Heat Kernel Measures and Critical Limits

Abstract

This article is an exposition of several questions linking heat kernel measures on infinite-dimensional Lie groups, limits associated with critical Sobolev exponents, and Feynman–Kac measures for sigma models. The first part of the article concerns existence and invariance issues for heat kernel measure classes. The main examples are heat kernel measures on groups of the form C0(X,F), where X is a Riemannian manifold and F is a finite-dimensional Lie group. These measures depend on a smoothness parameter \(s \downarrow {\rm dim}(X)/2\). The second part of the article concerns the limit s ↓ dim(X)/2, especially dim(X) ≤ 2, and how this limit is related to issues arising in quantum field theory. In the case of X = S1, we conjecture that heat kernel measures converge to measures which arise naturally from the Kac–Moody–Segal point of view on loop groups, as \(s \downarrow 1/2\).