Chern-Weil forms associated with superconnections, Analysis, geometry and topology of elliptic operators, World Sci

Dedicated to our friend Krzysztof Wojciechowski We define Chern-Weil forms ck ( IA) associated to a superconnection IA using ζ-regularisation methods extended to ΨDOvalued forms. We show that they are cohomologous in the de Rham cohomology to tr ( IA2k piP) involving the projection piP onto the kernel of the elliptic operator P to which the superconnection IA is associated. A transgression formula shows that the corresponding Chern-Weil cohomology classes are independent of the scaling of the superconnection. When P is a differential operator with scalar leading symbol, the k-th Chern-Weil form corresponds to the regularised k-th derivative at t = 0 of the Chern character ch(t IA) and it has a local description ck ( IA) = −