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

We are grateful to the Referees for the careful reading of the manuscript, and for the several precise and detailed comments. All of them have been taken into account and incorporated into the new version.
We have added several clarifications about the details of the RG flow and how the gauge group is Higgsed in the process. We have emphasized that this behaviour is related to the non-generic choice of mass deformation. More details on the relation with the Hanany-Witten brane setups have been added.
We have also corrected two typos in two formulas, that led to an apparent mismatch by an overall factor of $\pi$ between AdS and CFT in the previous version.
We consider that the new version includes all the improvements requested by the Referee and satisfies the requests. See below, "List of changes" for a detailed and point-by-point answer.

*** Referee 1 ***

We are grateful to the Referee for the careful reading of the manuscript, and for the several precise and detailed comments. All of them have been taken into account and incorporated into the new version.
We have carefully and seriously addressed the points raised by the Referee one by one, improving the overall clarity of the presentation. Below we answer in more detail, listing the modifications in the text.

-- Main comments.
1) We have clarified the discussion about mass deformation for 3d N=4 quivers. In order to unify the language with the 5d case, and because only the real component of the triplet of masses enters the matrix model computation, we have used the nomenclature ``real mass''. The deformation we consider does not break supersymmetry. It corresponds to the usual (supersymmetric) coupling of the dynamical hypermultiplets to a background vector multiplets for the flavour symmetry. We have improved the text to avoid confusion in the 3d N=4 case. In particular, at the beginning of Subsection 3.1 we have changed the sentence ``It can be readily exported to the 3d N=4 case, because we only consider real masses'' with ``It can be readily exported to the 3d N=4 case, as we spell out in Subsection 3.1.3''. We have then added Subsection 3.1.3, explaining the setup in the 3d case. We hope there is no confusion now that the deformation is perfectly analogous to what happens in 5d, and preserves the eight supercharges and the $SU(2) \times SU(2)$ R-symmetry. This is consistent with the supergravity background.

3) We thank the Referee for pointing out a possible lack of clarity in the previous version. The answers to the Referee's second and third bullet points are related, so we start by addressing the third bullet point.
The Referee's analysis of the Hanany--Witten configuration is correct. However, that one is not the only configuration consistent with the mass deformation. The Referee considers the case $N_2=0$ in our language. We, instead, list all the consistent Hanany--Witten configurations, namely all choices of $N_2$. All of them leave a certain amount $N_1 F_1$ of massless strings at the position of the first set of flavour and colour branes, plus leave the amount $N_2 F_2$ of massless strings at the position of the second set of flavour and colour branes. They also give mass to W-bosons, by stretching the strings between the two stacks of colour branes.
We stress that, in the Hanany--Witten setup, one is free to move the branes and each given configuration in which the colour branes are separated corresponds to a different point of the Coulomb branch of the gauge theory.
At this point, we ask which of the configurations survives the RG flow. The answer we find is $N_2 = F_2 /2$. Incidentally, this result guarantees that the IR quivers are good and moreover no imaginary FI terms are generated.
In other words, to take a bunch of flavour branes and send them to infinity keeping all colour branes fixed, gives a perfectly valid Hanany--Witten configuration. However, this is not the configuration that lives at the end of the RG flow we consider.

We have modified the text to more accurately reflect these observations. We have added a new Subsection 3.1.4, together with an accompanying Appendix (Appendix C.1 of the new version) in which we elaborate on the Referee's observation.

As per the Referee's comment on adding FI parameters by moving the NS5-branes, note that our entire setup, both in 3d and 5d, assumes that the theory has a Coulomb branch. For the brane motion suggested by the referee, one should first move the colour branes into a Higgs branch configuration, and then displace the NS5 branes to turn on a FI parameter. This displacement would lift the Coulomb branch, obstructing the deformations of interest (in the geometric sense of obstructing partial resolutions of a singular variety).

2) We acknowledge that the exposition in the previous version may have obscured the mechanism leading to the Higgsing of the gauge group. We have incorporated the following improvements in the text:
- We have added Subsection 3.1.4 in which we write down explicitly our prescription, and claim that correctly captures the IR fixed point.
- We have added a discussion about how to determine the value of the Higgsed gauge ranks in Subsection 3.4.3.
- We have showcased the explicit calculation in the SQCD example in Appendix C.2.3.
The detailed description of Gauge$_1 \oplus$Gauge$_2$ is explained in the recipe of Subsection 3.1.4. The reasoning by which the mass deformation breaks part of the gauge group is written down more explicitly in Subsection 3.4.3. The simple example of 3d N=4 $U(N)$ with $2N$ flavours is worked out in great detail in Appendix C.2.3.

We would also like to stress that giving a mass controlled by a single parameter $m$ to a large number of hypermultiplets across the quiver, as we do, is radically different than giving a large mass to one hypermultiplet at a time. This is why the Higgsing can take place this case. We agree that, had we given a large mass to the hypermultiplets one at a time, nothing interesting would happen.

For the two concrete points raised by the Referee, we have edited the text accordingly
- We have added a caption to the figure of the Higgsing, now Figure 9 in the text. In the caption we refer to Subsection 3.4.3 for more details.
- We have improved the explanation of Eq.(C.3a)-(C.3b) in Appendix C.2. To answer directly to the Referee's comment ``In other words, the expression including (C.3a) and (C.3b) should not be simply understood as the partition function of an alternative 3d N = 4 theory, rather a formal change of variables of the matrix integral.'' We agree that it does not describe an alternative theory, but, as we have clarified in Appendix C.2.3, the change of variables is a useful tool to explore the different regions of the Coulomb branch. We disagree with the word ``formal'', since we can map each term precisely to its field theory origin and to the Hanany--Witten setup.

We consider that all these modifications together address the concerns of the Referee.

4) We thank the Referee for the comment. We have added a discussion on the FI parameters in the first bullet point in Subsection 3.2.3, and also at the end of Subsection 3.1.3, as well as a sentence in the last paragraph of the Conclusions section. We have moreover explained at the end of Subsection 3.3.3 why, under the assumption of long and balanced, they do not change the free energy for generic quivers, and how this is consistent with mirror symmetry. Albeit certainly non-generic, this behaviour is checked explicitly in the class of quivers we consider. We believe that this answer the Referee's question, which concerned the QFT side.
On the other hand, we admit that it is unclear to us how to incorporate the FI parameters in the holographic dual picture in full generality. As alluded to in a previous answer, we lack a prescription when the brane configuration lifts the Coulomb branch. Therefore, we turn off all FI parameters to treat the two sides of the holographic duality equally.

5) The goal of our analysis is to relate the mass deformation of the SCFT with the electrostatic problem with two (or more) rank functions. The latter requires to take an AdS$_{d+1}$ anstaz. Even if it is known that it does not reflect the accurate holographic dual, we show that this very simple (albeit approximate) setup is a valuable tool to compute certain quantities, such as the free energy.
We have added clarifying comments at the end of Subsection 2.7 emphasizing this approach.

To answer to the second part of the Referee's comment, we thank for pointing out the interesting relation with the AdS$_2$ subsector of AdS$_4$. However, it is our understanding that the mass being related to the $\Omega$-background parameter is the one of the adjoint field, from the twisted holography perspective of the worldvolume theory on $N$ M2-branes. This is shown for instance in Eq.(2.44) of 2004.13810. There are no adjoint hypermultiplets in our quivers. In turn, the mass of the fundamental hypermultiplet couples to them on the same footing as the gauge scalar $\sigma$. Since we are not computing correlation functions of local operators, which is when the 1d TQM becomes manifest, our setup should be unable to detect the AdS$_2$ subsector of AdS$_4$.
At present, we do not know how to reconcile the Referee's comment with the fact that the topological subsector only exists in 3d N=4, whereas our approach is completely uniform in 3d and 5d. It would be extremely interesting to understand how the electrostatic problem is modified in 3d N=4 to describe the AdS$_2$ of Mezei--Pufu--Wang 1703.08749, but it would entail the analysis of correlation functions, which is beyond the scope of our work.

-- Minor points.

1. We thank the Referee for pointing out the references, which have been added to the bibliography.
2. We have edited the sentence, now reads ``when the mass is very large compared to the scale set by the inverse of the radius of the sphere''.
3. Corrected, thanks.
4. We thank the Referee for spotting the mistake. The previous formula had a wrong factor 1/2 in the $N_0 N_P$ term. We have corrected it, and also added a brief explanation of the regularisation.
5. We have added a comment and an explanatory footnote to confirm the match in that case as well.
6. We thank the Referee for the suggestion. We have added a clarifying sentence. Note that the expression relating $\ell$ and $\sigma_{\ast}$ is valid both at large $N$ with $\ell$ fixed as well as with scaling. We have clarified this point immediately below the equation. The Wilson loop is a probe, even when $\ell$ grows linearly in $N$, and the back-reaction on the geometry is subleading both in $N$ and $P$, thus negligible.
7. We thank the Referee for the suggestion. We have added a paragraph ``To study the situation in which the two rank functions are far away from each other [...]'' explaining how to obtain the Wilson loops at $\sigma_0 /P \to \infty$.
8. The setup starts with two ``overlapping'' rank functions at the origin in the $\sigma$-axis and one of them is moved away by a distance $\sigma_0$. The Poisson equation is solved at finite $\sigma_0$, with the boundary infinitely far away. We then send $\sigma_0 \to \infty$ while keeping the boundary infinitely far from both $\sigma_0$ and the origin. This is a well-defined procedure. One may start solving the problem in a box, with boundary in the $\sigma$-direction at $\sigma_0+b$. We may solve in this situation, and then take the two limits in order: first send the distance from the boundary $b \to \infty$, then $\sigma_0 \to \infty$ afterwards. Alternatively, the Referee may take a sequence of such solutions and write $b=e^{1000n} s$, $\sigma_0 = n s$, sending $n \to \infty$ with fixed $s$.
9. We have edited the sentence into ``real scalar $\vec{\sigma} \in \R^{\lvert \vec{N} \rvert }$ in the $\lvert \vec{N} \rvert$-dimensional Cartan subalgebra''.
10. We thank the Referee for the suggestion. References [105-109] of the previous version were meant to include papers that originally derived the sphere partition function from localization. We have added a footnote in Appendix C.1, were the finite $N$ sphere partition function in 3d is discussed, citing the two papers suggested by the Referee, and two more.
11. We have added a comment on the fact that the FI parameters do not modify the saddle point equation at the end of Subsection 3.3.3.
12. We have elucidated the limit $\lvert \mu \rvert \to \infty$ at the end of Subsections 3.4.5 and 3.4.6, leaving a more exhaustive discussion to Subsection 3.6.
13. We agree with the Referee that the usage of ``wall-crossing phenomenon'' in the previous version was misleading. In that subsection/paragraph, we have changed the word ``wall'' with ``interface'' and ``wall-crossing phenomenon'' with the more neutral ``jump across the interface''.
14. We are grateful to the Referee for pointing out the missing factor of $\pi$. The calculations were correct on both sides, but there was a discrepant factor of $\pi$ between the definitions of Wilson loops in supergravity and in QFT. We have corrected the definition of the Wilson loop at the very beginning of Subsection 2.6, and the factor of $\pi$ in all the ensuing expressions. Now the two results really match. Thanks.
15. As per our previous answer, the eigenvalue density is not modified by the possibility of FI parameters, for this specific setup.
16. We have commented on the $\lvert \mu_{\alpha} \rvert \to \infty$ limit of the Wilson loop vev, in parallel with point 7 above.

*** Referee 2 ***

We thank the Referee for the careful reading of the manuscript and for the suggested improvements. We have carefully taken all of them into account, as we now detail.

1. We are grateful to the Referee for pointing out the mistaken factor of 1/2. We have corrected the regularised value of the sum and also checked and corrected the same factor in the subsequent formula. We have also added a brief comment and an explanatory footnote to confirm the match with the 3d N=4 free energy.

2. Again, we are grateful to the Referee for spotting the discrepancy. Actually, the result on former page 46 was the correct one. We were missing a factor $\pi$ in the definition of the Wilson loop in supergravity. We have corrected the definition of the Wilson loop at the very beginning of Subsection 2.6, and the factor of $\pi$ in all the ensuing expressions. Now the two results really match. Thanks.

3. We thank the Referee for the suggestions, as we acknowledge that the exposition in the previous version may have obscured the mechanism leading to the Higgsing of the gauge group. To amend this problem, we have incorporated the following improvements in the text: (all numbers refer to the latest version)
- We have added Subsection 3.1.4 in which we write down explicitly our prescription, and claim that correctly captures the IR fixed point.
- We have added a discussion about how to determine the value of the Higgsed gauge ranks in Subsection 3.4.3.
- We have showcased the explicit calculation in the SQCD example in Appendix C.2.3.
- We have added Appendix C.1 in which we explain in more detail the Hanany--Witten brane configurations.
To more directly answer the Referee's comment, we have also
(i) Added a caption to the Higgsing Figure in Subsection 3.5, with some explanation and referring to the pertinent part of the text for more details;
(ii) Expanded the explanation of Eq.(C.3a)-(C.3b) and the relation with the Figure of the brane system, in Appendix C.2.

The idea underlying our work is that to give a mass controlled by a single parameter $m$ to a large number of hypermultiplets across the quiver, as we do, is different than to give a large mass to one hypermultiplet at a time. This is what allows the Higgsing to take place, and is alluded to in the new caption to Figure 9 of the latest version. We hope that, with the adjustments listed above as suggested by the Referee, this idea will now transpire even more clearly.

We believe that these changes address the Referee's remarks and requests, and improve the overall quality of the paper.