A twisted local index formula for curved noncommutative two tori

Fathizadeh, F; Luef, F; Tao, J

doi:10.48550/arXiv.1904.03810

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A twisted local index formula for curved noncommutative two tori

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Fathizadeh, F
Department Physiology of Cognitive Processes, Max Planck Institute for Biological Cybernetics, Max Planck Society;

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Citation

Fathizadeh, F., Luef, F., & Tao, J. (submitted). A twisted local index formula for curved noncommutative two tori.

Cite as: https://hdl.handle.net/21.11116/0000-0003-65EB-3

Abstract

We consider the Dirac operator of a general metric in the canonical conformal class on the noncommutative two torus, twisted by an idempotent (representing the K-theory class of a general noncommutative vector bundle), and derive a local formula for the Fredholm index of the twisted Dirac operator. Our approach is based on the McKean-Singer index formula, and explicit heat expansion calculations by making use of Connes' pseudodifferential calculus. As a technical tool, a new rearrangement lemma is proved to handle challenges posed by the noncommutativity of the algebra and the presence of an idempotent in the calculations in addition to a conformal factor.