Spectral flow and the Atiyah-Patodi-Singer index theorem

27.11.2025, 16:15  –  Raum 1.22, Haus 9
Forschungsseminar Differentialgeometrie

Christian Bär (UP)

We establish a formula for the spectral flow of a smooth family of twisted Dirac operators on a closed odd-dimensional Riemannian spin manifold. The spectral flow is expressed in terms of the Â-form of the manifold, the odd Chern character form of the family of connections, and the ξ-invariants of the initial and final operators. This slightly generalizes a result by Getzler. Our proof is based on a reduction to the Atiyah-Patodi-Singer index theorem for manifolds with boundary, which provides a conceptually very simple approach to the problem. As an application, we give a streamlined proof of Llarull's rigidity theorem in odd dimensions.

This is based on joint work with Remo Ziemke.