The local version of the Lorentzian index theorem

30.11.2017, 16:15  –  Haus 9, Raum 1.10
Forschungsseminar Differentialgeometrie

Christian Bär

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

<tex>\mathrm{ind}(D) = \mathrm{Tr}(e^{-tD^*D}) - \mathrm{Tr}(e^{-tDD^*})</tex>

for any t>0. Inserting the short time asymptotics 

<tex>\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)</tex>

and similarly for DD*, yields

<tex>\mathrm{ind}(D) = (4\pi)^{-n/2} \int_M \left( a_{n/2}^{D^*D}(x) - a_{n/2}^{DD^*}(x) \right)</tex>.

The local index theorem states that <tex>a_{n/2}^{D^*D}(x) - a_{n/2}^{DD^*}(x)</tex> coincides pointwise with the <tex>\widehat A</tex>-integrand and that <tex>a_{j}^{D^*D}(x) - a_{j}^{DD^*}(x) </tex>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.