The local version of the Lorentzian index theorem

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.