Conformal Field Theory — Operator Spectrum

A 2D CFT is completely determined by its spectrum of primary operators (Δ, s) and OPE coefficients C_{ijk}. Radial quantization maps the cylinder to the plane: states ↔ operators via |Δ,s⟩ = O(0)|0⟩. The conformal bootstrap constrains which spectra are consistent.

c = 0.50
Δ₁ = 0.12
C = 0.60
Virasoro algebra: [L_n, L_m] = (n−m)L_{n+m} + c/12 · n(n²−1)δ_{n+m,0}
Primary ↔ highest-weight state: L_n|h,h̄⟩=0 (n>0), L₀|h,h̄⟩=h|h,h̄⟩
Descendants: L_{-n}|h⟩ at level n | Unitarity: Δ ≥ |s|, Δ ≥ c/24 for degenerate reps