Equivalent complex and real fermions in heterotic superstring solutions

As a first step towards a classification of the heterotic superstring solutions in the fermionic approach, we study a class of solutions characterized by complex fermions with ZN boundary conditions. The same solutions can be obtained by an equivalent modular invariant system of real fermions with Z2 boundary conditions. The proof is supplied by Riemann identities for Jacobi Θ-functions that we derive. The patterns of the gauge groups of the D-dimensional solutions are determined for 4 ≤ D ≤ 10, and their uniqueness in 10-dimensions is checked for the different ZN boundary conditions. A construction of the E8 x E8 supersymmetric solution based on nine complex fermions with Z3 periodicity is exhibited