w 1+ ∞ and Carrollian Holography

: In a 1+2D Carrollian conformal field theory, the Ward identities of the two local fields S +0 and S +1 , entirely built out of the Carrollian conformal stress-tensor, contain respectively up to the leading and the subleading positive helicity soft graviton theorems in the 1+3D asymptotically flat space-time. This work investigates how the subsubleading soft graviton theorem can be encoded into the Ward identity of a Carrollian conformal field S +2 . The operator product expansion (OPE) S +2 S +2 is constructed using general Carrollian conformal symmetry principles and the OPE commutativity property, under the assumption that any time-independent, non-Identity field that is mutually local with S +0 , S +1 , S +2 has positive Carrollian scaling dimension. It is found that, for this OPE to be consistent, another local field S +3 must automatically exist in the theory. The presence of an infinite tower of local fields S + k ≥ 3 is then revealed iteratively as a consistency condition for the S +2 S + k − 1 OPE. The general S + k S + l OPE is similarly obtained and the symmetry algebra manifest in this OPE is found to be the Kac-Moody algebra of the wedge sub-algebra of w 1+ ∞ . The Carrollian time-coordinate plays the central role in this purely holographic construction. The 2D Celestial conformally soft graviton primary H k ( z, ¯ z ) is realized to be contained in the Carrollian conformal primary S +1 − k ( t, z, ¯ z ). Finally, the existence of the infinite tower of fields S + k is shown to be directly related to an infinity of positive helicity soft graviton theorems.


Introduction
The thriving research program to understand the holographic principle [1,2] for the case of the 1 + 3D asymptotically flat space-times (AFS) has been approached mainly via two1 seemingly different avenues.The currently most well-developed one is known as the Celestial holography [3] where the dual Celestial CFT [4] is thought to live on a Celestial sphere S 2 at the null-boundary of the AFS.The Celestial conformal fields depend on the two stereographic coordinates (z, z) on the Celestial S 2 and are labeled by a Celestial conformal weight ∆ c which is a continuous parameter [5]; thus, these fields effectively are functions of three variables.In the other framework known as the Carrollian holography, gaining attention in the recent years [6][7][8][9][10][11][12], the dual Carrollian CFT (CarrCFT) resides on the three-dimensional null-infinity (with topology R×S 2 ) where the Carrollian conformal fields also depend on three variables: (z, z) as well as the null-coordinate or the Carrollian time t.
The central idea of the Celestial holography [4,5] is that the null-momentum space scattering amplitude of a mass-less scattering process in the 1 + 3D bulk AFS can be re-expressed as a 2D Euclidean CFT correlation function, via a Mellin transformation that trades the energies {ω} of the external bulk scattering particles for the (continuous) Celestial conformal weights {∆ c }.Even before the explicit examples presented in [4,5], that the holographic dual of the (quantum) gravitytheory in the 1 + 3D AFS might be a 2D Euclidean CFT was partly motivated by the realization that the Weinberg (leading) soft graviton theorem [13] and the Cachazo-Strominger subleading soft graviton theorem [14] can be respectively recast into the 2D CFT U(1) Kac-Moody [15] and the 2D CFT energy-momentum (EM) tensor Ward identities [16].These works were in turn inspired by linking up the following four observations: that the asymptotic symmetry group at the null-infinity of the 1 + 3D AFS is the BMS 4 group [17,18] in the presence of the gravitational radiation, that the Weinberg soft graviton theorem [13] is completely equivalent to the BMS 4 super-translation Ward identity [15,19], that the Lorentz SL(2, C) subgroup of the original BMS 4 group should be infinitely enhanced to include the 2D singular conformal transformations (super-rotations) on the Celestial S 2 [20][21][22] and finally, that the Cachazo-Strominger subleading soft graviton theorem [14] implies a Virasoro symmetry of the 1 + 3D AFS gravitational S-matrix [23].
On the other hand, the starting point of the Carrollian holography is the observation [24,25] that the original BMS 4 group [17,18] is isomorphic to the 1 + 2D conformal Carroll group (at level 2) on a Carrollian manifold with topology R × S 2 (with S 2 equipped with the round metric).Inspired by this isomorphism, it was shown in [10] how the EM tensor Ward identities of a 1 + 2D source-less CarrCFT on a Carrollian background with topology R × S 2 with assumed Weyl invariance can encode the 1 + 3D bulk AFS leading [13,19] and the subleading [14,23] soft graviton theorems as well as the 2D Celestial CFT EM tensor [16] and the (BMS 4 ) super-translation Kac-Moody [15] Ward identities.
In addition to the leading [13] and subleading [14] soft graviton theorems, another soft graviton theorem at the subsubleading order was found in [14] explicitly for tree-level Einstein gravity in the 1 + 3D bulk AFS.But, unlike the soft theorems at the two more leading orders, this subsubleading soft theorem is not universal [26,27].Equivalently in the Carrollian picture, the fact that the special Carrollian conformal fields S + 0 , S + 1 and T , whose Ward identities contain the two universal AFS soft graviton theorems can be constructed purely out of the CarrCFT EM tensor [10], points towards an universality in the sense that every CarrCFT is expected to have an EM tensor.Consistently, the non-universal AFS subsubleading soft graviton theorems were not captured by those CarrCFT Ward identities.
In this work, we aim to understand how the 1 + 3D bulk AFS subsubleading soft graviton theorem [14] can be encoded in the framework of the 1 + 2D CarrCFT.Obviously, to provide a holographic description of a non-universal phenomenon in the bulk, the dual boundary theory also must possess some non-universal features.For this purpose, taking cue from the field contents of a 2D Euclidean CFT [28] or a 1 + 1D CarrCFT [29] enjoying infinite additional symmetries, we postulate that, in a 1 + 2D CarrCFT that carries the imprint of the positive-helicity subsubleading soft graviton theorem, there is a local quantum field S + 2 in addition to the three universal local generators S + 0 , S + 1 and T .To avoid the potentially problematic hologram of the bulk ambiguity associated with the double soft-limits of opposite helicities [30], we do not attempt to simultaneously describe the negativehelicity subsubleading soft theorem in this work.By ignoring one helicity, we are not inviting inconsistency per se -we are focusing only on the holomorphic-sector of the 1 + 2D CarrCFT; so, our holographic analysis will correspond only to the positive-helicity sectors of the bulk theories of gravitons.An exception is the MHV sector [31] of the tree-level Einstein gravity, where the opposite helicity soft sectors decouple and our conclusions are rendered applicable to both sectors (with trivial modifications).
Following the methods, elaborated in [10], to completely determine the singular parts of the mutual operator product expansions (OPEs) of the generators S + 0 and S + 1 , we derive the mutual OPEs of S + 2 and the two aforementioned generators.This algorithm only makes use of the general Carrollian symmetry principles and the OPE commutativity property under the assumption that there are no time-independent (in the OPE limit) local fields with negative Carrollian scaling dimension in the CarrCFT.Similar symmetry arguments (together with similar assumptions) fixed the singularities of the mutual OPEs of the symmetry generators in the cases of 2D Euclidean CFT in [28,32] and of 1 + 1D CarrCFT in [29,33].
While trying to find the S + 2 S + 2 OPE, we realize that for this OPE to be consistent, another local Carrollian conformal field S + 3 must automatically appear in the theory.Extending the algorithm, we find in general that for the CarrCFT OPE S + 2 S + k≥2 to be consistent, there must already exist an infinite tower of local fields S + k+1 (t, z, z).The general CarrCFT OPE S + k S + l (4.21) is the main result of this work.
From this general S + k (x)S + l (x p ) OPE (4.21), we immediately recover the 2D Celestial conformal OPE of two conformally soft [34,35] primary gravitons H 1−k (z, z)H 1−l (z p , zp ) of [36], just by inspection.This conformally soft graviton OPE in [36] was directly obtained by taking the conformally soft limit of the general OPE between two general 2D Celestial conformal primary gravitons of arbitrary weights, derived in [37].But the derivation of the general graviton primary OPE in [37] was explicitly for the tree-level (linearized) Einstein theory in the bulk AFS and required some hints from this specific bulk theory to fix the singular structure of the said OPE.In another method, this OPE was obtained via a Mellin transformation from the bulk graviton scattering amplitude in the collinear limit in [36,37].
It is to be emphasized that we have not obtained the general Celestial conformal primary graviton OPE of [37] since that is a theory-specific result.Rather, we have given a completely holographic, Carrollian conformal symmetric derivation of the OPEs between the symmetry generators [36] that is expected to contain the asymptotic symmetry algebra of any (quantum) gravity theory in the 1 + 3D bulk AFS.Another important difference is that while the above mentioned Celestial CFT OPEs in [36,37] are valid only at tree-level Einstein theory (but exact for quantum self-dual gravity [38]), the Carrollian conformal derivation of the S + k S + l OPE (under the aforementioned assumption) involves no (Carrollian) perturbation theory analysis; its starting point is the CarrCFT Ward identities derived in [10] using a Carrollian path-integral formalism [39].
Using the 1 + 2D CarrCFT OPE ←→ commutation-relation prescription developed in [10], we find that the (local) symmetry algebra manifest in the CarrCFT OPE (4.21) is the Kac-Moody algebra of the wedge subalgebra [40] of the w 1+∞ algebra [41], in perfect agreement with the symmetry algebra derived in [42] from the Celestial conformally soft primary graviton OPE of [36].
Finally, we shed light on the direct connection between the existence of the infinite tower of Carrollian fields S + k and an infinity of the soft graviton theorems discussed in [43,44] in the context of the tree-level (linearized) Einstein theory.We find that the Ward identity of the Carrollian field S + 2 does indeed encode up to the positive helicity subsubleading soft graviton theorems [14].While it is hinted that the Ward identities of the other fields S + k>2 contain the soft theorems in more subleading orders, it is also demonstrated that the fields beyond S + 2 does not generate any new (independent) global symmetries of the theory.This is the CarrCFT analogue of the fact that the Celestial conformally soft gravitons H k<−1 do not impose any new constraints on the 1 + 3D bulk AFS S-matrices [36].
The rest of the paper is organized as follows.In section 2, we review the main results of [10] on the universal features of a 1 + 2D CarrCFT.We introduce the Carrollian conformal field S + 2 whose Ward identity (supposedly) encodes the 1 + 3D bulk AFS positive-helicity subsubleading soft graviton theorem, in section 3. We state the assumptions on the field content of the CarrCFT in section 3.1 and find that for the S + 2 S + 2 OPE to be consistent, a local field S + 3 must already exist in the theory.Our purely symmetry-based algorithm to find the OPEs reveal the automatic existence of the infinite tower of Carrollian fields S + k≥3 in section 4. We obtain the general S + k S + l OPE in section 4.3, from which the quantum symmetry algebra is determined in section 5. Finally, in section 6 we relate the infinite tower of fields S + k≥2 with an infinite number of soft graviton theorems before concluding in section 7.

Review
In [10], it was shown how the EM tensor Ward identities of an honest (i.e.source-less) Carrollian CFT on a 1 + 2D flat Carrollian background (with topology R × S2 and the assumption of the Weyl invariance) can be recast into the forms resembling to the 1 + 3D bulk AFS leading and subleading conformally soft [34,35] graviton theorems [31,45].The 1 + 2D Carrollian conformal fields S ± 0 and S ± 1 containing respectively the leading and the subleading conformally soft graviton primaries of the 2D Celestial CFT, as well as the fields T and T that contain respectively the holomorphic and the anti-holomorphic components of the 2D Celestial CFT EM tensor [16] were constructed purely out of the Carrollian EM tensor components T µ ν .Below we note the generic Ward identities 2 of the 1 + 2D CarrCFT EM tensor, derived in [10] using the Carrollian path-integral formalism [39]: where X is a string of Carrollian conformal primary multiplets; (ξ i ) p is the Carrollian boost (in the i-th direction) representation-matrix, s p is the spin (i.e. the eigenvalue of the spatial rotation) and ∆ p is the Carrollian conformal weight of the p-th primary field.

2.1
The fields S ± 0 Subtraction of the spatial divergence of (2.2) from (2.1) ν=t leads to: Choosing the following initial condition: the solution to the above temporal partial differential equation is obtained as: The S 2 contact-term singularities in (2.7) can be converted into pole singularities (but avoiding branch-cuts) if we note that: Inverting the ∂2 operator in (2.8) and the ∂ 2 operator in (2.9), the Carrollian fields S ± 0 were respectively defined in [10] as: where the descendant fields P = ∂S + 0 and P = ∂S − 0 consist respectively of the modes generating the holomorphic and the anti-holomorphic super-translations.The scaling dimension ∆ and the spin m of the fields S ± 0 are (∆, m) = (1, ±2).So, the defining relations (2.10) and (2.11) imply that S ± 0 are the 2D shadow-transformations (on S 2 ) of each other [46] by construction.Since, a field and its shadow (being a highly non-local integral transformation) can not both be treated as local fields in a theory [47], only one among S ± 0 is to be chosen as a local field while relegating the other merely to its non-local shadow.
We treat S + 0 as the local field and S − 0 as its shadow in this work.This corresponds to investigating the holomorphic sector of the 1 + 2D Carrollian CFT.The S + 0 Ward identity, for a string X of n mutually local Carrollian conformal primary multiplet fields {Φ p (t p , z p , zp )}, reads [10]: where ξ, ξ := ξ x ± iξ y denote the matrix-representation of the classical Carrollian boost under which a Carrollian multiplet Φ transforms3 .
The above correlator was derived from the (Carrollian) super-translation Ward identity in [10].The temporal step-function appearing in this Carrollian correlator captures the essence of the 1 + 3D bulk AFS super-translation memory effect [50,52].Consistently, temporal-Fourier transforming (2.12) to (positive) ω-space and then making the S + 0 field energetically soft [50], one recovers the Weinberg (leading) positive-helicity soft graviton theorem [13,19] as the residue of the leading 1 ω pole when all of the primaries in X have ξ = ξ = 0 [10].Thus, explicitly at the t → ∞ limit, the Ward identity (2.12) is same as the positive-helicity leading conformally soft graviton theorem [45] when all ξ p = ξp = 0.
The S 2 stereographic coordinates z and z are now treated as independent variables [53] so that terms like (z−zp) r (z−zp) s with r ≥ 0, s ≥ 1 have (meromorphic) pole singularity (avoiding phase ambiguity when r = s).Together with the form of the Ward identity (2.12), this suggests that, inside the correlator, S + 0 can be decomposed as [31]: These relations reminisce holomorphic Kac-Moody like Ward identities in a 2D Euclidean CFT.Clearly, P i and S + 0 have the same holomorphic weight h = 3 2 .Let us next discuss on the Carrollian conformal OPEs.As explained in [10,33], it is convenient to convert the temporal step-function appearing in the correlators like (2.12) into a jϵ-prescription for this purpose, with j being a second complex unit.The starting point is to hyper-complexify the (t, z, z) coordinates as below: ẑ := z + jt ; ẑ := z + jt ; t := t While z is a complex number on the x − y plane, ẑ can be thought of as a complex number on a y = ax + b plane of the 3D t − x − y space.t > 0 is the upper-half of this plane.

It can be shown that all of
ẑ, ẑ can be treated as independent variables.Thus, in most cases, we choose the point of insertion of a Carrollian field to be at ( t, ẑ, ẑ) = (t, z, z).E.g. the jϵ-form of the Ward identity (2.12) is (with ∆z p := z − z p − jϵ(t − t p ) ): The main application of this jϵ-prescription is to establish the relation between the CarrCFT OPEs and the corresponding commutation relations while allowing for a straightforward utilization of the OPE commutativity property.
It is important to remember that 1 ∆zp reduces to 1 z−zp only when t − t p > 0 and to 0 when t − t p < 0 in the sense of distributions [33] encoding the property of the temporal step-function.Thus, 1  ∆zp ≡ 1 z−zp when t → ∞, and 1 ∆zp ≡ 0 when t → −∞, thus providing a justification to the initial condition (2.6).
The OPE4 of S + 0 with a general (non-primary) Carrollian conformal field Φ (that is mutually local with S + 0 ) is noted below [10]: Thus, subtraction of ∂ z ⟨T z z (x)X⟩ from (2.1) ν=z results into: Choosing an initial condition similar to (2.6), one obtains the following solution to the above temporal partial differential equation: (2.17) Its 'complex-conjugated' version can be obviously derived in an exactly similar way.
The S + 1 Ward identity with a string of mutually local Carrollian conformal primaries was obtained as [10]: that for all ξ p = ξp = 0 resembles (in the limit t → ∞) the positive-helicity subleading conformally soft graviton theorem as presented (but very differently interpreted) in [31].More appropriately, the Ward identity (2.20) is the 1+2D Carrollian conformal manifestation of the 2D Celestial subleading conformally soft graviton theorem.
This Ward identity was derived from the (Carrollian) super-rotation Ward identity in [10].There, it was also shown that, upon temporal-Fourier transforming (2.20) and then taking the ω → 0 + limit only for S + 1 [51], one obtains a Laurent expansion around ω = 0, the coefficient of the leading 1 ω 2 pole of which is the Weinberg positive-helicity soft-graviton theorem [13,19] while the subleading 1 ω pole's coefficient is recognized to be the Cachazo-Strominger subleading positive-helicity softgraviton theorem [14,23] when each of the primaries in X has ξ = ξ = 0 and (Carrollian) scaling dimension ∆ = 1.Therefore, the temporal step-function in (2.20) is the Carrollian manifestation of the super-rotation memory effect [51,52].
The last condition on the scaling dimension of a Carrollian conformal primary whose temporal-Fourier transformation can correspond to a 1 + 3D bulk AFS null momentum-space field (see also [54,55]) describing a mass-less external hard scattering particle, was discovered in [7,9] by analyzing the radiative fall-off conditions of the bulk mass-less fields.The higher dimensional counterpart of this condition was obtained more recently in [12].
(2.21) and (2.14) motivate us to re-express the S + 1 field inside the correlator as below [10]: It should be noted that S + 1e is not a local Carrollian field but merely a collection of the modes not appearing in the S + 0 field.These modes generate the following local infinitesimal Carrollian diffeomorphisms: with f (z) being a meromorphic function and q = 0, ±1.It is the 'Ward identity' ⟨S + 1e (t → ∞, z, z)X⟩ that directly gives rise to the Cachazo-Strominger subleading energetically soft graviton theorem [14,23].
Since, z and z are treated independently, the form of (2.20) allows us to decompose S + 1e inside a correlator as [31,56]: Since, all three j a e have holomorphic weight h = 1 like S + 1 , their Ward identities are effectively same as the holomorphic Kac-Moody Ward identities in a usual 2D CFT.
Finally, we note down the general OPE of the S + 1 field in the jϵ-form [10]: For a Carrollian conformal primary multiplet Φ, we have: whereas an SL(2, C) or Lorentz covariant quasi-primary must satisfy: j Obviously, the corresponding correlators and OPEs involving the S − 1 field (replacing S + 1 ) are just the complex conjugations (z → z, h → h, ξ → ξ) of the above mentioned equations.But as shown in [10], following [47], the fields S + 0 and S − 1 can not be simultaneously treated as local fields i.e. they are not mutually local.The reason is that while ∂2 S + 0 S − 1 ∼ 0 respects (2.15), S − 1 ∂2 S + 0 contains anti-meromorphic pole singularities, thus violating the OPE commutativity property.This is a Carrollian manifestation of the ordering ambiguity in the double soft limit involving two opposite helicity particles [30].On the other hand, the OPE of S + 0 with S + 1 does not suffer from this problem; hence, S + 0 and S + 1 can both be simultaneously taken as local fields that we do.As we shall see, the OPE conditions like (2.15), (2.16), (2.25) and (2.26) will play very crucial roles in this work.

The fields T and T
All the S 2 contact terms in the Ward identity (2.17) (or, its conjugate version) can also be converted into pole singularities by extracting from the R.H.S. a ∂ (or, ∂) -derivative.The Carrollian fields T and T were then constructed in [10] by inverting these derivatives as: Since, the field T has dimensions (∆, m) = (2, 2) and T has (∆, m) = (2, −2), the above relations, together with the fact that S ± 1 have (∆, m) = (0, ±2), imply that (S + 1 , T ) and (S − 1 , T ) automatically are two shadow pairs (on S 2 ).Since, we have chosen S + 1 as a local field, T now has to be treated as its non-local shadow.
With mutually local primaries, the T Ward identity was derived from the super-rotation Ward identity to be [10]: Inspired by the last relation, we decompose the T field inside the correlator as [10]: where T e is not a local Carrollian field but contains the modes generating the holomorphic superrotations.It is the object T e (t → ∞, z, z) that corresponds to the 2D Celestial stress-tensor [16]; by construction, it is the 2D shadow transformation of the negative-helicity energetically soft graviton The ⟨T e (t, z, z)X⟩ 'Ward identity' is the Carrollian conformal analogue of that of the 2D Celestial holomorphic stress-tensor [57,58] when all ξ p = ξp = 0.
The generic OPE involving the T field is given in the jϵ-form as [10]: For a Carrollian conformal primary, (L n Φ) = 0 for n ≥ 1.An SL(2, C) quasi-primary on the other hand needs to satisfy only (L 1 Φ) = 0.
In this work, we shall simultaneously treat the three Carrollian fields S + 0 , S + 1 and T as SL(2, C) quasi-primary (non-descendant) local fields, as was shown to be allowed in [10].

The Carrollian Conformal Field S + 2
The features reviewed in the previous section are the generic properties of any (Weyl invariant) 1 + 2D Carrollian CFT on flat (Carrollian) background.The Carrollian conformal Ward identities that were shown [7,9,10] to contain the equivalent information as the bulk AFS leading [13] and subleading [14] soft graviton theorems were obtained as a consequence of only the Poincaré, the super-translation and the Weyl invariance in [10].In view of the putative AFS/CarrCFT duality [24], this is in perfect agreement with the conclusions of [27] that in a generic theory of quantum gravity, only the leading and the subleading soft graviton theorems are universal.Consistently in [70], only these two soft graviton theorems were reached via a 'large AdS radius' limit from the AdS 4 /CFT 3 correspondence.The non-universality of the subsubleading soft graviton [14] theorem was discussed in [26,27].
To probe the non-universal subsubleading soft graviton theorem that occurs, e.g. at the tree-level (linearized) Einstein gravity [14], it then naturally appears that additional Carrollian conformal fields beyond the usual Carrollian generators S ± 0 , S ± 1 , T and T are required to be present in the CarrCFT.This situation is similar in spirit to those considered in [28,29] where, besides the conformal EM tensor, extra symmetry generators were postulated to exist in the 2D theory.
As we reviewed, the Carrollian fields S ± 1 encode both the ±ve helicity leading and the subleading energetically soft graviton theorems while S ± 0 account for only the leading ones.The OPEs of the same spin fields among them are related by e.g.(2.26) while the S ± 0 OPEs satisfy e.g.(2.16).Moreover, it is the temporal step-function (and its time-integrals) appearing in the Ward identities, from which the energetically-soft pole structures are arising.Inspired by these observations, we assume that, in the theory, there exists a Carrollian conformal field S + 2 that, in the OPE limit, satisfies: Since S + 2 is postulated to be a Carrollian field, it must contain some new modes.In view of (2.22), this suggests that inside a correlator, S + 2 can be decomposed as: where the S + 2e part consists of the new modes.Thus, the dimensions of the field Clearly, an analogously introduced Carrollian field S − 2 with spin m = −2 can not be treated as a mutually local field with S + 0 , S + 1 and T , following the argument presented in [10,47].While its shadow or light transformations (on S 2 ) may be fine in this regard, we leave this possibility for a future work.Thus, in this work, we refrain from introducing the S − 2 field or its integral transformations.
It is also evident that the shadow or the light transformations of the S + 2 field can not be mutually local with S + 0 , S + 1 and T .So, we would like S + 2 itself to fit in as a local field in the holomorphic sector of the 1 + 2D CarrCFT.It will be possible only if the S + 2 Φ OPEs have the similar singularity structure as those of the above three generators, i.e. having meromorphic pole singularities while being anti-analytic.We shall proceed by assuming this to be true.

The OPEs of S +
2 with the generators In [10], the singular parts of mutual (self and cross) OPEs between the three generators S + 0 , S + 1 and T were completely determined by demanding the OPE commutativity property to hold, after making appropriate ansatz respecting the general forms (2.14), (2.24) and (2.31) and truncating those ansatz by assuming that: 1. no local field in the theory has negative scaling dimension ∆ < 0, following the 2D Euclidean [28,32] and Carrollian [29,33] CFT cases.
2. the fields S + 0 and S + 1 are Lorentz quasi-primaries, following the Celestial CFT case [53].The first assumption is clearly not respected by the field S + 2 with ∆ = −1.So, it needs to be relaxed into a weaker one that is stated below: • no time-independent local field in the theory has negative scaling dimension; moreover, the time-independent local field with ∆ = 0 = m is unique and it is the identity operator.
(We will see that there are several time-dependent fields with ∆ = 0 = m.) Fortunately, the modified assumption does not change any of the results obtained in [10] for the mutual OPEs between S + 0 , S + 1 and T .This statement can be verified by using the restrictions (2.16) and (2.26), then repeating the steps to derive those mutual OPEs as elaborated in [10] and finally, keeping in mind that the global space-time translation invariance must remain unbroken.Below we collect the mutual OPEs between S + 0 and S + 1 in the jϵ-form [10]: where K is a constant not fixed by symmetry.
We now proceed to find the mutual OPEs involving the S + 2 field.The general form of the S + 0 OPE (2.14), the modified first assumption, the relation (3.1) and the form of the S + 0 S + 1 OPE together completely fix the singular part of the S + 0 S + 2 OPE to be: Using the OPE (bosonic) commutativity property, the S + 2 S + 0 OPE is readily obtained as: It is now evident that the ⟨S + 1 (x)S + 2 (x p )⟩ correlator can not be time-translation invariant unless K = 0 ; so, the final form of the above OPE reduces to: To find the S + 2 S + 1 OPE using the OPE commutativity property from (3.6), we first need to know if the resultant Taylor series gets truncated into a polynomial in (z − zp ).Recalling (2.25), we observe that the OPE (3.4) satisfies: Now, we consider an arbitrary operator product ∂3+N 1 S + 2 (x 1 )S + 0 (x 2 )Φ(x 3 ) (where the field Φ is mutually local with both S + 2 and S + 0 ) and apply the OPE associativity property5 in two different ways such that both the resulting series are convergent for the following ordering of the flat Carrollian norms: ) and (z 1 − z 3 ) as the two independent variables, we easily reach the following restriction on S + 2 , analogous to (2.15) and (2.25): implying that an S + 2 Φ OPE will be at most a cubic order polynomial in (z − zp ).As a consequence and remembering that we have also postulated that the S + 2 Φ OPEs have only meromorphic pole singularities while being anti-analytic so that S + 2 , S + 1 , S + 0 are mutually local and that z, z are treated independently, inside a correlator the S + 2e part can be decomposed as: which is also seen to be consistent with the decompositions (3.2) and (3.8).
Finally, we proceed to find the S + 2 S + 2 OPE with all the above information at our disposal.

3.2
The S + 2 S + 2 OPE We first make the following ansatz directly in the jϵ-form for the S + 2 S + 2 OPE that is consistent with the decomposition (3.2) and the restriction (3.7): where A r,s (x p ) are yet undetermined fields mutually local with S + 2 , S + 1 , S + 0 .Now, due to the restriction (3.1), singular part of S + 2 (x)∂ tp S + 2 (x p ) obtained from this ansatz must completely match with that of the S + 2 (x)S + 1 (x p ) OPE given by (3.9).Only the O (t − t p ) 0 terms, while comparing, give non-trivial constraints that are listed below (with ˙representing time-derivative): The first line says that all the local fields A r,s≥2 and A 0,1 are time-independent (in the OPE limit).Moreover, since the l.h.s. of the OPE (3.10) has a total scaling dimension ∆ = −2 , all of these fields have negative scaling dimensions.Hence, by the modified first assumption, we set all of them zero.
More interestingly, the requirements (3.11) reveal that, for the S + 2 S + 2 OPE to be consistent, there must exist a local field A 1,1 that, in the OPE limit, satisfies Ȧ1,1 ∼ 6S + 2 which is a condition analogous to (3.1).Let us denote the local field as S + 3 that obeys: The dimensions of the field S + 3 then are (∆, m) = (−2, 2).It needs to be emphasized that, unlike S + 2 , we did not need to postulate the existence of the local field S + 3 .Rather its existence is automatically demanded if the S + 2 S + 2 OPE is to be consistent.This can be interpreted as the non-closure of the mode-algebra of the three fields S + 2 , S + 1 , S + 0 alone.Finally, we write down the S + 2 S + 2 OPE below: with K 1 being a constant not yet fixed.It is clear from this OPE that the commutators of only the modes contained in S + 2e will involve new modes appearing in S + 3e that is defined below analogously to (3.2) and consistently with (3.12): We shall now investigate on the properties of the local field S + 3 and discover that a tower of local fields S + k (k ≥ 4) will be required to automatically exist for the consistency of the OPEs.
4 The Tower of Fields S + k We shall begin by looking into the mutual OPEs of the four local fields S + k with 0 ≤ k ≤ 3 and observe that, for the OPE S + 2 S + 3 to be consistent, a new local field S + 4 must exist.Similarly, for the consistency of the S + 2 S + 4 OPE, another new field S + 5 is required to automatically exist.As is anticipated, this sort of argument will recursively generate the whole tower of the fields S + k (k ≥ 4).

The field S + 3
Following the steps taken (and remembering the modified first assumption) in the previous section to construct the OPEs involving S + 2 , we directly write down the following OPEs for S + 3 (it has h = −2): Together with the restriction (3.7), the first OPE implies, similarly as derived for S + 2 , that: which means that an S + 3 Φ OPE will be at most a quartic polynomial in (z − zp ).Since, the singularity structure of such an OPE must be the same as that of the S + 0 Φ and S + 1 Φ OPEs, the condition (4.3) allows us to decompose S + 3e into five objects l a e (t, z, z) whose correlators are both time-independent and holomorphic in the OPE limit, analogous to (3.8).Now, to find the S + 2 S + 3 OPE, let us first construct the following ansatz consistent with the conditions (3.2) and (3.7): where B r,s (x p ) are yet undetermined local fields.Now, obeying the restriction (3.12), we match the singular part of S + 2 (x)∂ tp S + 3 (x p ) obtained from this ansatz with that of the S + 2 (x)S + 2 (x p ) OPE given by (3.13) to find that, again, only the O (t − t p ) 0 terms give non-trivial constraints collected below: Clearly, all of the time-independent local fields B r,s≥2 and B 0,1 have negative scaling dimensions; hence, all of them are set zero by the modified first assumption.But, the conditions (4.4) demand that there must exist a local field S + 4 such that: The dimensions of S + 4 hence must be (∆, m) = (−3, 2).
The S + 2 S + 3 OPE then finally is: with K 1 = 0 also, to keep the time-invariance of the ⟨S + 2 (x)S + 3 (x p )⟩ correlator intact.
It is by now apparent that this procedure will recursively reveal the automatic existence of a tower of local fields S + k+1 (k ≥ 2) just from the requirement of the consistency of the S + 2 S + k OPE such that: if we merely postulate that the 1 + 2D CarrCFT contains the field S + 2 in the first place.
We shall now provide a recursive construction of the general S + 2 S + k OPE.

The fields S + k
The relation (4.7) between the fields S + k and S + k+1 leads to: . So, the following OPEs are immediately constructed: As can be deduced by induction from the knowledge of (3.7), the first OPE implies that: which translates into the fact that an S + k Φ OPE will be at most a (k + 1)-th order polynomial in (z − zp ).
The relation (4.7) allows for the following decomposition of S + k inside a correlator: while the condition (4.10) permits further decomposition of S + k(e) inside a correlator as shown below: Such a decomposition is possible because of the holomorphic singularity structure and z, z being independent.As before, S + k(e) are local Carrollian fields but are merely collections of modes.The holomorphic weight of H k k+1 2 −s as well as the field Now, we make an ansatz for the S + 2 S + k OPE following the same steps as before but omitting the terms that will be eventually set zero by the modified first assumption: being some local fields.The S + 2 (x) Ṡ+ k (x p ) OPE obtained from this ansatz must be the same as the ansatz for the S + 2 (x)S + k−1 (x p ) OPE due to the relation (4.7).Again, the non-trivial constraints come only from the O (t − t p ) 0 terms.These give rise to the following recursive system for the local fields We demonstrate the solution of the A 1 recursive system below: The inversion of the time-derivatives are unique because of the severe restrictions imposed by the modified first assumption.This can also be motivated from the specific examples studied above.
The unique solutions to all the three recursions are similarly obtained to be: consistent with all the previously considered specific cases.The S + 2 S + k OPE is finally noted below: With the above derivation as a warm-up, we finally attempt to find the most general S + k S + l OPE.

4.3
The general S + k S + l OPE We shall find the general S + k S + l OPE via a recursive method analogous to the one demonstrated above.To find the seeds of the recursions, we first note down the three cases with l = 0, 1, 2. This can be achieved using the OPE commutativity property from the known OPEs S + l S + k with l = 0, 1, 2, i.e. from the OPEs (4.8), (4.9) and (4.13) respectively.While employing the OPE commutativity property, we need to recall that an S + k Φ OPE is a (k + 1)-th order polynomial in (z − zp ) and a k-th order polynomial in (t − t p ), due to respectively (4.10) and (4.11).Also, since in an S + k S + l OPE only the O (t − t p ) 0 terms are new in the sense that the O ((t − t p ) r ) (with r ∈ N) terms' coefficients have already been the O (t − t p ) 0 term in the S + k−r S + l OPE, we shall only explicitly write the O (t − t p ) 0 term from now on in a general OPE.The results are (with h.o.t.denoting the O ((t − t p ) r ) terms with r ≥ 1): With this knowledge, we shall now find the S + k S + 3 OPE in exactly the same way we derived the S + 2 S + 3 OPE exploiting the S + 2 S + 2 OPE.Let the ansatz for the S + k S + 3 OPE be: m (x p ) are the local fields to be determined via recursion.Now, due to the relation (3.12), the iS + k (x) Ṡ+ 3 (x p ) OPE derived from this ansatz must be the same as the OPE (4.16).This gives rise to the following recursion relation: To solve this recursion, we need a seed.Since we are comparing the coefficients of the (t − t p ) 0 (z−zp) m+1 (∆zp) terms, we need to remember that such terms only occurs in the iS + r S + 3 OPEs with r ≥ m.This means that B (m−1) m = 0 and that is the seed of the recursion.Thus, we have: So, the S + k S + 3 OPE is given by: where (. ..) 3 is an upward Pochhammer symbol.
In the exactly similar way, we find the S + k S + 4 OPE from the knowledge of the S + k S + 3 OPE to be: That this is indeed true can be quickly verified by noticing that the OPE S + k Ṡ+ l is exactly the same as the OPE S + k S + l−1 , as must hold because of the relation (4.7).For the sake of completeness, the full form of the above OPE, consistent with the decomposition (4.11), is written below: Having derived the general S + k S + l OPE, we now take on the final goal of this paper: uncovering the symmetry algebra manifest in the mutual OPEs of the tower of fields S + k .

The Symmetry Algebra
To find the quantum symmetry algebra from the OPEs, one needs to first identify the contributions of the modes to the OPEs.As is implied by the decomposition (4.11), the new modes in a field S + k that do not appear in the fields S + k−r with 1 ≤ r ≤ k are all contained in the part S + k(e) .In this sense, the part S + k(e) is the unique signature of the field S + k .Similarly, the 'new' information content of the S + k S + l OPE is described by the S + k(e) S + l(e) part.We recall that the part S + k(e) is not a Carrollian conformal field.So, strictly speaking, the S + k(e) S + l(e) part is not a Carrollian conformal OPE.Nevertheless, we shall call it an 'OPE' from now on.It can be readily extracted from the S + k S + l OPE (4.21) by comparing the O(t 0 t 0 p ) terms from both sides to be: In the explicit t → ∞ limit where 1 ∆zp ≡ 1 z−zp (and then recalling that any S + r(e) is time-independent in the OPE limit), this 'OPE' is exactly the same as the OPE of the positive-helicity conformally soft gravitons at the tree level of the Einstein gravity in the bulk AFS [36].There, it was obtained by taking the conformally soft limit from the general OPE of two Celestial conformal primary gravitons derived in [37] that needed some crucial inputs from the explicit bulk gravity theory.Here, we have reached the conformally soft graviton OPE directly by instead exploiting only the constraints imposed by the Carrollian conformal symmetry along with the two assumptions stated in section 3.1.No hint from the underlying bulk theory was needed in our field theory analysis.The Carrollian time-coordinate t played the central role in our construction.
Thus, the conformally soft graviton operator H k (z, z) of [36] that is a primary field in 2D Celestial CFT is the two-dimensional object iS + 1−k(e) (∞, z, z) appearing in the Carrollian conformal field iS + 1−k (t → ∞, z, z).Moving back to our goal of finding the symmetry algebra, we now substitute the anti-holomorphic mode-expansion (4.12) into the 'OPE' (5.1) and match the powers of both z and zp from both hand sides to find an 'OPE' H k k+1 2 −r (z)H l l+1 2 −s (z p ).The dependence on the time-and the antiholomorphic coordinates are removed since, by (4.11) and (4.12), the Carrollian 'modes' H k s ′ (t, z, z) are independent of respectively t and z in the OPE limit, thus making them effectively holomorphic inside an OPE.While it is straightforward to match the powers of z, to do the same with zp we must first insert the anti-holomorphic mode-expansion for ∂m p S + k+l−1(e) (z p , zp ) on the r.h.s.. Doing these, one finds the following 'OPE' (with ∆z p = z − z p − jϵ0 + from now on): The summation over m at the r.h.s. of this 'OPE' can be done as below (for r ≥ 1): that also holds true for r = 0.
Thus, the above 'OPE' reduces to the following simple form: This has precisely the same appearance as a Kac-Moody current OPE in a holomorphic 2D CFT (after explicitly putting ϵ → 0 + ).But the infinite-dimensional Lie algebra underlying such a Kac-Moody current algebra is not standard in the sense that it has no name!Fortunately, as we see below, this Lie algebra can be recast into a standard form.
For this purpose, we relabel the 'modes' H k a as: and get the following 'OPE' for the modes w p a from (5.2): that resembles a 2D CFT Kac-Moody current OPE with the Lie algebra being (a sub-algebra of) the w 1+∞ algebra [41].The underlying Lie algebra would actually be the 'wedge sub-algebra' defined by the restriction (5.3) on a [40] of the full w 1+∞ algebra with unconstrained a ∈ Z [41].
In the context of 2D Celestial CFT, the algebra of conformally soft gravitons was re-expressed as this same Kac-Moody algebra in [42].In this work, the central term w 1 0 of [42] is set zero to respect the time-translation invariance.But more importantly, we do not interpret the mode-redefinition (5.3) as a (discrete) light-transformation as opposed to the descriptions presented in [42,60].
We shall now explicitly show that the 'OPE' (5.3) actually gives rise to the above said currentalgebra symmetry in the CarrCFT (rather than the suggestive resemblance) as the algebra of the modes.This check is important because the CarrCFT technology has some differences with those in the usual 2D CFT.In particular, the mode-commutation relation is shown in [10,33] to be related with the corresponding OPE via a complex-contour integral in CarrCFT without the need to perform any radial-quantization.It is the temporal step-function-or the jϵ-prescriptions of the CarrCFT OPEs instead that play the crucial role in establishing such a relation.
First we notice that the objects H k a (z) can be holomorphic mode-expanded inside a correlator as below, by remembering that its holomorphic weight is h = 3−k 2 : facilitated by the property that H k a is independent of z (and t) in the OPE limit.This modeexpansion is inverted to recover the modes as following [10]: where C ′ u is the counterclockwise contour on a y = ax + b plane in the 1 + 2D Carrollian space-time that encloses the entire upper half plane t > 0 as well as the line t = 0. Finally, we shall now find the algebra of the modes H k a;n .This will be (part of) the symmetry algebra manifest at the level of the OPEs of a CarrCFT that contains the field S + 2 .For that, we first compute the following commutator: where the contour C u encloses only the upper half plane t > 0 but not the t = 0 line; passing from the C ′ u to the C u is valid only in the ϵ → 0 + limit.Clearly, the C u contour does not enclose the singularity at ẑ = z p + jϵ0 − coming from the H k a (t − p , ẑ)H l b (t p , z p ) term; so it has no contribution to the above commuator.
Next, using the mode-expansions (5.5) for H l b (z p ) and H k+l−1 a+b (z p ) and comparing the powers of z p on both h.s. of the above commutator, we get the following mode-algebra: or, in terms of the relabeled modes from (5.3): w p a;n ≡ i 2 H 2p−3 a;n with p + n ∈ Z as: confirming that the (wedge sub-algebra of) w 1+∞ Kac-Moody algebra indeed arises as the algebra of the modes from the OPEs of a CarrCFT containing the field S + 2 .Let us have a closer look into the algebra (5.7) or equivalently the 'OPE' (5.2).As discussed earlier, the existence of the fields S + 1 and S + 0 is a universal feature of any 1 + 2D CarrCFT since they are constructed purely in terms of the Carrollian EM tensor.The 'modes' H 1 a (or w 2 a ) and H 0 b (or w b ) are their respective unique signatures.In [10], from the mutual 'OPE's (that remains unchanged even under the modified first assumption) of these five 'modes', the symmetry algebra was derived to be the Kac-Moody extension of the sl(2, R) algebra with an abelian super-translation ideal.This is thus the 'universal' sub-algebra of the symmetry algebra (5.7).The Carrollian conformal modes H 1 a;n generate the ŝl(2, R) sub-algebra and the modes H 0 ± 1 2 ;m , the ideal.On the space of the Carrollian quantum fields, the three zero-modes H 1 a;0 generate the three sl(2, R) Lorentz transformations while the four modes H 0 The special 'OPE's involving the 'universal modes' H 1 a contain some information on representation theoretic properties of the tower of fields S + k or, rather, their signature 'modes' H k a .From the general 'OPE' (5.2), we readily find that: implying that the (k + 2) 'modes' H k b transform under the (k + 2)-dimensional representation of the group SL(2, R).Consistently, the SL(2, R) generator 'modes' H 1 a transform under the 3-dimensional adjoint-representation of the said group.
We recall that there is another 'universal' Carrollian conformal field T , built out of the Carrollian EM tensor, that can be treated as a local field simultaneously with S + 0 and S + 1 [10].Since the T Φ OPEs (2.31) have only meromorphic pole-singularities but are anti-analytic, these have the same singularity structures as that of all the S + k Φ OPEs.So, the field T and the tower of fields S + k are mutually local.
Even with the modified first assumption, the OPEs T S + 0 and T S + 1 remain unchanged from the following, as derived in [10]: while the S + 0 T and the S + 1 T OPEs were readily obtained from the above by using the OPE commutativity property and remembering the conditions (2.15), (2.16), (2.25), (2.26) and (2.32) to be: On the other hand, the T T OPE was derived to be [10]: where c is a constant not fixed by symmetry arguments alone.
The (singular parts of the) general T S + k OPEs can be completely determined by remembering the general form (2.31) of the T Φ OPEs and the modified first assumption to be: that is easily checked to be consistent with the relation (4.7) and the restriction (4.10).The S + k T OPEs can be immediately derived from this, using the OPE commutativity property and applying these two conditions.Thus, together with the S + 0 S + k and the S + 1 S + k OPEs (4.8), (4.9), the T S + k OPE (5.13) implies that: But, this is not the case with the field T .From the S + 0 T , S + 1 T and the T T OPEs above, we can immediately conclude that T is not even an ISL(2, C) quasi-primary field; it is only an SL(2, C) or Lorentz quasi-primary [10].Moreover, both (ξ • T ) and ξ • T are non-zero.
We now proceed to find the mode-algebra from the T S + k OPE (5.13).Just as the 'modes' H k a , contained in the S + k(e) part of the field S + k is its unique signature, the object T e introduced in (2.30), that corresponds to the 2D Celestial CFT EM tensor [16], is the unique signature of the field T .So, from (5.13), one can easily find the following 'OPE': ∆z p (t p , z p , zp ) (5.14) In the limit t → t + p and t p → ∞, this is actually a 2D Celestial CFT OPE saying that the Celestial conformally soft graviton field S + k(e) (∞, z, z) is a Celestial conformal primary [36].We then note the following holomorphic mode-expansion for T e in a CarrCFT OPE [10]: Using this and then, first the anti-holomorphic decomposition (4.12) for S + k(e) and next the holomorphic mode-expansion (5.5), we find the following commutator from the 'OPE' (5.14) in a manner similar to the derivation of the algebra (5.7): We also note the [L n , L m ] commutator derived in [10] from the T e T e 'OPE': which is the holomorphic Virasoro algebra Vir.
Thus, the complete symmetry algebra manifest in the OPEs of a 1 + 2D CarrCFT that contains a local field S + 2 obeying the relation (3.1) is the semi-direct product of Vir and the wedge sub-algebra [40] of ŵ1+∞ with the semi-direct product structure given by (5.16).The 'universal' sub-algebra of this algebra, i.e.Vir ⋉ ŝl(2, R) with an abelian super-translation ideal, is the OPE-level symmetry (in the holomorphic sector) of any 1 + 2D CarrCFT [10].
Unlike the Celestial CFT literature [31,36,42,59, 60], we reached the above conclusion solely from the Carrollian conformal symmetry arguments and the general properties of OPEs, under the two assumptions stated in section 3.1, without requiring any hint from the explicit (quantum) theory of gravitation in the 1 + 3D bulk AFS.Thus, our analysis is purely holographic in nature.

An Infinity of Soft Theorems
We shall now uncover, in the current framework of the 1 + 2D CarrCFT, the direct connection between the existence of the infinite tower of conformally soft graviton fields, as described in [36] in the context of the 2D Celestial CFT and in section 4 of this work and an infinite number of soft graviton theorems manifest as the Ward identities of large diffeomorphisms, as presented in [43,44].
As suggested in section 3, for a Carrollian conformal field S + 2 to encode the subsubleading energetically soft graviton theorem [14] in its Ward identity, it should obey a relation like (3.1).As we have seen in sections 4 and 5, recursive iteration of this suggestion reproduces the correct Carrollian conformal OPEs containing the conformally soft graviton OPEs [36] of 2D Celestial CFT.Providing a direct justification to this suggestion is the first step towards the goal of this section.We shall closely follow the argument presented in [10] that showed the relation between the sourceless 1 + 2D CarrCFT EM tensor Ward identities and the leading [13] and the subleading [14] soft graviton theorems.
We begin by noting down the S + 2 S + k OPE (4.13) below: and recall that all the fields S + k are Carrollian conformal primaries with ξ . Since valid for an infinite number of fields S + k , we postulate that the field S + 2 has the following OPE with a special Carrollian conformal primary6 Φ with dimensions (h, h) and ξ that is the Carrollian conformal counterpart of the subsubleading conformally soft graviton OPE with a 2D Celestial conformal primary as in [61], since clearly, ∆ Φ1 = ∆ Φ − 1 and m Φ1 = m Φ .This observation is consistent with the earlier interpretation in section 5 of the object S + 2(e) (∞, z, z) as the Celestial conformally soft graviton field H −1 (z, z) in [36].This 'OPE' is further decomposed according to the anti-holomorphic 'mode-expansion' (4.12) as: where it is understood that the fields on the r.h.s. are at x p .Finally, the holomorphic modeexpansion of the 'modes' H 2 k are given by (5.5) as: that implies the following transformation generated by, e.g. the modes H 2 − 1 2 ;n , from (6.2): derived using the CarrCFT OPE ←→ commutator prescription developed in [10].
Thus, the four Carrollian conformal modes H 2 a; 1 2 generate four global transformations on the quantum field Φ, that can be read off of the numerators of the 'OPE's (6.2).These are the four global transformations generated by the Carrollian conformal field S + 2 .Consequently, the existence of the field S + 2 in the theory demands that the CarrCFT correlators must be invariant under these four global transformations, in addition to the ten global Poincaré constraints imposed by the three 'universal' generators S + 1 , S + 0 and T .The most striking feature of the transformations generated by S + 2 is their non-locality in time (but locality in space): they involve a time-integral of the original primary field.This is consistent with the conclusion of an Einstein-gravity analysis in [62] that the subsubleading soft graviton theorem arises as a consequence of conservation of a spin-2 charge generating a non-local spacetime symmetry at null infinity; these symmetry transformations are also non-local only in (retarded-)time.
That in the O (t − t p ) 0 terms of the S + 2 Φ OPE (6.1), it is indeed h that appears instead of h can be checked by considering e.g., a Jacobi identity involving the primary Φ and the two Carrollian conformal modes H 2 − 1 2 ; 1 2 and H 0 1 2 ;− 1

2
. There is no doubt about the terms linear and quadratic in (t − t p ) that are already fixed by the restriction (3.1).Also, by considering another Jacobi identity with H 0 , it becomes apparent that the form of the O (t − t p ) 0 terms in the OPE (6.1) must be modified when the primary Φ has non-zero ξ • Φ and/or (ξ • Φ) or Φ is a non-primary.An important example of the second case is the S + 2 T OPE whose O (t − t p ) 0 terms are very different from those in (6.1).In this work, we do not consider the case of the general primaries because only the Carrollian primaries with ξ • Φ = 0 = (ξ • Φ) can describe mass-less scattering in the bulk AFS [10].
Restoring the temporal step-function factor, the S + 2 Ward identity corresponding to the OPE (6.1) is given by: with X being a string of Carrollian conformal primaries, all with ξ • Φ = 0 = (ξ • Φ).The terms regular in z − z p all vanish because we demand that the ⟨S + 2 (t, z, z)X⟩ correlator be finite whenever z ̸ = z p ; in particular, this correlator must be finite when z = ∞ (remembering that the CarrCFT primary S + 2 has h > 0).The complete argument is analogous to the ones elaborated in [10] for the finite-ness of the correlators ⟨S + 1 X⟩, ⟨S + 0 X⟩ and ⟨T X⟩.Since the integral operator ∂ −1 tp clearly changes (∆, m) → (∆ − 1, m), the S + 2(e) part of this Ward identity is the Carrollian conformal version of the positive-helicity subsubleading conformally soft graviton Ward identity in [37].
In [10], it was shown how to recover the 1 + 3D bulk AFS leading [13] and the subleading [14] energetically soft graviton theorems by simply taking a temporal Fourier transformation [50][51][52] of the S + 1 Ward identity (2.20) and then imposing an energetically soft limit only for the field S + 1 .To perform the temporal Fourier transformation, opposite phase conventions were required [9] for the Carrollian conformal primaries with ∆ = 1 that were to describe the outgoing or incoming bulk AFS mass-less particles in null-momentum space [7,9,12]; this convention was chosen in [10] as (with ω ≥ 0): We now temporal Fourier transform the S + 2 Ward identity (6.4) according to the above convention while choosing the outgoing convention for S + 2 , complexify the energy ω of the field S + 2 , set all ∆ p = 1 , m p = s p (the helicity of the bulk mass-less particle) and finally, take the energetically soft ω → 0 limit to obtain 7 : with ϵ p = +1 for outgoing and ϵ p = −1 for incoming fields.As expected, the r.h.s. is recognized as the soft factorization and energetically (holomorphic 8 [14]) soft expansion of the soft factor of a 1 + 3D bulk AFS mass-less scattering amplitude involving an outgoing soft external graviton up to the subsubleading order.The leading O( 1 ω 3 ) term is the Weinberg universal soft factor [13,19], the subleading O( 1 ω 2 ) term is the Cachazo-Strominger universal soft factor [14,23] and the subsubleading O( 1 ω ) is a non-universal [26,27] soft factor; its form agrees with the one in [63] explicitly calculated for the example of tree-level Einstein-gravity [14].Within the Carrollian context, this subsubleading soft factor is non-universal since it is present only in those 1 + 2D CarrCFTs that contain the field S + 2 , unlike the two in the more leading orders [10].Thus, we have shown that the Ward identity of the Carrollian conformal field S + 2 obeying the relation (3.1), indeed encodes the bulk AFS soft graviton theorems up to the subsubleading order.In section 5, it was also concluded that this S + 2 field contains the positive-helicity subsubleading conformally soft graviton primary [36,37] of the 2D Celestial CFT.Inspired by the above conclusions for the field S + 2 , one may get tempted to draw a blind analogy for the whole infinite tower of fields S + k containing, as shown in section 5, the whole family of the 2D Celestial conformal primary soft gravitons of [36].We shall find out that while this infinite tower of fields S + k indeed encodes, as Ward identities, an infinity of projected energetically soft theorems described in [43,44], the fields S + k>2 do not generate any additional global symmetries and hence, impose no new constraints on the CarrCFT correlators, consistent with [36].
and recall that dimensions of the field S + l are (h, h) = 3−l 2 , − 1+l 2 .In exact similarity with the case of S + 2 , we postulate the following S + k Φ OPE: for a special Carrollian conformal primary Φ with dimensions (h, h) and with ξ • Φ = 0 = (ξ • Φ).The unique local fields {Φ r } satisfy: Following the discussion on S + 2 , we can extract the 'OPE's H k a (z)Φ(x p ) analogous to (6.2), by appealing to the decomposition (4.11) and (4.12) and from there, derive the space-time transformations inflicted by the Carrollian conformal modes H k a;n on the quantum fields, just like (6.3).It is clear that all of these transformations are non-local in time but spatially local as they involve time-integrals of various order of the original primary field.For a discussion on the Einstein-gravity dual of these transformations, see [64].
We are now in a position to conclude that the infinite number of S + k Φ OPEs will thus generate an infinite number of global space-time symmetry transformations under all of which the CarrCFT correlators must be invariant, just as we did for the S + 2 case.But that is not the case!To show this, let us start with the case of S + 3 .The transformations that its modes generate is derived as H 3 a;n , Φ(x p ) via the OPE ←→ commutator prescription starting from the S + 3 Φ OPE.But, from the symmetry algebra (5.7), we see that: Thus, the transformation H 3 a;n , Φ(x p ) can be directly found out using a Jacobi identity involving the field Φ and two appropriate modes H 2 r;l and H 2 s;m and the knowledge of the transformations H 2 b;k , Φ(x p ) , without any need to learn the S + 3 Φ OPE at all.We can iterate this process to extract the transformations generated by the modes of the field S + k>2 from the knowledge of the transformations inflicted by the modes of S + 2 and S + k−1 without ever appealing to the S + k Φ OPE.Hence by induction, all we require to find the transformations generated by the tower of fields S + k>2 is the knowledge of only the transformations that S + 2 generate.Thus, the seemingly infinite number of global symmetries are not independent at all from the four generated by S + 2 .Thus, the correlators in such a CarrCFT are subject to a merely four additional global symmetry constraints in addition to the ten 'universal' Poincaré constraints.This conclusion resonates with the observation in section 4 that in a CarrCFT containing a field S + 2 obeying (3.1), an infinite tower of fields S + k≥3 are required to automatically exist to render the S + 2 S + k−1 OPEs consistent.Since the temporal Fourier transformation of a 1 + 2D position-space CarrCFT correlator of primaries all with ξ • Φ = 0 = (ξ • Φ) [10] and ∆ = 1 [7,9,12] gives the 1 + 3D bulk AFS nullmomentum space S-matrix [9], the above discussion implies that no new global symmetry constraint besides the ten 'universal' Poincaré plus the four generated by the field S + 2 is imposed on this Smatrix by the tower of fields S + k≥3 .The 2D Celestial CFT counterpart of this statement is proved in [36].
We shall now finally establish that the infinite tower of Carrollian conformal primaries S + k does indeed imply the existence of an infinity of projected (energetically) soft graviton theorems [43,44] as their Ward identities.
Restoring the temporal step-function, the Ward identity corresponding to the CarrCFT OPE (6.7) is given as the following: up to a (k − 3)-th degree polynomial in z that can not be fixed by symmetry considerations alone.Obviously, it will now receive the same treatment as the S + 2 Ward identity (6.4).For that, we first note the following identity: where ω is a complex quantity.
Following the convention (6.5), we now temporal Fourier transform the S + k Ward identity (6.8) choosing the outgoing convention for S + k , set all ∆ p = 1, m p = s p , use the identity (6.9) (and explicitly put the limits for a, b) and finally impose the energetically soft ω → 0 limit to obtain the following schematic Laurent series around ω = 0: where F (0) is the Weinberg leading soft factor [13,19], F (1) is the Cachazo-Strominger subleading soft factor [14,23] and F (2) is the subsubleading soft factor [14,63] that appear in (6.6); F (k≥3) are the next order soft factors.A soft factor F (k) has (z − z p ) in the denominator and a (k + 1)-th degree polynomial in (z − zp ) in the numerator.It is possible (but tedious) to derive its explicit form following our derivation of F (2) .From the explicit examples of the S + 0 , S + 1 , S + 2 cases, it is clear that the soft factor F (k) first appears from the Ward identity of the field S + k at the order O( 1 ω ).In this manner, an infinite number of projected energetically soft graviton theorems [43,44] arises from the Ward identities of the infinite number of Carrollian conformal primaries S + k that contain the 2D Celestial conformal primary soft gravitons H 1−k of [36].Furthermore, the undetermined terms polynomial in z in the ⟨S + k X⟩ Ward identity correspond to the homogeneous part of the graviton amplitude that are projected out to obtain the infinite-order soft factorization [43,44].

Discussion
Building on the direct relation between the EM tensor Ward identities of a 1 + 2D source-less CarrCFT on a flat Carrollian background (with R × S 2 topology) and the universal leading [13] and the subleading [14] soft graviton theorems that was uncovered in [10], in this work we investigated how the non-universal [26,27] subsubleading soft graviton theorem [14] can be holographically encoded into a CarrCFT if at all.We found out that in addition to the following three universal local generator fields in any generic 1 + 2D CarrCFT: S + 0 (and its non-local shadow S − 0 ) whose Ward identity contains the leading soft graviton theorems and S + 1 and T encoding the subleading ones (of both helicities) [10], a local Carrollian conformal field S + 2 must be postulated to exist in the theory such that it obeys the relation (3.1), for capturing the positive-helicity subsubleading soft graviton theorem.To avoid the ambiguity associated with double soft limits of opposite helicities [30], here we have refrained from looking into the case of the negative-helicity non-universal subsubleading soft theorem.
field theory [26], that the semi-direct product of the (chiral) Virasoro algebra and the wedge subalgebra of ŵ1+∞ is an exact quantum symmetry of the positive-helicity sector of any gravity-theory in 1 + 3D bulk AFS, just like the specific case of the quantum self-dual gravity [38].
In [65], it was shown that there is a discrete infinite family of 2D Celestial CFTs possessing the (wedge sub-algebra of) w 1+∞ symmetry.Two known examples of such theories are the MHV gravitons [31,61] and the quantum self-dual gravity [38].Since any 1 + 2D CarrCFT containing the local field S + 2 will enjoy the above said symmetry at the level of the OPEs, the conclusion of [65] suggests that there also exists an infinite number of 1 + 2D CarrCFT of this type.It will be very interesting to construct such an explicit CarrCFT example.
Finally, we showed that the CarrCFT Ward identity of the field S + 2 with a special class of primaries [7,9,10,12] does indeed encode up to the subsubleading energetically soft graviton theorem [14,63].Following this method, the infinite number of soft graviton theorems of [43,44] were then directly interpreted as the Ward identities of the members of the infinite tower of Carrollian conformal primaries S + k .In [10], it was found that the three universal CarrCFT generators S + 0 , S + 1 , T inflict the ten global ISL(2, C) Poincaré transformations on the quantum fields.Here, we showed that S + 2 generates four additional global symmetry transformations that, unlike the Poincaré ones, are non-local in time (but local in space).The Einstein-gravity analogue of this result is described in [62].We further clarified that the other primaries S + k≥3 do not generate any further independent global symmetries.Recalling that the global symmetries constrain the correlators of a theory and the CarrCFT correlators can be mapped to the bulk AFS null-momentum space S-matrices [9], our results provided a Carrollian justification of the statement [36] that the infinite number of soft graviton theorems of [43,44] beyond the subsubleading order does not impose any additional constraints on the bulk AFS mass-less S-matrices.
An obvious future direction that can be pursued following the methodology presented in [10] and in this work would be to figure out how the soft theorems of the gauge theories [43,44,66] in the 1 + 3D bulk AFS and the tower of conformally soft gluons of 2D Celestial CFT [36] can arise in the framework of 1 + 2D CarrCFT.By now it is apparent that some additional Carrollian conformal field(s) besides the three universal generators S + 0 , S + 1 , T must be postulated to exist in the theory, the Ward identities of which would encode the soft gluon theorems.Similar situations have been considered in [28] for 2D chiral CFTs and in [29] for 1 + 1D CarrCFTs.
More important is to try to find a resolution to the problem of the double soft limits of opposite helicities [30] within the CarrCFT framework.Extending the current work, one needs to start by assuming the existence of an opposite-spin counterpart of S + 2 .But as discussed in section 3, this field S − 2 can not be simultaneously treated as mutually local with S + 0 , S + 1 , S + 2 , T .The findings of the work [47] in the context of Celestial CFT are expected to play a very crucial role in this endeavour.We hope to report on this in a very near future.
hp ) r−m (r − m)! • m! ∂m p ∂ 1−r tp ⟨X⟩ (6.8) dt e −iωt θ(t − t p ) (t − t p ) Carrollian field Φ 1 satisfying Φ1 ∼ Φ.It is important to note that the OPE of the local field Φ 1 must not be completely determinable in terms of that of the field Φ; more precisely, it should hold that: Φ 1 (t, z, z) ∼