Geometric Aspects of Covariant Phase Space Formalism: Solution Space Slicings and Surface Charge Integrability

TL;DR

This paper develops a geometric framework for the covariant phase space formalism, analyzing solution space slicings and surface charge integrability, and introduces slicing-independent criteria for flux and charge definitions in diffeomorphism-invariant theories.

Contribution

It establishes a parallel geometric formulation for spacetime and solution phase space, extending the Wald-Zoupas criterion to be slicing-independent and defining fundamental geometric quantities on the solution space.

Findings

Slicing dependence of surface charge integrability is resolved.

Defines SPS connection, torsion, and curvature geometrically.

Distinguishes between fake and genuine fluxes in gravitational theories.

Abstract

The Covariant Phase Space Formalism (CPSF) provides a robust framework for deriving symplectic structures and surface charges in diffeomorphism-invariant theories. By construction, the CPSF operates on two distinct manifolds: the spacetime and the Solution Phase Space (SPS). In this paper, we advance the formalism by establishing a strictly parallel geometric formulation for both manifolds. Within this framework, we systematically analyze diffeomorphisms and frame changes on both spaces. While spacetime diffeomorphisms have been extensively studied in the literature, transformations on the SPS have been largely overlooked; we rigorously define and investigate these as changes of slicing on SPS. We demonstrate that the standard Wald-Zoupas criterion for the integrability of surface charge variations is inherently slicing-dependent. To resolve this issue, we develop the Frobenius theorem on the SPS and use it to extends the Wald-Zoupas condition into an inherently slicing-independent criterion for integrability. The Frobenius theorem on the SPS also yields a rigorous and natural definition of fundamental geometric quantities on the solution space, specifically the SPS connection, torsion, and curvature. Furthermore, this geometric machinery naturally distinguishes between fundamentally different surface fluxes: "fake" fluxes are identified mathematically as pure gauge artifacts of the SPS connection, while "genuine" fluxes manifest as non-vanishing SPS torsion, which directly relates to the physical gravitational News tensor. Finally, we present a geometric formulation of the Liouville theorem on the SPS, offering a unified classification scheme for theories with and without propagating bulk degrees of freedom.