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On the covariant formulation of gauge theories with boundaries

This paper reviews the covariant formulation of Yang-Mills theory and general relativity with boundaries, arguing that the necessity of edge modes arises not from gauge invariance but from the requirement that the presymplectic form be degenerate on the initial field space to enable its relation to the symplectic form on the gauge-reduced space.

Original authors: Mehdi Assanioussi, Jerzy Kowalski-Glikman, Ilkka Mäkinen, Ludovic Varrin

Published 2026-09-18

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CC BY 4.0

Original authors: Mehdi Assanioussi, Jerzy Kowalski-Glikman, Ilkka Mäkinen, Ludovic Varrin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Physics often deals with the invisible rules that govern the universe, from the tiny particles inside an atom to the vast curvature of space itself. A central idea in modern physics is that many of these rules are symmetries, meaning the laws of nature look the same even if you change certain things, like shifting your position or rotating your view. For decades, physicists believed that a specific type of symmetry, called a gauge symmetry, was purely a mathematical trick. They thought it was just a way to describe the same physical reality in different languages, carrying no real physical weight. If you changed the language, the physics didn't change, so the "charge" associated with that change was thought to be zero. However, a puzzle emerged when scientists looked at systems with edges or boundaries, such as the surface of a material or the edge of a black hole. In these places, the rules seemed to change, and new physical effects appeared that couldn't be explained by the old view. This suggests that boundaries might force these mathematical symmetries to become real, physical things with their own energy and behavior.

A team of researchers has now taken a fresh look at this problem using a powerful mathematical tool called the covariant phase space formalism. This method allows physicists to study the behavior of fields, like light or gravity, without having to pick a specific moment in time or a specific direction to measure, keeping the description of the universe consistent no matter how you look at it. The researchers applied this tool to two major theories: Yang–Mills theory, which describes the forces holding atomic nuclei together, and Einstein's theory of gravity. Their goal was to understand exactly what happens at the boundaries of these systems and why new physical entities, known as edge modes, seem to appear there.

The study begins by revisiting a long-standing debate about whether gauge symmetries are physical. In the past, physicists argued that because the charge for these symmetries vanished in empty space, they were just redundant mathematical features. The new work confirms that in a system without boundaries, this is true: the mathematical description remains degenerate, meaning it has directions where it doesn't change, which is a sign of redundancy. However, the researchers found that when a boundary is introduced, this degeneracy is broken. The mathematical form that describes the system's energy and motion develops a non-zero value right at the edge. This means the symmetry is no longer just a redundancy; it has generated a real, measurable charge localized at the boundary.

To fix this and restore the proper mathematical structure, the researchers showed that one must introduce a new field living on the boundary. They call this an edge mode. In the case of the forces inside an atom, this edge mode is a field that takes values in a group of symmetries, effectively acting as a new degree of freedom that cancels out the unwanted charge at the edge. This restores the system to a state where the mathematical description is consistent and can be reduced to the true physical degrees of freedom. The researchers emphasize that this need for an edge mode comes from a requirement of mathematical consistency—specifically, that the system must be degenerate in the right way—rather than from a need to keep the equations invariant under changes that depend on the fields themselves. This distinction is crucial, as it clarifies that the edge mode is a fundamental necessity for the theory to work correctly at a boundary, not just a patch for a specific type of mathematical transformation.

The paper then moves to the realm of gravity, where the symmetries involve moving and twisting space and time itself. Here, the situation is even more complex because the boundaries of a region in space can move. When the researchers analyzed the gravitational field, they found that if a transformation moves the boundary, it creates a "flux," a flow of energy or information across that edge. This flux prevents the charge from being well-defined. To solve this, they proposed treating the boundary itself as a dynamic object, defined by an embedding map that tells us where the boundary is located in the larger universe. By including this map as a new variable in the theory, the flux is absorbed, and the charge becomes well-defined again. Remarkably, the researchers discovered that in gravity, this new variable—the embedding map—plays the exact same role as the edge mode found in the other theories. The edge mode in gravity is simply the map that defines where the boundary is.

This unification is a significant finding. It suggests that the mysterious edge modes that appear in various theories are not arbitrary additions but are the natural consequence of how we define the boundaries of our physical systems. Whether we are looking at the forces inside an atom or the curvature of spacetime, the presence of a boundary forces the theory to acknowledge the location of that boundary as a physical entity. The researchers also tested their ideas on a specific, complex solution to Einstein's equations known as the Kerr–Newman–de Sitter spacetime, which describes a rotating, charged black hole in a universe with a cosmological constant. They calculated the charges associated with this system at a finite distance and found that their extended framework correctly accounted for the boundary effects, producing results that matched known physical quantities like mass and angular momentum.

The study concludes by highlighting a gap in our current understanding. While the researchers have successfully shown how to introduce these edge modes to make the mathematics consistent and to define the charges correctly, the actual physical laws that govern the dynamics of these edge modes are still unknown. In some condensed matter systems, like the quantum Hall effect, the behavior of edge states is well understood because they arise from a specific term added to the energy equation. In the theories of gravity and atomic forces, however, the edge modes have been introduced at the level of the mathematical description of motion, but the underlying energy equation that would generate their behavior has not yet been found. The researchers suggest that finding this equation is the next great challenge, as it would provide a complete picture of how the bulk of the universe and its boundaries interact, potentially unlocking new insights into black holes, quantum gravity, and the nature of entanglement.

Technical Summary: On the covariant formulation of gauge theories with boundaries

Problem Statement
The paper addresses the formulation of gauge theories, specifically Yang–Mills theory and General Relativity (Einstein–Hilbert gravity), in the presence of spacetime boundaries. A central issue in this context is the behavior of the presymplectic form—a closed two-form on the space of field configurations—when boundaries are introduced. In the absence of boundaries, the presymplectic form is known to be degenerate along gauge orbits, a property required for it to be the pullback of a non-degenerate symplectic form on the gauge-reduced physical field space. However, the presence of boundaries often leads to non-vanishing boundary charges and flux terms, which can break this degeneracy or introduce non-integrable charges.

A specific point of confusion addressed is the role of "edge modes." Previous literature (e.g., Donnelly and Freidel) introduced edge modes to ensure the gauge invariance of the presymplectic form under field-dependent gauge transformations. This paper investigates the necessity of edge modes when considering only field-independent gauge transformations, where the presymplectic form is already gauge invariant. The authors ask: if gauge invariance is satisfied, why are edge modes still required, and what is their precise physical and mathematical justification in this context?

Methodology
The authors utilize the covariant phase space formalism, a geometric framework that describes Hamiltonian dynamics and symplectic structures without breaking covariance by selecting a specific time coordinate. The methodology relies on:

Cartan Calculus on Field Space: The authors generalize the standard Cartan calculus (exterior derivative $d$ , interior product $\iota_v$ , Lie derivative $L_v$ ) from spacetime to the infinite-dimensional space of field configurations ( $\mathcal{F}$ ). This allows for the definition of field space forms (symplectic potentials $\Theta$ and presymplectic forms $\Omega$ ) and the computation of Noether charges via contraction with vector fields tangent to gauge orbits.
Analysis of Degeneracy: The core methodological step involves testing the condition $i_V \Omega = 0$ for vector fields $V$ tangent to gauge orbits. The authors argue that for the physical phase space to be a symplectic manifold, the presymplectic form on the initial field space must be the pullback of the physical symplectic form. This necessitates that $\Omega$ be degenerate along gauge directions ( $i_V \Omega = 0$ ).
Extended Phase Space: To handle symmetries that do not preserve the boundary (such as global spacetime translations or general diffeomorphisms), the authors employ the extended phase space formalism. This involves treating spacetime embedding maps $\phi$ as independent dynamical variables (phase space fields). This extension allows the variation of the action to include terms that cancel fluxes arising from the movement of the boundary.

Key Contributions and Results

Re-evaluation of Edge Modes in Yang–Mills Theory:
- The authors demonstrate that for Yang–Mills theory with field-independent gauge transformations, the standard presymplectic form is already gauge invariant, even with boundaries. Thus, edge modes are not required for gauge invariance in this specific context.
- However, they show that the standard presymplectic form generates non-vanishing boundary charges ( $H_{YM}[\epsilon]$ ), meaning $i_V \Omega \neq 0$ . This violates the degeneracy condition required for the pullback relation to the reduced phase space.
- To restore degeneracy, they introduce a group-valued edge mode field $\phi$ on the boundary. This field contributes a boundary term to the presymplectic form that exactly cancels the emergent boundary charges, thereby satisfying $i_V \Omega_{total} = 0$ .
Global Spacetime Symmetries and Fluxes:
- In the context of global Poincaré symmetries (e.g., translations and Lorentz transformations in Maxwell theory), the authors show that the presymplectic form contraction yields non-integrable terms known as fluxes. These fluxes arise because global spacetime transformations move the location of the boundary.
- By extending the phase space to include embedding maps, the authors show that the variation of the extended action produces a boundary term that cancels these fluxes. This restores the integrability of the charges (energy-momentum and angular momentum).
Gravity (Einstein–Hilbert Action):
- Similar to Yang–Mills, the authors analyze the contraction of the presymplectic form along diffeomorphism orbits. They find that for general diffeomorphisms (which do not preserve the boundary), flux terms appear, and the contraction is not a total differential.
- Restricting to boundary-preserving diffeomorphisms recovers the standard Komar charges.
- By introducing the embedding map as an edge mode, the authors demonstrate that the extended presymplectic form yields the exact differential of the Komar charge, eliminating the flux.
- Crucially, they show that the additional edge mode term required to satisfy the degeneracy condition ( $i_V \Omega = 0$ ) in gravity can be expressed entirely in terms of the embedding map. This leads to the conclusion that the edge mode in gravity is the embedding map itself.
Distinction Between Edge Mode Types:
The paper clarifies a conceptual distinction between two types of edge modes:
1. Internal Gauge Edge Modes: Required to satisfy the degeneracy condition for field-independent internal symmetries (Yang–Mills) or to ensure invariance for field-dependent symmetries.
2. Spacetime Edge Modes (Embedding Maps): Required to eliminate fluxes and restore integrability for spacetime symmetries (diffeomorphisms) that move the boundary. In gravity, these two roles coincide because the gauge group is the diffeomorphism group.

Significance and Claims
The paper claims to provide a unified and clarified understanding of edge modes in gauge theories with boundaries, specifically within the context of field-independent gauge transformations. Its primary significance lies in shifting the justification for edge modes:

From Invariance to Degeneracy: The authors argue that the introduction of edge modes is not primarily driven by the need for the presymplectic form to be gauge invariant (which is often already true for field-independent transformations), but rather by the degeneracy condition. The presymplectic form must be degenerate along gauge orbits to ensure it is the pullback of a symplectic form on the gauge-reduced space.
Unification in Gravity: The work establishes that in General Relativity, the "edge mode" necessary to handle boundary fluxes and the "edge mode" necessary to satisfy the degeneracy condition are the same object: the spacetime embedding map.
Accessibility: The authors aim to make these advanced concepts accessible by using a simplified mathematical language, avoiding the heavy abstract geometry often found in the literature, while maintaining rigorous derivations.

The paper concludes by noting that while the necessity of these edge modes is established at the level of the presymplectic form, the dynamics of these edge modes (i.e., the specific Lagrangian that generates them) remains an open question for future work. The current formulation introduces them at the level of the symplectic structure to ensure consistency, but a full path integral formulation with a dynamical boundary Lagrangian is not yet provided.

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