Stable Wild Vafa-Witten Bundles on the Projective Plane

This work explores the geometry of stable wild Vafa-Witten bundles over the complex projective plane ${\mathbb{P}}^2$. Specifically, we consider stable rank-two pairs $(E,\mathrm{\Phi })$, with $E\to {\mathbb{P}}^2$ a rank-two holomorphic vector bundle and $\mathrm{\Phi }\in H^0({\mathbb{P}}^2,{\mathrm{E}\mathrm{n}\mathrm{d}}_{0}E\otimes \mathcal{O}(d))$ for $d\ge 0$, and compute the dimension of the moduli space of such stable pairs. Moreover, we classify stable pairs $(E,\mathrm{\Phi })$ when the underlying rank-two bundle $E$ splits or is the push-forward of a line bundle on ${\mathbb{P}}^1\times {\mathbb{P}}^1$. Lastly, we examine the fixed point locus of the natural ${\mathbb{C}}^{\ast }$-action on the moduli space.