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Signature for flat unitary bundles over surfaces with boundary

Inkang Kim, Pierre Pansu, Xueyuan Wan

Abstract

This paper deals with the representations of the fundamental groups of compact surfaces with boundary into classical simple Lie groups of Hermitian type. We relate work on the signature of the associated local systems of Atiyah-Patodi-Singer, to Burger-Iozzi-Wienhard's Toledo invariant. To measure the difference, we extend Atiyah-Patodi-Singer's rho invariant, initially defined on $\mathrm{U}(p)$, to discontinuous class functions, first on $\mathrm{U}(p,q)$, and then on other classical groups via embeddings into $\mathrm{U}(p,q)$. In this way, we present three different invariants -- signature, Toledo and rho invariant -- in a unifying way, which is a version of the classical signature formula of Atiyah-Patodi-Singer for manifolds with boundary.

Signature for flat unitary bundles over surfaces with boundary

Abstract

This paper deals with the representations of the fundamental groups of compact surfaces with boundary into classical simple Lie groups of Hermitian type. We relate work on the signature of the associated local systems of Atiyah-Patodi-Singer, to Burger-Iozzi-Wienhard's Toledo invariant. To measure the difference, we extend Atiyah-Patodi-Singer's rho invariant, initially defined on , to discontinuous class functions, first on , and then on other classical groups via embeddings into . In this way, we present three different invariants -- signature, Toledo and rho invariant -- in a unifying way, which is a version of the classical signature formula of Atiyah-Patodi-Singer for manifolds with boundary.
Paper Structure (66 sections, 51 theorems, 611 equations, 1 figure)

This paper contains 66 sections, 51 theorems, 611 equations, 1 figure.

Table of Contents

  1. Motivation
  2. Main result
  3. Other classical groups of Hermitian type
  4. Final remarks
  5. Organization of the paper
  6. Flat Hermitian vector bundles and eta invariants over a circle
  7. The signature of a flat Hermitian vector bundle
  8. Definition of signature
  9. Relation to indices of operators
  10. A formula for signature
  11. The Atiyah-Patodi-Singer index theorem
  12. The Atiyah-Singer integrands
  13. Limiting values of extended $\mathrm{L}^2$-sections
  14. Toledo invariants
  15. Definition of Toledo invariant
  16. ...and 51 more sections

Key Result

Theorem 1

Let $\Sigma$ be a compact oriented surface with nonempty boundary. Let $E$ be a complex vector space equipped with a (possibly indefinite) Hermitian form $\Omega$. Let $\phi:\pi_1(\Sigma)\to \mathrm{U}(E,\Omega)$, the unitary group of the Hermitian space $(E,\Omega)$, be a homomorphism, and $\mathca Furthermore, where $\boldsymbol{\rho}:\mathrm{U}(E,\Omega)\to\mathbb R$ is a discontinuous real-va

Figures (1)

  • Figure 1: Ideal triangulation of a pair of pants consisting of two ideal triangles

Theorems & Definitions (126)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Corollary 1
  • proof
  • Proposition 2
  • Proposition 1.1
  • proof
  • Remark 1.2
  • ...and 116 more