Abstract
We study the partition function of odd-dimensional conformal field theories placed on spheres with a squashed metric. We establish that the round sphere provides a local extremum for the free energy which, in general, is not a global extremum. In addition, we show that the leading quadratic correction to the free energy around this extremum is proportional to the coefficient, C T , determining the two-point function of the energy-momentum tensor in the CFT. For three-dimensional CFTs, we compute explicitly this proportionality constant for a class of squashing deformations which preserve an SU(2) × U(1) isometry group on the sphere. In addition, we evaluate the free energy as a function of the squashing parameter for theories of free bosons, free fermions, as well as CFTs holographically dual to Einstein gravity with a negative cosmological constant. We observe that, after suitable normalization, the dependence of the free energy on the squashing parameter for all these theories is nearly universal for a large region of parameter space and is well approximated by a simple quadratic function arising from holography. We generalize our results to five-dimensional CFTs and, in this context, we also study theories holographically dual to six-dimensional Gauss-Bonnet gravity.
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Bobev, N., Bueno, P. & Vreys, Y. Comments on squashed-sphere partition functions. J. High Energ. Phys. 2017, 93 (2017). https://doi.org/10.1007/JHEP07(2017)093
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DOI: https://doi.org/10.1007/JHEP07(2017)093