Higher-Dimensional CFT and Controlled Regimes

Higher-dimensional conformal symmetry fixes the kinematics of correlators but rarely determines an interacting theory by itself. Progress comes from choosing two things correctly: a representation language adapted to spin, and a parameter that controls the dynamics. This chapter moves from exact free and generalized-free anchors through embedding-space and spinning methods to perturbative, large-$N$, large-charge, weakly broken higher-spin, and complex-fixed-point regimes.

Helpful background. Crossing symmetry and positivity provide the consistency equation to be solved. Fixed points and linearized RG flow relate RG eigenvalues to scaling dimensions. Large-$N$ limits and normalizations and controlled EFT expansions supply two of the principal small-parameter limits.

Two choices organize the chapter

The first choice is kinematical. Scalar correlators can be written directly in terms of cross-ratios; tensor and spinor correlators benefit from the projective null cone, polarization variables, and weight-shifting operators. The second choice is dynamical. A calculation may be controlled because a coupling, $\epsilon=4-d$, $1/N$, $1/Q$, or a current nonconservation coefficient is small. A long walking interval near colliding fixed points is different: it can be controlled by proximity to complex CFT data without furnishing a real unitary fixed point.

Target question	Start here	What must be fixed before calculating
Which exact correlator is a useful benchmark?	Free and generalized free CFTs	Field normalization, equation-of-motion shortening, locality, and stress-tensor content
How should tensor or spinor indices be represented?	The embedding-space formalism	Embedding signature, projective weight, transversality, spin cover, and physical section
How are spinning blocks generated and checked?	Spinning operators and blocks	Three-point basis, seed normalization, statistics, shortening, parity, and chirality
Which coefficients characterize conserved currents and stress tensors?	Current and stress-tensor CFT data	Two-point normalization, Ward normalization, improvements, contact terms, and dimension
What follows near a Gaussian fixed point?	Perturbative CFT data near free fixed points	Coupling convention, fixed-point substitution, mixing basis, order, and remainder estimate
What follows when $N$ is large?	Large-$N$ CFT data and vector models	Operator normalization, saddle, singlet taxonomy, $N$ scaling, and nonuniform limits
How should different controlled methods be compared?	Controlled families of interacting CFTs	An invariant observable, independent uncertainties, overlap regime, and existence assumptions
What follows in a fixed sector with $Q\gg1$?	Large-charge EFT and fixed-charge data	Symmetry-breaking pattern, charge density, sphere radius, EFT hierarchy, and competing sectors
How does approximate higher-spin conservation constrain data?	Weakly broken higher-spin symmetry	Nonconservation equation, large-$N$ or coupling order, mixing, contact terms, and parity
What does a fixed-point collision imply?	Nonunitary sectors, complex CFTs, and fixed-point collisions	Real versus complex couplings, beta-function convention, observable, walking window, and evidence cutoff

The general conformal-block and positivity framework used by these routes is reviewed in Poland, Rychkov, and Vichi 2019, §§III–IV.

Invariant data versus calculation choices

A primary dimension $\Delta_i$ and a three-point coefficient in a fixed unit-normalized operator basis are CFT data. Intermediate quantities are often not: a coupling depends on its coordinate in theory space, an anomalous-dimension matrix depends on the operator basis, and an EFT Wilson coefficient depends on field normalization. A reliable calculation records the map from those choices to invariant observables.

For a set of operators $\mathcal O_i$ with two-point matrix $G_{ij}$ and three-point tensor $C_{ijk}$, a basis change $\mathcal O'_i=M_i{}^j\mathcal O_j$ gives

$$G'=MGM^{\mathsf T}, \qquad C'_{ijk}=M_i{}^{i'}M_j{}^{j'}M_k{}^{k'}C_{i'j'k'}.$$

Eigenvalues of the dilatation operator and basis-invariant contractions built with $G^{-1}$ can be compared across methods. Raw matrix entries cannot be compared until the bases and normalizations are matched.

The same distinction applies to error estimates. If

$$X(g)=\sum_{k=0}^{K}x_k g^k+R_{K+1}(g),$$

then “known to order $g^K$” states an algebraic truncation, not a numerical uncertainty at $g=1$. A controlled claim also needs a domain for $g$, an estimate or bound for $R_{K+1}$, and a test under changing $K$ or an independent method. Perturbative series can be asymptotic, and limits such as $N\to\infty$, spin $\ell\to\infty$, and $d\to4$ need not commute.

Exact anchors and controlled deformations

Three exact structures recur throughout the chapter:

1. Gaussian factorization. Wick contractions give crossing-symmetric correlators and an exact tower of bilinear primaries.
2. Representation theory. Embedding-space homogeneity, Casimir equations, and shortening conditions are exact and independent of a dynamical expansion.
3. Ward identities. Conserved charges fix relations among two- and three-point coefficients after normalization.

Controlled regimes deform these anchors. In the expansion of Wilson and Fisher 1972, pp. 240–243, a free equation-of-motion multiplet recombines and operator dimensions become series in $\epsilon$. In a vector model, generalized-free factorization receives $1/N$ corrections. At large charge, a finite-density state creates a derivative expansion for a Goldstone mode even when the vacuum theory is strongly coupled. Weakly broken higher-spin currents have small anomalous dimensions fixed by their nonconservation operators. Near a fixed-point collision, the small parameter controls a long RG time but can move the fixed points off the real coupling space.

A common comparison record

Every controlled result in this chapter can be recorded with the same scientific fields:

Field	Required content
Theory and sector	Dimension, global symmetry, operator representation, and vacuum or fixed-charge sector
Observable	A normalized dimension, OPE coefficient, current coefficient, or correlator component
Control parameter	$\epsilon$, $1/N$, inverse charge, weak breaking, or distance from a collision
Computed order	The highest retained power, loop order, or derivative order
Remainder	A bound, asymptotic estimate, scale variation, or explicit statement that no numerical error follows
Independent check	Exact limit, Ward identity, crossing equation, another expansion, lattice observable, or numerical bootstrap
Nonuniform limit	Which large or small limits were held fixed and which may fail to commute
Supported conclusion	Exact identity, controlled asymptotic statement, extrapolation, or qualitative evidence

This table is not a device for averaging unlike methods. An $O(\epsilon^3)$ truncation and a Monte Carlo covariance matrix describe different uncertainties and must remain separate until a justified statistical model relates them.

Scope and handoffs

General beta-function construction and EFT matching remain in Renormalization and effective field theory. Primary large-$N$ saddle dynamics remain in Nonperturbative methods, and regulator-specific uncertainty belongs in Lattice and Hamiltonian methods. This chapter uses those methods to extract and compare higher-dimensional CFT data. Mutable claims about the existence or interpretation of candidate fixed points require dated evidence rather than an undated classification.

Review the chapter

After completing the leaves, you should be able to:

• decompose a generalized-free scalar four-point function and identify the bilinear spectrum;
• lift and pull back a spinning correlator without retaining projective gauge artifacts;
• generate a spinning block from seed data and verify its OPE, Casimir, exchange, and shortening conditions;
• convert current and stress-tensor coefficients between declared normalizations;
• diagonalize a perturbative mixing matrix at a fixed point;
• state the $N$ scaling of normalized singlet correlators and a representative $1/N$ correction;
• derive the leading fixed-charge power law on the cylinder;
• relate a current nonconservation equation to a small anomalous dimension;
• integrate the fixed-point-collision normal form and distinguish walking from a real fixed point.

In every case, the answer should include the expansion parameter, normalization, computed order, limitation, and at least one independent check.

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
• Wilson, Kenneth G., and Michael E. Fisher. “Critical Exponents in 3.99 Dimensions.” Physical Review Letters 28 (1972): 240–243. DOI.