Refined Vafa-Witten invariants for toric surfaces from supersymmetric localization in 5D gauge theory

Abstract

We study the partition function of five-dimensional $\mathcal{N}=1$ $U(N)$ supersymmetric Yang-Mills (SYM) theory with an adjoint hypermultiplet of mass $m_{\rm adj}$ on a toric K\"ahler surface $S$ times a circle of radius $\boldsymbol{\beta}$. Extending earlier work in $\mathcal{N}=2^*$ SYM theory on $S$, and in pure $\mathcal{N}=1$ SYM on $S\times \mathbb{S}^1_{\boldsymbol{\beta}}$, we find that the path integral localizes to an integral along the Cartan torus of the product of Nekrasov 5D partition functions for each affine patch. Restricting to the gauge group $U(2)$ for simplicity, the integrand has an infinite set of poles of degree at most $\chi(S)-2$. With a natural prescription for integrating around such poles, we find that the contributing poles are in one-to-one correspondence with the torus-fixed points in the moduli space of semi-stable torsion-free sheaves on $S$. Moreover, for non-even first Chern class, their contributions are independent of the equivariant parameters $\epsilon_1,\epsilon_2$ and add up to the $\chi_{y^2}$-genus of that moduli space, where $y^2=e^{-\boldsymbol{\beta} m_{\rm adj}}$, and hence coincide with the refined Vafa-Witten invariants. For even Chern class, the partition function depends on the equivariant parameters $\epsilon_1,\epsilon_2$ as well as $y$, and its relation to rational, refined Vafa-Witten invariants remains unclear.