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 N=2N=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 cc anomaly receives corrections proportional to the supersymmetric completion of the (Weyl)^22 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 Kähler 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 N=4N=4 super Yang-Mills with any gauge group and at all values of the coupling, extending previous results.
Highlights
If we have an N = 2 superconformal field theory (SCFT), in order to preserve supersymmetry we must turn on other background fields in the supergravity multiplet when deforming the metric [9,10,11]
This leads to additional contributions to the conformal anomaly, which results in the supersymmetric completion of the Euler density and the (Weyl)2 terms
We study the dependence of log Z on the marginal couplings of the SCFT and derive the results in eqs. (1.4) and (1.5)
Summary
The free energy of a conformal field theory on a compact four-manifold M is ambiguous due to ultraviolet divergences. If we have an N = 2 superconformal field theory (SCFT), in order to preserve supersymmetry we must turn on other background fields in the supergravity multiplet when deforming the metric [9,10,11] This leads to additional contributions to the conformal anomaly, which results in the supersymmetric completion of the Euler density and the (Weyl) terms. We study the scale dependence of the two-point functions of the tensor multiplet to fix all but one coefficient This generalizes the result in [13, 14] for the supersymmetrized Weyl anomaly by including the contribution of all background fields in the supergravity multiplet. In the appendix we derive Ward identities for two-point functions
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