The Free Scalar as a Thermal Envelope for Three-Dimensional CFTs

Abstract

We conjecture that every unitary three-dimensional conformal field theory obeys $c_S/C_T\leq16πζ(3)$, where $s=c_ST^2$ is the thermal entropy density and $C_T$ the stress-tensor two-point coefficient, with equality for free scalars. We embed this in a stronger conjecture: on $S^1\times H^2$, the $C_T$-normalized hyperbolic heat capacity is bounded above by that of the free scalar for all $T\geq(2πR)^{-1}$. Near the conformal temperature, for theories smooth from the high-temperature side and without boundary terms in the modular Hamiltonian, the heat-capacity bound follows from conformal-collider positivity. We test the conjectures in free, interacting large-$N$, and holographic theories. Monte Carlo data for the three-dimensional Ising and $O(N)$ models also obey the flat-space bound.