Conformal field theories on deformed spheres, anomalies, and supersymmetry

1 - Clear presentation and useful review of background material, especially in Sections 1-3.

2 - Interesting new formula for the partition function of four-dimensional $\mathcal{N}=2$ SCFT's on supersymmetric backgrounds, with a focus on deformed four-spheres.

Section 4 is more technical and harder to follow. Some more explanations could have helped the reader.

The authors use model-independent techniques, based on conformal symmetry and supersymmetry, to constrain the partition function $Z$ of $\mathcal{N}=2$ SCFT's on supersymmetric backgrounds, with a focus on the case of a deformed four-sphere. First they study the logarithmically divergent part of $\log Z$, that is the super-Weyl anomaly, providing an explicit expression for the bosonic part of the super-Weyl$^2$ invariant, up to one coefficient that is left to be determined. Then they turn to the finite part of $\log Z$ and study its dependence on the marginal couplings. For the case of a deformed four-sphere they obtain a nice formula where the partition function is given in terms of a few functions involving the marginal couplings and the parameter deforming the geometry. Finally, the result is checked and made even more explicit in a few examples based on $\mathcal{N}=4$ SYM. A set of relations between integrated correlators of $\mathcal{N}=4$ SYM is also obtained.

The results are general, rigorous and interesting. The paper is clearly written, especially Sections 1-3. I recommend publication in SciPost after the minor requested changes below have been addressed.

- It may be useful to specify that the variations $\delta\alpha$, $\delta\sigma$ are assumed independent of the coordinates in Eq. (3.2).

- In Section 4, it is not entirely clear which parts are a review of known results and which parts are original. The authors may add some comments clarifying this.

- Minor typos: page 3 Zamalodchikov $\to$ Zamolodchikov; position of index $\nu$ in last term of Eq. (3.9); repeated "any" in the text under (3.39).

1-clear exposition, including nice review on previous results.

2-new general result for supersymmetric 4d N=2 partition functions on deformed sphere (and other similar backgrounds)

1-the discussion of the main example in section 4 could be clearer.

In this clearly written paper, the authors study (N=2 super)conformal field theories on deformed 4-spheres. On general grounds, small deformations of the metric (and other background fields) around the round sphere capture integrated correlation functions of the stress tensor (and its susy partners). For supersymmetric backgrounds, one can directly extract the two-point function of marginal deformations. This is nicely reviewed. In section 3.2, the authors study the supersymmetrization of the Weyl anomaly by studying the structure of the two-point functions. In section 3.3, they also comment on the generalization of the famous dependence of the finite part of the S4 partition function on marginal couplings, to general Pestun-like backgrounds. Section 3.4 presents the general form of the free energy (finite part of log Z) on 4-spheres that are not round.

Section 4 deals with one main example, N=2 SU(N) with one massive adjoint, which is the N=4 SYM in the UV. It would be useful if the authors could comment on why, apparently, m=0 is not the proper SCFT point "on curved space". The confusion is that at m=0 we have an SCFT, and the fact that I couple it to an arbitrary background does not change the fact that it's a CFT. Presumably, it has to do with the precise form of the Pestun-like background, which modify the UV behavior at the poles, introducing some sort of UV contact terms? (If so, the issue is already present in the flat-space Omega background.) Some clarification would be appreciated.

Finally, using that claimed conformal value of the mass, they derive some identities amongst correlators at large N which follows from their general formula. This reproduces and generalizes recent results in the literature.

In conclusion, this clearly written paper presents new, interesting and topical results. I recommend it for publication in SciPost after a small revision addressing the question above.

-Small typo to be corrected in intro: on p3, ref. to (3.39) is presumably to (3.40).

-comment on why m=0 is not the conformal value.

1- Important novel formula for the dependence of partition function of SCFT's in deformed spheres in terms of marginal couplings

1- Some points in section 4.3 need clarification.

This paper investigates properties of the free energy in four-dimensional CFT's in
deformed spheres, with the main focus on N=2 supersymmetric CFT's.
The authors derive a new very interesting formula describing the dependence of the free energy
on the marginal couplings of the theory, in particular, the relation to the K\" ahler potential.
The new general formula (3.52) is then checked by specializing to N=2 theories on the ellipsoid, where one can compute the free energy explicitly by using supersymmetric localization.

In the last part, the paper considers the case of N=2 theory with a massive adjoint
hypermultiplet, pointing out that, for a special value of the hypermultiplet mass,
the theory simplifies: the instanton partition function becomes trivial and the one-loop
determinant cancels out. The resulting theory is called $N=4_2$ theory. The
"triviality" of the free energy of $N=4_2$ theory implies an infinite number of identities
involving correlation functions of integrated operators in N=4 theory (some of these
are consistent with results that already appeared in the literature).
In the case $b=1$, this theory seems to coincide with the theory pointed out by Pestun
in eq. (5.13) in [33]. The resulting partition function still has a non-trivial dependence on the couppling from the sum over instanton sectors. Maybe the authors should clarify
the discrepancy with their eq. (4.30).

The $N=4_2$ theory is here used as a tool to derive constraints in the N=4 theory, but otherwise it remains obscure.
It would be useful for readers if the authors also add more explanations on the expected structure
of the $N=4_2$ theory, in particular, which sectors of correlators are expected to differ from those of N=4 theory.
The authors should consider the above points before the paper is accepted for publication.