Renormalised Chern–Weil forms associated with families of Dirac operators

Abstract

We provide local expressions for Chern–Weil type forms built from superconnections associated with families of Dirac operators previously investigated in [S. Scott, Zeta–Chern forms and the local family index theorem, Trans. Amer. Math. Soc. (in press). arXiv: math.DG/0406294] and later in [S. Paycha, S. Scott, Chern–Weil forms associated with superconnections, in: B. Booss-Bavnbeck, S. Klimek, M. Lesch, W. Zhang (Eds.), Analysis, Geometry and Topology of Elliptic Operators, World Scientific, 2006]. When the underlying fibration of manifolds is trivial, the even degree forms can be interpreted as renormalised Chern–Weil forms in as far as they coincide with regularised Chern–Weil forms up to residue correction terms. Similarly, a new formula for the curvature of the local fermionic vacuum line bundles is derived using a residue correction term added to the naive curvature formula. We interpret the odd degree Chern–Weil type forms built from superconnections as Wodzicki residues and establish a transgression formula along the lines of known transgression formulae for η -forms.