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Hidden Conformal Symmetries from Killing Towers with an Application to Large-D/CFT
by Cynthia Keeler, Victoria Martin, Alankrita Priya
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Submission summary
| Authors (as registered SciPost users): | Victoria Martin |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202201_00019v1 (pdf) |
| Date submitted: | 2022-01-17 18:11 |
| Submitted by: | Martin, Victoria |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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Abstract
We generalize the notion of hidden conformal symmetry in Kerr/CFT to Kerr-(A)dS black holes in arbitrary dimensions. We build the SL(2, R) generators directly from the Killing tower, whose Killing tensors and Killing vectors enforce the separability of the equations of motion. Our construction amounts to an explicit relationship between hidden conformal symmetries and Killing tensors: we use the Killing tower to build a novel tensor equation connecting the SL(2,R) Casimir with the radial Klein-Gordon operator. For asymptotically flat black holes in four and five dimensions we recover previously known results that were obtained using the "near-region" limit and the monodromy method. We then perform a monodromy evaluation of the Klein-Gordon scalar wave equation for all Kerr-(A)dS black holes, finding explicit forms for the zero mode symmetry generators. We also extend this analysis to the large-dimensional Schwarzschild black hole as a step towards buliding a Large-D/CFT correspondence.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 1) on 2022-3-27 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202201_00019v1, delivered 2022-03-27, doi: 10.21468/SciPost.Report.4788
Strengths
The manuscript is very well written. Computations are explicit and it is easy to follow. It summarises clearly prior results and embeds with existing literature.
Weaknesses
For the Kerr-AdS$_{D+1}$ BHs studied, it would be interesting to connect the hidden symmetries of the wave equation with the dual CFT$_{D}$. One expects the thermodynamic properties and greybody factors of the black hole to be tied to the dual CFT, so I was expecting more comments or discussion about how this ties in the more traditional context of AdS/CFT.
Report
This is an interesting manuscript that studies and unveils new connections between the analytic properties of the wave equation to the Killing tower. These new structures and corrections are important to the development of methods that would allow more control on dynamical properties of black holes, and therefore I find this manuscript valuable.
Requested changes
I think it might be of interest to the authors to cite https://arxiv.org/abs/2105.01069 which also explores hidden symmetries that are well defined for Schwarzschild. (There is also a subsequent paper https://arxiv.org/abs/2203.08832, but this one came out after this manuscript, so I don't expect it to be cited.)
Report #1 by Anonymous (Referee 2) on 2022-3-27 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202201_00019v1, delivered 2022-03-27, doi: 10.21468/SciPost.Report.4781
Strengths
1. The article connects two different techniques for studying hidden symmetries (Killing tensors and monodromy method).
2. The article identifies the algebra of conformal symmetries in the near-horizon limit of rotating black holes in arbitrary dimensions.
Weaknesses
None.
Report
This interesting article explores hidden conformal symmetries of Kerr black holes in arbitrary dimensions. Specifically, the authors combine two different methods associated with Killing tensors and monodromy, which have been used in different contexts in the past, to construct the algebra of hidden symmetries underlying the dynamics of scalar fields in Kerr geometries.
Over the last two decades, hidden symmetries of rotating black holes have been studied in two different contexts. Using the standard techniques of general relativity, separation of dynamical equations has been demonstrated in all dimensions, and the Killing tensors underlying such separation have been identified. Separately, in the context of AdS/CFT correspondence, new conformal symmetries have been found in the near-horizon limits of black holes in 4 and 5 dimensions. This article combines these two lines of research and uses the Killing tensors to identify the conformal symmetries in arbitrary dimensions and to analyze their algebra.
The article is very well written. It starts with a self-contained review of methods associated with Killing tensors and with monodromy used in studies of conformal symmetry. Then in sections 3 and 4 the authors combine these techniques to derive the algebraic structure of hidden conformal symmetry in arbitrary dimensions, and in section 5, the large D limit is analyzed. This article will be interesting to a wide audience of researchers working on general relativity and AdS/CFT correspondence, so I recommend it for publication.
I would suggest adding a couple of clarifications on page 19.
1. From the sentence "We want to propose a tensor equation T_{ab} = 0, which enforces the result (nabla T nabla Phi)." it is not clear what result is being enforced. If tensor T vanishes in a region, then all its derivatives vanish as well, so perhaps the last condition should read (nabla T nabla Phi)=0.
2. Before equation (3.42) the authors say "We thus propose the tensor equation." They should clarify whether this relation is derived or it is a new constraint imposed for convenience.
I leave implementation of these suggestions to authors' discretion, and I recommend the article for publication.