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

1) We thank the Referee for the suggestion. We have shown in the last paragraphs of Subsection 3.3.3 that the complexification of the real mass does not modify the free energy we compute. We note, however, that in this report the Referee is talking about the complexification of the real mass, which is a different thing than the complex masses the Referee alluded to in their previous reports. A complex mass combines with the real $m$ to form an $SU(2)$ triplet. The holomorphy in $m$ the Referee discusses in their latest report, on the contrary, was first discovered by Jafferis [1012.3210] in the context of 3d $\mathcal{N}=2$ theories, which do not admit complex masses at all, since the R-symmetry is $U(1)$ and not $SU(2)$. Notwithstanding, we have seriously and carefully addressed the latest Referee's concern in the manuscript.

2) We thank the Referee for the comment. We have added clarifications on the scheme-dependence at the end of Subsection C.2.4, where the counterterm is explicitly discussed, and therein we refer to the works of Jafferis-Klebanov-Pufu-Safdi (Ref. [98,99]) and Gerchkovitz-Gomis-Komargodski (Ref.[125]). We have also added further clarifications around Eq. (C.6), including a comment on the case of odd $F_1$.

The other comments of the Referee are based on certain misguided assumptions. -- The Referee writes "If it vanishes, it may indicate that the limit $m \to \infty$ of the partition function does not make sense." We do not understand this comment, since the $m \to \infty$ limit of every partition function vanishes. It would seem that the Referee suggests that it does not make sense to give a large mass to hypermultiplets, in any theory. In the answer to the report on the previous submission, we attached some simple examples (even beyond the linear quivers considered in the paper), and encourage the Referee to plot the partition function of their favourite supersymmetric theory. They will observe that it is exponentially damped at $m \to \infty$. -- In the second sentence of the report, the Referee writes "As the original matrix integral is well-defined without any counterterm, I cannot see any reason to introduce it." This is not entirely true. The localized partition function is defined assuming some regularization scheme. For instance, it is known that there is a parity anomaly in 5d. When computing the partition function, one needs to make a choice of whether to preserve parity, at the expense of invariance under background gauge transformation, or save the latter at the expense of parity. Both choices are well-defined, and differ by a counterterm. This argument applies to every anomaly of global symmetries, mixed gravitational-global and mixed R-global anomaly. These facts are discussed in the references [121,122] (for the cases of interest to us), which we cited in the main text since v1. Phrased differently, the choice of regularisation of the one-loop determinants that leads to the localized partition function is a choice of regularisation scheme. Therefore, the partition function has a certain scheme-dependence built in. When the Referee writes that "the original matrix integral is well-defined without any counterterm" probably means that it is well-defined with the counterterms assumed by the authors who computed the localized partition function. This is true, but it does not exclude that any other choice of regularisation scheme is equally valid. -- The Referee also writes "once the additional term is introduced, it may modify the original theory." We do not understand this comment. The fact that we can cancel the exponential suppression with a scheme-dependent counterterm was shown in the references [98,99,125], which we adequately cite at various points in the manuscript. We have added a further clarifying comment in Appendix C.2.4. The fact that adding scheme-dependent counterterms does not change the physical theory is textbook material, and it essentially boils down to the definition of counterterm. -- The sentence "The situation does not seem to be so simple, as the factor $e^{- \frac{\pi m}{2} [...]}$ is not overall factor which may be cancelled by the counterterm but rather it differs for each term labeled by distinct $N_2$". We totally agree: that one is not an overall factor. This was the whole point of the discussion: to figure out which quivers survive in the deep IR. As we have already explained below Eq. (C.7) as well as in our previous reply, we can save the least-suppressed term with a counterterm. The terms with other choices of $N_2 \ne F_2/2$ are more suppressed and vanish in the limit $m \to \infty$. We are left with a pair of balanced quivers, as claimed. That sentence of the Referee's report is correct and indeed supports our claim that only one choice of $N_2$ survives at $m \to \infty$.

1) We have shown in Subsection 3.3.3 that the complexification of the real mass does not modify the free energy.
2) We have added clarifications on the scheme-dependence at the end of Subsection C.2.4, where the counterterm is explicitly discussed. We have also added further clarifications around Eq. (C.6), including a comment on the case of odd $F_1$.
3) Corrected two typos.