Toroidal Compactifications: 10-Dimensional Heterotic String

Abstract

So far we discussed the 26-dimensional bosonic string and three kinds of 10-dimensional superstring theories, the type IIA/B theories and the type I theory. One option to make contact with the four-dimensional world is to compactify the closed string theories on compact spaces. We first study the simplest examples, toroidal compactifications of the bosonic string and the type II superstring theories. These feature a new symmetry, called T-duality. To break supersymmetry, however, one has to compactify on non-flat spaces. The simplest such class are toroidal orbifolds. Moreover, we introduce two additional superstring theories in ten-dimensions, which are hybrid theories of a right-moving superstring and a left-moving bosonic string, whose additional sixteen dimensions are compactified on the weight-lattice of \(\mathrm{Spin}(32)/{\mathbb{Z}}_{2}\)or \({E}_{8} \times {E}_{8}\). We then study D-branes on toroidal spaces and how they transform under T-duality. We introduce intersecting D-branes, their T-dual images and simple orientifolds on such toroidal spaces.