Matrix Models and Holography: Mass Deformations of Long Quiver Theories in 5d and 3d

We thank both Referees for their careful reading of our work.
We have edited the manuscript, amended the typos and introduced additional clarifications, following the input of one of the Referees. We answer to the Referee's points in the section "List of changes".

We thank the Referee for the review and the additional comments and corrections. We have edited the manuscript taking them into account. We answer point by point below.

1) Our choice of mass deformation, and the way it is enforced, is explained in detail in Subsection 3.1.3.
Our claim is that we couple the hypers to a background 3d $\mathcal{N}=4$ vector multiplet in the canonical way. The scalar in this background vector multiplet is given a vev. With respect to the $\mathcal{N}=2$ subalgebra fixed by localization, the vev splits into a real mass and a complex mass. Our choice of mass deformation is that the real mass is $m$, and the complex mass is 0, where $m$ is a real number and 0 is a complex number.
The quantities we aim to compute are functions of the real mass $m$, and are insensitive to the complex mass. The dependence on the complex mass appears only through to the constraints imposed by a superpotential. This fact can be seen for instance in the exact evaluations given in the papers of Benvenuti-Pasquetti (ref. [135]) or Gaiotto-Okazaki (ref. [138]), or in Willett's review (ref. [108]) cited in Subsection 3.1.3.
The goal of our work is to study a particular deformation by real masses, not a superpotential deformation. This can be done without breaking any supersymmetry. The study of superpotential deformations is a separate problem that we do not address in the present paper.

If the Referee disagrees with the explanation of Subsection 3.1.3, it would be helpful for us if they could pinpoint what they believe is wrong there. If the Referee is still unsatisfied with our explanation, it would be helpful for us if they could explain to us in more concrete terms what they mean by studying $\mathcal{F}$ as a function of the complex masses.
2) In Figures 1 and 2, the horizontal arrows are meant to be the same in the CFT and gauge theory. They pictorially indicate the entire flow, which starts at $m=0$ and ends at $m=\infty$ in both cases. We have made it more explicit in the figures, indicating these values as suggested by the Referee.
3) Following the Referee's suggestion, we have indicated the $m \to \infty$ limit at the end of the arrow that represents the RG flow in Figure 8.
4) Misprint corrected, thanks for noticing it.
5) We have fixed the factors of $1/N!$ in Eq.(C.1) and subsequent. Thank you.
6) To answer to this point, we would like to emphasize that the detour in Appendix C is to exhaustively and explicitly explain in a simple example how to figure out what is the CFT at the endpoint of the RG flow. We of course do not claim any factorization at finite $m$.
As usual, we start with a CFT and deform it by a relevant operator, triggering an RG flow. At every finite $m$, the way of thinking of this model is as a QFT which is the deformation of the UV CFT. In general, one would like to know what is the endpoint of an RG flow. Here, we use the power of supersymmetry and localization to answer this question. The idea of exploring different candidate IR fixed points using changes of variables in the matrix integral was first used in Ref. [93].

This is the philosophy of Appendix C. To answer the question of what happens at $m\to \infty$, we do not need to carefully keep track of all the terms. We only need to understand which ones are not killed by the limit and keep track of them.
To use the approximation in the equation for $S_{\text{int}}$ below (C.5) we split the integral into a sum of contributions. One of them is the integral over $-m < \sigma_a < m$ and $-m < \phi_{\dot{a}} < m$, which becomes an integral over the two Cartan subalgebras of $\mathfrak{u}(N_1)$ and $\mathfrak{u}(N_2)$ as $m \to \ifnty$. All the other terms vanish in the large $m$ limit, due to the domain shrinking to zero measure. These terms vanish faster than the one we retain: terms $e^{- \pi \lvert \sigma -m \rvert - \pi \lvert \sigma +m \rvert }$ are more suppressed than $e^{-2\pi m}$ if $\sigma >m$ or $\sigma <-m$, and, in addition, the integration domain shrinks. These terms are doubly-suppressed and can be safely neglected in the limit $m \to \infty$.
We added a comment below (C.5) to clarify this point.
7) Eq. (C.6) is obtained as follows: We are making $(N+1)$ different changes of variables in the same integral, which is the UV partition function. We thus obtain $(N+1)$ different-looking integrals that are in fact the same partition function. We sum all of them and divide by $N+1$, and get again the same result. Eq (C.6) reads $\mathcal{Z}= (\mathcal{Z}+\mathcal{Z}+...+\mathcal{Z})/(N+1)$, where the notation $\mathcal{Z}[N_2]$ on the r.h.s. of (C.6) is to remember which change of variables we use in each summand. $\mathcal{Z}[N_2]$ is given in (C.5), and the $[N_2]$ notation is to insist that we are making one choice of $N_2$. We have improved the definition and notation in Eq. (C.5) in accordance with the Referee's comment.
No approximation whatsoever is made in (C.6).
8) We have corrected the missing factor of 1/2 in passing from the equation below (C.5) to the equation above (C.6). Thanks a lot for pointing it out.
9) Yes, (C.7) becomes 0 in the limit $m \to \infty$. This is a generic feature of massive QFTs, and is cured by appropriately adjusting the regularization scheme. It is straightforward to check, for instance, that also the partition function of SQED3 with $F$ flavours vanishes if $m\to \infty$. This follows immediately from expressions (2.13)-(2.16) of Benvenuti-Pasquetti (ref. [135]) or (4.19) of Gaiotto-Okazaki (ref. [138]).
The various terms in the sum (C.7) will in general go to 0 in different ways as $m \to \infty$. Thanks to our rewriting, we identify the term that is less suppressed.

What we want to conclude is that we must focus on this term as $m \to \infty$, since we can get a finite partition function out of it by adding a counterterm in the standard way.
The other way around: imagine we pick a summand randomly in the sum in (C.7) and add a counterterm to the action so that the randomly chosen term is finite in the limit $m \to \infty$. Then, the $F_2=2N_2$ term will diverge as $m \to \infty$ due to the counterterm.
9bis) The factor of 2 is due to the fact that we are giving mass $+m$ to one hyper and mass $-m$ to another hyper, as per the first equation of Appendix C.2.5. Also, we are not subtracting the free energy $2\ln (2\cosh (\pi m))$, but subtracting the scheme-dependent part using a gravitational counterterm with coefficient $m$ if $m>0$ and $-m$ if $m<0$.
Subtracting $2\ln (2\cosh (\pi m))$ would not correspond to such a counterterm (see e.g. ref. [124] of our manuscript), and moreover would not interpolate between the correct values. Our definition of $\mathcal{F}_{\text{EFT}}$ is of the form $\mathcal{F} $ minus local, gauge-invariant counterterms, which is what we are after.
10) We have improved the caption to Figure 12 to amend the lack of clarity pointed out by the Referee. The UV and IR CFTs in the plot are ($g_{YM}^2 \to \infty$ fixed points of) SQED with $F$ and $F-2$ flavours, respectively. We denote by $\mathcal{F}_{\text{UV}}$ and $\mathcal{F}_{\text{IR}}$ the respective free energies.