Modular Properties of Generalised Gibbs Ensembles

This is a report on the manuscript "Modular Properties of Generalised Gibbs Ensembles". This is a well-written paper, with novel and interesting results. I recommend it for publication in SciPost.

The paper studies two dimensional conformal field theory (CFT), which is known to have an infinite number of mutually commuting higher spin conserved charges $I_n$, which are called the quantum KdV charges. The main focus of the work is the modular property of the Generalized Gibbs Ensemble (GGE), in which these higher spin charges are inserted in the toroidal partition function. Most of the work is restricted to the Lee-Yang model, in which the fugacity $\alpha$ for a single charge $I_5$ is turned on, though there are some results also for GGEs in a Verma module. This work can be considered an extension and generalization of earlier work by one of the authors in [16,17], in which the question of modularity of the GGE was addressed in the context of the free fermion model.

In the first part of the paper, the GGE is expanded for small fugacity in an asymptotic series. Then, the first few terms essentially amounts to the calculation of thermal correlators of the form $\langle I_5^n \rangle$, where the angular brackets indicate the thermal vev in the vacuum module. For small $n$, these are calculated along with their $S$-transforms. The left hand side can be written as quasimodular differential operators acting on the characters of the model, and their S-transforms can be computed easily. These are then repackaged as linear combinations of correlators of quasiprimaries. A useful technical point here is the use of a basis that already takes into account the null vectors within the module of the minimal model, that leads to the evaluation of the vevs and their S-transforms up to $O(\alpha^3)$. The work done by the authors builds upon earlier literature on thermal correlators [7] and also the work specific to the Lee-Yang model in [40]. This is already a non-trivial calculation, and a rather striking result is obtained: the S-transform of the GGE involves not only higher KdV charges but also zero-modes of quasi-primary fields $J_k$ that are conserved but that are not in involution with the KdV charges $I_n$. This is a novel result.

The authors then propose to resum this asymptotic expansion into a transformed GGE, in which both the quantum KdV charges and the zero-modes of the quasi-primary operators are included. This is a formal sum of an infinite number of terms that diverges, which therefore needs regularization. This leads to the second part of the project in which the spectrum of the transformed GGE is obtained by means of the thermodynamic Bethe ansatz. This part is quite technical and there is an impressive amount of numerical work that is done to provide evidence for this proposal. The fundamental claim here is that if the spectrum of the original GGE (with non-vanishing fugacity for a single quantum KdV charge) is obtained by a TBA (and this has been known since the seminal work in [24]), then the spectrum of the S-transformed GGE (which includes non-commuting charges) can be obtained from the mirror-TBA of the Lee-Yang model. The resulting analysis is new and the authors clearly explain how to systematically compute the spectrum using asymptotic solutions to the mirror TBA.

The last point that is made is to show the existence of non-asymptotic solutions to the TBA. Based on previous and analogous work done for the free fermion model, this leads to a conjecture that terms arising from these non-asymptotic solutions also have to be added in order to find the full modular transformation of the original GGE. This last part is conjectural, with more work needed to prove the claims made here, but the authors are aware of this.

This is a well written paper that deserves to be published. The questions that they raise, namely the modular transformation of GGEs, is a worthwhile one to explore. The answers that they provide, using the analysis of the TBA for the Lee-Yang model, are compelling.

Publish (easily meets expectations and criteria for this Journal; among top 50%)

The paper addresses an interesting and timely question and makes a genuine attempt to answer the question by a straightforward brute-force calculation.

The naive answer to the main question -if the KdV GGE has nice modular properties - seems to be "no." The paper attempts to analyze this question within a restricted framework of eq. (57) with constant coefficient alpha_5, beta_9, etc. It is conceivable (though should not be taken for granted) that the "proper" way to transform the GGE would lead to a more complicated form of the RHS in (57).

The paper addresses an interesting question, if 2d CFT generalized partition function on a torus, decorated by quantum KdV charges admits (nice) modular properties. Previous studies in case of c=1/2 model as well as in c>>1 limit suggest that no simple modular properties exist. The example of c=1/2 though suggested that higher KdV charges can be treated as a defect, giving rise to an asymptotic expansion in the dual channel. This work extends these studies to general c, and analyzes the case of Lee-Yang model in great detail. One interesting observation of this paper that asymptotic expansion in the dual channel must include charges outside of the KdV hierarchy.

The paper is very detailed, it contains a number of interesting calculations, relying on different techniques (including TBA, which to my knowledge, was not applied to this problem before). I think results of this paper will be useful for future studies of the KdV GGE in 2d CFTs.

Some of the calculations (in Appendix B) of the zero modes of the quasi-primary fields on the cylinder were discussed in the literature before. For example eq. (211) is the same as eq. (4.4) in 1912.13444 (there are also other examples). It would be proper to acknowledge the overlap.

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As pointed out in my first report.

As pointed out in my first report.

My original report was very positive and only made very minor suggestions for changes. The authors have accepted one suggestion regarding additional references but argue that another change I had proposed may not be suitable. My proposal was really just a suggestion to move some cumbersome formulae to an Appendix, but the paper already has many appendices so leaving the formulae where they are is also acceptable. This is what the authors have done and that's fine.

I think the paper can be published in its current form.

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-The paper is clearly written
-The paper addresses a timely problem
-The paper contains original and rigorous results
-The paper cites the relevant literature
-The paper points to further developments and open problems

-Many of the analytical results are very specific to the model under consideration

In this paper the authors address a timely and interesting problem, namely how to analytically and numerically compute the expectation values of higher conserved quantities in 2D conformal field theories (CFTs) whose partition function is characterised as a generalised Gibbs ensemble.

The authors address this problem by employing several methods, notably, the thermodynamic Bethe ansatz approach and also by interpreting the GGE partition function, as the partition function of a Gibbs ensemble in the presence of a conformal defect.

These methods are then exemplified for a the simplest non-trivial non-unitary minimal model, the Lee-Yang CFT where in particular the TBA can be solved perturbatively in the higher spin charge coupling. It is conjectured that the TBA description of the GGE is complete in the sense that it encodes information about the averages of all higher conserved charges.

As I wrote elsewhere, this paper has many strengths and the potential weakness of containing many analytical and numerical results which are highly model specific. However, the methodology is rather general and the intention to generalise and extend the results is stated in the conclusion. It is my view that the paper deserves to be published in SciPost in its present form.

The paper is well written and I think that it could be published in its current form. I would like to suggest two minor changes.

1) I suggest citing the paper https://arxiv.org/abs/1203.1305 since it contains an early derivation of the TBA equations for GGEs. The fact that the driving term of the TBA should be modified by including all conserved quantities had already been pointed out in https://arxiv.org/pdf/0911.3345

2) I am not 100% sure about this since there are already a lot of Appendices in this paper. However, I would suggest that perhaps parts of subsections 5.4 and 5.5 could be moved to an Appendix too (some of the longer formulae and tables) and that this would make this section more readable.

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