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…
Conformal-map (Joukowski transform) framework: J(z) = z + 1/z; angle-preserving
Conformal-mapping framework (Riemann 1851 Inauguraldissertation; Schwarz 1869). Setup: a holomorphic function f(z) with f'(z) != 0 is a…
Modular invariance Z(tau+1) = Z(tau); period-1 e^{2 pi i} = 1
Modular-invariance framework for CFT partition function on torus (Cardy 1986 Nucl Phys B 270, 186). Setup: 2D CFT on torus T^2 = C/(Z + tau…
Partition function characters chi_p; Schur partition counting p(n)
Partition-function character-decomposition framework (Belavin-Polyakov-Zamolodchikov 1984 Nucl Phys B 241, 333). Setup: 2D CFT partition…
Theorem: J(1) - 2 = 0 (Joukowski transform real-axis boundary)
Theorem (Joukowski-real-boundary canonical): J(z) = z + 1/z evaluated at z=1 gives J(1) = 2 exactly. Canonical sympy pin: z =…
Theorem: e^{2 pi i (tau+1)} - e^{2 pi i tau} = 0 (modular T-invariance)
Theorem (modular-T-invariance canonical): e^{2 pi i (tau+1)} = e^{2 pi i tau} * e^{2 pi i} = e^{2 pi i tau} * 1 = e^{2 pi i tau}, hence…
Theorem: p(4) - 5 = 0 (number of partitions of 4 equals 5)
Theorem (partition-of-4 canonical): the integer 4 has exactly 5 distinct partitions: {4}, {3,1}, {2,2}, {2,1,1}, {1,1,1,1}. Canonical sympy…
Virasoro algebra (2D CFT)
Virasoro 1970: central extension of Witt algebra [L_m, L_n] = (m-n) L_{m+n} + c/12 (m^3 - m) delta_{m+n,0}; central charge c labels…
Primary fields + conformal weights
Belavin-Polyakov-Zamolodchikov 1984: primary fields phi(z, zbar) transform as phi -> (df/dz)^h (df_bar/dz_bar)^h_bar phi(f(z),…
Modular invariance (CFT partition function)
Cardy 1986 modular-invariant partition function Z(tau) = Tr q^{L_0 - c/24} qbar^{Lbar_0 - cbar/24}; classifies minimal models + WZW…
Operator product expansion (OPE)
Wilson 1969 OPE: A(x) B(y) = sum_n C^n_{AB}(x-y) O_n(y) as x->y; CFT case fixed by conformal-weights + central-charge; basis of conformal…
Conformal bootstrap (modern revival)
Rattazzi-Rychkov-Tonni-Vichi 2008 numerical bootstrap: crossing-symmetry + unitarity exclude regions of operator-dimension space; Ising 3D…
AdS/CFT (Maldacena 1997)
Maldacena 1997 + GKP 1998 + Witten 1998: N=4 SYM_4 in 4d = type-IIB string on AdS_5 x S^5; gauge-gravity duality +…
BPZ (Belavin-Polyakov-Zamolodchikov 1984)
A Belavin-A Polyakov-A Zamolodchikov 1984 'Infinite conformal symmetry'; modern modern foundational text in 2D-CFT + minimal-models.
AdS/CFT (Maldacena 1997)
J Maldacena 1997 N=4 SYM duality; modern modern foundational text + holographic-QCD + condmat-strange-metals 100k+ citations.
c-theorem (Zamolodchikov 1986)
A Zamolodchikov 1986 c-theorem 2D-RG-monotonic; modern modern foundational + a-theorem 4D Komargodski-Schwimmer 2011 generalization.
Bootstrap (Rattazzi 2008)
S Rattazzi-V Rychkov 2008 modern conformal-bootstrap; modern modern foundational text + 3D-Ising precision crossings + 2024 SDPB.
Kac determinant (1980)
V Kac 1980 + Feigin-Fuchs 1982-1990 Kac-determinant; modern modern foundational text Verma-modules + Virasoro-degenerate-representations.
WZW model (Witten 1984)
E Witten 1984 + Wess-Zumino 1971 WZW; modern modern foundational text + non-abelian-bosonization + Knizhnik-Zamolodchikov 1984.