The geometry of optimal functionals

Abstract

In this paper, we give a geometric interpretation of optimal functionals in the context of intersection of symmetry planes and cyclic polytopes. For 1D CFTs, we demonstrate that at given derivative order, the functional is given by a degenerate simplex of the cyclic polytope. More precisely the derivative functionals at $2N+1$-th order, is given by an unique $N$-dimensional simplex enclosing the origin. Taking the continuous limit, in the large $\Delta$ approximation this qualitatively agrees with that derived by Mazac et al. Remarkably similar construction applies to 2D CFT in the diagonal limit as well as the spin-less modular bootstrap. Finally we show that such geometric interpretation can be extended to functionals associated with bounds beyond the leading operator.

arXiv comment45 pages, 26 figures. Two mathematica notebooks included

References in corpus (7)

• EPFL Lectures on Conformal Field Theory in D>= 3 Dimensions
• Bootstrapping Conformal Field Theories with the Extremal Functional Method
• Analytic Bounds and Emergence of ${\text{AdS}}_2$ Physics from the Conformal Bootstrap
• Diagonal Limit for Conformal Blocks in d Dimensions
• On the Positive Geometry of Conformal Field Theory
• A Basis of Analytic Functionals for CFTs in General Dimension
• Positive geometry in the diagonal limit of the conformal bootstrap

Cited by in corpus (2)

• Positivity and Geometric Function Theory Constraints on Pion Scattering
• How to Succeed at Witten Diagram Recursions without Really Trying