I have a maybe simple question about even positive-definite lattices and lattices coming from binary codes. They seemed to be used in framed vertex operator algebras.

What is known about even lattices, which have $(2\mathbb{Z})^n$ (with standard scalar product) as a sublattice. I am particular interested in lattices, which are sublattices of the lattice $(1/2\mathbb{Z})^n$. I guess non-trivial examples are just possible for $n=8,16,\ldots$ and that these come from binary codes. Is this true? I have as examples $E_8$ and the Leech lattice in mind.

Is there maybe any correspondence between such lattices and binary codes?

  • $\begingroup$ You might be interested in reading this: mathoverflow.net/questions/17617/… $\endgroup$ – André Henriques May 5 '11 at 21:27
  • $\begingroup$ I assume you want your lattices to be unimodular (which gives the restriction to dimensions that are multiples of $8$); otherwise, $(2\mathbb{Z})^n$ is itself an example in any dimension. Chapters 5 and 7 of Sphere Packings, Lattices and Groups (by Conway and Sloane) give a lot of information on how to construct lattices from codes. $\endgroup$ – Henry Cohn May 6 '11 at 0:44
  • $\begingroup$ Thank you. I will check this book. $(2\mathbb Z)^n$ it self is what i considered as a trivial examples. So I should maybe say I want sublattices of $(1/2\mathbb{Z})^n$ which are a proper sublattices of $(2\mathbb Z)^n$. So I have to choose a subsets of $(1/2\mathbb Z)^n / (2\mathbb Z)^n \cong \mathbb Z_4^n$ which I add, and eveness gives restrictions. $\endgroup$ – Marcel Bischoff May 6 '11 at 9:59
  • $\begingroup$ Even lattices do not necessarily have $(2\mathbb Z)^n$ as a sublattice. The correct definition is: An integral lattice is even if all its vectors have even norm (where the norm is the square of the usual Euclidean norm). Example $\sqrt 2\mathbb Z$ is an even lattice (which is not unimodular, even unimodular lattice exist if and only if the dimension is divisible by $8$). $\endgroup$ – Roland Bacher May 6 '11 at 11:37
  • $\begingroup$ Yeah I know that not all even lattices are sublattices of $(2\mathbb{Z})^n$ for example $\sqrt{2}\ZZ$. I am wondering if this is also true for unimodular lattices. $\endgroup$ – Marcel Bischoff May 6 '11 at 19:57

I don't have any references (e.g., Conway-Sloane) in front of me, so this will be rather basic.

For all non-negative integers $n$, the set of even lattices $L$ satisfying $(2\mathbb{Z})^n \subseteq L \subseteq (\frac12 \mathbb{Z})^n$ is nonempty (and in particular contains $(2\mathbb{Z})^n$). Also, note that the second containment follows automatically from the first by the integrality of the inner product. There is a natural bijection between such lattices and the quotients $L/(2\mathbb{Z})^n$, which are subgroups of $(\frac12 \mathbb{Z}/2\mathbb{Z})^n$ such that the induced $\frac14 \mathbb{Z}/4\mathbb{Z}$-valued quadratic form takes even integral values. You can think of the quotient group as a $\mathbb{Z}/4\mathbb{Z}$-code, so this is one possible correspondence between lattices and codes.

The even lattice $L$ is unimodular if and only if the quotient has order $2^n$, and it is well-known that this can only happen when $n$ is a multiple of 8. You can think of this as a self-duality condition on the code, but I would hesitate to describe non-self-dual examples as "trivial". At any rate, for each non-negative integer $k$, the lattice $II_{8k,0}$ is a standard example of an even unimodular lattice that arises from this construction. The codewords are those elements of $(\frac12 \mathbb{Z}/2\mathbb{Z})^{8k}$ whose components are either all integers or all half-integers, and whose sum is even. When $k=1$, you get $E_8$.

There is a different way to make a lattice from a binary code, using the quotient $(\frac{1}{\sqrt2}\mathbb{Z})^n/(\sqrt{2}\mathbb{Z})^n$, which is isomorphic to $(\mathbb{Z}/2\mathbb{Z})^n$. One obtains a bijection between integral lattices $L$ satisfying $(\sqrt{2}\mathbb{Z})^n \subseteq L \subseteq (\frac{1}{\sqrt2}\mathbb{Z})^n$ and codes in $(\mathbb{Z}/2\mathbb{Z})^n$ such that every pair of codewords has even inner product. Evenness of the lattice then corresponds to the "doubly even" condition (namely, that every codeword has weight a multiple of 4).

As I understand it, the construction of the Leech lattice from the binary Golay code is a bit more complicated than the methods I outlined above. See Wikipedia for details.

I am not an expert in framed VOAs, but my impression is that such an object is built up from tensor products of the Virasoro minimal model $L(\frac12, 0)$ and its (twisted) modules using a code, not by constructing a vertex operator algebra corresponding to a lattice built from a code.

  • $\begingroup$ Thank you for the nice answer. I am just learning about Framed Vertex Operators. But if I take $L(1/2,0)\otimes L(1/2,0)$ and "extend" it with $L(1/2,1/2) \otimes L(1/2,1/2)$ I have exactly 4 "modules" with charges $1/2\mathbb{Z}_4$, right? So if I take 8 copies of this VOA I can for example realize E8 etc. and in general I should be able to realize all lattices of the above type $\endgroup$ – Marcel Bischoff May 6 '11 at 20:09

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