Positive geometry in the diagonal limit of the conformal bootstrap

• Published in: The journal of high energy physics Vol. 2019; no. 11; pp. 1 - 32
• Main Authors: Sen, Kallol, Sinha, Aninda, Zahed, Ahmadullah
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2019; Springer Nature B.V; SpringerOpen
• Subjects: Classical and Quantum Gravitation; Conformal Field Theory; Elementary Particles; High energy physics; Linear programming; Physics; Physics and Astronomy; Polytopes; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Regular Article - Theoretical Physics; Relativity Theory; Scattering Amplitudes; String Theory; Scattering Amplitudes; Conformal Field Theory
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP11(2019)059

A bstract 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 boot- strap 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.