Twenty-five years of two-dimensional rational conformal field theory

TL;DR

This article provides a concise, integrative panorama of 25 years of two-dimensional rational conformal field theory, weaving together algebraic, geometric, and topological perspectives. It traces the evolution from foundational chiral symmetries and vertex algebras through modular tensor categories and TFT formulations to boundary phenomena, D-branes, and newer categorical and probabilistic approaches such as SLE. Key contributions include the structural role of the Verlinde formula, the TFT construction of full RCFT correlators via special symmetric Frobenius algebras, and the central importance of modular invariants and boundary data for physical realizations. The survey highlights how RCFT connects with 3D topological field theory, higher category theory, and modern geometric insights, while signaling promising directions in non-semisimple theories, higher target-space geometry, and probabilistic methods that will shape future research.

Abstract

In this article we try to give a condensed panoramic view of the development of two-dimensional rational conformal field theory in the last twenty-five years.