Quantum field theory on projective modules

Abstract

We propose a general formulation of perturbative quantum field theory on (finitely generated) projective modules over noncommutative algebras. This is the analogue of scalar field theories with non-trivial topology in the noncommutative realm. We treat in detail the case of Heisenberg modules over noncommutative tori and show how these models can be understood as large rectangular $p\times q$ matrix models, in the limit $p/q\to \theta$, where $\theta$ is a possibly irrational number. We find out that the model is highly sensitive to the number theoretical aspect of $\theta$ and suffers from an UV/IR-mixing. We give a way to cure the entanglement and prove 1-loop renormalisability.