Quantum field theory on a projective module

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 pxq matrix models, in the limit p/q → θ, where θ is a possibly irrational number. We find out that the model is highly sensitive to the number-theoretical aspect of θ and suffers from an UV/IR-mixing. We give a way to cure the entanglement and prove one-loop renormalizability. (author)