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arXiv · 1904.03810

A twisted local index formula for curved noncommutative two tori

Farzad Fathizadeh·Franz Luef·Jim Tao
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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.

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Farzad Fathizadeh, Franz Luef, Jim Tao. 2019-04-08. A twisted local index formula for curved noncommutative two tori. https://arxiv.org/abs/1904.03810

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