SciPost logo

SciPost Submission Page

Classical Lie Bialgebras for AdS/CFT Integrability by Contraction and Reduction

by Niklas Beisert, Egor Im

This is not the latest submitted version.

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Niklas Beisert
Submission information
Preprint Link: scipost_202212_00020v1  (pdf)
Date submitted: 2022-12-07 23:20
Submitted by: Beisert, Niklas
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
  • Mathematical Physics
Approach: Theoretical


Integrability of the one-dimensional Hubbard model and of the factorised scattering problem encountered on the worldsheet of AdS strings can be expressed in terms of a peculiar quantum algebra. In this article, we derive the classical limit of these algebraic integrable structures based on established results for the exceptional simple Lie superalgebra d(2,1;𝜀) along with standard sl(2) which form supersymmetric isometries on 3D AdS space. The two major steps in this construction consist in the contraction to a 3D Poincaré superalgebra and a certain reduction to a deformation of the u(2|2) superalgebra. We apply these steps to the integrable structure and obtain the desired Lie bialgebras with suitable classical r-matrices of rational and trigonometric kind. We illustrate our findings in terms of representations for on-shell fields on AdS and flat space.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2023-2-6 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202212_00020v1, delivered 2023-02-06, doi: 10.21468/SciPost.Report.6686


This is a clearly presented and rigorous paper.




This paper provides some detailed and interesting results on the integrable origins of the AdS/CFT and Hubbard S matrices. The paper is easy to follow because of the clear presentation. It would be interesting to identify a string theory interpretation for the results. I have no hesitation in recommending publication.

Requested changes

1. Should the reference to (3.29) above (4.9) instead refer to (3.19)?
2. The ultra-short representations of d(2,1;a) that are discussed in section 6.2 are attributed to the 2019 paper [32]. As far as I can see they have been known for a long time following Kac's original list of shortening conditions. In the holographic integrability literature they are discussed in section 3.1 of 1106.2558 following the original discussion from 1985 by van der Jeugt (see reference [66] of 1106.2558). I would recommend adjusting the referencing in section 6.2 to reflect that.
3. Could the authors comment on how their construction may fit into worldsheet string theory? Can one hope to see any of the extended symmetries used in their construction to feature in some way in the string theory?

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

Author:  Niklas Beisert  on 2023-02-14  [id 3357]

(in reply to Report 2 on 2023-02-06)
reply to objection

We would like to thank the referee for the careful reading and useful suggestions. Regarding point 1, indeed, there was a typo with a wrong reference, the intended reference is (3.27) which is based on (3.19). We also agree with the citation issue raised in point 2. We will adjust the citation to refer to the original paper on the representation theory as well as some applications of the latter. Finally, the question of interpretation of the symmetry algebras discussed in the paper within the worldsheet picture is indeed interesting and important. We point out the second work in [26] in section 4 and in the conclusions, where some hints might be found. We will also comment separately on this question in the conclusions as it is an important point to draw one's attention to.

Anonymous Report 1 on 2023-1-1 (Invited Report)


Mathematical rigour




Please see document attached

Requested changes

Please see document attached


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

Author:  Niklas Beisert  on 2023-02-02  [id 3298]

(in reply to Report 1 on 2023-01-01)

We would like to thank the referee for the kind report. We fully agree that the two mentioned points are exciting questions for followup work. One thought that crossed our minds regaring item 1 was the second work in [26] which we mentioned at the beginning of section 4 and in the conclusions. We comment on item 2 in the second paragraph in the conclusions. We believe that there will be no obstacle towards quantisation, only that practical quantum constructions will be very challenging, as we probably cannot rely on established methods. Since the present classical consideration is already (too) complex, we considered it better to leave any further investigation or speculations in the quantum direction aside for the present paper. Nevertheless, we could once again consider the two suggestions and emphasise them a bit more in the conclusions because these are indeed interesting aspects!

Login to report or comment