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Classification of Self-Dual Vertex Operator Superalgebras of Central Charge at Most 24

Gerald Höhn, Sven Möller

Abstract

We classify the self-dual (or holomorphic) vertex operator superalgebras of central charge 24, or in physics parlance the purely left-moving, fermionic 2-dimensional conformal field theories with just one primary field. There are exactly 969 such vertex operator superalgebras under suitable regularity assumptions (essentially strong rationality) and the assumption that the shorter moonshine module $V\!B^\natural$ is the unique self-dual vertex operator superalgebra of central charge 23.5 whose weight-1/2 and weight-1 spaces vanish. Additionally, there might be self-dual vertex operator superalgebras arising as fake copies of $V\!B^\natural$ tensored with a free fermion $F$. We construct and classify the self-dual vertex operator superalgebras by determining the 2-neighbourhood graph of the self-dual vertex operator algebras of central charge 24 and also by realising them as simple-current extensions of a dual pair containing a certain maximal lattice vertex operator algebra. We show that all vertex operator superalgebras besides $V\!B^\natural\otimes F$ and potential fake copies thereof stem from elements of the Conway group $\mathrm{Co}_0$, the automorphism group of the Leech lattice $Λ$. By splitting off free fermions $F$, if possible, we obtain the classification for all central charges less than or equal to 24.

Classification of Self-Dual Vertex Operator Superalgebras of Central Charge at Most 24

Abstract

We classify the self-dual (or holomorphic) vertex operator superalgebras of central charge 24, or in physics parlance the purely left-moving, fermionic 2-dimensional conformal field theories with just one primary field. There are exactly 969 such vertex operator superalgebras under suitable regularity assumptions (essentially strong rationality) and the assumption that the shorter moonshine module is the unique self-dual vertex operator superalgebra of central charge 23.5 whose weight-1/2 and weight-1 spaces vanish. Additionally, there might be self-dual vertex operator superalgebras arising as fake copies of tensored with a free fermion . We construct and classify the self-dual vertex operator superalgebras by determining the 2-neighbourhood graph of the self-dual vertex operator algebras of central charge 24 and also by realising them as simple-current extensions of a dual pair containing a certain maximal lattice vertex operator algebra. We show that all vertex operator superalgebras besides and potential fake copies thereof stem from elements of the Conway group , the automorphism group of the Leech lattice . By splitting off free fermions , if possible, we obtain the classification for all central charges less than or equal to 24.
Paper Structure (35 sections, 30 theorems, 113 equations, 4 figures, 8 tables)

This paper contains 35 sections, 30 theorems, 113 equations, 4 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Vertex Operator Superalgebras
  3. Definition
  4. Modules
  5. Regularity
  6. Modular Tensor Categories and Evenness
  7. Representation Category
  8. Free Fermions
  9. Lattice Vertex Operator Superalgebras
  10. Self-Dual Vertex Operator Superalgebras
  11. Representation Category
  12. Free-Fermion Splitting
  13. Neighbourhood Graph Method
  14. Results in Small Central Charges
  15. Characters
  16. ...and 20 more sections

Key Result

Theorem 1

Up to isomorphism, there are exactly 968 nice, self-dual vertex operator superalgebras of central charge 24 for which the weight-1 space or that of the canonically twisted module is non-zero. In addition, there is the nice, self-dual vertex operator superalgebra $V\!B^\natural\otimes F$ of central c

Figures (4)

  • Figure 1: The notion of $2$-neighbourhood for positive-definite, even, unimodular lattices.
  • Figure 2: The notion of $2$-neighbourhood for nice, self-dual vertex operator algebras.
  • Figure 3: The $2$-neighbourhood graphs for the nice, self-dual vertex operator algebras of central charges $c=0$, $8$ and $16$.
  • Figure 4: The Heisenberg commutant $2$-neighbourhood graph for the nice, self-dual vertex operator algebras of central charge $24$.

Theorems & Definitions (46)

  • Theorem : Classification, \ref{['thm:class']}
  • Theorem : Heisenberg Commutants, \ref{['thm:commconway']}
  • Definition 2.1: Vertex Operator Superalgebra
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • proof
  • ...and 36 more