Patodi’s Proof of the Gauss-Bonnet-Chern Theorem and Superproofs of Index Theorems
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.
KeywordsQuantum Theory Quantum Information Dirac Operator Path Integral Simple Proof
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