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A Lorentzian Equivariant Index Theorem

Onirban Islam, Lennart Ronge

TL;DR

This work establishes a Lorentzian, Γ-equivariant Atiyah-Segal-Singer-type index theorem under APS boundary conditions on compact globally hyperbolic spacetimes with timelike boundary. The authors reduce the equivariant problem to the non-equivariant setting via a spacetime splitting M ≅ [0,1] × Σ, decomposing the group action into Γ-eigenspaces and relating the Lorentzian Dirac operator to an abstract model operator of the form ∂_t − iB, thereby linking the equivariant index to the equivariant spectral flow of the hypersurface family A(t). The main result expresses ind_γ(D_APS) as interior and boundary integrals involving the local fixed-point data 𝔞 and 𝔗𝔞 plus a boundary term 𝔟, matching the Riemannian equivariant fixed-point framework (via transgression forms) and recovering known non-equivariant Lorentzian index results. Two illustrative examples demonstrate how the equivariant index can be nontrivial even when the non-equivariant index vanishes and how boundary contributions regulate the total invariant. Overall, the paper extends fixed-point index theory to Lorentzian settings with symmetry, providing a practical method to convert equivariant results from the non-equivariant theory and highlighting potential generalisations to broader Dirac-type operators.

Abstract

We develop a formula for the equivariant index of a twisted Dirac operator on a compact globally hyperbolic spacetime with timelike boundary on which a group acts isometrically, subject to APS boundary conditions. The formula is the same as in the Riemannian case: the equivariant index for a group element is an integral over the fixed point set of that element plus some boundary terms. The proof uses a surprisingly simple technique for reducing from the equivariant to the non-equivariant regime in order to show an equivariant version of the Lorentzian "index $=$ spectral flow" formula.

A Lorentzian Equivariant Index Theorem

TL;DR

This work establishes a Lorentzian, Γ-equivariant Atiyah-Segal-Singer-type index theorem under APS boundary conditions on compact globally hyperbolic spacetimes with timelike boundary. The authors reduce the equivariant problem to the non-equivariant setting via a spacetime splitting M ≅ [0,1] × Σ, decomposing the group action into Γ-eigenspaces and relating the Lorentzian Dirac operator to an abstract model operator of the form ∂_t − iB, thereby linking the equivariant index to the equivariant spectral flow of the hypersurface family A(t). The main result expresses ind_γ(D_APS) as interior and boundary integrals involving the local fixed-point data 𝔞 and 𝔗𝔞 plus a boundary term 𝔟, matching the Riemannian equivariant fixed-point framework (via transgression forms) and recovering known non-equivariant Lorentzian index results. Two illustrative examples demonstrate how the equivariant index can be nontrivial even when the non-equivariant index vanishes and how boundary contributions regulate the total invariant. Overall, the paper extends fixed-point index theory to Lorentzian settings with symmetry, providing a practical method to convert equivariant results from the non-equivariant theory and highlighting potential generalisations to broader Dirac-type operators.

Abstract

We develop a formula for the equivariant index of a twisted Dirac operator on a compact globally hyperbolic spacetime with timelike boundary on which a group acts isometrically, subject to APS boundary conditions. The formula is the same as in the Riemannian case: the equivariant index for a group element is an integral over the fixed point set of that element plus some boundary terms. The proof uses a surprisingly simple technique for reducing from the equivariant to the non-equivariant regime in order to show an equivariant version of the Lorentzian "index spectral flow" formula.
Paper Structure (16 sections, 22 theorems, 165 equations)

This paper contains 16 sections, 22 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Spacetime splitting
  3. Group index and group spectral flow
  4. Spectral subspaces and equivariant operators
  5. Equivariant index
  6. Equivariant spectral flow
  7. Set-up: The Dirac bundle
  8. Relating to an abstract model operator
  9. Suitable isometries
  10. Relating $D$ to the model operator
  11. Equivariant index theorem
  12. From Lorentzian to Riemannian Dirac index
  13. A Riemannian equivariant index theorem
  14. Transgression forms
  15. The main theorem
  16. ...and 1 more sections

Key Result

Theorem 1.0

Let $(M, g)$ be an even-dimensional smooth, compact globally hyperbolic spin spacetime with spacelike boundary $\partial M = \Sigma_{0} \sqcup \Sigma_{1}$ where $\Sigma_{0}$ and $\Sigma_{1}$ are respectively the past and future boundary. Let $E \to M$ be a smooth complex vector bundle over $M$ and $ Here $M_{\gamma}$ is the fixed-point set of $\gamma$ with the orientation (see Remark rem: orientat

Theorems & Definitions (60)

  • Theorem 1.0
  • Definition 2.1
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • Remark 2.5
  • proof
  • Proposition 2.6
  • proof
  • Corollary 2.7
  • ...and 50 more