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\(\varvec{{\mathbb {Z}}_3}\)-orbifold construction of the Moonshine vertex operator algebra and some maximal 3-local subgroups of the Monster

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Abstract

In this article, we describe some maximal 3-local subgroups of the Monster simple group using vertex operator algebras (VOA). We first study the holomorphic vertex operator algebra obtained by applying the orbifold construction to the Leech lattice vertex operator algebra and a lift of a fixed-point free isometry of order 3 of the Leech lattice. We also consider some of its special subVOAs and study their stabilizer subgroups using the symmetries of the subVOAs. It turns out that these stabilizer subgroups are 3-local subgroups of its full automorphism group. As one of our main results, we show that its full automorphism group is isomorphic to the Monster simple group by using a 3-local characterization and that the holomorphic VOA is isomorphic to the Moonshine VOA. This approach allows us to obtain relatively explicit descriptions of two maximal 3-local subgroups of the shape \(3^{1+12}.2.{{\mathrm{Suz}}}{:}2\) and \(3^8.\Omega ^-(8,3).2\) in the Monster simple group.

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Abbreviations

\(\langle \cdot ,\cdot \rangle \) :

The (normalized) invariant bilinear form on a VOA.

\(a_{n}\) :

The n-th mode of an element \(a\in V\).

\({{\mathrm{Aut}}}V\) :

The automorphism group of a VOA V.

\(C_G(H)\) :

The centralizer of a subgroup H in a group G.

G.H :

A group extension with normal subgroup G such that the quotient by G is H.

\(K_{12}\) :

The Coxeter–Todd lattice of rank 12.

\({\mathbb {M}}\) :

The Monster simple group.

\(N_G(H)\) :

The normalizer of a subgroup H in a group G.

\(\Omega ^-_8(3)\) :

The commutator subgroup of the orthogonal group of 8-dimensional quadratic space over \({\mathbb {F}}_3\) of minus type.

\(O_3(G)\) :

The maximal normal 3-subgroup of a finite group G.

O(L):

The automorphism group of L preserving the inner product \(\langle \cdot |\cdot \rangle \).

O(R, q):

The orthogonal group of the quadratic space R with quadratic form q.

\(\tau \) :

A fixed-point free automorphism of a lattice of order 3 or its lift to an automorphism of the lattice VOA of order 3.

\(\Lambda \) :

The Leech lattice.

R(V):

The set of isomorphism classes of irreducible V-modules.

\(V_L\) :

The lattice VOA associated to an even lattice L.

\(V_\Lambda ^\tau \) :

The fixed-point subspace of \(\tau \) in \(V_\Lambda \), a subVOA of \(V_\Lambda \).

\(V^\natural \) :

The Moonshine VOA.

\(V^\sharp \) :

The holomorphic VOA of central charge 24 constructed as in (3.1).

\({{\mathrm{wt}}}(M)\) :

The lowest L(0)-weight of an irreducible (g-twisted) module M.

\(\omega \) :

The conformal vector of a VOA.

W :

The \(\tau \)-fixed-point subspace \(V_{K_{12}}^\tau \) of the Lattice VOA \(V_{K_{12}}\) associated to \(K_{12}\)

\(\xi \) :

\(\exp (2\pi \sqrt{-1}/3)\).

Z(G):

The center of a finite group G.

\({\mathbb {Z}}_n\) :

The cyclic group of order n.

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Acknowledgements

The authors wish to thank Xingjun Lin for helpful comments on unitary vertex algebras.

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Authors and Affiliations

  1. National University of Tainan, Tainan, 70005, Taiwan

    Hsian-Yang Chen

  2. Institute of Mathematics, Academia Sinica, Taipei, 10617, Taiwan

    Ching Hung Lam

  3. National Center for Theoretical Sciences, Hsinchu, Taiwan

    Ching Hung Lam

  4. Graduate School of Information Sciences, Tohoku University, Sendai, 980-8579, Japan

    Hiroki Shimakura

Authors
  1. Hsian-Yang Chen
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Correspondence to Hsian-Yang Chen.

Additional information

C. H. Lam was partially supported by MoST Grant 104-2115-M-001-004-MY3 of Taiwan.

H. Shimakura was partially supported by JSPS KAKENHI Grant Numbers 26800001.

C. H. Lam and H. Shimakura were partially supported by JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers “Development of Concentrated Mathematical Center Linking to Wisdom of the Next Generation”.

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Chen, HY., Lam, C.H. & Shimakura, H. \(\varvec{{\mathbb {Z}}_3}\)-orbifold construction of the Moonshine vertex operator algebra and some maximal 3-local subgroups of the Monster. Math. Z. 288, 75–100 (2018). https://doi.org/10.1007/s00209-017-1878-z

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