# Positive geometry in the diagonal limit of the conformal bootstrap

## TL;DR

This paper explores the geometric structures underlying the conformal bootstrap in various dimensions, revealing that in the large conformal dimension limit, the relevant polytopes approximate cyclic polytopes, providing new insights into the theory's geometric nature.

## Contribution

It demonstrates that the weighted Minkowski sum of cyclic polytopes describes the bootstrap geometry in higher dimensions and shows this sum approximates a cyclic polytope at large conformal dimensions.

## Key findings

- Weighted Minkowski sum of cyclic polytopes in higher dimensions
- Approximation of the sum by cyclic polytopes at large conformal dimensions
- Analytic formulas valid for O(1) conformal dimensions

## Abstract

We consider the diagonal limit of the conformal bootstrap in arbitrary dimensions and investigate the question if physical theories are given in terms of cyclic polytopes. Recently, it has been pointed out that in $d=1$, the geometric understanding of the bootstrap equations for unitary theories leads to cyclic polytopes for which the faces can all be written down and, in principle, the intersection between the unitarity polytope and the crossing plane can be systematically explored. We find that in higher dimensions, the natural structure that emerges, due to the inclusion of spin, is the weighted Minkowski sum of cyclic polytopes. While it can be explicitly shown that for physical theories, the weighted Minkowski sum of cyclic polytopes is not a cyclic polytope, it also turns out that in the large conformal dimension limit it is indeed a cyclic polytope. We write down several analytic formulae in this limit and show that remarkably, in many cases, this works out to be very good approximation even for $O(1)$ conformal dimensions. Furthermore, we initiate a comparison between usual numerics obtained using linear programming and what arises from positive geometry considerations.