Most vertex superalgebras associated to an odd unimodular lattice of rank 24 have an N=4 superconformal structure
Gerald Höhn, Geoffrey Mason
TL;DR
The paper investigates which odd Niemeier lattices $N$ of rank $24$ yield vertex superalgebras $V_N$ that contain an $N{=}4$ superconformal subalgebra of central charge $c'=6$. It leverages MTY's construction by embedding the rank-$6$ lattice $L^+$ into $N$, producing a chain $A\subseteq V_{L^+}\subseteq V_N$ with the same Virasoro element, to realize the $N{=}4$ structure. Through a computer-assisted lattice-theoretic classification, it shows that 116 lattices with $\mu=1$ and 152 lattices with $\mu=2$ admit such an embedding, totalling at least $267$ of the $273$ odd Niemeier lattices, while $\mu=3$ lattices (notably $\Lambda^{\rm odd}$) do not. The work documents a detailed neighborhood-search algorithm in MAGMA using stabilizers to test embeddings and discusses two open questions regarding exceptional lattices and non-lattice self-dual VOAs. Overall, the results reveal a broad presence of $N{=}4$ subconformal structure in VOAs associated to odd unimodular lattices and have implications for moonshine phenomena and lattice-VOA symmetries.
Abstract
Odd, positive-definite, integral, unimodular lattices N of rank 24 were classified by Borcherds. There are 273 isometry classes of such lattices. Associated to them are vertex superalgebras $V_N$ of central charge c=24. We show that at least 267 of these vertex operator superalgebras contain an N=4 superconformal subalgebra of central charge $c'=6$. This is achieved by studying embeddings $L+\subseteq N$ of a certain rank 6 lattice L+.