[Paper Review] The Automorphism Group of the Vertex Operator Algebra $V_L^+$ for an even lattice $L$ without roots

Hiroki Shimakura|ArXiv.org|Nov 10, 2003

Algebraic structures and combinatorial models13 references4 citations

TL;DR

This paper determines the automorphism group of the vertex operator algebra $V_L^+$ for even lattices $L$ without roots by analyzing its action on irreducible modules via fusion rules. It establishes that $V_L^+$ has extra automorphisms (beyond the stabilizer $H_L$) if and only if $L$ arises from a binary code via Construction B, and fully characterizes ${\rm Aut}(V_L^+)$ for key lattices, including the Barnes-Wall lattice $\Lambda_{16}$, yielding $2^{16} \cdot \Omega_{10}^+(2)$ as the group structure.

ABSTRACT

The automorphism group of the vertex operator algebra $V_L^+$ is studied by using its action on isomorphism classes of irreducible $V_L^+$-modules. In particular, the shape of the automorphism group of $V_L^+$ is determined when $L$ is isomorphic to an even unimodular lattice without roots, $\sqrt2R$ for an irreducible root lattice $R$ of type $ADE$ and the Barnes-Wall lattice of rank 16.

Motivation & Objective

• To determine the full automorphism group $\mathrm{Aut}(V_L^+)$ for even lattices $L$ without roots, especially when extra automorphisms exist beyond the stabilizer $H_L$.
• To establish a uniform method for computing $\mathrm{Aut}(V_L^+)$ using the action on isomorphism classes of irreducible $V_L^+$-modules.
• To clarify the condition under which $V_L^+$ admits extra automorphisms, showing it occurs precisely when $L$ is constructed from a binary code via Construction B.
• To compute the automorphism group explicitly for significant lattices, including the even unimodular lattice $\Lambda_{16}$, $\sqrt{2}R$ for $ADE$ root lattices $R$, and the Barnes-Wall lattice.

Proposed method

• The method uses the action of $\mathrm{Aut}(V_L^+)$ on the set $S_L$ of isomorphism classes of irreducible $V_L^+$-modules, focusing on the orbit of the $(-1)$-eigenspace $V_L^-$.
• A subset $P \subset S_L$ is constructed that is closed under fusion rules and preserved by $\mathrm{Aut}(V_L^+)$, forming a vector space over $\mathbb{F}_2$.
• A group homomorphism $\zeta_{V_L^+}: \mathrm{Aut}(V_L^+) \to \mathrm{GL}(P)$ is defined, leveraging the preservation of fusion rules by automorphisms.
• The kernel of $\zeta_{V_L^+}$ is analyzed via the stabilizer $H_L$, which is shown to be isomorphic to the centralizer of $\theta_{V_L}$ in $\mathrm{Aut}(V_L)$ modulo $\langle \theta_{V_L} \rangle$, with structure $\mathrm{Hom}(L, \mathbb{Z}_2) \cdot (O(L)/\langle -1 \rangle)$.
• The image of $\zeta_{V_L^+}$ is determined by analyzing the orbit size of $[0]^{-}$ and using group-theoretic constraints in $\mathrm{GL}(P)$, particularly in the case of $\Lambda_{16}$.
• For the Barnes-Wall lattice $\Lambda_{16}$, the orbit size of $[0]^{-}$ is shown to be 527, leading to the conclusion that $\mathrm{Im}(\zeta_{V_{\Lambda_{16}}^+}) \cong \Omega_{10}^+(2)$.

Experimental results

Research questions

• RQ1 When does $V_L^+$ admit automorphisms not in the stabilizer $H_L$, i.e., extra automorphisms?
• RQ2 What is the precise structure of $\mathrm{Aut}(V_L^+)$ for even unimodular lattices $L$ without roots, such as $\Lambda_{16}$?
• RQ3 How is the automorphism group of $V_L^+$ related to the lattice $L$ when $L$ is constructed via Construction B from a binary code?
• RQ4 What is the role of the fusion rules and module orbits in determining the automorphism group of $V_L^+$?
• RQ5 Is the exact sequence $1 \to F \to \mathrm{Aut}(V_L^+) \to \mathrm{Im}(\zeta) \to 1$ split for $L = \Lambda_{16}$?

Key findings

• The automorphism group $\mathrm{Aut}(V_L^+)$ is isomorphic to $H_L$ if and only if $L$ is not obtained from a binary code via Construction B; otherwise, extra automorphisms exist.
• For the Barnes-Wall lattice $\Lambda_{16}$, $\mathrm{Aut}(V_{\Lambda_{16}}^+)$ has the shape $2^{16} \cdot \Omega_{10}^+(2)$, with the $2^{16}$-normal subgroup $F$ being the kernel of the homomorphism $\zeta_{V_{\Lambda_{16}}^+}$.
• The orbit $Q_{\Lambda_{16}}$ of the isomorphism class $[0]^{-}$ consists of exactly 527 elements, corresponding to all non-zero isotropic vectors in the 10-dimensional $\mathbb{F}_2$-vector space $S_{\Lambda_{16}}$.
• The image of the homomorphism $\zeta_{V_{\Lambda_{16}}^+}$ is $\Omega_{10}^+(2)$, and the exact sequence $1 \to F \to \mathrm{Aut}(V_{\Lambda_{16}}^+) \to \Omega_{10}^+(2) \to 1$ is non-split.
• The stabilizer $H_{\Lambda_{16}}$ has shape $2^{16} \cdot 2^8 \cdot \Omega_8^+(2)$, and the kernel $F$ of $\zeta_{V_{\Lambda_{16}}^+}$ is isomorphic to $2^{16}$, generated by elements of order 2.
• The automorphism group $\mathrm{Aut}(V_{\Lambda_{16}}^+)$ is generated by $O(\hat{\Lambda}_{16})/\langle \theta_{V_{\Lambda_{16}}} \rangle$ and an extra automorphism $\sigma$ from Section 2.3, confirming the full group structure.