Conformal field theories on deformed spheres, anomalies, and supersymmetry

TL;DR

This paper investigates the free energy of four-dimensional conformal field theories on deformed spheres, analyzing anomalies, supersymmetry preservation, and correlator relations, with applications to $ ext{N}=2$ and $ ext{N}=4$ theories using localization.

Contribution

It provides a detailed analysis of anomalies, supersymmetric deformations, and correlator relations in 4D CFTs on deformed spheres, extending previous results to general gauge groups and couplings.

Findings

The coefficient of the logarithmic divergence in free energy is extremized on the round sphere.

Corrections to the $c$ anomaly are proportional to the supersymmetric completion of the (Weyl)$^2$ term.

Derived constraints between integrated correlators in $ ext{N}=4$ super Yang-Mills.

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.