Vertices, Vortices & Interacting Surface Operators

TL;DR

This work shows that non-abelian vortices in 2D $N=(2,2)$ gauge theories with adjoint and anti-fundamental matter form a holomorphic submanifold of the 4D instanton moduli space. It computes the vortex partition functions via equivariant localization, deriving explicit weights for adjoint and anti-fundamental sectors, and connects these counts to the field-theory limit of the topological vertex on the strip with column-boundary conditions. The authors then resum the vortex functions in terms of generalized hypergeometric functions, unveiling an AGT dual description in terms of degenerate Toda blocks and revealing a parallel q-deformed open-string formulation governed by finite-difference equations. Together, these results weave a coherent bridge among vortex counting, instanton calculus, topological strings on strips, and Toda CFT with interacting surface operators, suggesting rich integrable structures and multiple avenues for future extensions.

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.

Figures (4)

• Figure 1: Brane construction of surface operators
• Figure 2: Toric diagram engineering the 4D gauge theory and its classical limit to the strip
• Figure 3: Strip diagrams: (a) anti-fundamental, (b) adjoint
• Figure 4: The strip amplitude for matter in the anti-fundamental