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.

The name "no-ghost theorem" stems from the fact that in the original statement of the theorem, the vector space inner product is positive definite. Thus, there were no vectors of negative norm for r ≠ 0. The name "no-ghost theorem" is also a word play on the phrase no-go theorem.

Formalism

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.

Applications

The 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.