Navigator Function for the Conformal Bootstrap

TL;DR

This paper introduces a continuous navigator function for the conformal bootstrap, enabling efficient exploration of high-dimensional theory spaces and precise localization of allowed regions, such as the 3d Ising model.

Contribution

It proposes a novel navigator function that replaces binary allowed/excluded labels, facilitating smooth and efficient high-dimensional searches in conformal bootstrap studies.

Findings

Navigator function is well-defined everywhere

Can be computed with standard SDP methods

Gradient evaluation enables efficient optimization

Abstract

Current numerical conformal bootstrap techniques carve out islands in theory space by repeatedly checking whether points are allowed or excluded. We propose a new method for searching theory space that replaces the binary information "allowed"/"excluded" with a continuous "navigator" function that is negative in the allowed region and positive in the excluded region. Such a navigator function allows one to efficiently explore high-dimensional parameter spaces and smoothly sail towards any islands they may contain. The specific functions we introduce have several attractive features: they are everywhere well-defined, can be computed with standard methods, and evaluation of their gradient is immediate due to an SDP gradient formula that we provide. The latter property allows for the use of efficient quasi-Newton optimization methods, which we illustrate by navigating towards the 3d Ising island.