Kac-Moody Symmetry in Hosotani Model

TL;DR

This paper clarifies the Kac-Moody symmetry structure in higher-dimensional Yang-Mills theories compactified on M_d x S^1, revealing how spontaneous symmetry breaking affects the mass spectrum and gauge transformations.

Contribution

It demonstrates that the massive tower of fields exhibits a loop algebra (Kac-Moody algebra) symmetry and analyzes the effects of spontaneous symmetry breaking on field masses and gauge transformations.

Findings

The symmetry forms a loop algebra (Kac-Moody algebra).

Spontaneous symmetry breaking leads to vector fields acquiring masses.

Zero-mass modes can become massive through vacuum expectation values.

Abstract

The symmetry of the massive tower of fields in higher-dimensional Yang-Mills theory compactified on a space-time of the form M_d x S^1 is clarified. The transformations form a loop algebra, a class of Kac-Moody algebras. Since the symmetry is spontaneously broken, vector fields "eat" Goldstone bosons and acquire masses. The field of zero-mass mode can also become massive provided that the field of the internal component develops a vacuum expectation value. The relation between the "restoration" of the symmetry in massive modes and the gauge transformation of the zero-mode vacuum field is discussed.