# The local version of the Lorentzian index theorem

Posted in
Speaker:
Christian Bär
Affiliation:
University of Potsdam
Date:
Tue, 2017-09-19 09:30 - 10:30
Location:
MPIM Lecture Hall

The Atiyah-Singer index theorem for Dirac operators $D$ on compact Riemannian spin $n$-manifolds can be proved using the heat kernels of $D^*D$ and of $DD^*$.
Namely, one easily sees that
$$\mathrm{ind}(D) = \mathrm{Tr}(e^{-tD^*D}) - \mathrm{Tr}(e^{-tDD^*})$$
for any $t>0$. Inserting the short time asymptotics
$$\mathrm{Tr}(e^{-tD^*D}) \quad\stackrel{t\searrow 0}{\sim}\quad (4\pi t)^{-n/2} \sum_{j=0}^\infty t^j \int_M a_j^{D^*D}(x)$$
and similarly for $DD^*$, yields
$$\mathrm{ind}(D) = (4\pi)^{-n/2} \int_M \left( a_{n/2}^{D^*D}(x) - a_{n/2}^{DD^*}(x) \right).$$
The local index theorem states that $a_{n/2}^{D^*D}(x) - a_{n/2}^{DD^*}(x)$ coincides pointwise with the $\widehat A$-integrand and that $a_{j}^{D^*D}(x) - a_{j}^{DD^*}(x)$ vanishes pointwise for $j<n/2$. The index theorem for Dirac operators on Lorentzian manifolds due to A. Strohmaier and the speaker is not related to any heat equation but the talk will contain a local version of it based on the Hadamard expansion of solutions of wave equations.

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