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

Miyazawa, Jin¹

¹Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, Japan

Algebraic and Geometric Topology, Tome 25 (2025) no. 2, pp. 887-918

Résumé

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 Hn(s) structure defined by D Freed and 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.

Détail

DOI : 10.2140/agt.2025.25.887

Keywords: Witten deformation, localization of index, Seiberg–Witten theory, $K$-theory

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