Feynman–Kac formula for perturbations of order $$\le 1$$, and noncommutative geometry

Abstract

AbstractLet Q be a differential operator of order $$\le 1$$ ≤ 1 on a complex metric vector bundle $$\mathscr {E}\rightarrow \mathscr {M}$$ E → M with metric connection $$\nabla $$ ∇ over a possibly noncompact Riemannian manifold $$\mathscr {M}$$ M . Under very mild regularity assumptions on Q that guarantee that $$\nabla ^{\dagger }\nabla /2+Q$$ ∇ † ∇ / 2 + Q canonically induces a holomorphic semigroup $$\mathrm {e}^{-zH^{\nabla }_{Q}}$$ e - z H Q ∇ in $$\Gamma _{L^2}(\mathscr {M},\mathscr {E})$$ Γ L 2 ( M , E ) (where z runs through a complex sector which contains $$[0,\infty )$$ [ 0 , ∞ ) ), we prove an explicit Feynman–Kac type formula for $$\mathrm {e}^{-tH^{\nabla }_{Q}}$$ e - t H Q ∇ , $$t>0$$ t > 0 , generalizing the standard self-adjoint theory where Q is a self-adjoint zeroth order operator. For compact $$\mathscr {M}$$ M ’s we combine this formula with Berezin integration to derive a Feynman–Kac type formula for an operator trace of the form $$\begin{aligned} \mathrm {Tr}\left( \widetilde{V}\int ^t_0\mathrm {e}^{-sH^{\nabla }_{V}}P\mathrm {e}^{-(t-s)H^{\nabla }_{V}}\mathrm {d}s\right) , \end{aligned}$$ Tr V ~ ∫ 0 t e - s H V ∇ P e - ( t - s ) H V ∇ d s , where $$V,\widetilde{V}$$ V , V ~ are of zeroth order and P is of order $$\le 1$$ ≤ 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.