A symplectic approach to certain functional integrals and partition functions

We define the Liouville functional on the set of functions on an infinite-d imensional symplectic manifold which are Hamiltonian with respect to a toru s-action. In the case of finite-dimensional manifolds this functional is cl osely connected with the integral over the Liouville measure by a theorem d ue to Duistermaat and Heckman. The symplectic setup turns out to be natural for the calculation of partition functions of certain quantum field theori es. In particular, among other examples, we calculate the partition functio n of the Wess-Zumino-Witten model on an elliptic curve in terms of this fun ctional and deduce its modular invariance from its expression as a function al 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 measur e. (C) 2001 Elsevier Science B.V. All rights reserved.