Chern-Simons theory for the noncommutative 3-torus

TL;DR

This paper computes the Chern-Simons action for the noncommutative 3-torus, finds it to be always zero regardless of deformation parameters, and explores related quantum field theoretic aspects.

Contribution

It provides the first explicit calculation of the Chern-Simons action on the noncommutative 3-torus and analyzes its independence from deformation parameters.

Findings

Chern-Simons action for the noncommutative 3-torus is always zero.

The first coefficient in the loop expansion series is independent of the deformation matrix.

The computed action does not depend on the specific deformation parameters.

Abstract

We study the Chern-Simons action, which was defined for noncommutative spaces in general by the author, for the noncommutative 3-torus, the universal C*-algebra generated by 3 unitaries. D. Essouabri, B. Iochum, C. Levy, and A. Sitarz constructed a spectral triple for the noncommutative 3-torus. We compute the Chern-Simons action for this noncommutative space. In connection with this computation we calculate the first coefficient in the loop expansion series of the corresponding Feynman path integral with the Chern-Simons action as Lagrangian. The result is independent of the deformation matrix of the noncommutative 3-torus and always 0.