A generalized Lichnerowicz formula, the Wodzicki residue and gravity (Q1919291)

Given a Dirac operator, associated with a metric connection, the authors prove a formula which expresses the square of the Dirac operator \(\widetilde{D}\) of a metric connection as the sum of the Laplace operator of a suitable connection and a zeroth-order differential operator \(F\), in such a way that an explicit expression for \(F\) can be given. This is used to compute the subleading term of the expansion of the heat kernel of \(\widetilde{D}^2\), which in turn yields the Wodzicki residue of the operator \(\widetilde{D}^{-2n+2}\), where \(2n\) is the dimension of the base manifold. The residue corresponds to a gravitational Lagrangian which is the sum of the scalar curvature and a term quadratic in the torsion.