Goddard–Thorn theorem
In mathematics, and in particular, in the mathematical background of string theory, the Goddard–Thorn theorem (also called the no-ghost theorem) is a theorem about certain vector spaces. It is named after Peter Goddard and Charles Thorn.
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Formal version
Suppose that V is a vector space with a nondegenerate bilinear form (·,·).
Further suppose that V is acted on by the Virasoro algebra in such a way that 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 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.
Why "no-ghost" theorem?
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 definite lattice so that the inner product on Vi was positive definite; thus, had no ghosts (vectors of negative norm) for r ≠ 0. The name "no-ghost theorem" is also a word play on the phrase no-go theorem.
Applications
The no-ghost theorem can be used to construct some generalized Kac–Moody algebras, in particular the monster Lie algebra.
References
- P. Goddard and C. B. Thorn, Compatibility of the dual Pomeron with unitarity and the absence of ghosts in the dual resonance model, Phys. Lett., B 40, No. 2 (1972), 235-238.