Gravity and Yang-Mills Theory: Two Faces of the Same Theory?

TL;DR

This paper introduces a gauge-invariant theory linking gravity and Yang-Mills fields, showing their equivalence under certain conditions, and explores its properties, solutions, and limitations in different signatures and gauge groups.

Contribution

It presents a unified gauge-invariant framework that encompasses gravity and Yang-Mills theory, extending the Ashtekar formulation and analyzing its solutions and matter couplings.

Findings

The theory matches Ashtekar gravity with a cosmological constant for SO(3,C).

In Euclidean signature, the theory is real; in Lorentzian, it is complex with unresolved reality conditions.

A static, spherically symmetric solution exists for U(2) gauge group.

Abstract

We introduce a gauge and diffeomorphism invariant theory on the Yang-Mills phase space. The theory is well defined for an arbitrary gauge group with an invariant bilinear form, it contains only first class constraints, and the spacetime metric has a simple form in terms of the phase space variables. With gauge group $SO(3,C)$, the theory equals the Ashtekar formulation of gravity with a cosmological constant. For Lorentzian signature, the theory is complex, and we have not found any good reality conditions. In the Euclidean signature case, everything is real. In a weak field expansion around de Sitter spacetime, the theory is shown to give the conventional Yang-Mills theory to the lowest order in the fields. We show that the coupling to a Higgs scalar is straightforward, while the naive spinor coupling does not work. We have not found any way of including spinors that gives a closed constraint algebra. For gauge group $U(2)$, we find a static and spherically symmetric solution.