Compactification of the Ten-Dimensional Heterotic String to Four Dimensions
David Balin, A. Love · 1994
Any string theory that is to be a candidate theory of the world we live in will have to possess just four observable space-time dimensions, or, if there are extra spatial dimensions, they will have to be compactified on a sufficiently small scale as to be unobservable with the energies that are currently available to us. In Chapter 9 a heterotic string theory with ten space-time dimensions was constructed by either fermionizing 16 of the left-mover bosonic degrees of freedom, or equivalently by compactifying these degrees of freedom on a torus in such a way that an E 8 × E′ 8 gauge group arose from these 16 extra left-mover dimensions. To complete the construction of a four-dimensional theory it is necessary next to compactify six of these ten dimensions in some way for both right and left movers. The simplest possibility is a toroidal compactification. However, we shall see that such a compactification produces a theory with N = 4 space-time supersymmetry rather than the N = 1 space-time supersymmetry that we saw in Chapter 1 was required to obtain a chiral theory. Fortunately, theories with N = 1 space-time supersymmetry can be obtained by simple modifications of toroidal compactifications, referred to as orbifolds, in which points on the torus are identified by a symmetry of the lattice of the torus. Alternatively, compactification on a special class of manifolds, called Calabi—Yau manifolds, may be employed.