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Supersymmetric quantum mechanics and the index theorem. (English) Zbl 0702.58072
Geometry and Physics, Proc. Miniconf., Canberra/Aust. 1989, Proc. Cent. Math. Anal. Aust. Natl. Univ. 22, 50-81 (1989).
[For the entire collection see Zbl 0678.00018.]
From the introduction: “This review aims to give a pedagogical introduction to supersymmetric quantum mechanics and to establish its relevance to the index theorem. Finally,..., a discrete approximation is set up for the path integral representation of the supersymmetric quantum mechanics equivalent of the index, and used to provide a heuristic derivation of the index theorem itself.
It should be pointed out that the whole story of supersymmetry and the index theorem is a precursor to the exciting recent developments in “topological quantum field theory”... From this perspective this review should perhaps be subtitled “topological (supersymmetric) quantum mechanics”.”
The paper is organized as follows: §1. Supersymmetric quantum mechanics. SSQM and the Witten index. Example: the hydrogen atom. Example: Massless fields in curved space. §2. The index theorem and applications. The Atiyah-Singer index theorem for the twisted Dirac operator. Specializations. The index theorem and chiral anomalies. Anomaly cancellation in the standard electroweak model. §3. Path integral derivation of the index theorem. Fermionic path integrals. A discrete approximation to the index (flat space).
Reviewer: H.Boseck
MSC:
58J20 Index theory and related fixed-point theorems on manifolds
81T60 Supersymmetric field theories in quantum mechanics
81S40 Path integrals in quantum mechanics
81T50 Anomalies in quantum field theory