Yang-Mills moduli space in the adiabatic limit

TL;DR

This paper studies the behavior of Yang-Mills equations in a Lorentzian cone over hyperbolic space, showing that in the adiabatic limit, the dynamics reduce to geodesic motion in an infinite-dimensional gauge group manifold.

Contribution

It demonstrates that Yang-Mills dynamics on a Lorentzian cone over hyperbolic space simplifies to geodesic flow in an infinite-dimensional gauge group as the hyperbolic metric is scaled down.

Findings

Yang-Mills equations on Lorentzian cones relate to hyperbolic space geometry.

In the adiabatic limit, dynamics reduce to geodesic motion in a gauge group manifold.

Conformal equivalence links the Lorentzian cone to a cylinder, facilitating analysis.

Abstract

We consider the Yang-Mills equations for a matrix gauge group $G$ inside the future light cone of 4-dimensional Minkowski space, which can be viewed as a Lorentzian cone $C(H^3)$ over the 3-dimensional hyperbolic space $H^3$. Using the conformal equivalence of $C(H^3)$ and the cylinder $R\times H^3$, we show that, in the adiabatic limit when the metric on $H^3$ is scaled down, classical Yang-Mills dynamics is described by geodesic motion in the infinite-dimensional group manifold $C^\infty (S^2_\infty,G)$ of smooth maps from the boundary 2-sphere $S^2_\infty=\partial H^3$ into the gauge group $G$.