Coset conformal blocks and N=2 gauge theories

TL;DR

This paper explores the proposed relationship between coset conformal field theories and N=2 SU(N) gauge theories on orbifolds, providing explicit checks and evidence for the factorization of conformal blocks and their correspondence to instanton partition functions.

Contribution

It offers the first detailed verification of the coset/gauge theory correspondence for specific cases and suggests a factorization structure of conformal blocks matching instanton partition functions.

Findings

Verified the coset/gauge theory relation for (N,p)=(2,4).

Discovered factorization of conformal blocks into products of simpler blocks.

Provided evidence linking conformal blocks to instanton partition functions.

Abstract

It was recently suggested that the su(N)_k+su(N)_p/su(N)_{k+p} coset conformal field theories should be related to N=2 SU(N) gauge theories on R^4/Z_p. In this paper we study various aspects of this proposal. We perform explicit checks of the relation for (N,p)=(2,4), where the symmetry algebra of the coset is the so called S_3 parafermion algebra. Even though the symmetry algebra of the coset is unknown for generic (N,p) models, we manage to perform non-trivial checks in the general case by using knowledge of the Kac determinant of the coset CFT. We also find evidence that the conformal blocks of the (N,p) model should factorise into a certain product of p (N,1) conformal blocks. Precisely this structure is present in the instanton partition function on R^4/Z_p.