Bounding 3d CFT correlators

Abstract: We consider the problem of bounding CFT correlators on the Euclidean section. By reformulating the question as an optimization problem, we construct functionals numerically which determine upper and lower bounds on correlators under several circumstances. A useful outcome of our analysis is that the gap maximization bootstrap problem can be reproduced by a numerically easier optimization problem. We find that the 3d Ising spin correlator takes the minimal possible allowed values on the Euclidean section. Turning to the maximization problem we find that for d > 2 there are gap-independent maximal bounds on CFT correlators. Under certain conditions we show that the maximizing correlator is given by the generalized free boson for general Euclidean kinematics. In our explorations we also uncover an intriguing 3d CFT which saturates gap, OPE maximization and correlator value bounds. Finally we comment on the relation between our functionals and the Polyakov bootstrap.