Dynamics of Finite-Temperature CFTs from OPE Inversion Formulas

We apply the operator product expansion inversion formula to thermal two-point functions of bosonic and fermionic conformal field theories in general odd dimensions. This allows us to analyze in detail the operator spectrum of these theories. We find that nontrivial thermal conformal field theories arise when the thermal mass satisfies an algebraic transcendental equation that ensures the absence of an infinite set of operators from the spectrum. The solutions of these gap equations for general odd dimensions are in general complex numbers and follow a particular pattern. We argue that this pattern unveils the large-N vacuum structure of the corresponding theories at zero temperature.