Szegő quasiforms: a Schottky-scheme construction for vertex operator superalgebras

We study the half weight (N=) analogue of the Bers quasiform on a Schottky group $Γ$ of genus g: the Poincaré series S₍x,y)=∑>∈j(>,x)/(γx-y)\,(dx\,dy)1/2 over a lift ₂() of $Γ$. The :=2ᵍ lifts give trivial $α$-holonomy even spin structures . The remaining 2²ᵍ- spin structures are not treated in this paper. We prove, on an open set (H) of Schottky parameters and more generally whenever the critical exponent of $Γ$ is below (any genus g≥ 1) the following: absolute and locally uniform convergence on Ω₀×Ω₀, where Ω₀ is the full region of discontinuity including the point ∞ (treated in homogeneous coordinates); invariance without Eichler-type defect; skew-symmetry; a simple pole on the diagonal as the only singularity; H⁰(,)=0; and hence identification of S_ with the Szegő kernel of . At genus one the kernel is computed explicitly and the relations between lifts and theta characteristics are determined. We further prove that the diagonal coefficient defines a projective connection that differs from the Bergman projective connection by a quadratic differential constructed from $θ[δ](0)$, a residue-reconstruction formula for half-differentials with poles of any order, an expansion in the Schottky multipliers, and a formula for the free fermion stress tensor.