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On Reducible but Indecomposable Representations of the Virasoro Algebra

Falk Rohsiepe

TL;DR

This work extends Virasoro representation theory beyond semisimple lowest-weight modules by introducing Jordan lowest weight modules and staggered modules, characterizing their submodules and moduli spaces at logarithmic central charges $c=c_{1,q}$. It develops a framework of gradations, filtrations, and maximal preserving submodules, yielding precise constraints on extensions and their universal properties. The paper then applies these ideas to the ${\mathcal W}$-algebra ${\mathcal W}(2,3^3)$ at $c=-2$, showing a rational logarithmic model with a finite set of building blocks closed under fusion and with modular-transform properties that differ from ordinary rational CFTs yet remain tractable. Overall, the results provide a structured, workable approach to logarithmic Virasoro and ${\mathcal W}$-algebra representations, with clear implications for fusion, modularity, and potential applications in physics and mathematics.

Abstract

Motivated by the necessity to include so-called logarithmic operators in conformal field theories (Gurarie, 1993) at values of the central charge belonging to the logarithmic series c_{1,p}=1-6(p-1)^2/p, reducible but indecomposable representations of the Virasoro algebra are investigated, where L_0 possesses a nontrivial Jordan decomposition. After studying `Jordan lowest weight modules', where L_0 acts as a Jordan block on the lowest weight space (we focus on the rank two case), we turn to the more general case of extensions of a lowest weight module by another one, where again L_0 cannot be diagonalized. The moduli space of such `staggered' modules is determined. Using the structure of the moduli space, very restrictive conditions on submodules of `Jordan Verma modules' (the generalization of the usual Verma modules) are derived. Furthermore, for any given lowest weight of a Jordan Verma module its `maximal preserving submodule' (the maximal submodule, such that the quotient module still is a Jordan lowest weight module) is determined. Finally, the representations of the W-algebra W(2,3^3) at central charge c=-2 are investigated yielding a rational logarithmic model.

On Reducible but Indecomposable Representations of the Virasoro Algebra

TL;DR

This work extends Virasoro representation theory beyond semisimple lowest-weight modules by introducing Jordan lowest weight modules and staggered modules, characterizing their submodules and moduli spaces at logarithmic central charges . It develops a framework of gradations, filtrations, and maximal preserving submodules, yielding precise constraints on extensions and their universal properties. The paper then applies these ideas to the -algebra at , showing a rational logarithmic model with a finite set of building blocks closed under fusion and with modular-transform properties that differ from ordinary rational CFTs yet remain tractable. Overall, the results provide a structured, workable approach to logarithmic Virasoro and -algebra representations, with clear implications for fusion, modularity, and potential applications in physics and mathematics.

Abstract

Motivated by the necessity to include so-called logarithmic operators in conformal field theories (Gurarie, 1993) at values of the central charge belonging to the logarithmic series c_{1,p}=1-6(p-1)^2/p, reducible but indecomposable representations of the Virasoro algebra are investigated, where L_0 possesses a nontrivial Jordan decomposition. After studying `Jordan lowest weight modules', where L_0 acts as a Jordan block on the lowest weight space (we focus on the rank two case), we turn to the more general case of extensions of a lowest weight module by another one, where again L_0 cannot be diagonalized. The moduli space of such `staggered' modules is determined. Using the structure of the moduli space, very restrictive conditions on submodules of `Jordan Verma modules' (the generalization of the usual Verma modules) are derived. Furthermore, for any given lowest weight of a Jordan Verma module its `maximal preserving submodule' (the maximal submodule, such that the quotient module still is a Jordan lowest weight module) is determined. Finally, the representations of the W-algebra W(2,3^3) at central charge c=-2 are investigated yielding a rational logarithmic model.

Paper Structure

This paper contains 24 sections, 37 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Lowest weight modules revisited
  3. The general case
  4. Gradation and filtration by $\mathbf{L_0}$
  5. More filtrations
  6. Jordan lowest weight modules
  7. Shapovalov form and minimal JLWMs
  8. Submodules of JVMs
  9. Staggered modules
  10. Moduli spaces
  11. The maximal preserving submodules of JVMs
  12. Characters
  13. Embeddings
  14. Rational models
  15. Null field relations
  16. ...and 9 more sections

Key Result

Corollary 2.3

An $\mathcal{L}$-module $V$ is a lowest weight module if and only if it contains a singular vector $v \in V$ such that $V = \mathcal{U}.v$. The number $h$ in definition singulardef is then called the lowest weight and $v$ a lowest weight vector of the module.

Theorems & Definitions (67)

  • Definition 2.1
  • Definition 2.2
  • Corollary 2.3
  • Remark 2.4
  • Definition 2.5
  • Theorem 2.6
  • Corollary 2.7
  • Theorem 2.8
  • Theorem 2.9
  • Definition 3.1
  • ...and 57 more