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.

arXiv comment27 pages

References in corpus (15)

• The Gauss-Bonnet Theorem for the noncommutative two torus
• Modular curvature and Morita equivalence
• Holomorphic structures on the quantum projective line
• An Asymmetric Noncommutative Torus
• Twisted cyclic theory, equivariant KK theory and KMS States
• The Ricci flow on noncommutative two-tori
• A twisted spectral triple for quantum SU(2)
• Local index formula and twisted spectral triples
• The term a_4 in the heat kernel expansion of noncommutative tori
• The Theory of Pseudo-differential Operators on the Noncommutative n-Torus
• Scalar curvature in conformal geometry of Connes-Landi noncommutative manifolds
• On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for $C^{\ast }$-Dynamical Systems
• Pseudodifferential calculus on noncommutative tori, I. Oscillating integrals
• On modular semifinite index theory
• Solitons of general topological charge over noncommutative tori