Numerical Results for U(1) Gauge Theory on 2d and 4d Non-Commutative Spaces

TL;DR

This paper investigates non-perturbative properties of U(1) gauge theories on non-commutative spaces, revealing unique behaviors of Wilson loops and progress towards continuum limits in 2d and 4d cases.

Contribution

It provides the first non-perturbative analysis of U(1) gauge theories on non-commutative spaces, including Wilson loop behaviors and continuum extrapolation.

Findings

Small Wilson loops follow an area law in 2d.

Large Wilson loops exhibit a linear phase increase with area.

In 4d, non-commutative plane loops behave similarly to 2d results.

Abstract

We present non-perturbative results for U(1) gauge theory in spaces, which include a non-commutative plane. In contrast to the commutative space, such gauge theories involve a Yang-Mills term, and the Wilson loop is complex on the non-perturbative level. We first consider the 2d case: small Wilson loops follows an area law, whereas for large Wilson loops the complex phase rises linearly with the area. In four dimensions the behavior is qualitatively similar for loops in the non-commutative plane, whereas the loops in other planes follow closely the commutative pattern. In d=2 our results can be extrapolated safely to the continuum limit, and in d=4 we report on recent progress towards this goal.