Bounding 3d CFT correlators

TL;DR

This paper develops a numerical optimization approach to bound 3d CFT correlators, revealing minimal and maximal bounds, and connecting to the Polyakov bootstrap, with implications for understanding the structure of conformal theories.

Contribution

It introduces a numerical functional method to bound 3d CFT correlators, reproduces gap maximization, and identifies extremal correlators like the generalized free boson.

Findings

3d Ising correlator attains minimal allowed values

Existence of gap-independent maximal bounds for d > 2

Generalized free boson saturates bounds under certain conditions

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.