Toroidal Compactification of Closed Bosonic Strings

Abstract

The goal of this lecture is to show something of the algebraic structure of compactified closed bosonic strings. The works reported here was done in collaboration with Stuart Raby [1] and the recent work on Lorentzian lattices includes Louise Dolan. We study the boundary conditions of the first-quantized compactified closed bosonic string. Demanding that internal symmetries result from the Frenkel-Kac construction, we show at the tree level that the “left” and “right” lattices must be self-dual if all masses are integers; they can be chosen independently to be E₈ X E₈ or Spin(32)/ℤ₂ for 16 compactified dimensions. It is also shown how to get a group G X G on a slightly different lattice, which gives tachyons of two different mass values, including tachyons in nontrivial representations of GXG. We discuss here the closed bosonic string; it would be interesting to apply some of these considerations to the heterotic string.