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Journal of High Energy Physics

, 2019:59 | Cite as

Positive geometry in the diagonal limit of the conformal bootstrap

  • Kallol Sen
  • Aninda Sinha
  • Ahmadullah ZahedEmail author
Open Access
Regular Article - Theoretical Physics
  • 27 Downloads

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 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.

Keywords

Conformal Field Theory Scattering Amplitudes 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited

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Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Kavli Institute for the Physics and Mathematics of the Universe (WPI)University of TokyoChibaJapan
  2. 2.Centre for High Energy PhysicsIndian Institute of ScienceBangaloreIndia

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