Transcript: Celestial 1-form symmetries

Alex: Welcome to another episode of ResearchPod. Today, we're diving into a paper on symmetries in a special part of gauge theory. Sam, can you give us the lay of the land?

Sam: This work, by Laurent Freidel and Atul Sharma, looks at self-dual Yang-Mills theory—a simplified version of the theory that describes how gluons, the particles carrying the strong force, behave in flat space. The central puzzle is why two different ways of calculating charges in flat-space holography—celestial CFT modes and Carrollian corner charges—give the same numbers.

Alex: And the answer involves symmetries that extend from the edge of spacetime into the whole space?

Sam: Yes, exactly. The paper shows these symmetries, called the S-algebra, usually appear only at the boundary but here extend throughout as 1-form symmetries. They come from conserved 2-form currents made from special building blocks called charge aspects. Celestial approaches treat asymptotic symmetries as modes in a 2D theory on the celestial sphere, integrating over time and contours there. Carrollian geometry sees them as charges fixed at specific times on null infinity, integrated over full spheres. Both compute the same charges because they integrate identical conserved 2-form currents over surfaces that are topologically the same—homologous 2-cycles linking bulk and boundary.

Alex: Okay, let's unpack symmetries first. In physics, they're like rules that keep the laws of nature unchanged if you tweak things in certain ways—think rotating a round object without changing its motion. But these higher-form ones act differently. Instead of everyday conserved quantities like energy from 1D currents over time slices, they use 2D currents spread over surfaces, giving charges on codimension-2 boundaries. Researchers call these 1-form symmetries.

Sam: That's right. In standard symmetries, you integrate a current over a 3D slice in 4D space to get a charge. Here, the 2-form current gets integrated over a 2D surface, like a sphere, making the charge depend on that closed surface. The paper builds these from charge aspects—pieces of an anti-self-dual 2-form field B, expanded using a recursion operator from the Lax pair. That operator ensures consistency because the Lax operators commute on solutions.

Alex: Self-dual Yang-Mills is this integrable subsector where the anti-self-dual part of the curvature vanishes. It's like focusing on half the field equations, making it solvable through hierarchies of conserved forms. But why does this setup let the S-algebra move from the boundary into the bulk?

Sam: Self-dual Yang-Mills uses a connection A and a 2-form B that enforces self-duality via its equations. The integrability comes from a Lax pair—two operators L and M whose commutator vanishes exactly when self-duality holds. From a seed B, a recursion operator generates higher B_s: each new one combines previous steps into anti-self-dual 2-forms that satisfy the Bianchi identity. These radiation 2-forms B_s form the hierarchy. Pairing them with parameters tau_s that follow a dual recursion gives a conserved current J_tau = sum of traces of tau_s with B_s. And d J_tau = 0 only if those dual relations hold.

Celestial 1-form symmetries

Laurent Freidel, Atul Sharma

Apr 13, 2026·4:39·arxiv·Original Paper

1-form symmetriesSelf-dual Yang-Mills (sdYM)S-algebraCharge aspectsCelestial and Carrollian charges

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5 min

Summary

Overview

This paper bridges two exciting frontiers in quantum field theory: generalized higher-form symmetries and asymptotic symmetries in gauge theories. Traditionally separate, these meet beautifully in the self-dual sector of Yang-Mills theory (sdYM), where infinite-dimensional symmetries at infinity 'upgrade' to bulk 1-form symmetries. The authors, Laurent Freidel and Atul Sharma, show how the mysterious S-algebra—a noncommutative chiral algebra from asymptotic symmetries—emerges as genuine 1-form symmetries with conserved 2-form currents. A killer application: proving celestial and Carrollian charges are identical, unifying flat-space holography approaches.

Think of it intuitively: usual 0-form symmetries (like global U(1)) have 1-form currents integrated over codimension-1 surfaces. 1-form symmetries have 2-form currents over codimension-2 surfaces (like spheres). Asymptotic symmetries also live on codimension-2 boundaries but were boundary-glued. Here, they're freed into the bulk!

Self-dual Yang-Mills (sdYM): The Integrable Playground

sdYM imposes the self-duality constraint F⁻ = 0 on Yang-Mills fields, selecting an integrable subsector. Key features:

Lax pair formulation: Equations of motion as zero-curvature conditions, enabling soliton-like solutions.
Recursion operator: Generates infinite hierarchies of conserved quantities, specifically 2-form currents J^{(n)}.
Twistor connections: Links to Penrose's twistor space for holomorphic structures.

Intuition: Self-duality halves degrees of freedom, making the theory chiral and recursive—like a 1D integrable system embedded in 4D spacetime. Conserved 2-forms arise naturally from this hierarchy.

Charge Aspects: Building Blocks of Symmetries

From the anti-self-dual field B, extract charge aspects R_s via Laurent series and recursion: R_0 = B, R_{s+1} = ℛ R_s

These coefficients are the 'DNA' of conserved currents: J = ∑ ε_s R_s, where ε_s are symmetry parameters solving dual Lax equations. Parameters aren't arbitrary—they must satisfy integrability, mirroring the fields themselves.

Picture charge aspects as modes of a generating function, expandable like Fourier series but for nonlinear gauge fields. They encode all higher symmetries compactly.

1-form Symmetries and the Noncommutative S-algebra

The big reveal: S-algebra (from celestial holography) becomes a bulk 1-form symmetry algebra in sdYM. Charges Q_ε = ∫_Σ J_ε are:

Gauge invariant in bulk.
Noncommutative: [Q_ε, Q_η] ≠ 0, defying standard 1-form expectations.

Why noncommutative? Charges are corner charges surviving gauge constraints, depending implicitly on the codimension-1 slice connecting to Σ. Different slices → different 'gluings' → noncommutativity.

Example in sdMaxwell (section 5): Explicit currents and charges match asymptotic symmetries.

Unifying Celestial and Carrollian Holography

Celestial approach: Asymptotic symmetries → 2D CFT modes on celestial sphere, charges ∫ du ∮ dθ dφ J. Carrollian approach: Symmetries of 3D theory on null infinity, charges ∫ dθ dφ J at fixed u.

Proof of equality: Both integrate the same 2-form current J over homologous 2-cycles Σ_celestial ~ Σ_Carrollian. Pure spacetime argument, no twistors needed (though consistent with prior work).

This unifies flat-space holography: same symmetries, different slicings. Implications for soft theorems, black hole microstates, and infrared physics.

Abstract

The $S$-algebra originally arose as a chiral algebra of asymptotic symmetries of Yang-Mills theory. We show that in the self-dual sector of Yang-Mills, the $S$-algebra gets upgraded to an infinite-dimensional algebra of $1$-form symmetries in the bulk. The associated 2-form currents encode the integrability and hierarchies of self-dual Yang-Mills. As an application, we prove the equality of Carrollian corner charges with modes of the celestial chiral algebra by expressing them as integrals of the same 2-form currents over homologous 2-cycles.

Celestial 1-form symmetries | Laurent Freidel, Atul Sharma | ResearchPod

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