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Sparsity in the numerical six-point bootstrap

Sebastian Harris

TL;DR

This work develops a sparse semidefinite programming framework for the numerical six-point conformal bootstrap, exploiting banded matrix structure to recast one-dimensional problems as effectively two-dimensional mixed correlator four-point computations. By decomposing large banded SDPs into coupled small dense blocks and carefully choosing X and C matrices, the authors eliminate spectrum discretisation and enable efficient, parallelizable optimization with SDPB. The approach yields significantly higher-derivative bounds and enables a two-parameter extremal flow connecting Generalised Free Boson and Fermion, with perturbative AdS$_2$ compatibility guiding the interpretation of deformations as leading-order corrections without introducing new states. These advances enlarge the practical reach of higher-point bootstrap, offering a scalable path toward richer CFT data and potential extensions to higher dimensions and the flat-space limit. The paper also provides concrete numerical data and code to reproduce the results, strengthening the pipeline for exploring triple OPEs and multi-point crossing.

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$_2$.

Sparsity in the numerical six-point bootstrap

TL;DR

This work develops a sparse semidefinite programming framework for the numerical six-point conformal bootstrap, exploiting banded matrix structure to recast one-dimensional problems as effectively two-dimensional mixed correlator four-point computations. By decomposing large banded SDPs into coupled small dense blocks and carefully choosing X and C matrices, the authors eliminate spectrum discretisation and enable efficient, parallelizable optimization with SDPB. The approach yields significantly higher-derivative bounds and enables a two-parameter extremal flow connecting Generalised Free Boson and Fermion, with perturbative AdS compatibility guiding the interpretation of deformations as leading-order corrections without introducing new states. These advances enlarge the practical reach of higher-point bootstrap, offering a scalable path toward richer CFT data and potential extensions to higher dimensions and the flat-space limit. The paper also provides concrete numerical data and code to reproduce the results, strengthening the pipeline for exploring triple OPEs and multi-point crossing.

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.

Paper Structure

This paper contains 34 sections, 90 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Outline.
  3. Semidefinite programming and six-point crossing
  4. The crossing equation
  5. The functionals
  6. The bootstrap
  7. New numerics for the six-point bootstrap
  8. Banded SDP
  9. Setup and notation.
  10. Splitting the SDP in two.
  11. Dense decomposition of banded SDPs.
  12. Example.
  13. Improved six-point bootstrap
  14. Overview.
  15. X matrices.
  16. ...and 19 more sections

Figures (8)

  • Figure 1: The three different decompositions of a six-point correlator into products of four-point functions.
  • Figure 2: Allowed region for gap maximisation without identity.
  • Figure 3: Upper bound on solutions of the four-point maximisation problem for $\Delta_\phi = 0.01$.
  • Figure 4: Upper bound on solutions of the four-point maximisation problem for $\Delta_\phi = 0.1$.
  • Figure 5: Tree level Witten diagrams occurring in $\lambda_6\Phi^6$ perturbation theory.
  • ...and 3 more figures