  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.
