Symplectic charges in the Yang-Mills theory of the normal conformal Cartan connection: applications to gravity

TL;DR

This paper explores the symplectic structure and charges in the Yang-Mills formulation of the normal conformal Cartan connection, revealing connections to gravity, holography, and asymptotic symmetries.

Contribution

It introduces a conformally invariant presymplectic potential current and relates it to gravitational charges without holographic techniques.

Findings

Decomposition of the presymplectic potential into topological and Einstein-Hilbert parts.

Matching of the boundary current with holographically renormalized gravitational action.

Vanishing of the current at null infinity for BMS-induced variations.

Abstract

It is known that a source-free Yang-Mills theory with the normal conformal Cartan connection used as the gauge potential gives rise to equations of motion equivalent to the vanishing of the Bach tensor. We investigate the conformally invariant presymplectic potential current obtained from this theory and find that on the solutions to the Einstein field equations, it can be decomposed into a topological term derived from the Euler density and a part proportional to the potential of the standard Einstein-Hilbert Lagrangian. The pullback of our potential to the asymptotic boundary of asymptotically de Sitter spacetimes turns out to coincide with the current obtained from the holographically renormalized gravitational action. This provides an alternative derivation of a symplectic structure on scri without resorting to holographic techniques. We also calculate our current at the null infinity of asymptotically flat spacetimes and in particular show that it vanishes for variations induced by the BMS symmetries. In addition, we calculate the Noether currents and charges corresponding to gauge transformations and diffeomorphisms.