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Spectral flow and the Atiyah-Patodi-Singer index theorem

Christian Baer, Remo Ziemke

TL;DR

This work derives a precise spectral-flow formula for smooth families of twisted Dirac operators on closed odd-dimensional spin manifolds, expressing the flow in terms of the A-hat form, the odd Chern character of the connection family, and the ξ-invariants at the endpoints. The authors reduce the problem to the Atiyah-Patodi-Singer index theorem on a manifold with boundary, delivering a conceptually streamlined proof that unifies several prior results. They generalize Getzler's formula to arbitrary connection families and nontrivial twists, and they demonstrate an application to Llarull-type rigidity for scalar curvature on strictly convex hypersurfaces, valid in both even and odd dimensions. The approach highlights a deep connection between spectral flow, index theory, and geometric constraints, with implications for scalar curvature problems and related geometric analysis.

Abstract

We establish a formula for the spectral flow of a smooth family of twisted Dirac operators on a closed odd-dimensional Riemannian spin manifold, generalizing a result by Getzler. The spectral flow is expressed in terms of the $\hat{A}$-form of the manifold, the odd Chern character form of the family of connections, and the $ξ$-invariants of the initial and final operators. Our proof is based on a reduction to the Atiyah-Patodi-Singer index theorem for manifolds with boundary, which provides a conceptually very simple approach to the problem. As an application, we give a proof of Llarull's rigidity theorem for scalar curvature of strictly convex hypersurfaces in Euclidean space which works the same in even and odd dimensions.

Spectral flow and the Atiyah-Patodi-Singer index theorem

TL;DR

This work derives a precise spectral-flow formula for smooth families of twisted Dirac operators on closed odd-dimensional spin manifolds, expressing the flow in terms of the A-hat form, the odd Chern character of the connection family, and the ξ-invariants at the endpoints. The authors reduce the problem to the Atiyah-Patodi-Singer index theorem on a manifold with boundary, delivering a conceptually streamlined proof that unifies several prior results. They generalize Getzler's formula to arbitrary connection families and nontrivial twists, and they demonstrate an application to Llarull-type rigidity for scalar curvature on strictly convex hypersurfaces, valid in both even and odd dimensions. The approach highlights a deep connection between spectral flow, index theory, and geometric constraints, with implications for scalar curvature problems and related geometric analysis.

Abstract

We establish a formula for the spectral flow of a smooth family of twisted Dirac operators on a closed odd-dimensional Riemannian spin manifold, generalizing a result by Getzler. The spectral flow is expressed in terms of the -form of the manifold, the odd Chern character form of the family of connections, and the -invariants of the initial and final operators. Our proof is based on a reduction to the Atiyah-Patodi-Singer index theorem for manifolds with boundary, which provides a conceptually very simple approach to the problem. As an application, we give a proof of Llarull's rigidity theorem for scalar curvature of strictly convex hypersurfaces in Euclidean space which works the same in even and odd dimensions.

Paper Structure

This paper contains 8 sections, 8 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Setup and preliminaries
  3. The η- and ξ-invariants
  4. Characteristic forms
  5. The Atiyah-Patodi-Singer index theorem
  6. Spectral flow
  7. Proof of the theorem
  8. Examples and an application

Key Result

Theorem 1

Let $M$ be a closed Riemannian spin manifold of odd dimension and $E \to M$ a hermitian vector bundle. Let $\nabla^\bullet=(\nabla^s)_{s \in [a,b]}$ be a smooth 1-parameter family of metric connections on $E$ and $D^s$ the induced twisted Dirac operators acting on sections of $\Sigma M \otimes E$. T

Theorems & Definitions (18)

  • Theorem
  • Corollary 1
  • Theorem : Atiyah-Patodi-Singer index theorem APS1*Thm. 3.10
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Example
  • Remark
  • Lemma 3
  • ...and 8 more