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

Miyazawa, Jin

doi:10.2140/agt.2025.25.887

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

Jin Miyazawa

Algebraic & Geometric Topology 25 (2025) 887–918

DOI: 10.2140/agt.2025.25.887

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 and 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.

Keywords

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

Mathematical Subject Classification

Primary: 57K41, 58J20

References

Publication

Received: 17 September 2022

Revised: 22 September 2023

Accepted: 12 February 2024

Published: 16 May 2025

Authors

Jin Miyazawa
Graduate School of Mathematical Sciences The University of Tokyo Tokyo Japan

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Volume 25, issue 2 (2025)