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Anticommuting Variables, Fermionic Path Integrals and Supersymmetry
(Replacement because mailer changed `hat' for supercript into something
weird. The macro `\sp' has been used in place of the `hat' character in this
revised version.) Fermionic Brownian paths are defined as paths in a space
para\-metr\-ised by anticommuting variables. Stochastic calculus for these
paths, in conjunction with classical Brownian paths, is described; Brownian
paths on supermanifolds are developed and applied to establish a Feynman-Kac
formula for the twisted Laplace-Beltrami operator on differential forms taking
values in a vector bundle. This formula is used to give a proof of the
Atiyah-Singer index theorem which is rigorous while being closely modelled on
the supersymmetric proofs in the physics literature.Comment: 18 pages, KCL-TH-92-
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