A & B model approaches to surface operators and Toda theories

TL;DR

This paper explores the relationship between surface operators in 4d N=2 gauge theories and Toda theories using topological string theory, extending previous semi-classical results to a more general and exact framework.

Contribution

It introduces a topological string theory approach to analyze multiple surface operators in 4d N=2 gauge theories, providing exact calculations beyond semi-classical limits.

Findings

Multiple surface operators can be computed using B-model topological recursion.

The 5d lift of gauge theory partition functions matches A-model topological string with toric branes.

Results align with AGT conjecture and degenerate operator interpretations.

Abstract

It has recently been argued by Alday et al that the inclusion of surface operators in 4d N=2 SU(2) quiver gauge theories should correspond to insertions of certain degenerate operators in the dual Liouville theory. So far only the insertion of a single surface operator has been treated (in a semi-classical limit). In this paper we study and generalise this proposal. Our approach relies on the use of topological string theory techniques. On the B-model side we show that the effects of multiple surface operator insertions in 4d N=2 gauge theories can be calculated using the B-model topological recursion method, valid beyond the semi-classical limit. On the mirror A-model side we find by explicit computations that the 5d lift of the SU(N) gauge theory partition function in the presence of (one or many) surface operators is equal to an A-model topological string partition function with the insertion of (one or many) toric branes. This is in agreement with an earlier proposal by Gukov. Our A-model results were motivated by and agree with what one obtains by combining the AGT conjecture with the dual interpretation in terms of degenerate operators. The topological string theory approach also opens up new possibilities in the study of 2d Toda field theories.