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 P. Goddard and C. B. Thorn.
Formal version
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 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 <math>V_{II_{1,1}}<math> be the vertex algebra of the double cover <math>\hat{I}I_{1,1}<math> of the two-dimensional even unimodular Lorentzian lattice <math>II_{1,1}<math> (so that <math>V_{II_{1,1}}<math> is <math>II_{1,1}<math>-graded, has a bilinear form (·,·) and is acted on by the Virasoro algebra).
Furthermore, let P1 be the subspace of the vertex algebra <math>V\otimes V_{II_{1,1}}<math> of vectors v with L0(v) = v, Li(v) = 0 for i > 0, and let <math>P^1_r<math> be the subspace of P1 of degree r ∈ <math>II_{1,1}<math>. (All these spaces inherit an action of G from the action of G on V and the trivial action of G on <math>V_{II_{1,1}}<math> and R2).
Then, the quotient of <math>P^1_r<math> by the nullspace of its bilinear form is naturally isomorphic (as a G module with an invariant bilinear form) to <math>V^{1-(r,r)/2}<math> if r ≠ 0, and to <math>V^1 \oplus \mathbb{R}^2<math> 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, <math>P^1_r<math> had no "ghosts" (vectors of negative norm) for r ≠ 0.
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.