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 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}}

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.

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.

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.

<< Previous

Word Browser

Next >>

tachelhit language
john paul twitchell
nathan farragut twining
commodity price index
faustin twagiramungu
charles egbert tuttle
vlastimil tusar
stansfield turner
joseph turner
hanan ashrawi
fred turner

scissor sisters
carl friedrich heinrich graf von wylich und lottum
spatial tense
billboard 200
luigi colani
riyal
sokhna benga
man friday
canoeing at the 2000 summer olympics
argyll robertson pupil
auerbach's plexus

operation polarfuchs
zuber corners, ontario
enix america corporation
baker's cyst
boulevard
cedardale, ottawa, ontario
barlow's syndrome
scots guards
barlow's disease
boxing at the 2000 summer olympics
edwards, ontario

walter f. tichy
oakville
ish bosheth
blast processing
kawartha lakes
christopher mccandless
forer effect
windows on the world
bill laimbeer
leslie feinberg
judo at the 2000 summer olympics