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Automorphism groups of certain orbifold vertex operator algebras arising from coinvariant lattices associated with the Leech lattice

Takara Kondo

TL;DR

The paper determines Aut(V_{ olimits{Λ_{pX}}}^{ hat{g}}) for coinvariant Leech lattice lattices in the classes 3C, 5C, 11A, and 23A by analyzing the action of automorphisms on the irreducible modules Irr(V_L^{ hat{g}}) equipped with a nondegenerate quadratic form q. Central to the method is the μ-map to the orthogonal group O(Irr(V_L^{ hat{g}}), q) and a detailed kernel–image analysis, captured by a generalized exact sequence 1 → A → Ker μ → B → 1 with A = Hom(L/(1−g)L^*, Z_p and B tied to centralizers modulo the fixed sublattice. The authors compute the kernel and image explicitly for each Λ_{3C}, Λ_{5C}, Λ_{11A}, and Λ_{23A}, using lattice theory, automorphism-centralizer data, and MAGMA, and obtain the precise automorphism groups: (3^4.(3^4:2)).(Ω_7(3).2), (5^2.(5^2:2)).(2×Ω_5(5)), Ω^-_4(11).2, and Ω_3(23).2, respectively. These results contribute to the understanding of automorphisms in orbifold VOAs and reinforce their role in the holomorphic VOAs of central charge 24 classification.

Abstract

We determine the automorphism groups of the orbifold vertex operator algebras associated with the coinvariant lattices of isometries of the Leech lattice in the conjugacy classes 3C, 5C, 11A and 23A. These orbifold vertex operator algebras appear in a classification given by C.H. Lam and H. Shimakura.

Automorphism groups of certain orbifold vertex operator algebras arising from coinvariant lattices associated with the Leech lattice

TL;DR

The paper determines Aut(V_{ olimits{Λ_{pX}}}^{ hat{g}}) for coinvariant Leech lattice lattices in the classes 3C, 5C, 11A, and 23A by analyzing the action of automorphisms on the irreducible modules Irr(V_L^{ hat{g}}) equipped with a nondegenerate quadratic form q. Central to the method is the μ-map to the orthogonal group O(Irr(V_L^{ hat{g}}), q) and a detailed kernel–image analysis, captured by a generalized exact sequence 1 → A → Ker μ → B → 1 with A = Hom(L/(1−g)L^*, Z_p and B tied to centralizers modulo the fixed sublattice. The authors compute the kernel and image explicitly for each Λ_{3C}, Λ_{5C}, Λ_{11A}, and Λ_{23A}, using lattice theory, automorphism-centralizer data, and MAGMA, and obtain the precise automorphism groups: (3^4.(3^4:2)).(Ω_7(3).2), (5^2.(5^2:2)).(2×Ω_5(5)), Ω^-_4(11).2, and Ω_3(23).2, respectively. These results contribute to the understanding of automorphisms in orbifold VOAs and reinforce their role in the holomorphic VOAs of central charge 24 classification.

Abstract

We determine the automorphism groups of the orbifold vertex operator algebras associated with the coinvariant lattices of isometries of the Leech lattice in the conjugacy classes 3C, 5C, 11A and 23A. These orbifold vertex operator algebras appear in a classification given by C.H. Lam and H. Shimakura.
Paper Structure (21 sections, 79 theorems, 56 equations)

This paper contains 21 sections, 79 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lattices
  4. Groups
  5. VOAs, modules and automorphisms
  6. Lattice VOAs and their automorphism groups
  7. Irreducible $V_L^{\hat{g}}$-modules and their conjugates
  8. Irreducible $V_L^{\hat{g}}$-modules of untwisted type
  9. Irreducible $V_L^{\hat{g}}$-modules of twisted type
  10. Irreducible $V_L^{\hat{g}}$-modules and the quadratic form
  11. The structure of the kernel associated with the action of $\mathrm{Aut}\,(V_L^{\hat{g}})$
  12. structure of $\mathrm{Aut}\,(V_{L}^{\hat{g}})$ in the case $L=\Lambda_{3C}, \Lambda_{5C}, \Lambda_{11A}$, or $\Lambda_{23A}$.
  13. The case $L=\Lambda_{3C}$
  14. The case $L=\Lambda_{5C}$
  15. The case $L=\Lambda_{11A}$
  16. ...and 6 more sections

Key Result

Theorem 1.1

(LS) Let $L$ be a positive-definite rootless even lattice. Let $g$ be a fixed-point free isometry of $L$ of prime order $p$ and let $\hat{g}$ be a standard lift of $g$. Then the following are equivalent:

Theorems & Definitions (106)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Definition 2.7
  • ...and 96 more