Conformal blocks of Chiral fields in N=2 SUSY CFT and Affine Laumon Spaces

TL;DR

This paper develops a method to compute N=2 superconformal blocks using affine sl(2) algebra, establishing explicit relations with Nekrasov's instanton partition functions and providing a combinatorial representation for N=2 chiral blocks.

Contribution

It introduces a new explicit combinatorial representation for N=2 superconformal blocks via affine sl(2) algebra and AGT correspondence, clarifying their relation to gauge theory partition functions.

Findings

Derived an explicit affine sl(2) four-point conformal block

Established a direct AGT-type correspondence for N=2 chiral blocks

Provided a combinatorial formula for N=2 superconformal blocks

Abstract

We consider the problem of computing N=2 superconformal block functions. We argue that the Kazama-Suzuki coset realization of N=2 superconformal algebra in terms of the affine sl(2) algebra provides relations between N=2 and affine sl(2) conformal blocks. We show that for N=2 chiral fields the corresponding sl(2) construction of the conformal blocks is based on the ordinary highest weight representation. We use an AGT-type correspondence to relate the four-point sl(2) conformal block with Nekrasov's instanton partition functions of a four-dimensional N=2 SU(2) gauge theory in the presence of a surface operator. Since the previous relation proposed by Alday and Tachikawa requires some special modification of the conformal block function, we revisit this problem and find direct correspondence for the four-point conformal block. We thus find an explicit representation for the affine sl(2) four-point conformal block and hence obtain an explicit combinatorial representation for the N=2 chiral four-point conformal block.