Numerical Bootstrap
Numerical bootstrap converts crossing and positivity into optimization problems that exclude candidate spectra or bound CFT data. This chapter follows the conversion from exact equations through block approximations to independently checked results. It separates an exclusion under stated assumptions from reconstruction or identification of a physical theory.
Enter this chapter
Section titled “Enter this chapter”Begin with From Crossing Equations to Convex Optimization to understand what a separating functional proves.
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.
Topics
Section titled “Topics”- From Crossing Equations to Convex Optimization — Derives the convex problem from exact crossing and positivity.
- Linear Functionals and Positivity — Constructs linear functionals with the signs required for exclusion.
- Block Approximations and Semidefinite Programs — Represents conformal blocks and continuous positivity in a semidefinite program.
- Automated Crossing-System Generation — Generates complete crossing equations with consistent tensor and symmetry bases.
- Precision, Convergence, and Numerical Error Budgets — Separates arithmetic, block, functional, and spectral truncation errors.
- Solver Certificates and Independent Verification — Checks residuals and positivity independently of a solver’s status message.
- Numerical Boundary and Defect Bootstrap — Adapts numerical methods to the distinct signs in defect channels.
- Single-Correlator Bounds — Interprets bounds obtained from one external-operator system.
- Mixed-Correlator Islands — Explains how mixed correlators and spectral assumptions isolate allowed regions.
- Global Symmetry and Spinning Bootstrap Systems — Implements matrix positivity with spin, conservation, and global symmetry.
- Extremal Functionals, Navigators, and Spectrum Reconstruction — Extracts candidate data while tracking degeneracy and refinement stability.
- Benchmark Reproduction and Data Provenance — Makes benchmark inputs and results reproducible.
How the ideas fit together
Section titled “How the ideas fit together”A separating functional assigns a sign to each allowed contribution in the crossing equation. If the identity and every possible nonidentity contribution have incompatible signs with a zero sum, the proposed spectrum is excluded. A finite computation represents that functional and the block dependence in a tractable form; semidefinite programming handles the matrix positivity of mixed systems. Simmons-Duffin 2015, §§ 2.1–2.3 and 3.1–3.2.
The logical exclusion is only as strong as the positivity check and approximation control. More arithmetic precision does not repair a missing symmetry sector, and positivity at sampled dimensions does not establish positivity between them. An allowed region records what has not been excluded under the specified assumptions.
Review the chapter
Section titled “Review the chapter”An island shrinks and stabilizes as the derivative cutoff increases. What does this establish about CFT existence?
Solution
It strengthens the numerical restriction under the stated spectral and symmetry assumptions. It does not construct a theory realizing an interior point or prove uniqueness. Those claims need additional consistency data or a construction; truncation and numerical errors also remain separate checks.
Where to go next
Section titled “Where to go next”Continue to Analytic and Lorentzian Bootstrap for complementary constraints from discontinuities and asymptotic limits.