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Superspin chains from superstring theory
by Nafiz Ishtiaque, Seyed Faroogh Moosavian, Surya Raghavendran, Junya Yagi
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Submission summary
Authors (as registered SciPost users):  Nafiz Ishtiaque · Seyed Faroogh Moosavian 
Submission information  

Preprint Link:  scipost_202202_00033v1 (pdf) 
Date accepted:  20220823 
Date submitted:  20220218 20:02 
Submitted by:  Ishtiaque, Nafiz 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We present a correspondence between twodimensional N=(2,2) supersymmetric gauge theories and rational integrable gl(mn) spin chains with spin variables taking values in Verma modules. To explain this correspondence, we realize the gauge theories as configurations of branes in string theory and map them by dualities to brane configurations that realize line defects in fourdimensional ChernSimons theory with gauge group GL(mn). The latter configurations embed the superspin chains into superstring theory. We also provide a string theory derivation of a similar correspondence, proposed by Nekrasov, for rational gl(mn) spin chains with spins valued in finitedimensional representations.
Published as SciPost Phys. 13, 083 (2022)
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 2 on 2022816 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202202_00033v1, delivered 20220816, doi: 10.21468/SciPost.Report.5546
Strengths
1 very clear
2 very systematic
Weaknesses
1 collection of new and known results, not always clear which is which
Report
The paper certainly meets the requirements for publishing.
The authors are very clear and present precise statements establishing as facts statements that were more or less already present in the literature at the level of vague conjectures.
To my knowledge this is the most complete dictionary of the gaugeBethe correspondence. It is to be hoped (expected?) that it will lead to progress either in the study of supersymmetric gauge theories, or in integrable models, or both.
Anonymous Report 1 on 202246 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202202_00033v1, delivered 20220406, doi: 10.21468/SciPost.Report.4870
Strengths
1. The paper is extremely wellorganized and wellwritten. Whenever needed, concise and pedagogical reviews are given both on integrability and the brane construction of supersymmetric gauge theories, most of which I found more accessible than the ones that can be found in the literature.
2. Careful, detailed, and thorough analysis: detailed exposition is given on how the spinchain description arises from the brane construction and the string duality. The authors are also careful on every detail and various subtle points (e.g. the difference between compact and noncompact representations.)
Weaknesses
1. It certainly provides new perspectives on the Bethe/gauge correspondence and its relation to the 4d holomorphic ChernSimons theory. However the analysis in this paper does not seem to lead to new results either in spin chain or supersymmetric gauge theories, at least for the moment. Of course, having a complete understanding on the subject is a stepping stone for the future discovery (and I personally feel that this will be the case), but it could be seen as a weak point of this paper.
2. The main achievement of the paper is extending the previous results on the relation between the Bethe/gauge correspondence and the 4d holomorphic ChernSimons theory to spin chains with supergroup symmetries. It is certainly of interest from a mathematical point of view, but the superspin chain is a rather exotic object, not commonly found in the condensed matter system. So the relevance and the importance of the paper to a wide physics community could be debated.
Report
This paper explains how the Bethe/gauge correspondence for the super spin chain proposed by Nekrasov can be understood from the 4d holomorphic ChernSimons theory constructed by Costello. The key idea is to realize both in terms of intersecting branes and use the duality of string theory.
The results are new and important and the paper is wellwritten. So I think it is clearly above the scientific threshold for the publication in SciPost (and other journals). The only thing I would like to ask the authors is to clarify the point below.
Requested changes
1. In subsection 3.4.2, the authors explain the socalled fermionic duality of spin chains using the HananyWitten effects of branes. However, in the duality frame they use (which is given in (3.13)), there seem to be no NS5 branes although they use the NS5 branes in the discussion. I would like to ask the authors to clarify this point.
Author: Nafiz Ishtiaque on 20220409 [id 2367]
(in reply to Report 1 on 20220406)We thank the referee for the wellconsidered report and for pointing out important typos in section 3.4.2 discussing fermionic dualities.
Indeed, in section 3.4.2 the duality frame in which fermionic duality corresponds to HananyWitten transitions is not exactly the one in eq. 114 (this equation no. refers to the scipost submission, in the current arxiv version it is the eq. 3.13 as the referee pointed out), rather it is the Sdual of it. In the Sdual frame, we have D3 branes ending on NS5s. Fermionic duality corresponds to swapping the positions of two adjacent NS5 branes with opposite gradings (i.e. wrapping different omega deformation planes), along with the D3 branes attached to them. This causes D3 branes and NS5 branes to intersect creating D1 branes according to the HananyWitten mechanism. We note importantly, that in section 3.4.2 when we say F1 branes are being created/destroyed in the transition we should also clarify that these F1s are the Sduals of the aforementioned D1s. These changes will be made in the final version of the paper.
Finally, we would like to address the first weakness of the paper, as mentioned by the referee. While we agree that we have not presented any new results in the language of superspin chains in this paper, we believe that on the supersymmetric gauge theory side our proposal for the quiver gauge theories that are dual to the noncompact superspin chains is new (eq. 73 in the scipost submission, or eq. 2.71 in the arxiv version). This, in particular, suggests new quiver varieties (Higgs branches of these quiver gauge theories) whose equivariant cohomology carries representations of super Yangians (conjecture 1 and 2 in section 1)  this is a concrete new proposal that can be subjected to further mathematical investigations.