# nLab rational 2d conformal field theory

Contents

### Context

#### Functorial Quantum field theory

functorial quantum field theory

# Contents

## Idea

Rational 2d conformal field theories are that class of relatively simple 2d CFTs, such as the WZW model, which are characterized by the fact that their spaces of conformal blocks are finite dimensional vector spaces. Locally such 2d CFTs are given by a rational vertex operator algebra.

Interpreted as perturbative string theory vacua, rational 2d CFTs correspond to (non-geometric) target spaces which are compact (such as fibers of Kaluza-Klein compactifications, but not Minkowski spacetime, for instance). See e.g. Schomerus 05 for contrast.

## Properties

### Classification

The FRS-theorem on rational 2d CFT provides a complete classification of rational 2d CFT via the CS-WZW correspondence and the Reshetikhin-Turaev construction of 3d Chern-Simons theory.

## References

See the references at FRS-theorem on rational 2d CFT.

Lecture notes on modular tensor categories and rational CFT:

• Ingo Runkel, Algebra in Braided Tensor Categories and Conformal Field Theory (pdf, pdf)

The contrast with non-rational CFTs:

• Zhihao Duan, Kimyeong Lee, Kaiwen Sun, Hecke Relations, Cosets and the Classification of 2d RCFTs $[$arXiv:2206.07478$]$