Numerical Bootstrap

Numerical conformal bootstrap turns exact crossing and positivity into finite optimization problems. Its trustworthy output is never just a plot or solver label: it is a chain from a declared correlator system, through controlled approximations and a serialized conic problem, to a certificate checked independently of the solver Simmons-Duffin 2015, §§2–3. This chapter develops that chain and keeps mathematical exclusion, numerical convergence, interpretation, and model identification distinct.

Helpful background. Crossing equations and positivity provide the physical consistency condition. Convex cones, separation, and conic duality provide the optimization logic. Mixed correlators and symmetry sectors provide the matrix-positive systems used later.

Enter the numerical bootstrap

For an identical Hermitian scalar, crossing may be written schematically as

$$\mathbf F_{\mathbf1}+\sum_{\mathcal O\ne\mathbf1} \lambda_{\phi\phi\mathcal O}^{\,2}\mathbf F_{\Delta,\ell}=0, \qquad \lambda_{\phi\phi\mathcal O}^{\,2}\geq0.$$

The exact equation is infinite in the spectrum and functional domain. A computation chooses a finite functional space, approximates the blocks, represents positivity over continuous dimensions and spins, and solves the resulting linear or semidefinite program Poland, Rychkov, and Vichi 2019, §§VI.A–VI.C. Every arrow changes the statement that can be justified, so the approximation data must travel with the result.

Question	Continue to	Required output
Is a convex reduction valid?	From Crossing Equations to Convex Optimization	Positivity domain, normalization, assumptions, and finite projection
Which functional proves exclusion?	Linear Functionals and Positivity	Coefficients, basis, sector signs, and continuum check
How do blocks become an SDP?	Block Approximations and Semidefinite Programs	Block convention, poles, polynomial matrices, and tail treatment
Is the crossing system complete?	Automated Crossing-System Generation	Canonical representation and tensor-basis serialization
Is the result stable?	Precision, Convergence, and Numerical Error Budgets	Independent refinement sequences and an observable-level envelope
Does the solver output prove its claim?	Solver Certificates and Independent Verification	Primal/dual residuals, certificate, interval checks, and hashes
Is a boundary or defect involved?	Numerical Boundary and Defect Bootstrap	Channel-specific signs, defect blocks, and a free benchmark
What does one correlator constrain?	Single-Correlator Bounds	Conditional exclusion or bound, not automatic theory identification
How is an allowed island formed?	Mixed-Correlator Islands	Shared-operator and gap assumptions attached to every boundary
Are symmetry or spin present?	Global Symmetry and Spinning Bootstrap Systems	Projectors, tensor bases, conservation rank, and PSD sectors
Can candidate CFT data be recovered?	Extremal Functionals, Navigators, and Spectrum Reconstruction	Stability of zeros, primal data, degeneracies, and objective choice
Can another group reproduce it?	Benchmark Reproduction and Data Provenance	Frozen inputs, versions, hashes, clean rerun, and discrepancy report

The certification spine

A defensible computation preserves the following order:

1. Exact specification: operators, representations, tensor bases, crossing maps, identity normalization, gaps, and positivity hypotheses.
2. Finite approximation: derivative or integral basis, spin coverage, block representation, pole order, arithmetic precision, and any sampling or interpolation.
3. Solver problem: canonical sector ordering, scaled matrices, feasibility sign convention, stopping tolerances, and software build.
4. Certificate: serialized primal or dual variables with residuals, complementarity or objective gap where relevant, and positivity beyond the sampled points.
5. Independent verification: a separate evaluator reads the frozen specification and certificate, recomputes residuals, checks interval and asymptotic tails, and compares hashes.
6. Bounded interpretation: the conclusion states exactly which spectra are excluded or allowed under which assumptions and at which finite approximation.

No refinement axis substitutes for another. Raising derivative order does not repair an insufficient spin tail; high arithmetic precision does not correct a wrong crossing sign; a small residual does not prove positivity between sample points. Current functional, solver, and search strategies—and their distinct numerical controls—are reviewed in Rychkov and Su 2024, §§II–III.

Evidence classes are not interchangeable

Output	Direct conclusion	Additional step before stronger language
Verified separating functional	The declared finite problem is infeasible, subject to the verified continuum positivity representation	Control the relation between the finite representation and exact crossing
Stable exclusion curve	Trial spectra on one side fail the tested assumptions across stated refinements	Report all numerical and spectral assumptions
Kink	The boundary curve has a stable geometric feature	Supply independent operator-data or model evidence
Allowed island	Points outside the island are excluded under the mixed-system assumptions	Do not infer that every interior point is realized or that one unique CFT exists
Navigator minimum	A chosen continuous objective locates a candidate allowed region	Check objective dependence, feasibility, and convergence
Extremal reconstruction	A boundary solution suggests dimensions and OPE coefficients	Check zero multiplicity, degeneracy, primal-dual agreement, and cutoff stability

An exact generalized-free-field correlator is the durable synthetic thread: its crossing equation, nonnegative OPE coefficients, and known spectrum provide inputs that can be regenerated without importing a mutable phenomenological claim. A three-dimensional Ising reproduction requires a separate dated record with the precise publication, conventions, software, and target; this chapter does not claim such a run.

Review the chapter

Before accepting a numerical statement, ask whether the exact crossing file can be regenerated, whether the finite problem and certificate are serialized, whether at least two independent approximation axes were varied, whether another evaluator verifies the signs and residuals, and whether the prose stops at the conclusion those checks support.

References

• Poland, David, Slava Rychkov, and Alessandro Vichi. “The Conformal Bootstrap: Theory, Numerical Techniques, and Applications.” Reviews of Modern Physics 91 (2019): 015002. DOI. Open PDF
• Rychkov, Slava, and Ning Su. “New Developments in the Numerical Conformal Bootstrap.” Reviews of Modern Physics 96 (2024): 045004. DOI. Open PDF
• Simmons-Duffin, David. “A Semidefinite Program Solver for the Conformal Bootstrap.” Journal of High Energy Physics 06 (2015): 174. DOI. Open PDF