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.
1. Introduction 2. Spacetime splitting 3. Group index and group spectral flow 3.1. Spectral subspaces and equivariant operators 3.2. Equivariant index 3.3. Equivariant spectral flow 4. Set-up: The Dirac bundle 5. Relating to an abstract model operator 5.1. Suitable isometries 5.2. Relating D to the model operator 6. Equivariant index theorem 6.1. From Lorentzian to Riemannian Dirac index 6.2. A Riemannian equivariant index theorem 6.3. Transgression forms 6.4. The main theorem 6.5. Examples
The Atiyah and Singer [1968a,b] index theorem is the seminal work in geometric analysis that combines geometry, topology, and analysis. On an even-dimensional closed Riemannian spin manifold Σ, it states that the Fredholm index ind( / D) of a Dirac operator / D is given by the integral of the Â-class of Σ [Atiyah and Singer, 1968b, Theorem (5.3)]. There are various generalisations of their work over time, for which we refer to the textbook [Lawson and Michelsohn, 1989] and the recent survey [Freed, 2021]. Amongst those, an interesting case occurs when a compact Lie group Γ acts on Σ such that its action is compatible with all the relevant structures so that it commutes with / D. In this situation, the Fredholm index ind( / D) is replaced by the equivariant index, a function on Γ given by ind γ ( / D) := tr(γ| ker( / D) ) -tr(γ| ker( / D * ) ).
The Lefschetz fixed-point formula due to Atiyah and Segal [1968, Theorem 2.12] and Atiyah and Singer [1968b, (5.4)] then states that ind γ ( / D) (for any γ ∈ Γ) is given by the integral of a suitable form over the fixed-point set Σ γ of γ. The first proof of the Atiyah-Segal-Singer index theorem was essentially topological (K-theoretic and cohomological). Analytical proofs were developed later by [Bismut, 1984, Theorem 4.15] (probabilitic approach) and by Berline and Vergne [1985, Theorem 3.32] (heat kernel method).
The ellipticity of Dirac operators on a Riemannian (spin) manifold is arguably the quintessential ingredient in the Atiyah-(Segal-)Singer index theorem. Since Dirac operators on a Lorentzian (spin) manifold are hyperbolic, a straightforward Lorentzian generalisation of the Atiyah-(Segal-)Singer index theorem was not expected until Bär and Strohmaier [2019, theorems 3.3, 3.5, and 4.1] showed that a Dirac operator D on a spatially compact globally hyperbolic spacetime M , subject to the Atyiah-Patodi-Singer (APS) boundary condition at the spacelike boundary ∂M , is Fredholm. They have also proven that ind(D) is given by the same expression as in the Atiyah, Patodi, and Singer [1975, Theorem (3.10)] index theorem. So far, there are a few generalisations [Braverman, 2020;van den Dungen and Ronge, 2021;Damaschke, 2021;Shen and Wrochna, 2022;Bär and Strohmaier, 2024] of the Bär-Strohmaier global index theorem. Amongst these, Damaschke [2021] has considered a modified index in the presence of a group action, but he studied the L 2 -index theorem, where the group serves as a way of compensating for the non-compactness of the manifold and the index is still a single number. In contrast, we assume compactness but use a refined notion of index.
The purpose of this article is to put forward the Bär-Strohmaier global index theorem on an equivariant setting in order to obtain a Lorentzian analogue of the Atiyah-Segal-Singer index theorem. But the Lorentzian spin spacetime (see Corollary 2.7) M pertinent to this article has spacelike boundary ∂M . Hence we have an additional boundary contribution to the Atiyah-Segal-Singer index theorem. On a Γ-equivariant Riemannian spin manifold, such an index theorem was first derived by Donnelly [1978, Theorem 1.2]. His result can be considered as an equivariant generalisation of the Atiyah-Patodi-Singer index theorem. Recent results on this direction can be found, for instance, in [Braverman, 2015;Braverman and Maschler, 2019;Hochs et al., 2020Hochs et al., , 2023;;Hochs, 2024;Sadegh et al., 2024]. Therefore, our result can be perceived as an analogue of the Atiyah-Segal-Singer-Donnelly index theorem in a Lorentzian setting. To be precise, our main theorem is as follows.
Theorem 1.1. Let (M, g) be an even-dimensional smooth, compact globally hyperbolic spin spacetime with spacelike boundary ∂M = Σ 0 ⊔ Σ 1 where Σ 0 and Σ 1 are respectively the past and future boundary. Let E → M be a smooth complex vector bundle over M and D the twisted Dirac operator on the twisted right-handed spinor bundle S + M ⊗ E → M . We assume that there is a group Γ acting on M and E by isometries that preserve all relevant structures, so D is Γ-equivariant. Then, for each γ ∈ Γ, the equivariant index ind γ (D APS ) of the twisted Dirac operator D APS under the Atiyah-Patodi-Singer (APS) boundary condition is given by
, b := -1 2 tr(γ| ker A(0) ) + tr(γ| ker A(1) ) + η γ (A(0)) -η γ (A(1)) .
Here M γ is the fixed-point set of γ with the orientation (see Remark 6.14) induced by the spin-structure and γ on each connected component M n γ of M γ , ℓ := dim M n γ /2 a
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