Let $Q$ be a differential operator of order $\leq 1$ on a complex metric vector bundle $\mathscr{ E}\to \mathscr{ M}$ with metric connection $\nabla$ over a possibly noncompact Riemannian manifold $\mathscr{ M}$. Under very mild regularity assumptions on $Q$ that guarantee that $\nabla^{ \dagger}\nabla/2+Q$ generates a holomorphic semigroup $\mathrm{ e}^{ -zH^{ \nabla}_{ Q}}$ in $\Gamma_{ L^2}(\mathscr{ M},\mathscr{ E})$ (where $z$ runs through a complex sector which contains $[0,\infty)$), we prove an explicit Feynman-Kac type formula for $\mathrm{ e}^{ -tH^{ \nabla}_{ Q}}$, $t>0$, generalizing the standard self-adjoint theory where $Q$ is a self-adjoint zeroth order operator. For compact $\mathscr{ M}$'s we combine this formula with Berezin integration to derive a Feynman-Kac type formula for an operator trace of the form $$ \mathrm{ Tr}\left(\widetilde{ V}\int^t_0\mathrm{ e}^{ -sH^{ \nabla}_{ V}}P\mathrm{ e}^{ -(t-s)H^{ \nabla}_{ V}}\mathrm{ d} s\right), $$ where $V,\widetilde{ V}$ are of zeroth order and $P$ is of order $\leq 1$. These formulae are then used to obtain a probabilistic representations of the lower order terms of the equivariant Chern character (a differential graded extension of the JLO-cocycle) of a compact even-dimensional Riemannian spin manifold, which in combination with cyclic homology play a crucial role in the context of the Duistermaat-Heckmann localization formula on the loop space of such a manifold.
Categories: math-ph math.DG math.MP math.OA math.PR