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

TL;DR

The authors extend the AGT framework by proposing and testing a correspondence between affine sl(N) conformal blocks and instanton partition functions of 4d N=2 SU(N) quiver gauge theories in the presence of a full surface operator, and they extend the analysis to non-conformal theories. They develop explicit constructions for SU(2) and SU(N) cases, employing K and K† insertions to relate conformal blocks on spheres and tori to Nekrasov sums across conformal and asymptotically free regimes. The work provides detailed dictionaries between CFT data (isospin variables, affine levels, degenerate insertions) and gauge theory data (Coulomb moduli, masses, instanton counting), with hypergeometric structures emerging at low orders and across several topologies. It also discusses the interplay between 2d and 4d defects, the distinction between simple and full surface operators, and outlines future directions including proofs in special cases, topological string interpretations, and connections to broader W-algebras via Drinfeld–Sokolov reductions.

Abstract

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.