A low-energy limit of Yang-Mills theory on de Sitter space

TL;DR

This paper explores the low-energy behavior of Yang-Mills theory on de Sitter space, showing it reduces to geodesic motion on an infinite-dimensional moduli space of vacua, which is expected to be integrable.

Contribution

It introduces a novel low-energy approximation of Yang-Mills theory on de Sitter space, linking it to geodesic motion on a gauge-invariant moduli space of vacua.

Findings

Yang-Mills vacua form an infinite-dimensional moduli space.

Low-energy dynamics approximates geodesic motion on this space.

The moduli space is isomorphic to a group manifold, suggesting integrability.

Abstract

We consider Yang--Mills theory with a compact structure group $G$ on four-dimensional de Sitter space dS$_4$. Using conformal invariance, we transform the theory from dS$_4$ to the finite cylinder ${\cal I}\times S^3$, where ${\cal I}=(-\pi/2, \pi/2)$ and $S^3$ is the round three-sphere. By considering only bundles $P\to{\cal I}\times S^3$ which are framed over the temporal boundary $\partial{\cal I}\times S^3$, we introduce additional degrees of freedom which restrict gauge transformations to be identity on $\partial{\cal I}\times S^3$. We study the consequences of the framing on the variation of the action, and on the Yang--Mills equations. This allows for an infinite-dimensional moduli space of Yang--Mills vacua on dS$_4$. We show that, in the low-energy limit, when momentum along ${\cal I}$ is much smaller than along $S^3$, the Yang--Mills dynamics in dS$_4$ is approximated by geodesic motion in the infinite-dimensional space ${\cal M}_{\rm vac}$ of gauge-inequivalent Yang--Mills vacua on $S^3$. Since ${\cal M}_{\rm vac}\cong C^\infty (S^3, G)/G$ is a group manifold, the dynamics is expected to be integrable.