Higher-Dimensional CFT and Controlled Regimes
Higher-dimensional conformal symmetry organizes correlators but rarely solves an interacting theory. This chapter combines tools for spin and tensor structures with controlled limits that make CFT data calculable. Free theories anchor the calculations; weak coupling, large numbers of fields, and large charge each control a different regime.
Enter this chapter
Section titled “Enter this chapter”Begin with Free and Generalized Free CFTs to establish exact normalization and crossing checks.
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- limits and normalizations and controlled EFT expansions supply two of the principal small-parameter limits.
Topics
Section titled “Topics”- Free and Generalized Free CFTs — Distinguishes Wick-factorized correlators from the stronger structure of a local free theory.
- The Embedding-Space Formalism — Linearizes conformal transformations using projective null rays.
- Spinning Operators and Blocks in Higher Dimensions — Builds spinning blocks from specified tensor bases and seed data.
- Current and Stress-Tensor CFT Data — Fixes Ward-normalized current and stress-tensor coefficients.
- Perturbative CFT Data near Free Fixed Points — Extracts dimensions and OPE coefficients while treating operator mixing.
- Large-N CFT Data and Vector Models — Organizes singlet data and corrections around large-N factorization.
- Controlled Families of Interacting CFTs — Compares the domains and uncertainty of different calculational methods.
- Large-Charge EFT and Fixed-Charge CFT Data — Uses a large fixed charge to construct an effective description on the cylinder.
- Weakly Broken Higher-Spin Symmetry — Relates small current nonconservation to anomalous dimensions.
- Nonunitary Sectors, Complex CFTs, and Fixed-Point Collisions — Distinguishes real fixed points from complex continuations and walking flows.
How the ideas fit together
Section titled “How the ideas fit together”Exact representation theory and approximate dynamics play different roles. The conformal algebra fixes descendant contributions and conservation conditions; a dynamical calculation supplies primary dimensions and operator-product coefficients. To compare calculations, operators must have compatible two-point normalizations and any mixing must be resolved. These are the data entering conformal-block expansions and crossing. Poland, Rychkov, and Vichi 2019, §§ III.B–III.E.
A controlled limit changes the dynamical calculation, not these matching requirements. Large-N factorization can provide the leading answer while connected correlators carry the first interactions. A result derived at large N or large charge needs its own error control before extrapolation to small values.
Review the chapter
Section titled “Review the chapter”Two methods disagree on entries of an operator-mixing matrix but agree on its eigenvalues. Is that necessarily a physical disagreement?
Solution
No. Matrix entries change with the operator basis. Scaling dimensions are the eigenvalues of the fixed-point dilatation operator and survive an invertible basis change. To compare OPE coefficients as well, match the eigenoperators, resolve degeneracies, and fix their two-point normalization.
Where to go next
Section titled “Where to go next”Continue to Numerical Bootstrap to constrain these data without choosing a weak coupling expansion.