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Celestial Sector in CFT: Conformally Soft Symmetries
by Leonardo Pipolo de Gioia, Ana-Maria Raclariu
Submission summary
| Authors (as registered SciPost users): | Leonardo Pipolo de Gioia |
| Submission information | |
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| Preprint Link: | scipost_202308_00030v2 (pdf) |
| Date submitted: | 2024-04-25 19:40 |
| Submitted by: | Pipolo de Gioia, Leonardo |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We show that time intervals of width $\Delta \tau$ in 3-dimensional conformal field theories (CFT$_3$) on the Lorentzian cylinder admit an infinite dimensional symmetry enhancement in the limit $\Delta \tau \rightarrow 0$. The associated vector fields are approximate solutions to the conformal Killing equations in the strip labelled by a function and a conformal Killing vector on the sphere. An Inonu-Wigner contraction yields a set of symmetry generators obeying the extended BMS$_4$ algebra. We analyze the shadow stress tensor Ward identities in CFT$_d$ on the Lorentzian cylinder with all operator insertions in infinitesimal time intervals separated by $\pi$. We demonstrate that both the leading and subleading conformally soft graviton theorems in $(d-1)$-dimensional celestial CFT (CCFT$_{d-1}$) can be recovered from the transverse traceless components of these Ward identities in the limit $\Delta\tau \rightarrow 0$. A similar construction allows for the leading conformally soft gluon theorem in CCFT$_{d-1}$ to be recovered from shadow current Ward identities in CFT$_d$.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
{\bf Report 1.} We have added a detailed discussion addressing the relation to the conformal Carroll group. Regarding the references quoted: some of them have already been cited, while others focus either on the contraction of AdS$_3$ isometries, which are already infinite dimensional, or (as eg. in the appendix of 1203.5795) they show that the contraction of AdS$_4$ gives Poincare (note that all the modes in eq. B6 are restricted to $-1,0,1$). The sentence ``We again observe that the generators can be defined for any integer n and the algebra can be given an infinite dimensional lift'' appears on page 23 of that reference without justification. 1402.5894 seems to be the first place where a relation between the conformal Carroll algebra and the BMS algebra is established, thereby confirming the infinite-dimensional lift proposed in 1203.5795. Our construction gives a self-contained justification by deriving the same lift from a limit of the Conformal Killing Equation.
We have included these relevant references in our discussion, and further pointed out that the conformal Carroll stress tensor Ward identities seem to contain slightly less information than the Ward identities of a stress tensor in a CFT in the small time limit. This can for example be seen by looking at the relation between soft theorems and Carrollian stress tensor Ward identities vs. CFT stress tensor Ward identities. All components of the Carroll stress tensor Ward identities need to be used in order to obtain the leading and subleading soft theorems (see 2308.03673 for a compact review), while we only needed the transverse traceless component of the 3d CFT stress tensor in the flat limit. This seems puzzling, and calls for future understanding of the other components of the CFT stress tensor Ward identity, as the referee also remarks. This is beyond the scope of the present work, but we have added a sentence in the discussion emphasizing that it will be an interesting direction for future work.
{\bf Report 2.} We agree with the reviewer that the $1/R$ corrections will be extremely important to understand. We have expanded the third paragraph of the discussion section with a comment that in certain explicit examples of holography, large $R$ is related to large $N$ and hence $1/R$ corrections to conformal correlators may be linked to loop corrections in the bulk. This resonates with previous works relating large-$r$ corrections to the asymptotic charges with loop corrected soft theorems, but we believe that a complete understanding of these issues is beyond the scope of the present paper.
Regarding GBMS, we have added a discussion of this at the end of the paragraph containing eq. 4.38. In summary, for a fixed CFT background GBMS is not allowed. This can be seen already from eq. 4.11. In order to allow for gbms, one would have to allow for the metric in eq. 4.5 to become dynamical. This seems unnatural in the AdS/CFT correspondence, see however the added reference [50] where it is argued that one can impose Neumann boundary conditions (by instead fixing a subleading component of the metric) and still obtain a boundary CFT (not coupled to gravity).
Regarding the relation between the Carrollian $c \rightarrow 0$ and $R \rightarrow \infty$ limit, we have added an extensive discussion at the end of section 4.1 (see also the response to Report 1).
Finally, thank you for pointing out the missing references, we incorporated them appropriately.
List of changes
- We have added a detailed discussion addressing the relation to the conformal Carroll group at the end of section 4.1
- Included a comment pointing out that the conformal Carroll stress tensor Ward identities seem to contain slightly less information than the Ward identities of a stress tensor in a CFT in the small time limit
- Added a discussion on GBMS at the end of the paragraph containing eq. 4.38
- Expanded the third paragraph of the discussion section with a comment that in certain explicit examples of holography, large $R$ is related to large $N$ and hence $1/R$ corrections to conformal correlators may be linked to loop corrections in the bulk
- References suggested by the referees were incorporated