Stable sheaves with twisted sections and the Vafa–Witten equations on smooth projective surfaces

Abstract

This article describes a Hitchin–Kobayashi style correspondence for the Vafa–Witten equations on smooth projective surfaces. This is an equivalence between a suitable notion of stability for a pair ${(\mathcal{E}, \varphi)}$, where ${\mathcal{E}}$ is a locally-free sheaf over a surface X and ${\varphi}$ is a section of End${(\mathcal{E}) \otimes K_{X}}$; and the existence of a solution to certain gauge-theoretic equations, the Vafa–Witten equations, for a Hermitian metric on ${\mathcal{E}}$. It turns out to be a special case of results obtained by Álvarez-Cónsul and García-Prada on the quiver vortex equation. In this article, we give an alternative proof which uses a Mehta–Ramanathan style argument originally developed by Donaldson for the Hermitian–Einstein problem, as it relates the subject with the Hitchin equations on Riemann surfaces, and surely indicates a similar proof of the existence of a solution under the assumption of stability for the Donaldson–Thomas instanton equations described in Tanaka (2013) on smooth projective threefolds; and more broadly that for the quiver vortex equation on higher dimensional smooth projective varieties.