Symmetries of conformal correlation functions

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.

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