IC/92/17 (January 1992) SO(2,C) INVARIANT RING STRUCTURE OF BRST COHOMOLOGY AND SINGULAR VECTORS IN 2D GRAVITY WITH C < 1 MATTER N. Chair, V.K. Dobrev and H. Kanno ∗ 2 International Centre for Theoretical Physics 9 P.O.Box 586, 34100 Trieste, Italy 9 1 n a ABSTRACT J 9 We consider BRST quantized 2D gravity coupled to conformal matter with arbitrary 2 central charge cM = c(p,q) < 1 in the conformal gauge. We apply a Lian-Zuckerman 1 SO(2,CI) ((p,q) - dependent) rotation to Witten’s cM = 1 chiral ground ring. We show v that the ring structure generated by the (relative BRST cohomology) discrete states in 1 7 the (matter Liouville ghosts) Fock module may be obtained by this rotation. We 0 ⊗ ⊗ give also explicit formulae for the discrete states. For some of them we use new formulae 1 0 for c < 1 Fock modules singular vectors which we present in terms of Schur polynomials 2 generalizing the c = 1 expressions of Goldstone, while the rest of the discrete states we 9 obtain by finding the proper SO(2,CI)rotation. Our formulae give the extra physical states / h (arising from the relative BRST cohomology) on the boundaries of the p q rectangles of t - × p the conformal lattice and thus all such states in (1,q) or (p,1) models. e h : v i X Introduction. r a Recently the continuum approach to 2D gravity [1-4] has received a lot of attention. In particular, there was progress in the computation of the BRST cohomological extra discrete states of 2D gravity coupled to cM 1 conformal matter in the conformal gauge ≤ [5-14]. (For the calculation of correlators and fusion rules see [15].) More recently Witten [12] made the observation that for cM = 1 the discrete states of ghost number ( 1), − parametrized by two negative integers (r,s), give rise to a polynomial ring of ghost number zero operators. This ring is generated by two elements which correspond to the two cohomology states with (r,s) = (-1,-2), (-2,-1). The states of ghost number zero give rise to spin one currents which act as derivatives of this ring and obey the W - algebra of ∞ area preserving diffeomorphisms. permanent address : Bulgarian Academy of Sciences, Institute of Nuclear Research ∗ and Nuclear Energy, 72 Boul. Trakia, 1784 Sofia, Bulgaria. 1 Not much is known about this ring for cM < 1. (Some conjectures are made in [16] in the case of minimal conformal matter.) This is the first question we address in this letter. We use the SO(2,CI) rotation trick [6,17,10] to change cM = 1. The operators of BRST cohomologies for cM < 1 are obtained by applying the SO(2,CI) to those for cM = 1. We observe that the (chiral) ring structure, defined by the short distance behaviour of operator product, is preserved by this rotation. This includes also the spin one currents which again fall in SU(2) multiplets and generate a W - algebra. This is done in Section 1. ∞ The second question we address is the construction of explicit representatives of the relative BRST cohomology for cM < 1. In the case cM = 1 prominent role was played [7,8,13,11,14] by the singular vectors of Fock spaces [18]. We try to generalize these formu- lae, in particular, for the ghost number zero states. We generalize the Goldstone formula for Fock singular vectors for the cases (r,s) = (1,s),(2,s),(r,1),(r,2),(3,3) and arbitrary central charge, and we give a conjecture for the general case. The singular vectors are given as linear combinations of Schur polynomials S , (n = r or n = s), of all (in k1,...,kn general) ordered partitions k ... k 0, k +...+k = rs, in contrast to the c = 1 1 n 1 n ≥ ≥ ≥ case in which k = = k = s, or r s. Our formulae for the discrete states have 1 r ··· ←→ direct application for the (1,q) models, (or for the (p,1) models). In the general case our formulae are valid for (primary) fields on the boundaries of the Kac table. All this is presented in Section 2., while compact formulae for the action of Virasoro generators on the Schur polynomials (needed for the derivation of singular vectors) are derived in the Appendix. 1. SO(2,C) invariance of the ring structure of the discrete states 1.1. We can consider the BRST cohomology of the Virasoro algebra H (Vir, ) ∗ M for any Virasoro module with the energy momentum tensor T(z) of the central charge M c = 26. The BRST operator dz 1 d = : (T(z)+ TG(z))c(z) : (1.1) 2πi 2 Z acts on G, where G is a fermion Fock module of (b,c) ghosts with the energy M ⊗ F F momentum tensor TG(z). We will be concerned with the case in which is a tensor M product of two Feigin-Fuchs modules, = M L. In the free field realization of M F ⊗ F 2D quantum gravity coupled to conformal matter, M corresponds to the matter and L F F represents the Liouville field. We will distinguish between the quantities for matter and the Liouville fields by using the superscript M and L, respectively. Let us recall that the Feigin-Fuchs module is a boson Fock module of 2D scalar F field φ(z). is characterized by the two parameters: the background charge ξ and the F 2 momentum p of the vacuum. The energy momentum tensor 1 T(z) = : ∂φ(z)∂φ(z) : + iξ∂2φ((z) (1.2) −2 defines an action of Virasoro algebra. The central charge c and the conformal weight h of the vacuum are given by 1 1 c = 1 12ξ2 , h = p(p 2ξ) = (η +ξ)(η ξ) . (1.3) − 2 − 2 − Here we have introduced the ”shifted” momentum η = p ξ, which will be used in the − following instead of the ”unshifted” momentum p. We will denote the Feigin-Fuchs module with the parameters ξ and η, (ξ,η). F The BRST cohomology of the Virasoro algebra for = M(ξM,ηM) L(ξL,ηL) M F ⊗F was studied in [5,6,11,14]. Let us fix the matter central charge cM = 1 12(ξM)2. The − condition cM +cL = 26 gives ξ ξ = 1 , (1.4) + − − where we have defined the light-cone combination 1 ξ = (ξM iξL) , ± √2 ± (1.5) 1 η = (ηM iηL) . ± √2 ± ξM and ξL are fixed up to sign. In terms of two (real) parameters r and s defined by 1 1 r = ξ η+ , s = ξ η . (1.6) + − −2 − −2 one can parametrize the momenta in the following way: 1 ηM(r,s) = (r +s)ξM +(r s)iξL , 2 − (1.7) 1h i iηL(r,s) = (r s)ξM +(r +s)iξL . 2 − h i Then the result of [11] (for cM = 1 see [6]) is summarized as follows: Theorem. TherelativecohomologyH (Vir, )isnon-trivialonlyinthefollowing r∗el M cases. (i) If rs = 0, then H = δ CI . r∗el n,0 (ii) If both r and s are positive integers, then H = δ CI δ CI . r∗el n,0 ⊕ n,1 3 (iii) If both r and s are negative integers, then H = δ CI δ CI . r∗el n,0 ⊕ n,−1 In all cases the non-trivial cohomology states appear at level rs. The gravitationally dressed primary states of [3,4] are included in the case (i). The cases (ii) and (iii) give the ”extra” physical states. (In fact there is a doubling of the physical states since finally one has to take the absolute cohomology into account (cf. [11]).) Especially, when cM = 1 (ξM = 0), the physical states with ghost number ( 1) − are basic ingredients for the construction of the chiral ground ring by Witten [12]. At the (discrete) momenta ηM = (r s)/√2 , ηL = i(r + s)/√2 with r and s ± − ∓ being negative integers, there is an operator corresponding to the BRST cohomology r,s O class of ghost number ( 1). has conformal spin zero and ghost number zero, due r,s − O to a shift of quantum numbers corresponding to the ghost c(z) in the transition from states to operators. By this crucial property the short distance behavior of the product (z) (w) defines a ring structure of operators . Witten has proved that this r,s r′,s′ r,s O O O ring is the polynomial algebra generated by x = and y = and that we can 2, 1 1, 2 O− − O− − identify with x r 1 y s 1. Hence the ring structure is given by r,s − − − − O = . (1.8) r,s r′,s′ r+r′+1,s+s′+1 O O O (Note that the indeces are negative integers.) Furthermore, at the same momenta as we have a current W (z) with ghost number zero. W (z) is also an extra BRST r,s r,s r,s O cohomologyclass(seethecase(iii)above.). W (z) actsonthegroundring asa derivation. r,s The action is defined again in terms of the operator product expansion (OPE): (w) r+r′+1,s+s′+1 W (z) (w) O + , (1.9) r,s r′,s′ O ∼ z w ··· − where dots represent BRST exact terms. From the viewpoint of SU(2) symmetry it is natural to include the tachyonic currents W and W . The relation (1.9) is also valid 0,s r,0 for r = 0 or s = 0. Finally the symmetry algebra of the ground ring is identified with W ∞ - algebra of area preserving diffeomorphisms [19]: W (w) r+r′+1,s+s′+1 W (z)W (w) . (1.10) r,s r′,s′ ∼ z w − 1.2. The existence of extra BRST cohomologies at discrete values of momenta is independent of the value of background charges for Feigin-Fuchs modules, as long as the constraint (1.4) is satisfied. Therefore, one can obtain the operators and the currents r,s O W for any value of cM in the same manner as described above. To investigate the OPE r,s relations of the types (1.8) (1.10), we will make use of the SO(2,CI) rotation of the form ∼ 1 m+n i(m n) R(m,n) = − . (1.11) 2√mn i(m n) m+n (cid:18)− − (cid:19) 4 Any pair of background charges (ξM,ξL) satifying the condition (1.4) can be obtained by making an appropriate SO(2,CI) rotation from the ”reference” charges (0, i√2) for ± cM = 1 case. For example, the background charges corresponding to (p,q) model are (p q) ξM 0 ± − ξ~ = = R(q,p) = √2pq . (1.12) ξL i√2 i(p+q) ± ! (cid:18) (cid:19) (cid:18)± (cid:19) √2pq In the following we will use a vector notation like ξ~ = (ξM,ξL)t for a pair of quantities for matter and the Liouville systems. The discrete momenta where we have extra physical states are related to the background charges by (1.7) with integers r and s. It is quite remarkable that this relation is decomposed into the product of rotation and scaling: ~ ~ ~η(r,s) = √rs R(r,s) ξ , which is valid for any value of ξ. The discrete momenta ~η(r,s) are obtained from those for cM = 1 case (the ”reference” momenta) by the same rotation as (1.12): 0 1 (r s) ~η(r,s) = √rsR(r,s)R(q,p) = R(q,p) ±√2 − . (1.13) i√2 1 (r +s) (cid:18)± (cid:19) ±i√2 ! ~ (p~ = ~η +ξ is also rotated by the same R(q,p).) The operators and the currents W arising from BRST cohomology for = r,s r,s O M M L have dependence on the characterestic parameters ξ~ and ~η of , F ⊗F M ~ ~ = (ξ,~η), W = W (ξ,~η) . (1.14) r,s r,s r,s r,s O O This dependence is universal in the sense that the way of constructing BRST physical ~ ~ states is independent of ξ with ξ and ~η treated as free parameters. Let us define an action of R(m,n) by ~ ~ R(m,n) (ξ,~η) = (R(m,n)ξ,R(m,n)~η) , (1.15) r,s r,s ·O O and the same for W . Then the universality stated above implies r.s (p,q) = R(p,q) (1,1) , W(p,q) = R(p,q) W(1,1) , (1.16) r,s r,s r,s r,s O ·O · where the superscript (p,q) specifies the model. We will show some examples later on. In the calculation of OPE, two point function for free scalar fields φI(z) (I = M,L): φI(z) φJ(w) = δIJln(z w) (1.17) h i − − is used for contractions. Since two point function (1.17) is invariant under SO(2,CI) rota- tion,we can conclude that the coefficient of OPE are independent of (p,q), which means, 5 for example, W(p,q)(z) (p,q)(w) = R(p,q) W(1,1)(z)R(p,q) (1,1)(w) = r,s Or′,s′ · r,s ·Or′,s′ = R(p,q) W(1,1)(z) (1,1)(w) · r,s Or′,s′ ∼ 1 (1.18) (cid:0) (1,1) (cid:1) R(p,q) (w) = ∼ z w ·Or+r′+1,s+s′+1 − 1 (p,q) = (w) . z w Or+r′+1,s+s′+1 − Thus, SO(2,CI) rotation preserves the OPE structure of BRST cohomology classes of types (1.8) (1.10). The whole algebraic structure appearing in the cM = 1 case remains true ∼ (p,q) for any (p,q). For example, the algebra of ’s is identified with the polynomial ring r,s O generated by x(p,q) = (p,q) and y(p,q) = (p,q) . The currents W(p,q) acts on this O 2, 1 O 1, 2 r,s polynomial ring as derivat−ion−s like W(p,q) = ∂/∂−x(p−,q) and W(p,q) = ∂/∂y(p,q). 0, 1 1,0 − − 1.3. We show a few examples. The generators x(p,q) and y(p,q) are operators corresponding to the BRST cohomology class with ghost number 1 at level 2: − ξ b−2 − √−2 φM−1 +iφL−1 b−1 |~η(−2,−1)i , (1.19.a) (cid:20) (cid:21) (cid:0) (cid:1) and ξ b + φM iφL b ~η( 1, 2) , (1.19.b) −2 − √2 −1 − −1 −1 | − − i (cid:20) (cid:21) (cid:0) (cid:1) respectively. Noting a shift of momenta in going to operators, we obtain ξ ξ x(p,q) = cb − i ∂φM +i∂φL exp i + (φM iφL) , − √2 − √2 − (cid:20) (cid:21) (1.20) ξ (cid:0) (cid:1) ξ y(p,q) = cb + i ∂φM i∂φL exp i − (φM +iφL) . − √2 − − √2 (cid:20) (cid:21) (cid:0) (cid:1) The dependence on (p,q) only appears in the parameters ξ on which SO(2,CI) acts as ± scale transformation. Our next example is SO(2,CI) rotated SU(2) current algebra. In terms of a combination ξ ξ Φ(p,q) = + ∂φM i∂φL − ∂φM +i∂φL (1.21) √2 − − √2 (cid:0) (cid:1) (cid:0) (cid:1) the generators of SU(2) current algebra is given by T = W(p,q) = expiΦ(p,q) , + 0, 2 − T = W(p,q) = i∂Φ(p,q) , (1.22) 3 1, 1 − − T = W(p,q) = exp iΦ(p,q) . − −2,0 − 6 The SU(2) currents acts on the polynomial ring in x(p,q) and y(p,q) as derivations: ∂ T = x(p,q) , + y(p,q) 1 ∂ ∂ T = x(p,q) y(p,q) , (1.23) 3 2 ∂y(p,q) − ∂x(p,q) (cid:0) ∂ (cid:1) T = y(p,q) . − ∂x(p,q) The currents W(p,q) (r+s = n) constitutes an SU(2) multiplet of spin n. The following r,s − 2 is an example of spin 3 multiplet: 2 i W(p,q) = exp ξ (φM +iφL) 2(ξ (φM iφL) , −3,0 √2 − − + − (cid:18) (cid:19) 1 i W(p,q) = T2 ∂T exp ξ (φM iφL) , −2,−1 2 3 − 3 −√2 + − (1.24) 1(cid:0) (cid:1) i W(p,q) = T2 +∂T exp − ξ (φM +iφL) , −1,−2 2 3 3 √2 − (cid:0) i (cid:1) W(p,q) = exp ξ (φM iφL) 2(ξ (φM +iφL) . 0,−3 √2 + − − − (cid:18) (cid:19) With these examples at lower levels it is easy to check the OPE structure of type (1.8) ∼ (1.10) explicitly. The conditions necessary for this calculation are the constraint (1.4) and the fundamental two point functions (1.17). 2. Discrete states in (p,q) models and Fock modules singular vectors 2.1. In this Section we give explicit representatives of the discrete states of H (Vir, ) for arbitrary (p,q) models. We start with the ghost number zero case. r∗el M For cM = 1 it is well known [7,8,13,11,14] that explicit formulae for the physical states are provided by the singular vectors of the matter Fock spaces expressed in terms of the so called Schur polynomials which are recalled in the Appendix. Let ξM be given as in (1.12) with p,q IN. Then the Fock module M(ξM,ηM) is ∈ F reducible [20] iff ηM is given by (1.7) with r,s ZZ,rs > 0. ∈ Let us consider first the case r,s IN. For cM = 1 the singular vector of level rs is ∈ given by [18] (see also [11,14] whose notation we partly follow): √2 vc=1 = Ss,...,s ηM ηM , (2.1a) r,s n n | i − ! r | {z } 7 or, equivalently, by √2 vc=1 = Sr,...,r ηM ηM , (2.1b) r,s − n −n! | i s the two expressions differing by sign. We recall that a singular vector of a Virasoro highest | {z } weight module (here Fock modules) is a weight vector v different from the highest weight vector and obeying L v = 0, for n 1, where L are the Virasoro generators. Let us n n ≥ note that the Schur polynomials involved are exactly of length r, (resp. s), i.e., each term involves the product of exactly r, (resp. s), elementary Schur polynomials. We would like to generalize these formulae for c < 1. For r = 1 or s = 1 one may check that the singular vectors are given by √2 v = S ξ ηM ηM , r = 1 , (2.2a) 1,s s n + −n! | i √2 v = S ξ ηM ηM , s = 1 , (2.2b) r,1 r n − −n! | i which reduce to (2.1) for c = 1, p = q = 1, ξ = 1. The derivation of (2.2) and the ± ± formulae, necessary for the derivation of the singular vectors given below, are given in the Appendix. The cases r,s > 1 are much more interesting. In particular, for r = 2 or s = 2 we have: s √2 v = βs(ξ )S ξ ηM ηM , r = 2 , (2.3a) 2,s k + 2s−k,k n + −n! | i k=0 X r √2 v = βr(ξ )S ξ ηM ηM , s = 2 , (2.3b) r,2 k − 2r−k,k n − −n! | i k=0 X ( 1)kΓ(2(n+u )) 2(n+u ) 1 βkn(ξ±) = Γ(2n −k +u +1)Γ(±k +u ) = (−1)k k +u ± −1 , (2.3c) − ± ± (cid:18) ± − (cid:19) where u = γ (ηM ξM) 1 = ξ2 n, (cf. the Appendix), and only the expression ± ± − − ± − in terms of Γ - functions can be used if u is not integer. ± Our first observation is that for c < 1 and r = 2, (resp. s = 2), all Schur polynomials of degree 2n and of length 2 are involved, since the term with k = 0 involves the ≤ elementary Schur polynomial S = S . Of course, there are partial cases when not all 2n 2n,0 terms are present in (2.3a,b). Thus for p = 1, resp. q = 1, we have u = q s for (2.3a), + − resp. u = p r for (2.3b), and βn(ξ ) = 0 for k < 1 u . Finally, for c = 1,p = q = 1 − − k ± − ± 8 we have βn( 1) = 0 for k < n and (2.3a,b) collapse (up to signs) to (2.1a,b). Note that k ± βn = 0 in all cases. n 6 Let us give also the example of a singular vector in the case r = s = 3 : √2 v = β9 (ξ ) S ξ ηM ηM , (2.4a) 3,3 j,s + 9−j−s,j,s n + −n! | i 9 j s j s 0 − −X≥ ≥ ≥ ( 1)j+sβ β9 (ξ ) = − j,s , (2.4b) j,s + Γ(10 j s+u)Γ(j +u)Γ(s+u 1)u − − − β = β = (u 3)(u+8) , β = β = (u 1)(u+6) , 0,0 1,0 1,1 2,1 − − β = β = (u+2)(u+3) , β = β = β = u(u+5) , (2.4c) 4,0 4,1 2,2 3,2 3,3 − β = 18 , β = 6 , β = (u2 +5u+12) . 2,0 3,1 3,0 − − − where u = u = γ (ηM(3,3) ξM) 1 = 2ξ2 3 = 2q/p 3 > 3, (cf. the + + − − + − − − Appendix). Note that β9 have no poles (even for u = 0). For cM = 1, p = q = 1 = u, j,s − all coefficients are zero except β9 and (2.4) goes into (2.1). For u = 2,0,1,3 there are 3,3 − also some vanishing coeffiecients (always β9 ). Note that β9 = 0 in all cases. The same 0,0 3,3 6 expression is valid if we replace ξ ξ , u = u u = 2ξ2 3 = 2p/q 3. + + −→ − −→ − − − − Thus we are lead to the conjecture that the general expression for the singular vectors should be: ( 1)k1β (ξ ) √2 v = − k1,...,kr + S ξ ηM ηM r,s Γ(k +u +1)...Γ(k +u r +2) k1,...,kr n + n | i 1 + r + − ! k1≥.X..≥kr≥0 − k1+...+kr=rs (2.5a) or, up to sign, ( 1)k1β (ξ ) √2 vr,s = Γ(k1 +u −+1)..k.1Γ,.(..k,krs+u− s+2)Sk1,...,ks n ξ−η−Mn!|ηMi k1≥.X..≥ks≥0 − − − k1+...+ks=rs (2.5b) where u = γ (ηM(r,s) ξM) 1 = √2ξ (ηM(r,s) ξM) 1, (cf. the Appendix), ± ± − − ± − − and the coefficients obey a simple recursion relation (from the action of L ): 1 n β = β , (2.5c) k1,...,kn k1 1,...,kj+1,...,kn − j=2 X and a more complicated one (from L ) which we omit for the lack of space. Note that in 2 (2.5a), (resp. (2.5b)), s k rs, (resp. r k rs), 0 k s, (resp. 0 k r), 1 1 r s ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ β = 0, β = 0. Thus according to our conjecture Schur polynomials of degree rs, s,...,s r,...,r 6 6 9 and of arbitrary length r, (or s), are involved, i.e., every term involves the product of ≤ ≤ at most r, (resp. s), elementary Schur polynomials. (0) The Proof that the states given in (2.2) - (2.5) are representatives of H (Vir, ) is rel M analogous to the case cM = 1 [11]. Since these are singular vectors they are BRST closed. Consider (2.5b). It is enough to note that by (2.2b), (2.3b), (2.4b), and our conjecture, the term S (√2ξ ηM /n) ηM,ηL is always present which is the representative for r,...,r n − − | i cM = 1 and then we can repeat the proof of [11]. In the last step we are reduced to the term (η˜+ )s = (ξ )s(η+ )s which is representative of H(0)(Vir, ) [11]. Analogously is −r − −r rel M considered (2.5a). Let us note that we could have used also the singular vectors of the same degree in the Liouville sector. Finally, let us remark that representatives for the case cM < 1 may be obtained from (2.1) by the SO(2,CI) rotation of [6], however, these would not be singular vectors. 2.2. Next we consider the ghost number zero case with r,s IN. The physical ∈ − importance of these sates comes from the fact that they are associated to the currents (p,q) W = W discussed in Section 1. r,s r,s FirstwenotethatforcM = 1, ξM = 0, from(1.6)wehavethatηM(r,s) =ηM( s, r). − − Thus the cohomology representatives are singular vectors vc=1 as given in (2.1) (cf. also s, r − − [14]). For cM < 1 things are rather more complicated with one exception, namely the case r = kp , s = ℓq , k,ℓ IN. (2.6a) − − ∈ In this case we have (cf. (1.7)): ηM(r,s) = ηM(pℓ,kq) , (2.6b) and the cohomology representatives at level rs = kℓpq are singular vectors v from (2.5) pℓ,kq (and also by (2.2) - (2.4) whenever applicable). Let us remark that these r,s correspond to corners of the conformal lattice with dimension p,q [21,22,23,20], (see also [24] for an explicit parametrization of the reducible Virasoro highest weight modules). In the generic situation when (2.6a) is not fulfilled there is no singular vector of degree rs in the matter sector (and also in the Liouville sector). (There is only the so called cosingular vector [20].) Thus we can obtain representatives only by implementing the proper SO(2,CI) rotation of [6] as discussed in Section 1. In this case we have the 10