222

C. Klimˇcík

arbitrary element p ∈ P does not imply that the smooth mapping p(¯z, z) : G → P can be smoothly decomposed as p(¯z, z) = g(¯z, z)a(¯z, z)g(¯z, z)−1 ,

g(¯z, z) ∈ G,

a(¯z, z) ∈ A+ .

(115)

Of course, the mappings g(¯z, z) and a(¯z, z) do exist but it is by no means guaranteed that they be smooth. This fact has serious implications for the proper meaning of the formula (112). Strictly speaking, we cannot choose the gauge (111) smoothly. Then what do we mean by Eq. (112)? A hard work of Blau and Thompson [8] was needed for solving this problem in the case of the ordinary Yang–Mills theory (and also in the case of the G/G gauged WZNW model). Fortunately, we can fully rely on their results also in the LWS case because P is diffeomorphic to the Lie algebra G of G and this (exponential) diffeomorphism commute with conjugations by the elements of G. In fact, we have P = exp iG.

(116)

This means that we map p(¯z, z) ∈ P into G by taking the inverse of the mapping (116) and then we apply the Blau–Thompson diagonalization i.e. −i ln p(¯z, z) = g(¯z, z)t (¯z, z)g −1 (¯z, z),

t (¯z, z) ∈ C+ ⊂ T .

(117)

Here C+ = −iA+ is the Weyl chamber in T . By multiplying (117) by i and then exponentiating, we arrive at the seeken Cartan decomposition p(¯z, z) = g(¯z, z)a(¯z, z)g −1 (¯z, z),

a(¯z, z) = eit (¯z,z) ∈ A+ .

(118)

The analysis [8] of the diagonalization of the type (117) can be translated into our context along the lines above and it gives the following results: If lnp(¯z, z) is a smooth map from the Riemann surface g into a subset of regular elements of G then 1) The smooth decomposition (115) can always be achieved locally on g . 2) The diagonalized map a(¯z, z) can always be chosen to be smooth globally. 3) Non-trivial T -bundles on g are the obstructions to finding smooth functions g(¯z, z) globally. In particular, if there are no nontrivial principal G-bundles on g (like in our case), all isomorphism classes of torus bundles appear as obstructions. 4) The gauge field path integral should include a sum over the T -connections on all isomorphism classes of T -bundles on g . Actually, the point 4) shows in which sense we should understand the formula (112). It is not so difficult to understand intuitively, what is going on here. If the function g(¯z, z) is not smooth somewhere, then passing from p(¯z, z) to a(¯z, z) is a singular gauge transformation and the g(¯z, z)-transformed connection field η becomes singular. It is well-known that singular connections can be sometimes interpreted as connections on nontrivial bundles (see [2, 26] for examples). The condition that −i ln p(¯z, z) is a regular element of the Lie algebra G may seem inconspicuous but it is in fact crucial for the proper definition of the path integral. By restricting our space of fields p(¯z, z) to those verifying the condition of the regularity, we make a certain choice. We can certainly understand it simply as a part of a plausible definition of the path integral, because, as it was shown in [8] such regular maps with values in the Lie algebra G are generic. In order to corroborate this choice we give two arguments (a more detailed discussion is provided in [2, 8] and it is directly relevant also to our Poisson–Lie Yang–Mills case):


223

First of all, the restriction to the regular maps gives the correct answer for the ordinary Yang–Mills case, which is confirmed by alternative methods of calculation [1, 16–19]. Secondly, the non-regular φ’s are anyway automatically suppressed from the path integral since the Jacobian J (φ), originating from the generalized Weyl integral formula, vanishes for them. Indeed, the Jacobian J (φ) vanishes if and only if −iφ is not a regular element of T . This can be seen directly from the definition of the regularity; an element X of the Cartan subalgebra T is regular iff it satisfies a condition det T ⊥ (ad(X)) = 0 where the notation means that the ad(X) operator is restricted to T ⊥ . Of course, if this ad(X) determinant vanishes then there exists a root α such that α(X) = 0. The latter fact implies the vanishing of the Jacobian J (X). The opposite direction can be also easily proved. Following closely [2], we shall now perform the path integral over the affine space of Abelian connections Aj . The maximal torus bundles are parametrized by the monopole numbers (n1 , . . . , nrankG ) given by Fj (A) = 2π nj , (119) g

where A = Aj K j is a connection on the bundle and F (A) = Fj (A)K j is its curvature. For each set of the monopole numbers we choose one connection Acl which we call the classical monopole solution. It fulfils dAcl = 2π nj K j ω.

(120)

Now every connection A can be written as A = Acl + Aq ,

(121)

where the “quantum part” Aq of the connection is a 1-form on g with values in T . The path integral over A then becomes the sum over the monopole numbers and the path integral over Aq . The latter imposes a constraint dφ = 0.

(122)

The details of this procedure which, of course, must be accompanied by an appropriate gauge fixing and ghost integration are presented in [2] Eqs. (2.71)–(2.80). Their calculation applies to our case without any change. Thus the integral over φ reduces to a finite integral over the constant mode of φ and we shall note the corresponding measure dφ instead of Dφ. Of course, the integral over dφ is taken over the (interior of the) Weyl chamber A+ = −iC+ . The result is Z(ε, h, ¯ g) =

(dφDBDCDcr Dcj ) exp −ε

n1 ,...,nrankG

i exp {inj φ j − 2π

g α∈K +

[α(φ)]2h¯

α∈K

([α(φ)]h¯ Bα ∧ Cα + ω[α(φ)]2h¯ cαr cαj )}.

(123)

Here we have also used Eq. (74). The path integral over the B,C fields and over the ghosts was performed in [2] in generality, which covers not only the ordinary Yang–Mills and the G/G gauged WZNW

224

C. Klimˇcík

case but also our Poisson–Lie Yang–Mills case. The functions Mα defined in Eq. (B.5) of [2] are in our context Mα = [α(φ)]h¯ . We infer from (B.23) of [2] that Z(ε, h¯ , g) = dφ(;α∈K+ [α(φ)]h¯ )2−2g (124) exp inj φ j − ε [α(φ)]2h¯ . n1 ,...,nrankG

α∈K

Contrary to the G/G case and in accord with the ordinary Yang–Mills theory, there is no shift of a “level” 1/h. ¯ The last step of calculation consists in performing the dφ integral. We use the well-known formula 1 inj φ j e = δ(φ j − 2π mj ). (125) 2π n j m

j

Here on the right-hand side we recognize the periodic δ function. The usual 2π factor in this formula is understood to be hidden in the definition of the measure dφ. Substituting the expression (125) into (124), we arrive at Z(ε, h, ¯ g) =

1 |W |

(;α∈K+ [2π mj α(Hλj )]h¯ )2−2g

m1 ,...,mrankG

exp −ε

[2π m

α∈K

j

α(Hλj )]2h¯

(126) .

Note that here we have conveniently extended the domain of definition of φ from A+ to whole A and we compensated this by factoring the volume |W | of the Weyl group. We now interpret the summation over mj as summation over the weight lattice of G. The latter is defined as M = Z[λ1 , . . . , λr ],

(127)

where r = rankG and λi are the fundamental weights. We set λ = mj λj and we rewrite (126) as Z(ε, h, ¯ g) =

1 |W |

;α∈K+ ([B(α, λ + ρ)]2π h¯ )2−2g

λ+ρ∈Mr

· exp −ε

α∈K

[B(α, λ + ρ)]22π h¯

(128) .

Note the shift by the Weyl vector ρ = 21 K+ α = j λj . Since we anyway sum up over the whole weight lattice, this shift can be interpreted as a pure change of the summation variable. Another important remark concerns the notation Mr in (128). By this we mean that we sum only over the regular points of the weight lattice in accord with the discussion above. There is a simple criterion to decide whether an element of the weight lattice is regular or not. In fact, the non-regular elements are precisely those which are located on the walls of the Weyl chambers. The formula (128) is our final result for the partition function of the LWS Poisson– Lie Yang–Mills theory, or for ε = 0, of the Poisson σ -model corresponding to the LWS


225

˜ = AN . We see that, indeed, we can interpret this partition Poisson–Lie structure on G function as the series over the q-numbers. In the limit h¯ → 0, our result agrees with the ordinary Yang–Mills partition function Eq. (2.86) of [2]. To see this, we just have to use the well-known identity [23] B(x, y) = B(x, α)B(α, y), x, y ∈ G ∗ (129) α∈K

for x = y = λ + ρ. 4.5. The Verlinde formula. Let us rewrite our formula (128) for the partition function Z(ε, h, ¯ g) for the case ε = 0: Z(ε = 0, q, g) =

1 1−g ;α∈K |1 − q (α,λ+ρ) |1−g . A |W |

(130)

λ+ρ∈Mr

Here q = exp 2π hB(ψ, ψ), A is a normalization constant to be discussed later and ¯ the bilinear form (. , .) on the dual of T is defined by the following rescaling of the Killing–Cartan form: (X, Y ) =

2 B(X, Y ). B(ψ, ψ)

(131)

Here ψ denotes the longest root. In our construction, q was a real parameter, nevertheless, we shall now consider Z(0, q, g) as a function of a complex q. Strictly speaking, if q is complex, we cannot say apriori whether Z(0, q, g) is a partition function of some theory. We shall see, however, that for q being a root of unity, Z(0, q, g) can be (almost) interpreted as the partition function of the G/G gauged WZNW model. As it is well-known, the latter is given by the Verlinde formula [7] which is a finite sum and not a series. In which sense can we say that our series (130) gives the Verlinde formula for q being a root of unity? The point is that for q equal to a root of unity, say q k = 1, the expression under the summation symbol in (130) acquires the affine Weyl symmetry. The affine Weyl group [28] is a semidirect product of the standard Weyl group and of the coroot lattice. The action of the Weyl group on the weight lattice is standard and since the product ;α in (130) is taken over all roots, the expression (130) is Weyl invariant (this follows from an idempotency and Killing–Cartan orthogonality of Weyl reflections). An element β ∨ of the coroot lattice acts on the weight lattice as 1 λβ ∨ = λ + k B(ψ, ψ)B(β ∨ , .) = λ + k(β ∨ , .)∗ . 2

(132)

Here use the symbol (. , .)∗ for the form on T dual to (. , .). Recall that for a root β, the coroot β ∨ ∈ T is defined by β∨ =

2Hβ . B(β, β)

(133)

The affine Weyl symmetry of (130) for q k = 1 is now obvious because q (α,λβ ∨ ) = q (α,λ)+kα(β

∨)

= q (α,λ) .

(134)

226

C. Klimˇcík

The last equality follows from the fact that α(β ∨ ) is integer, as the result of the contraction of α (which is also an element of the weight lattice) with the coroot β ∨ . Actually, for q k = 1, the expression (130) makes sense only if the weights lying on affine Weyl orbits of the non-regular weights are excluded from Mr . It is precisely in this sense that we understand (130). We conclude that, for q k = 1, the series (130) can be written as a summation over the fundamental domain of the affine Weyl group multiplied by its (infinite) volume. It is because of this infinite volume renormalization that we have said above that Z(0, q, g) can be almost interpreted as the partition function of the G/G gauged WZNW model. The fundamental domain of the action of the affine Weyl group is often referred to as the (Weyl) alcove and it contains those elements λ of the standard Weyl chamber that fulfil λ(ψ ∨ ) < k.

(135)

ψ∨

Here is the coroot of the longest root. For G = SU (r + 1), the condition (135) translates into r

mj < k,

mj > 0,

(136)

j =1

where λ = j mj λj are the dominant weights sweeping the Weyl chamber. Equation (130) with the summation restricted to the alcove is nothing but the Verlinde formula. Its correct normalization

can be achieved by adding to the action (70) a suitable “counterterm” of the form const g R, where R is the curvature on the Riemann surface. As we have already remarked, it is not clear from our derivation whether Z(0, q, g) is a partition function of some theory if q is not real. On the other hand, the result for q equal to a root of unity suggests that it is indeed so because the partition function of the G/G gauged WZNW model is given by the Verlinde formula. It might be that this fact is related to an observation made by Alekseev, Schaller and Strobl [29]. They remarked that, modulo some δ-function relict of the WZNW term, the G/G gauged WZNW model can be represented as the Poisson σ -model where the Poisson structure on G is a complex bivector. If q and, hence, h¯ has a non-vanishing imaginary part then in ˜ g) our context the real Poisson bivector ;( ˜ gets rescaled by a complex Planck constant h¯ and it becomes complex too. However, there is a crucial difference between our theory and that of Alekseev et al. Our LWS Poisson σ -model (or the Poisson–Lie Yang–Mills ˜ = AN while theory with the vanishing coupling constant) lives on the dual group G the Poisson σ -model of [29] lives on the compact group G. Inspite of this, the partition functions for particular q’s happen to coincide! This suggests that a sort of a quantum Poisson–Lie T-duality [13, 30] between the two topological σ -models takes place here. 5. Outlook An open field for future investigations is the study of the correlation functions of the Poisson–Lie Yang–Mills theories. It seems to be very plausible that, like in the ordinary Yang–Mills case, one can identify observables whose correlators are insertion independent. It would be also interesting to calculate exactly the three-point boundary correlator of the LWS Poisson σ -model on the disc. It is known [5] that the perturbation expansion in h¯ gives the Kontsevich formula. If this correlator can be computed by a closed formula, we might attempt to consider a convergence in h. ¯ Thus h¯ would make sense not only as the formal expansion parameter.


227

A more algebraic problem consists in attempting a classification of all doubles D(G) of a connected simple compact group G. This would lead to a generalization of the classification [31] of the Poisson–Lie structures on G. The Verlinde formula was recently associated to the quantum double of a finite group [32]. Although we have dealt with with the classical double of a Lie group, it would be certainly worth looking at possible relations of the two constructions. It would be desirable to produce a derivation of the partition function by an alternative method. For example, the gluing and pasting procedure of [16]. It may seem problematic to use this approach because of problems with the definition of the WZNW action on surfaces with boundaries. On the other hand we note that in the LWS Poisson–Lie Yang– Mills case the WZNW term is cohomologically trivial therefore this problem should be accessible. Of course, this last remark is pertinent also for the very definition of the Poisson–Lie Yang–Mills theory on the Riemann surfaces with boundary and for the implications concerning the Kontsevich formula. Acknowledgement. I acknowledge discussions with P. Delorme.

References 1. Witten, E.: Commun. Math. Phys. 141, 153 (1991) 2. Blau, M. and Thompson, G.: Lectures on 2d gauge theories: Topological aspects and path integral techniques. In: Proceedings of the 1993 Trieste Summer School on High Energy Physics and Cosmology, eds. E. Gava et al., Singapore: World Scientific, 1994, p. 175; hep-th/9310144 3. Ikeda, N.: Ann. Phys. 235, 435 (1994) 4. Schaller, P. and Strobl, T.: Mod. Phys. Lett. A 9, 3129 (1994); Poisson-σ -models: A generalization of 2d Gravity-Yang–Mills systems. In: Proceedings of the International Workshop on “Finite Dimensional Integrable Systems”, eds. A.N. Sissakian and G.S. Pogosyan, Dubna 1995, 181–190, hep-th/9411163; A brief introduction to Poisson σ -models. In: Proceedings Schladming, 1995, hep-th/9507020 5. Cattaneo, A.S. and Felder, G.: A path integral approach to the Kontsevich quantization formula. math.QA/9902090 6. Kontsevich, M.: Deformation quantization of Poisson manifolds I. q-alg/9709040 7. Verlinde, E.: Nucl. Phys. B B300, 360 (1988) 8. Blau, M. and Thompson, G.: Commun. Math. Phys. 171, 639 (1995) 9. Klimˇcík C. and Ševera, P.: Nucl. Phys. B 488, 653 (1997) 10. Polyakov, A. and Wiegmann, P.B.: Phys. Lett. B 311, 549 (1983) 11. Lu, J.-H. and Weinstein, A.: J. Diff. Geom. 31, 510 (1990); Soibelman, Ya.S.: Algebra Analiz 2, 190 (1990); Drinfeld, V.G.: unpublished 12. Drinfeld, V.G.: Quantum groups. In: Proceedings ICM, Berkeley (1986) 708; Falceto, F. and Gaw¸edzki, K.: J. Geom. Phys. 11, 251 (1993); Alekseev, A.Yu. and Malkin, A.Z.: Commun. Math. Phys. 162, 147 (1994) 13. Klimˇcík, C. and Ševera, P.: Phys. Lett. B 351, 455 (1995) 14. Flaschka, H. and Ratiu, T.: A convexity theorem for Poisson actions of compact Lie groups. Preprint IHES/M/95/24 (1995) 15. Semenov-Tian-Shanski, M.A.: Dressing transformations and Poisson–Lie group actions. In: Publ. Res. Inst. Math. Sci. Kyoto Univ. 51, 1985, p. 1237 16. Blau,M. and Thompson, G.: Int. J. Mod.Phys. A 7, 3781 (1992) 17. Witten, E.: Commun. Math. Phys. 141, 153 (1991) 18. Rusakov, B., Mod. Phys. Lett. A 5, 693 (1990) 19. Hirshfeld, A. and Schwarzweller, T.: Path Integral Quantization of the Poisson σ -model. hep-th/9910178 20. Zhelobenko, D.P. and Stern, A.I.: Representations of Lie groups. Moscow: Nauka, 1983, p. 116, in Russian 21. Gaw¸edzki, K. and Kupianen, A.: Nucl. Phys. B 320, 625 (1989) 22. Harish-Chandra: J. Func. Anal. 19, (1975) 104 23. Cornwell, J.F.: Group theory in physics II. London–New York: Academic Press, 1984, Chapter 13 24. Helgason, S.: Groups and Geometric Analysis. London–New York: Academic Press, 1984, p. 186 25. Delorme, P.: Invent. Math. 105, 305 (1991) 26. Klimˇcík, C.: Commun. Math. Phys. 199, 257 (1998)

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27. Gaw¸edzki, K.: WZW conformal field theory. In: Constructive Quantum Field Theory II, eds. G. Velo and A. Wightman, New York: Plenum, 1990, p. 89 28. Pressley, A. and Segal, G.: Loop groups. London: Clarendon Press, 1986, p. 65; Gaw¸edzki, K.: Conformal Field Theory. Unpublished book 29. Alekseev, A.Yu., Schaller, P. and Strobl, T.: Phys.Rev. D 52, 7146 (1995) 30. Alekseev, A.Yu., Klimˇcík, C. and Tseytlin, A.A.: Nucl. Phys. B 458, 430 (1996) 31. Soibelman, Ya.S.: Dokl. AN SSSR 307, 41 (1989); Levendorskii, Z. and Soibelman, Ya.S.: Commun. Math. Phys. 139, 141 (1991) 32. Koornwinder, T.H., Schroers, B.J., Slingerland, J.K. and Bais, F.A.: Fourier transform and the Verlinde formula for the quantum double of a finite group. math. QA/9904029 Communicated by A. Connes

Commun. Math. Phys. 217, 229 – 248 (2001)

Communications in



A Discrete Density Matrix Theory for Atoms in Strong Magnetic Fields Christian Hainzl, Robert Seiringer Institut für Theoretische Physik, Universität Wien, Boltzmanngasse 5, 1090 Vienna, Austria. E-mail: [email protected]; [email protected] Received: 20 October 2000 / Accepted: 3 November 2000

Abstract: This paper concerns the asymptotic ground state properties of heavy atoms in strong, homogeneous magnetic fields. In the limit when the nuclear charge Z tends to ∞ with the magnetic field B satisfying B Z 4/3 all the electrons are confined to the lowest Landau band. We consider here an energy functional, whose variable is a sequence of one-dimensional density matrices corresponding to different angular momentum functions in the lowest Landau band. We study this functional in detail and derive various interesting properties, which are compared with the density matrix (DM) theory introduced by Lieb, Solovej and Yngvason. In contrast to the DM theory the variable perpendicular to the field is replaced by the discrete angular momentum quantum numbers. Hence we call the new functional a discrete density matrix (DDM) functional. We relate this DDM theory to the lowest Landau band quantum mechanics and show that it reproduces correctly the ground state energy apart from errors due to the indirect part of the Coulomb interaction energy.

1. Introduction The ground state properties of atoms in strong magnetic fields have been the subject of intensive mathematical studies during the last decade. This paper is based on the comprehensive work of Lieb, Solovej andYngvason [LSY94a, LSY94b], which we refer to for an extensive list of references concerning the history of this subject. The starting point of our investigation is the Pauli Hamiltonian for an atom with N electrons and nuclear charge Z in a homogeneous magnetic field B = (0, 0, B) with vector potential A(x) = 21 B × x, H =

σ j · −i∇j + A(xj )

2

1≤j ≤N

Z + − |xj |

1≤i<j ≤N

1 , |xi − xj |

(1.1)

230

C. Hainzl, R. Seiringer

which acts on the Hilbert space 1≤j ≤N L2 (R3 ; C2 ) of antisymmetric spinor-valued wave functions. Here σ denotes the usual Pauli spin matrices. The units are chosen such that h¯ = 2me = e = 1, so the unit of energy is four times the Rydberg energy. The magnetic field B is measured in units B0 = 2.35 · Gauß, the field strength for which the cyclotron radius B = ¯ equals the Bohr radius a0 = h¯ 2 /(me e2 ). The ground state energy is defined as

m2e e3 c h¯ 3

=

(hc/(eB))1/2

109

E Q (N, Z, B) = inf{, H : ∈ domain H, , = 1},

(1.2)

and if there is a ground state wave function , the corresponding ground state density ρ Q is given by |(x, s1 , . . . , xN , sN )|2 dx2 . . . dxN . (1.3) ρ Q (x) = ρ (x) ≡ N si =±1/2

Recall that the spectrum of the free Pauli Hamiltonian on L2 (R3 ; C2 ) for one electron in the magnetic field B, HA = [σ · (−i∇ + A(x))]2 ,

(1.4)

is given by pz2 + 2νB

ν = 0, 1, 2, ...,

pz ∈ R.

(1.5)

The projector 0 onto the lowest Landau band, ν = 0, is represented by the kernel i B 1 2 exp (x⊥ × x⊥ ) · B − (x⊥ − x⊥ ) B δ(z − z )P↓ , (1.6) 0 (x, x ) = 2π 2 4 where x⊥ and z are the components of x perpendicular and parallel to the magnetic field, and P↓ denotes the projection onto the spin-down (s = −1/2) component. With the decomposition L2 (R3 , dx; C2 ) = L2 (R2 , dx⊥ ) ⊗ L2 (R, dz) ⊗ C2 it can be written as |φm φm | ⊗ I ⊗ P↓ , (1.7) 0 = m≥0

where φm denotes the function in the lowest Landau band with angular momentum −m ≤ 0, i.e., using polar coordinates (r, ϕ),

2 m/2 B 1 Br 2 φm (x⊥ ) = e−imϕ e−Br /4 . (1.8) √ 2π m! 2 The projector onto the subspace of 1≤j ≤N L2 (R3 ; C2 ), where all the electrons are in the lowest Landau band is the N th tensorial power of 0 and will be denoted by N 0 . The ground state energy of electrons restricted to the lowest Landau band is defined as N Econf (N, Z, B) = inf , N 0 H 0 . Q

=1

(1.9)

The assertion that for B Z 4/3 the electrons are to the leading order confined to the lowest Landau band is confirmed by the following theorem.

Discrete Density Matrix Theory for Atoms in Strong Magnetic Fields

231

Theorem 1.1 (Lowest Landau band confinement). ([LSY94a, Theorem 1.2]) For any fixed λ = N/Z there is a δ(x) with δ(x) → 0 as x → ∞ such that

Q Q Econf ≥ E Q ≥ Econf 1 + δ(B/Z 4/3 ) . (1.10) We now define the functional we are considering. We are only interested in B Z 4/3 , so we can restrict ourselves to considering all particles in the lowest Landau band. Recalling that 0 HA 0 = |φm φm | ⊗ (−∂z2 ) ⊗ P↓ , (1.11) m≥0

it is natural to define the following discrete density matrix functional (DDM) as DDM ρ), EB,Z Tr[−∂z2 %m ] − Z Vm (z)ρm (z)dz + D(ρ, [%] = (1.12) m∈N0

where

ρ) = 1 D(ρ, 2 m,n

Vm,n (z − z )ρm (z)ρn (z )dzdz ,

and the potentials Vm and Vm,n are given by 1 Vm (z) = |φm (x⊥ )|2 dx⊥ , |x| )|2 |φm (x⊥ )|2 |φn (x⊥ Vm,n (z − z ) = dx⊥ dx⊥ . |x − x |

(1.13)

(1.14) (1.15)

Here % is a sequence of density matrices acting on L2 (R, dz), % = (%m )m∈N0 ,

(1.16)

DDM depends with corresponding densities ρ = (ρm )m , ρm (z) = %m (z, z). Note that EB,Z on B via the potentials Vm and Vm,n . This functional is defined for all % with the properties:

(i) 0 ≤ %m ≤ I (ii)

m

E

(1.17)

Tr[(1 − ∂z2 )%m ] < ∞.

The corresponding energy is given by DDM

for all m ∈ N0 ,

(N, Z, B) = inf

DDM EB,Z [%]

(1.18)

Tr[%m ] ≤ N .

(1.19)

m

(We do not require that N is an integer.) As we will show, E DDM correctly reproduces Q the confined ground state energy Econf apart from errors due to the indirect part of the Coulomb interaction energy:

232

C. Hainzl, R. Seiringer Q

Theorem 1.2 (Relation of E DDM and Econf ). For some constant cλ depending only on λ = N/Z, Q

0 ≥ Econf (N, Z, B) − E DDM (N, Z, B) ≥ −RL ,

(1.20)

RL = cλ min Z 17/15 B 2/5 , Z 8/3 (1 + [ln(B/Z 3 )]2 ) .

(1.21)

with

Q

The proof will be given in Sects. 5 and 6. Note that Econf is of order min{Z 7/3 (B/Z 4/3 )2/5 , Z 3 (1 + [ln(B/Z 3 )]2 )}, so RL is really of lower order. The functional (1.12) is in fact a reduced Hartree–Fock approximation (in the sense of [S91]) for the quantum mechanical many body problem, i.e. Hartree–Fock theory with the exchange term dropped. The upper bound in Theorem 1.2 holds quite generally for (reduced) Hartree–Fock approximations to many body quantum mechanics with positive two-body interactions. Hartree and Hartree–Fock approximations to H with restriction to the lowest Landau level were probably for the first time considered in [CLR70] and [CR73]. They have been studied numerically in [NLK86, NKL87]. In this paper we present a rigorous mathematical treatment of the functional (1.12). In Sect. 3 we show that there exists a unique solution to the minimization problem in (1.19). The corresponding minimizer is composed of eigenfunctions of one-dimensional effective mean-field Hamiltonians depending on m. Moreover, the superharmonicity of the effective mean-field potential implies monotonicity and concavity of the eigenvalues in m, which amounts to “filling the lowest angular momentum channels”. This fact is important for numerical treatments of the model, for it means that at most the N lowest angular momenta have to be considered. For B/Z 3 large enough, we will show that each angular momentum channel is occupied by at most one particle. In Sect. 4 we estimate the maximum number of electrons that can be bound to the nucleus. We use Lieb’s strategy to derive an upper bound analogous to [S00]. The DDM theory can also be considered as a discrete analogue of the DM functional introduced in [LSY94a]. To express this analogy we will recall its definition and main properties in the next section. 2. Comparison with the DM Functional In [LSY94a] Lieb, Solovej and Yngvason defined a density matrix (DM) functional as E DM [%] = Tr L2 (R) [−∂z2 %x⊥ ]dx⊥ − Z |x|−1 ρ% (x) + D(ρ% , ρ% ). (2.1) R2

Its variable is an operator valued function % : x⊥ → %x⊥ ,

(2.2)

where %x⊥ is a density matrix on L2 (R), given by a kernel %x⊥ (z, z ) and satisfying 0 ≤ %x⊥ ≤ (B/2π )I

(2.3)


233

as an operator on L2 (R). Here we denote ρ% (x) = %x⊥ (z, z)

and

D(f, g) =

1 2

f (x)g(x ) dxdx . |x − x |

(2.4)

The energy E DM (N, Z, B) = inf{E DM [%]| % satisfies (2.3) and

ρ% ≤ N }

(2.5)

turns out to be asymptotically equal to the confined quantum mechanical ground state Q energy Econf in the following precise sense: Q

Theorem 2.1 (Relation of E DM and Econf ). ([LSY94a, Sects. 5 and 8]) For some constants cλ and cλ , depending only on λ = N/Z, Q

RU ≥ Econf (N, Z, B) − E DM (N, Z, B) ≥ −RL ,

(2.6)

with RL given in (1.21) and RU = cλ min{Z 5/3 B 1/3 , Z 8/3 [1 + ln(Z) + (ln(B/Z 3 ))2 ]5/6 }.

(2.7)

The DM energy fulfills the simple scaling relation E DM (N, Z, B) = Z 3 E DM (λ, 1, η),

(2.8)

where we introduced the parameters λ = N/Z and η = B/Z 3 . In the limit η → ∞, E DM (λ, 1, η)/(ln η)2 converges to the so-called hyper-strong (HS) energy E HS (λ), which is the ground state energy of the functional 2 d 1 ρ(z)2 dz ρ(z) dz − ρ(0) + (2.9) E HS [ρ] = dz 2 R under the condition ρ(z)dz ≤ λ. The corresponding minimizer can be given explicitly, namely ρ HS (z) =

2(2 − λ)2 (4 sinh[(2 − λ)|z|/4 + c(λ)])2 tanh c(λ) = (2 − λ)/2,

ρ HS (z) = 2(2 + |z|)−2

for

λ < 2, (2.10)

for

λ ≥ 2.

DDM and E DM . The functional E DDM is the restriction We now discuss the relation of EB,Z B,Z of E DM to density matrices of the form %x⊥ (z, z ) = |φm (x⊥ )|2 %m (z, z ). (2.11) m

Therefore it is clear that E DDM (N, Z, B) ≥ E DM (N, Z, B).

(2.12)

234


In the limit N → ∞ they coincide, i.e. we will show that E DDM (N, Z, B) = E DM (λ, 1, η) N→∞ Z3 lim

(2.13)

for all fixed λ and η. In fact, (2.13) follows immediately from Theorem 1.2 and Theorem 2.1, but it could of course also be shown directly without referring to the relation to Q Econf . Note that in contrast to DM, the DDM energy depends non-trivially on all the three parameters N, Z and B, like the original QM problem. DDM 3. Properties of EB,Z

Theorem 3.1 (Existence of a minimizer). For each N > 0, B > 0 and Z > 0 there DDM under the condition exists a minimizer % DDM for EB,Z m Tr[%m ] ≤ N . DDM under the normalization condiProof. Let % (i) be a minimizing sequence for EB,Z (i) tion m Tr[%m ] ≤ N, with corresponding densities ρ (i) = (ρm )m . Using the onedimensional Lieb–Thirring inequality [LT76] we have (i) 3 (i) ]. (3.1) ρm ≤ const.Tr[−∂z2 %m

Moreover (cf. [LSY94a, Eq. (4.5b)]),

 2 (i) d ρm (z) (i)   dz ≤ Tr[−∂z2 %m ]. dz

(3.2)

Since the potential energy is relatively bounded with respect to the kinetic energy (compare Thm. 2.2 in [LSY94a]), the right-hand sides of (3.1) and (3.2) are uniformly (i)

(i)

bounded. Hence the sequence ρm is bounded in L3 (R, dz) ∩ L1 (R, dz), and ρm is bounded in H1 (R, dz). By the “diagonal sequence trick”, there is a subsequence, (i) (∞) again denoted by ρ (i) , such that ρm converges to some ρm for each m, weakly in 3 p L (R, dz) ∩ L (R, dz) for some 1 < p ≤ 3 and pointwise almost everywhere. By Fatou’s lemma, lim inf i→∞

m,n

(i) dzdz Vm,n (z − z )ρm (z)ρn(i) (z)

≥

m,n

(∞) dzdz Vm,n (z − z )ρm (z)ρn(∞) (z).

(3.3)

Observe now that Vm ≤ C for some C > 0 and for all m. Moreover, Vm (z) ≤ 1/|z|, so Vm ∈ Lp for all p > 1. By the weak convergence we can conclude that (i) (∞) dzVm (z)ρm (z) = dzVm (z)ρm (z) (3.4) lim i→∞


235

for each m. Moreover, by the dominated convergence theorem, we get (i) (∞) dzVm (z)ρm dzVm (z)ρm (z) = (z). lim i→∞

m

(3.5)

m

Using the “diagonal sequence trick” once more, we can use the trace class property of (i) DDM ) the %m ’s to conclude that there exists a subsequence of % (i) and a % DDM = (%m m such that (i) DDM %m 1 %m

(3.6)

in the weak operator sense, for each m. It follows from weak convergence that 0 ≤ DDM ≤ 1. Using Fatou’s lemma twice, we have %m DDM (i) Tr[%m ] ≤ lim inf Tr[%m ] ≤ N. (3.7) i→∞

m

By the same argument m

m

Tr[−∂z2 %zDDM ] ≤ lim inf i→∞

m

(i) Tr[−∂z2 %m ].

(3.8) (∞)

DDM denote the density of % DDM . It remains to show that ρ Let now ρm m m each m. From weak convergence it follows that

DDM for = ρm

(i) DDM (1 − ∂z2 )1/2 %m (1 − ∂z2 )1/2 1 (1 − ∂z2 )1/2 %m (1 − ∂z2 )1/2

(3.9) (i)

weakly on the dense set C0∞ (R). Since the operators are bounded by Tr[(1 − ∂z2 )%m ] ≤ C, we see that (3.9) holds weakly in L2 (R, dz). With η ∈ C0∞ (R) considered as a multiplication operator, it is easy to see that (1 − ∂z2 )−1/2 η(1 − ∂z2 )−1/2 is a compact operator (it is even Hilbert–Schmidt). Thus it can be approximated in norm by finite rank operators. Using (3.9) we can therefore conclude that (i) DDM lim Tr[%m η] = Tr[%m η],

(3.10)

i→∞ (i)

(i)

DDM in the sense of distributions. Since we already know that ρ converges i.e. ρm → ρm m (∞) (∞) DDM for each m. to ρm pointwise almost everywhere, we conclude that ρm = ρm DDM ] ≤ N and We have thus shown that there exists a % DDM with m Tr[%m DDM DDM DDM (i) DDM EB,Z [% ] ≤ lim inf i→∞ EB,Z [% ] = E . ! "

Lemma 3.2 (Uniqueness of the density). The density corresponding to the minimizer (1) (2) is unique, i.e., if there are two minimizers % (1) and % (2) , their densities ρm and ρm are equal, for all m. Proof. Observe that ρ) = D(ρ, D(ρ, ˜ ρ), ˜

(3.11)

2 where we set ρ(x) ˜ = m |φm (x⊥ )| ρm (z). Using the positive definiteness of the (1) (2) Coulomb kernel and the fact that ρ˜ (1) = ρ˜ (2) implies ρm = ρm for all m, we immediately get the desired result. ! "

236


Having established the uniqueness of the density, we can now define a linearized DDM functional by DDM Elin [%] = Tr[hDDM %m ], (3.12) m m

with the one-particle operators hDDM = −∂z2 − 2DDM (z), m m

(3.13)

where the potentials are given by 2DDM (z) = ZVm (z) − m

n

ρnDDM (z )Vm,n (z − z )dz .

(3.14)

DDM under Lemma 3.3 (Equivalence of linearized theory). A minimizer % DDM for EB,Z DDM the constraint m Tr[%m ] ≤ N is also a minimizer for the linearized functional Elin (with the same constraint).

Proof. We proceed essentially as in [LSY94a]. For any %, DDM DDM % − ρ DDM , ρ% − ρ DDM ) − D(ρ DDM , ρ DDM ). [%] = Elin [%] + D(ρ EB,Z

(3.15)

In particular, for all δ > 0, DDM DDM DDM DDM DDM , ρ DDM ) EB,Z [(1 − δ)% DDM + δ%] = (1 − δ)Elin [% ] + δElin [%] − D(ρ % − ρ DDM , ρ% − ρ DDM ). + δ 2 D(ρ (3.16)

(0) DDM [% (0) ] < E DDM [% DDM ] we Now if there exists a % (0) with m Tr[%m ] ≤ N and Elin lin can choose δ small enough to conclude that DDM DDM DDM DDM DDM DDM , ρ DDM ) = EB,Z [(1 − δ)% DDM + δ% (0) ] < Elin [% ] − D(ρ [% ], EB,Z (3.17) DDM . which contradicts the fact that % DDM minimizes EB,Z

" !

DDM by means of the eigenfunctions We now are able to construct the minimizer of EB,Z i of the one-dimensional operators hDDM . If µ1 < µ2 < . . . denote the corresponding em m m m eigenvalues, there is a µ ≤ 0 such that

DDM %m

=

I m −1 i=1

i i Im Im |em em | + λm |em em |,

(3.18)

where Im = max{i : µim ≤ µ}, and 0 ≤ λm ≤ 1 is the filling of the last level. Since λm DDM , we immediately get the following corollary: is determined by the unique density ρm Corollary 3.4 (Uniqueness of % DDM ). For any B > 0 and Z > 0 the minimizer of the DDM under the condition functional EB,Z m Tr[%m ] ≤ N is unique.


237

Note that % DDM really depends on the three parameters N , Z and B, but we suppress this dependence for the simplicity of the notation. Of course, the choice of µ is not unique, but Im is unique for every m. In the following, we will choose for µ the smallest possible, i.e. µ ≡ max{µImm }.

(3.19)

The energy E DDM (N, Z, B) is a convex, non-increasing function in N . Moreover, it has the following property. Theorem 3.5 (Differentiability of E DDM ). For N $ ∈ N the energy E DDM is differentiable in N , and the derivative is given by ∂E DDM /∂N = µ with µ given above. For N ∈ N, the right and left derivatives, ∂E DDM /∂N± , are given by ∂E DDM = µ, ∂N−

∂E DDM = min{µImm +1 }. ∂N+

(3.20)

DDM as above. Let E (N ) be the infimum of Proof. For fixed N, Z and B define Elin lin DDM Elin under the constraint m Tr[%m ] ≤ N . For 0 < N− < N < N+ let % ± be the corresponding minimizers. We have, with 0 < δ < 1, DDM [(1 − δ)% DDM + δ% ± ] E DDM (N + δ(N± − N ), Z, B) ≤ EB,Z

= E DDM (N, Z, B)

(3.21)

+ δ (Elin (N± ) − Elin (N )) + O(δ ). 2

Dividing by δ(N± − N ) and taking the limit δ → 0 followed by N± → N , we conclude that ∂E DDM ∂Elin ≤ , ∂N− ∂N−

∂E DDM ∂Elin ≤ . ∂N+ ∂N+

(3.22)

DDM , ρ DDM ) for all N , From (3.15) we infer that E DDM (N , Z, B) ≥ Elin (N ) − D(ρ with equality for N = N . Therefore we get the inverse inequalities in (3.22), so equality holds. The assertions of the theorem follow from Elin (N ) =

[N]

µ(j ) + (N − [N ]) µ([N ] + 1),

(3.23)

j =1

where [N ] is the largest integer ≤ N , and µ(j ) is the j th element of the set {µim } in increasing order. ! " Corresponding to the DDM minimizer we define the three-dimensional DDM density as ρ˜ DDM (x) =

m≥0

DDM |φm (x⊥ )|2 ρm (z).

(3.24)

238


Theorem 3.6 (Ordering of the µim ). Assume hDDM has at least M eigenvalues. Then, m for m ≥ 1, M i=1

mµim−1

+ (m + 1)µim+1

− (2m + 1)µim

M 2π i 2 ρ˜ DDM |φm em ≤− | (3.25) B i=1

DDM (with the understanding that µim−1 (µim+1 ) = 0 if hDDM m−1 (hm+1 ) has less than i eigenvalues). The analogous inequality for m = 0 is M M 2π ρ˜ DDM |φ0 e0i |2 . |e0i (0)|2 − µi1 − µi0 ≤ Z B

M i=1

i=1

(3.26)

i=1

Proof. With 6(2) the two-dimensional Laplacian, one easily computes that (2B)−1 6(2) |φm |2 = m|φm−1 |2 + (m + 1)|φm+1 |2 − (2m + 1)|φm |2

(3.27)

for m ≥ 1. Multiplying (3.27) with 2(x) = Z/|x| − ρ˜ DDM ∗ |x|−1 and integrating over x⊥ , we therefore get, for any density matrix γ (recall definitions (1.14), (1.15), (3.13) and (3.14)), DDM DDM Tr mhDDM γ m−1 + (m + 1)hm+1 − (2m + 1)hm = −(2B)−1 dx2(x)ργ (z)6(2) |φm (x⊥ )|2 (3.28) ∂2 2π 1 =− ργ , dxρ˜ DDM |φm |2 ργ + dz2DDM m B 2B ∂z2 where we used partial integration for the second step, and the fact that φm (0) = 0 for m ≥ 1. To treat the last term in (3.28), note that the function w→

M i=1

i dz2DDM (z)|em (z + w)|2 m

(3.29)

has its maximum at w = 0, because otherwise one could lower the energy by shifting the i . Therefore the second derivative of (3.29) at w = 0 is negative. Setting eigenvectors em M i ei | we see that the last term in (3.28) is negative, so we can conclude γ = i=1 |em m that 2π DDM DDM Tr mhDDM + (m + 1)h − (2m + 1)h γ ≤ − ρ˜ DDM |φm |2 ργ . m m−1 m+1 B (3.30) By the variational principle (3.25) holds. The proof of (3.26) is analogous, considering also the contribution from φ0 (0) in (3.28). ! " As a corollary, we immediately get M i i Corollary 3.7 (Monotonicity of M i=1 µm in m). i=1 µm is increasing in m for all DDM = 0 for m ≥ N . M, and strictly increasing as long as it is < 0. Moreover, %m


239

Proof. This follows immediately from Theorem 3.6, noting that limm→∞ µim = 0 for all i. ! " i Remark 3.8 (Concavity of M i=1 µm ). The result (3.25) (together with the monotonicity) 1 implies concavity of µm in m. More precisely, 1 m m+1 1 1 µ1m−1 + µm+1 + µm−1 − µ1m+1 2m + 1 2m + 1 2(2m + 1) ≤ µ1m (3.31) i (and the same holds for M i=1 µm ). This is the analogue of Prop. 2.3 in [LSY94a], which states that −µ1 (x⊥ ) is a increasing and concave function of |x⊥ | (note the different sign convention). 1 1 2 µm−1

+ 21 µ1m+1 =

We now introduce the parameters λ = N/Z and η = B/Z 3 . The next theorem deals with the η → ∞ limit of E DDM with fixed λ. To prove it we need the following lemma. Lemma 3.9 (Convergence to delta function). Let L = L(η) be the solution of the equation η1/2 = L(η) sinh(L(η)/2). (3.32) Let ψ ∈ H1 (R, dz), with λ = |ψ|2 and T = |dψ/dz|2 . Then, for all m ≥ 0, Z > 0 and η = B/Z 3 , 2 |ψ(0)|2 − 1 Vm (z/LZ)|ψ(z)| dz ZL2

1/4 1 π Z 1/2 1/4 3/4 (m + 1) ≤ . (3.33) + 16λ T λ L 2 (m + 1)1/2 Z 1/4 Proof. After an appropriate scaling, this is a direct consequence of [BSY00, Lemma 2.1], using the estimates

π B 1/2 2 −1 |φm (x⊥ )| |x⊥ | dx⊥ ≤ (3.34) 2 (m + 1)1/2 and

|φm (x⊥ )|2 |x⊥ |1/2 dx⊥ ≤ 2

(m + 1)1/4 . B 1/4

" !

(3.35)

Theorem 3.10 (The limit B Z 3 ). For all λ > 0, E DDM (λZ, Z, ηZ 3 ) = E HS (λ), η→∞ Z 3 (ln η)2 lim

(3.36)

uniformly in Z. Remark. The uniformity in Z will be important for the proof of Theorem 3.12. It is non-trivial in contrast to DM, where one has the scaling relation (2.8), which implies that the left-hand side of (3.36) (with DDM replaced by DM) is independent of Z.

240


Proof. The lower bound is quite easy, using the results of [LSY94a]. As shown in Sect. 2, we have E DM (λZ, Z, ηZ 3 ) E DDM (λZ, Z, ηZ 3 ) ≥ = E DM (λ, 1, η), Z3 Z3

(3.37)

where we have used the scaling properties of E DM . It is shown in [LSY94a] that the right-hand side of (3.37) divided by (ln η)2 converges to E HS (λ). For the upper bound we assume N ∈ N for the moment. We use as trial density matrices, Z 2 HS L ρ (LZz) ρ HS (LZz ), N %m (z, z ) = 0, m ≥ N, %m (z, z ) =

0 ≤ m ≤ N − 1,

(3.38)

where ρ HS is given in (2.10) and L = L(η) is defined in (3.32). The kinetic energy is easily computed to be Z 3 L2 |d ρ HS /dz|2 . (3.39) For the attraction term we use Lemma 3.9 to estimate N−1 1 1 Cλ (m + 1)1/4 Z 1/2 HS ρ (0) − Z Vm ρm ≥ + Z 3 L2 m N L (m + 1)1/2 Z 1/4 m=0

≥ ρ HS (0) −

Cλ L

(3.40)

for some constant Cλ depending on λ. For the repulsion term we first estimate √ √ 2 |) ;( 2N/B − |x⊥ |);( 2N/B − |x⊥ B Vm,n (z) ≤ dx⊥ dx⊥ , 2π 2 2 |x⊥ − x⊥ | + z 0≤m,n≤N−1 (3.41) which follows from monotonicity of 1/|x| in |x⊥ | and the fact that m |φm |2 ≤ B/2π . Therefore we have 1 Vm,n (z − z )ρm (z)ρm (z )dzdz Z 3 L2 0≤m,n≤N −1 1 ≤ ξ f (ξ(z − z ))ρ HS (z)ρ HS (z )dzdz , (3.42) L where we set ξ = (η/λ)1/2 /L, and the function f is given by √ √ |) ;( 2 − |x⊥ |);( 2 − |x⊥ −2 f (z) = (2π) dx⊥ dx⊥ . 2 2 |x⊥ − x⊥ | + z

(3.43)


241

We now claim that L−1 ξf (ξ z) → δ(z) as η → ∞. Since f (z) ≤ 1/|z| we have, for any χ ∈ H1 ∩ L1 , ξ −1 L ξ f (ξ z)χ (z)dz − χ (0)L−1 f (z)dz −ξ 1 −1 = L ξ f (ξ z)(χ (z) − χ (0))dz + L−1 ξ −1

≤ const.L−1 ξ

1

−1

f (ξ z)z1/2 dz + L−1

|z|≥1

|z|≥1

f (ξ z)χ (z)dz

|χ (z)|dz ≤ const.L−1 .

(3.44)

Now L−1 |z|≤ξ f → 1 as η → ∞, which proves our claim. And since L(η) ≈ ln η for large η, this finishes the proof of Theorem 3.10, in the case where N is an integer. The proof for N $ ∈ N is analogous, using (3.38) with the density matrix corresponding to m = [N ] multiplied by N − [N ] as trial density matrices. ! " Corollary 3.11 (HS limit of the density). For fixed λ = N/Z lim

1

η→∞ Z 2 ln η

m

DDM ρm (z/Z ln η) = ρ HS (z)

(3.45)

in the weak L1 sense, uniformly in Z. Proof. The convergence of the densities in (3.45) follows from the convergence of the energies in a standard way by considering perturbations of the external potential (cf. e.g. [LSY94a]). Moreover, since the convergence in (3.36) is uniform in Z, (3.45) holds for any function Z = Z(β), so we can conclude that (3.45) holds uniformly in Z, too. ! " Using the results above we can now prove the analogue of Theorem 4.6 in [LSY94a]. DDM has rank at most 1 for large η). There exists a constant C such Theorem 3.12 (%m DDM has rank at most 1 for all m. that η ≥ C implies that %m

Proof. We first treat the case λ < 2. From Theorems 3.5 and 3.10 and the fact that E DDM (λZ, Z, ηZ 3 ) is convex in λ, we get lim

µ

η→∞ Z 2 (ln η)2

=

dE HS (λ) < 0. dλ

(3.46)

Suppose that µ is not the ground state energy of some hDDM . Then µ ≥ −Z 2 /4, because m DDM the second lowest eigenvalue of hm is equal to the ground state energy of the threedimensional operator −6 − 2DDM (|x|). m

(3.47)

This follows because 2DDM (z) is reflexion symmetric, so the eigenvector corresponding m to the second lowest eigenvalue, um (z), has a node at z = 0. Therefore um (|x|)/|x| is an eigenvector of (3.47), and because it does not change sign, it must be a ground state. Since 2DDM (|x|) ≤ Z/|x|, the ground state energy of (3.47) is greater than −Z 2 /4. So m 2 2 µ/Z (ln η) would go to zero as η → ∞, in contradiction to (3.46). Therefore there exists a constant C such that η > C implies the assertion to the theorem. This constant

242


can be chosen independent of Z, because the limit (3.46) is uniform in Z (by the same argument as in the proof of Corollary 3.11). Now assume that λ = 1 + δ¯ for some δ¯ > 0. From Corollary 3.11 we infer that for large enough η there is some cδ > 0 such that DDM ρm ≥ (1 + 21 δ)Z, (3.48) m

|z|≤cδ (Z ln η)−1

¯ We now will show that for η large enough hDDM where δ = min{1, δ}. has at most one m eigenvalue, for all 0 ≤ m ≤ N. By the same argument as above, we need to show that the three-dimensional operator (3.47) has no eigenvalues. Using Vm (z) ≤ 1/|z| and (cf. the next section) √ 1 1 Vm,n (z) ≥ √ Vm+n (z/ 2) ≥ 2 z2 + 2(m + n + 1)B −1 (3.49) 1 ≥ , z2 + 2(2N + 1)B −1 we can use (3.48) to estimate 2DDM (|x|) ≤ m

Z(1 + 21 δ) . 2 −1 −1 + 2(2N + 1)B |x| + cδ (Z ln η)

Z − |x|

Therefore, for η large enough, 2DDM (|x|) ≤ m

|x| ≤ 3δ −1 cδ (Z ln η)−1

Z/|x|

for

0

otherwise.

(3.50)

(3.51)

By means of the Cwikel–Lieb–Rosenbljum bound [RS78] we can estimate the number of negative eigenvalues of (3.47) as 3/2 const. |2DDM (|x|)|+ dx ≤ cδ (ln η)−3/2 , (3.52) m which is less than 1 for η large enough.

" !

Remark 3.13 (Chemical potential for large B/Z 3 ). The theorem above, together with Corollary 3.7, shows that for B/Z 3 large enough, the chemical potential is given by the 1 ground state energy of hDDM , i.e. µ = µ , where < N > denotes the smallest integer ≥ N . 4. Maximal Negative Ionization ρ) is The DDM energy is convex and monotonously decreasing in N . Because D(ρ, DDM is strictly convex up to some N = Nc (Z, B), and constant strictly convex in ρ, E DDM for N ≥ Nc . By uniqueness of % DDM the minimizer for N > Nc is equal to the one with N = Nc . In particular, m Tr[%m ] = min{N, Nc }. The “critical” Nc measures the maximal particle number that can be bound to the nucleus. We will proceed essentially as in [S00] and use Lieb’s strategy [L84] to get an upper bound on Nc . In addition, the following lemma is needed. Throughout, we use various properties of Vm stated in [BRW99, Sect. 4.], and proven in [RW00].


Lemma 4.1 (Comparison of Vm and Vm,n ). 1 1 1 1 + + + Vm,n (z − z ) ≥ 1. Vm (z) Vn (z) Vm (z ) Vn (z )

243

(4.1)

Proof. Using the definition of Vm,n it can be shown (cf. [P82]) that m+n √ √ 2Vm,n ( 2z) = ci Vi (z)

(4.2)

i=0

for some coefficients ci ≥ 0 that fulfill 4b]) we get

i ci

= 1. Since Vm ≥ Vm+1 for all m ([BRW99,

√ 1 1 Vm,n (z) ≥ √ Vm+n (z/ 2) ≥ Vm+n (z/2), 2 2

(4.3)

where we used the fact that aVm (az) ≤ Vm (z) if a ≤ 1 ([BRW99, 4g]). Moreover, using convexity of 1/Vm+n ([BRW99, 4i]), we arrive at

−1 Vm,n (z − z ) ≥ Vm+n (z)−1 + Vm+n (z )−1 .

(4.4)

The assertion (4.1) follows if we can show that 1 Vm+n (z)

≤

1 1 + . Vm (z) Vn (z)

(4.5)

√ −1 This is of course trivial if n or m equals zero. If n, m ≥ 1 we use z2 + m ≥ Vm (z) ≥ √ −1 z2 + m + 1 ([BRW99, 4a]) to estimate √ √ 1 ≤ z2 + m + n + 1 ≤ 2 z2 + ( m + n)2 /4 Vm+n (z) (4.6) 1 1 + , ≤ z2 + m + z2 + n ≤ Vm (z) Vn (z) which finishes the proof. ! " Theorem 4.2 (Critical particle number). Z ≤ Nc ≤ 4Z −

1 ∂E DDM (Nc , Z, B) . Nc ∂Z

(4.7)

Remark. The factor 4 stems from the symmetrization of (4.1) in m and n. Due to this symmetrization one could expect that Lemma 4.1 holds with 1 replaced by 2 on the right-hand side. This would imply that 4Z could be replaced by 2Z in (4.7). i denote the eigenvectors of hDDM , i.e. Proof. Let em m i i em = µim em . hDDM m

(4.8)

244


i /V and integrating we get Multiplying (4.8) with em m 1 i (z)2 Vm,n (z − z )ρnDDM (z )dzdz em Z≥ V (z) m n i + em |

1 i (−∂z2 )|em , Vm

(4.9)

where we used that µim ≤ 0. Since 1/Vm is convex and |z|Vm (z) → 1 as |z| → ∞ we have |(1/Vm ) | ≤ 1. Using this and partial integration we can estimate the last term in (4.9) by 1 1 Vm 2 i −1/2 −1/2 i i 2 i 2 em | (−∂z )|em = em Vm | − ∂z − |em Vm Vm 4 Vm (4.10) 1 i 2 ≥− Vm (z)em (z) dz. 4 Summing over all m and i (and, according to Eq. (3.18), multiplying the factor corresponding to the largest µim by λm ), we arrive at 1 DDM ρm (z)Vm,n (z − z )ρnDDM (z )dzdz NZ ≥ V (z) m m,n (4.11) 1 DDM Vm ρm . − 4 m Note that the last term in (4.11) is equal to ∂E DDM /∂Z. To treat the first term in (4.11) we use symmetry and Lemma 4.1 to get 1 DDM (z)Vm,n (z − z )ρnDDM (z )dzdz ρm V (z) m m,n 1 1 1 1 1 (4.12) = + + + 4 m,n Vm (z) Vn (z) Vm (z ) Vn (z ) DDM × ρm (z)Vm,n (z − z )ρnDDM (z )dzdz ≥

1 2 N . 4

Inserting this into (4.11) and dividing by N/4 we arrive at N ≤ 4Z −

1 ∂E DDM . N ∂Z

(4.13)

The lower bound on Nc is quite easy. We just have to show that hDDM [N+1] has a bound state if N < Z. Using ψ(z) = exp(−a|z|) with a > 0 as a trial vector we compute ψ|hDDM [N+1] |ψ Vn,[N+1] (z − z )ρnDDM (z )e−2a|z| dzdz = a − Z V[N+1] (z)e−2a|z| dz + n

≤a−Z

V[N+1] (z)e

−2a|z|

dz + N max n≤N

Vn,[N+1] (z)e−2a|z| dz.

(4.14)


245

Since 1 a→0 ln(1/a) lim

1 a→0 ln(1/a)

V[N+1] e−2a|z| = lim

Vn,[N+1] e−2a|z| = 1,

ψ|hDDM [N+1] |ψ will be negative for small enough a, if N < Z.

(4.15)

" !

Remark 4.3 (Explicit bound on Nc ). The concavity of E DDM in Z implies that ∂E DDM (N, Z, B)/∂Z ≥ E DDM (N, 2Z, B)/Z. Using (2.12) and the bounds on E DM given in [LSY94a, Thm. 4.8], (4.7) implies the upper bound

Nc ≤ 4Z 1 + C min (B/Z 3 )2/5 , 1 + [ln(B/Z 3 )]2

(4.16)

for some constant C independent of Z and B. Remark 4.4 (Upper bound on NcDM ). The upper bound (4.16) holds also for NcDM , the critical particle number in the DM theory. In fact, the convergence in (2.13) implies that Nc (Z, ηZ 3 ) NcDM ≤ Z lim inf Z→∞ Z

(4.17)

for all fixed η = B/Z 3 . 5. Upper Bound to the QM Energy We now show that E DDM is an upper bound to the quantum mechanical ground state Q energy. In fact it is even an upper bound to Econf . By Lieb’s variational principle [L81], E Q ≤ Tr[(HA − Z|x|−1 )γ ] +

1 2

γ (x, x)γ (x , x ) − |γ (x, x )|2 dxdx |x − x |

(5.1)

for all density matrices 0 ≤ γ ≤ 1 with Tr[γ ] ≤ N . We choose γ (x, x ) =

m

φm (x⊥ )φm (x⊥ )%m (z, z ).

(5.2)

Q

Since 0 γ 0 = γ , we get an upper bound even for Econf . Omitting the negative “exchange term”, we compute Q

DDM Econf (N, Z, B) ≤ EB,Z [%],

where % = (%m )m . Note that Tr[γ ] = that Q

(5.3)

m Tr[%m ]. Therefore we immediately conclude

Econf (N, Z, B) ≤ E DDM (N, Z, B).

(5.4)

246


6. Lower Bound to the QM Energy To get a lower bound on the QM energy we need to estimate the two-body interaction potential in terms of one-body potentials. One way to do this is to use the Lieb–Oxford inequality [LO81] together with the positive definiteness of the Coulomb kernel: 4/3 −1 | |xi − xj | ≥ 2D(ρ, ρ ) − D(ρ, ρ) − 1.68 ρ , (6.1) i<j

where we can choose ρ = ρ˜ DDM . However, if is an (approximate) ground state wave 4/3 function in the lowest Landau band, the error term ρ will in general be greater than the energy itself. More precisely, if γ denotes the one-particle density matrix of , we can estimate 1/6 5/6 4/3 3 ρ ≤ ρ ρ ≤

1/6

3 2 B Tr[−∂z2 γ ] π2

N 5/6

(6.2)

≤ const. Z 1/5 N 14/15 B 2/5 , 3 where we used [LSY94a, Lemma 4.2], to estimate ρ and the kinetic energy. For large enough B this bound is of no use. That is why we follow the method of [LSY94a] to get an improved bound for large B. Their result is that for wave functions that satisfy |H < 0 the bound

| |xi − xj |−1 ≥ 2D ρ DM , ρ − D ρ DM , ρ DM − cλ Z 8/3 1 + (ln η)2 i<j

(6.3) holds. The proof of this result uses only properties of ρ DM which hold also for ρ˜ DDM . Therefore (6.3) holds also with ρ DM replaced by ρ˜ DDM , possibly with a different constant. So we can estimate ! " Q (i) Econf (N, Z, B) ≥ inf HA − Z/|xi | + ρ˜ DDM ∗ |xi |−1 (6.4) i DDM DDM − D ρ˜ , ρ˜ − RL , where the infimum is over all with N 0 = and = 1, and

RL = cλ min Z 17/15 B 2/5 , Z 8/3 1 + (ln η)2 .

(6.5)

Because HA − Z/|x| + ρ˜ DDM ∗ |x|−1 is a one-particle operator that is invariant under rotations around the z-axis, we can restrict ourselves to considering Slater determinants of angular momentum eigenfunctions, which leads to Q DDM ρ DDM , ρ DDM − RL Econf (N, Z, B) ≥ inf Elin [%] − D % (6.6) = E DDM (N, Z, B) − RL .


247

Remark 6.1 (Magnitude of the “exchange term”). One may ask for the optimal magnitude of RL such that a bound of the form (6.6) is valid. RL is an upper bound on the Q difference between Econf and E DDM , which is given by the exchange energy ! " I = D(ρ , ρ ) − |xi − xj |−1 , (6.7) i<j

where is an (approximate) ground state wave function of H . The exchange energy is roughly N times the self energy of the charge distribution of one particle. This distribution has the shape of a cylinder with diameter R ∼ B −1/2 and length L ∼ max{Z −2/5 B −1/5 , Z −1 ln(B/Z 3 )−1 }, if we assume N ∼ Z (see also [LSY95] for heuristic arguments). Note that L R is equivalent to B Z 4/3 . In this case, we expect an exchange energy of the order ZL−1 ln(L/R) ∼ Z 7/5 B 1/5 ln B/Z 4/3 for Z 4/3 ( B ≤ Z 3 , (6.8) ZL−1 ln(L/R) ∼ Z 2 ln B/Z 3 ln B/Z 2 for B Z 3 . 1/2 × some factor logarithmic Hence the exchange energy should be of order Z 1/2 E DDM in B. We conjecture that (at least for appropriate in the lowest Landau band) 1 2 I ≤ const. ρ × some factor logarithmic in B, (6.9) B which is precisely of the correct order (6.8). This is in accordance with results on the homogeneous electron gas in a magnetic field [DG71,FGPY92]. Acknowledgement. The authors would like to thank Bernhard Baumgartner and Jakob Yngvason for proofreading and valuable comments.

References [BRW99] Brummelhuis, R., Ruskai, M.B. and Werner, E.: One Dimensional Regularizations of the Coulomb Potential with Applications to Atoms in Strong Magnetic Fields. arXiv:math-ph/9912020 [BSY00] Baumgartner, B., Solovej, J.P. and Yngvason, J.: Atoms in strong magnetic fields: The high field limit at fixed nuclear charge. Commun. Math. Phys. 212, 703–724 (2000) [CLR70] Cohen, R., Lodenquai, J. and Ruderman, M.: Atoms in Superstrong Magnetic Fields. Phys. Rev. Lett. 25, 467–469 (1970) [CR73] Constantinescu, D.H. and Rehák, P.: Ground State of Atoms and Molecules in a Superstrong Magnetic Field. Phys. Rev. D 8, 1693–1706 (1973) [DG71] Danz, R.W. and Glasser, M.L.: Exchange Energy of an Electron Gas in a Magnetic Field. Phys. Rev. B 4, 94–99 (1971) [FGPY92] Fushiki, I., Gudmundsson, E.H., Pethick, C.J. and Yngvason, J.: Matter in a Magnetic Field in the Thomas–Fermi and Related Theories. Ann. Physics 216, 29–72 (1992) [L81] Lieb, E.H.: A variational principle for many-fermion systems. Phys. Rev. Lett. 46, 457–59 (1981); Erratum, Phys. Rev. Lett. 47, 69 (1981) [L84] Lieb, E.H.: Bound on the maximum negative ionization of atoms and molecules. Phys. Rev. A 29, 3018–28 (1984) [LO81] Lieb, E.H. and Oxford, S.: Improved Lower Bound on the Indirect Coulomb Energy. Int. J. Quant. Chem. 19, 427–439 (1981) [LSY94a] Lieb, E.H., Solovej, J.P. and Yngvason, J.: Asymptotics of Heavy Atoms in High Magnetic Fields: I. Lowest Landau Band Regions. Commun. Pure Appl. Math. 47, 513–591 (1994) [LSY94b] Lieb, J., Solovej, J.P. and Yngvason, J.: Asymptotics of Heavy Atoms in High Magnetic Fields: II. Semiclassical Regions. Commun. Math. Phys. 161, 77–124 (1994)

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[LSY95] [LT76] [NLK86] [NKL87] [P82] [RS78] [RW00] [S00] [S91]


Lieb, J., Solovej, J.P. and Yngvason, J.: Asymptotics of Natural and Artificial Atoms in Strong Magnetic Fields. In: The Stability of Matter: From Atoms to Stars, Selecta of E.H. Lieb, Berlin– Heidelberg–New York: Springer, 1997 Lieb, E.H. and Thirring, W.: A bound on the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities. In: Studies in Mathematical Physics, Princeton: Princeton University Press, 1976, pp. 269–303 Neuhauser, D., Langanke, K. and Koonin, S.E.: Hartree–Fock calculations of atoms and molecular chains in strong magnetic fields. Phys. Rev. A 33, 2084–86 (1986) Neuhauser, D., Koonin, S.E. and Langanke, K.: Structure of matter in strong magnetic fields. Phys. Rev. A 36, 4163–75 (1987) Pröschel, P., Rösner, W., Wunner, G., Ruder, H. and Herold, H.: Hartree–Fock calculations for atoms in strong magnetic fields. I: Energy levels of two-electron systems. J. Phys. B 15, 1959–76 (1982) Reed, M. and Simon, B.: Methods of Modern Mathematical Physics IV London–New York: Academic Press, 1978 Ruskai, M.B. and Werner, E.: Study of a Class of Regularizations of 1/|x| using Gaussian Integrals. SIAM J. Math. Anal. 32, 435–463 (2000) Seiringer, R.: On the maximal ionization of atoms in strong magnetic fields. arXiv:mathph/0006002 Solovej, J.P.: Proof of the ionization conjecture in a reduced Hartree–Fock model. Invent. Math. 104, 291–311 (1991)

Communicated by B. Simon

Commun. Math. Phys. 217, 249 – 284 (2001)

Communications in



Non-local Conservation Laws and Flow Equations for Supersymmetric Integrable Hierarchies Jens Ole Madsen, J. Luis Miramontes Departamento de Física de Partículas, Facultad de Física, Universidad de Santiago de Compostela, 15706 Santiago de Compostela, Spain. E-mail: [email protected]; [email protected] Received: 21 July 1999 / Accepted: 9 April 2000

Abstract: An infinite series of Grassmann-odd and Grassmann-even flow equations is defined for a class of supersymmetric integrable hierarchies associated with loop superalgebras. All these flows commute with the mutually commuting bosonic ones originally considered to define these hierarchies and, hence, provide extra fermionic and bosonic symmetries that include the built-in N = 1 supersymmetry transformation. The corresponding non-local conserved quantities are also constructed. As an example, the particular case of the principal supersymmetric hierarchies associated with the affine superalgebras with a fermionic simple root system is discussed in detail. 1. Introduction Arguably, one of the most important works in the subject of integrable systems was that of Drinfel’d and Sokolov [1], who showed how to associate integrable hierarchies of zerocurvature equations with the loop algebra of an affine Lie algebra. Their construction and its generalizations [2] provide a systematic approach to the study and classification of many integrable hierarchies previously described by means of pseudo-differential Lax operators [3]. It is not difficult to extend the generalized Drinfel’d–Sokolov construction to the case of superalgebras. One simply has to replace the loop algebra by a loop superalgebra and include fermionic (Grassmann-odd) anticommuting fields among the dynamical degrees of freedom. However, the resulting hierarchy will not necessarily be supersymmetric. The first authors who succeeded in finding a supersymmetric generalization of the DS construction were Inami and Kanno [4] who restricted their study to the class of affine superalgebras with a fermionic simple root system. Since they made use of the principal gradation of the loop superalgebra in an essential way, their work has to be viewed as the direct generalization of the original DS construction. More recently, Delduc and Gallot [5] realized that it is possible to associate a supersymmetric integrable hierarchy of the DS type with each constant graded odd element of the loop superalgebra whose

250

J. O. Madsen, J. L. Miramontes

square = [, ]/2 is semi-simple, a condition that is obviously satisfied in the cases considered by Inami and Kanno. A common feature of the hierarchies constructed in [4] and [5] is that they consist only of bosonic flow equations. This is in sharp contrast with the supersymmetric extensions of the KP hierarchy (SKP) which always include both Grassmann-odd and Grassmanneven flow equations [6]. This observation led Kersten to find an infinite set of fermionic non-local conservation laws [7] for the supersymmetric extension of the KdV equation obtained by Manin and Radul as a reduction of the SKP hierarchy [6]. A few years later, Dargis and Mathieu showed that they actually generate an infinite sequence of non-local Grassmann-odd flows [8] (see also [9]). The aim of this paper is to show that the same is true for the whole class of supersymmetric hierarchies of [5] by constructing an infinite series of non-local (bosonic and fermionic) flow equations and conserved quantities which generalize those obtained in [7, 8] for the supersymmetric KdV equation. The new flows close a non-abelian superalgebra that has to be regarded as an algebra of symmetry transformations for the hierarchy. It includes the built-in N = 1 supersymmetry transformation and, in some cases, extended supersymmetry transformations. Following ref. [5], a generalized supersymmetric hierarchy of equations can be associated with a fermionic Lax operator of the form L = D + q(x, θ ) + , where q(x, θ) is an N = 1 Grassmann-odd superfield taking values in a particular subspace of the loop superalgebra, D is the superderivative, and is a constant graded odd element whose square = [, ]/2 is semi-simple. Delduc and Gallot defined an infinite set of mutually commuting bosonic flows and conserved quantities associated with the elements in the centre of K = Ker(ad ), which contains only even elements. In contrast, in our construction there will be a non-local flow equation and a conserved quantity for each (fermionic or bosonic) element in K with non-negative grade. These flows close a non-abelian superalgebra isomorphic to the subalgebra of K formed by the elements with non-negative grade. The authors of [2] distinguished between (bosonic) generalized DS hierarchies of type-I and type-II. The generalized DS hierarchies are associated with bosonic Lax operators of the form L = ∂x + Q(x) − , where Q(x) is a bosonic field taking values in a subspace of a loop algebra and is a constant semi-simple graded element. Then, the hierarchy is of type-I or type-II depending on whether is regular or not, i.e., a type-II hierarchy is associated with a semi-simple element such that K = Ker(ad ) is nonˆ L]/2 = D 2 +Q(x, θ )−. abelian. In [5], the analogue of L is the even Lax operator [L, Therefore, from this point of view, all the supersymmetric hierarchies of [5] have to be considered as type-II: is in K but [, ] = 2 = 0, which proves that K is non-abelian. In fact, it is straightforward to extend our construction of non-local flow equations and conserved quantities to all the type-II bosonic hierarchies of [2]. The paper is organized as follows. In Sect. 2 we briefly summarize the supersymmetric hierarchies of Delduc and Gallot. In the next two sections we present the construction of non-local flows, Sect. 3, and conserved quantities, Sect. 4. We also characterise the flows which are compatible with the supersymmetry transformation, and the conserved quantities which are supersymmetric. As an example, in Sect. 5 we give a very detailed description of the non-local flows and conserved quantities for the principal hierarchies originally considered by Inami and Kanno, i.e., those associated with superalgebras with a fermionic simple root system. We show that the complete set of local and non-local flow equations are associated with a subalgebra of the superoscillator algebra constructed by Kac and van de Leur in [10], which is the principal “super Heisenberg algebra” of

Non-local Conservation Laws and Flow Equations

251

a∞|∞ [11]. This relationship generalizes the well known role of the principal Heisenberg algebra in the original DS hierarchies, which could be used to derive a τ -function formalism for these hierarchies following the method of [12–14] and, especially, to construct solutions for the equations of the hierarchy using super vertex operator representations. It is also remarkable that, in the particular case of the affine superalgebras A(m, m)(1) , there is a local Grassmann-odd flow D¯ 1 that, together with the built-in supersymmetry transformation, closes an N = 2 supersymmetry algebra, in agreement with the results of [15, 5]. In Sect. 6 we apply our construction to the supersymmetric KdV equation in order to show how the results of [8] are recovered. As a bonus, we get additional non-local conserved quantities not obtained in previous works. Our conventions and some basic properties of Lie superalgebras are presented in Appendix B. Since we are not aware of any reference where they are available, we have included in this appendix detailed expressions for the matrix-representations of the affine Lie superalgebras with fermionic simple root systems which are needed for the understanding of Sect. 5. Recall that this class of superalgebras plays an important role in supersymmetric Toda theories and supersymmetric hamiltonian reduction and, hence, these expressions should be useful beyond the scope of this paper. Our conclusions are presented in Sect. 7. 2. Review of the Delduc–Gallot Construction Following [5], a supersymmetric (partially modified) KdV system can be associated with four data (A, d1 , d0 , ). The first, A, is a twisted loop superalgebra A = L(G, τ ) ⊂ G ⊗ C[λ, λ−1 ]

(2.1)

attached to a finite dimensional classical Lie superalgebra G together with an automorphism τ of finite order. The second and third, d1 and d0 , are derivations of A that induce two compatible integer gradations: [d0 , d1 ] = 0 . Then, the subsets Anm = Am ∩ An , where Am = {X ∈ A | [d0 , X] = mX} and An = {X ∈ A | [d1 , X] = nX}, define a bi-grading of A. We will assume that A0 ⊂ A≥0 ,

(2.2)

which means, in the notation of [5], that we restrict ourselves to the “KdV type systems”. Finally, is a constant odd element of A with positive d1 -grade, i.e., [d1 , ] = k with k > 0, whose square = 21 [, ] is semi-simple: A = Ker(ad ) ⊕ Im(ad ).

(2.3)

Moreover, has to satisfy the non-degeneracy condition1 Ker(ad ) ∩ A det B > 0. In such a case, as is seen below, the eigenvalues are all positive and form a discrete set with finite multiplicity. We want to describe the set of eigenvalues and eigenfunctions in terms of some special functions, as is the scalar-valued harmonic oscillator. The operator above possesses two kinds of non-commutativity, non-commutativity with respect to multiplication of the matrices and that with respect to differential operators. The interaction of these two results in a non-trivial relation to the “connection problem” concerning an ordinary differential operator in a complex domain. Theorem 1. There exists a third-order differential operator P (z, Dz ) so that the eigenvalue problem (1.1) is equivalent to the existence of holomorphic solutions U (z) of the differential equation P (z, Dz )U (z) = 0 (1.2) on the unit disk. The explicit form of P (z, Dz ) is given in terms of A, B and µ (in Sect. 3.2). We note that every solution u(x) of Eq. (1.1) is the sum of an even function solution ue (x) and an odd function solution uo (x) since the differential operator is invariant under changing of the variable x → −x. We can make a stronger statement regarding the eigenvalue problem for even/odd functions. Theorem 2. There exist differential operators H (w, Dw ) and P e (w, Dw ) so that the eigenvalue problem (1.1) for an odd function is equivalent to the existence of the holomorphic solution of H (w, Dw )f (w) = 0, f ∈ O(), while that for an even function is equivalent to that of P e (w, Dw )f (w) = 0,

f ∈ O().

Here H (w, Dw ) is Heun’s differential operator (3.4), that is, a Fuchsian second-order differential operator with four regular singular points. The operator P e (w, Dw ) is also a Fuchsian differential operator of third-order with four regular singular points and has an expression (4.1) in terms of Heun’s operator. In [PW], they construct the eigenfunctions and eigenvalues in terms of continued fractions determined by some three-term recurrence relation. Then this expression is a limit in some Hilbert space and is functional-analytic. Our expression given above is more complex-analytic, or even topological in the sense that the eigenvalues and eigenfunctions are determined by the monodromy representations of the Heun’s operator.

Non-Commutative Harmonic Oscillators and Fuchsian Differential Operators

359

We note that the non-commutative harmonic oscillator introduced here gives a counterexample to the naive impression that an eigenfunction of the operator naturally arising in representation theory can be expressed in terms of hypergeometric functions. In fact, Heun’s operator for general parameters is far from “hypergeometric”. For example, its solution is not of hypergeometric type, and the structure of the monodromy representation is different from that of hypergeometric equations, which has a finite expression using Gamma functions. This paper is organized as follows. In Sect. 2.1, we define appropriate coordinates in C2 for the investigation. This step is fairly easy, but the transformation in Lemma 4(ii) is a key to the analysis. In Sect. 2.2, we introduce several representations of the three dimensional simple Lie algebra sl2 . We explain the transformation T6 in detail, though these seem standard in representation theory of the semisimple Lie algebra, in order to give a formula for the inner product, which plays an important role in determining the eigenfunction. Section 2.3 represents the standard process of obtaining a single equation from a system of differential equations. In the final subsection, we transform the third-order operator into a second-order operator. In the context of classical analysis, this transformation can be expressed in terms of the (modified) Laplace transformation. In terms of the representations of sl2 , the third-order operator exists in the harmonic oscillator representation, while the second-order operator exists in the flat picture of the principal series representation. This fancy terminology is explained in Sect. 2.4. In Sect. 3.1, we introduce Heun’s operator and its relation to the differential operator under consideration. In Sect. 3.2, the inhomogeneous second-order differential equation (2.12) is proved to be equivalent to the third-order homogeneous equation (1.2). In Sect. 3.3, which is central in this paper, we translate the L2 -condition of an eigenfunction u into a holomorphy condition of its Laplace transform u. ˆ The idea here is fairly simple, and the task is to obtain a precise formulation. We next complete the proof of Theorem 1. In the last subsection, we calculate the index of the operator. In Sect. 4, we study the ordinary differential equation (1.2). We can state this problem in several (equivalent) forms, but here we choose a formulation using Heun’s operator. In Sect. 4.1, we summarize the properties of the operator P e (w, Dw ), and prove Theorem 2 for even eigenfunctions. In Sect. 4.2, we prove the equivalence of the eigenvalue problem and the connection problem. In particular, we find that the spectrum is given by the zeros of the special connection coefficient ηij (µ) of the ordinary differential equation P e (w, Dw )f (w) = 0. In Sect. 4.3, we give a proof of Theorem 2 for odd eigenfunctions. In Sect. 4.4, we introduce a monodromy representation and prove that the eigenvalue problem is equivalent to the problem of determining the set of invariants of the restricted monodromy representation. This description of a spectrum depends only on the topological data, the monodromy, of the ordinary differential equations. 2. Several Equivalent Forms of the Problem 2.1. Parmeggiani and Wakayama’s trick. We begin with the eigenvalue problem ∂2 1 x2 A − x + + B x∂x + − µI u(x) = 0. 2 2 2 We denote the standard generators of the simple Lie algebra sl2 by 0 1 0 0 1 0 , X− = . H = , X+ = 0 0 1 0 0 −1

360

H. Ochiai

These satisfy the commutation relations [H, X + ] = 2X+ ,

[H, X− ] = −2X− ,

[X+ , X− ] = H.

Next we define the oscillator representation π of sl2 by π(H ) = x∂x + 1/2,

π(X + ) = x 2 /2,

π(X − ) = −∂x2 /2.

(2.1)

We also denote the algebra homomorphism from the universal enveloping algebra U (sl2 ) to the ring C[x, ∂x ] of differential operators by the same character π . Using this representation, problem (1.1) can be stated as Aπ(X+ + X − ) + Bπ(H ) − µI u(x) = 0. (2.2)

Remark 3. Let us define the matrix

J =

0 −1 . 1 0

Since B is a skew symmetric matrix of order two, it is a multiple of J : B = pf(B)J, where the Pfaffian pf(B) is the lower-left entry of the skew symmetric matrix B. We note pf(gB t g) = (det g) pf(B). Lemma 4. (i) Diagonalization of A. There is an orthogonal matrix g1 ∈ SO(2) such that g1 Ag1−1 is a diagonal matrix. Then we see that g1 Bg1−1 = B. (ii) Parmeggiani and Wakayama’s trick [PW, Corollary 4.2]. Let g2 = (g1 Ag1−1 )1/2 =

√ α 0 √ for g1 Ag1−1 = α0 β0 . Then β 0 g2−1 g1 Ag1−1 g2−1 = I,

g2−1 Bg2−1 = √

1 det A

B,

g2−1 Ig2−1 = (g1 Ag1−1 )−1 .

(iii) Cayley transformation. Define a unitary matrix 1 1 −i g3 := √ ∈ U (2). 2 1 i Then g3 J g3−1 Here we define δ :=

1 2

= −iH,

Tr(A−1 ) =

g3 g1 A−1 g1−1 g3−1 Tr(A) 2 det A ,

and ε := −

1 ε =δ . ε 1

Tr(g1 Ag1−1 H ) . Tr(A)

4 det A Now, since Tr(g1 Ag1−1 H )2 = Tr(A)2 −4 det A, we have the relations ε 2 = 1− Tr(A) 2,

and (1 − ε2 )δ 2 det A = 1.


361

Corollary 5. Let us define pf(B) S± := X+ + X − ± i √ H ∈ sl2 . det A Then problem (2.2) is equivalent to π(S− ) − µδ −εµδ u(3) (x) = 0. −εµδ π(S+ ) − µδ

(2.3)

Here u(3) = g3 g2 g1 u. As a corollary of this proposition, we prove the positivity of the eigenvalue, µ > 0, in Sect. 4.5. 2.2. Inner automorphisms of sl2 . We change the representation π of the Lie algebra sl2 into another representation appropriate √ for the purpose. Let us introduce the parameter κ defined by the relation pf(B)/ det A = tanh κ. In other words, κ = √ 1 2

log( √det A+pf(B) ). We consider det A−pf(B)

pf(B) S± = X + + X − ± i √ H = X+ + X − ± i(tanh κ)H ∈ sl2 . det A

(2.4)

Lemma 6. We define cosh κ 1 1 1 i sinh κ cosh κ 0 0 i−sinh κ g7 = , g5 = , g4 = . , g6 = −1 1 0 1 0 1 0 1 Then, for g = g7 g6 g5 g4 , we have gS− g

−1

= (sech κ)H,

gS+ g

−1

cosh 2κ − sinh 2κ = (sech κ) . sinh 2κ − cosh 2κ

Proof. This can be obtained easily by using the relations sinh κ − cosh κ cosh κ − sinh κ igH g −1 = , g(X+ + X − )g −1 = . cosh κ − sinh κ sinh κ − cosh κ

We define the representations π4 and π5 of sl2 by π4 (Y ) := π(g4 Y g4−1 ), π5 (Y ) := π(g5 g4 Y g4−1 g5−1 )

for Y ∈ sl2 .

Then, we define the linear transformations T4 , T5 : L2 (R) → L2 (R) by √ (T4 f )(x) := (cosh κ)1/4 f ( cosh κx), (T5 f )(x) := ei(sinh κ)x

2 /2

f (x).

The map T4 is induced from the matrix g4 ∈ GL(2, R), and represents the dilation of the variable. The map T5 corresponds to the matrix g5 = exp(i sinh κX+ ) ∈ GL(2, C),

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H. Ochiai

and is equivalent to multiplication by the function ei(sinh κ)x /2 . The maps T4 and T5 preserve the standard inner product on L2 (R). We see that the map T4 intertwines the action of sl2 with π and π4 . Similarly, T5 does that with π4 and π5 . Graphically, we have the commutative diagram for any Y ∈ sl2 , 2

T5

T4

L2 (R) −→ L2 (R) −→ L2 (R) π(Y ) ↓ π4 (Y ) ↓ π5 (Y ) ↓ T4

L2 (R) −→

L2 (R)

T5

−→

L2 (R).

Next, we consider the transformation T6 that corresponds to the matrix g6 ∈ SL(2, R). ∞ In the space L2 (R), we have the inner product (f, g) = −∞ f (x)g(x)dx. We define √ √ the annihilation operator ψ = (x + ∂x )/ 2, and creation operator ψ + = (x − ∂x )/ 2. 2 Then we have [ψ, ψ + ] = 1. The function ϕ0 (x) := e−x /2 is the ground state, and by + n definition, ψϕ0 = 0. In general, we define ϕn = (ψ ) ϕ0 . Then √ the set {ϕn | n ∈ Z+ } constitutes an orthogonal basis with inner product (ϕn , ϕn ) = π n!. We denote the set of all finite linear combinations of the ϕn by L2 (Rx )fin . We also denote the set of all polynomials in one variable y by C[y]. The transformation T6 mentioned above is a linear map T6 : L2 (R)fin → C[y] and we defined it by T6 (ϕn ) = y n . We see that T6 (ψ + ϕ) = yT6 (ϕ) and T6 (ψϕ) = ∂y T6 (ϕ). Recall the definition (2.1) of the representation π of the Lie algebra sl2 on L2 (R)fin π(H ) = x∂x + 1/2, π(X + ) = x 2 /2, π(X − ) = −∂x2 /2. We define the representation of sl2 on C[y] by π (H ) = y∂y + 1/2, π (X + ) = y 2 /2, π (X − ) = −∂y2 /2.

(2.5)

Then, we calculate  −1 + − 2 2 +   π(g6 Hg6 ) = π(X + X ) = (x − ∂x )/2 = ψ ψ + 1/2, π(g6−1 X + g6 ) = π((−H + X + − X − )/2) = (ψ + )2 /2,   π(g6−1 X − g6 ) = π((−H − X + + X − )/2) = −ψ 2 /2. This proves that

π (g6 Y g6−1 )T6 = T6 π(Y ),

which implies that the following diagram for any Y ∈ sl2 : T6

C[y] L2 (R)fin −→ π(Y ) ↓ ↓ π (g6 Y g6−1 ) T6

L2 (R)fin −→

C[y].

Next, we introduce an inner product on C[y] such that the map T6 is an isometry with respect to √ this inner product. To be more explicit, the inner product should be (y m , y n ) = δmn πn!. This can be realized as √ (f, g) = π(f (∂y )g(y))| ¯ for any f, g ∈ C[y]. y=0


363

If we denote the completion of C[y] with respect to this inner product by C[y], then the map T6 can be extended to the isometry between the Hilbert spaces L2 (R) and C[y]. Finally, we define the representation of sl2 on C[y] by π6 (Y ) := π (g6 g5 g4 Y g4−1 g5−1 g6−1 ), π7 (Y ) := π (gY g −1 ),

recall g = g7 g6 g5 g4 ,

and the intertwiner T7 : C[y] → C[y] by cosh κ (T7 f )(y) := f ( i−sinh κ y). The map T7 corresponds to the matrix g7 ∈ GL(2, C) and preserves the inner product on C[y]. We thus obtain the commutative diagram T7

T6

L2 (R)fin −→ C[y] −→ C[y] π5 (Y ) ↓ π6 (Y ) ↓ π7 (Y ) ↓ T6

L2 (R)fin −→

T7

−→

C[y]

C[y].

Corollary 7. With the notation above, we have the expression π7 (S− ) = (sech κ)π (H ), π7 (S+ ) = (sech κ)π ((cosh 2κ)H − (sinh 2κ)(X + − X − )). Equation (2.3) is equivalent to π7 (S− ) − µδ −εµδ u(7) (x) = 0, −εµδ π7 (S+ ) − µδ

(2.6)

(2.7)

where u(7) = T7 T6 T5 T4 u(3) . 2.3. Reduction of a system of equations to a single equation. This is a standard argument. Lemma 8. Suppose εµ = 0. Then the system of differential equations (2.7), (7) π7 (S− ) − µδ 0 −εµδ u− (x) = , 0 −εµδ π7 (S+ ) − µδ u(7) (x) + is equivalent to the single differential equation

(7) (π7 (S+ ) − µδ)(π7 (S− ) − µδ) − (εµδ)2 u− (x) = 0. Proof. u+ = (π7 (S− ) − µδ)(u− )/(εµδ).

(2.8)

Corollary 9. Let us define an element R of the universal enveloping algebra as µδ 2 R := (2(X+ − X − ) − 2(coth 2κ)H + sinh κ )(H − µδ cosh κ) + (εµδ) coth κ ∈ U (sl 2 ).

Then, (π7 (S+ ) − µδ)(π7 (S− ) − µδ)) − (εµδ)2 = −(tanh κ)π (R).

364

H. Ochiai

We define ν := µδ cosh κ, for simplicity. Then the operator R can be written as 2(εν)2 2ν (2.9) R = 2(X+ − X − ) − 2(coth 2κ)H + sinh 2κ (H − ν) + sinh 2κ ∈ U (sl 2 ). For convenience, we next give an expression for the coefficients of R in terms of the matrices A and B and the eigenvalue µ: √ 2(1 − ε 2 ) 4(det A − det B) det A = , sinh 2κ Tr(A)2 pf(B) det A + det B 2 coth 2κ = √ , det A pf(B) Tr(A) ν= √ µ. 2 (det A − det B) det A 2.4. Laplace transform. We recall the realization given in (2.5):   π (H ) = y∂y + 1/2, π (X + ) = y 2 /2,  π (X − ) = −∂ 2 /2. y We now introduce a second realization (representation) of the sl2 -triple:   7 (H ) = z∂z + 1/2 = θz + 1/2, 7 (X+ ) = z2 ( 21 z∂z + 1) = z2 ( 21 θz + 1),  7 (X − ) = − 1 ∂ + 1 = − 1 (θ − 1). 2z z 2z2 2z2 z

(2.10)

Here, θz = z∂z denotes Euler’s degree operator. The relation between these two realizations is given by the following. Proposition 10. We define the (modified) Laplace transform ∞ z∞ 2 2 2 u(z) ˆ := u(yz)e−y /2 ydy = z−2 u(y)e−y /(2z ) ydy. 0

0

This is a linear map from y-space to z-space with the following properties: √ (i) It gives y n → 9( n2 + 1)( 2z)n for any n ∈ Z+ . In particular, 1 → 1. ∞ ∞ √ n (ii) Suppose the expansion u = un y n ∈ C[y], then u(z) ˆ = un 9( + 1)( 2z)n . 2 n=0 n=0 (iii) The Laplace transform almost intertwines the action of sl2 :   ˆ (π (H )u)ˆ = 7 (H )(u) (2.11) (π (X + )u)ˆ = 7 (X + )(u) ˆ  (π (X − )u)ˆ = 7 (X − )(u) 1 −2 ˆ − 2 u(0)z . (iv) If we define the inner product in z-space such that {zn | n ∈ Z+ } forms an orthogonal n basis and (zn , zn ) = 9( n+1 2 )/ 9( 2 + 1), then the Laplace transformation is an isometry.


365

Proof. (iii) This is proved by the calculation

∞ 1 1 2 −y 2 /2 − u (yz)e ydy = u (yz)(e−y /2 y) dy, (π (X )u)ˆ(z) = 2 2z 0 0 ∞ 1 1 −y 2 /2 2 ˆ =− u (yz)e y dy, − ∂z u(z) 2z 2z 0 ∞ 1 1 2 u(z) ˆ = − u(yz)(e−y /2 ) dy 2 2 2z 2z 0

y=∞ ∞ 1 2 2 = − 2 u(yz)e−y /2 − zu (yz)e−y /2 dy . y=0 2z 0

−

(iv) Since (y n , y n ) = (zn , zn ) =

∞

√ πn!, we have √ 9( 21 )9(n + 1) 9( n+1 π n! 2 ) = = . n n n n 2 n 2 2 9( 2 + 1) 2 9( 2 + 1) 9( 2 + 1)

n Note that 9( n+1 2 )/ 9( 2 + 1) =

π/2

sinn xdx. Since 9(z + a)/ 9(z + b) 0 √ n n (z , z ) ∼ 2/n.

√2 π

∼ za−b ,

asymptotically for large n, we have The property (iii) means that the representations π and 7 do not intertwine, but almost. The precise relation is given as follows. In fact, the Casimir operator 4X + X − + H 2 − 2H ∈ U (sl2 ) takes the value −3/4 in both representations. Let us denote the set of all Laurent polynomials (finite linear combinations of {zn | n ∈ Z}) by C[z, z−1 ]. The subspace C[z2 , z−2 ] consists of the set of all even functions in C[z, z−1 ]. The set of even polynomials is written C[z2 ], while the set of odd polynomials is written zC[z2 ]. By the definition of 7 , (2.10), we see that the representation (7, C[z, z−1 ]) has a subrepresentation (7, z−2 C[z−2 ]). The representation (π , C[y 2 ]) is isomorphic to the quotient representation (7, C[z2 , z−2 ]/z−2 C[z−2 ]). In other words, the even part (π , C[y 2 ]) is the Langlands quotient of the representation (7, C[z2 , z−2 ]). On the other hand, (7, zC[z2 ]) is a subrepresentation of (7, C[z2 , z−2 ]). This subrepresentation is isomorphic to the odd part (π , yC[y 2 ]). Corollary 11. The operator π (R) satisfies the relation 1 −2 (π (R)u)ˆ = 7 (R)(u) ˆ + ( − ν)u(0)z . ˆ 2 It follows from this corollary that Eq. (2.8) is equivalent to 1 −2 7 (R)(u) ˆ + ( − ν)u(0)z = 0. ˆ 2

(2.12)

Remark. Parmeggiani and Wakayama [PW] have already obtained part of the results in this section, and stated it in different terminology. For example, their recurrence equations (13) in [PW] corresponds to the differential equation (2.12). However, the Laplace transformation seems new.

366

H. Ochiai

3. Laplace Transform of an Eigenfunction 3.1. Heun’s differential operator. Substituting the realization (2.10) into (2.9), we obtain the realization 2ν 7 (R) = z2 (θz + 2) + z−2 (θz − 1) − 2(coth 2κ)(θz + 21 ) + sinh 2κ 2

2(εν) · (θz + 21 − ν) + sinh 2κ , 7 (R) = (z2 + z−2 − 2 coth 2κ)(θz + 21 ) + 23 (z2 − z−2 ) +

· (θz +

1 2

− ν) +

2ν sinh 2κ

2(εν)2 sinh 2κ .

(3.2)

Conjugating by z, we have z−1 7 (R)z = (z2 + z−2 − 2 coth 2κ)(θz + 23 ) + 23 (z2 − z−2 ) + · (θz +

3 2

(3.1)

− ν) +

2ν sinh 2κ

2(εν)2 sinh 2κ .

Since the operators 7 (H ), 7 (X + ) and 7 (X − ) are invariant under the symmetry z → −z, the operator 7 (R) is invariant under the same symmetry. This can be seen in expression (3.1) or (3.2). This implies that the operator 7 (R) can be written in terms of the variable z2 . Now let us introduce the new variable w := z2 coth κ. Then, factoring by the leading coefficient, from the above expression, we obtain z−1 7 (R)z = 4(tanh κ)w(w − 1)(w − coth2 κ)H (w, Dw ),

(3.3)

where, H (w, Dw ) is Heun’s differential operator [H, (1.1.1)], H (w, Dw ) := ∂w2 + (

δ ; γ αβ w − q + + )∂ + , w w w − 1 w − a w(w − 1)(w − a )

(3.4)

with the parameters 7 − 2ν 3 − 2ν , δ = , 4 4 3 3 − 2ν α = , β = , 2 4

γ =

and q =

; =

3 + 2ν , 4

a = coth2 κ

4ν 2 (1 − ε 2 ) − 12ν cosh2 κ + 9 cosh 2κ . 16 sinh2 κ

Recall that 1 − ε2 =

4 det A , Tr(A)2

coth2 κ =

det A , det B

Tr(A) ν= √ µ. 2 (det A − det B) det A

In particular, H (w, Dw ) is a second-order linear differential operator with four regular singular points on the Riemannian sphere. In terms of the Riemann schema (a P -symbol), the list of the exponents [H, (1.1.3)] is given as     ∞ 0 1 coth2 κ 0 1 a ∞   3  0 0 0 α w q  =  0 0 0 w q  . 2 −2ν+1 −2ν+3 2ν−3 2ν+1 1−γ 1−δ 1−; β 4

4

4

4


367

In particular, the parameter κ designates the locations of the singular points, the parameter ν designates the exponents, and the parameter ε is an accessory parameter. The above discussion gives the following: Proposition 12. The operator 7 (R) has the following properties: (i) It is a second-order linear ordinary differential operator and has rational coefficients in z. √ √ (ii) It has six singular points z = 0, ± tanh κ, ± coth κ, ∞. (iii) All singularities are regular singularities. (iv) The exponents at z = 0 are 1 and √ (2ν −1)/2. Those at z = ∞ are 2 and −(2ν −1)/2. Those √ at the points z = ± tanh κ are 0 and (2ν + 1)/4. Those at the points z = ± coth κ are 0 and (−2ν + 1)/4. Proof. By the conjugation of w 1/2 , the operator 7 (R) has the P -symbol   0 tanh κ coth κ ∞   1 0 0 1 z2  .  2 2ν−1 4

2ν+1 4

−2ν+1 4

−2ν+1 4

By unfolding z2 → z, this P -symbol is equal to  √ √ √ √ tanh κ − tanh κ coth κ − coth κ 0  0 0 0 0  1 2ν−1 2

2ν+1 4

This proves the assertion (iv).

2ν+1 4

−2ν+1 4

−2ν+1 4

∞ 2 −2ν+1 2

  z .

3.2. The differential equation. We the third-order differential operator P (z, Dz ) define := (z∂z + 2)7 (R) = 7 H + 23 R . This is the operator in Theorem 1 in the Introduction. Lemma 13. The following conditions for a holomorphic function u(z) ˆ ∈ O0 at the origin are equivalent: −2 = 0 (that is, the condition (2.12)). (i) 7 (R)(u) ˆ + ( 21 − ν)u(0)z ˆ 1 2 ˆ = 0. (ii) (z 7 (R))uˆ + ( 2 − ν)u(0) 2 (iii) ∂z z 7 (R)uˆ = 0. (iv) P (z, Dz )uˆ = 0.

Proof. The relations (i) ⇔ (ii) ⇒ (iii) ⇔ (iv) are clear. Since the constant term on the left-hand side of (ii) vanishes for an arbitrary holomorphic function u(z) ˆ ∈ O0 , the condition (iii) implies (ii). Now we summarize the properties of the operator P (z, Dz ). We see that the operator P (z, Dz ) is holomorphic on C. This follows from (3.1) and the formula (θz +2)z−2 (θz − 1) = ∂z2 . Also, the coefficient of ∂z3 of P (z, Dz ) is z3 (z2 + z−2 − 2 coth 2κ) = z(z2 − 2 − coth κ). The order of the zeros of this coefficient at the singular points tanh κ)(z√ √ z = 0, ± tanh κ and ± coth κ is 1.

368

H. Ochiai

3.3. L2 -conditions and analytic continuation. We now consider the holomorphic solution u(z) ˆ ∈ O0 satisfying the equation P (z, Dz )uˆ = 0 (see Lemma 13). The solution can be analytically continued along a path avoiding the singular points of the differential equation. In particular, the solution is holomorphic on the (open) disk of √ radius √ | tanh κ|. We consider the behavior of the solution near the points z = ± tanh κ. There are two possibilities: √ • If a solution u(z) ˆ is holomorphic near the points z = ± tanh κ, then it is√continued to a (single-valued) holomorphic function on the disk {z ∈ C | |z| < | coth κ|} ∞ centered at the origin. In this case, the Taylor expansion u(z) ˆ = an zn satisfies the n=0

following asymptotic property: ∀ε1 > 0, ∃N such that ∀n > N,

|an | ≤ (| tanh κ| + ε1 )n/2 .

• If a solution u(z) ˆ cannot be holomorphically continued to at least one point z = ∞ √ ± tanh κ, then the radius of convergence of the Taylor series u(z) ˆ = an zn is n=0

√ | tanh κ|. In this case, ∀ε1 > 0 and ∀N, ∃n > N such that

|an | ≥ (| coth κ| − ε1 )n/2 .

Proposition 14. Consider a formal power series solution u(y) =

∞

un y n ∈ C[[y]] of

n=0

the equation π (R)u = 0. Its Laplace transform is written as u(z) ˆ ∈ C[[z]]. Then, we have the following: (i) Since the operator is regular singular at the origin, any formal power series solution u(z) ˆ ∈ C[[z]] of the equation P (z, Dz )uˆ = 0 converges to a holomorphic function near the origin. (ii) The following conditions are equivalent: (a) u(y) converges in the Hilbert space C[y]. (b) u(z) ˆ converges to a holomorphic function on the unit disk. (c) Both the even part uˆ e (z) and the odd part uˆ o (z) of u(z) ˆ can √ be holomorphically continued to a neighbourhood of the closed interval [0, tanh κ]. (iii) The following conditions are equivalent: (a) u(y) does not converge in the Hilbert space C[y]. (b) u(z) ˆ cannot be continued to a holomorphic function on the unit disk. (c) Either the even part uˆ e (z) or the odd part uˆ o (z) of u(z) ˆ cannot√be holomorphically continued to a neighbourhood of the closed interval [0, tanh κ]. Proof. We set an = un 9( n2 + 1)2n/2 . Then the condition (ii-b) and (iii-b) imply |un |2 (y n , y n ) =

√ |2 πn!

 n+1 ≤ (| tanh κ| + ε1 )n 9(n 2 )

|an 9( 2 +1) n+1 9( n2 + 1)2 2n ≥ (| coth κ| − ε1 )n 9(n 2 ) 9( +1) 2

for case (ii), for case (iii).

(3.5)


369

This proves the equivalences (ii-a) ⇔ (ii-b) and (iii-a) ⇔ (iii-b). Then, (ii-b) implies (ii-c) by the relations uˆ e (z) = (u(z) ˆ + u(−z))/2, ˆ uˆ o (z) = (u(z) ˆ − u(−z))/2. ˆ Conversely,√the condition (ii-c) implies that the even function uˆ e (z) is also holomorphic at z = − tanh κ. Since this function is a solution of the differential equation, it can be holomorphically continued to regular singular points. Thus, it is holomorphic on the unit disk. The same holds for the odd part uˆ o (z). This proves (ii-b) for uˆ = uˆ e + uˆ o . Remark 15. If we replace the Hilbert space L2 (R) by some Sobolev-type space, then the estimate (3.5) requires some additional factors such as (1 + n2 )σ . This change affects neither the statement nor the proof of the proposition. Remark 16. The unit disk can be replaced by a connected and simply-connected open √ subset of C √ which contains the three points 0, ± tanh κ and which contains neither of the points ± coth κ. In fact, Proposition 14 remains valid if we replace the unit disk by such a domain in the statements of conditions (ii-b) and (iii-b). We denote the set of holomorphic functions on by O( ). The germ of holomorphic functions at the origin is denoted by O0 . To summarize the above discussion, the spectral problem (1.1) is equivalent to that of finding all the holomorphic solutions U (z) ∈ O( ) of the differential equation P (z, Dz )U (z) = 0. 3.4. Index of the operator. Lemma 17. The index of the map P (z, Dz ) : O( ) → O( )

(3.6)

is zero. Proof. The index is, by definition, the difference of the dimensions of the kernel and cokernel. It is known (e.g., see [K]) that the map (3.6) is continuous, it has a closed range, the dimensions of its kernel and cokernel are both finite, and its index is equal to the difference of the order of the operator P (z, Dz ) and the number of zeros with multiplicity in the domain . Thus the index of the map in question is 3 − 1 × 3 = 0. Corollary 18. The following conditions are equivalent: (i) The map (3.6) is injective. (ii) The map (3.6) is surjective. (iii) The map (3.6) is bijective. If one takes a monomial basis {zn | n ∈ Z+ } of C[[z]], the condition (iii) of Corollary 18 can be expressed in terms of the “determinant” of a tridiagonal, by (3.1), matrix of infinite size. This determinant should coincide with the function f introduced in [PW], where this function is constructed with the limit of some continued fractions. If the determinant is zero, then we have an infinite series solution in C[[z]] = Oˆ 0 . Since the operator P (z, Dz ) has regular singularity at the origin (Proposition 14(i)), such a formal solution converges to a holomorphic solution at the origin.

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4. Connection Problem In this section, we discuss the relation between our problem and the connection problem of a linear ordinary differential equation in a complex domain. It is instructive to relate our work with Heun’s operator, and therefore we work in the variable w in this section. We set = {z ∈ C | |z| < 1}, = {w ∈ C | |w| < | coth κ|}. 4.1. Expression. Every solution f ∈ O( ) of P (z, Dz )f = 0 is the sum of an even solution f e (z) and an odd solution f o (z). By the map w = z2 coth κ, the set of even functions √ in O( ) is isomorphic to O(). The set of odd functions in O( ) is identified with wO(). We denote the differential operator P (z, Dz ) written in terms of w by √ P e (w, Dw ). This operator acts on O() and wO() as √ √ P e (w, Dw ) : O() → O(), P e (w, Dw ) : wO() → wO(). Using the variable w, we have 7 (R) = 2w(tanh κ)(θw + 1) + w −1 (coth κ)(2θw − 1) 2ν 1 − 2(coth 2κ)(2θw + 21 ) + sinh 2κ (2θw + 2 − ν) +

2(εν)2 sinh 2κ ,

where θw = w∂w = 21 θz . By this expression w7 (R) is holomorphic on w ∈ C. We see that P e (w, Dw ) = (θz + 2)7 (R) = 2(θw + 1)7 (R) = 2∂w w7 (R). In other words, P e (w, Dw ) can be factorized into a product of the two differential operators ∂w and w7 (R) holomorphic on C in the variable w. By (3.3), we have P e (w, Dw ) = 8(tanh κ)∂w · w 2 (w − 1)(w − coth2 κ) ·

√

√ −1 wH (w, Dw ) w . (4.1)

Thus the exponents of P e (w, Dw ) are given by the P -symbol   ∞ 0 1 coth2 κ      0  1 1 2   ν

2

1 2

−

1 4

ν 2

0 +

1 4

0 − ν2 +

1 4

1 − ν2 +

1 4

w  

.

The new exponent is 2 − 2 = 0 at w = 0, it is 2 − 1 = 1 at w = 1, coth2 κ, and it is −(2 − 4) = 2 at w = ∞. 4.2. Connection problem. We illustrate the corresponding connection problem briefly. For simplicity, we assume in this section that the parameter ν satisfies ν + 21 ∈ / Z+ . First, consider the differential equation w7 (R)U (w) = 0. Choose a basis {f01 , f02 } of solutions near w = 0 such that the exponent of f01 is 1/2 and that of f02 is (2ν −1)/4. √ −1 Then w f01 is holomorphic and f02 is not. If (2ν − 1)/4 ∈ / 21 − Z+ , then f02 is ambiguous, but this is irrelevant to the present discussion. Now, choose a basis {f11 , f12 } of solutions near w = 1 such that the exponent of f11 is 0 and that of f12 is (2ν + 1)/4. Then f11 is holomorphic and f12 is not.


371

Next we select a basis of the solution of the differential equation P e (w, Dw )U (w) = 0. Let f00 be a solution with exponent 0 near w = 0. This solution is holomorphic. Then, let f10 be a solution with exponent 1 near w = 1. This is also holomorphic. The space ker(P e , O0 ) of√holomorphic solutions at the origin is spanned by f00 . Similarly, the space ker(P e , wO0 ) is spanned by f01 . For the point w = 1, the space ker(P e , O1 ) is spanned by f10 and f11 . Now we consider the connection matrix along the open interval (0, 1): f0j =

2

f1i ηij ,

j = 0, 1, 2.

i=0

By definition, the function ηij = ηij (ν, κ, ε) does not depend on w, but it may depend on the parameters ν, κ, and ε. Since {fkj | j = 1, 2} forms a basis of solutions of the equation w7 (R)U (w) = 0, we have η01 = η02 = 0. The restriction map from to the origin gives a natural map ker(P e , O()) → ker(P e , O0 ). We characterize the image of this map by the connection coefficient. Proposition 19. (i) The (even) function f00 ∈ ker(P e , O0 ) comes from a function in ker(P e , O()) if and only if η20 = 0.√ e e (ii) The √ (odd) function f01 ∈ ker(P , wO0 ) comes from a function in ker(P , wO()) if and only if η21 = 0. In other words, provided that ν is “non-integral”, then ν is an eigenvalue of the original eigenvalue problem if and only if η20 η21 = 0. For an expression of the connection coefficients and other quantities, see, e.g., [F, KS]. 4.3. Odd case. For the odd case, the connection to Heun’s operator is more direct. Let us consider √ √ P e (z, Dz ) : wO() → wO(). Lemma 20. We have the isomorphism √ √ ker(P e , wO()) = w ker(H (w, Dw ), O()). √ √ Proof. First, we see that wf ∈ ker(P e , wO()) if and only if √ −1 √ f ∈ ker( w P e (w, Dw ) w, O()). Then, by (4.1), we have √ −1 e √ √ −1 √ w P (w, Dw ) w = 8(tanh κ) w ∂w w · w 2 (w − 1)(w − coth2 κ)H (w, Dw ). √ −1 √ Finally, since w · ∂w · w is bijective on O(), the kernel of the operator is given by the kernel of H (w, Dw ). Remark 21. The operator w(1 − w)H (w, Dw ) : O() → O() has index zero.

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The exponents of this operator are, as is in Sect. 3.1, given by   ∞ 1 coth2 κ  0  3 0 0 0 w . 2 ν  3 ν 1 ν 1 ν 3 − + − + − + 2 4 2 4 2 4 2 4 Then if (2ν + 1)/4 ∈ / Z+ , the eigenvalue problem remains equivalent to η21 = 0. In particular, this is the case for (2ν − 1)/4 ∈ Z+ , which was excluded in Sect. 4.2. / Z+ . 4.4. Monodromy representation. In this subsection, we assume that −(ν + 23 ) ∈ Since all the eigenvalues ν satisfy ν > 0, as seen in Sect. 4.5, this assumption is harmless. We choose a base point w0 such that 0 < w0 < 1 and fix a basis of ker(P , Ow0 ) ∼ = C3 . Then the monodromy defines the representation of the fundamental group π1 (C \ {0, 1, coth2 κ}) → GL(3, C). Now, we consider the subgroup F2 = π1 ( \ {0, 1}), which is isomorphic to the free group of two generators. Restricting the above representation to the subgroup F2 , we have a representation, ρ : F2 → GL(3, C), which is called a restricted monodromy representation. Proposition 22. The set of invariants of the restricted monodromy representation ρ is isomorphic to the solutions ∼

ker(P : O() → O()) → (C3 )ρ(F2 ) . We take two loops γ0 , γ1 ∈ π1 ( \ {0, 1}) around 0, 1, respectively, and consider the corresponding local monodromy matrices ρ(γ0 ), ρ(γ1 ) ∈ GL(3, C). Then the set of invariants is equivalent to the common kernel of these two matrices: (C3 )ρ(F2 ) = {f ∈ C3 | ρ(γ0 )f = f, ρ(γ1 )f = f }. Example 23. For a non-generic parameter ν, we have the following properties for local monodromy matrices: (i) The case (2ν − 1)/4 ∈ Z+ . The matrix ρ(γ1 ) is semisimple with eigenvalues 1, 1 and −1, while the matrix ρ(γ0 ) has eigenvalues 1, 1 and −1. Then the dimension of the common kernel is 0, 1, or 2. (ii) The case (2ν + 1)/4 ∈ Z+ . The matrix ρ(γ0 ) has eigenvalues 1, −1 and −1. The eigenspace for the eigenvalue 1 is one-dimensional. The matrix ρ(γ1 ) has eigenvalues 1, 1 and 1. The dimension of the common kernel is 0 or 1. (iii) The case of odd functions with (2ν + 1)/4 ∈ Z+ . We can define the restricted monodromy representation for the operator appearing in Sect. 4.3: ρ o : F2 → GL(2, C). We have a relation similar to that appearing in Proposition 22, ∼

ker(H (w, Dw ), O()) → (C2 )ρ

o (F ) 2

= {f ∈ C2 | ρ o (γ0 )f = ρ o (γ1 )f = f }.

Both the matrices ρ o (γ0 ) and ρ o (γ1 ) are unipotent (with eigenvalues 1, 1). The dimension of the kernel is 0, 1, or 2.


373

4.5. of eigenvalues. Assume Eq. (2.3) holds for the function u(3) (x) = Positivity u− (x) . Then u+ (x) u¯ − π(S− )u− + u¯ + π(S+ )u+ = µδ(|u+ |2 + |u− |2 − ε(u¯ + u− + u¯ − u+ )) = µδ((1 − ε)(|u+ |2 + |u− |2 ) + ε|u+ + u− |2 ). On the other hand, the integration by parts implies that ∞ 1 u¯ ± π(S± )u± dx = #xu± #2 + #∂x u± #2 2 −∞

± (tanh κ)($xu± , ∂x u± % + $∂x u± , xu± %) .

The right-hand side is non-negative, and nonzero unless u± = 0. Then for non-zero u(3) , we have the required positivity µδ > 0 and ν = µδ cosh κ > 0. Acknowledgement. The author thanks Professor Masato Wakayama for fruitful discussions, and the referee for valuable comments.

References [E] [F] [H] [HT] [KS] [K] [PW] [SS1] [SS2]

Erdelyi, A. et al.: Higher transcendental functions. Vol. III, New York: McGraw-Hill, 1955 Fodoryuk, M.V.: Asymptotics of the spectrum of the Heune equation and of Heune functions Izv. Akac. Nauk SSSR Ser. Mat. 55, 631–646 (1991) Heun’s Differential Equations. With contributions by F. M. Arscott, S. Yu. Slavyanov, D. Schmidt, G. Wolf, P. Maroni and A. Duval, Edited by A. Ronveaux, Oxford: Oxford University Press, 1995 Howe, R. and Tan, Eng-Chye: Nonabelian harmonic analysis. Applications of SL(2, R). Universitext, Berlin–Heidelberg–New York: Springer-Verlag, 1992 Kazakov, A.Ya. and Slavyanov, S.Yu.: Integral relations for special functions of the Heun class. Teoret. Mat. Fiz. 107, 388–396 (1996) Komatsu, H.: On the index of ordinary differential operators. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 18, 379–398 (1971) Parmeggiani, A. and Wakayama, M.: Non-commutative harmonic oscillators. To appear in Forum Math. Schäfke, R. and Schmidt, D.: Connection problems for linear ordinary differential equations in the complex domain. Lecture Notes in Math. 810. Berlin–Heidelberg–New York: Springer-Verlag, 1980, pp. 306–317 Schäfke, R. and Schmidt, D.: The connection problem for general linear ordinary differential equations at two regular singular points with applications in the theory of special functons. SIAM J. Math. Anal. 11, 848–862 (1980)

Communicated by T. Miwa

Commun. Math. Phys. 217, 375 – 382 (2001)

Communications in



On Minimal Eigenvalues of Schrödinger Operators on Manifolds Pedro Freitas Departamento de Matemática, Instituto Superior Técnico, Av.Rovisco Pais, 1049-001 Lisboa, Portugal. E-mail: [email protected] Received: 17 July 2000 / Accepted: 11 October 2000

Abstract: We consider the problem of minimizing the eigenvalues of the Schrödinger operator H = − + αF (κ) (α > 0) on a compact n-manifold subject to the restriction that κ has a given fixed average κ0 . In the one-dimensional case our results imply in particular that for F (κ) = κ 2 the constant potential fails to minimize the principal eigenvalue for α > αc = µ1 /(4κ02 ), where µ1 is the first nonzero eigenvalue of −. This complements a result by Exner, Harrell and Loss, showing that the critical value where the constant potential stops being a minimizer for a class of Schrödinger operators penalized by curvature is given by αc . Furthermore, we show that the value of µ1 /4 remains the infimum for all α > αc . Using these results, we obtain a sharp lower bound for the principal eigenvalue for a general potential. In higher dimensions we prove a (weak) local version of these results for a general class of potentials F (κ), and then show that globally the infimum for the first and also for higher eigenvalues is actually given by the corresponding eigenvalues of the Laplace– Beltrami operator and is never attained. 1. Introduction In the last years there has been a great interest in the study of optimal properties of eigenvalues of Schrödinger operators of the form H = − + V defined on compact manifolds, when some restrictions are imposed on the potential V . Some of these problems are related to several physical phenomena such as motion by mean curvature, electrical properties of nanoscale structures, etc. (see, for instance, [A,AHS, EI, EHL, HL, Ke] and the references therein). In [EHL], the authors considered the case of potentials depending on the curvature κ and studied the problem of minimizing the first eigenvalue of the operator H = Partially supported by FCT, Portugal

376

P. Freitas

−d 2 /ds 2 + ακ 2 defined on a closed planar curve with length one. They proved that for 0 < α < 1/4 the circle is the unique minimizer, while for α > 1 this is no longer the case, leaving open the question of the value of α where the transition takes place, and also what happens after this critical value. In this paper we consider an operator H defined on a compact n-manifold (M, g) by H = − + αF (κ) and with eigenvalues λ0 < λ1 ≤ . . . , and study the problem of determining j (α) = inf λj (κ), j = 0, 1, . . . , κ∈K

where

1 K = κ ∈ C(M; R) : κdvg = κ0 . |M| M

In particular, we are interested in knowing whether or not there exists a critical value of α, say αc , where the constant potential stops being a global minimizer for the first eigenvalue. We show that in the one-dimensional case studied in [EHL] this critical value equals 1/4, and that for α larger than αc the infimum is identically equal to π 2 and is not attained. Note that here we do not impose any restrictions on κ, and thus the problem of what happens over closed planar curves remains open. The first part of this result is a consequence of a more general result which provides an upper bound for αc holding in any dimension. Furthermore, we show that for potentials of the form κ = κ0 + εq, where q has zero average, this bound is in fact precise for sufficiently small values of ε, in the sense that for α smaller than the bound, the constant potential κ0 gives a smaller eigenvalue than κ, while for larger values of α this is not always the case. These results could lead us to expect that results similar to those in one dimension would also hold in higher dimensions, that is, that there would exist a nontrivial interval (0, αc ) where the constant potential was the unique minimizer. However, it turns out that for dimensions higher than the first there exist potentials satisfying the given restrictions and which make the principal eigenvalue as close to zero as desired. Thus, we see that in this case the constant potential is never a global minimizer. It remains an open question if it is a local minimizer. A similar statement also holds for higher eigenvalues and for minimizations subject to other types of integral restrictions – see Theorem 3 and the remarks that follow it. The reason for this different behaviour in dimensions higher than the first is directly related to the fact that in this case, given a manifold M and a geodesic ball Bδ of radius δ centred at a point x0 in M, the Dirichlet eigenvalues of the Laplacian in !δ = M \ Bδ converge to those of the Laplacian in M as δ approaches zero – for a more precise statement of this property see Sect. 4 and [CF]. Finally, we point out that the results in one dimension enable us to obtain a lower bound for the principal eigenvalue in the case of a general potential (Corollary 1) which, of course, corresponds also to the first eigenvalue of Hill’s equation. Note that one of the motivations behind the study of the minimization of eigenvalues when the potential is subject to integral restrictions was precisely to obtain lower bounds for eigenvalues – see [Ke]. 2. Notation and General Local Results Let (M, g) be a compact Riemannian n-manifold with metric g and let − denote the Laplace–Beltrami operator defined on M with eigenvalues 0 = µ0 < µ1 ≤ . . .

On Minimal Eigenvalues of Schrödinger Operators on Manifolds

377

repeated according to their multiplicities. Denote the corresponding orthonormal (with respect to the L2 (M) inner product induced by the Riemannian measure vg ) system of ∞ eigenfunctions by vj j =0 . Consider now the operator defined on M by H = − + αF (κ), where 1 κdvg = κ0 , |M| M and F : R → R is assumed to be of class C 3 in a neighbourhood of κ0 . The main result in this section is then the following Theorem 1. Assume that F (κ0 ) = 0 and define α∗ =

µ1 F (κ0 ) 2[F (κ0 )]2

.

Then, if q is a continuous real valued function with zero average and not identically zero, we have that for κ = κ0 + εq with sufficiently small ε (depending on q and α), the principal eigenvalue λ0 of H satisfies λ0 (κ) > αF (κ0 ), if 0 < α < α ∗ , while for α > α ∗ there exist functions q as above for which λ0 (κ) < αF (κ0 ). Proof. Consider the Schrödinger operator defined on M by Hε = − + αF (κ0 + εq). Since M is compact, the spectrum of Hε is discrete and its first (simple) eigenvalue and the corresponding (normalized) eigenfunction are analytic functions of the (real) parameter ε [Ka]. We thus expand λ0 and the corresponding eigenfunction u as a power series of ε around zero: λ0 = $0 + $1 ε + $2 ε 2 + . . . , u = φ0 + φ1 ε + φ2 ε 2 + . . . . On the other hand, we also have that F (κ0 + εq) = f0 + f1 qε + f2 q 2 ε 2 + o(ε 2 ), where 1 F (κ0 ). 2 Substituting these expressions in the equation giving the eigenvalues for Hε we obtain, equating like powers in ε, f0 = F (κ0 ), f1 = F (κ0 ), and f2 =

ε0 : −φ0 + αf0 φ0 = $0 φ0 , ε1 : −φ1 + αf0 φ1 + αf1 qφ0 = $0 φ1 + $1 φ0 , ε2 : −φ2 + αf0 φ2 + αf1 qφ1 + αf2 q 2 φ0 = $0 φ2 + $1 φ1 + $2 φ0 . From the first equation it follows that $0 = αf0 and that φ0 is constant, which we take to be one. Substituting this in the equation for ε1 and integrating over M gives that $1 vanishes and φ1 satisfies −φ1 = −αf1 q.

(1)

378

P. Freitas

Substituting now this in the last equation gives that φ2 satisfies −φ2 = −αf2 q 2 − αf1 qφ1 + $2 . Again integrating over M gives $2 =

αf2 |M|

q 2 dvg +

M

αf1 |M|

M

qφ1 dvg .

(2)

Taking squares on both sides of (1) we get [(φ1 )]2 = α 2 f12 q 2 . On the other hand, multiplying the same equation by φ1 and integrating over M gives that qφ1 dvg = − |∇φ1 |2 dvg . αf1 M

M

Substituting these two expressions into (2) we finally obtain f2 1 2 (φ ) dv − |∇φ1 |2 dvg , $2 = 1 g |M| M αf12 |M| M and it follows from Lemma 1 below that $2 is always positive for α < α ∗ . To give an example of a function q for which $2 becomes negative when α > α ∗ it is sufficient to take q = v1 . We obtain from (1) that in this case φ1 = c −

α f1 v1 , µ1

where c is an arbitrary constant. Substituting this into the expression for λ2 yields α α 2 f , f2 − $2 = |M| µ1 1 which is negative for α > α ∗ . An obvious consequence of this result is that for all F of the form above there exists a value of α, say α ∗∗ such that for α > α ∗∗ the constant potential is not a minimizer of the first eigenvalue. In the case where F is allowed to vanish, it is also clear that if κ0 = κ0∗ is a (local) minimizer (resp. maximizer) of F , it follows that, for positive values of α, κ(x) ≡ κ0∗ will be a (local) minimizer (resp. maximizer). This is the case, for instance, when F (κ) = κ 2 and κ0 = 0, where obviously κ = 0 is a global minimizer for all α. The result needed to prove that $2 > 0 for α < α ∗ is neither new nor difficult, but a specific reference could not be found in the literature and so, for the sake of completeness, we provide a proof here. Lemma 1. The functional Iα (u) = is nonnegative for α ≥ 1/µ1 .

M

α(u)2 − |∇u|2 dvg


379

Proof. The spectral problem corresponding to Iα is α2 u + u = γ u,

(3)

which has discrete spectrum γ0 ≤ γ1 ≤ . . . . We will prove that if α > 1/µ1 then γj ≥ 0 for all j = 0, 1, . . . . To this end rewrite (3) as (αu + u) = γ u. It is not difficult to see that u is an eigenfunction if and only if αu + u = βvj for some real number β different from zero. For α > 1/µ1 the operator α + I is invertible and thus this last equation has one and only one solution given by u = βvj /(1− αµj ). Substituting this into (3) gives γ = (αµj − 1)µj from which the result follows. 3. The One-Dimensional Case In this section we consider the particular case studied in [EHL] with F (κ) = κ 2 , and for which µ1 α∗ = 2 . 4κ0 As a consequence of Theorem 1 and the results in [EHL] we have the following Theorem 2. In the one dimensional case and for F as above, αc = α ∗ . Furthermore, for α > αc , 0 (α) ≡ µ1 /4. Proof. It only remains to show the result for α larger than αc . Clearly in this case 0 (α) ≥ µ1 /4. Consider now the family of potentials given by κ0 /δ, 0 < s < δ, κδ (s) = 0, δ < s < $. Note that although κδ is not continuous on the circle, it can be approximated by continuous functions without affecting our results. For this family of potentials we obtain the functional $ 2 ακ02 δ 2 Jδ (u) = u ds, u ds + 2 δ 0 0 √ where u is normalized. We now take u(s) = 2 sin(π s/$) to obtain µ1 2ακ02 δ 2 π s Jδ (u) = sin + 2 ds, 4 $ δ 0 and since

δ

sin2 (π s)ds

= 0, δ2 it follows that Jδ can be made to be arbitrarily close to µ1 /4. lim

0

δ→0+

380

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Remark 1. Clearly for α > αc the infimum is not attained, as was conjectured in [EHL] for the case of closed planar curves. A simple consequence of Theorem 2 is a lower bound for the principal eigenvalue of the Schrödinger operator on the circle. Corollary 1. Consider the operator H = −d 2 /dx 2 + V (x) defined on (0, L) with periodic boundary conditions, and define 1 L Vm = inf V (x) and I = [V (x) − Vm ]1/2 dx. x L 0 Then λ0 ≥

 Vm + I 2 , 2 Vm + π 2 , L

if I ≤ π L if I > π L,

with equality for I < π/L if and only if V is constant. Proof. The first inequality follows directly by writing the eigenvalue problem as −u + (V −Vm )u = (λ−Vm )u and applying the previous theorem with κ = [(V − Vm )/α]1/2 . The second part is a consequence of the fact that for α larger than αc the principal eigenvalue must be larger than αc κ02 . Remark 2. It follows from Theorem 2 that the given inequalities are sharp in both cases. 4. Higher Dimensions In [EHL], the proof of the fact that for α smaller than α ∗ the constant potential is the unique global minimizer of 0 relied on a result that is not available in higher dimensions. Namely, while in one dimension we have that

µ1 2 (u − um ) ds ≥ (u − um )2 ds, 4 S1 S1 where um is the minimum of u in S 1 , from the results in [CF] it is known that there is no similar result in higher dimensions. More precisely, if we impose that a function f be zero at a finite number of points of a compact manifold with dimension greater than or equal to two, then there is no relation of the form above with a positive constant on the right-hand side. This suggests that an argument similar to that used in the proof of Theorem 2 can now be used for all positive values of α, and not just for α larger than α ∗ . This is indeed the case, and we have the following Theorem 3. Assume that F (0) is a global minimum of F . Then, for n greater than one, j (α) ≡ µj − F (0) for all positive α and j = 0, 1, . . . . Proof. Fix a point x0 in M and denote by Bδ the geodesic ball centred at x0 with radius δ. Let now !δ = M \ Bδ and define the potential κ 0 , if x ∈ Bδ (x0 ), κδ (x) = |Bδ | 0, if x ∈ !δ (x0 ).


381

(As before, this is discontinuous but can be approximated by continuous functions without changing the results.) By subtracting F (0) on both sides of the equation for the eigenvalues, we can, without loss of generality, take F (0) to be zero. We are thus lead to the functional κ0 Jδ (u) = |∇u|2 dvg + αF u2 dvg . |B | δ M Bδ Consider now the auxiliary eigenvalue problem defined by −w = µw, x ∈ !δ , w = 0, x ∈ ∂!δ , and denote its eigenvalues by 0 < µ0 (δ) < µ1 (δ) ≤ . . . , with corresponding normalized eigenfunctions vj δ . From the results in [CF] we have that lim µj (δ) = µj , j = 0, . . . .

δ→0+

We now build test functions uj δ , j = 0, . . . defined by vj,δ (x), x ∈ !δ , uj δ (x) = 0, x ∈ Bδ , for which

Jδ (uj δ ) =

!δ

|∇vj,δ |2 dvg ,

and, by the result from [CF] mentioned above, this converges to µj , j = 0, . . . , as ∞ δ goes to zero. Finally, note that for each δ the set uj δ j =0 satisfies the necessary ∞ orthogonality conditions, since this is the case for vj δ j =0 . A similar result will also hold in other cases, such as manifolds with boundary with Dirichlet or Neumann boundary conditions, for instance. 5. Concluding Remarks As was pointed out in [EHL] for the one-dimensional case, it is not difficult to see that for negative α the constant potential still maximizes the principal eigenvalue. It is also possible to show that in this case there is no lower bound on this eigenvalue, in the sense that there exist potentials κ with fixed average κ0 for which this eigenvalue can be made as large (in absolute value) as desired. It is not completely clear what happens to the supremum of the first eigenvalue for positive values of α. Regarding higher dimensions, it was shown that integral restrictions of this and similar type actually impose no restrictions at all as far as minimization is concerned, in the sense that it is possible to approximate the eigenvalues of the Laplacian as much as desired by potentials satisfying the given restrictions. Although we have seen that in this case the constant potential is never a global minimizer for positive α, the results in Sect. 2 raise the question of whether or not it is a local minimizer for α < α ∗ . We end by remarking that similar results to those in Sects. 2 and 3 also hold in the case of manifolds with boundary and Neumann boundary conditions.

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P. Freitas

Acknowledgements. This work was carried out while I was visiting the Department of Mathematics of the Royal Institute of Technology in Stockholm, Sweden. I would like to thank the people there and, in particular, Ari Laptev, for their hospitality.

References Ashbaugh, M.S.: Optimization of the characteristic values of Hill’s equation subject to a p-norm constraint on the potential. J. Math. Anal. Appl. 143, 438–447 (1989) [AHS] Ashbaugh, M.S. and Harrell, E.M.: Maximal and minimal eigenvalues and their associated nonlinear equations. J. Math. Phys. 28, 1770–1786 (1987) [CF] Chavel, I. and Feldman, E.A.: Spectra of domains in compact manifolds. J. Funct. Anal. 30, 198–222 (1978) [EI] El Soufi, A. and Ilias, S.: Second eigenvalue of Schrödinger operators and mean curvature. Commun. Math. Phys. 208, 761–770 (2000) [EHL] Exner, P., Harrell, E.M. and Loss, M.: Optimal eigenvalues for some Laplacians and Schrödinger operators depending on curvature. Mathematical results in quantum mechanics (Prague 1998), Oper. Theory Adv. Appl. 108, pp. 47–58 [HL] Harrell, E.M. and Loss, M.: On the Laplace operator penalized by mean curvature. Commun. Math. Phys. 195, 643–650 (1998) [Ka] Kato, T.: Perturbation theory for linear operators. Berlin–Heidelberg–New York: Springer, 1966 [Ke] Keller, J.B.: Lower bounds and isoperimetric inequalities for eigenvalues of the Schrödinger equation. J. Math. Phys. 2, 262–266 (1961) [A]

Communicated by B. Simon

Commun. Math. Phys. 217, 383 – 407 (2001)

Communications in



Self Duality Equations for Ginzburg–Landau and Seiberg–Witten Type Functionals with 6th Order Potentials Weiyue Ding1 , Jürgen Jost2 , Jiayu Li1 , Xiaowei Peng2 , Guofang Wang1 1 Institute of Mathematics, Academia Sinica, 100080 Beijing, P.R. China.

E-mail: [email protected]; [email protected]; [email protected]

2 Max-Planck Institute for Mathematics in the Sciences, Inselstrasse 22–26, 04103 Leipzig, Germany.

E-mail: [email protected]; [email protected] Received: 13 October 1998 / Accepted: 21 October 2000

Abstract: The abelian Chern–Simons–Higgs model of Hong-Kim-Pac and Jackiw– Weinberg leads to a Ginzburg–Landau type functional with a 6th order potential on a compact Riemann surface. We derive the existence of two solutions with different asymptotic behavior as the coupling parameter tends to 0, for any number of prescribed vortices. We also introduce a Seiberg–Witten type functional with a 6th order potential and again show the existence of two asymptotically different solutions on a compact Kähler surface. The analysis is based on maximum principle arguments and applies to a general class of scalar equations. 0. Introduction Let be a compact Riemann surface with a line bundle L. For a unitary connection DA = d +A on L with curvature FA , and a section φ of L, we have the Ginzburg–Landau functional 1 GL(A, φ) = (|DA φ|2 + |FA |2 + (1 − |φ|2 )2 ) ∗ 1. 4 This functional can be rewritten as 1 GL(A, φ) = (|(D1 + iD2 )φ|2 + (FA − (1 − |φ|2 ))2 ) ∗ 1 + 2π deg L, 2 see e.g. [J; Sect. 9.1]1 for details. This reformulation shows that absolute minimizers satisfy the self duality equations (D1 + iD2 )φ = 0, 1 F = (1 − |φ|2 ). 2 1 Here, however, in agreement with the physics literature A = −iA dx α , F α αβ = ∂α Aβ − ∂β Aα , FA = − 2i Fαβ dx α ∧dx β , F = F12 , Dα = ∂α −iAα . We assume w.l.o.g. that the degree of L, deg L, is nonnegative.

384

W. Ding, J. Jost, J. Li, X. Peng, G. Wang

The first equation says that φ is a holomorphic section of the line bundle L. The self duality mechanism still works if we introduce a coupling parameter ε as follows: 1 (|DA φ|2 + ε 2 |FA |2 + 2 (1 − |φ|2 )2 ) ∗ 1 GLε (A, φ) = 4ε 1 = (|(D1 + iD2 )φ|2 + (εF − (1 − |φ|2 ))2 ) ∗ 1 2ε + 2π deg L. The self duality equations then are (D1 + iD2 )φ = 0, 1 ε2 F12 = (1 − |φ|2 ). 2 N := deg L is the degree of L and determines the number of zeroes p1 , . . . , pN (counted with multiplicity) of φ. With u := log |φ|2 , the equations are reduced to the single scalar equation N

u =

1 u (e − 1) + 4π δpi , 2 2ε i=1

where δpi is the Dirac functional based at pi . It follows from the analysis of Taubes [T1] that there exists εc > 0 such that for 0 < ε < εc , this equation has a unique solution uε for any prescribed set of vortices p1 , · · · , pN . Hong–Jost–Struwe [HJS] carried out the asymptotic analysis of uε for ε → 0. In the limit, |φε | tends to 1 away from the vortices, and the curvature FAε becomes a sum of delta distributions centered at the vortices. Thus, the line bundle is degenerated into a flat bundle with a covariantly constant section with N singular points where the curvature concentrates. As described in [J; Sect. 9.1], the self duality mechanism works in still more generality, namely, we may replace the parameter ε by an arbitrary real function γ (φ) of φ and consider 1 GLγ (A, φ) = (|DA φ|2 + γ (φ)2 |FA |2 + (1 − |φ|2 )2 ) ∗ 1 4γ (φ)2 1 = (|(D1 + iD2 )φ|2 + (γ (φ)F − (1 − |φ|2 ))2 ) ∗ 1 2γ (φ) + 2π deg L. For the choice γ (φ) =

ε , |φ|

Self Duality Equations

385

we obtain the Chern–Simons–Higgs functional introduced by Hong-Kim-Pac [HKP] and Jackiw–Weinberg [JW] for the time independent vortex solutions of an abelian Chern–Simons–Higgs model on R2,1 , namely ε2 1 CS(A, φ) = (|DA φ|2 + |F |2 + 2 |φ|2 (1 − |φ|2 )2 ) ∗ 1 2 A |φ| 4ε 1 ε = (|(D1 + iD2 )φ|2 + ( F − |φ|(1 − |φ|2 ))2 ) ∗ 1 |φ| 2ε + 2π deg L. Absolute minimizers satisfy the following self duality equations: (D1 + iD2 )φ = 0, 1 ε 2 F = |φ|2 (1 − |φ|2 ). 2

(0.1) (0.2)

The first authors to consider this problem on a compact Riemann surface, namely a torus, were Caffarelli–Yang [CY]. They introduced a sub/supersolution method to construct a solution (A1ε , φε1 ) for every positive ε below some critical threshold εc above which no solution exists. For ε → 0, this solution has the same asymptotic behavior as one of the Ginzburg–Landau model described above. Tarantello [Ta] then showed the existence of a second solution (A2ε , φε2 ) for 0 < ε < εc (as follows from [DJLW1], there may exist more than two solutions). For the case of one vortex, N = 1, she was able to analyze the asymptotic behavior of a second solution; φε2 converges to 0 uniformly for ε → 0, and after rescaling, one obtains a solution of an interesting mean field equation whose geometric significance remains to be explored. The method was restricted to N = 1 because it was of a variational nature and depended on the Moser–Trudinger inequality. The case N = 2 represents a borderline case for this inequality and was treated in [DJLW1, DJLW2] and [NT]. In the present paper, we construct a second solution for which we are able to perform the asymptotic analysis for an arbitrary number N of vortices, thereby completing this line of investigation. As in the quoted previous papers, by putting v := log |φ|2 , we reduce the above system to the single scalar equation N

v =

4 v v e (e − 1) + 4π δpj , ε2 j =1

or with u0 being the corresponding Green function, i.e. the solution of N 4π N δpj , + 4π || j =1 u0 = 0,

u0 = −

u = v − u0 satisfies u =

4 Keu (Keu − 1) + A, ε2

(0.3)

386


with K = eu0 , A = 4πN || . This is the equation we shall study in some generality, namely on an arbitrary compact Riemannian manifold. Our result for the Chern–Simons–Higgs problem then is Theorem 0.1. For N > 0, p1 , · · · , pN ∈ and 0 < ε < εc , there are solutions (A1ε , φε1 ) and (A2ε , φε2 ) of (0.1)–(0.2) such that for ε → 0, (1) |φε1 | → 1 on every " ⊂⊂ \ {p1 , · · · , pN }; (2) |φε2 | → 0 almost everywhere. (1) of course is the result of Caffarelli–Yang [CY]. The first solution corresponds to a topological, the second one to a non-topological solution of the field equations. As already indicated, our method works in any dimension. Therefore, we now introduce a functional on a 4-manifold, namely a generalization of the Seiberg–Witten functional with 6th order potential obtained by the same type of self duality mechanism as above to which our method also applies, at least if the manifold is Kähler. First, we recall some facts from the Seiberg–Witten theory (for more details, see [J, JPW] and [S]). Let (X, g) be a compact, oriented four-dimensional manifold with a Riemannian metric g, and PSO(4) → X its oriented orthonormal frame bundle. Let spinc (4) be the U (1) extension of SO(4), namely, 1 → U (1) → spinc (4) → SO(4) → 1. A spinc -structure on the Riemannian Manifold (X, g) is a lift of the structure group SO(4) to spinc (4), i.e. there is a principle spinc (4)-bundle Pspinc (4) → X such that there is a bundle map Pspinc (4) −→ PSO(4) ↓ X

↓ −→

X

It is well-known that any compact, oriented four-manifold admits a spinc -structure. Let Q = Pspinc (4) /spin(4) be a principle U (1)-bundle. W = Pspinc (4) ×spinc (4) C4 and L = Q ×U (1) C resp. is the associated spinor bundle and the line bundle resp.. W can be decomposed globally as W + and W − . Locally, W ± = S ± ⊗ L1/2 . Here S ± is a spinor bundle with respect to a local spin-structure on X. Both S ± and L1/2 are locally defined. There exists a Clifford multiplication T X × W+ → W− denoted by e · φ ∈ W − for e ∈ T X and φ ∈ W + . Here T X is the tangent bundle of X. A connection on the bundle W + can be defined by the Levi–Civita connection and a connection on L. The “twisted” Dirac operator DA : ,(W + ) → ,(W − ) is defined by DA =

4 i=1

ei · ∇ A .


387

Here, ,(W ± ) is the space of sections of W ± , {ei } is an orthonormal basis of T X and ∇A is a connection on W + induced by the Levi–Civita connection and a connection A on the line bundle L. Let A(L) be the space of the Hermitian connections of the line bundle L. The Seiberg–Witten functional is defined for pairs (A, φ) ∈ A(L) × ,(W + ): R 1 SW (A, φ) = (|∇A φ|2 + |FA+ |2 + |φ|2 + |φ|4 ) ∗ 1, 4 8 X where FA+ is the self-dual part of the curvature of A, and R is the scalar curvature of X. Using the Weitzenböck formula, this can be rewritten as 1 SW (A, φ) = (|DA φ|2 + |FA+ − ei ej φ, φei ∧ ej |2 ) ∗ 1, 4 X where {ei } is the dual basis of an orthonormal basis {ei }. From this reformulation, one directly sees the self duality involved: Absolute minimizers satisfy the Seiberg–Witten equations DA φ = 0, 1 FA+ = ei ej φ, φei ∧ ej . 4 Now, first of all, the Seiberg–Witten functional may be perturbed by adding 2-forms σ , η in the functional: 1 SWσ,η (A, φ) = (|DA φ|2 + |(FA+ − σ ) − (ei ej φ, φei ∧ ej − η)|2 ) ∗ 1 4 X R = (|∇A φ|2 + |FA+ |2 + |φ|2 4 X + |η − ei ej φ, φei ∧ ej |2 + 2FA+ , η − σ ) ∗ 1.

Secondly, the self duality mechanism still works if we insert a real-valued function γ (φ) of φ in the following manner: R SWσ,η,γ (A, φ) = (|∇A φ|2 + γ (φ)2 |FA+ |2 + |φ|2 4 X 1 + |η − ei ej φ, φei ∧ ej |2 + 2FA+ , η − σ ) ∗ 1 γ (φ)2 = (|DA φ|2 X

+ |γ (φ)(FA+ − σ ) −

1 (ei ej φ, φei ∧ ej − η)|2 ) ∗ 1. 4γ (φ)

In analogy with the Chern–Simons–Higgs functional discussed above, we choose γ (φ) =

ε |φ|

388


for a real parameter ε > 0. This choice seems to lead to the most natural and interesting theory, and so we study the following Seiberg–Witten type functional with 6th order potential 2 ε |φ| + 2 i j (ei ej φ, φe ∧ e − η) ) ∗ 1 (|DA φ| + (FA − σ ) − L(A, φ) = |φ| 4ε X 2 ε R = (|∇A φ|2 + |F + |2 + |φ|2 |φ|2 A 4 X 2 |φ| + 2 |η − ei ej φ, φei ∧ ej |2 + 2FA+ , η − σ ) ∗ 1. ε Given η ∈ ∧2 T ∗ X, we thus consider the self duality equations that are satisfied by minimizers of L(A, φ), namely DA φ = 0, 1 FA+ − σ = 2 |φ|2 (ei ej φ, φei ∧ ej − η). 4ε

(0.4)

The Seiberg–Witten functional as described above exhibits a strong structural similarity with the Ginzburg–Landau functional; namely it contains a squared covariant derivative of the scalar field, a squared curvature of the vector field and a 4th order potential term for the scalar field. In fact, the Ginzburg–Landau functional can be considered as a dimensional reduction of the Seiberg–Witten functional. The analogy goes further. In the Ginzburg–Landau functional with parameter ε, one sees that for ε → 0, the (unique) solution (Aε , φε ) concentrates at the prescribed vortices, in the sense that φε converges to 1 uniformly away from those vortices, and the curvature FAε tends to a delta distribution supported at the vortices, see [HJS]. Taubes [T2, T3, T4] showed that on a symplectic manifold, with η being the symplectic 2-form, the Seiberg–Witten functional with parameter ε (i.e. γ (φ) = ε in our above notation) for ε → 0 exhibits a similar limiting behavior in the sense that now a concentration along a set of pseudoholomorphic curves occurs. Recently, Lin and Rivière [LR] were able to obtain such a concentration analysis in a general context in arbitrary dimension. As we discussed above, the Chern–Simons–Higgs functional exhibits a richer asymptotic structure than the Ginzburg–Landau functional, in the sense that we are able to show in this paper the existence of two very different types of asymptotic solutions for ε → 0, for any number of vortices. As the structural relation between our functional L and the Chern–Simons–Higgs functional is completely analogous to the one between the Seiberg–Witten functional and the Ginzburg–Landau one, we also expect an analogously rich asymptotic behavior for L. In the present paper, we perform the corresponding analysis in the case where X is a Kähler surface. In this case, our self duality equations admit a reduction to a single scalar valued equation of the same type as (0.3), to be derived in Sect. 1. We shall prove Theorem 0.2. Let (X, ω) be a compact Kähler surface with a spinc -structure induced by a hermitian line bundle E → X, and let K be the canonical line bundle of the Kähler surface X. Let η = ω, and σ = FAcan , where Acan is the canonical connection on K ∗ 1 induced by the Kähler metric. There exists εc with ε12 > Vol(X) 64π c1 (E) · [ω] such that c

for any ε < εc Eq. (0.4) admits two solutions (A1ε , φε1 ) and (A2ε , φε2 ), with the following asymptotic behavior:


389

(1) |φε1 | → 1 almost everywhere, as ε → 0; (2) |φε2 | → 0 almost everywhere, as ε → 0. Technically, our approach will be based on maximum principle arguments. Variational arguments do not seem to work already in the case of the Chern–Simons–Higgs functional for more than two vortices, because the case of two vortices is the limiting case for the Moser–Trudinger inequality as explained above. For the functional L, a 6th order potential term can not be controlled by a squared derivative via a Sobolev type embedding theorem. In fact, in physical terms, our functional L will lead to a nonrenormalizable theory, and so no general approach applies. Our point here, however, is that although we are beyond the range of embedding theorems, there still exists a finer internal structure that allows to draw interesting consequences. We expect, however, that a similar result also holds in the general case of a symplectic 4-manifold X; necessarily, the analysis needs to be somewhat different as one has to deal with vector valued equations. We speculate that the expected two types of asymptotic regimes will lead to topological applications by allowing to relate topological quantities identified by the two different asymptotic solutions. The paper is organized as follows. In Sect. 1, we derive the reduction to a scalar valued equation of the equation (1.8), if X is a Kähler surface. In Sect. 2, we show the existence of two solutions. The first solution is obtained by the super/subsolution of Caffarelli– Yang [CY]. The second solution is constructed with the help of the mountain pass method for some associated functional. We use a heat flow to construct the required deformation. This constitutes the main technical innovation of the present paper compared to previous works on the Chern–Simons–Higgs functional. Section 3 then establishes the different asymptotic behavior of the two types of solutions.

1. The Self Duality Equations on a Kähler Surface In this section we shall derive the self duality equations for our generalized Seiberg– Witten functional on Kähler surfaces. Let (X, ω, J ) be a Kähler surface with Kähler metric g(v, w) = ω(v, J w). The tangent bundle of X carries a canonical spinc structure with Wcan = ∧0,∗ T ∗ X,

Lcan = K ∗ = ∧0,2 T ∗ X,

where K is the canonical line bundle of X. The Levi–Civita connection of the Kähler metric induces a canonical connection Acan on the line bundle Lcan , and the curvature tensor considered as a 2-form FAcan of type (1,1), represents the first Chern class of the line bundle, namely, i FAcan = c1 (Lcan ) = −c1 (K). 2π Let E → X be a hermitian line bundle over X, and consider the spinc -structure corresponding to the line bundle LE = K ∗ ⊗ E 2 .

390


Then we have the spinor bundles WE+ = (∧0,0 ⊕ ∧0,2 ) ⊗ E,

WE− = ∧0,1 ⊗ E,

where ∧p,q = ∧p,q T ∗ X. Let A(E) be the space of hermitian connections on E. For B ∈ A(E), we have an induced connection A = Acan + B 2 ∈ A(LE ) on the line bundle LE = K ∗ ⊗ E 2 , with curvature 2-form given by FA = FAcan + 2FB . The self duality equations (0.4) reduce to the following equations for the pair (B, 6),

4i(FAcan

∂¯B φ0 + ∂¯B∗ φ2 = 0, 1 4(FB − σ )0,2 = 2 φ¯ 0 φ2 |6|2 + η0,2 , ε 1 + 2FB − σ )ω = 2 |6|2 (|φ2 |2 − |φ0 |2 + ηω ), ε

(1.1)

where 6 = (φ0 , φ2 ) ∈ (∧0,0 ⊗ E) × (∧0,2 ⊗ E), and the perturbations σ , η ∈ i∧2,+ are self-dual 2-forms with respect to the Kähler metric g. ηω is the component of η in the direction of ω. Remark 1.1. Without the |6|2 term Eq. (1.1) is exactly the Seiberg–Witten equation on a Kähler manifold. As in that case (see [S]), we have Proposition 1.2. Let X be a connected Kähler surface, and σ, η ∈ ∧1,1 ∩ ∧2,+ . Then for any solution (B, 6) of Eq. (1.1) either φ0 = 0 or φ2 = 0. Proof. The proof is same as in the Seiberg–Witten case, since we need only the first two equations of (1.1) to get the conclusion. Applying the operator ∂¯B to the first equation of (1.1), and using ∂¯B ∂¯B = FB0,2 and the second equation of (1.1), we have 1 ∂¯B ∂¯B∗ φ2 = −∂¯B ∂¯B φ0 = −FB0,2 φ0 = − 2 |φ0 |2 |6|2 φ2 . 4ε Now take the L2 -product with φ2 to get 1 |∂¯B∗ φ2 |2 + 2 |φ0 |2 |φ2 |2 |6|2 = 0. 4ε X Then this yields ∂¯B∗ φ2 = 0,

∂¯B φ0 = 0,

and |φ0 |2 |φ2 |2 = 0.

By the unique continuation theorem for the Dirac operator, we obtain the conclusion. As in the Seiberg–Witten case, which one of the two sections φ0 , φ2 vanishes is determined by the topology of the line bundle LE , if σ = 0, and η = 0.


391

Proposition 1.3. Let X be a connected Kähler surface and let (B, 6) be a solution of Eq. (1.1) with σ = 0, η = 0. Then (2c1 (E) − c1 (K))[ω] < 0 ⇐⇒ φ0 = 0, (2c1 (E) − c1 (K))[ω] > 0 ⇐⇒ φ0 = 0,

φ2 = 0; φ2 = 0.

Proof. Integrating the third equation of (1.1) over X, 1 4 4 ω∧ω (|φ2 | − |φ0 | ) 2i(FAcan + 2FB ) ∧ ω = ε2 X 2 X = 4π(c1 (K ∗ ) + 2c1 (E)) ∧ ω X

= 4π(2c1 (E) − c1 (K))[ω], where ω∧ω 2 is the volume form of the Kähler metric g. The conclusion follows directly from the above equation. Remark 1.4. The situation for η = 0 is different from the one of Proposition 1.3. Let η = kω, if k >> 1 or ε 2 0. Then we shall get another type of solution of Eqs. (1.1), see Theorem 2.1. If we assume that φ2 = 0, η = kω ∈ ∧1,1 , and σ ∈ ∧1,1 , then Eqs. (1.1) reduce to the following equations, ∂¯B φ0 = 0,

4i(FAcan

FB0,2 = 0, 1 + 2FB − σ )ω = 2 |φ0 |2 (−|φ0 |2 + k). ε

(1.2)

From complex geometry, we know that the equations ∂¯B φ0 = 0, FB0,2 = ∂¯B ∂¯B = 0

(1.3)

always admit a solution (B, φ0 ). Equations (1.2) and (1.3) are invariant under a unitary gauge transformation, and Eqs. (1.3) are also invariant under a real gauge transformation, hence (1.3) is invariant under the complexified gauge group C ∞ (X, C∗ ). This fact can be seen from the following computation. Let u : X → C∗ ; u acts on the pair (B, φ0 ) by ¯ − u¯ −1 ∂ u, u∗ B = B + u−1 ∂u ¯ u∗ φ0 = u−1 φ0 . Then we have ∂¯u∗ B (u∗ φ0 ) = u−1 ∂¯B u(u−1 φ0 ) = u−1 ∂¯B (φ0 )

(1.4)

392


and Fu∗ B = d(u∗ B) ¯ − u¯ −1 ∂ u) = d(B + u−1 ∂u ¯ −1 ¯ ¯ − u¯ −1 ∂ u). = FB + (∂ + ∂)(u ¯ ∂u Let u be a real gauge, i.e. u = e−θ , θ : X → R. We have ¯ − u¯ −1 ∂ u¯ = −∂θ ¯ + ∂θ, u−1 ∂u and ¯ ¯ + ∂θ) Fu∗ B = FB + (∂ + ∂)(− ∂θ ¯ = FB + 2∂∂θ

(1.5)

¯ = FB − 2∂ ∂θ. From (1.4) and (1.5), (u∗ B, u∗ φ0 ) is a solution of (1.3), if (B, φ0 ) is a solution of (1.3). Then (u∗ B, u∗ φ0 ) satisfies (1.2) if and only if 1 ∗ 2 |u φ0 | (−|u∗ φ0 |2 + k). ε2

4i(FAcan + 2Fu∗ B − σ )ω =

(1.6)

From (1.5), 4i(FAcan + 2Fu∗ B − σ )ω = 4i(FAcan + 2FB − σ + 2(Fu∗ B − FB ))ω ¯ ω, = 4i(FAcan + 2FB − σ )ω − 16i(∂ ∂θ) on the other side

¯ ω = −d ∗ dθ = θ, 4i(∂ ∂θ)

where is the negative Laplace operator, i.e. = −d ∗ d. Using the above computations, we rewrite Eq. (1.6) in the following way: 4θ =

1 2θ e |φ0 |2 (e2θ |φ0 |2 − k) + 4i(FAcan + 2FB − σ )ω . ε2

(1.7)

To simplify Eq. (1.7), let us set v :=2θ, or eu0 = |φ0 |2 ,

u0 := ln |φ0 |2 ,

u0 is the Green function for the divisor D defined by the zero set of φ0 , namely (see [GH]) u0 = −4i(FB )ω + 4π δD . Set λ :=

1 . 2ε2

Equation (1.7) assumes the following form: v = λev+u0 (ev+u0 − k) + 2i(FAcan + 2FB − σ )ω .

For simplicity, we put

σ · [ω] :=

X

iσ ∧ ω. 2π

(1.8)


393

Proposition 1.5. A necessary condition for the existence of a solution for (1.8) is 16π (2c1 (E) − c1 (K) − σ ) · [ω]. Vol(X)

λk 2 >

Proof. Rewrite the left side of (1.8) v = λev+u0 (ev+u0 − k) + 2i(FAcan − σ + 2FB )ω k λ = λ(ev+u0 − )2 − k 2 + 2i(FAcan − σ + 2FB )ω . 2 4 Integrating the equation over X, we obtain k λ 0= λ(ev+u0 − )2 − k 2 Vol(X) + 4π(2c1 (E) − c1 (K) − σ )[ω]. 2 4 X Hence we have 16π (2c1 (E) − c1 (K) − σ )[ω]. Vol(X)

λk 2 >

2. Existence of Solutions In this section, we consider the equation u = λeu+u0 +v0 (eu+u0 +v0 − 1) + A

(2.1)

for a constant A and a smooth function v0 with v0 = 0 X

and the Green function u0 corresponding to some subvariety D of real codimension 2. For example, Eq. (0.3) is of this type. Also, Eq. (1.8) is of this form as we now wish to 2 explain: Let λ = κ2 , and put k = 1 for simplicity, A :=

4π (2c1 (E)[ω] − c1 (K)[ω] − σ [ω]), Vol(X)

and let v1 , v2 be the solutions of v1 = 4i(FB )ω − A1 , v2 = 2i(FAcan − σ )ω − A2 , with A1 = condition

8π Vol(X) c1 (E)[ω]

and A2 = vj = 0, X

and put

4π Vol(X) (−c1 (K)

− σ )[ω], normalized by the

for j = 1, 2,

v0 = v1 + v2 .

Returning to the general case, the Green function u0 satisfies an equation of the type u0 = −; + 4π δD ,

394


where ; is smooth with A1 := X ; = 4π Vol(D), and we let v1 be the solution of v1 = ; − A1 with X v1 = 0, A2 := A − A1 , v2 := v0 − v1 . Since u0 is the Green function for the subvariety D, the method of Caffarelli–Yang [CY] yields the existence of a first solution of (2.1): Theorem 2.1. For λ sufficiently large, Eq. (2.1) admits a maximal solution uλ with uλ + u0 + v0 < v, ¯ where v¯ is a smooth function defined below. Proof. Let v¯ be a smooth function satisfying (−v2 + v) ¯ ≤ A2 + λev¯ (ev¯ − 1).

(2.2)

Such v¯ exists, and in fact we can choose v¯ ≥ 0. Choose a constant K ≥ 2λe2v¯ . We want to use induction to construct a sequence wk that converges to a solution of Eq. (2.1). ¯ It is clear that w0 (x) → +∞, as x → x0 ∈ D. Put w0 = −(u0 + v1 ) + (−v2 + v). We have ( − K)w0 = w0 − Kw0 = −(u0 + v1 ) + (−v2 + v) ¯ − Kw0 ≤ −8πδD + A1 + A2 + λev¯ (ev¯ − 1) − Kw0 . Now set ( − K)wk = λeu0 +v0 +wk−1 (eu0 +v0 +wk−1 − 1) + A1 + A2 . Then we have ( − K)(w1 − w0 ) = ( − K)w1 − ( − K)w0 ≥ λeu0 +v0 +w0 (eu0 +v0 +w0 − 1) + A1 + A2 − (−8π δD + A1 + A2 + λev¯ (ev¯ − 1)) = 0, for any x ∈ X \ D. Let Bε (D) = {x ∈ X dist(x, D) ≤ ε} be the ε-neighborhood of D, and Xε = X \ Bε (D). Since w0 (x) → +∞, as x → x0 ∈ D, we have w1 − w0 < 0, on ∂Xε . The maximum principle implies that w1 − w0 < 0 on Xε . This implies the first step of the induction: w1 − w0 < 0 on X. Next, by induction assumption wk −wk−1 < 0, and we want to prove wk+1 −wk < 0.


395

We compute ( − K)(wk+1 − wk ) = λewk +u0 +v0 (ewk +u0 +v0 − 1) − λewk−1 +u0 +v0 (ewk−1 +u0 +v0 − 1) − K(wk − wk−1 ) = λe2(u0 +v0 ) (e2wk − e2wk−1 ) − λeu0 +v0 (ewk − ewk−1 ) − K(wk − wk−1 ) ≥ λe(2u0 +v0 ) (e2wk − e2wk−1 ) − K(wk − wk−1 ) since wk − wk−1 < 0 = 2λe2u0 +2v0 +2w (wk − wk−1 ) − K(wk − wk−1 ) for a w, with wk ≤ w ≤ wk−1 < · · · < w0 ≥ 2λe2u0 +2v0 +2w0 (wk − wk−1 ) − K(wk − wk−1 ) = 2λe2v¯ (wk − wk−1 ) − K(wk − wk−1 ) = (2λe2v¯ − K)(wk − wk−1 ) ≥ 0, and again by the maximum principle, we get wk+1 − wk < 0. We inductively get a monotonically decreasing sequence wk+1 < wk < · · · < w1 < w0 . Let w− be a subsolution of the equation w− ≥ λeu0 +v0 +w− (eu0 +v0 +w− − 1) + A1 + A2 . Such a subsolution exists for sufficient large λ, see Lemma 2.2 below. Now we want to show that the subsolution w− is a lower bound for the sequence wk . We proceed by induction as above. First we check that (w− − w0 ) = (w− + u0 + v1 + v2 − v) ¯ ≥ λeu0 +v1 +v2 +w− (eu0 +v1 +v2 +w− − 1) − λev¯ (ev¯ − 1) = λew− −w0 +v¯ (ew− −w0 +v¯ − 1) − λev¯ (ev¯ − 1) = λe2v¯ (e2(w− −w0 ) − 1) − λev¯ (ew− −w0 − 1). From the maximum principle, we have w− − w0 < 0. By induction, we suppose that w− − wk < 0. We want to prove that w− − wk+1 < 0. ( − K)(w− − wk+1 ) ≥ λeu0 +v0 +w− (eu0 +v0 +w− − 1) + A1 + A2 − Kw− − λeu0 +v0 +wk (eu0 +v0 +wk − 1) − A1 − A2 + Kwk = λe2(u0 +v0 ) (e2w− − e2wk ) − eu0 +v0 (ew− − ewk ) − K(w− − wk ) ≥ λe2(u0 +v0 ) (e2w− − e2wk ) − K(w− − wk ) = 2λe2(u0 +v0 )+2w (w− − wk ) − K(w− − wk )

396


for a w with w− ≤ w ≤ wk < · · · < w0 ≥ 2λe2(u0 +v0 )+2w0 (w− − wk ) − K(w− − wk ) = (2λe2v¯ − K)(w− − wk ) ≥ 0, for any x ∈ X\D, where the third and last inequalities are from the inductive assumption. From the maximum principle, we obtain the conclusion w− ≤ wk+1 . Combining the two inductions, we get a monotonically decreasing sequence that is bounded from both sides by smooth functions, namely w− < wk+1 < wk < · · · < w1 < w0 . Then by the standard bootstrap argument, wk converges to a solution uλ of Eq. (2.1) in C k , for any k ≥ 0. From the argument of Caffarelli–Yang and Tarantello [Ta; p. 3776], this solution is the maximal one. We now proceed to derive the lemma utilized above. Lemma 2.2. For λ sufficiently large, there exists a subsolution w− of w− ≥ λeu0 +v0 +w− (eu0 +v0 +w− − 1) + A. Proof. Recall that Bε (D) is the ε-neighborhood of D. Let fε be a smooth function with 0 ≤ fε ≤ 1, fε = 1 on Bε (D), and fε = 0 on X \ B2ε (D). Let c > 0 be a constant, and define a new function 1 gε,c = (A + c)fε − (A + c)fε . Vol(X) X The function gε,c has the following properties: (1) X gε,c = 0; (2) gε,c ≥ A, on Bε (D), for ε sufficiently small and c sufficiently large. The first one results from the definition of gε,c , and the second one can be seen from the following computation: Vol(B2ε (D)) Vol(X) Vol(B2ε (D)) Vol(B2ε (D)) = A + c(1 − )−A Vol(X) Vol(X) ≥ A,

gε,c ≥ A + c − (A + c)

if ε is sufficiently small, and c is sufficiently large. A solution w of the equation w = gε,c , is unique up to additive a constant, and we may therefore choose a solution w− with eu0 +v0 +w− < 1 on X.


397

On Bε (D), w− = gε,c ≥ A ≥ λeu0 +v0 +w− (eu0 +v0 +w− − 1) +A.

≤0

On X \ Bε (D), let µ0 = inf{eu0 +v0 +w− x ∈ X \ Bε (D)}, µ1 = sup{eu0 +v0 +w− x ∈ X \ Bε (D)}. Obviously 0 < µ0 < µ1 < 1. Let c0 = −µ1 (µ0 − 1), then eu0 +v0 +w− (eu0 +v0 +w− − 1) ≤ µ1 (µ0 − 1) = −c0 < 0. Choosing λ > 0 sufficiently large, we have gε,c ≥ λeu0 +v0 +w− (eu0 +v0 +w− − 1) + A. Hence, we get a subsolution w− for λ sufficiently large.

Corollary 2.3. If 2i(FAcan − σ )ω ≥ 0, then there exists a critical value λc ≥ 4A such that for every λ > λc Eq. (1.8) admits a maximal solution uλ with uλ + u0 + v0 < 0, while for λ < λc Eq. (1.8) admits no solution. Proof. If 2i(FAcan − σ )ω ≥ 0, then we can choose v¯ = 0, since v2 = 2i(FAcan − σ )ω − A2 ≥ −A2 . This is the inequality (2.2) for v¯ = 0. Let λc := inf{λ ≥ 4A | Eq. (2.1) is solvable}. If uλ is a solution of (2.1), then uλ is a subsolution of (2.1) for any λ1 > λ, since uλ = λeuλ +u0 +v0 (euλ +u0 +v0 − 1) + A = λ1 euλ +u0 +v0 (euλ +u0 +v0 − 1) + A + (λ − λ1 )euλ +u0 +v0 (euλ +u0 +v0 − 1)

≥ λ1 e

uλ +u0 +v0

(e

≥0 uλ +u0 +v0

− 1) + A.

From the proof of Theorem 2.1, the existence of the maximal solution of (2.1) depends on the existence of a subsolution of (2.1). By the definition of λc , Eq. (2.1) admits a maximal solution for any λ > λc , and admits no solution for any λ < λc .

398


Tarantello [Ta] proved that in the two-dimensional case, (2.1) has a second solution, and this solution (or else a third one) is known to have a different asymptotic behavior at least in the cases of one and two vortices, see [Ta, DJLW1, DJLW2, NT]. The method, however, does not extend to higher dimensions, because we then do not have a Palais– Smale condition anymore. In this section, we develop a heat equation method that yields a second solution of (2.1) in any dimension. We recall the equation as u = λeu+u0 +v0 (eu+u0 +v0 − 1) + A.

(2.3)

Let uλ be the solution obtained in Theorem 2.1 by using the super/subsolution method, or the solution obtained in Corollary 2.3 for any λ > λc . We choose a fixed subsolution ψ0 of Eq. (2.3) for λ sufficiently large, or λ > λc in the case of Corollary 2.3. We define a partial ordering in L1,2 (X) ∩ C 0 (X) by f1 > f2 (resp. f1 ≥ f2 ) if f1 (x) > f2 (x) (resp. f1 (x) ≥ f2 (x)) for all x ∈ X. If f1 > f2 , we define [f2 , f1 ] := {g ∈ L1,2 (X) ∩ C 0 (X) | f2 ≤ g ≤ f1 }, and [f2 , f1 ) := {g ∈ L1,2 (X) ∩ C 0 (X) | f2 ≤ g < f1 }. Here, possibly f1 = +∞ or f2 = −∞. Set Sλ = {u is a solution of (2.1) u ∈ (ψ0 , uλ ]}. Clearly, Sλ = ∅, since uλ ∈ Sλ . Lemma 2.4. There exists u1λ ∈ Sλ such that Sλ ∩(ψ0 , u1λ ] = {u1λ }, i.e. there is no solution u of (2.3) with u ∈ (ψ0 , u1λ ). Remark 2.5. We believe that Sλ = {uλ }, at least for large λ. Proof of Lemma 2.4. For any u ∈ Sλ , define µ(u) = minx∈X (u(x) − ψ0 (x)) and µ0 = inf u∈Sλ µ(u). By Lemma 2.9 below, it is easy to show that Sλ is compact, see also [Ta]. It follows that there is a u1λ ∈ Sλ such that µ0 = µ(u1λ ). Assume that µ0 = u1λ (x0 ) − ψ0 (x0 ). We claim that Sλ ∩ (ψ0 , u1λ ] = {u1λ }. Assume by contradiction that there is another solution v ∈ (ψ0 , u1λ ]. We have, by the definition of µ0 , v(x) ≤ u1λ (x) and v(x0 ) = u1λ (x0 ). The maximum principle implies that v = u1λ , a contradiction.

By Lemma 2.4, we may assume Sλ = {uλ }. Now we consider the following functional: 1 1 Jλ (u) = (2.4) |∇u|2 + λ(eu+u0 +v0 − 1)2 + Au 2 2 X in Xλ = (−∞, uλ ] ∩ C 1 (X).


399

We want to show that uλ is a strict local minimizer of Jλ in Xλ . We first show Lemma 2.6.

Jλ (uλ ) =

inf

g∈(ψ0 ,uλ ]

Jλ (g).

Proof. Minimizing Jλ in (ψ0 , uλ ], we can obtain a solution v by a standard method (see Appendix in [Ta]) such that v ∈ (ψ0 , uλ ]. From the discussion above, v = uλ . Hence, Jλ (uλ ) =

inf

g∈(ψ0 ,uλ ]

Jλ (g).

Remark 2.7. From Lemma 2.4, uλ is a local minimizer of Jλ in Xλ with respect to the C 1 -norm, i.e., there exists a δ0 such that if u ∈ Xλ with %uλ − u%C 1 < δ0 , then Jλ (uλ ) ≤ Jλ (u). Actually, we shall show in the sequel that uλ is a strict local minimizer of Jλ . To achieve this, we first discuss the heat equation with respect to (2.3), ut = u − λeu+u0 +v0 (eu+u0 +v0 − 1) − A u(·, 0) = g0

(2.5)

which will be also used to construct deformations below. Lemma 2.8. For any g0 ∈ Xλ , there exists a T ∈ (0, ∞] such that (2.5) admits a solution u(·, t) in [0, T ), and either limt→T Jλ (u(t)) = −∞, or Jλ (u(t)) ≥ c > −∞ for any t ∈ [0, T ), in this case T = +∞ and u(·, ∞) = limt→+∞ u(·, t) is a solution of Eq. (2.3). Moreover, solutions of Eq. (2.5) continuously depend on initial functions. To prove Lemma 2.8, we need the standard apriori estimates for parabolic equations. Here we first prove an auxiliary lemma. Lemma 2.9. For any u ∈ Xλ , let f = u−λeu+u0 +v0 (eu+u0 +v0 −1)−A. If %f %L2 < c1 , then %∇u%L2 ≤ c3 ; If, in addition, |Jλ (u)| < c2 , then %u%L1,2 ≤ c4 , for some constants c3 and c4 depending only on the geometry of the manifold X, the constants c1 , c2 , λ, A and %v% ¯ L∞ . Proof. For simplicity, we set h = λeu+u0 +v0 (eu+u0 +v0 − 1) + A. First we know ¯ for any u ∈ Xλ , u + u0 + v0 ≤ uλ + u0 + v0 < v, hence we have %h%L∞ = %λeu+u0 +v0 (eu+u0 +v0 − 1) + A%L∞ ≤ c, where c depends on λ, A and %v% ¯ L∞ . Taking the L2 -product of u with the equation f = u − h yields fu = − |∇u|2 + hu. X

X

400


Integrating the equation f = u − h, we have f + h = 0. X

Let

1 u¯ = Vol(X)

X

u

be the mean value of u. Combining the two equations, we have 2 |∇u| = (f + h)u X X = (f + h)(u − u) ¯ X 1 ≤ε |u − u| ¯ 2+ |f + h|2 ε X X ε 1 2 ≤ |∇u| + |f + h|2 , ε X λ1 X where ε is some positive constant, and λ1 is the first positive eigenvalue of the Laplace operator . Choosing an ε with λε1 ≤ 21 , we obtain 2 |∇u| ≤ c(ε, λ1 ) |f + h|2 ≤ c. X

On the other side, Jλ (u) =

X

X

1 λ |∇u|2 + 2 2

X

(eu+u0 +v0 − 1)2 + A

X

u,

and we rewrite the above equation

1 1 λ u¯ = Jλ (u) − |∇u|2 − (eu+u0 +v0 − 1)2 , A Vol(X) 2 X 2 X to get

|u| ¯ < c,

and our conclusion %u%L1,2 ≤ c

X

where c depends λ, A, %v% ¯ L∞ , c1 and c2 .

|∇u|2 + u¯ 2 ≤ c,

Proof of Lemma 2.8. As in the proof of Lemma 2.9, we set h = λeu+u0 +v0 (eu+u0 +v0 − 1) + A. Recall Eq. (2.5)

ut = u − h u(·, 0) = g0

for g0 ∈ Xλ .

(2.6)


401

First we know that if g0 ∈ Xλ , then ut ∈ Xλ , for any t ∈ [0, T ), where T is the maximal existence time of the solution, since uλ is a solution of the equation u − h = 0. If limt→T Jλ (u(·, t)) = −∞, we have shown the first statement of Lemma. So we assume that limt→T Jλ (u(·, t)) = c0 > −∞. Then by Lemma 2.9, we have sup %h(·, t)%L∞ ≤ c.

0≤t λc , there exists another solution u¯ λ of Eq. (2.1) with the property that u¯ λ ∈ (−∞, uλ ), but u¯ λ ∈ (ψ0 , uλ ). Proof. We want to prove the theorem by using the mountain pass argument. It is clear that the heat flow (2.5) preserves Xλ . It is trivial to see that Jλ (u + c) → −∞ as c → −∞. Take a ρ > 0, such that Jλ (uλ − ρ) < Jλ (uλ ). Let u(x, t; c) be the solution of the heat equation ∂u u+u0 +v0 (eu+u0 +v0 − 1) − A ∂t = u − λe u(x, 0; c) = uλ (x) − c, for c ∈ [0, ρ].

(2.11)

We note that

d Jλ (u(·, t; c)) ≤ 0, dt Jλ (u(·, t; c)) is monotonically decreasing in t. In particular, Jλ (u(·, t; ρ)) ≤ Jλ (u(·, 0; ρ)) = Jλ (u − ρ) < Jλ (uλ ) for all t,

and Jλ (u(·, t; 0)) = Jλ (uλ ) for all t, since u(x, t; 0) = uλ (x) for any t. We consider the curve u(·, t; s), where s ∈ [0, ρ] is variable, and t is the deformation parameter. By Lemma 2.10, there is a positive constant ε such that for any t there exists a ct ∈ [0, ρ] with Jλ (u(·, t; ct )) ≥ Jλ (uλ ) + ε.

(2.12)

For a sequence tn → +∞, we thus obtain a sequence ctn ∈ [0, ρ]. Since [0, ρ] is compact, we assume that ctn converges to c0 ∈ (0, ρ). Then we have Jλ (u(·, +∞; c0 )) = lim Jλ (u(·, tn ; ctn )) ≥ Jλ (uλ ) + ε. n→+∞

404


We claim that

Jλ (u(·, t; c0 )) ≥ Jλ (uλ ) + ε,

for all t. If the claim is not true, there is a t0 such that Jλ (u(·, t0 ; c0 )) < Jλ (uλ ) + ε. Jλ (u(·, t; c0 )) is monotonically decreasing in t, and thus for any t ≥ t0 , Jλ (u(·, t; c0 )) ≤ Jλ (u(·, t0 ; c0 )) < Jλ (uλ ) + ε. On the other side, u(·, t; c) is continuous in t and c, and thus for n large enough, Jλ (u(·, t0 ; ctn )) < Jλ (uλ ) + ε, and for tn > t0 , we have Jλ (u(·, tn ; ctn )) < Jλ (u(·, t0 ; ctn )) < Jλ (uλ ) + ε. This contradicts the inequality (2.12). Thus, we prove the claim. Let u¯ λ = limt→+∞ u(·, t; c0 ). By Lemma 2.8 u¯ λ is a solution of Eq. (2.1), and Jλ (u¯ λ ) = lim Jλ (u(·, t; c0 )) ≥ Jλ (uλ ) + ε. t→+∞

On the other side, from Lemma 2.4 u¯ λ ∈ (ψ0 , uλ ). This finishes the proof of Theorem 2.11. 3. The Asymptotic Behavior of the Solutions Let uλ be the solution of the equation uλ = λeuλ +u0 +v0 (euλ +u0 +v0 − 1) + A.

(3.1)

In this section, we will study the asymptotic behavior as λ → ∞ of the solutions of Eq. (3.1) obtained in Sects. 1 and 2. In this section for technical reasons, we choose the perturbation σ = FAcan . From Corollary 2.3, any solution uλ of (3.1) satisfies the inequality uλ + u0 + v0 < 0. We will show the following theorem. 1 µλ → ∞ then euλ +u0 +v0 → 1 almost Theorem 3.1. Let µλ = Vol(X) X uλ . If λe everywhere as λ → ∞; if λeµλ ≤ c then euλ +u0 +v0 → 0 almost everywhere as λ → ∞. We need the following lemma. Lemma 3.2. Let n = dim X. Then for any 1 < q < Proof. Let q & = %∇uλ %Lq

n n−1 ,

%∇uλ %Lq ≤ c.

q q−1

> n. Then & ≤ sup{| ∇uλ ∇φ| φ ∈ L1,q (X), φ = 0, %φ%L1,q & (X) = 1}. X

X

By the Sobolev embedding theorem we have for φ as in (3.2), %φ%L∞ (X) ≤ c.

(3.2)


405

It is clear that ∇uλ ∇φ = uλ φ X

X

≤ %φ%L∞ (X) λ ≤c

X

euλ +u0 +v0 (euλ +u0 +v0 − 1)

since uλ + u0 + v0 < 0. This proves Lemma 3.2.

Proof of Theorem 3.1. By the Sobolev embedding theorem, we may assume that uλ − µλ → u∞ in Lp (X) for some p > 1. Integrating Eq. (3.1) on both sides, we get (λeuλ +u0 +v0 (euλ +u0 +v0 − 1)) + A Vol(X) = 0, X

hence λe

µλ

X

euλ +u0 +v0 −µλ (1 − eµλ euλ +u0 +v0 −µλ ) = A · Vol(X).

If λeµλ ≤ c , we have µλ → −∞, so euλ +u0 +v0 = eµλ · e(uλ −µλ )+u0 +v0 → 0 a.e. as λ → 0. We consider now the case that λeµλ → ∞. Because uλ ≤ −u0 − v0 , by the maximum principle we have µλ ≤ 1. Hence 0 ≤ eµλ ≤ 1. We assume that eµλ → α. By Fatou’s Lemma, we have eu∞ +u0 +v0 (1 − αeu∞ +u0 +v0 ) = 0. X

So we have eu∞ +u0 +v0 =

1 α

a.e.

and consequently u∞ + u0 + v0 = log

1 . α

It is clear that X u∞ = X (u0 + v0 ) = 0. Hence α = 1 and u∞ = u0 + v0 . This proves the theorem. Theorem 3.3. There are two solutions uλ and u¯ λ of Eq. (3.1) with the following properties: (1) |euλ +u0 +v0 | → 1 a.e., as λ → ∞; (2) |eu¯ λ +u0 +v0 | → 0 a.e., as λ → ∞.

406


Proof. It is clear that the solution obtained by the super/sub-solution method satisfies |euλ +u0 +v0 | → 1

a.e., as λ → ∞.

It suffices to show that the second solution we obtained satisfies |eu¯ λ +u0 +v0 | → 0

a.e., as λ → ∞.

For simplicity, we just denote the second solution by uλ . By Theorem 3.1, we need only show that uλ does not converge to −u0 − v0 in Lp (X) for some p > 1. (Note that, in the first case of Theorem 3.1, uλ → u∞ in Lp (X) for some p > 1.) We will show in the sequel that, if uλ → −u0 − v0 in Lp (X) for some p > 1, then uλ ∈ [ψ0 , −u0 − v0 ] for large λ, where ψ0 is the subsolution used in the proof of the existence for the second solution. We first show that, for any ε > 0, uλ → −u0 − v0 in C 0 (X \ Bε (D)), where Bε (D) = {x ∈ X| dist(x, D) < ε}, and D is the zero set of φ0 . In X \ B 2ε (D), we have (uλ + u0 + v0 ) ≤ 0, since v¯ = 0 and v2 = 0. By Theorem 8.17 in [GT], we get (uλ + u0 + v0 )(x) ≥ −c(ε)%uλ + u0 + v0 %Lp (X\B ε (D)) , 2

for all x ∈ X \ Bε (D). Since uλ (x) ≤ −(u0 + v0 )(x) for all x ∈ X, we have %uλ + u0 + v0 %C 0 (X\Bε (D)) ≤ c(ε)%uλ + u0 + v0 %Lp (x) → 0,

as λ → ∞.

(3.3)

We set mλ,ε = min∂Bε (D) uλ (x). It is clear that limε→0 limλ→∞ mλ,ε = ∞. We then consider uλ (x) − mλ,ε in Bε (D). Since, for n = dim X, (uλ − mλ,ε −

A 2 |x| ) = λeuλ +u0 +v0 (euλ +u0 +v0 − 1) 2n ≤ 0,

and (uλ − mλ,ε −

A 2 A |x| )|∂Bε (D) ≥ − ε 2 , 2n 2n

from the maximum principle, we get uλ (x) − mλ,ε −

A 2 A |x| ≥ − ε 2 , 2n 2n

(3.4)

for all x ∈ Bε (D). This implies that A 2 (3.5) ε > ψ0 (x), 2n for all x ∈ Bε (D), provided that λ is large and ε is small. Equations (3.4) and (3.5) imply that uλ > ψ0 , for λ sufficiently large. This is in contradiction with our construction for the second solution, namely, uλ ∈ [ψ0 , uλ ]. This finishes the proof of the theorem. uλ (x) ≥ mλ,ε −


407

Theorem 0.1 and Theorem 0.2 are direct consequences of Corollary 2.3, Theorem 2.11 and Theorem 3.3. Acknowledgement. The research for this paper was carried out at the Max-Planck Institute for Mathematics in the Sciences in Leipzig. The first, third, and fifth author thank the institute for generous hospitality and good working conditions.

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Commun. Math. Phys. 217, 409 – 421 (2001)

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Transformations on the Set of All n-Dimensional Subspaces of a Hilbert Space Preserving Principal Angles Lajos Molnár Institute of Mathematics and Informatics, University of Debrecen, P.O.Box 12, 4010 Debrecen, Hungary. E-mail: [email protected] Received: 28 August 2000 / Accepted: 30 October 2000

To my wife for her unlimited(?) patience Abstract: Wigner’s classical theorem on symmetry transformations plays a fundamental role in quantum mechanics. It can be formulated, for example, in the following way: Every bijective transformation on the set L of all 1-dimensional subspaces of a Hilbert space H which preserves the angle between the elements of L is induced by either a unitary or an antiunitary operator on H . The aim of this paper is to extend Wigner’s result from the 1-dimensional case to the case of n-dimensional subspaces of H with n ∈ N fixed. 1. Introduction and Statement of the Main Result Let H be a (real or complex) Hilbert space and denote B(H ) the algebra of all bounded linear operators on H . By a projection we mean a self-adjoint idempotent in B(H ). For any n ∈ N, Pn (H ) denotes the set of all rank-n projections on H , and P∞ (H ) stands for the set of all infinite rank projections. Clearly, Pn (H ) can be identified with the set of all n-dimensional subspaces of H . As it was mentioned in the abstract, Wigner’s theorem describes the bijective transformations on the set L of all 1-dimensional subspaces of H which preserve the angle between the elements of L. It seems to be a very natural problem to try to extend this result from the 1-dimensional case to the case of higher dimensional subspaces (in our recent papers [11–13] we have presented several other generalizations of Wigner’s theorem for different structures). But what about the angle between two higher dimensional subspaces of H ? For our present purposes, the most adequate concept of angles is that of the so-called principal angles (or canonical angles, in a different terminology). This concept is a generalization of the usual notion of angles between 1-dimensional subspaces and reads as follows: If P , Q are finite dimensional projections, then the principal angles between them (or, equivalently, between their ranges as subspaces) is defined as the arccos of the square root of the eigenvalues (counted according to multiplicity) of the positive (self-adjoint) finite rank operator QP Q (see, for example, [1, Exercise VII.1.10] or [7, Problem 559]). We remark that this concept of

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angles was motivated by the classical work [3] of Jordan and it has serious applications in statistics, for example (see the canonical correlation theory of Hotelling [4], and also see the introduction of [9]). The system of all principal angles between P and Q is denoted by (P , Q). Thus, we have the desired concept of angles between finite rank projections. But in what follows we would also like to extend Wigner’s theorem for the case of infinite rank projections. Therefore, we also need the concept of principal angles between infinite rank projections. Using deep concepts of operator theory (like scalarvalued spectral measure and multiplicity function) this could be carried out, but in order to formulate a Wigner-type result we need only the equality of angles. Hence, we can avoid these complications saying that for arbitrary projections P , Q, P , Q on H we have (P , Q) = (P , Q ) if and only if the positive operators QP Q and Q P Q are unitarily equivalent. This obviously generalizes the equality of principal angles between pairs of finite rank projections. Keeping in mind the formulation of Wigner’s theorem given in the abstract, we are now in a position to formulate the main result of the paper which, we believe, also has physical interpretation. Main Theorem. Let n ∈ N. Let H be a real or complex Hilbert space with dim H ≥ n. Suppose that φ : Pn (H ) → Pn (H ) is a transformation with the property that

(φ(P ), φ(Q)) = (P , Q)

(P , Q ∈ Pn (H )).

If n = 1 or n = dim H /2, then there exists a linear or conjugate-linear isometry V on H such that φ(P ) = V P V ∗

(P ∈ Pn (H )).

If H is infinite dimensional, the transformation φ : P∞ (H ) → P∞ (H ) satisfies

(φ(P ), φ(Q)) = (P , Q)

(P , Q ∈ P∞ (H )),

and φ is surjective, then there exists a unitary or antiunitary operator U on H such that φ(P ) = U P U ∗

(P ∈ P∞ (H )).

As one can suspect from the formulation of our main result, there is a system of exceptional cases, namely, when we have dim H = 2n, n > 1. In the next section we show that in those cases there do exist transformations on Pn (H ) which preserve the principal angles but cannot be written in the form appearing in our main theorem above. 2. Proof This section is devoted to the proof of our main theorem. In fact, this will follow from the statements below. The idea of the proof can be summarized in a single sentence as follows. We extend our transformation from Pn (H ) to a Jordan homomorphism of the algebra F (H ) of all finite rank operators on H which preserves the rank-1 operators. Fortunately, those maps turn out to have a form and using this we can achieve the desired conclusion. On the other hand, quite unfortunately, we have to work hard to carry out all the details of the proof that we are just going to begin. From now on, let H be a real or complex Hilbert space and let n ∈ N. Since our statement obviously holds when dim H = n, hence we suppose that dim H > n.

Preserving Principal Angles Between Subspaces of a Hilbert Space

411

In the sequel, let tr be the usual trace functional on operators. The ideal of all finite rank operators in B(H ) is denoted by F (H ). Clearly, every element of F (H ) has a finite trace. We denote by Fs (H ) the set of all self-adjoint elements of F (H ). We begin with two key lemmas. In order to understand why we consider the property (1) in Lemma 1, we note that if (P , Q) = (P , Q ) for some finite rank projections P , Q, P , Q , then, by definition, the positive operators QP Q and Q P Q are unitarily equivalent. This implies that tr QP Q = tr Q P Q . But, by the properties of the trace, we have tr QP Q = tr P QQ = tr P Q and, similarly, tr Q P Q = tr P Q . So, if our transformation preserves the principal angles between projections, then it necessarily preserves the trace of the product of the projections in question. This justifies Condition (1) in the next lemma. Lemma 1. Let P be any set of finite rank projections on H . If φ : P → P is a transformation with the property that tr φ(P )φ(Q) = tr P Q

(P , Q ∈ P),

(1)

then φ has a unique real-linear extension onto the real-linear span spanR P of P. The transformation is injective, preserves the trace and satisfies (A, B ∈ spanR P).

tr (A)(B) = tr AB

(2)

Proof. For any finite sets {λi } ⊂ R and {Pi } ⊂ P we define

λ i Pi = λi φ(Pi ).

i

i

We have to show that is well-defined. If i λi Pi = k µk Qk , where {µk } ⊂ R and {Qk } ⊂ P are finite subsets, then for any R ∈ P we compute

tr

λi φ(Pi )φ(R) = λi tr(φ(Pi )φ(R)) = λi tr(Pi R)

i

i

= tr =

i

λi Pi R = tr

i

µk Q k R = µk tr(Qk R)

k

µk tr(φ(Qk )φ(R)) = tr

k

k

µk φ(Qk )φ(R) .

k

Therefore, we have tr

i

λi φ(Pi ) −

µk φ(Qk ) φ(R) = 0

k

for every R ∈ P. By the linearity of the trace functional it follows that we have similar equality if we replace φ(R) by any finite linear combination of φ(R)’s. This gives us that tr λi φ(Pi ) − µk φ(Qk ) λi φ(Pi ) − µk φ(Qk ) = 0. i

k

i

k

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2 The operator i λi φ(Pi )− k µk φ(Qk ) , being the square of a self-adjoint operator, is positive. Since its trace is zero, we obtain that

λi φ(Pi ) −

i

2

µk φ(Qk )

=0

k

which plainly implies that

λi φ(Pi ) −

i

µk φ(Qk ) = 0.

k

This shows that is well-defined. The real-linearity of now follows from the definition. The uniqueness of is also trivial to see. From (1) we immediately obtain (2). One can introduce an inner product on Fs (H ) by the formula A, B = tr AB

(A, B ∈ Fs (H ))

(the norm induced by this inner product is called the Hilbert-Schmidt norm). The equality (2) shows that is an isometry with respect to this norm. Thus, is injective. It follows from (1) that tr φ(P ) = tr φ(P )2 = tr P 2 = tr P

(P ∈ P)

which clearly implies that tr (A) = tr A

(A ∈ spanR P).

This completes the proof of the lemma. In what follows we need the concept of Jordan homomorphisms. If A and B are algebras, then a linear transformation : A → B is called a Jordan homomorphism if it satisfies (A2 ) = (A)2

(A ∈ A),

or, equivalently, if (AB + BA) = (A)(B) + (B)(A)

(A, B ∈ A).

Two projections P , Q on H are said to be orthogonal if P Q = QP = 0 (this means that the ranges of P and Q are orthogonal to each other). In this case we write P ⊥ Q. We denote P ≤ Q if P Q = QP = P (this means that the range of P is included in the range of Q). In what follows, we shall use the following useful notation. If x, y ∈ H , then x ⊗ y stands for the operator defined by (x ⊗ y)z = z, yx

(z ∈ H ).

Lemma 2. Let : Fs (H ) → Fs (H ) be a real-linear transformation which preserves the rank-1 projections and the orthogonality between them. Then there is an either linear or conjugate-linear isometry V on H such that (A) = V AV ∗

(A ∈ Fs (H )).


413

Proof. Since every finite-rank projection is the finite sum of pairwise orthogonal rank-1 projections, it is obvious that preserves the finite-rank projections. It follows from [2, Remark 2.2] and the spectral theorem that is a Jordan homomorphism (we note that [2, Remark 2.2] is about self-adjoint operators on finite dimensional complex Hilbert spaces, but the same argument applies for Fs (H ) even if it is infinite dimensional and/or real). We next prove that can be extended to a Jordan homomorphism of F (H ). To see ˜ : F (H ) → F (H ) this, first suppose that H is complex and consider the transformation defined by ˜ (A + iB) = (A) + i(B)

(A, B ∈ Fs (H )).

˜ )2 ˜ 2 ) = (T It is easy to see that is a linear transformation which satisfies (T ˜ (T ∈ F (H )). This shows that is a Jordan homomorphism. If H is real, then the situation is not so simple, but we can apply a deep algebraic result of Martindale as follows (cf. the proof of [10, Theorem 3]). Consider the unitalized algebra F (H ) ⊕ RI (of course, we have to add the identity only when H is infinite dimensional). Defining (I ) = I , we can extend to the set of all symmetric elements of the enlarged algebra in an obvious way. Now we are in a position to apply the results in [8] on the extendability of Jordan homomorphisms defined on the set of symmetric elements of a ring with involution. To be precise, in [8] Jordan homomorphism means an additive map which, besides (s 2 ) = (s)2 , also satisfies (sts) = (s)(t)(s). But if the ring in question is 2-torsion free (in particular, if it is an algebra), this second equality follows from the first one (see, for example, the proof of [15, 6.3.2 Lemma]). The statements [8, Theorem 1] in the case when dim H ≥ 3 and [8, Theorem 2] if dim H = 2 imply that can be uniquely extended to an associative homomorphism of F (H ) ⊕ RI into itself. To be honest, since the results of Martindale concern rings and hence linearity does not appear, we could guarantee only the additivity of the extension of . However, the construction in [8] shows that in the case of algebras, linear Jordan homomorphisms have linear extensions. To sum up, in every case we have a Jordan homomorphism of F (H ) extending . In order to simplify the notation, we use the same symbol for the extension as well. As F (H ) is a locally matrix ring (every finite subset of F (H ) can be included in a subalgebra of F (H ) which is isomorphic to a full matrix algebra), it follows from a classical result of Jacobson and Rickart [6, Theorem 8] that can be written as = 1 + 2 , where 1 is a homomorphism and 2 is an antihomomorphism. Let P be a rank-1 projection on H . Since (P ) is also rank-1, we obtain that one of the idempotents 1 (P ), 2 (P ) is zero. Since F (H ) is a simple ring, it is easy to see that this implies that either 1 or 2 is identically zero, that is, is either a homomorphism or an antihomomorphism of F (H ). In what follows we can assume without loss of generality that is a homomorphism. Since the kernel of is an ideal in F (H ) and F (H ) is simple, we obtain that is injective. We show that preserves the rank-1 operators. Let A ∈ F (H ) be of rank 1. Then there is a rank-1 projection P such that P A = A. We have (A) = (P A) = (P )(A) which proves that (A) is of rank at most 1. Since is injective, we obtain that the rank of (A) is exactly 1. From the conditions of the lemma it follows that φ sends rank-2 projections to rank-2 projections. Therefore, the range of contains an operator with rank greater than 1. We now refer to Hou’s work [5] on the form of linear rank preservers on operator algebras. It follows from the argument leading to [5, Theorem 1.3] that either

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there are linear operators T , S on H such that is of the form (x ⊗ y) = (T x) ⊗ (Sy)

(x, y ∈ H )

or there are conjugate-linear operators T , S on H such that is of the form (x ⊗ y) = (S y) ⊗ (T x)

(x, y ∈ H ).

(3)

Suppose that we have the first possibility. By the multiplicativity of we obtain that u, yT x ⊗ Sv = u, y(x ⊗ v) = (x ⊗ y · u ⊗ v) (x ⊗ y)(u ⊗ v) = T u, SyT x ⊗ Sv.

(4)

This gives us that T u, Sy = u, y for every u, y ∈ H . On the other hand, since sends rank-1 projections to rank-1 projections, we obtain that for every unit vector x ∈ H we have T x = Sx. These imply that T = S is an isometry and with the notation V = T = S we have (A) = V AV ∗ for every A ∈ Fs (H ). We show that the possibility (3) cannot occur. In fact, similarly to (4) we have u, yS v ⊗ T x = S v, T xS y ⊗ T u

(x, y, u, v ∈ H ).

Fixing unit vectors x = y = u in H and considering the operators above at T x, we find that S v = S v, T xT x, T uS y giving us that S is of rank 1. Since sends rank-2 projections to rank-2 projections, we arrive at a contradiction. This completes the proof of the lemma. We are now in a position to present a new proof of the nonsurjective version of Wigner’s theorem which is equivalent to the statement of our main theorem in the case when n = 1. For another proof see [16]. To begin, observe that if P , Q are finite rank projections such that tr P Q = 0, then we have tr(P Q)∗ P Q = tr QP Q = tr P QQ = tr P Q = 0 which implies that (P Q)∗ (P Q) = 0. This gives us that P Q = 0 = QP . Therefore, P is orthogonal to Q if and only if tr P Q = 0. Theorem 3. Let φ : P1 (H ) → P1 (H ) be a transformation with the property that tr φ(P )φ(Q) = tr P Q

(P , Q ∈ P1 (H )).

(5)

Then there is an either linear or conjugate-linear isometry V on H such that φ(P ) = V P V ∗

(P ∈ P1 (H )).

Proof. By the spectral theorem it is obvious that the real linear span of P1 (H ) is Fs (H ). Then, by Lemma 1 we see that there is a unique real-linear extension of φ onto Fs (H ) which preserves the rank-1 projections and, by (5), also preserves the orthogonality between the elements of P1 (H ). Lemma 2 applies to complete the proof.


415

As for the cases when n > 1 we need the following lemma. Recall that we have previously supposed that dim H > n. Lemma 4. Let 1 < n ∈ N. Then spanR Pn (H ) coincides with Fs (H ). Proof. Since the real-linear span of P1 (H ) is Fs (H ), it is sufficient to show that every rank-1 projection is a real-linear combination of rank-n projections. To see this, choose orthonormal vectors e1 , . . . , en+1 in H . Let E = e1 ⊗ e1 + . . . + en+1 ⊗ en+1 and define Pk = E − ek ⊗ ek

(k = 1, . . . , n + 1).

Clearly, every Pk can be represented by a (n + 1) × (n + 1) diagonal matrix whose diagonal entries are all 1’s with the exception of the k th one which is 0. The equation λ1 P1 + . . . + λn+1 Pn+1 = e1 ⊗ e1 gives rise to a system of linear equations with unknown scalars λ1 , . . . , λn+1 . The matrix of this system of equations is an (n + 1) × (n + 1) matrix whose diagonal consists of 0’s and its off-diagonal entries are all 1’s. It is easy to see that this matrix is nonsingular, and hence e1 ⊗ e1 (and, similarly, every other ek ⊗ ek ) is a real-linear combination of P1 , . . . , Pn+1 . This completes the proof. We continue with a technical lemma. Lemma 5. Let P , Q be projections on H . If QP Q is a projection, then there are pairwise orthogonal projections R, R , R such that P = R + R , Q = R + R . In particular, we obtain that QP Q is a projection if and only if P Q = QP . Proof. Let R = QP Q. Since R is a projection whose range is contained in the range of Q, it follows that R = Q − R is a projection which is orthogonal to R. If x is a unit vector in the range of R, then we have QP Qx = 1. Since P Qx is a vector whose norm is at most 1 and its image under the projection Q has norm 1, we obtain that P Qx is a unit vector in the range of Q. Similarly, we obtain that Qx is a unit vector in the range of P and, finally, that x is a unit vector in the range of Q. Therefore, x belongs to the range of P and Q. Since x was arbitrary, we can infer that the range of R is included in the range of P . Thus, we obtain that R = P − R is a projection which is orthogonal to R. Next, using the obvious relations P R = RP = R,

QR = RQ = R

we deduce (Q − R)(P − R)(Q − R) = QP Q − QP R − QRQ + QR − RP Q + RP R + RQ − R = R − R − R + R − R + R + R − R = 0.

(6)

Since A∗ A = 0 implies A = 0 for any A ∈ B(H ), we obtain from (6) that R R = (P − R)(Q − R) = 0. The second part of the assertion is now easy to check. We next prove the assertion of our main theorem in the case when 1 < n ∈ N and H is infinite dimensional.

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Theorem 6. Suppose 1 < n ∈ N and H is infinite dimensional. If φ : Pn (H ) → Pn (H ) is a transformation such that

(φ(P ), φ(Q)) = (P , Q)

(P , Q ∈ Pn (H )),

then there exists a linear or conjugate-linear isometry V on H such that φ(P ) = V P V ∗

(P ∈ Pn (H )).

Proof. By Lemma 1 and Lemma 4, φ can be uniquely extended to an injective real-linear transformation on Fs (H ). The main point of the proof is to show that preserves the rank-1 projections. In order to verify this, just as in the proof of Lemma 4, we consider orthonormal vectors e1 , . . . , en+1 in H , define E = e1 ⊗ e1 + . . . + en+1 ⊗ en+1 and set Pk = E − ek ⊗ ek

(k = 1, . . . , n + 1).

We show that the ranges of all Pk = φ(Pk )’s can be jointly included in an (n + 1)dimensional subspace of H . To see this, we first recall that has the property that tr (A)(B) = tr AB

(A, B ∈ Fs (H ))

(see Lemma 1). Next we have the following property of : if P , Q are orthogonal rank-1 projections, then (P )(Q) = 0. Indeed, if P , Q are orthogonal, then we can include them into two orthogonal rank-(n + 1) projections. Now, referring to the construction given in Lemma 4 and having in mind that preserves the orthogonality between rank-n projections, we obtain that (P )(Q) = 0. (Clearly, the same argument works if dim H ≥ 2(n + 1).) Since the rank-n projections Pk are commuting, by the preserving property of φ and Lemma 5, it follows that the projections (Pk ) are also commuting. It is well-known that any finite commuting family of operators in Fs (H ) can be diagonalized by the same unitary transformation (or, in the real case, by the same orthogonal transformation). Therefore, if we restrict onto the real-linear subspace in Fs (H ) generated by P1 , . . . , Pn+1 , then it can be identified with a real-linear operator from Rn+1 to Rm for some m ∈ N. Clearly, this restriction of can be represented by an m × (n + 1) real matrix T = (tij ). Let us examine how the properties of are reflected in those of the matrix T . First, is trace preserving. This gives us that for every λ ∈ Rn+1 the sums of the coordinates of the vectors T λ and λ are the same. This easily implies that the sum of the entries of T lying in a fixed column is always 1. As we have already noted, (ei ⊗ ei )(ej ⊗ ej ) = 0 holds for every i = j . For the matrix T this means that the coordinatewise product of any two columns of T is zero. Consequently in every row of T there is at most one nonzero entry. Since sends rank-n projections to rank-n projections, we see that this possibly nonzero entry is necessarily 1. So, every row contains at most one 1 and all the other entries in that row are 0’s. Since the sum of the elements in every column is 1, we have that in every column there is exactly one 1 and all the other entries are 0’s in that column. These now easily imply that if λ ∈ Rn+1 is such that its coordinates are all 0’s with the exception of one which is 1, then T λ is of the same kind. When it concerns , this means that sends every ek ⊗ ek (k = 1, . . . , n + 1) to a rank-1 projection. So, we obtain that preserves the rank-1 projections and the orthogonality between them. Now, by Lemma 2 we conclude the proof. We turn to the case when H is finite dimensional.


417

Theorem 7. Suppose 1 < n ∈ N, H is finite dimensional and n = dim H /2. If φ : Pn (H ) → Pn (H ) satisfies

(φ(P ), φ(Q)) = (P , Q)

(P , Q ∈ Pn (H )),

then there exists a unitary or antiunitary operator U on H such that φ(P ) = U P U ∗

(P ∈ Pn (H )).

(7)

Proof. First suppose that dim H = 2d, 1 < d ∈ N. If n = 1, . . . , d − 1, then we can apply the method followed in the proof of Theorem 6 concerning the infinite dimensional case. If n = d +1, . . . , 2d −1, then consider the transformation ψ : P → I −φ(I −P ) on P2d−n (H ). We learn from [7, Problem 559] that if (P , Q) = (P , Q ), then there exists a unitary operator U such that U P U ∗ = P and U QU ∗ = Q . It follows from the preserving property of φ that for any P , Q ∈ P2d−n (H ) we have φ(I − P ) = U (I − P )U ∗ ,

φ(I − Q) = U (I − Q)U ∗

for some unitary operator U on H . This gives us that

(ψ(P ), ψ(Q)) = (U P U ∗ , U QU ∗ ) = (P , Q).

In that way we can reduce the problem to the previous case. So, there is an either unitary or antiunitary operator U on H such that ψ(P ) = U P U ∗

(P ∈ P2d−n (H )).

It follows that φ(I − P ) = I − ψ(P ) = I − U P U ∗ = U (I − P )U ∗ , and hence we have the result for the considered case. Next suppose that dim H = 2d + 1, d ∈ N. If n = 1, . . . , d − 1, then once again we can apply the method followed in the proof of Theorem 6. If n = d + 2, . . . , 2d + 1, then using the “dual method” that we have applied just above we can reduce the problem to the previous case. If n = d, consider a fixed rank-d projection P0 . Clearly, if P is any rank-d projection orthogonal to P0 , then the rank-d projection φ(P ) is orthogonal to φ(P0 ). Therefore, φ induces a transformation φ0 between d + 1-dimensional spaces (namely, between the orthogonal complement of the range of P0 and that of the range of φ(P0 )) which preserves the principal angles between the rank-d projections. Our “dual method” and the result concerning 1-dimensional subspaces lead us to the conslusion that the linear extension of φ0 maps rank-1 projections to rank-1 projections and preserves the orthogonality between them. This implies that the same holds true for our original transformation φ. Just as before, using Lemma 1 and Lemma 2 we can conclude the proof. In the remaining case n = d + 1 we apply the “dual method” once again. We now show that the case when 1 < n ∈ N, n = dim H /2 is really exceptional. To see this, consider the transformation φ : P → I − P on Pn (H ). This maps Pn (H ) into itself and preserves the principal angles. As for the complex case, the preserving property follows from [1, Exercise VII.1.11] while in the real case it was proved already by Jordan in [3] (see [14, p. 310]). Let us now suppose that the transformation φ can be written in the form (7). Pick a rank-1 projection Q on H . We know that it is a real linear combination of some P1 , . . . , Pn+1 ∈ Pn (H ). It would follow from (7) that considering the same linear combination of φ(P1 ), . . . , φ(Pn+1 ), it is a rank-1 projection as well. But due to the definition of φ, we get that this linear combination is a constant minus Q. By the trace preserving property we obtain that this constant is 1/n. Since n > 1,

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the operator (1/n)I − Q is obviously not a projection. Therefore, we have arrived at a contradiction. This shows that the transformation above can not be written in the form (7). It would be a nice result if one could prove that in the present case (i.e., when 1 < n, n = dim H /2) up to unitary-antiunitary equivalence, there are exactly two transformations on Pn (H ) preserving principal angles, namely, P → P and P → I −P . This is left as an open problem. We now turn to our statement concerning infinite rank projections. In the proof we shall use the following simple lemma. If A ∈ B(H ), then denote by rng A the range of A. Lemma 8. Let H be an infinite dimensional Hilbert space. Suppose P , Q are projections on H with the property that for any projection R with finite corank we have RP = P R if and only if RQ = QR. Then either P = Q or P = I − Q. Proof. Let R be any projection on H commuting with P . By Lemma 5, it is easy to see that we can choose a monotone decreasing net (Rα ) of projections with finite corank such that (Rα ) converges weakly to R and Rα commutes with P for every α. Since Rα commutes with Q for every α, we obtain that R commutes with Q. Interchanging the role of P and Q, we obtain that any projection commutes with P if and only if it commutes with Q. Let x be any unit vector from the range of P . Consider R = x ⊗x. Since R commutes with P , it must commute with Q as well. By Lemma 5 we obtain that x belongs either to the range of Q or to its orthogonal complement. It follows that either d(x, rng Q) = 0, or d(x, rng Q) = 1. Since the set of all unit vectors in the range of P is connected and the distance function is continuous, we get that either every unit vector in rng P belongs to rng Q or every unit vector in rng P belongs to (rng Q)⊥ . Interchanging the role of P and Q, we find that either rng P = rng Q or rng P = (rng Q)⊥ . This gives us that either P = Q or P = I − Q. Theorem 9. Let H be an infinite dimensional Hilbert space. Suppose that φ : P∞ (H ) → P∞ (H ) is a surjective transformation with the property that

(φ(P ), φ(Q)) = (P , Q)

(P , Q ∈ P∞ (H )).

Then there exists a unitary or antiunitary operator U on H such that φ(P ) = U P U ∗

(P ∈ P∞ (H )).

Proof. We first prove that φ is injective. If P , P ∈ P∞ (H ) and φ(P ) = φ(P ), then by the preserving property of φ we have

(P , Q) = (P , Q)

(Q ∈ P∞ (H )).

(8)

Putting Q = I , we see that P is unitarily equivalent to P . We distinguish two cases. First, let P be of infinite corank. By (8), we deduce that for every Q ∈ P∞ (H ) we have Q ⊥ P if and only if Q ⊥ P . This gives us that P = P . As the second possibility, let P be of finite corank. Then P , P can be written in the form P = I − P0 and P = I − P0 , where, by the equivalence of P , P , the projections P0 and P0 have finite and equal rank. Let Q0 be any finite rank projection on H . It follows from

(I − P0 , I − Q0 ) = (I − P0 , I − Q0 )


419

that there is a unitary operator W on H such that W (I − Q0 )(I − P0 )(I − Q0 )W ∗ = (I − Q0 )(I − P0 )(I − Q0 ). This implies that W (−Q0 − P0 + P0 Q0 + Q0 P0 − Q0 P0 Q0 )W ∗ = −Q0 − P0 + P0 Q0 + Q0 P0 − Q0 P0 Q0 . Taking traces, by the equality of the rank of P0 and P0 , we obtain that tr P0 Q0 = tr P0 Q0 .

(9)

Since this holds for every finite rank projection Q0 on H , it follows that P0 = P0 and hence we have P = P . This proves the injectivity of φ. Let P ∈ P∞ (H ) be of infinite corank. Then there is a projection Q ∈ P∞ (H ) such that Q ⊥ P . By the preserving property of φ, this implies that φ(Q) ⊥ φ(P ) which means that φ(P ) is of infinite corank. One can similarly prove that if φ(P ) is of infinite corank, then the same must hold for P . This yields that P ∈ P∞ (H ) is of finite corank if and only if so is φ(P ). Denote by Pf (H ) the set of all finite rank projections on H . It follows that the transformation ψ : Pf (H ) → Pf (H ) defined by ψ(P ) = I − φ(I − P )

(P ∈ Pf (H ))

is well-defined and bijective. Since φ(I − P ) is unitarily equivalent to I − P for every P ∈ Pf (H ) (this is because (φ(I − P ), φ(I − P )) = (I − P , I − P )), it follows that ψ is rank preserving. We next show that tr ψ(P )ψ(Q) = tr P Q

(P , Q ∈ Pf (H )).

(10)

This can be done following the argument leading to (9). In fact, by the preserving property of φ there is a unitary operator W on H such that W (I − ψ(Q))(I − ψ(P ))(I − ψ(Q))W ∗ = (I − Q)(I − P )(I − Q). This gives us that W (−ψ(Q) − ψ(P ) + ψ(P )ψ(Q) + ψ(Q)ψ(P ) − ψ(Q)ψ(P )ψ(Q))W ∗ = −Q − P + P Q + QP − QP Q. Taking traces on both sides and referring to the rank preserving property of ψ, we obtain (10). According to Lemma 1, let : Fs (H ) → Fs (H ) denote the unique real-linear extension of ψ onto spanR Pf (H ) = Fs (H ). We know that is injective. Since Pf (H ) is in the range of , we obtain that is surjective as well. It is easy to see that Lemma 2 can be applied and we infer that there exists an either unitary or antiunitary operator U on H such that (A) = U AU ∗

(A ∈ Fs (H )).

Therefore, we have φ(P ) = U P U ∗

420

L. Molnár

for every projection P ∈ P∞ (H ) with finite corank. It remains to prove that the same holds true for every P ∈ P∞ (H ) with infinite corank as well. This could be quite easy to show if we know that φ preserves the order between the elements of P∞ (H ). But this property is far away from being easy to verify. So we choose a different approach to attack the problem. Let P ∈ P∞ (H ) be a projection of infinite corank. By the preserving property of φ we see that for every Q ∈ P∞ (H ) the operator φ(Q)φ(P )φ(Q) is a projection if and only if QP Q is a projection. By Lemma 5, this means that φ(Q) commutes with φ(P ) if and only if Q commutes with P . Therefore, for any Q ∈ P∞ (H ) of finite corank, we obtain that Q commutes with U ∗ φ(P )U (this is equivalent to that φ(Q) = U QU ∗ commutes with φ(P )) if and only if Q commutes with P . By Lemma 8 we have two possibilities, namely, either U ∗ φ(P )U = P or U ∗ φ(P )U = I − P . Suppose that U ∗ φ(P )U = I − P . Consider a complete orthonormal basis e0 , eγ (γ ∈ -) in the range of P and, similarly, choose a complete orthonormal basis f0 , fδ (δ ∈ /) in the range of I − P . Pick nonzero scalars λ, µ with the property that |λ|2 + |µ|2 = 1 and |λ| = |µ|. Define eγ ⊗ e γ + fδ ⊗ f δ . Q = (λe0 + µf0 ) ⊗ (λe0 + µf0 ) + γ

δ

Clearly, Q is of finite corank (in fact, its corank is 1). Since φ(Q)φ(P )φ(Q) = U QU ∗ φ(P )U QU ∗ is unitarily equivalent to QP Q, it follows that the spectrum of QU ∗ φ(P )U Q is equal to the spectrum of QP Q. This gives us that the spectrum of Q(I − P )Q is equal to the spectrum of QP Q. By the construction of Q this means that 0, 1, |µ|2 = 0, 1, |λ|2 which is an obvious contradiction. Consequently, we have U ∗ φ(P )U = P , that is, φ(P ) = U P U ∗ . Thus, we have proved that this latter equality holds for every P ∈ P∞ (H ) and the proof is complete. Acknowledgements. This research was supported from the following sources: (1) Hungarian National Foundation for Scientific Research (OTKA), Grant No. T030082, T031995; (2) a grant from the Ministry of Education, Hungary, Reg. No. FKFP 0349/2000.

References 1. Bhatia, R.: Matrix Analysis. Berlin-Heidelberg-New York: Springer-Verlag, 1997 2. Brešar, M., Šemrl, P.: Mappings which preserve idempotents, local automorphisms, and local derivations. Canad. J. Math. 45, 483–496 (1993) 3. Jordan, C.: Essai sur la géométrie á n dimensions. Bull. Soc. Math. France 3, 103–174 (1875) 4. Hotelling, H.: Relations between two sets of variates. Biometrika 28, 321-377 (1935) 5. Hou, J.C.: Rank-preserving linear maps on B(X). Sci. China Ser. A 32, 929–940 (1989) 6. Jacobson, N., Rickart, C.: Jordan homomorphisms of rings. Trans. Am. Math. Soc. 69, 479–502 (1950) 7. Kirillov, A.A., Gvishiani, A.D.: Theorems and Problems in Functional Analysis. Berlin–Heidelberg–New York: Springer-Verlag, 1982 8. Martindale, W.S.: Jordan homomorphisms of the symmetric elements of a ring with involution. J. Algebra 5, 232–249 (1967) 9. Miao, J., Ben-Israel, A.: On principal angles between subspaces in Rn . Linear Algebra Appl. 171, 81–98 (1992) 10. Molnár, L.: An algebraic approach to Wigner’s unitary-antiunitary theorem. J. Austral. Math. Soc. 65, 354–369 (1998)


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11. Molnár, L.: A generalization of Wigner’s unitary-antiunitary theorem to Hilbert modules. J. Math. Phys. 40, 5544–5554 (1999) 12. Molnár, L.: Generalization of Wigner’s unitary-antiunitary theorem for indefinite inner product spaces. Commun. Math. Phys. 201, 785–791 (2000) 13. Molnár, L.: A Wigner-type theorem on symmetry transformations in type II factors. Int. J. Theor. Phys. 39, 1463–1466 (2000) 14. Paige, C.C., Wei, M.: History and generality of the CS decomposition. Linear Algebra Appl. 208/209, 303–326 (1994) 15. Palmer, T.W.: Banach Algebras and The General Theory of *-Algebras, Vol. I. Cambridge: Cambridge University Press, 1994 16. Sharma, C.S., Almeida, D.F.: A direct proof of Wigner’s theorem on maps which preserve transition probabilities between pure states of quantum systems. Ann. Phys. 197, 300–309 (1990) Communicated by H. Araki

Commun. Math. Phys. 217, 423 – 449 (2001)

Communications in



Polynomial Invariants for Torus Knots and Topological Strings J. M. F. Labastida1 , Marcos Mariño2 1 Departamento de Física de Partículas, Universidade de Santiago de Compostela,

15706 Santiago de Compostela, Spain. E-mail: [email protected]

2 New High Energy Theory Center, Rutgers University, Piscataway, NJ 08855, USA.

E-mail: [email protected] Received: 1 May 2000 / Accepted: 6 November 2000

Abstract: We make a precision test of a recently proposed conjecture relating Chern– Simons gauge theory to topological string theory on the resolution of the conifold. First, we develop a systematic procedure to extract string amplitudes from vacuum expectation values (vevs) of Wilson loops in Chern–Simons gauge theory, and then we evaluate these vevs in arbitrary irreducible representations of SU (N ) for torus knots. We find complete agreement with the predictions derived from the target space interpretation of the string amplitudes. We also show that the structure of the free energy of topological open string theory gives further constraints on the Chern–Simons vevs. Our work provides strong evidence towards an interpretation of knot polynomial invariants as generating functions associated to enumerative problems. Contents 1. 2. 3.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extracting String Amplitudes from Chern–Simons Gauge Theory Polynomial Invariants for Torus Knots in Arbitrary Irreducible Representations of SU (N ) . . . . . . . . . . . . . . . . . . . . . 4. Explicit Results for fR . . . . . . . . . . . . . . . . . . . . . . . 5. A Conjecture for the Connected vevs . . . . . . . . . . . . . . . 6. Conclusions and Open Problems . . . . . . . . . . . . . . . . . Appendix A. The Functions fR (t, λ) for = 4 . . . . . . . . . . . . .

. . . . . . . . . .

423 425

. . . . .

430 437 444 446 447

. . . . .

. . . . .

. . . . .

. . . . .

1. Introduction Ever since the Jones polynomial and its generalizations were discovered [1], knot theorists have been searching for an interpretation of the integers entering these polynomials. Though it seems rather natural to regard these polynomials as generating functions associated to enumerative problems, not much progress has been achieved in this direction.

424

J. M. F. Labastida, M. Mariño

One of the main goals of this paper is to point out that the situation has changed dramatically after the recent work by Ooguri and Vafa in [2]. Based on their results, we will provide strong evidence to affirm that from the ordinary polynomial invariants associated to arbitrary irreducible representations of the group SU (N ) one can construct new ones whose integer coefficients can be interpreted as the solutions to specific enumerative problems in the context of string theory. Thus, with regard to a picture of polynomial invariants as generating functions, these new polynomials are more fundamental than the ordinary ones. At the heart of this development is Chern–Simons gauge theory [3] and the relationship between large N gauge theories and gravity in the light of the AdS/CFT correspondence (see [4] for a review). The proposal of [5–7, 2], which can be regarded as a simpler version of the AdS/CFT correspondence, relates Chern–Simons gauge theory on S3 to topological string theory whose target is the resolution of the conifold1 . This proposal is very interesting from the point of view of knot theory and three-manifold topology, since it reformulates the invariants obtained in the context of Chern–Simons gauge theory in terms of invariants associated to topological strings and related to the counting of BPS states. In particular, in [2] a generating function of vacuum expectation values (vevs) of Wilson loops was expressed in terms of certain integers counting the number of D2 branes ending on D4 branes. This reformulation makes some predictions about the structure of Chern– Simons vevs and provides an interpretation for the integer coefficients of some related polynomial invariants. It was verified in [2] that these predictions were true in the simple case of the unknot. One purpose of this paper is to make a precision test of the proposal of [2] for a wide class of nontrivial knots. As a preliminary step, we present a systematic procedure to extract from the vevs of Wilson loops a series of polynomials arising naturally in the context of topological strings. This is the content of Eq. (2.19) below. These polynomials, that we denote2 by fR , are labeled by the irreducible representations, R, of SU (N ), and according to the conjecture in [2] they have a very precise structure dictated by the BPS content of the “dual” theory. We then test this conjecture with actual computations in Chern–Simons gauge theory. The technical challenge associated to the conjecture in [2], on the Chern–Simons side, is that it involves vevs of Wilson loops in arbitrary irreducible representations of SU (N ), with N generic. For the fundamental representation, the vevs are related (up to a normalization) to the HOMFLY polynomial [12]. Not much is known about these vevs for other irreducible representations, except in the case of SU (2), where they are related to the Akutsu–Wadati polynomials (for a review, see [13]). There are also some sample computations for a few knots in [14], for representations of SU (N ) with only one row in their Young diagram3 . However, in the case of torus knots, one can compute these vevs using the formalism of knot operators introduced in [16]. Knot operators were used in [16–19] to compute the vevs of Wilson loops for torus knots and links in the fundamental representations of SU (N ) and SO(N ), and in arbitrary irreducible representations of SU (2). The computation of vevs for torus knots in arbitrary irreducible representations of SU (N ), as needed to test the conjecture of [2], is technically difficult, but fortunately many of the intermediate results were already obtained in [18]. This leads to a general 1 The relation between Chern–Simons gauge theory and string theory has been addressed also in [8–10]. The connection between Chern–Simons and topological open string theory was discovered by Witten in [11]. 2 Though we will refer to the f as polynomials, they are not. They are polynomials up to a common factor R as stated in (2.8) and (4.22). 3 There are also a few computations in [15] for the gauge group SU (3).

Torus Knots and Topological Strings

425

formula for these vevs, which can be found in (3.29) below. Despite its intimidating aspect, it is not difficult to implement it in a computer routine to obtain the vevs for any torus knot. Equation (3.29) is of course a result interesting in its own, and we hope that it will be helpful in exploring the generalizations of the HOMFLY polynomial to arbitrary irreducible representations of SU (N ). Using the general formula (3.29), we will test the conjecture presented in [2] for some nontrivial knots. We will find that, in all the examples that we have checked, the polynomials fR have in fact the structure predicted by [2]. This is a highly nontrivial fact from the point of view of Chern–Simons gauge theory, and we regard it as a strong evidence for the duality advocated in [5–7, 2]. There are in fact two different predictions in [2], which are in a sense complementary. The first one predicts the structure of the polynomials fR , it is based on a target space interpretation, and it is nonperturbative. The second one is perturbative and it is based on the worldsheet interpretation of the Chern–Simons vevs presented in [11]. These two predictions are related in a very interesting way. More precisely, it turns out that the perturbative structure of the free energy of the open string gives some “sum rules” on the integers that count BPS configurations. We have also found complete agreement with the perturbative prediction in all the examples that we have checked. The paper is organized as follows: in Sect. 2, we describe the conjecture presented in ref. [2], which expresses a generating functional of Chern–Simons gauge theory in terms of certain polynomials fR . We extract from the conjecture a “master equation” which allows us to obtain these functions from usual vevs in Chern–Simons gauge theory through a recursive procedure. In Sect. 3, we obtain a general formula for the vevs of torus knots in arbitrary irreducible representations of SU (N ). This section contains the arguments leading to formula (3.29), which are independent of the rest of the paper. It could be skipped in a first reading. In Sect. 4, we use formula (3.29) to obtain some of the polynomials fR , taking as an example the right-handed trefoil knot. The results are in full agreement with the conjecture of 2. In Sect. 5 we show that the perturbative point of view gives some nontrivial constraints among the integer invariants that appear in the polynomials fR , and we also show that the connected vevs of Chern–Simons have the structure dictated by these constraints. Finally, in Sect. 6, we conclude with some comments and open problems. An appendix collects the expressions of fR for the right-handed trefoil knot for all irreducible representations of SU (N ) whose associated Young Tableaux contains four boxes. 2. Extracting String Amplitudes from Chern–Simons Gauge Theory We first recall some basic aspects of Chern–Simons gauge theory, mainly to fix our notation. Chern–Simons gauge theory is a topological gauge theory whose action is, k 2 S= (2.1) Tr A ∧ dA + A ∧ A ∧ A , 4π M 3 where A is a gauge connection on some vector bundle over a three-manifold M, and k is the coupling constant. From the holonomy of the gauge field around a closed loop γ in M, U = P exp A, (2.2) γ

426


one can construct a natural class of topological observables, the gauge-invariant Wilson loop operators, which are given by γ

WR (A) = Tr R U,

(2.3)

where R denotes an irreducible representation of SU (N ). Some of the standard topological invariants that have been considered in the context of Chern–Simons gauge theory are vevs of products of these operators: γ WR11

γ · · · WRnn

1 = Z(M)

n

[DA]

i=1

γ WRii eiS ,

(2.4)

where Z(M) is the partition function of the theory. In this paper we will consider an enlarged set of operators which, to our knowledge, has not been studied from a Chern– Simons gauge theory point of view for non-trivial knots. These operators involve, besides the standard Wilson loops and their products, additional products with traces of powers of the holonomy (2.2). We will compute their vevs for the case of torus knots. In the process we will derive a formula for the vevs of Wilson loops in arbitrary irreducible representations of the gauge group SU (N ). The resulting vevs will be expressed in terms of the variables4 , t = exp

2π i , k+N

λ = tN .

(2.5)

In order to make a precise test of the conjecture presented in ref. [2], we will consider the vev of the operator, Z(U, V ) = exp

∞ n=1

1 n n Tr U Tr V , n

(2.6)

where U is the holonomy of the Chern–Simons SU (N ) gauge field (2.2), and V is an SU (M) matrix that can be regarded as a source term. In this operator the trace is taken over the fundamental representation. In what follows, when no representation is indicated in a trace, it should be understood that it must be taken in the fundamental representation. The main conjecture of [2] has two parts. First, it states that the vev of (2.6) can be written as, Z(U, V ) = exp

∞

fR (t n , λn )Tr R

n=1 R

Vn , n

(2.7)

where the sum over R is a sum over irreducible representations of SU (M). Second, it predicts the following structure for the functions fR (t, λ): fR (t, λ) =

NR,Q,s s,Q

1 2

t −t

− 21

λQ t s ,

(2.8)

4 A word of caution about notation: in [18], the variable λ is denoted t N−1 . Also, in order to compare to [2], notice that our t is their exp(iλ), and our λ is their exp t.


427

where NR,Q,s are integer numbers, and the Q and s are, in general, half-integers (however, for a given fR , the Q differ by integer numbers). In writing (2.7), and to be able to compare to the results in Chern–Simons gauge theory, we have performed an analytic continuation, as suggested in [2]. The prediction (2.8) is based on the duality between Chern–Simons theory and topological string theory. As explained in [2], given a knot K in S3 one constructs a Lagrangian submanifold CK in the noncompact Calabi–Yau O(−1) + O(−1) → S2 (the resolution of the conifold). The integers NR,Q,s count, very roughly, holomorphic maps from Riemann surfaces with boundaries to the Calabi–Yau, in such a way that the boundaries are mapped to CK . A more precise understanding of the integers NR,Q,s is given by the target space interpretation of the string amplitudes. In this interpretation, one reformulates the counting problem in terms of D-branes. One considers configurations of D2 branes ending on CK , in the presence of M D4 branes wrapping CK and filling an R2 in the uncompactified spacetime. The D2 branes are BPS particles from the two-dimensional point of view. These particles are characterized by their magnetic charge, their bulk D2 brane charge, and their spin, which correspond, respectively, to R, Q and s in (2.8). The integer NR,Q,s counts the number of BPS states with these quantum numbers. We then see that the conjecture of [2] makes a remarkable connection between knot invariants and an enumerative problem in the context of symplectic and algebraic geometry, and that the polynomials fR can be regarded as counting functions for this enumerative problem. In this section we will prove the first part of the conjecture. It follows from simple group theoretical arguments. Thus, it will be established that the vevs of Wilson loops in arbitrary irreducible representations of the gauge group can be encoded in the functions fR (t, λ). This also gives a concrete procedure to compute these functions from Chern– Simons vevs, and using this procedure we will present a highly nontrivial evidence for (2.8) in the case of torus knots. Our starting point is the construction of a set of linear equations for the functions fR (t, λ) in terms of vevs of standard Wilson loops in arbitrary irreducible representations. To carry this out, it is convenient to use the following basis of class functions (see, for example, [20–22]). Take a vector k with an infinite number of entries, almost all zero, and whose nonzero entries are positive integers. Given such a vector, we define: kj . (2.9) = j kj , |k| = We can associate to any vector k a conjugacy class C(k) of the permutation group S . This class has k1 cycles of length 1, k2 cycles of length 2, and so on. The number of elements of the permutation group in such a class is given by [23] |C(k)| =

! . kj ! j kj

(2.10)

Equivalently, the vectors k with j j kj = are in one-to-one correspondence with the partitions of . Given an ordered h-uple of positive integers (n1 , · · · , nh ), we can map it to a vector k by putting ki equal to the number of i’s in the h-uple. Notice that h = |k|, and that there are h!/ kj ! different h-uples giving the same vector k. We now introduce the following basis in the space of class functions, labeled by the vectors k: ∞ kj ϒk (U ) = Tr U j . (2.11) j =1

428


It is easy to see that: Z(U, V ) = 1 +

|C(k)| !

k

ϒk (U )ϒk (V ),

(2.12)

since we are assuming > 0. Let’s now consider the expansion of the exponent in (2.7) in terms of the basis (2.11). We first recall the Frobenius formula to express traces in an arbitrary irreducible representation of SU (M) in terms of the elements of the basis (2.11) referred to this group: Tr R (V ) =

|C(k)| !

k

χR (C(k))ϒk (V ).

(2.13)

In this formula, the irreducible representation R can be associated to a Young diagram in the standard way. The sum is then over conjugacy classes with equal to the number of boxes in the diagram. To analyze the expansion in (2.7), we have to write Tr R V n in terms of the basis (2.11). To do this it is convenient to define the following vector k 1/n . Fix a vector k, and consider all the positive integers that satisfy the following condition: n|j for every j with kj = 0. Notice that n = 1 always satisfies this condition. When this happens, we will say that “n divides k”, and we will denote this as n|k. We can then define the vector k 1/n whose components are: (k 1/n )i = kni .

(2.14)

The vectors which satisfy the above condition and are “divisible by n” have the structure (0, . . . , kn , 0, . . . , 0, k2n , . . . ), and the vector k 1/n is then given by (kn , k2n , . . . ). It is a simple combinatorial exercise to prove that the exponent in (2.7) is given by: |C(k)| k

!

n|k

n|k|−1

χR (C(k 1/n ))fR (t n , λn )ϒk (V ).

(2.15)

R

In this equation, the third sum is over representations of S . We now define a generalization of the cumulant expansion for the vevs we are considering. First, associate to k any k the polynomial pk (x) = j xj j in the variables x1 , x2 , . . . . We then define the (c)

“connected” coefficients ak as follows:

|C(k)| |C(k)| (c) ak pk (x) = ak pk (x). log 1 + ! ! k

(2.16)

k

One has, for example: (c)

2 a(2,0,... ) = a(2,0,... ) − a(1,0,... ), (c)

a(1,1,0,... ) = a(1,1,0,... ) − a(1,0,... ) a(0,1,0,... ) ,

(2.17)

(c)

a(0,...,0,1,0,... ) = a(0,...,0,1,0,... ) , and so on. For vectors of the form (n, 0, . . . ), this is just the cumulant expansion.


429

Define now the vevs: Gk (U ) = ϒk (U ).

(2.18)

Using (2.6), (2.12), (2.16) and (2.18), we find: log Z(U, V ) =

|C(k)| !

k

(c)

Gk (U )ϒk (V ).

(2.19)

Since the ϒk (V ) are a basis in the space of class functions, we find that Eq. (2.7) can be written as (c) n|k|−1 χR (C(k 1/n ))fR (t n , λn ). (2.20) Gk (U ) = R

n|k

This is our “master equation”. It allows us to obtain the functions fR (t, λ) once we compute the vevs that appear on the left-hand side. The way to do that is to consider all vectors k with a fixed , where will be considered as the “order” of the expansion. The number of these vectors is the number of partitions of , p(). At every order there (c) are then p() vevs Gk (U ) and also p() representations R of S . The relations (2.20) provide p() equations with p() unknowns, the functions fR (t, λ). The data to solve the equations are the vevs (2.18) and the fR (t, λ) for representations with < boxes. The procedure to find the polynomials is then recursive, and the structure one finds is very similar, in fact, to the recursive procedure which determines the integer invariants introduced in [24], as it is explained in [25]. The above system of linear equations has a unique solution. This follows from the fact that the associated matrix, |C(k)|χR (C(k)), is invertible due to orthonormality of the characters. Thus, the first part of the conjecture, Eq. (2.7), is proved. There are two cases of the above expression which are particularly interesting. The first one is for k = (, 0, . . . ). In this case, the corresponding conjugacy class in S is the identity, and one finds (Tr U ) (c) = (dim R)fR (t, λ), (2.21) R

where the sum is over the representations R of S . The left-hand side is the usual connected vev. The second example corresponds to the vector k = (0, . . . , 0, 1, 0, . . . ), where the nonzero entry is in the th position. In this case, we have to sum in (2.20) over all the divisors of , that we will denote by n. The vector k 1/n is then (0, . . . , 0, 1, 0, . . . ), where the nonzero entry is in the /nth position. The characters χ (C(k 1/n )) are different from zero only for the hook representations, i.e., those corresponding to Young diagrams with (/n) − s boxes in the first row, and one box in the remaining ones, for example, (2.22) The character is then (−1)s (see [23], 4.16). The formula (2.20) reads in this case:

Tr U =

/n−1 n|

s=0

(−1)s fhook,s (t n , λn ).

(2.23)

430


3. Polynomial Invariants for Torus Knots in Arbitrary Irreducible Representations of SU (N ) In order to obtain the functions fR from the master equation (2.20) one needs to compute (c) the connected functions Gk (U ). After using (2.18) and the inverse of the Frobenius formula (2.13), it turns out that these involve the computation of vevs of Wilson loops in arbitrary irreducible representations of SU (N ). As stated in the introduction these vevs are known only for some particular cases. In order to have a good testing ground of the conjecture (2.7) it would be desirable to have a formula for these vevs valid for any representation, at least for some particular class of knots. The goal of this section is to derive such a formula for torus knots. The result is contained in Eq. (3.29) below. The arguments leading to it are independent of the rest of the paper and thus this section could be skipped in a first reading. The techniques used to obtain the formula (3.29) are based on the application of the operator formalism to Chern–Simons gauge theory [26], that in the case of torus knots leads to the useful concept of knot operators [16]. 3.1. Knot operators. Knot operators for torus knots were introduced in [16]. They allow the computation of vevs of Wilson loops corresponding to this type of knots for arbitrary irreducible representations of the gauge group. The first piece of data we need to introduce these operators is the Hilbert space of Chern–Simons gauge theory on a torus [26]. This space has an orthonormal basis |p labeled by weights p in the fundamental chamber of the weight lattice of SU (N ), Fl , where l = k + N . We take as representatives of p the ones of the form p = i pi λi , where λi , i = 1, · · · , N − 1, are the fundamental weights, pi > 0 and i pi < l. The vacuum is the state |ρ, where ρ is the Weyl vector (i.e., the sum of all the fundamental weights). Torus knots are labeled by two coprime integers (n, m). They correspond to winding numbers around the two non-contractible classes of cycles, A and B on the torus. Let m be the number of times that the torus knot winds around the axis of the torus, and let , be the highest weight of an irreducible representation. Then, the Wilson loop corresponding to that torus knot is represented by the following operator: nm m (n,m) exp − iπ µ2 − 2π i p · µ |p + nµ. (3.1) W, |p = k+N k+N µ∈M,

In this equation, M, is the set of weights corresponding to the irreducible representation ,. To compute the vev of the Wilson loop around a torus knot in S3 , one proceeds as follows: first of all, one makes a Heegard splitting of S3 into two solid tori. Then, one puts the torus knot on the surface of one of the solid tori by acting with the knot operator (3.1) on the vacuum. Finally, one glues together the tori by performing an S-transformation. There is an extra subtlety related to the framing dependence in Chern–Simons gauge theory, since the vev computed in this way has to be corrected with a phase. In the standard framing the vev of the Wilson loop is given by: (n,m)

W,

(n,m)

= e2πinmhρ+,

where, hp =

ρ|SW, |ρ , ρ|S|ρ

p2 − ρ 2 , 2(k + N )

(3.2)

(3.3)

is the conformal weight of the primary fields in the associated WZW model at level k.


431

3.2. Vacuum expectation values from knot operators. Our next task is to provide a more precise expression for the vev (3.2). When acting with the knot operator (3.1) on the vacuum, we get the set of weights ρ + nµ, where µ ∈ M, . These weights will have representatives in the fundamental chamber, which can be obtained by a series of Weyl reflections. If the representative has a vanishing component, then the corresponding state in the Hilbert space is zero due to antisymmetry of the wave function under Weyl reflections. The set of weights that have a nonzero representative in Fl will be denoted by M(n, ,), and it depends on the irreducible representation with highest weight ,, and on the integer number n. The representative of ρ + nµ in M(n, ,) will be denoted by ρ + µn . The matrix elements of S have the explicit expression,

2π ip · w(p ) Sp,p = c(N, k) , (3.4) /(w) exp − k+N w∈W

where c(N, k) is a constant depending only on N and k, and the sum is over the Weyl group of SU (N ), W. Using this, the vev (3.2) can be written as:

2π i m nm − 2π i ρ · µ chµn − ρ . exp −iπ µ2 e2πinmhρ+, k+N k+N k+N µ∈M(n,,)

(3.5) In this expression, we have used the Weyl formula for the character: w(,+ρ)·a w∈W /(w)e ch, (a) = . w(ρ)·a w∈W /(w)e

(3.6)

Notice that, since the representatives µn live in Fl , they can be considered as highest weights for a representation, hence the above expression (3.5) makes sense. In practice, the main problem to compute this vevs explicitly is to find the nonzero representatives of the weights that appear in (3.1), and to find an expression for the characters in (3.5). Fortunately, this has been done in [18] in a slightly different context. In that paper, these problems were solved for all the weights in the product representation V ⊗s , where V is the fundamental representation of SU (N ) and s is any integer. Since all the representations of SU (N ) that correspond to Young diagrams with s boxes are in fact contained in the reducible tensor product V ⊗s , we only have to combine the results of [18] with some simple group theory. This will give an explicit expression for the vev value of Wilson loops for torus knots in arbitrary representations of SU (N ). 3.3. Group theory. To obtain the expression for the vev of the Wilson loop, we need the weight space associated to arbitrary representations of SU (N ). It is very convenient to regard this space as a subspace of the weight space associated to the reducible representation V ⊗s . Let’s denote by µi , i = 1, . . . , N the weights of the fundamental representation of SU (N ). Any weight in V ⊗s will have the form k1 µi1 + · · · + kr µir ,

1 ≤ i1 < · · · < ir ≤ N,

(3.7)

where (kλ ) = (k1 , . . . , kr ) is an ordered partition of s, i.e. an r-tuple that sums up to s. The kλ will be taken as strictly positive integers, therefore 1 ≤ r ≤ s. The corresponding unordered partition will be simply denoted by k. Unordered partitions for SU (N ) will

432


be written as N -tuples with nonincreasing components, as in [23]. The set of weights (3.7), for a fixed (kλ ), will be denoted by Mkλ . Consider now a irreducible representation R of SU (N ), associated to the highest weight ,=

N−1

ai λ i .

(3.8)

i=1

This representation can be labeled by a Young diagram with s = i iai boxes in the usual way. Equivalently, we can assign to the highest weight (3.8) an unordered partition of s: a = (a1 + · · · + aN−1 , a2 + · · · + aN−1 , · · · , aN−1 , 0). The weight space of this representation can always be written as follows M, = m, (kλ ) Mkλ ,

(3.9)

(3.10)

kλ

where the m, (kλ ) are nonnegative integers giving the multiplicities of the weights (3.7) in M, . This can be proved as follows (see [23] for more details). The irreducible representation associated to , is given by Sa (V ), where Sa is the Schur–Weyl functor. Any endomorphism of V will extend to Sa (V ), and its character will be given by the Schur polynomial Sa (x1 , · · · , xN ), where x1 , · · · , xN are the eigenvalues of g. The Schur polynomials can be expanded in terms of the symmetric polynomials Fk , which are also labeled by unordered partitions of s, k = (k1 , · · · , kN ) (with k1 ≥ · · · ≥ kN ). Fk is kN the sum of the elementary monomial Xk = x1k1 · · · xN and all the monomials obtained k from it by permuting the variables. The set of X and its permutations is then labeled by ordered partitions. The expansion of the Schur polynomials is given by: Sa = Nak Fk , (3.11) k

where the Nak are called the Kostka numbers. These numbers are nonnegative integers and can be also computed as the number of ways one can fill the diagram a with k1 1’s, k2 2’s, . . . , kr r’s in such a way that the entries in each row are nondecreasing and those in each column are strictly increasing. Since the xi , i = 1, · · · , N , correspond to the weights µi of the fundamental representation, each of the monomials in Fk corresponds to a one-dimensional weight space with a weight of the form (3.7). The different monomials in Fk are in one-to-one correspondence with the different ordered partitions associated to the unordered partition k. We have then proved the equality (3.10). From the proof above follows that in the decomposition (3.10) all the ordered partitions corresponding to the same unordered partition appear with the same multiplicity, and moreover that m, (kλ ) = Nak ,

(3.12)

where a is the partition associated to ,. We can then compute the multiplicities in (3.10) 1 very easily. For example, for R = Syms (V ), we have , = sλ1 , and msλ (kλ ) = 1 for every ordered partition of s. For R = ∧s V one has , = λs , and mλ(ksλ ) = 0 for every (kλ )


433

except for (kλ ) = (1, 1, · · · , 1), where the multiplicity is one. For the diagram can represent (3.10) as: = 2(1, 1, 1) + (2, 1) + (1, 2),

, we (3.13)

where the vectors in the r.h.s. represent ordered partitions, and the coefficients are the multiplicities. For representations with four boxes one has: = 3(1, 1, 1, 1) + (1, 3) + (3, 1) + (2, 2) + 2{(2, 1, 1) + (1, 2, 1) + (1, 1, 2)}, = 3(1, 1, 1, 1) + (2, 1, 1) + (1, 2, 1) + (1, 1, 2),

(3.14)

= 2(1, 1, 1, 1) + (2, 2) + (2, 1, 1) + (1, 2, 1) + (1, 1, 2).

3.4. General formula. We are now in a position to be more explicit about the expression (3.5). Using the decomposition (3.10), we can write all the weights for the irreducible representation R in the form (3.7). We have to find now which vectors of the form ρ +nµ have a representative in Fl , and the explicit structure of such a weight. This has been completely solved in Theorem 4.1 of [18]. The main output of this theorem is that the weights ρ + n(k1 µi1 + · · · + kr µir )

(3.15)

associated to a partition of cardinal r give a representative only for certain values of the indices i1 , . . . , ir . The procedure to get these indices, as well as the corresponding representative, is rather involved, but we will give it here for completeness. For further details, we refer the reader to [18]. The arrangement of indices iλ , λ = 1, · · · , r, producing a weight in Fl is contained in the set specified by the following conditions: (I) (II)

iλ ≤ kλ n, iλ = iµ + kλ n,

µ < λ,

(3.16)

in such a way that, in (II), no previous index iν , ν < λ, has the form iν = iµ + kν n, µ < ν. Given an arrangement of indices like this, with r − k indices verifying condition (I) (which will be called of type I) and k indices verifying condition (II) (which will be called of type II), a weight belonging to Fl is obtained if and only if: iµ − iν + (kν − kµ )n = 0,

(3.17)

for every pair of indices iµ , iν , verifying (I). The set of arrangements of indices selected in this way will be denoted by I(kλ ) (n), and the corresponding set of weights will be denoted by MI(kλ ) (n) . To each arrangement of indices in I(kλ ) (n) we will associate a canonical representative in Fl accompanied by a sign. This association is carried out by the following procedure: (1) For indices of type I, which will be denoted by iλ1 , · · · , iλr−k , one defines a total order relation according to: iλp iλq iff iλp − iλq + (kλq − kλp )n > 0.

(3.18)

434


This relation defines a permutation τ of the set of indices of type I under consideration with respect to their natural ordering:

iλ1 iλ2 · · · iλr−k (3.19) τ= iτ (λ1 ) iτ (λ2 ) · · · iτ (λr−k ) (2) For the k indices of type II, iν1 , · · · , iνk , iν1 < · · · < iνk , one takes the set of indices iνˆ 1 , · · · , iνˆ k , verifying iνp = iνˆ p + kνp n, and defines on it the order relation inherited from the natural ordering of the indices iνp : iνˆ p iνˆ q iff iνp > iνq .

(3.20)

This gives again a permutation σ with respect to the natural ordering of the set iνˆ p :

iσ −1 (ˆν1 ) iσ −1 (ˆν2 ) · · · iσ −1 (ˆνk ) σ = , iνˆ 1 iνˆ 2 ··· iνˆ k

(3.21)

with iσ −1 (ˆν1 ) < iσ −1 (ˆν2 ) < · · · < iσ −1 (ˆνk ) .

(3) Define r − k numbers ξ(λp ), p = 1, · · · , r − k, associated to type I indices as follows: ξ(λp ) is the number of type II indices preceding the type I index iλp in the original arrangement of indices i1 , · · · , ir , in (3.15). The canonical representative in Fl of the weight (3.15) is the weight: ρ + p1 λ1 + p2 λ2 + · · · + pr−k λr−k + λiµ1 +r−k−1 + λiµ2 +r−k−2 + · · · + λiµr−k , (3.22) where pi , i = 1, · · · , r − k, are given by: p1 = iτ (λ2 ) − iτ (λ1 ) + (kτ (λ1 ) − kτ (λ2 ) )n − 1, p2 = iτ (λ3 ) − iτ (λ2 ) + (kτ (λ2 ) − kτ (λ3 ) )n − 1, .. . pr−k = kτ (λr−k ) n − iτ (λr−k ) ,

(3.23)

and the indices iµp in (3.22) are the complementary ones to the indices {iνˆ p }p=1,··· ,k , i.e., those indices iµp such that no index iν > iµp has the form iν = iµp + kν n. They are ordered according to their natural ordering: iµ1 < · · · < iµr−k . Finally, the sign associated to this weight because of the Weyl reflections needed to obtain it is: /(τ )/(σ )(−1)

r−k

p=1 iµp −µp +ξ(λp )

.

(3.24)

This result gives then an explicit description of the set of weights M(n, ,) in (3.5): M(n, ,) =

(kλ )

m, (kλ ) MI(kλ ) (n) ,

and the representatives of these weights have the form (3.22).

(3.25)


435

The last ingredient in (3.5) is the character, which has also been computed in [18] for weights with the structure of (3.22). Before doing this, it is useful to introduce q-numbers and q-combinatorial numbers as follows: x

x

[x] = t 2 − t − 2 , (x) = t x − 1,

[x]! x . = [x − y]![y]! y

(3.26)

One can then easily prove that

−1

(λ − t j ) N +p − 21 p 21 p(p+1) j =−p =λ t , p (p)! i−1

j N j =0 (λ − t ) − 2i 2i . =λ t (i)! i

(3.27)

The character for the weight (3.22) is given by

r−1 r−1 2πi − [pk + 1] · · · pλ + r − 1 ρ = k+N

ch,

k=1

λ=k

r

[ik − ij ]

1≤j 0, and p = (j − 2l)n − 2, q = nl + 1 if 1 + n(2l − j ) < 0. In the first case, the sign is +1, and it is −1 in the second case. The character of this weight for λ = t 2 is simply p

chpλ1 +qλ2 = t − 2

t p+1 − 1 . t −1

(3.33)


437

jλ

Taking into account that m(kλ1) = 1 for all the ordered partitions, one finds, after a short calculation: (n,m)

Wj

j

= t− 2

j j t 2 (n−1)(m−1) m(1+nl)(j −l) 1+nl t − t n(j −l) , t t −1

(3.34)

l=0

where the summands with l = 0, j come from the partition (s), and the summands with 1 ≤ l ≤ j − 1 come from the partition (l, j − l). The expression in (3.34) is in fact the unnormalized Akutsu–Wadati polynomial for the (n, m) torus knot, in the representation of isospin j/2 [17]. 4. Explicit Results for fR The results of the previous section will allow us to compute the quantities on the left hand side of the master equation (2.20). These quantities are products of traces of powers of the holonomy associated to a given knot. Computations of this type are delicate from a field theory point of view because they involve products of operators evaluated for the same loop. The corresponding calculations are plagued with singularities which must be regularized. One way to do this, advocated in [15] and also suggested in [2], involves the use of Frobenius formula. In particular, what is needed is the inverse of (2.13): ϒk (U ) = χR (C(k))Tr R (U ). (4.1) R

All problems arising from products of operators evaluated for the same loop are avoided using this formula since one ends computing vevs of standard Wilson loops. Actually, it is rather simple to prove that the choice (4.1) leads to the following general form for the functions fR (t, λ): fR (t, λ) = Tr R (U ) + lower order terms,

(4.2)

where “lower order terms” involves fR (t, λ) for representations R with < . Thus the set of functions fR (t, λ) is equivalent to the set of vevs in arbitrary irreducible representations. This relation implies that the new polynomial invariants are basically the ordinary ones plus correction terms. As it follows from the master equation (2.20) these corrections terms are linear combinations of products of lower-order fR evaluated at different arguments. The remarkable consequence that follows from the validity of the conjecture (2.8) is that the corrected polynomials possess integer coefficients which can be interpreted as the solutions to counting problems in the context of string theory. Using (4.1) and the result for Tr R (U ) in (3.29) we will be able to obtain the functions fR (t, λ) for torus knots after solving the master equation (2.20). We will present in this section the computations up to third order, where the order is set by , as we explained in Sect. 2. 4.1. = 1. In this case, k = (1, 0, · · · ) and (2.21) just says that Tr U = f (t, λ).

(4.3)

438


The left-hand side of this equation is the unnormalized HOMFLY polynomial. To normalize it we have to divide it by the vev of the unknot: λ 2 − λ− 2 1

Tr U u =

1

t 2 − t− 2 1

1

.

(4.4)

Due to the skein relations, the normalized HOMFLY polynomial always has the structure [27]: Tr U = ps (λ)t s . Tr U u s

(4.5)

In this equation, the s take integer values, and ps (λ) = j as,j λj are Laurent polynomials in λ. The as,j are integer numbers. Therefore, f (t, λ) has indeed the structure predicted in (2.8). The integers N ,Q,s are given by: N

,j +1/2,s

= as,j − as,j +1 .

(4.6)

We then see that, for the fundamental representation, the integers introduced in [2] are simple linear combinations of the coefficients in the normalized HOMFLY polynomial. Notice that (4.6) is valid for any knot, since we have only used the general structure of the HOMFLY polynomial. As an example, let us consider the right-handed trefoil, which is the (2, 3) torus knot. One obtains that f (t, λ) =

1 1 2

t −t

(−2λ 2 + 3λ 2 − λ 2 ) + (t 2 − t − 2 )(−λ 2 + λ 2 ), 1

− 21

3

5

and from here one can easily extract the values of N

1

1

1

3

(4.7)

,Q,s .

4.2. = 2. In this case, there are two possible vectors corresponding to conjugacy classes: (2, 0, · · · ), and (0, 1, 0, · · · ). From (2.20), (2.21) and (2.23) we find two equations: (Tr U )2 − Tr U 2 = f Tr U 2 = f

(t, λ) + f (t, λ), (t, λ) − f (t, λ) + f (t 2 , λ2 ).

(4.8)

As argued above, the Frobenius formula (4.1) allows to express the new quantities appearing on the left of this equation in terms of vevs of Wilson loops. For the case under consideration it leads to: (Tr U )2 = Tr

U + Tr U ,

Tr U 2 = Tr

U − Tr U .

(4.9)

From these relations and Eq. (4.8) we obtain, after taking into account (4.3): 1 f (t, λ)2 + f (t 2 , λ2 ) , 2 1 f (t, λ) = Tr U − f (t, λ)2 − f (t 2 , λ2 ) . 2

f

(t, λ) = Tr

U −

(4.10)


439

We will now present explicit formulae for these functions in the simplest nontrivial case, namely the right-handed trefoil knot. The vev in the symmetric representation is given by: Tr

U =

(λ − 1)(λt − 1) λ(t 2 − t − 2 )2 (1 + t) × (λt −1 )2 (1 − λt 2 + t 3 − λt 3 + t 4 − λt 5 + λ2 t 5 + t 6 − λt 6 ) . 1

1

(4.11) In this equation, we have explicitly factored out the vev for the unknot in the symmetric representation. The polynomial multiplying the fraction in the right-hand side is then the normalized polynomial invariant in the symmetric representation. One can see that this expression agrees with the result presented in [14]. It can also be easily checked that, when we substitute λ → t 2 , we obtain the Akutsu–Wadati polynomial for the right-handed trefoil in the j = 2 representation, as it should be. For the antisymmetric representation we find: Tr U =

(λ − 1)(λ − t) λ(t 2 − t − 2 )2 (1 + t) × (λt −2 )2 (1 − λ − λt + λ2 t + t 2 + t 3 − λt 3 − λ t 4 + t 6 ) . 1

1

Notice that, when N = 2 (i.e., when λ = t 2 ), indeed (4.12) becomes 1. Using (4.10) one finds:

(4.12)

becomes the trivial representation and

t − 2 λ(λ − 1)2 (1 + t 2 ) (t + λ2 t − λ (1 + t 2 )) 1

f

(t, λ) =

1 f (t, λ) = − 3 f t

t 2 − t− 2 1

1

, (4.13)

(t, λ).

The structure of these functions is in perfect agreement with (2.8). This computation makes clear that the prediction (2.8) is far from being trivial from the Chern–Simons side. The vevs (4.11) and (4.12) have complicated denominators that have to cancel out 1 1 except for a single factor of t 2 − t − 2 when one subtracts the lower order contributions as in (4.10). Also notice that the coefficients of the functions in (4.13) are in fact integers, and again this is not obvious from (4.10) (which involves dividing by 2). These features become more and more remarkable as we increase the number of boxes of the representations, as we will see. 4.3. = 3. At this order there are three vectors that contribute, k = (3, 0, · · · ), k = (1, 1, 0, · · · ), and k = (0, 0, 1, 0, · · · ). From (2.21) we obtain: (Tr U )3 − 3Tr U (Tr U )2 + 2Tr U 3 =f (t, λ) + f (t, λ) + 2f

(t, λ),

(4.14)

440


while from (2.23) one has: Tr U 3 = f

(t, λ) + f (t, λ) − f

(t, λ) + f (t 3 , λ3 ).

(4.15)

(t, λ) − f (t, λ).

(4.16)

Finally, the vector (1, 1, 0, · · · ) gives us: Tr U Tr U 2 − Tr U Tr U 2 = f Using again Frobenius formula, we find: f

1 (t, λ) − f (t, λ)3 6 1 1 2 2 3 3 − f (t, λ)f (t , λ ) − f (t , λ ), 2 3 (t, λ) = Tr U − f (t, λ)(f (t, λ) + f (t, λ)) (t, λ) = Tr

f

U − f (t, λ)f

1 1 − f (t, λ)3 + f (t 3 , λ3 ), 3 3

(4.17)

1 1 f (t, λ) = Tr U − f (t, λ)f (t, λ) − f (t, λ)3 + f (t, λ)f (t 2 , λ2 ) 6 2 1 − f (t 3 , λ3 ). 3 Let’s now present some results for the right-handed trefoil knot. For the representation , we find: Tr

U =

(λ − 1)(λt − 1)(λt 2 − 1) 1 1 λ3/2 (t 2 − t − 2 )3 (1 + t) 1 + t + t 2 × (λt −1 )3 (1 − λ t 3 + t 4 − λ t 4 + t 5 − λ t 5 + t 6 − λ t 7 + λ2 t 7 + t 8 − 2 λ t 8 + λ2 t 8 + t 9 − 2 λ t 9 + λ2 t 9 + t 10 − λ t 10

(4.18) − λ t 11 + λ2 t 11 + t 12 − λ t 12 + λ2 t 12 − λ3 t 12 − λ t 13 + λ2 t 13 ) ,

which also agrees with the computation in [14]. For the representation Tr

U =

, we find:

(λ − 1)(λ − t)(λt − 1) 3 −5 1 − λ + 2 t 2 − 2 λ t 2 + λ2 t 2 λ t 1 1 − 3/2 3 2 λ (t 2 − t 2 ) 1 + t + t − t 3 + λ2 t 3 + 2 t 4 − 3 λ t 4 + λ2 t 4 − λ3 t 5 + 2 t 6

(4.19) − 3 λ t 6 + λ2 t 6 − t 7 + λ2 t 7 + 2 t 8 − 2 λ t 8 + λ2 t 8 + t 10 − λ t 10 .

Notice that, when λ → t 2 , the normalized polynomial (which is the polynomial inside the parentheses in (4.19)) goes to the Jones polynomial of the right-handed trefoil, since the representation reduces to the fundamental representation j = 1 when N = 2.


441

Finally, for the representation ∧3 V , with Young diagram , one has: Tr U =

(λ − 1)(λ − t)(λ − t 2 ) 1 λ3/2 (t − t − 2 )3 (1 + t) 1 + t + t 2 × λ3 t −10 − λ + λ2 + t − λ t + λ2 tλ3 t − λ t 2 1 2

(4.20)

+ λ2 t 2 + t 3 − λ t 3 + t 4 − 2 λ t 4 + λ2 t 4 + t 5 − 2 λ t 5

+ λ2 t 5 − λ t 6 + λ2 t 6 + t 7 + t 8 − λ t 8 + t 9 − λ t 9 − λ t 10 + t 13 . Using these vevs, one finds: λ 2 t −1 (λ−1)2 3 t 1 + t + t 3 + λ4 t 3 1 + t + t 2 + t 3 + t 4 + t 6 1 1 t 2 − t− 2 − λ t 1 + 3 t + 3 t2 + 4 t3 + 5 t4 + 2 t5 + 2 t6 + t7 − λ3 t 1 + 3 t + 3 t 2 + 5 t 3 + 5 t 4 + 4 t 5 + 2 t 6 + 3 t 7 + t 9 + λ2 1 + 2 t + 3 t 2 + 7 t 3 + 7 t 4 + 6 t 5 + 6 t 6 + 4 t 7 + t 8 + 2 t 9 , 3

f

(λ, t) = −

3

f

(t, λ) =

λ 2 t −4 (λ − 1)2 (1 + t + t 2 )

t 2 − t− 2 × t 3 + λ4 t 2 + t 4 − λ t 1 + 2 t + t 2 + 2 t 3 + t 4 − λ3 t 2 + t + 3 t 2 + t 3 + 2 t 4 + λ2 1 + t + 3 t 2 + 3 t 3 + 3 t 4 + t 5 + t 6 , (4.21) 1 −3 3 λ2 t (λ − 1)2 4 f (t, λ) = − t + t 6 + t 7 + λ4 t + t 3 + t 4 + t 5 + t 6 + t 7 1 1 − t2 −t 2 2 − λ t 1 + 2 t + 2 t2 + 5 t3 + 4 t4 + 3 t5 + 3 t6 + t7 + λ2 t 2 + t + 4 t 2 + 6 t 3 + 6 t 4 + 7 t 5 + 7 t 6 + 3 t 7 + 2 t 8 + t 9 − λ3 1 + 3 t 2 + 2 t 3 + 4 t 4 + 5 t 5 + 5 t 6 + 3 t 7 + 3 t 8 + t 9 . 1

1

Again, this is in perfect agreement with (2.8). In the appendix we list the fR (t, λ) for the right-handed trefoil knot, for representations with four boxes. 4.4. Structure of fR . The functions fR (t, λ) that we have listed in this section, as well as many other examples that we have explicitly computed, have the structure predicted by (2.8). In all cases they can be written as fR (t, λ) =

λ 2 − λ− 2 1

1

1 2

− 21

t −t

1 1 PR t 2 , λ 2 ,

(4.22)

1 1 1 1 where PR t 2 , λ 2 is a Laurent polynomial in t 2 , λ 2 with integer coefficients. Notice that we have factored out the vev of the unknot in the fundamental representation. The above

442


structure is far from being obvious from its definition, or from the explicit expressions given above: to get fR (t, λ) we have to add up functions with rather complicated denominators, however the result has the simpler structure given in (4.22). Also, to obtain fR (t, λ) we have to divide by !, however the coefficients of the resulting polynomial in (4.22) have integer coefficients. Notice that (4.22) has an extra piece of information 1 1 when compared to (2.8): namely, that one can extract a common factor λ 2 − λ− 2 from the functions fR (t, λ). It would be interesting to see if this is a general fact, and if it can be also predicted from the string side. It also follows from our computations that, for a given irreducible representation R, the integers NR,Q,s are only different from zero for a finite number of values of Q and s. However, the functions fR become more and more complicated as we increase the number of boxes, even for the trefoil knot (which is the simplest nontrivial knot). This seems to indicate that, given an irreducible representation R, there are always values of Q and s for which NR,Q,s = 0. Therefore, for every nontrivial knot there seems to be an infinite number of nonzero integers NR,Q,s . 4.5. The functions fR for the unknot. In [2] it was explicitly shown that, for the unknot, the functions fR (t, λ) vanish for all R but the fundamental representation. This property can be easily checked using the fact that for the unknot, Tr R U u = dimt R,

(4.23)

where dimt R is the quantum dimension of the representation R. Recall that this dimension is easily computed for SU (N ) using the hook rule to calculate the ordinary dimension of R. This rule assigns a quotient to dim R obtained in the following way: for the numerator, products of N ± i, i = 0, 1, 2, . . . , each coming from a box located on the parallel to the diagonal placed ±i times away from the diagonal, taking the plus sign for the upper part and the minus sign for the lower part; a denominator provided by the hook lengths. For example, for the Young tableau: R= the dimension is,

dimR =

,

(4.24)

N (N + 1)(N − 1) . 3·1·1

(4.25)

The corresponding quantum dimension is obtained after replacing each of the integers n appearing in the quotient by its corresponding quantum number, x

{x} = so that, dimt R =

x

t 2 − t− 2 t 2 − t− 2 1

1

,

(4.26)

{N }{N + 1}{N − 1} . {3}{1}{1}

(4.27)

Using (4.3) we obtain for the unknot, f (t, λ)u = Tr U u = dimt

= {N },

(4.28)


443

which is consistent with (4.4). This relation is also consistent with the results in [2]. Let us now test the rest of the expressions for the functions fR which we have obtained. Taking (4.10) one finds that, indeed, 1 f (t, λ)u = dimt − {N }2 + {N } 2 = 0, t→t 2 (4.29) 1 2 f (t, λ)u = dimt − {N } − {N } 2 = 0. t→t 2 Similarly, using these results and (4.17) one confirms that for the functions of third order: f

(t, λ)u = f

(t, λ)u = f (t, λ)u = 0.

(4.30)

Equations (4.29) and (4.30) constitute an important check of our calculations. 4.6. Perturbative expansions and Vassiliev invariants. Using the same arguments as in [28] to prove that the coefficients of the perturbative series expansion are Vassiliev invariants [29] one can easily show that the vevs (2.18) lead to a perturbative series expansion whose coefficients are also Vassiliev invariants. This implies that the functions fR (t, λ) share the same properties. In other words, if one considers the power series expansion, ∞ αi x i , (4.31) fR (ex , eNx ) = i=0

the coefficients αi , i = 0, 1, 2, . . . , are Vassiliev invariants of order i. The explicit form of the Vassiliev invariants for torus knots (n, m) are known [30, 31] up to order six. They turn out to be polynomials in n and m. At lowest orders, the form of these invariants imply the following structure for the polynomials PR in (4.22): PR (ex , eNx ) = g0 + g2 (R)β2,1 x 2 + g3 (R)β3,1 x 3 + O(x 4 ), where,

1 2 (n − 1)(m2 − 1), 24 1 = nm(n2 − 1)(m2 − 1), 144

(4.32)

β2,1 = β3,1

(4.33)

and g2 (R) and g3 (R) are constants (independent of n and m) which depend on the representation R. After computing PR for a variety of torus knots we find, P

(x) = 2N (N 2 − 1)β3,1 x 3 + O(x 4 ),

P (x) = −2N (N 2 − 1)β3,1 x 3 + O(x 4 ), P

(x) = 6N (N 2 − 1)β3,1 x 3 + O(x 4 ),

P

(4.34)

(x) = −6N (N − 1)β3,1 x + O(x ), 2

3

4

P (x) = 6N (N 2 − 1)β3,1 x 3 + O(x 4 ), in full agreement with (4.32). These results constitute a test of the fact that the coefficients of the perturbative series expansion associated to the polynomials PR must be Vassiliev invariants. But the

444


test indicates the existence of more structure. As argued above, the functions fR have a very simple structure, many cancellations occur in such a way that these functions have a simple denominator and the vev for the unknot factorizes. The results (4.34) indicate that they might satisfy more striking properties. Though the fact that g0 = 0 is a simple consequence of (4.29) and (4.30), there is no reason to expect that g2 (R) = 0 for the representations under consideration. This property implies that the second derivative with respect to t of PR (after replacing λ → t N ) vanishes at t = 1. This might be a first indication of the existence of some important properties shared by the polynomials PR . Further work is needed to study their general features and applications. In particular, it would be very interesting to understand in more detail the relation between the coefficients NR,Q,s of (2.8) and Vassiliev invariants. 5. A Conjecture for the Connected vevs In the previous sections we have shown how to extract string amplitudes from Chern– Simons vevs, and that the amplitudes computed in that way have in fact the structure predicted in (2.8) by using the target space interpretation. The worldsheet interpretation of the amplitudes is in principle more complicated, since it involves open string instantons. However, the structure of the free energy of topological string theory dictated by worldsheet perturbative considerations gives a remarkable set of constraints on the integers NR,Q,s or, equivalently, on the connected vevs of Chern–Simons gauge theory. As explained in [2], the arguments in [11] imply that the free energy F (V ) = − logZ(U, V ) is given by the expression F (V ) =

∞ ∞ g=0 h=1 n1 ,··· ,nh

x 2g−2+h Fg;n1 ,··· ,nh (λ) Tr V n1 · · · Tr V nh ,

(5.1)

where x is 2π i/(k + N ), and g and h denote the genus and number of boundaries of the string worldsheet. Let us now compare this expression with Eqs. (2.7) and (2.8), based on the target space interpretation. To do this we assume that there is some analytical continuation which turns the series (5.1) into a series involving only positive integers ni . If we then choose the basis (2.11) for the class functions, we find: F (V ) =

∞ |k|! Fg,k (λ)x 2g−2+|k| ϒk (V ). kj ! k

(5.2)

g=0

Comparing to (2.19) we immediately obtain: (c)

Gk (U ) = −|k|!

j kj

∞

Fg,k (λ) x 2g−2+|k| .

(5.3)

g=0

This makes a highly nontrivial prediction about the structure of the connected vevs: if we put t = ex , keeping λ as an independent variable, then the expansion in x should start with a power greater than or equal to |k| − 2. Moreover, the expansion should contain powers of the same parity (i.e. the powers should be all even or all odd, depending on (c) the parity of |k|). This implies that the functions Gk (U ) are even (odd) under t ↔ t −1 when |k| is even (odd).


445

Let us now analyze this prediction. For = 1 (and therefore |k| = 1), the left-hand side of (5.3) is the unnormalized HOMFLY polynomial. The fact that the expansion in x starts with x −1 is a consequence of (4.5). Since the normalized HOMFLY polynomial (c) is even under the exchange of t and t −1 [27], G(1,0,··· ) (U ) is odd, in agreement with the prediction. These two facts were already noted in [7] in this context (indeed, Eq. (5.3) generalizes Eq. (2.3) of [7] to more complicated vevs). For > 1, the prediction (5.3) is far from being obvious: the vevs of Wilson loops in the representation R start typically with the power x − when we do not expand λ, and they do not have any a priori symmetry under t ↔ t −1 . However, we have found that the prediction (5.3) is in fact true in all the cases that we have checked. For example, in the case of the right-handed trefoil knot, and for the connected vevs at order four, we have obtained: 1 2 λ (λ − 1)(−134 + 1498λ − 6278λ2 + 13146λ3 − 15129λ4 4 1 + 9735λ5 − 3289λ6 + 455λ7 ) + O(x), x (c) G(0,2,0,··· ) (U ) = 12λ2 (λ − 1)4 (9 − 72λ + 198λ2 − 176λ3 + 49λ4 ) + O(x 2 ),

(c)

G(0,0,0,1,0,··· ) (U ) =

(c)

G(1,0,1,0,··· ) (U ) = 9λ2 (λ − 1)4 (10 − 92λ + 233λ2 − 200λ3 + 55λ4 ) + O(x 2 ), (c)

G(2,1,0,··· ) (U ) = 72(λ − 1)5 λ2 (−3 + 27λ − 58λ2 + 28λ3 ) x + O(x 3 ), (c)

G(4,0,0,··· ) (U ) = 432(λ − 1)6 λ2 (1 − 9λ + 16λ2 ) x 2 + O(x 4 ). (5.4) In addition, one finds that the expansion only contains powers of x of the same parity, in agreement with (5.3). We think that this result gives another important check of the open string interpretation of Chern–Simons gauge theory. The prediction (5.3) can be stated in terms of the integer invariants NR,Q,s by using our master equation (2.20). Notice that, from (2.20), the most we can say about the expansion of the connected vevs is that they start with x −1 . However, more is true, as we have just seen. This means that there must be some constraints on the integer invariants NR,Q,s . Let us obtain these constraints. Using the definition of the Bernoulli polynomials, ∞ ext t m−1 = Bm (x) , t e −1 m!

(5.5)

m=0

we find the following equation for Fg,k : Fg,k (λ) =−

|k|!

1

j kj

n|k

n2g+2|k|−3

R,Q,s

χR (C(k 1/n ))NR,Q,s

B2g−1+|k| (s + 1/2) nQ λ . (2g − 1 + |k|)! (5.6)

This expression can be interpreted as a multicovering formula for open string instantons, in the spirit of [24]. Notice that the sum over representations in this equation is finite, as

446


in (2.20). The structure of the expansion in (5.2) also implies the following sum rules. Fix a vector k and a half-integer j . Then, one has: n|k|+m−2 χR (C(k 1/n ))NR,j/n,s Bm (s + 1/2) = 0, (5.7) n|k

R,s

when m ≡ |k| mod 2, and also when m = 0, 1, · · · , |k| − 2 (for |k| ≥ 2). NR,j/n,s is taken to be zero if j/n is not a half-integer. Notice that the sum in (5.7) involves only a finite number of terms. The sum rules (5.7) encode the properties about the perturbative expansion of the connected vevs that we discussed above, in terms of the integers NR,Q,s . 6. Conclusions and Open Problems In this paper we have presented strong evidence for the existence of new polynomial invariants, fR , whose integer coefficients NR,Q,s can be regarded as the solutions of a certain enumerative problem in the context of string theory. These polynomials are labeled by irreducible representations of SU (N ), and for the fundamental representation they correspond to the unnormalized HOMFLY polynomials. For other irreducible representations they have the form of the corresponding unnormalized ordinary polynomial invariants, plus a series of correction terms which involve representations whose associated Young tableaux have a lower number of boxes. Their existence would answer a basic question in knot theory which has remained open for many years: polynomial invariants, appropriately corrected, can indeed be regarded as generating functions. The evidence for the existence of the new polynomials is a consequence of the precision test of the correspondence between Chern–Simons gauge theory and topological strings carried out in this paper. We have proved that one can in fact extract the string amplitudes from the Chern–Simons vevs following a recursive procedure. This makes it possible to compute the integer invariants NR,Q,s starting from the Chern–Simons side. Using explicit results for torus knots, we have been able to give remarkable evidence for the predictions of [2], and we have also exploited the interplay between worldsheet and target results to give further checks of the string theory interpretation of Chern–Simons gauge theory. There are clearly two different avenues for future research. On the Chern–Simons side, it would be extremely interesting to extend these results to more general knots, ncomponent links, and/or other gauge groups. It would be also very interesting to explore in more detail the relations between the integers NR,Q,s and the two other sets of integer invariants of knots: the coefficients of the normalized polynomials, and the Vassiliev invariants. On the string side, the duality with Chern–Simons gauge theory opens the possibility of extracting information about open string instantons in the resolved conifold geometry. The procedure we have developed in this paper gives a very concrete strategy to compute the string amplitudes and obtain the relevant spectrum of BPS states associated to D2 branes ending on D4 branes. As a preliminary step, one should make more precise the geometry of the Lagrangian submanifold in the resolved geometry. We hope to report on these and other related issues in the near future. Acknowledgements. We would like to thank M. Bershadsky, M. Douglas and A.V. Ramallo for useful conversations. We are specially indebted to H. Ooguri and C. Vafa for discussions and correspondence, and for a critical reading of the manuscript. M.M. would like to thank the Departamento de Física de Partículas at the Universidade de Santiago de Compostela, where part of this work was done, for their hospitality. The work by J.M.F.L. was supported in part by DGICYT under grant PB96-0960. The work of M.M. is supported by DOE grant DE-FG02-96ER40959.


447

Appendix A. The Functions fR (t, λ) for = 4 In this appendix, we list the functions fR (t, λ) for representations with four boxes, in the case of the right-handed trefoil knot. The results are: f

(t, λ) =

f

f

t −3/2 1 2

− 21

2

−1+λ

λ2 λ − t − 1 + λ t

t −t t3 1 + t + 2 t2 + t3 + 2 t4 + t6 − λ t 1 + 3 t + 6 t 2 + 9 t 3 + 11 t 4 + 11 t 5 + 10 t 6 + 8 t 7 + 5 t 8 + 3 t 9 + 2 t 10 + t 11 (A.1) + λ4 t 3 1 + t + 3 t 2 + 2 t 3 + 4 t 4 + 2 t 5 + 4 t 6 7 8 9 10 11 12 14 + 2t + 3t + t + 2t + t + t + t + λ2 1 + 2 t + 7 t 2 + 10 t 3 + 18 t 4 + 19 t 5 + 24 t 6 + 19 t 7 + 20 t 8 + 13 t 9 + 12 t 10 + 6 t 11 + 5 t 12 + 2 t 13 + 2 t 14 − λ3 t 1 + 3 t + 6 t 2 + 10 t 3 + 13 t 4 + 15 t 5 + 15 t 6 + 14 t 7 + 12 t 8 + 10 t 9 + 8 t 10 + 6 t 11 + 4 t 12 + 2 t 13 + 2 t 14 + t 15 ,

(t, λ) = −

t −9/2 1 2

− 21

2

−1+λ

λ2 1 + t λ − t − 1 + λ t

t −t 2 3 t 1 + t + t 2 + t 3 + λ2 1 + t + t 2 + t 3 1 + t + 2 t 2 + t 4 (A.2) − λ t 1 + 3 t + 5 t2 + 8 t3 + 7 t4 + 6 t5 + 3 t6 + 2 t7 + λ4 t 2 1 + t + 3 t 2 + 2 t 3 + 3 t 4 + t 5 + 2 t 6 + t 8 − λ3 t 2 + 4 t + 8 t 2 + 10 t 3 + 11 t 4 + 10 t 5 + 7 t 6 + 4 t 7 + 3 t 8 + t 9 + t 10 ,

(t, λ) =

t −19/2 1 2

− 21

2

−1+λ

λ2 1 + t λ − t − 1 + λ t

t −t 2 t 5 1 + t + t 2 + t 3 + λ2 t 1 + t + t 2 + t 3 1 + 2 t 2 + t 3 + t 4 (A.3) − λ t3 2 + 3 t + 6 t2 + 7 t3 + 8 t4 + 5 t5 + 3 t6 + t7 4 3 4 5 6 7 8 9 + λ t + 2t + t + 3t + 2t + 3t + t + t − λ3 1 + t + 3 t 2 + 4 t 3 + 7 t 4 + 10 t 5 + 11 t 6 + 10 t 7 + 8 t 8 + 4 t 9 + 2 t 10 ,

448

f


(t, λ) = − t −6 (−1 + λ)2 λ2 (λ − t) (−1 + λ t) 1 + t + t 2 t 3 + t 5 − λ t (1 + t + t 2 + t 3 )2 − λ3 t(1 + t)2 1 + t + t 2 + t 3 + t 4 + λ4 t + 2 t 3 + t 4 + 2 t 5 + t 7 + λ2 2 + 2 t + 6 t 2 + 6 t 3 + 9 t 4 + 6 t 5 + 6 t 6 + 2 t 7 + 2 t 8 , (A.4)

f (t, λ) = −

t −35/2

(−1 + λ)2 λ2 (λ − t) (−1 + λ t) 1 t − t− 2 t8 1 + 2 t2 + t3 + 2 t4 + t5 + t6 − λ t5 1 + 2 t + 3 t 2 + 5 t 3 + 8 t 4 + 10 t 5 + 11 t 6 + 11 t 7 + 9 t 8 + 6 t 9 + 3 t 10 + t 11 + λ4 1 + t 2 + t 3 + 2 t 4 + t 5 1 2

+ 3 t 6 + 2 t 7 + 4 t 8 + 2 t 9 + 4 t 10 + 2 t 11 + 3 t 12 + t 13 + t 14 + λ2 t 3 2 + 2 t + 5 t 2 + 6 t 3 + 12 t 4 + 13 t 5 + 20 t 6 + 19 t 7 + 24 t 8 + 19 t 9 + 18 t 10 + 10 t 11 + 7 t 12 + 2 t 13 + t 14 − λ3 t 1 + 2 t + 2 t 2 + 4 t 3 + 6 t 4 + 8 t 5 + 10 t 6 + 12 t 7 + 14 t 8 + 15 t 9 + 15 t 10 + 13 t 11 (A.5) + 10 t 12 + 6 t 13 + 3 t 14 + t 15 .

References 1. Jones, V.F.R.: Hecke algebras representations of braid groups and link polynomials. Ann. of Math. 126, 335 (1987) 2. Ooguri, H. and Vafa, C.: Knot invariants and topological strings. hep-th/9912123, Nucl. Phys. B 577, 419 (2000) 3. Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351 (1989) 4. Aharony, O., Gubser, S., Maldacena, J., Ooguri H. and Oz, Y.: Large N field theories, string theory and gravity. hep-th/9905111, Phys. Rep. 323, 183 (2000) 5. Gopakumar, R. and Vafa, C.: Topological gravity as large N topological gauge theory. hep-th/9802016, Adv. Theor. Math. Phys. 2, 413 (1998) 6. Gopakumar, R. and Vafa, C.: M-theory and topological strings, I. hep-th/9809187 7. Gopakumar, R. and Vafa, C.: On the gauge theory/geometry correspondence. hep-th/9811131 8. Awada, M.A.: The exact equivalence of Chern–Simons theory with fermionic string theory. Phys. Lett. B221, 21 (1989) 9. Periwal, V.: Topological closed-string theory interpretation of Chern–Simons theory. hep-th/9305112, Phys. Rev. Lett. 71, 1295 (1993) 10. Douglas, M.R.: Chern–Simons-Witten theory as a topological Fermi liquid. hep-th/9403119 11. Witten, E.: Chern–Simons gauge theory as a string theory. hep-th/9207094, In: The Floer memorial volume, H. Hofer, C.H. Taubes, A. Weinstein and E. Zehner, eds., Bael–Boston: Birkhäuser, 1995, p. 637 12. Freyd, P., Yetter, D., Hoste, J., Lickorish, W.B.R., Millet, K. and Ocneanu, A.: A new polynomial invariant of knots and links. Bull. Am. Math. Soc. 12, 239 (1985) 13. Wadati, M., Deguchi, T. and Akutsu, Y.: Exactly solvable models and knot theory. Phys. Rep. 180, 247 (1989) 14. Ramadevi, P., Govindarajan, T.R. and Kaul, R.K.: Three-dimensional Chern–Simons theory as a theory of knots and links (III), Compact semi-simple group. Nucl. Phys. B 402, 548 (1993) 15. Guadagnini, E.: The universal link polynomial. Int. J. Mod. Phys. A 7, 877 (1992); The link invariants of the Chern–Simons field theory. Berlin: Walter de Gruyter, 1993 16. Labastida, J.M.F., Llatas, P.M. and Ramallo, A.V.: Knot operators in Chern–Simons theory. Nucl. Phys. B 348, 651 (1991)


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17. Isidro, J.M., Labastida, J.M.F. and Ramallo, A.V.: Polynomials for torus links from Chern–Simons gauge theories. hep-th/9210124, Nucl. Phys. B 398, 187 (1993) 18. Labastida, J.M.F. and Mariño, M.: The HOMFLY polynomial for torus links from Chern–Simons gauge theory. hep-th/9402093, Int. J. Mod. Phys. A 10, 1045 (1995) 19. Labastida, J.M.F. and Pérez, E.: A relation between the Kauffman and the HOMFLY polynomial for torus knots. q-alg/9507031, J. Math. Phys. 37, 2013 (1996) 20. Drouffe, J.M. and Zuber, J.B.: Strong-coupling and mean field methods in lattice gauge theories. Phys. Rep. 102, 1 (1983) 21. Cordes, S., Moore, G. and Ramgoolam, S.: Lectures on two-dimensional Yang-Mills theory, equivariant cohomology, and topological field theory. hep-th/9412210, Nucl. Phys. Proc. Suppl. 41, 184 (1995) 22. Douglas, M.R.: Conformal field theory techniques in large N Yang-Mills theory. hep-th/9311130 23. Fulton, W. and Harris, J.: Representation theory. A first course. Berlin–Heidelberg–New York: SpringerVerlag, 1991 24. Gopakumar, R. and Vafa, C.: M-theory and topological strings. II, hep-th/9812127 25. Katz, S., Klemm, A. and Vafa, C.: M-theory, topological strings, and spinning black-holes. hep-th/9910181 26. Elitzur, S., Moore, G., Schwimmer, A. and Seiberg, N.: Remarks on the canonical quantization of the Chern–Simons-Witten theory. Nucl. Phys. B 326, 108 (1989); Labastida, J.M.F. and Ramallo, A.V.: Operator formalism for Chern–Simons theories. Phys. Lett. B 227, 92 (1989); Chern–Simons theory and conformal blocks. Phys. Lett. B 228, 214 (1989); Axelrod, S., Della Pietra, S. and Witten, E.: Geometric quantization of Chern–Simons gauge theory. J. Diff. Geom. 33, 787 (1991) 27. Lickorish, W.B.R.: An introduction to knot theory. Berlin–Heidelberg–New York: Springer-Verlag, 1998 28. Labastida, J.M.F. and Pérez, E.: Gauge-invariant operators for singular knots in Chern–Simons gauge theory. hep-th/9712139, Nucl. Phys. B 527, 499 (1998) 29. Vassiliev, V.A.: Cohomology of knot spaces. In: Theory of singularities and its applications, Advances in Soviet Mathematics, Vol. 1, Providence, RI: Am. Math. Soc., 1990, pp. 23–69 30. Álvarez, M. and Labastida, J.M.F.: Vassiliev invariants for torus knots. q-alg/9506009, J. Knot Theory Ramifications 5, 779 (1996) 31. Willerton, S.: On universal Vassiliev invariants, cabling, and torus knots. University of Melbourne preprint (1998) Communicated by R. H. Dijkgraaf

Commun. Math. Phys. 217, 451 – 473 (2001)

Communications in



Braided Quantum Field Theory Robert Oeckl1,2 1 Centre de Physique Théorique, CNRS Luminy, 13288 Marseille, France. E-mail: [email protected] 2 Department of Applied Mathematics and Mathematical Physics, University of Cambridge,

Cambridge CB3 0WA, UK. E-mail: [email protected] Received: 3 July 1999 / Accepted: 10 November 2000

Abstract: We develop a general framework for quantum field theory on noncommutative spaces, i.e., spaces with quantum group symmetry. We use the path integral approach to obtain expressions for n-point functions. Perturbation theory leads us to generalised Feynman diagrams which are braided, i.e., they have non-trivial over- and undercrossings. We demonstrate the power of our approach by applying it to φ 4 -theory on the quantum 2-sphere. We find that the basic divergent diagram of the theory is regularised. 1. Introduction The idea that space-time might not be accurately described by ordinary geometry was expressed already a long time ago. It was then motivated by the problems encountered in dealing with the divergences of quantum field theories. An early suggestion was that spatial coordinates might in fact be noncommuting observables [27]. For a long time development has been hampered by the lack of proper mathematical tools. Only with the advent of noncommutative geometry [5] and quantum groups have such ideas taken a more concrete form. Quantum groups emerged in fact from the theory of integrable models in physics and were connected from the beginning to the idea of noncommutative symmetries in physical systems [7,11,30]. It was then also suggested that they might play a role in physics at very short distances [18]. The idea that quantum symmetry or noncommutativity might serve as a regulator for quantum field theories was emphasised in [19] and [12]. The persistent inability to unite quantum field theory with gravity is a main motivation behind such considerations. In this context it is interesting to note that noncommutative geometric structures are emerging also in string theory [6]. Despite progress in describing various physical models on noncommutative spaces (see e.g. [17, 10, 4, 3]), an approach general enough to be independent of a particular choice of noncommutative space has been lacking. We aim at taking a step in this direction by providing a framework for doing quantum field theory on any noncommutative space with quantum group symmetry.

452

R. Oeckl

The basic underlying idea of our approach is to take ordinary quantum field theory, formulate it in a purely algebraic language and then generalise in this formulation to noncommutative spaces. It turns out that this generalisation is completely natural. It involves no arbitrary additional input and no further choices (except for trivial choices like taking left or right actions). We start with two fundamental ingredients of quantum field theory, namely the space of fields together with the group of symmetries acting on it. Generalising to the noncommutative context, this means that we have a vector space of fields coacted upon by a quantum group (which we take to mean coquasitriangular Hopf algebra) of symmetries. Thus, the space of fields becomes an object in the category of representations (comodules) of the quantum group, which is braided 1 . I.e., we are naturally in the context of braided geometry [21, Chapter 10]. We emphasise that the braiding is forced on us by the requirement of covariance under the quantum group symmetry and not introduced by hand. It also turns out (at least for our example in Sect. 5) that the braiding rather than the noncommutativity itself is crucial to achieve regularisation of a conventional theory. This seems to have been missed out in previous works. For previous indications that noncommutativity is not necessarily sufficient for regularisation see e.g. [8]. We follow the path integral approach, going from Gaussian path integrals via perturbation theory to Feynman diagrams. In the noncommutative setting this procedure naturally leads us to generalised Feynman diagrams that are braid diagrams, i.e., they have nontrivial over- and under-crossings. For an algebraically rigorous treatment we require the quantum group of symmetries to be cosemisimple corresponding to compactness in the commutative case. However, when aiming to regularise UV-divergences this is not necessarily a disadvantage, since they should not be affected by the global properties of a space. We start out in Sect. 2 by defining normalised Gaussian integrals on braided spaces based on [13] naturally generalising Gaussian integration on commutative spaces. This provides us with the free n-point functions of a braided quantum field theory. Developing perturbation theory in analogy to ordinary quantum field theory we obtain the braided analogues of Feynman diagrams. It turns out that symmetry factors of ordinary Feynman diagrams are resolved into different (and not necessarily equivalent) diagrams in the braided case. In Sect. 3 we consider the case where the space of fields is a quantum homogeneous space under the symmetry quantum group. Inspired by the conventional commutative case this gives us a more compact description of n-point functions. Furthermore, it allows for simplifications in braided Feynman diagrams. While our approach is somewhat formal up to this point, Sect. 4 introduces a context that allows us to work algebraically rigorously in infinite dimensions. We need a further assumption to do this, which corresponds in the commutative case to the space-time being compact. Finally, in Sect. 5 we deliver on the promise to perform q-regularisation within braided quantum field theory. To this end we consider φ 4 -theory on the standard quantum 2-sphere [26]. We make use of all the machinery developed up to this point to show that the only basic divergence of φ 4 -theory in two dimensions, the tadpole diagram, becomes finite at q > 1. We identify the divergence in q-space and suggest that it would not depend on the conventional degree of divergence of a diagram. 1 Recall that a braiding means that for two representations V ,W the intertwiner of the tensor products V ⊗ W → W ⊗ V becomes nontrivial, i.e. different from the flip map.

Braided Quantum Field Theory

453

By a quantum group we generally mean a Hopf algebra equipped with a coquasitriangular structure (see e.g. [21]). We denote the coaction by , the counit by , and the antipode by S. We use Sweedler’s notation [28] a = a (1) ⊗ a (2) , etc., with summation implied. We apply the same notation to Hopf algebras in braided categories. The braiding is denoted by ψ. While working over a general field k in Sects. 2–4 we specialise to the complex numbers in Sect. 5.

2. Formal Braided Quantum Field Theory We start out in this section by developing normalised Gaussian integration on braided spaces leading to a braided generalisation of Wick’s Theorem. The less algebraically minded reader may find it convenient to proceed with Sect. 2.2 where braided path integrals are discussed in quantum field theoretic language, and accept the main result of Sect. 2.1 (Theorem 2.1 and its corollary) as given.

2.1. Braided Gaussian Integration. Braided categories arise as the categories of modules or comodules over quantum groups (Hopf algebras) with quasitriangular respectively coquasitriangular structure (see e.g. [21]). The latter case will be the one of interest to us later. We consider rigid braided categories, where we have for every object X a dual object X∗ and morphisms ev : X ⊗ X ∗ → k (evaluation) and coev : k → X ∗ ⊗ X (coevaluation) that compose to the identity in the obvious ways. Although rigidity usually implies finite dimensionality, we shall see later (Sect. 4) how we can deal with infinite dimensional objects. The differentiation and Gaussian integration on braided spaces that we require were developed by Majid [20] and Kempf and Majid [13] in an R-matrix setting. (The special case of Rnq was treated earlier in [9].) We need a more abstract and basis free formulation of the formalism so that we redevelop the notions here. Furthermore, our Theorem 2.1 goes beyond [13, Theorem 5.1]. Recall that a braiding on a category of vector spaces is an assignment to any pair of vector spaces V , W of an invertible morphism ψV ,W : V ⊗ W → W ⊗ V . These morphisms are required to be compatible with the tensor product such that ψU ⊗V ,W = (ψU,W ⊗ id) ◦ (id ⊗ψV ,W ) and ψU,V ⊗W = (id ⊗ψU,W ) ◦ (ψU,V ⊗ id). If the category is a category of modules or comodules of a quantum group the morphisms are the intertwiners. The braiding then generalises the trivial exchange map ψV ,W (v ⊗ w) = w ⊗ v which is an intertwiner for representations of ordinary groups. In the following we simply write ψ for the braiding if no confusion can arise as to the spaces on which it is defined. Suppose we have some rigid braided category B and a vector space X ∈ B. Essentially, we want to define the (normalised) integral of functions α in the “coordinate ring” on X multiplied by a Gaussian weight function w, i.e., we want to define αw Z(α) := . w

(1)

First, we need to specify this “coordinate ring”. We identify the dual space X ∗ ∈ B as the space of “coordinate functions” on X. This corresponds to the situation in Rn where a coordinate function is just a linear map from Rn into the real numbers. The polynomial

454

R. Oeckl

functions on X are naturally elements of the free unital tensor algebra over X ∗ , ∗ := X

∞

X∗ n ,

with X∗ 0 := 1

n=0

and

X ∗ n := X∗ ⊗ · · · ⊗ X∗ , n times

where 1 is the one-dimensional space generated by the identity. 1 plays the role of the constant functions and the tensor product corresponds to the product of functions. ∗ naturally has the structure of a braided Hopf algebra (a Hopf algebra in a braided X category, see [21]) via a = a ⊗ 1 + 1 ⊗ a,

(a) = 0,

S a = −a

∗ as braided (anti-)algebra maps. Explicitly, the for a ∈ X ∗ and , , S extend to X coproduct is defined inductively by the identity ◦· = (· ⊗ ·) ◦ (id ⊗ψ ⊗ id) ◦ ( ⊗ ) ∗ ⊗ X ∗ → X ∗ ⊗ X ∗ . The braided Hopf algebra structure can be thought of of maps X as encoding translations on X. To make the notion of “coordinate ring” more precise, one could perhaps consider ∗ in analogy with the observation that coordinates a kind of symmetrised quotient of X commute in ordinary geometry. There seems to be no obvious choice for such a quotient in the general braided case. Remarkably, however, such a choice is not necessary. In fact, the following discussion is entirely independent of any relations, as long as they preserve the (graded) braided Hopf algebra structure. The next step is the introduction of differentials [20]. The space of coordinate differentials should be dual to the space X ∗ of coordinate functions. We just take X itself and define differentiation on X∗ by the pairing ev : X ⊗ X∗ → k in B. To extend ∗ , we note that the coproduct encodes differentiation to the whole “coordinate ring” X coordinate translation. This leads to the natural definition that ∗ → X ∗ diff := ( ev ⊗ id) ◦ (id ⊗ ) : X ⊗ X ∗ . Here, ev is the trivial extension of ev to X ⊗ X ∗ → k, i.e., is differentiation on X ev |X⊗X∗ n = 0 for n = 1. We also use the more intuitive notation ∂(a) := diff(∂ ⊗ a) ∗ . Let ∂ ∈ X and α, β ∈ X ∗ . The definition of ev gives at once for ∂ ∈ X and a ∈ X ev(∂ ⊗ αβ) = ev(∂ ⊗ α) (β) + ev(∂ ⊗ β) (α). Using that the coproduct is a braided algebra map, we obtain the braided Leibniz rule ∂(αβ) = ∂(α)β + ψ −1 (∂ ⊗ α)(β).

(2)

Iteration yields ∂(α) = (ev ⊗ idn−1 )(∂ ⊗ [n]ψ α), where n is the degree of α and [n]ψ := idn +ψ ⊗ idn−2 + · · · + ψn−2,1 ⊗ id +ψn−1,1 is a braided integer. We adopt the convention of writing ψn,m for the braiding between −1 for the inverse braiding). X∗ n and X ∗ m (respectively ψn,m


455

∗ and define As in [13] we view the Gaussian weight w formally as an element of X its differentiation via an isomorphism γ : X → X∗

∂(w) = −γ (∂)w

so that

for

∂ ∈ X.

(3)

This expresses the familiar notion that differentiating a Gaussian weight yields a coordinate function times the Gaussian weight. γ should accordingly be thought of as defining a braided analogue of the quadratic form in the exponential of the weight. Also familiar from ordinary Gaussian integration is the fact that integrals of total differentials vanish. That is, we require ∂(αw) = 0

for

∗ . ∂ ∈ X, α ∈ X

(4)

It turns out that the three rules (2), (3), and (4) completely determine the integral (1). Remarkably, the statement that the Gaussian integral of a polynomial function can be expressed solely in terms of Gaussian integrals of quadratic functions still holds true in the braided case. This generalises what is known in quantum field theory as Wick’s Theorem. To state it, we need another set of braided integers [n]ψ : X∗ n → X∗ n with −1 [n]ψ := idn + idn−2 ⊗ψ −1 + · · · + ψ1,n−1 ,

(5)

−1 which are related to the original ones by [n]ψ = ψ1,n−1 ◦ [n]ψ . We also require the 2n ∗ corresponding braided double factorials [2n − 1]ψ !! : X → X∗ 2n with

[2n − 1]ψ !! := ([1]ψ ⊗ id2n−1 ) ◦ ([3]ψ ⊗ id2n−3 ) ◦ · · · ◦ ([2n − 1]ψ ⊗ id).

(6)

Theorem 2.1 (Braided Wick Theorem). Z |X∗ 2 = ev ◦ψ ◦ (id ⊗γ −1 ),

Z |X∗ 2n = (Z |X∗ 2 )n ◦ [2n − 1]ψ !!,

Z |X∗ 2n−1 = 0,

∀n ∈ N.

∗ and a ∈ X∗ we have Proof. For α ∈ X αaw = −α diff(γ −1 (a) ⊗ w) = − diff(ψ(α ⊗ γ −1 (a))w) + (diff ◦ψ(α ⊗ γ −1 (a)))w using the differential property (3) of w and the braided Leibniz rule (2). Applying Z, we can ignore the total differential and obtain Z(αa) = Z(diff ◦ψ(α ⊗ γ −1 (a))). This gives us immediately Z(a) = 0

and

Z(ab) = ev ◦ψ(a ⊗ γ −1 (b))

(7)

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R. Oeckl

for b ∈ X∗ . We rewrite (7) to find Z |X∗ n = Z |X∗ n−2 ◦ diff ◦(γ −1 ⊗ idn−1 ) ◦ ψn−1,1 = Z |X∗ n−2 ◦ (ev ⊗ idn−2 ) ◦ (γ −1 ⊗ [n − 1]ψ ) ◦ ψn−1,1 = (ev ⊗ Z |X∗ n−2 ) ◦ (γ −1 ⊗ [n − 1]ψ ) ◦ ψn−1,1 = (ev ⊗ Z |X∗ n−2 ) ◦ ψn−1,1 ◦ ([n − 1]ψ ⊗ γ −1 ) = (Z |X∗ 2 ⊗ Z |X∗ n−2 ) ◦ (id ⊗ψn−2,1 ) ◦ ([n − 1]ψ ⊗ id) −1 ◦ (id ⊗ψn−2,1 ) ◦ ([n − 1]ψ ⊗ id) = (Z |X∗ n−2 ⊗ Z |X∗ 2 ) ◦ ψ2,n−2 −1 = (Z |X∗ n−2 ⊗ Z |X∗ 2 ) ◦ (ψ1,n−2 ⊗ id) ◦ ([n − 1]ψ ⊗ id) = (Z |X∗ n−2 ⊗ Z |X∗ 2 ) ◦ ([n − 1]ψ ⊗ id), which gives us a recursive definition of Z leading to the formulas stated.

Another set of the braided integers −1 [n]ψ := idn +ψ −1 ⊗ idn−2 + · · · + ψ1,n−1

with [2n − 1]ψ !! := (id ⊗[2n − 1]ψ ) · · · (id2n−3 ⊗[3]ψ )(id2n−1 ⊗[1]ψ ) serves to formulate the dual version of the theorem. Corollary 2.2. Let Z k ∈ Xk denote the dual of Z |X∗ k . Then Z 2 = ψ ◦ (γ −1 ⊗ id) ◦ coev, Z 2n = [2n − 1]ψ !! (Z 2 )n ,

Z 2n−1 = 0,

∀n ∈ N,

Proof. This is obtained from Theorem 2.1 by reversing of arrows or equivalently by turning diagrams upside down in the diagrammatic language of braided categories. 2.2. Braided Path Integrals. The n-point function of an ordinary quantum field theory with action S, evaluated at (x1 , . . . , xn ) is given by the path integral2 Dφ φ(x1 ) · · · φ(xn )e−S(φ) φ(x1 ) · · · φ(xn ) = . Dφ e−S(φ) This is really the normalised integral of the functional φ → φ(x1 ) · · · φ(xn ) with weight w(φ) = e−S(φ) over the space X of classical fields of the theory. The parameters xi denote here points in space-time as well as additional internal field indices. For the non-interacting theory the action S is replaced by the free action S0 . The path integral is then a Gaussian integral and the decomposition of n-point functions into 2-point functions (propagators) is governed by Wick’s Theorem. Generalising to braided spaces (when the symmetry group is allowed to be a quantum group) we are in the framework of Sect. 2.1. Then, the value of an n-point function is still given in terms of values of 2-point functions (propagators). This is the result of Theorem 2.1 which generalises Wick’s Theorem. The (unevaluated) n-point function Z n itself is an element in the n-fold tensor product Xn of the space of fields X and we write Z n (x1 , . . . , xn ) = φ(x1 ) · · · φ(xn )0 , 2 The Euclidean signature of the action is chosen for definiteness and does not imply a restriction to Euclidean field theory.


457

the index 0 indicating that we deal with the free theory. The decomposition of Z n into propagators Z 2 is given by Corollary 2.2, which is Theorem 2.1 in dual form, i.e., for “unevaluated” functions. The connection between the map γ determining the (unevaluated) propagator according to Theorem 2.1 (Corollary 2.2) and the free action in ordinary quantum field theory is as follows. Let ∂ be some differential with respect to the space of fields. The definition of γ in (3) corresponds to (∂(e−S0 ))(φ) = −(γ (∂))(φ)e−S0 (φ) , in ordinary quantum field theory. Thus we obtain (γ (∂))(φ) = (∂S0 )(φ).

(8)

To determine interacting n-point functions, we use the same perturbative techniques as in ordinary quantum field theory. For S = S0 + λSint with coupling constant λ, we expand Z nint (x1 , . . . , xn ) = φ(x1 ) · · · φ(xn ) Dφ φ(x1 ) · · · φ(xn )(1 − λSint (φ) + . . . )e−S0 (φ) = Dφ (1 − λSint (φ) + . . . )e−S0 (φ) φ(x1 ) · · · φ(xn )0 − λφ(x1 ) · · · φ(xn )Sint (φ)0 + . . . = . 1 − λSint (φ)0 + . . . For Sint of degree k we can write φ(x1 ) . . . φ(xn )Sint (φ)0 = ((idn ⊗Sint ) Z n+k )(x1 , . . . , xn ) etc. by viewing Sint as a map Xk → k. Then, removing the explicit evaluations we obtain Z nint =

Z n −λ(idn ⊗Sint )(Z n+k ) + 21 λ2 (idn ⊗Sint ⊗ Sint )(Z n+2k ) + . . . 1 − λSint (Z k ) + 21 λ2 (Sint ⊗ Sint )(Z 2k ) + . . .

,

(9)

an expression for the interacting n-point function valid in the general braided case. Vacuum contributions cancel as usual. Note that we have used the ordinary exponential expansion for the interaction and not, say, a certain braided version. The latter might be more natural if, e.g., one wants to look at identities between diagrams of different order. However, we shall not consider this issue here. 2.3. Braided Feynman Diagrams. We are now ready to generalise Feynman diagrams to our braided setting. To do this we use and modify the diagrammatic language of braided categories appropriately: • An n-point function is an element in X ⊗ · · · ⊗ X (n-fold). Thus, its diagram is closed to the top and ends in n strands on the bottom. Any strand represents an element of X, i.e., a field. • The propagator Z 2 ∈ X ⊗ X is represented by an arch, see Fig. 1(a). • An n-leg vertex is a map X ⊗ · · · ⊗ X → k. It is represented by n strands joining in a dot, see Fig. 1(b). Notice that the order of incoming strands is relevant. • Over- and under-crossings correspond to the braiding and its inverse, see Fig. 2.

458

R. Oeckl ···

(a)

(b)

Fig. 1. Propagator (a) and vertex (b)

ψ

ψ −1

Fig. 2. The braiding and its inverse

+

+

Fig. 3. Free 4-point function

• Any Feynman diagram is built out of propagators, (possibly different kinds of) vertices, and strands with crossings, connecting the propagators and vertices, or ending at the bottom. Otherwise the usual rules of braided diagrammatics apply. Notice that in contrast to ordinary Feynman diagrams all external legs end on one line (the bottom line of the diagram) and are ordered. This is necessary due to the possible non-trivial braid statistics in our setting. For the case of trivial braiding we can relax this and shift the external legs around as well as change the order of strands at vertices so as to obtain ordinary Feynman diagrams in more familiar form. The diagrams for the free 2n-point functions can be read off directly from Corollary 2.2. The crossings are encoded in the braided integers [j ]ψ . Figure 3 shows for example the free 4-point function and Fig. 4 the free 6-point function. For the interacting n-point functions we use formula (9) to obtain the diagrams. Sint gives us the vertices. Consider for example the 2-point function in Euclidean φ 4 -theory. To order λ we get

Z 2int = Z 2 −λ (id2 ⊗Sint )(Z 6 ) − Z 2 ⊗Sint (Z 4 ) + O(λ2 ). (10) Sint is just the map φ1 ⊗ φ2 ⊗ φ3 ⊗ φ4 → φ1 φ2 φ3 φ4 . To obtain the diagrams at order λ we start by drawing the free 6-point function (Fig. 4) and attach to the 4 rightmost strands of each diagram a 4-leg vertex (Fig. 1(b)). Those diagrams are generated by the first term in brackets of (10). We realise that the first three of our diagrams are vacuum diagrams which are exactly cancelled by the second term in the brackets. The remaining 12 diagrams are shown in Fig. 5. In ordinary quantum field theory they all correspond to the same diagram: The tadpole diagram, see Fig. 6. However, not all of them are necessarily different, as we shall see in Sect. 3.2. 3. Braided QFT on Homogeneous Spaces In ordinary quantum field theory fixing one point of an n-point function still allows to recover the whole n-point function. Thus, we can reduce an n-point function to a


459

Fig. 4. Free 6-point function

Fig. 5. Interacting 2-point function of φ 4 -theory at order 1

Fig. 6. Tadpole diagram of ordinary φ 4 -theory

460

R. Oeckl

function of just n − 1 variables. This is simply due to the fact that any n-point function is invariant under the isometry group G of the space-time M and G acts transitively on M. In this case M is a homogeneous space under G and we can make the above statement more precise in the following way. Lemma 3.1. Let G be a group and K a subgroup of G. For any n ∈ N there is an isomorphism of coset spaces ρn : (K\G × · · · × K\G)/G ∼ = (K\G × · · · × K\G)/K n times

n−1 times

given by ρn : [a1 , . . . , an ] → [a1 an−1 , . . . , an−1 an−1 ] for ai ∈ K\G. Its inverse is given by ρn−1 : [b1 , . . . , bn−1 ] → [b1 , . . . , bn−1 , e] for bi ∈ K\G, where e denotes the equivalence class of the identity in K\G. If G is a topological group (i.e., it is a topological space and multiplication and inversion are continuous), then equipping the coset spaces with the induced topologies makes ρn into a homeomorphism. If space-time is an ordinary manifold we can obviously do the same trick in braided quantum field theory. More interestingly, however, we can extend it to noncommutative space-times.

3.1. Quantum Homogeneous Spaces. Lemma 3.1 generalises to the quantum group case. To see this we first recall the notion of a quantum homogeneous space. Suppose we have two Hopf algebras A and H together with a Hopf algebra surjection π : A → H . This induces coactions βL = (π ⊗ id) ◦ and βR = (id ⊗π ) ◦ of H on A, making A into a left and right H -comodule algebra. Define HA to be the left H invariant subalgebra of A, i.e., HA = {a ∈ A|βL (a) = 1 ⊗ a}. We have HA ⊆ HA ⊗ A since (βL ⊗ id) ◦ = (id ⊗ ) ◦ βL . This makes HA into a right A-comodule (and H -comodule) algebra. Observe also that π(a) = (a)1 for a ∈ HA. HA is called a right quantum homogeneous space. Define the left quantum homogeneous space AH correspondingly. Due to the anti-coalgebra property of the antipode we find S HA ⊆ AH and S AH ⊆ HA. If the antipode is invertible, the inclusions become equalities. Proposition 3.2. In the above setting with invertible antipode the map ρn : (H · · ⊗ HA)A → (H · · ⊗ HA)H A ⊗ · A ⊗ · n times

n−1 times

given by ρn = (idn−1 ⊗ ) for n ∈ N is an isomorphism. Its inverse is (idn−1 ⊗ S)◦β n−1 , where β n−1 is the right coaction of A on HA extended to the (n − 1)-fold tensor product. Proof. Let a 1 ⊗ · · · ⊗ a n be an element of (HA ⊗ · · · ⊗ HA)A . In particular, a 1 (1) ⊗ · · · ⊗ a n (1) ⊗ a 1 (2) · · · a n (2) = a 1 ⊗ · · · ⊗ a n ⊗ 1. Applying the antipode to the last component and multiplying with the nth component we obtain a 1 (1) ⊗ · · · ⊗ a n−1 (1) ⊗ (a n ) S(a 1 (2) · · · a n−1 (2) ) = a 1 ⊗ · · · ⊗ a n .

(11)


461

Thus, (idn−1 ⊗ S) ◦ β n−1 ◦ (idn−1 ⊗ ) is the identity on (HA ⊗ · · · ⊗ HA)A . On the other hand, applying the inverse antipode and then π to the last component of (11) we get a 1 (1) ⊗ · · · ⊗ a n−1 (1) ⊗ (a n )π(a 1 (2) · · · a n−1 (2) ) = a 1 ⊗ · · · ⊗ a n−1 ⊗ (a n )1. This is to say that a 1 ⊗ · · · ⊗ a n−1 (a n ) is indeed right H -invariant. Conversely, it is clear that (idn−1 ⊗ ) ◦ (idn−1 ⊗ S) ◦ β n−1 = (idn−1 ⊗ ) ◦ β n−1 is the identity. Now take b1 ⊗ · · · ⊗ bn−1 in (HA ⊗ · · · ⊗ HA)H . Its image under β n−1 is b1 (1) ⊗ · · · ⊗ bn−1 (1) ⊗ b1 (2) · · · bn−1 (2) .

(12)

Applying π to the last component we get b1 (1) ⊗ · · · ⊗ bn−1 (1) ⊗ π(b1 (2) · · · bn−1 (2) ) = b1 ⊗ · · · ⊗ bn−1 ⊗ 1 by right H -invariance. Applying β n−1 ⊗ id we arrive at b1 (1) ⊗ · · · ⊗ bn−1 (1) ⊗ b1 (2) · · · bn−1 (2) ⊗ π(b1 (3) · · · bn−1 (3) ) = b1 (1) ⊗ · · · ⊗ bn−1 (1) ⊗ b1 (2) · · · bn−1 (2) ⊗ 1. We observe that this is the same as applying (idn−1 ⊗βR ) to (12). Thus, the last component of (12) lives in AH and the application of the antipode sends it to HA as required. That the result is right A-invariant is also clear by the defining property of the antipode. To make use of the result we assume our space X of fields to be a quantum homogeneous space under a quantum group (coquasitriangular Hopf algebra) A of symmetries. (Note that coquasitriangularity implies invertibility of the antipode.) That is, together with A we have another Hopf algebra H and a Hopf algebra surjection A → H . We then assume that the algebra of fields is the right quantum homogeneous space X = HA living in the braided category MA of right A-comodules. 3.2. Diagrammatic Techniques. Proposition 3.2, to which we shall refer as invariant reduction, is not only useful to express n-point functions in a more compact way, but can also be applied in the evaluation of braided Feynman diagrams. For this we note that any horizontal cut of a braided Feynman diagram lives in some tensor power of X (since the only allowed strand lives in X) and is invariant (since the diagram is closed at the top). Thus, we can apply invariant reduction to it. We shall give three examples for this, assuming vertices that are evaluated by multiplication and subsequent integration. Here, any quantum group invariant linear map X → k is admissible as the integral. Vertex evaluation. Consider the evaluation of an n-leg vertex (the horizontal slice of an invariant diagram depicted in Fig. 7) with incoming elements a1 ⊗ · · · ⊗ ak+n . By invariant reduction this can be expressed in two ways, a1 ⊗ · · · ⊗ ak ak+1 · · · ak+n = a1 (1) ⊗ · · · ⊗ ak (1) (ak+1 ) · · · (ak+n ) S(a1 (2) · · · ak (2) ). Depending on the circumstances each side might be easier to evaluate.

462

R. Oeckl ···

···

··· Fig. 7. Vertex evaluation in a diagram slice

= 1 Fig. 8. Extracting a loop

···

···

=

Fig. 9. Separating a loop in an invariant slice

Loop extraction. Assume that the integral on HA is normalised, 1 = 1. Consider the diagram in Fig. 8 (left-hand side). It is obviously invariant. Thus, the single outgoing strand carries a multiple of the identity and we can replace it by the integral followed by the identity element (Fig. 8, right-hand side). Loop separation. We assume further that the coquasitriangular structure R : H ⊗ H → k is trivial on HAH in the sense R(a ⊗ b) = (a) (b),

if a ∈ HAH or b ∈ HAH .

(13)

Consider now the diagram in Fig. 9 (left-hand side) as a horizontal slice of an invariant diagram. According to invariant reduction we apply the counit to the rightmost outgoing strand. This makes the braiding trivial due to the assumed property of R. We can push the counit up to each of the joining strands and disentangle them. Then proceeding as in the previous example leads to the diagram in Fig. 9 (right-hand side). Note that this works the same way for an under-crossing.

6

+

6

Fig. 10. Simplified 2-point function of φ 4 -theory at order 1

Let us come back to the 2-point function of φ 4 theory that we considered at the end of Sect. 2.3. Assuming 1 = 1 and property (13) we can use loop extraction and loop


463

separation to simplify the order 1 diagrams of Fig. 5 considerably. The result is shown in Fig. 10. Instead of 12 different diagrams we only have 2 different and much simpler diagrams, each with a multiplicity of 6. 4. Braided QFT on Compact Spaces 4.1. Braided Spaces of Infinite Dimension. Up to now we have developed our approach on a formal level insofar, that we have not addressed the question how an infinite dimensional space (of fields) can be treated in a braided category. This is certainly necessary if we want to do quantum field theory, i.e., deal with infinitely many degrees of freedom. An obvious problem is the definition of the coevaluation. It seems that we need at least a completed tensor product for this. However, instead of introducing heavy functional analytic machinery, we can stick with our algebraic approach given a further assumption. Let us assume that the space of (regular) fields X decomposes into a direct sum i Xi of countably many finite dimensional comodules under the symmetry quantum group A. This corresponds roughly to the classical case of the space-time manifold being compact. In particular, it is the case if the symmetry quantum group A is cosemisimple (or classically the Lie group of symmetries is compact, see Sect. 4.2 below). Denote the projection X → Xi by τi . We now allow arbitrary sums of elements in X given that any projection τi annihilates all but finitely many summands. Similarly, we allow infinite sums in the n-fold tensor product Xn with the restriction that any projection τi1 ⊗ · · · ⊗ τin yields a finite sum. To define the dual of X, we take the dual of each Xi and set X ∗ = i Xi∗ . For each component Xi we have an evaluation map evi : Xi ⊗ Xi∗ → k and a coevaluation map coevi : k → Xi ⊗ Xi∗ in the usual way. We then formally define ev = i evi ◦(τi ⊗ τi∗ ) and coev = i coevi . Our definition is invariant under coactions of A as it should be, since the projections τi commute with the coaction of A. In particular, it is invariant under braidings. 4.2. Cosemisimplicity and Peter–Weyl Decomposition. We describe a context in which all comodules over a Hopf algebra decompose into finite dimensional (and even simple) pieces. The discussion here uses results of [28] but is more in the spirit of [2, II.9]. Assume k to be algebraically closed, e.g., k = C. Let C be a coalgebra, V a simple right C-comodule (i.e. V has no proper subcomodules) with coaction β : V → V ⊗ C. In particular, V is finite dimensional. The dual space V ∗ is canonically a (simple) left C-comodule. Denote a basis of V by {ei }, the dual basis of V ∗ by {f i }. Identify the endomorphism algebra on V , End V ∼ = V ⊗ V∗ j via (ei ⊗ f j )(ek ⊗ f l ) = δk (ei ⊗ f l ). We denote the dual coalgebra by (End V )∗ and

identify (End V )∗ ∼ = V ∗ ⊗ V via (f i ⊗ ej ) = k (f i ⊗ ek ) ⊗ (f k ⊗ ej ). Now consider the map (End V )∗ → C given by f i ⊗ ej → (f i ⊗ id) ◦ β(ej ). It is an injective (since V is simple) coalgebra map. We extend this to the direct sum of all inequivalent simple comodules. The resulting map (End V )∗ → C V

is a coalgebra injection. It is an isomorphism of coalgebras if and only if all C-comodules are semisimple (i.e. they are direct sums of simple ones) or equivalently if C is semisimple (i.e. it is a direct sum of simple coalgebras).

464

R. Oeckl

Assume now that A is a cosemisimple Hopf algebra, i.e., A is semisimple as a coalgebra. We write the above decomposition as A∼ (V ∗ ⊗ V ). (14) = V

It is also referred to as the Peter–Weyl decomposition, in analogy to the corresponding decomposition of the algebra of regular functions on a compact Lie group. There is a unique normalised left- and right-invariant integral (Haar measure) on A, given by the induced projection to the unit element in A. Note also that the antipode is invertible. Consider a second Hopf algebra H with a Hopf algebra surjection π : A → H . This induces a coaction of H on each A-comodule. For the right quantum homogeneous space we have H ∼ H A= (V ∗ ) ⊗ V (15) V

as right H -comodules. 5. φ 4 -Theory on the Quantum 2-Sphere In accordance with the motivation of braided quantum field theory as a way of regularising ordinary quantum field theory, we replace Lie groups of symmetries by corresponding parametric deformations. In order to have a well defined theory in the sense of Sect. 4 we make use of the Peter–Weyl decomposition and thus restrict to compact Lie groups. A natural choice are the standard q-deformations of Lie groups with compact ∗-structure. We specialise to k = C, although the discussion of the free action in Sect. 2.2 was in the spirit of real-valued scalar field theory. This is necessary since the standard qdeformations viewed as deformations of complexifications of compact Lie groups do not restrict to real subalgebras for q = 1. However, viewing q-deformation purely as a mathematical tool we can always restrict to R when considering physical quantities living at q = 1. In the following we consider perturbative φ 4 -theory on the quantum 2-sphere with SUq (2)-symmetry as an example of a quantum field theory on a braided space. Ordinary φ 4 -theory in 2 dimensions is super-renormalisable and has just one basic divergence: The tadpole diagram (Fig. 6). (See e.g. [31] for a treatment of ordinary φ 4 -theory.) We demonstrate that this diagram becomes finite for q > 1. Our Hopf algebra of symmetries is SUq (2) under which Sq2 is a homogeneous space as a right comodule. (We adopt the convention to denote the Hopf algebra of regular functions by the name of the (quantum) group or space.) 5.1. The Decomposition of SUq (2) and Sq2 . To prepare the ground we need to recall the construction of Sq2 as a quantum homogeneous space under SUq (2) and the Peter–Weyl decomposition of the latter [15, 25]. This will enable us to apply the machinery of the previous sections. Recall that SUq (2) is the compact real form of SLq (2) for q real which we assume in the following. (See Appendix A for the defining relations.) It is cosemisimple and there is one simple (right) comodule Vl for each integer dimension, conventionally labelled by


465

a half-integer l such that the dimension is 2l + 1. Thus, the Peter–Weyl decomposition (14) is (Vl∗ ⊗ Vl ). SUq (2) ∼ = l∈ 21 N0

There is a Hopf ∗-algebra surjection π : SUq (2) → U (1) corresponding to the diagonal inclusion in the commutative case. (See Appendix A for an explicit definition of π.) This defines the quantum 2-sphere Sq2 as the right quantum homogeneous ∗-space U (1) SU (2). Under the coaction of U (1) induced by π the comodules V decompose q l into inequivalent one-dimensional comodules classified by integers. (This is the usual (l) representation theory of U (1).) This determines up to normalisation a basis {vn } for Vl U (1) with half-integers n taking values −l, −l + 1, . . . , l. In particular, we find that Vl is one-dimensional if l is integer and zero-dimensional otherwise. Thus, (15) simplifies to Vl Sq2 ∼ = l∈N0

as right SUq (2)-comodules. We write the induced (normalisation independent) basis (l) (l) (l) (l) (l) vectors of SUq (2) as ti j = fi ⊗ id) ◦ β(ej , where fn is dual to en and β : Vl → (l)

Vl ⊗ SUq (2) is the coaction of SUq (2) on Vl . As a subalgebra Sq2 has the basis {t0 i }. U (1) Sq2

(l)

= U (1) SUq (2)U (1) has the basis {t0 0 }. The bi-invariant subalgebra Note that by construction

(l) (l) (l) (l) tm tm k ⊗ t k n . tm n = δm,n and n = k

The antipode and ∗-structure of SUq (2) in this basis are

∗ (l) (l) m−n (l) (l) S tm t−n −m , tm = S tn(l)m = (−q)n−m t−m −n , n = (−q) n as can be verified by direct calculation from the formulas in [14, 4.2.4]. The normalised (l) invariant integral (Haar measure) is simply ti j = δl,0 . We also need its value on the product of two basis elements, (−1)m−n q m+n (l) (l ) tm t = δl,l δm+m ,0 δn+n ,0 . (16) n m n [2l + 1]q This can be easily worked out considering the equation (a) = a (1) S a (2) and using the invariance of the integral in the form b(1) ab(2) = S a (1) a (2) b and S b(2) ab(1) = a (2) a (1) b on basis elements. The q-integers for q ∈ C∗ are defined as [n]q :=

n−1 k=0

q n−2k−1 =

q n − q −n . q − q −1

(The second expression is only defined for q 2 = 1). (l) (l) Denoting a dual basis of {tm n } by {t˜m n }, we observe that SUq (2)∗ becomes an object in MSUq (2) , the category of right comodules over SUq (2) by equipping it with the coaction

466

R. Oeckl

(l) (l) (l) t˜m n → k t˜m k ⊗S−1 tn k . We then have an evaluation map ev : SUq (2)⊗SUq (2)∗ → C and a coevaluation map coev : C → SUq (2)∗ ⊗ SUq (2) in the obvious way. (l) In the commutative case q = 1, the basis {tm n } becomes the usual basis of regular functions (i.e., matrix elements of representations) on SU (2) (see e.g. [29, Chapter 6] to (l) whose conventions we conform in this case). The restriction to {t0 n } recovers nothing 2 but (a version of) the spherical harmonics on S . In particular, we notice that the zonal (l) spherical functions can be expressed in terms of Legendre polynomials t0 0 (φ, θ, ψ) = Pl (cos θ ), where φ, θ, ψ are the Euler angles on SU (2) (see [29, Chapter 6]). From the orthogonality relation of the Legendre polynomials, the fact that their only common value is at Pl (1) = 1, and considering that θ = 0 denotes a pole of SU (2), we find that the delta function at the identity of SU (2) restricted to S 2 can be represented as (l) (2l + 1) Pl (cos θ) = (2l + 1) t0 0 (φ, θ ). (17) δ0 (φ, θ ) = l

l

Recall that a coquasitriangular structure R : H ⊗ H → k on a quantum group H determines a braiding between right comodules V and W via ψ(v ⊗ w) = w(1) ⊗ v (1) R(v (2) ⊗ w (2) ) for v ∈ V and w ∈ W . (We use here Sweedler’s coproduct notation for the coaction.) For calculations we need the functionals u and v defined with R as (see e.g. [21]) u(a) := R(a (2) ⊗ S a (1) ),

v(a) := R(a (1) ⊗ S a (2) )

(18)

for a ∈ H . For H = SUq (2) in our basis they are (l) −2l(l+1)+2m u(tm , n ) = δm,n q

(l) −2l(l+1)−2m v(tm . n ) = δm,n q

We also note that property (13) is satisfied, i.e.,

(l) (l) (l) (l) R t0 0 ⊗ ti j = δi,j = R ti j ⊗ t0 0 .

(19)

(20)

See Appendix B for a derivation of (19) and (20).

5.2. The Free Propagator. In ordinary quantum field theory the free propagator is defined by the free action. For a Euclidean massive real scalar field theory on a manifold M it takes the form 1 S0 (φ) = dx φ(x)(m2 − M )φ(x), 2 M where M is the Laplace operator on M and m is the mass of the field. Define L := m2 − M . Let {φi } be a basis of X and {φi∗ } a dual basis. Denote the differential with respect to φi by ∂i . We have dx φ(x)Lφi (x) = φk∗ (φ) dx φk (x)Lφi (x). (∂i S0 )(φ) = M

k

M


467

Comparing with Eq. (8) we obtain in the more abstract notation of Sect. 2.1 γ = id ⊗ ◦ (id ⊗·) ◦ (coev ⊗L), M

(21)

which we take as the defining equation for γ . While initially well defined only at q = 1 we extend it to the noncommutative realm in the following. First, note that at q = 1 we still have a well defined integral on our “manifold” M = Sq2 , namely the induced Haar measure of SUq (2). Next, we need an analogue of the Laplace operator. By the duality of SUq (2) with the quantum enveloping algebra Uq (sl2 ), a central element of the latter defines an invariant operator on SUq (2)-comodules. A natural choice is the quantum Casimir element which we define as Cq = EF +

(K − 1)q −1 + (K −1 − 1)q . (q − q −1 )2

Here K, K −1 , E, and F are the generators of Uq (sl2 ) (see Appendix B). Cq differs from quantum Casimir elements considered elsewhere (see e.g. [25] or [14]) only by a q-multiple of the identity. The eigenvalue of Cq on Vl is [l]q [l + 1]q so that we get exactly the (negative of the) usual Laplace operator for q = 1. Including a mass term we set L = Cq + m2 . Thus, the eigenvalue of L on Vl is Ll = [l]q [l + 1]q + m2 . We determine γ according to (21). Using (16) we find

(l) (m) (m) (l) −i (l) γ t0 i = t0 j L t0 i = [2l + 1]−1 t˜0 j q Ll (−q) t˜0 −i . m,j

Inverting we obtain

(l) (l) γ −1 t˜0 i = [2l + 1]q Ll−1 (−q)−i t0 −i . Now we are ready to determine the free propagator according to Corollary 2.2,

(l) (l) Z2 = (id ⊗γ −1 ) ◦ ψ t˜0 k ⊗ t0 k l,k

=

l,i,j,k

=

l,i,j

(l) (l) (l) (l) t0 i ⊗ γ −1 t˜0 j R S−1 tk j ⊗ ti k

(l) (l) (l) t0 i ⊗ γ −1 t˜0 j u ti j

(l) (l) = [2l + 1]q Ll−1 q −2l(l+1) (−q)i t0 i ⊗ t0 −i . l,i

468

R. Oeckl

Using invariant reduction (Proposition 3.2) we find (l) 2 = Z [2l + 1]q Ll−1 q −2l(l+1) t0 0

(22)

l

to be the reduced form of the propagator as an element of Sq2 case q = 1 we can rewrite (22) as

U (1)

. In the commutative

2 |q=1 = (m2 − )−1 δ0 Z by comparison with (17). This is the familiar expression from ordinary quantum field theory. 5.3. Interactions. We proceed to evaluate the order 1 contribution of the φ 4 -interaction to the 2-point function. The corresponding diagrams are depicted in Fig. 5 (see Sect. 2.3). Since the property (13) holds in SUq (2) the diagrams simplify to those of Fig. 10 (see Sect. 3.2). The disconnected loop comes out as δloop :=

=

l

[2l + 1]q q −2l(l+1) . [l]q [l + 1]q + m2

(23)

(Just apply the counit to (22).) The connected diagram in the right-hand summand of Fig. 10 is (in reduced form)

= id ⊗ ⊗

=

l,m,i,j

=

l

◦ (id2 ⊗·) ◦ (id ⊗ Z 2 ⊗ id) ◦ Z 2

(l) α l αm t 0 i

(m) t0 j

(m)

(l)

S tj 0 S ti 0

(l)

αl2 [2l + 1]−1 q t0 0 ,

2 with αl := [2l + 1]q Ll−1 q −2l(l+1) . We have used Z as reconstructed from its reduced form (22), the property ◦ S = of the integral, and (16). The connected diagram in the left-hand summand of Fig. 10 is (in reduced form)

= id ⊗ ⊗ ◦ (id2 ⊗·) ◦ (id ⊗ψ −1 ⊗ id) ◦ (Z 2 ⊗ Z 2 ) =

l,m,i,j,k,n

=

l,m,i,j,n

(l) αl αm t 0 i (l)

α l αm t 0 i

(m) t0 k

(l) (m) (m) (l) S tn 0 S tj 0 R−1 tk j ⊗ S ti n

(m) (l) (m) (l) tj 0 tn 0 R t0 j ⊗ ti n


=

469

(l)

l,m,i,j,k

=

l,m,i,k

=

l

(m) (l) (m) (m) tk 0 ti 0 R t0 j ⊗ S tj k

αl α m t 0 i (l)

α l αm t 0 i

(m) (l) (m) tk 0 ti 0 v t0 k (l)

−2l(l+1) αl2 [2l + 1]−1 t0 0 . q q

We have also used the invariance of the integral in the form ( ab(2) )b(1) = ( a (2) b) S a (1) in the third equality. Thus, the (reduced) 2-point function up to order 1 comes out as (l) 2int = Z [2l + 1]q Ll−1 q −2l(l+1) t0 0 l (24)

−1 −2l(l+1) −2l(l+1) 2 (1 + q ) + O(λ ) . 1 − 6 λ δloop Ll q In the commutative case (q = 1), we know that the order 1 contribution (given by the tadpole diagram in Fig. 6) is divergent. We can easily see where this divergence comes from. The loop contribution (23) δloop |q=1 =

l

2l + 1 l(l + 1) + m2

(25)

is infinite. However, at q > 1 it becomes finite. We are truly able to regularise the tadpole diagram. Let us identify the divergence in q-space. For q > 1 we can find both an upper and a lower bound for (23) of the form ∞ 2 2 const + dl q −2l , l 1 2

where const does not depend on q (but may depend on m2 ). Setting q = e2h with h > 0 we find δloop |q>1 =

1 + O(1). h

The conventional divergence of (25) is only logarithmic in l. What would happen with higher divergences? It seems natural to assume that they would give rise to terms like [l]nq q −2l(l+1) . l

But this converges in the domain q > 1 for any n. We can even apply the very same discussion of the divergence in q-space as above. The nature of the divergence in q-space does not seem to be affected by the degree of the ordinary (commutative) divergence at all. This suggests that q-regularisation in our framework is powerful indeed. Reviewing our calculations of Z 2 and Z 2int we find that the crucial factor of q −2l(l+1) is caused by the braiding. Thus, the braiding and not the mere noncommutativity appears to be essential for the regularisation.

470

R. Oeckl

5.4. Renormalisation. Ordinarily, φ 4 -theory in dimension 2 is super-renormalisable. The only basic divergent diagram is the tadpole (Fig. 6). Our approach yields a simple and diagrammatic way to renormalise it. We have used above the loop separation technique of Sect. 3.2 (Fig. 9) to factorise the single tadpole diagram(s) into φ 2 -vertex diagrams and the loop factor δloop . For any given diagram we can perform the same operation for all tadpole subdiagrams appearing in it. The remaining diagram (with the loop factors removed) is finite at q = 1, since the commutative theory has no further divergences. However, from a rigorous point of view this procedure can only be performed if the diagram we start out with is finite. While we have seen that the tadpole diagram alone becomes finite for q > 1, it is conceivable that certain diagrams that converge at q = 1 would diverge at q > 1. This might be due to the introduction of factors like q 2l(l+1) into summations over l. The expression (24) suggests, however, that this does not happen, but rather that all q-factors introduced in summations have negative exponent. We shall assume this in the following. Let us perform the usual mass renormalisation in our framework. We introduce an extra perturbative mass term which generates diagrams with φ 2 -vertices. These diagrams are then used to cancel the corresponding diagrams where the φ 2 -vertices are the remnants of the factorisation of tadpole subdiagrams. To effect the cancellation the perturbative mass term must carry the same factor δloop as the factorised tadpoles. To compensate for the different combinatoric multiplicity of quadratic and quartic vertices we need an extra factor of 6 in front of the φ 2 -vertex. Since a mass term carries an overall factor of 1/2 in the action, the effective mass shift is m2 → m2 − 12λ δloop . Performing this (finite) mass renormalisation at q > 1, only the divergence-free diagrams without tadpoles remain as q → 1 at any given order in perturbation theory. 6. Concluding Remarks We have presented a coherent framework for the treatment of quantum field theory on braided spaces. In particular, we have developed a quantum group covariant perturbation theory. The example of φ 4 -theory on the quantum 2-sphere has shown that quantum deformations of symmetries do lead to the regularisation of divergences in our approach. This method is superior to regularisation methods such as using a lattice or fuzzy spaces in that it does not resort to discrete approximations with only finitely many degrees of freedom. On the other hand it does not suffer from the crude breaking of symmetries as many quantum field theoretic methods do (e.g. momentum cut-off, dimensional regularisation, lattice). However, symmetries are not preserved as such, but deformed to quantum group symmetries. Our results also suggest that divergences of arbitrary order could be regularised in this way. A next step would be the investigation of quantum field theories on deformations of higher dimensional spaces to obtain more physically interesting models. We note in particular that quantum deformations of Minkowski space are available (see [1, 22] and [16, 23]). Further one would like to include internal (quantum group) symmetries as well. In particular, this might open new possibilities for the old idea of unifying internal and external symmetries. In a different direction, one might speculate that the braided Feynman diagrams obtained from theories with q-deformed symmetries have interesting number theoretic


471

properties related to modular functions. This is suggested by the observation of such properties for the quantum rank of q-deformed enveloping algebras [24]. Acknowledgements. I would like to thank S. Majid for valuable discussions during the preparation of this paper. I would also like to acknowledge the financial support by the German Academic Exchange Service (DAAD) and the Engineering and Physical Sciences Research Council (EPSRC).

A. Definition of SUq (2) This appendix recalls the defining relations of SUq (2) and the quantum Hopf fibration, see e.g., [21] or [14]. The matrix Hopf algebra SLq (2) is defined over C with generators a, b, c, d and relations ab = qba,

ac = qca,

bd = qdb,

cd = qdc,

bc = cb,

−1

ad − da = (q − q )bc, ad − qbc = 1, ab ab ˙ ab ab 10 = ⊗ , = , cd cd cd cd 01 ab d −q −1 b . S = cd −qc a

Matrix multiplication is understood in the definition of the coproduct. The ∗-structure defining the real form SUq (2) for real q is given by ∗ d − qc a b . = c d a −q −1 b As a Hopf ∗-algebra, U (1) has one generator g with inverse g −1 and relations and ∗-structure g = g ⊗ g,

S g = g −1 ,

g = 1,

g ∗ = g −1 .

There is a Hopf ∗-algebra surjection π : SUq (2) → U (1) defined by g 0 a b . → c d 0 g −1 This determines the quantum 2-sphere Sq2 as a right quantum homogeneous ∗-space under SUq (2). At q = 1 we recover the ordinary Hopf fibration. B. Coquasitriangular Structure of SUq (2) In this appendix we provide the formulas for the coquasitriangular structure of SUq (2) in the Peter–Weyl basis needed in Sect. 5. We use the context of Sect. 5.1. Definitions and results that are just stated are standard and can be found e.g. in [21] or [14]. The Hopf algebra Uq (sl2 ) is defined over C for q ∈ C∗ and q 2 = 1 with generators E, F, K, K −1 and relations KEK −1 = q 2 E,

KF K −1 = q −2 F,

472

R. Oeckl

KK −1 = K −1 K = 1,

S(K) = K −1 ,

K − K −1 , q − q −1

(F ) = F ⊗ 1 + K −1 ⊗ F,

(E) = E ⊗ K + 1 ⊗ E, (K) = K ⊗ K,

[E, F ] =

(K) = 1,

(E) = (F ) = 0,

S(E) = −EK −1 ,

S(F ) = −KF.

Uq (sl2 ) and SUq (2) are non-degenerately paired. Thus, actions of Uq (sl2 ) and coactions of SUq (2) on finite dimensional vector spaces are dual to each other. In particular, the simple comodule Vl of SUq (2) is a simple module of Uq (sl2 ). By the representation theory of Uq (sl2 ) it has a basis {wi }, i = −l, −l + 1, . . . , l such that K ! wm = q 2m wm ,

E ! wm = ([l − m]q [l + m + 1]q )1/2 wm+1 , F ! wm = ([l + m]q [l − m + 1]q )1/2 wm−1 .

(26)

Uq (sl2 ) has an h-adic version Uh (sl2 ) defined over C[[h]] correspondingly with q = eh and an additional generator H so that q H = K. It has the quasitriangular structure R = q (H ⊗H )/2

∞ q n(n+1)/2 (1 − q −2 )n

[n]q !

n=0

En ⊗ F n.

(27)

The elements (define R (1) ⊗ R (2) = R) u = (S R (2) )R (1) ,

v = R (1) S R (2)

(28)

act on Vl as [21, Prop. 3.2.7] u ! wm = q −2l(l+1)+2m wm ,

v ! wm = q −2l(l+1)−2m wm .

(29)

The coquasitriangular structure R of SUq (2) is given by the duality with Uq (sl2 ) from the quasitriangular structure R of Uh (sl2 ). Using u(a (1) )a (2) = S2 a (1) u(a (2) ) we find

and

(l) (l) 2(m−k) u tm u tk k , n = δm,n q

v(a (1) ) S2 a (2) = a (1) v(a (2) ),

(l) (l) 2(k−m) v tm v tk k . n = δm,n q

(30)

Since the definitions (18) and (28) are dual to each other we can use (l) g ! vn = vm g, tm g ∈ Uq (sl2 ) n , m

to compare (29) with (30). We find (19) and infer that wi is (a multiple of) vi . With the latter, the pairing between Uq (sl2 ) and SUq (2) comes out from (26) as (l) 2n K, tm n = δm,n q ,

(l) 1/2 E, tm , n = δm,n+1 ([l − n]q [l + n + 1]q )

(l) 1/2 F, tm . n = δm,n−1 ([l + n]q [l − n + 1]q ) (l)

Note also H, tm n = δm,n 2n in the h-adic version. With this pairing and (27) we easily verify the property (20).


473

References 1. Carow-Watamura, U., Schlieker, M., Scholl, M., Watamura, S.: Tensor representation of the quantum group SLq (2, C) and quantum Minkowski space. Z. Phys. C 48, 159–165 (1990) 2. Carter, R., Segal, G., MacDonald, I.: Lectures on Lie Groups and Lie Algebras. Cambridge: Cambridge University Press, 1995 3. Chaichian, M., Demichev, A., Prešnajder, P.: Quantum Field Theory on the Noncommutative Plane with Eq (2) Symmetry. J. Math. Phys. 41, 1647–1671 (2000) 4. Cho, S., Hinterding, R., Madore, J., Steinacker, H.: Finite Field Theory on Noncommutative Geometries. Internat. J. Modern Phys. D 9, 161–199 (2000) 5. Connes, A.: Noncommunicative Geometry. London: Academic Press, 1994 6. Connes, A., Douglas, M., Schwarz, A.: Noncommutative geometry and matrix theory: Compactification on tori. J. High Energy Phys. 9802, 003 (1998) 7. Drinfeld, V.G.: Quantum groups. In Gleason, A. (ed.) Proceedings of the ICM 1986, Providence, RI: AMS, 1987, pp. 798–820 8. Filk, T.: Divergencies in a field theory on quantum space. Phys. Lett. B 376, 53–58 (1996) 9. Fiore, G.: The SOq (N, R)-symmetric harmonic oscillator on the quantum Euclidean Space RN q and its Hilbert space structure. Internat. J. Modern Phys. A 8, 4679–4729 (1993) 10. Grosse, H., Klimˇcik, C., Prešnajder, P.: On Finite 4D Quantum Field Theory in Non-Commutative Geometry. Commun. Math. Phys. 180, 429–438 (1996) 11. Jimbo, M.: A q-difference analogue of U (g) and the Yang–Baxter equation. Lett. Math. Phys. 10, 63–69 (1985) 12. Kempf, A.: Noncommutative geometric regularization. Phys. Rev. D 54, 5174–5178 (1996) 13. Kempf, A., Majid, S.: Algebraic q-integration and Fourier theory on quantum and braided spaces. J. Math. Phys. 35, 6802–6837 (1994) 14. Klimyk, A., Schmüdgen, K.: Quantum Groups and Their Representations. Berlin: Springer Verlag, 1997 15. Koornwinder, T.H.: Representations of the twisted SU (2) quantum group and some q-hypergeometric orthogonal polynomials. Nederl. Akad. Wetensch. Proc. Ser. A 92, 97–117 (1989) 16. Lukierski, J., Nowicki, A., Ruegg, H.: New quantum Poincaré algebra and κ-deformed field theory. Phys. Lett. B 293, 344–352 (1991) 17. Madore, J.: An Introduction to Noncommutative Differential Geometry and its Physical Applications. Cambridge: Cambridge University Press, 1995 18. Majid, S.: Hopf algebras for physics at the Planck scale. Class. Quantum Grav. 5, 1587–1606 (1988) 19. Majid, S.: On q-Regularization. Internat. J. Modern Phys. A 5, 4689–4696 (1990) 20. Majid, S.: Free braided differential calculus, braided binomial theorem, and the braided exponential map. J. Math. Phys. 34, 4843–4856 (1993) 21. Majid, S.: Foundations of Quantum Group Theory. Cambridge: Cambridge University Press, 1995 22. Majid, S., Meyer, U.: Braided matrix structure of q-Minkowski space and q-Poincaré group. Z. Phys. C 63, 357–362 (1994) 23. Majid, S., Ruegg, H.: Bicrossproduct structure of κ-Poincaré group and non-commutative geometry. Phys. Lett. B 334, 348–354 (1994) 24. Majid, S. and Soibelman,Ya.S.: Rank of Quantized Universal Enveloping Algebras and Modular Function. Commun. Math. Phys. 137, 249–262 (1991) 25. Masuda, T., Mimachi, K., Nakagami, Y., Noumi, M., Ueno, K.: Representations of the Quantum Group SUq (2) and the Little q-Jacobi Polynomials. J. Funct. Anal. 99, 357–386 (1991) 26. Podle´s, P.: Quantum Spheres. Lett. Math. Phys. 14, 193–202 (1987) 27. Snyder, H.S.: Quantized Space-Time. Phys. Rev. 71, 38–41 (1947) 28. Sweedler, M.E.: Hopf Algebras. New York: W. A. Benjamin, 1969 29. Vilenkin, N.Ja., Klimyk, A.U.: Representation of Lie Groups and Special Functions. Vol. 1, Dordrecht: Kluwer Academic Publishers, 1991 30. Woronowicz, S.L.: Compact Matrix Pseudogroups. Commun. Math. Phys. 111, 613–665 (1987) 31. Zinn-Justin, J.: Quantum Field Theory and Critical Phenomena. Third Edition, Oxford: Oxford University Press, 1996 Communicated by A. Connes

Commun. Math. Phys. 217, 489 – 502 (2001)

Communications in



The Norm Convergence of the Trotter–Kato Product Formula with Error Bound Takashi Ichinose1, , Hideo Tamura2, 1 Department of Mathematics, Faculty of Science, Kanazawa University, Kanazawa, 920–1192, Japan.


2 Department of Mathematics, Faculty of Science, Okayama University, Okayama, 700–8530, Japan.

E-mail: [email protected] Received: 26 June 2000 / Accepted: 21 September 2000

Abstract: The norm convergence of the Trotter–Kato product formula with error bound is shown for the semigroup generated by that operator sum of two nonnegative selfadjoint operators A and B which is selfadjoint. 1. Introduction If A and B are selfadjoint operators bounded below in a Hilbert space H with domains D[A] and D[B] and if their sum A + B is essentially selfadjoint on D[A] ∩ D[B], then the exponential product formula lim (e−tB/2n e−tA/n e−tB/2n )n = lim (e−tA/n e−tB/n )n = e−tC

n→∞

n→∞

(1.1)

holds in strong operator topology, where C is the closure of A + B. The convergence in (1.1) is uniform on each compact t-interval in the closed half line [0, ∞). This is the celebrated result by Trotter [26]. It was extended by Kato [15] to the case for the form sum C of two arbitrary nonnegative selfadjoint operators A and B. The aim of the present paper is to prove that (1.1) holds even in operator norm, uniformly on each compact t-interval in the open half line (0, ∞), together with an error bound of order O(n−1/2 ), when the sum C := A + B is selfadjoint on D[C] = D[A] ∩ D[B]. To state our theorem, consider real-valued, Borel measurable functions f on [0, ∞) satisfying 0 ≤ f (s) ≤ 1, f (0) = 1, f (0) = −1. (1.2) Partially supported by the Grant-in-Aid for Scientific Research (B) No. 11440040, Japan Society for the Promotion of Science. Partially supported by the Grant-in-Aid for Scientific Research (B) No. 11440056, Japan Society for the Promotion of Science.

490

Takashi Ichinose, Hideo Tamura

Some examples of functions satisfying (1.2) are f (s) = e−s ,

f (s) = (1 + k −1 s)−k ,

k > 0.

(1.3)

In fact, it was also for f (tA), g(tB) in place of e−tA , e−tB with f and g being the functions satisfying (1.2) that Kato [15] proved the product formula (1.1) in strong operator topology. We are interested in those functions f which satisfy (1.2) and further that for every small ε > 0 there exists a positive constant δ = δ(ε) < 1 such that f (s) ≤ 1 − δ(ε),

s ≥ ε,

(1.4)

and that for some fixed constant κ with 1 < κ ≤ 2, [f ]κ := sup s>0

|f (s) − 1 + s| < ∞. sκ

(1.5)

A function f (s) satisfying (1.2) has property (1.4), if it is non-increasing. Of course, the functions in (1.3) have properties (1.4) and (1.5). Theorem. Let f and g be functions having properties (1.4) and (1.5) with κ ≥ 3/2 as well as (1.2). If A and B are nonnegative selfadjoint operators in a Hilbert space H with domains D[A] and D[B] such that the operator sum C := A + B is selfadjoint on D[C] = D[A] ∩ D[B], then it holds in operator norm that [g(tB/2n)f (tA/n)g(tB/2n)]n − e−tC = O(n−1/2 ), (1.6) [f (tA/n)g(tB/n)]n − e−tC = O(n−1/2 ), n → ∞. The convergence is uniform on each compact t-interval in the open half line (0, ∞) and further, if C is strictly positive, i.e. C ≥ η for some constant η > 0, uniform on the closed half line [T , ∞) for every fixed T > 0. The first original result of such a norm convergence of the Trotter–Kato product formula (1.1) was proved by Rogava [21] under an additional condition that A is strictly positive and B is A-bounded, with error bound of order O(n−1/2 log n). The next is a result by Helffer [6] for the Schrödinger operators H = H0 + V ≡ − 21 + V (x) with C ∞ nonnegative potentials V (x), roughly speaking, growing at most of order O(|x|2 ) for large |x| with error bound of order O(n−1 ). Each of these two results is independent of and does not cover the other. Then under some stronger or more general conditions, several further results are obtained. As for the abstract case, a better error bound O(n−1 log n) than Rogava’s is obtained by Ichinose–Tamura [13] (cf. [11]) when B is Aα -bounded for some 0 < α < 1, even though the B = B(t) may be t-dependent, and by Neidhardt–Zagrebnov [16, 17] (cf. [18, 19]) when B is A-bounded with relative bound less than 1. As for the Schrödinger operators, a different proof to Helffer’s result was given by Dia–Schatzman [3]. Further, more general results were proved for continuous nonnegative potentials V (x), roughly speaking, growing of order O(|x|ρ ) for large |x| with ρ > 0, together with error bounds dependent on the power ρ (for instance, of order O(n−2/ρ ), if ρ ≥ 2), by Ichinose–Takanobu [7, 8], Doumeki–Ichinose–Tamura [4], Ichinose–Tamura [12], Takanobu [24] and Ichinose–Takanobu [9, 10]. It should be noted (see Guibourg [5], Shen [22, 23]) that in all these cases of the Schrödinger operators the sum H = H0 + V is selfadjoint on the domain D[H ] = D[H0 ] ∩ D[V ].

Norm Convergence of Trotter–Kato Product Formula

491

Thus the present theorem not only extends Rogava’s result, but also can include all the results mentioned above. It should be emphasized that the error bound O(n−1/2 ) obtained, in fact, is even better than Rogava’s, and than the error bounds (e.g. [8, 4, 12, 9]) known for the Schrödinger operators with potentials V (x) growing of order O(|x|ρ ) when ρ > 4. We note here that unless the sum A + B is selfadjoint on D[A] ∩ D[B], the norm convergence of (1.1) does not always hold, even though the sum is essentially selfadjoint there and B is A-form-bounded with relative bound less than 1. This fact has recently been pointed out by Hiroshi Tamura [25] with a counterexample. To prove our theorem, in Sect. 2, we establish an operator-norm version of Chernoff’s theorem (cf. [1, 2]) with error bounds. The theorem is proved in Sect. 3. Section 4 remarks on conditions (1.4) and (1.5). 2. Operator-norm Version of Chernoff’s Theorem To prove the theorem, we shall use the following operator-norm version of Chernoff’s theorem (cf. [1, 2]) with error bounds. The case without error bounds was noted by Neidhardt–Zagrebnov [18]. Lemma 2.1. Let C be a nonnegative selfadjoint operator in a Hilbert space H and let {F (t)}t≥0 be a family of selfadjoint operators with 0 ≤ F (t) ≤ 1. Define St = t −1 (1 − F (t)). Then in the following two assertions, for 0 < α ≤ 1, (a) implies (b) . (a)

(1 + St )−1 − (1 + C)−1 = O(t α ),

t ↓ 0.

(2.1)

(b) For any δ > 0 with 0 < δ ≤ 1, F (t/n)n − e−tC = δ −2 t −1+α eδt O(n−α ),

n → ∞,

(2.2)

for all t > 0. Therefore, for 0 < α < 1 (resp. α = 1), the convergence in (2.2) is uniform on each compact t-interval in the open half line (0, ∞) (resp. in the closed half line [0, ∞)). Moreover, if C is strictly positive, i.e. C ≥ η for some constant η > 0, the error bound on the right-hand side of (2.2) can also be replaced by (1 + 2/η)2 t −1+α O(n−α ), so that, for 0 < α < 1 (resp. α = 1), the convergence in (2.2) is uniform on the closed half line [T , ∞) for every fixed T > 0 (resp. on the whole closed half line [0, ∞)). Proof. Assume (a). Let t > 0. We have F (t/n)n − e−tC = (F (t/n)n − e−tSt/n ) + (e−tSt/n − e−tC ).

(2.3)

To estimate the first term on the right-hand side of (2.3), let us note 0 ≤ e−n(1−λ) − λn ≤ e−1 /n,

for 0 ≤ λ ≤ 1.

(2.4)

Though this can be in fact shown with the upper bound 2e−2 /n in place of e−1 /n, we shall content ourselves with it. To see (2.4) is easy. Since the function ξ(λ) := e−n(1−λ) − λn attains its maximum at λ0 satisfying e−n(1−λ0 ) = λn−1 0 , we obtain 0 ≤ ξ(λ) ≤ ξ(λ0 ) = n = (1/n)n(1 − λ )e−n(1−λ0 ) ≤ e−1 /n. λn−1 − λ 0 0 0

492


Then by (2.4), we have by the spectral theorem for every t > 0, F (t/n)n − e−tSt/n = F (t/n)n − e−n(1−F (t/n)) ≤ e−1 n−1 .

(2.5)

To estimate the second term, we use a formula in Kato [14, IX.4, (2.27)] (1 + Sε )−1 [e−t (δ+Sε ) − e−t (δ+C) ](1 + C)−1 t = e−(t−s)(δ+Sε ) [(1 + Sε )−1 − (1 + C)−1 ]e−s(δ+C) ds 0

=

0

t/2

+

t t/2

(2.6)

≡ S1 + S2 ,

where δ > 0 and ε > 0. Putting D(ε) = (1 + Sε )−1 − (1 + C)−1 in the following, we are assuming D(ε) = O(εα ) by (2.1). For S1 we have by integration by parts s=t/2 S1 = −e−(t−s)(δ+Sε ) D(ε)e−s(δ+C) (δ + C)−1 s=0 t/2 −(t−s)(δ+Sε ) −s(δ+C) + (δ + Sε )e D(ε)e (δ + C)−1 ds 0

= − e−(t/2)(δ+Sε ) D(ε)e−(t/2)(δ+C) (δ + C)−1 + e−t (δ+Sε ) D(ε)(δ + C)−1 t/2 + (δ + Sε )e−(t−s)(δ+Sε ) D(ε)e−s(δ+C) (δ + C)−1 ds. 0

Then (1+Sε )S1 (1 + C) = − (1 + Sε )e−(t/2)(δ+Sε ) D(ε)e−(t/2)(δ+C) (δ + C)−1 (1 + C) + (1 + Sε )e−t (δ+Sε ) D(ε)(δ + C)−1 (1 + C) t/2 + (1 + Sε )(δ + Sε )e−(t−s)(δ+Sε ) D(ε)e−s(δ+C) (δ + C)−1 (1 + C)ds, 0

and similarly for S2 , (1+Sε )S2 (1 + C) = (1 + Sε )(δ + Sε )−1 D(ε)e−t (δ+C) (1 + C) − (1 + Sε )(δ + Sε )−1 e−(t/2)(δ+Sε ) D(ε)e−(t/2)(δ+C) (1 + C) t (1 + Sε )(δ + Sε )−1 e−(t−s)(δ+Sε ) D(ε)e−s(δ+C) (δ + C)(1 + C)ds. + t/2

We know e−δt (e−tSε − e−tC ) = (1 + Sε )S1 (1 + C) + (1 + Sε )S2 (1 + C).

(2.7)


493

Since λγ e−λ ≤ (γ /e)γ for λ ≥ 0 and γ ≥ 0, we can estimate (2.7) with assumption (2.1) by the spectral theorem as t/2 (t − s)−2 ds O(ε α )/δ 2 (1 + Sε )S1 (1 + C) ≤ 3e−1 /t + 4e−2 0

≤ 2O(ε α )/(δ 2 t), (1 + Sε )S2 (1 + C) ≤ 3e−1 /t + 4e−2

t t/2

s −2 ds O(ε α )/δ 2 ≤ 2O(ε α )/(δ 2 t).

Here we have needed that δ ≤ 1. Hence with (2.7), e−δt (e−tSε − e−tC ) = (1 + Sε )(S1 + S2 )(1 + C) ≤ 4O(ε α )/(δ 2 t).

(2.8)

It follows that with ε = t/n the second term of (2.3) obeys e−tSt/n − e−tC ≤ (δ 2 t)−1 eδt O((t/n)α ) = δ −2 t −1+α eδt O(n−α ).

(2.9)

Thus, combining (2.5) and (2.9) with (2.3), we have the assertion (b) or (2.2). In case C is strictly positive, that is, C ≥ η for some constant η > 0, we can show Sε ≥ η/2 or Sε−1 ≤ 2/η for sufficiently small ε > 0. In fact, by (2.1), (1 + Sε )−1 = (1 + C)−1 + O(εα ) ≤ (1 + η)−1 + O(ε α ) ≤ (1 + η/2)−1 < 1, so that Sε has bounded inverse Sε−1 = [(1 + Sε ) − 1]−1 = (1 + Sε )−1 [1 − (1 + Sε )−1 ]−1 with bound Sε−1 ≤ 2/η. Therefore in the above argument around (2.8), though (1 + Sε )(δ + Sε )−1 is bounded as well as (1 + C)(δ + C)−1 , with bound (1 + η/2)(δ + η/2)−1 in place of 1/δ, one can in turn use the formula (2.5) with δ = 0, and show, since both (1 + Sε )Sε−1 and (1 + C)C −1 are bounded with bound (1 + 2/η) for small ε > 0, that the right-hand side of (2.9) simply becomes of order (1 + 2/η)2 t −1+α O(n−α ). In particular, it is of order O(n−α ) uniformly on the closed half line [T , ∞) for every T > 0 for 0 < α < 1, and on the whole closed half line [0, ∞) for α = 1. This proves Lemma 2.1. 3. Proof of Theorem We are now in a position to prove the theorem. First note that since C = A + B is itself selfadjoint and so a closed operator, by the closed graph theorem there exist constants a1 and a2 such that Au + Bu ≤ a1 Cu + a2 u,

u ∈ D[C] = D[A] ∩ D[B].

Therefore we may assume for some constant a > 0 that (1 + A)u + (1 + B)u ≤ a(1 + C)u,

u ∈ D[C] = D[A] ∩ D[B].

(3.1)

For t > 0 define positive bounded operators At = t −1 [1 − f (tA)],

Bt = t −1 [1 − g(tB)],

Ct = t −1 [1 − e−tC ].

(3.2)

494


Note that tAt = 1 − f (tA) ≤ 1,

tBt = 1 − g(tB) ≤ 1.

(3.3)

The proof of the theorem is now divided into two cases, (a) the symmetric product case concerning F (t) = g(tB/2)f (tA)g(tB/2), (3.4) and (b) the non-symmetric product case concerning G(t) = f (tA)g(tB).

(3.5)

In the former case we shall use Lemma 3.1. The latter case will follow from the former case. (a) The symmetric product case. To prove the symmetric product case of the theorem, by Lemma 2.1 it suffices to show in operator norm that with St = t −1 (1 − F (t)), (1 + St )−1 − (1 + C)−1 = O(t 1/2 ),

t ↓ 0.

(3.6)

We should already know (cf. Chernoff [1, 2], Kato [15] and Reed–Simon [20]) that (1 + St )−1 → (1 + C)−1 in strong operator topology. Define a positive bounded operator 2 Kt = 1 + At + Bt/2 − 4t Bt/2

= 1 + At + 21 Bt/2 + 21 Bt/2 (1 − 2t Bt/2 )Bt/2 1/2 1/2 = 1 + At + Bt/2 1+g(tB/2) Bt/2 ≥ 1. 2 1/2

1/2

(3.7)

Rewrite 1 + St , by introducing Qt , as 2 1 + St = 1 + At + Bt/2 − 4t Bt/2 + 1/2

= Kt Qt =

t2 4

1/2

(1 + Qt )Kt

−1/2

Kt

t2 4 Bt/2 At Bt/2

− 2t (At Bt/2 + Bt/2 At ) (3.8)

, −1/2

Bt/2 At Bt/2 Kt

−1/2

− 2t Kt

−1/2

(At Bt/2 + Bt/2 At )Kt

.

Then we need that 1 + Qt has bounded inverse uniformly for t > 0. The proof of this fact in Reed–Simon [20] seems to contain a small flaw. So we prove it in the following lemma. At this stage note that differing from their proof, ours is exchanging the roles of A and B. Lemma 3.1. For t > 0,

(1 + Qt )−1 ≤ 2/(3 −

√

(3.9)

5).

If (3.9) is proved, then we can obtain that for t > 0, (1 + St )−1 Kt

1/2

−1/2

= Kt

(1 + Qt )−1 ≤ 2/(3 −

√

5).

(3.10)

Proof of Lemma 3.1. We shall use (3.3) and 1/2

−1/2

At Kt 2

−1/2

−1/2

≤ (1 + At )1/2 Kt

1/2 −1/2 Bt/2 Kt

≤ (1 + 21 Bt/2 )

≤ 1,

1/2

−1/2

Kt

≤ 1.

(3.11)


495

We see from the definition of Qt in (3.8), −1/2 t −1/2 ( 2 Bt/2 − r)At ( 2t Bt/2 − r)Kt −1/2 −1/2 − (1−r)t (At Bt/2 + Bt/2 At )Kt 2 Kt −1/2 −1/2 − (1−r)t (At Bt/2 + Bt/2 At )Kt 2 Kt

Qt = Kt ≥

−1/2

At K t

−1/2

At K t

− r 2 Kt − r 2 Kt

−1/2 −1/2

,

where r is a constant with 0 < r < 1 to be determined later. Hence we have for u ∈ H, (Qt u,u)

1/2 1+g(tB/2) 1/2 1/2 −1/2 1/2 −1/2 Bt/2 Kt u, At Kt u ≥ − 2(1 − r)Re (tAt )1/2 1−g(tB/2) 1+g(tB/2) 2 1/2 −1/2 2 u − r 2 At Kt 1/2 −1/2 1+g(tB/2) 1/2 1/2 −1/2 1/2 −1/2 2 ≥ − 2(1 − r)At Kt u Bt/2 Kt u − r 2 At Kt u 2 1/2 1/2 −1/2 2 1/2 −1/2 2 ≥ − (1 − r) p At Kt u + (1/p) 1+g(tB/2) B Kt u t/2

2

1/2 −1/2 2 u . − r 2 At Kt

Here p is an aribitray that (1 − r)p + r 2 = (1 − r)/p, √ positive constant. Choose p such√ namely, p =

−r 2 +

r 4 +4(1−r)2 . 2(1−r)

Then with β(r) =

r2+

r 4 +4(1−r)2 , 2

we have 1/2 1/2 −1/2 2 1/2 −1/2 2 u + 1+g(tB/2) Bt/2 Kt u (Qt u, u) ≥ −β(r) At Kt 2 −1/2 −1/2 = −β(r) At + 1+g(tB/2) Bt/2 Kt u, Kt u 2 ≥ −β(r)u2 . √

We can see β(r) attains its minimum at r = 5−1 2 :

√ √ √5−1 2 3−√5 2 5−1 4 5−1 1 β( 2 ) = 2 + +4 2 2 2 √ 1/2 1 √ √ √ = 41 3 − 5 + 70 − 30 5 = 4 3− 5+3 5−5 = It follows that (Qt u, u) ≥ − √ 3− 5 2 2 u .

√

5−1 2 2 u ,

so that ((1 + Qt )u, u) ≥ 1 −

This yields (3.9), showing Lemma 3.1.

√

√

5−1 2

5−1 2 .

u2 =

Now we have (1 + St )−1 − (1 + C)−1 = (1 + St )−1 A + B − (At + Bt/2 − 4t Bt/2 (1 − tAt )Bt/2

− 2t (At Bt/2 + Bt/2 At )) (1 + C)−1 = (1 + St )−1 (A − At )(1 + C)−1 + (1 + St )−1 (B − Bt/2 )(1 + C)−1

(3.12)

+ (1 + St )−1 [ 4t Bt/2 (1 − tAt )Bt/2 + 2t (At Bt/2 + Bt/2 At )](1 + C)−1 ≡ R1 (t) + R2 (t) + R3 (t). We are going to show in the following lemma that all the three Ri (t) in the last member of (3.12) converge to zero in operator norm of order O(t 1/2 ) as t ↓ 0.

496


Lemma 3.2. For small t > 0, R1 (t) ≤ cat 1/2 ,

R2 (t) ≤ cat 1/2 ,

R3 (t) ≤ cat 1/2 ,

(3.13)

with a constant c > 0 independent of t > 0. Proof. First note by the spectral theorem that 1 − f (tλ) < ∞, λ≥0 t (1 + λ) 1 − g(tλ/2) b0 := Bt/2 (1 + B)−1 = sup < ∞. λ≥0 t (1 + λ)/2 a0 := At (1 + A)−1 = sup

(3.14)

I. For R1 (t) we have R1 (t) = [(1 + St )−1 Kt

1/2

−1/2

][Kt

(1 + At )1/2 ]

× [(1 + At )−1/2 − (1 + At )1/2 (1 + A)−1 ](1 + A)(1 + C)−1 . Hence by (3.1), (3.10) and (3.11), R1 (t) ≤

2√ a(1 + At )−1/2 3− 5

− (1 + At )1/2 (1 + A)−1 .

Then by the spectral theorem we have (1 + At )−1/2 − (1 + At )1/2 (1 + A)−1 = sup |at (λ)|, λ≥0

1 − f (tλ) −1/2 1 − f (tλ) 1/2 at (λ) = 1 + − 1+ (1 + λ)−1 t t 1/2 t 1 1 − f (tλ) = 1− 1+ 1 + t − f (tλ) 1+λ t 1/2 t f (tλ) − 1 + tλ = . 1 + t − f (tλ) t (1 + λ) Since f satisfies f (0) = −1 by (1.2), there exists a small positive constant s1 such that for 0 ≤ s ≤ s1 , −s/2 ≤ f (s) − 1 + s ≤ s/2, Then

or

s/2 ≤ 1 − f (s) ≤ 3s/2.

sup |at (λ)| = sup |at (µ/t)| λ≥0

µ≥0

1/2 |f (µ) − 1 + µ| t t +µ µ≥0 1 + t − f (µ) = max sup |at (µ/t)|, sup |at (µ/t)| . = sup

0≤µ≤s1

µ≥s1

As for the first component in the last member above, we have, since f satisfies (1.5) with κ ≥ 3/2, 1/2 [f ] µκ √ t κ κ−3/2 1/2 t . ≤ 2[f ]κ s1 sup |at (µ/t)| ≤ sup t +µ 0≤µ≤s1 0≤µ≤s1 t + µ/2


497

As for the second component, since, by (1.4), for the same s1 as above there exists a positive constant δ = δ(s1 ) < 1 such that if s ≥ s1 then f (s) ≤ 1 − δ(s1 ), we have sup |at (µ/t)| ≤

µ≥s1

1/2 t (1 + a0 ) ≤ δ(s1 )−1/2 (1 + a0 )t 1/2 . t + δ(s1 )

This proves the estimate for R1 (t). II. The proof for R2 (t) is the same as for R1 (t). We have only to replace At , A and f by Bt/2 , B and g, and only note that

1/2 −1/2 R2 (t) = (1 + St )−1 Kt (1 + 21 Bt/2 )1/2 (1 + 21 Bt/2 )−1/2 (1 + Bt/2 )1/2 Kt

× (1 + Bt/2 )−1/2 − (1 + Bt/2 )1/2 (1 + B)−1 (1 + B)(1 + C)−1 . III. For R3 (t) we have R3 (t) =

√

1/2 −1/2 1/2 t Bt/2 ( 2 Bt/2 )1/2 (1 − tAt ) (1 + St )−1 Kt Kt

× Bt/2 (1 + B)−1 (1 + B)(1 + C)−1 1/2 −1/2 1/2 Kt + 21 t 1/2 (1 + St )−1 Kt At

× (tAt )1/2 Bt/2 (1 + B)−1 (1 + B)(1 + C)−1 √ 1/2 −1/2 1/2 + 22 t 1/2 (1 + St )−1 Kt Bt/2 Kt t

× ( 2 Bt/2 )1/2 At (1 + A)−1 (1 + A)(1 + C)−1 . 2 1/2 4 t

It follows by (3.1), (3.9), (3.10) and (3.14) that √ √ √ √ R3 (t) ≤ 42 2 √2 b0 + 21 2√ b0 + 22 2 √2 a0 at 1/2 ≤ 3− 5

3− 5

3− 5

2√ (a0 3− 5

+ b0 )at 1/2 .

This completes the proof of Lemma 3.2. Thus we have proved (3.6), so that by Lemma 3.1 with F (t) in (3.4), F (t/n)n − e−tC = δ −2 t −1/2 eδt O n−1/2 , n → ∞, (3.15) and in particular, the symmetric product case of the theorem. (b) The non-symmetric product case. What we have proved in the symmetric product case (a) of the theorem, namely, (3.15), is that F (t/n)n = e−tC + Op (t, n), where Op (t, n) is some bounded operator with norm of order δ −2 t −1/2 eδt O(n−1/2 ) for n large and t > 0 with 0 < δ ≤ 1. We are now going to show this implies that G(t/n)n = e−tC + Op (t, n); here it should be noted that the following proof is equally valid, even if Op (t, n) means some bounded operator with norm of such an order δ −2 t −1+α eδt O(n−α ) for some 0 < α ≤ 1 as we have had on the right-hand side of (2.2) in Lemma 2.1. Given g(t), put g1 (t) = g(2t)1/2 or g1 (t)2 = g(2t). We can see that g1 satisfies the same condition as f and g. Put F1 (t) = g1 (tB/2)f (tA)g1 (tB/2), similarly to (3.4). Then by the symmetric product case (a), we have F1 (t/n)n = [g1 (tB/2n)f (tA/n)g1 (tB/2n)]n = e−tC + Op (t, n).

(3.16)

498


Then we have by (3.5) and (3.16),

n G(t/n)n = [f (tA/n)g(tB/n)]n = f (tA/n)g1 (tB/2n)2 = f (tA/n)g1 (tB/2n)F1 (t/n)n−1 g1 (tB/2n)

= f (tA/n)g1 (tB/2n) e−(n−1)tC/n + Op (t, n) g1 (tB/2n)

(3.17)

= f (tA/n)g1 (tB/2n)e−(n−1)tC/n g1 (tB/2n) + Op (t, n). In the following lemma we are denoting by [U, V ] = U V − V U the commutator of bounded linear operators U and V . Lemma 3.3. For τ = t/n or τ = t/2n, [f (τ A), e−tC ] = δ −1 eδt O(n−1 ), [g(τ B), e−tC ] = δ −1 eδt O(n−1 ),

[g1 (τ B), e−tC ] = δ −1 eδt O(n−1 ),

with 0 < δ ≤ 1 a constant. Therefore the norm bounds on the right-hand side are of order O(n−1 ) uniformly on each compact t-interval in the closed half line [0, ∞). If C is strictly positive, i.e. C ≥ η for some constant η > 0, then these norm bounds are of order O(n−1 ) uniformly on the whole closed half line [0, ∞). Proof. We have only to prove the first one for f (τ A). We see by (3.2) for δ > 0, [f (τ A), e−tC ] = eδt f (τ A)e−t (δ+C) − e−t (δ+C) f (τ A) = eδt (1 − τ Aτ )e−t (δ+C) − e−t (δ+C) (1 − τ Aτ ) = −eδt τ (Aτ e−t (δ+C) − e−t (δ+C) Aτ ). Since by (3.1) and (3.14) the norm of Aτ e−t (δ+C) = t −1 [Aτ (1+A)−1 ][(1+A)(1+C)−1 ][(1+C)(δ+C)−1 ]t (δ+C)e−t (δ+C) is bounded by a0 ae−1 /(δt) and similarly for e−t (δ+C) Aτ , we have [f (τ A), e−tC ] = a0 ae−1 δ −1 eδt O(n−1 ). In case C is strictly positive, i.e. C ≥ η for η > 0, we may begin the above argument with δ = 0 to get the norm bound Aτ e−tC ≤ a0 ae−1 /(ηt), so that [f (τ A), e−tC ] = a0 ae−1 η−1 O(n−1 ). This proves Lemma 3.3. By Lemma 3.3, we obtain from (3.17), G(t/n)n = [f (tA/n)g(tB/n)]n = f (tA/n)g(tB/n)e−(n−1)tC/n + Op (t, n) = f (tA/n)e−(n−1)tC/2n g(tB/n)e−(n−1)tC/2n + Op (t, n) =e

−(n−1)tC/2n

f (tA/n)g(tB/n)e

−(n−1)tC/2n

+ Op (t, n).

Lemma 3.4. For τ = t/n, (1 + C)−1/2 [f (τ A)g(τ B) − e−τ C ](1 + C)−1/2 = O(τ ).

(3.18)


499

Proof. We have by (3.2),

(1 + C)−1/2 f (τ A)g(τ B) − e−τ C (1 + C)−1/2

= (1 + C)−1/2 (1 − τ Aτ )(1 − τ Bτ ) − e−τ C (1 + C)−1/2 = τ (1 + C)−1/2 Cτ (1 + C)−1/2 − τ (1 + C)−1/2 (Aτ + Bτ )(1 + C)−1/2 + τ 2 (1 + C)−1/2 Aτ Bτ (1 + C)−1/2 ≡ E1 (τ ) + E2 (τ ) + E3 (τ ). It is easy to see that E1 (τ ) ≤ τ . We have also E2 (τ ) ≤ (a0 +b0 )τ and E3 (τ ) ≤ (a0 b0 )1/2 τ , by (3.3), because

E2 (τ ) = − τ (1 + C)−1/2 (1 + A)1/2 (1 + A)−1/2 Aτ (1 + A)−1/2 × (1 + A)1/2 (1 + C)−1/2

− τ (1 + C)−1/2 (1 + B)1/2 (1 + B)−1/2 Bτ (1 + B)−1/2 × (1 + B)1/2 (1 + C)−1/2 ,

E3 (τ ) = τ (1 + C)−1/2 (1 + A)1/2 (1 + A)−1/2 A1/2 τ

× (τ Aτ )1/2 (τ Bτ )1/2 Bτ1/2 (1 + B)−1/2 (1 + B)1/2 (1 + C)−1/2 . This proves Lemma 3.4.

Finally, by Lemma 3.4 we obtain from (3.18),

n G(t/n)n = f (tA/n)g(tB/n) = e−(n−1)tC/2n (1 + C)1/2 (1 + C)−1/2 f (tA/n)g(tB/n)(1 + C)−1/2 × (1 + C)1/2 e−(n−1)tC/2n + Op (t, n)

= e−(n−1)tC/2n (1 + C)1/2 (1 + C)−1/2 e−tC/n (1 + C)−1/2 + Op (t/n) × (1 + C)1/2 e−(n−1)tC/2n + Op (t, n)

= e−tC + e−(n−1)tC/2n (1 + C)1/2 Op (t/n) (1 + C)1/2 e−(n−1)tC/2n + Op (t, n) = e−tC + δ −1 eδt Op (n−1 ) + Op (t, n) = e−tC + Op (t, n).

(3.19) Here Op (t/n) and Op (n−1 ) also mean some bounded operators with norm of order O(t/n) and O(n−1 ), respectively, for n large and t > 0. Therefore we can conclude from (3.19), (3.20) G(t/n)n − e−tC = O(n−1/2 ), n → ∞, uniformly on each compact t-interval in the open half line (0, ∞). If C is strictly positive, then we can see this norm bound O(n−1/2 ) on the right-hand side of (3.20) is uniform on the closed half line [T , ∞) for every T > 0, taking this case of both Lemma 2.1 and Lemma 3.3 into consideration. Thus we have proved the non-symmetric product case of the theorem.

500


4. Remarks on Conditions (1.4) and (1.5) In this section, we note that condition (1.4) is necessary, and make a remark on what both conditions (1.4) and (1.5) imply. First, let f and g be real-valued smooth functions satisfying (1.2) and (1.5) such that f (s) = g(s) = 1 for s > 1. Note that these f and g do not satisfy (1.4). Let H be a nonnegative selfadjoint operator in H. Assume that H has only discrete eigenvalues ∞ divergent to infinity. Let {λj }∞ j =1 be the eigenvalues with {ψj }j =1 the corresponding normalized eigenvectors. Take three operators A, B and C as A = B = 21 H, C = A + B = H. Fix n sufficiently large, and take N so large that λN > 2n. Then [f (A/n)g(B/n)]n ψN = [f (H /2n)g(H /2n)]n ψN = [f (λN /2n)g(λN /2n)]n ψN = ψN , which preserves the norm as vectors in the Hilbert space H. On the other hand, we have e−C ψN = e−H ψN = e−λN ψN → 0, strongly as N → ∞. This means that [f (A/n)g(B/n)]n never converges to e−C in operator norm. Next, in general, let f and g be real-valued smooth functions satisfying (1.2) and (1.5), but one of them, say, f not (1.4). We may suppose that f (1) = 1. Let H be a selfadjoint operator as above but with eigenvalues {λj = j }∞ j =1 . Take A = H, B = O, C = A + B = H . Then [f (A/n)g(B/n)]n ψn = f (H /n)n ψn = f (1)n ψn = ψn , which preserves the norm, while e−C ψn = e−H ψn = e−n ψn → 0, strongly as n → ∞. This means again that [f (A/n)g(B/n)]n never converges to e−C in operator norm. Thus, finally we arrive at the following remark on both conditions (1.4) and (1.5). Since the Theorem should also hold in both the special and trivial cases B = O or C = A, and A = O or C = B, we expect (2.1) in Lemma 2.1 to hold with F (t) = f (tA) and α = 1/2: (1 + At )−1 − (1 + A)−1 = O(t 1/2 ), t ↓ 0, (4.1) and similarly with F (t) = g(tB/2)2 . Here note that g(s/2)2 also have the same properties (1.2), (1.4) and (1.5) as g(s). The fact is, conditions (1.4) and (1.5) are giving sufficient conditions for (4.1) to hold. In fact, for t > 0 put 1 − f (tλ) −1 f (tλ) − 1 + tλ at (λ) = 1 + − (1 + λ)−1 = . t (1 + λ)(t + 1 − f (tλ)) Then the right-hand side of (4.1) is equal to t|f (µ) − 1 + µ| . µ≥0 (t + µ)(1 + t − f (µ))

sup |at (λ)| = sup |at (µ/t)| = sup λ≥0

µ≥0

(4.2)

Take the same s1 > 0 as in proof I of Lemma 3.2. Then, dividing the supremum over µ ≥ 0 in (4.2) into those over two parts 0 ≤ µ ≤ s1 and µ ≥ s1 , we have by (1.5), sup |at (µ/t)| ≤ sup

0≤µ≤s1

0≤µ≤s1

[f ]κ µκ t ≤ 2[f ]κ t κ−1 , (t + µ)(t + µ/2)


501

and by (1.4) with a0 in (3.14), sup |at (µ/t)| ≤ sup

µ≥s1

µ≥s1

t|f (µ) − 1 + µ| (1 + a0 )t ≤ sup ≤ (1 + a0 )δ(s1 )−1 t. (t + µ)(t + δ(s1 )) µ≥s1 (t + δ(s1 ))

Therefore, as for the bound of (4.1) we can conclude O(t κ−1 ), which, for small t > 0, is less than or equal to O(t 1/2 ) because 3/2 ≤ κ ≤ 2. Acknowledgement. Thanks are due to Hiroshi Tamura for a careful reading of the manuscript and some useful comments. The authors are also very grateful to an anonymous kind referee for pointing out some errors in the original manuscript.

References 1. Chernoff, P.R.: Note on product formulas for operator semigroups. J. Funct. Anal. 2, 238–242 (1968) 2. Chernoff, P.R.: Product Formulas, Nonlinear semigroups and Addition of Unbounded Operators. Mem. Amer. Math. Soc., 140, 1–121 (1974) 3. Dia, B.O. and Schatzman, M.: An estimate on the Kac transfer operator. J. Funct. Anal. 145, 108–135 (1997) 4. Doumeki, A., Ichinose, T. and Tamura, Hideo: Error bound on exponential product formulas for Schrödinger operators. J. Math. Soc. Japan 50, 359–377 (1998) 5. Guibourg, D.: Inégalités maximales pour l’opérateur de Schrödinger. C. R. Acad. Sci. Paris 316, Série I Math., 249–252 (1993) 6. Helffer, B.: Around the transfer operator and the Trotter–Kato formula. Operator Theory: Advances and Appl. 78, 161–174 (1995) 7. Ichinose, T. and Takanobu, S.: On the norm estimate of the difference between the Kac operator and the Schrödinger semigroup. Differential Equations, Asymptotic Analysis, and Mathematical Physics. In: (Proceedings of the International Conference on Partial Differential Equations, Potsdam, July 29 – Aug. 3, 1996), edited by M. Demuth and B.-W. Schulze, Berlin: Akademie-Verlag, 1997, pp. 165–173 8. Ichinose, T. and Takanobu, S.: Estimate of the difference between the Kac operator and the Schrödinger semigroup. Commun. Math. Phys. 186, 167–197 (1997) 9. Ichinose, T. and Takanobu, S.: The norm estimate of the difference between the Kac operator and the Schrödinger semigroup: A unified approach to the nonrelativistic and relativistic cases. Nagoya Math. J. 149, 53–81 (1998) 10. Ichinose, T. and Takanobu, S.: The norm estimate of the difference between the Kac operator and the Schrödinger semigroup II: The general case including the relativistic case. Electronic J. Probability 5, Paper no. 5, 1–47 (2000); http://www.math.washington.edu/˜ejpecp/paper5.abs.html 11. Ichinose, T. and Tamura, Hideo: Error estimates in operator norm for Trotter–Kato product formula. Integr. Equat. Oper. Theory 27, 195–207 (1997) 12. Ichinose, T. and Tamura, Hideo: Error bound in trace norm for Trotter–Kato product formula of Gibbs semigroups. Asymptotic Analysis 17, 239–266 (1998) 13. Ichinose, T. and Tamura, Hideo: Error estimates in operator norm of exponential product formulas for propagators of parabolic evolution equations. Osaka J. Math. 35, 751–770 (1998) 14. Kato, T.: Perturbation Theory for Linear Operators. 2nd edition, Berlin–Heidelberg–NewYork: SpringerVerlag, 1976 15. Kato, T.: Trotter’s product formula for an arbitrary pair of self-adjoint contraction semigroups. In: Topics in Functional Analysis (G.C. Rota, ed.), New York: Academic Press, 1978, pp. 185–195 16. Neidhardt, H. and Zagrebnov, V.: On error estimates for the Trotter–Kato product formula. Lett. Math. Phys. 44, 169–186 (1998) 17. Neidhardt, H. and Zagrebnov, V.: Fractional powers of selfadjoint operators and Trotter–Kato product formula. Integr. Equat. Oper. Theory 35, 209–231 (1999) 18. Neidhardt, H. and Zagrebnov, V.: Trotter–Kato product formula and operator-norm convergence. Commun. Math. Phys. 205, 129–159 (1999) 19. Neidhardt, H. and Zagrebnov, V.: Trotter–Kato product formula and symmetrically normed ideals. J. Funct. Anal. 167, 113–147 (1999) 20. Reed, M. and Simon, B.: Methods of Modern Mathematical Physics I: Functional Analysis. Revised and enlarged ed., New York–London: Academic Press, 1980 21. Rogava, Dzh.L.: Error bounds for Trotter–type formulas for self-adjoint operators. Functional Analysis and Its Applications, 27, 217–219 (1993)

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22. Shen, Z.: Lp estimates for Schrödinger operators with certain potentials. Ann. Inst. Fourier, Grenoble 45, 513–546 (1995) 23. Shen, Z.: Estimates in Lp for magnetic Schrödinger operators. Indiana Univ. Math. J. 45, 817–841 (1996) 24. Takanobu, S.: On the error estimate of the integral kernel for the Trotter product formula for Schrödinger operators. Ann. Probab. 25, 1895–1952 (1997) 25. Tamura, Hiroshi: A remark on operator-norm convergence of Trotter–Kato product formula. Integr. Equat. Oper. Theory 37, 350–356 (2000) 26. Trotter, H.F.: On the product of semi-groups of operators. Proc. Amer. Math. Soc. 10, 545–551 (1959) Communicated by H. Araki

Commun. Math. Phys. 217, 475 – 487 (2001)

Communications in



Quantum Cohomology and the Periodic Toda Lattice Martin A. Guest1 , Takashi Otofuji2 1 Department of Mathematics, Graduate School of Science, Tokyo Metropolitan University,

Minami-Ohsawa 1-1, Hachioji-shi, Tokyo 192-0397, Japan. E-mail: [email protected]

2 Department of Mathematics, Graduate School of Science and Engineering, Tokyo Institute of Technology,

Okayama 2-12-1, Meguro-ku, Tokyo 152-8551, Japan. E-mail: [email protected] Received: 20 April 1999 / Accepted: 12 April 2000

Abstract: We describe a relation between the periodic one-dimensional Toda lattice and the quantum cohomology of the periodic flag manifold (an infinite-dimensional Kähler manifold). This generalizes a result of Givental and Kim relating the open Toda lattice and the quantum cohomology of the finite-dimensional flag manifold. We derive a simple and explicit “differential operator formula” for the necessary quantum products, which applies both to the finite-dimensional and to the infinite-dimensional situations. Introduction The quantum cohomology of the full flag manifold Fn of SUn is known to be related to an integrable system, the open one-dimensional Toda lattice. This relation was established in [Gi-Ki], and a rigorous framework for the calculations was developed in [Ci1, Ki1, Lu], building on earlier fundamental work in quantum cohomology. We shall give – in the spirit of [Gi-Ki] – an analogous relation between the quantum cohomology of the periodic flag manifold F l (n) and the periodic one-dimensional Toda lattice. Such an extension to the periodic case is perhaps not unexpected, but we feel that it is worth noting, for two reasons. First, there are several new features of the quantum cohomology of the periodic flag manifold F l (n) , the most obvious one being that F l (n) is an infinite-dimensional Kähler manifold. Second, very few concrete examples of this phenomenon are known (cf. Sect. 2.3 of [Au]). Indeed, the full flag manifold Fn seems to be the only example so far, together with its generalization1 G/B which was accomplished in [Ki2]. Now, F l (n) is an infinite-dimensional flag manifold (of the loop group LSUn ), and is therefore a close relative of this family. However, the periodic one-dimensional Toda lattice is more complicated than the open one; for example its solutions generally involve theta functions, whereas those of the open Toda lattice are rational expressions of exponential functions. 1 Some comments on the case of partial flag manifolds and their relation with Toda lattices can be found in Sect. 5 of [Gi1].

476

M. A. Guest, T. Otofuji

The open one-dimensional Toda lattice is a (nonlinear) first-order differential equation L˙ n (t) = [Ln (t), Mn (t)], where Ln is the tri-diagonal matrix  X1 Q1  −1 X2    0 −1   . Ln =  .. . . .    0 ···   0 ···  0 ···

0

···

···

···

Q2

0

···

···

0 ..

··· .. .

X3 Q3 .. .. . . 0 ···

.

−1 Xn−2 Qn−2 0

··· ···

−1

Xn−1

0

−1

 0

 0    0   ..  .    0   Qn−1   Xn

and Mn is a certain modification of Ln . Here, X1 , . . . , Xn and Q1 , . . . , Qn−1 are functions of a real variable t with Qi < 0, and we assume that X1 + · · · + Xn = 0. Let det(Ln + µI ) = On =

n i=0

Oni µi .

Then the polynomials On0 , On1 , . . . , Onn−1 in X1 , . . . , Xn and Q1 , . . . , Qn−1 are “the conserved quantities” of the Toda lattice, which give rise to its integrability. (For further explanation of Toda lattices we refer to [Ol-Pe, Pe, Re-Se].) The result of [Gi-Ki] is that the (small) quantum cohomology algebra of Fn = {E1 ⊆ E2 ⊆ · · · ⊆ En = Cn | Ei is an i-dimensional linear subspace of Cn } is

QH ∗ Fn ∼ = C[X1 , . . . , Xn , Q1 , . . . , Qn−1 ]/On0 , On1 , . . . , Onn−1 , where X1 , . . . , Xn , Q1 , . . . , Qn−1 are regarded now as indeterminates. In other words, the conserved quantities of the open one-dimensional Toda lattice are precisely the defining relations for the quantum cohomology algebra of Fn . This remarkable fact has been explored in a number of very interesting papers (such as [Gi2, Ki2, Ko1, Ko2, Fo-Ge-Po]). The periodic one-dimensional Toda lattice is a differential equation of the form L˙ n (t) = [Ln (t), Mn (t)], where Ln is the matrix 

X1

  −1    0   . Ln =  ..    0   0  Qn /z

Q1

0

X 2 Q2

···

···

···

0

···

···

0 .. .

··· .. .

−1 X3 Q3 .. .. .. . . . ···

0

··· ···

−1 Xn−2 Qn−2 0

··· ··· ···

−1

Xn−1

0

−1

−z



 0    0   ..  .    0   Qn−1   Xn

Quantum Cohomology and Periodic Toda Lattice

477

and where z is a “spectral parameter” in S 1 = {z ∈ C | |z| = 1}. Thus, Ln may be interpreted as a function of the real variable t with values in the loop algebra Map(S 1 , Mn C). The variables X1 , . . . , Xn and Q1 , . . . , Qn here are functions of a real variable t with Qi < 0, and we assume that X1 + · · · + Xn = 0 and that Q1 Q2 · · · Qn is constant. Let det(Ln + µI ) = Pn =

n i=0

Pni µi + An

1 + Bn z, z

where Pnk , An , Bn are polynomials in X1 , . . . , Xn and Q1 , . . . , Qn . The Pn0 , Pn1 , . . . , Pnn−1 are “the conserved quantities” of the periodic Toda lattice. The loop group LSUn = Map(S 1 , SUn ) plays an analogous role here to that of the group SUn for the open Toda lattice, and the periodic flag manifold F l (n) is analogous to Fn (it is a complete flag manifold for an affine Kac–Moody group). For a precise definition of F l (n) we refer to Sect. 8.7 of [Pr-Se]; we just remark that it is related to the Grassmannian model Gr (n) of the based loop group SUn ∼ = LSUn /SUn as follows: F l (n) = {W0 ⊆ W1 ⊆ · · · ⊆ Wn | Wi ∈ Gr (n) , virt. dim Wi = i − n, λWn = W0 }. Here, Gr (n) is a certain subspace of the Grassmannian of all linear subspaces of the Hilbert space H = L2 (S 1 , Cn ) = λi C n , i∈Z

and λWn denotes the result of applying the linear “multiplication operator” λ (of H ) to Wn . The virtual dimension is defined by virt. dim W = dim(W ∩H− )−dim(W ⊥ ∩H+ ), where H+ = ⊕i≥0 λi Cn , H− = ⊕i 1, except on a periodic orbit {x0 , T x0 , . . . , T q−1 x0 } of period q, and also that |det(DT q (x0 ))| = 1. We call such a periodic orbit indifferent. Let λ denote the normalized Lebesgue measure on X. Although T fails to be uniformly expanding, we can still show, under some regularity conditions, the existence of an invariant probability measure µ equivalent to λ which is exact (see §1). Given functions f, g ∈ L2 (X, µ) we define the correlation function by ρ(n) = f ◦ T n gdµ − f dµ gdµ. The authors would like to thank the Japanese Ministry of Science, Culture and Sport for their support.

504

M. Pollicott, M. Yuri

We call the measure µ strong mixing if ρ(n) → 0, as n → +∞. Let us assume that f ∈ L∞ (X, µ), i.e., f is a bounded function. In this paper we shall study the rate at which ρ(n) → 0, for a family of maps with indifferent periodic points. We shall consider both extensions of earlier one dimensional results to higher dimensions and also to a broader class of functions g. More specifically, we establish for (T , µ) bounds on the convergence of iterates of the transfer operator in the L1 -norm (Theorem 1), and thus bounds on the correlation functions (Corollary), relative to a class of functions F (defined in Sect. 5) which contains all piecewise Lipschitz functions. In Sect. 6, we apply our results to the following representative examples. Example 1 (Inhomogeneous diophantine algorithm [7], [10–14]). Let X = {(x, y) : 0 ≤ y ≤ 1, −1 ≤ x < −y + 1} and define T : X → X by T (x, y) =

y y 1 1−y y − + − ,− − − , x x x x x

where [·] denotes the integer part. There is an absolutely continuous invariant measure µ and for any 0 < δ < 1 we may bound: 1 |ρ(n)| = O n− 2 +δ , whenever g ∈ L2 (X, µ) ∩ F; and |ρ(n)| = O n−1+δ , whenever g ∈ L∞ (X, µ) ∩ F. Example 2 (Manneville–Pomeau equation). Let 0 < α < 1 and define Sα : [0, 1) → [0, 1) by Sα (x) = {x + x 1+α } where {·} gives the fractional part. There is an absolutely continuous invariant measure µ and for any 0 < δ < 1 we may bound:

1 −min 21 + 2α −δ,1 |ρ(n)| = O n , whenever g ∈ L2 (X, µ) ∩ F; and

−min α1 −1 −δ,1 |ρ(n)| = O n , whenever g ∈ L∞ (X, µ) ∩ F.

(0.1)

One should compare the estimates in (0.1) with those of Isola and Liverani–Saussol– Vienti and Hu. Both Isola and Hu showed that for Lipshitz maps f, g ∈ L∞ one can estimate 1 (0.2) |ρ(n)| = O n− α +1 Liverani, Saussol and Vienti [6] presented an interesting alternative proof, which results in only a slightly weaker estimate, i.e., for f ∈ L∞ (X, µ) and g ∈ C 1 we can estimate 1 1 |ρ(n)| = O (log n) α n− α +1

(0.3)

In particular, observe that (0.1) is slightly weaker than both (0.2) and (0.3) although, as we shall see, it holds under quite modest assumptions on g.

Statistical Properties of Maps with Indifferent Periodic Points

505

1. The Existence of Exact Invariant Measures In this section we recall sufficient conditions for the existence of an absolutely continuous invariant measure µ which is exact. This provides us with the opportunity to introduce notation and hypotheses for later sections. As in the introduction, we let T : X → X denote a piecewise C 1 -invertible Bernoulli map on X ⊂ Rd . We suppose that |det(DT (x))| > 1 except on the indifferent periodic orbit {x0 , T x0 , . . . , T q−1 x0 } of period q. Given i = i1 . . . in , we let Xi = Xi1 ∩ T −1 Xi2 ∩ . . . ∩ T −(n−1) Xin , denote a typical element, or cylinder, of A(n) .We denote the local inverse to T n : Xi → X by ψi = (T n |Xi )−1 : X → Xi .

Definition. We say that Xi satisfies the Renyi condition (relative to a constant C > 1) if we have the distortion estimate 1 ≤ sup

x,y∈Xi

|det(DT n (x))| ≤ C, |det(DT n (y))|

where n = |i|. Let R(C, T ) be the set of those cylinders which satisfy the Renyi condition. Notation. Let R : X → R ∪ {∞} be the stopping time with respect to R(C, T ), i.e., R(x) = inf{n ∈ N : Xi1 ...in (x) ∈ R(C, T )} where Xi1 ...in (x) denotes the cylinder of length n containing x. For each n ≥ 0 define Dn = {x ∈ X : R(x) > n}. For each n > 0 we then define Bn = Dn−1 \Dn . Given two strings i and j we let ij denote the concatenation. If |i| = n and |j | = m then |ij | = n + m. For the two examples presented in the introduction, it is known that there is a T invariant exact probability measure equivalent to lebesgue measure. The following result gives a criteria for more general maps to have an exact T -invariant probability measure equivalent to the lebesgue measure λ. Proposition 1 (cf. [10, 11]). If there exists C > 1 such that: (C-1) R(C, T ) = ∅, and for any Xj ∈ R(C, T ) and any string i we have Xij ∈ R(C, T ); and (C-2) ∞ n=1 λ(Dn ) < +∞, then there exists an invariant probability measure µ equivalent to λ which is exact. Moreover, the indifferent periodic points are singular points of the density h := dµ dλ and are contained in ∩∞ n=1 Dn . Remark. Property (C-1) is sometimes referred to as the strong playback condition. Property (C-2) corresponds to the integrability of the stopping time to R(C, T ).

506


2. Transfer and Integral Operators In this section we shall consider the Perron–Frobenius type operators Pλ : L1 (X, λ) → L1 (X, λ) and Pµ : L1 (X, µ) → L1 (X, µ) defined by, respectively, Pλ f (x) =

|detDψi (x)|f (ψi x), for f ∈ L1 (X, µ),

i

and Pµ f (x) =

h(ψi x) i

h(x)

|detDψi (x)|f (ψi x), for f ∈ L1 (X, µ).

We shall write the operators Pλn and Pµn as integral operators. Towards this end, we introduce the following definitions. Definition. Define for n > m ≥ 1 (n,m)

Kλ

(x, z) =

Pλn [χXi ](x)

χXi (z) λ(Xi )

n−m |detDψi (x)| Pλ = χXi (z), λ(Xi ) |i|=m

|i|=m

and Kµ(n,m) (x, z) =

Pµn [χXi ](x)

χXi (z) µ(Xi )

n−m |detDψi |hψi Pµ (x)χXi (z), = µ(Xi )h |i|=m

|i|=m

where χXi (x) is the characteristic function for Xi . The following lemma relates Pλn and Pµn to integral operators. Lemma 1. Pλn Eλ (f |A(m) )(x)

=

and Pµn Eµ (f |A(m) )(x)

(n,m)

X

=

X

f (z)Kλ

(x, z)dλ(z)

f (z)Kµ(n,m) (x, z)dµ(z),

−i where A(m) = ∨m−1 i=0 T A.

Proof. This is a direct computation.

Remark. Each of these operators has a role to play in our approach. In Sect. 3, we shall (n,m) associated to the intergral operator Pλn by require lower bounds on the kernel Kλ choosing an appropriate second parameter n = n(m) with n(m) > m for each m ≥ 1.


507

(n,m)

3. Lower Bounds of the Kernel Kλ

Standing assumption. We shall assume in this note that ∩∞ n=1 Dn consists of a single indifferent periodic orbit {x0 , T x0 , . . . , T q−1 x0 }. Moreover, each point of the orbit is (i) contained in a unique element of the partition A. Let Dn denote the unique cylinder of i length n containing T x0 . The following conditions will prove useful in the sequel. (i)

(Neighbour property) ∃0 < π1 < 1 such that ∀Xi1 ...im ⊂ Dm , ∀m ≥ 1, inf y∈D1 |det(Dψi1 ...im (y))| ≥ π1 . supy∈B1 |det(Dψi1 ...im (y))| (i)

(ii) (Polynomial decay of cylinder measure near indifferent fixed point) Let Bn+1 = (i)

(i)

Dn \Dn+1 . Then λ(B1 )(i) > 0 and ∃p1 , p2 > 0, ∃l > 2 such that p1 n−l ≤ (i)

λ(Bn+1 ) ≤ p2 n−l (∀n ≥ 1). (iii) (Expansion on non-Renyi cylinders) ∃, > 0 such that for Xi ⊂ D1 inf |det(Dψi )(x)| ≥ ,.

x∈X

Lemma 2. Under the hypotheses of Proposition 1 and (i)–(iii) above ∃0 < γ1 < 1 and (n,m) (x, z) ≥ γ1 > 0, ∀x, z ∈ X. ∃0 < n0 < ∞ such that ∀n ≥ n0 m we have Kλ The following technical result is important in the proof of Lemma 2. Sublemma (Well ordered neighbour convention). If µ is exact, then ∃0 < π2 < 1, ∃0 < n0 < ∞ such that ∀n ≥ n0 m we have the lower bound λ(Xi1 ...im ∩ T −n B1 ) ≥ π2 > 0, λ(Xi1 ...im ∩ T −n D1 ) where Xi1 ...im ⊂ Dm , m ≥ 1. (i)

Proof of Sublemma. Let Dm = Xi1 ...im , for some 1 ≤ i ≤ q, then we can write  

  (i) λ(Dm ∩ T −n B1 ) = λ(Xi1 ...im j1 ...jn−m ∩ T −n B1 )  j2 ...jn−m

(i)

j1 :Xi1 ...im j1 ⊂Bm+1

 +  j2 ...jn−m





 λ(Xi1 ...im j1 ...jn−m ∩ T −n B1 ) . (i)

j1 :Xi1 ...im j1 =Dm+1

It follows from Renyi’s condition that the first term on the right-hand side is bounded (i) from below by C −1 j2 ...jn−m λ(Xj2 ...jn−m ∩ T −(n−m−1) B1 )λ(Bm+1 ). Hence we have (i)

(i) ∩ T −n B1 ) ≥ C −1 λ(T −(n−m−1) B1 )λ(Bm+1 ) λ(Dm  



 + λ(Xi1 ...im j1 ...jn−m ∩ T −n B1 ) .  j2 ...jn−m

(i)

j1 :Xi1 ...im j1 =Dm+1

508


Repeating this procedure allows one to establish the following inequality. (i) (i) ∩ T −n B1 ) ≥ C −1 inf λ(T −k B1 )(λ(Bm+1 ) + . . . + λ(Bn(i) )) λ(Dm k≥1 (i) = C −1 inf λ(T −k B1 )(λ(Dm ) − λ(Dn(i) )) . k≥1

By Condition (ii) we know that λ(B1 ) > 0. Clearly, inf k≥1 λ(T −k B1 ) > 0 by exactness of µ ∼ λ. Therefore, we have the lower bound (i)

λ(Dm ∩ T −n B1 ) (i)

λ(Dm ∩ T −n D1 )

≥

(i)

λ(Dm ∩ T −n B1 ) (i)

λ(D ) m

≥ C −1 inf λ(T −k B1 ) 1 − k≥1

(i)

λ(Dn )

(i)

λ(Dm )

.

(l)

By Condition (ii) we have that there exists p2 > p1 > 0 such that p1 ≤ λ(Dn )n(l−1) ≤ (i) )l−1 ), for n0 > 0 sufficiently large we may choose 0 < p2 . Since λ(Dn(i) ) = O ( m n λ(Dm ) (i) π2 < 1 such that ∀n ≥ n0 m we have C −1 inf k≥1 λ(T −k B1 )(1 − λ(Dn(i) ) ) ≥ π2 . λ(Dm )

Proof of Lemma 2. Consider a cylinder Xi1 ...im ⊂ Dm . We first note that Pλn−m (| det Dψi1 ...im |)(x) =

=





j1 ...jn−m−1

Xjn−m ⊂B1

+

 

j1 ...jn−m−1

| det Dψi1 ...im j1 ...jn−m (x)|

j1 ...jn−m



| det Dψi1 ...im j1 ...jn−m (x)|  | det Dψi1 ...im j1 ...jn−m−1 (ψjn−m x)|| det Dψjn−m (x)| .

Xjn−m ⊂D1

By Property (C-1) of R(C, T ) the first term of the right-hand side is bounded from below by

C −1

j1 ...jn−m−1

 

Xjn−m ⊂B1

 λ(Xi1 ...im j1 ...jn−m ) = C −1 λ(Xi1 ...im ∩ T −(n−1) B1 ).


509

By Hypothesis (i) and (iii), the second term on the right-hand side is bounded from below by  

 | det Dψi1 ...im j1 ...jn−m−1 (ψjn−m x)|| det Dψjn−m (x)| j1 ...jn−m−1

Xjn−m ⊂D1

≥

j1 ...jn−m−1

≥

π1 , C

π1 C

sup | det Dψi1 ...im j1 ...jn−m−1 | , B1

B1

| det Dψi1 ...im j1 ...jn−m−1 (y)|dλ(y) λ(B1 )

j1 ...jn−m−1

π1 , λ(Xi1 ...im ∩ T −(n−1) B1 ) Cλ(B1 ) π1 π2 , ≥ λ(Xi1 ...im ∩ T −(n−1) D1 ). Cλ(B1 ) =

Since X = B1 ∪ D1 is a disjoint union, combining the above observations gives π1 π2 , n−m | det Dψi1 ...im | (x) ≥ . Pλ λ(Xi1 ...im ) C More generally, if Xi1 ...im satisfies Xi1 ...il ∈ R(C, T ) and Xil+1 ...im ⊂ Dm−l , for some 0 ≤ l ≤ m then n−m | det Dψi1 ...im | −1 n−m | det Dψil+1 ...im | Pλ (x) ≥ C Pλ (x) λ(Xi1 ...im ) λ(Xil+1 ...im ) π1 π2 , ≥ , C2 by the previous inequality. Finally, setting γ1 := C −2 π1 π2 , gives the desired lower bound. (n,m)

4. Lower Bounds Involving the Kernel Kµ

We first recall the formula of T -invariant measure µ, µ(E) =

∞

µ∗ (Dn ∩ T −n E),

n=0

µ∗

is an invariant measure for the jump transformation T ∗ defined by T ∗ (x) = where R(x) T (x), for which density dµ∗ /dλ satisfies G−1 ≤ dµ∗ /dλ ≤ G, for some 1 < G < ∞. Lemma 3. Assume that (iv) ∃C˜ > 1 such that ∀k > 0 and any Xi ∈ A(n) , for any n ≥ 1, det(Dψi (x)) ≤ C. ˜ sup x,y∈Bk det(Dψi (y))

510


Then there exists 0 < H < ∞ such that for all n ≥ 1 inf x∈Dn h(x) ≥ H. (i) µ(Dn ) max1≤i≤q (i) λ(Dn )

Proof. We first claim that ∃1 < R < ∞ such that R −1 n ≤ h|Bn ≤ Rn.

(∗)

To see this, we first need to observe that Dn ∩T −n Bm ⊆ Bn+m ∪(Bn+1 ∩T −(n+1) Bm−1 ), for m ≥ 2, and Dn ∩ T −n B1 ⊂ Bn+1 . More precisely, assume that x ∈ Dn ∩ T −n Bm is contained in Xi1 ...in ...in+m then we can assume either Xi1 ...in ⊂ Dn , Xin+1 ...in+m−1 ⊂ Dm−1 and Xin+m ⊂ B1 or Xi1 ...in ⊂ Dn , Xin+1 ...in+m−1 ⊂ Dm−1 and Xin+m ⊂ D1 . Using assumption (C-1) we see that if Xin in+1 ⊂ D2 then Xi1 ...in+m−1 ⊂ Dn+m−1 . Similarly, if Xin in+1 ⊂ B2 then Xi1 ...in+m−1 ⊂ Bn+1 ∩ T −(n+1) Dm−2 . Hence, we have Xi1 ...in+m−1 ⊂ Dn+m−1 ∪ (Bn+1 ∩ T −(n+1) Dm−2 ). Furthermore, if Xin+m ⊂ B1 then Xi1 ...in+m ⊂ Bn+m , and if Xin+m ⊂ D1 then Xi1 ...in+m ⊂ Bn+1 ∩ T −(n+1) Bm−1 . Finally we have x ∈ Bn+m ∪ (Bn+1 ∩ T −(n+1) Bm−1 ). We have the bounds µ(Bm ) ≤ G

∞

λ(Dn ∩ T −n Bm )

n=0

≤G ≤G

∞

n=0 ∞

n=0

λ(Bn+m ) +

∞

λ(Bn+1 ∩ T

n=0 ∞

λ(Bn+m ) + C

−(n+1)

Bm−1 )

λ(Bn+1 )λ(Bm−1 )

n=0

= G (λ(Dm−1 ) + Cλ(Bm−1 )), and, consequently

µ(Bm ) λ(Dm ) ≤ G × (constant) × , λ(Bm ) λ(Bm ) where we use condition (ii) to write λ(Dm−2 )/λ(Dm ) ≤ Constant. The above inequalities then allow us to apply the argument in the proof of Theorem 3.1 in [14] to establish (*). Observe that since Dn = ∪∞ k=n+1 Bk , the lower bound in (*) implies inf x∈Dn h(x) supB h

≥ R −1 (n + 1). By Lemma 3.3 in [14], there exists H > 0 such that inf B k h ≤ H . We k can estimate (i) (i) (i) k≥n+1 (supB (i) h)λ(Bk ) µ(Dn ) k≥n+1 µ(Bk ) k ≤ = (i) (i) (i) λ(Dn ) k≥n+1 λ(Bk ) k≥n+1 λ(Bk ) (i) k≥n+1 (inf B (i) h)λ(Bk ) k ≤H , (i) k≥n+1 λ(Bk )


511

which we see to be bounded by Const × n using (*) and the bounds p1 n−l ≤ λ(Bn ) ≤ p2 n−l (assumed in (ii)). This allows us to establish the inequality in the statement. Lemma 4. Under the hypotheses of Proposition 1 and Assumptions (i)–(iv) there exists 0 < γ < 1 such that ∀n ≥ n0 m, h(x)Kµ(n,m) (x, z) ≥ γ > 0,

∀x, z ∈ X.

Proof of Lemma 4. First we observe that | det Dψi |h ◦ ψi (n−m) | det Dψi |h ◦ ψi h(x)Pµ(n−m) (x) = Pλ (x). µ(Xi )h µ(Xi ) We can bound the right-hand side of this identity by (n−m) | det Dψi |h ◦ ψi (n−m) | det Dψi |h ◦ ψi (x) (x) = Pλ Pλ µ(Xi ) Xi hdλ (n−m) | det Dψi | inf y∈X h ◦ ψi (y) ≥ Pλ (x) Xi hdλ (n−m) | det Dψi | inf y∈Xi h(y) ≥ Pλ (x). λ(Xi ) supy∈Xi h(y) c = ∪m B . Then by Lemma 3.3 from [14], we know that Suppose Xi ⊂ Dm j =1 j

inf y∈Xi h(y) supy∈Xi h(y)

≥

1 H

and so we can bound 1 (n−m) | det Dψi | (n−m) | det Dψi |h ◦ ψi (x) ≥ Pλ (x). Pλ µ(Xi ) H λ(Xi ) c = ∪m B , we have Hence for z ∈ Xi ⊂ Dm j =1 j (n,m) (n−m) | det Dψi |h ◦ ψi (x)h(x) h(x)Kµ (x, z) = Pµ µ(Xi )h (n−m) | det Dψi |h ◦ ψi = Pλ (x) µ(Xi ) 1 (n,m) 1 ≥ Kλ (x, z) ≥ γ1 . H H

For z ∈ Xi ⊂ Dm , h ◦ ψi (n−m) | det Dψi | (n−m) | det Dψi | inf y∈Xi h(y) (x) ≥ Pλ (x) Pλ λ(Xi ) µ(Xi )/λ(Xi ) λ(Xi ) µ(Xi )/λ(Xi ) (n−m) | det Dψi | ≥ H Pλ (x), λ(Xi )

512


by Lemma 3. In particular, we have that (n,m)

h(x)Kµ(n−m) (x, z) ≥ H Kλ

(x, z) ≥ H γ1 .

Since we can see that H ≤ 1 and H ≥ 1 we have that h(x)Kµ(n−m) (x, z) ≥

H (n−m) H Kλ (x, z) ≥ γ1 . H H

5. The Main Theorem First we give an upper bound on the rate of convergence of the iterates of the transfer operator in the L1 norm. We want to introduce the following technical bounds. The first is a quantification of the behaviour near the indifferent fixed point. (1) ∀m > 0 we denote

31 (m) := sup

sup

c ,|i|=m j Xi ⊂Dm

|detDψj (x)| sup 1 − |detDψj (y)| x,y∈Xi

and require that 31 (m) → 0 as m → +∞. The next condition describes the regularity of the density. (2) ∀m > 0 we denote

32 (m) :=

sup

c ,|i|=m Xi ⊂Dm

h(x) . sup 1 − h(y) x,y∈Xi

The Conditions (1) and (2) are readily verified for each of our examples. The regularity assumption which we actually need to impose on the function f is given by the following weak Lipschitz-type condition. Definition. Denote by F those f : X → R, for which there exists Lf > 0 such that sup

sup |f (x) − f (y)| ≤ Lf σ (m),

c x,y∈X Xi ⊂Dm i |i|=m

∀m ≥ 0

where σ (m) = sup{Diam(Xi ) : |i| = m}, i.e., a bound on the diameter of all cylinders of length m. Remark. To illustrate that this condition is milder than the usual Lipschitz condition, we can consider the Manneville Pomeau equation with T (x) = x + x 1+α . In this case, if we consider the (unbounded) function f (x) = x1γ , for any 0 < γ < α then we readily check that f ∈ F. We now come to the key theorem of this paper. Theorem 1. Let T : X → X be a piecewise C 1 -invertible Bernoulli map satisfying the assumptions of Proposition 1. Suppose that: (a) T satisfies Conditions (i)–(iv) and Conditions (1)–(2) ; and


(b) f ∈ L2 (X, µ) ∩ F with

513

f dµ = 0.

Then for any 0 < 4 < 1, ||Pµn f ||1 ≤ O max µ(D[n4 ] )1/2 , σ ([n4 ]), 31 ([n4 ]), 32 ([n4 ]) . Moreover, if f ∈ L∞ (X, µ) ∩ F with

f dµ = 0, then for any 0 < 4 < 1,

||Pµn f ||1 ≤ O max µ(D[n4 ] ), σ ([n4 ]), 31 ([n4 ]), 32 ([n4 ]) . Before giving the proof of Theorem 1, we observe the following corollary. Corollary. For f ∈ L∞ (X, µ) and g ∈ L2 (X, µ) ∩ F and any 0 < 4 < 1 we can bound |ρ(n)| ≤ O max µ(D[n4 ] )1/2 , σ ([n4 ]), 31 ([n4 ]), 32 ([n4 ]) . Moreover, for g ∈ L∞ (X, µ) ∩ F and any 0 < 4 < 1 we can bound |ρ(n)| ≤ O max µ(D[n4 ] ), σ ([n4 ]), 31 ([n4 ]), 32 ([n4 ]) .

Proof of Corollary. This follows immediately from the bound |ρ(n)| = | f Pµn gdµ| ≤ ||f ||∞ ||Pµn (g)||1 .

The proof of Theorem 1 will be based on the following two lemmas. Lemma 5. For f ∈ L1 (X, µ) with f dµ = 0, we have ||Pµn ◦ E(f |A(m) )||1 ≤ (1 − γ )||f ||1 , for all n ≥ n0 m and for all m ≥ 1. Proof. Using Lemma 1 we can write ||Pµn Eµ (f |A(m) )||1 = || f (z)Kµ(n,m) (x, z)dµ(z)||1 ≤2

X

−2

f (z) X

{x

:

Pµn Eµ (f |A(m) )≥0}

f (z)dµ(z)

Kµ(n,m) (x, z)h(x)dλ(x)

{x

: Pµn Eµ (f |A(m) )≥0}

dµ(z)

(5.1)

γ dλ(x) ,

(the last term being trivially zero) where X Eµ (f |A(m) )dµ(z) = X f (z)dµ(z) = 0 allows us to write n (m) ||Pµ Eµ (f |A )||1 = 2 | f (z)Kµ(n,m) (x, z)|dµ(x)dµ(z). {x : Pµn Eµ (f |A(m) )≥0}

514


Furthermore, the right hand side of (5.1) can be bounded by

2

Kµ(n,m) (x, z)h(x) − γ

dλ(x) dµ(z) : Pµn Eµ (f |A(m) )≥0} ≤2 Kµ (x, z)h(x) − γ dλ(x) dµ(z) f (z) {z : f ≥0} {x : Pµn Eµ (f |A(m) )≥0} X

f (z)

{x

≤ (1 − γ )||f ||1 , where we use that {x

:

Pµn Eµ (f |A(n) )≥0}

Kµ (x, z)h(x) − γ dλ(x) ≤

X

Kµ (x, z)h(x)dλ(x) − γ = 1 − γ .

This completes the poof of Lemma 5. Lemma 6. If f ∈ F with f dµ = 0 then for l ≥ 0 ||Pµl f − Eµ (Pµl f |A(m) )||1 ≤ O ||f ||1 max 3i (m) i=1,2

+ Lf σ (l + m) + ||f ||2 (µ(Dm ))1/2 . If, in addition, f ∈ L∞ (X, µ) then l l (m) ||Pµ f −Eµ (Pµ f |A )||1 ≤ O ||f ||1 max 3i (m) +Lf σ (l+m)+||f ||∞ (µ(Dm )) . i=1,2

Proof. For j with |j | = l we write 5j (x) = triangle inequality to bound ||Pµl f − Eµ (Pµl f |A(m) )||1 ≤



χXi (y) c Xi ⊂Dm

X

µ(Xi )



Xi

χXi (y) c Xi ⊂Dm

X

µ(Xi )

χXi (y)

Xi ⊂Dm X |i|=l

µ(Xi )

 

Xi

|i|=m

+

We can use the

 |f (ψj x)|.|5j (x) − 5j (y)| dµ(x)dµ(y)

|j |=l

|i|=m

+

h(ψj x) h(x) | det Dψj (x)|.

Xi

 |f (ψj x) − f (ψj y)|.|5j (y)| dµ(x)dµ(y)

|j |=l

|Pµl f (x)| + |Pµl f (y)| dµ(x)dµ(y). (5.2)


515

We can estimate the first term in (5.2) using conditions (1) and (2) as follows:  

χXi (y)  |f (ψj x)|.|5j (x) − 5j (y)| dµ(x)dµ(y) X µ(Xi ) Xi c Xi ⊂Dm |i|=m

|j |=l

≤ O maxi=1,2 3i (m) χXi (x)Pµl |f |(x)dµ(x) X c Xi ⊂Dm |i|=m

χT −l Xi (x)|f (x)|dµ(x) ≤ O maxi=1,2 3i (m) X c Xi ⊂Dm |i|=m

≤ O maxi=1,2 3i (m) ||f ||1 .

Using that f ∈ F and property (C-1) we can bound the second term in (5.2) by  

χXi (y)  |f (ψj x) − f (ψj y)|.|5j (y)| dµ(x)dµ(y) X µ(Xi ) Xi c |j |=l

Xi ⊂Dm |i|=m





    µ(T −l Xi ) ≤ Lf σ (l + m)    c Xi ⊂Dm |i|=m

≤ Lf σ (l + m). Finally, we can bound the third term in (5.1) by

χXi (y) Xi ⊂Dm X |i|=m

≤

µ(Xi )

Xi

Xi ⊂Dm |i|=m

|Pµl f (x)| + |Pµl f (y)| dµ(x)dµ(y)

X

χXi (x)Pµl |f |(x)dµ(x)

χXi (x) + dµ(x). Pµl |f |(y).χXi (y)dµ(y) X µ(Xi )

|f |dµ ≤ 2||f ||2 (µ(Dm ))1/2 ≤2 −l Xi ⊂Dm T Xi |i|=m

by Cauchy–Schwartz inequality. For the second part of the statement, we observe that we do not require the Cauchy– Schwartz inequality, since if f ∈ L∞ (X, µ) the upper bound of the final term immediately gives 2||f ||∞ µ(Dm ).

516


Proof of Theorem. Given 0 < 4 < 1, we denote m(N ) := [N 4 ], n(N ) := n0 m(N ) and k(N) := [N 1−4 /n0 ]. We can estimate ||Pµn(N)k(N) f ||1

k ≤ ||Pµn(N)k(N) f − Pµn(N) Eµ (·|A(m(N)) ) f ||1 k(N) + || Pµn(N) Eµ (·|A(m(N)) ) f ||1 ≤

k−1 j

|| Pµn(N) Eµ (·|A(m(N)) ) Pµn(N) (I − Eµ (·|A(m(N)) ))Pµn(N)(k−j −1) f ||1 j =0

+ (1 − γ )k(N) ||f ||1 k(N)−1

1/2 ≤ O max µ(Dm(N) ) , σ (m(N )), max 3i (m(N )) (1 − γ )j i=1,2

+ (1 − γ )

k(N)

j =0

||f ||1 ,

k(N) n(N) where || Pµ Eµ (·|A(m(N)) ) f ||1 is bounded using Lemma 5 and the contribun(N)(k−j −1)

tion ||(I − Eµ (·|A(m(N)) ))Pµ

f ||1 is bounded by Lemma 6. Therefore,

||PµN f ||1 ≤ O max µ(D[N 4 ] )1/2 , σ ([N 4 ]), maxi=1,2 3i ([N 4 ]) , where we can neglect the term (1−γ )k(N) ||f ||1 since it has stretched exponential decay, which is faster than polynomial. This completes the proof of Theorem 1.

6. Examples of Maps with Indifferent Period Points Example 3 (One parameter families of maps on the interval 0, 1]). Given 0 < α < 1, we can define a map Tα : [0, 1] → [0, 1] as follows: Tα (x) =

x (1−x α )1/α x (1/2)1/α −1

for x ∈ X0 := [0, (1/2)1/α ) +

1 1−(1/2)1/α

for x ∈ X1 := [(1/2)1/α , 1].

The map Tα was studied in [10–14], and the special case Tα with α = 1 first appeared in Lasota–Yorke’s paper [5]. It is closely related to the well-known Manneville–Pomeau transformation Sα : [0, 1) → [0, 1) defined by Sα (x) = x + x 1+α . Both of the maps Tα and Sα have an indifferent fixed point at x = 0, and have similar dynamical properties. In particular, they both show similar intermittent behaviour and for either of them we can take Dk = [0, ak ] and Bk+1 = ak+1 , ak , where ak is the smallest preimage of 1 under k th iteration. For Tα we have the exact equality ak = k −1/α , wheras for Sα we have only the estimate ak ∼ k −1/α [4]. Thus Condition (ii) holds with l = 1 + 1/α.


(-1,1)

517

(0,1) X

X

( -2-1 21 )

( -3-2 )

X -2 -1

( ) X

X

( -3-1 )

( 32 ) X

2

( 1)

X

2 -2

( 1 -1 )

X 3 1

( )

(0,0)

(1,0)

Fig. 1.

−1 We denote ψ0 = Tα |X0 . Then we can easily verify for l ≥ 1 x ψ0 := ψ0 ◦ . . . ◦ ψ0 (x) = , and ! " (1 + lx α )1/α ×l

ψ0 (x) =

1 . (1 + lx α )1+1/α

The above identities allows us to verify (i) and (iv) for Tα directly. Finally, (iii) holds for Tα by direct observation. Properties (1) and (2) were established for this map in [12]. The corresponding properties for Sα can be similarly verified. Example 4 (The inhomogeneous diophantine algorithm [10–14]). For the transformation T : X → X, defined in the introduction, the points (1, 0) and (−1,#1) aare indifferent periodic points with period 2, We can introduce an index set I = : a, b ∈ Z, b $ a > b > 0, or a < b < 0 and a partition % & X a : ab ∈ I b

as shown in Fig. 1. Dk consists of the two cylinders X 2 −2 2 −2 ... and X −2 2 −2 2 ... −1 1 −1 ! 1 ... " 1 −1 1!−1 ... "

k

k

y

Let us denote a(x, y) = − x and an (x, y) = a(T n (x, y)) then we can define inductively qn = an qn−1 +qn−2 , starting from q0 = 1 and q1 = 0. We can now compute 1−y x

1 |detDψ 2 −2 2 −2 ... (x, y)| = . |q2m + q2m−1 x|3 1 −1 1!−1 ... " 2m

518


Moreover, an easy calculation gives q2m = (−1)m (2m + 1), q2m+1 = (−1)m (2m + 2) and |q2m + q2m−1 x| = |2m(−1)m−1 (x − 1) + (−1)m | = |2m(x − 1) − 1| = 1 + 2m(1 − x) and so

1 |detDψ 2 −2 2 −2 ... (x, y)| = . |1 + 2m(1 − x)|3 1 −1 1 −1 ... ! " 2m

Similar observations finally allow us to establish 1 |detDψ 2 −2 2 −2 ... (x, y)| = . |1 + (2m + 1)(1 + x)|3 1 −1 1!−1 ... " 2m+1

1 . |detDψ −2 2 −2 2 ... (x, y)| = |1 + 2m(1 + x)|3 −1 1 −1 ! 1 ... " 2m

1 |detDψ −2 2 −2 2 ... (x, y)| = . |1 + (2m + 1)(1 − x)|3 −1 1 −1 ! 1 ... " 2m+1

By using the above equalities we can verify (i) and (iv) directly. Property (ii) was verified in [12, pp. 1454–1455], with l = 3. The Property (iii) is obvious on the two components of D1 by inspection. (1) and (2) were verified in [12]. Results on the decay of the function |ρ(n)| can be used to deduce rates of convergence in the Birkhoff ergodic theorem. We recall the following result from the survey of Kachurovskii [2, Theorem 16] in the case the case f = g: If ρ(n) = O(1/nβ ), for some 0 < β < 1, then n−1

1 f (T i x) − n i=0

f dµ = O

(log n)1/2 (log log n)1/2+4 nβ/2

, µ − a.e. for any 4 > 0.

In particular, we conclude the following. Proposition 2. For the inhomogenous diophantine algorithm, and any 0 < δ, we have: 1 n−1 1 i 2 (i) n i=0 f (T x) − f dµ = O 1 −δ , ∀f ∈ L (X, µ) ∩ F; n4 1 i ∞ (ii) n1 n−1 1 −δ , ∀f ∈ L (X, µ) ∩ F; i=0 f (T x) − f dµ = O n2

For the Manneville–Pomeau equation, and any 0 < δ we have

1 −δ, 21 −min 41 + 4α 1 n−1 i , ∀f ∈ L2 (X, µ) ∩ F (i) n i=0 f (T x) − f dµ = O n

1 1 1 i x) − f dµ = O n−min 2α − 2α −δ, 2 (ii) n1 n−1 , ∀f ∈ L∞ (X, µ) ∩ F, f (T i=0


519

7. An Application to Inhomogeneous Diophantine Approximations Example 2 is particularly interesting in light of its close connection with approximation of irrational numbers. Given an irrational number α, a homogeneous diophantine approximation is one of the infinitely many rational numbers pq which satisfies |q|. |qα − p| < z,

(7.1)

where z > 0 is a constant. A well-known approach to generating solutions # $is to study the continued fraction transformation T : [0, 1] → [0, 1] by T (x) = x1 . This preserves the Gauss mea1 sure dµ = h(x)dx with density h(x) = (log 2)(1+x) The natural extension of this transformation is denoted by T¯ : [0, 1] × [0, 1] → [0, 1] × [0, 1] and given by 1 T¯ (x, y) = (T x, [1/x]+y ).) If we write T¯ n (x, 0) = (T n x, qn−1 qn ), then we can define th θn (x) = qn |qn x − pn |. Here qn is the denominator of the n convergent of x. Given z > 0, the condition θn (x) ≤ z corresponds to % & xy T¯ n (x, 0) ∈ B(z) = (x, y) : ≤z . 1 + xy Consider a second irrational number β which is rationally independent of α and 1. An inhomogeneous diophantine approximation is a rational number pq which satisfies |q|. |qα + β − p| < z.

(7.2)

By a result of Minkowski [1], providing z > 41 there are infinitely many solutions to (7.2). The inhomogeneous transformation T plays a role in this problem akin to that of the continued fraction transformation for homogeneous diophantine approximations. Let Y = {(z, w) ∈ R2 : 0 ≤ w < 1, 0 ≤ w − z < 1}. We shall consider the following natural extension T¯ of T defined on X × Y by  y y y 1−y 1 T (x, y, z, w) =  − + − ,− − − , x x x x x

z+

1

1−y x

 w − − yx  , − − yx z + 1−y − − yx x

We also use µ to denote the extension of the T -invariant probability measure to X×Y . We write T¯ n (α, β, 0, 0) = (αn , βn , γn , δn ), n ≥ 0. Following Schweiger[8, pp. 202–203] we can associate a sequence of pairs of integers (un , vn ) satisfying β δ n n θn (α, β) := |un |.|un α + β − vn | = . 1 + α n γn Schweiger observes [8, Remark 24.3.12] that there exists a limit distribution g(z) such that 1 Card{1 ≤ k ≤ n : θk (α, β) ≤ z} → g(z), as n → +∞ (7.3) n

520


for almost all (α, β). Given z > 0 the condition θn (α, β) ≤ z corresponds to yv T¯ n (α, β, 0, 0) ∈ B(z) = {(x, y, u, v) : 1+xu ≤ z}. [n/2] [n/2] ¯ We can approximate B(z) by a union Bn of elements from A¯ −[n/2] = ∨i=−[n/2] T¯ −i A, where A = {Xa }a∈I is the canonical partition for X (cf Section 4) and A¯ = {Xa ×Y }a∈I is the corresponding partition for X × Y . [n/2] We choose the smallest collection of elements (or cylinders) from A¯ −[n/2] which cover B(z). Let us call the union of this collection Q0 , say. Next we choose the largest [n/2] collection of cylinders Q1 from A¯ −[n/2] which are contained in B(z). [n/2] (or equivalently on the There are bounds on the measure of elements of A¯ −[n/2]

[n/2] ¯ of the type O( 1 ). measure of elements of ∨i=−[n/2] T¯ −i A) n µ(∪C∈Q0 −Q1 C) = O(supC∈A¯ [n/2] µ(C)) tends to zero at the specific rate O( n1 ). −[n/2] Let us denote η([n/2]) := µ(B(z)$Bn ) then we claim that η([n/2]) = O n−1 . In particular, we can then estimate

ρ(n) = µ(χBn ◦ T n χBn ) + η([n/2]) = µ(χT −[n/2] Bn ◦ T [n/2] χT −[n/2] Bn ) + η([n/2]). = O n−1+δ By Proposition 2 we get an error term in the Birkhoff ergodic theorem of 0 n−1/2+δ . Finally, we see that |T¯ n (x, y, 0, 0) − T¯ n (x, y, z, w)| = O n−1+δ with the result that 1 1 , Card{1 ≤ k ≤ n : θk (α, β) ≤ z} = g(z) + O 1 n n 2 −δ for any 0 < δ
Uq so(N ), where we show that a frame has to transform under the action of the quantum op group Uq so(N ) with opposite coalgebra. We also find a dual set of inner derivations by decomposing in the frame basis the formal “Dirac operator”, which had already been found [10, 35] previously. It would be interesting, but requires some nontrivial

Geometrical Tools for Quantum Euclidean Spaces

523

handwork, to compare our results with the ones of Ref. [1, 2]. There multiparametric deformations of the inhomogeneous SOq (N ) quantum groups are considered, whereby multiparametric deformations (including as a particular case the one-parameter one at hand) of the Euclidean space are obtained by projection. The frame of the quantum group in the Woronowicz bicovariant differential calculi sense, i.e. the left- (or right-) invariant 1-forms, might also be projected and compared to ours. Then in Sect. 5 we show that it is possible to find homomorphisms ϕ ± : AN > ± Uq so(N ) → AN which act trivially on the factor AN on the left-hand side and which project the components of the frame and of the inner derivations from elements of AN > Uq so(N ) onto elements in AN . This implies that in the x i basis they satisfy the “RLL” and the “gLL” relations fulfilled by the L± [19] generators of Uq so(N ). In the case that N is odd it is possible to “glue” the homomorphisms together to an isomorphism from the whole of AN > Uq so(N ) to AN , an interesting and surprising result in itself. Finally in Sect. 6 we see that for each of the two calculi there is essentially a unique metric, and two torsion-free SOq (N )-covariant linear connections which are compatible with it up to a conformal factor. 2. The Cartan Formalism In this section we briefly review a noncommutative extension [13] of the moving-frame formalism of E. Cartan. We start with a formal noncommutative associative algebra A with a differential calculus ∗ (A). If A has a commutative limit and if this limit is the algebra of functions on a manifold M then we suppose that the limit of the differential calculus is the ordinary ∗ (AN )> Uq so(N )de Rham differential calculus on M. We shall concentrate on the case where the module of the 1-forms 1 (A) is free of rank N as a left or right module and admits a special basis {θ a }1≤a≤N , referred to as “frame” or “Stehbein”, which commutes with the elements of A: [f, θ a ] = 0.

(2.1)

This means that if the limit manifold exists it must be parallelizable. The integer N plays the role of the dimension of the manifold. We suppose further that the basis θ a is dual to a set of inner derivations ea = ad λa such that: df = ea f θ a = [λa , f ]θ a

(2.2)

for any f ∈ A. The formal “Dirac operator” [12], defined by the equation df = −[θ, f ],

(2.3)

θ = −λa θ a .

(2.4)

is then given by

We shall consider only the case where the center Z(A) of A is trivial: Z(A) = C. If the original algebra does not have a trivial center then we shall extend it an algebra which does. The (wedge) product π in ∗ (A) can be defined by relations of the form θ a θ b = P ab cd θ c ⊗ θ d

(2.5)

524

B.L. Cerchiai, G. Fiore, J. Madore

for suitable P ab cd ∈ Z(A) = C. It can be shown that consistency with the nilpotency of d requires that the λa satisfy a quadratic relation of the form 2λc λd P cd ab − λc F c ab − Kab = 0.

(2.6)

The coefficients of the linear and constant terms must also belong to the center. In the cases which interest us here they vanish. Notice that Eq. (2.6) has the form of the structure equation of a Lie algebra with a central extension. We define [17] the metric as a non-degenerate A-bilinear map g : 1 (A) ⊗A 1 (A) → A.

(2.7)

This means that it can be completely determined up to central elements once its action on a basis of 1-forms is assigned. For example set g(θ a ⊗ θ b ) = g ab .

(2.8)

The bilinearity implies that f g ab = g(f θ a ⊗ θ b ) = g(θ a ⊗ θ b f ) = g ab f and therefore g ab ∈ Z(A) = C: {θ a } is a special basis of 1-forms in which the coefficients of the metric are central elements, namely complex numbers in our assumptions. This is the property characterizing frames (vielbein) in ordinary geometry, and is at the origin of the name “frame” for this basis also in noncommutative geometry. To define a covariant derivative D which satisfies [17] a left and right Leibniz rule we introduce a “generalized flip”, an A-bilinear map σ : 1 (A) ⊗A 1 (A) → 1 (A) ⊗A 1 (A).

(2.9)

The flip is also completely determined once its action on a basis of 1-forms is assigned. For example set σ (θ a ⊗ θ b ) = S ab cd θ c ⊗ θ d .

(2.10)

As above, bilinearity implies that S ab cd ∈ Z(A) = C. Using the flip a left and right Leibniz rule can be written: D(f ξ ) = df ⊗ ξ + f Dξ, D(ξf ) = σ (ξ ⊗ df ) + (Dξ )f.

(2.11) (2.12)

& : 1 (A) → 2 (A)

(2.13)

& = d − π ◦ D.

(2.14)

The torsion map

is defined by

We shall assume that σ satisfies the condition π ◦ (σ + 1) = 0

(2.15)


525

in order that the torsion be bilinear. The usual torsion 2-form &a is defined as &a = dθ a − π ◦ Dθ a . It is easy to check [17] that if on the right-hand side of Eq. (2.6) the term linear in λa and the constant term vanish then a torsion-free covariant derivative can be defined by Dξ = −θ ⊗ ξ + σ (ξ ⊗ θ),

(2.16)

for any ξ ∈ 1 (A). The most general torsion-free D for fixed σ is of the form D = D(0) + χ

(2.17)

where χ is an arbitrary A-bimodule morphism χ

1 (A) −→ 1 (A) ⊗ 1 (A)

(2.18)

π ◦ χ = 0.

(2.19)

fulfilling

The compatibility of a covariant derivative with the metric is expressed by the condition [18] g23 ◦ D2 = d ◦ g.

(2.20)

For the covariant derivative (2.16) this condition can be written as the equation S ae df g fg S bc eg = g ab δdc

(2.21)

if one uses the coefficients of the flip with respect to the frame. Introduce the standard notation σ12 = σ ⊗ id, σ23 = id ⊗ σ , to extend to three factors of a module any operator σ defined on a tensor product of two factors. There is a natural continuation of the map (2.9) to the tensor product 1 (A) ⊗A 1 (A) given by the map D2 (ξ ⊗ η) = Dξ ⊗ η + σ12 (ξ ⊗ Dη).

(2.22)

We define formally the curvature as Curv ≡ D 2 = π12 ◦ D2 ◦ D.

(2.23)

We recover the standard definition of the frame components R a bcd of the curvature tensor from the decomposition 1 Curv(θ a ) = − R a bcd θ c θ d ⊗ θ b . 2

(2.24)

One can easily show [23] that the curvature associated to (2.16) is given by Curv(ξ ) = ξa θ 2 ⊗ θ a + π12 σ12 σ23 σ12 (ξ ⊗ θ ⊗ θ).

(2.25)

The algebra we shall consider is a ∗-algebra. We shall require that the involution ∗ be extendable to the algebra of differential forms in such a way that (ξ η)∗ = (−1)pq η∗ ξ ∗ ,

ξ ∈ p (A),

η ∈ q (A).

(2.26)

526


We recall that the elements of the algebra are considered as 0-forms. One would like to have a differential fulfilling the reality condition (df )∗ = df ∗

(2.27)

as in the commutative case. Neither of the two differential calculi we shall introduce in Sect. 3 satisfies this condition; the differential calculus ∗ (A) is mapped by ∗ into ¯ ∗ (A). As a consequence, the reality conditions on the covariant derivative a new one and curvature formulated in [24] cannot be satisfied. However we shall still suppose [18] that the extension of the involution to the tensor product is given by (ξ ⊗ η)∗ = σ (η∗ ⊗ ξ ∗ ).

(2.28)

A change in σ therefore implies a change in the definition of an hermitian tensor. The reality condition for the metric will be, as in [24], g ◦ σ (η∗ ⊗ ξ ∗ ) = (g(ξ ⊗ η))∗ .

(2.29)

We shall also continue to assume that σ satisfies the braid equation σ12 σ23 σ12 = σ23 σ12 σ23 ,

(2.30)

a condition implied [24] by the reality condition on the covariant derivative and the curvature. At the end of Sect. 6 we shall briefly consider the question how to modify reality condition on the covariant derivative and the curvature in the present case. 3. The Quantum Euclidean Spaces and Their q-Deformed Differential Calculi The starting point for the definition of the N -dimensional quantum Euclidean space RN q is the braid matrix Rˆ for SOq (N, C) a N 2 × N 2 matrix, whose explicit expression we give in Appendix 7.1. Certain properties of Rˆ which we shall use follow immediately from the definition. First, it fulfills the braid equation Rˆ 12 Rˆ 23 Rˆ 12 = Rˆ 23 Rˆ 12 Rˆ 23 .

(3.1)

ˆ Here we have used again the conventional tensor notation Rˆ 12 = Rˆ ⊗ id, Rˆ 23 = id ⊗ R. By repeated application of Eq. (3.1) one finds f (Rˆ 12 ) Rˆ 23 Rˆ 12 = Rˆ 23 Rˆ 12 f (Rˆ 23 )

(3.2)

for any polynomial function f (t) in one variable. Equations (3.1) and (3.2) are evidently satisfied also after the replacement Rˆ → Rˆ −1 . Second, Rˆ is invariant under transposition of the indices: ij Rˆ kl = Rˆ ijkl .

(3.3)

Here and in the sequel we use indices with values N −1 for N odd, 2 N with n ≡ for N even 2

i = −n, . . . , −1, 0, 1, . . . n, with n ≡ i = −n, . . . , −1, 1, . . . n,

(3.4)


527

ij and n to denote the rank of SO(N, C). The matrix element Rˆ kl vanishes unless the indices satisfy the following condition:

either i = −j or i = −j The R-matrix, defined by

and k = i, l = j or l = i, k = j and k = −l. ij

(3.5)

ji

Rkl = Rˆ kl ,

is lower-triangular. ˆ There exists also [19] a projector decomposition of R: Rˆ = qPs − q −1 Pa + q 1−N Pt .

(3.6)

The Ps , Pa , Pt are SOq (N )-covariant q-deformations of the symmetric trace-free, antisymmetric and trace projectors respectively and satisfy the equations Pµ = 1, µ, ν = s, a, t. (3.7) Pµ Pν = Pµ δµν , µ

The Pt projects on a one-dimensional sub-space and therefore it can be written in the form ij

Pt kl = (g sm gsm )−1 g ij gkl =

k g ij gkl , ωn (q −ρn +1 − q ρn −1 )

(3.8)

where gij is the N × N matrix gij = q −ρi δi,−j .

(3.9)

We have here introduced the notation (n − 21 , . . . , 21 , 0, − 21 , . . . , 21 − n) for N odd, ρi = (n − 1, . . . , 0, 0, . . . , 1 − n) for N even and we have set

k ≡ q − q −1 ,

ωi ≡ q ρi + q −ρi .

The matrix gij is a SOq (N )-isotropic tensor and is a deformation of the ordinary Euclidean metric in a set of coordinates pairwise conjugated to each other under complex conjugation. It is easily verified that its inverse g ij is given by g ij = gij .

(3.10)

The metric and the braid matrix satisfy the relations [19] ˆ ∓1hl gil Rˆ ±1lh j k = R ij glk ,

jk ij g il Rˆ ±1 lh = Rˆ ∓1 hl g lk .

(3.11)

The N -dimensional quantum Euclidean space is the associative algebra RN q generated i by elements {x }i=−n,... ,n with relations ij

Pa kl x k x l = 0.

(3.12)

528


These relations are preserved by the (right) action of the quantum group Uq so(N ), which is defined on the generators by x i g = ρji (g)x j ,

ρji (g) ∈ C,

(3.13)

where ρ is the N-dimensional vector representation of Uq so(N ), and extended to the rest of RN q so that the latter becomes a Uq so(N ) module algebra. That is, for arbitrary g, g ∈ Uq so(N ) and a, a ∈ RN q , we have a (gg ) = (a g) g

(3.14)

(aa ) g = (a g(1) ) (a g(2) ).

(3.15)

Here we have used Sweedler notation (with lower indices) for the coproduct, = 6(g) I ⊗ g(1) ⊗g(2) ; the right-hand side is actually a short-hand notation for a finite sum I g(1) I . g(2) Relations (3.12) can be written more explicitly in the form [30] x i x j = qx j x i i

−i

[x , x ] = [x , x 1

−1

for i < j, i = −j,

−1 2 kωi−1 ri−1

for i > 1, for N even, for N odd.

0 ]= hr02

(3.16)

We have here introduced h defined by h ≡ q 2 − q− 2 , 1

1

(3.17)

and we have defined as well a sequence of numbers ri , r by ri2 =

i k,l=−i

gkl x k x l ,

r 2 ≡ rn2

(3.18)

where i ≥ 0 for N odd, whereas for N even i ≥ 1 and of course in the sum only k, l = 0 actually occur. The element r 2 is SOq (N )-invariant and generates the center of the algebra RN q . It can be easily checked that  2 j  for |j | ≤ i, ri x j 2 2 2 j x ri = q ri x (3.19) for j < −i,  q −2 r 2 x j for j > i. i As this will be necessary for the construction of the elements λa be introduced in 2 Sect. 4, we now extend the algebra RN q by adding the square root ri of ri for i = 0 . . . n as well as the inverses ri−1 of these elements. As the relations (3.19) contain only q ±2 it is consistent to set for i ≥ 0  j  for |j | ≤ i, ri x j x ri = qri x j (3.20) for j < −i,  q −1 r x j for j > i. i


529

We shall be mainly interested in the case q ∈ R+ . In this case a conjugation (x i )∗ = x j gj i

(3.21)

can be defined on RN q to obtain what is known as real quantum Euclidean space. The elements ri are then real. There are [3] two differential calculi which are covariant with respect to Uq so(N ), obtained by imposing the condition α ∈ ∗ (RN q )

(dα) g = d(α g)

(3.22)

on the differential. We denote the two exterior derivatives by d and d¯ and the correspondi i ¯ i ¯∗ N ¯i ing exterior algebras by ∗ (RN q ) and (Rq ). If we introduce ξ = dx and ξ = dx , then they are characterized respectively by ij

x i ξ j = q Rˆ kl ξ k x l , ij

x i ξ¯ j = q −1 Rˆ −1 kl ξ¯ k x l .

(3.23) (3.24)

¯∗ N For q ∈ R+ neither ∗ (RN q ) nor (Rq ) possesses an involution. However, one can ¯1 N introduce a ∗-structure on the direct sum 1 (RN q ) ⊕ (Rq ) by setting (ξ i )∗ = ξ¯ j gj i .

(3.25)

ˆ Using the properties (3.11, 3.3) of the R-matrix one sees that the two calculi are conjugate; Eqs. (3.23) and (3.24) are exchanged. By taking the differential of (3.23) and (3.24) the ξ ξ -commutation relations are determined ij

ij

Ps kl ξ k ξ l = 0,

Pt kl ξ k ξ l = 0,

Ps kl ξ¯ k ξ¯ l = 0,

Pt kl ξ¯ k ξ¯ l = 0.

ij

ij

¯∗ N These relations define the algebraic structure of ∗ (RN q ) and (Rq ). It is useful to introduce a set of gradings degi , i = 1, . . . n on ∗ (RN q ) by   if i = j, 1 j j degi (ξ ) = degi (x ) = −1 if i = −j,  0 otherwise.

(3.26)

(3.27)

ˆ All these gradings are preserved by the commutation relations (3.12), since the Rmatrix, and therefore any polynomial function of it like Pa , fulfills (3.5). The n-ple (deg1 , . . . , degn ) coincides with the weight vector of the fundamental vector representation of so(N ). The Dirac operator [12], defined by Eq. (2.3), ξ i = −[θ, x i ]

(3.28)

is easily verified to be given by N

θ = ωn q 2 k −1 r −2 gij x i ξ j ,

(3.29)

530


¯ ∗ (RN ¯ as pointed out in [10, 35]. For the barred calculus q ) the “Dirac operator” θ (2.3) is N

θ¯ = −ωn q − 2 k −1 r −2 gij x i ξ¯ j .

(3.30)

¯ θ ∗ = −θ.

(3.31)

If q ∈ R+ it satisfies

In order to construct the λa and θ a satisfying the conditions described in Sect. 2 we first must solve the following problem. In Sect. 2 we assumed the center of the algebra A to be trivial, which makes possible the construction of elements λa and θ a with the features described there. But the algebra generated by the x i and rj has a nontrivial center. With a general Ansatz of the type θ a = θia ξ i

(3.32)

the condition [θ a , rn2 ] = 0 can be rewritten as (rn2 θia − q −2 θia rn2 )ξ i = 0,

(3.33)

which has no solution since rn2 ∈ Z(RN q ). To find a solution to (3.33) we further enlarge the algebra by adding a unitary element , the “dilatator”, which satisfies the commutation relations x i = q x i .

(3.34)

We also add its inverse −1 . In the case N odd we can now follow the scheme previously proposed for N = 3 [23] but in the case of even N the situation is slightly 1 more complicated. We have added the elements r1±1 = (x −1 x 1 )± 2 and as a consequence the center is non trivial even after addition of . The elements ± 1 2 r1−1 x ±1 = x 1 (x −1 )−1 commute also with . (We recall that x −1 )−1 is the inverse of the element x i with i = −1.) In other words, since the algebra generated by ( , r1±1 , x ±1 , . . . x ±n ) is completely symmetric in the exchange of x 1 and x −1 , there is no way to distinguish between these two elements. To have N linearly independent θ a , instead of fewer, we shall need to add yet another element to the algebra. We choose to add a “Drinfeld–Jimbo” generator H1

K = q 2 and its inverse K −1 , where H1 belongs to the Cartan subalgebra of Uq so(N ) and represents the component of the angular momentum in the (−1, 1)-plane. This new element satisfies the commutation relations Kx ±1 = q ±1 x ±1 K, Kx ±i = x ±i K,

for i > 1,

(3.35)

as well as K = K.

(3.36)


531

When q ∈ R+ it is compatible with the commutation relations to extend the ∗-structure (3.21) , K as ∗ = −1 ,

K ∗ = K.

(3.37)

We must decide now which commutation relations , K should satisfy with the ξ i . As already observed [23] there are different possibilities. A first possibility is to set [30] ξ i = ξ i ,

d = qd .

(3.38)

This choice has the disadvantage that cannot be considered as an element of the quantum space, because due to (3.38) it does not satisfy the Leibniz rule d(fg) = f dg + (df )g ∀ f, g ∈ RN q . Nevertheless, it can be interpreted in a consistent way as an element of the Heisenberg algebra, because −2 can be constructed [30] as a simple polynomial in the coordinates and derivatives. Alternatively, what was considered also in [2], one could ask the Leibniz rule d(fg) = f dg + (df )g to hold also if f = . By differentiating (3.34) one obtains that ξ i + x i d = qd x i + q ξ i .

(3.39)

A solution would be to require that x i (d ) = q(d )x i ,

ξ i = q ξ i .

(3.40)

In particular it would then be possible to set d = d, which implies that (d ) = 0. This choice is not completely satisfactory either since we would like the relation df = 0 implies f ∝ 1

(3.41)

to hold, and this would not be the case if d = 0. As a consequence the general formalism is still not strictly applicable and there will be a conformal ambiguity in the choice of metric. We shall see below that with a procedure similar to the one described previously [23] for N = 3, we would recover R × S N−1 as geometry rather than RN in the commutative limit. Therefore, in the sequel we shall impose the first condition (3.38). As will be shown in the next section, this allows us to normalize the θ a and λa in such a way as to obtain RN as geometry in the commutative limit. The above discussion with can be repeated to determine the commutation relations between K and the 1-forms ξ i . We choose dK = 0. Then consistency with (3.35) requires that Kξ ±1 = q ±1 ξ ±1 K, Kξ i = ξ i K, for i > 1,

(3.42)

To summarize, we shall consider the algebra AN , an extension of RN q defined for odd N as AN = {x i , rj , rj−1 , , −1 : −n ≤ i ≤ n, 0 ≤ j < n} with generators which satisfy the relations (3.12), (3.20), (3.34) and for even N as AN = {x i , rj , rj−1 , , −1 , K, K −1 : −n ≤ i ≤ n, 1 ≤ j < n} with generators which satisfy the relations (3.12), (3.20), (3.34), (3.35). The algebra of differential forms ∗ (AN ) is generated by the one-forms ξ i satisfying relations (3.23), (3.26), (3.38) when N is odd, and (3.23), (3.26), (3.38), (3.42) when N is even. However one must bear in mind that the additional elements and K are rather exceptional since dK = 0 and either d = 0, or it does not satisfy the Leibniz rule. These elements would be better interpreted as elements of the Heisenberg algebra.

532


4. Inner Derivations and Frame We would like to construct a frame θ a and the associated inner derivations ea = ad λa satisfying the conditions in Sect. 2 for the case of the algebra AN . We first solve this problem in a larger algebra, which we now define. It is possible to extend ∗ (AN ) to the cross-product algebra ∗ (AN )> Uq so(N ) by postulating the cross-commutation relations ξg = g(1) (ξ g(2) )

(4.1)

for any g ∈ Uq so(N ) and ξ ∈ ∗ (AN ). The algebra ∗ (AN )> Uq so(N ) can be made into a module algebra under the action of Uq so(N ) by extending the latter on the elements of Uq so(N ) as the adjoint action, h g = Sg(1) hg(2) ,

g, h ∈ Uq so(N ).

The S here denotes the antipode of Uq so(N ). op Let us introduce Uq so(N ) the Hopf algebra with the same algebra structure of Uq so(N ), but opposite coalgebra, and by 6op (g) = g(2) ⊗ g(1) its coproduct. On any op

op

module algebra M of Uq so(N ) the corresponding action will thus fulfill the relations op

op

op

a (gg ) = (a g) g ,

op

op

op

(aa ) g = (a g(2) ) (a g(1) ).

(4.2) (4.3)

These are to be compared with (3.14) and (3.15). It is immediate to show that definition (4.1) implies that one can realize the action in the “adjoint-like way” η g = Sg(1) η g(2)

(4.4)

on all of ∗ (AN )> Uq so(N ). On the other hand, one can realize also a corresponding op

action by op

η g = (S −1 g(2) ) η g(1) ,

(4.5)

where S −1 is the antipode of 6op . We return now to the problem of the construction of a frame and of a set of dual inner derivations for the differential calculus (∗ (AN ), d). As a first step, we must find N independent solutions ϑ a to the equation [f, ϑ a ] = 0

∀ f ∈ AN .

(4.6)

We shall look first for solutions ϑ a in ∗ (AN )> Uq so(N ). The reason is the following. For each solution ϑ a of (4.6) and for any g ∈ Uq so(N ) we can consider the image ϑg a ∈ ∗ (AN )> Uq so(N ) of ϑ a , defined by op

ϑg a := ϑ a g = (S −1 g(2) ) ϑ a g(1) .

(4.7)


533

Now we show that its commutator with any element f ∈ AN vanishes: [f, ϑg a ] = [f, (S −1 g)(1) ϑ a S(S −1 g)(2) ] = f (S −1 g)(1) ϑ a S(S −1 g)(2) − (S −1 g)(1) ϑ a S(S −1 g)(2) f (4.1)

= (S −1 g)(1) [f (S −1 g)(2) ]ϑ a S(S −1 g)(3) − (S −1 g)(1) ϑ a [f (S −1 g)(2) ] S(S −1 g)(3)

= (S −1 g)(1) f (S −1 g)(2) , ϑ a S(S −1 g)(3) = 0.

In the last equality we have used the fact that f (S −1 g)(2) ∈ AN and (4.6). In other words ϑg a also behaves as a frame element. Moreover by its very definition ϑg a will in general belong to 1 (AN )> Uq so(N ) even if ϑ a ∈ 1 (AN ) (unless the n-plet {ϑ a } op builds an irreducible representation of Uq so(N )). We can summarize the above results in the Proposition 1. The subspace F of 1 (AN )> Uq so(N ) spanned by frame elements op carries a representation of Uq so(N ). We shall call a set {ϑ a }a=−n,...,n a generalized frame if ϑ a are elements of 1 (AN )> Uq so(N ) fulfilling (4.6), and any ξ ∈ 1 (AN )> Uq so(N ) can be uniquely decomposed in the form ϑ a ξa with ξa ∈ AN > Uq so(N ). We now look for a generalized frame in the form of a basis of an irreducible N op dimensional representation of Uq so(N ). Recall that the universal R-matrix R is a special element R ≡ R (1) ⊗ R (2) ∈ Uq so(N ) ⊗ Uq so(N )

(4.8)

intertwining between 6 and 6op , and so does also R −1 21 : R 6(·) = 6op (·)R ,

−1 op R −1 21 6(·) = 6 (·)R 21 .

(4.9)

In (4.8) we have used a Sweedler notation with upper indices: the right-hand side is a (1) (2) short-hand notation for a sum I R I ⊗ R I of infinitely many terms. The other main properties of R are recalled in the appendix 7.2. Proposition 2. Let (u5 )−1 = R (1) (SR (2) ), (u7 )−1 = R −1(2) (SR −1(1) ). The elements of 1 (AN )> Uq so(N ) (2) SR ξ a R (1) ϑ a := α u−1 7 SR −1(1) ξ a R −1(2) ϑ¯ a := α¯ u−1 5

(4.10) (4.11)

are covariant under the action (4.5), more precisely op

ϑ a g = ρba (g)ϑ b

op

ϑ¯ a g = ρba (g)ϑ¯ b .

(4.12)

In the previous definitions we have inserted two scalar factors α, α¯ to be fixed later.

534


Proof. We prove the first formula. We find op (4.5) (4.10) (2) ϑ a g = (S −1 g(2) )ϑ a g(1) = (S −1 g(2) )α u−1 SR ξ a R (1) g(1) 7 (7.11) (2) ξ a R (1) g(1) = α u−1 7 (Sg(2) ) SR (4.9) SR (2) (Sg(1) )ξ a g(2) R (1) = α u−1 7 (3.13) a (4.10) (2) = ρb (g)α u−1 SR ξ b R (1) = ρba (g)ϑ b . 7 The proof of the second formula is completely analogous.

We can give an alternative and very useful expression for ϑ a , ϑ¯ a . It is convenient to introduce the Faddeev–Reshetikin–Taktadjan generators [19] of Uq so(N ): L+al := R (1) ρla (R (2) )

L−al := ρla (R −1(1) )R −1(2) .

(4.13)

In our conventions R ∈ Uq+ so(N ) ⊗ Uq− so(N )

(4.14)

where Uq+ so(N ), Uq− so(N ) denote the positive and negative Borel subalgebras; hence we see that L+al ∈ Uq+ so(N ) and L−al ∈ Uq− so(N ). From formulae (7.8), (7.9) in the Appendix 7.2 one finds that the coproducts are given by 6(L+ij ) = L+ih ⊗ L+hj ,

6(L−ij ) = L−ih ⊗ L−hj .

(4.15)

Their commutation relations will be given in following section. Using them and (4.1) one easily proves the Proposition 3. Let u3 = R (2) S −1 R (1) , u4 = R −1(1) S −1 R −1(2) . l ϑ a = αL−al ηl = αηm ρm (u4 )L−al l ¯ +al ηl = αη ¯ m ρm (u3 )L+al , ϑ¯ a = αL

(4.16) (4.17)

with ηi = ξ i or ηi = ξ¯ i . Thus ϑ a and ϑ¯ a belong to 1 (AN )> Uq− so(N ) and ¯ 1 (AN )> Uq− so(N ) and 1 (AN )> Uq+ so(N ) respectively for ηi = ξ i , or 1 + i i ¯ (AN )> Uq so(N ) respectively for η = ξ¯ . We now show that (ϑ a , ϑ¯ a ) constitute a frame. Recall that the braid matrix can be written ij j as Rˆ = (ρ ⊗ ρ i )R . Using (4.1), (4.15) one can easily prove hk

h

k

Lemma 1. x i L±ab = L±ac x j Rˆ ±1ci jb,

(4.18)

ξ i L±ab = L±ac ξ j Rˆ ±1ci jb,

(4.19)

L±ac ξ¯ j Rˆ ±1ci jb.

(4.20)

¯i

ξ

L±ab

=


535

We should note that the commutation relations we have just found are different from (in a sense opposite to) the ones which we would have found by imposing, instead of (4.1), the condition gξ = g(1)

ξg(2) ,

(4.21)

with a left action . This is true also in the commutative case (q = 1), and in a sense is unpleasant since the latter definition of the commutation relations is what we are usually more familiar to. For instance, if h is a primitive generator in the Cartan subalgebra then [h, x i ] defined in the first way is opposite to the one defined in the second; or if g + is a positive root, then [g + , x i ] defined in the first way is proportional (up to a Cartan subalgebra factor) to the commutator [g − , x i ] defined in the second way, where g − is the negative root opposite to g + . The reason why we have adopted (4.1) is that this is necessary in order that the coordinates x i carry upper indices (as it is conventional in general relativity) and that the representation ρ defined by (3.13) can be considered as the fundamental (vector) j one, rather than its contragradient. This follows from ρhi (gg ) = ρji (g)ρh (g ). Had we used lower indices to label the coordinates, or replaced ρ(g) with ρ(Sg), we could have adopted (4.21). Proposition 4. Assume ξ i , ξ¯ i are the differentials (3.23), (3.24), of the previous section. If we choose α¯ = α −1 =

(4.22)

then {ϑ a }, {ϑ¯ a } are generalized frames respectively in 1 (AN )> Uq so(N ) and ¯ 1 (AN )> Uq so(N ). Proof. We prove this in the first case. (4.16)

(4.19)

(3.23)

(4.19)

j −1 l x i ϑ a = x i L−al −1 ξ l = L−ac Rˆ −1ci = L−ac −1 ξ c x i = ϑ a x i . jlx ξ

The proof in the second case is completely analogous.

In the commutative limit {ξ i }i=1,...,n is a frame and all other frames can be obtained from it by a G ⊂ U so(N ) transformation. Formulae (4.16) and (4.17) say that in the noncommutative case one frame can be obtained from {ξ i } by a very particular Uq so(N ) transformation, the one with matrix elements L±ab . Note that by the choice (4.22) ϑ a , ϑ¯ a commute not only with the coordinates x i , rj , but also with the special element , [ , ϑ a ] = 0

[ , ϑ¯ a ] = 0,

(4.23)

under the assumption (3.38); had we assumed instead (3.40), then the same would be true by choosing α = −1 r, α¯ = r. On the other hand, if we choose α¯ = f¯(r) , α = f (r) −1 (with f, f¯ =const), then ϑ a , ϑ¯ a will still commute with the coordinates x i , rj , but in general not with . The commutation relations between the frame elements are given by the

536


Proposition 5. The commutation relations among the ϑ a (resp. ϑ¯ a ) are as the ones among the ξ i (resp. ξ¯ i ), except for the opposite products: cd b a b a Ps cd ab ϑ ϑ = 0, Pt ab ϑ ϑ = 0 cd ¯ b ¯ a ¯b ¯a Ps cd ab ϑ ϑ = 0, Pt ab ϑ ϑ = 0.

(4.24)

Proof. We prove the claim in the first case (in the second the proof is completely analogous). For both P = Ps , Pt cd b a (4.16) ab −1 −b l −1 −a m (4.19) −2 cd −b −a k ˆ −1hl m Pab ϑ ϑ = Pcd L l ξ L m ξ = Pab L l L h ξ R mk ξ (3.6)

(5.6)

(3.26)

−d −c −2 l n mh −d −c = −2 ξ l ξ n (P Rˆ −1 )mh ln L h L m ∝ ξ ξ Pln L h L m = 0.

¯ in terms both of differentials ξ i = dx i and We now decompose d (and similarly d) Uq so(N )-covariant derivatives ∂i and of frame elements ϑ a and derivations @a : d = ∂i ξ i = @a ϑ a

d = ∂¯i ξ¯ i = @¯a ϑ¯ a .

(4.25)

∂¯i = @¯a L+ai .

(4.26)

Looking at (4.16), (4.17) we find that ∂i = @a L−ai −1

Now assume that the “Dirac operator” exists; it must necessarily be Uq so(N )invariant. We can decompose it as well both in the basis of differentials and in the basis of frame elements: θ = −wi ξ i = −ya ϑ a ,

θ¯ = −w¯ i ξ¯ i = −y¯a ϑ¯ a ,

(4.27)

with wi , w¯ i ∈ AN and ya ∈ AN > Uq− so(N ), y¯a ∈ AN > Uq+ so(N ); in this case @a = [ya , ·]

@¯a = [y¯a , ·].

(4.28)

The commutation relations among the wi , w¯ i will be of the form Pa hk ij wk wh = 0,

Pa hk ¯ k w¯ h = 0. ij w

(4.29)

¯ θ and θ¯ it follows that From the Uq so(N ) invariance of d, d, µi g = µl ρil (Sg),

µi = wi , w¯ i , ∂i , ∂¯i .

(4.30)

From (4.27), (4.16), (4.17) we find ya = wi SL−ia ,

(4.31)

y¯a =

(4.32)

w¯ i SL+ia −1 .

op ¯ θ, θ¯ we find the transformation rules From the Uq so(N )-invariance of d, d, op

νa g = νb ρab (S −1 g),

νa = ya , y¯a , @a , @¯a .

(4.33)


537

Proposition 6. The commutation relations among the ya and among the y¯a are Pa cd ab yc yd = 0,

Pa cd ab y¯c y¯d = 0.

(4.34)

Proof. Pa cd ab yc yd

j

∝

−i − 2 Pa cd ab wi (SL c )wj (SL d )

=

−i −h − 2 Pa cd ab wi (wj L h )(SL c )(SL d )

j

j

k −i −h − 2 Pa cd ab wi wk ρj (SL h )(SL c )(SL d ) j ki 2 = wi wk Rˆ hj S L− d L−hc Pa cd ab (5.6),(4.13) hj ki −l 2 = wi wk Rˆ hj S Pa lm L−m L a b −m −l 2 (4.29) ∝ wi wk Pa ki = 0. lm S L b L a

=

The proof is completely analogous for y¯a .

From ξ i = [x i , θ ] = [wj ξ j , x i ], ξ¯ i = [x i , θ¯ ] = [w¯ j ξ¯ j , x i ] it follows ji x i wk = Rˆ −1 hk wj x h − δhi

ji ¯ h − δhi −1 x i w¯ k −1 = Rˆ hk w¯ j x

using these relations and (4.1), (4.15) one can easily prove Proposition 7. j

j

[ya , x j ] = SL− a ,

[y¯a , x j ] = −1 SL+ a .

(4.35)

These formulae can be seen as the inverse of (4.31), (4.32), since they give SL+aj , SL−aj in terms of ya , y¯a . In next section we find algebra homomorphisms ϕ + : AN > Uq+ so(N ) → AN , −

ϕ :

AN > Uq− so(N )

→ AN ,

(4.36) (4.37)

defined as the identity on AN , ϕ ± (a) = a

if a ∈ AN .

(4.38)

Then the 1-forms θ a = θla ξ l , θ¯ a = θ¯la ξ¯ l ,

θla := ϕ − ( −1 L−al ), θ¯la := ϕ + ( L+al )

(4.39) (4.40)

¯ 1 (AN ) and still fulfil the property (4.6), in other words build up a belong to 1 (AN ), frame. Similarly the elements of AN defined by λa = ϕ − (ya ),

λ¯ a = ϕ + (y¯a )

(4.41)

e¯a = [λ¯ a , ·].

(4.42)

will yield the dual inner derivations, ea = [λa , ·],

538


It is clear that since Z(AN ) = C then the frame and the dual set of inner derivations are uniquely determined up to a linear transformation (with numerical coefficients). By definition, the λa , λ¯ a will still fulfill the commutation relations (4.34), ¯ ¯ P(a) ab cd λa λb = 0.

P(a) ab cd λa λb = 0,

(4.43)

After application of ϕ ± Eqs. (4.26) become ∂¯i = e¯a ϕ + (L+ai ) .

∂i = ea ϕ − (L−ai ) −1 ,

(4.44)

Their interest lies in the fact that they relate the SOq (N )-covariant derivatives ∂i , ∂¯i [3, 31], introduced following the approach of Woronowicz and Wess–Zumino [37–39, 36], to the inner derivations ea , e¯a dual to the frame and which are defined using ordinary commutation relations, following the approach of Connes [11, 12]. On the other hand, ϕ + , ϕ − cannot be extended to homomorphisms of ∗ (AN )> ± Uq so(N ) → ∗ (AN ), since the commutation relations between the elements of Uq so(N ) and the 1-forms do not map into the commutations of the elements of AN with the 1-forms; this is immediate to check if we use the frame basis: the frame elements don’t commute with Uq± so(N ), but do commute with ϕ + (Uq+ so(N )), ϕ − (Uq− so(N )) ⊂ AN . As a consequence, the commutation relations among the θa , θ¯a differ from (4.24) by a reversal of the product order. That is, Proposition 8. The commutation relations among the θ a (resp. θ¯ a ) are as the ones among the ξ i (resp. ξ¯ i ): a b Ps cd ab θ θ = 0, ¯a ¯b Ps cd ab θ θ = 0,

a b Pt cd ab θ θ = 0 ¯a ¯b Pt cd ab θ θ = 0

(4.45)

Proof. We prove the claim in the first case. For both P = Ps , Pt we have (4.39)

(4.6)

cd a b cd a b n cd b a n Pab θ θ = Pab θ θn ξ = Pab θn θ ξ (4.39)

cd −2 − −b − −a l n cd −b −a l n = Pab ϕ (L n )ϕ (L l )ξ ξ ∝ −2 ϕ − (Pab L n L l )ξ ξ

(5.6)

(3.26)

ab l n = −2 ϕ − (L−db L−ca Pln )ξ ξ = 0

The proof is completely analogous for y¯a .

The θ a , θ¯ a , λa , λ¯ a do not inherit from ϑ a , ϑ¯ a , ya , y¯a the transformation properties op op (4.12), (4.33) under the action of Uq so(N ). For the θ a , θ¯ a this is clear since the op commutation relations (4.45) are no longer compatible with the action of Uq so(N ). As a consequence, this is true also for the λa , λ¯ a , because θ = −θ a λa , θ¯ = −θ¯ a λ¯ a are invariant. op op One could in principle define an Uq so(N ) action on AN by postulating instead op

op

λ¯ a g = λ¯ b ρab (S −1 g);

λa g = λb ρab (S −1 g) op

but the latter would differ from the action fulfilling (4.5). We shall therefore not do so.


539

5. Homomorphism AN > Uq so(N ) → AN It is known [19] that a set of generators of Uq so(N ) is provided by {L+ij , L−ij } and some further elements obtained by introducing square roots and inverses of the elements L±ii , which is always possible because they are invertible elements belonging to the Cartan subalgebra. The set {L+ij , L−ij } has 2N 2 elements, but N (N − 1) of them vanish due to the upper and lower triangularity of the matrices L± , whereas their diagonal elements are the inverses of each other: L+ij = 0,

if i > j

(5.1)

L−ij = 0, L+ii L−ii = 1, ±n L±−n −n . . . .L n

if i < j

(5.2)

∀i

(5.3)

= 1.

(5.4)

To these relations we have to add the relations characterizing Uq so(N ), j

L±ij L±hk g kj = g hi

L± i L±kh gkj = ghi ,

(5.5)

which further reduce to N (N − 1)/2 the number of independent generators, and finally the commutation relations dc ab +d +c Rˆ cd L f L e = L+bc L+ad Rˆ ef

(5.6)

ab −d −c dc Rˆ cd L f L e = L−bc L−ad Rˆ ef

(5.7)

ab Rˆ cd

(5.8)

L+df L−ce

=

L−bc L+ad

dc . Rˆ ef

Note that (5.4) yields no constraint for N even since it is a consequence of (5.5), whereas for N odd it yields one further constraint. In fact (5.5) implies that (L+00 )2 = 1 which together with (5.4) implies that L+00 = 1. The antipode on the generators takes the form SL±ij = g hi L±kh gj k

(5.9)

To construct a homomorphism ϕ : AN > Uq so(N ) → AN acting as the identity on AN it is therefore sufficient to define it on L+ij , L−ij and to verify that all of relations (4.19) and (5.1) to (5.8) are satisfied. Applying ϕ to (4.35) and using (5.9) it follows that (5.9)

(4.35)

(5.9)

(4.35)

ϕ(L−ij ) = g ih ϕ(SL−kh )gkj = g ih −1 [λh , x k ]gkj ϕ(L+ij ) = g ih ϕ(SL+kh )gkj = g ih [λ¯ h , x k ]gkj .

(5.10) (5.11)

The problem is thus equivalent to the construction of N objects λa and N objects λ¯ a such that relations (5.10), (5.11) define elements ϕ(L±kh ) in AN which satisfy all of relations (4.19)1 and (5.1) to (5.8). Actually for the construction of a frame and a set of dual inner derivations in ¯ ∗ (AN )) we just need a homomorphism ϕ − : Uq− so(N ) 1. The proof is given in Appendix 7.3. The relation (5.15) fixes only the product γa γ−a . Notice that γ02 for N odd and γ1 γ−1 for N even are positive real numbers, while all the remaining products γa γ−a are negative. Notice that the embedding AN E→ AN+2 defined by (7.4) automatically induces an embedding for the corresponding λa . Similarly one can prove Theorem 2. One can define a homomorphism AN > Uq+ so(N ) → AN by setting on the generators ϕ + (a) = a,

∀ a ∈ AN ,

ϕ + (L+ij ) = g ih [λ¯ h , x k ]gkj ,

(5.16) (5.17)

with λ¯ 0 = γ¯0 −1 (x 0 )−1

for N odd,

λ¯ ±1 = γ¯±1 −1 (x ±1 )−1 K ±1 for N even, −1 −1 r|a|−1 x −a otherwise, λ¯ a = γ¯a −1 r|a|

(5.18)

and γ¯a ∈ C normalization constants fulfilling the conditions γ¯0 = q 2 h−1 −qh−2 γ¯1 γ¯−1 = −2 k 1

for N odd, for N odd, for N even,

γ¯a γ¯−a = −qk −2 ωa ωa−1 for a > 1.

(5.19)


541

To “glue” ϕ + , ϕ − into a unique homomorphism ϕ : AN > Uq so(N ) → AN we still need to satisfy the image under ϕ of equations (5.8) and (5.3). As we shall prove in Appendix 7.5, this is possible only in the case of odd N and completely fixes the coefficients γa , γ¯a in the previous Ansätze: Theorem 3. In the case of odd N we one can define a homomorphism Uq so(N ) 1, γa = qγ−a for a ≤ 1, γ¯a = −qγa .

(5.23)

Notice that the γa , γ¯a for a = 0 are imaginary and fixed only up to a sign. This has as a consequence that the homomorphism ϕ does not preserve the star structure of Uq so(N ), in other words it is not a star-homomorphism. In the case N = 3 this is the same homomorphism which is also constructed in [6]. Alternatively, we can fix the normalizations of the barred objects so that for q ∈ R+ the involution gives λ∗a = −g ab λ¯ b ,

(θ a )∗ = θ¯ b gba .

(5.24)

Then the coefficients γ¯a will be related to the γa by γ¯0 = −qγ0∗ , ∗ γ¯±1 = −γ∓1 , ∗ γ¯a = −γ−a

if N odd, if N even, 1 q2

if a > 0 if a < 0

(5.25)

otherwise.

We summarize our results for the frames θ a , θ¯ a : θ a = θla ξ l = −2 g ab [λb , x j ]gj l ξ l θ¯ a = θ¯la ξ¯ l = 2 g ab [λ¯ b , x j ]gj l ξ¯ l .

(5.26) (5.27)

These objects commute both with x i , rj and with . Had we adopted the commutation rule (3.40), instead of (3.38), then the same would be true by introducing an additional factor r in the right-hand side of both the previous equations. The matrix elements θ¯ia , θia fulfill the ϕ + , ϕ − images of Eq. (5.6), (5.7) ab d c Rˆ cd θj θi = θlb θka Rˆ ijkl ,

ab ¯ d ¯ c Rˆ cd θj θi = θ¯lb θ¯ka Rˆ ijkl .

(5.28)

542


6. Metrics and Linear Connections on the Quantum Euclidean Space In this section we construct the covariant derivative, the corresponding linear connection and the metric associated with the frames introduced in Sect. 5. We follow for RN q the same scheme proposed previously [23] for R3q . ¯ 1 (AN ), we recall, are free modules, the covariant derivatives Since the 1 (AN ) and can be defined by their actions on the frame Dθ a = −ωa bc θ b ⊗ θ c , D¯ θ¯ a = −ω¯ a bc θ¯ b ⊗ θ¯ c .

(6.1)

For the generalized permutation σ we can write ab c θ ⊗ θd. σ (θ a ⊗ θ b ) = Scd

(6.2)

For the same reasons as for the coefficients of the metric the requirement of bilinearity ab c ab f Scd θ ⊗ θ d = σ (f θ a ⊗ θ b ) = σ (θ a ⊗ θ b f ) = Scd f θc ⊗ θd

(6.3)

forces S ab cd ∈ Z(AN ) = C. According to the consistency condition (2.15) for the torsion and the commutation relations (4.45) for the frame we find that the matrix S has the form S = Cs Ps − Pa + Ct Pt ,

(6.4)

with Cs and Ct complex N 2 × N 2 matrices. As the next step we would like to define the metric according to (2.8). g(θ a ⊗ θ b ) = g ab .

(6.5)

But we have a problem. Due to the form of (3.6) it is not possible to satisfy simultaneously the metric compatibility condition and (6.4). Similarly to what was previously done for N = 3 [23] the best we can do is to weaken the compatibility condition to a condition of proportionality. In this way like in the case N = 3 we find the two solutions ˆ S = q R,

ˆ −1 , S = (q R)

(6.6)

corresponding to the choices Cs = q 2 , Ct = q 2−N or Cs = q −2 , Ct = q N−2 respectively. As Rˆ does, also both these solutions for S satisfy the Yang–Baxter equation (3.1). ˆ Now, using the property (3.11) for the R-matrix, we see that ae fg cb g Seg = q ±2 g ac δdb . Sdf

(6.7)

This has to be compared with the metric compatibility condition (2.21), which in a basis becomes ae fg cb g Seg = g ac δdb . Sdf

(6.8)

The metric is in fact compatible with the linear connection only up to a conformal factor. We can compute the action of σ on the basis ξ i ⊗ ξ j ij

σ (ξ i ⊗ ξ j ) = Shk ξ h ξ k .

(6.9)


543

To see this we have started with the definition (6.2) of the action on θ a ⊗ θ b , and used (3.32). Then the result follows from (5.28). Notice that (6.9) coincides with (6.2). In a similar way, from (6.5), (3.32) and (7.24) we can determine the action of g on the ξ . g(ξ i ⊗ ξ j ) = g ij 2 .

(6.10)

This expression contains , therefore the addition of this element to the algebra is a necessary condition for the construction of the metric, even if we would perform the calculation directly in the basis given by ξ i . According to (2.16) a covariant derivative can be defined by Dξ = −θ ⊗ ξ + σ (ξ ⊗ θ).

(6.11)

As θ is SOq (N )-invariant, so is D. This is true for both choices of σ . One can show [23] that the expression (6.11) is the most general torsion-free, SOq (N )-invariant linear connection. The explicit action of the covariant derivative on ξ i can be computed. We apply (3.11), (3.23) and obtain for S = q −1 Rˆ −1 Dξ i = 0.

(6.12)

Analogously, from (3.11), (3.23), (7.3) it turns out that N

ji

Dξ i = −θ ⊗ ξ i + q 2+ 2 ωn k −1 r −2 ghj (Rˆ 2 )lm x h ξ l ⊗ ξ m N N mi l h = (q 2 − 1)θ ⊗ ξ i − q 2 ωn r −2 glm q 1− 2 x i ξ l ⊗ ξ m + q 2 Rˆ hj x ξ ⊗ ξj N

= (q 2 − 1)(θ ⊗ ξ i + ξ i ⊗ θ) − q 3− 2 ωn r −2 x i ξ l ⊗ ξ m glm .

(6.13)

The coordinates are well adapted to the first linear connection, but not to the second one, because in the latter case Dξ i = 0. However, for both of these covariant derivatives the corresponding curvature (2.23) vanishes. Curv(ξ ) = 0

(6.14)

This can be seen by performing the same calculation done previously [23] for N = 3. Curv(ξ ) = π12 σ012 σ023 σ012 (θ a ⊗ θ b ⊗ θ c ) ξa λb λc = π12 (S12 S23 S12 )abc def (θ d ⊗ θ e ⊗ θ f ) ξa λb λc = (S12 S23 S12 )abc def (θ d θ e ⊗ θ f ) ξa λb λc = −(S12 S23 Pa 12 )abc def (θ d θ e ⊗ θ f ) ξa λb λc = −(Pa 23 S12 S23 )abc def (θ d θ e ⊗ θ f ) ξa λb λc = 0. We have used here the braid relation (3.1) for S, (2.15) and (4.43).

544


¯ 1 (AN ). To conclude this section we repeat the above construction for the calculus Again, there is a unique metric g and two torsion-free SOq (N )-covariant linear connections compatible with it up to a conformal factor. In the θ¯ a basis the actions of g and σ are respectively g(θ¯ a ⊗ θ¯ b ) = g ab , σ (θ¯ a ⊗ θ¯ b ) = S¯ ab cd θ¯ c ⊗ θ¯ d .

(6.16)

ˆ S¯ = q R,

ˆ −1 . S¯ = (q R)

(6.17)

S¯ ae df g fg S¯ cb eg = q ±2 g ac δdb .

(6.18)

(6.15)

The two choices for σ are or

This implies

In the ξ¯ i basis the actions of g and σ become g(ξ¯ i ⊗ ξ¯ j ) = g ij −2 , σ (ξ¯ i ⊗ ξ¯ j ) = S¯ ij hk ξ¯ h ⊗ ξ¯ k .

(6.19) (6.20)

The two covariant derivatives, one for each choice of σ , are ¯ D¯ ξ¯ = −θ¯ ⊗ ξ¯ + σ (ξ¯ ⊗ θ).

(6.21)

The associated linear curvatures Curv vanish. Since in the commutative limit q → 1 we have → 1, we have also

g(ξ i ⊗ ξ j ) → δ i,−j

K → 1, g(ξ¯ i ⊗ ξ¯ j ) → δ i,−j .

The right-hand side is the matrix of coefficients of the flat metric in the complex cartesian coordinates x i , and we recover RN as geometry (at least formally). Had we adopted the commutation rule (3.40), instead of (3.38), then we would have found an additional factor limq→1 r 2 in the right-hand side, which would have corresponded to the coefficients of the metric of R × S N−1 . 7. Appendix 7.1. Miscellanea. In this appendix we give some miscellanea formulae on RN q . The braid matrix of SOq (N ) is given by [19] j i Rˆ = q δii ⊗ δii + δi ⊗ δji + q −1 δi−i ⊗ δ−i (7.1) i=0

i=j,−j or i=j =0

i=0

j −j j + k( δii ⊗ δj − q −ρi +ρj δi ⊗ δ−i ) i<j

i<j


545

where here δji is the N × N matrix with all elements equal to zero except for a 1 in the ij ji ith column and j th row. Clearly, R := Rˆ is a lower-triangular matrix. Moreover, hk

hk

under substitution of q by q −1 in (7.1) we find that

ˆ −1 )ij = R(q) ˆ −1 −i−j . R(q −k−l kl

(7.2)

Using the projector decomposition (3.6) and the expression (3.8) for Pt the square of ˆ the R-matrix can be computed to be ij ij j (Rˆ 2 )kl = k Rˆ kl + δki δl − q 1−N kg ij gkl .

(7.3)

There is an embedding AN E→ AN+2 given by x i → x i for − n ≤ i ≤ n, ri → ri for 0 ≤ i ≤ n → , K → K. It follows from (3.16) that one can rewrite (3.18) as −1 2 −1 2 ri2 = ωi q −1 ωi−1 ri−1 + x i x −i = ωi qωi−1 ri−1 + x −i x i ,

(7.4)

(7.5)

for i > 1 and for N odd and i = 1. This implies that for i ≥ 1 x −i x i − q 2 x i x −i = −qkωi−1 ri2 .

(7.6)

Finally, we recall a useful property of the fundamental representation ρ of Uq so(N ), namely ρba (Sg) = g ad ρdc (g)gcb .

(7.7)

7.2. Universal R-matrix. In this appendix we recall the basics about the universal Rmatrix [15] of a quantum group Uq g, while fixing our conventions. We recall some useful formulae (6 ⊗ id)R = R 13 R 23 (id ⊗ 6)R = R 13 R 12 (S ⊗ id)R = R −1 = (id ⊗ S −1 )R S

−1

−1

(g) = u

S(g)u.

(7.8) (7.9) (7.10) (7.11)

Here u is any of the elements u1 , u2 , ..u8 defined below: u1 := (SR (2) )R (1)

u2 := (SR −1(1) )R −1(2)

u3 := R (2) S −1 R (1)

u4 := R −1(1) S −1 R −1(2)

(u5 )−1 := R (1) SR (2)

(u6 )−1 := (S −1 R (1) )R (2)

(u7 )−1 := R −1(2) SR −1(1)

(u8 )−1 := (S −1 R −1(2) )R −1(1)

(7.12)

546


In fact, using the results of Drinfeld [15, 16] one can show that u1 = u3 = u7 = u8 = vu2 = vu4 = vu5 = vu6 ,

(7.13)

where v is a suitable element belonging to the center of Uq so(N ). From (4.9) and (7.8,7.9) it follows the universal Yang–Baxter relation R 12 R 13 R 23 = R 23 R 13 R 12 ,

(7.14)

whence the other two relations follow R −1 12 R −1 13 R −1 23 = R −1 23 R −1 13 R −1 12 , R 13 R 23 R

−1

12

=R

−1

12 R 23 R 13 .

(7.15) (7.16)

By applying id ⊗ ρca ⊗ ρdb to (7.14), ρca ⊗ ρdb ⊗ id to (7.15) and ρca ⊗ id ⊗ ρdb to (7.16) we respectively find the commutation relations (5.6), (5.7), (5.8).

7.3. Proof of Theorem 1. We divide the proof in various steps. First note that, because of (3.11), Eq. (4.19) for ϕ − (L− ) is equivalent to x h [λa , x i ] = q Rˆ jhik [λa , x j ]x k .

(7.17)

Proposition 9. N independent solutions to Eq. (7.17) are given by (5.14) where γa ∈ C are arbitrary normalization constants. Proof. As a first step in the proof that Eq. (7.17) is satisfied by the λa of (5.14), we calculate the commutation relations between the x i and the λa : ρa −ρi ϕ(L−−a [λa , x i ] = 0 −i ) = q

ϕ(L−−a −i )

=q

ρa −ρi

i

for i < a, ρa −ρi +1

i

[λa , x ] = −q kλa x −1 ra −γa qkωa−1 ra−1 a ϕ(L−−a −1 −1 −a ) = [λa , x ] = γa kω|a|−1 r|a| r|a|−1

ϕ(L−00 ) = [λ0 , x 0 ] = −q 2 hγ0 −qkλ1 x 1 1 ϕ(L−−1 −1 ) = [λ1 , x ] = −γ qh(x 0 )−1 r 1 1 −qkλ−1 x −1 ϕ(L−11 ) = [λ−1 , x −1 ] = γ−1 h(r1 )−1 x 0 1

for i > a, for a > 1, for a < −1, for N odd

(7.18)

for N even, for N odd, for N even, for N odd.

To obtain these relations we have used (3.20), (3.34) and (3.16). In the case a = i we also need (7.6), and for even N , |a| = 1 (3.35). It will be noticed that although complicated in appearance the system of Eqs. (7.18) is actually mainly the first two equations, which are quite simple, plus a series of special cases when i = a. The commutation relations between the x i and the λa are independent of the normalization of the latter, so that they impose no restriction on the γa .


547

ˆ Writing down the explicit expression for the R-matrix (7.1) and using the fact that [λa , x i ] = 0 for i < a, one finds that the relation (7.17) becomes: x h [λa , x i ] =q[λa , x i ]x h + kq[λa , x h ]x i x h [λa , x i ] = q[λa , x i ]x h x i [λa , x i ] = q 2 [λa , x i ]x i

for h < i, h = −i h > i, h = −i, for h = i = 0;

(7.19)

for h = i = 0;

finally, when h = −i, x −i [λa , x i ] = [λa , x i ]x −i + kq [λa , x −i ]x i − q −ρk +ρi [λa , x k ]x −k ,

for i > 0,

k

x −i [λa , x i ] = [λa , x i ]x −i − kq

q −ρk +ρi [λa , x k ]x −k

for i < 0.

k

These relations can be checked one by one with a lengthy but straightward calculation by substituting the explicit expression (7.18) for [λa , x i ] and commuting x h through it. More particularly, in the case i < a both sides of the equations are identically 0, because both [λa , x i ] and R = P Rˆ are lower-triangular matrices. In the case i > a we first use (7.18) again to commute x h with λa . Then we need (3.16) to commute x h with x i if h = −i, while we need to apply (7.5) to the expressions of the type x k x −k to write 2 and r 2 them in terms of r|k| |k|−1 if h = −i. In the case i = a we need (3.20) to commute h x through r|a| and r|a|−1 . In the particular case i = h = 0 (7.19) follows from the x commutation relation (3.34). Next, look for γa such that equations (5.5), (5.4) for ϕ − (L− ) with i = −h, ϕ − (L−ii L−−i −i ) = 1,

ϕ − (L−00 ) = 1,

(7.20)

are fulfilled. Using (7.18), we easily find the γa ’s given in equations (5.15). Lemma 2. With the γa given in equations (5.15) the elements λa are also solutions to equations (4.43) and equations i λa [λb , x i ] = q −1 (Rˆ −1 )cd ab [λc , x ]λd .

(7.21)

We shall occasionally use the short-hand notations eai := [λa , x i ]

e¯ai := [λ¯ a , x i ]

(7.22)

Note that, because of (3.11), the ϕ − images of equations (5.7) and (5.5) are respectively equivalent to the “RTT-relations” ij j cd Rˆ kl eak ebl = eci ed Rˆ ab

(7.23)

and the “gTT-relations” j

g ab eai eb = g ij 2 ,

j

gij eai eb = gab 2 .

(7.24)

548


Proposition 10. With the γa given in equations (5.15) the matrices eai fulfill (7.23), (7.24), and the ϕ − image of (5.4). This will conclude the proof of Theorem 1. Proof of Lemma 2. It is interesting to note that the commutation relations (4.43) between cd the λa are the same as those (3.12) satisfied by the x i , because Pa ab cd = Pa ab . As (3.12) is equivalent to (3.16), so will equations (4.43) be equivalent to λa λb = qλb λa

for a < b, a = −b,

−1 2 sa−1 [λa , λ−a ] = kωa−1 0 [λ1 , λ−1 ] = hs02

for a > 1, for N even, for N odd,

(7.25)

where the quantities sa2 are defined by the equation sa2 =

a

g cd λc λd

(7.26)

c,d=−a

for a ≥ 0 in the case N odd, and for a ≥ 1 in the case N even (in the latter case the sum of course runs over c, d = 0). It is easy to show the commutation relations (for i ≥ 0)  2  q λa ri for a < −i, (7.27) ri λa = qλa ri for |a| ≤ i,  λ r for a > i, a i λa = q −1 λa ,

(7.28)

and, for N even, [K, λb ] = 0 Kλ±1 = q

for |b| = 1, ∓1

(7.29)

λ±1 K

which follow from (3.36). To show now the relation (4.43)1 we consider first the case a < b, excluding the cases N odd and a = 0, and N even and a = ±1. By using (7.18)2 , (7.27) and (7.28) we obtain respectively the identities −1 −1 −1 −1 r|a|−1 λb x −a = γa λb r|a| r|a|−1 x −a = qλb λa . λa λb = γa r|a|

If N is even and a = ±1 we obtain λ±1 λb = γ±1 (x ±1 )−1 λb K ∓1 = γ1 λb (x ±1 )−1 K ∓1 = qλb λ±1 ,

(7.30)

using respectively the identities (7.29), (7.18) and (7.28). We proceed similarly in the case |b| = 1 when N is even. The calculation is analogous for the other cases b = −a. Summing up, for b = −a the λa and λb q-commute, so that there is no restriction on the normalization constants γa .


549

We now consider the cases a = −b. It follows from (5.14), (5.15) that s02 = 2 q −2 h−2 (x0 )−2

for N odd,

2 q −2 k −2 ω12 r1−2

for N even.

s12

=

(7.31)

We use these two relations as initial steps to show by induction that sa2 = q −2 2 ωa2 k −2 ra−2

for a ≥ 1.

(7.32)

In fact (7.26)

2 + q −ρa λa λ−a + q ρa λ−a λa sa2 = sa−1

(5.14) 2 −2 = sa−1 + 2 γa γ−a ra−2 ra−1 q −2−ρa x −a x a + q ρa x a x −a

(7.5) 2 −2 −1 2 = sa−1 + 2 γa γ−a ra−2 ra−1 ra−1 ) q −2−ρa (ωa−1 ra2 − qωa−1 −1 2 +q ρa (ωa−1 ra2 − q −1 ωa−1 ra−1 )

−2 −1 2 2 = sa−1 + 2 γa γ−a ra−2 ra−1 ra−1 q −1 ωa−1 ωa−1 ra2 − q −1 ωa ωa−1 (5.15) 2 = sa−1

−2 2 − 2 q −2 k −2 ωa−1 ra−1 + 2 q −2 k −2 ωa2 ra−2 .

Assuming that (7.32) holds for a = b − 1, the first two terms in the last line are opposite and therefore cancel, and the third gives (7.32) for a = b, as claimed. We now consider the commutators [λa , λ−a ] with a ≥ 1. For N even we find the claim [λ1 , λ−1 ] = 0 by a straightforward calculation. In all other cases we proceed as follows, [λa , λ−a ]

(5.14),(3.20)

=

−2 γa γ−a q −1 2 ra−2 ra−1 (q −1 x −a x a − qx a x −a )

(7.6)

−2 −2 −q −1 2 γa γ−a ωa−1 kra−1 = q −2 k −1 ωa−1 2 ra−1

(7.32)

−1 2 kωa−1 sa−1 ,

(5.15)

=

=

as claimed. With the choice (5.15) for the normalization constants γa , the algebra generated by x i , λi , ±1 , K ±1 , ri±1 is symmetric, i.e. is invariant, with respect to the following transformation S 1 2 λ±1 → γ±1 (q −1 )γ∓1 (q)−1 x ∓1 for N even, 1 1 2 λa → q − 2 γa (q −1 )γ−a (q)−1 x −a x ±1 → γ±1 (q −1 )γ∓1 (q)−1 λ∓1 1 2

a

x →q

− 21

→ , K → K −1 , q → q −1 .

γa (q

−1

)γ−a (q)

−1

1 2

λ−a

otherwise, for N even, otherwise,

(7.33)

550


Notice that S is an involution, S 2 = id. Because of (3.3), (7.2) and (3.5) under S the xxcommutation relations (3.12) and the λλ-commutation relations (4.43) are exchanged. The x a (3.34) and the λa relations (7.28) are exchanged as well, while the xλ relations (7.18) are invariant. We can immediately check that for N odd under S ra2 →

a b=−a

q ρb q −1 λ−b λb (γb (q −1 )γ−b (q −1 )γb (q)−1 γ−b (q)−1 ) 2 = sa2 , 1

(7.34)

where we have used (5.15). In the case of even N the same calculation holds for the terms with |b| > 1, but we have to treat the term with |b| = 1 separately r12 → 2λ−1 λ1 (γ1 (q −1 )γ−1 (q −1 )γ1 (q)−1 γ−1 (q)−1 ) 2 = s12 . 1

(7.35)

The relation (5.14) expressing λa in terms of x −a is invariant as well. To see this we first −1 in (5.14) through sa and sa−1 respectively, then we use (7.28) to express ra−1 and ra−1 move −1 to the left. In this way we are able to rewrite (5.14) in the form λa = γa (q)q 2 −1 sa sa−1 x −a .

(7.36)

Taking into account (5.15) and (3.34), under S (7.36) becomes x −a = γa (q)−1 ra ra−1 −1 λa ,

(7.37)

i.e. we recover (5.14). Again, in the case of even N the special case |a| = 1 has to be treated separately, but due to (7.33)6 and (7.35), it is easily checked that (5.14) is invariant in this case, too. This transformation is useful, because it enables us to get (7.21) by applying S to ˆ (7.17) and then using the properties (3.3), (7.2) and (3.5) of the R-matrix and (5.15). For N odd and N even, h = −i λh [x a , λi ] =

j,k

=

j,k

q

−1

ˆ −1 )−h−i R(q −j −k

γh (q) γi (q) γ−j (q −1 ) γ−k (q −1 ) a [x , λj ]λk −1 γ−h (q ) γ−i (q −1 ) γj (q) γk (q)

ˆ −1 j k [x a , λj ]λk . q −1 R(q) hi

(7.38)

In the particular case that N is even and h = −i from γ1 (q)γ−1 (q) = γ1 (q −1 )γ−1 (q −1 ), γb (q)γ−b (q) = q −2 γb (q −1 )γ−b (q −1 )

for b = ±1

ˆ and the property (3.5) of the R-matrix, it is easily seen that (7.21) still holds. This concludes the proof of Lemma 2.


551

Proof of Proposition 10. To prove (7.23), we use (7.21) and (7.17) cd cd [λc , x i ][λd , x j ] = Rˆ ab (λc x i − x i λc )[λd , x j ] Rˆ ab ij

ef

cd = Rˆ ab (q Rˆ kl λc [λd , x k ]x l − q −1 (Rˆ −1 )cd x i [λe , x j ]λf ) cd −1 ˆ −1 ef ˆ ij = Rˆ ab q (R )cd q Rkl [λe , x k ](λf x l − x l λf ) ij

= Rˆ kl [λa , x k ][λb , x l ], i.e. the “RTT”-relations for eai . By repeated application of the “RTT”-relations it is an ˆ immediate result that for any polynomial f (R) ij j cd f (Rˆ kl )eak ebl = eci ed f (Rˆ ab ).

(7.39)

In particular the projectors Ps , Pa , Pt are of this form. If we write Pt explicitly using (3.8), this yields the gT T -relations (7.24) [which are equivalent to the ϕ − -image of equations (5.5)] also for h = −i, which we had not proved yet.

7.4. Proof of Theorem 2. The proof of 2 is similar to the one of Theorem 1. The explicit expressions for −1 ϕ + (L+ −a −i ) are ρa −ρi ¯ [λa , x i ] = 0 −1 ϕ(L+−a −i ) = q −1

ϕ(L+−a −i )

=q

ρa −ρi

for i > a,

[λ¯ a , x ] = q i

a ¯ −1 ϕ(L+−a −a ) = [λa , x ] =

ρa −ρi −1

k λ¯ a x

i

−1 −1 ra ra−1 −γ¯a −1 kωa−1 −1 −1 −1 −1 γ¯a kq ωa r|a|−1 r|a|

−1 ϕ(L+00 ) = [λ¯ 0 , x 0 ] = γ¯0 −1 q − 2 h q −1 k λ¯ 1 x 1 −1 +−1 1 ¯ ϕ(L −1 ) = [λ1 , x ] = −γ¯1 −1 hr1−1 x 0 q −1 k λ¯ −1 x −1 −1 ϕ(L+11 ) = [λ¯ −1 , x −1 ] = γ¯−1 −1 hq −1 r1 (x 0 )−1 1

for i < a, for a > 1, for a < −1, for N odd

(7.40)

for N even, for N odd, for N even, for N odd.

7.5. Proof of Theorem 3. Using relations (3.11), it is easy to show that the image under ϕ of (5.8) is equivalent to ij cd i j Rˆ ab e¯c ed = Rˆ kl eak e¯bl

(7.41)

in the notation (7.22). Theorem 1 (2) fixes the coefficients γa (γ¯a ) for a < 0 in terms of γa (γ¯a ) for a ≥ 0. We can use the remaining freedom in the choice of γa , γ¯a to find further conditions on γa for a > 0, and relations relating the coefficients γ¯a to γa so that (5.3) and (5.8) are fulfilled. We start with the observation that in the case of odd N (5.14) and (5.18) imply λ¯ a = γ¯a γa−1 −2 λa

(7.42)

552


and use the equations (7.21) and (7.17). In this way we see that cd ¯ [λc , x i ][λd , x j ] Rˆ ab c,d

=

c,d

=

c,d

=

c,d

=

cd γ¯c γc−1 ( −2 λc x i − x i −2 λc )[λd , x j ] Rˆ ab cd Rˆ ab γ¯c γc−1 −2 (λc x i − q −2 x i λc )[λd , x j ] cd Rˆ ab γ¯c γc−1 −2 (q

c,d e,f k,l

=

c,d e,f k,l

k,l

ij Rˆ kl λc [λd , x k ]x l − q −3

e,f

cd ˆ −1 ef ˆ ij Rˆ ab R cd Rkl γ¯c γc−1 [λe , x k ]( −2 λf x l

ef Rˆ −1 cd x i [λe , x j ]λf )

− x l −2 λf )

cd ˆ −1 ef ˆ ij Rˆ ab R cd Rkl γ¯c γc−1 γf γ¯f−1 [λe , x k ][λ¯ f , x l ].

If the further condition holds, γ¯a γa−1 = c ≡ const,

(7.43)

then in the last line the γ ’s cancel with each other, so do Rˆ and Rˆ −1 , and we find that (7.41) is actually satisfied. To see that (7.43) is not only a sufficient, but also a necessary condition for (7.41), we write down the latter for the particular values of the indices i = a, j = b = a + 1 for a = −n, . . . n − 1. = [λ¯ a+1 , x a ][λa , x a+1 ] + k[λ¯ a , x a ][λa+1 , x a+1 ] [λa , x a+1 ][λ¯ a+1 , x a ] + k[λa , x a ][λ¯ a+1 , x a+1 ]

(7.44)

We plug in the expressions (7.18),(7.40) for eai and e¯ai and apply the relations (7.5). For a > 1, (7.44) implies −1 −1 k 3 q(γ¯a γa+1 − γ¯a+1 γa )ra−1 ra+1 ra−2 ωa−1 ωa+1 = 0.

(7.45)

This means that in order for (7.41) to hold, we have to require −1 γ¯a γa−1 = γ¯a+1 γa+1

(7.46)

for every a > 1. A similar reasoning can be repeated for a ≤ 1 to show that (7.46) has to hold for any value of a. Therefore (7.43) is necessary. When N is even, it is not possible to satisfy (7.41). This is a consequence of the fact that (7.42) does not hold for |a| = 1 in this case, due to the particular form of λ±1 and λ¯ ±1 . Choose the indices in (7.41) to be e.g. i = a = 1, j = b = 2. Plug in (7.18), (7.40) for eai and e¯ai and the definitions (5.14), (5.18) for λa and λ¯ a , then apply (3.35) and (3.16). In this way (7.44) becomes − k 2 γ¯2 γ1 r2−1 r1−1 x −2 x 2 K −1 + k γ¯1 γ2 qω2−1 r2 r1−1 K − k 2 γ¯2 γ1 r2−1 r1−1 x 2 x −2 − kω1−1 r12 K −1


553

Due to the commutation relation (3.16) between x 2 and x −2 the terms which are proportional to K −1 cancel and the following equation should be satisfied k 3 q γ¯1 γ2 ω2−1 r2 r1−1 K = 0. But this would mean that either γ¯1 or γ2 should vanish, which is not possible, if we want to have N independent objects λa instead of fewer. That is the reason why the theorem does not hold for N even. In the case N odd (7.43) is consistent with (5.15), (5.19). We can determine c e.g. by applying (7.43) to a = 0 and by recalling (5.15)1 , (5.19)1 . We thus find c = −q. In this way we get the last of the equations (5.23), which completely fixes the coefficients γ¯a in terms of the γa . As the last step, we require (5.3). This imposes another condition relating γ¯a to γa : γ¯a γa = k −2 ωa−1 ωa q −1

(7.47)

q

(7.48)

−2

(7.49)

γ¯1 γ1 = h

γ¯0 γ0 = −h

−2

q

−2

ωa−1 ωa q

γ¯−1 γ−1 = h γ¯a γa = k

for a > 1,

−2 −1

(7.50) for a < −1.

(7.51)

a Here we have used the expression (7.18) for eaa = ϕ(L− −a −a ) and (7.40) for e¯a = ϕ(L+ −a −a ) Eqs. (7.47) to (7.51) are compatible with (5.15), (5.19). If we replace γ¯a = −qγa we find the remaining equations in (5.23).

References 1. Aschieri, P., Castellani, L.: Bicovariant Calculus on Twisted I SO(N ), Quantum Poincaré Group and Quantum Minkowski Space. Int. J. Mod. Phys. A 11, 4513 (1996) 2. Aschieri, P., Castellani, L., Scarfone, A.M.: Quantum Orthogonal Planes: I SOq,r (N ) and SOq,r (N ) – Bicovariant Calculi and Differential Geometry on Quantum Minkowski Space. Eur. Phys. J. C 7, 159–175 (1999) 3. Carow-Watamura, U., Schlieker, M. and Watamura, S.: SOq (N ) covariant differential calculus on quantum space and quantum deformation of Schroedinger equation. Z. Physik C - Particles and Fields 49, 439 (1991) 4. Castellani, L.: Differential calculus on I SOq (N ), quantum Poincaré algebra and q-gravity. Commun. Math. Phys. 171, 383 (1995) 5. Cerchiai, B.L., Hinterding, R.. Madore, J. and Wess, J.: The geometry of a q-deformed phase space. Eur. Phys. J. C 8, 533 (1999) 6. Cerchiai, B.L., Madore, J., Schraml, S. and Wess, J.: Structure of the three-dimensional Quantum Euclidean Space. Eur. Phys. J. C 16, 169–180 (2000) 7. Cerchiai, B. and Wess, J.: q-deformed Minkowski space based on a q-Lorentz algebra. Euro. Phys. J. C 5, 553 (1998) 8. Chaichian, M., Demichev, A.,and Prešnajder, P.: Quantum field theory on noncommutative space-times and the persistence of ultraviolet divergences. Helsinki Preprint HIP-1998-77/TH, http://xxx.lanl.gov/abs/hep-th/9812180hep-th/9812180 9. Cho, S., Hinterding, R., Madore, J., Steinacker, H.: Finite field theory on noncommutative geometries. Int. J. Mod. Phys. D9, 161–199 (2000) 10. Chu, C.S., Ho, P.M., Zumino, B.: Some complex quantum manifolds and their geometry. In: Quantum Fields and Quantum Space Time, Cargese, August 1996, G.’t Hooft, A.Jaffe, G.Mack, P. Mitter, R. Stora (Eds.) New York: Plenum Press, NATO ASI Series 364, 1997, p. 283 11. Connes, A.: Non-commutative differential geometry. Publications of the I.H.E.S. 62, 257 (1986) 12. Connes, A.: Noncommutative Geometry. London: Academic Press, 1994 13. Dimakis,A. and Madore, J.: Differential calculi and linear connections. J. Math. Phys. 37 no. 9, 4647–4661 (1996)

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14. Doplicher, S., Fredenhagen, K. and Roberts, J.: The quantum structure of spacetime at the Planck scale and quantum fields. Commun. Math. Phys. 172, 187 (1995) 15. Drinfeld, V.: Quantum groups. In: I.C.M. Proceedings, Berkeley, 1986, p. 798 16. Drinfeld, V.: Quasi hopf algebras. Leningrad Math. J. 1, 1419 (1990) 17. Dubois-Violette, M., Madore, J., Masson, T. and Mourad, J.: On curvature in noncommutative geometry. J. Math. Phys. 37 no. 8, 4089–4102 (1996) 18. Dubois-Violette, M., Madore, J., Masson, T. and Mourad, J.: Linear connections on the quantum plane. Lett. Math. Phys. 35 no. 4, 351–358 (1995) 19. Faddeev, L., Reshetikhin, N., and Takhtajan, L.: Quantization of Lie groups and Lie algebras. Leningrad Math. J. 1, 193 (1990) 20. Fichtmüller, M., Lorek, A. and Wess, J.: q-deformed phase space and its lattice structure. Z. Physik C Particles and Fields 71, 533 (1996) 21. Filk, T.: Divergencies in a field theory on quantum space. Phys. Lett. 376, 53 (1996) 22. Fiore, G.: SOq (N )-isotropic harmonic oscillator on the quantum euclidean space RN q . In: Nonlinear, Deformed and Irreversible Quantum Systems, Proceedings of an International Symposium on Mathematical Physics at the Arnold Sommerfeld Institute, Clausthal, August 1994. Singapore: World Scientific Publishing, 1995 23. Fiore, G. and Madore, J.: The geometry of Quantum Euclidean Spaces. J. Geom. Phys. 33, 257–287 (2000) 24. Fiore, G. and Madore, J.: Leibniz rules and reality conditions. Dip. Matematica e Applicazioni, Università di Napoli, 98–13; http://xxx.lanl.gov/abs/math/9806071math/9806071. 25. Hebecker, A., Schreckenberg, S., Schwenk, J. Weich, W. and Wess, J.: Representations of a q-deformed heisenberg algebra. Z. Physik C - Particles and Fields 64, 355 (1994) 26. Kempf, A. and Mangano, G.: Minimal length uncertainty relation and ultraviolet regularization. Phys. Rev. D 55, 7909 (1997) 27. Lukierski, J., Nowicki, A. and Ruegg, H.: New quantum Poincaré, algebra and κ-deformed field theory. Phys. Lett. B 293, 344 (1992) 28. Madore, J. and Steinacker, H.: Propagators on the h-deformed Lobachevsky plane. Ludwig-Maximilian Universität, Preprint, LMU-TPW 99-08, http://xxx.lanl.gov/abs/q-alg/9907023q-alg/9907023 29. Martín C. and Sánchez-Ruiz, D.: The one-loop UV-divergent structure of U (1)-Yang-Mills theory on noncommutative R4 . Phys. Rev. Lett. 83, 476-479 (1999) 30. Ogievetsky, O.: Differential operators on quantum spaces for GLq (n) and SOq (n). Lett. Math. Phys. 24, 245 (1992) 31. Ogievetsky, O., Schmidke, W., Wess, J. and Zumino, B.: q-deformed Poincaré algebra. Commun. Math. Phys. 150, 495 (1992) 32. Seiberg, N. and Witten, E.: String theory and noncommutative geometry. J. High Energy Phys. 09, 032 (1999) 33. Snyder, H.: Quantized space-time. Phys. Rev. 71, 38 (1947) 34. Snyder, H.: The electromagnetic field in quantized space-time. Phys. Rev. 72, 68 (1947) 35. Steinacker, H.: Integration on quantum euclidean space and sphere in N dimensions. J. Math. Phys. 37, 7438 (1996) 36. Wess, J. and Zumino, B.: Covariant differential calculus on the quantum hyperplane. Nucl. Phys. B (Proc. Suppl.) 18, 302 (1990) 37. Woronowicz, S.: Pseudospaces, pseudogroups and Pontriagin duality. In: Mathematical problems in theoretical physics, Lausanne, August, 1979, K. Osterwalder, ed., Lect. Notes in Phys. 116, Berlin–Heidelberg– New York: Springer-Verlag, 1980, pp. 407–412 38. Woronowicz, S.: Twisted SU (2) group, an example of a non-commutative differential calculus. Publ. RIMS, Kyoto Univ. 23, 117 (1987) 39. Woronowicz, S.: Compact matrix pseudogroups. Commun. Math. Phys. 111, 613 (1987) Communicated by T. Miwa

Commun. Math. Phys. 217, 555 – 577 (2001)

Communications in



Phase Transitions for Countable Markov Shifts Omri M. Sarig Department of Mathematics, University of Warwick, Coventry, CV4 7AL, UK. E-mail: [email protected] Received: 6 March 2000 / Accepted: 17 October 2000

Abstract: We study the analyticity of the topological pressure for some one-parameter families of potentials on countable Markov shifts. We relate the non-analyticity of the pressure to changes in the recurrence properties of the system. We give sufficient conditions for such changes to exist and not to exist. We apply these results to the Manneville– Pomeau map, and use them to construct examples with different critical behavior. 1. Introduction A well known theorem of Ruelle [Ru2, Ru1] states that for every topologically mixing topological Markov shift X with a finite number of states, the topological pressure Ptop is analytic on the space of Hölder continuous functions. That is, ∀φ, ψ ∈ C(X) Hölder continuous, t → Ptop (φ + tψ) is real analytic in a neighborhood of t = 0 (whence for every t). In ferromagnetism, this is sometimes interpreted as “lack of phase transitions” (see [E]). If the number of states is countable, this theorem is no longer true. [S3] contains an example of a φ which depends on a finite number of coordinates (“finite range potential”) for which Ptop (φ +tφ) has a positive Lebesgue measure set of critical points. Other finite range examples with critical behavior can be found in [Hof, Lo,W1,W2]. Infinite range examples include the Manneville–Pomeau map (see e.g. [PM, Lo]) and the Farey map [PS] (see also [LSV]). The purpose of this paper is to study critical phenomena for some smooth oneparameter families of infinite range potentials on countable Markov shifts. The critical phenomena we consider are non-analyticity of the pressure, changes in the existence of an equilibrium measure, and changes in its finiteness, when it exists. It was observed in [S2], that there are three modes of recurrence for potentials on countable Markov shifts: positive recurrence, null recurrence and transience. Positive Part of a dissertation prepared in the Tel-Aviv University

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recurrent potentials admit finite equilibrium measures. Null recurrent potentials admit conservative infinite equilibrium measures. Transient potentials do not have conservative conformal measures. A change in the mode of recurrence of a one-parameter family affects, therefore, the existence or finiteness of the equilibrium measure. We show that a change in the mode of recurrence is also related with non-analyticity of the pressure, and give conditions governing the existence of such changes (Theorems 2 and 3). We use these results to derive some of the properties of the Manneville–Pomeau map, and show that all systems with the same symbolic structure have similar properties. This explains why the Manneville–Pomeau map has the same critical behavior as that of the examples considered in [PS, Lo,W1] and [W2, Theorem 5]. We also construct examples with different critical behavior, using the methods of [S3]. Among these examples is a potential which is “intermittent” (i.e., admits infinite conservative equilibrium measure) for a whole interval of “temperatures” (example 4). This is different than the Manneville–Pomeau example, which is intermittent only for a specific “temperature”. The structure of the paper is as follows. Section 2 contains a survey of relevant results on the thermodynamic formalism of countable Markov shifts. Section 3 contains the statement of our main results, Theorems 2, 3 and 4. Section 4 contains an application of these results to the study of the renewal shift and the closely related Manneville–Pomeau map. Section 5 contains other examples. Section 6 contains the proof of Theorem 2. Section 7 contains the proof of Theorems 3 and 4. 2. Survey of the Thermodynamic Formalism for Countable Markov Shifts In this section we survey some results from [S1, S2] concerning the thermodynamic formalism of some infinite range potentials on countable Markov shifts. For a survey on finite range potentials see [GS] (see also [G1]). 2.1. Basic definitions and notational conventions. Let S be a countable set and A = tij S×S a matrix of zeroes and ones with no columns or rows which are all zeroes. Let X be the set X := x ∈ S N∪{0} : txi xi+1 = 1 , ∀i ≥ 0 endowed with the relative product topology, which is also given by the base of cylinders [a0 , . . . , an−1 ] := {x ∈ X : xi = ai , 0 ≤ i ≤ n − 1} , where n ∈ N and a0 , . . . , an−1 ∈ S. An admissible word is a a ∈ S n such that [a] = ∅. Its length is |a| = n. Let T : X → X be the left shift (T x)i := xi+1 . The topological dynamical system (X, T ) is called a (one sided) topological Markov shift. We say that X is topologically mixing if (X, T ) is topologically mixing. The members of S are called the states of the shift, and the matrix A is called the transition matrix. The sets [a] where a ∈ S are called the partition sets. Let φ : X → R be some real function (also called potential). The variations of φ are Vn (φ) := sup{|φ(x) − φ(y)| : x, y ∈ X , xi = yi , 0 ≤ i ≤ n − 1}. φ is said to have summable variations if n≥2 Vn (φ) < ∞. φ is called weakly Hölder continuous (with parameter θ ) if there exist A > 0 and θ ∈ (0, 1) such that for all n ≥ 2, Vn (φ) ≤ Aθ n . Note that in both cases the quantification begins with n = 2 so φ may be unbounded or may satisfy V1 (φ) = ∞.

Phase Transitions for Countable Markov Shifts

557

Foreveryφ with summable variations we associate the corresponding Ruelle operator [Ru2] Lφ f (x) := T y=x eφ(y) f (y). If Lφ 1∞ < ∞ this is a bounded operator on the Banach space CB (X) = {f ∈ C(X) : f ∞ < ∞} (complex valued functions). k One checks that (Lnφ f )(x) = T n y=x eφn (y) f (y) where φn := n−1 k=0 φ ◦ T . We use the following notational conventions. All logarithms are natural logarithms. The indicator functions of sets A ⊆ X are denoted by 1A , and 1 := 1X . a = B ±n b means that B ≥ 1, a, b > 0 and B −n b ≤ a ≤ B n b; an bn means that ∃B∀n, an = B ±1 bn ; an ∝ bn means that ∃c = 0 such that an /bn → c; and an ∼ bn means that an /bn → 1. 2.2. Pressure and recurrence. For a ∈ S set ϕa (x) := 1[a] (x)inf{n ≥ 1 : T n (x) ∈ [a]} (where inf ∅ := ∞ and 0 · ∞ = 0). Set eφn (x) 1[a] (x) and Zn∗ (φ, a) := eφn (x) 1[ϕa =n] (x). Zn (φ, a) := T n x=x

T n x=x

It is known that if X is topologically mixing, φ has summable variations1 , and Lφ 1∞ is finite then the following limit exists, is finite and is independent of the choice of a ∈ S PG (φ) := lim

n→∞

1 log Zn (φ, a). n

(1)

PG (φ) is called the Gurevich pressure of φ ([S1, G1, G2, G3]) and satisfies the following variational principle PG (φ) = sup hµ (T ) + φdµ : µ ∈ P T (X); − φdµ < ∞ , where PT (X) is the set of T -invariant Borel probability measures. λ := exp[PG (φ)]. We say that φ is recurrent if for some a ∈ S, Let −n transient if it converges. We say that φ is positive n≥1 λ Zn (φ, a) diverges and recurrent if it is recurrent and n≥1 nλ−n Zn∗ (φ, a) < ∞ and null recurrent if it is re current and n≥1 nλ−n Zn∗ (φ, a) = ∞. It turns out that these definitions do not depend on the choice of a ∈ S and that [S2]: Theorem 1. Let X be a topologically mixing countable Markov shift, and let φ be some real function on X with summable variations. If φ has finite Gurevich pressure, then φ is recurrent if and only if there exist λ > 0, a conservative measure ν finite and positive on cylinders, and a positive continuous function h such that L∗φ ν = λν and Lφ h = λh. In this case λ = exp PG (φ) and there exist an ↑ ∞ such that for every cylinder [a] and x∈X n 1 −k k λ Lφ 1[a] (x) −→ h(x)ν[a]. n→∞ an k=1

The sequence {an }n>0 satisfies an ∼

[a] hdν

−1

n −k k=1 λ Zk (φ, a) for every a

∈ S.

1 The following results, including Theorem 1 below, were stated in [S1] and [S2] under stronger continuity assumptions on φ, but the proofs given there are also valid for φ with summable variations.

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Furthermore, 1. if φ is positive recurrent then ν(h) < ∞, an ∝ n, and for every [a] λ−n Lnφ 1[a] −→

n→∞

hν[a] uniformly on compacts, where h is normalized so that ν(h) = 1. 2. if φ is null recurrent then ν(h) = ∞, an = o(n), and for every [a] λ−n Lnφ 1[a] tends to zero uniformly on cylinders.

It is easy to check that hdν is T -invariant, that h is bounded away from zero and infinity on partition sets and that Vn (log h) ≤ k≥n+1 Vk (φ). It is also clear from the convergence part of the theorem that ν and h are unique up to a multiplicative constant. As a corollary we obtain, Lemma 1. Let X be topologically mixing and let φ be a function with summable variations and finite Gurevich pressure. Then there exist two continuous functions φ ! and ϕ such that φ ! ≤ 0, PG (φ ! ) = 0 and φ ! = φ + ϕ − ϕ ◦ T − PG (φ). The function ϕ is bounded on partition sets. If φ is recurrent then Lφ ! 1 = 1, and if φ is transient then Lφ ! 1 ≤ 1. If φ is weakly Hölder continuous then so are φ ! and ϕ. Proof.Set λ := exp PG (φ). Assume that φ is transient. Fix some state a ∈ S and set h := n≥1 λ−n Lnφ 1[a] . By transience, topological mixing and summable variations h is finite. Also, if φ is weakly Hölder, then so are log h and log h ◦ T . It is easy to check that λ−1 Lφ h ≤ h. Set ϕ := log h and φ ! := φ + ϕ − ϕ ◦ T − PG (φ). Clearly, Lφ ! 1 ≤ 1 whence φ ! ≤ 0 as required. The case when φ is recurrent is handled by replacing h in the last argument by the h given by theorem 1 (see [Wal] for a similar normalization procedure). " # 3. Statement of Main Results We recall the well-known process of inducing in the context of topological Markov shifts (see [S2] and [A, Sect. 1.5]). Fix some state a ∈ S. Set S := {[a] : |a| ≥ 1 ; ai = N∪{0} and let T : X → X be the left shift. For every a iff i = 0 ; [a, a] = ∅}, X := S φ : X → R set   ϕ a −1 φ :=  φ ◦ T k  ◦ π, k=0

where π : X → [a] is given by π([a 0 ], [a 1 ], . . . ) := (a 0 , a 1 , . . . ). The pair (X, φ) is called the induced system and φ is called the induced potential (on [a]). Induced systems are in many cases easier to handle than the original systems, as shown by the following example: A system (X, φ) is called Bernoulli if X = S N∪{0} and if φ(x) = φ(x0 ). A potential is called Markov if φ(x) = φ(x0 , x1 ). If φ is a Markov potential, then (X, φ) is a Bernoulli system.2 If φ is weakly Hölder continuous, so is φ. Summable variations alone, however, is not enough: φ may not have summable variations, even if φ does. The existence of pressure, however, is always guaranteed: 2 This is also true for the larger class of potentials φ for which ∃a ∈ S such that φ(x) = φ(x , . . . , x 0 ϕa (x) ) as long as the inducing is done with respect to [a]. The state a can be viewed as a “gap” between non-interacting clusters of interacting particles. Analogous potentials are studied in a different mathematical setting in [FF].


559

Lemma 2. Let X be topologically mixing and let φ : X → R be some function with summable variations. Let a ∈ S be some fixed state and let X and φ be the induced system and induced potential. Then the following limit exists for all [a] ∈ S (although it may be infinite) and is independent of the choice of [a]: PG (φ) = lim

n→∞

1 log Zn (φ, [a]). n

Proof. Follows from the proof of Theorem 1 in [S1] and the standard estimation n−1 ]≤ ∞ # Vn [φ + φ ◦ T + . . . + φ ◦ T k=2 Vk [φ]. " To state our main results, we need the following definition. Definition 1. Let X be topologically mixing and let φ : X → R have summable variations and finite Gurevich pressure. Fix a ∈ S and let (X, φ) be the induced system. Set pa∗ [φ] := sup{p : PG (φ + p) < ∞}. The a-discriminant of φ is ,a [φ] := sup{PG (φ + p) : p < pa∗ [φ]} ≤ ∞. As we shall see later (Sect. 6, Proposition 3), ,a [φ] = PG (φ + pa∗ [φ]). (2) Both ,a [φ] and pa∗ [φ] are determined by ξ n Zn∗ (φ, a) in the following way. Let R be the radius of convergence of this series. Then ∞ ∞ k ∗ R Zk (φ, a) ≤ Vk (φ) (3) ,a [φ] − log k=1

k=2

1 pa∗ [φ] = − lim sup log Zn∗ (φ, a) n→∞ n Both relations follow from the stronger statement (Sect. 6, Proposition 3): ∞ ∞ ekp Zk∗ (φ, a) ≤ Vk (φ). PG (φ + p) − log k=1

(4)

(5)

k=2

Note that when φ is a Markov potential, both (5) and (3) are equalities (because for Markov potentials k≥2 Vk (φ) = 0). Our basic result is: Theorem 2 (Discriminant theorem). Let X be a topologically mixing countable Markov shift and let φ : X → R be some function with summable variations and finite Gurevich pressure. Let a ∈ S be some arbitrary fixed state. 1. The equation PG (φ + p) = 0 has a unique solution p(φ) if ,a [φ] ≥ 0, and no solution if ,a [φ] < 0. The Gurevich pressure of φ is given by −p(φ) ,a [φ] ≥ 0 PG (φ) = ; (6) −pa∗ [φ] ,a [φ] < 0 2. φ is positive recurrent if ,a [φ] > 0 and transient if ,a [φ] < 0. In the case ,a [φ] = 0, φ is either positive recurrent or null recurrent.

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Theorem 2 should be understood in the context of one-parameter families of potentials. Given such a family {φβ }, let {,β } be the corresponding one-parameter family of discriminants. When ,β changes sign, {φβ } changes its recurrence properties and the case in (6) changes. A change in the mode of recurrence implies, by Theorem 1, a change in the qualitative properties of the equilibrium measure (existence and finiteness). A change of case in (6) may imply non-smoothness for β → PG (φβ ). This suggests that the search for critical phenomena for one-parameter families may be done by studying the sign changes of the discriminant. This can sometimes be done with the aid of (3), as we shall see in Sects. 4 and 5. The proof of Theorem 2 is given in Sect. 6. We now discuss the case when the discriminant does not change sign and remains positive. Let φ be some function with summable variations and finite pressure. We say that φ is strongly positive recurrent if for some state a ∈ S ,a [φ] > 0. (This generalizes the notion of stable positivity for Markov potentials discussed in [GS].) The Discriminant Theorem implies that every strongly positive recurrent function is positive recurrent. The opposite statement is false (Example 2 below). We are interested in differentiability of the pressure functional, i.e. in the existence d of directional derivatives dt P (φ + tψ). We restrict ourselves to the following set t=0 G of directions: ∞ Dir(φ) := ψ : Vn (ψ) < ∞, ∃ε > 0 s.t. ∀|t| < ε, PG (φ + tψ) < ∞ . n=2

The following theorem completes Theorem 2 by saying that if the discriminant is positive, then there is no critical phenomena of the sort that can be encountered when , changes sign. Its proof is given in Sect. 7. Theorem 3. Let X be a topologically mixing and φ be a weakly Hölder continuous function such that PG (φ) < ∞. If φ is strongly positive recurrent then ∀ψ ∈ Dir(φ) weakly Hölder continuous, ∃ε > 0 such that φ + tψ is positive recurrent for all |t| < ε and such that t → PG (φ + tψ) is real analytic in (−ε, ε). The case ψ = φ is particularly interesting, as it appears in the study of the oneparameter family {βφ}β≥β0 .3 If PG (β0 φ) < ∞, then PG (βφ) < ∞ for all β > β0 , because by lemma 1, φ is cohomologous to a non-positive function. Therefore, ∀β > β0 , φ ∈ Dir(βφ). This may not be true for β = β0 : Example 1. Let X = NN∪{0} and φ(x) := − log x0 (log 2x0 )2 . Then PG (βφ) < ∞ for β ≥ 1, and PG (φ) = ∞ for β < 1. # Proof. PG (βφ) = log k≥1 1/[k β (log 2k)2β ]. " Corollary 1. Let X be a topologically mixing and φ be weakly Hölder continuous function such that PG (φ) < ∞ and φ ∈ Dir(φ). The following conditions are equivalent: 1. φ is strongly positive recurrent. 2. for every weakly Hölder continuous ψ ∈ Dir(φ) there exists ε > 0 such that φ + tψ is positive recurrent for every real t such that |t| < ε. 3. for every a ∈ S ,a [φ] > 0. 3 Such families appear in models for systems whose inverse temperature β is changed [E].


561

Proof. The first statement implies the second by Theorem 3. The third statement trivially implies the first. It remains to show that the second statement implies the third. Assume that the second statement is true, but that the third statement is false. Then for some a ∈ S, ,a [φ] is not positive. Since by our assumptions φ is positive recurrent, ,a [φ] cannot be negative, so ,a [φ] = 0. Set ψ := 1[a] . Since ψ = 1 on X, ,a [φ + tψ] = t. This contradicts the second statement because if t < 0 then φ + tψ is transient. " # We remark that the assumptions of Theorem 3 can be weakened: Theorem 4. Let X be a topologically mixing and φ be a function with summable variations, such that PG (φ) < ∞, ,a [φ] > 0 and such that the induced potential on a, φ, is weakly Hölder. Then ∀ψ ∈ Dir(φ) such that ψ is weakly Hölder continuous, ∃ε > 0 such that φ + tψ is positive recurrent ∀|t| < ε, and such that t → PG (φ + tψ) is real analytic in (−ε, ε). In Sect. 7 we prove this stronger version.

4. The Renewal Shift The examples studied in [PM, Hof, GW,W1,W2, PS] and [Lo] share the same critical behavior: for some potential φ, the function β → PG (βφ) has one point of nondifferentiability βc , and is constant for β > βc . A close look at these examples shows that they can be represented as different potentials on the same countable Markov shift, the renewal shift. This is the shift with set of states S := N ∪ {0} and transition matrix (tij )S×S whose 1 entries are t00 , t0i and ti,i−1 (i = 1, 2, 3, . . . ). The main result of this section is Theorem 5. Let X be the renewal shift and let φ : X → R be a function with summable variations such that sup φ < ∞ and such that φ is weakly Hölder continuous. Then there exists 0 < βc ≤ ∞ such that: 1. βφ is strongly positive recurrent for 0 < β < βc and transient for β > βc . 2. PG (βφ) is real analytic in (0, βc ) and linear in (βc , ∞). It is continuous but not analytic at βc (in case βc < ∞). 3. Set An := esup{φn (x):x∈[0,n−1,... ,0]} and let R(β) be the radius of convergence of β Fβ (ξ ) := n≥1 An ξ n . If Fβ (R(β)) is infinite for every β then βc = ∞. If ∃β > 0 such that Fβ (R(β)) < 1 then βc < ∞. Proof. It is easy to check that X it topologically mixing. Also, βφ has finite pressure for all β ≥ 0, since PG (βφ) ≤ log Lβφ 1 ≤ log(2eβ sup φ ). One can easily check that for every function f , n ∈ N and β > 0 Zn∗ (βf, 0) = Zn∗ (f, 0)β

and

p0∗ [βf ] = βp0∗ [f ].

(7)

Henceforth (X, φ) denotes the induced system on [0], P (β) := PG (βφ) and ,[β] := ,0 [βφ]. If p0∗ [φ] = ∞ then ,[β] = sup{PG (φ + p) : p < ∞} = ∞ because PG (φ + p) ≥ PG (φ) + p. In this case parts 1 and 2 follow with βc = ∞ from Theorem 4 and the discussion after Theorem 3. We therefore restrict ourselves to the case pa∗ [φ] < ∞.

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Without loss of generality, assume that p0∗ [φ] = 0 (else pass to φ + p0∗ [φ] and use (4)). By (7), p0∗ [βφ] = 0 for all β > 0, whence by (2) ,[β] = PG (βφ).

(8)

As before, if ,[β] > 0 for every β, parts 1 and 2 follow with βc = ∞. Assume ∃β > 0 such that ,[β] ≤ 0 and set βc := inf{β > 0 : ,[β] ≤ 0}. Note that βc > 0 because according to (3) and (7) ,[β] ≥ log

∞ n=1

∗

β

enp0 [φ] Zn∗ (φ, 0)

−β

∞

Vn (φ) −→ +∞.

n=2

β→0+

We claim that ,[β] → −∞ as β ↑ +∞. Fix some β0 such that ,[β0 ] ≤ 0. By (8) β0 φ has finite pressure, whence by Lemma 1, β0 φ is cohomologous to ψ + PG (β0 φ), where ψ is weakly Hölder continuous (in X) such that Lψ 1 ≤ 1. Since φ has summable variations, V1 (φ) < ∞. It follows from Lemma 1 that V1 (ψ) < ∞ as well. By (8), for all t > 1, ,[tβ0 ] = PG (tβ0 φ) = PG (tψ) + tPG (β0 φ). Since PG (β0 φ) = ,[β0 ] ≤ 0, we have for all t > 1, ,[tβ0 ] ≤ PG (tψ) ≤ log Ltψ 1∞ . By construction, Lψ 1 ≤ 1. Therefore, since every x ∈ X has more than one pre-image, ψ is strictly negative. It follows from this and V1 (ψ) < ∞ that Ltψ 1∞ → 0 as t ↑ ∞. This implies that ,[β] → −∞ as β ↑ ∞. We show that ,[β] < 0 in (βc , +∞). By the definition of βc there are βn ↓ βc such that ,[βn ] ≤ 0. By what we just showed there are βn! ↑ ∞ such that ,[βn! ] < 0. By (8) ,[β] = PG (βφ), so ,[β] is convex in (βn , βn! ). By convexity, ,[β] < 0 in (βn , βn! ]. Since βn ↓ βc and βn! ↑ ∞, ,[β] is strictly negative in (βc , +∞). We have shown that ,[β] < 0 in (βc , ∞). It is obvious that ,[β] > 0 in (0, βc ). Part 1 now follows from the discriminant theorem. We prove part 2. The analyticity of P (β) in (0, βc ) follows from Theorem 4 and that fact that P (β) < ∞. The discriminant theorem and (7) imply that ∀β > βc , PG (βφ) = p0∗ [βφ] = βp0∗ [φ] and ∀β ∈ (0, βc ), PG (βφ) > p0∗ [βφ] = βp0∗ [φ]. Thus PG (βφ) is linear in (βc , ∞), but not in (0, ∞). This implies that βc is a point of nonanalyticity. The continuity of P (β) in βc follows from the convexity of this function. To prove part 3,recall that ,[β] > 0 for β > 0 small, and note that by (8) that log Fβ (R(β)) − β ∞ # n=2 Vn (φ) ≤ ,[β] ≤ log Fβ (R(β)). " Example 2. βc φ can be positive recurrent, null recurrent or transient. Proof. Let {fn }n≥1 be a sequence such that fn > 0 and log fn = o(n). Set φ := ∗ (φ, 0) = f , p ∗ [φ] = 0 and 1 log f . Then, Z [0,n−1] n n n n≥1 n≥2 Vn (φ) = 0, whence 0 β −s by (2), ,0 [βφ] = log n≥1 fn . Let ζ (s) := n≥1 n . 1 β 1. Positive recurrence. Set fn := ζ (3)n 3 . Then ,0 [βφ] = log[ζ (3β)/ζ (3) ] whence ∗ βc = 1. Note that ,0 [βc φ] = 0 whence PG (βc φ) = −p0 [βc φ] = 0. It also follows that βc φ is recurrent. Positive recurrence follows from n≥1 ne−nPG (φ) Zn∗ (φ, 0) = 3 n≥1 n/(ζ (3)n ) < ∞. Note that βc φ is positive recurrent but not strongly positive recurrent.


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2. Null recurrence. The same calculations with fn = 1/(ζ (2)n2 ). 3. Transience. Set fn := C/n[log(2n)]2 , where C is a constant such that n≥1 fn = 21 . Similar calculations show that ,0 [βφ] is infinite for β < 1 and ,0 [βφ] ≤ − log 2 for β ≥ 1. Thus βc = 1 and βc φ is transient. " # For an example of the possible applications of theorem 5, consider the Manneville– Pomeau map T : [0, 1] → [0, 1] given by T (x) = x + x 1+s (mod 1), where the value of T at its discontinuity is 0, T (1) = 1 and s > 0 [PM]. The following proposition, is a generalization of results which are known for f = − log |T ! | (see [PM] and [Lo]) to other potentials, whose equilibrium measure is not necessarily equivalent to Lebesgue’s measure. Proposition 1 (The Manneville–Pomeau Model). Let T be the Manneville–Pomeau map and let f : [0, 1] → R be C[0, 1] ∩ C 1 (0, 1] such that f ! (x) ∼ cαx α−1 as x & 0, where c = 0 and α > 0. Set P (β) := sup hm (T ) + β f dm : m ∈ PT ([0, 1]); − f dm < ∞; m{0} = 0 . 1. There exists 0 < βc ≤ ∞ such that P (β) is real analytic in (0, βc ) and linear in (βc , ∞). It is continuous but not real-analytic at βc . 2. βc is finite if and only if α ≤ s and c < 0. In particular, it is finite for f := − log T ! . Proof. It is common knowledge that T can be described symbolically as a renewal shift. We check that the symbolic representation of f has summable variations and apply theorem 5. To do this we recall some facts on the natural Markov partition of T (see [I], Lemma 4.8.6 in [A] and [T1]). 1+s −s Define by induction c0 := 1 and cn = cn+1 + cn+1 . Rewriting this as cn+1 = −s s )s = c−s (1 + scs s )) we see that c −s ∼ s whence + o(c − c cn−s (1 + cn+1 n n n+1 n+1 n+1 cn ∼ (sn)−1/s . It follows from the recursive relation which defines {cn } that 1 . (sn)1+1/s n−1 −k Set I [n] := (cn+1 , cn ] and I [a0 , . . . , an−1 ] := I [ak ]. One checks that k=0 T T I [0] = (0, 1] and T I [n + 1] = I [n], whence I [a0 , . . . , an−1 ] is not empty if and only if (a0 , . . . , an−1 ) is an admissible word of the renewal shift. cn − cn+1 ∼

Claim 1. The diameter of I [a0 , . . . , an−1 ] satisfies for every ε > 0 1 |I [a0 , . . . , an−1 ]| = O . n1+1/s−ε

(9)

Proof. By the previous discussion, I [a0 , . . . , an−1 ] = I [a0 , . . . , an−1 ; an−1 − 1, . . . , 0], so we may assume that an−1 = 0. Set M := 1 + sup{ak } and N := |{k : ak = 0}|. Since (a0 , . . . , an−1 ) is admissible with respect to the transition matrix of the renewal shift, MN ≥ n. Thus, for every β ∈ (0, 1) either M ≥ nβ or N ≥ n1−β .

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Set mn := 'nβ ( + 1. If M ≥ nβ then for some power k, T k I [a0 , . . . , an−1 ] ⊆ I [mn ] whence since T ! ≥ 1 |I [a0 , . . . , an−1 ]| ≤ |I [mn ]| = cmn − cmn +1 = O(n−β(1+1/s) ). If N ≥ n1−β then for every x ∈ I [a0 , . . . , an−1 ] n !

(T ) (x) =

n−1

!

i

T (T x) ≥

i=0

!

N

inf T (x)

x∈I [0]

whence for θ := 1/inf x∈I [0] T ! (x) |I [a0 , . . . , an−1 ]| ≤ θ N .

(10)

Since θ < 1 and N ≥ n1−β we have again that I [a0 , . . . , an−1 ] = O(n−β(1+1/s) ). Since β ∈ (0, 1) was arbitrary, the claim is proved. " # Let (X, σ ) be therenewal shift and π0 : X → [0, 1] be the map defined by the equation {π0 (x)} = n≥0 I [x0 , . . . , xn−1 ] = n≥0 I [x0 , . . . , xn−1 ]. By (9) π0 is well defined. It is easy to check that π0 ◦ σ = T ◦ π0 , that π0 is 1-1 and that π0 (X) = [0, 1] \ ∪n≥0 T −n {0}. Claim 2. Let f be C[0, 1] ∩ C 1 (0, 1] in [0, 1] such that f ! (x) ∼ cαx α−1 as x ↓ 0, where c = 0 and α > 0. Then φ := f ◦ π0 has summable variations and φ, the induced potential on [0], is weakly Hölder continuous. Proof. Fix x, y ∈ [a0 , . . . , an−1 ], where without loss of generality an−1 = 0. Fix ε > 0 (to be determined later). Then there exists ξ ∈ I [a0 , . . . , an−1 ] such that α−1 ξ . |φ(x) − φ(y)| = |f ! (ξ )| · |I [a0 , . . . , an−1 ]| = O n1+1/s−ε Since ξ ∈ I [a0 , . . . , an−1 ] ⊆ (ca0 +1 , ca0 ), and since by the structure of the renewal shift a0 ≤ n − 1, we have that ξ α−1 = O(1 + cnα−1 ) whence 1 + n−(α−1)/s . Vn (φ) = O n1+1/s−ε If α ≥ 1 the nominator is bounded and choosing ε < 1/(2s) we see that Vn (φ) is (1−α)/s ) and we have that V (φ) = summable. If α < 1 then the nominator is O(n n Vn (φ) is summable. In any case, φ O(n−(1+α/s−ε) ). Choosing ε < α/s we see the has summable variations. The weak Hölder continuity of φ can be proved in a similar way. Claim 3. P (β) = PG (βφ), where φ := f ◦ π0 . Proof. Since sup f < ∞ and ∀x |T −1 x| = 2, Lβφ 1∞ < ∞. It follows as in ([S1, Theorem 3],) that PG (βφ) = sup hµ (σ ) + β φdµ : µ ∈ Pσ (X); − φdµ < ∞ (the argument there works also for functions with summable variations). The claim follows because m ↔ m ◦ π0 is a 1-1 onto correspondence between the sets of measures which define P (β) and PG (βφ).


565

Claims 2 and 3 show that we can apply Theorem 5 to φ = f ◦ π0 and deduce the existence of βc . We check the conditions for the finiteness of βc . Let An be as in Theorem 5. Since φ has summable variations, An exp fn (dn ), where dn ∈ I [0] are defined by T (dn ) = cn . It is easy to check that dn ↓ c1 , whence An exp nk=1 f (ck ). Without loss of generality, f (0) = 0 (addition of constants does not affect the finiteness of βc ). Then by the assumptions f , f (x) ∼ cx α . Since ck ∼ (sk)−1/s , f (ck ) ∼ c(sk)−α/s .

n on−α/s n Thus k=1 f (ck ) c 1 x dx. It follows that there exist constants K1 , K2 , K3 , K4 such that n n 1 1 β dx ≤ An ≤ K3 exp K4 βc dx . K1 exp K2 βc α/s α/s 1 x 1 x Let Fβ (ξ ) and R(β) be as in Theorem 5. Using the above, β

1. If α > s then An 1 for every β > 0. In this case Fβ (R(β)) = ∞ for every β, so βc = ∞. β 2. If α = s then K1 nK2 βc ≤ An ≤ K3 nK4 βc . It follows that R(β) = 1 and that F (R(β)) is infinite for every β if c > 0, and F (R(β)) −→ 0 if c < 0. Thus for α = s, if β→∞

c > 0 then βc is infinite, and if c < 0 then βc < ∞. a 3. If α ∈ (0, s) and a := 1−(α/s) then for some constants C1 , C2 , C3 , C4 , C1 eC2 βcn ≤ a β An ≤ C3 eC4 βcn . Since a < 1, R(β) = 1 for every β. It follows that if c > 0 then F (R(β)) = ∞ for every β, and if c < 0 then F (R(β)) −→ 0. Thus for 0 < α < s, β→∞

if c > 0 then βc is infinite and if c < 0 then βc is finite. Thus βc < ∞ if and only if 0 < α ≤ s and c < 0.

# "

5. Other Examples In this section we construct examples whose critical behavior is different than that of potentials on the renewal shift. Our constructions are based on the tools of [S3] which we now explain. We say that a one parameter family of functions Fβ (ξ ) is an exponent β power series if it is of the form Fβ (ξ ) = n,k≥0 ank ξ n , where ank ≥ 0. Clearly, if Fβ and Gβ are exponent power series, then so are Fβ Gβ , Fβ ◦ Gβ and c1 Fβ + c2 Gβ , where c1 , c2 are positive integers. We say that an exponent power series Fβ is aperiodic if the power expansion of Fβ contains two co-prime powers of ξ . We say that Fβ is adequate if it is of the form cβ ξ + ξ 2 Gβ (ξ ), where c ≥ 0 and Gβ is an exponent power series. The following theorem was essentially proved in [S3]. We include its proof for completeness. Theorem 6. For every adequate exponent power series Fβ there exists an irreducible topological Markov shift X and a Markov potential φ = φ(x0 , x1 ) such that for all β, PG (βφ + p) = log Fβ (ep ). If Fβ is aperiodic, X is topologically mixing. Proof. Write Fβ (ξ ) = cβ ξ +

∞ n=2

ξn

Nn k=1

β

ank ,

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O. M. Sarig

where 0 ≤ Nn ≤ ∞. Let S be a countable set indexed in the following way S := {a} ∪

Nn ∞

{bnk (1), . . . , bnk (n − 1)}.

n=2 k=1

Let (tij )i,j ∈S be the transition matrix whose non zero entries are exactly ta,bnk (1) , tbnk (i)bnk (i+1) , tbnk (n−1)a for all n, k ≥ 1 and i = 1, . . . , n − 1 with the addition of taa if and only if c = 0. Let X be the corresponding topological Markov shift. Define φ(x) by φ(x) := log ank if x ∈ [a, bnk (1)], φ(x) := log c if x ∈ [a, a] and φ(x) := 0 otherwise. One checks that Fβ (ξ ) =

∞ n=1

ξ n Zn∗ (βφ, a)

whence by (5) and the fact that ∀k ≥ 2 Vk (φ) = 0, PG (βφ + p) = log Fβ (ep ). Note that X is irreducible, because all states connect to a and a connects to all states. It is topologically mixing if and only if there are two words of co-prime lengths which connect a to a. This can be easily seen to be equivalent to the aperiodicity of ξ n Zn∗ (βφ, a), hence to that of Fβ . " # The following example shows that {βφ}β>0 can change from recurrent to transient an infinite number of times. (This is different than the example with infinite number of non differentiability points in [S3], which is always transient.) Example 3. There exists X topologically mixing and φ = φ(x0 , x1 ) such that for some βn ↓ 0, βφ is recurrent in (βi+1 , βi ) for i even and transient for i odd. Proof. Consider the following sequence of numbers 2 2n − 2 Nn := . n n−1 A calculation with Stirling’s formula shows that Nn ∼ π −1/2 n−3/2 22n−1 . Another calculation shows that 1 2n 1 2n − 2 4−n Nn = n−1 − n . (11) n−1 n 4 4 Multiplying both sides of (11) by 4n we see that Nn are all natural numbers. Summing both sides of (11) over n we see that n≥2 Nn 4−n = 21 . 1/β θ n < ∞ for all θ ∈ (0, 1) (e.g. Fix some βn ↓ 0 with the property that 1 β/βn βn := 1/n). Set αn (β) := −2( 2 ) and p(β) := n≥1 (1 + αn (β)). Then for all β > 0, p(β) is well defined, non zero for β ∈ {βn }n and satisfies p(β) = 1 + α1 (β) + [1 + α1 (β)] α2 (β) + . . . , where the convergence on the right is absolute. Collecting summands with the same β β sign write p(β) = A(β) − B(β), where A(β) = an and B(β) = bn for some an , bn ≥ 0. If β ∈ (βi+1 , βi ) then β > βn iff n ≥ i + 1 whence i ∞ 1−β/βn 1−β/βn sgn (A(β) − B(β)) = sgn (1 − 2 ) (1 − 2 ) = (−1)i . n=1

n=i+1


567

Thus, A(β) > B(β) iff i is even. Now construct X and φ = φ(x0 , x1 ) such that PG (βφ + p) = log Fβ (ep ), where Fβ (ξ ) is the exponent power series Fβ (ξ ) = 8A(β)B(β)ξ 2 +

∞

Nn B(β)n ξ n .

n=2

Since Nn

4n n3/2

the radius of convergence of Fβ (ξ ) is R(β) = 1/4B(β) whence ∞ A(β)B(β) −n ,a [βφ] = log Fβ (R(β)) = log 8 + Nn 4 16B(β)2 n=2

whence ,a [βφ] = log 21 (1 + A(β)/B(β)). This is positive iff A(β) > B(β). Thus βφ is recurrent for β ∈ (βi+1 , βi ) and i even, and transient for β ∈ (βi+1 , βi ) and i odd. # " We have seen that for potentials φ on the renewal shift, βφ can be null recurrence for at most one value of β (the critical point). Our next example shows that for other topological Markov shifts null recurrence can occur in an entire interval. A trivial example would be a Markov shift for which the potential φ ≡ 0 is null recurrent. We therefore restrict ourselves to examples where φ is not cohomologous to a constant. Example 4. There exist a topologically mixing topological Markov shift X and a function φ = φ(x0 , x1 ) not cohomologous to a constant such that βφ is null recurrent for every β. Proof. Let Nn be as in Example 3 and set fβ (p) := 2β (ep + e2p ). Construct X topologically mixing and φ = φ(x0 , x1 ) such that ∞ n Nn fβ (p) . PG (βφ + p) = log 2 n=2

4n n−3/2 ,

Since Nn pa∗ [βφ] is determined by the equation fβ (pa∗ [βφ]) = 1/4. It follows from this that ,a [βφ] = 0. By the Discriminant theorem for all β, βφ is recurrent and PG (βφ) = −pa∗ [βφ]. It also follows that φ is not cohomologous to a constant, since PG (βφ) is not a linear function of β (it is given by the equation fβ [−PG (βφ)] = 1/4). We show that βφ is null recurrent for all β. Since V2 (φ) = 0, PG (βφ + p) = log n≥1 enp Zn∗ (βφ, a) whence ∞ n=1

ne−nPG (βφ) Zn∗ (βφ, a) =

∞ d PG (βφ+p) ! ∗ e = 2f (p [βφ]) nNn 4−(n−1) β a dp p=pa∗

and this diverges, because Nn 4n n−3/2 .

n=2

# "

Our last example shows that all modes of recurrence can co-exist for interval ranges of inverse temperatures. Example 5. There exist X topologically mixing and φ = φ(x0 , x1 ) such that for some 1 < β1 < β2 < ∞, βφ is null recurrent for β ∈ (1, β1 ), positive recurrent for β ∈ (β1 , β2 ) and transient for β ∈ (β2 , ∞).

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Proof. Fix some positive an ∼ 1/[2n(log n)2 ] such that a1 = 43 , β β A(β) = n≥1 an , Fβ (ξ ) := n≥1 an A(β)n ξ n+1 and

n≥1 an

= 1 and set

Gβ (ξ ) = Fβ (2Fβ (ξ )). This is an adequate aperiodic exponent power series. Let X and φ be the corresponding shift and potential. Let RF (β) and RG (β) denote the radii of convergence of Fβ (ξ ) and Gβ (ξ ). Note that RF (β) = 1/A(β) and Fβ (RF (β)) = 1. Let β2 be the solution of RF (β2 ) = 2. Clearly, RF (β) < 2 for β < β2 and RF (β) > 2 for β > β2 . Thus, 1. if β ∈ (1, β2 ) then 2Fβ (RF (β)) = 2 > RF (β) so RG (β) = Fβ−1 ( 21 RF (β)). In this case Gβ (RG (β)) = Fβ (2 · 21 RF (β)) = Fβ (RF (β)) = 1; 2. if β > β2 then 2Fβ (RF (β)) = 2 < RF (β) so RG (β) = RF (β). In this case Gβ (2Fβ (RF (β)) < Fβ (RF (β)) = 1. Since ,a [βφ] = log Gβ (RG (β)), βφ is transient for β > β2 and recurrent for β ∈ (1, β2 ). We check positive recurrence and null recurrence for β ∈ (1, β2 ). Fix some β ∈ ∗ e−PG (βφ) = e−pa [βφ] = RG (β). (1, β2 ). Since Gβ (RG (β)) = 1, ,a [βφ] = 0 whence d Thus n≥1 ne−nPG (βφ) Zn∗ (φ, a) = RG (β) dξ Gβ (ξ ). Now, since RG (β) = Fβ−1 ( 21 RF (β)),

ξ =RG (β)

d Fβ (2Fβ (ξ )) = Fβ! (RF (β)) · 2Fβ! (RG (β)). dξ ξ =RG (β)

Since RG (β) < RF (β), this is finite iff Fβ! (RF (β)) < ∞, which is comparable to ∞

1 n . A(β) 2β nβ (log n)2β n=2

This sum is infinite for β ∈ (1, 2) and finite for β ∈ (2, β2 ). (Note that β2 > 2 since β a12 > 21 , whereas a1 2 < A(β2 ) = 21 .) " # 6. Proof of Theorem 2 The proof of Theorem 2 is based on a generalization of certain renewal theoretic ideas. These are presented in the following subsection. The proof of Theorem 2 is given in the subsection following it. 6.1. A renewal sequence of operators. Let a ∈ S be some fixed state. Let CB [a] be the Banach space CB [a] := {f ∈ CB (X) : f (x) = 0 for x ∈ [a]} equipped with the supremum norm. Let 1, 0 : CB [a] → CB [a] be the operators defined by 1f = f, 0f = 0 ∀f ∈ CB [a]. Consider the operators Tn , Rn : CB [a] → CB [a] given by T0 := 1 , R0 := 0 and Tn f := 1[a] Lnφ f Rn f := 1[a] Lnφ (f 1[ϕa =n] )


569

(see also [FL] and [PS].) A direct calculation shows that these operators satisfy the following “renewal equation” for n ≥ 1, Tn = R1 Tn−1 + R2 Tn−2 + . . . + Rn T0 , Tn = Tn−1 R1 + Tn−2 R2 + . . . + T0 Rn . Set Ta [φ](z) := 1 +

∞

n

z Tn ,

Ra [φ](z) :=

n=1

∞

(12)

z n Rn .

n=1

These are well defined bounded linear operators on CB [a] for |z| < λ−1 . To see this usethe summable variations property toprove that Ta [φ](z) = Ta [φ](z)1[a] ∞ ≤ B n≥0 |z|n Zn (φ, a), where B := exp n≥2 Vn (φ), and note that the radius of conver gence of the series zn Zn (φ, a) is λ−1 by (1). In terms of these generating functions, we can restate (12) in the following form ∀|z| < λ−1 , Ta [φ](z) = [1 − Ra [φ](z)]−1 . It also follows from (12) that for all |z| < λ−1 , Ta [φ](z) = 1 +

∞

Ra [φ](z)n .

(13)

n=1

Note that (13) is also valid for all z real such that z ≥ λ−1 , as long as both sides are applied to positive functions. For every bounded linear operator S on CB [a] let ρ(S) denote the spectral radius of S (with respect to the supremum norm), with the convention that the ‘operator’Sf = ∞1[a] has an infinite spectral norm. The following two propositions relate the renewal sequence to the discriminant. Proposition 2. Let φ denote the induced potential on [a]. Then PG (φ + p) = log ρ Ra [φ](ep ) . Proof. Let π : X → [a] be the natural embedding. A calculation shows that for every p and f ∈ CB [a]

Ra [φ](ep )f ◦ π = Lφ+p (f ◦ π ).

Fix some [a] ∈ S. Since X = S follows from this and (14). " #

N∪{0}

(14)

, Zn (φ + p, [a]) Lnφ+p 1∞ . The proposition

Proposition 3. Let X be topologically mixing countable Markov shift, let φ be a function with summable variations and finite Gurevich pressure. Let X and φ denote the induced pair with respect to a ∈ S. Then PG (φ + p) is convex, strictly increasing and continuous in (−∞, pa∗ [φ]]. Also, (2)–(5) hold.

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Proof. Fix a ∈ S and set γ (p) := PG (φ + p), p ∗ := pa∗ [φ], z := ep and R(z) := Ra [φ](z). Proposition 2 and summable variations imply that for all x ∈ [a], 1 log R(ep )n 1[a] n→∞ n 1 = lim log R(ep )n 1[a] (x). n→∞ n We use this formula to prove that γ is convex, strictly increasing and continuous in (−∞, p∗ ) and that (5) and(4) hold. The expressions R(ep )n 1[a] (x) are convex in p, since they are of the form ai ebi p , where ai ≥ 0. Thus γ (p) is convex in p, being a limit of convex functions. Clearly, ∀p1 < p2 < p ∗ and all 0 ≤ f ∈ CB [a], R(ep2 )f ≥ ep2 −p1 R(ep1 )f . Iterating this and using the above formula for γ (p) we have that γ (p) is strictly increasing. It follows that γ is finite in (−∞, p ∗ ), whence by convexity it is continuous there. Standard estimations show that for every 0 ≤ f ∈ CB [a], R(ep )f = ±1 ∗ B n≥1 Zn (φ, a)f , where B = exp k≥2 Vk (φ). Iterating this, and using the above formula for γ (p), we have (5). Clearly, (5) implies (4), so (4) is also proved. It remains to prove that γ (p) is continuous on the left in p ∗ . We prove this under the assumption that γ (p∗ ) < ∞ (the proof for the infinite case is essentially the same). By monotonicity, it is enough to prove that for every ε > 0 there exists p < p ∗ such that ∗ p γ (p) > γ (p ) − ε. Fix x ∈ [a]. Setting R(e ) = n≥1 enp Rn in the above formula for γ , we have γ (p) = lim

1 log n→∞ n

γ (p) = lim

∞

ep(k1 +...+kn ) Rk1 · . . . · Rkn 1[a] (x).

(15)

k1 ,... ,kn =1

By the definition of p∗ , there are N and p and n > log B/ε such that an := log

N

ep(k1 +...+kn ) Rk1 · . . . · Rkn 1[a] ≥ n(γ (p ∗ ) − ε).

k1 ,... ,kn

By the summable variations property, am1 + am2 ≤ am1 +m2 + log B for every m1 , m2 . Write m = kn + r, where 0 ≤ r ≤ n − 1. Then am an kan + ar − (k + 1) log B log B ≥ −→ − ≥ γ (p ∗ ) − 2ε m→∞ m kn + r n n whence γ (p) ≥ γ (p∗ ) − 2ε. This proves that γ is continuous in (−∞, p ∗ ]. This also implies that (2). This and (5) imply (3). " # We will need the following version of the Kac formula, whose proof follows in a standard way from Theorem 1 and general results for Markov operators (see Theorem VI.C in [F]). Lemma 3. Let X be a topologically mixing Markov shift, assume φ has summable variations and fix some a ∈ S. Let (X, φ) be the induced system on [a], and assume that both φ and φ have summable variations. Then φ is recurrent with pressure zero if and only if φ is positive recurrent with pressure zero. In this case, if L∗φ ν = ν, Lφ h = h, L∗φ ν = ν, Lφ h = h then up to a multiplicative constant ν = ν ◦ π ,

∞ n ∞ n−1 k h = h◦π, ν(A) = n=0 @ 1A ◦π dν and h = n=1 k=0 Lφ (ha 1[ϕa =n] ) mod ν, where @ is the operator @(f ) := Lφ (f · 1[a]c ) and ha := 1[a] h ◦ π −1 .


571

6.2. Proof of Theorem 2. Throughout the proof let a and φ be fixed. Let T (z) := Ta [φ](z), R(z) := Ra [φ](z). Let B := exp k≥2 Vk [φ] and set , := ,a [φ], γ (p) := PG (φ + p), and p ∗ := pa∗ [φ]. Part 1. Proof of (6). Assume , ≥ 0. According to Proposition 3, γ is continuous and strictly increasing in (−∞, p∗ ], γ (p ∗ ) ≥ 0 and γ (p) → −∞. Therefore, there exists a unique pa (φ) for which γ (pa (φ)) = 0. strictly increasing, We claim that pa (φ) = −PG (φ). Fix some p < pa (φ). Since γ is By (13), T (ep )1[a] ∞ ≤ 1 + R(ep )n < ∞ γ (p) < 0, whence ρ(R(ep )) < 1. whence, by summable variations, enp Zn (φ, a) converges. The radius of convergence of this series is exp[−PG (φ)]. Therefore p ≤ −PG (φ). Taking p ↑ pa (φ) we have pa (φ) ≤ −PG (φ). Assume by way of contradiction that ∃pa (φ) < p! < ! ! −P (φ). Then T (ep ) ≤ B enp Zn (φ, a) < ∞ whence by (13), the series 1 + G p! n R(e ) 1[a] (x) converges for every x. Summable variations imply that ∀x ∈ [a] and ! ! ! ! ∀n ≥ 1, R(ep )n ≤ BR(ep )n 1[a] (x). Thus R(ep )n < ∞ whence ρ[R(ep )] ≤ ! 1, or equivalently, γ (p ) ≤ 0. This, however, is impossible because γ is strictly increasing, γ (pa (φ)) = 0 and p ! > pa (φ). This proves that pa (φ) = −PG (φ) and settles (6) for the case , ≥ 0. Assume now that , < 0. In this case there is no solution for the equation γ (p) = 0, because for p ≤ p∗ γ (p) ≤ , < 0 and for p > p ∗ γ (p) = ∞. We show that in this case PG (φ) = −p∗ . By (4) and the inequality Zn∗ (φ, a) ≤ Zn (φ, a), p∗ ≥ −PG (φ). Assume by way contradiction that p ∗ > −PG (φ). Then x ∈ [a], for every of ∗ ∗ ∗ p −1 np e Zn (φ, a) = ∞, whence by (13), 1 + R(ep )n diverges T (e )1[a] (x) ≥ B ∗ everywhere on [a]. Thus ρ[R(ep )] ≥ 1, whence , = γ (p ∗ ) ≥ 0 in contradiction to our assumptions. This settles the case , < 0. Part 2. Proof that recurrence is equivalent to , ≥ 0. Assume that , ≥ 0. By the first part of the theorem γ (−PG (φ)) = 0 so the spectral norm of R(e−PG (φ) ) is iξ R(e−PG (φ) ) does not equal to 1. Therefore, there exists ξ ∈ [0, 2π ) such that 1 − e have a bounded inverse operator. In particular, the series 1 + k≥1 eikξ R(e−PG (φ) )k does not converge in the strong norm. It follows that there exists some ε > 0 such that for every N there exists n = n(N ) > N and gn ∈ CB [a] such that gn ∞ = 1 and k≥n eikξ R(e−PG (φ) )k gn ∞ > ε. Now, on [a] ∞

R(e

k=n

−PG (φ) k

) 1[a] ≥

∞ k=n

R(e

∞ ikξ −PG (φ) k ) |gn | ≥ e R(e ) gn ≥ ε

−PG (φ) k

k=n

whence k≥n R(e−PG (φ) )k 1[a] ∞ ≥ ε for every n. By the summable variations prop erty this is only possible if R(e−PG (φ) )k 1[a] diverges on [a] whence T (e−PG (φ) )1[a] ∞ = ∞. This is equivalent to e−kPG (φ) Zk (φ, a) = ∞, so φ is recurrent. ∗ Assume that , < 0. Set ρ := ρ[R(ep )]. Then ρ = exp , < 1. By the definition of the spectral radius, there exists some C and ρ0 ∈ (ρ, 1) such that for every n, ∗ ∗ R(ep )n ∞ < Cρ0n . The renewal equation implies that T (ep ) ≤ C/(1 − ρ0 ). It ∗ ∗ follows that T (ep ) is bounded, whence ekp Zk (φ, a) is convergent. By the first part of the theorem, and since , < 0, p ∗ = −PG (φ). It follows that φ is transient.

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Part 3. Proof that , > 0 implies positive recurrence. Assume that , > 0. By what we have just proved φ is recurrent. Let ν and h be the eigenmeasure and eigenfunction given by theorem 1, and set dm = hdm. Recall that m an invariant measure, and that m(X) < ∞ if and only if φ is positive recurrent. We will prove positive recurrence by showing that m(X) < ∞. Let νa , ma be the measures νa (A) := ν(A ∩ [a])/ν[a] and ma (A) := m(A ∩ [a])/m[a]. Let Ta := T ϕa be the induced transformation. Since m is T -invariant, ma is Ta -invariant. Note that the transfer operator of Ta with respect to νa is R(λ−1 ). To see this note that ∀g ∈ L∞ (νa ), ∀f ∈ L1 (νa ), νa [gR(λ−1 )f ] =

∞ ν λ−n Lnφ g ◦ T n · f 1[ϕa =n] = νa (g ◦ Ta · f ). n=1

Set A(x) := R(λ−1 )ϕa (x). By Kac’s formula, the fact the R(λ−1 ) acts as the transfer operator of νa and the boundness of h = dm dν away from zero and infinity on partition sets, ∃C1 > such that A(x)dν(x). m(X) = ϕa dma = C1±1 ϕa dνa = [a]

Clearly, A(x) =

∞

nλ−n

n=1

T n y=x

eφn (y) 1[ϕa =n] (y) = B ±1

∞ n=1

nλ−n Zn∗ (φ, a).

∗ −1 is smaller than the radius of convergence of the Since , >n 0,∗ p > −PG (φ) so λ series z Zn (φ, a). It follows that nλ−n Zn∗ (φ, a) < ∞ whence A(x)∞ < ∞. Since ν[a] < ∞, m(X) < ∞ as required. " #

7. Proof of Theorem 4 In this section we prove Theorem 4, a strengthened version of Theorem 3. 7.1. Preparatory lemmas. Let X be topologically mixing and let a ∈ S be some fixed state. Let φ be some function with summable variations and finite pressure and let (X, φ) be the induced system on [a]. For every x, y ∈ X set t(x, y) := inf{n ≥ 0 : x n = y n }. Fix θ ∈ (0, 1) and set for every function f : X → C , Df := sup{|f (x) − f (y)|/θ t(x,y) : x = y}. Let L = L(θ, a) be the space L(θ, a) := {f ∈ CB (X) : f L := f ∞ + Df < ∞}.

(16)

A standard argument shows that L is a Banach space and that if Lφ 1∞ < ∞ then Lφ (L) ⊂ L and Lφ B(L) < ∞, where B(L) is the space of bounded operators on L equipped with the strong operator norm. The following lemma says that the induced system has a spectral gap, and is similar to well-known results in the theory of interval maps with indifferent fixed points ([T1,T2,A,ADU, B, PS]).


573

Lemma 4. Let X be topologically mixing, φ some function on X and a ∈ S a fixed state. Let (X, φ) be the induced system on [a] and assume that φ is weakly Hölder continuous with exponent θ ∈ (0, 1) and that Lφ 1∞ < ∞. Then φ is positive recurrent if and only if the spectrum of Lφ : L(θ, a) → L(θ, a) consists of a simple eigenvalue λ and a subset of {z : |z| < τ λ}, where τ < 1. In this case, λ = ePG (φ) .

Proof. Assume that φ is positive recurrent with finite pressure and set λ := exp PG (φ). X has the structure of a full shift. It is known ([S1, Sect. 5], see also [A, Theorem 4.7.7] and [Yu]) that for such systems there exists K > 0 and τ ∈ (0, 1) such that for every f ∈L −n λ (L )n f − h f dν < Kτ n , φ L

where Lφ h =

λ h, L∗φ ν

= λν and ν(h) = 1. This implies the required spectral property. −n n Lφ

The opposite direction is trivial, since the spectral property implies that λ

has a non

trivial limit (the eigenprojection of λ), and this is only possible if φ is positive recurrent with pressure log λ. " # Lemma 5. Let X be topologically mixing and let φ be a function with summable variations such that PG (φ) < ∞ and ,a [φ] > 0. For every ψ ∈ Dir(φ), ∃ε > 0 ∃r > exp[−PG (φ)] such that n≥1 nr n Zn∗ (φ + ε|ψ|, a) < ∞. Proof. Without loss of generality, PG (φ) = 0 (else pass to φ−PG (φ)). Since ,a [φ] > 0, ∃r G (φ)] = 1 such that Ra [φ](r)1[a] ∞ is finite, or equivalently, > nexp[−P ∗ n≥1 r Zn (φ, a) < ∞. Without loss of generality lim sup n→∞

1 log Zn∗ (φ, a) < − log r. n

Set fn (t) := (1/n) log Zn∗ (φ + tψ, a) and f (t) := lim supn→∞ fn (t). By Hölder’s inequality, fn are convex, whence so is f . Since ψ ∈ Dir(φ), there is some ε > 0 such that ∀|t| < 2ε, −∞ ≤ f (t) ≤ PG (φ + tψ) < ∞. By convexity and since f < ∞, either f (t) = −∞ everywhere in (−2ε, 2ε), or |f (t)| < ∞ everywhere in (−2ε, 2ε). In the first case the radius of convergence of k ∗ k≥1 x Zk (φ + tψ, a) is infinite for t = ±ε and we are done. In the second case, by convexity and finiteness, f (t) is continuous in (−2ε, 2ε). Thus, since r was chosen so ! f (t) < − log r. It follows that f (0) < − log r, there exists ε ! < ε such that ∀|t| < 2ε that r is strictly smaller than the radius of convergence of k≥1 x k Zk∗ (φ + tψ, a) for # t = ±ε! and again, we are done. " Recall that function F : C × C → B(L) is called analytic in a neighborhood of (z0 , w0 ) if ∃Fnk ∈ B(L) such that F (z, w) = n,k≥0 (w − w0 )k (z − z0 )k Fnk and the series converges in the strong operator norm in a neighborhood of (z0 , w0 ). Lemma 6. Let X be topologically mixing, let φ be some function with summable variations such that PG (φ) < ∞ and let ψ ∈ Dir(φ). Let a ∈ S be some state such that ,a [φ] > 0 and assume that φ and ψ, the induced potentials on [a], are weakly Hölder continuous with parameter θ. Then F : C×C → B(L) given by F (z, w) = Lφ+zψ+log w is analytic in a neighborhood of (z, w) = (0, e−PG (φ) ).

574

O. M. Sarig

Proof. Throughout this proof · denotes the strong operator norm in B(L). We assume, without loss of generality, that PG (φ) = 0 and prove analyticity in (0, 1). For every function g : X → C let Mg be the operator Mg f = gf . Set En := {x ∈ X : ϕa (π(x)) = n}. This is a union of partition sets in X. Set R n := Lφ M1En . Then, F (z, w) = =

∞

w n R n Mezψ

n=1 ∞ ∞ n=1 k=0

w n zk R n Mψ k . k!

We show that this converges in B(L) in some open ball containing (z, w) = (0, 1). 2 N Fix some N and set AN (x) := ex − (1 + x + x2! + . . . , xN! ). Then 1 n k n z R M |w| M w k ≤ R . n n A (zψ) N ψ k! n,k>N n>N We estimate the summands of the last series. For every p ∈ S set Qp f (x) := f (p x). Let E!n := {p ∈ S : [p] ⊆ En }. By definition, R n MAN (zψ) = p∈E!n Qp MA (zψ)eφ . Since for every f, g ∈ L, f gL ≤ N f L gL , Qp MAN (zψ) exp φ ≤ Qp eφ Qp AN (zψ)L . L

It is standard to check that ∀x, y ∈ C, |AN (x) − AN (y)| ≤ |x − y|(e|x| + e|y| ) and that ∀x, y ∈ R, |ex − ey | ≤ |x − y|(ex + ey ). Using this and the inequality |AN (x)| ≤ e|x| , it is easy to show that there is some constant K1 (independent of n and N ) such that ∀|z| < 1, Qp MAN (zψ) exp φ ≤ K1 eφ 1[p] e|zψ| 1[p] . ∞

∞

Summing over all p ∈ E!n and using weak Hölder continuity, we have that for some K independent of n and N , R n MAN (zψ) ≤ KZn∗ (φ + |z| · |ψ|, a). Let ε > 0 and r > 1 be as in Lemma 5. Without out loss of generality re−ε > 1 and ε < 1. Then for all |z| < ε and |w| < r, ∞

∞ ∞ |w|n |z|k Rn Mψnk ≤ K r n Zn∗ (φ + ε|ψ|, a) −→ 0 N→∞ k!

n=N+1 k=N+1

n=N+1

whence F (z, w) is analytic in a neighborhood of (0, 1).

# "


575

7.2. Proof of Theorem 4. Let φ be a function with summable variations and finite pressure and let ψ ∈ Dir(φ). Assume that ∃a ∈ S such that ,a [φ] > 0 and such that the induced potentials on [a], φ, ψ are weakly Hölder continuous with exponent θ ∈ (0, 1). Without loss of generality, assume that PG (φ) = 0. Set G(z, w) := PG (φ + zψ − w). By the discriminant theorem, ∀z ∈ R, if ∃w ∈ R such that G(z, w) = 0, then w = PG (φ + zψ). Thus, PG (φ + zψ) is given implicitly by G(z, PG (φ + zψ)) = 0.

(17)

We will show that G has a complex holomorphic extension to a neighborhood of (z, w) = (0, 0) in C×C, and apply the complex implicit function theorem ([Boch], p. 39) to deduce that (17) defines PG (φ +zψ) real analytically in a neighborhood of z = 0. (This theorem applies since ∀h > 0, G(0, h) ≤ PG (φ − h) = PG (φ) − h whence Gw (0, 0) = 0.) By Theorem 2 and Lemma3, since ,a [φ] > 0 and PG (φ) = 0, φ is positive recurrent with pressure zero. By (5), Zn∗ (φ, a) < ∞ whence Lφ < ∞. By Lemma 4 the spectrum of Lφ : L → L consists of the simple isolated eigenvalue 1 and a compact subset of {z : |z| < τ } for some τ < 1. By standard analytic perturbation theory [Ka], there exists δ > 0 such that if L − Lφ B(L) < δ then L has a (unique) simple eigenvalue λ(L) of maximal magnitude, this eigenvalue is simple, has magnitude larger than (1 + τ )/2, and the rest of the spectrum is contained in {z : |z| < (1 + τ )/2}. Furthermore, the map L → λ(L) is holomorphic in {L ∈ B(L) : L − Lφ B(L) < δ}. By lemma 6, ∃ε > 0 such that (z, w) → Lφ+zψ−w is holomorphic in U := {(z, w) ∈ C2 : |z|, |w| < ε} and such that Lφ+zψ−w − Lφ B(L) < δ for all |z|, |w| < ε. In this neighborhood we define ! G (z, w) := log λ Lφ+zψ−w . ! G is holomorphic in U . For every z, w real such that (z, w) ∈ U , the spectrum of Lφ+zψ−w consists of a simple eigenvalue λ(z, w) and a compact subset of {λ : |λ| < |λ(z, w)|}. By Lemma 4, φ + zψ − w is positive recurrent with pressure log λ(z, w) = ! G (z, w). It follows that ! G is a holomorphic extension of G. This proves that t → PG (φ + tψ) is real analytic in (−ε, ε). We show that φ + tψ is positive recurrent for |t| small. Real analyticity implies continuity, so ∃δ ! > 0 such that ∀|t| < δ ! , PG (φ + tψ) ∈ (− 2ε , 2ε ). Set w := −PG (φ + tψ). Then |w − 3ε | < ε whence PG (φ + tψ − w + 3ε ) = G(t, w − 3ε ) < ∞. Since PG (φ + tψ + p) is increasing in p, PG (φ + tψ + ( 3ε − w)) > G(t, w) = 0 whence # ,a [φ + tψ] > 0. " Acknowledgements. This paper is part of a dissertation prepared in the Tel-Aviv university under the supervision of Jon Aaronson. I wish to express my deep gratitude to Jon Aaronson for his support and countless useful suggestions.

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O. M. Sarig

References [A] [ADU] [Boch] [B] [E] [F] [FF] [FL] [GS]

[G1] [G2] [G3] [GW] [Hof] [I] [Ka] [LSV] [Lo] [PM] [PP] [PS] [Ru1] [Ru2] [S1] [S2] [S3] [T1] [T2] [Wal] [W1] [W2]

Aaronson, J.: An introduction to infinite ergodic theory. Math. Surv. and Monographs 50, Providence, RI: AMS, 1997 Aaronson, J., Denker, M., Urbanski, M.: Ergodic theory for Markov fibered systems and parabolic rational maps. Trans. Am. Math. Soc. 337, 495–548 (1993) Bochner, S., Martin, W.T.: Several Complex Variables. Princeton, NJ: Princeton Univ. Press, 1948 Bowen, R.: Invariant measures for Markov maps of the interval. Commun. Math. Phys. 69, 1–17 (1979) Ellis, R.S.: Entropy, Large Deviations and Statistical Mechanics. Berlin–Heidelberg–New York: Springer Verlag, 1985 Foguel, S.R.: The ergodic theory of Markov processes. New-York: Van-Nostrand, 1969 Fisher, M.E., Felderhof, B.U.: Phase Transitions in One-Dimensional Cluster Interaction Fluids: IA. Thermodynamics, IB. Critical Behavior. Ann. of Phys. 58, 176–280 (1970). Foguel, S.R., Lin, M.: Some Ratio Limit Theorems for Markov Operators. Z. Wahrschein. verw. Geb. 23, 55–66 (1972) Gurevich, B.M., Savchenko, S.V.: Thermodynamic formalism for countable symbolic Markov chains. Uspekhi. Mat. Nauk. 53 2, 3–106 (1998); Engl. Transl. in Russian Math. Surv. 53 2, 245–344 (1998) Gurevich, B.M.:AVariational Characterization of One-Dimensional Countable State Gibbs Random Fields. Z. Wahrscheinlichkeitstheorie verw. Gebiete 68, 205–242 (1984) Gurevic, B.M.: Shift entropy and Markov measures in the path space of a denumerable graph. Dokl. Akad. Nauk. SSSR 192 (1970); Engl. Transl. in Soviet Math. Dokl. 11, 744–747 (1970) Gurevic, B.M.: Topological entropy for denumerable Markov chains. Dokl. Akad. Nauk. SSSR 187 (1969); Engl. Transl. in Soviet Math. Dokl. 10, 911–915 (1969) Gaspard, P., Wang, X.-J.: Sporadicity between periodic and chaotic dynamical behaviors. Proc. Math. Acad. Sci. USA 85, (13), 4591–4595 (1988) Hofbauer, F.: Examples for the nonuniqueness of the equilibrium state. Trans. Am. Math. Soc. 228, 133–241 (1977) Isola, S.: Dynamical Zeta Functions and Correlation Functions for Non-Uniformly Hyperbolic Transformations. Preprint Kato, T.: Perturbation Theory of Linear Operators. Berlin–Heidelberg–New York: Springer Verlag, 1966 Liverani, C., Saussol, B., Vaienti, S.: A probabilistic approach to intermittency. Erg. Th. Dyn. Sys. 19, 671–685 (1999) Lopes, A.O.: The Zeta function, Non-differentiability of Pressure, and the Critical Exponent of Transition. Adv. in Math. 101, 133–165 (1993) Pomeau, Y., Manneville, P.: Intermittent transition to turbulence in dissipative dynamical systems. Commun. Math. Phys. 74, 189–197 (1980) Parry, W., Pollicott, M.: Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque 187–188 (1990) Prellberg, T., Slawny, J.: Maps of intervals with indifferent fixed points: Thermodynamic formalism and phase transitions. J. Stat. Phys. 66, (No. 1/2), 503–514 (1992) Ruelle, D.: Thermodynamic Formalism. Encycl. of Math. and its App. Vol 5, Reading, MA: AddisonWesley, 1978 Ruelle, D.: Statistical Mechanics of a one-dimensional lattice gas. Commun. Math. Phys. 9, 267–278 (1968) Sarig, O.M.: Thermodynamic Formalism for Countable Markov Shifts. Erg. Th. Dyn. Sys. 19, 1565–1593 (1999) Sarig, O.M.: Thermodynamic Formalism for Null Recurrent Potentials. To appear in Isreal J. Math. Sarig, O.M.: On an example with topological pressure which is not analytic. C. R. Acad. Sci. Paris série I 330, 311–315 (2000) Thaler, M.: Transformation on [0, 1] with infinite invariant measures. Isr. J. Math. 46, 67–96 (1983) Thaler, M.: Estimates of the invariant densities of endomorphisms with indifferent fixed points. Isr. J. Math. 37, (4), 303–314 (1980) Walters, P.: Ruelle’s operator theorem and g-measures. Trans. Am. Math. Soc. 214, 375–387 (1978) Wang, X.-J.: Abnormal Fluctuations and thermodynamic phase transition in dynamical systems. Phys. Rev. A 39, (No. 6), 3214–3217 (1989) Wang, X.-J.: Statistical Physics of temporal intermittency. Phys. Rev. A 40, (No. 11), 6647–6661 (1989)


[Ya] [Yu]

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Young, L.-S.: Statistical properties of dynamical systems with some hyperbolicity. Ann. Math. 147, 585–650 (1998) Yuri, M.: Multidimensional maps with infinite invariant measures and countable state sofic shifts. Indag. Math. 6, 355–383 (1995)

Communicated by Ya. G. Sinai

Commun. Math. Phys. 217, 579 – 593 (2001)

Communications in



Equivalence of Projections as Gauge Equivalence on Noncommutative Space Kazuyuki Furuuchi Laboratory for Particle and Nuclear Physics, High EnergyAccelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan Received: 8 June 2000 / Accepted: 18 October 2000

Abstract: Projections play crucial roles in the ADHM construction on noncommutative R4 . In this article a framework for the description of equivalence relations between projections is proposed. We treat the equivalence of projections as “gauge equivalence” on noncommutative space. We find an interesting application of this framework to the study of the U (2) instanton on noncommutative R4 : A zero winding number configuration with a hole at the origin is “gauge equivalent” to the noncommutative analog of the BPST instanton. Thus the “gauge transformation” in this case can be understood as a noncommutative resolution of the singular gauge transformation in ordinary R4 . 1. Introduction The concept of a smooth space-time manifold should be modified at the Planck scale due to quantum fluctuations, and we expect that the short scale structure of space-time has noncommutative nature. When the coordinates of the space are noncommutative, we except the appearance of short scale cut off at the noncommutative scale. For example, instantons on noncommutative R4 constructed by the ADHM method [1] never become singular [2], due to the cut off in the size of instanton.1 Although the noncommutativity in this case is quite simple, the construction reveals deep insights into the nature of gauge theory on noncommutative space. Indeed, the precise mechanism that leads to the absence of singularity is quite nontrivial. In order to construct instantons on noncommutative R4 , one needs to project out some states in Hilbert space, where the Hilbert space is introduced to represent the algebra of noncommutative R4 .2 Since noncommutative R4 is defined by the whole Hilbert space and projection removes some of the states in this Hilbert space, projection can be interpreted as a change of topology of the base 1 This is the case when the noncommutativity of the coordinates has self-dual part (and instantons are anti-self-dual) [5]. 2 This necessity of the projection depends on the choice of “Murray–von Neumann gauge” introduced in this article. There is a choice of “gauge” that does not need projection.

580

K. Furuuchi

manifold. More precisely, projection removes some points from R4 and creates holes. Hence instantons on noncommutative R4 indicate the necessity for the unified description of gauge fields and geometry [2, 3]. In this article a framework for the description of equivalence relations between projections is proposed. We treat the equivalence of projections as a kind of gauge equivalence. Hence the formalism of this framework is similar to the gauge theory. However since the projection contains information of the Hilbert space which represents noncommutative R4 , the transformation between equivalent projections may be regarded as a noncommutative analog of coordinate transformation. Therefore this is a possible framework for the unified description of gauge fields and geometry. We find an interesting application of this framework to the study of U (2) instanton on noncommutative R4 . 2. Equivalence of Projections as Gauge Equivalence on Noncommutative Space In this section we explain the notion of the equivalence of projections in a concrete example, the gauge theory on noncommutative R4 . However it is obvious that the following arguments can be extended to gauge theory on more general noncommutative space. Reviews on Gauge Theory on Noncommutative R4 . The noncommutative R4 we shall consider is described by an algebra generated by the noncommutative coordinates x µ (µ = 1, · · · , 4) which satisfy the following commutation relations: [x µ , x ν ] = iθ µν ,

(2.1)

where θ µν is real and constant. In this article we consider the case where the θ µν is self-dual, and set ζ , ζ > 0 (others: zero), 4 for simplicity. Next we introduce the complex noncommutative coordinates by θ 12 = θ 34 =

z1 = x2 + ix1 ,

z2 = x4 + ix3 .

(2.2)

(2.3)

Their commutation relations become [z1 , z¯ 1 ] = [z2 , z¯ 2 ] = −

ζ 2

(others: zero).

(2.4)

We start with the algebra End H of operators acting in the Hilbert space C |n1 , n2 , H= (n1 ,n2 )∈Z2≥0

where z and z¯ are represented as creation and annihilation operators: √ 2 2 z1 |n1 , n2 = n1 + 1 |n1 + 1, n2 , z¯ 1 |n1 , n2 = n1 |n1 − 1, n2 , ζ ζ √ 2 2 z2 |n1 , n2 = n2 + 1 |n1 , n2 + 1 , z¯ 2 |n1 , n2 = n2 |n1 , n2 − 1 . (2.5) ζ ζ

Equivalence of Projections as Gauge Equivalence on Noncommutative Space

581

The commutation relations in (2.1) have automorphisms of the form x µ → x µ + cµ , where cµ is a commuting real number. These automorphisms are generated by a unitary operator Uc : Uc := exp[cµ ∂ˆµ ],

(2.6)

where we have introduced a derivative operator ∂ˆµ by ∂ˆµ := iBµν x ν .

(2.7)

Here Bµν is an inverse matrix of θ µν . ∂ˆµ satisfies following commutation relations: [∂ˆµ , x ν ] = δµν ,

[∂ˆµ , ∂ˆν ] = iBµν .

(2.8)

One can check the following equation: Uc x µ Uc† = x µ + cµ . We define a derivative of operators Oˆ ∈ End H by 1 ˆ † µ − Oˆ = [∂ˆµ , O]. ˆ µ OU ∂µ Oˆ := lim U δc δc δcµ →0 δcµ

(2.9)

(2.10)

The action of two derivatives commutes: ˆ − (µ ↔ ν) = 0. ∂µ ∂ν Oˆ − ∂ν ∂µ Oˆ = [∂ˆµ , [∂ˆν , O]]

(2.11)

Operator Oˆ is called a bounded operator if ˆ ∀ |φ ∈ Dom(O),

||Oˆ |φ || ≤ C|| |φ ||,

(2.12)

ˆ is a domain of operator O. ˆ The norm of for some constant C > 0, where Dom(O) bounded operators are defined by ˆ := sup ||O||

|| Oˆ |φ || ˆ , φ = 0, |φ ∈ Dom(O), || |φ ||

(2.13)

where sup means the supremum. We call the operator smooth when the derivative of the operator is a bounded operator. We shall consider the algebra of smooth bounded operators and denote this algebra by A. The U (n) gauge field on noncommutative R4 is defined as follows. First we consider an n-dimensional vector space An := Cn ⊗ A. The elements of An can be thought of as n-dimensional vectors with their entries in A. Let us consider the unitary action on the element of An : φ → U φ.

(2.14)

Here U ∈ Mn (A) (Mn (A) denotes the algebra of n × n matrices with their entries in A) and satisfying U U † = U † U = IdMn (A) , where IdMn (A) is the identity operator in Mn (A). In general U depends on z and z¯ , and hence we regard this unitary transformation as a gauge transformation. We define the action of an exterior derivative d by da := (∂µ a) dx µ ,

a ∈ A.

(2.15)

582

K. Furuuchi

We define the covariant derivative of φ ∈ An as a derivative which transforms covariantly under the gauge transformation (2.14), i.e. Dφ → U Dφ,

D = d + A.

(2.16)

Here the U (n) gauge field A is introduced to ensure the covariance. A is a matrix valued one-form: A = Aµ dx µ and Aµ ∈ Mn (A) is anti-Hermitian. dx µ commute with x µ and anti-commute among themselves, and hence d 2 a = 0 for a ∈ A. From (2.14) and (2.16), the covariant derivative transforms as D → U DU † .

(2.17)

Hence the gauge field A transforms as A → U AU † + U dU † .

(2.18)

The field strength is defined by F := D 2 = dA + A2 .

(2.19)

We can construct a gauge invariant action S as follows: S=−

√ 1 2 det θ Tr F ∧ ∗F, (2π ) g2

(2.20)

where Tr denotes the trace over Hn := Cn ⊗ H and ∗ is the Hodge star.3 If we use the operator symbols and the star product, (2.20) can be rewritten as4 1 S = − 2 d 4 x tr Fµν F µν . (2.21) 4g Here tr denotes the trace over the U (n) gauge group. In the above, and throughout this article, we use the same letters for operators and corresponding operator symbols for notational simplicity. Next let us consider a gauge theory with projection [2].5 A projection p is an Hermitian idempotent element in Mn (A): p† = p, p2 = p. We consider the vector space pAn := {φp ∈ An : φp = pφp }. We can consider a unitary action on pAn (which is unitary in the restricted vector space pAn ): φ p → U φp ,

U † U = U U † = p.

(2.22)

We can construct a covariant derivative Dp for pAn by Dp = pd + Ap ,

Ap = pAp p.

(2.23)

We require Dp φp to transform covariantly under the unitary transformation: D p φp → U Dp φp . 3 In this paper we only consider the case where the metric on R4 is flat: g µν = δµν . 4 For the explicit form of the map from operators to operator symbols, see for example [6, 2]. 5 For the roles of projections in noncommutative geometry, see for example [12, 13].

(2.24)


583

Then the covariant derivative Dp must transform as Dp → U Dp U † .

(2.25)

For any φ p ∈ pAn , the following equation holds U Dp U † φ p = U (pd + Ap )U † φ p = U d(U † φ p ) + U Ap U † φ p = U dU † φ p + U (U † dφ p ) + U Ap U † φ p = pdφ

p

(U = Up = pU )

+ (U dU + U Ap U )φ p . †

†

(2.26)

Hence the gauge field Ap transforms as Ap → U Ap U † + U (dU † )p.

(2.27)

The field strength becomes F := Dp2 = p(dAp )p + A2p + pdpdp.

(2.28)

Indeed, for arbitrary φp ∈ pAn , F φp = (pd + Ap )(pdφp + Ap φp ) = pd(pdφp ) + pd(Ap φp ) + Ap pdφp + A2p φp = pd(pdφp ) + pdAp φp + A2p φp ,

(2.29)

and since φp = pφp and p 2 = p, the term pd(pdφp ) in (2.29) becomes pd(pdφ) = pd(pd(pφp )) = pd(pdp φp + pdφp ) = pdpdp φp − pdpdφp + pdpdφp = pdpdp φp .

(2.30)

We can construct action S which is invariant under the unitary transformation (2.22): S=−

√ 1 (2π )2 detθ Tr F ∧ ∗F. 2 g

(2.31)

Equivalence of Projections6 . However, there exists a larger class of transformations under which the action (2.31) is invariant. In this subsection we will describe these transformations. We start from the definition of the equivalence of projections, and then we treat the equivalence relation as gauge equivalence. Projections p and q in the algebra Mn (A) are said to be equivalent, or Murray–von Neumann equivalent when7 ∃

U ∈ Mn (A),

p = U †U

and

q = U U †,

(2.32)

6 For detailed explanations on the equivalence of projections, see [12, 14]. 7 Here we consider M (A) as an example, but Murray–von Neumann equivalence can be considered in n any C ∗ -algebra.

584

K. Furuuchi

and denoted as p ∼ q. These operators satisfy following equations: U † = p U † = U † q,

U = Up = q U, Ker U = IdMn (A) − U U, †

n

n

n

U H = U U H = pH , †

†

(2.33)

Ker U = IdMn (A) − U U , †

n

†

n

n

U H = U U H = qH . †

(2.34) (2.35)

By choosing the orthonormal basis of pHn and qHn , it is easily seen that p ∼ q ⇔ dim pHn = dim qHn .

(2.36)

Note that p can be equivalent to the identity if p has infinite rank. From (2.35), U can be regarded as a map from pAn to qAn : φp → φq = U φp ,

φp ∈ pAn , φq ∈ qAn .

(2.37)

We also require the covariant derivative of φp to be mapped in the same form as φp : Dp φp → U Dp φp = Dq φq ,

(2.38)

where Dq = qd +Aq and Aq = qAq q is a transform of Ap . This requirement determines the transformation rule of gauge fields Ap → Aq uniquely: Dq U φp = (qd + Aq )U φp = U U † d(U φp ) + U † Aq U φp = U (pd + U † (dU ) + U † Aq U )φp . (2.39) Hence Ap = U † Aq U + U † (dU )p.

(2.40)

Then, U Ap U † = Aq + U U † (dU )pU † = Aq + q(dU )U † q = Aq + q(d(U U † ) − U dU † )q = Aq + q(dq − U dU † )q = Aq − U (dU † )q.

(2.41)

Here we have used the basic identity for projections: q(dq)q = 0. Hence we obtain the reversal formula of (2.40) consistently: Aq = U Ap U † + U (dU † )q.

(2.42)

The transformation rule (2.42) is similar to the usual gauge transformation, and therefore we also call it gauge transformation, or Murray–von Neumann gauge transformation (MvN gauge transformation) if we stress the difference from the usual gauge transformation on noncommutative space. The MvN gauge transformation contains the


585

transformation proposed in [4] as a special case.8 The transformation rule for the field strength is obtained as Fp = Dp2 → Fq = Dq2 = U Dp U † U Dp U † = U Dp pDp U † = U Dp2 U † = U Fp U † .

(2.43)

The important point is that under the MvN gauge transformation the action (2.31) is invariant. This is because Tr Fp ∧ ∗Fp → Tr Fq ∧ ∗Fq = Tr U Fp ∧ ∗Fp U † = Tr pFp ∧ ∗Fp p = Tr Fp ∧ ∗Fp .

(2.44)

Here we have used Eq. (2.35). The noncommutative R4 is represented by the operators End H. Hence a one-to-one map between the Hilbert space may be regarded as a noncommutative analog of a coordinate transformation. The MvN gauge transformation U can be regarded as a map from pHn to qHn , and thus it can be understood as a mixture of gauge transformation and coordinate transformation on noncommutative R4 . 3. Application to Instanton on Noncommutative R4 U (2) One-Instanton Solution on Ordinary R4 . In order to illustrate the similarity and difference between the commutative and noncommutative cases, let us first construct the U (2) one-instanton solution by the ADHM method in the case of ordinary commutative R4 . In this subsection, z and z¯ represent ordinary commuting coordinates. In order to construct instantons by the ADHM method [7], we start from the following data: 1. A pair of complex hermitian vector spaces V = Ck and W = Cn , 2. The operators B1 , B2 ∈ H om(V , V ), I ∈ H om(W, V ), J = H om(V , W ) satisfying the equations µR = µC = 0, where µR = [B1 , B1† ] + [B2 , B2† ] + I I † − J † J, µC = [B1 , B2 ] + I J.

(3.1) (3.2)

Next we define a Dirac-like operator Dz : V ⊕ V ⊕ W → V ⊕ V by τz , Dz = σz† τz = ( B2 − z2 , B1 − z1 , I ), σz† = ( −(B1† − z¯ 1 ), B2† − z¯ 2 , J † ).

(3.3)

8 However we regard that the rank of the projection does not change under this transformation as opposed to [4]. For example, IdH and IdH − |0, 0 0, 0| can be Murray–von Neumann equivalent since both have infinite rank (see Eq. (2.36)).

586

K. Furuuchi

The equation µR = µC = 0 is equivalent to the set of equations τz τz† = σz† σz := ✷z ,

τz σz = 0.

(3.4)

The second equation means Im σz ∈ Ker τz , and therefore dim Ker τz /Im σz = (2k + n − k) − k = n. Hence there are n zero-eigenvalue-vectors (we call them zero-modes for short) of Dz : Dz . (a) = 0,

a = 1, . . . , n.

(3.5)

We can choose orthonormal basis of the space of the zero-modes: . (a)† . (b) = δ ab . There is a freedom in the choice of the basis:  . → .U † ,

(3.6) 

. =  . (1) · · · . (n)  ,

(3.7)

where U is an n × n unitary matrix. U may depend on z and z¯ , and this change of basis will become a U (n) gauge symmetry after we construct gauge fields from the zero-modes. An anti-self-dual U (n) gauge field is constructed by the formula Aab = . (a)† d. (b) ,

(3.8)

where a and b are indices of the U (n) gauge group. There is an action of U (k) that does not change (3.8): (B1 , B2 , I, J ) −→ (uB1 u−1 , uB2 u−1 , uI, J u−1 ),

u ∈ U (k).

(3.9)

Therefore the moduli space of the anti-self-dual U (n) gauge field with instanton number k is given by −1 M0 (k, n) = µ−1 R (0) ∩ µC (0)/U (k),

(3.10)

where the action of U (k) is the one given in (3.9). The fixed points of the U (k) action −1 in µ−1 R (0) ∩ µC (0) become singularities after the U (k) quotients. These singularities correspond to the instantons shrinking to zero size, and often are called small instanton singularities in the physical literature. Let us check that the field strength constructed from (3.8) is really anti-self-dual: F = dA + A2 = d(. † d.) + (. † d.)(. † d.) = d. † (1 − .. † )d..

(3.11)

In the above we have suppressed the U (n) indices. One of the important points in the ADHM construction is that (1 − .. † ) is a projection acting on V ⊕ V ⊕ W ≈ C2k+n and it projects out the space of zero-modes (≈ Cn ). Hence it can be rewritten as 1 − .. † = Dz†

1

Dz Dz Dz† 1 1 = τz† τ + σz † σz† † z τ z τz σ z σz 1 1 † = τz† τz + σ z σ , ✷z ✷z z

(3.12)


587

where we have used the notations in (3.4). Since τz . = σz† . = 0 by definition (3.5), it follows that τz d. = −dτz ., σz† d. = −dσz† .. Hence F = d. † (1 − .. † )d. 1 1 † τz + σ z σz d. = d. † τz† ✷z ✷z 1 1 = . † dτz† dτz + dσz dσz† . ✷z ✷z   1 dz1 ✷z d z¯ 1 + d z¯ 2 ✷1 z dz2 −dz1 ✷1 z d z¯ 2 + d z¯ 2 ✷1 z dz1 0 = . †  −dz2 ✷1 d z¯ 1 + d z¯ 1 ✷1 dz2 dz2 ✷1 d z¯ 2 + d z¯ 1 ✷1 dz1 0  . z z z z 0 0 0 − := FADHM .

(3.13)

− FADHM

− FADHM

− + ∗FADHM

is anti-self-dual: = 0. Now let us construct a U (2) one-instanton solution by the ADHM method. A solution to the ADHM equations is given by B1 = B2 = 0,

I = (ρ 0), J † = (0 ρ).

Then the Dirac-like operator Dz becomes −z2 −z1 ρ 0 Dz = . z¯ 1 −¯z2 0 ρ We can find following zero-mode: .BPST



ρ  0 = z2 −¯z1

 0 1 ρ . z1  r 2 + ρ 2 z¯ 2

(3.14)

(3.15)

(3.16)

The gauge field constructed from this zero-mode is nothing but the well known BPST instanton [8]: † AµBPST = .BPST ∂µ .BPST

= where r =

µ x xµ and x µ σµ 1 g(x) = = r r

r2 g −1 ∂µ g, r 2 + ρ2

z2 z1 −¯z1 z¯ 2

(3.17)

,

σµ = (iτ1 , iτ2 , iτ3 , 1).

(3.18)

Here τi (i = 1, 2, 3) are Pauli matrices. The instanton number is classified by the winding number π 3 (U (2)): 1 1 4 µν ˜ d x tr Fµν F = d 4 x ∂µ K µ 16π 2 16π 2 1 =− tr g −1 dgg −1 dgg −1 dg 24π 2 S 3 = −1. (3.19)

588

K. Furuuchi

Here F˜µν = 21 4µν ρσ Fρσ and µ

K = 2 tr 4

µνρσ

2 A ν ∂ρ A σ + A ν A ρ A σ . 3

(3.20)

For a later purpose let us consider the following zero-mode which is not well defined at the origin r = 0:   ρ z¯ 2 −ρz1 1  ρ z¯ ρz2  . (3.21) .sing =  1 ρ 0  r r 2 + ρ2 0 ρ .sing and .BPST are related by the “singular” gauge transformation .sing = .BPST g −1 (x).

(3.22)

Note that this transformation is not continuous at the origin. Therefore .sing is not an appropriate zero-mode for the ADHM construction. However in the next subsection we will observe that in the noncommutative case, we can construct a zero-mode similar to .sing , but well defined everywhere! The gauge field constructed from .sing is given by Aµ sing = gAµBPST g −1 + gg −1 ∂µ gg −1 =(

r2

r2 ρ2 − 1)g∂µ g −1 = 2 g∂µ g −1 , 2 +ρ r + ρ2

(3.23)

which is singular at the origin. Note that the winding of AµBPST is resolved by the singular gauge transformation g. U (2) One-Instanton Solution on Noncommutative R4 and MvN Gauge Transformation. As we have seen in the previous subsection, the moduli space of instantons M0 (k, n) in (3.10) has small instanton singularities. The resolution of these singularities is given in [9]. The fixed points of U (k) action are removed when we add a constant to the right-hand side of (3.1): µR = ζ,

µC = 0.

(3.24)

Then the quotient space −1 Mζ (k, n) = µ−1 R (ζ IdV ) ∩ µC (0) /U (k)

(3.25)

is no longer singular. The modification in (3.24) modifies the key equation (3.4) if we use ordinary commutative coordinates on R4 . However it was found in [1] that if we use noncommutative coordinates zi , z¯ i (i = 1, 2) which satisfies [z1 , z¯ 1 ] + [z2 , z¯ 2 ] = −ζ , τz and σz do satisfy (3.4). We define operator Dz : (V ⊕ V ⊕ W ) ⊗ A → (V ⊕ V ) ⊗ A by the same formula (3.3): τz Dz = , σz† τz = ( B2 − z2 , B1 − z1 , I ), σz† = ( −(B1† − z¯ 1 ), B2† − z¯ 2 , J † ).

(3.26)


589

The operator Dz Dz† : (V ⊕ V ) ⊗ A → (V ⊕ V ) ⊗ A has a block diagonal form, ✷z 0 † Dz Dz = , ✷z ≡ τz τz† = σz† σz , (3.27) 0 ✷z which is a consequence of (3.4) and important for the ADHM construction. Next we look for solutions of the equation Dz . (a) = 0

(a = 1, . . . , n),

. (a)

(3.28)

. (a)

are operators: : A → (V ⊕ V ⊕ W ) ⊗ A. There is where the components of an important property that . must satisfy (see (3.12)): 1 − .. † = Dz†

1 Dz Dz†

Dz .

(3.29)

This equation contains the following two requirements. First, . must contain all the vector zero-modes [2] on the left: The vector zero-mode |U is an element of H⊕k ⊕ H⊕k ⊕ H⊕n which satisfies Dz |U = 0.

(3.30)

The operator zero-mode . can be constructed from vector zero-modes. In the case when the gauge group is U (1), the vector zero-modes are fully classified [10, 11]. Second, (3.29) imposes normalization condition for .. The feature peculiar to the noncommutative case is that there may be some states in H which are annihilated by . (a) for some a [1, 2]. More precisely, all the components of . (a) annihilate those states. Then we can normalize . (a) only in the subspace of H which is not annihilated by . (a) . It means that . is normalized as . † . = p,

(3.31)

where p ∈ Mn (A) is a projection to the states which are not annihilated by .. If the operator zero-mode satisfies (3.29) and normalized as (3.31), we can construct anti-selfdual gauge field Ap by the formula Ap = . † (d.). † ..

(3.32)

Ap is anti-self-dual as a gauge connection for pAn , i.e. if we consider the covariant derivative for pAn : Dp = pd + Ap . When the gauge group is U (1), there is a natural choice for the normalization condition: . † . = pI ,

(3.33)

where pI is a projection to the ideal states described in [2]. We call the zero-mode normalized minimal operator zero-mode when it is normalized as in (3.33). Then the covariant derivative for pI A: DpI = pI d + ApI

(3.34)

gives anti-self-dual field strength. Because of the associativity of the operator multiplication, there is a freedom for the choice of the operator zero-mode: . → .U † ,

(3.35)

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K. Furuuchi

where U † U = p, U U † = q and q ∈ Mn (A) is a projection. It is apparent that .U † satisfies (3.29) if . satisfies (3.29): .U † (.U )† = .U † U . † = .p. † = .. † .

(3.36)

It is also easily seen that this change of zero-modes corresponds to the MvN gauge transformation. Indeed, Ap = . † (d.)(. † .) → (U . † )(d(.U † ))(U . † .U † ) = U (. † d.)U † (U . † .U † ) + U . † .(dU † )(U . † .U † ) = U Ap U † + U (dU † )q,

(3.37)

which is nothing but the MvN gauge transformation (2.42). Although we can choose arbitrary “MvN gauge” (or arbitrary projection), there are not so many gauge choices which are convenient or physically interesting. In the case of the U (1) instanton, the most natural choice may be the one that corresponds to the projection to the ideal states. However, in the case of the U (2) instanton, there is another choice which is physically interesting, as will be explained below. Let us construct the U (2) one-instanton solution by the ADHM method. From hereafter we set ζ = 2. A solution to the ADHM equation (3.24) is given by B1 = B2 = 0, I = ( ρ 2 + 2 0 ), J † = ( 0 ρ ). (3.38) Then the Dirac-like operator Dz becomes ρ2 + 2 0 . Dz = −z2 −z1 z¯ 1 −¯z2 0 ρ

(3.39)

The operator zero-mode can be obtained as      .min =  (2) (1) .  min .min

(1)

.min

(2)

.min

  ρ 2 + 2 z¯ 2  ρ 2 + 2 z¯  1 1 = ,  z1 z¯ 1 + z2 z¯ 2  Nˆ (Nˆ + 2 + ρ 2 ) 0   −ρz1 1 ρz2   = ,  0 2 ˆ ˆ (N + 2)(N + 2 + ρ ) z z¯ + z z¯ + 2 1 1

2 2

(3.40) where √1 is defined as Nˆ

1 1 = |n1 , n2 n1 , n2 |, √ n1 + n 2 Nˆ (n1 ,n2 )=(0,0)

(3.41)


591

i.e. when we consider the inverse of Nˆ we omit the kernel of Nˆ , that is, |0, 0, from the Hilbert space. Hence √1 is a well defined operator. This is an essential point in the Nˆ

construction of instantons on noncommutative R4 [2]. When ρ = 0, the contribution (2) (1) of .min to the field strength vanishes whereas .min reduces to the normalized minimal operator zero-mode in U (1) one-instanton solution [2]. The operator zero-mode .min is normalized as † .min = p, .min

where p is a projection in M2 (A): IdH − |0, 0 0, 0| 0 . p= 0 IdH

(3.42)

(3.43)

Although in the case where the gauge group is U (2) the vector zero-modes have not been classified at the moment, we can directly check that Eq. (3.29) holds in this case: Dz†

1 Dz Dz†

† Dz = 1 − .min .min .

(3.44)

Therefore the connection Dp = pd + Ap ,

† † Ap = .min .min d.min .min

(3.45)

gives anti-self-dual field strength. Since the projection p has infinite rank as an operator in M2 (A), it is the Murray–von Neumann equivalent to the identity operator IdM2 (A) . So let us MvN gauge transform p to IdM2 (A) . In order to do so one seeks for the operator U ∈ M2 (A) which satisfies U † U = p,

U U † = IdM2 (A) .

(3.46)

Of course there are (infinitely) many choices for such U . However there is a choice which has a physically interesting interpretation. Let us consider the following operator U which satisfies (3.46):   √1 z2 √1 z1 Nˆ Nˆ . U† =  (3.47) − √ 1 z¯ 1 √ 1 z¯ 2 ˆ N+2

ˆ N+2

Notice the similarity between this operator and (the inverse of) the singular gauge transformation (3.22) in the commutative case. However the operator U is well defined unlike (3.18) (remember that √1 is defined as in (3.41)). Hence the MvN gauge transformaNˆ

tion in this case can be understood as a noncommutative resolution of the singular gauge transformation (3.22)! After the MvN gauge transformation the covariant derivative takes the familiar form, i.e. without the projection operator on the left side of the derivative: D = d + A.

(3.48)

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K. Furuuchi

Here the gauge field A is constructed from the gauge transformed zero-mode .BPST∗ = .min U † : † A = ABPST∗ = .BPST ∗ d.BPST∗ .

(3.49)

If we express the gauge fields using operator symbols, the long r behavior of ABPST∗ is the same as that of the BPST instanton ABPST in the commutative case, and the instanton number 8π1 2 d 4 x tr F ∧ F is classified by π 3 (U (2)), as in (3.19). On the other hand the large r behavior of Ap which is constructed from .min is the same as the one in the singular configuration Asing in the commutative case. Therefore the instanton number is not classified by π 3 (U (2)) in this gauge. However the instanton number itself does not change under the MvN gauge transformation, and in this case the instanton number counts the dimension of the projection (1 − p), as described below. We define a new gauge field A µ for notational convenience: p(∂ˆµ + Aµ p )p = ∂ˆµ − (1 − p)∂ˆµ − (1 − p)∂ˆµ + (1 − p)∂ˆµ (1 − p) + Aµ p = ∂ˆµ + A µ . (3.50) Here ∂ˆµ is the derivative operator (2.7). A is not an MvN gauge transform of Ap . The field strength of A µ is given as F µν = [∂ˆµ + A µ , ∂ˆν + A ν ] − [∂ˆµ , ∂ˆν ] − = p(iBµν + Fµν min )p − iBµν

− = (1 − p)iBµν (1 − p) + Fµν min ,

(3.51)

− where F min = Dp2 . It can be shown that the operator symbol of Ap decays like O(r −3 ) for large r (recall the resemblance between the minimal zero-mode (3.40) and the singular

zero-mode (3.21)). The operator symbol of (1 − p) decays like ∼ e− ζ r . θ µν in the star product appear as a multiplication of the combination θ/r 2 for large r and hence does not contribute to the surface integral at large r. Taking all these accounts, the instanton number of A vanishes: √ 1 1 µν 2 ˜ d 4 x tr F µν F˜ µν F detθ Tr F = (2π) µν 16π 2 16π 2 1 d 4 x ∂µ K µ = 0. = (3.52) 16π 2 2 2

Here µ

K = 2 tr 4

µνρσ

Aν

∂ρ Aσ

2 + Aν A ρ A σ . 3

(3.53)

On the other hand, √ 1 (2π)2 det θ Tr F µν F˜ µν 2 16π √ 1 2 ˜ µν (1 − p) + F − ˜ µν− , = B F det θ Tr (1 − p)B (2π) µν min µν min 16π 2

(3.54)


593

(Bµν is self-dual as we have set θ µν self-dual). Thus the instanton number counts the dimension of the projection (1 − p): −

√ 1 − ˜ µν− (2π)2 det θ Tr Fµν min F min 2 16π √ 1 2 = det θ Tr (1 − p)Bµν B˜ µν (1 − p) (2π ) 16π 2 = dim (1 − p).

(3.55)

4. Conclusion In this article the formalism that describes the equivalence of projections as a kind of gauge equivalence on noncommutative space is given. We apply this formalism to the U (2) one-instanton solution on noncommutative R4 . The gauge equivalence between the BPST type configuration with winding number one and the configuration without winding but with projection is shown. In this case the gauge transformation can be understood as a noncommutative resolution of the singular gauge transformation in ordinary R4 . Recall that the projection describes holes on noncommutative R4 [2]. Hence this formalism gives a unified description to the intriguing mixing of gauge fields and geometry in noncommutative space [2, 3]. Acknowledgements. I would like to thank N. Ishibashi and S. Iso for useful discussions. I would also like to thank T. Hirayama and Y. Okada for suggestions and encouragements.

References 1. Nekrasov, N. and Schwarz, A.: Instantons on Noncommutative R4 and (2, 0) Superconformal Six Dimensional Theory. Commun. Math. Phys. 198, 689 (1998) 2. Furuuchi, K.: Instantons on Noncommutative R4 and Projection Operators. Prog. Theor. Phys. 103 (No.5), 1043 (2000) 3. Braden, H. and Nekrasov, N.: Space-Time Foam From Non-Commutative Instantons. hep-th/9912019 4. Ho, P.-M.: Twisted Bundle on Noncommutative Space and U (1) Instanton. hep-th/003012 5. Seiberg, N. and Witten, E.: String Theory and Noncommutative Geometry. J. High Energy Phys. 9909, 032 (1999) 6. Aoki, H., Ishibashi, N., Iso, S., Kawai, H., Kitazawa, Y., Tada, T.: Noncommutative Yang–Mills in IIB Matrix Model. Nucl. Phys. B 565, 176 (2000) 7. Atiyah, M., Hitchin, N., Drinfeld, V. and Manin, Y.: Construction of Instantons. Phys. Lett. 65A, 185 (1978) 8. Belavin, A.A., Polyakov, A.P., Schwartz, A.S. and Tyupkin, Yu.S.: Pseudoparticle Solutions of the Yang– Mills Equations. Phys. Lett. B59, 85 (1975) 9. Nakajima, H.: Resolutions of moduli spaces of ideal instantons on R4 . Singapore: World Scientific, 1994, p. 129 10. Nakajima, H.: Heisenberg algebra and Hilbert schemes of points on projective surfaces. Ann. of Math. 145, 379 (1997), alg-geom/9507012; Instantons and affine Lie algebra. Nucl. Phys. Proc. Suppl. 46, 154 (1996), alg-geom/9510003 11. H. Nakajima, Lectures on Hilbert scheme of points on surfaces. AMS University Lecture Series Vol. 18, Providence, RI: Am. Math. Soc. 1999 12. Connes, A.: Noncommutative Geometry. London–New York: Academic Press (1994) 13. Landi, G.: An Introduction to Noncommutative Space and their Geometry. Lecture Notes in Physics: Monographs, 51. Berlin–Heidelberg: Springer-Verlag, 1997, hep-th/9701078 14. Wegge-Olsen, N.E.: K-Theory and C ∗ -Algebras. Oxford: Oxford University Press, 1993 Communicated by A. Connes

Commun. Math. Phys. 217, 595 – 622 (2001)

Communications in



Holographic Reconstruction of Spacetime and Renormalization in the AdS/CFT Correspondence Sebastian de Haro1,2 , Kostas Skenderis3 , Sergey N. Solodukhin1 1 Spinoza Institute, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands.

E-mail: [email protected]; [email protected]

2 Institute for Theoretical Physics, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands. 3 Physics Department, Princeton University, Princeton, NJ 08544, USA.

E-mail: [email protected] Received: 6 April 2000 / Accepted: 6 November 2000

Abstract: We develop a systematic method for renormalizing the AdS/CFT prescription for computing correlation functions. This involves regularizing the bulk on-shell supergravity action in a covariant way, computing all divergences, adding counterterms to cancel them and then removing the regulator. We explicitly work out the case of pure gravity up to six dimensions and of gravity coupled to scalars. The method can also be viewed as providing a holographic reconstruction of the bulk spacetime metric and of bulk fields on this spacetime, out of conformal field theory data. Knowing which sources are turned on is sufficient in order to obtain an asymptotic expansion of the bulk metric and of bulk fields near the boundary to high enough order so that all infrared divergences of the on-shell action are obtained. To continue the holographic reconstruction of the bulk fields one needs new CFT data: the expectation value of the dual operator. In particular, in order to obtain the bulk metric one needs to know the expectation value of stress-energy tensor of the boundary theory. We provide completely explicit formulae for the holographic stress-energy tensors up to six dimensions. We show that both the gravitational and matter conformal anomalies of the boundary theory are correctly reproduced. We also obtain the conformal transformation properties of the boundary stress-energy tensors.

1. Introduction and Summary of the Results Holography states that a (d+1)-dimensional gravitational theory (referred to as the bulk theory) should have a description in terms of a d-dimensional field theory (referred to as the boundary theory) with one degree of freedom per Planck area [45, 41]. The arguments leading to the holographic principle use rather generic properties of gravitational physics, indicating that holography should be a feature of any quantum theory of gravity. Nevertheless it has been proved a difficult task to find examples where holography is realized, let alone to develop a precise dictionary between bulk and boundary physics.

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S. de Haro, K. Skenderis, S. N. Solodukhin

The AdS/CFT correspondence [31] provides such a realization [47, 42] with a rather precise computational framework [24, 47]. It is, therefore, desirable to sharpen the existing dictionary between bulk/boundary physics as much as possible. In particular, one of the issues one would like to understand is how spacetime is built holographically out of field theory data. The prescription of [24, 47] gives a concrete proposal for a holographic computation of physical observables. In particular, the partition function of string theory compactified on AdS spaces with prescribed boundary conditions for the bulk fields is equal to the generating functional of conformal field theory correlation functions, the boundary value of fields being now interpreted as sources for operators of the dual conformal field theory (CFT). String theory on anti-de Sitter (AdS) spaces is still incompletely understood. At low energies, however, the theory becomes a gauged supergravity with an AdS ground state coupled to Kaluza–Klein (KK) modes. On the field theory side, this corresponds to the large N and strong ’t Hooft coupling regime of the CFT. So in the AdS/CFT context the question is how one can reconstruct the bulk spacetime out of CFT data. One can also pose the converse question: given a bulk spacetime, what properties of the dual CFT can one read off? The prescription of [24, 47] equates the on-shell value of the supergravity action with the generating functional of connected graphs of composite operators. Both sides of this correspondence, however, suffer from infinities – infrared divergences on the supergravity side and ultraviolet divergences on the CFT side. Thus, the prescription of [24, 47] should more properly be viewed as an equality between bare quantities. Ones needs to renormalize the theory to obtain a correspondence between finite quantities. It is one of the aims of this paper to present a systematic way of performing such renormalization. The CFT data1 that we will use are: which operators are turned on, and what is their vacuum expectation value. Since the boundary metric (or, more properly, the boundary conformal structure) couples to the boundary stress-energy tensor, the reconstruction of the bulk metric to leading order involves a detailed knowledge of the way the energymomentum tensor is encoded holographically. There is by now an extended literature on the study of the stress-energy tensor in the context of the AdS/CFT correspondence starting from [3, 34]. We will build on these and other related works [15, 32, 30]. Our starting point will be the calculation of the infrared divergences of the on-shell gravitational action [26]. Minimally subtracting the divergences by adding counterterms [26] leads straightforwardly to the results in [3, 15, 30]. After the subtractions have been made one can remove the (infrared) regulator and obtain a completely explicit formula for the expectation value of the dual stress-energy tensor in terms of the gravitational solution. We will mostly concentrate on the gravitational sector, i.e. on the reconstruction of the bulk metric, but we will also discuss the coupling to scalars. Our approach will be to build perturbatively an Einstein manifold of constant negative curvature (which we will sometimes refer to as an asymptotically AdS space) as well as a solution to the scalar field equations on this manifold out of CFT data. The CFT data we start from is what sources are turned on. We will include a source for the dual stress-energy tensor as well as sources for scalar composite operators. This means that in the bulk we need to solve the gravitational equations coupled to scalars given a conformal structure at infinity and appropriate Dirichlet boundary conditions for the scalars. It is well-known that if one 1 We assume that the CFT we are discussing has an AdS dual. Our results only depend on the spacetime dimension and apply to all cases where the AdS/CFT duality is applicable, so we shall not specify any particular CFT model.

Holographic and Renormalization in the AdS/CFT Correspondence

597

considers the standard Euclidean AdS (i.e., with isometry SO(1, d + 1)), the scalar field equation with Dirichlet boundary conditions has a unique solution. In the Lorentzian case, because of the existence of normalizable modes, the solution ceases to be unique. Likewise, the Dirichlet boundary condition problem for (Euclidean) gravity has a unique (up to diffeomorphisms) smooth solution in the case the bulk manifold is topologically a ball and the boundary conformal structure is sufficiently close to the standard one [21]. However, given a boundary topology there may be more than one Einstein manifold with this boundary. For example, if the boundary has the topology of S 1 × S d−1 , there are two possible bulk manifolds [25, 47]: one which is obtained from standard AdS by global identifications and is topologically S 1 × R d , and another, the Schwarzschild-AdS black hole, which is topologically R 2 × S d−1 . We will make no assumption on the global structure of the space or on its signature. The CFT should provide additional data in order to retrieve this information. Indeed, we will see that only the information about the sources leaves undetermined the part of the solution which is sensitive on global issues and/or the signature of spacetime. To determine that part one needs new CFT data. To leading order these are the expectation values of the CFT operators. In particular, in the case of pure gravity, we find that generically a boundary conformal structure is not sufficient in order to obtain the bulk metric. One needs more CFT data. To leading order one needs to specify the expectation value of the boundary stress-energy tensor. Since the gravitational field equation is a second order differential equation, one may expect that these data are sufficient in order to specify the full solution. In general, however, non-local observables such as Wilson loops may be needed in order to recover global properties of the solution and reconstruct the metric in the deep interior region. Furthermore, higher point functions of the stress-energy tensor may be necessary if higher derivatives corrections such as R 2 terms are included in the action. We emphasize that we make no assumption about the regularity of the solution. Under additional assumptions the metric may be determined by fewer data. For example, as we mentioned above, under certain assumptions on the topology and the boundary conformal structure one obtains a unique smooth solution [21]. Another example is the case when one restricts oneself to conformally flat bulk metrics. Then a conformally flat boundary metric does yield a unique, up to diffeomorphisms and global identifications, bulk metric [40]. Turning things around, given a specific solution, we present formulae for the expectation values of the dual CFT operators. In particular, in the case the operator is the stress energy tensor, our formulae have a “dual” meaning [3]: both as the expectation value of the stress-energy tensor of the dual CFT and as the quasi-local stress-energy tensor of Brown and York [11]. We provide very explicit formulae for the stress-energy tensor associated with any solution of Einstein’s equations with negative constant curvature. Let us summarize these results for spacetime dimension up to six. The first step is to rewrite the solution in the Graham–Fefferman coordinate system [16] ds 2 = Gµν dx µ dx ν =

l2 2 dr + gij (x, r)dx i dx j , 2 r

(1.1)

where g(x, r) = g(0) + r 2 g(2) + · · · + r d g(d) + h(d) r d log r 2 + O(r d+1 ).

(1.2)

The logarithmic term appears only in even dimensions and only even powers of r appear up to order r [(d−1)] , where [a] indicates the integer part of a. l is a parameter of dimension

598


of length related to the cosmological constant as = − d(d−1) . Any asymptotically AdS 2l 2 metric can be brought in the form (1.1) near the boundary ([21], see also [22, 20]). Once this coordinate system has been reached, the stress-energy tensor reads Tij =

dl d−1 g(d)ij + Xij [g(n) ], 16π GN

(1.3)

where Xij [g(n) ] is a function of g(n) with n < d. Its exact form depends on the spacetime dimension and it reflects the conformal anomalies of the boundary conformal field theory. In odd (boundary) dimensions, where there are no gravitational conformal anomalies, Xij is equal to zero. The expression for Xij [g(n) ] for d = 2, 4, 6 can be read off from (3.10), (3.15) and (3.16), respectively. The universal part of (1.3) (i.e. with Xij omitted) was obtained previously in [34]. Actually, to obtain the dual stress-energy tensor it is sufficient to only know g(0) and g(d) as g(n) with n < d are uniquely determined from g(0) , as we will see. The coefficient h(d) of the logarithmic term in the case of even d is also directly related to the conformal anomaly: it is proportional to the metric variation of the conformal anomaly. It was pointed out in [3] that this prescription for calculating the boundary stressenergy tensor provides also a novel, free of divergences2 , way of computing the gravitational quasi-local stress-energy tensor of Brown and York [11]. This approach was recently criticized in [2], and we take this opportunity to address this criticism. Conformal anomalies reflect infrared divergences in the gravitational sector [26]. Because of these divergences one cannot maintain the full group of isometries even asymptotically. In particular, the isometries of AdS that rescale the radial coordinate (these correspond to dilations in the CFT) are broken by infrared divergences. Because of this fact, bulk solutions that are related by diffeomorphisms that yield a conformal transformation in the boundary do not necessarily have the same mass. Assigning zero mass to the spacetime with boundary R d , one obtains that, due to the conformal anomaly, the solution with boundary R × S d−1 has non-zero mass. This parallels exactly the discussion in field theory. In that case, starting from the CFT on R d with vanishing expectation value of the stress-energy tensor, one obtains the Casimir energy of the CFT on R × S d−1 by a conformal transformation [12]. The agreement between the gravitational ground state energy and the Casimir energy of the CFT is a direct consequence of the fact that the conformal anomaly computed by weakly coupled gauge theory and by supergravity agree [26]. It should be noted that, as emphasized in [3], agreement between gravity/field theory for the ground state energy is achieved only after all ambiguities are fixed in the same manner on both sides. A conformal transformation in the boundary theory is realized in the bulk as a special diffeomorphism that preserves the form of the coordinate system (1.1) [28]. Using these diffeomorphisms one can easily study how the (quantum, i.e., with the effects of the conformal anomaly taken into account) stress-energy tensor transforms under conformal transformations. Our results, when restricted to the cases studied in the literature [12], are in agreement with them. We note that the present determination is considerably easier than the one in [12]. The discussion is qualitatively the same when one adds matter to the system. We discuss scalar fields but the discussion generalizes straightforwardly to other kinds of matter. We study both the case when the gravitational background is fixed and the case when gravity is dynamical. 2 We emphasize, however, that one has to subtract the logarithmic divergences in even dimensions in order for the stress-energy tensor to be finite.


599

Let us summarize the results for the case of scalar fields in a fixed gravitational background (given by a metric of the form (1.1)). We look for solutions of massive scalar fields with mass m2 = ( − d) that near the boundary have the form (in the coordinate system (1.1)) (x, r) = r d− φ(0) + r 2 φ(2) + · · · + r 2−d φ(2−d) + r 2−d log r 2 ψ(2−d) + O(r +1 ).

(1.4)

The logarithmic terms appear only when 2 − d is an integer and we only consider this case in this paper. We find that φ(n) , with n < 2 − d, and ψ(2−d) are uniquely determined from the scalar field equation. This information is sufficient for a complete determination of the infrared divergences of the on-shell bulk action. In particular, the logarithmic term ψ(2−d) in (1.4) is directly related to matter conformal anomalies. These conformal anomalies were shown not to renormalize in [37]. We indeed find exact agreement with the computation in [37]. Adding counterterms to cancel the infrared divergences we obtain the renormalized on-shell action. We stress that even in the case of a free massive scalar field in a fixed AdS background one needs counterterms in order for the on-shell action to be finite (see (5.9)). The coefficient φ(2−d) is left undetermined by the field equations. It is determined, however, by the expectation value of the dual operator. Differentiating the renormalized on-shell action one finds (up to terms contributing contact terms in the 2-point function) O(x) = (2 − d)φ(2−d) (x).

(1.5)

This relation, with the precise proportionality coefficient, has first been derived in [29]. The value of the proportionality coefficient is crucial in order to obtain the correct normalization of the 2-point function in standard AdS background [17]. In the case when the gravitational background is dynamical we find that, for scalars that correspond to irrelevant operators, our perturbative treatment is consistent only if one considers single insertions of the irrelevant operator, i.e. the source is treated as an infinitesimal parameter, in agreement with the discussion in [47]. For scalars that correspond to marginal and relevant operators one can compute perturbatively the backreaction of the scalars to the gravitational background. One can then regularize and renormalize as in the discussion of pure gravity or scalars in a fixed background. For illustrative purposes we analyze a simple example. This paper is organized as follows. In the next section we discuss the Dirichlet problem for AdS gravity and we obtain an asymptotic solution for a given boundary metric (up to six dimensions). In Sect. 3 we use these solutions to obtain the infrared divergences of the on-shell gravitational action. After renormalizing the on-shell action by adding counterterms, we compute the holographic stress-energy tensor. Section 4 is devoted to the study of the conformal transformation properties of the boundary stress-energy tensor. In Sect. 5 we extend the analysis of Sects. 2 and 3 to include matter. In appendices A and D we give the explicit form of the solutions discussed in Sects. 2 and 5. Appendix B contains the explicit form of the counterterms discussed in Sect. 3. Finally, in Appendix C we present a proof that the coefficient of the logarithmic term in the metric (present in even boundary dimensions) is proportional to the metric variation of the conformal anomaly.

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2. Dirichlet Boundary Problem for AdS Gravity The Einstein–Hilbert action for a theory on a manifold M with boundary ∂M is given by3 √ 1 √ d+1 Sgr [G] = [ d x G (R[G] + 2) − dd x γ 2K], (2.1) 16πGN M ∂M where K is the trace of the second fundamental form and γ is the induced metric on the boundary. The boundary term is necessary in order to get an action which only depends on first derivatives of the metric [18], and it guarantees that the variational problem with Dirichlet boundary conditions is well-defined. According to the prescription of [24, 47], the conformal field theory effective action is given by evaluating the on-shell action functional. The field specifying the boundary conditions for the metric is regarded as a source for the boundary operator. We therefore need to obtain solutions to Einstein’s equations, 1 Rµν − RGµν = Gµν , 2

(2.2)

subject to appropriate Dirichlet boundary conditions. Metrics Gµν that satisfy (2.2) have a second order pole at infinity. Therefore, they do not induce a metric at infinity. They do induce, however, a conformal class, i.e. a metric up to a conformal transformation. This is achieved by introducing a defining function r, i.e. a positive function in the interior of M that has a single zero and non-vanishing derivative at the boundary. Then one obtains a metric at the boundary by g(0) = r 2 G|∂M 4 . However, any other defining function r = r exp w is as good. Therefore, the metric g(0) is only defined up to a conformal transformation. We are interested in solving (2.2) given a conformal structure at infinity. This can be achieved by working in the coordinate system (1.1) introduced by Feffermam and Graham [16]. The metric in (1.1) contains only even powers of r up to the order we are interested in [16] (see also [22, 20]). For this reason, it is convenient to use the variable ρ = r 2 [26],5 2 1 2 µ ν 2 dρ i j ds = Gµν dx dx = l + gij (x, ρ)dx dx , 4ρ 2 ρ (2.3) g(x, ρ) = g(0) + · · · + ρ d/2 g(d) + h(d) ρ d/2 log ρ + · · · , where the logarithmic piece appears only for even d. The sub-index in the metric expansion (and in all other expansions that appear in this paper) indicates the number of derivatives involved in that term, i.e. g(2) contains two derivatives, g(4) four derivatives, 3 Our curvature conventions are as follows R l = ∂ ) l + ) l ) p − i ↔ j and R = R k . ij k i jk ip j k ij ikj With these conventions the curvature of AdS comes out positive, still use the terminology “space d+1but wewill x = dd x 0∞ dr and the boundary is at of constant negative curvature”. Notice also that we take d r = 0 (in the coordinate system (1.1)). The minus sign in front of the trace of the second fundamental form is correlated with the choice of having r = 0 in the lower end of the radial integration. 4 Throughout this article the metric g (0) is assumed to be non-degenerate. For studies of the AdS/CFT correspondence in cases where g(0) is degenerate we refer to [9, 43]. 5 Greek indices, µ, ν, .. are used for d + 1-dimensional indices, Latin ones, i, j, .. for d-dimensional ones. To distinguish the curvatures of the various metrics introduced in (2.3) we will often use the notation Rij [g] to indicate that this is the Ricci tensor of the metric g, etc.


601

etc. It follows that the perturbative expansion in ρ is also a low energy expansion. We set l = 1 from now on. One can easily reinstate the factors of l by dimensional analysis. One can check that the curvature of G satisfies Rκλµν [G] = (Gκµ Gλν − Gκν Gλµ ) + O(ρ).

(2.4)

In this sense the metric is asymptotically anti-de Sitter. The Dirichlet problem for Einstein metrics satisfying (2.4) exactly (i.e. not only to leading order in ρ) was solved in [40]. In the coordinate system (2.3), Einstein’s equations read [26] ρ [2g − 2g g −1 g + Tr (g −1 g ) g ] + Ric(g) − (d − 2) g − Tr (g −1 g ) g = 0, ∇i Tr (g −1 g ) − ∇ j gij = 0, 1 Tr (g −1 g ) − Tr (g −1 g g −1 g ) = 0, 2 (2.5) where differentiation with respect to ρ is denoted with a prime, ∇i is the covariant derivative constructed from the metric g, and Ric(g) is the Ricci tensor of g. These equations are solved order by order in ρ. This is achieved by differentiating the equations with respect to ρ and then setting ρ = 0. For even d, this process would have broken down at order d/2 if the logarithm was not introduced in (2.3). h(d) is −1 h(d) = 0, and covariantly conserved, ∇ i h(d)ij = 0. We show in Aptraceless, Tr g(0) pendix C that h(d) is proportional to the metric variation of the corresponding conformal anomaly, i.e. it is proportional to the stress-energy tensor of the theory with action the conformal anomaly. In any dimension, only the trace of g(d) and its covariant divergence are determined. Here is where extra data from the CFT are needed: as we shall see, the undetermined part is specified by the expectation value of the dual stress-energy tensor. We collect in Appendix A the results for g(n) , h(d) as well as the results for the trace and divergence g(d) . In dimension d the latter are the only constraints that Eqs. (2.5) yield for g(d) . From this information we can parametrize the indeterminacy by finding the most general g(d) that has the determined trace and divergence. In d = 2 and d = 4 the equation for the coefficient g(d) has the form of a conservation law ∇ i g(d)ij = ∇ i A(d)ij ,

d = 2, 4,

(2.6)

where A(d)ij is a symmetric tensor explicitly constructed from the coefficients g(n) , n < d. The precise form of the tensor A(d)ij is given in Appendix A (Eq. (A.4)). The integration of this equation obviously involves an “integration constant” tij (x), a symmetric covariantly conserved tensor the precise form of which can not be determined from Einstein’s equations. In two dimensions, we get [40] (see also [6]) g(2)ij =

1 (R g(0)ij + tij ), 2

(2.7)

where the symmetric tensor tij should satisfy ∇ i tij = 0,

Tr t = −R.

(2.8)

602


In four dimensions we obtain6 g(4)ij =

1 1 2 1 2 ] + (g(2) )ij − g(2)ij Tr g(2) + tij , g(0)ij [(Tr g(2) )2 − Tr g(2) 8 2 4

(2.9)

The tensor tij satisfies ∇ i tij = 0,

1 2 Tr t = − [(Tr g(2) )2 − Tr g(2) ]. 4

(2.10)

In six dimensions the equation determining the coefficient g(6) is more subtle than the one in (2.6). It is given by 1 ∇ i g(6)ij = ∇ i A(6)ij + Tr(g(4) ∇j g(2) ), 6

(2.11)

where the tensor A(6)ij is given in (A.4). It contains a part which is antisymmetric in the indices i and j . Since g(6)ij is by definition a symmetric tensor the integration of Eq. (2.11) is not straightforward. Moreover, it is not obvious that the last term in (2.11) takes a form of divergence of some local tensor. Nevertheless, this is indeed the case as we now show. Let us define the tensor Sij , 1 (∇i ∇j B − g(0)ij ∇ 2 B) 10 2 2 4 3 3 2 + g(2)ij B + g(0)ij (− Tr g(2) − (Tr g(2) )3 + Tr g(2) Tr g(2) ), 5 3 15 5 (2.12)

Sij = ∇ 2 Cij − 2R ki lj Ckl + 4(g(2) g(4) − g(4) g(2) )ij +

where 1 2 1 1 Cij = (g(4) − g(2) + g(2) Tr g(2) )ij + g(0)ij B, B = Tr g22 − (Tr g2 )2 . 2 4 8 The tensor Sij is a local function of the Riemann tensor. Its divergence and trace read ∇ i Sij = −4Tr(g(4) ∇j g(2) ),

TrS = −8Tr(g(2) g(4) ).

(2.13)

With the help of the tensor Sij Eq. (2.11) can be integrated in a way similar to the d = 2, 4 cases. One obtains g(6)ij = A(6)ij −

1 Sij + tij . 24

(2.14)

Notice that tensor Sij contains an antisymmetric part which cancels the antisymmetric part of the tensor A(6)ij so that g(6)ij and tij are symmetric tensors, as they should. The symmetric tensor tij satisfies ∇ i tij = 0,

1 1 3 1 2 3 Tr t = − [ (Trg(2) )3 − Trg(2) Trg(2) + Trg(2) − Trg(2) g(4) ]. 3 8 8 2 (2.15)

−1 −1 6 From now on we will suppress factors of g . For instance, Tr g g (0) (2) (4) = Tr [g(0) g(2) g(0) g(4) ]. Unless we explicitly mention to the contrary, indices will be raised and lowered with the metric g(0) , all contractions will be made with this metric.


603

Notice that in all three cases, d = 2, 4, 6, the trace of tij is proportional to the holographic conformal anomaly. As we will see in the next section, the symmetric tensors tij are directly related to the expectation value of the boundary stress-energy tensor. When d is odd the only constraint on the coefficient g(d)ij (x) is that it is conserved and traceless ∇ i g(d)ij = 0,

Tr g(d) = 0.

(2.16)

So that we may identify g(d)ij = tij .

(2.17)

3. The Holographic Stress-Energy Tensor We have seen in the previous section that given a conformal structure at infinity we can determine an asymptotic expansion of the metric up to order ρ d/2 . We will now show that this term is determined by the expectation value of the dual stress-energy tensor. According to the AdS/CFT prescription, the expectation value of the boundary stressenergy tensor is determined by functionally differentiating the on-shell gravitational action with respect to the boundary metric. The on-shell gravitational action, however, diverges. To regulate the theory we restrict the bulk integral to the region ρ ≥ 1 and we evaluate the boundary term at ρ = 1. The regulated action is given by √ 1 √ Sgr,reg = dd+1 x G (R[G] + 2) − dd x γ 2K 16πGN ρ≥1 ρ=1 1 d (3.1) = dd x dρ d/2+1 det g(x, ρ) 16πGN ρ 1 1 + d/2 (−2d det g(x, ρ) + 4ρ∂ρ det g(x, ρ))|ρ=1 . ρ Evaluating (3.1) for the solution we obtained in the previous section we find that the divergences appear as 1/1 k poles plus a logarithmic divergence [26], l dd x det g(0) Sgr,reg = 16πGN × 1 −d/2 a(0) + 1 −d/2+1 a(2) + . . . + 1 −1 a(d−2) − log 1 a(d) + O(1 0 ), (3.2) where the coefficients a(n) are local covariant expressions of the metric g(0) and its curvature tensor. We give the explicit expressions, up to the order we are interested in, in Appendix B. We now obtain the renormalized action by subtracting the divergent terms, and then removing the regulator, 1

Sgr,reg − dd x det g(0) Sgr,ren [g(0) ] = lim 1→0 16πGN (3.3) × 1 −d/2 a(0) + 1 −d/2+1 a(2) + . . . + 1 −1 a(d−2) − log 1 a(d) .

604


The expectation value of the stress-energy tensor of the dual theory is given by ∂Sgr,ren ∂Sgr,ren 1 2 2 Tij = = lim T [γ ] , = lim √ ij ij 1→0 det g(x, 1) ∂g ij (x, 1) 1→0 1 d/2−1 det g(0) ∂g(0) (3.4) where Tij [γ ] is the stress-energy tensor of the theory at ρ = 1 described by the action in (3.3) but before the limit 1 → 0 is taken (γij = 1/1 gij (x, 1) is the induced metric at ρ = 1). Notice that the asymptotic expansion of the metric only allows for the determination of the divergences of the on-shell action. We can still obtain, however, a formula for Tij in terms of g(n) since, as (3.4) shows, we only need to know the first 1 d/2−1 orders in the expansion of Tij [γ ]. The stress-energy tensor Tij [γ ] contains two contributions, reg

Tij [γ ] = Tij + Tijct ,

(3.5)

reg

Tij comes from the regulated action in (3.1) and Tijct is due to the counterterms. The first contribution is equal to reg

1 (Kij − Kγij ) 8πGN 1 −∂1 gij (x, 1) + gij (x, 1) = − 8πGN 1−d × Tr[g −1 (x, 1)∂1 g(x, 1)] + gij (x, 1) . 1

Tij [γ ] = −

(3.6)

The contribution due to counterterms can be obtained from the results in Appendix B. It is given by 1 1 1 Tijct = − (Rij − Rγij ) (d − 1)γij + 8πGN (d − 2) 2 1 d −2 d − ∇ i ∇j R − RRij [−∇ 2 Rij + 2Rikj l R kl + 2 (d − 4)(d − 2) 2(d − 1) 2(d − 1) 1 d 1 − γij (Rkl R kl − (3.7) R2 − ∇ 2 R)] − Tija log 1 , 2 4(d − 1) d −1 √ where Tija is the stress-energy tensor of the action dd x det γ a(d) . As it is shown in Appendix C, Tija is proportional to the tensor h(d)ij appearing in the expansion (2.3). The stress tensor Tij [g(0) ] is covariantly conserved with respect to the metric g(0)ij . To reg see this, notice that each of Tij and Tijct is separately covariantly conserved with respect reg to the induced metric γij at ρ = 1: for Tij one can check this by using the second equation in (2.5), for Tijct this follows from the fact that it was obtained by varying a local covariant counterterm. Since all divergences cancel in (3.4), we obtain that the finite part in (3.4) is conserved with respect to the metric g(0)ij . We are now ready to calculate Tij . By construction (and we will verify this below) the divergent pieces cancel between T reg and T ct .


605

3.1. d = 2. In two dimensions we obtain Tij =

l tij , 16π GN

(3.8)

where we have used (2.7) and (2.8) and the fact that Tija = 0 since R is a topological invariant (and reinstated the factor of l). As promised, tij is directly related to the boundary stress-energy tensor. Taking the trace we obtain Tii = −

c R, 24π

(3.9)

where c = 3l/2GN , which is the correct conformal anomaly [10]. Using our results, one can immediately obtain the stress-energy tensor of the boundary theory associated with a given solution G of the three dimensional Einstein equations: one needs to write the metric in the coordinate system (2.3) and then use the formula Tij =

2l (g(2)ij − g(0)ij Tr g(2) ). 16π GN

(3.10)

From the gravitational point of view this is the quasi-local stress energy tensor associated with the solution G. 3.2. d = 4. To obtain Tij we first need to rewrite the expressions in T ct in terms of g(0) . This can be done with the help of the relation Rij [γ ] = Rij [g(0) ] 1 1 1 + 1 2Rik R k j −2Rikj l R kl − ∇i ∇j R+∇ 2 Rij − ∇ 2 Rg(0)ij +O(1 2 ). 3 6 4 (3.11) After some algebra one obtains, 1 1 1 1 Tij [g(0) ] = − lim − g(2)ij + g(0)ij Tr g(2) + Rij − g(0)ij R 8πGN 1→0 1 2 4

a + log 1 − 2h(4)ij − Tij 1 2 −2g(4)ij − h(4)ij − g(2)ij Tr g(2) − g(0)ij Tr g(2) 2 1 1 1 Rik R k j − 2Rikj l R kl − ∇i ∇j R+∇ 2 Rij − ∇ 2 Rg(0)ij 8 3 6 1 1 1 (3.12) − g(2)ij R + g(0)ij Rkl R kl − R 2 . 4 8 6 Using the explicit expression for g(2) and h(4) given in (A.1) and (A.6) one finds that both the 1/1 pole and the logarithmic divergence cancel. Notice that had we not subtracted the logarithmic divergence from the action, the resulting stress-energy tensor would have been singular in the limit 1 → 0.

606


Using (2.9) and (2.10) and after some algebra we obtain Tij = −

1 [−2tij − 3h(4) ]. 8π GN

(3.13)

Taking the trace we get Tii =

1 (−2a(4) ), 16π GN

(3.14)

which is the correct conformal anomaly [26]. Notice that since h(4)ij = − 21 Tija the contribution in the boundary stress energy tensor proportional to h(4)ij is scheme dependent. Adding a local finite counterterm proportional to the trace anomaly will change the coefficient of this term. One may remove this contribution from the boundary stress energy tensor by a choice of scheme. Finally, one can obtain the energy-momentum tensor of the boundary theory for a given solution G of the five dimensional Einstein equations with negative cosmological constant. It is given by

1 4 1 1 2 2 Tij = g(4)ij − g(0)ij (Tr g(2) )2 − Tr g(2) − (g(2) )ij + g(2)ij Tr g(2) , 16πGN 8 2 4 (3.15) where we have omitted the scheme dependent h(4) terms. From the gravitational point of view this is the quasi-local stress energy tensor associated with the solution G. 3.3. d = 6. The calculation of the boundary stress tensor in the d = 6 case goes along the same lines as in d = 2 and d = 4 cases although it is technically involved. Up to a local traceless covariantly conserved term (proportional to h(6) ) the result is 3 1 Tij = g(6)ij − A(6)ij + Sij , (3.16) 8πGN 24 where A(6)ij is given in (A.4) and Sij in (2.12). It is covariantly conserved and has the correct trace 1 Tii = (−a(6) ), (3.17) 8π GN reproducing correctly the conformal anomaly in six dimensions [26]. Given an asymptotically AdS solution in six dimensions Eq. (3.16) yields the quasilocal stress energy tensor associated with it. 3.4. d = 2k + 1. In this case one can check that the counterterms only cancel infinities. Evaluating the finite part we get Tij =

d g(d)ij , 16π GN

(3.18)

where g(d)ij can be identified with a traceless covariantly conserved tensor tij . In odd boundary dimensions there are no gravitational conformal anomalies, and indeed (3.18) is traceless. As in all previous cases, one can also read (3.18) as giving the quasi-local stress energy tensor associated with a given solution of Einstein’s equations.


607

3.5. Conformally flat bulk metrics. In this subsection we discuss a special case where the bulk metric can be determined to all orders given only a boundary metric. It was shown in [40] that, given a conformally flat boundary metric, Eqs. (2.5) can be integrated to all orders if the bulk Weyl tensor vanishes7 . We show that the extra condition in the bulk metric singles out a specific vacuum of the CFT. The solution obtained in [40] is given by g(x, ρ) = g(0) (x) + g(2) (x)ρ + g(4) (x)ρ 2 , g(4) =

1 (g(2) )2 , 4

(3.19)

where g(2) is given in (A.1) (we consider d > 2), and all other coefficients g(n) , n > 4 vanish. Since g(4) and g(6) are now known, one can obtain a local formula for the dual stress energy tensor in terms of the curvature by using (2.9) and (2.14). In d = 4, using (2.9) and g(4) = 41 (g(2) )2 , one obtains

1 1 1 2 tij = tijcf ≡ − (g(2) )2ij + g(2)ij Tr g(2) − g(0)ij (Tr g(2) )2 − Tr g(2) . 4 4 8

(3.20)

It is easy to check that trace of tijcf reproduces (2.10). Furthermore, by virtue of Bianchi’s, one can show that tijcf is covariantly conserved. It is well-known that the stress-energy tensor of a quantum field theory on a conformally flat spacetime is a local function of the curvature tensor (see for example [8]). Our Eq. (3.20) reproduces the corresponding expression given in [8]. In d = 6, using (2.14) and g(6) = 0 we find

1 3 1 1 1 g − g 2 Tr g(2) + g(2) (Trg(2) )2 − g(2) Tr g(2) 4 (2) 4 (2) 8 8 1 1 1 2 3 3 + g(0) Tr g(2) Tr g(2) − Tr g(2) − (Tr g(2) ) . 8 12 24 ij

tij = tijcf ≡

(3.21)

One can verify that the trace of tijcf reproduces (2.15) (taking into account that g(4) = 1 2 cf 4 g(2) ) and that tij is covariantly conserved (by virtue of Bianchi’s). Following the analysis in the previous subsections we obtain Tij =

d t cf . 16π GN ij

(3.22)

So, we explicitly see that the global condition we imposed on the bulk metric implies that we have picked a particular vacuum in the conformal field theory. Note that the tensors tijcf in (3.20), (3.21) are local polynomial functions of the Ricci scalar and the Ricci tensor (but not of the Riemann tensor) of the metric g(0)ij . It is perhaps an expected but still a surprising result that in conformally flat backgrounds the anomalous stress tensor is a local function of the curvature. 7 In [40] it was proven that if the bulk metric satisfies Einstein’s equations and it has a vanishing Weyl tensor, then the corresponding boundary metric has to be conformally flat. The converse is not necessarily true: one can have Einstein metrics with non-vanishing Weyl tensor which induce a conformally flat metric in the boundary.

608


4. Conformal Transformation Properties of the Stress-Energy Tensor In this section we discuss the conformal transformation properties of the stress-energy tensor. These can be obtained by noting [28] that conformal transformations in the boundary originate from specific diffeomorphisms that preserve the form of the metric (2.3). Under these diffeomorphisms gij (x, ρ) transforms infinitesimally as [28] δgij (x, ρ) = 2σ (1 − ρ∂ρ ) gij (x, ρ) + ∇i aj (x, ρ) + ∇j ai (x, ρ), where aj (x, ρ) is obtained from the equation 1 ρ ij a i (x, ρ) = dρ g (x, ρ )∂j σ (x). 2 0 This can be integrated perturbatively in ρ, a i (x, ρ) =

k=1

i a(k) ρk .

(4.1)

(4.2)

(4.3)

We will need the first two terms in this expansion, i a(1) =

1 i ∂ σ, 2

1 ij i a(2) = − g(2) ∂j σ. 4

(4.4)

We can now obtain the way the g(n) ’s transform under conformal transformations [28] δg(0)ij = 2σg(0)ij , δg(2)ij = ∇i a(1)j + ∇j a(1)i , δg(3)ij = − σg(3)ij , k ∇k g(2)ij δg(4)ij = − 2σ (g(4) + h(4) ) + a(1)

(4.5)

k k + ∇i a(2)j + ∇j a(2)i + g(2)ik ∇j a(1) + g(2)j k ∇i a(1) ,

δg(5)ij = − 3σg(3)ij , where the term h(4) in g(4) is only present when d = 4. One can check from the explicit expressions for g(2) and g(4) in (A.1) that they indeed transform as (4.5). An alternative way to derive the transformation rules above is to start from (A.1) and perform a conformal variation. In [28] the variations (4.5) were integrated leading to (A.1) up to conformally invariant terms. Equipped with these results and the explicit form of the energy-momentum tensors, we can now easily calculate how the quantum stress-energy tensor transforms under conformal transformations. We use the term “quantum stress-energy tensor” because it incorporates the conformal anomaly. In the literature such transformation rules were obtained [12] by first integrating the conformal anomaly to an effective action. This effective action is a functional of the initial metric g and of the conformal factor σ . It can be shown that the difference between the stress-energy tensor of the theory on the manifold with metric ge2σ and the one on the manifold with metric g is given by the stress-energy tensor derived by varying the effective action with respect to g. In any dimension the stress-energy tensor transforms classically under conformal transformations as δTµν = −(d − 2) σ Tµν .

(4.6)


609

This transformation law is modified by the quantum conformal anomaly. In odd dimensions, where there is no conformal anomaly, the classical transformation rule (4.6) holds also at the quantum level. Indeed, for odd d, and by using (3.18) and (4.5), one easily verifies that the holographic stress-energy tensor transforms correctly. In even dimensions, the transformation (4.6) is modified. In d = 2, it is well-known that one gets an extra contribution proportional to the central charge. Indeed, using (3.10) and the formulae above we obtain δTij =

l c (∇i ∇j σ − g(0)ij ∇ 2 σ ), (∇i ∇j σ − g(0)ij ∇ 2 σ ) = 8πGN 12

(4.7)

which is the correct transformation rule. In d = 4 we obtain, δTij = − 2σ Tij

1 1 1 1 −2σ h(4) + ∇ k σ ∇k Rij − (∇i Rj k + ∇j Rik ) − ∇k Rg(0)ij + 4πGN 4 2 6 1 1 + (∇i σ ∇j R + ∇i σ ∇j R) + R(∇i ∇j σ − g(0)ij ∇ 2 σ ) 48 12 1

2 k k k l + Rij ∇ σ − (Rik ∇ ∇j σ + Rj k ∇ ∇i σ ) + g(0)ij Rkl ∇ ∇ σ . 8 (4.8) The only other result known to us is the result in [12], where they computed the finite conformal transformation of the stress-energy tensor but for a conformally flat metric g(0) . For conformally flat backgrounds, h(4) vanishes because it is the metric variation of a topological invariant. The terms proportional to a single derivative of σ vanish by virtue of Bianchi identities and the fact that the Weyl tensor vanishes for conformally flat metrics. The remaining terms, which only contain second derivatives of σ , can be shown to coincide with the infinitesimal version of (4.23) in [12]. One can obtain the conformal transformation of the stress energy tensor in d = 6 in a similar fashion but we shall not present this result here. 5. Matter In the previous sections we examined how spacetime is reconstructed (to leading order) holographically out of CFT data. In this section we wish to examine how field theory describing matter on this spacetime is encoded in the CFT. We will discuss scalar fields but the techniques are readily applicable to other kinds of matter. The method we will use is the same as in the case of pure gravity, i.e. we will start by specifying the sources that are turned on, find how far we can go with only this information and then input more CFT data. We will find the same pattern: knowledge of the sources allows only for determination of the divergent part of the action. The leading finite part (which depends on global issues and/or the signature of spacetime) is determined by the expectation value of the dual operator. We would like to stress that in the approach we follow, i.e. regularize, subtract all infinities by adding counterterms and finally remove the regulator to obtain the renormalized action, all normalizations of the physical correlation functions are fixed and are consistent with Ward identities. Other papers that discuss similar issues include [1, 36, 35, 44].

610


5.1. Dirichlet boundary problem for scalar fields in a fixed gravitational background. In this section we consider scalars on a fixed gravitational background. This is taken to be of the generic form (2.3). In most of the literature the fixed metric was taken to be that of standard AdS, but with not much more effort one can consider the general case. The action for the massive scalar is given by 1 2

SM =

dd+1 x

√ µν G G ∂µ ∂ν + m2 2 ,

(5.1)

where Gµν has an expansion of the form (2.3). We take the scalar field to have an expansion of the form (x, ρ) = ρ (d−)/2 φ(x, ρ),

φ(x, ρ) = φ(0) + φ(2) ρ + . . . ,

(5.2)

where is the conformal dimension of the dual operator. We take the dimension to be quantized as = d2 + k, k = 0, 1, . . . . This is often the case for operators of protected dimension. For the case of scalars that correspond to operators of dimensions d d 2 − 1 ≤ < 2 we refer to [29]. Inserting (5.2) in the field equation, (− where

G

=

√

√1 ∂µ ( G

G

+ m2 ) = 0,

(5.3)

GGµν ∂ν ), we obtain that the mass m2 and the conformal

dimension are related as m2 = ( − d), and that φ satisfies [−(d − )∂ρ log g φ + 2(2 − d − 2)∂ρ φ −

g φ]

+ ρ[−2∂ρ log g ∂ρ φ − 4∂ρ2 φ] = 0.

(5.4)

Given φ(0) one can determine recursively φ(n) , n > 0. This is achieved by differentiating (5.4) and setting ρ equal to zero. We give the result for the first couple of orders in Appendix D. This process breaks down at order − d/2 (provided this is an integer, which we assume throughout this section) because the coefficient of φ(2−d) (the field to be determined) becomes zero. This is exactly analogous to the situation encountered for even d in the gravitational sector. Exactly the same way as there, we introduce at this order a logarithmic term, i.e. the expansion of now reads, = ρ (d−)/2 (φ(0) + ρφ(2) + . . . ) + ρ /2 (φ(2−d) + log ρ ψ(2−d) + . . . ). (5.5) Equation (5.4) now determines all terms up to φ(2−d−2) , the coefficient of the logarithmic term ψ(2−d) , but leaves undetermined φ(2−d) . This is analogous to the situation discussed in Sect. 2 where the term g(d) was undetermined. It is well known [4, 5, 29] that precisely at order ρ /2 one finds the expectation value of the dual operator. We will review this argument below, and also derive the exact proportionality coefficient. Our result is in agreement with [29].


611

We proceed to regularize and then renormalize the theory. We regulate by integrating in the bulk from ρ ≥ 1,8 √ 1 SM,reg = dd+1 x G Gµν ∂µ ∂ν + m2 2 2 ρ≥1 1 = − dd x g(x, 1)1 −+d/2 [ (d − )φ 2 (x, 1) + 1 φ(x, 1)∂1 φ(x, 1)] 2 ρ=1 √ M M = dd x g(0) [1 −+d/2 a(0) + 1 −+d/2+1 a(2) + ... M + 1 a(2−d+2) − log 1 a(2−d) ] + O(1 0 )

(5.6) Clearly, with − d/2 a positive integer there are a finite number of divergent terms. The logarithmic divergence appears exactly when = d/2 + k, k = 0, 1, .., in agreement with the analysis in [37], and is directly related to the logarithmic term in (5.5). The first few of the power law divergences read 1 M 2 a(0) = − (d − )φ(0) , 2

1 M 2 a(2) = − Tr g(2) φ(0) + (d − + 1) φ(0) φ(2) . 4

(5.7)

Given a field of specific dimension it is straightforward to compute all divergent terms. We now proceed to obtain the renormalized action by adding counterterms to cancel the infinities,

√ M M SM,ren = lim SM,reg − dd x g(0) [1 −+d/2 a(0) + 1 −+d/2+1 a(2) + ... 1→0 (5.8) M + 1 a(2−d+2) − log 1 a(2−d) . Exactly as in the case of pure gravity, and since the regulated theory lives at ρ = 1, one needs to rewrite the counterterms in terms of the field living at ρ = 1, i.e. the induced metric γij (x, 1) and the field (x, 1), or equivalently gij (x, 1) and φ(x, 1). This is straightforward but somewhat tedious: one needs to invert the relation between φ and φ(0) and between gij and g(0)ij to sufficiently high order. This then allows to express all M φ(n) , and therefore all a(n) , in terms of φ(x, 1) and gij (x, 1) (the φ(n) ’s are determined in terms of φ(0) and g(0) by solving (5.4) iteratively). Explicitly, the first two orders read, √ 1 SM,ren = lim dd+1 x G Gµν ∂µ ∂ν + m2 2 1→0 2 ρ≥1 √ (d − ) 2 + γ (5.9) (x, 1) 2 ρ=1 1 d − + ((x, 1) γ (x, 1) + R[γ ]2 (x, 1)) + . . . . 2(2 − d − 2) 2(d − 1) The addition of the first counterterm was discussed in [29]. The action (5.9) with only the counterterms written explicitly is finite for fields of < d/2 + 2. As remarked 8 This regularization for scalar fields in a fixed AdS background was considered in [33, 17]. In these papers the divergences were computed in momentum space, but no counterterms were added to cancel them. Addition of boundary counterterms to cancel infinities for scalar fields was considered in [13], and more recently in [29].

612


above, it is straightforward to obtain all counterterms needed in order to make the action finite for any field of any mass. These counterterms contain also logarithmic subtractions that lead to the conformal anomalies discussed in [37]. For instance, if = 21 d + 1, the coefficient [2(2 − d − 2)]−1 in (5.9) is replaced by − 41 log 1. An alternative way to derive the counterterms is to demand that the expectation value O is finite. This holds in the case of pure gravity too, i.e. the counterterms can also be derived by requiring finiteness of Tµν [3]. The expectation value of the dual operator is given by O(x) = −

1 δSM,ren 1 δSM,ren = − lim √ . 1→0 det g(x, 1) δφ(x, 1) det g(0) δφ(0)

(5.10)

Exactly as in the case of pure gravity, the expectation value receives a contribution both from the regulated part and from the counterterms. We obtain, O(x) = (2 − d) φ(2−d) + F (φ(n) , ψ(2−d) , g(m) ),

n < 2 − d,

(5.11)

where we used that φ(2−d) is linear in φ(0) (notice that the action (5.1) does not include interactions). F (φ(n) , ψ(2−d) , g(m) ) is a local function of φ(n) with n < 2−d, ψ(2−d) and g(m) . These terms are related to contact terms in correlation functions of O with itself and with the stress-energy tensor. Its exact form is straightforward but somewhat tedious to obtain (just use (5.9) and (5.10)). As we have promised, we have shown that the coefficient φ(2−d) is related with the expectation value of the dual CFT operator. In the case that the background geometry is the standard Euclidean AdS one can readily obtain φ(2−d) from the unique solution of the scalar field equation with given Dirichlet boundary conditions. One finds that φ(2−d) is proportional to (an integral involving) φ(0) . Therefore, φ(2−d) carries information about the 2-point function. The factor ( − d/2) is crucial in order for the 2-point function to be normalized correctly [17]. We refer to [29] for a detailed discussion of this point. We finish this section by calculating the conformal anomaly associated with the scalar fields and in the case the background is (locally) standard AdS (i.e. g(n) = 0, for 0 < n < d). Equation (5.4) simplifies and can be easily solved. One gets 1 0 φ(2n−2) , 2n(2 − d − 2n) 1 1 =− ( 0 φ(2−d−2) = − 2k 2(2 − d) 2 )(k))(k + 1)

φ(2n) = ψ(2−d)

k 0 ) φ(0) ,

(5.12)

where k = − d2 and 0 is the Laplacian of g(0) . The regularized action written in terms of the fields at ρ = 1 contains the following explicit logarithmic divergence: d √

SM,reg = − dd x γ log 1 − (5.13) φ(x, 1) ψ(2−d) (x, 1) + · · · , 2 ρ=1 where the dots indicate power law divergent and finite terms, ψ(2−d) (x, 1) is given by (5.12) with g(0) replaced by γ and φ(0) by φ(x, 1). Using the same argument as in [26] we obtain the matter conformal anomaly, 1 1 AM = φ(0) ( 0 )k φ(0) . (5.14) 2 22k−2 ()(k))2 This agrees exactly with the anomaly calculated in [37] (compare with formulae (10), (37) in [37]).


613

5.2. Scalars coupled to gravity. In the previous section we ignored the back-reaction of the scalars to the bulk geometry. The purpose of this section is to discuss this issue. The action is now the sum of (2.1) and (5.1), S = Sgr + SM .

(5.15)

The gravitational field equation in the presence of matter reads 1 Rµν − (R + 2)Gµν = −8π GN Tµν . 2

(5.16)

In the coordinate system (2.3) and with the scalar field having the expansion in (5.5), these equations read ρ [2gij − 2(g g −1 g )ij + Tr (g −1 g ) gij ] + Rij (g) − (d − 2) gij − Tr (g −1 g ) gij ( − d) 2 = −8π GN ρ d−−1 φ gij + ρ ∂i φ∂j φ , (5.17) d −1

d − ∇i Tr (g g = − 16π GN ρ φ∂i φ + ρ ∂ρ φ∂i φ , 2 d( − d)( − d + 1) 2 1 Tr (g −1 g ) − Tr (g −1 g g −1 g ) = − 16π GN ρ d−−2 φ 2 4(d − 1) + (d − ) ρ φ∂ρ φ + ρ 2 (∂ρ φ)2 . −1

) − ∇ j gij

d−−1

If > d, the right-hand side diverges near the boundary whereas the left-hand side is finite. Operators with dimension > d are irrelevant operators. Correlation functions of these operators have a very complicated singularity structure at coincident points. As remarked in [47], one can avoid such problems by considering the sources to be infinitesimal and to have disjoint support, so that these operators are never at coincident points. Requiring that the equations in (5.17) are satisfied to leading order in ρ yields 2 φ(0) = 0,

(5.18)

which is indeed the prescription advocated in [47]. If ≤ d, which means that we deal with marginal or relevant operators, one can perturbatively calculate the back-reaction of the scalars to the bulk metric. At which order the leading back-reaction appears depends on the mass of the field. For fields that correspond to operators of dimension = d − k the leading back-reaction appears at order ρ k , except when k = 0 (marginal operators), where the leading back-reaction is at order ρ. Let us see how conformal anomalies arise in this context. The logarithmic divergences are coming from the regulated on-shell value of the bulk integral in (5.15). The latter reads √ d m2 Sreg (bulk) = (5.19) 2 dρ dd x G − 8π GN d −1 ρ≥1

d 1 m2 φ 2 (x, ρ) ρ −k , dρ dd x g(x, ρ) ρ −d/2 − = ρ 16π GN 2(d − 1) ρ≥1

614


where k = − d/2. We see that gravitational conformal anomalies are expected when d is even and matter conformal anomalies when k is a positive integer, as it should. In the presence of sources the expectation value of the boundary stress-energy tensor is not conserved but rather it satisfies a Ward identity that relates its covariant divergence to the expectation value of the operators that couple to the sources. To see this consider the generating functional

1 √ ij ZCFT [g(0) , φ(0) ] = exp dd x g(0) (5.20) g(0) Tij − φ(0) O . 2 Invariance under infinitesimal diffeomorphisms, δg(0)ij = ∇i ξj + ∇j ξi ,

(5.21)

∇ j Tij = O ∂i φ(0) .

(5.22)

yields the Ward identity,

As we have remarked before, Tij has a dual meaning [3], both as the expectation value of the dual stress-energy tensor and as the quasi-local stress-energy tensor of Brown and York. The Ward identity (5.22) has a natural explanation from the latter point in view as well. According to [11] the quasi-local stress-energy tensor is not conserved in the presence of matter but it satisfies ∇ j Tij = −τiρ ,

(5.23)

where τiρ expresses the flow of matter energy-momentum through the boundary. Evidently, (5.22) is of the form (5.23). Solving the coupled system of equations (5.17) and (5.4) is straightforward but somewhat tedious. The details differ from case to case. For illustrative purposes we present a sample calculation: we consider the case of two-dimensional massless scalar field (d = = 2, k = 1). The equations to be solved are (5.4) and (5.17) with d = = 2 and the expansion of the metric and the scalar field are given by (2.3) and (5.5) (again with d = = 2), respectively. Equation (5.4) determines ψ(2) , ψ(2) = −

1 4

0 φ(0) .

(5.24)

Equations (5.17) determine h(2) , the trace of the g(2) and provide a relation between the divergence of g(2) and φ(2) , 1 2 h(2) = −4πGN ∂i φ(0) ∂j φ(0) − g(0)ij (∂φ(0) ) , 2 1 (5.25) Tr g(2) = R + 4π GN (∂φ(0) )2 , 2 ∇ i g(2)ij = ∂i Tr g(2) + 16π GN φ(2) ∂i φ(0) . Notice that g(2) and φ(2) are still undetermined and are related to the expectation values of the dual operators (3.4) and (5.11), respectively. Notice that h(2) is equal to the stressenergy tensor of a massless two-dimensional scalar.


615

Going back to (5.19), we see that the second term drops out (since m2 = 0) and one can use the result already obtained in the gravitational sector, 1 1 (−2a(2) ) = (−2Tr g(2) ) 16πGN 16π GN 1 1 1 =− R + φ(0) 0 φ(0) − ∇i (φ(0) ∇ i φ(0) ), 16πGN 2 2

A=

(5.26)

which is the correct conformal anomaly [26, 37] (the last term can be removed by adding a covariant counterterm). The renormalized boundary stress tensor reads Tij (x) =

1 g(2)ij + h(2)ij − g(0)ij Tr g(2) (x). 8π GN

(5.27)

Its trace gives correctly the conformal anomaly (5.26). On the other hand, taking the covariant derivative of (5.27) we get ∇ j Tij = O(x) ∂i φ0 (x), O(x) = 2(φ2 (x) + ψ2 (x)).

(5.28)

in agreement with Eqs. (5.22) and (5.11).

6. Conclusions Most of the discussions in the literature on the AdS/CFT correspondence are concerned with obtaining conformal field theory correlation functions using supergravity. In this paper we started investigating the converse question: how can one obtain information about the bulk theory from CFT correlation functions? How does one decode the hologram? Answering these questions in all generality, but within the context of the AdS/CFT duality, entails developing a precise dictionary between bulk and boundary physics. A prescription for relating bulk/boundary observables is already available [24, 47], and one would expect that it would allow us to reconstruct the bulk spacetime from the boundary CFT. The prescription of [24, 47], however, relates infinite quantities. One of the main results of this paper is the systematic development of a renormalized version of this prescription. Equipped with it, and with no other assumption (except that the CFT has an AdS dual), we then proceeded to reconstruct the bulk spacetime metric and bulk scalar fields to the first non-trivial order. Our approach to the problem is to start from the boundary and try to build iteratively bulk solutions. Within this approach, the pattern we find is the following: • Sources in the CFT determine an asymptotic expansion of the corresponding bulk field near the boundary to high enough order so that all infrared divergences of the bulk on-shell action can be computed. This then allows to obtain a renormalized on-shell action by adding boundary counterterms to cancel the infrared divergences. • Bulk solutions can be extended one order further by using the 1-point function of the corresponding dual CFT operator.

616


In the case the bulk field is the metric, our results show that a conformal structure at infinity is not in general sufficient in order to obtain a bulk metric. The first additional information one needs is the expectation value of the boundary stress energy tensor. As a by-product, we have obtained ready-to-use formulae for the Brown–York quasilocal stress-energy tensor for arbitrary solution of Einstein’s equations with negative cosmological constant up to six dimensions. The six-dimensional result is particularly interesting because, via AdS/CFT, provides new information about the still mysterious (2, 0) theory. Furthermore, we have obtained the conformal transformation properties of the stress-energy tensors. These transformation rules incorporate the trace anomaly and provide a generalization to d > 2 of the well-known Schwartzian derivative contribution in the conformal transformation rule of the stress-energy tensor in d = 2. Our discussion extends straightforwardly to the case of different matter. We expect that in all cases obstructions in extending the solution to the deep interior region will be resolved by additional CFT data (including data about non-local observables such as Wilson loops, Wilson surfaces etc.).An interesting case to study in this framework is point particles [14]. Reconstructing the trajectory of the bulk point particle out of CFT data will present a model of how holography works with time dependent processes. Furthermore, following [27], one could study the interplay between causality and holography. Another extension is to study renormalization group flows using the present formalism. This amounts to extending the discussion in Sect. 5.2 by adding a potential for the scalars. Another application of our results is in the context of Randall–Sundrum (RS) scenarios [38]. Incorporating such a scenario in string theory, in the case the bulk space is AdS, may yield a connection with the AdS/CFT duality [46, 48]. As advocated in [48], one may view the RS scenario as 4d gravity coupled to a cut-off CFT. The regulated theory in our discussion provides a dual description of a cut-off CFT. In this context, the counterterms are re-interpreted as providing the action for the bulk modes localized in the brane [39, 23, 19]. We see, for instance, that the counterterms in (5.9) can be re-interpreted as an action for a bulk scalar mode localized on the brane.

Note added. As this paper was being finalized, [7] appeared with some overlap with the results of Sect. 2. Acknowledgements. We would like to thank G. ’t Hooft for reading the manuscript and his useful remarks. This research is supported in part by NSF grants PHY94-07194 and PHY-9802484. KS would like to thank ITP in UCSB for hospitality during initial stages of this work. SS would like to thank the Theory Division at CERN for the hospitality extended to him while this work was in progress.

Appendix A. Asymptotic Solution of Einstein’s Equations In this appendix we collect the results for the solution of Eqs. (2.5) up to the order we are interested in. From the first equation in (2.5) one determines the coefficients g(n) , n = d, in terms of g(0) . For our purpose we only need g(2) and g(4) .


617

There are given by g(2)ij g(4)ij

1 1 = Rij − R g(0)ij , d −2 2(d − 1) 1 1 1 = − D i Dj R + Dk D k Rij d −4 8(d − 1) 4(d − 2) 1 1 − Dk D k Rg(0)ij − R kl Rikj l 8(d − 1)(d − 2) 2(d − 2) d −4 1 + Ri k Rkj + RRij 2 2(d − 2) (d − 1)(d − 2)2 1 3d kl 2 + R Rkl g(0)ij − R g(0)ij . 4(d − 2)2 16(d − 1)2 (d − 2)2

(A.1)

The expressions for g(n) are singular when n = d. One can obtain the trace and the divergence of g(n) for any n from the last two equations in (2.5). Explicitly, 1 2 , Tr g(2) 4 = 0,

2 1 3 , Tr g(2) g(4) − Tr g(2) 3 6 = 0,

Tr g(4) =

Tr g(6) =

Tr g(3)

Tr g(5)

(A.2)

and ∇ i g(2)ij = ∇ i A(2)ij ,

∇ i g(3)ij = 0,

∇ i g(4)ij = ∇ i A(4)ij , i

i

∇ g(6)ij = ∇ A(6)ij

∇ i g(5)ij = 0, 1 + Tr (g(4) ∇j g(2) ), 6

(A.3)

where A(2)ij = g(0)ij Tr g(2) , 1 1 2 1 2 − (Tr g(2) )2 ] g(0)ij + (g(2) )ij − g(2)ij Tr g(2) , A(4)ij = − [Tr g(2) 8 2 4 1 1 3 2 A(6)ij = − (Tr g(2) )2 ] g(2)ij 2(g(2) g(4) )ij + (g(4) g(2) )ij − (g(2) )ij + [Tr g(2) 3 8 1 2 1 2 − Tr g(2) [g(4)ij − (g(2) )ij ] − [ Tr g(2) Tr g(2) (A.4) 2 8 1 1 1 3 − (Tr g(2) )3 − Tr g(2) + Tr (g(2) g(4) )] g(0)ij . 24 6 2 For even n = d the first equation in (2.5) determines the coefficients h(d) . They are given by h(2)ij = 0, 1 2 1 2 − g(0)ij Tr g(2) h(4)ij = g(2)ij 2 8 1 k ∇ ∇i g(2)j k + ∇ k ∇j g(2)ik − ∇ 2 g(2)ij − ∇i ∇j Tr g(2) + 8 1 1 1 1 = Rikj l R kl + ∇i ∇j R − ∇ 2 Rij − RRij 8 48 16 24

(A.5)

(A.6)

618


1 1 1 ∇ 2 R + R 2 − Rkl R kl g(0)ij , 96 96 32 2 1 3 = (g(4) g(2) + g(2) g(4) )ij − g(2)ij 3 3 1 1 3 − g(4)ij Tr g(2) + g(0)ij (3Trg(6) − 3Trg(2) g(4) + Trg(2) ) 6 6 1 1 2 − ∇ k ∇i g(4)j k − ∇ k ∇j g(4)ik + ∇ 2 g(4)ij − [− ∇i ∇j Trg(2) 12 4 kl + g(2) [∇l ∇i g(2)j k + ∇l ∇j g(2)ik − ∇l ∇k g(2)ij ] 1 + ∇ k Trg(2) (∇i g(2)j k + ∇j g(2)ik − ∇k g(2)ij ) 2 1 kl + ∇k g(2)il ∇ l g(2)j k − ∇k g(2)il ∇ k g(2)j l ]. + ∇i g(2)kl ∇j g(2) 2 +

h(6)ij

(A.7)

B. Divergences in Terms of the Induced Metric In this appendix we rewrite the divergent terms of the regularized action in terms of the induced metric at ρ = 1. This is needed in order to derive the contribution of the counterterms to the stress energy tensor. The coefficients a(n) of the divergent terms in the regulated action (3.2) are given by a(0) = 2(1 − d), a(2) = b(2) (d) Tr g(2) , 2 a(4) = b(4) (d) [(Tr g(2) )2 − Tr g(2) ], 1 3 1 3 2 3 − Tr g(2) Tr g(2) + Tr g(2) − Tr g(2) g(4) , Tr g(2) a(6) = 8 8 2

(B.1)

where a(6) is only valid in six dimensions and the numerical coefficients in a(2) and a(4) are given by

b(2) (d = 2) = − b(2) (d = 2) = 1,

(d − 4)(d − 1) , d −2

−d 2 + 9d − 16 , 4(d − 4) 1 b(4) (d = 4) = . 2

b(4) (d = 4) =

(B.2)

Notice that the coefficients a(n) are proportional to the expression for the conformal anomaly (in terms of g(n) ) in dimension d = n [26].


619

The counterterms can be rewritten in terms of the induced metric by inverting the relation between γ and g(0) perturbatively in 1. One finds 1 √ −1 d/2 1 − 1 Tr g(0) g(0) = 1 g(2) 2 1 2

−1 −1 2 2 3 √ + 1 (Tr g(0) g(2) ) + Tr (g(0) g(2) ) + O(1 ) γ, 8 1 1 Tr g(2) = R[γ ] (B.3) 2(d − 1) 1 1 1 + (Rij [γ ]R ij [γ ] − R 2 [γ ]) + O(R[γ ]3 ) , d −2 2(d − 1) 1 1 −3d + 4 2 2 ij 3 Tr g(2) = 2 Rij [γ ]R [γ ] + R [γ ] + O(R[γ ] ) . 1 (d − 2)2 4(d − 1)2 The terms cubic in curvatures in (B.3) give vanishing contribution in (3.4) up to six dimensions. Putting everything together we obtain that the counterterms, rewritten in terms of the induced metric, are given by 1 1 √ S ct = − R γ 2(1 − d) + 16πGN ρ=1 d −2 (B.4) 1 d ij 2 − Rij R − R − log 1 a(d) + . . . , (d − 4)(d − 2)2 4(d − 1) where all quantities are now in terms of the induced metric, including the one in the logarithmic divergence. These are exactly the counterterms in [3,15,30] except that these authors did not include the logarithmic divergence. Equation (B.4) should be understood as containing only divergent counterterms in each dimension. This means that in even dimension d = 2k one should include only the first k counterterms and the logarithmic one. In odd d = 2k + 1, only the first k + 1 counterterms should be included. The logarithmic counterterms appear only for d even. The counterterms in (B.4) render the renormalized action finite up to d = 6. This covers all cases relevant for the AdS/CFT correspondence. It is straightforward but tedious to compute the necessary counterterms for d > 6. From (B.4) one straightforwardly obtains (3.7). C. Relation Between h(d) and the Conformal Anomaly a(d) We show in this appendix that the tensor h(d) appearing in the expansion of the metric in (2.3) when d is even is a multiple of the stress tensor derived from the action a(d) . (a(d) is, up to a constant, the holographic conformal anomaly.) This can be shown by deriving the stress-energy tensor of the regulated theory at ρ = 1 in two ways and then comparing the results. In the first derivation one starts from reg (3.1) and obtains the regulated stress-energy tensor as in (3.6). Expanding Tij [γ ] in 1 (keeping g(0) fixed) we find that there is a logarithmic divergence, reg

Tij [γ ; log] =

3 1 log 1 d − 1 h(d)ij . 8π GN 2

(C.1)

620

S. de Haro, K. Skenderis, S. N. Solodukhin reg

On the other hand, one can derive Tij [γ ] starting from (3.2). One has to first rewrite the terms in (3.2) in terms of the induced metric. This is done in the previous appendix. Once reg Tij [γ ] has been derived, we expand in 1. We find the following logarithmic divergence: 1 log 1 (1 − d)h(d)ij − Tija , (C.2) 8π GN where Tija is the stress-energy tensor of the action dd x det g(0) a(d) . If follows that reg

Tij [γ ; log] =

2 h(d)ij = − Tija . d

(C.3)

We have also explicitly verified this relation by brute-force computation in d = 4. D. Asymptotic Solution of the Scalar Field Equation We give here the first two orders of the solution of Eq. (5.4):

1 0 φ(0) + (d − )φ(0) Tr g(2) , 2(2 − d − 2) 1 1 2 = (D.1) 0 φ(2) − 2 Tr g(2) φ(2) − (d − ) [Tr g(2) φ(0) 4(2 − d − 4) 2 1 1 √ µν −2Tr g(2) φ(2) ] − √ ∂µ ( g(0) g(2) ∂ν φ(0) ) + ∂ i Tr g(2) ∂j φ(0) , g(0) 2

φ(2) = φ(4)

where in 0 the covariant derivatives are with respect to g(0) . If 2−d −2k = 0 one needs to introduce a logarithmic term in order for the equations to have a solution, as discussed in the main text. For instance, when = 21 d + 1, φ(2) is undetermined, but instead one obtains for the coefficient of the logarithmic term, 1 d ψ(2) = − (D.2) − 1) φ φ + ( Tr g 0 (0) (0) (2) . 4 2

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Communications in



Cohomologies of Affine Hyperelliptic Jacobi Varieties and Integrable Systems A. Nakayashiki1 , F. A. Smirnov2, 1 Graduate School of Mathematics, Kyushu University, Ropponmatsu 4-2-1, Fukuoka 810-8560, Japan 2 Laboratoire de Physique Théorique et Hautes Energies , Université Pierre et Marie Curie, Tour 16 1er étage, 4 place Jussieu, 75252 Paris cedex 05, France

Received: 2 February 2000 / Accepted: 15 November 2000

Abstract: We study the affine ring of the affine Jacobi variety of a hyperelliptic curve. The matrix construction of the affine hyperelliptic Jacobi varieties due to Mumford is used to calculate the character of the affine ring. By decomposing the character we make several conjectures on the cohomology groups of the affine hyperelliptic Jacobi varieties. In the integrable system described by the familly of these affine hyperelliptic Jacobi varieties, the affine ring is closely related to the algebra of functions on the phase space, classical observables. We show that the affine ring is generated by the highest cohomology group over the action of the invariant vector fields on the Jacobi variety. 1. Introduction Our initial motivation is the study of integrable systems. Consider an integrable system with 2n degrees of freedom, by definition it possesses n integrals in involution. The levels of these integrals are n-dimensional tori. This is a general description, but the particular examples of integrable models that we meet in practice are much more special. Let us explain how they are organized. The phase space M is embedded algebraically into the space RN . The integrals are algebraic functions of coordinates in this space. This situation allows complexification; the complexified phase space MC is an algebraic affine variety embedded into CN . The levels of integrals in the complexified case allow the following beautiful description. The systems that we consider are such that with every one of them one can identify an algebraic curve X of genus n whose moduli are defined by the integrals of motion. On the Jacobian J (X) of this curve there is a particular divisor D (in this paper we consider the case when this divisor coincides with the theta divisor, but more complicated situations are possible). The level of integrals is isomorphic to the affine variety J (X) − D. The Membre du CNRS

Laboratoire associé au CNRS.

624

A. Nakayashiki, F. A. Smirnov

real space RN ⊂ CN intersects with every level of integrals by a compact real subtorus of J (X) − D. This structure explains why the methods of algebraic geometry are so important in application to integrable models. Closest to the present paper account of these methods is given in Mumford’s book [1]. Let us describe briefly the results of the present paper. We study the structure of the ring A of algebraic functions (observables) on the phase space of certain integrable models. The curve X in our case is hyperelliptic. As is clear from the description given above this ring of algebraic functions is, roughly, a product of the functions of integrals of motion by the affine ring of hyperelliptic Jacobian. The commuting vector fields defined by taking Poisson brackets with the integrals of motion are acting on A. We shall show that by the action of these vector-fields the ring A is generated from a finite number of functions corresponding to the highest nontrivial cohomology group of the affine Jacobian. We conjecture the form of the cohomology groups in every degree and demonstrate the consistence of our conjectures with the structure of the ring A. Finally we would like to say that this relation to cohomology groups became clear analyzing the results of papers [2] and [3] which deal with quantum integrable models. Very briefly the reason for that is as follows. It has been mentioned that the algebra of classical observables is nothing but the ring A. The quantum observables are in one-toone correspondence with the classical ones, the algebra of quantum observables being a certain non-commutative deformation of A. The quantum Hamiltonians constitute a commutative sub-algebra. In the quantum mechanics the Hilbert space must be presented in which the observables act. This Hilbert space can be realized in the case under consideration as the space of functions of “coordinates”. The latter are defined as one half of the separated variables [5] which correspond to angle variables of classical integrable model. The first problem is that of diagonalization of Hamiltonians. This problem is solved by means of Baxter equation (see references in [3]). The second problem is that of calculation of matrix elements of observables. These matrix elements are written as integrals with respect to the “coordinates”. Semi-classically, the product of two eigenfunctions gives a volume form on the torus, and the observable itself can be considered as a multiplier in front of this volume form, i.e. as a coefficient of some differential top form on the torus which is the same as the form of one-half of the maximal dimension on the phase space. Some observables vanish due to quantum equations of motion. These observables correspond to the “exact form” which means that the integrals defining their matrix elements vanish. This is how the relation to the cohomologies appears. The paper, after the introduction, consists of five sections and six appendices which contain technical details and some proofs. In Sect. 2 we recall the standard construction of the Jacobi variety which is valid for any Riemann surface. An algebraic construction of the affine Jacobi variety J (X) − of a hyperelliptic curve X is reviewed in Sect. 3 following the book [1, Vol. II]. This construction is specific to hyperelliptic curves or more generally spectral curves [6]. In Sect. 4 we study the affine ring of J (X) − using the description in Sect. 3. The main ingredient here is the character of the affine ring. To be precise we consider the ring A0 corresponding to the most degenerate curve y 2 = z2g+1 . The ring A and the affine ring Af of J (X) − for a non-singular X can be studied using A0 . It is important that A0 is a graded ring and the character ch(A0 ) is defined. We calculate it by determining

Cohomologies of Affine Hyperelliptic Jacobi Varieties and Integrable Systems

625

explicitly a C-basis of A0 . The relation between A0 and Af for a non-singular X is given in Appendix E. A set of commuting vector fields acting on A is introduced in Sect. 5. This action descends to the quotients A0 and Af . If X is non-singular, the action of these vector fields on Af coincides with that of invariant vector fields of J (X). With the help of these vector fields we define the de Rham type complexes (C∗ , d), (C∗0 , d), (C∗f , d) with the coefficients in A, A0 , Af respectively. The complex (C∗f , d) is nothing but the algebraic de Rham complex of J (X) − whose cohomology groups are known to be isomorphic to the singular cohomology groups of J (X) − . What is interesting for us is the cohomology groups of (C∗0 , d). We calculate the q-Euler characteristic of (C∗0 , d). Then we show that it coincides with ch(A0 )/ch(D), where D is the ring generated by commuting vector fields. Then, by the Euler–Poincaré principle, ch(A0 )/ch(D) is found to be expressible as the alternating sum of the characters of cohomology groups of (C∗0 , d). Decomposing independently the explicit formula of ch(A0 ) into the alternating sum, we make conjectures on the cohomology groups of (C∗0 , d) which are formulated in the next section. In Sect. 6 and in Appendix B we study the singular homology and cohomology groups of J (X) − . The Riemann bilinear relation plays an important role here. We formulate conjectures on the cohomology groups of (C∗ , d), (C∗0 , d), (C∗f , d). 2. Hyperelliptic Curves and Their Jacobians Consider the hyperelliptic curve X of genus g described by the equation: y 2 = f (z), where f (z) = z2g+1 + f1 z2g + · · · + f2g+1 .

(1)

The hyperelliptic involution σ is defined by σ (z, y) = (z, −y). The Riemann surface X can be realized as a two-sheeted covering of the z-sphere with the quadratic branch points which are zeros of the polynomial f (z) and ∞. A basis of holomorphic differentials is given by: µj = zg−j

dz , y

j = 1, . . . , g.

Choose a canonical homology basis of X: α1 , . . . , αg , β1 , . . . , βg . The basis of normalized differentials is defined as ωi =

g

(M −1 )ij µj ,

j =1

where the matrix M consists of α-periods of holomorphic differentials µi : Mij = µi , i, j = 1 . . . , g. αj

(2)

626


The period matrix

Bij =

ωj βi

defines a point B in the Siegel upper half space: Bij = Bj i ,

Im(B) > 0.

The Jacobi variety of X is a g-dimensional complex torus: J (X) =

Zg

Cg . + BZg

The Riemann theta function associated with J (X) is defined by θ (ζ ) = exp 2π i 21 t mBm + t mζ , m∈Zg

where ζ ∈ Cg . The theta function satisfies θ (ζ + m + Bn) = exp 2π i − 21 t nBn − t nζ θ(ζ ), for m, n ∈ Zg . Consider the symmetric product of X, the quotient of the product space by the action of the symmetric group: X(n) = X n /Sn . The Abel transformation defines the map a

X(g) −→ J (X) explicitly given by pk

wj =

g

ωj + ',

k=1 ∞

where p1 , . . . , pg are points of X, ' is the Riemann characteristic corresponding to the choice of ∞ for the reference point. In the present case ' is a half-period because ∞ is a branch point [1]. The divisor is the (g − 1)-dimensional subvariety of J (X) defined by

= {w | θ(w) = 0}.

(3)

The main subject of our study is the ring A of meromorphic functions on J (X) with singularities only on . The simple way to describe this ring is provided by theta functions: A=

∞

k , θ(w)k

k=0

(4)


627

where k is the space of theta functions of order k, i.e. the space of regular functions on Cg satisfying θk (w + m + Bn) = exp 2kπ i − 21 t nBn − t nw θk (w). There are k g linearly independent theta functions of order k. Let us discuss the geometric meaning of the ring A. It is well known that with the help of theta functions one can embed the complex torus J (X) into the complex projective space as a non-singular algebraic subvariety. It can be done, for example, using theta functions of third order: 1. 3g theta functions of third order define an embedding of J (X) into the complex g projective space P3 −1 , 2. a set of homogeneous algebraic equations for these theta functions can be written, which allows us to describe this embedding as an algebraic one. Now consider the functions

3 . θ(w)3 Obviously, with the help of these functions, we can embed the non-compact variety g J (X) − into the complex affine space C3 −1 . Denote the coordinates in this space by x1 , . . . , x3g −1 , the affine ring of J (X) − is defined as the ring C [x1 , . . . , x3g −1 ]/(gα ), where (gα ) is the ideal generated by the polynomials {gα } such that {gα = 0} defines the embedding. It is known that the affine ring is the characteristic of the non-compact variety J (X)− independent of a particular embedding of this variety into affine space. Obviously the ring A defined above is isomorphic to the affine ring. We remark that the above argument on the embedding J (X) − into an affine space is valid if (J (X), ) is replaced by any principally polarized abelian variety. Consider X(g) which is mapped to J (X) by the Abel map a. Riemann’s theorem says that p g−1 j θ (w) = 0 iff w = ω + ' in J (X) j =1 ∞

which allows to describe in terms of the symmetric product. One easily argues that the preimage of under the Abel map is described as D := D∞ ∪ D0 , where D∞ = {(p1 , . . . , pg ) ∈ X(g)| pi = ∞ for some i}, D0 = {(p1 , . . . , pg ) ∈ X(g)| pi = σ (pj ) for some i = j }.

(5)

The Abel map is not one-to-one, and the compact varieties J (X) and X(g) are not isomorphic. However, the affine varieties J (X) − and X(g) − D are isomorphic since the Abel map a X(g) − D −→ J (X) −

is an isomorphism. In what follows we shall study the affine variety X(g) − D J (X) − .

628


3. Affine Model of Hyperelliptic Jacobian Consider a traceless 2 × 2 matrix

a(z) b(z) m(z) = , c(z) −a(z)

where the matrix elements are polynomials of the form: a(z) = a 3 zg−1 + a 5 zg−2 + · · · + ag+ 1 , 2

2

2

(6)

b(z) = zg + b1 zg−1 + · · · + bg , c(z) = zg+1 + c1 zg + c2 zg−1 + · · · + cg+1 . We shall explain the strange way of counting of coefficients by integers and half-integers in the next section. Later we shall set b0 = c0 = 1. Consider the affine space C3g+1 with coordinates a 3 , . . . , a g+1 , b1 , . . . , bg , c1 , . . . , cg+1 . Fix the determinant of m(z): 2

2

a 2 (z) + b(z)c(z) = f (z),

(7)

where the polynomial f (z) is the same as used above (1). Comparing each coefficient of zi (i = 0, 1, . . . , 2g) of (7) one gets 2g + 1 different equations. In fact Eq. (7) defines a g-dimensional sub-variety of C3g+1 . This algebraic variety is isomorphic to J (X) −

as shown in the book [1]. We shall briefly recall the proof. Consider a matrix m(z) satisfying (7). Take the zeros of b(z): b(z) =

g

(z − zj )

j =1

and set

yj = a(zj ).

Obviously zj , yj satisfy the equation yj2 = f (zj ), which defines the curve X. So, we have constructed a point of X(g) for every m(z) which satisfies Eq. (7). Conversely, for a point (p1 , . . . , pg ) of X(g), construct the matrix m(z) as g g

z − zk b(z) = , (z − zj ), a(z) = yj zj − z k j =1

c(z) =

j =1

k=j

+ f (z) , b(z)

−a(z)2

where zj = z(pj ) is the z-coordinate of pj . Considering the function b(z) as a function on X(g) one finds that it has singularities when one of zj equals ∞. The function a(z) is singular at zj = ∞ and also at the points where zi = zj but yi = −yj . This is exactly the description of the variety D. The function c(z) does not add new singularities. Thus we have the embedding of the affine variety X(g) − D into the affine space: X(g) − D ,→ C3g+1 .


629

Therefore we can profit from the wonderful property of the hyperelliptic Jacobian: it allows an affine embedding into a space of very small dimension equal to 3g+1 (compare with 3g − 1 which we have for any abelian variety). Actually, the space C3g+1 occurs foliated with generic leaves isomorphic to the affine Jacobians. 4. Properties of Affine Ring Consider the free polynomial ring A: A = C [a 3 , . . . , ag+ 1 , b1 , . . . , bg , c1 , . . . , cg+1 ]. 2

2

On the ring A one can naturally introduce a grading. Prescribe the degree j to any of the generators aj , bj , cj and extend this definition to all monomials in A by deg(xy) = deg(x) + deg(y). Now it is clear that the integer and half-integer degrees of aj , bj , cj were chosen in order that the coefficients of the determinant (7) are homogeneous. Every monomial of the ring has positive degree (except for 1 whose degree equals 0). Thus, as a linear space, A splits into A= A(p) , 2p∈Z+

where A(j ) is the subspace of degree j and Z+ = {0, 1, 2, . . . }. Define the character of A by q p dim(A(p) ). ch(A) = 2p∈Z+

Since the ring A is freely generated by ap , bp , cp one easily finds 1 ch(A) =

g+

1 2

where, for k ∈ Z+ , [k] = 1 − q k ,

[k]! = [1] . . . [k],

2

! [g]! [g + 1]!

,

(8)

k + 21 ! = 21 23 . . . k + 21 .

This important formula allows us to control the size of the ring A. The relation of the ring A to the affine ring A is obvious. The latter is the quotient of A by the ideal generated by the relations −det(m(z)) = f (z), where the coefficients of f are considered fixed constants. From the point of view of integrable models, it is more natural to see f1 , . . . , f2g+1 as variables than complex numbers. If we assign degree j to the variables fj , all the equations in (7) are homogeneous. Consider the polynomial ring F = C [f1 , . . . , f2g+1 ]. The ring F is graded and its character is ch(F) =

1 . [2g + 1]!

630


The ring F acts on A, that is, f (z) acts by the multiplication of the left hand side of (7). Consider the space A0 which consists of F-equivalence classes: A0 = A / (F× A),

F× =

2g+1

Ffi .

i=1

Since F× A is a homogeneous ideal of A, A0 is a graded vector space: (p) A0 = A0 . 2p∈Z+

One can consider the space A0 as a subspace of A taking a set of homogeneous representatives of the equivalence classes (being homogeneous they are automatically of smallest possible degree). Consider any homogeneous x ∈ A. One can write x as x = x (0) +

2g+1

f i xi ,

i=1

where x (0) ∈ A0 and xi is a homogeneous element in A satisfying deg xi = deg x − i. Since the degree of xi is less than the degree of x, repeating the same procedure for xi one arrives, by a finite number of steps, at (0) x= hj xj , (9) (0)

where xj

∈ A0 , hi ∈ F and the summation is finite. There is an F-linear map: m

F ⊗C A0 −→ A, which corresponds to multiplying the elements of A0 by elements from F and taking linear combinations. The above reasoning shows that Im(m) = A. Hence ch(A0 ) ≥

ch(A) . ch(F)

(10)

The equality takes place iff Ker(m) = 0. We shall see that this is indeed the case. Informally the equality Ker(m) = 0 is a manifestation of the fact that the space C3g+1 is foliated into g-dimensional sub-varieties, the coordinates fj describe transverse direction. The pure algebraic proof of this fact is given by the following proposition. Proposition 1. The set of elements g

j =1

where up =

i

j u 1+j

ap , bp ,

2

g

k=1

k ulg+1+k , 2

p = half-integer p = integer,

is a basis of A0 as a vector space, where i1 , . . . , ig are non-negative integers and l1 , . . . , lg are 0 or 1.


The proof of Proposition 1 is given in Appendix A. Proposition 1 shows that 1 g g

g+1+k 1 ch(A) 2 [2g + 1]! 2 , ch(A0 ) = 1+q = =

1 1+j ch(F) g + 2 ! [g]! [g + 1]! j =1

2

631

(11)

k=1

which means that Ker(m) = 0. We summarize this in the following: Proposition 2. As an F module, A is a free module, A F ⊗C A0 . In other words every element x ∈ A can be uniquely presented as a finite sum: (0) x= hj xj , (0)

where {xj } is a basis of the C-vector space A0 and hj ∈ F. 5. Poisson Structure and Cohomology Groups The affine model of the hyperelliptic Jacobian is interesting for its application to integrable models. The ring A that we introduced in the previous section can be supplied with Poisson structure. This fact is also important because introducing the Poisson structure is the first step towards the quantization. The Poisson structure in question is described in r-matrix formalism as follows: {m(z1 ) ⊗ I, I ⊗ m(z2 )} = [r(z1 , z2 ), m(z1 ) ⊗ I ] − [r(z2 , z1 ), I ⊗ m(z2 )].

(12)

The r-matrix acting in C2 ⊗ C2 is r(z1 , z2 ) =

z2 1 3 σ ⊗ σ 3 + σ + ⊗ σ − + σ − ⊗ σ + + z2 σ − ⊗ σ − , z1 − z 2 2

where σ 3 , σ ± are Pauli matrices. The variables z1 , . . . , zg (zeros of b(z)) and yj = a(zj ) have dynamical meaning of separated variables [4, 5]. The Poisson brackets (12) imply the following Poisson brackets for the separated variables: {zi , yj } = δi,j zi . The determinant f (z) of the matrix m(z) generates a Poisson commutative subalgebra: {f (z1 ), f (z2 )} = 0. It can be shown that the coefficients f1 , f2 , . . . , fg and f2g+1 belong to the center of Poisson algebra. The Poisson commutative coefficients fg+1 , . . . , f2g are the integrals of motion. Introduce the commuting vector-fields Di h = {fg+i , h},

i = 1, . . . g.

For completeness let us describe explicitly the action of these vector-fields on m(z). Define g D(z) = zj −1 Dg+1−j . j =1

632


Then the Poisson brackets (12) imply: D(z1 )m(z2 ) =

1 [m(z1 ), m(z2 )] − [σ − m(z1 )σ − , m(z2 )]. z1 − z 2

(13)

One can think of these commuting vector-fields as Dj = ∂τ∂ j , where τj are “times” corresponding to the integrals of motion fg+j . The “times” τj are coordinates on the Jacobi variety, they are related to w as follows: τ = 21 Mw, where M is the matrix defined in (2). We remark that Di here coincides with −2Di in Mumford’s book [1, Vol. II]. Earlier we have introduced on the ring A. We a gradation can prescribe the degrees to the vector-fields Dj as deg Dj = j − 21 because it is clear from (13) that: 1 Dj A(p) ⊂ A(p+j − 2 ) . Consider the differential forms fi1

... ik dτi1

∧ · · · ∧ dτik ,

(14)

with fi1 ,...,ik ∈ A. These forms span the linear spaces Ck for k = 0, . . . , g. The differential g d= dτj Dj , j =1

acts from Ck to Ck+1 . As usual applying d we first apply the vector fields Dj to the coefficients of the differential form and then take the exterior product with dτj . We have the complex d

d

d

d

d

0 −→ C0 −→ C1 −→ · · · −→ Cg−1 −→ Cg −→ 0. The k th cohomology group of this complex is denoted by H k (C∗ ). Consider the problem of grading of the spaces Cj . Clearly we have to prescribe the degree to dτj as deg(dτj ) = −j +

1 2

in order that d has degree zero. Consider the spaces Ck0 spanned by (14) with fi1 ... ik ∈ A0 . Since the elements of F are “constants” (commute with Di ), Di acts on A0 . So, we have the complex C∗0 : d

d

d

g−1

0 −→ C00 −→ C10 −→ · · · −→ C0

d

This complex is graded. One easily calculates that 1 2 2 g g−j ch(C0 ) = q 2 (j −g ) ch (A0 ) , j where the q-binomial coefficient is defined as [g]! g = . j [j ]! [g − j ]!

g

d

−→ C0 −→ 0.

(15)


633

The differential d respects the grading. In this case the q-Euler characteristic can be introduced by g χq C∗0 = ch C00 − ch C10 + · · · + (−1)g ch C0 . (16) Using the formula (15) one finds 1 2 ! ch (A0 ) χq C∗0 = (−1)g q − 2 g 2g−1 2 = (−1)g q − 2 g

[2g + 1]! [ 21 ]

1 2

[g + 21 ] [g]! [g + 1]!

(17)

.

Consider the cohomology groups H k (C∗0 ). The vector spaces H k (C∗0 ) inherit a gradj ing from C0 . Then χq (C∗0 ) = ch(H 0 (C∗0 )) − ch(H 1 (C∗0 )) + · · · + (−1)g ch(H g (C∗0 )). The q-number in (17) has a finite limit for q → 1: lim χq (C∗0 ) = (−1)g

q→1

(2g)! 2g = (−1)g 2g g − g−1 . (g)! (g + 1)!

(18)

Certainly the fact that the q-Euler characteristic has a finite limit does not mean that cohomology groups are finite-dimensional, but we believe that this is the case. So, we put forward Conjecture 1. The spaces H k (C∗0 ) are finite-dimensional. More explicitly the cohomology groups will be discussed in the next section. In the situation under consideration there is an important connection between the algebra A and the highest cohomology group H g (C∗0 ). Proposition 3. Consider some homogeneous representatives of a basis of the space H g (C∗0 ): hα dτ1 ∧ · · · ∧ dτg . Arbitrary x ∈ A0 can be presented in the form Pα (D1 , . . . Dg )hα , (19) x= α

where Pα (D1 , . . . Dg ) are polynomials in D1 , . . . , Dg with C-number coefficients. Proof. For x ∈ A0 construct g

8 = x dτ1 ∧ · · · ∧ dτg ∈ C0 . By the definition of the cohomology group we have 8 = 80 + d8 ,

80 ∈ H g (C∗0 ),

which implies that x =h+

g−1

8 ∈ C0

,

Di x i ,

with h such that 80 = h dτ1 ∧ · · · ∧ dτg , xi ∈ A0 . Apply the same procedure to xi and go on along the same lines. The resulting representation (19) will be achieved in a finite number of steps for the reason of grading.

634


Let us introduce the notation D = C [D1 , . . . , Dg ]. We shall call the expressions of the type (19) the D-descendents of {hα }. The interesting question concerning the formula (19) is whether such a representation is unique for any x. The answer is that it is not the case, and to understand why it is so we have to return to formula (17) which can be rewritten as follows: 1 2 1 q − 2 g ch (A0 ) =

ch(H g (C∗0 )) − g − 21 ! 1 1 −

ch(H g−1 (C∗0 )) +

ch(H g−2 (C∗0 )) − . . . . (20) 1 g− 2 ! g − 21 ! Obviously, the first term in the RHS represents the character of the space of all Ddescendents of {hα } (recall that the degree of Dj equals j − 21 ). This is equivalent to saying that the first term has the same character as the space generated freely over D by H g (C∗0 ):

ch(H g (C∗0 )) = ch(D)ch(H g (C∗0 )) = ch D ⊗C H g (C∗0 ) . g− !

1

1 2

The existence of the second term of the RHS of (20) implies that, in A0 , there are linear relations among D-descendents of {hα } and they are parametrized by the second term. The third term explains that there are relations among linear relations counted by the second term of the RHS of (20) and so on. This is nothing but the usual argument of constructing a resolution of a module. In the present case it is actually possible to construct a free resolution of A0 as a D module assuming some conjectures. This construction is given in Appendix F. Combining Proposition 2 and Proposition 3 one arrives at Proposition 4. Let {hα } be the same as in Proposition 3. Then every x ∈ A can be presented as x= Pα D1 , . . . , Dg hα , (21)

α

where Pα D1 , . . . , Dg are polynomials in D1 ,...,Dg with coefficients from F. Proof. We shall prove the proposition by the induction on the degree of x. Since A(p) = {0} for p < 0, the beginning of induction obviously holds. Suppose that the proposition is true for all elements of degree less than deg(x). By Proposition 2, there exist xj such that 2g+1 x = x0 + fi xi , xj ∈ A0 , i=1

where deg(x) = deg(x0 ) and deg(xi ) = deg(x) − i < deg(x) for i > 0. By Proposition 3, there exist polynomials Pα (D1 , . . . , Dg ) with the coefficients in C such that x0 =

Pα (D1 , . . . , Dg )hα +

2g+1

fi y i ,

yi ∈ A,

i=1

where deg yi = deg x0 − i. Since deg(yi ) < deg(x0 ), x can be written in the form (21) by the induction hypothesis.


635

Proposition 4 represents the most important result of this paper. The possibility of presenting every algebraic function on the phase space of the integrable model in the form (21) starting from a finite number of functions {hα }, which are representatives of the highest cohomology group, is important both in the classical and in the quantum case. The description of null-vectors follows from the one given above because Di commute with fi . 6. Conjectures on Cohomology Groups In the previous section we have seen that the cohomology group H g (C∗0 ) is important for describing the algebra A. This cohomology group is rather exotic, since the complex C∗0 corresponds to the case when the algebraic curve X is singular, that is, y 2 = z2g+1 . In this section we first discuss the relation between H k (C∗0 ) and the singular cohomology groups of the non-singular affine Jacobi variety J (X)− . For a set of complex numbers 0 f 0 = (f10 , . . . , f2g+1 ) we set Af 0 = A ⊗F Cf 0 ,

Cf 0 = F/

2g+1 i=1

F(fi − fi0 ),

0 and f 0 (z) = z2g+1 +f10 z2g +· · ·+f2g+1 . In the case when all fi0 = 0, Af 0 = A0 . If the 2 0 curve X: y = f (z) is non-singular, Af 0 is isomorphic to the affine ring of J (X) − . Since d commutes with F, the complex (C∗ , d) induces the complex (C∗f 0 , d), where

Ckf 0 = Ck ⊗F Cf 0 =

Af 0 dτi1 ∧ · · · ∧ dτik .

i1 ul . 2. If u2k/2 = xul um + . . . , then u2k/2 > ul um . 3. If u2k/2 = xul um un + . . . , then u2k/2 > ul um un . Proof. 1. Since deg(u2k/2 ) = k ≥ g + 2 > g + 1/2 ≥ l = deg(ul ), the claim follows. 2. If k > l + m, there is nothing to be proved. Suppose that k = l + m. Then l < k/2 < m. Thus comparing by the lexicographical order we have u2k/2 > ul um . The statement of 3 is similarly proved. Starting from any element P = [−m1 , . . . , −mg ] we shall show that P can be reduced to the desired form. If some mj ≥ 2, then rewrite it using (28). By Lemma A.3 every term in the resulting expression is less than P . Repeating this procedure we finally arrive at the linear combinations of [−n1 , . . . , −ng ], n1 , . . . , ng = 0, 1 with the coefficients in B0 . Thus Proposition A.1 is proved. By Proposition A.1 we have 1 g g

g+1+k 1 ch(A) 2 [2g + 2]! ch(A0 ) ≤ 1+q 2 = . (29) =

1 1+j ch(F) g + 2 ! [g]! [g + 1]! j =1

2

k=1

Thus from (10) we conclude ch(A) , ch(F) which completes the proof of Proposition 1. ch(A0 ) =

Appendix B. Proof of Proposition 5 We define W k by the LHS of (25):

W k = H k (Xaff (g), C)/ ω ∧ H k−2 (Xaff (g), C) .

We first show that the map i : H k (Xaff (g), C) −→ H k (X(g) − D, C) satisfies

i ω ∧ H k−2 (Xaff (g), C) = 0

and thereby it induces the map i : W k −→ H k (X(g) − D, C). Next we shall construct a subspace Wk of the homology group Hk (X(g) − D, C) such that the pairing between Wk and i (W k ) is non-degenerate. This proves Proposition 5. Let us study the properties of the differential form ω defined in (24). Consider some differentials λj from H 1 (Xaff , C), j = 1, . . . , k − 2, and construct the g-form: 8 = d κ˜ ∧ λ˜ 1 ∧ · · · ∧ λ˜ k−2 , (30)


where the one form κ˜ is given by κ (ij ) , κ˜ =

κ (ij ) =

i<j

1 yi −yj 4 zi −zj

dzi yi

+

dzj yj

641

.

The form under d in RHS of (30) belongs to Ck−1 f , that is, it has singularity on the divisor D = D0 ∪ D∞ , but after the differential is applied the singularities on D0 disappear. Indeed, one easily shows that − dκ (ij ) =

1 4

g

(j ) (i) (i) (j ) (2k − 1) νk ν−k+1 − νk ν−k+1 ,

k=1

(31)

where νj are defined as νj = qj (z)

dz , y

j = −(g − 1), . . . , g − 1, g,

and qj is the following polynomial of degree g − j : −j y1 z1 √ qj (z) = resp1 =∞ d z1 . z1 − z The differentials νj are normalized at infinity as √ νj ∼ −2 z−j + O(z−g−1 ) d z

for

p → ∞.

The differentials νj for j = 1, . . . , g are holomorphic and νj for j = −g + 1, . . . , 0 are of the second kind. It is easy to verify that ν i ◦ νj =

4 δi+j,1 . j −i

This means, in particular, that, for two cycles γ1 , γ2 on X, g dκ (12) = 41 (2k − 1) γ1 ν−k+1 γ2 νk − γ2 ν−k+1 γ1 νk γ1 ×γ2

k=1

= γ1 ◦ γ2 ,

(32)

due to the Riemann bilinear relation, where γ1 ◦ γ2 is the intersection number. This is an important property of dκ (ij ) . Equation (31) means that 8 = d κ˜ ∧ λ˜ 1 ∧ · · · ∧ λ˜ k−2 = −ω ∧ λ˜ 1 ∧ · · · ∧ λ˜ k−2 , where we have to remind that ω=

1 4

g k=1

(2k − 1) ν˜ k ∧ ν˜ −k+1 .

This proves that i ω ∧ H k−2 (Xaff (g), C) = 0.

642


Consider the homology groups of Xaff (g). Taking dual to the relation (23) we obtain a similar relation for the homology groups: Hk (Xaff (g), C)

k

H1 (Xaff (g), C).

The first homology group H1 (Xaff (g), C) is isomorphic to H1 (Xaff , C). To have an element δ˜ from H1 (Xaff (g), C) one takes a cycle δ from H1 (Xaff , C) and symmetrizes it over g copies of Xaff . In other words, if we fix a point p0 in Xaff , then δ˜ = δ (1) + · · · + δ (g) , ⊗(i−1)

δ (i) = p0 ⊗ · · · ⊗ δ ⊗ · · · ⊗ p0 ∈ H0

⊗(g−i)

⊗ H1 ⊗ H 0

g

,→ H1 (Xaff , C),

where Hj = Hj (Xaff , C). There is an obvious embedding X(g) − D into Xaff (g). It induces a map between the homology groups i

Hk (X(g) − D, C) −→ Hk (Xaff (g), C).

(33)

The meaning of this map is simple: every cycle on X(g) − D is at the same time a cycle on Xaff (g). There are two subtleties: 1. The nontrivial cycle on X(g) − D can be trivial on Xaff (g), i.e. the map (33) can have kernel. 2. On Xaff (g) there are cycles that intersect with D0 which means that they are not cycles on X(g) − D, so, the map (33) can have cokernel. Let us study the image of the map (33). A k-cycle from Hk (Xaff (g), Z) is a linear combination of elements of the form: ' = δ˜1 ∧ · · · ∧ δ˜k , where δj ∈ H1 (Xaff , Z). The product ' = δ1 × · · · × δk × p0 × · · · × p0 g

g

defines an element of Hk (Xaff , Z). Let π be the projection map Xaff −→ Xaff (g) and π∗ g the induced g map on the homology groups, π∗ : Hk (Xaff , Z) −→ Hk (Xaff (g), Z). Then ' = k! k π∗ (' ) in Hk (Xaff (g), Z). Thus the cycle ' belongs to Im(i) if ' does not intersect π −1 (D). Recall that D = D0 ∪ D∞ . By construction ' has no intersection with π −1 (D∞ ). One easily realizes that ' does not intersect with π −1 (D0 ) iff δi ◦ σ (δj ) = 0

∀ i, j.

It is a rather obvious property of the hyperelliptic involution that δi ◦ σ (δj ) = −δi ◦ δj . Hence we come to the following


643

Proposition B.1. The Im(i) contains a linear combination of cycles ' = δ˜1 ∧ · · · ∧ δ˜k ,

δ1 , . . . , δk ∈ H1 (Xaff , Z),

such that δi ◦ δj = 0 ∀ i, j. Let Wk denote the space obtained as a C-linear span of cycles in Hk (X(g) − D, Z) corresponding to '’s in this proposition. Take canonical cycles αi , βj and set Ai+g = β˜i ,

Ai = α˜ i ,

1 ≤ i ≤ g.

Define 2g

VZ = H1 (Xaff , Z) = ⊕i=1 ZAi ,

2g

VC = H1 (Xaff , C) = ⊕i=1 CAi .

The symplectic form on VC is defined by Ai ◦ Aj = ±δj i±g . By definition i(Wk ) = SpanC (Uk ),

Uk = {γ1 ∧ · · · ∧ γk |γ1 , . . . , γk ∈ VZ , γi ◦ γj = 0 ∀i, j }.

Define W˜ k = SpanC (U˜ k ),

U˜ k = {γ1 ∧ · · · ∧ γk |γ1 , . . . , γk ∈ VC , γi ◦ γj = 0 ∀i, j }.

Consider the map ϕk : ∧k VC −→ ∧k−2 VC , ϕk (γ1 ∧ · · · ∧ γk ) = (−1)i+j −1 (γi ◦ γj )γ{ij } ,

(34)

i<j

where γ{ij } is obtained from γ1 ∧· · ·∧γk removing γi and γj . It is known that, for k ≤ g, ϕk is surjective and its kernel Kerϕk is isomorphic to the k th fundamental irreducible representation of Sp(2g, C) (cf. Theorem 17.5 [10]). In particular dk := dim Ker(ϕk ) =

2g k

−

2g

The following lemma can be easily proved. Lemma B.1. Suppose that k ≤ g. Then i(Wk ) = W˜ k = Ker(ϕk ).

k−2

.

644


As a consequence of the lemma one has in particular 2g dim Wk ≥ dim i(Wk ) = 2g k − k−2 .

(35)

Let us show that the pairing between Wk and i (W k ) is non-degenerate. The pairings
1 : Hk (X(g) − D, C) ⊗ H k (X(g) − D, C) −→ C

(36)

2 : ∧k H1 (Xaff (g), C) ⊗ ∧k H 1 (Xaff (g), C) −→ C

(37)

and

are related by

< γ , i (η) >1 =< i(γ ), η >2 ,

for γ ∈ Hk (X(g) − D, C), η ∈ ∧k H 1 (Xaff (g), C). The pairing (37) is given by the integral: < γ˜1 ∧ · · · ∧ γ˜k , η˜ 1 ∧ · · · ∧ η˜ k >2 g = η˜ 1 ∧ · · · ∧ η˜ k = k! k det( ηj )1≤i,j ≤k . γ˜1 ∧···∧γ˜k

By (32) we have

γi

< i(Wk ), ω ∧k−2 H 1 (Xaff (g), C) >2 = 0.

Thus the pairing (36), (37) induce pairings < , < ,

>1 : Wk ⊗ i (W k ) −→ C, >2 : i(Wk ) ⊗ W k −→ C.

(38) (39)

Since (37) is non-degenerate and dim i(Wk ) = dim W k , the pairing (39) is non-degenerate. It easily follows from this that the pairing (38) is also non-degenerate. Thus we have proved Proposition 5. Corollary B.1. We have Wk i(Wk ). In particular 2g dim Wk = 2g k − k−2 . Appendix C. The Case of Generic Abelian Variety Let (J, ) be a principally polarized Abelian variety such that is non-singular. Then Proposition C.1. The dimensions of cohomology groups of J − are given by 2g 2g dimH k (J − , C) = − , k ≤ g − 1, k k−2 2g 2g (2g)! = − + g! − , k = g, g g−2 g!(g + 1)! = 0, k > g.


645

Proof. Consider the inclusions ⊂ J ⊂ (J, ) and the induced homology exact sequence: . . . −→ Hk ( , C) −→ Hk (J, C) −→ Hk (J, ) −→ Hk−1 ( , C) −→ . . . . Taking the dual sequence of this and using the Poincaré–Lefschetz duality we get . . . −→ Hk−1 ( , C) −→ Hk (J − , C) −→ Hk (J, C) −→ −→ Hk−2 ( , C) −→ . . . .

(40)

Since J − is affine Hk (J − , C) = 0,

k > g.

Then we have Hk ( , C) Hk+2 (J, C),

k ≥ g,

Hk ( , C) Hk (J, C),

k ≤ g − 2.

It is easy to check that, for k ≤ g, the dual map of Hk (J, C) −→ Hk−2 ( , C) is given by wedging the fundamental class [ ] of : [ ]∧ : H k−2 ( , C) H k−2 (J, C) −→ H k (J, C).

(41)

Using the representation theory of sl2 as in the proof of the hard Lefschetz theorem (cf. [7]), the map (41) is injective for k ≤ g. Thus by (40) the following exact sequences hold: [ ]∧

0 → H k−2 ( , C) −→ H k (J, C) → H k (J − , C) → 0,

k < g,

[ ]∧

0 → H g−2 ( , C) −→ H g (J, C) → H g (J − , C) → → H g−1 ( , C) → H g+1 (J, C) → 0. Proposition C.1 follows from these exact sequences and the fact χ ( ) = (−1)g−1 g!.

By Proposition F.3 the fundamental class of coincides with ω in Proposition 5 in the hyperelliptic case. Thus, if we define W k in a similar formula to (25), we have W k H k (J − , C), k ≤ g − 1, W g ,→ H g (J − , C).

646


Appendix D. The Proof of Conjecture 3 for k = 1 Notice that

H k (Xaff (g), C) ∧k H 1 (X, C) ∧k H 1 (J (X), C),

and X(g) − D J (X) − . In particular W 1 H 1 (J (X), C). Proposition D.1. For any principally polarized Abelian variety (J, ) such that is irreducible we have the isomorphism H 1 (J, C) H 1 (J − , C). Proof. Following [9] we shall use the following notations: O(n ): the sheaf of meromorphic functions on J which have poles only on of order at most n, O(∗ ): the sheaf of meromorphic functions on J which have poles only on , 8k (n ): the sheaf of meromorphic k-forms on J which have poles only on of order at most n, 8k (∗ ): the sheaf of meromorphic k-forms on J which have poles only on , Fk (n ): the sheaf of closed meromorphic k-forms on J which have poles only on

of order at most n, Fk (∗ ): the sheaf of closed meromorphic k-forms on J which have poles only on , R k (n ) = Fk (n )/d 8k−1 ((n − 1) ) , R k (∗ ) = Fk (∗ )/d 8k−1 (∗ ) . In particular 80 (n ) = O(n ), 80 (∗ ) = O(∗ ). We first recall the description of H 1 (J, C) in terms of the differentials of the first and second kind. Consider the sheaf exact sequence: d 0 −→ C −→ O(∗ ) −→ d O(∗ ) −→ 0. Since

H k (J, O(∗ )) = 0,

k ≥ 1,

we have H 1 (J, C) H 0 (J, dO(∗ ))/dH 0 (J, O(∗ )).

(42)

The numerator in the right-hand side of (42) is nothing but the space of differential one forms of the first and the second kind on J and the denominator is the space of globally exact meromorphic one forms. On the other hand, by the algebraic de Rham theorem, the first cohomology group of the affine variety J − is described as H 1 (J − , C) H 0 (J, F1 (∗ ))/dH 0 (J, O(∗ )).

(43)

Comparing (42) and (43) what we have to prove is H 0 (J, dO(∗ )) H 0 (J, F1 (∗ )). Consider the exact sequence 0 −→ dO(∗ ) −→ F1 (∗ ) −→ R 1 (∗ ) −→ 0.

(44)


647

The cohomology sequence of this is 0 −→ H 0 (J, dO(∗ )) −→ H 0 (F1 (∗ )) −→ H 0 (R 1 (∗ )) −→ . . . . From this what should be proved is that the map H 0 (F1 (∗ )) −→ H 0 (R 1 (∗ )) is a 0-map. To study this map we refer the lemma from [9]. Lemma D.1 (Lemma 8 [9]). (1) R 1 (∗ ) C , where C is the constant sheaf on and the isomorphism is given by dθ [ ] ←− [1 ] θ at any stalk. (2) R 1 (∗ ) R 1 (n ), n = 1, 2, . . . . In the proof of Lemma D.1 (1) we use our assumption that is irreducible. Using this lemma we reduce the problem from “∗ ” to “n ” with finite n. Consider the exact sequence 0 −→ dO(n ) −→ F1 ((n + 1) ) −→ R 1 (∗ ) −→ 0,

n = 0, 1, . . . .

(45)

From the cohomology sequence of it we have the map H 0 (J, R 1 (∗ )) −→ H 1 (J, dO(n )) which we denote by πn . Let us prove Kerπn = 0,

n = 0, 1, 2, . . . .

To this end we study H 1 (J, dO(n )). Using the exact sequence d

0 −→ C −→ O(n ) −→ dO(n ) −→ 0, we easily have H 1 (J, dO(n )) H 2 (J, C),

n ≥ 1,

H (J, dO) ,→ H (J, C). 1

2

(46) (47)

The natural maps dO(n ) −→ dO((n + 1) ), F1 (n ) −→ F1 ((n + 1) ), R 1 (n ) R 1 ((n + 1) ), and the sequence (45) induce a commutative diagram of cohomology groups. It follows from this commutative diagram and (46), (47) that Kerπ0 = 0 implies Kerπn = 0 for n ≥ 1. Now Kerπ0 = 0 follows from H 0 (J, dO) H 0 (J, 81 ),

H 0 (J, F1 ( )) H 0 (J, 81 ).

The second isomorphism follows from the fact that a meromorphic function on J which has poles only on of order at most one is a constant. Thus Proposition D.1 is proved.

648


Appendix E 0 ), the ring Af 0 is defined Recall that, for a set of complex numbers f 0 = (f10 , . . . , f2g+1 by

Af 0 = A ⊗F Cf 0 = 2g+1 i=1

A A(fi − fi0 )

,

where fi is the coefficient of f (z). If all fi0 = 0, then Af 0 = A0 . The ring A0 is graded while Af 0 is not graded unless all fi = 0. Instead Af 0 is a filtered ring for any f 0 . Let Af 0 (n) be the set of elements of Af 0 represented by ⊕k≤n A(n) . Then Af 0 = ∪n≥0 Af 0 (n),

Af 0 (0) = C ⊂ Af 0

1 2

⊂ Af 0 (1) ⊂ . . . .

Consider the graded ring associated with this filtration: grAf 0 = ⊕grn Af 0 ,

grn Af 0 =

Af 0 (n) Af 0 (n − 21 )

.

Since deg(fi0 ) = 0, grAf 0 becomes a quotient of A0 . In other words there is a surjective ring homomorphism A0 −→ grAf 0 .

(48)

We shall prove that this map is injective. Proposition E.1. There is an isomorphism of graded rings: A0 grAf 0 . Proof. Notice that the map (48) respects the grading and it is surjective at each grade. By Proposition 2, A F ⊗C A0 as an F-module. Thus we have A0 Af 0 as a C-vector space for any f 0 . In particular the basis of A0 given in Proposition 1 is also a basis of (n) (n) (n) Af 0 . Denote by {xj } the basis of the degree n part A0 . Let us prove that {xj } are linearly independent in grAf 0 by the induction on n. For n = 0 the statement is obvious. (m)

We assume that the statement is true for all m satisfying m < n. This means that {xj } (m)

is a basis of the degree m part of grAf 0 for all m < n. In particular {xj |m < n} is a basis of Af 0 (n − 21 ). Suppose that the relation j

(n)

αj x j

∈ Af 0 (n − 21 ) (m)

holds, where some αj = 0. Then this means that {xj |m ≤ n} are linearly dependent (k)

(n)

in Af 0 . This contradicts the fact that {xj } is a basis of Af 0 . Thus {xj } are linearly independent in grAf 0 .


649

Appendix F. Construction of a Free Resolution of A0 Recall that D = C [D1 , . . . , Dg ] is the polynomial ring generated by the commuting vector fields D1 , . . . , Dg . Then A, A0 and Af 0 are D modules. We shall construct a free D-resolution of A0 assuming Conjecture 2 and 3. To avoid the notational confusion we shall describe the space W k using the abstract vector space V of dimension 2g with a basis vi , ξi (1 ≤ i ≤ g): g

g

V = ⊕i=1 Cvi ⊕i=1 Cξi . Set ω=

g

vi ∧ ξi ∈ ∧2 V

i=1

and define Wk =

∧k V . ω ∧k−2 V

(49)

Assign degrees to the basis elements by deg(vi ) = −(i − 21 ),

deg(ξi ) = i − 21 .

Then W k defined by (49) has the same character as W k defined in Sect. 6. This justifies the use of the same symbol. Consider the free D-module D ⊗C W k generated by W k . We shall construct an exact sequence of D-modules of the form d

d

d

0 ← A0 dτ1 ∧ · · · ∧ dτg ← D ⊗C W g ← D ⊗C W g−1 ← · · · ← D ⊗C W 0 ← 0. Define the map d

D ⊗C ∧k V −→ D ⊗C ∧k+1 V , by d(P ⊗ vI ∧ ξJ ) =

g

Di P ⊗ v i ∧ v I ∧ ξ J ,

i=1

where for I = (i1 , . . . , ir ), vI = vi1 ∧ · · · ∧ vir etc., P ∈ D and Di P is the product of Di and P in D. Since the map d commutes with the map taking the wedge with Q ⊗ ω for any Q ∈ D: d(QP ⊗ ω ∧ vI ∧ ξJ ) =

g

QDi P ⊗ ω ∧ vi ∧ vI ∧ ξJ = (Q ⊗ ω) ∧ d(P ⊗ vI ∧ ξJ ),

i=1

it induces a map d

D ⊗C W k −→ D ⊗C W k+1 . It is easy to check that the map d satisfies d 2 = 0.

650


Proposition F.1. The complex d

d

d

0←−D ⊗C W g ←− D ⊗C W g−1 ←− · · · ←− D ⊗C W 0 ←− 0

(50)

is exact at D ⊗C W k except k = g. Proof. Notice that the following two facts: 1. The following complex is exact at D ⊗ ∧k V except k = g: d

d

d

0 ←− D ⊗C ∧g V −→ · · · ←− D ⊗C V ←− D ←− 0.

(51)

2. The map ω∧ : ∧k V −→ ∧k+2 V is injective for k ≤ g − 1. Property 1 is the well known property of the Koszul complex. Property 2 is also well known and easily proved using the representation theory of sl2 . Now suppose that x ∈ D ⊗C ∧k V , k < g and dx = ω∧y for some y ∈ D ⊗C ∧k−1 V . Then ω ∧ dy = 0 and thus dy = 0 by Property 2. Then y = dz for some z by Property 1. Thus we have d(x − ω ∧ z) = 0 and x = dw + ω ∧ z for some w again by Property 1. Next we shall define a map from D ⊗C W g to A0 dτ1 ∧ · · · ∧ dτg . To this end we need to identify W k in this section and that of Sect. 6, which is defined as the quotient of the cohomology groups of a Jacobi variety. For this purpose we describe the cohomology groups of a Jacobi variety in terms of theta functions. Define ∂ ζi (w) = Di log θ(w) = log θ(w). ∂τi Then, for each i, the differential dζi (w) defines a meromorphic differential form on J (X) which has double poles on and which is locally exact. This means that dζi is a second kind of differential on J (X). By (42) first and second kinds of differential one forms define elements of H 1 (J (X), C). The pairing with H1 (J (X), C) is given by integration. The following proposition can be easily proved by calculating periods. Proposition F.2. The first and the second kinds of differentials dτ1 ,...,dτg , dζ1 ,...,dζg give a basis of the cohomology group H 1 (J (X), C). Thus we can identify V with H 1 (J (X), C) H 1 (Xaff (g), C) by vi = dτi ,

ξi = dζi .

The next proposition can be easily proved by calculating integrals over two cycles in a similar way to [1, Vol. I, p. 188]. Proposition F.3. Let ω be defined in Proposition 5. Then we have 1 √

2π −1

g

dτi ∧ dζi = ω

i=1

as elements in ∧2 H 1 (J (X), C). Moreover ω represents the fundamental class of the theta divisor in H 2 (J (X), C).


651

From this proposition it is possible to identify W k in this section and that in the previous sections. Define the map ∧g V → Af 0 dτ1 ∧ · · · ∧ dτg , vI ∧ ξJ % → dτI ∧ dζJ , where for I = (i1 , . . . , ir ), dτI = dτi1 ∧ · · · ∧ dτir etc. and f 0 is any set of complex numbers such that y 2 = f 0 (z) defines a non-singular curve X. This map extends to the map of D modules D ⊗C ∧g V → Af 0 dτ1 ∧ · · · ∧ dτg , in the KfollowingK manner. Let P ∈ D and consider P ⊗ (vI ∧ ξJ ). Write dζJ = K FJ dτK , FJ ∈ Af 0 . Then we define P (FJK )dτI ∧ dτK . P ⊗ (vI ∧ ξJ ) −→ Since, as a meromorphic differential form on J (X), g

dτi ∧ dζi =

i=1

g ∂ 2 log θ(w) dτi ∧ dτj = 0, ∂τi ∂τj

i,j =1

the above map induces the map ev

D ⊗C W g −→ Af 0 dτ1 ∧ · · · ∧ dτg . Denote by (D ⊗C W g )n the subspace of elements with degree n and by evn the restriction of ev to (D ⊗C W g )n . By the definition, for x ∈ (D ⊗C W g )n , evn (x) ∈ Af 0 (n + g 2 /2). By Proposition E, there is a natural isomorphism A0 grAf 0 (see Appendix E for the filtration of Af 0 ). Composing the map evn with the natural projection map Af 0 (k) → (k)

grk Af 0 A0 we obtain the map, which we denote by evn too, evn

(D ⊗C W g )n −→ (A0 dτ1 ∧ · · · ∧ dτg )n , where the RHS means the degree n subspace. Taking the sum of evn we finally have the map ev = ⊕n evn : D ⊗C W g −→A0 dτ1 ∧ · · · ∧ dτg , which we also denote by the symbol ev. If we assume Conjecture 2 and 3, H k (C∗0 ) W k . If this holds, then ev is surjective by Proposition 3. Lemma F.1. Suppose that Conjecture 2 and 3 are true. Then the kernel of ev is given by Ker(ev) = d D ⊗ W g−1 . Proof. It is easy to check that

Ker(ev) ⊃ d D ⊗ W g−1 .

Since ev is surjective and

ch(A0 ) = ch the claim of the lemma follows.

D ⊗C W g , d D ⊗C W g−1

652


We summarize the result as Theorem F.1. Suppose that Conjecture 2 and 3 are true. Then the following complex gives a free resolution of A0 dτ1 ∧ · · · ∧ dτg as a D-module: ev

d

d

0←−A0 dτ1 ∧ · · · ∧ dτg ←− D ⊗C W g ←− · · · ←− D ⊗C W 0 ←− 0. Acknowledgements. This work was begun during the visit of one of the authors (A.N.) to LPTHE of Université Paris VI and VII in 1998–1999. We express our sincere gratitude to this insitution for generous hospitality. A.N. thanks K. Cho for helpful discussions.

References 1. 2. 3. 4. 5. 6. 7.

Mumford, D.: Tata Lectures on Theta. Vol. I and II, Boston: Birkhäuser, 1983 Babelon, O., Bernard, D., Smirnov, F.A.: Commun. Math. Phys. 186, 601 (1997) Smirnov, F.A.: J. Phys. A: Math. Gen. 31, 8953 (1998) Flaschka, H., McLaughlin, D.W.: Progr. Theor. Phys. 55, 438 (1976) Sklyanin, E.K.: Separation of Variables. Progress in Theoretical Physics Supplement 118, 35 (1995) Beauville, A.: Acta Math. 164, 211 (1990) Griffiths, P. and Harris, J.: Principles of Algebraic Geometry. New York: Wiley-Interscience Publication, 1978 8. Macdonald, I.G.: Topology 1, 319 (1962) 9. Atiyah, M. and Hodge, W.V.D.: Ann. of Math. 62, 56 (1955) 10. Fulton, W. and Harris, J.: Representation Theory. New York: Springer, 1991 Communicated by T. Miwa

Commun. Math. Phys. 217, 653 – 696 (2001)

Communications in



Certain Extensions of Vertex Operator Algebras of Affine Type Haisheng Li Department of Mathematical Sciences, Rutgers University, Camden, NJ 08102, USA Received: 7 March 2000 / Accepted: 10 November 2000

Abstract: We generalize Feigin and Miwa’s construction of extended vertex operator (super)algebras Ak (sl(2)) for other types of simple Lie algebras. For all the constructed extended vertex operator (super)algebras, irreducible modules are classified, complete reducibility of every module is proved and fusion rules are determined modulo the fusion rules for vertex operator algebras of affine type. 1. Introduction In the development of vertex operator algebra theory, one of the most important problems is to construct new solvable vertex operator (super)algebras in the sense that irreducible modules and fusion rules can be completely determined and that intertwining operators can be explicitly constructed. To a certain extent, such algebras give rise to solvable physical models. One of many ways to get such vertex operator (super)algebras is to consider certain extensions of some well known algebras. For example ([MS, Li5]), the vertex operator algebra L(k, 0) associated to the affine Lie algebra sˆ l2 with a positive even integral level k can be extended to a vertex operator (super)algebra L(k, 0) + L(k, k). When k is odd, L(k, 0) + L(k, k) does not have an extended vertex operator (super)algebra structure because of the failure of the locality. (For a certain class of vertex operator algebras, e.g., vertex operator algebras associated to positive-definite even lattices, as proved in [DL], the sum of a copy of each irreducible module does have a nice structure, called an abelian intertwining algebra. See [DL, FFR, M] and [Hua2] for notions of various generalized structures.) In [FM], Feigin and Miwa constructed a family of extended vertex operator (super)algebras Ak from the vertex operator algebras L(k, 0) associated to the affine Lie algebra sˆ l2 with an arbitrary positive integral level k, and they classified all irreducible modules and determined all fusion rules. In addition, they obtained very interesting results on the monomial basis for irreducible modules. This paper was mainly motivated Partially supported by NSF grant DMS-9970496

654

H. Li

by [FM] and [DLM2]. As the main results of this paper we generalize their results except the monomial basis result to affine Lie algebras of other types by using a different approach. The algebras Ak were defined in [FM] by a set of mutually local vertex operators (or fields). On the other hand, in terms of vertex operator algebra language, Ak are extensions (by an infinite sum of irreducible modules) of vertex operator algebra L(k, 0)⊗M(1, 0), where M(1, 0) is the vertex operator algebra associated to an infinitedimensional Heisenberg Lie algebra of rank one, or a single free bosonic field. The essential building block of Ak is the irreducible L(k, 0) ⊗ M(1, 0)-module L(k, k) ⊗ M(1, α) for some α ∈ C. The L(k, 0)-module L(k, k) has been known to be a simple current ([FG, GW, SY]) in the sense that the left multiplication of the equivalence class [L(k, k)] in the Verlinde algebra gives rise to a permutation on the standard basis. It seems to be known to physicists (cf. [MS, SY]) that a simple current with integer weights can be included to generate an extended vertex operator algebra. In [Li5], as an exercise by using an explicit construction of simple currents given in [Li4] we studied the extension of certain vertex operator algebras by a self-dual (or order 2) simple current where L(k, 0) + L(k, k) is a special case. For such extended algebras, all their irreducible modules were classified and the complete reducibility of every module was proved. A little bit later, the results of [Li5] were greatly extended in [DLM2]. The construction of simple currents given in [Li4] was based on a result of [Li2]. Let V be a vertex operator algebra and let h be a weight one primary vector in V such that the component operators h(m) of the vertex operator Y (h, z) satisfy the Heisenberg algebra relation and such that h(0) is semisimple on V with only rational eigenvalues. Clearly, σh := e2πih(0) is an automorphism of V . Define ∞ h(n) (−z)−n , (h, z) = zh(0) exp −n n=1

an element of (End V ){z}. It was proved in [Li2] that for any V -module W , (W (h) , Yh (·, z)) := (W, Y ((h, z)·, z)) is a σh -twisted V -module. In particular, this gives an (untwisted) V -module if h(0) acting on V has only integral eigenvalues. It3 was furthermore proved in [Li4] that if I (·, z) is an intertwining operator of type WW (in the sense of [FHL]), then I ((h, z)·, z) is 1 W2 W3(h) an intertwining operator of type (h) . By using this result and the invertability of V (h)

W1 W2

(h, z), it was proved that is a simple current. As a matter of fact, for certain well known vertex operator algebras almost all simple currents can be constructed in this way. For example, when V is the vertex operator algebra VL associated with a positive-definite even lattice L, it was proved that all irreducible VL -modules can be constructed in this way. In this case, this construction is intimately related to the construction of twisted modules by using shifted vertex operators in [Le]. When V = L(, 0) associated to an affine Lie algebra gˆ with g = E8 and with a positive integral level , all the simple currents can also be constructed in this way. The merit of this construction of a simple current is on the canonicalness of the vector space and the vertex operator map (in terms of the algebra V and the element h). With this construction, certain intertwining operators can be constructed canonically. The canonical construction of intertwining operators of certain types is the basis of [Li5] and [DLM2].

Certain Extensions of Vertex Operator Algebras of Affine Type

655

The essential results of [DLM2] can be described as follows: Let V be a vertex operator algebra and H be a subspace of V(1) such that the components h(n) of vertex operators Y (h, z) for h ∈ H, n ∈ Z satisfy the Heisenberg algebra relation. Let L be a subgroup of H such that for every α ∈ L, α(0) acts semisimply on V with only integral eigenvalues. Consider the space V [L] = C[L]⊗V , where for α ∈ L, eα ⊗V is identified with V (α) equipped with the V -module structure Yα . Extend the V -module structure on V [L] in a certain canonical way to a vertex operator map Y on V [L]. Then it was proved that V [L] equipped with the defined vertex operator map Y is a generalized vertex algebra in the sense of [DL]. It was proved that V [L] is a vertex operator (super)algebra when L satisfies certain conditions. When V = Mh (1, 0) with h = C ⊗Z L, where L is an integral lattice, we have V [L] = VL ([FLM, DL]). Then we may view V [L] as a generalization of VL . In [DLM2], we were mainly interested in the case V = L(k, 0) associated to an affine algebra gˆ with a positive integral level k. Because L(k, 0) has only finitely many irreducible modules up to equivalence, each irreducible V -module V (α) in V [L] is not multiplicity-free. Having noticed this we proved that V [L] has a quotient algebra V [L] such that every irreducible V -module in V [L] is multiplicity-free and that V [L] and V [L] contain the same number of non-isomorphic irreducible V -modules. An irreducible V -module W was also extended to W [L] on which V [L] acts and it was proved that W [L] is in general a twisted V [L]-module with respect to a certain automorphism of V [L]. With this result, under the assumption of complete reducibility of V -modules, all irreducible V [L]-modules were classified and a complete reducibility theorem for V [L]-modules was proved. In this paper, to generalize Feigin and Miwa’s construction we apply the results of [DLM2] by taking V = L(k, 0) ⊗ Mh (1, 0) and by choosing h and L appropriately, depending on the type of g. In this case, each irreducible V -module in V [L] is multiplicity-free and V [L] is a simple algebra. On the other hand, since the category of V -modules is not semisimple, the results of [DLM2] for the complete reducibility of V [L]-modules do not apply to this case directly. The complete reducibility of V [L]modules is proved here. We also naturally extend intertwining operators for modules in the category of V -modules to intertwining operators for modules in the category of V [L]-modules. Using this result we are able to derive a formula of fusion rules for V [L]-modules in terms of fusion rules for V -modules. This paper is organized as follows: In Sect. 2, we recall the classification of irreducible modules for certain vertex operator algebras and recall a construction of simple currents. Most part of this section is preliminary and the only new result is about the fusion rules for simple currents for L(k, 0). In Sect. 3 we recall and refine some of the results of [DLM2] on the extended vertex (super)algebra V [L]. We furthermore study the multiplicity-free case. In Sect. 4, we apply the results of Sect. 3 to construct extended vertex operator (super)algebras associated to an affine algebra gˆ .

2. Vertex Operator Algebras and Simple Currents The extended vertex operator (super) algebras we shall construct are based on vertex operator algebras Lg (, 0), Mh (1, 0), VL and their representations. For this reason, in this section we shall recall the relevant information about these algebras and we also recall from [Li4] a construction of simple currents for a certain type of vertex operator algebras, including Lg (, 0), Mh (1, 0), VL . For the vertex operator algebra Lg (, 0) (1) associated to an affine algebra gˆ (not type E8 ) with a positive integral level , we prove

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H. Li

that the equivalence classes of the simple currents form an abelian group isomorphic to P ∨ /Q∨ , where P ∨ and Q∨ are the co-weight and co-root lattices of g. 2.1. Vertex operator algebras Lg (, 0), Mh (1, 0) and VL . We shall use standard definitions and notations as given in [FHL] and [FLM] and we also use [K] and [H] as our references for (Kac–Moody) Lie algebras. Following [DL] we use the term “vertex (super)algebra” for an object that satisfies all the axioms defining the notion of vertex operator (super)algebra except the two grading restrictions. Let g be a finite-dimensional simple Lie algebra, h a Cartan subalgebra, and · , · the normalized killing form such that the squared length of a long root is 2. Let gˆ = g ⊗ C[t, t −1 ] ⊕ Cc

(2.1)

be the affine Lie algebra. Let be a complex number such that = −h∨ , where h∨ is the dual Coxeter number of g. Let C be the one-dimensional (g ⊗ C[t] + Cc)-module on which c acts as scalar and g ⊗ C[t] acts as zero. Form the generalized Verma gˆ -module Mg (, 0) = U (ˆg) ⊗U (g⊗C[t]+Cc) C .

(2.2)

It was well known (cf. [FF, FZ, Li1, MP]) that Mg (, 0) has a natural vertex operator algebra structure. Furthermore, the category of weak Mg (, 0)-modules in the sense that all the axioms defining the notion of module except those involving grading hold is canonically equivalent to the category of restricted (cf. [K]) gˆ -modules of level in the sense that for every vector w of the module, (g ⊗ t n C[t])w = 0 for n sufficiently large. Remark 2.1. More generally, let W be an arbitrary vector space. An element a(z) of (End W )[[z, z−1 ]] is called a vertex operator if a(z)w ∈ W ((z)) for every w ∈ W . Two vertex operators a(z) and b(z) are said to be mutually local if there exists a nonnegative integer N such that (z1 − z2 )N [a(z1 ), b(z2 )] = 0

(2.3)

(cf. [DL], (1.4)). It was proved in [Li1] (Corollary 3.2.11) that any set of mutually local vertex operators on W automatically generates a vertex algebra, which is a canonical vector subspace of (End W )[[z, z−1 ]], and that W is a natural module for this vertex algebra. For λ ∈ h∗ , denote by Lg (, λ) the irreducible highest weight gˆ -module of level with highest weight λ. Each Lg (, λ) is an irreducible Mg (, 0)-module possibly with infinite-dimensional homogeneous subspaces. Denote by θ the highest long root of g. For a positive integer , set P = {λ ∈ P+ | λ, θ ≤ },

(2.4)

where P+ is the set of dominant integral weights of g. Then L(, λ) is an integrable gˆ module if and only if λ ∈ P [K]. The following result was known (cf. [DL, Proposition 13.17], [FZ, Theorem 3.1.3], [Li1, Propositions 5.2.4 and 5.2.5], [MP, Theorems 5.9, 5.14 and 5.15]):


657

Proposition 2.2. Let be a positive integer. (1) The set of irreducible Lg (, 0)-modules is exactly the set of irreducible highest weight integrable (or standard) gˆ -modules of level . (2) Every Lg (, 0)-module is completely reducible. The following stronger result was obtained in [DLM1] (Theorem 3.7): Proposition 2.3. Let be a positive integer. Then every weak Lg (, 0)-module is a direct sum of irreducible highest weight integrable (or standard) gˆ -modules of level . In particular, every irreducible weak Lg (, 0)-module is an (ordinary) Lg (, 0)-module. A vertex operator algebra with the property that every weak module is a direct sum of irreducible (ordinary) modules is said to be regular [DLM1]. Let h be a finite-dimensional vector space equipped with a nondegenerate symmetric bilinear form · , ·, i.e, a finite-dimensional abelian Lie algebra equipped with a nondegenerate symmetric invariant bilinear form. Let hˆ = h ⊗ C[t, t −1 ] + Cc be the affine Lie algebra. We have hˆ = hˆ Z ⊕ h,

(2.5)

where hˆ Z = n=0 h ⊗ t n + Cc is a Heisenberg algebra and h is central. Similar to the construction of Mg (, 0) we construct a space Mh (1, 0), and just like Mg (, 0), Mh (1, 0) is a vertex operator algebra. For any α ∈ h∗ (= h), let Ceα be a one-dimensional (h ⊗ C[t] + Cc)-module on which h ⊗ tC[t] acts as zero, c acts as 1 and h = h(0) acts as scalar α, h for h ∈ h. Form the induced module ˆ ⊗U (h⊗C[t]+Cc) Ceα Mh (1, 0) ⊗ Ceα (linearly). Mh (1, α) = U (h)

(2.6)

As in the case of Mg (, 0), Mh (1, α) is an irreducible Mh (1, 0)-module. Furthermore, from [FLM] the lowest L(0)-weight of Mh (1, α) is α =

1 α, α. 2

(2.7)

On the other hand, clearly every irreducible Mh (1, 0)-module is isomorphic to Mh (1, α) for some α. It follows from the complete reducibility of certain hˆ Z -modules ([LW, K]) that an Mh (1, 0)-module on which h semisimply acts is completely reducible. In general, an Mh (1, 0)-module may not be completely reducible. Let P be a rational lattice of finite rank with the Z-bilinear form · , ·. Set h = C ⊗Z P ,

(2.8)

and extend · , · to a C-bilinear form on h. Denote by C[P ] the group algebra. Set VP = C[P ] ⊗ Mh (1, 0),

(2.9)

equipped with the standard Mh (1, 0)-module (or hˆ Z -module) structure. That is, VP is a direct sum of irreducible Mh (1, 0)-modules Mh (1, α) Ceα ⊗ Mh (1, 0) for α ∈ P . It was proved in [FLM] (cf. [B]) that when P = L is even and positive-definite, VL has a natural simple vertex operator algebra structure which extends the Mh (1, 0)module structure. (It follows from Proposition 3.22 that such a vertex operator algebra structure is unique up to equivalence.) Furthermore, let Lo be the dual lattice of L. Then VLo is a natural VL -module and Vβ+L is an irreducible VL -module for β ∈ Lo .

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Proposition 2.4. (1) Let {β1 , . . . , βr } be a complete set of representatives of cosets of L in Lo . Then {Vβ1 +L , . . . , Vβr +L } is a complete set of representatives of equivalent classes of irreducible VL -modules. (2) Every VL -module is completely reducible. (3) VL is regular, i.e., every weak VL module is a direct sum of irreducible (ordinary) VL -modules. The assertion (1) was proved in [FLM] and [D1], (2) was proved in [Guo] and (3) was proved in [DLM1]. Remark 2.5. For a general rational lattice P , VP is not a vertex (operator) algebra because of the involvement of non-local vertex operators. A notion of generalized vertex algebra was introduced in [DL] with a generalized Jacobi identity as one of the main axioms and it was proved in [DL, Theorem 5.1 and Remark 9.11] that VP is a generalized vertex algebra. 2.2. Simple currents and a construction. We first recall from [FHL] the definition of an intertwining operator. Definition 2.6. Let V be a vertex operator algebra 3 and let W1 , W2 and W3 be V modules. An intertwining operator of type WW is a linear map I from W1 to 1 W2 (Hom(W2 , W3 )){z}, (where for a vector space U , U {z} is defined to be the vector space of U -valued formal series in z with arbitrary complex powers of z), satisfying the following conditions: for w1 ∈ W1 , w2 ∈ W2 , I (w1 , z)w2 ∈ zγ1 W3 [[z]] + · · · + zγn W3 [[z]]

(2.10)

for some (finitely many) complex numbers γ1 , . . . , γn , and for v ∈ V , w1 ∈ W1 , [L(−1), I (w1 , z)] =

(2.11)

z2 − z1 I (w1 , z2 )Y (v, z1 ) −z0 z1 − z0 = z2−1 δ I (Y (v, z0 )w1 , z2 ). (2.12) z2 3 W3 All intertwining operators of type WW form a vector space denoted by IW . The 1 W2 1 W2

z0−1 δ

z1 − z2 z0

d I (w1 , z), dz

Y (v, z1 )I (w1 , z2 ) − z0−1 δ

W3 W3 dimension of IW is called the fusion rule, denoted by NW . Clearly, the fusion rule 1 W2 1 ,W2 only depends on the equivalence class of each Wi . The following are among the immediate consequences of the Jacobi identity (2.12): n [vn , I (w1 , z2 )] = zn−i I (vi w1 , z2 ) (2.13) i 2 i≥0

(the commutator formula) and I (vn w1 , z2 ) = Resz1 (z1 − z2 )n Y (v, z1 )I (w1 , z2 ) − (−z2 + z1 )n I (w1 , z2 )Y (v, z1 ) (2.14) (the iterate formula) for n ∈ Z. Conversely, the commutator and iterate formulas imply the Jacobi identity.


659

Remark 2.7. If V = Lg (, 0) and W1 = Lg (, λ), for w1 ∈ L(λ) (the lowest L(0)weight subspace of Lg (, λ)), the commutator formula (2.13) gives [an , I (w1 , z2 )] = z2n I (aw1 , z2 )

(2.15)

for a ∈ g ⊂ Lg (, 0) and for n ∈ Z. In many literatures such as [TK] (and [FM]), an intertwining operator was defined on L(λ) with Eqs. (2.11) and (2.15) as the defining axioms. However, it can be proved (cf. [TK, Li3, Li6]) that the two definitions give rise to the same fusion rules. Let V be a vertex operator algebra. We denote by Irr(V ) the set of equivalence classes of irreducible modules and for an irreducible V -module W , denote by [W ] the equivalence class. V is said to be quasi-rational [MS] if all fusion rules associated with [W3 ] irreducible modules are finite and if for any [W1 ], [W2 ] ∈ Irr(V ), N[W = 0 for all 1 ],[W2 ] but finitely many [W3 ] ∈ Irr(V ). For a quasi-rational vertex operator algebra V , the Verlinde algebra A(V ) is defined to be an algebra (over C) with Irr(V ) as a basis and with the fusion rules as the structural constants, i.e., [Wk ] [Wi ] · [Wj ] = N[W [Wk ]. (2.16) i ],[Wj ] [Wk ]∈Irr(V )

When V is simple, it is easy to show that [V ] is the unit. It follows immediately from [FHL, HL2] that A(V ) is commutative. Under certain conditions, Huang [Hua1] established the associativity, but for a general V the associativity is still an unsolved problem. Let V = Lsl(2) (k, 0) with a positive integer k. It is well known ([GW, TK, FZ]) that the Verlinde algebra has the following relations: [L(k, i)] · [L(k, j )] =

min(i+j,2k−i−j )

[L(k, r)].

(2.17)

r=max(i−j,j −i)

The following definition is due to Schellekens and Yankielowicz [SY]. Definition 2.8. An irreducible V -module W is called a simple current if the associated matrix of the left multiplication of [W ] with respect to the standard basis of the Verlinde algebra is a permutation. The order of the associated matrix as a group element is called the order of W . We now recall a construction of simple currents from [Li4]. In the following, one may think of V as one of, or more generally, any tensor product of the following vertex operator algebras: Lg (, 0),

Mh (1, 0),

VL ,

Lg (, 0) ⊗ Mh (1, 0),

Lg (, 0) ⊗ VL .

Let α ∈ V(1) satisfying the following conditions: L(n)α = δn,0 α, α(n)α = δn,1 γ 1 for n ∈ Z+ , (2.18) where Y (α, z) = n∈Z α(n)z−n−1 , i.e., α(n) = αn , and γ is a fixed complex number. Notice that condition (2.18) implies that α is a primary vector and that operators α(n) satisfy the Heisenberg algebra relation [α(m), α(n)] = mγ δm+n,0

for m, n ∈ Z.

(2.19)

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H. Li

Furthermore, assume that α(0) acts semisimply on V . It is clear that e2πiα(0) is an automorphism of V and that e2πiα(0) = 1 if and only if α(0) has only integral eigenvalues on V . If each α(0) has only rational eigenvalues on V and if V is finitely generated, then e2πiα(0) is of finite order. Define ∞ α(k) α(0) −k . (2.20) exp (−z) (α, z) = z −k k=1

This is a well defined element of (End W ){z} for any weak V -module W on which α(0) semisimply acts. The following result is a special case of Proposition 5.4 of [Li2]. Proposition 2.9. Let α, (α, z) be given as before. Assume that α(0) has only integral eigenvalues on V . Let W be any (irreducible) weak V -module. Set (W (α) , Yα (·, z)) = (W, Y ((α, z)·, z)).

(2.21)

Then (W (α) , Yα ) carries the structure of an (irreducible) weak V -module. As a convention, by V -module W (α) we mean the V -module (W (α) , Yα ). Recall the following result from [Li4] (Proposition 2.5): Proposition 2.10. Let α, (α, z) be as in Proposition 2.9. Let W1 and W2 be weak V -modules and f be a V -homomorphism from W1 to W2 . Then f is also a V -homo(α) (α) morphism from W1 to W2 . Remark 2.11. In view of Propositions 2.9 and 2.10, we obtain a canonical functor Fα from the category of weak V -modules to itself in the obvious way. Since (α, z)(−α, z) = (−α, z)(α, z) = 1,

(2.22)

we easily see that F−α is the inverse functor of Fα . Therefore, Fα is an isomorphism. The following result [Li4, Proposition 2.4] is a generalization of Proposition 2.9: Proposition 2.12. Let α, (α, z) be given as in Proposition 2.9. Let I be an intertwining 3 . Define operator of type WW 1 W2 Iα (w1 , z)w2 = I ((α, z)w1 , z)w2 for w1 ∈ W1 , w2 ∈ W2 . Then Iα is an intertwining operator of type

(2.23)

(α)

W3

(α)

W1 W2

.

The following results [Li4, Corollary 2.12, Theorem 2.15, Proposition 3.2] give a construction of simple currents: Theorem 2.13. Let V be a simple vertex operator algebra and let α ∈ V(1) be such that (2.18) holds and such that α(0) has only integral eigenvalues on V . If for each irreducible (ordinary) V -module W , the weak module W (α) is an (ordinary) V -module, then V (α) is a simple current. Furthermore, for any irreducible V -module W , [W ] · [V (α) ] = [W (α) ]. The following lemma gives more information about V (α) .

(2.24)


661

Lemma 2.14. Let V , α be as in Theorem 2.13. Then L(0) acts semisimply on V (α) with eigenvalues in 21 γ + Z, where α(1)α = γ 1. Proof. From [Li5, (3.18)], we have 1 1 (α, z)ω = ω + αz−1 + α(1)αz−2 = ω + αz−1 + γ 1z−2 , 2 2

(2.25)

where ω is the Virasoro element. Then 1 Yα (ω, z) = Y ((α, z)ω, z) = Y (ω, z) + z−1 Y (α, z) + γ z−2 . 2

(2.26)

In terms of components we have 1 Lα (m) = L(m) + α(m) + γ δm,0 (2.27) 2 for m ∈ Z, where Yα (ω, z) = m∈Z Lα (m)z−m−2 . Since L(0) and α(0) act semisimply on V with integral eigenvalues, Lα (0) acts semisimply on V with eigenvalues in 21 γ +Z. That is, L(0) acts semisimply on V (α) with eigenvalues in 21 γ + Z. Since any irreducible weak module is an ordinary module for a regular vertex operator algebra, from Theorem 2.13 we immediately have: Corollary 2.15. Let V be a regular vertex operator algebra and let α ∈ V(1) be given as in Theorem 2.13. Then V (α) is a simple current. In particular, this is true when V is a tensor product from the following algebras: Lg (, 0),

VL ,

Lg (, 0) ⊗ VL ,

where is a positive integer and L is a positive-definite even lattice. The following result was obtained in [Li4]: Proposition 2.16. Let L be a positive definite even lattice. (β)

(1) For β ∈ Lo , as a VL -module, VL is isomorphic to VL+β . (2) Every irreducible VL -module is a simple current. (3) The Verlinde algebra is canonically isomorphic to the group algebra C[Lo /L]. Previously, intertwining operators for VL were explicitly constructed, fusion rules were calculated and (3) was proved in [DL]. (Of course, (3) implies (2).) Though the vertex operator algebra Mh (1, 0) is not regular, the same proof of Proposition 2.16 gives the following result: Proposition 2.17. (1) For h ∈ h, as an Mh (1, 0)-module, Mh (1, 0)(h) is isomorphic to Mh (1, h). (2) Every irreducible Mh (1, 0)-module is a simple current. (3) The Verlinde algebra is canonically isomorphic to the group algebra C[h].

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Remark 2.18. Suppose that V = V 1 ⊗ V 2 is a tensor product vertex operator algebra. 1 and V 2 are canonical subspaces of V . Suppose α = α 1 + α 2 , where Then V(1) (1) (1) 1 , a 2 ∈ V 2 . Then α satisfies (2.18) if and only if both α 1 and α 2 satisfy (2.18). α 1 ∈ V(1) (1) Furthermore, α(0) acting on V has only integral eigenvalues if and only if α i (0) acting on V i has only integral eigenvalues for i = 1, 2. Let W = W1 ⊗ W2 , where W1 , W2 are V1 and V2 -modules, respectively. Since [α 1 (m), α 2 (n)] = 0 for m, n ∈ Z, we have (α, z) = (α 1 , z)(α 2 , z).

(2.28)

Then we have (α 1 )

W (α) = W1

(α 2 )

⊗ W2

.

(2.29)

Let g, h, · , · be as in Sect. 2.1. Let {ei , fi | i = 1, . . . , n} be the Chevalley generators with simple roots α1 , . . . , αn and simple coroots α1∨ , . . . , αn∨ . Let λi (i = 1, . . . , n) be the fundamental weights for g. Let Q = ⊕ni=1 Zαi be the root lattice and let P = ⊕ni=1 Zλi be the weight lattice. Let h(1) , . . . , h(n) ∈ h be the fundamental co-weights, i.e., αi (h(j ) ) = δi,j

for i, j = 1, . . . , n.

(2.30)

Set Q∨ = Zα1∨ + · · · + Zαn∨ , P

∨

(1)

= Zh

(n)

+ · · · + Zh

(2.31) ,

(2.32)

the co-root lattice and the co-weight lattice. Let θ=

n

a i αi

(2.33)

i=1

be the highest long root. Then θ (h(i) ) = ai

for i = 1, . . . , n.

(2.34)

We shall need to know which ai equal 1. The following is a list for such ai (cf. [K]): An Bn Cn Dn E6 E7

: a1 , . . . , an , : a1 , : an , : a1 , an−1 , an , : a1 , a5 , : a6 .

(2.35)

Remark 2.19. Simple roots for type E (E6 , E7 , E8 ) were numbered differently in [H] and [K]. Here, we use the numbering system of [K].


663

Let 70 , . . . , 7n be the fundamental weights of gˆ [K]. Then each λi for 1 ≤ i ≤ n is naturally extended to 7i . From the Dynkin diagram ([K], TABLE Aff 1) we find that ai = 1 if and only if the vertices 0 and i are in the same orbit under the automorphism group of the affine Dynkin diagram. We point out that if ai = 1, then 7i is of level one. The following proposition was proved in [Li4] (Proposition 3.5, Remark 3.8): Proposition 2.20. Let be a complex number with = −h∨ , where h∨ is the dual Coxeter number of g. If the coefficient ai of αi in θ is 1, then as an L(, 0)-module (i) )

L(, 0)(h

L(, λi ).

(2.36)

Furthermore, if is a positive integer, L(, λi ) is a simple current for L(, 0). Remark 2.21. It was known ([FG, F]) that L(, λi ) with ai = 1 are all the simple currents except the level 2 simple current L(77 ) for g of type E8 . Remark 2.22. The element (h(i) , z) gives rise to an automorphism ψi of affine Lie algebra gˆ via ψi (Y (a, z)) = Yh(i) (a, z) = Y ((h(i) , z)a, z) for a ∈ g, where Y (a, z) =

n∈Z a(n)z

−n−1

(2.37)

is the generating series of a. That is,

ψi (αi∨ (n)) = αi∨ (n) + δn,0 , ψi (ei (n)) = ei (n + 1), ψi (fi (n)) = fi (n − 1);

ψi (αj∨ (n)) = αj∨ (n),

(2.38)

ψi (ej (n)) = ej (n),

ψi (fj (n)) = fj (n) for j = i, n ∈ Z,

(2.39)

and ψi (fθ (n)) = fθ (n − 1) for n ∈ Z.

(2.40)

(i)

The vector 1 in L(, 0)(h ) is a highest weight vector of weight 7i . More general automorphisms of this type were recently studied in [FS]. Note that the composition of the representation on L(, 0) with the corresponding Dynkin diagram automorphism of gˆ also gives L(, λi ). But ψi are not Dynkin diagram automorphisms. Dynkin diagram automorphisms played important roles in [FG, F, SY] and [FM]. Remark 2.23. For α ∈ h, from (2.25) we have 1 Lα (m) = L(m) + α(m) + α, αδm,0 (2.41) 2 for m ∈ Z, recalling that Yα (ω, z) = m∈Z Lα (m)z−m−2 with ω being the Virasoro element. Then the Lh(i) (0)-weight of 1 is 21 h(i) , h(i) . Since under the new action Y ((h(i) , z)·, z) on L(, 0), 1 is still a highest weight vector for gˆ , the lowest L(0)weight of L(, λi ) is 21 h(i) , h(i) . Of course, there is a formula for the lowest weight of any irreducible module (cf. [DL]).

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From now on we assume that k is a positive integer. Let h ∈ h. Note that for any root vector eα of g and for m ∈ Z, [h(0), eα (m)] = α(h)eα (m).

(2.42)

Since h(0)1 = 0 and L(k, 0) is generated by gˆ from 1, using (2.42) we see that h(0) has only integral eigenvalues on L(k, 0) if and only if h ∈ P ∨ . Define a map π from P ∨ to the Verlinde algebra V(L(k, 0)) of L(k, 0) by π(α) = [L(k, 0)(α) ]

for α ∈ P ∨ .

(2.43)

The map π naturally extends to a linear map from the group algebra C[P ∨ ] to V(L(k, 0)). We abuse the notation π for this extension also. Proposition 2.24. The linear map π is an algebra homomorphism. Proof. For α, β ∈ P ∨ , since [α(r), β(s)] = 0 for r, s ≥ 0, we have (α + β, z) = (α, z)(β, z),

(2.44)

recalling (2.20). Then we get L(k, 0)(α+β) (L(k, 0)(α) )(β)

(2.45)

as L(k, 0)-modules. In view of Theorem 2.13 we have [L(k, 0)(α) ] · [L(k, 0)(β) ] = [(L(k, 0)(α) )(β) ] = [L(k, 0)(α+β) ].

(2.46)

Thus π is an algebra homomorphism. It follows immediately that π(P ∨ ) is an abelian group. The following result gives important kernel elements of π as a group homomorphism on P ∨ . Proposition 2.25. We have [L(k, 0)(α) ] = [L(k, 0)]

for α ∈ Q∨ .

(2.47)

Furthermore, for any irreducible L(k, 0)-module W , we have [W (α) ] = [W ]

for α ∈ Q∨ .

(2.48)

Proof. It suffices to prove (2.47) for α = αi∨ , i = 1, . . . , n. For 1 ≤ i ≤ n, set r = αi2,αi , a positive integer. From [H] or [K] we have αi∨ , αi∨ =

4 = 2r. αi , αi

(2.49)

Since 1 [ei (1), fi (−1)] = αi∨ (0) + ei , fi c = αi∨ (0) + αi∨ , αi∨ c = αi∨ (0) + rc, (2.50) 2 Li := Cei (1) + Cfi (−1) + C(αi∨ + rc) is a subalgebra of gˆ isomorphic to sl(2). Furthermore, L(k, 0), being an integrable gˆ -module, is an integrable Li -module. Clearly,


665

1 is a highest weight vector of weight rk for Li . Then fi (−1)rk 1 = 0. From (2.25) we have 1 Lαi∨ (0) = L(0) + αi∨ (0) + kαi∨ , αi∨ = L(0) + αi∨ (0) + kr, 2 where Yαi∨ (ω, z) = n∈Z Lαi∨ (n)z−n−2 . Then Lαi∨ (0)fi (−1)rk 1 = (L(0) + αi∨ (0) + kr)fi (−1)rk 1

= (kr − 2kr + kr)fi (−1)rk 1 = 0.

(2.51)

(2.52)

∨

That is, fi (−1)rk 1 is a non-zero element of L(k, 0)(αi ) of weight zero. On the other hand, with L(k, 0) being regular, every irreducible module, in particular, ∨ L(k, 0)(αi ) , is an integrable gˆ -module of level k, which is unitary. Thus the lowest ∨ ∨ weights of L(k, 0)(αi ) are nonnegative. Consequently, the lowest weight of L(k, 0)(αi ) is 0. Because L(k, 0) is the only irreducible L(k, 0)-module with 0 being the lowest ∨ weight, we must have L(k, 0)(αi ) L(k, 0) as L(k, 0)-modules. For an irreducible L(k, 0)-module W , using the first part and Theorem 2.13 we get [W ] = [W ] · [V ] = [W ] · [V (α) ] = [W (α) ]

for α ∈ Q∨ .

(2.53)

This completes the proof. The following result generalizes Proposition 2.20 with L = Q: Theorem 2.26. The algebra homomorphism π gives rise to an algebra isomorphism from the group algebra C[P ∨ /Q∨ ] onto π(C[P ∨ ]). Furthermore, π(P ∨ ) = {[L(k, kλi )] | ai = 1}.

(2.54)

Proof. In view of Proposition 2.25, π gives rise to an algebra homomorphism π¯ from C[P ∨ /Q∨ ] onto π(C[P ∨ ]). From [H] (Sect. 13.1), we have |P /Q| = n + 1, 2, 2, 4, 3, 2, 1, 1, 1

(2.55)

for g of type An+1 , Bn , Cn , Dn , E6 , E7 , E8 , F4 , G2 , respectively. Note that Bn and Cn are dual to each other and the others are self-dual. Then |P ∨ /Q∨ | = |P /Q| for all types. On the other hand, from Proposition 2.20, all [L(k, kλi )] for i with ai = 1, which are distinct basis elements of V(L(k, 0)), are images of π . Then it follows immediately that (2.54) holds and π¯ is an algebra isomorphism. The group structure of P /Q for the nontrivial cases was given in [H] (Exercise 4 on p. 71). Then we have: For An+1 , P ∨ /Q∨ Z/(n + 1)Z with h(1) + Q∨ as a generator such that ¯ mh(1) + Q∨ = h(m) + Q∨ ,

(2.56)

where m ¯ is the least nonnegative residue of m modulo n + 1. For Dn with n being odd, P ∨ /Q∨ Z/4Z with h(n) + Q∨ as a generator such that 2h(n) + Q∨ = h(1) + Q∨ ,

3h(n) + Q∨ = h(n−1) + Q∨ .

(2.57)

666

H. Li

For Dn with n being even, P ∨ /Q∨ Z/2Z×Z/2Z with h(n−1) +Q∨ and h(n) +Q∨ as generators such that h(1) + Q∨ = h(n−1) + h(n) + Q∨ .

(2.58)

Then with Theorem 2.26 we immediately have the following results which seem to be known to physicists: Corollary 2.27. The following fusion algebra relations hold: For An+1 , for 0 ≤ i, j ≤ n, [L(k, kλi )] · [L(k, kλj )] = [L(k, kλi+j )].

(2.59)

[L(k, kλ1 )] · [L(k, kλ1 )] = [L(k, 0)].

(2.60)

[L(k, kλn )] · [L(k, kλn )] = [L(k, 0)].

(2.61)

For Bn ,

For Cn ,

For Dn with odd n, [L(k, kλn )]2 = [L(k, kλ1 )],

[L(k, kλn )]3 = [L(k, kλn−1 )],

[L(k, kλn )]4 = [L(k, 0)].

(2.62)

For Dn with even n, [L(k, kλ1 )]2 = [L(k, kλn−1 )]2 = [L(k, kλn )]2 = [L(k, 0)], [L(k, kλn−1 )] · [L(k, kλn )] = [L(k, kλ1 )].

(2.63) (2.64)

[L(k, kλ1 )]2 = [L(k, kλ5 )], [L(k, kλ1 )]3 = [L(k, 0)].

(2.65)

[L(k, kλ6 )]2 = [L(k, 0)].

(2.66)

For E6 ,

For E7 ,

We shall need the number h(i) , h(i) and the explicit expression of h(i) in terms of αj∨ . The expression of each λi in terms of simple roots αj was known ([H], p. 69, Table 1). Suppose that for 1 ≤ i ≤ n, λi (i)

h

= ai1 α1 + · · · + ain αn , =

bi1 α1∨

+ · · · + bin αn∨ .

(2.67) (2.68)

Then aij = λi (h(j ) ) = bj i .

(2.69)

Furthermore, from [H] we get h(i) , αj∨ = h(i) , tαj

2 2 2 = αj (h(i) ) = δi,j . αj , αj αj , αj αj , αj

(2.70)


667

Then using (2.68) we get h(i) , h(i) = bii

2 2 = aii . αi , αi αi , αi

(2.71)

With a glance of Table Aff in [K] we see that if ai = 1, αi is a long root, hence αi , αi = 2. Therefore, we have obtained: Lemma 2.28. If λi = ai1 α1 + · · · + ain αn , for 1 ≤ i ≤ n, then h(i) = a1i αi∨ + · · · + ani αi∨ , 2 . h(i) , h(i) = aii αj , αj

(2.72) (2.73)

In particular, if ai = 1, we have h(i) , h(i) = aii . 3. Extension of Vertex Operator Algebras by Simple Currents We shall first recall some of the results from [DLM2] on V [L], an extension of a certain vertex operator algebra V by simple currents V (α) parametrized by α ∈ L, where L is a lattice carrying an intrinsic structure of V . We then extend and refine those results. In Sect. 3.3, we concentrate a special class of V [L]. We classify all irreducible V [L]modules, prove a complete reducibility theorem and we give a formula of fusion rules in the category of V [L]-modules in terms of fusion rules in the category of V -modules. 3.1. Extension of algebras. We shall first establish some basic assumptions which will remain in force throughout this section. Let V be a finitely generated simple vertex operator algebra. As in Sect. 2, one may think of V as a tensor product from the following vertex operator algebras: Lg (, 0),

Mh (1, 0),

VL .

Let H be a (finite-dimensional) subspace of V(1) satisfying the following conditions: L(n)h = δn,0 h,

h(n)h = B(h, h )δn,1 1 for n ∈ Z+ , h, h ∈ H,

(3.1)

where Y (h, z) = h(n)z−n−1 for h ∈ H , and B(· , ·) is assumed to be a nondegenerate symmetric bilinear form on H . We then identify H with its dual H ∗ . We also assume that for any h ∈ H , h(0) acts semisimply on V . Then V = ⊕α∈H V (0,α) ,

where V (0,α) = {v ∈ V | h(0)v = B(α, h)v for h ∈ H }. (3.2)

Set P = {α ∈ H | V (0,α) = 0}.

(3.3)

As V is simple, P is a subgroup of H (cf. [LX]). With V being finitely generated, we see that P is finitely generated. Then P equipped with the bilinear form B is a lattice.

668

H. Li

Let L be a subgroup of H such that for each α ∈ L, α(0) acting on V has only integral eigenvalues. This amounts to L ⊂ P o , where P o = {h ∈ H | B(h, α) ∈ Z for α ∈ P }

(3.4)

is the dual lattice of P . Note that the rank of P may be less than dim H . Let W be a V -module. By Proposition 2.9, for α ∈ L, we have a (weak) V -module (W (α) , Yα (·, z)) := (W, Y ((α, z)·, z)).

(3.5)

For convenience, we reformulate the construction of the V -module W (a) as follows: Set W (α) = Ceα ⊗ W W (linearly),

(3.6)

where eα is a symbol for now and Ceα is a one-dimensional vector space with eα as a pre-chosen basis element. Then define Yα (v, z)(eα ⊗ w) = eα ⊗ Y ((α, z)v, z)w

for v ∈ V , w ∈ W.

(3.7)

Set W [L] = ⊕α∈L W (α) = C[L] ⊗ W,

(3.8)

equipped with the direct sum V -module structure. Now, the symbol eα in the definition of W (α) is considered as an element of the group algebra C[L]. For α ∈ L, we define a linear endomorphism ψα 1 of W [L] by ψα (eβ ⊗ w) = eα+β ⊗ w

for β ∈ L, w ∈ W.

(3.9)

Then ψ0 = 1,

ψα+β = ψα ψβ

for α, β ∈ L.

That is, ψ gives rise to a representation of L on W [L]. For α ∈ L, we set ([LW, FLM]) ∞ α(±n) ± ∓n . E (α, z) = exp z ±n

(3.10)

(3.11)

n=1

Then (α, z) = zα(0) E + (−α, −z).

(3.12)

Next, we extend the domain of Yα from V to V [L]. Definition 3.1. For u ∈ V (α) , v ∈ V (β) with α, β ∈ L, we define Yα (u, z)v = ψα+β E − (−α, z)Y (ψ−α (β, z)u, z)(α, −z)ψ−β (v) ∈ V (α+β) {z}. (3.13) We then define a linear map Y˜ (·, z) from V [L] to (End V [L]){z} via Y˜ (u, z) = Yα (u, z) for u ∈ V (α) . 1 The map ψ here is the inverse of the map ψ in [DLM2]. α α


669

Note that the E − (α, z) defined in [DLM2] is the E − (−α, z) defined in (3.11) ([LW, FLM]). Then the definition of Yα is exactly the same as the one defined in [DLM2]. For α ∈ P , h ∈ H , set V (α,h) = ψα (V (0,h) ).

(3.14)

V (α) = ⊕h∈P V (α,h) .

(3.15)

Then

Definition 3.2. We define C-valued functions η and C on (L × H ) × (L × H ) by η((α1 , h1 ), (α2 , h2 )) = −B(α1 , α2 ) − B(α1 , h2 ) − B(α2 , h1 ) ∈ C, C((α1 , h1 ), (α2 , h2 )) = e(B(α1 ,h2 )−B(α2 ,h1 ))πi ∈ C×

(3.16) (3.17)

for (αi , hi ) ∈ L × H, i = 1, 2. Then we have ([DLM2], Theorem 3.5): Theorem 3.3. Let u ∈ V (α,h1 ) , v ∈ V (β,h2 ) , w ∈ V (γ ,h3 ) with α, β, γ ∈ L, h1 , h2 , h3 ∈ P . Then z1 − z2 η((α,h1 ),(β,h2 )) ˜ z1 − z2 −1 Y (u, z1 )Y˜ (v, z2 )w z0 δ z0 z0 z2 − z1 η((α,h1 ),(β,h2 )) ˜ z2 − z1 Y (v, z2 )Y˜ (u, z1 )w − C((α, h1 ), (β, h2 ))z0−1 δ −z0 z0 z2 + z0 η((α,h1 ),(γ ,h3 )) ˜ ˜ z1 − z0 −1 Y (Y (u, z0 )v, z2 )w. = z2 δ z2 z1 (3.18) Furthermore, for all v ∈ V [L], [L(−1), Y˜ (v, z)] =

d ˜ Y (v, z). dz

(3.19)

Now we express Y˜ more explicitly. Lemma 3.4. For α, β ∈ L, u, v ∈ V , we have Y˜ (eα ⊗ u, z)(eβ ⊗ v) = eα+β ⊗ zB(α,β) E − (−α, z) × Y ((β, z)u, z)E + (−α, z)(−z)α(0) v.

(3.20)

In particular, Y˜ (eα ⊗ 1, z)(eβ ⊗ v) = eα+β ⊗ zB(α,β) E − (−α, z)E + (−α, z)(−z)α(0) v.

(3.21)

Proof. From Lemma 3.2 of [DLM2] we have ψ−α (β, z) = zB(α,β) (β, z)ψ−a .

(3.22)

Note that the map ψα defined here is the map ψ−α defined in [DLM2]. Then (3.20) follows immediately.

670

H. Li

Remark 3.5. Let G = L × P be the product group. Suppose that there exists a positive integer T such that η restricted on G is T1 Z/2Z-valued. The original theorem states that (V [L], 1, ω, Y˜ , T , G, η(· , ·), C(· , ·)) is a generalized vertex algebra in the sense of [DL]. This result is similar to Theorem 5.1 of [DL], which states that if L is a rational lattice, then VL has a canonical generalized vertex algebra structure. In fact, by taking V = Mh (1, 0) with h = C ⊗Z L, we have P = 0 and V [L] = VL . In order to get vertex (super)algebras from V [L], we shall restrict ourselves to special L. We have already assumed that L ⊂ P o , or what is equivalent to, α(0) acting on V has only integral eigenvalues for every α ∈ L. Now we furthermore assume that (L, B) is an integral lattice. Then B(α, α),

B(α, β) ∈ Z

for α ∈ L, β ∈ P .

(3.23)

Recall from Lemma 2.14 that for α ∈ L, the weights of V (α) are contained in 21 B(α, α)+ Z. Then V [L] is 21 Z-graded by L(0)-weights. Furthermore, the function η restricted to (L × P ) × (L × P ) is Z-valued and C((α1 , h1 ), (α2 , h2 )) = (−1)B(α1 ,h2 )−B(α2 ,h1 ) ,

(3.24)

(recall (3.16) and (3.17)). Then we have ([DLM2], (3.59)): Corollary 3.6. Assume that L ⊂ P 0 and (L, B) is an integral lattice. For a ∈ V (α) , b ∈ V (β) with α, β ∈ L, we have z0−1 δ

z1 − z2 z0

z2 − z1 ˜ B(α,β) −1 ˜ ˜ Y (a, z1 )Y (b, z2 ) − (−1) Y (b, z2 )Y˜ (a, z1 ) z0 δ −z0 z1 − z0 ˜ ˜ −1 = z2 δ Y (Y (a, z0 )b, z2 ). (3.25) z2

Remark 3.7. Set Le = {α ∈ L | B(α, α) ∈ 2Z}.

(3.26)

Clearly, Le is a subgroup of L of index 2. If Le ⊂ 2Lo , i.e., B(α, β) ∈ 2Z

for α ∈ Le , β ∈ L,

(3.27)

we easily see that V [L] = ⊕β∈L V (β) is a vertex superalgebra with V [L]0 = V [Le ] = ⊕α∈Le V (α) ,

V [L]1 = ⊕α∈L−Le V (α) .

(3.28)

In particular, if L is of rank one, clearly (3.27) holds, hence V [L] is a vertex (super)algebra. However, without assuming (3.27) V [L] equipped with the vertex operator map Y˜ may not be a vertex superalgebra. Even if L is even, V [L] may not be a vertex algebra unless B is 2Z-valued. Corollary 3.13 of [DLM2], which states that V [L] equipped with Y˜ is a vertex superalgebra if L is integral (without the condition (3.27), is incorrect.


671

As in [FLM] and [DL] for VL , we shall need a 2-cocycle = on L. Suppose that (L, B) is an integral lattice of finite rank. Let {α1 , . . . , αd } be a Z-basis for L. Define = to be the (uniquely determined) {±1}-valued multiplicative function on L × L such that =(αi , αj ) = (−1)B(αi ,αj )+B(αi ,αi )B(αj ,αj )

if i ≥ j, and 1 otherwise

(3.29)

(cf. [DL, FLM]). Note that =(αi , αi ) = 1 because B(αi , αi ) + B(αi , αi )B(αi , αi ) is even. Then =(α, β)=(β, α)−1 = (−1)B(α,β)+B(α,α)B(β,β)

for α, β ∈ L.

(3.30)

Define a vertex operator map Y on V [L] by Y (eα ⊗ u, z)(eβ ⊗ v) = =(α, β)Y˜ (eα ⊗ u, z)(eβ ⊗ v)

(3.31)

for α, β ∈ L, u, v ∈ V . By Lemma 3.4, Y (eα ⊗ u, z)(eβ ⊗ v) = =(α, β)eα+β ⊗ zB(α,β) E − (α, z)Y ((β, z)u, z)E + (−α, z)(−z)α(0) v.

(3.32)

From Corollary 3.6 we immediately obtain: Proposition 3.8. Suppose that L ⊂ P o and that (L, B) is an integral lattice of finite rank. Let = be the {±1}-valued multiplicative function on L × L defined in (3.29). Then V [L] equipped with the vertex operator map Y defined in (3.32) is a vertex (super)algebra with the even and odd parts being defined in (3.28). Remark 3.9. It was proved in [DL] (Theorem 6.7 and Remarks 6.17 and 12.38) that if L is an integral lattice, VL is a vertex superalgebra. Since we in this paper are mainly interested in vertex operator (super)algebras, for the rest of Sect. 3 we shall assume that L ⊂ P o and (L, B) is integral, and we fix the multiplicative function =. 3.2. Extensions of modules and intertwining operators. We continue Sect. 3.1 to study the extension of an irreducible V -module, following [DLM2]. The extension W [L] of an irreducible V -module W is in general a twisted V [L]-module with respect to an automorphism of V [L]. We shall also study the extension of an intertwining operator. Let W be an irreducible V -module. By definition, homogeneous subspaces of W are finite-dimensional. Since [L(0), h(0)] = 0 for h ∈ H , H preserves each homogeneous subspace of W so that there exist 0 = w ∈ W, h ∈ H = H ∗ such that h(0)w = B(h, h )w for h ∈ H . Since H acts semisimply on V (by assumption) and w generates W by V (from the irreducibility of W ), H also acts semisimply on W . For any h ∈ H , we define W (0,h) = {w ∈ W | h(0)w = B(h, h )w for h ∈ H }.

(3.33)

P (W ) = {h ∈ H | W (0,h) = 0}.

(3.34)

Set

Since W is irreducible, P (W ) is an irreducible P (V )-set. Then P (W ) = h + P (V ) for any h ∈ P (W ).

672

H. Li

Definition 3.10. Let W be an irreducible V -module and h ∈ P (W ). Define a linear endomorphism σW of V [L] by σW (a) = e−2πiB(α,h) a

for a ∈ V (α) with α ∈ L.

(3.35)

P (V )o

Because L ⊂ and P (W ) = h + P (V ), σW is well defined, i.e., it does not depend on the choice of h. Lemma 3.11. The defined linear endomorphism σW of V [L] is an automorphism of the vertex (super)algebra and σW = 1 if and only if α(0) has only integral eigenvalues on W for every α ∈ L, i.e., P (W ) ⊂ Lo . Furthermore, σW is of finite order if and only if α(0) has rational eigenvalues on W for every α ∈ L, or equivalently, B(α, h) ∈ Q for α ∈ L. Proof. In view of (3.13), clearly, σW is an automorphism of the vertex (super)algebra and σW = 1 if and only if α(0) has only integral eigenvalues on W for every α ∈ L. Furthermore, since by assumption V is finitely generated, σW is of finite order if and only if α(0) has rational eigenvalues on W for every α ∈ L, or equivalently, B(α, h) ∈ Q for α ∈ L. Recall that W [L] = C[L] ⊗ W = ⊕α∈L W (α) . Definition 3.12. Let W be an irreducible V -module. For a ∈ V (α) , w ∈ W (β) , α, β ∈ L, we define YW [L] (a, z)w ∈ W (α+β) {z} by YW [L] (a, z)w = =(α, β)ψα+β E − (α, z)Y (ψ−α (β, z)a, z)(α, −z)ψ−β (w) (3.36) (cf. (3.13)). In terms of the notion of twisted module as defined in [Le, D2] and [FFR] we have ([DLM2], Theorem 3.6, Corollary 3.14): Proposition 3.13. Let W be an irreducible V -module such that α(0) has rational eigenvalues on W for every α ∈ L. Then the following twisted Jacobi identity holds for u ∈ V (α) , v ∈ V (β) , w ∈ W (γ ,h) , z1 − z2 −1 z0 δ Y (u, z1 )Y (v, z2 )w z0 z2 − z1 B(α,α)B(β,β) −1 − (−1) Y (v, z2 )Y (u, z1 )w z0 δ (3.37) −z0 z1 − z0 B(α,h) z1 − z0 Y (Y (u, z0 )v, z2 )w. δ = z2−1 z2 z2 Moreover, W [L] is a σW -twisted V [L]-module. In particular, if P (W ) ⊂ Lo , i.e., α(0) acting on W has only integral eigenvalues for α ∈ L, W [L] is an (untwisted) V [L]module. Next, we prove a functorial property. Proposition 3.14. Let σ be an automorphism of V [L] of finite order such that σ (V (α) ) = V (α) σ (v) = v

for α ∈ L,

(3.38)

for v ∈ V = V (0) .

(3.39)

Let M be a σ -twisted weak V [L]-module which is a direct sum of irreducible V -modules and let W be an irreducible V -submodule of M. Then σ = σW and there is a canonical V [L]-homomorphism φ[L] from W [L] to M extending the embedding φ of W into M.


673

Proof. The following arguments are essentially the ones of [DLM2], Corollary 3.15 and Lemma 4.3. Note that for α ∈ L, YW [L] restricted to V (α) × W is a nonzero intertwining operator (α) of type VW(α) W . Since YM (·, z)φ restricted to V (α) × W is an intertwining operator of M type V (α) W and [V (α) ] · [W ] = [W (α) ], it follows from the Schur lemma (cf. [FHL]) that there exists a unique V -homomorphism φα from W (α) to M such that YM (a, z)w = φα (YW [L] (a, z)w)

(3.40)

for a ∈ V (α) , w ∈ W . From the definition of a twisted module we have zrα YM (a, z)w ∈ M((z)), φα (zB(α,h) YW [L] (a, z)w) ∈ M((z)).

(3.41)

Then it follows from (3.40) that rα − B(α, h) ∈ Z, hence σ (a) = σW (a). Thus σ = σW . Define a V -homomorphism φ[L] from W [L] to M by φ[L] = φα on W (α) for α ∈ L. Now we show that φ[L] is a V [L]-homomorphism. Let w ∈ W, a ∈ V (α) , b ∈ V (β) with α, β ∈ L. Let k0 be a positive integer such that (z0 + z2 )k0 +B(α,h) YW [L] (a, z0 + z2 )YW [L] (b, z2 )w = (z2 + z0 )k0 +B(α,h) YW [L] (Y (a, z0 )b, z2 )w, (z0 + z2 ) =

(3.42)

k0 +B(α,h)

YM (a, z0 + z2 )YM (b, z2 )φ(w) k0 +B(α,h) (z2 + z0 ) YM (Y (a, z0 )b, z2 )φ(w).

(3.43)

Then using (3.40) we get (z0 + z2 )k0 +B(α,h) φα+β YW [L] (a, z0 + z2 )YW [L] (b, z2 )w = (z2 + z0 )k0 +B(α,h) φα+β YW [L] (Y (a, z0 )b, z2 )w = (z2 + z0 )k0 +B(α,h) YM (Y (a, z0 )b, z2 )φ(w) = (z0 + z2 )k0 +B(α,h) YM (a, z0 + z2 )YM (b, z2 )φ(w) = (z0 + z2 )k0 +B(α,h) YM (a, z0 + z2 )φβ YM (b, z2 )w.

(3.44)

Multiplying both sides by (z0 + z2 )−k0 −B(α,h) we get φα+β YW [L] (a, z0 + z2 )YW [L] (b, z2 )w = YM (a, z0 + z2 )φβ YM (b, z2 )w,

(3.45)

that is, φ[L]Y (a, z0 + z2 )YW [L] (b, z2 )w = YM (a, z0 + z2 )φ[L]YM (b, z2 )w.

(3.46)

Since W (β) is linearly spanned by bn W for b ∈ V (β) , n ∈ Z, we have φ[L](YW [L] (a, z)u) = YM (a, z)u for a ∈ V (α) , u ∈ W (β) . Thus φ[L] is a V [L]-homomorphism.

(3.47)

Our next result gives a characterization of the equivalence relation on (twisted) V [L]modules W [L] in terms of the equivalence of V -modules:

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Proposition 3.15. Let W1 and W2 be irreducible V -modules on which α(0) has rational eigenvalues for each α ∈ L. Then σW1 = σW2 and W1 [L] W2 [L] if and only if (α) W2 W1 for some α ∈ L. Proof. The “only if” part is clear. Note that σW (α) = σW for any irreducible V -module (α) W and α ∈ L because P (W (α) ) = α + P (W ). Assume W2 W1 for some α ∈ L. (α) Then σW1 = σW2 . Let φ be a V -isomorphism from W2 to W1 ⊂ W1 [L]. It follows from Proposition 3.14 that φ extends to a V [L]-homomorphism φ[L] from W2 [L] into (β) (α+β) for β ∈ L. With each W (β) being an irreducible W1 [L] with φ[L](W2 ) = W1 V -module, φ[L] is an isomorphism. Next, we shall extend an intertwining operator I in the category of V -modules to an intertwining operator I [L] in the category of V [L]-modules. Definition 3.16. Let W1 , W2 and W3 be irreducible V -modules and I be an intertwining 3 operator of type WW . We define a linear map 1 W2 I [L] : W1 [L] → (Hom(W2 [L], W3 [L])){z}

(3.48)

by I [L](a, z)w = =(α, β)ψα+β E − (α, z)I (ψ−α (β, z)a, z)(α, −z)ψ−β (w) (α)

(β)

(cf. (3.13) and (3.36)) for a ∈ W1 , w ∈ W2

(3.49)

with α, β ∈ L.

The same proof of Theorem 3.5 in [DLM2] gives: I [L](L(−1)a, z) =

d I [L](a, z) dz

for a ∈ W1 [L],

(3.50)

and z1 − z2 η((α,h),(β,h1 )) YW (a, z1 )I [L](b, z2 )u z0 z2 − z1 η((α,h),(β,h1 )) z2 − z1 −1 I [L](b, z2 )YW (a, z1 )u − C((α, h), (β, h1 ))z0 δ −z0 z0 z2 + z0 η((α,h),(γ ,h2 )) z1 − z0 I [L](Y (a, z0 )b, z2 )u = z2−1 δ z2 z1 (3.51) z0−1 δ

z1 − z2 z0

(β,h )

(γ ,h )

for a ∈ V (α,h) , b ∈ W1 1 , u ∈ W2 2 , with α, β, γ ∈ L, h ∈ P , h1 ∈ P (W1 ), h2 ∈ P (W2 ). If the extensions Wi [L] are (untwisted) V [L]-modules, then P (Wi ) ⊂ P o , so that η and C have integer values. Then we conclude: Proposition 3.17. Let W1 , W2 , W3 be irreducible V -modules such that for α ∈ L, α(0) has only integral eigenvalues on Wi for i = 1, 2, 3, or what this is equivalent to, the extensions W3 Wi [L] are (untwisted) V [L]-modules. Let I be an intertwining operator of type W1 W2 in the category of V -modules. Then I [L] is an intertwining operator of type W3 [L] W1 [L]W2 [L] in the category of V [L]-modules.


675

Note that various V -submodules V (α) of V [L] may be V -isomorphic to each other. It was proved in [DLM2] that V [L] contains an ideal I such that each irreducible V -module V (α) is of multiplicity-one in the quotient algebra V [L] modulo I . In the following we present an abstract reformulation of this result. Consider an (abstract) vertex operator (super)algebra U = ⊕g∈G U g graded by a (finite or infinite) abelian group G satisfying the following conditions: (C1) U 0 is a vertex operator subalgebra and U g for g ∈ G are simple currents for U 0 . (C2) For u ∈ U g , v ∈ U h with g, h ∈ G, un v ∈ U g+h for n ∈ Z. (C3) For g, h ∈ G, un v = 0 for some u ∈ U g , v ∈ U h , n ∈ Z. It is easy to see that under these conditions, U is a simple G-graded algebra, i.e., there is no nontrivial G-graded ideal. From Conditions (2) and (3), Y restricted to U g × U h g+h is a nonzero intertwining operator of type UUg U h . Then from Condition (1) we have [U g ] · [U h ] = [U g+h ]

for g, h ∈ G.

(3.52)

as U 0 -modules }.

(3.53)

Set G0 = {g ∈ G | U g U 0

Using (3.52) by a routine argument we easily get (cf. [DLM2], Lemma 3.7): Lemma 3.18. The defined subset G0 of G is a subgroup and for g, h ∈ G, [U g ] = [U h ] if and only if g − h ∈ G0 . For each g0 ∈ G0 , fix a U 0 -isomorphism fg0 from V 0 to U g0 . We particularly define f0 = 1. Let g0 ∈ G0 , h ∈ G. Then Y (·, z) ◦ fg0 is a nonzero intertwining operator of g+h type UUh U 0 . On the other hand, for any U 0 -isomorphism ψ from U h to U g+h , ψ ◦Y (·, z) g+h is a nonzero intertwining operator of type UUh U 0 . Because U h , U 0 , U g+h are simple currents and [U g+h ] = [U g ] · [U h ] = [U 0 ] · [U h ] = [U h ], there exists a unique U 0 -isomorphism fg0 ,h from U h to U g0 +h such that fg0 ,h (Y (u, z)v) = Y (u, z)fg0 (v)

for v ∈ U 0 , u ∈ U h .

(3.54)

Define a U 0 -endomorphism f¯g0 of U via f¯g0 = fg0 ,h on U h for h ∈ G. Next we define I to be the linear span of f¯g0 (u) − u

for g0 ∈ G0 , u ∈ U.

(3.55)

Lemma 3.19. The defined subspace I is an ideal of U with I ∩ U 0 = 0. Furthermore, I = 0 if and only if G0 = 0.

Proof. Let g0 ∈ G0 , h, h ∈ G and let u ∈ U h , v ∈ U h . Then f¯g0 (u) = Resz2 f¯g0 (Y (u, z2 )1) = Resz2 Y (u, z2 )f¯g0 (1).

(3.56)

Y (v, z)f¯g0 (1) = f¯g0 (Y (v, z)1) ∈ U [[z]],

(3.57)

Since

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H. Li

we have (cf. [Li2]) Y (v, z0 + z2 )Y (u, z2 )fg0 ,h (1) = Y (Y (v, z0 )u, z2 )fg0 ,h (1).

(3.58)

Using (3.56)–(3.58) we obtain Y (v, z0 )(f¯g0 (u) − u) = Resz2 Y (v, z0 + z2 )Y (u, z2 )fg0 ,h (1) − Y (v, z0 )u = Resz2 Y (Y (v, z0 )u, z2 )fg0 ,h (1) − Y (v0 , z)u = fg0 ,h (Y (v, z0 )u) − Y (v, z0 )u = f¯g0 (Y (v, z0 )u) − Y (v, z0 )u.

(3.59)

It follows immediately that I is an ideal. Clearly, I = 0 if G0 = {0}, so I = 0 implies G0 = 0. If G0 = 0, with f0 = 1 from the definition of I we have I = 0. Proposition 3.20. The algebra U is simple if and only if G0 = 0. Furthermore, the quotient algebra U¯ = U/I is simple. Proof. From Lemma 3.19, U is not simple if G0 = 0. Now we prove that U is simple if G0 = {0}. In view of Lemma 3.18, all U g for g ∈ G are non-isomorphic irreducible U 0 -modules. Then any nonzero ideal of U as a U 0 -module must be a sum of some U g . Then it follows from the conditions (1)–(3) that any nonzero ideal of U must be U . That is, U is simple. Note that U¯ = U/I is a vertex operator (super)algebra graded by group G/G0 with all the conditions (1)–(3) being satisfied. Furthermore, U¯ is a direct sum of non-isomorphic simple current V 0 -modules. From Part one, U¯ must be simple. Applying Proposition 3.20 to V [L] we immediately have (cf. [DLM2], Corollary 3.13): Corollary 3.21. Let V , L be as before. Set L0 = {α ∈ L | V (α) V

as V -modules}.

(3.60)

Then V [L] has an ideal I such that V [L]/I is simple with V as a subalgebra and such that as a V -module V /I ⊕α∈S V (α) ,

(3.61)

where S is a complete set of representatives of cosets of L0 in L. 3.3. Multiplicity-free extension V [L]. In this subsection we consider the extended vertex (super)algebra V [L] in which each V -module V (α) is multiplicity-free. We shall classify all irreducible V [L]-modules in terms of irreducible V -modules and determine the fusion rules of V [L]-modules by the fusion rules of V -modules. Our concrete examples we shall construct in Sect. 4 are of this type, so that all the results of this subsection apply to those examples. Throughout this subsection we assume that for any irreducible V -module W and for α, β ∈ L, W (α) W (β) as V -modules if and only if α = β.


677

We shall need the following result: Proposition 3.22. The vertex (super)algebra V [L] is simple. Furthermore, if Y1 and Y2 are two simple vertex operator (super)algebra structures on V [L] extending the V module structure YV , then vertex operator (super)algebras (V [L], Y1 ) and (V [L], Y2 ) are isomorphic. Proof. First, we prove that V [L] is simple. Notice that in the definition (3.13) of the vertex operator map Y , ψα+β , ψ−α , ψ−β , E − (α, z), (β, z) and (α, −z) are invertible elements and that Y (a, z)b ∈ V (α+β)

for a ∈ V (α) , b ∈ V (β)

(cf. (3.13)). Since V is simple, Y (u, z)v = 0 for 0 = u, v ∈ V ([DL], Proposition 11.9, or [FHL], Remark 5.4.6). Then it follows from Proposition 3.20 immediately that V [L] is simple. For α, β ∈ L, because V (α) and V (β) are simple currents, there exists = (α, β) ∈ C× such that Y2 (a, z)b = = (α, β)Y1 (a, z)b

for a ∈ V (α) , b ∈ V (β) .

(3.62)

It follows from weak associativity of vertex operators (cf. (3.43)) that = (α, β + γ )= (β, γ ) = = (α, β)= (α + β, γ )

(3.63)

for α, β ∈ L. That is, = is a (C× -valued) 2-cocycle on L. We also have = (0, α) = = (α, 0) = 1.

(3.64)

Since V = V (0) is even for both superalgebra structures, each structure corresponds to a sublattice Li of L of index 2 with V [Li ] being the even parts. Now we claim that L1 = L2 . Otherwise, suppose L1 − L2 = ∅ and let β ∈ L1 − L2 . Then we have the following skew-symmetry: Y1 (a, z)b = ezL(−1) Y1 (b, −z)a, Y2 (a, z)b = −e

zL(−1)

Y2 (b, −z)a

(3.65) (3.66)

for a, b ∈ V (β) . Since both Y1 and Y2 extend the V -module structure, the two vertex superalgebra structures have the same Virasoro vector. Consequently, = (β, β) = −= (β, β),

(3.67)

which is impossible because = (β, β) = 0. With L1 = L2 , using the skew-symmetry we obtain = (α, β) = = (β, α)

for α, β ∈ L.

(3.68)

It follows from the proof of Propositions 5.1.2 and 5.2.3 in [FLM] (with Z/sZ being replaced by C× ) that = is a 2-coboundary. Then the two vertex superalgebra structures on V [L] are equivalent.

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Remark 3.23. More generally, let G be a (finite or infinite) abelian group and let V be a simple vertex operator algebra and V [G] = ⊕g∈G V g be a V -module with each V g being an irreducible V -submodule. Furthermore, assume that each V g is a simple current of V . Then the set of equivalence classes of simple vertex operator (super)algebra structures on V [G] which extend the V -module structure one-to-one corresponds to the set of equivalence classes of symmetric C× -valued 2-cocycles of G. Similar to Proposition 3.22 we have (cf. [DLM2], Lemma 4.2): Proposition 3.24. Let W be an irreducible V -module. Then W [L] is irreducible. The following theorem gives the complete reducibility for every V [L]-module under certain conditions: Theorem 3.25. Assume that there is a sublattice L1 of L such that V [L1 ] is regular and that every irreducible V [L1 ]-module is a direct sum of irreducible V -modules. Let σ be an automorphism of V [L] such that σ fixes V [L1 ] point-wise. Then any σ -twisted weak V [L]-module is a direct sum of irreducible σ -twisted V [L]-modules of type W [L] with σ = σW . In particular, V [L] is regular. Proof. Let M be a σ -twisted weak V [L]-module. Since σ fixes V [L1 ] point-wise, M is a weak V [L1 ]-module. With V [L1 ] being regular, M is a direct sum of irreducible (ordinary) V [L1 ]-modules. With the assumption that each irreducible V [L1 ]-module is a direct sum of irreducible V -modules, M is a direct sum of irreducible V -modules. Let W be an irreducible V -submodule of M. By Proposition 3.14, σ = σW and there exists a V [L]-homomorphism φ[L] from W [L] to M extending the embedding φ of W into M. Since W [L] is V [L]-irreducible (Proposition 3.24), the V [L]-submodule of M generated by W , which is the image of φ[L], is an irreducible σ -twisted V [L]-module. Now, it follows that M is a direct sum of irreducible σ -twisted V [L]-modules of type W [L]. Let W1 , W2 , W3 be irreducible V -modules such that σWi = 1, or equivalently, the (α) extensions W1 [L], W2 [L] and W3 [L] are V [L]-modules. For each α ∈ L, with W3 being a V -submodule of W3 [L], by Proposition 3.14, there is a V [L]-homomorphism (α) gα from W3 [L] to W3 [L]. Then with Proposition 3.17 we obtain a natural linear map W

(α)

W3 [L] defined by fα from IW13W2 to IW 1 [L]W2 [L]

fα (I ) = gα ◦ I [L].

(3.69)

The next result gives a precise connection between the fusion rules for V -modules and the fusion rules for V [L]-modules: Theorem 3.26. Let W1 , W2 , W3 be irreducible V -modules such that σWi = 1, which implies that W1 [L], W2 [L], W3 [L] are irreducible V [L]-modules. In addition, we assume (α) W that V is quasi-rational. Then f = α∈L fα is a linear isomorphism from α∈L IW13W2 W3 [L] to IW . In particular, 1 [L]W2 [L]

W3 [L] NW = 1 [L]W2 [L]

α∈L

W

(α)

NW13W2 .

(3.70)

Certain Extensions of Vertex Operator Algebras of Affine Type (α)

679

(β)

Proof. Since W3 W3 only if α = β, clearly, fα is one-to-one. On the other hand, W3 [L] if Y is an intertwining operator of type W1 [L]W , then by restricting Y to W1 × W2 2 [L] W (α) we have an intertwining operator I of type W 3W for a unique α ∈ L. It is clear that 1 2 Y = I [L]. This completes the proof. We now describe the Verlinde algebra of V [L] in terms of the Verlinde algebra of V explicitly. Let A(V ) be the Verlinde algebra of V . The Verlinde algebra A(V [L]) with a basis [W [L]] for [W ] ∈ A(V ) with σW = 1. First, we have: Lemma 3.27. All [W ] with σW = 1 linearly span a subalgebra A(V , L) of A(V ). Proof. Suppose that [W1 ], [W2 ] ∈ Irr(V ) with σW1 = σW2 = 1. Let h1 ∈ P (W1 ), h2 ∈ P (W2 ). Then σW1 = σW2 = 1 amounts to h1 , h2 ∈ Lo . Let Y be a nonzero intertwining 3 operator of type WW for some irreducible V -module W3 . Then h1 + h2 ∈ P (W3 ) 1 W2 (0,hi )

because for h ∈ H, w(i) ∈ Wi

,

h(0)Y(w(1) , z)w(2) = Y(h(0)w(1) , z)w(2) + Y(w(1) , z)h(0)w(2) = B(h, h1 + h2 )Y(w(1) , z)w(2) . Since h1 + h2 ∈ Lo , σW3 = 1. Then [W3 ] ∈ A(V , L). The proof is complete.

(3.71)

Define a subspace R of A(V , L) linearly spanned by [W ] − [W (α) ]

for α ∈ L.

(3.72)

Then R is a two-sided ideal of A(V , L). Indeed, let W1 and W2 be irreducible V -modules with σWi = 1 for i = 1, 2. For α ∈ L, by Proposition 2.10, W

(α)

W2 IW IW 2W (α) . 1W

(3.73)

1

Thus [W1 ] · ([W ] − [W (α) ]) =

[W2 ]∈I rr(V )

(α)

W2 NW ([W2 ] − [W2 ]) ∈ R. 1W

(3.74)

Since A(V ) is a commutative algebra, R is a two-sided ideal. Furthermore, by Proposition 3.15, [W1 ] = [W2 ] in the quotient algebra A(V , L)/R if and only if [W1 [L]] = [W2 [L]] in A(V [L]). Then in view of Theorem 3.26 we immediately have: Proposition 3.28. The subspace R is a two-sided ideal of A[V , L] and the Verlinde algebra A(V [L]) is canonically isomorphic to the quotient algebra of A(V , L) modulo R. 4. Extended Vertex Operator (Super)Algebras of Affine Types In this section we shall specialize the vertex (super)algebra V [L] constructed in Sect. 3 from a pair (V , L) to obtain extensions of vertex operator algebras associated with affine Lie algebras gˆ . In the case g = sl(2), we shall obtain Feigin–Miwa’s extended vertex operator (super)algebras Ak [FM]. To apply the results of Sect. 3 we need to define V and L explicitly and check the necessary conditions. For each type, V will be the tensor product vertex operator algebra Lg (k, 0) ⊗ Mh (1, 0), where h is a 1 or 2-dimensional vector space equipped with a nondegenerate symmetric bilinear form. When defining h , we follow two basic principles:

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H. Li

(1) To include all the L(k, 0)-simple currents in the construction of V [L], (2) To make dim h , or equivalently, to make the rank of V [L] as small as possible. After h is chosen, we still have plenty of choices for the bilinear form on h . Another principle to follow is to make V [L] as large as possible. We shall define h and L type by type. 4.1. A complete reducibility theorem for a certain family of V [L]. In Sect. 3, for a general pair (V , L), under a certain assumption we proved a complete reducibility theorem (Theorem 3.25) for the extended algebra V [L]. In this section, we shall consider a family of V [L] such that the assumption of Theorem 3.25 holds. All the extended algebras we shall construct later belong to this family, so the complete reducibility theorem holds for all of them. Let g = g1 ⊕ · · · ⊕ gr be a semisimple Lie algebra with a Cartan subalgebra h = h1 + · · · + hr ,

(4.1)

where gi are simple factors with Cartan subalgebras hi , equipped with the normalized ∨ ∨ ∨ ∨ Killing forms. Let Q∨ = Q∨ 1 + · · · + Qr and P = P1 + · · · + Pr be the coroot ∨ are the coroot lattice and P lattice and coweight lattice of g, respectively, where Q∨ i i and coweight lattice of gi . Let k = (k1 , . . . , kr ) be an r-tuple of nonnegative integers. Set Lg (k, 0) = Lg1 (k1 , 0) ⊗ · · · ⊗ Lgr (kr , 0),

(4.2)

equipped with the standard tensor product vertex operator algebra structure. Set Pk = {(λ1 , . . . , λr ) | λi ∈ Pki (gi )},

(4.3)

where Pki (gi ) stands for Pki for the Lie algebra gi . Then Lg (k, 0) is regular [DLM1], i.e., every weak module is a direct sum of irreducible (ordinary) modules Lg (k, λ), where Lg (k, λ) = Lg1 (k1 , λ1 ) ⊗ · · · ⊗ Lgr (kr , λr )

(4.4)

for λ = (λ1 , . . . , λr ) ∈ Pk . Let h be a finite-dimensional vector space equipped with a nondegenerate symmetric bilinear form · , ·. Associated to h is the vertex operator algebra Mh (1, 0). Set V = Lg (k, 0) ⊗ Mh (1, 0),

(4.5)

equipped with the standard tensor product vertex operator algebra structure. The algebra V can be considered as the vertex operator algebra associated to the affine algebra of the reductive Lie algebra g + h . Set H = h + h ⊂ g + h = V(1) .

(4.6)

Then H is a Cartan subalgebra of the reductive Lie algebra g + h . Clearly, (3.1) holds and B(h1 , h2 ) = δi,j ki h1 , h2

for h1 ∈ hi , h2 ∈ hj

(4.7)


681

because h1 (1)h2 = h1 (1)h2 (−1)1 = δi,j ki h1 , h2 1 (and h1 (1)h2 = B(h1 , h2 )1). We also have B(h1 , h2 ) = h1 , h2

for h1 , h2 ∈ h .

(4.8)

h = h + h ,

(4.9)

For h ∈ H , we write h = h1 + · · · + hr + h ,

where hi ∈ hi , h ∈ h , h ∈ h. For λ = (λ1 , . . . , λr ), γ ∈ h with λi ∈ Pki (gi ), we set W (λ, γ ) = Lg1 (k1 , λ1 ) ⊗ · · · ⊗ Lgr (kr , λr ) ⊗ Mh (1, γ ).

(4.10)

Then from [FHL] all such W (λ, γ ) form a complete set of representatives of equivalence classes of irreducible V -modules. Let L be a subgroup of H such that (L, B) is an integral lattice of finite rank. To have a vertex (super)algebra V [L], we shall also need the condition that α(0) has only integral eigenvalues on V for every α ∈ L. For α ∈ L, since α (0) acts as zero on V , α(0) has only integral eigenvalues on V if and only if α (0) has only integral eigenvalues on Lg (k, 0). Then we immediately have: Lemma 4.1. For α ∈ L, α(0) has only integral eigenvalues on V if and only if α ∈ P ∨ , the coweight lattice of g. Set L = {α | α ∈ L},

L = {α | α ∈ L}.

(4.11)

Lemma 4.2. Assume that the projection of L onto L is one-to-one. Then for λ ∈ Pk , γ ∈ h and for α, β ∈ L, W (λ, γ )(α) W (λ, γ )(β) if and only if α = β. Proof. Because (α, z) = (α , z)(α , z) and (α , z) = 1 on Mh (1, γ ) and (α , z) = 1 on Lg (k, λ) for α ∈ L, we have

W (λ, γ )(α) = Lg (k, λ)(α ) ⊗ Mh (1, γ )(α ) Lg (k, λ)(α ) ⊗ Mh (1, γ + α ). (4.12) We know Mh (1, γ + α ) Mh (1, γ + β ) if and only if α = β . Then it follows immediately. Now we have: Proposition 4.3. Let V , H be defined as in (4.5) and (4.6) and let L be a subgroup of H such that (L, B) is an integral lattice of finite rank. Assume that L ⊂ P ∨ , the projection of L onto L is one-to-one and L is a positive definite lattice. Then V [L] equipped with the vertex operator map Y defined in (3.13) is a simple vertex operator (super)algebra.

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H. Li

Proof. Since L ⊂ P ∨ , by Lemma 4.1, for α ∈ L, α(0) acting on V has only integral eigenvalues. With Lemma 4.2, in view of Corollary 3.6 and Proposition 3.22, V [L] is a simple vertex (super)algebra with all L(0)-weights being half integers. Now we only need to verify the two grading restrictions [FLM]. For α ∈ L, we have

V (α) = Lg (k, 0)(α ) ⊗ Mh (1, 0)(α ) = Lg (k, 0)(α ) ⊗ Mh (1, α ).

(4.13)

With Lg (k, 0) being regular, Lg (k, 0)(α ) is an irreducible Lg (k, 0)-module, which is unitary. Then the L(0)-weights of Lg (k, 0)(α ) are nonnegative. On the other hand, the lowest weight of Mh (1, α ) is 21 α , α . Thus each V (α) satisfies the two grading restrictions with the lowest weight at least 21 α , α . Since the projection of L onto L (α) is one-to-one and L is positive definite, for every n ∈ 21 Z, V(n) = 0 only for finitely many α. Then the two grading restrictions follow immediately. Furthermore, we have: Theorem 4.4. Let V , L be as in Proposition 4.3 with all the assumptions. In addition we assume that dim h = rank L . Let σ be an automorphism of V [L] of finite order which fixes V point-wise. Then every σ -twisted weak V [L]-module is a direct sum of irreducible (ordinary) σ -twisted V [L]-modules isomorphic to W (λ, γ )[L] with σ = σW (λ,γ ) . In particular, every weak V [L]-module is a direct sum of irreducible (ordinary) V [L]modules isomorphic to W (λ, γ )[L] for λ ∈ Pk , γ ∈ (L )o (the dual of L ) with λ1 (α 1 ) + · · · + λr (α r ) + γ , α ∈ Z

for α ∈ L.

(4.14)

Proof. Denote by o(σ ) the order of σ . In view of Lemma 4.2, V (α) for α ∈ L are nonisomorphic irreducible V -modules. Then σ (V (α) ) = V (α) for α ∈ L and σ acts on V (α) as a scalar, which is an o(σ )th root of unity. Therefore, σ acts trivially on V (mo(σ )α) for α ∈ L, m ∈ Z. Because Lg (k, 0) has only finitely many irreducible modules up to equivalence, there exists a positive integer d1 such that as Lg (k, 0)-modules, Lg (k, 0)(d1 α

)

Lg (k, 0)

for all α ∈ L.

(4.15)

Let d2 be another positive integer such that d2 L is an even lattice. Set d = o(σ )d1 d2 and L1 = dL. Then V [L1 ] = ⊕α∈L V (dα) .

(4.16)

Then V [L1 ] is a simple vertex operator subalgebra of V [L] and σ fixes V [L1 ] point-wise. Furthermore, as a V -module,

V (dα) Lg (k, 0)(dα ) ⊗ Mh (1, dα ) Lg (k, 0) ⊗ Mh (1, dα )

(4.17)

for α ∈ L, hence V [L1 ] Lg (k, 0) ⊗ VdL .

(4.18)

(Here we used the fact that dim h = rank L .) Note that Lg (k, 0) ⊗ VdL is a natural simple vertex operator algebra, which is regular. It follows from Proposition 3.22 that V [L1 ] is regular. Clearly, each irreducible V [L1 ]-module is a direct sum of irreducible


683

V -modules. Then it follows from Theorem 3.25 immediately that every σ -twisted weak V [L]-module is a direct sum of irreducible (ordinary) σ -twisted V [L]-modules of type W (λ, γ ) with σ = σW (λ,γ ) . From Lemma 3.11, σW (λ,γ ) = 1 if and only if for α ∈ L, α(0) has only integral eigenvalues on W (λ, γ ). With (λ1 , . . . , λr , γ ) being an H -weight of W (λ, γ ), we see that σW (λ,γ ) = 1 if and only if λ1 (α 1 ) + · · · + λr (α r ) + γ , α ∈ Z

for α ∈ L,

(4.19)

which furthermore implies that γ ∈ (L )o because λ ∈ Pk . This completes the proof. Remark 4.5. From the proof of Theorem 4.4, one can easily see that the regularity result still holds for V [L] if we replace Lg (k, 0) by any regular vertex operator algebra U . Because

W (λ, γ )(α) = L(k, λ)(α ) ⊗ Mh (1, γ + α ) and Mh (1, γ ) Mh (1, γ ) if and only if γ = γ , in view of Proposition 3.15, we see that W (λ, γ )[L] W (λ , γ )[L] if and only if there is α ∈ L such that γ = γ + α,

L(k, λ )(α

)

L(k, λ).

(4.20)

To describe explicitly the equivalence relation on the set of W (λ, γ )[L], or to get a complete set of equivalence classes of irreducible V [L]-modules, we need to know more about L(k, λ )(α ) as a gˆ -module. Of course, from Theorem 2.13,

[L(k, λ )(α ) ] = [L(k, λ )] · [L(k, 0)(α ) ].

(4.21)

Nevertheless, in view of Proposition 3.28 we immediately have: Proposition 4.6. The subspace A of V(L(k, 0)) ⊗ C[(L )o ], linearly spanned by [L(k, λ)] ⊗ eγ

(4.22)

for λ ∈ Pk , γ ∈ (L )o satisfying (4.14), is a subalgebra. Furthermore, the Verlinde algebra V(V [L]) is canonically isomorphic to the quotient algebra of A modulo the relations:

[L(k, λ)] ⊗ eγ − [L(k, λ)(α ) ] ⊗ eγ +α for α ∈ L.

(4.23)

684

H. Li

4.2. Extended vertex operator (super)algebras Ak of type sl(n + 1). Starting from this subsection we shall work on the setting of Sect. 4.1 and we shall consider a simple Lie algebra g, i.e., r = 1. For g of each type, we take V = L(k, 0) ⊗ Mh (1, 0)

(4.24)

and we define Ak (g) to be the extended algebra V [L] for a certain L. We shall case by case define the pair (h , · , ·) and the lattice L, and then verify that (L, B) is an integral lattice, L ⊂ P ∨ , L is positive-definite and the projection of L onto L is one-to-one, so that Proposition 4.3 and Theorem 4.4 hold. In this subsection we shall consider the case g = sl(n + 1). For a fixed positive integer k, L(k, 0) has n + 1 simple currents L(k, kλi ) for i = 0, . . . , n. By Corollary 2.27 the equivalence classes of the (n + 1) simple currents form a cyclic group of order (n + 1) with [L(k, kλ1 )] as a generator. Recall that h(i) ∈ h with αj (h(i) ) = δi,1 for i, j = 1, . . . , n. From [H] (Table 1 on p. 69) and Lemma 2.28, we have 1 (n + 1 − i)α1∨ + 2(n + 1 − i)α2∨ + · · · n+1 ∨ + (i − 1)(n + 1 − i)αi−1 1 ∨ + + · · · + iαn∨ , i(n + 1 − i)αi∨ + i(n − i)αi+1 n+1 i(n + 1 − i) h(i) , h(i) = . n+1 h(i) =

(4.25) (4.26)

Define h = Cα to be a one-dimensional vector space equipped with a symmetric bilinear form · , · such that k . n+1

(4.27)

where α = h(1) + α .

(4.28)

α , α = Set L = Zα, Because

B(α, α) = B(h(1) , h(1) ) + B(α , α ) =

kn k + = k ∈ Z, n+1 n+1

(4.29)

(L, B) is an integral lattice. Clearly, L = Zh(1) ⊂ P ∨ , L is positive-definite and the projection of L onto L is one-to-one. By Proposition 4.3, we have a simple vertex operator (super)algebra V [L]. Definition 4.7. We define Ak (sl(n + 1)) to be the simple vertex operator (super)algebra V [L] with V and L being defined in (4.24) and (4.28). Remark 4.8. There are many ways to define α , α such that V [L] is a vertex operator nk , where nk superalgebra. For examples, one may define h with α , α = 1 − n+1 is the least nonnegative residue of nk modulo n + 1. One may also define h with k α , α = s + n+1 , where s is any nonnegative integer.


685

For λ ∈ Pk , γ ∈ C, set

γ W (λ, γ ) = L(k, λ) ⊗ Mh 1, α . k

(4.30)

Since L = Z(h(1) + α ), (4.14) amounts to λ(h(1) ) +

γ ∈ Z, n+1

(4.31)

which from (4.25) is equivalent to nλ(α1∨ ) + (n − 1)λ(α2∨ ) + · · · + λ(αn∨ ) + γ ∈ (n + 1)Z.

(4.32)

Note that (4.32) implies γ ∈ Z because λ ∈ Pk . We also see that in general, σW (λ,γ ) is of finite order if and only if γ ∈ Q. Then Propositions 3.13 and 3.24 immediately give: Proposition 4.9. For λ ∈ Pk , γ ∈ Q, σW (λ,γ ) is of finite order and W (λ, r)[L] is an irreducible σW (λ,γ ) -twisted Ak (sl(n + 1))-module. In particular, if (4.32) holds, W (λ, γ )[L] is an irreducible V [L]-module. From Theorem 4.4 we also have: Proposition 4.10. Let σ be an automorphism of Ak (sl(n + 1)) of finite order which fixes V = L(k, 0)⊗Mh (1, 0) point-wise. Then every σ -twisted weak Ak (sl(n+1))-module is a direct sum of irreducible (ordinary) σ -twisted Ak (sl(n + 1))-modules W (λ, γ )[L] for some λ ∈ Pk , γ ∈ Q with σ = σW (λ,γ ) . In particular, every weak Ak (sl(n+1))-module is a direct sum of W (λ, γ )[L] for some λ ∈ Pk , γ ∈ Z that satisfy (4.32). Let us consider the case n = 1. Then we can make our results more explicit. We have Pk = {0, 1, . . . , k} and h(1) = 21 α1∨ . From Corollary 2.27 we have (1) )

L(k, 0)(2mh

(1) )

L(k, 0)((2m+1)h

L(k, 0),

L(k, k)

(4.33)

for m ∈ Z. Since (1) )

V (mα) = L(k, 0)(mh

(1) )

⊗ Mh (1, 0)(mα ) = L(k, 0)(mh

⊗ Mh (1, mα ),

(4.34)

it follows from Proposition 3.15 that W (i, γ )[L] W (i , γ )[L] if and only if there exists m ∈ Z such that (1) )

L(k, i ) L(k, i)(mh

,

γ γ = + m. k k

(4.35)

Recall that [L(k, k)]·[L(k, i)] = [L(k, k −i)]. If m is even, we have i = i and γ −γ ∈ 2kZ. If m is odd, we have i = k − i and γ = γ + mk. Then W (i, γ )[L] = W (i , γ )[L] if and only if either i = i and γ ≡ γ mod 2k, or i = k − i and γ ≡ γ + k mod 2k. Then Propositions 4.9 and 4.10 give (cf. [FM], Proposition 3): Corollary 4.11. Every weak Ak (sl(2))-module is completely reducible and W (i, j )[L]

for 0 ≤ i ≤ k, 0 ≤ j ≤ k − 1

with i + j ∈ 2Z

(4.36)

form a complete set of representatives of equivalence classes of irreducible Ak (sl(2))modules.

686

H. Li

Note that W (i, j )[L] was denoted by R(i, j ) in [FM]. Using the fusion rules for L(k, 0) we have the following relations in the Verlinde algebra A(V ): [W (i1 , j1 )] · [W (i2 , j2 )] =

min(i1 +i 2 ,2k−i1 −i2 )

[W (i, j1 + j2 )].

(4.37)

[W (i, j1 + j2 )[L]].

(4.38)

i=max(i1 −i2 ,i2 −i1 )

Then in the Verlinde algebra of Ak , we have [W (i1 , j1 )[L]] · [W (i2 , j2 )[L]] =

min(i1 +i 2 ,2k−i1 −i2 ) i=max(i1 −i2 ,i2 −i1 )

Note that when j1 + j2 ≥ k, we have W (i, j1 + j2 )[L] = W (k − i, j1 + j2 − k)[L]. Remark 4.12. Set L = Zα . Then, as a V -module, Ak L(k, 0) ⊗ V2L + L(k, k) ⊗ V2L +α .

(4.39)

Furthermore, using more general fusion rules we get W (i, γ )[L] L(k, i) ⊗ V2L + γ α + L(k, k − i) ⊗ V2L + γ +k α k

k

(4.40)

for i = 0, . . . , k; γ ∈ Q. With this, one can easily write down the characters of W (i, j )[L] in terms of the characters of L(k, j ) and the theta functions of 2L . Remark 4.13. For g = sl(n + 1), from Corollary 2.27, we have (1) )

L(k, 0)(mh

L(k, kλm¯ )

for m ∈ Z.

(4.41)

Then Ak (sl(n + 1))

n

L(k, kλi ) ⊗ V2(n+1)L +iα

(4.42)

i=0

as a V -module. More general fusion rules are needed to express W (λ, j )[L] explicitly. 4.3. Generating property for the extended algebras Ak (sl(n+1)). First, we review some properties for a general vertex operator superalgebra U . Recall Borcherds’ commutator formula [B]: m [um , vn ]± = (ui v)m+n−i (4.43) i i≥0

for u, v ∈ U and m, n ∈ Z, where [· , ·]± refers to the super commutator. Thus, the super commutator [Y (u, z1 ), Y (v, z2 )]± is uniquely determined by ui v for i ≥ 0. From this we have (z1 − z2 )r [Y (u, z1 ), Y (v, z2 )]± = 0

(4.44)


687

if r is a nonnegative integer such that ui v = 0 for i ≥ r. For homogeneous vectors u, v ∈ U and for m ∈ Z, we have (cf. [FLM]) wt (um v) = wt u + wt v − m − 1,

(4.45)

where wt u stands for the L(0)-weight of u. Let U = n∈ 1 Z U(n) be such that U(0) = C (= C1) and U(n) = 0 for n < 0. Then 2

[um , vn ]+ = (u0 v)m+n−1 = δm+n,1 u0 v

(4.46)

for u, v ∈ U( 1 ) , m, n ∈ Z, where u0 v ∈ U(0) = C. That is, the component operators 2 um for u ∈ U( 1 ) , m ∈ Z give rise to a Clifford algebra. 2 It is well known ([B, FLM]) that the weight-one subspace U(1) is a Lie algebra with [u, v] = u0 v and with a symmetric invariant bilinear form (u, v) = u1 v ∈ C. We have [um , vn ] = (u0 v)m+n + mδm+n,0 (u, v)

(4.47)

for u, v ∈ U(1) , m, n ∈ Z. Then operators um for u ∈ U(1) , m ∈ Z give rise to a natural representation of affine Lie algebra Uˆ(1) . Now we consider Ak (sl(n + 1)), which is a vertex operator algebra when k is even and which is a vertex operator superalgebra when k is odd. It is easy to see that vertex (β) operator (super)algebra Ak (sl(n + 1)) is generated by V (α) and V (−α) . Denote by Vlow the lowest L(0)-weight subspace of V (β) for β ∈ L. Because V as a vertex operator algebra is generated by g + Cα and both V (α) and V (−α) are irreducible V -modules, Ak (sl(n + 1)) is furthermore generated by (α)

(−α)

S := (g + Cα ) + Vlow + Vlow .

(4.48)

Since (Corollary 2.27) V (α) = L(k, kλ1 ) ⊗ Mh (1, α ), V (−α) = L(k, kλn ) ⊗ Mh (1, −α ), we have

(α)

Vlow = L(kλ1 ) ⊗ eα ,

(−α)

Vlow

= L(kλn ) ⊗ e−α .

(4.49)

From Remark 2.23 and (2.7), we find that the lowest L(0)-weights of V -modules V (α) and V (−α) are 21 B(α, α) = 2k . Now we are ready to prove our main result of this subsection. Proposition 4.14. The algebra Ak (sl(n + 1)) is generated by the subspace (α)

(−α)

(g + Cα ) + Vlow + Vlow , (α)

(−α)

where Vlow = L(kλ1 ) ⊗ eα and Vlow the following relations hold:

= L(kλn ) ⊗ e−α are of weight 2k . Furthermore,

Y (u, z1 )Y (v, z2 ) = (−1)k Y (v, z2 )Y (u, z1 ),

(4.50)

Y (u , z1 )Y (v , z2 ) = (−1)k Y (v , z2 )Y (u , z1 )

(4.51)

688

H. Li (α)

(−α)

for u, v ∈ Vlow , u , v ∈ Vlow and us v ∈ V(k−s−1) ,

(4.52)

k

(z1 − z2 ) [Y (u, z1 ), Y (v , z2 )]± = 0 (α)

(4.53)

(−α)

for u ∈ Vlow , v ∈ Vlow , s ∈ Z. Proof. First we calculate the lowest L(0)-weight of V (mα) . From Theorem 2.26 and Corollary 2.27 we have (1) )

V (mα) = L(k, 0)(mh

⊗ Mh (1, mα ) = L(k, kλm¯ ) ⊗ Mh (1, mα ),

where m ¯ is the least nonnegative residue of m modulo n + 1. From Remark 2.23, we see ¯ m)k ¯ (mα) is that the lowest weight of L(k, kλm ) is m(n+1− 2(n+1) . Then the lowest weight of V m(n ¯ + 1 − m)k ¯ m2 k mk ¯ (m2 − m ¯ 2 )k + = + , 2(n + 1) 2(n + 1) 2 2(n + 1) which is at least k if |m| ≥ 2. (α) Let u, v ∈ Vlow . Thus wt u = wt v = 2k . Then for i ≥ 0, ui v ∈ V (2α) and wt (ui v) = k − i − 1 < k. Since the lowest weight of V (2α) is at least k, we obtain ui v = 0

for i ≥ 0.

(4.54)

Then (4.50) follows immediately from (4.43). Similarly, (4.51) holds. Equation (4.52) directly follows from the definition of the vertex operator map and the weight formula (4.45). Since us v ∈ V (0) = V for s ∈ Z and the lowest weight of V is zero, we have ui v = 0

for i ≥ k.

Then (4.53) follows immediately from (4.44).

(4.55)

Remark 4.15. In the case k = 1, L(λ1 ) is the vector representation of sl(n + 1) on Cn+1 . In this case, the algebra A1 (sl(n + 1)) is generated by L(λ1 ) ⊗ eα + L(λ1 )∗ ⊗ e−α , which generates a Clifford algebra. The algebra A1 (sl(n + 1)) is exactly the spinor representation of Dn+1 [FFR], which is isomorphic to L(1, 0) + L(1, λ1 ) as a Dˆ n+1 module. Remark 4.16. When k = 2,

L(2λ1 )∗ ⊗ e−α + (g + Cα ) + L(2λ1 ) ⊗ eα

is exactly the weight-one subspace of A2 (sl(n + 1)) and it has a natural Lie algebra structure with the obvious Z-grading. Using the fact that L(2λ1 )∗ ⊗e−α and L(2λ1 )⊗eα are non-isomorphic irreducible (g+Cα )-modules one easily shows that this Lie algebra is simple and of rank n+1. Consider the standard Dynkin diagram embedding of sl(n+1) into Cn+1 . Then we see Cn+1 = sl(n + 1) + L(2λ1 ) + L(2λn ). Thus the weight one subspace of A2 (sl(n + 1)) as a Lie algebra is isomorphic to Cn+1 . (For n = 1, this was pointed out in [FM].) Then A2 (sl(n + 1)) is a vertex operator algebra associated to the affine Lie algebra Cˆ n+1 .


689

Remark 4.17. For k ≥ 3, since wt (u0 v ) = k − 1 ≥ 2, [Y (u, z1 ), Y (v , z2 )]± involves nonlinear normal ordered products of vertex operators (or fields) Y (a, z) for a ∈ g+Cα . This type of algebras are commonly referred to by physicists as nonlinear W -algebras. Remark 4.18. The following consideration was motivated by [GH1-2] and [Gun]. In the k construction of Ak , let us define h = Cα with α , α = 1 + n+1 and keep the rest unchanged. Then B(α, α) = 1 + k. With (L, B) being an integral lattice, V [L] is a vertex operator (super)algebra (cf. Remark 4.8). Furthermore, V [L] is generated by the subspace (−α)

(α)

Vlow + (g + Cα ) + Vlow = L(kλ1 )∗ ⊗ e−α + (g + Cα ) + L(kλ1 ) ⊗ eα ,

where L(kλ1 ) ⊗ eα and L(kλ1 )∗ ⊗ e−α are of weight k+1 2 . In particular, when k = 2, α V [L] is a vertex operator superalgebra and L(2λ1 )⊗e and L(2λ1 )∗ ⊗e−α are of weight 3 2 . In view of this and Remark 4.16, we may view V [L] with k = 2 as a superization of the vertex operator algebra V [L] with k = 2 defined in Remark 4.16. In [Gun], an N = 2 vertex operator superalgebra was constructed from a simple Lie algebra g equipped with a Z-grading such that gm = 0 for |m| > 1. From Remark 4.16, the symplectic Lie algebra C = Cn+1 is naturally Z-graded with only three homogeneous subspaces being nonzero. A further study on the connection between V [L] and the N = 2 vertex operator superalgebra constructed in [Gun] will be conducted in a future paper. 4.4. Extended algebras Ak of type Dn . We consider g of Dn type for n ≥ 3. From Corollary 2.27, the equivalence classes of simple currents L(k, kλ1 ), L(k, kλn−1 ), L(k, kλn ) and L(k, 0) form a group which is cyclic for an odd n and which is isomorphic to Z/2Z × Z/2Z for an even n. We shall define the extended algebra separately for the two cases. From [H] and Lemma 2.28 we have 1 ∨ ∨ + (α + 2α2∨ + · · · + (n − 2)αn−2 2 1 1 ∨ = (α1∨ + 2α2∨ + · · · + (n − 2)αn−2 + 2

h(n−1) = h(n)

1 ∨ 1 + (n − 2)αn∨ ), nα 2 n−1 2 1 1 ∨ + nαn∨ ) (n − 2)αn−1 2 2

(4.56) (4.57)

and h(n−1) , h(n−1) = h(n) , h(n) =

n . 4

(4.58)

∨ − 1 α ∨ we get Using the relation h(n−1) = h(n) + 21 αn−1 2 n

h(n−1) , h(n) =

n−2 . 4

Case I: n is odd. Define h = Cα with α , α = L = Zα,

3nk 4 .

(4.59) Set

where α = h(n) + α .

(4.60)

Then L = Zα ,

L = Zh(n) .

(4.61)

690

H. Li

Since B(α, α) = kh(n) , h(n) + α , α = nk,

(4.62)

(L, B) is a positive definite integral lattice. By Proposition 4.3 (with the other assumptions being obvious) V [L] is a simple vertex operator (super)algebra. We define Ak (g) to be V [L]. Then we have the following results with the same proof as that of Propositions 4.9 and 4.10: Proposition 4.19. For g of type Dn with an odd n, the extended algebra Ak (g) is regular. Furthermore, for λ ∈ Pk , j ∈ Q, set

j α . W (λ, j ) = L(k, λ) ⊗ Mh 1, 3nk Then any irreducible Ak (g)-module is isomorphic to W (λ, j )[L] for some λ ∈ Pk , j ∈ Z with ∨ 2λ(α1∨ ) + 4λ(α2∨ ) + · · · + 2(n − 2)λ(αn−2 )

(4.63)

∨ + (n − 2)λ(αn−1 ) + nλ(αn∨ ) + j ∈ 4Z.

Furthermore, Ak (g) is generated by (α)

(−α)


(−α)

where Vlow = L(kλn ) ⊗ eα and Vlow = L(kλn−1 ) ⊗ e−α are of weight relations (4.50)–(4.53) with k being replaced by nk hold.

nk 2 ,

and the

Case II: n is even. Define h = Cα1 + Cα2 to be a two-dimensional vector space with a symmetric bilinear form · , · such that α1 , α1 = α1 , α1 =

3 nk, 4

(4.64)

1 k(n − 2). 4

(4.65)

L = Zα1 + Zα2 ,

(4.66)

α1 , α2 = Define

where α1 = h(n−1) + α1 ,

α2 = h(n) + α2 .

(4.67)

L = Zh(n−1) + Zh(n) .

(4.68)

Then L = Zα1 + Zα2 , We have B(α1 , α1 ) = B(α2 , α2 ) = kn, 1 1 1 B(α1 , α2 ) = k(n − 2) + k(n − 2) = k(n − 2). 4 4 2

(4.69) (4.70)


691

Since n is even, (L, B) is a positive-definite even lattice. Clearly, L = Zh(n−1) + Zh(n) ⊂ P ∨ , L is positive-definite, and the projection of L onto L is one-to-one. By Proposition 4.3, V [L] is a simple vertex operator algebra. Now we define Ak (g) = V [L], as a simple vertex operator algebra. We just mention that this is a regular vertex operator algebra and a set of generators and relations can be worked out similarly but with some extra work. Remark 4.20. Note that L(k, kλ1 ) is a simple current of order 2 and we have h(1) , h(1) = 1. Let V = L(k, 0) and L = Zh(1) . Then in view of Corollary 3.21, L(k, 0) + L(k, kλ1 ) has a natural simple vertex operator superalgebra structure (cf. Remark 4.15). 4.5. Extended vertex operator (super)algebras Ak (E6 ). Let g be of type E6 . From Sect. 2.2, for any positive integer k, L(k, kλ1 ) and L(k, kλ5 ) are (the only) nontrivial simple currents for L(k, 0). From [H] and Lemma 2.28, we have 1 ∨ 4α1 + 3α6∨ + 5α2∨ + 6α3∨ + 4α4∨ + 2α5∨ , 3 1 ∨ 2α1 + 3α6∨ + 4α2∨ + 6α3∨ + 5α4∨ + 4α5∨ = 3

h(1) =

(4.71)

h(5)

(4.72)

and h(1) , h(1) = h(5) , h(5) =

4 . 3

(4.73)

Define h = Cα to be a one-dimensional vector space equipped with the bilinear form · , · such that 2k . 3

(4.74)


(4.75)

α , α = Set L = Zα, Then L = Zh(1) ⊂ P ∨ ,

L = Zα .

(4.76)

Since B(α, α) =

4k 2k + = 2k, 3 3

(4.77)

(L, B) is a positive-definite even lattice. By Proposition 4.3 (with the other assumptions being obvious), we have a simple vertex operator algebra V [L]. We define Ak (g) to be the vertex operator algebra V [L]. For λ ∈ Pk , j ∈ Q, set W (λ, j ) = L(k, λ) ⊗ Mh (1, an irreducible V -module. Then we have:

1 α ), 2k

(4.78)

692

H. Li

Proposition 4.21. For g of type E6 , the extended algebra Ak (g) is regular and any irreducible module is isomorphic to W (λ, j )[L] for some λ ∈ Pk , j ∈ Z with 4λ(α1∨ ) + 3λ(α6∨ ) + 5λ(α2∨ ) + 6λ(α3∨ ) + 4λ(α4∨ ) + 2λ(α5∨ ) + j ∈ 3Z.

(4.79)

Similar to the case g = sl(n + 1), Ak (E6 ) as a vertex operator algebra is generated by (α)

(−α)

(g + Cα ) + Vlow + Vlow . We have

(α)

Vlow = L(k, kλ1 )low ⊗ eα = L(kλ1 ) ⊗ eα , (−α) Vlow

=

L(k, kλ1 )∗low

−α

⊗e

∗

= L(kλ1 ) ⊗ e

−α

(4.80) .

(4.81)

Since the L(0)-weights of L(k, kλ1 )low and L(k, kλ1 )∗low are 1 k 2k B(h(1) , h(1) ) = h(1) , h(1) = , 2 2 3 the lowest weights of V (α) and V (−α) are

2k 3

+

k 3

= k.

For m ∈ Z, the lowest L(0)-weight of Mh (1, mα ) is 21 mα , mα = km2

V (mα)

km2 3 .

Then

is at least 3 . If |m| ≥ 3, the lowest L(0)-weight of the lowest L(0)-weight of (mα) V is at least 3k. The lowest L(0)-weight of V (2α) is the sum of the lowest L(0)(1) (2h(1) ) L(k, kλ ) whose lowest weight of L(k, 0)(2h ) and 4k 5 3 . We know that L(k, 0) (2α) is 8k , which is greater than 2k. L(0)-weight is 4k . Then the lowest L(0)-weight of V 3 3 With this information, using the same proof of Proposition 4.14 we immediately have: Proposition 4.22. The extended vertex operator algebra Ak (E6 ) is generated by (α)

(−α)


(−α)

(4.82)

where Vlow = L(kλ1 ) ⊗ eα and Vlow = L(kλ1 )∗ ⊗ e−α are of weight k. Furthermore, the relations (4.50)–(4.53) with k being replaced by 2k hold. (−α)

(α)

Remark 4.23. When k = 1, Vlow +(g+Cα )+Vlow is exactly the weight-one subspace of A1 (g), which is a natural Lie algebra with

[L(λ1 ) ⊗ eα , L(λ1 ) ⊗ eα ] = 0, α

∗

[L(λ1 ) ⊗ e , L(λ1 ) ⊗ e

−α

[L(λ1 )∗ ⊗ e−α , L(λ1 )∗ ⊗ e−α ] = 0,

] ⊂ g + Cα .

(4.83) (4.84)

These relations give rise to a Z-grading for the Lie algebra. One can easily show that this Lie algebra is simple and of rank 7. Using the standard Dynkin diagram embedding of E6 into E7 we can show that it is really E7 .


693

4.6. Extended vertex operator (super)algebras Ak (E7 ). Let g be of type E7 . From Sect. 2.2, for any positive integer k, L(k, kλ6 ) is a (and the only nontrivial) simple current for L(k, 0). Using [H] (Table 1 on p. 69) and Lemma 2.28 we have h(6) =

1 ∨ 2α1 + 3α7∨ + 4α2∨ + 6α3∨ + 5α4∨ + 4α5∨ + 3α6∨ , 2

3 . 2 (4.85)

h(6) , h(6) =

Define h = Cα to be a one-dimensional vector space equipped with a bilinear form · , · such that k . 2

(4.86)


(4.87)

α , α = Set L = Zα,

k Then L = Zα and L = Zh(6) . Since B(α, α) = 3k 2 + 2 = 2k, (L, B) is a positivedefinite even lattice. By Proposition 4.3 (with the other assumptions being obvious), V [L] is a simple vertex operator algebra. We define Ak (E7 ) to be the simple vertex operator algebra V [L]. For λ ∈ Pk , j ∈ Q, we set

j W (λ, j ) = L(k, λ) ⊗ Mh (1, α ). k In view of Theorem 4.4 we immediately have:

(4.88)

Proposition 4.24. For g of type E7 , the extended algebra Ak (g) is regular and any irreducible module is isomorphic to W (λ, j )[L] for λ ∈ Pk , j ∈ Z with 2λ(α1∨ ) + 3λ(α7∨ ) + 4λ(α2∨ ) + 6λ(α3∨ )

(4.89)

+ 5λ(α4∨ ) + 4λ(α5∨ ) + 3λ(α6∨ ) + j ∈ 2Z.

(6)

The lowest weights of V (α) and V (−α) are 21 B(α, α) = k. Since [L(k, 0)(2h ) ] = [L(k, 0)], the lowest weights of V (2α) and V (−2α) are 21 B(2α , 2α ) = k also. For

|m| ≥ 3, the lowest weight of V (mα) is at least 21 B(mα , mα ) = than 2k. With this information we immediately have:

m2 k 4 ,

which is greater

Proposition 4.25. The vertex operator algebra Ak (E7 ) is generated by (−2α)

Vlow

(−α)

(α)

(2α)

+ Vlow + (g + Cα ) + Vlow + Vlow ,

where

(α)

(−α)

Vlow = L(kλ6 ) ⊗ eα , Vlow (2α) Vlow

=C⊗e

2α

,

(−2α) Vlow

= L(kλ6 ) ⊗ e−α , =C⊗e

(4.90)

−2α

(4.91)

are of weight k. Remark 4.26. When k = 1,

C ⊗ e−2α + L(λ6 ) ⊗ e−α + (g + Cα ) + L(λ6 ) ⊗ eα + C ⊗ e2α

is exactly the weight-one subspace of A1 (g). It is a Z-graded Lie algebra with the obvious grading. Similarly, we can show that it is E8 .

694

H. Li

4.7. Extended algebras of types Bn and Cn . For g of type Bn , L(k, kλ1 ) is the only nontrivial simple current and for g of type Cn , L(k, kλn ) is the only nontrivial simple current. For Bn , from [H] and Lemma 2.28 we have 1 ∨ + αn∨ , h(1) = α1∨ + · · · + an−1 2

h(1) , h(1) = 1

(4.92)

and for Cn we have h(n) =

1 ∨ (α + 2α2∨ + · · · + nαn∨ ), 2 1

h(n) , h(n) =

n . 2

(4.93)

Remark 4.27. We here correct an error of [DLM2] (Examples 5.12 and 5.13) where it was stated that L(k, kλn ) was the nontrivial simple current for g of type Bn and that L(k, kλ1 ) was the nontrivial simple current for g of type Cn . For g of type Bn , it follows from [DLM2] or Proposition 3.20 with V = L(k, 0) and L = Zh(1) that for any positive integral level k, L(k, 0) + L(k, kλ1 ) has a natural simple vertex operator (super)algebra structure. However, for Cn , L(k, 0) + L(k, kλn ) is a vertex operator (super)algebra only for a positive integral level k with nk being even. For g of type Cn , we define h = Cα with α , α = L = Zα,

nk 2 .

Set

where α = h(n) + α .

(4.94)

Then L = Zh(n) and L = Zα . Furthermore, B(α, α) = kh(n) , h(n) + α , α = nk.

(4.95)

Then (L, B) is a positive definite integral lattice. Hence V [L] is a simple vertex operator (super)algebra. Furthermore, we have: Proposition 4.28. For g of type Cn , Ak (g) is regular and any irreducible Ak (g)-module is isomorphic to W (λ, j )[L] for λ ∈ Pk , j ∈ Z with λ(α1∨ ) + 2λ(α2∨ ) + · · · + nλ(αn∨ ) + j ∈ 2Z,

(4.96)

where W (λ, j ) = L(k, λ) ⊗ Mh (1,

j α ). nk

(4.97)

Furthermore, Ak (g) is generated by (α)

(−α)

(g + Cα ) + Vlow + Vlow

(α)

and the the relations (4.50)–(4.53) with k being replaced by nk hold, where Vlow and (−α) Vlow are of weight nk 2 .


695

References [B] [D1] [D2] [DL] [DLM1] [DLM2] [DMZ] [F] [FF] [FFR] [FG] [FHL] [FLM] [FM] [FS] [FZ] [GH1] [GH2] [GW] [Gun] [Guo] [Hua1] [Hua2]

[HL1] [HL2] [HL3] [H] [K] [Le] [Li1] [Li2] [Li3]

Borcherds, R.E.: Vertex algebras, Kac–Moody algebras, and the Monster. Proc. Natl. Acad. Sci. USA 83, 3068–3071 (1986) Dong, C.: Vertex algebras associated with even lattices. J. Alg. 161, 245–265 (1993) Dong, C.: Twisted modules for vertex operator algebras associated with even lattices. J. Alg. 165, 91–112 (1993) Dong, C. and Lepowsky, J.: Generalized Vertex Algebras and Relative Vertex Operators. Progress in Math. Vol. 112, Boston: Birkhäuser, 1993 Dong, C., Li, H. and Mason, G.: Simple currents and extensions of vertex operator algebras. Commun. Math. Phys. 180, 671–707 (1996) Dong, C., Li, H. and Mason, G.: Regularity of rational vertex operator algebras. Adv. Math. 132, 148–166 (1997) Dong, C., Mason, G. and Zhu, Y.-C.: Discrete series of the Virasoro algebra and the moonshine module. Proc. Symp. Pure Math., Amer. Math. Soc. 56 II, 295–316 (1994) Fuchs, J.: Simple WZW currents. Commun. Math. Phys. 136, 345–356 (1991) Feigin, B. and Frenkel, E.: Duality in W -algebras. Internal. Math. Res. Notices No. 6, 75–82 (1991) Feingold, A.J., Frenkel, I.B. and Ries, J.F.X.: Spinor Construction of Vertex Operator Algebras, (1) Triality, and E8 . Contemp. Math. 121 (1991) Fuchs, J. and Gepner, D.: On the connection between WZW and free field theories. Nucl. Phys. B 294, 30–42 (1988) Frenkel, I., Huang, Y.-Z. and Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. Memoirs Am. Math. Soc. 104, 1993 Frenkel, I., Lepowsky, J. and Meurman, A.: Vertex Operator Algebras and the Monster. Pure and Appl. Math. Vol. 134, Boston: Academic Press, 1988 Feigin, B. and Miwa, T.: Extended vertex operator algebras and monomial bases. math.QA/9901067 Fuchs, J. and Schweigert, C.: The action of outer automorphisms on bundles of chiral blocks. Commun. Math. Phys. 206, 691–736 (1999) Frenkel, I. and Zhu, Y.-C.: Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J. 66, 123–168 (1992) Günaydin, M. and Hyun, S.: Ternary algebraic constructions of extended superconformal algebras. Mod. Phys. Lett. A 6, 1733–1743 (1991) Günaydin, M. and Hyun, S.: Ternary algebraic approach to extended superconformal algebras. Nucl. Phys. B 373, 688–712 (1992) Gepner, D. and Witten, E.: String theory on group manifold. Nucl. Phys. B 278, 493–549 (1986) Günaydin, M.: N = 2 superconformal algebras and Jordan triple systems. Phys. Lett. B 255, 46–50 (1991) Guo, H.: On abelian intertwining algebras and modules. Ph.D thesis, Rutgers University, 1994 Huang, Y.-Z.: A theory of tensor product for module category of a vertex operator algebra, IV. J. Pure Appl. Alg. 100, 173–216 (1995) Huang, Y.-Z.: Intertwining operator algebras, genus-zero modular functors and genus-zero conformal field theories. In: Operads: Proceedings of Renaissance Conferences, ed. J.-L. Loday, J. Stasheff, and A. A. Voronov, Contemp. Math. 202, Providence, RI: Amer. Math. Soc., 1997, pp. 335–355 Huang, Y.-Z. and Lepowsky, J.: A theory of tensor product for module category of a vertex operator algebra, I. Selecta Mathematica, New Series 1, 699–756 (1995) Huang, Y.-Z. and Lepowsky, J.: A theory of tensor product for module category of a vertex operator algebra, II. Selecta Mathematica, New Series 1, 757–786 (1995) Huang, Y.-Z. and Lepowsky, J.: A theory of tensor product for module category of a vertex operator algebra, III. J. Pure Appl. Alg. 100, 141–171 (1995) Humphreys, J.E.: Introduction to Lie algebras and representation theory. Graduate Texts in Mathematics 9, New York: Springer-Verlag, 1984 Kac, V.G.: Infinite dimensional Lie algebras. 3rd ed., Cambridge: Cambridge Univ. Press, 1990 Lepowsky, J.: Calculus of twisted vertex operators. Proc. Natl. Acad Sci. USA 82, 8295–8299 (1985) Li, H.-S.: Local systems of vertex operators, vertex superalgebras and modules. J. Pure Appl. Alg. 109, 143–195 (1996) Li, H.-S.: Local systems of twisted vertex operators, vertex superalgebras and twisted modules. Contemp. Math. 193, 203–236 (1995) Li, H.-S.: Representation theory and tensor product theory for vertex operator algebras. Ph.D. thesis, Rutgers University, 1994

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H. Li

Li, H.-S.: The physics superselection principle in vertex operator algebra theory. J. Alg. 196, 436– 457 (1997) Li, H.-S.: Extension of vertex operator algebras by a self-dual simple module. J. Alg. 187, 236–267 (1997) Li, H.-S.: An analogue of the hom functor and a generalized nuclear democracy theorem. Duke Math. J. 93, 73–114 (1998) Lepowsky, J. and Wilson, R.L.: A new family of algebras underlying the Rogers-Ramanujan identities and generalization. Proc. Natl. Acad. Sci. USA 78, 7254–7258 (1981) Li, H.-S. and Xu, X.-P.: A characterization of vertex algebras associated to even lattices. J. of Alg. 173, 253–270 (1995) ˜ Meurman, A. and Primc, M.: Annihilating fields of standard modules of sl(2, C) and combinatorial identities. Preprint, 1995; Memoirs Amer. Math. Soc. 652, 1999 Moore, G. and Seiberg, N.: Classical and quantum conformal field theory. Commun. Math. Phys. 123, 177–254 (1989) Mossberg, G.: Axiomatic vertex algebras and the Jacobi identity. Ph. D dissertation, University of Lund, Sweden, 1993; Axiomatic vertex algebras and the Jacobi identity. J. Algebra 170, 956–1010 (1994) Schellekens, A.N. and Yankielowicz, S.: Extended chiral algebras and modular invariant partition functions. Nucl. Phys. 327, 673–703 (1989) Tsuchiya, A. and Kanie, Y.: Vertex operators in conformal field theory on P1 and monodromy representations of braid group. In: Conformal Field Theory and Solvable Lattice Models, Advanced Studies in Pure Math., Vol. 16, Tokyo: Kinokuniya Company Ltd., 1988, pp .297–372

Communicated by T. Miwa

Commun. Math. Phys. 217, 697 – 698 (2001)

Communications in



Erratum

Superconformal Algebras and Transitive Group Actions on Quadrics Victor G. Kac Department of Mathematics, M.I.T., Cambridge, MA 02139, USA. E-mail: [email protected] Received: 4 December 2000 / Accepted: 4 December 2000 Commun. Math. Phys. 186, 233–252 (1997)

In this erratum I make corrections to my paper [K]. 1. It escaped my attention that in the simple superconformal algebra [K4 , K4 ] one should replace the field ν(z), where ν = ξ1 ξ2 ξ3 ξ4 , by the field ∂z ν(z), which has conformal weight 1. Hence [K4 , K4 ] is a physical superconformal algebra and therefore should be added to the list given by Theorem 4.1 of [K] to make it complete. This algebra was missed in the proof since Lemma 4.1(b) is false as stated: the representation π is not necessarily faithful. But it is easy to show that dim Ker π is commutative and with a little extra work to show that [K4 , K4 ] is the only physical superconformal algebra for which π is not faithful. This error has been pointed out in [Y], where also a different proof of the corrected version of Theorem 4.1 is given. 2. Formulas for the central extension of S2 , given in Remark 2.2, should be corrected as follows: α3 (L, L) = c/2,

α1 ((ξ1 )2 , (ξ2 )1 ) = c/6,

α1 ((ξ1 )1 − (ξ2 )2 , (ξ1 )1 − (ξ2 )2 ) = c/3, α2 ((1)i , (ξi )0 − ∂((ξi ξ1 )1 + (ξi ξ2 )2 )) = −c/3,

i = 1, 2.

Formulas for the central extension of KN , N ≤ 3, given in Remark 2.3 should be corrected as follows: (i = j = k), α0 (ξi ξj ξk , ξi ξj ξk ) = −c/12 α1 (ξi ξj , ξi ξj ) = −c/12 (i = j ), α2 (ξi , ξi ) = c/6, α3 (1, 1) = c/2.

(1) (2) (3)

Furthermore the conformal superalgebra associated to [K4 , K4 ] has exactly two linearly independent central extensions. The corresponding 2-cocycles α and β are as follows

698

V. G. Kac

(cf. [KL, S, STP]). The cocycle α is given by (1)–(3) and α1 (∂ν, ∂ν) = −c/12 .

(4)

The cocycle β is given by: β2 (1, ∂ν) = 2c , β1 (ξi , ∂i ν) = c , β1 (ξi ξj , ∂i ∂j ν) = −c . Remark. Except for the central term, all superalgebras of the family considered in [S] (resp. [STP]) are isomorphic to K4 (resp. [K4 , K4 ]). 3. In [K], a classification of all simple conformal superalgebras which are finitely generated as C[∂]-modules has been announced. (This is, of course, a far reaching generalization of Theorem 4.1.) The method of its proof is the same as in the non“super” case developed in [DK], and it uses heavily the results of [K1] . A complete list consists of current conformal superalgebras associated to finite-dimensional simple Lie superalgebras, and the following conformal superalgebras (N ≥ 0): WN , SN+2,a (a ∈ C), S N+2 (N even), KN (N = 4), [K4 , K4 ] and CK6 . Here the conformal superalgebras are denoted by the same letters as the corresponding formal distribution Lie superalgeSN bras in [K] and SN,a , S N are the derived algebras of the following deformations of respectively: {D ∈ WN | div eat D = 0},

{D ∈ WN | div(1 + ξ1 . . . ξN )D = 0} .

(The first of these two series coincides with its derived subalgebra if a = 0, and in the remaining cases the derived subalgebra has codimension 1.) The details of the proof of this theorem will appear in [FK]. References [DK] [FK] [K] [K1] [KL] [S] [STP] [Y]

D’Andrea, A. and Kac, V.G.: Structure theory of finite conformal algebras. Selecta Mathematica 4, 377–418 (1998) Fattori, D. and Kac, V.G.: Structure theory of finite conformal superalgebras. To appear Kac, V.G.: Conformal superalgebras and transitive group actions on quadrics. Commun. Math. Phys. 186, 233–252 (1997) Kac, V.G.: Classification of infinite-dimensional simple linearly compact Lie superalgebras. Adv. Math. 139, 1–55 (1998) Kac, V.G. and van de Leur, J.: On classification of superconformal algebras. In: Strings 88, S.J. Gates et al., eds., Singapore: World Sci, 1989, pp. 77–106 Schoutens, K.: O(N )-extended superconformal field theory in superspace. Nucl. Phys. B 295, 634– 652 (1988) Severin, A., Troost, W. and Van Proeyen, A.: Superconformal algebras in two dimensions with N = 4. Phys. Lett. B 208, 447–450 (1988) Yamamoto, G.: Algebraic structures on quasiprimary states in superconformal algebras. Preprint

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