Unification of Gravity and Yang-Mills Theory in (2+1)-Dimensions

TL;DR

This paper presents a unified gauge-invariant theory in (2+1)-dimensions that combines gravity and Yang-Mills fields, showing it reproduces Einstein gravity and Yang-Mills equations, with specific solutions analyzed.

Contribution

It unifies gravity and Yang-Mills theory in (2+1) dimensions within a gauge-invariant framework, including explicit solutions and weak-field correspondence.

Findings

The theory describes Einstein gravity with a cosmological constant for gauge group SO(1,2).

Yang-Mills equations emerge naturally for the combined gauge group.

Static and rotation symmetric solutions reveal that charges must be spatially extended, not point-like.

Abstract

A gauge and diffeomorphism invariant theory in (2+1)-dimensions is presented in both first and second order Lagrangian form as well as in a Hamiltonian form. For gauge group $SO(1,2)$, the theory is shown to describe ordinary Einstein gravity with a cosmological constant. With gauge group $G^{tot}=SO(1,2)\otimes G^{YM}$, it is shown that the equations of motion for the $G^{YM}$ fields are the Yang-Mills equations. It is also shown that for weak $G^{YM}$ Yang-Mills fields, this theory agrees with the conventional Einstein-Yang-Mills theory to lowest order in Yang-Mills fields. Explicit static and rotation symmetric solutions to the Einstein-Maxwell theory are studied both for the conventional coupling and for this unified theory. In the electric solution to the unified theory, point charges are not allowed, the charges must have spatial extensions.