Givental J-functions, quantum integrable systems, AGT relation with surface operator

Abstract

We study 4d N = 2 gauge theories with a co-dimension two full surface operator, which exhibit a fascinating interplay of supersymmetric gauge theories, equivariant Gromov-Witten theory and geometric representation theory. For pure Yang-Mills and N = 2* theory, we describe a full surface operator as the 4d gauge theory coupled to a 2d N = (2, 2) gauge theory. By supersymmetric localizations, we present the exact partition functions of both 4d and 2d theories which satisfy integrable equations. In addition, the form of the structure constants with a semi-degenerate field in SL(N, R) WZNW model is predicted from one-loop determinants of 4d gauge theories with a full surface operator via the AGT relation.