Singular Spin Structures and Superstrings

There are two main problems in finding the higher genus superstring measure. The first one is that for $g\geq 5$ the super moduli space is not projected. Furthermore, the supermeasure is regular for $g\leq 11$, a bound related to the source of singularities due to the divisor in the moduli space of Riemann surfaces with even spin structure having holomorphic sections, such a divisor is called the $\theta$-null divisor. A result of this paper is the characterization of such a divisor. This is done by first extending the Dirac propagator, that is the Szeg\"o kernel, to the case of an arbitrary number of zero modes, that leads to a modification of the Fay trisecant identity, where the determinant of the Dirac propagators is replaced by the product of two determinants of the Dirac zero modes. By taking suitable limits of points on the Riemann surface, this {\it holomorphic Fay trisecant identity} leads to identities that include points dependent rank 3 quadrics in $\mathbb{P}^{g-1}$. Furthermore, integrating over the homological cycles gives relations for the Riemann period matrix which are satisfied in the presence of Dirac zero modes. Such identities characterize the $\theta$-null divisor. Finally, we provide the geometrical interpretation of the above points dependent quadrics and shows, via a new $\theta$-identity, its relation with the Andreotti-Mayer quadric.