[Paper Review] Local Index Theory for Lorentzian Manifolds

Christian Baer, Alexander Strohmaier|arXiv (Cornell University)|Dec 2, 2020

Advanced Operator Algebra Research4 citations

TL;DR

This paper establishes a local index formula for Lorentzian Dirac-type operators on globally hyperbolic spacetimes, extending index theory beyond elliptic operators by using microlocal analysis and Hadamard expansions. The key contribution is a general index theorem for spatially compact spacetimes with Atiyah-Patodi-Singer boundary conditions, valid without requiring a positive definite inner product or self-adjointness of the spatial Dirac operator.

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.

Motivation & Objective

• To develop a local index formula for Lorentzian Dirac-type operators on globally hyperbolic spacetimes, overcoming the limitations of standard elliptic index theory.
• To generalize existing Lorentzian index theorems by removing the requirement for a positive definite inner product on the vector bundle over spacelike hypersurfaces.
• To establish an index theorem for general Dirac-type operators on spatially compact spacetimes with Atiyah-Patodi-Singer boundary conditions, even when the spatial Dirac operator is not self-adjoint with respect to a positive definite metric.
• To extend the applicability of index theory to relativistic quantum field theory and anomalies, such as the chiral anomaly, in settings where Wick rotation or separation of variables fails.
• To provide a rigorous microlocal foundation for index theory in Lorentzian geometry, using propagation of singularities and Hadamard expansions of Green's functions.

Proposed method

• Derives a local index formula via the asymptotic expansion of the supertrace of the difference between retarded and advanced Green's operators.
• Employs Hadamard parametrix techniques to compute the short-time behavior of the heat kernel and Green's functions, extracting local invariants.
• Uses microlocal analysis and Hörmander's propagation of singularities theorem to establish Fredholm properties of the Dirac operator with boundary conditions.
• Applies the Atiyah-Patodi-Singer boundary condition in the Lorentzian setting, treating the Cauchy hypersurface as a boundary with a well-defined spectral projection.
• Introduces a refined Hadamard expansion involving the Feynman propagator and boundary contributions, incorporating the $ ilde{C}(eta,n, u)$ and $C(eta,n)$ functions for analytic continuation.
• Computes the local index density by taking the trace of the wave front set and using the $eta$-regularization of the Green's function at critical dimensions.

Experimental results

Research questions

• RQ1 Can a local index formula be established for Lorentzian Dirac-type operators on globally hyperbolic spacetimes, analogous to the Atiyah-Singer local index theorem in the elliptic case?
• RQ2 How can the index theorem be extended to spacetimes with boundary when the spatial Dirac operator is not self-adjoint with respect to a positive definite inner product?
• RQ3 What is the role of microlocal analysis and Hadamard expansions in defining the index for hyperbolic operators in Lorentzian geometry?
• RQ4 How do boundary contributions in the index formula relate to the $ar{ heta}$-invariant and the $ ilde{C}(eta,n, u)$ functions in the non-elliptic setting?
• RQ5 Can the index formula be made independent of the choice of positive definite metric on the bundle, thereby generalizing previous results?

Key findings

• The local index formula is derived from the asymptotic expansion of the supertrace of the difference between retarded and advanced Green's operators, yielding a distributional index density.
• The index formula includes logarithmic and power-law divergences in time $ t $, with coefficients involving $ ilde{C}(eta,n, u) $, $ C(eta,n) $, and their derivatives at $ eta = -n/2 $.
• The leading term in the expansion is proportional to $ ext{tr}( abla_{(1)} ilde{D} ullet V_{rac{n-2}{2}}(0,y,0,y) ullet ot{n}_ ext{Σ}) imes rac{-i 2^{1-n} u^{-rac{n}{2}}}{(rac{n-2}{2})!} ig( ext{Euler--Mascheroni constant} + ext{log}|t| + ext{imaginary shift} ig) $, capturing the non-trivial dynamics.
• The formula remains valid even when the spatial Dirac operator is not self-adjoint with respect to a positive definite inner product, generalizing prior results.
• The index is shown to be finite and well-defined via $ eta $-regularization, with the $ eta $-derivative of $ ilde{C}(eta,n,2 u) $ at $ eta = -n/2 $ contributing to the $ t^{1-n} $ term.
• The result implies a global index theorem for spatially compact spacetimes with Atiyah-Patodi-Singer boundary conditions, even in the absence of a positive definite metric on the bundle.