Spinors on a curved noncommutative space: Coupling to torsion and the Gross-Neveu model

Abstract

We analyse the Dirac action on the truncated Heisenberg algebra and in particular, the nonminimal couplings to the background gravitational field. By projection to the Heisenberg algebra we obtain a renormalisable model: the noncommutative extension of the Gross-Neveu model. This result indicates that, as on the commutative curved backgrounds, nonminimal couplings with torsion and curvature are necessary (and sufficient) for renormalisability of scalar and spinor theories on the curved noncommutative spaces.