Two-Dimensional CFT: Algebraic and Rational Core

Two-dimensional conformal field theory is unusually tractable because local conformal transformations split into holomorphic and antiholomorphic parts. That local symmetry generates Virasoro or larger chiral algebras; their representations organize states, null vectors turn symmetry into differential equations, and sewing tests whether the chiral data define a local theory on complete surfaces. This algebra-to-sewing structure is developed systematically in Di Francesco, Mathieu, and Sénéchal 1997, chs. 5–10. This chapter develops that chain and uses the critical Ising model as one exact benchmark from Kac table to torus partition function.

Helpful background. Conformal generators and their algebra fix the finite-dimensional conformal subalgebra; descendant Gram matrices supply the radial-quantization inner product; and crossing and positivity explain why a channel expansion must agree with its crossed expansions.

Enter the algebraic core

Start by deciding which object is actually at issue. A local coordinate question belongs to Complex Coordinates and Local Conformal Symmetry. A question about the stress tensor, central charge, or the plane–cylinder map belongs to The Virasoro Algebra and the Stress Tensor. A representation question begins with Highest-Weight Modules, Null States, and the Kac Determinant.

Exact theories then divide by construction:

Question	Primary route	Output to carry forward
Finite Virasoro spectrum or BPZ equation	Minimal Models and Fusion Rules	Kac labels, null equations, fusion coefficients
Momentum, winding, or T-duality	Free Bosons and Vertex Operators	charge lattice, cocycles, neutral correlators
Current algebra or group-valued field	Affine Current Algebras and WZW Models	level, integrable modules, Sugawara and KZ data
Quotient or discrete gauging	Cosets and Orbifolds	branching, projections, twisted sectors, fixed-point resolution
Full correlator or torus consistency	Chiral Blocks, Sewing, and Modular Invariance	monodromy-invariant pairings, modular matrices, factorization checks

The order matters. A list of chiral representations is not yet a full two-dimensional CFT. One must choose left–right pairings, make correlators single-valued, include every sector required by factorization, and verify modular consistency. Conversely, modular invariance of a torus partition function is necessary but does not by itself prove all higher-genus or local sewing identities.

Conventions that connect the pages

Work in Euclidean signature on an oriented surface. In a local flat patch,

$$z=x^1+i x^2, \qquad \bar z=x^1-i x^2, \qquad ds^2=dz\,d\bar z.$$

For genuine Euclidean configurations, $\bar z=z^*$; analytic continuation may temporarily treat $z$ and $\bar z$ as independent variables, but branch choices and the return to a physical domain must then be stated. A field of weights $(h,\bar h)$ has scaling dimension $\Delta=h+\bar h$ and Euclidean spin $s=h-\bar h$. Locality on the plane requires integer spin for ordinary bosonic fields, with the expected graded qualification for fermionic or more general braided sectors.

The holomorphic stress tensor is normalized by

$$T(z)T(0) \sim \frac{c/2}{z^4}+\frac{2T(0)}{z^2}+\frac{\partial T(0)}{z}, \qquad T(z)=\sum_{n\in\mathbb Z}L_n z^{-n-2}.$$

Contours are counterclockwise, and radial conjugation in a unitary theory gives $L_n^\dagger=L_{-n}$. The antiholomorphic algebra has independent generators $\bar L_n$ and central charge $\bar c$; most diagonal examples below have $c=\bar c$.

For the cylinder coordinate $w=\tau+i\sigma$ with $\sigma\sim\sigma+2\pi$ and $z=e^w$, the Schwarzian convention used here gives

$$T_{\mathrm{cyl}}(w) =\left(\frac{dz}{dw}\right)^2T_{\mathrm{plane}}(z) +\frac{c}{12}\{z,w\} =z^2T_{\mathrm{plane}}(z)-\frac{c}{24}.$$

Thus a chiral torus character is

$$\chi_i(\tau)=\operatorname{Tr}_{\mathcal H_i} q^{L_0-c/24}, \qquad q=e^{2\pi i\tau}.$$

For a spatial circle of circumference $L$, the full Hamiltonian is

$$H=\frac{2\pi}{L} \left(L_0+\bar L_0-\frac{c+\bar c}{24}\right).$$

This Casimir shift is consistent with the volume convention $W=-\log Z$ and the fixed-point Weyl response $\langle T^\mu{}_{\mu}\rangle=+cR/(24\pi)$ for the book’s curvature convention: the local trace relation and the finite Schwarzian transformation encode the same anomaly, while their displayed signs refer to different geometric operations.

The Ising benchmark

The diagonal minimal model $M(3,4)$ has $c=\bar c=1/2$ and three scalar primaries; its null vectors, fusion rules, and characters provide a compact exact test of the entire chapter Ginsparg 1990, §§4–5:

Field	Kac label	$(h,\bar h)$	Null levels in each chiral module
$\mathbf 1$	$(1,1)\sim(2,3)$	$(0,0)$	$1$ and $6$
$\sigma$	$(1,2)\sim(2,2)$	$(1/16,1/16)$	$2$ and $4$
$\epsilon$	$(1,3)\sim(2,1)$	$(1/2,1/2)$	$3$ and $2$

With unit two-point functions, the nontrivial fusion rules are

$$\sigma\times\sigma=\mathbf1+\epsilon, \qquad \sigma\times\epsilon=\sigma, \qquad \epsilon\times\epsilon=\mathbf1,$$

and the conventional real choice has $C_{\sigma\sigma\epsilon}=1/2$. Later pages derive the level-two null equation, its two four-spin blocks, the Virasoro characters, and the modular $S$ and $T$ matrices. Each is a separate check: agreement of the central charge alone would not determine the spectrum, OPE coefficients, or sector completion.

Scope and limits

The focus is the algebraic and rational core. Unitary compact examples make positivity and finite modular representations especially transparent, but neither Virasoro symmetry nor the null-vector construction assumes unitarity. Continuous spectra, distribution-valued correlators, nonsemisimple fusion, and Jordan blocks require modified completeness and modular statements and are treated in Nonunitary, Logarithmic, and Noncompact Two-Dimensional CFT.

The modular-bootstrap chapter studies bounds from torus consistency, while this chapter constructs exact modular data. General anomaly definitions and renormalization-scheme issues remain with the earlier symmetry and background-field volumes. Here the goal is narrower and testable: every quoted spectrum, null relation, fusion product, correlator, and modular transformation must use one declared normalization and must survive its own algebraic checks.

References

• Di Francesco, Philippe, Pierre Mathieu, and David Sénéchal. Conformal Field Theory. Graduate Texts in Contemporary Physics. New York: Springer, 1997. DOI.
• Ginsparg, Paul. “Applied Conformal Field Theory.” In Fields, Strings and Critical Phenomena, Les Houches Session XLIX, edited by Édouard Brézin and Jean Zinn-Justin, 1–168. Amsterdam: North-Holland, 1990. arXiv.