Symmetries of conformal correlation functions

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 paper 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 $\mathcal{G}$, the crossing equations can be linearly converted into the $SO(N)$ vector crossing equations associated with the $SO(N)\ensuremath{\rightarrow}\mathcal{G}$ branching rules and the transformations satisfy positivity condition. The dynamics constrained by the $\mathcal{G}$-symmetric crossing equations combined with the 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.