A symplectic approach to certain functional integrals and partition functions

Abstract

We define the Liouville functional on the set of functions on an infinite-dimensional symplectic manifold which are Hamiltonian with respect to a torus-action. In the case of finite-dimensional manifolds this functional is closely connected with the integral over the Liouville measure by a theorem due to Duistermaat and Heckman. The symplectic setup turns out to be natural for the calculation of partition functions of certain quantum field theories. In particular, among other examples, we calculate the partition function of the Wess–Zumino–Witten model on an elliptic curve in terms of this functional and deduce its modular invariance from its expression as a functional integral. In the case that the symplectic manifold is given as a generic coadjoint orbit of a loop group, the Liouville functional can be shown to give the same result as usual integration with respect to the Wiener measure.