The goal of this paper is to give a new and short proof of the local Atiyah-Singer index theorem by using approximations of heat semigroups. The heat equation approach to index theorems is not new: It was suggested by Atiyah-Bott [1] and McKean-Singer [18], and first carried out by Patodi [21] and Gilkey [13]. Bismut in [8] introduces stochastic methods based on Feynman-Kac formula. For probabi...

A version of the Atiyah Patodi Singer index theorem is proved for general families of Dirac operators on compact manifolds with boundary The van ishing of the analytic index of the boundary family inK of the base allows us to de ne through an explicit trivialization a smooth family of bound ary conditions of generalized Atiyah Patodi Singer type The calculus of b pseudodi erential operators is ...

We express explicitly the analytic torsion functions associated with Rumin complex on lens spaces in terms of Hurwitz zeta function. In particular, we find that vanish at origin and determine torsions. Moreover, have a formula between this Ray-Singer torsion.

In the spirit of Ray and Singer we define a complex valued analytic torsion using non-selfadjoint Laplacians. We establish an anomaly formula which permits to turn this into a topological invariant. Conjecturally this analytically defined invariant computes the complex valued Reidemeister torsion, including its phase. We establish this conjecture in some non-trivial situations. Mathematics Subj...

Let w be an Abelian differential on compact Riemann surface of genus g ≥ 1. We obtain an explicit holomorphic factorization formula for ζ-regularized determinant of the Laplacian in flat conical metrics with trivial holonomy |w|2, generalizing the classical Ray-Singer result in g = 1.

We derive a formula for the index of a Dirac operator on an incomplete edge space satisfying a “geometric Witt condition.” We accomplish this by cutting off to a smooth manifold with boundary, applying the Atiyah-Patodi-Singer index theorem, and taking a limit. We deduce corollaries related to the existence of positive scalar curvature metrics on incomplete edge spaces.

We derive a formula for the index of a Dirac operator on a compact, evendimensional incomplete edge space satisfying a “geometric Witt condition”. We accomplish this by cutting off to a smooth manifold with boundary, applying the Atiyah–Patodi–Singer index theorem, and taking a limit. We deduce corollaries related to the existence of positive scalar curvature metrics on incomplete edge spaces.