Supersymmetry and the generalized Lichnerowicz formula

Details Supersymmetry and the generalized Lichnerowicz formula Supersymmetry and the generalized Lichnerowicz formula

1996-01-14

arxiv.org/abs/dg-ga/9601004v1

Mathematics

Differential Geometry

20 pages, plain tex

Scientific paper

A classical result in differential geometry due to Lichnerowicz [8] is concerned with the decomposition of the square of Dirac operators defined by Clifford connections on a Clifford module ${\cal E}$\ over a Riemannian manifold $M$. Recently, this formula has been generalized to arbitrary Dirac operators [2]. In this paper we prove a supersymmetric version of the generalized Lichnerowicz formula, motivated by the fact that there is a one-to-one correspondence between Clifford superconnections and Dirac operators. We extend this result to obtain a simple formula for the supercurvature of a generalized Bismut superconnection. This might be seen as a first step to prove the local index theorem also for families of arbitrary Dirac operators.

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