$L^2$-cohomology of quasi-fibered boundary metrics

TL;DR

This paper develops techniques to compute the weighted L^2-cohomology of quasi-fibered boundary metrics, enabling the verification of conjectures in geometric analysis and mathematical physics for specific moduli spaces.

Contribution

It introduces new methods for calculating weighted L^2-cohomology of QFB-metrics and applies these to verify the Vafa-Witten and Sen conjectures in particular geometric contexts.

Findings

Computed reduced L^2-cohomology for various QFB-metrics.

Confirmed the Vafa-Witten conjecture for the Nakajima metric.

Proved the Sen conjecture for monopole moduli space of charge 3.

Abstract

We develop new techniques to compute the weighted $L^2$-cohomology of quasi-fibered boundary metrics (QFB-metrics). Combined with the decay of $L^2$-harmonic forms obtained in a companion paper, this allows us to compute the reduced $L^2$-cohomology for various classes of QFB-metrics. Our results applies in particular to the Nakajima metric on the Hilbert scheme of $n$ points on $\mathbb{C}^2$, for which we can show that the Vafa-Witten conjecture holds. Using the compactification of the monopole moduli space announced by Fritzsch, the first author and Singer, we can also give a proof of the Sen conjecture for the monopole moduli space of magnetic charge 3.