On the positive geometry of conformal field theory

A bstract It has long been clear that the conformal bootstrap is associated with a rich geometry. In this paper we undertake a systematic exploration of this geometric structure as an object of study in its own right. We study conformal blocks for the minimal SL(2 , R ) symmetry present in conformal field theories in all dimensions. Unitarity demands that the Taylor coefficients of the four-point function lie inside a polytope U determined by the operator spectrum, while crossing demands they lie on a plane X . The conformal bootstrap is then geometrically interpreted as demanding a non-empty intersection of U ∩ X . We find that the conformal blocks enjoy a surprising positive determinant property. This implies that U is an example of a famous polytope — the cyclic polytope. The face structure of cyclic polytopes is completely understood. This lets us fully characterize the intersection U ∩ X by a simple combinatorial rule, leading to a number of new exact statements about the spectrum and four-point function in any conformal field theory.