Vertex Operator Algebra - Vertex Operator Algebra Defined By A Lattice

Vertex Operator Algebra Defined By A Lattice

Let Λ be an integral lattice in Euclidean space X = RN, i.e. a subgroup isomorphic to Zn with m ≤ N and such that (α,β) lies in Z for α,β in Λ . The lattice is said to be even if (α,α) is even for each α in Λ. Setting

there is an essentially unique normalised cocycle ε(α,β) with values ±1 such that

and the cocycle identity

is satisfied along with the normalisation conditions

A cocycle representation can be defined on C, with basis e_α (α in Λ), by

Thus

and

The operators U_α are unitary if the e_α are taken to be orthonormal.

There is a bosonic system associated to X, namely operators v_n depending linearly on v in X such that

In addition the system has a derivation D satisfying with

There is a unique irreducible representation of this system characterised by the existence of a vacuum vector Ω with v_n Ω = 0 for n ≥ 0. The underlying space S has a unique inner product structure for which v_n* = v_–n. The vector space V of the vertex superalgebra is defined by

The operators v_n with n non-zero act on S ⊗ e_α exactly as they act on S,

The operators v₀ act as scalars on S ⊗ e_α:

For each v in X define the field

For each α in Λ define

where

If v(i) is an orthonormal basis of X, define

where the normal ordering is given by

Then the vertex operators v(z) and Φ_α(z) generate a vertex operator superalgebra with underlying space V. The operators D and T are given by L₀ and L_–1 respectively.