# Conformal field theories on deformed spheres, anomalies, and supersymmetry

### Submission summary

 As Contributors: Joseph Minahan · Usman Naseer Arxiv Link: https://arxiv.org/abs/2012.01781v1 (pdf) Date submitted: 2021-01-05 17:47 Submitted by: Naseer, Usman Submitted to: SciPost Physics Academic field: Physics Specialties: High-Energy Physics - Theory Approach: Theoretical

### Abstract

We study the free energy of four-dimensional CFTs on deformed spheres. For generic nonsupersymmetric CFTs only the coefficient of the logarithmic divergence in the free energy is physical, which is an extremum for the round sphere. We then specialize to $\mathcal{N}=2$ SCFTs where one can preserve some supersymmetry on a compact manifold by turning on appropriate background fields. For deformations of the round sphere the $c$ anomaly receives corrections proportional to the supersymmetric completion of the (Weyl)$^2$ term, which we determine up to one constant by analyzing the scale dependence of various correlators in the stress-tensor multiplet. We further show that the double derivative of the free energy with respect to the marginal couplings is proportional to the two-point function of the bottom components of the marginal chiral multiplet placed at the two poles of the deformed sphere. We then use anomaly considerations and counter-terms to parametrize the finite part of the free energy which makes manifest its dependence on the K\"ahler potential. We demonstrate these results for a theory with a vector multiplet and a massless adjoint hypermultiplet using results from localization. Finally, by choosing a special value of the hypermultiplet mass where the free energy is independent of the deformation, we derive an infinite number of constraints between various integrated correlators in $\mathcal{N}=4$ super Yang-Mills with any gauge group and at all values of the coupling, extending previous results.

###### Current status:
Editor-in-charge assigned

### Submission & Refereeing History

Submission 2012.01781v1 on 5 January 2021

## Reports on this Submission

### Strengths

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

### Weaknesses

1- Some points in section 4.3 need clarification.

### Report

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.

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