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

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.