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.

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 L_i 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 $V_{II_{1,1}}$ be the vertex algebra of the double cover $\hat{I}I_{1,1}$ of the two-dimensional even unimodular Lorentzian lattice II_1,1 (so that $V_{II_{1,1}}$ is II_1,1-graded, has a bilinear form (·,·) and is acted on by the Virasoro algebra).

Furthermore, let P¹ be the subspace of the vertex algebra $V\otimes V_{II_{1,1}}$ of vectors v with L₀(v) = v, L_i(v) = 0 for i > 0, and let $P^1_r$ be the subspace of P¹ of degree r ∈ II_1,1. (All these spaces inherit an action of G from the action of G on V and the trivial action of G on $V_{II_{1,1}}$ and R²).

Then, the quotient of $P^1_r$ by the nullspace of its bilinear form is naturally isomorphic (as a G-module with an invariant bilinear form) to V^{1 − (r,r) / 2} if r ≠ 0, and to $V^1 \oplus \mathbb{R}^2$ 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, $P^1_r$ 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.

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