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# Advances in Theoretical and Mathematical Physics

## Volume 19 (2015)

### Number 6

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

Pages: 1277 – 1338

DOI: http://dx.doi.org/10.4310/ATMP.2015.v19.n6.a4

#### Author

#### Abstract

We study 4d $\mathcal{N}=2$ 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 $\mathcal{N}=2^{*}$ theory, we describe a full surface operator as the 4d gauge theory coupled to a 2d $\mathcal{N}=(2,2)$ 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 $\mathrm{SL}(N,\mathbb{R})$ WZNW model is predicted from one-loop determinants of 4d gauge theories with a full surface operator via the AGT relation.