Generalized Rogers Ramanujan Identities Motivated by AGT Correspondence

Abstract

AGT correspondence and its generalizations attracted a great deal of attention recently. In particular it was suggested that U (r) instantons on R 4 /Z p describe the conformal blocks of the coset A(r,where n is a parameter. Our purpose here is to describe Generalized Rogers Ramanujan (GRR) identities for these cosets, which expresses the characters as certain q series. We propose that such identities exist for the coset A(r, p) for all positive integers n and all r and p. We treat here the case of n = 1 and r = 2, finding GRR identities for all the characters.

Level two string functions and Rogers Ramanujan type identities

“…The case of A r was already discussed in ref. [6], where GRR identities were found for all the characters of H. Our aim here is to generalize these results to all of the simply laced algebras.…”

On the Characters of Parafermionic Field Theories

“…Also, for the level two simply laced algebras a GRR expression was described in refs. [7,8], and our formula here specialises precisely to these level two results. Thus our formula, eq.…”

“…Several examples of characters in CFT were studied, and were shown to be expressible as Rogers Ramanujan type sums [2,3,4,5,6,7,8]. The origin of these identities is somewhat perplexing, though, they were shown to be connected with local state probabilities in solvable lattice models [9,10,11], and with thermodynamic Bethe ansatz equations (see, e.g., [6] and refs.…”

The SU ( r ) 2 string functions as q -diagrams

“…This already implies that the SU (r + 1) q-diagrams indeed have the right symmetry structure as to the describe the SU (r + 1) level two string functions. In our work regarding the SO(2r) diagrams it was evident that the diagrammatic picture for the 2 The reader might recall that the mentioned correspondence, refers strictly to characters associated with fundamental weights. For the case at hand we note that these provide all the characters of the theory due to identifications via the external automorphism of SU (r).…”

“…This was followed by [7] where this program was generalized to all simply laced affine Lie algebras and the characters of ADE generalized parafermions at level two and any rank were also found. More specifically, the authors of [2] considered the coset A(k, r) = H × SU (r) k × SU (k) r × SU (k) n SU (k) r+n (1.3) corresponding to the construction described above for the k M 5-branes on R 4 /Z N instanton partition function where n is given in terms of the Nekrasov parameters 1,2 [10]. For two M 5-branes on R 4 /Z N it was observed that, up to U (1) factors which enter trivially in the characters, the coset theory A(k, r) is described in a more illuminating fashion via the use of level-rank duality and the ladder coset construction A(2, r) m r+1 k 1 ,...,kr,k r+1 = m 2 ,...,mr m i +m i+1 =k i+1 mod 2 r i=2 SU (2)…”