The geometric interpretation of fermionic superstrings and modular invariance

The author reviews some recent results obtained concerning the geometric reinterpretation of four-dimensional superstrings, as obtained in the free fermion construction. By geometric the author means that the two-dimensional quantum theory, by means of which the superstring model is put into existence, corresponds indeed to the canonical quantization of a classical field theory, processing the same local in variances that are advocated at the quantum level. It is shown how this, seemingly obvious, condition is not guaranteed a priori in the free fermion construction. Enforcing it leads to additional constraints on the permissible spin-structures of the free fermions and henceforth, to an actual reduction of the allowed superstring theories. Preliminary investigations have encouraged the author to conclude that the 4D-superstring models with target space supersymmetry are, under these restrictions, a few tens rather than a plethora