Non-convergent sequences of solutions to the massive Vafa-Witten equations with 'interesting' $\mathbb{Z}/2\mathbb{Z}$ self-dual harmonic 2-form limits

TL;DR

This paper constructs divergent sequences of solutions to the Vafa-Witten equations with small mass on a product of a torus and a Riemann surface, which converge after renormalization to a harmonic 2-form data set with interesting topological features.

Contribution

It introduces a novel construction of non-convergent solution sequences that converge to a harmonic 2-form data set with a non-trivial zero locus, revealing new geometric phenomena.

Findings

Sequences of solutions diverge but converge after renormalization

The limit involves a harmonic 2-form with a non-extendable line bundle

The zero locus of the 2-form is a codimension 2 submanifold

Abstract

This paper constructs sequences of solutions to the Vafa-Witten equations with non-zero (but small) mass term on the product of a 2-dimensional torus with a Riemann surface of genus greater than 1. These are divergent sequences (modulo principle bundle automorphisms) that converge after renormalization to define an 'interesting' $\mathbb{Z}/2\mathbb{Z}$ harmonic 2-form data set. This data set consists of a non-empty, codimension 2 submanifold, a real line bundle defined on the complement of that submanifold with no extension across it, and a self-dual, harmonic 2-form with values in that line bundle that does not extend over the submanifold. Even so, the norm of this 2-form does extend as a H\"older continuous function with that submanifold being its zero locus.