Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Equivariant Spectral Flow and Equivariant $η$-invariants on Manifolds With Boundary

Johnny Lim, Hang Wang

Abstract

In this article, we study several closely related invariants associated to Dirac operators on odd-dimensional manifolds with boundary with an action of the compact group $H$ of isometries. In particular, the equality between equivariant winding numbers, equivariant spectral flow, and equivariant Maslov indices is established. We also study equivariant $η$-invariants which play a fundamental role in the equivariant analog of Getzler's spectral flow formula. As a consequence, we establish a relation between equivariant $η$-invariants and equivariant Maslov triple indices in the splitting of manifolds.

Equivariant Spectral Flow and Equivariant $η$-invariants on Manifolds With Boundary

Abstract

In this article, we study several closely related invariants associated to Dirac operators on odd-dimensional manifolds with boundary with an action of the compact group of isometries. In particular, the equality between equivariant winding numbers, equivariant spectral flow, and equivariant Maslov indices is established. We also study equivariant -invariants which play a fundamental role in the equivariant analog of Getzler's spectral flow formula. As a consequence, we establish a relation between equivariant -invariants and equivariant Maslov triple indices in the splitting of manifolds.

Paper Structure

This paper contains 9 sections, 35 theorems, 166 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Equivariant spectral flow
  4. Equivariant Maslov index
  5. Equivariant winding number
  6. Equivariant Maslov triple index
  7. Equivariant $\zeta$-determinants
  8. Equivariant Scott-Wojciechowski Theorem
  9. Equivariant $\eta$-invariants

Key Result

Lemma \oldthetheorem

Let $h \in H.$ Let $P \in {\mathrm{Gr} }_h(A).$ By Assumption assump1, the isometry $h$ can be characterized by the form

Figures (14)

  • Figure 1: A path with crossings, and endpoints in the same quadrant.
  • Figure 2: A path homotopy relative endpoints in the same quadrant.
  • Figure 3: The resulting path without crossings.
  • Figure 4: A path with crossings, and endpoints in different quadrants.
  • Figure 5: A path homotopy relative endpoints in different quadrants.
  • ...and 9 more figures

Theorems & Definitions (89)

  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • ...and 79 more