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Patodi’s Proof of the Gauss-Bonnet-Chern Theorem and Superproofs of Index Theorems

Part of the Theoretical,Mathematical Physics book series (TMP)

Abstract

Witten`s remarkable paper (370) and a companion paper (369) contained only the last chapter. but also extensive remarks on the form of the Atiyah-Singer index theorem in supersymmetric quantum theory. this suggested that supersymmetry might provide the framework for a simple proof of the classical Atiyah-Singer theorem. This hope was realized by Alvarez-Gaumez (12) and subsequently by Friedan-Windy (113) (see also Windy (368)). These theoretical physicistists relied on formal manipulations inside path integrals(including "fermion" path integrals) so theirproofs were certainly not rigorous. Getzler (129) found a rigorous version of their arguments which, since it relied ojn some pseudodifferential operator machinery and the theory of supermanifold, was still rather involved. More recently, Getzler (130) found a proof whose gometric and algebraic parts are especially elemantary and transparent. Independentof this work, Bismut (49) found a related proof. The analytic part of these two proofs (49,130) is somewhat sophisticated, relying on Brownian motioon estimates on manifolds.

Keywords

Quantum Theory Quantum Information Dirac Operator Path Integral Simple Proof 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

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