Conformal Field Theory and Bootstrap
Conformal field theory (CFT) and bootstrap research studies quantum field theories constrained by conformal symmetry, unitarity, locality, and operator-product consistency. The field includes analytic and numerical bootstrap, exact two-dimensional structures, defects and boundaries, Lorentzian observables, and classification or reconstruction programs. “Bootstrap” names a family of consistency methods, not a guarantee that every feasible crossing solution is an actual CFT.
Evidence cutoff. 11 August 2026.
Required background. Primaries, descendants, and multiplets supplies representation data, the state–operator correspondence turns local operators into Hilbert-space states, and crossing equations and positivity provides the core consistency problem.
Helpful background. OPE convergence and associativity controls the domain of crossing manipulations, crossing as convex optimization explains numerical bounds, solver certificates supplies reproducibility standards, and conformal nets gives a rigorous operator-algebraic comparison.
CFT consistency space, theories, and observables
Section titled “CFT consistency space, theories, and observables”| Program | Objects constrained | Characteristic output | What the output does not establish |
|---|---|---|---|
| numerical bootstrap | dimensions and OPE coefficients in selected correlators | exclusion bounds, kinks, islands, extremal spectra | existence or uniqueness of a full CFT without further reconstruction |
| analytic Lorentzian bootstrap | double discontinuities, large-spin families, Regge behavior | asymptotic data and inversion formulae | low-spin data outside convergence and boundedness assumptions |
| two-dimensional exact methods | chiral algebras, modular data, vertex operator algebras | classifications and exact correlators in controlled classes | completeness outside the specified rationality/unitarity hypotheses |
| defects and boundaries | defect operators, displacement data, mixed correlators | defect spectra, central charges, interface constraints | a unique ambient theory from a small correlator set |
| rigorous reconstruction | nets, Euclidean correlators, positivity and locality axioms | existence/equivalence theorems under explicit hypotheses | automatic applicability to every numerical crossing solution |
The modern numerical program began by converting crossing and positivity into linear-functional bounds Rattazzi et al. 2008, foundational. Analytic constraints, conformal-block technology, and numerical optimization now form a shared toolkit across dimensions Poland, Rychkov, and Vichi 2019, field review. Islands arise only after declared assumptions—often symmetry, a gap, or information from several correlators—are imposed. Their small geometric size is not itself a statistical confidence interval.
Bootstrap validation cultures
Section titled “Bootstrap validation cultures”Numerical claims depend on derivative or functional truncation, spin cutoffs, block approximations, arithmetic precision, solver tolerances, and assumptions about spectral gaps. A strong result records all of them and supplies a functional or primal/dual certificate that an independent implementation can check. SDPB made large semidefinite bootstrap problems practical, but solver convergence messages are not mathematical certificates unless residuals and rational/interval validation support the claimed sign conditions Simmons-Duffin 2015, method.
Analytic results have different ceilings: the Lorentzian inversion formula reconstructs OPE data analytic in spin from a double discontinuity under Regge and convergence assumptions Caron-Huot 2017, method. Large-spin expansions require error estimates before extrapolation to low spin. Exact supersymmetric or integrable data provide high-value benchmarks but can overrepresent protected sectors.
The three-dimensional Ising model is the principal cross-method benchmark: an early modern bootstrap analysis constrained critical exponents and low-lying operator data from crossing and unitarity El-Showk et al. 2012, 3D Ising benchmark, while Monte Carlo, high-temperature expansion, and experiment constrain overlapping quantities. Agreement is strongest where systematics differ; repeated bootstrap studies using the same conformal blocks and optimization stack are not fully independent. A convention benchmark should evaluate the same conformal blocks and crossing vectors at shared points after explicitly translating normalizations. A numerical reproduction should publish truncations, precision, primal and dual residuals, and a checkable exclusion certificate; a defect benchmark should additionally recover the displacement-operator Ward identities and any exactly known limits.
Obstructions and serious alternatives
Section titled “Obstructions and serious alternatives”Crossing, positivity, and a finite set of correlators are necessary but may be insufficient for a complete local CFT. A candidate must support a consistent operator algebra across all correlators, appropriate stress tensor and symmetry sectors, locality, and reconstruction of states/observables. Extremal-functional spectra are powerful hypotheses for identifying a theory, not existence proofs by default.
Numerical kinks can arise from changes in the extremal solution without a known physical theory. Assumed gaps can carve out an island while excluding an unsuspected nearby theory. Nonunitary, logarithmic, noncompact, and continuous-spectrum CFTs require different positivity or spectral treatments. In two dimensions, modular invariance and chiral algebra can dominate; in higher dimensions, there is no comparable universal classification.
Entering the field
Section titled “Entering the field”Begin with a fully specified correlator system and reproduce a published low-cost bound. Test convergence under functional dimension, spin cutoff, precision, and block approximation before interpreting a feature. Pair the mathematical foundations pathway with reproduce and validate a result.
This guide omits conformal perturbation theory away from fixed points except where it checks CFT data, and it does not catalog every supersymmetric localization or integrability result. Those belong in their field guides unless they function as bootstrap input or an independent benchmark.
Evidence scope and related assessments
Section titled “Evidence scope and related assessments”The finite search used arXiv, INSPIRE, journal/DOI records, numerical-bootstrap repositories, and citation chaining, with targeted searches for certification failures and formal crossing solutions without known theories. Sources public through 11 August 2026 were eligible. Reassess when a certified computation materially changes a benchmark island, a reconstruction theorem closes a known sufficiency gap, or independent methods resolve a disputed kink.
Continue to crossing solutions and actual CFTs, numerical bootstrap certification, the analytic and numerical bootstrap method map, or the three-dimensional Ising benchmark.
References
Section titled “References”- S. Caron-Huot, “Analyticity in Spin in Conformal Theories,” JHEP 09 (2017) 078. DOI.
- S. El-Showk et al., “Solving the 3D Ising Model with the Conformal Bootstrap,” Physical Review D 86 (2012) 025022. DOI.
- D. Poland, S. Rychkov, and A. Vichi, “The Conformal Bootstrap: Theory, Numerical Techniques, and Applications,” Reviews of Modern Physics 91 (2019) 015002. DOI.
- R. Rattazzi, V. S. Rychkov, E. Tonni, and A. Vichi, “Bounding Scalar Operator Dimensions in 4D CFT,” JHEP 12 (2008) 031. DOI.
- D. Simmons-Duffin, “A Semidefinite Program Solver for the Conformal Bootstrap,” JHEP 06 (2015) 174. DOI.