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Nonrational conformal field theory

J. Teschner

TL;DR

The work argues for extending conformal-field-theory mathematics beyond rational models by developing a gluing-based framework for conformal blocks on general Riemann surfaces and a stable, unitary modular functor. It emphasizes a holomorphic factorization of correlation functions, a canonical Hermitian form on block spaces, and a careful treatment of moduli-space boundary behavior via gluing, all aimed at enabling harmonic-analysis-like methods on block spaces. The Virasoro nonrational example (c>25) is used to illustrate genus-zero blocks, free-field realizations, and factorization, while outlining how unitary fusion and modular geometry could underpin a generalized Verlinde-like structure. The paper sketches how W-algebras, Langlands duality, and boundary CFT fit into this nonrational paradigm, suggesting deep connections to automorphic-harmonic analysis and potential new invariants beyond the classical Verlinde formula.

Abstract

We discuss the problem to develop a mathematical theory of a certain class of nonrational conformal field theories (CFT) which contain the unitary CFT. A variant of the concept of a modular functor is proposed that appears to be suitable for such CFT.

Nonrational conformal field theory

TL;DR

The work argues for extending conformal-field-theory mathematics beyond rational models by developing a gluing-based framework for conformal blocks on general Riemann surfaces and a stable, unitary modular functor. It emphasizes a holomorphic factorization of correlation functions, a canonical Hermitian form on block spaces, and a careful treatment of moduli-space boundary behavior via gluing, all aimed at enabling harmonic-analysis-like methods on block spaces. The Virasoro nonrational example (c>25) is used to illustrate genus-zero blocks, free-field realizations, and factorization, while outlining how unitary fusion and modular geometry could underpin a generalized Verlinde-like structure. The paper sketches how W-algebras, Langlands duality, and boundary CFT fit into this nonrational paradigm, suggesting deep connections to automorphic-harmonic analysis and potential new invariants beyond the classical Verlinde formula.

Abstract

We discuss the problem to develop a mathematical theory of a certain class of nonrational conformal field theories (CFT) which contain the unitary CFT. A variant of the concept of a modular functor is proposed that appears to be suitable for such CFT.

Paper Structure

This paper contains 52 sections, 3 theorems, 93 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Constraints from conformal symmetry
  3. Motivation: Chiral factorization of physical correlation functions
  4. Vertex algebras
  5. Representations of vertex algebras
  6. Conformal blocks
  7. Definition
  8. Insertions of the vacuum representation
  9. Deformations of the complex structure of $X$
  10. Correlation functions vs. hermitian forms
  11. Behavior near the boundary of moduli space
  12. Gluing of Riemann surfaces
  13. Gluing of conformal blocks
  14. Conformal Ward identities
  15. Decorated marking graphs
  16. ...and 37 more sections

Key Result

Theorem 1

--- [RS] --- A nodal punctured Riemann surface $X_d$ admits a universal unfolding if and only if it is stable, i.e. iff $n>2-2g$.

Figures (8)

  • Figure 1: Standard marking of a three holed sphere.
  • Figure 2: The B-move
  • Figure 3: Two representations for the decoration on a marking graph
  • Figure 4: Diagrammatical representation for chiral vertex operators.
  • Figure 5: Diagrammatic representation for the compositions in (\ref{['compo']}).
  • ...and 3 more figures

Theorems & Definitions (12)

  • Definition 1
  • Theorem 1
  • Remark 1
  • Proposition 1
  • Theorem 2
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Claim 1
  • ...and 2 more