Celestial Sector in CFT: Conformally Soft Symmetries

The flat space S-matrix with external massless particles exhibits infinite-dimensional symmetries in the form of soft theorems. The authors show how to recover some of these from correlation functions in AdS/CFT when the AdS radius is taken to infinity. This builds on earlier work where the authors showed that correlators in ($d-1$)-dimensional celestial CFT (CCFT${}_{d-1}$) can be obtained from a suitable flat space limit of correlators in CFT${}_d$. Given the much earlier seminal works on recovering the flat space S-matrix from AdS/CFT correlators it is not surprising that one should recover the local symmetry enhancement of the flat space S-matrix in the limit where the AdS scale $R\to \infty$. However, the details of how this works out are interesting. The flat space symmetries are broken by $O(R^{-1})$ terms but the explicit form of these corrections is not given. I would like to ask the authors to add a discussion of these corrections since they encode non-trivial information about how CFT${}_3$ and CCFT${}_2$ (or their higher-dimensional cousins) are related.

To arrive at the central result of this paper, the authors develop the relation between CFT${}_d$ and CCFT${}_{d-1}$. This includes, as an important piece of the correspondence, how to extract a continuum of operators in CCFT from the large $R$ expansion of the bulk-to-boundary propagators in AdS, and how the antipodal map must enter. Another interesting result is how the conformally soft graviton operators are obtained from certain components of the CFT${}_d$ stress tensor after a $u$-projection that mimics the $\omega$-projection of the momentum-space soft graviton operators.

The authors discuss how from the shadow stress tensor one recovers extended BMS symmetry, which includes the enhancement of Lorentz symmetry to Virasoro symmetry. Alternatively, it has been argued in 1408.2228 that there is a further enhancement to generalized BMS symmetry - the enhancement of Lorentz symmetry to Diff($S^2$) symmetry - which can be realized from bulk asymptotic symmetries. I would like to ask the authors to add a comment on if and how from the flat space limit of AdS/CFT one only recovers extended BMS (as opposed to generalized BMS). It is emphasized in the paper that the shadow transform is needed to recover ''ordinary" CFT Ward identities in general dimensions - this was already pointed out in 1711.04371 (as is referenced), and was further examined in 2109.00073 and 2302.10222 which derive conserved (shadow) CCFT${}_{d-1}$ operators/Ward identities and show using CFT techniques that there is no infinite enhancement in $d-1>2$; given the related results of the present work it may be worth commenting on their connection.

The authors show that the $R\to \infty$ limit relates correlators in CFT${}_3$ to those in CCFT${}_2$. On the other hand the latter are related by an integral transform to correlators in Carrollian CFT${}_3$ which in turn arise as the $c\to 0$ limit of relativistic CFT${}_3$. The authors mention references [21,22] as source for the relation between 3D Carrollian and 2D celestial operators, but it appears that the transform referred by the authors as ''time Mellin transform", e.g. (4.49), was discussed before in 2105.09792 and which thus would be appropriate to cite in that context. The authors briefly comment that the two limits, $R\to \infty$ and $c\to 0$ should be related but without detail - could the authors add a paragraph discussing the connection? In particular, the isomorphism between Carrollian symmetries and BMS symmetries in one higher dimension has been discussed extensively in earlier literature and it would seem rather natural that the small time interval limit is related to the $c\to 0$ limit. The latter is the basis of 2202.08438 which appeared around the same time as [21,22] and also discusses the celestial and Carrollian connection; it would be appropriate to add it to the list of references and include it in the discussion on how the $R\to \infty$ and $c\to 0$ limits are related.

After these requests are incorporated I am happy to recommend publication. While the central result of the paper - that the flat space symmetries emerge in the flat space limit of AdS/CFT and are broken when the AdS scale is taken into account - is not surprising, the details of how this comes about are interesting and nicely worked out. I am confident that they will be quite useful in further studies of the connection between AdS/CFT and flat space holography.

This paper attempts to connect the large radius limit of AdS /CFT with an intrinsic formulation of flat space holography, known as Celestial holography. The authors compute the large radius expansion of AdS Witten diagrams and consider the implication of this limit on the field theory side. Ultimately, they derive celestial ward identities as a limit of a parent higher dimensional CFT. While the results of this paper are interesting, it also contains several shortcomings. Thus, it can be accepted for publication after proper modifications.

Sections 2 and 3 discuss the large radius limit of AdS Witten diagram. The results of these two sections are slight generalisations of a previous paper, 2206.10547 by the same authors, where they extend the results of the former by introducing the spin of external particles. In the previous article, the authors recovered the conformal primary wave functions as the leading term in the large R expansion of bulk to boundary AdS propagators. As the large R limit maps the boundary of AdS to the null boundary of flat spacetimes, some restriction of the null coordinate u is required to relate to celestial amplitude. In 2206.10547, the choice of the insertion points of boundary operators was taken to be $\tau_p=\frac{\pi}{2}$, which amounts to restricting the null boundary to $u=0$ slice. However, in contrast to this approach, in this paper the regime of the boundary insertion points are extended to include the subleading term, i.e. $\tau_p=\frac{\pi}{2}+\frac{u}{R}$. Furthermore, the conformal primary wavefunctions are obtained via an integral transformation, introduced in eq (2.12). The authors introduce this integral transformation abruptly and don’t clarify the relationship between these two seemingly different methods of arriving at celestial amplitudes from the large radius expansion. If they are unrelated, it begs the question, which of these two is more natural to go forward with?

Section 4 discusses the symmetries and Ward identities of a CFT zooming into a region of small time. This contraction, in essence, is the Carrollian contraction, where the speed of light goes to zero. Algebraically, this amounts to taking an Inonu- Wigner contraction. Going forward with this prescription, the authors claim to discover an infinite enhancement of symmetries in this regime. In particular, they find the extended BMS group three dimensions. However, the BMS algebra has been studied extensively in the past decade as a group contraction of conformal algebra. For example, to the best of my knowledge, in 1203.5795 , Carrollian contraction of CFTs in general dimensions was first considered. Subsequently, conformal killing equations on Carrollian manifold in general dimension were studied, and an isomorphism with BMS was established in 1402.5894. Realisation of this symmetry group in a field theory has been considered in 1609.06203, more recently in 2202.08438 and 2212.12553. Although 2212.12553 has been cited, the authors ignore several of these works and claim the results to be original. This is a major drawback of this article.

Although the technologies developed in section 4 is not novel, it has been put to good use in section 5, where the authors derive the conformal soft theorems as a limit of CFT Ward identities. The analysis is carried out in embedding space formalism. The authors first set up the framework in generic dimension and provides explicit results in 3 dimensions. The Ward identities of shadow transformed spin 1 current and stress tensor reduce to soft gluon and graviton theorems respectively. This is a nice result. However, while the Ward of some components of these vector and tensors boils down to the known soft theorems, the same for others remain unidentified in terms of celestial currents. The authors mentioned that the Ward identity associated with the u component of the spin 1 field resembles the scalar soft theorem (5.39). Why this effectively behave like a scalar is not clear to me. I would appreciate some comments in this regard along with comments on the fate of other stress tensor Ward identities. Would they for eg. Correspond to any of the soft theorems, existing in the literature ?

All in all, although this paper has drawbacks, in the end it provides some relevant results that will have a role to play in a better understanding of flat space holography. I recommend this paper for publication when the questions are satisfactorily answered and relevant previous works are mentioned properly.