Theorem in string theory
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 describing properties of a functor that quantizes bosonic strings. It is named after Peter Goddard and Charles Thorn.
The name "no-ghost theorem" stems from the fact that in the original statement of the theorem, the natural inner product induced on the output vector space is positive definite. Thus, there were no so-called ghosts (Pauli–Villars ghosts), or vectors of negative norm. The name "no-ghost theorem" is also a word play on the no-go theorem of quantum mechanics.
Statement.
This statement is that of Borcherds (1992).
Suppose that formula_1 is a unitary representation of the Virasoro algebra formula_2, so formula_1 is equipped with a non-degenerate bilinear form formula_4 and there is an algebra homomorphism formula_5 so that 
formula_6
where the adjoint is defined with respect to the bilinear form, and 
formula_7
Suppose also that formula_1 decomposes into a direct sum of eigenspaces of formula_9 with non-negative, integer eigenvalues formula_10, denoted formula_11, and that each formula_11 is finite dimensional (giving formula_1 a formula_14-grading). Assume also that formula_1 admits an action from a group formula_16 that preserves this grading.
For the two-dimensional even unimodular Lorentzian lattice II1,1, denote the corresponding lattice vertex algebra by formula_17. This is a II1,1-graded algebra with a bilinear form and carries an action of the Virasoro algebra. 
Let formula_18 be the subspace of the vertex algebra formula_19 consisting of vectors formula_20 such that formula_21 for formula_22. Let formula_23 be the subspace of formula_18 of degree formula_25. Each space inherits a formula_16-action which acts as prescribed on formula_1 and trivially on formula_17.
The quotient of formula_23 by the nullspace of its bilinear form is naturally isomorphic as a formula_16-module with an invariant bilinear form, to formula_31 if formula_32 and formula_33 if formula_34.
II1,1.
The lattice II1,1 is the rank 2 lattice with bilinear form
formula_35
This is even, unimodular and integral with signature (+,-).
Formalism.
There are two naturally isomorphic functors that are typically used to quantize bosonic strings. In both cases, one starts with positive-energy representations of the Virasoro algebra of central charge 26, equipped with Virasoro-invariant bilinear forms, and ends up with vector spaces equipped with bilinear forms. Here, "Virasoro-invariant" means "Ln" is adjoint to "L"−"n" for all integers "n".
The first functor historically is "old canonical quantization", and it is given by taking the quotient of the weight 1 primary subspace by the radical of the bilinear form. Here, "primary subspace" is the set of vectors annihilated by "Ln" for all strictly positive "n", and "weight 1" means "L"0 acts by identity. A second, naturally isomorphic functor, is given by degree 1 BRST cohomology. Older treatments of BRST cohomology often have a shift in the degree due to a change in choice of BRST charge, so one may see degree −1/2 cohomology in papers and texts from before 1995. A proof that the functors are naturally isomorphic can be found in Section 4.4 of Polchinski's "String Theory" text.
The Goddard–Thorn theorem amounts to the assertion that this quantization functor more or less cancels the addition of two free bosons, as conjectured by Lovelace in 1971. Lovelace's precise claim was that at critical dimension 26, Virasoro-type Ward identities cancel two full sets of oscillators. Mathematically, this is the following claim:
Let "V" be a unitarizable Virasoro representation of central charge 24 with Virasoro-invariant bilinear form, and let π be the irreducible module of the R1,1 Heisenberg Lie algebra attached to a nonzero vector "λ" in R1,1. Then the image of "V" ⊗ π under quantization is canonically isomorphic to the subspace of "V" on which "L"0 acts by 1-("λ","λ").
The no-ghost property follows immediately, since the positive-definite Hermitian structure of "V" is transferred to the image under quantization.
Applications.
The bosonic string quantization functors described here can be applied to any conformal vertex algebra of central charge 26, and the output naturally has a Lie algebra structure. The Goddard–Thorn theorem can then be applied to concretely describe the Lie algebra in terms of the input vertex algebra.
Perhaps the most spectacular case of this application is Richard Borcherds's proof of the monstrous moonshine conjecture, where the unitarizable Virasoro representation is the monster vertex algebra (also called "moonshine module") constructed by Frenkel, Lepowsky, and Meurman. By taking a tensor product with the vertex algebra attached to a rank-2 hyperbolic lattice, and applying quantization, one obtains the monster Lie algebra, which is a generalized Kac–Moody algebra graded by the lattice. By using the Goddard–Thorn theorem, Borcherds showed that the homogeneous pieces of the Lie algebra are naturally isomorphic to graded pieces of the moonshine module, as representations of the monster simple group.
Earlier applications include Frenkel's determination of upper bounds on the root multiplicities of the Kac–Moody Lie algebra whose Dynkin diagram is the Leech lattice, and Borcherds's construction of a generalized Kac–Moody Lie algebra that contains Frenkel's Lie algebra and saturates Frenkel's 1/∆ bound.