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Dynamically restoring conformal invariance in (integrable) $σ$-models

by Rigers Aliaj, Konstantinos Sfetsos, Konstantinos Siampos

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Authors (as registered SciPost users): Konstantinos Siampos
Submission information
Preprint Link:  (pdf)
Date submitted: 2022-05-25 09:55
Submitted by: Siampos, Konstantinos
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


Integrable $\lambda$-deformed $\sigma$-models are characterized by an underlying current algebra/coset model CFT deformed, at the infinitesimal level, by current/parafermion bilinears. We promote the deformation parameters to dynamical functions of time introduced as an extra coordinate. It is conceivable that by appropriately constraining them, the beta-functions vanish and consequently the $\sigma$-model stays conformal. Remarkably, we explicitly materialize this scenario in several cases having a single and even multiple deformation parameters. These generically obey a system of non-linear second-order ordinary differential equations. They are solved by the fixed points of the RG flow of the original $\sigma$-model. Moreover, by appropriately choosing initial conditions we may even interpolate between the RG fixed points as the time varies from the far past to the far future.Finally, we present an extension of our analysis to the Yang-Baxter deformed PCMs.

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Reports on this Submission

Anonymous Report 2 on 2022-9-26 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2205.09143v1, delivered 2022-09-26, doi: 10.21468/SciPost.Report.5768


1- The paper suggests a possible embedding of integrable $\sigma$-models into 1-loop conformal ones by adding a time direction and inserting time-dependence in the couplings and the dilaton background.

2- It is checked in various examples that time-dependence may be chosen to successfully engineer a 1-loop conformal theory.

3- The resulting theories seem to be some previously unknown conformal $\sigma$-models.


1- It is not addressed whether the resulting 1-loop conformal theories are integrable (e.g. after gauge fixing).

2- The paper focuses on examples related to rank-1 groups. Would it be possible to consider $\lambda$-models based on general groups or cosets?

3- The generality of this picture is not discussed. It would be good to consider adding a time direction and inserting time-dependence in a general metric to see in which way the time-dependence must be inserted for conformal invariance. In this way, it could be possible to prove, or give evidence for, the claim of this paper in greater generality.



The authors consider several integrable $\sigma$-models based on rank-1 groups and their products and cosets. They add an extra "time" direction to the target space and insert time-dependence in the models' couplings (the deformation parameter, etc.), as well as the dilaton. They make the interesting observation that, in these examples, the time-dependence may be chosen so that the resulting $\sigma$-models are 1-loop conformally invariant. In some cases the correct pattern of time-dependence is explicitly solved, while in others it is solved approximately near fixed points.

This is an interesting idea since it has the potential to embed integrable $\sigma$-models into conformally invariant ones that make contact with string theory. The paper is, for the most part, clearly written, and I recommend its publication subject to minor revisions set out below.

General comments:

It would be good to address whether the resulting time-dependent theories are, in any sense, integrable. This seems like a crucial question: the $\lambda$- and $\eta$-deformed models considered here are interesting principally because they are classically integrable, so a useful embedding in string theory should somehow preserve this property.

It would also be good to consider more general examples (e.g. not based on rank-1 groups), and to provide evidence that this picture extends beyond models related to $\lambda$-deformations. (Note that the $\eta$-deformation considered in Section 7 is Poisson-Lie dual to a $\lambda$-deformed model, so might not be seen as an independent example. Does this Poisson-Lie duality commute with the insertion of time-dependence?)

Below equation 1.4 when the main idea is introduced, it may be helpful to include an equations demonstrating the general idea: promoting a metric $G_{MN} dX^M dX^N$ to a time-dependent one $-dt^2 + G_{MN}(t) dX^M dX^N$.

Requested changes

The authors are asked to add a brief discussion (and if possible a careful treatment) of the 3 "weaknesses" stated above.

  • validity: good
  • significance: ok
  • originality: ok
  • clarity: good
  • formatting: good
  • grammar: good

Anonymous Report 1 on 2022-7-20 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2205.09143v1, delivered 2022-07-20, doi: 10.21468/SciPost.Report.5421


Please see below


Please see below


This paper contains a very interesting study of how to restore conformal invariance in sigma-models. The technique is worked out in great detail and showed in action in a large wealth of examples.

The paper is written clearly, the study is very thorough and extremely convincing, and of definite interest. I recommend its publication.

Here below are a few naive observations which the authors may or may not decide to take into account. I consider them merely as an opportunity for interesting discussion.

Requested changes

1. One of the key ideas around which the method gravitates seems to me to be an identification of the RG scale with the target-space time, so that the authors can speak of an RG flow occurring "as time progresses". It would perhaps be useful to spend some comments at the beginning of the paper highlighting this point and showing explicitly (or commenting in some detail) how the dynamical identification of scales work.

2. By adding a term to the Lagrangian and promoting a target-space time-dependence in some of the parameters the authors have achieved the goal of preserving conformal invariance. This seems to naively suggest that, as a net effect, the authors have found a complicated way of writing (the most general?) marginal deformations. Is this ultimately one way to interpret these results? Once again a few more comments might clarify or expand on this naive point.

3. In the Conclusions, the authors comments on the extension to all loops, and how this clearly generates a more complicated set of differential equations, to be solved in order to restore conformality. It would be useful to perhaps provide a more explicit justification as to whether a nontrivial solution is still expected to exist, given that this is crucial for the success of the method.

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

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