Algebras, Lattices and Strings

Abstract

A unified construction is given of various types of algebras, including finite dimensional Lie algebras, affine Kac-Moody algebras, Lorentzian algebras and extensions of these by Clifford algebras. This is done by considering integral lattices (i.e. ones such that the scalar product between any two points is an integer) and associating to the points of them the square of whose length is 1 or 2, the contour integral of the dual model vertex operator for emitting a “tachyon”. If the scalar product is positive definite, the algebra of these quantities associated with the points of length 2 closes, when the momenta are included, to form a finite dimensional Lie algebra. If the scalar product is positive semi definite, this algebra closes to an affine Kac-Moody algebra when the vertex operators for emitting “photons” are added. If the scalar product is Lorentzian, the algebra closes if the vertex operators for all the emitted states in the dual model are added. Special lattices in 10, 18, and 26 dimensional Lorentzian space are discussed and implications of the dual model no ghost theorem for these algebras are mentioned. This framework links many physical ideas, including concepts in magnetic monopole theory and the fermion-boson equivalence as well as the dual model. (Knowledge of dual models is not assumed but familiarity with aspects of the theory of Lie algebras is presumed in the latter part of this paper.)