Quantum field theory on projective modules

TL;DR

This paper develops a perturbative quantum field theory framework on projective modules over noncommutative algebras, revealing sensitivity to number theory and addressing UV/IR mixing with renormalization techniques.

Contribution

It introduces a novel formulation of quantum field theory on projective modules, especially Heisenberg modules over noncommutative tori, and demonstrates renormalizability.

Findings

Models are equivalent to large matrix models in a specific limit

The theory exhibits UV/IR mixing influenced by number theory

One-loop renormalizability is established

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 pxq matrix models, in the limit p/q->theta, where theta is a possibly irrational number. We find out that the modele 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 one-loop renormalizability.