A Riemann-Roch theorem for the noncommutative two torus

TL;DR

This paper establishes a Riemann-Roch theorem for the noncommutative two torus, linking index theory with scalar curvature, using spectral triples, pseudodifferential calculus, and heat kernel methods.

Contribution

It introduces a noncommutative Riemann-Roch formula for the two torus, connecting index theory with scalar curvature in a novel noncommutative geometric setting.

Findings

Derived the explicit form of the Riemann-Roch formula for the noncommutative two torus.

Connected the curvature term in the formula with the scalar curvature previously defined.

Utilized heat kernel techniques to compute asymptotic expansions of spectral traces.

Abstract

We prove the analogue of the Riemann-Roch formula for the noncommutative two torus $ A_{\theta} = C(\mathbb{T}_{\theta}^2)$ equipped with an arbitrary translation invariant complex structure and a Weyl factor represented by a positive element $k\in C^{\infty}(\mathbb{T}_{\theta}^2)$. We consider a topologically trivial line bundle equipped with a general holomorphic structure and the corresponding twisted Dolbeault Laplacians. We define an spectral triple ($A_{\theta}, \mathcal{H}, D)$ that encodes the twisted Dolbeault complex of $ A_{\theta}$ and whose index gives the left hand side of the Riemann-Roch formula. Using Connes' pseudodifferential calculus and heat equation techniques, we explicitly compute the $b_2$ terms of the asymptotic expansion of $\text{Tr} (e^{-tD^2})$. We find that the curvature term on the right hand side of the Riemann-Roch formula coincides with the scalar curvature of the noncommutative torus recently defined and computed in \cite{CM1} and \cite{FK2}.