Geometrical tools for embedding fields, submanifolds, and foliations

TL;DR

This work develops an embedding-field formalism to study background structures—embedded submanifolds, boundaries, and foliations—in diffeomorphism-invariant theories, unifying their treatment under a covariant map $X:M_0\to M$. It derives the full local geometry of foliations, including intrinsic/extrinsic data, Gauss–Codazzi–Ricci–Voss identities, and variational formulas, and shows how the Jacobi equation governs perturbations of extremal surfaces such as RT surfaces. The formalism cleanly separates changes of the ambient metric from deformations of the embedding, enabling rigorous analysis of boundary terms in gravitational Hamiltonians, edge modes, and holographic perturbations, and it demonstrates broad applicability to brane dynamics, fluid dynamics, RT perturbations, and edge-mode physics in finite subregions. The framework also clarifies invariants and obstructions (e.g., normal extrinsic curvature) under foliation changes and offers pathways to handle null surfaces, field-dependent symmetries, and applications to MHD/FFE via a two-dimensional foliation perspective.

Abstract

Embedding fields provide a way of coupling a background structure to a theory while preserving diffeomorphism-invariance. Examples of such background structures include embedded submanifolds, such as branes; boundaries of local subregions, such as the Ryu-Takayanagi surface in holography; and foliations, which appear in fluid dynamics and force-free electrodynamics. This work presents a systematic framework for computing geometric properties of these background structures in the embedding field description. An overview of the local geometric quantities associated with a foliation is given, including a review of the Gauss, Codazzi, and Ricci-Voss equations, which relate the geometry of the foliation to the ambient curvature. Generalizations of these equations for curvature in the nonintegrable normal directions are derived. Particular care is given to the question of which objects are well-defined for single submanifolds, and which depend on the structure of the foliation away from a submanifold. Variational formulas are provided for the geometric quantities, which involve contributions both from the variation of the embedding map as well as variations of the ambient metric. As an application of these variational formulas, a derivation is given of the Jacobi equation, describing perturbations of extremal area surfaces of arbitrary codimension. The embedding field formalism is also applied to the problem of classifying boundary conditions for general relativity in a finite subregion that lead to integrable Hamiltonians. The framework developed in this paper will provide a useful set of tools for future analyses of brane dynamics, fluid mechanics, and edge modes for finite subregions of diffeomorphism-invariant theories.

Figures (2)

• Figure 1: The embedding field $X$ is a map between a reference space $M_0$ and the spacetime manifold $M$. The metric $g_{ab}$ and other dynamical fields live on $M$, and can be mapped to pulled-back fields $X^*g_{ab}$ on the reference space using the embedding map.
• Figure 2: The fixed foliation on $M_0$ is specified through the normal form ${\bm{ \nu} }$ by requiring all tangent vectors to the foliation annihilate it. $X$ maps ${\bm{ \nu} }$ to the form $\nu = X_* {\bm{ \nu} }$, which defines a foliation in $M$. The $M$ foliation can be varied by changing the embedding map $X$.