Two-Dimensional CFT: Algebraic and Rational Core

In two dimensions, local conformal symmetry expands into an infinite-dimensional algebra. Its representations make exact spectra and correlators accessible, but a consistent theory also needs compatible left–right pairings and sewing of surfaces. This chapter develops that chain through minimal models, free bosons, current algebras, and their constructions.

Enter this chapter

Begin with Complex Coordinates and Local Conformal Symmetry to see why local holomorphic transformations supply more symmetry than global Möbius maps.

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.

Topics

1. Complex Coordinates and Local Conformal Symmetry — Separates local holomorphic symmetry from globally invertible coordinate maps.
2. The Virasoro Algebra and the Stress Tensor — Derives the central extension and its effect on stress-tensor transformations.
3. Highest-Weight Modules, Null States, and the Kac Determinant — Uses descendant pairings to identify irreducible modules and null equations.
4. Minimal Models and Fusion Rules — Combines null-vector decoupling, fusion, and the exact Ising example.
5. Free Bosons and Vertex Operators — Connects oscillator modes, momentum, winding, and vertex-operator products.
6. Affine Current Algebras and WZW Models — Relates affine symmetry and its level to spectra and differential equations.
7. Cosets and Orbifolds — Constructs theories using commutants or discrete gauging with the required sectors.
8. Chiral Blocks, Sewing, and Modular Invariance — Tests whether chiral data combine into local correlators and consistent torus amplitudes.

How the ideas fit together

Null states connect representation theory directly to correlation functions: their decoupling becomes differential equations whose solutions are chiral blocks. Fusion restricts which intermediate modules occur. A full correlator must then pair holomorphic and antiholomorphic blocks so their monodromies are compatible with locality. The Ising model makes these steps concrete. Ginsparg 1990, §§ 4.5 and 5.2–5.3.

Global consistency adds another test. On a torus, characters count states within chiral modules, and the full partition function must transform correctly under changes of cycles. That test constrains sector choices, but a modular-invariant list of states alone does not specify all local OPE coefficients.

Review the chapter

Why does constructing an orbifold require more than keeping invariant states of the original theory?

Solution

A modular transformation exchanges the torus cycles. A symmetry insertion in the trace is then related to a twisted spatial boundary condition. Keeping only projected untwisted states omits the corresponding twisted sectors, so the proposed partition function generally fails modular closure. Ginsparg 1990, § 8.3.

Where to go next

Continue to Nonunitary, Logarithmic, Noncompact, and Nonrational Two-Dimensional CFT to see which structures change when positivity, diagonalizability, or a discrete rational spectrum is absent.

References

• 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.