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

The submitted article constitutes a detailed analysis of the holographic duality between specific classes of 3d $\mathcal{N}$=4 and 5d $\mathcal{N}$=1 superconformal theories and warped AdS$_{4}$ and AdS$_{6}$ IIB supergravity backgrounds, respectively. These families of 3d and 5d theories are balanced linear unitary quivers and the core of the work regards their deformations under activation of real mass parameters, the formulation of these deformations in the dual, supergravity description and holographic computations of observables, engaging matrix model techniques, which provide tests of the duality in points along the RG flow.

The paper is very well structured: after a detailed abstract and an introduction describing clearly the motivation, importance and stages of the implementation of the presented analysis, the parts regarding the supergravity solutions & quivers (sufficiently self contained) and the matrix model formulation are organised in two respective chapters, which are followed by a well written conclusion, including a summary of the methods and the results, while underlying a wide variety of future research directions. It should be noted that the authors took great care of the clarification of the content and of important points of each chapter, by providing a brief introduction before proceeding in the central analysis of each chapter. Four rich appendices conclude the paper, including explicit computations supplementing the ones of the corresponding sections, serving in the self-consistency, clarity and structure of the paper. Finally, the paper provides a rich reference list, covering as sufficiently as possible the relevant background analyses and more recent relevant developments.

Overall, the main idea and motivation of this analysis is original and highly interesting: I would suggest this paper for publication on SciPost Physics, provided certain adjustments are carried out, as proposed in the points below (see "Requested changes" part ).

1. Clarifications for eqn (2.24): the second infinite sum should converge to -log2; insert description of matching of the result in (2.24) with the 3d $\mathcal{N}$=4 free energy, analogously to (2.21)

2. Control for typo in page 46, eqn. (2.36) ($\pi^{3}$)

3. Clarification of the gauge group breaking: caption below the figure on page 39 with explanatory information should be inserted. Clarifying the breaking of the gauge group would help the both specialists and non-specialists who will read this work to have a clear picture of the connection between gauge theory, the matrix model computation (cf. eqn C.1, C.3a, C.3b) and the brane configuration (cf. Fig.9 p.56).

The authors consider the supergravity solutions corresponding to the dual CFTs realized from 5d $\mathcal{N}=1$ (resp. 3d $\mathcal{N}=4$) linear and balanced quiver gauge theories with $(P-1)$ gauge nodes of unitary gauge groups and $(P-1)$ flavor nodes. As the supergravity solution can be found by solving the 2d Laplace equation, they generalize the equation and discuss that it corresponds to a certain real mass deformation of the gauge theory.

As the proposal is new and the problem is certainly interesting for many experts working on the holography as well as the gauge theories, this work may be worth to be published. However, having read the manuscript, I am confused with several unclear points which I list in the attachment. In my opinion they need to be improved for the publication. I would like the authors to clarify them more carefully and seriously.