Higher-spin algebras from soft theorems I: the wedge condition

Abstract: In this article we use the sub$^n$-soft graviton theorems to construct the map $\Top$ from the spin-graded set of holomorphic functions on local celestial sphere patches to differential operators acting on the asymptotic data for massless particles at $\scrip$, in analogy with previous results in the literature for the sub$^n$-soft photon theorems. The result is an explicit closed-form formula. We show that the wedge subalgebras for both Yang-Mills and gravity are the natural domain on which $\Top$ becomes a representation.

• The paper demonstrates that the map T converts spin-graded holomorphic functions into differential operators that generate higher-spin symmetry charges.
• It establishes the wedge condition as the unique maximal subalgebra where these operators, derived from subleading soft theorems, form a closed Lie algebra.
• The study unifies subleading soft graviton theorems with asymptotic symmetry structures, providing explicit formulas for both gauge and gravitational cases.

Higher-Spin Algebras from Soft Theorems and the Wedge Condition

Introduction

The paper "Higher-spin algebras from soft theorems I: the wedge condition" (2605.07864) develops a systematic and algebraic perspective on the connection between subleading ($\text{sub}^n$) soft graviton theorems and higher-spin symmetry algebras in asymptotically flat gauge and gravity theories. The work constructs an explicit map $T$, generalizing prior constructions for soft photons, that translates spin-graded holomorphic functions on celestial sphere patches into differential operators acting on the asymptotic (null infinity) data of massless particles. A central result is the identification of the so-called wedge subalgebra as the unique maximal subalgebra on which $T$ is a representation; this wedge condition turns the soft theorem-generated differential operators into a closed Lie algebra structure.

Theoretical Framework and Motivation

The study is situated within the ongoing program relating three deep features of low-energy gauge and gravitational physics: the IR (soft limit) behavior of scattering amplitudes, the structure of asymptotic symmetries, and the memory effect. This "IR triangle" has elucidated the link between soft theorems and large gauge/diffeomorphism symmetries at null infinity, providing an avenue for extending the scope of known symmetries. While these themes are well-established for leading soft theorems and their associated charges, the algebraic structure associated with higher-order (sub$^n$) soft terms—especially for non-abelian Yang-Mills and gravity—has been less clear, particularly with respect to the domain on which symmetry representations are well-defined.

Building on previous analyses for QED [Campiglia & Laddha, JHEP 2019], the authors extend the formalism to arbitrary spins and focus on the explicit algebraic content of the higher-spin symmetry generators, with particular attention to their realization as differential operators via the soft expansion. Importantly, the construction is developed in explicit holomorphic coordinates familiar from celestial holography and memory effect calculations.

Construction of the T Map and Differential Hard Charges

Central to the analysis is the map $T$, which assigns to each spin-$-s$ holomorphic function $\tau_s$ on a local patch of the celestial sphere a differential operator acting on asymptotic particle data. For QED and Yang-Mills, the explicit form of $T$ is

$$T(\tau_s) = \sum_{k=0}^s \frac{(-1)^{k+1}}{(s-k)!} \partial_z^{s-k} \tau_s\, \partial_E^{s-k} E^{-k} \partial_z^k$$

where $E$ is the energy variable of the hard particle and $T$0 the celestial sphere coordinate. For gravity, the paper provides the first closed form for arbitrary spin,

$T$1

This formula resolves prior open questions about the all-spin structure of higher-spin hard charges in general relativity, encapsulating all known lower-spin results ($T$2) as special cases.

One notable feature is the explicit energy scaling: in gravity, $T$3 has $T$4-weight $T$5, as opposed to $T$6 for QED/Yang-Mills. This distinction underlies significant differences in closure properties and representations of the charge algebra.

The Wedge Condition and Algebra Closure

A principal result is the demonstration that $T$7, as an operator-valued map, is generically not a Lie algebra homomorphism on the full algebra of holomorphic functions. To identify where $T$8 does furnish a representation, the authors analyze the closure condition

$T$9

where the right-hand side is defined via the appropriate symmetry bracket (trivial, Lie, or Schouten-Nijenhuis) dictated by particle spin and theory. The wedge subalgebra is then identified as

$T$0

where $T$1 for Yang-Mills and $T$2 for gravity. This wedge condition restricts parameters $T$3 to those annihilated by sufficiently high derivatives, corresponding to polynomials of bounded degree (related to the global part of the symmetry algebra). The wedge subalgebra turns out to be the maximal domain of closure: only for $T$4 does $T$5 yield a faithful representation reflecting the underlying Lie algebra of asymptotic symmetries.

Notably, this closure proof is accomplished without requiring prior knowledge of the detailed algebraic structure of gauge generators, instead relying on the operator relations derived from the soft theorem expansion and the properties of the celestial functions. The analysis also makes manifest a tight algebraic and operational connection to celestial light-ray operators and subsequent conjectures in celestial holography [Himwich & Pate, 2025].

Algebraic Implications and Generalizations

The result supplies a purely algebraic route to higher-spin charge algebras long recognized in the celestial approach to asymptotic symmetries [Strominger, 2021]. For gravity, the wedge subalgebra corresponds to the global part of the $T$6 algebra; for Yang-Mills, to the wedge algebra of the $T$7-algebra. The construction, together with explicit commutator derivations, clarifies how the infinite-dimensional symmetries observed in celestial OPEs and large gauge/diffeomorphism studies are realized on the physical asymptotic phase space.

The work also sets the stage for future generalizations. Outside the wedge, the map $T$8 must be deformed, analogous to Stueckelberg extensions, introducing additional Goldstone-type degrees of freedom to accommodate larger algebras. The formalism anticipates the incorporation of Diff$T$9 covariance and the extension to nonlocal or logarithmic corrections (e.g., log soft theorems, superrotations).

Relation to Celestial Holography and Light-Ray Operators

The analysis highlights a concrete link between the map $^n$0 and the structure of light-ray operators in the celestial holography program. Integration of $^n$1 over null infinity against hard particle data is shown to produce weighted light-ray operators, and the wedge closure condition can be interpreted as a constraint on the local OPE structure of such operators for both gauge and gravity sectors.

This provides strong evidence for conjectures proposing a universal identification of light-ray operators with matter contributions to hard charges generating asymptotic symmetries, helping to unify several perspectives on celestial amplitudes, memory effects, and symmetry algebras.

Conclusion

This work rigorously establishes the wedge subalgebra as the maximal domain on which higher-spin symmetry charges, derived from subleading soft theorems, close under commutation as represented by the map $^n$2 to differential operators on null infinity (2605.07864). The explicit all-spin closed-form formulas for both gauge and gravity cases offer a definitive answer regarding the algebraic structure of higher-spin asymptotic symmetries and provide tools for further investigations into the algebraic underpinnings of celestial holography and advanced IR phenomena. By laying out a purely algebraic derivation and explicitly proving closure conditions, the paper clarifies longstanding ambiguities in the realization of infinite-dimensional symmetry algebras and establishes a basis for systematic extensions to broader contexts in gauge and gravitational scattering.