Conformal field theories at nonzero temperature: Operator product expansions, Monte Carlo, and holography

We compute the nonzero temperature conductivity of conserved flavor currents in conformal field theories (CFTs) in 2+1 space-time dimensions. At frequencies much greater than the temperature, ℏ ω ≫k_BT , the ω dependence can be computed from the operator product expansion (OPE) between the currents and operators, which acquire a nonzero expectation value at T >0 . Such results are found to be in excellent agreement with quantum Monte Carlo studies of the O(2) Wilson-Fisher CFT. Results for the conductivity and other observables are also obtained in vector 1 /N expansions. We match these large ω results to the corresponding correlators of holographic representations of the CFT: the holographic approach then allows us to extrapolate to small ℏ ω /(k_BT ) . Other holographic studies implicitly only used the OPE between the currents and the energy-momentum tensor, and this yields the correct leading large ω behavior for a large class of CFTs. However, for the Wilson-Fisher CFT, a relevant "thermal" operator must also be considered, and then consistency with the Monte Carlo results is obtained without a previously needed ad hoc rescaling of the T value. We also establish sum rules obeyed by the conductivity of a wide class of CFTs.