# Integrable hierarchies, Hurwitz numbers and a branch point field in critical topologically massive gravity

### Submission summary

 As Contributors: Yannick Mvondo-She Arxiv Link: https://arxiv.org/abs/2109.03595v1 (pdf) Date submitted: 2021-09-21 13:23 Submitted by: Mvondo-She, Yannick Submitted to: SciPost Physics Core Academic field: Physics Specialties: High-Energy Physics - Theory Approach: Theoretical

### Abstract

We discuss integrable aspects of the logarithmic contribution of the partition function of cosmological critical topologically massive gravity. On one hand, written in terms of Bell polynomials which describe the statistics of set partitions, the partition function of the logarithmic fields is a generating function of the potential Burgers hierarchy. On the other hand, the polynomial variables are solutions of the Kadomtsev-Petviashvili equation, and the partition function is a KP $\tau$ function, making more precise the solitonic nature of the logarithmic fields being counted. We show that the partition function is a generating function of Hurwitz numbers, and derive its expression. The fact that the partition function is the generating function of branched coverings gives insight on the orbifold target space. We show that the logarithmic field $\psi^{new}_{\mu \nu}$ can be regarded as a branch point field associated to the branch point $\mu l =1$.

###### Current status:
Has been resubmitted

### Submission & Refereeing History

#### Published as SciPost Phys. 12, 132 (2022)

Resubmission 2109.03595v2 on 1 February 2022

Submission 2109.03595v1 on 21 September 2021

## Reports on this Submission

### Report 2 by Luca Ciambelli on 2021-12-30 (Invited Report)

• Cite as: Luca Ciambelli, Report on arXiv:2109.03595v1, delivered 2021-12-30, doi: 10.21468/SciPost.Report.4119

### Strengths

1- well written and detailed

2- Important review section of mathematical tools

3- Interesting new results on the logarithmic contribution to the partition function of critical TMG

### Weaknesses

In my view, the main weakness is that the paper contains almost 8 pages of (very well-written) review of mathematical background material, whereas the main result is presented in a quick and dismissal way in the last 2 pages. I suggest to keep the first review part as it is, but to spend considerably more time explaining the original result on the logarithmic contribution to the partition function. What is the state of art on this topic? Why the author's new original result is important? While this paper is supposed to be about critical TMG, the main body of the paper is on an a priori different topic, so I believe it is important to dedicate more time to the ''TMG side''.

The author uses a lot of technical expressions and words, which sometimes make the manuscript not accessible and hard to follow. The paper is dense, with a lot of technical concepts dropped (see the paragraph below eq. (18), as an indicative example). The author should clarify, expand more, and make this work more accessible to the reader.

### Report

I believe this paper is well written and meets all the criteria to be published in this journal, once the minor revisions requested below are addressed. Therefore, I am happy to review the revised version of this paper.

### Requested changes

1- Dilute the paper and spend more time on the new original results.

2- Beginning of page 2: the comment on ''deforming pure Einstein gravity'' is confusing. First of all, Einstein gravity with negative cosmological constant has still no propagating degrees of freedom in 3d. Second, the presence of black holes is not relating to propagating degrees of freedom, and indeed also Einstein gravity with vanishing cosmological constant has black holes solutions (Banados solutions). I would refrain from putting on equal footing Einstein gravity with negative cosmological constant and TMG.

3- The terminology used by the author is often not the standard one (e.g. ''cosmological TMG'', or ''Fefferman-Graham-Brown-Henneaux'', or ''Lie Heisenberg algebra''). In a such a dense paper (see weaknesses), this makes it harder to read.

4- Please define q and qbar in eq. (2).

5- Please define in the Introduction Hurwitz numbers and Burgers hierarchies, since they are the core of the paper

6- Please add references in the first paragraph of section 2 and the first paragraph of section 2.1.

7- Please define notation in equation (14)

8- Please explain more below (16)

9- Please expand the first paragraph in section 2.2.2

10- Please explain better how the assignment (25) defines the embedding (26)

11- Unclear notation in (29) and (30) for the ''z''. In the previous equations, ''z'' had a superscript, while in these equations it has a subscript.

12- The symbol for the bosonic Fock space is undefined

13- Please expand the paragraph below eq. (42)

14- Since, as already pointed out, the manuscript is dense, I suggest to write a recap at the end of section 4 of the main equations and main conclusions.

15- Please provide at the end of the conclusions some possible outlook of this work, to put it in context and localize it in the vast literature on the subject. This is also important to show the author's far-reaching research lines and goals.

• validity: high
• significance: good
• originality: good
• clarity: high
• formatting: excellent
• grammar: perfect

### Report 1 by Daniel Grumiller on 2021-9-28 (Invited Report)

• Cite as: Daniel Grumiller, Report on arXiv:2109.03595v1, delivered 2021-09-28, doi: 10.21468/SciPost.Report.3581

### Strengths

1. The main result that the partition function associated with log excitations in cTMG is a KP tau-function is clear, succinct, yet unexpected (at least to me) and novel.

2. Relatedly, also the interpretation of this partition function as generating function for Hurwitz numbers is both surprising and intriguing.

3. The introduction does an excellent job in reviewing essential aspects of AdS/log CFT in the context of cTMG.

### Weaknesses

1. Some pertinent questions remain unanswered. For instance, it is unclear whether or not the partition function in cTMG is 1-loop exact. Perhaps this is the case, and if so, the structures uncovered in this paper may be a first indication for 1-loop exactness. However, this is far from clear.

2. Similarly, it would be nice to also understand the instanton contributions to the full partition function. Is it possible to simply take a sum over modular images, like in Ref. [6], or not? As the paper makes not statement about 1-loop exactness or instanton contributions, this is not a criticism of the way the paper is written, but more an encouragement to eventually take the next step.

3. Some of the jargon was unfamiliar to me. While the paper gave all the required pointers to the literature and explained all expressions that were new to me, it is still somewhat difficult to read.

### Report

The paper meets all the acceptance criteria.

It addresses an important problem in cTMG, namely to get better insight into the 1-loop partition function, with potential applications for AdS/log CFT. The main research result, that this partition function is the KP tau-function, is a significant and unexpected advance in this research field.

My only criticism is that some of the jargon was unfamiliar to me (and by extrapolation could be unfamiliar to many readers), so we might have benefited from additional explanations or simple examples - for instance, the discussion in section 3 on Hurwitz numbers requires a couple of additional definitions (like "ramification" or "branched covering"), and they just follow one after the other without providing more context. While the clarity of the paper is still good, I think with a bit of additional effort it could be made even clearer.

### Requested changes

1. Go as easy on the jargon as possible and try to find simple examples or provide additional explanations for less familiar terms. To give just one example, a sentence like "By definition, the Plücker embedding of the Grassmannian is given by quadratic equations called Hirota bilinear equations." is only understandable if it is clear to the reader what a Plücker embedding is. Granted, this is briefly defined before, but for a reader encountering this term for the first time it could be helpful to give a bit more context. There are similar examples throughout the text.

2. Optionally, perhaps add some outlook in section 5 pointing to some of the unresolved issues concerning the partition function (see points 1. and 2. under "Weaknesses")

• validity: top
• significance: high
• originality: top
• clarity: good
• formatting: excellent
• grammar: excellent