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 S3 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 cT 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.
Versions
Related
- The Free Scalar as a Thermal Envelope for Three-Dimensional CFTs
- Dimensional uplifts of thermal two-point functions and higher-derivative CFTs
- Scalar QED effective action in dS: implications for primordial magnetogenesis and the running of coupling constants
- BF effective action topological implications
- Hadronic resonances from the 4PI effective action of QCD
BibTeX
@misc{parmentier2026cfts,
title = {{CFTs on Squashed Spheres and the Thermal Effective Action}},
author = {Klaas Parmentier and Nikolay Bobev},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.30716},
eprint = {2606.30716},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.30716},
}
RIS
TY - UNPB
TI - CFTs on Squashed Spheres and the Thermal Effective Action
AU - Klaas Parmentier
AU - Nikolay Bobev
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2606.30716
UR - https://arxiv.org/abs/2606.30716
AB - 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.
ER -