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

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.