A Riemann-Roch theorem for the noncommutative two torus

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 $\text{Tr}(e^{-t{D}^2})$. 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}.

15 pages

References in corpus (2)

• The Gauss-Bonnet Theorem for the noncommutative two torus
• Curved noncommutative torus and Gauss--Bonnet

Cited by in corpus (5)

• Modular curvature and Morita equivalence
• Pseudodifferential calculus on noncommutative tori, I. Oscillating integrals
• On Certain Spectral Invariants of Dirac Operators on Noncommutative Tori
• Pseudodifferential calculus on noncommutative tori, II. Main properties
• The Curvature of the Determinant Line Bundle on the Noncommutative Two Torus