Localization of indices and orientations on $G$-instanton moduli spaces

TL;DR

This paper explores the orientations of $G$-instanton moduli spaces on 4-manifolds, relating different $ ext{Spin}^c$ structures and providing alternative proofs and constructions using Witten localization.

Contribution

It establishes the relationship between orientations from various $ ext{Spin}^c$ structures and offers new proofs and methods for orientability of instanton moduli spaces.

Findings

Relation between orientations from different $ ext{Spin}^c$ structures clarified

Alternative proof of orientability provided

Orientation constructed using Witten localization

Abstract

Joyce, Tanaka, and Upmeier give an orientation of the $G$-instanton moduli spaces on a closed four manifolds which is canonically defined using the the $\mathrm{Spin}^c$ structure on the $4$-manifold. In this note, we describe the relation between the orientations given by other choices of the $\mathrm{Spin}^c$ structures, in a slightly more general setting. Furthermore, we give an alternative proof of the orientability of the instanton moduli spaces and an alternative construction of the orientation by Joyce, Tanaka and Upmeier, by using Witten localization of the index of a Dirac type operator.