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Local Index Theory for Lorentzian Manifolds

Christian Baer, Alexander Strohmaier

TL;DR

The paper develops a local index formula for Lorentzian Dirac-type operators on globally hyperbolic spacetimes, linking Fredholm properties to boundary data via Feynman parametrices rather than elliptic heat kernels. By constructing and analyzing Feynman propagators, including a Hadamard-type expansion and a careful treatment of product-structure near Cauchy hypersurfaces, it derives a Lorentzian index theorem that expresses the index as a sum of a local spectral current (xi/eta-invariants) and a bulk Hadamard term. This Lorentzian framework generalizes APS-type results to non-elliptic settings and to non-selfadjoint operators, enabling index-theoretic interpretations of relativistic phenomena without Wick rotation. The results unify boundary contributions with local Hadamard data, giving a robust mechanism to compute indices in physically relevant spacetimes and clarifying the role of eta-/xi-type invariants in a Lorentzian context. In the twisted setting, the bulk term recovers familiar characteristic forms, establishing a broad, local, and physically meaningful index theory on Lorentzian manifolds.

Abstract

Index theory for Lorentzian Dirac operators is a young subject with significant differences to elliptic index theory. It is based on microlocal analysis instead of standard elliptic theory and one of the main features is that a nontrivial index is caused by topologically nontrivial dynamics rather than nontrivial topology of the base manifold. In this paper we establish a local index formula for Lorentzian Dirac-type operators on globally hyperbolic spacetimes. This local formula implies an index theorem for general Dirac-type operators on spatially compact spacetimes with Atiyah-Patodi-Singer boundary conditions on Cauchy hypersurfaces. This is significantly more general than the previously known theorems that require the compatibility of the connection with Clifford multiplication and the spatial Dirac operator on the Cauchy hypersurface to be selfadjoint with respect to a positive definite inner product.

Local Index Theory for Lorentzian Manifolds

TL;DR

The paper develops a local index formula for Lorentzian Dirac-type operators on globally hyperbolic spacetimes, linking Fredholm properties to boundary data via Feynman parametrices rather than elliptic heat kernels. By constructing and analyzing Feynman propagators, including a Hadamard-type expansion and a careful treatment of product-structure near Cauchy hypersurfaces, it derives a Lorentzian index theorem that expresses the index as a sum of a local spectral current (xi/eta-invariants) and a bulk Hadamard term. This Lorentzian framework generalizes APS-type results to non-elliptic settings and to non-selfadjoint operators, enabling index-theoretic interpretations of relativistic phenomena without Wick rotation. The results unify boundary contributions with local Hadamard data, giving a robust mechanism to compute indices in physically relevant spacetimes and clarifying the role of eta-/xi-type invariants in a Lorentzian context. In the twisted setting, the bulk term recovers familiar characteristic forms, establishing a broad, local, and physically meaningful index theory on Lorentzian manifolds.

Abstract

Index theory for Lorentzian Dirac operators is a young subject with significant differences to elliptic index theory. It is based on microlocal analysis instead of standard elliptic theory and one of the main features is that a nontrivial index is caused by topologically nontrivial dynamics rather than nontrivial topology of the base manifold. In this paper we establish a local index formula for Lorentzian Dirac-type operators on globally hyperbolic spacetimes. This local formula implies an index theorem for general Dirac-type operators on spatially compact spacetimes with Atiyah-Patodi-Singer boundary conditions on Cauchy hypersurfaces. This is significantly more general than the previously known theorems that require the compatibility of the connection with Clifford multiplication and the spatial Dirac operator on the Cauchy hypersurface to be selfadjoint with respect to a positive definite inner product.

Paper Structure

This paper contains 32 sections, 29 theorems, 241 equations, 2 figures.

Table of Contents

  1. The setup
  2. Normally hyperbolic and Dirac-type operators
  3. Restriction to hypersurfaces
  4. Odd Dirac-type operators
  5. The Cauchy problem
  6. The formally dual operator
  7. Laplace-type and Dirac-type operators on Sigma
  8. Laplace-type operators
  9. Dirac-type operators
  10. Distinguished parametrices for normally hyperbolic operators
  11. The Feynman propagator in the product case
  12. Feynman propagators on general globally hyperbolic spacetimes
  13. Feynman propagators for Dirac-type operators
  14. APS boundary conditions
  15. The Feynman propagator in the product case
  16. ...and 17 more sections

Key Result

Proposition 1

Let $P$ be a normally hyperbolic operator acting on sections of a vector bundle $\mathscr{S}$ over a $2m$-dimensional globally hyperbolic manifold $X$. Then every Feynman parametrix $G$ of $P$ is of the form where $G^\mathrm{reg}$ is a distribution on $X\times X$ which is $C^2$ on a neighborhood of the diagonal and near the diagonal we have

Figures (2)

  • Figure 1: Strip containing the spectrum of $\Delta_\theta^{1/2}$
  • Figure 2: $\omega$-sectoriality of $\Delta_\theta^{1/2}+C'$

Theorems & Definitions (71)

  • Proposition
  • Theorem
  • Definition 1.1
  • Example 1.2
  • Remark 1.3
  • Definition 1.4
  • Definition 1.5
  • Example 1.6
  • Example 1.7
  • Definition 1.8
  • ...and 61 more