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

TL;DR

This work examines the dimensional reduction of pure six-dimensional Einstein gravity on the spacetime $M_4\times S^2$ and shows that a Yang–Mills sector in four dimensions emerges from the internal geometry via normalized Killing vectors on $S^2$. Using a Cartan-formalism KK ansatz, the authors derive a 4D Lagrangian that couples gravity to a scalar $\phi$ from the internal volume and to gauge-field strengths built from the $SO(3)$ isometries, ultimately diagonalizing the gauge-kinetic sector. They find a degenerate gauge-kinetic matrix with eigenvalues $\lambda_1=0$ (multiplicity 1) and $\lambda_2=1$ (multiplicity 2), meaning only two dynamical gauge directions survive; the zero mode is a non-dynamical direction tied to the coset structure $S^2\simeq SO(3)/SO(2)$. An orthogonal rotation yields two propagating gauge combinations with canonical normalization, and the scalar sector from the internal metric has a healthy kinetic term, confirming the consistency of the reduction. The results illuminate the geometric origin of gauge degrees of freedom in coset-based Kaluza–Klein reductions and provide a purely gravitational route to Einstein–Yang–Mills theory in four dimensions, with potential extensions to other coset spaces.

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.