Heat trace asymptotics and the Gauss-Bonnet Theorem for general connections

TL;DR

This paper investigates the local super trace asymptotics for the de Rham complex with arbitrary super connections, revealing that these invariants are generally non-zero and differ from classical cases involving Levi-Civita connections.

Contribution

It extends the understanding of super trace invariants for general super connections, contrasting with classical Levi-Civita cases and highlighting new non-zero invariants.

Findings

Super trace invariants are generally non-zero for arbitrary super connections.

The critical term in the asymptotics is not the Pfaffian.

Results differ from classical Levi-Civita connection cases.

Abstract

We examine the local super trace asymptotics for the de Rham complex defined by an arbitrary super connection on the exterior algebra. We show, in contrast to the situation in which the connection in question is the Levi-Civita connection, that these invariants are generically non-zero in positive degree and that the critical term is not the Pfaffian.