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

by Yannick Mvondo-She

Submission summary

As Contributors: Yannick Mvondo-She
Arxiv Link: (pdf)
Date submitted: 2021-09-21 13:23
Submitted by: Mvondo-She, Yannick
Submitted to: SciPost Physics Core
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


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:
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Submission & Refereeing History

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Submission 2109.03595v1 on 21 September 2021

Reports on this Submission

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


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.


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.


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

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