An Explicit Kaluza-Klein Reduction of Einstein's Gravity in $6D$ on $S^2$

TL;DR

This paper explicitly constructs the gauge sector in a 6D Kaluza-Klein reduction on $S^2$, revealing that only two gauge fields are dynamical due to the coset structure, clarifying geometric origins of gauge degrees of freedom.

Contribution

It provides an explicit reduction of Einstein gravity on $S^2$, analyzing the gauge kinetic matrix and identifying the origin of gauge degeneracy from the coset structure.

Findings

Only two gauge fields propagate in 4D despite $SO(3)$ symmetry.

The gauge kinetic matrix has rank two with a zero eigenvalue.

Degeneracy stems from the coset structure $S^2 \,\simeq\, SO(3)/SO(2)$.

Abstract

We study a six-dimensional Kaluza-Klein theory with spacetime topology $M_4 \times S^2$ and analyze the gauge sector arising from dimensional reduction. Using normalized Killing vectors on $S^2$, we explicitly construct the reduced Yang-Mills action and determine the corresponding gauge kinetic matrix. Despite the $SO(3)$ isometry of $S^2$, we show that only two physical gauge fields propagate in four dimensions. The gauge kinetic matrix therefore has rank two and possesses a single zero eigenvalue. We demonstrate that this degeneracy is a direct consequence of the coset structure $S^2 \simeq SO(3)/SO(2)$ and reflects a non-dynamical gauge direction rather than an inconsistency of the reduction. Our results clarify the geometric origin of gauge degrees of freedom in Kaluza-Klein reductions on coset spaces.