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Dirac Operator on Complex Manifolds and Supersymmetric Quantum Mechanics

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arxiv 1012.2069 v2 pith:WFSBIGWO submitted 2010-12-09 hep-th math-phmath.MP

E.A. Ivanov , A.V. Smilga This is my paper · ORCID
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keywords diraccomplexoperatortwistedmodelahlerdolbeaultinterpreted
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We explore a new simple N=2 SQM model describing the motion over complex manifolds in external gauge fields. The nilpotent supercharge Q of the model can be interpreted as a (twisted) exterior holomorphic derivative, such that the model realizes the twisted Dolbeault complex. The sum Q + \bar Q can be interpreted as the Dirac operator: the standard Dirac operator if the manifold is K\"ahler and a certain "truncated" Dirac operator for a generic complex manifold. Focusing on the K\"ahler case, we give new simple physical proofs of the two mathematical facts: (i) the equivalence of the twisted Dirac and twisted Dolbeault complexes and (ii) the Atiyah-Singer theorem.

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  1. Index and localization for type B superconformal mechanics on singular spaces

    hep-th 2024-12 conditional novelty 7.0 of 10

    A localization method for type B superconformal indices on singular spaces is proposed, with a self-adjointness caveat that can make the index ambiguous.

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