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 P. Goddard and C. B. Thorn.

Contents

Formal version

Why "no-ghost" theorem?

Applications

References

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 ''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 ''P''¹ be the subspace of the vertex algebra $Votimes\; V\_\{II\_\{1,1\}\}$ of vectors ''v'' with ''L''₀(''v'') = ''v'', ''L''_i(''v'') = 0 for ''i'' > 0, and let $P^1\_r$ be the subspace of ''P''¹ 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 'R'²).
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.