Quantum field theories invariant under the conformal group — in 2D, the infinite-dimensional Virasoro algebra — governing critical phenomena, worldsheet string theories, and exact-solvable lattice models. Virasoro algebra — the Laurent…
conformal-field-theory
Virasoro: [L_m,L_n] = (m−n)L_{m+n} + (c/12)m(m²−1)δ_{m+n,0}
Virasoro algebra — the infinite-dimensional central extension of the Witt algebra of holomorphic vector fields on the punctured complex…
Cardy formula: S = 2π·√(c·L₀/6)
Cardy formula (Cardy 1986, Nucl. Phys. B 270:186) — in a 2D CFT of central charge c, the asymptotic density of states at conformal weight Δ…
Free boson TT OPE: T(z)T(w) ~ c/(2(z−w)⁴) + …
Free massless boson φ on a 2D cylinder — holomorphic stress tensor T(z) = −½ :(∂φ)²: (normal-ordered, ∂ ≡ ∂_z). Quantisation of the mode…
[L₂,L₋₂] = 4·L₀ + c/2 (exact; at c=1: 4L₀ + 1/2)
Closed-form evaluation of the Virasoro central term at m = 2. [L_m, L_{−m}] = 2m·L_0 + (c/12)·(m³−m); substituting m = 2 gives 4·L_0 +…
Cardy S = 2π·√(c·L₀/6); c=1, L₀=24 → 4π (exact)
Closed-form evaluation of the Cardy entropy formula at two canonical (c, L_0) points. (i) Free-boson c = 1, L_0 = 24: S = 2π·√(1·24/6) =…
Free boson: leading TT-OPE coefficient c/2 = 1/2 at c=1
Closed-form pinned evaluation of the leading TT-OPE singular coefficient. T(z)T(w) = c/(2(z−w)⁴) + 2T(w)/(z−w)² + ∂T(w)/(z−w) + reg; the…