Sparsity in the numerical six-point bootstrap

TL;DR

This paper advances the numerical six-point conformal bootstrap by employing sparse matrix techniques to improve scalability, enabling the derivation of new bounds and interpolations between known correlators in conformal field theories.

Contribution

It introduces a sparse matrix approach to efficiently solve six-point bootstrap SDPs, transforming them into more manageable four-point problems and expanding the scope of numerical bootstrap applications.

Findings

Derived novel bounds on CFT data from six-point bootstrap

Interpolated six-point functions between free fermion and boson models

Matched extremal correlators with perturbative deformations in AdS2

Abstract

The paper contributes to an ongoing effort to extend the conformal bootstrap beyond its traditional focus on systems of four-point correlation functions. Recently, it was demonstrated that semidefinite programming can be used to formulate a six-point generalisation of the numerical bootstrap, yielding qualitatively new, rigorous bounds on CFT data. However, the numerical six-point bootstrap requires solving SDPs involving infinite-dimensional matrices, which has so far limited its applicability and hindered scalability in early implementations. This work overcomes the challenges by using sparse matrix decompositions to exploit the banded structure of the underlying SDP. The result is a rewriting of one-dimensional six-point bootstrap problems as effectively two-dimensional standard mixed correlator four-point bootstrap computations. As application, novel bounds whose extremal correlators interpolate between the six-point functions of the generalised free fermion and boson are derived. The extremal interpolations are matched with perturbative deformations of the massive free boson in AdS$_2$.