Asymptotically Free $\hat{U}(1)$ Kac-Moody Gauge Fields in $3+1$ dimensions

TL;DR

This paper introduces a nonlinear, nonlocal pure gauge sector based on $$ Kac-Moody symmetry in 3+1 dimensions, demonstrating its one-loop renormalizability and asymptotic freedom, with potential applications in condensed matter physics.

Contribution

It presents the first analysis of a pure $$ Kac-Moody gauge theory in 3+1 dimensions, establishing its renormalizability and asymptotic freedom.

Findings

The theory is one-loop renormalizable in 3+1 dimensions.

The gauge theory exhibits asymptotic freedom.

Adding fermions introduces multiple couplings and richer structure.

Abstract

$ \hat {U}(1)$ Kac-Moody gauge fields have the infinite dimensional $ \hat{U}(1)$ Kac-Moody group as their gauge group. The pure gauge sector, unlike the usual $U(1)$ Maxwell lagrangian, is nonlinear and nonlocal; the Euclidean theory is defined on a $d+1$-dimensional manifold $ {\cal{R}}_d \times {\cal{S}}^1 $ and hence is also asymmetric. We quantize this theory using the background field method and examine its renormalizability at one-loop by analyzing all the relevant diagrams. We find that, for a suitable choice of the gauge field propagators, this theory is one-loop renormalizable in $3+1$ dimensions. This pure abelian Kac-Moody gauge theory in $3+1$ dimensions has only one running coupling constant and the theory is asymptotically free. When fermions are added the number of independent couplings increases and a richer structure is obtained. Finally, we note some features of the theory which suggest its possible relevance to the study of anisotropic condensed matter systems, in particular that of high-temperature superconductors.