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 × q 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 1-loop renormalisability.
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Victor Gayral, Jan-Hendrik Jureit, Thomas Krajewski, Raimar Wulkenhaar, Quantum field theory on projective modules. J. Noncommut. Geom. 1 (2007), no. 4, pp. 431–496DOI 10.4171/JNCG/13