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 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.

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

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