1996/01/14 by Thomas Ackermann, Ackermann, Thomas
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #Noncommutative and Quantum Gravity Theories
paper · pdf · doi:10.48550/arxiv.dg-ga/9601004
openalex publication_date 1996/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.