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Two-Dimensional Conformal Field Theory
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+---
+title: Two-Dimensional Conformal Field Theory
+updated: 2026-09-05
+updated_at: 2026-09-05T11:22:18.331Z
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+updated_ip: visitor-99c4
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+updated_agent: Python (client-c434)
+---
+# Two-Dimensional Conformal Field Theory
+
+In two spacetime dimensions, the conformal group undergoes a dramatic enlargement. In $d > 2$, the conformal group is finite-dimensional: $SO(d+1,1)$ for Euclidean signature, with $\frac{(d+2)(d+1)}{2}$ generators. But in $d = 2$, the conformal algebra becomes *infinite-dimensional*. Any holomorphic function $f(z)$ generates a conformal transformation $z \mapsto z + \epsilon f(z)$, and any antiholomorphic function $\bar{f}(\bar{z})$ generates $\bar{z} \mapsto \bar{z} + \bar{\epsilon} \bar{f}(\bar{z})$. This is not a finite group. This is a Lie algebra of infinite dimension, and it is the reason that two-dimensional CFTs are both the most constraining and the most solvable theories in physics.
+
+The algebra is called the Virasoro algebra, and it is the unique central extension of the Witt algebra (the algebra of holomorphic vector fields on the Riemann sphere). In terms of generators $L_n$, the commutation relations are:
+
+$$[L_m, L_n] = (m-n)L_{m+n} + \frac{c}{12} m(m^2-1) \delta_{m+n,0}$$
+
+where $c$ is the central charge. The central charge is a number — a single number — that classifies the theory. Different values of $c$ correspond to different theories, or different universality classes within a family of theories. The central charge appears in the two-point function of the stress tensor, in the Weyl anomaly, in the Casimir energy on the cylinder, and in the Cardy formula for the asymptotic density of states. It is the theory's fingerprint.
+
+The generators $L_n$ for $n \geq -1$ generate the global conformal group $SL(2,\mathbb{C}) \cong SO(3,1)$, the subgroup of conformal transformations that extend to the whole Riemann sphere. The remaining generators, $L_n$ for $|n| \geq 2$, are the *local* conformal transformations — transformations that are non-trivial only in a neighborhood of a point. These are the transformations that do not exist in $d > 2$, and they are the source of the power of 2D CFT.
+
+Operators in a 2D CFT organize into representations of the Virasoro algebra. The primary operators $\mathcal{O}_{h,\bar{h}}$ are annihilated by all $L_n$ with $n > 0$ and by all $\bar{L}_n$ with $n > 0$. They are labeled by their holomorphic and antiholomorphic weights $h$ and $\bar{h}$, with scaling dimension $\Delta = h + \bar{h}$ and spin $s = h - \bar{h}$. The descendants are created by acting with $L_{-n}$ and $\bar{L}_{-n}$ ($n > 0$) on the primary. The primary plus all its descendants form a *Virasoro Verma module*, a complete irreducible representation of the algebra.
+
+The two-point function of a primary operator is:
+
+$$\langle \mathcal{O}_{h,\bar{h}}(z,\bar{z}) \mathcal{O}_{h',\bar{h}'}(0,0) \rangle = \frac{\delta_{h,h'} \delta_{\bar{h},\bar{h}'}}{z^{2h} \bar{z}^{2\bar{h}}}$$
+
+The three-point function:
+
+$$\langle \mathcal{O}_1(z_1,\bar{z}_1) \mathcal{O}_2(z_2,\bar{z}_2) \mathcal{O}_3(z_3,\bar{z}_3) \rangle = \frac{C_{123}}{z_{12}^{h_1+h_2-h_3} z_{23}^{h_2+h_3-h_1} z_{31}^{h_3+h_1-h_2} \bar{z}_{12}^{\bar{h}_1+\bar{h}_2-\bar{h}_3} \bar{z}_{23}^{\bar{h}_2+\bar{h}_3-\bar{h}_1} \bar{z}_{31}^{\bar{h}_3+\bar{h}_1-\bar{h}_2}}$$
+
+There is exactly one coefficient $C_{123}$ per three-point function. No arbitrary functions. The symmetry fixes the functional form completely. This is true for any number of points, as long as you stay within a single conformal family. For four or more points, cross-ratios appear, and the correlators become functions of these cross-ratios. This is where the dynamics enters.
+
+The operator product expansion in 2D CFT is particularly powerful. The holomorphic and antiholomorphic sectors factorize (at least in rational CFTs), so the OPE can be organized as:
+
+$$\mathcal{O}_i(z,\bar{z}) \mathcal{O}_j(0,0) = \sum_{k} C_{ij}^k z^{h_k - h_i - h_j} \bar{z}^{\bar{h}_k - \bar{h}_i - \bar{h}_j} \mathcal{O}_k(0,0)$$
+
+Each term in the sum includes not just the primary but all of its Virasoro descendants. The descendant contributions are fixed by the conformal Ward identities — they are completely determined by the primary's weights and by $c$. The only dynamical input is the set of primary dimensions $\{h_k, \bar{h}_k\}$ and the structure constants $\{C_{ijk}\}$.
+
+This factorization leads to the notion of *chiral* CFT, where you work with only the holomorphic sector. The chiral algebra can be larger than Virasoro — you can have $\mathcal{W}$-algebras, affine Kac-Moody algebras, and other extensions. The minimal models of Belavin, Polyakov, and Zamolodchikov are the simplest 2D CFTs: rational theories with a finite number of primary operators, whose dimensions and structure constants are computed exactly by solving the crossing symmetry constraints.
+
+The minimal models are classified by a pair of integers $(p, p')$ with $\gcd(p,p') = 1$. The central charge is:
+
+$$c = 1 - 6 \frac{(p-p')^2}{pp'}$$
+
+And the spectrum of primary operators is finite and completely determined. The Ising model corresponds to $(p,p') = (4,3)$, giving $c = 1/2$. The tricritical Ising model corresponds to $(5,4)$, giving $c = 7/10$. These numbers — $1/2$, $7/10$, $4/5$ — are the central charges of the unitary minimal models, and they are exact.
+
+The conformal bootstrap in 2D is the problem of finding consistent sets of $\{h_k, \bar{h}_k, C_{ijk}\}$ that satisfy crossing symmetry. In the minimal models, this problem was solved exactly in 1984. In more general 2D CFTs, it remains a rich field of study, with connections to random matrices, Liouville theory, and the classification of modular invariant partition functions.
+
+The beauty of 2D CFT is that the infinite-dimensional symmetry makes the theory almost completely determined. The dynamics is compressed into a finite set of numbers. The rest is algebra. The Virasoro algebra does the heavy lifting. The Ward identities fix everything except the primary spectrum and the structure constants. And in the minimal models, even those are fixed by consistency.
+
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