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Fermionic BogomolnyiPrasadSommerfield Wilson loops in fourdimensional $\mathcal{N}=2$ superconformal gauge theories
by Hao Ouyang, JunBao Wu
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Submission summary
Authors (as registered SciPost users):  Hao Ouyang · JunBao Wu 
Submission information  

Preprint Link:  https://arxiv.org/abs/2205.01348v2 (pdf) 
Date submitted:  20220606 11:44 
Submitted by:  Wu, JunBao 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
In this paper we construct fermionic Bogomol\`nyiPrasadSommerfield (BPS) Wilson loops in $\mathcal{N}=2$ superconformal $SU(N)\times SU(N)$ quiver theory and $\mathcal{N}=4$ super YangMills theory. The connections of these fermionic BPS Wilson have a supermatrix structure. We construct timelike BPS Wilson lines in Minkowski spacetime and circle BPS Wilson loops in Euclidean space. These Wilson loops involve dimensionful parameters. For generic values of parameters, they preserve one real (complex) supercharge in Lorentzian (Euclidean) signature. Supersymmetry enhancement for Wilson loops happens when the parameters satisfy certain constraints.
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Reports on this Submission
Anonymous Report 1 on 202288 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2205.01348v2, delivered 20220808, doi: 10.21468/SciPost.Report.5242
Strengths
1) the authors apply to 4dimensional theories an algorithm to construct fermionic BPS Wilson loops that was previously only used in the case of 3dimensional theories, so this novelty is a strength of the paper
2) the methodology used in the construction is sound
3) the computations, to the extent I could check them, are correct
4) the paper is clearly written, albeit quite technical
Weaknesses
1) some novelties appearing in the 4dimensional case should be further discussed: for example, the appearance of dimensionful couplings in the definition of the fermionic Wilson loops
2) the discussion of what is expected to happen at the quantum level could be improved
3) maybe some more details about the computation of the cohomological equivalence could be given (for example, the expression for V at the lowest orders in the expansion)
4) the paper is quite technical, albeit clearly written
Report
The authors apply to 4dimensional theories an algorithm to construct novel supersymmetric Wilson loops, that was previously discovered in the context of 3dimensional quiver theories, like ABJ(M) and N=4 ChernSimonsmatter theories. This is an interesting attempt, resulting in a few novel Wilson loops defined in terms of superconnections. An obvious difference with the 3dimensional case is that the couplings to the fields appearing in this superconnection are dimensionful, as can be immediately seen by dimensional analysis. The construction is quite interesting and the computations are correct, for the extent I could check them.
Requested changes
1) I would improve the discussion of the first 3 points mentioned above in the "Weaknesses" section.
Anonymous Report 2 on 2022724 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2205.01348v2, delivered 20220724, doi: 10.21468/SciPost.Report.5449
Report
The results are relevant but the way they are presented can be improved. I suggest a revision before publishing. I attach a PDF with detailed comments and suggestions that I believe would increase the quality of the paper.
Author: JunBao Wu on 20220920 [id 2832]
(in reply to Report 2 on 20220724)
We thank the referee for his/her report and careful reading of our manuscript.
1) We thank the referee for pointing out many typos, which we have corrected in the revised version.
2) We added a paragraph on page 3 and a footnote on page 4 to address the referee's comments on our introduction. We also added a paragraph on page 4 to stress that this is the first construction of DrukkerTrancanelli type fermionic BPS Wilson loops in 4d.
3) We added a sentence above (48) to mention how the $G_s^2$ piece is obtained in section 2.3.
4) The number of preserved supercharges is equal to the number of free parameters in $u_\alpha$. In another word, it is just the dimension of the linear space of the preserved supercharges. Detailed clarifications were added on page 12.
5) It should be $QF$ above (44), as pointed out by the referee.
6) We added a paragraph at the end of section 4 to discuss the consequences of scale invariance breaking due to fermions in the WL.
7) We changed the order of B.1 and B.2.
Author: JunBao Wu on 20220920 [id 2831]
(in reply to Report 1 on 20220808)We thank the referee very much for his/her valuable comments. We address the requests:
1) We added a paragraph at the end of section 4 to discuss the consequences of the dimensionful couplings.
2) We added a comment about the framing issue at the end of the second paragraph in section 4 which is related to the perturbative computation of the WL vev.
3) We gave the first few order result of V in Eq (53). The complete construction of V can be performed following the procedure in appendix D of 1506.06192.