nLab Leech lattice

1. Idea

A unimodular integral lattice in $\mathbb{R}^{24}$. Related to the monster group.

2. Properties

It is the unique unimodular rank $24$ lattice with minimum norm greater than $2$. The minimum norm is $4$.

It is equivalent to the sum $\Lambda_{23} \oplus A_1$, where $\Lambda_{23}$ is a rank $23$ lattice with minimum norm of $3$, called the shorter Leech lattice (Conway & Sloane 82).

3. Related concepts

• E8 lattice, Niemeier lattice

• bosonic string theory

4. References

General

• Richard Borcherds, The Leech lattice and other lattices (arXiv:math.NT/9911195)

• John Horton Conway, Neil J.A. Sloane, The Unimodular Lattices of Dimension up to 23 and the Minkowski-Siegel Mass Constants. European Journal of Combinatorics Volume 3, Issue 3, September 1982, Pages 219-231. (doi80034-6)).

See also:

• Wikipedia, Leech lattice

Relation to the octonions:

• Robert A. Wilson, Octonions and the Leech lattice, Journal of Algebra, Volume 322, Issue 6, 15 September 2009, Pages 2186-2190, (pdf, slides)

• Robert A. Wilson, A new approach to the Leech lattice, talk at University of Cambridge, 21st October 2009 (slides pdf)

Relation to string theory

Relation to heterotic string theory

• Bert Schellekens, Classification of Ten-Dimensional Heterotic Strings, Phys.Lett. B277 (1992) 277-284 (arXiv:hep-th/9112006)