Hidden Conformal Symmetry in Higher Derivative Dynamics for the Kerr Black Hole

Round 1

Reviewer 1 Report

The authors studied the hidden conformal symmetry for the higher derivative theories for a probe scalar field. 

 

The crucial point in establishing the hidden conformal symmetry is to build the radial equation for the probe, using the generators of an SL(2,R) algebra. The authors have an example (based on well known ref. 18) in review section 2.

 

Such a direct construction has not presented for the higher derivative theories of the scalar field, which are supposed to be dual to AdS or other black holes.

 

The only superficial indication is in section 4.2.1 where the authors showed that the roots of secular equation has multiplicity 2 and so must contain a log as in equation (77). 

They never establish the correspondence explicitly.

 

Again when they consider the Kerr metric, they solely focus on the roots of secular equation to argue there is no log CFT. 

 

In general terms the paper is well written. On the other hand, for the reasons I mentioned, I believe that the work in its present form would probably better fit on a journal more oriented towards mathematical physics. Therefore I cannot recommend the paper for publication. 

 

I kindly invite the authors to consider submission to a different journal, or make amendments to make the results of the paper more appealing and certain.

Author Response

Dear Referee,

Please see the attachment. 

Author Response File: Author Response.pdf

Reviewer 2 Report

This interesting paper shows how to make manifest the hidden conformal symmetry of a higher derivative generalization of the Klein-Gordon field. This problem is dealt with by using the monodromy properties of the eqs. of motion. More interestingly this study has been performed in the spirit of Kerr/CFT correspondence and its black hole applications.

 

The paper is comprehensive, very well written and understandable in all its details and calculations. For all these reasons I highly recommend it for publication in your notable Journal.    

Author Response

Dear Referee 2,

We thank you very much for the positive feedback. 

Reviewer 3 Report

The authors nicely show that the higher-derivative Klein-Gordon equation has manifest SL(2, R) symmetry using the monodromy method. I have several comments regarding their proof: - In illustrating the monodromy method, the authors cite reference [23] in arguing that the solution must be invariant when going around twice around the singular point (in the paragraph starting with line 172). Why is that? - The authors then use the form of the coordinates t_L and t_R to find the zero mode generator in equation (33). They then suggest that H_{\pm 1} and \bar H_{\pm 1} can be found using equation (13). However, these killing vectors are obtained by taking the near-region limit which the authors want to explicitly avoid in section 2.2. Can the authors clarify if all SL(2, R) generators can be obtained without considering the near-region limit? - The mapping between equation (43) and (44) is clear. For reader's convenience can the authors clarify the mapping between solution of (43) and those of (45). - In the paragraph on line 290 the authors remark that it might be possible to identify the fourth-order Klein-Gordon equation with the quartic Casimir (which I do not find to be a conceptual issue as the authors remark). Can the authors expand on whether this is indeed a way of finding the SL(2, R) hidden symmetry? Since the authors remark that the generators of SL(2, R) remained unchanged in the higher derivative example why is this reasoning invalid? - For reader convenience, can the authors better motivate why the monodromy data does not depend on this choice of $\theta$ slice. Additionally I have several comments regarding the broader perspective of the paper: - Can the authors motivate why the higher derivative actions in (1) are relevant in a theory of quantum gravity, perhaps through some effective field theory arguments? - Since the authors did not identify a Kerr/logCFT correspondence I feel the authors dedicate to much of the material in this paper towards this aim. Perhaps several of the sections which discuss the Kerr/logCFT correspondence can be shortened. In particular, some of the points in the discussion (such as the last paragraph) I found to be to distant from the results presented in the paper.

Author Response

Dear Referee 3,

Please see the attachment. 

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

N/A