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Center for Theoretical Physics

arXiv:hep-th/9110063v1 22 Oct 1991FUSION POTENTIALS FOR GkGkGkAND HANDLE SQUASHING*Michael CrescimannoCenter for Theoretical PhysicsLaboratory for Nuclear Scienceand Department of PhysicsMassachusetts Institute of TechnologyCambridge, Massachusetts 02139U.S.A.Submitted to: Nuclear Physics BTypeset in TEX by Roger L. GilsonCTP#2021October 1991* This work is supported in part by funds provided by the U.S. Department of Energy(D.O.E.) under contract #DE-AC02-76ER03069, and by the Division of Applied Mathematicsof the U.S. Department of Energy under contract #DE-FG02-88ER25066.0

ABSTRACTUsing Chern–Simons gauge theory, we show that the fusion ring of the conformal fieldtheory Gk (G any Lie algebra) is isomorphic toP [u](∇V ) where V is a polynomial in u and (∇V )is the ideal generated by conditions ∇V = 0. We explicitly construct V for all Gk.Wealso derive a residue-like formula for the correlation functions in the Chern–Simons theorythus providing an RCFT version of the residue formula for the topological Landau–Ginzburgmodel.

An operator that acts like a measure in this residue formula has the interpretation ofa handle-squashing operator and explicit formulae for this operator are given.1

I.INTRODUCTIONIn the study of rational conformal field theories (RCFT), the fusion algebra plays a centralrole. For example, Gepner et al.1 characterized the fusion rules of Gk Wess–Zumino–Witten(WZW) models and Verlinde2 displayed the connection between modular transformationsand the fusion coefficients.

Over the last few years there has been much progress in betterunderstanding the fusion rules of Gk theory.3−8, 32 A good introduction to RCFT and thefusion algebra may be found in Ref. [9].Recently, Gepner10 has conjectured that the fusion ring R of any RCFT is isomorphic toP [u](∇V ) where P[u] is a polynomial ring in ui over Z, the components of some finite-dimensionalvector u, and (∇V ) is the ideal generated by the ∇V = 0 where V is a polynomial in theui’s.V is called the fusion potential of the ring R.Gepner was able to show that thisconjecture was true for the RCFT SU(N)k. In this note we show this conjecture is true foran arbitrary Gk.

Furthermore, the extension of these techniques to arbitrary coset modelsappears promising.11 Since there is yet no classification of RCFT’s this seems to be about asfar as one can come at the present time to verifying the conjecture.In a related and somewhat parallel development, Witten12 has shown that Chern–Simonsgauge theory in three dimensions is closely connected with conformal field theory in twodimensions. This connection has been explored in many ways.

Understanding the holomorphicquantization and the connection to the KZ equation13 has been studied in Refs. [14 – 16] andunderstanding the modular transformations of the Hilbert space was detailed in Refs.

[17,8].Also, understanding how to explictly compute fusion algebra via Chern–Simons theory wasstudied in Ref. [8].2

In this paper we use Chern–Simons field theory to explicitly construct the fusion poten-tials of Gk. We then explore some of the ideas of Ref.

[10] in the context of Chern–Simonstheory to derive an interesting formula for the correlation functions of Chern–Simons gaugetheory. Perhaps not surprisingly, the correlators are given by a residue-like formula, reminis-cent of the correlators of topological models (see Ref.

[18–20,33]). This formula is interestingbecause it relates the measure in Gepner’s approach to the K matrix2, 21, 22 of a RCFT.This paper is organized as follows.

Section II is a short review of the canonical quantiza-tions of Chern–Simons gauge theory for the case of three-fold T 2 × R (T 2 is the torus). Thistechnique will be used in Section III where we first describe, in general, how to compute thefusion potential for Gk and then simply write them all down.

Finally, Section IV describescorrelation functions in Chern–Simons theory from Gepner’s point of view and it is there weencounter a connection between the measure in Gepner’s paper and the K matrix of RCFT.Section V is a short conclusion.II.CANONICAL QUANTIZATION OF THE CHERN–SIMONS THEORYON T 2 × RT 2 × RT 2 × RIn Ref. [12] Witten identified the Hilbert space of Chern–Simons theory as the space ofconformal blocks of the associated RCFT.

We will use this idea to reduce the computationof the fusion algebra of a Gk conformal field theory to a quantum mechanical computationon the Hilbert space of the associated Gk Chern–Simons theory. In what follows we assumefamiliarity with Refs.

[12,14,15,16,17,8] and use the conventions of Ref. [8] throughout.To begin, let us recall that we wish to canonically quantize the actionICS = k4πZNTrA ∧dA + 23A ∧A ∧A.

(2.1)3

A is the g-valued one-form on N, g the Lie algebra of G, where N will be taken to be thethree-fold Σ×R (Σ is a two-dimensional [Riemann] surface), k is, as usual by gauge invariance,an integer referred to as the level and Tr is the symmetric bilinear form of the Lie algebra gnormalized here so that in terms of the generators τ a of the G action Trτ aτ b= 2δab. Thisis the Chern–Simons theory related to the Gk RCFT.

To canonically quantize this action wemust first fix a time direction and choose a gauge. We choose the time axis to be along theR in N and choose the axial gauge At = 0.

The local coordinates on Σ are x1 and x2. Then,as discussed in the above references, we find that this gauge choice implies the superselectionrule (in analogy with electromagnetism it is called the Gauss’ law constraint)F12 = ∂1A2 −∂2A1 + [A1, A2] = 0(2.2)and also the action in this gauge (with ∂Σ = 0) isICS = k2πZNd3x Tr (A1∂tA2).

(2.3)Finally the observables of the theory are simply Wilson lines around non-trivial one-cycles inΣ,Oµ,c = Tr µP eRc A(2.4)where c is a one-cycle corresponding to some element of π1(Σ), µ is a representation of g inwhich the Tr is to be taken.We are not yet ready to quantize the action Eq. (2.3) because we need to include factorsthat come from the measure of the path integral.

As shown for example in Ref. [12,15] onecan very simply include these factors.

They lead to an additional term in the Lagrangian andare proportional to the original action Eq. (2.1).

Indeed when combined with Eq. (2.1) they4

result in simply shifting k to k + c where c is the quadratic Casimir of g, Thus again choosinggauge At = 0 and proceeding as before we see that the action we wish to quantize is simplythat of Eq. (2.3) with k replaced with k + c.For the remainder of the paper we specialize to the case Σ = T 2.

It is convenient to firstsatisfy the constraint Eq. (2.2) classically and quantize the remaining degrees of freedom.Note also that the gauge choice At = 0 does not fix the gauge completely: We may use a timeindependent and single-valued gauge transformation to make the gauge field on T 2 a constantvector field.

Then, classically, the constraint Eq. (2.2) implies that the two components ofthe vector field must commute in g, and so, in general, both are in the Cartan subalgebra ofg.

As such defineZc1A = aiνiZc2A = biνii = 1, 2, . .

. ,n = Rank G(2.5)where c1, c2 are the cycles associated to the two generators of π1 (T 2), νi are simple roots ofg,.

We may now plug Eq. (2.5) into Eq.

(2.3) with k shifted to k + c and using (c1, c1) = 0 =(c2, c2), (c1, c2) = 1 we haveICS = −k + c2πZdt aiCij∂tbj(2.6)where Cij = Tr (νiνj). We have now reduced Chern–Simons theory in N to just a quantummechanics problem where the ai and bj are just the coordinates and the momenta.Thecommutator in terms of ai, bj is,[ai, bj] = −2πik + cC−1ij[ai, aj] = 0 = [bi, bj].

(2.7)Rather than represent these operators on a Hilbert space we find a more convenient set ofoperators is suggested by the observables of the theory Eq. (2.4).

Whatever representation µ5

is chosen, Oµc will involve only exponentials of the ai’s and bi’s. It is thus natural to find aHilbert space realization ofAj = eiaj ,Bj = eibj(2.8)and so Eq.

(2.7) impliesAiBjA−1iB−1j= e2πik+c(C−1)ijAiAj = AjAi ,BiBj = BjBi . (2.9)This is analogous to a Weyl basis for the CCR.

We now simply define a vacuum |0⟩byAi|0⟩= |0⟩∀i (note A, B are unitary operators), and from Eq. (2.9) use the Bi’s as raisingoperators to generate the Fock space of states.

It is easy to see that due to the fact that thecommutator in Eq. (2.9) is idempotent, the spectrum of eigenvalues of the Ai’s will repeatafter some number of applications of the raising operators Bi’s.

We may thus consistentlytruncate this Fock space to a finite-dimensional Hilbert space, which we call the “Hilbertspace of the Gaussian model” because of the strong resemblance of Eq. (2.9) with those of asystem of n free bosons.This resulting finite-dimensional Hilbert space is not yet to be identified with the spaceof conformal blocks: the operators Ai are not invariant under the residual gauge invarianceassociated with Weyl transformations.As shown in Ref.

[8], under the Weyl action theGaussian Hilbert space described above breaks into Weyl covariant subspaces. Finally, imple-menting this remaining gauge invariance we project all the operators of the theory onto thecompletely Weyl-odd sector.

* It is natural from the point of the characters23, 24, 17 that oneshould identify this completely Weyl-odd sector with the conformal blocks. For more detailssee Ref.

[8]. * We mean states ψ s.t.

∀ω ∈W (W is the Weyl group) with ω2 = 1, that ωψ = −ψ.6

In the basis where Ai|0⟩= |0⟩∀i we have finally that the Oµ,c1 are diagonal on thesecompletely Weyl-odd states. Furthermore, because Oµ,c2 is composed of raising operatorsand is, by definition, even under all ω ∈W (W is the Weyl group) it is easy to see that Oµ,c2is a map from completely Weyl-odd states to themselves.

It is not hard to show that thesemaps are precisely the fusion matrices. In notation, let H denote the Hilbert space of statescorresponding to the conformal blocks.

They are labelled by representations sinceψ0 ∈H ,andψµ = Oµ,c2ψ0 . (2.10)one hasOµ,c2ψν = N τµνψτ(2.11)where N τµν are the fusion coefficients.

The proof of Eq. (2.11) is found by comparing themanipulations above to Refs.

[3,7]. Note the Oµ,c1 are just the diagonalization of the fusionmatrices Oµ,c2 under S;Oµ,c1 = S Oµ,c2S−1 .

(2.12)This is the Verlinde theorem.2, 26 For more details on the method described here see Ref. [8].III.FUSION POTENTIALSHaving described how one can compute the fusion rules of a Gk RCFT using Chern–Simons theory, we will now proceed to derive the fusion potential of any Gk.It is worth mentioning that there are other ways to characterize the fusion algebra, suchas finding the generating functions first introduced in Refs.

[27,28]. However, in this note wewill follow the spirit of Ref.

[10] where it is conjectured that the fusion ring of any RCFT isisomorphic toP [u](∇V ) where V (u) a polynomial is called the fusion potential.7

We will show that this is true for the conformal field theories Gk.Our strategy isas follows: We first show that there is a potential for the Gaussian model (described inSection II). We then demonstrate that the fusion rules of Gk are realized on a subvariety ofthe variety M defined by the ∇xV = 0 conditions of the Gaussian model, and show how theWeyl action naturally removes from M all the points except those on the subvariety.

Manyof the ideas in this section come from Ref. [10].

Although the Gaussian model of Eq. (2.9)is something like a “free field” decomposition of the theory (in that it has such a strongresemblance to a system of n free bosons) this author sees no firm connection between thisapproach and that of Ref.

[29]. As the reader will see below, the idempotency and “free-field”Gestalt of Eq.

(2.9) are the key notions that allow one to integrate the fusion rules to a singlepotential.Before displaying the potentials for Gk, we pause to more explicitly describe the method.Imagine assigning to each Ai operator of Eq. (2.8) a complex variable xi.

We will find apotential V (x) such that the ring of fusions in the Gaussian model, given by Eq. (2.9), willbe given byP [x](∇xV ).

In accordance with the general ideas of Gepner,10 the solutions x(⃗ℓ) of∇xV = 0 will be an affine variety M whose points will be in a 1 −1 correspondence with thestates |⃗ℓ⟩in the Hilbert space of the Gaussian model, the 1 −1 map being given byAi|⃗ℓ⟩= x(⃗ℓ)i |⃗ℓ⟩. (3.1)We thus see that the variety ∇xV = 0 is essentially isomorphic to ΛW /(k + c)ΛR.

We wishto ultimately only discuss the ring of fusions of Weyl-even operators, like the Wilson line ofEq. (2.4).

Thus consider the mapxi −→ui(x)(3.2)8

where ui are invariant under the action of the Weyl group (under the Weyl group, take thexi’s to transform as the Ai’s) and are taken to be the defining representations of g. Obviously,P[u] contains all the operators of Eq. (2.4).

We write the potential of the Gaussian model asa function of the ui’s. Then the ring R =P [u](∇uV ) is the fusion ring of Gk.

We can show thisas follows. Since∇xV =∂u∂x∇uV = 0on M we ask what subvariety is picked out by ∇uV = 0.

It is simply M minus the points forwhich det ∂u∂x= 0. Now it is simple to show that det ∂u∂xis completely Weyl-odd and thatit is, up to some trivial factors, the vacuum state (as viewed as an operator) of the conformalfield theory Gkψ0 ∝det∂u∂xxi=Bi|0⟩,(3.3)Thus det ∂u∂x= 0 at precisely those points on M that correspond to Weyl orbits of lengthless than |W|, the order of the Weyl group.

For those points of M on the Weyl orbits thathave length |W| we see that the map Eq. (3.2) maps all points in that orbit to the same point.Thus ∇uV = 0 corresponds to a subvariety with a point for each integral representation ofGk.

This may be readily seen by comparing the above construction to the construction ofthe space of conformal blocks for Gk described in Ref. [8].

Indeed, remembering that theOµ,c1Ai=xi ∈P[u] and writing Eq. (2.11) on the variety M;Oµ,c1Oν,c1S ψ0 = N τµνOτ,c1S ψ0we see that this equation only gives a condition on the product of polynomials in P[u] whenψ0 ̸= 0 which, as described above (see Eq.

(3.3)), is precisely at the points ∇uV = 0. Thisshows that R, the fusion ring of Gk, is given by byR =P[u](∇uV ) .9

We next construct the potentials V for Gk. As described above it will be enough to computethe V (x) of the Gaussian model.

The xi’s are the natural variables to write the potential inand are just the qi’s of Gepner10 in the case G = SU(N). For clarity of exposition we divideG into two classes; G simply-laced and G non-simply-laced.G Simply-LacedIf G is simply-laced then the matrix Cij of Eq.

(2.6) is just the Cartan matrix. Werecall that in a Gaussian fusion ring the “vanishing” conditions are just statements of theidempotency of the operators, for example in U(1)k, A2k = 1l = B2k.

(Recall that we haveconvention that Tr (τ aτ b) = 2δab.) It is not difficult to recognize that idempotency of theoperators in Eq.

(2.8) and Eq. (2.9) implies the “vanishing” conditions (and thereby the fusionrules) of the Gk theory.

We study idempotency in the Gaussian model of Eq. (2.9) in thefollowing way: We find a list of n(= Rank G) linearly independent vectors ⃗ri of Zn withsmallest integer components such that for each ⃗ri,nQjArijj= 1l (1l in the Hilbert space of theGaussian model).

Since Aℓ|0⟩= |0⟩∀ℓ= 1, . .

. , n it is enough (by Schur’s lemma) to requireQjArijjcommutes with all other operators of the theory, namely we requirenYjArijj , Bℓ= 0∀ℓ, i .For simply-laced G it is very easy to characterize this set of vectors.

By virtue of Eq. (2.9) apossible set of the ⃗ri is just given by the rows of the Cartan matrix Aij multiplied by k + c,rji = (k + c) Aij .

(3.4)10

Now returning to the notion of the vanishing conditions as specifying a variety we wish tosolve the n simultaneous conditionsnYjxrijj= 1∀i = 1, . .

. , n .This may be easily done and here we simply write down the potential whose gradient ∇xV = 0are the conditions above.

* Details of the particular case of SU(N) is in Appendix A, addedas an aid to the reader. (AN−1)k = SU(N)k(AN−1)k = SU(N)k(AN−1)k = SU(N)k(Rank = N −1).

The fusion potential isV =xN(k+N)+11N(k + N) + 1 −x1 +N−1Xi=2 α(k+N+1)ik + N + 1 −αi! (3.5)in which the xj’s (that correspond to eigenvalues of the Aj’s of Eq.

(2.8)) are given byxj = αjxj1, 2 ≤j < N −1. (Note that the Jacobian in going from xi’s to αi’s is alwaysnon-zero.

)(Dℓ)k(Dℓ)k(Dℓ)kLet κ = k + c. We find that one must distinguish the two cases ℓ=even and ℓ=odd.Thus we find,V = xRκ+11Rκ + 1 −x1 + xRκ+12Rκ + 1 −x2 +ℓ−2Xj=3 ακ+1jκ + 1 −αj! (3.6)where R = 2 is ℓis even and R = 4 is ℓis odd.

The other xj, 3 < j ≤ℓare:xj = αjx−(j−2)mod R1xj mod R2. (3.7)* Care must be taken when solving these equations not to remove points or introduceadditional images of the variety.

See Appendix A.11

(E6)k(E6)k(E6)kAgain let κ = k + c. The potential is,V = xκ+11κ + 1 −x1 + x3κ+123κ + 1 −x2 +6Xj=3 ακ+1jκ + 1 −αj! (3.8)where the xj 3 ≤j ≤6 arex3 = α3x22x4 = α4x5 = α5x2x6 = α6x22 .

(3.9)(E7)K(E7)K(E7)KLet κ = k + c. The fusion potential for (E7)k isV = x2κ+112κ + 1 −x1 +7Xj=2 ακ+1jκ + 1 −αj! (3.10)where the xj 2 ≤j ≤7 arex2 = α2x3 = α3x4 = α4x5 = α5x1x6 = α6x7 = α7x1 .

(3.11)(E8)k(E8)k(E8)kWe let κ = k + c. The fusion potential is,V = x2κ+112κ + 1 −x1 +8Xj=2 ακ+1jκ + 1 −αj! (3.12)where the xj, 2 ≤j ≤8 arex2 = α2x3 = α3x4 = α4x5 = α5x1x6 = α6x8 = α8x7 = α7x1(3.13)12

G Non-Simply-LacedIf G is non-simply-laced then the matrix Cij of Eq. (2.6) will not be the Cartan matrix.However, the “vanishing” conditions are still a result of the idempotency of the Ai’s and Bj’sof Eq.

(2.8) and Eq. (2.9) and one may modify the argument for the simply-laced case.

Nowthe ⃗ri vectors will correspond to rows in the matrix Cij.rji = (k + c)Cji . (3.14)We will call k + c = κ throughout.

We will now write down the Cji matrix for each non-simply-laced group and also write down the corresponding potential for Gk. (Bℓ)k(Bℓ)k(Bℓ)k(ℓ≥2)The Cij matrix is (ℓ× ℓ),4−2−24−2−2...4−2−22(3.15)and the corresponding potential is,V =x2(ℓ−1)κ+112(ℓ−1)κ + 1 −x1 +ℓXi=2 α2κ+1j2κ + 1 −αj!

(3.16)where the xj, 2 ≤j ≤ℓare given as xj = xj1.13

(Cℓ)k(Cℓ)k(Cℓ)k(ℓ≥3)The Cij matrix is (ℓ× ℓ),2−1−12−1−1...2−2−24. (3.17)In writing the potential down it is convenient to distinguish the cases ℓeven and ℓodd.

Wefind:ℓeven:V = x2κ+112κ + 1 −x1 + x2κ+1ℓ2κ + 1 −xℓ+ℓ−1Xj=2 ακ+1jκ + 1 −αj! (3.18)where for 2 ≤j ≤ℓ−1, xj = αxj mod 21andℓodd:V = x4κ+1ℓ4κ + 1 −xℓ+ℓ−1Xi=1 ακ+1jκ + 1 −αj!

(3.19)where x1 = α1x2ℓand xj = αjxj mod 22for 2 ≤j ≤ℓ−1. (F4)k(F4)k(F4)kThe Cij matrix is,2−1−12−2−24−2−24(3.20)and the corresponding potential isV = xκ+11κ + 1 −xi + xκ+12κ + 1 −x2 + xκ+132κ + 1 −x3 + x2κ+142κ + 1 −x4 .

(3.21)14

(G2)k(G2)k(G2)kThe Cij matrix is,"2−3−36#(3.22)and the potential is,V = xκ+11κ + 1 −x1 + x3κ+123κ + 1 −x2 . (3.23)IV.CORRELATION FUNCTIONS AND HANDLE-SQUASHINGIn this section we use the ideas of Ref.

[10] and some elementary facts about the Gaussianmodel to suggest an interesting connection between the measure (for inner products andcorrelators) used by Gepner10 and the K matrix of Verlinde2, 34 (see Bott22). The Gaussianmodel will suggest a simple formula for K−1.We begin by noting that all the potentials of the last section look simply like the potentialof a theory made by tensoring some number of “free fields.” The many Maxwell conditionsone would have if one tried to combine the vanishing conditions of Gk into one potentialmight look very restrictive but in these “free field” variables the Maxwell conditions aretrivial and have no content.

Indeed, integrating the “vanishing” conditions seems artificialand only serves to make contact with what was done for Landau–Ginzburg models:19, 20, 30integration and differentiation of a C-valued variable xi with respect to xi makes sense but(thinking of the map Eq. (3.2) above) such an integration or differentiation in the space ofintegrable representations does not seem to have a natural interpretation in the conformalfield theory.

* So, instead of proceeding as Gepner10 has by defining an inner product (and* Note that the Jacobian of partials of the map Eq. (3.2) does have the loose interpretationas a map from the Wilson line representations to the space of states in Gk.15

thereby correlators) as integrations of polynomials with respect to some measure, we seek away of defining the inner product that is more natural vis-a-vis the conformal field theory.To motivate our method recall that in the variety ∇uV = 0 each point corresponds toan integrable representation (and therefore to a state) of the rational conformal field theory.Furthermore, by Eq. (3.1) each point’s x(ℓ)i -value is the entry of the matrix Ai along thediagonal in the (ℓ, ℓ)th positions.

Also recall that the inner product on the Hilbert spaceof Gk really came from the inner product on the Hilbert space of the Gaussian model, thenorm of which was set by ⟨0|0⟩= 1. Finally, we know that in terms of the raising operators(the Bi’s) of the Gaussian model there exists a distinguished polynomial Γ(B) such that thevacuum state ψ0 of the Gk may be written as,ψ0 = Γ(B)|0⟩.

(4.1)Now combining all these ideas we have a description of the inner product in terms of theoperators of the conformal field theory,δij = (ψi, ψj) =ψ0, O¯i,c2Oj,c2ψ0=0Γ+(B)O¯i,c2Oj,c2Γ(B) 0=0S+O¯i,c1Oj,c1Γ+(A)Γ(A)S 0. (4.2)Now, for a Gaussian model S|0⟩=1√RRPℓ|ℓ⟩(sum runs over all states of the Gaussian model)and thus(ψi, ψj) = δij = |W|R Tr HO¯i,c1Oj,c1Γ+(A)Γ(A)(4.3)where the trace is taken over, H, just the states in Gk (the Γ’s project out everything in theGaussian Hilbert space except the states in Gk).

|W| is the order of the Weyl group and R isthe total number of states in the Gaussian Hilbert space, i.e.R =ΛW(k + c)ΛR.16

Further, we may write this as(ψi, ψj) = δij = |W|RX∇uV =0O¯iOjΓ+Γ(x)(4.4)where the sum is over the variety ∇uV = 0. Thus “integration” may be understood as asum of the values of a function evaluated on the points of the variety.

(The Oi and Oj inEq. (4.4) are “polynomials” of Gepner as described in Section III).

Equation (4.4) bears astriking resemblance to the residue formulae of Refs. [18 – 20] and is, in a sense, the RCFTversion of those formulae.

Note that this is a genus one formula and its generalization to thespace of operators in higher genus is not immediately obvious.Note that although Γ is completely Weyl-odd (see Eq. (4.1)) the operator Γ+Γ is Weyl-even.

Γ+Γ is also positive and real. Furthermore, by constructionΓ+Γ(B), Oµ,c2= 0∀µ .

(4.5)Γ+Γ is the conformal field theory analogue of Gepner’s measure.10 It may be expressed as avector in the space of the operators Oµ. This follows by virtue of Eq.

(4.3).It is not difficult to show that |W|R Γ+Γ = K−1, the handle squashing operator.23, 33, 34For example, Fig. 1 is a diagrammatic picture of Eq.

(4.3). One may prove |W|R Γ+Γ = K−1directly by using the Verlinde formula Eq.

(2.12). Indeed suppose there exists an M such that[Oµ,c1, M] = 0∀µ(4.6)and,δij = Tr HO¯i,c1 M Oj,c1.17

By Eq. (4.6) such an M defines a Hermitian bilinear form on the operators Oµ,c1 Thus wehaveδij =XℓψℓO¯i,c1Ok,c1M ψℓ=XℓN m¯ij (ψℓ|Om,c1M| ψℓ)(4.8)and usingOiclψj = S+jiSj0ψjand Eq.

(2.12) we find that(ψℓ|M|ψℓ) = Mℓℓ=Sℓ02.Thus MK = 1l.One final remark is in order. We can use the above descriptions of Gk to give an expressionfor K−1.

From the method described in Section II it is clear that Γ(B) is simply characterizableas the operator in the Gaussian model that is associated to the completely Weyl-odd state(ψ0 the vacuum of Gk, see Eq. (4.1)) containing the vector |ρ⟩where ρ = 12Pα∈∆+ α. Thisgives one an explicit way of computing K−1 in the Hilbert space of Gk.

As an example wegive the following formulae for the case G = SU(2) and G = SU(3);G = SU(2)Γ(B) = B −B−1√2K−1 =12(k + 2) (3O1 −O3)G = SU(3)Γ(B) =1√6B1B2 −B−11 B22 + B−21 B2 −B21B−12+ B1B−22−B−11 B−12K−1 =ΛW(k + 3)Λr−1 9O1 −6O8 + 3O10 + O10−O27(4.9)These formulae are true for all k. It is intriguing that K−1 has this universal description.18

V.CONCLUSIONSIn this note we have shown that there is a fusion potential for all Gk and have used theideas of Ref. [10] and Ref.

[8] to motivate a rather explicit description of the handle squashingoperator K−1.In closing we note that many of these notions seem to allow simple generalizations tocoset models11 and that a more group-theoretic approach to the fusion potentials is beingpursued by Schnitzer,31 and that recently there has been progress in studying the fusion rulesfrom the N = 2 Landau–Ginzburg approach.3519

REFERENCES1. D. Gepner and E. Witten, Nucl.

Phys. B278 (1986) 493.2.

E. Verlinde, Nucl. Phys.

B300 (1988) 389.3. M. A. Walton, Nucl.

Phys. B340 (1990) 777.4.

M. Spiegelglas, Phys. Lett.

B247 (1990) 36.5. M. Spiegelglas, Phys.

Lett. B245 (1990) 169.6.

V. G. Kaˇc, talk presented at the Canadian Mathematical Society Meeting on Lie Algebrasand Lie Groups (University of Montr´eal, August 1989).7. J. Fuchs and P. van Driel, Nucl.

Phys. B346 (1990) 632.8.

M. Crescimanno and S. A. Hotes, “Monopoles, Modular Invariance and Chern–SimonsField Theory,” preprint UCB-PTH-90/38.9. P. Ginsparg, in Les Houches Lectures (1988).10.

D. Gepner, “Fusion Rings and Geometry,” preprint NSF-ITP-90-184.11. M. Crescimanno and S. A. Hotes, in preparation.12.

E. Witten, Commun. Math.

Phys. 121 (1989) 351.13.

V. Knizhnik and A. B. Zamolodchikov, Nucl.

Phys. B247 (1984) 83.14.

M. Bos and V. P. Nair, Int. Journ.

Mod. Phys.

A5 (1990) 959; Phys. Lett.

B223(1989) 61.15. S. Elitzer, G. Moore, A. Schwimmer and N. Seiberg, Nucl.

Phys. B326 (1989) 108.20

16. A. P. Polychronakos, Ann.

Phys. 203 (1990) 231.17.

T. R. Ramadas, I. M. Singer and J. Weitsman, Commun. Math.

Phys. 126 (1989) 409.18.

R. Dijkgraff, H. Verlinde and E. Verlinde, Nucl. Phys.

352 (1991) 59.19. C. Vafa, Mod.

Phys. Lett.

A 6 (1991) 337.20. K. Intrilligator, “Fusion Residues,” Harvard University preprint HUTP-91/A041.21.

H. Verlinde and E. Verlinde, “Conformal Field Theory and Geometric Quantization,”Princeton University preprint PUPT-89/1149.22. R. Bott, Surveys in Diff.

Geom. 1 (1991) 1.23.

V. G. Kaˇc and D. H. Peterson, Adv. in Math.

53 (1984) 125.24. V. G. Kaˇc and M. Wakimoto, Ad.

in Math. 70 (1988) 156.25.

C. Imbimbo, Phys. Lett.

B258 (1991) 353.26. G. Moore and N. Seiberg, Phys.

Lett. B212 (1988) 451.27.

C. J. Cummins, P. Mathieu and M. A. Walton, Phys. Lett.

B254 (1991) 386.28. L. B´egin, P. Mathieu and M. A. Walton, “New Example for a Generating Function forWZWN Fusion Rules,” preprint LAVAL-PHY-23/91.29.

D. Gepner and R. Cohen, Mod. Phys.

Lett. A6 (1991) 2249.30.

M. Spiegelglas, “Setting Fusion Rings in Topological Landau–Ginzburg,” Technionpreprint PH-8-91.31. H. Schnitzer, private communication.21

32. M. Bourdeau, E. Mlawer, H. Riggs and H. Schnitzer, “The Quasirotational Fusion Struc-ture of SO(M|N) Chern–Simons and QZW Theories,” Brandeis preprint BRX–TH–319.33.

E. Witten, Nucl. Phys.

B340 (1990) 281.34. E. Verlinde and H. Verlinde, “Conformal Field Theory and Geometric Quantization,”Princeton preprint PUPTH–89–1149.35.

D. Nemeschansky and N. P. Warner, “Topological Matter, Integrable Models and FusionRings,” USC preprint USC-91/031.22

APPENDIX AIn this Appendix we describe how one solves the conditionsnQjxrjij= 1. The ⃗ri’s corre-spond to rows in some matrix, Cij.

A little thought indicates that the algebraic manipulationsused in solvingnQjxrjij= 1 correspond to finding a matrix aℓm with aℓm ∈Z and det |aℓm| = 1such that a new Cij defined throughCnewij = Ciℓaℓj(A.1)has a simpler form than Ciℓ. That is, if ⃗ri’s are taken to be the rows of Cnewij then wewish to find aℓj’s such that in these new ⃗ri’s the conditionsnQjxrjij= 1 all involve at mosttwo different xi’s each.

Note the conditions that aℓm ∈Z and det |aℓm| = 1 are just therequirement that the new system (in the ⃗rs’s taken as rows in Cnewij) has neither added tonor removed points from the variety specified by the original set of conditions. An aℓm mayvery simply be found for all the Cij’s in the text by careful row reductions Cij.For example, for SU(N) it is not difficult to show that with anaiℓ=1xx′.

. .01x′...001.........1(N −1) × (N −1)(A.2)(where the x’s means some integers) that the Cijnew (= the Cartan matrix for the case ofSU(N)) may be put into the formCijnew =2−1.........0−1000−10N −10−1N00.

. .0(A.3)from which follows the potential Eq.

(3.5).23

FIGURE CAPTIONFig. 1:The operator |W|R Γ+Γ is K−1, the handle-squashing operator.24

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