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Line Operators in 4d Chern-Simons Theory and Cherkis Bows

by Nafiz Ishtiaque, Yehao Zhou

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Submission summary

Authors (as registered SciPost users): Nafiz Ishtiaque
Submission information
Preprint Link:  (pdf)
Date submitted: 2023-10-03 04:33
Submitted by: Ishtiaque, Nafiz
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
  • Mathematical Physics
Approach: Theoretical


We show that the phase spaces of a large family of line operators in 4d Chern-Simons theory with $\text{GL}_n$ gauge group are given by Cherkis bow varieties with $n$ crosses. These line operators are characterized by Hanany-Witten type brane constructions involving D3, D5, and NS5 branes in an $\Omega$-background. Linking numbers of the five-branes and mass parameters for the D3 brane theories determine the phase spaces and in special cases they correspond to vacuum moduli spaces of 3d $\mathcal{N}=4$ quiver theories. Examples include line operators that conjecturally create T, Q, and L-operators in integrable spin chains.

List of changes

Main changes:
1. Clarified the meaning of phase space, second paragraph of section 1.
2. Clarified the meaning of "monodromy lines", 3rd paragraph of page 2.
3. Added a paragraph at the end of section 1 summarizing the new results of the paper.
4. Added a paragraph at the beginning of section 3.2 explaining the reason for considering the Kapustin-Witten twist of 4d N=4 in the given context, as opposed to some other topological twist.
5. Added footnote 18.

Additionally, several typos have been corrected and minor modifications to some sentences have been made to clarify their meaning based on the referee report on the previous submission.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 1 on 2023-11-23 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2211.00049v3, delivered 2023-11-23, doi: 10.21468/SciPost.Report.8182


See the attached pdf file.


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Author:  Nafiz Ishtiaque  on 2024-01-13  [id 4245]

(in reply to Report 1 on 2023-11-23)

We thank the referee for the careful review. We also thank the referee for bringing three relevant references, especially Cherkis, O'Hara, Saemann, Phys. Rev. D, 83:126009 to our attention. This latter one clearly derives the relation between the bow varieties and supersymmetric vacuum equations of 4d N=4 theories with impurity walls. We presented this relation in our paper as part of our results -- but clearly, the paper by Cherkis, O'Hara, and Saemann is in fact the original reference for this result, of which we were regrettably unaware. We also present the equivalence between the vacuum equations and bow varieties in the context of 2d BF theory, which we hope will be of interest to researchers working with topological twists of supersymmetric gauge theories. This and the other references will be added to the new version of the paper. For more detailed responses to the other points mentioned in the review please see the attached document.



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