Comments on Squashed-sphere Partition Functions

Understand

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 $SU(2)\times U(1)$ isometry group on the sphere.