Direct Construction of Four-Dimensional Heterotic String Theories
David Balin, A. Love · 1994
In Chapter 10 we saw how it was possible to construct heterotic string theories with only four space-time dimensions by compactifying six of the dimensions of the ten-dimensional heterotic string of Chapter 9 on an orbifold or Calabi—Yau manifold. It is also possible to construct four-dimensional heterotic string theories in a more direct fashion without the intermediate state of a ten-dimensional theory. One approach (1) returns to the original heterotic string, with 10 dimensions for the superstring right movers and 26 dimensions for the bosonic string left movers, bosonizes the fermionic degrees of freedom, other than those associated with four-dimensional space-time, and compactifies all the right- and left-moving bosonic degrees of freedom on a torus, with the exception of those associated with four-dimensional space-time. An alternative, essentially equivalent, approach (2) , (3) which we shall pursue here exploits the possibility discussed in Chapter 9 of fermionizing toroidally compactified bosonic degrees of freedom. In this approach, all bosonic degrees of freedom other than the four-dimensional space-time degrees of freedom are replaced by fermionic degrees of freedom. The boundary conditions for all the internal fermionic degrees of freedom (i.e. other than four-dimensional space-time) are then chosen in such a way as to be consistent with the fundamental constraint of modular invariance. In the next two sections we shall describe the way in which modular invariance enters a consistent string theory.