Conformal Field Theory and Bootstrap

Conformal field theory describes quantum field theories at scale-invariant fixed points through local operators, their correlation functions, and exact consistency conditions. The conformal bootstrap turns those conditions into a nonperturbative method: specify the theory class and observable, expand the same correlator in compatible operator-product channels, add positivity or analyticity only when their hypotheses hold, and extract the strongest conclusion supported by a proof, controlled approximation, or checked certificate. A systematic account of this data-and-consistency formulation is given by Poland, Rychkov, and Vichi 2019, §§2–4.

Helpful background. The Wilson OPE supplies the short-distance expansion specialized here to conformal families. Reflection Positivity supplies the Hilbert-space sign conditions. Linearized RG Flow explains fixed points and deformations. Multiplets and Selection Rules supplies symmetry-sector language.

Choose an entry route

The volume has four connected layers.

Symmetry, states, and conformal data

Chapter	Use it to…
Conformal Symmetry and Representations	derive the global algebra, primary modules, unitarity bounds, currents, and characters
Radial Quantization and State–Operator Correspondence	turn insertions into states, adjoints, Gram matrices, and completeness relations
Correlators, OPE, and Conformal Blocks	build normalized tensor structures, convergent OPEs, blocks, crossing, and positivity

Dimensions, models, defects, and deformations

Chapter	Use it to…
CFT in One Dimension	control ordering sectors, exact $SL(2,\mathbb R)$ blocks, functionals, and generalized-free data
Two-Dimensional CFT	use Virasoro and affine symmetry, rational models, sewing, and modular consistency
Nonunitary, Logarithmic, and Noncompact 2D CFT	replace positivity or discrete sums with indefinite pairings, Jordan modules, and continuous measures
Higher-Dimensional CFT and Controlled Regimes	organize spinning data, free anchors, $\epsilon$, $1/N$, large charge, and higher-spin breaking
Boundaries, Defects, and Interfaces	formulate bulk/defect channels, displacement identities, interfaces, and defect flows
Weyl Anomalies, Deformations, and Flow Constraints	separate universal curvature response, contact terms, conformal perturbation, manifolds, and monotonicity

Numerical, Lorentzian, detector, and thermal methods

Chapter	Use it to…
Numerical Bootstrap	compile crossing into a convex problem and verify conditional bounds or reconstructed data
Analytic and Lorentzian Bootstrap	continue to causal sheets, invert discontinuities, control Regge arcs, and derive sum rules
Energy Flow and Light-Ray Operators	normalize detectors, apply ANEC and collider positivity, and bootstrap energy correlators
Modular and Thermal Bootstrap	distinguish torus modular crossing from local thermal KMS crossing and inversion

Protected and large-N interfaces

Chapter	Use it to…
Superconformal Bootstrap Interfaces	convert versioned protected data into Ward-reduced, recombination-complete crossing
Large-N, Mellin, and Holographic CFT Interfaces	state CFT-side factorization, Mellin, gap, and limit criteria without assuming a bulk conclusion

If your goal is a current numerical bound, software comparison, or active open problem, continue from the durable method chapter to the dated conformal-bootstrap research map.

The data-to-consistency map

Every valid path begins by declaring the spacetime, state, operators, and conventions. It ends at a consistency equation or observable, then passes through a proof or error-controlled inference step before reaching a claim.

For Euclidean correlators, the radial-quantization argument makes the OPE an absolutely convergent expansion inside the appropriate separating sphere, rather than merely a formal short-distance series; the precise domain and quantitative tail bounds are developed by Pappadopulo et al. 2012, §§2–4.

In the diagram, begin with the declared theory class and normalized data at the center, then distinguish the exact consistency and observable branches from the dashed interface where further interpretation requires new assumptions.

Conformal bootstrap starts with a declared theory class and normalized operator data, compares convergent or analytically continued channel representations, and adds positivity, modularity, causality, or asymptotics only under their own hypotheses. A method can exclude or constrain data without proving that a CFT exists or identifying a model. The diagram is schematic.

The full relationship remains available without the figure:

Stage	Required input	Output	Claim boundary
Theory class	$d$, signature, global setting, sectors, unitarity or replacement	allowed representations and correlators	no dynamics yet
Conformal data	spectrum, two-point metric, three-point structures, OPE normalization	channel expansion	completeness and convergence domain stated
Consistency	crossing, sewing, modular covariance, KMS, Ward identities	exact functional equation	equation alone does not prove existence
Additional structure	positivity, causality, Regge bounds, gaps, sparsity, protection	convex cone or analytic constraints	every hypothesis remains attached
Method	exact solution, controlled expansion, functional, inversion, or certified optimization	bound, sum rule, or reconstructed candidate	truncation and ambiguity retained
Interpretation	comparison with independent data	bounded model or interface statement	kink, island, or large-N pattern is not automatic identification

A scalar crossing problem in one page

For identical real scalar primaries with unit two-point function in Euclidean signature,

$$\langle\phi(x)\phi(0)\rangle=(x^2)^{-\Delta_\phi}.$$

Define

$$u=\frac{x_{12}^2x_{34}^2}{x_{13}^2x_{24}^2}=z\bar z, \qquad v=\frac{x_{14}^2x_{23}^2}{x_{13}^2x_{24}^2} =(1-z)(1-\bar z),$$

and write

$$\langle\phi_1\phi_2\phi_3\phi_4\rangle =\frac{\mathcal G(u,v)}{(x_{12}^2x_{34}^2)^{\Delta_\phi}}.$$

In the $12\to34$ OPE domain,

$$\mathcal G(u,v) =\sum_{\mathcal O\in\phi\times\phi} \lambda_{\phi\phi\mathcal O}^2 g_{\Delta,\ell}(u,v),$$

where the identity is included and $g_{\Delta,\ell}\sim u^{\Delta/2}$ in the chosen normalization. Crossing gives

$$\sum_{\mathcal O}\lambda_{\phi\phi\mathcal O}^2 \left[v^{\Delta_\phi}g_{\Delta,\ell}(u,v) -u^{\Delta_\phi}g_{\Delta,\ell}(v,u)\right]=0.$$

The coefficients are nonnegative only because the external operator is identical and Hermitian, the theory is reflection positive, the two-point metric is positive, and the channel uses the compatible radial adjoint. Mixed, nonunitary, logarithmic, and Lorentzian systems require their own matrix, indefinite, or ordered replacements.

The representation-theoretic construction of the radial adjoint and the standard functional-separation formulation of this crossing equation are reviewed, with compatible conventions, in Rychkov 2017, §§2–3 and Simmons-Duffin 2017, §§2–4.

This simple equation already displays the volume’s recurring pattern: symmetry fixes the blocks; dynamics is the spectrum and OPE data; associativity gives crossing; positivity turns it into separation; and a functional exclusion still does not construct a theory.

Common conventions and translation checks

The table below fixes the shared starting convention. A page may override a row when its subject requires it, but must state the conversion and reproduce an invariant quantity after converting.

Axis	Shared convention	Required translation check
Lorentzian signature	$\eta_{\mu\nu}=\operatorname{diag}(+,-,\ldots,-)$	scalar product and $i\epsilon$ prescription agree
Euclidean continuation	positive $x_E^2$ with continuation path stated for Lorentzian use	same ordered boundary value is recovered
Conformal algebra	$[D,P_\mu]=P_\mu$, $[D,K_\mu]=-K_\mu$, $[K_\mu,P_\nu]=2\delta_{\mu\nu}D-2M_{\mu\nu}$	a Casimir eigenvalue or Jacobi identity agrees
Radial adjoint	$D^\dagger=D$, $P_\mu^\dagger=K_\mu$, $M_{\mu\nu}^\dagger=-M_{\mu\nu}$ in the no-$i$ generator convention	level-one and scalar level-two Gram norms agree
Scalar two-point function	unit coefficient, $(x^2)^{-\Delta}$	rescaling is inverted in every OPE coefficient
Degenerate operators	positive two-point matrix retained until a declared whitening	basis-invariant quadratic OPE weight agrees
OPE coefficient	coefficient multiplies a unit-normalized primary and its fixed descendants	a two- or three-point function round-trips
Cross ratios	$u=z\bar z$, $v=(1-z)(1-\bar z)$ with channel and ordering stated	all six permutation images agree
Scalar block	leading $12$-channel behavior $g_{\Delta,\ell}\sim u^{\Delta/2}$ with angular tensor declared locally	Casimir and OPE-limit checks agree
One-dimensional block	$g_\Delta(z)=z^\Delta{}_2F_1(\Delta,\Delta;2\Delta;z)$	series, integral, and Casimir forms agree
Two-dimensional stress tensor	$T(z)T(0)\sim(c/2)z^{-4}+2T(0)z^{-2}+\partial T(0)z^{-1}$	plane–cylinder shift and character vacuum energy agree
Four-dimensional stress tensor	$\langle TT\rangle=C_T\mathcal I/x^8$ and $C_T=40c/\pi^4$ in the displayed anomaly convention	a free-field anchor agrees
Weyl anomaly in 4D	$\langle T^\mu{}_{\mu}\rangle=(cW^2-aE_4+b\nabla^2R)/(4\pi)^2$	$S^4$ gives $dF/d\log R=-4a$
Fourier transform	$\widetilde{\mathcal O}(p)=\int d^dx\,e^{-ip\cdot x}\mathcal O(x)$, inverse measure $d^dp/(2\pi)^d$	position/momentum Ward identity round-trips
Discontinuity	$\operatorname{Disc}f(x)=f(x+i0)-f(x-i0)$; dDisc phases declared on the page	a frozen continuation path gives the same result
Modular variable	$q=e^{2\pi i\tau}$ with spin structure, vacuum shift, and sector sum explicit	$S$ and $T$ actions close in the chosen sector basis
Numerical output	crossing basis, derivative order, precision, approximation, feasibility sign, and solver version stated	independent residual and certificate agree
Mellin or detector data	contour/measure/normalization and Regge or smearing domain declared locally	pole residues or total-energy sum rule agrees

The translation procedure is short: record the source formula, write the forward transformation, write its inverse, and test an invariant such as a norm, correlator, crossing residual, anomaly response, displacement identity, or Mellin residue. An unresolved sign, phase, branch, contact term, or degeneracy remains visible in the conclusion.

Strength of conclusions

Use the narrowest description that fits the evidence:

Evidence	Supported statement	Unsupported promotion
Algebraic identity or exact solution	equality under the stated definitions	existence of a microscopic realization unless constructed
Theorem	conclusion under all listed hypotheses	extension beyond dimension, positivity, or regularity assumptions
Controlled expansion	coefficient and remainder in a declared regime	extrapolation outside that regime
Certified numerical exclusion	no solution inside the specified ansatz and assumptions	existence at an allowed point
Convergent numerical pattern	evidence with precision and truncation study	theorem or exact endpoint
Kink, island, or reconstructed spectrum	candidate boundary structure	unique model identification
Large-N, protected, or holographic interface	conditional CFT statement	bulk dynamics, locality, or duality without its separate construction

Current benchmark values, software versions, and open-problem status belong in dated resources. Durable pages retain the equations, assumptions, verification workflow, and claim boundary.

Review your starting point

Build a crossing vector

Using the scalar conventions above, define a crossing vector whose positive sum vanishes.

Solution

For each exchanged primary define $F_{\Delta,\ell}(u,v)=v^{\Delta_\phi}g_{\Delta,\ell}(u,v)-u^{\Delta_\phi}g_{\Delta,\ell}(v,u)$. Crossing is $\sum_{\mathcal O}\lambda_{\phi\phi\mathcal O}^2F_{\Delta,\ell}=0$. Positivity of the weights requires the identical-Hermitian reflection-positive hypotheses stated above.

Diagnose a model claim

A finite numerical search finds a small allowed island near known CFT data. What is established?

Solution

The search conditionally excludes data outside the island at its stated derivative order, approximations, precision, spectrum assumptions, and symmetry sectors. Agreement with a known model is evidence for an interpretation, not proof of existence, uniqueness, or identification. Those require convergence and independent CFT data.

References

• Pappadopulo, D., Rychkov, S., Espin, J., and Rattazzi, R. “OPE Convergence in Conformal Field Theory.” Physical Review D 86, 105043 (2012). arXiv. DOI.
• Poland, D., Rychkov, S., and Vichi, A. “The Conformal Bootstrap: Theory, Numerical Techniques, and Applications.” Reviews of Modern Physics 91, 015002 (2019). arXiv. DOI.
• Rychkov, S. EPFL Lectures on Conformal Field Theory in $D\ge3$ Dimensions. SpringerBriefs in Physics. Springer, 2017. arXiv. DOI.
• Simmons-Duffin, D. “TASI Lectures on the Conformal Bootstrap.” In New Frontiers in Fields and Strings, 1–74. World Scientific, 2017. arXiv. DOI.