A proof of Bismut local index theorem for a family of Dirac operators (Q1098144)

The authors give a different proof of the local index theorem for a family of Dirac operators by an extension of their equivariant computation, i.e., the heat kernel of the Laplacian of a vector bundle is expressed as an average over the holonomy group and the \(\hat A\)-genus naturally enters as the Jacobian of the exponential map on a frame bundle [see the authors, Bull. Soc. Math. Fr. 113, 305-345 (1985; Zbl 0592.58044)]. The basic starting point is Bismut's extension of Quillen's Chern-Weil theory of superconnections to infinite dimension. This is presented here with an interesting proof by a homotopy argument. The authors then proceed to give a description of Bismut's Levi-Civita superconnection as a Dirac-like operator. Finally a deformation argument along the line of \textit{E. Getzler} [Topology 25, 111-117 (1986; Zbl 0607.58040)] is used to overcome the difficulty presented by the noncompact holonomy. A closely related proof is given by \textit{W. P. Zhang} in Res. Rep. Nankai Inst. Math., No.6.