Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An algebraic approach to logarithmic conformal field theory

Matthias R Gaberdiel

TL;DR

Gaberdiel develops an algebraic framework for logarithmic conformal field theory, emphasizing indecomposable representations and Zhu's algebra as essential tools. He works through the $c=-2$ triplet model to show how logarithmic structure arises in fusion, modular properties, and the partition function, and demonstrates an explicit local theory equivalent to the bosonic sector of symplectic fermions. He then extends the discussion to fractional level WZW models, notably $k=-4/3$ for $\mathfrak{su}(2)$, where logarithmic representations appear in fusion and are organized by an outer automorphism symmetry. The work argues that many non-rational CFTs are naturally logarithmic and provides a general, algebraic pathway to understanding their representation theory and modular behavior.

Abstract

A comprehensive introduction to logarithmic conformal field theory, using an algebraic point of view, is given. A number of examples are explained in detail, including the c=-2 triplet theory and the k=-4/3 affine su(2) theory. We also give some brief introduction to the work of Zhu.

An algebraic approach to logarithmic conformal field theory

TL;DR

Gaberdiel develops an algebraic framework for logarithmic conformal field theory, emphasizing indecomposable representations and Zhu's algebra as essential tools. He works through the triplet model to show how logarithmic structure arises in fusion, modular properties, and the partition function, and demonstrates an explicit local theory equivalent to the bosonic sector of symplectic fermions. He then extends the discussion to fractional level WZW models, notably for , where logarithmic representations appear in fusion and are organized by an outer automorphism symmetry. The work argues that many non-rational CFTs are naturally logarithmic and provides a general, algebraic pathway to understanding their representation theory and modular behavior.

Abstract

A comprehensive introduction to logarithmic conformal field theory, using an algebraic point of view, is given. A number of examples are explained in detail, including the c=-2 triplet theory and the k=-4/3 affine su(2) theory. We also give some brief introduction to the work of Zhu.

Paper Structure

This paper contains 20 sections, 183 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The baby example
  3. The analytic approach
  4. The algebraic approach
  5. The triplet model
  6. The triplet algebra
  7. Interlude 1: Representations of a conformal field theory
  8. The representations of the triplet algebra
  9. Interlude 2: Zhu's algebra
  10. Interlude 2 (ctd): Zhu's algebra for the Yang-Lee model
  11. The fusion rules of the triplet theory
  12. Characters and modular transformations
  13. The local triplet theory
  14. The partition function
  15. Symplectic fermions
  16. ...and 5 more sections

Figures (2)

  • Figure 1: The representation ${\cal R}_{{1\over 3}}$.
  • Figure 2: The representation ${\cal R}_0$.