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

Jin Miyazawa

TL;DR

This work develops a unified framework, the $G^{\pm}(n,s^+,s^-)$ structures, to interpolate Spin, Spin^c, Pin^{\pm}, and Freed–Hopkins’ $H_s(n)$ settings and defines a $KO^{s^- - n - s^+}(pt)$-valued index $\text{ind}(\mathfrak s)$. The authors prove a localization (via Witten deformation) of this index to a generalized characteristic submanifold, yielding a commutative diagram that links $G^{\pm}$ bordism to KO-theory and thereby enabling a topological criterion for orientability of the $Pin^-(2)$ monopole moduli space. They instantiate the theory with Freed–Hopkins’ $H_n(s)$ and the special case $G^{+}(5,0,4)$, and they derive a mod 2 obstruction criterion for orientation in the $Pin^-(2)$-monopole setting, including explicit nonorientable examples. The construction promises a broad analytic-topological toolkit for gauge-theoretic moduli spaces and suggests potential applications to other moduli problems via the same localization mechanism.

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.

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

TL;DR

This work develops a unified framework, the structures, to interpolate Spin, Spin^c, Pin^{\pm}, and Freed–Hopkins’ settings and defines a -valued index . The authors prove a localization (via Witten deformation) of this index to a generalized characteristic submanifold, yielding a commutative diagram that links bordism to KO-theory and thereby enabling a topological criterion for orientability of the monopole moduli space. They instantiate the theory with Freed–Hopkins’ and the special case , and they derive a mod 2 obstruction criterion for orientation in the -monopole setting, including explicit nonorientable examples. The construction promises a broad analytic-topological toolkit for gauge-theoretic moduli spaces and suggests potential applications to other moduli problems via the same localization mechanism.

Abstract

It is known that the Dirac index of a structure is localized to the characteristic submanifold. We introduce the notion of structure on a manifold as a common generalization of the structure and the structure defined by D.~Freed--M.~Hopkins, and formulate a version of characteristic submanifold for the structure. We show that the -valued index associated with the structure is localized to the characteristic submanifold. As an application, we give a topological sufficient condition for the moduli space of monopoles to be orientable.

Paper Structure

This paper contains 17 sections, 31 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Definition of the index
  4. The $G^{\pm}(n,s^+,s^-)$ structures
  5. Definition of spinor bundles
  6. Proof of the Main Theorem
  7. Statement of the Main Theorem
  8. Localization by Witten deformation
  9. Step.1
  10. Step.2
  11. Examples
  12. Freed-Hopkins' $H_n(s)$
  13. The case of $G^+(5,0,4)$ structure
  14. The orientability of $Pin^-(2)$ monopole moduli space
  15. The orientability and mod $2$ indeces
  16. ...and 2 more sections

Key Result

Lemma 2.1

Assume $a \ge b$. Let $(s,\epsilon,Cl_{(b,a)},H)$ be a representative element of $KO^{(b,a)}(X)$. Then there exists an element $(s',\epsilon',Cl_{(0,a-b)},H')$ of $KO^{(0.a-b)}(X)$ which satisfies the following properties : There is a isomorphism between the Hilbert spaces such that $s=f \circ s' \otimes \epsilon^b \circ f^{-1}$, $\epsilon=f \circ \epsilon' \otimes \epsilon^b \circ f^{-1}$, $\eps

Theorems & Definitions (95)

  • Lemma 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Remark 2.9
  • ...and 85 more