Quantum Harmonic Analysis and Geometric Invariants

Abstract

We develop from scratch a theory of invariants within the framework of non-commutative geometry. Given an operator Q (a supercharge in physics language) and an operator a (whose square equals the identity I), we derive a general formula for an invariant Z(Q,a) depending on Q and a. In case a=I, our formula reduces to the McKean-Singer representation of the index of Q. The function Z is invariant in the following sense: if Q=Q(s) depends on a parameter s, and if Z(Q(s),a) is differentiable in s, then in fact Z(Q(s),a) is independent of s. We give detailed conditions on Q(s) for which Z(Q(s),a) is differentiable in s. At the end of this paper, we consider a 2-dimensional generalization of our theory motivated by space-time supersymmetry. In the case that expectations are given by functional integrals, Z(Q,a) has a simple integral representation. We also explain in detail how our construction relates to Connes' entire cyclic cohomology, as well as to other frameworks.