Level two string functions and Rogers Ramanujan type identities

Abstract

The level two string functions are calculated exactly for all simply laced Lie algebras, using a ladder coset construction. These are the characters of cosets of the type G/U (1) r , where G is the algebra at level two and r is its rank. This coset is a theory of generalized parafermions. A conjectured Rogers Ramanujan type identity is described for these characters. Using the exact string functions, we verify the Rogers Ramanujan type expressions, that are the main focus of this work.

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

“…Interestingly, it was shown that the A type parefermions theories of level two and any rank can be realised by a product theory of minimal models with particular combinations of the representing fields taken to insure modular invariance is preserved. 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].…”

“…With this in mind the authors of [2] conjectured and numerically verified GRR identities for the A type generalized parafermion characters. Furthermore, an ADE generalization for level two parafermions soon followed and also verified numerically in [7]. Finally although, only ADE level two generalized parafermions characters were given exact analytical expressions the corresponding GRR identities led to a conjecture for the characters of generalized parafermions associated with any Lie algebras at any level and rank level [8].…”

“…More specifically, for level two simply laced Lie algebras, exact expressions for all characters were found in ref. [13] via the ladder coset construction. In addition, a conjectured expression for H(G r ) Λ β , for some cases of Λ, was given and verified numerically.…”