3d Large $N$ Vector Models at the Boundary

Dear editor,

we would like to thank both referees for reading our manuscript and for their useful comments.

Before addressing their specific requests, we would like to reply to the comment from both referees about only having fixed points with bulk and boundary decoupled (though we bring to the attention of referee 1 that actually in the appendix B we do consider an example of a fixed point with bulk-boundary interactions). We certainly agree that interacting conformal boundary conditions are of prime interest. On the other hand we think that the interesting aspect of our models is that they provide examples of non-trivial boundary RG flows that can be followed from the UV to the IR, and admit dual descriptions from the two endpoints. In our opinion it is surprising and noteworthy that taking the limit of infinite bulk/boundary coupling one finds decoupling. As we tried to stress in the introduction, we find especially interesting that certain local interacting CFTs emerge from a boundary RG because they decouple from the bulk. Since we wanted to give more attention to this aspect we only studied the example with the theta term in the appendix B, and we decided to leave for the future a more thorough analysis of these interacting fixed points, including also the case with boundary fermions.

Let us now address one by one the more specific points the referees raised.

Reply to referee 1:

• 1: We agree with the referee and we added a comment above equation (3.17);
• 2: We agree that our terminology “matrix-like degrees of freedom” was poor given that the CS gauge fields do not propagate, we just meant to refer to the fact that the diagrams are considerably more involved than those of ungauged vector models, and indeed in a generic gauge one needs to resum all planar diagrams, like in a matrix-like large N limit. In any case to avoid imprecision we erased “due to the presence of matrix-like degrees of freedom”;
• 3: We agree that this is an interesting direction. The theory with a CS level for the U(1) gauge field in 3d can be obtained by adding a bulk theta term for the 4d Maxwell field and going to infinite gauge coupling in the bulk (through EM duality). An initial study of this was performed in appendix B in the case of scalar matter on the boundary, as mentioned above. We did not consider the fermionic case because in the absence of parity there are more interactions to consider and the analysis changes substantially, so we decided to leave it for future work;
• 4: Monopole operators in this setup would appear as endpoints of bulk ’t Hooft lines. Following the suggestion by the referee, we added this statement in the footnote 7, in which we were already discussing how operators charged under the gauge group can appear as endpoints of Wilson lines. We agree that analyzing such line operators is an interesting direction but we believe it goes beyond the scope of the present paper.

Reply to referee 2:

We added the definition of the coupling lambda in the introduction. Indeed there was a typo in the bullet of pag. 15 and we meant to say that the normal derivative of Phi goes to 0, we amended that.

Besides the requests from the referees, we made the following change: We realized that we had an inconsistency in the normalization of the kinetic term/propagator for the Majorana fermions compared to the quartic interaction, and therefore we had to correct various factors of 2, in their action and in their propagator. However this does not affect any physical result.