Celestial Soft Current Algebra Explained

• Celestial soft current algebra is defined by the OPE of conformally soft operators derived from collinear limits of scattering amplitudes.
• It organizes soft gauge and gravitational scattering into structures like level-zero Kac–Moody, Virasoro/BMS, and w₁+∞-type algebras through precise differential constraints.
• Loop corrections, supersymmetry, and multi-particle extensions refine the algebra, revealing consistency conditions, logarithmic multiplet behavior, and extended symmetry structures.

to=arxiv_search 天天彩票提现_code 天天中彩票充值ლობა json {"query":"celestial soft current algebra supersymmetry Jacobi identity one loop arXiv", "max_results": 10} to=arxiv.search 聚利_code аҵjson {"query":"celestial soft current algebra", "limit": 10} Celestial soft current algebra is the algebra defined by the OPE of celestial soft currents, obtained from the collinear limit of scattering amplitudes. In celestial CFT, Mellin-transformed amplitudes are interpreted as correlators of celestial conformal primaries, and conformally soft operators arise at special integer conformal dimensions where the Mellin transform develops poles. Their singular OPEs organize the leading and subleading soft behavior of gauge and gravitational scattering, producing structures that range from level-zero Kac–Moody algebras and Virasoro/BMS generators to $w_{1+\infty}$-type wedges and their supersymmetric extensions. The subject is technically subtle: the same collinear data that suggests a current algebra also exposes Jacobi constraints, double residue conditions, branch-cut contributions, non-closure of some subleading generators, and loop-level logarithmic or multi-particle refinements (Ball, 2024).

1. Definition from celestial amplitudes and collinear limits

The standard celestial transform writes an amplitude as a Mellin transform over external energies. For gluons, one representative formula is

$$\mathcal{M}_n(\{z_i,\bar z_i,h_i,\bar h_i,a_i\}) = \prod_{i=1}^n \int_0^\infty d\omega_i \,\omega_i^{\Delta_i-1}\, \mathcal{S}_n(\{\omega_i,z_i,\bar z_i,\sigma_i,a_i\}),$$

with $\Delta_i=h_i+\bar h_i$, and with null momenta parametrized by

$$p_i^\mu = \epsilon_i \,\omega_i\, \bigl(1+z_i\bar z_i,\ z_i+\bar z_i,\ -i(z_i-\bar z_i),\ 1-z_i\bar z_i\bigr)$$

(Banerjee et al., 2020). In this basis, the OPE on the celestial sphere is extracted from holomorphic collinear limits such as $z_{12}\to 0$, and the resulting pole terms define current-like operator products.

For soft applications, the relevant operators are celestial representatives of positive-helicity soft bosons, such as soft gluons and soft gravitons. A generic tree-level celestial OPE in Yang–Mills takes the form

$$\mathcal O^{a,+}_{\Delta_1}(z_1,\bar z_1)\mathcal O^{b,+}_{\Delta_2}(z_2,\bar z_2) \sim \frac{-if^{ab}{}_c}{z_{12}} \sum_{m=0}^\infty B(\Delta_1-1+m,\Delta_2-1)\, \frac{\bar z_{12}^m}{m!}\, \partial_{\bar z_2}^m\,\mathcal O^{c,+}_{\Delta_1+\Delta_2-1}(z_2,\bar z_2),$$

while mixed-helicity OPEs carry the shifted Beta function $B(\Delta_1-1+m,\Delta_2+1)$ (Ball, 2024). Closely related formulas appear in Einstein-Yang-Mills, where same-helicity gauge boson and graviton OPEs are extracted directly from collinear singularities of four-dimensional amplitudes (Fotopoulos et al., 2019).

Conformally soft operators are residues at special integer dimensions. A standard definition is

$$R^{(k,1),a}\equiv \lim_{\varepsilon\to 0}\varepsilon\,H^{a}_{k+\varepsilon,+},$$

and analogous soft-current definitions occur for gravitons and for the infinite tower $k=1,0,-1,\dots$ or $k=2,1,0,-1,\dots$, depending on spin (Bhardwaj et al., 2024, Jiang, 2021). This residue construction is the celestial counterpart of extracting Laurent coefficients in the bulk soft-energy expansion.

2. Tree-level gauge and gravitational current algebras

For gluons, the leading conformally soft positive-helicity current is

$$\mathcal{M}_n(\{z_i,\bar z_i,h_i,\bar h_i,a_i\}) = \prod_{i=1}^n \int_0^\infty d\omega_i \,\omega_i^{\Delta_i-1}\, \mathcal{S}_n(\{\omega_i,z_i,\bar z_i,\sigma_i,a_i\}),$$0

and it obeys a level-zero Kac–Moody algebra,

$$\mathcal{M}_n(\{z_i,\bar z_i,h_i,\bar h_i,a_i\}) = \prod_{i=1}^n \int_0^\infty d\omega_i \,\omega_i^{\Delta_i-1}\, \mathcal{S}_n(\{\omega_i,z_i,\bar z_i,\sigma_i,a_i\}),$$1

(Banerjee et al., 2020). In the purely holomorphic soft sector one also finds the OPE

$$\mathcal{M}_n(\{z_i,\bar z_i,h_i,\bar h_i,a_i\}) = \prod_{i=1}^n \int_0^\infty d\omega_i \,\omega_i^{\Delta_i-1}\, \mathcal{S}_n(\{\omega_i,z_i,\bar z_i,\sigma_i,a_i\}),$$2

together with the soft-hard OPE

$$\mathcal{M}_n(\{z_i,\bar z_i,h_i,\bar h_i,a_i\}) = \prod_{i=1}^n \int_0^\infty d\omega_i \,\omega_i^{\Delta_i-1}\, \mathcal{S}_n(\{\omega_i,z_i,\bar z_i,\sigma_i,a_i\}),$$3

(Fan et al., 2020). These relations support a level-zero affine interpretation of the conformally soft gluon subsector.

The same tree-level data can be organized as differential constraints. For $$\mathcal{M}_n(\{z_i,\bar z_i,h_i,\bar h_i,a_i\}) = \prod_{i=1}^n \int_0^\infty d\omega_i \,\omega_i^{\Delta_i-1}\, \mathcal{S}_n(\{\omega_i,z_i,\bar z_i,\sigma_i,a_i\}),$$4-point tree-level MHV celestial gluon amplitudes in pure Yang–Mills, there is a system of $$\mathcal{M}_n(\{z_i,\bar z_i,h_i,\bar h_i,a_i\}) = \prod_{i=1}^n \int_0^\infty d\omega_i \,\omega_i^{\Delta_i-1}\, \mathcal{S}_n(\{\omega_i,z_i,\bar z_i,\sigma_i,a_i\}),$$5 linear first-order PDEs, one for each positive-helicity gluon. The first two terms are KZ-like, while a third term arises from subleading soft gluon symmetry and has no direct analogue in standard WZW current algebra (Banerjee et al., 2020). This allows extraction of the leading gluon-gluon OPE and some subleading mixed-helicity coefficients directly from symmetry.

For gravity, the subleading positive-helicity soft graviton theorem becomes a Ward identity for $$\mathcal{M}_n(\{z_i,\bar z_i,h_i,\bar h_i,a_i\}) = \prod_{i=1}^n \int_0^\infty d\omega_i \,\omega_i^{\Delta_i-1}\, \mathcal{S}_n(\{\omega_i,z_i,\bar z_i,\sigma_i,a_i\}),$$6 currents on the celestial sphere. The corresponding currents $$\mathcal{M}_n(\{z_i,\bar z_i,h_i,\bar h_i,a_i\}) = \prod_{i=1}^n \int_0^\infty d\omega_i \,\omega_i^{\Delta_i-1}\, \mathcal{S}_n(\{\omega_i,z_i,\bar z_i,\sigma_i,a_i\}),$$7 satisfy

$$\mathcal{M}_n(\{z_i,\bar z_i,h_i,\bar h_i,a_i\}) = \prod_{i=1}^n \int_0^\infty d\omega_i \,\omega_i^{\Delta_i-1}\, \mathcal{S}_n(\{\omega_i,z_i,\bar z_i,\sigma_i,a_i\}),$$8

and the leading positive-helicity soft graviton produces supertranslation currents $$\mathcal{M}_n(\{z_i,\bar z_i,h_i,\bar h_i,a_i\}) = \prod_{i=1}^n \int_0^\infty d\omega_i \,\omega_i^{\Delta_i-1}\, \mathcal{S}_n(\{\omega_i,z_i,\bar z_i,\sigma_i,a_i\}),$$9 that close with the $\Delta_i=h_i+\bar h_i$0 into an extended algebra (Banerjee et al., 2020). In Einstein-Yang-Mills, the celestial stress tensor is identified with the shadow transform of the conformally soft graviton operator of dimension $\Delta_i=h_i+\bar h_i$1,

$\Delta_i=h_i+\bar h_i$2

and it acts on celestial primaries through the standard primary-field OPE

$\Delta_i=h_i+\bar h_i$3

(Fotopoulos et al., 2019).

The soft graviton and soft gluon towers can also be organized into an infinite-dimensional chiral algebra generated by positive-helicity soft currents after summing $\Delta_i=h_i+\bar h_i$4 descendants. In the bosonic graviton sector the commutator takes the form

$\Delta_i=h_i+\bar h_i$5

which is the $\Delta_i=h_i+\bar h_i$6-type structure emphasized in supersymmetric Einstein-Yang-Mills and related holographic chiral-algebra constructions (Jiang, 2021).

3. Extended symmetry structures: BMS, Sugawara, and $\Delta_i=h_i+\bar h_i$7

In celestial Einstein-Yang-Mills, the OPEs of BMS-generating operators are obtained from soft and collinear theorems. The operator

$\Delta_i=h_i+\bar h_i$8

packages all supertranslations into a single primary conformal field of dimension $\Delta_i=h_i+\bar h_i$9, and it acts on a primary through

$$p_i^\mu = \epsilon_i \,\omega_i\, \bigl(1+z_i\bar z_i,\ z_i+\bar z_i,\ -i(z_i-\bar z_i),\ 1-z_i\bar z_i\bigr)$$0

(Fotopoulos et al., 2019). In this language, supertranslations are realized as a flow of conformal dimensions, while the shadow soft graviton gives the Virasoro stress tensor.

A separate construction identifies conformally soft positive-helicity gluons as holomorphic conserved currents and builds a Sugawara tensor

$$p_i^\mu = \epsilon_i \,\omega_i\, \bigl(1+z_i\bar z_i,\ z_i+\bar z_i,\ -i(z_i-\bar z_i),\ 1-z_i\bar z_i\bigr)$$1

which obeys

$$p_i^\mu = \epsilon_i \,\omega_i\, \bigl(1+z_i\bar z_i,\ z_i+\bar z_i,\ -i(z_i-\bar z_i),\ 1-z_i\bar z_i\bigr)$$2

(Fan et al., 2020). The same analysis also states that $$p_i^\mu = \epsilon_i \,\omega_i\, \bigl(1+z_i\bar z_i,\ z_i+\bar z_i,\ -i(z_i-\bar z_i),\ 1-z_i\bar z_i\bigr)$$3 is insufficient for generic hard states and mixed-helicity sectors, and proposes an alternative Einstein-Yang-Mills or double-copy-like construction that acts correctly on both soft and hard insertions (Fan et al., 2020).

The $$p_i^\mu = \epsilon_i \,\omega_i\, \bigl(1+z_i\bar z_i,\ z_i+\bar z_i,\ -i(z_i-\bar z_i),\ 1-z_i\bar z_i\bigr)$$4 perspective becomes more explicit once one introduces light-transformed currents

$$p_i^\mu = \epsilon_i \,\omega_i\, \bigl(1+z_i\bar z_i,\ z_i+\bar z_i,\ -i(z_i-\bar z_i),\ 1-z_i\bar z_i\bigr)$$5

whose modes satisfy a deformed $$p_i^\mu = \epsilon_i \,\omega_i\, \bigl(1+z_i\bar z_i,\ z_i+\bar z_i,\ -i(z_i-\bar z_i),\ 1-z_i\bar z_i\bigr)$$6-like algebra with structure function

$$p_i^\mu = \epsilon_i \,\omega_i\, \bigl(1+z_i\bar z_i,\ z_i+\bar z_i,\ -i(z_i-\bar z_i),\ 1-z_i\bar z_i\bigr)$$7

(Mago et al., 2021). In the minimal theory this reduces to the wedge algebra familiar from positive-helicity graviton and gluon soft currents; with non-minimal couplings it becomes a physically constrained deformation rather than a formal $$p_i^\mu = \epsilon_i \,\omega_i\, \bigl(1+z_i\bar z_i,\ z_i+\bar z_i,\ -i(z_i-\bar z_i),\ 1-z_i\bar z_i\bigr)$$8 with a free deformation parameter (Mago et al., 2021).

4. Consistency: Jacobi identity, double residues, and supersymmetry

A recurrent issue is whether the current algebra suggested by soft OPEs is genuinely associative. The review literature formulates the problem as the equivalence between the Jacobi identity for soft currents and a double residue condition on hard amplitudes or hard celestial correlators. The obstruction arises from three-particle factorization poles in momentum space and from branch cuts after Mellin transform (Ball, 2024). In this framework, the naive current algebra is consistent only if the OPE is purely factorizing and the corresponding four-point amplitude has no problematic angle-bracket-weight $$p_i^\mu = \epsilon_i \,\omega_i\, \bigl(1+z_i\bar z_i,\ z_i+\bar z_i,\ -i(z_i-\bar z_i),\ 1-z_i\bar z_i\bigr)$$9 part (Ball et al., 2023).

Supersymmetry imposes a strong simplification. In $z_{12}\to 0$0 supergravity and in $z_{12}\to 0$1 global SUSY EFTs around a stable vacuum, the tree-level bosonic celestial OPEs satisfy Jacobi automatically because supersymmetric Ward identities remove the amplitudes that would have generated the anomalous OPE terms. The potentially dangerous three-point amplitudes

$z_{12}\to 0$2

vanish in $z_{12}\to 0$3 supergravity,

$z_{12}\to 0$4

and the analogous $z_{12}\to 0$5 is absent from the most general renormalizable global SUSY Lagrangian (Ball et al., 2023). Vacuum stability is part of the argument because it excludes scalar cubic couplings of the form that would destabilize the vacuum and generate unwanted three-point amplitudes.

The same theme appears in non-minimal theories, but there the conclusion is different. Deformations of the soft-current algebra by couplings such as $z_{12}\to 0$6, $z_{12}\to 0$7, and scalar interactions lead to a deformed $z_{12}\to 0$8-type algebra whose Jacobi identity imposes strong constraints on the couplings $z_{12}\to 0$9 (Mago et al., 2021). In particular, once non-minimal couplings are allowed, the algebra involving only gravitons and gluons is generally not closed, and soft scalar currents are required for closure (Mago et al., 2021). A plausible implication is that the celestial algebra detects consistency conditions on the bulk EFT spectrum and its allowed cubic interactions.

5. Loop corrections, logarithmic structures, and hard-current refinements

Loop effects modify the celestial soft current algebra in several distinct ways. In Yang–Mills, one-loop collinear behavior introduces logarithms and derivatives with respect to conformal dimensions, so tree-level conformally soft operators are no longer sufficient. The enlarged set of one-loop soft operators is

$$\mathcal O^{a,+}_{\Delta_1}(z_1,\bar z_1)\mathcal O^{b,+}_{\Delta_2}(z_2,\bar z_2) \sim \frac{-if^{ab}{}_c}{z_{12}} \sum_{m=0}^\infty B(\Delta_1-1+m,\Delta_2-1)\, \frac{\bar z_{12}^m}{m!}\, \partial_{\bar z_2}^m\,\mathcal O^{c,+}_{\Delta_1+\Delta_2-1}(z_2,\bar z_2),$$0

$$\mathcal O^{a,+}_{\Delta_1}(z_1,\bar z_1)\mathcal O^{b,+}_{\Delta_2}(z_2,\bar z_2) \sim \frac{-if^{ab}{}_c}{z_{12}} \sum_{m=0}^\infty B(\Delta_1-1+m,\Delta_2-1)\, \frac{\bar z_{12}^m}{m!}\, \partial_{\bar z_2}^m\,\mathcal O^{c,+}_{\Delta_1+\Delta_2-1}(z_2,\bar z_2),$$1

$$\mathcal O^{a,+}_{\Delta_1}(z_1,\bar z_1)\mathcal O^{b,+}_{\Delta_2}(z_2,\bar z_2) \sim \frac{-if^{ab}{}_c}{z_{12}} \sum_{m=0}^\infty B(\Delta_1-1+m,\Delta_2-1)\, \frac{\bar z_{12}^m}{m!}\, \partial_{\bar z_2}^m\,\mathcal O^{c,+}_{\Delta_1+\Delta_2-1}(z_2,\bar z_2),$$2

with schematic Laurent expansion

$$\mathcal O^{a,+}_{\Delta_1}(z_1,\bar z_1)\mathcal O^{b,+}_{\Delta_2}(z_2,\bar z_2) \sim \frac{-if^{ab}{}_c}{z_{12}} \sum_{m=0}^\infty B(\Delta_1-1+m,\Delta_2-1)\, \frac{\bar z_{12}^m}{m!}\, \partial_{\bar z_2}^m\,\mathcal O^{c,+}_{\Delta_1+\Delta_2-1}(z_2,\bar z_2),$$3

(Bhardwaj et al., 2024). These operators transform as a logarithmic multiplet, and the paper identifies the one-loop soft currents as forming a rank-2 logarithmic multiplet (Bhardwaj et al., 2024).

The same work emphasizes a subtle obstruction: at loop level, the OPE of two conformally soft operators is not canonically defined because different orders of taking the soft limits give different answers. Even mixed soft-soft OPEs $$\mathcal O^{a,+}_{\Delta_1}(z_1,\bar z_1)\mathcal O^{b,+}_{\Delta_2}(z_2,\bar z_2) \sim \frac{-if^{ab}{}_c}{z_{12}} \sum_{m=0}^\infty B(\Delta_1-1+m,\Delta_2-1)\, \frac{\bar z_{12}^m}{m!}\, \partial_{\bar z_2}^m\,\mathcal O^{c,+}_{\Delta_1+\Delta_2-1}(z_2,\bar z_2),$$4 show explicit dependence on the regularization path. The proposed interpretation is that one should not expect a naive local OPE for arbitrary pairs of soft currents at one loop; consecutive soft limits are the better-defined operation (Bhardwaj et al., 2024).

Supersymmetry again simplifies the loop story. In the supersymmetric theories studied in (Ball et al., 2023), the loop-corrected OPE

$$\mathcal O^{a,+}_{\Delta_1}(z_1,\bar z_1)\mathcal O^{b,+}_{\Delta_2}(z_2,\bar z_2) \sim \frac{-if^{ab}{}_c}{z_{12}} \sum_{m=0}^\infty B(\Delta_1-1+m,\Delta_2-1)\, \frac{\bar z_{12}^m}{m!}\, \partial_{\bar z_2}^m\,\mathcal O^{c,+}_{\Delta_1+\Delta_2-1}(z_2,\bar z_2),$$5

has vanishing double-pole coefficient,

$$\mathcal O^{a,+}_{\Delta_1}(z_1,\bar z_1)\mathcal O^{b,+}_{\Delta_2}(z_2,\bar z_2) \sim \frac{-if^{ab}{}_c}{z_{12}} \sum_{m=0}^\infty B(\Delta_1-1+m,\Delta_2-1)\, \frac{\bar z_{12}^m}{m!}\, \partial_{\bar z_2}^m\,\mathcal O^{c,+}_{\Delta_1+\Delta_2-1}(z_2,\bar z_2),$$6

because the potentially dangerous splitting configurations are incompatible with the available SUSY three-point couplings (Ball et al., 2023). This is the one-loop counterpart of the tree-level Jacobi simplification.

Another refinement is the introduction of hard currents. A recent proposal constructs an infinite-dimensional hard current algebra from subleading operators such as $$\mathcal O^{a,+}_{\Delta_1}(z_1,\bar z_1)\mathcal O^{b,+}_{\Delta_2}(z_2,\bar z_2) \sim \frac{-if^{ab}{}_c}{z_{12}} \sum_{m=0}^\infty B(\Delta_1-1+m,\Delta_2-1)\, \frac{\bar z_{12}^m}{m!}\, \partial_{\bar z_2}^m\,\mathcal O^{c,+}_{\Delta_1+\Delta_2-1}(z_2,\bar z_2),$$7 and $$\mathcal O^{a,+}_{\Delta_1}(z_1,\bar z_1)\mathcal O^{b,+}_{\Delta_2}(z_2,\bar z_2) \sim \frac{-if^{ab}{}_c}{z_{12}} \sum_{m=0}^\infty B(\Delta_1-1+m,\Delta_2-1)\, \frac{\bar z_{12}^m}{m!}\, \partial_{\bar z_2}^m\,\mathcal O^{c,+}_{\Delta_1+\Delta_2-1}(z_2,\bar z_2),$$8, with modes

$$\mathcal O^{a,+}_{\Delta_1}(z_1,\bar z_1)\mathcal O^{b,+}_{\Delta_2}(z_2,\bar z_2) \sim \frac{-if^{ab}{}_c}{z_{12}} \sum_{m=0}^\infty B(\Delta_1-1+m,\Delta_2-1)\, \frac{\bar z_{12}^m}{m!}\, \partial_{\bar z_2}^m\,\mathcal O^{c,+}_{\Delta_1+\Delta_2-1}(z_2,\bar z_2),$$9

and with the explicit relations

$B(\Delta_1-1+m,\Delta_2+1)$0

(Liu et al., 15 Jan 2026). The paper states that the soft algebra is not independent but is reconstructed from the hard one by repeated commutators with $B(\Delta_1-1+m,\Delta_2+1)$1. This suggests a logarithmic pairing of soft and hard sectors rather than a purely infrared algebra (Liu et al., 15 Jan 2026).

6. Multi-particle extensions, dimensional variants, and open structural issues

Celestial soft current algebra is not exhausted by single-particle OPEs. In $B(\Delta_1-1+m,\Delta_2+1)$2 supergravity, multi-particle OPEs of a single-particle celestial operator with a two-particle operator produce higher-order poles and an extended mode algebra. The analysis yields ninety-five $B(\Delta_1-1+m,\Delta_2+1)$3anti$B(\Delta_1-1+m,\Delta_2+1)$4commutators for the single-particle contributions obtained by contour integrals on these multi-particle OPEs (Ahn, 6 Jul 2026). The generalized three-particle celestial OPE involves Beta-function coefficients and descendant sums, while the leading single-particle-exchange channel has the schematic form

$B(\Delta_1-1+m,\Delta_2+1)$5

(Ahn, 6 Jul 2026). The proposed $B(\Delta_1-1+m,\Delta_2+1)$6-particle generalization has leading singularity $B(\Delta_1-1+m,\Delta_2+1)$7 and mode coefficients of degree $B(\Delta_1-1+m,\Delta_2+1)$8 (Ahn, 6 Jul 2026).

A complementary $B(\Delta_1-1+m,\Delta_2+1)$9 construction describes the single-particle soft-current algebra itself as a supersymmetric $$R^{(k,1),a}\equiv \lim_{\varepsilon\to 0}\varepsilon\,H^{a}_{k+\varepsilon,+},$$0-type wedge with manifest $$R^{(k,1),a}\equiv \lim_{\varepsilon\to 0}\varepsilon\,H^{a}_{k+\varepsilon,+},$$1 symmetry. It states that the twenty five couplings in this celestial algebra can be written in terms of eight arbitrary couplings via the Jacobi identity (Ahn et al., 8 Sep 2025). The algebra includes graviton, gravitinos, graviphotons, graviphotinos, and scalars, and generalizes the classical $$R^{(k,1),a}\equiv \lim_{\varepsilon\to 0}\varepsilon\,H^{a}_{k+\varepsilon,+},$$2 extended superconformal algebra of Ademollo et al. into a manifest $$R^{(k,1),a}\equiv \lim_{\varepsilon\to 0}\varepsilon\,H^{a}_{k+\varepsilon,+},$$3 form (Ahn et al., 8 Sep 2025).

The subject also has variants beyond standard celestial amplitudes. Leaf amplitudes associated with an AdS$$R^{(k,1),a}\equiv \lim_{\varepsilon\to 0}\varepsilon\,H^{a}_{k+\varepsilon,+},$$4 foliation of flat spacetime realize the same infinite-dimensional soft $$R^{(k,1),a}\equiv \lim_{\varepsilon\to 0}\varepsilon\,H^{a}_{k+\varepsilon,+},$$5-algebra as celestial MHV amplitudes, despite not being translation invariant (Melton et al., 2024). In a different direction, conformal representation theory in general dimensions shows that the infinite local symmetry enhancement of CCFT$$R^{(k,1),a}\equiv \lim_{\varepsilon\to 0}\varepsilon\,H^{a}_{k+\varepsilon,+},$$6 is special: in $$R^{(k,1),a}\equiv \lim_{\varepsilon\to 0}\varepsilon\,H^{a}_{k+\varepsilon,+},$$7, the directly inherited charges from conformally soft operators are trivial, while non-trivial conserved charges arise from shadow transforms and are finite-dimensional (Pano et al., 2023). This places the standard two-dimensional celestial soft current algebra in a dimension-specific setting.

Several open structural issues are explicit in the literature. Branch-cut terms in celestial OPEs indicate new primary content and prevent a symmetry interpretation while remaining fully compatible with a consistent OPE (Ball, 2024). In non-abelian loop-level soft radiation, mixed-helicity OPEs suggested by multiple-emission currents involve coefficients depending on gluon energy fractions, break holomorphic factorization, and break associativity when double limits are taken; strongly-ordered soft limits recover associativity, but suffer from ambiguities already discussed in earlier literature (Magnea et al., 26 Dec 2025). A plausible implication is that the full celestial soft current algebra, beyond special sectors, is more general than a conventional local chiral current algebra and may require non-holomorphic, logarithmic, or multi-particle operator structures to encode the complete infrared data.