On Vafa-Witten equations over Kaehler manifolds

TL;DR

This paper investigates the analytic properties and compactness of solutions to the Vafa-Witten equations on compact Kähler manifolds, revealing obstructions, behavior of spectral covers, and connections to complex geometry and monopole moduli spaces.

Contribution

It provides new insights into the compactness and limits of solutions to Vafa-Witten equations, including a simplified proof of Taubes' results and a geometric interpretation involving spectral covers.

Findings

Obstructions to nontrivial solutions identified.

Gauge-theoretic compactness similar to Hermitian-Yang-Mills connections.

Limits of solutions characterized by spectral cover convergence.

Abstract

In this paper, we study the analytic properties of solutions to the Vafa-Witten equation over a compact Kaehler manifold. Simple obstructions to the existence of nontrivial solutions are identified. The gauge theoretical compactness for the $\mathbb{C}^*$ invariant locus of the moduli space is shown to behave similarly as the Hermitian-Yang-Mills connections. More generally, this holds for solutions with uniformly bounded spectral covers such as nilpotent solutions. When spectral covers are unbounded, we manage to take limits of the renormalized Higgs fields which are intrinsically characterized by the convergence of the associated spectral covers. This gives a simpler proof for Taubes' results on rank two solutions over Kaehler surfaces together with a new complex geometric interpretation. The moduli space of $SU(2)$ monopoles and some related examples are also discussed in the final section.