5d SCFTs and their non-supersymmetric cousins

The paper studies supersymmetry-breaking deformations of five-dimensional superconformal field theories. Combining the analysis of the 5d prepotential of ref [30] with the methods developed in references [20] and [21], the authors are able to study numerous families of theories. Of particular interest is the case of Sp(N) SYM with an antisymmetric hyper, which has a known holographic dual [46]. The study of the deformation from field theory thus highlights the importance of the study of non-SUSY deformations in gravity.

Some presentation issues in the introduction and minor typos, but easily solvable with minor revisions.

The paper uses field-theoretic and geometric methods to investigate the existence of non-supersymmetric conformal field theories in five dimensions. This is argued via a chain of arguments: first one studies deformations of the SCFT leading to SYM phases, which are described by the prepotential. The authors stress the importance of integration constants in the prepotential, which represent Chern-Simons levels for background vector multiplets. When different deformations are related by a symmetry of the fixed point, the gauge theory phases are related as well, and are sometimes referred to as ``UV duals''. Focusing on these, the authors are able to fix the Chern-Simons levels for the background vector multiplets. One then deforms breaking supersymmetry and, in presence of a particular change of levels, argue in favour of the existence of a phase transitions and a non-supersymmetric fixed point, of which the authors study the stability as well.

The paper represents a significant contribution to the literature, and I would recommend it for publication after the authors address the minor revisions described below.

1) Presentation. One of the key concepts used in the paper is that of "UV duality". This is something of a misnomer, accepted because of its common usage in the literature. I believe that it should be emphasized in the introduction that "UV dualities" between apparently inequivalent gauge theories are simply due to related susy-preserving mass deformations. This point is already present in the article at p. 6, but to avoid imprecisions I believe that it would be better to bring this forward to the introduction, taking the place of the sentence about the "continuation past infinite coupling of the weakly coupled gauge theory description" (p. 1 and p. 3). The RG flow is unidirectional, and the "infinite coupling limit" is not a well-defined operation, as stressed, for instance in [1812.10451].

2) Typos and clarifications.
- p. 1 +4: Is there a reason why ref [46] is not grouped together with [4-6] in the introduction?
- p. 1 +14: that the global symmetry of $E_1$ is $SO(3)_I$ instead of $SU(2)_I$ was first suggested in [36]
- p. 1 +17 (and elsewhere): it should be "Cartan subgroup" rather than "Cartan"
- p. 1 -17: I believe that the sentence would be clearer as "is however related to the original weakly coupled description by"
- p. 2 +2: "on a generic" $\to$ "at a generic"
- p. 2 -2: "the background CS levels for the instantonic symmetry"
- p. 6 - 18: Could the authors please clarify, perhaps in a footnote, what they mean by a "non-perturbative hypermultiplet"?
- p. 8 +5: "in terms"
- p. 9 +12: that the global symmetry of the fixed point of $SU(N)_N$ is $SO(3)_I$ instead of $SU(2)_I$ was first suggested in [31]
- p. 11 (33): I believe that it would be clearer if $\alpha$ was introduced in the text, e.g. "this integration constant $\alpha$ into the IMS prepotential"
- p. 12 +4: is there a proof of the fact that the Weyl group of the global symmetry of the SCFT should be S-duality in the $(p,q)$ web, apart from the fact that they have a related action? If not, perhaps "implemented" is too strong compared to "related"?
- p. 14 +4: "the gauginos being in the adjoint"
- p. 14 footnote 5: what group are the authors referring to? $d_{abc}$ for $SU(N)$ is not vanishing for $N\geq 3$.
- p. 15 - 3: "a shift of the CS level"
- p. 28 -8: "reach"
- p. 32 -8: I suppose that the authors here restricting to holographic supersymmetric fixed points?