05

Index theory on Lorentzian manifolds

Index theory for hyperbolic equations on Lorentzian manifolds has recently been initiated and has already had applications in quantum field theory. The index theorem of C. Bär and A. Strohmaier applies to Lorentzian manifolds with compact Cauchy hypersurfaces.

In this project, Lorentzian index theory will be developed further and, in particular, the assumption on spatial compactness will be relaxed. This will allow for further physical applications.

The work program can be structured as follows.

1. Determine the class of admissible boundary conditions in the spatially compact case.
2. Relative index theory.
3. Spatial preiodicity.
4. Spatially bounded geometry.
5. Decay conditions at spatial infinity.

## Publications

Let M(n,D) be the space of closed n-dimensional Riemannian manifolds (M,g) with diam(M)≤D and |secM|≤1. In this paper we consider sequences (Mi,gi) in M(n,D) converging in the Gromov–Hausdorff topology to a compact metric space Y. We show, on the one hand, that the limit space of this sequence has at most codimension one if there is a positive number r such that the quotient vol(BMir(x))/injMi(x) can be uniformly bounded from below by a positive constant C(n, r, Y) for all points x∈Mi. On the other hand, we show that if the limit space has at most codimension one then for all positive r there is a positive constant C(n, r, Y) bounding the quotient vol(BMir(x))/injMi(x) uniformly from below for all x∈Mi. As a conclusion, we derive a uniform lower bound on the volume and a bound on the essential supremum of the sectional curvature for the closure of the space consisting of all manifolds in M(n,D) with C≤vol(M)/inj(M).

 Journal Journal of Geometric Analysis Publisher Springer Volume 28, no. 3 Pages 2707-2724 Link to preprint version Link to published version

Related project(s):
5Index theory on Lorentzian manifolds

We consider rough metrics on smooth manifolds and corresponding Laplacians induced by such metrics. We demonstrate that globally continuous heat kernels exist and are H\"older continuous locally in space and time. This is done via local parabolic Harnack estimates for weak solutions of operators in divergence form with bounded measurable coefficients in weighted Sobolev spaces.

Related project(s):
5Index theory on Lorentzian manifolds

We prove that the Atiyah-Singer Dirac operator ${\mathrm D}_{\mathrm g}$ in ${\mathrm L}^2$ depends Riesz continuously on ${\mathrm L}^{\infty}$ perturbations of complete metrics ${\mathrm g}$ on a smooth manifold. The Lipschitz bound for the map ${\mathrm g} \to {\mathrm D}_{\mathrm g}(1 + {\mathrm D}_{\mathrm g}^2)^{-\frac{1}{2}}$ depends on bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius. Our proof uses harmonic analysis techniques related to Calder\'on's first commutator and the Kato square root problem. We also show perturbation results for more general functions of general Dirac-type operators on vector bundles.

Related project(s):
5Index theory on Lorentzian manifolds

We discuss the chiral anomaly for a Weyl field in a curved background and show that a novel index theorem for the Lorentzian Dirac operator can be applied to describe the gravitational chiral anomaly. A formula for the total charge generated by the gravitational and gauge field background is derived in a mathematically rigorous manner. It contains a term identical to the integrand in the Atiyah-Singer index theorem and another term involving the η-invariant of the Cauchy hypersurfaces.

 Journal Commun. Math. Phys. Publisher Springer Volume 347 Pages 703-721 Link to preprint version Link to published version

Related project(s):
5Index theory on Lorentzian manifolds

## Team Members

Prof. Dr. Christian Bär
Universität Potsdam
cbaer(at)uni-potsdam.de

Prof. Dr. Carla Cederbaum
Universität Tübingen
cederbaum(at)math.uni-tuebingen.de

## Former Members

Dr. Lashi Bandara
Researcher
Brunel University London
lashi.bandara(at)brunel.ac.uk

Sebastian Hannes
Doctoral student
Universität Potsdam
shannes(at)math.uni-potsdam.de

PD Dr. habil. Olaf Müller
Researcher
Humboldt-Universität zu Berlin
mullerol(at)math.hu-berlin.de