Communications in Mathematical Physics - Volume 185

Commun. Math. Phys. 185, 1 – 36 (1997) Communications in Mathematical Physics c Springer-Verlag 1997 Motion by Mean ...

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Commun. Math. Phys. 185, 1 – 36 (1997)

Communications in

Mathematical Physics c Springer-Verlag 1997

Motion by Mean Curvature from the Ginzburg-Landau ∇φ Interface Model T. Funaki1 , H. Spohn2,? 1 Department of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153 Japan. E-mail: [email protected] 2 Theoretische Physik, Ludwig-Maximilians-Universit¨ at, Theresienstr. 37, D-80333 München, Germany. E-mail: [email protected]

Received: 1 February 1996 / Accepted: 2 July 1996

Abstract: We consider the scalar field φt with a reversible stochastic dynamics which is defined R d by the standard Dirichlet form relative to the Gibbs measure with formal energy d xV (∇φ(x)). The potential V is even and strictly convex. We prove that under a suitable large scale limit the φt -field becomes deterministic such that locally its normal velocity is proportional to its mean curvature, except for some anisotropy effects. As an essential input we prove that for every tilt there is a unique shift invariant, ergodic Gibbs measure for the ∇φ-field. 1. Introduction and Main Results There has been a considerable effort to prove that particle models with stochastic dynamics behave deterministically and are governed by a suitable partial differential equation on a sufficiently coarse scale [21,32]. Almost exclusively, the models are constructed in such a way that they have at least one local conservation law, like the number of particles. The locally conserved fields vary slowly in space and therefore also slowly in time. It is this slow motion which persists on a sufficiently coarse space-time scale and which is governed by a partial differential equation. Local degrees of freedom relax quickly under given constraints and have a statistics as defined by the corresponding Gibbs measure. As pointed out long ago, physically, slow motion arises also from broken symmetry. The prime example is the ferromagnetic Ising model at low temperatures and zero external field, which has then the two distinct shift invariant pure phases (ergodic Gibbs measures) µ+ and µ− . We assume reversible spin-flip dynamics - no conservation law. Then in the pure phases the relaxation is (essentially) exponentially fast. If however we prepare µ+ and µ− as spatially coexisting and glued together at a fairly sharp interface, then such a situation will persist (essentially) forever. Of course, in general the interface ?

H.S. is partially supported by DFG

2

T. Funaki, H. Spohn

is not stationary and will relax slowly provided it has some intrinsic curvature. It is then natural to expect that locally the interface velocity is proportional to the mean curvature, except for some anisotropy due to the underlying lattice. To prove such a behavior on the basis of the stochastic Ising model seems to be out of reach at present. To make nevertheless some progress we follow the physics lore which states that the pure phases play a spectator role. It is therefore sensible to model directly the interface degrees of freedom. This approach leads quite naturally to the Ginzburg-Landau ∇φ interface model which will be the subject of our paper. We consider the scalar field {φ(x) ∈ R, x ∈ 3 ⊂ Zd }. x 7→ φ(x) is interpreted as a (discretized) surface embedded in the d + 1 dimensional space (the dimension of physical interest would be d = 2). Therefore we refer to φ(x) also as height variable. The interface energy, H3 , should not change under a uniform translation φ(x) → φ(x) + a for all x ∈ 3 . This leads to X V (φ(x) − φ(y)). (1.1) H3 (φ) = x,y∈3 ,|x−y|=1

The sum is here over nearest neighbor bonds. As will be explained below, H3 has to be supplemented with suitable boundary conditions. We require Conditions on V (i) V ∈ C 2 (R), (ii) (symmetry) V (−η) = V (η), η ∈ R, (iii) (strict convexity) c− ≤ V 00 (η) ≤ c+ , η ∈ R, for some c− , c+ > 0.

(1.2)

The symmetry of V ensures that the model is of gradient type. Strict convexity is used in a coupling argument and also to provide a uniform bound on the second moments through the Brascamp-Lieb inequality. Before defining the dynamics we first have to explain a few static properties of the ∇φ model. If we think of φ as describing some elastic sheet, it is natural to fix a finite volume 3 ⊂ Zd , | 3 | < ∞, and to prescribe the field along its boundary ∂ 3 . As we will see a zero curvature interface remains stationary. Thus the boundary condition forcing a definite tilt is singled out from the rest. Let 3 ` = [−`, `]d ∩ Zd , ` ∈ N be a cube of side length 2` + 1 with boundary ∂ 3 ` = {x = (x1 , ..., xd )||xα | = ` for at least one α, α = 1, ..., d}. We enforce a fixed tilt u = (u1 , ..., ud ) ∈ Rd by setting φ(x) = u · x for x ∈ ∂ 3 ` . The finite volume surface tension, σ3 ` , is then defined by Z Y Y 1 log dφ(x) δ(φ(y) − u · y) exp[−H3 ` (φ)], (1.3) σ3 ` (u) = − | 3 `| x∈3 ` y∈∂ 3 ` i.e. the height variables along the boundary are fixed as φ(x) = u · x. Some qualitative properties are stated in Proposition 1.1. (i) The following limit exists lim σ3 ` (u) = σ(u).

`→∞

(ii) σ is convex. (iii) σ ∈ C 1 (Rd ) and there exists a constant C > 0 such that for all u, v ∈ Rd , |∇σ(u)| ≤ C(1 + |u|),

|∇σ(u) − ∇σ(v)| ≤ C|u − v|,

Motion by Mean Curvature from Ginzburg-Landau Model

and

3

u · ∇σ(u) ≥ c− |u|2 − 1.

The proof of (i),(ii) follows in essence from [27] and, for sake of completeness, will be explained in Appendix II. (iii) is part of a whole package which deals with the uniqueness of shift invariant Gibbs measures for the ∇φ-field, cf. Sects. 2 and 3. In passing we should mention that in the early 80’s static properties of ∇φ field theories were studied in fair detail, in particular for the quartic potential V3 (∇φ) = (∇φ)2 + λ(∇φ)4 , λ > 0. Bricmont et. al. [4, 5] establish bounds on the decay of correlations. More detailed information has been obtained [18, 19, 26, 6] through a multi-scale expansion which requires λ to be small. Recently Naddaf and Spencer [28] prove, for conditions on the potential identical to (1.2), that under rescaling the φ-field converges to a Gaussian limit with covariance κ(−1)−1 (massless Gaussian field) with κ some effective stiffness coefficient. If there is no conservation law, then except for noise the interface will merely try to lower its energy. Therefore we assume that the φ-field is governed by the stochastic differential equations (SDE) X √ V 0 (φt (x) − φt (y))dt + 2dwt (x), (1.4) dφt (x) = − y∈3 ,|x−y|=1

x ∈ 3 , where {wt (x), x ∈ Zd } is a collection of independent standard Brownian motions. As for the energy, the SDE (1.4) may have to be supplemented with appropriate boundary conditions. Clearly, if 3 is connected and | 3 | < ∞, then the SDE (1.4) admit Y dφ(x) exp[−H3 (φ)] x∈3

as the unique invariant (in fact infinite mass) measure and the dynamics is reversible with respect to this measure. As will become clear from our proof, the basic mathematical object is really the ∇φ-field. It must be rotation free, i.e. the sum along every closed loop has to vanish. This constraint creates a dependence over long distances. The broken symmetry referred to above is reflected by the fact that in the infinite volume limit the SDE (1.4) considered as equations for ∇φt have as stationary measures a d-parameter family of (extremal) Gibbs measures labeled by the average tilt u. In dimension d ≥ 3 the φ-field itself admits a Gibbs measure with an additional parameter for the average height, a fact which plays no role here. Our goal is to investigate the interface dynamics in a limit where the average curvature is small as measured on the scale of the lattice. For sake of simplicity we want to disregard boundary conditions and define the φ-field on the (lattice) torus 0 N = [1, ..., N ]d = (Z/N Z)d . The φt process is then governed by the SDE (1.4) where the sum is over bonds in 0 N . In its rescaled version it is convenient to think of the φ-field as a step function on the (continuum) torus Td = S 1 × ... × S 1 = [0, 1]d . We define then φ (θ, t) = φ−2 t (x)for − /2 ≤ θα − xα < /2, α = 1, ..., d

(1.5)

with θ ∈ Td , x ∈ 0 N , N = [−1 ]. Note that both x- and φ-axis are rescaled by a factor . Also, anticipating motion by mean curvature, we speeded up time diffusively as −2 . As we will see, the natural function space for our problem is the Hilbert space L2 (Td , dθ) with norm || · ||. Let then h0 ∈ L2 (Td ) be prescribed as a “macroscopic” height profile

4

T. Funaki, H. Spohn

at the initial time t = 0. We choose a sequence of initial measures, µ0 , for the φt process such that (1.6) lim E µ0 [||h0 − φ (0)||2 ] = 0, →0

where E refers to expectation with respect to the specified measure and we shall often denote φ (t) for φ (·, t). By construction, the initial φ -field approximates h0 and becomes deterministic as → 0. The problem is then to prove the existence of lim φ (θ, t) = h(θ, t)

→0

and to identify the limit height h(θ, t). Since there is no conservation law, on physical grounds one expects that the surface free energy serves as a potential for h(t). More precisely for h ∈ C 1 (Td ) we define the free energy Z dθσ(∇h(θ)) F = Td

with σ defined in Proposition 1.1. Then, in general, δF ∂ h = −µ(∇h) . ∂t δh µ is the mobility which depends on the underlying dynamics. In [33] we argued that µ = 1 for Ginzburg-Landau ∇φ interface models. The height should therefore be governed by the partial differential equation (PDE) X ∂ ∂ h(θ, t) = [σ,α (∇h(θ, t))] ∂t ∂θα d

(1.7)

α=1

(u) = ∂σ(u)/∂uα . For isotropic motion by mean curvature one would have with σ,αp σ(u) = 1 + |u|2 . In our case this is likely to hold for small |u|, however σ(u) ' |u|2 for large |u|, which reflects the constraints due to the underlying lattice. Proposition 1.2. Let (ii),(iii) of Proposition 1.1 hold. Then for h0 ∈ L2 (Td ) Eq. (1.7) has a unique solution t 7→ h(t) ∈ L2 (Td ) such that h(0) = h0 . In Appendix I we will explain how Eq. (1.7) fits into the general framework of coercive evolution equations and along with it the notion of solution. As our main result we state Theorem 1.1. Let µ0 be a sequence of initial measures for the SDE (1.4) on 0 N such that (1.6) holds with given h0 ∈ L2 (Td ) and let t 7→ h(t) be the corresponding unique solution of PDE (1.7). Then for t > 0, lim E[||h(t) − φ (t)||2 ] = 0.

→0

(1.8)

The proof of Theorem 1.1 relies on techniques developed in the context of GinzburgLandau equations with a conserved order parameter [21]. As hinted at already by the existence theory of the PDE (1.7), cf. Appendix I, the natural notion of a distance is the L2 -norm. In fact local smoothness is not required. It suffices to establish local equilibrium. In other words, we have to make sure that locally the measure is a mixture of ∇φ Gibbs measures. From the local space-time averaging one can merely conclude that


5

locally the entropy production vanishes. Therefore the local measure is shift invariant and satisfies the DLR equations for ∇φ. We are thus forced to solve the uniqueness problem for Gibbs measures in our context. Because of long range dependence Dobrushin type methods do not seem to work and the dynamics will be used in an essential way: By a suitable coupling we prove that the SDE for ∇φt have at most one stationary measure which is extreme shift invariant and has a definite mean, cf. Sect. 2. The construction of Gibbs measures implies that there is at least one such measure, cf. Sect. 3. This construction requires some tightness which comes from the Brascamp-Lieb inequalities [2]. We believe that our results nicely illustrate how stochastic dynamics can help to establish purely static properties of Gibbs measures. Once local equilibrium is established, the canonical procedure is to prove a oneblock and two-blocks estimate. In fact we avoid the two-blocks estimate by considering directly the time-change of the L2 -norm appearing in (1.8). If d = 1, the L2 -norm coincides with the H −1 -norm for the gradient field. Thus our strategy is an extension of the H −1 -method first used by Chang and Yau [7]. The one-block estimate seems to be unaccessible in our context because the local functions in question are not bounded. To overcome this difficulty we introduce in Sect. 4 the notion of coupled local equilibria. We couple the solution of a discretized version of the PDE (1.7) to the distribution of the stochastic dynamics. We can then prove Lp -integrability for suitable p > 2 from the solution of the PDE. Unfortunately, these bounds hold only provided the diffusion coefficient in (1.7) is strictly elliptic, i.e. the surface tension is strictly convex. Such a property cannot be inferred from the pyramid inequality of Appendix II. Thus one further level of approximation is introduced, in which the surface tension σ is replaced by a smoothened and strictly convex version σ a . This cut-off is then removed at the end. In fact the approximation works so well that we do not even need an entropy bound for the initial measure and the proof of Theorem 1.1 will be concluded in Sect. 5. We add a few miscellaneous Remarks. (i) For d = 1 the ∇φ interface model is identical to the Ginzburg-Landau model of [21]. The limit (1.8) holds then under much weaker assumptions on the potential V . (ii) Although not needed for our proof it would be nice to have some smoothness of the surface tension σ. The conventional approach through expansion around the quadratic potential is blocked because of slow decay of correlations, see (3.5) and [28]. (iii) The strict convexity and symmetry of the potential is used in an essential way. Without these assumptions, even on the level of the static Gibbs measure there are only very partial results. Could σ have flat pieces, corresponding to a macroscopic phase segregation into distinct orientations, or could σ have cusps, corresponding to roughening transitions? E.g. without symmetry it is not even clear how to estimate the average of φ(x) for zero boundary conditions. 2 (iv) A related interface model is Pthe sine-Gordon field theory. In this case V (η) = η , but one adds to H3 the potential λ x cos φ(x) in order to take the lattice structure transverse to the reference plane into account. One could instead also assume that φ(x) ∈ Z which is the discrete Gaussian model. The diffusion process (1.4) would have to be replaced then by a jump process. These models undergo a roughening transition in d = 2 as λ becomes large [13]. Nevertheless we still expect (1.8) to hold [33]. Much more seriously in our context, these models are of nongradient type and therefore considerably harder to handle. (v) In [17] a random interface motion in d = 1 is derived from the continuum GinzburgLandau equation in the low temperature limit.

6

T. Funaki, H. Spohn

2. Stochastic Dynamics and its Stationary Measures In this section, we will prove that there is at most one stationary probability measure for the dynamics of height differences defined on an infinite lattice which is an extreme shift invariant and has a definite mean. But first we should establish some convenient lattice notation and introduce the basic sequence spaces. 2.1. Basic notation. Let (Zd )∗ be the set of all directed bonds b = (x, y), x, y ∈ Zd , |x − y| = 1 in Zd . Each bond b = (x, y) is directed from y to x. We also write xb = x and yb = y for b = (x, y). −b := (yb , xb ) stands for the bond b reversely directed. Note that each undirected bond appears twice in (Zd )∗ . For α = 1, ..., d, eα ∈ Zd denotes the αth unit vector given by (eα )β = δαβ . The bond (eα , 0) will sometimes also be denoted by eα . For 3 ⊂ Zd , we set 3 ∗ = {b ∈ (Zd )∗ |xb , yb ∈ 3 }, ∂ 3 ∗ = {b ∈ (Zd )∗ |xb ∈ / 3 , yb ∈ 3 } and ∂ 3 = {y ∈ 3 |y = yb for some b ∈ ∂ 3 ∗ }. We shall also denote 3 ∗ = {b ∈ (Zd )∗ |xb ∈ 3 or yb ∈ 3 }. 3 b Zd means that 3 is a finite set in Zd . For ` ∈ N, 3 ` = [−`, `]d ∩ Zd . A sequence of bonds C = {b(1) , b(2) , . . . , b(n) } is called a chain connecting y and x, y, x ∈ Zd , if yb(1) = y, xb(i) = yb(i+1) for 1 ≤ i ≤ n − 1 and xb(n) = x. The chain C is called a closed loop if xb(n) = yb(1) . A plaquette is a closed loop P = {b(1) , b(2) , b(3) , b(4) } such that {xb(i) , i = 1, .., 4} consists of four different points. d ∗ The field η = {η(b)} ∈ R(Z ) is said to satisfy the plaquette condition if X η(b) = 0 for all plaquettes Pin Zd . (2.1) η(b) = −η(−b)for all b ∈ (Zd )∗ and b∈P

Note that, if for φ = (φ(x))x∈Zd ∈ RZ and b ∈ (Zd )∗ we define η(b) = ∇φ(b) = φ(xb ) − φ(yb ), then ∇φ = {∇φ(b)} satisfies the plaquette condition. Let X be the set of d ∗ all η ∈ R(Z ) which satisfy the plaquette condition and let L2r , r > 0, be the set of all d ∗ η ∈ R(Z ) such that X |η(b)|2 e−2r|xb | < ∞. |η|2r := d

b∈(Zd )∗

We denote Xr = X ∩ L2r equipped with the norm | · |r . 2.2. Dynamics. According to (1.4) the dynamics of the height variables φt = {φt (x)} ∈ d RZ are governed by the SDE X √ V 0 (∇φt (b)) dt + 2dwt (x), x ∈ Zd , (2.2) dφt (x) = − b:xb =x

where {wt (x), x ∈ Zd } is a family of independent Brownian motions. The potential V is assumed to satisfy the conditions (1.2). The dynamics for height differences ηt = d ∗ {ηt (b)} ∈ R(Z ) are then determined by the SDE    X  X √ V 0 (ηt (b0 )) − V 0 (ηt (b0 )) dt + 2dwt (b), b ∈ (Zd )∗ , dηt (b) = − 0  0 b :xb0 =xb

where wt (b) = wt (xb ) − wt (yb ).

b :xb0 =yb

(2.3)


7

Height differences η φ are associated with the heights φ by η φ (b) := ∇φ(b),

b ∈ (Zd )∗ ,

and, conversely, the heights φη,φ(0) can be constructed from height differences η and the height variable φ(0) at x = 0 as X η(b) + φ(0), (2.4) φη,φ(0) (x) := b∈C0,x

where C0,x is an arbitrary chain connecting 0 and x. Note that φη,φ(0) is well-defined if η = {η(b)} ∈ X . The relationship between the solutions of (2.2) and (2.3) is stated in the next lemma. We always assume η0 ∈ X for the initial data of (2.3). Lemma 2.1. (i) The solution of (2.3) satisfies ηt ∈ X for all t > 0. (ii) If φt is a solution of (2.2), then ηt := η φt is a solution of (2.3). (iii) Conversely, let ηt be a solution of (2.3) and define φt (0) through (2.2) for x = 0 and ∇φt (b) replaced by ηt (b) with arbitrary initial condition φ0 (0) ∈ R. Then φt := φηt ,φt (0) is a solution of (2.2). The proof is straightforward and hence omitted. The condition (1.2)–(iii) on V implies global Lipschitz continuity in Xr , r > 0, of the drift term of the SDE (2.3). Therefore, a standard method of successive approximations yields the existence and uniqueness of solutions to (2.3). Lemma 2.2. For each η ∈ Xr , r > 0, the SDE (2.3) has a unique Xr -valued continuous solution ηt starting at η0 = η. 2.3. Stationary measures. We first state an energy inequality for φt . Lemma 2.3. Let φt and φ¯ t be two solutions of (2.2) and set φ˜ t (x) := φt (x) − φ¯ t (x). Then, for every 3 b Zd , we have 2 ∂ X ˜ φt (x) = It3 + Bt3 , ∂t x∈3 where It3 = −

X b∈3 ∗

Bt3

=2

X

(2.5)

∇φ˜ t (b) V 0 (∇φt (b)) − V 0 (∇φ¯ t (b)) , φ˜ t (yb ) V 0 (∇φt (b)) − V 0 (∇φ¯ t (b)) .

b∈∂3 ∗

The interior term It3 and the boundary term Bt3 admit the following bounds, respectively, X 2 (2.6) ∇φ˜ t (b) , It3 ≤ −c− b∈3 ∗

Bt3

≤ 2c+

X

b∈∂3 ∗

|φ˜ t (yb )| |∇φ˜ t (b)|.

(2.7)

8

T. Funaki, H. Spohn

Proof. From Eq. (2.2), ∂ ˜ φt (x) ∂t

2

X

= −2

Φt (b) · φ˜ t (x) = −

b:xb =x

 X 

Φt (b) −

b:xb =x

X

  Φt (b)

b:yb =x



φ˜ t (x),

where Φt (b) := V 0 (∇φt (b)) − V 0 (∇φ¯ t (b)). The second equality uses the symmetry of V which implies V 0 (∇φ(b)) = −V 0 (∇φ(−b)). The right-hand side summed over x ∈ 3 becomes −

X

φ˜ t (x)

x∈3

X

=−

X

Φt (b) +

x∈3

b:xb =x

=

+

φ˜ t (x)

X

∇φ˜ t (b)Φt (b) −

X

Φt (b)

b:yb =x

b:xb ∈3 ,yb ∈ /3

b∈3 ∗

It3

X

φ˜ t (xb )Φt (b) +

X

φ˜ t (yb )Φt (b)

b:yb ∈3 ,xb ∈3 /

Bt3 ,

which proves (2.5). To obtain the term Bt3 we again used the symmetry of V . The two bounds (2.6), (2.7) follow from the condition (1.2)–(iii) on V . We now state the main result of this section. Let P(X ) be the set of all probability measures on X and let P2 (X ) be those µ ∈ P(X ) satisfying E µ [|η(b)|2 ] < ∞ for each b ∈ (Zd )∗ . The set P(Xr ), r > 0, is defined correspondingly and P2 (Xr ) stands for the set of all µ ∈ P(Xr ) such that E µ [|η|2r ] < ∞. We denote by S the class of all shift invariant µ ∈ P2 (X ) which are stationary for the SDE (2.3) and by ext S those µ ∈ S which are ergodic with respect to shifts. Furthermore, for each u ∈ Rd , (ext S)u denotes the family of all µ ∈ ext S such that E µ [η(eα )] = uα , α = 1, ..., d. Note that S ⊂ P2 (Xr ) for every r > 0. Theorem 2.1. For every u ∈ Rd there exists at most one µ ∈ (ext S)u . We prepare the proof of this theorem by a proposition which also implies the Lipschitz continuity of the derivative of the surface tension σ(u), see Theorem 3.4 below. Suppose that there exist µ ∈ (ext S)u and µ¯ ∈ (ext S)v for u, v ∈ Rd . Let us construct two independent Xr -valued random variables η = {η(b)} and η¯ = {η(b)} ¯ on a common probability space (Ω, F, P ) in such a manner that η and η¯ are distributed by µ and µ¯ ¯ under P , respectively. We define φ0 = φη,0 and φ¯ 0 = φη,0 using the notation in (2.4). ¯ Let φt and φt be the two solutions of the SDE (2.2) with common Brownian motions having initial data φ0 and φ¯ 0 . In view of Lemma 2.1 and 2.2 such solutions certainly ¯ exist. Since µ, µ¯ ∈ S, we conclude that ηt := η φt and η¯t := η φt are distributed by µ and µ, ¯ respectively, for all t ≥ 0. Our claim is then Proposition 2.1. There exists a constant C > 0 independent of u, v ∈ Rd such that 1 lim T →∞ T

Z

T 0

d X α=1

E P [(ηt (eα ) − η¯t (eα ))2 ] ≤ C|u − v|2 .

(2.8)


9

Once this proposition is proved, Theorem 2.1 immediately follows. Indeed, suppose that there exist two measures µ, µ¯ ∈ (ext S)u . Then Proposition 2.1 with u = v implies Z |η − η| ¯ 2r PT (dηdη) ¯ = 0, (2.9) lim T →∞

where PT is a shift-invariant probability measure on Xr × Xr , r > 0, defined by Z 1 T ¯ := P {ηt (b), η¯t (b)}b ∈ dηdη¯ dt. PT (dηdη) T 0 The first marginal of PT is µ and the second one is µ. ¯ Thus (2.9) implies that the Vaserstein distance between µ and µ¯ vanishes and hence µ = µ, ¯ see, e.g., [14,p.482] for the Vaserstein metric on the space P2 (Xr ). This proves Theorem 2.1. Proof of Proposition 2.1. Step 1. We apply Lemma 2.3 to the differences {φ˜ t (x) := φt (x) − φ¯ t (x)} and obtain, with the choice 3 = 3 ` ,   " # Z T X X 2 2 EP + c− EP  φ˜ T (x) ∇φ˜ t (b)  dt (2.10) x∈3 `

≤ EP

0

"

X

2

φ˜ 0 (x)

b∈3 ∗ `

#

Z

T

+ 2c+ 0

x∈3 `



EP 



X

|φ˜ t (yb )| |∇φ˜ t (b)| dt

b∈∂3 ∗ `

for every T > 0 and ` ∈ N. Set g(t) =

d X

EP

h

2 i

∇φ˜ t (eα )

.

α=1

Then, noting that the distribution of (ηt , η¯t ) = (∇φt , ∇φ¯ t ) on Xr × Xr is shift-invariant, RT the second term on the left-hand side of (2.10) coincides with c− d−1 | 3 ∗` | 0 g(t) dt. On the other hand, estimating |φ˜ t (yb )| |∇φ˜ t (b)| ≤ {`γ|∇φ˜ t (b)|2 + `−1 γ −1 |φ˜ t (yb )|2 }/2 for arbitrary γ > 0, the second term on the right-hand side is bounded by Z T Z T g(t) dt + c+ `−1 γ −1 |∂ 3 ∗` | sup ||φ˜ t (y)||2L2 (P ) dt. c+ `γd−1 |∂ 3 ∗` | 0 0 y∈∂ 3 ` Then, choosing γ = c− /2c+ c0 with c0 := sup`≥1 {`|∂ 3 ∗` |/| 3 ∗` |} < ∞, we obtain from (2.10), " # Z T X 2 2d P ˜ E g(t) dt ≤ φ0 (x) c− | 3 ∗` | 0 x∈3 ` Z (2c+ c0 )2 d T + sup ||φ˜ t (y)||2L2 (P ) dt, (2.11) (c− `)2 y∈∂3 0 ` where we have dropped the non-negative first term on the left-hand side of (2.10). Step 2. Here we derive the following bound on the boundary term: For each > 0 there exists an `0 ∈ N such that

10

T. Funaki, H. Spohn

Z t 2 2 2 2 2 −2 ˜ sup ||φt (y)||L2 (P ) ≤ C1 ` + ` |u − v| + ` t g(s) ds

y∈∂3 `

(2.12)

0

for every t > 0 and ` ≥ `0 , where C1 > 0 is a constant independent of , `, and t. To this end, as an immediate consequence of the mean ergodic theorem applied to µ ∈ (ext S)u , we have 1 lim ||φη,0 (x) − x · u||L2 (µ) = 0 (2.13) |x|→∞ |x| and correspondingly for µ¯ with v in place of u. Taking 3 0 = 3 [`/2] one obtains ||φ˜ t (y)||L2 (P )

1 X φt (x) − y · u||L2 (P ) | 3 0| x∈3 0 1 X ¯ φt (x) − y · v||L2 (P ) + ||φ¯ t (y) − | 3 0| x∈3 0 √ 1 X ˜ φt (x)||L2 (P ) + d`|u − v| + || 0 |3 | 0

≤

||φt (y) −

x∈3

=: I1 + I2 + I3 + I4 , for y ∈ ∂ 3 ` . However, since

P x∈3 0

x = 0 and using (2.13),

1 X ||φt (y) − φt (x) − (y − x) · u||L2 (P ) | 3 0| x∈3 0 1 X = ||φη,0 (y − x) − (y − x) · u||L2 (µ) ≤ ` , | 3 0| 0

I1 ≤

x∈3

provided ` is sufficiently large; recall that ∇φt is distributed by µ for all t ≥ 0. Similarly, I2 ≤ ` for sufficiently large `. Finally, since as in the proof of Lemma 2.3 ∂ ∂t

(

X

) φ˜ t (x)

x∈3 0

=−

X X

X

Φt (b) =

Φt (b),

b∈(∂ 3 0 )∗

x∈3 0 b:xb =x

I3 is bounded as I3 ≤ ||

Z t 1 X ˜ 1 2 φ (x)|| + 0 L (P ) 0| | 3 0| | 3 0 0 x∈3

X

||Φs (b)||L2 (P ) ds.

b∈(∂3 0 )∗

The right-hand side can be further estimated as X

||Φs (b)||L2 (P ) ≤ c+ d−1 |(∂ 3 0 )∗ |

b∈(∂3 0 )∗

and, using again (2.13),

d X α=1

||∇φ˜ s (eα )||L2 (P ) ,


||

11

1 X ˜ φ0 (x)||L2 (P ) | 3 0| x∈3 0 1 X η,0 ¯ ≤ (x) − x · v||L2 (µ) ||φ (x) − x · u||L2 (µ) + ||φη,0 ¯ + |x| · |u − v| 0 |3 | x∈3 0 √ ≤ ` + d`|u − v|,

for sufficiently large `. Therefore, I3 ≤ ` +

Z tX d √ d`|u − v| + c+ d−1 | 3 0 |−1 |(∂ 3 0 )∗ | ||∇φ˜ s (eα )||L2 (P ) ds 0

α=1

for sufficiently large `. This completes the proof of (2.12). Step 3. Using (2.13), one can choose `1 ∈ N such that 2 i 1 X Ph ˜ φ0 (x) E | 3 ` | x∈3 ` o 3 X n η,0 ¯ ||φ (x) − x · u||2L2 (µ) + |x · u − x · v|2 + ||φη,0 (x) − x · u||2L2 (µ) ≤ ¯ | 3 ` | x∈3 `

≤ 2 `2 + 3d`2 |u − v|2 ,

` ≥ `1 .

(2.14)

Inserting the estimates (2.12) and (2.14) into (2.11), we have Z T g(t) dt ≤ C2 (2 `2 + `2 |u − v|2 ) 0

+C2 `−2

Z

T

2 `2 + `2 |u − v|2 + `−2 t

Z

0

≤ C2 (2 + |u − v|2 )(`2 + T ) + C2 `−4 T 2

t

g(s) ds

dt

0

Z

T

g(t) dt 0

for every T > 0 and ` ≥ `2 := max{`0 , `1 }, which may depend on u, v, and > 0. C2 is a constant independent of u, v and . Choosing ` = (2C2 T 2 )1/4 and letting T → ∞, we obtain Z p 1 T lim g(t) dt ≤ 2C2 ( 2C2 + 1)(2 + |u − v|2 ) T →∞ T 0 for every > 0. Finally, letting → 0, the desired estimate (2.8) is shown.

2.4. Finite volume approximation. As a final topic we establish the approximation of the solutions of (2.3) by the corresponding finite volume equations, cf. [9,31,36] for related results. For every ξ ∈ X and 3 b Zd , we consider the SDE nP o  P 0 0 ¯ − ¯ ¯  V V (b) = − (η ( b)) (η ( b)) dt dη  ¯ t t t b:x b:x =x ∈ 3 =y ∈ 3 ¯ ¯ b b  b b  √  3 + 2dwt (b), b ∈ 3 ∗ , (2.15)   ∗,  η (b) = ξ(b), b ∈ / 3 t   b ∈ (Zd )∗ , η0 (b) = ξ(b),

12

T. Funaki, H. Spohn

where wt3 (b) = 1{xb ∈3 } wt (xb ) − 1{yb ∈3 } wt (yb ). The solution ηt and its distribution on C([0, T ], X ) are sometimes denoted by ηt3 and Pξ3 , respectively. The distribution ∞ denote the of the solution of the SDE (2.3) starting at ξ is denoted by Pξ . Let Cloc ,b family of all functions F on X of the form F (η) = F˜ ({η(b)}b∈3 ∗ ) for some 3 b Zd ∗ and F˜ ∈ Cb∞ (R3 ). ∞ , Proposition 2.2. For every ξ ∈ Xr and F ∈ Cloc ,b 3

lim E Pξ [F (ηt )] = E Pξ [F (ηt )].

3 %Zd

As preparation we state Lemma 2.4. There exists C = CT > 0 independent of 3 b Zd such that (i) E[sup0≤t≤T |ηt3 |4r ] ≤ C(1 + |ξ|4r ), (i) E[|ηt3 − ηs3 |4r ] ≤ C(1 + |ξ|4r )(t − s)2 , 0 ≤ s ≤ t ≤ T, r > 0. Proof. The proof is fairly standard. Computing the stochastic differential d|ηt3 − ζ|2r for ζ = {ζ(b)} ∈ Xr by Itô’s formula, it can be seen that Z t |ηt3 − ζ|2r ≤ |η03 − ζ|2r + C1 (1 + |ηs3 |2r + |ηs3 − ζ|2r ) ds + m3 (2.16) t (ζ), 0

for some C1 > 0, where

√ X −2r|x | Z t 3 b m3 (ζ) = 2 2 e (ηs (b) − ζ(b)) dws3 (b). t 0

b∈3 ∗

Recall that V 0 (η) is linearly bounded. By Doob’s inequality for martingales, Z t 3 2 3 2 E[ sup (ms (ζ)) ] ≤ 4E[(mt (ζ)) ] ≤ C2 E[|ηs3 − ζ|22r ] ds 0≤s≤t

0

Z

t

≤ C2 0

(a + a−1 E[|ηs3 − ζ|4r ]) ds

(2.17)

for some C2 > 0 and arbitrary a > 0. Note that |η|2r ≤ |η|r . Therefore, squaring both sides of (2.16), using (2.17) and Schwarz’s inequality, we obtain E[ sup | ηs3 − ζ|4r ] ≤ C3 |η03 − ζ|4r + C3 (t2 + at) 0≤s≤t

Z

t

+ C3 t 0

E[|ηs3 |4r ] ds

+ C3 (t + a

−1

Z

t

) 0

E[|ηs3 − ζ|4r ] ds

(2.18)

for some C3 > 0. Now assertion (i) follows as an application of Gronwall’s lemma to (2.18) taking ζ = 0 and a = 1. To show assertion (ii), we take ζ = η03 in (2.18). Then, by (i), the third term in the right-hand side is bounded by C4 t2 (1 + |η03 |4r ) if 0 ≤ t ≤ T and therefore Gronwall’s lemma proves that E[|ηt3 − η03 |4r ] ≤ (C3 (t2 + at) + C4 t2 (1 + |η03 |4r )) exp[C3 (t + a−1 )t]. Taking a = t in this estimate, we have E[|ηt3 − η03 |4r ] ≤ C5 t2 (1 + |η03 |4r ),

0 ≤ t ≤ T.

Hence, assertion (ii) follows from the Markov property of ηt3 and the bound (i).


13

Proof of Proposition 2.2. Fix ξ ∈ Xr , r > 0. It is sufficient to prove that Pξ3 converges weakly to Pξ on C([0, T ], Xr˜ ) for some r˜ > 0 as 3 % Zd . However, the family {Pξ3 }3 of probability measures on C([0, T ], Xr˜ ) is tight for all r˜ > r. Indeed, the embedding L2r ⊂ L2r˜ is compact if r˜ > r and therefore tightness follows from Lemma 2.4 by using Holley-Stroock’s criterion, see [16, Prop. 3.1]. Let Q be an arbitrary limit point of {Pξ3 }3 as 3 % Zd . Note that such a Q exists. If one could show Q = Pξ , the proof would be concluded. To this end, it is enough to show the uniqueness of the solutions Q˜ to the martingale problem corresponding to the SDE (2.3), since Q is one of such solutions. However, this uniqueness assertion can be shown similarly to the method used in [15,p.514]. In fact, let Cξ be the space of all ηt ∈ C([0, T ], Xr ) satisfying η0 = ξ equipped with the metric determined by the usual uniform norm and consider a map Rt Θ : Cξ −→ Cξ defined by (Θη)t = ηt − 0 9 (ηs ) ds, η ∈ Cξ , where 9 (η) is the drift term of the SDE (2.3). Since 9 : Xr −→ Xr is globally Lipschitz continuous, one can show that Θ : Cξ −→ Cξ is bijective and continuous with measurable inverse Θ−1 ; cf. [15,Lemma 4.4]. Then, Q˜ ◦ Θ−1 turns out to be a solution to the martingale problem corresponding to the SDE (2.3) with V = 0.√But such a measure is uniquely characterized. It is precisely the distribution of {ξ(b) + 2wt (b)}b on the space Cξ . This proves the uniqueness of Q˜ and concludes that Q = Pξ . 3. Uniqueness of Shift Invariant Gibbs Measures 3.1. Definition and DLR equations. For every ξ ∈ X and 3 b Zd the space of height differences on 3 ∗ for given boundary condition ξ is defined as X3 ∗ ,ξ := {η = (η(b))b∈3 ∗ |η ∨ ξ ∈ X }, where η ∨ ξ ∈ X is determined by (η ∨ ξ)(b) = η(b) for b ∈ 3 ∗ and = ξ(b) for b ∈ / 3 ∗ . Note that X3 ∗ ,ξ is an affine space such that dim X3 ∗ ,ξ = | 3 | (at least / 3 when Zd \ 3 is connected). Indeed, fixing a point x0 ∈ P we consider the map J : X3 ∗ ,ξ 3 η 7−→ φ = {φ(x)}x∈3 ∈ R3 defined by φ(x) = b∈Cx ,x (η ∨ ξ)(b) for a 0 chain Cx0 ,x connecting x0 and x ∈ 3 . J is then well-defined and diffeomorphic. The finite volume Gibbs measure in 3 with boundary condition ξ is defined by 1 X V (η(b))]dη3 ,ξ ∈ P(X3 ∗ ,ξ ), µ3 ,ξ (dη) := Z3−1,ξ exp[− 2 ∗ b∈3

where dη3 ,ξ denotes a uniform measure on the affine space X3 ∗ ,ξ and Z3 ,ξ is the normalization; note that the factor 1/2 appears here in the exponential but not in (1.1). This difference is caused since we now count each (undirected) bond twice. µ ∈ P(X ) is called a Gibbs measure for the height differences if it satisfies the DLR equations µ(·|F

(Zd )∗ \ 3 ∗

)(ξ) = µ3 ,ξ (·),

µ-a.e. ξ,

for every 3 b Zd , where F(Zd )∗ \3 ∗ is the σ-algebra generated by {η(b)|b ∈ (Zd )∗ \ 3 ∗ }. The finite volume Gibbs measure for the heights themselves is defined as follows. d For every boundary condition ψ ∈ RZ and 3 b Zd , let

14

T. Funaki, H. Spohn

ν3 ,ψ (dφ) := Z3−1,ψ exp[−

Y 1 X V (∇(φ ∨ ψ)(b))] dφ(x) ∈ P(R3 ), 2 ∗ x∈3 b∈3

where φ∨ψ ∈ RZ is determined by (φ∨ψ)(x) = φ(x) for x ∈ 3 and (φ∨ψ)(x) = ψ(x) for x ∈ / 3 . For every ξ ∈ X and a ∈ R, let ψ = φξ,a be defined by (2.4) and consider the measure ν3 ,ψ . Then µ3 ,ξ is the image measure of ν3 ,ψ under the map {φ(x)}x∈3 7→ {η(b) := ∇(φ ∨ ψ)(b)}b∈3 ∗ . Note that the image measure is determined only by ξ and is independent of the choice of a. d

3.2. Uniqueness. Let G be the family of all shift invariant Gibbs measures µ ∈ P2 (X ) for the height differences and let ext G be its subfamily consisting of all µ ergodic with respect to shifts. For each u ∈ Rd , (ext G)u denotes the class of all µ ∈ ext G such that E µ [η(eα )] = uα , α = 1, ..., d. Note that, if µ ∈ P2 (X ) is shift-invariant, then µ ∈ P2 (Xr ) for all r > 0. We first establish that every Gibbs measure is reversible for the SDE (2.3). Proposition 3.1. Every µ ∈ G is reversible under the dynamics (2.3), namely, Z Z F (ξ)E Pξ [G(ηt )] µ(dξ) = E Pξ [F (ηt )]G(ξ) µ(dξ) Xr

(3.1)

Xr

∞ for every t ≥ 0 and F, G ∈ Cloc . Here Pξ denotes the distribution of the solution ηt ,b of (2.3) starting at ξ ∈ Xr .

Proof. For every ψ ∈ RZ and 3 b Zd we consider the SDE for φt ∈ RZ ,  √ P  dφt (x) = − b:xb =x V 0 (∇φt (b)) dt + 2dwt (x), x ∈ 3 , x∈ / 3, φ (x) = ψ(x),  t x ∈ Zd . φ0 (x) = ψ(x), d

d

(3.2)

Clearly, ν3 ,ψ is reversible under (3.2). Therefore, since ηt = ∇φt satisfies (2.15) provided ψ = φξ,0 , µ3 ,ξ is reversible under (2.15), i.e. Z Z 3 3 Pη∨ξ F (ξ)E [G(ηt )] µ3 ,ξ (dη) = E Pη∨ξ [F (ηt )]G(ξ) µ3 ,ξ (dη) (3.3) X

X

3 ∗ ,ξ

3 ∗ ,ξ

for all ξ ∈ X if both F and G are supported in 3 . For given µ ∈ G, integrating both sides of (3.3) with respect to µ(dξ) we have Z Z 3 3 F (ξ)E Pξ [G(ηt )] µ(dξ) = E Pξ [F (ηt )]G(ξ) µ(dξ). Xr

Xr

Hence, (3.1) follows from Proposition 2.2 by letting 3 % Zd .

Remark . Results similar to the above proposition together with its converse were obtained for lattice systems by [9,31,37] and for continuum systems by [22,14]. Theorem 3.1. For every u ∈ Rd there exists at most one µ ∈ (ext G)u . Proof. By Proposition 3.1, µ ∈ ext G implies µ ∈ ext S. Consequently the conclusion follows from Theorem 2.1.


15

3.3. Construction. To automatically ensure shift invariance it is convenient to construct a Gibbs measure through the use of periodic boundary conditions. For this reason let 0 N , N ∈ N, be the periodic lattice (Z/N Z)d . As before, 0 ∗N denotes the set of all ∗ directed bonds in 0 N and X0 N denotes thePset of all η ∈ R0 N which satisfy the plaquette condition, i.e. η(b) = −η(−b) and b∈C η(b) = 0 for every closed loop in 0 N . On the torus shift invariance of the measure always implies E[η(b)] = 0 because of the plaquette condition. Therefore it seems that only the state with tilt u = 0 could be constructed. To avoid such a restriction we note that boundary conditions with definite tilt u are identical to boundary conditions u = 0 but with the shifted potential V (· + uα ) for a bond directed along eα . Thus a Gibbs measure for arbitrary u is constructed from the torus with potential V (· + uα ). Theorem 3.2. For every u ∈ Rd there exists at least one µu ∈ (ext G)u . Proof. We consider the finite volume Gibbs measures µ˜ N,u ∈ P(X0 N ) with periodic boundary conditions which, for each u ∈ Rd , are defined by −1 µ˜ N,u (dη) ˜ = ZN,u exp[−

1 X V (η(b) ˜ + ub )]dηÑ ∈ P(X0 N ). 2 ∗ b∈0 N

Here dηÑ is the uniform measure on the affine space X0 N , ZN,u is the normalization, and ub is defined by ub = ±uα for b = (x ± eα , x), x ∈ Zd , α = 1, ..., d. The law of {η(b) := η(b) ˜ + ub } under µ˜ N,u is denoted by µN,u . By the Brascamp-Lieb inequality, cf. [12, Prop.1.1.6], there exists a β > 0 such that the uniform bound 2

sup E µN,u [eβ(η(b)−ub ) ] < ∞

(3.4)

N,u∈Rd

holds. In particular, (3.4) implies the tightness of the family {µN,u }N . Therefore a limiting measure exists by taking N → ∞ along a suitable subsequence. Every limit µ has the following properties, cf. [4;5,p.365,Theorem 2;12,1.1.7]: µ ∈ G, E µ [η(eα )] = 2 uα , α = 1, ..., d, E µ [eβ|η(b)| ] < ∞ for some β > 0, and the correlations of µ decay as µ E (η(b) − E µ [η(b)])(η(b0 ) − E µ [η(b0 )]) ≤

C , dist (b, b0 )

b, b0 ∈ (Zd )∗ .

(3.5)

By Theorem 14.17 of [20, p. 298] the measure µ has the ergodic decomposition Z µ(·) = ν(·) wµ (dν), extG for some wµ ∈ P(extG). Now let us fix some sequence 3 % Zd and let E := {η| lim

1 X η(bx,α ) = uα , α = 1, ..., d} | 3 | x∈3

along that sequence with bx,α = (x + eα , x) ∈ (Zd )∗ . Then from (3.5) we conclude that µ(E) = 1 and therefore ν(E) = 1 for wµ -a.e. ν. Since ν is shift-invariant, this implies E ν [η(eα )] = uα . Thus, there exists a measure ν =: µu ∈ ext G having the properties stated in Theorem 3.2. Because of the uniqueness of ergodic Gibbs measures for each u, in fact, µN,u itself converges weakly to µu ∈ ext G.

16

T. Funaki, H. Spohn

The uniform bound (3.4) implies 2

sup E µu [eβ(η(b)−ub ) ] < ∞.

(3.6)

u∈Rd

Theorems 3.1 and 3.2 are summarized by Theorem 3.3. The set G is the closed convex hull of {µu |u ∈ Rd }. 3.4. Thermodynamic identities. For the proof of Theorem 1.1 we will need some identities which are best established at this stage. Let σN (u) = −| 0 N |−1 log ZN,u ,

(3.7)

and ∇σN = (σN,1 , ..., σN,d ), σN,α = ∂σN /∂uα , α = 1, ..., d. Lemma 3.1. (i) E µN,u [η(b)] = ub , (ii) E µN,u [V 0 (η(eα ))] = σN,α , Pd (iii) E µN,u [ α=1 η(eα )V 0 (η(eα ))] = u · ∇σN + 1. P Proof. η˜ ∈ X0 N implies b∈Cα η(b) ˜ = 0 for the closed loop Cα along the α-axis. Since µ˜ N,u is shift invariant we conclude that E µ˜ N,u [η(b)] ˜ = 0 for every b. This implies (i). The identity (ii) is shown by differentiating Z 1 X 1 ∂ ∂σN −1 ZN,u exp[− =− V (η(b) ˜ + ub )]dηÑ ∂uα | 0 N| ∂uα 2 b∈0 ∗ N Z X 1 1 X 1 −1 0 ZN,u V (η(b = ˜ 0 ) + ub0 ) exp[− V (η(b) ˜ + ub )]dηÑ | 0 N| 2 b0 ∈0 ∗ : 2 ∗ N

-directed

b∈0 N

α

=

X 1 E µ˜ N,u [ V 0 (η(b ˜ x,α ) + uα )] = E µN,u [V 0 (η(eα ))]. | 0 N| x∈0 N

We used V 0 (η) = −V 0 (−η) in the third and the shift invariance of µN,u in the fourth equality. Since the left-hand side of (iii) can be rewritten as X 1 E µ˜ N,u [(η(b) ˜ + ub )V 0 (η(b) ˜ + ub )], 2| 0 N | ∗ b∈0 N

the proof of (iii) is concluded if one can show that I = 2|0 N | with X 0 I= E µ˜ N,u [η(b)V ˜ (η(b) ˜ + ub )]. b∈0 ∗ N

We have


I=

X Z b∈0 ∗ N

= −2

y∈0 N

=2

X Z y∈0 N

0 ˜ ˜ + ub )F (φ) ˜ ∇φ(b)V (∇φ(b)

R0 N \{0}

X Z

17

Y

˜ dφ(x)

x∈0 N \{0}

Y ∂F ˜ ˜ ˜ φ(y) dφ(x) (φ) ˜ ∂ φ(y) R0 N \{0} x∈0 \{0}

R0 N \{0}

Y

˜ F (φ)

N

˜ dφ(x) = 2| 0 N |,

x∈0 N \{0}

˜ where φ(0) = a is arbitrary but fixed and ˜ := Z −1 exp[− 1 F (φ) N,u 2

X

˜ + ub )], V (∇φ(b)

˜ φ˜ = {φ(x)} x∈0 N \{0} .

b∈0 ∗ N

The infinite volume properties are summarized in σ ∈ C 1 (Rd ), Theorem 3.4. (0) limN →∞ σN (u) = σ(u), (i) σis convex, (ii) E µu [η(b)] = ub , (iii) E µu [V 0 (η(eα ))] = σ,α (u), Pd (iv) E µu [ α=1 η(eα )V 0 (η(eα ))] = u · ∇σ(u) + 1, (v) |∇σ(u)| ≤ C(1 + |u|), |∇σ(u) − ∇σ(v)| ≤ C|u − v|, for some C > 0, (vi) u · ∇σ(u) ≥ c− |u|2 − 1. Proof. Noting that µN,u converges weakly to µu as N → ∞ and using the uniform bound (3.4), we see that E µN,u [V 0 (η(eα ))] converges to E µu [V 0 (η(eα ))] as N → ∞ and is uniformly bounded for |u| ≤ U, U > 0. Therefore, applying Lebesgue’s dominated convergence theorem to Z

1

σN (u) − σN (u) ¯ =

(u − u) ¯ · ∇σN (tu + (1 − t)u) ¯ dt,

u, u¯ ∈ Rd ,

0

we establish the existence of the limit in (0) and obtain, with Lemma 3.1-(ii), Z

1

σ(u) − σ(u) ¯ = 0

d X

(uα − u¯ α )E µtu+(1−t)u¯ [V 0 (η(eα ))] dt.

α=1

In Appendix II, we will prove that the σ(u) as defined in (0) agrees with σ(u) as defined in Eq. (1.3). This yields then also the convexity (i). E µu [V 0 (η(eα ))] is continuous in u, which can be seen from the fact that µun =⇒ µu as un → u. Therefore, σ ∈ C 1 (Rd ) and the identity (iii) is established. The identities (ii) and (iv) follow from Lemma 3.1 by letting N → ∞. The second bound in (v) follows from (iii) and Proposition 2.1 noting ¯ α ))| ≤ c+ |η(eα ) − η(e ¯ α )|. The first bound in (v) is an immediate that |V 0 (η(eα )) − V 0 (η(e consequence of the second one taken v = 0. Finally, since ηV 0 (η) ≥ c− η 2 ,

18

T. Funaki, H. Spohn

u · ∇σ(u) =

d X

E

µu

0

[η(eα )V (η(eα ))] − 1 ≥ c−

α=1

d X

E µu [η(eα )2 ] − 1

α=1

≥ c−

d X

(E µu [η(eα )])2 − 1 = c− |u|2 − 1.

α=1

4. Coupled Local Equilibria We move now towards the proof of Theorem 1.1 and consider the SDE (2.2) on the lattice torus 0 N with N = [−1 ]. If the initial measure µ0 satisfies (1.6), one would expect that at later times the interface has locally a definite tilt u and a statistics as specified by the Gibbs measure µu . Such a strong property will come out only indirectly. However for the space-time averaged measure we will establish that it is some mixture of Gibbs measures. In fact such a property will be established for the measure coupled to the solution of a discretized version of the PDE (1.7), see Theorem 4.1 for a precise statement. We first have to explain the discretization scheme for the PDE and supply two preparatory lemmas. 4.1. Uniform bound on second moments. Let µt ∈ P(X0 N ) be the distribution of ∇φt on X0 N and AvT (µ ) be its space-time average, Z −2 T X 1 (−2 T )−1 µt ◦ τx dt, T > 0. Av(µ ) = T | 0 N | x∈0 0 N

Here τx denotes the shift by x on 0 N . µ ∈ P(X0 N ) is always regarded as µ ∈ P(X ) by extending it periodically. We shall simply denote by µN = µN,0 ∈ P(X0 N ) the finite volume Gibbs measure with periodic boundary conditions and tilt u = 0 (see proof of Theorem 3.2). To obtain uniform L2 -bounds, we again use a coupling argument for the SDE (2.2) on 0 N . Assume that two initial data (R0 N -valued random variables) φ0 = {φ0 (x)}x∈0 N and φ¯ 0 = {φ¯ 0 (x)}x∈0 N are given and let φt and φ¯ t be the corresponding two solutions of the SDE (2.2) on 0 N with common Brownian motions. The macroscopic φ-fields obtained from φt and φ¯ t by scaling in space, time and magnitude as in (1.5) are denoted by φ (θ, t) and φ¯ (θ, t), θ ∈ Td , respectively. Recall that || · || denotes the norm of the space L2 (Td ). Lemma 4.1. (i)

We have

E[||φ (t) − φ¯ (t)||2 ] ≤ E[||φ (0) − φ¯ (0)||2 ].

(ii) Assume the condition (1.6) on the distribution µ0 of φ0 . Then, sup E AvT (µ ) [η(b)2 ] < ∞,

b ∈ (Zd )∗ .

0 1,

|η(eα )(σ a )0α (u)|q ≤ C(1 + |u|p + |η(eα )|2 ), |V 0 (η(eα ))uα |q ≤ C(1 + |u|p + |η(eα )|2 ).

q = 2p/(2 + p) > 1,

Therefore uα (σ a )0α (u), η(eα )(σ a )0α (u), and V 0 (η(eα ))uα are uniformly integrable with respect to the probability measures {p,a }0 0, such that for any 1 , 2 ≤ 0 , L0 (p, 1 ) L0 (p, 2 )

(1.6)

as p → pc , for two-dimensional independent short-range percolation models. From now on, we will suppress in the notation the dependence of L0 on and write simply L0 (p). Another result from [16], which will be important for us is:

Non-Gaussian Limiting Behavior of Percolation Threshold in Large System

77

Theorem 1.5. In two dimensions, and for small , L0 (pc − y, ) L0 (pc + y, )

(1.7)

as y ↓ 0. From now on, whenever we discuss two-dimensional models, we will assume that is small enough for the two above asymptotic relations to hold. We will also use a very recent result: Theorem 1.6. For d = 2,

L0 (p) ξ(p)

(1.8)

as p → pc . This new result has been communicated to the authors by K. Alexander [2]. We emphasize that for d > 2 no analogous result is known. In general, different definitions of correlation length are known to be equivalent only up to possible logarithmic corrections (see [4] and also [19]). We shall now discuss some assumptions on the behavior of the correlation length(s) as p → pc . Rigorous mathematical study of the function ξ (or L0 ) is a very hard problem. At the same time a powerful physical theory – the renormalization group method [18] – offers predictions concerning critical behavior in percolation and other models. While only some of these predictions have been rigorously verified, many others are virtually universally trusted. Among such noncontested statements is existence of critical exponents. In what follows, we restrict the discussion to the critical exponent ν and to the correlation length L0 . Roughly speaking, it is believed that as p → pc , L0 (p) behaves as a power of |p − pc |, possibly with some corrections of an order of magnitude lower than any power. More precisely, it is believed that both limits log L0 (p) p→pc − log(pc − p)

(1.9)

log L0 (p) p→pc + log(p − pc )

(1.10)

ν− = − lim and

ν+ = − lim

exist. Their values have been calculated numerically for various models ([20, 23]) and found to depend only on the dimensionality of the system (universality); moreover with high accuracy ν+ = ν− (note that in two dimensions the last statement follows from the relation L( pc − y, ) L0 (pc + y, )); this common value is denoted by ν and called the correlation length exponent. While existence of ν has not been rigorously proven, we think it is useful to assume it in some form and prove rigorous results contingent on this assumption. The assumption we make implies existence of critical exponents. We 1 , do not want to favor any explicit form of L0 (p) (like |p − pc |−ν or |p − pc |−ν log |p−p c| etc.), so we will make a much less restrictive assumption, consistent with all explicit formulas postulated or heuristically derived in the literature. See [24] for a recent study of critical behavior of percolation models, where some results use assumptions related to ours. Assumption 1.1. (regularity) L0 (p + δ) is a regularly varying function of δ, as δ → 0 from above and from below. This means, by definition [9] that for every x 6= 0, the limit limt→0 LL00(p(pcc+tx) +t) exists.

78

L. Berlyand, J. Wehr

Regular variation implies [9] that there exist two exponents, ν− and ν+ , such that L0 (pc + δ) = δ −ν+ S(δ); and

δ>0

L0 (pc + δ) = (−δ)−ν− S(δ);

δ < 0,

(1.11) (1.12)

where S is a slowly varying function of δ at 0, i.e. for every x 6= 0, lim

t→0

S(tx) = 1. S(t)

(1.13)

The last statement clearly implies that for any > 0. t− < S(t) < t ,

(1.14)

if |t| is sufficiently small (depending on ). It also follows from representations (1.11) and (1.12) that the limit of LL00(p(pcc+tx) +t) (whose existence is postulated in the definition of a regularly varying function) equals (−x)−ν− for x < 0 and x−ν+ for x > 0. In view of (1.11) and (1.12) we say that L0 (p + δ) is a regularly varying function of δ with the exponent ν− as δ → 0− and with the exponent ν+ as δ → 0+. The existence of critical exponents thus follows from the regularity assumption, which in particular admits all power laws with logarithmic corrections. Such corrections are indeed known to appear in critical phenomena, especially at upper critical dimensions [18]. For example, the correlation length in weakly coupled four-dimensional φ4 1 1 lattice field theory satisfies the asymptotic relation ξ(δ) δ − 2 (log δ) 6 [13]. We remark here that Assumption 1.1 takes into account not only the size but also the nature of the possible correction to a power law. All such corrections discussed in physical literature (logarithms and their powers) satisfy the assumption. However, as pointed out by the referee of this paper, from the mathematical point of view not all power laws with corrections of at most logarithmic order satisfy the assumption. For example, the monotone function δ −ν (2 + cos δ1 ) is not a regularly varying function of δ as δ → 0, even though it differs from a pure power only by a factor bounded from below and from above (by 1 and 3). While there are no physical grounds to expect such form of the correlation length, no rigorous argument excluding it is presently available. The concept of a correlation length leads naturally to that of a critical interval: Definition 1.5. For a positive L, define p− c (L) = inf{p : L0 (p) > L}

(1.15)

p+c (L) = sup{p : L0 (p) > L}.

(1.16)

and + We call [p− c (L), pc (L)] the critical interval and its width,

δ0 (L) = p+c (L) − p− c (L)

(1.17)

– the critical width corresponding to the length scale L. We will also use the one-sided critical widths (1.18) δ− (L) = pc − p− c (L); δ+ (L) = p+c (L) − pc . Of course, δ0 (L) = δ− (L) + δ+ (L).

(1.19)


79

p+c (L) and p− c (L) play the role close to that of inverse functions of L0 on the intervals p < pc and p > pc respectively. They are defined for arbitrary positive real, and not only for integer values of L. Note that L0 is not a strictly increasing function (it assumes only integer values) and therefore it has no inverse function. Moreover, since L0 does not have to assume all integer values, we cannot even guarantee that L0 (p± c (L)) = L. The following proposition shows, however, that the last equation is at least approximately correct and shows that δ± inherit the regular variation property from L0 . Proposition 1.2. For any integer L,

As L → ∞,

L0 (p± c (L)) ≤ L.

(1.20)

L0 (p± c (L)) → 1. L

(1.21)

Moreover, δ± is a regularly varying function of L with the exponent − ν1± , i.e. for any t > 0, δ± (tL) − 1 = t ν± . (1.22) lim L→∞ δ± (L) The last statement is equivalent to saying that (compare (1.11) and (1.12)) − ν1

δ± (L) = L

±

S± (L),

(1.23)

where S± are slowly varying functions of L as L → ∞ [9]. We prove the proposition in Appendix 2. The next theorem estimates fluctuations of p(L) c in terms of the critical width δ0 (L). Theorem 1.7. Let d = 2 and suppose Assumption 1.1. holds. There exist positive constants c and C, independent of L, such that for every L, 2 2 cδ02 (L) ≤ E[(p(L) c − pc ) ] ≤ Cδ0 (L).

(1.24)

Proof will be given in Sect. 3. Even assuming their existence, not much is known about the values of the critical exponents ν− and ν+ (one known inequality, ν > 1, rigorously established in two dimensions, will be used in the proof of Theorem 1.7). On the other hand, extensive numerical studies give consistent values. As mentioned above, these studies strongly suggest that ν− = ν+ . We emphasize that in two dimensions, once the critical exponents ν± exist, they have to be identical, by virtue of Theorem 1.5. Also, the numerical value of ν = ν− = ν+ in two dimensions is 1.33, a result further confirmed by conformal field theory calculations [7] (see also [22]), which yield an exact value ν = 43 (there is, however, no rigorous proof of the last equality). The following statement, used in the proof of the next theorem is thus a generally believed assumption: Assumption 1.2. In two dimensions ν < 2. Remark 1.1. It is clear from the definition of L0 that ν is a monotone decreasing function of the dimension (if it exists) and therefore Assumption 1.2 implies that ν < 2 in all dimensions d ≥ 2.

80


q Theorem 1.8. Suppose Assumptions 1.1 and 1.2 hold. Let σL = V ar[p(L) c ] be the (L) 2 standard deviation of pc (V ar[X] = E[(X − E[X]) ] denotes the variance of a p(L) −E[p(L) ]

random variable X). Then, in d = 2 the sequence of random variables c σL c does not converge in distribution to a nondegenerate Gaussian variable (nor does it have a subsequence convergent in distribution to such a variable). The proof is given in Sect. 4. Notation. Throughout the paper we follow the usual custom of denoting arbitrary positive constants in estimates by c. The actual value of c may vary from one equation to another and, sometimes, when this does not cause a confusion, also within one equation. c may depend on the dimensionality of the model, but not on the size of the system L or on the value of bond density p. Also, it may change its value when the arguments are applied to models other than the independent bond model.

2. Law of Large Numbers and Large Deviation Bounds In this section we prove Theorem 1.3. The usual coupling of percolation models with different values of p ([10], pp. 10–11) could be used to obtain an alternative proof. The advantage of the proof below is that it yields explicit bounds on large deviation probabilities, which are also used later in Sect. 3. The idea of the proof is to estimate the probability that p(L) c deviates from pc by probabilities of crossing events in the independent bond percolation model with appropriately chosen bond density p. Notation. We will use the following notation for events in the space of bond configurations in the box ΛL : Γ will denote the event that there exists an L-R open crossing of ΛL , Ak – the event that there are exactly k open bonds and A¯ k – the event that there are at least k open bonds, i.e. A¯ k is the disjoint union of Al with l = k, k + 1, . . . , n. Pp will denote the probability measure on bond configurations in ΛL in the independent bond model, i.e. a product of n Bernoulli measures with probability p of any given bond being open. P will denote the uniform measure on the set Ωn of ordered bond sequences introduced in Sect. 1. P [E|F ] will denote the probability of E conditioned on F and similarly for Pp . A complement of a set S will be denoted by S c . Finally, #(S) will denote the number of elements in a set S. Proof of Theorem 1.3. Fix a δ > 0. We will only prove that lim P [p(L) c < pc − δ] = 0.

(2.1)

lim P [p(L) c > pc + δ] = 0

(2.2)

L→∞

The proof that L→∞

is analogous. Choose m so that pc − δ ≤

m 3δ < pc − . n 4

(2.3)

m This is always possible for large L (here n = n(L)). Now, p(L) c ≤ n for a given ordered sequence of bonds implies that the first m bonds in the sequence contain an L-R crossing of ΛL . Since all ordered sequences have the same probability, we obtain


(L) P [p(L) c ≤ pc − δ] ≤ P [pc ≤

#(Am ∩ Γ ) m ]= n #(Am )

81

(2.4)

and multiplying the numerator and the denominator by pm (1 − p)n−m , we obtain P [p(L) c ≤

Pp [Γ ∩ Am ] m ]= = Pp [Γ |Am ]. n Pp [Am ]

(2.5)

We emphasize that (2.5) holds with any choice of p ∈ (0, 1). Let us choose p = pc − δ2 . Then Pp [C] = πL (pc − δ2 ). In order to obtain from here a good estimate on Pp [Γ |Am ], let us note first that Pp [Γ |Am ] ≤ Pp [Γ |A¯ m ]. (2.6) This is intuitively obvious, since conditioning on an event with more open bonds should make an open crossing more likely. For a formal proof, note that for any i all con figurations in Ai have the same probability pi (1 − p)(n−i) and since #(Ai ) = ni , i#(Ai ) = (n − i + 1)#(Ai−1 ). Since Γ is an increasing event, i.e. adding open bonds to a configuration in Γ we obtain another element of Γ , we have Pp [Γ |Ai ] =

(n − i + 1)#(Γ ∩ Ai−1 ) #(Γ ∩ Ai ) ≥ = Pp [Γ |Ai−1 ]. #(Ai ) i#(Ai )

(2.7)

TakingSa convex combination of inequalities (2.7) with i = m, . . . , n we get (using n A¯ m = i=m Ai ): Pn i=m Pp [Γ ∩ Ai ] ¯ Pp [Γ |Am ] = P n j=m Pp [Aj ] n X Pp [Γ ∩ Ai ] P [A ] Pn p i = Pp [Ai ] j=m Pp [Aj ] i=m n X Pp [Ai ] ≥ Pp [Γ |Am ] = Pp [Γ |Am ] Pp [A¯ m ] i=m

which proves (2.6). With our choice of p, a standard bound on the probability of a large deviation for a sum of i.i.d. random variables based on the exponential Chebyshev inequality (see [8], Ch.1, Sect. 9) implies Pp [A¯ m ] ≥ 1 − e−cδ

2

n

(2.9)

with a strictly positive constant c. It follows that (using (2.4), (2.5), (2.6) and (2.9)) ¯ P [p(L) c ≤ pc −δ] ≤ Pp [Γ |Am ] ≤ Pp [Γ |Am ] ≤

Pp [Γ ] ≤ cPp [Γ ] = cπL (p), (2.10) Pp [A¯ m ]

with p = pc − δ2 , where c is an absolute constant. Since the last quantity goes to zero when L → ∞ (Theorem 1.2), the theorem is proven. It follows from the presented proof (using Proposition A1.1) that convergence to pc takes place with exponential bounds on large deviation probabilities. These exponential bounds will be used in the next section. The following corollary shows that a law of large numbers also holds when we subtract the means from p(L) c .

82


(L) Corollary 2.1. p(L) c − E[pc ] converges to 0 in probability.

Proof. For any η > 0, (L) (L) |pc − E[p(L) c ]| ≤ E[|pc − pc |] ≤ ηP [|pc − pc | ≥ η] + η.

(2.17)

This implies that lim E[p(L) c ] = pc

L→∞

and the corollary follows.

(2.18)

3. Order of Fluctuations In this section we prove Theorem 1.7. Our proof of the lower bound in Theorem 1.7 does not use the assumption d = 2 except for the inequality ν > 1. It can be carried out in any dimension, in which the inequality ν > d2 is satisfied and we will present it in this generality. The condition ν > d2 was introduced by Harris in [14], in relation to fluctuations of the critical temperature of finite magnetic systems with site dilution – a quantity closely related to the percolation threshold studied here. The weak inequality ν ≥ d2 was subsequently proven for a large class of systems, including percolation models in all dimensions d ≥ 2 [5]. In two dimensions the inequality ν > 1 has been rigorously established in [16]. However, since ν is known to be 21 in high dimensions [12], the strict inequality holds when d is large. The numerical value of ν in three dimensions is 0.9 [23], which would make the inequality strict also in three dimensions. The following assumption is therefore well-founded: ν > d2 .

Assumption 3.1. (Harris condition)

The following proposition contains the lower bound claimed in Theorem 1.4: Proposition 3.1. Suppose Assumption 3.1 holds (as mentioned above, this is known to be true in two dimensions). Then there exists a constant c > 0, independent of L, such that for all L, 2 2 (3.1) E[(p(L) c − pc ) ] ≥ c[δ0 (L)] . Proof. We will use the following integral representation of the second moment of a random variable X in terms of its distribution function (see [8], Ch.1, Lemma 5.7): Z ∞ 2 2yP [|X| ≥ y] dy. (3.2) E[X ] = 0

With X =

p(L) c

− pc , we obtain E[(p(L) c

Z

pc 0

Z

− pc ) ] = 2

0

∞

2δP [|p(L) c − pc | ≥ δ] dδ = Z

2δP [p(L) c ≤ pc − δ] dδ +

1−pc 0

2δP [p(L) c ≥ pc + δ] dδ.

(3.3)

Notation. In what follows the first and the second term on the right-hand side of (3.3) will be denoted by I1 and I2 respectively. Let 0 < δ < pc . Take an m so that


83

m 5 ≤ pc − δ. pc − δ ≤ 4 n

(3.4)

4 This is possible for a fixed L, whenever δ > n(L) (note that n(L) behaves like cLd ). Proceeding as in the proof of Theorem 1.3. (using (2.5) and (3.4)), we obtain for an arbitrary p, ˜ (3.5) P [p(L) c ≤ pc − δ] ≥ Pp [Γ |Am ] ≥ Pp [Γ |Am ],

where A˜ m denotes the event that a configuration in ΛL has at most m open bonds. The inequality Pp [Γ |Am ] ≥ Pp [Γ |A˜ m ] is proven similarly to (2.6). Using the obvious inequality Pp [A ∩ B] ≥ Pp [A] − Pp [B c ], we get Pp [Γ |A˜ m ] ≥

Pp [Γ ] − Pp [A˜ cm ] . Pp [A˜ m ]

(3.6)

Let us choose p = pc − 23 δ. Just like in (2.9), a standard large deviation estimate implies that with this choice 2 (3.7) Pp [A˜ m ] ≥ 1 − e−ncδ with a positive constant c independent of n and δ. Combining (3.4) and (3.6) and noting that Pp [C] = πL (p), we arrive at the inequality 3 0 −ncδ 2 ], P [p(L) c ≤ pc − δ] ≥ c [πL (pc − δ) − e 2

(3.8)

where c0 > 0 is an absolute constant. Thus Z pc −cL−d 2 3 δ[πL (pc − δ) − e−ncδ ] dδ. I1 ≥ c0 2 0

(3.9)

The reason for subtracting cL−d from pc in the upper limit of integration is that we have 4 to prove (3.8). Since for p > p− used δ > n(L) c (L) (see (1.15)), πL (p) ≥ (where is chosen as described in Definition 1.4.), the last expression is bounded below by c0

Z

2 3 δ− (L)

cL−d

δ dδ − c0

Z

pc

e−cL

d 2

δ

δ dδ ≥ c00 [δ− (L)]2 − c0

0

Z

pc

e−cL

d 2

δ

δ dδ. (3.10)

0

To estimate the first term we used the fact that L−d δ− (L) → 0, which follows from (1.23) together with the Harris condition ν > d2 . Changing the variable of integration to 1 d u = c 2 L 2 δ, we estimate the integral in the second term as follows Z pc Z ∞ d 2 1 1 e−cL δ δ dδ ≤ e−u u du = O( d ) (3.11) d cL 0 L 0 as L → ∞. The Harris condition implies now that the first term dominates the second one, so we obtain (3.12) I1 ≥ c[δ− (L)]2 . A similar proof shows that

I2 ≥ c[δ+ (L)]2 .

(3.13)

We just sketch the argument, which is analogous to the bound on I1 : choosing m so that 5 3 this time pc + δ ≤ m n < pc + 4 δ, we have, with p = pc + 2 δ,

84


Pp [Γ c ] − Pp [A¯ cm ] c c ¯ P [p(L) . c ≥ pc + δ] ≥ Pp [Γ |Am ] ≥ Pp [Γ |Am ] ≥ Pp [A¯ m ]

(3.14)

From here, using steps analogous to (3.7)-(3.11), we obtain the desired bound (3.13), which, together with (3.12) proves the proposition. Remark 3.1. We have actually proven two separate bounds – on I1 and on I2 . In two dimensions, in view of the equivalence relation (1.6), we have δ− (L) δ+ (L) and, consequently, both bounds have the form c[δ0 (L)]2 . This will be essential in Sect. 4. The next proposition takes care of the upper bound in Theorem 1.7. Unlike the previous one, it is strictly limited to two dimensions. Proposition 3.2. Suppose Assumption 1.1 holds. Then there exists a constant C < +∞, independent of L, such that for all L, 2 2 E[(p(L) c − pc ) ] ≤ C[δ0 (L)] .

(3.15)

Proof. We use again the integral representation (3.3) of the second moment. Again, we shall just prove the bound for I1 ; the bound for I2 can be handled similarly. In Appendix 1 we show that for p < pc , −c L L(p)

πL (p) ≤ e

0

,

(3.16)

with a constant c > 0 independent of p. It follows (using first the inequality (2.10)) that Z

pc

I1 = 0

Z

Z 2δP [p(L) c

≤ pc − δ] dδ ≤ c Z

δ− (L)

≤c

pc

0 pc

δ dδ + c

δe

−c L (pLc −δ) 0

δ δπL (pc − ) dδ 2 dδ.

(3.17)

δ− (L)

0

The first term is proportional to δ0 (L)2 and we just need to estimate the second term. Using the regularity assumption, together with (1.11), (1.12) and (1.20), we can bound it by Z

L0 (pc − δ− (L)) ] dδ = c δ exp[−c c L0 (pc − δ) δ− (L) pc

Changing the variable to z =

δ δ− (L) ,

Z cδ− (L)

1

pc

δ exp[−c( δ− (L)

δ ν S(δ− (L)) ) ] dδ. δ− (L) S(δ) (3.18)

we obtain

pc δ− (L)

2

Z

z exp[−cz ν

S(δ− (L)) ] dz. S(zδ− (L))

(3.19)

It follows from the proposition proven in Appendix 2 that the integral in the last formula is bounded as L → ∞ and therefore, Proposition 3.2. is proven.


85

4. The Distribution of p(L) c for Large L In this section we study possible limit theorems for the sequence of finite-volume percolation thresholds p(L) c and, in particular, we prove Theorem 1.8. Under assumptions made in Sect. 1, we show that no nontrivial Gaussian limiting behavior is possible in two dimensions (of course, choosing an appropriate normalization, we can obtain in the limit a degenerate distribution, which is, by definition, Gaussian; this is the case, e.g. in Theorem 1.3). Let p(L) − pc . (4.1) ηL = c δ0 (L) Our strategy is to prove that this sequence of random variables is relatively compact and therefore contains a subsequence converging in distribution. Next, we will show that the limit of each such convergent subsequence has a nondegenerate distribution, which is not normal. Finally, we will extend the result about non-Gaussian limiting behavior to other normalizations of p(L) c using the Convergence of Types Theorem ([8], Ch. 2, Theorem 7.16). Proposition 4.1. The sequence ηL contains a subsequence convergent in distribution. Proof. It follows from Proposition 3.2 that second moments of ηL are bounded by C. The proposition follows from a well known relative compactness criterion (see [9], Vol. II, Ch. 8 or [8], Ch. 2, Theorem 2.7). Proposition 4.2. No subsequence of ηL converges in distribution to a constant random variable. Proof. Proceeding exactly as in the proof of Proposition 3.2, we can show that the sequence of fourth moments of ηL is bounded by a constant (using Proposition A2.1 with k = 3; in fact a similar proof shows that for any m the sequence E[|ηL |m ] is d bounded). This implies that if ηLK → η, then for any function f (η) which grows at ±∞ 4 slower than |η| , (4.2) E[f (ηL )] → E[f (η)] (see [8], Ch.2, Exercise 2.5). In particular, taking f (η) = η 2 , f (η) = η 2 IR+ and f (η) = η 2 IR− (where IR+ and IR− denote the indicator functions of the positive and of the negative half-line respectively), we see that 2 E[η 2 ] = lim E[ηL ],

(4.3)

2 IR+ (ηL )] E[η 2 IR+ (η)] = lim E[ηL

(4.4)

2 IR− (ηL )]. E[η 2 IR− (η)] = lim E[ηL

(4.5)

L→∞

L→∞

and L→∞

We have seen in the proof of Proposition 3.1 that the right-hand sides of the three above equations are strictly positive and, therefore, so are the left-hand sides. Equation (4.3) clearly implies that η cannot be almost surely equal to 0, and (4.4) together with (4.5) show that its distribution cannot be concentrated at any other single point. η is thus a nondegenerate random variable. Proposition 4.3. No subsequence ηL converges in distribution to a normal random variable.

86

L. Berlyand, J. Wehr d

Proof. Let ηLk → η. For any y which is a continuity point of the distribution function of η we have, using (3.8), P [η < y] = lim P [ηL < y] = P [p(L) c < pc + δ0 (L)y] L→∞

3 ≥ c[πL (pc + δ0 (L)y) − exp(−cLd (δ0 (L)y)2 )]. 2

(4.6)

We need to assume that y is a continuity point of the limiting distribution function, since we do not know that the distribution of η is continuous (see Remark 4.1). Of course, all y except for countably many satisfy this condition. Let us choose a negative y with a large absolute value. Using the lower bound in Proposition A1.1 (Appendix 1), the right-hand side of (4.6) can be bounded below by c[exp(−c

L ) − exp(−cLd (δ0 (L)y)2 )]. L0 (pc + 23 δ0 (L)y)

(4.7)

Let us recall that in two dimensions δ0 (L) δ− (L) δ+ (L) (see Remark 3.1). Using (1.21) and the representation (1.12) of the regularly varying function L0 , we obtain: L0 (pc − δ0 (L)) S(δ0 (L)y) ). ) = exp(−cy ν 3 S(δ0 (L)) L0 (pc + L0 (pc + 2 δ0 (L)y) (4.8) Since S is a νslowly varying function, the limit of the last expression as L → ∞ equals Ke−cy . On the other hand, Proposition 1.2 implies (in two dimensions) that 1 δ0 (L) = L− ν S1 (L), where S1 is a slowly varying function of L. Consequently, 2 Ld δ02 (L) = Ld− ν S1 (L) → ∞ and, therefore, for sufficiently large L, exp(−c

L

3 2 δ0 (L)y)

) exp(−c

P [ηL ≤ y] ≥

c exp(−2cy ν ). 2

(4.9)

The inequality ν < 2 (Assumption 1.2) implies now that the distribution of η cannot be Gaussian. Note, that the right tail of the distribution of η can be estimated in the same way, using an appropriate analog of Proposition A1, satisfied in two dimensions by duality. Theorem 4.1. Let bL and aL be arbitrary sequences of real numbers, with bL > 0. No p(L) −a subsequence of c bL L converges in distribution to a nondegenerate normal random variable. Proof. This is a standard application of the Convergence of Types Theorem (see [8], Ch. 2, Theorem 7.16). We have k) pc − a L k − aLk δ0 (Lk ) p(L c = ηLk + , bL k bL k bL k

p

(Lk )

(4.10)

−a

so that the random variables c bL Lk are related to ηL by affine transformations. If both k sequences converge in distribution to nondegenerate random variables, the Convergence p −a k) of Types Theorem implies that δ0b(L converges to a nonzero constant and cbL Lk L k

converges to a constant and, consequently, if the limit of of ηLk .

(Lk )

pc

k

−aLk

bL k

is normal, so is that


87

Theorem 1.8 is an immediate corollary of Theorem 4.4. Remark 4.1. We briefly discuss here the issue of continuity of the limiting distribution function and its relation to assumptions made in literature. If we assume that in addition to the relation ξ L0 (proven in two dimensions) the correlation length ξ behaves as a pure power, ξ(p) |p − pc |−ν , and that 1

lim L ν

L→∞

dπL dp

(4.11)

exists (these assumption are not used in the rest of this paper; see [17] for discussion of numerical results), then equicontinuity of the functions πL (δ0 (L)y) implies continuity of the limit F (y) of this sequence (or any limit of its subsequence) by the Arzelà-Ascoli theorem. On the other hand, bounds of Sect. 2 show that 3y y F ( ) ≤ lim inf P [ηL ≤ y] ≤ lim sup P [ηL ≤ y] ≤ F ( ), L→∞ 2 2 L→∞

(4.12)

and an obvious modification shows that for an arbitrary α > 0, F ((1 − α)y) ≤ lim inf P [ηL ≤ y] ≤ lim sup P [ηL ≤ y] ≤ F ((1 + α)y), L→∞

(4.13)

L→∞

which implies that any limit of the distribution functions P [ηL ≤ y] is continuous. While Theorems 1.7 and 1.8 are limited to two dimensions, because of the use of the Rescaling Lemma and duality, large parts of their proofs remain valid in higher dimensions. Plausible consequences for the behavior of the finite-volume percolation threshold, based on available rigorous and numerical information about the value of the correlation length exponent ν were discussed by the present authors in [3]. As pointed out there, since the numerical value of ν in three dimensions is close to 0.9 and smaller than that in higher dimensions (in particular, ν = 21 above 5 dimensions, according to a well known prediction, see [12]), one may expect the limiting distribution to be even more radically non-Gaussian in higher dimensions (see Remark 1.1). In [21] the moments of the limiting distributions have been studied numerically and found to behave in a non-Gaussian way. In three dimensions the limiting distribution was even found (numerically) to be asymmetric [11]. Both results support the predictions of [3]. Such a non-Gaussian behavior in high dimensions would stand in a sharp contrast to the behavior of such extensive (additive) quantities like magnetization or energy of a spin system, which are known to be Gaussian high dimensions. However, the finite-volume percolation threshold is a highly nonlocal function of the bond variables, to which an analogy to the central limit theorem for sums of weakly dependent random variables needs not apply. Appendix 1. Exponential bounds on crossing probabilities In this appendix we use the technique of rescaling [1] to prove bounds on crossing probabilities, used in Sects. 3 and 4. Proposition A1.1. Let d = 2. There exist strictly positive constants k, c1 and c2 , independent of p, such that for p < pc ke

−c1 L L(p) 0

−c2 L L(p)

≤ πL (p) ≤ e

0

.

(A1.1)

88


Remark. The lower bound was communicated to us together with its proof by L. Chayes. Proof. Let πL,M denote the probability that there is an open crossing in a given rectangle L×M (L units in the horizontal and M units in the vertical direction). Consider the dual percolation model (see [15, 10]). This is also a two-dimensional independent percolation model (bond percolation is self-dual, in the sense that its dual is also a bond percolation model; the dual of site percolation is known as star percolation, etc.). The dual of a model with density p (of open bonds, occupied sites, etc.) has density 1 − p. If we denote the ∗ , then crossing probabilities of the dual model by πL,M ∗ πL,M (p) + πM,L (1 − p) = 1.

(A1.2)

It is convenient to use crossing probabilities of elongated boxes, in view of the following Rescaling Lemma. There exists an 0 > 0 such that for p < pc . Then, if πL,2L (p) ≤ 0 , we have 2 . (A1.3) π2L,4L ≤ πL,2L A proof of the Rescaling Lemma can be found, e.g. in [1]. It is written there for bond percolation, but no changes are necessary to cover other independent two-dimensional models considered here. Clearly, πL (p) ≤ πL,2L (p). On the other hand, using the FKG inequality and the Russo-Seymour-Welsh lemma [10] for the dual model, we have p ∗ ∗ ∗ ∗ (1 − p))6 (1 − p) ≥ πL (1 − p)(π ∗3 L,L (1 − p))2 ≥ πL (1 − p)(1 − 1 − πL π2L,L 2 p ∗ (1 − p))7 . ≥ (1 − 1 − πL (A1.4) We have used here a well-known technique of creating and L-R crossing of a 2L × L box from L-R crossings of two overlapping 23 L × L boxes and a vertical crossing of a √ √ (middle) L × L box [10]. Since (1 − πL (p))7 ≥ 1 − 7 πL (p) (Bernoulli inequality), using the duality relation (A1.3), we obtain p πL,2L (p) ≤ 7 πL (p). (A1.5) We now introduce an analog of L0 using crossing probabilities of elongated boxes: def

L1 (p, ) = min{L : πL,2L (p) ≤ }.

(A1.6)

Comparing (A1.6) to Definition 1.4, we see, using (A1.5) that L0 (p, ) ≤ L1 (p, ) ≤ L0 (p,

2 ), 49

(A1.7)

which, in view of Theorem 1.4, proves that L0 (p) L1 (p).

(A1.8)

Provided that in the definition of L1 has been chosen sufficiently small, the Rescaling lemma implies 2k , (A1.9) π2k L1 (p),2k+1 L1 (p) (p) ≤ πL 1 (p),2L1 (p)


89

which yields an upper bound −c L L(p)

πL,2L (p) ≤ e

1

(A1.10)

for L of the form 2k L1 (p) (with c = − log > 0) and hence also (perhaps with a different constant c) for arbitrary L. Since πL (p) ≤ πL,2L (p) this proves the desired upper bound, in view of the asymptotic equivalence of L0 and L1 . To prove the lower bound, note first that the above inequalities between πL (p) and πL,2L (p) imply that the correlation length ξ1 (p) defined by log πL,2L (p) (A1.11) ξ1−1 (p) = − lim sup L L→∞ is asymptotically equivalent to ξ defined in (1.5) (as for ξ, the choice of lim inf would also yield an asymptotically equivalent quantity). It follows from (A1.11) that for large L log πL,2L (p) ≥ −2ξ1−1 (p), (A1.12) L which implies −2 ξ L(p)

πL,2L (p) ≥ e

1

(A1.13)

for large L. In view of the asymptotic equivalence of ξ1 to ξ (and, therefore, also to L0 ), this implies the desired lower bound in (A1.1) (the constant k may be necessary to accommodate the finite number of values of L for which (A1.13) does not hold). Appendix 2. Regularly Varying Functions In this Appendix we state a fundamental representation theorem for slowly varying functions and use it to prove a bound necessary in the proof of Proposition 3.2. We then give a proof of Proposition 1.2. Theorem (Karamata). Let S(x) be a slowly varying function, x ≥ 0. Then there exist functions a and , such that limx→0 (x) = 0; limx→0 a(x) = c < ∞ and for x ≤ 1, Z

1

S(x) = a(x) exp( x

(y) dy). y

(A2.1)

This theorem is proven in [9]; one has to change variables from x to x1 to obtain the present formulation. It also follows from the proof in [9] that a and can be chosen so that c > 0. Proposition A2.1. Let S(z) be a slowly varying function, (z ≥ 0), bounded above on intervals [z0 , ∞] for all z0 > 0. c and ν – positive constants and k – a positive integer. Let Z ∞ S(t) ] dz. (A2.2) z k exp[−cz ν J(t) = S(zt) 0 Then lim sup J(t) < ∞. t→0

(A2.3)

90


Proof. We use the representation (A2.1), where with no loss of generality we can assume that (x) ≥ C where C is a (negative) constant. Fix a µ ∈ (0, ν). There exists x0 such that for x ≤ x0 we have (x) ≥ −µ. Now let z ≤ xt0 . We have Z 1 Z 1 a(t) S(t) (y) (y) = exp( dy − dy) S(tz) a(tz) y t tz y Z tz a(t) (y) exp( ) dy) ≥ K exp(−µ log z), = a(tz) y t where K is a positive constant. Hence, for z ≤ zν

(A.2.4)

x0 t ,

S(t) ≥ Kz ν−µ . S(tz)

(A2.5)

As a result we obtain a finite, t-independent bound on a part of our integral: Z

x0 t

1

S(t) ) dz ≤ z exp(−z S(tz) k

Z

ν

Z

1

≤

x0 t

∞

z k exp(−Kz ν−µ ) dz z k exp(−Kz ν−µ ) dz.

(A.2.6)

1

The remaining part of the integral can be estimated by a similar application of the Karamata theorem or directly as follows: when zt > x0 , we have S(zt) < c−1 , where c is a positive constant (by assumption). Also, for small t we have S(t) ≥ tµ ≥ xµ0 z −µ . Hence for small t the remaining part of the integral (from xt0 to ∞ is bounded by Z ∞ z k exp(−cxµ0 z ν−µ ) dz < ∞ (A2.7) 1

and the proof is finished.

Proof of Proposition 1.2. To prove (1.20), let pn ↑ p− c (L) with pn < p. Then, by definition of p− c (L), L0 (pn ) ≤ L, i.e. (see Definition 1.4) ∃Ln ≤ L : πLn (pn ) ≤ . Since there are only finitely many integers between 0 and L, Ln is equal to some M ≤ L for infinitely many n. Taking a subsequence and letting n to infinity, we obtain, using continuity of πL as a function of p, that πM (p− c (L)) ≤ , which, in view of Definition + 1.4 implies (1.20) for p− c (L). The proof for pc (L) is analogous. Equation (1.21) will be proven for δ− ; the proof for δ+ is analogous. For any θ < 1, the definition of δ− (see (1.15) and (1.18)) implies that L0 (pc − θδ− (L)) > L.

(A2.8)

Hence, using (1.13), lim inf L→∞

L0 (p− L0 (pc − δ− (L)) c (L)) ≥ lim inf = θν . L→∞ L0 (pc − θδ− (L)) L L (p− (L))

(A2.9)

≥ 1, which in combination with Taking the limit θ → 1 implies lim inf L→∞ 0 Lc (1.20) proves (1.21). To prove the last statement of the proposition, we need to show that for any t > 0,

Non-Gaussian Limiting Behavior of Percolation Threshold in Large System 1 δ− (tL) → t− ν . δ− (L)

91

(A2.10)

We argue by contradiction. If (A2.10) does not hold, then either lim sup L→∞

or lim inf L→∞

1 δ− (tL) ≥ (1 + α)t− ν δ− (L)

(A2.11)

1 δ− (tL) ≤ (1 − α)t− ν δ− (L)

(A2.12)

for some α > 0. In the first case, regular variation of L0 with the exponent −ν implies that L0 (pc − δ− (tL)) L0 (pc − (1 + α)t− ν δ− (L)) ≤ lim inf = (1+α)−ν t. (A2.13) lim inf L→∞ L0 (pc − δ− (L)) L→∞ L0 (pc − δ− (L)) 1

We have used here the fact that for p < pc , L0 is a (not strictly) increasing function of p, which follows from Definition 1.4 and monotonicity of πL in p. But (1.21) implies that lim inf L→∞

L0 (pc − δ− (tL)) = t, L0 (pc − δ− (L))

(A2.14)

so (A2.11) leads to a contradiction. Similarly, (A2.12) implies that lim sup L→∞

L0 (pc − δ− (tL)) ≥ (1 − α)−νt, L0 (pc − δ− (L))

(A2.15)

which again contradicts (A2.14). This proves (A2.10) and the last part of the theorem for δ− . The proof for δ+ is similar. Acknowledgement. We are pleased to thank K. Alexander, L. Chayes, H. Kesten, A. Kupiainen and D. Stauffer for very helpful discussions and for providing us with necessary references. We are particularly grateful to H. Kesten who read the manuscript and made several valuable suggestions. We are also very grateful to the referee whose comments and constructive criticism of the first version of the paper led to significant improvements in organization of proofs and helped us clarify the status of Assumption 1.1. Part of the work was done while L.B. was visiting the Department of Mathematics of the University of Arizona and he is grateful for the hospitality received there.

References 1. Chayes, J.T. and Chayes, L.: Percolation and random media. In: Osterwalder, K. and Stora R. (eds.): Critical Phenomena, Random Systems, Gauge Theories. Amsterdam: North Holland, 1986 2. Alexander, K.: Private communication 3. Berlyand, L., Wehr, J.: The probability distribution of the percolation threshold in a large system. J. Phys. A: Math. Gen. 28, 7127 (1995) 4. Chayes, J.T., Chayes, L. and Fröhlich, J.: The low-temperature behavior of disordered magnets. Commun. Math. Phys. 100, 399 (1985) 5. Chayes, J.T., Chayes, L., Fisher, D. and Spencer, T.: Finite-size scaling and correlation lengths for disordered systems. Phys. Rev. Lett. 57, 3002 (1986) 6. Chayes, L.: Private communication 7. Saleur, H.: Conformal invariance for polymers and percolation. J. Phys. A: Math. Gen. 20, 455 (1987) 8. Durrett, R.: Probability. Theory and Examples. Second Edition, Wadsworth, 1996 9. Feller, W.: An Introduction to Probability Theory and Its Applications, v.2. New York: Wiley, 1971

92


10. Grimmett, G.: Percolation. Berlin–Heidelber–New York: Springer, 1989 11. Haas, U.: The distribution of percolating concentrations in finite systems. Physica A 215, 247 (1995) 12. Hara, T. and Slade, G.: Mean-field critical behaviour for percolation in high dimensions. Commun. Math. Phys. 128, 333 (1991) 13. Hara, T. and Tasaki, H.: A rigorous control of logarithmic corrections in four-dimensional φ4 spin systems II. J. Stat. Phys. 47, 98 (1987) 14. Harris, A.B.: Upper bounds for the transition temperatures of generalized Ising models. J. Phys. C 7, 3082 (1974) 15. Kesten, H.: Percolation Theory for Mathematicians. Basel–Boston: Birkhäuser, 1982 16. Kesten, H.: Scaling relations for 2D-percolation. Commun. Math. Phys. 109, 109 (1987) 17. Langlands, R., Pouliot, P., Saint-Aubin, Y.: Conformal invariance in 2D percolation. Bull. Am. Math. Soc. 30, 1 (1994) 18. Ma, S.-k.: Modern Theory of Critical Phenomena. New York: Benjamin, 1976 19. Nguyen, B.G.: Correlation length and its critical exponent for percolation processes. J. Stat. Phys. 46, 517 (1987) 20. Stauffer, D. and Aharony, A.: Introduction to Percolation Theory. Taylor and Francis, 1992 21. Ziff, R.: Spanning probabilities in 2D percolation. Phys. Rev. Lett. 69, 2670 (1992); see also Phys. Rev. Lett. 72, 1942 (1994) 22. Nienhuis, B.: Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas. J. Stat. Phys. 34, 731 (1984) 23. Isichenko, M.B.: Percolation, statistical topography and transport in random media. Rev. Mod. Phys. 64, 961 (1992) 24. Borgs, C., Chayes, J.T., Kesten, H., Spencer, J.: The birth of the infinite cluster: Finite-size scaling in percolation. In preparation Communicated by A. Kupiainen

Commun. Math. Phys. 185, 93 – 127 (1997)

Communications in


The Universal Vassiliev Invariant for the Lie Superalgebra gl(1|1) José M. Figueroa-O’Farrill1,? , Takashi Kimura2,?? , Arkady Vaintrob3,??? 1 Department of Physics, Queen Mary and Westfield College, London E1 4NS, UK. E-mail: [email protected] 2 Department of Mathematics, Boston University, 111 Cummington Street, Boston, MA 02215, USA. E-mail: [email protected] 3 Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA. E-mail: [email protected]

Received: 4 July 1996 / Accepted: 21 August 1996

Abstract: We compute the universal weight system for Vassiliev invariants coming from the Lie superalgebra gl(1|1) applying the construction of [13]. This weight system is a function from the space of chord diagrams to the center Z of the universal enveloping algebra of gl(1|1), and we find a combinatorial expression for it in terms of the standard generators of Z. The resulting knot invariants generalize the Alexander-Conway polynomial. Introduction Vassiliev in [15] initiated a study of a new class of knot invariants which attracted a lot of interest because of their ability to distinguish knots as well as their aesthetic beauty and numerous connections with other classical (as well as quantum) fields of mathematics and physics. The space of Vassiliev invariants has a natural filtration V0 ⊂ V1 ⊂ V2 · · · whose (adjoint) quotients Vn /Vn−1 can be described in terms of combinatorial objects called weight systems (as it was shown in the works of Birman–Lin [4], Bar-Natan [1] and Kontsevich [8]). Weight systems of order n are functions on the set Dn of chord diagrams — circles with n chords (unordered pairs of points) satisfying certain relations. They form a finite-dimensional vector space Wn and to describe this space (and thus to find all Vassiliev invariants of order n) we just need to solve a system of linear equations. But the numbers of unknowns and equations in this system grow extremely fast with n (cf. [1]) and therefore we need a better way to approach Wn . Motivated by perturbative Chern–Simons theory, Bar-Natan [2] and Kontsevich [8] gave a construction of weight systems using a Lie algebra L with an invariant inner ? ?? ???

Supported by the EPSRC under contract GR/K57824. This research was supported in part by an NSF postdoctoral research fellowship. On leave from New Mexico State University, Las Cruces, NM 88003.

94

J.M. Figueroa-O’Farrill, T. Kimura, A. Vaintrob

product and a module. In [13] one of us generalized this construction to so-called selfdual Yang–Baxter Lie algebras. This class of algebras includes Lie algebras and Lie superalgebras with invariant inner products. Each such algebra produces a sequence of weight systems WL : Dn → U (L) with values in the center ZL of the universal enveloping algebra U (L). We call this family of weight systems a universal weight system for L, since weight systems corresponding to L-modules are obtained from WL by taking traces in the corresponding representation. In [1, 2] Bar-Natan computed weight systems for the defining representations of classical Lie algebras. Although for L = sln the center of U(L) is dual to the space of functionals on U spanned by the characters of the exterior powers of the defining representations and the corresponding weight systems are known (cf. [1]), the problem of finding a direct combinatorial expression for WL in terms of standard generators of ZL is highly nontrivial. The first universal weight system was studied by Chmutov and Varchenko [5] who considered the case where L = sl2 . The center of U (sl2 ) is isomorphic to the polynomial algebra C[c], where c is the Casimir element of U (sl2 ), and the main result of [5] is a recursive formula for computing values of Wsl2 on chord diagrams. In this paper we are studying the universal weight system for gl(1|1), the simplest interesting example of a self-dual Lie superalgebra. The case of gl(1|1) is different from that of sl2 in several ways. First, the center of U(gl(1|1)) has two generators h and c. Second, all the invariant functionals on U (sl2 ) are linear combinations of traces in irreducible representations, which is not true for gl(1|1) (since its superdimension as well as the superdimension of its generic modules is 0). Third, a choice of values for h and c in our weight system gives rise to the sequence of coefficients of the Alexander-Conway polynomial. This once again confirms the relationship between gl(1|1) supersymmetry and the Alexander polynomial (cf. [7, 10]). Weight systems coming from Lie superalgebras other than gl(1|1) can be used to explain relations between classical knot invariants (cf. [13, 14]) and to construct invariants that are stronger than all invariants corresponding to semi-simple Lie algebras (cf. [16]). Let us formulate our answer. Let c be the quadratic Casimir for gl(1|1), and h ∈ gl(1|1), the identity matrix. Then Z(U(gl(1|1))) = C[h, c], and our main result is the following recursive formula for values of the weight system W = Wgl(1|1) . Theorem. Let D be a chord diagram, “a” a fixed chord in D and b1 , b2 , . . . are all the chords of D intersecting a. Denote by Da (resp. Da,i , Da,ij ) the diagram D − a (resp. D − a − bi , D − a − bi − bj ). Then W (D) is a polynomial in c and y = −h2 satisfying W (D)

=

c W (Da ) − y +y

X

X

W (Da,i )

(1)

i

+− −+ l r W Da,ij + W Da,ij − W Da,ij − W Da,ij ,

i<j +− −+ l r where Da,ij (resp. Da,ij ; Da,ij ; Da,ij ) is the diagram obtained by adding to Da,ij a new chord connecting the left end of bi and the right end of bj (resp. the right end of bi and the left end of bj ; the left ends of bi and bj ; the right ends of bi and bj ) assuming that the chord a is drawn vertically. Pictorially, if D denotes the diagram below and a is the vertical chord and i, j are chords which intersect a so that i is the upper chord and j is the lower chord then

Universal Vassiliev Invariant for the Lie Superalgebra gl(1|1)

D=

+− Da,ij =

r

r

i

r

j

r

a

r b

r b

r

b br

r Da =

−+ Da,ij =

r

r

" r"

95

r r "

"

r Da,i =

r

r

r l Da,ij =

r

r

Da,j =

r r Da,ij =

r (2)

Since all the diagrams in the right-hand side of (1) have less chords than D, this allows us to compute the value of the weight system Wgl(1|1) on any chord diagram recursively (and quite effectively); see the examples in Subsect. 3 and the tables in the appendix. The outline of the paper is as follows. In Sect. 1 we collect preliminary facts about Vassiliev invariants, chord diagrams and weight systems, and recall the construction from [13] of weight systems based on Lie superalgebras (or, more generally, on Yang–Baxter Lie algebras). In Sect. 2 we present the necessary algebraic information about gl(1|1), its representations, invariant tensors and the center of the universal enveloping algebra. An identity between invariant tensors of fourth order, which is pivotal for proving formula (1), is also established here. In Sect. 3 we prove the recursive formula (1) for the universal weight system Wgl(1|1) . We also prove that deframing the universal weight system consists simply in evaluating it at c=0. We conclude the section with an alternative way of producing the same weight system. In Sect. 4 we show how the Alexander-Conway polynomial can be obtained from our weight system. 1. Vassiliev Invariants and Yang–Baxter Lie Algebras Here we recall some concepts and results related to Vassiliev invariants which we will need later. In particular, we review their relationship with Lie algebra-type structures. For more details see [1, 8, 13]. 1.1. Vassiliev invariants, chord diagrams and weight systems. Definition 1.1. A singular knot is an immersion K : S 1 → R3 with a finite number of transversal double self-intersections (or double points). The set of singular knots with n double points is denoted by Kn . A chord diagram of order n is an oriented circle with n non-intersecting pairs of points (chords) on it, up to an orientation preserving diffeomorphism of the circle. Denote by Dn the set of all chord diagrams with n chords. Every K ∈ Kn has a chord diagram ch(K) ∈ Dn whose chords are the inverse images of the double points of K. Vassiliev showed that every knot invariant I with values in an abelian group k extends to an invariant of singular knots by the rule I(K0 ) = I(K+ ) − I(K− ),

(3)

96


where K0 , K+ , and K− are singular knots which differ only inside a small ball as shown below: '$ '$ '$ I @ I @ @ r I @ @ @ @ @ @ @ &% &% &% K0

K+

K−

(4)

Definition 1.2. A knot invariant I is called an invariant of order (≤) n if I(K) = 0 for any K ∈ Kn+1 . The k-module of all invariants of order n is denoted by Vn . We have the obvious filtration V0 ⊂ V1 ⊂ V2 · · · ⊂ Vn ⊂ · · · . Elements of V∞ = ∪n Vn are called invariants of finite type or Vassiliev invariants. Analogously, Vassiliev invariants of framed knots (and links) can be defined. An immediate corollary of the definition of Vassiliev invariants is that the value of an invariant I ∈ Vn on a singular knot K with n self-intersections depends only on the diagram ch(K) of K. In other words, I descends to a function on Dn which we will still denote by I slightly abusing notation. These functions satisfy two groups of relations. r   =0

I

(5)

r and



I

  r r −I

r r r

r   r r −I

  r r +I r r

r

r  r =0

(6)

r

(When drawing a chord diagram we always assume that the circle is oriented counterclockwise. Chords whose endpoints lie on the solid arcs are shown explicitly. The diagram may contain other chords whose endpoints lie on the dotted arcs, provided that they are the same in all diagrams appearing in the same relation.) Definition 1.3. A function W : Dn → k is called a weight system of order n if it satisfies the conditions (6). If, in addition, W satisfies the relations (5), then we call it a strong weight system. Denote by Wn (resp. by W n ) the set of all weight systems (resp. strong weight systems) of order n. It is easy to see that the natural map Vn /Vn−1 → W n is injective. The remarkable fact proved by Kontsevich and Bar-Natan is that this map is also surjective (at least when k ⊃ Q). In other words, each strong weight system of order n is a restriction to Dn of some Vassiliev invariant. Let An (resp. A¯ n ) be the dual space to Wn (resp. W n ), i.e., the space of formal linear combinations of diagrams from Dn modulo four-term relations1 (6) (resp. fourand one-term (5) relations), and 1

which are now considered as relations in the space of diagrams, i.e., without I.


97

A¯ = ⊕n A¯ n .

A = ⊕ n An ,

Denote by Vnf the space of Vassiliev invariants of order ≤ n of framed knots. Theorem 1.4. [1, 8]. If k ⊃ Q then (1) Vn /Vn−1 ' W n ' A¯ ∗n . f ' Wn ' A∗n . (2) Vnf /Vn−1 (3) The operation of connected sum of diagrams induces on A and A¯ structures of commutative graded k-algebras. The comultiplication dual to this product makes the graded algebras W = ⊕n Wn ' A∗ and W = ⊕n W n ' A¯ ∗ into commutative and co-commutative Hopf algebras. ¯ (4) A = A[Θ], where Θ is the primitive element in A1 represented by the only chord diagram of order 1, i.e.,

An = A¯ 0 ⊗ Θn + A¯ 1 ⊗ Θn−1 + . . . A¯n ⊗ 1 . Part (4) of the theorem shows that there is a canonical projection (“deframing”) Wn → W n . Therefore, for every weight system there is a canonical strong weight system (and consequently, a knot invariant). We will not be concerned with the one-term relations in the bulk of the paper, but will return to it briefly in Sect. 3.4 when we discuss the deframed universal weight system. Remark 1.5. Similar results hold for links. The only difference is that in this case we will have to consider several circles. 1.2. Weight systems coming from Yang–Baxter algebras . Here we recall a construction from [13] that assigns a family of weight systems to every Yang–Baxter Lie algebra with an invariant inner product. First, we introduce a more general class of diagrams. Definition 1.6. A Feynman diagram of order p is a graph with 2p vertices of degrees 1 or 3 with a cyclic ordering on the set of its univalent (external) vertices and on each set of 3 edges meeting at a trivalent (internal) vertex.2 Let Fp denote the set of all Feynman diagrams with 2p vertices (up to the natural equivalence of graphs with orientations). The set Dp of chord diagrams with p chords is a subset of Fp . We draw Feynman diagrams by placing their external vertices (legs) on a circle which is oriented in the counterclockwise direction called the Wilson loop. The edges of a Feynman diagram are called propagators. We assume that the propagators meeting at each internal vertex are oriented counterclockwise. Denote by Bp the vector space generated by Feynman diagrams of order p modulo relations

r DY

=

T TTr Tr D||

−

b

br r

(7)

DX

More precisely, Bp = hFp i/hDY − D|| − DX i, where the diagrams D|| and DX are obtained from the diagram DY by replacing its Y -fragment by the ||- and X- fragments f )∗ = A p . respectively. This gives another description of the space (Vpf /Vp−1 2 A Feynman diagram after forgetting the ordering of external vertices is called a Chinese Character diagram in [1].

98


Proposition 1.7. [1] (1) The embedding Dp ,→ Fp induces an isomorphism Bp ' Ap . (2) The following local relations hold for internal vertices in Feynman diagrams:

=−

and

=

b

T TT T

−

b

Definition 1.8. A Feynman graph is a graph with 1- and 3-valent vertices such that the set of its legs (univalent vertices) is a disjoint union of two linearly ordered sets: incoming and outgoing legs and the set of propagators meeting at each of its trivalent (internal) vertices has a cyclic ordering. We denote by Fa,b the set of Feynman graphs with a incoming and b outgoing legs and by F∗ the set of all Feynman graphs. We draw Feynman graphs by putting all incoming legs on a horizontal segment, outgoing legs on a parallel segment below, and all the internal vertices between these two segments (with the counterclockwise orientation at each vertex). Notice that closing these segments into a circle, we turn a Feynman graph into a Feynman diagram. Feynman graphs can be regarded as morphisms in a tensor category FG with objects 0, 1, 2, . . . and natural operations of composition and tensor product. If A ∈ Fb,c and B ∈ Fa,b , then their composition A ◦ B ∈ Fa,c is the Feynman graph obtained by attaching the outgoing vertices of B to the corresponding incoming vertices of A. The tensor product of A ∈ Fa,b and C ∈ Fc,d is the graph A ⊗ C ∈ Fa+c,b+d obtained by placing C to the right of A. A natural way to assign invariants to Feynman graphs (and, ultimately, to Feynman and chord diagrams) is to consider representations of the category FG. Recall that a representation of a tensor category C is a tensor functor F from C to the category of vector spaces, i.e., an assignment of a vector space F (A) to each object A of C and a linear map F (g) : F (A) → F (B) to each morphism g ∈ Mor(A, B) such that it respects composition and tensor products. In the case C = FG a representation is specified by a choice of a vector space L and a k-linear map F (Γ ) : L⊗a → L⊗b for every Γ ∈ Fa,b (where L⊗0 = k). Every element of F∗ = Mor(FG) can be obtained by means of composition and tensor product from the following elementary graphs:

I=

∈ F1,1 ,

b=

∈ F2,0 ,

c=

f=

∈ F2,1 ,

g=

∈ F1,2 ,

S=

∈ F0,2 ,

A

A

A A

∈ F2,2 .


99

Therefore, to specify a representation of FG we need to fix a vector space L and six tensors corresponding to the generators I, b, c, f, g, and S. In [13] the defining relations between these generators were found and it was shown that representations of the tensor category FG on the category of vector spaces are in one-to-one correspondence with the set of self-dual Yang–Baxter algebras. Definition 1.9. A vector space L over a field k with an operator S :L⊗L→L⊗L and a multiplication f : L ⊗ L → L satisfying conditions (1-3) below is called a Yang–Baxter algebra or, simply, an S-algebra. If, in addition, L is finite-dimensional and is equipped with an inner product b : L ⊗ L → k satisfying (4-7) below, it is called a symmetric self-dual S-algebra3 . A (self-dual) Yang–Baxter Lie algebra is a (self-dual) S-algebra L, such that the multiplication f : L ⊗ L → L obeys (8-9) below. We use the standard notation: if T ∈ Hom(V ⊗ V, W ), then T12 denotes the operator T ⊗ id : V ⊗ V ⊗ U → W ⊗ U, etc. 1. The operator S is a symmetry: S 2 = idL⊗L

=

2. S satisfies the quantum Yang–Baxter equation: S12 S23 S12 = S23 S12 S23

@

@

@ @

= @

@ @

@

3. The multiplication f is compatible with the symmetry S: S f = f23 S12 S23

=

4. The bilinear form b is S-symmetric: bS = S =

3 This nomenclature is not standard. Such algebras have been termed “Euclidean” in [13] and “metric” in [1, 2]. Throughout this paper we will refer to such algebras simply as self-dual, leaving implicit the fact that b is S-symmetric.

100


5. b is compatible with S: b12 S23 = b23 S12

=

6. b is a non-degenerate bilinear form with inverse c: b23 c12 = idL = b12 c23

=

=

7. b is f -invariant: b ◦ f12 = b ◦ f23

=

8. f is S-skew-symmetric: f ◦ S = −f

=−

9. f satisfies the S-Jacobi identity: f ◦ f12 = f ◦ f23 − f ◦ f23 ◦ S12

=

−

Every self-dual S-algebra L gives a representation of the category FG on the vector space L (or, more accurately, a tensor functor FL from FG to the category T (L) of tensor powers of L). Lie S-algebras arise when we attempt to construct representations of FG satisfying relations (2) of Proposition 1.7. Denote by Ga,b the quotient space of the space hFa,b i of formal linear combinations of elements of Fa,b by relations 1.7(2) for S internal vertices. It is clear that F∗ induces on G∗ = Ga,b the structure of a tensor category with operations extended by linearity. Proposition 1.10. There is a one-to-one correspondence between representations of the category G∗ and self-dual Yang–Baxter Lie algebras. To go from the category G∗ to weight system we need to take care of the relations (7) for external vertices of Feynman diagrams. This leads us to the following notion. Definition 1.11. The universal enveloping algebra U(L) of the S-Lie algebra L is the quotient algebra of the tensor algebra T ∗ (L) by the ideal generated by the expressions a ⊗ b − S(a ⊗ b) − f (a, b) for a, b ∈ L.


101

Denote by WL the composition of FL and the projection T ∗ (L) → U(L). The following fact makes it possible to pull WL down to the space B = ⊕Bp of Feynman diagrams modulo relations (7). Proposition 1.12. For every Γ ∈ G0,∗ the element WL (Γ ) ∈ U(L) is invariant with respect to the canonical action of L on U(L). In other words, WL (Γ ) belongs to the S-center of U (L). Now, given a Feynman diagram Γ and a self-dual Lie S-algebra L, we can define WL (Γ ) ∈ U (L) to be WL (Γ 0 ), where Γ 0 is any element of G0,∗ whose closure (i.e., a Feynman diagram obtained by placing the legs of Γ 0 on the Wilson loop) is Γ . The previous proposition guarantees that the result does not depend on the choice of Γ 0 . Summarizing we come to the following result of [13]. Theorem 1.13. For every self-dual Yang–Baxter Lie algebra L there exists a natural homomorphism of algebras WL : A → U (L). The image of WL belongs to the S-center ZL of U (L). The weight system WL is called the universal weight system corresponding to L. 1.3. Examples and particular cases. 1. Lie (super)algebras as S-Lie algebras. Perhaps the most familiar example of an S-Lie algebra, L, is that of a Lie (super)algebra when L is a Z2 -graded vector space L = L0¯ ⊕ L1¯ . Here, the S-structure, S : L ⊗ L → L ⊗ L, is a linear map given by S(u ⊗ v) = (−1)|u||v| v ⊗ u for homogeneous elements u and v with degrees |u| and |v|, respectively. (In the case of an ordinary Lie algebra, L1¯ = 0 and S is nothing more than the usual transposition of factors in the tensor product.) Relation (3) in the definition of an S-Lie algebra then says that the Lie bracket is Z2 -graded (i.e., the map f : L ⊗ L → L preserves the parity). Relations (8) and (9) become the usual (super-)skew-symmetry and the (super-)Jacobi identity. Finally, relations (7), (8) and (5) say, respectively, that the bilinear form b is invariant, (super-)symmetric and of even parity with respect to the grading. 2. Evaluating a diagram (an example). Consider the following chord diagram D: '$ r r r

.r

r r &% If it is cut open, it becomes the following:

The associated element in U(L) is WL (D) =

dimL X a1 ,...,a10 =1

C a1 a7 C a8 a9 C a10 a6 Saa72aa83 Saa94aa105 ea1 ea2 ea3 ea4 ea5 ea6 .

102


Notice that although we know from Proposition 1.12 that WL (D) belongs to ZL , the precise expression of WL (D) in terms of the generators of ZL requires a computation in U (L). Thus it is a computationally involved matter to determine, for example, given two diagrams D and D0 , whether or not WL (D) and WL (D0 ) are the same. A better method to compute WL (D) is thus required, and our recursion formula provides just such a method in the case where L is gl(1|1). 3. Some special diagrams. Henceforth, and in order to simplify diagrams, when displaying the value of WL on a chord diagram, we will simply draw the diagram. (a) The identity element 1 in U(L) is given by '$ 1= &% since WL : A → U(L) is a homomorphism of algebras with unit. (b) The Casimir, c, (dual to the inner product b) is given by r '$ .

c=

(8)

r &% (c) The image in U(L) of the standard totally S-antisymmetric element in T 3 (L) is defined by '$ r r bb"" . (9) y= r &% (d) Finally, the following “bubble” diagram B is the image in U (L) of the dual to the Killing form for L: r '$ B= and can be shown to obey B = 2y.

i

,

r &%

4. The leading order terms. For D ∈ Dn the leading term of WL (D) (with respect to the filtration in U(L) inherited from T (L)) is equal to cn . In some cases, the subleading term can also be readily computed. In those cases for which B is proportional to c (e.g., when L a simple Lie algebra) the numerical coefficient of the cn−1 term is proportional to the number of intersections of the chords in D when the chords are placed in generic position. 5. Multiplicativity. Because WL : A → U (L) is a homomorphism of algebras, we need only consider indecomposable diagrams when computing universal weight systems, that is, diagrams whose chords cannot be split into two non-intersecting subsets. For example, we see that r r r '$ '$ '$ '$ r r r r r = =c . r r r r r r r r &% &% &% &%


103

2. Representations and Invariant Tensors of gl(1|1) In this section we present algebraic facts about the Lie superalgebra gl(1|1) necessary for the analysis of the corresponding universal weight system. We define gl(1|1), examine the subring of its representation ring generated by cyclic modules, and determine its invariant tensors. We will also prove an identity between invariant tensors of fourth order which is crucial in the proof of the recursion relation (1). We assume that the ground field k is C. 2.1. Preliminaries on gl(1|1). We follow the standard conventions about linear superspaces and Lie superalgebras which can be found for example in [3, 6]. Definition 2.1. Let V ∼ = C1|1 be a (1|1)-dimensional vector superspace. (Recall that an (m|n)-dimensional vector superspace, V , is a Z2 -graded vector space V = V0¯ ⊕ V1¯ where dim V0¯ = m and dim V1¯ = n.) The Lie superalgebra of endomorphisms of V is called gl(1|1). The bilinear form

hx, yistr = str(xy)

(10)

on gl(1|1) is invariant and nondegenerate. (Recall that the supertrace of an (m|n)×(m|n) A B matrix M = is defined as str M = tr A−tr D.) Therefore, gl(1|1) is a self-dual C D Lie superalgebra. Relative to a homogeneous basis (e0 , e1 ) for V , with e0 ∈ V0¯ and e1 ∈ V1¯ , a convenient basis for gl(1|1) is given by the matrices: 0 0 0 1 1 0 0 0 H= , Q− = . (11) , G= , Q+ = 1 0 0 0 0 1 0 1 The even part L0¯ of gl(1|1) = L = L0¯ ⊕ L1¯ is spanned by {H, G} and the odd part L1¯ is spanned by {Q+ , Q− }. The Lie bracket [−, −] : L⊗2 → L in this basis is [G, Q± ] = ±Q±

and

[Q+ , Q− ] = H,

(12)

and zero everywhere else. The nonvanishing inner products between elements of the basis (11) are hH, Gistr = −1,

hG, Gistr = −1,

and

hQ+ , Q− istr = −1 .

(13)

The element H belongs to the center of gl(1|1) and the quotient Lie superalgebra gl(1|1)/hHi is called pgl(1|1). 2.2. The ring of cyclic modules. The structure of the representation ring of gl(1|1) is not trivial, since not every finite dimensional module is completely reducible. The situation for the cyclic modules (modules generated by one element) is much simpler and luckily this is all we need for our purposes. We will only consider those modules V in which G and H act diagonally. Since H is in the center of gl(1|1), Lit must be a scalar operator in any cyclic module: Hv = λ v for all v ∈ V . Let V = γ Vγ be the decomposition of V into eigenspaces of G. We will call any γ for which Vγ 6= 0 a weight of V . We will now study the cyclic modules. Because Q2± = 21 [Q± , Q± ] = 0, any cyclic module is at most (2|2)-dimensional. Indeed, let v be a cyclic vector. Then the module is spanned by the following four

104


elements: v, Q+ v, Q− v, and Q+ Q− v, which need not all be different from zero. We can distinguish the following types of cyclic modules, which are depicted in Table 1.

Table 1. Indecomposable cyclic modules. Right arrows denote the action of Q+ and left arrows that of Q−

Iγ

qγ

IIλ,γ

λ,γ−1 - λ,γ

II+γ

γ−1 q

- qγ

II− γ

γ−1 q

qγ

YH γ * qH IIIγ γ−2q YH H *q q γ−1

q q

q γ−1 q - qγ III0γ γ−2

Type I. Both Q+ v and Q− v vanish. In this case λ = 0, since λv = Hv = [Q+ , Q− ]v = 0. If Gv = γ v, then we say that the module is of type Iγ . Such modules are irreducible. They are either (0|1)- or (1|0)-dimensional, depending on the parity of v. Type II. Only one of Q− v and Q+ v is zero and λ 6= 0. If, say Q− v 6= 0, then the module V is spanned by v and w = Q− v or, equivalently, by w and Q+ w. Both v and w are cyclic vectors, but v in this case is the highest-weight vector. If v has weight γ, then we say that the module is of type IIλ,γ . These modules are (1|1)-dimensional and also irreducible. Type II+ . In this case Q− v = 0 and w = Q+ v 6= 0 but λ = 0. Because of this condition, Q− w = 0. If v has weight γ we say that the module is of type II+γ . These modules are (1|1)-dimensional, reducible but indecomposable. Type II− . Now Q+ v = 0 and w = Q− v 6= 0 and λ is still zero. In this case the highest-weight vector is not the cyclic vector. If w has weight γ (so that v has weight γ − 1) we say that the module is of type II− γ . Again such a module is (1|1)-dimensional, reducible and indecomposable. Type III. Both Q+ v and Q− v are nonzero. Here we have to distinguish two cases depending on whether λ is zero or not. If λ 6= 0, then Q+ Q− v is also nonzero. In this case we say that the module V is of type IIIλ,γ , where γ is the weight of the highest weight vector Q+ v. Such modules are (2|2)-dimensional and reducible; but they are also decomposable: IIIλ,γ ∼ = IIλ,γ ⊕ IIλ,γ−1 . When λ = 0, the vector Q+ Q− v coincides with −Q− Q+ v, but this vector might vanish. If it does not, then V is a (2|2)-dimensional reducible module which, if the highest-weight vector Q+ v has weight γ, we denote by IIIγ . It is indecomposable. Finally, if Q+ Q− v = 0, then we have a (1|2)- or (2|1)-dimensional reducible indecomposable module which, if it has highest-weight γ, is denoted by III0γ . Note that the defining representation is of type II1,1 whereas the adjoint representation is of type III1 . The Lie superalgebra pgl(1|1) as a gl(1|1)-module is of type III01 .


105

The following is a summary of our case study. Proposition 2.2. Each diagonalizable indecomposable cyclic gl(1|1)-module V is isomorphic to one of the modules Iγ ,

IIλ,γ (λ 6= 0),

II± γ,

III0γ ,

IIIγ .

(14)

Notice that there is a two-fold ambiguity implicit in our notation. Every module type described above comes in two flavors depending on the choice of parity of the highestweight vector. When it is necessary to specify this parity, we will do it explicitly, for , etc. example IIIodd γ Direct sums of the cyclic modules considered above form a subring of the representation ring of gl(1|1), as the next proposition shows. Proposition 2.3. The above cyclic modules have the following multiplication table under tensor product. Iγ ⊗ Iγ 0 Iγ ⊗ IIλ,γ 0

∼ = ∼ =

Iγ ⊗ III0γ 0 Iγ ⊗ IIIγ 0

∼ = ∼ = ∼ =

IIλ,γ ⊗ IIλ0 ,γ 0

∼ =

IIλ,γ ⊗ II± γ0

∼ = ∼ =

Iγ ⊗ II± γ0

II± γ II+γ IIλ,γ II+γ II− γ

⊗ II± γ0 ⊗ II− γ0 ⊗ III0γ 0 ⊗ III0γ 0 ⊗ III0γ 0

III0γ ⊗ III0γ 0 IIλ,γ ⊗ IIIγ 0 II± γ ⊗ IIIγ 0 III0γ ⊗ IIIγ 0 IIIγ ⊗ IIIγ 0

∼ = ∼ =

Iγ+γ 0 IIλ,γ+γ 0 , II± γ+γ 0 , III0γ+γ 0 , IIIγ+γ 0 , IIλ+λ0 ,γ+γ 0 ⊕ IIλ+λ0 ,γ+γ 0 −1 IIIγ+γ 0

for λ + λ0 6= 0, for λ = −λ0 .

IIλ,γ+γ 0 ⊕ IIλ,γ+γ 0 −1 , ± II± γ+γ 0 ⊕ IIγ+γ 0 −1 ,

IIIγ+γ 0 IIλ,γ+γ 0 ⊕ IIλ,γ+γ 0 −1 ⊕ IIλ,γ+γ 0 −2 ,

∼ = ∼ =

II+γ+γ 0 ⊕ IIIγ+γ 0 −1 ,

∼ = ∼ =

II+γ+γ 0 ⊕ III0γ+γ 0 −1 ⊕ 2Iγ+γ 0 −2 ⊕ II− γ+γ 0 −3 , 0 0 0 IIλ,γ+γ ⊕ 2 IIλ,γ+γ −1 ⊕ IIλ,γ+γ −2 ,

∼ = ∼ = ∼ =

(15)

IIIγ+γ 0 ⊕ II− γ+γ 0 −2 ,

IIIγ+γ 0 ⊕ IIIγ+γ 0 −1 , IIIγ+γ 0 ⊕ 2IIIγ+γ 0 −1 , IIIγ+γ 0 ⊕ 2 IIIγ+γ 0 −1 ⊕ IIIγ+γ 0 −2

(16)

Proof. Let V and V 0 be two cyclic modules with highest vectors v and v 0 , weights γ and γ 0 , and H acting by scalars λ and λ0 respectively. The module W = V ⊗ V 0 has highest vector w = v ⊗ v 0 with Hw = (λ + λ0 )w and Gw = (γ + γ 0 )w. Together with the classification of cyclic modules before Proposition 2.2, this gives the first nine statements of our multiplication table. The products of IIλ,γ and modules with λ = 0 will have λ 6= 0 and, therefore, will be sums of modules of the same type.

106


0 0 0 When V = II− γ and V = IIIγ 0 with cyclic vectors v and v respectively, the vectors 0 0 v ⊗ v and v ⊗ Q− v generate two non-intersecting cyclic submodules in V ⊗ V 0 of − − 0 ∼ types IIIγ+γ 0 and II− γ+γ 0 −2 respectively. Therefore, IIγ ⊗ IIIγ 0 = IIIγ+γ 0 ⊕ IIγ+γ 0 −2 . The cases II+γ ⊗ III0γ 0 and III0γ ⊗ III0γ 0 are handled similarly. The last four formulas follow from the previous ones using IIIγ ∼ = II+0 ⊗ II− γ (15) and the associativity of the tensor product.

2.3. Invariant tensors. Let L denote the adjoint module of gl(1|1). The tensor powers L⊗n have a natural gl(1|1)-module structure. Since L is of type III1 , it follows from (16) that L⊗n decomposes into a direct sum of modules of types IIIp for integer p. In fact, iterating (16) we obtain Proposition 2.4. ∼ L⊗n ∼ = III⊗n = 1

n M 2n − 2 III` . n−`

(17)

`=−n+2

Now we will compute the number of linearly independent invariant tensors of order n, i.e., the dimension of the subspace Inv(L⊗n ) = {v ∈ L⊗n | gv = 0

for all g ∈ gl(1|1)} .

Nonzero invariants in cyclic modules can exist only when λ=0, and the following lemma tells us for which values of γ they appear in the indecomposable cyclic modules (14). Lemma 2.5. Let V be one of the cyclic modules (14). Then the dimension of the space Inv V is either 0 or 1 and dim Inv V = 1 if and only if λ = 0 and  for V of type Iγ or II+γ 0 1 for V of type II− γ = γ or IIIγ .  0 or 2 for V of type III0γ of III1 in the direct sum It follows that dim Inv(L⊗n ) is equal to the multiplicity decomposition of L⊗n , which from (17) equals 2n−2 . In summary, n−1 Proposition 2.6. The number of linearly independent gl(1|1)-invariant tensors of order n on L is given by 2n−2 . n−1 For n = 1 there is only one independent invariant tensor H. For n = 2 there are two4 : one is H 2 , and the other one is given by C = G H + H G + Q− Q+ − Q+ Q− .

(18)

Notice that they are both (super)symmetric. Together they span the space of gl(1|1)invariants in S 2 L. In particular, C is nondegenerate and therefore induces a gl(1|1)isomorphism L ∼ = L∗ . This isomorphism allows us to identify S 2 L with the space of 4 In order to avoid notational clutter, we will hereafter omit the ⊗ from the notation for gl(1|1)-tensors. Therefore we will understand x y to mean x ⊗ y and x2 = x ⊗ x, etc.


107

invariant symmetric bilinear forms on L. Under this identification, we associate with the invariant tensor αC + βH 2 the bilinear form hQ+ , Q− i = α , hH, Gi = α and hG, Gi = β.

(19)

Provided α 6= 0 this bilinear form is nondegenerate; that is, it defines an invariant metric on gl(1|1). Therefore there are several ways to make gl(1|1) a self-dual Lie superalgebra. The metric (19) coincides with h−, −istr when α = −1 and β = −1. Weight systems WL corresponding to different choices of α 6= 0 and β in (19) are equivalent, since the transformation β 1 H, Q± 7→ √ Q± 2α α

H 7→ α−1 H, G 7→ G −

is an automorphism of gl(1|1) that takes the bilinear form (19) to the metric dual to C (it corresponds to the choice α = 1 and β = 0). From now on we will use this metric on gl(1|1) in our computations of the universal weight system Wgl(1|1) . 2.4. Tensor diagrammar. The language of Feynman graphs of Subsect. 1 provides a convenient tool to treat (invariant) tensors in Lie superalgebras graphically. We will use such pictures for denoting components of tensors as well. Let L be any vector superspace with a fixed homogeneous basis {ei }. Relative to this basis, every linear ···jn ) defined by map ϕ : L⊗m → L⊗n is determined by the (dim L)m+n numbers (ϕji11ij22···i m ···jn e e · · · e jn , ϕ(ei1 ei2 · · · eim ) = ϕji11ij22···i m j1 j2

where we use the summation convention (and all the ⊗ between ei are omitted). We will represent it graphically by the following diagram: 1 2

···

m

ϕ 1 2

···

n

.

For an arbitrary vector superspace L, there are two canonical maps: the identity and the symmetry. The identity map id : L → L, with components δij , is represented by i

δi j ↔

; j

k` = (−1)|i||j| δi` δjk , where |i| whereas the symmetry S : L⊗2 → L⊗2 has components Sij is the parity of ei ; and is represented by i

k` ↔ Sij

j

A

A

A A

k

`

.

108


Now let L be a Lie superalgebra. The Lie bracket is a linear map L⊗2 → L, or in k ek . It will be depicted as component form [ei , ej ] = fij i

j

k ↔ fij

.

(20)

k

If, in addition, L is self-dual, there is a canonical quadratic invariant tensor. If we let Cij = hei , ej i denote the coefficients of the invariant metric in this basis,5 the following is an invariant tensor in L⊗2 : C ≡ C ij ei ej ,

(21)

where (C ij ) is the matrix inverse of (Cij ); that is, C ij Cjk = δki . The diagram which represents this invariant tensor is C ij ↔

.

(22)

j

i

The invariant metric itself can be represented diagrammatically as follows: i

j

Cij ↔

.

Just as in the case of Feynman graphs considered in Sect. 1, there are two ways to combine diagrams of invariant tensors together to make a third diagram: tensor product and composition. Because the identity, the symmetry, the Lie bracket, the metric and the quadratic tensor are L-invariant, so will be any diagram obtained by gluing these together. For example, gluing two copies of (22) and (20) we can obtain a cubic invariant. Concretely, we define the invariant tensor F ∈ L⊗3 by the diagram

= =

i f`m C mj C `k ei ej ek .

(23)

k Although many invariant tensors can be constructed from C ij and fij by gluing, it is important to keep in mind that not every invariant tensor is of this form. Indeed, returning now to gl(1|1) we see this immediately. Already for order one, the only linearly-independent gl(1|1)-invariant tensor, H, cannot be written in terms of the quadratic (22) and cubic (23) invariants. If we wish to be able to depict H graphically we must introduce a new symbol: 5

In the previous section, the metric was denoted by b.


109

u H↔ For n=2 we saw that there are two linearly independent invariant tensors: H 2 and the quadratic invariant C (22), which in the case of gl(1|1) is given by (18). But these are not all the invariant tensors of order two that we can draw. Take, for example, the “bubble” B, defined by the diagram

B=

$ ' .

This is clearly an invariant tensor, since it is constructed out of gluing invariant tensors (in fact, it is dual to the Killing form)

$ ' = =

i j C kn C `m fk` fmn ei ej ;

but since according to Proposition 2.6 there are only two linearly independent quadratic invariant tensors, there must be a linear relation between H 2 , C and B. Indeed, bubbles burst: Proposition 2.7. u u $ ' ' = −2

$ or B = −2H 2 .

(24)

Proof. From the fact that the Killing form of pgl(1|1) vanishes, or equivalently from the fact that G is not a linear combination of commutators (12), it follows that B is proportional to H 2 . It is then a simple matter of finding the constant of proportionality. According to Proposition 2.6 there are six linearly independent invariant tensors of order three. We can write five of them using the invariant tensors H, C and F that we have introduced before: u u u u u

=

H3

=

CH

=

C ij ei H ej

110


u =

HC

=

−H Q− Q+ − H Q+ Q− − Q− H Q+ + Q− Q+ H − Q + H Q− + Q+ Q− H ;

but the sixth invariant is of a different kind: T

=

G2 H + G H G + H G2 + G Q− Q+ − G Q+ Q− + Q− G Q+ + Q− Q+ G − Q+ G Q− − Q+ Q− G.

We will not attempt to write down twenty linearly independent invariant tensors that exist in L⊗4 . We simply note that five tensors in Inv(L⊗4 ) which can be constructed from the quadratic and cubic tensors satisfy a linear relation. 2.5. A fundamental relation. The following result is crucial in the proof of the recursion formula (1). Theorem 2.8. Let K be the invariant tensor corresponding to the following diagram:

= =

j k C im C np C q` fmn fpq ei ej ek e`

and let

M

i $ ' $ ' i i i + − − = = B13 C24 + C13 B24 − C14 B23 − B14 C23 ,

then K=

1 M 2

(25)

(where C24 B13 = C j` B ik ei ej ek e` , etc.) V2 Proof. Notice that both K and M belong to Inv(S 2 ( L)) ⊂ L⊗4 . Let L0 be the quotient superalgebra pgl(1|1) = gl(1|1)/hHi considered as gl(1|1)module, and p : Inv(S 2 (

^2

L)) → Inv(S 2 (

^2

L0 ))

the map induced by the projection L → L0 . From Lemma 2.9 below, it follows that the V2 space Inv(S 2 ( L)) is two-dimensional and, in fact, spanned by the two tensors


M

=

and N

= =

111

B13 C24 + C13 B24 − C14 B23 − B14 C23 , − C13 C24 − C14 C23 ,

such that M is a generator of Ker(p) and p(N ) spans Inv(S 2 ( On the other hand, the invariant tensor K is given by K

=

V2

L0 )).

−H Q− H Q+ + H Q− Q+ H + H Q+ H Q− − H Q+ Q− H + Q − H 2 Q + − Q − H Q + H − Q+ H 2 Q − + Q + H Q − H .

Therefore K belongs to Ker(p) and has to be proportional to M . By comparing the coefficients at, say, Q− H 2 Q+ in K and M we find that the constant of proportionality is equal to 21 . It remains to prove the lemma used above. Lemma 2.9. dim Inv(S 2 (

(i)

^2

L)) = 2;

^2

L0 )) = 1; dim Inv(S 2 ( ^2 ^2 dim Ker Inv(S 2 ( L)) → Inv(S 2 ( L0 ) = 1.

(ii) (iii)

Proof. Paying closer attention to the calculation behind (16), it follows that ^2 ^2

even ∼ IIIodd = IIIeven g 2g ⊕ III2g−2

∼ IIIeven = 2IIIodd g 2g−1

and

∼ S 2 IIIodd = 2IIIodd g 2g−1 ,

even ∼ and S 2 IIIeven = IIIeven g 2g ⊕ III2g−2 .

Since the adjoint module is of type IIIodd 1 , it follows that S2

^2

IIIodd 1

∼ = ∼ =

⊕ IIIeven S 2 IIIeven 2 0

even ⊕ 2IIIeven ⊕ 2IIIodd ⊕ IIIeven IIIeven 4 2 1 ⊕ 2III0 −2 .

Part (i) of the lemma now follows from Lemma 2.5. The case (ii) is proved similarly by using the fact that the module L0 is of type III01 . The statement (iii) follows from (i), (ii) and the fact that p is surjective (since p(N ) 6= 0). By Proposition 2.7 the “bubble” vanishes in L⊗2 0 , therefore M ∈ Ker(p). Remark 2.10. A similar (and even slightly simpler) argument can be used to prove the key relation between invariant sl2 -tensors needed for computation of the universal sl2 V2 weight system in [5]. In this case dim Inv(S 2 ( L)) = 1.

112


3. U (gl(1|1))-Valued Weight Systems

3.1. Fundamental relation on the level of weight systems. Now we will rewrite the relation (25) in terms of the universal U(gl(1|1))-valued weight system W . The weight system W is defined as in Example 1.3.2. Take a Feynman diagram F and cut open the Wilson loop anywhere but at an external leg. This gives us an invariant tensor, which is not unique, since it depends on where we cut the diagram open. However, as it was explained in Sect. 1, the image W (F ) of this tensor in the universal enveloping algebra U(gl(1|1)) is well-defined and by Proposition 1.12 belongs to the center of U (gl(1|1)). It is known (cf. [3]) that the center of U (gl(1|1)) is the polynomial algebra C [h, c] where h and c are the images in U(gl(1|1)) of the invariant tensors H and C, respectively. However since Feynman diagrams only possess trivalent vertices and propagators, the image of W is not all of C [c, h] but the subalgebra C [c, y] where y was defined in Eq. (9). This fact is an immediate corollary from the recursion relation (1). With our choice of invariant metric, it is easy to check that y = −h2 .

(26)

It follows from the previous discussion that any linear relation in the tensor algebra like the one in (25) can be inserted in any Feynman diagram to yield a linear relation for the corresponding weight system. Indeed, in any fixed chord diagram of order n−2, we can insert (25) to obtain the following relation for the universal weight system W : r

r  r r r  r r r r g g b b 1 =−  − b − b g  , g r+ b r r r 2 br r r r r r r

(27)

where the n−2 original chords are not pictured but are the same for all five diagrams. The following identity for W is an immediate consequence of Proposition 2.7 and Eq. (26): r i

= 2y

.

(28)

r Finally, as a direct corollary of the relations (27) and (28) we have: Proposition 3.1. The universal U(gl(1|1))-valued weight system W satisfies the following relation: r r r   r r r = y  bb  . + − − b r r br r r r

3.2. The recursion formula. We now prove the recursion formula stated in the Introduction, but we shall first need a lemma.


113

Lemma 3.2. (The Eight-Term Relations) Let R1 , . . . , R4 be formulas: r r r r r r r r r − − + R1 = r r r r r r r r r r r r r r r r r r " b " b R2 = − " − + b r r r r r" br r r r r r r r r r r r r " " − R3 = − + " r r r" r r r r r r r r r r r r r r r b b R4 = − − + b r r r r br r r r r and let S1 , . . . , S4 be defined by: S1

S2

=

" r"

r T

r

"

"

=

+

S3

=

=

−

TTr r r

" r"

r "

T

TTr

r T +

r

r

r r r r r b " b " " b r" br r r r r r r r r r T r T , r TTr r

r

r

r r −

r

r

r

+

"

r

− r

−

r T

r S4

T

r r

r

r

+ r

defined by the following

T

TTr

−

r −

r

r

r −

r

r −

. r

Then for all i = 1, . . . , 4, Ri = ySi . Proof. The left-hand sides of the first, third, and fourth equations are obtained by applying the three-term relations to the diagram r r . r

r

114


On the other hand, applying Proposition 3.1 to the same diagram gives rise to the right-hand sides. The left-hand side of the second equation is obtained by applying the transposition operator, S, to the right two ends of the preceding diagram: r r . r r Doing the same to the right-hand sides of Proposition 3.1 gives rise to its right-hand side. Theorem 3.3. Let D be a chord diagram, “a” a fixed chord in D, and b1 , b2 , . . . be all the chords of D intersecting a. Denote by Da (resp. Da,i , Da,ij ) the diagram D − a (resp. D − a − bi , D − a − bi − bj ). Then W (D)

=

cW (Da ) − y +y

X

X

W (Da,i )

i

+− −+ l r W Da,ij + W Da,ij − W Da,ij − W Da,ij ,

i<j +− −+ l r (resp. Da,ij ; Da,ij ; Da,ij ) is the diagram obtained by adding to Da,ij a where Da,ij new chord connecting the left end of bi and the right end of bj (resp. the right end of bi and the left end of bj ; the left ends of bi and bj ; the right ends of bi and bj ) assuming that the chord a is drawn vertically. See Eq. (2) for a pictorial description of the diagrams appearing in the recursion relation.

Proof. The proof is a double induction similar to the one done by Chmutov-Varchenko [5] for sl2 . Let D be a chord diagram of order ordD = n and let “a” be a chord in D. Draw D so that a is a vertical line where the number ` = `(D, a) of chord endpoints to the left of a is less than or equal to the number of chord endpoints to the right of a. Assume that ` ≥ 2. We take as the induction hypothesis that the recursion formula holds for all pairs (E, b) consisting of a chord diagram E and a chord b in E, such that ordE ≤ n and λ(E, b) < `. For the pair (D, a) described above, there are seven possible configurations of chords which have endpoints immediately to the left of a: r r r r

r

r

r

r

r r r r

r r r r

r

r

r

r

r r

r r r r

r r r

r r

r r

r r r

r r

r r

The vertical line is the chord a in each case. In the first six cases, we denote by u and v respectively the upper and lower chords adjacent to a in the diagram. We will work through the induction step for a diagram, D, that looks like the first configuration. Configurations two to six follow similarly. (The induction step for the last configuration is trivial because a chord intersects a if and only if it intersects the adjacent chord.) Apply the first eight-term relation of Lemma 3.2 to D to obtain:


r r r r



r r

r

r

y 

=

115

r

+

r

r r r

r r

r

+

r

r r

−

r

r r

r r r

r

+

−

− r r

 r  r

r r

Let us denote the terms of the right-hand side of this equation by X1 , X2 , . . . , X7 , respectively. Since the terms on the right-hand side have fewer endpoints to the left of a than D, by the induction hypothesis, we obtain the following: −+ 1. X1 = y W Da,uv , +− 2. X2 = y W Da,uv , l 3. X3 = −y W Da,uv , r 4. X4 = −y W Da,uv , P P 5. X5 = cW (Da ) − y i W Da,i + W Da,v − i<j W (Λa,ij ) P − i W Λa,iv P P 6. X6 = cW (Da ) − y i W Da,i + W Da,u − i<j W (Λa,ij ) P − j W Λa,uj , P P W D W (Λ ) , 7. X7 = −cW (Da ) + y − a,i a,ij i i<j where Λa,ij is given by +− −+ l r + Da,ij − Da,ij − Da,ij . Λa,ij = Da,ij

(29)

Adding all the terms X1 , . . . , X7 , we obtain the right-hand side of the recursion formula. This completes the induction step. It remains to show the induction base: ` = 0 and ` = 1. If ` = 0, then no chord in D intersects a, so that D is a connected sum of the chord diagram Θ of order 1 and a chord diagram Da of order n − 1. Therefore, W (D) = c W (Da ) which agrees with the recursion formula. If ` = 1, there is only one chord i, say, intersecting a: ra r r

r

i

Using the three-term relation (7), r r r

r r = r r

=

c

r−

r

r

r

r

r−

1 2

r

i

r= c

r

r− y

,

116


which is just c W (Da ) − y W (Da,i ), in agreement with the recursion formula.

Corollary 3.4. The universal weight system Wgl(1|1) corresponding to gl(1|1) takes values in the polynomial algebra Z [c, y]. Furthermore, the polynomial corresponding to a chord diagram, D, of order n is a weighted homogeneous polynomial of order n, where c and y are assigned weights 1 and 2, respectively. The leading term in c of Wgl(1|1) (D) is equal to cn , where n is the number of chords in D. The coefficient of the subleading term cn−1 y is the negative of the number of intersections of the chords in D when the chords are placed in generic position. Proof. The corollary follows from a simple induction on the number of chords.

3.3. Examples of computations. We will now illustrate the use of the recursion formula on a few examples. Normalizing W by setting it equal 1 on the diagram with no chords, the recursion formula yields the value of W on any other diagram. Eq. (8) is an immediate consequence of the recursion relation. For the unique indecomposable chord diagram of order 2 we obtain r '$ '$ '$ r

r =

cr

r− y

r &% &% &% =

c2 − y .

There are two indecomposable chord diagrams of order 3: r '$ '$ '$ '$  r r r r r r  = c −y  + r r r r r r r &% &% &% &% '$ '$ '$ '$   r r r r b "  + y  bb + "" − − r r br r" &% &% &% &% = and

c3 − 2yc ,

'$ '$ '$ r r r r r r '$  T T T r T r = c  T T −y  + T T T Tr r Tr r Tr r &% &% &% &% '$ '$ '$ r r r r '$  +y 

+

−

−



r r r r &% &% &% &% =

c3 − 3yc .


117

These results are summarized in the following table. Table 2. Indecomposable chord diagrams of order ≤ 3

D

Wgl(1|1) (D)

'$

D

r '$ c

1 &% r '$ r

r

c2 − y

r &% '$ r r T r T r c3 − 3cy T r Tr &%

Wgl(1|1) (D)

r &% '$ r r T r T r c3 − 2cy r TTr &%

We will now use these results to perform a more complex computation: '$ '$ '$ r r r r r  '$ r r r r r r  = c −y  + r r r r r r r r r r r &% &% &% &% '$ '$ '$ r r r  r r '$ r r r b "  + y  bb + "" − − r br r" r r r r r &% &% &% &% =

c(c3 − 2cy) − 2yc2 = c4 − 4c2 y

In the appendix we list the values of the universal weight system Wgl(1|1) for all indecomposable chord diagrams of orders 4 and 5. 3.4. Deframing the universal weight system. In this subsection we will prove that the strong weight system obtained from the universal weight system Wgl(1|1) under the deframing projection W −→ W (see Sect. 1.1) is the same as evaluating it at c = 0.6 We start by defining a linear map on the algebra A of chord diagrams s : A → A as follows. If D is a nonempty chord diagram, then X Da , s(D) = a

where the sum runs over all chords a in D and where Da denotes D − a; and if D is the empty chord diagram s(D) = 0. We extend s linearly. It is clear that s respects the four-term relation, so it is well defined on A. 6

The results of this subsection owe much to conversations with Bill Spence.

118


¯ Recall (see Theorem 1.4) that A = A[Θ], where Θ is the unique chord diagram with one chord. Let ψ be the restriction to A¯ of the map λ : A → A defined by λ(D) =

X

(−1)k

k≥0

1 k k Θ · s (D) . k!

Notice that the sum has only a finite number of nonzero terms when acting on any chord diagram D. Proposition 3.5. [1, 9]. The deframing projector W → W is dual to ψ. In other words, if w ∈ Wn is any weight system, and D ∈ An is a chord diagram, then the deframed weight system w ∈ W n is given by: w(D) = ψ ∗ w(D) = w(ψ(D)) . Theorem 3.6. Let W = Wgl(1|1) denote the universal weight system of gl(1|1), i.e., W : A → Z(U(gl(1|1))) = C[h, c], and let W = ψ ∗ (W ) be its deframing. Then if D is any chord diagram, W ∈ C[h] and W (h) = W (h, 0) . To prove this we start with a lemma. Lemma 3.7. For any chord diagram D, W (s(D)) =

∂ W (D) . ∂c

Proof. We proceed by induction on the order of the chord diagram. The induction base is clear for diagrams of order 1. Assume that the lemma holds for all chord diagrams of order < n. P Let D be a chord diagram of order n, and let a be a chord in D. Then s(D) = Da + i6=a Di . We will use the recursion relation (1) to compute W (s(D)) = P W (Da )+ i6=a W (Di ). Since a is a chord in every Di , for i 6= a, we can use the recursion relation on a to obtain X X W (Dia,j ) + y W (Λia,jk ) , W (Di ) = cW (Dia ) − y j6=i

j 2, then all five terms in (33) vanish. 2. If a = d, then by Lemma 4.8 the l.h.s. is zero, and in the r.h.s. the second and the third terms, as well as the first and the fourth, cancel out due to Lemma 4.6. 3. If a = b 6= c = d, the last two terms of the r.h.s. vanish, and the first two together give 2, which is the value of the l.h.s. by Lemma 4.8. 4. At last, if a = c 6= b = d, then the l.h.s. is equal to −2, and in the r.h.s. the first two terms vanish, and the last two give −2. A. Values of Wgl(1|1) for Chord Diagrams of Orders 4 and 5 In this appendix we tabulate the values of the universal weight system Wgl(1|1) for chord diagrams of orders 4 and 5. These values have been obtained by computer, after implementing the recursion relation (1) in Mathematica7 . Our results agree with the direct computation of the weight system using the Lie superalgebraic definition, which was implemented on a computer by Craig Snydal as part of his undergraduate thesis [12]. Because of 1.3.5, only indecomposable diagrams are tabulated. Table 3. Indecomposable chord diagrams of order 4

D

Wgl(1|1) (D)

'$ r r r r r c4 − 3c2 y + y 2 r r &% rr '$ r r r r c4 − 3c2 y r r r &% '$ r r r r c4 − 4c2 y r r r r &%

D

Wgl(1|1) (D)

'$ r r r r

r

c4 − 4c2 y + y 2

r r r &% '$ r r r r b " " b c4 − 5c2 y " b r" br r r &% '$ r r r r r c4 − 6c2 y + y 2 r r r &%

In fact, we have computed the values of Wgl(1|1) on all chord diagrams of order ≤ 7, and they are available upon request. 7


Tabelle 4.

D

125

Indecomposable chord diagrams of order 5 Wgl(1|1) (D)

'$ r r r r r r c5 − 4c3 y + 3cy 2 r r r r &% '$ r r r r r r c5 − 4c3 y + 2cy 2 r r r r &% '$ r r r r r r c5 − 5c3 y + 3cy 2 r r r r &% '$ r r r r r r c5 − 5c3 y + 3cy 2 r r r r &% '$ r r r r r r c5 − 5c3 y + 2cy 2 r r r r &% '$ r r r r r r c5 − 7c3 y + 4cy 2 r r r r &% '$ r r r r r r c5 − 6c3 y + 2cy 2 r r r r &% '$ r r r r r T r T c5 − 5c3 y + 2cy 2 r T r r Tr &%

D

Wgl(1|1) (D)

'$ r r r r r r c5 − 5c3 y + 4cy 2 r r r r &% '$ r r r r r r c5 − 4c3 y + 3cy 2 r r r r &% '$ r r r r r r c5 − 6c3 y + 3cy 2 r r r r &% '$ r r r r r r c5 − 4c3 y + 2cy 2 r r r r &% '$ r r r r r T r T c5 − 4c3 y + 2cy 2 r T r r Tr &% '$ r r r r r r c5 − 6c3 y + 2cy 2 r r r r &% '$ r r r r r r c5 − 5c3 y + 2cy 2 r r r r &% '$ r r r r r T r T c5 − 4c3 y r T r r Tr &%

126


Table 5.

D

Indecomposable chord diagrams of order 5 (cont’d) Wgl(1|1) (D)

'$ r r r r r r c5 − 6c3 y + 3cy 2 r r r r &% '$ r r r r r r c5 − 5c3 y + 4cy 2 r r r r &% '$ r r r r r r c5 − 6c3 y + 5cy 2 r r r r &% '$ r r r r r r c5 − 6c3 y + 4cy 2 r r r r &% '$ r r r r r r c5 − 7c3 y r r r r &% '$ r r r r r r c5 − 6c3 y r r r r &% '$ r r r r r r c5 − 9c3 y + 2cy 2 r r r r &% '$ r r r r r r c5 − 10c3 y + 5cy 2 r r r r &%

D

Wgl(1|1) (D)

'$ r r r r r r c5 − 5c3 y + 2cy 2 r r r r &% '$ r r r r r r c5 − 4c3 y + 3cy 2 r r r r &% '$ r r r r r r c5 − 7c3 y + 3cy 2 r r r r &% '$ r r r r r r c5 − 8c3 y + 3cy 2 r r r r &% '$ r r r r r r c5 − 7c3 y + 2cy 2 r r r r &% '$ r r r r r r c5 − 5c3 y + 5cy 2 r r r r &% '$ r r r r r r c5 − 8c3 y r r r r &%


127

Acknowledgement. The work described in this paper has been done when the first and the third authors visited the second at the University of North Carolina at Chapel Hill. We are all grateful to the Department of Mathematics of UNC and to Jim Stasheff for their hospitality. Additional thanks for hospitality go from JMF to Louise Dolan and the Department of Physics of UNC and from AV to the Max-Planck Institut für Mathematik where a large part of the paper was written up. It is a pleasure to thank Bill Spence for helpful conversations. We would like to convey our gratitude to Craig Snydal without whom this paper would not have been written. In his thesis [12], Craig wrote computer programs which computed universal Vassiliev invariants associated to gl(1|1) up to order 5 before the recursion relation was discovered. It was while supervising Craig’s thesis that TK learned a great deal about this subject. In particular, the fundamental relation (25) was initially discovered using software written by TK which was incorporated into Craig’s programs. Craig’s computations proved to be useful in checking our results, as well. To him we give our special thanks.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

11. 12. 13. 14. 15. 16.

Bar-Natan, D.: On the Vassiliev knot invariants. Topology, 34, 423-472 (1995) Bar-Natan, D.: Weights of Feynman diagrams and the Vassiliev knot invariants. Preprint, 1991 Berezin, F. A.: Introduction to Superanalysis. Dordrecht: Reidel 1987 Birman, J.S., Lin, X.-S.: Knot polynomials and Vassiliev’s invariants. Invent. math. 111, 225–270 (1993) Chmutov, S.V. Varchenko, A.N.: Remarks on the Vassiliev knot invariants coming from sl2 . Topology 36, 153–178 (1997) Kac, V.G.: Lie superalgebras. Adv. Math. 26, 8–96 (1977) Kauffman, L. H., Saleur, H.: Free fermions and the Alexander-Conway polynomial. Comm. Math. Phys. 141, 293–327 (1991) Kontsevich, M.: Vassiliev’s knot invariants. Adv. in Soviet Math. 16, 137–150 (1993) Kricker, A., Spence, B., Aitchinson, I.: Cabling the Vassiliev invariants. q-alg/9511024. Rozansky, L. Saleur, H.: S and T matrices for the super U (1, 1) WZW model. Application to surgery and 3-manifolds invariants based on the Alexander-Conway polynomial. Nucl. Phys. B389, 365–423 (1993) Rozansky, L. Saleur, H.: Quantum field theory for the multi-variable Alexander-Conway polynomial. Nucl. Phys. B376, 461–509 (1992) Snydal, C. T.: Vassiliev knot invariants arising from the Lie superalgebra gl(1|1). Undergraduate thesis, Univ. of North Carolina, Chapel Hill, April 1995 Vaintrob, A.: Vassiliev knot invariants and Lie S-algebras. Mathematical Research Letters 1, 579–595 (1994) Vaintrob, A.: Algebraic structures related to Vassiliev knot invariants. Preprint, 1996 Vassiliev, V.A.: Cohomology of knot spaces, Theory of singularities and its applications. (V.I.Arnold, ed.), Providence: Amer. Math. Soc., 1990, pp. 23–69 Vogel, P. Algebraic structures on modules of diagrams. Preprint JUSSIEU, 1995

Communicated by G. Felder

Commun. Math. Phys. 185, 129 – 154 (1997)

Communications in


q -Gaussian Processes: Non-commutative and Classical Aspects 2 ¨ Marek Bo˙zejko1,? , Burkhard Kummerer , Roland Speicher3,?? 1 Instytut Matematyczny, Uniwersytet Wrocławski, Plac Grunwaldzki 2/4, 50-384 Wrocław, Poland. E-mail: [email protected] 2 Mathematisches Institut A, Pfaffenwaldring 57, D-70569 Stuttgart, Germany. E-mail: [email protected] 3 Institut f¨ ur Angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld 294, D-69120 Heidelberg, Germany. E-mail: [email protected]

Received: 25 July 1996 / Accepted: 17 September 1996

Abstract: We examine, for −1 < q < 1, q-Gaussian processes, i.e. families of operators (non-commutative random variables) Xt = at + a∗t – where the at fulfill the q-commutation relations as a∗t − qa∗t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expectation state. We show that there is a q-analogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on q-Gaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of q-Gaussian processes possesses a non-commutative kind of Markov property, which ensures that there exist classical versions of these non-commutative processes. This answers an old question of Frisch and Bourret [FB]. Introduction What we are going to call q-Gaussian processes was essentially introduced in a remarkable paper by Frisch and Bourret [FB]. Namely, they considered generalized commutation relations given by operators A(t) and a vacuum vector 90 with A(t)A∗ (t0 ) − qA∗ (t0 )A(t) = 0(t, t0 )1 and

A(t) 90 = 0

for some real covariance function 0 (i.e. positive definite function). The aim of the authors was to study the probabilistic properties of the “parastochastic” process M (t) = A(t) + A∗ (t). The basic problems arising in this context were the following two types of questions: ? ??

Partially supported by Polish National Grant, KBN 4233 Supported by a Heisenberg fellowship from the DFG

130

M. Bo˙zejko, B. Kümmerer, R. Speicher

(I) (realization problem) Do there exist operators on some Hilbert space and a corresponding vacuum vector in this Hilbert space which fulfill the above relations, i.e. are there non-commutative realizations of the q-Gaussian processes? (II) (random representation problem) Are these non-commutative processes of a classical relevance, i.e. do there exist classical versions of the q-Gaussian processes (in the sense of coinciding timeordered correlations, see our Definition 4.1)? Frisch and Bourret could give the following partial answers to these questions. (I) For q = ±1 the realization is of course given by the Fock space realization of the bosonic/fermionic relations. The case q = 0 was realized by creation and annihilation operators on the full Fock space (note that this was before the introduction of the Cuntz algebras and their extensions [Cun, Eva]). For other values of q the realization problem remained open. (II) The q = 1 processes are nothing but the Fock space representations of the classical Gaussian processes. For q = −1 a classical realization by a dichotomic Markov process could be given for the special case of exponential covariance 0(t, t0 ) = exp(−|t − t0 |). A classical realization for q = 0 could not be found, but the authors were able to show that there is an interesting representation in terms of Gaussian random matrices. The authors started also the investigation of parastochastic equations (i.e. the coupling of parastochastic processes to other systems), but – probably because of the open problem on the mere existence and classical relevance of these q-processes – there was apparently no further work in this direction and the paper of Frisch and Bourret fell into oblivion. Starting with [AFL] there has been another and independent approach to noncommutative probability theory. This wide and quite inhomogenous field – let us just mention as two highlights the quantum stochastic calculus of Hudson-Parthasarathy [HP] and the free probability theory of Voiculescu [VDN] – is now known under the name of “quantum probability”. At least some of the fundamental motivations for undertaking such investigations can be compared with the two basic questions of Frisch and Bourret: (I) Non-commutative probability theory is meant as a generalization of classical probability theory to the description of quantum systems. Thus first of all their objects are operators on some Hilbert spaces having a meaning as non-commutative analogues of the probabilistic notions of random variables, stochastic processes, etc. (II) In many investigations in this area one also tries to establish connections between non-commutative and classical concepts. The aim of this is twofold. On one side, one hopes to get a better understanding of classical problems by embedding them into a bigger non-commutative context. Thus, e.g., the Azéma martingale, although classically not distinguished within the class of all martingales, behaves in some respects like a Brownian motion [Par1]. The non-commutative “explanation” for this fact comes from the observation of Schürmann [Sch] that this martingale is one component of a non-commutative process with independent increments. In the other direction, one hopes to get a classical picture (featuring trajectories) of some aspects of quantum problems. Of course, a total reduction to classical concepts is in general not possible, but partial aspects may sometimes allow a classical interpretation.

q-Gaussian Processes

131

It was in this context of quantum probability where two of the present authors [BSp1] reintroduced the q-relations – without knowing of, but much in the same spirit as [FB]. Around the same time the q-relations were also proposed by Greenberg [Gre] as an example for particles with “infinite statistics”. The main progress in connection with this renewed interest was the solution of the realization problem of Frisch and Bourret. There exist now different proofs for the existence of the Fock representation of the q-relations for all q with −1 ≤ q ≤ 1 [BSp1, Zag, Fiv, Spe1, BSp3, YW]. In [NSp], the idea of Frisch and Bourret to use the q-relations as a model for a generalized noise was pursued further and the Greens function for such dynamical problems could be calculated for one special choice of the covariance function – namely for the case of the exponential covariance. We will call this special q-process in the following q-Ornstein-Uhlenbeck process. It soon became clear that the special status of the exponential covariance is connected with some kind of (non-commutative) Markovianity – as we will see the q-Ornstein-Uhlenbeck process is the only stationary q-Gaussian Markov process. But using the general theory of Kümmerer on non-commutative stationary Markov processes [Kum1, Kum2] this readily implies the existence of a classical version (being itself a classical Markov process) of the q-Ornstein-Uhlenbeck process. Thus we got a positive solution of the random representation problem of Frisch and Bourret in this case. However, the status of the other q-Gaussian processes, in particular q-Brownian motion, remained unclear. Motivated by our preliminary results, Biane [Bia1] (see also [Bia2, Bia3]) undertook a deep and beautiful analysis of the free (q = 0) case and showed the remarkable result that all processes with free increments are Markovian and thus possess classical versions (with a quite explicit calculation rule for the corresponding transition probabilities). This includes in particular the case of free Brownian motion. Inspired by this work we could extend our investigations from the case of the qOrnstein-Uhlenbeck process to all q-Gaussian processes. The results are presented in this paper. Up to now there is only one strategy for establishing the existence of a classical version of a non-commutative process, namely by showing that the process is Markovian. That this implies the existence of a classical version follows by general arguments, the main point is to show that we have this property in the concrete case. Whereas Biane could use the quite developed theory of freeness [VDN] to prove Markovianity for processes with free increments, there is at the moment (and probably also in the future [Spe2]) no kind of q-freeness for general q. Thus another feature of our considered class of processes is needed to attack the problem of Markovianity. It is the aim of this paper to convince the reader of the fact that the q-analogue of Gaussianity will do this job. The essential idea of Gaussianity is that one can pull back all considerations from the measure theoretic (or, in the non-commutative frame, from the operator algebraic) level to an underlying Hilbert space, thus in the end one essentially has to deal with linear problems. The main point is that this transcription between the linear and the algebraic level exists in a consistent way. The best way to see and describe this is by presenting a functor (“second quantization”) which translates the Hilbert space properties into operator algebraic properties. Our basic considerations will therefore be on the existence and nice properties of the q-analogue of this functor. Having this functor, the rest is mainly linear theory on the Hilbert space level. It turns out that all relevant questions on our q-Gaussian processes can be characterized totally in terms of the corresponding covariance function. In particular, it becomes quite easy to decide whether such a process is Markovian or not.

132


The paper is organized as follows. In Sect. 1 we recall some basic facts about the qFock space and its relevant operators. Furthermore we collect in this section the needed combinatorial results, in particular on q-Hermite polynomials. Section 2 is devoted to the presentation of the functor 0q of second quantization. The main results (apart from the existence of this object) are the facts that the associated von Neumann algebras are in the infinite dimensional case non-injective II1 -factors and that the functor maps contractions into completely positive maps. Having this q-Gaussian functor the definition and investigation of properties of q-Gaussian processes (like Markovianity or martingale property) is quite canonical and parallels the classical case. Thus our presentation of these aspects, in Sect. 3, will be quite condensed. Sect. 4 contains the classical interpretation of the q-Gaussian Markov processes. As pointed out above general arguments ensure the existence of classical versions for these processes. But we will see that we can also derive quite concrete formulas for the corresponding transition probabilities. 1. Preliminaries on the q-Fock Space Let q ∈ (−1, 1) be fixed in the following. For a complex Hilbert space H we define its q-Fock space Fq (H) as follows: Let F f inite (H) be the linar span of vectors of the form f1 ⊗ . . . ⊗ fn ∈ H⊗n (with varying n ∈ N0 ), where we put H⊗0 ∼ = C for some distinguished vector , called vacuum. On F f inite (H) we consider the sesquilinear form h·, ·iq given by a sesquilinear extension of X q i(π) hf1 , gπ(1) i . . . hfn , gπ(n) i, hf1 ⊗ . . . ⊗ fn , g1 ⊗ . . . ⊗ gm iq := δnm π∈Sn

where Sn denotes the symmetric group of permutations of n elements and i(π) is the number of inversions of the permutation π ∈ Sn defined by i(π) := #{(i, j) | 1 ≤ i < j ≤ n, π(i) > π(j)}. Another way to describe h·, ·iq is by introducing the operator Pq on F f inite (H) by a linear extension of , Pq = P Pq f1 ⊗ . . . ⊗ fn = π∈Sn q i(π) fπ(1) ⊗ . . . ⊗ fπ(n) . Then we can write hξ, ηiq = hξ, Pq ηi0

(ξ, η ∈ F f inite (H)),

where h·, ·i0 is the scalar product on the usual full Fock space F0 (H) =

M

H⊗n .

n≥0

One of the main results of [BSp1] (see also [BSp3, Fiv, Spe1, Zag]) was the strict positivity of Pq , i.e. hξ, ξiq > 0 for 0 6= ξ ∈ F f inite (H). This allows the following definitions.


133

Definition 1.1. 1) The q-Fock space Fq (H) is the completion of F f inite (H) with respect to h·, ·iq . 2) Given f ∈ H, we define the creation operator a∗ (f ) and the annihilation operator a(f ) on Fq (H) by a∗ (f ) = f, a (f )f1 ⊗ . . . ⊗ fn = f ⊗ f1 ⊗ . . . ⊗ fn , ∗

and

a(f ) = 0, Pn a(f )f1 ⊗ . . . ⊗ fn = i=1 q i−1 hf, fi if1 ⊗ . . . ⊗ fˇi ⊗ . . . ⊗ fn , where the symbol fˇi means that fi has to be deleted in the tensor. Remark 1.2. The operators a(f ) and a∗ (f ) are bounded operators on Fq (H) with √ kf k/ 1 − q, 0 ≤ q < 1 ∗ ka(f )kq = ka (f )kq = kf k, −1 < q ≤ 0,

and they are adjoints of each other with respect to our scalar product h·, ·iq . Furthermore, they fulfill the q-relations a(f )a∗ (g) − qa∗ (g)a(f ) = hf, gi · 1

(f, g ∈ H).

Notation 1.3. For a linear operator T : H → H0 between two complex Hilbert spaces we denote by F(T ) : F f inite (H) → F f inite (H0 ) the linear extension of F(T ) = , F(T )f1 ⊗ . . . ⊗ fn = (T f1 ) ⊗ . . . ⊗ (T fn ). In order to keep the notation simple we denote the vacuum for H and the vacuum for H0 by the same symbol . It is clear that F(T ) can be extended to a bounded operator F0 (T ) : F0 (H) → F0 (H0 ) exactly if T is a contraction, i.e. if kT k ≤ 1. The following lemma ensures that the same is true for all other q ∈ (−1, 1), too. Lemma 1.4. Let T : F f inite (H) → F f inite (H0 ) be a linear operator which fulfills Pq0 T = T Pq , where Pq and Pq0 are the operators on F f inite (H) and F f inite (H0 ), respectively, which define the respective scalar product h·, ·iq . Then one has kT kq = kT k0 . Hence, if kT k0 < ∞, then T can, for each q ∈ (−1, 1), be extended to a bounded operator from Fq (H) to Fq (H0 ). Proof. Let ξ ∈ F f inite (H). Then kT ξk2q = = = ≤ = which implies

hT ξ, T ξiq hT ξ, Pq0 T ξi0 1/2 1/2 hPq ξ, T ∗ T Pq ξi0 1/2 1/2 ∗ kT T k0 hPq ξ, Pq ξi0 ∗ 2 kT T k0 kξkq ,

kT k2q ≤ kT ∗ T k0 ≤ kT ∗ k0 kT k0 = kT k20 ,

and thus kT kq ≤ kT k0 . Since we can estimate in the same way, by replacing Pq by 0 Pq−1 and Pq0 by Pq −1 , also kT k0 ≤ kT kq , we get the assertion.

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Notation 1.5. For a contraction T : H → H0 , we denote the extension of F(T ) from F f inite (H) → F f inite (H0 ) to Fq (H) → Fq (H0 ) by Fq (T ). Remark 1.6. 1) One might call Fq (T ) the second quantization of T , but we will reserve this name for the restriction of Fq (T ) to some operator algebra lying in Fq (H) – see the next section, where we will also prove some positivity properties of this restricted version. 2) The operator Fq (T ) and its differential version (in particular the number operator) were also considered in [Wer] and [Sta, Mol], respectively. 3) It is clear that Fq (·) behaves nicely with respect to composition and taking adjoints, i.e. Fq (ST ) = Fq (S)Fq (T ), Fq (T ∗ ) = Fq (T )∗ , Fq (1) = 1, but not with respect to the additive structure, i.e. Fq (T + S) 6= Fq (T ) + Fq (S)

in general.

In the context of the q-relations one usually encounters some kind of q-combinatorics. Let us just recall the basic facts. Notation 1.7. We put for n ∈ N0 , [n]q :=

1 − qn = 1 + q + . . . + q n−1 1−q

([0]q := 0).

Then we have the q-factorial [n]q ! := [1]q . . . [n]q ,

[0]q ! := 1,

and a q-binomial coefficient n k

:= q

n−k Y 1 − q k+i [n]q ! = . [k]q ![n − k]q ! 1 − qi i=1

Another quite frequently used symbol is the q-analogue of the Pochhammer symbol (a; q)n :=

n−1 Y

(1 − aq j )

in particular

(a; q)∞ :=

j=0

∞ Y

(1 − aq j ).

j=0

The importance of these concepts in connection with the q-relations can be seen from the following q-binomial theorem, which is by now quite standard. Proposition 1.8. Let x and y be indeterminates which q-commute in the sense xy = qyx. Then one has for n ∈ N, (x + y)n =

n X n k=0

k

y k xn−k . q

Proof. This is just induction and the easily checked equality n n n+1 + qk = . k+1 q k+1 q k q


135

In the same way as the usual Hermite polynomials are connected to the bosonic relations, the q-relations are linked to q-analogues of the Hermite polynomials. Definition 1.9. The polynomials Hn(q) (n ∈ N0 ), determined by H0(q) (x) = 1,

H1(q) (x) = x,

and (q) (q) xHn(q) (x) = Hn+1 (x) + [n]q Hn−1 (x)

(n ≥ 1)

are called q-Hermite polynomials. We recall two basic facts about these polynomials which will be fundamental for our investigations on the classical aspects of q-Gaussian processes. √ √ Theorem 1.10. 1) Let νq be the measure on the interval [−2/ 1 − q, 2/ 1 − q] given by ∞ Y 1p 1 − q sin θ (1 − q n )|1 − q n e2iθ |2 dx, νq (dx) = π n=1

where x= √

2 cos θ 1−q

with θ ∈ [0, π].

Then the q-Hermite polynomials are orthogonal with respect to νq , i.e. Z √ 2/

−2/

1−q

√

Hn (x)Hm (x)νq (dx) = δnm [n]q !. 1−q

√ √ 2) Let r > 0 and x, y ∈ [−2/ 1 − q, 2/ 1 − q]. Denote by p(q) r (x, y) the kernel p(q) r (x, y) :=

∞ X rn H (q) (x)Hn(q) (y). [n]q ! n n=0

Then we have with x= √

2 cos ϕ, 1−q

y=√

2 cos ψ 1−q

the formula p(q) r (x, y) =

(r2 ; q)∞ . i(ϕ+ψ) |(re ; q)∞ (rei(ϕ−ψ) ; q)∞ |2

In particular, for q = 0, we get p(0) r (x, y) =

1 − r2 . (1 − r2 )2 − r(1 + r2 )xy + r2 (x2 + y 2 )

As usual in q-mathematics these formulas are quite old, namely the orthogonalizing measure νq was calculated by Szego [Sze], whereas the kernel p(q) r (x, y) goes even back to Rogers [Rog]. For more recent treatments, see [Bre, ISV, GR], in connection with the q-Fock space also [LM1, LM2].

136


2. Second Quantization – The Functor 0q An abstract way of dealing with classical Gaussian processes is by using the Gaussian functor 0. This is a functor from real Hilbert spaces and contractions to commutative von Neumann algebras with specified trace-state and unital trace preserving completely positive maps [Nel1, Nel2, Gro, Sim1, Sim2]. Essentially, this point of view can be traced back to Segal [Seg]. Fermionic and free analogues of this functor are also known, see, e.g., [Wil, CL, Voi, VDN]. In this section we will present a q-analogue of the Gaussian functor. Namely, to each real Hilbert space, H, we will associate a von Neumann algebra with specified tracestate, ( 0q (H), E), and to every contraction T : H → H0 a unital completely positive trace preserving map 0q (T ) : 0q (H) → 0q (H0 ). Definition 2.1. Let H be a real Hilbert space and HC its complexification HC = H⊕iH. Put, for f ∈ H, ω(f ) := a(f ) + a∗ (f ) ∈ B(Fq (HC )) and denote by 0q (H) ⊂ B(Fq (HC )) the von Neumann algebra generated by all ω(f ), 0q (H) := vN(a(f ) + a∗ (f ) | f ∈ H). Notation 2.2. We denote by

E : 0q (H) → C

the vacuum expectation state on 0q (H) given by E[X] := h , X iq

(X ∈ 0q (H)).

We recall some basic facts about 0q (H) in the following proposition. Proposition 2.3. The vacuum is a cyclic and separating trace-vector for 0q (H), hence the vacuum expectation E is a faithful normal trace on 0q (H) and 0q (H) is a finite von Neumann algebra in standard form. Proof. See Theorems 4.3 and 4.4 in [BSp3].

The first part of the proposition yields in particular that the mapping 0q (H) → Fq (HC ), X 7→ X is injective, in this way we can identify each X ∈ 0q (H) with some element of the q-Fock space Fq (HC ). Notation 2.4. 1) Let us denote by L∞ q (H) := 0q (H) the image of 0q (H) under the mapping X 7→ X . 2) We also put L2q (H) := Fq (HC ). ∞ Definition 2.5. Let 9 : L∞ q (H) → 0q (H) be the identification of Lq (H) with 0q (H) given by the requirement

9(ξ) = ξ

for

2 ξ ∈ L∞ q (H) ⊂ Lq (H) = Fq (HC ).


137

Remark 2.6. 1) Of course, not each element of the q-Fock space comes from an X ∈ 0q (H), but the main relation for observing the cyclicity of , namely f1 ⊗ . . . ⊗ fn = ω(f1 ) . . . ω(fn ) − η

with η ∈

n−1 M

H⊗l ,

l=0

L∞ q (H).

yields that we have at least f1 ⊗ . . . ⊗ fn ∈ 2) In a quantum field theoretic context [Sim1, Sim2] the operator 9(f1 ⊗ . . . ⊗ fn ) would be called “Wick product” and denoted by 9(f1 ⊗ . . . ⊗ fn ) =: ω(f1 ) . . . ω(fn ) : . 3) In a quantum probabilistic context [Par2, Mey] 9 would correspond to taking an iterated quantum stochastic integral: For HC = L2 (R) and ξ = f1 ⊗ . . . ⊗ fn with ξ(t1 , . . . , tn ) = f1 (t1 ) . . . fn (tn ) one would denote Z 9(ξ) = ξ(t1 , . . . , tn )dω(t1 ) . . . dω(tn ) and call ξ the “Maassen kernel” of 9(ξ). The explicit form of our Wick products is given in the following proposition. Proposition 2.7. We have for n ∈ N and f1 , . . . , fn ∈ H the normal ordered representation fn ) = 9(f P1 ⊗ . . . ⊗P = k,l=0,...,n a∗ (fi(1) ) . . . a∗ (fi(k) )a(fj(1) ) . . . a(fj(l) ) · q i(I1 ,I2 ) , I ={i(1),...,i(k)} k+l=n

where

1 I2 ={j(1),...,j(l)} with I1 ∪I2 ={1,...,n} I1 ∩I2 =∅

i(I1 , I2 ) := #{(p, q) | 1 ≤ p ≤ k, 1 ≤ q ≤ l, i(p) > j(q)}.

Denote by X the right-hand side of the above relation. It is clear that X = f1 ⊗ . . . ⊗ fn , the problem is to see that X can be expressed in terms of the ω’s. Proof. Note that the formula is true for 9(f ) = ω(f ) = a(f ) + a∗ (f ) and that the definition of a∗ (f ) and of a(f ) gives 9(f ⊗f1 ⊗. . .⊗fn ) = ω(f ) 9(f1 ⊗. . .⊗fn )−

n X

q i−1 hf, fi i 9(f1 ⊗. . .⊗ fˇi ⊗. . .⊗fn ).

i=1

From this the assertion follows by induction.

Note that 9(f1 ⊗ . . . ⊗ fn ) is just given by multiplying out ω(f1 ) . . . ω(fn ) and bringing all appearing terms with the help of the relation aa∗ = qa∗ a into a normal ordered form – i.e. we throw away all normal ordered terms in ω(f1 ) . . . ω(fn ) which have less than n factors. Thus, for the special case f1 = . . . = fn , we are in the realm of the q-binomial theorem and we have the following nice formula.

138


Corollary 2.8. We have for n ∈ N and f ∈ H, 9(f ⊗n ) =

n X n k=0

k

a∗ (f )k a(f )n−k . q

Instead of writing 9(f ⊗n ) in a normal ordered form we can also express it in terms of ω(f ) with the help of the q-Hermite polynomials. Proposition 2.9. We have for n ∈ N0 and f ∈ H with kf k = 1 the representation 9(f ⊗n ) = Hn(q) (ω(f )). Proof. This follows by the fact that the 9(f ⊗n ) fulfill the same recurrence relation as the Hn(q) (ω(f )), namely ω(f ) 9(f ⊗n ) = 9(f ⊗(n+1) ) + [n]q 9(f ⊗(n−1) ) and that we have the same initial conditions 9(f ⊗0 ) = 1,

9(f ⊗1 ) = ω(f ).

We know [Voi, VDN] that for q = 0 the von Neumann algebra 00 (H) is isomorphic to the von Neumann algebra of the free group on dim H generators – in particular, it is a non-injective II1 -factor for dim H ≥ 2. We conjecture non-injectivity and factoriality in the case dim H ≥ 2 for arbitrary q ∈ (−1, 1), but up to now we can only show the following. Theorem 2.10. 1) For −1 < q < 1 and dim H > 16/(1 − |q|)2 the von Neumann algebra 0q (H) is not injective. 2) If −1 < q < 1 and dim H = ∞ then 0q (H) is a II1 -factor. Proof. 1) This was shown in a more general context in Theorem 4.2 in [BSp3]. 2) Let {ei }i∈N be an orthonormal basis of H. Fix n ∈ N0 and r(1), . . . , r(n) ∈ N and consider the operator X := 9(er(1) ⊗ . . . ⊗ er(n) ). (For n = 0 this will be understood as X = 1.) We put 1 X ω(ei )Xω(ei ) m m

φm (X) :=

(m ∈ N)

i=1

and claim that φm (X) converges for m → ∞ weakly to φ(X) := q n X. Because of the m-independent estimate kφm (X)kq ≤ kXkq kω(e1 )k2q it suffices to show lim hξ, φm (X)ηiq = hξ, φ(X)ηiq

m→∞

for all ξ, η ∈ Fq (HC ) of the form ξ = ea(1) ⊗ . . . ⊗ ea(u) ,

η = eb(1) ⊗ . . . ⊗ eb(v)


139

with u, v ∈ N0 , a(1), . . . , a(u), b(1), . . . , b(v) ∈ N (for u = 0 we put ξ = ). To see this, put m0 := max{a(1), . . . , a(u), b(1), . . . , b(v), r(1), . . . , r(n)}. Since |hξ, ω(ei )Xω(ei )ηiq | ≤ M for some M (independent of i), we have Pm 1 lim hξ, φm (X)ηiq = lim m i=m +1 hξ, ω(ei )Xω(ei )ηiq n→∞ n→∞ Pm 0 1 = lim m i=m0 +1 hξ, a(ei ) 9(er(1) ⊗ . . . ⊗ er(n) )a∗ (ei )ηiq . n→∞

By Prop. 2.7, 9(er(1) ⊗ . . . ⊗ er(n) ) is now a linear combination of terms of the form Y = Y1 Y2 with Y1 = a∗ (er(i(1)) ) . . . a∗ (er(i(k)) )

and

Y2 = a(er(j(1)) ) . . . a(er(j(l)) )

with k + l = n. Each such term gives, for i > m0 , a contribution hξ, a(ei )Y a∗ (ei )ηiq = = = = =

hξ, a(ei )Y1 Y2 a∗ (ei )ηiq q k+l hξ, Y1 a(ei )a∗ (ei )Y2 ηiq q n hξ, Y1 (1 + qa∗ (ei )a(ei ))Y2 ηiq q n hξ, Y1 Y2 ηiq q n hξ, Y ηiq ,

and hence limm→∞ hξ, φm (X)ηiq = limm→∞

1 m

Pm i=m0 +1

q n hξ, 9(er(1) ⊗ . . . ⊗ er(n) )ηiq

= hξ, q n Xηiq . Thus we have shown w-lim φm (X) = φ(X). m→∞

Let now tr be a normalized normal trace on 0q (H). Then tr[φ(X)] = limm→∞ tr[φm (X)] Pm 1 = limm→∞ m tr[ω(ei )Xω(ei )] Pi=1 m 1 = limm→∞ m i=1 tr[Xω(ei )ω(ei )] Pm 1 = tr[X · limm→∞ m i=1 ω(ei )ω(ei )] = tr[Xφ(1)] = tr[X]. Since φk (X) = q kn X converges, for k → ∞, (even in norm) to 0, n≥1 E[X] · 1 = , X = 1, n = 0 we obtain tr[X] = lim tr[φk (X)] = tr[ lim φk (X)] = E[X] tr[1] = E[X]. k→∞

k→∞

Thus tr coincides on all operators of the form X = 9(er(1) ⊗ . . . ⊗ er(n) )

(n ∈ N0 , r(1), . . . , r(n) ∈ N)

with our canonical trace E. Since the set of finite linear combinations of such operators X is weakly dense in 0q (H), we get the uniqueness of a normalized normal trace on 0q (H), which implies that 0q (H) is a factor.

140


The second part of our q-Gaussian functor 0q assigns to each contraction T : H → H0 a map 0q (T ) : 0q (H) → 0q (H0 ). The idea is to extend 0q (T )ω(f ) = ω(T f ) in a canonical way to all of 0q (H). In general, the q-relations prohibit the extension as a homomorphism, i.e. 0q (T )ω(f1 ) . . . ω(fn ) 6= ω(T f1 ) . . . ω(T fn )

in general.

But what can be done is to demand the above relation for the normal ordered form, i.e. 0q (T ) 9(f1 ⊗ . . . ⊗ fn ) = 9(T f1 ⊗ . . . ⊗ T fn ) = 9(Fq (T )f1 ⊗ . . . ⊗ fn ), or ( 0q (T )X) = Fq (T )(X ). Thus our second quantization 0q (T ) is the restriction of Fq (T ) from Fq (H) = L2q (H) to 0q (H) ∼ = L∞ q (H) and the question on the existence of 0q (T ) amounts to the problem ∞ 0 whether Fq (T )(L∞ q (H)) ⊂ Lq (H ). We know that Fq (T ) can be defined for T a contraction and we will see in the next theorem that no extra condition is needed to ensure its nice behaviour with respect to L∞ q . The case q = 0 is due to Voiculescu [Voi, VDN]. Theorem 2.11. 1) Let T : H → H0 be a contraction between real Hilbert spaces. There exists a unique map 0q (T ) : 0q (H) → 0q (H0 ) such that ( 0q (T )X) = Fq (T )(X ). The map 0q (T ) is linear, bounded, completely positive, unital and preserves the canonical trace E. 2) If T is isometric, then 0q (T ) is a faithful homomorphism, and if T is the orthogonal projection onto a subspace, then 0q (T ) is a conditional expectation. Proof. Uniqueness of 0q (T ) follows from the fact that is separating for 0q (H0 ). To prove the existence and the properties of 0q (T ) we notice that any contraction T can be factored [Hal] as T = P OI where – I : H → K = H ⊕ H0 is an isometric embedding – O : K → K is orthogonal – P : K = H ⊕ H0 → H0 is an orthogonal projection onto a subspace. Thus if we prove our assertions for each of these three cases then we will also get the general statement for 0q (T ) = 0q (P ) 0q (O) 0q (I). a) Let I : H → K = H ⊕ H0 be an isometric embedding and Q : K → K the orthogonal projection onto H. Then Fq (Q) is a projection in Fq (KC ) and Fq (HC ) can be identified with Fq (Q)Fq (KC ). Let us denote by ωK (f ) the sum of the creation and annihilation operator on Fq (KC ). If we put 0K q (H) := vN(ωK (f ) | f ∈ H) ⊂ B(Fq (KC )), then

0K q (H)Fq (HC ) ⊂ Fq (HC ),

and we have the canonical identification 0q (H) ∼ = 0K q (H)Fq (Q), which gives a homomorphism (and thus a completely positive)


141

0q (I) : 0q (H) → 0q (K). Faithfulness is clear since Fq (Q) = and separating. This yields also that the trace is preserved. b) Let P : K = H ⊕ H0 → H0 be an orthogonal projection, i.e. P P ∗ = 1H0 , where P ∗ : H0 → K is the canonical inclusion. Then 0q (P )X := Fq (P )XFq (P ∗ )

(X ∈ 0q (K))

gives the right operator, because we have for k, l ∈ N0 and f1 , . . . , fk , g1 , . . . , gl ∈ K, Fq (P )a∗ (f1 ) . . . a∗ (fk )a(g1 ) . . . a(gl )Fq (P ∗ ) = = a∗ (P f1 ) . . . a∗ (P fk )Fq (P )Fq (P ∗ )a(P g1 ) . . . a(P gl ) = a∗ (P f1 ) . . . a∗ (P fk )a(P g1 ) . . . a(P gl ). By its concrete form, 0q (P ) is a conditional expectation and E[Fq (P )XFq (P ∗ )] = hFq (P ∗ ) , XFq (P ∗ ) iq = h , X iq = E[X] shows that it preserves the trace. c) Let O : K → K be orthogonal, i.e. OO∗ = O∗ O = 1K . Then, as in b), 0q (O)X = Fq (O)XFq (O∗ ), which is, by

Fq (O∗ )Fq (O) = Fq (1K ) = 1Fq (KC )

also a faithful homomorphism.

Instead of working on the level of von Neumann algebras we could also consider the C ∗ -analogues of the above constructions. This would be quite similar. We just indicate the main points. Definition 2.12. Let H be a real Hilbert space and HC its complexification HC = H ⊕ iH. Put, for f ∈ H, ω(f ) := a(f ) + a∗ (f ) ∈ B(Fq (HC )), and denote by 8q (H) ⊂ B(Fq (HC )) the C ∗ -algebra generated by all ω(f ), 8q (H) := C ∗ (a(f ) + a∗ (f ) | f ∈ H). Clearly, the vacuum is also a separating trace-vector for 8q (H) and, by Remark 2.6., it is also cyclic and 9(f1 ⊗ . . . ⊗ fn ) ∈ 8q (H) for all n ∈ N0 and all f1 , . . . , fn ∈ H. The most important fact for our later considerations is that 0q (T ) can also be restricted to the C ∗ -level. Theorem 2.13. 1) Let T : H → H0 be a contraction between real Hilbert spaces. There exists a unique map 8q (T ) : 8q (H) → 8q (H0 ) such that ( 8q (T )X) = Fq (T )(X ). The map 8q (T ) is linear, bounded, completely positive, unital and preserves the canonical trace E. 2) If T is isometric, then 8q (T ) is a faithful homomorphism, and if T is the orthogonal projection onto a subspace, then 8q (T ) is a conditional expectation. 3) We have 8q (T ) = 0q (T )/ 8q (H).

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Proof. This is analogous to the proof of Theorem 2.11.

We can now also prove the analogue of the second part of Theorem 2.10. The analogue of factoriality for C ∗ -algebras is simplicity. Theorem 2.14. If −1 < q < 1 and dim H = ∞ then 8q (H) is simple. Proof. Again, this is similar to the proof of the von Neumann algebra result. We just indicate the main steps. We use the notations from the proof of Theorem 2.10. First, by norm estimates, one can show that the convergence limm→∞ φm (X) = φ(X) for X of the form X := 9(er(1) ⊗ . . . ⊗ er(n) ) is even a convergence in norm. Since φ(X) is nothing but φ(X) = 0q (q)X, where q is regarded as a multiplication operator on H, we have, by 2.13, the bound kφ(X)kq ≤ kXkq . This together with the m-independent bound kφm (X)kq ≤ kXkq kω(e1 )k2q implies that lim φm (X) = 0q (q)X

m→∞

uniformly for all X ∈ 8q (H).

Now assume we have a non-trivial ideal I in 8q (H) and consider a positive nonvanishing X ∈ I. Then φm (X) ∈ I for all m ∈ N and thus 0q (q)X ∈ I. Iterating shows 0q (q n )X ∈ I for all n ∈ N and because of the uniform convergence limn→∞ 0q (q n )X = E[X]1, we obtain E[X]1 ∈ I. The faithfulness of E implies then I = 8q (H). Remark 2.15. One might be tempted to conjecture that, for fixed H, the von Neumann algebras 0q (H) or the C ∗ -algebras 8q (H) are for all q ∈ (−1, 1) isomorphic. At the moment, no results in this direction are known. One should note that there exist partial answers [JSW1, JSW2, JW, DN] to the analogous question for the C ∗ -algebra generated by a(f ), a∗ (f ) (not the sum) showing that at least for small values of q and n := dim HC < ∞ these algebras are isomorphic to the (q = 0)-algebra, which is an extension of the Cuntz algebra On by compact operators [Cun, Eva]. However, the methods used there do not extend to the case of 0q (H) or 8q (H). 3. Non-commutative Aspects of q-Gaussian Processes Before we define the notion of a q-Gaussian process, we want to present our general frame on non-commutative processes. By T we will denote the range of our time parameter t, typically T will be some interval in R. Definition 3.1. 1) Let A be a finite von Neumann algebra and ϕ : A → C a faithful normal trace on A. Then we call the pair (A, ϕ) a (tracial) probability space. 2) A random variable on (A, ϕ) is a self-adjoint operator X ∈ A. 3) A stochastic process on (A, ϕ) is a family (Xt )t∈T of random variables Xt ∈ A (t ∈ T ). 4) The distribution of a random variable X on (A, ϕ) is the probability measure ν on the spectrum of X determined by Z n for all n ∈ N0 . ϕ(X ) = xn dν(x)


143

We should point out that there are also a lot of quantum probabilistic investigations in the context of more general, non-tracial situations, see e.g. [AFL, Kum1]. Of course, life becomes much harder there. We will only consider centered Gaussian processes, thus a q-Gaussian process will be totally determined by its covariance. Since we would like to have realizations of our processes on separable Hilbert spaces, our admissible covariances are not just positive definite functions, but they should admit a separable representation. Definition 3.2. A function c : T × T → R is called a covariance function, if there exists a separable real Hilbert space H and vectors ft ∈ H for all t ∈ T such that c(s, t) = hfs , ft i

(s, t ∈ H).

Definition 3.3. Let c : T × T → R be a covariance function corresponding to a real Hilbert space H and vectors ft ∈ H (t ∈ T ). Then we put for all t ∈ T , Xt := ω(ft ) ∈ 0q (H) and call the process (Xt )t∈T on ( 0q (H), E) the q-Gaussian process with covariance c. Remark 3.4. 1) Of course, the q-Gaussian process depends, up to isomorphism, only on c and not on the special choice of H and (ft )t∈T . 2) We can characterize our q-Gaussian process also by the q-relations in the form Xt = at + a∗t

and

E[ · ] = h , · i,

where for all s, t ∈ T , as a∗t − qa∗t as = c(s, t) · 1

and

at = 0.

In this form our q-Gaussian processes were introduced by Frisch and Bourret [FB]. We can now define q-analogues of all classical Gaussian processes, just by choosing the appropriate covariance. In the following we consider three prominent examples. Definition 3.5. 1) The q-Gaussian process (XtqBM )t∈[0,∞) with covariance c(s, t) = min(s, t)

(0 ≤ s, t < ∞)

is called q-Brownian motion. 2) The q-Gaussian process (XtqBB )t∈[0,1] with covariance c(s, t) = s(1 − t)

(0 ≤ s ≤ t ≤ 1)

is called q-Brownian bridge. 3) The q-Gaussian process (XtqOU )t∈R with covariance c(s, t) = e−|t−s|

(s, t ∈ R)

is called q-Ornstein-Uhlenbeck process. Remark 3.6. 1) That the three examples for c are indeed covariance functions is clear by the existence of the respective classical processes, for a direct proof see, e.g., [Sim2]. 2) The Ornstein-Uhlenbeck process is often also called an oscillator process, see [Sim2].

144


Let (A, ϕ) be a tracial probability space and let B be a von Neumann subalgebra of A. Then we have (see, e.g., [Tak]) a unique conditional expectation (“partial trace”) from A onto B which preserves the trace ϕ – which we will denote in a probabilistic language by ϕ[ · |B]. Thus in the frame of tracial probability spaces we always have the following canonical generalization of the classical Markov property (which says that the future depends on the past only through the present). Definition 3.7. Let (A, ϕ) be a probability space and (Xt )t∈T a stochastic process on (A, ϕ). Denote by At] := vN(Xu | u ≤ t) ⊂ A, ⊂ A, A[t] := vN(Xt ) A[t := vN(Xu | u ≥ t) ⊂ A. We say that (Xt )t∈T is a Markov process if we have for all s, t ∈ T with s ≤ t the property for all X ∈ A[t] . ϕ[X|As] ] ∈ A[s] Note that another canonical definition for the Markov property would be the requirement for all X ∈ A[s . ϕ[X|As] ] ∈ A[s] In the classical case this latter condition is equivalent to the one we use in Definition 3.7, but in the non-commutative case there is in general a difference. We have chosen the weaker condition, since this is sufficient to ensure the existence of transition operators (see Definition 4.3 and Theorem 4.4). Now, the conditional expectations E[ · |As] ] in the case of q-Gaussian processes are quite easy to handle because they are nothing but the second quantization of projections in the underlying Hilbert space. Namely, consider a q-Gaussian process (Xt )t∈T corresponding to the real Hilbert space H and vectors ft (t ∈ T ). Let us denote by Ht] := span(fu | u ≤ t) ⊂ H, ⊂ H, H[t] := Rft H[t := span(fu | u ≥ t) ⊂ H, the respective Hilbert space analogues of At] , A[t] , A[t . Then we have At] ∼ A[t] ∼ A[t ∼ = 0q (Ht] ), = 0q (H[t] ), = 0q (H[t ), and E[ · |At] ] = 0q (Pt] ) is the second quantization of the orthogonal projection Pt] : H → Ht] . Thus we can translate the Markov property for q-Gaussian processes into the following Hilbert space level statement. Proposition 3.8. Let (Xt )t∈T be a q-Gaussian process as above. It has the Markov property if and only if Ps] H[t] ⊂ H[s]

for all s, t ∈ T with s ≤ t.

Note that the stronger form of Markovianity, ϕ[A[s , As] ] ⊂ A[s] , corresponds for the q-Gaussian processes on the linear level to Ps] H[s ⊂ H[s] . But this is clearly equivalent to the condition of Proposition 3.8. Thus, for q-Gaussian processes the apriori possibly different definitions for “Markovianity” are all equivalent. As Proposition 3.8. shows, Markovianity is a property of the underlying Hilbert space and does not depend on q. Thus we get as in the classical case the following characterization in terms of the covariance.


145

Proposition 3.9. A q-Gaussian process with covariance c is Markovian if and only if we have for all triples s, u, t ∈ T with s ≤ u ≤ t that c(t, s)c(u, u) = c(t, u)c(u, s). Proof. See the proof of Theorem 3.9 in [Sim2].

Corollary 3.10. The q-Brownian motion (XtqBM )t∈[0,∞) , the q-Brownian bridge (XtqBB )t∈[0,1] , and the q-Ornstein-Uhlenbeck process (XtqOU )t∈R are all Markovian. Analogously, we have all statements of the classical Gaussian processes which depend only on Hilbert space properties. Let us just state the characterization of the Ornstein-Uhlenbeck process as the only stationary Gaussian Markov process with continuous covariance and the characterization of martingales among the Gaussian processes. Proposition 3.11. Let (Xt )t∈T be a q-Gaussian process which is stationary, Markovian qOU for suitable and whose covariance c(s, t) = c0 (t − s) is continuous. Then Xt = αXβt α, β > 0. Proof. See the proof of the analogous statement for classical Gaussian processes, Corollary 4.10 in [Sim2]. Definition 3.12. Let (Xt )t∈T be a stochastic process on a probability space (A, ϕ) and let the notations be as in Definition 3.7. Then we say that (Xt )t∈T is a martingale if ϕ[Xt |As] ] = Xs

for all s ≤ t.

Proposition 3.13. A q-Gaussian process is a martingale if and only if Ps] ft = fs for all s ≤ t – which is the case if and only if c(s, t) = c(s, s) for all s ≤ t. Proof. We have ω(fs ) = Xs = E[Xt |As] ] = 0q (Ps] )ω(ft ) = ω(Ps] ft ), implying Ps] ft = fs .

4. Classical Aspects of q-Gaussian Processes In this section we want to address the question whether our non-commutative stochastic processes can also be interpreted classically. Definition 4.1. Let (Xt )t∈T be a stochastic process on some non-commutative probability space (A, ϕ). We call a classical real-valued process (X˜ t )t∈T on some classical probability space ( , A, P ) a classical version of (Xt )t∈T if all time-ordered moments of (Xt )t∈T and (X˜ t )t∈T coincide, i.e. if we have for all n ∈ N, all t1 . . . , tn ∈ T with t1 ≤ . . . ≤ tn , and all bounded Borel functions h1 , . . . , hn on R the equality Z ϕ[h1 (Xt1 ) . . . hn (Xtn )] = h1 (X˜ t1 (ω)) . . . hn (X˜ tn (ω))dP (ω).

146


Remark 4.2. Most calculations in a non-commutative context involve only time-ordered moments, see, e.g., the calculation of the Green function of the q-Ornstein-Uhlenbeck process in [NSp]. Thus, results of such calculations can also be interpreted as results for the classical version – if such a version exists. It is clear that there is at most one classical version for a given non-commutative process (Xt )t∈T . The problem consists in showing the existence. If we denote by 1B the characteristic function of a measurable subset B of R, then we can construct the classical version (X˜ t )t∈T of (Xt )t∈T via Kolmogorov’s existence theorem from the collection of all µt1 ,...,tn (n ∈ N, t1 ≤ . . . ≤ tn ) – which are for B1 , . . . , Bn ⊂ R defined by µt1 ,...,tn (B1 × . . . × Bn ) = P (X˜ t1 ∈ B1 , . . . , X˜ tn ∈ Bn ) = ϕ[1B1 (Xt1 ) . . . 1Bn (Xtn )] – if and only if all µt1 ,...,tn are probability measures. Whereas this is of course the case for µt1 and, in our tracial frame because of µt1 ,t2 (B1 × B2 ) = ϕ[1B1 (Xt1 )1B2 (Xt2 )] = ϕ[1B1 (Xt1 )1B2 (Xt2 )1B1 (Xt1 )], also for µt1 ,t2 , there is no apriori reason why it should be true for bigger n. And in general it is not. It is essentially the content of Bell’s inequality that there are examples of non-commutative processes which possess no classical version – for a discussion of these subjects see, e.g., [KM]. But for special classes of non-commutative processes classical versions might exist. One prominent example of such a class are the Markov processes. Definition 4.3. Let (Xt )t∈T be a Markov process on a probability space (A, ϕ). Let, for t ∈ T , spect(Xt ) and νt be the spectrum and the distribution, respectively, of the self-adjoint operator Xt . Denote by L∞ (Xt ) := vN(Xt ) = L∞ (spect(Xt ), νt ). The operators

Ks,t : L∞ (Xt ) → L∞ (Xs )

(s ≤ t),

determined by ϕ[h(Xt )|As] ] = ϕ[h(Xt )|A[s] ] = (Ks,t h)(Xs ) are called transition operators of the process (Xt )t∈T , and, looked upon from the other side, the process (Xt )t∈T is called a dilation of the transition operators K = (Ks,t )s≤t . The following theorem is by now some kind of folklore in quantum probability, see, e.g. [AFL, Kum2, BP, Bia1]. We just indicate the proof for sake of completeness. Theorem 4.4. If (Xt )t∈T is a Markov process on some probability space (A, ϕ), then there exists a classical version (X˜ t )t∈T of (Xt )t∈T , which is a classical Markov process. Proof. One can express the time-ordered moments of a Markov process in terms of the transition operators via ϕ[h1 (Xt1 ) . . . hn (Xtn )] = = = = = =

ϕ[h1 (Xt1 ) . . . hn (Xtn )|Atn−1 ] ] ϕ[h1 (Xt1 ) . . . hn−1 (Xtn−1 )ϕ[hn (Xtn )|Atn−1 ] ]] ϕ[h1 (Xt1 ) . . . hn−1 (Xtn−1 )(Ktn−1 ,tn hn )(Xtn−1 )] ϕ[h1 (Xt1 ) . . . hn−2 (Xtn−2 )(hn−1 · Ktn−1 ,tn hn )(Xtn−1 )] ... ϕ[(h1 · Kt1 ,t2 (h2 · Kt2 ,t3 (h3 · . . .)))(Xt1 )],


147

from which it follows – because Ks,t preserves positivity – that the corresponding µt1 ,...,tn are probability measures. That the classical version is also a classical Markov process follows by the same formula. Corollary 4.5. There exist classical versions of all q-Gaussian Markov processes. In particular, we have classical versions of the q-Brownian motion, of the q-Brownian bridge, and of the q-Ornstein-Uhlenbeck process. Our aim now is to describe these classical versions more explicitly by calculating their transition probabilities in terms of the orthogonalizing measure νq and the kernel p(q) r (x, y) of Theorem 1.10. Theorem 4.6. Let (Xt )t∈T be a q-Gaussian Markov process with covariance c and put λt :=

p c(t, t)

and

λs,t := √

c(t, s) . c(s, s)c(t, t)

1) We have p p L∞ (Xt ) = L∞ ([−2λt / 1 − q, 2λt / 1 − q], νq (dx/λt )). (q) is given by 2) If λs,t = ±1, then the transition operator Ks,t (q) h)(x) = h(±xλt /λs ). (Ks,t (q) is given by If |λs,t | < 1, then the transition operator Ks,t Z (q) (q) h)(x) = h(y)ks,t (x, dy), (Ks,t (q) are Feller kernels which have the explicit form where the transition probabilities ks,t (q) (x, dy) = p(q) ks,t λs,t (x/λs , y/λt )νq (dy/λt ).

In particular, for q = 0 and |λs,t | < 1, we have the following transition probabilities for the free Gaussian Markov processes: p (1 − λ2s,t ) 4λ2t − y 2 dy 1 (0) . ks,t (x, dy) = 2πλ2t (1−λ2s,t )2 −λs,t (1+λ2s,t )(x/λs )(y/λt ) + λ2s,t ((x2 /λ2s ) + (y 2 /λ2t )) Recall that a kernel k(x, dy) is called Feller, if the map x 7→ k(x, dy) is weakly continuous and k(x, ·) → 0 weakly as x → ±∞ – or equivalently that the corresponding operator K sends C0 (R) to C0 (R), see, e.g., [DM]. Proof. 1) This was shown in [BSp2]; noticing the connection between q-relations and q-Hermite polynomials the assertion reduces essentially to part 1) of Theorem 1.10. 2) By Prop. 2.9, we know 9(f ⊗n ) = kf kn Hn(q) (ω(f )/kf k). Let our q-Gaussian process (Xt )t∈T now be of the form Xt = ω(ft ). Markovianity implies

148


Ps] ft = µfs , Because of

µ=

where

hft , fs i c(t, s) = . hfs , fs i c(s, s)

E[ 9(ft⊗n )|As] ] = 9((Ps] ft )⊗n ) = µn 9(fs⊗n )

we obtain with λt := kft k =

p c(t, t)

λs,t := µ

and

the formula E[Hn(q) (Xt /λt )|As] ] = =

λs c(t, s) =√ λt c(s, s)c(t, t)

1 E[ 9(ft⊗n )|As] ] λn t µn λn t

9(fs⊗n )

= (µ λλst )n Hn(q) (Xs /λs ) = λns,t Hn(q) (Xs /λs ), implying (q) Ks,t (Hn(q) (·/λt )) = λns,t Hn(q) (·/λs ).

Let us now consider the canonical extension of our transition operators from the L∞ spaces to the L2 -spaces, i.e. (q) : L2 (Xt ) → L2 (Xs ). Ks,t

√ If we use the fact that the rescaled q-Hermite polynomials (Hn(q) (·/λt )/ [n]!)n∈N0 2 constitute an orthonormal basis of L (Xt ), we get directly the assertion in the case (q) (q) λs,t = ±1. (For λs,t = −1 one also has to note that H2k and H2k+1 are even and odd polynomials, respectively.) (q) is a Hilbert-Schmidt operator, In the case |λs,t | < 1, our formula implies that Ks,t (q) thus it has a concrete representation by a kernel ks,t , which is given by (q) ks,t (x, dy) =

P∞

λn s,t (q) (q) n=0 [n]q ! Hn (x/λs )Hn (y/λt )νq (dy/λt )

= pλs,t (x/λs , y/λt )νq (dy/λt ). That our kernels are Feller follows from the fact that, by Theorem 2.13, our second quantization (i.e. our transition operators) restrict to the C ∗ -level (i.e. to continuous functions). (0) follows from the concrete form of p(0) The formula for ks,t r of Theorem 1.10 and the fact that 1 p 4 − y 2 dy for y ∈ [−2, 2]. ν0 (dy) = 2π The main formula of our proof, namely the action of the conditional expectation on the q-Hermite polynomials, says that we have some quite canonical martingales associated to q-Gaussian Markov processes – provided the factor λs,t decomposes into a quotient λs,t = λ(s)/λ(t). Since this can be assured by a corresponding factorization property of the covariance function – which is not very restrictive for Gaussian Markov processes, see Theorem 4.9 of [Sim2] – we get the following corollary.


149

Corollary 4.7. Let (Xt )t∈T be a q-Gaussian process whose covariance factorizes for suitable functions g and f as for s ≤ t.

c(s, t) = g(s)f (t)

Then, for all n ∈ N0 , the processes (Mn (t))t∈T with n/2

Mn (t) := g(t)/f (t)

Hn(q) (Xt /λt )

are martingales. Note that the assumption on the factorization of the covariance is in particular fulfilled for the q-Brownian motion, for the q-Ornstein-Uhlenbeck process, and for the q-Brownian bridge. Proof. Our assumption on the covariance implies s g(s)/f (s) , λs,t = g(t)/f (t) hence our formula for the action of the conditional expectation on the q-Hermite polynomials (in the proof of Theorem 4.6) can be written as g(t)/f (t)

n/2

n/2

E[Hn(q) (Xt /λt )|As] ] = g(s)/f (s)

Hn(q) (Xs /λs ),

which is exactly our assertion.

Remark 4.8. Consider the q-Brownian motion (XtqBM )t∈[0,∞) . Then the corollary states that √ Mn(q) (t) := tn/2 Hn(q) (XtqBM / t) is a martingale. In terms of quantum stochastic integrals these martingales would have the form Z Z dXtqBM . . . dXtqBM . 1 n

···

Mn(q) (t) =

0≤t1 ,...,tn ≤t ti 6=tj (i6=j)

Since at the moment, for general q, no rigorous theory of q-stochastic integration exists, this has to be taken as a purely formal statement. For q = 0, however, such a rigorous theory was developed in [KSp], and the above representation by stochastic integrals was established by Biane [Bia2]. In this case, he could put this representation into the form of the stochastic differential equation Mn(0) (t)

=

n−1 XZ t k=0

0

(0) Mk(0) (s)dXs0BM Mn−k−1 (s),

which should be compared with the classical formula Z Mn(1) (t)

=n 0

t

(1) Mn−1 (s)dXs1BM .

150


(0) Example 4.9 (Free Gaussian processes). We will now specialize the formula for ks,t to the case of the free Brownian motion, the free Ornstein-Uhlenbeck process and the free Brownian bridge. The transition probabilities for the two former cases were also derived by Biane [Bia1] in the context of processes with free increments. 1) Free Brownian motion: We have c(s, t) = min(s, t), thus p √ and λs,t = s/t. λt = t

This yields (t − s) ks,t (x, dy) = (t − s)2 − (t + s)xy + x2 t + y 2 s for

√ √ x ∈ [−2 s, 2 s]

and

p 4t − y 2 dy 2π

√ √ y ∈ [−2 t, 2 t].

2) Free Ornstein-Uhlenbeck process: We have c(s, t) = e−|t−s| , thus λt = 1

λs,t = e−|t−s| .

and

Since this process is stationary, it suffices to consider the transition probabilities for s = 0: p (e2t − 1) 4 − y 2 dy for x, y ∈ [−2, 2]. k0,t (x, dy) = 2π 4 sinh2 t − 2xy cosh t + x2 + y 2 Let us also calculate the generator N of this process – which is characterized by Ks,t = e−(t−s)N . It has the property N Hn(0) = nHn(0)

(n ∈ N0 ),

and differentiating the above kernel shows that it should be given formally by a kernel −2/(y − x)2 with respect to ν0 . Making this more rigorous [vWa] yields that N has on functions which are differentiable the form Z f (y) − f (x) − f 0 (x)(y − x) 0 ν0 (dy). (N h)(x) = xf (x) − 2 (y − x)2 3) Free Brownian bridge: We have c(s, t) = s(1 − t) for s ≤ t, thus s p s(1 − t) . and λs,t = λt = t(1 − t) t(1 − s) This yields ks,t (x, dy) = = for x ∈ [−2

p

√

(t−s) 1−s 1−t (t−s)2 −(s+t−2st)xy+t(1−t)x2 +s(1−s)y 2

s(1 − s), 2

p s(1 − s)]

and

4t(1−t)−y 2 dy , 2π

y ∈ [−2

p p t(1 − t), 2 t(1 − t)].


151

Example 4.10 (Fermionic Gaussian processes). For illustration, we also want to consider the fermionic (q = −1) analogue of Gaussian processes. Although this case has not been included in our frame everything works similarly, the only difference is that in the Fock space we get a kernel of our scalar product consisting of anti-symmetric tensors. This is responsible for the fact that the corresponding (−1)-Hermite polynomials collapse just to H0(−1) (x) = 1

and

H1(−1) (x) = x.

The corresponding measure ν−1 is not absolutely continuous with respect to the Lebesgue measure anymore, but collapses to ν−1 (dx) =

1 (δ−1 (dx) + δ+1 (dx)). 2

This yields (x, y) = H0(−1) (x)H0(−1) (y) + rH1(−1) (x)H1(−1) (y) = 1 + rxy, p(−1) r giving as transition probabilities (−1) ks,t (x, dy) =

c(s, t) 1 (1 + xy)(δ−√c(t,t) (dy) + δ+√c(t,t) (dy)). 2 c(s, s)c(t, t)

√ √ 1) Fermionic Brownian motion: Xt can only assume the values + t and − t and the transition probabilities are given by the table √ √ t − pt ks,t p √ 1 1 s 2 (1 + ps/t) 2 (1 − p s/t) . √ − s 21 (1 − s/t) 21 (1 + s/t) This case coincides with the corresponding c = −1 case of the Azéma martingale, see [Par1]. 2) Fermionic Ornstein-Uhlenbeck process: This stationary process lives on the two values +1 and −1 with the following transition probabilities ks,t 1 −1

1 1 −(t−s) ) 2 (1 + e 1 −(t−s) (1 − e ) 2

−1

1 −(t−s) ). 2 (1 − e 1 −(t−s) (1 + e ) 2

This classical two state Markov realization of the corresponding fermionic relations has been known for a long time, see [FB]. √ 3)√Fermionic Brownian bridge: Xt can only assume the values + t(1 − t) and − t(1 − t) and the transition probabilities are given by the table √ √ t(1 − t) − t(1 ks,t q q − t) √ s(1−t) s(1−t) 1 1 s(1 − s) 2 (1 + t(1−s) ) 2 (1 − t(1−s) ) . q q √ s(1−t) 1 − s(1 − s) 21 (1 − s(1−t) t(1−s) ) 2 (1 + t(1−s) )

152


Example 4.11 (Hypercontractivity). Consider the q-Ornstein-Uhlenbeck process with qOU . Note that this q-Ornstein-Uhlenbeck semistationary transition operators Kt(q) := Ks,s+t group is nothing but the second quantization of the simplest contraction, namely with the one-dimensional real Hilbert space H = R and the corresponding identity operator 1 : R → R we have p p (q) and 0q (e−t 1) ∼ 0q (R) ∼ = L∞ ([−2/ 1 − q, 2/ 1 − q], νq (dx)) = Kt . We have seen that the Kt(q) are, for all t > 0, contractions on L2 and on L∞ (and thus, by duality and interpolation, on all Lp ). In the classical case q = 1 (and also for q = −1) it is known [Sim1, Nel1, Nel2, Gro, CL] that much more is true, namely the OrnsteinUhlenbeck semigroup is also hypercontractive, i.e. it is bounded as a map from L2 to L4 for sufficiently large t. Having the concrete form of the kernel kt(q) (x, dy) = p(q) e−t (x, y)νq (dy) of Kt(q) , it is easy to check that we also have hypercontractivity for all −1 < q < 1. Even more, we can show that Kt(q) is bounded from L2 to L∞ for t > 0, i.e. we have what is called “ultracontractivity” [Dav] – which is, of course, not given for q = ±1. This ultracontractivity follows from the estimate kKt(q) hk∞ ≤ α(t, q)1/2 khk2

where

α(t, q) :=

sup

sup p(q) e−t (x, y),

x∈[−2,2] y∈[−2,2]

and from the explicit form of p(q) r from Theorem 1.10, which ensures that α(t, q) is finite for t > 0 and −1 < q < 1 (comp. also [Dav], Lemma 2.1.2). One may also note that for small t the leading term of α(t, q)1/2 is of order t−3/2 . 4.12 Open Problems. 1) The situation concerning classical versions of non-Markovian q-Gaussian processes is not clear at the moment. 2) Consider a symmetric measure µ on R with compact support. Then there exist a sequence of polynomials (Pn )n∈N0 such that Pn is of degree n and such that these polynomials are orthogonal with respect to µ. Let us define a semigroup Ut on L2 (µ) by Ut Pn = e−nt Pn . If these Ut are positivity preserving then they constitute the transition operators of a stationary Markov process, whose stationary distribution is given by µ. Our q-OrnsteinUhlenbeck process is an example of this general construction for the measure νq . The existence of the functor 0q “explains” the fact that the q-Ornstein-Uhlenbeck semigroup is positivity preserving from a more general (non-commutative) point of view – note that although Theorem 2.11 is for dim H = dim H0 = 1 a purely commutative statement, its proof is even in this case definitely non-commutative. Of course, not for all measures µ the semigroup Ut is positivity preserving. But one might wonder whether it is possible to find for each measure with this property –or at least for some special class of such measures – some analogous kind of functor. See also [BSp4] for related investigations. Acknowledgement. We thank Philippe Biane for stimulating discussions and remarks. We also thank the referee for pointing out some misprints and for suggesting the remark following Definition 3.7.


153

References [AFL] [BP] [Bia1] [Bia2] [Bia3] [BSp1] [BSp2] [BSp3] [BSp4] [Bre] [CL] [Cun] [Dav] [DM] [DN] [Eva] [Fiv] [FB] [GR] [Gre] [Gro] [Hal] [HP] [ISV] [JSW1] [JSW2] [JW] [Kum1] [Kum2]

[KM] [KSp] [LM1] [LM2]

Accardi, L., Frigerio, A., Lewis, J.T.: Quantum stochastic processes. Publ. RIMS 18, 97–133 (1982) Bhat, B.V.R., Parthasarathy, K.R.: Markov dilations of nonconservative dynamical semigroups and a quantum boundary theory. Ann. Inst. Henri Poincaré 31, 601–651 (1995) Biane, Ph.: On processes with free increments. Mfath. Z. To appear Biane, Ph.: Free Brownian motion, free stochastic calculus and random matrices. In: Free probability theory (Fields Institute Communications 12), ed. D. Voiculescu, Providence: AMS, 1997, pp. 1–19 Biane, Ph.: Quantum Markov processes and group representations. Preprint, 1995 Bo˙zejko, M., Speicher, R.: An example of a generalized Brownian motion. Commun. Math. Phys. 137, 519–531 (1991) Bo˙zejko, M., Speicher, R.: An example of a generalized Brownian motion II. Quantum Probability and Related Topics VII, ed. L. Accardi. Singapore: World Scientific, 1992, pp. 219–236 Bo˙zejko, M., Speicher, R.: Completely positive maps on Coxeter groups, deformed commutation relations, and operator spaces. Math. Ann. 300, 97–120 (1994) Bo˙zejko, M., Speicher, R.: Interpolations between bosonic and fermionic relations given by generalized Brownian motions. Math. Z. 222, 135–160 (1996) Bressoud, D.M.: A simple proof of Mehler’s formula for q-Hermite polynomials. Indiana Univ. Math. J. 29, 577–580 (1980) Carlen, E.A., Lieb, E.H.: Optimal hypercontractivity for Fermi fields and related non-commutative integration inequalities. Commun. Math. Phys. 155, 27–46 (1993) Cuntz, J.: Simple C ∗ -algebras generated by isometries. Commun. Math. Phys 57, 173–185 (1977) Davies, E.B.: Heat kernels and spectral theory, Cambridge: Cambridge University Press, 1989 Dellacherie, D., Meyer, P.A.: Probabilités et potentiel, Paris: Hermann, 1975 Dykema, K., Nica, A.: On the Fock representation of the q-commutation relations. J. reine angew. Math. 440, 201–212 (1993) Evans, D.E.: On On . Publ. RIMS 16, 915–927 (1980) Fivel, D.I.: Interpolation between Fermi and Bose statistics using generalized commutators. Phys. Rev. Lett. 65, 3361–3364 (1990), Erratum 69, 2020 (1992) Frisch, U., Bourret, R.: Parastochastics. J. Math. Phys. 11, 364–390 (1970) Gasper, G., Rahman, M.: Basic hypergeometric functions, Cambridge: Cambridge U.P., 1990 Greenberg, O.W.: Particles with small violations of Fermi or Bose statistics. Phys. Rev. D 43, 4111– 4120 (1991) Gross, L.: Existence and uniqueness of physical ground states. J. Funct. Anal. 10, 52–109 (1972) Halmos, P.R.: Normal dilations and extensions of operators. Summa Brasiliensis Math. 2, 125–134 (1950) Hudson, R.L., Parthasarathy, K.R.: Quantum Ito’s formula and stochastic evolution. Commun. Math. Phys. 93, 301–323 (1984) Ismail, E.M., Stanton, D., Viennot, G.: The combinatorics of q-Hermite polynomials and AskeyWilson integral. Europ. J. Comb. 8, 379–392 (1987) Jørgensen, P.E.T., Schmitt, L.M., Werner, R.F.: q-canonical commutation relations and stability of the Cuntz algebra. Pac. J. Math. 165, 131–151 (1994) Jørgensen, P.E.T., Schmitt, L.M., Werner, R.F.: Positive representations of general commutation relations allowing Wick ordering. J. Funct. Anal. 134, 3–99 (1995) Jørgensen, P.E.T., Werner, R.F.: Coherent states on the q-canonical commutation relations. Commun. Math. Phys. 164, 455–471 (1994) Kümmerer, B.: Markov dilations on W ∗ -algebras. J. Funct. Anal. 63, 139–177 (1985) Kümmerer, B.: Survey on a theory of non-commutative stationary Markov processes. Quantum Probability and Applications III, L. Accardi, W.v. Waldenfels, ed., Berlin: Springer-Verlag, 1988, pp. 228–244 Kümmerer, B., Maassen, H.: Elements of Quantum Probability. Quantum Probability Communications X. To appear Kümmerer, B., Speicher, R.: Stochastic integration on the Cuntz algebra O∞ . J. Funct. Anal. 103, 372–408 (1992) van Leeuwen, H., Maassen, H.: A q-deformation of the Gauss distribution. J. Math. Phys. 36, 4743– 4756 (1995) van Leeuwen, H., Maassen, H.: An obstruction for q-deformation of the convolution product. J. Phys. A 29, 1–8 (1996)

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Commun. Math. Phys. 185, 155–175 (1997)

Communications in


Field Theory on a Supersymmetric Lattice H. Grosse1,? , C. Klimˇc´ık2 , P. Preˇsnajder3 1

Institute for Theoretical Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria Theory Division CERN, CH-1211 Geneva 23, Switzerland 3 Department of Theoretical Physics, Comenius University, Mlynsk´ a dolina F1, SK-84215 Bratislava, Slovakia

2

Received: 1 September 1996 / Accepted: 23 September 1996

Abstract: A lattice-type regularization of the supersymmetric field theories on a supersphere is constructed by approximating the ring of scalar superfields by an integer-valued sequence of finite dimensional rings of supermatrices and by using the differencial calculus of non-commutative geometry. The regulated theory involves only finite number of degrees of freedom and is manifestly supersymmetric. 1. Introduction The idea that a fine structure of space-time should be influenced by quantum gravity phenomena is certainly not original but so far there was a little success in giving it more quantitative expression. String theory constitutes itself probably the most promising avenue to a consistent theory of quantum gravity. It is therefore of obvious interest to study the structure of spacetime from that point of view. Though string theory incorporates a minimal length the physical quantities computed in its framework reflect the symmetry properties of continuous space-time. The situation is somewhat analogous to ordinary quantum mechanics: though the phase space acquires itself a cell-like structure its symmetries remain intact, in general. In a sense the space-time possesses the cell-like structure also in string theory, e.g. the quantum WZNW model for a compact group has as effective target, perceived by a string center of mass, a truncated group manifold or, in other words, a “manifold” with a cell-like structure (see [1]). Indeed, the zero-modes’ subspace of the full Hilbert space contains only the irreducible representations of a spin lower than the level k. Because this subspace describe the scalar excitations, it is clear that high frequency (or spin) modes in an effective field theory are absent. In this way string theory leads to the UV finite behaviour of physical amplitudes as was probably realized by several researchers in the past (e.g.[2]). ? Part of Project No. P8916-PHY of the “Fonds zur F¨ orderung der wissenschaftlichen Forschung in ¨ Ostereich”

156

H. Grosse, C. Klimˇc´ık, P. Preˇsnajder

In our contribution we would like to initiate an investigation of similar regularization in pure a field theory context. That is, we wish to consider fields living on truncated compact manifolds, endow them with dynamics and establish rules of their quantization. Among advantages of such a development, there would be not only the manifest preservation of all symmetries of a theory but also an expected compatibility with quantum gravity and string phenomena. In some sense we shall construct a lattice-type of regularization but the “lattice” will not approximate the underlying spacetime (and hence the ring of functions on it) but directly the ring. As the starting point of our treatment we choose a 2d field theory on a truncated two-sphere1 . The truncated sphere was extensively studied in past two decades for various reasons. Apparently, the structure was introduced by Berezin in 1975 [3] who quantized the (symplectic) volume two-form on the ordinary two-sphere. He ended up with a series of possible quantizations parametrized by the size of quantum cells. In 1982, Hoppe [4] investigated properties of spherical membranes. As a technical tool he introduced the truncation of high frequency excitations which effectively lead to the quantum sphere. In 1991 the concept was reinvented by Madore [5] (see also [6]). His motivation originated in the so-called non-commutative geometry, i.e. the generalization of the ordinary differential geometry to non-commutative rings of “functions”. The truncated algebra of ordinary functions is just the example of such a non-commutative ring. For our purposes, we shall use the results of all those previous works, however, we shall often put emphasis on different aspects of formalism as compared to the previous investigations. Our main concern will consist in developing basic differential and integral structures for a non-commutative sphere which are needed to define a classical (and quantum) field dynamics. We shall require that the symmetries of the undeformed theory are preserved in the non-commutative deformation; such as space-time supersymmetry, global isospin, local (non)abelian gauge or chiral symmetry2 ; and, obviously, that the commutative limit should recover the standard formulation of the dynamics of the field theory. In many respects a canonical procedure for endowing non-commutative rings with differential and integral calculus is known for several years from basic studies of A. Connes [7]. From his work it follows that geometrical properties of a non-commutative manifold are encoded in a fundamental triplet (A, H, D) where A is the representation of a non-commutative algebra A of “functions” on the manifold in some Hilbert space. Elements of A are linear operators acting on H in such a way that the multiplication of elements of the “abstract” algebra A is represented by the composition of the operators from A which represent them. D is a self-adjoint operator (called the Dirac operator) odd with respect to an appropriate grading,3 H is interpreted as a spinor bundle over the non-commutative manifold and the action of the algebra A on it makes it possible to define the action of a (truncated) gauge group on spinors. Noncommutative geometry has been already applied in theoretical physics by providing the nice geometrical description of the standard model action including the Higgs fields [7, 8]. The latter were interpreted as the components of a noncommutative gauge connection. Starting in this paper, we hope to provide another relevant application of non-commutative geometry with the aim to understand the short distance behaviour of 1

Also referred to as “fuzzy”, “non-commutative” or “quantum” sphere in literature [5, 4, 3]. An attempt to formulate a field theory on the fuzzy sphere was published in [5, 9, 10]. However, the crucial concept of chirality was not studied there. 3 We ignore in this paper aspects concerning the norms of the operators from A and commutators of the form [D, A] because all algebras we consider are finite-dimensional. 2

Field Theory on a Supersymmetric Lattice

157

field theory. We believe that non-commutative geometry can provide powerful technical tools for performing new and nontrivial relevant calculations. In the present contribution, we construct the fundamental triplet (A, H, D) and use the construction for developing the supersymmetric regularization of field theories. Though the uniqueness of (A, H, D) for a given fundamental algebra A is by no means guaranteed we give a highly natural choice stemming from the following construction. First we give a suitable description of spinors on the ordinary sphere as components of a scalar superfield on a supersphere. Then we represent the standard Dirac operator on the sphere in terms of the superdifferential generators of the OSp(2, 1) algebra which is the supersymmetry superalgebra of the supersphere. The standard Dirac operator on the sphere turns out to be nothing but the fermionic part of the Casimir of OSp(2, 1) written in the superdifferential representation (the bosonic part is the standard Laplace operator on the sphere). Then we shall mimick the same construction for the non-commutative sphere. We describe spinors on the non-commutative sphere as the suitable components of a scalar superfield on a non-commutative supersphere. In other words, we perform the supergeometric Berezin-like quantization of the supersphere4 but in the language of Madore. The resulting quantized ring of scalar superfields will reveal a cell-like structure of the non-commutative supersphere. The algebra A will be the enveloping algebra of OSp(2, 1) in its irreducible representation with a spin j/2. As j → ∞ one recovers the standard ring of superscalar functions on the supersphere. The quantized ring constitutes itself the representation space of the adjoint action of OSp(2, 2) in the irreducible representation with the OSp(2, 1) superspin j/2. We postulate that the fermionic part of the OSp(2, 1) Casimir in this adjoint representation is the Dirac operator on the noncommutative sphere. We shall find that it is selfadjoint and odd. We shall compute its complete spectrum of eigenvalues and eigenfunctions and find a striking similarity with the commutative case. Namely, the non-commutative Dirac operator turns out simply to be a truncated commutative one!5 We then construct both Weyl (chiral) and Majorana fermions. The building of the supersymmetric theories requires even more structure. We shall demonstrate that enlarging the superalgebra OSp(2, 1) to OSp(2, 2) the additional odd generators can be identified with the supersymmetric covariant derivatives and the additional even generator with the grading of the Dirac operator. All encountered representations of OSp(2, 1) will turn out to be also the representations of OSp(2, 2). In the following section (which does not contain original results) we repeat the known construction of the standard non-commutative sphere in a language suitable for SUSY generalization. In Sect. 3 we give the full account of the spectrum of the standard Dirac operator on the commutative sphere. Though not the results themselves, but the (algebraic) method of their derivation is probably new and very suitable for the later non-commutative analysis. From the fourth section we present original results. We start with the description of the (untruncated) Dirac operator in terms of the fermionic part of the OSp(2, 1) Casimir acting on the ring of superfields on the supersphere and we quantize that ring. Then we identify the Dirac operator on the non-commutative sphere, give a full account of its spectrum and describe the grading of the non-commutative spinor bundle, completing thus the construction of the fundamental triplet (A, H, D). In Sect. 5 we apply the developed constructions in (supersymmetric) field theories. We shall construct (super)symmetric action functionals of the deformed theories containing only 4 Recently several papers have appeared dealing with supergeometric quantization of the Poincar´ e disc [11, 12, 13]. 5 This suggests, in turn, that in the regulated field theory one should avoid the problem of fermion doubling [14].


158

finite number of degrees of freedom. We finish with conclusions and outlook concerning the construction of a noncommutative de Rham complex, a non-commutative gauge connection, chiral symmetry, dynamics of gauge fields and construction of twisted bundles over the non-commutative sphere needed for the description of “truncated” monopoles. 2. The Non-commutative Sphere

2.1. The commutative warm-up. A very convenient manifestly SU (2) invariant description of the (L2 -normed) algebra of functions A∞ on the ordinary sphere can be obtained by factorizing the algebra B of analytic functions of three real variables by its ideal I, P 2 consisting of all functions of a form h(xi )( xi − ρ2 ). The scalar product on A∞ is given by6 Z 1 2 (f, g)∞ ≡ d3 xi δ(xi − ρ2 )f ∗ (xi )g(xi ), f, g ∈ A∞ (1) 2πρ R3 Here f (xi ), g(xi ) ∈ B are some representatives of f and g. The algebra A∞ is obviously generated by functions7 xi , i = 1, 2, 3 which commute with each other under the usual pointwise multiplication. Their norms are given by ||xi ||2∞ =

ρ2 . 3

(2)

Consider the vector fields in R3 generating SU (2) rotations of B. They are given by explicit formulae ∂ (3) Rj = −ijkl xk l ∂x and obey the SU (2) Lie algebra commutation relations [Ri , Rj ] = iijk Rk

(4)

The action of Ri on B leaves the ideal I invariant hence it induces an action of SU (2) on A∞ . The generators xi ∈ A∞ form a spin 1 irreducible representation of the SU (2) algebra under the action (hence they are linear combinations of the spherical functions with l = 1). They fulfill an obvious relation 2

xi = ρ2 .

(5)

Higher powers of xi can be rearranged into irreducible multiplets corresponding to higher spins. For instance, the multiplet of spin l is conveniently constructed subsequently applying the lowering operator R− ≡ R1 − iR2 on the highest weight vector x+ l . It is well-known (cf. any textbook on quantum mechanics) that the full decomposition of A∞ into the irreducible representations of SU (2) is given by the infinite direct sum A∞ = 0 + 1 + 2 + . . . , where the integers denote the spins of the representations. 6 7

The normalization ensures that the norm of the unit element of A∞ is 1. Speaking more precisely, xi denote the corresponding equivalence classes in B.

(6)


159

2.2. The truncation of A∞ . We define the family of non-commutative spheres Aj by furnishing the truncated sum of the irreducible representations Aj = 0 + 1 + . . . + j,

(7)

with an associative product and a scalar product which in the limit j → ∞ give the standard products in A∞ . To do this, consider the space L(j/2, j/2) of linear operators from the representation space of the irreducible representation with the spin j/2 into itself. Clearly, the SU (2) algebra acts on L(j/2, j/2) by the adjoint action. This “adjoint” representation is reducible and the standard Clebsch-Gordan series for SU (2) [15] gives its decomposition (8) L(j/2, j/2) = 0 + 1 + . . . + j ≡ Aj . The scalar product on Aj is defined by8 (f, g)j ≡

1 Tr(f ∗ g), j+1

f, g ∈ Aj ,

(9)

and the associative product is defined as the standard composition of operators from the space L(j/2, j/2). Now we make more precise the notion of the commutative limits of the scalar product and the associative product. There is a natural chain of the linear embeddings of the vector spaces A1 ,→ A2 ,→ . . . ,→ Aj ,→ . . . ,→ A∞

(10)

Any (normalized) element from Aj of the form p Xj+ cj,lp R−

l

(11)

is mapped in an (normalized) element from Ak given by l

p Xk+ . ck,lp R−

(12)

Here Xjα are representatives of the SU (2) generators in the irreducible representation α with spin j/2 (X∞ ≡ xα ). They are normalized so that [Xjm , Xjn ] = i q

ρ j j 2(2

mnp X p ,

(13)

+ 1)

and cj(k),lp are the (real) normalization coefficients given by the requirement that the embedding conserves the norm. Note that Xj+ l are the highest weight vectors in Aj . Because the adjoint action of the SU (2) algebra is hermitian for arbitrary Aj (as it can be easily seen from the definitions of the scalar products (1),(9)) the embeddings are in fact isometric. Indeed, the scalar product of the eigenvectors of the hermitian operator vanishes if the corresponding eigenvalues are different. Obviously different l’s give different eigenvalues of the (hermitian) adjoint Casimir. The commutative limit of the associative product is more involved, however9 . Clearly, the embeddings cannot be (and should not be) the homomorphisms of the associative products! For instance the product of two elements from Aj with the maximal spin j has again a maximal spin j 8

The normalization ensures that the norm of the identity matrix is 1. The nice establishment of the correct commutative limit of the product was given in [6] using the coherent states for SU (2). 9


160

because it is from Aj but could have a spin 2j component if the product is taken in a sufficiently larger algebra Ak . Consider more closely the behaviour of the product as the function of k. According to (10), two arbitrary elements f, g of Aj can be canonically considered as the elements of Ak for whatever k > j (including k = ∞). Their product in every Ak can also be embedded in A∞ . Denote the corresponding element of A∞ as (f g)k . We shall argue that (14) lim (f g)k = f g, k→∞

where f g is the standard commutative pointwise multiplication in A∞ . Before plunging into the proof of this statement we try to formulate its meaning more “physically”. It is not true that the algebra Aj tends to be commutative for large j (as the matrix algebra it, in fact, cannot.) What is the case is that for large j the elements with much lower spins than j almost commute. In the field theory language: the long distance limit corresponds to the standard commutative theory but for short distances the structure is truly non-commutative. This non-commutativeness, however, preserves the symmetry of the space-time. The algebra Aj is finite-dimensional with the dimension being (j + 1)2 . That means that the sphere is effectively divided in (j + 1)2 cells of an 4πρ2 average area (j+1) 2 . A theory based on the non-commutative ring Aj has, therefore, a 2ρ incorporated. minimal length j+1 Now it is easy to prove (14). Actually because of relation (13), which ensures the commutativity of the limit, it is enough to show that the normalization coefficients cj,lp defined in (11,12) have the property

lim ck,lp = c∞,lp .

k→∞

(15)

Due to the rotational invariance of the inner products in all Ak (k = 1, . . . , ∞), it is enough to demonstrate it just for the highest weight element Xk+ l . Then + + 2l c−2 k,l0 = (Xk , Xk )k = ρ l

l

(2l)!! (k + l + 1)! . (2l + 1)!! (k + 1)(k)l (k + 2)l (k − l)!

(16)

The last equality follows from a formula derived in [16] (p. 618, Eq. (36)). The relation (15) then obviously holds since the last fraction tends to 1 and it can be simply computed from (1) that 2l c−2 ∞,l0 = ρ

(2l)!! . (2l + 1)!!

(17)

Note that the generators Xki are themselves normalized as (Xki , Xki )k =

ρ2 3

(18)

and the standard relation defining the surface S2 holds in the non-commutative case 2

Xki = ρ2 .

(19)

We observe from (2) and (18) that for every j Xji ∈ Aj are embedded in A∞ as just the standard commutative generators xi and in Ak , k > j as Xki ∈ Ak . The notation is therefore justified and in what follows we shall often write just X i in the non-commutative case and xi in the commutative one.


161

3. The Dirac Operator on S2 and its Spectrum The construction of the spinor bundle10 over S2 is a standard part of any textbook of quantum field theory (e.g. see [17]) though, perhaps, it is not stressed explicitly. Also the spectrum of the Dirac operator acting on this bundle is known in that context, the eigenfunctions are nothing but the so-called spinorial harmonics [17]. We present the manifestly rotation invariant description of the spectrum in the spirit of the previous section. Consider the trivial spinor bundle SB over R3 . Its sections are ordinary quantum mechanical two-component spinorial wave-functions of the form Ψ+ (20) , Ψ+ , Ψ− ∈ B. Ψ− The action of the SU (2) algebra is described by the generators 1 Ji ≡ Ri + σi , 2

(21)

where σi are the standard Pauli matrices. Hence, SB is the representation space of some (reducible) representation of SU (2) . Now R3 can be viewed as the fibration of S2 by the half-lines in R3 starting in its centre. The position of a point on the fiber we measure by the radial coordinate r. The subbundle SA∞ of the sections of SB independent on the fiber coordinate r can be interpreted as the spinor bundle over the base manifold S2 of the fibration. Clearly, SA∞ is the SU (2) subrepresentation of SB . The decomposition of SA∞ into anirreducible representation follows from the standard Clebsch-Gordan series [15] for the tensor product of the representations A∞ and 1/2, SA∞ = 2(1/2 + 3/2 + 5/2 + . . .).

(22)

Here the factor 2 in front of the bracket means that each representation in the bracket occurs in the direct sum twice. This doubling may be interpreted as the sum of the left and right chiral spinor bundles. We shall argue that the standard Dirac operator corresponding to the round metric on S2 can be written solely in terms of the SU (2) generators as follows11 1 D = (σi Ri + 1). (23) ρ Here ρ is the radius of the sphere. This operator is self-adjoint with respect to the scalar product on SA∞ given by Z 1 2 ∗ d3 xi δ(xi − ρ2 )(Ψ+∗ Ξ+ + Ψ− (Ψ, Ξ) ≡ Ξ− ), Ψ, Ξ ∈ SA∞ . (24) 2πρ The easy way of deriving (23) consists in comparing a three dimensional flat Dirac operator D3 on SB written in the spherical coordinates with the two dimensional round Dirac operator D2 on the sphere in the same coordinates. Due to the rotational invariance the choice of a coordinate chart is irrelevant and we may proceed by choosing (and fixing) the poles of the sphere. The Dirac operator D in arbitrary coordinates in a general (curved) Riemannian manifold is given by 10 We have in mind the trivial bundle, twists by U (1) bundles needed for the inclusion of monopoles will be considered in a forthcoming paper. 11 The same formula was already given in [18, 19]. We give different evidence of its validity, however.


162

1 D2 = −iγ a eµa (∂µ + ωµab [γ a , γ b ]), 4

(25)

where γ a are generators of the flat Clifford algebra {γ a , γ b } = 2δ ab ,

γ a 2 = 1,

γa† = γa,

(26)

eµa is the vielbein and ωµab the spin connection defined by λ a eλ + ωµab ebν = 0. ∂µ eaν − Γµν

(27)

For S2 in the spherical coordinates eθ1 =

1 , ρ

eφ2 =

1 , ρ sin θ

ωφ12 = −ωφ21 = − cos θ.

(28)

All remaining components of the vielbein and the connection vanish. For R3 in the spherical coordinates eθ1 =

1 , r

eφ2 =

1 , r sin θ

er3 = 1

(29)

and ωφ21 = −ωφ12 = cos θ, Thus

ωφ23 = −ωφ32 = sin θ,

ωθ13 = −ωθ31 = 1.

1 1 1 ∂φ , D2 = −iγ 1 (∂θ + ctgθ) − iγ 2 ρ 2 ρ sin θ

(30)

(31)

and

1 1 1 1 ∂φ + −iγ 3 (∂r + ). D3 = −iγ 1 (∂θ + ctgθ) − iγ 2 r 2 r sin θ r We observe a simple relation between D3 restricted on SA∞ and D2 namely − iγ 3 D3 |restr. + 1/ρ = D2 .

(32)

(33)

(Note that −iγ 3 γ a , a = 1, 2 fulfil the defining relations of the Clifford algebra (26).) D3 can be expressed also in the flat coordinates in R3 D3 = −iσi ∂i ,

(34)

where σi are the Pauli matrices which also generate the Clifford algebra (26). A simple algebra gives 1 σ k x k 2 σk xk xi ∂ i − σ i Ri . D3 = −i (35) D3 = r r r r Because xri ∂i = ∂r and the vector fields Ri have no radial component it follows from (32) and (35) that σk xk . (36) γ3 = r Inserting γ3 from (36) and D3 from (35) into Eq.(33) we get the SU (2) covariant form (23) of the round Dirac operator on S2 . The spectrum of D2 readily follows from the group representation considerations. Consider a (normalized) spinor


163

Θ+ 1 = . 0 ρ

(37)

It is obviously the eigenvector of D2 with an eigenvalue 1. Moreover it is the highest weight state of one of the spin 1/2 representations in the decomposition (22) as it can be directly checked using the generators Ji from (21). Indeed J+ Θ+ = 0,

Ji Ji Θ+ = 3/4.

(38)

The construction of the other (normalized) highest weight states in the irreducible representations with the higher spins is obvious. They are given by s (2l + 1)!! + l + x Θ . (39) Ψl,h.w. = ρ−l−1 (2l)!! Here l is the spin of the irreducible representation. A direct computation shows D2 Ψl,h.w. = (l + 1)Ψl,h.w. .

(40)

Due to the rotational invariance of D2 the other eigenvectors within the irreducible representation are obtained by the action of the lowering generator J− , i.e. s (2l + 1 − m)! (2l + 1)!! m + l + J x Θ . (41) Ψl,m = ρ−l−1 (2l + 1)!m! (2l)!! − The eigenvalue corresponding to the eigenvector Ψl,m , m = 0, . . . , 2l is obviously l + 1. So far we have constructed only one branch of the spectrum. However, due to an obvious relation (42) D 2 γ 3 + γ 3 D2 = 0 also spinors γ 3 Ψl,m are the eigenvectors of D2 with the eigenvalues −(l + 1). In this way we found the complete spectrum because all eigenvectors Ψl,m and γ 3 Ψl,m form the basis of the spinor bundle SA∞ . 4. Non-commutative Supersphere Having in mind the goal of constructing a non-commutative spinor bundle, we have to look for a language to describe the commutative case which would be best suited for performing the non-commutative deformation. We shall argue that the very structure to be exploited is the OSp(2, 2) superalgebra which is somewhat hidden in the presentation given in the previous section. We shall proceed conceptually as follows: The non-commutative sphere, described in Sect: 2, emerged naturally from the quantization of the algebra of the scalar fields on the ordinary sphere. Hence, it is natural to expect that the quantization of the supersphere would give a deformed ring of the scalar superfields on the supersphere. Those superfields contain as their components the ordinary fermion fields on the sphere, therefore the deformation of the algebra of the superfield should give (and it does give) the non-commutative spinor bundle on the non-commutative sphere, i.e. the structure we are looking for. 4.1. (Super)commutative supersphere. Consider a three-dimensional superspace SR3 with coordinates xi , θα ; the super-coordinates are the SU (2) Majorana spinors. Consider


164

an algebra SB of analytic functions on the superspace with the Grassmann coefficients in front of the odd monomials in θ. SB can be factorized by its ideal SI, consisting of P 2 all functions of a form h(xi , θα )( xi + Cαβ θα θβ − ρ2 ). Here C = iσ 2 .

(43)

We refer to the quotient SA∞ as to the algebra of superfields on the supersphere. An OSp(2, 2) invariant inner product of two elements Φ1 , Φ2 of SA∞ is given by12 Z ρ 2 (Φ1 , Φ2 )∞ ≡ d3 xi dθ+ dθ− δ(xi + Cαβ θα θβ − ρ2 )Φ‡1 (xi , θα )Φ2 (xi , θα ) : (44) 2π R3 Here Φ1 (xi , θα ), Φ2 (xi , θα ) ∈ SB are some representatives of Φ1 and Φ2 and the (graded) involution [20, 21] is defined by ‡

‡

θ+ = θ− , θ− = −θ+ , (AB)‡ = (−1)degA degB B ‡ A‡ .

(45)

The algebra SA∞ is obviously generated by (the equivalence classes) xi (i = 1, 2, 3) and θ α (α = +, −) which (anti)commute with each other under the usual pointwise multiplication, i.e. xi xj − xj xi = xi θα − θα xi = θα θβ + θβ θα = 0. Their norms are given by

||xi ||2∞ = ||θα ||2∞ = ρ2 .

(46) (47)

Consider the vector fields in SR generating OSp(2, 2) superrotations of SB. They are given by explicit formulae 3

1 1 3 x ∂θ− − (x1 + ix2 )∂θ+ + − θ+ ∂x3 − θ− (∂x1 + i∂x2 ) , 2 2 1 3 1 v− = − x ∂θ+ + (x1 − ix2 )∂θ− + θ− ∂x3 − θ+ (∂x1 − i∂x2 ) , 2 2 2 + − 1 θ− θ+ d+ = − r(1 + 2 θ θ )∂− + R+ − (xi ∂i − R3 ), 2 r 2r 2r 2 + − 1 θ+ θ− i d− = r(1 + 2 θ θ )∂+ + R− − (x ∂i + R3 ) 2 r 2r 2r θ − x3 θ + x3 θ − x+ θ + x− + )∂+ + ( − )∂− ≡ 2(θ− v+ − θ+ v− ); Γ∞ = ( r r r r

v+ = −

r+ = x3 (∂x1 + i∂x2 ) − (x1 + ix2 )∂x3 + θ+ ∂θ− , −

r− = −x (∂x1 − i∂x2 ) + (x − ix )∂x3 + θ ∂θ+ , 3

1

2

(48) (49) (50) (51) (52) (53) (54)

1 (55) r3 = −ix1 ∂x2 + ix2 ∂x1 + (θ+ ∂θ+ − θ− ∂θ− ); 2 and they obey the OSp(2, 2) Lie superalgebra graded commutation relations [13, 21] [r3 , r± ] = ±r± ,

[r+ , r− ] = 2r3 ,

(56)

The normalization ensures that the norm of the unit element of SA∞ is 1. The inner product is supersymmetric but it is not positive definite. However, such a property of the product is not needed for our purposes. 12


1 [r3 , v± ] = ± v± , 2

165

[r± , v± ] = 0,

[r± , v∓ ] = v± ,

(57)

1 1 {v± , v± } = ± r± , {v± , v∓ } = − r3 ; 2 2 [Γ∞ , v± ] = d± , [Γ∞ , d± ] = v± , [Γ∞ , ri ] = 0, 1 [r3 , d± ] = ± d± , 2

[r± , d± ] = 0,

(58) (59)

[r± , d∓ ] = d± ,

(60)

1 {d± , v∓ } = ± Γ∞ , 4 1 1 {d± , d± } = ∓ r± , {d± , d∓ } = r3 . 2 2 {d± , v± } = 0,

(61) (62) 2

Note, that all introduced generators do annihilate the quadratic form xi + Cαβ θα θβ ; hence they induce the action of OSp(2, 2) on SA∞ 13 . In order to demonstrate the OSp(2, 2) invariance of the inner product (44) we have to settle the properties of the OSp(2, 2) generators with respect to the graded involution. It holds (63) (Φ1 , ri Φ2 )∞ = (ri Φ1 , Φ2 )∞ ; (Φ1 , v∓ Φ2 )∞ = ±(v± Φ1 , Φ2 )∞ ;

(64)

(Φ1 , d∓ Φ2 )∞ = ∓(d± Φ1 , Φ2 )∞ ;

(65)

(Φ1 , Γ∞ Φ2 )∞ = (Γ∞ Φ1 , Φ2 )∞ .

(66)

Consider now the variation of a superfield Φ δΦ = i(ε+ v+ + ε− v− )Φ,

(67)

where εα is a constant Grassmann Majorana spinor, i.e. ε‡+ = ε− ,

ε‡− = −ε+

(68)

and, much in the same manner, a variation δΦ = i(ε− d+ + ε+ d− )Φ.

(69)

Using the relations (63–66) it is straightforward to observe the invariance of the inner product with respect to the defined variations. As it is well known [21] the typical irreducible representations of OSp(2, 2) consist of quadruplets of the SU (2) irreducible representations j ⊕ j − 21 ⊕ j − 21 ⊕ j − 1. The number j is an integer or a half-integer and it is referred to as the OSp(2, 2) superspin. The generators xi , θα ∈ SA∞ together with 1 + 3 (θ x + θ− x+ ), ρ2

j=

1 1 , j3 = , 2 2

(70)

1 + − (θ x − θ− x3 ), ρ2

j=

1 1 , j3 = − , 2 2

(71)

13 The appearance of r in Eqs. (50–52) may seem awful because we have considered the ring of superanalytic functions on SR3 . However, this is only a formal drawback, which can be cured by a completion of the space of superanalytic functions with respect to an appropriate inner product. In fact, we need not even do that for our purposes because the terms involving r become anyway harmless after the factorization by the ideal SI.


166

1+

1 + − θ θ , ρ2

j = 0, j3 = 0;

(72)

indeed form the (typical) superspin 1 irreducible representation of OSp(2, 2) algebra under the action of the vector fields (48–55). The numbers j, j3 in (70–72) correspond to the total SU (2) spin and its third component. The supermultiplet with the superspin 1 can be conveniently constructed applying subsequently the lowering operators v− and d− on the highest weight vector x+ . Supermultiplets with higher superspins can be obtained in the same way starting with the highest weight vectors x+ l . Thus the full decomposition of SA∞ into the irreducible representations of OSp(2, 2) can be written as the infinite direct sum (73) SA∞ = 0 + 1 + 2 + . . . , where the integers denote the OSp(2, 2) superspins of the representations14 . From the point of view of the SU (2) representations, the algebra of the superfields consists of two copies of A∞ and the spinor bundle 21 ⊗ A∞ (see Eq. (22)): Note that the generators of SA∞ fulfill the obvious relation 2

xi + Cαβ θα θβ = ρ2 .

(74)

The big algebra SB has a natural grading as the vector space, given by the parity of the total power of the Grassmann coordinates θα . Because we factorized over the quadratic surface in the superspace, this grading induces the grading in SA∞ . It is easy to see that the odd elements of SA∞ with respect to this grading can be identified with the fermion fields on the sphere. Indeed, they can be written as Ψ = Ψα (xi )

θα , ρ

(75)

where the (Grassmann) components Ψα belong to A∞ 15 . But this is the standard spinor bundle on the sphere Ψ+ (xi ) , (76) i Ψ− (x ) described in Sect. 3. The scalar product on the bundle is inherited from the inner product (44) Z ρ 2 d3 xi δ(xi − ρ2 )dθ+ dθ− Ψ ‡ Ξ, (77) (Ψ, Ξ) ≡ 2π and (up to a sign) it coincides with the scalar product (24). The Pauli matrices, as the operators acting on the two-component spinors, can be expressed in the superfield formalism as follows σ 3 = θ + ∂θ + − θ − ∂θ − ,

σ ± = 2θ± ∂θ∓ .

(78)

In what follows we shall refer to the odd (even) elements with respect to the described grading as to the fermionic (bosonic) superfields in order to make a difference with the even and odd superfields in the standard (Grassmann) sense. The OSP (2, 1) superalgebra generated by ri , v± has a quadratic Casimir 14

The “baryon” number of those representations, in the sense of Ref.[21], is zero. P i2 P i2 The factorization by the relation x − ρ2 = 0 and the relation x + Cαβ θ α θ β − ρ2 = 0 is effectively the same in this case because the term quadratic in θ is killed upon the multiplication by another θ in Eq. (75). 15


167

1 K2 = (r32 + {r+ , r− }) + (v+ v− − v− v+ ) ≡ B2 + F2 . 2

(79)

Using Eqs. (78), it is easy now to check that the fermionic part F2 of the Casimir is directly related to the Dirac operator (23) ρD = σ i Ri + 1 = 2F2 −

1 1 = 2(v+ v− − v− v+ ) − . 2 2

(80)

The grading γ 3 of the Dirac operator is just the OSp(2, 2) generator Γ∞ . Its eigenfuctions are obviously the Weyl spinors. A Majorana spinors are given by the restriction ψ+‡ = ψ− ,

‡ ψ− = −ψ+ ,

(81)

which can be easily derived from the reality condition on the superfield Φ. 4.2. The truncation of SA∞ . We define the family of non-commutative superspheres SAj by furnishing the truncated sum of the irreducible representations of OSp(2, 2) , SAj = 0 + 1 + . . . + j,

j∈Z

(82)

with an associative product and an inner product which in the limit j → ∞ give the standard products in SA∞ . In order to do this consider the space L(j/2, j/2) of linear operators from the representation space of the OSp(2, 1) irreducible representation with the OSp(2, 1) superspin j/2 into itself. (Note that the OSp(2, 1) irreducible representation with the OSp(2, 1) superspin j has the SU (2) content j ⊕ j − 21 [21]). The action of the superalgebra OSp(2, 2) itself on L(j/2, j/2)16 is described by operators Ri , Vα , Dα , γ ∈ L(j/2, j/2) given by [23] ! j 0 Ri2 −j Id 0 Ri = , γ = . (83) j 1 − 0 −(j + 1)Id 0 R2 2 i

Vα = where

j

Vα2

0

j j 1 2 ,2 −2

− 21 , j2

Vα

!

j

, 0

Dα =

0

j

−Vα2

− 21 , j2

p (l − l3 )(l + l3 + 1), p l hl, l3 − 1|R− |l, l3 i = (l + l3 )(l − l3 + 1), hl, l3 + 1|R+l |l, l3 i =

hl, l3 |R3l |l, l3 i

= l3 , r 1 j2 , j2 − 21 1 j 1 + l3 + , |l3 i = − hl3 + |V+ 2 2 2 2 r j j 1 1 1 j 1 , − − l3 + , hl3 − |V−2 2 2 |l3 i = − 2 2 2 2 r 1 j2 − 21 , j2 1 j − l3 , |l3 i = − hl3 + |V+ 2 2 2

Vα2

, j2 − 21

! ,

(84)

0 (85) (86) (87) (88) (89) (90)

16 The so-called non-typical irreducible representation of OSp(2, 2) [21, 22] is at the same time also the OSp(2, 1) irreducible representation with the OSp(2, 1) superspin j/2.


168

r 1 j2 − 21 , j2 1 j + l3 . hl3 − |V− |l3 i = 2 2 2 Every Φ ∈ L(j/2, j/2) can be written as a matrix φ R ψR Φ= , ψL φ L

(91)

(92)

where φR and φL are square (j + 1) × (j + 1) and j × j matrices respectively and ψR and ψL are respectively rectangular (j + 1) × j and j × (j + 1) matrices. The meaning of the indices R and L will become clear in the next subsection. A fermionic element is given by a supermatrix with vanishing diagonal blocks and a bosonic element by one with vanishing off-diagonal blocks. Clearly, the OSp(2, 2) superalgebra acts on L(j/2, j/2) by the superadjoint action Ri Φ ≡ [Ri , Φ],

Γ Φ ≡ [γ, Φ],

(93)

Vα Φeven ≡ [Vα , Φeven ],

Vα Φodd ≡ {Vα , Φodd },

(94)

Dα Φeven ≡ [Dα , Φeven ],

Dα Φodd ≡ {Dα , Φodd }.

(95)

This ‘superadjoint’ representation is reducible and, in the spirit of Ref.[21, 22], it is easy to work out its decomposition into OSp(2, 2) irreducible representations L(j/2, j/2) = 0 + 1 + . . . + j.

(96)

The associative product in L(j/2, j/2) is defined as the composition of operators and the OSp(2, 2) invariant inner product on L(j/2, j/2) is defined by17 (Φ1 , Φ2 )j ≡ STr(Φ‡1 , Φ2 ),

Φ1 , Φ2 ∈ L(j/2, j/2).

(97)

Here ST r is the supertrace and ‡ is the graded involution. Although these concepts are quite standard in the literature it is instructive to work out their content in our concrete example. The supertrace is defined as usual ST rΦ ≡ T rφR − T rφL , and the graded involution as [20] ‡

Φ ≡

φ†R † ±ψR

† ∓ψL † φL

(98)

.

(99)

† means the standard hermitian conjugation of a matrix and the upper (lower) sign refers to the case when the entries consists of odd (even) elements of a Grassmann algebra. Note that V+‡ = V− , V−‡ = −V+ . (100) Ri‡ = Ri , Now we identify SAj with even elements of L(j/2, j/2) which means that the entries of the (off)-diagonal matrices are (anti)-commuting variables. This correspond to the similar requirement in the untruncated case because in the truncated case the spinors form the off-diagonal part of the superfield. We can demonstrate the OSp(2, 2) invariance of the inner product (97) again by settling the properties of the OSp(2, 2) generators with respect to the graded involution (99). They read 17

The normalization ensures that the norm of the identity matrix is 1.


169

(Φ1 , Ri Φ2 )j = (Ri Φ1 , Φ2 )j ,

(101)

(Φ1 , V∓ Φ2 )j = ±(V± Φ1 , Φ2 )j ,

(102)

(Φ1 , D∓ Φ2 )j = ∓(D± Φ1 , Φ2 )j ,

(103)

(Φ1 , Γ Φ2 )j = (Γ Φ1 , Φ2 )j .

(104)

Consider now the variation of a superfield Φ δΦ = i(+ V+ + − V− )Φ, where α is given by

α =

εα 0

0 −εα

(105)

(106)

and εα are the usual Grassmann variables with the involution properties ε‡− = −ε+ .

ε‡+ = ε− ,

(107)

Much in the same manner, consider also a variation δΦ = i(− D+ + + D− )Φ.

(108)

Using the relations (101-104) it is straightforward to observe the invariance of the inner product with respect to the defined variations. Note that α do anticommute with Dα and Vα as they should. We can choose a basis in SAj formed by eigenvectors of the Hermitian operators Q2 ≡ R2i + Cαβ Vα Vβ ,

(109)

R2i and R3 . The spectrum of (the OSp(2, 1) Casimir) Q2 consists of numbers q(q + 1/2) where the OSp(2, 1) superspin q runs over all integers and half-integers from 0 to j [23]; the remaining two operators have the standard spectra known in the SU (2) context. Now we make more precise the notion of the commutative limits of the inner product and the associative product. There is a natural chain of the linear embeddings of the vector spaces (110) SA1 ,→ SA2 ,→ . . . ,→ SAj ,→ . . . ,→ SA∞ Any (normalized) element from SAj of a form l

(111)

l

(112)

sj,lpq V− p D− q Xj+ is mapped into an element from SAk of the form

sk,lpq V− p D− q Xk+ .

Here Xji (and Θj± ≡ −V∓ Xj± ) are the representatives of the OSp(2, 1) generators in i ≡ xi the OSp(2, 1) irreducible representation with the OSp(2, 1) superspin j/2 (X∞ α α and Θ∞ ≡ θ ). They are normalized so that [X m , X n ] = i q

ρ j j 2(2

+

1 2)

mnp X p ,

(113)


170

ρ βα [X i , Θα ] = q σi Θβ , 2 j2 ( j2 + 21 )

(114)

ρ (Cσ i )αβ X i . {Θα , Θβ } = q j j 1 2 2(2 + 2)

(115)

(Xji , Xji )j = (Θjα , Θjα )j = ρ2 .

(116)

Hence sj,lpq are (real) normalization coefficients given by the requirement that the embedding is norm-conserving. Because the operators Q2 , R2i and R3 are hermitian for arbitrary SAj (as it can be easily seen from the definitions of the inner products (44),(97)) the embeddings are in fact isometric. Indeed, the inner product of the eigenvectors of hermitian operators vanishes if the corresponding eigenvalues are different. The commutative limit of the associative product is more involved, however. We proceed in an analogous way as in the purely bosonic case SU (2). Consider more closely the behaviour of the product as the function of k. According the relation (110), two arbitrary elements Φ1 , Φ2 of SAj can be canonically considered as the elements of SAk for whatever k > j (including k = ∞). Their product in every SAk can also be embedded in SA∞ . Denote the corresponding element of SA∞ as (Φ1 Φ2 )k . We shall argue that lim (Φ1 Φ2 )k = Φ1 Φ2 ,

k→∞

(117)

where Φ1 Φ2 is the standard supercommutative pointwise multiplication in SA∞ . For proving the relation (117), it is convenient to realize that SAj can be generated by taking products of generators Xji and Θjα of OSp(2, 1) in the irreducible representation with the OSp(2, 1) superspin j/2. This statement follows from the Burnside lemma [24], but its validity can be seen directly. Indeed, from the OSp(2, 2) commutation relations it follows easily that every element of the form (111) can be expressed in terms of Xji and Θjα . Hence the relations (113-115) ensure the (graded) commutativity in the limit j → ∞, and it is therefore sufficient just to show that the normalization coefficients sj,lpq defined in (112) have the property lim sk,lpq = s∞,lpq .

k→∞

(118)

Because of the OSp(2, 2) invariance of the inner products in all SAk (k = 1, . . . , ∞), it is in fact enough to demonstrate it just for the highest weight elements Xk+ l . Then it is a straighforward computation to check that −2 + + lim s−2 k,l00 ≡ lim (Xk , Xk )k = (2l + 1)c∞,l0 , l

k→∞

l

k→∞

(119)

−2 where c−2 ∞,l0 have been given in Eq. (17). But s∞,l00 can be directly computed from (44) giving −2 (120) s−2 ∞,l00 = (2l + 1)c∞,l0 .

We have thus proven the commutative limit relation (117). Note that the normalization of Xji and Θjα is such that the value of the Casimir in the j2 OSp(2, 1) irreducible representation is equal to ρ2 , i.e. 2

Xji + Cαβ Θjα Θjβ = ρ2 .

(121)


171

Thus the relation defining the supersphere is preserved also in the truncated case. We observe from Eqs. (47) and (116) that for every j Xji , Θjα ∈ SAj are embedded in SA∞ as just the standard (super)commutative generators xi , θα and in SAk , k > j as Xki , Θkα ∈ SAk . The notation is therefore justified and in what follows we shall often write just X i and Θα . 4.3. Dirac operator on the truncated sphere. In an analogy with the (super)commutative case, we define the non-commutative spinor bundle on the sphere S2 as the odd part of the truncated superfield Φ ∈ SAj and the Dirac operator we define as 1 ρD ≡ 2(V+ V− − V− V+ ) − . (122) 2 This operator is manifestly self-adjoint, SU (2) invariant and it is also odd with respect to the grading Γ given by Eqs. (93) and (83) or simply, if the diagonal part of a superfield vanishes, by Id 0 (123) Φfermionic . Γ Φfermionic = 0 −Id This explains the notation in Eq. (92): in the first (second) line there are right (left) objects with respect to the chiral grading Γ . Hence, a fermionic superfield of the upper(lower)triangular form will be referred to as the right (left) chiral spinor on the truncated sphere. The spectrum of D readily follows from the group representation considerations. Consider a normalized spinor Θ+ /ρ. It follows directly from OSp(2, 1) graded commutation relations (56–58) that this is the eigenvector of D with an eigenvalue 1. Moreover it is the highest weight state of one of the SU (2) spin 1/2 representations in the decomposition (82). This can be directly checked using the generators (93–95): R+ Θ+ = 0,

R2i = 3/4.

(124)

The construction of the other (normalized) highest weight states in the irreducible representations with the higher spins is obvious. They are given by s (2l + 1)!! + l + −l−1 X Θ . (125) Ψl,h.w. = bjl ρ (2l)!! Here l is the spin of the SU (2) irreducible representation and bjl is a normalization coefficient. A direct computation shows l ≤ j − 1. (126) DΨl,h.w. = (l + 1)Ψl,h.w. , Due to the rotational invariance of D the other eigenvectors within the irreducible representation are obtained by the action of the lowering generator R− , i.e. s (2l + 1 − m)! (2l + 1)!! m + l + R− X Θ . (127) Ψl,m = bjl ρ−l−1 (2l + 1)!m! (2l)!! The eigenvalue corresponding to the eigenvector Ψl,m , m = 0, . . . , 2l is obviously l + 1. So far we have constructed only one branch of the spectrum. However, due to an obvious relation DΓ + Γ D = 0 (128) are the eigenvectors of D with the eigenvalues −(l + 1). In this way also spinors Γ Ψl,m we found the complete spectrum because all eigenvectors Ψl,m and Γ Ψl,m form the basis of the space of the fermionic superfields from SAj . Thus, we have obtained precisely the truncation of the commutative Dirac operator D.


172

5. Supersymmetric Field Theories 5.1. The bosonic preliminaries. Consider the following action for a real scalar field living on the sphere S2 : Z 1 1 2 2 d3 xi δ(xi − ρ2 )φ(x)Ri2 φ(x). (129) S(φ) = (φ, Ri φ)∞ ≡ 2 4πρ It is easy to show that this is just the action of a free massless field on S2 , i.e. Z 1 dΩφ 4Ω φ, S(φ) = − 8π

(130)

where 4Ω is the Laplace-Beltrami operator on the sphere or, simply, the angular part of the flat Laplacian in R3 . Adding a mass and an interaction term is easy, e.g. the P (φ)-models [25, 10] are described by the action S∞ =

1 (φ, Ri2 φ)∞ + (1, P (φ))∞ , 2

(131)

where P (φ) is a polynomial in the field variable. The non-commutative analogue of the action (129) is now obvious: Sj =

1 1 1 (φ, R2i φ)j + (1, P (φ))j = T rj (φR2i φ) + T rj P (φ). 2 2j + 2 j+1

(132)

The truncated action is manifestly SU (2)invariant with respect to the infinitesimal transformation of the scalar field δφ = εi Ri φ ≡ εi [Ri , φ].

(133)

Another interesting class of Lagrangians consists of the nonlinear σ-models describing the string propagation in curved backgrounds. The (truncated) action reads Sj =

1 (Ri φA , g AB (φ)Ri φB )j 2

(134)

with the obvious commutative limit. It is not difficult, in fact, to define a quantization of the truncated system via the path integral because the space of field configurations is finite-dimensional. We gave the details in a separate publication [10] with the aim to develop the efficient nonperturbative regularization of field theories which could (hopefully in many aspects) compete with the traditional lattice approach. 5.2. The supersymmetric actions. The supersymmetric case is somewhat more involved than the bosonic one not only because of the enlargement of the number of degrees of freedom. Starting from the undeformed case one could suspect that the standard free OSp(2, 1)-supersymmetric action for a real superfield on the sphere should be written in our three dimensional formalism as Ssusp =

1 (Φ, (R2i + Cαβ Vα Vβ )Φ)∞ . 2

(135)

Though the OSp(2, 1) Casimir sitting within the brackets does give the SUSY invariance it does not yield the correct two dimensional “world-sheet” action containing just the


173

free massless bosonic field and free massless Majorana fermion. To get out of trouble we may use the philosophy used about a decade ago where supersymmetric models on the homogeneous spaces have been intensively studied [26]. In particular, Fronsdal has considered the spinors on anti-de Sitter spacetime and has constructed the OSP (4, 1) invariant supersymmetric actions by introducing another set of odd generators [26]. They were analogues of the standard supersymmetric covariant derivatives needed to build up the super-Poincaré invariant Lagrangians. The same approach applies in our case. The new odd generators are nothing but the additional OSp(2, 2) generators Dα . The standard Lagrangian of the free OSp(2, 1) supersymmetric theory can be written solely in terms of the “covariant derivatives” Dα and the grading Γ . Let us begin with the detailed quantitative account first in the non-deformed case. It is easy to check that the operator 1 2 Cαβ dα dβ + Γ∞ 4

(136)

is invariant with respect to OSp(2, 1) supersymmetry generated by ri and v± . Hence we may consider the action

ρ ≡ 2π

Z

1 2 Φ)∞ ≡ S = (Φ, Cαβ dα dβ Φ)∞ + (Φ, Γ∞ 4

1 2 d3 xi dθ+ dθ− δ(xi + Cαβ θα θβ − ρ2 )Φ(xi , θα )(Cαβ dα dβ + Γ 2 )Φ(xi , θα ), 4 R3 (136a) where Φ is a real superfield, i.e. Φ‡ = Φ. Consider now the variation of the real superfield Φ, δΦ = iεα vα Φ,

(137)

which preserves the reality condition. Now Eqs.(63–66) hold also when Φ1 is an even and Φ2 an odd superfield in the standard Grassmann sense. Using this and the fact that εα vα commutes with the operator (136), the supersymmetry of the action S obviously follows. It is straightforward to work out the action (136a) in the two-dimensional component language. It reads Z 1 1 1 1 dΩ(− φ 4Ω φ + ρ4 F 2 − ψ † ρ3 DΩ ψ), (138) S= 4π 2 2 2 where DΩ is the Dirac operator on S 2 and the superfield ansatz is Φ(xi , θα ) = φ(xi ) + ψα θα + (F +

xi ∂i φ)θ+ θ− . r2

(139)

Of course, ψα are anticommuting objects and the reality condition Φ‡ = Φ makes the fields φ and F real and the spinor ψα becomes Majorana18 , i.e. ψ+‡ = ψ− , 18

‡ ψ− = −ψ+ .

Note that we consider the graded involution defined by Eq.(45) (see also [20]).

(140)


174

We recognize in the expression (138) the standard free supersymmetric action in two dimensions. Adding a (real) superpotential W (Φ) we may write down a supersymmetric action with the interaction term. It reads 1 2 )Φ)∞ + (1, W (Φ))∞ . (141) S∞ = (Φ, (Cαβ dα dβ + Γ∞ 4 The truncated version of the action S∞ , 1 Sj = (Φ, (Cαβ Dα Dβ + Γ 2 )Φ)j + (1, W (Φ))j 4 is manifestly supersymmetric with respect to the variations δΦ = iα Vα Φ.

(142)

(143)

It remains to prove that Sj approaches S∞ for j → ∞. In order to do that it is convenient to rewrite both the truncated and untruncated action as follows:: 1 (144) Sj = (D+ Φ, D+ Φ)j + (D− Φ, D− Φ)j + (Γ Φ, Γ Φ)j + (1, W (Φ))j , 4 where the index j can be both finite and infinite and we have used the formulas (63–66) and (101–104). Now it is enough to show that lim (Dα Φ)k = dα Φ,

k→∞

lim (Γ Φ)k = Γ∞ Φ

k→∞

(145)

(The embedding (Φ)k was defined in Eqs. (111, 112).) But this is true almost by definition because Dα Φ can be written as a linear superposition of the vectors of the form (111, 112). As in the bosonic case we may write down the regularized action for the supersymmetric σ-models describing the superstring propagation in curved backgrounds 1 Sj = (D+ ΦA , gAB (Φ)D+ ΦB )j + (D− ΦA , gAB (Φ)D− ΦB )j + (Γ ΦA , gAB (Φ)Γ ΦB )j . 4 (146) The OSp(2, 1) supersymmetry and the commutative limit is obvious. The regularized action (146) can be used as the base for the path integral quantization manifestly preserving supersymmetry and still involving only the finite number of degrees of freedom. Particularly this aspect of our approach seems to be very promising both in comparison with the lattice physics as well as in general. Indeed so far we are not aware of any nonperturbative regularization which would possess all those properties. 6. Conclusions and Outlook We have regulated in the manifestly supersymmetric way the actions of the field theories on the supersphere, involving scalar and spinor fields. As a next step we plan to include in the picture the topologically non-trivial bundles and the gauge fields [27] and to study the chiral symmetry in the context. From the purely mathematical point of view we have to build up the non-commutative de Rham complex and understand the notions of oneand two-forms. It would be also interesting to establish a connection between previous works on supercoherent states [28, 12, 13] and our present treatment. In the future we shall attempt to reach two challenging goals in our programme, namely the truncation of the four-dimensional sphere and the inclusion of gravity.


175

´ ˇ y, T. Acknowledgement. We are grateful to A. Alekseev, L. Alvarez-Gaum´ e, M. Bauer, A. Connes, V. Cern´ Damour, J. Fröhlich, J. Ftácˇ nik, K. Gawe¸dzki, J. Hoppe, B. Jurˇco, E. Kiritsis, C. Kounnas, M. Rieffel, R. Stora and D. Sullivan for useful discussions. Part of the research of C.K. has been done at I.H.E.S. at Buressur-Yvette and of C.K. and P.P. at the Schrödinger Institute in Vienna. We thank both these institutes for hospitality.

References 1. Fröhlich, F. and Gawe¸dzki, K.: Conformal Field Theory and Geometry of Strings. Preprint IHES/P/93/59, Proceedings of the Mathematical Quantum Theory Conference, UBC-Vancouver, August 1993, hepth/9310187 2. Kiritsis, E. and Kounnas, C.: Nucl. Phys. B442, 472 (1995) 3. Berezin, F.A.: Commun. Math. Phys. 40, 153 (1975) 4. Hoppe, J.: MIT PhD Thesis, 1982 and Elem. Part. Res. J. (Kyoto) 80, 145 (1989) 5. Madore, J.: J. Math. Phys. 32, 332 (1991) and Class. Quant. Grav. 9, 69 (1992) 6. Grosse, H. and Preˇsnajder, P.: Lett. Math. Phys. 28, 239 (1993) 7. Connes, A.: Noncommutative Geometry, Academic Press London, 1994 8. A. Connes and J. Lott, Nucl. Phys. Proc. Suppl. B18, 29 (1991) 9. H. Grosse and J. Madore Phys. Lett. B283, 218 (1992) 10. H. Grosse, Klimˇc´ık, C. and Preˇsnajder, P.: Int. J. Theor. Phys. 35, 231 (1996) 11. Borthwick, D., Klimek, S., Lesniewski, A. and Rinaldi, M.: Comm. Math. Phys. 153, 49 (1993) 12. El Gradechi, A.: On the Supersymplectic Homogeneous Superspace Underlying the OSp(2, 1) Coherent States. Montreal preprint CRM-1850, 1994 13. El Gradechi, A. and Nieto, L.: Supercoherent States, Super-Kähler Geometry and Geometric Quantization. Montreal preprint CRM-1876, 1994, hep-th/9403109 14. Montvay, I. and Muenster, G.: Quantum fields on a lattice. Cambridge: Cambridge Univ. Press, 1994 and references therein 15. Landau, L.D. and Lifshitz, E.M.: Quantum Mechanics, Moscow: Nauka, 1989 (in Russian) 16. Prudnikov, A.P., Brytshkov, Yu.A. and Maritshev, O.I.: Integrals and Series, Moscow: Nauka, 1981, (in Russian) 17. Itzykson, I. and Zuber, J.B.: Quantum Field Theory, Moscow: Mir, 1984 (in Russian) 18. Jayewardena, J.: Helv. Phys. Acta 61, 636 (1988) 19. Grosse, H. and Preˇsnajder, P.: Lett. Math. Phys. 33, 171 (1995) 20. Scheunert, M., Nahm, W. and Rittenberg, V.: J. Math. Phys. 18, 146 (1977) 21. Scheunert, M., Nahm, N. and Rittenberg, V.: J. Math. Phys. 18, 154 (1977) 22. Marcu, M.: J. Math. Phys. 21, 1284 (1980) 23. Pais, A. and Rittenberg, V.: J. Math. Phys. 16, 2062 (1975) 24. Naimark, M.A.: Appendix III.. In Linear Representations of the Lorentz Group. Oxford: Pergamon Press, 1964 25. Glimm, J. and Jaffe, A.: Quantum Physics; A Functional Integral Point of View. Heidelberg: SpringerVerlag, 1981 26. Fronsdal, C.: 3+2 de Sitter Superfields. In: Essays on Supersymmetry, ed. by C. Fronsdal, Dordrecht: D. Reidel Publishing Company, 1986 27. Grosse, H., Klimˇc´ık, C. and Preˇsnajder, P.: Gauge Symmetry and Non-commutative Geometry. To appear 28. Chaichian, M., Ellinas, D. and Preˇsnajder, P.: J. Math. Phys. 32, 3381 (1991) Communicated by A. Connes

Commun. Math. Phys. 185, 177 – 196 (1997)

Communications in


Levi-Civita Connections on the Quantum Groups SLq (N ), Oq (N ) and Spq (N ) ¨ István Heckenberger, Konrad Schmudgen Universität Leipzig, Fakultät für Mathematik und Informatik und NTZ, Augustusplatz 10, 04109 Leipzig, Germany. E-mail: [email protected], [email protected] Received: 28 February 1996 / Accepted: 1 October 1996

Abstract: For bicovariant differential calculi on quantum groups various notions on connections and metrics (bicovariant connections, invariant metrics, the compatibility of a connection with a metric, Levi-Civita connections) are introduced and studied. It is proved that for the bicovariant differential calculi on SLq (N ), Oq (N ) and Spq (N ) from the classification in [18] there exist unique Levi-Civita connections.

0. Introduction The seminal work of S. L. Woronowicz [20] was the starting point to study noncommutative bicovariant differential calculi on quantum groups (Hopf algebras). Woronowicz has developed a general theory of such calculi which in many aspects can be considered as a non-commutative version of the classical Lie group theory. Bicovariant differential calculi on the quantum matrix groups SLq (N ), Oq (N ) and Spq (N ) have been classified (under natural assumptions) in two recent papers [18] and [19]. An outcome of this classification is that except for finitely many values of q there are precisely 2N such calculi on SLq (N ) for N ≥ 3 and two on Oq (N ) and Spq (N ) for N ≥ 4. See Sect. 1 for a brief review. It is clear that these calculi are basic tools of non-commutative geometry on the corresponding quantum groups. The aim of this paper is to define and to study invariant metrics and Levi-Civita connections for the bicovariant differential calculi on SLq (N ), Oq (N ) and Spq (N ). The Killing metric of a compact Lie group and the Levi-Civita connection of a Riemannian manifold are fundamental notions of (commutative) differential geometry, so it seems that extensions of these concepts to the bicovariant differential calculi are necessary steps toward the development of non-commutative differential geometry on quantum groups. It turns out that these generalizations are by no means straightforward and that phenomena occur which are absent in classical differential geometry. We briefly mention some of these new features. Firstly, no ad-invariant metric for these bicovariant

178

I. Heckenberger, K. Schmüdgen

differential calculi is symmetric in the usual sense if q is not a root of unity. However, the ad-invariant metrics are symmetric with respect to the corresponding braiding, see Corollary 2.5 below. In our opinion, this fits nicely into the concepts of the braided geometry, see [15] and the references therein. Secondly, ad-invariant metrics for SLq (N ) resp. Oq (N ), Spq (N ) depend on two resp. three complex parameters. Roughly speaking, this stems from the fact that the dimension of these bicovariant calculi is N 2 rather than the dimension of the corresponding Lie groups as in case of the classical differential calculus. Thirdly, if we generalize the notion of a Levi-Civita connection in an obvious straightforward manner, then there exist many Levi-Civita connections for a given metric. For our bicovariant differential calculi they depend on a number of free complex parameters, see e. g. Appendix B. The main purpose of this paper is to propose notions of compatibility of a connection and an ad-invariant metric (see formula (9) and Definition 3.3 below) and of a LeviCivita connection (see Definition 3.5) which ensure that the important result in classical differential geometry of uniqueness of the Levi-Civita connection remains valid in the present setting. If the braiding map σ is the flip operator of the tensor product as in case of the “ordinary” differential calculus on compact Lie groups, our definitions give just the corresponding classical concepts. This paper is organized as follows. In Sect. 1 we briefly recall the bicovariant differential calculi on SLq (N ), Oq (N ) and Spq (N ) studied in [18] and we collect some basic facts needed later. In Sect. 2 we define metrics and invariant metrics. The invariant metrics for these calculi and their restrictions to the invariant subspaces for the right coaction and the right adjoint action, respectively, are determined. For some of the calculi the quantum Lie algebra contains the corresponding classical Lie algebra as an ad-invariant subspace when q → 1. In these cases the limits of the invariant metrics exist and their restrictions to these subspaces give multiples of the Killing forms. In Appendix A we show how a variant of the Rosso form of the quantized enveloping algebra Uq (sl(N )) gives an ad-invariant metric for SLq (N ). Sect. 3 is concerned with connections. After reviewing some generally known definitions on connections (see e. g. [4]) we define bicovariant connections, the compatibility of a connection with a metric and Levi-Civita connections and we discuss some of their properties. The main results of this paper (Theorems 4.2 and 5.2) are stated and proved in Sects. 4 and 5, respectively. They assert that for each of the bicovariant differential calculi on SLq (N ), Oq (N ) and Spq (N ) described in Sect. 1 there is precisely one Levi-Civita connection. Moreover, it is shown that these Levi-Civita connections admit limits when q → 1 in an appropriate way. In Appendix B of this paper we show that if we define Levi-Civita connections by taking the “usual” compatibility with a metric, then the set of Levi-Civita connections for SLq (N ) depends on three free parameters. Let us fix some notation and assumptions which are needed in the sequel. Throughout we use Sweedler’s notation 1(a) = a(1) ⊗ a(2) and the Einstein convention to sum over repeated indices. The antipode of a Hopf algebra is denoted by κ and the counit by ε. Let Mor(v, w) be the space of intertwiners of corepresentations v and w and let Mor(v) := Mor(v, v). We use the definitions of the quantum groups SLq (N ), Oq (N ) and Spq (N ) and their basic properties established in [8]. Let u = (uij ) denote the corresponding fundamental representation and let uc be the contragredient representation of u. In the case where A is Oq (N ) or Spq (N ), let C = (C ij ) denote the matrix of the metric of A and B the inverse matrix of C. We abbreviate Q = q − q −1 . Also we shall freely use the general theory of bicovariant differential calculi developed in [20]. The abbreviation FODC means a first order differential calculus. As noted

Levi-Civita Connections on the Quantum Groups SLq (N ), Oq (N ) and Spq (N )

179

above, our main intention is to study the bicovariant differential calculi 0±,z (see Sect. 1) which occurred in the classification of [18, 19]. But the corresponding concepts and general facts apply to an arbitrary bicovariant differential calculus over a Hopf algebra. ˜ is always a general In order to avoid confusion, let us adopt the following notations: 0 0 bicovariant FODC over a Hopf algebra A, A is the Hopf dual of A, X˜ is the quantum ˜ {χi | i ∈ I} is the ˜ {ηi | i ∈ I} is a finite vector space basis of inv 0, Lie algebra of 0, ˜ as defined by Proposition ˜ ⊗A 0 corresponding dual basis of X˜ and σ˜ is the braiding of 0 3.1 in [20]. In this paper we suppose that the deformation parameter q is not a root of unity and q 6= 0. Then, roughly speaking, the representation theory of A is similar to the classical case [13, 17]. We shall need this assumption only in order to ensure that the decompositions of certain tensor product representations of u and uc can be labelled by Young tableaus similar to the classical case. The corresponding results are Lemmas 2.3, 2.4, 4.1 and 5.1. All other considerations are valid without this assumption. 1. Review of Some Facts on Bicovariant Differential Calculi on Quantum Groups SLq (N ), Oq (N ) and Spq (N ) Firstly we repeat the construction of bicovariant differential calculi, see [18] for some missing details of proofs in the following discussion. Let z be a nonzero complex number. We assume that z N = q for A = SLq (N ) and that z 2 = 1 for A = Oq (N ) and A = ±i ±i Spq (N ). Let L± z = (lz j ) denote the N ×N matrix of linear functionals lz j on A defined in [8], Sect. 2, by taking the matrix z −1 P Rˆ as R. By definition, we then have l+z ij (unm ) = z −1 Rˆ in mj

and

i n ˆ −1in l− z j (um ) = z R mj .

(1)

Let Di := q 2i for A = SLq (N ) and Di := (C(C −1 )t )ii for A = Oq (N ), Spq (N ), where C is the matrix given by the metric of Oq (N ) and Spq (N ), cf. [8], Sect. 1. Then we have κ2 (uij ) = Di uij Dj−1 (no summation). There are 2N bicovariant FODC 0±,z , z N = q 2 on SLq (N ), N ≥ 3, and 2 bicovariant FODC 0+ = 0+,1 and 0− = 0+,−1 on Oq (N ), Spq (N ) and SLq (2). Except for the quantum group Oq (3), these FODC exhaust the bicovariant first order differential calculi which appeared in the classification of [18, 19]. They are the objects of our study in the paper. In what follows we assume that 0 always denotes such a FODC 0±,z . The FODC 0±,z on A is given by N X (χij ∗ a)ηij , a ∈ A, da = i,j=1

where χij =

N X

∓ j −1 n Dn−1 l± z 0 i κ(lz 00 n ) − Di δij ε,

i, j = 1, . . . , N

(2)

n=1

and {ηij | i, j = 1, . . . , N } is a basis of the space inv (0±,z ) of left-invariant elements of 0±,z . The right and left A-module operations satisfy the equations ηij a = (f ij mn ∗a)ηmn , a ∈ A, and the right coaction of A on this basis is given by 1R (ηij ) = ηmn ⊗ v mn ij , i, j = 1, . . . , N , where ±i ∓ n f ij (3) mn = lz 0 m κ(lz 00 j ),

180

I. Heckenberger, K. Schmüdgen c mn i n v mn ij = (u u)ij = κ(um )uj .

(4)

The linear span X of the linear functionals χij , i, j = 1, . . . , N , equipped with the bracket [·, ·] : X × X → X defined in [20], Sect. 5, is called the quantum Lie algebra of the bicovariant FODC 0±,z . ˜ and X˜ given by There is a duality < ·, · > between inv 0 + * X X X ai ηi , χ j bj = a k bk (5) i

j

k

˜ X˜ ⊗B → A⊗A B for all ak , bk ∈ C. This definition extends to a map < ·, · >: A⊗ inv 0× ∧ ˜ ˜ ∧ by taking ak ∈ A ˜ for linear subspaces A of 0 resp. X ⊗A and linear subspaces B of 0 and bk ∈ B in (5) for k ∈ I. Let σ denote the braiding map of 0±,z ⊗A 0±,z (see Proposition 3.1 in [20]) and mnrs be the matrix coefficients of σ with respect to the basis {ηij ⊗A ηkl } of let σijkl mnrs inv (0±,z ⊗A 0±,z ), i. e. σ(ηij ⊗ ηkl ) = σ ijkl ηmn ⊗ ηrs . ˆ ±tn ˆ ∓xi ˆ ± ys = Dk Dx−1 Rˆ ∓ pk Lemma 1.1. σ mnrs xj R yr R tm R pl , where the upper and lower ijkl signs refer to 0+,z and 0−,z , respectively. − s + t Proof . We carry out the proof for 0+,z . From (3) we see that f tp rs = lz 0 r κ(lz 00 p ) are the linear functionals of [20], Theorem 2.1, for the FODC 0+,z . Equation (1) yields ij n −1 ˆ tn ˆ ys n k k tp 2 n Ryr Rpl and so δ ir δ js δ nm = f ij f tp rs (ul ) = z rs (κ(κ(uk )um )) = f tp (κ(um ))f rs (κ (uk )). ij ˆ xs From κ2 (unk ) = Dn Dk−1 unk we easily derive δ ir δ js δ nm = f tp (κ(ukm ))z −1 Dn Dk−1 Rˆ tn xr Rpk , ij k −1 ˆ −1 pk ˆ −1xi so the latter yields f tp (κ(um )) = zDk Dx R xj R tm . By the general theory (cf. ij ij c m n k tp n formula (3.15) in [20]) and (4), we have σ mnrs ijkl = f rs ((u )k ul ) = f tp (κ(um ))f rs (ul ) from which the above formula follows.

Recall that the matrix Rˆ satisfies the Hecke relation (Rˆ − qI)(Rˆ + q −1 I) = 0 for A = SLq (N ) and the cubic equation (Rˆ − qI)(Rˆ + q −1 I)(Rˆ − q −N I) = 0 for A = Oq (N ), Spq (N ), where = 1 for A = Oq (N ) and = −1 for A = Spq (N ). From these equations and Lemma 1.1 it follows that (σ − I)(σ + q −2 I)(σ + q 2 I) = 0 for A = SLq (N ) and (σ − I)(σ + q −2 I)(σ + q 2 I) (σ − q N −+1 I)(σ − q −N −1 I) ×(σ + q N −−1 I)(σ + q −N +1 I) = 0 for A = Oq (N ), Spq (N ) (cf. [3]). The above formulas show that the set of for q > 0 positive eigenvalues of σ is {1} for SLq (N ) and {1, q N , q −N } for Oq (N ) and Spq (N ). In the following we use the abbreviations p := q N − , s := 1 + Q−1 (p − p−1 ) for PN Oq (N ) and Spq (N ) and s := i=1 q −2i for SLq (N ). Let ωij := κ(uin )dunj , i, j = 1, . . . , N . In [18] and [19] it is proved that if q is not a root of unity and apart from finitely many other values of q the set {ωij | i, j = 1, . . . , N } is a basis of the vector space inv 0. Let {Xij | i, j = 1, . . . , N } be the corresponding dual basis of X with respect to the duality (5). We call the sets {ωij } and {Xij } standard


181

bases of inv 0 and X , respectively. For this basis of inv 0 the right coaction also fulfills kl (cf. (4)). 1R (ωij ) = ωkl ⊗ vij ˜ over a Hopf algebra A such Let us briefly return to a general bicovariant FODC 0 ˜ < ∞. Then the quantum Lie algebra X˜ of 0 ˜ is contained in the Hopf dual that diminv 0 A0 and the bracket [·, ·] of X˜ can be written as [x, y] = adR y(x). Here adR denotes the right adjoint action of A0 given by adR f (g) = κ(f(1) )gf(2) for f, g ∈ A0 . Moreover, we have [x, f ] := adR f (x) ∈ X˜ for all f ∈ A0 and x ∈ X˜ . A linear subspace Y of X˜ is called ad-invariant if [y, f ] ∈ Y for all y ∈ Y and f ∈ A0 . Next we are looking for ad-invariant subspaces of X˜ . For this we need the following simple ˜ i. e. Aij v˜ kj = v˜ li Alk for all i, k ∈ I. Then Lemma 1.2. Suppose that A = (Aij ) ∈ Mor(v), i ˜ and im At = lin{Aij χj | i ∈ im A = lin{Aj ηi | j ∈ I} is a 1R -invariant subspace of inv 0 I} is an ad-invariant subspace of X˜ . Proof . Since A ∈ Mor(v), ˜ we have 1R (Aij ηi ) = Aij ηk ⊗ v˜ ik = Akl ηk ⊗ v˜ jl , so that 1R (im A) ⊂ im A ⊗ A. From the general theory [20] we easily derive that [χi , f ] = f (v˜ ki )χk for all f ∈ A0 . Therefore, we compute adR f (Aik χk ) = Aik f (v˜ lk )χl = f (v˜ ji )Ajl χl . That is, adR f (im At ) ⊂ im At for all f ∈ A0 . 1 −2i ij δ δkl , For SLq (N ) the projections P0 , P1 : inv 0 → inv 0 defined by P0 ij kl = s q ij i j c 0 = δk δl − P0 kl belong to Mor(u ⊗ u). Hence the subspaces Υ = im P0 = lin{ω 0 := kl 1 1 P0 (ω11 ) = P0 kl 11 ωkl } and Υ = im P1 = lin{ωij := P1 (ωij ) = P1 ij ωkl } of inv 0 are 1R invariant. The corresponding ad-invariant subspaces of X are Y 0 = im P0t = lin{Y 0 := P ij 1 1 t 1 −2i ij 0 δ Y } respectively. ij − q k Xkk } and Y = im P1 = lin{Yij := P1 kl Xkl = XP s Since {ωij } and {Xij } are dual bases, we also have da = ij (Xij ∗ a)ωij for all a ∈ A. Because of P0 + P1 = id and P0 P1 = P1 P0 = 0, the latter leads to the formula

P1 ij kl

da =

N X

1 (Yij1 ∗ a)ωij + s(Y 0 ∗ a)ω 0 ,

a ∈ A.

i,j=1

Now we turn to the quantum groups Oq (N ) and Spq (N ). Then the projections P0 , P+ , P− : inv 0 → inv 0 given by P0 ij kl =

1 t i mj B C δkl , s m P− ij kl =

P+ ij kl =

1 tn −1 (q −1 δki δlj + B t im Rˆ mj + p−1 )P0 ij kl ), nl C k − (q q + q −1

1 tn −1 (qδ i δ j − B t im Rˆ mj − q)P0 ij kl ) nl C k + (p q + q −1 k l

belong to Mor(uc ⊗ u). Let denote P1 := P− , P2 := P+ for Oq (N ) and P1 := P+ , P2 := P− for Spq (N ). Then, by Lemma 1.2, the subspaces Υ 0 , Υ 1 and Υ 2 of inv 0 spanned kn 1 2 by the sets {ω 0 := s1 B t m ωmn }, {ωij := P1 (ωij ) | i, j = 1, . . . , N } and {ωij := k C P2 (ωij ) | i, j = 1, . . . , N } respectively, are 1R -invariant. Moreover, the subspaces Y 0 , P Y 1 and Y 2 of the quantum Lie algebra X generated by the sets {Y 0 := s1 m Xmm }, ij 2 {Yij1 := P1 ij kl Xkl | i, j = 1, . . . , N } and {Yij := P2 kl Xkl | i, j = 1, . . . , N }, respectively, are ad-invariant. Similarly as in the case of SLq (N ), we obtain the following formula for the differentiation:

182


da =

N X

(Yij1 i,j=1

∗

1 a)ωij

+

N X

2 (Yij2 ∗ a)ωij + s(Y 0 ∗ a)ω 0 ,

a ∈ A.

i,j=1

To investigate the classical limit of the structures appearing in this article we keep the basis {ωij | i, j = 1, . . . , N } fixed. Firstly let A = SLq (N ). We always consider the classical limit in the sense that z → 1 and q → 1, where z is the N th root of q 2 . (This is not the only possibility, see [10].) For simplicity we shall write limq→1 for this classical limit. Then, as shown in [10], all functionals Xij , Y 0 and Yij1 have limits when q → 1. It is easily seen that the 1 and the projections P0 and P1 have limits as well for q → 1. We 1-forms ω 0 and ωij * * * * ω 0, * ω 1ij , P 0 and P 1 , respectively. It was proved in [10] denote these limits by Y 0 , Y 1ij , * *1 that the functionals Y ij equipped with the limit of the bracket [·, ·] are generators of the Lie algebra sl(N ). Now let A = Oq (N ) or A = Spq (N ). As proved in [10], for both calculi 0+ and 0− all functionals Xij , Y 0 , Yij1 and Yij2 , i, j = 1, . . . , N admit limits when q → 1. Also, 1 2 , ωij and the projections Pk , k = 0, 1, 2 have limits as q → 1. We the 1-forms ω 0 , ωij *0 * * * shall use the notations Y := limq→1 Y 0 , Y 1ij := limq→1 Yij1 , Y 2ij := limq→1 Yij2 , P k := 1 2 ω 0 := limq→1 ω 0 , * ω 1ij := limq→1 ωij and * ω 2ij := limq→1 ωij for limq→1 Pk , k = 0, 1, 2, * *1 i, j = 1, . . . , N . For the FODC 0 = 0+ the functionals Y ij , i, j = 1, . . . , N equipped with the limit of the bracket [·, ·] span the Lie algebras o(N ) and sp(N ), respectively.

2. Metrics ˜ over a Hopf algebra We begin with some definitions for a general bicovariant FODC 0 A. ˜ → A is called a metric on 0 ˜ if g is ˜ ⊗A 0 Definition 2.1. A bilinear map g : 0 ˜ ˜ nondegenerate (i. e. g(ξ ⊗ ζ) = 0 for all ζ ∈ 0 implies ξ = 0, g(ξ ⊗ ζ) = 0 for all ξ ∈ 0 implies ζ = 0) and if g(aξ ⊗ ζ) = ag(ξ ⊗ ζ)

for any a ∈ A,

˜ ξ and ζ ∈ 0.

(6)

We call a metric g symmetric if g σ˜ = g. For the “ordinary” differential calculus on Lie groups the braiding map σ˜ is just the flip operator, so we obtain the usual notion of a symmetric metric in this case. From ˜ is already condition (6) in the preceding definition it follows easily that a metric g on 0 completely determined by the elements g(ηi ⊗ ηj ), i, j ∈ I, of A. ˜ we have for all ξ, ζ ∈ 0 ˜ and a ∈ A, For any metric g on 0 (id ⊗ εg)(1L (aξ ⊗ ζ)) = a(id ⊗ εg)(1L (ξ ⊗ ζ)), (εg ⊗ id)(1R (aξ ⊗ ζ)) = a(εg ⊗ id)(1R (ξ ⊗ ζ)).

(7)

˜ is called invariant if for all ξ, ζ ∈ 0, ˜ Definition 2.2. A metric g on 0 (id ⊗ εg)(1L (ξ ⊗ ζ)) = g(ξ ⊗ ζ)

and

(εg ⊗ id)(1R (ξ ⊗ ζ)) = g(ξ ⊗ ζ). (8)


183

˜ A 0. ˜ By (7), the above definition is compatible with the left A-module structure of 0⊗ Using the representation theory of quantum groups we now show that the invariant metrics on 0 form a 2-parameter family for SLq (N ) and a 3-parameter family for Oq (N ) and Spq (N ). Lemma 2.3. Let A = SLq (N ) and let 0 be as in Sect. 1. A metric g on 0 is invariant if and only if with complex parameters α and β such that α 6= 0, α + sβ 6= 0, g(ηij ⊗ ηkl ) = q 2j αδil δjk + βδij δkl . Proof . Suppose g is an invariant metric on 0. From the first equation in (8) it follows that g(ξ ⊗ ζ) = εg(ξ ⊗ ζ) for ξ, ζ ∈ inv 0. From (4) and the second equation in (8) we conclude that g ∈ Mor(uc ⊗ u ⊗ uc ⊗ u, 1). Since q is not a root of unity by assumption, the multiplicities of irreducible components in the decomposition of the tensor product representation uc ⊗ u ⊗ uc ⊗ u are the same as in the classical case. Therefore, dim Mor(uc ⊗ u ⊗ uc ⊗ u, 1) = 2. Since q 2j ujn κ(urj ) = q 2n δnr by [8], the transformations T = (Tijkl ) and S = (Sijkl ) with Tijkl := q 2j δil δjk and Sijkl := δij δkl belong to Mor(uc ⊗ u ⊗ uc ⊗ u, 1). Thus we get g(ηij ⊗ ηkl ) = q 2j αδil δjk + βδij δkl for some complex numbers α and β. This map is nondegenerate if and only if α 6= 0 and α + sβ 6= 0. Conversely, it is easily seen that the above formula defines an invariant metric g on 0. Lemma 2.4. Let A = Oq (N ) or A = Spq (N ) and 0 denote one of the FODC from Sect. 1. A metric g on 0 is invariant if and only if with complex parameters α, β and γ such that α + pβ + sγ 6= 0, α − qβ 6= 0, α + q −1 β 6= 0, g(ηij ⊗ ηkl ) = ((αB14 B23 + βB12 B34 Rˆ 23 + γB12 B34 )C t1 C t3 )ijkl . In Lemma 2.4 we used the following notation. Let C = (Cji ) be the matrix of the metric which occurs in the definition of Oq (N ) resp. Spq (N ) (see [8]) and let B = (Bji ) its inverse matrix. Set C t := (C t ij ) = (C ji ) (= (Cij )) and B t := (B t ij ) = (Bji ) (= (Bij )). Then the notation in Lemma 2.4 is the usual leg numbering notation, i. e. the equation therein reads as tm tn g(ηij ⊗ ηkl ) = (αBml Bjn + βBmr Bsl Rˆ rs jn + γBmj Bnl )C i C k .

Proof of Lemma 2.4. The proof is similar to the proof of Lemma 2.3. The decomposition of the tensor product gives now dim Mor(uc ⊗ u ⊗ uc ⊗ u, 1) = 3. The conditions on the coefficients ensure the nondegeneracy of the metric g. Some straightforward computations based on Lemma 1.1 and the particular form of the invariant metrics in Lemma 2.3 and 2.4 prove the following Corollary 2.5. Any invariant metric on 0 is symmetric. A.

˜ over an arbitrary Hopf algebra Let us consider again a general bicovariant FODC 0

˜ and let gij be the matrix coefficients of g, i. e. g(ηi ⊗ηj ) = ˜ A0 Let g be a metric on 0⊗ ˜ Recall that {χi } is the dual basis of {ηi }. gij with respect to a fixed basis {ηi } of inv 0. We suppose that {aj | j ∈ I} is a finite subset of A such that χi (aj ) = δij .

184


P Definition 2.6. A map g ∗ : X˜ ⊗ X˜ → A such that g ∗ (χi ⊗χj ) = g ∗ij and k gik g ∗kj = δ ji for all i, j ∈ I is called the dual metric of g. The metric g ∗ is called ad-invariant if g ∗ ([χ0 , f(1) ] ⊗ [χ00 , f(2) ]) = f (1)g ∗ (χ0 ⊗ χ00 )

for all χ0 , χ00 ∈ X˜ , f ∈ A0 .

(9)

Note that the nondegeneracy of g corresponds to the nondegeneracy of g ∗ . There is an interesting link between invariant and ad-invariant metrics given by ˜ then the dual metric g ∗ of g is adProposition 2.7. If g is an invariant metric on 0 ∗ ˜ ˜ invariant. Conversely, let g : X ⊗ X → C be an ad-invariant metric and suppose that A0 separates the points of A. Then g is an invariant metric. ˜ and X˜ described above. Proof . In the proof we take the bases of inv 0 Let g be invariant. From the first equation in (8) it follows that g(ηi ⊗ ηj ) = gij ∈ C for all i, j ∈ I. Since g is nondegenerate, the dual metric g ∗ is defined and is a map to i j v˜ n = gmn which yields g ∗ij v˜ m ˜ nj = C · 1. The second equation in (8) means that gij v˜ m i v ∗mn ∗ 0 for the dual metric g . Applying functionals f ∈ A to the last equation we get g g ∗ (f(1) (v˜ im )χi ⊗f(2) (v˜ jn )χj ) = f (1)g ∗ (χm ⊗χn ). Using [χi , f ] = f (v˜ ki )χk for all f ∈ A0 the latter is equivalent to g ∗ ([χm , f(1) ] ⊗ [χn , f(2) ]) = f (1)g ∗ (χm ⊗ χn ) for all m, n ∈ I and all f ∈ A0 , i. e. the dual metric is ad-invariant. Suppose that g ∗ : X˜ ⊗ X˜ → C is ad-invariant. Then the matrix elements of g are also complex numbers. This implies the first equation in (8). Reversing the reasoning from ˜ nj − the preceding paragraph, it follows from the ad-invariance of g ∗ that f (g ∗ij v˜ m i v ∗mn 0 0 ) = 0 for all m, n ∈ I and f ∈ A . Since A separates the points of A, we obtain g ˜ nj = g ∗mn for all m, n ∈ I from which the second equation in (8) follows. This g ∗ij v˜ m i v proves the invariance of g. Next we specialize again to the bicovariant FODC 0 = 0±,z over A = SLq (N ), Oq (N ), Spq (N ). We compute the dual metrics g ∗ of the invariant metrics g from Lemmas 2.3 and 2.4 and their restrictions to the ad-invariant subspaces of the quantum Lie algebra X . Let us say that two subspaces Υ and Υ 0 of inv 0 are mutually orthogonal with respect to a metric g on inv 0 if g(x ⊗ x0 ) = g(x0 ⊗ x) = 0 for all x ∈ Υ , x0 ∈ Υ 0 . A similar notion is used for the dual metric g ∗ on X . For SLq (N ) we have (see Lemma 2.3) g(ηij ⊗ ηkl ) = q 2j αδil δjk + βδij δkl and we get β ∗ −2i −1 −2i−2k g (χij ⊗ χkl ) = q α δil δjk − q (10) δij δkl α(α + sβ) for all i, j, k, l = 1, . . . , N . For Oq (N ) and Spq (N ) the dual metric of the metric in Lemma 2.4 is g ∗ (χij ⊗ χkl ) =

B t im B t k n (α−qβ)(α+q −1 β)

αCml Cjn −β Rˆ −1jn rs Cmr Csl +

−Qαβ−αγ+p−1 βγ Cmj Cnl α+pβ+sγ

for i, j, k, l = 1, . . . , N . From Eqs. (2) and (1) we get the transformation formula between the bases {Xij } and {χij }. Then we can express the generators of the ad-invariant subspaces in terms of the basis {χij }. For the quantum group SLq (N ) and the FODC 0 = 0±,z we obtain P ij −1 1 groups the formulas Y 0 = s−1 µ−1 ±,z k χkk and Yij = ν±,z P1 kl χkl . For the quantum P Oq (N ) and Spq (N ) and the calculus 0 = 0±1 we get Y 0 = s−1 µ0± −1 m χmm ,


185

−1 2 Yij1 = µ1± −1 P1 ij P2 ij kl χkl and Yij = µ2± kl χkl . Here the constants are defined by −1 −1 −1 µ+,z = s(z − 1) + z q Q, µ−,z = s(z − 1) − zq −2N −1 Q, ν+,z = z −1 q −1 Q, ν−,z = −zq −2N −1 Q, µ0+ = (p − p−1 )Q, µ0− = (p − p−1 )Q + 2s, µ1± = ±p(1 + q −N )Q and µ2± = ±(p − p−1 q N )Q. One easily verifies that Y 0 and Y 1 are orthogonal subspaces of X with respect to the dual metric g ∗ . The restrictions of g ∗ to Y 0 ⊗ Y 0 and Y 1 ⊗ Y 1 are given by 1 1 q −2i−2k ∗ 1 1 −2i δ g ∗ (Y 0 ⊗Y 0 ) = q , g (Y ⊗Y ) = δ δ − δ il jk ij kl ij kl 2 s s(α + sβ)µ2±,z αν±,z

for all i, j, k, l = 1, . . . , N . Then Υ 0 and Υ 1 are orthogonal subspaces of inv 0 with respect to all invariant metrics g. If g has the form as in Lemma 2.3, then we have (α + sβ)µ2±,z 1 1 1 2 , g(ωij q 2j δil δjk − δij δkl ⊗ ωkl ) = αν±,z g(ω 0 ⊗ ω 0 ) = s s for all i, j, k, l = 1, . . . , N . The ad-invariant subspaces Y 0 , Y 1 and Y 2 of the quantum Lie algebras X of Oq (N ) and Spq (N ) are mutually orthogonal with respect to g ∗ . The restrictions of g ∗ to Y i ⊗Y i , i = 0, 1, 2 are described by the formulas g ∗ (Y 0 ⊗ Y 0 ) =

1 µ0 ± 2 sα0 ,

g ∗ (Yij1 ⊗ Ykl1 ) =

B t im B t k n µ1± 2 (p−1 q N+pq −N )α1

mr sl p p−1 q N C ml C jn − Rˆ −1jn C + rs C

g ∗ (Yij2 ⊗ Ykl2 ) =

−1

(1−q N ) mj nl C C s

B t im B t k n µ2 ± 2 (p−1 q N+pq −N )α2

mr sl C −p pq −N C ml C jn − Rˆ −1jn rs C

−1

+pq −N s

,

C mj C nl ,

where we use the abbreviations α0 := α + pβ + sγ, α1 := α − p−1 q N β, α2 := α + pq −N β. Note that all denominators in the above formulas are non-zero, since the metric is nondegenerate and 0 is a FODC as in [18]. The corresponding subspaces Υ 0 , Υ 1 and Υ 2 of inv 0 are mutually orthogonal with respect to all invariant metrics g. If g is as in Lemma 2.4, then we have g(ω 0 ⊗ ω 0 ) = 1 1 g(ωij ⊗ ωkl )=

µ0 ± 2 α0 , s µ1± 2 α1 p−1 q N +pq −N

pq −N Bml Bjn − Bmr Bsl Rˆ rs jn +

2 2 ⊗ ωkl )= g(ωij

p(1−q −N ) Bmj Bnl s

µ2 ± 2 α2 p−1 q N +pq −N

p−1 q N Bml Bjn + Bmr Bsl Rˆ rs jn −

for all i, j, k, l = 1, . . . , N . Now we want to examine the classical limits.

p+p−1 q N s

tn Ctm i C k,

tn Bmj Bnl C t m i C k

186


Let A = SLq (N ) and let g be an invariant metric as described in Lemma 2.3. The complex numbers α and β may of course depend on the parameters q and z. Let us assume that the functions α = α(q, z) and β = β(q, z) are chosen such that the limits c1 := limq→1 Q2 α(q, z) and c0 := limq→1 Q4 (α(q, z) + sβ(q, z)) exist and are non-zero. Then it follows immediately from the existence of the classical limits as discussed at the end of Sect. 1 that the invariant metric g and its dual metric admit limits when * q → 1 and z → 1. The restriction of the limit of g ∗ to the linear functionals Y 1ij , i, j = 1, . . . , N , is just a complex multiple of the Killing form for sl(N ). Similar results are valid for the quantum groups Oq (N ) and Spq (N ) and for the FODC 0+ if we suppose that for the functionals αk = αk (q), k = 0, 1, 2 the limits c0 := limq→1 µ0+ 2 α0 (q), c1 := limq→1 µ1+ 2 α1 (q) and c2 := limq→1 µ2+ 2 α2 (q) exist and are non-vanishing. 3. Connections We begin with some general definitions (cf. [4]) which, of course, apply to any differential L ˜ n over an arbitrary algebra A. ˜∧ = ∞ 0 calculus 0 n=0 ˜ ⊗A E Let E be a left A-module. A left connection on E is a linear map ∇ : E → 0 such that ∇(aζ) = da ⊗ ζ + a∇(ζ) for all a ∈ A and ζ ∈ E. (11) ˜ := 0 ˜ ∧ ⊗A E be the “E-valued differential forms”. A connection ∇ on E admits Let 0E ˜ → 0E ˜ of degree one such that a unique extension to a linear map ∇ : 0E ∇(αζ) = (dα)ζ + (−1)n α∇(ζ)

˜ n and ζ ∈ 0E. ˜ for α ∈ 0

2

˜ ⊗A E is called the curvature of the connection ∇. The mapping R(∇) = ∇2 : E → 0 Clearly, R(∇) is A-linear, i. e. R(∇)(aζ) = aR(∇)(ζ) for a ∈ A and ζ ∈ E. Similar concepts can be defined for a right A-module E. A right connection on E ˜ satisfying ∇(ζa) = ζ ⊗ da + ∇(ζ)a for a ∈ A is then a linear map ∇ : E → E ⊗A 0 ˜ ∧ → E ⊗A 0 ˜ ∧ such and ζ ∈ E. It extends uniquely to a linear map ∇ : E ⊗A 0 ˜ ∧ and ζ ∈ E ⊗A 0 ˜ n . The curvature of ∇ is that ∇(ζα) = (−1)n ζdα + ∇(ζ)α, α ∈ 0 2 ˜ . R(∇) := ∇2 : E → E ⊗A 0 If ∇ is a connection on a left A-module E, then there is a connection ∇∗ on the right A-module E ∗ , called the dual connection of ∇, defined by < ξ, ∇∗ (ζ) >= d < ξ, ζ > − < ∇(ξ), ζ >

for ξ ∈ E,

ζ ∈ E ∗.

By a right (resp. left) connection on an A-bimodule E we mean a connection on E when E is considered as a right (resp. left) A-module. ˜ is considered as a We now specialize to the case of our main interest where E = 0 ˜ ˜ Then the ˜ ˜ left A-module. Suppose that ∇ : 0 → 0 ⊗A 0 is a (left) connection on 0. 2 ˜ ⊗A 0 ˜ →0 ˜ denotes the torsion T (∇) is defined by T (∇) = d − m∇, where m : 0 ˜ The torsion T (∇) is A-linear, multiplication map, i. e. m(ξ ⊗ ζ) = ξ ∧ ζ for ξ, ζ ∈ 0. since T (∇)(aξ) = d(aξ) − m∇(aξ) = da ∧ ξ + adξ − da ∧ ξ − am∇(ξ) = aT (∇)(ξ) for ˜ by the Leibniz rule and (11). a ∈ A, ξ ∈ 0 Definition 3.1. The connection ∇ is called bicovariant if 1L ∇ = (id ⊗ ∇)1L

and

1R ∇ = (∇ ⊗ id)1R .

(12)


187

˜ forms A simple computation shows that the set of all bicovariant connections on 0 ˜ a complex affine space BC(0). L∞ n ˜ ˜∧ = Let 0 n=0 0 be a bicovariant differential calculus over a Hopf algebra A. ˜ 1R (ηi ) = ηj ⊗ v˜ ij , and let D(∇)jk Let {ηi | i ∈ I} be a basis in inv 0, i denote arbitrary ˜ →0 ˜ ⊗A 0, ˜ defined by ∇(ηi ) = D(∇)jk elements of A. Then the map ∇ : inv 0 i ηj ⊗ η k ˜ and any connection on 0 ˜ is of this form. extends uniquely to a connection on 0 ˜ The Lemma 3.2. Let v˜ denote the corepresentation of A defined by 1R on inv 0. jk ˜ connection ∇ on 0 is bicovariant if and only if D(∇)i ∈ C for all i, j, k and D(∇) = (D(∇)jk ˜ v˜ ⊗ v). ˜ i ) ∈ Mor(v, jk ˜ and ∇(ηi ) = D(∇)jk Proof . Let ∇ be a bicovariant connection on 0 i ηj ⊗ ηk , D(∇)i ∈ jk jk A. Because of the first formula of (12) we have 1(D(∇)i ) = 1 ⊗ D(∇)i and so D(∇)jk i ∈ C · 1. The second equation of (12) tells us that (∇ ⊗ id)1R (ηi ) = ηj ⊗ ηk ⊗ xy D(∇)jk ˜ xi = 1R ∇(ηi ) = ηj ⊗ ηk ⊗ (v˜ ⊗ v) ˜ jk ˜ = (v˜ ⊗ v)D(∇). ˜ x v xy D(∇)i , i. e. D(∇)v Let now D(∇)jk ∈ C and D(∇) ∈ Mor( v, ˜ v ˜ ⊗ v). ˜ Then the above equations i written in reversed order show that (12) is true for the connection ∇ defined by ∇(ηi ) = D(∇)jk i ηj ⊗ ηk .

˜ cf. Definition 2.2. Let us introduce some new Now let g be an invariant metric on 0, concepts. ˜ is compatible with the metric g if Definition 3.3. We say that the connection ∇ on 0 (id⊗g)(∇(ξ)⊗ζ)+(g ⊗id)(id⊗ σ)(ξ ˜ ⊗∇(ζ))−dg(ξ ⊗ζ) = 0

˜ ˜ ζ ∈ inv 0. for any ξ ∈ 0,

˜ is compatible Lemma 3.4. Let g denote an invariant metric. Then a connection ∇ on 0 with g if and only if ˜ (id ⊗ g)(∇(ηi ) ⊗ ηj ) + (g ⊗ id)(ηi ⊗ σ∇(η j )) = 0 for all i, j ∈ I.

(13)

Proof . In both directions we use only that the metric g is left-invariant, i. e. the first equation in (8) is fulfilled. This is equivalent to gij = g(ηi ⊗ ηj ) ∈ C. Suppose ∇ is compatible with g. Because of the first formula in (8) and Definition 3.3 equation (13) is valid. Let us now assume equation (13) is fulfilled and introduce ˜ ⊗A 0, ˜ aij ∈ A. Then the assertion follows arbitrary elements ξ ⊗ ζ = aij ηi ⊗ ηj ∈ 0 from the computation ˜ (id ⊗ g)(∇(aij ηi ) ⊗ ηj ) + (g ⊗ id)(aij ηi ⊗ σ∇(η j )) − dg(aij ηi ⊗ ηj ) = ˜ j) − = (id ⊗ g)((daij ⊗ ηi + aij ∇ηi ) ⊗ ηj ) + aij (g ⊗ id)(ηi ⊗ σ∇η −(d(aij )g(ηi ⊗ ηj ) + aij dg(ηi ⊗ ηj )) = ˜ j )) − aij dg(ηi ⊗ ηj ) = aij ((id ⊗ g)(∇ηi ⊗ ηj ) + (g ⊗ id)(ηi ⊗ σ∇η and the assumption gij ∈ C.

˜ is called Definition 3.5. Let g be an invariant metric. A bicovariant connection ∇ on 0 a Levi-Civita connection (with respect to the metric g) if ∇ is compatible with g and has vanishing torsion T (∇).

188


˜ is a free left A-module basis of 0. ˜ Recall that any basis of the vector space inv 0 ˜ its curvature and its torsion are uniquely determined Therefore, any connection on 0, by its values on such a basis. ˜ and X˜ ⊗A as a right A-module (see formula Now we consider the pairing between 0 ˜ (5)). Since for a connection ∇ on 0 there is a dual connection ∇∗ on X˜ ⊗ A, it is natural to ask what the conditions in Definition 3.5 mean in terms of ∇∗ . For this we assume ˜ ⊗A 0)/ ˜ ker(σ˜ − id). ˜ 2 = (0 that 0 ˜ and let ∇∗ be its dual connection. For arbitrary Suppose that ∇ is a connection on 0 ∗ ∗ ˜ ˜ and X˜ ξ, ζ ∈ X we define ∇ξ (ζ) =< ∇ (ζ), ξ > and mean the pairing between X˜ ⊗ 0 jk ˜ we write ∇(ηi ) = 0jk (cf. (5)). For the basis elements ηi of inv 0 i ηj ⊗ ηk with 0i ∈ A. jk Recall that by Lemma 3.2, we have 0i ∈ C for all i, j, k ∈ I. Then the pairing gives ∇∗χi (χj ) = −0ij k χk for the elements of the dual basis. Let η denote the left- and right˜ used in the definition of the differentiation, i. e. da = ηa − aη for invariant element of 0 a ∈ A. Recall that the torsion is A-linear. Therefore, because of dηi = η ∧ ηi + ηi ∧ η, the torsion of ∇ is vanishing if and only if dηi = m∇(ηi ) or equivalently (id − σ)(η ˜ ⊗ ηi + ηi ⊗ η − ∇(ηi )) = 0

for all i ∈ I.

Using the notation above the latter is equivalent to the equation ij ∗ ∇∗χi (χj ) − σ˜ kl ∇χk (χl ) = [χi , χj ] for all i, j ∈ I.

Suppose that g is an invariant metric. By dualizing the condition in Lemma 3.4, it follows that ∇ is compatible with the metric g if and only if ∗ g ∗ (χi ⊗ ∇∗χj (χk )) + g ∗ (σ˜ ij mn ∇χm (χn ) ⊗ χk ) = 0 for all i, j, k ∈ I.

The above equations show that our concepts are analogous to the corresponding notions in classical differential geometry. 4. Levi-Civita Connections on SLq (N ) In this section we examine the differential calculi 0±,z for A = SLq (N ). After a short lemma we will prove our main results. Lemma 4.1. We have dim BC(0) = 5 for N = 2 and dim BC(0) = 6 for N ≥ 3. Proof . Since q is not a root of unity, decompositions of tensor product representations of A = SLq (N ) can be labelled by Young tableaus as in the classical case. Therefore, we obtain uc ⊗ u = [0] ⊕ [2, 1N −2 ] and uc ⊗ u ⊗ uc ⊗ u = 2[0] ⊕ k[2, 1N −2 ]⊕ other terms with k = 3 for N = 2 and k = 4 for N ≥ 3. Then by the general representation theory we conclude that dim Mor(v, v ⊗v) = 5 for N = 2 and dim Mor(v, v ⊗v) = 6 for N ≥ 3. By Lemma 3.2, a connection ∇ on 0±,z is bicovariant if and only if D(∇) ∈ Mor(v, v ⊗ v), where v = uc ⊗ u. Thus the assertion of the lemma follows. L ∞ Let 0∧ = n=0 0n be a differential calculus over A which contains 0±,z as its first order differential calculus 0. As in [20] we suppose that 02 = (0 ⊗A 0)/ ker(σ − id). Then we have the following


189

Theorem 4.2. For any invariant metric g on 0 there exists precisely one Levi-Civita connection ∇. If g(ηij ⊗ ηkl ) = q 2j αδil δjk + βδij δkl with α, β ∈ C, α 6= 0, α + sβ 6= 0 then for 0+,z this Levi-Civita connection is given by Q 2p−2a−2c ˆ bn ˆ dp ˆ im Ram Rcn Rjp ηab ⊗ ηcd − q −1 ηia ⊗ ηaj − (q 2 −Qηij ⊗ η − Qη ⊗ ηij + δij Q(1 + sα−1 β)q −2a ηab ⊗ ηba −

∇(ηij ) =

−δij (q 2N +1 Q2 + Qα−1 β)η ⊗ η) and for 0−,z by Q −2N −1 ˆ dp ˆ im (−q 2p−2a−2c Rˆ bn ηia ⊗ ηaj + am Rcn Rjp ηab ⊗ ηcd + q 2 +δij (2Q + Qsα−1 β)q −2a ηab ⊗ ηba + δij (q 2N +1 Q2 − Qα−1 β)η ⊗ η).

∇(ηij ) =

Proof . By Lemma 2.3, any invariant metric g is of the form g(ηij ⊗ ηkl ) = q 2j αδil δjk + βδij δkl with α, β ∈ C, α 6= 0, α + sβ 6= 0. From the proof of Lemma 4.1 we know that a connection ∇ on 0 is bicovariant if and only if there are complex numbers λ1 , . . . , λ6 such that 6 X λn (An )abcd (14) ∇(ηij ) = ij ηab ⊗ ηcd , n=1

where {A1 , . . . , A6 } generates the vector space Mor(v, v ⊗ v). For our calculi we have v = uc ⊗ u. By explicit decompositions of the tensor product representations uc ⊗ u and uc ⊗ u ⊗ uc ⊗ u it can be shown that the following 6 morphisms A1 , . . . , A6 c c c (Ak = (Ak abcd ij )) span the vector space Mor(u ⊗ u, u ⊗ u ⊗ u ⊗ u): A1 = q −2a−2c δij δ ab δ cd , A2 = q −2a δij δ ad δ bc , A3 = δ ai δ bj q −2c δ cd , ˆ dp ˆ im A5 = δ ai δ bc δ dj , A6 = q 2n−2a−2d Rˆ bn A4 = q −2a δab δ ci δ dj , am Rcn Rjp . For our differential calculus we have dηij = η ∧ ηij + ηij ∧ η. Since 02 = (0 ⊗A 0)/ ker(σ − id), the torsion of ∇ vanishes if and only if we have (σ − id)(η ⊗ ηij + ηij ⊗ η −

6 X

λk Ak mnrs ηmn ⊗ ηrs ) = 0 ij

k=1

in the tensor product 0 ⊗A 0. Comparing the coefficients of basis elements the latter is equivalent to the equations λ 3 = λ4 = 1 +

λ5 q −1 Q

λ 3 = λ4 = 1 −

−

(Q2 + 1)λ6 Q

λ6 λ5 + q −2N −1 Q Q

for 0+,z , for 0−,z .

(15) (16)

Using Lemma 3.4 it follows that ∇ satisfies the compatibility condition with the metric g if and only if the following equations are fulfilled: λ4 + Qλ6 = 0,

λ5 + q −1 λ6 = 0,

(α + sβ)λ1 + βλ2 + βλ4 + q

2N

αλ2 + (α + sβ)λ3 = 0,

Q(β + qQα)λ6 = 0

(17)

190


for 0+,z and λ4 = 0,

λ5 + q −2N −1 λ6 = 0,

(α + sβ)λ1 + βλ2 + (q 2N +1 Q2 α + q 2N Qβ)λ6 = 0, αλ2 + (α + sβ)λ3 + (−q

2N +1

Qα + β)λ5 + (Qα + q

−1

(18)

β)λ6 = 0

for 0−,z . Some straightforward computations show that Eqs. (15) and (17) resp. (16) and (18) have unique solutions λi for 0+,z resp. 0−,z . Inserting these solutions into (14) we obtain the formulas given in the theorem. It is not difficult to check that the connection defined by the above formulas fulfills all conditions for a Levi-Civita connection. Let us look closer at the Levi-Civita connection on 0 = 0+,z . Using Theorem 4.2 for 2 −2a ηab ⊗ the left and right invariant element η ∈ inv 0 we compute ∇(η) = Q 2α (α + sβ)(sq ηba − η ⊗ η). Transforming the basis we obtain ∇(ω 0 ) =

z 2 q 2 µ+,z (α + sβ) −2a 1 µ+,z 1 ∇(η) = q ωab ⊗ ωba . s 2α

(19)

ˆ −1dp ˆ im −1 δ ai δ bc δ dj −Qδ ab δic δ dj + Moreover, ∇(ηij −δij s−1 η) = Q2 (q 2n−2a−2d Rˆ bn am R cn Rjp −q Qs−1 q −2a−2c δij δ ab δ cd )ηab ⊗ ηcd . Transforming the basis once again we get 1 ∇(ωij ) = ν+,z ∇(ηij − δij s−1 η) = 1 1 ˆ −1dp ˆ im 1 = (zqq 2n−2a−2d Rˆ bn am R cn Rjp ωab ⊗ ωcd − 2 1 1 1 0 0 1 ⊗ ωaj − Q2 µ−1 −zωia +,z (ωij ⊗ ω + ω ⊗ ωij )).

(20)

What happens with the Levi-Civita connection in the classical limit? As explained in Sect. 1, we consider the classical limit in the sense that z → 1 and q → 1. Retaining the notation introduced in Sects. 1 and 2, formulas (19) and (20) show that the limit of the Levi-Civita connection exists and takes the form *0 ) = lim ∇(ω 0 ) = ∇cl (ω q→1

*1 ∇cl (ω ij ) =

1 2

(N 2 − 1)c0 *1 ω ab ⊗ *1 ω ba , 4N c1

*1 ω aj

⊗ *1 ω ia − *1 ω ia ⊗ *1 ω aj −

(21)

2N *1 *0 *0 *1 (ω ⊗ ω + ω ⊗ ω ) . (22) ij N 2 − 1 ij

5. Levi-Civita connections on Oq (N ) and Spq (N ) Now we turn to the differential calculi 0+ and 0− on the quantum groups Oq (N ) and Spq (N ). We begin with Lemma 5.1. We have dim BC(0± ) = 14 for Spq (4) and dim BC(0± ) = 15 for Oq (N ), N ≥ 3 and Spq (N ), N > 4.


191

Proof . The proof is similar to the proof of Lemma 4.1. In the present cases A = Oq (N ) and A = Spq (N ) we obtain the decompositions of the tensor product representations uc ⊗ u = [0] ⊕ [2] ⊕ [1, 1], uc ⊗ u ⊗ uc ⊗ u = 3[0] ⊕ 6[2] ⊕ k[1, 1]⊕ other terms, where k = 5 for Spq (4) and k = 6 for Oq (N ), N ≥ 3 and Spq (N ), N > 4. L ∞ n 1 Let 0∧ = n=0 0 be a differential calculus over A such that 0 = 0 = 0± . In 2 contrast to the SLq (N ) case we assume that 0 = (0 ⊗A 0)/K, where K = ker(σ − id) ⊕ ker(σ − q N id) ⊕ ker(σ − q −N id) (see the last remarks in Sect. 1). Such an assumption for the higher order calculus has already been used in [3]. In case of the “ordinary” classical differential calculus the 2-forms are the quotient of the tensor product of 1-forms by the eigenspace of the flip operator with eigenvalue 1. Since the eigenvalues q N and q −N of the braiding map σ tend to 1 when q → 1, the assumption 02 = (0 ⊗A 0)/K means that the higher order calculus 0∧ is some sense nearer to the corresponding construction in the classical case. Moreover, this assumption is essential in order to prove the following Theorem 5.2. Suppose that g is an invariant metric. There is exactly one Levi-Civita connection on 0 with respect to g. Proof . The proof is similar to that of Theorem 4.2. In Lemma 2.4 we proved that all invariant metrics have the form g(ηij ⊗ ηkl ) = (αB14 B23 C t1 C t3 + βB12 B34 Rˆ 23 C t1 C t3 + γB12 B34 C t1 C t3 )ijkl with α + pβ + sγ 6= 0, α 6= qβ, α 6= −q −1 β. By Lemma 5.1, the dimension of the vector space Mor(uc ⊗ u, uc ⊗ u ⊗ uc ⊗ u) is at most 15. A closer investigation of the proof of Lemma 5.1 shows that the following 15 morphisms ) generate the vector space. (In case N ≥ 5 they form a basis of this Ak = (Ak mnrs ij space.) A1 = B t1 B t3 C12 C34 B12 C t1 , A3 = B t1 B t3 C23 C14 B12 C t1 , A5 = B t1 B t3 Rˆ 12 C34 C t1 , A7 = B t1 B t3 C12 Rˆ 34 C t1 , A9 = B t1 B t3 C23 Rˆ 14 C t1 , −1 t A11 = B t1 B t3 Rˆ 12 C23 Rˆ 14 C 1, t t ˆ −1 ˆ A13 = B 1 B 3 R23 C34 R12 C t1 , −1 t A15 = B t1 B t3 Rˆ 12 Rˆ 23 C34 Rˆ 12 C 1.

−1 A2 = B t1 B t3 Rˆ 23 C12 C34 B12 C t1 , t A4 = B 3 C34 , A6 = B t1 C12 , A8 = B t3 C23 , A10 = B t1 B t3 Rˆ 12 C23 C t1 , −1 A12 = B t3 Rˆ 23 C34 , t t ˆ A14 = B 1 B 3 R12 Rˆ 23 C34 C t1 ,

Therefore, by Lemma 3.2, we can make the following ansatz for our Levi-Civita connection ∇: 15 X λk Ak mnrs ηmn ⊗ ηrs , λk ∈ C. ∇(ηij ) = ij k=1

The condition for the vanishing torsion takes the form (σ − id)(σ − q N id)(σ − q −N id)(η ⊗ ηij + ηij ⊗ η −

15 X

λk Ak mnrs ηmn ⊗ ηrs ) = 0. ij

k=1

This leads to the equations λ4 = λ6 − Qλ15 ,

λ5 = λ7 − Qλ14 ,

λ8 = pQλ6 + pλ15 − pQ,

λ9 = pQλ7 + pλ14 ,

192


λ11 − q −N pλ10 = Q(q −N pλ6 + q −N (−q −N p2 + 1)λ7 + λ15 −q −N p − λ13 = −Qλ7 + q N p−1 λ12 − q N Qλ15 +

1 p2 +q N

p2

(pλ6 + λ7 + pλ14 + p2 λ15 − p)),

qN (pQλ6 + Qλ7 + pQλ14 + p2 Qλ15 − pQ). + qN

By Lemma 3.4, the connection is compatible with the metric if and only if the following equations are satisfied: λ6 β + λ7 α = 0, λ6 (α − Qβ) + λ7 β = 0, λ1 (α + pβ + sγ) + λ2 p−1 γ + λ3 (Qβ + γ) = 0, λ2 α + λ3 β + λ5 (α + pβ + sγ) + λ9 (Qβ + γ) + λ10 γ + λ13 p−1 γ + λ14 p(Qβ + γ) = 0, λ2 β + λ3 (α − Qβ) + λ4 (α + pβ + sγ) + λ8 (Qβ + γ) +λ11 γ + λ12 p−1 γ + λ15 p(Qβ + γ) = 0, λ8 (α − Qβ) + λ11 pβ + λ12 β + λ15 p(α − Qβ) = 0, λ8 β + λ10 pβ + λ12 α + λ14 p(α − Qβ) = 0, λ9 (α − Qβ) + λ11 pα + λ13 β + λ15 pβ = 0, λ9 β + λ10 pα + λ13 α + λ14 pβ = 0. Set p˜ = (q N + 1)−1 α1−1 (p − p−3 q 3N ). Some computations show that the above system of equations admits a unique solution 2λ4 = −Q2 (1 + pβ), ˜ 2λ5 = −Q2 pα, ˜ λ6 = λ7 = 0, ˜ 2λ9 = pQpα, ˜ 2λ10 = Q(1 − pβ), ˜ 2λ8 = pQ(−1 + pβ), 2λ11 = −Qp(α ˜ − Qβ), 2λ12 = −pQp(α ˜ − Qβ), 2λ13 = −pQ(1 + pβ), ˜ 2λ14 = Qpα, ˜ 2λ15 = Q(1 + pβ), ˜ λ3 =

(23)

Q2 α 0 α Q2 α 0 β Q 2 pα ˜ 0 − Qpp(Qβ + Qpγ, ˜ λ2 = − + ˜ + γ), 2α1 α2 2α1 α2 2 Q2 −Qαβ − αγ + p−1 βγ − p−1 pγ ˜ . λ1 = 2 α1 α2

From the preceding considerations it is clear that the corresponding connection ∇ is indeed a Levi-Civita connection for g. In order to examine the classical limit, we have to rewrite the Levi-Civita connection from Theorem 5.2 in terms of the standard basis. Using the projections P0 , P1 and P2 defined in Sect. 2 some straightforward computations yield the formulas Qα0 −N (pq + p−1 q N )(p2 q −N − p−2 q N )B t am C md P1 ηab ⊗ P1 ηbd + 2α1 Qα0 Qs N N −2 + + p(1 ˜ − q )(1 − q p ) B t am C md P2 ηab ⊗ P2 ηbd , 2 α2

∇(η) =

∇(P1 ηij ) =

Q(pq −N + p−1 q N ) 2

p−2 q N − p2 q −N (η ⊗ P1 ηij + P1 ηij ⊗ η) − s

−2 2N −(p2 q −N + 1)P1 kl − p2 q −N )P1 kl ij P1 ηkm ⊗ P1 ηml + (p q ij P2 ηkm ⊗ P1 ηml − −N −1 N kl ˜ 1 (p q − p)P1 kl −ppα ˜ 1 (1 + q )P1 ij P1 ηkm ⊗ P2 ηml + pα ij P2 ηkm ⊗ P2 ηml ,


193

Q(pq −N +p−1 q N ) q −N −q N η ⊗ P2 ηij ∇(P2 ηij ) = 2 s (q N −1)(1−p−2 q N )pα ˜ 2 −Qs P2 ηij ⊗ η − −N −1 N (pq +p q )s N kl −(q N + p2 q −2N )P2 kl ij P1 ηkm ⊗ P2 ηml + (1 − q )P2 ij P2 ηkm ⊗ P2 ηml + +

+ppα ˜ 2 (1 + q −N )P2 kl ij P1 ηkm ⊗ P1 ηml + −1 N kl pα ˜ 2 (p − p q )P2 ij P2 ηkm ⊗ P1 ηml . From these formulas it follows that the classical limits of the Levi-Civita connections ∇ for both calculi 0+ and 0− exist. For the subspaces Υ 1 of 1-forms of 0+ which corresponds to the classical differential calculus (see [10]) we obtain N − 2 *0 *1 *1 * *1 *1 *1 1 ∇cl (ω (ω ⊗ ω ij + ω ij ⊗ *0 ω ) − P 1 kl ij ) = lim ∇(ωij ) = − ij ω km ⊗ ω ml − q→1 N − N − 4 * kl *1 P 1 (ω ⊗ *2 ω ml + *2 ω km ⊗ *1 ω ml + *2 ω km ⊗ *2 ω ml ). − N − 2 ij km A. The Rosso Form of Uq (sl(N )) In Sect. 2 we defined invariant metrics and we have seen that such metrics are not uniquely determined. On the other hand, Rosso showed in [17] that there is a unique ad-invariant bilinear form for the quantum universal enveloping algebra Uq (g) for a simple Lie algebra g. In this appendix we define such a form adapted to the preceding considerations and we compute the corresponding ad-invariant metric on the quantum Lie algebra X of the FODC 0+,z . Let q be a complex number, q 6= 0, q k 6= 1 for all k ∈ N and let (aij ) be the Cartanmatrix for sl(N ). Let (Uq (sl(N )), 1, κ, ε) be the Hopf algebra over C generated by the ˜ i, K ˜ N, K ˜ −1 , K ˜ −1 | i = 1, . . . , N −1} with relations set of elements {Ei , Fi , Ki , Ki−1 , K i N K i K j = Kj K i , ñ = K ˜ m, ˜ nK ˜ mK K

Ki Ki−1 = Ki−1 Ki = 1, ˜ n−1 = K ˜ n = 1, ˜ nK ˜ n−1 K K

Ki Ej = q aij Ej Ki , ˜ n, ˜ n Ej = q 2(δn,j+1 −δnj ) Ej K K

X

˜ i+1 = Ki2 , ˜ −1 K K i

Ki Fj = q −aij Fj Ki , ˜ n, ˜ n Fj = q 2(δnj −δn,j+1 ) Fj K K

Ei Fj − q −aij Fj Ei = δij 1−aij

ñ = K ˜ n Ki , Ki K

Ki2 − 1 , q2 − 1

1−aij −k

(−1)k b−aij ,k Eik Ej Ei

(24)

=0

(i 6= j),

=0

(i 6= j)

k=0

X

1−aij

k=0

1−aij −k

(−1)k b−aij ,k Fik Fj Fi

194


and coproduct 1, antipode κ, counit ε defined by 1(Ei ) = Ei ⊗ Ki + 1 ⊗ Ei , 1(Ki ) = Ki ⊗ Ki , κ(Ei ) = −q

2

Ki−1 Ei ,

1(Fi ) = Fi ⊗ Ki + 1 ⊗ Fi , ˜ n) = K ñ ⊗K ˜ n, 1(K

κ(Fi ) = −Fi Ki−1 ,

ε(Ei ) = ε(Fi ) = 0,

κ(Ki ) = Ki−1 ,

˜ n) = K ˜ n−1 , κ(K

(25)

˜ n ) = 1, ε(Ki ) = ε(K

for all i, j = 1, . . . , N − 1 and all m, n = 1, . . . , N . The constants in (24) are b0,0 = b0,1 = b1,0 = b1,2 = 1, b1,1 = q + q −1 . A realization of the Hopf algebra Uq (sl(N )) in terms of the L-functionals is obtained by setting Ei = Q−1 l−ii l+ii+1 ,

+i+1 Fi = −q −1 Q−1 l−i+1 i l i+1 ,

Ki = l−ii l+i+1 i+1 ,

˜ i = (l+ii )2 . (26) K

This Hopf algebra is ZN −1 -graduated with grading ∂ given by ∂Ei := αi , ∂Fi = ˜ i = 0. Let < ·, · > denote the symmetric bilinear form such that −αi , ∂Ki = 0, ∂ K < αi , αj >= aij . It is well-known that all elements of Uq (sl(N )) can be written as finite linear combinations of terms of the form F KE, where F and E are finite products of elements Fi ˜ i and K ˜ −1 . and Ei , respectively, and K is a product of the generators Ki , Ki−1 , K i Let us recall that a map (·, ·) : B × B → C for a Hopf algebra B is called ad-invariant if (27) (adR ξ(1) (ζ1 ), adR ξ(2) (ζ2 )) = ε(ξ)(ζ1 , ζ2 ) for ξ, ζ1 , ζ2 ∈ B. Then one can prove the following proposition which is essentially Rosso’s result adapted to the present setting. Note that in [17] the algebra Uq (sl(N )) is different from ours and the left adjoint action is used. Proposition A.1. There is a unique ad-invariant bilinear map (·, ·) : Uq (sl(N )) × Uq (sl(N )) → C such that (F KE, F 0 K 0 E 0 ) = (F, E 0 )(K, K 0 )(E, F 0 ), (K, K 0 ) = ˜ i , Kj ) = q δij −δi,j+1 (K 0 , K), (KK 0 , K 00 ) = (K, K 00 )(K 0 , K 00 ), (Ki , Kj ) = q −aij /2 , (K −2δij ˜ ˜ and (Ki , Kj ) = q . Proof . The proof is similar to that of Theorem 6 in [17]. We omit the details. Howeover, we want to stress that we deal with another adjoint action and with different commutation relations of the generators of Uq (sl(N )). In what follows we use the abbreviations Ei,i+1 := Ei , Fi+1,i := Fi , Ei,j+1 := Ei+1,j+1 Ei − q −1 Ei Ei+1,j+1 , Fj+1,i := Fi Fj+1,i+1 − q −1 Fj+1,i+1 Fi for i < j. Using Eq. (27) and Proposition A.1 the bilinear form (·, ·) for these elements of Uq (sl(N )) can be computed. The result is given by the formulas (Ei , Fj ) = −q −1 Q−1 δij , (Eij , Fkl ) = −q 2i−2j+1 Q−1 δil δjk , (Fi , Ej ) = −qQ−1 δij , (Fij , Ekl ) = −qQ−1 δil δjk . From Eqs. (2) and (26) the generators χij of the quantum Lie algebra X of the FODC ˜ i: 0+,z can be expressed in terms of the elements Eij , Fji , Ki and K ˜ i + q −1 Q2 χij = q −1 QFji K

X r

˜ r Eri , Fjr K

(i < j),


˜ j Eji + q −1 Q2 χij = q −1 QK

X

˜ r Eri , Fjr K

195

(i > j),

r<j

˜ i − q −1 Q χii = q −2 K

i−1 X

˜ m + q −2 Q2 q 2m−2i K

− q −1 Q3

˜ n Eni − Fin K

n

m=1 i−1 m−1 X X

X

˜ n Enm − q −2i ε. q 2m−2i Fmn K

m=1 n=1

Combining both sets of the preceding formulas it follows that the bilinear form in Proposition A.1 takes the following form on the generators χij of the quantum Lie algebra: (28) (χij , χkl ) = −q −2i−1 Qδil δjk + q −2i−2j Q2 δij δkl . This is precisely the dual metric g ∗ described by formula (10) for the parameter values α = −qQ−1 , β = −q 2N +2 (cf. (10)). B. Another Way to Levi-Civita Connections The results in Theorem 4.2 and 5.2 indicate that the concepts defined above are useful. Nevertheless, there are various other possibilities to define a metric and the compatibility of a metric with a connection. Here we pick out one of these possibilities which is similar to the classical case. Let 0 denote one of the first order differential calculi 0±,z of SLq (N ) defined in Sect. 1. Suppose that q is real. Then 0 is a ∗-calculus for the Hopf-∗-algebra A = SUq (N ). The involutions of A and of 0 are given by (uij )∗ = κ(uji ) and ηij ∗ = −ηji , respectively. We call a map g : 0 × 0 → A a metric if g(aξ, bζ) = ag(ξ, ζ)b∗

for all a, b ∈ A,

ξ, ζ ∈ 0.

We say a (left) connection ∇ on 0 is compatible with the metric g if g(∇(ξ), ζ) + g(ξ, ∇(ζ)) = dg(ξ, ζ)

for all ξ, ζ ∈ 0.

This is well defined because of g(∇(aξ), bζ) + g(aξ, ∇(bζ)) − dg(aξ, bζ) = d(a)g(ξ, ζ)b∗ + ag(∇(ξ), ζ)b∗ + ag(ξ, ζ)d(b∗ )+ + ag(ξ, ∇(ζ))b∗ − d(a)g(ξ, ζ)b∗ − adg(ξ, ζ)b∗ − ag(ξ, ζ)d(b∗ ) = a (g(∇(ξ), ζ) + g(ξ, ∇(ζ)) − dg(ξ, ζ)) b∗ . A bicovariant left connection ∇ is called Levi-Civita connection in respect to the metric g, if ∇ has vanishing torsion and if it is compatible with the metric g. As above, such a Levi-Civita connection is uniquely determined by its values on the generators ηij ∈ inv 0. Let g 0 be a metric from Lemma 2.3 with α, β ∈ R and let g defined by g(ξ, ζ) = 0 g (ξ ⊗ ζ ∗ ) with ξ ∈ 0, ζ ∈ inv 0. Then we have g(ηij , ηkl ) = q 2j αδik δjl + βδij δkl , α 6= 0, P6 α + sβ 6= 0. Let us assume that ∇(ηij ) = k=1 λk Ak mnrs ηmn ⊗ ηrs as in the proof of ij Theorem 4.2. In order to simplify the calculation, let λk be real. The torsion of ∇ has to vanish and so we get (as in Theorem 4.2)

196


Qλ3 = Qλ4 = qλ5 − (Q2 + 1)λ6 + Q. The compatibility with the metric g gives only one equation λ2 α − λ3 (α + sβ) − λ5 β − q −1 λ6 β = 0. Hence there is a three parameter family of Levi-Civita connections. Moreover, we have ∇(η) = (λ2 s + λ5 + q −1 λ6 )q −2m ηmn ⊗ ηnm + (λ1 s + λ3 + λ4 + q 2N Qλ6 )η ⊗ η for the bi-invariant element η ∈ 0. That is, even if we require in addition that ∇(η) = λη ⊗ η for some λ ∈ C we do not get a unique Levi-Civita connection. References 1. Bresser, K., Dimakis, A., Müller-Hoissen, F. and Sitarz, A.: Noncommutative Geometry of Finite Groups. GOET-TP 95/95 (1995) 2. Brzezinski, T. and Majid, S.: Quantum Group Gauge Theory on Quantum Spaces. Commun. Math. Phys. 157, 591–638 (1993) 3. Carow-Watamura, U., Schlieker, M., Watamura, S. and Weich, W.: Bicovariant Differential Calculus on Quantum Groups SUq (N ) and SOq (N ). Commun. Math. Phys. 142, 605–641 (1991) 4. Connes, A.: Non-Commutative Geometry. San Diego: Academic Press (1995) 5. Cuntz, J. and Quillen, D.: Algebra Extensions and Nonsingularity. J. Am. Math. Soc. 8(2), 251–289 (1995) 6. Drinfeld, V.G.: Quantum groups. In: Proceedings ICM 1986, Providence, RI: Amer. Math. Soc. 1987, pages 798–820 7. Dubois-Violette, M. and Michor, W.: Connections on central bimodules. LPTHE-Orsay 94/100 (1995) 8. Faddeev, L.D., Reshetikhin, N.Yu. and Takhtajan, L.A.: Quantization of Lie Groups and Lie Algebras. Algebra and Analysis 1, 178–206 (1987) 9. Georgelin, Y., Madore, J., Masson, T. and Mourad, J.: On the non-commutative Riemannian geometry of GLq (N ). LPTHE-Orsay 95/51 (1995) 10. Heckenberger, I., Schmüdgen, K. and Schüler, A.: Classical limits of bicovariant differential calculi on the quantum groups SLq (N ), Oq (N ) and Spq (N ). To appear 11. Jimbo, M.: A q-analog of U (gl(N + 1)) Hecke algebras, and the Yang-Baxter equation. Lett. Math. Phys. 11, 247–252 (1986) 12. Kassel, C.: Quantum groups. Grad. Texts in Math. Heidelberg: Springer-Verlag, 1995 13. Lusztig, G.: Quantum deformations of certain simple modules over enveloping algebras. Adv. Math. 70, 237–249 (1988) 14. Majid, S.: Quantum and Braided Lie Algebras. J. Geom. Phys. 13, 307–356 (1994) 15. Majid, S.: Foundations of Quantum Group Theory. Cambridge: Cambridge University Press, 1995 16. Mourad, J.: Linear Connections in non-commutative geometry. Class. Quantum Grav. 12, 965–974 (1995) 17. Rosso, M.: Analogues de la forme de Killing et du théorème d’Harish-Chandra pour les groupes ´ Norm. Sup. 23, 445–467 (1990) quantiques. Ann. scient. Ec. 18. Schmüdgen, K. and Schüler, A.: Classification of Bicovariant Differential Calculi on Quantum Groups of Type A, B, C and D. Commun. Math. Phys. 167, 635–670 (1995) 19. Schmüdgen, K. and Schüler, A.: Classification of Bicovariant Differential Calculi on Quantum Groups. Commun. Math. Phys. 170, 315–335 (1995) 20. Woronowicz, S.L.: Differential Calculus on Quantum Matrix Pseudogroups (Quantum Groups). Commun. Math. Phys. 122, 125–170 (1989) Communicated by M. Jimbo

Commun. Math. Phys. 185, 197–209 (1997)

Communications in


Elliptic Genera of Symmetric Products and Second Quantized Strings Robbert Dijkgraaf 1 , Gregory Moore2 , Erik Verlinde3 , Herman Verlinde4 1

Mathematics Dept, Univ. of Amsterdam, 1018 TV Amsterdam, The Netherlands Physics Dept, Yale University, New Haven, CT 06520, USA 3 Theory Division, CERN, CH-1211 Geneva 23, Switzerland and Inst. for Theor. Physics, University of Utrecht, 3508 TA Utrecht, The Netherlands 4 Inst. for Theor. Physics, Univ. of Amsterdam, 1018 XE Amsterdam, The Netherlands

2

Received: 16 August 1996 / Accepted: 3 October 1996

Abstract: In this note we prove an identity that equates the elliptic genus partition function of a supersymmetric sigma model on the N -fold symmetric product M N /SN of a manifold M to the partition function of a second quantized string theory on the space M × S 1 . The generating function of these elliptic genera is shown to be (almost) an automorphic form for O(3, 2, Z). In the context of D-brane dynamics, this result gives a precise computation of the free energy of a gas of D-strings inside a higher-dimensional brane. 1. The Identity Let M be a Kähler manifold. In this note we will consider the partition function of the supersymmetric sigma model defined on the N -fold symmetric product S NM of M , which is the orbifold space (1.1) S NM = M N /SN with SN the symmetric group of N elements. The genus one partition function depends on the boundary conditions imposed on the fermionic fields. For definiteness, we will choose the boundary conditions such that the partition function χ(S NM ; q, y) coincides with the elliptic genus [1, 2], which is defined as the trace over the Ramond-Ramond sector of the sigma model of the evolution operator q H times (−1)F y FL . Here q and y are complex numbers and F = FL + FR is the sum of the left- and right-moving fermion number. (See the Appendix for background.) In particular, χ(M ; q, y) = TrH(M ) (−1)F y FL q H with H = L0 − the trace.

c 24 .

(1.2)

Of the right-moving sector only the R-ground states contribute to

198

R. Dijkgraaf, G. Moore, E. Verlinde, H. Verlinde

We will prove here an identity, conjectured in [3], that expresses the orbifold elliptic genera of the symmetric product manifolds in terms of that of M as follows:1 ∞ X N =0

Y

pN χ(S NM ; q, y) =

n>0,m≥0,`

1 c(nm,`) , (1 − pn q m y ` )

(1.3)

where the coefficients c(m, `) on the right-hand side are defined via the expansion χ(M ; q, y) =

X

c(m, `)q m y ` .

(1.4)

m≥0,`

The proof of this identity follows quite directly from borrowing standard results about orbifold conformal field theory [6], and generalizes the orbifold Euler number computation of [7] (see also [8]). Before presenting the proof, however, we will comment on the physical interpretation of this identity in terms of second quantized string theory. 1.1. String Theory Interpretation. Each term on the left-hand side with given N can be thought of as the left-moving partition sum of a single (non-critical) supersymmetric string with space-time S NM × S 1 × R. This string is wound once around the S 1 direction, and in the light-cone gauge its transversal fluctuations are described by the supersymmetric sigma-model on S NM . The right-hand side, on the other hand, can be recognized as a partition function of a large Fock space, made up from bosonic and fermionic (depending on whether c(nm, `) is positive or negative) creation operators I with I = 1, 2, . . . , |c(nm, `)|. This Fock space is identical to the one obtained by αn,m,` second quantization of the left-moving sector of the string theory on the space M × S 1 . I create string states with winding number In this correspondence, the oscillators αn,m,` 1 n and momentum m around the S . The number of such states is easily read off from the single string partition function (1.4). In the light-cone gauge we have the level matching condition (1.5) L0 − L0 = mn, and since L0 = 0, this condition implies that the left-moving conformal dimension is equal to h = mn. Therefore, according to (1.4) the number of single string states with winding n, momentum m and FL = ` is indeed given by |c(nm, `)|. (Strictly speaking, the elliptic genus counts the number of bosonic minus fermionic states at each oscillator level. Because of the anti-periodic boundary condition in the time direction for the fermions, only the net number contributes in the space-time partition function (1.3). ) The central idea behind the proof of the above identity is that the partition function of a single string on the symmetric product S NM decomposes into several distinct topological sectors, corresponding to the various ways in which a once wound string on S NM × S 1 can be disentangled into separate strings that wind one or more times around M × S 1 . To visualize this correspondence, it is useful to think of the string on S NM × S 1 as a map that associates to each point on the S 1 a collection of N points in M . By following the path of these N points as we go around the S 1 , we obtain a collection of strings on M × S 1 with total winding number N , that reconnect the N 1 In case we have more than one conserved quantum number such as F , the index ` becomes a multiL index and the denominator on the RHS of (1.3) becomes a general product formula as appears in the work of Borcherds [4], see also [5].

Elliptic Genera of Symmetric Products and Second Quantized Strings

199

points with themselves. Since all permutations of the N points on M correspond to the same point in the symmetric product space, the strings can reconnect in different ways labeled by conjugacy classes [g] of the permutation group SN . The factorization of [g] into a product of irreducible cyclic permutations (n) determines the decomposition into several strings of winding number n. (See Fig. 1). The combinatorical description of the M

S1 Fig. 1. The string configuration corresponding to a twisted sector by a given permutation g ∈ SN . The string disentangles into separate strings according to the factorization of g into cyclic permutations

conjugacy classes, as well as the appropriate symmetrization of the wavefunctions, are both naturally accounted for in terms of a second quantized string theory. 2. The Proof The Hilbert space of an orbifold field theory [6] is decomposed into twisted sectors Hg , that are labelled by the conjugacy classes [g] of the orbifold group, in our case the symmetric group SN . Within each twisted sector, one only keeps the states invariant under the centralizer subgroup Cg of g. We will denote this Cg invariant subspace by C Hg g . Thus the total orbifold Hilbert space takes the form M HgCg . (2.1) H(S NM ) = [g]

For the symmetric group, the conjugacy classes [g] are characterized by partitions {Nn } of N X nNn = N, (2.2) n

where Nn denotes the multiplicity of the cyclic permutation (n) of n elements in the decomposition of g, (2.3) [g] = (1)N1 (2)N2 . . . (s)Ns . The centralizer subgroup of a permutation g in this conjugacy class takes the form Ns 2 Cg = SN1 × (SN2 n ZN 2 ) × . . . (SNs n Zs ).

(2.4)

200


Here each subfactor SNn permutes the Nn cycles (n), while each subfactor Zn acts within one particular cycle (n). Corresponding to the above decomposition of [g] into irreducible cyclic permutaC tions, we can decompose each twisted sector Hg g into the product over the subfactors Zn (n) of Nn -fold symmetric tensor products of appropriate smaller Hilbert spaces H(n) , HgCg =

O

Zn S Nn H(n) ,

(2.5)

n>0

where we used the following notation for (graded) symmetric tensor products 

S N

S N H = H ⊗ . . . ⊗ H | {z }

.

(2.6)

N times

Here the symmetrization is assumed to be compatible with the grading of H. In particular VN . for pure odd states S N corresponds to the exterior product Zn The Hilbert spaces H(n) in (2.5) denote the Zn invariant subsector of the Hilbert space H(n) of a single string on M × S 1 with winding number n. We can represent H(n) as the Hilbert space of the sigma model of n coordinate fields Xi (σ) ∈ M with the cyclic boundary condition Xi (σ + 2π) = Xi+1 (σ),

i ∈ (1, . . . , n).

(2.7)

The group Zn , acting on the Hilbert space H(n) , is generated by the cyclic permutation ω : Xi → Xi+1 .

(2.8)

We can glue the n coordinate fields Xi (σ) together into one single field X(σ) defined on the interval 0 ≤ σ ≤ 2πn. Hence, relative to the string with winding number one, the oscillators of the long string that generate H(n) have a fractional n1 moding. The Zn Zn consists of those states in H(n) for which the fractional oscillator invariant subspace H(n) numbers combined add up to an integer. We will make use of this observation in the next subsection. 2.1. Partition Function of a Single String. The elliptic genus of S NM can now be computed by taking the trace over the Hilbert space in the various twisted sectors. We introduce the following notation: χ(H; q, y) = Tr H (−1)F y FL q H

(2.9)

for every (sub)Hilbert space H of a supersymmetric sigma-model. Note that χ(H ⊕ H0 ; q, y) = χ(H; q, y) + χ(H0 ; q, y), χ(H ⊗ H0 ; q, y) = χ(H; q, y) · χ(H0 ; q, y). These identities will be used repeatedly in the following.

(2.10)


201

As the first step we will now compute the elliptic genus of the twisted sector H(n) . This is the left-moving partition sum of a single string with winding n on M ×S 1 . As we have explained, its elliptic genus can be simply related to that of a string with winding 1 number one via a rescaling q → q n , X 1 m χ(H(n) ; q, y) = χ(H; q n , y) = c(m, `)q n y ` . (2.11) m≥0,`

This rescaling accounts for the fractional n1 moding of the string oscillation numbers. The projection onPthe Zn invariant sector is implemented by insertion of the projection operator P = n1 k ω k , with ω as defined in (2.8), 1X Tr H ω k (−1)F y FL q H . (n) n n−1

Zn ; q, y) = χ(H(n)

(2.12)

k=0

Since the boundary condition (2.7) on the Hilbert space H(n) represents a Zn -twist by ω along the σ direction, the operator insertion of ω in the genus one partition sum can in fact be absorbed by performing a modular transformation τ → τ + 1, which amounts 1 1 2πi to a redefinition q n → q n e n .2 Thus we can write 1 2πik 1X χ(H; q n e n , y) n k=0 X c(mn, `)q m y ` . =

n−1

Zn ; q, y) = χ(H(n)

(2.13)

m≥0,`

2.2. Symmetrized products. The next step is to consider the partition function for the Zn . We need the following result: symmetrized tensor products of the Hilbert spaces H(n) If χ(H; q, y) has the expansion X d(m, `)q m y ` , (2.14) χ(H; q, y) = m,`

then we want to show that the partition function of the symmetrized tensor products of H is given by the generating function X N ≥0

pN χ(S N H; q, y) =

Y m,`

1 d(m,`) . (1 − pq m y ` )

(2.15)

This identity is most easily understood in terms of second quantization. The sum over symmetrized products of H is described by a Fock space with a generator for every state in H, where states with negative “multiplicities” d(m, `) are identified as fermions. The usual evaluation of the partition function in a Fock space then results in the RHS of equation (2.15). 2 The redefinition τ → τ + 1 means that the periodic boundary condition in the time direction is composed with a space-like translation σ → σ + 2π. According to (2.7) and (2.8) this indeed results in an extra insertion of the operator ω into the trace.

202


In more detail, we can interpret the elliptic genus as computing the (super)dimension3 of vector spaces Vm,` d(m, `) = dim Vm,` . (2.16) We then evaluate X pN χ(S N H; q, y) N ≥0

=

X

pN

N ≥0

=

X

X

S N dim Vm1 ,`1 ⊗ · · · ⊗ VmN ,`N q m1 +...+mN y `1 +...+`N

m1 ...,m N `1 ,...,` N

pN

N ≥0

X

Y

PNm,`

m,`

qm y`

Nm,`

dim S Nm,` Vm,`

Nm,` =N

=

Y X

pN q m y `

N

dim S N Vm,` .

(2.17)

m,` N ≥0

Using the identity

d(m, `) + N − 1 dim S Vm,` = , (2.18) N where the RHS is defined as (−1)N |d(m,`)| for negative d(m, `), gives the desired N result.

N

2.3. Combining the ingredients. The proof of our main identity follows from combining the results of the previous two subsections. Our starting point has been the fact that the Hilbert space of the orbifold field theory has a decomposition in terms of twisted sectors as M O Zn S NnH(n) . (2.19) H(S NM ) = P n>0 nNn =N

Physically speaking, the right-hand side describes the Hilbert space of a second quantized string theory with Nn the number of strings with winding number n. With this form of the Hilbert space H(S NM ), we find for the partition function X X Y X Zn pN χ(S NM ; q, y) = pN χ(S Nn H(n) ; q, y) n>0 N ≥0 N ≥0 P Nn nNn =N

=

YX

Zn pnN χ(S N H(n) ; q, y).

(2.20)

n>0 N ≥0

Here we used repeatedly the identities (2.10). In order to evaluate the elliptic genera of the symmetric products, we apply the result (2.15) of the previous subsection to the Zn , which gives Hilbert space H(n) 3 We define dim V = Tr (−1)F = d+ − d− , where d± are the dimensions of the even and odd subspaces V V ± in the decomposition V = V + ⊕ V − .


X

Zn pN χ(S N H(n) ; q, y) =

N ≥0

Y m≥0,`

1 c(mn,`) . (1 − pq m y ` )

203

(2.21)

If we insert this into (2.20) we get our final identity X

Y

pN χ(S NM ; q, y) =

N ≥0

n>0,m≥0,`

1 c(mn,`) , (1 − pn q m y ` )

(2.22)

which concludes the proof.

3. One-Loop Free Energy In this section we will discuss some properties of our identity. For convenience we will assume here that the space M is a Calabi-Yau manifold, so that the sigma-model defines a N = 2 superconformal field theory. For the elliptic genus this implies that it transforms as a modular form. We have argued that the quantity on the right-hand side of (2.22) Y

Z(p, q, y) =

n>0,m,`

1 c(mn,`) (1 − pn q m y ` )

(3.1)

has an interpretation as the partition function of a second quantized string theory with target space M ×S 1 . This identification was based on the fact that Z has the form of the trace I with I = 1, . . . , |c(nm, `)|, over free field Fock space generators by oscillators αn,m,l i.e. one oscillator for each first quantized string state. We will now comment on the path integral derivation of this expression. Since we are dealing with a free string theory, we should be able to take the logarithm of the partition sum F (p, q, y) = log Z(p, q, y)

(3.2)

and obtain an interpretation of F as the one-loop free energy of a single string. From a path-integral perspective, this free energy is obtained by summing over irreducible oneloop string amplitudes. The time coordinate of the target space is taken to be compactified (since the partition function is defined as a trace) and thus the irreducible one loop string amplitudes are described in terms of all possible maps of T 2 into the Euclidean target space-time M × T 2 . From this point of view the parameters p, q, y obtain the interpretation as moduli of the target space two-torus. We can introduce parameters ρ, σ, υ via p = e2πiρ ,

q = e2πiσ ,

y = e2πiυ .

(3.3)

Here ρ and σ determine the complexified Kähler form and complex structure modulus of T 2 respectively, whereas υ parametrizes the U (1) bundle on T 2 corresponding to FL . 3.1. Instanton sums and Hecke operators. We will now show that the logarithm F of the partition function (3.1) indeed has the interpretation of a one-loop free energy for a string on M × T 2 . First we compute

204


X

F(p, q, y) = −

c(nm, `) log 1 − pn q m y `

n>0,m,`

X

=

n>0,m,`,k>0

=

X

X 1X c(nm, `)q km y k` . k

pN

N >0

1 c(nm, `)pkn q km y k` k

kn=N

(3.4)

m,`

To write this expression in a more convenient form, it is useful to recall the definition of the Hecke operators TN . (For more details on Hecke operators see e.g. [9].) In general, the Hecke operator TN acting on a weak Jacobi form4 φ(τ, z) of weight zero and index r produces a weak Jacobi form TN φ of weight zero and index N r, defined as follows X 1 aτ + b φ , az . N d ad=N

TN φ(τ, z) =

(3.5)

b mod d

Hence if φ(τ, z) has a Fourier expansion X

φ(τ, z) =

c(m, `)q m y ` ,

(3.6)

m≥0,`

then TN φ(τ, z) takes the form TN φ(τ, z) =

X 1 X c(md, `)q am y a` . a

ad=N

(3.7)

m≥0,`

Comparing with the expression (3.4) for the free energy F, we thus observe that it can be rewritten as a sum of Hecke operators acting on the elliptic genus of M , F(p, q, y) =

X

pN TN χ(M ; q, y).

(3.8)

N >0

(See also [4, 10] for similar expressions.) This representation has a natural interpretation that arises from the geometric meaning of the Hecke operators TN . The expression on the right-hand side of (3.7) that defined TN φ can be reformulated as the sum of pullbacks for all holomorphic maps f : T 2 → T 2 of degree N , 1 X ∗ f φ. (3.9) TN φ = N f

These maps f act as linear transformations on the two-torus and can be represented by the matrices ! ab f= , (3.10) 0d 4

See the Appendix for the definition of a Jacobi form.


205

where ad = N and 0 ≤ b ≤ d − 1. The factor 1/N in (3.9) is natural because of the automorphisms of the torus. With this interpretation, the free energy is represented as a sum over holomorphic maps X 1 Nf ∗ p f χ(M ; q, y) (3.11) F(p, q, y) = N f 2 2 f : T →T

with Nf the degree of the map f . The right-hand side can be recognized as a summation over instanton sectors. 3.2. Automorphic properties. As suggested by its form, the above expression can indeed be reproduced from a standard string one-loop computation. To make this correspondence precise, we notice that the partition function Z is in fact almost equal to an automorphic form for the group SO(3, 2, Z) of the type discussed in [4]. The precise form of this automorphic function has been worked out in detail in [12]. It is defined by the product Y (1 − pn q m y ` )c(nm,`) , (3.12) Φ(p, q, y) = pa q b y c (n,m,`)>0

where the positivity condition means: n, m ≥ 0 with ` > 0 in the case n = m = 0. The “Weyl vector” (a, b, c) is defined by a=b=

1 χ(M ), 24

c=

X `

−

|`| c(0, `). 4

(3.13)

One can then show that the expression Φ is an automorphic form of weight c(0, 0)/2 for the group O(3, 2, Z) for a suitable quadratic form of signature (3, 2), see [12]. The form Φ follows naturally from a standard one-loop string amplitude defined as an integral over the fundamental domain [5, 13]. The integrand consists of the genus one partition function of the string on M × T 2 and has a manifest O(3, 2, Z) T-duality invariance. We will not write down the explicit form of this partition function, but refer to [12] for the specific details. For our purpose it is sufficient to mention the final result of the integration (3.14) I = − log Y c(0,0)/2 |Φ(p, q, y)|2 with Y = ρ2 σ2 − 21 d υ22 , d = dim M , in the notation (3.3). Since the integral I is by construction invariant under the T-duality group O(3, 2, Z), this determines the automorphic properties of Φ. The factor Y transforms with weight −1, which fixes the weight of the form Φ to be c(0, 0)/2. The holomorphic contribution in I is recovered by taking the limit p → 0. In the sigma model this corresponds to the localization of the path-integral on holomorphic instantons and in this way one makes contact with the description of the free energy F in the previous subsection. We note however that log Φ contains extra terms that do not appear in F . Apart from a log p contribution that arises from degree zero maps5 these terms are independent of p and have no straightforward interpretation in terms of instantons. 5 For degree zero the two-torus gets mapped to a point in M , and the moduli space of such maps is the product M ×M1 , with M1 the moduli space of elliptic curves. Weighting this contribution by the appropriate ) log p, in accordance with (3.13). characteristic class [11], we obtain − χ(M 24

206


4. Concluding Remarks Our computation of the elliptic genus of the symmetric product space S NM can be seen as a refinement of the calculations in [7, 8] of the orbifold Euler number. In fact, if we restrict to y = 1, the elliptic genus reduces to the Euler number and our identity takes the simple form Y X 1 pN χ(S NM ) = (4.1) χ(M ) . (1 − pn ) N ≥0

n>0

Here the Kähler condition is not necessary. If M is an algebraic surface, it can be shown that this formula also computes the topological Euler characteristic of the Hilbert scheme M [N ] of dimension zero subschemes of length n [15]. This space is a smooth resolution of the symmetric product S NM . (In complex dimension greater than two the Hilbert scheme is unfortunately not smooth.) It is natural to conjecture that in the case of a two-dimensional Calabi-Yau space, i.e. a K3 or an abelian surface, the orbifold elliptic genus of the symmetric product also coincides with elliptic genus of the Hilbert scheme. The left-hand side of our identity (1.3) can be seen to compute the superdimension of the infinite, graded vector space M

Vm,` (S NM ),

(4.2)

N,m,`

where Vm,` are the index bundles (A.12). Our result suggests that this space forms a natural representation of the oscillator algebra generated by string field theory creation I . This statement is analogous to the assertion of Nakajima [16] (see operators αn,m,` also [17]) that the space ⊕N H ∗ (M [N ] ) forms a representation of the Heisenberg algebra generated by αnI , where I runs over a basis of H ∗ (M ). It would be interesting to explore possible applications to gauge theories along the lines of [8]. On a K3 manifold the moduli space of Yang-Mills instantons takes (for certain instanton numbers) the form of a symmetric product of K3. This fact was used in [8] to relate the partition function of N = 4 Yang-Mills theory on K3 to the generating function of Euler numbers (4.1). Our formula gives an explicit expression for the elliptic genus of these instanton moduli spaces. It seems a natural conjecture that the analysis of [8] can be generalized to show that the generating function of the elliptic genera is the partition function of an appropriately twisted version of N = 2 Yang-Mills theory on K3 × T 2 . For some interesting recent work in this direction, see [18]. Finally, our calculation is likely to be relevant for understanding the quantum statistical properties of D-branes [19] and their bound states [20]. Particularly useful examples of such possible bound states are those between D-strings with one (or more) higher dimensional D-branes. In type II string compactifications on manifolds of the form M × S 1 , we can consider the configuration of a D-string wound N times around the S 1 bound to a (dimM +1)-brane. (For the case where M is a K3 manifold, this situation was first considered by Vafa and Strominger [21] in their D-brane computation of the 5-dimensional black hole entropy.) As argued in [22, 21], the quantum mechanical degrees of freedom of this D-brane configuration are naturally encoded in terms of a two-dimensional sigma model on the N -fold symmetric tensor product of M , that describes the transversal fluctuations of the D-string. As was also pointed out in [23], this description implies that a multiply wound D-string can carry fractional oscillation numbers. Our result shows that the resulting quantum statistical description of these first


207

quantized “fractional” D-strings is in fact equivalent to a description in terms of second quantized “ordinary” strings. In this correspondence the extra degrees of freedom that arise from the fractional moding are used to assign to each individual string a momentum along the S 1 direction. This result may be a useful clue in explaining some of the miraculous non-perturbative dualities between strings and D-branes. Acknowledgement. We thank D. Neumann, J-S. Park, G. Segal, W. Taylor and C. Vafa for discussions, and the Aspen Center of Physics for hospitality during the final stage of this work. This research is partly supported by a Pionier Fellowship of NWO, a Fellowship of the Royal Dutch Academy of Sciences (K.N.A.W.), the Packard Foundation and the A.P. Sloan Foundation.

A. Appendix: Elliptic Genus We summarize some facts about the elliptic genus for a Kähler manifold M of complex dimension d [1, 2]. We start with an elliptic curve E with modulus τ and a line bundle labeled by z ∈ Jac(E) ∼ = E. We define q = e2πiτ , y = e2πiz . The elliptic genus is defined as d d (A.1) χ(M ; q, y) = Tr H(M ) (−1)F y FL q L0 − 8 q L0 − 8 , where F = FL + FR and H(M ) is the Hilbert space of the N = 2 supersymmetric field theory with target space M . For a Calabi-Yau space the elliptic genus is a weak Jacobi form of weight zero and index d/2. Recall that a Jacobi form φ(τ, z) of weight k and index r (possibly half-integer) transforms as [24] rcz 2 z aτ + b , = (cτ + d)k eπi cτ +d φ(τ, z), φ cτ + d cτ + d φ(τ, z + mτ + n) = e−πir(m

2

τ +2mz)

φ(τ, z),

(A.2)

and is called weak if it has a Fourier expansion of the form φ(τ, z) =

X

c(m, `)q m y ` .

(A.3)

m≥0,`

The coefficients of such a form depend only on 4rm − `2 and on ` mod 2r. The elliptic genus has the following properties: First of all, it is a genus; that is, it satisfies the relations χ(M t M 0 ; q, y) = χ(M ; q, y) + χ(M 0 ; q, y), χ(M × M 0 ; q, y) = χ(M ; q, y) · χ(M 0 ; q, y), χ(M ; q, y) = 0,

(A.4)

if M = ∂N ,

where the last relation is in the sense of complex bordism. Furthermore, for q = 0 it reduces to a weighted sum over the Hodge numbers, which is essentially the Hirzebruch χy -genus, X d (−1)p+q y p− 2 hp,q (M ), (A.5) χ(M ; 0, y) = p,q

208


and for y = 1 its equals the Euler number of M , χ(M ; q, 1) = χ(M ).

(A.6)

For smooth manifolds, the elliptic genus has an alternative definition in terms of characteristic classes, as follows. For any vector bundle V one defines the formal sums V

qV

=

M

qk

Vk

V,

Sq V =

M

k≥0

q k S k V,

(A.7)

k≥0

Vk and S k denote the k th exterior and symmetric product respectively. One then where has an equivalent definition of the elliptic genus as Z χ(M ; q, y) = ch(Eq,y )td(M ) (A.8) M

with Eq,y = y − 2

d

OV

−yq n−1 TM

⊗

V

−y −1 q n T M

⊗ S q n T M ⊗ Sq n T M ,

(A.9)

n≥1

where TM denotes the holomorphic tangent bundle of M . Expanding the bundle Eq,y as M Eq,y = q m y ` Em,` , (A.10) m,`

one can define the coefficients c(m, `) as c(m, `) = indexD /Em,`

(A.11)

with D /E the Dirac operator twisted with the vector bundle E. So c(m, `) computes the dimension of the virtual vector space /Em,` cokD /Em,` . Vm,` (M ) = kerD

(A.12)

References 1. Landweber, P.S. (ed.): Elliptic Curves and Modular Forms in Algebraic Topology. Berlin–Heidelberg– New York: Springer-Verlag, 1988; Witten, E.: Commun. Math. Phys. 109, 525 (1987); Schellekens, A. and Warner, N.: Phys. Lett. B177, 317 (1986); Nucl. Phys. B287, 317 (1987) 2. Kawai, T., Yamada, Y. and Yang, S.-K.: Elliptic Genera and N=2 Superconformal Field Theory. Nucl. Phys. B414, 191-212 (1994); Eguchi, T., Ooguri, H., Taormina, A., Yang, S.-K.: Superconformal Algebras and String Compactification on Manifolds with SU (N ) Holonomy. Nucl.Phys. B315, 193 (1989) 3. Dijkgraaf, R., Verlinde, E. and Verlinde, H.: Counting Dyons in N=4 String Theory. hep-th/9607026; BPS Spectrum of the Five-Brane and Black Hole Entropy. hep-th/9603126 4. Borcherds, R. E.: Automorphic Forms on Os+2,2 (R) and Infinite Products. Invent. Math. 120, 161 (1995) 5. Harvey, J. and Moore, G.: Algebras, BPS States, and Strings. Nucl. Phys. B463, 315-368, (1996), hepth/9510182 6. Dixon, L., Harvey, J., Vafa, C. and Witten, E. : Nucl. Phys. B261, 620 (1985); Nucl. Phys. B274, 285 (1986)


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7. Hirzebruch, F. and Höfer, T.: On the Euler Number of an Orbifold. Math. Ann. 286, 255 (1990) 8. Vafa, C. and Witten, E.; A Strong Coupling Test of S-Duality. Nucl. Phys. B431, 3-77 (1994), hepth/9408074 9. Lang, S.: Introduction to Modular Forms. Berlin–Heidelberg–New York: Springer-Verlag, 1976 10. Gritsenko, V.A. and Nikulin, V.V.: Siegel Automorphic Form Corrections of Some Lorentzian KacMoody Algebras. alg-geom/9504006 11. Bershadsky, M., Cecotti, S., Ooguri, H. and Vafa, C.: Holomorphic Anomalies in Topological Field Theories. Nucl. Phys. B405, 279–304 (1993), hep-th/9302103 12. Neumann, C.D.D.: The Elliptic Genus of Calabi-Yau 3- and 4-Folds, Product Formulae and Generalized Kac-Moody Algebras. hep-th/9607029 13. Kawai, T.: N = 2 Heterotic String Threshold Correction, K3 Surface and Generalized Kac-Moody Superalgebra. hep-th/9512046 14. Dixon, L., Kaplunovsky, V. and Louis, J.: Moduli-dependence of string loop corrections to gauge coupling constants. Nucl. Phys. B307, 145 (1988) 15. Göttsche, L.: The Betti numbers of the Hilbert Scheme of Points on a Smooth Projective Surface. Math. Ann. 286, 193–297 (1990) 16. Nakajima, H.: Heisenberg Algebra and Hilbert Schemes of Points on Projective Surfaces. alggeom/9507012 17. Grojnowski, I.: Instantons and Affine Algebras I: The Hilbert Scheme and Vertex Operators. alggeom/9506020 18. Nekrasov, N.: Princeton PhD-thesis, May 1996, unpublished 19. Polchinski, J.: Dirichlet-Branes and Ramond-Ramond Charges. hep-th/9510017; Polchinski, J., Chaudhuri, S. and Johnson, C.: Notes on D-Branes. hep-th/9602052 20. Witten, E.: Bound States Of Strings And p-Branes. hep-th/9510135 21. Strominger, A., Vafa, C.: Microscopic Origin of the Bekenstein-Hawking Entropy. hep-th/9601029 22. Vafa, C.: Gas of D-Branes and Hagedorn Density of BPS States. hep-th/9511026, Instantons on D-branes. hep-th/9512078; Bershadsky, M., Sadov, V. and Vafa, C.: D-Branes and Topological Field Theories. hepth/9511222 23. Maldacena, J. and Susskind, L. : D-branes and Fat Black Holes. hep-th/9604042 24. Eichler, M. and Zagier, D.: The Theory of Jacobi Forms. Basel–Boston: Birkhäuser, 1985 Communicated by A. Jaffe

Commun. Math. Phys. 185, 211 – 230 (1997)

Communications in


Equivalence of Two Approaches to Integrable Hierarchies of KdV type Benjamin Enriquez1 , Edward Frenkel2 1 2

Centre de Mathématiques, URA 169 du CNRS, Ecole Polytechnique, 91128, Palaiseau, France Department of Mathematics, Harvard University, Cambridge, MA 02138, USA

Received: 1 July 1996 / Accepted: 21 October 1996

Abstract: The equivalence between the approaches of Drinfeld-Sokolov and FeiginFrenkel to the mKdV and KdV hierarchies is established. A new derivation of the mKdV equations in the zero curvature form is given. Connection with the Baker-Akhiezer function and the tau-function is also discussed. 1. Introduction To each affine Kac-Moody algebra g one can associate a modified Korteweg-de Vries (mKdV) hierarchy of non-linear partial differential equations. The mKdV hierarchy, which can be viewed as a refined form of a generalized KdV hierarchy (see [4]), is a completely integrable hamiltonian system. The equations of the hierarchy can be written in hamiltonian form, and the corresponding hamiltonian flows commute with each other. It is known that the equations of an mKdV hierarchy can be represented in the zero curvature form (1.1) [∂tn + Vn , ∂z + V ] = 0, where tn ’s are the times of the hierarchy, and t1 = z. Here V and Vn are certain time dependent elements of the centerless affine algebra g. To write V explicitly, consider the principal abelian subalgebra a of g (the precise definition is given below). It has a basis pi , i ∈ ±I, I being the set of all exponents of g modulo the Coxeter number. Then V = p−1 + u(z), where u(z) lies in the Cartan subalgebra h of g. The element p−1 has degree −1 with respect to the principal gradation of g, while u has degree 0. This makes finding an element Vn that satisfies (1.1) a non-trivial problem. Indeed, equation (1.1) can be written as ∂tn u = [∂z + p−1 + u, Vn ].

(1.2)

212

B. Enriquez, E. Frenkel

The left hand side of (1.2) has degree 0. Therefore Vn should be such that the expression in the right hand side of (1.2) is concentrated in degree 0. Such elements can be constructed by the following trick (see [18, 4, 16]). Suppose we found some Vn ∈ g which satisfies [∂z + p−1 + u, V\ ] = 0.

(1.3)

We can split Vn into the sum V+ + V− of its components of positive and non-positive degrees with respect to the principal gradation. Then Vn = V− has the property that the right hand side of (1.2) has degree 0. Indeed, from (1.3) we find [∂z + p−1 + u(z), V− ] = −[∂z + p−1 + u(z), V+ ], which means that both commutators have neither positive nor negative homogeneous components. Therefore Eq. (1.2) makes sense. Now we have to find solutions of Eq. (1.3). Drinfeld and Sokolov [4] proposed a powerful method of finding solutions of (1.3), which is closely related to the dressing method of Zakharov and Shabat [18]. Another approach was proposed by Wilson [16] (see also [14]). Let us briefly explain the Drinfeld-Sokolov method. Recall that g has the decomposition g = n+ ⊕ b− , where n+ is the nilpotent subalgebra of g. Let N+ be the corresponding Lie group. In [4] it was proved that there exists an N+ –valued function M (z), which is called the dressing operator, such that X hi (z)pi , M (z)−1 ∂z + p−1 + u(z) M (z) = ∂z + p−1 + i∈I

where hi ’s are certain functions. The dressing operator M (z) is defined up to right multiplication by a z–dependent element of the subgroup A+ of N+ corresponding to the Lie algebra a+ = a ∩ n+ . Thus, M (z) represents a coset in N+ /A+ . The element Vn = M (z)p−n M (z)−1 clearly satisfies (1.3) and by substituting Vn = (M (z)p−n M (z)−1 )− in Eq. (1.1) for n ∈ I one obtains the mKdV hierarchy. Recently, another approach to mKdV hierarchies was proposed by Feigin and one of the authors [7, 8]. In this approach, the flows of the mKdV hierarchy are considered as vector fields on the space of jets of the function u(z). Let π0 = C[u(n) i ]i=1,...,l;n≥0 , where ui = (αi , u) and uni = ∂zn ui , be the ring of differential polynomials in ui ’s. In [8], π0 was identified with the ring of algebraic functions on the homogeneous space N+ /A+ . Thus, each function u(z) gives rise to a function K(z) with values in N+ /A+ . The Lie algebra a− = a ∩ b− naturally acts on N+ /A+ from the right. Consider the derivation ∂n on π0 which corresponds to the infinitesimal action of p−n on N+ /A+ . These derivations clearly commute with each other. Moreover, it was shown in [8] that ∂1 coincides with ∂z and therefore ∂n ’s are evolutionary (i.e. commuting with ∂z ) derivations. In this work we prove that the cosets M (z) and K(z), obtained by the constructions of [4] and [8], coincide. We then show that the derivation ∂n satisfies Eq. (1.1) with Vn = (K(z)p−n K(z)−1 )− . Thus, we establish an equivalence between the two constructions. b 2 based on Note that another approach to establishing this equivalence in the case of sl KdV gauge fixing [4] was proposed by one of the authors in [5].

Equivalence of Two Approaches to Integrable Hierarchies of KdV Type

213

This gives us a direct identification of the flows corresponding to ∂n and ∂tn .1 Thus we obtain a new derivation of the zero curvature representation of the mKdV hierarchies. We remark that there exist generalizations of the mKdV hierarchies which are associated to abelian subalgebras of g other than a. It is known that the Drinfeld-Sokolov approach can be applied to these generalized hierarchies [1, 10]. On the other hand, the approach of [8] can also be applied; in the case of the non-linear Schrödinger hierarchy, which corresponds to the homogeneous abelian subalgebra of g, this has been done by Feigin and one of the authors [9]. The results of our paper can be extended to establish the equivalence between the two approaches in this general context. The paper is arranged as follows. In Sect. 2 we recall the construction of [8] and derive the zero curvature equations. In Sect. 3 we prove that the cosets M and K coincide and that the derivations ∂n and ∂tn coincide. We also explain the connection with the KdV hierarchies, and discuss a possible generalization to the case of an arbitrary curve. In Sect. 4 we construct a natural system of coordinates on the group N+ and using it, we give another proof of the equivalence of two formalisms. Finally, in Sect. 5 we obtain explicit formulas for the one-cocycles defined in [8] and the densities of the hamiltonians of the mKdV hierarchy. We also discuss a connection between the formalism of [8] and the τ –functions.

2. Unipotent Cosets 2.1. Notation. Let e g be an affine algebra. It has generators ei , fi , αi∨ , i = 0, . . . , l, and d, which satisfy the standard relations [11]. The Lie algebra e g carries a non-degenerate invariant inner product (·, ·). One associates to e g the labels ai , a∨ i , i = 0, . . . , l, the exponents di , i = 1, . . . , l, and the Coxeter number h, see [11]. We denote by I the set of all positive integers, which are congruent to the exponents of e g modulo h (with multiplicities). The elements ei , i = 0, . . . , l, and fi , i = 0, . . . , l, generate the nilpotent subalgebras n+ and n− of e g, respectively. The elements αi∨ generate the Cartan subalgebra e h of e g. We have: e g = n+ ⊕ b− , where b− = n− ⊕ e h. Each x ∈ e g can be uniquely written as x+ + x− , where x+ ∈ n+ and x− ∈ b− . The element l X ∨ a∨ C= i αi i=0

of e h is a central element of e g. Let g be the quotient of [e g, e g] by CC. We identify e g with the direct sum g ⊕CC ⊕Cd. The Lie algebra g has a Cartan decomposition g = n+ ⊕ h⊕ n− , where h is spanned by αi∨ , i = 1, . . . , l. l X ai ei . Let a be the centralizer of p1 in g. This is an abelian subalgebra of g Set p1 = i=0

which we call the principal abelian subalgebra. We have a decomposition: a = a+ ⊕ a− , where a+ = a ∩ n+ , and a− = a ∩ b− . It is known that a± is spanned by elements pi , i ∈ ±I, which have degrees deg pi = i with respect to the principal gradation of g. In particular, we choose 1 In [8] the following indirect proof of this fact was given: the derivations ∂ were identified in [8] with the n symmetries of the affine Toda equation corresponding to g. But it is known that mKdV equations constitute all symmetries of the affine Toda equation, see [4, 14, 16].

214


p−1 =

l X (αi , αi ) i=0

2

fi ,

where αi ’s are the simple roots of g, considered as elements of h using the inner product. (1) , each exponent occurs exactly once. In the Remark 1. For all affine algebras except D2n (1) case of D2n , the exponent 2n−1 has multiplicity 2. In this case, there are two generators of a, p1i and p2i , for i congruent to 2n − 1 modulo the Coxeter number 4n − 2.

Let N+ be the Lie group of n+ . This is a prounipotent proalgebraic group (see, e.g., [8]). The exponential map exp : n+ → N+ is an isomorphism of proalgebraic varieties. Let A+ be the image of a+ under this map. The Lie algebra g acts on N+ from the right because N+ can be embedded as an open subset in the flag manifold B− \G of g. Therefore the normalizer of a+ in g acts on N+ /A+ from the right. In particular, a− acts on N+ /A+ , and each p−n , n ∈ I gives rise to a derivation of C[N+ /A+ ], see [8]. These derivations commute with each other. 2.2. Actions on the space of jets. Consider the space U of jets of smooth function u(z) : A1 → h. The space U is the inverse limit of the finite-dimensional vector spaces (n) = ∂zn ui . Thus, the ring UN = span{u(n) i }i=1,...,l;n=1,...,N , where ui = (αi , u), and ui (n) π0 of regular functions on U is C[ui ]i=1,...,l;n≥0 . The derivative ∂z gives rise to a derivation of π0 . Theorem 1 ([8], Theorem 2). There is an isomorphism of rings C[N+ /A+ ] ' C[u(n) i ], under which p−1 gets identified with ∂z . Let ∂n be the derivation of C[u(n) i ] corresponding to p−n under this isomorphism. The theorem shows that the derivations ∂n are evolutionary, i.e. commuting with ∂z . We would like to represent the action of these derivations on u(z) explicitly in the zero curvature form (1.1). For g ∈ G and x ∈ g we will write gxg −1 for Adg (x). Proposition 1. For K ∈ N+ /A+ , [∂m + (Kp−m K −1 )− , ∂n + (Kp−n K −1 )− ] = 0,

∀m, n ∈ I.

(2.1)

Let us explain the meaning of formula (2.1). For each K ∈ N+ /A+ , Kp−n K −1 is a well-defined element of g. The Lie algebra g can be realized as g ⊗ C((t)) (or a subalgebra thereof if g is twisted) for an appropriate finite-dimensional Lie algebra g. If we choose a basis in g, we can consider an element of g as a set of Laurent power series. In particular, for Kp−n K −1 , any Fourier coefficient of each of these power series gives us an algebraic function on N+ /A+ . Hence, by Theorem 1, each coefficient corresponds to a differential polynomial in ui ’s, and we can apply ∂m to it. In order to prove formula (2.1), we need to find an explicit formula for the action of ∂n on Kp−m K −1 . Let us first obtain a formula for the infinitesimal action of an element of g on N+ . Recall from [8] that since N+ embeds as an open subset in the flag manifold B− \G, the Lie algebra g infinitesimally acts on N+ from the right by vector fields. Therefore g acts


215

by derivations on C[N+ ]. Since N+ acts on g, we obtain a homomorphism from N+ to the group of automorphisms of g over the ring C[[t]]. Now we can consider each element of N+ as a matrix, whose entries are Taylor power series. Each Fourier coefficient of such a series defines an algebraic function on N+ , and the ring C[N+ ] is generated by these functions. Hence any derivation of C[N+ ] is uniquely determined by its action on these functions. We can write this action concisely as follows: ν · x = y, where x is a “test” matrix representing an element of N+ , and y is another matrix, whose entries are the results of the action of ν on the entries of x. For a ∈ g, let aR be the derivation of C[N+ ] corresponding to the right infinitesimal action of a on N+ . For b ∈ n+ , let bL be the derivation of C[N+ ] corresponding to the left infinitesimal action of b on N+ . Lemma 1. aR · x = (xax−1 )+ x, ∀a ∈ g, L ∀b ∈ n+ . b · x = bx,

(2.2) (2.3)

Proof. Consider a one-parameter subgroup a() of G, such that a() = 1 + a + o(). We have: x · a() = x + xa + o(). For small we can factor x · a() into a product (1) ∈ B− . We then find that y− y+ , where y+ = x + y+(1) + o() ∈ N+ and y− = 1 + y− (1) x + y+(1) = xa, from which we conclude that y+(1) = (xax−1 )+ x. This proves formula y− (2.2). Formula (2.3) is obvious. It follows from formula (2.2) that aR · xvx−1 = [(xax−1 )+ , xvx−1 ],

a, v ∈ g.

(2.4)

If a and v are both elements of a, then formula (2.4) does not change if we multiply x from the right by an element of A+ . Denote by K the coset of x in N+ /A+ . Then we can write: v ∈ a. (2.5) ∂n · KvK −1 = [(Kp−n K −1 )+ , KvK −1 ], Proof of Proposition 1. Substituting v = p−m into formula (2.5), we obtain: ∂n · Kp−m K −1 = [(Kp−n K −1 )+ , Kp−m K −1 ]. Hence ∂n · (Kp−m K −1 )− = [(Kp−n K −1 )+ , Kp−m K −1 ]− = [(Kp−n K −1 )+ , (Kp−m K −1 )− ]− . Therefore we obtain: [∂m + (Kp−m K −1 )− , ∂n + (Kp−n K −1 )− ] = ∂m · (Kp−n K −1 )− − ∂n · (Kp−n K −1 )− + [(Kp−m K −1 )− , (Kp−n K −1 )− ] = [(Kp−m K −1 )+ , (Kp−n K −1 )− ]− − [(Kp−n K −1 )+ , Kp−m K −1 ]− +[(Kp−m K −1 )− , (Kp−n K −1 )− ]− . Adding up the first and the last terms, we obtain [Kp−m K −1 , (Kp−n K −1 )− ]− − [(Kp−n K −1 )+ , Kp−m K −1 ]− =

216


= [Kp−m K −1 , Kp−n K −1 ]− = 0, and the proposition is proved.

2.3. Zero curvature form. In order to write the action of ∂n in the zero curvature form, we will use Proposition 1 in the case m = 1. But first we determine (Kp−1 K −1 )− explicitly. Lemma 2. (Kp−1 K −1 )− = p−1 + u.

(2.6)

It is clear that (Kp−1 K −1 )− = p−1 + x, where x ∈ h. Hence we need to show that x = u, or, equivalently, that (αi , x) = ui , i = 1, . . . , l. We can rewrite the latter formula as ui = (αi , Kp−1 K −1 )− , and hence as ui = (αi , Kp−1 K −1 ). To establish the last formula, recall the interpretation of ui from [8]. Consider the module Mλ∗ contragradient to the Verma module Mλ over g with highest weight λ. This module can be realized in the space C[N+ ] in such a way that the highest weight vector vλ corresponds to the constant function. For a ∈ g denote by fλ (a) the function on N+ which corresponds to a · vλ . Then ui = fαi (p−1 ) [8]. But in fact there is a general formula for fλ (a) due to Kostant [13] (Konstant proved this result when g is a finite-dimensional simple Lie algebra, but his proof can be generalized to an arbitrary Kac–Moody algebra). Proposition 2 ([13], Theorem 2.2). Consider λ ∈ h∗ as a functional on g which is trivial on n± . Let h·, ·i be the pairing between g∗ and g. Then fλ (a)(x) = hλ, xax−1 i. The formula above immediately implies that the function ui on N+ /A+ takes value (αi , Kp−1 K −1 ) at K ∈ N+ /A+ . This completes the proof of Lemma 2. Now specializing m = 1 in formula (2.1) and using Lemma 2 we obtain the zero curvature representation of the equations. Theorem 2. [∂z + p−1 + u, ∂n + (Kp−n K −1 )− ] = 0.

(2.7)

This equation can be rewritten as ∂n u = ∂z (Kp−n K −1 )− + [p−1 + u, (Kp−n K −1 )− ].

(2.8)

The map K → Kp−n K −1 defines an embedding of N+ /A+ into g as an N+ – orbit. The entries of the matrix Kp−n K −1 are Laurent series in t whose coefficients are differential polynomials in ui , i = 1, . . . , l (see the paragraph after Proposition 1). Equation (2.8) expresses ∂n ui in terms of differential polynomials in ui ’s. Since, by construction, ∂n commutes with ∂1 ≡ ∂z , formula (2.8) uniquely determines ∂n as an evolutionary derivation of C[u(n) i ]. Thus, we have now derived the zero curvature equations for the mKdV hierarchy using the formalism of [8].


217

3. Equivalence with the Drinfeld-Sokolov Formalism

3.1. Identification of cosets. For v ∈ a and K ∈ N+ /A+ set Vv = KvK −1 .

(3.1)

Since a is commutative, this is a well-defined element of g. Proposition 3. [∂z + p−1 + u, Vv ] = 0,

∀v ∈ a.

(3.2)

Proof. Using formula (2.5) and Lemma 2 we obtain: ∂z Vv = [(Kp−1 K −1 )+ , Vv ] = −[(Kp−1 K −1 )− , Vv ] = −[p−1 + u, Vv ].

(3.3)

Now we define the Drinfeld-Sokolov dressing operator M . Proposition 4 ([4], Proposition 6.2). There exists an element M = M (z) ∈ N+ , such that X h i pi , (3.4) M −1 ∂z + p−1 + u(z) M = ∂z + p−1 + i∈I

where hi ’s are functions. M is defined uniquely up to right multiplication by a (possibly z–dependent) element of A+ . One can choose M in such a way that all entries of its matrix and all hi ’s are polynomials in u(n) i , i = 1, . . . , l; n ≥ 0. The proposition defines a map from the space of smooth functions u(z) : A1 → h to the space of smooth functions A1 → N+ /A+ , u(z) → M (z). On the other hand, Theorem 1 also defines such a map u(z) → K(z). The following lemma will allow us to identify these two maps. Remark 2. Note that both maps are local in the following sense. For each z, M (z) and K(z) depend only on the jet of u at z. In particular, for each v ∈ a, all entries of the matrices M (z)vM (z)−1 and K(z)vK(z)−1 are Taylor series whose coefficients are differential polynomials in ui ’s. Lemma 3 ([4]). Let V be an element of g of the form V = p−n + terms of degree higher than −n with respect to the principal gradation on g, such that [∂z + p−1 + u, V] = 0.

(3.5)

Then V = M vM −1 , where M ∈ N+ satisfies (3.4) and v ∈ a is such that v = p−n + constant terms of degree higher than −n. The proof of the lemma requires the following important result. Proposition 5 ([12], Proposition 3.8). The Lie algebra g has the decomposition g = a ⊕ Im(ad p−n ) for each n ∈ I. Moreover, Ker(ad p−n ) = a.

218


Proof of Lemma 3. If V satisfies (3.5), then we obtain from Proposition 4: X hi pi , M −1 VM ] = 0. [∂z + p−1 +

(3.6)

i∈I

P We can write M −1 VM as a sum j vj of its homogeneous components of principal degree j. According to Proposition 5, each vj can be split into the sum of vj0 ∈ a+ and vj1 ∈ Im(ad p−1 ). Suppose that M −1 VM does not lie in a. Let j0 be the smallest number such that 1 vj0 6= 0. Then the term of smallest degree in (3.6) is [p−1 , vj10 ] which is non-zero, because Ker(ad p−n ) = a. Hence (3.6) can not hold. Therefore M −1 VM ∈ a. But then (3.6) gives: ∂z vj = 0 for all j. This means that each vj is a constant element of a, and the lemma follows. Theorem 3. The cosets M (z) and K(z) in N+ /A+ assigned in [4] and [8], respectively, to the jet of function u : A1 → h at z, coincide. Proof. According to Proposition 3, [∂z + p−1 + u, Kp−n K −1 ] = 0. Since Kp−n K −1 = p−n + terms of degree higher than −n with respect to the principal gradation, we obtain from Lemma 3 that Kp−n K −1 = M vM −1 , where M ∈ N+ satisfies (3.4) and v ∈ a. This implies that v = p−n and that M = K in N+ /A+ . Indeed, from the equality Kp−n K −1 = M vM −1 we obtain that (M −1 K)p−n (M −1 K)−1 lies in a. We can represent M −1 K as exp y for some y ∈ n+ . Then (M −1 K)p−n (M −1 K)−1 = v can be expressed as a linear combination of multiple commutators of y and p−n : ey p−n (ey )−1 =

X 1 (ad y)n · p−n . n!

n≥0

P

We can write y = j>0 yj , where yj is the homogeneous component of y of principal degree j. It follows from Proposition 5 that n+ = a+ ⊕ Im(ad p−n ). Therefore each yj can be further split into a sum of yj0 ∈ a+ and yj1 ∈ Im(ad p−1 ). Suppose that y does not lie in a+ . Let j0 be the smallest number such that yj10 6= 0. Then the term of smallest degree in ey p−n (ey )−1 is [yj10 , p−n ] which lies in Im(ad p−n ) and is non-zero, because Ker(ad p−n ) = a+ . Hence ey p−n (ey )−1 can not be an element of a+ . Therefore y ∈ a+ and so M −1 K ∈ A+ , which means that M and K represent the same coset in N+ /A+ , and that v = p−n . 3.2. Identification of the equations. As was explained in the previous section, Theorem 1 allows us to define a set of commuting derivations ∂n , n ∈ I, of π0 , or equivalently, vector fields on the space of jets U . These derivations can be represented in the zero curvature form (2.7).


219

On the other hand, in [4] another set of derivations ∂tn , n ∈ I, of π0 was defined in the zero curvature form. Set (3.7) Vn = M (z)p−n M (z)−1 − , where M (z) is defined in Proposition 4. In particular, formula (3.4) shows that V1 = V = p−1 + u. The nth zero curvature equation is Eq. (1.1) for Vn given by (3.7). Now we obtain from Theorem 3: Theorem 4. The derivations ∂n and ∂tn coincide. Remark 3. This theorem together with Theorem 1 implies that solutions of the mKdV hierarchy are just the integral curves of the vector fields of the infinitesimal action of the Lie algebra a− on N+ /A+ . Remark 4. The variable t−1 appearing in the affine algebra g is often denoted by λ, and is called the spectral parameter. Remark 5. For each n ∈ I ∪ −I, the map K 7→ Kp−n K −1 defines an embedding N+ /A+ → g, because the stabilizer of p−n in N+ is A+ . In practice, it is convenient to find Kp−n K −1 using the equation [∂z + p−1 + u(z), Kp−n K −1 ] = 0,

(3.8)

which follows from formula (3.2). We can split Kp−n K −1 into the sum of homogeneous components lying in a and in Im(ad p−1 ). These homogeneous components can then be determined recursively using Eq. (3.8) as explained in [16], Sect. 3. This recursion is actually non-trivial: at certain steps one has to take the antiderivative of a differential polynomial. But we know from Proposition 3 that the element Kp−n K −1 satisfies (3.8) and that its entries are differential polynomials (see the paragraph after Proposition 1). Therefore whenever an anti-derivative occurs, it can be resolved in the ring of differential polynomials. Another proof of this fact has been given by Wilson [16]. Every time we compute the anti-derivative, we have the freedom of adding an arbitrary constant. This corresponds to adding to Kp−n K −1 a linear combination of Kpm K −1 with m > −n. Remark 6. The map which attaches to ui ’s a coset in N+ /A+ can be viewed as a universal feature in various approaches to soliton equations. In this section we have explained how these maps arise in the formalisms of [4] and [8] and proved that these maps coincide. But a map to N+ /A+ can also be found, in a somewhat disguised form, in the SegalWilson approach to the soliton equations based on Sato’s Grassmannian [15, 17] (see also [2]). One can associate to ui ’s their Baker-Akhiezer function 9 which is a solution of the equation (∂z + p−1 + u(z))9 = 0, (3.9) and more generally the equations (∂n + (Kp−n K −1 )− )9 = 0,

∀n ∈ I.

(3.10)

In our notation, Segal and Wilson [15, 17] attach in the case of g = sln a BakerAkhiezer function 9 to each point x of the flag manifold B− \G using its realization via an infinite Grassmannian. The flows of the mKdV hierarchy then correspond to the right infinitesimal action of a− on the flag manifold. As x moves along the integral curves

220


of the vector fields ∂n , the Baker-Akhiezer function evolves according to the mKdV hierarchy and so does the function u. One shows [15] that 9 is regular at a given set of times of the hierarchy if the corresponding point of the flag manifold lies in the big cell (which is isomorphic to N+ ). Moreover, u does not change under the right action of A+ on x [17]. Thus, one obtains a map which assigns to u an element of N+ /A+ . In [17] the equivalence between the dressing method and the Grassmannian approach was established (see also [10]). Therefore this map coincides with the map studied in our paper. We will derive an explicit formula for the Baker-Akhiezer function in Sect. 4. 3.3. From mKdV to KdV. First let us recall the definition of the KdV hierarchies from [4]. Consider the operator (3.11) ∂z + p−1 + u(z), where now u(z) lies in the finite-dimensional Borel subalgebra h ⊕ n+ , where n+ is generated by ei , i = 1, . . . , l. Drinfeld and Sokolov construct in [4] the dressing operator and the zero-curvature equations (1.1) for this operator in the same way as for the mKdV hierarchy using formulas (3.4) and (3.7). The Lie group N + of n+ acts naturally on the space of operators (3.11) and these equations preserve the corresponding gauge equivalence classes [4]. Thus one obtains a system of compatible evolutionary equations on the gauge equivalence classes, which is called the generalized KdV hierarchy corresponding to e g. Let n0+ be a subspace of n+ that is transversal to the image in n+ of the operator ad p−1 , where p−1 =

l X (αi , αi ) i=1

2

fi .

It is shown in [4] that each equivalence class contains a unique operator (3.11) satisfying the condition that u ∈ n0+ . The space n0+ is l–dimensional. If we choose coordinates v1 , . . . , vl of n0+ , then the KdV equations can be written as partial differential equations on vi ’s. On the other hand, the dressing operator M (z) corresponding to a gauge class of operators (3.11) should now be considered as a double coset in N + \N+ /A+ . Thus, a smooth function v(z) = (v1 (z), . . . , vl (z)) : A1 → n0+ gives rise to a smooth function A1 → N + \N+ /A+ . Denote by L the space of all operators (3.11) where u ∈ h, and by Le the space of e which sends all operators (3.11) where u ∈ n0+ . We obtain a surjective map L → L, e This map an operator from L to the unique representative of its gauge class lying in L. is called the Miura transformation. It induces a homomorphism of differential rings C[vi(n) ] → C[u(n) i ]. It was shown in [7] that the image of C[vi(n) ] in C[u(n) i ] coincides with the invariant subspace of C[u(n) ] under the left action of the group N + . Hence we obtain from Thei orem 1 that C[vi(n) ] ' C[N + \N+ /A+ ]. Thus, we obtain a local map which assigns to each smooth function v(z) : A1 → n0+ a smooth function A1 → N + \N+ /A+ . According to the results of this section, this map coincides with the Drinfeld-Sokolov map defined above. We also see that the KdV flows on N + \N+ /A+ correspond to the right infinitesimal action of a− on it considered as an open subset of G[t−1 ]\G/A+ . Thus, the passage from mKdV hierarchy to the KdV hierarchy simply consists of projecting from the flag manifold B− \G to the loop space G[t−1 ]\G.


221

Remark 7. Drinfeld and Sokolov attached in [4] a generalized KdV hierarchy to each vertex of the Dynkin diagram of e g; the hierarchy considered above corresponds to the 0th node. In general, we obtain the following picture. Fix j between 0 and l. Let nj+ be the finite-dimensional Lie subalgebra of n+ generj ated by ei , i 6= j. Let N + be the corresponding Lie subgroup of N+ . The dressing operj ator of the j th generalized KdV hierarchy gives rise to a double coset in N + \N+ /A+ . j On the other hand, there is an isomorphism between C[N + \N+ /A+ ] and the ring of nj+ –invariants of C[u(n) i ] (with respect to the left action). The latter is itself a ring of differential polynomials in l variables. Note that it coincides with the intersection of kernels of the operators eL i , i 6= j, which are classical limits of the so-called screening operators (see [7]). Remark 8. Our results and those of [8, 9] suggest the following possible generalizations of the hierarchies of KdV type (compare with Cherednik’s construction [2]). Let G be a complex semisimple Lie group, H be a Cartan subgroup of G, and h be the Lie algebra of H. Let X be a compact complex curve, and S a finite set of points of X. For each s ∈ S, denote by Os and Ks the completed local ring of s and its field of fractions, respectively. Let O = ⊕s∈S Os , and K = ⊕s∈S Ks . Denote by R the subring of K, formed by the Taylor expansions of meromorphic functions on X which are regular outside S. Then the double quotient G(R)\G(K)/H(O)

(3.12)

fibers naturally over the moduli space BunG (X) of G-bundles over X, and it has a natural action of h(K)/h(O) on the right by infinitesimal right translations. It would be interesting to write explicitly the action of the right vector fields generated by elements of h(K) on (3.12). Of special interest is the case where X is an elliptic curve; in this case, an open part of the moduli space BunG (X) is isomorphic to the symmetric power of X. There is a variant of this construction, where H(Os ) is replaced by another commutative subgroup of G(Os ), conjugated to H(Os ) over a finite extension of Ks . The nonlinear Schrödinger equation corresponds to the case where X = CP 1 , and S is one point. In the case of the KdV hierarchies, X and S are the same, and H(O) is replaced by A+ , which is conjugated to it over C((t1/h )). 4. Realization of C[N+ ] as a Polynomial Ring The approach to the mKdV and affine Toda equations used in [8] is based on Theorem 1 which identifies C[N+ /A+ ] with the ring of differential polynomials C[u(n) i ]i=1,...,l;n≥0 . In this section we add to the latter ring new variables corresponding to A+ and show that the larger ring thus obtained is isomorphic to C[N+ ]. An analogous construction has been given in [6] in the lattice case. 4.1. Coordinates on N+ . Consider u(n) i , i = 1, . . . , l; n ≥ 0, as A+ –invariant regular functions on N+ . Recall that ui (x) = (αi , xp−1 x−1 ),

x ∈ N+ .

222


Recall that C is the central element of e g. Now choose an element χ of e h, such that (χ, C) 6= 0. Introduce the regular functions χn , n ∈ I, on N+ by the formula: χn (x) = (χ, xp−n x−1 ),

x ∈ N+ .

(4.1)

g and the pairing on e g. Note that here we consider the action of N+ on e Theorem 5. C[N+ ] ' C[u(n) i ]i=1,...,l;n≥0 ⊗ C[χn ]n∈I . Proof. Let us show that the functions u(n) i ’s and χn ’s are algebraically independent. In order to do that, let us compute the values of the differentials of these functions at the origin. Those are elements of the cotangent space to the origin, which is isomorphic to the dual space n∗+ of n+ . It follows from Proposition 5 that n∗+ can be written as n∗+ = a∗+ ⊕ e n∗+ , where a∗+ = n∗+ is the annihilator of a+ with respect to the pairing between n+ and Ker(ad∗ p−1 ) and e n∗+ with respect to the principal gradation as ⊕∞ n∗+ . Moreover, if we decompose e n∗,j + , j=1 e ∗,j ∗,j ∗,j−1 ∗ n+ isomorphically to e n+ for then dim e n+ = l for all j > 0, and ad p−1 maps e j > 1. By construction of ui ’s given in [8], dui |1 , i = 1, . . . , l, form a basis of e n+∗,1 , and hence ∗,n (n) (n) n+ . Thus, the covectors dui |1 , i = 1, . . . , l; n ≥ dui |1 , i = 1, . . . , l, form a basis of e 0, are linearly independent. Let us show now that the covectors dχn |1 are linearly independent from them and among themselves. For that it is sufficient to show that the pairing between dFm |1 and pn is non-zero if and only if n = m. But we have: −1 −1 pR n · (χ, xp−m x ) = (χ, x[pn , p−m ]x )

(4.2)

= (χ, n(pn , p−n )C)δn,−m = n(pn , p−n )(χ, C)δn,−m , where h is the Coxeter number. Therefore this pairing equals n(pn , p−n )(χ, C)δn,−m . This satisfies the condition above. Thus, the functions u(n) i ’s and χn ’s are algebraically independent. Hence we have an embedding C[u(n) ] i=1,...,l;n≥0 ⊗ C[χn ]n∈I → C[N+ ]. But the characters of the two i spaces with respect to the principal gradation are both equal to Y Y (1 − q n )−l (1 − q i )−1 . i∈I

n≥0

Hence this embedding is an isomorphism.

4.2. Another proof of Theorem 3. Now we will explain another point of view on the equivalence between the formalisms of [4] and [8] established in Sect. 3. In any finite-dimensional representation of N+ , each element x of N+ is represented by a matrix whose entries are Taylor series in t with coefficients in C[N+ ]. Denote cn = (n(pn , p−n )(χ, C))−1 . Proposition 6. Let x be an element of N+ . We associate to it another element of N+ , ! X x = x exp − cn pn χn (x) . n∈I


223

In any finite-dimensional representation of N+ , x is represented by a matrix whose entries are Taylor series with coefficients in the ring of differential polynomials in ui , i = 1, . . . , l. The map N+ → N+ which sends x to x is constant on the right A+ –cosets, and hence defines a section N+ /A+ → N+ . Proof. Each entry of x = x exp −

X

! cn pn χn (x)

n∈I

is a function on N+ . According to Theorem 1 and Theorem 5, to prove the proposition it is sufficient to show that each entry of x is invariant under the right action of a+ . By formula (4.2) we obtain for each m ∈ I: −1 pR m · χn = cn δn,−m ,

and hence pR m xpm exp −

X

x exp −

X

!! cn pn χn (x)

!

+ x exp −

c n pn χ n

=

n∈I

n∈I

X

! c n pn χ n

(−pm ) = 0.

n∈I

Therefore x is right a+ –invariant. To prove the second P statement, let a be an element of A+ and let us show that xa = x. We can write: a = exp i∈I αi pi . Then according to formulas (4.1) and (4.2), −1 χn (xa) = (χ, xap−n a−1 x−1 ) = (χ, xp−n x−1 ) + c−1 n αn = χn (x) + cn αn .

Therefore xa = xa exp −

X

α n pn −

n∈I

X

! cn pn χn (x)

= x.

n∈I

Consider now the matrix x. According to Proposition 6, the entries of x are Taylor series with coefficients in differential polynomials in ui ’s. Hence we can apply to x any R derivation of C[u(n) i ], in particular, ∂n = p−n . In the following proposition we consider pi and u as matrices acting in a finite-dimensional representation. Lemma 4. In any finite-dimensional representation of N+ , the matrix of x satisfies: X x−1 (∂n + (xp−n x−1 )− )x = ∂n + p−n − ci (pR (4.3) −n · χi )pi . i∈I

Proof. Using formula (2.2), we obtain: −1 −1 x−1 (∂n + (xp−n x−1 )− )x = ∂n + x−1 (pR −n x) + x (xp−n x )− x X −1 −1 ci (pR = ∂n + x−1 (xp−n x−1 )+ x − −n · χi )pi + x (xp−n x )− x

= ∂n + p−n −

X

i∈I

ci (pR −n

i∈I

which coincides with (4.3).

· χi )pi ,

224


Second proof of Theorem 3. Let K(z) be the map A1 → N+ /A+ assigned to a smooth function u : A1 → h by Theorem 1. Let K(z) be the element of N+ corresponding to K(z) under the map N+ /A+ → N+ defined in Proposition 6. By construction, K(z) lies in the A+ –coset of K(z). According to Lemma 2, (K(z)p−1 K(z)−1 )− = p−1 + u(z). Setting n = 1 in formula (4.3) we obtain: X −1 K (∂z + p−1 + u(z))K = ∂z + p−1 − ci (pR −1 · χi )pi . i∈I

This shows that K(z) gives a solution to Eq. (3.4), and hence lies in the A+ –coset of the Drinfeld-Sokolov dressing operator M (z). Therefore the cosets of K(z) and M (z) coincide. It is possible to lift the map u(z) → N+ /A+ constructed in [4] and [8] to a map u(z) → N+ . We can first attach to u(z) the coset K(z) and then an element K(z) of N+ defined as in the proof of Theorem 3. In the next section we will show that (n) (n) R H n = pR −1 · Rχn ∈ C[ui ] ⊂ C[N+ ] (recall that χn 6∈C[ui ]). Since p−1 ≡ ∂z , we can z view χn as −∞ Hn dz. Hence we can construct the image of u(z) in N+ by the formula ! Z z X e c n pn Hn dz . K(z) = K(z) exp −∞

n∈I

Comparing (4.3) and (3.4), we can write an equivalent formula ! X Z z pn hn (z)dz , M (z) exp − −∞

n∈I

where hn (z) are defined by formula (3.4). Note that the last formula does not depend on the choice of M (z). We see that in contrast to the map to N+ /A+ , which is local, i.e. depends only on the jet of u(z) at z, the map to N+ is non-local. Remark 9. In [4] it was proved that hn is the hamiltonian of the nth equation of the mKdV hierarchy. In the next section we will prove in a different way that pR −1 · χn is proportional to the hamiltonian of the nth equation of the mKdV hierarchy. Remark 10. Now we can write an explicit formula for the Baker-Akhiezer function associated to u. Recall that this function is a formal solution of eqs. (3.10). From formula (4.3) we obtain the following solution: ! Z z X X p−i ti − c i pi Hi (z)dz 9(t) = K(t) exp − i∈I

e exp − = K(t)

X

i∈I

! p−i ti

−∞

,

i∈I

where t = {ti }i∈I and ti ’s are the times of the hierarchy (in particular, t1 = z). On the other hand, by construction, the action of the vector field ∂n of the mKdV hierarchy


225

e ∈ N+ corresponds to the right action of p−n ∈ a− on N+ ⊂ B− \G. Hence if on K e when all ti = 0, then e K0 ∈ N+ is the initial value of K, e = K e 0 0 (t) , K(t) +

where 0 (t) = exp

X

! p−i ti

i∈I

and g+ denotes the projection of g ∈ B− · N+ ⊂ G on N+ (it is well-defined for almost all ti ’s). Finally, we obtain: e 0 0 (t) 0 (t)−1 . 9(t) = K +

Note that this formula differs slightly from the one given in [17, 10] because in those papers another realization of the flag manifold was chosen: G/B− instead of our B− \G. 5. One-Cocycles, Hamiltonians and τ -Functions In the previous section we established the equivalence between the approaches of [4] and [8] to the mKdV hierarchies. In both papers the mKdV equations were proved to be hamiltonian. In this section we will discuss explicit formulas for the hamiltonians of the mKdV equation and for some closely related cohomology classes of n+ . Note also that both in [4] and [8] it was shown that the hamiltonians of the mKdV equations are integrals of motion of the corresponding affine Toda equation (see also [14, 16]). 5.1. Connection between the hamiltonians and the n+ –cohomology. In [7, 8] the space spanned by the hamiltonians of the mKdV equations was identified with the first cohomology of n+ with coefficients in π0 , H 1 (n+ , π0 ). Let us briefly recall how to assign an mKdV hamiltonian to a cohomology class. The cohomology of n+ with coefficients in π0 can be computed using the Koszul V∗ complex π0 ⊗ (n∗+ ). A cohomology class from H 1 (n+ , π0 ) is represented in the Koszul complex by a functional f on n+ with coefficients in π0 , which satisfies the cocycle condition f ([a, b]) − a · f (b) + b · f (a) = 0. This condition uniquely determines f by its values fi ∈ π0 on the generators ei , i = 0, . . . , l, of n+ . Now set gi = ∂z fi − ui fi , i = 0, . . . , l. As shown in [8], there exists h ∈ π0 , such that gi = eL i · h, i = 0, . . . , l. It was proved in [7, 8] that H 1 (n+ , π0 ) ' a∗+ . Using the invariant inner product, we can identify a∗+ with a− . Let fn be the cohomology class corresponding to p−n ∈ a− . Then hn ∈ π0 constructed from fn is, by definition, the density of the hamiltonian of the nth mKdV equation (i.e. the projection of hn onto the space of local functionals π0 /(Im ∂z ⊕ C) is an mKdV hamiltonian). Below we give explicit formulas for fn (ei ) and hn as functions on N+ /A+ . To R simplify notation we will simply write ei for eL i and p−n for p−n . 5.2. Formulas for one-cocycles. Now recall that π0 ' C[N+ /A+ ]. Hence the values of a one-cocycle of n+ with coefficients in π0 can be viewed as a regular function on N+ /A+ .

226


Proposition 7. There exists a one-cocycle φn , such that (φn (ei )) (K) = (ei , Kp−n K −1 ),

K ∈ N+ /A+ .

(5.1)

The cohomology classes corresponding to these cocycles span H 1 (n+ , π0 ). In particular, if n is a multiplicity free exponent, then the cohomology classes defined by φn and fn coincide up to a constant multiple. Proof. There exists a unique element ρ∨ ∈ e h'e h∗ , such that (αi , ρ∨ ) = 1, ∀i = 0, . . . , l, ∨ ∗ e e and (d, ρ ) = 0. But h is isomorphic to h via the non-degenerate inner product (·, ·). Let us use the same notation for the image of ρ∨ in e h under this isomorphism. Then ρ∨ ∨ ∨ ∨ satisfies: [ρ , ei ] = ei , [ρ , hi ] = 0, [ρ , fi ] = −fi , i = 0, . . . , l. Thus, the adjoint action of ρ∨ on e g coincides with the action of the principal gradation. Any function F ∈ C[N+ ] can be viewed as an element of the zeroth group of the Koszul complex of the cohomology of n+ with coefficients in C[N+ ]. The coboundary of this element is a (trivial) one-cocycle, whose value on ei is ei · F ∈ C[N+ ], i = 0, . . . , l. Consider a function Fn = ρ∨ n on N+ defined by the formula Fn (x) = (ρ∨ , xp−n x−1 )

(5.2)

g and we consider the adjoint action Note that here p−n is considered as an element of e g. The value of the corresponding one-cocycle on ei is equal to ei · Fn . We of N+ on e have: (ei · Fn )(x) = (ρ∨ , [ei , xp−n x−1 ]) = ([ρ∨ , ei ], xp−n x−1 ) = (ei , xp−n x−1 ). Thus, there exists a one-cocycle f of n+ with coefficients in C[N+ ], such that f (ei ) = (ei , xp−n x−1 ), i = 0, . . . , l.

(5.3)

Moreover, f (ei ) is A+ –invariant for all i = 0, . . . , l. Indeed, (pm · f (ei ))(x) = (ei , x[pm , p−n ]x−1 ) = n(pn , p−n )(ei , xCx−1 )δn,−m = n(pn , p−n )(ei , C)δn,−m = 0. Therefore formula (5.3) defines a one-cocycle of n+ with coefficients in C[N+ /A+ ] ' π0 . This is the cocycle φn . By construction, φn is a trivial one-cocycle of n+ with coefficients in C[N+ ]. But it is non-trivial as a one-cocycle of n+ with coefficients in C[N+ /A+ ]. Indeed, if it were a coboundary, there would exist an A+ –invariant function Fen on N+ , such that φn (ei ) = ei · Fen . But then ei · (Fen − Fn ) = 0 for all i, and Fen − Fn is N+ –invariant, and hence constant. However, by (4.2), pn · Fn = nh(pn , p−n ) 6= 0, where h is the Coxeter number of e g. Hence the function Fn is not A+ –invariant. e Thus, Fn − Fn can not be a constant function. Therefore φn defines a non-zero cohomology class. Let us compute its degree with respect to the principal gradation. We have: ρ∨ · (φn (ei )) (x) = (ei , [(xρ∨ x−1 )+ , xp−n x−1 ]) = (ei , [xρ∨ x−1 , xp−n x−1 ]) − (ei , [(xρ∨ x−1 )− , xp−n x−1 ]) = (ei , x[ρ∨ , p−n ]x−1 ) − ([ei , ρ∨ ], xp−n x−1 ]) = (−n + 1)φn (ei ). Hence the degree of φn equals −n.


227

For the multiplicity free exponent n this implies that the cohomology class of φn is proportional to that of fn . For the multiple exponents i which occur in the case of (1) (see Remark 1), we need to show that the cocycles φ1i and φ2i , corresponding to two D2n linearly independent elements p1−i and p2−i of a− of degree −i, are linearly independent. But a linear combination αφ1i + βφ2i of these cocycles is just the cocycle corresponding to αp1−i + βp2−i ∈ a− . The argument that we used above can be applied to the cocycle αφ1i + βφ2i to show that it is non-trivial unless both α and β equal 0. Remark 11. Homogeneous functions on N+ /A+ are necessarily algebraic. Thus, φn (ei ) ∈ C[N+ /A+ ]. Remark 12. One can show in the same way as above that for any χ ∈ e h, such that (χ, C) 6= 0, there exists a one-cocycle χ en of n+ with coefficients in C[N+ /A+ ], which en (ei ) satisfies the following property: considered as an A+ –invariant function on N+ , χ en equals ei · χn , where χn is the function on N+ defined in Sect. 4.1. The one-cocycle χ is homologous to Fn , suitably normalized. 5.3. Formulas for hamiltonians. Now we can find a formula for the density of the nth mKdV hamiltonian using Proposition 7 and the procedure of Sect. 5.1. Proposition 8. The function Hn on N+ /A+ , such that Hn (K) = (p−1 , Kp−n K −1 ),

K ∈ N+ /A+

is a density of the nth hamiltonian of the mKdV hierarchy. Proof. We have to show that ei · Hn = p−1 φn (ei ) − ui φn (ei ),

i = 0, . . . , l.

(5.4)

Let us consider functions on N+ /A+ as A+ –invariant functions on N+ . Recall from Sect. 2 that there is a unique up to a constant isomorphism λ between C[N+ ] and the contragradient Verma module Mλ∗ , which commutes with the left action of n+ . For R a ∈ g the operator λ a−1 λ on C[N+ ] is the first order differential operator a + fλ (a). −1 Here fλ (a) = λ (a · vλ ) We know that ui = fαi (p−1 ), see [8] and Sect. 2. Hence R −1 −αi p−1 −αi = p−1 − ui , and hence formula (5.4) can be rewritten as −αi (ei · Hn ) = p−1 · −αi (φn (ei ))

i = 0, . . . , l.

(5.5)

Let us show that Hn = p−1 · Fn , where the function Fn ∈ C[N+ ] is defined by formula (5.2). Indeed, (p−1 · Fn )(x) = (ρ∨ , [(xp−1 x−1 )+ , xp−n x−1 ]) = −(ρ∨ , [(xp−1 x−1 )− , xp−n x−1 ]) = −([ρ∨ , (xp−1 x−1 )− ], xp−n x−1 ) = ([ρ∨ , p−1 ], xp−n x−1 ) = (p−1 , xp−n x−1 ). The fact that Hn is A+ –invariant can be proved in the same way as for φn (ei ). ∗ commutes with the action of g, Now recall that the map −αi ei : C[N+ ] → M−α i where g acts on C[N+ ] from the right by vector fields, see [8], Sect. 4. Therefore we obtain −αi ei (p−1 · Fn ) = p−1 · −αi ei (Fn ). This implies formula (5.5) if we take into account that Hn = p−1 ·Fn and φn (ei ) = ei ·Fn .

228


Remark 13. Our formula for the hamiltonians is equivalent to the formula given by Wilson [16], (4.10). Remark 14. One can also construct the density of the nth hamiltonian as p−1 · χn where χn was defined in Sect. 4.1. For different χ, these densities, suitably normalized, differ by total derivatives, and hence define the same hamiltonian. 5.4. Involutivity of the hamiltonians. Now we want to prove that the Poisson bracket between two mKdV hamiltonians vanishes. This is equivalent to showing that p−n ·Hm = p−1 Hn,m for some Hn,m ∈ C[N+ /A+ ] (see [8]). Proposition 9. Define Hn,m ∈ C[N+ /A+ ] by formula Hn,m (K) = −([ρ∨ , (Kp−n K −1 )− ], Kp−m K −1 ). Then p−n · Hm = p−m · Hn = p−1 · Hn,m . Proof. We have: (p−n · Hm )(K) = (p−1 , [(Kp−n K −1 )+ , Kp−m K −1 ]) = (p−1 , [(Kp−n K −1 )+ , (Kp−m K −1 )− ]), because (p−1 , [y1 , y2 ]) = 0 if y1 , y2 ∈ n+ . On the other hand, (p−m · Hn )(K) = (p−1 , [(Kp−m K −1 )+ , Kp−n K −1 ]) = −(p−1 , [(Kp−m K −1 )− , Kp−n K −1 ]) = −(p−1 , [(Kp−m K −1 )− , (Kp−n K −1 )+ ]), because (p−1 , y) = 0 if y ∈ b− . Therefore p−n · Hm = p−m · Hn . Consider now Hm as an A+ –invariant function on N+ . Then we have: Hm = p−1 ·Fm . Hence p−n · Hm = p−1 · (p−n · Fm ). Let Hn,m = p−n · Fm . We obtain: (p−n · Fm )(x) = (ρ∨ , [(xp−n x−1 )+ , xp−m x−1 ]) = −(ρ∨ , [(xp−n x−1 )− , xp−m x−1 ]) = −([ρ∨ , (xp−n x−1 )+ ], xp−m x−1 ). The latter expression is A+ –invariant, which can be shown in the same way as in the proof of Proposition 7. Hence Hn,m ∈ C[N+ /A+ ] and p−n · Hm = p−m · Hn = p−1 Hn,m . 5.5. Connection with τ –functions. The τ –functions have the following meaning from our point of view. For λ ∈ e h∗ , consider the contragradient Verma module Mλ∗ over g. This module can be realized in the space of sections of a line bundle ξλ over N+ , considered as a big cell of the flag manifold B− \G. By definition, the τ –function τλ corresponding to λ is the unique up to a constant N+ –invariant section of ξλ over N+ . Remark 15. This should be compared with the definition of the τ –functions in the framework of the Grassmannian approach [3, 15, 17, 2].


229

Note that ξλ can be trivialized over N+ , and so there exists a unique up to a nonzero constant isomorphism between the space of sections of ξλ and C[N+ ]. Under this isomorphism, τλ corresponds to a constant function on N+ . Let Λi , i = 0, . . . , l, be the fundamental weights of the affine algebra g. We call τΛi the ith τ –function of g and denote it by τi . Let us also set τ = τρ∨ . According to Proposition 2, for any a ∈ g, a · τλ = fλ (a)τλ , where fλ (a)(x) = hλ, xax−1 i. In particular, we see that eϕi = ταi , and p−1 eϕi = ∂z eϕi = ui eϕi . Note that eϕi can b N we have: be expressed in terms of τj ’s. For example, for g = sl −1 2 −1 τi τi+1 , eϕi = τi−1

which is well-known. Now we can interpret the functions Fn as logarithmic derivatives of τ . Indeed, we obtain ∂n τ = Fn τ , so that we can formally write: Fn = ∂n log τ . Further, Hn = ∂n ∂z log τ , and, more generally, Hn,m = ∂n ∂m log τ , which coincides with known n+1 log ταi . results. Similarly, we can write: u(n) i = ∂z Remark 16. More generally, we have the following formula for the function χn defined in Sect. 4.1: χn = ∂n τχ /τχ . To summarize, the group N+ has natural coordinates u(n) and Fn , which can be i obtained as logarithmic derivatives of τ –functions. The vector fields pR −n written in terms of these coordinates provide the flows of the mKdV hierarchy, and the vector field Pl L i=0 ei written in terms of these coordinates gives the affine Toda equation [8]. b 2 . Here we will write explicit formulas for the action of the generators 5.6. Example of sl b 2 and mKdV hamiltonians on the corresponding unipotent of the nilpotent subalgebra of sl subgroup. According to the results of this section, we have an isomorphism C[N+ ] ' C[u(n) , Fm ]n≥0,m odd . The left action of the generators e0 and e1 of n+ on C[N+ ] is given by e0 = −

X n≥0

e1 = −

X

n≥0

Pn+

X ∂ ∂ + φm (e0 ) , (n) ∂u ∂Fm m odd

Pn−

X ∂ ∂ + φm (e1 ) , (n) ∂u ∂Fm m odd

± = s where Pn± are elements of C[u(n) ], defined recursively as follows: P0± = 1, Pn+1 ± ± ∂Pn ± uPn , and φm (ei ) are the values of a one-cocycle φm of n+ with coefficients in C[u(n) i ] of degree m. The right action of pk , k positive odd, is given by 4k∂/∂Fk , and the action of p−k , k positive odd, is given by

p−k =

X n≥0

(∂ n+1 qk )

X ∂ ∂ + Hk,m . (n) ∂u ∂Fm m odd

230


The mth mKdV equation now reads: ∂ m u = qm . This equation is hamiltonian with the hamiltonian (1/m)Hm,1 , and hence qm =

1 δHm,1 . m δu

The involutivity of the hamiltonians means that X n≥0

(∂ n+1 qk )

∂Hm,1 = ∂Hk,m ∂u(n)

(note that Hk,m = Hm,k and Hm,1 = H1,m = Hm ). The KdV variable is v = 21 u2 + u0 , and C[v (n) ]n≥0 ⊂ C[u(n) ]n≥0 coincides with the e1 –invariant subspace of C[u(n) ]n≥0 . Acknowledgement. We would like to thank B. Feigin for his collaboration in [6, 7, 8] and useful discussions. The second author also thanks D. Ben-Zvi for interesting discussions. The research of the second author was supported by grants from the Packard Foundation, NSF and the Sloan Foundation.

References 1. De Groot, M.F., Hollowood, T.J., Miramontes, J.L.: Commun. Math. Phys. 145, 57–84 (1992) 2. Cherednik, I.: Funct. Anal. Appl. 17, 243–245 (1983); Russ. Math. Surv. 38, N 6, 113–114 (1983) 3. Date, E., Jimbo, M., Kashiwara, M. Miwa, T.: In Non-linear Integrable Systems – Classical Theory and Quantum Theory. M. Jimbo, T. Miwa (eds.), Singapore: World Scientific, 1983, pp. 39–120 4. Drinfeld, V.G., Sokolov, V.V.: Sov. Math. Dokl. 23, 457–462 (1981); J. Sov. Math. 30,1975–2035 (1985) 5. Enriquez, B.: Theor. Math. Phys. 98, 256–258 (1994) 6. Enriquez, B., Feigin, B.: Theor. Math. Phys. 103, 738–756 (1995) (hep-th/9409075) 7. Feigin, B., Frenkel, E.: In Lect. Notes in Math. 1620, pp. 349–418, Springer Verlag, 1995 (hepth/9310022) 8. Feigin, B., Frenkel, E.: Invent. Math. 120, 379–408 (1995) (hep-th/9311171) 9. Feigin, B., Frenkel, E.: Non-linear Schrödinger equations and Wakimoto modules. To appear 10. Hollowood, T., Miramontes, J.L.: Commun. Math. Phys. 157, 99–117 (1993) 11. Kac, V.G.: Infinite-dimensional Lie Algebras. 3rd Edition, Cambridge: Cambridge University Press, 1990 12. Kac, V.G.: Adv. Math. 30, 85–136 (1978) 13. Kostant, B.: In: Lect. Notes in Math. 466, Berlin–Heidelber–New York: Springer Verlag, 1974, pp. 101–128 14. Kupershmidt, B.A., Wilson, G.: Commun. Math. Phys. 81, 189–202 (1981) 15. Segal, G., Wilson, G.: Loop groups and equations of KdV type. Publ. Math. IHES 63, 5–65 (1985) 16. Wilson, G.: Ergod. Th. and Dynam. Syst. 1,361–380 (1981) 17. Wilson, G.: C. R. Acad. Sc. Paris 299, Serie I, 587-590 (1984); Phil. Trans. Royal Soc. London A 315, 393–404 (1985) 18. Zakharov, V.E., Shabat, A.B.: Funct. Anal. Appl. 13, 166–174 (1979) Communicated by G. Felder

Commun. Math. Phys. 185, 231 – 256 (1997)

Communications in


On the Stability of Stationary Wave Maps Jalal Shatah1,? A. Shadi Tahvildar-Zadeh2,?? 1 2

Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012, USA Department of Mathematics, Princeton University, Princeton, NJ 08544, USA

Received: 19 September 1995 / Accepted: 1 November 1996

Abstract: Equivariant wave maps from S2 ×R into S2 have smooth, stationary solutions which are critical points of the energy subject to constant charge. These solutions are globally stable under equivariant perturbations. Consequently, there exists a large set of initial data, with no degree or energy restrictions, for which the Cauchy problem is globally well-posed.

1. Introduction A wave map is a map U from a Lorentzian manifold (M m+1 , g) into a Riemannian manifold (N n , h) which is a critical point of the Lagrangian Z Z Z 1 1 1 ∗ 2 L[U ] = Trg U h = kDU k = g αβ hab ∂α U a ∂β U b . 2 M 2 M 2 M In local coordinates U i satisfy a system of hyperbolic, semi-linear equations i (U )∂α U j ∂ α U k = 0, ∂µ ∂ µ U i + Γjk

(1.1)

where Γ ’s are the Christoffel symbols of the target manifold N . Wave maps are the hyperbolic analogue of harmonic maps between two Riemannian manifolds, and were first introduced by Gell-Mann and Levy (see [5]) in the context of a nonlinear sigmamodel, with M = R3,1 and N = S3 . Other examples showed up later in other field theories, and led to the generalization of the concept.1 ?

Supported in part by the National Science Foundation grant DMS-9401558. Supported in part by the National Science Foundation grants DMS-9203413 and DMS-9504919. 1 See [11] for a survey of the history and the role of sigma models and harmonic maps in physics. For the connection between General Relativity and wave maps see [1] and [9]. ??

232

J. Shatah, A.S. Tahvildar-Zadeh

The natural problem to study in connection with Eq.(1.1) is the Cauchy problem, specifying data on a space-like Cauchy hypersurface Σ in M , U |Σ = U 0 ,

∂ t U |Σ = U1 .

(1.2)

A major open question for the Cauchy problem of wave maps is whether in two space dimensions (m = 2), smooth initial data imply global existence, uniqueness, and regularity of solutions for (1.1,1.2). For the case where the domain M is the Minkowski space R2,1 the answer is known under certain symmetry assumptions, namely radial and co-rotational invariance. There are also analytical restrictions on the target that exclude the interesting case when N = S2 . In this paper we consider the case where M = S2 ×R and N = S2 . First we will prove the existence of stationary maps generated by the symmetry groups, and then show that these maps are stable under equivariant perturbations. A consequence of the stability result is the existence of a large set of initial data, with no degree or energy restrictions, for which the Cauchy problem is globally well posed. 1.1. Symmetry and Reduction. Let M and N be two manifolds and U a map between them. Let G and H be two symmetry groups acting on M and N respectively. U is called equivariant with respect to the actions of G and H if there exists a homomorphism ρ : G → H such that for all g ∈ G and x ∈ M , U (gx) = ρ(g)U (x). If ρ is trivial, then U is said to be invariant. Equivariant solutions exist, for example, when the domain M = M0 × R, where M0 is a rotationally symmetric manifold, and when the target N = S2 . The metric on M is given by −dt2 + dr2 + γ 2 (r)dθ2 , where (r, θ) ∈ R+ × S1 are local coordinates on M0 and γ a smooth, odd function such that γ 0 (0) 6= 0. By embedding S2 ⊂ R3 we have U as a unit vector in R3 satisfying the equation U + (|Ut |2 − |∇U |2 )U = 0,

(1.3)

0

where = ∂t2 − ∆M0 = ∂t2 − ∂r2 − γγ ∂r − γ12 ∂θ2 is the wave operator on M , Ut = ∂U ∂t etc., and |∇U |2 = |Ur |2 + γ −2 |Uθ |2 . For this problem we will consider the following group actions and the corresponding equivariant solutions: Spatial Rotations. Let G = SO(2) be the group of spatial rotations, and H = SO(3). It is easy to see that the ansatz x w ∈ R3 , |w| = 1, R ∈ SO(3), U (x, t) = R( )w(|x|, t), |x| makes U equivariant with respect to these actions. From (1.3) we obtain equations for R and w. The R equation is easily solved to give R(θ) = eA`θ with ` ∈ Z and A a 3 × 3 skew-symmetric matrix which without any loss of generality can be taken to be   0 −1 0 A = 1 0 0, (1.4) 0 00 i.e., Av = k × v for any v ∈ R3 , where k = (0, 0, 1). We then obtain the following reduced equation for w(r, t):

On the Stability of Stationary Wave Maps

wtt − wrr −

γ0 `2 wr + (|wt |2 − |wr |2 )w = 2 (A2 w + |Aw|2 w). γ γ

233

(1.5)

The invariant case corresponds to ` = 0 and is also referred to as “spherical symmetry.” The global existence in this case was shown in [2]. Another case of equivariant maps corresponds to the unit vector w remaining in a fixed plane, i.e. w = (0, sin φ(r, t), cos φ(r, t)). This will from now on be referred to as the “co-rotational” case. The global existence problem under this type of symmetry was solved in [13].2 Time translations. Let G = (R, +) and H as before. The corresponding equivariance ansatz is ω ∈ R+ , u ∈ R3 , |u| = 1. (1.6) U (x, t) = eAωt u(x), These are the so-called “stationary waves” (meaning that they are stationary in a rotating frame). u will satisfy a nonlinear elliptic equation in this case: ∆M0 u + |∇u|2 u = ω 2 (A2 u + |Au|2 u). As a further reduction, one can take G to be SO(2) × (R, +) and H as before. Then one possible ansatz is U (x, t) = eA(ωt+`θ) u(r), where we can assume u(r) = (0, sin f (r), cos f (r)) and f satisfies the ODE which was first derived by Duff and Isham [3] in the flat case γ(r) = r as a special, time-periodic solution of the nonlinear sigma-model: f 00 +

γ0 0 `2 f + (ω 2 − 2 ) sin f cos f = 0. γ γ

2. Existence of Stationary Wave Maps In this section we prove the existence of stationary solutions for the wave map problem in the special case when N = M0 = S2 . Let (α, β) be the spherical coordinates on M0 . Then the line element of the Lorentzian domain is −dt2 + dα2 + sin2 αdβ 2 . The wave map Lagrangian for a map U : S2 × R → S2 is Z 1 L= −|Ut |2 + |∇U |2 , 2 S2 ×R where |∇U |2 = |Uα |2 + sin12 α |Uβ |2 . The wave map equations are the Euler-Lagrange equations of L subject to the constraint |U |2 = 1: U + (|Ut |2 − |∇U |2 )U = 0. The stationary solution ansatz (1.6) has a variational characterization that is common to Hamiltonian systems with symmetry: Such solutions can be obtained by “minimizing energy subject to constant charge.” In our case we can define the conserved quantities energy E and charge Q associated with a wave map U as follows: 2 It should be noted however that the meanings of the words “equivariant” and “co-rotational” in [13] and [14] are different from those in this paper.

234


E(U ) =

1 2 Z

Z

Q(U ) = S2

S2

|Ut |2 + |∇U |2 ,

(k × U ) · Ut .

(2.1) (2.2)

If we now set P = Ut , then using the method of Lagrange multipliers, we need to minimize the quantity (E − ωQ)(U, P ) on a suitable domain in a function space such as H 1 × L2 , for ω ∈ R. It is easy to see that Z 1 |P − ωk × U |2 + |∇U |2 − ω 2 |k × U |2 . E − ωQ = 2 S2 From here it is clear that one needs P = ωk × U for a minimum, which is equivalent to (1.6). Therefore the problem reduces to minimizing the following functional: Z 1 |∇u|2 − ω 2 |k × u|2 Gω (u) = 2 S2 over the following subset of the Sobolev space H 1 (S2 ): X` = {u : S2 → R3 | u ∈ H 1 (S2 ), |u| = 1, degS2 →S2 u = `}. Note that the degree is well-defined for an H 1 map of S2 into itself since we are in two space dimensions. It is easy to see that the functional Gω is bounded below. In fact in the static case ω = 0 it is well-known that Z 1 |∇u|2 = 4π|`|, inf X ` 2 S2 so that we have the lower bound Gω (u) > 4π|`| − 2πω 2 := G∗ .

(2.3)

Let m = inf u∈X` Gω (u) and let {un } be a minimizing sequence in X` . From the definition of Gω we have that un is a uniformly bounded sequence in H 1 (S2 ). Thus there is a subsequence, also denoted by un , that converges weakly in 1 2 H 1 , strongly in L2 and almost everywhere to a map R v ∈2 H (S ). Moreover, |v(x)| = 2 1, a.e. x ∈ S , and by the lower semicontinuity of |∇u| we have that m = lim inf Gω (un ) ≥ Gω (v). n→∞

However, as the following proposition illustrates, m = G∗ and v does not belong to X` because of a degree change: Proposition 1. There is a sequence un in X` such that Gω (un ) & G∗ and un converges pointwise to a constant map. Proof. Let un = (cos ρn , sin ρn cos `β, sin ρn sin `β), with

α |`| ) ), 2 R so that limn→∞ ρn = π. It is easy to see that un ∈ X` and S2 |∇un |2 = 8π|`| while |k × un |2 = sin2 ρn cos2 `β + cos2 ρn → 1 as n → ∞, and thus Gω (un ) → G∗ , while un → (−1, 0, 0). ρn = ρn (α) = 2 tan−1 (n(tan


235

One way of finding stationary solutions is to minimize Gω on a subset X`0 ⊂ X` of maps satisfying the additional equivariance condition u(eAs x) = eA`s u(x),

∀s ∈ R,

a.e. x ∈ S2 ,

with matrix A as in (1.4). Note that X`0 is the set of all fixed points of the following group action on X` : Let G := {eAs }s∈R be a fixed representation of the rotation group SO(2) in SO(3). Let g ∈ G and v ∈ X` . Then we define the action πg : G × X` → X` by (πg v)(x) = g −` v(gx), for all x ∈ S2 . Moreover, the functional Gω is invariant under this action, i.e. Gω (πg v) = Gω (v). In this way we reduce the problem to finding the minima of Gω restricted to X`0 . Remark 1. However, a key question remains: Assuming for the moment that Gω attains its infimum on a smooth function in X`0 , will this minimum be at least a local minimum for Gω on the original domain X` ? The answer to this question, as we shall see later in Proposition 3 is no, at least for ` > 1 and ω small. In other words, Coleman’s Principle (see e.g. [6]) does not hold. Nevertheless, since the absence of any useful regularity result for wave maps in the general non-symmetric case precludes any discussion of stability of stationary solutions outside the symmetry class, the study of the symmetric case is still justified. The rest of this section is devoted to the proof of the existence of a minimizer in X`0 . We begin by describing the elements in X`0 . 2.1. Equivariant Maps. We begin with some remarks on the notation we use to describe spherical coordinates on S2 . Let U denote the unit circle in the complex plane C, and let Φ : S2 → [0, π], Z : S2 → U denote spherical coordinate functions defined for y ∈ R3 with |y| = 1: y1 + iy2 Φ(y) = cos−1 y3 , Z(y) = p 2 . y1 + y22 These functions are smooth except at the poles P := (0, 0, 1)

− P := (0, 0, −1),

where Z is undefined and ∇Φ is singular. Let K denote the cylinder [0, π] × S2 , with π1 and π2 denoting the projections on each factor. Let I : K → S2 ⊂ R3 be the mapping I(φ, z) = z sin φ + k cos φ, where φ ∈ R and z ∈ U. Then y = I(Φ(y), Z(y)) for y ∈ R3 , |y| = 1. Let u be an H 1 map of S2 into S2 and let G := {eAs }s∈R be a fixed representation of SO(2) in SO(3), corresponding to a choice of a 3 × 3 skew-symmetric matrix A. Suppose u is `-equivariant with respect to the action of G, i.e, that there exists an integer ` 6= 0 such that u(gx) = g ` u(x), ∀g ∈ G, a.e. x ∈ S2 . Then there is clearly a choice of basis for the domain and the target spheres such that the fixed points of the action of G on these two spheres are the “poles” P and −P . With this choice of basis we have Lemma 1. Suppose u is as in the above. Then,

236


1. u is Hölder continuous away from the poles: u ∈ C 1/2 (S2 \ {P, −P }, R3 ), and u can be defined at the poles in such a way that it becomes a continuous map of S2 into S2 , mapping poles into poles. 2. Let α and z be spherical coordinates on the domain sphere, i.e. α = Φ(x) and z = Z(x), and let C± := (u◦I)−1 (±P ) ⊂ K. Then there exists a bounded continuous function φ : [0, π] → R, φ(0) = 0, φ(π) = 0 mod π, and a map ζ : (0, π) → U continuous except possibly on π1 (C+ ∪ C− ), such that u(x) = I(φ(α), ζ(α)z ` ). Moreover, degS2 →S2 u =

` φ(π). π

Proof. Let (α1 , z1 ) and (α2 , z2 ) be two nearby points of S2 . Then, |u(α2 , z2 ) − u(α1 , z1 )| ≤ |u(α2 , z2 ) − u(α2 , z1 )| + |u(α2 , z1 ) − u(α1 , z1 )| Z α2 ∂u 0 (α , z1 ) dα0 | ≤ |z2 − z1 | + | α1 ∂α ≤ |z2 − z1 | + (min{sin α2 , sin α1 })−1/2 kukH 1 |α2 − α1 |1/2 establishing the Hölder continuity away from P or −P . To prove continuity of u, we first note that since u is `- equivariant we have |∇u|2 = |

∂u 2 `2 | + |Au|2 . ∂α sin2 α

that |Au|2 = u21 + u22 → 0 at least along a sequence xk → ±P , otherwise RThis implies 2 |∇u| will not be finite. Moreover, Z α+h ∂u hAu, A i dα0 |Au(α + h, z)|2 − |Au(α, z)|2 = 2 ∂α α Z α+h ∂u |Au|2 0 ≤ sin α0 | |2 + dα ∂α sin α0 α = o(1), √ as h → 0, and hence |Au(x)|2 → 0 as x → ±P . Now let f (x) = sin−1 x + x 1 − x2 , √ so that f 0 (x) = 2 1 − x2 . Then Z α+h ∂u3 0 dα | f 0 (u3 ) |f (u3 (α + h, z)) − f (u3 (α, z))| ≤ | ∂α α Z α+h ∂u3 2 4 | sin α0 + | (1 − u23 ) dα0 ≤ ∂α sin α0 α = o(1), as h → 0. Therefore u3 is also continuous everywhere, and as x → P , u(x) must approach only one of the poles in the target. Similarly for x → −P . Let γ : [0, π] → S2 be the continuous curve γ(α) = u(α, 0). Since u is equivariant, the closed sets C+ and C− each consist of a disjoint union of “bands” in K. Let us first consider the case where these bands do not interleave, i.e., suppose for the moment


237

that a map u in addition to being H 1 and `-equivariant, also has the property that if γ(α1 ) = γ(α2 ) = P , then γ(α) 6= −P for any α ∈ [α1 , α2 ], and likewise for −P . On K \ (C+ ∪ C− ) we then define v := Φ ◦ u ◦ I and w := Z ◦ u ◦ I, where Φ, Z, and I as in the above. We extend v to all of K by setting v = 0 on C+ and v = π on C− . Thus we have that 0 ≤ v ≤ π and u = I(v, w). By equivariance of u, for all g ∈ SO(2) ∼ = U we have that v(α, gz) = v(α, z)

and

w(α, gz) = g ` w(α, z),

for all α ∈ [0, π] and z ∈ U. It follows that v(α, 1) = v(α, z) and w(α, 1) = z −` w(α, z) for all z ∈ U, so that we can define φ(α) := v(α, 1),

ζ(α) := w(α, 1).

We then have u(α, z) = I(φ(α), ζ(α)z ` ). The non-interleaving assumption about u implies that the continuous curve γ contains at most one arc joining P and −P on the target sphere. It is then clear that there are only three possibilities for the degree of u as a map between spheres, each corresponding to a specific set of boundary values for φ: 1. φ(0) = φ(π) = 0 or π and deg u = 0, 2. φ(0) = 0, φ(π) = π and deg u = `, 3. φ(0) = π, φ(π) = 0 and deg u = −`. Now consider a general `-equivariant H 1 map u with a nonzero degree. Without loss of generality we can assume γ(0) = P . We define a sequence of numbers αj ∈ [0, π] as follows: α1 := the smallest α > 0 such that γ(α) = −P, αj := the smallest α > αj−1 such that γ(αj ) = (−1)j P. Clearly, there exists an integer k > 0, depending only on kukH 1 , such that αk = π. Let α0 = 0 and let ui : S2 → S2 be defined as follows:   (−1)i−1 P 0 ≤ α ≤ αi−1 ui (α, z) := u(α, z) αi−1 ≤ α ≤ αi  (−1)i P α ≤ α ≤ π. i Then ui is of the non-interleaving type considered above, and hence there exists a pair (φi , ζi ) of functions such that ui (α, z) = I(φi (α), ζi (α)z ` ),

φi (αi−1 ) =

π (1 + (−1)i ), 2

φi (αi ) =

π (1 − (−1)i ). 2

Let di = deg ui . Then by construction, di = ` or −`. Moreover, deg u =

k X

deg ui = m`,

i=1

for some integer m. We need to define the function φ such that it remains continuous on [0, π] and that its final value φ(π) determines the degree. It is easy to see that under these conditions we can no longer keep φ between 0 and π. Instead we do the following:

238


For i ≤ k − 1 let χi be the characteristic function of the interval [αi−1 , αi ) and let χk be the characteristic function of the interval [αk−1 , αk ]. Let 2[j/2] k X X di dj , mj π + (−1)j−1 φj (α) χj (α), mj := φ(α) := ` ` j=1

i=1

where [·] denotes the integer part. One can easily check that φ is continuous and satisfies the degree formula as claimed. Moreover, n1 π ≤ φ ≤ n2 π for two integers n1 , n2 . 2.2. Minimization. Let (α, z) denote local coordinates on the domain sphere, and let u ∈ X`0 . Then by Lemma 1 u can be expressed in local coordinates as (2.4) u(x) = I(φ(α), ζ(α)z ` ), where φ is a continuous function such that φ(0) = 0 and φ(π) = π. In terms of these local coordinates the energy functional Gω can be written as Z π `2 Gω (u) = π (∂α φ)2 + sin2 φ( 2 − ω 2 ) sin α dα sin α 0 Z π 2 2 sin φ|∂α ζ| sin α dα +π 0

:= H`,ω (φ) + J(u). For every function u ∈ X`0 given by (2.4), we associate a new function u˜ ∈ X`0 given by u(x) ˜ = I(φ(α), z ` ). It is clear that ˜ ≤ Gω (u) , H`,ω (φ) = Gω (u) and that def d(`, ω) = inf Gω (u) = inf H`,ω (φ). u∈X`0

φ(0)=0 φ(π)=π

Therefore we start by studying the minima of the reduced functional Z π dφ `2 ( )2 + sin2 φ( 2 − ω 2 ) sin α dα, (2.5) H`,ω (φ) = π dα sin α 0 subject to the boundary conditions φ(0) = 0, φ(π) = π. The parameters ` and ω are taken to be nonzero, and without loss of generality we can assume ` > 0, ω > 0. The structure of the functional H`,ω , in particular the noncompactness, becomes tractable if we introduce a variable s = ln tan α2 . Then, Z ∞ dφ ω2 ( )2 + (`2 − H`,ω (φ) = π ) sin2 φ ds. cosh2 s −∞ ds Note that H`,0 is translation invariant in s. As a result, in the static case, with ω = 0, the infimum is achieved on a non-compact, one-parameter family of functions φa∗ (s) = 2 tan−1 e`(s−a) . (These correspond to the harmonic mappings ζ 7→ λζ ` of the Riemann sphere.) The following proposition implies that all minima of H`,ω , if they exist, must necessarily be monotone nondecreasing.


239

Proposition 2. Let φ be such that H`,ω (φ) < ∞, φ(−∞) = 0, φ(∞) = π. Then there ˜ ˜ exists a function φ˜ such that φ˜ is monotone nondecreasing, φ(−∞) = 0, φ(∞) = π and ˜ < H`,ω (φ), with equality holding iff φ is monotone. H`,ω (φ) Proof. First observe that since H`,ω (kπ ± φ) = H`,ω (φ) for k ∈ Z, by successive reflections along the lines φ = kπ we can assume that 0 ≤ φ ≤ π. Define s0 ≥ 0 by cosh−1 (ω/`) ` < ω s0 := , 0 `≥ω and let I0 denote the interval [−s0 , s0 ]. To rearrange φ into a monotone function we perform the following: 1. Outside I0 : If φ is not monotone on (−∞, −s0 ], then φ has a local maximum at smax < −s0 . Choose two points m∗ < smax < m ≤ −s0 such that φ(m) = φ(m∗ ) < φ(smax ). Let ˜ := φ(s − m + m∗ ) s ≤ m φ(s) φ(s) s>m (see Fig.1). Then Z Z m 2 ˜ ds = φ˜ 0 (s) + `2 sin2 φ(s)

m∗

φ0 (s)2 + `2 sin2 φ(s) ds,

−∞

−∞

while, Z

m −∞

˜ sin2 φ(s) ds = cosh2 s ≥

Z

m∗

Z

−∞ m∗ −∞

sin2 φ(s) ds cosh2 (s + m − m∗ ) sin2 φ(s) ds, cosh2 s

since m > m∗ and cosh s is decreasing for s < 0. Hence, Z m ω2 ˜ ≥ φ0 (s)2 + (`2 − H`,ω (φ) − H`,ω (φ) ) sin2 φ(s) ds > 0. 2 cosh s m∗

π

0

−s0

−∞

................................ ..... ...... . .... ..... . .... .... . ... . ... . . ... .. . . ... .. . . ... . .. ... . . . . . ... . . .. . . . . .... ..... ..... ..... ..... ...... ..... ..... ..... ..... ...... . . . . . ... . . . . . . .... .. . . . . . . ..... . . ..... . . ...... . . ..... . ..... . . . . . . . . . ...... . . . . . . . . . . . . . . . . . . . . . ...................................... . . . .

q qqqq qqqq q q q q qqqqqq qqqqqqqq q q q q q q q q q q q q q q q q q q q q q q qqqqqqqqqqqqqqqqqqqqq

m∗

smax

Fig. 1. Rearrangement on (−∞, −s0 ]

m

240


−s0

π

s0

q qqq qqq qq qqq qqqq q q qqqq qqqq q q q q q q q q q q qqqqqqqqqqqqqqqqqqq

π/2

0

qqq qqqqqqqqqqqqqqqqq qqqqqqqq q q q q q q q qqqqqqqq qqqqqqqq qqqqqqqqq

..... ..... ..... ... . .... ... .. .... .. . ........ . .......... . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

a smax b Fig. 2. Rearrangement on [−s0 , s0 ]

The last inequality holds since we assumed m < −s0 and thus both terms in the integrand are positive. Continuing in this way, all interior local maxima of φ can be eliminated and thus φ can be rearranged into a monotone non-decreasing function on (−∞, −s0 ], of smaller energy. In the same manner as above, φ can be rearranged to be monotone non-decreasing on [s0 , ∞). 2. Inside I0 : If φ has an interior local maximum at smax ∈ I0 , with φ(smax ) > π/2, let [a, b] be the largest interval contained in I0 and containing smax such that φ(a) = φ(b) ≥ π/2 and φ(s) > φ(a) for s ∈ (a, b). Let ˜ := φ(s) s 6∈ [a, b] φ(s) φ(a) s ∈ [a, b]. ˜ < H(φ) (See Fig.2). Then H(φ) Likewise, if φ has an interior local minimum smin ∈ I0 with φ(smin ) < π/2, then a similar flattening procedure can be applied. Hence, after rearrangement, φ is either monotone on I0 , or has one local minimum in I0 with φmin ≥ π/2, or has one local maximum in I0 with φmax ≤ π/2. We now distinguish four cases: a) φ(−s0 ) < π/2 < φ(s0 ): It is clear that in this case φ has to be monotone on I0 , and therefore monotone on R. b) φ(−s0 ) > π/2 > φ(s0 ): Then ∃s1 ∈ I0 such that φ(s1 ) = π/2. Moreover, since φ(∞) = π, there exists s2 > s0 such that φ(s2 ) = π/2 also. Since the functional enjoys the symmetry H`,ω (π − φ) = H`,ω (φ), we can reflect the part of the graph of φ between s1 and s2 across the line φ = π/2 without changing the functional, i.e., let s 6∈ [s1 , s2 ] ˜ := φ(s) φ(s) π − φ(s) s ∈ [s1 , s2 ]. ˜ = H(φ). This introduces a local minimum at s = s2 which can be reThen H(φ) moved as before, decreasing H in the process (see Fig.3). Thus by rearrangement we have reduced this case to the next: c) φ(−s0 ) ≥ π/2 and φ(s0 ) ≥ π/2: Suppose φ is not monotone. Let a ∈ I0 be the largest possible point where the absolute minimum of φ on I0 is attained. By the


241

reflection argument above, we can assume φ(a) ≥ π/2. Since φ(−∞) = 0, there exists a∗ < −s0 such that φ(a∗ ) = φ(a). Let   φ(s + s0 + a∗ ) s ≤ −s0 ˜ := φ(a) −s0 ≤ s ≤ a φ(s)  φ(s) s > a. ˜ < H(φ). (See Fig. 3.) It is then clear that φ˜ is monotone on R, and H(φ)

−s0

π

s0

qqqqqqqqqqqqqqqqqqqqqqqq qqqqqqq qqqqq

................................... ......... .... .... ... ... ... .................. . . . . . . . . . ..... ... .. ..... ..... .. ....... ..... ..... ..... ..... .... .... .... .... ... .. ... .... ... ... . .. . . . . . .. ... .. .... ... ... .... .... ....... ... ... .. .... ....... .. ...... . ......... . . . ..... ...... . ... . . ..... . ..... .. ... .... . .. .. ... . . ... .... . ... . . ... .. . . .. . . . . . ... .. . . . . . . .. ... ... .. . . . . . . . . ... .. . . . ..... . . . . . . . . . . . . ....... . . . . . . . . . . . . . . . . ........ . . . . . . . . . . . . . . . . . . . . .

π/2

qq qqqq qq qqq q q q q q q q q q qqqqqqqq qqqqqqqqqqqqqqqqqq

0

a∗

qqq qqq q q q qq qqqqqqqq

s1 a

s2

Fig. 3. Reflection and rearrangement

d) φ(−s0 ) ≤ π/2 and φ(s0 ) ≤ π/2: This is treated in the same manner as in the above. Theorem 1. For ω 6= 0, every minimizing sequence φn for the functional (2.5) has a subsequence converging strongly to a smooth, monotone function φ with φ(−∞) = 0 and φ(∞) = π. Moreover the solution set of this minimization problem is compact. Proof. We are going to apply the concentration-compactness principle ([7, 8]). In our case, there are only two ways a minimizing sequence can lose compactness: (1) bubblingoff and (2) splitting. We rule them out in the following manner: Let φn be a minimizing sequence for (2.5). Again by reflecting along φ = kπ we can assume that 0 ≤ φn ≤ π. From the bound on H`,ω (φn ) we have that the sequence φn has a subsequence also denoted by φn such that 2

L φ0n * φ¯ 0 ,

and that φn → φ¯ uniformly on any compact interval [−R, R]. By lower semicontinuity of weak limits we have ¯ ≤ d(ω, `), H`,ω (φ) ¯ ¯ which implies that φ(±∞) = 0 or π . There are three possibilities for the limit φ: ¯ ¯ 1. φ(−∞) = 0 and φ(∞) = π, i.e. boundary conditions are preserved. This immediately implies that the convergence is strong in H 1 and the minimum is achieved. 2. φ¯ ≡ 0. This is bubbling-off, i.e., the boundary condition is carried away to infinity by the sequence, and φn * 0 so that nothing is left behind. The case φ¯ ≡ π is also possible but because of the symmetry of the functional it can be treated in a similar way.

242


¯ ¯ 3. φ(−∞) = φ(∞) = 0 but φ¯ 6≡ 0. This is splitting: The boundary condition is carried away but something is left behind. This is a possibility here because the functional H`,ω is not positive definite for ` < ω. The same thing can also happen at π and is treated similarly. The bubbling-off is ruled out by observing that the infimum of the functional for ω > 0 is strictly less than the infimum for ω = 0: Clearly, d(`, ω) ≤ H`,ω (φ0∗ )

Z

∞

sin2 φ0∗ ds 2 −∞ cosh s = 4π` − c1 (`, ω) = d(`, 0) − c1 (`, ω),

= 4π` − πω 2

(2.6)

2

with c1 ≥ πω `+1 . Now suppose φ¯ ≡ 0. Then given > 0 small and R > 0 large, there exists N > 0 such that φn (s) < for all n > N and |s| < R. At the same time φn (∞) = π for all n. Therefore, Z ∞ (φ0n )2 + `2 sin2 φn ≥ 4π` − c. R

Hence, Z H`,ω (φn ) =

Z

−R −∞

Z

R

+

∞

+ −R

R

≥ 4π` − c(R) − cω 2 e−2R . But for R large enough and small enough this will contradict (2.6). Hence bubbling-off cannot occur. Splitting cannot occur, since otherwise φn would not be monotone: Suppose ¯ ¯ ¯ φ(−∞) = φ(∞) = 0, and φ¯ 6≡ 0. Let M := maxs∈R φ(s) > 0. Let smax be a point ¯ max ) = M , and let R > 0 be large enough so where this maximum is achieved, φ(s ¯ that φ(s) < M/4 for |s| ≥ R. Given > 0 small, there exists N > 0 such that ¯ < for all n > N and |s| ≤ R. Therefore φn (smax ) > M − while |φn (s) − φ(s)| φn (R) < M/4+. Meanwhile, φn (∞) = π, which means that for n > N , every φn has a departure from monotonicity which is independent of n. By Proposition 2, H`,ω (φn ) can be reduced by at least a fixed amount, which contradicts the fact that φn is a minimizing sequence. Therefore every minimizing sequence has a subsequence that converges to a minimizer with the proper boundary conditions, and this implies that the solution set S`,ω = {φ : [0, π] → [0, π] | H`,ω (φ) = d(`, ω)} is compact. To prove the smoothness of the minima it is more convenient to go back to the original variable α ∈ [0, π]. The limit φ¯ of the minimizing sequence φn satisfies the Euler-Lagrange equation associated with H`,ω , φ00 + cot α φ0 + (ω 2 −

`2 ) sin φ cos φ = 0 sin2 α

(2.7)

in the weak sense. Since φ¯ is Hölder continuous away from 0 and π, it follows easily that it is actually smooth on (0, π) with the only possible singularities being the endpoints 0


243

and π. The following lemma establishes the regularity at α = 0. Regularity at α = π is by symmetry exactly the same. Notice that if we define a map u : S2 → S2 by setting u(α, z) = I(φ(α), z ` ), and if ` ≥ 2, then in order for u to be smooth as a map between spheres we must have, in addition to φ(0) = 0 and φ(π) = π, that the derivatives of φ up to the order ` − 1 vanish at α = 0 and α = π. Lemma 2. The variationally obtained solution φ to (2.7) is in C ∞ ([0, π]), and in fact sin φ(α)/ sin` α is smooth near α = 0 and α = π. Proof. We want to view (2.7) as ∆S2 φ = (

`2 1 − ω 2 ) sin 2φ 2 2 sin α

∞ and use elliptic regularity R 2 theory. Obviously φ ∈ L , and from the finiteness of H`,ω (φ) we can deduce that (φ / sin α)dα is finite. We need a better estimate for φ near α = 0, namely, that there exist a b > 0 such that

Z

b 0

φ2 (α) dα < C(b). sin2 α

(2.8)

The technique of obtaining this estimate is essentially that of [12], Lemma 4.1. We first multiply (2.7) by φ and integrate the result on [a, b] for 0 < a < b < δ, where δ is a small number, to obtain Z b 1 `2 1 φ02 − cot αφφ0 + ( 2 − ω 2 ) φ sin 2φdα = ((φ2 )0 (b) − (φ2 )0 (a)). 2 2 sin α a Now let a & 0. Since φ(0) = 0 and φ(α) > 0 for α > 0, we have that (φ2 )0 (a) > 0 and thus Z b `2 1 φ02 − cot αφφ0 + ( 2 − ω 2 ) φ sin 2φdα < C(b) < ∞. 2 sin α 0 Now, 21 φ sin 2φ ≥ 21 φ2 as long as φ remains small, which is guaranteed by taking b < δ. Moreover, φ02 − cot αφφ0 = (φ0 − so that 2`2 − 1 4

1 1 φ2 cot αφ)2 − φ2 cot2 α ≥ − , 2 4 4 sin2 α Z

b 0

φ2 dα < C(b), sin2 α

which proves the estimate since ` ≥ 1. Now let φ = ψ sin` α. Then ψ satisfies 2

∆S2`+2 ψ =

1 1 `2 ω2 2 ( ((` sin 2φ − φ) + sin 2φ) := R(ψ). + `)φ − 2 sin`+2 α 2 sin` α

Bearing in mind that φ is small close to the north pole where α = 0, we have that |R(ψ)| ≤ C(sin2`−2 α)|ψ|3 + C 0 |ψ| ≤ C 00 |ψ|1+2/` + C 0 |ψ|.

244


Let Bb be the geodesic ball of radius b centered at the north pole of the 2`+2-dimensional sphere S2`+2 . We need to find p > 1 such that R(ψ) ∈ Lp (Bb ) and such that the Lp estimates for the Laplacian allow us to bootstrap the regularity. I.e., suppose ∆S2`+2 ψ ∈ 1 . Then R(ψ) ∈ Lpn+1 with Lpn (Bb ). Then ψ ∈ W 2,pn (Bb ) ,→ Lqn for q1n = p1n − `+1 pn+1 = qn /(1 + 2/`) and thus we have the recursion formula `+2 1 `+2 1 . = − pn+1 ` pn `(` + 1) The solution to this is `+2 2 1 `+2 1 )+ , = (1 + )n ( − pn ` p0 2` + 2 2` + 2 which means that pn can be made larger than ` + 1 provided that initially we have R(ψ) ∈ Lp0 with p0 > (2` + 2)/(` + 2). We now show that using the estimate (2.8) we have in fact p0 = (2` + 3)/(` + 2): Z b |ψ|(1+2/`)p0 sin2`+1 α dα kR(ψ)kpL0p0 (Bb ) ≤ C Z

0

b

|φ|2+3/`

= C 0

dα sin2 α

≤ C(b). Once we have ∆ψ ∈ Lp with p > ` + 1 the Sobolev imbedding theorem implies that ψ is Hölder continuous and we can easily jack up the regularity to smoothness. This completes the proof of Lemma 2, as well as the proof of Theorem 1. We can now state the main result of this section: Theorem 2. Let ` and ω be fixed and nonzero. Then every minimizing sequence {un } in X`0 for the functional Gω defined above has a subsequence that converges strongly to a smooth map in X`0 . Thus Gω attains its global minimum over X`0 on a compact subset S of smooth functions. Remark 2. The set S contains at least a circle due to rotation invariance. Proof. Let {un } be a minimizing sequence in X`0 of Z 1 |∇u|2 − ω 2 |k × u|2 . Gω (u) = 2 S2 There exists a subsequence, also denoted by {un }, and a map u¯ ∈ H 1 (S2 ) with |u(x)| ¯ =1 for a.e. x ∈ S2 , such that L2

∇un * ∇u, ¯

L2

un → u, ¯

a.e. un → u. ¯

The map u¯ is equivariant, since for any g ∈ G, ¯ ≤ |u(gx) ¯ − un (gx)| + |g ` un (x) − g ` u(x)|, ¯ |u(gx) ¯ − g ` u(x)| which converge to zero. By Lemma 1, there exist functions φ¯ and ζ¯ such that


245

` ¯ ¯ u(α, ¯ z) = I(φ(α), ζ(α)z ),

(2.9)

and since un is also equivariant, and deg un = `, then by Lemma 1 we have un (α, z) = I(φn (α), ζn (α)z ` ),

(2.10)

where {φn } is a sequence of bounded continuous functions with φn (0) = 0 and φn (π) = π for all n. The sequence {φn } defined above is a minimizing sequence for H`,ω . Thus by Theorem 1 it has a subsequence converging strongly to a φ∞ ∈ S`,ω . By uniqueness ¯ Thus the limiting map u¯ of the minimizing sequence un for Gω will of limits, φ∞ = φ. have degree equal to ` and belongs to X`0 . By lower semicontinuity of Gω we have that ¯ ≥ d(`, ω). d(`, ω) = lim inf Gω (un ) ≥ Gω (u) n→∞

¯ = d(`, ω) and the convergence is actually strong. Now it follows that Therefore Gω (u) ζ¯ has to be constant. Otherwise, by replacing ζ¯ with a constant, Gω (u) ¯ can be reduced, ¯ which is impossible. Therefore u(α, ¯ z) = I(φ(α), cz ` ), for a constant c ∈ U, and thus u¯ : S2 → S2 is a smooth map. Finally since S`,ω is compact, then the set S of all possible minima of Gω on X`0 S = {I(φ(α), cz ` )}φ∈S,c∈U ,

(2.11)

is also compact in X`0 . We conclude this section by addressing the issue raised in Remark 1. Proposition 1 shows that the infimum of Gω on X` is not achieved, and therefore points in S are no longer global minima of Gω if we enlarge the domain from X`0 to X` . We now show that, at least for ω small and ` > 1, these points are not even local minima of Gω on X` . Proposition 3. Let ` be an integer larger than one. Then there is an ω0 > 0 such that for all ω < ω0 and for any u ∈ S a minimizer of Gω in X`0 there exists a map u ∈ X` arbitrarily close to u such that Gω (u ) < d(`, ω). Proof. Let uω ∈ S, i.e. uω ∈ X`0 and Gω (uω ) = d(`, ω). Our plan is to move out of the equivariant symmetry class by tilting the axis of rotation: For ∈ [0, π/2] let u (α, z) := e−B uω (α, z), where B is the 3 × 3 skew-symmetric matrix such that Bv = j × v for v ∈ R3 . Thus |∇u |2 = |∇uω |2 while

Z

Z

3 |k × u | = |k × uω | + sin (4π − 2 2 2 S S 2

so that

3 Gω (u ) = Gω (uω ) − ω sin (2π − 4

2

Z

2

2

S2

|k × uω |2 ),

Z

2

S2

|k × uω |2 ).

Z

Let us define h(`, ω) := sup u∈S

S2

|k × u|2 .

Since S is compact, this supremum is achieved at a u∗ ∈ S. Then

246


3 Gω (u ) ≤ d(`, ω) − 2πω 2 sin2 + ω 2 sin2 h(`, ω). 4 We can extend the definition of h to ω = 0 by setting Z π h(`, 0) := sup 2π sin2 φλ∗ sin αdα, λ>0

(2.12)

0

with φλ∗ (α) = 2 tan−1 (tan(α/2)/λ)` . Let u0 ∈ X`0 to be the map corresponding to φ1∗ , i.e. u0 (α, z) = I(φ1∗ (α), z ` ). It is not hard to check that R (i) h(`, 0) = S2 |k × u0 |2 , (ii) h(1, 0) = 8π/3, (iii) h(`, 0) is a decreasing function of ` for ` ≥ 1, (iv) h(`, 0) ≤ h(`, ω) for ω > 0. (i) holds because the supremum in (2.12) is achieved at λ = 1; (ii) is a simple computad h(`, 0) explicitly and seeing that it is negative; and tion; (iii) is verified by computing d` to get (iv), we observe that Z 1 1 |k × uω |2 = Gω (uω ) ≤ Gω (u0 ) = G0 (u0 ) − ω 2 h(`, 0), G0 (uω ) − ω 2 2 2 S2 R 2 and G0 (u0 ) ≤ G0 (uω ), so that h(`, 0) ≤ S2 |k × uω | , taking the supremum of which over S yields (iv). It thus follows that for ` ≥ 2, h(`, 0) = 8π 3 −c` for some positive increasing sequence c` . If we assume that h(`, ·) is continuous at ω = 0, we will have 3 Gω (u ) ≤ d(`, ω) − ω 2 sin2 (c` − o(1)), 4 so that for ω small and ` ≥ 2 we will have the desired result. To show the continuity of h, let ωn be a sequence of positive numbers tending to zero, and let un denote the corresponding minimizers in X`0 , i.e. Gωn (un ) = d(`, ωn ). Since un is a uniformly bounded sequence in H 1 (S2 ), passing to a subsequence, there exists u¯ ∈ H 1 (S2 ) such that L2

L2

¯ ∇un * ∇u,

un → u, ¯

a.e. un → u. ¯

Once again, this implies that |u(x)| ¯ = 1 a.e. x ∈ S2 and that u¯ is also equivariant. By (2.6) we have 4π` ≤ G0 (un ) ≤ Gωn (un ) + 2πωn2 ≤ 4π` + c ωn2 , which implies G0 (un ) → 4π` as n → ∞, and un is a minimizing sequence for G0 . Moreover since the minimum of Gωn is achieved at un and that of G0 at u0 , we have Z Z 1 1 4π` − ωn2 |k × un |2 ≤ Gωn (un ) ≤ Gωn (u0 ) = 4π` − ωn2 |k × u0 |2 , 2 2 S2 S2 for all n, and this implies that Z Z |k × un |2 ≥ |k × u0 |2 = h(`, 0) = 8π/3. S2

(2.13)

S2

This inequality holds for u¯ since un converges strongly to u¯ in L2 . Therefore the sequence {un }, which is equivariant, cannot concentrate at P or −P . This implies that the convergence of un to u¯ is actually strong in H 1 , and by (2.13) we have u¯ = u0 and φ¯ = φ1∗ . This establishes the continuity of h.


247

3. Regularity of Wave Maps with Data of Small Energy In this section we will show that the Cauchy problem for equivariant wave maps U : S2 × R → S2 with smooth initial data of small energy, has globally regular solutions. The proof is based on combining techniques of [2] and [13]. The equivariant maps that we consider have rotation number ` 6= 0 and can be represented as U (α, z, t) = (z ` (u1 + iu2 ), u3 ), where (α, z, t) are coordinates on S2 × R and u : S2 × R → S2 is radial in the sense that u = u(α, t). The Cauchy problem for U is equivalent to the following problem for u(α, t):  

utt − uαα − cot α uα + (|ut |2 − |uα |2 )u =

 u(α, 0) = f (α),

ut (α, 0) = g(α),

`2 (A2 u + |Au|2 u), sin2 α

(3.1)

where f, g : [0, π] → R3 are smooth functions such that f (α) = (0, 0, cos α) + f˜(cos α) sin` α, `

g(α) = g(cos ˜ α) sin α, ∞ ˜ f , g˜ ∈ C ([−1, 1], R3 ).

|f | ≡ 1,

f · g ≡ 0,

The hypothesis of small energy on the initial data in terms of f and g is: ! Z π 2 df + |g(α)|2 sin αdα < 2 . dα 0

(3.2)

(3.3)

Theorem 3. There exists an 0 > 0 such that for ≤ 0 the Cauchy problem (3.1,3.2,3.3) has a unique smooth solution that exists for all time. The first step in the proof of this theorem consists of getting Hölder estimates on the solution u using smallness of the energy. To accomplish this we need to obtain sharp pointwise estimates on the fundamental solution of the equation utt − uαα − cot α uα +

`2 u = 0, sin2 α

for ` ∈ Z. For the wave equation on flat space-time, i.e. on R2 × R, these estimates were obtained in [2] (see also [13]). Here we apply the argument given in [2] for invariant maps on R2 × R to the equivariant case ` 6= 0, and on the curved space S2 × R. The modification needed due to the fact that the space is S2 × R rather than R2 × R can be handled in several ways. One way is to use the formulas for the fundamental solution of the wave equation given in [4]. Another way, which is more specific to our case and which we elect to use here, is to derive the fundamental solution explicitly using the Penrose conformal transformation [10]. One should observe that all our regularity estimates are local, and therefore the fundamental solution estimates that are needed can be easily derived from the flat case. For this reason we will give only an outline of the proof and carry out the details of showing that the solution is Hölder continuous. The higher regularity estimates are an easy consequence from the flat space estimates carried out in complete details in [2].

248


Proof of the theorem. The local existence of smooth solutions follows from standard estimates for nonlinear wave equations. To show that the solution is globally regular, we will assume that the solution is regular for t < t0 , with possible singularity at t0 , and then show that in fact the solution is regular for t ≥ t0 . Note that by the invariance of equations with respect to time translation, we may assume t0 = 0; moreover since the solution is co-rotational and smooth initial data, the first possible singularity has to appear where sin α = 0, i.e. α = 0 or α = π. Therefore without loss of generality we can assume that the first possible singularity is located at t0 = 0, α = 0. For t < 0, i.e. when the map u is regular, the image of α = 0 under the map u is the point (0, 0, 1); and conservation of energy gives Z 1 `2 |Au|2 1 |ut |2 + |∇u|2 + ≤ ε, 2 2 2 sin2 α which implies that for α close to zero we have Z α (u1 ∂α u1 + u2 ∂α u2 ) u21 + u22 = 2 0

Z ≤

|∇u|2 sin α dα

21 Z

Au|2 / sin α dα

21

≤ energy ≤ ε2 . Thus for α close p to zero we can use v := (u1 , u2 ) as local coordinates on the target sphere and set u3 = 1 − |v|2 . To compute the fundamental solution of the v equation `2 v = q(v, ∂v), sin2 α

vtt − vαα − cot α vα + where

q = (−|vt |2 + |vα |2 +

`2 |v|2 )v, sin2 α

we use the Penrose transform, t = tan−1 (τ + r) + tan−1 (τ − r), α = tan−1 (τ + r) − tan−1 (τ − r), z = eiΘ , which maps Minkowski space into the diamond-shaped region D = (t, α, z) ∈ R × S2 | |t| + |α| < π . Let Ω := cos α + cos t. Then the inverse transform is given by r=

sin α , Ω

τ=

sin α . Ω

Ω is the conformal factor, and if we let 1

v˜ := Ω 2 v, then by the conformal invariance of the D’Alembertian τ,r v˜ = Ω 5/2 (t,α v + v/4) we have that v˜ satisfies the wave equation on the flat spacetime,


249

1 `2 ˜ v˜ τ τ − v˜ rr − v˜ r + 2 v˜ = h, r r 5

where h˜ = Ω 2 (q + v/4). We invert the above wave operator using the fundamental solution that was computed in [2] (see also [13, 14]), and then we compute the fundamental solution of the v equation by inverting the Penrose transform3 to obtain: r ZZ sin α0 0 J(µ)h(α0 , t0 ) dα0 dt0 , (3.4) v(α, t) = v (α, t) + c sin α K(α,t) where K(α, t) = {(α0 , t0 ) | − 1 ≤ t0 ≤ t, max{0, α − t + t0 } ≤ α0 ≤ α + t − t0 } , cos(t − t0 ) − cos α cos α0 r2 + r0 − (τ − τ 0 )2 = , sin α sin α0 2rr0 Z λ cos `θ dθ √ J(µ) = , cos θ − µ −λ −1 cos µ −1 ≤ µ ≤ 1 λ= , π µ < −1 v h=q+ , 4 2

µ=

and v 0 is the solution to the linear homogeneous equation v 0 = 0 with the same initial data as v. (Since v is regular for t < 0, without loss of generality we can assume that the initial data is prescribed at t = −1.) It is easy to see (cf. [2], Lemma 3.3) that J(µ) ≤

1 c log(1 + p ), 1 + |µ| |µ + 1|

|J 0 (µ)| ≤

c (|µ| +

1)3/2 |µ

+ 1|

.

We may now proceed as in [2], Sect. 3.1: Let (ξ, η) be the null coordinates ξ=

t−α , 2

η=

t+α , 2

and let e and m be the energy and momentum density for u: e=

`2 1 (|ut |2 + |uα |2 + |Au|2 ), 2 sin2 α

m = ut · u α .

Then if we define A and B as: sin α `2 (|∂η u|2 + u21 + u22 ) , 2 2 sin α sin α `2 (|∂ξ u|2 + B 2 = sin α(e − m) = u21 + u22 ) , 2 2 sin α

A2 = sin α(e + m) =

we can bound the nonlinearity in (3.4) by: 3 We return to the original variables to avoid the complications in obtaining the analogous estimate to (3.5) ˜ in terms of v. for h ˜

250


|h| ≤ c1

AB + c2 . sin α

(3.5)

The plan is to obtain an estimate for A(α, t) by differentiating (3.4) with respect to η and using the above bounds on J and h. Notice that conservation of energy gives a bound on B for any fixed η < 0, Z 2η+1 Z η B 2 (ξ 0 , η) dξ 0 = e(t = −1, α) sin αdα ≤ 2 . (3.6) −η−1

0

This bound on B is not sufficient to give a pointwise estimate for A, however it will allow us to estimate: n o 1 X(t¯ ) := sup (sin α) 2 −δ A(t, α) , Q(t¯ )

where δ ∈ (0, 21 ) is fixed , t¯ ≤ 0, and where Q(t¯ ) is defined to be the truncated cone region Q(t¯ ) = {(t0 , α0 ) | − 1 ≤ t0 < t¯, 0 ≤ α0 ≤ α¯ + t¯ − t0 } , where α¯ > 0 is fixed and small. For (α, t) ∈ Q(t¯) we have K(α, t) ⊂ Q(t¯). Differentiating (3.4) and setting all irrelevant constants equal to 1, ∂η v(α, t) = ∂η v 0 (α, t) + (sin α)−5/2

ZZ

Λ(α, α0 , t, t0 )h(α0 , t0 )dα0 dt0 K(α,t)

+ boundary term := I + II + III,

(3.7)

where √ Λ = {2[cos α0 − cos(t0 − t + α)]J 0 (µ) + cos α sin α sin α0 J(µ)}/ sin α0 . Note that the fundamental solution in the radial case is nonsingular on the characteristic portion of ∂K, where µ = 1, so estimating the boundary term III in (3.7) is not difficult: Z ξ 1 0 1/2 0 0 (sin α ) J(1)h(α , t ) III = 0 dξ 0 . η =η (sin α)1/2 −1−η By (3.5), |h| ≤ c(sin α0 )−1 AB ≤ c(sin α0 )δ−3/2 XB, and thus Z sin α|III| ≤ cJ(1)(sin α)1/2 X Z ≤ cX(sin α)1/2 ≤ Cδ X(sin α)δ .

ξ −1−η

B 2 dξ 0

(sin α0 )δ−1 B η0 =η dξ 0

1/2 Z

(sin α0 )2δ−2 dξ 0

1/2


ξ 0.........................

251

η0

.... ................. ..... ..... ..... ..... . ..... . . . ..... ... ..... ..... ... ..... ..... ..... ........ ..... ..... ..... ..... ........ ..... ..... .. ........ .... ..... ......... ..... ..... . . . . ... ..... .... . . . . . .. ....... ..... ......... ... ..... ... . . . . . . . . . . . . ....... .... . .... ..... ......... ..... ..... .......... ..... ..... . ...... ..... ..... . . . . . . .... ... ..... ... . . . .. . ..... ... .... . ..... ........ . . . . ..... ..... .. ..... ... . .. . . . . ... ..... ... ..... . . . . ..... . . . . . . . .... ..... ..... ..... ...... ..... ..... ..... ..... .... . . . ..... . ... . . ..... . . . . . . . ..... ... . . . . . . ..... ... ... . . ..... . . ..... ... ..... . . . . ..... ..... ... . . . ..... . ..... ... . ..... . . . ..... ..... ... . . . . ..... .. ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ... ..... ..... ......... ..... ..... .. . . . ..... ..... . . ..... . . . ..... ..... ..... ..... ..... ....

α

0

α

η

ξ

η

........... ......

t0

ξ

p p p p pppp p p p p ppppppppp pppp pp p ppppp ppppp pp ppppp ppp p ppppp p K1 ppppppppppp ppppp ppppp ppppp K2 pppp −1

t

−1 − η

Fig. 4. Null coordinates and the region of integration

To estimate II however, since the kernel J is singular at µ = −1, it is necessary to break up the region of integration K into two parts (See Fig. 4): K1 (α, t) := K(α, t) ∩ {−1 ≤ µ ≤ 1}, R and set II = II1 + II2 where IIi := Ki . In II1 , we have

K2 (α, t) := K(α, t) ∩ {µ ≤ −1},

|µ + 1| = 2 |sin(ξ − η 0 ) sin(η − ξ 0 )/ sin α sin α0 | ≥ C |sin(ξ − η 0 )/ sin α| , √ 1 |Λ| ≤ C log 1 + √ sin α sin α0 , µ+1 which implies II1 ≤ C(sin α)−3/2

ZZ

1 AB √ log 1 + √ dα0 dt0 . µ+1 sin α0 K1

Now we estimate II1 by taking the energy norm of B on the characteristic line η 0 = constant and bound the term A by X to obtain s !Z Z η ξ sin α δ−1 dη 0 log 1 + dξ 0 |sin(η 0 − ξ 0 )| B II1 ≤ CX(sin α)−3/2 0) sin(ξ − η ξ −1−η s ! Z η sin α ≤ CεX(sin α)−3/2 dη 0 | sin(ξ − η 0 )|δ−1/2 log 1 + sin(ξ − η 0 ) ξ ≤ Cε| sin α|δ−1 X.

252


Similarly in II2 we have |µ + 1| ≥ |µ − 1| = ≥ |Λ| ≤ which implies

Z

II2 ≤ C K2

sin α + sin α0 , C sin(ξ − η ) sin α0 sin α 2| sin(ξ − ξ 0 ) sin(η − η 0 )/ sin α sin α0 | C sin(η − η 0 )/ sin α, √ C sin α sin α0 , |µ − 1| |µ + 1|1/2 0

AB √ dα0 dt0 . sin(η − η 0 ) sin(ξ − η 0 )(sin α + sin α0 )

Again we estimate II2 by taking the energy norm of B on the characteristic line η 0 = constant and bound the term A by X to obtain Zξ II2 ≤ C(sin α) εX δ

−1

≤ Cε (sin α)

δ−1

These estimates imply

dη 0 1/2 sin(η − η 0 ) sin(ξ − η 0 )

X.

| sin α|1−δ |∂η v| ≤ C0 + CεX.

To obtain that X is bounded we still have to estimate |v|/ sin α. This is however identical to estimating ∂η v and thus we have | sin α|−δ |v| ≤ C0 + CεX, which implies that X is bounded, provided the energy ε is small. Since these bounds are independent of t¯ we can let t¯ → 0 and maintain the same bound on X on Q(0). By conservation of energy the bound on A implies energy decay of B in Q(0): Z

η

B ξ

2

η 0 =η

0

Z

η

dξ ≤

A ξ

2

ξ 0 =ξ

0

Zt

dη ≤ C

| sin(α − t + t0 )|2δ−1 dt0 ≤ Cα2δ ,

t−α

for α close to zero; and by integrating by part this implies that for any 0 < δ 0 < δ, Z η 0 0 (sin α0 )−2δ B 2 0 dξ 0 ≤ C|α|2(δ−δ ) , ξ

η =η

provided α is small. To obtain Hölder continuity of v we note that on the ξ = constant characteristic we have Z η2 0 ∂η v dη |v(ξ, η2 ) − v(ξ, η1 )| = η1 Z η2 | sin(ξ − η 0 )|δ−1 dη 0 ≤ CX η1 δ

≤ C | α2 − α1 | ;


253

and on η = constant characteristic we have Zξ2 B 0 v(ξ2 , η) − v(ξ1 , η) ≤ | sin(ξ 0 − η)|1/2 dξ ξ1 1/2  1/2  −2δ0 2 δ0 −1 Zξ2 Zξ2     B 2   sin(ξ 0 − η) ≤ C  sin(ξ 0 − η)  ξ1

≤ C|α2 |

ξ1

δ−δ 0

1/2 2δ0 α2 − α12δ0

≤ C|α2 − α1 |

δ

for α1 and α2 small (cf. [2, Lemma 3.4]). Therefore we conclude that v is C δ in Q(0). Higher regularity of v can be obtained by differentiating the equation once and repeating a similar argument (see [2] for complete details for the flat case and ` = 0).

4. Global Stability and Regularity In this section we will show that equivariant maps with smooth initial data that are close to stationary maps in the energy norm, are globally regular. This will be accomplished by showing that stationary maps are stable under such a perturbation and by applying the small energy regularity theorem of the previous section. The following notation will be used: For a function v defined on S2 ×R, let v(t) be the induced function on S2 × {t}, i.e., v(t)(x) = v(x, t) for all x ∈ S2 . Let ` and ω be a fixed nonzero integer and a fixed nonzero real number respectively. Let X = H 1 (S2 )×L2 (S2 ), with the norm kuk2X = ku1 k2H 1 +ku2 k2L2 for u = (u1 , u2 ) ∈ X. Let d denote the associated distance function on X: d(u, v) = ku − vkX . Also for K ⊂ X let d(u, K) be defined as usual: d(u, K) = inf v∈K d(u, v). Let S := {(v, ωAv)}v∈S , where S is as in (2.11). Thus S is a compact subset of X. For u : S2 × R → S2 a finite energy map, let ut := (u(t), ∂t u(t)) ∈ X. First we are going to prove Theorem 4. There exists η > 0 such that for all T ∗ > 0, if u is a classical solution to the Cauchy problem (3.1,3.2) on [0, T ∗ ) such that sup d(ut , S) < η,

t∈[0,T ∗ ]

then u is smooth in S2 × [0, T ∗ ]. Proof. We know that a local-in-time classical solution always exists. Because Eqs.(3.1) are radial, the first singularity, if there is one, will develop at time T ∗ at either the north or the south pole. Without loss of generality we can assume it occurs at the north pole. For t ∈ [0, T ∗ ) and R ∈ [0, π/2) let DR (t) = {(α0 , z 0 , t) ∈ S2 × R | 0 ≤ α0 ≤ R}

254


be the geodesic disk of radius R centered at the north pole and lying in the time slice S2 × {t}. Let Z 1 |∇u(t)|2 + |∂t u(t)|2 ER (u(t)) = 2 DR (t) be the energy carried by DR (t). The local conservation of energy implies that ER (u(T1 )) ≥ ER−T2 +T1 (u(T2 )),

(4.1)

for 0 < T1 ≤ T2 ≤ T ∗ and T2 − T1 < R < π/2. From the hypothesis of the theorem we have that there exist a v t ∈ S such that ku(t) − v t k2H 1 (S2 ) + k∂t u(t) − ωAv t k2L2 (S2 ) < η 2 . The same inequality obviously holds if we replace S2 by DT ∗ −t (t). Therefore we have 2ET ∗ −t (u(t)) ≤ k∇v t k2L2 (DT ∗ −t (t)) + kωAv t k2L2 (DT ∗ −t (t)) + η 2 . Since S is compact then for any small disk, the local energy of v ∈ S is uniformly small. Therefore there exist an η > 0 small enough and a T 0 < T ∗ such that for t ∈ [T 0 , T ∗ ) we have ET ∗ −t (u(t)) < 0 /2, where 0 is as in Theorem 3. Furthermore, since the integral defining the energy depends continuously on the domain of integration, there exists δ > 0 such that ET ∗ −T 0 +δ (u(T 0 )) < 0 , and thus by the energy inequality (4.1) we have that for all T ∈ [T 0 , T ∗ ), Eδ (u(T )) ≤ ET ∗ −T +δ (u(T )) ≤ ET ∗ −T 0 +δ (u(T 0 )) < 0 . But in that case, since T is arbitrary in [T 0 , T ∗ ), the regularity Theorem 3 applies to show that no singularity can develop at time T ∗ . This completes the proof of the theorem. Now we can prove global stability which will imply that in a neighborhood of stationary maps the hypothesis of Theorem 4 holds. Theorem 5. Let u0 ∈ S be fixed and let u˜ 0 : S2 × R → S2 be the corresponding stationary, G-equivariant wave map, u˜ 0 (x, t) = eAωt u0 (x). Then u˜ 0 is globally stable in the following sense: Let η > 0 be as in Theorem 4. For every ∈ (0, η) there exists a δ = δ() > 0 such that if (f, g) is an initial data pair for (3.1) satisfying conditions (3.2) and d((f, g), (u0 , ωAu0 )) < δ, then the classical solution u to (3.1) defined on [0, T ∗ ) × S2 remains -close to S in the energy norm for all t ∈ [0, T ∗ ), i.e., d(ut , S) < ,

∀t ∈ [0, T ∗ ).

In particular, it follows from Theorem 4 that T ∗ = ∞, i.e., u is a global smooth solution.


255

Proof. We proceed by contradiction: Assume u not stable, then there must exist an > 0, a sequence of pairs of functions (fn , gn ) satisfying the conditions (3.2), and a sequence of times tn such that d((fn , gn ), (u0 , ωAu0 )) < n1 while the corresponding solutions un to (3.1) with data (fn , gn ) exit the -neighborhood of S at time tn , i.e., inf {kun (tn ) − vkH 1 + k∂t un (tn ) − ωAvkL2 } = .

v∈S

(4.2)

Without loss of generality we can assume < η, where η is as in the previous theorem. We claim that {un (tn )} is a minimizing sequence for the functional Gω on X`0 . If so, according to Theorem 2 it will have a subsequence converging to a point in S, contradicting (4.2). To prove this claim, we note that by conservation of energy we have: c E(un (tn )) = E(un (0)) ≤ E(u˜ 0 (0)) + n c = Gω (u0 ) + ωQ(u˜ 0 (0)) + , n where c denotes a generic constant. On the other hand, by the conservation of charge Q we have: E(un (tn )) = Gω (un (tn )) + ωQ(un (tn )) + D(un (tn )) = Gω (un (tn )) + ωQ(un (0)) + D(un (tn )) c ≥ Gω (un (tn )) + ωQ(u˜ 0 (0)) − + D(un (tn )), n where D(u(t)) :=

1 2

Z S2

|∂t u(t) − ωAu(t)|2 .

Thus we obtain d(`, ω) ≤ Gω (un (tn )) ≤ D(un (tn )) + Gω (un (tn )) c c ≤ Gω (u0 ) + = d(`, ω) + , n n which implies that {un (tn )} is a minimizing sequence for Gω and that D(un (tn )) → 0. Theorem 2 now implies that by passing to a subsequence, there exists v ∈ S such that kun (tn ) − vkH 1 → 0. Moreover k∂t un (tn ) − ωAvk2L2 ≤ D(un (tn )) + kωAun − ωAvk2L2 → 0, thus a contradiction is obtained and the proof of stability is complete. Finally, Theorem 4 applies to conclude that the solution u is smooth for all time. Remark 3. The above theorems can be stated for finite energy weak solutions provided we restrict the class of finite energy weak solutions to mean weak solutions of Eq.(3.1), with finite energy initial data, that are strong limits of classical solutions (in the energy norm)4 . This class of solutions has the property that if u is a classical solution, and v a weak solution to the Cauchy problem (3.1,3.2) with the same initial data, then u = v. 4 Global existence of finite-energy weak solutions in all dimensions to the Cauchy problem of sphere-valued wave maps has been established in [12].

256


This property is a simple consequence of the energy estimate. To see this we use the form of the equation given in (1.3) and let ω = u − v to obtain ω · ωt = |ut |2 − |∇u|2 ][∂t |u − v|2 /2 + (|ut |2 − |vt |2 ) − (|∇u|2 − |∇v|2 ) ut · v . Since u is smooth we can integrate this on S2 and use Gronwall’s inequality to conclude that ω = 0. If the initial data is only of finite energy we can approximate it by smooth functions and use the above theorem and a limiting argument to conclude that such finite energy weak solutions are stable. Acknowledgement. We wish to thank Prof. Eric Séré for many helpful discussions, and the referee for suggestions on how to improve our presentation.

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Commun. Math. Phys. 185, 257 – 284 (1997)

Communications in


The “Two and One–Half Dimensional” Relativistic Vlasov Maxwell System? Robert Glassey1 , Jack Schaeffer2 1 Department of Mathematics, Indiana University, Bloomington, IN 47405–5701, USA. E-mail: [email protected] 2 Department of Mathematics, Carnegie Mellon University, Pittsburgh, PA 15213, USA. E-mail: [email protected]

Received: 18 March 1996 / Accepted: 29 July 1996

Abstract: The motion of a collisionless plasma is modeled by solutions to the Vlasov– Maxwell system. The Cauchy problem for the relativistic Vlasov–Maxwell system is studied in the case when the phase space distribution function f = f (t, x, v) depends on the time t, x ∈ R2 and v ∈ R3 . Global existence of classical solutions is obtained for smooth data of unrestricted size. A sufficient condition for global smooth solvability is known from [12]: smooth solutions can break down only if particles of the plasma approach the speed of light. An a priori bound is obtained on the velocity support of the distribution function, from which the result follows.

1. Introduction The relativistic Vlasov–Maxwell system in three dimensions is given by ∂t fα + vˆ α · ∇x fα + eα (E + c−1 vˆ α × B) · ∇v fα = 0, ∂t E = c∇ × B − j, ∂t B = −c∇ × E,

∇ · E = ρ, ∇ · B = 0.

(1.1) (1.2) (1.3)

Here x ∈ R3 is position, v ∈ R3 is momentum, and c is the speed of light. fα expresses the number density in phase space of particles of species α (with mass mα and charge eα ) and the velocity of these particles is given by vˆ α = (m2α + c−2 |v|2 )−1/2 v. The charge and current densities are given by ?

Supported in part by NSF DMS 9321383

(1.4)

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R. Glassey, J. Schaeffer

ρ(t,x) = 4π

Z X

eα fα dv,

(1.5)

eα fα vˆ α dv

(1.6)

α

j(t,x) = 4π

Z X α

respectively. Consistent initial data for fα , E and B are prescribed at t = 0. Since it has no effect on the results of this work, we will simplify the discussion by taking the case of one species. Hence the subscript α will be omitted. We will also take m = e = c = 1. We will consider the following “two and one–half dimensional” version of the problem: we denote x = (x1 , x2 ) and v = (v1 , v2 , v3 ) and seek f = f (t, x, v), E = E1 (t, x), E2 (t, x), E3 (t, x) , B = B1 (t, x), B2 (t, x), B3 (t, x) , such that ∂t f + vˆ · ∇x f + (E + vˆ × B) · ∇v f = 0, ∂t E = ∇ × B − j, ∂t B = −∇ × E, where vˆ ρ j vˆ · ∇x ∇×B

= = = ≡ ≡

(1.7)

∂x1 E1 + ∂x2 E2 = ρ,

(1.8)

∂x1 B1 + ∂x2 B2 = 0,

(1 +R v12 + v22 + v32 )−1/2 v, 4π R f dv, 4π vf ˆ dv, vˆ 1 ∂x1 + vˆ 2 ∂x2 , ∂x2 B3 , −∂x1 B3 , ∂x1 B2 − ∂x2 B1 ,

(1.9)

(1.10) etc.

The solution is to satisfy the initial conditions f (0, x, v) = f0 (x, v) ∈ C 1 , E(0, x) = E0 (x) ∈ C 2 , B(0, x) = B0 (x) ∈RC 2 , ∇ · E0 = ρ0 = 4π f0 (x, v) dv, ∇ · B0 = 0,

(1.11)

and f0 (x, v) is to have compact support in v. There is an invariant group action on solutions to the three–dimensional problem which has the consequence that if initially f , E and B are independent of x3 , then so are the solutions at any later time t. This is analogous to the “one and one–half dimensional” problem analyzed in [9], [21]. A representation for the fields E, B in the three dimensional problem (1.1), (1.2), (1.3) was derived in [12]. There it was shown that a bound on the v support of f implies a bound on the derivatives of f , E and B (in the spirit of [3]). In the recent paper [10] the same agenda was carried out for the two dimensional problem, and global existence was established for large data in [11]. The problem studied in this paper is a special case of the three–dimensional situation, and therefore it suffices to find an a priori bound on the v support of f .

“Two and One–Half Dimensional” Relativistic Vlasov Maxwell System

259

There are three key ingredients in the present paper. The first is the improved representation for the fields as exposed in [10]. The second is the “cone estimate” which is obtained from the standard energy conservation law by integrating over a backward characteristic light cone. The relevant quantities are independent of x3 , and we exploit this to obtain a much stronger estimate than is available in the full three–dimensional problem. Lastly, we derive and employ a new conservation law for the characteristic system of ordinary differential equations which is associated with the Vlasov equation (1.7). At this time the a priori bound on v support is known in the full three dimensional case only when the size of the Cauchy data are restricted (see [8, 15, 17, 24]). In lower dimensions such a bound is known for the two dimensional problem ([10], [11]; see also [7]) as mentioned above, and for solutions independent of x2 ([9, 21]). We mention also the works [1, 5, 13, 14, 19, 25, 28] relating to the above problems. We make estimates below on an arbitrary interval of time [0, T ] on which a local solution exists. Constants depending on T and the initial data only are written cT . When a, b are 3-vectors, a × b denotes the usual cross product. Below we will use the following notation. Let w = (w1 , w2 ) ∈ R2 . When w is used as a 3-vector, we understand that w3 = 0. Further we will write for w = (w1 , w2 ) ∈ R2 and v ∈ R3 , w∧ v = third component of w × v = w1 v 2 − w 2 v 1 . Thus for such w

w × v = (w2 v3 , −w1 v3 , w∧ v).

Denote by Cbk (R2 ) the space of functions whose derivatives up through order k are bounded and continuous on R2 , with a similar definition for Cbk (R2x × R3v ). Our major result is the following: Main Theorem. Consider the (RVM) system (1.7)–(1.9) and assume that the Cauchy data f (0, x, v) = f0 (x, v), E(0, x) = E0 (x), B(0, x) = B0 (x) satisfy the following: R (i) 0 ≤ f0 (x, v) ∈ Cb1 ; ρ0 (x) ≡ 4π f0 dv ∈ L∞ (R2 ) ∩ L1 (R2 ); (1 + |v|)f0 ∈ L1 (R2x × R3v ); there exists k > 0 such that f0 (x, v) = 0 for |v| ≥ k. (ii) E0 (x) ∈ Cb2 (R2 ); ∇x · E0 = ρ0 ; E0 ∈ L2 (R2 ). (iii) There exists a gauge potential A0 (x) = A01 (x), A02 (x), A03 (x) (x ∈ R2 ) such that A0k ∈ Cb3 (R2 ) for k = 1, 2, 3; B0 (x) = ∇ × A0 (x) ≡ (∂x2 A03 , −∂x1 A03 , ∂x1 A02 − ∂x2 A01 )

(1.12)

and B0 ∈ L2 (R2 ). Then there exists a uniquely determined, global C 1 solution (f, E, B) to the (RVM) system (1.7)–(1.9) satisfying the initial conditions (1.11) and the properties

1 2

for all (t, x, v) ∈ R+ × R2x × R3v ; Z q 1 + |v|2 f (t, x, v) dvdx = const., |E(t, x)|2 + |B(t, x)|2 dx + 4π f (t, x, v) ≥ 0

Z R2

R2x ×R3v

(1.13) and there is a continuous function Q(t) such that f (t, x, v) = 0 for |v| > Q(t) and all x.

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R. Glassey, J. Schaeffer

Remark . 1. The assumptions (i)–(iii) imply that the initial energy ((1.13) at t = 0) is finite. In what follows one may see that we actually need only locally finite energy. 2. The representation of B0 in (1.12) automatically gives ∇ · B0 = 0. 3. The hypotheses on A0 (in particular, on A03 ) are needed to estimate the gauge function A3 (t, x) which arises in the conservation law below.

2. A New Conservation Law * Denote by A= (A0 , A1 , A2 , A3 ) the gauge potentials. Each Aν is a function of t, x1 , x2 for ν = 0, 1, 2, 3. Then we may write the Maxwell fields E, B as follows: E = (E1 , E2 , E3 ), Ek = ∂xk A0 − ∂t Ak

B = (B1 , B2 , B3 ); (k = 1, 2, 3);

B = ∇ × A = (∂x2 A3 , −∂x1 A3 , ∂x1 A2 − ∂x2 A1 ). Thus E1 E2 E3 B1 B2 B3

= ∂x1 A0 − ∂t A1 , = ∂x2 A0 − ∂t A2 , = −∂t A3 ; = ∂x2 A3 , = −∂x1 A3 , = ∂x1 A2 − ∂x2 A1 .

(2.1) (2.2)

(2.3)

We may choose any convenient gauge, and will take the temporal gauge A0 = 0. The characteristics for the Vlasov equation (1.7) are x˙ 1 x˙ 2 v˙ 1 v˙ 2 v˙ 3

= vˆ 1 , = vˆ 2 , = E1 + vˆ 2 B3 − vˆ 3 B2 , = E2 + vˆ 3 B1 − vˆ 1 B3 , = E3 + vˆ 1 B2 − vˆ 2 B1 .

(2.4)

Lemma 2.1. For fields E, B and potentials A independent of x3 as above, we have along characteristics v3 + A3 = const. Proof. By the equation for v˙3 and (2.3), v˙ 3 = E3 + vˆ 1 B2 − vˆ 2 B1 = −∂t A3 + vˆ 1 (−∂x1 A3 ) − vˆ 2 (∂x2 A3 ) = −∂t A3 − x˙ 1 ∂x1 A3 − x˙ 2 ∂x2 A3 d = − ds A3 (s, X (s, t, x, v)) , where X(s, t, x, v), V (s, t, x, v) are the solutions of (2.4) with initial values X(t, t, x, v) = x, V (t, t, x, v) = v. This completes the proof. Remark . There is a similar invariant for the “one and one-half dimensional” model, as well as for the cylindrically symmetric three dimensional problem.

“Two and One–Half Dimensional” Relativistic Vlasov Maxwell System

261

3. The Cone Estimate Let e be the energy density: 1 e = (|E|2 + |B|2 ) + 4π 2

Z q 1 + |v|2 f dv.

(3.1)

Then as is well-known [7, 11, 14, 19] Z ∂t e + ∇x · −B × E + 4π vf dv = 0. Given x ∈ R2 , t > 0 we integrate this over a backward truncated cone {(τ, y) : 0 ≤ τ ≤ t, |y − x| < R + t − τ } and apply Green’s Theorem. In the standard manner we find the identity (as in [11, 14]) R R − |y−x| R . We clearly have

Z

Z S1

On S2 ,

|vˆ 3 |f dv ≤ kf k∞

|vˆ 3 |

p

=

1 + |v|2

On S3 there holds

S2

|vˆ 3 |f dv ≤ Rr

|vˆ 3 | (ω1 v2 −ω2 v1 )2 +v32 √

(3.7)

|v3 | R ≤ , ≤ Rr−2 1 + |v|2 1 + r2

Z

and hence

S1

dv ≤ crR2 .

−2

Z f

q 1 + |v|2 dv ≤ Rr−2 e.

|v3 | (ω1 v2 − ω2 v1 )2 + v32

=

1+|v|2

1 ≤q ≤ R−1 , 2 (w1 v2 − w2 v1 )2 + v3 so that

Z S3

|vˆ 3 |f dv ≤ R−1

Z f

(ω1 v2 − ω2 v1 )2 + v32 p dv ≤ 2R−1 J . 1 + |v|2

Combining these estimates we have |j3 | ≤ c(rR2 + Rr−2 e + R−1 J ). Choosing r = J −1/8 e3/8 and R = J 3/8 e−1/8 we obtain the lemma.

Our next goal is to bound the extent of the v3 -support of f . In view of the conservation law from Lemma 2.1, we want to estimate A3 . By Maxwell’s equations, ∂t E3 = (∇ × B)3 − j3 . From (2.3), −E3 = ∂t A3 and (∇ × B)3 = ∂x1 B2 − ∂x2 B1 = ∂x1 (−∂x1 A3 ) − ∂x2 (∂x2 A3 ) = −∆A3 . Hence ∂t2 A3 = ∆A3 + j3 , so that

Z tZ 2πA3 = 2π A˜ 3 + 0

where A˜ 3 denotes a data term.

|y−x| 1 we still have a nontrivial set of differentials dA acting on the Cartan algebra. 3.4. N = 2 Weil and Cartan algebra. We now focus on the case N = 2, as is appropriate bb to X. The N = 2 Weil model W 2 (g) is the unique BDGA with generators (see also [8] for a somewhat different definition of a bigraded version of the Weil algebra) φ++ ω+ connections :

η+ curvatures:

φ+−

(3.16) η−

ω− φ−−

Balanced Topological Field Theories

421

Here we indicated the ghost charges graphically. The generators ωA , φAB , , ηA have the following degrees and sl2 representations (1, 2), (2, 3), (2, 10 ), (3, 20 ). (The primes indicate pseudo-representations.) They satisfy the relations = d+ ω− − d− ω+ , φAB = 21 [DA , DB ], ηA = − 16 [DB , [DC , DA ]]BC .

(3.17)

We can define the Cartan model by putting ωA , = 0. This leaves us just the variables φAB , ηA in (3.16). The transformation laws become now dA φBC = AB ηC + AC ηB , dA ηB = − 21 [φAC , φBD ]CD ,

(3.18)

reproducing the transformation laws of [4]. Let us make a comment about the object ηA , because it illustrates very well the new features of the extended algebras. Indeed, let us see why these objects appear according to the general definition given in the previous subsection. In the N = 2 case we have two covariant derivatives DA in degree one and three curvatures φAB = 21 [DA , DB ] in degree two. In degree three we have the triple commutators [DA , [DB , DC ]]. However, for N = 2 the six independent triple commutators [DA , [DB , DC ]] are not all determined by the Jacobi identity. This should be compared to the N = 1 case, where the JacobiBianchi identity gives us [D, [D, D]] = [D, φ] = 0. In fact, for N = 2 there are only four Jacobi-Bianchi identities which are given by [D+ , φ++ ] = 0, [D− , φ−− ] = 0,

2[D+ , φ−+ ] + [D− , φ++ ] = 0, 2[D− , φ+− ] + [D+ , φ−− ] = 0.

(3.19)

This implies that there are two (six minus four) new generators ηA at degree three. Equation (3.17) implies that these are explicitly given by η+ = −[D+ , φ+− ],

η− = [D− , φ+− ].

(3.20)

One easily verifies that at degree four and higher no new generators appear. So we learn that a ), A, B = ±, a = 1, . . . , dim g. (3.21) C 2 (g) = S ∗ (φaAB , ηA 3.5. N = 2 extended equivariant cohomology. We are now in a position to discuss the equivariant cohomology of iterated superspaces. Suppose X has a G action generated by vector fields Va , where ea denote a basis of the Lie algebra g of G. By lifting these bb vector fields as described in Sect. 2.5, we obtain a G action on the space X, together bb with the derivations dA , L(Va ), ιA (Va ), I(Va ) of C ∞ (X). As in the case N = 1, there are several models for the equivariant cohomology. We discuss here briefly the Weil model, BRST model and Cartan model. For the Weil model bb and the differential is simply the sum of the we consider the complex W 2 (g) ⊗ C ∞ (X) two differentials as defined above: dW A = dA ⊗ 1 + 1 ⊗ dA , acting on the basic forms, that are now defined to satisfy

(3.22)

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R. Dijkgraaf, G. Moore

ιA α = Lα = Iα = 0. The BRST model is defined analogously as in (3.6), BRST A W A ≡ exp ι (ωA ) + I( ) dA exp −ι (ωA ) − I( ) . dA

(3.23)

(3.24)

bb Finally, the Cartan model is based on the G-invariant subalgebra of C 2 (g) ⊗ C ∞ (X) with equivariant differential W a B a dC A = dA + φAB ι (Va ) + ηA I(Va ).

(3.25)

This gives the explicit transformation laws (3.18) together with the following action of bb b the Cartan differential on functions on X (or, equivalently, differential forms on X) i , d A ui = ψA dA ψB = L(φAB )u + AB H, dA H = −L(φAB )ψC BC − L(ηA ) · u,

(3.26)

where we have dropped the superscript on dA and used the compressed notation φAB = φaAB Va , etc. This again reproduces the transformation laws in [4]. The extended equivariant cohomology is not very different from the ordinary equivariant cohomology. One can show that the N = 2 equivariant cohomology of X is actually isomorphic to that of X, at least outside of degrees (a, b) for a, b = 0, 1. To prove this we introduce the homotopy operator K = K X + K C , where K C for the Cartan model is defined by K C η+ = −φ+− , K C η− = − 21 φ−− with K = 0 on all other generators and K X = K− is defined in (2.25). A short calculation shows that a [d+ , K] = L− + ηA

∂ ∂ ∂ + φa+− a + φa−− a , a ∂ηA ∂φ+− ∂φ−−

(3.27)

from which the result follows. 4. Balanced Topological Field Theory We now introduce a new class of topological field theories, which include the “cofield construction” of [1, 2] as a special case. One natural name for these theories would be NT = 2 topological field theories. Here NT denotes the number of topological supercharges or BRST operators. This should not be confused with extended supersymmetric theories. In fact, the twisting procedure will typically relate models with N = 2 supersymmetry to NT = 1 topological symmetry and models with N = 4 supersymmetry to NT = 2 topological field theories. Since this nomenclature has perhaps too many misleading connotations and since the ghosts and antighosts are perfectly matched in these theories we propose to call them “balanced topological field theories” (BTFT’s). 4.1. Review of the standard construction of TFT. The basic data for a TFT are (i) a space C of fields, (ii) a bundle E → C of equations equipped with a metric (, )E and connection ∇ compatible with the metric, and (iii) a section s ∈ 0 (E) such that its zero locus M = Z(s) defines a moduli problem of interest. (This is reviewed in detail in [2], see e.g. also [9].)


423

The construction of the topological field theory can be phrased in terms of the b ∗ . As in Sect. 2.1 we wish to distinguish the fiber coordinates from supergeometry of E the field space coordinates coordinates uµ , ψ µ . The fiber coordinates are the “antighosts,” coordinates on the dual to the bundle of equations: ρa ∈ 0 (M ; ΠE ∗ ), Ha ∈ 1 (M ; ΠE ∗ ).

(4.1)

The bundle E ∗ carries a connection ∇ with curvature R and the BRST operator Q = d is defined by: ∇ρ = H, (4.2) ∇H = R · ρ. The topological field theory action I is defined in terms of the gauge fermion

in the form

Ψ = ihρ, si − (ρ, ∇ρ)E ∗

(4.3)

I = QΨ = iHa sa − (H, H) − ihρ, ∇si + (ρ, Rρ)E ∗ ,

(4.4)

where we use the compatibility of the metric and connection. General arguments show R that the path integral Z = e−I computes the Euler character of the bundle of antighost zero modes over the moduli space Z(s): Z χ(cok ∇s). (4.5) Z= Z(s)

The above story becomes a little more intricate in the presence of a gauge symmetry G. The basic topological multiplet (A, B) takes values in an equivariant bundle over field space with connection ∇ and has transformation laws: ∇A = B, ∇B = R · A,

(4.6)

where the combination R = R + L(φ) is the equivariant curvature [10]. In order to construct the Poincaré dual to the moduli space Z(s)/G one introduces the extra multiplet λ, η ∈ g = Lie(G) of degree −2, −1. We will write here the Lie algebra indices as λx , η x . Let us denote the vertical vector fields associated with the gauge group action by: (L(λ)u)I = λx VxI (u), (4.7) VxI (u) = C : g → Tu C and define the projection gauge fermion: Ψproj = i(ψ, L(λ) · u) to project out the redundant gauge degrees of freedom. The resulting term in the action is: QΨproj = (λ, C † Cφ + C † Ru + ∂J (C † )xI ψ I ψ J ) − (ψ, L(η)u).

(4.8)

Note that λ is a Lagrange multiplier and the resulting delta function fixes φ away from fixed points of the gauge group. The fermion kinetic terms may be written as: iρa ∇I sa ψ I + iη x (C † )xI ψ I = ( ρ η ) Oψ, where the operator O is defined by:

(4.9)

424


TC

O=∇s⊕C †

1 (C; E) ⊕ g∗ ,

−→

(4.10)

and is associated to the deformation complex 0

→

g

C

−→

TC

∇s

−→ E

→0

(4.11)

by using the metric. Equation (4.11) is a complex if the equations are gauge invariant. The complex is exact at degree −1, if the group action is free. Again general arguments show that the path integral is just: Z χ(cok O/G). (4.12) Z= Z(s)/G

4.2. Balanced topological field theories: Field content. In a balanced or NT = 2 topobb logical field theory, the fields in the model are the generators of functions on X. We will i denote coordinates on X by u . Sometimes we will divide up the coordinates into fiber and basic coordinates. As usual the generators form a quartet:

ui

% &

ψ+i

i ψ−

& %

Hi

(4.13)

i where we note that all of ψA , H i should be regarded as (even or odd) sections of a vector bundle. These bundles have connections so we can define the differentials as in (2.17). We will assume a group G acts on X and introduce the Cartan multiplet φAB , ηA as in (3.16). The G-equivariant BRST differentials are now defined to act by

∇ A u = ψA , ∇A ψB = RAB · u + AB H, ∇A H = −RAB ψC BC + PA · u,

(4.14)

where the geometrical operators are defined by RAB = RAB + L(φAB ), PA = 13 ∇B (RCA )BC = PA + L(ηA ), P± = ±∇± R±∓ .

(4.15)

Here RAB and PA are the NT = 2 extended equivariant curvatures. 4.3. Balanced topological field theories: The action potential. A topological field theory b with field space of the form Cb is called balanced if the action is an sl2 invariant and bb d+ , d− closed function on C. Let us characterize the most general action of a BTFT. The bb action I, being a function in C(X), carries a bigrading (q+ , q− ). According to Theorem 2.1 the action is both d+ and d− exact and is, in fact, of the form: I = I0 + d + d − F ,

(4.16)


425

where the “topological term” I0 (u) is constant on the components of C. We refer to F as the “action potential” since it is analogous to the Kähler potential of a Kähler form. Note that F is not uniquely defined; we can always shift F → F + d+ Φ− + d− Φ+ .

(4.17)

Note further that if H 1 (C) 6= 0, then F need not be globally well-defined on field space. So the analogy to a Kähler potential is quite good. By sl2 invariance, action potentials must be of total ghost charge zero. The most natural action potentials are of the form F = F0 (u) + (ψ+ , ψ− ) + β(H, u) + γ(η+ , η− ),

(4.18)

where (·, ·) is a metric on the bundles over field space which is compatible with the connections and F0 (u) is a function on field space. Locally, this is the most general cation potential which is at most first order in ψ± , H, η± . Let us discuss the separate terms individually. The gauge fermions Ψ− = d− F and actions S = d+ Ψ− = d+ d− F associated with these terms are: • F0 (u). We will assume that F0 (u) is a G-invariant function. Then: I , Ψ− = ∇I F0 ψ− I I J d+ d− F0 (u) = −H ∇I F0 + 21 AB ψA ψ B ∇J ∇I F 0 .

(4.19)

• (ψ+ , ψ− ). The fermion bilinear gives rise to Ψ− = (H, ψ− ) + (R−+ u, ψ− ) − (R−− u, ψ+ ), d+ d− (ψ+ , ψ−) = −(H, H) + 2(PA u, ψB )AB − 21 AC BD (RAB u, RCD u) + 2(ψA , RBC ψD ) .

(4.20)

• (H, u) is equivalent to (ψ+ , ψ− ). This follows from the identity (H, u) = (ψ− , ψ+ ) − d− (ψ+ , u).

(4.21)

• (η+ , η− ). This equivariant term gives the following contributions to the gauge fermion and action Ψ− = 21 ([φ−− , φ++ ], η− ) − (η+ , [φ−− , φ−+ ]), d+ d− (η+ , η− ) = ([φ++ , φ+− ], [φ−− , φ−+ ]) + ([φ++ , φ−− ], [φ++ , φ−− ]) +AB CD ([ηA , φBC ], ηD ).

(4.22)

In Sect. 5 below we will show that under good conditions the path integral for the theory (4.18) localizes to the critical submanifold of F0 modulo gauge transformations: M = {u : ∇F0 (u) = 0}/G

(4.23)

and that, moreover, the partition function computes the Euler number of this moduli space, Z = χ(M). (4.24) Thus, balanced topological field theories compute Morse theory on field space, with the action potential serving as a Morse function.

426


4.4. Viewing BTFT as a standard TFT. The transformation laws (4.14) are not standard TFT transformations. But we may make the redefinition H 0 = R+− u − H and then view the theory as a standard one with the following familiar field content: • Matter multiplets: ∇+ u = ψ+ , ∇+ ψ+ = R++ u, (4.25) ∇+ φ+− = −η+ , ∇+ η+ = −[φ++ , φ+− ]; • Antighosts:

∇+ ψ− = H 0 , ∇+ H 0 = R++ ψ− ;

(4.26)

∇+ φ−− = −2η− , ∇+ η− = − 21 [φ++ , φ−− ];

(4.27)

• Projection multiplet:

• Gauge fermion for equations I iψ− ∇I F0 + 2α(ψ− , L(φ+− )u) + α(H 0 , ψ− );

(4.28)

• Projection gauge fermion − α(ψ+ , L(φ−− )u) − 21 γ(η+ , [φ−− , φ+− ]).

(4.29)

The rest of the gauge fermion following from the action potential is then declared an irrelevant Q-exact modification. 4.5. The cofield construction. The “co-field construction” described in [1, 2, 4] is a map by which we can assign a BTFT to any TFT. Under good conditions this will compute the Euler character of the original moduli space to which the TFT localizes. We return to the original moduli problem in Sect. 4.1 defined by the vanishing of a section sa (u) in the “bundle of equations. The basic idea is to take X = E ∗ as the field ba ). The degree 0 part of the action potential is then space with coordinates U I = (uµ ; u simply ba sa (u). (4.30) F0 (U ) = u Clearly, the critical points of this Morse function are: ∇µ s a u ba = 0.

sa (u) = 0,

(4.31)

If s is sufficiently nondegenerate the second equation implies u ba = 0 and the solutions to the equations is the same moduli space as in the original TFT. By (4.24) we see that the BTFT will calculate the Euler character of this moduli space. c c∗ . They It is straightforward to implement this idea in detail. The fields generate E may be arranged into two basic quartets: % uµ

&

χµ

ρbµ

& %

bµ H

(4.32)


427

filling out the “fields” of the original moduli problem, and

u ba

%

χ ba

&

& Ha

%

(4.33)

ρa filling out the “antighosts” of the original moduli problem. The BTFT action potential is: Gµν Gµb ψ+ν µ ψ−,a ) F = iF0 (U ) − ( ψ− ψ+,b (4.34) Gaν Gab , F0 (U ) = u ba sa (u), where G is a metric. The construction is easily “equivariantized” by using the equivariant differentials. 4.6. Summary: The deformation complexes. The various classes of topological field theories are nicely summarized by their associated deformation complexes: • For a general topological field theory we have the usual complex →

0

g

C

−→

TC

∇s

−→ E

→0

(4.35)

of symmetries, fields, and equations [11]. • For a balanced topological field theory we get the complex 0

→

(C,0)

−→

g

TC ⊕ g

(∇2 F0 ,C)

−→

TC

→ 0,

(4.36)

where the maps act as η− → (Cη− , 0) (ψ+ , η+ ) → ∇2 F0 ψ+ + Cη+ .

(4.37)

• The cofield construction is associated with the complex 3 0

→

g

−→

E∗ ⊕ T C ⊕ g

−→

E∗ ⊕ T C

→ 0,

(4.38)

where the maps are defined as η− → (Cη− , 0, 0) (ψb+a , ψ+i , η+ ) → ∇2 F0 ψ+ + Cη+ .

(4.39)

3 The rolled-up complex of the cofield construction suggests a role for quaternionic vector spaces. Moreover, these equations suggest a duality between equations and symmetries. We thank Andrei Losev for an interesting discussion about this.

428


5. Localization of BTFT In this section we justify the localization result (4.23) and (4.24) more fully. As we have seen, the general action potential can be taken to be a sum of a function F0 on field space and quadratic terms in the fermions ψA and ηA : F = iF0 (u) + α(ψ+ , ψ− ) + γ(η+ , η− ),

(5.1)

where α, γ are constants. Putting α to zero results in a singular Lagrangian and an illdefined path integral. The coefficient γ is subtle and is related to the introduction of mass terms into topological field theory. The general discussion of the localization of the theory based on the action potential (5.1) is quite involved. We will simply illustrate it for the following situation: (i) All the curvatures and connections can be set to zero. This is the case for topological Yang-Mills, where C is an affine space and for 2D topological gravity in the Beltrami formulation. (ii) The gauge group G acts without fixed points. (iii) The coefficient γ = 0. (Otherwise the action is not quadratic in φ++ , φ−− . ) (iv) All the zero modes of the Hessian of F0 on critical submanifolds are associated with gauge symmetries or tangent directions to the moduli space. In the case that the conditions (i)–(iv) are satisfied, we can justify the localization to (4.23) above, as we will now demonstrate. Let us introduce the notation (ψ1 , L(φ) · ψ2 )T ∗ C ≡ (φ, K(ψ1 , ψ2 ))g .

(5.2)

The action (5.1) becomes: d+ d− F = L1 + L2 + αL3 + αL4 , with L1 L2 L3 L4

= −ih∇F0 , Hi − α(H, H), J = ψ+I (∇2 F0 )IJ ψ− + 2(ψ+ , Cη− ) − 2(ψ− , Cη+ ), † = (φ+− , C Cφ+− ) − 2(φ+− , K(ψ− , ψ+ )), = −(φ−− , C † Cφ++ − K(ψ+ , ψ+ )) + (φ++ , K(ψ− , ψ− )).

(5.3)

(5.4)

The four terms of the Lagrangian play distinguished roles in the evaluation of the path integral, and can be discussed separately: • L1 is the familiar localization to the critical points of the action potential. The evaluation of the path integral near these critical points gives 1 . | Det ∇2 F0 | 0

(5.5)

• L2 is the fermion Lagrangian associated with the deformation complex (4.36) and (4.39) of the equations ∇F0 = 0. Note that gauge invariance of F0 guarantees that this is a complex since ∇2 FCη = 0 at the critical points. Note that the virtual dimension of the moduli space is automatically zero: in the balanced theory there are as many ghost zero modes as antighost zero modes, and they live in the same bundle. The fermion operator is thus: 2 ∇ F0 C . (5.6) 0 C†


429

Because we assume that F0 is a nondegenerate Morse function we can block diagonalize into the kernel of ∇2 F0 and its orthogonal subspace: ! (∇2 F0 )0 0 0 0 0 C . (5.7) 0 C† 0 There is also a finite-dimensional space of fermion zero modes associated to the tangent to moduli space, or, better, to the cohomology of the complex (4.36). The determinant of the fermion non-zero modes is det 0 (∇2 F0 ) · det(C † C).

(5.8)

• L3 : The integral is gaussian, so that φ+− effectively localizes to zero. (More precisely, it localizes to an even nilpotent.) The path integral gives: 1 1 1 √ (K(ψ− , ψ+ ), † K(ψ− , ψ+ )) . (5.9) exp α C C detC † C • L4 : This is also a gaussian integral and gives: 1 1 1 exp (K(ψ+ , ψ+ ), † K(ψ− , ψ− )) . detC † C α C C

(5.10)

Notice that the determinants of C’s do not cancel. The reason is that we have not fixed the gauge. This can be very elegantly solved using the differential topology that we introduced in Sect. 3. We can include naturally the ghosts as well as the antighosts of G-gauge fixing by passing to the Weil model (instead of the Cartan model) of equivariant cohomology, and introducing a gauge-noninvariant term in the action potential. Recall that in the case NT = 2 the Weil multiplet consists of a triplet (ω+ , ω− , ) of connections, see (3.16). Here the connection ω+ appears as the ghost. The connections (ω− , ) represent the antighost multiplet. The gauge fixing Lagrangian is written as d+ d− (u2 ) = d+ (ω− , C † u) + d+ (u, ψ− ) (5.11) = ( , C † u) + (ω− , C † Cω+ ) + d+ (u, ψ− ). √ The integrals over the first two terms provide the missing detC † C. The last term adds some gauge-noninvariant pieces to the “matter” Lagrangian, but we can invoke -independence to argue that these terms make no contribution. The net result of the path integral is an integral over collective coordinates: Z Y 1 0,I duI0 dψ+0,I dψ− exp (K(ψ+ , ψ+ ), † K(ψ− , ψ− ) . (5.12) C C M Finally, let us recall that if E1,2 are trivial hermitian vector bundles and A is a linear fiber map A : E2 → E1 then there is a natural connection on ker A† ⊂ E1 given by P ◦ d, where P is the projection operator. The curvature is just R = P dA

1 dA† P. A† A

(5.13)

In our case C : g → T C and the tangent bundle to the moduli space is T M ∼ = ker C † . We recognize this form in the remaining integral (5.12). Putting all this together we obtain the result (4.24).

430


5.1. Localization of the cofield model: “counting without signs”. The cofield model can be put into the standard framework by taking the field space to be E ∗ → M and the ba ) antighost bundle to be π ∗ (E ⊕ T ∗ M ) → E ∗ . We choose the section s = (s, ∇µ sa u and localize to (4.31). For simplicity suppose ∇µ sa has no kernel and the index is all cokernel. Then we localize to u b = 0. Note that the fermionic operator is 0 ∇µ sa ∇ 2 F0 = . (5.14) (∇µ sa )† 0 For this reason the fermionic path integral is always positive semidefinite and we are “counting without signs” [4]. In any case, the result is: Z = χ(Z(s)), which was, of course, the original motivation for the cofield construction [1].

6. Examples of BTFT’s In this section we briefly mention some important examples of balanced topological field theories in various dimensions. Note that in principle we have a map that associates to any local QFT action F a BTFT, by simply using F as action potential. Of course, to get a reasonable action for the BTFT, for example quadratic in derivatives, the action potential should satisfy certain constraints. Typically it will be first order in derivatives. Fortunately, there are quite a few interesting candidates of that form. 6.1. Morse theory. Take X to be a finite dimensional Riemannian manifold and F0 to be a Morse function. This is the standard example to which supersymmetric quantum mechanics on X (SM Q(X)) reduces. The path integral becomes: Z (6.1) Z = exp −iH µ ∇µ F0 − Gµν H µ H ν + [(ψ+ , ∇2 F0 ψ− ) + · · · . P λi u2i , then where the ellipses indicate various curvature terms. Note that if F0 = 21 P 2 2 2 the quadratic term in the Lagrangian is (∇F0 ) = λi ui . This is indeed the canonical b n 2n|2n b with F(U ) = 2i uAu + ψ+ Bψ− example. If we choose U = (u, ψ± , H) ∈ R = R and A, B quadratic forms, the fundamental gaussian integral is Z 1 exp d+ d− F = sign(detA). (6.2) (2πi)n b b Rn The determinants cancel, and the result does not depend on the choice of A and B, up to a sign. So we see that Z reduces to the sum of the indices of the critical points, and indeed equals the Euler number χ(X). 6.2. Balanced quantum mechanics. Ironically, one cannot obtain SQM (X) as a balanced theory, in spite of the fact that Z = χ(X) for SQM (X) [12]. The balanced theory must necessarily have an action of the form: Z ν dt ωµν [x˙ µ H ν − ψ˙ +µ ψ− − H µ H ν ], (6.3) S1

where ω = ωµν dxµ dxν is a closed two-form.


431

A very natural class of such theories is provided by a symplectic target space (X, ω). Our field space is in that case LX, the space of closed unbased loops. The action potential leading to (6.3 is just I Z µ αµ x˙ = x∗ ω, (6.4) F0 = S1

D

where dα = ω and we consider the circle to be the boundary of a disk D. Moreover, if H(x(t), t) is a time-dependent Hamiltonian then it is natural to consider the more general action potentials: Z I x∗ ω + H(x(t), t)dt. (6.5) F0 = D

Morse theory based on this functional is the subject of symplectic Floer homology [13]. 6.3. Balanced σ-models. There are many natural action potentials R one might want to consider in the context of sigma-models. For example F0 = (∇f )2 would lead to a theory which calculates the Euler character of the moduli space of harmonic maps. Closely related actions have appeared in [14, 15]. Other obvious choices are the Nambu action F0 = Area(f (Σ)). Such actions lead to nonrenormalizable actions. For example, the harmonic map choice leads to an action fourth-order in derivatives. (N.B. The theory is easily generalized to four dimensions.) For this reason we focus on a particular case, described in the next section. 6.4. Cofield σ-models. We describe the cofield construction for topological sigma models [1, 2]. Begin with the standard moduli problem from holomorphic maps: E → M AP (Σ, X). The fields bb in E fit into two quartets:

xi

% &

and x¯ ı¯

% &

ψi

π¯ i ψ¯ ı¯

π ı¯

& %

& %

H¯ i

H ı¯

pz¯i

p¯zı¯

% &

% &

π z¯i

ψ¯ z¯i π¯ zı¯

ψ zı¯

& %

& %

H¯ z¯i

(6.6)

H zı¯ ,

(6.7)

where i, ¯i are holomorphic (anti-holomorphic) indices on the target space X and z, z¯ are holomorphic (anti-holomorphic) on the worldsheet. As we will discuss in Sect. 7 these fields will describe a conformal field theory. The action potential is: F BT σ = iF0 + F metric , where

(6.8)

432


F0 = F

metric

=

R √ Σ

R √ Σ

h

pz¯i ∂¯z¯ xi

hh

( ψ¯ z¯i

z z¯

+

p¯zı¯ ∂z x¯ ı¯ −π¯ ) L i

,

π¯ z¯ ψ¯ ¯

+

( ψ z¯

−π ) L ¯

Tr

π z¯i ψi

(6.9) ,

where h is a metric on Σ and L is a metric related to the hyperkähler metric on T ∗ X, as described in Sect. 7 below. If there is a moduli space but no antighost zero modes, i.e. if dim cok Dz¯ = 0, dim ker Dz¯ > 0, where Dz¯ = ∂¯T ∗ X : 0,0 (Σ; T ∗ X) → 0,1 (Σ; T ∗ X), then we localize to fˆµα = 0, f ∈ HOL(Σ, X), the space of holomorphic maps. Furthermore, the path integral is given by Z = χ(HOL(Σ, X)). This situation is uncommon. 6.5. Balanced topological 2D gravity: Beltrami formulation. There is also a balanced version of topological gravity. We can give two (equivalent) definitions, either using the language of metrics or of complex curves. We start with the latter point of view. In that case the relevant moduli problem is a pair (C, V ) with C a complex curve of genus g and V a holomorphic vector field on C. Since for g > 1 such a vector field is generically zero, the moduli space reduces to the moduli space Mg of curves. However, the virtual dimension of the moduli problem is zero and the theory is thus balanced. So, by definition this model computes χ(Mg ). In more detail: We fix a complex structure and consider the Beltrami differentials µzz¯ ∈ B (−1,1) which modify the Dolbeault operator to ∂¯ (µ) = ∂¯z¯ + µ∂z .

(6.10)

The deformation complex becomes: 0

−→

V ect1,0

with

C

−→

Cη− =

B (−1,1) ⊕ V ect1,0

∂¯ (µ) η− [fˆ, η− ]

D

−→ B (−1,1)

−→

0,

(6.11)

,

D(µ, η+ ) = ([fˆ, µ], ∂¯ (µ) η+ ),

(6.12)

where the “cofield” fˆ is a vector field, and one must take care to write: ∂ (µ) z ∂ z z ¯ ¯ (∂ V ) ∂z ≡ (∂z¯ + µ∂z )V + [µ, V ] ) ∂z ,

(6.13)

[µ, V ]z = µ∂z V z − V z ∂z µ. As we mentioned above, interpreted as an ordinary topological field theory we have the moduli problem of a holomorphic vector field and a complex curve (C, V ). For g > 1 there are no nonsingular holomorphic vector fields. Thus, the localization to fˆ = 0 makes sense and the path integral computes the orbifold Euler character of moduli space. 6.6. Balanced topological 2D gravity: Metric formulation. An alternative formulation of balanced topological gravity starts from metrics. Let M ET denote the space of \ \ ET will be Riemannian metrics hαβ on a topological surface Σ. The basic quartet of M denoted as


433

% hαβ

ψαβ,+

&

& kαβ .

%

(6.14)

ψαβ,− We continue to take Diff (Σ) as the gauge group, since there are no Weyl-invariant metrics on M ET . A natural choice of action is Z √ hhαβ kαβ ), (6.15) I = d + d− ( but an equivalent and more convenient choice of action potential is: F BT G = (ψ+ , ψ− ), d− F BT G = (H, ψ− ) + (R−+ u, ψ− ) − (R−− u, ψ+ ) R√ α β R√ h ρzz¯ (Dz fˆz¯ − Hzz¯ ) + c.c. + hλ ∇ ψαβ . =

(6.16)

Translating the fields to the standard notation for 2D gravity (see, e.g. [2], sec. 16.2) we have: uαβ → δhαβ ,

ψ− → ρ,

φ+− → fˆα ,

φ−− → λα ,

φ++ → γ α .

(6.17)

Again we recognize the gauge fermion appropriate to the moduli problem of a pair (C, V ), C a curve and V a holomorphic vector field. 4 6.7. Balanced topological strings. The coupling of the sigma model to gravity is simply summarized by taking the sum of the action potentials F = F BT G + F BT σ and using the Diff-equivariant version of dA . This completely encodes the coupling to balanced topological gravity ! Let us study the resulting coupling to gravity. We separate the BRST operator into the part that varies the graviton and the rest: dA = d0A + ψA,αβ

δ . δgαβ

(6.18)

The action may then be expressed as

d + d− F = ψB,αβ δ2 F + δgδF (kαβ + · · ·) + δgαβ δgγδ ψ+,αβ ψ−,γδ . αβ

d0+ d0− F

+AB d0A

δF δgαβ

(6.19)

Thus, the auxiliary field kαβ couples to the stress tensor Kαβ of the action potential, while the two partners ψA,αβ of the graviton couple to the variations of the gauge fermions: δΨA0 δF 0 . (6.20) = GA,αβ ≡ dA δgαβ δgαβ The four currents Kαβ , GA,αβ , Tαβ fit into a quartet: 4 We still must choose an action potential that fixes the Weyl mode. In principle any Diff (Σ)-invariant functional of the metric F0 [hαβ ] which takes a unique minimum in each conformal class can serve as F0 .

434


% Kαβ

G+,αβ

&

& %

Tαβ .

(6.21)

G−,αβ ¯ ı¯ Specifically, for the cofield sigma model: Kzz = πzi ∂xi + . . . , Kz¯ z¯ = π¯ z¯ ¯ ı ∂z¯ x + . . .. 6.8. 2D Yang-Mills. There is an obvious choice for an action potential that is first order in derivatives for a two-dimensional gauge theories. Consider a connection A together with a Lie-algebra valued scalar field φ on a Riemann surface Σ. Choose the action potential Z F = Tr(φF ), (6.22) which has its critical points onRthe moduli space of flat connections on Σ. The resulting 2 + · · ·. In fact, this model is rather familiar, since it action will be of the form I = Fµν corresponds directly to the reduction to two dimensions of four-dimensional Donaldson theory [16]. 6.9. 3D Chern-Simons. For a three-dimensional gauge theory there is also a canonical choice for a first-order action potential: the famous Chern-Simons term. Note that quite generally, for any gauge theory we have the field quartet % Aµ

ψ+,µ

&

& %

Hµ

(6.23)

ψ−,µ while the Cartan multiplet φAB , ηA are g-valued fields on spacetime X. If we choose the three-dimensional action potential Z 2 Tr(AdA + A3 ) + Tr ψ+ ψ− , F= 3 X

(6.24)

the resulting action is, according to our general formulae: R I = X Tr F H − H 2 + 2(Dη+ ψ− − Dη− ψ+ ) +ψ+ [φ−− , ψ+ ] + ψ− [φ++ , ψ− ]− 2ψ− [φ+− , ψ+ ]

(6.25)

+(Dφ+− ) − Dφ++ Dφ−− . 2

This turns out to be the reduction to three dimensions of Donaldson theory. The Morse theory problem in this case defines Floer’s 3-manifold homology theory. The theory computes the Euler number of the moduli space of flat connections on X. One can similarly discuss the IG theories of [17]. See also the work of [5]. 6.10. 4D Yang-Mills. This is the context of the twisting of N = 4 supersymmetric Yang-Mills discussed in [4]. We now consider the cofield construction applied to Donaldson theory. This should calculate the Euler character of the moduli space of self-dual instantons. According to the cofield construction we should have two quartets:


% fields

Aµ

ψ+,µ

&

435

& %

% Hµ

equations :

Bµν

ψ+,µν

&

ψ−,µ

& %

Hµν .

ψ−,µν

(6.26) In addition we have the Cartan quintet for YM gauge symmetry, as above. R The naive cofield construction would suggest the action potential F0 = X BF + , but to match with the twisted action of N = 4 SYM one must take a modified action potential. The correct choice is F F1 F2 F3

=F R 1 + F2 + F3 ,+ 1 = RX Tr (B µν Fµν + 12 B µν [Bµλ , Bνλ ]), µν µ = RX Tr (ψ− ψ+,µν + ψ− ψ+,µ ), = X Tr (η+ η− )

(6.27)

The balanced 4D YM theory may be identified with a twist of the N = 4 SYM theory as described in [4]. We embed SU (2)R into the internal SU (4) symmetry of N = 4 SYM so that 2 + 2 = 4. The fermion multiplets then become: ¯ = (2, 2) ⊕ (2, 2) ⊕ (1, 1) ⊕ (1, 3) ⊕ (1, 1) ⊕ (1, 3) (2, 1, 4) ⊕ (1, 2, 4) = (ψ+,µ ) ⊕ (ψ−,µ ) ⊕ (η+ ) ⊕ (ψ+,µν ) ⊕ (η− ) ⊕ (ψ−µν ).

(6.28)

The unbroken internal SU (2) symmetry is the sl2 symmetry of the balanced theory. Similarly we obtain the scalars from ψ IJ = (4 × 4)antisymm → 3 × (1, 1) + (1, 3) giving Bµν , φAB . Adding F3 makes the twisted theory closer to the physical theory, by giving a potential energy to the scalars. 7. A New Twist on the Topological Sigma Model Just the way the balanced four-dimensional Yang-Mills theory is related by a twist to the N = 4 supersymmetric Yang-Mills theory, the balanced topological string on a Kähler target space X is closely related to an N = 4 string with target space T ∗ X. These strings are quite interesting, since the balancing property implies that they are critical in any dimension.5 7.1. Free N = 4 Multiplets. As a warmup we that we write as x XAB˙ = −p ψ ψAB˙ = −π

consider a single free N = 4 multiplet p¯ , x¯ (7.1) π¯ . ¯ ψ

When we generalize to d such multiplets we should regard it as defining a hyperkähler sigma model with target space T ∗ Cd . On shell, this theory has a large N = 4 superconformal symmetry. We focus on the aspects that generalize to a more general target T ∗ X with X a Kähler manifold. The four supercurrents are π∂x − ψ∂p π∂ p¯ + ψ∂ x¯ . (7.2) GA˙ B˙ = ψAA˙ ∂XB B˙ AB = ¯ − π∂p ¯ p¯ + π∂ −ψ∂x ¯ −ψ∂ ¯ x¯ 5

The work in this section was done in collaboration with K. Intriligator and R. Plesser.

436


We furthermore have an SU (2)L × SU (2)R current algebra. The right currents are given by JA˙ B˙ = 21 ψAA˙ ψB B˙ AB .

(7.3)

∗ in the case of a These three currents correspond to the three Kähler forms ωC , ωR , ωC general hyperkähler manifold. We will be interested in targets for which there is an additional U (1) isometry of the metric. In the present case the U (1) isometry current is:

Jzisom = π+,i π¯ +,¯ı − pi ∂ p¯ı¯ , ¯ i, J˜zisom = π¯ −,¯ı π−,i − p¯ı¯ ∂p ¯

(7.4)

Note that the isometry current is not a conformal current, and the conservation law ¯ z − ∂ J˜z¯ = 0. Nevertheless, if one proceeds naively and evaluates the OPE’s for is ∂J on-shell fields, one finds that the charges of the fields under this current are: 1 π(w), Jzisom (z) · π(w) ∼ z−w 1 isom (z) · p(w) ∼ z−w p(w), Jz 1 ¯ ∼ − z−w ∂ p(w) ¯ Jzisom (z) · ∂ p(w)

(7.5)

Now recall that the standard topological twist of an N = 4 multiplet is defined as the following modification of the stress-tensor T 0 = T + ∂J+˙ −˙ , T˜ 0 = T − ∂¯ J˜+˙ −˙ .

(7.6)

This gives the standard A-model for T ∗ Cd [18]. We now describe the new twist, which we call the “isometry twist” or “I-twist.” In terms of conformal field theory the isometry twisted model is related to the T ∗ Cd A-model by the twists: T 00 = T + ∂J+˙ −˙ − ∂J isom = T 0 − ∂J isom , T˜ 00 = T˜ 0 − ∂¯ J˜isom .

(7.7)

The field content of the I-twisted model off shell is described by the bosonic fields xi , x¯ ı¯ , pzi¯ , pzı¯ , the ghost number one fields ψ i , ψ¯ ı¯ , πiz¯ , π¯ ız¯ , and the ghost number −1 fields ψ¯ zı¯ , ψzi¯ , πi , π¯ ı¯ . On shell, we have holomorphic fields: pzi , p¯zı¯ , ψ i , ψ¯ zı¯ , πzi , π¯ ı¯ and similarly for anti-holomorphic fields. In particular, the anti-holomorphic bosonic fields include ˜¯zi¯ , π˜¯ z¯ ˜ i . We summarize a comparison of dimensions for holomorphic conformal p˜z¯ ¯ı, p ¯ı, π fields in the following table. Here 1 0 and 1 00 indicate the conformal dimensions in the usual A-twist and the new I-twist respectively.


437

operator ψ ψ¯ π π¯ p ∂ p¯ π∂x − ψ∂p π∂ p¯ + ψ∂ x¯ ¯ − π∂p −ψ∂x ¯ ¯ p¯ + π∂ −ψ∂ ¯ x¯ J+˙ +˙ J+˙ −˙ J+˙ +˙

10 0 1 0 1 0 1 1 1 2 2 0 1 2

1 00 0 1 1 0 1 0 2 1 2 1 1 1 1

Note the unusual feature that in the I-twist a bosonic current gets twisted. This is one of the most interesting aspects of the isometry twist. Note also that the isometry current is BRST exact: I ¯ −pπ}. ¯ (7.8) J isom = { π∂ p¯ + ψ∂ x, Thus, even though it is not a good conformal current, the resulting model is well-defined. The currents from the isometry twist couple to gravity as in (6.21) with: K G+ G− T

= p∂x, = π∂x + p∂ψ, ¯ − p∂ π, = ψ∂x ¯ ¯ = ∂ x∂x ¯ + p∂(∂ p) ¯ + π∂ π¯ + ψ∂ψ.

(7.9)

7.2. Hyperkähler metric on T ∗ X. Suppose X is a Kähler manifold with metric Gi¯ dxi dx¯ and corresponding Kähler form ω. Let K0 be the Kähler potential. The noncompact manifold T ∗ X has a hyperkähler metric (of signature (n, n)) G on T ∗ X [19]. To make this plausible note that c1 (T ∗ X) = 0 and that, in terms of local holomorphic coordinates (z i , pi ) on T ∗ X, there is a very natural nonvanishing holomorphic 2-form: ωC = dz i ∧ dpi . We denote the components of this hyperkähler metric on T ∗ X as: ds2 = Gi¯ dxi dx¯ ¯ + Gi¯ Dpi dx¯ ¯ + Gi¯ dxi Dp¯¯ + Gi¯ Dpi Dp¯¯ .

(7.10)

The Kähler potential is of the form K = f (ξ),

ξ = Gi¯ (x)pi p¯¯ =k p k2 ,

(7.11)

and hence the metric has the required U (1) isometry in the tangent directions. Example 1. One example of this construction has appeared in the theory of the N = 2 string [20]. Let X be the upper half plane with Poincaré metric and ξ = (Imz)2 k w k2 . Then the Kähler potential for the hyperkähler metric is: 6 p p cξ + e2 − e 2 . (7.12) K = 2 cξ + e + e log p cξ + e2 + e 6 Actually, the signature is (2, 2) so the metric is hypersymplectic. See [21] for a careful discussion of the signs involved.

438


To avoid a singularity we must take c > 0. The construction is SL(2, R) invariant and thus defines a hyperkähler metric on the cotangent bundles to Riemann surfaces of genus g > 1. 7.3. Isometry twisted σ-model in the general case. Using the isometry we twist in the manner described above. In order to do this with the sigma model action one must first add a topological Kähler term to make the bosonic part of the action chiral. The momentum coordinates pick up conformal spins ±1. Consequently the off-diagonal parts of the metric G obtain conformal spin. We then proceed as follows. Define zz¯ h Gi¯ Gzi ¯ . (7.13) G= Gzi¯ ¯ Gi¯ The gauge fermion of the I-twisted model will be: Z √ ¯ i ∂x − Hzi¯ Ψ− = h ( ψ¯ z¯ π¯ ¯ ) G + ( ψzi¯ z¯ Dz¯ pi − Hi

πi ) Gtr

∂x¯ − Hz¯ Dz pz¯ − H¯

. (7.14)

To relate this model to the balanced σ model we must relate the fields. We take: z i ψ ¯ ψz¯ = G , ¯ πi −πz H ¯ Hzi¯ (7.15) =G , ¯ Hi H ¯ z¯ ¯ ¯ ψi ψz = Gtr i −π¯ π¯ ¯ and L = G−1 . In order to relate this theory to the actual action written in [1, 2] we need to use that for ξ → 0 the hyperkähler potential has the form: K → K0 + aξ + O(ξ 2 ) + F (z i ) + F (z i )∗

(7.16)

so that near ξ = 0 the metric becomes a product metric. Since the theory localizes to ξ = 0 (thus effectively killing half the bosonic degrees of freedom) the theories are effectively the same. 7.4. Relation to the N = 2 String. The matter systems described above appear in the N = 2 string. Indeed, the twisting of the N = 4 theory was used in [22] to produce topological field theory formulae for certain N = 2 string amplitudes. However, the gravitational sector of the N = 2 string and the balanced topological string appears to be different. The string theory of large N 2D Yang-Mills theory is a balanced topological string, and that lead to a conjecture that the balanced topological string for the 4D balanced topological string is related to the large N limit of the Donaldson invariants [23, 2]. A slightly different conjecture has been put forward in [24] relating the N = 2 string to the large N limit of “holomorphic Yang-Mills” [25]. A better understanding of relation of the gravitational sector of balanced topological gravity and the gravitational sector of the N = 2 string might shed some light on the compatibility of these conjectures, and even on the nature of 4D topological gauge theories.


439

8. Concluding Remarks Some aspects of the above discussion deserve further investigation. For example, the isometry twist provides a novel method of eliminating bosonic zero modes, and thus provides a novel means of dimensional reduction. Also, there are subtle issues related to the fact that the current used in the twist is not a conformal current. bb Naively, the absence of interesting cohomology on X suggests that there are no interesting observables. Moreover, the cancellation of the anomaly reinforces this. However, this is probably too naive since the action is itself d+ d− exact and yet the path integral is not zero. This point remains to be clarified. The fact that balanced topological strings exist in any dimension is quite curious. In view of this it would be exciting to introduce observables into the theory. Recently there has been intense study of “Dirichlet branes” or D-branes [26]. BPS states associated to D-branes are counted by Euler characters of certain moduli spaces. It would be interesting to see if one can apply BTFT’s to the study of D-branes. Acknowledgement. We thank K. Intrilligator and R. Plesser for some initial collaboration on this project. We also thank A. Losev, N. Nekrasov, J.-S. Park, S. Shatashvili, I. Singer, G. Thompson and G. Zuckerman for useful discussions. R.D. likes to thank the Yale Physics Department for hospitality during part of this work. This work was supported by DOE grants DE-FG02-92ER40704, DOE-91ER40618, by NSF grant PHY 91-23780 and by a Presidential Young Investigator Award.

References 1. Cordes, S., Moore, G. and Ramgoolam, S.: Large N 2D Yang-Mills Theory and Topological String Theory. hep-th/9402107. To appear in Commun. Math. Phys. 2. Cordes, S., Moore, G. and Ramgoolam, S.: in Les Houches Session LXII on http://xxx.lanl.gov/lh94 3. Gross D. and Taylor, W.: Two Dimensional QCD is a String Theory. Nucl. Phys. B400, 181–210 (1993), hep-th/9301068 4. Vafa C. and Witten, E.: A Strong Coupling Test of S-Duality. Nucl. Phys. B431, 3–77 (1994), hepth/9408074 5. Blau, M. and Thompson, G.: N = 2 topological gauge theory, the Euler characteristic of moduli spaces, the Casson invariant. Commun. Math. Phys. 152, 41–72 (1993) 6. Blau M. and Thompson, G.: Topological Gauge Theories from Supersymmetric Quantum Mechanics on Spaces of Connections. Int. J. Mod. Phys. A8, 573–585 (1993) 7. Kalkman, J.: BRST Model for Equivariant Cohomology and Representatives for the Equivariant Thom Class. Commun. Math. Phys. 153, 447 (1993); BRST Model Applied to Symplectic Geometry. PRINT93-0637 (Utrecht), hep-th/9308132 8. Dubois-Violette, M.: A bigraded version of the Weil alegbra and of the Weil homomorphism for Donaldson invariants. J. Geom. Phys. 19, 18–30 (1996), hep-th/9402063 9. Blau, M., Thompson, G.: Localization and Diagonalization: A review of functional integral techniques for low-dimensional gauge theories and topological field theories. J. Math. Phys. 36, 2192–2236 (1995), hep-th/9501075 10. Berline, N., Getzler, E. and Vergne, M.: Heat Kernels and Dirac Operators, Berlin–Heidelberg–New York: Springer-Verlag, 1992 11. Witten, E.: Introduction to Cohomological Field Theories. Lectures at Workshop on Topological Methods in Physics, Trieste, Italy, June 11–25, 1990, Int. J. Mod. Phys. A6, 2775 (1991) 12. Witten, E.: Constraints on Supersymmetry Breaking. Nucl. Phys. B202, 253 (1982) 13. Floer, A.: Symplectic Fixed Points and Holomorphic Spheres. Commun. Math. Phys. 120, 575 (1989) 14. Polyakov, A. M.: Fine Structure of Strings. Nucl. Phys. B268, 406 (1986) 15. Horava, P.: Topological Strings and QCD in Two Dimensions. hep-th/9311156; Topological Rigid String Theory and Two Dimensional QCD. hep-th/9507060 16. Witten, E.: Two Dimensional Gauge Theories Revisited. J. Geom. Phys. G9, 303 (1992), hep-th/9204083

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17. Witten, E.: Topology Changing Amplitudes in (2 + 1)-Dimensional Gravity. Nucl. Phys. B323, 113 (1989) 18. Witten, E.: In: Essays on Mirror manifolds. Ed. S.-T. Yau, Hong Kong: International Press, 1992 19. Calabi, E.: Ann. Sci. Ecole Norm. Sup. 12, 269 (1979); See also, Cecotti, S., Ferrara, S. and Girardello, L.: Geometry of Type II Superstrings and the moduli of superconformal field theories. Int. J. Mod. Phys. A4, 2475 (1989) 20. Ooguri, H. and Vafa, C.: Geometry of N = 2 Strings. Nucl. Phys. B361, 469–518 (1991) 21. Barrett, J., Gibbons, G.W., Perry, M.J., Pope, C.N. and Ruback, P.J.: Kleinian geometry and the N=2 string. Int. J. Mod. Phys. A9, 1457–1494 (1994), hep-th/9302073 22. Berkovits, N. and Vafa, C.: N=4 Topological Strings, Nucl. Phys. B433, 123–180 (1995), hep-th/9407190 23. Moore, G.: 2D Yang-Mills Theory and Topological Field Theory, hep-th/9409044, Proceedings of ICM94 24. Ooguri, H. and Vafa, C.: All Loop N=2 String Amplitudes. Nucl. Phys. B451, 121–161 (1995), hepth/9505183 25. Park, J.-S.: Holomorphic Yang-Mills theory on compact Kahler manifolds, hep-th/9305095; Nucl. Phys. B423, 559 (1994); Park, J.-S.: N = 2 Topological Yang-Mills Theory on Compact Kähler Surfaces, Commun. Math, Phys. 163, 113 (1994); Park, J.-S.: N = 2 Topological Yang-Mills Theories and Donaldson Polynomials, hep-th/9404009 26. Chaudhuri, S., Johnson C. and Polchinski, J.: Notes on D-branes. hep-th/9602052 Communicated by A. Jaffe

Commun. Math. Phys. 185, 441 – 465 (1997)

Communications in


Skew Young Diagram Method in Spectral Decomposition of Integrable Lattice Models Anatol N. Kirillov1,? , Atsuo Kuniba2 , Tomoki Nakanishi3,?? 1 2 3

Department of Mathematical Sciences, University of Tokyo, Komaba, Meguro-ku, Tokyo 153, Japan Institute of Physics, University of Tokyo, Komaba, Meguro-ku, Tokyo 153, Japan Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599, USA

Received: 28 July 1996 / Accepted: 11 October 1996

Abstract: The spectral decomposition of the path space of the vertex model associb n ) is studied. We ated to the vector representation of the quantized affine algebra Uq (sl give a one-to-one correspondence between the spin configurations and the semistandard tableaux of skew Young diagrams. As a result we obtain a formula of the characters for the degeneracy of the spectrum in terms of skew Schur functions. We conjecture that our result describes the sln -module contents of the Yangian Y (sln )-module structures b n . An analogous result is obof the level 1 integrable modules of the affine Lie algebra sl tained also for a vertex model associated to the quantized twisted affine algebra Uq (A(2) 2n ), where Y (Bn ) characters appear for the degeneracy of the spectrum. The relations to the spectrum of the Haldane-Shastry and the Polychronakos models are also discussed.

1. Introduction The corner transfer matrix (CTM) has been attracting much attention in the recent study of integrable lattice models based on the Yang-Baxter equation [3]. The CTM acts on the space of paths, which is often identified with the semiinfinite tensor product of a finite-dimensional quantum group module. It is well known that the trace of the CTM g ) is related to affine of a vertex model associated to the quantized affine algebra Uq (b Lie algebra characters. We call this correspondence the DJKMO (Date-Jimbo-KunibaMiwa-Okado) correspondence. In [2] a fine structure of the CTM spectrum, called the spectral decomposition, is b 2 ) vertex models. The idea behind it is as follows: The logarithmic studied in the Uq (sl derivative of the CTM is regarded as the energy operator, or the Hamiltonian, of the path space. As a nature of an integrable system, we expect that there exists a family of ? ??

Permanent address: Steklov Mathematical Institute, Fontanka 27, St. Petersburg, 191011, Russia Permanent address: Department of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464, Japan

442

A. N. Kirillov, A. Kuniba, T. Nakanishi

commuting operators (the integrals of motion) which act on the path space and commute with the Hamiltonian. The spectral decomposition is the simultaneous diagonalization of these integrals of motion. The degeneracy of the spectrum, then, reflects the non-abelian symmetry which commutes with these integrals of motion. In the meanwhile, the action of the Yangian algebra Y (sl2 ) is defined on the level b 2 in [11], and their Y (sl2 )1 integrable modules of the untwisted affine Lie algebra sl module structures are determined [6]. It turns out that the degeneracy of the spectrum of b 2 ) vertex model exactly describes the Y (sl2 )-module the CTM Hamiltonian in the Uq (sl structure of the level 1 integrable modules [2]. We believe this coincidence is a universal phenomena. Namely, we expect that a similar coincidence occurs between the spectrum of the commutant of the non-abelian, probably some quantum group, symmetry of a conformal field theory and the CTM spectrum in the corresponding lattice model. Motivated by this expectation, in this paper we study the spectral decomposition b n ). The counter-part of of the vertex model of the vector representation of the Uq (sl b n . The action of the the DJKMO correspondence is the level 1 integrable modules of sl Yangian Y (sln ) on these modules is defined in [22], but the Y (sln )-module structure is not fully studied yet (however, see [5] for a related result). In this paper we determine the characters of the degeneracy of the spectrum, and show that they are the characters of irreducible Y (sln )-modules as expected. Therefore, we conjecture that our spectral decomposition exactly describes the Y (sln )-module structure of the level 1 integrable modules at the character level. Conceptually, the spectral decomposition in the sln case is formulated just as in the case of sl2 . However, due to the complexity of the irreducible modules of Y (sln ) for n ≥ 3, the incidence matrix technique used in [2] is not very efficient. A key to overcome this difficulty is the observation that there exists a natural one-to-one correspondence between the paths of the vertex model and the semistandard tableaux of certain skew Young diagrams. The appearance of the skew Young diagrams is not quite unexpected, because they label a family of Y (sln )-modules, called the tame modules in [20]. Thanks to this correspondence, we can express the characters of the degeneracy of the spectrum in terms of the skew Schur functions. This enables us to identify them as irreducible Y (sln ) characters. It is possible to extend our analysis for a vertex model associated to the quantized twisted affine algebra Uq (A(2) 2n ). The characters of the degeneracy of the spectrum are analogues of the skew Schur functions. They are conjectured in [18] to be irreducible Y (Bn ) characters. The content of the paper is as follows: In Sect. 2 we review the DJKMO correspondence for the vertex model in the sln case. In Sect. 3 we formulate the spectral decomposition of the model. In Sect. 4 some properties of skew Young diagrams and skew Schur functions are given. In Sect. 5 we describe the correspondence between the configurations of the vertex model and the semistandard tableaux, and determine the characters of the degeneracy of the spectrum. In Sect. 6 the identifications with irreducible Y (gln ) and Y (sln ) characters are given. In Sect. 7 we discuss the relation with the spectrum of other type of spin models, such as the Haldane-Shastry model and the Polychronakos model. In Sect. 8 an analogous result for an Uq (A(2) 2n ) vertex model is presented. In Appendix A the equality between the skew Schur functions and the Rogers-Szegö polynomials is proved. In Appendix B a new combinatorial description of the Kostka-Foulkes polynomials is given. In Appendix C we describe the level 1 character of A(2) 2n .

Skew Young Diagram Method

443

2. DJKMO Correspondence We review the correspondence between the CTM spectrum of the vertex models of the b n ) and the affine Lie algebra characters of sl b n [8]. vector representation of Uq (sl ~ For given two infinite sequences, ~a = (a1 , a2 , . . .) and b = (b1 , b2 , . . .), of any kind of objects ai , bi , we write ~a ≈ ~b if ai 6= bi only for finitely many i. We often use a shorthand notation ~a = (a1 , . . . , ak , (ak+1 , . . . , ak+m )∞ ) for such a periodic sequence as ~a = (a1 , . . . , ak , ak+1 , . . . , ak+m , ak+1 , . . . , ak+m , . . .). Let 31 , . . . , 3n−1 be the fundamental weights of the Lie algebra sln , and let i = 3i − 3i−1

(2.1)

for i = 1, . . . , n with 30 = 3n = 0. Then B(31 ) = {1 , . . . , n } is the set of all the weights of the irreducible representation (vector representation) of sln whose highest weight is 31 . We give a total ordering in B(31 ) as 1 ≺ 2 ≺ . . . ≺ n . We define the energy function H : B(31 ) × B(31 ) → {0, 1} as 0 if i ≺ j , (2.2) H(i , j ) = 1 if i j . The function H will play an essential role in our study. It is the logarithm of the R-matrix b n ) in the limit q → 0. associated to the vector representation of Uq (sl An infinite sequence ~s = (si ), si ∈ B(31 ), is called a spin configuration if it has a form ~s = (s1 , . . . , sm , (1 , 2 , . . . , n )∞ ), where s1 , . . . , sm is an arbitrary finite sequence. Equivalently, ~s is a spin configuration if ~s ≈ ~s(k) for some k = 0, . . . , n − 1, where ~s(k) = (1 , 2 , . . . , k , (1 , 2 , . . . , n )∞ ).

(2.3)

The set of all the spin configurations S has a natural decomposition S=

n−1 G

S (k) ,

S (k) = {~s | ~s ≈ ~s(k) }.

(2.4)

k=0

For ~s = (si ) ∈ S (k) we define its energy E(~s) and sln -weight wt(~s) as ∞ n o X (k) E(~s) = i H(si , si+1 ) − H(s(k) i , si+1 ) , i=1

wt(~s) = 3k +

∞ X

si − s(k) . i

(2.5a) (2.5b)

i=1

Proposition 2.1. Let ~s = (s1 , . . . , sm , (1 , . . . , n )∞ ) be any element of S (k) . Then wt(~s) =

m X i=1

si .

444


Proof. Since m ≡ k modulo n, m X i=1

si = 3k +

m X

Pm

(k) i=1 si

= 3k . Thus

(si − s(k) i ) = 3k +

i=1

∞ X

(si − s(k) s). i ) = wt(~

i=1

There is a remarkable connection between the partition function of S (k) and an affine Lie algebra character. Theorem 2.2 (DJKMO correspondence [8, 15]). For k = 0, 1, . . . , n − 1, let L(3k ) b n whose highest be the level 1 integrable module of the untwisted affine Lie algebra sl b weight is the kth fundamental weight 3k of sln . Then the following equality holds: X q E(~s) ewt(~s) (2.6a) ch L(3k ) = q 1k −c/24 ~ s∈S (k)

X

= q 1k −c/24

q E(~s) e−wt(~s) ,

(2.6b)

~ s∈S (n−k)

where ch L(3k ), 1k = k(n − k)/2n, and c = n − 1 are the (normalized) character, the conformal dimension, and the Virasoro central charge of L(3k ), respectively [13]. In (2.6b) S (n) = S (0) . Remark . Often spin configurations are described by an alternative notion, paths. An b n -weights is called a path if it satisfies the conditions: infinite sequence p~ = (pi ) of sl (i) pi+1 − pi ∈ B(31 ), (ii) p~ = (p1 , . . . , pm , (30 , 31 , . . . , 3n−1 )∞ ). It is clear that the map p~ = (pi ) 7→ ~s = (pi+1 − pi ) is a bijection from the set of all the paths P to S. By this identification our spin configurations are also called paths in some literature. The sln -weight of a path p~ = (pi ) is defined as wt(~ p) = p1 . Then, under the bijection p~ 7→ ~s, wt(~ p) = −wt(~s) holds. It is this context where (2.6b) is proved in [8]. The expression (2.6a) follows from (2.6b) by the Dynkin diagram automorphism αi ↔ αn−i of sln . 3. Spectral Decomposition We introduce the local energy map h : S → {0, 1}N such that h : ~s = (si ) 7→ ~h = (hi ),

hi = H(si , si+1 ).

(3.1)

Each number hi is called the ith local energy of ~s. We call the image Sp = h(S) the Fn−1 spectrum of S. Let Sp(k) = h(S (k) ). Then we have the decomposition Sp = k=0 Sp(k) . An element of Sp(k) is characterized as follows. Proposition 3.1. An element ~h = (hi ) ∈ {0, 1}N belongs to Sp(k) if and only if it satisfies the conditions, hi + hi+1 + · · · + hi+n−1 ≥ 1 for any i. (k) ~ ~ (ii) h ≈ h , where ~h(k) := h(~s(k) ) = (0, . . . , 0, 1, (0, . . . , 0, 1)∞ ). | {z } | {z } (i)

k

n

(3.2a) (3.2b)


445

One can paraphrase the condition (3.2a) as “There are at most n − 1 consecutive 0’s in ~h”. Here we prove only the necessity of the conditions. The sufficiency will be proved after Prop. 5.1 in Sect. 5. Proof. Let us assume hi = hi+1 = · · · = hi+n−2 = 0. Then we have si ≺ si+1 ≺ · · · ≺ si+n−1 , from which si+n−1 = n follows. Therefore, hi+n−1 = H(si+n−1 , si+n ) = 1 regardless of the value of si+n . The condition (ii) is an immediate consequence of (2.3). Any element ~h of Sp(k) is uniquely written in the form [m1 , . . . , mr ] := (0, . . . , 0, 1, . . . , 0, . . . , 0, 1, (0, . . . , 0, 1)∞ ), | {z } | {z } | {z } m1

mr

1 ≤ mi ≤ n, mr 6= n.

n

(3.3) Obviously

Sp

(k)

r X

= {[m1 , . . . , mr ] | r ≥ 0, 1 ≤ mi ≤ n, mr 6= n,

mi ≡ k mod n}.

i=1

The surjection h : S (k) → Sp(k) induces the decomposition of S (k) , S (k) =

G

S~h = h−1 (~h).

S~h ,

(3.4)

~ h∈Sp(k)

We call this decomposition the spectral decomposition of S (k) . Let us introduce the character of the degeneracy of the spectrum at ~h, X ch S~h = q 1k −c/24 q E(~s) ewt(~s) ~ s∈S~h

P∞ (k) = q 1k −c/24+ i=1 i(hi −hi ) χ~h ,

χ~h =

X

ewt(~s) .

(3.5)

~ s∈S~h

As is standard in the character theory of sln , we regard χ~h = χ~h (x) as a function of the variables x1 = e1 , x2 = e2 , . . . , xn = en with the relation x1 x2 · · · xn = 1. Due to Theorem 2.2, the character of L(3k ) is decomposed as ch L(3k )(q, x) = q 1k −c/24

X

P∞ q

i=1

i(hi −h(k) ) i

χ~h (x).

(3.6)

~ h∈Sp(k)

The main purpose of the paper is to calculate the characters χ~h and to show that they are irreducible characters of Y (sln ). To do it, we make use of a hidden relation between spin configurations and skew Young diagrams.

446


6

m1

6? m2

? 6 ?

mr

Fig. 1. A border strip hm1 , . . . , mr i

4. Skew Diagrams and Skew Schur Functions Let us recall the definitions of a skew diagram, a semistandard tableau, and the skew Schur function. We basically follow the definitions and notations of [19]. A partition λ = (λ1 , λ2 , . . . , λm ) is a non-increasing Pm sequence of non-negative integers, λ1 ≥ λ2 ≥ · · · ≥ λm ≥ 0. We let |λ| = i=1 λi . The length l(λ) of λ is the number of the non-zero elements in λ. As usual, a partition λ is represented by a (Young) diagram, which is denoted by the same symbol λ. We conveniently identify the partitions (λ1 , . . . , λm ), (λ1 , . . . , λm , 0), (λ1 , . . . , λm , 0, 0), etc. The conjugate of a partition λ is a partition λ0 whose diagram is the transpose of the diagram of λ along the main diagonal. For example, if λ = (4, 3, 2), then its conjugate is λ0 = (3, 3, 2, 1). For a pair of partitions λ and µ, we write λ ⊃ µ if λi ≥ µi for any i. If λ ⊃ µ, the diagram µ is naturally embedded inside the diagram λ. Then the skew (Young) diagram λ/µ (denoted by λ−µ in [19]) is obtained by subtracting the diagram µ from the diagram λ. For example, if λ = (5, 4, 4, 1) and µ = (4, 3, 2, 0), then λ/µ looks as follows:

We set |λ/µ| = |λ| − |µ|. We say λ/µ is a skew diagram of rank n if the length of any column of λ/µ does not exceed n. Two boxes in a skew diagram are adjacent if they share a common side. A skew diagram λ/µ is connected if, for any pair of boxes a and a0 in λ/µ, there exits a series of boxes b1 = a, b2 , . . . , bj = a0 in λ/µ such that bi and bi+1 are adjacent. A skew diagram is called a border strip if it is connected, and contains no 2 × 2 block of boxes. Let hm1 , . . . , mr i denote the border strip of r columns such that the length of ith column (from the right) is mi (Fig. 1). For a skew diagram λ/µ, we now define the skew Schur function sλ/µ . In each box of a given skew diagram λ/µ, let us inscribe one of the numbers 1, 2, . . . , n. We call such an arrangement of numbers a semistandard tableau T of shape λ/µ, if it satisfies the following condition: Let a and b be the inscribed numbers in any pair of adjacent boxes. Then, (i) (ii)

a < b if b is lower-adjacent to a. a ≥ b if b is left-adjacent to a.

(4.1a) (4.1b)


447

The sln -weight of a semistandard tableau T is defined as wt0 (T ) =

n X

ma · a ,

(4.2)

a=1

where ma is the number counting how many a’s are in T , and a is given in (2.1). Definition 4.1. The skew Schur function sλ/µ is defined as X

sλ/µ (x) =

0

ewt (T ) ,

x i = e i ,

(4.3)

T ∈SST(λ/µ)

where SST(λ/µ) is the set of all the semistandard tableaux of shape λ/µ. The following proposition is well-known. See Sect. 5 of [19] for a proof. Proposition 4.1. The skew Schur function sλ/µ is also expressed as sλ/µ (x) = det(eλ0i −µ0j −i+j (x))1≤i,j≤r ,

(4.4)

where r ≥ l(λ0 ), and em = em (x) is the mth elementary symmetric polynomial of variables x1 , . . . , xn for m = 0, . . . , n, and em = 0 for other m. We impose the relation x1 x2 · · · xn = 1 throughout the paper. Then em is the character of the mth fundamental representation of sln with the highest weight 3m for m = 1, . . . , n − 1. The following properties of sλ/µ follow either from (4.3) or from (4.4): (i) If λ/µ is not a skew diagram of rank n, then sλ/µ = 0. (ii) When µ = (0), the expression (4.4) reduces to the Jacobi-Trudi formula of the ordinary Schur function sλ . (iii) Let cλµν , |λ| = |µ| + |ν|, be the Littlewood-Richardson coefficient, i.e., sµ sν = P λ P λ λ cµν sλ . Then, sλ/µ = ν cµν sν . The conjugate s∗λ/µ of the skew Schur function sλ/µ is defined as s∗λ/µ = det(en−λ0i +µ0j +i−j )1≤i,j≤r X 0 = e−wt (T ) .

(4.5) (4.6)

T ∈SST(λ/µ)

It is also possible to express s∗λ/µ as a skew Schur function. Suppose λ/µ is of rank n with µ = (µ1 , . . . , µm ). Then we have a new pair µ˜ = (λ1 , . . . , λ1 , µ1 , . . . , µm ) ⊃ λ, | {z } n

and the compliment of λ/µ, (λ/µ)c := µ/λ, ˜ is also a skew diagram of rank n. The picture below illustrates the example of n = 4, λ = (5, 4, 3, 1), µ = (3, 2):

448


λ/µ →

← (λ/µ)c

Proposition 4.2. Let λ/µ be a skew diagram of rank n. Then s∗λ/µ = s(λ/µ)c . Proof. s(λ/µ)c = det(en+µ0i −λ0j −i+j ) = det(en−λ0i +µ0j +i−j ) = s∗λ/µ .

5. Correspondence between Spin Configurations and Semistandard Tableaux We proceed to calculate the character χ~h of (3.5) using the language of skew diagrams and tableaux. For a given ~h ∈ Sp, we associate a skew diagram κ( ˜ ~h) of infinite-size in the following procedure: 1. Write the first box. 2. Attach the second box under (resp. left to) the first box if h1 = 0 (resp. h1 = 1). 3. Similarly attach the i + 1th box under (resp. left to) the ith box if hi = 0 (resp. hi = 1) for i = 2, 3, . . . Then κ( ˜ ~h) has the following properties. (i)

It is a border strip.

(ii) It is of rank n, i.e., the length of any column of κ( ˜ ~h) does not exceed n, due to (3.2a). (iii) Due to (3.2b), it has a periodic tail which consists of length-n columns. ˜ ~h) is the border strip hm1 , . . . , mr , n, n, . . .i. Equivalently, for ~h = [m1 , . . . , mr ], κ( ˜ A semistandard tableau T of shape κ( ˜ ~h) is an arrangement of numbers 1, . . . , n in ~ the boxes of κ( ˜ h) obeying the conditions (4.1a) and (4.1b), just in the same way as in the finite-size case. See Fig. 2. Notice that the arrangements in the length-n columns are uniquely determined, or “frozen”, because of the semistandard condition (4.1a). Since κ( ˜ ~h) is a border strip, one can give a total ordering of the boxes in it from the right to the left and from the top to the bottom in the unique way. Let (ai ) = (a1 , a2 , . . . , (1, 2, . . . , n)∞ ) be the sequence of the content of T˜ along the total ordering of the boxes. Now we have a natural map ϕ˜ from the set of the semistandard tableaux of shape κ( ˜ ~h) to the set of the spin configurations S defined by ϕ( ˜ T˜ ) := (ai ) = (a1 , a2 , . . . , (1 , 2 , . . . , n )∞ ). A key observation of this paper is

(5.1)


449

q

q

q

a1 a4 a3 a2 a6 a5

1 n 2 .. . 1 n

q

q

q

Fig. 2. A semistandard tableau T˜ of shape κ( ˜ ~h) for ~h = (0, 1, 1, 0, 1, . . . , (0, . . . , 0, 1)∞ )

Proposition 5.1. The map ϕ˜ gives a one-to-one correspondence between the semistandard tableaux of shape κ( ˜ ~h) and the spin configurations in S~h . Proof. A necessary and sufficient condition for a sequence (a1 , a2 , . . .), ai ∈ {1, . . . , n} to be the sequence of the content of a semistandard tableau of shape κ( ˜ ~h) is ai+1 > ai , (resp. ai+1 ≤ ai ) ⇐⇒ hi = 0, (resp. hi = 1).

(5.2)

This is an immediate consequence of the construction of κ( ˜ ~h) and (4.1a,b). On the other hand (5.2) is also a necessary and sufficient condition for a sequence (a1 , a2 , . . .) to belong to S~h because of (2.2), (3.1), and (3.4). The first application of Prop. 5.1 is to prove the sufficiency part of Prop. 3.1. Proof. By Prop. 5.1 we have only to show that there exists at least one semistandard tableau of shape κ( ˜ ~h) for any ~h. In fact, for a given ~h, a semistandard tableau of shape ~ κ( ˜ h) is obtained by filling the boxes by 1, 2, 3, . . . from the top to the bottom in each column. This completes the proof of Prop. 3.1. For our purpose it is convenient to define a “finite part” κ(~h) of the infinite diagram ~ κ( ˜ h) by cutting off its periodic tail. Namely, for ~h = [m1 , . . . , mr ] we define κ(~h) = hm1 , . . . , mr i. (For ~h = ~h(0) , κ(~h(0) ) is the empty diagram ∅.) It is clear that the map κ : ~h 7→ κ(~h) is injective, and we get the following description of the space of the spectrum Sp in terms of border strips. Theorem 5.2. The space Sp is parametrized by the border strips of rank n such that the lengths of their leftmost columns are less than n. The following lemma is obvious. Lemma 5.3. There is a one-to-one correspondence between the semistandard tableaux of shape κ( ˜ ~h) and the ones of shape κ(~h). The correspondence is given by the restriction of a semistandard tableau of shape κ( ˜ ~h) on κ(~h).

450


Combining the bijection in Lemma 5.3 with the bijection ϕ˜ of (5.1), we obtain a bijection ϕ : SST(κ(~h)) → S~h . Proposition 5.4. The bijection ϕ : SST(κ(~h)) → S~h is weight-preserving, i.e., for any T ∈ SST(κ(~h)), wt(ϕ(T )) = wt 0 (T ) holds. Proof. Let (a1 , . . . , am ) be the content of T aligned along our total order of the boxes in κ(~h). Then ) = (a1 , . . . , am , (1 , . . . , n )∞ ). From Prop. 2.1 and (4.2), we have Pϕ(T m wt(ϕ(T )) = i=1 ai = wt0 (T ). Now we state the first half of our main theorem. Theorem 5.5. (i) The character χ~h of S~h is equal to the skew Schur function sκ(~h) . (ii) Let ~h = [m1 , . . . , mr ] ∈ Sp. Then

sκ(~h) = shm1 ,...,mr i

e mr 1 0 =

···

emr +mr−1 emr−1 1 0

.. .. . . 0 1 e m2 0 1

emr +···+m1 .. . . em2 +m1 e m1

(5.3)

b n decomposes as (iii) The character of the level 1 integrable module L(3k ) of sl X 1 1 ch L(3k )(q, x) = q − 24 c q 2n |κ|(n−|κ|)+t(κ) sκ (x) (5.4a) κ∈BS |κ|≡k mod n

= q − 24 c 1

X

q 2n |κ|(n−|κ|)+t(κ) sκc (x) , 1

(5.4b)

κ∈BS |κ|≡n−k mod n

where BS is the set of all the border strips κ = hm1 , . . . , mr i of rank n with mr < n, Pr−1 and t(κ) = i=1 (r − i)mi . Proof. The property (i) is an immediate consequence of Prop. 5.4. To show (ii), notice that the skew diagram hm1 , . . . , mr i is represented as λ/µ with a pair λ ⊃ µ such that λ0i = m1 + · · · + mr+1−i − r + i,

µ0i = m1 + · · · + mr−i − r + i.

Substituting them into (4.4) we obtain the formula (5.3). The property (iii) follows from (3.6), Prop. 4.2, the property (i), and the fact that for ~h = [m1 , . . . , mr ] ∈ Sp(k) 1k +

∞ X i=1

X 1 m(n − m) + (r − i)mi , 2n r−1

i(hi − h(k) i )=

i=1

m=

r X i=1

Two expressions (5.4a) and (5.4b) differ from each other when n ≥ 3.

mi .


451

6. Yangian Characters In this section we show, based on the result of [20], that the characters χ~h = sκ(~h) are irreducible characters of the Yangian algebras Y (gln ) and Y (sln ). The Yangian of gln , Y (gln ), is an algebra generated by tij (r), i, j = 1, . . . , n, r ∈ Z≥0 with the relations [tij (r), tkl (s − 1)] − [tij (r − 1), tkl (s)] = tkj (s − 1)til (r − 1) − tkj (r − 1)til (s − 1), where tij (−1) = δij 1, tij (−2) = 0. The elements tij (0) generate the universal enveloping algebra of gln . Therefore sln acts on Y (gln )-modules. Consider a pair of partitions λ ⊃ µ with λ = (λ1 , . . . , λN +n ), µ = (µ1 , . . . , µN ), N ≥ 1. Let Vλ be the irreducible glN +n -module associated to λ, Vµ be the irreducible glN -module associated to µ, and Vλ,µ be the space of the multiplicity of Vµ in Vλ under the standard embedding glN ⊂ glN +n . There is an irreducible action of Y (gln ) on the space Vλ,µ , having a remarkable property (A module with such property is called a tame module [20]): Proposition 6.1 ([7, 20]). A maximal commutative subalgebra of Y (gln ), called the Gelfand-Zetlin (GZ) algebra, acts on Vλ,µ in a semi-simple way. Furthermore, a basis diagonalizing the GZ algebra is labeled by the GZ schemes of Vλ,µ . A GZ scheme \3/ of Vλ,µ is an array of integers λmi , λ1 λ2 ··· λn,N +n ··· λN +n ··· λn−1,N +n−1 λn−1,1 ··· λn−1,N +n−1 = \3/ = ··· ··· λ01 · · · λ0N µ 1 · · · µN satisfying the condition λmi ≥ λm−1,i ≥ λm,i+1 . The sln -weight of the basis vector labeled by \3/ is (N +m ) n NX +m−1 X X λmi − λm−1,i m . λn1 λn2 λn−1,1

m=1

i=1

i=1

Lemma 6.2. There is a weight-preserving, one-to-one correspondence between the GZ schemes of Vλ,µ and the semistandard tableaux of shape λ/µ. The correspondence is described as follows. For a given GZ scheme \3/, we have a sequence of partitions, λ(0) = µ ⊂ λ(1) ⊂ λ(2) ⊂ · · · ⊂ λ(n) = λ,

λ(m) = (λm1 , λm2 , . . . , λm,N +m ).

A semistandard tableau of shape λ/µ is obtained by inscribing the number m in λ(m) /λ(m−1) , which is a part of the diagram λ/µ. For example, 5

4 5

4 4

4

1 4

4

0 0

2

0 0

0

2 7→

1 2 2

3 4 3 2 It is easy to check that this map is bijective and weight-preserving. It follows from Prop. 6.1 and Lemma 6.2 that the sln -character of Vλ,µ is equal to sλ/µ . In particular, we have

452


Theorem 6.3. The character χ~h = sκ(~h) is the sln -character of the irreducible Y (gln )module Vλ,µ with λ/µ = κ(~h). The Yangian of sln , Y (sln ), is generated by x± ik , hik , i = 1, . . . , n − 1, k ∈ Z≥0 with the relations [hik , hjl ] = 0,

± [hi0 , x± jl ] = ±Aij xjl ,

[x+ik , x− jl ] = δij hik+l ,

1 ± ± ± [hik+1 , x± jl ] − [hik , xjl+1 ] = ± Aij (hik xjl + xjl hik ), 2 1 ± ± ± ± ± ± ± [x± ik+1 , xjl ] − [xik , xjl+1 ] = ± Aij (xik xjl + xjl xik ), 2 i ii h h X h ± ± ± x± = 0, , x , . . . , x , x ikσ(1) ikσ(2) ikσ(1−A ) jl . . . ij

σ:permutation

where Aij is the Cartan matrix of sln . An irreducible finite-dimensional module of Y (sln ) is characterized by n − 1 monic polynomials (the Drinfel’d polynomials), P1 (u), . . . , Pn−1 (u). The polynomial Pi (u) describes the action of hi (u) = 1 + P ∞ −k−1 on a highest weight vector v as hi (u)v = (Pi (u + 1)/Pi (u))v [9]. k=0 hik u Proposition 6.4 ([20]). There is a one-parameter family of irreducible Y (sln )-module structures on Vλ,µ with a parameter b ∈ C, whose Drinfel’d polynomials are [20]1 Pi (u) =

λ1 Y j=1 λ0 −µ0 =i j j

1 0 1 0 u + (λj + µj ) − j + + b . 2 2

(6.1)

These Y (sln )-module structures on Vλ,µ are the ones induced by a one-parameter family of embeddings of Y (sln ) into Y (gln ). There is a simple pictorial interpretation of the zeros of Pi (u) as shown in Fig. 3. As a corollary of Prop. 6.4, we obtain the second half of our main theorem: Theorem 6.5. The character χ~h = sκ(~h) is the sln -character of the one-parameter family of the irreducible Y (sln )-modules whose Drinfel’d polynomials are given by (6.1) with λ/µ = κ(~h). We have shown that the characters χ~h of the degeneracy of the spectrum are irreducible Y (sln ) characters. Furthermore, the Y (sln )-module structure of L(3k ) partially studied in [22] agrees with (5.4a). Based on these strong evidences, we conjecture that Conjecture 6.6. The decomposition (5.4a) describes the Y (sln )-module structure on L(3k ) of [22]. 7. The Relation with the spectrum in other spin models

7.1. The Haldane-Shastry model. The sln Haldane-Shastry (HS) model is a lattice model with the Hamiltonian 1 The Drinfel’d polynomials here are the one in [9]. The convention in [20] is slightly different. Their P (u) i is equal to (−1)degPi Pi (−u − n/4 + i/2) here.


453

4−b

•

−b

•

−b

•

6

@

@

3 2

@

@

@

@

@

@

−3 − b

@

•

@

@

@

@

@

@

@

@

@•

@ @•

@ @ @ @• •

x=b Fig. 3. An example: λ = (5, 4, 4, 1), µ = (4, 3, 2, 0). The Drinfel’d polynomials are P1 (u) = (u + 3 + b)(u + b)(u − 4 + b), P2 (u) = u − 23 + b, and Pi (u) = 1 for i ≥ 3. The zeros of Pi (u) are identified with the intersections of the line x = b and the diagonal lines passing through the middle points of the columns of length i

HHS =

X 1≤j6=k≤N

xj xk (Pjk − 1), (xj − xk )(xk − xj )

xj = e2π

√

−1j/N

acting on the vector space V ⊗N , V = Cn , where Pjk exchanges the j th and k th components of V ⊗N . There is an action of Y (sln ) on V ⊗N which commutes with the Hamiltonian HHS [11]. By this action the space V ⊗N decomposes as [5] M V ⊗N ' Wd , (7.1) d∈MN

where Wd are certain irreducible Y (sln )-modules described below. MN is the set of the binary sequences d = (d1 , . . . , dN −1 ), di ∈ {0, 1}, such that there are at most n − 1 consecutive 1’s. The eigenvalue of HHS on the eigenspace Wd is given by PN −1 i=1 idi (idi − N ). We notice that the condition for d ∈ MN turns into (3.2a) through the identification hi = 1 − di . 2 For a given d ∈ MN , let ~hd = (1 − d1 , 1 − d2 , . . . , 1 − dN −1 , 1, (0, . . . , 0, 1)∞ ) ∈ Sp. | {z } n

We translate the description of the Y (sln )-module structure of Wd in [5] into our language as follows: Proposition 7.1 ([5]). As a Y (sln )-module, the eigenspace Wd is isomorphic to the irreducible module whose Drinfel’d polynomials are given by (6.1), where in (6.1) λ1 and b are certain constants independent of d, and λ/µ = κ(~hd ). 2 This intriguing relation between the spectrum of the HS model and the vertex model was first indicated by Bernard et al. [4, 6] in the sl2 case.

454


Comparing Prop. 7.1 with Prop. 6.4 and Theorem 6.5, we obtain the following proposition, which answers the questions of the sln -module content of Wd and its factorizability asked in [5, 10, 11, 12]. Proposition 7.2. Let ~hd = [m1 , . . . , mr ]. Then (i) ch Wd = shm1 ,...,mr i . (ii) shm1 ,...,mr i = shm1 ,...,mi i shmi+1 ,...,mr i if mi + mi+1 ≥ n + 1. Proof. We only need to prove the property (ii), which follows from (5.3).

7.2. The Polychronakos model. There is another relevant spin model, called the Polychronakos model. The sln Polychronakos model has the Hamiltonian HP =

X 1≤j n. To complete the proof of Theorem A.1, we have only to show that Lemma A.6. For 1 ≤ m ≤ min(n, N ), AN,m (q) = (−1)m+1

(q)N −1 . (q)N −m

(A.7)

Proof. For any N ≥ 1, AN,1 (q) = q cN,1 (1) = 1. Below we show AN,m (q) = −(1 − q N −1 )AN −1,m−1 (q), for 2 ≤ m ≤ N . From these, the lemma follows. Consider m X X (−1)j+1 q cN,m (k1 ,...,kj ) AN,m (q) = j=1

(A.8)

(A.9)

ki ≥1 k1 +···+kj =m

for 2 ≤ m ≤ N . Notice that we have dropped the upper inequality ki ≤ n in the summation, because it is automatically satisfied under the assumption m ≤ n. Let Im = {(k1 , . . . , kj ) | j, ki ≥ 1, k1 + · · · + kj = m} (1) and be the set of all the ordered partitions of m. Then Im is the disjoint union of Im (2) Im , where (1) = {(k1 , . . . , kj , 1) | (k1 , . . . , kj ) ∈ Im−1 }, Im (2) = {(k1 , . . . , kj + 1) | (k1 , . . . , kj ) ∈ Im−1 }. Im (1) (2) and Im , separately. The contribution Let us perform the summation in (A.9) over Im (1) from Im is

Pm−1 P =−

Pm−1 P j=1

j=1

j+2 cN,m (k1 ,...,kj ,1) q (k1 ,...,kj )∈Im−1 (−1)

j+1 cN −1,m−1 (k1 ,...,kj ) q (k1 ,...,kj )∈Im−1 (−1)

= −AN −1,m−1 (q).

(2) is The contribution from Im m−1 X

X

(−1)j+1 q cN,m (k1 ,...,kj +1)

j=1 (k1 ,...,kj )∈Im−1

=

m−1 X

X

(−1)j+1 q cN −1,m−1 (k1 ,...,kj )+N −1 = q N −1 AN −1,m−1 (q).

j=1 (k1 ,...,kj )∈Im−1

Putting the both together, we get (A.8).

462


B. A New Combinatorial Formula for Kostka-Foulkes Polynomials As another corollary of Theorem 7.4, or Theorem A.1, we get a new formula for the Kostka-Foulkes polynomials in terms of the Littlewood-Richardson tableaux of border strips. See [16, 19] for further information of the material discussed here. For a border strip κ = hm1 , . . . , mr i we define t(κ) as t(κ) =

r−1 X

(r − i)mi .

i=1

Further, for a partition λ, we denote by C(κ, λ) the number of the semistandard tableaux that form lattice permutations (cf. [19]) and are of shape κ and content λ. The number C(κ = µ/ν, λ) has an interpretation as cµνλ = Mult Vλ (Vµ,ν ), i.e., the multiplicity of the irreducible representation Vλ of sln in the restriction of the representation Vµ,ν of Y (sln ) to sln , where n ≥ l(λ). Proposition B.1. Let λ be a partition, then X q t(κ) C(κ, λ) = Kλ,(1|λ| ) (q),

(B.1)

κ

where the summation is taken over all the border strips κ with |κ| = |λ|. Proof. Let n ≥ l(λ). Theorem A.1 can be rewritten as Pn 1 X X (q)N q t(κ) sκ (x) = q i=1 2 ki (ki −1) xk1 1 · · · xknn . (q) · · · (q) k1 kn κ:border strip k ∈Z i ≥0 k1 +···+kn =N

|κ|=N

On the other hand, the degree-N part of Corollary 6 of [16] reads as Pn 1 X X (q)N Kλ,(1|λ| ) (q)sλ (x) = q i=1 2 ki (ki −1) xk1 1 · · · xknn . (q) · · · (q) k1 kn λ:Young diagram k ∈Z i ≥0 k1 +···+kn =N

|λ|=N

P Equate the left hand sides of two equalities, and expand sκ (x) as λ C(κ, λ)sλ (x), where |λ| = |κ| in the summation. Using the independence of sλ ’s, we have (B.1). Example . Let λ = (3, 2, 1). Then it is known [19] that K(3,2,1),(16 ) (q) = q 4 (1 + q)2 (1 + q 2 )(1 + q 3 ). On the other hand, there exist 14 border strips which give non-zero contribution to the left hand side of (B.1) with the following tableaux: 11 12 2 3 4

1 112 2 3 5

11 2 13 2 5

1 12 12 3 6

1 12 13 2 6

1 12 2 13 7

1 2 113 2 7

111 22 3

1 2 13 12 8

111 2 23

11 122 3

11 12 23

11 22 13

11 2 123

1 112 23

1 12 123

8

8

9

9

10

10

7

11


463

where the number t(κ) is attached below each diagram. Hence the left hand side of (B.1) is q 4 (1 + 2q + 2q 2 + 3q 3 + 3q 4 + 2q 5 + 2q 6 + q 7 ) = K(3,2,1),(16 ) (q). k In the same spirit we define the branching function b3 λ (q) of L(3k ) (as an sln module) as X k b3 ch L(3k )(q, x) = λ (q)sλ (x). λ:Young diagram l(λ) is the Verma module bosonic highest weight state, with quantum numbers h, 21 h− , 21 h+ given by L0 |Λ >= h|Λ >,

J03 |Λ >=

1 h− |Λ >, 2

U0 |Λ >=

1 h+ |Λ >, 2

(4.61)

we have, Class I.|Λ > has conformal weight h ≥ 0, arbitrary real charge 21 h+ and isospin 21 h− obeying (4.56) h− + (k + 1)m − n = 0, 0 ≤ m ≤ q − 1, 0 ≤ n ≤ p − 1.

(4.62)

All singular vectors are bosonic descendants of the highest weight state, and therefore have the same fixed arbitrary real charge 21 h+ . They are organised in four families labeled by a positive integer a ≥ 0, with quantum numbers Za0 Ta0 Za+1 Ta+1

Ha = h + a2 pq + a(qn − pm), (h− )a = n + 2ap − m(k + 1), : Ha = h + mn + a2 pq + a(qn + pm), (h− )a = −n − 2ap − m(k + 1), : Ha+1 = h + mn + (a + 1)2 pq − (a + 1)(qn + pm), (h− )a+1 = −n + 2(a + 1)p − m(k + 1), : Ha+1 = h + (a + 1)2 pq − (a + 1)(qn − pm), (h− )a+1 = n − 2(a + 1)p − m(k + 1). :

(4.63)

Note that at the edge of the Kac table, when n = 0, one has the following identification: 0 0 ≡ Za+1 , and Ta+1 ≡ Ta+1 . (4.64) Za+1 We refer to this case as the collapsed version of the generic case n 6= 0. The corresponding embedding diagrams (Fig. 2a and Fig. 2b) are constructed in an iterative way by using the lemmas above. One has, in terms of modified Weyl transformations, Za0

=

(w0(q−m−1) w1(m) )a Z00 ,

Ta0

=

(w1(m) w0(q−m−1) )a w1(m) Z00 ,

Za+1

=

(w0(q−m−1) w1(m) )a w0(q−m−1) Z00 ,

Ta+1

=

(w1(m) w0(q−m−1) )a+1 Z00 .

(4.65)

Class II. |Λ > has conformal weight h ≥ 0, but the charge and isospin obey the following constraints, h− + (k + 1)m − n = 0 which implies

and h− − h+ = −2(k + 1)m0

h− + h+ = 2(k + 1)(m0 − m) + 2n.

(4.66) (4.67)

484

P. Bowcock, A. Taormina

Fig. 5. Class III Embedding Diagrams for n = 1

ˆ Representation Theory of sl(2/1; C) at Fractional Level

Fig. 6. Class III Embedding Diagram for p other than 1

485

486


Here, and

0 ≤ m ≤ q − 1, 1 ≤ n ≤ p − 1, m0 ∈ Z+ m0 − m = (σ − 1)q + m, ˜

σ ∈ Z+ ,

0≤m ˜ ≤ q − 1.

(4.68) (4.69)

0

The case where m is a negative integer is totally similar and leads to embedding diagrams which are mirror images of the ones presented in Figure 3 and Figure 4. The singular vectors have charge 21 h+ when they are bosonic descendants of the highest weight state, and their quantum numbers are those of (4.63). The fermionic descendants have charge 1 1 2 H+ − 2 , and their other quantum numbers are obtained from (4.63) by shifting n → n − 1,

h → h + m0 .

(4.70)

The value n = 1 corresponds to a degenerate (collapsed) situation. The embedding diagrams for this case and the case n 6= 1 are given in Fig. 3 and Fig. 4 when m ˜ = 0. If m ˜ 6= 0, one must distinguish between the cases when 0 ≤ m + m ˜ ≤ q − 1 and q ≤ m+m ˜ ≤ 2q − 1. However, the diagrams have the same structure as in the m ˜ =0 case. The only difference is in the singular vector sitting at the annihilation node in the fermionic sector. Unlike Class I, Class II possesses bosonic and fermionic singular vectors in the same Verma module. The nilpotency of the fermionic generators in A(1, 0)(1) has a crucial impact on the way the singular vectors are related. In Fig. 3 for instance, the singular 0 : the “path” between these two vectors is formally vector Tσ is not a descendant of Tσ−1 given by 0 , (4.71) Tσ = (w1(m) w0(q−m−1) )σ w1(m) (w0(q−m−1) w1(m) )σ−1 Tσ−1 0 0 0 = w1(m) Zσ−1 . It can be shown that w1(m) w1(m) Zσ−1 is zero, using but one also has Tσ−1 (3.41). In order to connect singular vectors of charge h+ to singular vectors of charge h+ −1, one uses the transformations w0(q−m−1) and w1(m) as well as two fundamental fermionic −0 0 transformations. The first one relates Zσ−1 and Zσ−1 using Lemma 2, namely, 0

0

˜ ˜ − 0 Zσ−1 = w˜ 0(m+m) j0− w˜ 1(m+m) Zσ−1 , 0

(4.72)

0

− 0 which reduces to Zσ−1 = j0− Zσ−1 when m + m ˜ = 0. The second one is not of the kind given in the lemmas. Although it connects two singular vectors which correspond to zeros of the Kac determinant, the latter does not encode the fact that one is the −0 descendant of the other. This second basic fermionic transformation relates Zσ−1 and 0 Tσ−1 as follows: ˜ ˜ −0 0 Tσ−1 = w˜ 1(m) (J0− )h+ −1 j0− w˜ 1(m+m) Zσ−1 . (4.73)

Class III. |Λ > has conformal weight h ≥ 0, but the charge and isospin obey the following constraints, h− + (k + 1)m = 0 which implies

and h− − h+ = −2(k + 1)m0

h− + h+ = 2(k + 1)(m0 − m).

(4.74) (4.75)


487

Here, 0 ≤ m ≤ q − 1, m0 ∈ Z+ ,

(4.76)

and m0 − m = (σ − 1)q + m ˜ ≥ 1,

σ ∈ Z+ ,

0≤m ˜ ≤ q − 1.

(4.77)

0

The case where m is a negative integer produces embedding diagrams which are mirror images of the diagrams in Fig. 5 and Fig. 6. The bosonic singular vectors have charge 1 2 h+ , with the other quantum numbers given by (4.63) when n = 0, Za0

:

0 Ta+1

:

Ha = h + a2 pq − apm, (h− )a = 2ap − m(k + 1), Ha+1 = h + (a + 1)2 pq + (a + 1)pm, (h− )a+1 = −2(a + 1)p − m(k + 1),

(4.78)

with a ≥ 0. The fermionic singular vectors have charge 21 h+ − 21 , with quantum numbers 0

Za− 0

Ta− − Za+1 − Ta+1

:

Ha = h + m0 − m + a2 pq + a(q − pm), (h− )a = 1 + 2ap − m(k + 1),

Ha = h + m0 + a2 pq + a(q + pm), (h− )a = −1 − 2ap − m(k + 1), : Ha+1 = h + m0 + (a + 1)2 pq − (a + 1)(q + pm), (h− )a+1 = −1 + 2(a + 1)p − m(k + 1), : Ha+1 = h + m0 − m + (a + 1)2 pq − (a + 1)(q − pm), (h− )a+1 = 1 − 2(a + 1)p − m(k + 1). :

(4.79)

The two diagrams in Fig. 5 and Fig. 6 correspond to the cases p = 1 and p 6= 1 respectively, with m ˜ = 0. If m ˜ 6= 0, one must, as in Class II, distinguish between the cases 0 ≤ m+m ˜ ≤ q − 1 and q ≤ m + m ˜ ≤ 2q − 1. However, the diagrams have the same structure as the ones given here, and we omit them. Class IV. |Λ > has conformal weight h ≥ 0, but the charge and isospin obey the following constraints, h− + (k + 1)m = 0

and h− − h+ = −2(k + 1)m0

(4.80)

which implies h− + h+ = 2(k + 1)(m0 − m).

(4.81)

0 ≤ m ≤ q − 1, m0 ∈ Z+ ,

(4.82)

Here, but m0 − m = (σ − 1)q + m ˜ ≤ 0,

σ ∈ Z+ , 0 ≤ m ˜ ≤ q − 1.

(4.83)

The bosonic singular vectors have charge 21 h+ , with the other quantum numbers given by (4.78). The fermionic singular vectors have either charge 21 h+ − 21 or 21 h+ + 21 .

488


Z’0 T 0+

T −0 Z’1

+

−

Z1

Z1 T’1

+

−

T1

T1 Z’2

+

−

Z2

Z2 T’2

−

+

T2

T2

Fig. 7. Class IV Embedding Diagram

Their other quantum numbers are respectively, − Za+1

Ta−

Ha+1 = h + m0 − m + (a + 1)2 pq + (a + 1)(q − pm), (h− )a+1 = 1 + 2(a + 1)p − m(k + 1), : Ha = h + m0 + a2 pq + a(q + pm), (h− )a = −1 − 2ap − m(k + 1).

(4.84)

Ha+1 = h − m0 + (a + 1)2 pq + (a + 1)(q − pm), (h− )a+1 = 1 + 2(a + 1)p − m(k + 1), : Ha = h + m − m0 + a2 pq + a(q + pm), (h− )a = −1 − 2ap − m(k + 1)

(4.85)

:

and + Za+1

Ta+

:


489

with a ≥ 0. The corresponding embedding diagram is given in Fig. 7. 0 0 and Za+2 for a ≥ 0 is a rather new and The double multiplicity of the vectors Ta+1 remarkable feature. Until recently ([17], [12]), it was common belief that the singular vectors appearing in embedding diagrams all had multiplicity one. Our analysis for ˆ sl(2/1; C) confirms the presence of singular vectors of higher multiplicities for particular choices of highest weight state quantum numbers (namely class IV). We can indeed generalise the MFF construction in a highly nontrivial way in order to properly take into account the complications due to the presence of nilpotent fermionic generators. Given the importance of the higher multiplicities of singular vectors, in particular in deriving character formulas, we now illustrate our technique and construct two singular vectors 0 0 T1 (1) and T1 (2) with the quantum numbers of T10 as they appear in (4.78). We restrict ourselves to the case m0 < m/2, m odd. Similar ideas can be used for 0 m even, and/or any m0 such that m0 − m ≤ 0. The vector T1 (1) is easily constructed as 0

T1 (1) = w1(m) w0(q−m−1) Z00 ,

(4.86)

with the help of the generalised Weyl transformations introduced in the previous section. The second vector is far from being trivial as a descendant of the highest state Z00 . A reasonable starting point would be to construct the state w1(m) Z00 . However, with the class IV choice of quantum numbers for Z00 in particular, this state identically vanishes due to its internal fermionic structure, as can be checked by using the definition (3.38). One therefore needs to “improve” the state w1(m) Z00 to avoid its vanishing. As described (m) in the following expressions, the improved state, called w c1 Z00 , is given by a linear 0 + and (J0− )(−1) j0− j0− . combination of the appropriately dressed neutral objects logJ−1 0

Explicitly, the singular vector T1 (2) is given by 0

T1 (2) = w1(m) w0(q−m−1) w c1

(m)

Z00 ,

(4.87)

with w c1

(m)

Z00

=

4h2+

0 w˜ 0(m )

(m/2−m0 −1/2)

0

(J0− )−h+ −1 j0− j0− w˜ 1

(m/2−m0 −1/2)

+ αlogJ−1 w˜ 0

0

0

(J0− )h+ −1 j0− j0− + β(J0− )−1 j0− j0−

0

w˜ 1(m ) Z00 ,

(4.88)

(h+ + (2j − 1)(k + 1)) (h+ − (2j − 1)(k + 1)).

(4.89)

where α = h+ and m/2−m0 −1/2

β=

Y j=1

0

+ allows to write up vectors such as T1 (2) in a concise way, making The function logJ−1 it easy to check they are singular. Indeed, J1− commutes with w˜ 0(M ) and w˜ 1(M ) for any value of M , and because the commutator + + −1 ] = (J−1 ) (k + 1 − 2J03 ) [J1− , logJ−1

(4.90)

490


vanishes when evaluated on a state whose quantum number H− satisfies k + 1 − H− = 0 0 (it is always the case in our construction), one concludes that, in particular, J1− T1 (2) = 0 + , it is interesting to note that checking 0. Although j0+ and j0+ commute with log J−1 0

0

0

+ is not an eigenvector of J03 j0+ T1 (2) = j0+ T1 (2) = 0 is not straightforward. Indeed, log J−1 (although (4.88) is), i.e., + ] = 1, (4.91) [J03 , logJ−1 0

and hence the corrective term β(J0− )−1 j0− j0− is introduced in (4.88) in order to ensure 0 that j0+ T1 (2) = 0. + appears in the formal expression of some of our singular vectors, Although log J−1 it does actually not survive in any evaluation of the vector, since it can be commuted through with the help of the following relations, + ] [j0− , logJ−1

0

=

+ + −1 j−1 (J−1 ) ,

+ [j0− , logJ−1 ]

=

+ + −1 −j−1 (J−1 ) ,

+ [J0− , logJ−1 ]

=

+ −1 3 + −2 + −2(J−1 ) J−1 + (J−1 ) J−2 ,

0

(4.92)

and disappears because the first term in the square bracket of (4.88) vanishes identically + is removed. when log J−1 We have concentrated here on the case where m is odd. If m is even, one substitutes 0 + . appropriately the expression [h+ log J0− + (J0− )−1 j0− j0− ] for log J−1 We now proceed to indicate the relation between the fermionic singular vector T0+ 0 0 and the two uncharged singular vectors T1 (1) and T1 (2) . The construction given in Lemma 5 for T0+ , namely, 0 0 T0+ = w˜ 0(m−m ) j0− w˜ 1(m−m ) Z00 , (4.93) leads to a state which vanishes identically in this class, again because of its internal fermionic nature. The actual fermionic vector is obtained by using an “improved” version of Lemma 5, inspired by the idea above, namely, 0 (m/2−m0 −1/2) (m/2−m0 −1/2) (m−m0 ) − + + αlogJ−1 w˜ 0 (J0− )h+ −1 j0− j0− j0 w˜ 1 T0 = 2h+ w˜ 0 0 − h+ − −β(J0 ) j0 w˜ 1(m ) Z00 , (4.94) with α and β given as before. It is now almost straightforward to identify which particular 0 0 linear combination of T1 (1) and T1 (2) is a descendant of T0+ , 0

0

N1 (−4h2+ βN2 T1 (1) + T1 (2) ) = 0

0

0

w1(m) w0(q−m−1) w˜ 0(m ) (J0− )−h+ −1 (−2h+ j0− ) w˜ 1(m−m ) T0+ , (4.95) where N1 is a nonzero normalisation constant given by (3.42) for M = m0 − m, H+ = −H− = h+ + 1, and N2 is similarly given by (3.43) for M = m0 , H− = h− and H+ = h+ . 0 Finally, the uncharged singular vector T1 (2) can be seen as a descendant of the fermionic singular vector T0− in the following way. By Lemma 2, one constructs T0− as,

ˆ Representation Theory of sl(2/1; C) at Fractional Level 0

0

491 0

T0− = w˜ 0(m ) j0− w˜ 1(m ) Z00

(4.96)

and 0

0

N 0 T1 (2) = 4h2+ w˜ 0(m ) 0 (m/2−m0 −1/2) (m/2−m0 −1/2) + αlogJ−1 w˜ 0 (J0− )h+ −1 j0− (J0− )−h+ −1 j0− j0− w˜ 1 0 − −1 − β(J0 ) j0 w˜ 1(m ) T0− . (4.97) The normalisation factor N 0 is, 0

N =

0 m −1 Y

(h+ + h− + 2i(k + 1) − 2) (h+ − h− − 2i(k + 1)).

(4.98)

i=0

Once more, this detailed analysis illustrates very well the power of our analytic expressions in relating singular vectors between themselves within a given embedding diagram. 5. Conclusions The Lie superalgebra A(1, 0) and its affinisation A(1, 0)(1) play a crucial role in the description of noncritical N = 2 superstrings. In order to study the space of physical states of the latter theory, using the tool provided by topological G/G WZNW models, a detailed analysis of various modules over A(1, 0)(1) is needed. Many Lie superalgebras share with A(1, 0) the property that two sets of simple roots may not be equivalent up to Weyl tranformations, which are generated by reflections with respect to bosonic simple roots. An added technical complication in A(1, 0) is the fact that the fermionic roots are lightlike, which prevents one from defining coroots and fundamental weights in a straightforward way. These properties are emphasized in Sect. 2. The classical and ˆ quantum free field Wakimoto representations of sl(2/1; R) built with two inequivalent sets of simple roots are given in [10]. It is shown there that there exists a set of field transformations which relate the two Wakimoto representations in the classical and the quantum case. Section 3 organises the information provided by the Kac-Kazhdan determinant formula relevant to the Lie superalgebra A(1, 0)(1) in five lemmas. The Malikov-Feigin-Fuchs construction is generalised to incorporate transformations which relate bosonic and fermionic singular vectors within a Verma module. Section 4 provides vital information for the construction of admissible representations of A(1, 0)(1) , namely the quantum numbers and embedding diagrams of the singular vectors appearing in highest weight Verma modules when the level k of the algebra satisfies the necessary condition k + 1 = p/q with p, q nonzero positive relatively prime integers. We believe that the extra conditions leading to the seven embedding diagrams of Classes I,II,III, IV are sufficient to determine all admissible representations, whose characters should provide finite representations of the modular group. Our analysis clearly shows a very close link between the embedding diagrams of Sect. 4 and those of some completely degenerate representations of the N = 2 superconformal algebra [17]. This striking \ similarity is reminiscent of the link between the admissible sl(2; C) modules and the \ degenerate Virasoro modules, and between the admissible osp(1/2; C) modules and the

492


N = 1 degenerate N = 1 superconformal modules. A recent paper [38] offers some explanation of this similarity. Acknowledgement. We would like to thank Gerard Watts for sharing with us his insight on N = 2 singular vectors, and for making one of his symbolic computer programmes available to us. We also thank Alexei Semikhatov for discussions on the relation between N = 2 and A(1, 0)(1) singular vectors, and Jonathan Evans for pointing out reference [28]. Anne Taormina acknowledges the U.K. Engineering and Physical Sciences Research Council for the award of an Advanced Fellowship. She also thanks CERN for its hospitality, where part of this work was done.

References 1. Abdalla, E., Zadra, A.: Noncritical superstrings: A comparison between continuum and discrete approaches. Nucl. Phys. B432, 163 (1994) 2. Abdalla, E., Abdalla, M.C.B., Dalmazi, D.:On the amplitudes for noncritical N = 2 superstrings. Phys. Lett. B291, 32 (1992) 3. Aharony, G., Ganor, O., Sonnenschein, J., Yankielowicz, S., Sochen, N.: Physical states in G/G models and 2d gravity Nucl. Phys. B399, 527 (1993) 4. Aharony, G., Sonnenschein, J., Yankielowicz, S.: G/G models and WN strings. Phys. Lett. B289, 309 (1992) 5. Antoniadis, I., Bachas, C., Kounnas, C.: N = 2 super-Liouville and noncritical strings. Phys. Lett. B242, 185 (1990) 6. Bars, I., Gunaydin, M.: Unitary representations of noncompact supergroups. Commun. Math. Phys. 91, 31 (1983) 7. Bershadsky,M., Ooguri, H.: Hidden Osp(N, 2) symmetries in superconformal field theories. Phys. Lett. B229, 374 (1989) 8. Bouwknegt, P., Mc Carthy, J., Pilch, K.: Semi-infinite cohomology in conformal field theory and 2d gravity. Commun. Math. Phys. 145, 541 (1992) 9. Bowcock, P., Taormina, A.: Noncritical N = 2 strings. Replace by Phys. Lett. B388, 303 (1996) ˆ 10. Bowcock, P., Koktava, R-L. K., Taormina, A.: Wakimoto modules for the affine superalgebra sl(2/1; C) and noncritical N = 2 strings. To appear 11. Cornwell J.F.: Group Theory in Physics. Volume III. New York: Academic Press, 1989 12. de Vos, K., van Driel, P.: The Kazdhan-Lusztig conjecture for W-algebras. hep-th/9508020 13. D’Hoker, E., Vinet, L.: Dynamical supersymmetry of the magnetic monopole and the 1/r2 potential. Commun. Math. Phys. 97, 391 (1985) 14. Distler, J., Hlousek, Z., Kawai, H.: Super-Liouville theory as a two-dimensional, superconformal supergravity theory. Int. J. Mod. Phys. A5, 391 (1990) 15. Dobrev, V.K., Petkova, V.B.: Group theoretical approach to extended conformal supersymmetry: Function space realisations and invariant differential operators. Fortschr. Phys. 35, 7, 537 (1987) 16. Dobrev, V.K.: Multiplets of Verma Modules over the osp(2/2)(1) super Kac-Moody algebra. In: Topological and Geometrical Methods in Field Theory, Proceedings, eds. J. Hietarinta and J. Westerholm (Espoo 1986), Singapore: World Scientific, 1986 17. Dorrzapf, M.: Analytic expressions for singular vectors of the N = 2 superconformal algebra. Preprint DAMTP-94-53 (May 1995), hep-th/9601056; Superconformal field theories and their representations. Cambridge Thesis 1995 18. Fan, J-B., Yu, M.: Modules over affine Lie superalgebras. Preprint AS-ITP-93-14 (1993), hep-th/9304122 19. Fan, J-B., Yu, M.: G/G gauged supergroup valued WZWN field theory. Preprint AS-ITP-93-22 (1993), hep-th/9304122 20. Gawedski, K., Kupiainen A.: Coset construction from functional integrals. Nucl. Phys. B320, 649 (1989) 21. Goulian, M., Li, M.: Correlation functions in Liouville theory. Phys. Rev. Lett. 66, 2051 (1991) 22. Hu, H.L.,Yu, M.: On the equivalence of noncritical strings and G/G topological field theories. Phys. Lett. B289, 302 (1992); On BRST cohomology of sl(2, R)p/q−2 /sl(2, R)p/q−2 gauged WZNW models. Nucl. Phys. B391, 389 (1993) 23. Ito, K.: Quantum hamiltonian reduction and N = 2 cosets. Phys. Lett. B259, 73 (1991) 24. Kac, V.G.: Lie superalgebras. Adv. Math. 26, 8 (1977)


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25. Kac, V.G., Wakimoto, M.: Integrable highest weight modules over affine superalgebras and number theory. hep-th/9407057 (1994) 26. Kac, V.G., Wakimoto M.: Modular invariant representations of infinite-dimensional Lie algebras and superalgebras. Proc. Nat. Acad. Sci. 85, 4956 (1988) 27. Kac, V.G., Kazhdan D.A.: Structure of representations with highest weight of infinite-dimensional Lie algebras. Adv. Math. 34, 97 (1979) 28. Kac, V.G.: Highest weight representations of conformal current algebras. In: Topological and Geometrical Methods in Field Theory, Proceedings, eds. J. Hietarinta and J. Westerholm (Espoo 1986), Singapore: World Scientific, 1986 29. Karabali, D., Schnitzer, H.J.: BRST quantization of the gauged WZW action and coset conformal field theories. Nucl. Phys. B329, 625 (1990) 30. Kimura, K.: Hamiltonian reduction of super Osp(1, 2) and sl(2, 1) Kac-Moody algebras. Int. Journ. Mod. Phys. A7, suppl.1B, 533 (1992). 31. Koktava, R-L. K.: Field transformations of the Lie superalgebra sl(2/1). Phys. Lett. B351, 476 (1995) 32. Parker, M.: Classification of real simple Lie superalgebras of classical type. J. Math. Phys. 21, 689 (1980) 33. Penkov, I., Serganova, V.: Indag. Math. N.S.3(4), 419 (1992). 34. Petersen, J.L., Rasmussen, J., Yu, M.: Conformal blocks for admissible representations in SL(2) current algebra. Nucl. Phys. B457, 309 (1995) 35. Polyakov, A.M., Wiegmann, P.B.: Theory of nonabelian Goldstone bosons. Phys. Lett. B131, 121 (1983); Goldstone fields in two dimensions with multivalued actions. Phys. Lett. B141, 223 (1984) 36. Sadov, V.: On the spectra of SL(N )k /SL(N )k cosets and W(N) gravities. Int. J. Mod. Phys. A8, 5115 (1993). 37. Scheunert, M., Nahm, W., Rittenberg, V.: Irreducible representations of the osp(2, 1) and spl(2, 1) graded Lie algebras. J. Math. Phys. 18, 155 (1977) 38. Semikhatov, A.: The non-critical N=2 string is an sl(2/1) theory, hep-th/9604105. 39. Van der Jeugt, J., Hughes, J.W.B., King, R.C., Thierry-Mieg, J.: Character formulae for irreducible modules of the Lie superalgebras sl(M/N ) J. Math. Phys. 31, 274 (1990). Communicated by R.H. Dijkgraaf

Commun. Math. Phys. 185, 495 – 508 (1997)

Communications in


On the Classification of Reflexive Polyhedra Maximilian Kreuzer? , Harald Skarke?? Institut für Theoretische Physik, Technische Universität Wien, Wiedner Hauptstraße 8–10, A-1040 Wien, Austria Received: 11 January 1996 / Accepted: 26 November 1996

Abstract: Reflexive polyhedra encode the combinatorial data for mirror pairs of Calabi– Yau hypersurfaces in toric varieties. We investigate the geometrical structures of circumscribed polytopes with a minimal number of facets and of inscribed polytopes with a minimal number of vertices. These objects, which constrain reflexive pairs of polyhedra from the interior and the exterior, can be described in terms of certain non-negative integral matrices. A major tool in the classification of these matrices is the existence of a pair of weight systems, indicating a relation to weighted projective spaces. This is the cornerstone for an algorithm for the construction of all dual pairs of reflexive polyhedra that we expect to be efficient enough for an enumerative classification in up to 4 dimensions, which is the relevant case for Calabi–Yau compactifications in string theory. 1. Introduction In the framework of toric geometry, it is possible to encode properties of algebraic varieties in terms of fans or polyhedra defined on integer lattices. In particular, it has been shown by Batyrev that the Calabi–Yau condition for hypersurfaces of toric varieties is equivalent to reflexivity of the underlying polyhedron [1]. Moreover, the duality of reflexive polyhedra corresponds to the mirror symmetry of the resulting class of Calabi– Yau manifolds (see for example [2, 3, 4, 5] and references therein). This is the main motivation for the interest in a classification of 4-dimensional reflexive polyhedra in the context of string theory. It is known that the total number of reflexive polyhedra is finite in any given dimension, because various bounds on the volume and the number of points have been derived as a function of the dimension and the number of interior lattice points [6, 7, 8]. The ? ??

E-mail: [email protected] E-mail: [email protected]

496

M. Kreuzer, H. Skarke

case of n = 2 dimensions is the easiest because all polygons with one interior point are reflexive (this is no longer true for n > 2). There are 16 such polygons, which were constructed in [9, 10] (we will rederive this result in the last section to illustrate the application of our tools). In 4 dimensions we expect at least some 104 reflexive pairs and the known bounds for general lattice polytopes [6, 7, 8] are not very useful for explicit constructions. What we need is an efficient algorithm, which probably should rely on reflexivity in an essential way. It is the purpose of the present paper to provide such an algorithm. Our approach is partly motivated by experience with transversal polynomials in weighted projective spaces [11, 12, 13, 14] and by the orbifold construction of mirror pairs [15, 16, 17, 18], but this will become clear only at a later stage. The basic strategy is to find minimal integral polytopes M that are spanned by vertices of 1 and that still have 0 in the interior (the generic case is a simplex). By duality, M ∗ bounds 1∗ and its facets carry facets of 1∗ . If we have minimal polytopes M and M for 1 and 1∗ , respectively, then the pairing matrix of the respective vertices turns out to be strongly ∗ constrained. Such a matrix encodes the structures of M and M , which bound 1 from the interior and the exterior. The final step in the classification is the reconstruction of the complete pairing matrix of vertices of 1 and 1∗ . The pairings of all vertices characterize the reflexive pair up to a finite number of possible choices of dual pairs of lattices. In the simplex case the barycentric coordinates of the interior point correspond to the weights in the context of weighted projective spaces. Indeed, the authors of [19] tried to interpret toric Calabi-Yau manifolds as non-transverse hypersurfaces in weighted P4 . Our results imply that, even without transversality, only a finite number of weight systems makes sense in the toric context. Moreover, the large ambiguity in the generalized transposition rule of [19] is constrained by our rules for the selection of vertices, which may be regarded as rules about which transpositions make sense. In Sect. 2 we give some basic definitions and deduce geometrical properties of minimal polytopes. In general we may need a number of lower dimensional simplices containing 0 in the interior to span a neigborhood of 0. Then we have several weight systems and the toric variety can only be related to sort of a (non-direct) product of weighted spaces. In Sect. 3 we discuss the properties of (minimal) pairing matrices and the relations among pairings in higher-dimensional lattices that we use to embed a reflexive pair. We illustrate our concepts using an example of a 4-dimensional polyhedron that was analysed in the toric context in [5]. This completes the setup that we need in Sect. 4 to state the classification algorithm and to prove its finiteness. As an illustration we rederive the 2-dimensional case. 2. Reflexive Polyhedra and Minimal Polytopes We first recall some elementary definitions about polytopes [20]. A rational polyhedron is an intersection of finitely many halfspaces {x ∈ Qn : ai xi ≥ b} with ai , b ∈ Z. A polytope is a bounded polyhedron or, equivalently, the convex hull of a finite number of points. A lattice (or integral) polytope is a polytope whose vertices belong to some lattice 0 ∼ = Zn . We will identify Qn with the rational extension 0 Q of 0 , i.e. 0 = 0 ⊗ Qn ∼ = Q Z Q. The distance of a lattice point x ∈ 0 to a lattice hyperplane H(ai , b) = {x ∈ 0 Q : ai xi = b}, where the integers ai have greatest common divisor 1, is defined by d(H, x) := |ai xi − b|. This number is 1 plus the number of lattice hyperplanes between x and H. These definitions are invariant under changes of the

Classification of Reflexive Polyhedra

497

lattice basis, so we can write ha, xi instead of ai xi whenever we do not want to refer to a specific basis. Then the condition that the ai have no common divisor means that a ∈ 0 ∗ is primitive. A reflexive polyhedron 1 is a polytope with one interior point P whose bounding hyperplanes are all at distance 1 from P . If an arbitrary convex set 1 contains the origin in its interior we define the dual (or polar) set 1∗ := {y ∈ 0 ∗Q : hy, xi ≥ −1 ∀x ∈ 1}.

(1)

Assuming that the interior point P = 0 is the origin it is easy to see that a polytope 1 is reflexive if and only if 1∗ is integral, i.e. if all vertices of 1∗ belong to 0 ∗ . Consider an n-dimensional reflexive pair of polyhedra 1 and 1∗ defined on lattices 0 and 0 ∗ . For each of these polyhedra we choose a set of k (k) hyperplanes Hi , j i = 1, · · · , k (H , j = 1, · · · , k) carrying facets in such a way that these hyperplanes define a bounded convex body Q (Q) containing the original polyhedron and that k and k are minimal. We define a redundant coordinate system where the ith coordinate of a point is given by its integer distance to Hi (nonnegative on the side of the polyhedron). This is just the degree of the homogeneous coordinate [21] corresponding to Hi in the monomial determined by the point. Note that the vertices of Q and Q need not have integer coordinates. All coordinates of the interior points are equal to 1, each coordinate of any point of a polyhedron is nonnegative. Whenever we use this sort of coordinate system we will label the interior points by 1 and 1. Note that 1 is the only integer point in the interior of Q: For all other points one coordinate must be smaller than one so that they belong to some Hi . We have thus shown that any such polytope Q has all lattice points of 1, except for 1, at its boundary. i

The duals of these hyperplanes are two collections of k (k) vertices Vj (V ) spanning ∗ polyhedra M = Q and M = Q∗ that contain the interior points of 1 (1∗ ). M and M are minimal in the sense that there are no collections of less than k (k) vertices of 1 (1∗ ) containing 1 (1) in the interior. Let us first obtain some information on the general structure of minimal polytopes. Here we will not use the affine structure (labelling the interior point by 1), but instead we will use a linear structure, calling the interior point 0 and identifying vertices V with vectors. Then the fact that M has 0 in its interior is equivalent to the fact that any point in Qn can be written as a nonnegative linear combination of vertices. Considering all triangulations where every simplex contains a specific vertex V of M , we see that there is at least one simplex of dimension n with this vertex containing 0, i.e. 0 lies in the interior of this simplex or one of its simplicial faces containing V . So we have a collection of vertices and a collection of subsets of this set of vertices defining lower dimensional simplices with 0 in their interiors (we will call such simplices “good simplices”), in such a way that each vertex belongs to at least one good simplex. Now we note that if we have a collection of good simplices, then 0 is also in the interior of the polytope spanned by all the vertices of these simplices (of course, “interior” here means interior w.r.t. the linear subspace spanned by these vertices). Lemma 1. A minimal polytope M = ConvexHull{V1 , · · · , Vk } in Qn is either a simplex or contains an n0 -dimensional minimal polytope M 0 := ConvexHull{V1 , · · · , Vk0 } and a good simplex S := ConvexHull(R ∪ {Vk0 +1 , · · · , Vk }) with R ⊂ {V1 , · · · , Vk0 } such 0

that k − k = n − n0 + 1 and dim S ≤ n0 .

498


Proof. If M is a simplex, there is nothing left to prove. Otherwise, consider the set of all good simplices consisting of vertices of M . Any subset of this set will define a lower dimensional minimal polytope. Among these, take one (call it M 0 ) with the maximal 0 0 0 dimension n0 smaller than n. Qn factorizes into Qn and Qn /Qn ∼ = Qn−n (equivalence 0 classes in Qn ). The remaining vertices define a polytope Mn−n0 in Qn /Qn . If Mn−n0 were not a simplex, it would contain a simplex of dimension smaller than n − n0 which would define, together with the vertices of M 0 , a minimal polytope of dimension s with n0 < s < n, in contradiction with our assumption. Therefore Mn−n0 is a simplex. Because of minimality of M , each of the n − n0 + 1 vertices of Mn−n0 can have only 0 one representative in Qn , implying k − k = n − n0 + 1. The equivalence class of 0 can be described uniquely as a positive linear combination of these vertices. This 0 linear combination defines a vector in Qn , which can be written as a negative linear combination of ≤ n0 linearly independent vertices of M 0 . These vertices, together with those of Mn−n0 , form the simplex S. By the maximality assumption about M 0 , dimS cannot exceed dimM 0 . Corollary 1. A minimal polytope M = ConvexHull{V1 , · · · , Vk } in Qn allows a structure {Vj } = {V1 , · · · , Vk1 , Vk1 +1 , · · · , Vk2 , · · · , Vkλ−1 +1 , · · · , Vkλ } with the following properties: (a) Mµ := ConvexHull{V1 , · · · , Vkµ } is a (k µ − µ) – dimensional minimal polytope, Mλ = M . (b) For each µ, there is a subset Rµ of {V1 , · · · , Vkµ−1 } such that Sµ := ConvexHull(Rµ ∪ {Vkµ−1 +1 , · · · , Vkµ }) defines a simplex with dim Sµ ≤ dim Mµ−1 for µ > 1. Proof. If M is a simplex, λ = 1 and k = k 1 = n + 1. Otherwise one can proceed inductively using Lemma 1. Corollary 2. n + 1 ≤ k ≤ 2n. 0

Proof. If M is a simplex, the lower bound is satisfied. Otherwise, k = k + n − n0 + 1 and induction gives k ∈ {n + 2, · · · , n0 + n + 1} ⊂ {n + 1, · · · , 2n}. S Lemma 2. Denote by {Sλ } a set of good simplices spanning M . Then Sµ − ν6=µ Sν never contains exactly one point. Proof. A simplex with 0 in its interior contains line segments V V 0 with V 0 = −εV , where ε is a positive number. If a simplex S = ConvexHull{V1 , · · · , Vs+1 } has all of its vertices except one (Vs+1 ) in common with other simplices, then all points in the linear span of S are nonnegative linear combinations of the Vj and the −εj Vj with j ≤ s, thus showing that Vs+1 violates the minimality of M . Example . n = 5, M = ConvexHull{V1 , · · · , V8 } with V1 = (1, 1, 0, 0, 0)T , V2 = (1, −1, 0, 0, 0)T , V3 = (−1, 0, 1, 0, 0)T , V4 = (−1, 0, −1, 0, 0)T , V5 = (−1, 0, 0, 1, 0)T , V6 = (−1, 0, 0, −1, 0)T , V7 = (1, 0, 0, 0, 1)T , V8 = (1, 0, 0, 0, −1)T .

(2)

M contains the good simplices S1234 = V1 V2 V3 V4 (in the x1 x2 x3 –plane), S1256 (in the x1 x2 x4 –plane), S3478 (in the x1 x3 x5 –plane), S5678 (in the x1 x4 x5 –plane) and the


499

4-dimensional minimal polytopes M123456 , M123478 , M125678 , M345678 . Each of these 4dimensional minimal polytopes spans a hyperplane of codimension 1. In order to span Q5 , we need two additional points which may belong to one of two possible simplices. For example, if we choose M = M123456 , then we require V7 and V8 and may choose S = S3478 or S = S5678 . M123456 again has to fulfill Lemma 1, and indeed it contains the two simplices S1234 and S1256 . The structure of Corollary 1 can be realised, for example, by S1 = M1 = S1234 , S2 = S1256 (implying M2 = M123456 ), and S3 = S3478 . Note that S5678 does not occur in this structure and that S1 − (S2 ∪ S3 ) is empty (compare with Lemma 2).

3. Pairing Matrices Let the elements Ai j of the integer k × k matrix A denote the ith coordinate of Vj in the coordinate system defined by Q, i.e. Ai j = (Vj )i is the distance of Vj to the hyperplane Hi . Because of reflexivity of 1∗ (all facets are at distance 1 from 1) this is related to the i i pairing of Vj with V = Hi∗ by Ai j = hV , Vj i+ with h , i+ := h , i + 1, where h , i is the original lattice pairing. The definition of the affine pairing h , i+ might seem awkward at first sight, but it has two advantages: On the one hand, it is nonnegative for any pairing between 1∗ and 1, and on the other hand, we will see later that it is a natural linear pairing for a higher dimensional pair of lattices into which we will embed 0 and 0 ∗ . i By duality Ai j also denotes the j th coordinate of V in the coordinate system defined i by Q, i.e. Ai j = (V )j . In other words, the columns of A correspond to the vertices of M whereas the lines correspond to the vertices of M . We will label all points of 1 by column vectors and all points of 1∗ by line vectors, in particular 1 = (1, · · · , 1)T and 1 = (1, · · · , 1). If M and M are simplices, then A is an (n + 1) × (n + 1) matrix. We denote by the “weights” qi and q j the barycentric coordinates of 1 and 1, respectively: X X X X i 1= qi V , 1= q j Vj , qi = q j = 1. (3) i

This implies X qi Ai j = 1, i

j

X

q j Ai j = 1,

j

i

X

j

X

qi (Vj )i = 1,

i

i

q j (V )j = 1.

(4)

j

We can now give a new interpretation to our coordinate systems as coordinates in n+1 ∼ (n + 1)–dimensional lattices 0 n+1 ∼ = Zn+1 and 0 = Zn+1 and their rational exP n+1 ∼ tensions 0 n+1 qi xi = 1 defines an = Qn+1 and 0 Q ∼ = Qn+1 . Then the equation Q n n–dimensional affine hyperplane 0 Q spanned by the Vj , which obviously contains 1. Linear independence of the Vj , i.e. of the columns of A, implies regularity of A. We can invert Eqs. (4) to get qi =

n X j=0

(A−1 )j i ,

qj =

n X

(A−1 )j i .

i=0 n+1

Defining arbitrary point pairings h , in+1 between 0 n+1 Q and 0 Q by

(5)

500


hP , P in+1 := P k (A−1 )k l P l

(6)

n

allows us to identify 0 nQ and 0 Q as n n+1 0Q ∼ = {P ∈ 0 Q | hP , 1in+1 = 1}. (7)

0 nQ ∼ = {P ∈ 0 n+1 Q | h1, P in+1 = 1},

At this point it is easy to see the relation of our framework to the orbifold mirror construction that works for minimal polynomials in weighted projective spaces [15, 16]: That construction relates a monomial with exponent vector W to a twist group element whose diagonal action on the homogeneous coordinates is exp(2πi diag(W A−1 )) [18]. Even in our more general context the lines of A−1 provide the phases for generators of the phase symmetry group of the n + 1 monomials whose exponents are the columns of the matrix A (this does not mean, however, that all such symmetries can be used for an orbifold construction, because transversality requires additional monomials in the non-minimal case; we will soon give an example for how this manifests itself in the context of toric geometry). n

Lemma 3. Let P ∈ 0 nQ , P ∈ 0 Q . Then (a) hP , P in+1 = hP , P i+ . (b) hP − 1, P − 1in+1 = hP , P i. Proof. For vertices we have by definition i

i

i

hV , Vj in+1 = (V )k (A−1 )k l (Vj )l = Ai k (A−1 )k l Al j = Ai j = hV , Vj i+ and

i

i

i

hV − 1, Vj − 1in+1 = hV , Vj i+ − 1 − 1 + 1 = hV , Vj i. For general P , P (b) follows from linearity in 0 nQ and because hP , 1in+1 = h1, P in+1 = h1, 1in+1 = 1.

n 0Q

(8) (9)

and (a) follows from (b)

The first statement of this lemma shows us that h , in+1 is a natural extension of h , i+ n+1 n+1 n+1 to 0 n+1 Q × 0 Q . We will use this fact to define h , i+ in 0 Q × 0 Q , thus showing that our originally affine pairing is indeed a linear pairing in the higher dimensional context. n n n+1 Let us also define the n-dimensional sublattices 0 n = 0 nQ ∩ 0 n+1 and 0 = 0 Q ∩ 0 carrying 1 and 1∗ , respectively. i n Corollary 3. There is a natural identification (0 n )∗ ∼ = 1 + Span{V − 1} ⊆ 0 .

Proof. By the embedding of 0 n into Zn+1 an element of (0 n )∗ becomes an equivai lence class of points in the dual lattice Zn+1 modulo 1. Since (V )k (A−1 )k l = δli the i vertices V are representatives of equivalence classes that generate (0 n )∗ . Using the i mod 1 ambiguity we may always choose a representative in 1 + Span{V − 1} because h1, P in+1 = 1. Given a pairing matrix A for our simplices M and M , let us see how we can obtain all corresponding dual pairs 1, 1∗ : First we choose some sublattice 0 ⊆ 0 n that contains n 1 and all vectors Vj . The dual lattice 0 ∗ is a sublattice of 0 , which obviously contains i the vectors V . Then


501

i Q = {P ∈ 0 n+1 Q | h1, P i+ = 1 ∧ P ≥ 0 ∀i}, n+1

Q = {P ∈ 0 Q | hP , 1i+ = 1 ∧ P j ≥ 0 ∀j}.

(10)

Defining the finite point sets 0 + = {P ∈ 0 | P i ≥ 0 ∀i} and (0 ∗ )+ and their convex hulls 1max and 1max , respectively, we may choose polyhedra 1 and 1 with {Vj } ⊆ 1 ⊆ 1max i and {V } ⊆ 1 ⊆ 1max and check for duality. In practice, the following algorithm will n be far more efficient: Calculate all points P, P in 0 n+ and 0 + and the corresponding pairing matrix (w.r.t. h , i+ ), which may have rational entries. Then we can create a list of possible vertices V by noting that any vertex is dual to a hyperplane, i.e. for any vertex V there must be n linearly independent points P with hP , V i+ = 0. Creating a list of possible vertices V , we use the same argument, working only with our list of possible vertices in 0 n+ . This procedure may be iterated, reducing the respective lists in each step. In particular we can drop a model whenever our original vertices Vj or i V don’t show up in the resulting lists of possible vertices. In a last step we may then choose subsets of these lists, making sure that each coordinate hyperplane contains n linearly independent vertices. Choosing a particular point P to be a vertex of 1 implies that we can eliminate all points P with rational or negative pairings with P from our list of candidates for vertices of 1∗ . Example . The following example is motivated by the non-degenerate Landau–Ginzburg potential 8 3 3 3 3 2 W = x25 1 + x2 x1 + x3 x5 + x4 x2 + x5 x1 + λx2 x5 ,

(11)

to which we assign the matrix 

25 1  0 8  A= 0 0  0 0 0 0 with q =

1 75 (3, 9, 17, 22, 24)

25

A

0 1 0 3 0

 1 0  0 0 3

(12)

1 36 (1, 3, 12, 12, 8). It is easy to points in 0 4+ ) and the 100 points

and q =

points allowed by the q system (the help of  1 −1

0 0 3 0 1

0  = 0 0 0

1 − 200 1 8

0 0 0

1 225

0 1 3

0 − 19

1 600 1 − 24

0 1 3

0

1  − 75 0   0   0 

construct all 33 4

in 0 + . With the

(13)

1 3

we get the 33×100 matrix of point pairings, which turns out to have half–integer entries. After eliminating all lines and columns with less than 4 zeroes we get the pairing matrix for candidates for vertices shown in Table 1. The first five lines and columns indicate

502


  25       0      0      0       0      0       0      0      0        0       0        0       0       0        1   1     1      2      3       3      4        8      9       9      12        16      16       24      24       27      33    36

1

0

0

1

0

0

2

2

3

3

3

4

4

5

8

0

1

0

3

4

2

5

0

2

3

0

7

1

0

3

0

0

0

1

3

0

0

0

1

1

0

3

0

0

3

0

0

1

0

0

3

0

1

1

0

0

0

1

0

3

2

0

0

1

0

2

0

1

0

0

0

0

1

3

2

0

−1

1

1

2

0

1

0

−1

0

1

2

0

0

1

1

0

2

0

1

1

0

1

0

2

1

0

0

1

2

0

1

0

1

1

0

2

3 2

0

7 2

− 21

3

1

1 2

7 2

− 21

4

0

0

3

7 2

4

0

2

0

3 2

5 2

1

5 2

3 2

1

2

1 2

7 2

1 2

4

1

1

0

3 2

5 2

2

5 2

1 2

1

2

1 2

7 2

3 2

4

2

0

0

3 2

5 2

3

5 2

− 21

1

2

1 2

7 2

5 2

8

1

0

0

3

4

3

5

−1

2

3

0

7

2

11 2

3

15 2

− 23

3

4

− 21

21 2

3 2

12

0

0

0

9 2

1

0

0

4

3

0

−1

2

0

3

0

1

1

−1

1

0

2

1

1

1

0

1

2

1

1

1

1

0

9

0

0

1

4

4

2

6

−1

3

3

0

8

1

2

0

1

2

2

1

0

2

1

2

1

1

2

0

3

0

0

3

3

1

0

3

0

3

1

1

3

0

11

0

0

0

4

5

3

7

−1

3

4

0

10

2

0

0

0

4

5 2

− 21

−1

3 2

1 2

3

0

3 2

1 2

− 21

0

0

1

2

1

0

0

1

2

2

1

2

1

1

1

0

0

3

2

0

0

2

1

3

1

2

2

1

9

0

0

0

3

4

3

6

0

3

4

1

9

3

− 21

0

3 2

3 2

3

1

5 2

3 2

3 2

0

0

0

3

3 2

0

0

1

1

0

0

1

1

3

2

2

3

2

3

0

1

0

1

0

0

2

1

2

2

2

3

2

4

0

0

1

0

−1

0

2

1

4

2

3

4

3

5

0

1

0

0

−1

0

3

1

3

2

3

4

3

6

3

0

0

0

0

1

3

3

3

3

4

4

6

6

1

0

0

0

−1

0

3

2

4

3

4

5

5

7

0

− 23

− 21

3

3 2

9 2

4

11 2

9 2

15 2

0

0

0

3

Table 1: Pairing matrix for candidates for vertices, part I


503

5

6

7

8

8

9

9

10

11

12

13

14

16

17

19

4

1

6

0

3

0

1

5

2

0

4

1

3

0

2

0

0

0

3

0

0

1

0

0

1

0

0

0

0

0

0

0

0

0

0

0

1

0

0

1

0

0

0

0

0

1

2

0

0

1

2

0

0

1

0

0

1

0

1

0

1

2

0

−1

1

2

0

0

1

0

0

1

0

1

0

0

0

0

1

0

0

1

0

0

1

0

0

0

0

0

0

0

0

2

0

0

1

0

0

1

0

0

0

0

0

3

5 2

3

−1

5 2

2

0

5 2

2

− 21

2

3 2

3 2

1

1

2

1 2

3

0

3 2

0

1

5 2

1

1 2

2

1 2

3 2

0

1

2

1 2

3

1

3 2

0

1

5 2

1

1 2

2

1 2

3 2

0

1

2

1 2

3

2

3 2

0

1

5 2

1

1 2

2

1 2

3 2

0

1

4

1

6

1

3

0

1

5

2

0

4

1

3

0

2

6

3 2

9

0

9 2

0

1

15 2

3

− 21

6

3 2

9 2

0

3

2

3

1

−1

2

3

0

1

2

0

1

2

1

2

1

1

1

1

0

1

1

1

1

1

1

1

1

1

1

1

5

2

7

0

4

1

1

6

3

0

5

2

4

1

3

2

2

2

0

2

2

1

2

2

1

2

2

2

2

2

3

3

3

0

3

3

1

3

3

1

3

3

3

3

3

6

2

9

1

5

1

2

8

4

1

7

3

6

2

5

2

7 2

1

0

5 2

4

1

3 2

3

3 2

2

7 2

5 2

4

3

2

3

2

2

3

4

3

3

4

4

4

5

5

6

6

3

4

3

2

4

5

3

4

5

4

5

6

6

7

7

6

3

9

3

6

3

4

9

6

4

9

6

9

6

9

3

9 2

3

3

9 2

6

4

9 2

6

11 2

6

15 2

15 2

9

9

3

4

4

5

5

6

6

6

7

8

8

9

10

11

12

3

4

4

6

5

6

6

6

7

8

8

9

10

11

12

4

5

6

8

7

8

9

9

10

12

12

13

15

16

18

4

5

6

9

7

8

9

9

10

12

12

13

15

16

18

6

6

9

9

9

9

10

12

12

13

15

15

18

18

21

6

7

9

11

10

11

12

13

14

16

17

18

21

22

25

6

15 2

12

21 2

13

27 2

15

35 2

18

39 2

45 2

24

27

9

12

Table 1: Pairing matrix for candidates for vertices, part II

  22      1       0      0      0       0      0       0     1   2     1    2     1   2     1    2      1     3    2      1    1    2      2       3      4     7    2      7      8       9     21   2       14      14      21       21      24       29     63 2

504


coordinates w.r.t. the coordinate systems defined by the q and q system. All entries (ith P5 i line, j th column) are k,l=1 (P )k (A−1 )k l (Pj )l . The occurrence of half integers means that we still have a Z2 freedom in choosing sublattices. Eliminating all columns with non– 4 integer entries corresponds to choosing 0 = 0 4 /Z2 and 0 ∗ = 0 , whereas eliminating 4 all lines with non–integer entries corresponds to choosing 0 = 0 4 and 0 ∗ = 0 /Z2 . In the first case we would eliminate P6 and P7 which would result in a first line with 1 only two entries of 0, in contradiction with our requirement that V is a vertex of 1∗ . Transversality of the polynomial (11) requires the presence of its last monomial x32 x25 , which is not invariant under this Z2 twist (see the column corresponding to P6 in Table 1). In our context, the Z2 twist is forbidden by the requirement that the vertices of M remain vertices of 1∗ (dropping this requirement, the Z2 twist may and does lead to reflexive pairs). In the case 0 = 0 4 the full matrix of point pairings is a 33 × 52 matrix. i The convex hulls of the points Pj and P are polytopes 1max and 1max , respectively, which are obviously not dual to one another, as the entries −1 show. We can now choose any subset of our candidates of vertices (containing the vertices in A), thus defining some polytope 1. Then we have to eliminate all points of 1max which have negative pairings with vertices of 1, resulting in 0 ∗ ∩ 1∗ ⊆ 1∗ . Then 1 is reflexive if and only if each of its vertices has pairings of 0 with 4 linearly independent points of 0 ∗ ∩ 1∗ . If we keep all points of 1max , for example, we have to delete all lines containing −1. It turns out that this indeed leads to a reflexive pair [5]. In fact, it was checked numerically for all transversal polynomials that 1max is reflexive [19, 25]. In [26] there is also an explicit proof (again, for n ≤ 4) that 1max is always reflexive even for a larger class of weight systems. If the minimal polytope M is not a simplex we define a weight system for each of the lower dimensional simplices Sµ (µ = 1, · · · , λ) occurring in Corollary 1. Then k we have lattices 0 k ∼ = Zk and 0 ∼ = Zk and their rational extensions 0 k ∼ = Qk and Q

k 0Q

k ∼ = Qk , and we can interpret our coordinate systems as coordinates in 0 kQ and 0 Q . P j We get λ = k − n equations of the type q xj = 1. Due to the structure given in the lemma, we can solve this system by successively eliminating the xkµ , µ = 1, · · · , λ. n

Therefore these k − n equations define an n–dimensional affine hyperplane 0 Q spanned i

by the V , which P obviously contains 1 again. In the same way we also get k−n equations of the type qi xi = 1 defining an n–dimensional affine hyperplane 0 nQ spanned by the Vj .

4. A Classification Algorithm The classification of dual pairs of reflexive polyhedra can be done in 3 steps: (1) Classification of possible structures of minimal polytopes, (2) Classification of weight systems, (3) Construction of complete vertex pairing matrices for dual pairs of polytopes and choice of a lattice. Let us first discuss the classification of possible structures of minimal polytopes. With the help of Lemma 1 of Sect. 2, it is easy to construct all possible structures recursively. For a given dimension n, one either has the n-dimensional simplex or one


505

has to consider all minimal polytopes of dimension n0 with n/2 ≤ n0 < n, add n−n0 +1 points and consider all possible structures of S compatible with the lemmata. For n = 2 this allows the triangle V1 V2 V3 and the “1 simplex” V1 V2 with 2 additional points V10 , V20 , which can only form another 1 simplex. In n = 3 dimensions we can either have a 3 simplex V1 V2 V3 V4 or a 2 dimensional minimal polytope with 2 more points. The latter case allows the possibilities S1 = V1 V2 V3 , S2 = V10 V20 ; S1 = V1 V2 V3 , S2 = V1 V20 V30 ; S1 = V1 V2 , S2 = V10 V20 , S3 = V100 V200 . In n = 4 dimensions we can have a 4 simplex, a 3 dimensional minimal polytope with 2 more points or a 2 dimensional minimal polytope with 3 more points defining a 2 simplex. The complete list of possible structures is the following: With 5 points there only is the 4 simplex M = S1 = V1 V2 V3 V4 V5 . With a total of 6 points we have the 4 minimal configurations {S1 = V1 V2 V3 V4 , S2 = V1 V2 V30 V40 }, {S1 = V1 V2 V3 V4 , S2 = V10 V20 },

{S1 = V1 V2 V3 V4 , S2 = V1 V20 V30 }, {S1 = V1 V2 V3 , S2 = V10 V20 V30 }.

With 7 points there are the 3 possibilities {S1 = V1 V2 V3 , S2 = V10 V20 , S3 = V100 V200 }, {S1 = V1 V2 V3 , S2 = V1 V20 V30 , S3 = V100 V200 }, {S1 = V1 V2 V3 , S2 = V1 V20 V30 , S3 = V1 V200 V300 }, and with 8 points we can have only 1 simplices S1 = V1 V2 , S2 = V10 V20 , S3 = V100 V200 , S4 = V1000 V2000 . The next step in the classification program, namely the classification of weight systems, was done in a different paper [26]. There, weight systems with up to 5 weights and with the property that 1 is in the interior of 1max were completely classified. All weight systems occurring in our scheme (whether alone or in combination with other weight systems) obviously must have this “interior point property”. The fact that Q consists of hyperplanes carrying facets of 1 leads to another property of weight systems which we may call the “span property”. It asserts that the facets of Q must actually be affinely spanned by points of 1max . According to [26], there are the following weight systems with the interior point property: With two weights, there is only (1/2, 1/2), which also has the span property; with three weights there are the systems (1/3, 1/3, 1/3), (1/2, 1/4, 1/4) and (1/2, 1/3, 1/6), which all have the span property as well. There are 95 systems of four weights (58 of them with the span property), and there are 184026 systems of five weights (38730 with the span property). With this information, there are essentially two ways to construct all reflexive polyhedra for a given n. We can either pick a specific structure and a combination of weight systems both for M and M . Then it is easy to write a computer program that finds all k × k matrices A that are compatible with such structures, and we can proceed as in the previous section. Alternatively, we may give up the symmetry between 0 and 0 ∗ and simply construct the polyhedron 1max corresponding to some combination of weight systems. Next, we would consider all of its subpolyhedra 1 such that the facets of Q are affinely spanned by points of 1. Finally, we must classify all sublattices of 0 n that contain all the vertices of 1 and check for reflexivity of 1 w.r.t. any of these lattices. In both approaches it is important to calculate and store the pairing matrices for the vertices of 1 and 1∗ because this is the information required for identifying or distinguishing polyhedra. As an illustration for our concepts and methods, we will now rederive the well-known (see, e.g., [9, 10]) classification of reflexive polyhedra for n = 2 in the “asymmetric” approach.

506


In two dimensions there are only two minimal polytopes, namely the triangle V1 V2 V3 and the parallelogram V1 V2 V10 V20 . Thus we either have a single weight system of one of the types (1/3, 1/3, 1/3), (1/2, 1/4, 1/4), (1/2, 1/3, 1/6), or the combination of weight systems (1/2, 1/2, 0, 0; 0, 0, 1/2, 1/2). The points (except 1) allowed in the systems of type (q1 , q2 , q3 ) can be arranged as columns of the matrices ! 0 0 0 0 1 2 3 2 1 0 1 2 3 2 1 0 0 0 , (14) 3 2 1 0 0 0 0 1 2 0 0 4

0 1 3

0 2 2

0 3 1

0 4 0

1 2 0

2 0 0

0 0 6

0 1 4

0 2 2

0 3 0

2 0 0

1 0 3

and

1 0 2

! (15)

! ,

(16)

respectively (see Fig. 1). r A AA A8 r A r6 A A A 9A r A r A r5 A A A A x1 A r A r A r A r 6 1A A A A 4 3 2

x2 5 @@ r @@ @@@ 6@ @ r @@ r @@@@@ @@@@@@ x1 @@ r @@ r @@ r @r 6 4 1 3 2 @@@@@@@

x3

7 x2 x3 Aj

x3= 7~x2

Sr S SS S r S r6 8Sr S S S S x SrS rS rS rS r 61 1SSSSS5 2 3 4

Fig. 1: The bounding simplices Q for the weight systems ( 13 , 13 , 13 ), ( 21 , 41 , 41 ) and ( 21 , 13 , 16 )

In the first case there is a Z3 sublattice defined by x1 = x2 mod 3, which reduces the set of allowed points to ! 0 0 3 0 3 0 , (17) 3 0 0 and in the second case there is a Z2 sublattice defined by x2 = x3 mod 4, which reduces the allowed points to ! 0 0 0 2 0 2 4 0 (18) 4 2 0 0 (see Fig. 2), whereas in the case of (1/2, 1/3, 1/6) there is no allowed sublattice. x3 Aj r x2 A AA A A A A A A A r A A A A A x1 A r A A A r 6 A A A A

x3=

x2

~ Sr SS S S r S S S S S S x SrS S rS S r 61 SSSSS

Fig. 2: Alternative lattices for the weight systems ( 13 , 13 , 13 ) and ( 21 , 41 , 41 )


507

For (1/3, 1/3, 1/3) we get the triangle P1 P4 P7 twice (both on the original and the reduced lattice), and in addition we get, on the original lattice, the polygons P1 P4 P6 P8 , P1 P4 P6 P9 , P2 P3 P5 P7 P9 , P2 P3 P5 P7 P8 , P2 P3 P6 P7 P8 and P2 P3 P5 P6 P8 P9 (of course, there are more polygons which are related to the ones given above by the permutation symmetry in the coordinates). For (1/2, 1/4, 1/4) we get the triangle P1 P5 P7 twice (both on the original and the reduced lattice), and in addition we get the polygons P1 P4 P6 P7 , P1 P3 P6 P7 , P1 P2 P6 P7 , P2 P4 P6 P7 P8 and P2 P3 P6 P7 P8 . For (1/2, 1/3, 1/6) we get the polygons P1 P4 P5 , P2 P4 P5 P6 and P3 P4 P5 P6 . The only case we have not considered so far is the case of two q systems with q1 = q2 = 1/2. Allowed points can be encoded by   0 0 0 1 2 2 2 1 2 2 2 1 0 0 0 1 (19) 0 1 2 2 2 1 0 0 2 1 0 0 0 1 2 2 (see Fig. 3). If we drop any of the vertices P1 , P3 , P5 , P7 , we can find a triangle containing 1, so we get only two new polygons, namely P1 P3 P5 P7 on the original lattice and on the sublattice defined by x1 = x3 mod 2 (see Fig. 4) x4

r x2 ?

r

r

r

r

r

r - x3

r

x1 r 6

Fig. 3: 2 + 2 full lattice

x4

r x2 ?

r

r r - x3

x1 r 6

Fig. 4: 2 + 2 alternative lattice

We have constructed some polygons more than once. For example, P1 P2 P6 P7 in the (1/2, 1/4, 1/4) system is (up to a reflection) equivalent to P2 P4 P5 P6 in the (1/2, 1/3, 1/6) system. Here this can be seen by inspection. In our approach this redundancy will be sorted out when we bring the complete pairing matrices into a normal form by permutations of columns and lines. Taking this into account, we arrive at the known 16 reflexive polygons [9, 10]. Acknowledgement. We would like to thank Victor V. Batyrev, Albrecht Klemm and Mahmoud Nikbakht Tehrani for helpful discussions. This work is supported in part by the Austrian Science Foundation (FWF) under grant number P10641-PHY.

References 1. Batyrev, V.V.: Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties. alg-geom/9310003. J. Alg. Geom. 3, 493 (1994) 2. Aspinwall, P.S., Greene, B.R., Morrison D.R.: Space-time topology change and stringy geometry. J. Math. Phys. 35, 5321 (1994) 3. Morrison, D.R., Plesser, M.R.: Summing the instantons: Quantum cohomology and mirror symmetry in toric varieties. hp-th/9412236, Nucl. Phys. B440, 279 (1995)

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4. Candelas, P., de la Ossa, X., Font, A., Katz, S., Morrison, D.R.: Mirror symmetry for two parameter models (I). hep-th/9308083, Nucl. Phys. B416, 481 (1994); hep-th/9403187, Nucl. Phys. B429, 626 (1994) 5. Hosono, S., Klemm, A., Theisen, S., Yau, S.-T.: Mirror symmetry, mirror map and applications to Calabi Yau hypersurfaces. hep-th/9308122, Commun. Math. Phys. 167, 167 (1995); hep-th/9406055, Nucl. Phys. B433, 501 (1994) 6. Batyrev, V.: Boundedness of the degree of higher dimensional toric Fano varieties. Bull. Moscow Math. Univ. 37, n.1, 28 (1982) 7. Hensley, D.: Lattice vertex polytopes with interior lattice points. Pacific J. Math. 105, n.1, 183 (1983) 8. Borisov, A.A., Borisov, L.A.: Singular toric Fano varieties. Mat. Sbornik 183, n.2, 134 (1992) 9. Batyrev, V.V.: Higher-dimensional toric varieties with ample anticanonical class. Moscow State Univ., Thesis, 1985 (Russian) 10. Koelman, R.J.: The number of moduli of families of curves on toric varieties. Katholieke Universiteit Nijmegen, Thesis, 1990 11. Fletcher, A.R.: Working with complete intersections. Bonn preprint MPI/89–35 (1989) 12. Kreuzer, M., Skarke, H.: On the classification of quasihomogeneous functions. hep-th/9202039, Commun. Math. Phys. 150, 137 (1992) 13. Kreuzer, M., Skarke, H.: No mirror symmetry in Landau-Ginzburg spectra! hep-th/9205004, Nucl. Phys. B388, 113 (1992) 14. Klemm, A., Schimmrigk, R.: Landau–Ginzburg string vacua. hep-th/9204060, Nucl. Phys. 411, 559 (1994) 15. Greene, B.R., Plesser, M.R.: Duality in Calabi–Yau moduli space. Nucl. Phys. 338, 15 (1990) 16. Berglund, P., Hübsch, T.: A generalized construction of mirror manifolds. hep-th/9201014, Nucl. Phys. 393, 377 (1993) 17. Kreuzer, M., Skarke, H.: All abelian symmetries of Landau–Ginzburg potentials. hep-th/9211047, Nucl. Phys. 405, 305 (1993) 18. Kreuzer, M.: The mirror map for invertible LG models. hep-th/9402114, Phys. Lett. 328, 312 (1994) 19. Candelas, P., de la Ossa, X., Katz, S.: Mirror symmetry for Calabi–Yau hypersurfaces in weighted P4 and extensions of Landau–Ginzburg theory. hep-th/9412117, Nucl. Phys. 450, 267 (1995) 20. Ziegler, G.M.: Lectures on polytopes. GTM 152, Berlin: Springer, 1995 21. Cox, D.: The homogeneous coordinate ring or a toric variety. J. Alg. Geom. 4, 17 (1995) 22. Fulton, W.: Introduction to toric varieties. Princeton: Princeton Univ. Press, 1993 23. Oda, T.: Convex bodies and algebraic geometry. Berlin–Heidelberg: Springer, 1988 24. Danilov, V.I.: The geometry of toric varieties. Russ. Math. Surv. 33, n.2, 97 (1978) 25. Klemm, A.: Unpublished 26. Skarke, H.: Weight systems for toric Calabi–Yau varieties and reflexivity of Newton polyhedra. Mod. Phys. Lett. A11, 1637 (1996) Communicated by S.-T. Yau

Commun. Math. Phys. 185, 509 – 541 (1997)

Communications in


Secondary Quantum Hamiltonian Reductions Jens Ole Madsen, Eric Ragoucy

L

Laboratoire de Physique Théorique?? ENS APP, groupe d’Annecy, LAPP, Chemin de Bellevue, B.P. 110, F-74941 Annecy-le-vieux Cedex, France. E-mail: [email protected], [email protected] Received: 18 May 1995 / Accepted: 16 January 1996

Abstract: Recently, it has been shown how to perform the quantum hamiltonian reduction in the case of general s`(2) embeddings into Lie (super)algebras, and in the case of general osp(1|2) embeddings into Lie superalgebras. In another development it has been shown that when H and H0 are both subalgebras of a Lie algebra G with H0 ⊂ H, then classically the W(G, H) algebra can be obtained by performing a secondary hamiltonian reduction on W(G, H0 ). In this paper we show that the corresponding statement is true also for quantum hamiltonian reduction when the simple roots of H0 can be chosen as a subset of the simple roots of H. As an application, we show that the quantum secondary reductions provide a natural framework to study and explain the linearization of the W algebras, as well as a great number of new realizations of W algebras.

1. Introduction In recent years, we have seen great activity in the area of extended conformal algebras, i.e. algebras that contain the Virasoro algebra as a subalgebra, see e.g. [1]. Examples of this are the well known Kac-Moody algebras and superconformal algebras, but also the more complicated W algebras, introduced by Zamolodchikov in 1985, [2]. Apart from their inherent interest as mathematical objects, such algebras are of interest in many different fields of physics, such as the study of integrable hierarchies [3], string theory [4], Toda theories [5], quantum Hall effect [6], etc. Different methods have been developed for the construction of extended conformal algebras. One method is the direct construction, essentially the solution, using algebraic computation, of a set of consistency equations for a prescribed set of fields, see e.g. [7]. Another method for constructing W -algebras is the coset method, the generalization of the well-known coset construction in conformal field theory, for a review see e.g. [1]. However, one of the most powerful methods of constructing extended conformal algebras

510

J.O. Madsen, E. Ragoucy

is the hamiltonian reduction. The starting point here is an affine Lie (super)algebra, on which one imposes a suitably chosen set of constraints. The reduced algebra is then an extended conformal algebra [8–11]. The quantum version of this reduction has recently been done using BRST techniques [12–18]. It has recently been shown [19] that in certain cases one can impose extra constraints on a W algebra obtained by hamiltonian reduction, and get yet another W algebra; this procedure is called secondary hamiltonian reduction. In this paper we show how to quantize this procedure, thus performing the secondary quantum hamiltonian reduction. As an application of the technique developed, we find the generalized quantum Miura transformation corresponding to the secondary reduction. This secondary quantum Miura transformation leads to a large number of new realizations of W algebras. As another application, we show that the secondary quantum hamiltonian reduction yields a general method of constructing linearizations of W algebras; For a large class of W algebras, we find that we can use the secondary quantum hamiltonian reduction to construct linearizations in a systematic way. Recall that a linearization of a W algebra, introduced in [20] is the embedding of that W algebra into a larger algebra which is equivalent to a linear algebra. This paper is organized in the following way: In Sect. 2, we briefly remind the reader of the primary hamiltonian reduction; Sect. 2 is a description of the classical reduction, while Sect. 2 gives a brief account of the quantum reduction. In Sect. 2 we briefly recall the classical secondary reduction. These sections are included with the purpose of keeping the paper reasonably self-contained. Sections 3 and 4 are the two main sections of the paper. In these sections we find the cohomology corresponding to the secondary quantum hamiltonian reduction. In Sect. 3 we use the theory of spectral sequences to show that for triples G, H0 and H with H0 ⊂ H, and satisfying certain supplementary conditions, the secondary reduction of W(G, H0 ) gives as a result W(G, H). In Sect. 4 we give a method to find explicitly the cohomology corresponding to W(G, H), i.e. to find expressions for the generators of W(G, H) in terms of the generators of W(G, H0 ). Furthermore we describe the generalized quantum Miura transformation corresponding to the secondary quantum hamiltonian reduction, a transformation which gives us numerous new realizations of W algebras. Section 4 is an example of the secondary quantum hamiltonian reduction. The main results of these sections are collected in Theorem 2, Sect. 3, Theorem 3, Sect.4.1, and Theorem 4, Sect. 4.3. In Sect. 5 we show how to use the technique, developed in the preceding sections, to linearize W algebras. Using the secondary quantum hamiltonian reduction we show that we can embed many W algebras into larger algebras, which are equivalent to linear algebras. Finally we have included two appendices, one on spectral sequences and one on the use of modified gradings in the hamiltonian reduction. For all explicit calculations, we have used the OPE mathematica package of K. Thielemans [21]. 2. Hamiltonian Reductions: A Reminder 2.1. The Classical Case. Let us briefly recall the construction of classical W(G,H) algebras as they appear in the context of Hamiltonian reduction [9]. We start with a Lie algebra G with generators ta and inner product g ab = hta tb i. Furthermore we consider a

Secondary Quantum Hamiltonian Reductions

511

regular subalgebra H ⊂ G, and we denote the generators of the principal s`(2) of H by {M+ , M0 , M− }. M0 defines a grading gr(·) of G X G = G− + G0 + G− = Gm, m

where m is the eigenvalue under the operator ad(M0 ). We denote the affine Lie algebra corresponding to G as G (1) , and we write the affine current J in the form J(z) = J a (z) ta . ta = gab tb is an element of the dual algebra, gab is defined by gab g bc = δac . We will use greek letters as indices for currents with negative grades, and barred greek letters for currents with non-negative grades, i.e. J α : tα ∈ G − and J α¯ : tα¯ ∈ G 0 ∪ G + . The constraints that we want to impose are (J α (z) − χα ) = 0

(tα ∈ G − ).

(2.1)

χα = χ(J α ) define the set of constraints and M− = χα tα . These constraints are chosen such that they are first class. This means that the Poisson bracket of two constraints is weakly zero (i.e. one finds zero when imposing the constraints after computation of the Poisson bracket). We can then apply the general technique developed by Dirac to take care of these constraints: The first class constraints generate gauge transformations, i.e. they are associated to a group of symmetries of the physical states of the theory. To eliminate this symmetry, one imposes new constraints (gauge fixing constraints) in such a way that the set of all constraints becomes second class (i.e. it is no more first class), and the matrix formed by the Poisson brackets of two constraints Cij (t1 , t2 ) = {Φi (t1 ), Φj (t2 )} is invertible. As these constraints must not interfere with the physical contents of the theory, one constructs new brackets (called Dirac brackets) with the help of the Poisson brackets and the inverse C ij of Cij : Z {X(z), Y (w)}D = {X(z), Y (w)}− dt1 dt2 {X(z), Φi (t1 )}C ij (t1 , t2 ){Φj (t2 ), Y (w)}, (2.2) where Φi (t) are the (second class) constraints. These Dirac brackets are defined such that any quantity has (strongly) zero Dirac brackets with any of the constraints. In other words, we have decoupled the constraints from the other physical quantities. In the case of the constraints (2.1), it can be shown that one can choose gaugefixing constraints such that the remaining generators correspond to the highest weight components with respect to the embedded s`(2): The gauge-fixed current Jgf is of the form Jgf = χα tα + J ı¯ tı¯ with [M+ , tı¯ ] = 0. As the constraints are decoupled from the other physical quantities, it is clear that the Dirac bracket of two J ı¯ ’s will close (polynomially) on the J ı¯ ’s. These Dirac brackets realize the W algebra W(G,H). To get a realization of the W generators as polynomials of the currents J a , one uses the gauge transformations generated by the first class constraints R + dz α (z){J α (z), J a (w)} [J a (w)]g = J a (w) R (2.3) + 2!1 dzdx α (z)β (x){J β (x), {J α (z), J a (w)}} + . . . to fix [J(w)]g to be of the form [J(z)]g = χα tα + W ı¯ (z)tı¯ ,

(2.4)

512


where the W ı¯ (z) are polynomials in the J a ’s and realize also the W algebra when using the Poisson brackets. It can also be shown that one can realize the W algebra by using the zero grade generators only: starting with J(z) = χα tα + J α¯ 0 (z)tα¯ 0 , where [M0 , tα¯ 0 ] = 0 and doing the gauge transformations as above, one gets the W ı¯ (z) as polynomials in the J α¯ 0 ’s. This transformation is called the (classical) Miura transformation. Finally, let us note that the choice (2.1) of constraints is not unique. We introduce a new grading operator H = M0 + U , where U is an element of the Cartan subalgebra. H defines a grading which we write as X G 0n . G = G 0− + G 00 + G 0+ = n

In [22, 11] it was shown that the constraints obtained by replacing G − by G 0− in Eq- (2.1) leads to the same classical algebra, W(G, H), if U commutes with the s`(2) algebra and “respects” the highest weights, i.e. satisfies the non-degeneracy condition ker ad(M+ ) ∩ G 0− = 0. 2.2. Primary Quantum Hamiltonian Reduction. This section is intended as a brief recapitulation of the method developed in [16] of quantum hamiltonian reduction. We want to quantize the Hamiltonian reduction which we have presented in the previous section. To do this, we will use the BRST formalism, which is a standard procedure in the framework of constrained systems, see e.g. [12]. For each constraint we introduce a ghost-antighost pair (cα , bα ). Corresponding to these constraints we define a BRST operator s: I dz j(z)φ(w), s(φ)(w) = w

1 j(z) = (J α (z) − χα )cα (z) + f αβ γ bγ (z)cβ (z)cα (z). 2

(2.5)

The quantized W algebra W(G, H) is then W(G, H) = H 0 (Ω; s), where Ω is the operator product algebra generated by the affine currents, the ghosts and anti-ghosts, and their derivatives and normal ordered products. As usual when using BRST quantization, one defines a set of modified generators Jˆα (z) = s(bα )(z) + χα = J α (z) + f αβ γ bγ (z)cβ (z), and it turns out that it is useful to modify in a similar way the non-constrained generators: ¯ γ Jˆα¯ (z) = J α¯ (z) + f αβ γ (b cβ )0 (z),

where we have introduced the normal ordered product of fields Aj (z) (j = 1, 2, ...): I dz (A1 A2 )0 (w) = A1 (z)A2 (w) w z−w (2.6) and (An . . . A1 )0 (w) = (An (An−1 . . . A1 )0 )0 (w).


513

With these definitions the space Ω α generated by Jˆα and bα is actually a subcomplex (i.e. s(Ω α ) ⊂ Ω α ) with trivial cohomology H n (Ω α ; s) = δn,0 C . The space Ωred generated by the “hatted” non-constrained generators Jˆα¯ and the ghosts cα is also a subcomplex, and in fact using the version of the Künneth formula which was shown in [16], one can show that we need not consider the trivial subcomplexes Ω α since H ∗ (Ω; s) ∼ = H ∗ (Ωred ⊗ (⊗α Ω α ); s) ∼ = H ∗ (Ωred ; s) ⊗ (⊗α H ∗ (Ω α ; s)) ∼ (2.7) = H ∗ (Ωred ; s). In order to actually calculate this cohomology and find explicit expressions for the generators of W(G, H), one splits the BRST operator s into two nilpotent anticommuting operators s0 and s1 defined by the currents j0 (z) = −χα cα (z), 1 j1 (z) = J α (z)cα (z) + f αβ γ bγ (z)cβ (z)cα (z). 2

(2.8)

Corresponding to these two operators, we can define a bigrading of the complex Ω as a combination of the ghostnumber and the grading gr(·): J a : (m, −m), bα : (m, −m − 1), cα : (−m, m + 1),

(2.9)

where m is the grade of ta , respectively tα . With this definition s0 has bigrade (1, 0) and s1 has bigrade (0, 1). Using the technique of spectral sequences (see Sect. 3 and Appendix A), one can show that there is a vector space isomorphism H n (Ωred ; s) ∼ = H n (Ωred ; s0 ) ∼ = δn,0 Ωhw , where Ωhw is generated by the hatted generators that are highest weight under the embedded s`(2), i.e. Jˆα¯ ∈ Ωhw iff [M+ , tα¯ ] = 0. In order to actually find the elements in H 0 (Ωred ; s), i.e. the generators of W(G, H), one can use the tic-tac-toe construction with Ωhw as starting point. For every highest α ¯ weight generator Jˆhw we can construct the generator W α¯ (z): W α¯ (z) =

m X

W`α¯ (z),

`=0 α ¯ α ¯ α ¯ where m is the grade of Jˆhw , W0α¯ (z) = Jˆhw (z), and s1 (W`α¯ ) + s0 (W`+1 ) = 0. We find that the bi-grade of W`α¯ is (m − `, ` − m), and since s1 vanishes on terms with bigrade (0, 0) we see that the sequence stops at ` = m. It is easy to verify that s(W α¯ ) = 0. In principle the operator product algebra of the W ’s close only modulo s-exact terms. However, there are no elements in Ωred with negative ghostnumber, thus there are no s-exact terms with zero ghostnumber, and therefore the operator products of the W ’s close exactly. The operator product expansion (ope) preserves the grading, which implies that the operator product expansions of the zero grade part of the generators Pm must giveα¯the same must be algebra as the ope’s of the full generators, i.e. the map W α¯ = `=0 W`α¯ → Wm

514


an algebra isomorphism1 . This defines a realization of W(G, H) in terms of the algebra (1) Gˆ 0 , the algebra generated by the “hatted” grade zero affine currents. This is known as the quantum Miura transformation, and can be used to define a free field realization (1) of W(G, H) by using the Wakimoto construction [23] to write the generators of Gˆ 0 in terms of free fields. Just as in the classical hamiltonian reduction, we can modify the grading operator M0 by adding a U (1) current obeying the non-degeneracy condition. In that case, the modified grading operator H = M0 + U will lead to a modification of the BRST operator (2.5). One finds that the calculation of the cohomology leads to an equivalent but different (“twisted”) realization of W(G, H). 2.3. Classical Secondary Reductions. First, we briefly recall the framework of secondary reductions as they appear in the classical case. We start with a W(G, H0 ) algebra (defined as in Sect. 1), with H0 a regular subalgebra of G. We suppose now that there is another regular subalgebra H such that H0 ⊂ H. Since H0 is embedded in H, it is natural to wonder whether the W(G, H0 ) algebra can be related to W(G, H). In fact, considering the constraints associated to both W-algebras, it is clear that we have to impose more constraints on W(G, H0 ) to get W(G, H); for instance, the number of primary fields (which is directly related to the number of constraints) is lower in W(G, H) than in W(G, H0 ). These (further) constraints will be imposed on W fields themselves, so that we will gauge a part of the W(G, H0 ) algebra. In [19], it has been proved: Theorem 1. Let G = s`(N ) and let H0 and H be two regular subalgebras of G such that H0 ⊂ H.

(2.10)

Then, there is a set of constraints on the W(G, H0 ) algebra such that the (associated) Hamiltonian reduction of this algebra leads to the W(G, H) algebra. We will represent this secondary reduction as W(G, H0 ) → W(G, H).

(2.11)

The proof of this theorem relies on a general property of the Dirac brackets, which can be stated as follows: We start with a Hamiltonian theory on which we impose constraints. Instead of considering directly the complete set of second class constraints, we can divide this set into several subsets (of second class constraints) and compute the Dirac brackets at each step (using the Dirac bracket of the previous steps as initial Poisson brackets). Then the last Dirac brackets do not depend on the partition of the second class constraints set we have used. Thus, coming back to our W-algebras, it is sufficient to find a gauge fixing for the W(G, H0 ) algebra such that the corresponding set of second class constraints is embedded into the set of second class constraints for W(G, H) as soon as H0 ⊂ H. Indeed, with such an embedding, it is clear that the constraints one will impose on the W(G, H0 ) generators will just be the constraints related to H that are not in the subset associated to H0 . Such a gauge has been explicitly constructed in [19] for G = s`(N ). Because of the generality of the property of Dirac brackets, and considering the construction of 1 Actually this argument shows only that the map is an algebra homomorphism, but one can prove that the map is also injective.


515

orthogonal and symplectic algebras from the (folding of) unitary ones [24], it is clear that the theorem is also true for the other classical Lie algebras. We present below a quantization of the secondary reduction using the BRST formalism. Note that the BRST operator involves only first class constraints, so that the quantization is not straightforward: in the classical case, we have to embed the sets of second class constraints one into the other, while in the quantized version it is the sets of first class constraints that we will embed. 3. Quantum Secondary Reductions: Algebra Isomorphism In order to show that W(G, H) can be obtained from a secondary hamiltonian reduction of W(G, H0 ), we will use the theory of spectral sequences. For a good introduction see e.g. [25]; in Appendix A we give a brief description of some main points in the theory. Assume that we have a Lie algebra G and two regular subalgebras H0 and H with 0 H ⊂ H, as in the previous section. The principal s`(2) subalgebra of H0 is denoted 0 , M00 , M+0 }, while the principal s`(2) subalgebra of H is {M− , M0 , M+ }. The by {M− eigenvalues of the operator ad(M0 ) defines a natural grading of G: X Gm. (3.1) G = G− + G0 + G+ = m

A second grading is defined by M00 , but as described in Appendix B we can use the more general, but equivalent, grading operator ad(H 0 ) = ad(M00 + U ). We write the corresponding grading as: X G 0n . (3.2) G = G 0− + G 00 + G 0+ = n

We assume the gradings to be integer, and call H the corresponding grading operator2 . We wish to constrain the negative grade parts of the two algebras, G 0− and G − respectively. To each generator tα ∈ G − corresponds one of the first class constraints of the type φα (z) = J α (z) − χα = 0, where the χα are constants. We denote by Φ the set of these first class constraints, and similarly Φ0 denotes the set of constraints corresponding to G 0− . We assume that we can choose the constraints in such a way that Φ0 ⊂ Φ, and we note that this implies that the following two conditions are satisfied: 1) the set of simple roots of H0 can be chosen to be a subset of the set of simple roots of H, and 2) G 0− ⊂ G − . The classification of triples G, H0 , and H satisfying these conditions are given in Appendix B. Note that although the “usual” constraints (defined by the grading ad(M0 )) obeying Φ0 ⊂ Φ are very few, the use of modified gradings as described in the appendix gives us a large class of triples that satisfy Φ0 ⊂ Φ. Let us introduce the notation for the indices: tA ∈ G − ta ∈ G 0−

tA¯ ∈ G 0 ∪ G + , ta¯ ∈ G 00 ∪ G 0+ ,

tα ∈ G − \ G 0− .

2

For the algebras which we consider, it is always possible to choose an integer grading.

(3.3)

516


Note that the generators tα must have grade zero with respect to H 0 : tα ∈ G 00 (if tα ∈ G 0+ , then we can find a tα¯ ∈ G 0− , corresponding to a generator in G 0− \ G − ; but this is in contradiction with the condition 2) above). Note also that tα is a highest weight 0 , M00 , M+0 }. To show this, assume a root basis, and generator under the embedding of {M− let α1 , α2 , . . . , αn , β1 , β2 , . . . , βm let α1 , α2 , . . . , αn denote the simple roots of H0 , andP denote the simple roots of H. We can write M+0 = a i tα i . P On the other hand, since 0 tα has grade zero under H we find that we can write α = − nj βj . This shows that [M+0 , tα ] = 0. We can write the constraints in the form: φa (z) = φA (z) =

J a (z) − χa = 0, J A (z) − χA = 0,

φa ∈ Φ0 , φA ∈ Φ,

(3.4) (3.5)

0 where M− = χa ta and M− = χA tA . Corresponding to the constraints (3.4) we introduce ghosts ca and anti-ghosts ba , and we define the BRST current j 0 by (see Eq. (2.5))

1 j 0 (z) = (J a (z) − χa )ca (z) + f ab c bc (z)cb (z)ca (z). 2 Similarly we introduce ghosts cA and anti-ghosts bA corresponding to (3.5). The set of ghosts ca is a subset of the set cA and ba is a subset of bA ; in fact the set cA is the union of the set ca and the set cα , and the set bA is the union of the set ba and the set bα . The BRST current j is defined by 1 j(z) = (J A (z) − χA )cA (z) + f AB C bC (z)cB (z)cA (z). 2 We define the current j 00 by j = j 0 + j 00 , and we find 1 j 00 (z) = j(z) − j 0 (z) = (J α (z) − χα )cα (z) + (f AB C bC (z)cB (z)cA (z) 2 − f ab c bc (z)cb (z)ca (z)). 0

00

0

(3.6) (3.7)

00

Corresponding to the currents j, j , and j we define operators s, s , and s by I sφ(w) = dz j(z)φ(w), w

and similarly for s0 and s00 . Using tα ∈ G 00 and ta ∈ G 0− we can show that terms of the form f ab γ bγ cb ca , f aβ γ bγ cβ ca , and f αβ c bc cβ cα vanish. This means that we can write: j 00 (z) = (J α (z) − χα )cα (z) 1 + (f αβ γ bγ (z)cβ (z)cα (z) + f aβ c bc (z)cβ (z)ca (z) + f αb c bc (z)cb (z)cα (z)) 2 1 (3.8) = (J˜α (z) − χα )cα (z) + f αβ γ bγ (z)cβ (z)cα (z), 2 where in the last line J˜α is defined by J˜α (z) = s0 (bα )(z) + χα = J α (z) + f αb c bc (z)cb (z). Note that since J α is a highest weight generator with grade zero under the grading operator H 0 , J˜α is actually a generator W α of the algebra W(G, H0 ), so we can alternatively write


517

1 j 00 (z) = (W α (z) − χα )cα (z) + f αβ γ bγ (z)cβ (z)cα (z). (3.9) 2 Let A denote the algebra generated by the currents as well as their derivatives and normal ordered products. Λ0 is the algebra generated by the ghosts ca and the anti-ghosts ba (and their derivatives and normal ordered products), Λ00 is generated by cα and bα , and Λ is generated by cA and bA . Note that Λ = Λ0 ⊗ Λ00 . The algebras A ⊗ Λ0 and A ⊗ Λ are graded by ghost numbers, and we know [12, 13, 16] that W(G, H) ∼ = H 0 (A ⊗ Λ; s), 0 ∼ W(G, H ) = H 0 (A ⊗ Λ0 ; s0 ). We can define a bigrading on the algebra Ω = A ⊗ Λ : Ω = has bigrading (1, 0) and s00 has bigrading (0, 1), namely: J ca cα ba bα

(3.10)

P p,q

Ω p,q such that s0

: (0, 0), : (1, 0), : (0, 1), : (−1, 0), : (0, −1).

(3.11)

Be careful that this bigrading is not the bigrading used in the previous section: it is based on two ghostnumbers, while for primary reductions, the bigrading is based on the gradation of G, and on one ghostnumber (see Eq. (2.9)). Define JÂ = s(bA ) + χA = J A + f AB C bC cB , ¯ ¯ ¯ JÂ = J A + f AB C bC cB , A Jghost ≡ f AB C bC cB is the ghost realization of the constrained part of the algebra. We will use as basis of Ω the set of “hatted” currents and the ghosts and anti-ghosts ˆ c, b}. For each index A the algebra Ω A generated by JÂ and bA is an s-subcomplex {J, with trivial cohomology [16]

H n (Ω A ; s) ∼ = δn,0 C . ¯

Define Ω¯ red to be the algebra generated by {JÂ , cA }. As in Eq. (2.7) we find that ! ! ! O O ∗ ∗ A ∗ ∗ A ∼ Ω H (Ω ; s) ;s H (Ω; s) ∼ = H Ω¯ red ⊗ = H (Ω¯ red ; s) ⊗ A

∼ = H ∗ (Ω¯ red ; s) ⊗

O

A

! C

∼ =

H ∗ (Ω¯ red ; s),

(3.12)

A

i.e. we can reduce the problem to finding the cohomology of Ω¯ red , ignoring the trivial subcomplexes Ω A . In our case it will turn out to be convenient to perform this reduction only partly, in the sense that we will use the Künneth formula only to extract the subcomplexes Ω a . We will therefore define Ωred to be the subcomplex generated by {Jâ¯ , ca } and Λ00 . The full complex Ω can be written in the form ! O a ∼ Ω , Ω = Ωred ⊗ a

518


and we use the Künneth formula to find H n (Ω; s) ∼ = H n (Ωred ; s).

(3.13)

Now we make a change of basis. As new basis we choose the currents Jã¯ = J a¯ + f c bc cb , the ghosts ca and cα , and anti-ghosts bα . We denote the space generated by Jã¯ and ca by Γ , so we have Ωred = Γ ⊗ Λ00 . Note that the cohomology of Γ with respect to the operator s0 is the W(G, H0 ) algebra: ab ¯

H n (Γ ; s0 ) ∼ = δn,0 W(G, H0 ). Note also that we have (Γ ⊗ Λ00 )p,q = Γ p,0 ⊗ (Λ00 )0,q . We can now consider the spectral sequence corresponding to the double complex (Ωred ; s0 ; s00 ). The spectral sequence is a sequence of complexes (Erp,q ; sr ), such that E0p,q = (Ωred )p,q p,q Er+1 = H p,q (Er ; sr ) =

Erp,q ∩ ker (sr ) , Erp,q ∩ im (sr )

(3.14)

where sr is a nilpotent operator of bigrade (1−r, r), s0 = s0 and s1 = [s00 ]. The operators sr for r ≥ 2 are defined in Appendix A. The notation s1 = [s00 ], is to be interpreted as s1 ([x]) = [s00 (x)] of a given [x] ∈ E1 . This is well-defined because [s00 (x + s0 (y))] = [s00 (x) + s00 (s0 (y))] = [s00 (x) − s0 (s00 (y))] = [s00 (x)]. It is now possible to show that if the spectral sequence collapses, i.e. if there exists R such that Er = ER for r ≥ R, then3 we have p,q ∼ E∞ = F q H p+q /F q+1 H p+q ,

(3.15)

where E∞ = ER and F q H is a filtration on the cohomology H(Ωred ; s) defined by L M ( i≥0 (Ωred )p−i,q+i ) ∩ ker s q p+q p+q p−i,q+i ; (3.16) = H ( (Ωred ) ; s) = L F H ( i≥0 (Ωred )p−i,q+i ) ∩ im s i≥0

thus we can in principle reconstruct the cohomology H(Ωred ; s) from the spectral sequence, on the condition that we can reconstruct H(Ωred ; s) from the quotient spaces F q H p+q /F q+1 H p+q . The first element in the spectral sequence is E0p,q = (Ωred )p,q = (Γ ⊗ Λ00 )p,q . For the second element we find E1p,q = H p,q (E0 ; s0 ) ∼ = H p (Γ ; s0 ) ⊗ (Λ00 )0,q ∼ = δp,0 W(G, H0 ) ⊗ (Λ00 )0,q .

(3.17)

The third element in the spectral sequence is E2 = H(E1 , s1 ). We find E2p,q = H p,q (E1 ; s1 ) = δp,0 H p,q (W(G, H0 ) ⊗ Λ00 ; [s00 ]). 3

A.

(3.18)

Actually this condition is sufficient but not necessary; it can in fact be relaxed considerably, see Appendix


519

The spectral sequence collapses here, i.e. [s00 ] is the last non-trivial operator in the sequence. In fact E2p,q is nontrivial only for p = 0, and since sr has bigrade (1 − r, r) it is clear that sr is trivial for r ≥ 2. We conclude that Er = E2 for any r ≥ 2, and so E∞ = E2 . Note that from Eq. (3.9) it follows that if we restrict s00 to W(G, H0 ) ⊗ Λ00 then it maps into W(G, H0 ) ⊗ Λ00 , which means that we can replace [s00 ] by s00 in Eq. (3.18). From Eq. (3.15) it follows that we have p,q F q H p+q /F q+1 H p+q ∼ = E2 ∼ = δp,0 H p,q (W(G, H0 ) ⊗ Λ00 ; s00 ),

where F q H p+q is defined in Eq. (3.16). L (Ωred )p,q is trivial for p < 0, and therefore i>0 (Ωred )p−i,q+i is trivial for p < 0. Using Lemma 2 of Appendix A, one can verify that this implies that H(A ⊗ Λ; s) ∼ = E2 , and that this isomorphism is in fact an algebra isomorphism. Let us collect the results of this section in the following Theorem 2. Given two W algebras W 0 = W(G, H0 ) and W = W(G, H) with H0 ⊂ H. If we can find sets of first class constraints Φ0 and Φ (where W 0 is the result of imposing the set of constraints Φ0 on G (1) and W is the result of imposing Φ on G (1) ) such that Φ0 ⊂ Φ, then: 1) It is possible to perform a secondary quantum hamiltonian reduction on W 0 . This secondary reduction consists of imposing a set Φ00 of first class constraints on W 0 . There is a simple one-to-one correspondence between the constraints Φ00 imposed on W 0 , and the “missing” constraints Φ \ Φ0 . 2) Let A be the algebra generated by currents in G (1) and their derivatives and normal ordered products, and let Λ0 be the algebra generated by the ghosts and anti-ghosts corresponding to the constraints Φ0 . If we denote by s0 and s the BRST operators corresponding to the quantum hamiltonian reduction leading to W 0 and W respectively, then the BRST operator that corresponds to the secondary quantum hamiltonian reduction of W 0 is [s−s0 ] ≡ [s00 ]. Considering W 0 as the cohomology H 0 (A⊗Λ0 ; s0 ), [s00 ] on an element [x] ∈ W 0 is defined by [s00 ]([x]) ≡ [s00 (x)]. 3) Let Λ and Λ00 be the algebras generated by the ghosts corresponding to Φ and Φ00 respectively. The result of the secondary hamiltonian reduction of W 0 is H 0 (W(G, H0 )⊗Λ00 ; [s00 ]) ∼ = H 0 (H 0 (Γ ⊗Λ00 ; s0 ); s00 ) ∼ = H 0 (A⊗Λ; s) ∼ = W(G, H). 4. Quantum Secondary Reduction: Direct Calculation In Sect. 2 we explained briefly the primary quantum hamiltonian reduction of the affine Lie algebra G (1) that results in the W algebra W(G, H). Let us recall some of the main points of the procedure: 1) The cohomology H 0 (A ⊗ Λ; s) ≡ H 0 (Ω; s) is isomorphic to the cohomology H 0 (Ωred ; s), where Ωred is the space generated by the “hatted” unconstrained generators Jˆα¯ and the ghosts cα . 2) There is a vector space isomorphism between the space Ωhw , generated by the hatted highest weight generators, and W(G, H). 3) We can use the tic-tac-toe construction to construct explicit realizations of the generators of W in terms of the hatted unconstrained generators. The starting points for the tic-tac-toe construction are the elements of Ωhw .

520


4) We can show that there exists an algebra isomorphism between the algebra W(G, H), and the algebra generated by the zero-grade part of the W -generators as constructed by the tic-tac-toe method. This is the generalized quantum Miura transformation. In this section we will take the corresponding steps for the secondary quantum hamiltonian reduction. Among the consequences will be the secondary quantum Miura transformation and a systematic method of linearization of W algebras. Note that we have already found the BRST cohomology H 0 (W 0 ⊗Λ00 ; s00 ) to be identical to the algebra W(G, H); the aim of this section is to construct concrete realizations of W(G, H)-generators from the generators of W(G, H0 ). 4.1. Isomorphism between W(G, H) and H0 (Γred , s). The reduction of the W 0 algebra is defined in terms of the grading (H − H 0 ). The fact that this is actually a well-defined grading of the algebra follows from the fact that the simple roots of H0 has grade 1 both ˜ − H˜ 0 is a under H and H 0 , which implies that [M+0 , (H − H 0 )] = 0 and therefore H 0 α α ˜ generator of W . The generators to be constrained are W = J , which are just the generators with negative grade. Define Γ = W(G, H0 ) ⊗ Λ00 . Just as in the case of the primary reduction, we define “hatted” constrained generators by ˆ α (z) = s00 (bα )(z) + χα , W ˆα ˆ α = Jˆα . For each α, define Γ α to be space generated by W and we find that in fact W and bα . We note that Γ α is a subcomplex with trivial cohomology: H n (Γ α ; s00 ) = δn,0 C . In the primary quantum hamiltonian reduction, we saw in Sect. 2 that it was possible to split the complex Ω into a product of subcomplexes Ω = Ωred ⊗ (⊗α Ω α ) , where the subcomplex Ωred was generated by the “hatted” unconstrained generators Jˆα¯ and the ghosts cα . We want to show that we can split the complex Γ in a similar way: Property 1. It is possible to define a subcomplex Γred generated by modified unˆ A¯ and ghosts cα , such that Γred is a subcomplex (i.e. constrained generators W s00 (Γred ) ⊂ Γred ). We will do the proof by a double induction, using the conformal dimension and the (H − H 0 )-grade of the generators as induction parameters. We consider the “twisted” algebra, i.e. the algebra where the conformal dimensions are given by the H 0 -grade + 1. In this case the conformal dimensions of all the constrained generators is 1 and the (H − H 0 )-grade of the constrained generators is less than zero. We will need a lemma: Lemma 1. Consider an unconstrained generator W α¯ with conformal dimension h and grade n: all unconstrained generators occurring in s00 (W α¯ ) has either conformal dimension strictly less than h or conformal dimension h and grade less than n.


521

In the expression s00 (W α¯ ), all generators are the result of OPEs between a constrained generator in j 00 , and W α¯ . Thus all monomials4 of generators occurring in s00 (W α¯ ) must have conformal dimension h and grade less than n. Write: s00 (W α¯ ) = Pβ¯ (c)W β + Qαγ¯ (c)W α W γ¯ + · · · , ¯

¯

then we see that the conformal dimension of W β is h and the grade is less than n. The conformal dimension of W γ¯ is h − 1, etc..., which proves the lemma. Assume that we have already found hatted generators for all generators with conforh−1 to be the space generated by these hatted mal dimension less than h, and define Γred α ¯ generators and the c’s. Assume that W is any generator with conformal dimension h c α¯ ) ∈ Γ h−1 . Consider c α¯ such that s00 (W and grade 0, we will show that we can define W red 00 α ¯ s (W ). According to the lemma, all unconstrained generators occurring in s00 (W α¯ ) must have conformal dimension less than h. We can therefore write X h−1 Aij Bj , Aij ∈ B = ⊗β Γ β , Bj ∈ Γred , s00 (W α¯ ) = i,j

where the Bj ’s are chosen to be linearly independent. Since j 00 is linear in the constrained currents, each of the terms Aij are monomials in the constrained currents, the W α ’s. Let us consider only those terms that have the highest grade, considered as monomials in W α , X α s00 (W α¯ ) = Am lower orders terms , Am ij Bj + ij is order m in W . i,j

Now apply s00 once again. We get: X m 00 0= s00 (Am ij )Bj ± Aij s (Bj ) . i,j h−1 00 m We know that s00 (Bj ) ∈ Γred , and s00 (Am ij ) ∈ B. We also know that s (Aij ) is of order α m + 1 in the W ’s, and these are the only possible terms of order m + 1; and since the expression must vanish order by order in the W α ’s, we find X 0= s00 (Am ij )Bj . i,j

Since the Bj ’s are linearly independent we find that X 0= s00 (Am ij ). i

P 00 Now we use the fact that B has trivial cohomology: since i Am ij is in the kernel of s 00 α it must be P in the image of s , so we can find Xj (of grade m − 1 in the W ’s) such that s00 (Xj ) = i Am ij . Define X W(1) = W − Xj Bj . j

We find that: 4

We use the word “monomial”, even though what we have is actually a normal-ordered product.

522


s00 (W(1) ) =

X

Am ij Bj +

lower orders terms

−

i,j

=

X

X (s00 (Xj )Bj ± Xj s00 (Bj )) j

Am ij Bj

+

−

lower order terms

i,j

= lower order terms ∓

X

X

Am ij Bj ∓

i,j

X

Xj s00 (Bj )

j

00

Xj s (Bj )

j

(the ± depends on the Grassmann parity of Xj ). All these terms are of order at most ˆ α¯ such that s00 (W ˆ α¯ ) is a m − 1 in the W α ’s. By induction we see that we can define W polynomial of degree 0 in the constrained currents. ˆ α¯ ) either. Actually this is quite We want to show that in fact no b’s appear in s00 (W simple: write X X ˆ α¯ ) = B + B α bα + Bαβ bα bβ + · · · . s00 (W α

α,β

00

Apply s again to get 0 = s00 (B) +

X

s00 (Bα )bα ± Bα (Jˆα − χα ) + · · · .

α

P Since s00 (Bα ) does not contain any constrained currents, we must have 0 = α Bα Jˆα , ˆ α¯ ) ∈ Γred . but this can only be true if Bα = 0 for all α. We see that indeed s00 (W Now assume that we have found hatted generators for all generators with conformal dimension less than h, and with conformal dimension h and grade less than n, and define h to be the space generated by these hatted generators and the c’s. Assume that Γred,n−1 α ¯ W is any generator with conformal dimension h and grade n, we want to show that 00 α ¯ c α¯ ) ∈ Γ h c α¯ such that s00 (W we can define W red,n−1 . Consider s (W ). According to the ¯ lemma, any unconstrained generator W β that occurs in s00 (W α¯ ) has either conformal dimension less than h or conformal dimension h and grade less than n. We can therefore write X h Aij Bj , Aij ∈ B, Bj ∈ Γred,n−1 . s00 (W α¯ ) = i,j

ˆ α¯ such We can therefore repeat the arguments from above word by word to define W 00 ˆ α ¯ that s (W ) ∈ Γred . ˆ α¯ such that s00 (W ˆ α¯ ) ∈ We have shown that to any generator W α¯ we can construct W Γred . We have therefore shown that Γred is a sub-complex. Thus, s00 (Γred ) ⊂ Γred , and we can then use the Künneth theorem (see Eq. (2.7) to find ! O ∗ 00 ∼ ∗ α 00 H (Γ ; s ) = H Γred ⊗ ( Γ ); s α

∼ = H ∗ (Γred ; s00 ) ⊗ ⊗α H ∗ (Γ α ; s00 ) ∼ = H ∗ (Γred ; s00 ) ∗

00

(4.1)

Thus in order to calculate the cohomology H (Γ ; s ) it is in fact enough to calculate H ∗ (Γred ; s00 ).


523

Next step is to split s00 into two anti-commuting nilpotent operators s000 and s001 defined by the currents j000 and j100 respectively, j000 (z) = −χα cα (z), 1 j100 (z) = W α (z)cα (z) + f αβ γ bγ (z)cβ (z)cα (z). 2

(4.2)

In order to verify that s000 and s001 are indeed nilpotent and anti-commuting one can either directly calculate the operator products j100 (z)j100 (w) etc., or one can use the fact that s000 = s0 − s00 and s001 = s1 − s01 , where s0 , s00 , s1 , and s01 are all nilpotent and anti-commuting. Corresponding to this split, we can define a bigrading of Γ : W α , W i : (m, −m), bα : (m, −m − 1), cα : (−m, m + 1),

(4.3)

where m is the grade of W α or W i defined by the grading (H − H 0 ); with these definitions s000 has bigrading (1, 0), while s001 has bigrading (0, 1). We can now define the spectral sequence corresponding to the double complex (Γred ; s000 ; s001 ). The first element of the spectral sequence is p,q , E0p,q = Γred

while the second element is the cohomology of s000 : E1p,q = H p,q (E0 ; s000 ) =

p,q Γred ∩ ker (s000 ) . p,q Γred ∩ im (s000 )

(4.4)

In the primary hamiltonian reduction one can show that for each ghost cA , we can ¯ find a linear combination of generators aAA¯ JÂ , such that ¯ ¯ ¯ s0 (aAA¯ JÂ (z)) = s0 aAA¯ f AB C (bC cB )0 (z) = −aAA¯ f AB C χC cB (z) = cA (z). If we replace the index A by α in this equation, then since the index α corresponds to a generator with H 0 -grade zero and A¯ has non-negative H 0 -grade, then we find that also ¯ B and C has H 0 -grade zero. This implies that s00 (aαA¯ JÂ ) = 0, and since s000 = s0 − s00 we find ¯ s000 (aαA¯ JÂ (z)) = cα (z). This shows that the ghosts cα are s000 -exact, and the cohomology of s000 is only non-trivial at ghostnumber zero, i.e. we find E1p,q = H p,q (E0 ; s000 ) ∼ = δp+q,0 Γ0 ,

(4.5)

where Γ0 is the cohomology of s000 at ghostnumber zero. We recall that in the primary quantum hamiltonian reduction, the zeroth cohomology of s0 is Ωhw . In the secondary hamiltonian reduction there is no notion of highest weights, but Γ0 can be considered to be the secondary hamiltonian reduction analogue of Ωhw . Equation (4.5), together with the fact that the bigrade of the operator sr is (1 − r, r) implies that sr is trivial for r ≥ 1. Thus the spectral sequence collapses already here, and we have

524

J.O. Madsen, E. Ragoucy p,q E∞ = E1p,q = H p,q (E0 ; s000 ) ∼ = δp+q,0 Γ0 .

Using Eq. (3.15) of the theory of spectral sequences, we find that F q (H p+q )/F q+1 (H p+q ) = δp+q,0 Γ0 , where F q (H p+q ) is defined as in Eq. (3.16). p,q p+q is trivial for q > 0, so F q Γred = Note that with the bi-gradings defined in (4.3), Γred p−i,q+i ⊕i≥0 Γred is trivial for q > 0, so we can use Lemma 2 in Appendix A and find H n (Γred ; s00 ) ∼ = δn,0 Γ0 ;

(4.6)

however, this isomorphism is a vector space isomorphism but not an algebra isomorphism. We have proven the following: Theorem 3. The BRST operator s00 defined by I 00 dz j 00 (z)φ(w) s (φ)(w) = w

j 00 (z) = (W α (z) − χα )cα (z) + f αβ γ (bγ cβ cα )0 (z)

(4.7)

corresponding to the secondary hamiltonian reduction W(G, H0 ) → W(G, H) can be split into two anticommuting, nilpotent operators s000 and s001 defined by j000 (z) = −χα cα (z) j100 (z) = W α (z)cα (z) + f αβ γ (bγ cβ cα )0 (z). The cohomology of s00 is

H n (Γred ; s00 ) ∼ = δn,0 Γ0 ,

where Γ0 = H 0 (Γred ; s000 ). This isomorphism is a vector space isomorphism, but not an algebra isomorphism. Note that Γred does not contain any elements of negative ghostnumber, and consequently Γ0 ∼ = ker (s000 ). This, together with the fact that we know the number of generators of Γ0 (it is equal to the number of generators of W) considerably simplifies the problem of finding Γ0 in concrete examples. 4.2. Explicit Construction of Generators.. Once we have found the generators V0k of Γ0 , we can use the tic-tac-toe construction to find the generators of H 0 (Γred , s00 ), i.e. the generators of W. These take the form V k (z) =

p X

V`k (z),

`=0 k ) = 0 and p is given by V0k ∈ where V`k is defined inductively by s001 (V`k ) + s000 (V`+1 p,−p k k Γred (i.e. it is the grade of V0 ). Note that if V0 has bigrade (p, −p), V1k has bigrade (p − 1, −p + 1), etc., Vpk has bigrade (0, 0), and the construction stops here because s001 vanishes on generators with grade zero. It is easy to verify that with this construction, s00 (V k ) is indeed zero.


525

The resulting generators V k constitutes a basis of the algebra W. In principle the operator product expansion of these generators close only modulo s00 -exact terms; however, since there are no elements in Γred with negative ghostnumber, there can be no s00 -exact terms with zero ghostnumber, and therefore the algebra of the V k closes exactly. 4.3. Generalized Quantum Miura Transformation. Because the operator product expansion preserves the grading, it is clear that the grade zero part of the generators gives a copy of the W-algebra, or more precisely: the map V k → Vpk (V k = V0k +V1k +· · ·+Vpk ) is an algebra homomorphism. In order to show that this map is in fact an algebra isomorphism, we need to show that the map is an injection. To show this, one can consider the so-called “mirror spectral sequence”, the spectral sequence obtained by inverting the role of s000 and s001 . Thus for the mirror spectral sequence, we define p,q E¯ 0p,q = Γred (= E0p,q ),

Γ p,q ∩ ker (s001 ) E¯ 1p,q = H p,q (E0 ; s001 ) = red , p,q Γred ∩ im (s001 ) E¯ p,q ∩ ker (s000 ) E¯ 2p,q = H p,q (E1 ; s000 ) = ¯1p,q , E ∩ im (s00 ) 1

(4.8)

0

etc. We already know that H ∗ (Γred ; s00 ) is nontrivial only at ghostnumber zero. This p,q implies that also E¯ ∞ is nontrivial only at ghostnumber zero, i.e. at q = −p. We find ¯ ¯ 00 ˆ A ˆ that s1 (W ) = 0 iff W A has bi-grade (0, 0). To see this, note that s001 has bigrade (0,1), 0,1 = {0}. This shows that: and that Γred ˆ A¯ ) = 0. ˆ A¯ has bi-grade (0,0) ⇒ s001 (W W ¯

To see that the opposite is also true, note that for each W A with grade larger than zero, there is a W α such that ¯

¯

W α (z)W A (w) =

g αA + ···, (z − w)(1+hA¯ )

where · · · denotes less singular terms. This gives rise to a term proportional to ∂ hA¯ cα ˆ A¯ ); and one can show that this term will not be cancelled by other terms in in s001 (W 00 ˆ A¯ ˆ A¯ ) 6= 0. s1 (W ), thus showing that s001 (W It follows that 0,0 E¯ 1p,−p = δp,0 Γred . Therefore there is an isomorphism of vector spaces H 0 (Γred ; s00 ) ∼ = E¯ 0,0 , ∞

0

00

and therefore the map from H (Γred ; s ) to its zero grade component is injective, and therefore indeed an isomorphism of algebras. This proof is essentially identical to the one given in [16] for the case of the primary hamiltonian reduction. We have shown: Theorem 4. For generators V k of W, constructed using the tic-tac-toe construction defined above, the mapping V k = V0k + V1k + · · · + Vpk → Vpk of the generator to the zero grade part of the generator is an algebra isomorphism.

526


This mapping is the generalization of the quantum Miura transformation to the case of the secondary hamiltonian reduction. This theorem means that we can realize the generators of the algebra W in terms of ˆ 00 , generated by the “hatted” grade zero generators the generators of the simpler algebra W 0 ˆ 0 always includes the energy-momentum tensor Tˆ , since T is always part of of W 0 . W the grade zero subspace of W 0 . This construction gives us an impressive variety of new realizations of W algebras: for every possible secondary hamiltonian reduction, written in the form G (1) → W(G, H0 ) → W(G, H), we get a realization of the generators of W(G, H) in terms of the hatted generators of the grade zero subalgebra of W(G, H0 ). Similar realizations of W algebras in terms of simpler W algebras have been constructed before, see e.g. [26, 27]; however, the present construction gives a systematic method for constructing a large number of such realizations. 4.4. Example: W(s`(3), s`(2)) → W 3 . Let us consider the simplest possible example of the secondary quantum hamiltonian reduction, namely the reduction of the Bershadsky algebra W(s`(3), s`(2)) to the W 3 algebra. We consider the regular embedded s`(2) subalgebra {Eα1 , Hα1 , E−α1 }. The corresponding standard grading of s`(3) is   0 1 21  −1 0 − 1  . 2 − 21 21 0 As described in Sect. 2, we can modify the grading operator Hα1 by adding a U (1) current. If we choose  1 0 6 0 U =  0 16 0  , 0 0 − 13 then we get the modified integer gradings   011  −1 0 0  . −1 0 0 The constrained current corresponding to these gradings are  α  H 1 J α1 J α1 +α2 Jred =  1 H α2 − H α1 J α2  0 J −α2 −H α2

(4.9)

(to simplify the notation we suppress the z dependence). The grading and constraints for the W 3 –algebra are:   α   0 12 H 1 J α1 J α1 +α2  −1 0 1  (4.10) Jred =  1 H α2 − H α1 J α2  . 0 1 −H α2 −2 −1 0


527

We introduce ghosts c−α1 , c−α2 , c−α1 −α2 , and anti-ghosts b−α1 , b−α2 , b−α1 −α2 . Corresponding to the constraints (4.9) and (4.10) we have the BRST currents j 0 and j respectively: (4.11) j 0 = (J −α1 − 1)c−α1 + J −α1 −α2 c−α1 −α2 j = (J −α1 − 1)c−α1 + (J −α2 − 1)c−α2 + J −α1 −α2 c−α1 −α2 + b−α1 −α2 c−α1 c−α2 . We define “improved” generators J˜−α1 = J −α1 , J˜α1 = J α1 , −α −α −α −α J˜ 2 = J 2 + b 1 2 c−α1 , J˜α2 = J α2 + b−α1 c−α1 −α2 , −α −α −α −α α J˜ 1 2 = J 1 2 , J˜ 1 +α2 = J α1 +α2 , α α −α −α −α H˜ 1 = H 1 − 2b 1 c−α1 − b 1 2 c−α1 −α2 , H˜ α2 = H α2 + b−α1 c−α1 − b−α1 −α2 c−α1 −α2 , and we find that W(s`(3), s`(2)) = H 0 (A ⊗ Λ0 ; s0 ) is generated by ˜ α1 + 2 H ˜ α2 , J =H G− = J˜−α2 , G+ = J˜α1 +α2 + (k + 2)∂ J˜α2 − (H˜ α1 J˜α2 )0 − (H˜ α2 J˜α2 )0 , 1 + k ˜ α1 1 J˜α1 − ∂ H + (J˜α2 J˜−α2 )0 T = k+3 2 1 ˜ α1 ˜ α1 ˜ α1 H˜ α2 )0 + (H˜ α2 H˜ α2 )0 . H )0 + (H + (H 3 Here J is a U (1) field and G± are primary bosonic spin chosen, we have

3 2

(4.12)

fields. With the normalizations

9 + 6k + ···, (4.13) (z − w)2 ±3G± (w) + ···, J(z)G± (w) = z−w 1 (k + 1)(2k + 3) (k + 1)J(w) (k + 3)T − k+1 2 ∂J − 3 (JJ)0 G+ (z)G− (w) = − + ···, − + (z − w)3 (z − w)2 z−w J(z)J(w) =

where · · · denotes non-singular terms. The central charge is c = − (2k+3)(3k+1) . (k+3) The BRST current for the secondary hamiltonian reduction can now be written in the form: j 00 = (J −α2 − 1)c−α2 + b−α1 −α2 c−α1 c−α2 . = (G− − 1)c−α2

(4.14)

The operator s00 is found to act on the fields as follows: 3 − 1 G c−α2 , + ∂G− c−α2 , 2 2 s00 (J) = 3G− c−α2 ,

s00 (T ) =

s00 (G+ ) = −(k + 3)T c−α2 − (k + 1)J∂c−α2 − s00 (b−α2 ) = G− − 1,

k+1 1 ∂Jc−α2 + (JJc−α2 )0 , 2 3 (4.15)

528


and s00 (G− ) = s00 (c−α2 ) = 0. The “hatted” operators are: 3 1 Tˆ = T − b−α2 ∂c−α2 + c−α2 ∂b−α2 , 2 2 Gˆ − = G− , Gˆ + = G+ , Jˆ = J − 3b−α2 c−α2 , ˆ = 3c−α2 . and we find that s00 (Tˆ ) = 23 ∂c−α2 and s00 (J) in terms of the currents j000 and j100 respectively:

(4.16)

The operators s000 and s001 are given

j000 = −c−α2 , j100 = G− c−α2 . We find that the generators of Γ0 = H 0 (Γred ; s000 ) are T2 = Tˆ − 21 ∂ Jˆ and G+ . T2 is already in the cohomology of s00 , and using the tic-tac-toe construction with G+ as the starting point, we find W : 1 ˆ T2 = Tˆ − ∂ J, 2 1 ˆ 0 + 1 + k (J∂ ˆ J) ˆ 0 + (k + 3) (Tˆ J) ˆ0 W = G+ − (JˆJˆJ) 27 6 3 (k + 3)(k + 2) ˆ (k + 3)k + 4 2 ˆ ∂T − ∂ J. − 2 12

(4.17)

ˆ 23 )≥0 , the This gives us a realization of the W 3 algebra in terms of the generators of (W “hatted” generators of W 23 with non-negative grade. Using the primary hamiltonian reduction, we can find expressions for G+ , T , and J in terms of the currents of the affine algebra s`(3)(1) . Inserting these expressions into Eq. (4.17) gives us a realization of W 3 in terms of the currents of the s`(3)(1) . Note, however, that this is not identical to the realization we would get by doing the hamiltonian reduction to W 3 in one step, using the primary hamiltonian reduction. 5. Linearization of W-Algebras Very recently, the construction of linearized W algebras [20] have attracted some attention. The idea in this construction is to add some extra generators to an algebra W, such that the resulting larger algebra is equivalent to a linear algebra. We will show that the secondary quantum hamiltonian reduction gives us a general method to find such linearizations of W algebras. In the specific case of the linearization of W 3 , we find the same result as [20]. The basic idea of our construction is very simple. Define W 0− to be the subalgebra of W 0 with negative grading, i.e. the constrained subalgebra of W 0 , and define W 0≥0 to ˆ 0≥0 to be the algebra be the subalgebra with nonnegative grading. Define furthermore W generated by the “hatted” generators in W ≥0 . We have above shown (Property 1) that we ˆ 0≥0 . can construct a realization of W as differential polynomials in the generators of W Let us denote the number of generators of an algebra5 A by |A|, and define 5

It remains to show that A is indeed an algebra.


529

ˆ 0≥0 | = |W 0− |. n = |W| − |W ˆ 0≥0 to W, in such a way We will show that it is possible to add n of the generators of W ˆ 0≥0 . that there is an invertible transformation between the resulting algebra W ext and W ˆ 0≥0 in the form Let us write W

ˆ 0≥0 = Γ0 ⊕ V. W (V is not uniquely defined.) It follows from the tic-tac-toe construction that the generators of W have the form: W = W 0 + W1 ,

W0 ∈ Γ0 , W1 ∈ V.

It is now clear that if we extend W with a basis of generators in V , then the transformation ˆ 0≥0 ↔ W ext is invertible. We have W ˆ 0≥0 in the form W ˆ 0≥0 = Γ0 ⊕ V , where Γ0 = ker (s00 ). Theorem 5. Write the algebra W Define W ext to be the algebra Wextended with a basis of generators in V . Then there is an invertible mapping ˆ 0≥0 → W ext . φ:W We see that every secondary hamiltonian reduction gives rise to an embedding W ,→ ˆ 0≥0 that will in general be simpler than W ext , where W ext is equivalent to an algebra W W. ˆ 0≥0 is linear, the result of this procedure is a linearization of W. Generically, If W ˆ 0≥0 is not linear, but we find that it is actually linear for a large class of reductions: W Property 2. All algebras of the form W(s`(N ), ⊕ln=1 s`(pn ) ),

p1 > pn + 1,

∀n≥2

can be linearized by the secondary hamiltonian reduction W(s`(N ), s`(2)) → W(s`(N ), ⊕ln=1 s`(pn ) ). The tic-tac-toe construction gives an algorithmic method for the explicit construction of these linearizations. Let us restrict ourselves to showing this in the case of W(s`(n), s`(m)) – the general case is a straightforward generalization. So we consider the secondary reduction W(s`(n), s`(2)) → W(s`(n), s`(m)). The constraints and highest weight gauge corresponding to W(s`(n), s`(2)) are 

∗ 1 0  .  ..

∗ ··· ∗ ··· ∗ ··· .. .

 ∗∗ ∗ ∗ ∗ ∗  .. ..  . .

0 ∗ ··· ∗ ∗

 U T G1 G2 · · · Gn−2 0 0 ··· 0   1 U   0 G¯   1 .  0 G¯   2 2U   . . s`(n − 2) + n−21   .. .. 0 G¯ n−2 

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The G’s are bosonic spin 23 fields. The U (1) operator U commutes with the s`(n−2) Kac¯ negative Moody subalgebra in W(s`(n), s`(2)), while the G’s have positive and the G’s U (1)-charge. This shows that the only possible operator product expansions containing nonlinear terms are Gi (z)G¯ j (w). The constraints corresponding to W(s`(n), s`(m)) are 

∗ 1  0 . . . 0  0  .  .. 0

∗ ∗ 1 .. . 0 0 .. . 0

 ∗ ∗  ∗ ..   . . ∗   ∗ ..  . ··· 0 ∗ ··· ∗

··· ∗ ∗ ··· ··· ∗ ∗ ··· ··· ∗ ∗ ··· . .. . .. ··· 1 ∗ ··· ··· 0 ∗ ··· .. .. . .

We find that the secondary reduction is made by constraining G¯ 1 = 1, G¯ 2 = 0, . . . , G¯ n−2 = 0, in general in addition to constraining also a number of the Kac-Moody currents. Since ˆ ≥0 is linear. all the fields G¯ i are constrained, it follows that W For the so(n) algebras, due to the few cases that allow the secondary reductions (in our framework), it is clear that we will not be able to linearize most of the corresponding W-algebras. In fact, demanding that the starting W-algebra is built on H = s`(2) and reasoning as above, it is easy to see that only the algebras6 WBC2 and WD3 can be linearized (the last one being in fact identical with WA3 ). For sp(2n) algebras, the complete classification of linearizable W(sp(2n), H) algebras is quite heavy and beyond the scope of the present article: we refer to [28] for an exhaustive classification. Let us just remark that the secondary reduction W(sp(2n), sp(2)) → W(sp(2n), H) with H simple always provide a linearization of the W(sp(2n), H) algebra. Let us remark that the spin of the new fields we add to linearize the algebra are always positive, since we take the positive grade part of a given W-algebra7 . The most popular W-algebras are the W(G, G) ≡ WG ones: it is natural to see whether one can linearize these algebras. From the above property, it is easy to deduce: Property 3. The W-algebras WAn and WCn can be linearized by the secondary reductions through the schemes: W (s`(n + 1), s`(2)) → WAn , W (sp(2n), sp(2)) → WCn . For the WBCn , and WDn algebras, our techniques allow to linearize only the WBC2 and WD3 algebras through W (so(5), so(3)) → WBC2 , W (so(6), so(3)) → WD3 . 6 We denote by WBC the algebra W(B , B ) obtained from Hamiltonian reduction of B to distinguish n n n n field. WBCn as the same spin contents as WCn it from the Casimir algebra WBn that contains a spin n+1 2 but different structure constants. 7 Actually the spin is at least 1.


531

The above method of linearizing W algebras is not limited to the secondary quantum hamiltonian reduction – it can also be used in the primary quantum hamiltonian reduction. Following the above procedure in that case, we find any W algebra W(G, H) (1) (1) can be extended by adding the generators in Gˆ ≥0 which are not highest weight, Gˆ ≥0 are the hatted affine currents with non-negative grade: (1) (1) W ext = W + Gˆ ≥0 \ Gˆ hw . (1) The algebra W(G, H)ext is then equivalent to Gˆ ≥0 . As explicit examples of the linearization using secondary hamiltonian reduction, we 2 c )≥0 = will give the two simplest: the linearization of W 3 = W(s`(3), s`(3)) using (W 3 c W(s`(3), s`(2))≥0 (this linearization was already given in [20]), and the linearization of c s`(2))≥0 . W 4 = W(s`(4), s`(4)) using (W(s`(3), Note also that the secondary reduction W(so(5), so(3)) → W(so(5)) will provide the linearization of the W2,4 algebra [20]. For the linearization of the WB2 algebra (containing a spin 25 field), a secondary reduction of super algebras will have to be performed [28].

5.1. Linearization of W 3 . As we already saw in example in Sect. 4, W 3 can be realized ˆ in terms of the generators Tˆ , Gˆ + , and J: 1 ˆ T = Tˆ − ∂ J, 2 1 + ˆ 0 + 1 + k (J∂ ˆ J) ˆ 0 + (k + 3) (Tˆ J) ˆ0 W = Gˆ − (JˆJˆJ) 27 6 3 (k + 3)(k + 2) ˆ (k + 3)k + 4 2 ˆ ∂T − ∂ J. − 2 12

(5.1)

(5.2)

If we add the current J = Jˆ to the W 3 algebra, then it is clear that the transformation ˆ is invertible. The new operator product expansions of the {T, W, J} ↔ {Tˆ , Gˆ + , J} extended W 3 algebra are: 18 + 6k + ···, (z − w)2 12 + 6k J ∂J T (z)J(w) = + ···, + + 3 2 (z − w) (z − w) z−w (k 2 + 5k + 6)J J(z)W (w) = (z − w)3 (2k 2 + 12k + 18)T − 13 (k + 3)(JJ)0 + 21 (3k 2 + 15k + 18)∂J + (z − w)2 3 2 3W + 2 (k + 5k + 6)∂T + 21 (2k 2 + 9k + 11)∂ 2 J − (k + 3)(T J)0 + z−w 1 (JJJ) − (k + 2)(J∂J) 0 0 +9 + ···, (5.3) z−w J(z)J(w) =

while the (equivalent) nontrivial operator product expansions of the linear algebra generated by Tˆ , Gˆ + , and Jˆ are

532

J.O. Madsen, E. Ragoucy 3k2 +11k+18 k+3 (z − w)4

2Tˆ ∂ Tˆ + ···, + 2 (z − w) z−w 3 ˆ+ G ∂ Gˆ + Tˆ (z)Gˆ + (w) = 2 + ···, + (z − w)2 z − w Jˆ −6 ∂ Jˆ ˆ Tˆ (z)J(w) = + ···, + + (z − w)3 (z − w)2 z − w ˆ+ ˆ Gˆ + (w) = 3G + · · · , J(z) z−w Tˆ (z)Tˆ (w) = −

+

(5.4)

ˆ J(w) ˆ and of course J(z) = J(z)J(w). 5.2. Linearization of W 4 . In order to show an example where the linearization has not been done before, we take the linearization of the W 4 -algebra. In this case, the algebra W(s`(4), s`(2)) contains T , a U (1) subalgebra generated by U , an affine s`(2) algebra generated by K 0 and K ± , and 4 spin 23 fields Gσ , = ±, σ = ±. Gσ has U (1)-charge 1 and the eigenvalue under K 0 is σ 21 . ˆ In the secondary reduction we constrain G−± and K − , so the algebra W(s`(4), ˆ 0, K ˆ + , and Uˆ . Gˆ +± are primary Virasoro and Kacs`(2))≥0 is generated by Tˆ , Gˆ +± , K ˆ 0. K ˆ + is a primary Moody fields, with spin 23 , U (1)-charge 1, and eigenvalue ± 21 under K ˆ 0 (and U (1)-charge 0). The central charge is spin 1 field with eigenvalue 1 under K 3(2k2 +11k+32) cˆ = − , and the rest of the nontrivial operator product expansions are: k+4 −4 Uˆ ∂ Uˆ + ···, + + 3 2 (z − w) (z − w) z−w ˆ0 ˆ0 −1 K ∂K ˆ 0 (w) = Tˆ (z)K + ···, + + 3 2 (z − w) (z − w) z−w k+4 Uˆ (z)Uˆ (w) = + ···, (z − w)2 Tˆ (z)Uˆ (w) =

k+4 2

+ ···, (z − w)2 ˆ ++ ˆ + (z)Gˆ +− (w) = −G + · · · . K z−w ˆ 0 (w) = ˆ 0 (z)K K

(5.5)

The tic-tac-toe construction gives us the expressions for the generators of W 4 : ˆ 0 − 2∂ Uˆ , T = Tˆ − ∂ K ˆ + Uˆ )0 + (2k + 6)∂ K ˆ + + (4 + k)(Tˆ Uˆ )0 − 1 (Uˆ Uˆ Uˆ )0 W3 = Gˆ +− − 2(K 2 ˆ 0 )0 + (k + 1)(Uˆ ∂ Uˆ 0 )0 + (k + 2)(Uˆ ∂ K ˆ 0K ˆ 0 )0 + 4(k + 3)(K ˆ 0∂K ˆ 0 )0 −2(Uˆ K (3k + 8) 2 ˆ ˆ 0 − (k + 3)(k + 4))∂ Tˆ , ∂ U − (k + 2)(k + 3)∂ 2 K + 2 ˆ + )0 + 1 (Gˆ +− Uˆ )0 + (Gˆ +− K ˆ 0 )0 − (k + 4)(Tˆ K ˆ +K ˆ + )0 W4 = Gˆ ++ + (K 2 1 ˆ 0K ˆ + )0 − (k + 3)(Uˆ ∂ K ˆ + )0 + 2(K ˆ 0K ˆ + )0 + (Uˆ Uˆ K 2


533

2 ˆ + )0 + (22 + 13k + 2k ) ∂ 2 K ˆ+ ˆ + )0 − 2(K ˆ 0∂K ˆ + )0 − k(∂ K ˆ 0K −(k + 1)(∂ Uˆ K 2 (4 + k)2 (952 + 643k + 108k 2 ) ˆ ˆ k+4 ˆ ˆ ˆ ˆ 0 )0 ˆ 0K (T T )0 + (T U U )0 − (k + 4)(Tˆ K + 4(2552 + 1763k + 300k 2 ) 4 3 1 ˆ 0 )0 + ( K ˆ 0K ˆ 0K ˆ 0 )0 + k + 1 (Uˆ Uˆ ∂ Uˆ )0 ˆ 0K ˆ 0K − (Uˆ Uˆ Uˆ Uˆ )0 + (Uˆ Uˆ K 16 2 4 (4 + k)(40 + 793k + 513k 2 + 84k 3 ˆ ˆ 3k + 10 ˆ ˆ ˆ 0 (T ∂ U )0 − ( U U ∂ K )0 + 2(2552 + 1763k + 300k 2 ) 4 (4 + k)(11504 + 12158k + 4251k 2 + 492k 3 ) ˆ ˆ 0 ˆ 0∂K ˆ 0 )0 (T ∂ K )0 − 2(3 + k)(Uˆ K + 2(2552 + 1763k + 300k 2 ) 2 ˆ 0∂K ˆ 0K ˆ 0 )0 − 3(4 + k)(8 + 3k)(13 + 4k)(184 + 121k + 20k ) ∂ 2 Tˆ −(k + 2)(K 2 8(2552 + 1763k + 300k ) 7336 + 9073k + 4814k 2 + 1265k 3 + 132k 4 ˆ ˆ ˆ 0 )0 + (∂ U ∂ U )0 + (k 2 + 6k + 9)(Uˆ ∂ 2 K 4(2552 + 1763k + 300k 2 ) 15152 + 27782k + 18163k 2 + 5077k 3 + 516k 4 ˆ ˆ 0 (∂ U ∂ K )0 + 2(2552 + 1763k + 300k 2 ) 168352 + 235972k + 123812k 2 + 28811k 3 + 2508k 4 ˆ 0 ˆ 0 (∂ K ∂ K )0 + 4(2552 + 1763k + 300k 2 ) ˆ 0 )0 + (2k 2 + 13k + 22)(K ˆ 0 )0 − k + 4 (Uˆ ∂ 2 Uˆ )0 ˆ 0K ˆ 0∂2K −(k + 1)(∂ Uˆ K 4 244688 + 354290k + 201124k 2 + 55477k 3 + 7326k 4 + 360k 5 3 ˆ ∂ U + 12(2552 + 1763k + 300k 2 ) 1179328 + 1976920k + 1325876k 2 + 445043k 3 + 74808k 4 + 5040k 5 3 ˆ 0 − ∂ K 24(2552 + 1763k + 300k 2 ) k+3 ∂W3 . − (5.6) 2

ˆ +, K ˆ 0 , and Uˆ to the W 4 We define the algebra (W 4 )ext by adding the generators K algebra. It is obvious that there is an invertible transformation between this extended ˆ +, K ˆ 0 , and Uˆ . algebra, and the linear algebra generated by Tˆ , Gˆ ++ Gˆ +− K 6. Conclusion In this paper, we have considered secondary quantum hamiltonian reductions, i.e. hamiltonian reductions that can be described by the diagram: G (1) W(G, H0 ) @ @ R ? @ W(G,H)

534


or in words: starting with a Lie algebra G, and two regular subalgebras H0 and H with H0 ⊂ H satisfying certain conditions as described in Appendix B, we carry out the hamiltonian reduction of the W algebra W(G, H0 ) with suitable constraints, and show that the result is the W algebra W(G,H). Note that for G=s`(N ), the conditions that we impose on H0 and H in order to perform the secondary quantum hamiltonian reduction are more restrictive than the conditions necessary for the classical secondary hamiltonian reduction, see [19]. This should not be taken as a sign that not all classical secondary hamiltonian reductions can be quantized; it simply reflects the fact that the method that we have used for the quantum secondary reduction in this paper cannot be applied to all possible secondary reductions. On the other hand, we have been able to explicitly do some quantum reductions when G = so(N ) or G = sp(2N ), while the techniques have not been developed for the classical case. The quantum secondary reductions show that the W algebras W(G,H) that can be obtained by the hamiltonian reduction of a certain affine Lie algebra G (1) are not only related by their common “ancestor” G (1) , but that they are mutually directly connected by the hamiltonian reduction. As a simple example, consider this diagram of the possible hamiltonian reductions connecting the algebras W(s`(4), H); the simple lines symbolize the quantum reductions we have been able to perform, the double lines symbolize secondary reductions that give rise to linearizations, and the dashed line the classical secondary reduction that is not quantized by ou r method: s`(4)(1) AHH A HH j H A - W(s`(4), 2 s`(2)) W(s`(4), s`(2)) A Q Q Q Q Q A Q A Q A Q A Q QQ AU ? ? s - W4 W(s`(4), s`(3)) There are two important consequences that follow from the secondary quantum hamiltonian reduction. One of these is the secondary quantum Miura transformation. The usual quantum Miura transformation can be used to find free field realizations of the W algebras, and in a similar way the seondary quantum Miura transformation can be used to find realizations of W algebras in terms of subalgebras of other W algebras. For example, in the diagram above there are 4 (5 if the dashed line is included) possible secondary reductions, and the secondary quantum Miura transformation corresponding to these gives us realizations of W 4 in terms of W(s`(4), s`(2)) or W(s`(4), s`(3)), and of W(s`(4), s`(3)) and W(s`(4), 2 s`(2)) in terms of W(s`(4), s`(2)) (and W 4 in terms of W(s`(4), 2 s`(2)) if the dashed line is included). The other consequence that follows from the seondary quantum hamiltonian reduction is the linearization of W algebras. For a large class of algebras W(G, ⊕ln=1 Hn ), where the possible Hn ’s are given in Sect. 5, we can find a secondary hamiltonian reduction and a corresponding extended algebra W(G, ⊕`n=1 Hn )ext which is equivalent to a linear algebra with new generators of positive spin. In particular, we are able to linearize the WAn , WCn and WBC2 algebras. To take once again the diagram above as example, this procedure can give us linearizations of W 4 and W(s`(4), s`(3)). This linearization of W algebras could be very useful in the study of the representation theory of W algebras. In fact one could use the linearization to reduce the representation theory


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of the non-linear W algebras to the representation theory of the corresponding linear algebras. Note that, as mentioned above, we have not in this paper exhausted the possible secondary reductions; a number of classical secondary reductions cannot be quantized using the present methods, and it would be of interest to find a method to quantize these remaining secondary reductions. Besides these problems, there are other open questions about the secondary quantum hamiltonian reduction. It would be interesting to generalize the procedure to supersymmetric W algebras, and to do the secondary quantum Miura transformation and the linearization also in that case [28]. Another interesting possibility is to study in more detail the linearization of W algebras, and to what extent we can actually reduce the analysis of the non-linear W algebras to the analysis of the matching linear algebra. Acknowledgement. The authors would like to thank R. Stora for stimulating discussions. One of the authors (JOM) would like to thank the Niels Bohr Institute, where this work was started, for financial support.

A. Spectral Sequences In this appendix, we will give a few key definitions that are used in the theory of spectral sequences. For a good introduction to the theory of spectral sequences see Pe.g. [25]. We assume that we have a complex (Ω, s), i.e. a graded space Ω = n Ω n and a nilpotent derivation s : Ω n → Ω n+1 . We assume furthermore that it is possible to define a filtration on the space, i.e. a sequence of subspaces F q Ω such that {0} ⊂ · · · ⊂ F q+1 Ω ⊂ F q Ω ⊂ F q−1 Ω ⊂ · · · ⊂ Ω. We define a sequence of “generalized co-cycles" Zrp,q by Zrp,q = F q Ω p+q ∩ s−1 (F q+r Ω p+q+1 ) = {x ∈ F q Ω p+q |s(x) ∈ F q+r Ω p+q+1 }.

(A.1)

We note that it is natural to define p,q Z∞ = F q Ω p+q ∩ ker s.

We also define a sequence of “generalized co-boundaries" Brp,q by Brp,q = F q Ω p+q ∩ s(F q−r Ω p+q−1 ), p,q = F q Ω p+q ∩ im s, B∞

(A.2)

and we see that (suppressing the (p, q) indices) · · · ⊂ Br ⊂ Br+1 ⊂ · · · ⊂ B∞ ⊂ Z∞ ⊂ · · · ⊂ Zr+1 ⊂ Zr ⊂ · · · . From these generalized cocycles and coboundaries, we can now define a sequence of “generalized cohomologies" Erp,q by E0p,q = F q Ω p+q /F q+1 Ω p+q , p−1,q+1 p,q Erp,q = Zrp,q / Zr−1 , + Br−1 p,q p,q p−1,q+1 p,q = Z∞ / Z∞ + B∞ . E∞

(A.3)

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It is now possible to show that for every space Er , we can define a nilpotent derivative sr . sr is defined by the commutative diagram: Zrp,q

s −→

Zrp+1−r,q+r

η↓ Erp,q

↓η

,

(A.4)

−→ Erp+1−r,q+r sr

where η is the canonical projection operator from Zr onto Er . In other words, for [x] ∈ Erp,q , sr ([x]) = [s(x)]. With all these definitions, we are now finally in a position to state the main theorems of the theory of spectral sequences: The “generalized cohomologies” Er that we have introduced are in fact cohomologies, namely p,q ∼ Er+1 = H p,q (Er ; sr ).

(A.5)

If the filtration exhausts all of the space Ω, and if the generalized co-cycles Zrp,q conp,q p,q , i.e. if Ω = ∪n F n Ω and Z∞ = ∩r Zrp,q , then verges to Z∞ p,q ∼ E∞ = F q H p+q /F q+1 H p+q ,

(A.6)

where F q H is the filtration on the cohomology H(Ω; s) induced by the filtration on Ω: F q H p+q = H p+q (F q Ω; s). This is the principal result of the theory of spectral sequences. It gives us a way to find the cohomology H(Ω; s), supposing that we are able to use the knowledge of the spaces F q H p+q /F q+1 H p+q to reconstruct H(Ω; s). The usefulness of the spectral sequences rests on the fact that in practical application the spectral sequence often collapses after a few steps, i.e. sr is identically zero for r > r0 where r0 is some low number. Let us show the following Lemma 2. If F q Ω = 0 for q > 0, then H(Ω; s) ∼ = E∞ . Namely F q Ω = 0 for q > 0 implies that F q H p+q = 0 for q > 0. This means that p,0 ∼ E∞ = F 0 H p /F 1 H p ∼ = F 0H p, E p+1,−1 ∼ = F −1 H p /F 0 H p , ∞

(A.7) (A.8)

etc..., and we can use this to show that p+r,−r p,0 ⊕ · · · ⊕ E∞ F −r H p ∼ = E∞ p,q or equivalently (since E∞ = 0 for q > 0)

H p (Ω; s) ∼ =

X

p+r,−r E∞ ,

r∈ZZ

which proves Lemma 2. p,q ∼ Let us mention here that the isomorphism E∞ = F q H p+q /F q+1 H p+q is in general a vector space isomorphism. If the space Ω in addition to being a vector


537

space is also an algebra (as in the case that we are interested in here) then if we can define algebras on all the cohomologies in the spectral sequence such that the algebra on E0p,q = F q Ω p+q /F q+1 Ω p+q is induced by the algebra on F q Ω p+q , i.e. [a], [b] ∈ E0 : [a] ◦ [b] = [a ◦ b] (where ◦ denotes the algebra composition), and the algebra on Er+1 = H(Er , sr ) is induced by the algebra on Er , then the isop,q ∼ morphism E∞ = F q H p+q /F q+1 H p+q is an algebra isomorphism. However, even if p,q ∼ E∞ = F q H p+q /F q+1 H p+q is an algebra isomorphism, it may be nontrivial to reconstruct the algebra of H n (Ω; s). In the case where the complex (Ω; s) can be given the structure of a double complex , s00 ) with two anti-commuting nilpotent operators s0 and s00 , and with a structure (Ω; s0P bigrading Ω = p,q Ω p,q , the spectral sequence simplifies somewhat. Define s = s0 +s00 . The filtration is defined in terms of the bi-grading as M Ω i,j , F qΩ = i∈ZZ,j≥q q

F Ω

p+q

=

M

Ω p−i,q+i .

(A.9)

i≥0

The first element in the spectral sequence, E0 , is defined by E0p,q = F q Ω p+q /F q+1 Ω p+q ∼ = Ω p,q ,

(A.10)

and the nilpotent bigrade (1, 0)-operator s0 on E0 is defined by the commutative diagram F q Ω p+q

s −→ F q Ω p+q+1

η↓ E0p,q

↓η −→ s0

,

(A.11)

E0p+1,q

where η is the canonical projection operator η F q Ω p+q → F q Ω p+q /F q+1 Ω p+q ∼ = Ω p,q . This means that if we identify x ∈ E0p,q with x ∈ F q Ω p+q , then s0 (x) = η(s(x)) – and since η here is the projection operator on Ω p+1,q−1 , s0 is simply the bigrade (1, 0) part of s: s0 = s0 . The second element in the spectral sequence is E1p,q = Z1p,q /(Z0p−1,q+1 + B0p,q ) Ω p,q ∩ s−1 (Ω p,q+1 ) ∼ . = Ω p,q ∩ s(Ω p−1,q )

(A.12)

Note that Ω p,q ∩ s−1 (Ω p,q+1 ) = Ω p,q ∩ ker s0 and Ω p,q ∩ s(Ω p−1,q ) = Ω p,q ∩ im s0 ; so, in agreement with Eq. (A.5), we can also write E1 as E1p,q = H p,q (E0 ; s0 ).

(A.13)

538


The operator s1 on E1 is again defined by the commutative diagram Z1p,q

s −→

η↓ E1p,q

Z1p,q+1 ↓η

(A.14)

−→ E1p,q+1 s1

and η is again the canonical projection operator. Consider x ∈ ker s0 and let [x] = η(x) be the corresponding equivalence class in E1 . Then s1 ([x]) = η(s(x)) is just the bigrade (0, 1)-part of s(x), projected on E1 , i.e.: s1 ([x]) = [s0 (x)]. B. Shift of the Constraints Using a U (1) Generator We are looking for couples of W(G, H) algebras such that the sets of first class constraints are embedded one into the other. We first consider the case G = s`(N ), and to clarify the presentation, we focus on the secondary reductions of type W(G, H) → W(G, G) ≡ W(G). In s`(N ), the regular subalgebras H can always be chosen in such a way that the simple roots of H are also simple roots of G. Let H = ⊕`n=1 Hn , where Hn are simple subalgebras of rank rn = rank(Hn ), ordered in such a way that rn ≤ rm if n > m. We define as simple roots Simple roots of G α1 , ..., αr1 ; αρ1 ; αρ1 +1 , ..., αρ1 +r2 ; αρ2 ; αρ2 +1 , ..., αρ2 +r3 ; αρ3 ; ... ...; αρ`−1 ; αρ`−1 +1 , ..., αρ`−1 +r` ; αρ` +1 , ..., αN −1 , ; αρ1 +1 , ..., αρ1 +r2 ; ; αρ2 +1 , ..., αρ2 +r3 ; ; ... Simple roots of H α1 , ..., αr1 ; ...; P ; αρ`−1 +1 , ..., αρ`−1 +r` n with ρn = i=1 (ri + 1). (B.1) In the fundamental representation of s`(N ), this simply means that we have divided the N × N matrix into rj × rj blocks of decreasing size, plus (when it exists) a block (N − ρ` ) × (N − ρ` ). The gradation associated to the Cartan generator of the principal s`(2) in H attributes a grade 1 to each simple root of H, but the grade of the simple roots of type αρn is gr(αρn ) = −

rn + rn+1 < 0, 2

(B.2)

where we have set r`+1 = 0. This implies that the root generators Eρn are constrained in W(G, H) although they are not in W(G). Thus, it is clear that we have to introduce a new gradation such that (B.3) gr(αρn )0 ≥ 0 while not changing the resulting W-algebra. Let H be the gradation we are looking for. Then, if M0 is the Cartan generator of the s`(2) embedding we are considering (it has not been changed because we want the W-algebra to be the same), the new gradation is characterized by the generator U = H − M0 which commutes with the s`(2) algebra and


539

which ”respects" the highest weight gauge. Thus, classifying the different gradations H is the same as classifying the different U (1) generators submitted to the non-degeneracy condition (B.4) ker ad(M+ ) ∩ G 0− = 0, where G 0− denotes the subalgebra of G-generators which have negative grade w.r.t. H. This technique has been developed in [11, 22], where all the possible gradations leading to the same W-algebra have been classified. The procedure goes along the following lines. We start with the decomposition of the fundamental of s`(N ) w.r.t. the principal s`(2) in H: with jµ 6= jν when µ 6= ν (B.5) N = ⊕Iµ=1 nµ Djµ and add the following U (1) eigenvalues N = ⊕Iµ=1 nµ Djµ (yµ ).

(B.6)

Then, computing the adjoint representation from this decomposition of the fundamental, G = ⊕k Dk (Yk ),

(B.7)

where the Yk ’s are differences of two yµ ’s. The eigenvalues of the allowed U (1) generators will be characterized by the equations |Yk | ≤ k

and

1 Yk ∈ ZZ , ∀ Dk (Yk ). 2

(B.8)

Then, the different gradations will be M0 + U , with M0 the Cartan generator of the s`(2) under consideration, and U one of the allowed U (1) generators. Now, to get a gradation satisfying both Eqs. (B.3) and (B.8), we have to impose |yµ − yν | ≤ |jµ − jν |

and

yµ − y ν ≥ j µ + j ν

(B.9)

which is clearly satisfied only if one of the two j’s is zero, ie if H is simple8 . In that case, the s`(2) ⊕ U (1) decomposition j(N − 2j − 1) N

j(2j + 1) N (B.10) indeed gives a gradation where all the simple roots of s`(N ) have positive grades, and whose associated W-algebra is W(s`(N ), H). For the general secondary reduction W(s`(N ), H0 ) → W(s`(N ), H), the reasoning follows along the same lines. We however have to look at the grade of all the roots (since some simple roots have negative grades in the general case). Then, one asks the gradations to satisfy G − ⊂ G 0− . This necessary condition is sufficent in the case of s`(N ) because the simple roots of H can always be chosen among the simple roots of H0 (and thus the constraints J αi = 1 for H are a subset of the constraints J αi = 1 for H0 ). After a tedious calculation, and using non-degenerated U (1) generators both for H and H0 , one gets the following property: Dj (y) ⊕ (N − 2j − 1)D0 (z)

8

with

y=

and

z=−

If H is simple, there will be only one Dj representation with j 6= 0 in the fundamental of G.

540


Property 4. In the case of secondary reductions of type W(s`(N ), H0 ) → W(s`(N ), H), we have the following necessary and sufficient condition for the existence of a U (1) generator which satisfies both the non-degeneracy (B.4) and the embedding of the set of constraints associated to H into the set of constraints associated to H0 : If H0 decomposes as H0 = ⊕lα=1 mα s`(pα ) (B.11) the U (1) generator exists iff H decomposes as: H=

⊕lα=1 mα

s`(qα )

with

2 ≤ pα ≤ qα ∀α . |pα − pβ | ≤ |qα − qβ | ∀α, β

(B.12)

Now, turning to the case of orthogonal and symplectic algebras, we can do the same calculation. However, for these algebras, the U (1) generator is much more constrained (see [11], Sects. 5.2 and 5.3) so that there are less U (1) generators satisfying both the non-degeneracy and the embedding conditions. Note that one has really to check in each case that the sets of currents constrained to 1 are also embedded one into the other, since the simple roots of H0 are not always simple roots of H. Apart from these restrictions, the calculation is the same as for s`(N ) algebras, so that one is led to Property 5. In the case of secondary reductions of type W(G, H0 ) → W(G, H) with G = so(N ) or sp(N ), we have the following necessary and sufficient condition for the existence of a U (1) generator which satisfies both the non-degeneracy (B.4) and the embedding of the sets of constraints. − For so(N ), H and H0 must be of the form

H0 = (n + 1) so(p) H = n so(p) ⊕ so(p + 2)

with

N = (n + 1)p + 2 ; n ≥ 0 N ≡ p [mod 2]

− For sp(N ), H and H0 must be of the form

H0 = s`(2) ⊕µ s`(2pµ ) H = sp(4) ⊕µ s`(2pµ )

with

pµ ∈ N

or of the form

H0 = s`(2) H = ⊕j sp(2qj ) ⊕µ s`(pµ )

  either p1 ∈ 2 N , p1 ≥ pµ + 1 ∀µ ≥ 2 and p1 ≥ 2qj + 1 ∀j with or p1 ∈ (2 N + 1), p1 ≥ pµ + 2 ∀µ ≥ 2 and p1 ≥ 2qj + 1 ∀j .  or q ≥ q + 1 ∀j ≥ 2 and q ≥ 1 (p + 1) ∀µ 1 j 1 2 µ Let us remark that in the case G = so(5), the U (1) generator exists when considering the reduction W(so(5), so(3)) → W(so(5)), while in the case G = sp(4) the U (1) generator exists for the reduction W(sp(4), s`(2)) → W(sp(4)) which is in agreement with the isomorphism between the so(5) and sp(4) algebras.


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References 1. Bouwknegt, P. and Schoutens, K.: Phys. Rep. 223, 183 (1993) 2. Zamolodchikov, A.B.: Theor. Math. Phys. 65, 1205 (1985) 3. Burroughs, N., de Groot, M., Hollowood, T. and Miramontes, L.: Phys. Lett. B277 89 (1992); Spence, B.: Phys. Lett. B276, 311 (1992); Feher, L.: Generalized Drinfeld-Sokolov Hierarchies and W-Algebras. Proceedings, Quantum Groups, Integrable Models and Statistical Systems, Kingston, 1992 p. 71 4. Lu, H., Pope, C.N., Schrans, S. and Wang, X.J.: Mod. Phys. Lett. A7; 1835 (1992); Pope, C.N.: Review of W Strings, Proc. of Int. Symp. on Black Holes, Worm Holes, Membranes and Superstrings, Woodlands TX, 1992; Lu, H., Pope, C.N., Thielemans, K., Wang, X.J. and Xu, K.W.: Quantizing Higher Spin String Theories. Preprint CTP-TAMU-24-94, hep-th/9410005 5. Aratyn, H.,Constantinidis, C.P., Ferreira, L.A., Gomes, J.F. and Zimerman, A.H.: Phys. Lett. B281, 245 (1992); Feher, L., O’Raifeartaigh, L., Ruelle, P., Tsutsui, I. and Wipf, A.: Ann. Phys. 213, 1 (1992); Delduc, F., Ragoucy, E. and Sorba, P.: Commun. Math. Phys. 146, 403, (1992); Gervais, J.-L., O’Raifeartaigh, L., Razumov, A.V. and Saveliev, M.V.: Phys. Lett. B301, 41 (1993) 6. Azuma, H.: Progr. Theor. Phys. 92, 293 (1994); Karabali, D.: Nucl. Phys. B428, 531 (1994) 7. Kausch, H.G. and Watts, G.M.T.: Nucl. Phys. B354, 740 (1991); Kausch H.G.: Phys. Lett. B259, 448 (1991) 8. Bais, F.A., Tjin, T. and van Driel, P.: Nucl. Phys. BB357, 632 (1991) 9. Fehér, L., O’Raifeartaigh, L., Ruelle, P. and Tsutsui, I.: Phys. Rep. 222, 1 (1992) and references therein 10. Bershadsky, M. and Ooguri, H.: Commun. Math. Phys. 126, 49 (1989) 11. Frappat, L., Ragoucy, E. and Sorba, P.: Commun. Math. Phys. 157, 499 (1993) 12. Kostant, B. and Sternberg, S.: Ann. Phys. 176, 49 (1987) 13. Feigin, B.L. and Frenkel, E.: Phys. Lett. B246, 75 (1990) 14. Bouwknegt, P., McCarthy, J. and Pilch, K.: In: Proceedings Strings and Symmetries, N. Berkovitz et al. (eds.), Singapore: World Scientific, 1991 15. Frenkel, E., Kac, V. and Wakimoto, M.: Commun. Math. Phys. 147, 295 (1992) 16. de Boer, J. and Tjin, T.: Commun. Math. Phys. 160,317 (1994); Commun. Math. Phys. 158, 485 (1993) 17. Sevrin, A., Thielemans, K. and Troost, W.: Nucl. Phys. B407, 459 (1993); Sevrin, A. and Troost, W.: Phys. Lett. B315, 304 (1993) 18. Madsen, J.O. and Ragoucy, E.: Nucl. Phys. B429, 277 (1994) 19. Delduc, F., Frappat, L., Ragoucy, E. and Sorba, P.: Phys. Lett. B335, 151 (1994) 20. Krivonos, S. and Sorin, A.: Phys. Lett. B335, 45 (1994); Belluci, S., Krivonos, S. and Sorin, A.: Phys. Lett. B347, 260 (1995) 21. Thielemans, K.: Int. Journ. Mod. Phys. C2, 787 (1991) 22. Delduc, F., Ragoucy, E. and Sorba, P.: Phys. Lett. B279, 319 (1992) 23. Wakimoto, M.: Commun. Math. Phys. 104, 605 (1986); Bernard, D. and Felder, G.: Commun. Math. Phys. 127, 145(1990); Feigin, B. and Frenkel, E.: In “Physics and Mathematics of Strings", eds. L. Brink et al., Singapore: World Scientific, 1990 24. Frappat, L., Ragoucy, E. and Sorba, P.: Nucl. Phys. B404, 805 (1993) 25. McCleary, J.: User’s Guide To Spectral Sequences, Wilmington;DE: Publish or Perish, 1985 26. Das, S.R., Dhar, A. and Rama, S.K.: Int. J. Mod. Phys. A7, 2295 (1992) 27. Deckmyn, A.: Phys. Lett. B298, 318 (1993) 28. Madsen, J.O. and Ragoucy, E.: Linearization of W algebras and W superalgebras. Preprint ENS APP-A-520/95, hep-th/9510061, to be published in Proceedings SIMI 96

L

Communicated by R.H. Dijkgraaf

Commun. Math Phys. 185, 543 – 619 (1997)

Communications in


Large N 2D Yang-Mills Theory and Topological String Theory Stefan Cordes? , Gregory Moore?? , Sanjaye Ramgoolam??? Dept. of Physics, Yale University, New Haven, CT 06511, USA. E-mail: [email protected], [email protected], [email protected] Received: 3 March 1994 / Accepted: 2 February 1995

Abstract: We describe a topological string theory which reproduces many aspects of the 1/N expansion of SU (N ) Yang-Mills theory in two spacetime dimensions in the zero coupling (A = 0) limit. The string theory is a modified version of topological gravity coupled to a topological sigma model with spacetime as target. The derivation of the string theory relies on a new interpretation of Gross and Taylor’s “−1 points”. We describe how inclusion of the area, coupling of chiral sectors, and Wilson loop expectation values can be incorporated in the topological string approach.

1. Introduction The possibility that the strong interactions might be described by a theory of strings has been an enduring source of fascination and frustration to particle theorists for the past twenty-five years [3–9]. In the early 80’s some interesting progress on this question was made in the case of large N Yang-Mills theory in two dimensions (Y M2 ) [10, 11]. Recently this work has been revived, considerably extended, and deepened. Exact results are now available for partition functions Z(G, ΣT ) and Wilson loop averages for a compact gauge group G on two-dimensional spacetimes ΣT of arbitrary topology [12–14]. 1 Building on the results [10, 12–14] D. Gross and W. Taylor returned to the problem of strings and Y M2 in a beautiful series of papers [15] (see also [16]). In particular, [15] derives the N → ∞ asymptotic expansion for the partition function Z(A, G, N ). ?

Current address: World Financial Center, North Tower, New York, NY 10281-1316 USA Currently visiting the Rutgers University, Dept. of Physics ??? Current address: Dept. of Physics, Judwin Hall, Princeton University, Princeton, NT 08544 USA 1 Y M has area-preserving diffeomorphism symmetry so Z only depends on the gauge group, topology 2 and total area, of ΣT . For gauge group G = SU (N ) and ΣT of genus G we denote the partition function by Z(A, G, N ). ??

544

S. Cordes, G. Moore, S. Ramgoolam

Moreover many aspects of the expansion in 1/N have a natural geometrical explanation in terms of weighted sums over maps from a worldsheet ΣW to the spacetime ΣT . In some cases (e.g. when ΣT is a torus) Gross and Taylor were able to write the weighted sum explicitly as a sum over covering maps with weights given by symmetry factors for the cover. Given the results of [15] no one could seriously doubt that Y M2 is equivalent to a string theory. Nevertheless, [15] left untied some loose ends, such as the following two problems: 1) The problem of the true meaning of the “−1 points”. 2) The problem of finding the appropriate string action. With regard to problem (1), the geometrical interpretation of the 1/N expansion necessitated the introduction of |2 − 2G| “twist points”, (“ points”, or “−1 points”). In contrast to the clear and natural geometrical interpretation of all the other aspects of the series, the nature of the −1 points was fraught with mystery. Related difficulties had already presented themselves ten years earlier in the work of Kazakov and Kostov [11]. Several authors have emphasized the importance of a proper understanding of the ±1 points. As for problem (2), one of the key motivations for Gross and Taylor’s work was that the action for a string interpretation of Y M2 might have a natural generalization to four-dimensional targets, or might suggest essential features of a string interpretation of Y M4 . However, difficulties associated with problem (1) presented a serious obstacle to finding the action for Y M2 . Indeed, after the appearance of the first papers of [15] it was quickly noted in [17, 18] that, for the case of the partition function of a toroidal target, (where there are no −1 points) the interpretation in terms of covering maps naturally suggests that the string action principle for Y M2 will involve a topological sigma model, with ΣT as target, coupled to topological gravity. However, even in the genus one case, the evaluation of Wilson loops necessitates consideration of −1 -points. No theory of Y M2 can go very far without an understanding of these factors. In the present paper we will solve problem (1). The solution of this problem allows us to make some definite progress on problem (2). The solution to problem (1) is simple: there is no such thing as an “−1 point”! We have not completely solved problem (2) in the sense that we have not reproduced all known results on Y M2 from the string approach. Nevertheless, we have reproduced enough to say that (a) a description in terms of topological string theory is possible but (b) the action is more elaborate than the standard coupling of topological gravity to the topological sigma model for ΣT , and (c) a careful analysis of contact terms is needed to reproduce the Y M2 results. In more detail, the paper is organized as follows. We review some aspects of [15] and establish notation in Sect. 2. We will discuss both the “chiral” partition function Z + (A, G, N ) (Eq. (2.4)) as well as the “nonchiral” partition function (Eq. (2.2)), both of which we view as asymptotic expansions in 1/N . In Sects. 3, 4 we review some necessary background material from mathematics, in particular, we describe the Hurwitz moduli space H(h, G) of holomorphic maps ΣW → ΣT from a connected Riemann surface ΣW of genus h to a Riemann surface ΣT of genus G with fixed complex structure. In Sects. 4.3, 4.4 we explain how H(h, G) can be thought of as the base of a principal fibre bundle for Dif f + (ΣW ) × W eyl(ΣW ). In Sect. 5 we begin with the simplest quantity in Y M2 : the chiral partition function at zero area: Z + (A = 0, G, N ). This expansion can be interpreted as a sum over branched covers [15]. Taking proper account of the −1 factors leads to our first main result, stated as Proposition 5.2 (Sect. 5.2): Z + (A = 0, G, N ) is the generating function for the orbifold Euler characteristic Z of the compactified Hurwitz moduli space, H(h, G) of branched covers of ΣT :

Large N 2D Yang-Mills Theory and Topological String Theory

2h−2 X ∞ 1 Z + (0, N, G) = exp χ H(h, G) . N

545

(1.1)

h=0

A branched cover of surfaces ΣW → ΣT can always be interpreted as a holomorphic map for appropriate complex structures on ΣW , ΣT . Thus, the appropriate category of maps with which to formulate the chiral 1/N expansion of Y M2 is the category of holomorphic maps. This is precisely the situation best suited to an introduction of topological sigma models. Accordingly, in Sect. 6 we introduce a topological string theory which counts holomorphic (and antiholomorphic) maps ΣW → ΣT . Our central claim is that this string theory is the underlying string theory of Y M2 . The action is schematically of the form: Ichiral YM2

string

= Itg + Itσ + Icσ .

(1.2)

The first two terms give the (standard) action of 2D topological gravity coupled to a topological σ-model with ΣT as target. The action Icσ turns out to be complicated but can be deduced using a procedure which is in principle straightforward. This procedure is based on the point of view that topological field theory path integrals are related to infinite-dimensional generalisations of the Mathai-Quillen representative of equivariant Thom classes[19–21]. In Sect. 7, we use this point of view to construct Icσ explicitly. In Sects. 8 and 9 we show how many of the results of chiral Y M2 can be derived from the topological string theory (1.2). In Sect. 8 we describe how the area can be restored by a perturbation of the topological action (1.2) by the area operator: Z (1.3) A = f ∗ ω, where ω is the Kähler class of the target space Riemann surface. This is, of course, just the standard Nambu action, as one might well expect. The novelty in the present context is that the Nambu action is regarded as a perturbation of a previously constructed theory. Calculating this perturbation of the topological string theory is rather subtle because of contact terms. These are responsible for the polynomial dependence on the area in Y M2 . In Sect. 8.1 we isolate what we believe are “the most important” area polynomials and, after some preliminary analysis of the contact terms between the area operator (1.3) and curvature insertions arising from Icσ in (1.2), as well as those between area operators themselves, we show in Sect. 8.7 how these polynomials follow from the string picture. In Sect. 9 we discuss Wilson loop expectation values in the case of nonintersecting Wilson loops. Following the lead of [15] we show that these may be incorporated in the string approach by computing macroscopic loop amplitudes. The data of the representation index on the Wilson line 0 is translated into covering data of the boundary of the worldsheet over the lines 0. In Sect. 10 we repeat the discussion of Sect. 5 for the partition function of the “full nonchiral Y M2 ”. We follow closely the geometrical picture introduced in [15]. In order to state the analog of (1.1) it is necessary to introduce both holomorphic and antiholomorphic maps, as well as “degenerating coupled covers” (see Definitions 10.3, 10.4). We introduce a Hurwitz space for such maps, called coupled Hurwitz space and, in Proposition 10.3, we state the result for the nonchiral theory analogous to (1.1). In Sect. 11 we explain how the nonchiral partition function can be incorporated in topological string theory. The path integral localizes on both holomorphic and antiholomorphic maps. It also localizes on singular maps (“degenerated coupled covers”) and the

546


contributions from these singular geometries must be defined carefully. The answer for the partition function of the string theory (1.2) depends on how we choose our contact terms for these singular geometries. We discuss two choices which lead to two distinct answers: X 2h−2 1 Zstring (ΣW → ΣT ) exp N h≥0

=

(1.4) +

−

Z (A = 0, N )Z (A = 0, N ) . Z(A = 0, N )

In the first case we choose contact terms so that singular geometries make no contribution (we “set all contact terms to zero”). This reproduces answers of the chiral theory. A more non-trivial choice of contact terms reproduces the full zero-area theory. Some technical arguments are contained in appendices. Finally the reader should note that while we were preparing this paper a closely related paper appeared on hep-th [22]. In this paper P. Horava proposes a formulation of Y M2 in terms of topological string theory. The theory in [22] is based on counting of harmonic maps, rather than holomorphic maps, (or degenerated coupled covers) and, at least superficially, appears to be different from the proposal of this paper. 2. The Gross-Taylor Asymptotic Series 2.1. Partition Functions. The partition function of two dimensional Yang-Mills theory on an orientable closed manifold ΣT of genus G is [10, 12] Z Z p −1 Z(SU (N ), ΣT ) = [DAµ ]exp[ 2 d2 x detGij T rFij F ij ] 4e ΣT X 2−2G − λA = (dim R) e 2N C2 (R) , (2.1) R

where the gauge coupling λ = e2 N is held fixed in the large N limit, the sum runs over all unitary irreducible representations R of the gauge group G = SU (N ), C2 (R) is the second casimir, and A is the area of the spacetime in the metric Gij . We will henceforth absorb λ into A. Using the Frobenius relations between representations of symmetric groups and representations of SU (N ) Gross and Taylor derived an expression for the 1/N asymptotics of (2.1) in terms of a sum over elements of symmetric groups. The result of [15] is: Z(A, G, N ) ∞ X ∼

X

X

n± ,i± =0 p± ,...,p± ∈T2 ⊂S ± s± ,t± ,...,s± ,t± ∈S ± n n ± 1 1 1 G G

1 (n+ +n− )(2G−2)+(i+ +i− ) N

i

+

−

1 1 + − + 2 − 2 + − 2 + − (−1)(i +i ) (A)(i +i ) e− 2 (n +n )A e 2 ((n ) +(n ) −2n n )A/N + − + − i !i !n !n ! G G Y Y − 2−2G − − + + · · · p [s , t ] [s , t ] , δSn+ ×Sn− p+1 · · · p+i+ p− j j 1 k k i− n+ ,n− j=1

k=1

(2.2)


547

where [s, t] = sts−1 t−1 . Here δ is the delta function on the group algebra of the product of symmetric groups Sn+ × Sn− , T2 is the class of elements of Sn± consisting of transpositions, and −1 n+ ,n− are certain elements of the group algebra of the symmetric group Sn+ × Sn− with coefficients in R((1/N )). These will be discussed in detail below. One of the striking features of (2.2) is that it nearly factorizes, splitting into a sum over n+ , i+ , · · · and n− , i− , · · ·. Gross and Taylor interpreted the contributions of the (+) and (−) sums as arising from two “sectors” of a hypothetical worldsheet theory. These sectors correspond to orientation reversing and preserving maps, respectively. One views the n+ = 0 and n− = 0 terms as leading order terms in a 1/N expansion. At higher orders the two sectors are coupled via the n+ n− term in the exponential and via terms in n+ n− . The latter are described by a simple set of rules in [15], and will be addressed in detail in Sect. 10 below. The expression (2.2) simplifies considerably if we concentrate on one chiral (or antichiral) sector. In general we define chiral expectation amplitudes in Y M2 by translating SU (N ) representation theory into representation theory of symmetric groups and making the replacement: X X X → , (2.3) R∈Rep(SU (N ))

n≥0 R∈Yn

where Yn stands for the set of Young Tableaux with n boxes. For example, in the case of the partition function we may define n ≡ n,0 and write the “chiral Gross-Taylor series” (CGTS) as [15]: Z + (A, G, N ) ∞ X (A)i+t+h 1 n(2G−2)+2h+i+2t nh (n2 − n)t = e−nA/2 (−1)i i!t!h! N 2t+h n,i,t,h=0

X

X

p1 ,...,pi ∈T2 s1 ,t1 ,...,sG ,tG ∈Sn

G Y 1 −1 −1 δ(p1 · · · pi 2−2G s t s t ) . j j n j j n!

(2.4)

j=1

2.2. factors. Let us now define . We postpone describing n+ ,n− to Sect. 10.4 and concentrate on the “chiral factors” n . This is the element of the group algebra of Sn defined by the equation Nn χY (n ). dim RY = n! Here RY is an SU (N ) representation associated to a Young tableaux Y with n boxes, and χY is the character in the corresponding representation of Sn . Explicitly, n is given by X 1 n−Kv n = v. (2.5) N v∈Sn

Here, Kv is the number of cycles in the permutation v. When G > 1 one must introduce −1 which is the inverse of in the group algebra2 . For example, we have 2 Note that for some values of N , will fail to have an inverse. This does not happen when N > n. n Hence may always be considered as invertible in the 1/N expansion.

548


2 = 1 + −1 2 =

∞ X

1 v, N (−1/N )i v i

i=0

=

∞ X i=0

=

(1/N ) − v 2i

∞ X

(1/N )(2i+1)

(2.6)

i=0

1/N 1 ·1− ·v (1 − 1/N 2 ) (1 − 1/N 2 )

for n = 2. In the first line we have written it as an element of the free algebra generated by elements of S2 . In the second we have reduced it to to an element of the group algebra of S2 whose coefficients are expansions in 1/N . 3. Maps and Coverings We would like to interpret the terms in the 1/N expansion as weighted sums of maps ΣW → ΣT between compact orientable surfaces without boundary, of genus h, G, respectively. In the next two sections we summarize some relevant mathematical results pertaining to such maps. 3.1. Homotopy groups. We will use heavily the properties of homotopy groups of punctured Riemann surfaces. As abstract groups these are “F-groups”. The group FG,L may be described in terms of generators and relations by FG,L ≡

{αi , βi }i=1,G , {γs }s=1,L |

G Y

[αi , βi ]

i=1

L Y

γs = 1 .

(3.1)

s=1

(The product is ordered, say, lexicographically.) Consider a compact orientable surface ΣT of genus G. If we remove L distinct points, and choose a basepoint y0 , then there is an isomorphism FG,L ∼ = π1 (ΣT − {P1 , . . . PL }, y0 ).

(3.2)

This isomorphism is not canonical. The choices are parametrized by the infinite group Aut(FG,L ). On several occasions we will make use of a set of generators αi , βi , γi of π1 so that, if we cut along curves in the homotopy class the surface looks like Fig. 1. The conjugacy class of the curves γ(P ) can be characterised intrinsically as follows. The process of filling in a point P1 defines an inclusion i : ΣT − {P1 , P2 , · · · PL } → ΣT − {P2 , · · · PL } with an induced map i∗ on π1 . [γ(P1 )] is the kernel of i∗ .

(3.3)


549

Fig. 1. A choice of generators for the homotopy group of a punctured surface. The curves γ(P ) become trivial if we fill in the puncture P

3.2. Branched coverings. Maps of particular importance to us are branched coverings. Definition 3.1. a) A continuous map f : ΣW → ΣT is a branched cover if for any open set U ⊂ ΣT , the inverse f −1 (U ) is a union of disjoint open sets on each of which f is topologically equivalent to the complex map z 7→ z n for some n. b) Two branched covers f1 and f2 are said to be equivalent if there exists a homeomorphism φ : ΣW → ΣW such that f1 ◦ φ = f2 . For Q ∈ ΣW , the integer n will be called the ramification index of Q and will be denoted Ram(f, Q). For any P ∈ ΣT the sum X Ram(f, Q) Q∈f −1 (P )

is independent of P and will be called the index of f (sometimes the degree). Points Q for which the integer n in condition (a) is bigger than 1 will be called ramification points. Points P ∈ ΣT which are images of ramification points will be called branch points. The set of branch points is the branch locus S. The branching number at P is X [Ram(f, Q) − 1]. BP = Q∈f −1 (P )

P The branching number of the map f is B(f ) = P ∈S(f ) BP . A branch point P for which the branching number is 1 will be called a simple branch point. Above a simple branch point all the inverse images have ramification index = 1, with the exception of one point Q with index = 2.3 We will often use the Riemann-Hurwitz formula. If f : ΣW → ΣT is a branched cover of index n and branching number B, ΣW has genus h, ΣT has genus G, then : 3

Unfortunately, several authors use these terms in inequivalent ways.

550


2h − 2 = n(2G − 2) + B.

(3.4)

Equivalence classes of branched covers may be related to group homomorphisms through the following construction. Choose a point y0 which is not a branch point and label the inverse images f −1 (y0 ) by the ordered set {x1 , . . . xn }. Following the lift of elements of π1 (ΣT − S, y0 ) the map f induces a homomorphism f# : π1 (ΣT − S, y0 ) → Sn . Suppose γ(P ) is a curve surrounding a branch point P as in Fig. 1. There is a close relation between the cycle structure of vP = f# (γ(P )) and the topology of the covering space over a neighbourhood of P . If the cycle decomposition of vP has r distinct cycles then f −1 (P ) has r distinct points. Moreover, a cycle of length k corresponds to a ramification point Q of index k. With an appropriate notion of equivalence the homomorphisms are in 1-1 correspondence with equivalence classes of branched covers. Definition 3.2. Two homomorphisms ψ1 , ψ2 : π1 (ΣT − S, y0 ) → Sn are said to be equivalent if they differ by an inner automorphism of Sn , i.e., if ∃g such that ∀x, ψ1 (x) = gψ2 (x)g −1 . Theorem 3.1. [23, 24] Let S ⊂ ΣT be a finite set and n a positive integer. There is a one to one correspondence between equivalence classes of homomorphisms ψ : π1 (ΣT − S, y0 ) → Sn and equivalence classes of n-fold branched coverings of ΣT with branching locus S. Proof. We outline the proof which is described in [24]. The first step shows that equivalent homomorphisms determine equivalent branched coverings. Given a branched cover, we can delete the branch points from ΣT and the inverse images of the branch points from ΣW giving surfaces Σ W and Σ T respectively. The branched cover restricts to a topological (unbranched) cover of Σ T by Σ W . To this map we can apply the theorem [25] which establishes a one-to-one correspondence between conjugacy classes of subgroups of π1 (Σ T ) and equivalence classes of topological coverings of Σ T . Similarly the second step proves that equivalent covers determine equivalent homomorphisms. The restriction of φ to the inverse images of y0 determines the permutation which conjugates one homomorphism into the other. Finally one proves that the map from equivalence classes of homomorphisms to equivalence classes of branched covers is onto. We cut n copies of ΣT along chosen generators of π1 (ΣT − S) (illustrated in Fig. 1), and we glue them together according to the data of the homomorphism. This theorem goes back to Riemann. Since the Y M2 partition function sums over covering surfaces which are not necessarily connected we do not restrict to homomorphisms whose images are transitive subgroups of Sn . Definition 3.3. An automorphism of a branched covering f is a homeomorphism φ such that f ◦ φ = f .


551

It follows from the above that the number of such automorphisms of a given equivalence class of branched coverings is equal to the order of the centraliser of the subgroup generated by the image of π1 (ΣT − S, y0 ) in Sn . For a homomorphism ψ we call this |C(ψ)|. Then n!/|C(ψ)|, the number of cosets of this subgroup, is the number of distinct homomorphisms related to the given homomorphism by conjugation in Sn . 3.3. Continuous maps. The space of branched coverings is not the only space of maps of surfaces one might wish to consider in rewriting Y M2 as a string theory. Another natural choice of category is the category of continuous maps. We mention here two theorems concerning the classification of these maps, and their relation to the category of branched coverings. Since Riemann surfaces have a contractible universal cover, continuous maps between Riemann surfaces are topologically classified by their action on homotopy groups: Theorem 3.2. [26] Homotopy classes of continuous maps f : (ΣW , x0 ) → (ΣT , y0 ) of fixed degree are in 1-1 correspondence with homomorphisms f∗ : π1 (ΣW , x0 ) → π1 (ΣT , y0 ).

Fig. 2. Example of a pinch map

A map ΣW → ΣT is said to be a pinch map if there is a compact connected submanifold H ⊂ ΣW , with boundary consisting of a simple closed curve in the interior of ΣW , such that the ΣT is ΣW /H, the quotient of ΣW with H identified to a point, and such that f is the quotient map. Pinch maps can collapse entire regions of surface to a single point as in Fig. 2. Theorem 3.3. [27] In each homotopy class of maps f : ΣW → ΣT there is a representative f = p ◦ π, where p is a pinch map and π is a branched covering. Notice that pinch maps can only decrease the Euler character of ΣW . It therefore follows from Theorem 3.3 and (3.4) that the existence of a nonconstant f : ΣW → ΣT implies that 2h − 2 ≥ n(2G − 2). This is the now-famous Kneser bound.

552


4. The Hurwitz Space H of Branched Coverings 4.1. Definition. The Hurwitz space of branched coverings is nicely described in [23, 28]. Let H(n, B, G; S) be the set of equivalence classes of branched coverings of ΣT , with degree n, branching number B, and branch locus S, where S is a set of distinct points on ΣT . H(n, B, G; S) is a finite set. The union of these spaces over sets S with L elements is the Hurwitz space H(n, B, G, L) of equivalence classes of branched coverings of ΣT with degree n, branching number B and L branch points. Finally let CL (ΣT ) be the configuration space of ordered L-tuples of distinct points on ΣT , that is CL (ΣT ) = {(z1 , . . . , zL ) ∈ ΣTL |zi ∈ ΣT , zi 6= zj for i 6= j}. The permutation group SL acts naturally on CL and we denote the quotient CL (ΣT ) = CL (ΣT )/SL . There is a map p : H(n, B, G, L) → CL (ΣT )

(4.1)

which assigns to each covering its branching locus. This map can be made a topological (unbranched) covering map [23] with discrete fiber H(n, B, G; S) over S ∈ CL . The lifting of closed curves in CL will in general permute different elements of the fibers H(n, B, G, S). Note however that Autf is invariant along any lifted curve so that Autf is an invariant of the different components of H(n, B, G, L). 4.2. H as an analytic variety. One great advantage of branched covers is that they allow us to introduce the powerful methods of complex analysis, which are crucial to introducing ideas from topological field theory. Choose a complex structure J on ΣT . Then given a branched cover f : ΣW −→ ΣT there is a unique complex structure on ΣW making f holomorphic (use the complex structure f ∗ (J) on ΣW [29]). (This is far from true for pinch maps.) Conversely, any nonconstant holomorphic map f : ΣW −→ ΣT defines a branched cover. It follows that we can consider the Hurwitz space H(n, B, G, L) as a space of holomorphic maps. The complex structure J on ΣT induces a complex structure on H(n, B, G, L) such that p is a holomorphic fibration. Moreover, the induced complex structure on ΣW defines a holomorphic map m : H(n, B, G, L) −→ Mh,0 , where Mh,0 is the Riemann moduli space of curves of genus h, where h is given by (3.4) . The image of H is a subvariety of Mh,0 . 4.3. Fiber Bundle approach to Hurwitz space. For comparison with topological field theory we will need another description of Hurwitz space as the base space of an infinitedimensional fiber bundle. Let ΣW be an orientable surface, and suppose ΣT is a Riemann surface with a choice of Kähler metric and complex structure J. Let us begin with the configuration space f = {(f, h)| f ∈ C ∞ (ΣW , ΣT ), h ∈ Met(ΣW )} , M

(4.2)

where C ∞ (ΣW , ΣT ) is the space of smooth (C ∞ ) maps, f : ΣW → ΣT and Met(ΣW ) is the space of smooth metrics on ΣW . A choice of metric h induces a complex structure: (h) ∈ 0[End(T ΣW )], 2 = −1. If we choose a basepoint h0 in the space of metrics, and choose oriented isothermal coordinates relative to h0 then we can define a basepoint complex structure to be the standard antisymmetric tensor ˆαγ ,


ˆαγ =

0 1 −1 0

553

(4.3)

in the isothermal coordinates. In these terms we define αβ (h) = h1/2 ˆαγ hγβ . The subspace of pairs defining a holomorphic map ΣW → ΣT is then given by f F˜ = {(f, h) : df = Jdf } ⊂ M. (4.4) The defining equation df = Jdf is an equation in 0 End(Tx ΣW , Tf (x) ΣT ) . Let Dif f + (ΣW ) × W eyl(ΣW ) be the semidirect product of the group of orientation preserving diffeomorphisms of ΣW and the group of Weyl transformations on ΣW . f There is an action of Dif f + (ΣW ) × There is a natural action of this group on M. e W eyl(ΣW ) on F . The quotient space ˜ Dif f + (ΣW ) × W eyl(ΣW ) (4.5) F ≡ F/ parametrizes holomorphic maps ΣW −→ ΣT . We have now provided two descriptions of the space of holomorphic maps: Hurwitz space H and (4.5). Let H(h, G) = qn(2−2G)−B=2−2h H(n, B, G, L), where the disjoint union runs over n, B, L ≥ 0 compatible with (3.4). There is a map σ : H(h, G, L) → F which is generically smooth and one-one. The space F has orbifold singularities where the group Dif f + (ΣW ) fails to act freely. Because we divide by Dif f + (ΣW ), the local orbifold group at (f, h) ∈ F is Autf . This will be important when we introduce the f −→ M et(ΣW ) obtained from (f, h) → h orbifold Euler character of H. The map M induces the map m : H(h, G) → Mh,0 of Sect. 4.2 and relates the bundle description of Hurwitz space to the bundle description of Mh,0 . Since Autf ⊂ Aut(ΣW ) , orbifold points of F map to orbifold points of Mh,0 . 4.4. Geometry of F. Our discussion of the topological string theory approach to Y M2 requires a brief discussion of the geometry of F . In particular, we need to define a connection on T F and compute its curvature. f is Let us first make the tangent space to Fe more explicit. The tangent space to M f = 0[f ∗ (T ΣT )] ⊕ 0[S ⊗2 (T ∗ ΣW )], Tf,h M

(4.6)

where 0 is the space of C ∞ sections and S ⊗n is the nth symmetric power. The tangent space to Fe is the subspace of pairs (δf, δh) which preserve the equation df = Jdf . In order to characterize this subspace by a differential equation we identify the differential df with a section of T ∗ ΣW ⊗ f ∗ (T ΣT ) (∼ = 0 End(Tx ΣW , Tf (x) ΣT ) ). Then, in order to vary with respect to f we must compare T ΣT at different points. We do this using the Kähler metric on ΣT to define a pullback connection 4 on f ∗ (T ΣT ): ∇ : 0[f ∗ (T ΣT )] → 0[T ∗ ΣW ⊗ f ∗ (T ΣT )]. Finally, let k(δh) = δ be the variation of complex structure on ΣW induced from a variation of metric δh ∈ S ⊗2 (T ∗ ΣW ). The tangent space Fe at (f, h) is the subspace of f defined by TM Tf,h Fe = {(δf, δh) : ∇(δf ) + J∇(δf ) + Jdf k(δh) = 0}. 4

See Appendix B for a careful derivation of this connection.

(4.7)

554


See Appendix B for the proof. We now separate out “pure gauge” deformations. The action of the gauge group on Fe defines a subbundle T vert Fe ⊂ T Fe with fibers isomorphic to imC where C: T [Dif f + (ΣW ) × W eyl(ΣW )] −→ T vert Fe α ξ ξ α ∂α f i C 7−→ δσ (P ξ)αβ + (δσ + ∇ · ξ)hαβ and

(P ξ)αβ = ∇(α ξβ) − hαβ ∇ · ξ,

(4.8) (4.9)

as is familiar from string theory. In general ker C = 0 and we have an isomorphism e T[f,h] F ∼ = Tf,h F/imC. f (hence on Fe), To go further we use the natural Dif f + (ΣW )-invariant metric on M given by h(δf1 , δh1 ), (δf2 , δh2 )iT M e Z n o = d2 z h1/2 Gij δf1i δf2j + (hαγ hβδ + c hαβ hγδ )δh1 αβ δh2 γδ , (4.10) with c ∈ R+ arbitrary. This allows us to define adjoints and orthogonal projections. We now introduce the operator: D Jdf k f → 0[T ΣW ⊗ f ∗ (T ΣT )] ⊕ 0[T ΣW ], : Tf,h M (4.11) O= † ∂f P where the components are given by: Dχi = ∇χi + J(∇χi ), (4.12) i

∂f χ = h

αβ

i

j

(∂β f )Gij χ .

Consider deformations (δf, δh) in the kernel of O. The first line of (4.11) ensures that (δf, δh) ∈ T Fe and the second ensures that (δf, δh) 6∈ T vert Fe. An index theorem shows that dimC ker O−dimC cokerO = B = 2h−2−n(2G−2) is the total branching number. On the other hand, a generalisation of Kodaira-Spencer theory described in Appendix A shows that dimC F = B. In the generic case (G, h > 2), dimC cokerO = 0. Moreover we have an orthogonal decomposition Tf,h Fe = T (Dif f + (ΣW ) × W eyl(ΣW )) ⊕ ker O.

(4.13)

Even though the metrics are not W eyl(ΣW ) invariant, the orthogonal decomposition (4.13) is invariant. Therefore ker O is isomorphic to the tangent space T[f,h] F . The natural metric (4.10) on Fe also defines connections on the principal Dif f + (ΣW ) × W eyl(ΣW ) bundle π : Fe → F as well as on the tangent T Fe → Fe. In the first case we define a lift of a curve γ(t) ⊂ F to be γ(t) ˜ ⊂ Fe defined by the conditions: d γ˜ ∈ ker O, dt dπ

d γ˜ dt

(4.14)

=

d γ. dt


555

˜ on T Fe by declaring ker O to be the horizontal subspace In the second case we define ∇ of the fiber. Finally we use these connections to define a connection ∇ on T F → F . It suffices to define the parallel transport X(t) of X(0) ∈ Tγ(0) F along a path γ(t) ⊂ F. e ˜ ˜ Choose lifts γ(t) ˜ ⊂ Fe and X(0) ∈ Tγ(0) ˜ F. Use ∇ to define the parallel-transported e ˜ ˜ ˜ X(t) ∈ Tγ(t) ˜ F and define X(t) = dπ X(t) . Since ∇ preserves the orthogonal decomposition (4.13) our definition is independent of the choices made in lifting. See Fig. 3.

Fig. 3. Construction of a connection on T F → F : δ(t) is the lift of γ(t), determined by choosing as initial point a lift Y (0) of X(0)

˜ on T Fe more explicitly. In Finally, let us describe the curvature of the connection ∇ local coordinates we may describe the connection on Tf,h Fe as follows. Introduce local tangent vector fields i i χ˜ χ ˜ = , X X= ψ˜ αβ . ψαβ of as elements of ker O. Then ˜ X X˜ = δX X˜ + O† 1 δX OX, ˜ ∇ OO†

(4.15)

where δ X˜ = X ◦ X˜ and δO = X · O. Equation (4.15) makes sense since OO† is invertible. A simple calculation, using repeatedly the fact that OX˜ = 0 shows that the curvature on horizontal vectors is given by e 2) e 1 , R [X1 , X2 ]X (X e2 − X e 2 .(4.16) e 1 , (δ2 O† )(OO† )−1 (δ1 O)X e 1 , (δ1 O† )(OO† )−1 (δ2 O)X = X When we descend to F we are working on an analytic space, and, from (4.10) we see that T F is a holomorphic Hermitian vector bundle. Thus if we choose a local

556


holomorphic framing GI we may use (4.10) to form the positive definite matrix of inner products hIJ = hGI , GJ i. In these terms the connection and curvature are given by [30] ∇ = ∂logh,

¯ R = ∂ ∂logh.

(4.17)

We will return to this formula in our discussion of contact terms.

4.5. Compactification of Hurwitz Space. Consider H(n, B, G) the space of branched coverings with a given degree and branching number of a surface of genus G. A Bdimensional open subset of this space is H(n, B, G, L = B), which consists of maps where all the branch points are simple. We will refer to this space as simple Hurwitz space. Simple Hurwitz space can be (partially) compactified to form the Hurwitz space H(n, B, G) by adding L-dimensional compactification varieties of the form H(n, B, G, L < B):

H(n, B, G) =

[

H(n, B, G, L).

(4.18)

L

We thus have in mind the following schematic description of Hurwitz space:

Fig. 4. Simple Hurwitz spaces with other Hurwitz spaces as compactification varieties

Compactified Hurwitz space is a bundle over the compactified configuration space where the branch points on the target are allowed to collide. The compactification subvarieties may be described in terms of basic degenerations of branched coverings. We will use the facts in Sects. 3.1 and 3.2 to describe some properties of degenerations that happen when two branch points collide. The following observation is basic . Let f ∈ H(n, B, G; S). Choosing a set of generators for π1 we may then associate a cut surface as in the proof of Theorem 3.1, and as in the LHS of Fig. 5. Choose two generators of π1 (ΣG − S, y0 ), γj and γj+1 (see (3.1)). Consider a closed path γ ∗ homotopic to the product γj γj+1 . As we let the points enclosed by γj and γj+1 approach each other, the image of γ ∗ in Sn does not change and remains the product uj uj+1 , where uj is the image of γj and uj+1 is the image of γj+1 under f# .


557

Fig. 5. A collision of neighboring branch cuts (for an appropriate choice of generators) produces the product of the monodromy data

Fig. 6. Three types of Collisions

558


Using the above rule we see that there are three types of collisions of simple branch points. They are classified by the behaviour of the inverse ramification points, and illustrated in Fig. 6. Type 1. A collision of type 1 produces a single ramification point of index 3. For example if uj = (12), uj+1 = (23) then collision of Pj , Pj+1 produces a ramification point with u = (123). Starting from the space of generic branched covers successive collisions can lead to multiple branch points; a collision of ` simple branch points can lead to a ramification point of index ` + 1. Type 2. A collision of type 2 produces two ramification points. This occurs when uj and uj+1 are disjoint transpositions. Type 3. A collision of type 3 produces no ramification point but instead produces a double point. This occurs when uj and uj+1 are the same transposition. Collisions of any two branch points in a hyperelliptic curve produce degenerations of the third type. Collisions of types 1, 2 and their generalisations explain why H(n, B, G, L < B) are used as some of the compactification varieties in the compactification of Hurwitz space. We now describe a class of collisions of branch points generalising collisions of type 3, which will be useful in the discussion of the nonchiral theory. Let uj = (1, 2 · · · k, k + 1)(k + 2) · · · (n) and

uj+1 = (1) · · · (k − 1)(k, k + 1, · · · , 2k)(2k + 1) · · · (n).

The monodromy u∗ around γ ∗ is (1, 2, · · · , k − 1, k + 1)(k, k + 2, · · · , 2k)(2k + 1) · · · (n). The product permutation has two cycles of length k and remaining cycles of length 1. Before collision the total branching number at the ramification points is (n − Kuj ) + (n − Kuj+1 ) = 2k. After collision the branching occurs at a single point so the branching number is Ku∗ − n = 2k − 2. But the genus of the worldsheet does not change during the collision. So there is a collapsed tube connecting the two cycles of length k, the other sheets labelled 2k + 1, · · · , n do not participate in the collision. The fact that collision of branch points can produce tubes connecting ramification points of equal index will be used in Sect. 11.3. (More complicated collisions can occur but do not seem to contribute to the Y M2 partition function.) Note that the deficiency in total branching number can only be even when branch points collide, which is clear geometrically. This also follows from the fact that (−1)Ku −n is equal to the parity of the permutation u. Remark . Beware. Compactifications of moduli spaces are not unique. Moreover, the construction of compactifications is a tricky business. Compactifications of the base of Hurwitz space are described in [31], The complete mathematical description of families of maps associated with collisions of arbitrary branch points is rather complicated. A compactification of Hurwitz spaces and its relation to the Deligne-Mumford compactification of moduli spaces of complex structures [32] is discussed in detail in [28]. 5. The CGTS and the Space of Branched Coverings In this section we make our first connection between the topology of Hurwitz space and Y M2 amplitudes. Consider the CGTS (2.4). As in 2D gravity, relations to topological


559

field theory become most transparent in the limit A → 0. Accordingly in this section we will study the series +

Z (0, G, N ) =

∞ X

N

X

n(2−2G)

s1 ,t1 ,...,sG ,tG ∈Sn

n=0

G Y 1 −1 −1 2−2G δ(n s j t j s j tj ) . n!

(5.1)

j=1

5.1. Recasting the CGTS as a sum over branched coverings. The first step in rewriting (5.1) is to count the weight of a given power of 1/N . To this end we expand the −1 point as an element of the free algebra generated by elements of the symmetric group, Pk n−Kvj ∞ X X j=1 0 1 (v1 v2 · · · vk )(−1)k , (5.2) −1 n =1+ v1 ···vk ∈Sn N k=1

where the primed sum means no vi = 1. We could rewrite (5.1) by imposing relations of the symmetric group of Sn as in (2.6). However, we decline to do this and rather substitute the expansion (5.2) into (5.1) to obtain Z + (0, G, N ) =

∞ ∞ X X

s1 ,t1 ,...,sG ,tG ∈Sn v1 ,v2 ,...,vL ∈Sn

n=0 L=0

PL N

j=1

(Kvj −n)

0 X

X

N n(2−2G)

d(2 − 2G, L) δ(v1 v2 · · · vL n!

G Y

(5.3)

−1 sj tj s−1 j tj ) ,

j=1

where d(m, L) is defined by (1 + x)m =

∞ X

d(m, L)xL .

(5.4)

L=0

Explicitly we have (2G + L − 3)! , (2G − 3)!L! d(0, L) = 0, unless L = 0, d(2, L) = 0, unless L = 0, 1, 2.

d(2 − 2G, L) = (−1)L

for

G > 1, (5.5)

For G > 1, d(2 − 2G, L) is the number of ways of collecting L objects into 2G − 2 distinct sets. Equation (5.3) correctly gives the the partition function for any G including zero and one. For example the vanishing of d(0, L) for L > 0 means that in the zero area limit only maps with no branch points contribute to the torus partition function. And for genus zero the vanishing of d(2, L) for L > 2 means that only maps with no more than two branch points contribute to the CGTS for the sphere. To each nonvanishing term in the sum (5.3) we may associate a homomorphism ψ : FG,L → Sn , where FG,L is an F group (3.1), since, if the permutations v1 , . . . , vL , QG −1 s1 , t1 , . . . , sG ,tG in Sn satisfy v1 · · · vL i=1 si ti s−1 i ti = 1 we may define ψ : α i → si

ψ : βi → ti

ψ : γi → vi .

(5.6)

560


Moreover, if there exists a g ∈ Sn such that g{v1 , · · · , vL ; s1 , t1 · · · sG , tG }g −1 = 0 0 0 0 0 0 {v1 , · · · vL ; s1 , t1 , · · · sG , tG } as ordered sets, then by Definition 3.2 the induced homomorphisms are equivalent. Evidently the class of ψ will appear in the sum n!/|C(ψ)| times in (5.3). Therefore, we may write (5.3) as Z + (0, G, N ) =

∞ ∞ X X

N n(2−2G)−B

B X

n=0 B=0

d(2 − 2G, L)

X ψ∈9(n,B,G,L)

L=0

1 , |C(ψ)|

(5.7)

where 9(n, B, G, L) is the set of equivalence classes of homomorphisms FG,L → Sn , with the condition that the γi all map to elements of Sn not equal to the identity. We have collected terms with fixed value of: B≡

L X

(n − Kvi ).

(5.8)

i=1

Now we use Theorem 3.1 to rewrite the sum (5.7) as a sum over branched coverings. To do this we must make several choices. We choose a point y0 ∈ ΣT and for each n, B, L, ψ we also make a choice of : 1. some set S of L distinct, points on ΣT . 2. an isomorphism (3.2). To each ψ, S we may then associate a homomorphism π1 (ΣT − S, y0 ) → Sn . By Theorem 3.1 we see that, given a choice of S, to each class [ψ] we associate the equivalence class of a branched covering f ∈ H(n, B, G; S), where f : ΣW → ΣT . The genus of the covering surface h = h(G, n, B) is given by the Riemann-Hurwitz formula (3.4). Note that the power of N1 in (5.7) is simply 2h − 2. Finally, the centralizer C(ψ) ⊂ Sn is isomorphic to the automorphism group of the associated branched covering map f . The order of this group, |Aut(f )|, does not depend on the choice of points S used to construct f . Accordingly, we can write Z + as a sum over equivalence classes of branched coverings: 2h−2 ∞ X B ∞ X X X 1 1 . (5.9) d(2 − 2G, L) Z + (0, G, N ) = N |Autf | f ∈H(n,B,G;S)

n=0 B=0 L=0

5.2. Euler characters. We have now expressed the CGTS as a sum over equivalence classes of branched coverings. We now interpret the weights in terms of the Euler characters of the Hurwitz space H. To begin we write (χG )(χG − 1) · · · (χG − L + 1) , (5.10) L! where χG = 2 − 2G. The RHS of (5.10) is the Euler character of the space CL (ΣT ) = CL (ΣT )/SL . This may be easily proved in two ways. Recall that it is general property of fibre bundles with connected base that their Euler character is the product of Euler characters of base and fibre [26, 34]. Let MG,L be the uncompactified moduli space of complex structures of a surface of genus G with L punctures. The fibration: d(2 − 2G, L) =

CL (ΣT )

−→

MG,L  y MG,0

(5.11)


561

together with the celebrated Harer-Zagier-Penner formula: [35] χ(MG,L ) = (−1)L

(2G − 3 + L)!(2G − 1) B2G , L!(2G)!

(5.12)

(where G ≥ 0, 2G − 2 + L > 0, and B2G is a Bernoulli number) gives the result: χ(MG,L ) χ(MG,0 ) (2G + L − 3)! = (−1)L . (2G − 3)!L!

χ(CL (ΣT )) =

(5.13)

An alternative proof, (which also covers the cases of interest at genus zero) uses the fibration of configuration spaces described in [36]. Let Cm,n (ΣT ) be the configuration space of n labelled points on a surface ΣT of genus G with m fixed punctures. There is a fibration CL−1,1 (ΣT )

−→

C0,L(ΣT )  y C0,L−1 (ΣT ).

(5.14)

Using the product formula for Euler characters of a fibration we get χ C0,L (ΣT ) = (2 − 2G − (L − 1))χ(C0,L−1 (ΣT ))

(5.15)

This recursion relation together with χ(C0,1 (ΣT )) = χ(ΣT ) gives χ(C0,L (ΣT )) = (χG )(χG − 1) · · · (χG − L + 1). But C0,L (ΣT ) is a topological covering space of CL (ΣT ) of degree L! so this leads to χ(CL (ΣT )) = d(2 − 2G, L).

(5.16)

Using (5.16), we can further rewrite the CGTS as Z + (0, G, N ) = ∞ B ∞ X X X N n(2−2G)−B χ(CL (ΣT )) n=0 B=0

L=0

X f ∈H(n,B,G,S)

1 . |Autf |

(5.17)

Let us now return to the fibration [4.1]. Consider first the case where the covering surface ΣW has no automorphisms. From the results of [37], it follows that this will happen for primitive branched coverings of surfaces with G > 2 with B > n/2 simple branch points. In such cases we can identify χ(CL (ΣT ))

X f ∈H(n,B,G,S)

1 = χ(CL (ΣT ))|H(n, B, G; S)| |Autf | = χ(H(n, B, G, L)),

(5.18)

where we have again used the fact that the Euler character of a bundle is the product of that of the base and that of the fibre [26](the Euler character of the fiber is χ(H(n, B, G; S)) = |H(n, B, G; S)|). When H(n, B, G, L) contains coverings with automorphisms the corresponding space F has orbifold singularities. We introduce the orbifold Euler character χorb (H)

562


as the Euler character of χ(F) calculated by resolving its orbifold singularities. The division by the factor |Aut(f )| is the correct factor for calculating the orbifold Euler characteristic of the subvariety H(n, B, G, L) since Aut(f ) is the local orbifold group of the corresponding point in F. With this understood we naturally define: X 1 (5.19) χorb ((H(n, B, G, L))) ≡ χ(CG (L)) |Autf | f ∈H(n,B,G,S)

in the general case. Thus we finally arrive at our first main result: Proposition 5.1. The CGTS is the generating functional for the orbifold Euler characters of the Hurwitz spaces: 2h−2 X ∞ ∞ X B X 1 χorb (H(n, B, G, L)), (5.20) Z + (0, G, N ) = N n=0 B=0

L=0

where h is determined from n, G and B via the Riemann-Hurwitz theorem. The L = B contribution in the sum is the Euler character of the space of generic branched coverings. As described in Sect. 4.4 compactification of this space involves addition of boundaries corresponding to the space of maps with higher branch points, i.e., where L < B. Quite generally, 5 suppose X is a closed manifold with boundary ∂X. The inclusion ∂X ,→ X gives rise to a long exact sequence in homology · · · → Hi (∂X) → Hi (X) → Hi (X, ∂X) → Hi−1 (∂X) → · · · .

(5.21)

By Lefschetz duality we may write the relative cohomology groups in terms of the homology of the interior X 0 : H i (X, ∂X) = Hn−i (X 0 ),

(5.22) 0

where n is the dimension of X [38]. If n is even then χ(X, ∂X) = χ(X ). Applying the above discussion to Hurwitz space we see that we can interpret B X

χorb (H(n, B, G, L))

L=0

as the Euler character of a partially compactified Hurwitz space (H(h, B, G)) obtaining by adding degenerations of type 1 and 2 and their generalizations. Proposition 5.2. The CGTS is the generating functional for the orbifold Euler characters of the analytically compactified Hurwitz spaces: 2h−2 ∞ ∞ X X 1 χorb ((H(n, B, G))) Z + (0, G, N ) = N n=0 B=0

X 2h−2 ∞ 1 = exp χorb ((H(h, G))) . N h=0

5

We thank E. Getzler for these clarifying remarks.

(5.23)


563

Remarks. 1. Recall that we allow for the possibility of disconnected worldsheets. Expressing the result in terms of connected coverings leads to the final equation. 2. The importance of the high-codimension compactification varieties in the Y M2 partition sum ( Hurwitz spaces with L < B − 1) is extremely intriguing from the point of view of the topological field theory discussed in the remainder of this paper. Y M2 appears to be an example where the “higher contact terms” of the toplogical field theory are extremely important in getting the correct answer. This will be a recurring theme throughout this paper. 3. There is more than one way to interpret the expansion Z + of (5.1) as a sum over maps. The ambiguity arises from the treatment of the −1 terms. Had we used the relations of the symmetric group in writing a formula for −1 we would have found that the coefficient of any permutation v multiplies an infinite series (1/N )n−Kv + · · ·. All powers in the series, but the leading one, would be too large to be accounted for by branching alone. We could still describe the Y M2 partition function in terms of maps but we would need to invoke the pinch maps of Sect. 3.3. As an example, consider the expansion of −1 in (2.6) . In the first line we have written the inverse omega point as a sum where i transpositions come with a factor N −i . Interpreting each factor v as the data of some branch point leads to a description in terms of branched covers. Using the relations of the symmetric group we obtain the last line of (2.6). The higher powers of 1/N must be accounted for by collapsed handles, for example, by pinch maps. The advantage of excluding pinch maps and associating each term in the CGTS with branched coverings is that, as in Sect. 4.2, when the target is equipped with a complex structure such maps can always be interpreted as holomorphic or antiholomorphic. This is an encouraging sign because topological sigma models count (anti)-holomorphic maps [39]. This remark is the first step on the road to the construction of the equivalent string theory in Sect. 6. 4. The paper of Gross and Taylor already showed that by expanding −1 one could interpret all contributions to Z + in terms of branched covers. However, in the picture of [15] there are |2−2G| special points: “twist points. on the target space, and one imagines that all the branch points v1 · · · vL are somehow “anchored” to these special twist points, for all values of L. In this paper we allow the branch points to “sail” over the entire target space ΣT . From the latter point of view the combinatorial factors d(2 − 2G, L) are more natural.

6. Synopsis of Topological Field Theory In this section we briefly review topological field theory. A number of reviews of this subject already exist [41–45] and it is to these that we refer the interested reader for more detail and further references. Topological Field Theories (TFT) study the topology of moduli spaces. There are usually a number of different descriptions of a moduli space, M. For the purpose of formulating a TFT, it is convenient to characterize M as a subspace within a G-manifold6 , e C: 6 Recall that for any group G, a G-manifold is a manifold on which the action of G is defined at every point. In the context of TFTs, Ce and G are typically infinite dimensional. We will remain formal and largely ignore this fact in our discussion.

564


f M = M/G, f = {ϕ ∈ Ce | s(ϕ) = 0}. M e Here s is a G-equivariant map between Ce and an auxiliary G-manifold, V: e s: Ce −→ V. e depend on the moduli The precise natures of the map, s, and the auxiliary G-manifold, V, space under consideration. Remarks. 6.1.1. We may regard

Ee ←−   ye π e C

e V

as an equivariant G-bundle which may be either trivial (as in case of topological Yang-Mills theory [46]) or non-trivial (as in case of topological sigma models e [39]). s induces a section, se, of E. e is 6.1.2. The action of G on Ee may fail to be free, in which case the quotient E/G not a manifold. For example, let Ee = A be the space of gauge connections on a principal G-bundle, P . The group of gauge transformations acts on A. For a reducible connection, A ∈ A, G possesses a non-trivial isotropy group. f This is the case for the 6.1.3. G may, of course, be trivial, in which case M = M. topological sigma model [39]. e Since the quotient space, E/G, is potentially a singular space, we cannot in general study e its cohomology directly. Instead one must examine the G-equivariant cohomology of E. 6.1. Topological Description of Equivariant Cohomology. A sketch of equivariant cohomology necessitates a brief discussion of universal bundles. Imagine that we are given a contractible G-manifold, X, on which G acts freely, then 1. Since EG is contractible, Ee and Ee × X have the same homotopy type and thus have identical de Rham cohomology [34]. 2. Since G acts freely on X, Ee × X inherits a free G action: g · (e, x) = (ge, xg −1 ).

(6.1)

Ee × X G

(6.2)

Then the quotient Ee ×G X

def

=

defines a manifold. The case when X is a principal G-bundle is of particular importance. Such spaces are to a large degree unique:


565

Definition 6.1.1. To a compact, finite dimensional group, G we can associate a universal G-bundle, EG, EG  ←− G  πG (6.3) y BG def

EG is a contractible principal G-bundle over the classifying space, BG = EG/G. EG is unique up to equivariant homotopy type, while BG is unique up to homotopy type. e have the same homotopy type e then Ee ×G EG and E/G Note that if G acts freely on E, and therefore have identical de Rham cohomologies. More generally, Ee ×EG  πG ye Ee ×G EG

←−

G (6.4)

may be viewed as a G-bundle over BG. These properties motivate the following Definition 6.1.2. The G-equivariant cohomology of Ee is defined to be • e def HG,top (E) = H • (Ee ×G EG), d .

(6.5)

Though this definition offers a perfectly sensible way in which to define the topology e of “E/G", the manifold Ee ×G EG is generally quite difficult to study directly. In certain cases, e.g. G finite dimensional and compact, the cohomology of this space may be studied very effectively via indirect means. The pullback by the projection, π eG , induces an injective homomorphism: π eG∗ : • (Ee ×G EG) −→ • (Ee × EG).

(6.6)

e G e Since E×EG is topologically a simpler space, it is very useful to characterize π eG∗ H • (E× • e e EG) as a subset of H (E × EG). Since G acts freely on E × EG, there is at each point x ∈ Ee × EG a map of G into the fiber over y = π eG (x), Rx : G −→ Ee × EG|y ,

(6.7)

where Ee × EG|y = π eG−1 (y). The differential of Rx defines a map from g = Lie G into the vertical tangent space def

def Cx = dRx : g −→ Tx (Ee × EG)vert

(6.8) def

Cx : g 7−→ Xg |x = Cx g. This defines two actions of g on • (Ee × EG): 1. Contraction: For all g ∈ g, i(g): k (Ee × EG) i(g): ω

−→ 7−→

k−1 (Ee × EG) iXg ω,

where iXg is the usual interior product with a vector field Xg .

566


2. Lie Derivative: For all g ∈ g, L(g): k (Ee × EG) L(g): ω

−→ 7−→

k (Ee × EG), LXg ω,

where LXg ≡ [iXg , d]+ is the usual Lie derivative with the vector field Xg . These derivations characterize a special subcomplex of • (Ee × EG). Definition 6.1.3. Let {Ti } be a basis for g. A form η ∈ • (Ee × EG) is called g-basic, if it is both Tdim G 1. Horizontal, i.e. η ∈ i=1 ker i(Ti ), and Tdim G 2. Invariant, i.e. η ∈ i=1 ker L(Ti ). The g-basic subcomplex will be denoted by (Ee × EG)g−basic . Theorem 6.1.1. In the case that G is finite dimensional and compact, the de Rham cohomology of Ee ×G EG is precisely the basic cohomology of Ee × EG.

Proof. See Mathai and Quillen [47]

It follows from Definition 6.1.2 and Theorem 6.1.1, that the G-equivariant cohomology of Ee can be computed as follows: • e ∼ HG,top (E) = H • (Ee ×G EG), d ∼

=H

•

(6.9)

(Ee × EG)g−basic , d .

Remarks. 6.1.4. The local gauge groups, G, encountered in TFT are neither finite dimensional nor compact. Nevertheless the cohomologies found are remarkably similar to those of related compact groups. π e e and fiber metric, (·, ·) , 6.1.5. If Ee−→Ce is a non-trivial bundle with standard fiber V, e V e bundle, Fe, the bundle of all orthonormal then Ee is associated to a principal SO(V) e frames on E: e (6.10) Ee = Fe ×SO(Ve) V. It is then convenient to express the cohomology of Ee in terms of the basic cohoe i.e. from Theorem 6.1.1 mology of Fe × V, e d ∼ e d = H • (Fe ×SO(Ve) V), H • (E), (6.11) ∼ e = H • (Fe × V) e)−basic , d . so(V Note also that in this case the G-equivariant cohomology of Ee is given by

Large N 2D Yang-Mills Theory and Topological String Theory • e HG,top (E)

=H

•

567

e × EG) (Fe × V e)−basic , d g⊕so(V

(6.12)

e and G. so that effectively there are two gauge groups SO(V) 6.2. Algebraic Description of Equivariant Cohomology. There are also algebraic models e These are far from unique, a fact which partially for the G-equivariant cohomology of E. accounts for the abundance of TFT for a given moduli space. For reasons of brevity we shall direct most of our attention to the Cartan model. As was shown by Kalkman [21], this model is closely related to the BRST model which is often used in physics. We shall discuss other models only briefly; we refer the interested reader to [21, 45, 47] for a fuller description of the various algebraic models and their equivalence to one another. The Cartan model proceeds from the complex e S • (g∗ ) ⊗ • (E),

(6.13)

where g∗ is the dual to g and S • (g∗ ) is the symmetric algebra on g∗ , which is freely generated by {φi }i=1,...,dim g . A differential, dC , may be defined via its action on the generators of the complex d C φi = 0

∀φi ∈ S 2 (g∗ )

dC ω = (1 ⊗ d − φi ⊗ i(Ti ))ω

e ∀ω ∈ • (C)

(6.14)

In analogy to the geometric Lie derivative, one may define an algebraic Lie derivative e to be Li def = [1 ⊗ i(Ti ), dC ]+ . Note that d2C = −φi ⊗ L(Ti ) = φi Li ⊗ 1; so on S(g∗ ) ⊗ • (E) that dC is an nilpotent only on the subcomplex of equivariant differential forms, defined as G e def e = S • (g∗ ) ⊗ • (E) , (6.15) G (E) e where the superscript (·)G denotes the G-invariant subcomplex. That is η ∈ S(g∗ )⊗• (E) Tdim G is an equivariant differential form if η ∈ i=1 ker Li . This subcomplex corresponds to the basic subcomplex [21, 45]. This motivates the following Definition 6.2.1. The algebraic definition of the G-equivariant cohomology of Ee is • e def e dC . HG,alg (6.16) (E) = H • G (E), Theorem 6.2.1. For G finite dimensional and compact: ∼

• e = H • (E). e (E) HG,top G,alg

Proof. Please see [47, 21, 45].

(6.17)

The relationship between the de Rham and Cartan models can be made very concrete. Given a connection, ∇, and its curvature, F∇ , we may define the Chern-Weil homomorphism7 , wC,∇ : 7 The subscript C indicates that this homomorphism acts on the Cartan complex. We will in Sect. 6.3.2 define an analogous homomorphism for the Weil model.

568


e −→ • (E) e g−basic , wC,∇ : •G (E) (6.18) wC,∇ : P(φ) 7−→ (P(F∇ ))

hor

.

The superscript (· · ·)hor indicates projection onto the horizontal subcomplex via ∇. 6.3. The Mathai-Quillen Representative of the Thom Class. The Thom class is central to the construction of TFT actions [20]. In this subsection we shall outline constructions of representatives of this class in the context of various models of equivariant cohomology. Let Ee be an orientable vector bundle Ee ←−   ye π Ce

e V

• e (E), the cohomology of forms that are rapidly decreasing in the vertical We consider Hvrd 8 direction On such forms integration along the fiber is well-defined:

e(E) E e → • (C). e π e∗ : •+rank vrd

(6.19)

In fact, the cohomologies of these two complexes are isomorphic. Theorem 6.3.1 (Thom Isomorphism). Integration along the fiber defines an isomorphism e(E) •+rank E e ∼ e = H • (C) (6.20) π e∗ : Hvrd Proof. Please see Bott and Tu [34].

The Thom class, of Ee is defined as h i e def = (e π∗ )−1 (1) 8(E)

∈

e(E). rank E e Hvrd

(6.21)

e i.e. a particular representative of the Thom class, will in general A Thom form, 8∇ (E), e the Thom isomorphism is e In terms of such a 8∇ (E), depend on a connection, ∇, on E. explicitly given by: e −→ H •+rank Ee(E), e T : H • (C) vrd e T (ω) 7−→ π e∗ (ω) ∧ 8∇ (E). e bundle, F. e A From Remark 6.1.5 we know that Ee is associated to a principal SO(V) e Thom form may therefore be constructed in the context of an algebraic model of SO(V)9 equivariant cohomology. An element UC−SO(Ve) is called a universal Thom form, if it is related via the Chern-Weil homomorphism to a Thom form. The relevant complexes fit together as indicated in the diagram below: 8 This is equivalent to H • (E e cv ), the cohomology of forms that are compactly supported along the (vertical) fiber direction [47] 9 The subscript C − SO(V e) indicates that we are working within the Cartan model for SO(Ve).


SO(Ve)

e ∗ ) ⊗ • (V) e S(so(V)

569

e × E SO(V) e • V e)−basic so(V x  ∗ π e & w¯ C,∇  SO(e V) • e e V ×SO(Ve) E SO(V)

wC,∇

−→

Diagram 6.3.2

and the various forms are related as follows: UC−SO(Ve)

e wC,∇ (UC−SO(Ve) ) = π e∗ (8∇ (E)) x SO(Ve) e ∗ & w¯ C,∇ πSO(e V) e = w¯ C,∇ U 8∇ (E) e) C−SO(V

wC,∇

−→

Diagram 6.3.3

Mathai and Quillen [47] constructed an explicit representative of the universal Thom e and introduce anticommuting orthonormal class. Let xi be orthonormal coordinates for V e ∗ . Here Π is the parity change functor [48] and indicates that we coordinates, ρi , for Π V e ∗ as being anti-commuting. Then a representative of are to regard the coordinates of Π V the universal Thom form may be written as 1 rank Ee 1 2 UC−SO(Ve) = 4πt Z 1 dρ exp − (x, x)e + ihρ, dxi + t(ρ, φρ) , V V∗ e 4t e∗ ΠV

(6.22)

where (·, ·)X denotes the inner product on X, while h·, ·i denotes the dual pairing. Note that we actually have a one parameter family of representatives that depend on t ∈ R. Remarks. 6.3.1. We are often interested in constructing the Thom class of Ee ×G EG → Ce×G EG. e × GFrom Remark 6.1.5 we know that in this case we need to consider SO(V) equivariant cohomology. In the following we shall construct a number of other universal Thom forms; we shall leave the total local symmetry group unspecific, to allow for the possibility of non-trivial G. 6.3.2. Mathai-Quillen representatives of Thom classes play a central role in the construction of topological field theories [20]. The argument of the exponential in the Mathai-Quillen representative is interpreted as the action of a TFT. For example, the term hρ, dxi is the kinetic term for the ghost/anti-ghost system. 6.3.3. ker ∇s are the ghost zero modes, while coker ∇s are the anti-ghost zero modes. When coker ∇s 6= 0, the Grassmann integral over the anti-ghosts in the MathaiQuillen representative of the Thom class brings down powers of the curvature from the argument of the exponential. This fact will be made precise in the localization theorems. The main use of Thom classes stems from the following

570


e by any section, se: Ce → E, e Proposition 6.3.1. The pullback of the Thom class, 8∇ (E), e is the Euler class of E.

Proof. Please see [34, 35].

We conclude this subsection by briefly indicating a few other representatives of the Thom class. Please see [45] for more details. 6.3.1. Another Representative of the Universal Thom Class in the Cartan Model. If we e ∗ , and extend the Cartan differential introduce further commuting coordinates 10 , π for V • e • • ∗ ∗ e to S (g ) ⊗ (V) ⊗ (Π V ), via QC = 1 ⊗ d ⊗ 1 + 1 ⊗ 1 ⊗ d − φi ⊗ i(Ti ) ⊗ 1 − φi ⊗ 1 ⊗ i(Ti ), ρ 0 QC = π −φi ⊗ L(Ti )

1 0

(6.23)

ρ , π

then we may compactly write UC−G =

1 2π

rank E Z e∗ ×Π Ve∗ V

∗ dπ dρ exp −QC −i hρ, xi − t(ρ, π)V e .

(6.24)

The significant feature of this formulation is that the argument of the exponential (the TFT action) is QC -exact. 6.3.2. A Representative of the Universal Thom Class in the Weil Model. The Weil model of the G-equivariant cohomology of Ee starts from the complex: W(g) = 3(g∗ ) ⊗ S(g∗ ), where 3(g∗ ) is the exterior algebra of g∗ which is freely generated by {θi }i=1,...,dim G . The differential of the Weil complex need not concern us here. A fuller discussion may be found in [45, 21, 47]. The universal Thom form within the Weil model is by:

UW−G =

1 π 2 rank Ee 1

e

−(x,x)

e V

Z e∗ ΠV

dρ exp

1 (ρ, φρ)e + ih∇x, ρi , ∗ V 4

(6.25)

where ∇x = dx + θ · x. In the context of the Weil model, the Chern-Weil homomorphsim, wW,∇ , simply θ A makes the replacement , where A is a connection on Ee and FA is → φ FA its curvature. The absence of horizontal projection is often convenient. On the other hand the Chern-Weil homomorphism explicitly introduces a connection which in many situations is non-local. In these cases the Weil model is unsuitable for the construction of TFT actions. 10 Notation: Conventions and the paucity of alphabets force us to use π for both the Lagrange multiplier fields and the projections. Projections between (in general) infinite dimensional spaces will have a tilde; projections between (in general) finite dimensional spaces have a bar. Lagrange multiplier fields have neither accent.


571

6.3.3. A Representative of the Universal Thom Class in Hybrid Cartan and Weil Models. To construct a universal Thom form on Ee × EG → Ee ×G EG, we know from Remark e × G-equivariant cohomology. It is useful to 6.1.5 that it is useful to work within SO(V) use different algebraic models for the equivariant cohomologies of these groups. Since e connection is local, it is convenient to work in the context of the Weil model, the SO(V) thereby explicitly introducing the connection, but obviating the horizontal projection. On the other hand, the G-connection is generally non-local, so that for purposes of constructing a TFT action, it is essential that we work in the context of the Cartan model for G. The diagram depicting the interrelation of the various complexes is the natural generalization of Diagram 6.3.2. G wW,∇ ˜ G SO(V) ˜ ˜ ˜ S(g ∗ )⊗ W(so(V))⊗ (V) −→ S(g ∗ )⊗(F˜ ×V) ˜ so(V)−basic ˜ so(V)−basic x π˜ ∗ ˜ &w¯ W,∇SO(V)  SO(V)˜ G ∗ ˜ S(g )⊗(E) wC,∇

˜ (E×EG) x g−basic π˜ ∗  G ˜ G EG) (E×

G

−→

&w¯ C,∇G

Diagram 6.3.3

The universal Thom form, UW−SO(V),C−G , takes its values in the complex (S(g∗ ) ⊗ ˜ e ⊗ (V)) e )G . Applying the Chern-Weil homomorphism we obtain (W(so(V)) ˜ so(V)−basic wW,∇SO(V)˜ UW−SO(V),C−G = π˜ ∗ e ΥC−G , ˜ SO(V ) where ΥC−G ∈ G (E) will play an important role in the general localization theorem (Proposition 6.5.1). Altogether, we have wW,∇

UW−SO(V),C−G ˜

˜ SO(V)

−→

wW,∇

˜ &w¯ W,∇SO(V)

˜ SO(V)

∗ (UW−SO(x )=π˜ SO( ˜ ˜ (ΥC−G ) V),C−G V) π˜ ∗  SO(V)˜ wC,∇

G

ΥC−G −→

&w¯ C,∇G

wC,∇G (ΥC−G )

x π˜ ∗  G

˜ G EG) 8∇ (E×

Diagram 6.3.4 Explicitly

Z ΥC−G =

V˜ ∗ ×Π V˜ ∗

dπ dρ exp −QC 9Loc ,

(6.26)

e analogous to where QC is the Cartan differential for G-equivariant cohomology of E, (6.23); and where 9Loc = −i hρ, xi + t(ρ, 0SO(Ve) · ρ)Ve∗ − t(ρ, π)Ve∗ .

(6.27)

572


6.4. The Localization Formula for trivial G. We shall now apply the construction of Thom classes to sketch a localization formula for the simpler case when G is trivial. Consider an orientable vector bundle e Ee ←− V   ye π e C Let se: Ce → Ee be a section. In terms of a local trivialization, we may write this as se = (id, s)

where

e s: Ce −→ V.

The subspace of interest is characterized by f = {ϕ ∈ Ce | s(ϕ) = 0}. M

(6.28)

e the differential of s is a map For every ϕ ∈ C, e ds|ϕ : Tϕ Ce → Ts(ϕ) V.

(6.29)

∼

e and we may view ds|ϕ as a linear operator: e is a linear space, T V e=V Actually, since V e ds|ϕ : Tϕ Ce → V. Clearly

f ker ds|ϕ = ker ∇s|ϕ ⊂ Tϕ M

f ∀ϕ ∈ M.

Moreover, if ds|ϕ is injective, then [45] ∼ f ker ∇s|ϕ = Tϕ M

It is also clear that

Im ds|ϕ = Im ∇s|ϕ

f ∀ϕ ∈ M.

(6.30)

f ∀ϕ ∈ M.

f Now consider the exact sequence of bundles over M: 0 → Im ∇s → Ee → coker ∇s → 0.

(6.31)

Proposition 6.4.1. Let se = (id, s) be a section of the orientable vector bundle Ee as f → C. e If P ⊂ M f is the Poincaré dual to above. Let i denote the inclusion, i: M e in C. e f then i(P ) ⊂ Ce is Poincaré dual to e(Ee → C) e = se∗ 8∇ (E) e(coker ∇s → M), Proof. For a physical proof, please see [49, 50].

It follows from Propositions 6.3.1 and 6.4.1 that e ∈ H index ∇s (C), e we have Proposition 6.4.2. For O Z Z e ∧O = se∗ 8∇ (E) e(coker ∇s → M) ∧ i∗ O, e C M f = M. where we have used the fact that for trivial G, M Proof. For a discussion, please see Sect. 11.10.3 of [45].

(6.32)


573

Proposition 6.4.2 is the basis for the construction of TFTs without local symmetries. There are a number of applications of this localization theorem to super quantum mechanics and topological sigma models. For a survey of such applications as well as a more extensive list of references, please see [45]. 6.5. The Localization Formula for Non-trivial G. In order to give a description of Y M2 as a topological string theory, we need to consider a more general localization theorem with G non-trivial. Let Ee be an orientable G-equivariant vector bundle e Ee ←− V   ye π e C and se: Ce → Ee a G-equivariant section. Then se induces a section, s: ¯ s e Ee ×EG ←− Ce ×EG  π¯ πG ye y G s¯ e e E ×G EG ←− C ×G EG These maps induce pullback maps between the corresponding de Rham complexes: e s∗ × EG) −→ • (Cex × EG) • (Eex π¯ ∗  ∗ π e  G  G s¯ ∗ • e e (E ×G EG) −→ (C ×G EG) Though Ee × EG → Ee ×G EG is a principal G-bundle, we may define an analogue of the Thom isomorphism for vector bundles [45] e TG : H • (E) → H •+dim G (E), (6.33) TG (ω) 7→

π eG∗ (ω)

e ∧ 8G (E),

e ∈ G (E) e is partially characterized by the fact that for all x ∈ Ee × EG, where 8G (E) ∗ e is the normalized Haar measure of G. Please see [45]for a fuller discussion. Rx 8G (E) e Now π e∗ (ω) ∈ (Ee×EG)g−basic , so that by Theorem 6.1.7 it is related to $ ∈ G (E) G

via the Chern-Weil homomorphism: eG∗ (ω). wC,∇G ($) = π

(6.34)

We may then define a map e → • (Ee × EG), SG : •G (E) (6.35) SG ($) 7→ wC,∇G ($) ∧ 8G . The virtue of this map is that it may be readily interpreted in the context of a TFT. Introduce λa and ηa as generators of S(g) and 3(g), respectively and extend the action of the Cartan differential on these generators as follows: 0 1 λ QC ( λη ) = . (6.36) i −φ ⊗ L(Ti ) 0 η Then we have the following

574


e Proposition 6.5.1. For all $ ∈ •G (E), ( dim Z 1 SG ($) = dφ $ ∧ 2πi where

G

)

Z dλ dη exp −QC 9Proj

,

(6.37)

g×Πg

9Proj = −i(λ, C † )g .

(6.38)

Proof. Please see [45). Remarks.

6.5.1 We have used the metric on Ee to define the adjoint, Cx† for all x ∈ Ee × EG: Cx : g −→ Tx Ee Cx† : Tx Ee −→ g. e We may view C † as a g-valued 1-form and so (λ, C † )g ∈ Tx∗ E. 6.5.2 This procedure is distinct from Faddeev-Popov gauge fixing. Note that no section of Ee×EG → Ee×G EG enters into our discussion. For a more careful comparison, please see [45). Again it is useful to depict the interrelation of the various complexes and maps diagramatically: G

˜ S(g ∗ ) ⊗ (E)

∗

SG

s˜ ˜ EG) (E˜ × x EG) −→ (C ×  T (π¯ ) & w¯ ∇G  G y G∗ (E˜ ×G EG) −→ (C˜ ×G EG) ∗

−→

s¯

Diagram 6.5.4

e which we constructed in (6.26) is related to the The object Υ∇SO(V)˜ ,C−G ∈ G (E) equivariant Thom class, 8∇ (Ee ×G EG), via the following diagram Υ∇SO(V)˜ ,C−G

SG Υ∇SO(V)˜ ,C−G ˜ ×G EG) = TG 8(Ex T & w¯ C,∇G  G 8(E˜×G EG) = w¯ C,∇G Υ∇SO(V)˜ ,C−G SG

−→

s˜ ∗

−→

s˜∗ SG Υ∇SO(V)˜ ,C−G  (π¯ ) y G∗

−→ ∗ s¯

s¯∗ 8(E˜ ×G EG) = (π¯ G )∗ se∗ SG Υ∇SO(V)˜ ,C−G

Diagram 6.5.5

From this it is apparent that s¯∗ 8(Ee ×G EG) = se∗ SG Υ∇SO(V)˜ ,C−G ,

(6.39)


575

where Υ∇SO(V)˜ ,C−G is given by (6.26). e We shall assume that G acts freely on Ee and C: e E = E/G, e C = C/G in order that M be a manifold11 . Then we know from Proposition 6.4.2 that for O ∈ H index∇s¯ (C), Z Z s¯∗ 8(Ee ×G EG) ∧ O = e(coker ∇s¯ → M) ∧ i∗ O (6.40) C

and from (6.39)

M

Z

s¯ 8(Ee ×G EG) ∧ O = ∗

C

Z e C

se∗ SG ΥC−G ∧ π eG∗ O.

(6.41)

Combining (6.40) and (6.41) we arrive at the following Proposition 6.5.2. For O ∈ H index∇s¯ (C), Z Z eG∗ O = SG Υ∇SO(V)˜ ,C−G ∧ π e(coker ∇s¯ → M ∧ i∗ O e C M

(6.42)

Remarks. b whose odd coordinates are generated from 6.5.3. If we introduce a supermanifold C, ∗e the fibers of T C, then b ∼ e = • (C). C ∞ (C) On the other hand, Cb has a natural measure µ b = dϕ1 ∧ · · · ∧ dϕn dψ 1 ∧ · · · ∧ dψ n , b If ω b corresponds to the b ∈ C ∞ (C) where (ϕi , ψi ) are local coordinates on C. • e differential form ω ∈ (C), then Z Z ω= µ bω b b e C C so that we can rewrite the integral over Ce in superspace form. 6.5.4. The vector bundle coker ∇s¯ → M, though crucial in the general localization e vert is a simpler formula, is difficult to work with directly. ∇s: Tϕ Ce → (Ts(ϕ) E) operator. Since s is G-equivariant, ker ∇s and coker ∇s are in general infinite dimensional. However the operator def ∇s e vert ⊕ g O= : Tϕ Ce −→ (Ts(ϕ) E) (6.43) C+ f defines equivariant vector bundles of finite rank, ker O and coker O, over M. These descend to vector bundles over M. The operator O is of direct importance to TFT as it appears as the fermionic kinetic term of the complete lagrangian. 11 Hurwitz space is, in fact, not smooth but possesses orbifold singularities. In this case, we actually compute orbifold Euler characters. For more singular spaces, we do not know a general prescription.

576


Finally using Remarks 6.5.3 and 6.5.4 we may rewrite (6.42) in a way that makes the TFT action more apparent: Z e(coker ∇s¯ → M) ∧ i∗ O M Z f e((coker O → M)/G) ∧ i∗ O (6.44) = M Z Z Z [dφ] [dϕ] [dψ] [dπ] [dρ] ∼ e×Π Ce e∗ ×Π Ve∗ g∗ C V Z [dλ] [dη] exp −QC 9Loc + 9Proj g×Πg

where we have absorbed the normalizations into the measures [d · · ·]. The TFT action may be identified with ITop = QC (9Loc + 9Proj ) where 9Loc is given by (6.27) and 9Proj is given by (6.38). 7. Topological String Theory and the Chiral Theory

7.1. Standard topological string theory. The basic configuration space is given by12 Ce = {(f, h) | f ∈ C ∞ (ΣW , ΣT ) and h ∈ Met −1 (ΣW )},

(7.5)

where Met −1 (ΣW ) is the space of metrics on ΣW with constant Ricci scalar curvature13 −1 Hurwitz space may be described by the Gromov equation for (pseudo-) holomorphic maps: The localization to Met −1 (ΣW ) ⊂ Met(ΣW ) is standard. (For a review and more extensive references, please see [45).) For the localization Lagrangian we introduce a scalar antighost ρ and its Lagrange multiplier π. Then the gauge fermion for localizing to Met(Σ)k is: 12

Z

9Weyl Loc = Using the following two relations:

√ d2 z h ρ (R + 1).

(7.1)

α

QC 0βγ = 21 hαδ Dβ Qhγδ + Dγ Qhβδ − Dδ Qhβγ , (7.2) QC R = − 21 Dα D α (hγβ Qhβγ ) + D α D β Qhαβ − 21 R hαβ Qhαβ we may write this action as

Z IWeyl Loc =

where

13

√ d2 z h

π (R + 1) − ρ Lαβ ψαβ

,

Lαβ = D α D β − 21 hαβ D 2 + 21 hαβ .

We assume for simplicity that the genus of the world sheet is greater than one.

(7.3)

(7.4)


577

+ f M = M/Diff (ΣW ),

(7.6) f = {(f, h) ∈ Ce | df + J df [h] = 0}. M f the tangent space is described by (see Appendix B for a derivation) At (f, h) ∈ M, f = {(δf, δh) ∈ T(f,h) Ce | D(δf ) + J D(δf ) [h] + J df k[δh] = 0}, T(f,h) M

(7.7)

where D is the pulled-back connection (Dα δf )µ = ∂α δf µ + 0µκλ ∂α f κ δf λ ; and k[δh] is the variation of the complex structure. Equation (7.7) suggests that we introduce an operator e(f,h) , D(f,h) : T(f,h) Ce −→ V (7.8) D(f,h) (δf, δh) 7−→ D(δf ) + J D(δf ) [h] + J df k[δh], e(f,h) will be defined shortly. where V To construct a topological string theory action, we regard the Gromov equation, (7.6), as a Diff + (ΣW )-equivariant section e s: Ce −→ E, (7.9) s(f, h) 7−→ df + J df [h], where Ee is a Diff(ΣW )-equivariant vector bundle whose fiber above (f, h) ∈ Ce is given by e(f,h) : = 0[T ∗ ΣW ⊗ f ∗ (T ΣT )]+ . (7.10) V The superscript (· · ·)+ indicates that the sections must satisfy the self-duality constraint: ρ ∈ 0[T ∗ ΣW ⊗ f ∗ (T ΣT )]+ ⇐⇒ ρ − J ρ [h] = 0.

(7.11)

e Ee admits an SO(V)-connection, ∇SO(Ve) , characterized by (6.30), f ker ∇SO(Ve) s|(f,h) = T(f,h) M

f ∀(f, h) ∈ M.

e and define Let sα µ [f, h] be a local section of Ee → C, Z √ δsµα δsµα µ 2 κ (∇SO(Ve) s)α = d σ h δf δhβγ (σ) (σ) + δf κ (σ) δhβγ (σ)

(7.12)

− 0µκλ [f (σ), h(σ)]sα κ δf λ (σ) . Then for s[f, h] = df + J df , one may readily check that ∇SO(Ve) s = D(δf, δh). Having determined ∇SO(Ve) , we may construct Υ∇

e ,C−G

SO(V )

(7.13) :

578


Z ΥC−G =

V˜ ∗ ×Π V˜ ∗

dπ dρ exp −ITop σ ,

where from (6.26) and (6.27): h i ITop σ = QC i hρ, s(f, h)i − t(ρ, 0SO(Ve) · ρ)Ve∗ + t(ρ, π)Ve∗ . e there is a canonical isomorphism Ce is a Diff(ΣW )-manifold, so that for all (f, h) ∈ C, vert e between diff(ΣW ) and (T C) given by e vert C(f,h) : diff(ΣW ) −→ (T C) C(f,h) (γ) 7−→

Lγ f . Lγ h

(7.14)

Hence from (6.38) the projection fermion is given by 9Proj = −i(λ, C † )g . 9Weyl Loc + 9Top σ + 9Proj , and therefore the action we have produced thus far, is Diff(ΣW ) invariant. Therefore, in order to compute anything using the standard methods of local quantum field theory, we still need to fix this symmetry [51]. This may be done by adding the (gauge-fixing) action Z √ (7.15) 9GF = d2 z h bαβ (hαβ − h(0) αβ ), where the action of QC extends to the symmetric tensor fields, bαβ and dαβ as QC bαβ = dαβ . Altogether the action of standard topological string theory is given by ITS = IWeyl Loc + ITop σ + IGF + IProj .

(7.16)

Our objective is to find a string theory whose connected partition function is Zstring ∼ χorb (M) (7.17)

Z = M

e(T M → M).

If we pursue the construction outlined in Sect. 6, it is apparent that we indeed obtain a theory that localizes to Hurwitz space. From (6.44) the measure is given by e(coker O/G), where (7.12) and (7.14) together define O(f,h) . Unfortunately, however [45), ∼

ker O(f,h) /G = T(f,h) M, (7.18) ∼

coker O(f,h) = {0}, so that the standard topological string theory clearly does not produce the desired measure, e(T M → M). It is clear that we have to modify this theory somewhat. The following gives a clue about what this modification ought to entail.


579

e is endowed with a metric, we may define the adjoint of O(f,h) , which we Since V may view as an operator e ⊕ g −→ T(f,h) C. e O†(f,h) : Ts(f,h) V From (7.18) it follows that ∼

ker O† = {0}, (7.19) ∼

coker O† /G = T(f,h) M. Clearly we want to produce a TFT wherein the fermion kinetic term is OTotal = O ⊕ O† .

(7.20)

In this case, ker OTotal = coker OTotal = T M, so that such a theory would produce the correct measure. 7.2. “Cofields”. In order to obtain (7.20) as the fermion kinetic operator, we must extend the field space relative to that of the standard topological string theory. The new fields are completely determined by two requirements 1. O† maps ghosts to antighosts, 2. QC extends to act on the new fields as the Cartan differential of Diff(ΣW )-equivariant cohomology. b we To describe the additional fields, it is easiest to begin with the new set of ghosts, G; shall sometimes refer to these as the “Co-Ghosts”. These take values in the domain of O† , so that b ∈ 0(T ΣW ⊗ f ∗ (T ∗ ΣT ))+ ⊕ 0(T ΣW ) G (7.21) and their index structure is given by b= G

χ bµ α ψbα

.

bµ α satisfy a self-duality constraint: As usual, the superscript (· · ·)+ indicates that the χ χ bw z = 0. e This The Co-Ghosts represent differential forms on an enlarged field space, D. e → C, e enlarged field space may itself be viewed as the total space of a vector bundle, D where the fibre at (f, h) ∈ Ce is given by e(f,h) = 0 T ΣW ⊗ f ∗ (T ∗ ΣT ) + ⊕ 0(T ΣW ). D We refer to the additional fields as “Co-Fields”; like the Co-Ghosts, their index structure is given by b = fbµ α b (7.22) F hα , where fbw z = 0. e forms the base space of a Diff(ΣW )-equivariant vector bundle, In turn D

580


Ee ⊕Eecf  ye π e D

←−

e⊕V ecf V

Now consider the following section: e −→ V e⊕V ecf , S: D (7.23) b e 7−→ (df + J df [h], O† F). S: (F, F) The zero set of this section is still Hurwitz space b ∈D b = 0} = M × {0}, e | s(F, F) {(F, F)

(7.24)

since ker O† = {0}. Moreover, when restricted to M × {0}, the operator appearing in the total fermion kinetic term is given by OTotal = O ⊕ O† . Our choice of section dictates e is defined in (7.10), while the e⊕V ecf , where V that the antighost bundle be dual to V range of O† defines ecf = 0(f ∗ (T ΣT )) ⊕ 0(Sym(T Σ ⊗2 )), V W or

e= A

ρbµ ηbαβ

(7.25)

.

(7.26)

QC extends to the Co-Fields as the Cartan differential for Diff(ΣW )-equivariant cohomology, b b b b 0 1 0 1 F A A . = = Q QC F C b b b b −Lγ 0 −Lγ 0 G G I˙ I˙ The addition of the cofields does not change the QC -cohomology, so we expect to have the same observables as in topological string theory. (Please see Sect. 7.3). The Lagrangian for the YM2 string will be a sum of a Lagrangian for the topological b = 0: string theory ΣW → ΣT plus a Lagrangian for localizing to F IY M2 = ITS + Icf .

(7.27)

Following Sect. 6, we write down the gauge fermion for the co-fields: E D b b b O† F b − t A, I˙ , 9cf = A, where t ∈ R and O† =

−(δµ ν ∇γ + Jµ ν ∇β β γ ) −δ α γ J ν µ ∂δ f µ δβ

−Gµν ∂γ f ν 1 α β β α 2 (δγ D + δγ D )

(7.28)

.

(7.29)

To derive the localization theorem for this theory, we must analyze the exact bundle sequence analogous to (6.31). For the choice of section, S, and connection, ∇SO(Ve) , we have e⊕V ecf −→ coker (O ⊕ O† ) → 0 0 −→ Im (O ⊕ O† ) −→ V


581

as a sequence of bundles over M×{0}. Then by the general principles we have explained in the previous section, we see, by combining (7.18) and (7.19), with (6.44), that the path integral computes the Euler character of the cokernel bundle, T M, which is the problem we set out to solve. 7.3. Observables in the theory. The most obvious observables in topological string theories are made from gravitational descendents of the primaries of the corresponding topological sigma model. The observables in topological string theories are of two types: (a) homology observables and (b) homotopy observables [46, 52–54]. Homology Observables. These observables are built from cohomology classes of the target space. For a target space a Riemann surface of genus G, the cohomology classes are described by: {1} ∈ H 0 (ΣT ), ¯

{ξ A , ξ A } ∈ H 1 (ΣT )

A = 1, . . . , G,

(7.30)

{ω} ∈ H (ΣT ), 2

where ω is the Kähler class. So the homology observables of the topological string theory are given by:

σn (1), ¯ A i σn (ξi χ ), σn (ξiA χi ) A = σn (ωij χi χj ),

1, . . . , G,

(7.31)

where σn (· · ·) represents the gravitational dressing of the operator. In essence [55] σn (O) = (αβ ∂ α γ β )n O. Homotopy observables. There is a ring of homotopy observables with 2G generators: ( Ok = exp 2πi

X G A=1

Z

Z

f (z)

kA ·

ξ A + k¯ A¯ · w0

)

f¯(z)

ξ

A¯

,

(7.32)

w¯ 0

where k = (k[1] , . . . , k[G] ) are vectors in the dual to the period lattice, 3. We expect that these operators will form a ring related to the group ring of the fundamental group of the target manifold. (Since the matter is not a topological conformal field theory we expect it will involve a nontrivial deformation of that ring.) Via the descent equations we may obtain 1-form versions of the operators (7.23), which generate an algebra of symmetries which we naturally expect to be related to w∞ . The operators relevant to the topological string theory are, of course, the gravitational dressings of the above. Indeed, 2D string theories are famous for having w∞ -type symmetries in the target space theory. This should be true in our case and should explain the area-preserving diffeomorphism invariance of Y M2 from the string perspective. We have not carried this out in detail.

582


8. Turning on the Area 8.1. Area polynomials in Y M2 . The same basic reasoning we have used in the A = 0 case can be applied to the A > 0 case. We begin with the 1/N expansion of the chiral Y M2 partition function. Manipulations identical to those leading to (5.7) give Z + (A, G, N ) ∞ ` X 1 n2 (−A) = N n(2−2G)−` e− 2 A(n− N 2 ) `! =

n,`>0 ∞ ∞ X X

1

n2

e− 2 A(n− N 2 )

n=0 `=0

X

s1 ,t1 ,...,sG ,tG ∈Sn

G

si ,ti ∈Sn

1

(−A) `!

p1 ,...,pk ∈T2 ⊂Sn L0 =0 v1 ,...,vL0 ∈Sn

X

Y 1 ` δ(2−2G T2,n [si , ti ]) n n!

`

0 X

∞ X

X

1 n(2G−2)+`+ N

PL0 j=1

(Kvj −n)

Y d(2 − 2G, L0 ) δ(p1 · · · pk v1 · · · vL0 [si , ti ]), n! G

(8.1)

i=1

where T2,n ∈ C[Sn ] is the sum of transpositions. Recall that to establish a correspondence between homomorphisms from FG,L to Sn and branched coverings over a set S we make a choice of generators of π1 (ΣT − S, y0 ). This choice leads to an association of a set of k points with the permutations p1 · · · pk . After collecting powers of N we get Z + (A, G, N ) =

∞ ∞ X X

n2

e−nA/2 e 2N 2 A

n=0 B=0

1 N

2h−2 X B

Pn,B,L (A),

(8.2)

1 , |C(ψ)|

(8.3)

L=0

where Pn,B,L (A) is a polynomial defined by: Pn,B,L (A) =

B B X (−A)k X k=0

k!

X

χ(CL−k (ΣT ))

9(n,B,G,L,k)

L=k

where 9(n, B, G, L, k) is the set of homomorphisms FL,G → Sn which, via Theorem 3.1, correspond to branched coverings over some set S of branch points, with the property that over a fixed subset of k points the branching is simple. For example, if ψ has only simple branch points, i.e., corresponds to a map in the simple Hurwitz space then the formula becomes: ψ ∈ 9(n, B, G, L = B) =⇒ Pψ (A) =

B X (−A)k k=0

k!

χ(CB−k (ΣT )).

(8.4)

In order to obtain the full 1/N expansion we must also expand the factor n2

e 2N 2 A

(8.5)

in (8.2). This has been interpreted in [16] and in [15] in terms of contributions of “collapsed tubes and handles”.


583

8.2. Area polynomials from perturbations. In the topological field theory there is a natural mechanism by which the area can be included: perturbation by BRST closed but nonexact operators. This is how, e.g., one explores the more physical phases of 2d gravity, studied in the double scaling limit of matrix models, in the framework of topological 2D gravity. In the present case we will study the perturbation of the action by a BRST R invariant operator 21 A(2) . Here A(2) fits into the area operator descent multiplet: A(0) = σ0 ωij (f (x))χi χj , (8.6) A(1) = σ0 dxα ωij (f (x))∂α f i χj , (2) α β i j A = σ0 dx ∧ dx ωij (f (x))∂α f ∂β f . Here ω is the Kähler two-form from the target space. The form degree 0, ghost number 2 member of this multiplet has a geometric interpretation as a 2-form on F. Thus insertions of A(0) compute intersection numbers on F. The deformed action is Z (8.7) I0 −→ I0 + 21 A(2) . 1 R Naively, the contribution of 21 A(2) in a path integral over maps f of index n is e− 2 nA . This accounts nicely for the genus-independent exponential factors in (8.2), but fails to explain the polynomial of A in (8.2). We can understand some features of the polynomial in A in (8.2) by considering more carefully the “conformal perturbation series” in question:

Z + (A, G, N ) =

1

e− 2

R

A(2)

= A=0

Z ∞ X (−1)` `=0

`!

1 2

f ∗ω ΣW

` .

(8.8)

A=0

The measure h · · ·i implicitly contains further operators, such as the four fermion terms in the curvature, which arise from the co-sigma model. It is important to understand that the expression (8.() is ill-defined. Evaluation of the terms in the series involves integration over operators inserted at coincident points. As in all theories of gravity, merely identifying the operators as in (8.6) does not fully specify their correlators, because we must choose contact terms, i.e., we must carefully specify the terms in the e R[G, G] Ai e which have delta-function support on two or correlators of A(2) and hA, more points. In the following subsections we will show how a consideration of contact terms can account for the area polynomials (8.4) which arise from the contributions of simple Hurwitz space. We will not try to account for the other types of coverings in the sum over 9 in (8.2). Similarly we do not try to account for the the terms arising from expanding (8.5). We firmly believe that these more complicated polynomials can also be explained by looking at more complicated contact terms from the higher codimension boundaries of Hurwitz space and the space of Maps× Metrics. 8.3. Measure on the space of simple covers. Let F (1) be the simple Hurwitz space of maps with B simple branch points. Denote these simple branch points by PI with corresponding ramification points of index 2 at RI : these are the unique ramification points above PI .We can choose a basis {GI }I=1,...,2B for T F , such that G2I−1 and G2I have support only at the I th ramification point. The analogue in ordinary string theory

584


is a choice of Beltrami differentials which have support only at punctures. This is a well-defined choice away from the boundary of moduli space. Now consider the curvature insertions in these local coordinates: Z B e I RIJ A e J = (−1) Pfaff(RIJ ), e exp − 1 A (8.9) D[A] 4 2B where B is even and the matrix, RIJ , takes the following form in an oriented orthonormal basis   0 R12 0   −R12   .. (8.10) RIJ =  , .   0 R2B−1 2B −R2B−1 2B 0 so that B Y Pfaff(RIJ ) = R2I−1 2I [G2I−1 , G2I ](RI ) (8.11) I=1

and the full measure for the topological string theory is (−1)B (2π)B = =

Z F (1)

D[F, G]

B Y

R2I−1 2I [G

2I−1

, G ](RI ) exp 2I

I=1

Z − 21

∗

f ω ΣW

Z k Z ∞ B Y 1 X (−1)k 2I−1 2I ∗ 1 D[F, G] R [G , G ] f ω 2I−1 2I 2 (2π)B k! F (1) ΣW 1 (2π)B

k=0 ∞ X k=0

I=1

k

(−1) hhA(2) · · · A(2) iiF (B,k) . 2k k!

(8.12)

In the last line we have introduced a space F(B, k), which is the product space F(B, k) = F (1) × (ΣW )k

.

(8.13)

The integral over this space, hh· · ·iiF (B,k) is formally defined by (8.12). In order to define the integrated correlators we have to describe possible delta-function supported contributions at places where area operators collide with curvature insertions, and where area operators collide with themselves. Collisions of curvature operators Ri = Rj belong to higher codimension boundaries outside the space of simple covers and contribute to other terms in (8.12). In the next three sections we analyze the other collisions. 8.4. Plumbing fixtures. Suppose a simple ramification point R and a marked unramified point S collide on the worldsheet. The corresponding images of these points P and Q collide on the target space. Let U1 ⊂ ΣT be a disk containing P and Q. Let the disk V1 ⊂ ΣW be the preimage of U1 . V1 contains R and S. It is a double covering of the disk U1 ⊂ ΣT . The disk U1 glues into the annulus U2 on ΣT . V2 is the double-covering preimage of U2 . The figure below describes this collision In the plumbing fixture description of the degeneration, the points are kept fixed in their local coordinates. The transition functions depend on the modulus which describe the relative position of R and S. To describe the plumbing fixture more explicitly, we shall need to specify transition maps between coordinate patches both on the worldsheet as well as on the target.


585

Fig. 7. Collision of Area and Curvature Operators

h

21 V1 −→ f y1 g21 U1 −→

V2 f y2 U2

(8.14)

We introduce local coordinates w1,2 on U1,2 and z1 , z2 on V1 , V2 , respectively. Then the transition functions of the holomorphic family of degenerations are given by g21 (w1 ) =

q , w1 (8.15) 1/2

h21 (z1 ) = η

q , z1

where η = ±. Note that the transition function g21 does not change the complex structure of the target Riemann surface without marked points. Similarly the function f : ΣW → ΣT is locally defined by f1 (z1 ) = z12 , (8.16) f2 (z2 ) =

z22 .

The modular deformation which leads to this degeneration corresponds to a diffeomorphism generated by a quasi conformal vector field with support on V2 and a discontinuity along C. This quasi conformal vector field is given by V2 = z2

δq ∂ . q ∂z2

(8.17)

586


Integrating against the stress tensor, we obtain δq L0 . q We exponentiate this to obtain the plumbing fixture: R 2 2 ˆ 0 ) exp −2πi(τ¯ L¯ 0 + σ¯ˆ l¯0 ) d τ d σˆ exp 2πi(τ L0 + σl Z =

(8.18)

(8.19) d2 q L0 L¯ 0 ¯ q q¯ l0 l0 , |q|2

where l0 and l¯0 are modes of the topological superstress tensor, t, which is the BRST partner to the ordinary stress energy tensor, T , {Q, t} = T.

(8.20)

8.5. Area-curvature contact terms. To begin, let us note that simple considerations of quantum numbers and scaling constrain the contact term to be of the form: Z (8.21) R2I−1,2I [G, G]A(2) = cA(0) for some constant c. The integral is over an infinitesimal disc surrounding the curvature insertion, which is nonzero because of a delta function contribution from the collision of two operators. Since R is a two form on F, it has ghost number two. A(2) is a two form on the target pulled back to the worldsheet. After doing the integral, it is clear that the result should have ghost number two and should be a zero form on the worldsheet. Moreover, the LHS scales like the area of the target space. The unique operator in Sect. 6.5 which satisfies all these criteria is A(0) . We now describe how the above contact term can be directly derived using ideas along the lines of those used in the case of pure topological gravity [53]. The derivation is only heuristic. The contact term we wish to compute here is the collision of an integrated area operator and a curvature insertion Z d2 q L0 L¯ 0 ¯ 1 q q¯ l0 l0 {A(0) (1)RIJ [G, G](0)} 2π |q|≤ |q|2 (8.22) Z n o 1 ¯ d2 q ∂q ∂q¯ q L0 q¯L0 A(0) (1)8IJ (0) , = 2π |q|≤ where A(0) is BRST invariant, 0-form descendant of A(2) . 1, so that (8.24) becomes Z −→ δ (2) (q)A(0) (1),

(8.27)

|q|≤

which corresponds to an area operator inserted at a ramification point. In conclusion, we have derived (8.21) with c = −2. 8.6. Area-area contact terms. The argument (8.22) to (8.27) can be repeated for the collision of two integrated area operators. The plumbing fixture in this case is the one familiar from bosonic string theory Z d2 q L0 L¯ 0 ¯ 1 q q¯ l0 l0 {A(0) (1)A(0) (0)} 2π |q|≤ |q|2 (8.28) Z o n 1 ¯ d2 q ∂q ∂q¯ q L0 q¯L0 A(0) (1)k(0) , = 2π |q|≤ where k is the Kähler potential,

¯ ω = ∂ ∂k.

(8.29)

The calculation now proceeds as in the previous subsection. The analog of (8.26) is (8.30) (L0 + L¯ 0 ) A(0) k = 0 since both A(0) and k are invariant under worldsheet scale transformations. Remark. The absence of A · A contact terms may at first seem a bit counter-intuitive. Indeed, the collisions of analogous operators in conformal field theory are well known to play an important role [56] For example, when changing the “compactification data” of a product of Gaussian models by conformal perturbation theory it is exactly these 14 The extra minus sign arises because we have computed the change in 8 IJ under an active transformation ¯ 0 measures the response of an operator under a passive transformation on hαβ whereas the eigenvalue of L0 + L z → λz.

588


contact terms which account for the dependence of the conformal weights and operator product coefficients on the compactification data (see e.g. [57]). In the present case we have made implicit choices in our evaluation of contact terms. Our choices are related to the preservation of BRST invariance of the theory. Conjecture 8.1. Within the family of contact terms preserving the BRST Ward identities the area polynomials will remain unchanged and will be given by (8.3) above. 8.7. Recursion relations and calculation of an area polynomial. We now combine the above results on contact terms to derive recursion relations for the integrated area correlators. We attempt to remove the area operators A(2) successively. Each operator contributes a bulk term and a contact term. If r such operators have collided with curvature operators producing a contact term of type (8.21) then the remaining correlator is integrated over a space F(B, `; r) = F (1) × (ΣW )` , where B − r copies of A(0) are inserted at simple ramification points, r copies of the curvature operator are inserted at the remaining simple ramification points, and ` area operators A(2) are integrated over the worldsheet ΣW . If we try to remove an area operator A(2) we obtain the recursion relation: B−r

k

z }| { z }| { hhA(0) · · · A(0) A(2) · · · A(2) iiF (B,k;r) B−r

k−1

z }| { z }| { = nAhhA(0) · · · A(0) A(2) · · · A(2) iiF (B,k−1;r) B−r+1

(8.31)

k−1

z }| { z }| { −2rhhA(0) · · · A(0) A(2) · · · A(2) iiF (B,k−1;r−1) . The first term represents the bulk contribution. In the second term there is one extra insertion of A(0) which has replaced a curvature operator at a ramification point, and there is one fewer A(2) operator. The coefficient r in the second term comes from the fact that for each area integral there are r collisions with curvature insertions at ramification points. The factor of −2 comes from the normalization of the contact term. Iterating this recursion relation, we are led to the following: hhA(2) · · · A(2) iiF (B,k) (8.32) =

k X l=0

k l

2l B!(−1)l (B − l)!

(nA)k−l hhA(0) (R1 ) · · · A(0) (Rl )iiF (B,0;B−l) .

When l > B it is clear that the correlation function on the right vanishes, by ghost number counting. So that altogether 1 (2π)B =

∞ X k=0

Z F (1)

D[F, G]

B! k!(B − k)!

B Y

Z R2I−1 2I [G

2I−1

, G ](QI )

I=1 min[k,B] X l=0

2I

exp − 21

f ∗ω

(8.33)

ΣW

k (− 21 nA)k−l hhA(0) (R1 ) · · · A(0) (Rl )iiF (B,0;B−l) . l

Substituting in the RHS of (8.33) we obtain


e

B X

− 21 nA

k=0

589

B! hhA(0) (R1 ) · · · A(0) (Rk )iiF (B,0;r=B−k) . k!(B − k)!

(8.34)

So we are left with the integral Z F (1)

D[F, G]

B−k Y

R2I−1 2I [G2I−1 , G2I ](RI ) A(0) (RB−k+1 ) · · · A(0) (RB ).

(8.35)

I=1

Now we use again the fact that we are only interested in the contribution of simple Hurwitz space. This space is a bundle over C0,B /SB with discrete fiber the set 9(n, B, G, L = B). Further the measure on Hurwitz space inherited from the path integral divides out by diffeomorphisms. Therefore the correlator in (8.34 is: X ψ∈9(n,B,G,L=B)

1 1 × |C(ψ)| B!

Z h C0,B

B−k Y

R2I−1 2I [G2I−1 , G2I ](RI )

I=1

(8.36) A (RB−k+1 ) · · · A (RB )i. (0)

(0)

The correlation function has singularities when any two ramification points RI collide. In isolating the contributions of simple Hurwitz space we must ignore the singularities from the collisions of RI , I ≤ B − k with RJ , J ≥ B − k + 1. Thus we replace (8.36) by the expression: X ψ∈9(n,B,G,L=B)

1 1 × × |C(ψ)| B! (8.37)

Z C0,B−k ×(ΣT

h )k

B−k Y

R2I−1 2I [G2I−1 , G2I ](RI ) i ∧ ω(PB−k+1 ) ∧ · · · ∧ ω(PB ),

I=1

where PJ ∈ ΣT are the images of the simple ramification points RJ . We can do the integrals over the the wedge product of Kahler classes separately to get Ak (the area of ΣT ×k ). The remaining integral over the B −k curvature insertions is the same correlator appearing in the partition function. Thus we have: hhA(0) (R1 ) · · · A(0) (Rk )iiF (B,0;r=B−k) (−A)k χ(CB−k (ΣT )) = B!

X ψ∈9(n,B,G,L=B)

(8.38) 1 . |C(ψ)|

Substituting in (8.34), and comparing with (8.4), we see that the contribution of simple Hurwitz space to the Path integral perturbed as in (8.7) agrees with the conjecture that the perturbation in (8.7) is equivalent to 2D Yang Mills at finite area. Remark. Our discussion has only focused on the contact terms needed to reproduce the area polynomial of simple Hurwitz space. Higher contact terms will be affected by the presence of gravitational descendents of the area operator in the action. Thus we should consider perturbations generalizing (8.7) like:

590


I0 −→ I0 +

X

Z τn

σn (A(2) ).

(8.39)

n≥0

In the original Y M2 theory there is a similar class of deformations of the theory obtained by adding higher Casimirs to the heat kernel Boltzman weight: X tn Cn (R) (8.40) C2 (R) −→ n≥2

It is natural to conjecture that these classes of deformed theories are in fact equivalent. Experience from 2D gravity [58] leads us to expect that the change of variables {tn } → {τn } can involve complicated nonlinear terms. Indeed nothing in the present discussion precludes the possibility that the pure C2 (R) Y M2 theory is equivalent to a perturbation of type [8.39] with τn 6= 0 for n > 0. 9. Wilson Loops The techniques of [15] extend to Wilson loop expectation values. In general the answer is expressed in terms of rather intricate gluing rules [15]. In this section we will restrict attention to the simplified case of the chiral theory. The string interpretation of these quantities is given by macroscopic loop amplitudes (familiar from gravity) with certain Dirichlet boundary data on the boundary of the worldsheet. 9.1. Observables. The natural observables in gauge theory are the Wilson loops. Let R be a finite-dimensional representation of SU (N ) and let 0 be a piecewise-differentiable oriented curve 0 : S 1 → ΣT . Such curves generically have at most double points as self-intersections and we will assume this to be the case. We define: W (R, 0) ≡ trR (U0 ), (9.1)

I U0 = P exp

A, 0

we will often denote the image 0 ⊂ ΣT by the same symbol. As pointed out in [15] a more natural basis of observables for the 1/N expansion are the loop functions: ∞ Y j (trU0j )k0 . (9.2) Υ (k0 , 0) ≡ j=1

The vector k0 = (k01 , k02 , . . .) determines a conjugacy class (via cycle decomposition) in P j jk0 . By Frobenius reciprocity we have Sm0 where m0 = X χR (p0 )W (R, 0), (9.3) Υ (k0 , 0) = R∈Ym0

where p0 is any element in the conjugacy class k0 . 9.2. Exact answer: nonintersecting loops. Suppose that we have a collection {0} of nonintersecting curves in ΣT . Let


591

ΣT − q0 = qc ΣTc

(9.4)

be the decomposition into disjoint connected components. Each component has Gc handles and bc boundaries. Since ΣT and 0 are each oriented, each curve 0 can be deformed into two curves 0± as in Fig. 8.

Fig. 8. Using the orientation of the surface and of the Wilson line we can define two infinitesimal deformations ± of the Wilson line 0 ± c We let c± 0 denote the label of the component ΣT which contains 0 . The exact answer for correlation functions of Wilson loops is easily obtained from standard cutting and gluing techniques. One finds: XY Y χ(Pc ) − 1 Ac c2 (R(c))/N Y R(c+0 ) T e 2 W (R0 , 0) = NR(c− ),R , (9.5) dim R(c) R(c) c

0

0

0

0

where we sum over unitary irreps R(c) for each component c, NRR13,R2 are the “fusion numbers” defined by the decomposition of a tensor product into irreducible representations (9.6) R1 ⊗ R2 = ⊕R3 NRR13,R2 R3 , and Ac denote the areas of the components ΣTc . Note that ΣTc are open manifolds. When we speak of the Euler character we glue back in the bc boundary circles. 9.3. Chiral expansion: nonintersecting loops. The chiral expansion of Wilson loop averages may be obtained directly from (9.5) without recourse to the gluing rules of [15]. To begin one derives a formula for the fusion numbers in terms of a sum over the symmetric group. This may be done by expressing them as integrals of characters, passing to the loop function basis, and then expressing in answer in terms of the symmetric group. The result, in a form useful for us, is: X

R(c+ )

χR (p0 )NR(c−0 ),R = δn(c+ ),n(c− )+m0 0

R∈Ym0

0

0

dR(c+0 ) dR(c− 0 ) (n(c+0 )!) (n(c− 0 )!) (9.7)

X

X

u+ ,x0 ∈Sn(c+ ) u− 0 ∈Sn(c− ) 0 0

−

(u0 ) χR(c+0 ) (u0 ) χR(c− −1 0 ) δn(c+0 ) u+0 , x0 (u− , 0 · p0 )x0 + dR(c0 ) dR(c− ) +

0

0

where dR is the dimension of the representation of the symmetric group, and, in the last factor (u− 0 · p0 ) is the image under the natural embedding of symmetric groups

592

S. Cordes, G. Moore, S. Ramgoolam + − ι± 0 : Sn(c ) × Sm0 −→ Sn(c0 )

(9.8)

0

which takes the first permutation to a permutation of the first n(c− 0 ) entries and the second permutation to a permutation of the last m0 entries. As usual we obtain the chiral sum by making the replacement in (2.3). Using the standard set of identities from [15] together with (9.7), the chiral expansion of (9.5) becomes: Y X X X |k0 | Υ (k0 , 0) = m0 ! 0 n(c)≥0 `(c)≥0 L(c)≥0 X X X X X v1 (c),...vL(c) (c)∈Sn(c) s1 (c),...tGc (c)∈Sn(c) u± ∈S ± x0 ∈Sn(c+ ) p0 ∈Sm0 0 0 n(c )

Y c

e

1 − 2 Ac (n(c)−

n(c)2 N2

`(c)

)

−Ac `(c)!

0

1 N

n(c)χ(ΣTc )−P

i

(n(c)−kvi (c) )

χ(CL(c) (ΣTc ))

Y L(c) Gc Y Y Y Y 1 `(c) − + δn(c) u0 u0 vi (c)T2,n(c) [si (c), ti (c)] n(c)! − c 1 1 0:c+ 0:c0 =c 0 =c Y 1 −1 δn(c+ ),n(c− )+m0 δn(c+0 ) u+0 , x0 (u− . 0 · p0 )x0 0 0 m0 ! 0

(9.9)

The normalization factors in front of Υ are chosen for later convenience; |k0 | is the order of the conjugacy class determined by p0 . Despite its extremely cumbersome appearance, this expression has an elegant geometrical content as we shall see. The fourth line defines the coverings ΣW c of components ΣTc . The last line describes how these covering spaces ΣW c are glued together. 9.4. Chiral expansion: intersecting loops. If the loops q0 have intersections (including self-intersections) then the exact answer for Y M2 is much more complicated than (9.5) and involves summing over 6j symbols at the intersection vertices of the loops [13]. Nevertheless the chiral 1/N expansion for intersecting Wilson loops has a relatively simple set of rules which have been worked out in [15]. The only modification of (9.9) is the replacement: (9.10) ΣTc → Σ˜ Tc , where ΣTc is constructed from the open manifold ΣTc by gluing in open intervals and vertices along ∂ΣTc . The rules for constructing Σ˜ Tc are as follows. Consider q0 as a graph. It has open edges Ej and vertices vj . Using the orientation we can define deformations Ej+ and vj++ . The edge Ej+ is the deformation of the edge in the direction of 0+ , the vertex is obtained by deforming into the + region for each of the two intersecting curves. We glue the edges Ej to the boundary of the component containing Ej+ and we glue the vertices vj to the boundary of the component containing vj++ . We may define the Euler character of Σ˜ Tc to be χ(Σ˜ Tc ) = χ(ΣTc ) +

X ˜c Ej ∈ Σ T

(−1) +

X ˜c vj ∈ Σ T

(+1).

(9.11)


593

This is not a homotopy invariant, but it is a homeomorphism invariant. In the previous case of nonintersecting Wilson loops the modification ΣTc → Σ˜ Tc makes no difference since χ(S 1 ) = 0. Gross and Taylor’s rule says that the only change we must make in (9.9) is the change ΣTc → Σ˜ Tc of (9.10)! 9.5. String interpretation. The string interpretation of the chiral nonintersecting Wilson loop averages in the Υ basis is stated very simply. The vectors k0 may be thought of as specifying the homotopy class of a map from a disjoint union of circles to 0: We have k0j j-fold coverings of the circle by the circle.15 The only change that is needed in the path integral of Sects. 6, 7 is that we have a macroscopic loop amplitude: The worldsheet ΣW has a boundary. Data specifying the Wilson loops is encoded in the boundary conditions on f : ΣW → ΣT . These boundary conditions state that f : ∂ΣW → q0 is in the homotopy class {k0 }. Boundary conditions on the metric are standard [59] and follow from the requirements that 1) The loop 0 is unparametrized, and 2) P † is the adjoint of P . Let n denote a normal vector and t a tangent vector to ∂ΣW . We take g(n, t) = 0 on ∂ΣW . Correspondingly, vector fields ξ generating diffeomorphisms satisfy n.ξ = 0 and na tb ∇(a ξb) = 0. Boundary conditions for other fields follow from BRST invariance and invariance of the action. This string interpretation will be justified in the next section. Conjecture 9.1. The string interpretation for the case of chiral intersecting Wilson loop amplitudes is obtained by the boundary condition that f : ∂ΣW → q0 is in the homotopy class {k0 } q0 (9.12) ∂ΣW −→ q0 S 1 −→ q 0. The first arrow describes a covering of circles by circles. The second is the homotopy class of the curves defining the Wilson loops. 9.6. Hurwitz spaces for surfaces with boundary. We now give an argument for the claim of the previous subsection. Definition 9.6.1. A boundary-preserving branched covering is a map f : ΣW , ∂ΣW → ΣT , ∂ΣT

(9.13)

such that 1. f : ∂ΣW → ∂ΣT is a covering map. 2. f : ΣW − ∂ΣW → ΣT − ∂ΣT is a branched covering. Equivalence and automorphism of such maps are defined in the obvious way. Note that the boundary components ∂ΣW are unlabelled so φ : ΣW , ∂ΣW → ΣW , ∂ΣW can permute the boundaries. By (1) f determines a class k0 for each component 0 of ∂ΣT . Let us assume that ΣT − ∂ΣT is connected. Then, by (2) f determines a branch locus S(f ) ⊂ ΣT − ∂ΣT , an index n, and an equivalence class of a homomorphism ψf : π1 (ΣT −S(f ), y0 ) → Sn . We have the direct analog of the Riemann existence theorem Theorem 3.1: Proposition 9.6.1. Let ΣT be a connected, closed surface with boundary. Let S ⊂ ΣT − ∂ΣT be a finite set, and let n be a positive integer. There is a one-one correspondence between equivalence classes of boundary-preserving branched covers (9.13) with branch locus S and equivalence classes of homomorphisms ψ : π1 (ΣT − S(f ), y0 ) → Sn . 15

Since j ≥ 1 the homotopy class has an orientation preserving representative.

594


Proof. The proof proceeds as before. We choose a representative for ψ and a basis of generators for π1 . Then we glue together n copies ΣT according to the data given by the homomorphism. The maps that we will need for nonintersecting Wilson loops are considerably more complicated than boundary-preserving covers: we must allow for the possibility that the inverse images of the loops 0 contain loops which lie in the interior of ΣW . This leads us to introduce: Definition 9.6.2. Suppose ΣW is a closed oriented surface with boundary and {0} is a collection of nonintersecting oriented closed curves in ΣT . By a covering map f : ΣW → ΣT with boundaries over {0} we mean a continuous orientation-preserving map f such that 1. f : ∂ΣW → q0 is a covering map. 2. f −1 (q0) is a disjoint union of circles. 3. f : ΣW − f −1 [q0] → ΣT − q0 is a branched covering. Equivalence and automorphism are defined as before. We now describe these maps in some detail. As before, by (1) f determines a homotopy class {k0 } of ∂ΣW → q0. By (3), we have a covering f c : ΣW c → ΣTc ,

(9.14)

where qc ΣW c = ΣW − f −1 (q0). The number of sheets of a covering will be different for different components of ΣTc . An elementary example is Fig. 9.

Fig. 9. Different components of the target space can be covered by different number of sheets

In general, the coverings f c are boundary-preserving branched coverings, the boundaries of ΣW c covering the boundaries of ΣTc . From each covering f c we obtain a


595

branch locus Sc ⊂ ΣTc , index n(c), and equivalence classes of homomorphisms: ψc : π1 (ΣTc − Sc , y0,c ) → Sn(c) . By (2) the inverse image under f of any loop 0 may be divided into interior loops and boundary loops, that latter living in ∂ΣW . The different surfaces ΣW c must be smoothly glued together along the interior loops of f −1 (0). This requirement results in the gluing conditions (9.5) to (9.20) below. P First, above a loop 0 there are j k0j components with m0 sheets belonging to ∂ΣW . P j jk0 .) The remaining components lie in the interior of ΣW . Using (Recall that m0 = the orientation we see that if we perturb the curves f −1 (0) in the plus direction we get an n(c+0 )-sheeted covering of 0+ . On the other hand, since a perturbation of the boundary curves of ΣW in the minus direction takes us off the surface ΣW we can only perturb the interior curves of f −1 (0) in the minus direction. Thus we get an n(c− 0 )-sheeted covering of 0− . The situation may be summarized in Fig. 10.

−

Fig.10. Locally the covering map looks like this. Above 0 are interior and boundary curves

there are only interior curves. Above 0 there +

596


From Fig. 10 the first gluing condition n(c+0 ) = n(c− 0 ) + m0

(9.15)

becomes evident. We obtain the second gluing condition by starting from Proposition 9.6.1. As in Fig. 1 of Sect. 3.1 we can choose generators αi (c), βi (c), σi (c), γi+ , γi−

(9.16)

of π1 (ΣTc − Sc , y0,c ), such that αi (c), βi (c), run around handles, σi (c) become trivial if we fill in branch points, γi± become trivial if we fill in 0± , we have the relation: Y Y Y Y γi+ γi− σi (c) [αi (c), βi (c)] = 1 (9.17) and the covering ΣW c → ΣTc can be constructed by glueing together copies of ΣTc using the homomorphism ψc as in the LHS or RHS of Fig. 11.

Fig. 11. Figure showing construction of branched covers with boundary using the data D. 0

Consider now two components ΣTc and ΣT c which must be glued along a loop 0, 0 ± as in Fig. 11. Suppose that c+0 = c, c− surrounds 0± . The homotopy 0 = c , and that γ − type of the covering of the interior circles above 0 is given by the conjugacy class of


597

ψ0− = ψc− (γ − ),

(9.18)

0

while the homotopy type of the covering of the interior circles above 0+ is given by the conjugacy class of (9.19) ψ0+ = ψc+0 (γ + ). Because the interior curves must be smoothly glued together there must exist a re+ labelling x0 ∈ Sn(c+0 ) of the sheets above ΣT c0 such that −1 − ψ0+ = x0 ι± 0 (ψ0 p0 ) x0 ,

(9.20)

where ι± 0 is the embedding (9.8) and p0 is any element in the conjugacy class k 0 . Finally two equivalent coverings f˜ = f ◦ φ will lead to the same data but with ψc , ψ˜ c differing by an inner automorphism of Sn(c) . In summary, to any map satisfying Definition 9.6.2 we can unambiguously associate an equivalence class of covering data D. This data is composed of D1. Branch loci Sc ⊂ ΣTc , basepoints y0,c , indices n(c), boundary data k0 . D2. Generators (9.16) of π1 (ΣTc − Sc , y0,c ) D3. Homomorphisms ψc : π1 (ΣTc −Sc , y0,c ) → Sn(c) . These are obtained after choosing a labelling of the inverse images of y0,c . D4. Elements p0 ∈ Sm0 . These are obtained after choosing basepoints yA 0 ∈ 0 and choosing a labelling of the points lying on the boundaries of ΣW in f −1 (y0 ). D5. Elements x0 ∈ Sn(c+0 ) These data are required to satisfy the conditions: C1. n(c+0 ) = n(c− 0 ) + m0 − −1 C2. ∀0 ∃p0 ∈ Sm0 such that [p0 ] = k0 and ψ0+ = x0 ι± 0 (ψ0 p0 )x0 Two sets of data D and D˜ satisfying these conditions will be considered to be equivalent if the following relations holds: E1. The bases αi (c), . . ., α˜ i (c), . . . differ by Aut(π1 (ΣTc − Sc , y0,c )). E2. α˜ i = αi , . . ., and ∃w(c) ∈ Sn(c) , w(0) ∈ Sm0 such that ψ˜ c (·) = w(c)ψc (·)w(c)−1 −1 x˜ 0 = w(c+0 )x0 w(c− w(0)−1 0 ) −1

p˜0 = w(0)p0 w(0)

(9.21)

.

We define C(ψc , x0 , p0 ) to be the subgroup of {ψc , x0 , p0 } fixed under the action defined above. The above discussion has proved half of:

Q

Snc

Q

Sm0 which leaves the set

Proposition 9.6.2. There is a one-one correspondence between equivalence classes of covering maps with boundaries over q0 with prescribed data: a1. f defines an n(c)-fold cover of ΣTc , with branch locus Sc a2. ∂ΣW → q0 is of homotopy type k0 and equivalence classes of covering data D as defined above. Proof. Given the data D we first construct boundary-preserving branched coverings f c : ΣW c → ΣTc in the standard way. Then, with the labelling of the sheets specified by the data we glue the loops covering 0− to the interior loops covering 0+ using the relabelling x0 . This gluing is smooth by the conditions Q D. Equivalent data Q defining induce equivalent coverings. Note in particular that c Sn(c) 0 Sm0 acts transitively

598


on equivalence classes of relation E2 and that the number of distinct elements in a class is given by Q Q n(c)! m0 ! . (9.22) |C(ψc , x0 , p0 )| It is also clear from this construction that the automorphism group of the covering is Aut(f ) = C(ψc , x0 , p0 ). Definition 9.6.3. The Hurwitz space H(h, b, n(c), Sc , k0 ) is the space of equivalence classes of covers f with boundaries above q0, where ΣW has h handles, b boundaries, such that f is an n(c)-fold covering of ΣTc with branch locus Sc , and restricts to ∂ΣW → q0 of homotopy type k0 . The union of these spaces over sets of branch points Sc with L(c) elements defines the Hurwitz space H(h, b, n(c), L(c), k0 ). The compactification of this space with different L(c) defines the Hurwitz space H(h, b, n(c), k0 ). AQcorollary of Proposition 9.6.2 is that H(h, b, n(c), L(c), k0 ) is a discrete fibration over c CL(c) (ΣTc ). Therefore, the orbifold Euler characteristic is defined as usual: Y X 1 , χ(CL(c) (ΣTc )) χorb H(h, b, n(c), L(c), k0 ) = |Aut(f )| c [f ]∈H(h,b,n(c),Sc ,k0 )

(9.23) where, on the RHS we may choose any set Sc of L(c) distinct points in ΣTc . Now finally let us compare with the chiral expansion (9.9). If we first put A(c) = 0 then the effects of homotopically trivial loops 0 can be shown to be trivial, simply contributing overall powers of 1/N . Nevertheless, homotopically nontrivial loops have nontrivial correlators. Combining (9.9) with Proposition 9.6.2 we see that the chiral expansion becomes: X 1 2h+b−2 χorb H(h, b, n(c), k0 ) (9.24) N n(c),h,b≥0

as in the previous sections. Thus, with an appropriate choice of contact terms decoupling the ± sectors 16 the macroscopic loop path integral described in Sect. 9.4 will produce the product of chiral Wilson loop averages, in complete analogy to the partition function. Finally, we include the effects of the area in the Wilson loop averages, as computed in (9.9). The structure is exactly the same as that found in Sect. 8 and the same discussion shows that – at least on the simple Hurwitz sub-space H(h, b, n(c), L(c) = B(c), k0 ) the contact terms account for the area. 9.7. Divers remarks on wilson loop averages. We gather several miscellaneous remarks in this subsection. 1. First, even in the chiral theory the other Ac -dependent polynomials in (9.9) associated with non-simple Hurwitz spaces remain to be analyzed. 2. The discussion should be generalized to the chiral case allowing intersections of the graph q0. The conjectural string formulation (Conjecture 9.1) will follow from Conjecture 9.2. The chiral intersecting loop amplitudes may be discussed as in Sect. 9.6 with by simply modifying condition 2 in Definition 9.2 by allowing f −1 (q0) to be a graph and modifying Proposition 9.2 and Definition 9.3 by allowing the branch locus Sc to intersect 0. In order to avoid double-counting we must let Sc approach from the + side only. The only effect on Proposition 9.2 and (9.23) is the replacement: ΣTc → Σ˜ Tc . 16

See Sect. 11 below.


599

3. The extension of our discussion to the coupled theory is very nontrivial. The rules of [15] become considerably more elaborate. Some of the issues arising in this extension appear to be quite relevant to establishing a string picture of four-dimensional QCD along the lines of [60]. 4. An interesting open problem is the derivation of the Migdal-Makeenko loop equations from the topological string theory point of view. Given our experience with 2D gravity, we may guess that there is an analog of W∞ constraints on the partition function which is equivalent to the loop equations, and that these W∞ constraints follow from a contact term analysis. 5. Further investigation of Wilson loop amplitudes also promises to yield some extremely interesting insights in mathematics. Recently, V.I. Arnold has discovered new invariants of plane curves (immersions) [61]. We remark Q that if Sk (x1 , x2 , . . .) ∂ )| h Υ i are also invariants of are elementary symmetric polynomials then Sl ( ∂A A =0 c c immersions (by the area-preserving diffeomorphism invariance of Y M2 .) The relations of these invariants to the mathematics of covering spaces may well be very rich. 6. Finally we mention that, following [62, 63, 15] one should be able to incorporate dynamical quarks into the present framework. One must modify the string theory by turning it into an open-closed (Dirichlet) string. We hope to return to these issues in future work. 10. The Coupled Theory We have seen that some aspects of the chiral Y M2 theory are related to the topological string theory of Sect. 6. In the next two sections we will show how the full ("coupled") theory also fits into the framework of the theory of a topological string theory. We will restrict attention to reproducing the partition function (2.2). The key observation is that Z in (2.2) differs from a product of chiral partition functions Z + Z − through the contribution of boundaries of the space Maps × Met(ΣW ). In Sects. 10.1–10.5 we will use the geometrical picture of [15] to show that the A = 0 partition function (2.2) computes Euler characters of spaces of maps from singular surfaces ΣW to ΣT . In 10.1, 10.2 we describe in detail the maps and worldsheets in question. In 10.3 we develop a combinatoric approach to the space of maps. In 10.4 we describe the relation of the space of maps to configuration spaces. In 10.5 we write the zero area QCD sum in terms of Euler characters of the spaces of maps. 10.1. Degenerated coupled covers. Let ΣW be a smooth surface, perhaps with double points. Topologically this means that there is a set of points {P1 , . . . , Pd } ⊂ ΣW such that a local neighborhood Di of Pi is the one-point union of disks Di(1) , Di(2) : Di = Di(1) q Di(2) /(Pi(1) ∼ Pi(2) ).

(10.1)

The normalization of ΣW , N (ΣW ) is the smooth surface obtained by replacing Di → Di(1) q Di(2) . N (ΣW ) may be connected or disconnected. A map f : ΣW → ΣT defines a normalized map N (f ) in a natural way. Definition 10.1.1. Let ΣW be a surface with double points. A degenerate branched cover f : ΣW → ΣT is a continuous map such that 1. N (f ) : N (ΣW ) → ΣT is a branched cover, and 2. If Pi(1) , Pi(2) are the normalizations of the double points Pi then

600


Ram N (f ), Pi(1) = Ram N (f ), Pi(2) = ei ,

(10.2)

where Ram is the ramification index. The cover in the neighborhood of the double point may be thought of as a degeneration of a family of 2ei -sheeted covers of annuli by annuli degenerating to the cover of one disk by two disks. One of the very strange aspects of the Y M2 partition function is that it appears to involve maps which are branched covers which are neither holomorphic nor antiholomorphic. Definition 10.1.2. A coupled map f : ΣW → ΣT of Riemann surfaces is a continuous map such that there are circles Si which separate ΣW into two disjoint surfaces ΣW = ΣW + q ΣW − /(Si+ ∼ Si− )

(10.3)

and such that f : ΣW + → ΣT is holomorphic while f : ΣW − → ΣT is antiholomorphic. An example of such a map is a mapping of the complex plane to a closed disk given by f (z) = z n for |z| ≤ 1 and = 1/z¯ n for |z| ≥ 1. Note that df is discontinuous along the unit circle. Our main object of interest combines the above two notions and will be called a degenerated coupled cover. It is a coupled map, where the circles Si have been shrunk to points. More formally, we state Definition 10.1.3. A degenerated coupled cover (dcc) f : ΣW → ΣT of Riemann surfaces is a map such that if we take the normalization of the double points {Q1 , . . . Qd } then we have a disjoint decomposition into smooth surfaces N (ΣW ) = N + (ΣW ) q N − (ΣW ) such that f + : N + (ΣW ) → ΣT is holomorphic, f − : N − (ΣW ) → ΣT is antiholomorphic and such that Ram(f + , Q+i ) = Ram(f − , Q− i ).

(10.4)

Two dcc’s f1 and f2 are said to be equivalent if there is a homeomorphism φ : ΣW → ΣW such that f1 ◦ φ = f2 . A homeomorphism φ of ΣW is an automorphism of the dcc if f ◦ φ = f . If ΣW has no double points then the map is just a branched cover, either holomorphic or antiholomorphic. As with branched covers we may associate several natural quantities to a dcc. f ± define indices n± (f ), branch loci: S ± (f ) = {f ± (Q)|Ram(f ± , Q) > 1}

(10.5)

S T (f ) = {f (Q)|Q is a double point}

(10.6)

d(f, P ) = Card[{Q|P = f (Q), Q is a double point}]

(10.7)

double point locus: tube number

ramification vectors:17 17

These vectors are considered to be infinite tuples of positive integers with almost all entries zero.


601

r± (f, P ) = (r1± , r2± , . . .),

(10.8)

rj± (f, P ) = Card {Q|f ± (Q) = P, Ram(f ± , Q) = j} ,

and, finally, homomorphisms : ψ ± : π1 (ΣT − S ± (f ), y0 ) → Sn± .

(10.9)

In the ordinary case the specification of the branch locus and the homomorphism ψ essentially specified the equivalence class of the map f as in Theorem 3.1. This is no longer the case for dcc’s because there are many ways in which the “double points” connecting the different ramification points can be introduced. This leads us to the combinatoric discussion of Sect. 10.3.

10.2. Degenerating coupled covers. The answer provided by Y M2 demands a further refinement of the above ideas. We must take into account the way in which a family of covering maps has degenerated to a dcc. We will define a local degenerating family of coupled covers to be specified by the following data. 1. We have a plumbing fixture degenerating to the double point of ΣW : Uq = {(z1 , z2 )|z1 z2 = ηq, q ≤ |z1 |, |z2 | < 1},

(10.10)

where 0 ≤ q < 1 and η is an nth root of unity for a positive integer n. 2. On the plumbing fixture we have a family of covering maps f q,n (z) =

z1n z¯2n

for q 1/2 ≤ |z1 | < 1 . for q 1/2 ≤ |z2 | < 1

(10.11)

Notice that n different degenerating complex structures on ΣW determine maps f q,n projecting to the same target space. Figure 12 illustrates the model in the case where n = 2. We will want to distinguish different f ’s corresponding to these n different degenerations, so we introduce: Definition 10.2.1. A degenerating coupled cover (Dcc) is a dcc equipped with a choice of a locally degenerating family of coupled covers ((10.10), (10.11)) for each double point.

602


Fig. 12. Local model for a degenerating coupled cover with n=2. The region between stripes single-covers the annulus

Remarks. 1. In a degenerating family f q,n is not differentiable along |zi | = q 1/2 because the normal derivative is discontinuous. It lives in the space of piecewise differentiable n−1 maps rather than in the space of C ∞ maps. The discontinuity is proportional to q 2 and goes to zero when q goes to zero, for n > 1. 2. Our definition is admittedly somewhat ad hoc and could be considerably improved. The partition function of Y M2 computes the Euler characters of spaces of “degenerating coupled covers”, as opposed to those of dcc’s (see Proposition 10.3.1). The introduction of the discrete choice of degenerating family accounts for an important combinatoric factor proportional to the product over all the double points, of the index of the ramification points being joined at each double point. 3. Much has been made on the suppression of “folds” and “fold degrees of freedom” in Y M2 . It is perhaps worth noting that at q 6= 0 the map (10.11) has a fold, but this fold disappears for q → 0. In general, in the formulation of this paper folds are suppressed because they are incompatible with holomorphy of the map f . 10.3. Combinatoric description of degenerated coupled covers. We give now a combinatoric description of dcc’s, establishing a 1-1 correspondence between data defined in terms of symmetric groups and maps defined geometrically in the previous subsection. We will discuss here dcc’s with parameters n± , B ± , S = S + ∪S − ∪S T fixed. We denote L = |S|. When we wish to emphasize the cardinality of S we write S(L).


603

Consider a dcc. Pick a base point y0 on the target space and label the inverse images − x+1 to x+n+ on the holomorphic side and x− 1 to xn− on the antiholomorphic side. Inclusion gives natural maps: j∗

∗ ± π1 (ΣT − S(L), y0 )−→π 1 (ΣT − S (f ), y0 )−→π1 (ΣT , y0 ).

i

(10.12)

± ψL

The map i∗ naturally defines homomorphisms : π1 (ΣT − S(L), y0 ) → Sn± which factor through the homomorphisms ψ ± of the previous section. Now choose a set of generators αi , βi , γ(P ), P ∈ S(L) of π1 (ΣT − S(L), y0 ). Once we have chosen a set of generators, each loop αi , βi , γ(P ) defines a pair of permutations (si , s˜i ), (ti , t˜i ), and (v(P ), w(P )) in Sn+ × Sn− . The behavior of a dcc at a double point determines some further data from the following construction. Let us choose a representation of ΣT −S as in Fig. 1 of Sect. 3.1. If γ(y0 , P ) is a curve from y0 to P then we may lift this curve with f ± . We denote the ± endpoint of the lifted curve by x± a .γ(y0 , P ), where we choose xa as the lift of the initial + point. If v(P ) has a cycle (a1 , · · · , ak ) of length k then xa .γ(y0 , P ) will be ramification point Q+ over P of index k. Thus, a choice of curves γ(y0 , P ) allows us to define a pairing of the cycles in v(P ) with those in w(P ). To be precise, we introduce the following definition. Definition 10.3.1. Let v ∈ Sn+ and w ∈ Sn− . Let Cyc(v) be the set of cycles in the cycle decomposition of v, and Cyc(w) be the set of cycles of w. A pairing of (v, w) is a subset K ⊆ Cyc(v) × Cyc(w) such that 1. (α, β) ⊂ K only for cycles α, β of equal length. 2. Projections K → Cyc(v) and K → Cyc(w) are injective. The second condition expresses the fact that a ramification point in the holomorphic sector can be connected to at most one ramification point in the antiholomorphic sector, i.e. there are only double points and no higher singularities. The cardinality |K| is the number of pairings. Let Jvw denote the set of all pairings of (v, w). Example 1. Suppose v = (12)+ (3)+ (4)+ and w = (12)− (34)− . If K = { (12)+ , (12)− }, then one pairing has been made and the other cycles have been left unpaired. This pairing is illustrated by Fig. 13. This means that in the degenerate branched cover inducing this map the point x+1 .γ(y0 , P ) is a double point coinciding with x− 1 .γ(y0 , P ). Thus, in summary, a dcc, together with a choice of basis of π1 and curves γ(y0 , P ), P ∈ S(L) describes elements (1) (si , ti ) ∈ Sn+ (2) (s˜i , t˜i ) ∈ Sn− (3) v(P ), w(P )) ∈ Sn+ × Sn− ∀P ∈ S(L). (4) KP ∈ Jv(P ),w(P ) ∀P ∈ S T This data we call a configuration. Moreover, conjugation by Sn+ × Sn− acts on the data 1 − − 4, and defines an equivalence relation. We let CFG stand for the set of equivalence classes. The subgroup of Sn+ × Sn− which leaves an element e of CFG invariant is called C(e). Example 2. Suppose L is 1, n+ = n− = 4, and there is one tube over the branch point P . Let v(P ) = v and w(P ) = w, where v and w are the same as in the previous example. A possible CFG, e, has as representative the configuration defined by K = ˜ t˜’s. Conjugating by the permutation { (12)+ , (12)− } together with the s’s, t’s and s’s, − − (12) (34) leaves this pairing invariant. Suppose it also leaves the s’s ˜ and t˜’s invariant.

604


Fig. 13. Glueing together ramification points according to the data of a pairing

Then 1, (12)− (34)− is an element of C(e). The permutation 1, (14)− (23)− , on the other hand, does not leave the pairing K invariant although it does leave w invariant. So it cannot be an element of C(e). Proposition 10.3.1. There is a one-one correspondence between elements of CFG and equivalence classes of degenerated coupled covers (dcc’s). Proof. We describe the proof in three steps. 1. Equivalent configurations come from equivalent dcc’s. Suppose two maps f1 : Σ1 → ΣT and f2 : Σ2 → ΣT determine equivalent configurations related by a permutation g in Sn+ × Sn− . Delete the set S(L) from ΣT and its inverse images from Σ1 and Σ2 to give Σ¯ 1 and Σ¯ 2 . Then f1 ,f2 restrict to unbranched covers of ΣT − S(L) by Σ¯ 1 and Σ¯ 2 respectively. The proof that f1± and f2± give equivalent branched covers follows from Theorem 3.1. In the inverse image of y0 , φ restricts to the permutation g which conjugates the configuration associated to f1 into that associated with f2 . In the case of ordinary branched covers we just use continuity to complete the homeomorphism φ over the deleted points. In the case of dcc’s we have to prove that when two points of Σ¯ 1 over P get identified their images under the homeomorphism φ also get identified as Σ2 is reconstructed from Σ¯ 2 . This follows from the uniqueness of lifted paths [25] with given starting point on the cover, which implies that for any xi , 0

0

(φ(xi )).γ(y0 , P ) = φ(xi .γ(y0 , P )),

(10.13)

0

where P is a point infinitesimally close to P. 2. Equivalent dcc’s determine equivalent configurations. This part again uses the proof in the case of ordinary branched covers together with uniqueness of lifted paths. 3. The map from CFG to equivalence classes of dcc’s is onto. This third part follows from the usual proof in the case of branched covers together with our choice of allowed pairings in CFG. A corollary of Proposition 10.3.1 is that the group Autf is isomorphic to C(e), where e ∈ CFG represents the equivalence class of f . Remarks.


605

1. Each element e of CFG corresponds to one degenerated coupled cover. We denote by m(e), the number of degenerating coupled covers associated to it. m(e) is the product of the common cycle lengths over all the pairings. 2. Autf is a subgroup of Autf + × Autf − , and therefore C(e) is a subgroup of C(ψ + ) × C(ψ − ), where e is an equivalence class of dcc’s corresponding to a pair of equivalence classes ψ + , ψ − of branched covers. In general C(e) is a proper subgroup. As in Sect. 4.2 we see that Autf is a subgroup of Aut(ΣW ), where ΣW can now have double points. 10.4. Coupled Hurwitz space. Now let CHS(h, G) be the space of degenerating coupled covers from a surface of genus h to a surface of genus G. Let S be the union of the branch and tube loci. Define CHS(h, G, L) to be the space of Dcc’s for which the set S has L points. We isolate the subspace of coupled branched covers where n+ is the total degree of the map from holomorphic-sector, B + is the holomorphic branching number, n− , B − are corresponding quantities for the anti-holomorphic sector; and D is the total number of double points. We call this CHS(n± , B ± , L, D). If we specify the locus S(L) then we define the finite set CHS(n± , B ± , S(L), D). The space CHS(n± , B ± , L, D) is a bundle over the configuration space of L points on ΣT , with discrete fibre CHS(n± , B ± , S(L), T ). Following the reasoning of Sect. 5 we may write a formula for the orbifold Euler characteristic of CHS(n± , B ± , L, D). By definition this may be taken to be: ± ± (10.14) χ CHS(n , B , L, D) orb

X

≡ χ(CL (ΣT ))

[f ]∈CHS(n± ,B ± ,S(L),D)

1 . |Aut(f )|

We first show how to count equivalence classes of degenerating coupled covers [f ] inducing the data (10.5)–(10.9) compatible with n± , B ± , L, D. It will be convenient to introduce the following quantities. Given two vectors r± , of nonnegative integers with almost all entries zero, we define the polynomial ℘(r+ , r− , x) =

=

∞ Y j=1 ∞ X

1+

∞ X `=1

+ − rj rj j` ` `

x` `!

(10.15)

xt ℘(t, r+ , r− ).

t=0 r± The binomial coefficient `j is defined to be zero for rj± < `. If v and w are in + − ± symmetric groups we also define p(d) v⊗w to be equal to ℘(d, r , r ) − δ(v, w), where r encode the cycle decompositions of v, w and δ(v, w) = 1 if (v = 1, w = 1) and zero otherwise.

Proposition 10.4.1. Suppose we are given n± > 0, B ± , D and the set S(L). Then, we have: X m(e) X 1 = |C(e)| |Autf | ± ± e∈CF G

[f ]∈CHS(n ,B ,S(L),D)

606


=

X

X

d1 ,d2 ···dL s± ,t± ,vi ,wi i i

1 n+ !n− !

(10.16)

d1 +···dL =D

δ

Y L

vi ⊗ wi

i=1

G Y

Y L

− + [s+i ⊗ s− i , t i ⊗ ti ]

i=1

i) p(d vi ⊗wi ,

i=1

where the first sum is over elements of CFG compatible with the data, the second sum is over equivalence classes of degenerating coupled covers inducing the specified data, and the vi , wi in the third sum are compatible with total branching numbers B ± . Proof. The first equality follows from Proposition 10.1. To derive the second equality we note that for a given pair of homomorphisms which determine (anti-)holomorphic maps f ± : N ± (ΣW ) → ΣT , there are many ways to introduce double points to define a dcc. We sum over all possible numbers of double points d1 · · · dL compatible with d1 + d2 · · · dL = D. Now there are a number of ways of introducing double points d1 , · · · dL to define dccs compatible with the maps f ± . Let d(f, P ) denote the tube number above P . The case d(P ) = 2 is illustrated in Fig. 14.

Fig. 14. d(P ) is the number of tubes above the point P

Because we count distinct degenerations separately, a double Qpoint joining ramification points of index j is counted j-times. Thus there are exactly P pd(f,P ) (v(P ), w(P )) ways of introducing double points above points in S(L) compatible with the specified homomorphisms. Therefore each pair of homomorphisms in the third line is weighted by the total number configurations compatible with it, multiplied by the multiplicity appropriate for counting degenerating coupled covers. Sn+ × Sn− acts on the set of + !n− ! . This configurations and the number of times a given equivalence class occurs is n|C(e)| establishes the second equality. Now, combining Proposition 10.2 and (10.14), we may write the Euler character as χ CHS(n± , B ± , L, D) orb

X

X

= χ(CL (ΣT ))

1

d1 ,d2 ···dL s± ,t± ,vi ,wi i i

Y L i=1

d1 +···dL =D

vi ⊗ wi

G Y i=1

[s+i

⊗

+ s− i , ti

⊗

(10.17)

n+ !n− ! Y L

t− i ]

i=1

i) p(d vi ⊗wi .


607

10.5. The nonchiral Y M2 sum and Euler characters of CHS. Having completed our geometrical preliminaries we return to the 1/N expansion of Y M2 . The full partition function (2.2) can be written, in the zero area limit, as ∞ X

Z(0, G, N )

N (n

+

+n− )(2−2G)

n± =0

X s± ,t± ,...,s± ,t± ∈Sn 1 1 G G

(10.18)

G Y 1 2−2G − − + + δ n+ ,n− [sj , tj ] ⊗ [sj , tj ] . n+ !n− ! j=1

The delta function is over the group Sn+ × Sn− . The element n+ ,n− , introduced in [15), related to the dimension of SU (N ) representations by −

+

dim(RS) =

N n +n χ ( + − ), n+ !n− ! RS n n

(10.19)

where R has n+ boxes and S has n− boxes, and (RS) is the irreducible representation of largest dimension in the tensor product of R with the complex conjugate of S. Explicitly, n+ ,n− is an element of the group algebra of Sn+ × Sn− given by X

n+ ,n− =

(v ⊗ w)Pv,w (

v∈Sn+ ,w∈Sn−

1 1 (n+ −Kv )+(n− −Kw ) ) . N2 N

(10.20)

The polynomials Pv,w ( N12 ) are given by Pv,w ( N12 ) = ℘(r(v), r(w), −1/N 2 ), where r is the vector of non-negative integers describing the cycle decomposition of the permutations v, w and ℘ was defined in (10.15). We write X 1 1 (n+ −Kv )+(n− −Kw ) n+ ,n− = 1 ⊗ 1 + v⊗w pv,w ( 2 ) N N v∈Sn+ ,w∈Sn−

X −1 ( 2 )d =1⊗1+ N d

X v∈Sn+ ,w∈Sn−

(10.21) 1 (n+ −Kv )+(n− −Kw ) (d) v⊗w pv,w , N

where we have pulled out the leading term of 1 ⊗ 1 so that pv,w = Pv,w − δ(v, w). In the second line we have collected terms with a given power of 1/N . Using (10.21) and expanding the inverse point as in Sect. 5 leads to the following expression for the partition function: X X N 2−2h (−1)D Z(0, G, N ) = n± ,B ± ,D

h

X

(n+ +n− )(2G−2)+B + +B − +2D=2h−2

χ(CL (ΣT ))

X

(10.22)

d1 ,d2 ···dL

L

d1 +···+dL =D

X s± ,t± ,vi ,wi i i

Y Y Y (d ) 1 − + − + δ( v ⊗ w [s ⊗ s , t ⊗ t ]) pvii⊗wi . i i i i i i n+ !n− ! L

G

L

i=1

i=1

i=1

608


Since we have collected together the contributions with fixed branching number B + and − sum over the permutations in the last line is required to obey the condition B P , the ± (n − Kvi ) = B ± . Note that the sum on L is actually finite, and can be bounded i above by B + + B − + D. In the above, the sum appearing after the Euler character of the configuration space of L points in ΣT is a sum over a discrete set which we have described in the previous subsection as a sum over equivalence classes of dcc’s. Indeed, using (10.17) we finally arrive at Proposition 10.5.1. The full A = 0 partition function of Y M2 is a generating functional for the orbifold Euler characters of coupled Hurwitz spaces: X X N 2−2h (−1)D Z(0, G, N ) = n± ,B ± ,D

h

(n+ +n− )(2G−2)+B + +B − +2D=2h−2

(10.23) − B + +B X +D

χorb (CHS(n± , B ± , L, D)).

L=0

Note that the Euler character of configuration spaces which appears involves the configurations of both branch points and images of double points on the target.

11. The Nonchiral Topological String Theory The nonchiral analog of the theory of Sect. 7 must localize on both the space of holomorphic and antiholomorphic maps. When we regard the topological string path integral as an infinite dimensional version of an equivariant Thom class, it becomes clear that we f± of Ce defined need a section T of some bundle which localizes on the submanifolds M by df ± J df [h] = 0. It is therefore natural to choose a section of the form: e cf , enc ⊕ V T: Ce −→ V nc (11.1) T (f, h) 7−→ (df + J df [h]) ⊗ (df − J df [h]). Following the considerations for the construction of a general TFT, we have the following fields, ghosts, antighosts, and Lagrange-multipliers: F=

fµ hαβ

A = ρµν αβ

χµ ψαβ µν . Π = παβ G=

Only the anti-ghosts and Lagrange multipliers of the sigma model have changed relative to the chiral theory. In particular, the appropriate bundle for the antighosts ρ has fiber: nc e(f,h) = 0 (T ∗ ΣW )⊗2 ⊗ (f ∗ (T ΣT ))⊗2 ± , (11.2) V where the subscript ± indicates that the sections must satisfy “self-duality" constraints:


( e nc

ρ ∈ V(f,h)

⇐⇒

ρ − (J ⊗ 1) ρ ( ⊗ 1) = 0 or ρ + (1 ⊗ J) ρ (1 ⊗ ) = 0

609

.

(11.3)

The BRST transformations are the same as above. The nonchiral theory has an action nc nc + Icofield . IYM2 string = Itg + Itσ

(11.4)

The gravity part of the action is the same as before. The topological sigma model part is Z √ n µν o µ nc λ ρν ν λ µρ 1 µν Itσ = QC d2 z h ραβ , (11.5) µν itαβ − 0λρ χ ραβ − 0λρ χ ραβ + 2 παβ where the indices on ρ and π are raised and lowered with the metrics on the worldsheet (h), and target space (G) . If we expand (11.5) and integrate out the Lagrange multiplier then the bosonic term becomes (in local conformal coordinates) Z nc = hzz¯ G2ww¯ |∂z f w |2 |∂z¯ f w |2 + · · · , (11.6) Itσ thus clearly localizing on both holomorphic and antiholomorphic maps. Moreover, when we work out the quadratic terms in the fermions we find that many components of ρ do not enter the Lagrangian. These components are eliminated by the constraints (11.3). In locally conformal coordinates the only non-trivial components of µν nc ¯ w¯ w¯ ww ww¯ e(f,h) are ρww ρ∈V zz , ρz z¯ , ρzz ¯ , and ρz¯ z¯ . (Note that ραβ is not symmetric in interchanging {(αβ)(µν)} ↔ {(βα)(νµ)}.) The kinetic term for the fermions is given by Z √ nc (11.7) Itσ = i d2 z h ( ρ η ) Onc ( χψ ) + · · · , where Onc is a 2 × 2 matrix operator with entries: + − Onc 11 = D ⊗ [df − J df ] + [df + J df ] ⊗ D , Onc 12 = J df k ⊗ [df − J df ] − [df + J df ] ⊗ J df k,

(11.8) O21 = ∂f, nc

† Onc 22 = P ,

where (P † δh)β = Dα δhαβ , D± χµ = Dχµ ± J(Dχµ ) and, as usual, k[δh] is the variation of the complex structure on ΣW induced from a variation of the metric δh. The co-model is introduced using the same principles as before. 11.1. Singular gometries. The full path integral of the topological string theory involves field configurations (f, h) ∈ M ap(ΣW , ΣT ) × M et(ΣW ), which are not necessarily C ∞ . An appropriate completion of this space will introduce, among other things, piecewise continuous or even singular maps and geometries. Usually in field theory such considerations of analysis are of interest only in constructive quantum field theory [64] but in the present case the contribution of singular field configurations (f, h) in the path integration domain becomes an issue of great importance because the path integral can localize on subspaces of singular geometries. By allowing piecewise differentiable maps

610


and metrics we can incorporate the coupled Hurwitz space in the fiber bundle picture we described for the ordinary Hurwitz space in Sect. 4.4. In the theory of analytic functions one can show18 that the weak assumption of differentiability of a solution to the Cauchy Riemann equations implies the function is C ∞ and even analytic [65]. Thus it might seem that one gains nothing by replacing the C ∞ assumption by the assumption of differentiability when searching for zeroes of df ± Jdf . This reasoning breaks down if the place where derivatives of f are discontinuous is also a singular point in the geometry of the worldsheet. This is precisely what happens in a dcc. Indeed, the following simple reasoning shows the presence of solutions of w˜ = 0 in (11.1) which are not in Fe± . Consider the space Ξ = M ap(ΣW , ΣT ) × M et(ΣW ) where we have added singular geometries to form a “boundary.”19 The situation is illustrated schematically in Fig. 15.

e+ and Fe− in M ap × M et Fig. 15. F The Hurwitz spaces Fe± lie in the interior of Ξ, but extend out to the boundary because, as we have already seen in Sect. 4.5, type 3 collisions between branch points 18

This is sometimes called Goursat’s theorem. We have sketched how dcc’s arise in the bundle approach to Hurwitz space when we consider the full space of maps and metrics. It would be very interesting to construct a careful compactification of the space of Maps × Metrics in such a way that it is manifest why degenerated coupled covers should be counted with the degeneracy m(e). 19


611

can give rise to singular worldsheets. We saw there that we can obtain ramification points of equal index on each of the components which are joined by a tube. Such degenerations of holomorphic covers are labeled by (++) on the LHS of Fig. 15. Now, the theory of Sect. 11.1 is invariant only under the group of orientations preserving diffeomorphisms Dif f + (ΣW ), so configurations related by diffeomorphisms of type (±, ∓), which are orientation preserving on one component but orientation reversing on the other are considered gauge-inequivalent. The 4 points we have indicated on the boundary of Ξ which lie on the outer circle are related to one another by such diffeomorphisms. The configurations indicated at the top and bottom of Fig. 15 are additional, singular configurations which can contribute to the localisation of the path integral. The way in which we handle these surface contributions corresponds to a choice of contact terms, similar to the contact terms that accounted for area polynomials in Sect. 8. One choice of contact terms simply declares that the dcc’s do not contribute: we cut out all singular field configurations and define the integral by a limiting procedure. From the results of Sect. 7 we can immediately conclude that with this choice of contact terms the partition function of the topological string theory becomes: X 2h−2 1 Zstring (ΣW → ΣT ) = Z + (A = 0, N )Z − (A = 0, N ). (11.9) exp N h≥0

That is, we produce the “chiral” Y M2 theory obtained by replacing: n+ ,n− −→ n+ ,0 0,n− in the full Y M2 sum. (Note this is a product of chiral theories in the sense that “chiral” is usually employed.) It is not at all clear that this is a sensible (e.g., BRST invariant) choice of contact terms from the point of view of the topological string theory. A contact term analysis similar to that used in Sect. 8 is needed to explain why dcc’s contribute whereas degenerations of (++) or (−−) type do not contribute.

12. Conclusions

12.1. General remarks. In this paper we have used the results of [15] to make some progress towards a formulation of Y M2 as a topological string theory. We have seen that the 1/N expansion of Y M2 may always be formulated in terms of quantities associated to branched covers, provided we admit sufficiently singular geometries. In Sects. 6–9 we formulated a string theory which reproduces chiral Y M2 when proper account is taken of singular geometries. In Sects. 10, 11 we have partially extended the results to nonchiral Y M2 . Reproducing all Y M2 in complete detail, involves highly complicated considerations of the contributions of multiple contact terms. In some cases, for example, for the area polynomials associated with simple Hurwitz space a heuristic analysis of contact terms allows us to derive the singular contributions to the string path integral, as in Sect. 8. Amplitudes we have not explicitly calculated from the string theory picture are : 1) Area polynomials associated with non-simple coverings, and the contributions of handles and tubes (from expanding (8.5)). 2) Modifications of area polynomials from including dcc’s , and expanding exp(−n+ n− A/N 2 ) in (2.2).

612


3) Nonchiral Wilson loop amplitudes. There is an interesting analogy between the topological string formulation of Y M2 and the topological approach to 2D gravity. The exact Y M2 answers like (2.1) (9.5) play a role analogous to the results of the double-scaled matrix models. As in 2D gravity, these nontrivial and exact answers show the importance of singular geometries to any topological formulation. We are lucky to have these answers, since they guide us through the dense thicket of singular boundary contributions which are, presumably, ubiquitous in all theories of gravity.20 Contact terms are at once the Achilles heel and the rock of salvation of topological gravity. They make the truly interesting theories nearly impossible to analyze, yet provide a mechanism whereby the theory can be nontrivial enough to be worthy of attention. 12.2. Open problems, future directions. The present work suggests several possible generalizations and further directions for research. One important generalization is the string-theoretic version of Y M2 based on the other series of classical compact gauge groups Sp(N ), O(N ). The 1/N expansion of these theories has been worked out in [66, 67]. For O(N ) and Sp(N ) Yang Mills one has to deal with new subtleties associated with string theories on non-orientable surfaces. A given branched cover (possibly with double points) in these theories appears only once. There is no analog of the two sectors in the SU (N ) theory. In a bundle description of the corresponding Hurwitz space analogous to Sect. 4.3, one would replace Dif f + (ΣW ) with Dif f (ΣW ) . Douglas [8] has observed that for ΣT of genus one there is a “near” A → 1/A duality of Y M2 since the amplitudes are expressed in terms of Eisenstein series in q = e−A . As noted by many physicists, target space duality is quite natural for a topological string theory based on a topological conformal field theory (TCFT). In order to make the answer truly modular covariant it is necessary [8] to modify Y M2 in seemingly unmotivated ways. One may search for an explanation of this in the string formulation. Further, when ΣT does not have genus one we are coupling topological gravity to a topological σ-model which is not a TCFT. As is well known, an important challenge to the string approach is the derivation of quantities such as the meson spectrum. It is nontrivial to rederive the standard results of the ’t Hooft model. Finally, the construction of the anomaly-cancelling co-model Sc , can be applied to a wide class of topological field theories. One may wonder if the resulting theories are of any interest. For example, the analogous construction with Donaldson theory would compute the Euler character of the moduli space of instantons. What is the physical interpretation of this theory? 12.3. What about QCD4? Aside from the intrinsic beauty of the subject, one of the main reasons we are interested in a string action for Y M2 is the hope that this action might contain essential features of a hypothetical string action for Y M4 . In this respect our construction is disappointing since it relies on topological field theory. Our construction does have a natural generalization to 4 dimensions: Choose an almost complex structure on the four-dimensional target and form the appropriate topological sigma model. Then follow the construction of the co-model to cancel the anomalous R-symmetry and guarantee that we get the Euler density for the moduli space of curves in the target. Following the general ideas outlined above one will find that the topological string theory localizes 20 An interesting related example where the exact answers have not been previously available is curvecounting in Calabi-Yau three-folds [18].


613

onto the family of holomorphic maps, F(Σh , X). A formula for the dimension of this moduli space is given in Eq. (A.16) below. Of course, any given topological theory admits several different formulations, and it might be that these different formulations admit different generalizations to four dimensions. Perhaps one of these will teach us something about Y M4 . Some indication that this might be possible is given by the construction of [60] which also has a natural generalization to four dimensions. Acknowledgement. We would like to thank V.I. Arnold, M. Bershadsky, P. Bouwknegt, S. Cecotti, R. Dijkgraaf, M. Douglas, E. Getzler, D. Gross, M. Guest, J. Harris, J. Horne, V. Kazakov, M. Khovanov, I. Kostov, B. Lian, V. Mathai, J. McCarthy M. Newman, H. Ooguri, R. Rudd, J. Segert, W. Taylor, A. Wilkins and G. Zuckerman for discussions and correspondence. We would like to thank R. Dijkgraaf, R. Plesser, and W. Taylor for useful comments on the manuscript. This work is supported by DOE grants DE-AC02-76ER03075, DEFG02-92ER25121, DE-FG05-900ER40559, and by a Presidential Young Investigator Award. GM is grateful to the Rutgers Dept. of Physics for hospitality while this paper was being finished.

Appendix A. The Deformation Theory Approach to Hurwitz Space In this appendix we derive a formula for the dimension of the space of families of holomorphic maps. Our treatment here is valid also for the case of maps into higher dimensional target spaces. The deformation theory of holomorphic maps is a subject developed by Horikawa [69], Mijayima [69] and Namba [70]. Let U be a fixed compact complex manifold. A family of holomorphic maps into U is, by definition a family (X, π, S) = {Vs }s∈S of compact complex manifolds, together with a holomorphic map F : X → U . We denote it by (X, π, S, F). Set fs = F|Vs : Vs −→ U. Sometimes this family of maps is denoted by {Vs , fs }s∈S . Next we define an infinitesimal deformation of a family (X, π, S, F ) = {Vs , fs }s∈S at o ∈ S. Let f = fo . The data which characterize F locally are: α and transition func(a) An open covering of V , V = {Vm } with local coordinates z(m) tions gmn : Vn × S → Vm which vary with s ∈ S, i (b) An open covering of U , U = {Um } with local coordinates w(m) and fixed21 transition functions hmn : Un → Um , (c) A family of holomorphic maps f(m) : Vm × S → Um which vary with s ∈ S.

The {fm }, {gmn }, and {hmn } must satisfy the compatibility condition hmn ◦ fn = fm ◦ gmn ,

(A.1)

which ensures commutativity of the following: Un ∩ x Um f n V n ∩ Vm

h

mn −→

gmn

−→

Um x ∩ Un f m Vm ∩ Vn

21 There is a natural extension of the deformation theory of holomorphic maps developed by Namba [70], wherein the complex structures of both V and U are varied. This, however, would only become relevant if we studied YM2 coupled to 2d (spacetime) gravity.

614


Differentiating (A.1) with respect to s and contracting with X

i ∂s f(m)

α

=−

X

∂ , i ∂w(m)

we find

j X ∂ ∂ i ∂hmn − ∂s f(n) ◦ fn j i i ∂w(m) ∂w(n) ∂w(m) α,β α

β ∂s gmn

αβ

∂f(m) β

∂z(m)

◦ gmn

∂ . i ∂w(m)

(A.2)

˘ Now define the Cech 1-cocycle, θ, valued in the sheaf of germs of holomorphic sections of T V to be: θ = {θmn } ∈ Z 1 (V, ΘV ),

θmn = ∂s gmn .

(A.3)

˘ Further define a Cech 0-cochain, η, valued in the inverse image sheaf22 , f ∗ ΘU , by η = {ηi } ∈ C 0 (V, f ∗ ΘU ),

η m = ∂ s fm .

(A.5)

Then (A.2) expresses the fact that (δη)mn = −f∗ θmn .

(A.6)

The deformation theory of holomorphic maps has a very concise formulation in ∂ ∈ To S to a certain cohomology terms of a characteristic map from a tangent vector ∂s class. This is the analogue of the Kodaira-Spencer map which arises in the study of deformations of the complex structures of complex manifolds. In order to present this description of the tangent space to F, we need to introduce a few more notions: First we introduce the following complex of sheaves [70]: L∗ :

f∗

f∗

f∗

0 −→ L0 = ΘV −→ L1 = f ∗ ΘU −→ 0,

(A.7)

where f∗ is the sheaf map which tautologically satisfies (f∗ )2 = 0. Associated to this complex of sheaves [71] are the cohomology sheaves Hq = Hq (L). Setting Lq (U) = H 0 (V, Lq ), the presheaf V 7−→

ker {f∗ : Lq (V ) → Lq+1 (V )} f∗ Lq−1 (V ),

gives rise to a sheaf Hq whose stalk is ker {f∗ : Lq (V ) → Lq+1 (V )} . V 3x f∗ Lq−1 (V )

Hxq = lim

A section η of Hq over V is given by a covering {Vm } of V and ηm ∈ Lq (Vm ) such that f∗ ηm = 0.

(A.8)

A section is zero in the case when 22 Recall that if f : V → U is a continuous map of topological spaces, then the inverse image sheaf, f ∗ Θ , U of ΘU by the map f is defined as f ∗ ΘU = O(f ∗ T U ), (A.4)

the sheaf of germs of holomorphic sections of the pullback f ∗ T U of the holomorphic tangent bundle over f . f∗ : ΘV → ΘU is the push-forward map.


615

θm ∈ Lq−1 (Vm ),

η m = f∗ θ m ,

(A.9)

after perhaps refining the cover. ˘ cochains valued in Lq . Then we have the two operators: Let C p (V, Lq ) be Cech δ : C p (V, Lq ) −→ C p+1 (V, Lq ), (A.10) f∗ : C (V, L ) −→ C (V, L p

q

p

q+1

),

which satisfy (f∗ )2 = δ 2 = {f∗ , δ} = 0, so we have a double complex {C p,q = C p (V, Lq ), f∗ , δ}. The associated single complex (C ∗ , D) is defined by M Cn = C p,q , D = f∗ + δ. (A.11) p+q=n

We define the hypercohomology as follows: H∗ (V, L∗ ) = lim H ∗ (C ∗ (V), D). V

(A.12)

The pair (η, θ) ∈ H1 (V, L∗ ) is called an infinitesimal deformation of the family ∂ ∈ To S. {Vs , fs }s∈S at s ∈ S in the direction ∂s ∂ One denotes αo ( ∂s ) ≡ (η, θ). αo is a linear map αo : To S −→ H1 (U, L∗ ).

(A.13)

called the characteristic map. Definition . The family {Vs , fs }s∈S is said to be effectively parametrized at o ∈ S if α0 is injective. Definition . A morphism of (X 0 , π 0 , S 0 , F 0 ) to (X, π, S, F) is by definition a morphism (h, e h) which makes the following diagram commutative ˜ h

−→ X0 0 .F & F   0 U yπ h

S0

−→

X  π y S

Definition . A family {Vs , fs }s∈S is said to be complete at o ∈ S if for every family {Vs0 , fs0 }s0 ∈S 0 with a point o0 ∈ S 0 and a biholomorphic map i : Vo00 → Vo which makes the following diagram commutative: Vo00

i

& fo0 0

−→

. fo

Vo

U there is an open neighborhood U 0 of o0 ∈ S 0 and a morphism (h, e h) of {Vs0 , fs0 }s0 ∈S 0 such that (i) h(o0 ) = o,

616


(ii) e ho0 = i : V00 → V0 . If a family is complete at o ∈ S, then it contains all small deformations of f . {Vs , fs }s∈S is complete, if it is complete at every point of S. If a family {Vs , fs }s∈S is complete and effectively parametrized at o ∈ S, then it is said to be versal. In this case, it is the smallest among complete families. One important property of F which is of obvious interest is its dimension. For this purpose the following two theorems due to Horikawa [68] and Namba [70] are useful: Theorem (Horikawa). For a holomorphic map f : V → U , if (A) H 1 (V, ΘV ) → H 1 (V, f ∗ ΘU ) is surjective, (B) H 2 (V, ΘV ) → H 2 (V, f ∗ ΘU ) is injective, there exists a complete family {Vs , fs }s∈S of holomoprhic maps into U with a point o ∈ S, such that (1) (2) (3) (4)

Vo = V , fo = f , it is effectively parametrized at o, o is a non-singular point of S and dimo S = dim IH1 (V, L∗ ).

Theorem (Namba). The following sequence is exact: 0

−→ H0 (V, L) −→ H 0 (V, ΘV ) −→ H 0 (V, f ∗ ΘU ) −→ −→ H1 (V, L) −→ H 1 (V, ΘV ) −→ H 1 (V, f ∗ ΘU ) −→ . −→ H2 (V, L) −→ H 2 (V, ΘV ) −→ H 2 (V, f ∗ ΘU ) −→

(A.14)

For h ≥ 2, H0 (V, L∗ ) = 0, we find for versal families F (Σh , ΣG , J) that dim T F (Σh , ΣG , J) = dim H1 (V, L∗ ) (A.15) = −dim H 0 (V, ΘV ) + dim H 0 (V, f ∗ ΘU ) +dim H 1 (V, ΘV ) − dim H 1 (V, f ∗ ΘU ) = B, where B is precisely the branching number! As another application, one can use (A.15) to determine the dimension of the moduli space of holomorphic maps, f : ΣW → X, from a Riemann surface, ΣW , into a higher dimensional target space, X. If ΣW has genus h, then it is easy to establish that dim F (Σh , X) = 3(h − 1) + dim H 0 (Σh , f ∗ ΘX ) − dim H 0 (Σh , Kh ⊗ f ∗ KX ). (A.16)

Appendix B. Derivation of the Variation of Gromov’s Equation We shall remain general and consider the target space to be an arbitrary complex manifold, X. It is important to note that the Gromov equation is non-linear in f . We can make this clear by explicitly indicating that J is evaluated at f (σ): df (σ) + J[f (σ)]df (σ)(σ) = 0. Now consider a one parameter family of holomorphic maps

(B.1)


617

F : Σ × I → X, F (σ; t) 7→ ft (σ) with f0 (σ) = f (σ). This family must also satisfy the Gromov equation dft (σ) + J[ft (σ)]dft (σ)(σ) = 0

∀σ ∈ Σ and ∀t ∈ I.

Now take the derivative with respect to t and evaluate at t = 0. We suppress worldsheet indices where they are obvious, µ df˙t (σ) + ∂κ J µν [ft (σ)]f˙tκ (σ)dftν (σ)(σ) + J µν [ft (σ)]df˙tν (σ)(σ) t=0 = 0. Now consider the covariant derivative of J ∇κ J µν = ∂κ J µν + 0µκλ J λν − 0λκν J µλ . Then we may write (setting δf µ = f˙tµ |t=0 ) ∂κ J µν δf κ df ν = −0µκλ J λν δf κ df ν + 0λκν J µλ δf κ df ν + ∇κ J µν δf κ df ν . Now since ft (σ) is, by fiat, a family of holomorphic maps, dftµ = J µν dftν , for all t, so that ∂κ J µν δf κ df ν = 0λκν J µλ δf κ df ν − 0µκλ δf κ df λ + ∇κ J µν δf κ df ν . In the case that X is a complex manifold, ∇κ J µ ν = 0 and we deduce the following equation for the tangent space: D(δf ) + J D(δf ) [h] + J df k[δh] = 0,

(B.2)

where D is the pulled-back connection (Dα δf )µ = ∂α δf µ + 0µκλ ∂α f κ δf λ . References 1. Veneziano, G.: Construction of a crossing-symmetric, Regge-behaved amplitude for linearly rising trajectories. Nuovo Cim. 57A, 190 (1968) 2. Wilson, K.G.: Confinement of quarks. Phys. Rev. D10, 2445 (1974) 3. t Hooft, G.: A planar diagram theory for string interactions. Nucl. Phys. B72, 461 (1974) 4. Migdal, A.A.: Loop equations and 1/N expansion. Physics Reports (Review section of Physics Letters) 102, 199–290 (1983) 5. Polchinski, J.: Strings and QCD? Talk presented at the Symposium on Black Holes, Wormholes, Membranes and Superstrings. Houston, 1992; hep-th/9210045 6. Gross, D.: Some new/old approaches to QCD. Published in String Theory Workshop, Rome 1992, pp. 251–268; hep-th/9212148 7. Gross, D.J., Taylor, W.: Two-Dimensional QCD and Strings. Published in Strings ’93, Berkeley, 1993, pp. 214–225; hep-th/9311072 8. Douglas, M.R.: Conformal Field Theory Techniques in Large N Yang-Mills Theory. hep-th/9303159 9. Bars, I.: QCD and Strings in 2d. Talk given at International Conference on Strings 93, Berkeley, CA, Published in Strings 93: 175–189, hep-th/9312018 10. Migdal, A.: Recursion equations in gauge theories. Zh. Eskp. Teor. Fiz. 69, 810 (1975) (Sov. Phys. JETP. 42, 413) 11. Kazakov, V.A., Kostov, I.: Non-linear strings in two dimensional U (∞) gauge theory. Nucl. Phys. B176, 199–215 (1980); Kazakov, V.A.: Wilson Loop average for an arbitrary contour in two-dimensional U (N ) gauge theory. Nucl. Phys. B179, 283–292 (1981) 12. Rusakov, B.: Loop Averages and Partition Functions in U (N ) gauge theory on two-dimensional manifolds. Mod. Phys. Lett. A5, 693 (1990)

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13. Witten, E.: On gauge theories in two dimensions. Commun. Math. Phys. 141, 153 (1991) 14. Blau, M., Thompson, G.:Lectures on 2d Gauge Theories: Topological Aspects and Path Integral Techniques. Published in Trieste HEP and Cosmology 1993, pp. 175-244; hep-th/9310144 15. Gross, D.: Dimensional QCD as a String Theory. PUPT-1356, LBL-33415; hep-th/9212149; Gross, D., Taylor, W.: Two-dimensional QCD is a String Theory. Nucl. Phys. B400, 161–180 (1993); hepth/9301068; Gross D., Taylor, W.: Twists and Loops in the String Theory of Two Dimensional QCD. Nucl. Phys. B403, 395-452 (1993); hep-th/9303046 16. Minahan, J.: Summing over inequivalent maps in the string theory interpretation of QCD. Phys. Rev. D47, 3430 (1993) hep-th/930/005 17. Dijkgraaf, R., Rudd, R.: Unpublished 18. Bershadsky, M., Cecotti, S., Ooguri, H., Vafa, C.: Holomorphic anomalies in topological field theories. Nucl. Phys. B405, 279–304 (1993); hep-th/9302103; Kodaira-Spencer theory of gravity and exact quantum string amplitudes. Commun. Math. Phys. 165, 311–428 (1994); hep-th/9309140 19. Kanno, H.: Weil algebra structure and geometrical meaning of BRST transformation in topological quantum field theory. Z. Phys. C43, 477 (1989) 20. Atiyah M.F., Jeffrey, L.: Topological Lagrangians and Cohomology. Jour. Geom. Phys. 7, 119 (1990) 21. Kalkman, J.: BRST Model for Equivariant Cohomology and Representatives for the Equivariant Thom Class. Commun. Math. Phys. 153, 447 (1993) 22. Horava, P.: Topological Strings and QCD in Two Dimensions. EFI-93-66, hep-th/9311156. To appear in Proc. of The Cargese Workshop, 1993 23. Fulton, W.: Hurwitz schemes and irreducibility of moduli of algebraic curves. Annals of Math. 90, 542 (1969) 24. Ezell, C.L.: Branch point structure of covering maps onto nonorientable surfaces. Trans. Am. Math. Soc. 243 (1978) 25. Massey, W.S.: A basic course in Algebraic Topology. Berlin–Heidelberg–New York: Springer-Verlag, 1991 26. Spanier, E.H.: Algebraic Topology, New York: Mc. Graw-Hill 1966 27. Edmonds, A.L.: Deformation of maps to branched coverings in dimension two. Annals of Mathematics 110, 113–125 (1979) 28. Harris, J., Mumford, D.: On the Kodaira dimension of the Moduli Space of Curves. Invent. math. 67, 23–86 (1982) 29. Ahlfors, L.V., Sario, L.: Riemann Surfaces. Princeton: Princeton University Press, 1960 30. Wells, R.O.: Differential Analysis on Complex Manifolds. Berlin–Heidelberg–New York: Springer, 1980 31. Beilinson, A., Ginzburg, V.: Infinitesimal structure of moduli spaces of G-Bundles. International Mathematics Research Notices 1992, No. 4 32. Mumford, D.: Towards an enumerative geometry of the moduli space of Curves. In: Arithmetic and Geometry Basel–Boston: (Birkhauser). 33. Knudsen, F.: The projectivity of the moduli space of stable curves. Math. Scand. 52, 161 (1983) 34. Bott, R., Tu, L.: Differential Forms in Algebraic Topology. Berlin–Heidelberg–New York: Springer Verlag, 1982 35. Penner, R.C.: Perturbative Series and the moduli space of Riemann surfaces. J. Diff. Geom. 27, 35–53 (1988) 36. Birman, J.S.: Braids, Links and Mapping Class Groups. Princeton: Princeton University Press, 1975 pp. 11–12 37. Berstein, I., Edmonds, A.L.: On the classification of generic branched coverings of surfaces. Illinois J. Math. 28, number 1 (1984) 38. Vick, J.: Homology Theory. New York: Academic Press, 1973 39. Witten, E.: Topological sigma models. Commun. Math. Phys. 118, 411–419 (1988) 40. van Baal, P.: An Introduction to Topological Yang-Mills Theory. Acta Physica Polonica, B21, 73 (1990) 41. Witten, E.: Introduction to Cohomological Field Theories. Lectures at Workshop on Topological Methods in Physics, Trieste, Italy, Jun 11-25, 1990, Int. J. Mod. Phys. A6, 2775 (1991) 42. Birmingham, D., Blau, M., Rakowski, M., Thompson, G.: Topological Field Theories. Phys. Rep. 209, 129 (1991) 43. Blau, M.: The Mathai-Quillen Formalism and Topological Field Theory. Notes of Lectures given at the Karpacz Winter School on Infinite Dimensional Geometry in Physics, Karpacz, Poland, Feb. 17–29, 1992; J. Geom. and Phys. 11, 129 (1991) 44. Dijkgraaf, R., Verlinde, H., Verlinde, E.: Notes on Topological String Theory and 2-d Quantum Gravity. Lectures given at Spring School on Strings and Quantum Gravity, Trieste, Italy, Apr. 24–May 2, 1990


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45. Cordes, S., Moore, G., Ramgoolam, S.: Lectures on 2d Yang-Mills Theory, equivariant Cohomology and Topological Field Theories. http://xxx.lanl.gov/lh94. Published in two parts. Part I: Proceedings String Theory, Gauge Theory and Quantum Gravity, Trieste 1994, pp. 184–244. Part II: Proceedings of the 1994 Les Houches summer school, session 62, Fluctuating Geometries in Statistical Mechanics and Field Theory 46. Witten, E.: Topological Quantum Field Theory. Commun. Math. Phys. 117, 353 (1988) 47. Mathai, V., Quillen, D.: Superconnections, Thom Classes, and Equivariant Differential Forms. Topology 25, 85 (1986) 48. Manin, Yu.I.: Quantized Fields and Complex Geometry, Berlin–Heidelberg–NewYork: Springer Verlag 49. Witten, E.: Algebraic Geometry Associated with Matrix Models of Two-Dimensional Gravity. IASSNSHEP-91/74 50. Witten, E: “The N-Matrix Model and gauged WZW models. Nucl. Phys. B371, 191 (1992) 51. Baulieu, L. Singer, I.: Topological Yang-Mills Symmetry. Nucl. Phys. B. Proc. Suppl. 5B 12 (1988) 52. Witten, E.: On the Structure of the Topological Phase of Two-Dimensional Gravity. Nucl. Phys. B340, 281 (1990) 53. Verlinde, E., Verlinde, H., Nucl. Phys. B348, 457 (1991); Dijkgraaf, R., Verlinde; E., Verlinde H.: Nucl. Phys. B348, 435 (1991); B352, 59 (1991). In String Theory and quantum Gravity, Proc. Trieste Spring School, April 1990 (World Scientific, Singapore, 1991) 54. Horava, P.: Two dimensional string theory and the topological torus. Nucl. Phys. B386, 383–404 (1992) 55. Baulieu, L., Singer, I.: The Topological Sigma Model. Commun. Math. Phys. 125, 227–237 (1989) 56. Kutasov, D.: Geometry on the space of conformal field theories and contact terms. Phys. Lett. 220B, 153 (1989) 57. G. Moore: Finite in All Directions. hep-th/9305139 58. Moore, G. Seiberg, Staudacher, M.: From Loops to States in sD Quantum Gravity. Nucl. Phys. 362, 665 (1991) 59. Alvarez, O.: Theory of strings with boundaries: fluctuations, topology and quantum geometry. Nucl. Phys 216, 125 (1983) 60. Kostov, I.K.: Continuum QCD2 in terms of Discretized Random Surfaces with Local weights. SaclaySPht-93-050, Jun. 1993, hep-th/9306110 61. Arnold, V.: Remarks on enumeration of plane curves. Plane curves, their invariants, Perestroikas and classifications 62. Strominger, A.: Loop space solution of Two-Dimensional QCD. Phys. Lett. 101B, 271 (1981) 63. Bars, I. Hanson, A.: Phys. Rev. D13, 1744 (1976); Nucl. Phys. B111, 413 (1976) 64. See, e.g., Glimm, J., Jaffe, A.: Quantum Physics, Berlin–Heidelberg–New York: Springer 1981 65. Conway, J.H.: Functions of One Complex Variable, Berlin–Heidelberg–New York: Springer 66. Naculich, S.G., Riggs, H.A., Schnitzer, H.G.: 2D Yang Mills theories are string theories. Mod. Phys. Lett. A8, 2223 (1993) 67. Ramgoolam, S.: Comment on two dimensional O(N ) and Sp(N ) Yang Mills theories as string theories. hep-th/9307085: Nucl. Phys. B 418, 30 (1994) 68. Horikawa, E.: On deformations of holomorphic maps, I, II, III. J. Math. Soc. Japan, 25, 647 (1973) ibid 26, 372 (1974); Math. Ann. 222, 275 (1976) 69. Mijayima, M.: On the existence of Kuranishi Family for deformations of holomorphic maps. Science Rep. Kagoshima Univ., 27, 43 (1978) 70. Namba, M.: Families of Meromorphic Functions on Compact Riemann Surfaces. Lecture Notes in Mathematics, Number 767, New York,: Springer Verlag 1979 71. Griffiths, P., Harris, J.:Principles of Algebraic Geometry. New York: J.Wiley and Sons, 1978, p. 445 Communicated by R.H. Dijkgraaf

Commun. Math. Phys. 185, 621 – 640 (1997)

Communications in


Variational Derivation of Exact Skein Relations from Chern–Simons Theories Rodolfo Gambini1 , Jorge Pullin2 1

Instituto de F´ısica, Facultad de Ciencias, Tristan Narvaja 1674, Montevideo, Uruguay Center for Gravitational Physics and Geometry, Department of Physics, 104 Davey Lab, The Pennsylvania State University, University Park, PA 16802, USA

2

Received: 15 March 1996 / Accepted: 8 October 1996

Abstract: The expectation value of a Wilson loop in a Chern–Simons theory is a knot invariant. Its skein relations have been derived in a variety of ways, including variational methods in which small deformations of the loop are made and the changes evaluated. The latter method is only allowed to obtain approximate expressions for the skein relations. We present a generalization of this idea that allows to compute the exact form of the skein relations. Moreover, it requires to generalize the resulting knot invariants to intersecting knots and links in a manner consistent with the Mandelstam identities satisfied by the Wilson loops. This allows for the first time to derive the full expression for knot invariants that are suitable candidates for quantum states of gravity (and supergravity) in the loop representation. The new approach leads to several new insights in intersecting knot theory, in particular the role of non-planar intersections and intersections with kinks.

1. Introduction Witten [1] realized some years ago that the expectation value of a Wilson loop W (γ) in a Chern–Simons theory was a knot invariant. This follows from the fact that Chern– Simons theories are diffeomorphism invariant and that the Wilson loops are observables for such theories, having therefore diffeomorphism invariant expectation values. The resulting knot invariant is the Kauffman bracket [2] for the case of an SU (2) Chern– Simons theory and for the case of SU (N ) is a regular isotopic polynomial associated with the HOMFLY [3] polynomial. These results were derived based on the calculations of Moore and Seiberg [4] for the monodromies of rational conformal field theories. The knowledge of the Yang-Baxter relations satisfied by the monodromies translates immediately into skein relations for the polynomial in question. Independently, Smolin [5] and later Cotta-Ramusino, Guadagnini, Martellini and Mintchev [6] noted that a simpler heuristic derivation of the skein relations was possible.

622

R. Gambini, J. Pullin

The idea is similar to the Makeenko-Migdal [7] approach to Yang–Mills theories. It is based on studying the changes in the expectation value of the Wilson loop when one performs small deformations. This calculation can be done explicitly to first order in the deformation. The results can be interpreted as skein relations to first order in the inverse coupling constant of the theory, which is tantamount to determining the knot polynomial to first order. Other variational calculations in related contexts can be found in [9, 10]. The variational method is quick and computationally efficient, and has a simple generalization to the case of intersecting loops [8, 11]. The main drawback, especially in the case where the result is not known by other methods, is that one only gets the skein relations to first order in the inverse of the coupling constant of the theory. It is therefore of interest to find a suitable generalization that would yield the exact skein relations to all orders. This is the main purpose of this paper. On the other hand, the subject of intersecting knot invariants has received little attention and is of paramount importance for the construction of quantum states of gravity in the loop representation [12, 13]. In this approach, based on the canonical quantization of general relativity in terms of Ashtekar variables [14], wavefunctions are knot invariants due to the diffeomorphism symmetry of general relativity [15]. The Hamiltonian constraint has only a non-trivial action at intersections [16]. Only intersecting knots are associated with non-degenerate spacetimes [17]. Whenever one generalizes an invariant of smooth loops to take values on intersecting loops there is generically freedom in how the invariant is defined, as long as it is compatible with the Reidemeister moves. However, in the case of quantum gravity, wavefunctions have to be compatible with a set of constraints among functions of loops known as the Mandelstam identities. These identities naturally involve intersecting loops and severely limit the possible generalizations of invariants to intersections. We will show in this paper how to generalize the invariants stemming from Chern–Simons theory to be compatible in an exact way with the Mandelstam identities. This in particular also defines the values of the invariants for multicomponent links. This is of particular relevance for quantum gravity since it is known that the exponential of the Chern–Simons form built from the Ashtekar connection is an exact solution to all the constraints of quantum gravity [19, 8]. If one wishes to find the counterpart of this state in the loop representation one ends up computing exactly the same integral as the expectation value of a Wilson loop in a Chern–Simons theory. One additional motivation for the construction we present is that the Chern–Simons state not only arises in canonically quantized vacuum general relativity but also in other contexts, like Einstein–Yang–Mills theories [18] and supergravity [20]. In these cases the (super)gauge group of the associated Chern–Simons theory differs from that of gravity and therefore so do the resulting invariants. In the particular case of supergravity the resulting invariant had not been computed by other means, and turns out to be associated with the Dubrovnik–Kauffman polynomial [21]. A first attempt to obtaining a finite prescription from variational calculations was made by Brügmann [22]. In particular, the idea of exponentiating the infinitesimal transformation was presented there. Because of ambiguities in the formulation presented in that paper, one could only check, given the exact skein relations, that they were compatible with the formulation. Here we add two key elements that make the construction a well defined prescription: on the one hand we offer a justification of why the infinitesimal results can be exponentiated; on the other hand we make crucial use of the Mandelstam identities to uniquely fix a prescription for the exponentiation. The prescription now allows, in a case where one does not know the result beforehand, to compute the resulting polynomial.

Variational Derivation of Exact Skein Relations

L+

623

L-

^ L+

L0

L0

^ L-

^ L0

Fig. 1. The different crossings involved in the skein relations for invariants of non-intersecting loops

The organization of this article is as follows. In the next section we will discuss, using the non-Abelian Stokes theorem, how to perform a finite deformation of the expectation value of the Wilson loop.In Sect. III we discuss the deformation of twists and kinks and in Sect. IV of planar and non-planar intersections. We discuss the implications of the results in Sect. V. 2. The Non-Abelian Stokes Theorem and Finite Deformations of Loops One is interested in establishing skein relations for the following function of a loop, Z ik SCS W (γ), (1) < W (γ) >= dA exp 4π I W (γ) = Tr P exp i dy a Aa (y) ,

where

(2)

where γ is a closed curve in a three manifold, Aa is a connection in a semi-simple Lie algebra and SCS is the action of a Chern–Simons theory, Z 2 (3) SCS = d3 x abc Tr[Aa ∂b Ac − iAa Ab Ac ]. 3 Finding skein relations involves relating the value of the < W (γ) > for different loops. These loops are determined by the replacement of over-crossings by undercrossings in a planar projection. Skein relations also arise between a loop with and without a twist, over-crossings and intersections. The usual notation for the several types of crossings is shown in Fig. 1. We will see later on that the consideration of invariants with intersections requires several other crossings. For instance, the skein relations that define the Kauffman bracket knot polynomial on loops without intersections are, K(Lˆ + ) K(Lˆ − ) q

1/4

K(L+ ) − q

−1/4

= =

q 3/4 K(Lˆ 0 ), q −3/4 K(Lˆ 0 ),

K(L− ) = (q K(unknot) = 1.

1/2

−q

−1/2

(4) (5) )K(L0 ),

(6) (7)

624


With these relations the polynomial is completely characterized for any link. What Witten [1] showed using conformal field theory techniques is that 21 < W (γ) > satisfies the above relations. Here we will show it by performing directly deformations of the loops on the expression of the expectation value. In order to do this, we need to study deformations of Wilson loops. Wilson loops are traces of holonomies. It turns out that the information needed to compute a holonomy is less than that present in a closed curve. Several closed curves yield the same holonomy. In this paper we will use the word “loop” to denote the equivalence class of curves that yield the same holonomy for all connections 1 . Loops form a group structure called the group of loops [18]. It is well known that if one adds to a loop another loop of infinitesimal area, the change in the Wilson loop can be coded in terms of an infinitesimal operator in loop space called the loop derivative [18]. This is shown in Fig. 2. The concrete expression is, 1 W (γ ◦ δγ) = (1 + σ ab ∆ab (πox ))W (γ), 2

(8)

where ◦ denotes composition of loops, σ ab is the infinitesimal element of area of the loop δγ and πox is an open path connecting the basepoint of the loop γ to the point x at which one adds the infinitesimal loop with infinitesimal element of area σ ab . Notice that the loop derivative ∆ab depends on the path used to compute it and we denote it so in its expression. The loop derivative is related to the infinitesimal generators of the group of loops [18].

Fig. 2. The infinitesimal loop that defines the loop derivative.

One can write [18] an expression for an operator U (γ) that adds a finite loop γ to a Wilson loop in terms of the loop derivative. Let γ(s) be a parameterized curve belonging to the equivalence class defining the finite loop γ with s ∈ [0, 1]. Consider a one-parameter family of parameterized loops η(s, t) interpolating smoothly between γ(s) and the identity loop, such that η(s, 0) is in the equivalence class of the identity loop and η(s, 1) = γ(s). Consider the curves η(s, 1) (= γ(s)) and η(s, 1 − ). The two curves are drawn in Fig. 3 and differ by an infinitesimal element of area. The whole purpose of our construction will be to cover the infinitesimal area separating the two mentioned curves with a “checkerboard” of infinitesimal closed curves such that along each of them one can define a loop derivative. One can therefore express the curve γ(s) as (9) γ(s) = lim η(s, 1 − ) ◦ δη1 ◦ · · · ◦ δηn , n→∞

1

Other authors call these objects “hoops” to denote “holonomic loops” [23].


625

where the δηi are shown in Fig. 3. Analytically, in terms of differential operators on functions of loops we can write2 Ψ (η(1)) =

Ψ (η(1 − )) I 1 b + dsη˙ a (1 − , s)η 0 (1 − , s)∆ab (η(1 − )so )Ψ (η(1 − )), 0

(10) where η(t, ˙ s) ≡ d η(t, s)/dt and η 0 (t, s) ≡ d η(t, s)/ds. It is immediate to proceed from

Fig. 3. The construction of a finite loop from the loop derivative. The curves δηi are determined by two elements in the family η(t).

η(1 − , s) inwards just by repeating the same construction, and so continuing until the final curve is the identity. The end result is ! I Z 1

1

dt

Ψ (γ) = T exp 0

0

dsη˙ a (t, s)η 0 (t, s)∆ab (η(t)so ) Ψ (η(0)) ≡ U (γ)Ψ (η(0)), b

(11) where the outer integral is ordered in t (T-ordered). This result is the loop version of the non-Abelian Stokes theorem of gauge theories [24] (see also [25] for an extension to intersecting loops) and it shows that the loop derivative is a generator of loop space, i.e., it allows us to generate any finite loop homotopic to the identity. Due to the properties of the group of loops [18], the construction is independent of the particular family of loops used to go from the identity element to the final loop γ. It is useful to rewrite the expression of the operator U as, Z 1 dtδ(t), (12) U (γ) = T exp 0

where

I

1

δ(t) = 0 2

dsη˙ a (t, s)η 0 (t, s)∆ab (η(t)so ). b

We drop the s dependence of η where it is not relevant.

(13)

626


The operator δ is closely related to the unparameterized “connection derivative” [18]. We would now like to apply the above deformation to a portion of a loop in order to construct the elements that appear in the skein relations.

3. Skein Relations Associated to Twists and Kinks

3.1. Twists. We start with the simplest skein relations, those that relate the value of the invariant with and without a twist. Starting from a regular portion of a loop ηox going from the origin to x, one can add a twist by considering a family of loops of the form, η a (s, t) = η a (s) + tua (s),

(14)

a

where u (s) is a vector along the loop that materializes the deformation shown in Fig. 4.

η(s)

η(1,s) u(s)

Fig. 4. Adding a twist to a loop using the finite deformation operator

To compute the deformation, we first evaluate the action of the loop derivative on the expectation value of the Wilson loop, Z σ ab ∆ab (η(t)xo ) < W (γ) >= i

ik

k dAσ ab Fab (x)Tr(τ k H(η(t)ox ◦γ◦η(t)xo ))e 4π SCS , (15)

where we have taken into account that the deformation is along one element of the family of loops η(t). τ k are the generators of the algebra su(2). In this calculation we have assumed that the loop derivative acts on < W (γ) > by simply acting inside the functional integral W (γ). This is a strong hypothesis, notice that the integral < W (γ) > is diffeomorphism invariant and therefore the limit involved in the definition of the loop derivative is singular. We are assigning a value to that limit by permuting the functional integral and the limit involved in the loop derivative. We now use the fact that the field tensor is the dual of the magnetic field, which can in turn be obtained through a functional derivative of the exponential of the Chern–Simons state,


627

ik

ik

l iFab (x)e 4π SCS = iabc B c l e 4π SCS =

4π δ ik abc l e 4π SCS . k δAc

(16)

Substituting this expression in (15) and integrating the functional derivative by parts we get, Z ik 4π δ dAσ ab e 4π SCS abc l Tr(τ l H(η(t)ox ◦ γ ◦ η(t)xo )), σ ab∆ab (η(t)xo ) < W (γ) >= − k δAc (17) and acting with the functional derivative on the holonomy we get, Z I ik 4πi ab x ab 4π SCS dAσ e abc dy c δ 3 (x − y) (18) σ ∆ab (η(t)o ) < W (γ) > = − k γ l o y l ×Tr τ H(η(t)x )H(γo )τ H(γyo )H(η(t)xo ) . It should be noticed that the previous expression is distributional, involving a onedimensional integral of a three-dimensional Dirac delta. It is remarkable that one can use it to obtain an expression for the operator U that generates finite deformations. In order to see this, we first write an expression for the operator δ associated with the deformation induced by the vector field u(s), Z I 4πi 1 a 0b 0b dsu (s)(η (s) + tu (s))abc dy c δ 3 (η(s, t)−y) δu (t) < W (γ) > = − k 0 γ

(19) × Tr τ l H(η(t)ox )H(γoy )τ l H(γyo )H(η(t)xo ) . In order to construct the finite deformation operator we need to exponentiate δ. As is shown in Appendix A, one can find a regularization such that δ’s at different points commute when acting on the expectation value of the Wilson loop in Chern-Simons theory (in general they do not commute, see [18]). Therefore the T -ordered exponential reduces to an ordinary exponential and we get, ! Z 1

δu (t)dt

U (γ) = exp

(20)

0

when acting on < W (γ) > and the integral in the exponent can be computed explicitly, Z 1 3iπ < W (γ) >, (21) dtδu (t) < W (γ) >= ∓ k 0 and the sign ∓ corresponds to the kind of crossing generated in the loop by the vector field u. Positive sign corresponds to the right hand rule. To go from 19 to the last expression we have made use of the Fierz identity for SU (2), kC τ kA Bτ D =

1 A C 1 A C δ δ − δ δ , 2 D B 4 B D

(22)

and we performed explicitly the three one-dimensional integrals. The result is the normalized oriented volume subtended by the deformation. Strictly speaking, this result implies a choice of regularization, since the type of integral that one is left with is of the R1 form 0 dtδ(t) (see [22] for details). This choice in the regularization is tantamount to introducing a new parameter in the derivation, corresponding to the value of the above

628


integral. Throughout this paper we will take it to be unity. Otherwise, it would imply a multiplicative shift in the value of k. The above expression can be summarized in terms of the notation for skein relations we introduced before as, 3iπ < W (Lˆ o ) >, (23) < W (Lˆ + ) > = exp − k 3iπ < W (Lˆ − ) > = exp < W (Lˆ o ) >, (24) k which coincide with the skein relations for the Kauffman bracket (4,5) if one makes the identification 4iπ q = exp(− ). (25) k It is well known that perturbative techniques like the ones we are using here fail to capture the additive shift in the coupling constant first observed by Witten [1]. A discussion, and a proposal to amend perturbative techniques to capture this effect through a recourse to the semi-classical approximation can be found in Awada [26]. We could proceed in the same fashion here and modify the value of the coupling constant, but we will leave it as it is for simplicity. 3.2. Kinks. A kink (discontinuity of the tangent) is a diffeomorphism invariant feature of a loop. We will now show that the expectation value of a Wilson loop in a Chern– Simons theory is not sensitive to the presence of kinks in the loop under the kind of regularization we are using in our calculations. This is an important result because it directly relates to the value of the invariant when one has intersections with kinks, which are crucial to implement the Mandelstam identities. We will discuss this at the end of this section and will see later that the choice we make for the treatment of the kinks is central to establishing consistency. Let us consider a loop with a kink like that shown in

η(t,s)

η(s)

u

Fig. 5. Deforming a kink into a smooth section

Fig. 5. The kink defines a plane. We can deform the kink into a smooth section of the loop through a deformation in the plane. The calculation is exactly the same as that of a smooth section, so we will not repeat it here. The only precaution is to consider the appropriate tangent vector at both sides of the kink. The reason why we do not need


629

more details is that since the deformation is planar, the contribution vanishes. Basically one gets a contribution similar to (19) that has three coplanar vectors contracted with the Levi–Civita symbol. The case of intersections with kinks is essentially similar. There

LI

LW

LW

Fig. 6. Double intersections without and with kinks

are two different types of (double) planar intersections with kinks, as shown in Fig. 6. In all cases the operator U (γ) deforms independently each line at the intersection. For the cases with kinks that means that each kink can be deformed into a smooth section. That means that in cases LW and LW¯ the intersection can be removed by the deformation and we get the skein relations < W (LW ) > < W (LW¯ ) >

= < W (L0 ) >, = < W (L0¯ ) >

(26) (27)

that are shown in Fig. 7. It should be noted that given an LI , there exist two possibilities for LW¯ , the one shown in Fig. 6 or that same diagram rotated 90 degrees. A similar consideration holds for L0 . Which of these is appropriate to use depends on the connectivity of the loop (i.e., if we are talking about a single loop or a link). The skein relations satisfied by these additional elements are the same as those shown in Fig. 7, with all elements rotated 90 degrees.

LW

L0

LW

L0

Fig. 7. Skein relations for intersections with kinks

Notice that in all cases the removal of kinks at planar intersections can be accomplished given a straightforward regularization of the deformation operator. This provides a justification for the identification of LW with L0 that has been used in other works [22, 27]. We will return to discuss intersections with kinks at the end of the next section, where we will analyze a deformation that produces double lines in the loop that is of interest in the context of the exponentiation of the skein relation for straight through intersections, which we discuss now. 4. Skein Relations for Intersections 4.1. Infinitesimal skein relations. In the case of a “straight through” intersection (no kinks), the calculation is different than in the previously discussed cases. In this case,

630


the Wilson loop is the trace of the product of the holonomies along the petals defined by the intersection. For instance in the case of Fig. 8 one could write it as W (γ) = Tr(H12 H34 ). The action of the loop derivative on the holonomy is independent of the 1 η (1,s)

u

2

3 I

4 o

Fig. 8. Deformation of a “straight through” intersection. The deformation vector u is perpendicular to the plane determined by the intersection.

presence of the intersection, the expression is exactly the same as (15). The difference in the construction arises when one integrates by parts. Since the loop has a double point at the intersection, the functional derivative with respect to Aa has two contributions at that point, corresponding to its action on the holonomy when it traverses that point the first and the second time. One of the contributions vanishes since it produces a term proportional to the tangent of the loop in the same plane as the deformation and therefore spans no volume. The other contribution is the one of interest. It can be written as, Z Z ik 4πi dAσ ab e 4π SCS abc dy c δ 3 (x − y) σ ab ∆ab (η(t)xo ) < W (γ) >= − k γ23 (28) ×Tr τ l H(η(t)ox )H34 (γoI )H12 (γIy )τ l H12 (γyI )H34 (γIo )H(η(t)xo ) , where we have assumed that the origin o of the loop is in the petal 34. We now use the Fierz identity (22) and compute the integral of the δ operator, Z 1 iπ dtδu (t) < Tr(H12 (γ)H34 (γ)) > = − Tr(H12 (γ)H34 (γ)) (29) k 0 2iπ Tr(H12 (γ))Tr(H34 (γ)), + k which can be rewritten as Z 1 iπ 2iπ < W (LW ) > . dtδu (t) < W (LI ) >= − < W (LI ) > + k k 0

(30)

A couple of remarks are in order. First, there is a difference between Eqs. (29) and (30), since Eq. (29) refers to the whole loop and Eq. (30) only to its intersection. The former has information about the connectivity of the loop, the latter does not. By this we mean that we are slightly abusing the notation in the above equations, in the sense that we are applying them in some cases to multiple loops (links), in which case the Wilson


631

loop has to be replaced by a product of Wilson loops of the individual components. By connectivity we mean that a same graph can either correspond to a single intersecting loop or an intersecting link, and different formulae apply to both cases. Therefore, if one is to claim that Eq. (30) follows from Eq. (29) one should prove that it does so for any connectivity. For the case of double intersections, there are only two different possible connectivities. It is straightforward to check that Eq. (30) holds independently of the connectivity chosen. Second, in spite of similarities to Eq. (6), Eq. (30) is only valid to first order in 1/k and needs to be exponentiated to obtain the skein relation. Since Eq. (30) shows that the action of δ mixes LI and LW , in order to exponentiate it we need to compute the action of the deformation operator on LW . Notice that this is a different deformation than the one computed in the last section, which was coplanar with the intersection. We discuss the resulting calculation at the end of this section. Before doing that computation, it is worthwhile observing that one could combine the Fierz identity (22) with the following identity: A C A C δD − δD δB AC BD = δB

(31)

to get, 1 A C (δ δ − AC BD ), 4 D B and using this expression in (28) we get Z 1 iπ Tr(H12 (γ))Tr(H34 (γ)) dtδu (t) < Tr(H12 (γ)H34 (γ)) > = k 0 iπ −1 + Tr(H12 (γ)H34 (γ)). k kC τ kA Bτ D =

(32)

(33)

In contrast to (29) this expression leads to a different first-order skein relation depending on the connectivity of the knot. If the connectivity is such that the original loop has a single component, then the skein relation is, Z 1 iπ iπ < W (L0 ) > + < W (L0¯ ) >, dtδu (t) < W (LI ) >= (34) k k 0 where the element L0¯ ) is defined in Fig. 1. The interest of this skein relation stems from the fact that the original definition of the Kauffman bracket [21] was given in terms of relations of this kind. Moreover, Major and Smolin [27] used this identity to derive the binor identity from the Kauffman bracket. The original definition of the bracket differs from the one we use here in a factor (−1) to the power of the number of connected components of the loop, which fixes the connectivity difference we encountered above. To conclude, we discuss the deformation of an LW as mentioned before. We need to compute the deformation of an intersection with a kink that is shown in Fig. 9. The loop starts at the origin, goes through the deformation determined by the vector u, which we denote with a dotted line and then traverses the petal 12 going through the kink and ends with the petal 34. The resulting loop has double lines. In ordinary knot theory double lines are not considered. They could be incorporated through additional sets of skein relations, as we discuss in Subsect. 4.6. If one pursues a calculation similar to the ones we have been doing up to now for this case, one encounters two contributions, stemming from the volumes spanned by the deformation and the tangents to the loop at the intersection in lines 2 or 3. The integrals involve terms of the form δ 3 (y)Θ(y). The Heaviside function Θ(y) limits the integral in the contribution to the corresponding

632


4

η (1,s)

u

3 2

1 o

Fig. 9. The skein relation for intersections requires the analysis of this type of deformation of an intersection with kinks.

petal. Although these kinds of expressions can be regularized, there does not appear any natural way of assigning relative weights to the two contributions from the petals. The way we handle it here is to write the contribution in terms of two arbitrary factors α and β, Z 1 iπ dtδ± < W (LW ) >= ± (α < W (LI ) > +β < W (LW ) >) . (35) k 0 We will see in the next section that the arbitrary factors can be uniquely determined by the Mandelstam identities when we exponentiate the operator. 2

2

2 4

4 3

4 3

3 1

η

1

1

1234

η

I

W

1

1

,η

34 I

14 W

2 4 3

3

3

I

W

4

4

12

,η

23

2

2

η

η

1324

η

1432 W

1

η

1342 W

Fig. 10. The different loops that arise in the Mandelstam identities involved in double intersections. The grey boxes denote possible knottings and interlinkings with other portions of the loop (including possible additional intersections and kinks). In order to unclutter the figures, we only denoted partial connectivities, in all cases the loops have to be closed by connecting the open strands, with no particular restrictions, i.e. the resulting petal could have knottings and interlinkings with the rest of the loop.

4.2. Exponentiation and Mandelstam identities. The skein relations involving L± and LI are obtained studying the deformation of an intersection. In order to obtain these


633

deformations in a finite form we need to exponentiate the differential expressions we obtained in the previous section. Specifically, we have (eliminating the T-ordering as before), Z 1 < W (LI ) > < W (L∓ ) > dtδ±u (t) = exp (36) < W (LW ∓ ) > < W (LW ) > 0 iπ −1 2 < W (LI ) > = exp ± . < W (LW ) > α β k We are interested only in computing < W (L∓ ) >. We are not interested in the value of < W (LW ∓ ) > since as we explained before it involves double lines, which would require a separate discussion involving extra skein relations (see Subsect. 4.6). We do not know the values of α and β. However, since we are computing the invariant as an expectation value of a Wilson loop, we expect it to satisfy the same relations that Wilson loops satisfy: the Mandelstam identities. We will see that imposing the Mandelstam identities is enough to determine the coefficients α and β uniquely and therefore to find the skein relation. + η

= η

1

η 2

3

η

4

Fig. 11. Schematic depiction of the first identity used to determine α and β.

We therefore need to discuss the content of the Mandelstam identities. As we mentioned before, these identities relate the behavior of reroutings of the loops at intersections and involve non-trivial information about the connectivity of the loop. We will restrict the explicit discussion to double intersections, but it is immediate to see that suitable generalizations can be found for more complex intersections. For double intersections, one can always consider a planar diagram, and two possible routings exist, as shown in Fig. 10. Each routing leads to an identity. The identities, 1324 < W (ηI1234 ) > + < W (ηW ¯ )>

+

= =

23 14 < W (ηW )W (ηW ) >,

,

(37) (38)

involve different loops, as shown in the Fig. (10). Moreover, they are valid for a completely arbitrary loop that has a double intersection. In order to use these identities to allow us to fix the arbitrary parameters in the exponentiation, we will consider —for simplicity— its expression for a particular set of loops. We will then have to check that the resulting invariant is consistent with the identities for all possible loops. The loops we wish to consider are the ones obtained by reconnecting the loops of Fig. (10) with direct strands, i.e. adding no knottings or interlinkings and are shown in Figs. 11 and 12. The Mandelstam identities for these loops are, < W (η1 ) > + < W (η2 ) >=< W (η3 )W (η4 ) >, < W (γ3 ) > + < W (γ4 ) >=< W (γ1 )W (γ2 ) > .

(39) (40)

We now make use of the skein relations for intersections with kinks (26,27). In Eq. (39) this allows to replace η2 by the unknot and η3 and η4 by the two component unlinked link. Therefore < W (η2 ) >= 1.

634


= γ

1

+

γ

γ

2

γ

3

4

=

+ γ

γ

5

6

Fig. 12. Schematic depiction of the second identity used to determine α and β.

In Eq. (40) the use of the skein relations is shown in Fig. 12 to transform γ3 and γ4 to 3 the unknot with a twist added. Using the skein relations (23,24) one gets, < W (γ5 ) >= q 4 3 and < W (γ6 ) >= q − 4 . We now use (36) in combination with these expressions to determine the values of α and β. It turns out we only need to use the expression of the exponential that appears in (36) expanded up to second order in 1/k. Using this expression to determine < W (η1 ) >, < W (γ1 )W (γ2 ) > and < W (η3 )W (η4 ) > one gets that α = 0 and β = 1. This completely characterizes the skein relation for the intersections (36). From there we conclude that, ±1 1 1 < W (L∓ ) > < W (LI ) > q 4 (q ∓ 4 − q ± 4 ) = . (41) 1 < W (LW ∓ ) > < W (LW ) > 0 q∓ 4 >From this expression, adding and subtracting with appropriate weights the expressions with the different signs, we can work out the usual skein relation for the Kauffman bracket without intersections, q 4 < W (L+ ) > −q − 4 < W (L− ) >= (q 2 − q − 2 ) < W (L0 ) >, 1

1

1

1

(42)

and also the definition of the expectation value for a planar straight through intersection, < W (LI ) >=

1 (q + q − 4 ) 1 4

1

[< W (L+ ) > + < W (L− ) >].

(43)

We therefore conclude that the expectation value of the Wilson loop in an SU (2) Chern–Simons theory is, up to a factor of 2 with the conventions of this paper, identical to the Kauffman bracket knot polynomial. 4.3. Consistency for all loops and to all orders in 1/k. We have just shown that considering the exponential of the infinitesimal deformation to second order in 1/k and requiring consistency with the Mandelstam identity for a particular set of loops uniquely fixes the values of the indeterminate coefficients α and β of the infinitesimal deformation. We need to check that the construction is consistent to all orders in 1/k and for all possible sets of loops with planar intersections. We will now show that this is the case. In order to do this, the finite version of the form of the skein relation (34), which again can be derived in the same form as the finite skein relation we derived above is useful,


635

< W (L± ) >= q ± 4 < W (L0 ) > −q ∓ 4 < W (L0¯ ) > . 1

1

(44)

As we mentioned before, this relation depends on the connectivity of the loop, we are assuming it is such that LI has one independent component, as shown in loop ηI1234 of Fig. (10). Notice that by subtracting the + and − sign versions of (44) we obtain (42). The sum contains the information needed to derive the Mandelstam identity from the skein relations. Combining (43) with (44) and (26,27) we get, < W (LI ) > + < W (LW¯ ) >=< W (LW ) > .

(45)

Notice that we are stretching the notation here, since this identity is not purely local, a given connectivity of the loop was assumed to derive it, as mentioned above. Therefore one has to include the connectivity we mentioned above in its interpretation. With that connectivity, Eq. (45) is exactly the Mandelstam identity (37). This completes the proof. One could have chosen a different connectivity when deriving (44) and then one would have arrived at the Mandelstam identity (38). 4.4. Non-planar intersections. Traditionally, in knot theory invariants have been formulated through planar projections of knots. When generalizations to intersecting knots are considered one may need to consider non-planar intersections as inequivalent. This has already been noticed for triple intersections [28]. In this section we will discuss some skein relations satisfied by non-planar intersections. There are many types of non-planar intersections, we will restrict ourselves to some examples that have three of the four strands at the intersection in a single plane. We will show that there exists a one-parameter family of regularizations of the expectation value of the Wilson loop that is compatible with the Mandelstam identities for the case of non-planar double intersections. For one particular value of the parameter the intersections behave as planar ones. For other values of the parameter the intersections acquire common elements with under- and over-crossings and therefore imply the existence of distinct nontrivial generalizations of the usual invariants for the case on non-planar intersections. We consider a non-planar intersection as shown in Fig. 13, which we represent with two new types of crossings, labelled L+I and WI+ with obvious counterparts with a minus sign. z x

y

L

+ I

L

+ W

Fig. 13. The non-planar intersections we consider. The top row are three dimensional views and the bottom row are the associated crossings that arise in the skein relations.

636


In order to compute the skein relations we consider a deformation of a straightthrough planar intersection as shown in Fig. 14. The resulting integral is exactly the same as Eq. (28). The difference comes from the vector u. That vector vanishes in the z > 0 part of the loop. This implies that when one wants to rederive Eq. (29) one encounters ambiguities of the type δ(y)Θ(y) exactly as when we deformed an LW intersection in a direction perpendicular to the plane. The way to handle this is again to introduce an indeterminate parameter α. The resulting expression then reads,

4

z

1

x

y

2

u

3 Fig. 14. The deformation of a straight through planar intersection into a crossing of the type L+I .

Z

1

dtδu (t) < Tr(H12 (γ)H34 (γ)) >

iπ Tr(H12 (γ)H34 (γ)) k 2iπ − Tr(H12 (γ))Tr(H34 (γ)) ; k

= −α

0

this can again be reinterpreted as, Z 1 2iπ iπ < W (LW ) > − < W (LI ) >]. dtδu (t) < W (LI ) >= α[ k k 0

(46)

(47)

In order to exponentiate, since the action mixes LI and LW we need to compute, Z 1 < W (LI ) > < W (L∓ I )> dtδ (t) = exp ±u < W (LW ) > < W (L∓ W) > 0 iπ −α 2α < W (LI ) > = exp ± , (48) β1 β2 < W (LW ) > k and one can show that in order to have consistency with the Mandelstam identity, β1 = 0, β2 = α. Notice that α is undetermined by the Mandelstam identities and we therefore have a one-parameter family of definitions of the skein relation for the crossing. Exponentiating explicitly and identifying the variable k as before, we get, < W (L± I )>

=

α

α

α

q ∓ 4 < W (LI ) > +(q ± 4 − q ∓ 4 ) < W (L0 ) >, q

±α 4

< W (L0 ) > .

(49) (50)


637

± If α = 0 we get that L± I ≡ LI and LW ≡ LW , so the non-planar and planar intersections are treated in the same way. It is also immediate to prove from the skein relations (49,42,43) that,

< W (L± I ) > |α=1

< W (L± I ) > |α=0 ± < W (LI ) > |α=−1

=

< W (L+ ) >,

(51)

=

< W (LI ) >,

(52)

=

< W (L− ) >,

(53)

so we see that for different values of the free parameter associated with the non-planar intersections we can have them play the same role as over-crossings, intersections and under-crossings. This highlights the relation between the value of the α parameter and the different types of regularizations it implies for the intersections. As in the non-intersecting case, all the regular invariant information of the < W (γ) > is concentrated in a multiplicative “phase factor.” One can divide by it and construct an ambient isotopic invariant, which for SU (2) is the Jones polynomial. In the nonintersecting case, the writhe w(γ) can be computed by evaluating the expectation value of a Wilson loop in a U (1) Chern–Simons theory, < W (γ) > |U (1) = q w(γ) ,

(54)

with skein relations, w(Lˆ ± ) w(L± ) w(LI ) w(L± I )

= = = =

±1, ±1, w(L0 ) = 0, ±α.

(55) (56) (57) (58)

If one now defines a polynomial J(γ) through J(γ) =

1 − 3 w(γ) q 4 < W (γ) >, 2

(59)

the result is ambient isotopic invariant. The above definition of the Jones polynomial is also valid for multiloops, with a suitable generalization of the Abelian calculation of the writhe for multiloops. From the expression of the resulting Jones polynomial one can get an expression for the Gauss linking number of two loops with intersections. The resulting expression is consistent with lattice definitions of the linking number with intersections [29, 30]. 4.5. Triple and higher intersections. We will not discuss in detail the generalization to triple intersections in this paper, we sketch in this subsection how one performs the generalization of the construction to that case. In the case of triple intersections, there are many types of possible independent vertices. In the case of planar intersections it is easy to see that one can deform them to double intersections, using the same techniques as for the LW ’s. For the case of non-planar triple intersections, there are 10 independent vertices. All of them can be related to double intersections through deformations similar to the one that connects LI with L± . In order to exponentiate the infinitesimal deformations, one again has to exponentiate a matrix that connects the different intersection types. It will be a sparse 10 × 10 matrix. Again regularization issues will leave many coefficients undetermined and one would restrict them using the Mandelstam identities. It is not clear if the resulting polynomial will be completely

638


determined by the Mandelstam identities or if new free parameters will appear. As in the double case, the Mandelstam identities are non-local and there are now three different possible connectivities of intersections that are needed to implement the identities. 4.6. Loops with multiple lines. Throughout this paper we have assumed we were dealing with loops that have each line traversed only once. If one would like the polynomial that is being derived to take values on the complete set of loops that is of interest in quantum gravity and gauge theories, one needs to consider the case of loops with multiply traversed section. This is also of importance if one is to view the integral of the exponential of the Chern–Simons form as a rigorous measure on the space of connections modulo gauge transformations. We present here a brief discussion of how one could consider double lines, but a complete description again requires further study. Let us consider a loop γ and study the Wilson loop along γ 2 ≡ γ ◦ γ. Consider a generic direction in space w such that the tangent vector to γ is never parallel to w. Consider an continuous infinitesimal displacement of all the points of γ along w such that an arbitrary point on γ is kept fixed (we will call this point the origin of the loop). This produces a second copy of γ. If one deforms back and forth along w in this way one ends up with two copies of γ that are connected at the origin through an intersection of the type LI , as shown in Fig. 15. Because this is a planar deformation, the value of the expectation value of the Wilson loop does not change. We have therefore reduced the problem of computing the expectation value for a loop traversed twice to the problem of computing the expectation value along a simply traversed loop with an intersection of one of the types studied before. It is worthwhile noticing that applying the Mandelstam identity (37) at the resulting intersection one gets the identity < W (γ)W (γ) >=< W (γ ◦ γ) > +2.

o

o

Fig. 15. The deformation of a loop traversed twice into two loops with an intersection.

If one wants to consider a more general situation, in which a loop could have portions that are traversed twice, one can extend the above result in a relatively straightforward manner. Even if there are planar intersections of the multiply traversed segment with another segment one can separate them without additional contributions. However, if there are triple non-planar intersections involving sections multiply traversed, the deformation will give non-trivial contributions. These contributions can be evaluated explicitly given a specific loop.

5. Conclusions We have shown how to use variational techniques to obtain exact expressions for the knot invariant associated with the expectation value of the Wilson loop in Chern–Simons


639

theory. The method is completely general in the sense that it can be used in Chern– Simons theory with any semisimple gauge group with small generalizations. We have worked out explicitly the value of the invariant not only for smooth loops but also for loops with double planar and non-planar intersections. The resulting invariant is compatible with the Mandelstam identities of the gauge group and therefore is suitable for providing invariants of interest as quantum states of topological field theories and quantum gravity. Having a well defined linear function on the space of loops compatible with the Mandelstam identities may allow also, using the techniques of Ashtekar and collaborators [31] to define in a rigorous way a measure in the space of connections modulo gauge transformations dµ(A). Such a measure wouldR allow to give rigorous meaning in a mathematical sense to expression of the form dAekSCS f (A) for any gauge invariant function f (A). The generalization of the work described in this paper to triple intersections is straightforward and relevant for quantum gravity applications. The possibility of computing explicitly the knot polynomials associated with Chern–Simons theory for any group is clearly of relevance in other physics applications, like the recent discovery of Chern–Simons states in supergravity has shown [20]. It is expected that these results will be extendible to N > 1 supergravity where this method will provide a simple method of characterizing the potentially new invariants that may arise. Acknowledgement. We wish to thank Abhay Ashtekar, Leonardo Setaro and Daniel Armand-Ugón for discussions and Cayetano di Bartolo for several comments about the manuscript. This work was supported in part by grants NSF-INT-9406269, NSF-PHY-9423950, NSF-PHY-9396246, research funds of the Pennsylvania State University, the Eberly Family research fund at PSU and PSU’s Office for Minority Faculty development. JP acknowledges support of the Alfred P. Sloan foundation through a fellowship. We acknowledge support of Conicyt and PEDECIBA (Uruguay).

A. Commutativity of the δ Operators We wish to evaluate the successive action of two δ operators on < W (γ) >. We only discuss the case of a regular (non-intersecting) point of the loop. The first δ acts as indicated in (19), and using the Fierz identity we get, δu (t) < W (γ) >= −

3πi k

Z

1

ds η˙ a (s, t)η 0 (s, t)abc b

I dy c δ 3 (η(s, t) − y) < W (γ) > . γ

0

(60) The second δ has two contributions, stemming from the action of the loop derivative H on the loop dependence of < W (γ) > and γ respectively. The term that contributes to the commutator is the latter, since the contribution on the Wilson loop is the same in both orders. Let us therefore evaluate explicitly the non-trivial contribution, [δ(t1 ), δ(t2 )]

=

3πi − k

Z

Z

1

1

ds1 0

ds2 η˙ a (s1 , t1 )η 0 (s1 , t1 )η˙ c (s2 , t2 )η 0 (s2 , t2 ) b

d

0

×ab[c ∂d] δ 3 (x − η(s1 , t1 ))|x=η(s2 ,t2 ) − (t1 ↔ t2 ) Z Z 1 3πi 1 b ds1 ds2 η˙ a (s1 , t1 )η 0 (s1 , t1 )η˙ c (s2 , t2 )abc × = − k 0 0 × ∂s2 δ 3 (η(s2 , t2 ) − η(s1 , t1 ))∂t2 δ 3 (η(s2 , t2 ) − η(s1 , t1 )) −(t1 ↔ t2 ).

(61)

640


One can now pick a coordinate chart in which δ 3 (η(s2 , t2 ) − η(s1 , t1 )) = Zδ(s2 − s1 )δ(t2 − t1 ), where Z is a regularization dependent factor that does not depend on s2 or t2 . Using the distributional identity, f (y)∂y δ(y − x) = f (x)∂y δ(y − x) − δ(y − x)∂x f (x),

(62)

it is immediate to check that the contributions vanish. The result is obviously dependent on a regularization choice. References 1. Witten, E.: Commun. Math. Phys 121, 351 (1989) 2. Kauffman, L, “On knots”. Annals of Mathematics Studies, Princeton, NJ: Princeton University Press, 1987 3. Hoste, J., Ocneanu, A., Millet, A., Freyd, P., Lickorish, W., Yetter, D.: Bull. Am. Math. Soc. 129, 239 (1985) 4. Moore, G., Seiberg, N.: Phys. Lett. B212, 451 (1988) 5. Smolin, L.: Mod. Phys. Lett. A4 1091 (1989) 6. Cotta-Ramusino, P., Guadagnini, E., Martellini, M., Mintchev, M.: Nuc. Phys. B330, 557 (1990) 7. Makeenko, Yu., Migdal, A.: Phys. Lett. B88, 135 (1979); Nucl. Phys. B188, 269 (1981) 8. Brügmann, B., Gambini, R., Pullin, J.: Nucl. Phys. B385, 587 (1992) 9. Broda, B., Mod. Phys. Lett. A5, 2747 (1990) 10. Cattaneo, A., Cotta-Ramusino, P., Fröhlich, J., Martellini, M.: J. Math. Phys. 36, 6137 (1995) 11. Kauffman, L.: “Knots and quantum gravity”, editor J. Baez, Oxford: Oxford University Press, 1993 12. Gambini, R., Trias, A.: Nucl. Phys. B278, 436 (1986) 13. Rovelli, C., Smolin, L.: Nucl. Phys. B331, 80 (1990) 14. Ashtekar, A.: Phys. Rev. Lett. 57, 2244 (1986); Phys. Rev. D36, 1587 (1987) 15. Rovelli, C., Smolin, L.: Phys. Rev. Lett. 61, 1155 (1988) 16. Jacobson, T., Smolin, L.: Nucl. Phys. B299, 295 (1988) 17. Brügmann, B., Pullin, J.: Nucl. Phys. B363, 221 (1991) 18. Gambini, R., Pullin, J.: “Loops, knots, gauge theories and quantum gravity”. Cambridge: Cambridge University Press, Cambridge 1996 19. Kodama, H.: Phys. Rev. D42, 2548 (1990) 20. Armand-Ugon, D., Gambini, R., Obregón, O., Pullin, J.: Nucl. Phys. B460, 615 (1996) 21. Kauffman, L.: “Knots and physics”, World Scientific Series on Knots and Everything 1, Singapore: World Scientific, 1991 22. Brügmann, B., Int. J. Theor. Phys. 34, 145 (1995) 23. Ashtekar, A., Lewandowski, J.: “Knots and quantum gravity”, J. Baez editor, Oxford: Oxford University Press, Oxford 1993 24. Aref’eva, I.: Theor. Math. Phys 43, 353 (1980) (Teor. Mat. Fiz. 43, 111 (1980)) 25. Bralic, N.: “A generalized surfaceless Stokes’ theorem". Preprint hep-th:9311188 (1993); Phys. Rev. D22, 3090 (1980) 26. Awada, M.: Comm. Math. Phys. 129, 329 (1990) 27. Major, S., Smolin, L.: Preprint gr-qc:9512020 (1995) 28. Armand-Ugon, D., Gambini, R., Mora, P.: Phys. Lett. B305, 214 (1993); J. Knot. Theor. Ramif. 4, 1 (1995) 29. Polikarpov, A.: Moscow preprint ITEF-91-049 (1991) 30. Fort, H., Gambini, R., Pullin, J.: Preprint gr-qc/9608033 (1996) 31. Ashtekar, A., Lewandowski, J., Marolf, D., Mourao, J., Thiemann, T.: J. Math. Phys. 36, 6456 (1995) Communicated by R. H. Dijkgraaf

This article was processed by the author using the LaTEX style file pljour1 from Springer-Verlag.

Commun. Math. Phys. 185, 641 – 670 (1997)

Communications in


WZW Fusion Rings in the Limit of Infinite Level ¨ Jurgen Fuchs1,? , Christoph Schweigert2 1 2

DESY, Notkestraße 85, D – 22603 Hamburg, Germany IHES, 35, Route de Chartres, F – 91440 Bures-sur-Yvette, France

Received: 23 September 1996 / Accepted: 8 October 1996

Abstract: We show that the WZW fusion rings at finite levels form a projective system with respect to the partial ordering provided by divisibility of the height, i.e. the level shifted by a constant. From this projective system we obtain WZW fusion rings in the limit of infinite level. This projective limit constitutes a mathematically well-defined prescription for the “classical limit” of WZW theories which replaces the naive idea of “sending the level to infinity.” The projective limit can be endowed with a natural topology, which plays an important rôle for studying its structure. The representation theory of the limit can be worked out by considering the associated fusion algebra; this way we obtain in particular an analogue of the Verlinde formula.

1. Fusion Rings Fusion rings constitute a mathematical structure which emerges in various contexts, for instance in the analysis of the superselection rules of two-dimensional quantum field theories; they describe in particular the basis independent contents of the operator product algebra of two-dimensional conformal field theories (for a review see [1]). By definition, a fusion ring R is a unital commutative associative ring over the integers Z which possesses the following properties: there is a distinguished basis B = {ϕa } which contains the unit and in which the structure constants Na,bc are non-negative integers, and the evaluation at the unit induces an involutive automorphism, called the conjugation of R. A fusion ring is referred to as rational iff it is finite-dimensional. A rational fusion ring is called modular iff the matrix S that diagonalizes simultaneously all fusion matrices Na (i.e. the matrices with entries (Na )bc = Na,bc ) is symmetric and together with an appropriate diagonal matrix T generates a unitary representation of SL(2, Z) (see e.g. [2, 1]). ?

Heisenberg fellow

642

J. Fuchs, C. Schweigert

In this paper we consider the fusion rings of (chiral, unitary) WZW theories. A WZW theory is a conformal field theory whose chiral symmetry algebra is the semidirect sum of the Virasoro algebra with an untwisted affine Kac--Moody algebra g, with the level k ∨ of the latter a fixed non-negative integer. To any untwisted affine Kac--Moody algebra g we can thus associate a family of fusion rings, parametrized by the level k ∨ . The issue that we address in this paper is to construct an analogue of the WZW fusion ring for infinite level, which is achieved by giving a prescription for “sending the level to infinity” in an unambiguous manner. In view of the Lagrangian realization of WZW theories as sigma models, this procedure may be regarded as taking the “classical limit” of WZW theories. Performing a classical limit of a parametrized family of quantum field theories is a rather common concept in the path integral formulation of quantum theories; it simply corresponds to sending Planck’s constant to zero, and hence provides a kind of correspondence principle. In the Lagrangian description of WZW theories as principal sigma models with Wess--Zumino terms, the rôle of Planck’s constant is played by the inverse of the level k ∨ of the underlying affine Lie algebra g. However, it is known that the path integral of a WZW sigma model strictly makes sense only if the level k ∨ is an integer. In contrast to the path integral description, in the framework of algebraic approaches to quantum theory so far almost no attempts have been made to investigate limits of quantum field theories. In this paper we address this issue for the case of WZW theories. Now in an algebraic treatment of WZW theories the integrality requirement just mentioned is immediately manifest. Namely, one observes that the structure of the theory depends sensitively on the value k ∨ of the level. For non-negative integral k ∨ the state space is a direct sum of unitary irreducible highest weight modules of the algebra g, but its structure changes quite irregularly when going from k ∨ to k ∨ + 1; at intermediate, non-integral, values of the level there do not even exist any unitarizable highest weight representations. These observations indicate that it is a rather delicate issue to define what is meant by the classical limit of a WZW theory, and it seems mandatory to perform this limit in a manner in which the level k ∨ is manifestly kept integral (actually, treating the level formally as a continuous variable is potentially ambiguous even in situations where one deals with expressions which superficially make sense also at a non-integral level). It must also be noted that a priori it is by no means clear whether the so obtained limit will be identical with or at least closely resemble the structures which originally served to define the quantum theory in terms of some classical field theory; in the case of WZW fusion rings, this underlying classical structure is the representation ring of the finite-dimensional simple Lie algebra g¯ that is generated by the zero modes of the affine Lie algebra g. Indeed, it seems to be a quite generic feature of quantum theory that the classical limit does not simply reproduce the classical structure one started with. (Compare for instance the fact that in the path integral formulation of quantum field theory the classical paths are typically of measure zero in the space of all paths that contribute to the path integral. Similar phenomena also show up when the continuum limit of a lattice theory is constructed as a projective limit; see e.g. [3, 4].) However, it is still reasonable to expect that the original classical structure is, in a suitable manner, contained in the classical limit; as we will see, this is indeed the case for our construction. The desire of being able to perform a limit in which the level tends to infinity stems in part from the fact that WZW theories and their fusion rings can be used to define a regularization of various systems (such as two-dimensional gauge theories or the Ponzano--Regge theory of simplicial three-dimensional gravity), with the unregularized system corresponding, loosely speaking, to the classical theory. As removing the regulator is always a subtle issue, it is mandatory that the limit of the regularized theory

WZW Fusion Rings in Limit of Infinite Level

643

is performed in a well-defined, controllable manner, which, in addition, should preserve as much of the structure as possible. The basic idea which underlies our construction of the limit of WZW fusion rings is to interpret the collection of WZW fusion rings as a category Fus(g) within the category of all commutative rings and identify inside this category a projective system. By a standard category theoretic construction we can then obtain the limit (also known as the projective limit) of this projective system. The partial ordering underlying the projective system is based on a divisibility property of the parameter k ∨ + g ∨ that together with the choice of the horizontal subalgebra g¯ characterizes the WZW theory (g ∨ denotes the dual Coxeter number of g¯ ; the sum k ∨ + g ∨ is called the height). In contrast, in the literature often a purely formal prescription “ k ∨ → ∞ ” is referred to as the classical limit of WZW theories. In that terminology it is implicit that the standard ordering on the set of levels is chosen to give it the structure of a directed set. Now the projective limit is associated to a projective system as a whole, not just to the collection of objects that appear in the system; in particular it depends on the underlying directed set and hence on the choice of partial ordering. Our considerations show, as a by-product, that it is not possible to associate to the standard ordering any well-defined limit of the fusion rings. The rest of this paper is organized as follows. We start in Subsect. 2.1 by introducing the category Fus(g) of WZW fusion rings associated to an untwisted affine Lie algebra g; in Subsect. 2.2 conditions for the existence of non-trivial morphisms of this category are obtained. In Subsect. 2.3 we define the projective system, and in the remainder of Sect. 2 we check that the morphisms introduced by this definition possess the required properties. The projective limit of the so obtained projective system is a unital commutative associative ring of countably infinite dimension. This ring (∞)R is constructed in Sect. 3; there we also gather some basic properties of (∞)R and introduce a natural topology on (∞)R. In Sect. 4 a concrete description of a distinguished basis (∞)B for the projective limit is obtained. This basis is similar to the distinguished bases of the fusion rings at finite level; in order to show that (∞)R is indeed generated by (∞)B, the topology on (∞)R plays an essential rôle. In Sect. 5 we demonstrate that the representation ring of the horizontal subalgebra g¯ ⊂ g is contained in the projective limit (∞)R as a proper subring. In the final Sect. 6 we study the representation theory of (∞)R, respectively of the associated fusion algebra over C . In particular, we determine all irreducible representations and show that (∞)R possesses a property which is the topological analogue of semi-simplicity, namely that any continuous (∞)R-module is the closure of a direct sum of irreducible modules. To obtain these results it is again crucial to treat the projective limit as a topological space. Finally, we establish an analogue of the Verlinde formula for (∞)R. 2. The Projective System of WZW Fusion Rings

2.1. WZW fusion rings. The primary fields of a unitary WZW theory are labelled by integrable highest weights of the relevant affine Lie algebra g, or what is the same, by the value k ∨ of the level and by dominant integral weights Λ of g¯ (the horizontal subalgebra of g) whose inner product with the highest coroot of g¯ is not larger than k ∨ . We denote by g ∨ the dual Coxeter number of the simple Lie algebra g¯ and define I := {i ∈ Z | i ≥ g ∨ } .

(2.1)

644


Thus I is the set of possible values of the height h ≡ k∨ + g ∨ of the WZW theory based on g. For any h ∈ I the fusion rules of a WZW theory at height h define a modular fusion ring, with the elements of the distinguished basis corresponding to the primary fields. We denote this ring by (h)R and its distinguished basis by (h)B, and the corresponding generators of SL(2, Z) by (h)S and (h)T . The distinguished basis (h)B of the ring (h)R can be labelled as B = {(h)ϕa | a ∈ (h)P }

(2.2)

P := {a ∈ L | ai ≥ 1 for i = 1, 2, ... , r; (a, θ∨ ) < h}

(2.3)

(h)

by the set

w

(h)

of integral weights in the interior of (the horizontal projection of) the fundamental Weyl w chamber of g at level h; here r, θ∨ and L denote the rank, the highest coroot and the weight lattice of g¯ , respectively. Note that from here on we use shifted g¯ -weights a = Λ + ρ, which have level h = k ∨ + g ∨ , in place of unshifted weights Λ which are at level k ∨ . Here ρ is the Weyl vector of g¯ ; in particular, a = ρ is the label of the unit element of (h)R. This convention will simplify various formulæ further on. The ring product of (h)R will be denoted by the symbol “ ? ”; thus the fusion rules are written as X (h) (h) ϕa ? (h)ϕb = Na,bc (h)ϕc . (2.4) c∈(h)P

The collection ( R)h∈I of such WZW fusion rings forms a category, more precisely a subcategory of the category of commutative rings, which we denote by Fus(g). The objects of Fus(g) are the rings (h)R, and the morphisms (arrows) are those ring homomorphisms (which are automatically unital and compatible with the conjugation) which map 0 the basis (h )B up to sign factors to (h)B. These are the natural requirements to be imposed on morphisms. Namely, one preserves precisely all structural properties of the fusion ring, except for the positivity of the structure constants; the latter is not an algebraic property, so that one should be prepared to give it up. (h)

2.2. Existence of morphisms. It is not a priori clear whether the category Fus(g) as defined above has any non-trivial morphisms at all. To analyze this issue, we consider the quotients (h)S a,b /(h)S a,ρ (a, b ∈ (h)P ) of S-matrix elements. These are known as the (generalized) quantum dimensions, or more precisely, as the ath quantum dimension of the element (h)ϕb , of the modular fusion ring (h)R. The generalized quantum dimensions furnish precisely all inequivalent irreducible representations of (h)R [2]. We denote by (h)

πa :

R → C,

ϕb 7→

(h)

(h)

S a,b S a,ρ

(h) (h)

(2.5)

the irreducible representation of (h)R which associates to any element its ath generalized quantum dimension. 0 Assume now that f : (h )R → (h)R is a non-trivial morphism, i.e. a ring homomor0 0 phism which maps the distinguished basis (h )B of (h )R to the basis (h)B of (h)R. Then the composition (h) πa ◦ f provides us with a one-dimensional, and hence irreducible, 0 0 0) . Let now (h)L representation of (h )R, i.e. we have (h) πa ◦ f = (h ) πa0 for some a0 ∈ (h P denote the extension of the field Q of rational numbers by the quantum dimensions (h) S a,b /(h)S a,ρ of all elements of (h)B; the observation just made then implies that 0

L ⊆ (h )L

(h)

(2.6)


645

(when f is surjective, one gets in fact the whole field (h)L). As we will see, this result 0 puts severe constraints on the existence of morphisms from (h )R to (h)R. It follows from the Kac--Peterson formula [5] for the S-matrix that L ⊆ Q(ζM h ) ,

(h)

(2.7)

with ζm := exp(2πi/m) and M the smallest positive integer for which all entries of the metric on the weight space of g¯ are integral multiples of 1/M (except for g¯ = Ar , where M = r + 1, M satisfies M ≤ 4). The inclusion (2.6) therefore implies that (h)L lies in the intersection Q(ζM h ) ∩ Q(ζM h0 ) = Q(ζM lcd(h,h0 ) ), and that this intersection is strictly larger than Q unless (h)L = Q. Here lcd(m, n) stands for the largest common divisor of m and n. In the specific case that h and h0 are coprime, lcd(h, h0 ) = 1, it follows that L ⊆

(h)

(h0 )

L ∩ Q(ζM h ) ⊆ Q(ζM ) .

(2.8)

Now typically the field L is quite a bit smaller than Q(ζM h ), i.e. the inequality (2.7) is not saturated (e.g. if the ring is self-conjugate, (h)L is already contained in the maximal real subfield of Q(ζM h )); nevertheless, inspection shows that the requirement (h)L ⊆ Q(ζM ) is fulfilled only in very few cases (for instance, for g¯ of type B2n , Cr , D2n , E7 , E8 or F4 , one has M ≤ 2 so that the requirement is just (h)L = Q). In addition, the main quantum dimensions (h)S a,ρ /(h)S ρ,ρ lie in fact in Q(ζ2h ), and hence the above requirement would restrict them to lie in Q(ζ2h ) ∩ Q(ζM ) = Q(ζlcd(2h,M ) ), and thus to be rational whenever 2h and M are coprime. It follows that for almost all pairs h, h0 of coprime heights there cannot exist any 0 morphism from (h )R to (h)R. The same arguments also show that the existence of nontrivial morphisms becomes the more probable the larger the value of lcd(h, h0 ) is. The most favourable situation is when h0 is a multiple of h; in the next section we will show 0 that in this case a whole family of morphisms from (h )R to (h)R (with h ∈ I arbitrary) can be constructed in a natural way. The considerations above indicate in particular that the naive way of taking the limit “ k → ∞ ” with the standard ordering on the set I cannot correspond to any well-defined limit of the WZW fusion rings. In contrast, as we will show, when replacing the standard ordering by a suitable partial ordering, a limit can indeed be constructed, namely as the projective limit of a projective system that is associated to that partial ordering. Let us also mention that the required ring homomorphism property implies that any morphism of Fus(g) maps simple currents to simple currents. (By definition, simple currents are those elements the distinguished basis (h)B which have inverses in the P ϕa of c fusion ring; they satisfy c Na,b = 1 for all b. Such elements are sometimes also called units of the ring, not to be confused with the unit element of the fusion ring.) (h)

2.3. The projective system. On the set I (2.1) of heights one can define a partial ordering “ ” by i j :⇔ i | j , (2.9) where the vertical bar stands for divisibility. For any two elements i, i0 ∈ I there then exists a j ∈ I (for example, the smallest common multiple of i and i0 ) such that i j and i0 j. Thus the partial ordering (2.9) endows I with the structure of a directed set. We will now show that when the set I is considered as a directed set via the partial ordering (2.9), the collection ((h)R)h∈I of WZW fusion rings can be made into a projective system, that is, for each pair i, j ∈ I satisfying i j there exists a morphism fj,i :

R → (i)R ,

(j)

(2.10)

646


such that fi,i is the identity for all i ∈ I and such that for all i, j, k ∈ I which satisfy i j k, the diagram (k) R A A

fk,j R

A

(j)

fj,i

fk,i

A

(2.11)

AU - (i)R

commutes. We have to construct the maps fi,j for all pairs i, j with i|j. Writing i = h and j = `h with ` ∈ N, the construction goes as follows. The horizontal projection (h)W of the affine ∨ Weyl group at height h has the structure of a semidirect product (h)W = W × hL , with ∨ (h) W the Weyl group and L the coroot lattice of g¯ , so that in particular W is contained as a finite index subgroup in (`h)W , the index having the value ` r . Thus any orbit of (h)W decomposes into orbits of (`h)W , and each Weyl chamber at height `h is the union of ` r Weyl chambers at height h. As a consequence, we find that the following statement holds for the set (`h)P defined according to (2.3). To any a ∈ (`h)P there either exists a unique element wa ∈ (h)W such that a0 := wa (a)

(2.12)

belongs to the set (h)P , or else a lies on the boundary of some affine Weyl chamber at height h. In the former case we define f`h,h ((`h)ϕa ) := ` (a) · (h)ϕa0

(2.13)

with ` (a) = sign(wa ), while in the latter case we set f`h,h ((`h)ϕa ) := 0. It is convenient to include this latter case into the formula (2.13), which is achieved by setting  if a lies on the boundary of an  0 affine Weyl chamber at height h , ` (a) := (2.14)  sign(wa ) else . 2.4. Proof of the morphism properties. We have to prove that fi,j defined this way is a ring homomorphism and that it satisfies the composition property (2.11). It is obvious from the definition that fi,i = id (and also that fi,j is surjective). To show the homomorphism property, we write fi,j in matrix notation, and for convenience use capital letters for the fusion ring (`h)R and lower case letters for the fusion ring (h)R. Thus the elements of the basis (`h)B of (`h)R are denoted by φA ≡ (`h)ϕA with A ∈ (`h)P , while for the elements of (h) B we just write ϕa with a ∈ (h)P , and we use the notation S and s for the S-matrices in place of (`h)S and (h)S, respectively. The mapping is then defined on the preferred basis (`h) B as X FA,b ϕb (2.15) f`h,h (φA ) = b∈(h)P

with FA,b ≡ (`h,h)FA,b := ` (A) δwA (A),b ,

(2.16)


and extended linearly to all of satisfies the relations 1

647

R. As has been established in [6], the matrix (2.16)

(`h)

S F = ` r/2 D s ,

F s = ` r/2 S D ,

(2.17)

with DA,b ≡ (`h,h)DA,b := δA,`b .

(2.18)

Furthermore, from the Kac--Peterson formula [5] for the modular matrix S, one deduces the identity sa,b = ` r/2 Sà,b (2.19) for all a, b ∈ (h)P . Combining the relations (2.16) – (2.19) and the Verlinde formula [7], we obtain for any pair A, B ∈ (`h)P , X (`h) C f`h,h (φA ? φB ) = NA,B f`h,h (φC ) C∈(`h)P

=

X

X X SA,D SB,D S∗C,D FC,e ϕe = Sρ,D (h) (`h)

C,D∈(`h)P e∈ P

= ` r/2 ·

X

D∈

X SA,D SB,D (S∗F )D,e ϕe Sρ,D (h)

P e∈ P

X SA,`d SB,`d s∗d,e X SA,D SB,D (Ds∗ )D,e ϕe = ` r/2 · ϕe Sρ,D Sρ,`d (h) (h)

D∈(`h)P e∈ P

d,e∈ P

X SA,`d SB,`d s∗d,e X (SD)A,d (SD)B,d s∗d,e = `r · ϕe = ` r · ϕe sρ,d sρ,d (h) (h) d,e∈ P

d,e∈ P

X (F s)A,d (F s)B,d s∗d,e X sa,d sb,d s∗e,d = ϕe = FA,a FB,b ϕe sρ,d sρ,d d,e∈(h)P a,b,d,e∈(h)P X = FA,a FB,b (h)Na,bc ϕc a,b,c∈(h)P

=

X

FA,a FB,b ϕa ? ϕb = f`h,h (φA ) ? f`h,h (φB ).

a,b∈(h)P

(2.20) Thus f`h,h is indeed a homomorphism. As a side remark, let us mention that an analogous situation arises for the conformal field theories which describe a free boson compactified on a circle of rational radius squared. These theories are labelled by an (even) positive integer h, and for each value of h the fusion ring is just the group ring ZZh of the abelian group Zh = Z/hZ. The modular S-matrix is given by (h)S p,q = h−1/2 exp(2πipq/h), where the labels p and q which correspond to the primary fields are integers which are conveniently considered 1

In [6], mappings of the type (2.13) were encountered as so-called quasi-Galois scalings. In that setting, the level of the WZW theory is not changed, while the weights A are scaled by a factor of `, followed by an appropriate affine Weyl transformation to bring the weight À back to the Weyl chamber (`h)P or to its boundary. Since what matters is only the relative “size” of weights and the translation part of the Weyl group, these mappings are effectively the same as in the present setting where there is no scaling of the weights but the extension from (`h)W to (h)W scales the translation lattice down by a factor of `. Note that in [6] the letter P was used for the matrix (2.16) in place of F , and D was defined as the transpose of the matrix (2.18).

648


as defined modulo h. It is straightforward to check that the identities (2.17) are again (h) (`h) valid (with r set to 1) if one defines (`h,h)FA,b := δA,b and (`h,h)DA,b := δA,`b , where the (m) superscript on the δ-symbol δa,b indicates that equality needs to hold only modulo m. As a consequence, this way we obtain again a projective system based on the divisibility of h (the composition property is immediate). Moreover, precisely as in the case of WZW theories, with a different partial ordering of the set {h} = Z>0 it is not possible to define a projective system. 2.5. Proof of the composition property. Finally, the composition property (2.11) of the homomorphisms (2.13) is equivalent to the relation X 0 0 (`` h,`h) FA,B (`h,h)FB,c = (`` h,h)FA,c (2.21) B∈(`h)P

among the projection matrices F that involve the three different heights h, `h and ``0 h. Here as before the elements of (h)P and (`h)P are denoted by lower case and capital letters, 0 respectively, while for the elements of (`` h)P we use sans-serif font. The relation (2.21) is in fact an immediate consequence of the definition of the homomorphisms fi,j . The explicit proof is not very illuminating; the reader who wishes to skip it should proceed directly to Sect. 3. To prove (2.21), let us first assume that the left-hand side does not vanish. Then there ∨ exist unique Weyl transformations w1 , w2 ∈ W and unique vectors β1 , β2 ∈ hL in the (`h) coroot lattice scaled by h, and a unique weight B ∈ P , such that w1 (A) + `β1 = B ,

w2 (B) + β2 = c ,

(2.22)

and the left-hand side of (2.21) takes the value ``0 (A) ` (B) = sign(w1 ) sign(w2 ) = sign(w1 w2 ) .

(2.23)

By combining the two relations (2.22), it follows that w2 w1 (A) + β = c ,

(2.24) ∨

where β = ` w2 (β1 ) + β2 . Since β is again an element of hL , this means that (2.24) describes, up to sign, the mapping corresponding to the right-hand side of (2.21). Further, the sign of the right-hand side is then given by sign(w2 w1 ) and hence equal to (2.23); thus (2.21) indeed holds. We still have to analyze (2.21) when its left-hand side vanishes. Then either the Ath 0 row of (`` h,`h)F or the cth column of (`h,h)F must be zero. In the former case, the weight (``0 h) P belongs to the boundary of some Weyl chamber with respect to (`h)W , and thus A∈ ∨ 0 there exist w ∈ W and β ∈ hL such that w(A) + `β = A. But this means that A ∈ (`` h)P also lies on the boundary of some Weyl chamber with respect to (h)W ⊃ (`h)W , and hence also the right-hand side of (2.21) vanishes as required. In the second case, there are unique elements w1 ∈ (`h)W and w2 ∈ (h)W satisfying w1 (A) = B and w2 (B) = B. Because of (`h) W ⊂ (h)W , w1 can also be considered as an element of the Weyl group (h)W at height h. By assumption, w2 is a non-trivial element of (h)W . The product w0 := w1−1 w2 w1 ∈ (h)W is then non-trivial, too, and satisfies w0 (A) = w1−1 w2 w1 (A) = w1−1 w2 (B) = w1−1 (B) = A .

(2.25)


649

Thus the weight A is invariant under a non-trivial element of (h)W and hence lies on the boundary of some Weyl chamber with respect to (h)W ; this implies again that the right-hand side of (2.21) vanishes as required. This concludes the proof of (2.21), and hence of the claim that ((h)R)h∈I together with the maps fi,j constitutes a projective system. 3. The Projective Limit (∞)R We are now in a position to construct the projective limit (∞)R of the projective system that we introduced in Subsect. 2.3. 3.1. Projective limits and coherent sequences. A projective system ((h)R)h∈I in some category C is said to possess a projective limit (L, f ) (also called the inverse limit, or simply the limit) if there exist an object L as well as a family f of morphisms fh : L → (h)R (for all h ∈ I) which satisfy the following requirements (see e.g. [8]). First, for all h, h0 ∈ I with h h0 the diagram

fh 0

L A A A

(3.1)

AU - (h)R

(h0 )

R

fh

A

fh0 ,h

commutes; and second, the following universal property holds: for any object O of the category for which a family of morphisms gh : O → (h)R (h ∈ I) exists which possesses a property analogous to (3.1), i.e. for h h0 ,

fh0 ,h ◦ gh0 = gh

(3.2)

there exists a unique morphism g : O → L such that the diagram

L A A

fh 0

A

AU - (h) R

fh0 ,h

K A g 0 h

g

A

A A

commutes for all h, h0 ∈ I with h h0 .

fh

A

(h0 )

R

A O

gh

(3.3)

650


To be precise, in the above characterization of the projective limit (L, f ) it is implicitly assumed that L is an object in C and that the fi are morphisms of C. But in fact such an object and such morphisms need not exist. In the general case one must rather employ a definition of the projective limit as a certain functor from the category C to the category of sets, and then the question arises whether this functor is “representable” through an object L and morphisms fi as described above. In this language the crucial issue is the existence of a representing object L (see e.g. [9–11]). Now one and the same projective system can frequently be regarded as part of various different categories. For instance, when describing the projective system of our interest one can restrict oneself to the category Fus(g). As we will see, when doing so a projective limit of the projective system does not exist. But one can also consider it, say, in the category of commutative rings, or in the still bigger category of vector spaces, or even in the category of sets. The existence and the precise form of the projective limit usually depend on the choice of category. In our case, however, the category C = Fus(g) we start with is small, i.e. its objects are sets, and as a consequence there exists a natural construction by which the object L and the morphisms fi can be obtained in a concrete manner (in particular, L is again a set). Moreover, it turns out that the projective limit we obtain in the category of sets is exactly the same as the limit that we obtain in the category of commutative rings or vector spaces, which also indicates that this way of performing the limit is quite natural. This construction proceeds as follows. Given a projective system of objects (h)R and morphisms fh0 ,h of a small category C, one regards the objects (h)R ∈ C as sets and Q considers the infinite direct product h∈I (h)R of all objects of C. The elements of this set are those maps . [ (h) R (3.4) ψ: I → h∈I

from the index set I to the disjoint union of all objects (h)R which obey ψ(h) ∈ (h)R for all h ∈ I; they are sometimes called “generalized sequences” (ordinary sequences can be formulated in this languageQby considering the index set N with the standard ordering ≤ ). The subset (∞)R ⊂ h∈I (h)R consisting of coherent sequences, i.e. of those generalized sequences for which fh0 ,h ◦ ψ(h0 ) = ψ(h)

(3.5)

for all h, h0 ∈ I with h h0 , is isomorphic to the projective limit. More precisely, (∞)R is isomorphic to L as a set, and the morphisms fh are the projections to the components, i.e. (3.6) fh (ψ) := ψ(h) . For the projective system introduced in Subsect. 2.3 where C is the (small) category Fus(g), the projective limit is clearly not contained in the original category, because no object (i)R of Fus(g) can possess morphisms to all objects (j)R. In order to identify nevertheless a projective limit associated to the projective system defined by (2.13), it is therefore necessary to consider the set (∞)R of coherent sequences. In accordance with the remarks above, for definiteness from now on we will simply refer to (∞)R as “the” projective limit of the system (2.11) of WZW fusion rings. 3.2. Properties of (∞)R. Let us list a few simple properties of the projective limit (∞)R. First, (∞)R is a ring over Z. The product ψ1 ? ψ2 in (∞)R is defined pointwise, i.e. by the requirement that


651

(ψ1 ? ψ2 )(h) := ψ1 (h) ? ψ2 (h)

(3.7)

for all h ∈ I. This definition makes sense, i.e. for all ψ1 , ψ2 ∈ (∞)R also their product is in (∞)R, because fh0 ,h ◦ (ψ1 ? ψ2 )(h0 ) = (fh0 ,h ◦ ψ1 (h0 )) ? (fh0 ,h ◦ ψ2 (h0 )) = ψ1 (h) ? ψ2 (h) = (ψ1 ? ψ2 )(h) ;

(3.8)

here in the first line the morphism property of the maps fh0 ,h is used. From the definition (3.7) it is clear that the product of (∞)R is commutative and associative, and that (∞)R is unital, with the unit element being the element ψ◦ ∈ (∞)R that satisfies ψ◦ (h) = (h)ϕρ

(3.9)

for all h ∈ I. Second, a conjugation ψ 7→ ψ + can be defined on (∞)R by setting ψ + (h) := (ψ(h))+

(3.10)

for all h ∈ I. The conjugation (h)ϕ 7→ ((h)ϕ)+ on the rings (h)R commutes with the projections fh0 ,h . As a consequence, indeed ψ + ∈ (∞)R whenever ψ ∈ (∞)R, conjugation is an involutive automorphism of (∞)R, and the unit element ψ◦ is self-conjugate. In Sect. 4 we will construct a countable basis (∞)B of the ring (∞)R; this basis contains in particular the unit element ψ◦ . For any ψ ∈ (∞)B and any h ∈ I, ψ(h) is either zero or, up to possibly a sign, an element of the basis (h)B of (h)R. Also, while by construction the structure constants in the basis (∞)B are integers, there seems to be no reason why they should be non-negative. Accordingly, an interpretation of the limit (∞)R as the representation ring of some underlying algebraic structure is even less obvious than in the case of the fusion rings (h)R. 2 In particular, in Sect. 5 we will see that (∞)R does not coincide with the representation ring R of the simple Lie algebra g¯ ⊂ g, but rather that it contains R as a tiny proper subring. As it turns out, the fusion product of two elements of (∞)B is generically not a finite linear combination of elements of (∞)B, or in other words, (∞)B does not constitute an ordinary basis of (∞)R. Rather, it must be regarded as a topological basis. For this interpretation to make sense, a suitable topology on (∞)R must be defined. This will be achieved in the next subsection. 3.3. The limit topology of (∞)R. The fusion rings (h)R can be considered as topological spaces by simply endowing them with the discrete topology, i.e. by declaring every subset to be open (and hence also every subset to be closed). The projective limit (∞)R then becomes a topological space in a natural manner, namely by defining its topology as the coarsest topology in which all projections fh : (∞)R → (h)R are continuous; this will be called the limit topology on (∞)R. More explicitly, the limit topology on (∞)R is described by the property that any open set in (∞)R is an (arbitrary, i.e. not necessarily finite nor even countable) union of elements of (3.11) Ω := {fh−1 (M ) | h ∈ I, M ⊆ (h)R} , i.e. of the set of all pre-images of all sets in any of the fusion rings (h)R. Note that we need not also require to take finite intersections of these pre-images. This 2 The latter can e.g. be regarded as the representation rings of the “quantum symmetry” of the associated WZW theories. However, so far there is no agreement on the precise nature of those quantum symmetries.

652


is because Ω is closed under taking finite intersections, as can be seen as follows. Let (Mi ) ωi ∈ Ω for i = 1, 2, ... , N ; by definition, each of the ωi can be written as ωi = fh−1 i for some heights hi ∈ I and some subsets Mi ⊆ (hi )R. Denote then by h the smallest common multiple of the hi for i = 1, 2, ... , N . Because of (3.1) we have fhi = fh,hi ◦fh , so that −1 ˜ i) , fh−1 (Mi ) = fh−1 (fh,h (Mi )) = fh−1 (M (3.12) i i ˜ i := f −1 (Mi ) are subsets of (h)R. Because of where for all i = 1, 2, ... , N the sets M h,hi TN ˜ i ⊆ (h)R, it thus follows that M i=1

N \

ωi =

i=1

N \

˜ i ) = f −1 [ fh−1 (M h

i=1

N \

˜ i] M

(3.13)

i=1

is an element of the set Ω (3.11). Thus Ω is closed under taking finite intersections, as claimed. As a consequence of this property of Ω, in particular any non-empty open set in (∞)R contains a subset which is of the form fh−1 (M ) for some h ∈ I and some M ⊆ (h)R; for later reference, we call this fact the “pre-image property” of the non-empty open sets in (∞) R. Note that the limit topology on (∞)R is not the discrete one, but finer. To see this, suppose the limit topology were the discrete one. Then for any ψ ∈ (∞)R the one-element set {ψ} would be open and hence a union of sets in Ω (3.11); but as {ψ} just contains one single element, this means that it even has to belong itself to Ω. This in turn means that there would exist h ∈ I and M ⊆ (h)R such that {ψ} = fh−1 (M ), and hence simply M = {ψ(h)}. This, however, would imply that each element ψ ∈ (∞)R would already be determined uniquely by the value ψ(h) for a single height h. From the explicit description of (∞)R as a space of coherent generalized sequences, it follows that this is definitely not true. Thus the assumption that the limit topology is the discrete one leads to a contradiction. Whenever two elements ψ, ψ 0 ∈ (∞)R are distinct, there exists some height h ∈ I such that ψ(h) 6= ψ 0 (h). The open subsets ω := fh−1 ({ψ(h)}) and ω 0 := fh−1 ({ψ 0 (h)}) then satisfy ψ ∈ ω and ψ 0 ∈ ω 0 as well as ω ∩ ω 0 = ∅. This means that when endowed with the limit topology, (∞)R is a Hausdorff space. ) of (∞)R 4. A Distinguished Basis (∞B

In this section we construct a (topological) basis (∞)B of the projective limit (∞)R of WZW fusion rings. 4.1. A linearly independent subset of (∞)R. We start by defining the subset (∞)B ⊂ as the set of all those elements ψ ∈ (∞)R which for every h ∈ I satisfy ψ(h) = h · (h)ϕa

R

(∞)

(4.1)

for some a ∈ (h)P

and

h ∈ {0, ±1}

(4.2)

(i.e. for each height h the fusion ring element ψ(h) ∈ R is either zero or, up to a sign, an element of the distinguished basis (h)B), and for which in addition not all of the prefactors (h)


653

h vanish and h = 1 for the smallest h ∈ I for which h 6= 0. The latter requirement ensures that −ψ 6∈ (∞)B for all ψ ∈ (∞)B. Note that at this point we cannot tell yet whether the set (∞)B is large enough to generate the whole ring (∞)B; in fact, it is even not yet clear whether (∞)B is non-empty. These issues will be dealt with in Subsects. 4.2 to 4.4 below, where we will in particular see that the set (∞)B is countably infinite. However, what we already can see is that the set (∞)B is linearly independent. To prove this, consider any set of finitely many distinct elements ψi , i = 1, 2, ... , N of (∞)B. We first show that to any pair i, j ∈ {1, 2, ... , N } there exists a height hij ∈ I such that (i) ψi (hij ) 6= 0 and ψj (hij ) 6= 0 and (ii) ψi (hij ) 6= ±ψj (hij ) . To see this, assume that the statement is wrong, i.e. that for each height h either one of the elements ψi (h) and ψj (h) of (h)R vanishes, or one has ψi (h) = ±ψj (h). Now because of ψi 6= 0 and ψj 6= 0 there exist heights hi and hj with ψi (hi ) 6= 0 and ψj (hj ) 6= 0. This implies that also ψi (h˜ ij ) 6= 0 and ψj (h˜ ij ) 6= 0 for h˜ ij := hi hj . By our assumption it then follows that ψj (h˜ ij ) = ±ψi (h˜ ij ), which in turn implies that ψj (hi ) = ±ψi (hi ) 6= 0. Now this conclusion actually extends to arbitrary heights h. Namely, from the previous result we know that for any h the elements ψi (hh˜ ij ) and ψj (hh˜ ij ) must both be non-zero. By our assumption this implies that ψj (hh˜ ij ) = ±ψi (hh˜ ij ). Projecting this equation down to the height h, it follows that ψj (h) = ±ψi (h). Since h was arbitrary, it follows that in fact ψj = ±ψi , and hence (as −ψi is not in (∞)B) that ψj = ψi . This is in contradiction to the requirement that all ψi should be distinct. Thus our assumption must be wrong, which proves that (i) and (ii) are fulfilled. Applying now the propertiesQ (i) and (ii) for any pair i, j ∈ {1, 2, ... , N } with i 6= j, it follows that at the height h := i,j;i<j hij we have (i) ψi (h) 6= 0 for all i = 1, 2, ... , N and (ii) ψi (h) 6= ±ψj (h) for all i, j ∈ {1, 2, ... , N }, i 6= j . Thus all the elements ψi (h) of (h)R are distinct and, up to sign, elements of the distinguished basis (h)B. This implies in particular that the only solution of the equation PN ξi ψi (h) = 0 is ξi = 0 for i = 1, 2, ... , N , which in turn shows that also the equation Pi=1 N i=1 ξi ψi = 0 has only this solution. Thus, as claimed, the ψi are linearly independent elements of (∞)R. 4.2. (∞)B generates all of (∞)R. Next we claim that the set (∞)B spans (∞)R in the sense that the closure (in the limit topology) of the linear span of (∞)B in (∞)R, i.e. of the set h(∞)Bi ≡ spanZ ((∞)B)

(4.3)

of finite Z-linear combinations of elements of (∞)B, is already all of (∞)R. To prove this, assume that the statement is wrong, or in other words, that the set S := (∞)R \ h(∞)Bi

(4.4)

is non-empty. By definition, the set S is open, and hence because of the pre-image property it contains a subset M ⊆ S of the form M = fh−1 (M ) for some h ∈ I and some M ⊆ (h)R. Further, as an immediate consequence of the construction that we will present in Subsects. 4.3 and 4.4, for each a ∈ (h)P there exists an element (in fact, infinitely many elements) ψa ∈ (∞)B such that fh (ψa ) = (h)ϕa (namely, we need to prescribe the value of ψa (p) only for the finitely many prime factors p of h). Now choose some y ∈ M ,

654


P (h) (h) (h) decompose it with P respect to the basis B of R, i.e. y = a∈(h)P na ϕa with na ∈ Z, and define η := a∈(h)P na ψa . Then, on one hand, by construction we have fh (η) = y, i.e. η ∈ M, and hence η ∈ S, while on the other hand η is a finite linear combination of elements of (∞)B (since y is a finite linear combination of elements of (h)B), and hence η ∈ h(∞)Bi ⊆ h(∞)Bi. By the definition (4.4) of S, this is a contradiction, and hence our assumption must be wrong. Together with the result of the previous subsection we thus see that (∞)B is a (topological) basis of (∞)R. 4.3. Distinguished sequences of integral weights. We will now construct all elements of the projective limit (∞)R which belong to the subset (∞)B as introduced in Subsect. 4.1. These are obtained as generalized sequences ψ satisfying both (3.5) and the defining relation (4.1) of (∞)B. More specifically, we construct sequences (ah )h∈I of labels ah ∈ (h)P and associated signs η(ah ) such that all those ψ which are of the form ψ(h) = η(ah ) (h)ϕa

(4.5)

h

belong to the subset (∞)B ⊂ (∞)R. When applied to (4.5), the requirement (3.5) amounts to (4.6) f`h,h ((`h)ϕa ) = η(a`h )η(ah ) · (h)ϕa , `h

h

which in view of the definition (2.13) of f`h,h is equivalent to a`h = w(ah )

for some w ∈ (h)W

(4.7)

and η(a`h )η(ah ) = ` (a`h ) .

(4.8)

To start the construction of the elements of B, we first concentrate our attention to integral weights of the height h theory which are not necessarily integrable and which ∨ are considered as defined only modulo hL ; we denote these weights by bh . Suppose then that we prescribe for each prime p such a weight bp and that these weights satisfy in addition the restriction that for any two primes p, p0 they differ by an element of the coroot lattice, ∨ (4.9) b p − bp 0 ∈ L . (∞)

We claim that there then exists a sequence (bh )h∈I which for prime heights takes the prescribed values bp and for which the relation bh0 = bh mod hL

∨

(4.10)

holds for all h, h0 with h h0 . To prove this assertion, we display such a sequence explicitly. To this end, let Y n pj j (4.11) h =: j pj |h

denote the decomposition of h into prime factors, and define hi :=

h pni i

(4.12)


and

655

ni −1 [hi ]−1 pni := (hi ) mod pi .

(4.13)

i

Then we set bh := bp +

X

1

−1

hi [hi ]pni (bpi − bp ) .

(4.14)

1

i

i6=1 pi |h ∨

Recall that bh is defined only modulo hL . In (4.14) p1 is any of the prime divisors of h; it has been singled out only in order to make the formula for bh to look as simple ∨ as possible, and in fact bh does not depend (modulo hL ) on the choice of p1 . To see (2) this, let bh denote the number obtained analogously as in (4.14), but with p1 replaced by some other prime factor p2 of h. Then X −1 bh − b(2) = b − b + hi [hi ]pni (bp − bp ) p p h 1 2 2 1 i i6=1,2 (4.15) pi |h −1

−1

+ h2 [h2 ]pn2 (bp − bp ) − h1 [h1 ]pn1 (bp − bp ) . 2

2

1

1

1

2

n

Using the fact that hi is divisible by pj j for all primes pj dividing h except for j = i, −1

and that hi [hi ]pni = 1 mod pni i , it is easily checked that the right-hand side of this i

n

∨

expression vanishes modulo pj j L for all pj dividing h, and hence, using (4.9), also ∨ vanishes modulo hL . To establish the coherence property (4.10), we now consider two heights h, h0 such that h|h0 . Then we set Y n0 h0 =: pj j (4.16) j pj |h0 n0

and h0i := h0 /pi i , and without loss of generality we can assume that p1 divides h as well as h0 . By the definition (4.14) we then have X X −1 ni } (bp − bp ) + bh0 − bh = {h0i [h0i ]−1n0i − hi [hi ]−1 h0i [h0i ] n0i (bpi − bp ) ; p i 1 1 i p p i

i6=1 pi |h

i pi |h0 , pi 6 | h

i

(4.17) ∨ n again it is straightforward to verify that this vanishes modulo pj j L for all pj dividing h. This shows that the property (4.10) is satisfied for the sequence defined by (4.14) as claimed. Next we note that we did not require that the prescribed values bp lie on the Weyl orbit of an integrable weight at height p, but rather they may also lie on the boundary of some Weyl chamber of (p)W . However, if bp does belong to the Weyl orbit of an integrable weight, then also each weight bh with p|h is on the Weyl orbit of an integrable weight at height h. Namely, because of the property (4.10) we have in particular bh = bp + pn β ∨

(4.18)

for some β ∈ L . Hence, assuming that bh is left invariant by some w ∈ (h)W , i.e. that bh = w(bh ) ≡ w(bh ) + hγ for some element γ of the coroot lattice, it follows that

656


w(bp ) = w(bh − pn β) = w(bh ) − pn w(β) = bh − hγ − pn w(β) = bp + pn (β − w(β)) − hγ .

(4.19)

Since by assumption the only element of (p)W which leaves the weight bp invariant is the identity, it follows that w = id and γ = 0, implying that also the only element of (h)W that leaves bh invariant is the identity, which is equivalent to the claimed property. Our next task is to investigate to what extent the sequence (bh )h∈I is characterized by the prescribed values bp at prime heights and by the requirement (4.10). To this end let (b˜ h )h∈I be another such sequence, i.e. a sequence such that b˜ p = bp for all primes p ∨ and b˜ h0 − b˜ h ∈ hL for h|h0 . First we observe that for h and h0 coprime, the properties ∨ b˜ hh0 = b˜ h mod hL

∨ b˜ hh0 = b˜ h0 mod h0 L

and

(4.20)

∨ fix b˜ hh0 already uniquely (modulo hh0 L ), so that the whole freedom is parametrized by the freedom in the choice of b˜ h at heights which are a prime power. Concerning the latter freedom, we claim that for any prime p there is a sequence of elements βp(j ) of the ∨ ∨ coroot lattice L which are defined modulo pL such that the most general choice of b˜ pn reads

b˜ pn = bpn +

n−1 X

βp(j ) pj

(4.21)

j=1

with bpn defined according to (4.14), i.e. simply bpn = bp . This statement is proven by induction. For n = 1 it is trivially fulfilled. Further, assuming that (4.21) is satisfied for ∨ some n ≥ 1 and setting γ := b˜ pn+1 − bpn+1 (defined modulo pn+1 L ), one has

b˜ pn+1 − b˜ pn = γ −

n−1 X

βp(j ) pj .

(4.22)

j=1

By the required properties of the sequence (b˜ h ), the left-hand side of this formula must ∨ vanish modulo pn L , and hence we have

γ = βp(n) pn +

n−1 X

n X

j=1

j=1

βp(j ) pj =

βp(j ) pj

(4.23)

∨ for some βp(n) ∈ L . This shows that b˜ pn+1 is again of the form described by (4.21); ∨ ∨ furthermore, as γ is defined modulo pn+1 L , βp(n) is defined modulo pL as required, and hence the proof of the formula (4.21) is completed. With these results we are now in a position to give a rather explicit description of the allowed sequences b˜ h . Namely, we can parametrize the general form of b˜ h in terms of the freedom in b˜ pn according to


X

b˜ h = b˜ pn1 + 1

= bp +

−1 hi [hi ]pni (b˜ pni − b˜ pn1 )

X

−1

i

βp(ji) pji −

j=1 −1

i6=1 pi |h

nX 1 −1

(4.24)

βp(j1) pj1 ]

j=1

nX i −1

nX 1 −1

nX 1 −1

j=1

j=1

j=1

βp(ji) pji −

hi [hi ]pni [ i

1

i

i6=1 pi |h nX i −1

−1

X

1

hi [hi ]pni (bpi − bp )

hi [hi ]pni [

i6=1 pi |h

= bh +

X

βp(j1) pj1 +

j=1

+

i

i

i6=1 pi |h nX 1 −1

1

657

βp(j1) pj1 ] +

βp(j1) pj1 .

∨ Further, any such sequence fulfills the consistency requirement that b˜ h0 − b˜ h ∈ hL for heights h, h0 with h|h0 . Namely, in this case the formula (4.24) yields

b˜ h0 − b˜ h =

X

{ [ ] h0i

i6=1 pi |h

−1 h0i n0i pi

n0i −1

[

X

β

i

+

−

X

βp(ji) pji −

j=1 −1 h0i h0i n0i pi

i pi |h0 , pi 6 | h 0 n1 −1 X βp(j1) pj1 j=n1

βp(j1) pj1 ]

j=1

nX i −1

−1

+

pji

j=1

− hi [hi ]pni [ X

n01 −1 (j ) pi

[ ]

nX 1 −1

βp(j1) pj1 ]}

n0i −1

[

X

j=1

βp(ji) pji

(4.25)

n01 −1

−

j=1

X

βp(j1) pj1

]

j=1

∨

mod hL . n

∨

Once more one can easily check that this expression vanishes modulo pj j L for all ∨ primes pj that divide h, and hence vanishes modulo hL . Thus the consistency requirement indeed is satisfied. 4.4. Distinguished sequences of integrable weights and the basis (∞)B. What we have achieved so far is a characterization of all sequences (bh )h∈I of integral weights defined ∨ modulo hL that satisfy (4.10). We now use this result to construct sequences (ah )h∈I of highest weights which satisfy the requirement (4.7) and of which infinitely many are integrable weights, with all non-integrable weights being equal to zero. We start by prescribing integrable weights ap ∈ (p)P for all primes p with p ≥ g ∨ , and set bp = ap for p ≥ g ∨ , while for all primes p < g ∨ we choose arbitrary weights bp (which are necessarily non-integrable). Next we employ the previous results to find the sequences (bh )h∈I . Finally we define ah for any arbitrary height h as follows. If bh lies on the boundary of a Weyl chamber with respect to (h)W , then we set ah = 0. Otherwise there ∨ are a unique element wh ∈ W and a unique 3 element γh ∈ L such that wh (bh ) + hγh is integrable, and in this case we set ∨

To be precise, because the weights ah are defined only modulo hL , γh is only unique once a definite representative of the equivalence class of weights that is described by ah is chosen. 3

658


ah := wh (bh ) + hγh .

(4.26)

By construction, the weights ah have the following properties. If ah0 = 0 for some 0 height h0 , then bh0 is on the boundary of a Weyl chamber with respect to (h )W ; for any ∨ h dividing h0 , it then follows from bh0 − bh ∈ hL that bh is on the boundary of a Weyl chamber with respect to (h)W , and hence we also have ah = 0. On the other hand, if 0 0) , then it is a bh0 is equivalent with respect to (h )W to an integrable weight ah0 ∈ (h P (h) fortiori equivalent to ah0 with respect to the larger group W , and then the property ∨ bh0 − bh ∈ hL implies that also bh is equivalent with respect to (h)W to ah0 , and hence that the associated weight ah is integrable at height h and is equivalent with respect to (h) W to ah0 , too. Thus ah and ah0 are on the same orbit with respect to (h)W whenever h divides h0 , and hence (4.7) holds as promised. Note that by construction for all g¯ except g¯ = A1 the sequences so obtained contain some zero weights. However, any sequence which contains at least one non-zero weight contains in fact infinitely many non-zero (and hence integrable) weights. The final step is now to define ψ(h) := η(ah ) (h)ϕa

h

as in (4.5), where ah is as constructed above, and where sign(wh ) for ah ∈ (h)P , η(ah ) := 0 for ah = 0 .

(4.27)

(4.28)

To show that ψ is an element of the projective limit, it only remains to check the property (4.8) of the prefactor η(ah ). For ah = 0 (4.8) just reads 0 = 0 and is trivially satisfied. For ah ∈ (h)P , the previous results show that a`h = w`h ◦ w0 ◦ wh−1 (ah ), where w0 is the Weyl translation relating bh and b`h , so that ` (a`h ) = sign(w`h ◦ w0 ◦ wh ) = sign(w`h ) · sign(wh ) .

(4.29)

In view of the definition (4.28) of η(ah ), this is precisely the required relation (4.8). We conclude that the basis (∞)B of (∞)R precisely consists of the elements (4.27). In particular, (∞)B is countably infinite. 5. The Fusion Ring of g¯ As already pointed out in the introduction, it is expected that in the limit of infinite level of WZW theories somehow the simple Lie algebra g¯ which is the horizontal subalgebra of g and its representation theory should play a rôle. More specifically, one might think that the representation ring R of g¯ emerges. As we will demonstrate below, indeed this ring shows up, but it is only a proper subring of the projective limit (∞)R we constructed, and almost all elements of (∞)R are not contained in the ring R. Let us describe R and its connection with the category Fus(g) in some detail. R is defined as the ring over Z of all isomorphism classes of finite-dimensional g¯ representations, with the ring product the ordinary tensor product of g¯ -representations (or, equivalently, the pointwise product of the characters of these representations). This ring R is a fusion ring with an infinite basis. The elements ϕ¯ a of a distinguished basis of R are labelled by the (shifted) highest weights of irreducible finite-dimensional g¯ -representations, i.e. by elements of the set


659

w P¯ := {a ∈ L | 0 < ai for i = 1, 2, ... , r} .

(5.1)

Now for any h ∈ I let us define the map f¯h : R → (h)R as follows. If a ∈ P¯ lies on the boundary of some Weyl chamber with respect to (h)W , we set f¯h (ϕ¯ a ) := 0; otherwise there exist a unique a0 ∈ (h)P and a unique w ∈ (h)W such that w(a) = a0 , and in this case we set (5.2) f¯h (ϕ¯ a ) := (a) · (h)ϕa0 with (a) = sign(w). As in the case of the maps f`h,h (2.13), we will consider (5.2) as covering all cases, i.e. set (a) = 0 if a lies on the boundary of a Weyl chamber at height h. To analyze the relation between the ring R and the category Fus(g), we first recall the expressions X c Na,b = sign(w) multb (w(c) − a) (5.3) w∈W

for the Littlewood--Richardson coefficients (or tensor product coefficients) of g¯ [12, 13] and X (h) Na,bc = sign(w) multb (w(c) − a) (5.4) w∈(h) W

for the fusion rule coefficients, i.e. the structure constants of the WZW fusion ring R [5, 14–16]. Here multa (b) denotes the multiplicity of the (shifted) weight b in the g¯ -representation with (shifted) highest weight a. It will be convenient to extend the c validity of (5.3) by adopting it as a definition of Na,b for arbitrary (i.e., not necessarily lying in P¯ ) integral weights a and c, and also extend it to arbitrary integral weights b that do not lie on the boundary of any Weyl chamber with respect to W by setting (h)

mult b (c) := sign(wb ) multw

b

(b) (c) ,

(5.5)

with wb the unique element of W such that wb (b) ∈ P¯ . The multiplicities mult a (b) are invariant under the Weyl group W , i.e. mult a (w(b)) = c mult a (b) for all w ∈ W . As a consequence, the numbers Na,b and (h)Na,bc are related by 4 Na,bc =

(h)

1 |W |

X

w(c)

sign(w) Na,b

.

(5.6)

w∈(h) W

The invariance of mult a (b) under W also implies that for arbitrary integral weights a, b and c the symmetry property c c Na,b = Nb,a (5.7) follows from the analogous property of the Littlewood--Richardson coefficients with a, b, c ∈ P¯ , and that X w2 (c) Nw1 (a),b = sign(w) multb (w w2 (c) − w1 (a)) w∈W

=

X

c

sign(w) multb (w−1 1 w w 2 (c) − a) = sign(w 1 w 2 ) · Na,b .

(5.8)

w∈W c

In the formulation of [5,14–16] the factor of |W |−1 is absent because there Na,b is taken to be non-zero only if a, b ∈ P¯ . 4

660


When combined with the symmetry property (5.7), the latter formula yields w3 (c)

c

Nw1 (a),w2 (b) = sign(w1 w2 w3 ) · Na,b .

(5.9)

To obtain information about the effect of affine Weyl transformations on the labels c of Na,b , we consider an alternating sum over the Weyl group (h)W . We have X w2 (c) sign(w2 ) Nw1 (a),b w2 ∈(h) W

=

X

sign(w)sign(w2 ) multb (w w2 (c) + hw(β2 ) − w1 (a) − hβ1 )

w,w2 ∈W ¯∨ β2 ∈L

=

X

X

−1 sign(w w2 ) multb (w−1 1 w w 2 (c) + hw 1 w(β) − a)

¯∨ w,w2 ∈W β∈L

X

= sign(w1 ) ·

w2 (c)

sign(w2 ) Na,b

.

w2 ∈(h) W

(5.10) Here β := β2 − w−1 (β1 ). Together with the symmetry property (5.7) it then follows that X X w3 (c) w3 (c) sign(w3 ) Nw1 (a),w2 (b) = sign(w1 ) sign(w2 ) · sign(w3 ) Na,b (5.11) w3 ∈(h) W

w3 ∈(h) W

for all w1 , w2 ∈ (h)W . We can rewrite this as X X w(c) w(c) ` (A)` (B) sign(w) NwA (A),wB (B) = sign(w) NA,B , w∈(h) W

(5.12)

w∈(h) W

which by interpreting A and B as elements of P¯ rather than (`h)P yields, after summation over c ∈ (h)P , f¯ (ϕ¯ A ) ? f¯ (ϕ¯ B ) = f¯ (ϕ¯ A ? ϕ¯ B ) , (5.13) h

h

h

and hence shows that the maps f¯h defined by (5.2) are ring homomorphisms. Now for all A ∈ (`h)P we have f`h,h (φA ) = ` (A)(φwA (A) ) =: ` (A) · ϕa . Then owing to (5.6) we obtain, after dividing (5.12) by |W |, the relation X (h) c C Nf`h,h (A),f`h,h (B) = ` (A) ` (B) (h)Na,bc = ` (C) (`h)NA,B (5.14) −1 C: φC ∈f`h,h (ϕc )

(on the left-hand side, we use the shorthand notation f`h,h (A) ∈ (h)P to indicate the label that corresponds to the element ` (A)f`h,h (φA ) of (h)R). Summation over c ∈ (h)P then yields f`h,h (φA ) ? f`h,h (φB ) = f`h,h (φA ? φB ), so that (5.14) is just the homomorphism property of the maps fi,j which were defined by (2.13) in terms of the fusion rule coefficients. (Thereby we have also obtained an alternative proof of the homomorphism property of those maps.) To investigate further the relation between R and the projective limit (∞)R, we introduce the linear mappings ¯h : h,h0 :

R → R,

(h)

R→

(h)

(h0 )

R,

ϕa 7→ ϕ¯ a ,

(h)

0

ϕa 7→ (h )ϕa

(h)

(5.15)


661 0

which map each basis element (h)ϕa of (h)R to that basis element of R and (h )R (h0 ≥ h), 0) ⊂ P¯ . For h h0 respectively, which is labelled by the same weight a ∈ (h)P ⊆ (h P 00 h , these maps satisfy ¯h0 ◦ h,h0 = ¯h , as well as

f¯h ◦ ¯h = idh ,

h0 ,h00 ◦ h,h0 = h,h00 ,

(5.16)

fh0 ,h ◦ h,h0 = idh ,

(5.17)

and

f¯h ◦ ¯h0 = fh0 ,h , fh00 ,h ◦ h0 ,h00 = fh0 ,h . We say that a generalized sequence ψ in the projective limit constant iff there exists a h◦ ∈ I such that

(5.18) R is ultimately

(∞)

ψ(h) = h◦ ,h ◦ ψ(h◦ )

(5.19)

(and hence for basis elements in particular ah = ah◦ ) for all heights h ≥ h◦ . Now assume that ψ1 and ψ2 are elements of (∞)R which are ultimately constant, with associated heights h◦,1 and h◦,2 , respectively. Then in particular for all heights h larger than h◦ := 2 max(h◦,1 , h◦,2 ) the fusion product ψ1 (h) ? ψ2 (h) in (h)R is isomorphic to the product ψ 1 ? ψ 2 in R, where ψ 1 := ¯h◦ ◦ ψ1 (h◦ ), and analogously for ψ 2 . This implies that (h◦ ,h ◦ ψ1 (h◦ )) ? (h◦ ,h ◦ ψ2 (h◦ )) = h◦ ,h ◦ (ψ1 (h◦ ) ? ψ2 (h◦ ))

(5.20)

even though h◦ ,h is not a ring homomorphism, and hence (ψ1 ?ψ2 )(h) ≡ ψ1 (h)?ψ2 (h) = h◦ ,h ◦ (ψ1 ? ψ2 )(h◦ ) for all h ≥ h◦ . Thus the product ψ1 ? ψ2 is again ultimately constant. Also, the property of being ultimately constant is preserved upon taking (finite) linear combinations and conjugates. The set of ultimately constant elements therefore constitutes a subring of (∞)R. The following consideration shows that this subring is isomorphic to the fusion ring R. First, any ultimately constant element is a linear combination of ultimately constant elements ψ (a) for which ψ (a) (h◦ ) is an element of the canonical basis of (h◦ )R, ) ⊂ P¯ . But there is a unique element ψ of (∞)R with ψ (a) (h◦ ) = (h◦ )ϕa for some a ∈ (h◦P the latter property, because at all heights h smaller than h◦ the value ψ(h) is already fixed by imposing the requirement (4.6). Thus there is a bijective linear map between the subring of ultimately constant elements and the fusion ring R, defined by ϕ¯ a 7→ ψ (a) for a ∈ P¯ . Moreover, the same argument which led to (5.20) shows that this map is in fact an isomorphism of fusion rings. As this map is provided in a canonical manner, we can actually identify the two rings. A generic element of (∞)R is not ultimately constant, so that the subring of ultimately constant elements is a proper subring of (∞)R. Thus what we have achieved is to identify the fusion ring R as a proper sub-fusion ring of the projective limit ring (∞)R. To conclude this section, let us remark that of course we could have enlarged by hand the category Fus(g) to a larger category Fus(g) by just including one additional object into the category, namely the ring R, together with the morphisms f¯h . This essentially amounts to cutting the category of rings in such a way that one is able to identify the ring R as the projective limit of this category Fus(g). We do not regard this as a viable alternative to our construction, though, since when doing so one performs manipulations which are suggested merely by one’s prejudice on what the limit should look like. (Also, phenomena like level-rank dualities in fusion rings require to consider various rings for different algebras g on the same footing; the category Fus(g) cannot accommodate such phenomena.) In contrast, our construction of the limit employs only the description in terms of coherent sequences, which is a natural procedure for any small category, and does not presuppose any desired features of the limit.

662


6. Representation Theory of (∞)R A basic tool in the study of fusion rings is their representation theory. Of particular importance are the irreducible representations, which lead in particular to the notion of (generalized) quantum dimensions. In this section we show that an analogous representation theory exists for the projective limit as well. In our considerations the limit topology will again play an essential rôle. 6.1. One-dimensional representations. Let us consider for any two h, h0 ∈ I with h0 = `h 0) that is defined the injection of the label set (h)P (defined as in (2.3)) into the label set (h P by multiplying the weights a by a factor of `: a 7→ à

(6.1)

for all a ∈ (h)P . (This induces an injection (h)ϕa 7→ (`h)ϕà of the distinguished basis (h)B of (h)R into the distinguished basis (`h)B of (`h)R. However, when this map is extended linearly to all of (h)R, it does not provide a homomorphism of fusion rings.) We can use these injections to perform an inductive limit of the set ((h)P )h∈I of label sets, where the set I (2.1) is again considered as directed via the partial ordering (2.9). We denote this inductive limit by (∞)P . An element α of (∞)P can be characterized by an integrable weight α(h) ∈ (h)P at some suitable height h; at any multiple `h of this height, the same element α of (∞)P is then represented by the weight α(`h) = ` α(h). In particular, quite unlike as in the case of the projective limit, each element of the inductive limit (∞)P is already determined by its representative at a single height. Also note that an element α ∈ (∞)P is not defined at all heights h; in particular, for any h ∈ I the set of those α ∈ (∞)P which have a representative at height h is in one-to-one correspondence with the elements of (h)P , and hence is in particular finite. We will use the notation α ↓(h)P to indicate that α ∈ (∞)P has a representative α(h) ∈ (h)P at height h. We claim that any element of (∞)P gives rise to a one-dimensional representation of the projective limit (∞)R of the fusion rings. To see this, we choose for a given α ∈ (∞)P a suitable height h ∈ I such that α ↓ (h)P . To any coherent sequence (ψ(l))l∈I in the projective limit (∞)R we then associate the number Dα (ψ) :=

S ψ(h),α(h) , S ρ,α(h)

(h)

(6.2)

(h)

i.e. the α(h)th quantum dimension of the element ψ(h) of the ring (h)R. Here we use the short-hand notation X X (h) S ψ(h),b := ζa (h)S a,b for ψ(h) = ζa (h)ϕa (6.3) a∈(h)P

a∈(h)P

for linear combinations of S-matrix elements. Using the identities (2.18) and (2.17) as well as α(`h) = `α(h) and the defining properties of ψ, we have S ψ(`h),α(`h) ((`h)S D)ψ(`h),α(h) (F (h)S)ψ(`h),α(h) = (`h) = = S ρ,α(`h) ( S D)ρ,α(h) (F (h)S)ρ,α(h)

(`h)

(`h)

S ψ(h),α(h) ; S ρ,α(h)

(h)

(h)

(6.4)

this shows that the formula (6.2) yields a well-defined map from (∞)R to C , i.e. it does not depend on the particular choice of h. Using the knowledge about the representation theory of the rings (h)R, it then follows immediately that


Dα (ψ) Dα (ψ 0 ) =

X

663

ζa ζb0 (h)Na,bc

a,b∈(h)P

S c,α(h) = Dα (ψ ? ψ 0 ) . S ρ,α(h)

(h) (h)

(6.5)

Thus the prescription (6.2) indeed provides us with a one-dimensional representation of (∞) R. Let us now associate to any element ψ of (∞)R the infinite sequence of quantum dimensions (6.2), labelled by (∞)P ; this way we obtain a map D:

ψ 7→ (Dα (ψ))α∈(∞)P

(6.6)

X := {(ξα )α∈(∞)P | ξα ∈ C }

(6.7)

from the ring (∞)R to the algebra

of all countably infinite sequences of complex numbers. Since we are now dealing with complex numbers rather than only integers, it is natural to consider instead of the fusion ring (∞)R the corresponding algebra over C , to which we refer as the fusion algebra (∞)A. (For simplicity we regard (∞)A as an algebra over C . In principle it would be sufficient to consider it over a certain subfield of C generated by appropriate roots of unity.) It is then evident that the map D : (∞)A → X defined by (6.6) is an algebra homomorphism. (We continue to use the symbol D. More generally, below we will always assume that the various maps to be used, such as the projection (2.15), are continued C -linearly from the fusion rings (h)R to the associated fusion algebras (h)A, and use the same symbols for these extended maps as for the original ones.) 6.2. An isomorphism between (∞)A and X . In this subsection we show that the map D introduced above even constitutes an isomorphism between the complex algebras (∞)A and X : ∼ = (6.8) D : (∞)A −→ X . Injectivity of D is easy to check. Suppose we have D(ψ) = 0. Fix any h ∈ I; then all quantum dimensions of the element ψ(h) of (h)R vanish. From the properties of the fusion ring (h)R it then follows immediately that ψ(h) = 0. This is true for all h ∈ I, and hence we have ψ = 0. This proves injectivity. To show also surjectivity requires more work. We first need to introduce the elements X (h) ∗ S a,b (h)ϕb ∈ (h)A (6.9) ea ≡ (h)ea := (h)S ρ,a b∈(h)P

of the fusion algebras at height h. These elements are idempotents, i.e. obey ea ? eb = δa,b ea .

(6.10)

Owing to the unitarity of the modular transformation matrix S, the idempotents {(h)ea | a ∈ (h)P } form a basis of the fusion algebra (h)A, and they constitute a partition of the unit element, in the sense that X (h) ea = (h)ϕρ . (6.11) a∈(h)P

Also, for any element ψ ∈ (∞)A with ψ(h) = ea and any α ∈ (∞)P with α ↓(h)P we have Dα (ψ) = δa,α(h) .

(6.12)

664


We now study how the idempotents eα(h) behave under the projection (2.15). First, when α ∈ (∞)P has a representative α(h) at height h, then for every positive integer ` we have, using the first of the identities (2.17), X (`h) ∗ S α(`h),A f`h,h (φA ) f`h,h (eα(`h) ) = (`h)S ρ,α(`h) A∈(`h)P

X

= (`h)S ρ,`α(h)

X

S ∗α(`h),A FA,b ϕb

(`h)

A∈(`h)P b∈(h)P

X

= `−r/2 · (h)S ρ,α(h) = (h)S ρ,α(h)

X

((`h)S ∗ F )α(`h),b ϕb

(6.13)

b∈(h)P

D`α(h),c (h)S ∗c,b ϕb

b,c∈(h)P

= (h)S ρ,α(h)

X

S ∗α(h),b ϕb = eα(h) .

(h)

b∈(h)P

On the other hand, when α has a representative at height h, but not at height h0 , we can compute as follows. Since hh0 is a multiple of h, α has a representative α(hh0 ) at height hh0 . Thus we can repeat the previous calculation to deduce that X 0 0 0 (D (h )S)α(hh0 ),b (h )ϕb fhh0 ,h0 (eα(hh0 ) ) = (hh )S ρ,α(hh0 ) · hr/2 0

b∈(h )P

= (h/h0 )r/2 · (h)S ρ,α(h)

X

0

0

δh0 α(h),hc (h )S c,b (h )ϕb .

(6.14)

0

b,c∈(h )P

Now in the sum over c on the right-hand side one has a contribution only if c = 0) at height h0 . But in this case we h0 α(h)/h = α(hh0 )/h is an element of the label set (h P would conclude that α has in fact a representative at height h0 , namely α(h0 ) = c, which contradicts our assumption. Therefore we conclude that in the case under consideration we have fhh0 ,h0 (eα(hh0 ) ) = 0. Together with the result (6.13) it follows that by setting 0 if α ∈ (∞)P has no representative at height h , eα (h) := (6.15) eα(h) else , we obtain an element eα of the projective limit (∞)A. Moreover, according to the relation (6.12) the map (6.8) acts on eα ∈ (∞)A as

(D(eα ))β = δα,β

(6.16)

for all α, β ∈ (∞)P , and the eα provide a partition of the unit element, analogously as in (6.11), X eα = ψ◦ ; (6.17) α∈(∞)P

here the sum is to be understood as a limit of finite sums in the limit topology. us define the map gh : X → (h)A by (ξα )α∈(∞)P 7→ P Now for each h ∈ I let (`h) (∞) ξ e . Since β ↓ P if β ↓(h)P , we then have β∈ P β β(h) β↓(h)P


f`h,h ◦ g`h ((ξα )) =

X

665

X

ξβ eβ (h) =

β∈(∞)P β↓(`h)P

ξβ eβ(h) = gh ((ξα ))

(6.18)

β∈(∞)P β↓(h)P

for all positive integers `. Analogously we can define a map g:

X → (∞)A ,

(ξα )α∈(∞)P 7→

X

ξβ eβ

(6.19)

β∈(∞)P

with similar properties. As a consequence of the relation (6.16) one finds that this map satisfies D ◦ g = idX . (6.20) This implies that the injective map D is also surjective (and that g is injective). Thus we have proven the isomorphism (6.8). 6.3. Semi-simplicity. It is known [2] that the fusion algebras (h)A at finite heights h are semi-simple associative algebras. In this subsection we show that in a suitable topological sense the same statement holds for the projective limit (∞)A, too. We first combine the identity (6.10) and the definition (6.15) of the element eα of (∞) A with the fact that the idempotents ea form a basis of (h)A. This way we learn that for all ψ ∈ (∞)A and all heights h ∈ I the fusion product (eα ? ψ)(h) = eα (h) ? ψ(h) is proportional to eα (h). Thus for each α ∈ (∞)P the span Iα := heα i

(6.21)

of {eα } is a one-dimensional two-sided ideal of the projective limit, i.e. we have (∞)A Iα = Iα (∞)A ⊆ Iα . We claim that when we endow the algebra with the limit topology, then in fact (∞)A is the closure of the direct sum of the ideals (6.21) in this topology: A=

(∞)

M

Iα .

(6.22)

α∈(∞)P

(In particular, the idempotents eα form a topological basis of (∞)A.) To prove this, we first recall from Subsect. 3.3 that in the limit topology each open set in (∞)A is a union of elements of the set Ω = {fh−1 (M ) | h ∈ I, M open in (h)A} of all pre-images of all open sets in any of the fusion algebras (h)A. Here we assume that we have already chosen a topology on each of the fusion algebras (h)A. (Actually the choice of this topology on (h)A will not be important; for definiteness, we may take the discrete one, as in the case of fusion rings, or also the metric topology of (h)A as a finite-dimensional complex vector space.) Consider now an arbitrary element ξ ∈ (∞)A, which because of the isomorphism (∞)A ∼ = X we can write as ξ = (ξα )α∈(∞)P . Adopting some definite numbering (∞)P = {αm | m ∈ N} of the countable set (∞)P , for n ∈ N we define X M ξˆn := ξαm eαm ∈ Iα . (6.23) m≤n

α∈(∞)P

To prove our assertion, we must then show that for every h ∈ I and every open set M ⊆ (h)A which satisfy fh (ξ) ∈ M we have ξˆn ∈ fh−1 (M ), i.e. fh (ξˆn ) ∈ M ,

(6.24)

666


for all but finitely many n. Now by direct calculation we obtain X ξαm (h)eαm (h) ; fh (ξˆn ) =

(6.25)

m≤n αm ↓(h)P

this is a finite sum, and for sufficiently large n it becomes independent of n because only finitely many α ∈ (∞)P have a representative in (h)P . In fact, for sufficiently large n we simply have X ξα (h)eα(h) ≡ fh (ξ) . (6.26) fh (ξˆn ) = α↓(h)P

Since fh (ξ) ∈ M , this immediately shows that indeed fh (ξˆn ) ∈ M for almost all n, and hence the proof is completed. (Note that the fact that fh (ξˆn ) ultimately becomes equal to fh (ξ) holds for any chosen topology of (h)A, and hence the conclusion is indeed independent of that topology.) 6.4. Simple and semi-simple modules. The representation theory of (∞)A can now be developed by following the same steps as in the representation theory of semi-simple algebras. However, when considering modules V over (∞)A, it is natural to restrict one’s attention from the outset to continuous modules, i.e. to modules which are topological vector spaces and on which the representation of (∞)A is continuous (in particular, every element of (∞)A is represented by a continuous map). We will do so, and suppress the qualification “continuous” from now on. The one-dimensional ideals Iα are simple modules over (∞)A under the (left or right) regular representation. Our first result is now that these one-dimensional modules already provide us with all simple modules, i.e. that every simple (∞)A-module L satisfies L ∼ = Iα

(6.27)

for some α ∈ (∞)P . To show this, we first observe that if L ∼ 6 = Iα , then Iα L = 0. Namely, since Iα is an ideal of (∞)A, we have (∞)A Iα L ⊆ Iα L; thus Iα L is a submodule of L, which by the simplicity of L implies that either Iα L = L or Iα L = 0. In the former case, Iα L = L, we can find a vector y ∈ L such that the space Iα y is not zero-dimensional. Indeed, because of (∞)A Iα y ⊆ Iα y ⊆ L this space is a submodule of L, and hence by the simplicity of L it must be equal to L. It follows that the map from Iα to L defined by λ 7→ λy is surjective. Since L is simple, by Schur’s lemma this implies that it is even an isomorphism. This shows that L ∼ = Iα when Iα L = L, and hence Iα L = 0 when L 6∼ = Iα . Suppose now that L is a non-zero simple module and is not isomorphic to any Iα . Then L α Iα L = 0; since L is a continuous module, we can take the closure of this relation, so as to find that (∞)A L = 0. But we have L ⊆ (∞)A L, and hence this would imply that L = 0, which is a contradiction. Hence we learn that indeed, up to isomorphism, the ideals Iα of (∞)A exhaust all the simple modules over (∞)A. Next we consider modules V over (∞)A which can be obtained from families of simple modules. Similarly as in [17, §XVII.2] one can show that the following conditions are equivalent: (i) V is the closure of the sum of a family of simple submodules. (ii) V is the closure of the direct sum of a family of simple submodules.


667

(iii) Every closed submodule W of V is a direct summand, i.e. there exists a closed submodule W 0 such that V = W ⊕ W 0 . Any (continuous) module fulfilling these equivalent conditions will be referred to as a semi-simple module. P The equivalence of (i) -- (iii) is proven as follows. First, if V = i∈J Li is the closure 0 subset of a (not necessarily direct) P sum of simple submodules Li , denote by J a maximal 0 of J such that V := j∈J 0 Lj is a direct sum. Since the intersection of V 0 with any of the simple modules Li is a submodule of Li , the maximality of J 0 implies that i ∈ J 0 and hence in fact J 0 = J. Thus (i) implies (ii). Second, ifP W is a submodule of V , let J 00 be the maximal subset of J such that the sum W + j∈J 00 Lj is direct. Then the same arguments as before show that V = L L W ⊕ j∈J 00 Lj . If, furthermore, W is closed, then it follows that V = W ⊕ j∈J 00 Lj = L W ⊕ j∈J 00 Lj . This shows that(ii) implies (iii). 5 Third, assume that V is a non-zero module which satisfies (iii), and let v be a nonzero vector in V . The kernel of the homomorphism (∞)A → (∞)A v is a closed ideal of (∞)A, which in turn is contained in a maximal closed ideal J ⊂ (∞)A that is strictly contained in (∞) A. One then has V = J v ⊕ W and (∞)A v = J v ⊕ (W ∩ (∞)A v) with some submodule W ⊂ V . Now W ∩ (∞)A v is simple because J v is maximal in (∞)A v; thus V contains a simple submodule. Next let V 0 6= 0 be the submodule of V that is the closure of the sum of all simple submodules of V . If V 0 were not all of V , then one would have V = V 0 ⊕ V 00 with V 00 6= 0; but by the same reasoning as before, V 00 then would contain a simple submodule, in contradiction to the definition of V 0 . Thus V 0 = V , so we see that (iii) implies (i). 6.5. Arbitrary modules. With the characterization of semi-simple modules above, we are now in a position to study arbitrary modules of (∞)A, in an analogous manner as in [17, §XVII.4]. Let us first assume that W is a closed submodule of a semi-simple module V , and denote by W 0 the closure of the direct sum of all simple submodules of W . Then there is a submodule V 0 of V such that V = W 0 ⊕ V 0 . Every w ∈ W can be uniquely written as w = w0 + v 0 with w0 ∈ W 0 and v 0 ∈ V 0 . Because of v 0 = w − w0 ∈ W we thus have W = W 0 ⊕(W ∩V 0 ). The module W ∩V 0 is a closed submodule of W . If it were non-zero, it would therefore (by the same reasoning as in the proof of “(iii) → (i)” in Subsect. 6.4, contain a simple submodule, in contradiction with the definition of W . Thus we learn that W = W 0 , or in other words: Every closed submodule of a semi-simple (∞)A-module is semi-simple. Next we consider again a closed submodule W of a semi-simple module V , and investigate the quotient module V /W . There is a closed submodule W 0 such that V is the direct sum V = W ⊕ W 0 . Now the projection V → V /W induces a continuous isomorphism from W 0 to V /W . Furthermore, according to the result just obtained, W 0 is semi-simple. Thus we have shown: Every quotient module of a semi-simple (∞)A-module with respect to a closed submodule is semi-simple. It is indeed necessary to require W to be closed. Consider e.g. the case V = The submodule W is neither closed nor does it have a complement. 5

A and W =

(∞)

L α

Iα .

668


Now any arbitrary (∞)A-module can be regarded as a quotient module of a suitable free module modulo a closed submodule. Moreover, every free (∞)A-module is the closure of a direct sum of countably many copies of (∞)A and hence is a semi-simple module. The two previous results therefore imply: Every (∞)A-module is semi-simple. Finally we consider again an arbitrary (∞)A-module V . We denote by Vα the closure of the direct sum of all those submodules of V which are isomorphic to the simple (∞) A-module Iα . Since each simple module over (∞)A is isomorphic to some Iα , any simple submodule of V is contained in Vβ for some β ∈ (∞)P . Now every (∞)A-module is semi-simple and hence the closure of the direct sum of its simple submodules. Thus we learn that M Vα . (6.28) V = α∈(∞)P

Moreover, we have eβ Vα = δα,β Vα for all α, β ∈ (∞)P , and hence Vα = eα V = Iα V .

(6.29)

As a consequence, we see that: Every (∞)A-module V can be written as M M Iα V = eα V , V = α∈(∞)P

(6.30)

α∈(∞)P

and for each α ∈ (∞)P the submodule Iα V is the closure of the direct sum of all submodules of V that are isomorphic to Iα . We can conclude that the structure of any arbitrary (continuous) module over (∞)A is known explicitly, i.e. we have developed the full (topological) representation theory of (∞) A. 6.6. Diagonalization. From the definition (6.15) of eα ∈ (∞)A and the basic property (6.10) of the idempotents ea ∈ (h)A it follows that the elements eα of (∞)A are again idempotents: (6.31) eα ? eβ = δα,β eα for all α, β ∈ (∞)P . In other words, by the basis transformation from the distinguished basis (∞)B of the fusion algebra (∞)A to the basis of idempotents one diagonalizes the fusion rules of (∞)A, precisely as in the case of the algebras (h)A at finite level. Indeed, by combining the definitions (6.9) and (6.15) we can describe the transformation from (∞)B to the basis of idempotents eα explicitly. Namely, for any ψ = (ψ(h))h∈I ∈ (∞) B we have X (∞) Qψ,α eα , (6.32) ψ= α∈(∞)P

with Qψ,α :=

(∞)

S ψ(h),α(h) , S ρ,α(h)

(h)

(h)

(6.33)

where h ∈ I is a height at which α has a representative. Note that owing to the relation (6.4) the quotient (∞)Qψ,α does not depend on the particular choice of h. (This just rephrases the fact that the map D is an isomorphism.)


669

For any ψ, ψ 0 ∈ (∞)B we thus have X

S ψ(h),α(h) (h)S ψ0 (h),α(h) eα S ρ,α(h) (h)S ρ,α(h) (∞) α∈ P X X (∞) = ( Qψ,α (∞)Qψ0 ,α (∞)Q− α,χ ) χ ,

ψ ? ψ0 =

(h)

(h)

(6.34)

χ∈(∞)B α∈(∞)P

where (∞)Q− is the matrix6 for the inverse basis transformation, X (∞) − eα = Qα,ψ ψ .

(6.35)

ψ∈(∞)B

In other words, the fusion rule coefficients of the projective limit (∞)A can be written as 00

ψ Nψ,ψ 0 =

(∞)

X

Qψ,α (∞)Qψ0 ,α (∞)Q− α,ψ 00 .

(∞)

(6.36)

α∈(∞)P

This is nothing but the analogue of the Verlinde formula [7] that is valid for the fusion rule coefficients of the fusion algebras (h)A. Note that already at finite height h the two indices which label the rows and columns, respectively, of the matrix (h)S which diagonalizes the fusion rules are a priori of a rather different nature. Namely, one of them labels the elements of the distinguished basis (h) B, while the other labels the inequivalent one-dimensional irreducible representations of (h)A. It is a quite non-trivial property of the fusion algebras which arise in rational conformal field theory (and is a prerequisite for the modularity of those fusion algebras) that nevertheless the diagonalizing matrix can be chosen such that it is symmetric, so that in particular the two kinds of labels can be treated on an equal footing [1]. Our results clearly display that this nice feature of the finite height fusion algebras (h)A is not shared by their non-rational limit (∞)A; in the case of (∞)A, there seems to be no possibility to identify the two sets (∞)B and (∞)P which label the elements of the distinguished basis and the one-dimensional irreducible representations, respectively, with each other. On the other hand, our results show that the projective limit (∞)R that we constructed in this paper still possesses all those structural properties of a modular fusion ring which can reasonably be expected to survive in the limit of infinite level. Acknowledgement. We are grateful to I. Kausz and B. Pareigis for helpful comments.

Note added in proof In Eq. (6.35) we assume that there is not only a sequence of finite linear combinations of elements ψ ∈ (∞)B that converges to the element eα , but that also the corresponding coefficients converge elementwise to the entries of a matrix (∞)Q− . While the former statement follows directly from the fact that (∞)B is a topological basis, the latter should be considered as a conjecture. We thank K. Fredenhagen and K.-H. Rehren for a discussion on this issue. 6

See note added in proof.

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References 1. Fuchs, J.: Fusion rules in conformal field theory. Fortschr. Phys. 42, 1 (1994) 2. Kawai, T.: On the structure of fusion algebras. Phys. Lett. B 287, 247 (1989) 3. Ashtekar, A., Lewandowski, J.: Projective techniques and functional integration for gauge theories. J. Math. Phys. 36, 2170 (1995) 4. Bimonte, G., Ercolessi, E., Landi, G., Lizzi, F., Sparano, G., Teotonio-Sobrinho, P.: J. Geom. Phys. 20, 318 (1996) Lattices and their continuum limits 5. Kac, V.G.: Infinite-dimensional Lie Algebras. Third edition, Cambridge: Cambridge University Press, 1990 6. Fuchs, J., Schellekens, A.N., Schweigert, C.: Quasi-Galois symmetries of the modular S-matrix. Commun. Math. Phys. 176, 447 (1996) 7. Verlinde, E.: Fusion rules and modular transformations in 2D conformal field theory. Nucl. Phys. B 300 [FS22], 360 (1988) 8. Encyclopaedia of Mathematics. Dordrecht: Kluwer Academic Publishers, 1991, p. 333 9. Artin, M.: Grothendieck Topologies. Boston: Harvard University, 1962 (unpublished script) 10. Pareigis, B.: Categories and Functors. New York: Academic Press, 1970 11. Hilton P.J., Stammbach, U.: A Course in Homological Algebra. New York: Springer Verlag, 1970 12. Racah, G.: In: Istanbul Summer School on Group Theoretical Concepts and Methods in Elementary Particle Physics, Gursey, F., ed., New York: Gordon and Breach, 1964, p. 1 13. Speiser, D.: Fundamental representations of Lie groups. Helv. Phys. Acta 38, 73 (1965) 14. Walton, M.: Fusion rules in Wess--Zumino--Witten models. Nucl. Phys. B 340, 777 (1990) 15. Furlan, P., Ganchev, A.Ch., Petkova, V.B.: Quantum groups and fusion rules multiplicities. Nucl. Phys. B 343, 205 (1990) 16. Fuchs, J., van Driel, P.: WZW fusion rules, quantum groups, and the modular matrix S. Nucl. Phys. B 346, 632 (1990) 17. Lang, S.: Algebra. Reading, MA: Addison--Wesley, 1984 Communicated by R.H. Dijkgraaf

Commun. Math. Phys. 185, 671 – 688 (1997)

Communications in


A Characterization of Affine Kac-Moody Lie Algebras Bruce N. Allison1,?, , Stephen Berman2,? , Yun Gao1 , Arturo Pianzola1,? 1 2

Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1 Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Sask., Canada S7N 5E6

Received: 17 April 1996 / Accepted: 11 October 1996

Abstract: We give a new characterization of the affine Kac-Moody algebras in terms of extended affine Lie algebras. We also present new realizations of the twisted affine Kac-Moody algebras. Introduction The purpose of this paper is twofold. We present a new characterization of the affine Kac-Moody Lie algebras and then go on to give some new realizations for the twisted affine algebras. This work grew out of our study, in [AABGP], of extended affine Lie algebras (EALA’s, for short) and their root systems. EALA’s were first introduced in [H-KT] by R. Høegh-Krohn and B. Torresani (under the name irreducible quasisimple Lie algebras) as a generalization of finite dimensional simple Lie algebras and affine Kac-Moody Lie algebras. Thereafter classifications of tame EALA’s of simply-laced type (except A1 ) were carried out in [BGK] and [BGKN]. In [AABGP], we developed the basic structure theory of EALA’s; gave a satisfying picture and many classification results for their root systems; and introduced many new examples of these algebras. The characterization of affine Lie algebras proved in this paper says that a Lie algebra L over the complex field is an affine Kac-Moody Lie algebra if and only if it is a tame EALA with nullity ν = 1. Moreover the realizations we give for the twisted algebras show how they can be viewed as Lie algebras arising from some of the usual and wellknown constructions of finite dimensional simple Lie algebras in characteristic 0 studied by N. Jacobson, G. Seligman, and J. Tits, among others. Thus, for example, we are able to show that the twisted affine algebra F4(2) (using the notation from [MP]) has structure tied up with a Jordan algebra while that of G(3) 2 is connected with a Cayley algebra. ? The author gratefully acknowledges the support of the Natural Sciences and Engineering Research Council of Canada.

672

B.N. Allison, S. Berman, Y. Gao, A. Pianzola

The algebras Bl(2) (l ≥ 2), Cl(2) (l ≥ 3) and BCl(2) (l ≥ 1) are dealt with using matrix constructions. To describe our main result in more detail, we briefly recall some terminology from [AABGP]. We begin with a complex Lie algebra L which has a non-degenerate invariant symmetric bilinear form and a self-centralizing finite dimensional diagonalizable abelian subalgebra H. One then gets a root space decomposition of L and assumes that if x is an element in a root space of a non-isotropic root then ad x acts locally nilpotently on L. One also assumes that the set R of roots of L is a discrete subset of H∗ , that the set R× of non-isotropic roots is indecomposable and that there are no isolated isotropic roots. Such an algebra is called an extended affine Lie algebra. Let V be the real span of the roots. This is a positive semi-definite space and the nullity ν of R is by definition the dimension of the radical of this space. Finally let Lc be the subalgebra of L generated by the non-isotropic root spaces of L and call this the core of L. Then Lc is in fact an ideal of L, and L is called tame if the kernel of the representation ρ : L → End(Lc ) given by ρ(x) = ad x|Lc is just the center of Lc . Our main result then says that such an algebra (i.e., a tame EALA of nullity one) is an affine Kac-Moody Lie algebra. (The converse of this result follows from well known properties of affine Lie algebras.) Our proof relies on some of the general results in [AABGP] dealing with EALA’s as well as some results about the root systems involved. We think this result is interesting in its own right but note here that it also clearly identifies the affine Kac-Moody algebras within the class of tame EALA’s and hence shows how the latter algebras can be considered as natural generalizations of the former. Other characterizations of the affine Kac-Moody Lie algebras are known. For example, from the deep and beautiful work of O. Mathieu in [Ma1-3] which extends earlier work of Kac in [K3] one finds the following result. If L = ⊕n∈Z Ln is a simple Z-graded Lie algebra which is infinite dimensional in both directions and if the dimensions of Ln are uniformly bounded then either L is a Witt algebra or an affine Lie algebra. Moreover, the Witt algebra does not admit a non-degenerate invariant form. Of course, here, in this characterization, one means by affine Lie algebra the loop version, so there is no center and no degree derivation added. There are some other interesting characterizations within Kac-Moody Lie algebras. One, given in [BC], is in terms of universal enveloping algebras: If g is a symmetrizable Kac-Moody Lie algebra then its universal enveloping algebra is an Ore domain which is not Noetherian if and only if g is an affine Kac-Moody Lie algebra. Another characterization, given in [G], is in terms of the second homology group of Lie algebras: If g is a Kac-Moody Lie algebra (not necessarily symmetrizable) then the second homology group H2 (g ⊗C C[t, t−1 ]) of the Lie algebra g ⊗C C[t, t−1 ] is infinite dimensional if and only if g is affine. The paper is organized as follows. In Sect. 1 we will recall the terminology, notation and results that we need from [AABGP]. Our main result, the characterization of affine Lie algebras, is contained in Sect. 2. This depends on numerous results about root systems as well as some results about sl2 -triples which appear in our algebras. Finally in Sect. 3 we present our realizations following along the lines in [AABGP]. Throughout the paper, all algebras will be over the field C of complex numbers. By an affine Kac-Moody Lie algebra we will mean a minimally realized affine Kac-Moody Lie algebra as defined in [MP, Sects. 4.1–4.3] or, equivalently, an affine Kac-Moody Lie algebra as defined in [K2, Chapter 6]. Such a Lie algebra then has a 1-dimensional center and a derived algebra of codimension 1.

Characterization of Affine Kac-Moody Lie Algebras

673

1. Extended Affine Lie Algebras and their Root Systems In this section, we recall the definitions and facts that we will need from [AABGP]. An extended affine Lie algebra (EALA, for short) is a triple (L, (·, ·), H) consisting of two complex Lie algebras H ⊆ L and a symmetric bilinear form (·, ·) : L × L → C such that the axioms (EA1) through (EA5) described below hold. The first two axioms are (EA1)

The form (·, ·) is non-degenerate and invariant.

(EA2) H 6= (0) is finite dimensional, abelian, and self-centralizing. Moreover ad h ∈ End(L) is diagonalizable for all h ∈ H. From (EA2) we obtain the usual root space decomposition L = ⊕α∈H∗ Lα ,

(1.1)

where Lα = {x ∈ L|[h, x] = α(h)x for all h ∈ H}. The root system R = {α ∈ H∗ |Lα 6= (0)}

(1.2)

is divided into isotropic and non-isotropic roots, R = R0 ] R× where R0 = {α ∈ R|(α, α) = 0} and R× = {α ∈ R|(α, α) 6= 0}. Note that 0 ∈ R0 and that L0 = H. Reasoning along standard lines we obtain that (Lα , Lβ ) = (0)

unless α + β = 0.

(1.3)

In particular the restriction of (·, ·) to H×H is non-degenerate. This allows us to identify H with H∗ . For α ∈ H∗ let tα ∈ H be given by for all h ∈ H.

α(h) = (tα , h)

(1.4)

The map α → tα is an isomorphism and allows us to transfer the form to H∗ by setting (α, β) = (tα , tβ )

for all α, β ∈ H∗ .

(1.5)

It is easy to see that [xα , x−α ] = (xα , x−α )tα for xα ∈ Lα , x−α ∈ L−α . Thus [Lα , L−α ] = Ctα

for all α ∈ R.

(1.6)

The next axiom (EA3)

adL (x) is locally nilpotent for all x ∈ Lα and α ∈ R×

allows us to construct automorphisms of L of the form exp ad teα exp ad(−t−1 fα ) exp ad teα

(1.7)

whenever eα ∈ Lα , fα ∈ L−α , α ∈ R× , and t ∈ C× . Armed with this we obtain the following (see Theorem I.1.29 in [AABGP]):

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Theorem 1.8. Suppose that L satisfies (EA1)–(EA3). Let α ∈ R× be non-isotropic. Then • Z for all β ∈ R. (1.9) • The map 2(x, α) wα : x → x − α (1.10) (α, α) is a reflection in α of H∗ stabilizing R. • Cα ∩ R = {α, 0, −α}. (1.11) • dimC Lα = 1 and L−α ⊕ Ctα ⊕ Lα is a Lie algebra isomorphic to sl2 (C). (1.12) • For any β ∈ R there exist two non-negative integers u and d such that for n ∈ Z we have β + nα ∈ R ⇔ −d ≤ n ≤ u. (1.13) Moreover, d − u =

2(β,α) (α,α) .

Let W = WL be the subgroup of GL(H∗ ) generated by the reflections wα for α ∈ R× . W is called the Weyl group of our algebra L. The remaining axioms relate the geometry associated with the form (·, ·) to the topology of Euclidean spaces. (EA4)

R is discrete subset of H∗ .

(EA5) R× is indecomposable (i.e., R× cannot be decomposed as a disjoint union R1 ] R2 , where R1 and R2 are nonempty subsets of R× satisfying (R1 , R2 ) = {0}) and R0 has no isolated roots (i.e., given σ ∈ R0 there exists α ∈ R× such that α + σ ∈ R). Assume now that L is an EALA, or in other words that L satisfies the axioms (EA1)– (EA5). From these axioms we obtain the following crucial fact about R0 (see Proposition I.2.1 in [AABGP]): (1.14) (R, R0 ) = (0). Further if we let

V=

X

Rα.

(1.15)

α∈R

then Proposition 1.16. The form (·, ·) can be scaled so that its restriction to V is real valued and positive semidefinite. This fact was an assumption about (·, ·) made in [H-KT]. Therein the authors reported that V. Kac had conjectured that it follows from (EA1)–(EA5). This was proved in Theorem I.2.14 of [AABGP]. From now on we assume that our form (·, ·) is scaled so that (1.16) holds. At the outset we thus have (R0) V is a finite dimensional real space, (·, ·) is a positive semidefinite symmetric bilinear form on V, and R = R× ] R0 , where R× = {α ∈ R : (α, α) 6= 0} and R0 = {α ∈ R : (α, α) = 0}. Moreover it follows from the axioms (EA1)–(EA5) and Theorem 1.8 that we have: (R1) (R2)

0 ∈ R. −R = R.


675

R spans V. α ∈ R× ⇒ 2α ∈ / R. R is discrete in V. If α ∈ R× and β ∈ R, then there exist non-negative integers d and u so that (α, β) . {β + nα : n ∈ Z} ∩ R = {β − dα, ..., β + uα} and d − u = 2 (α, α) (R7) R× cannot be decomposed as a disjoint union R1 ] R2 , where R1 and R2 are nonempty subsets of R× satisfying (R1 , R2 ) = {0}. (R8) For any σ ∈ R0 , there exists α ∈ R× such that α + σ ∈ R. (R3) (R4) (R5) (R6)

The properties just listed suggest the following definition. An extended affine root system (EARS for short) is defined to be a subset R of a real vector space V with a form (·, ·) so that the axioms (R0)–(R8) hold. Thus, we have just seen that the root system of an EALA is an EARS. Assume now that R is an arbitrary EARS in a real vector space V. So in particular our discussion will apply to the root system of an EALA L. We now describe how R can be constructed from a finite root system and semilattices [AABGP, Chapter II]. Let V 0 = {x ∈ V|(x, V) = (0)}. Set V¯ = V/V 0 and let − : V → V¯ be the canonical map. Since our form is positive semidefinite we have V 0 = {α ∈ V|(α, α) = 0}

and

R0 = R ∩ V 0 .

We define the nullity of R to be the (real) dimension ν of V 0 . If R is the root system of an EALA L, ν is also called the nullity of L. Next one has that R¯ = {α|α ¯ ∈ R} is a finite irreducible (not necessarily reduced) root system [AABGP, Prop. II.2.9]. (Here we depart from [Bou] in assuming that an irreducible finite root system contains 0.) We define the finite type of R, which we ¯ If usually refer to simply as the type of R, to be the type Xl of the finite root system R. R is the root system of an EALA L, Xl is also called the type of L. We now want to lift R¯ to a root system in V. Fix a base Π = {α¯ 1 , . . . , α¯ l } for R¯ and choose α˙ i in R so that α¯˙ i = α¯ i . Let V˙ be the real span of the α˙ i ’s. Then, V = V˙ ⊕ V ◦ . ¯ If we let and − restricts to an isometry of V˙ onto V. R˙ = {α˙ ∈ V˙ : α˙ + σ ∈ R for some σ ∈ V ◦ }, then R˙ ' R¯ (isomorphism of root systems). For α˙ ∈ R˙ × = R˙ \ {0}, we define Sα˙ = {σ ∈ V 0 : α + σ ∈ R}. Then, R = R0 ∪ (

[

(α˙ + Sα˙ )).

˙× α∈ ˙ R

By means of the Weyl group one can see that Sα˙ depends only on the length of α˙ (Prop. II.2.15 in [AABGP]). Set S (respectively L, E) to be Sα˙ whenever α˙ is short (respectively long, extra-long). (Here, α˙ ∈ R˙ × is said to be short if it has minimal

676


length in R˙ × , extra long if it is twice a short root of R˙ × , and long if it is neither short nor extra long.) Then [ [ [ (α˙ + S)) ∪ ( (α˙ + L)) ∪ ( (α˙ + E)), (1.17) R = (S + S) ∪ ( ˙ sh α∈ ˙ R

˙ lg α∈ ˙ R

˙ ex α∈ ˙ R

where R˙ × = R˙ sh ] R˙ lg ] R˙ ex is the decomposition of R˙ × according to length. Note that R0 = S + S. One also knows that L ⊆ S and E ⊆ S and so S, L and E consist of isotropic roots. The key feature here is that S and L are semilattices. That is, they are subsets S of V 0 such that (S1) 0 ∈ S. (S2) −S = S. (S3) S + 2S ⊆ S. (S4) S spans V 0 . (S5) S is discrete in V 0 . As for E it is a translated semilattice (i.e., E is nonempty and (S2)–(S5) hold) which satisfies E ∩ 2S = ∅. Up to this point, we have shown how to decompose any EARS as in (1.17) using a finite root system and up to 3 semilattices or translated semilattices of isotropic roots. Conversely, we can use a finite root system and semilattices to construct EARS: Construction 1.18. Suppose that R˙ is an irreducible finite root system of type Xl in a finite dimensional real vector space V˙ with positive definite symmetric bilinear form (·, ·). We decompose the set R˙ × of nonzero elements of R˙ according to length as R˙ × = R˙ sh ] R˙ lg ] R˙ ex . Let V 0 be a finite dimensional real vector space, let V = V˙ ⊕ V 0 , and extend (·, ·) to V in such a way that (V, V 0 ) = {0}. (a) (The simply laced construction). Suppose that Xl is simply laced, i.e. Xl = Al (l ≥ 1), Dl (l ≥ 4), E6 , E7 or E8 . Suppose that S is a semilattice in V 0 . If Xl 6= A1 suppose further that S is a lattice in V 0 . Put [ R = R(Xl , S) := (S + S) ∪ ( (α˙ + S)). ˙× α∈ ˙ R

(b) (The reduced nonsimply laced construction). Suppose that Xl is reduced and nonsimply laced, i.e. Xl = Bl (l ≥ 2), Cl (l ≥ 3), F4 or G2 . Suppose that S and L are semilattices in V 0 so that L + kS ⊆ L

and S + L ⊆ S,

where k = 2 if Xl = Bl (l ≥ 2), Cl (l ≥ 3) or F4 , and k = 3 if Xl = G2 . Further, if Xl = Bl (l ≥ 3) suppose that L is a lattice, if Xl = Cl (l ≥ 3) suppose that S is a lattice, and if Xl = F4 or G2 suppose that both S and L are lattices. Put [ [ (α˙ + S)) ∪ ( (α˙ + L)). R = R(Xl , S, L) := (S + S) ∪ ( ˙ sh α∈ ˙ R

˙ lg α∈ ˙ R

(c) (The BCl construction, l ≥ 2). Suppose that Xl = BCl (l ≥ 2). Suppose that S and L are semilattices in V 0 and E is a translated semilattice in V 0 such that E ∩ 2S = ∅, and


677

L + 2S ⊆ L, S + L ⊆ S, E + 2L ⊆ E and L + E ⊆ L. If l ≥ 3, suppose further that L is a lattice. Put R = R(BCl , S, L,SE) S S ˙ + S)) ∪ ( α∈ ˙ + L)) ∪ ( α∈ ˙ + E)). := (S + S) ∪ ( α∈ ˙ sh (α ˙ lg (α ˙ ex (α ˙ R ˙ R ˙ R (d) (The BC1 construction). Suppose that Xl = BC1 . Suppose that S is a semilattice in V 0 and E is a translated semilattice in V 0 such that E ∩ 2S = ∅ and E + 4S ⊆ E and S + E ⊆ S. Put R = R(BC1 , S, E) := (S + S) ∪ (

[

(α˙ + S)) ∪ (

˙ sh α∈ ˙ R

[

(α˙ + E)).

˙ ex α∈ ˙ R

We can now state the main result on the structure of EARS (Theorem II.2.37 in [AABGP]). Theorem 1.19 Let Xl be one of the types for a finite root system. Starting from a finite root system R˙ of type Xl and up to three semilattices or translated semilattices (as indicated in the construction), Construction 1.18 produces an extended affine root system of type Xl . Conversely, any extended affine root system of type Xl is isomorphic to a root system obtained from the part of Construction 1.18 corresponding to type Xl . Using Theorem 1.19, it is easy to classify EARS of nullity 1. (This classification can also be deduced from the more general arguments in Sect. II.4 of [AABGP]. However, these more general arguments are not necessary in nullity 1.) Indeed, suppose that R is ˙ S, L an EARS of nullity 1. Then, R is obtained as in Construction 1.18 from some R, and E satisfying the assumptions in the construction. Since dim(V 0 ) = 1, any semilattice in V 0 is a lattice in V 0 (see Corollary II.1.7 of [AABGP]). So S = Zδ

(1.20)

for some 0 6= δ ∈ S. Next, if Rex 6= ∅, we have E = δ + 2Zδ.

(1.21)

Indeed, we have E+4S ⊆ E and S+E ⊆ S and so E is the union of cosets of 4S in S. But then since E ∩2S = ∅ and E +2E ⊆ E, it follows that E = (δ +4S)∪(3δ +4S) = δ +2Zδ as claimed. Also, if Rlg 6= ∅, we have L = Zpδ

(1.22)

for some integer p > 0 (since L is a lattice contained in S). Moreover, if R has type Bl (l ≥ 2), Cl (l ≥ 3), F4 or G2 , we have kS ⊆ L ⊆ S and so p = 1 or k (where k is as in Construction 1.18 (b)). On the other hand if R has type BCl (l ≥ 2), we have 2S ⊆ L ⊆ S and L + E ⊆ L which forces p = 1. Summarizing: If Rlg 6= ∅, ( 1 or 2 if R has type Bl (l ≥ 2), Cl (l ≥ 3) or F4 . (1.23) p = 1 or 3 if R has type G2 1 if R has type BCl (l ≥ 2) So R is one of the EARS in the following table:

678


Table 1.24: EARS when ν = 1 EARS R(Xl , Zδ) R(Xl , Zδ, Zδ) R(Xl , Zδ, 2Zδ) R(G2 , Zδ, 3Zδ) R(BC1 , Zδ, δ + 2Zδ) R(BCl , Zδ, Zδ, δ + 2Zδ)

Finite type Xl Al (l ≥ 1), Dl (l ≥ 4), E6 , E7 , E8 Bl (l ≥ 2), Cl (l ≥ 3), F4 , G2 Bl (l ≥ 2), Cl (l ≥ 3), F4 G2 BC1 BCl (l ≥ 2)

Affine label Xl(1) Xl(1) Xl(2) G(3) 2 BC1(2) BCl(2)

We note that in the last column of Table 1.24 we have attached to each root system R an affine label (or affine type) of the form Xl(t) , where Xl is the type of R and t is an integer between 1 and 3 called the tier number of R. (The affine labels are those used in [MP]. See the note below.) We will see in the next section that if L is a tame EALA of nullity 1 with root system R then L is isomorphic to an affine Kac-Moody Lie algebra constructed from an affine matrix A of type Xl(t) . For the convenience of the reader, we note that the correspondence between the affine labels in [MP] and the affine labels from V. Kac’ book [K2] is as follows (with the affine labels from [K2] listed second): Xl(1) ↔ Xl(1) if Xl is reduced , (2) (l ≥ 2), Bl(2) ↔ Dl+1

F4(2) ↔ E6(2) ,

(3) G(3) 2 ↔ D4

Cl(2) ↔ A(2) 2l−1 (l ≥ 3), and

BCl(2) ↔ A(2) 2l (l ≥ 1).

2. Tame EALA’s of Nullity One In this section, we obtain the characterization of affine Kac-Moody Lie algebras as tame EALA’s of nullity 1. We begin by considering EALA’s with arbitrary nullity. So let L, or more precisely (L, (·, ·), H), be an EALA with root system R and nullity ν. The subalgebra Lc of L generated by the non-isotropic subspaces Lα , α ∈ R× , is called the core of L. It is easy to see that Lc is an ideal of L. Its orthogonal complement L⊥ c = {x ∈ L|(x, Lc ) = (0)} is nothing but the centralizer of Lc . Thus L⊥ c = {x ∈ L|[x, Lc ] = (0)}.

(2.1)

L is said to be tame if L⊥ c equals the center Z(Lc ) of Lc . Equivalently L is tame if and only if L⊥ c ⊆ Lc .

(2.2)

We assume for the rest of the section that the EALA L is tame. We use the notation of Sect. 1 for the root system R. So R is decomposed as in (1.17) ˙ S, L and E. By Prop. II.1.11 of [AABGP] the group hSi generated by S is a using R, lattice in V 0 . In fact (2.3) hSi = Zδ1 ⊕ · · · ⊕ Zδν ,


679

where the elements δ1 , . . . , δν ∈ S form a basis for V 0 over R. Also, thanks to (II.2.34) and (II.2.35) of [AABGP] we have S, L, E ⊆ hSi. Furthermore, V = V˙ ⊕ V 0

(2.4)

with V˙ = ⊕li=1 Rα˙ i and V 0 = ⊕νi=1 Rδi . Let Q˙ C =

l X

Ctα˙ i

and

Q0C =

i=1

ν X

Ctδi .

i=1

P Then by (1.6) and (1.17) and the fact that Lc ∩ H = α∈R× [Lα , L−α ], one can easily show that (2.5) Lc ∩ H = Q˙ C ⊕ Q0C , Our next objective is to show that dimC H = l + 2 dimC Q0C .

(2.6)

Since (Q˙ C + Q0C , Q0C ) = (0) and the restriction of the form to both H and Q˙ C is nondegenerate, it follows that there exists a subspace D of H with the following properties: • H = Q˙ C ⊕ Q0C ⊕ D; • dim D ≥ dim Q0C ; • (Q˙ C , D) = (0). If dim D > dim Q0C then we can find x ∈ D \ {0} such that (Q˙ C ⊕ Q0C , x) = (0). By (1.3) and (2.5) we get x ∈ L⊥ / Lc . This contradicts tameness. Hence we have c and x ∈ dim D = dim Q0C and (2.6) follows. Assume now that the nullity ν of L is 1. (We are still assuming that L is tame.) Then, we have S = Zδ, E = δ + 2Zδ and L = Zpδ (2.7) as in (1.20)–(1.23). Moreover, R is one of the root systems in Table 1.24. Let Xl be the (finite) type of R and let Xl(t) be the affine label of R from Table 1.24. For convenience, from now on we write αi instead of α˙ i , i = 1, . . . , l. Then, with respect to the fixed base {α1 , . . . , αl } for R˙ there exists a unique root ξl of maximal ˙ If R˙ has roots of different lengths then in addition to ξl (which is either long height in R. or extra-long) there exists a unique short root ξs of maximal height. Note that if R˙ has type BCl (l ≥ 1), we have ξl = 2ξs . We use these maximal roots ξl and ξs to define α0 = −ξ + δ, where ξ ∈ R˙ is defined according to the affine label of R as follows: Table 2.9 Affine label Xl(1) or BCl(2) (2) G(3) 2 or Xl , Xl 6= BCl

ξ ξl = the highest root of R˙ ξs = the highest short root of R˙

The maximality of height, (1.9) and (2.7) yield

(2.8)

680


2(α0 , αi ) ∈ Z≤0 (αi , αi )

for all 1 ≤ i ≤ l.

(2.10)

Next we establish two results (2.11) and (2.12) which will eventually allow us to prove the main theorem of this section without having to appeal to the realizations of the different affine Kac-Moody Lie algebras. First of all: Pl Lemma 2.11 Let α ∈ R. Then α = i=0 ai αi for some unique a0 , a1 , . . . , al ∈ Z. Furthermore the ai ’s are all non-negative or all non-positive. Proof. Indeed if the ai ’s exist they are unique as the αi ’s are linearly independent. To establish existence, let α ∈ R. By (1.17) we can write α = α˙ + nδ for some α˙ ∈ R˙ and n ∈ Z. Substituting α by −α if necessary we may assume that n ≥ 0. Now (2.11) follows from (1.17) and the following observation about finite root systems: α˙ + ξl ∈

l X

Z≥0 αi ,

α˙ + 2ξs ∈

i=1

l X

Z≥0 αi for α˙ ∈ R˙

and

i=1

α˙ + ξs ∈

l X

Z≥0 αi for α˙ ∈ R˙ sh .

i=1

Lemma 2.12 If α ∈ R× then α = wαi , for some i, 0 ≤ i ≤ l, and some w in the subgroup of W generated by the fundamental reflections wα0 , . . . , wαl . Proof. We reason by induction on ht(α) where ht(·), is the height function defined by (2.11). We may assume that ht(α) > 1. By (1.17) we see that (α, αj ) > 0 for some 0 ≤ j ≤ l. Thus 1 ≤ ht(wαj α) < ht(α). By induction wαj α = wαi , and therefore α = wαj wαi is as desired. Next, since the αi ’s are non-isotropic, each of the 3-dimensional spaces sl(i) 2 = L−αi ⊕ [L−αi , Lαi ] ⊕ Lαi is a Lie algebra isomorphic to sl2 (C) (see (1.21)). We can thus find 3(l + 1) elements ei , hi , fi , 0 ≤ i ≤ l of L such that for all i, ei ∈ Lαi , fi ∈ L−αi , [hi , ei ] = 2ei , [ei , fi ] = h

and

[hi , fi ] = −2fi .

(2.13)

The elements hi above are unique since hi is the unique element of [Lαi , L−αi ] = Ctαi satisfying αi (hi ) = 2. By (1.4) we have hj =

2tαj . (αj , αj )

(2.14)

So by (1.9) we have αi (hj ) =

2(αi , αj ) ∈ Z. (αj , αj )

(2.15)

Using some standard facts about finite root systems, (2.10) and (2.15), one can show that the (l + 1) × (l + 1) matrix A given by


A = (Aij ),

681

Aij = αi (hj )

is an indecomposable (generalized) Cartan matrix. If we write (see Table 2.9) ξ=

l X

ni α i ,

i=1

and set n0 = 1, then for n = (n0 , n1 , · · · , nl ) we have nA = 0. Indeed, the j th entry of nA is l l X X ni Aij = A0j + ni αi (hj ) = α0 (hj ) + ξ(hj ) = δ(hj ) = 0. i=0

i=1

We have established that A is an affine Cartan matrix with null root n. It is easy to verify that A is of type Xl(t) according to the list in Chapter 3.5 of [MP]. We call A the affine Cartan matrix associated to (L, (·, ·), H). We will see that L is isomorphic to an affine Kac-Moody algebra with Cartan matrix A. Observe that in L the following familiar looking relations hold: (r1) (r2) (r3)

[hi , ej ] = Aji ej , [hi , fj ] = −Aji fj for 0 ≤ i, j ≤ l; [ei , fj ] = δij hi for 0 ≤ i, j ≤ l; (ad ej )−Aij +1 ei = 0, (ad fj )−Aij +1 fi = 0 for 0 ≤ i 6= j ≤ l.

(2.16)

Indeed, (r1) is a consequence of (2.13) and so is (r2) in the case i = j. The case i 6= j of (r2) follows from (2.11). To prove (r3), assume j 6= i. We have [fi , ej ] = 0, since / R by (2.11). Then by Lemma I.1.21 in [AABGP] we get (r3) (or one can −αi + αj ∈ use the standard sl2 theory to easily show this fact). Now let g = g(A) be the affine Kac-Moody Lie algebra constructed from the matrix A and a minimal realization of A. In the notation of [MP, Sect. 4.2], g is the Lie algebra g(A, R), where R = (h, Π, Π ∨ ) is a minimal realization of A. (See also [K2, Chapter 6], where g is denoted by g(A).) So h is a fixed Cartan subalgebra of g, Π = {α0 , α1 , . . . , αl } ⊆ h∗ is a base for the root system 1 of (g, h) and Π ∨ = {α0∨ , . . . , αl∨ } ⊆ h is a corresponding co-base. Observe that we are using the same notation αi for a root of (g, h) and for a root of (G, H), i = 0, . . . , l. We regard this Pl as an identification which we extend to an identification of the root lattice i=0 Zαi of Pl g in h∗ with i=0 Zαi in H∗ . For later use we note the following consequence of (2.12): Pl ∗ × Corollary 2.17. Let α = i=0 ai αi ∈ H . If α ∈ R , then α, when viewed as an ∗ element of h , is a real root of g. If Dg is the derived algebra of g (i.e., Dg = [g, g]) we have the following information (see Proposition 4.1.12 in [MP]): (Dg)α = gα

for all α ∈ h∗ \ {0},

(Dg)0 = Dg ∩ h = ⊕li=0 Cαi∨ .

(2.18 (2.19)

Since dim h = l + 2 (the realization being minimal) it follows from (2.18)-(2.19) that there exists d ∈ h such that (2.20) h = (⊕li=0 Cαi∨ ) ⊕ Cd, δ(d) 6= 0, where δ ∈ 1 is null,

(2.21)

682


g = Dg ⊕ Cd.

(2.22)

By (2.16), [MP, Proposition 4.3.3] and the Gaber-Kac theorem (see [MP, Theorem 4.6.4]) there exists a natural Lie algebra homomorphism ρ : Dg → L satisfying

ρ((Dg)αi ) = Lαi , ρ(αi∨ )

(2.23)

= hi ,

(2.24)

for all 0 ≤ i ≤ l. Choose D ∈ H so that αi (D) = αi (d) for all 0 ≤ i ≤ l. By (1.14) and (2.21) it follows that D∈ / Q˙ C ⊕ Q0C = ⊕li=0 Chi . Now (2.6) yields

H = (⊕li=0 Chi ) ⊕ CD.

We can now in view of (2.22) extend ρ to a Lie algebra homomorphism ρ : g → L, so that ρ(d) = D. Since h ∼ = H under ρ we conclude from (2.23) and [MP, Prop. 4.3.9] that ρ is injective. Note that ρ is graded: ρ(gα ) ⊆ Lα

for all α ∈

l X

Zαi .

(2.25)

i=0

Next we show that ρ(Dg) = Lc

and

ρ(g) = Lc + CD.

(2.26)

(The sum Lc + CD is actually direct by (2.5)). For this we must show that Lα ⊆ ρ(Dg) whenever α ∈ R× . By (2.17) α is a real root of g so gα 6= (0). Now (2.26) follows from the fact that ρ is injective, graded and dim Lα = 1. Let M = Lc + CD. This is a subalgebra of L and we know that H ⊆ M.

(2.27)

Consequently, by [MP, Prop. 2.1.1], M is a graded subalgebra: M = ⊕α∈H∗ Mα where Mα = M ∩ Lα .

(2.28)

Let us now show that M⊥ := {x ∈ L|(x, M) = (0)} = (0). Indeed, we have (as Lc ⊆ M) M⊥ ⊆ L ⊥ c = Z(Lc ) (by tameness) ⊆ H (use ρ and Z(Dg) ⊆ h [MP, Prop. 4.3.4]).

(2.29)


Now (1.3) and (2.27) yield M⊥ = {0}. Finally we can show that L = M.

683

(2.30)

By (2.28) it suffices to show that Mα = Lα for all α ∈ H∗ . Let α ∈ H∗ , and set L(α) = Lα ⊕ L−α and M(α) = Mα ⊕ M−α . Consider the canonical map

χ : L(α) → M(α)∗

given by χ(x)(y) = (x, y) for all x ∈ L(α), y ∈ M(α). By (1.3) and (2.29), χ is injective. On the other hand M(α) is finite dimensional because of (2.25) and (2.26). Thus (2.30) holds true. We have therefore proved Theorem 2.31 Let (L, (·, ·), H) be a tame extended affine Lie algebra of nullity one. Then there is a graded isomorphism L∼ = g(A), where A is the affine Cartan matrix associated to (L, (·, ·), H) and g(A) is a (minimally realized) Kac-Moody Lie algebra constructed from A. Thus we have proved the implication “⇐=” in the following characterization of affine Kac-Moody Lie algebras. The reverse implication “=⇒” follows from well known properties of affine algebras. Theorem 2.32 A Lie algebra L over C is isomorphic to an affine Kac-Moody Lie algebra if and only if L is isomorphic to a tame extended affine Lie algebra of nullity 1. 3. Constructions As we have seen in Sect. 1, the EARS of nullity 1 are: (a) (b) (c) (d) (e) (f)

R(Xl , Zδ) and R(Xl , Zδ, Zδ), where Xl is a reduced type, R(Bl , Zδ, 2Zδ) (l ≥ 2), R(Cl , Zδ, 2Zδ) (l ≥ 3), R(BC1 , Zδ, δ + 2Zδ) and R(BCl , Zδ, Zδ, δ + 2Zδ) (l ≥ 2), R(F4 , Zδ, 2Zδ) and R(G2 , Zδ, 3Zδ).

In this section, for each of the above root systems R, we describe a construction of a tame EALA with root system R. It follows from Theorem 2.31 that these Lie algebras are affine Kac-Moody Lie algebras. Also, all affine types are obtained, and so the Lie algebras that we construct are precisely the affine Kac-Moody Lie algebras. (This gives another proof of the implication “=⇒” in Theorem 2.32.) The constructions that we describe are special cases of the constructions given in Chapter III of [AABGP]. We present them here since they take on a much simpler form than in [AABGP], where EALA’s of arbitrary nullity were considered. We do not give proofs of any of the facts that we describe. The interested reader can either directly check the assertions or read the more general proofs in Chapter III of [AABGP].

684


We begin by recalling the nullity 1 case of the general P construction of EALA’s described in [AABGP, Chapter III, §1]. Assume that G = n∈Z G n is a Z-graded Lie algebra over C which possesses a nondegenerate invariant symmetric bilinear form (·, ·) and a nontrivial finite dimensional ad-diagonalizable abelian subalgebra H˙ such that the restriction of (·, ·) to H˙ is nondegenerate. Then as is usual we can transfer (·, ·) to a form ˙ Let on the dual space H˙ ∗ of H. X ˙ G= Gα˙ , where Gα˙ = {x ∈ G : [h, x] = α(h)x ˙ for all h ∈ H}, ˙∗ α∈ ˙ H

˙ and put R˙ = {α˙ ∈ H˙ ∗ : Gα˙ 6= {0}}. be the root space decomposition of G relative to H, We suppose further that the following conditions hold: P Gα˙ . • G is generated as a Lie algebra by α∈ ˙ ˙ R\{0} • The restriction of the form (·, ·) to the real space V˙ spanned by R˙ is a positive definite real valued form such that R˙ is an irreducible finite root system (including 0) in the ˙ Euclidean P space (V, (·, ·)). ˙ • Gα˙ = n∈Zn (G n ∩ Gα˙ ) for α˙ ∈ R. 0 ˙ • G ∩ Gα˙ 6= {0} for each α˙ ∈ R˙ \ {0} such that 21 α˙ ∈ / R. 0 ˙ • H = G ∩ G0 . • G n 6= {0} for at least one nonzero n ∈ Z, and • m, n ∈ Z, m + n 6= 0 =⇒ (G m , G n ) = {0}. Using this data, we can construct a tame EALA L of nullity 1. To do this, let L = G ⊕ Cc ⊕ Cd with anti-commutative product [·, ·]0 defined by [L, c]0 = {0}, [d, x]0 = nx for all x ∈ G n , and [x, y]0 = [x, y] + δm,−n (x, y)c for x ∈ G m , y ∈ G n . Next we define a form (·, ·) on L such that (·, ·) extends the form (·, ·) on G and (c, c) = (d, d) = 0,

(c, d) = 1

and

(c, G) = (d, G) = 0.

Finally, put H = H˙ ⊕ Cc ⊕ Cd. Then, it follows from [AABGP, Prop. III.1.20] that L is a tame EALA of nullity 1. Hence, by Theorem 2.31, L is an affine Kac-Moody Lie algebra. We are now ready to present, for each EARS R in the list at the beginning of this section, a construction of a tame EALA L of nullity 1 with root system R. In each case we will specify a Lie algebra G with a Z-grading, a form (·, ·) and a subalgebra H˙ as above; and use the general construction just described to construct the EALA L = G ⊕ Cc ⊕ Cd. In each of the constructions, we will use the ring S = C[t, t−1 ]

P of Laurent polynomials over C. Note that S has a natural Z-grading S = n∈Z S n , where S n = Ctn , n ∈ Z. Also we will use the linear map : S → S defined by linear extension of 1 if n = 0 n (t ) = 0 if n 6= 0.


685

Finally, in each case, at the beginning of the construction we will list the EARS R being considered, the affine type of the resulting Kac-Moody Lie algebra L and the relevant section of [AABGP, Chapter III] which contains the more general construction. (a) R = R(Xl , Zδ) or R(Xl , Zδ, Zδ), where Xl is a reduced type. Affine type = Xl(1) . [AABGP, Sect. III.1] This is the classical construction of the nontwisted affine Kac-Moody Lie algebras ([K1] and [M]). Let Xl be a reduced type, let G˙ be a finite dimensional simple Lie algebra ˙ We define a ˙ Let G = S⊗C G. of type Xl over C and let H˙ be a Cartan subalgebra of G. Z-grading on G by putting G n = S n ⊗C G˙ for n ∈ Z. The form (·, ·) on G is defined by (a⊗C x, b⊗C y) = (ab)κ(x, y) ˙ where κ is the Killing form on G. ˙ Finally, we identify H˙ = for a, b ∈ S and x, y ∈ G, ˙ 1⊗C H as a subalgebra of G. Then, applying the general construction, L = G ⊕ Cc ⊕ Cd is a tame EALA with root system R(Xl , Zδ) if Xl is simply laced and R(Xl , Zδ, Zδ) otherwise. So, by Theorem 2.31, L is an affine Kac-Moody Lie algebra of affine type Xl(1) . (b) R = R(Bl , Zδ, 2Zδ), l ≥ 2. Affine type = Bl(2) . [AABGP, Sect. III.3] Let l ≥ 2. We begin by letting   0 Il 0 0 I 0 0 0 G = {X ∈ M2l+2 (S) : G−1 X t G = −X}, where G =  l ∈ M2l+2 (S). 0 0 1 0 0 0 0 t Then, G is a Lie algebra over C under the commutator product, and G is the set of all (2l + 2) × (2l + 2)-matrices over S of the form   A S −C t −tE t  T −At −B t −tDt  , B C 0 −ta  D E a 0 where A, S, T ∈ Ml (S) = Ml×l (S), B, C, D, E ∈ M1×l (S), a ∈ S, S t = −S and T t = −T . We define a Z-grading on M2l+2 (S), and hence by restriction on G, by putting deg(tn epq ) = 2n + δp,2l+2 − δq,2l+2 for n ∈ Z and 1 ≤ p, q ≤ 2l + 2. (Here of course the elements epq are the matrix units.) The form (·, ·) on G is defined by (X, Y ) = (tr(XY )) for X, Y ∈ G. Finally the subalgebra H˙ of G is defined by l X αi (eii − el+i,l+i ) : αi ∈ C}. H˙ = { i=1

With this input, the general construction produces a tame EALA L = G ⊕ Cc ⊕ Cd with root system R(Bl , Zδ, 2Zδ). Hence, again by Theorem 2.31, L is an affine Kac-Moody Lie algebra of affine type Bl(2) .

686


(d) R = R(Cl , Zδ, 2Zδ), l ≥ 3. Affine type = Cl(2) . [AABGP, Sect. III.4]. Let l ≥ 3. We first let − be the involution of S such that t¯ = −t. We then define h i 0 Il ∈ M2l (S). G = {X ∈ M2l (S) : G−1 X¯ t G = −X, tr(X) = 0}, where G = −Il 0 G is a Lie algebra over C under the commutator product, and G is the set of all 2l × 2lmatrices over S of the form i h A S t ¯ T −A where A, S, T ∈ Ml (S), tr(A) = tr(A), S t = S and T t = T . We define a Z-grading on M2l (S), and hence by restriction on G, by putting deg(tn epq ) = n for n ∈ Z and 1 ≤ p, q ≤ 2l. The form (·, ·) and the subalgebra H˙ are defined exactly as in (b) above. This time the general construction produces a tame EALA L = G ⊕Cc⊕Cd with root system R(Cl , Zδ, 2Zδ), and therefore L is an affine Kac-Moody Lie algebra of affine type Cl(2) . (d) R = R(BCl , Zδ, δ + 2Zδ), l = 1, and R = R(BCl , Zδ, Zδ, δ + 2Zδ), l ≥ 2. Affine type = BCl(2) . [AABGP, Sect. III.3] Let l ≥ 1. Let − be the involution of S defined in (c) above, and put G = {X ∈ M2l+1 (S) : G−1 X¯ t G = −X, tr(X) = 0}, " # 0 Il 0 G = Il 0 0 ∈ M2l+1 (S). 0 0 1

where

Again G is a Lie algebra over C under the commutator product, and this time G is the set of all (2l + 1) × (2l + 1)-matrices over S of the form # " A S −C¯ t T −A¯ t −B¯ t , B C a where A, S, T ∈ Ml (S), B, C ∈ M1×l (S), a ∈ S, tr(A) − tr(A) + a = 0, S¯ t = −S and T¯ t = −T . We define a Z-grading on M2l (S), and hence by restriction on G, by putting deg(tn epq ) = n for n ∈ Z and 1 ≤ p, q ≤ 2l + 1. The form (·, ·) and the subalgebra H˙ are defined exactly as in (b) above. Then the general construction produces a tame EALA L = G ⊕ Cc ⊕ Cd with root system R(BCl , Zδ, δ + 2Zδ) if l = 1 and R(BCl , Zδ, Zδ, δ + 2Zδ) if l ≥ 2. Therefore L is an affine Kac-Moody Lie algebra of affine type BCl(2) . (e) R = R(F4 , Zδ, 2Zδ). Affine type = F4(2) . [AABGP, Sect. III.5] Let J be the 27–dimensional exceptional simple Jordan algebra over C with product denoted by · (see [S, Chapter IV]). Let T : J → C be the normalized trace on J; that is T is the usual trace normalized so that T (1) = 1. Then J = C1 ⊕ J0 , where J0 = {x ∈ J : T (x) = 0}. Further, let G˙ = [LJ , LJ ], where Lx is the left multiplication


687

operator by x ∈ J. Then, G˙ is the Lie algebra of all derivations of J and G˙ is the simple Lie algebra of type F4 over C. Put ˙ G = (Rt⊗C J0 ) ⊕ (R⊗C G), where R = C[t2 , t−2 ] in S. We define an anticommutative multiplication [ , ] on G by [at⊗C x, bt⊗C y] = abt2 ⊗C [Lx , Ly ],

[a⊗C D, bt⊗C x] = (ab)t⊗C Dy

and

[a⊗C D, b⊗C E] = ab⊗C [D, E] ˙ Then, G is a Lie algebra over C. The Z-grading for a, b ∈ R, x, y ∈ J0 and D, E ∈ G. on G is defined by deg(t2n+1 ⊗C x) = 2n + 1

and

deg(t2n ⊗C D) = 2n

˙ The form (·, ·) on G is the unique symmetric bilinear form such for x ∈ J0 and D ∈ G. ˙ that Rt⊗C J0 is orthogonal to R⊗C G, (at⊗C x, bt⊗C y) = (abt2 )T (x · y)

(a⊗C D, b⊗C [Lu , Lv ]) = (ab)T ((Du) · v)

and

for a, b ∈ R, D ∈ G˙ and x, y ∈ J0 and u, v ∈ J. Finally, we obtain a subalgebra H˙ of G ˙ Then the general construction by identifying a Cartan subalgebra H˙ of G˙ with 1⊗C H. produces a tame EALA L = G ⊕Cc⊕Cd with root system R(F4 , Zδ, 2Zδ), and therefore L is an affine Kac-Moody Lie algebra of affine type F4(2) . (f) R = R(G2 , Zδ, 3Zδ). Affine type = G(3) 2 . [AABGP, Sect. III.5] Let A be the 8-dimensional Cayley algebra over C (see [S, Chapter III]). Let T : A → C be the normalized trace on A, in which case we have A = C1 ⊕ A0 , where A0 = {x ∈ A : T (x) = 0}. Moreover, if x, y ∈ A, we have xy = T (xy)1 + x ∗ y for some unique x ∗ y ∈ A0 . Next, let G˙ = DA,A , where Dx,y = 41 (L[x,y] − R[x,y] − 3[Lx , Ry ]) for x, y ∈ J. (Here Lx and Rx denote the left and right multiplication operators by x in A.) Then G˙ is the Lie algebra of all derivations of A and G˙ is the simple Lie algebra of type G2 over C. Put ˙ G = (Rt⊗C A0 ) ⊕ (Rt2 ⊗C A0 ) ⊕ (R⊗C G), where R = C[t3 , t−3 ] in S. We define an anticommutative multiplication [ , ] on G by [at ⊗C x, bt ⊗C y] = (ab)t2 ⊗C x ∗ y,

[at2 ⊗C x, bt2 ⊗C y] = (abt3 )t ⊗C x ∗ y, a ⊗C D, bt2 ⊗C x = (ab)t2 ⊗C Dx,

[a ⊗C D, bt ⊗C x] = (ab)t ⊗C Dx, at ⊗C x, bt2 ⊗C y = abt3 ⊗C Dx,y ,

[a ⊗C D, b ⊗C E] = ab⊗C [D, E]

˙ Then, G is a Lie algebra over C. The Z-grading for a, b ∈ R, x, y ∈ A0 and D, E ∈ G. on G is defined by deg(t3n+1 ⊗C x) = 3n + 1,

deg(t3n+2 ⊗C x) = 3n + 2

and

deg(t3n ⊗C D) = 3n

˙ Next the form (·, ·) on G is the unique symmetric bilinear for x ∈ A0 and D ∈ G. ˙ Rt2 ⊗C A0 is orthogonal to form such that Rt⊗C A0 is orthogonal to Rt⊗C A0 + R⊗C G, 2 ˙ Rt ⊗C A0 + R⊗C G,

688


(at⊗C x, bt2 ⊗C y) = (abt3 )T (xy)

and

(a⊗C D, b⊗C Du,v ) = (ab)T ((Du)v)

for a, b ∈ R, D ∈ G˙ and x, y ∈ A0 and u, v ∈ A. Again, we obtain a subalgebra ˙ This time the general H˙ of G by identifying a Cartan subalgebra H˙ of G˙ with 1⊗C H. construction produces a tame EALA L = G ⊕Cc⊕Cd with root system R(G2 , Zδ, 3Zδ), and therefore L is an affine Kac-Moody Lie algebra of affine type G(3) 2 . References [AABGP] Allison, B., Azam, S., Berman, S., Gao, Y., Pianzola, A.: Extended affine Lie algebras and their root systems. Mem. Amer. Math. Soc. 126, 1–122 (1997) [BC] Berman, A., Cox, B.: Enveloping algebras and representations of toroidal Lie algebras. Pacific J. Math. 165, 239–267 (1994) [BGK] Berman, S., Gao, Y., Krylyuk, Y.: Quantum tori and the structure of elliptic quasi-simple Lie algebras. J. Funct. Anal. 135, 339–389 (1996) [BGKN] Berman, S., Gao, Y., Krylyuk, Y., Neher, E.: The alternative torus and the structure of elliptic quasi-simple Lie algebras of type A2 . Trans. Amer. Math. Soc. 347, 4315–4363 (1995) [Bou] Bourbaki, N.: Groupes et algèbres de Lie Chap. IV, V, VI. Paris: Hermann 1968 [G] Gao, Y. Central extensions of nonsymmetrizable Kac-Moody algebras over commutative algebras. Proc. Amer. Math. Soc. 121, 67–76 (1994) [H-KT] Høegh-Krohn, R., Torresani, B.: Classification and construction of quasi-simple Lie algebras. J. Funct. Anal. 89, 106–136 (1990) [K1] Kac, V.: An algebraic definition of compact Lie groups. Trudy MIEM (1968), 36–47 (Russian) [K2] Kac, V.: Infinite dimensional Lie algebras. 3rd ed., Cambridge: Cambridge University Press, 1990 [K3] Kac, V.: Simple graded Lie algebras of finite growth. Math. USSR Izv 2, 1271–1311 (1968) [Ma1] Mathieu, O.: Sur un problème of V. G. Kac: La classification de certain algèbres de Lie graduées simples. J. Algebra 102, 505–536 (1986) [Ma2] Mathieu, O.: Classification des algèbres de Lie graduées simples de croissance ≤ 1. Invent. Math. 86, 371–426 (1986) [Ma3] Mathieu, O.: Classification of simple graded Lie algebras of finite growth. Invent. Math. 108, 455–519 (1992) [M] Moody, R.V.: Euclidean Lie algebras. Canad. J. Math. 21, 1432–1454 (1969) [MP] Moody, R. V., Pianzola, A.: Lie algebras with triangular decomposition. New York: John Wiley, 1995 [S] Schafer, R.D.: An introduction to nonassociative algebras. New York: Academic Press, 1966 Communicated by T. Miwa

Commun. Math. Phys. 185, 689 – 707 (1997)

Communications in


Selfadjoint Extensions of the Neumann Laplacian in Domains with Cylindrical Outlets S.A. Nazarov1,? , M. Specovius-Neugebauer2 1 2

State Maritime Academy, Kosaya Liniya, 15-A, St.-Petersburg, 199026, Russia Universität Paderborn, 33095 Paderborn, Germany. E-mail: [email protected]

Received: 10 June 1996 / Accepted: 16 October 1996

Abstract: Let ⊂ Rn be a domain with N cylindrical outlets to infinity. The solutions of the Neumann Problem for the Poisson equation are characterized within the theory of self-adjoint extensions of the operator L. Here L is the symmetric operator associated to the problem − 1 u = f in , ∂ν u = 0 on ∂ , in weighted L2 −spaces. The results are applied to examples in the theory of continuum mechanics.

1. Preliminaries 1.1. Introduction. Let ⊂ Rn , n ≥ 2, be a domain with several cylindrical outlets Q1 , . . . , QN to infinity. We consider the Neumann problem − 1u = f

in ,

∂ν u = 0

on ∂ ,

(1.1)

where ν denotes the exterior normal vector on the boundary ∂ . A natural question is which asymptotic conditions should be prescribed at infinity for each outlet. Mechanics are often treating some conditions at infinity and investigating general properties of the posed problems without formulating them mathematically while presenting only formulas for solutions of concrete physical problems with a simple geometry (see, e.g. [16, 9]). The aim of this paper is to describe the solutions to the problem (1.1) within the theory of selfadjoint extensions of the operator L associated to the problem (1.1) in weighted L2 -spaces. In the scattering theory such appropriate selfadjoint extensions of differential operators are used, for example, to describe defects and their interactions (see, e.g. [2, 14, 5]). ? This research was supported by the DFG research group “Equations of Hydrodynamics”, Universities of Bayreuth and Paderborn.

690

S.A. Nazarov, M. Specovius-Neugebauer

The Neumann problem can be regarded as a model problem for more complicated problems arising in the theories of fluid mechanics and elasticity. Of course, if we consider − 1 : D(L) → L2 ( ), where D(L) contains all smooth functions with bounded supports and ∂ν u = 0 on ∂ , then there exists Friedrichs’ extension LF as a selfadjoint extension with the domain D(LF ) = {u ∈ H 2 ( ), ∂ν u = 0 on ∂ }. D(LF ) excludes functions with polynomial growth (even constants) in the direction of the axis of Qj , but in continuum mechanics especially those functions are of the most interest: In the theory of fluids such functions drive the fluxes, in the theory of elasticity vector polynomials describe deformations of cylinders resulting into non-trivial main vectors of forces and moments on cross-sections. Also von Neumann’s theorem on selfadjoint extensions is of less use, since it only gives an abstract definition of the extension. In this paper we outline another approach, which follow the same principle as indicated in [10, 11]. We enlarge the underlying Hilbert space by using weighted L2 -spaces with a weight µ ∼ |x|−γ for large x, γ > 0. Then the operator L0γ = −µ−2 1 with domain D(L) as above preserves the symmetry in L2 ( , µ) while the domain of the closure Lγ diminishes with increasing γ. Thus, we blow up the domain of the adjoint operator L∗γ . By calculating the factor spaces D(L∗γ )/D(Lγ ) and using general basic facts of operator theory it turns out that the problem of finding all selfadjoint extensions reduces to determining all maximal neutral subspaces of a bilinear form in RN . The fact that a nonvoid family of selfadjoint extensions appears, enables us to choose one among them which describes the behavior at infinity of a solution corresponding to a certain physical phenomenon. This paper is organized as follows: In Sect. 1.2 we introduce basic notations and function spaces. In Sect. 2 we prove existence, regularity and decay results for the problem (1.1) in weighted L2 -spaces. These results are applied in Sect. 3 to determine the domain of Lγ (Theorem 3.1) and L∗γ (Sect. 3.2). We obtain Lγ = L∗γ for γ ∈ [1, 3/2) (Theorem 3.2) and D(L∗γ )/D(Lγ ) ∼ = R2N for γ > 3/2 (Theorem 3.3). Sect. 3.3 is devoted to the representation of all selfadjoint extensions of Lγ for γ > 3/2 (Theorem 3.8 and 3.9). In Sect. 4 we give some examples and primary applications to mathematical physics. 1.2. Basic notations and function spaces. We recall some usual notations. Let G ⊂ Rn be an open nonvoid set with closure G and boundary ∂G. C0∞ (G) denotes the set of all smooth functions with compact support in G. For l ∈ N, H l (G) is the Sobolev space up consisting of all (real valued) functions in L2 (G) having all (distributional) derivatives R to the order l in L2 (G), the scalar product on L2 (G) we indicate by brackets: G vu dx = l l (G) (Hloc (G)) characterizes the set of all v with v|K is contained in H l (K) (u, v)G . Hloc for every bounded open set with K ⊂ G (K ⊂ G). We characterize the underlying domain with N cylindrical outlets in the following way. We assume that ∂ is at least of class C 2 , = 0 ∪ Q1 ∪ . . . ∪ QN , where 0 is the intersection of with a ball B(0, R0 ) of radius R0 , and Qi ∩ Qj = ∅ for i 6= j. For each outlet Qj = ωj × [1, ∞) we introduce a system of local coordinates (yj , zj ) with yj ∈ ωj , zj ∈ [1, ∞), where the cross section ωj ⊂ Rn−1 is a bounded domain with boundary ∂ωj of class C 2 at least (for n ≥ 3). In the following we omit the index j in the proofs while considering a fixed outlet. For β ∈ R, l ∈ N0 we introduce weighted Sobolev spaces Vβl ( ) on . We choose a basic weight function ρ ∈ C ∞ ( ) with ρ(x) = zj on Qj \B(0, R0 ) and define Vβl ( )

Neumann Laplacian in Domains with Cylindrical Outlets

691

as the completion of C0∞ ( ) in the norm  kv; Vβl ( )k = 

l X

1/2 kρβ−l+|α| ∂ α v; L2 ( )k2 

(1.2)

|α|=0 α1

αn

with ∂ α = ∂∂x1 · · · ∂∂xn in the usual multi-index terminology. First we observe that Vβl ( ) coincides with the space of all v ∈ L2loc ( ) such that the right hand side of (1.2) is finite. One inclusion is obvious. To see the other direction, let v ∈ L2loc ( ) be given with kv; Vβl ( )k < ∞. By using a partition of unity we may assume that supp v is contained in a single outlet Qj = Q. Let χR = χR (z) be a system of cut-off functions such that χR (z) = 0 for z > 2R, χR (z) = 1 for z < 1 and dσ χR (z) ≤ CR−σ , dz σ

0 ≤ σ ≤ l.

(1.3)

Furthermore, we set QR = ω × [R, ∞). Then 0 0 (Q)k ≤ kv; Vβ−l (QR )k → 0, kv − χR v; Vβ−l

as R → ∞,

0 because kv; Vβ−l (Q)k ≤ ∞. With the same argument we treat the derivatives in the direction of y, since ∂y χR = 0. For l ≥ α > 0 we obtain from Leibniz’ rule and (1.3) 0 k∂zα (1 − χR )v; Vβ−l+α (Q)k

≤ ≤

0 (QR )k + k∂zα v; Vβ−l+α

C

X σ

X

cβ Rσ−α kρα−σ ∂zσ v ρβ−l+σ ; L2 (QR \Q2R )k

0 0 is a suitable constant, hence for each fixed y ∈ ωj the function (vχj )(y, ·) ∈ C0∞ (1, ∞). Since β 6= 1/2, we may apply Hardy’s inequality (see, e.g., [12, p.38 and p. 127ff]) and obtain Z ∞ Z ∞ 2β 2 2 z |∂z (vχj )(y, z)| dz ≥ (β − 1/2) z 2(β−1) |(vχj )(y, z)|2 dz. (1.7) 1

1

Integrating over ωj and observing that ∂z χj has a compact support in G we get with (1.6): kv; Vβ1 ( )k ≤ C|||v|||β and the lemma is proved. 1 ˜ Remark 1.2. Let Vβ ( ) be the completion of C0∞ ( ) in ||| · |||β . From Lemma 1.1 we have Vβ1 ( ) = V˜β1 ( ) for β 6= 1/2. At β = 1/2, Hardy’s inequality takes the form Z ∞ Z ∞ dw 2 z −1 | log z|−2 |w(z)|2 dz ≤ C z dz dz 1 1 for all w ∈ C ∞ (1, ∞); hence V 1 ( ) is a proper (not closed) subspace of V˜ 1 ( ). 0

1/2

1/2

Let Hβ1 ( ) be the space of all functions v ∈ L2loc ( ) such that |||v|||β < ∞. Then for β < 1/2, Hβ1 ( ) = Vβ1 ( ): Suppose v ∈ Hβ1 ( ), multiplying v by χj we may R assume that v(y, 1) = 0 for y ∈ ωj and we obtain Qj |v|2 z 2(β−1) dz dy < ∞ by Hardy’s inequality. The other inclusion is always true due to (1.5). For β > 1/2, Vβ1 ( ) is a closed subspace of Hβ1 ( ) and the factor space 1 Hβ ( )/Vβ1 ( ) is isomorphic to RN ,which can be seen in the following way. Let v ∈ Hβ1 ( ) be fixed. We show, that for each outlet Qj there exists some constant j 0 such that kv − v∞ ; Vβ−1 (Qj )k < ∞. By using the same partition of unity as v ∞ = v∞ before and the usual mollifying procedure we may assume that v ∈ C ∞ (Q) ∩ Hβ1 (Q) R ¯ = |ω|−1 ω v(y, z) dy, where |ω| is the for some Qj = Q = ω × [1, ∞). We set v(z) d v(z) ¯ we have (n-1)- dimensional measure of ω. Then for the derivative v¯ 0 (z) = dz R R∞ 0 R R ∞ |v¯ (z)|2 z 2β dz ≤ |ω|−2 1 ω dy ω |∂z v(y, z)|2 dyz 2β dz 1 |ω|−1 k∂z v; Vβ0 (Q)k2 < ∞. Rz ¯ = limz→∞ 1 v¯ 0 (t) dt − v(1) ¯ = v∞ exists. Now we use Since β > 1/2, limz→∞ v(z) Poincaré’s inequality on H 1 (ω) for every fixed z and Hardy’s inequality for v(z) ¯ − v∞ and obtain Z Z ∞ 0 kv − v∞ ; Vβ−1 (Q)k2 = z 2(β−1) |v(y, z) − v∞ |2 dy dz ω 1 Z Z ∞ Z ∞ 2 z 2(β−1) |v(y, z) − v(z)| ¯ dy dz + |ω| z 2(β−1) |v(z) ¯ − v∞ |2 dz ≤ ω 1 1 Z Z ∞ Z ∞ 2(β−1) 2 z |∇v(y, z)| dy dz + z 2β |v¯ 0 (z)|2 dz ≤ C =

1

≤

ω

k∇v; Vβ0 (Q)k2 < ∞,

and the assertion is proved.

1


693

2. The Neumann Problem in a Domain with Cylindrical Outlets

2.1. Elementary existence results. In treating the Neumann problem (1.1) we start with some elementary existence results. Although, with a view to our aim to construct selfadjoint extensions of the operator L we only deal with homogeneous boundary data, we outline that all results of this section with appropriate modifications (of function spaces, compatibility conditions, e.g.) hold true for inhomogeneous boundary conditions. We recall the problem (1.1). We look for a function u with − 1u = f

in ,

∂ν u = 0

on ∂ ,

(2.1)

where ∂ν = ∇ · ν denotes the derivative in the direction of the exterior normal vector on ∂ . We assume for a moment that f is smooth with bounded support in . Multiplying (2.1) with η ∈ C0∞ ( ) and integrating by parts (over ) lead to (∇u, ∇η) = (f, η) .

(2.2)

1 The equality (2.2) has a sense even for u ∈ Hloc ( ) and f ∈ L2loc ( ). So we call 1 u ∈ Hloc ( ) a weak solution of (2.1) if (2.2) is satisfied for every η ∈ C0∞ ( ). In order to show the Fredholm property of the system (2.1) in V01 ( ) we add the term µ(u, η)G (G is chosen like in Sect. 1.2) on the left-hand side of (2.2) and assume that f ∈ V10 ( ) holds (which implies f ∈ L2 ( )), then we extend (2.2) to all η ∈ V01 ( ). So finally we look for u ∈ V01 ( ) such that

(∇u, ∇η) + µ(u, η)G = (f, η)

(2.3)

is valid for all η ∈ V01 ( ). We obtain the following basic solvability result: Theorem 2.1. (i) If µ > 0 then for every f ∈ V10 ( ) there exists a unique solution u of (2.3) with (2.4) ku; V01 ( )k ≤ Cµ kf ; V10 ( )k. (ii) If µ = 0 then there exists a solution of (2.3) if and only if (f, 1) = 0.

(2.5)

This solution is unique up to an additive constant. Proof. (i) According to Lemma 1.1 the left-hand side of (2.3) defines a bounded positive symmetric quadratic form on V01 ( ) while the right-hand side gives a continuous linear functional on V01 ( ) if f ∈ V10 ( ). Hence the Riesz theorem yields the assertion. (ii) If we put η = 1 and µ = 0 in (2.3) we see that (2.5) is necessary for the existence of a weak solution. Now we denote by A : V01 ( ) → V01 ( )∗ the operator associated with the form (∇p, ∇η) and by B : V01 ( ) → V01 ( )∗ the operator which assigns to p ∈ V01 ( ) the linear functional (p, ·)G . Then according to (i) the operator A + µB : V01 ( ) → V01∗ ( ) is an isomorphism for µ > 0. Since the embedding V01 ( ) → L2 (G) is compact, we gain that B is compact. Hence A is a Fredholm operator of index zero. For u ∈kerA we obtain (∇u, ∇u) = 0, hence u = const. Therefore the range of A has codimension 1 which proves that (2.5) is also sufficient for the existence of a solution.

694


2.2. The estimate for the highest derivatives. From the classical regularity theory it 2 ( ). A natural is clear that the weak solution u of Theorem 2.1 is contained in Hloc question in this context is which decay properties will be inherited from the data f to the solution and its derivatives. A partial answer is given in the following lemma. Lemma 2.2. Assume f ∈ Vβ0 ( ) and u ∈ Vβ1 ( ) is a solution of (2.1) in the sense of (2.2). Then the collection ∇2 u of all second order derivatives belongs to Vβ0 ( ) and the following estimate holds: k∇2 u; Vβ0 ( )k ≤ Ckf ; Vβ0 ( )k + k∇u; Vβ0 ( )k.

(2.6)

Proof. It is sufficient if we restrict ourselves to a single outlet Qj = Q, which we divide into portions of constant length in the following way: we set qk = {(y, z) ∈ Q : z ∈ (k, k + 1)} = ωR × (k, k + 1), qk0 = ω × (k − 1/2, k + 3/2). We define uk (y, z) = u(y, z) − |qk0 |−1 q0 u(x) dx, where |qk0 | is the n-dimensional measure of qk0 . k It is clear that − 1 uk = f

in qk0 ,

∂ ν uk = 0

on 0 k = ∂qk0 ∩ ∂Q.

Then due to the regularity theory (see, e.g. [3, Ch. 6]) kuk ; H 2 (qk )k2 ≤ C kf ; L2 (qk0 )k2 + kuk ; L2 (qk0 )k2 . Taking into account that u − uk = const, we arrive at

Since

k∇2 u; L2 (qk )k2 = k∇2 uk ; L2 (qk )k2 ≤ kuk ; H 2 (qk )k2 .

R 0 qk

uk dy dz = 0, Poincaré’s inequality leads to kuk ; L2 (qk0 )k2 ≤ Ck∇uk ; L2 (qk0 )k2 = Ck∇u; L2 (qk0 )k2 .

Thus

k∇2 u; L2 (qk )k2 ≤ C kf, L2 (qk0 )k2 + k∇u; L2 (qk0 )k2 .

(2.7)

¯ β with constants c¯, C¯ > 0 We multiply (2.7) by k 2β and note that c¯k β ≤ ρβ ≤ Ck independent of k. As a result, we obtain kρβ ∇2 u; L2 (qk )k2 ≤ C1 kρβ f ; L2 (qk0 )k2 + kρβ ∇u; L2 (qk0 )k2 where C1 is independent of k. Now we sum this inequality over k and arrive at p kρβ ∇u; L2 (Qj )k ≤ 2C1 kρβ f, L2 (Qj ) + kρβ ∇u; L2 (Qj )k .

(2.8)

The factor 2 appears, because Q is covered twice by the union of the qj0 . Finally we apply the usual local estimates to the bounded domain ∩ B(0, R), where the weight function is inessential, and gain k∇u; L2 ( ∩ B(0, R))k ≤ C kf ; L2 ( ∩ B(0, 2R))k + k∇u; L2 ( ∩ B(0, 2R))k . Together with (2.8) this leads to (2.6).


695

Remark 2.3. With the same arguments one can prove the following assertion for the higher order derivatives. For l ≥ 1, let ρβ f ∈ H l ( ), which can be easily seen to be equivalent to ∇k f ∈ Vβ0 ( ) for k = 0, . . . , l, and u as in Lemma 2.2. Then ∇k u ∈ Vβ0 ( ) for k = 2, . . . , l + 2 and we have the estimate: l+2 X

k∇k u; Vβ0 ( )k ≤ C kρβ f ; H l ( )k + k∇u; Vβ0 ( )k .

(2.9)

k=2

Corollary 2.4. (i) For f ∈ V10 ( ) fulfilling the compatibility condition (2.5) there exists a solution u of the Neumann problem with ∇u ∈ L2 ( ), hence ∇2 u ∈ L2 ( ) and (2.6) holds for β = 0. (ii) If f ∈ H l ( ) in addition, then we obtain from (2.9): (2.10) k∇u; H l+1 ( )k ≤ C kf ; H l ( )k + k∇u; L2 ( )k . Inequality (2.9), however, is not enough to describe other selfadjoint extensions than Friedrichs’ extension in L2 ( ). In order to prove more precise estimates we need another technique which will be described in the following section. 2.3. The slicing procedure. The main idea to obtain better estimates is to divide the solution of the Neumann problem into two parts, u = u¯ + u⊥ , where u⊥ can be estimated using Lemma 2.2 and u¯ drives the weaker decay properties of u in the direction of the cylinder axis in the outlet Qj . In principle this follows the method developed in [7, 13] for more complicated situations. For this purpose we fix some outlet Qj = ω × [1, ∞), 2 with and w ∈ Hloc − 1 w = f, ∂ν w(x) = 0,

x = (y, z) ∈ ω × [1, ∞), x ∈ ∂Q+ = ∂ω × [1, ∞).

(2.11) (2.12)

Moreover, we assume w = 0 for z < 1 + δ and a suitable constant δ > 0. We introduce the functions Z 1 w(y, z) dy for z ∈ (1, ∞), W (z) = |ω| ω w⊥ (y, z)

=

w(y, z) − W (z)

for

(y, z) ∈ Q,

where |ω| is the (n − 1)-dimensional measure of ω. From (2.11), the definition of W R and Gauss’ theorem applied to ω 1 y w(y, z) dy, it follows with (2.12), −|ω|

d2 W (z) = dz 2

Z

f (y, z) dy. ω

Thus we obtain the following ordinary differential equation for W : − where

d2 W (z) = F (z), dz 2 1 F (z) = |ω|

z ∈ (1, ∞),

W (1) = 0,

(2.13)

Z f (y, z) dy. ω

(2.14)

696


By subtracting (2.13) from (2.11) and using the obvious fact that ∂ν W = 0 we derive the following problem for w⊥ : − 1 w⊥ (y, z) ∂ν w⊥ (y, z)

= =

f ⊥ (y, z) = f (y, z) − F (z), 0, (y, z) ∈ ∂Q+ . Z

Moreover, we have

(y, z) ∈ Q,

w⊥ (y, z) dy = 0

(2.15) (2.16)

(2.17)

ω

for all z ∈ (1, ∞). First we estimate w⊥ . Proposition 2.5. Suppose, w⊥ ∈ Vγ1 (Q) is a weak solution of (2.15), (2.16) while f ⊥ ∈ Vβ0 (Q),

w⊥ = 0

for

z 1: 1 |∂z R(z)| ≤ max{|β|, |γ − |}εR(z), 2

z 6= t.

(2.22)

By virtue of (2.17) we can apply Poincaré’s inequality on the domain ω ⊂ Rn−1 to reach Z Z R |w⊥ |2 dy dz ≤ cω R|∇y w⊥ |2 dy dz. (2.23) Q

Q

Thus, with Hölder’s inequality, (2.22) and (2.23), we arrive at |I2 | ≤ Cε|I1 |.


697

For I3 we use Hölder’s and Young’s inequalities and gain with (2.23) : Z Z 4 ⊥ 2 |I3 | ≤ ε R |w | dx + R |f ⊥ |2 dx ε Q Q Z 4 R |f ⊥ |2 dx. ≤ cω ε|I1 | + ε Q

(2.24)

Fixing ε, ε small enough and collecting (2.21)-(2.24) we arrive at Z Z R(|∇w⊥ |2 + |w⊥ |2 ) dx ≤ C(ε, ε, ω) R|f ⊥ |2 dx, q

(2.25)

Q

where the constant does not depend on t chosen in the definition of R. Hence for fixed ε > 0 we may send t → ∞ in (2.25) if we observe that R(t, ε) tends to R(∞, ε) = (1 + ε2 z 2 )β ∼ ρβ monotonically from below. Finally we obtain Z Z (1 + ε2 z 2 )β |∇w⊥ |2 + |w⊥ |2 dx ≤ C(ε, ε, ω) (1 + ε2 z 2 )β |f ⊥ |2 dx, Q

Q

which, of course, is equivalent to (2.19).

To estimate the function W , we need some results on ordinary differential equations in weighted spaces. 2.4. Excursus on ordinary differential equations. We introduce the space Vβl = Vβl (1, ∞) of functions on the ray (1, ∞) with finite norm kv; Vβl k =

l X

kz β−l+k

k=0

1/2 dk 2 2 v; L (1, ∞)k . dz k ◦

Then C0∞ ([1, ∞)) is dense in Vβl for every l ∈ N, β ∈ R. For l = 1 let V 1β indicate the subspace of Vβ1 with v(1) = 0. With W fixed as in Sect. 2.3 we can prove the following a priori estimate. 2 ([1, ∞)) ∩ Proposition 2.6. Let σ ∈ R be an arbitrary weight exponent. If W ∈ Hloc 0 Vσ−2 is a solution of

− W 00 ≡ −

d2 W = F, dz 2

W (1) = 0

(2.26)

with F ∈ Vσ0 , then W ∈ Vσ2 and 0 k). kW ; Vσ2 k ≤ C(kF ; Vσ0 k + kW ; Vσ−2

(2.27)

Proof. The idea of the proof is well known, we repeat it for the reader’s convenience. S∞ We divide the interval [1, ∞) = k=0 Ik , Ik = [2k , 2k+1 ]. For z ∈ Ik we use the transformation ζ = 2−k z, ζ ∈ [1, 2]. We set Wk (ζ) = W (ζ2k ) and Fk (ζ) = 22k F (ζ2k ). Of course, −Wk00 = Fk . We use the interpolation inequality (2.28) kWk ; H 2 ([1, 2])k ≤ C kFk ; L2 ([1, 2])k + kWk ; L2 ([1, 2])k .

698


Here C is a constant independent of k. We multiply (2.28) by 2k(σ−2+1/2) , use the inverse transform z = 2k ζ and the fact that z ∼ 2k on Ik . Thus we obtain 0 (Ik )k , kW ; Vσ2 (Ik )k ≤ C kF ; Vσ0 (Ik )k + kW ; Vσ−2 where C is independent of k. Summing over k gives (2.27).

2

d l The next proposition describes the Fredholm properties of the operator − dz 2 in Vβ spaces. We recall first the following R ∞simple conclusion of the Cauchy-Schwarz inequality: If F ∈ Vσ0 for σ > 1/2, then 1 F (z) dz < ∞, hence

Z∞ F (t) dt = 0.

lim

z→∞

(2.29)

z

Proposition 2.7. The operator D2 defined by ◦ d2 : Vσ2 ∩ V 1σ−1 → Vσ0 , W 7→ −W 00 , (2.30) 2 dz is Fredholm if and only if σ 6= 1/2, 3/2. For σ < 1/2, the operator D2 is surjective with kernel kerD2 = {W : W = c(1−z), c ∈ R}. For 1/2 < σ < 3/2, D2 is an isomorphism, and for σ > 3/2 the map is injective with closed range Rσ , Z ∞ Rσ = {F ∈ Vσ0 : F (z)(1 − z) dz = 0}. (2.31)

−

1

Thus for σ > 1/2, σ 6= 3/2, it holds: kW ; Vσ2 k ≤ CkF ; Vσ0 k for any W ∈

Vσ2

(2.32)

00

with −W = F , W (1) = 0.

Proof. First we look at the kernel. The conditions W 00 = 0, W (1) = 0 lead to W (z) = a(z − 1) with a suitable a ∈ R, the condition W ∈ Vσ2 implies a = 0 for σ > 1/2. To RzRt see the surjectivity for σ < 1/2, we fix F ∈ Vσ0 and set W (z) = − 1 1 F (τ ) dτ dt. Applying Hardy’s inequality twice we get W ∈ Vσ2 while −W 00 = F and W (1) = 0 is The same argument remains valid for 1/2 < σ < 3/2 if we take W (z) = R∞ R zobvious. F (τ ) dτ dt and recall (2.29). 1 t It remains to show (2.31) for σ > 3/2. For V, W ∈ C0∞ ([1, ∞)) we easily see Z ∞ Z ∞ V 00 W dz = −V 0 (1)W (1) + V (1)W 0 (1) − W 00 V dz. (2.33) − 1

1

By density arguments, (2.33) is valid for all W ∈ ◦1 V σ−1

Vβ2 ,

2 V ∈ V2−β , β ∈ R. For β = σ >

and V (z) = z − 1, thus 3/2 we apply (2.33) to W ∈ Vσ2 ∩ R the condition in (2.31) 0 is necessary for f R∈ Rσ . Now let f ∈ Vσ be given with F (z)(z − 1) dz = 0. We R∞ ∞ set W (z) = − z t F (τ ) dτ dt, hence −W 00 = F . With (2.29) we may again apply Hardy’s inequality twice to see W ∈ Vσ2 . Finally we use (2.33) with V (z) = z and obtain R∞ W (1) = WR0 (1) − 1 zWR00 dz R∞ ∞ ∞ = − 1 F (z) dz + 1 F (z)z dz = 1 F (z)(z − 1) dz = 0, which proves the assertion.


699

The next result characterizes the asymptotic behavior of a solution W ∈ Vσ2 to the problem (2.26) if F ∈ Vκ0 with some κ > σ. Proposition 2.8. Let W ∈ Vσ2 be a solution of (2.26) with F ∈ Vκ0 while κ > σ. Then the following assertions hold true. (i) If σ > 3/2 or κ < 1/2 or 1/2 < σ < κ < 3/2, then W ∈ Vκ2 . (ii) If κ > 3/2 > σ > 1/2, then W (z) = c0 + W (z) with W (z) ∈ Vκ2 and |c0 | + kW ; Vκ2 k ≤ CkF ; Vκ0 k. (iii) If κ > 1/2 > σ then W (z) = C1 z + W (z) with W ∈ Vκ2 and |C1 | + kW ; Vκ2 k ≤ C kF ; Vκ0 k + kW ; L2 ([1, 2])k . Proof. R ∞Part (i) is an easy consequence of Proposition 2.7: σ > 3/2 implies also κ > 3/2 and 1 F (z)(1 − z) dz = 0, hence we obtain a solution v ∈ Vκ2 ⊂ Vσ2 which must be equal to W due to the uniqueness. A similar argument works for 3/2 > κ > σ > 1/2. For κ < 1/2 we also have σ < 1/2 and a solution V ∈ Vκ2 of (2.26). Since V ∈ Vσ2 , too, V (z) − W (z) = C(z − 1) for a suitable C ∈ R. But C(· − 1) ∈ Vκ2 hence W ∈ Vκ2 . Now let κ > 3/2 > σ > 1/2. Since F ∈ Vκ0 ⊂ Vσ0 , the unique solution W ∈ Vσ2 has the representation W (z) = W (z) − W (1), where Z ∞Z ∞ W (z) = − F (τ ) dτ dt. z

t

kW ; Vκ2 k ≤ CkF ; Vκ0 k follows again from Hardy’s inequality applied twice, and, by using Cauchy-Schwarz’ and Hardy’s inequalities it also follows: Z ∞ 1/2 Z ∞ 1/2 0 −κ+1 2 (t ) dt |W |2 t(κ−1)2 dt ≤ CkF ; Vκ0 k. |W (1)| ≤ 1

1

For 3/2 > κ > 1/2 > σ let V ∈ Vκ2 be the unique solution of (2.26) in Vκ2 . Then R2 W − V = C1 (z − 1), where C1 can be calculated by C1 = 1 [W (z) − V (z)] dz, e.g. We set W (z) = V (z) − C1 z, then (2.32) leads to the desired inequality. 3. Selfadjoint Extensions 3.1. The Neumann operator as a symmetric operator in weighted spaces. We want to use the results of the previous section, first, to characterize a densely defined closed operator Lγ related to the Neumann problem in weighted L2 −spaces. 0 ( ) provided with the scalar For γ ∈ R we define the Hilbert space Hγ = V−γ product Z ρ−2γ uv dx. hu, viγ = Then the operator L0γ defined by the differential expression L0γ u = −ρ2γ 1 u on the domain D(L0γ ) = {u ∈ C0∞ ( ) : ∂ν u = 0 on ∂ } is a densely defined symmetric operator on Hγ . Let us suppose that

700


γ ≥ 1,

γ 6= 3/2.

(3.1)

These conditions arise while applying the results of Sect. 2 to examine the closure Lγ of L0γ : For γ < 1 the operator L0γ is not closable in Hγ , since the completion of the graph of L0γ in the graph norm produces a domain which is not contained in Hγ . For γ = 3/2 we achieve a situation where Proposition 2.7 and 2.8 are not applicable, so we do not calculate the domain of Lγ in this case. We choose a sequence (un )n∈N ⊂ D(L0γ ) such that limn→∞ un = u and limn→∞ L0γ un = limn→∞ fn = f in Hγ with u, f ∈ Hγ . Obviously the differences un − um fulfil − 1 (un − um ) ∂ν (un − um )

= =

ρ−2γ (fn − fm ) 0 on ∂ .

in

Hence for each bounded subdomain K with K ⊂ we may use the apriori estimate kun − um ; H 2 ( ∩ K)k ≤ C kρ−2γ (fn − fm ); L2 ( ∩ K 0 )k + kun − um ; L2 ( ∩ K 0 )k ≤ C kfn − fm ; Hγ k + kun − um ; Hγ k , (3.2) since γ > 0. Here K 0 is a bounded domain with K ⊂ K 0 ⊂ . Thus, un converges 2 ( ). From the trace theorem we obtain ∂ν u = 0. Moreover, for every to u in Hloc η ∈ C0∞ ( ) ⊂ D(L0γ ) we have hL0γ uj , ηiγ = huj , L0γ ηiγ . Passing to the limit on both sides leads to Z Z hf, ηiγ = ρ−2γ fη dx = hu, L0γ ηiγ = − u 1 η dx, which implies − 1 u = ρ−2γ f. Collecting the arguments we have already proved: 2 D(Lγ ) ⊂ {u ∈ Hloc ( ) : ∂ν u = 0

on ∂ }.

To fill in more details, we have to describe the behaviour of u at infinity. Let {χj }j≥1 PN be the system of cut-off functions of Lemma 1.1 and χ0 = 1 − j χj . Then w = χj (un − um ) satisfies (2.11) and (2.12) with f = χj ρ−2γ (fn − fm ) − [ 1 , χj ](un − um ),

(3.3)

where [ 1 , χj ] denotes the commutator of the operators. Since it is possible to choose χj with ∂ν χj = 0 on ∂ we may also assume ∂ν w = 0 on ∂ . Furthermore, since supp[ 1 , χj ](un − um ) is compact, we have f ∈ Vγ0 (Qj ) for all j = 1, . . . , N. We introduce W, w⊥ , F and f ⊥ as in Sect. 2.3 and fix again one outlet Qj = Q. Equations (3.3) and (3.2) lead to the following estimate of f ⊥ and F : kf ⊥ ; Vγ0 (Q)k + kF ; Vγ0 (1, ∞)k ≤ C kfn − fm ; Hγ k + kun − um ; Hγ k . Since γ ≥ 1, γ 6= 3/2, (2.32) gives us kW ; Vγ2 (1, ∞)k ≤ CkF ; Vγ0 (1, ∞)k.

(3.4)


701

Moreover, Proposition 2.5 and Lemma 2.2 guarantee kργ w⊥ ; H 2 (Q)k ≤ Ckf ⊥ ; Vγ0 (Q)k.

(3.5)

Now we recall that w = χj (un − um ). Passing to the limit we see that (3.2), (3.4) and (3.5) are also valid for u which means that we have the following representation for u: u=

N X

χ j u = u0 +

j=0

with Wj ∈ Vγ2 ; Wj |Qj

N X

Wj (zj ) + wj⊥ (yj , zj )

(3.6)

j=0

u0 ∈ H 2 ( 0 ), u0 = χ0 u; Z = |ωj |−1 χj u dyj , Wj = 0 for x ∈ / suppχj ;

(3.7) (3.8)

ωj

ργ wj⊥ |Qj ∈ H 2 (Qj ), wj⊥ = χj u − Wj

(3.9)

for j = 1 . . . , N . 2 ( ) such that (3.6)-(3.9) holds. Dγ ( ) By Dγ ( ) we denote the set of all u ∈ Hloc is a Banach space provided with the norm ku; Dγ ( )k = ku0 ; H 2 ( 0 )k +

N X

kWj ; Vγ2 k +

j=1

N X

kργ wj⊥ ; H 2 (Qj )k.

(3.10)

j=1

We now have the following result on the closure Lγ of L0γ . Theorem 3.1. Let γ ∈ R fulfill (3.1). The closure Lγ of L0γ is defined by the differential expression Lγ u = −ρ2γ 1 u on the domain D(Lγ ) = {u ∈ Dγ ( ) : ∂ν u = 0 on ∂ }.

(3.11)

Proof. We have already proved that D(Lγ ) is contained in the right-hand side of (3.11). To see the other inclusion we have to show that Dγ ( ) ⊂ Hγ and that D(L0γ ) is dense in the right-hand side of (3.11) with respect to the norm (3.10). For the first step we need the condition (3.1): Let u ∈ Dγ ( ) be fixed, by definition d 0 z γ−2 Wj ∈ L2 (1, ∞), ργ wj⊥ ∈ L2 (Qj ); hence χj u ∈ Vγ−2 (Qj ). Further ργ−1 dz Wj ∈ 2

d 0 2 L2 (1, ∞), ργ ∇wj⊥ ∈ L2 (Qj ) give ∇(χj u) ∈ Vγ−1 (Qj ), and ργ dz 2 Wj ∈ L (1, ∞), γ 2 ⊥ 2 2 0 ρ ∇ wj ∈ L (Qj ) lead to ∇ (χj u) ∈ Vγ (Qj ) for j = 1, . . . , N . Summing up we get 0 ( ) = Hγ if and only if γ ≥ 1. Dγ ( ) ⊂ Vγ2 ( ), but Vγ2 ( ) ⊂ V−γ 0 To see that D(Lγ ) is dense in this set we use the same set of cut-off functions χR as in Sect. 1.2 and similar calculations, then apply the customary mollifying procedure, which finishes the proof.

3.2. The adjoint to Lγ . We are now going to calculate the adjoint operator of Lγ . Assume v ∈ Hγ and f ∈ Hγ satisfy hLγ u, viγ = hu, f iγ

for all

u ∈ D(Lγ ),

(3.12)

then v ∈ D(L∗γ ) and L∗γ v = f . Since C0∞ ( ) ⊂ D(Lγ ) using [6, Ch. II.6] we obtain once more

702

S.A. Nazarov, M. Specovius-Neugebauer 2 v ∈ Hloc ( ),

− 1 v = ρ−2γ f

in ,

∂ν v = 0

on ∂ .

(3.13)

We decompose v in the manner of (3.6), i.e. v = v0 +

N X

Vj (zj ) + vj⊥ (yj , zj ),

(3.14)

j=1

where v0 , Vj and vj⊥ are defined as in (3.7) - (3.8), and look at which decay properties for Vj and vj⊥ can be inherited from f ∈ Hγ , v ∈ Hγ . It turns out that for γ ∈ [1, 3/2), v ∈ D(Lγ ), i.e. Lγ is selfadjoint, while for γ > 3/2, L∗γ is a proper extension of Lγ . For the following calculations we recall the notations: Fj (z) = −V 00 (z),

fj⊥ = − 1 vj⊥ .

Theorem 3.2. If γ ∈ [1, 3/2) then Lγ = L∗γ , where Lγ is the operator defined in Theorem 3.1. Proof. We have to show that for v defined by (3.12), Vj ∈ Vγ2 (1, ∞) and ργ vj⊥ |Qj ∈ H 2 (Qj ), j = 1, . . . , N . The second assertion follows immediately from Proposition 2.5, since fj⊥ ∈ Vγ0 (Qj ). 0 0 2 . From Vj ∈ V−γ we obtain Vj ∈ V2−γ with Due to γ ≥ 1 we have Fj ∈ Vγ0 ⊂ V2−γ Proposition 2.6. Now we apply part (i) of Proposition 2.8 with κ = γ and σ = 2 − γ, which is possible, since for γ ∈ [1, 3/2) we have 1/2 < σ ≤ κ < 3/2. This gives us Vj ∈ Vγ2 , hence v ∈ D(Lγ ). For γ > 3/2 we may still use Proposition 2.5 and 2.6 to get ργ vj⊥ ∈ H 2 (Qj ) and 2 . However, it is not possible to gain Vj ∈ Vγ2 because now part (i) and (iii) of Vj ∈ V2−γ Proposition 2.6 have to be applied. We arrive at Vj = aj + b¯ j zj + v¯

with v¯ ∈ Vγ2 j

(3.15)

aj , b¯ j are suitable constants. Changing b¯ j and v¯ j leads to Vj = χj (aj +

bj zj ) + v˜ j |ωj |

with some v˜ j ∈ Vγ2 . Summing up we deduce the following result: Theorem 3.3. Let γ > 3/2, with Lγ the closed operator in Hγ defined in Theorem 3.1. Then the adjoint operator has the domain 2 D(L∗γ ) = {u ∈ Hloc () : u =

N X

χj (aj +

j=1

bj zj ) + u, ˜ u˜ ∈ Dγ ( ), ∂ν u = 0 on ∂ } |ωj |

and is defined by L∗γ u = −ρ2γ 1 u. Remark 3.4. Since ∂ν χj = 0 on ∂ we have u˜ ∈ D(Lγ ). Moreover, from D(Lγ ) ⊂ 2 2 ( ), we obtain D(L∗γ ) ⊂ V2−γ ( ). Vγ2 ( ) ⊂ V2−γ


703

3.3. Selfadjoint extensions of Lγ for γ > 3/2. In the following we assume γ > 3/2. We will see , by using some elementary results of operator theory, that the problem of determining all selfadjoint extensions of Lγ reduces to the problem to determine subspaces of R2N with specific properties. We recall a well-known result on closed operators in Hilbert spaces (see [1, Sec. 51], e.g.). Lemma 3.5. Let H be a Hilbert space, H × H the product space provided with the canonical product norm. Assume, L : D(L) ⊂ H → H is a densely defined closed operator with graph G(L) = {(u, Lu) : u ∈ D(H)} and adjoint L∗ . Then H × H can be decomposed into the orthogonal subspaces: H × H = U G(L) ⊕ G(L∗ ) where U : H × H → H × H is the unitary mapping with U (u, v) = (−v, u), u, v ∈ H. From Lemma 3.5 we draw the following conclusions: Corollary 3.6. Let q : D(L∗γ )× D(L∗γ ) → R be the antisymmetric bilinear form defined by q(u, v) = hL∗γ u, viγ − hu, L∗γ viγ . Then it holds q(u, v) = 0

u ∈ D(Lγ ), v ∈ D(L∗γ ).

for

(3.16)

Corollary 3.7. Let L : D(L) → Hγ be a selfadjoint extension of Lγ , then D(Lγ ) ⊂ D(L) ⊂ D(L∗γ ) and q(u, v) = 0 for all u, v ∈ D(L). From (3.16) we see that the crucial point is to determine q(u, v) on the factor space D(L∗γ )/D(Lγ ), which is isomorphic to R2N via the mapping P : D(L∗γ ) → R2N , u =

N X

χj (aj + bj

j=1

zj ) + u˜ 7→ (a, b) |ωj |

(3.17)

PN z with a = (a1 , . . . , aN ), b = (b1 , . . . , bN ). We pick up u = j=1 χj (aj + bj |ωjj | ) + u, ˜ PN zj ∗ ˆ v = j=1 χj (âj + bj |ωj | ) + v˜ ∈ D(Lγ ). With (3.16) and suppχj ∩ suppχi = ∅ for j 6= i, we calculate N X zj zj q(u, v) = ), χj (âj + bˆ j ) . q χj (aj + bj |ωj | |ωj | j=1

Now we use Green’s formula and obtain N Z X

zj zj ) 1 (χj (âj + bˆ j ) |ωj | |ωj | j=1 Qj zj zj ))(χj (âj + bˆ j ) dy dzj − 1 (χj (aj + bj |ωj | |ωj | N Z X d zj zj ) ) (aj + bj = lim (âj + bˆ j t→∞ |ω | dz |ω j j j| ωj

q(u, v) =

j=1

χj (aj + bj

704


d zj zj )(âj + bˆ j ) (aj + bj dy dzj |ωj | |ωj | zj =t N X t ˆ t ˆ bj − bj aˆ j − bj bj aj bˆ j + bj lim t→∞ |ωj | |ωj | −

=

j=1

=

ˆ a · bˆ − b · aˆ =: l((a, b), (â, b)),

(3.18)

while using the notation of (3.17), here the dot · indicates the scalar product in RN . ¯ ⊂ D(L∗γ ), and Lu ¯ = L∗γ u for u ∈ D(L). ¯ Let L¯ be any closed extension of Lγ with D(L) Then, via (3.17), L¯ is uniquely determined by a suitable subspace K ⊆ R2N , where ¯ is isomorphic to the factor space D(L)/D(L ¯ K = P(D(L)) γ ). Due to Lemma 3.5, (3.16) and (3.18), the adjoint L¯ ∗ is determined in the same manner by K ⊥ = {(a⊥ , b⊥ ) ∈ R2N , a · b⊥ − b · a⊥ = 0 for all (a, b) ∈ K}, moreover, dim K ⊥ = 2N − dim K.

(3.19)

Of course, for a symmetric extension we get K ⊂ K ⊥ . If L¯ is selfadjoint, then K = K ⊥ and dim K = N by (3.19). Hence we have proved the following theorem: Theorem 3.8. Let L : D(L) → Hγ be any closed symmetric extension of Lγ and let K = P(D(L)) ⊂ R2N be defined by (3.17). Then L is selfadjoint if and only if dim K = N and a · bˆ − b · aˆ = 0

for all

ˆ ∈ K. (a, b), (â, b)

(3.20)

From Theorem 3.8 we see that the task to find all selfadjoint extensions of Lγ is now reduced to the problem of determining all N −dimensional subspaces K ⊂ R2N which fulfill condition (3.20), which was, e.g., done in [5] (for N = 6). This leads to the final result: Theorem 3.9. Let γ > 3/2. Let L be any symmetric extension of Lγ . Then L is selfadjoint if and only if P(D(L)) has the following representation: P(D(L)) = {(a, b) = (a+ + a0 , Sa0 + b− ) : a+ ∈ N+ , a0 ∈ N0 , b− ∈ N− }, where

N− ⊕ N0 ⊕ N+ = RN

(3.21) (3.22)

is an orthogonal decomposition of RN , and S : N0 → N0 is a symmetric invertible operator. Remark 3.10. dim N = 0 is admissible for each component of the decomposition. Proof. We repeat the main idea of the proof. Let P(D(L)) admit the representation (3.21). We choose orthogonal fundamental systems for each subspace {a1 , . . . , aj+ } ⊂ N+ , {aj+ +1 , . . . , aj0 } ⊂ N0 , {aj0 +1 , . . . , aN } ⊂ N− . Then {(a1 , 0) . . . , (aj+ , 0), (aj+ +1 , S(aj+ +1 ), . . . , (aj0 , S(aj0 )), (0, aj0 +1 ), . . . , (0, aN )} is a fundamental system of P(D(L)). Obviously, (3.20) is fulfilled, since the subspaces in (3.22) are mutually orthogonal and S is symmetric, thus L is selfadjoint by Theorem 3.8.


705

To prove the necessity of the representation (3.21), let L be selfadjoint with K = P(D(L)). From Theorem 3.8, it follows dimK = N and K = {(x, y) ∈ R2N ; x · v = y · u for every (u, v) ∈ K}.

(3.23)

We set N+ = {x ∈ RN ; (x, 0) ∈ K},

N− = {y ∈ RN ; (0, y) ∈ K}.

By (3.23), N+ is perpendicular to N− in RN , moreover, regarded as subspaces of R2N in the canonical way, N+ and N− are perpendicular in R2N by definition. Let K0 be the orthogonal complement of N+ ⊕ N− in K and denote by N0 = {x ∈ RN ; (x, y) ∈ K0 } and N00 = {y ∈ RN ; (x, y) ∈ K0 }. For every element (x0 , y0 ) ∈ K0 the mapping x0 → y0 is well defined, since K0 is orthogonal to N− in R2N . Furthermore, the mapping is injective, since K0 is orthogonal to N+ , so we can define an isomorphism S from N0 onto N00 by setting y0 = Sx0 , and we arrive at: dimN0 = dimN00 = dimK0 = N − dimN+ − dimN− .

(3.24)

Let (x, y), (u, v) ∈ K be arbitrary. Then we have (x, y) = (x+ + x0 , x− + Sx0 ) and (u, y) = (u+ + u0 , u− + Su0 ) with x± , u± ∈ N± , x0 , u0 ∈ N0 . x · v = y · u implies x+ · u− + x0 · u− + x+ · Su0 + x0 · Su0 = x− · u+ + x− · u0 + Sx0 · u+ + Sx0 · u0 . (3.25) Arbitrariness of x± , x0 , u± and u0 leads to N− ⊥N0 and N+ ⊥N00 whereas N+ ⊥N0 and N− ⊥N00 by the definition of K0 . From (3.24) we obtain now RN = N+ ⊕ N0 ⊕ N− as well as RN = N+ ⊕ N00 ⊕ N− , thus N0 = N00 . Inserting x± = 0 = u± and arbitrary x0 , u0 ∈ N0 into (3.25) shows the symmetry of S, which completes the proof. 4. Examples and Applications

4.1. Friedrichs’ R extension. First we look at Friedrichs’ extension Lγ,F of Lγ . Since hLγ u, uiγ = ∇u · ∇u dx ≥ 0 for all u ∈ D(Lγ ), the domain D(Lγ,F ) is given by D(Lγ,F ) = Hγ,E ∩ D(L∗γ ), here Hγ,E is the energetic space defined as the completion of D(Lγ ) with respect to the norm ku; Hγ,E k = hLγ u, uiγ + hu, uiγ Thus, for u=

N X

1/2

= k∇u; L2 ( )k2 + ku; Vγ0 ( )k2

χj (aj + bj

j=1

1/2

.

zj ) + u, ˜ |ωj |

we have u ∈ D(Lγ,F ) if and only if bj = 0 for j = 1, . . . , N. In Theorem 3.9 this corresponds to the case N+ = RN ,

N0 = N− = {0},

which is of clear physical meaning, since this extension describes the solutions with finite energy functional, i.e. the Dirichlet integral in this case. Let us list other applications to physical problems.

706


4.2. Antiplane deformation problem in elasticity. In the theory of elasticity the antiplane problem is characterized by a single displacement component u(x1 , x2 ) normal to the (x1 , x2 )−plane and independent of the coordinate perpendicular to this plane. Hence we regard ⊂ R2 as the cross section of the isotropic homogeneous elastic cylinder Z = × R. If we assume that f (x1 , x2 ) means the volume force in the x3 direction and the boundary surfaces are free of stresses, then the Lamé system reduces to the Neumann problem −µ 1 u = f in , ∂ν u = 0 on ∂ , where µ is the shear modulus, i.e. a material constant. In the case N− = N+ = {0} and S : N0 = RN → RN is defined by a negative diagonal matrix, S = −diag{s1 , . . . , sn } we obtain D(L) = {u = u˜ +

N X j=1

aj χj (1 − sj

zj ), a ∈ R}. |ωj |

It means that a shift aj of the infinitely remoted end of the demistrip Qj leads to the force −µsj aj acting in the opposite directions. Thus, the corresponding conditions at infinity can be interpreted as an an elastic clamping (see, e.g. [12, Sec. 5.7]). 4.3. Flux conditions for ideal fluids. Imagining that v is the velocity potential in an ideal fluid then the choice N0 = N− = {0}, N+ = RN R ∂v gives one example of asymptotic conditions where the fluxes ωj ∂z dy are distributed j without fixing them in the outlets, but in accordance with the restriction that the limit constants aj of the velocity potential u vanish. We consider an example for N = 2. We assume that the outlets Q1 , Q2 are connected at a large distance from 0 . We choose 1 1 N0 = {0}, N− = R , N+ = R . 1 −1 Let L denote the corresponding selfadjoint extension and P(u) = (a1 , a2 , b1 , b2 ) for u ∈ D(L). Then Theorem 3.9 implies a1 = a2 , which may be interpreted from the physical point of view as continuity condition for the velocity potential at the inifinitely removed ends of the tubes Q1 and Q2 . In the same manner we obtain b1 + b2 = 0 which means that the total flux over the cross-sections of Q1 and Q2 is zero. References 1. Achiezer, N.I., Glazman, I.M.: Theorie der linearen Operatoren im Hilbert-Raum. Frankfurt a. Main: Harri Deutsch Verlag, 1977 2. Berezin, F.A., Fadeev, L.D.: A remark on Schrödinger’s equation with a singular potential. Dokl. Akd. Nauk SSSR 137, 1011–1014 (1961). Engl. Trans. in Soviet Math. Dokl.2, 373–375 (1961) 3. Folland, G.B., Introduction to partial differential equations. Princeton, NJ: Princeton University Press, 1976 4. Friedman, A.: Partial differential equations. New York: Academic Press, 1969 5. Karpeshina, Yu. E., Pavlov, B. S.: Zero radius interaction for the biharmonic and polyharmonic equations. Mat. Zametki. 40, 49–59 (1986) Engl. Transl. in Math. Notes 40, 528–533 (1986) 6. Lions, J. L., Magenes, E.: Nonhomogeneous boundary value problems I. Berlin: Springer Verlag, 1972


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7. Maz’ja, V. G., Plamenevski¨ı B. A.: The asymptotic behavior of solutions of differential equations in Hilbert space. Izv. Akad. Nauk SSSR Ser. Mat. 36, 1080–1133 (1972); Erratum, ibid. 37, 709–710 (1973), Engl. Transl.: Math. USSR-Izv 36, 1067–1116 (1972). 8. Morrey, C.B. jr.: Multiple integrals in the calculus of variations. Berlin: Springer Verlag, 1966 9. Loitsanskii, L.G.: Mechanics of liquids and gases. Oxford: Pergamon Press, 1966. 10. Nazarov, S.A.: Selfadjoint extensions of the operator of the Dirichlet problem in weighted function spaces. Mat. Sb. 187, 224–241 (1988). Engl. Transl. in Math. USSR-Sb. 65, 229–247 (1990) 11. Nazarov, S.A.: The methods of matched asymptotic expansions, asymptotic conditions at points, and selfadjoint extensions of differential operators. To appear in Proc. St.-Petersburg Mat. Soc. 7 (1996) 12. Nazarov, S. A., Plamenevski¨ı, B. A.: Elliptic problems in domains with piecewise smooth boundaries. Berlin: de Gruyter Verlag, 1994 13. Plamenevski¨ı, B. A.: The existence and asymptotic behavior of solutions of differential equations with unbounded operator coefficients in a Banach space. Izv. Akad. Nauk SSSR Ser. Mat. 36, 1348–1401 (1972); erratum, ibid. 37, 953 (1973). Engl. Transl.: Math. USSR-Izv 36, 1327–1379 (1972) 14. Pavlov, B.S.: The theory of extensions and explicitly-soluble models. Uspekhi Mat. Nauk 42, 99–131 (1987). Engl. Transl. in Russion Math. Surveys 42, 127–168 (1987) 15. Rofe-Beketov, F.S.: Selfadjoint extensions of differential operators in a space of vector-valued functions. Dokl. Akad. Nauk SSSR 184, 1034–1037 (1969) 16. Timoshenko, S.P., Goodier, J.N.: Theory of elasticity. New York: McGraw-Hill, 1970 Communicated by H. Araki

Commun. Math. Phys. 185, 709 – 722 (1997)

Communications in


Quantum Lie Algebras, Their Existence, Uniqueness and q -Antisymmetry Gustav W. Delius1,? , Mark D. Gould2 1 2

Department of Mathematics, King’s College London, Strand, London WC2R 2LS, Great Britain Department of Mathematics, University of Queensland, Brisbane Qld 4072, Australia

Received: 23 May 1996 / Accepted: 17 October 1996

Abstract: Quantum Lie algebras are generalizations of Lie algebras which have the quantum parameter h built into their structure. They have been defined concretely as certain submodules Lh (g) of the quantized enveloping algebras Uh (g). On them the quantum Lie product is given by the quantum adjoint action. Here we define for any finite-dimensional simple complex Lie algebra g an abstract quantum Lie algebra gh independent of any concrete realization. Its h-dependent structure constants are given in terms of inverse quantum Clebsch-Gordan coefficients. We then show that all concrete quantum Lie algebras Lh (g) are isomorphic to an abstract quantum Lie algebra gh . In this way we prove two important properties of quantum Lie algebras: 1) all quantum Lie algebras Lh (g) associated to the same g are isomorphic, 2) the quantum Lie product of any Lh (g) is q-antisymmetric. We also describe a construction of Lh (g) which establishes their existence.

1. Introduction Lie algebras play an important role in the description of many classical physical theories. This is particularly pronounced in integrable models which are described entirely in terms of Lie algebraic data. However, when quantizing a classical theory the Lie algebraic description seems to be destroyed by quantum corrections. It is conceivable that in some cases the Lie algebraic structure of the theory is deformed rather than destroyed. The quantum theory may be describable by a quantum generalization of a Lie algebra which has higher order terms in ~ built into its structure. These speculations were prompted by the beautiful structure found in affine Toda quan? Present address: Department of Physics, University of Bielefeld, Postfach 100131, 33501 Bielefeld, Germany, Fax: +49-521/106-2961

710

G.W. Delius, M.D. Gould

tum field theories [1]. They are the physical motivation for this work on quantum Lie algebras. As a preliminary step towards physical applications it is necessary to identify the natural quantum generalizations of Lie algebras and to study their properties. Quantum generalizations Uh (g) of the enveloping algebras U (g) of Lie algebras g have been known since the work of Drinfeld [2] and Jimbo [3] and they have been found to play a central role in quantum integrable models. This has lead us in [4] to define quantum Lie algebras Lh (g) as certain submodules of Uh (g), modelling the way in which ordinary Lie algebras are naturally embedded in U (g). Explicit examples of quantum Lie algebras were constructed in [4] using symbolic computer calculations, in particular for Lh (sl3 ), Lh (sl4 ), Lh (sp4 ) and Lh (G2 ). It was found empirically that in these quantum Lie algebras the quantum Lie products satisfy an intriguing generalization of the classical antisymmetry property. They are qantisymmetric. This can be exhibited already in the simple example of Lh (sl2 ). This quantum Lie algebra is spanned by three generators Xh+ , Xh− and Hh with the quantum Lie product relations − + + − Xh , Xh h = −Hh , X h , X h h = Hh , ± ± Xh± , Hh h = ∓2q ∓1 Xh± , Hh , Xh h = ±2q ±1 Xh , ± ± (1.1) Xh , Xh h = 0. [Hh , Hh ]h = 2(q − q −1 )Hh , Here q = eh is the quantum parameter. Clearly for q = 1 the above reduces to the ordinary sl2 Lie algebra. For q 6= 1 the Lie product is antisymmetric if the interchange of the factors is accompanied by q → q −1 . To convincingly establish that the quantum Lie algebras Lh (g) defined in [4] are the natural quantum generalizations of Lie algebras, three questions in particular should be answered: 1. Do the Lh (g) exist for all g? 2. Are all Lh (g) associated to the same g isomorphic? 3. Do all Lh (g) have q-antisymmetric quantum Lie products? These questions will be answered in the affirmative in this paper. The paper is organized as follows: Section 2 contains preliminaries about quantized enveloping algebras Uh (g) and defines the concept of q-conjugation. In Sect. 3 we give a new definition of quantum Lie algebras gh which is independent of any realization as submodules of Uh (g). We study the properties of the gh . In Sect. 4 we recall the definition of the quantum Lie algebras Lh (g) and then show that all Lh (g) are isomorphic to gh . It is in this way that we arrive in Theorem 1 at the answers to questions 2) and 3) above. In Sect. 5 we describe a construction for quantum Lie algebras Lh (g) for any finite-dimensional simple complex Lie algebra g, thus establishing their existence. There are many natural questions about quantum Lie algebras which we do not address in this paper. These are questions of representations, of the enveloping algebras, of exponentiation to quantum groups, of applications to physics and many more which we hope will be addressed in the future. We do not wish to reserve the term quantum Lie algebra only for the particular algebras defined in this paper. Rather we view the algebras gh and Lh (g) which are defined in Definitions 3.1 and 4.1 in terms of Uh (g) as particular examples of a more general concept of quantum Lie algebras. What a quantum Lie algebra should be in general is not yet known, i.e., there are not yet any satisfactory axioms for quantum Lie

Quantum Lie Algebras, Their Existence, Uniqueness and q-Antisymmetry

711

algebras. Finding such an axiomatic definition is an important problem. We hope that our study of the quantum Lie algebras arising from Uh (g) will help to provide the ideas needed to formulate the axioms. In particular we expect that the q-antisymmetry of the product discovered here will be an important ingredient. There has been an important earlier approach to the subject of quantum Lie algebras. It was initiated by Woronowicz in his work on bicovariant differential calculi on quantum groups [5]. He defined a quantum Lie product on the dual space to the space of left-invariant one-forms. This has been developed further by several groups [6]. These quantum Lie algebras are n2 -dimensional, where n is the dimension of the defining representation of g and thus they do not have the same dimension as the classical Lie algebra except for g = gln . It has never been shown how to project them onto quantum Lie algebras of the correct dimension. Only recently Sudbery [7] has defined quantum Lie algebras for g = sln which have the correct dimension n2 − 1. These are isomorphic to our (sln )h (0) (set s = −1, t = 0 in Proposition 3.3). Schüler and Schmüdgen [8] have defined n2 −1 dimensional quantum Lie algebras for sln using left-covariant differential calculi. In [9] we explained how our quantum Lie algebras lead to bicovariant differential calculi of the correct dimension. Up to date information on quantum Lie algebras can be found on the World Wide Web at http://www.mth.kcl.ac.uk/˜delius/q-lie.html 2. Preliminaries We recall the definition of quantized enveloping algebras Uh (g) [2, 3, 10] in order to fix our notation. Uh (g) is an algebra over C[[h]], the ring of formal power series in an indeterminate h. In applications of quantum groups in physics, the parameter h does not need to be identified with Planck’s constant. In general it will depend on a dimensionless combination of coupling constants and Planck’s constant. We use the notation q = eh . The formal power series in h form only a ring, not a field. It is not possible to divide by an element of C[[h]] unless the power series contains a term of order h0 . We will have to work with modules over this ring, rather than with vector spaces over a field as would be more familiar to physicists like ourselves. However C[[h]] is a principal ideal domain and thus many of the usual results of linear algebra continue to hold [11]. In the physics literature on quantum groups it is quite common to treat q not as an indeterminate but as a complex (or real) number. It is our opinion that in doing so, physicists lose much of the potential power of quantum groups. Keeping h as an indeterminate in the formalism will, when applied to quantum mechanical systems, lead to deeper insight. Definition 2.1. Let g be a finite-dimensional simple complex Lie algebra with symmetrizable Cartan matrix aij . The quantized enveloping algebra Uh (g) is the unital associative algebra over C[[h]] (completed in the h-adic topology) with generators 1 x+i , x− i , hi , 1 ≤ i ≤ rank(g) and relations hi hj = hj hi ,

± ± hi x± j − xj hi = ±aij xj ,

− + x+i x− j − xj xi = δij

−hi /2

± + Our x± i are related to the Xi of [10] by xi = qi Hopf-algebra structure. 1

qihi − qi−hi , qi − qi−1

(2.1) h /2

− i Xi+ and x− i = X i qi

and we use the opposite

712


X

1−aij

k=0

k

(−1)

1 − aij k

qi

k ± ± 1−aij −k (x± =0 i ) xj (xi )

i 6= j.

a are the q-binomial coefficients. We have defined qi = edi h where di are the Here b q coprime integers such that di aij is a symmetric matrix. The Hopf algebra structure of Uh (g) is given by the comultiplication ∆ : Uh (g) → ˆ denotes the tensor product over C[[h]], completed in the h-adic topol^ Uh (g) (⊗ Uh (g)⊗ ogy when necessary) defined by 2 ˆ 1 + 1⊗ ˆ hi , ∆(hi ) = hi ⊗ ∆(x± i )

=

ˆ −hi /2 x± i ⊗ qi

+

(2.2)

h /2 ˆ x± qi i ⊗ i ,

(2.3)

and the antipode S and counit defined by ∓1 ± ± S(hi ) = −hi , S(x± i ) = −qi xi , (hi ) = (xi ) = 0.

(2.4)

^ Uh (g). The adjoint action Uh (g) is quasitriangular with universal R-matrix R ∈ Uh (g)⊗ of Uh (g) on itself is given, using Sweedler’s notation [12], by (ad x) y =

X

x(1) y S(x(2) ),

x, y ∈ Uh (g).

(2.5)

If the Dynkin diagram of g has a symmetry τ which maps node i into node τ (i) then ± Uh (g) has a Hopf-algebra automorphism defined by τ (x± i ) = xτ (i) , τ (hi ) = hτ (i) . Such τ are referred to as diagram automorphisms and except for rescalings of the x± i they are the only Hopf-algebra automorphisms of Uh (g). Proposition 2.1 (Drinfel’d [13]). There exists an algebra isomorphism ϕ : Uh (g) → U (g)[[h]] such that ϕ ≡ id (mod h) and ϕ(hi ) = hi . Note. This is not a Hopf-algebra isomorphism however. Proposition 2.2. By (V µ , π µ ) denote the U (g)-representation with highest weight µ, carrier space V µ and representation map π µ . Let {(V µ , π µ )}µ∈D+ be the set of all finite-dimensional irreducible representations of U (g). D+ is the set of dominant weights. Let mµν λ denote the multiplicities in the decomposition of tensor product representations into irreducible U (g) representations Vµ⊗Vν =

M

λ mµν λ V .

(2.6)

λ∈D+

Then 1) {(V µ [[h]], π µ ◦ ϕ)}µ∈D+ is the set of all indecomposable representations of Uh (g) which are finite-dimensional, i.e., topologically free and of finite rank. Here ϕ is the isomorphism of Proposition 2.1. 2

Interchanging q and q −1 gives an alternative Hopf algebra structure, which is the one chosen in [4 ,10].


713

2) The decomposition of Uh (g) tensor product representations into indecomposable Uh (g) representations is described by the classical multiplicities mµν λ , M µν ˆ V ν [[h]] = mλ V λ [[h]]. (2.7) V µ [[h]]⊗ λ∈D+

Proof. 1) is from Drinfel’d [13]. It follows immediately from the isomorphism property of ϕ and from the fact that the finite dimensional representations of U (g) have no non-trivial deformations. 2) The decomposition can be achieved by the same method as classically. A careful analysis shows that working over C[[h]] does not lead to complications. The reason is that all expressions appearing have a non-vanishing classical term. Note. The Uh (g) modules V [[h]] are not irreducible. Their submodules are of the form c V [[h]] with c ∈ C[[h]] not invertible. In this setting Schur’s lemma takes the following form: Lemma 2.1 (Schur’s lemma). Let V [[h]] and W [[h]] be two finite-dimensional indecomposable Uh (g)-modules and let f : V [[h]] → W [[h]] be a Uh (g)-module homomorphism. Then if f 6= 0, then f = c g with c ∈ C[[h]] and g an isomorphism. A central concept in the theory of quantum Lie algebras [4] is q-conjugation which in C[[h]] maps h 7→ −h, i.e. q 7→ q −1 . Definition 2.2. (i) q-conjugation ∼: C[[h]] → C[[h]], a 7→ a˜ is the C-linear ring automorphism defined by h˜ = −h. (ii) Let M, N be C[[h]]-modules. An additive map φ : M → N is said to be q-linear if φ(λ a) = λ˜ φ(a), ∀a ∈ M, λ ∈ C[[h]]. (iii) A q-conjugation on a C[[h]] module M is a q-linear involutive map ∇ : M → M with ∇ = id (mod h). Note the analogy between the concepts of q-conjugation and complex conjugation and between q-linear maps and anti-linear maps. Remark. If M is a finite-dimensional C[[h]]-module then a q-conjugation ∇ on M is uniquely specified by giving a basis {bi } which is invariant. Then the q-conjugation takes P P ∇ the form ( i λi bi ) = λ˜ i bi . Conversely, for any q-conjugation on M there exists an invariant basis. It can be constructed from an arbitrary basis by adding correction terms order by order in h. The unique q-linear algebra automorphism ∼: Uh (g) → Uh (g) which extends qconjugation on C[[h]] by acting as the identity on the generators x± i and hi is a qconjugation on Uh (g). It exists because the relations (2.1) are invariant under q 7→ q −1 . We choose the isomorphism ϕ in Proposition 2.1 such that ∼ ◦ ϕ = ϕ ◦ ∼. This qconjugation is a coalgebra q-antiautomorphism of Uh (g), i.e., ◦ ∼=∼ ◦ , ∆ ◦ ∼=∼ ◦ ∆T and it satisfies S ◦ ∼=∼ ◦ S −1 . The map ∼ was introduced already in [13]. If in physical applications q were identified with a combination of a coupling constant and Planck’s constant, then q-conjugation would correspond to the strong-weak coupling duality3 . It has been observed in several quantum field theories, that such a duality 3 In some applications of quantum groups the relation between q and the coupling constant is not linear but exponential and then q-conjugation is not related to strong-weak duality

714


transformation can form a symmetry of the theory. Affine Toda field theories in two dimensions [1] as well as supersymmetric Yang-Mills theory in four dimensions provide examples of this phenomenon. It is thus very desirable to have an algebraic structure, in which q-conjugation is incorporated. We hope that the study of this structure will one day enhance our understanding of the origin of strong-weak coupling duality in physics. 3. Quantum Lie Algebras gh The quantized enveloping algebra Uh (g) is an infinite dimensional algebra. It is our aim to associate to it in a natural way a finite dimensional algebra which would be the quantum analog of the Lie algebra. Here our approach is based on the observation that classically a Lie algebra g is also the carrier space of the adjoint representation ad(0) of U (g). The superscript 0 is to remind us that this is the classical adjoint representation. It is defined by (ad(0) a) b = [a, b] ∀a, b ∈ g. It follows from the Jacobi identity

that

[a, [b, c]] = [[a, b], c] + [b, [a, c]] ∀a, b, c ∈ g

(3.1)

(ad(0) x) ◦ [ , ] = [ , ] ◦ (ad(0) 2 x),

(3.2)

∀x ∈ U (g),

x) = (ad ⊗ad ) ∆(x) is the tensor product representation carried by g⊗g. Equation (3.2) states that the Lie product [ , ] of g is a U (g)-module homomorphism from g ⊗ g to g. Because of Proposition 2.2 we know that g[[h]] is an indecomposable module of Uh (g). Let us denote the representation of Uh (g) on g[[h]] by ad(h) . Note that at this point there is no relation between the representation ad(h) of Uh (g) on g[[h]] and the adjoint action ad of Uh (g) on Uh (g) defined in (2.5). Generalizing the above classical observation we obtain a natural definition for a quantum Lie algebra 4 . where (ad(0) 2

(0)

(0)

ˆ g[[h]] → g[[h]] be a Uh (g)-module homomorphism Definition 3.1. Let [ , ]h : g[[h]]⊗ which satisfies [ , ]h = [ , ] (mod h). [ , ]h gives g[[h]] the structure of a non-associative algebra over C[[h]]. We call this algebra gh = (g[[h]], [ , ]h ) a quantum Lie algebra and the product [ , ]h a quantum Lie product. For each Lie algebra g this definition potentially gives many different quantum Lie algebras gh , one for each choice of the homomorphism [ , ]h . This would be unsatisfactory were it not for the fact that such a Uh (g)-module homomorphism is almost unique. Proposition 3.1. For a given g 6= sln>2 the quantum Lie algebra gh is unique (up to a rescaling of the product by an invertible element of C[[h]]). For g = sln with n > 2 there is a family of quantum Lie algebras (sln )h (χ) depending on a parameter χ ∈ C((h)) (see Proposition 3.3). Proof. The idea of the proof is simple: For g 6= sln>2 the adjoint representation appears in the tensor product of two adjoint representations with unit multiplicity. This is an ˆ g[[h]] into g[[h]] with the empirical fact. Thus the homomorphism [ , ]h from g[[h]]⊗ requirement that [ , ]h (mod h) = [ , ] is unique by the weak form of Schur’s lemma. 4 As Ding has informed us, he and Frenkel have been pursuing similar ideas for some time. See also their paper [14] in which the utility of defining algebraic structures using Uh (g) module homomorphisms is stressed.


715

In the case g = sln with n > 2 however, the adjoint representation appears with multiplicity two in the tensor product. Any module arising from a linear combination of the highest weight vectors of two adjoint modules is also an adjoint module and this leads to a one-parameter family of non-isomorphic weak quantum Lie algebras (sln )h (χ). We find it helpful to be more explicit here than necessary and to explain how the homomorphism [ , ]h is obtained from inverse Clebsch-Gordan coefficients. We begin with g 6= sln>2 and with the classical situation. Let {va } be a basis for g which contains a highest weight vector v0 , i.e., (ad(0) x+i ) v0 = 0,

(ad(0) hi ) v0 = ψ(hi )v0 , ∀i,

(3.3)

where ψ is the highest root of g. Let Pa (x− ) be the polynomials in the x− i such that va = (ad(0) Pa (x− )) v0 . The adjoint representation matrices π in this basis are defined by (3.4) (ad(0) x) va = vb π b a (x). In this paper we use the summation convention according to which repeated indices are summed over their range. g ⊗ g contains a highest weight state vˆ 0 such that + (ad(0) ˆ 0 = 0, 2 xi ) v

(ad(0) ˆ 0 = ψ(hi )vˆ 0 , ∀i. 2 hi ) v

(3.5)

For g 6= sln>2 this state is unique up to rescaling. The vectors − vˆ a = (ad(0) ˆ 0 = Ka bc vb ⊗ vc 2 Pa (x )) v

(3.6)

form a basis for g inside g ⊗ g such that ˆ a = vˆ b π b a (x) (ad(0) 2 x) v

(3.7)

with the same representation matrices π as in (3.4). Thus the map β : va 7→ vˆ a = Ka bc vb ⊗ vc

(3.8)

is a U (g)-module homomorphism β : g → g ⊗ g. The coefficients Ka bc are called the Clebsch-Gordan coefficients. g and Im(β) are irreducible modules and thus by Schur’s lemma the homomorphism β is invertible on its image. Define [ , ] : g⊗g → g to be zero on the module complement of the image of β and on the image of β define [ , ] = β −1 . Then [ , ] is the U (g) homomorphism from g ⊗ g to g, unique up to rescaling. It is the Lie product of g. On the basis it is given by [va , vb ] = fab c vc ,

where Ka bc fbc d = δa d .

(3.9)

Thus the structure constants are given by the inverse Clebsch-Gordan coefficients. ˆ g[[h]] We turn to the quantum case. Let vˆ 0 be a highest weight state inside g[[h]]⊗ satisfying the analog of (3.5) + ˆ 0 = 0, (ad(h) 2 xi ) v

(ad(h) ˆ 0 = ψ(hi )vˆ 0 , ∀i, 2 hi ) v

(3.10)

where ad(h) is the deformed adjoint representation ad(h) = ad(0) ◦ϕ and vˆ 0 (mod h) 6= 0. vˆ 0 ˆ g[[h]]. vˆ 0 must be unique up to rescalgenerates the Uh (g) module g[[h]] inside g[[h]]⊗ ˆ g[[h]]. ing, otherwise g[[h]] would appear with multiplicity greater than one in g[[h]]⊗ We construct a basis {vˆ a } as in (3.6) using Pa (x− ) ∈ Uh (g) with the same polynomials

716


Pa as in (3.6). This leads to quantum Clebsch-Gordan coefficients Ka bc (h) ∈ C[[h]]. ˆ g[[h]] as in (3.8). We obtain a Uh (g)-module homomorphism β : g[[h]] → g[[h]]⊗ β is invertible by the weak form of Schur’s lemma. A homomorphism [ , ]h : ˆ g[[h]] → g[[h]] is obtained as above (3.9) g[[h]]⊗ [va , vb ]h = fab c (h) vc ,

where Ka bc (h)fbc d (h) = δa d .

(3.11)

Up to rescaling it is the unique such homomorphism with the property that [ , ]h (mod h) 6= 0. We now turn to g = sln with n > 2 and again begin by considering the classical situation. There are two linearly independent highest weight vectors vˆ 0(+) and vˆ 0(−) in g ⊗ g which satisfy (3.5). They can be chosen so that σ vˆ 0(±) = ± vˆ 0(±) ,

(3.12)

where σ is the bilinear map acting as σ (va ⊗ vb ) = vb ⊗ va . Expressed differently, the Clebsch-Gordan coefficients Ka(±) bc defined as in (3.6) satisfy Ka(±) bc = ±Ka(±) cb . Any linear combination of vˆ 0(+) and vˆ 0(−) is a highest weight state and leads to a homomorphism as described above but clearly only vˆ 0(−) leads to an antisymmetric Lie product. In the quantum case too there are two linearly independent highest weight states satisfying (3.10). We can choose any linear combination and thus have a one-parameter ˆ vc ). We impose vˆ 0 (χ) (mod h) 6= 0 as before. In family of vˆ 0 (χ) = K0 bc (χ, h) (vb ⊗ this way we obtain the family (sln )h (χ) of quantum Lie algebras. We will give these explicitly in Proposition 3.3. Certain values for χ will lead to a q-antisymmetric quantum Lie product (see Proposition 3.5). Some important properties of g carry over immediately to gh . Define root subspaces g(α) of g by g(α) = {x ∈ g|(ad(0) hi ) x = α(hi ) x ∀i}. (3.13) g possesses a gradation g=

M

g(α) ,

[g(α) , g(β) ] ⊂ g(α+β) ,

(3.14)

α∈R∪{0}

where R is the set of non-zero roots of g. Proposition 3.2. A quantum Lie algebra gh possesses a gradation M (α) gh = g(α) [[h]], g [[h]], g(β) [[h]] h ⊂ g(α+β) [[h]].

(3.15)

α∈R∪{0}

Proof. According to Proposition 2.1 the algebra isomorphism ϕ : Uh (g) → U (g)[[h]] leaves the hi invariant and thus g(α) [[h]] = {x ∈ g[[h]]|(ad(h) hi ) x = α(hi ) x ∀i}.

(3.16)

Let Xα ∈ g(α) [[h]] and Xβ ∈ g(β) [[h]]. From the homomorphism property of [ , ]h and ˆ 1 + 1⊗ ˆ hi it follows that the coproduct ∆(hi ) = hi ⊗ h h i i (ad(h) hi ) [Xα , Xβ ]h = (ad(h) hi ) Xα , Xβ + Xα , (ad(h) hi ) Xβ h

= (α(hi ) + β(hi )) [Xα , Xβ ]h , and thus [Xα , Xβ ]h ∈ g(α+β) [[h]].

h

(3.17)


717

Choosing basis vectors Xα ∈ g(α) and Hi ∈ g(0) Proposition 3.2 implies that the quantum Lie product relations are of the form [Hi , Xα ]h = lα (Hi ) Xα , Hi , Hj h = fij k Hk , Xα , Xβ h = Nαβ Xα+β

[Xα , Hi ]h = −rα (Hi ) Xα , Xα , X−α h = gα k Hk , (3.18) 0 otherwise.

for α + β ∈ R,

This is similar in form to the Lie product relations of ordinary Lie algebras. The most important differences are 1. The structure constants are now elements of C[[h]], i.e., they depend explicitly on the quantum parameter. 2. [Hi , Hj ]h does not have to be zero. Thus the grade zero subalgebra g(0) [[h]] of gh is not abelian. We will nevertheless refer to it as the quantum Cartan subalgebra. 3. Each classical root α splits up into a “left” root lα and a “right” root rα . Classically they are forced to be equal because of the antisymmetry of the Lie product. The quantum Clebsch-Gordan coefficients which describe the homomorphism ˆ gh → gh can be calculated directly by decomposing the tensor product [ , ] h : gh ⊗ representation. This is however very tedious in general. In [15] it was done for (sln )h in an indirect way by using the R-matrix of Uq (sln ). The method is based on realizing the quantum Lie algebra as a particular submodule of Uh (g) as explained in Sect. 4. The particular submodule used in [15] gives the quantum Lie algebra (sln )h (χ = 1) but the method can be extended and gives the following result. Proposition 3.3. The parameter χ ∈ C((h)) of (sln )h (χ) is a fraction χ = t/s with s, t ∈ C[[h]] and with the restriction that (s + t)−1 ∈ C[[h]]. The Lie product relations for (sln )h (χ) are Xij , Hk h = −rij (Hk ) Xij , Hk , Xij h = lij (Hk ) Xij , Hi , Hj h = fij k Hk , Xij , Xji h = gij k Hk , Xij , Xkl h = δjk δi6=l Nijl Xil − δil δj6=k Mkij Xkj ,

(3.19)

where {Xij }i,j=1···n ∪ {Hi }i=1···n−1 is a basis and the structure constants are explicitly given by lij (Hk ) = (q 1−k δki − q −1−k δk,i−1 )(s + t q n ) −(q k−1 δkj − q k+1 δk,j−1 )(s + t q −n ), rij (Hk ) = −lji (Hk ),

(3.20) (3.21)

fij k = δij δki s (q k+1 − q −k−1 ) + t (q n+1−i − q −n−1+i ) +s δki (q + q −1 )(q n−k − q −n+k )

+δi,j−1 s δk≤i (q −k − q k ) + t δk>i (q k−n − q −k+n )

gij k Nijl

+δj,i−1 s δk≤j (q −k − q k ) + t δk>j (q k−n − q −k+n ) , (3.22) i−j k −k n−k k−n =q δk≥i − q δk≥j s q δk<j − q δk 2 possesses an automorphism which is due to the symmetry of the Dynkin diagram. It would be natural to require that this automorphism survives also at the quantum level. By inspecting the above Lie product relations we find Proposition 3.4. The quantum Lie algebra (sln )h (χ) possesses the Dynkin diagram automorphism (3.24) τ (Xij ) = −Xn+1−j,n+1−i , τ (Hi ) = Hn−i iff χ = 1. This is the reason why in [15] we focused our attention on the case of χ = 1. The most basic property of a Lie product is its antisymmetry. In quantum Lie algebras this has found an interesting generalization. Proposition 3.5. The quantum Lie product of gh for g 6= sln>2 and of (sln )h (χ) with χ˜ = χ is q-antisymmetric, i.e., there exists a q-conjugation ∇ : gh → gh consistent with the gradation (3.15) such that ∇ ∇ [a, b]∇ h = −[b , a ]h .

(3.25)

Thus, choosing the basis in (3.18) so that Xα∇ = Xα , Hi∇ = Hi , the structure constants satisfy (3.26) rα = l˜α , fij k = −f˜ji k , gα k = −g˜ −α k , Nαβ = −N˜ βα . Proof. For (sln )h the statement can be verified directly from the expressions in Proposition 3.3. For g 6= sln we use the same notation as in the proof of Proposition 3.1. The adjoint representation appears with multiplicity one in the tensor product and thus we ˆ g[[h]] satisfying know that the highest weight state vˆ 0 = K0 ab (h) va ⊗ vb in g[[h]]⊗ T ba (3.10) is unique up to rescaling. v˜ˆ 0 = K0 (−h) va ⊗ vb also satisfies the highest weight condition (3.10), T (h) (h) + ˜T + (ad(h) x ) v ˆ = (ad ⊗ ad )∆(x ) v˜ˆ 0 0 i i 2 h i = ∼ (ad(h) ⊗ ad(h) )∆T (x+i ) vˆ 0T iT h + x ) v ˆ = ∼ (ad(h) 0 i 2 = 0.

(3.27)

˜ ◦ ∼ (which follows from ∼ ◦ ϕ = ϕ ◦ ∼), that v˜ a = We used that ∼ ◦ (ad(h) x) = (ad(h) x) T T va and that ∼ ◦ ∆ = ∆ ◦ ∼. Thus vˆ 00 = 21 (vˆ 0 −v˜ˆ 0 ) is a highest weight state (proportional to vˆ 0 by uniqueness). It is non-zero because it is non-zero classically. Following a similar calculation to the above one finds that it leads to Clebsch-Gordan coefficients Ka0 bc (h) = 21 Ka bc (h) − Ka cb (−h) . These are manifestly q-antisymmetric. Following 0 c 0 c through the construction of the structure constants one finds fab (h) = −fba (−h).


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4. Quantum Lie Algebras Lh (g) Inside Uh (g) In Definition 3.1 quantum Lie algebras are defined abstractly, i.e., independently of any specific realization. In [4] quantum Lie algebras were defined as concrete objects, namely as certain submodules of the quantized enveloping algebras Uh (g). This definition is based on the observation that an ordinary Lie algebra g can be naturally viewed as a subspace of its enveloping algebra U (g) with the Lie product on this subspace given by the adjoint action of U (g). Thus it is natural to define a quantum Lie algebra as an analogous submodule of the quantized enveloping algebra Uh (g) with the quantum Lie product given by the adjoint action of Uh (g). Before we can state the precise definition we need some preliminaries. The Cartan involution θ : Uh (g) → Uh (g) is given by the same formulas as in ∓ the classical case: θ(x± i ) = xi , θ(hi ) = −hi . It is an algebra automorphism and a ˆ θ) ◦ ∆T and S ◦ θ = θ ◦ S −1 . We define coalgebra antiautomorphism, i.e., ∆ ◦ θ = (θ⊗ a tilded Cartan involution by composing the Cartan involution with q-conjugation, i.e., θ˜ =∼ ◦θ. Similarly we define a tilded antipode as S˜ =∼ ◦S. With respect to the adjoint ˜ ˜ ˜ ˜ ˜ action defined in (2.5) they satisfy (ad θ(a)) θ(b) = θ((ad a) b) and (ad S(a)) S(b) = −1 ˜ S((ad S (a)) b) for all a, b ∈ Uh (g). Definition 4.1. A quantum Lie algebra Lh (g) inside Uh (g) is a finite-dimensional indecomposable ad - submodule of Uh (g) endowed with the quantum Lie product [a, b]h = (ad a) b such that 1. Lh (g) is a deformation of g, i.e., there is an algebra isomorphism Lh (g) ∼ = g (mod h) ˜ S˜ and any diagram automorphism τ . 2. Lh (g) is invariant under θ, A weak quantum Lie algebra lh (g) is defined similarly but without the requirement 2. The existence of a Cartan involution and an antipode on Lh (g) plays an important role in the investigations into the general structure of quantum Lie algebras in [4]. In particular it allows the definition of a quantum Killing form. The invariance under the diagram automorphisms τ is less important but is clearly a natural condition to impose. It is shown in [4] that given any weak quantum Lie algebra lh (g) inside Uh (g), one can always construct a true quantum Lie algebra Lh (g) which satisfies property 2 as well. Thus this extra requirement is not too strong. We now come to the relation between the abstract quantum Lie algebras gh of Definition 3.1 and the concrete weak quantum Lie algebras lh (g) of Definition 4.1. Proposition 4.1. All weak quantum Lie algebras lh (g) inside Uh (g) are isomorphic to the quantum Lie algebra gh as algebras (or to (sln )h (χ) for some χ in the case of g = sln ). Proof. By definition lh (g) is a finite-dimensional, indecomposable Uh (g) module. Condition 1 of the definition implies that the representation of Uh (g) carried by this module is a deformation of the representation of U (g) carried by g. There is only one such deformation, namely the adjoint representation ad(h) carried by g[[h]]. Thus lh (g) is isomorphic to g[[h]] as a Uh (g) module. The identity X (ad (ad x(1) ) a) ((ad x(2) ) b) = (ad x) ((ad a) b) (4.1) can be rewritten using that, when restricted to lh (g) ⊂ Uh (g), [a, b]h = (ad a) b = (ad(h) a) b,

720


(ad(h) x) ◦ [ , ]h = [ , ]h ◦ (ad(h) ∀x ∈ U (g). (4.2) 2 x), This states that the quantum Lie product on lh (g) is a Uh (g)-module homomorphism and thus is a quantum Lie product in the sense of Definition 3.1. Remark. One should not confuse the adjoint action ad with the adjoint representation ad(h) . The adjoint action ad is defined using the coproduct and the antipode as (ad x) y = x(1) yS(x(2) )

∀x, y ∈ Uh (g).

The adjoint representation ad(h) is defined using the algebra isomorphism ϕ : Uh (g) → U (g)[[h]] of Proposition 2.1 as (ad(h) x) a = (ad(0) ϕ(x)) a

∀x ∈ Uh (g), a ∈ g[[h]].

Thus the adjoint action is determined by the h-deformed Hopf-algebra structure whereas the adjoint representation is determined by only the h-deformed algebra structure. From this point of view it is surprising that the two ever coincide. But the weak quantum Lie algebras lh (g) are exactly those embeddings of g[[h]] into Uh (g) on which ad and ad(h) coincide and we will establish their existence in the next section. Proposition 4.1 allows us to answer two important questions about the concrete quantum Lie algebras Lh (g) inside Uh (g) which were left unanswered in [4]. Theorem 1. Given any finite-dimensional simple complex Lie algebra g. 1. All quantum Lie algebras Lh (g) are isomorphic as algebras. 2. All quantum Lie algebras Lh (g) have q-antisymmetric Lie products. Proof. 1. For g 6= sln>2 this is obvious from Proposition 4.1 and the uniqueness of gh according to Proposition 3.1. For g = sln>2 the requirement of τ -invariance in Definition 4.1 implies through Proposition 3.4 that Lh (sln ) can be isomorphic only to (sln )h (χ = 1). 2. This is obvious because gh and (sln )h (χ = 1) have q-antisymmetric Lie products according to Proposition 3.5. 5. Construction of Quantum Lie Algebras Lh (g) There is a general method for the construction of weak quantum Lie algebras lh (g) and quantum Lie algebras Lh (g) inside Uh (g). The method was presented in [15] for g = sln but it works for any finite-dimensional simple complex Lie algebra g as we will discuss here. We begin with a lemma giving a construction of ad-submodules of Uh (g). ^ Uh (g) satisfying A ∆(x) = ∆(x) A, ∀x ∈ Lemma 5.1. Let A be any element of Uh (g)⊗ Uh (g). Let V [[h]] be any finite-dimensional indecomposable Uh (g) module and let πij be the corresponding representation matrices. Then the elements Aij = (πij ⊗ id) A ∈ Uh (g)

(5.1)

span an ad-submodule of Uh (g) which is isomorphic to a submodule of ˆ V [[h]], i.e., V [[h]]∗ ⊗ ∗ (ad x) Aij = Akl πki (x(1) ) πlj (x(2) ), ∀x ∈ Uh (g).

(5.2)

∗

Here π denotes the dual (contragredient) representation to π defined by ∗ πki (x) = πik (S(x)).

(5.3)


721

Proof. We first calculate x Aij = πij ⊗ id (1 ⊗ x) A = πij ⊗ id (S(x(1) ) ⊗ 1) A (x(2) ⊗ x(3) ) = πik (S(x(1) )) Akl πlj (x(2) ) x(3) .

(5.4)

Then, using (5.3) (ad x) Aij = x(1) Aij S(x(2) ) ∗ (x(1) ) πlj (x(2) ) x(3) S(x(4) ) = Akl πki ∗ (x(1) ) πlj (x(2) ) = Akl πki

(5.5)

This lemma can be applied to construct weak quantum Lie algebras. Proposition 5.1. Let A = h−1 RT R − 1 , where R is the universal R-matrix of Uh (g) P T and RP the same with the tensor factors interchanged (i.e., if R = ai ⊗ bi then RT = bi ⊗ ai ). Let {ei } be a basis for the Uh (g) module V [[h]] and let πij be the corresponding representation matrices. Choose a basis {va } for the adjoint representation ˆ V [[h]], va 7→ vˆ a = Ka ij (e∗i ⊗ ej ) be a g[[h]] of Uh (g) and let K : g[[h]] → V [[h]]∗ ⊗ Uh (g)-module homomorphism, i.e., the Ka ij are quantum Clebsch-Gordan coefficients. Then the elements (5.6) Aa = Ka ij πij ⊗ id A ∈ Uh (g) span a weak quantum Lie algebra lh (g) = spanC[[h]] {Aa }. Proof. The expression A = h−1 RT R − 1 is well defined because R = 1 (mod h). It follows from the defining property R ∆(x) = ∆T (x) R ∀x ∈ Uh (g) of the R-matrix that A ∆(x) = ∆(x) A, ∀x ∈ Uh (g). It is then clear from Lemma 5.1 that the Aa span an adsubmodule of Uh (g). It follows from the definition of the Clebsch-Gordan coefficients Ka ij that this ad-submodule is either isomorphic to the adjoint representation or zero. R satisfies R = 1 + h r + O(h2 ), where r ∈ g ⊗ g is the classical r-matrix. Thus A = r +rT (mod h) ∈ g⊗g and Aa (mod h) ∈ g. It follows that spanC[[h]] {Aa } = g (mod h). Using the fact, established in [4], that given a weak quantum Lie algebra lh (g) one can always construct a true quantum Lie algebra Lh (g), we arrive at the announced existence result. Theorem 2. For any finite-dimensional simple complex Lie algebra g there exists at least one quantum Lie algebra Lh (g) inside Uh (g). Acknowledgement. We thank Andrew Pressley, Vyjayanthi Chari, Manfred Scheunert and Chris Gardner for discussions and helpful comments.

Note added in proof A definition for quantum Lie algebras very similar to Definition 3.1 was given by Bremner in “Quantum deformations of simple Lie algebras”, Canad. Math. Bull. (to appear).

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References 1.

2. 3. 4. 5. 6.

7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

Braden, H.W., Corrigan, E., Dorey, P.E., Sasaki, R.: Affine Toda field theory and exact S-matrices. Nucl. Phys. B338, 689 (1990); Delius, G.W., Grisaru, M.T., Zanon, D.: Exact S-matrices for nonsimply-laced affine Toda theories. hep-th/9201067, Nucl. Phys. B382, 365 (1992) Drinfel’d, V.G.: Hopf algebras and the quantum Yang-Baxter equation. Sov. Math. Dokl. 32, 254 (1985) Jimbo, M.: A q-Difference Analogue of U(g) and the Yang-Baxter Equation. Lett. Math. Phys. 10, 63 (1985) Delius, G.W., Hüffmann, A.: On Quantum Lie Algebras and Quantum Root Systems. q-alg/9506017, J. Phys. A 29, 1703 (1996) Woronowicz, S.L.: Differential Calculus on Compact Matrix Pseudogroups (Quantum Groups). Commun. Math. Phys. 122, 125 (1989) Aschieri, P., Castellani, L.: An introduction to noncommutative differential geometry on quantum groups. Int. J. Mod. Phys. A8, 1667 (1993); Bernard, D.: Quantum Lie Algebras and Differential Calculus on Quantum Groups. Prog. Theo. Phys. Suppl. 102, 49 (1990); Jurco, B.: Differential Calculus on Quantized Simple Lie Groups. Lett. Math. Phys. 22, 177 (1991); Schupp, P., Watts, P., Zumino, B.: Bicovariant Quantum Algebras and Quantum Lie Algebras. Commun. Math. Phys. 157, 305 (1993); Schupp, P.: Quantum Groups, Non-Commutative Differential Geometry and Applications. hep-th/9312075 (1993) Sudbery, A., Lyubashenko, V.: Quantum Lie Algebras of Type An . q-alg/9510004 Schmüdgen, K., Schüler, A.: Left-covariant Differential Calculi on SLq (N ). Leipzig preprint Delius, G.W.: The Problem of Differential Calculus on Quantum Groups. q-alg/9608019 Chari, V., Pressley, A.: A Guide to Quantum Groups. Cambridge: Cambridge University Press, 1994 Curtis, C.W., Reiner, I.: Representation theory of finite groups and associative algebras. New York: Interscience Publishers, 1962 Sweedler, M.E.: Hopf algebras. New York: Benjamin, 1969 Drinfel’d, V.G.: On almost cocommutative Hopf algebras. Leningrad Math. J. 1, 321 (1990) Ding, J., Frenkel, I.B.: Spinor and Osciallator Representations of Quantum Groups. Prog. Math. 123, 127 (1994) Delius, G.W., Gould, M.D., Hüffmann, A., Zhang, Y.-Z.: Quantum Lie algebras associated to Uq (gln ) and Uq (sln ). q-alg/9508013, J. Phys. A 29, 5611 (1996) Reshetikhin, N.Yu.: Quantized Universal Enveloping Algebras, the Yang-Baxter Equation and Invariants of Links. I. LOMI-preprint E-4-87

Communicated by T. Miwa

Commun. Math. Phys. 185, 723 – 751 (1997)

Communications in


Reducibility and Gribov Problem in Topological Quantum Field Theory Roberto Zucchini Dipartimento di Fisica, Università degli Studi di Bologna, V. Irnerio 46, I-40126 Bologna, Italy Received: 22 July 1996 / Accepted: 21 October 1996

Abstract: In spite of its simplicity and beauty, the Mathai–Quillen formulation of cohomological topological quantum field theory with gauge symmetry suffers two basic problems: i) the existence of reducible field configurations on which the action of the gauge group is not free and ii) the Gribov ambiguity associated with gauge fixing, i. e. the lack of global definition on the space of gauge orbits of gauge fixed functional integrals. In this paper, we show that such problems are in fact related and we propose a general completely geometrical recipe for their treatment. The space of field configurations is augmented in such a way to render the action of the gauge group free and localization is suitably modified. In this way, the standard Mathai–Quillen formalism can be rigorously applied. The resulting topological action contains the ordinary action as a subsector and can be shown to yield a local quantum field theory, which is argued to be renormalizable as well. The salient feature of our method is that the Gribov problem is inherent in localization, and thus can be dealt within a completely equivariant setting, whereas gauge fixing is free of Gribov ambiguities. For the stratum of irreducible gauge orbits, the case of main interest in applications, the Gribov problem is solvable. Conversely, for the strata of reducible gauge orbits, the Gribov problem cannot be solved in general and the obstruction may be described in the language of sheaf theory. The formalism is applied to the Donaldson–Witten model.

0. Introduction Topological quantum field theories are complicated often fully interacting field theories having no physical degrees of freedom. Yet they can be solved exactly and the solution is rather non-trivial. Expectation values of topological observables provide topological invariants of the manifolds on which the fields propagate. These invariants are independent from the couplings and to a large extent from the interactions between the fields. At the same time topological field theories are often topological sectors of ordinary field

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R. Zucchini

theories. In this way they are convenient testing grounds for subtle non perturbative field theoretic phenomena such as duality. See refs. [1–10] for comprehensive reviews on the subject and complete referencing. As is well-known, there are two basic kinds of topological field theories: those of Schwarz type and those of cohomological type. The prototype of the Schwarz type topological field theories is the Chern–Simon model [11]. To the second group there belong the Donaldson–Witten model [12], the topological sigma model [13] and topological two-dimensional gravity [14]. In this paper, we shall concentrate on topological field theories of cohomological type. A great impetus to the development of cohomological topological quantum field theory has come from the realization that they may be understood in the framework of equivariant cohomology of infinite dimensional vector bundles [1, 15–19] and realized as Mathai–Quillen integral representations of Euler classes [20–23]. This has lead to a clear geometric interpretation of such field theoretic models providing a rather general framework for their understanding and has furnished a simple tool kit for the construction of other models. The formalism has been extended also to gauge theories in a way that respects general principles of field theory such as locality and renormalizability. In spite of its simplicity and beauty, the Mathai–Quillen formulation of cohomological topological quantum field theory with gauge symmetry suffers two basic problems. The first consists in the existence of reducible field configurations on which the action of the gauge group is not free. Various attempts at its solution in various models have appeared [24–27]. The second is the Gribov ambiguity associated with gauge fixing, i. e. the lack of global definition on the space of gauge orbits of gauge fixed functional integrals. It has been tackled in ref. [28–30] in topological two-dimensional gravity. In this paper, we propose a general recipe for the treatment of the problems just mentioned and explore its consequences. The space of field configurations A carrying the non free right action of a group Gb is substituted by the larger space P = N × A × G, where N is a stratum of the Gb orbit space of A and G is the subgroup of Gb of the elements path connected to the identity. G acts freely to the right on P in a natural fashion. In this way, the standard Mathai–Quillen formalism can be rigorously applied to P. The Mathai–Quillen localization sector is then suitably augmented to eliminate the extra degrees of freedom associated with the factor A of the G orbit space of P, N × A. A local topological action, which contains the customary action as a subsector and is argued to be renormalizable, is produced in this way. The topological quantum field theory yields a map from the equivariant cohomology of A to the cohomology of each element of an open covering {Uα } of N . The Gribov problem is solved if, for a given equivariant cohomology class O of A, the corresponding cohomology class of each Uα is the restriction on Uα of a unique cohomology class ϕO of N depending only on O. The salient feature of our method is that the Gribov problem is inherent in localization, and thus can be dealt with in a completely equivariant setting, whereas gauge fixing is free of Gribov ambiguities. If N is the stratum of Gb regular irreducible orbits, the case of main interest in applications, the Gribov problem is shown to be solvable under certain rather general assumptions. If N is instead a stratum of Gb singular reducible orbits, then the Gribov problem cannot be solved in general and the obstruction may be characterized in a suitable sheaf theoretic framework. This shows that reducibility and Gribov ambiguity are related aspects of topological quantum field theory. The formalism is applied to the Donaldson–Witten model as an illustration. It can also be applied to two-dimensional topological Yang–Mills and to topological QCD. In

Reducibility and Gribov Problem in Topological QFT

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principle, it should work also for two-dimensional topological gravity and topological string theory, though in these latter cases, a reexamination of gauge fixing is necessary.

1. Cohomological Topological Quantum Field Theory The field content of a topological quantum field theory of cohomological type is organized according to a certain algebraic structure called an operation. Recall that an operation is a quintuplet (g, Z, j, l, s), where g is a Lie algebra, Z is a graded associative algebra and j(ξ), l(ξ), ξ ∈ g, and s are graded derivations on Z of degree −1, 0, +1, respectively, satisfying Cartan’s algebra: [j(ξ), j(η)] = 0, [l(ξ), l(η)] = l([ξ, η]), [s, l(η)] = 0,

[l(ξ), j(η)] = j([ξ, η]), [s, j(η)] = l(η), [s, s] = 0,

(1.1)

where the above are graded commutators [31]. Since s2 = 0, one can define the cohomology of the differential complex (Z, s). This is called ordinary s cohomology. More importantly, one may consider the differential complex (Zbasic , s), where Zbasic is the s invariant subalgebra of Z annihilated by all j(ξ), l(ξ), ξ ∈ g. The corresponding cohomology is referred to as basic s cohomology. The main ingredients entering the construction of any cohomological topological quantum field theory are the following: i) A Lie group G with Lie algebra Lie G, ii) A principal G bundle πP : P → M, iii) A vector space U with a left G action. Using these, one can construct the relevant operation ( Lie G, W, j, l, s). Here, W is the graded tensor algebra 1 ˆ ∗ (P)⊗W ˆ −1 (U ∨ )⊗W ˆ −2 ( Lie G)⊗W ˆ −1 ( Lie G). W = W1 ( Lie G)⊗Ω

(1.2)

The generators of the tensor factors in the given order are 2 ω, Ω p, φ ρ, π ¯ ψ¯ Ω, ¯ χ¯ λ,

deg ω deg p deg ρ deg Ω¯ deg λ¯

= 1, = 0, = −1, = −2, = −1,

deg Ω deg φ deg π deg ψ¯ deg χ¯

= 2, = 1, = 0, = −1, = 0.

(1.3)

The action of the derivations j(ξ), l(ξ), ξ ∈ Lie G, and s on the generators is given by 1 For any vector space V , W (V ) = S(V ∨ ) ⊗ A(V ∨ ), where S(V ∨ ) and A(V ∨ ) are the symmetric and p antisymmetric algebras of V ∨ , respectively. S 1 (V ∨ ) ' V ∨ , A1 (V ∨ ) ' V ∨ carry degree p, p + 1, for p even, and p + 1, p, for p odd, respectively. For any manifold X, Ω ∗ (X) is essentially the exterior algebra of ˆ denotes graded tensor product. See ref. [9]. X. ⊗ 2 The generators α, β of W (V ) are of the form αi t , β i t , where {t } is a basis of V and {αi }, {β j } p i i i are the bases of S 1 (V ∨ ), A1 (V ∨ ) dual to {ti }. The generators q, ψ of Ω ∗ (X) correspond to x, dx for any local coordinate x of X. See ref. [9].

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R. Zucchini

j(ξ)ω = ξ, l(ξ)ω = −[ξ, ω], sω = Ω − (1/2)[ω, ω], j(ξ)p = 0, l(ξ)p = C(p)ξ, sp = φ + C(p)ω, j(ξ)ρ = 0, l(ξ)ρ = ξU ∨ ρ, sρ = π + ωU ∨ ρ, j(ξ)Ω¯ = 0, ¯ l(ξ)Ω¯ = −[ξ, Ω], ¯ ¯ ¯ = ψ − [ω, Ω], sΩ j(ξ)λ¯ = 0, ¯ l(ξ)λ¯ = −[ξ, λ], = χ, ¯ sλ¯

j(ξ)Ω = 0, l(ξ)Ω = −[ξ, Ω], sΩ = −[ω, Ω],

(1.4)

j(ξ)φ = 0, l(ξ)φ = φ∂C(p)ξ, sφ = −C(p)Ω − φ∂C(p)ω,

(1.5)

j(ξ)π = 0, l(ξ)π = ξU ∨ π, sπ = −ΩU ∨ ρ + ωU ∨ π,

(1.6)

j(ξ)ψ¯ = 0, ¯ l(ξ)ψ¯ = −[ξ, ψ], ¯ ¯ ¯ sψ = [Ω, Ω] − [ω, ψ], ¯ j(ξ)χ¯ = −[ξ, λ], l(ξ)χ¯ = −[ξ, χ], ¯ sχ¯ = 0,

(1.7)

(1.8)

in the so called intermediate or BRST model [15–18, 23]. 3 Above, [·, ·] denotes the Lie bracket of Lie G and C(p)ξ is the vertical vector field of P associated with ξ ∈ Lie G. The cohomology of the complex (W, s) will be called ordinary s cohomology below. Besides ( Lie G, W, j, l, s), one can consider the operation ( Lie G, Wequiv (P), j, l, s), where Wequiv (P) is given by the right-hand side of (1.2) with the last three tensor factors deleted and the derivations j, l and s are the same as above. The basic cohomology of the complex (Wequiv (P), s) is called equivariant s cohomology of P. To construct the topological field theory action, it is necessary to provide the relevant field spaces with invariant metrics as follows. An Ad G invariant metric (·, ·) on Lie G together with the induced right G invariant metric (·, ·)G , 4 ii) A G invariant metric (·, ·)P on P, iii) A G invariant metric (·, ·)U on U. i)

The action of cohomological topological field theory consists of three sectors: the Mathai–Quillen localization sector, the Weil projection sector and the Faddeev–Popov gauge fixing sector: (1.9) S = SM Q + S W + S F P . The three contributions are all s-exact, i.e. obtained by applying s to gauge fermions ΦM Q , ΦW , ΦF P of degree −1, respectively: 3 For a right action R X : X × G → X of a group G on a manifold X, we shall write interchangeably RX (x, g), RXx (g), RXg (x) and xgX , for x ∈ X and g ∈ G. Similarly, we shall denote by xtX the tangent map T RXx (1)t with t ∈ g and by ugX the tangent map T RXg (x)u with u ∈ Tx X. Recall that, for fixed t ∈ g, the map x ∈ X → xtX ∈ Tx X is the vertical vector field associated with t. 4 For a group G with Lie algebra g equipped with an Ad G invariant metric (·, ·) , the induced metric g (·, ·)G on G is defined by (δg1 , δg2 )G,g = (ζ(g)δg1 , ζ(g)δg2 )g , for g ∈ G and δg1 , δg2 ∈ Tg G, where ζ is the Maurer–Cartan form of G.


727

−SM Q = sΦM Q , l(ξ)ΦM Q = 0,

j(ξ)ΦM Q = 0,

−SW = sΦW , j(ξ)ΦW = 0,

(1.11) (1.12)

l(ξ)ΦW = 0,

−SF P = sΦF P , j(ξ)ΦF P 6= 0

(1.10)

l(ξ)ΦF P 6= 0.

(1.13) (1.14) (1.15)

The fact that ΦF P is not annihilated by j(ξ), l(ξ) is actually required by gauge fixing. In the standard formulation of cohomological topological field theory, the Mathai– Quillen gauge fermion is given by 1 ΦM Q = −ihρ, K(p)iU − (ρ, π)U ∨ . 4

(1.16)

Here, (·, ·)U ∨ is the metric on U ∨ canonically induced by (·, ·)U , h·, ·iU is the duality pairing of U ∨ and U and K : P → U is a G equivariant map. K defines a section of the vector bundle E = P ×G U on M. In practice, K may be defined only for a subspace of P of the form πP −1 (O) for some open neighborhood O of M. This will eventually entail Gribov type problems in the topological quantum field theory. From (1.5), (1.6), (1.10) and (1.16), one finds 1 1 −SM Q = − (π, π)U ∨ − ihπ, K(p)iU + ihρ, φ∂K(p)iU − (ρ, ΩU ∨ ρ)U ∨ . 4 4 As is well known,

Z IM Q =

dρdπe−SM Q

(1.17)

(1.18)

defines an equivariant s cohomology class of P independent from the metric (·, ·)U . Rescaling (·, ·)U into u(·, ·)U , u > 0, and taking the limit u → +∞, one has IM Q ≡ δ(K(p))δ(φ∂K(p)),

(1.19)

where ≡ denotes equivalence in equivariant s cohomology [23]. 5 The Weil gauge fermion is given by ¯ C † (p)φ), ΦW = −i(Ω,

(1.20)

where C † (p) denotes the adjoint of C(p) with respect to the metrics (·, ·) on Lie G and (·, ·)G on G. A straightforward calculation using standard properties of C(p), (1.5), (1.7), (1.12) and (1.20) yields ¯ C † (p)φ) + i(Ω, ¯ C † C(p)Ω − φ∂C † (p)φ). −SW = −i(ψ,

(1.21)

The integral 5 According to the conventions used in this paper, for two boson/fermion pairs of fields l, λ and a, α valued respectively in the n dimensional vector spaces E and F related by a pairing h·, ·i, the measures dldλ

R

R

n(n−1)

and dadα are such that dldλeihl,ai+ihλ,αi = δ(a)δ(α) and dadαδ(a)δ(α) = (2πi)−n (−1) 2 . If L : E → E is an invertible linear map, then d(Ll)d(Lλ) = sgn det Ldldλ. Similarly, if M : F → F is an invertible linear map, then δ(M a)δ(M α) = sgn det M δ(a)δ(α).

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R. Zucchini

Z

¯ −SW ¯ ψe dΩd

IW =

(1.22)

defines an equivariant s cohomology class of P independent from (·, ·). Indeed, IW is distributional [23], 1 1 † † C (p)φ δ Ω − † φ∂C (p)φ . (1.23) IW ≡ δ − † C C(p) C C(p) The Faddeev–Popov gauge fermion is given by ¯ Σ(p)) − ΦF P = −i(λ,

1 ¯ (λ, χ). ¯ 4t

(1.24)

Here, t > 0 is a gauge fixing parameter. Σ : πP −1 (O) → Lie G is a gauge fixing function, O being some open neighborhood of M. It is in general defined only locally in P because of the usual Gribov problem. From (1.5), (1.8), (1.14) and (1.24), one has −SF P = −

1 ¯ (φ + C(p)ω)∂Σ(p)). (χ, ¯ χ) ¯ − i(χ, ¯ Σ(p)) + i(λ, 4t

The Faddeev–Popov gauge fixing factor in the functional integral is Z ¯ χe IF P = dλd ¯ −SF P .

(1.25)

(1.26)

It defines an ordinary s cohomology class on P independent from (·, ·) and t. Taking the limit t → ∞, one finds IF P ' δ(Σ(p))δ((φ + C(p)ω)∂Σ(p)),

(1.27)

where ' denotes equivalence in ordinary s cohomology [23]. Let O = O(p, φ, Ω) be an equivariant s cohomology class of P. Then, the product IM Q O is an equivariant s cohomology class of P as well, as IM Q is. By the Weil homomorphism theorem [31], it yields a de Rham cohomology class of M upon replacing ω, Ω by υ, Υ , where υ is a connection on P and Υ = sυ + 21 [υ, υ] is its curvature. Further, by Cartan’s third theorem, such a class does not depend on the choice of υ. Now, consider the functional integral Z Z (1.28) IO = dpdφ dωdΩIM Q IW IF P O. It can be shown that IW can be recast as IW ≡ δ(ω − υ(p))δ(Ω − Υ (p)),

(1.29)

where υ is the metric connection on P associated with the metrics (·, ·) and (·, ·)P , υ(p) =

1 C † (p)sp, C † C(p)

(1.30)

and Υ is its curvature. In this way the ω, Ω integration implements the Weil homomorphism. Assuming that the Gribov problems can be solved, one finds then that Z ϑ K ∧ ϕO , (1.31) IO = M


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where ϑK is the Poincaré dual of the submanifold MK of M defined by the equivariant condition K(p) = 0 and ϕO is a form on M corresponding in one-to-one fashion to O. Of course, the above formula is only formal since M is in general infinite dimensional and non compact. For the reader familiar with Donaldson–Witten topological Yang–Mills theory, which we shall discuss in more detail in Sect. 5, but less familiar with the formalism used here, it may be useful to keep in mind the following identifications. P corresponds to the space A of all gauge connections a. G is the gauge group. K(p) is the antiself-dual part F− (a) of the curvature of a. C(p) is the gauge covariant derivative D(a) acting on Lie valued scalar fields. C(p)† and ∂K(p) are respectively the covariant divergence D(a)† and the antiself-dual part D− (a) of the gauge covariant derivative acting on Lie valued vector fields. 2. The Case of Non Free Group Actions The standard framework expounded in the previous section assumes that the action of G on P is free. In that case, the operator C(p) has no zero modes and the standard connection υ, given by (1.30), is well defined. In field theory such an ideal situation rarely occurs and most group actions are not free. The following is a proposal for a general recipe for the treatment of this problem. b For a given Let A be a manifold carrying a non free right action of a group G. b a ∈ A, its Gb stability subgroup G(a) is therefore generally non-trivial. In this way, A may be partitioned into a disjoint union of Gb invariant subspaces Ax in one-to-one correspondence with the conjugacy classes x of the stability group [32]. By definition, the Gb orbit space of A is b B = A/G. (2.1) The space B is not a manifold, since the Gb action is not free. It is rather a stratified space consisting of strata B x corresponding to the Gb invariant subspaces Ax in the quotient by the Gb action [32]. Each stratum is a true manifold. Let Zb be the invariant subgroup of Gb acting trivially on A. Then, for any a ∈ A, b b The elements a ∈ A for which G(a) b = Zb exactly are called irreducible and G(a) ⊃ Z. ∗ form a subspace A [32]. The corresponding Gb orbits n are called regular and span a stratum B ∗ . The elements a ∈ A \ A∗ are called reducible and correspond to strata of singular orbits n ∈ B \ B ∗ [32]. The manifold P and its structure. In order to apply the framework of Sect. 1, one cannot use A. We propose to substitute A by the space P = N × A × G,

(2.2)

where N is a stratum of B fixed once and for all and G is the subgroup of Gb of the elements path connected to the identity. P carries a natural right G action. This is the trivial action on N , the given right action on A and the right multiplication action on G. Since the latter is free, the action on P is free as well. P is thus a principal G bundle. Its base M is M = P/G ' N × A. (2.3) From (2.2), it follows that, for p = (n, a, g) ∈ P, Tp (P) = Tn (N ) ⊕ Ta (A) ⊕ Tg (G). Correspondingly, the vertical vector fields of P are of the form

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R. Zucchini

C(p)ξ = 0 ⊕ D(a)ξ ⊕ c(g)ξ,

(2.4)

for ξ ∈ Lie G. Here, D(a)ξ and c(g)ξ are the vertical vector fields of A and G, the latter seen as a principal G bundle on the singleton manifold. Recall that c(g)ξ is also the left invariant vector field of G corresponding to ξ. The graded algebra Ω ∗ (P) is given by ˆ ∗ (A)⊗Ω ˆ ∗ (G). Ω ∗ (P) = Ω ∗ (N )⊗Ω

(2.5)

So, the generators p and φ of Ω ∗ (P) have the following structure p = (n, a, g),

φ = θ ⊕ ψ ⊕ ,

(2.6)

where the pairs n, θ, a, ψ and g, are the generators of Ω ∗ (N ), Ω ∗ (A) and Ω ∗ (G), respectively. Equation (1.5) further implies j(ξ)n= 0, l(ξ)n = 0, sn = θ, j(ξ)a = 0, l(ξ)a = D(a)ξ, sa = ψ + D(a)ω, j(ξ)g = 0, l(ξ)g = c(g)ξ, sg = + c(g)ω,

j(ξ)θ = 0, l(ξ)θ = 0, sθ = 0, j(ξ)ψ = 0, l(ξ)ψ = ψ∂D(a)ξ, sψ = −D(a)Ω − ψ∂D(a)ω, j(ξ) = 0, l(ξ) = ∂c(g)ξ, s = −c(g)Ω − ∂c(g)ω,

(2.7)

(2.8)

(2.9)

for ξ ∈ Lie G. Localization and the structure of the vector space U. The G orbit space of P, M, is given by (2.3). The relevant orbit space is however N , the chosen stratum of the Gb orbit space of A. The extra degrees of freedom contained in M are eliminated by suitably modifying localization by adding extra equivariant localization conditions, which localize M to N . This is achieved by, roughly speaking, mimicking background gauge fixing in A as follows. An open covering {Uα } of N and a lift aα : Uα → A of each Uα to A are chosen. The lift must be such that, for n ∈ Uα , aα (n) belongs to the gauge orbit n. An equivariant localization condition driving a ∈ A to aα (n) for a fixed n ∈ N is then imposed. It is not obvious that this can be done at all but, as will be shown in due course, it can. The extra localization conditions being defined only locally on N , a Gribov problem is involved. To implement localization in the way indicated according to the general method described in Sect. 1, one needs therefore two localizing functionals. The first functional is valued in some vector space E and is employed to implement ordinary localization. The second functional is valued in another vector space F and serves the purpose of localizing M to N . Thus, the space U has a direct sum decomposition U = E ⊕ F.

(2.10)

E and F carry respectively a left Gb action and a left G action inducing the left G action on U . The vector space E may be quite arbitrary. F can be identified with the typical fiber of the tangent bundle T A. In fact, in order to drive a ∈ A to the background value


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aα (n), one needs an F valued functional on A that vanishes and is locally invertible near aα (n). This entails immediately the existence of a linear isomorphism of T A and F. The graded algebra W−1 (U ∨ ) is given by ˆ −1 (F ∨ ). W−1 (U ∨ ) = W−1 (E ∨ )⊗W

(2.11)

Correspondingly, the generators ρ, π of W−1 (U ∨ ) have the structure ρ = % ⊕ ω, ¯

π = $ ⊕ τ¯ ,

(2.12) ∨

∨

where the pairs %, $ and ω, ¯ τ¯ are the generators of W−1 (E ) and W−1 (F ), respectively. From (1.6), one has j(ξ)% = 0, l(ξ)% = ξE ∨ %, s% = $ + ωE ∨ %,

j(ξ)$ = 0, l(ξ)$ = ξE ∨ $, s$ = −ΩE ∨ % + ωE ∨ $,

(2.13)

j(ξ)ω¯ = 0, ¯ l(ξ)ω¯ = ξF ∨ ω, ¯ = τ¯ + ωF ∨ ω, sω¯

j(ξ)τ¯ = 0, l(ξ)τ¯ = ξF ∨ τ¯ , = −ΩF ∨ ω¯ + ωF ∨ τ¯ , sτ¯

(2.14)

for ξ ∈ Lie G. The natural gauge fixing. The G orbit space of P, M, is naturally realized as a gauge slice in P as M ' {(n, a, g) ∈ P|g = 1}, (2.15) on account of (2.3). There is therefore a natural gauge fixing condition free of Gribov problems: g = 1 for (n, a, g) ∈ P. (2.16) This will be used below. Metric structures. By the remark above (2.4), P can be given a G invariant metric of the form (δp1 , δp2 )P,p = (δn1 , δn2 )N ,n + (δa1 , δa2 )A,a + µ2 (δg1 , δg2 )G,g ,

(2.17)

for p = (n, a, g) ∈ P and δp1 = δn1 ⊕ δa1 ⊕ δg1 , δp2 = δn2 ⊕ δa2 ⊕ δg2 ∈ Tp (P). Above, (·, ·)N is any metric on N , whose choice will not matter. (·, ·)A is a right Gb invariant metric on A. (·, ·)G is the right G invariant metric on G induced by an Ad Gb invariant metric (·, ·) on Lie G (see footnote 4). µ2 > 0 is a parameter and, as perhaps suggested by the notation, will work as an infrared cutoff. By (2.4), the adjoint of C(p) with respect to the metrics (·, ·) and (·, ·)P is given by C † (p)δp = D† (a)δa + µ2 c† (g)δg,

(2.18)

for δp = δn ⊕ δa ⊕ δg ∈ Tp (P). From (2.4) and (2.18), one has thus C † C(p)ξ = D† D(a)ξ + µ2 ξ,

(2.19)

for ξ ∈ Lie G. In deducing this relation, one uses the fact that the metric (·, ·)G is such that (c(g)ξ, c(g)η)G,g = (ξ, η) (see footnote 4), so that c† c(g) = 1 Lie G .

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R. Zucchini

It should be noted that the left invariant Maurer–Cartan form ζ = g −1 sg of G is given by ζ = c† (g)sg = c† (g) + ω,

(2.20)

by (2.9) and the relation c† c(g) = 1 Lie G . Hence, j(ξ)ζ = ξ, l(ξ)ζ = −[ξ, ζ], sζ = −(1/2)[ζ, ζ],

(2.21)

from standard Lie group theory. The spaces E and F carry respectively a Gb invariant metric (·, ·)E and a G invariant (·, ·)F . To these, there corresponds a G invariant metric on U in obvious fashion (u1 , u2 )U = (e1 , e2 )E + (f1 , f2 )F ,

(2.22)

for u1 = e1 ⊕ f1 , u2 = e2 ⊕ f2 ∈ U . 3. The Basic Construction The task facing us now consists in applying the general formalism of Sect. 1 to the setting described in Sect. 2. To this end the use of the intermediate model of topological symmetry, in which the fields, the functional measures and the δ functions always appear in “supersymmetric” boson/fermion pairs, is crucial [33]. The Mathai–Quillen localization sector. Localization is achieved via a collection of maps Kα : Oα → U associated with an open covering {Oα } of P. The sets Oα are of the form Oα = Uα × A × G, where {Uα } is an open covering of N . As described in Sect. 2, there are two types of localization conditions, implying the direct sum decomposition (2.10). Correspondingly, one has the decomposition K α = F ⊕ Hα .

(3.1)

F : P → E is a G equivariant function constant on N and G. In fact, we shall assume more restrictively that F (aγA ) = γ −1 E F (a),

b γ ∈ G.

(3.2)

The specific form of F depends on the model considered. Hα : Uα × A × G → F has the following properties. Hα is G equivariant, Hα (n, aγA , gγ) = γ −1 F Hα (n, a, g),

γ ∈ G,

(3.3)

and satisfies sHα (n, a, g) = ιa (D(a)c† (g) − ψ) + θ∂n Hα (n, a, g) + hα (n, a, g, ψ + D(a)ω, + c(g)ω), (3.4) where (3.5) hα (n, a, g, ψ + D(a)ω, + c(g)ω) = 0 if Hα (n, a, g) = 0.


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Here, ι : T A → F is a map with the following properties. For any a ∈ A, ιa ≡ ι|Ta A is a linear isomorphism of Ta A onto F. ι is equivariant, i.e. ιaγA T RAγ (a) = γ −1 F ιa , b ι is also orthogonal, so that (ιa δa1 , ιa δa2 )F = (δa1 , δa2 )A,a for any two for γ ∈ G. δa1 , δa2 ∈ Ta A. Finally, ι is orientation preserving, i.e. sgn det ιa = 1. It is easy to check that the right-hand side of (3.4) depends on ψ and only through the combinations ψ + D(a)ω and + c(g)ω, as required by (2.8) and (2.9). At g = 1, Hα (n, a, g) localizes a to a gauge slice aα (n) defined on Uα : Hα (n, a, g)|g=1 = 0 ⇒ a = aα (n),

(3.6)

∂n Hα (n, a, g)|g=1 = ιaα (n) (∂n aα (n)).

(3.7)

The above hypotheses are technical and are motivated only by their consequences. By (1.17), (2.12), (2.22), (3.1) and (3.4), the Mathai–Quillen action is 1 1 −SM Q = − (π, π)U ∨ − ihπ, Kα (p)iU + ihρ, φ∂Kα (p)iU − (ρ, ΩU ∨ ρ)U ∨ 4 4 1 1 = − ($, $)E ∨ − ih$, F (a)iE + ih%, ψ∂F (a)iE − (%, ΩE ∨ %)E ∨ 4 4 1 − (τ¯ , τ¯ )F ∨ − ihτ¯ , Hα (n, a, g)iF + ihω, ¯ ιa (D(a)c† (g) − ψ) 4 1 ¯ ΩF ∨ ω) ¯ F ∨. (3.8) +θ∂n Hα (n, a, g) + hα (n, a, g, ψ, )iF − (ω, 4 R The functional integral dρdπe−SM Q defines an equivariant s cohomology class independent from (·, ·)E and (·, ·)F . By rescaling (·, ·)F into u(·, ·)F with u > 0 and taking the limit u → +∞ in the integral, we obtain a representative of such class. The result is Z Z 1 dρdπe−SM Q ≡ d%d$ exp − ($, $)E ∨ − ih$, F (a)iE + ih%, ψ∂F (a)iE 4 1 − (%, ΩE ∨ %)E ∨ δ(Hα∗ (n, a, g))δ(−ψ + D(a)c† (g) + θ∂n Hα∗ (n, a, g)), (3.9) 4 where ≡ denotes equivalence in equivariant s cohomology and one has set Hα∗ (n, a, g) = ιa −1 Hα (n, a, g). Above, the formal rules stated in footnote 5 have been used. The Weil projection sector. The action of the Weil sector is, according to (1.21), (2.6), (2.18) and (2.19), ¯ C † (p)φ) + i(Ω, ¯ C † C(p)Ω − φ∂C † (p)φ) −SW = −i(ψ, ¯ D† (a)ψ + µ2 c† (g)) + i(Ω, ¯ (D† D(a) + µ2 )Ω = −i(ψ, Ψ ∂D † (a)Ψ − µ2 ∂c† (g)).

(3.10)

R ¯ Ωe ¯ −SW is given by Hence, the functional integral dψd Z ¯ −SW ≡ δ((D† D(a) + µ2 )Ω − ψ∂D† (a)ψ ¯ ψe dΩd − µ2 ∂c† (g))δ(−(D† (a)ψ + µ2 c† (g))).

(3.11)

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This expression defines an equivariant s cohomology class independent from (·, ·) and µ2 . Then, its limit as µ2 → 0, if it exists, provides a representative of such a class. Let q(a) be the orthogonal projector of Lie G onto ker D(a). Since D(a)q(a) = 0 and q(a) = q(a)† , one has q(a)D† (a) = 0. Considering D† (a) as a Lie G valued one form, † † one has a q(a) ∧ D (a) + q(a) ∧ da D (a) = 0. Using this relation, one can show R thus d−S W ¯ Ωe ¯ has a formal limit as µ2 → 0 and compute it. In this way, one finds that dψd that Z ¯ −SW ≡ δ(q(a)(Ω − ψ∂q(a)c† (g) − ∂c† (g)))δ(−q(a)c† (g)) ¯ ψe dΩd × δ((1 − q(a))(D† D(a)Ω − ψ∂D† (a)ψ))δ(−(1 − q(a))D† (a)ψ)

(3.12)

in equivariant s cohomology. The Faddeev–Popov gauge fixing sector. As indicated in Sect. 2, a suitable Gribov free gauge fixing condition is given by (2.16). We assume that the gauge fixing function Σ : P → Lie G is of the form Σ(p) = C † (p)Ξ(p),

(3.13)

where Ξ : P → T P is a section of T P constant on N with the following structure: Ξ(a, g) = 0 ⊕ W (a, g) ⊕ c(g) ln g.

(3.14)

∗

Here, W : P → pA (T A) is a section of the pull-back of T A via the natural projection pA : P → A constant on the factor N of P and such that W (a, g)|g=1 = 0,

(3.15)

and sW (a, g) = D(a)(c† (g) + ω) + w(a, g)((ψ + D(a)ω) ⊕ ( + c(g)ω)),

(3.16)

where w(a, g)|g=1 = 0.

(3.17)

ln : G → Lie G is the inverse of the exponential map exp : Lie G → G. It is defined only near g = 1 so that K is not defined far away from g = 1. The solution of this problem will be provided later in this section when discussing renormalizability. Using (1.25), (2.18)–(2.19), (3.13), (3.14), (3.16) and the relations c† c(g) = 1 Lie G and s(c(g) ln g) = c(g)(c† (g) + ω) + o(g)( + c(g)ω), where o(g)|g=1 = 0, one has 1 ¯ (φ + C(p)ω)∂Σ(p)) (χ, ¯ χ) ¯ − i(χ, ¯ Σ(p)) + i(λ, 4t 1 ¯ (D† D(a) + µ2 )(c† (g) + ω) ¯ χ) ¯ − i(χ, ¯ D† (a)W (a, g) + µ2 ln g) + i(λ, = − (χ, 4t

−SF P = −

+ D † (a)w(a, g)((ψ + D(a)ω) ⊕ ( + c(g)ω)) + (ψ + D(a)ω)∂D† (a)W (a, g) + µ2 (c† (g)o(g)( + c(g)ω) + ( + c(g)ω)∂c† (g)c(g) ln g)).

(3.18)

R ¯ χe On general grounds, the integral dλd ¯ −SF P defines an s cohomology class indepen2 dent from (·, ·), t and µ . One can obtain a representative of such a class by taking the


735

limit t → +∞ (Landau gauge). The limit concentrates the integration on the zero set of the gauge fixing function Σ(p) and, since Σ(a, g) = 0 implies g = 1, it makes several terms in expression (3.18) vanish, by (3.15), (3.17) and the relations c(g) ln g|g=1 = 0 and o(g)|g=1 = 0. One has then Z

¯ χe dλd ¯ −SF P ' δ((D† D(a) + µ2 )(c† (g) + ω))δ(D† (a)W (a, g) + µ2 ln g),

(3.19)

where ' denotes equivalence in ordinary s cohomology. As an s cohomology class, this expression is still independent from µ2 . One can thus obtain a representative of the class by taking its limit as µ2 → 0, provided it exists. Proceeding in this way, one finds Z

¯ χe dλd ¯ −SF P ' δ(q(a)(c† (g) + ω))δ(q(a) ln g) × δ((1 − q(a))D† D(a)(c† (g) + ω))δ((1 − q(a))D† (a)W (a, g)),

(3.20)

where q(a) is defined above. Putting the three sectors together. From (3.9), (3.12) and (3.20), after some simple rearrangements, one finds that Z

Z Z ¯ χe ¯ ψ¯ dλd dρdπ dΩd ¯ −SM Q −SW −SF P Z 1 1 ∨ ∨ ∨ ' d%d$ exp − ($, $)E − ih$, F (a)iE + ih%, ψ∂F (a)iE − (%, ΩE %)E 4 4 × δ(Hα∗ (n, a, g))δ(−ψ + D(a)c† (g) + θ∂n Hα∗ (n, a, g)) × δ(q(a)(Ω − (1/2)[ω, ω] + θ∂n Hα∗ (n, a, g)∂q(a)ω))δ(q(a)ω) × δ((1 − q(a))(D† D(a)Ω − ψ∂D† (a)ψ))δ(−(1 − q(a))D† (a)ψ) × δ(q(a)(f (g)(c† (g) + ω) + θ∂n Hα∗ (n, a, g)∂q(a) ln g))δ(q(a) ln g) × δ((1 − q(a))D† D(a)(c† (g) + ω))δ((1 − q(a))D† (a)W (a, g))

(3.21)

in s cohomology. Here, f (g) is defined by the relation f (g)(c† (g) + ω) = s ln g and is explicitly given by f (g) = ad ln g/(1 − exp(− ad ln g)). To show the above relation, one uses the fact that the δ functions of the Faddeev–Popov sector enforce the constraints g = 1 and c† (g) + ω = 0 and the identities (c(g)ω)∂c† (g)(c(g)ω) = −(1/2)[ω, ω] and (D(a)ω)∂q(a)ω = −[ω, q(a)ω]+q(a)[ω, ω]. These follow respectively from the Maurer– Cartan equation dg c† (g) = −(1/2)[c† (g), c† (g)] and the relation l(ξ)q(a) = −[ ad ξ, q(a)], ξ ∈ Lie G. The δ functions appearing in the right-hand side of (3.21) have functional integral representations. The Mathai–Quillen factor is given by

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R. Zucchini

δ(Hα∗ (n, a, g))δ(−ψ + D(a)c† (g) + θ∂n Hα∗ (n, a, g)) Z 1 = lim dωd ¯ τ¯ exp − (τ¯ , τ¯ )F ∨ − ihτ¯ , Hα (n, a, g)iF b→+∞ 4b +ihω, ¯ ιa (D(a)c† (g) − ψ)

1 ∨ ∨ + θ∂n Hα (n, a, g) + hα (n, a, g, ψ, )iF − (ω, ¯ ΩF ω) ¯ F . 4b

(3.22)

The Weil factor is δ(q(a)(Ω − (1/2)[ω, ω] + θ∂n Hα∗ (n, a, g)∂q(a)ω))δ(q(a)ω)δ((1 − q(a))(D† D(a)Ω − ψ∂D † (a)ψ))δ(−(1 − q(a))D† (a)ψ) Z † † † ¯ ¯ ¯ ¯ = dΩdψ exp − i(ψ, D (a)ψ) + i(Ω, D D(a)Ω − ψ∂D (a)ψ) × (2πi)d(a) (−1)

d(a)(d(a)−1) 2

¯ ¯ + θ∂n Hα∗ (n, a, g)∂q(a)Ω)) ¯ δ(q(a)Ω)δ(q(a)( ψ¯ − [ω, Ω]

× δ(q(a)ω)δ(q(a)(Ω − (1/2)[ω, ω] + θ∂n Hα∗ (n, a, g)∂q(a)ω)),

(3.23)

where d(a) = dim ker D(a). Finally, the Faddeev–Popov factor has the representation δ(q(a)(f (g)(c† (g) + ω) + θ∂n Hα∗ (n, a, g)∂q(a) ln g))δ(q(a) ln g) × δ((1 − q(a))D† D(a)(c† (g) + ω))δ((1 − q(a))D† (a)W (a, g)) Z 1 ¯ ¯ D† D(a)(c† (g) + ω) = lim dλdχ¯ exp − (χ, ¯ χ) ¯ − i(χ, ¯ D† (a)W (a, g)) + i(λ, t→+∞ 4t + D † (a)w(a, g)((ψ + D(a)ω) ⊕ ( + c(g)ω)) + (ψ + D(a)ω)∂D† (a)W (a, g)) × (2πi)d(a) (−1)

d(a)(d(a)−1) 2

¯ ¯ δ(q(a)λ)δ(q(a)( χ¯ + θ∂n Hα∗ (n, a, g)∂q(a)λ))

δ(q(a) ln g)δ(q(a)(f (g)(c† (g) + ω) + θ∂n Hα∗ (n, a, g)∂q(a) ln g)).

(3.24)

In deducing the above relations, one uses the standard properties of the boson/fermion δ functions, the orthogonal factorization of the integration measures, the relation q(a)D† (a) = 0 and (3.15) and (3.17). The signs are determined according the conventions stated in footnote 5 above. Next, Eqs. (3.22)–(3.24) are to be substituted into Eq. (3.21). Now, the δ functions δ(q(a) ln g)δ((1 − q(a))D† (a)W (a, g)) arising in the Faddeev–Popov sector enforce the identity g = 1, by (3.13) and (3.14). The δ function δ(Hα∗ (n, a, 1)) resulting in this way in the Mathai–Quillen sector allows one to set a = aα (n) everywhere else, on account of (3.6). The finite dimensional δ functions so yielded in the integral representations (3.23) and (3.24) define s cohomology classes having in turn the following integral


737

representations. Set Kα (n) = ker D(aα (n)) for n ∈ Uα . Consider the auxiliary graded tensor algebra ˆ 1 (Kα (n))⊗W ˆ −1 (Kα (n))⊗W ˆ 0 (Kα (n)). Zα (n) = W−2 (Kα (n))⊗W

(3.25)

The generators of the factors in the given order are deg y¯α (n) = −2, deg xα (n) = 1, deg uα (n) = −1, deg rα (n) = 0,

y¯α (n), x¯ α (n) xα (n), yα (n) uα (n), vα (n) rα (n), zα (n)

deg x¯ α (n) = −1, deg yα (n) = 2, deg vα (n) = 0, deg zα (n) = 1.

(3.26)

The derivations j(ξ), l(ξ), ξ ∈ Lie G, and s are extended as follows: j(ξ)fα (n)= 0, = 0, l(ξ)fα (n) sfα (n) = gα (n) + θ∂n fα (n),

j(ξ)gα (n) = 0, l(ξ)gα (n) = 0, sgα (n) = θ∂n gα (n),

(3.27)

with (f, g) = (y, ¯ x), ¯ (x, y), (u, v), (r, z). One considers next the gauge fermions Ψequiv = i(y¯α (n), ω), ¯ Ψpr = −i(xα (n), Ω), Ψgau Ψfix

= −i(uα (n), ln g), ¯ = i(rα (n), λ),

(3.28) (3.29) (3.30) (3.31)

and defines s cohomology classes ∆equiv , ∆pr , ∆gau and ∆fix by: −Sequiv = sΨequiv

∆equiv

= i(x¯ α (n) + θ∂n y¯α (n), ω) + i(y¯α (n), Ω − (1/2)[ω, ω])), Z = dy¯α (n)dx¯ α (n)e−Sequiv

(3.32)

= δ(q(aα (n))ω)δ(q(aα (n))(Ω − (1/2)[ω, ω] + θ∂n (q(aα (n)))ω)),

(3.33)

−Spr = sΨpr ¯ + i(xα (n), ψ¯ − [ω, Ω]), ¯ = −i(yα (n) + θ∂n xα (n), Ω) Z ∆pr = dxα (n)dyα (n)e−Spr ¯ ¯ ¯ ¯ = δ(q(aα (n))Ω)δ(q(a α (n))(ψ − [ω, Ω] + θ∂n (q(aα (n)))Ω)),

(3.34)

(3.35)

−Sgau = sΨgau

∆gau

= −i(vα (n) + θ∂n uα (n), ln g) + i(uα (n), f (g)(c† (g) + ω)), Z = duα (n)dvα (n)e−Sgau

(3.36)

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R. Zucchini

= δ(q(aα (n)) ln g)δ(q(aα (n))(f (g)(c† (g) + ω) + θ∂n (q(aα (n))) ln g)),(3.37) −Sfix = sΨfix

∆fix

¯ + i(rα (n), χ), = i(zα (n) + θ∂n rα (n), λ) ¯ Z = drα (n)dzα (n)e−Sfix

(3.38)

¯ ¯ = δ(q(aα (n))λ)δ(q(a ¯ + θ∂n (q(aα (n)))λ)). α (n))(χ

(3.39)

where f (g) is defined below Eq. (3.21). From the above discussion and from (3.33), (3.35), (3.37) and (3.39), it follows that the product of the finite dimensional δ functions appearing in (3.23) and (3.24) with a = aα (n) may be substituted by the product ∆equiv ∆pr ∆gau ∆fix . Proceeding in this way, one finds the formula Z Z Z ¯ ¯ χe ¯ dρdπ dΩdψ dλd ¯ −SM Q −SW −SF P Z Z Z Z dy¯α (n)dx¯ α (n) dxα (n)dyα (n) duα (n)dvα (n) drα (n)dzα (n) ' lim t,b→+∞ Z Z Z Z ¯ χe ¯ ψ¯ dλd × (2πi)2dN d%d$ dωd ¯ τ¯ dΩd ¯ −Stop , (3.40) where dN = d(aα (n)) with n ∈ N is a non negative integer constant characterizing N and − Stop = − 41 ($, $)E ∨ − ih$, F (a)iE + ih%, ψ∂F (a)iE − 41 (%, ΩE ∨ %)E ∨ −

1 ∨ 4b (τ¯ , τ¯ )F

− ihτ¯ , Hα (n, a, g)iF + ihω, ¯ ιa (D(a)ζ − ψ − D(a)ω) + θ∂n Hα (n, a, g) + hα (n, a, g, ψ, c(g)(ζ − ω))iF −

1 ¯ ΩF ∨ ω) ¯ F∨ 4b (ω,

¯ D† (a)ψ) − i(ψ,

¯ D† D(a)Ω − ψ∂D† (a)ψ) + i(Ω, 1 ¯ D† D(a)ζ (χ, ¯ χ) ¯ − i(χ, ¯ D† (a)W (a, g)) + i(λ, − 4t

+ D† (a)w(a, g)((ψ + D(a)ω) ⊕ c(g)ζ) + (ψ + D(a)ω)∂D† (a)W (a, g)) + i(x¯ α (n) + θ∂n y¯α (n), ω) + i(y¯α (n), Ω − (1/2)[ω, ω]) ¯ −i(yα (n) + θ∂n xα (n), Ω) ¯ − i(vα (n) + θ∂n uα (n), ln g) + i(uα (n), f (g)ζ) + i(xα (n), ψ¯ − [ω, Ω]) ¯ + i(rα (n), χ), + i(zα (n) + θ∂n rα (n), λ) ¯

(3.41)


739

ζ being the Maurer–Cartan form given by (2.20). By construction, the action (3.41) is s exact. Hence, in s cohomology, the right-hand side of (3.40) is independent from t, b and the limit may be dropped. Note, however, that the right-hand sides of (3.21) and (3.40) are exactly equal and not merely equivalent in s cohomology. Stop is the effective topological action produced by the present method. It must be stressed that it contains the customary topological action as a subsector. On–shell analysis. Relation (3.21) can be cast in a more transparent form as follows: Z Z Z ¯ χe ¯ ψ¯ dλd dρdπ dΩd ¯ −SM Q −SW −SF P Z 1 1 ∨ ∨ ∨ ' d%d$ exp − ($, $)E − ih$, F (a)iE + ih%, ψ∂F (a)iE − (%, ΩE %)E 4 4 × δ(Hα∗ (n, a, g))δ(−sa + θ∂n Hα∗ (n, a, g))δ(ω − GD† D (a)D† (a)sa) × δ(sω − s(GD† D (a)D† (a)sa))δ(ln g)δ(s ln g),

(3.42)

where sa and sω are given by (2.8) and (1.4), respectively, s ln g = f (g)ζ and GD† D (a) is the Green function of D† D(a) uniquely defined by the relations D† D(a)GD† D (a) = GD† D (a)D† D(a) = 1 − q(a), = 0. q(a)GD† D (a) = GD† D (a)q(a)

(3.43)

Equation (3.42) is obtained using the relations listed below Eq. (3.21), (3.43) and the identities (D(a)ω)∂D(a)ω = (1/2)D(a)[ω, ω] and (sa−D(a)ω)∂D† (a)(sa−D(a)ω) = (sa)∂D† (a)sa − (sa)∂(D† D)(a)ω − [ω, D† D(a)ω − D† (a)sa] + (1/2)D† D(a)[ω, ω] and (3.43). These follow from the relations l(ξ)D† (a) = −[ξ, D† (a)] and D† (a)[ξ, η] = [D † (a)ξ, D† (a)η]A for ξ, η ∈ Lie G, [·, ·]A being the Lie bracket on vector fields on A. From (3.42), using (3.6) and (3.7), one deduces immediately that Z Z Z ¯ χe ¯ ψ¯ dλd dρdπ dΩd ¯ −SM Q −SW −SF P Z 1 1 ∨ ∨ ∨ ' d%d$ exp − ($, $)E − ih$, F (a)iE + ih%, ψ∂F (a)iE − (%, ΩE %)E 4 4 × δ(Hα∗ (n, a, 1))δ(−ψ|a=aα (n) + (1 − D(aα (n))GD† D (aα (n))D† (aα (n)))θ∂n aα (n)) × δ(ω − υα (n))δ(Ω − Υα (n))δ(ln g)δ(f (g)ζ), where

(3.44)

υα (n) = GD† D (aα (n))D† (aα (n))θ∂n aα (n),

(3.45)

Υα (n) = θ∂n υα (n) + (1/2)[υα (n), υα (n)].

(3.46)

Equation (3.44) shows clearly the on-shell structure of the functional integral and is the starting point for the analysis of the Gribov problem below. The functional integral of the topological field theory. According to the general framework described in Sect. 1, in a cohomological topological quantum field theory, one must

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R. Zucchini

integrate all the fields with action SM Q + SW + SF P and insertions of an equivariant s cohomology class O of P. In the present case, one has that, in s cohomology, the ρ, π, ¯ λ, ¯ χ¯ integral is given by (3.40)–(3.41). The resulting functional of p, φ and ω, Ω ¯ ψ, Ω, should be further integrated with respect to those fields. The n, θ integration involved by the p, φ integration is problematic for two main reasons: the non-compactness and infinite dimensionality of N and the lack of global definition on N of the integrand. Hence, it is omitted at this stage. In view of applications to concrete models, the discussion below is restricted to the family of equivariant s cohomology classes O that are constant on the factors N and G of P. These are of the form O = O(a, ψ, Ω) and thus represent equivariant s cohomology classes of A. In field theory, it is more natural to use instead of g, the integration variables χ = ln g,

λ = f (g)ζ.

(3.47)

This functional change of variables is not globally defined because the group logarithm ln is defined only near g = 1 as mentioned earlier. However, in spite of this shortcoming, it is suitable for the field theoretic analysis below. The topological action Stop given by (3.41) can be expressed in terms of χ, λ. Since λ = sχ and the boson/fermion measures are invariant under functional changes of variables with positive jacobian, one has that dgd = dχdλ. So, denoting by Iα (n, θ, a, ψ, χ, λ, ω, Ω) the functional integral given by the equivalent expressions (3.40)–(3.41) and (3.44), we shall consider the functional integral Z Z Z IOα (n, θ) = dadψ dχdλ dωdΩIα (n, θ, a, ψ, χ, λ, ω, Ω)O(a, ψ, Ω). (3.48) It is now time to discuss the locality and renormalizability of the resulting quantum field theory. Locality. The quantum field theory described by (3.48) is manifestly local provided the functionals Hα (n, a, g) and W (a, g) are judiciously chosen in such a way to yield local contributions to the action. In fact, if this is the case, the topological action Stop is a local functional of g and ζ which, in turn, are given by local expressions in terms of χ and λ, namely g = eχ and ζ = f (eχ )−1 λ. Had one used as functional variables g, instead of χ, λ, one would have had an action containing explicit occurrences of the non local functional c† (g), spoiling locality. Renormalizability. The quantum field theory yielded by (3.48) is not manifestly renormalizable for two basic reasons. First, the action contains the non linear combination eχ which is generally incompatible with renormalizability. Second, the available Ward identities may not be sufficient to prevent the generation in the quantum theory of terms allowed by locality and power counting but not contained in the action. The following formal argument shows that the field theory considered here is equivalent in s cohomology to one in which the group G is “flattened” into its Lie algebra Lie G. To begin with, one performs the following change of functional variables: = (1/k)χ, χ0 0 rα (n) = (1/k)rα (n),

λ0 = (1/k)λ, 0 zα (n) = (1/k)zα (n),

(3.49)

where k > 0. Since the boson/fermion measures are invariant under changes of coordinates with positive jacobian, one has that dχ0 dλ0 = dχdλ and drα0 (n)dzα0 (n) =


741

drα (n)dzα (n). This being a mere change of variables, the resulting functional integral is independent from k. Next, one rescales the metric (·, ·) of the χ, ¯ λ¯ and uα (n), vα (n) sectors into (1/k 0 )(·, ·) and replaces t by t/k 0 , where k 0 > 0, leaving the s cohomology unchanged. Now, nothing forbids setting k = k 0 and taking the limit k = k 0 → 0, provided the limit exists and is non singular. Using (3.15)–(3.17) and the fact that f (1) = 1, 0 it is easy to see that W (a, ekχ ) = kD(a)χ0 + O(k 2 ) and that ζ = kλ0 + O(k 2 ). The limit affects the τ¯ , ω, ¯ χ, ¯ λ¯ and uα (n), vα (n) sectors of the action (3.41), which now becomes −Sflat = −

1 (τ¯ , τ¯ )F ∨ − ihτ¯ , Hα (n, a, 1)iF − ihω, ¯ ιa (ψ + D(a)ω) − θ∂n Hα (n, a, 1) 4b 1 1 − hα (n, a, 1, ψ, −ω)iF − (ω, ¯ ΩF ∨ ω) ¯ χ) ¯ − i(χ, ¯ D† D(a)χ0 ) ¯ F ∨ − (χ, 4b 4t ¯ D † D(a)λ0 + (ψ + D(a)ω)∂D† (a)D(a)χ0 + i(λ, + D† (a)(T2 w(a, 1)χ0 )((ψ + D(a)ω) ⊕ 0)) − i(vα (n) + θ∂n uα (n), χ0 ) ¯ + i(rα0 (n), χ) + i(uα (n), λ0 ) + i(zα0 (n) + θ∂n rα0 (n), λ) ¯ + ···,

(3.50)

where T2 w(a, 1) is the tangent map of w(a, g) with respect to g at g = 1 and the ellipses denote the remaining terms of the action unchanged in the limit. In this way, one eliminates the troublesome exponential eχ present in the topological action without changing the topological content of the theory. It is remarkable that by this procedure the problem of the lack of global definition on G of the map g → ln g is simultaneously cured. Note however that the localization sector of the action Sflat is still of the form (1.10) but with (1.11) no longer holding. This somewhat obscures the topological origin of Sflat . In refs. [18, 23], it has been argued that, in order to ensure the renormalizability of the topological quantum field theory, the gauge fermion ΦF P should be of the form wΨF P , where w is a nilpotent operation anticommuting with s and annihilating all equivariant functionals such as ΦM Q and ΦW (see Sect. 1). w rather than s would be the true counterpart of the BRST operator. To w there are associated extra Ward identities which might be necessary for renormalizability. The gauge fermion which has been used above does not have this property. However, following the procedure of refs. [18, 23] would simply add to the Faddeev–Popov action which used further terms depending on extra ghostly fields, which, in the functional integral of the quantum field theory considered here, would yield a trivial insertion 1. While these terms may be necessary for the manifest renormalizability of the quantum field theory, they are not expected to affect its infrared properties which are the main object of our analysis. From the above discussion, it seems plausible that the topological quantum field theory constructed above is renormalizable, provided the functionals Hα (n, a, g) and W (a, g) are properly chosen. Of course, these arguments can not be considered in any way a conclusive proof. The regular irreducible stratum B ∗ . Suppose that N is the stratum B ∗ of regular Gb orbits (see Sect. 2). In this case, the topological action Stop and the flattened action Sflat , given respectively by Eqs. (3.41) and (3.50), simplify considerably. Indeed, since, for b = Z, b the auxiliary fields y¯α (n), x¯ α (n), xα (n), yα (n), uα (n), vα (n), any a ∈ A∗ , G(a) 0 rα (n), zα (n) or rα (n), zα0 (n) are valued in Lie Z with Z = G ∩ Zb and so can be chosen

742

R. Zucchini

independent from α and n. In this way, the only n dependent object appearing in the actions is the local lift aα (n). Moreover, if Z is trivial, the terms containing the auxiliary fields are identically zero.

4. The Gribov Problem and its Treatment The functional integral IOα (n, θ) of Eq. (3.48) is defined on the local patch Uα of N . Let us now find out under which conditions IOα (n, θ) is the local restriction on Uα of a globally defined functional IO (n, θ) on N . This is in essence the Gribov problem and amounts to checking whether IOα (n, θ) = IOβ (n, θ)

(4.1)

on Uβ ∩ Uα 6= ∅. The lifts aα and aβ of overlapping neighborhoods Uα and Uβ of N to A must satisfy a relation of the form (4.2) aα (n) = aβ (n)κβα (n)A on Uβ ∩ Uα . The transition functions κβα : Uβ ∩ Uα → Gb are defined up to right multib α (n)) and left multiplication by an element of G(a b β (n)). plication by an element of G(a b For this reason, in general, the G valued 1-cochain κ = {κβα } is not a 1-cocycle and thus it does not define a principal Gb bundle over N [34]. Instead of the 1-cocycle condition, the κβα satisfy the more general condition κγβ (n)κβα (n) = kβαγ (n)κγα (n)

(4.3)

b γ (n)). on any triple intersection Uβ ∩ Uα ∩ Uγ 6= ∅, where kβαγ (n) ∈ G(a The choice of the local lifts aα : Uα → A is conventional. If a0α : Uα → A is another choice of local lifts, then there exist maps να : Uα → Gb such that a0α (n) = aα (n)να (n)A .

(4.4)

With the lifts a0α , there is associated a new set of transition functions κ0βα satisfying a relation of the same form as (4.3). To carry out the Gribov analysis, one has to make some technical assumptions verified in concrete models. First, the invariant subgroup Zb of Gb acting trivially on A acts trivially on E as well: γE = 1E ,

b γ ∈ Z.

(4.5)

b Second, the insertion O(a, ψ, Ω) is flat in the direction of Lie Z, where Z = G ∩ Z: O(a, ψ, Ω + εξ) = O(a, ψ, Ω),

ξ ∈ Lie Z,

(4.6)

where ε is a parameter of degree 2. One needs also a matching assumption on the localizing functional Hα (n, a, g), namely that Hβ (n, a, g)|g=1 = κβα (n)F Hα (n, aκβα (n)A , g)|g=1 on Uβ ∩ Uα 6= ∅.

(4.7)


743

The Gribov analysis invokes repeatedly certain matching relations collected below. Using the definition of Hα∗ (n, a, g) given below (3.9), the properties of the map ι listed below Eq. (3.5) and (4.7), it is straightforward to show that Hα∗ (n, a, 1) = T RAκβα (n) (aκβα (n)−1 A )Hβ∗ (n, aκβα (n)−1 A , 1)

(4.8)

on Uβ ∩ Uα 6= ∅. Using (4.2) and the basic identities D† D(aγA ) = Ad γ −1 D† D(a) Ad γ and D(aγA ) = T RAγ (a)D(a)Ad γ for γ ∈ Gb and recalling that q(aγA ) = Ad γ −1 q(a) Ad γ, the following relations are easily obtained: θ∂n aα (n) = T RAκβα (n) (aβ (n))θ∂n aβ (n) + D(aα (n))(κβα (n)−1 θ∂n κβα (n)) and

υα (n) = κβα (n)−1 θ∂n κβα (n) + Ad κβα (n)−1 υβ (n) − σβα (n), Υα (n) = Ad κβα (n)

where

−1

Υβ (n) − Σβα (n),

(4.9) (4.10) (4.11)

σβα (n) = q(aα (n))(κβα (n)−1 θ∂n κβα (n)),

(4.12)

Σβα (n) = (θ∂n + ad υα (n))σβα (n) + (1/2)[σβα (n), σβα (n)],

(4.13)

on Uβ ∩ Uα 6= ∅. Now, from (3.44), it is clear that Iα (n, θ, a, ψ, χ, λ, ω, Ω) = I(a, ψ, χ, λ, ω, Ω; aα (n), θ∂n aα (n), υα (n), Υα (n)). (4.14) Using (3.2), (4.8)–(4.11) and the Gb invariance of the insertion O(a, ψ, Ω) and recalling the formal rules stated in footnote 5, it is straightforward to show that I(a, ψ, χ, λ, ω, Ω; aα (n), θ∂n aα (n), υα (n), Υα (n))O(a, ψ, Ω) = I(aκβα (n)−1 A , ψκβα (n)−1 A , χ, λ, Ad κβα (n)(ω − κβα (n)−1 θ∂n κβα (n) + σβα (n)), Ad κβα (n)Ω; aβ (n), θ∂n aβ (n), υβ (n), Υβ (n) − Ad κβα (n)Σβα (n)) ×O(aκβα (n)−1 A , ψκβα (n)−1 A , Ad κβα (n)Ω) × sgn det (κβα (n)E )sgn det (κβα (n)−1 A ) sgn det ( Ad κβα (n)).

(4.15)

From here, it follows immediately that Z Z Z dadψ dχdλ dωdΩ I(a, ψ, χ, λ, ω, Ω; aα (n), θ∂n aα (n), υα (n), Υα (n))O(a, ψ, Ω) Z Z Z = sgn det (κβα (n)E ) dadψ dχdλ dωdΩ I(a, ψ, χ, λ, ω, Ω; aβ (n), θ∂n aβ (n), υβ (n), Υβ (n) − Ad κβα (n)Σβα (n))O(a, ψ, Ω),

(4.16)

since d(aγ −1 A )d(ψγ −1 A ) = sgn det (γ −1 A )dadψ and d( Ad γω)d( Ad γΩ) = b From (3.48), (4.14) and (4.16), it appears that (4.1) sgn det ( Ad γ) dωdΩ for γ ∈ G.

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R. Zucchini

is not fulfilled in general. A possible mechanism by which the Gribov ambiguity may nevertheless be solved is the following. Let q Lie Z be the orthogonal projector of Lie G onto Lie Z. Let us assume that q Lie Z Σβα (n) = Σβα (n).

(4.17)

Then, recalling that Iα (n, θ, a, ψ, χ, λ, ω, Ω) is the functional integral given by (3.44) and that (4.14) holds, it is easy to show using (4.5), (4.6) and (4.17) that I(a, ψ, χ, λ, ω, Ω; aβ (n), θ∂n aβ (n), υβ (n), Υβ (n) − Ad κβα (n)Σβα (n))O(a, ψ, Ω), = I(a, ψ, χ, λ, ω, Ω + Ad κβα (n)Σβα (n); aβ (n), θ∂n aβ (n), υβ (n), Υβ (n)) (4.17) × O(a, ψ, Ω + Ad κβα (n)Σβα (n)). Substituting this relation in (4.16) and taking into account (3.48) and (4.14), one finds that (4.19) IOα (n, θ) = sβα IOβ (n, θ) on Uβ ∩ Uα 6= ∅, where sβα = sgn det (κβα (n)E ).

(4.20)

sβα = ±1 is independent from n if Uβ ∩ Uα is connected, as is assumed for simplicity. Equation (4.19) is weaker than (4.1) because of the sign ambiguity associated with sβα . We shall come back to this point shortly. In general, (4.17) does not hold. However, one may change the local lifts according to 0 satisfy (4.17). Using (4.4), (4.12) and (4.13), recalling that (4.4) and impose that the Σβα 0 −1 κβα (n) = νβ (n) κβα (n)να (n) and going through steps analogous to those involved in the derivation of (4.10), it is straightforward to show that this amounts to solving the equation 0 Σβα (n) = Ad να (n)−1 Σβα (n) − (θ∂n + ad υα (n))χα (n) − (1/2)[χα (n), χα (n)] +Ad κβα (n)−1 (θ∂n + ad υβ (n))χβ (n) + (1/2)[χβ (n), χβ (n)]) , χα (n) = −q(aα (n))(θ∂n να (n)να (n)−1 )

(4.21)

0 for the να with Σβα fulfilling (4.17). Before posing the question of the existence of a solution of Eq. (4.21), one has to check whether it is consistent with (4.3) and with the matching relation (4.11). From (4.3) and (4.11), one has

Σγα (n) − Σβα (n) − Ad κβα (n)−1 Σγβ (n) = − Ad κγα (n)−1 ( Ad kβαγ (n)−1 − 1)Υγ (n). (4.22) If Σβα satisfies (4.17), then the left hand side of the above equation is valued in Lie Z, as b So must be the right-hand side. It seems Lie Z is invariant under the adjoint action of G. unlikely that this can come about unless the kβαγ are Zb valued. This conclusion remains unchanged even if Σβα does not satisfy (4.17), but Eq. (4.21) can be solved, since, under 0 the replacement (4.4), kβαγ (n) = νγ (n)−1 kβαγ (n)νγ (n) and Zb is an invariant subgroup b of G. From now on, we shall thus assume that the kβαγ are Zb valued. If this is not the case, the analysis below can still be carried out, but it becomes much messier and has no apparent sheaf theoretic interpretation.


745

b Zb valued 1From (4.3), it follows then that the Gb valued 1-cochain κ defines a G/ V2 ∗ cocycle κ. ¯ Equation (4.22) states then Σ = {Σβα } is a 1-cocycle of T N ⊗ Ad κ, ¯ which, on account of (4.11), is trivial [34]. This, however, is not sufficient to guarantee the solvability of Eq. (4.21) for the να . b Zb valued 1-cocycle κ¯ defines a principal G/ b Zb bundle AN on N . The existence The G/ of a solution ν = {να } of Eq. (4.21) is a condition much stronger than the simple triviality of the 1-cocycle Σ; it is strongly reminiscent of a flatness condition and may entail topological restrictions for AN . Recall that, in finite dimension, for any principal G bundle P on X with transition functions gba , the local g valued 1-forms gba −1 dgba V1 ∗ always define a trivial 1-cocycle of T X ⊗ Ad g, a fact indeed equivalent to the existence of a connection on P . So, gba −1 dgba = ma − Ad gba −1 mb for certain local g valued 1-forms mc . In order for the bundle P to be flat, it is necessary and sufficient that the mc can be chosen of the form mc = −dgc gc −1 for certain local G valued functions gc . This is a non-trivial requirement with topological implications for P . Let us come to the problem of the sign sβα in (4.19). With the action of Gb on E there is associated the vector bundle VN = AN ×Gb/Zb E on N . The transition functions of such a bundle are precisely the κβαE . From the definition of sβα , Eq. (4.20), it appears that requiring that sβα = 1 is tantamount to demanding that VN is oriented. From the above discussion, we draw the following conclusions. The Gribov problem b Zb bundle AN allows for the solution is solvable provided the topology of the principal G/ of Eq. (4.21) and the associated vector bundle VN is oriented. Let us consider the important case where N is the stratum B ∗ of regular Gb orbits. b as G(a) b = Zb for any irreducible In this case, the kβαγ are necessarily valued in Z, ∗ b Zb bundle element a ∈ A , so that our basic assumption is fulfilled. The principal G/ ∗ ∗ AN associated with κ ≡ κ¯ may be identified with A [32]. Correspondingly, the vector bundle VN is V ∗ = A∗ ×Gb/Zb E [32]. For N = B ∗ , (4.17) is automatically satisfied. This follows from (4.12)–(4.13), the fact that q(a) = q Lie Z for any irreducible a ∈ A∗ and the invariance of the Lie subalgebra Lie G. Therefore, there is no need to solve Eq. (4.21)! This shows that for the regular stratum B ∗ of the Gb orbit space B, the Gribov problem is solved, provided V ∗ is oriented. When N 6= B ∗ very little can be said. Though we do not have conclusive evidence, it seems unlikely that for a singular stratum N eq. (4.21) is consistent with (4.22). When the Gribov problem is solvable for the stratum N of B, the map O → IO ∗ defines a homomorphism of the equivariant cohomology Hequiv (A) of A into the coho∗ mology H (N ) of N . This is not difficult to show. Let X denote the collection of all fields but n, θ. For any insertion K(X), the relevant functional integrals are of the form Z (4.23) IK (n, θ) = dXesΨ (n,θ,X) K(X), where Ψ (n, θ, X) is the appropriate gauge fermion. Then, denoting by s|X the restriction of s to the fields X, one has Z θ∂n IK (n, θ) = dX(s − s|X )esΨ (n,θ,X) K(X) Z Z = dXesΨ (n,θ,X) s|X K(X) − dXs|X esΨ (n,θ,X) K(X) Z (4.24) = dXesΨ (n,θ,X) sK(X).

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R. Zucchini

This relation shows that IK (n, θ) is θ∂n closed (exact) whenever K is s closed (exact). Assume that N has no boundary to have a sensible intersection theory. For a given ∗ (A), its image IO ∈ H ∗ (N ) factorizes as O ∈ Hequiv IO (n, θ) = P DNF (n, θ) · JO (n, θ).

(4.25)

Here, P DNF is formally the Poincaré dual of the submanifold NF of N defined by the equivariant condition F (a) = 0 and is independent from O and JO is an element of H ∗ (N ). This is fairly evident from the local expression (3.48) of IO (n, θ) and from the structure of the integrand Iα (n, θ, a, ψ, χ, λ, ω, Ω) given in (3.44). Indeed, the Mathai– Quillen %, $ integral is just a regularized version of the boson/fermion δ function pair δ(F (a))δ(ψ∂F (a)) [23]. In concrete models NF is finite dimensional and JO has finite degree in H ∗ (N ). Were one able to define integration on N , the above formula would be the starting point for the study of the intersection theory of NF using field theory. This is however a technically non-trivial problem, since N is generally infinite dimensional. 5. The Donaldson–Witten Model In this section, we shall apply the formalism developed in the previous section to the Donaldson Witten model as an illustration. Let X be an oriented connected compact 4-manifold equipped with a metric h. Let G be a reductive compact Lie group. Finally, let B → X be a principal G bundle over X. Consider the space A = Conn (B) (5.1) of connections of B. A is an affine space modelled on Ω 1 (X, Ad B). Hence, for any a ∈ A, one has the canonical identification Ta A ' Ω 1 (X, Ad B). Thus, 6 ψ ∈ Π1 Ω 1 (X, Ad B).

(5.2)

Ta A carries a natural metric, the standard metric on Ω 1 (X, Ad B) associated with h and a negative definite Ad G invariant extension Tr of the Cartan–Killing form tr of g.7 Consider the gauge group Gb = Gau (B) (5.3) of the principal G bundle B and the subgroup G of Gb of the elements homotopic to the identity. One has the canonical isomorphism Lie G ' Ω 0 (X, Ad B). Thus, ζ ∈ Π1 Ω 0 (X, Ad B).

(5.4)

Lie G carries the standard metric on Ω 0 (X, Ad B) associated with h and Tr , which is Ad Gb invariant. To this there is associated a metric on G in standard fashion (cf. footnote 4 above). Πr Ω p (X, Ad B) denotes the space of Ad B valued R p forms of Grassmann degreep r. For any p, this metric is given by (α, β) = − Tr (α ∧ ∗β) for α, β ∈ Ω (X, Ad B). ∗ is the X Hodge star operator associated with h. Tr may be constructed as follows. Let z be the center of g. Let < ·, · > be a negative definite symmetric bilinear form on z. Now, g = g/z ⊕ z with g/z semisimple. Correspondingly, every x ∈ g decomposes uniquely as x0 + x0 , where x0 ∈ g/z and x0 ∈ z. Then, for x, y ∈ g, Tr (xy) = tr (x0 y 0 )+ < x0 , y0 >. Tr is manifestly Ad G invariant. If G has a discrete center Z, then g is semisimple and Tr reduces to the usual Cartan–Killing form. 6 7


747

b aγA = γ −1 dγ + Gb acts on A by gauge transformations: for a ∈ A and γ ∈ G, Ad γ −1 a. The vertical vector fields of A are of the form D(a)ξ for ξ ∈ Lie G, where D(a) is the usual gauge covariant derivative of the connection a ∈ A: D(a)ξ = dξ+[a, ξ]. The subgroup Zb of Gb acting trivially on A consists of the constant elements of Gb valued in the center Z of G, so that Zb ' Z. Correspondingly, the Lie algebra Lie Z of Z = Zb ∩ G consists of the constant elements of Lie G valued in the center z of g and Lie Z ' z. Lie Z is clearly contained in the kernel of D(a) for any a ∈ A. Since Lie G ' Ω 0 (X, Ad B), one has ω ∈ Π1 Ω 0 (X, Ad B),

(5.5)

Ω ∈ Π2 Ω 0 (X, Ad B). The Mathai–Quillen localization sector. The localization sector consists of two subsectors based on the infinite dimensional vector spaces 2 (X, Ad B), E = Ω−

(5.6)

F = Ω (X, Ad B).

(5.7)

1

E and F carry the standard metrics associated with h and Tr and the standard adjoint b The localization functions are the antiselfdual part of the curvature F− : action of G. A→E, (5.8) F− (a) = (da + (1/2)[a, a])− , as usual, and the map Hα : Uα × A × G → F, Hα (n, a, g) = Ad g −1 (D(a)gg −1 − a + aα (n)),

(5.9)

where aα (n) is a local gauge slice on Uα . It is a straightforward matter to check that F− (a) satisfies (3.2) and that Hα (n, a, g) satisfies (3.3), (3.4)–(3.5), (3.6)–(3.7) and (4.7). (ιa is just the isomorphism Ta A ' Ω 1 (X, Ad B) = F .) The Mathai–Quillen fields are 2 % ∈ Π−1 Ω− (X, Ad B), 2 $ ∈ Π0 Ω− (X, Ad B), ω¯ ∈ Π−1 Ω 3 (X, Ad B), τ¯ ∈ Π0 Ω 3 (X, Ad B).

(5.10)

The Weil projection sector. The Weil projection sector is identical to that of the customary Donaldson theory. The Weil fields are Ω¯ ∈ Π−2 Ω 0 (X, Ad B), ψ¯ ∈ Π−1 Ω 0 (X, Ad B).

(5.11)

The Faddeev–Popov gauge fixing sector. The gauge fixing function has the general form Σ(a, g) = D† (a)W (a, g) + µ2 ln g,

(5.12)

where W (a, g) is chosen to be W (a, g) = D(a)gg −1 .

(5.13)

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R. Zucchini

By an infinitesimal argument, it is easy to see that for µ2 > 0, the constraint Σ(a, g) = 0 implies g = 1 at least for g near 1. It is conceivable that solutions other than g = 1 exist. This would yield non perturbative effects that remain to be explored. It is simple to check that W (a, g) satisfies (3.15)–(3.17). The Faddeev–Popov fields are λ¯ ∈ Π−1 Ω 0 (X, Ad B), χ¯ ∈ Π0 Ω 0 (X, Ad B).

(5.14)

The zero mode sector. The field content of the zero mode sector is y(n) ¯ ∈ Π−2 Ω 0 (X, Ad B),

D(a(n))y(n) ¯ = 0,

x(n) ¯ ∈ Π−1 Ω 0 (X, Ad B),

D(a(n))x(n) ¯ = 0,

x(n) ∈ Π1 Ω (X, Ad B),

D(a(n))x(n) = 0,

0

y(n) ∈ Π2 Ω (X, Ad B),

D(a(n))y(n) = 0,

u(n) ∈ Π−1 Ω 0 (X, Ad B),

D(a(n))u(n) = 0,

v(n) ∈ Π0 Ω 0 (X, Ad B),

D(a(n))v(n) = 0,

r(n) ∈ Π0 Ω 0 (X, Ad B),

D(a(n))r(n) = 0,

z(n) ∈ Π1 Ω 0 (X, Ad B),

D(a(n))z(n) = 0.

0

(5.15)

The Lagrangian. From (3.43), the topological action Stop of the Donaldson–Witten model is given by R R R Stop =R− 41 X Tr ($ ∧ ∗$) − i RX Tr ($ ∧ F− (a))R− i X Tr (% ∧ (D(a)ψ)− ) 1 Tr (τ¯ ∧ ∗τ¯ ) − i X Tr (τ¯ ∧ Ad g −1 (D(a)gg −1 + 41 X Tr (% ∧ ∗[Ω, %]) − 4b X R − a + a(n))) − i X Tr (ω¯ ∧ (D(a)ζ + ψ − D(a)ω − Ad g −1 θ∂n a(n) R 1 + [ζ − ω, Ad g −1 (D(a)gg −1 − a + a(n))])) − 4b Tr (ω¯ ∧ ∗[Ω, ω]) ¯ X R R R ¯ +i ¯ ψ]) − i X Tr (D(a)ψ¯ ∧ ∗ψ) + i X Tr (D(a)Ω ∧ ∗D(a)Ω) Tr (ψ ∧ ∗[Ω, X R R 1 ¯ ψ − D(a)ω]) ∧ ∗D(a)gg −1 ) − 4t Tr (χ¯ ∧ ∗χ) ¯ − i X Tr ((D(a)χ¯ + [λ, X R − i X Tr (D(a)λ¯ ∧ ∗(D(a)ζ + ( Ad g − 1)(D(a)ζ + ψ − D(a)ω))) R R ¯ +i ¯ − i X Tr ((y(n) + θ∂n x(n)) ∧ ∗Ω) Tr (x(n) ∧ ∗(ψ¯ − [ω, Ω])) X R R ¯ + θ∂n y(n)) ¯ ∧ ∗ω) + i X Tr (y(n) ¯ ∧ ∗(Ω − (1/2)[ω, ω])) + i X Tr ((x(n) R R − i X Tr ((v(n) + θ∂n u(n)) ∧ ∗ ln g) + i X Tr (u(n) ∧ ∗f (g)ζ) R R ¯ +i + i X Tr ((z(n) + θ∂n r(n)) ∧ ∗λ) Tr (r(n) ∧ ∗χ). ¯ X (5.16) To write the renormalizable action, one needs to introduce the rescaled fields χ0 , λ0 , defined by (3.47) and (3,49), and the rescaled fields r0 (n), z 0 (n). Clearly, χ0 ∈ Π0 Ω 0 (X, Ad B), λ0 ∈ Π1 Ω 0 (X, Ad B),

(5.17)


749

whereas r0 (n), z 0 (n) are of the same type as r(n), z(n). Then, by (3.50), the flattened action Sflat of the Donaldson–Witten model reads R R R 1 Sflat = − 4b Tr (τ¯ ∧ ∗τ¯ ) + i X Tr (τ¯ ∧ (a − a(n))) + i X Tr (ω¯ ∧ (D(a)ω − ψ X R R 1 1 +θ∂n a(n) − [ω, a − a(n)])) − 4b Tr (ω¯ ∧ ∗[Ω, ω]) ¯ − 4t Tr (χ¯ ∧ ∗χ) ¯ X X R R ¯ ψ − D(a)ω]) ∧ ∗D(a)χ0 ) − i −i X Tr ((D(a)χ¯ + [λ, Tr (D(a)λ¯ ∧ ∗(D(a)λ0 X R R +[χ0 , ψ − D(a)ω])) − i X Tr ((v(n) + θ∂n u(n)) ∧ ∗χ0 ) + i X Tr (u(n) ∧ ∗λ0 ) R R ¯ +i +i X Tr ((z 0 (n) + θ∂n r0 (n)) ∧ ∗λ) Tr (r0 (n) ∧ ∗χ) ¯ + ···, X (5.18) where the ellipses denote the remaining terms of Sflat which are the same as the corresponding terms of Stop . The topological observables of the Donaldson–Witten model, obtained from the descent equation [12], are given by the well-known expressions R O0 (a, ψ, Ω) = 8π1 2 C0 tr (Ω 2 ), R O1 (a, ψ, Ω) = 8π1 2 C1 tr (2Ωψ), R O2 (a, ψ, Ω) = 8π1 2 C2 tr (2ΩF (a) + ψ ∧ ψ), (5.19) R 1 O3 (a, ψ, Ω) = 8π2 C3 tr (2ψ ∧ F (a)), R O4 (a, ψ, Ω) = 8π1 2 C4 tr (F (a) ∧ F (a)), where Ck is a k-cycle of X. (Recall that C4 = X and that C0 is a finite set of points of X.) Note that here the true Cartan–Killing form tr rather than its extension Tr appears. The basic assumptions of the Gribov analysis of Sect. 4 are fulfilled in the present model. Equation (4.5) is obviously satisfied, since Gb acts on E by the adjoint action and b Equation (4.6) is satisfied, this is trivial when restricted to the Z valued elements of Z. as appears from (5.19) by inspection, observing that the Cartan–Killing form tr vanishes on the z valued elements of Lie Z. Equation (4.7) is also satisfied as noticed earlier in this section. So, from the analysis of Sect. 4, it follows that not only for the standard case G = SU (2), SO(3) but also for a general compact group G the Gribov problem is solvable for the regular irreducible stratum B ∗ , provided V ∗ = A∗ ×Gb/Zb E is oriented. This conclusion, albeit based on formal manipulations of functional integrals rather than on rigorous mathematics, is the main result of this paper.

6. Concluding Remarks The real challenge of cohomological topological field theory is expressing intersection theory on moduli spaces in the language of local renormalizable field theory. This is not an easy task. While it is relatively easy to cook up local topological actions, showing their renormalizability is a non-trivial problem. The point is that locality and renormalizability are essentially field theoretic notions to which geometry may be quite indifferent. Our experience in gauge theories has taught us that it is difficult carry out gauge fixing salvaging renormalizability at the same time. However, if one cannot accommodate such principles into the framework, the so called topological field theories will just remain formal functional integrals.

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R. Zucchini

In the method presented here, one views the moduli space NF of self-dual connections as a finite dimensional submanifold of the infinite dimensional space N of gauge orbits. Correspondingly, integration of forms on NF is reduced to integration on N by wedging with the formal Poincaré dual χNF of NF . Needless to say, this procedure is rather formal and the possibility of its concrete implementation is unclear at the present moment. The natural question arises about whether our approach for the treatment of reducible connections and the analysis of the Gribov ambiguity may be used in the study of ordinary gauge theories where similar problems occur. Indeed this can be done upon performing obvious modifications: the s cohomologies relevant in topological and ordinary gauge field theories are respectively equivariant and BRS cohomology and these are essentially different. The application of the corresponding method in ordinary gauge theory would yield the same gauge fixing sector and the bosonic half the Mathai–Quillen τ¯ , ω¯ sector containing the equivariant functional Hα . It remains to be seen if this is going to provide useful insight. Acknowledgement. We are greatly indebted to R. Stora for providing his invaluable experience and relevant literature. We also thank the referee of the paper for useful suggestions and improvements.

References 1. Witten, E.: Introduction to Cohomological Field Theories, Lectures at the Trieste Workshop on Topological Methods in Physics, Trieste, Italy, June 1990, Int. J. Mod. Phys. A6, 2775 (1991) 2. Dijkgraaf, R.: Notes on Topological String Theory and 2-D Quantum Gravity. Presented at the Trieste Spring School School in String Theory and Quantum Gravity, Trieste, Italy, April 1990, proceedings, M. Green, R. Iengo, S Randjbar–Daemi, H. Verlinde editors, 3. Dijkgraaf, R.: Introduction to Topological Field Theories. In: Recent Directions in Particle Theory. Proceedings Boulder 1992, p. 689 4. Birmingham, D., Blau, M., Rakowski, M. and Thompson, G.: Topological Field Theories. Phys. Rep. 209 4 & 5, 129 (1991) 5. Blau, M. and Thompson, G.: 1992 Lectures on Topological Gauge Theory and Yang–Mills Theory. Presented at the Trieste Summer School in High Energy Physics and Cosmology, Trieste, Italy, June– July 1992, proceedings, E. Gava, K. Narain, S. Randjbar–Daemi, E. Sezgin and Q. Shafi editors, hepth/9305120 6. Blau, M. and Thompson, G.: Lectures on 2-D Gauge Theories; Topological Aspects and Path Integral Techniques. Presented at the Trieste Summer School in High Energy Physics and Cosmology, Trieste, Italy, June–July 1993, proceedings, E. Gava, A. Masiero, K. Narain, S. Randjbar–Daemi, and Q. Shafi editors, hep-th/9310144 7. Blau, M. and Thompson, G.: Localization and Diagonalization, a Review of Functional Integral Techniques for Low Dimensional Gauge Theories and Topological Field Theories. J. Math. Phys. 36, 2192 (1995) 8. Thompson, G.: New Results in Topological Gauge Theory and Abelian Gauge Theory. Presented at Trieste Summer School in High Energy Physics and Cosmology, Trieste, Italy, June–July 1995, hepth/9511038 9. Cordes, S., Moore, G. and Ramgoolam, S.: Lectures on 2-D Yang–Mills Theory, Equivariant Cohomology and Topological Field Theory. Presented at the Trieste Spring School in String Theory, Gauge Theory and Quantum Gravity, Trieste, Italy, April 1994, and to the Les Houches Summer School on Fluctuating Geometries in Statistical Mechanics and Field Theory, NATO Advanced Study Institute, Les Houches, France August–September 1994, Nucl. Phys. Proc. Suppl. 141, 184 (1995) (part 1), Les Houches Proceedings (part 2), hep-th/9411210 10. Labastida, J. M. F.: Topological Quantum Field Theory: A Progress Report. Presented at the 4th Fall Workshop on Differential Geometry and its Applications, Santiago de Compostela, Spain, September 1995, hep-th/9511037


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