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3d Large $N$ Vector Models at the Boundary
by Lorenzo Di Pietro, Edoardo Lauria, Pierluigi Niro
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Submission summary
Authors (as registered SciPost users):  Lorenzo Di Pietro 
Submission information  

Preprint Link:  https://arxiv.org/abs/2012.07733v2 (pdf) 
Date submitted:  20210122 09:59 
Submitted by:  Di Pietro, Lorenzo 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We consider a 4d scalar field coupled to large $N$ free or critical $O(N)$ vector models, either bosonic or fermionic, on a 3d boundary. We compute the $\beta$ function of the classically marginal bulk/boundary interaction at the first nontrivial order in the large $N$ expansion and exactly in the coupling. Starting with the free (critical) vector model at weak coupling, we find a fixed point at infinite coupling in which the boundary theory is the critical (free) vector model and the bulk decouples. We show that a strong/weak duality relates one description of the renormalization group flow to another one in which the free and the critical vector models are exchanged. We then consider the theory with an additional Maxwell field in the bulk, which also gives decoupling limits with gauged vector models on the boundary.
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Reports on this Submission
Report #2 by Anonymous (Referee 2) on 202172 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2012.07733v2, delivered 20210702, doi: 10.21468/SciPost.Report.3171
Strengths
The authors provide a new and detailed analysis of an interesting quantum field theory with a boundary. The theory has a free scalar in the bulk that couples to either fermions or scalars on the boundary in a fundamental representation of O(N).
Weaknesses
The ratio of work required to information gained is large. A dozen pages of appendices provide details of the Feynman diagram calculations performed, while for the main characters in the paper no new interacting boundary conformal field theory is found. Instead, the bulk and boundary decouple at the fixed points, leading to well known interacting purely 3d CFTs. The one exception is their theory involving boundary scalars coupled to both a bulk scalar and photon, with a theta angle. In this case, in appendix B, they claim to find such interacting examples. I wonder if appendix B should have been the central focus of the paper. (Note that a bulk theta term is equivalent in this boundary context to a boundary ChernSimons term.)
Report
There is a sense in which this paper can be considered as a followup of an observation in Witten's 2001 paper [9] of a duality involving two equal mass scalar fields in antide Sitter space with different boundary conditions. After a Weyl transformation and a replacement of one of the bulk fields with its boundary degrees of freedom, one finds the type of large N field theories studied in this paper. It is very satisfying to see all the details of this old observation worked out in this more familiar boundary QFT context.
Without hesitation, I recommend publication.
Requested changes
There were a couple of minor points:
In the introduction $\lambda$ is introduced without stating that it is the gauge coupling.
I lost the thread of the argument in the first bullet point on p 15. The authors state ``the operators $\Phi$ vanishes in the limit, as expected for a decoupled bulk free scalar with Neumann boundary condition''. Naively, I would want $\partial_\perp \Phi$ on the boundary to vanish for Neumann, and so I was not sure what was being said. It might help to reference some equations earlier in the draft, (2.24) and (2.25) for example.
Report #1 by Anonymous (Referee 1) on 202141 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2012.07733v2, delivered 20210401, doi: 10.21468/SciPost.Report.2751
Strengths
1. The paper is very clearly written.
2. The results match previous results in various limits, so are probably correct.
Weaknesses
1. It's a bit disappointing that only the standard noninteracting conformal boundary conditions were found in the theories considered, tho this matches numerical evidence from a previous bootstrap study. The authors mention that generalizing their work to ChernSimons matter theories could possibly give interacting boundary conditions, which would be very interesting.
Report
This paper presents results on various bosonic and fermionic vector models in 3d coupled to a free scalar in 4d, studied in the limit of many flavors. In particular, they find that the coupling of various theories come in dual pairs, which they prove using a path integral argument. The paper is well written and correct, and deserves to be published.
Requested changes
1. When the authors consider the fermionic theory coupled to a U(1) gauge field, they should keep in mind that the parity anomaly requires that the number of complex 2 component fermions be even. This will not qualitatively change the main results tho.
2. In the conclusion the authors say "due to to the presence of matrixlike degrees of freedom in the large N limit". When the number of colors and ChernSimons level are both simultaneously large, the theory in fact still has vector degrees of freedom, which is in part why it is still believed to be dual to Vasiliev theory (with a parity breaking term).
3. The authors consider generalizing their result to ChernSimons matter theories with many colors. They should also consider adding a ChernSimons term just to the U(1) gauge field, which is much simpler, and should give different boundary conditions for each integer value of the ChernSimons level.
4. The authors should also address monopole operators, which appear in QED3 in general. How are these operators affected by the 4d bulk theory? Is it clear how they would map across the duality? Similarly, it would be interesting to see how nonlocal operators like Wilson loops map across the duality.