then has an n parameter family of Virasoro vectors, and when those parameters are zero, the character is qn/24 times the weight −n/2 modular form η−n. Vertex operator algebra defined by an even lattice The lattice vertex algebra construction was the original motivation for defining vertex algebras. It is constructed by taking a sum of irreducible modules for the free boson corresponding to lattice vectors, and defining a multiplication operation by specifying intertwining operators between them. That is, if is an even lattice, the lattice vertex algebra decomposes into free bosonic modules as: Lattice vertex algebras are canonically attached to double covers of even integral lattices, rather than the lattices themselves. While each such lattice has a unique lattice vertex algebra up to isomorphism, the vertex algebra construction is not functorial, because lattice automorphisms have an ambiguity in lifting. The double covers in question are uniquely determined up to isomorphism by the following rule: elements have the form for lattice vectors (i.e., there is a map to sending to α that forgets signs), and multiplication satisfies the relations eαeβ = (–1)(α,β)eβeα. Another way to describe this is that given an even lattice , there is a unique (up to coboundary) normalised cocycle with values such that , where the normalization condition is that ε(α, 0) = ε(0, α) = 1 for all . This cocycle induces a central extension of by a group of order 2, and we obtain a twisted group ring with basis , and multiplication rule – the cocycle condition on ensures associativity of the ring. The vertex operator attached to lowest weight vector in the Fock space is where is a shorthand for the linear map that takes any element of the α-Fock space to the monomial . The vertex operators for other elements of the Fock space are then determined by reconstruction. As in the case of the free boson, one has a choice of conformal vector, given by an element s of the vector space , but the condition that the extra Fock spaces have integer L0 eigenvalues constrains the choice of s: for an orthonormal basis , the vector 1/2 xi,12 + s2 must satisfy for all λ ∈ Λ, i.e., s lies in the dual lattice. If the even lattice is generated by its "root vectors" (those satisfying (α, α)=2), and any two root vectors are joined by a chain of root