A Riemann-Roch theorem for the noncommutative two torus

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 $k\in C^{\mathrm{\infty }}({\mathbb{T}}_{\theta }^2)$. 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}.