Ising vectors in the vertex operator algebra VΛ+V_{\Lambda}^+ associated with the Leech lattice Λ\Lambda

In this article, we study the Ising vectors in the vertex operator algebra $V_\Lambda^+$ associated with the Leech lattice $\Lambda$. The main result is a characterization of the Ising vectors in $V_\Lambda^+$. We show that for any Ising vector $e$ in $V_\Lambda^+$, there is a sublattice $E\cong \sqrt{2}E_8$ of $\Lambda$ such that $e\in V_E^+$. Some properties about their corresponding $\tau$-involutions in the moonshine vertex operator algebra $V^\natural$ are also discussed. We show that there is no Ising vector of $\sigma$-type in $V^\natural$. Moreover, we compute the centralizer C_{\aut V^\natural}(z, \tau_e) for any Ising vector $e\in V_\Lambda^+$, where $z$ is a 2B element in \aut V^\natural which fixes $V_\Lambda^+$. Based on this result, we also obtain an explanation for the 1A case of an observation by Glauberman-Norton (2001), which describes some mysterious relations between the centralizer of $z$ and some 2A elements commuting $z$ in the Monster and the Weyl groups of certain sublattices of the root lattice of type $E_8$ .Comment: 22 page

Moonshine paths for 3A and 6A nodes of the extended E8-diagram

We continue the program to make a moonshine path between a node of the extended $E_8$-diagram and the Monster. Our theory is a concrete model expressing some of the mysterious connections identified by John McKay, George Glauberman and Simon Norton. In this article, we treat the 3A and 6A-nodes. We determine the orbits of triples $(x,y,z)$ in the Monster where $z\in 2B$, $x, y \in 2A \cap C(z)$ and $xy\in 3A \cup 6A$. Such $x, y$ correspond to a rootless $EE_8$-pair in the Leech lattice. For the 3A and 6A cases, we shall say something about the "half Weyl groups", which are proposed in the Glauberman-Norton theory. Most work in this article is with lattices, due to their connection with dihedral subgroups of the Monster. These lattices are $M+N$, where $M, N$ is the relevant pair of $EE_8$-sublattices, and their annihilators in the Leech lattice. The isometry groups of these four lattices are analyzed

A moonshine path from E8E_8 to the monster

One would like an explanation of the provocative McKay and Glauberman-Norton observations connecting the extended $E_8$-diagram with pairs of 2A involutions in the Monster sporadic simple group. We propose a down-to-earth model for the 3C-case which exhibits a logic to these connections.Comment: this manuscript is the same as the 11 October 2009 arxiv version except for a change of titl

Rootless pairs of EE8EE_8-lattices

We describe a classification of pairs $M, N$ of lattices isometric to $EE_8:=\sqrt 2 E_8$ such that the lattice $M + N$ is integral and rootless and such that the dihedral group associated to them has order at most 12. It turns out that most of these pairs may be embedded in the Leech lattice. Complete proofs will appear in another article. This theory of integral lattices has connections to vertex operator algebra theory and moonshine.Comment: 87 pages, many figure

Rank 72 high minimum norm lattices

Given a polarization of an even unimodular lattice and integer $k\ge 1$, we define a family of unimodular lattices $L(M,N,k)$. Of special interest are certain $L(M,N,3)$ of rank 72. Their minimum norms lie in $\{4, 6, 8\}$. Norms 4 and 6 do occur. Consequently, 6 becomes the highest known minimum norm for rank 72 even unimodular lattices. We discuss how norm 8 might occur for such a $L(M,N,3)$. We note a few $L(M,N,k)$ in dimensions 96, 120 and 128 with moderately high minimum norms.Comment: submitte

The Alternating Group of degree 6 in Geometry of the Leech Lattice and K3 Surfaces

The alternating group of degree 6 is located at the junction of three series of simple non-commutative groups : simple sporadic groups, alternating groups and simple groups of Lie type. It plays a very special role in the theory of finite groups. We shall study its new roles both in a finite geometry of certain pentagon in the Leech lattice and also in a complex algebraic geometry of $K3$ surfaces.Comment: 24 pages, 6 figure

Automorphisms and Periods of Cubic Fourfolds

We classify the symplectic automorphism groups for cubic fourfolds. The main inputs are the global Torelli theorem for cubic fourfolds and the classification of the fixed-point sublattices of the Leech lattice. Among the highlights of our results, we note that there are 34 possible groups of symplectic automorphisms, with 6 maximal cases. The six maximal cases correspond to 8 non-isomorphic, and isolated in moduli, cubic fourfolds; six of them previously identified by other authors. Finally, the Fermat cubic fourfold has the largest possible order (174,960) for the automorphism group (non-necessarily symplectic) among all smooth cubic fourfolds.Comment: 41 pages; minor update following some comments from S. Mukai and G. Ouch

Delaunay polytopes derived from the Leech lattice

Given a lattice L of R^n, a polytope D is called a Delaunay polytope in L if the set of its vertices is S\cap L where S is a sphere having no lattice points in its interior. D is called perfect if the only ellipsoid in R^n that contains S\cap L is exactly S. For a vector v of the Leech lattice \Lambda_{24} we define \Lambda_{24}(v) to be the lattice of vectors of \Lambda_{24} orthogonal to v. We studied Delaunay polytopes of L=\Lambda_{24}(v) for |v|^2<=22. We found some remarkable examples of Delaunay polytopes in such lattices and disproved a number of long standing conjectures. In particular, we discovered: --Perfect Delaunay polytopes of lattice width 4; previously, the largest known width was 2. --Perfect Delaunay polytopes in L, which can be extended to perfect Delaunay polytopes in superlattices of L of the same dimension. --Polytopes that are perfect Delaunay with respect to two lattices $L\subset L'$ of the same dimension. --Perfect Delaunay polytopes D for L with |Aut L|=6|Aut D|: all previously known examples had |Aut L|=|Aut D| or |Aut L|=2|Aut D|. --Antisymmetric perfect Delaunay polytopes in L, which cannot be extended to perfect (n+1)-dimensional centrally symmetric Delaunay polytopes. --Lattices, which have several orbits of non-isometric perfect Delaunay polytopes. Finally, we derived an upper bound for the covering radius of \Lambda_{24}(v)^{*}, which generalizes the Smith bound and we prove that it is met only by \Lambda_{23}^{*}, the best known lattice covering in R^{23}.Comment: 16 pages, 1 tabl

An even unimodular 72-dimensional lattice of minimum 8

An even unimodular 72-dimensional lattice $\Gamma$ having minimum 8 is constructed as a tensor product of the Barnes lattice and the Leech lattice over the ring of integers in the imaginary quadratic number field with discriminant $-7$. The automorphism group of $\Gamma$ contains the absolutely irreducible rational matrix group (\PSL_2(7) \times \SL _2(25)) : 2.Comment: New version: Replaced Griess' offender construction by a shorter and more sophisticated description of the vectors of norm 6 that also allows to compute the vectors of norm 8 and hence the kissing configuration of Gamma. I thank Noam Elkies for asking this question. Also included remark that Mark Watkins independently verified extremality of the lattic

Lifts of automorphisms of vertex operator algebras in simple current extensions

In this article, we study isomorphisms between simple current extensions of a simple VOA. For example, we classify the isomorphism classes of simple current extensions of some VOAs. Moreover, we consider the same simple current extension and describe the normalizer of the abelian automorphism group associated with this extension. In particular, we regard the moonshine module as simple current extensions of five subVOAs V_L^+ for 2-elementary totally even lattices L, and describe corresponding five normalizers of elementary abelian 2-group in the automorphism group of the moonshine module in terms of V_L^+. By using this description, we show that three of them form a Monster amalgam.Comment: 20 page