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, CT , 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.
Highlights
Extract many exact results for large classes of QFTs — see [1] for a review
1In this work we focus on the real part of the free energy of odd-dimensional CFTs and avoid the subtleties associated with contact terms that can affect its imaginary part, see [2, 3]. 2A similar formula has been derived for entanglement entropies of deformed spherical and planar regions
We discuss the difference between the supersymmetric free energy evaluated by supersymmetric localization and the non-supersymmetric free energy studied in this work
Summary
Where φ stands schematically for the set of dynamical fields in the theory. We wish to understand how the free energy F ≡ − log Z changes under small deformations of the metric. The one on the second line depends on the details of the CFT at hand and, in particular, on OPE coefficients of local operators with the energy momentum tensor This leads to the conclusion that it seems hard to obtain an explicit expression for F (0) valid for general CFTs. In the examples discussed below, we will be able to estimate F (0) for squashed spheres in some specific CFTs, including free theories as well as theories with a weakly curved holographic dual. We expect that these counterterms will in general have an imaginary coefficient in Euclidean signature and they will not affect the real part of the squashed sphere free energy which is the main focus of our work
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