Fermionic Dyson expansions and stochastic Duistermaat-Heckman localization on loop spaces

Abstract

Given a self-adjoint operator $H\geq 0$ and (appropriate) densely defined and closed operators $P_{1},\dots, P_{n}$ in a Hilbert space $\mathscr{H}$, we provide a systematic study of bounded operators given by iterated integrals \begin{align}\label{oh} \int_{\{ 0\leq s_1\leq \dots\leq s_n\leq t\}}\mathrm{e}^{-s_1H}P_{1}\mathrm{e}^{-(s_2-s_1)H}P_{2}\cdots \mathrm{e}^{-(s_n-s_{n-1})H}P_{n} \mathrm{e}^{-(t-s_n)H}\, \mathrm{d} s_{1} \ldots \mathrm{d} s_{n},\quad t>0. \end{align} These operators arise naturally in noncommutative geometry and the geometry of loop spaces. Using Fermionic calculus, we give a natural construction of an enlarged Hilbert space $\mathscr{H}^{(n)}$ and an analytic semigroup $\mathrm{e}^{-t (H^{(n)}+P^{(n)} )}$ thereon, such that $\mathrm{e}^{-t (H^{(n)}+P^{(n)} )}$ composed from the left with (essentially) a Fermionic integration gives precisely the above iterated operator integral. This formula allows to establish important regularity results for the latter, and to derive a stochastic representation for it, in case $H$ is a covariant Laplacian and the $P_{j}$'s are first-order differential operators. Finally, with $H$ given as the square of the Dirac operator on a spin manifold, this representation is used to derive a stochastic refinement of the Duistermaat-Heckman localization formula on the loop space of a spin manifold.