A twisted local index formula for curved noncommutative two tori

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.

27 pages

References in corpus (4)

• The Gauss-Bonnet Theorem for the noncommutative two torus
• Local index formula and twisted spectral triples
• The term a_4 in the heat kernel expansion of noncommutative tori
• Solitons of general topological charge over noncommutative tori