Localization of a $KO^{\ast}(\text{pt})$-valued index and the orientability of the $Pin^-(2)$ monopole moduli space

TL;DR

This paper generalizes the localization of Dirac indices to a new class of structures called $G^{ ext{±}}(n,s^+,s^-)$, and applies this to determine when the $Pin^-(2)$ monopole moduli space is orientable.

Contribution

It introduces the $G^{ ext{±}}(n,s^+,s^-)$ structure, generalizing $Spin^c$, and formulates a characteristic submanifold concept for it, leading to a localization of the $KO^*(pt)$-valued index.

Findings

The $KO^*(pt)$-valued index localizes to the characteristic submanifold for the new structure.

Provides a topological criterion for orientability of the $Pin^-(2)$ monopole moduli space.

Extends index localization techniques beyond classical $Spin^c$ structures.

Abstract

It is known that the Dirac index of a $Spin^c$ structure is localized to the characteristic submanifold. We introduce the notion of $G^{\pm}(n,s^+,s^-)$ structure on a manifold as a common generalization of the $Spin^c$ structure and the $H_n(s)$ structure defined by D.~Freed--M.~Hopkins, and formulate a version of characteristic submanifold for the $G^{\pm}(n,s^+,s^-)$ structure. We show that the $KO^*(pt)$-valued index associated with the $G^{\pm}(n,s^+,s^-)$ structure is localized to the characteristic submanifold. As an application, we give a topological sufficient condition for the moduli space of $Pin^-(2)$ monopoles to be orientable.