On the Positive Geometry of Conformal Field Theory

TL;DR

The paper develops a geometric reformulation of the conformal bootstrap in one dimension by mapping unitarity to a convex unitarity polytope ${\bf U}$ spanned by conformal-block vectors and crossing to a fixed plane ${\bf X}$; the consistency condition is ${\bf F}\in{\bf U}\cap{\bf X}$. A key result is that the positivity of conformal-block determinants makes ${\bf U}$ a cyclic polytope, enabling a complete combinatorial description of ${\bf U}\cap{\bf X}$ and yielding exact statements about operator spectra and four-point functions at finite Taylor truncations (e.g., up to six terms). The authors provide a constructive roadmap to refine this analysis by increasing resolution (adding more Taylor coefficients) and discuss how the framework naturally extends toward higher-dimensional conformal symmetry. They also connect this positive-geometry viewpoint to broader amplitudes program concepts (positive Grassmannian/Amplituhedron), highlighting deep structural parallels and potential for rigorous analytic progress in CFT data.

Abstract

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 and 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 and 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.

Figures (10)

• Figure 1: A point $\vec{A}$ inside a polytope is given by a non-negative sum of the vertices $\vec{v}_i$.
• Figure 2: The "bad" and "good" configurations.
• Figure 3: A polygon. While the line segment $({\bf V}_n{\bf V}_2)$ is not an edge of the polygon, $({\bf V}_1{\bf V}_2)$ is.
• Figure 4: Signs of brackets involving the point ${\bf V}_i$ depend on its position relative to the other indices. In the top figure we see that for $d=2$, the signs can be positive or negative depending on whether there is a gap between $a,b$. When there is no gap, the signs are always positive. The middle figure shows the same phenomenon for $d=4$. The bottom figure shows the novelty for odd $d$, where ${\bf V}_1$ and -${\bf V}_n$ must appear.
• Figure 5: Both the tetrahedron and the triangular bipyramid are cyclic polytopes.