Bosonization of Bosons in Vertex Operator Representations of Affine Kae-Moody Algebras

In compactified closed string theories, our naive geometrical interpretation of space-time fails. For instance, the spectrum of closed strings in a box of size R is identical to that of closed strings in a box of size l/R.l} This symmetry comes from the duality between momentum modes and winding modes. This implies that there are two notions of position: One is defined to be canonically conjugate to momentum modes and the other is defined to be canonically conjugate to winding modes. Another surprising example is that there exist orbifold-compactifiedstring theories) which are equivalently described as strings on tori although orbifolds are topologically quite different from tori.)-6) In these theories, different definitions of string coordinates lead to different descriptions of strings. In this paper, we shall show that various compactified closed string theories on orbifolds and tori are connected with one another through the change of bases of affine Kac-Moody algebras in vertex operator representations. The essential idea is as follows: Consider the su(2) affine Kac-Moody algebra,