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CERN-TH.6761/92 December, 1992 NONPERTURBATIVE SOLUTION OF THE SUPER-VIRASORO CONSTRAINTS 3 9 9 1 n a J 6 ⋆ K. Becker 1 v and 7 1 M. Becker 0 † 1 0 Theory Division, CERN 3 9 CH-1211 Geneva 23, Switzerland / h t - p e h : ABSTRACT v i X r We present the solution of the discrete super-Virasoro constraints to all orders a of the genus expansion. Integrating over the fermionic variables we get a representation of the partition function in terms of the one-matrix model. We also obtain thenonperturbativesolutionofthesuper-Virasoroconstraints inthedoublescaling limit but do not find agreement between our flows and the known supersymmetric extensions of KdV. ⋆ †Permanentaddress: UniversitätBonn,PhysikalischesInstitut, Nussallee12,W-5300Bonn 1, Germany. 1.Introduction. Matrix models provide us with powerful methods to evaluate the sum over ge- ometries in two dimensional quantum gravity. Forc 1 conformalfield theories [1] ≤ coupled to gravity, n-point correlation functions of gravitationally dressed primary fields have been calculated within this formulation on world sheets with arbitrary topology, a result that is far from being obtained from the continuum Liouville approach [2]. It turns out that matrix models have a close connection to integrable systems [3], which is manifest through the appearence of the Korteweg-de Vries hierarchy in the double scaling limit [4] [5]. The recent discovery of Kontsevich matrix integrals has shown a natural connection to topological gravity [6] [7]. Despite of this important results, the generalization to the supersymmetric case is con- nected with difficulties. So for example, the role which the known supersymmetric extensions of KdV [8] play in this context remains obscure [9][10] . For N = 1 superconformal field theories coupled to two dimensional supergravity correlation functions for spherical topologies have been obtained in [11], using the continuum super-Liouville approach. The torus path integral and the spectrum of physical operators for c 1 superconformal models coupled to 2D-supergravity have been ≤ considered in [9] [12]. However the more powerful counterpart in terms of super- b matrix models has not yet been found [13]. To shed some light on this situation, the authors of [14] proposed an “eigenvalue model”, which is described in terms of N even and N-Grassmann odd variables. The basic idea for the construction of the model was the generalization of the Virasoro constraints known from the ordinary one matrix model [15] [16] to the N = 1 supersymmetric case. Gener- alizing Kazakov’s loop equations [17] to N = 1 superloop equations , the model was evaluated for spherical topologies in [14], obtaining agreement with correlation functions [11] and critical exponents [18] [19] of (2,4m) superconformal field theories coupled to world-sheet supergravity. Further analysis of the model carried out in [20] [21] showed, that the continuum super-Virasoro constraints are described in terms of a c = 1 theory, with a Z -twisted scalar and a Weyl Majorana fermion in 2 the Ramond sector. The solution of the continuum superloop equations in the first b 1 few orders of the genus expansion and a general heuristic argument showed [20], that there might be a close relation between the bosonic piece of the free energy and the free energy of the one matrix model. In this letter we analyze the solution of this model to all orders of the genus expansion. After a brief review about its construction (section two), we analyze the solution of the discrete super-Virasoro constraints in section three. Integrating over the fermionic variables, we get a representation of the partition function in terms of the one matrix model. In section four we analyze the solution of the super-Virasoro constraints in the double scaling limit and compare our flows with the known supersymmetric extensions of KdV [8]. 2.Construction of the model. The bosonic one matrix model is defined as: N (g ,N) = dN2Φexp[ tr g Φk] , Λ = e−µ (1) B k k B Z −Λ B Z k≥0 X where Φ is a Hermitian N N matrix and µ is the bare cosmological constant. It × satisfies the Virasoro constraints [15]: L = 0 , n 1 ; [L ,L ] = (n m)L n B n m n+m Z ≥ − − Λ2 n ∂2 ∂ L = B + kg , (2) n N2 ∂g ∂g k∂g n−k k k+n k=0 k≥0 X X which can be derived from (1) by implementing invariance under the shift Φ → Φ+εΦn+1. We obtain a relation to a free massless boson, if we introduce an infinite set of creation and annihilation operators related to the coupling constants N Λ √2 ∂ B α = ng , n > 0 ; α = , n 0 . (3) −n n n −ΛB√2 − N ∂gn ≥ The Virasoro generators L = 1 : α α : are the modes of the energy n 2 −k k+n momentum tensor T(p) = L p−n−2 of the scalar field ∂X(p) = α p−n−1 = n P n P P 2 ∂X+(p) + ∂X−(p). From (1) and (3) we see that the potential of the model is related to the negative modes of the scalar field 1 lim −1∂X− V′(p) = kg pk−1, (4) B B k N→∞ NZ Z ∼ k>0 X As a generalization to the N = 1 supersymmetric case, we take a c = 1 free massless superfield X(p,Π) = x(p) +Πψ(p) with energy momentum tensor T(p,Π) = b DX∂X = ψ∂ x + Π : (∂ x∂ x + ∂ψψ) : and the mode expansion ∂X(p) = p p p α p−n−1, ψ(p) = b p−r−1/2. The relation to the c = 1 super-Virasoro al- n r gebra: P P b (m3 m) [L ,L ] = (n m)L + − δ , (5) m n n+m n+m,0 − 8 m [L ,G ] = ( r)G , (6) m r m+r 2 − 1 1 G ,G = 2L + (r2 )δ , (7) r s r+s r+s,0 { } 2 − 4 and the coupling constants of the model is introduced, if we define the modes as: Λ ∂ N S α = ; α = pg , p = 0,1,2,... p −p p − N ∂g −Λ p S Λ ∂ N S b = ; b = ξ , p = 0,1,2,... . (8) p+21 − N ∂ξp+1 −p−21 −ΛS p+21 2 In terms of the coupling constants the Virasoro generators then become: 3 ∞ ∂ ∞ ∂ Λ2 n−1 ∂ ∂ G = ξ + kg + S n 0 , (9) n−21 k+12∂gk+n k∂ξk+n−1 N2 ∂ξk+1 ∂gn−1−k ≥ Xk=0 Xk=0 2 Xk=0 2 ∞ ∂ ∞ n ∂ Λ2 n ∂2 L = kg + ( +r)ξ + S + n k∂g 2 r∂ξ 2N2 ∂g ∂g k+n n+r k n−k Xk=1 rX=21 Xk=0 n−1 Λ2 2 n ∂2 S ( +r) n 1 . (10) 2N2 2 ∂ξ ∂ξ ≥ − r n−r rX=21 In analogy to the one matrix model the potential is determined from the negative modes of the superfield DV(p,Π) DX−: ∼ V(p,Π) = (g pk +Πξ pk) . (11) k k+1/2 k≥0 X This suggests the introduction of N even-and N Grassmann-odd eigenvalues, obtaining for the partition function: N N N Z (g ,ξ ,N) = dλ dθ ∆(λ,θ)exp (g λk +ξ θ λk) . S k k+21 i i −ΛS k i k+21 i i  Z i=1 k≥0 i=1 Y XX  (12) The generalization of the usual Vandermonde determinant, ∆(λ,θ), is determined by implementing invariance of (12) under the super-Virasoro constraints (9) (10), which yields a differential equation for the measure: ∂ ∂ λn λn λn( +θ )∆ = ∆ θ i − j , (13) i i i −∂θ ∂λ λ λ i i i j i i6=j − X X whose unique solution up to an irrelevant constant is ∆ = (λ λ θ θ ). i<j i− j− i j The model that we want to solve in the large N limit is thus: Q 4 N N N Z = dλ dθ (λ λ θ θ )exp (g λk +ξ θ λk) . S i i i − j − i j −ΛS k i k+21 i i  Z i=1 i<j k≥0 i=1 Y Y XX  (14) In (14) N is taken to be even in order to get a Grassmann even partition function. 3. Solution of the discrete model. We analyze the solution of the discrete superloop equations [14] [20] or equiv- alently the discrete super-Virasoro constraints through the representation (14) of the partition function. We first consider the case without fermionic couplings ξ = 0. The partition function can be written as a function of the Pfaffian of k+1 2 the antisymmetric matrix M = λ−1, where then λ = λ λ , i = j, and the ij ij ij i − j 6 usual Vandermonde ∆(λ) = (λ λ ) [14]: i<j i − j Q 2N 2N 2N Z (g ,ξ = 0,2N) = dλ ∆(λ)Pf2N(λ−1)exp( g λk) . (15) S k k+21 i ij −ΛS k i Z i=1 k i=1 Y XX Here the product of the Vandermonde and the Pfaffian is totally symmetric under exchange of two eigenvalues. We want to show, that the relation of the partition function (15) to the bosonic partition function of the one matrix model (1) is given by: 1 2N Z (g ,ξ = 0,2N) = 2(g ,N) . (16) S k k+21 2N N ZB k (cid:18) (cid:19) To prove (16) we show, that the following equivalence for the measures holds: 5 2N 1 Pf2N(λ−1) (λ λ ) = SI,J∆2(I)∆2(J) , (17) ij i − j 2N i<j I,J Y X Here we have introduced the notation I = (λ ,...,λ ), i < ... < i (same for i1 iN 1 N J), and SI,J means symmetric permutations between the elements of I and J. The equality (16) is a consequence of (17), since the partition function (15) with 2N 2N eigenvalues will then factorize into products of two independent one matrix N models with N eigenvalues. (cid:0) (cid:1) Obviously (16) holds for the case of two eigenvalues, so that to illustrate the situation we start with the first nontrivial case of four eigenvalues, N = 2. We will consider the right and left hand side of (17) as polynomials Q(λ), P(λ) respectively in λ = λ : 4 P(λ) = (λ λ )(λ λ )λ λ (λ λ )(λ λ )λ λ +(λ λ )(λ λ )λ λ (18) 2 3 12 13 1 3 12 23 1 2 13 23 − − − − − − − 1 Q(λ) = [(λ λ )2λ2 +(λ λ )2λ2 +(λ λ )2λ2 ] . (19) 3 12 2 13 1 23 2 − − − From P(λ ) = Q(λ ) for i = 1,2,3 it follows P(λ) = Q(λ), λ. i i ∀ We show that (17) is correct by induction over N. ForN > 2 we set λ = λ 2N+2 so that P(λ), Q(λ) become polynomials of degree 2N 2 in λ. It is enough to − show that they coincide for λ = λ since the symmetry under the interchange 2N+1 of λ and λ guarantees P(λ ) = Q(λ ) for i = 1,...,2N and thus Q(λ) = P(λ) i j i j N. If we consider both polynomials of (17) for N N + 1 and λ = λ we 2N+1 ∀ → obtain the desidered result by applying the induction hypothesis. 1 Q(λ ) = SI,J∆2(I)∆2(J) (λ λ )2(λ λ )2 2N+1 2N i − 2N+1 j − 2N+1 XI,J i1<Y...<iN j1<...<jN 6 2N = Pf2N(λ−1) (λ λ ) (λ λ )2 (λ λ )2 = P(λ ) . ij i− j i− 2N+1 j− 2N+1 2N+1 Yi<j i1<Y...<iN j1<Y...<jN (20) We conclude that the bosonic part of the supersymmetric (2N 2N) model is × proportional to the square of the corresponding (N N) bosonic one matrix model × (16). With this result we can already find the form of the discrete string equation for arbitrary genus. Writing Z = eN2FS, equation (16) implies, that the fermionic independent (0) piece of the supersymmetric free energy F is related to the free energy of the S one matrix model (up to an irrelevant additive constant): B F (0) B F = F . (21) S 2 Since the dependence of the free energy on g is trivial ∂F = 1/Λ, (21) implies 0 ∂g0 − a rescaling of the cosmological constant Λ = 2Λ . This relation can also be S B obtained, if we evaluate the bosonic piece of the L constraint (10): 0 ∞ ∂ Λ = 2 kg ∂ Λ2 FB . (22) S k ΛB − B ∂g k k=1 (cid:18) (cid:19) X Up to a factor of two the cosmological constant satisfies the same string equation as in the one matrix model. Including the fermionic couplings, the following equality for the second order in fermions holds: ∂2Z (2N) 1 2N ∂2 (N) ∂ (N)∂ (N) S B B B = Z (N) Z Z (k n) . ∂ξ ∂ξ 2N−1 N ∂g ∂g ZB − ∂g ∂g − ↔ k+21 n+21 (cid:18) (cid:19)(cid:18) k+1 n k+1 n (cid:19) (23) The proof of this relation is straightforward, using the relation: 7 2N 2N λkλn dθ ∆(λ,θ)θ θ = α β i α β α,β=1 Z i=1 X Y N 1 SI,J∆2(I)∆2(J) (λk+1λn λk+1λn ) (k n) , (24) 2N−1 iα iβ − iα jβ − ↔ I,J α,β=1 X X which can be shown using equation (17). Formula (23) is purely bosonic in the sense, that after derivation we set the fermionic couplings to zero. Equations (21) and (23) imply that the free energy until the second order in fermionic couplings is given by: 1 ∂2 B B F = F ξ ξ F . (25) S 2 − 2 k+21 n+21∂gk+1∂gn k,n X Since thesolutiontothecontinuum super-Virasoroconstraints willbeanalyzed in a moment, it will then become clear that (25) is the complete expansion of the free energy and no dependence on more than two fermionic couplings appears. Equation (25) gives a representation of the model (12) in terms of the one matrix model since the fermionic variables θ have been integrated out. We think i that further analysis of this interesting relation to the one matrix model for the discrete theory could lead to the geometrical interpretation of the model and the derivation of Feynman rules. We hope to report on this results elsewhere and turn now to the solution of the super-Virasoro constraints in the double scaling limit. 4.Solution to the super-Virasoro constraints in the double scaling limit. The continuum super-Virasoro constraints [20][21] are described by the super- energy momentum tensor of a c = 1 superconformal field theory b 8 1 1 1 1 T = ∂ϕ(z)ψ(z) , T = : ∂ϕ(z)∂ϕ(z) : + : ∂ψ(z)ψ(z) : + , F 2 B 2 2 8z2 (27) where ϕ(z) is a bosonic scalar field with antiperiodic boundary conditions and ψ(z) is a Weyl-Majorana Fermion in the Ramond sector. The modes of the fermionicpartoftheenergymomentumtensorT = 1 G z−n−2 aredescribed F 2 n+1 2 in terms of the coupling constants of the model for n 1: P ≥ − t τ 1 ∂ τ ∂ ∂ n ∂2 G = 0 0δ + (k+ )t + 0 + τ +κ2 n+12 4κ2 n,−1 2 k∂τn+k+1 2 ∂tn k+1∂tn+k+1 ∂tk∂τn−k k≥0 k≥0 k=0 X X X (28) and the bosonic modes areobtained by theanticommutation relations (7). The constraints imposed on the partition function Z (t ,τ ) = exp(F ) are: S k k S G Z (t ,τ ) = 0 , n 1 . (29) n+21 S k k ≥ − The equations which describe the model (14) in the double scaling limit are determined by (28) (29): κ2DF = D−1u 2τDτ , (30) S − t = (2k +1)t R [u] , τ = τ R [u] , (31) k k k k − − k≥1 k≥0 X X where D = ∂ , τ = τ + τ0 = ∂FS is the first fermionic scaling variable, t is ∂t0 2 ∂τ0 the renormalized cosmological constant and u is the bosonic piece of the two point b function of the puncture operator σ σ , satisfying the KdV-flow equation 0 0 h i 9