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Stochastic calculus in superspace. II. Differential forms, supermanifolds and the Atiyah-Singer index theorem

A Rogers

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For pt.I see ibid., vol.25, p.447-68, (1992). Starting with vector bundles over manifolds, supermanifolds are constructed whose function algebras correspond to twisted differential forms. Stochastic calculus for bosonic and fermionic Brownian paths is used to provide a geometric construction of Brownian paths on these supermanifolds. A Feynman-Kac formula for the heat kernel of the Laplace-Beltrami operator is then derived. This is used to provide a simple, rigorous version of the supersymmetric proofs of the Atiyah-Singer index theorem.


PACS

02.40.Vh Global analysis and analysis on manifolds

05.30.Jp Boson systems

02.40.Ky Riemannian geometries

05.30.Fk Fermion systems and electron gas

03.65.Vf Phases: geometric; dynamic or topological

02.50.Fz Stochastic analysis

MSC

60H07 Stochastic calculus of variations and the Malliavin calculus

58A50 Supermanifolds and graded manifolds (See also 14A22, 32C11)

58B20 Riemannian, Finsler and other geometric structures (See also 53C20, 53C60)

81S25 Quantum stochastic calculus

58C50 Analysis on supermanifolds or graded manifolds

53C20 Global Riemannian geometry, including pinching (See also 31C12, 58B20)

Subjects

Quantum gases, liquids and solids

Mathematical physics

Computational physics

Quantum information and quantum mechanics

Statistical physics and nonlinear systems

Dates

Issue 22 (21 November 1992)



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