Goddard-Thorn theorem
Encyclopedia
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 space
s. It is named after Peter Goddard and Charles Thorn
.
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.
so that the inner product on Vi was positive definite; thus,
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
.
s, in particular the monster Lie algebra
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, and in particular, in the mathematical background of string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
, the Goddard–Thorn theorem (also called the no-ghost theorem) is a theorem about certain vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s. It is named after Peter Goddard and Charles Thorn
Charles Thorn
Charles Thorn is a Professor of Physics at University of Florida in Gainesville, Florida. He played an important role in the development of Dual Models and string theory. Among his contributions is the proof of the non-existence of ghosts in string theory. The Goddard–Thorn theorem is a...
.
Formal version
Suppose that V is a vector spaceVector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
with a nondegenerate bilinear form (·,·).
Further suppose that V is acted on by the Virasoro algebra
Virasoro algebra
In mathematics, the Virasoro algebra is a complex Lie algebra, given as a central extension of the complex polynomial vector fields on the circle, and is widely used in conformal field theory and string theory....
in such a way that the adjoint
Adjoint
In mathematics, the term adjoint applies in several situations. Several of these share a similar formalism: if A is adjoint to B, then there is typically some formula of the type = .Specifically, adjoint may mean:...
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
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
G which preserves all of its structure.
Now let
be the vertex algebra of the double cover of the two-dimensional even unimodularUnimodular lattice
In mathematics, a unimodular lattice is a lattice of determinant 1 or −1.The E8 lattice and the Leech lattice are two famous examples.- Definitions :...
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
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
of
by the nullspace of its bilinear form is naturally isomorphic (as a G-moduleG-module
In mathematics, given a group G, a G-module is an abelian group M on which G acts compatibly with the abelian group structure on M. This widely applicable notion generalizes that of a representation of G...
with an invariant bilinear form) to
if r ≠ 0, and to 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 latticeLattice (group)
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
so that the inner product on Vi was positive definite; thus,
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 theoremNo-go theorem
In theoretical physics, a no-go theorem is a theorem that states that a particular situation is not physically possible.-Examples of no-go theorems:* Bell's theorem* Coleman–Mandula theorem* Haag-Lopuszanski-Sohnius theorem* Earnshaw's theorem...
.
Applications
The no-ghost theorem can be used to construct some generalized Kac–Moody algebraGeneralized Kac–Moody algebra
In mathematics, a generalized Kac–Moody algebra is a Lie algebra that is similar to aKac–Moody algebra, except that it is allowed to have imaginary simple roots.Generalized Kac–Moody algebras are also sometimes called GKM algebras,...
s, in particular the monster Lie algebra
Monster Lie algebra
In mathematics, the monster Lie algebra is an infinitedimensional generalized Kac–Moody algebra acted on by the monster group, which was used to prove the monstrous moonshine conjectures.- Structure :...
.