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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 accepted: 2024-02-06
Date submitted: 2024-01-17 05:56
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

Based on the last referee report:
1. New citation added for existing result relating bow equations to the vacuum equations of 4d N=4 with impurity walls (second paragraph, p. 3).
2. References cited for:
a) The result that 2d B-model with superpotential W localizes to 0d theory with action W upon omega deformation, concerning eq. 4.46.
b) Quantization of $gl_n^*$ to the universal enveloping algebra $U_\hbar(gl_n)$ upon turning on twisted masses (paragraph containing eq. 5.5).
c) Localization of 3d N=4 theory to topological quantum mechanics by omega deformation (second paragraph of section 2.1).
3. In the second paragraph on p.2: The statement regarding coupling a quantum mechanics with global symmetry to a gauge theory changed slightly to make it clearer.
4. Support of $\vec \omega$ corrected after eq. 2.1.
5. Mentioned $K_i$ and $L_p$ to be integers, before eq. 2.2.
6. Spelled out the meaning of "unsymmetrization" in Remark 5.3.
7. Corrected several spelling errors.

Published as SciPost Phys. 16, 052 (2024)

Reports on this Submission

Anonymous Report 2 on 2024-1-21 (Invited Report)


After a long refereeing process, the authors have addressed a lot of problems that were present in the original manuscript, and answered many questions asked by the referees. There may still be some minor issues here and there (see, e.g., my comment to the author's reply to the first referee report). However, at this point I can confidently recommend this article for publication in the current form.

  • validity: -
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  • originality: -
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Anonymous Report 1 on 2024-1-20 (Invited Report)


I recommend this paper for publication in its current form.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

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