Dmitry Gerenrot | Searcharxiv

arXiv2005

A note on the residue Chern character

The aim of this note is to improve upon our earlier result which translates Weyl's (curvature) formulation of Chern character of a smooth vector bundle into the language of residues. The dualized Chern character is the functional on smooth differential forms on M. In our previous paper, this functional has been expressed as a sum of certain residues in the spirit of the Local Index formula due to Connes and Moscovici. The present note provides a stronger and more effective formulation.

math.DG↗

arXiv2005

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 σbe 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+σis 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+σ)^2) in terms of residues of Γ(z)trace(\nabla+σ)^{-2z}. We also compute two examples.

math.DG↗