On infinite symmetry algebras in Yang-Mills theory

Similar to gravity, an infinite tower of symmetries generated by higher-spin charges has been identified in Yang-Mills theory by studying collinear limits or celestial operator products of gluons. This work aims to recover this loop symmetry in terms of charge aspects constructed on the gluonic Fock space. We propose an explicit construction for these higher spin charge aspects as operators which are polynomial in the gluonic annihilation and creation operators. The core of the paper consists of a proof that the charges we propose form a closed loop algebra to quadratic order. This closure involves using the commutator of the cubic order expansion of the charges with the linear (soft) charge. Quite remarkably, this shows that this infinite-dimensional symmetry constrains the non-linear structure of Yang-Mills theory. We provide a similar all spin proof in gravity for the so-called global quadratic (hard) charges which form the loop wedge subalgebra of w1+∞.


Introduction
It was shown in [1] that gravity and gauge theories possess an infinite tower of symmetries generated by increasingly subleading soft modes.For example, the negative helicity modes organize in finite dimensional representations of the SLp2, Rq L component of the Lorentz algebra (after analytic continuation to p2, 2q signature).In the case of pure Yang-Mills (YM) theory, the symmetry generators were related to the soft modes by a light-transform and were found to obey a simple algebra [2] 1 rS p,a m pzq, S q,b n pzqs " ´if ab c S p`q´1,c m`n pzq. (1) Here p, q are half integers bigger than 1 and m, n satisfy the restriction 1 ´p ď m ď p ´1, while m `p is restricted to be integer-and similarly for the pair pq, nq.These soft modes admit a further mode expansion2 [3] S p,a m pzq " The dictionary relating bulk asymptotic scattering states and operators in the celestial CFT [4,5,6] suggests that celestial symmetries like (1) should also be realized on the phase space of the theory.Progress in this direction was made in [7] where such a claim was established in Einstein gravity.Specifically, it was shown [8] that the asymptotic Einstein equations truncate to two towers of recursive differential equations for charges defined as appropriate combinations of the asymptotic metric components that transform covariantly under the homogeneous subgroup of the Weyl extension of BMS 4 [9].After appropriate regularization, half of the non-linear charges (whose linear part corresponds to the same helicity gravitons) were shown to obey the gravitational analog of (1), namely a w 1`8 algebra to linear order.Surprisingly, no restriction to the global subalgebra was necessary.One may therefore hope that the loop algebra continues to hold upon including higher non-linear contributions to the charges.
This paper extends the analysis of [7] to Yang-Mills theory, beyond the linear level.In particular, we construct for each helicity an infinite tower of charges R s pτ s q labelled by a Lie algebra valued function τ s pz, zq P g on the celestial sphere S (alternatively on the celestial torus T " S 1 ˆS1 ).Here pz, zq are complex coordinates on S (or circle coordinates on T ).It can be shown that τ s is an element of an SLp2, Cq (or SLp2, Rq ˆSLp2, Rq) representation of weight and spin p∆, Jq " p0, ´sq.These charges are constructed explicitly as operators acting on the Yang-Mills Fock space and are obtained after regularization from components of the gauge field in a large-r expansion.The latter satisfy a tower of recursive differential equations obtained from a large-r expansion of the Yang-Mills equations We prove that the charges R s pτ s q satisfy the algebra rR s pτ q, R s 1 pτ 1 qs " ´g2 YM R s`s 1 prτ, τ 1 s g q, (4) up to quadratic order in the creation and anihilation operators and for arbitrary functions τ on S (or T ).We label the Lie algebra bracket with a subscript g to avoid any confusion with the quantum commutators of operators.
The global S-subalgebra (1) appearing in celestial holography consists of the subalgebra generated by the τ s solution of D s`1 τ s " 0, where D is the covariant derivative with respect to z on S or T .The fact that this forms a subalgebra follows directly from the fact that D s`s 1 `1rτ s , τ s 1 s g " 0, when D s`1 τ s " 0 and D s 1 `1τ s 1 " 0. The algebra (1) and its modes (2) are recovered by choosing the smearing function on T to be a polynomial of degree s in z.More precisely, for τ s " z m`s 2 zn´s 2 T a , we have where r a s is the local charge aspect of spin s and T a is a basis element for the Lie algebra g.The main result of the present work is that (4) continues to hold for the local, nonlinear charges parameterized by arbitrary functions τ s pz, zq.In particular, at quadratic order, the commutator (4) receives contributions from the cubic component of the charges, rR s`p τ q, R s 1 `pτ 1 qs 2 " rR 2 s`p τ q, R 2 s 1 `pτ 1 qs `rR 1 s`p τ q, R 3 s 1 `pτ 1 qs `rR 3 s`p τ q, R 1 where R k s is the degree k term in the charge regarded as a polynomial in the gauge fields, The last two terms on the right-hand-side of (6) vanish for global τ s but conspire to ensure that the quadratic order algebra (4) is satisfied for arbitrary τ s .We also generalize the computation of the w 1`8 phase-space algebra in [7] to the global quadratic charges in gravity.Finally, we explicitly show that the local spin-2 charges also obey a w 1`8 algebra at the quadratic order.As in the YM case the inclusion of the cubic components in the charges is necessary to recover the correct commutation relations.
The relevance of the cubic component of the symmetry charge for the quadratic order w 1`8 algebra for spin-2 charges in the matter sector was already pointed out in [10].We are also aware of a forthcoming paper [11] which presents complementary results on higher-helicity fields.

Preliminaries
We consider non-Abelian gauge theory with gauge group G in 4-dimensional Minkowski spacetime.The Yang-Mills equations take the form d ‹ F " 0, F " dA `iA ^A, (8) where A " A µ dx µ is a one-form valued in the adjoint representation of the Lie algebra g of G.
We begin by describing the construction of the charge aspects in terms of the asymptotic phase space variables of Yang-Mills theory.We work in Bondi coordinates 3 where ds 2 " ´du 2 ´2dudr `r2 4dzdz p1 `|z| 2 q 2 , ( and assume an expansion of the field strength given by In the radial gauge A r " 0, this corresponds to the following fall-off conditions on the components of the gauge potential We further specify the gauge on the initial slice to be such that A p0q u " 0. The radiative field which carries information about the gluonic creation and annihilation operators is given 4 by A p0q z pu, zq.Our convention is such that the connection and field strength fields are hermitian.In other words we choose A µ " A a µ T a , where T a is a Hermitian generator satisfying the algebra rT a , T b s " if ab c T c . 5 The trace is normalized to TrpT a T b q " δ ab .The first few charge aspects are identified as the dominant elements in the radial expansion of the field strength: The higher spin charge aspects R s of conformal dimension and spin p∆, Jq " p2, sq are constructed recursively by solving a system of differential equations given by These evolution equations are consequences of the Yang-Mills evolution for R 0 and R 1 .For higher spin s ě 1 they correspond to a truncation of the full Yang-Mills equations expanded in 1{r.These equations parallel the ones extracted from the asymptotic Einstein equations in [7,8]. A complete derivation of this result starting from the Yang-Mills equations (8) will be provided elsewhere [14].The spin-0 charge is the leading while the spin-1 charge is the subleading one.The recursion relations (13) are formally solved in terms of R ´1 by 3 It can be convenient to work in retarded flat coordinates where ds 2 " ´2dudr`2r 2 dzdz and asymptotic infinity has the topology of a celestial plane.Removing the origin we get a celestial cylinder which can be compactified into a celestial torus T with respect to which we express the mode expansions (5). 4 In order to lighten the notation, we shall commonly indicate only z in the functional dependence of the fields on the coordinates on the sphere, e.g.A pnq z pu, zq, but it should be understood that in general the dependence is on both z, z. 5 Our conventions here are such that the structure constants differ by a sign compared to those in [12,13].
where AdpXqY " rX, Y s g denotes the adjoint action and pB ´1 u Oqpuq :" ş u `8 du 1 Opu 1 q for functionals that satisfy the boundary condition Op`8q " 0. This expression is non-linear in the radiation field A and it will therefore be convenient to expand the charge aspects as where R k s is homogeneous of degree k6 in the gauge fields A, A ˚.In the following we use that R k s " 0 unless k ď s `2.At linear order we simply have while the higher order components are recursively determined in terms of the lower order ones by It turns out that (17) suffer from divergences in the limit u Ñ ´8, the past boundary I `of future null infinity I `.This can be remedied by defining the renormalized charge aspects whose action on the corner phase space at I `is finite.The first two non-linear components of these charges will be central to our analysis and are given explicitly by (see Appendix A) It will prove useful to express these charge aspects in a discrete basis where all the integrals over I disappear, in analogy to the gravitational case recently considered in [15].We perform the discrete basis charge construction in Section 3 and use it in Section 4 to derive the algebra (4).
Motivated by the holographic calculation in [1], the linear component of the analogous commutator in gravity was first computed in [7].A simpler derivation will be given in Section 6 using the expansion of the asymptotic fields in terms of a discrete tower of modes recently derived in [15] (see also [16] for a complementary analysis) and reviewed in Sections 5 and 5.3, where the cubic charges are derived for the first time; in addition, we will also provide evidence for the gravitational w 1`8 loop algebra at quadratic order as well.

Discrete basis
In order to introduce the discrete basis, we first decompose the vector potential in terms the positive and negative energy fields defined as These fields enter the decomposition of the leading components A p0q and F p0q as follows7 A p0q z pu, zq" A `pu, zq `A˚p u, zq , We introduce also the Mellin transforms p A ˘p∆q and p F ˘p∆q " i p A ˘p∆ `1q of r A ˘pωq and r F ˘pωq respectively p A ˘p∆q :" This equation implies that By demanding the vector potential field to belong to the Schwartz space S [17], we can then introduce the YM memory observables These can be computed from the integrals ´8 du e iωu u n F p0q zu puq where in the first integral we take the limit ω Ñ 0 from above and in the second U is the upper half plane contour.The negative modes F ´pnq are defined by similar integrals but with F p0q zu replaced by F p0q zu .These memory observables can be understood as the coefficients in a Taylor expansion of r F ˘pωq around ω " 0, namely At the same time, the Goldstone fields are defined by evaluating p F ˘p∆q at positive integer ∆, namely They correspond to Taylor coefficients in the analytic expansion of A ˘puq around u " 0 Following the gravity analysis in [15], it can be shown that ( 27) and (30) form a basis for asymptotic gauge potentials that belong to the Schwartz space.

Phase space
The YM phase space at asymptotic infinity is characterized by the symplectic potential where Tr denotes the Cartan-Killing form for the Lie algebra associated to the YM theory.Modulo a canonical transformation, this can be rewritten as By means of ( 28), (31), we can rewrite the two symplectic potential components in terms of the YM memory and Goldstone modes as In the quantum theory, the only non-trivial commutator is then given by rF a ˘pm, zq, A b: ˘pn, z 1 qs " 2πg 2 YM δ ab δ n,m δ 2 pz, z 1 q. (35)

YM corner charges
In this section we rewrite YM higher spin charge operators in terms of the soft variables introduced in the previous section.We then compute their action on the discrete modes and review the connection with the celestial OPE [18,1].Finally we demonstrate that the global subalgebra of quadratic charges is precisely (4).

Charge aspects
All the charge operators of level k can be decomposed as sum of a positive helicity charge and the conjugate of a negative helicity charge operator according to The decomposition of the linear, quadratic and cubic charges follows straightforwardly from (19), (20), (21).One finds that r These charge aspects are valued in the Lie algebra g.

Charges
Given the charge aspects expressed as corner integrals of the memory and Goldstone variables, we introduce the symmetry charges labeled by Lie algebra valued generators of spin ´s denoted τ s " τ a s T a 8 R s˘p τ q :" ż S Tr pτ pzqr s,˘p zqq " Explicitly, the positive/negative helicity linear, quadratic and cubic charges read

Quadratic charge action
The action of the quadratic charge operator can be conveniently written in terms of the Lie algebra valued operator An essential property of this operator, proven in Appendix A.
From the expressions (42) and ( 46) for the charges we can evaluate the quantum commutator of charges when acting on the discrete fields.One finds that rR 2 s˘p τ q, F :b ˘pn, zqs " i ´sg 2 YM " P s pn `1; τ q, F : ˘ps `n, zq rR 2 s˘p τ q, A b ˘pn, zqs " i s g 2 YM rP s p´pn `1q; τ q, A `pn ´s, zqs b g .
The quantum commutators of R 2 s˘p τ q with A b ¯and F :b ¯obviously vanish.Given that A b `pn, zq9 lim ∆"1`n Âb p∆q and that F b `pn, zq9Res ∆"1´n Âb p∆q one can deduce from this the action on the Mellin transform of the asymptotic field.It is simply given by

Celestial OPE from charge action
From (49) we get that the commutator between the charge aspect and the Mellin transform of the radiative field is The correspondence between the Fock space commutator and the OPE is obtained through the identification Now given that we obtain the OPE Finally, one uses that F :a ˘ps, zq " ´iRes ∆"1´s Â:a ˘p∆q and the evaluation Res ∆"1´s Γp∆ ´1 `nq " p´1q s´n ps´nq! for n ď s, to see that the previous OPE is the residue at (55) This is the complex conjugate of the tree level OPE for positive-helicity gluons derived in [18, 1]. 9

Global charge
YM global charges are characterized by the condition By means of the generalized Leibniz rule 10 we conclude that, for the global charges, the second sum in (57) drops out and we can write the operator (44) simply as a conjugation where the product is simply the composition of the operations of differentiation and multiplication by τ s .This means that we can write the quantum commutator (50) in terms of the adjoint action defined around (14) as From this we see that the double quantum commutator action on Âp∆q is given by When τ and τ 1 are parameters of global symmetry we simply have that The antisymmetrization of this action with respect to s Ø s 1 gives the action of the commutator rR 2 s˘p τ q, R 2 s 1 ˘pτ 1 qs on Âp∆q.Using that rAdrτ 1 s, Adrτ ss " Adprτ 1 , τ sq, we thus obtain that the algebra of global charges satisfies for each helicity the global S-algebra: 9 Recall that the celestial operators O ∆ are related to Â:a 10 This follows from the expansion px `yq α " ř 8 n"0 pαqn n! x n y α´n valid when x ă y.
We can also conclude from our definition that the commutator of global charges of opposite helicities commute Since the total charge is the sum R s pτ q " R s`p τ q `R: s´p τ q we have that the charge algebra for R s is identical to (62).
On the torus T , the global algebra is a loop algebra parametrized by global charge parameters τ pn, mq " z m`s 2 zn´s 2 T a , where n, m P N and ´s 2 ď m ď s 2 .In this case we define S s˘p τ pn, mqq.This is the algebra revealed by [2] from the study of the OPE.It is the analog for Yang-Mills of the w 1`8 loop algebra.This global algebra also arises naturally in the study of self-dual Yang-Mills in the twistor formulation [19,20].
It is important to appreciate that on the complex sphere S 2 the set of global charges vanish if we insist that τ is a regular function on S 2 .11Non-trivial charges can be obtained by allowing for poles in τ at isolated points on the sphere and (56) will only hold away from these points.As we will see, the non-linear contributions to the charges such as (43) will be crucial in this case to ensure that the charge algebra closes.

YM corner algebra for the local charges
Given the charges derived in Section 3.2 and the commutator (35), we are now ready to compute their algebra at linear and quadratic order in the same helicity sector.We present the calculations for the positive helicity sector, however similar results hold for the negative one as well.

Linear order
In this section we compute the linear charge algebra in the positive helicity sector.We start by evaluating where in the first line we used the definition of the linear charge (41), in the second line we used (47), and in the third line we used (44).Integrating by parts and using the binomial expansion, we find where in the last line we shifted variables p Ñ p ´n, switched sums ř s n"0 ř s p"n " ř s p"0 ř p n"0 and evaluated the sum over n.
The linear contribution to the charge commutators is found by adding the term with s Ø s 1 , τ Ø τ 1 , namely This can be immediately evaluated by noting that the binomial coefficient is invariant under p Ñ ´p `s `s1 `1 while As a result, it follows that

Quadratic order
The quadratic commutator receives two types of contributions, namely We will show that, quite miraculously, the local contribution to the quadratic-quadratic charge commutator that spoils the algebra is precisely cancelled by the cubic-linear commutators.The remaining pieces of the cubic-linear commutators ensure that the global algebra (62) is promoted to a local one.In the following sections we evaluate the quadratic-quadratic and linear-cubic contributions.We present the main steps leading to the cancellation and defer the details to Appendix B.

Quadratic charge commutator
We start by computing the quadratic charge commutator where I 1 , I 2 arises from the action of R 2 s`o n F : `and A `using the charge actions (47) and (48).They read ¯, Tr ´rP s 1 pℓ `1; τ 1 q, F : `ps 1 `ℓqs g rP s p´ℓ ´1; τ q, A `pℓ ´sqs g ¯.
The transposition property (45) simply implies (after shifting ℓ Ñ ℓ `s in I 2 ) that It therefore suffices to evaluate I 1 and then antisymmetrize in ps, τ ; s 1 , τ 1 q.We have where in the second line we used the binomial expansion.We now change variables p Ñ p ´m ´n and perform the sum over m upon changing sums The different cases are worked out in Appendix B.1, the result being that the sum splits into two contributions where In (76) we defined the hypergeometric function The term J 1 is present for all ranges of admissible p, while the second term J 1 1 arises only for p ą n `s.This means that the term J 1  1 vanishes for global transformation parameters.We conclude that the commutator is

Cubic charge commutators
Next we compute the cubic-linear charge commutator where I 3 and I 4 are respectively associated with the commutator of R 1 s`i n (41) with A `pℓq and A `pmq in (43).For the first contribution, using that rR 1 s`p τ q, A `pℓqs " ´is 2πg 2 Y M δ s,ℓ D s`1 τ , we find After a straightforward series of changes of variables and sum switches that we detail in Appendix B.2, this can be shown to simplify to For the second contribution, direct binomial expansion yields After a short series of straightforward manipulations detailed in Appendix B.3 we find I 4 ps, τ ; s 1 , τ 1 q " J 4 ps, τ ; s 1 , τ 1 q `J1 4 ps, τ ; s 1 , τ 1 q, (83) where Tr ´rD n τ 1 , rD p´n τ, D s`s 1 ´pF : `ps `s1 `ℓqs g s g A `pℓq ¯(84) and `ps `s1 `ℓqs g s g A `pℓq ¯. ( Here we defined From this we conclude that rR 1 s`p τ q, R 3 s 1 `pτ 1 qs " I 3 ps, τ ; s 1 , τ 1 q `J4 ps, τ ; s 1 , τ 1 q `J1 4 ps, τ ; s 1 , τ 1 q. (87) From ( 81) and (82) it is easy to see that (79) vanishes provided that τ, τ 1 obey the global charge condition (56).

Full commutators
We can now put everything together.We first notice that, quite remarkably, the contributions from the hypergeometric functions cancel each other!In particular we find that due to the hypergeometric identity proven in Appendix B.4.It then follows that Tr ´rD n τ 1 , rD p´n τ, D s`s 1 ´pF : `ps `ℓ `s1 qs g s g A `pℓq ¯. ( This equality follows from the cancellations of sums In Appendix B.4 we show as well that the anti-symmetrization of J 1 `J4 under the exchange ps, τ q Ø ps 1 , τ 1 q simplifies into J 1 ps, τ ; s 1 , τ 1 q `J4 ps, τ ; s 1 , τ To evaluate the commutator at quadratic order we have to add this contribution to I 3 ps, τ ; s 1 τ 1 qÍ 3 ps 1 , τ 1 ; s, τ q given by (81).As a result, we find as shown in Appendix B.4 that After a series of straightforward manipulations described in Appendix B.4, the sum can be simply repackaged as ř s`s 1 p"0 ř p n"0 .The sum over n can be reabsorbed into D p rτ, τ 1 s using the Leibniz rule and we remarkably find that (93) reduces to rR s`p τ q, R s 1 `pτ 1 qs 2 " ´g2 YM R 2 s`s 1 ,`p rτ, τ 1 s g q. (94)

Gravity corner charges
Similarly to the YM case, in gravity the vacuum asymptotic Einstein's equations (EE) around null infinity can be recast as a set of recursive differential equations for higher spin gravitational charge aspects Q s given by [8,22,7] with Cpu, zq representing the shear field encoding radiation data, N " B u C ˚representing the news field and Q 0 the Bondi mass.The relation between (95) and the vacuum EE is exact up to s " 3 [23]; for s ě 4 corrections in higher powers of the shear field are expected to appear.Initially neglecting those corrections-that do not affect the linear order same helicity algebra-it was shown in [7] that the dynamical system defined by (95) provides a representation of the w 1`8 loop algebra at linear order.This established a direct relation between the celestial OPE [24,18,25,1] of two conformal primary gravitons in the collinear and soft limit with the commutator action of the quadratic order (hard) charge contribution on the shear field.This clarified the gravitational origin of the w 1`8 symmetry originally discovered through celestial OPE techniques in [1,2].
In this second part of the paper, we are going to employ the newly introduced discrete basis for celestial holography [15] in order to investigate the fate of such symmetry beyond the linear level.

Discrete basis and phase space
We introduce the shear decomposition with the positive and negative helicity graviton components C ´puq :" 1 2π Their Fourier and Mellin transforms are respectively given by r C `pωq :" Similarly, we can decompose the news field N puq " N ´puq `N ˚puq, where N ˘puq :" B u C ˘puq.
The the higher spin, positive and negative energy memory observables are then defined as These can be conveniently written also as Note that, in analogy with the YM case, we also have12 The memory observables M ˘pnq provide a Taylor expansion coefficients of r N ˘pωq around ω " 0, as On the other hand, by evaluating the news Mellin transform at positive integer conformal dimension ∆ " n, we obtain the Goldstone operators The Goldstone modes provide a Taylor expansion of C ˘puq around u " 0, as The gravitational phase space at asymptotic infinity is characterized by the radiative symplectic potential [26,27,28 with κ " ?32πG.This can be decomposed (up to a canonical transformation) into positive and negative helicity components Θ GR " Θ GR ``Θ

GR
´, each parametrized by the respective infinite tower of memory and their conjugate (complex conjugate) Goldstone operators as At the quantum level then, the only non-trivial commutators are rM ˘pn, zq, S : ˘pm, z 1 qs " πκ 2 δ n,m δ 2 pz, z 1 q. (108)

Charge aspects
The charge aspects solving (95) can again be expanded in powers of radiation fields as At a given order k in powers of radiation fields, the renormalized aspects can be expressed as qk s pu, zq " As clear from the expression above, the higher spin charges can be recursively expressed as a nested product of integrals over I.The discrete basis introduced in [15] allows one to eliminate all the time integrals and obtain expressions for the charges as a single integral over a corner at arbitrary value of retarded time u.In the rest of the paper we will concentrate on the case u " 0, but formulas for generalization to arbitrary u " u 0 can be found in [15].Let us first provide a brief review of the main ingredients of the new discrete basis.

Charges
As shown in [7,15], the renormalized higher spin charges in gravity are defined as where q s pzq " lim uÑ´8 qs pu, zq.
By performing a decomposition into positive and negative helicity components, for each order we can define The expressions of the positive and negative helicity parts in terms of the memory and Goldstone variables Mpnq, Spnq for the linear and quadratic orders were derived in [15] and are respectively given by and Q 2s ´pτ q " ´1 4π In Appendix C we compute the cubic charges in the discrete basis and their action on the Goldstone operators.These are given respectively by

Gravity corner algebra
As an application of the new discrete basis, we can verify that the expressions (114), ( 115), (116) for the linear and quadratic charges in terms of the memory observables and the Goldstone modes reproduce the Lw 1`8 symmetry (loop) algebra at linear order, as previously computed in [7].Furthermore, we will exploit the computational advantages of the new basis to prove the validity of the Lw 1`8 loop algebra also at quadratic order, when restricting to wedge sector, and in the general case of local charges for the choice of spins s " s 1 " 2 .More precisely, the wedge subalgebra W Lw 1`8 Ă Lw 1`8 is characterized by the following restriction of the transformation parameters D s`2 τ s pzq " 0 . (120) Note that the linear (soft) charges vanish for this choice of parameters.While on the plane, solutions to (120) are polynomials of degree s `1 in z, on the sphere (120) admits no non-trivial global solutions.Instead, (120) can only hold away from points z s where D s`2 τ s pzq " D p δpz ´zs q.
The corresponding charge aspects are associated to the global components (in a spherical harmonic decomposition) Ψ ps´2q 0 in the asymptotic expansion of the Ψ 0 Weyl scalar (see [7] for more details on this relation).These also represent the relevant symmetry sector of the twistor formulation of self-dual gravity [29].

Linear order commutator
As shown in Appendix D.1, the new basis considerably simplifies the calculations and, by means of the commutation relations (108), it allows us to recover the Lw 1`8 loop algebra for the positive helicity piece of the charges (113).Explicitly, the commutator at linear order yields rQ s pτ q, Q s 1 pτ 1 qs 1 `" rQ 1 s`p τ q, Q 2 The same result holds for the negative helicity piece.Some of the intricacies for the mixed helicity sector were pointed out in [7].We expect the computational simplifications brought along by the new basis to help investigate them.

Quadratic order commutator of global charges
In order to compute the quadratic order of the bracket rQ s pτ q, Q s 1 pτ 1 qs in the global sector (denoted by the subscript G), we use the Jacobi relation Let us introduce the useful operatorial relation which can be proven in terms of the generalized Leibniz rule (57).For the global charges (120), this reduces to (see Appendix D.2) and we have the quadratic charge global action rQ 2 s`p τ q, S `pn, zqs G " ´p´q s κ 2 4 D n`2 rps `1qτ D `ps ´n ´2qDτ s D s´n´3 S `pn ´s `1, zq.(125) From this we obtain (see Appendix D.2) where rτ, τ 1 s :" ps `1qτ Dτ It can easily be checked (see again Appendix D.2) that this matches exactly the action ´κ2 4 " ps 1 `1qrQ 2 s`s 1 ´1,`p τ 1 Dτ q, S `pn, zqs ´ps `1qrQ 2 s`s 1 ´1,`p τ Dτ 1 q, S `pn, zqs where notice that in this case we do not need to restrict to the global charges.Hence, from the Jacobi relation (122), we immediately obtain the quadratic commutator of the global charges 6.3 Quadratic order commutator of the local charges s " s 1 " 2 As a final step of this paper towards the full proof of the validity of the local Lw 1`8 algebra, we show it here for the simpler case s " s 1 " 2. We begin with general considerations and then specialize to this restriction on the spins.Consider the Jacobi identity rQ s`p τ q, rQ s 1 `pτ 1 q, S `pn, zqss ´rQ s 1 `pτ 1 q, rQ s`p τ q, S `pn, zqss " rrQ s`p τ q, Q s 1 `pτ 1 qs, S `pn, zqs, which at quadratic order gives rQ 2 s`p τ q, rQ 2 s 1 `pτ 1 q, S `pn, zqss ´rQ 2 s 1 `pτ 1 q, rQ 2 s`p τ q, S `pn, zqss `rQ 1 s`p τ q, rQ 3 s 1 `pτ 1 q, S `pn, zqss ´rQ 1 s 1 `pτ 1 q, rQ 3 s`p τ q, S `pn, zqss " rrQ s`p τ q, Q s 1 `pτ 1 qs p2q , S `pn, zqs , where we used the fact that rQ 3 s 1 `pτ 1 q, rQ 1 s`p τ q, S `pn, zqss " 0, (132) and rQ s`p τ q, Q s 1 `pτ 1 qs p2q " rQ 2 s`p τ q, Q 2 As shown in the derivation of (129), the restriction (120) to the wedge sector of the nested commutators involving quadratic charges in ( 131) is sufficient to yield the desired result.Therefore, the goal is to show that the remaining contributions (the 'remainders') to those commutators are cancelled exactly by those on the LHS of (131) involving linear and cubic charges.

Conclusions
In this paper we extracted a tower of non-linear operators from the asymptotic Yang-Mills equations and demonstrated that they form a representation of a higher-spin symmetry loop algebra on Fock space.This algebra contains a global subalgebra, which at the linear order is the phase space realization of the celestial algebra of soft gluon operators found in [1,2].Remarkably, we found that this algebra admits a local enhancement and continues to hold at quadratic order upon inclusion of new cubic terms as dictated by the equations of motion.The steps leading to this result involved a series of miraculous cancellations, which we believe hints at a deeper connection between symmetry and the asymptotic Yang-Mills equations.It would be great to find an elegant way of deriving the loop algebra (4) directly from the recursive towers of non-linear differential equations (3), perhaps by employing or generalizing the methods of [30].From a physical perspective, the implications of the infinite-dimensional symmetry for scattering remain rather unclear.As a first step in this direction, one should try to understand the signatures of the cubic and higher order components of the charges in scattering amplitudes.Moreover, the full Yang-Mills equations will include further non-linear corrections which deserve a better understanding (see [14] for progress in this direction).It would be very interesting to understand in what way these corrections, as well as coupling to matter, would affect the symmetry structures found in this work.
Central to this work was the algebra of quadratic charges, also known as hard charges.On the other hand, the celestial symmetry algebras of [1,2] were associated with soft operators.We would like to have a better understanding of the dictionary between symmetry generators in celestial CFT and realization of the symmetry algebras on the bulk Fock space.
In gravity, the non-linear charges were shown to correspond to the higher multipole moments of the gravitational field and hence directly related to gravitational observables such as the memory effect [31,32].It would be fascinating to explore the role of symmetry in constraining observables of gauge theory and gravity.We leave this to future work.

A YM charges
From the general expression (17) for the YM higher spin charge aspects, the quadratic and cubic charge aspects read respectively and Let us introduce the Leibniz rule for pseudo-differential calculus where we used For k " 1, the renormalized charges (18) can be expressed as where in the last line and below we recall that B ´1 u " ş u 8 du.For k ě 2, we can rewrite from which where we used (143).
From the general expression (148), we can also write the recursion relation where in the first line we applied the binomial expansion and switched the sums over ℓ and n.

A.1 Corner charge aspects
Let us first of all list the useful the relations valid @ n ě 0 , α P Z and where we defined the operator p ∆ :" B u u and the requirement of the potential field to be Schwartzian in order to integrate the p ∆ contributions to zero.The quadratic charge aspects (20) can be expressed as In the manipulations above we used u s`ℓ´n pB ´1 u q n " u s`ℓ p ∆ `n ´1q ´1 n " p ∆ `n ´s ´ℓ ´1q ´1 n u s`ℓ , which follows from the list (152), and the identity 1 pn ´s ´ℓ ´1q n " p´1q n ps `ℓ ´nq! ps `ℓq! .
We now perform this series of manipulations: we replace k Ñ k ´1, we then switch sums ř s n"1 where in the second equality we used the cyclicity of the trace.
In terms of the operator (44), the quadratic charges can be written as R 2 s`p τ q " ´p´iq s 2π where we used Using the identity (165), this can also be written as R 2 s`p τ q " ´is 2π The quadratic charge action on F :b `pn, zq can then be computed from the commutator (35) as rR 2 s`p τ q, F :b `pn, zqs " γ rA a `pℓ, z 1 q, F :b `pn, zqs " P s pℓ `1; τ q, F : `ps `ℓ, z 1 q ı ga " g 2 YM i ´s " P s pn `1; τ pzqq, F : `ps `n, zq

B YM algebra
In this appendix we provide the proofs of the results presented in Section 4.2.We make use of several identities involving the falling factorial First we use that p´x ´1q n " p´1q n px `nq n .
We make use of the fundamental binomial identity for the falling factorial where a, b P N. Finally taking x to be a negative integer gives the identity

B.1 Quadratic charge commutator
In this appendix we present a detailed computation of I 1 in Section 4.2.1.After changing variables p Ñ p ´m ´n (73) becomes In this parametrization we see that the integral factor does not depend on m.The goal is therefore to perform the sum over m.We can do that after interchanging sums The sum involves the binomial coefficient ˆs1 ´n p ´n ´m˙w hich vanishes when m ă p ´s1 .This means that we can replace the lower bound m " maxr0, p ´s1 s simply by m " 0. Therefore we have two cases to evaluate.where we defined and 3 F 2 is the generalized hypergeometric function.
We conclude that the final result of (182) includes two contributions.We will denote the common contribution to the two cases, namely the one proportional to ˆℓ `p p ´n˙b y J 1 .We then find that I 1 " J 1 `J1 1 where Note that the sum over n in J 1 1 only goes up to s 1 ´1 due to the fact that p ě n `s `1.In the first equality we used that

B.2 First term in cubic algebra
In this appendix we evaluate I 3 .We start with (80) (with s Ø s 1 and τ Ø τ 1 ) and shift or equivalently where we defined This equality which is similar to (185) can also be checked by direct evaluation of the sum in (198) with mathematica.We recover (84) from the first term in (203), while for (85) we use the second term in (203).In total we obtain (83).In both cases we acquire a sign upon cycling the terms in the trace and reordering a commutator.

B.4 Putting everything together
We first prove an important identity relating hypergeometric functions.This will allow us to show that the hypergeometric functions appearing in I 1 and I 4 cancel.The relevant hypergeometric identities are Repb 1 `b2 ´a1 ´a2 ´a3 q ą 0, Repb 1 ´a1 q ą 0, where pαq pnq " Γpα `nq{Γpαq is the raising factorial.The second of these can be used to rewrite F 1 ps, s 1 q " ps `2 ´1q pp´1´n´sq ps `2q pp´1´n´sq 3 F 2 r1, 1 ´p `s `n, ´p ´ℓ; 2 ´p `s `s1 , 1 `n ´p; 1s. ( 206) We could apply it because 1 `n ´p `s ă 0 for the relevant summation range.We now use the first identity in (205) to rewrite F 1 ps, s 1 q " ps `2 ´1q pp´1´n´sq ps `2q pp´1´n´sq pn ´pq p1 `s1 `ℓq ˆ3F 2 r1, 1 `s1 ´n, 2 `s `s1 `ℓ; 2 ´p `s `s1 , 2 `s1 `ℓ; 1s " ps `1q pp ´nq pn ´pq p1 `s1 `ℓq F 2 ps, s 1 q " ´ps `1q p1 `s1 `ℓq F 2 ps, s 1 q, which is exactly what we need for these terms to cancel in the sum.

C Gravity cubic charges
We focus on the positive helicity charge (same result can be derived for the negative one).We start from the general relation (110) to write the cubic charge as We now use (105) again together with and the expression for the quadratic charge to write We can now introduce the operator p ∆ :" B u u, which integrates to zero due to our choice of boundary conditions, and use the property pu ´1q ℓ´s´n pB ´1 u q ℓ´m´1 " u s`n´m´1 pu ´1B ´1 u q ℓ´m´1 " u s`n´m´1 p ∆ `ℓ ´m ´2q ´1 ℓ´m´1 " p ∆ `ℓ ´s ´n ´1q ´1 ℓ´m´1 u s`n´m´1 , pu ´1q m`1´s´n´k pB ´1 u q m´1 " u s`n`k´2 p ∆ `m ´2q ´1 m´1 " p ∆ `m ´s ´n ´kq ´1 m´1 u s`n`k´2 , (217) from which rQ s pτ q, Q s 1 pτ 1 qs 1 `" rQ 1 s`p τ q, Q 2 s 1 `pτ 1 qs `rQ 2 s`p τ q, Q 1 s 1 `pτ 1 qs " κ 2 4 " ps 1 `1qQ 1 s`s 1 ´1`p τ 1 Dτ q ´ps `1qQ If we demand that D s`2 τ s " 0, (232) we see that the sum can be restricted to the range s ď n.In this case all the derivative operators appear with a positive power.There is no longer any non-locality.This means that the charge action (117) can therefore be simply written as (125).

" ´κ2 4 p´q s D n` 2 z" κ 4 p´q s`s 1 16 D n` 2 z
rps `1qτ D z `ps ´n ´2qDτ s D s´n´3 z rQ 2 s 1 ,`p τ 1 q, S `pn ´s `1, zqs G rps `1qτ D z `ps ´n ´2qDτ s rps 1 `1qτ 1 D z `ps `s1 ´n ´3qDτ 1 s ˆDs`s 1 ´n´4 z S `pn ´s ´s1 `2, zq , s κ 2 4 s´k S `pn ´s `1, zq, (135) which is the complementary contribution to the action (117) when the global condition (120) is relaxed (see derivation of (125) in Appendix D.2).The two remainder contributions (134) are computed in Appendix D.3 and they are respectively and we replace n Ñ s ´n to obtain STr prP s pℓ `1; τ q, Bpzqs g Apzqq This identity can be proven by recurrence from the shift identity satisfied by the falling factorial ∆pxq n " npxq n´1 , ∆f pxq :" f px `1q ´f pxq (177)and the normalisation conditions p0q n " δ n0 .This identity is valid for ´s P CzN and it can be shown to be equivalent to the Gauss hypergeometric identity.When s P N it becomes more simply where b P N. Taking x to be a positive integer and interchanging n Ñ s ´n gives the identities Case I: When p ´n ď s we find Ñ ´ps 1 `ℓ `1q, y Ñ ps 1 ´nq and s Ñ pp ´nq.
1s`s 1 ´1`p τ Dτ 1 q In this section with give the proof of the relation (126).We start with the proof of (124) for global charges.Using the generalised Leibniz rule (57), we find that αD α´1 pDτ D s´α q " Summing these two contributions, we find that we have the key identity D α´1 rps `1qτ D `ps `1 ´αqDτ s D s´α " n τ qD s´n .