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Conformal field theories on deformed spheres, anomalies, and supersymmetry

by Joseph A. Minahan, Usman Naseer, Charles Thull

Submission summary

As Contributors: Joseph Minahan · Usman Naseer · Charles Thull
Arxiv Link: (pdf)
Date accepted: 2021-03-03
Date submitted: 2021-02-15 10:28
Submitted by: Naseer, Usman
Submitted to: SciPost Physics
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


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:
Publication decision taken: accept

Editorial decision: For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)

Author comments upon resubmission

We thank the three referees for their careful reading of the manuscript and for valuable suggestions. We have made several changes in the new version following the points made by the referees. These changes, are listed in the the appropriate section of our resubmission page.

List of changes

1.On page 3 we changed the reference from eq. (3.39) to eq. (3.40).
2. We added a footnote before eq. (3.2) clarifying that $\delta\sigma$ and $\delta\alpha$ are space independent.
3. We have added extra references in section 4 to clarify what parts of the material were already known.
4. We fixed the spellings of Zamolodchikov.
5. We deleted the repeated “any” in the text after eq. (3.39).
6. We have modified language to avoid the confusion about $m=0$ being the conformal value of the mass. For $m=0$ only eight supercharges are preserved at the poles while for m= I/2 (b-1/b), all 32 supercharges are preserved. We now call this the $\mathcal{N}=4$ value.
7. We added a footnote on page.27 addressing one of the referees’ comments regarding the theory pointed out by Pestun in eq. (5.13) of his paper.
8. We have dropped the names ${\mathcal{N}=4_1$ and $\mathcal{N}=4_2$ as we believe that this causes unnecessary confusion.

Apart from the above changes directly related to referee’s suggestions we have taken the opportunity
to fix various typos and to add further clarifications which are summarized below:

1. We added a few sentences before eq. (4.19) to clarify the subtlety regarding the cutoff. This leads to minor changes in equations (4.20)-(4.23) and (4.25)-(4.27).
2. We added a third relation between various fourth-derivatives of the free energy in eq. (4.42), which agrees with our large-N results.

Reports on this Submission

Anonymous Report 2 on 2021-2-24 Invited Report


The authors have incorporated the necessary changes and clarifications.
I now recommend this paper for publication in SciPost.

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Anonymous Report 1 on 2021-2-16 Invited Report


The authors have addressed my remarks in a satisfactory way. I recommend publication in SciPost.

  • validity: -
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