Analytic and Numerical Conformal Bootstrap

Conformal bootstrap is a family of methods, not one algorithm. Analytic tools extract consequences of crossing in controlled kinematic or spectral regimes; numerical optimization excludes data under explicit positivity and gap assumptions; extremal reconstruction turns a boundary solution into a candidate spectrum. Their strongest use is together: analytics controls tails and interpretation, numerics explores finite-rank constraints, and independent constructions test whether an allowed solution is an actual CFT.

Evidence cutoff. 11 August 2026.

Required background. Crossing equations and positivity supplies the common constraints; optimization from crossing supplies convex exclusion. Helpful background. The lightcone OPE and large spin supplies asymptotic control, block approximations and semidefinite programs supplies finite numerics, solver certificates supplies witness logic, and precision and convergence budgets supplies numerical claim limits.

Crossing constraints and complementary methods

The target classes are unitary CFT spectra and operator-product-expansion (OPE) coefficients; central charges and current data; defect or boundary spectra; thermal and modular data; and large-spin asymptotics. Inputs always include spacetime dimension, symmetry representations, correlator set, normalization, and any gap, uniqueness, parity, or supersymmetry assumptions. Without those inputs, a “bootstrap bound” has no defined theory class.

Method	Best target and required inputs	Error or approximation ceiling
Lightcone OPE and large-spin perturbation	Families at $\ell\gg1$ from crossed-channel low-twist data	Asymptotic in spin; low-spin extrapolation and operator mixing require extra control
Lorentzian inversion and dispersion	Analytic-in-spin OPE data from double discontinuities under Regge bounds	Subtractions, low-spin ambiguities, arc terms, and Regge assumptions
Analytic functionals and extremal sum rules	Sharp bounds in low-dimensional or specially structured crossing problems	Completeness of the functional basis and contact-term ambiguities
Linear/semidefinite optimization	Universal exclusion bounds and islands from positivity	Derivative, spin, block, and arithmetic truncations; conditional gaps
Extremal functional or navigator reconstruction	Candidate spectra and smooth navigation near boundaries	Near-degeneracy, finite-resolution drift, and no automatic existence or uniqueness theorem

Komargodski and Zhiboedov established universal large-spin additivity under broad CFT assumptions (Komargodski and Zhiboedov 2013). Caron-Huot’s Lorentzian inversion formula turns causal data into analytic-in-spin OPE coefficients with stated Regge conditions (Caron-Huot 2017). These methods are particularly effective when a small set of crossed-channel operators controls an asymptotic regime. They do not normally determine the entire low-spin spectrum.

Numerical bootstrap applies a functional $\alpha$ to crossing so that positivity would imply a contradiction in an excluded region, following the founding optimization formulation of Rattazzi et al. 2008. OPE convergence bounds control the truncation for an already existing CFT but do not prove that every feasible point defines one (Pappadopulo et al. 2012). SDPB solves the resulting polynomial matrix programs at arbitrary precision (Simmons-Duffin 2015). Increasing derivative order $\Lambda$ , maximum spin, block-approximation order, and arithmetic precision tests convergence, but these variations are correlated; a defensible error budget records each separately.

Errors, benchmarks, and independent checks

The numerical error decomposition should include conformal-block evaluation, pole or radial-series truncation, derivative basis, spin tail, optimization residual, rounding, scan coverage, and sensitivity to assumed gaps. Analytic errors include asymptotic remainder, omitted exchanges, mixing, Regge/subtraction uncertainties, and continuation from noninteger spin or dimension. Statistical error bars are usually inappropriate: most uncertainty is deterministic truncation or assumption sensitivity.

Common benchmarks serve different roles:

generalized free fields and one-dimensional solvable crossing test signs, normalization, and asymptotic tails;
two-dimensional minimal models test exact spectra but use special Virasoro structure;
the three-dimensional Ising and $O(N)$ models test mixed-correlator islands against Monte Carlo, high-temperature series, and experiments;
Wilson–Fisher and large- $N$ expansions provide perturbative held-out data;
supersymmetric localization supplies exact protected observables that can be withheld from the bootstrap input.

The three-dimensional Ising island is a model example of cross-method agreement, but derivative-order scans using the same block generator and solver are one computational lineage, not independent replications (Kos et al. 2016). Strong independence comes from different correlator systems, clean-room block and solver implementations, Monte Carlo or experiment, perturbative expansions, and exact protected data.

Known failure modes

An allowed point need not correspond to a CFT; an exclusion applies only to the declared assumptions. A kink can arise from decoupling, a change of the extremal solution, or a fake-primary pole rather than a distinguished theory. An island can be artificially small because an operator was assumed unique. Spectrum extraction can split or merge nearly degenerate operators. Analytic large-spin formulae can be excellent at high spin and misleading at $\ell=0,2$ . A numerically stable result can still carry a shared implementation error.

Claim ceiling. A certified positive functional can exclude a scoped theory class. Stable extremal data can identify a compelling candidate. Neither an allowed region nor a reconstructed finite spectrum proves existence, locality, or uniqueness of a CFT; those require the additional conditions discussed in from crossing solutions to actual CFTs.

Choosing a bootstrap route

Research need	Start with	Escalate when
Universal one-sided bound	Numerical linear or semidefinite optimization	Add correlators or assumptions only when physically justified
High-spin spectrum or perturbation around known data	Lightcone OPE or Lorentzian inversion	Use numerics for low spin and nonasymptotic mixing
Model identification	Mixed-correlator numerics plus navigator/extremal reconstruction	Demand independent observables withheld from the fit
Formal numerical claim	Dual witness and independent verifier	Add interval/exact block and tail bounds for end-to-end certification
Existence of a theory	Constructive or axiomatic method	Use bootstrap as constraints and consistency checks, not the sole existence argument

Evidence cutoff and change criteria

The finite selection covers the founding numerical method, OPE convergence, large-spin and inversion methods, high-precision mixed-correlator work, and modern optimization. Targeted arXiv, journal, software, and citation-chain searches covered public evidence through 11 August 2026. Reassessment is warranted by a new end-to-end certificate, a benchmark reproduction failure, a conformal-block or solver correction, a proved convergence theorem, or a counterexample to model identification.

References

Caron-Huot, Simon. “Analyticity in Spin in Conformal Theories.” Journal of High Energy Physics 2017, no. 9 (2017): 078. DOI.
Komargodski, Zohar, and Alexander Zhiboedov. “Convexity and Liberation at Large Spin.” Journal of High Energy Physics 2013, no. 11 (2013): 140. DOI.
Kos, Filip, David Poland, David Simmons-Duffin, and Alessandro Vichi. “Precision Islands in the Ising and $O(N)$ Models.” Journal of High Energy Physics 2016, no. 8 (2016): 036. DOI.
Pappadopulo, Duccio, et al. “OPE Convergence in Conformal Field Theory.” Physical Review D 86 (2012): 105043. DOI.
Rattazzi, Riccardo, et al. “Bounding Scalar Operator Dimensions in 4D CFT.” Journal of High Energy Physics 2008, no. 12 (2008): 031. DOI.
Simmons-Duffin, David. “A Semidefinite Program Solver for the Conformal Bootstrap.” Journal of High Energy Physics 2015, no. 6 (2015): 174. arXiv.