Localization of Vortex Partition Functions in $\mathcal{N}=(2,2) $ Super Yang-Mills theory

TL;DR

Localization of vortex partition functions in $N=(2,2)$ $U(N)$ SYM with $N_f=N$ is achieved by mass-deforming the $N=(4,4)$ theory and applying equivariant localization. The authors formulate the vortex matrix model, establish a $Q_ extepsilon$-exact structure, and identify fixed points with $N$-tuple one-dimensional partitions, yielding explicit contour integrals for $Z_k$. They extend to K-theoretic counting via the equivariant character of the vortex moduli space and show the two-dimensional limit reproduces residue formulas, including abelian cases linked to refined topological strings. The work connects vortex counting to Grassmannian geometry and $J$-functions, suggesting a geometric interpretation of nonabelian vortex counts and guiding future exploration for general flavor content.

Abstract

In this article, we study the localizaiton of the partition function of BPS vortices in $\mathcal{N}=(2,2)$ $U(N)$ super Yang-Mills theory with $N$-flavor on $\R^2$. The vortex partition function for $\mathcal{N}=(2,2)$ super Yang-Mills theory is obtained from the one in $\mathcal{N}=(4,4)$ super Yang-Mills theory by mass deformation. We show that the partition function can be written as $Q$-exact form and integration in the partition functions is localized to the fixed points which are related to $N$-tuple one dimensional partitions of positive integers.