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arXiv · 1605.06982

Path integrals, SUSY QM and the Atiyah-Singer index theorem for twisted Dirac

Dana Fine·Stephen Sawin
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Abstract

Feynman's time-slicing construction approximates the path integral by a product, determined by a partition of a finite time interval, of approximate propagators. This paper formulates general conditions to impose on a short-time approximation to the propagator in a general class of imaginary-time quantum mechanics on a Riemannian manifold which ensure these products converge. The limit defines a path integral which agrees pointwise with the heat kernel for a generalized Laplacian. The result is a rigorous construction of the propagator for supersymmetric quantum mechanics, with potential, as a path integral. Further, the class of Laplacians includes the square of the twisted Dirac operator, which corresponds to an extension of N=1/2 supersymmetric quantum mechanics. General results on the rate of convergence of the approximate path integrals suffice in this case to derive the local version of the Atiyah-Singer index theorem.

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Dana Fine, Stephen Sawin. 2016-05-23. Path integrals, SUSY QM and the Atiyah-Singer index theorem for twisted Dirac. https://doi.org/10.1063/1.4973368

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