Localization of a KO^∗(pt)-valued index and the orientability of the Pin^-(2) monopole moduli space

2021/09/22 by Miyazawa, Jin
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2109.10579

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^±(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^±(n,s⁺,s^-) structure. We show that the KO^*(pt)-valued index associated with the G^±(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.

Related

• Localization of indices and orientations on G-instanton moduli spaces
• Localization of Dirac Operators on 4n+2 Dimensional Open Spin^c Manifolds
• A mod 2 index theorem for pin^- manifolds
• Non-abelian Seiberg-Witten theory and projectively stable pairs
• Index theory on Pin manifolds