Goddard–Thorn theorem

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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 so-called Faddeev–Popov ghosts, or vectors of negative norm, for r ≠ 0. The name "no-ghost theorem" is also a word play on the phrase no-go theorem.

Formalism[edit]

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.

Applications[edit]

The theorem can be used to construct some generalized Kac–Moody algebras, in particular the monster Lie algebra.

References[edit]