# A twisted local index formula for curved noncommutative two tori

## TL;DR

This paper derives a local index formula for twisted Dirac operators on noncommutative two tori with general metrics, using heat kernel methods and pseudodifferential calculus, advancing noncommutative geometry techniques.

## Contribution

It introduces a new local index formula for twisted Dirac operators on noncommutative tori with general metrics, incorporating a novel rearrangement lemma for noncommutative analysis.

## Key findings

- Derived a local index formula for twisted Dirac operators
- Developed a new rearrangement lemma for noncommutative calculus
- Extended heat kernel techniques to noncommutative geometry

## 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.