Vertices, Vortices & Interacting Surface Operators

TL;DR

This paper connects vortex moduli spaces in 2D supersymmetric gauge theories with 4D instanton moduli spaces, computes vortex partition functions, and relates them to topological string amplitudes and hypergeometric functions, revealing dualities and finite difference equations.

Contribution

It introduces a novel description of vortex moduli spaces as submanifolds of instanton moduli spaces and links vortex partition functions to topological vertex and hypergeometric functions.

Findings

Vortex partition functions match the field theory limit of the topological vertex.

Resummation of vertex partition functions yields generalized hypergeometric functions.

Topological open string amplitudes satisfy finite difference equations.

Abstract

We show that the vortex moduli space in non-abelian supersymmetric N=(2,2) gauge theories on the two dimensional plane with adjoint and anti-fundamental matter can be described as an holomorphic submanifold of the instanton moduli space in four dimensions. The vortex partition functions for these theories are computed via equivariant localization. We show that these coincide with the field theory limit of the topological vertex on the strip with boundary conditions corresponding to column diagrams. Moreover, we resum the field theory limit of the vertex partition functions in terms of generalized hypergeometric functions formulating their AGT dual description as interacting surface operators of simple type. Analogously we resum the topological open string amplitudes in terms of q-deformed generalized hypergeometric functions proving that they satisfy appropriate finite difference equations.