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