Residue Formulation of Chern Character on Smooth Manifolds

The Chern character of a complex vector bundle is most conveniently defined as the exponential of a curvature of a connection. It is well known that its cohomology class does not depend on the particular connection chosen. It has been shown by Quillen that a connection may be perturbed by an endomorphism of the vector bundle, such as a symbol of some elliptic differential operator. This point of view, as we intend to show, allows one to relate Chern character to a non-commutative sibling formulated by Connes and Moscovici. The general setup for our problem is purely geometric. Let \sigma be the symbol of a Dirac-type operator acting on sections of a \Z_2-graded vector bundle E. Let \nabla be a connection on E, pulled back to T^*M. Suppose also that \nabla respects the Z_2-grading. The object \nabla+\sigma is a superconnection on T^*M in the sense of Quillen. We obtain a formula for the H_*(M)-valued Poincare dual of Quillen's Chern character ch(D)=trace(exp(\nabla+\sigma)^2) in terms of residues of \Gamma(z)trace(\nabla+\sigma)^{-2z}. We also compute two examples.