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Conformal field theory and a new geometry

Liang Kong

TL;DR

The paper advances the program of stringy algebraic geometry (SAG) by arguing that 2D conformal field theories, formalized through vertex operator algebras, provide a natural, stringy generalization of classical geometry where the spectrum is the category of D-branes. It surveys the mathematical foundations of closed CFTs (Segal/Huang frameworks, VOAs, and modular tensor categories) and develops the open-closed CFT formalism via Cardy algebras, including their modular-invariance constraints and Morita theory; it shows how centers encode bulk theories and how D-branes classify boundary data. A central theme is the holographic principle: the boundary theory determines the bulk, with a functorial center construction linking boundary and bulk data and shedding light on defects and dualities. The work posits SAG as a unifying paradigm that blends algebraic geometry, spectral geometry, and quantum geometry, with potential applications to string topology and geometric Langlands, contingent on constructing explicit examples and extending foundations to supersymmetric settings.

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.

Conformal field theory and a new geometry

TL;DR

The paper advances the program of stringy algebraic geometry (SAG) by arguing that 2D conformal field theories, formalized through vertex operator algebras, provide a natural, stringy generalization of classical geometry where the spectrum is the category of D-branes. It surveys the mathematical foundations of closed CFTs (Segal/Huang frameworks, VOAs, and modular tensor categories) and develops the open-closed CFT formalism via Cardy algebras, including their modular-invariance constraints and Morita theory; it shows how centers encode bulk theories and how D-branes classify boundary data. A central theme is the holographic principle: the boundary theory determines the bulk, with a functorial center construction linking boundary and bulk data and shedding light on defects and dualities. The work posits SAG as a unifying paradigm that blends algebraic geometry, spectral geometry, and quantum geometry, with potential applications to string topology and geometric Langlands, contingent on constructing explicit examples and extending foundations to supersymmetric settings.

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.

Paper Structure

This paper contains 21 sections, 18 theorems, 37 equations, 2 figures.

Table of Contents

  1. Introduction and summary
  2. Geometry and physics
  3. Closed conformal field theories
  4. Basic definitions
  5. Operads and Huang's Theorems
  6. Vertex operator algebras
  7. Holomorphic CFTs
  8. Full field algebras
  9. Modular tensor categories
  10. Non-holomorphic CFTs over $V^L\otimes V^R$
  11. Open-closed CFTs
  12. Basic definitions
  13. Operads
  14. Open-string vertex operator algebras
  15. Open-closed field algebras
  16. ...and 6 more sections

Key Result

Theorem 3.3

A VOA $V$ canonically gives a holomorphic projective $K$-algebra or equivalently a structure $(V_\mathrm{cl}, \{ \mathcal{F}([\Sigma])\}_{[\Sigma]\in K})$ for $V_\mathrm{cl}=V$ as a substructure of a CFT.

Figures (2)

  • Figure 1: Geometric intuition of the category of D-branes associated to a target manifold $M$: a closed CFT $A_\mathrm{cl}$ models the free loop space; a boundary condition, i.e. a chiral $A_\mathrm{cl}$-module $X$, might be able to model the space of $[0,1)$-paths starting from a submanifold associated to $X$ (also denoted by $X$); an open CFT $[X,X]$ models the space of paths from $X$ to $X$; and the $[Y,Y]$-$[X,X]$-bimodule $[X,Y]$ models the space of paths from $X$ to $Y$.
  • Figure 2: Sewing a negatively oriented puncture $(p_i, f_{p_i}, -)$ to a positively oriented puncture $(q_i, f_{q_i}, +)$.

Theorems & Definitions (49)

  • Definition 3.1
  • Remark 3.2
  • Theorem 3.3: h-book
  • Remark 3.4
  • Theorem 3.5: h-funvoa1h-funvoa2
  • Remark 3.6
  • Definition 3.7
  • Remark 3.8
  • Theorem 3.9: h-mtc
  • Definition 3.10
  • ...and 39 more