mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, and in particular in the mathematical background of

string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interac ...

, the Goddard–Thorn theorem (also called the no-ghost theorem) is a theorem describing properties of a

functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...

that quantizes

bosonic strings Bosonic string theory is the original version of string theory, developed in the late 1960s and named after Satyendra Nath Bose. It is so called because it contains only bosons in the spectrum. In the 1980s, supersymmetry was discovered in the con ...

. 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 natural

inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...

induced on the output vector space is positive definite. Thus, there were no so-called ghosts ( Pauli–Villars ghosts), or vectors of negative norm. The name "no-ghost theorem" is also a word play on the

no-go theorem In theoretical physics, a no-go theorem is a theorem that states that a particular situation is not physically possible. Specifically, the term describes results in quantum mechanics like Bell's theorem and the Kochen–Specker theorem that cons ...

of quantum mechanics.

Formalism

There are two naturally isomorphic functors that are typically used to quantize bosonic strings. In both cases, one starts with positive-energy representations of the

Virasoro algebra In mathematics, the Virasoro algebra (named after the physicist Miguel Ángel Virasoro) is a complex Lie algebra and the unique central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string the ...

of central charge 26, equipped with Virasoro-invariant bilinear forms, and ends up with vector spaces equipped with bilinear forms. Here, "Virasoro-invariant" means ''L_n'' is adjoint to ''L''_−''n'' for all integers ''n''. The first functor historically is "old canonical quantization", and it is given by taking the quotient of the weight 1 primary subspace by the radical of the bilinear form. Here, "primary subspace" is the set of vectors annihilated by ''L_n'' for all strictly positive ''n'', and "weight 1" means ''L''₀ acts by identity. A second, naturally isomorphic functor, is given by degree 1 BRST cohomology. Older treatments of BRST cohomology often have a shift in the degree due to a change in choice of BRST charge, so one may see degree −1/2 cohomology in papers and texts from before 1995. A proof that the functors are naturally isomorphic can be found in Section 4.4 of Polchinski's ''String Theory'' text. The Goddard–Thorn theorem amounts to the assertion that this quantization functor more or less cancels the addition of two free bosons, as conjectured by Lovelace in 1971. Lovelace's precise claim was that at critical dimension 26, Virasoro-type Ward identities cancel two full sets of oscillators. Mathematically, this is the following claim: Let ''V'' be a unitarizable Virasoro representation of central charge 24 with Virasoro-invariant bilinear form, and let be the irreducible module of the R^1,1 Heisenberg Lie algebra attached to a nonzero vector ''λ'' in R^1,1. Then the image of ''V'' ⊗ under quantization is canonically isomorphic to the subspace of ''V'' on which ''L''₀ acts by 1-(''λ'',''λ''). The no-ghost property follows immediately, since the positive-definite Hermitian structure of ''V'' is transferred to the image under quantization.

Applications

The bosonic string quantization functors described here can be applied to any conformal vertex algebra of central charge 26, and the output naturally has a Lie algebra structure. The Goddard–Thorn theorem can then be applied to concretely describe the Lie algebra in terms of the input vertex algebra. Perhaps the most spectacular case of this application is

Richard Borcherds Richard Ewen Borcherds (; born 29 November 1959) is a British mathematician currently working in quantum field theory. He is known for his work in lattices, group theory, and infinite-dimensional algebras, for which he was awarded the Fields M ...

's proof of the

monstrous moonshine In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group ''M'' and modular functions, in particular, the ''j'' function. The term was coined by John Conway and Simon P. Norton in 1979. ...

conjecture, where the unitarizable Virasoro representation is the

monster vertex algebra The monster vertex algebra (or moonshine module) is a vertex algebra acted on by the monster group that was constructed by Igor Frenkel, James Lepowsky, and Arne Meurman. R. Borcherds used it to prove the monstrous moonshine conjectures, by ap ...

(also called "moonshine module") constructed by Frenkel, Lepowsky, and Meurman. By taking a tensor product with the vertex algebra attached to a rank-2 hyperbolic lattice, and applying quantization, one obtains the

monster Lie algebra In mathematics, the monster Lie algebra is an infinite-dimensional generalized Kac–Moody algebra acted on by the monster group, which was used to prove the monstrous moonshine conjectures. Structure The monster Lie algebra ''m'' is a ''Z2' ...

, which is a

generalized Kac–Moody algebra In mathematics, a generalized Kac–Moody algebra is a Lie algebra that is similar to a Kac–Moody algebra, except that it is allowed to have imaginary simple roots. Generalized Kac–Moody algebras are also sometimes called GKM algebras, Borch ...

graded by the lattice. By using the Goddard–Thorn theorem, Borcherds showed that the homogeneous pieces of the Lie algebra are naturally isomorphic to graded pieces of the Moonshine module, as representations of the

monster simple group In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order 246320597611213317192329314147 ...

. Earlier applications include Frenkel's determination of upper bounds on the root multiplicities of the Kac-Moody Lie algebra whose Dynkin diagram is the

Leech lattice In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space, which is one of the best models for the kissing number problem. It was discovered by . It may also have been discovered (but not published) by ...

, and Borcherds's construction of a generalized Kac-Moody Lie algebra that contains Frenkel's Lie algebra and saturates Frenkel's 1/∆ bound.

References

* * * I. Frenkel, ''Representations of Kac-Moody algebras and dual resonance models'' Applications of group theory in theoretical physics, Lect. Appl. Math. 21 A.M.S. (1985) 325–353. * * * {{DEFAULTSORT:Goddard-Thorn theorem Theorems in linear algebra String theory Theorems in mathematical physics No-go theorems