Symmetries of conformal correlation functions

TL;DR

This paper reveals a new algebraic property of conformal crossing equations, linking various symmetries to $SO(N)$ vector crossing equations, which enhances the understanding of symmetry constraints in conformal bootstrap studies.

Contribution

It introduces a novel algebraic property that transforms symmetry-constrained crossing equations into $SO(N)$ vector crossing equations, providing a new approach to analyze conformal field theories.

Findings

Crossing equations can be linearly converted into $SO(N)$ vector crossing equations.

The transformations satisfy the positivity condition.

Non-$SO(N)$ symmetric theories require symmetry-breaking assumptions for analysis.

Abstract

A program of wide interest in modern conformal bootstrap studies is to numerically solve general conformal field theories, based on a critical assumption that the dynamics is encoded in the conformal four-point crossing equations and positivity condition. In this letter we propose and verify a novel algebraic property of the crossing equations which provides strong restriction for this program. We show for various types of symmetries $\cal G$, the crossing equations can be linearly converted into the $SO(N)$ vector crossing equations associated with the $SO(N)\rightarrow \cal G$ branching rules and the transformations satisfy positivity condition. The dynamics constrained by the $\cal G$-symmetric crossing equations combined with positivity condition degenerates to the $SO(N)$ symmetric cases, while the non-$SO(N)$ symmetric theories are not directly solvable without introducing the $SO(N)$ symmetry breaking assumptions on the spectrum.