Two-loop vacuum energy for Calabi-Yau orbifold models

A precise evaluation of the two-loop vacuum energy is provided for certain Z_2 x Z_2 Calabi-Yau orbifold models in the Heterotic string. The evaluation is based on the recent general prescription for superstring perturbation theory in terms of integration over cycles in supermoduli space, implemented at two-loops with the gauge-fixing methods based on the natural projection of supermoduli space onto moduli space using the super-period matrix. It is shown that the contribution from the interior of supermoduli space (computed with the procedure that has been used in previous two-loop computations) vanishes identically for both the E_8 x E_8 and Spin (32)/Z_2 Heterotic strings. The contribution from the boundary of supermoduli space is also evaluated, and shown to vanish for the E_8 x E_8 string but not for the Spin (32)/Z_2 string, thus breaking supersymmetry in this last model. As a byproduct, the vacuum energy in Type II superstrings is shown to vanish as well for these orbifolds.