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 θ = C ( T θ 2 ) equipped with an arbitrary translation invariant complex structure and a Weyl factor represented by a positive element k ∈ C ∞ ( T θ 2 ) . We consider a topologically trivial line bundle equipped with a general holomorphic structure and the corresponding twisted Dolbeault Laplacians. We define a spectral triple ( A θ , H , D ) that encodes the twisted Dolbeault complex of A θ 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 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 Connes and Moscovici (2014) and independently computed in Fathizadeh and Khalkhali (2014).