Algebraic structures in two-dimensional conformal field theory

A review for the Encyclopedia of Mathematical Physics of the algebraic structures underlying two-dimensional conformal field theory. It covers vertex operator algebras and nets of observables, the categorical structure of their representations, and the tools used to build consistent correlators for full conformal field theories with boundaries and defects.

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.

Recommended citation: J. Fuchs, C. Schweigert, S. Wood and Y. Yang, Encl. Math. Phys. (2nd) 3, 604-617 (2025)
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