Extremal bootstrapping: go with the flow

Sheer El-Showk,Miguel F Paulos

doi:10.1007/jhep03(2018)148

Abstract

The extremal functional method determines approximate solutions to the constraints of crossing symmetry, which saturate bounds on the space of unitary CFTs. We show that such solutions are characterized by extremality conditions, which may be used to flow continuously along the boundaries of parameter space. Along the flow there is generically no further need for optimization, which dramatically reduces computational requirements, bringing calculations from the realm of computing clusters to laptops. Conceptually, extremality sheds light on possible ways to bootstrap without positivity, extending the method to non-unitary theories, and implies that theories saturating bounds, and especially those sitting at kinks, have unusually sparse spectra. We discuss several applications, including the first high-precision bootstrap of a non-unitary CFT.

Highlights

At this point we must stop: we have hit a boundary of parameter space, and to go any further would require us to relinquish positivity and with it unitarity
The extremal functional method determines approximate solutions to the constraints of crossing symmetry, which saturate bounds on the space of unitary CFTs
We shall show that for these theories it is possible to write down a set of extremality equations that fully characterize the solution to the crossing symmetry constraints

Summary

Introduction

At this point we must stop: we have hit a boundary of parameter space, and to go any further would require us to relinquish positivity and with it unitarity. One begins by reformulating crossing symmetry of conformal four point functions as linear or semidefinite optimization problems which can be solved numerically This allows us to rule out regions of CFT parameter space rigorously, and to construct approximate solutions to crossing symmetry in certain cases. We extend the philosophy and observations of [12] and will examine in more detail what characterizes the CFTs that lie on the boundaries of parameter space We call such CFTs extremal: they have sparse spectra, or more precisely, such theories contain correlation functions receiving contributions from as few operators as possible below any given cutoff in conformal dimension. Using our method a single point can be used to generate the entire plot within hours on a single laptop

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Discussion

Conclusion