# Charting the space of 3D CFTs with a continuous global symmetry

## TL;DR

This paper advances the understanding of three-dimensional conformal field theories with a continuous global symmetry by deriving new constraints on correlation functions, developing recurrence relations for conformal blocks, and applying numerical bootstrap techniques to establish bounds on operator dimensions and central charges.

## Contribution

It introduces a recurrence relation for conformal blocks of spin-1 operators and applies numerical bootstrap to derive bounds and constraints in 3D CFTs with global symmetry.

## Key findings

- Universal bounds on operator dimensions and central charge.
- Numerical evidence supporting the conformal collider bound.
- New constraints on the critical O(2) model parameters.

## Abstract

We study correlation functions of a conserved spin-1 current $J_\mu$ in three dimensional Conformal Field Theories (CFTs). We investigate the constraints imposed by permutation symmetry and current conservation on the form of three point functions $\langle J_\mu J_\nu \mathcal O_{\Delta,\ell}\rangle $ and the four point function $\langle J_\mu J_\nu J_\rho J_\sigma \rangle $ and identify the minimal set of independent crossing symmetry conditions. We obtain a recurrence relation for conformal blocks for generic spin-1 operators in three dimensions. In the process, we improve several technical points, facilitating the use of recurrence relations. By applying the machinery of the numerical conformal bootstrap we obtain universal bounds on the dimensions of certain light operators as well as the central charge. Highlights of our results include numerical evidence for the conformal collider bound and new constraints on the parameters of the critical $O(2)$ model. The results obtained in this work apply to any unitary, parity-preserving three dimensional CFT.