Arthur Jaffe
Quantum Harmonic Analysis and Geometric Invariants
(206K, Latex)
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), a symmetry group G, and an
operator a (whose square equals the identity I), we derive a
general formula for an invariant Z(Q,a,g) depending on Q, on a,
and on g in G. 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,g) is differentiable in s, then
in fact Z(Q(s),a,g) is independent of s. We give detailed
conditions on Q(s) for which Z(Q(s),a,g) 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,g) 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.