Conformal field theory and a new geometry

TL;DR

This paper reviews open-closed rational conformal field theory through vertex operator algebras and proposes a new geometric framework based on CFTs and D-branes, linking algebraic structures to geometric insights.

Contribution

It introduces a novel geometric perspective on CFTs grounded in VOAs and D-branes, and classifies rational CFTs using VOA representation theory.

Findings

Classification of open-closed rational CFTs

Derivation of the Holographic Principle properties

Proposal of a new geometry based on CFTs and D-branes

Abstract

This paper is a review of open-closed rational conformal field theory (CFT) via the theory of vertex operator algebras (VOAs), together with a proposal of a new geometry based on CFTs and D-branes. We will start with an outline of the idea of the new geometry, followed by some philosophical background behind this vision. Then we will review a working definition of CFT slightly modified from Segal's original definition and explain how VOA emerges from it naturally. Next, using the representation theory of rational VOAs, we will discuss a classification result of open-closed rational CFTs, from which some basic properties of a rational CFT, such as the Holographic Principle, can be derived. They will also serve as supporting evidences for the vision of a new geometry. In the end, we briefly discuss the connection between our vision of a new geometry and other topics.