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Suppose that V is a vector space with a non-singular bilinear form (·,·).
Further suppose that V is acted on by the Virasoro algebra in such a way tha the adjoint of the operator Li is L-i, that the central element of the Virasoro algebra acts as multiplication by 24, that any vector of V is the sum of eigenvectors of L0 with non-negative integral eigenvalues, and that all eigenspaces of L0 are finite-dimensional.
Let Vi be the subspace of V on which L0 has eigenvalue i. Assume that V is acted on by a group G which preserves all of its structure.
Now let be the vertex algebra of the double cover of the two-dimensional even unimodular Lorentzian lattice (so that is -graded, has a bilinear form (·,·) and is acted on by the Virasoro algebra).
Furthermore, let P1 be the subspace of the vertex algebra of vectors v with L0(v) = v, Li(v) = 0 for i > 0, and let be the subspace of P1 of degree r ∈ . (All these spaces inherit an an action of G from the action of G on V and the trivial action of G on and R2.
Then, the quotient of by the nullspace of its bilinear form is naturally isomorphic (as a G module with an invariant bilinear form) to if r ≠ 0, and to if r = 0.
The name "no-ghost theorem" stems from the fact that in the original statement of the theorem by Goddard and Thorn, V was part of the underlying vector space of the vertex algebra of a positive definiteLet K be the field R or C V is a vector space over K and B : V × V → K is a bilinear map which is Hermitian in the sense that B ''x ''y is always the complex conjugate of B ''y ''x . Then B is positive-definite if B ''x ''x > 0 for every nonzero x in latticeIn colloquial usage, a lattice is a structure of crossed laths with open spaces left between them. The term is used in various technical senses, all of which have some geometrical relation to the dictionary definition. In one mathematical usage, a lattice so that the inner product on Vi was positive definite; thus, had no "ghosts" (vectors of negative norm) for r ≠ 0.
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