A Riemann-Roch theorem for the noncommutative two torus

We prove the analogue of the Riemann-Roch formula for the noncommutative two torus Aθ=C(Tθ2) equipped with an arbitrary translation invariant complex structure and a Weyl factor represented by a positive element k∈C∞(Tθ2). We consider a topologically trivial line bundle equipped with a general holomorphic structure and the corresponding twisted Dolbeault Laplacians. We define a spectral triple (Aθ,H,D) that encodes the twisted Dolbeault complex of Aθ 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 b2 terms of the asymptotic expansion of Tr(e−tD2). 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 Connes and Moscovici (2014) and independently computed in Fathizadeh and Khalkhali (2014).