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Asymptotic T-duality in three dimensions

by Stéphane Detournay, José Figueroa, Alejandro Vilar López

This is not the latest submitted version.

Submission summary

Authors (as registered SciPost users): Alejandro Vilar López
Submission information
Preprint Link: scipost_202504_00021v1  (pdf)
Date submitted: April 12, 2025, 8:18 p.m.
Submitted by: Vilar López, Alejandro
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
Specialties:
  • Gravitation, Cosmology and Astroparticle Physics
  • High-Energy Physics - Theory
Approach: Theoretical

Abstract

In (super)gravity theories, T-duality relates solutions with an exact isometry which can have wildly different asymptotic behaviors: a well-known example is the duality between BTZ black holes and (non-extremal) three-dimensional black strings. Using this dual pair, we show how the knowledge of a phase space which includes one set of solutions (here, BTZ black holes embedded in the Brown-Henneaux phase space) allows to obtain a phase space for the dual set via an asymptotic notion of T-duality. The resulting asymptotic symmetry algebras can be very different. For our particular example, we find a large algebra of symmetries for the black string phase space which includes as subalgebras $\mathfrak{bms}_2$, $\mathfrak{bms}_3$, and a twisted warped conformal algebra. On the way, we show that a chiral half of the Brown-Henneaux boundary conditions are dual to the Compère-Song-Strominger ones.

Author indications on fulfilling journal expectations

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  • Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Has been resubmitted

Reports on this Submission

Report #2 by Luca Ciambelli (Referee 2) on 2025-6-12 (Invited Report)

Report

This paper is well written, and explores an interesting and timely topic.

However, there are some points that should be improved and clarified for the paper to be publishable in SciPost:

1) While it is touched upon throughout the manuscript, the effect on the phase space of the T-duality map in the presence of an isometry remains unclear and mysterious. It is well-explained how such a transformation can be used as a solution-generating technique, but on the other hand it is claimed to be useful at the phase space level (see e.g. top of page 11). This is not so obvious, as it leads to a non-residual symmetry transformation, and thus it maps inequivalent phase spaces. One could (as the authors do) start again in the new solution defining new boundary conditions, but then this is not a procedure, rather an ad hoc -- case by case -- construction. It would be certainly useful to spend more time on this delicate point.

2) Section 2 starts abruptly with a list of equations and quantities that are not sufficiently-well introduced. It is necessary to improve this section, recalling conventions and quantities. Please be more specific about what "the theory" above eq. (1) exactly is, what are the dynamical fields, and the physical intuition.

3) The vocabulary around equations (5a-5c) is confusing. Indeed, these equations are fields falloffs, not boundary conditions. Boundary conditions are something of the form $\delta \alpha\stackrel{S}{=}0$, for a field $\alpha$ and a boundary $S$. Can the authors improve this part?

4) Related to point 1) above, is the vector field (33) the "T-dual" of the original residual symmetries generator? This is not clear, but I believe it is interesting and worth understanding better. A similar question pertains to the vector field (54).

5) From (34a) and (34b), one deduces that T-duality is a physical transformation, generating new non-vanishing charges. However, as we recently explored in 2412.14992, some new charges may well be "kinematical" charges. This means that they have trivial (or unconstrained) fluxes, and/or they come from a corner symplectic potential. This resonates with the discussion around equation (50), in which a new term is added to the symplectic potential, as well as around equations (62a-62d), in which, indeed, the new charges ${\cal R}$ and ${\cal Q}$ have trivial associated fluxes . So these new charges seem to be kinematical, in the language of the afore-mentioned paper. This is further supported by the fact that they do not enter in the entropy formula (70). Then, perhaps, one can make them vanishing using a different boundary term, making T-duality a map that does not generate new charges. This is worth clarifying.

On top of these conceptual questions, some minor typos/comments:

i) Please add references after the sentence "which has played a pivotal role in many developments of AdS/CFT" ii) Below eqs. (29a-29c), the word "together" is perhaps replaceable with the word "semi-directly"? iii) The "well-known fact" mentioned on top of page 10 should come with accompanying references iv) Perhaps a picture at the end of the discussion on page 15 could be clarificatory v) In the paragraph below (62a-62d), importantly, in the sentence "For ${\cal R}$, the implicit and explicit dependences can be shown..." the authors are referring to the charge ${\cal T}$, and not the charge ${\cal R}$. vi) Paragraph before Discussion on page 17: were the authors expecting such a result from T-duality? A bit of physical intuition and contextualization might be useful. vii) Top of page 17 "interesting and new asymptotic symmetry algebras". Aren't those known in 3d? Can the authors explain better what they have in mind related to this point?

Once these comments have been addressed, I will gladly reconsider this paper for publication in SciPost.

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Author:  Alejandro Vilar López  on 2025-07-06  [id 5624]

(in reply to Report 2 by Luca Ciambelli on 2025-06-12)

We would like to start by thanking the referee by his many insightful comments and interesting questions. We will split our reply into the main conceptual points made in the report and the minor typos and comments, and we will use in both cases the numbering introduced by the referee. We will also implement several changes in our manuscript, which will be available in a new resubmission

Main conceptual questions

  1. Both in the two final paragraphs of the introduction, the introductory paragraph to section 4, and the first two paragraphs of the discussion, we always refer to T-duality as a tool to generate new boundary conditions from existing ones. This is what we called asymptotic T-duality. This does not provide a direct transformation from one phase space to another one, and it was not our intention to claim such a result. Using words from the discussion, "T-duality transformations can inform the construction of a dual phase space that includes the dual solutions". So, they can inform the construction, but they don't produce such a construction of a dual phase space immediately. We tried to emphasize this more by adding a couple of sentences at the end of the first paragraph of the discussion. That said, it is true that when the original phase space has an exact isometry (as in our section 3), the relation is somewhat more direct. Our appendix A actually proves that in that case the asymptotic symmetry transformations are preserved.
  2. Following the suggestion made by the referee, we will include more details around equation (1) in our resubmission, in particular clarifying what the dynamical fields are and what the action describes (the low energy string effective theory governing the NS-NS sector).
  3. We acknowledge the distinction made by the referee, but we believe using "boundary conditions" to denote falloffs at infinity is a common abuse of language in the field. Note that even classic papers in the subject such as Brown-Henneaux (the following boundary conditions are generated) or Strominger's obtention of the BTZ entropy (In order to define the quantum theory on AdS3, we must specify boundary conditions at infinity) use this terminology. Taking this into account, and given that we are using "boundary conditions" throughout the paper in many different places, we believe changing the terminology to "falloffs" would be a major modification which would not add much to the clarity of the message being sent.
  4. It is unclear what is meant by the T-dual of a vector field in this question, as T-duality is a transformation which affects the dynamical fields on the manifold $(\Phi, G, B)$ but not other objects defined on it as, e.g., vector fields. It is true that (33) generates the asymptotic symmetry transformations of the metric (31), which is the result of applying T-duality to (27). It is also true that, in a situation like the one of Section 3 where there is an exact $U(1)$ isometry, there exists some relation between the asymptotic symmetry generators before and after duality: this is mentioned in footnote 9 and discussed in appendix A. The relation mixes diffeomorphisms and gauge transformations of the $B$-field though. Finally, this relation depends on having an exact isometry and it seems unlikely that one can use similar logic in a situation like the one discussed in section 4, where we only have an asymptotic Killing vector. Thus, (54) cannot be read from the symmetry transformations before duality, the asymptotic Killing vectors have to be found from scratch after having dualized the metric. Due to all of this, we believe introducing any notion of T-dual of a vector field would add more confusion than clarity: T-duality can be used to generate new boundary conditions from existing ones, but the analysis of the asymptotic symmetry transformations for each set of boundary conditions has to be done independently.
  5. This is a very interesting point. We were unaware of 2412.14992, which addresses the subtle question of large diffeomorphisms between gauges, and we are grateful to the referee for pointing this out. We believe properly answering this question requires very careful work, to an extent which would probably justify an independent project. Therefore, at this point, we will edit the draft to draw attention to this question, and we allow ourselves to make some preliminary comments here relating to our charges being or not kinematical:
    1. Our charges have indeed trivial flux, contrary to unconstrained. We believe this is related to the fact that the Weyl factor in 2412.14992 does not obey any equation of motion, while our function $A(z,w)$ does ($\partial_w^2 A = 0$). This makes us doubt that the charges we found can be seen as kinematical charges, in the language of that paper.
    2. From (51)-(53) in our paper, the symplectic form goes to zero at the boundary with the boundary conditions we impose in order to have a good variational principle. This seems to imply that there cannot be corner contributions, and it should be contrasted with equation (3.12) of 2412.14992, which is a key result in order to get unconstrained fluxes. From a different perspective, it seems very difficult to make our charges to vanish by adding a boundary term (although of course we have not ruled out this possibility). Even if it is possible, it seems T-duality is not going to be a map that does not change the charges, because we only get a single Virasoro tower, contrary to the two originally present in the Brown-Henneaux construction.
    3. It is true that at this point we do not know what happens when changing the gauge, and it is a question worth exploring about which we expect to report in the future (in particular, looking at our construction in Bondi gauge may shed light about several aspects, possibly one of them whether or not we are dealing with kinematical charges, as the referee suggests).

Minor points and typos

i) We will include more references about the role of the BTZ black hole in AdS/CFT in our resubmission.
ii) The modification will be implemented as suggested.
iii) It is difficult to identify a precise reference because it is a fairly evident fact that can be seen from the Buscher rules (30). We will explain it better in our resubmission so that this is clear.
iv) We will add a figure to help understanding our point.
v) This is indeed a typo which will be appropriately corrected. Thank you for pointing it out!
vi) If anything, we were expecting a result in which the asymptotic symmetry group would not change under T-duality, which is not what happened in the end. This is because T-duality in string theory defines equivalent worldsheet theories, and if the asymptotic symmetry group of the background is expected to capture the symmetry structure of a purportedly dual quantum theory, we would expect it to stay the same. That said, it is true that our theory is far from capturing the full structure of string theory, and the background asymptotics is heavily affected by T-duality, so it is also not completely surprising that we get different results at this level of the analysis. This is why we are not sure about making very clear comments about what to expect or not expect on the basis of physical intuition: we believe exploring the construction in new and different cases can actually help to build a better understanding (which we currently lack) of what is really going on.
vii) We do not think the algebra found in our paper was known, so that is an example of an interesting and new structure. The referee may be referring to the fact that in pure gravity some very general results exist (see 1608.01308 or 1704.07419), but our theory has also matter fields (corresponding to the massless states of the NS-NS sector of a string), and these can significantly modify the analysis. We are not aware of general results in such context. Furthermore, the construction itself can certainly be used beyond three dimensions, so we will add a comment in our resubmission clarifying this.

Report #1 by Anonymous (Referee 1) on 2025-4-24 (Invited Report)

Report

In this work, the phase space of black string solutions in three-dimensional asymptotically flat spacetime with a linear dilaton is claimed to have an intriguing large asymptotic symmetry algebra. It is obtained by applying asymptotic T-duality to their T-dual phase space of BTZ black holes. Although the asymptotic symmetries of such three-dimensional black strings were investigated previously (e.g. in ref. [34] of the present paper), the results that follow from asymptotic T-duality are new, raise questions, and are sufficient for publication.

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