CFTs on Squashed Spheres and the Thermal Effective Action

Abstract

We study three-dimensional CFTs on compact Euclidean manifolds in two complementary limits: small deformations of the round $S^3$ and the small-fiber, large-squashing limit of Seifert manifolds. Near the round sphere, conformal perturbation theory expresses the free-energy response through integrated stress-tensor correlators. We derive a harmonic-space formula for the universal quadratic response to arbitrary metric squashing and show that it is proportional to the $c_T$ coefficient of the stress-tensor two-point function. For unitary CFTs this establishes that the sphere free energy is a local maximum in the space of metric deformations. This result extends to conserved spin-$s$ currents, whose quadratic response alternates in sign. For the specific case of squashing the Hopf fiber, we find an explicit form for the cubic response, and in addition obtain the leading correction to the two-point function of scalar operators. In the small-fiber limit, corresponding to the large temperature regime of the CFT, the partition function is governed by a two-dimensional thermal effective action constructed out of the Weyl-rescaled base metric and a Kaluza--Klein field strength. The thermal effective action relates CFT free energies on different backgrounds, including squashed Lens spaces, which we discuss in detail. We also explicitly determine the Wilson coefficients of this effective action, to various orders in the high-temperature expansion, for free fields, the large-$N$ critical O($N$) model, and holographic CFTs.