Most Vertex Superalgebras Associated to an Odd Unimodular Lattice of Rank 24 Have an Superconformal Structure

Abstract

The classification of odd positive-definite unimodular lattices M of rank 24 was completed by Borcherds. There are 273 isometry classes of such lattices. Associated to them are vertex superalgebras V_M of central charge $c=24$. We show that at least 267 (but not all) of these vertex superalgebras contain an $N=4$ superconformal subalgebra of central charge $c\prime =6$. This is achieved by studying embeddings $L^{+}\subseteq M$ of a certain rank 6 lattice L⁺.

KEYWORDS:

• Niemeier lattice
• $N=4$ algebra
• vertex superalgebra

MATHEMATICS SUBJECT CLASSIFICATION::

• MSC(2010): 17B69
• 17B81

Acknowledgments

The authors are indebted to the referees for their careful consideration and helpful comments on a prior version of this paper. We also thank Philipp Höhn for proof reading.

Additional information

Funding

The first author was supported by the Simons Foundation Collaboration Grant #355294. The second author was supported by the Simons Foundation Grant #427007.