Conformal Field Theory — Mathematics Knowledge System

Overview

Conformal field theory (CFT) studies quantum field theories invariant under conformal transformations (angle-preserving maps). In 2D, the conformal group is infinite-dimensional (the Witt algebra / Virasoro algebra), making 2D CFTs exceptionally constrained and exactly solvable. Key structures include primary fields, operator product expansions (OPE), the Virasoro algebra with central charge c, and highest-weight representations. 2D CFTs appear in string theory (worldsheet theories), statistical mechanics at criticality, and condensed matter physics. The classification by central charge and spectrum is a major achievement.

Intuition

At a critical point (e.g., 2D Ising model at T_c), the system looks the same at all length scales -- it is scale invariant. In 2D, scale invariance plus locality implies conformal invariance. The conformal group in 2D is the group of holomorphic functions (complex variable z -> f(z)), which is infinite-dimensional. This infinite symmetry severely constrains correlators: a 2-point function of primary fields must be <phi_1(z) phi_2(0)> = delta_{h_1,h_2}/z^{2h} where h is the conformal weight. The OPE encodes how operators 'collide' and is the algebraic heart of CFT.

Formal Definition

Definition

A 2D CFT is specified by: (1) a Hilbert space H with an action of two commuting Virasoro algebras (holomorphic and antiholomorphic), (2) a set of primary operators phi_h(z,z-bar) with conformal weights (h, h-bar), (3) an OPE phi_i(z) phi_j(0) = sum_k C_{ij}^k z^{h_k-h_i-h_j} phi_k(0) + ... The Virasoro algebra Vir is generated by {L_n : n in Z} union {c} with [L_m, L_n] = (m-n)*L_{m+n} + (c/12)*(m^3-m)*delta_{m+n,0}. The central charge c characterizes the CFT.

[L_m, L_n] = (m-n)L_{m+n} + \frac{c}{12}(m^3-m)\delta_{m+n,0}

Virasoro algebra

\langle \phi(z,\bar{z})\phi(0,0)\rangle = \frac{1}{z^{2h}\bar{z}^{2\bar{h}}}

2-point function of a primary

\phi_i(z)\phi_j(0) = \sum_k C_{ij}^k z^{h_k-h_i-h_j}\bar{z}^{\bar{h}_k-\bar{h}_i-\bar{h}_j}\phi_k(0)+\cdots

Operator product expansion (OPE)

c = \frac{12k}{k+2} \text{ for }\widehat{\mathfrak{su}(2)}_k \text{ Wess-Zumino-Witten model}

Central charge of WZW model

Notation

Notation	Meaning
$c$	Central charge of the CFT
$L_n$	Virasoro generator
$h$	Holomorphic conformal weight of a primary
$C_{ij}^k$	OPE coefficient (structure constant)

Theorems

Theorem 1: Operator-State Correspondence

In a 2D CFT on the cylinder (= complex plane with radial quantization), there is a bijection between states |phi> in the Hilbert space and local operators phi(z,z-bar). The state corresponding to a primary operator phi of weight (h,h-bar) is a highest-weight state of the Virasoro algebra with L_0|phi> = h|phi> and L_n|phi> = 0 for n > 0. This bijection is the foundation of the CFT state-operator correspondence.

Theorem 2: Ward Identity and Virasoro Primary

An operator phi is primary (of weight h) if under conformal transformations z -> w(z) it transforms as phi(z) -> (dw/dz)^{-h} phi(w(z)). Equivalently, in the OPE with the stress-energy tensor T(z): T(z) phi(w) = h/(z-w)^2 * phi(w) + 1/(z-w) * partial_w phi(w) + regular. Descendants are obtained by acting with negative Virasoro generators L_{-n}, n > 0.

Theorem 3: Cardy Formula for Entropy

For a 2D CFT with central charge c on a torus of modulus tau, the asymptotic density of states rho(E) at energy E is rho(E) ~ exp(2*pi*sqrt(c*E/6)) as E -> inf. This is the Cardy formula. In the AdS/CFT correspondence, the Cardy formula matches the Bekenstein-Hawking entropy of BTZ black holes, providing one of the earliest checks of holography.

Worked Examples

1
The free boson has action S = (1/4pi) integral (partial X)^2. The 2-point function is <X(z) X(w)> = -log|z-w|^2.
$\langle X(z)X(w)\rangle = -\log|z-w|^2$
2
The stress-energy tensor is T(z) = -(1/2)(partial X)^2. Central charge c=1.
$T(z) = -\frac{1}{2}(\partial X)^2,\quad c=1$
3
Normal-ordering: the OPE of partial X with itself is: partial X(z) * partial X(w) = -1/(z-w)^2 + regular.
$\partial X(z)\partial X(w) = -\frac{1}{(z-w)^2} + \text{reg.}$
4
Vertex operators V_k(z) = :e^{ikX(z)}: are primary operators of weight h = k^2/2.
$h(V_k) = \frac{k^2}{2}$

✓ Answer

Free boson: partial X(z) partial X(w) = -1/(z-w)^2 + reg. Vertex operators e^{ikX} are primary with weight k^2/2. Central charge c=1.

Practice Problems

Hardfree response

Explain what the central charge c of a CFT measures and why c=1/2 for the 2D Ising model.

Common Mistakes

Common Mistake

Thinking 2D CFT and higher-dimensional CFT are essentially the same.

In 2D, the conformal algebra is infinite-dimensional (the Witt/Virasoro algebra), which makes the theory exceptionally constrained and often exactly solvable (rational CFTs). In D > 2 dimensions, the conformal group is finite-dimensional (SO(D+1,1)), giving much weaker constraints. The BPZ approach (1984) works specifically in 2D; higher-dimensional CFTs are studied by different methods (bootstrap program).

Quiz

An operator phi(z) is called 'primary' of weight h in a 2D CFT if:

It commutes with all Virasoro generatorsIt transforms as phi -> (dw/dz)^{-h} phi under z -> w (conformal primaries)It has dimension h = 0It is annihilated by the stress-energy tensor

Historical Background

Conformal invariance in statistical mechanics was recognized in the 1960s-70s (Wilson's renormalization group). The systematic study of 2D CFT was launched by Belavin, Polyakov, and Zamolodchikov's 1984 paper, which showed that rational CFTs are classified by their Virasoro representations. The Verlinde formula (1988) for fusion coefficients, Cardy's formula for entropy, and Kontsevich-Segal-Witten gauge theory connections followed. Schramm-Loewner evolution (SLE, 2000) gave a rigorous probabilistic construction of CFT correlators.

1970
Wilson introduces the renormalization group and conformal invariance at criticality
Kenneth Wilson
1984
BPZ paper establishes systematic 2D CFT with Virasoro algebra
Alexander Belavin, Alexander Polyakov, Alexander Zamolodchikov
1988
Verlinde formula for fusion coefficients from modular data
Erik Verlinde
2000
Schramm introduces SLE, rigorous mathematical framework for 2D CFT interfaces
Oded Schramm

Summary

2D CFT: a QFT invariant under conformal maps z -> f(z); symmetry algebra is the Virasoro algebra.
Primary operators transform as phi -> (dw/dz)^{-h} phi; OPE encodes how operators 'collide'.
Central charge c characterizes the CFT: c=1/2 (Ising), c=1 (free boson), c=26 (bosonic string).
Cardy formula: entropy ~ exp(2*pi*sqrt(c*E/6)), matching BTZ black hole entropy in AdS_3/CFT_2.

References

BookDi Francesco, P., Mathieu, P., and Senechal, D. Conformal Field Theory. Springer, 1997.
BookGinsparg, P. Applied Conformal Field Theory. Les Houches Lectures, 1988.
WebsiteWikipedia -- Conformal field theory