Affine sl(N) conformal blocks from N=2 SU(N) gauge theories

Recently Alday and Tachikawa proposed a relation between conformal blocks in a two-dimensional theory with affine sl(2) symmetry and instanton partition functions in four-dimensional conformal N=2 SU(2) quiver gauge theories in the presence of a certain surface operator. In this paper we extend this proposal to a relation between conformal blocks in theories with affine sl(N) symmetry and instanton partition functions in conformal N=2 SU(N) quiver gauge theories in the presence of a surface operator. We also discuss the extension to non-conformal N=2 SU(N) theories.Comment: 40 pages. v2: minor changes and clarification

A quantum isomonodromy equation and its application to N=2 SU(N) gauge theories

We give an explicit differential equation which is expected to determine the instanton partition function in the presence of the full surface operator in N=2 SU(N) gauge theory. The differential equation arises as a quantization of a certain Hamiltonian system of isomonodromy type discovered by Fuji, Suzuki and Tsuda.Comment: 15 pages, v2: typos corrected and references added, v3: discussion, appendix and references adde

On "Dotsenko-Fateev" representation of the toric conformal blocks

We demonstrate that the recent ansatz of arXiv:1009.5553, inspired by the original remark due to R.Dijkgraaf and C.Vafa, reproduces the toric conformal blocks in the same sense that the spherical blocks are given by the integral representation of arXiv:1001.0563 with a peculiar choice of open integration contours for screening insertions. In other words, we provide some evidence that the toric conformal blocks are reproduced by appropriate beta-ensembles not only in the large-N limit, but also at finite N. The check is explicitly performed at the first two levels for the 1-point toric functions. Generalizations to higher genera are briefly discussed.Comment: 10 page

Affine sl(N) conformal blocks from N=2 SU(N) gauge theories

Recently Alday and Tachikawa proposed a relation between conformal blocks in a two-dimensional theory with affine sl(2) symmetry and instanton partition functions in four-dimensional conformal N=2 SU(2) quiver gauge theories in the presence of a certain surface operator. In this paper we extend this proposal to a relation between conformal blocks in theories with affine sl(N) symmetry and instanton partition functions in conformal N=2 SU(N) quiver gauge theories in the presence of a surface operator. We also discuss the extension to non-conformal N=2 SU(N) theories