Algebraic structures in two-dimensional conformal field theory

TL;DR

This paper reviews algebraic structures in two-dimensional conformal field theory, focusing on symmetries, representation categories, and the construction of correlators, providing a mathematical framework for understanding chiral and full CFTs.

Contribution

It introduces algebraic structures that enable the construction of consistent correlator systems in full 2D conformal field theories, connecting symmetry, representation theory, and correlator construction.

Findings

Classification of chiral algebra symmetries in 2D CFTs

Description of representation categories with additional structures

Methods for constructing systems of correlators in full CFTs

Abstract

This is an invited contribution to the 2nd edition of the Encyclopedia of Mathematical Physics. We review the following algebraic structures which appear in two-dimensional conformal field theory (CFT): The symmetries of two-dimensional conformal field theories (CFTs) can be formalised as chiral algebras, vertex operator algebras or nets of observable algebras. Their representation categories are abelian categories having additional structures, which are induced by properties of conformal blocks, i.e. of vector bundles over the moduli space of curves with marked points, which can be constructed from the symmetry structure. These mathematical notions pertain to the description of chiral CFTs. In a full local CFT one deals in addition with correlators, which are specific elements in the spaces of conformal blocks. In fact, a full CFT is the same as a consistent system of correlators for arbitrary conformal surfaces with any number and type of field insertions in the bulk as well as on boundaries and on topological defect lines. We present algebraic structures that allow one to construct such systems of correlators.