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

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 $text{End} (mathcal{E}) otimes K_{X}$; 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 Alvarez-Consul and Garcia-Prada. 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 arXiv:0805.2192 on smooth projective threefolds; and more broadly that for the quiver vortex equation on higher dimensional smooth projective varieties.