W-algebras and surface operators in N=2 gauge theories

TL;DR

This paper links W-algebras derived from affine sl(N) algebras to surface operators in N=2 gauge theories, proposing a method to compute instanton partition functions using these algebras, and verifies it for specific cases.

Contribution

It introduces a novel approach to compute instanton partition functions in N=2 gauge theories with surface operators via associated W-algebras, establishing new algebraic relations.

Findings

Agreement with known partition functions for SU(3) with simple surface operators

Establishment of a correspondence between surface operators and W-algebras

Derivation of relations between W_3^(2) and W_3 algebras

Abstract

A general class of W-algebras can be constructed from the affine sl(N) algebra by (quantum) Drinfeld-Sokolov reduction and are classified by partitions of N. Surface operators in an N=2 SU(N) 4d gauge theory are also classified by partitions of N. We argue that instanton partition functions of N=2 gauge theories in the presence of a surface operator can also be computed from the corresponding W-algebra. We test this proposal by analysing the Polyakov-Bershadsky W_3^(2) algebra obtaining results that are in agreement with the known partition functions for SU(3) gauge theories with a so called simple surface operator. As a byproduct, our proposal implies relations between the W_3^(2) and W_3 algebras.