THE 3-STATE POTTS MODEL AND ROGERS-RAMANUJAN SERIES

Transcription

1 THE 3-STATE POTTS MODEL AND ROGERS-RAMANUJAN SERIES ALEX J FEINGOLD AND ANTUN MILAS Abstract We explain the appearance of Rogers-Ramanujan series inside the tensor product of two basic A 2 2 -modules, previously discovered by the first author in [F] The key new ingredients are 5, 6 Virasoro minimal models and twisted modules for the Zamolodchikov W 3-algebra Introduction Virasoro minimal models and their characters have a long and fruitful history in conformal field theory, string theory and of course in vertex algebra theory It is by now well-known that the character χ m,n s,t q of the minimal model Lc s,t, h m,n s,t is given by χ m,n s,t q = qh m,n s,t cs,t/2 q q stk2 q kmt ns q mt+nsk+mn, k Z where 2 s, t Z are coprime, m < s, n < t, q = φq = i= qi and 6s t2 c s,t =, h m,n s,t = mt ns2 s t 2 st st In particular, 2 χ, 2,5 q = q/60 q 5n+2 q 5n+3, 3 χ,2 2,5 q = q /60 n=0 n=0 q 5n+ q 5n+ give the product sides of the famous Rogers-Ramanujan series which appear in many papers on representation theory of Virasoro and affine Lie algebras The results in this note arose as an attempt to find a representation theoretic explanation of the following theorem obtained in 983 by the first author [F]: Theorem Denote by LΛ the basic level one highest weight module for the twisted affine Kac-Moody algebra A 2 2 Then LΛ LΛ = L2Λ V LΛ 0 V 2,

2 2 ALEX J FEINGOLD AND ANTUN MILAS such that suitably normalized and scaled characters of the multiplicity spaces V and V 2 coincide with the product sides of Rogers-Ramanujan series 2-3 This result was obtained in the principal mod 6 realization of A 2 2 also studied in [Ca], [X], [Bo], so it is perhaps not obvious what is its reformulation it in the language of vertex algebras and twisted modules The above result is somewhat unexpected because the coset spaces, which are clearly unitary modules for the Virasoro algebra, are essentially given by the characters of the s, t = 2, non-unitary minimal models! It is also interesting to notice that the principally specialized characters of L2Λ and LΛ 0 are also given by above Rogers-Ramanujan series again, suitably normalized and scaled To explain the appearance of Rogers-Ramanujan q-series we recall first a pair of identities which can be traced back to Bytsko and Fring [BF] For further identities, see [BF] and also [Mu], [Mi]: 6 χ,2 5,6 χ 2,2 5,6 q + χ,q = χ, 5,6 q + χ2,q = χ,2 5,6 We also mention the related identities 7 8 χ 2, 5,6 χ, 5,6 q χ2,5 5,6 2,5 q/2, 2,5 q/2 q = χ, 2,5 q2, q χ,5 5,6 q = χ,2 2,5 q2 Although all these formulas can be checked directly see the Appendix they are very interesting for several reasons Their right hand sides are clearly scaled Rogers-Ramanujan series Also, unlike the identities studied in [Mu] and [Mi], the above formulas are among characters of modules of different central charges in our case, central charge 22 5 and 5 The key observation is now that the central charge of the cosets spaces V and V 2 equals 5, the central charge of 5, 6 minimal models, So we are immediately led to the following conclusion: relations -6 should be interpreted as decomposition formulas of the coset spaces V and V 2 into irreducible Virasoro characters of central charge 5 As we shall see this is indeed the case In fact, there is something even deeper going on It turns out that V and V 2 are in fact the only twisted irreducible modules for a larger rational vertex algebra which we end up calling W 3, also known as the Zamolodchikov W 3 -algebra This algebra has already appeared in the physics literature under the name 3-State Potts model and more recently in the vertex algebra theory [KMY] We first show that there are at least four different ways of thinking about W 3 : Theorem 2 The following vertex operator algebras are isomorphic: a The parafermionic space K sl2 3, 0 L sl2 3, 0 [DW] b A certain subalgebra M 0 of the lattice vertex algebra V L, where L = 2Q and Q is the root lattice of type A2 see [KMY], [Miy]

3 THE 3-STATE POTTS MODEL AND ROGERS-RAMANUJAN SERIES 3 c The coset vertex algebra W L sl 3 2Λ 0 L sl3 Λ 0 L sl3 Λ 0 [KW] d The simple affine W-algebra L 5, obtained via Drinfeld-Sokolov reduction [Ara], [FKW] In the physics literature equivalence of a, c and d is more or less known Construction b is more recent again, see [KMY] Having enough knowledge about W 3 we can now return to the cosets V and V 2 These are not ordinary modules for W 3, but rather τ-twisted W 3 -modules Our main result is the following theorem about W3 Theorem 3 For the rational vertex algebra W 3 the following holds: a AutW 3 = Z2 b If we denote by τ the nontrivial automorphism of W 3, then W3 5 is τ-rational ie W 3 has finitely many irreps and every τ- twisted module is completely reducible c The algebra W 3 has precisely two inequivalent representations, W τ /0 = L 5, L 0 5, 2, and 0 W τ /8 = L 5, L 8 5, 3 8 d χw τ /0 = χ,2 2,5 q/2 and χw τ /8 = χ, 2,5 q/2 Now we connect results from Theorem with those in Theorem 3 Denote by σ the principal automorphism of order 6 of sl 3, and let L sl3 Λ 0 be the affine vertex algebra associated to sl ˆ 3 of level one The basic A 2 2 -module L σ sl 3 Λ can be viewed as a σ-twisted L sl3 Λ 0 -module, and its character is given by tr LΛ q Lσ 0 c/2 = q /72 n=0 q 6n+/6 q 6n+/6, where L σ 0 is the σ-twisted Virasoro operator Similarly, the characters of the two standard level two A 2 2 -modules if viewed as σ-twisted L sl3 2Λ 0 -modules are given by and tr L2Λ q Lσ 0 c/2 = q /72 tr LΛ0 q Lσ 0 c/2 = q /6 /72 n=0 n=0 q 6n+/6 q 6n+/6 χ,2 2,5 q/3 q 6n+/6 q 6n+/6 χ, 2,5 q/3

4 ALEX J FEINGOLD AND ANTUN MILAS Now, formula implies a q-series identity q /72 n=0 q 6n+/6 q 6n+/6 = χ,2 2,5 q/3 χ,2 2,5 q/2 + χ, 2,5 q/3 χ, 2,5 q/2, after we canceled tr LΛ q Lσ 0 c/2 from both sides of This formula can be traced back to Ramanujan This gives the main result of our paper Theorem As σ τ-twisted L sl3 2Λ 0 W 3 -module: 9 L σ sl 3 Λ L σ sl 3 Λ = L σ sl 3 2Λ W τ /0 L σ sl 3 Λ 0 W τ /8 We also have 0 tr V q L0 c/2 = χ,2 2,5 q/2 tr V2 q L0 c/2 = χ, 2,5 q/2 where L0 is the coset Virasoro operator for the tensor product Finally, we mention that 7-8 can be also explained in terms of representations of W 3 This requires a modular invariance theorem for τ-twisted modules [DLM2] This was pursued in the last section Remark The algebra W 3 and some of its modules also appear in [Ma], where the coset Virasoro construction is applied to the investigation of the branching rule decomposition of level- irreducible E 6 -modules with respect to the affine subalgebra F 2 The 3-state Potts model vertex algebra Let us recall a few basic facts about the Virasoro algebra and its representation theory We use Mc, h to denote the Virasoro Verma module of central charge c and lowest conformal weight h, and denote its lowest weight vector by v c,h We let V c, 0 = Mc, 0/ L v c,0, the vacuum vertex algebra We denote by Lc, h the unique irreducible quotient of Mc, h Recall We will focus on the central charge c 5,6 = 5 It is well-known that L 5, 0, viewed as a vertex algebra, has up to equivalence precisely 0 irreducible modules [W]: L 5, 0, L 5, 8 L 5, 2 2, L 5 5, 0, L, L 5, 2 3 5, 5, L, L 5, 3 8 5, 2 0, L, L 5, 3, 5, 7 5 It is also known that L 5, 0 L 5, 3 can be equipped with a simple vertex operator algebra structure cf [KMY] Of course, one can always define a vertex operator algebra structure on V M, where M is any module with integral grading by defining the action of M on M to be trivial But

5 THE 3-STATE POTTS MODEL AND ROGERS-RAMANUJAN SERIES 5 such vertex algebra is not simple As we shall see, there is a unique vertex operator algebra structure on L 5, 0 L 5, 3 Let us also recall that the space of irreducible characters of L 5, 0 is 0-dimensional ie the characters are linearly independent Proposition 2 Let V, Y, be a vertex algebra such that V = L 5, 0 L 5, 3 as a module for the Virasoro algebra and such that Y L 5,3 L 5,3 0 If another vertex algebra W = L 5, 0 L 5, 3 satisfies the same property, then W = V Proof We certainly have an isomorphism f = f L 5,0 f 2 L,3 between 5 V and W viewed as Virasoro modules f is unique up to a choice of two nonzero scalars Because of Y, x = id, the map f is uniquely determined sending the vacuum of V to the vacuum of W Observe that in V, Ỹ V := Y L 5,3 L 5,3 defines an intertwining operator of type L 5,0 L 5,3 L 5,3 otherwise this would contradict I L 5,3 L = 0 The same holds 5,3 L 5,3 for ỸW According to [FHL], this intertwining operator is unique up to a nonzero constant Thus, after identification via f, we can find ν 0 such that νỹv = ỸW Therefore u, v f u, λ f 2v, λ 2 = ν defines the wanted automorphism between V and W Existence of the vertex operator algebra satisfying conditions in Proposition 2 has been established in [KMY] We shall denote it by W 3 More precisely, we have [KMY] Theorem 2 The vertex algebra W 3 is rational with the following irreducible modules we also write their decompositions viewed as V ir-modules: W0 = L 5, 0 L 5, 3, W2/ = L 5, 2 L 5 5, 7, 5 W2/5, + = L 5, 2, W2/5, = L 3 5, 2, 3 W/5, + = L 5,, W/5, = L 5 5, 5 Now we prove the equivalence of b and d in Theorem 2 Theorem 22 Vertex algebras W 3 and L 5 are isomorphic Proof Recall first that L 5 is the unique irreducible quotient of the universal affine W-algebra M sl3 5, modulo the maximal ideal We can think of M sl3 5 as the vertex algebra obtained via Drinfeld-Sokolov reduction from a universal affine vertex algebra associated to ŝl 3, or constructed by using free fields via Miura transformation Either way, it is known that M sl3 5 is freely generated by the conformal vector ω and another primary vector of degree three, w, that is L0w = 3w and Lnw = 0 for

6 6 ALEX J FEINGOLD AND ANTUN MILAS n It is also known that Y w, xw is nonzero both in M sl3 5 and in the quotient L vac 5 Let us record tr Msl3 5 ql0 = q 2 ; q q 3, ; q where a; q = i=0 aqi More importantly, by using Frenkel -Kac- Wakimoto s character formula [FKW] proven by Arakawa [Ara], we compute the character of L 5 as tr L 5 ql0 /2 = q /2 q 2 m,n Z w S 3 ɛ w q 5wρ+20nα +20mα 2 ρ 2 0, where S 3 is the Weyl group of sl 3, α and α 2 are simple roots, ρ is the half-sum of positive roots and λ 2 = λ, λ is normalized such that α 2 = 2 for each simple root α An easy computation shows that tr L 5 ql0 /2 = q /30 + q 2 + 2q 3 + 3q + In fact we can show by expanding both sides in q-series that the following identity holds for m 50: 22 tr L 5 ql0 /2 χ, 5,6 q + χ,5 5,6 q mod qm, Thus, the character of L 5 equals the character of W 3 up to degree 50 Let us consider the Virasoro submodule M L 5 generated by ω and w By Proposition 2, it is sufficient to show UV ir = L 5, 0 L 5 and UV ir w = L 5, 3 Indeed, if that is the case, then L 5, 0 is a rational subalgebra of L 5, which decomposes as a sum of L 5, 0-modules with integral conformal weights But classification of L 5, 0-modules and 22 implies L 5 = L 5, 0 L 5, 3, and the rest of the proof now follows from Proposition 2 Denote by M the cyclic Virasoro module UV ir and by M 2 the cyclic module UV ir w By the universal property of Verma modules, these cyclic modules are quotients of M 5, 0 even V 5, 0 and M 5, 3, respectively We claim that M M 2 = {0} If not, M M 2 is a nontrivial submodule of a quotient of M 5, 0 and of M 5, 3 Embedding structure for Verma modules among the minimal series shows that this is impossible the two modules belong to different blocks If we let χ W q = tr W q L0 c/2, we get χ L q χ M q + χ M2 q χ, 5,6 q + χ,5 5,6 q, 5 where has an obvious meaning for two q-series with non-negative integer coefficients Now, relation 22 implies χ M q + χ M2 q χ, 5,6 q + χ,5 5,6 q mod q50 Thus χ M q = χ, 5,6 q and χ M 2 q = χ, 5,6 q, and hence UV ir = L 5, 0 and UV irw = L 5, 3

7 THE 3-STATE POTTS MODEL AND ROGERS-RAMANUJAN SERIES 7 Remark 2 The previous theorem gives a representation theoretic proof of the q-series identity q /2 q 2 5wρ+20nα +20mα 2 ρ 2 ɛ w q 0 = χ, 5,6 q + χ,5 5,6 q, w S 3 m,n Z which presumably can be checked directly by applying methods similar to those used in the Appendix Theorem 23 Denote by W sl 3 sl 3 sl 3 2 the coset vertex algebra obtained via the embedding L sl3 2, 0 L sl3, 0 L sl3, 0 Then, as vertex operator algebras, W 3 = sl 5 W 3 sl 3 sl 3 2 Proof Observe first that the central charge of the coset vertex algebra is 6 5 = 5, with the Virasoro generator ω + ω ω 2, where ω k stands for the Sugawara generator of level k 3 This coset is unitary so as a Virasoro module, or L 5, 0-module it decomposes as a direct sum of irreducible modules of central charge /5 The graded dimension can be now easily computed by using the rationality of L 5, 0 Alternatively, we can recall a result from [KW], where the character of W sl 3 sl 3 sl 3 2 is computed by using modular invariance Either way, W sl 3 sl 3 sl 3 2 = L 5, 0 L 5, 3, as Virasoro modules Finally to finish the proof we need Y L 5,3 L 5,3 0 in the coset algebra This was proven by Bowknegt et al in [BBSS] [BS], where it was shown that W sl 3 sl 3 sl 3 2 has a degree 3 generator w such that the brackets [w n, w m ] satisfy the relations as in L 5 3 τ-twisted W 3 -modules and the structure of Aτ W 3 In this part V is a vertex operator algebra and τ AutV has order two Recall the notion of a weak τ-twisted V -module M By definition we require a decomposition V = V 0 V, and the twisted vertex operator map Y τ, x Y τ u, x End M[[x /2, x /2 ]] such that 33 Y τ u, x = n Z+ r 2 unz n, u V r

8 8 ALEX J FEINGOLD AND ANTUN MILAS and the Jacobi identity holds x 0 δ x x 0 3 unv = 0, n >> 0 Y τ, x = Id M Y τ u, x Y τ v, x 0 δ x2 x x 0 x x 0 = x 2 r/2 x x 0 δ Y τ v, Y τ u, x Y τ Y u, x 0 v,, where u V r and v V s Clearly, from the Jacobi identity it follows that M is an untwisted weak V 0 -module If we also require M to be graded M = ν M ν, where the grading is induced by the spectrum of L0 and is bounded from below, and has finite dimensional graded components, we say that M is a τ-twisted V -module 2 If we instead require M to be 2 N-gradable, such that M = n NMn, 2 ummn Mn + m degu, then M is called admissible [DLM] Let M be an irreducible τ-twisted V -module Then there is λ such that we have the following decomposition with respect to the spectrum of L0: 3 M = M λ+n M λ+n+, n=0 so that if we let M i = M λ+n+ i, i {0, }, the vertex operator map Y τ, x 2 is compatible with this Z 2 -decomposition, a consequence of umm ν M ν+m degu, for homogeneous u Again, M i is a V 0 -module for i {0, } A vertex operator algebra V is said to be τ-rational if every admissible τ-twisted V -module is completely reducible [DLM] We also discuss intertwining operators among irreducible ordinary V - modules [FHL] Without giving the full definition, let us record that an intertwining operator of type W 3 W 2 is a linear map Yu, x HomW2, W 3 {x}, W u W such that among other things the following Jacobi identity holds: x 0 δ x Y u, x Yv, w x 0 x δ x2 x Yv, Y u, x w 0 x 0 = x 2 δ x x 0 YY u, x 0 v, w, where u V, v W and w W 2 Let us now focus on the twisted Jacobi identity 3 when u V, and v V 0 We shall see that the corresponding identity is essentially intertwining operator map between V 0 -modules Because u V, then Y τ u, x restricted on M i is mapped to M i+ where we use the mod 2 exponent notation Denote by Iu, x this restriction Since v V 0 its action on M i is the usual V 0 -module action, so we write Y 0 u, x instead 2

9 THE 3-STATE POTTS MODEL AND ROGERS-RAMANUJAN SERIES 9 of Y τ u, x The twisted Jacobi identity now reads after we apply it on a vector w: x 0 δ x Iu, x Y 0 v, w x 0 x δ x2 x 0 x 0 x x /2 0 x x 0 δ 36 x 0 δ x x 0 = x 2 Y 0 v, Iu, x w IY u, x 0 v, w, Now, apply the substitution x and x 0 x 0 The identity is then Y 0 v, x Iv, w x 0 δ x2 x Iu, Y 0 v, x w x 0 37 = x x2 + x /2 0 x2 + x 0 δ x x IY 0 u, x 0 v, x w Because of Y u, xv = e xl Y 0 v, xu the skew-symmetry, the right hand-side can be rewritten as x 38 = x x2 + x /2 0 x2 + x 0 δ x x x2 + x /2 0 x2 + x 0 δ x Ie x 0L Y 0 v, x 0 u, x w x IY 0 v, x 0 u, x x 0 w To finish the proof observe first that, by twisted weak associativity, we can always choose positive k N such that x k+/2 2 + x 0 k IY 0 v, x 0 u, w involves only positive integral! powers of the variable Consider 39 x 2 δ x x 0 x k x x k+/2 2 IY 0 v, x 0 u, w, 2 which also equals 320 x 2 δ x x 0 + x 0 k x k+/2 2 IY 0 v, x 0 u, w, Now, we are allowed to replace in 39 the variable with x x 0, so we get 32 x 2 δ x x 0 x k x x k 2x x 0 /2 IY 0 v, x 0 u, x x 0 w 2 Finally, we multiply the last expression with x k /2 x 2 δ x x 0 IY 0 v, x 0 u, w, = x 2 δ x x 0 2 x k and we obtain x x /2 0 IY 0 v, x 0 u, x x 0 w

10 0 ALEX J FEINGOLD AND ANTUN MILAS Consequently, 37 and the last formula imply the following result Proposition 3 The map I, x defines an intertwining operator among V 0 -modules of type M i+ V M The identity 36 is equivalent to the Jacobi i identity for I, x We also have the following useful result Lemma 3 Let M be an irreducible τ-twisted V -module for a simple vertex algebra V Then for every nonzero u V, m M Y τ u, xm 0 The proof is clear otherwise Y τ a, xm = 0 for every a V, which is impossible because M is cyclic This statement, in particular, yields M 0 and M 0 0 We shall need a few results about τ-twisted Zhu s algebra A τ V = V/OV following [DLM], where OV is the span of vectors of the form + x dega +δr+r/2 a b = Res x Y a, xb, x +δr where a V r and where δ 0 = and δ = 0, and the multiplication on A τ V is induced via + x dega a b = Res x Y a, xb x For every a V 0, and b V we have a b = 0 mod OV, so b = 0 as an element in A τ V Now we specialize V = W 3 = L 5, 0 L 5, 3 Lemma 32 The τ-twisted Zhu algebra A τ W 3 is a quotient of the polynomial algebra k[x] Proof Denote by π the natural projection from W 3 to Aτ W 3 Then πv = 0 for v AW 3 [DLM] If we denote by A 0 W 3 = AW 3, where AW3 is the usual Zhu s algebra of W3, we clearly have an isomorphism A τ W 3 = AW3 0 /I where I is a certain ideal We are primarily interested in irreducible τ-twisted W 3 -modules Every such module is an ordinary module for L 5, 0, so it decomposes as a direct sum of ordinary modules given on the list in 2 Because of 3, any irreducible τ-twisted W 3 -module is heavily constrained with respect to the spectrum of L0 More precisely, Proposition 32 Let M be an irreducible τ-twisted W 3 -module as in 3 Then M, viewed as a L 5, 0-module, is isomorphic to either L 5, L 0 5, 2 0 j J i I

11 or THE 3-STATE POTTS MODEL AND ROGERS-RAMANUJAN SERIES k K where I, J, K, L are finite sets L 5, L 8 5, 3, 8 l L Proof The spectrum of L0 on M = M 0 M must be contained inside the set λ + N 2, where λ is the lowest weights of an irreducible L 5, 0-modules 2 Also, in addition M i 0 This implies, that the absolute value of the difference between the lowest conformal weights of M 0 and of M must lie within the set N + 2 Easy inspection of allowed weights gives two possibilities: λ = 3 8 or λ = 0 The next result follows from the fusion rules for L 5, 0 [KMY], [W] see also [IK] Lemma 33 The module L 5, 3 is a simple current ie it permutes equivalence classes of irreducible modules under the fusion product In particular, L 5, 3 L 5, 3 L 5, 3 8 L 5, 0 = L 5, 8 = L 5, 2 0, L, L 5, 3 L 5, 3 L 5, 8 5, 2 0 = L = L, 5, 3 8 5, 0 Now let us examine the number of irreducible summands in decompositions in Proposition 32 Proposition 33 Let M be as in Proposition 32 Then, viewed as a V irmodule, M = L 5, L 0 5, 2 0 or M = L 5, L 8 5, 3 8 Proof We prove discuss the first assertion Because the twisted module in question is irreducible and A τ W 3 is commutative the top level must be an irreducible one-dimensional module Therefore the set I must be a singleton If J 2, the direct sum j J L 5, 2 0 decomposes into at least two irreducible L 5, 0-modules To rule out this case we apply the same argument as in the proof of Lemma 53 [KMY] Their argument and Lemma 33 yields a W 3 -module on L 5, 2 0, such that L 5, 0 acts trivially on it But this is clearly a contradiction, so J = Theorem 3 Let M be an irreducible τ-twisted W 3 -module such that M = L 5, L 0 5, 2 0

12 2 ALEX J FEINGOLD AND ANTUN MILAS or M = L 5, L 8 5, 3 8 Then such M is unique up to isomorphism Proof Recall that irreducible W 3 -modules are in one-to-one correspondence with the modules for the Zhu algebra AW 3, which is isomorphic to a quotient of the polynomial algebra k[x], where x = [ω] Every such module is one-dimensional so the top level of an irreducible W 3 -module must be one-dimensionalthe rest follows from Proposition 33 In the next section cf Proposition 2 we construct two irreducible τ- twisted W 3 -modules W τ 0 and Wτ 8, which decompose as modules in Theorem 3 Consequently, combined with Lemma 32 we immediately obtain Corollary 3 The vertex algebra W 3 has precisely two τ-twisted irreducible modules Moreover, the twisted Zhu algebra A τ W 3 is isomorphic to C[x]/ x /0x /8 Theorem 32 The vertex algebra W 3 is C2 -cofinite and τ-rational Proof The vertex algebra W 3 contains a C2 -cofinite subalgebra ie L 5, 0, with the same conformal vector thus it is C 2-cofinite itself To prove the rationality we follow the standard arguments as in Theorem 56 in [KMY], which we essentially repeat here We only have to prove complete reducibility So let M be an arbitrary admissible W 3 -module We may split M = M /8 M /0, where the weights of M /8 are contained inside /8 + 2 N 0, and the weights of M /0 are in /0 + 2 N 0 Indeed, this follows from complete reducibility with respect to L 5, 0, Lemma 33, Proposition 3 and the fact that allowed lowest weights are 8 and 0 Our proof of complete reducibility of M /8 is essentially the same as the proof for M /0, so let us assume M = M /8 for simplicity Consider the top weight /8 subspace M/8 of M /8 This is also an AW 3 -module Easy analysis shows that, as L 5, 0-module, M = L 5, 8 m L 5, 3 8 n, with some multiplicities m and n The multiplicity m is precisely the dimension of the weight /8 subspace To finish the proof we have to argue that m = n and M = L 5, 8 L 5, 3 8 m, as W 3 -modules Choose 0 v M/8 and consider W 3 v We claim that W3 v = L 5, 8 L 5, 3 8 Clearly there could be only one copy of L 5, 8 inside W 3 v Also, from the fusion rules Lemma 33, restriction of the module map Y L 5,3 L 5, 8 where L 5, 8 W 3 5 v, must land inside W, where W is isomorphic to L 5, 3 8, or is plainly zero If the image is zero then L 5, 8 becomes an irreducible module for W 3, contradicting our classification of W 3 -modules in Proposition 33 We conclude W 3 v = L 5, 8 L 5, 3 8

13 THE 3-STATE POTTS MODEL AND ROGERS-RAMANUJAN SERIES 3 Now, take a nonzero vector v M/8 in the complement of Cv and repeat the procedure Then W 3 v W 3 v must be trivial or L 5, 3 8 The latter case cannot occur because of Proposition 33 This way we obtain a decomposition of submodule of M isomorphic to L 5, 8 L 5, 3 8 m The condition m = n must be satisfied, otherwise we could quotient M with the submodule L 5, 8 L 5, 3 8 m and obtain a module of lowest weight, again a contradiction 3 8 Standard A 2 2 -modules and construction of τ-twisted W 3 -modules In this section, our focus is the Kac-Moody Lie algebra of type A 2 2 and its standard modules We will denote by σ a principal order 6 automorphism of ŝl 3, such that 5 sl 3 = sl 3 [j], sl 3 [j] = {a sl 3 : σa = ξ j a}, j=0 where ξ is a primitive 6th root of unity Let Λ be a dominant weight of level l for A 2 2 We denote by Lσ sl 3 Λ the σ-twisted A 2 2 -module, which is also an L sl3 lλ 0 -module [Li] In particular, L σ sl 3 Λ 2 which is of level 2 and L σ sl 3 2Λ are σ-twisted L sl3 2Λ 0 -modules Theorem gives construction of both modules inside the tensor product of the basic module L σ sl 3 Λ L σ sl 3 Λ Indeed, it is easy to see that v Λ v Λ is a highest weight module for L σ sl 3 2Λ and the vector f v Λ v Λ v Λ f v Λ generates the module L σ sl 3 Λ 0 Here {e 0, f 0, h 0, e, f, h } is the canonical set of generators of A 2 2 We denote the twisted module map with Y σ, x In particular, for any x sl 3 [j], we have Consider ω k=l = Y σ x, x = n Z xn + j 6 x n j 6 2l u i ū i L sl3 lλ 0, i= the Sugawara conformal vector of central charge 8l l+3, where {u i} and {ū i } are conveniently chosen orthogonal bases such that σu i = σ ū i, so that u i ū i is fixed under the automorphism we can actually choose {ū i } to be a permutation of {u i } [Bo] Thus Denote by Y ω l, x = n Z L σ k=l 0x n 2 ω = ω k= ω k= ω k=2 L sl3 Λ 0 L sl3 Λ 0

14 ALEX J FEINGOLD AND ANTUN MILAS the coset Virasoro generator of central charge = 5, and we let Y ω, x = n Z Lnx n 2 For related coset constructions see [AP] crucial technical fact Lemma We have and L σ k= 0v Λ = 5 72 v Λ, The following lemma is the L σ k=2 0v 2Λ = 360 v 2Λ, L σ k=2 0v Λ 0 = 3 80 v Λ 0 Consequently, L0v 2Λ = 0 v 2Λ, L0v Λ0 = 8 v Λ 0 Proof We prove only the first formula L0v 2Λ = 0 v 2Λ, the other formula is proven along the same lines Recall that for the Virasoro algebra operator L0 we picked generators u i, ū i, i {,, 8} such that σu i = ξ j u i, Thus we have to compute expressions σū i = ξ 6 j ū i Coeff x 2Y σ u i ū i, x, contributing to L0, acting on the highest weight vectors v Λ and on v 2Λ = v Λ v Λ For that we use a version of the Jacobi identity 3 with u = u i and v = ū i, where the automorphism τ is now σ In this setup Res x2 CT x0 Res x RHS of 3 = Res x2 CT x0 Res x x x x j/6 0 x x 0 2 δ Y σ Y u i, x 0 ū i,, which equals by [Bo], [Li] j/6 22 Res x2 Y σ u i ū i, + j/6[u i, ū i ]0 + l, 2 where l is the level Taking the same residues, now of the left hand side of the Jacobi identity, gives 23 Res x2 CT x0 Res x LHS of 3 = CT x2 Y σ u i, Y σ ū i,, the constant term of a twisted normally ordered product Comparing the formulas 22 and 23, and summing over i, gives an expression for L σ k=l 0 It is now easy to get Lσ k= 0v Λ = 5 72 v Λ Next, we use the Sugawara operator L σ k=2 0 and act on v 2Λ The only nonzero contributions when calculating these operators come from j/6[u i, ū i ]0 and j/6 2 l After

15 THE 3-STATE POTTS MODEL AND ROGERS-RAMANUJAN SERIES 5 we sum over i we get L σ k=2 0v 2Λ = 360 v 2Λ and finally L0v 2Λ = 0 v 2Λ Remark Presumably the computation in Lemma can be carried out via an explicit realization of A 2 2 on the twisted Fock space obtained in [F] Next result comes immediately from Theorem 23 and [KW]: Proposition We have the following decomposition of L sl3 2, 0 W 3 - modules: L sl3 Λ 0 L sl3 Λ 0 = L sl3 2Λ 0 W0 L sl3 Λ W2/ The next goal is to find a twisted version of Proposition The automorphism σ acts diagonally on L sl3 Λ 0 L sl3 Λ 0, and is denoted by σ σ We have to see how it behaves when restricted to the subalgebra L sl3 2Λ 0 W 3 Lemma 2 The automorphism σ preserves W 3 More precisely, we have σ W3 = τ Thus, σ σ L sl3 2Λ 0 W 3 = σ τ Proof The automorphism σ acts diagonally on the tensor product L sl3 Λ 0 L sl3 Λ 0 Therefore, σ also preserves L sl3 2Λ 0 Since W 3 5 is the commutant of L sl3 2Λ 0, by definition a m b = 0 for all a L sl3 2Λ 0 and b W 3, m 0 But then σam b = σa m σb = 0, and hence σb W 3 Recall AutW3 = Z2 If σ W3 =, then V would be an ordinary W 3 -module Easy inspection of modules in Theorem 2 implies that this is impossible The proof follows Consequently, we reached a desired decomposition analogous to the one in Proposition Proposition 2 As σ τ-twisted L sl3 2, 0 W 3 -modules, L σ sl 3 Λ L σ sl 3 Λ = L σ sl 3 2Λ W τ 0 and W τ 8 have lowest conformal weights 0 and 8, respec- where W τ tively 0 L σ sl 3 Λ 0 W τ, 8 Proof The vertex algebra L sl3 2Λ 0 is σ-rational, thus L σ sl 3 Λ L σ sl 3 Λ decomposes as a direct sum of σ-twisted L sl3 2Λ 0 -modules This decomposition is described in Theorem As W 3 is the commutant of L sl3 2, 0 L sl3 Λ 0 L sl3 Λ 0, and L σ sl 3 Λ L σ sl 3 Λ is a twisted σ τ- modules by Lemma 2, the multiplicities spaces in Theorem are naturally τ-twisted W 3 -modules The proof now follows from Lemma

16 6 ALEX J FEINGOLD AND ANTUN MILAS 5 Modular invariance Now we are ready to explain relations 7 and 8, and in particular the negative sign appearing in both identities First we recall a result from [DLM2], where a version of Zhu s of modular invariance theorem was extended to general C 2 -cofinite rational τ-twisted vertex algebras here τ is at first an automorphism of finite order The setup is as following Pick a pair of commuting automorphisms g, h of finite order of V, where V is C 2 -cofinite, and satisfies all the rationality and finiteness conditions as in [DLM2] Then we let T M g, h, v, q = tr M φhovq L0 c/2, where M is h-stable g-twisted sector This holomorphic function in q <, [ q = ] e 2πiy, satisfies the modular transformation property under γ = a b, ad bc = : c d T M g, h, v, q γ q = W σ W T W g, hγ, q where the summation goes over all g a h c -twisted sectors W which are g b h d - stable, and g, hγ = g a h c, g b h d Observe that the modular invariance mixes several twisted sectors Now specialize g = h = τ, where τ is of order two Claim: τ M = M for every irreducible τ-twisted module M The τ-twisted module τ M is defined via Ỹ u, x = Y τ τu, x Because τ M is M = M 0 M as a vector space, we let σ : M τ M, σ M 0 =, σ M = We claim that σ is the wanted isomorphism If u V 0, σy τ u, xm = Y τ u, xτm = Y τ u, xσm, and if u V, σy τ u, xm = Y τ u, xσm = Ỹ u, xσm This proves the claim Notice that not every ordinary V -module is τ-stable For instance, untwisted W 3 -modules W2/5, is not isomorphic W2/5, +, although the former is obtained as a τ-twist from the later cf Theorem 2 Thus when g b h d we can omit the stability condition always satisfied From now on we are only interested in vacuum twisted characters so we let u = Consider the standard generators S and T of the modular group, corresponding to y /y and y y +, respectively For the S matrix a = 0, b =, c = and d = 0, and for the T -matrix a =, b =, c = 0 and d = Under the S transformation T M τ,,, q q S = W c W T W, τ,, q,

17 THE 3-STATE POTTS MODEL AND ROGERS-RAMANUJAN SERIES 7 where the summation is over untwisted modules which are τ-stable Similarly, T M τ,,, q q T = d W T W τ, τ,, q, W where the summation is over τ-twisted modules We also have T M τ, τ,, q q S = e W T W τ, τ,, q, W where the summation is over τ-twisted modules which are τ-fixed, and T M τ, τ,, q q T = W f W T W τ,,, q, where the summation is over τ-twisted modules Moreover, T M, τ,, q q S = g W T W τ,,, q, W where the summation is over τ-twisted modules, and T M, τ,, q q T = W h W T W, τ,, q, where the summation is over τ-stable W 3 -modules In the above formulas c W, d W,, h W are some constants We summarize all these relations as Corollary 5 The vector space spanned by and is modular invariant {T M, τ,, q, M is untwisted, and τ stable} {T M τ, τ ɛ,, q, M is τ twisted, and ɛ = 0, } Going back to W 3 There are two τ-stable irreducible untwisted W 3 -modules, namely W0 and W2/ So the relevant modular invariant space has a basis: where ɛ {0, }, and T W τ /0τ, τ ɛ,, q = χ,2 5,6 q + ɛ χ, 5,6 q T W τ /8τ, τ ɛ,, q = χ 2,2 5,6 q + ɛ χ 2, 5,6 q, T W0, τ,, q = χ 2, 5,6 q χ2,5 5,6 q, T W2/, τ,, q = χ, 5,6 q χ,5 5,6 q Thus, the left hand-sides in 7 and 8 are simply the twisted characters of irreducible untwisted W 3 -modules Combined with the expected identities given on the right hand-sides in 7 and 8, or simply by using modular invariance arguments, we easily get another more natural basis:

18 8 ALEX J FEINGOLD AND ANTUN MILAS Proposition 5 For V = W 3 and τ as above, the vector space spanned by expressions in Corollary 5 is 6-dimensional with a basis {χ, 2,5 q2, χ,2 2,5 q2, χ, 2,5 q/2, χ,2 2,5 q/2, χ, 2,5 q/2, χ,2 2,5 q/2 } 6 Appendix In this section we discuss q-series identities underlying -8 The idea is relatively simple so we only prove here, and leave the rest to the reader Similar methods can be used to prove other identities Proposition 6 We have 62 χ,2 5,6 q + χ, 5,6 q = χ, 2,5 q/2 Proof The left hand side of 62 is equal to q /20 q 30m2 m q 30m2 +6m+2 + q 30m2 m+3/2 q 30m2 +26m++3/2 n Z q Thus it is sufficient to prove q 30m2 m q 30m2 +6m+2 + q 30m2 m+3/2 q 30m2 +26m++3/2 n Z q 62 = q 5n+2/2 q 5n+3/2 n=0 Recall the quintuple product identity 626 m Z m q 3m2+m z 3m+ + m q 3m2 m z 3m m Z = + z q 2n q n 2 z 2 q n 2 z 2 + q 2n z + q 2n z n= We rewrite the left hand-side as 627 q 2m2+2m z 6m+ q 2m2 +m+ z 6m+ m Z m Z + q 2m2 2m z 6m q 2m2 +0m+2 z 6m+3 m Z m Z Substitute now in 626 q for q 5/2 and z for q 3/2, and multiply the resulting expression with q 3/2 Then 626 turns into the left hand-side of 62 The

19 THE 3-STATE POTTS MODEL AND ROGERS-RAMANUJAN SERIES 9 proof now follows from an easy identity + q 3/2 q 5n q 0n 8 q 0n 2 + q 5n 3/2 + q 5n+3/2 = n= q q 5n+2/2 q 5n+3/2 n=0 References [AP] Adamović, D and Perše, O, On coset vertex algebras with central charge Math Commun 5 200, no, 357 [Ara] Arakawa, T, Representation theory of W -algebras, Invent Math , no 2, ; [BBSS] Bais,F, Bouwknegt, P, Surridge M, and Schoutens,K, Coset construction for extended Virasoro algebras Nuclear Phys B , no 2, [Bo] Borcea, J, Dualities, affine vertex operator algebras, and geometry of complex polynomials, PhD thesis, Lund University, 998 [BS] Bouwknegt, P and Schoutens, K, W -symmetry in conformal field theory, Phys [BF] Rep , no, Bytsko, A and Fring, A, Factorized combinations of Virasoro characters Comm Math Phys , no, [Ca] Capparelli, S, A combinatorial proof of a partition identity related to the level 3 representations of a twisted affine lie algebra, Comm Algebra, 23 99, [DLM] [DLM2] [DW] [F] [FHL] Dong, C, Li, H and Mason, G, Twisted representations of vertex operator algebras and associative algebras Internat Math Res Notices 998, no 8, Dong, C, Li, H and Mason, G, Modular invariance of trace functions in orbifold theory and generalized Moonshine, Comm Math Phys , 56 Dong, C and Wang, Q, On C 2-cofiniteness of parafermion vertex operator algebras, J Algebra , 203 Feingold, A, Some applications of vertex operators to Kac-Moody algebras Vertex operators in mathematics and physics Berkeley, Calif, 983, , Math Sci Res Inst Publ, 3, Springer, New York, 985 Frenkel, I, Huang, Y-Z and Lepowsky, J, On axiomatic approaches to vertex operators and modules, Memoirs Amer Math Soc 0, 993 [FKW] Frenkel, E, Kac, V and Wakimoto, M, Characters and Fusion Rules for W - Algebras via Quantized Drinfeld-Sokolov Reduction, Comm Math Phys 7 992, [IK] [KMY] [KW] [Li] Iohara, K and Koga, Y, Representation theory of the Virasoro algebra, Springer Monographs in Mathematics, 20 Kirazume, M, Miyamoto, M and Yamada, H, Ternary codes and vertex operator algebras J Algebra , no 2, Kac, V and Wakimoto, M, Modular and conformal invariance constraints in representation theory of affine algebras Adv in Math , no 2, Li, H, Local systems of twisted vertex operators, vertex operator superalgebras and twisted modules Moonshine, the Monster, and related topics South Hadley, MA, 99, , Contemp Math, 93, Amer Math Soc, Providence, RI, 996

20 20 ALEX J FEINGOLD AND ANTUN MILAS [Ma] Mauriello, C, Branching rule decomposition of irreducible level- E 6 -modules with respect to the affine subalgebra F, Dissertation in preparation, 202 [Mi] Milas, A, Modular forms and almost linear dependence of graded dimensions Lie algebras, vertex operator algebras and their applications, -2, Contemp Math, 2, Amer Math Soc, Providence, RI, 2007 [Miy] Miyamoto, M, 3-state Potts model and automorphisms of vertex operator algebras of order 3 J Algebra , no, 5676 [Mu] Mukhin, E, Factorization of alternating sums of Virasoro characters J Combin Theory Ser A 2007, no 7, 65-8 [W] Wang, W, Rationality of Virasoro vertex operator algebra, IMRN, 7, 993, 97-2 [X] Xie, C, Structure of the level two Standard Modules for the affine Lie algebra, Comm Algebra, 7 990, A 2 2 Department of Mathematical Sciences, SUNY-Binghamton Department of Mathematics and Statistics, SUNY-Albany address: alex@mathbinghamtonedu address: amilas@albanyedu