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

Abstract

We study the half weight ($N=\half$) analogue of the Bers quasiform on a Schottky group $Γ$ of genus $g$: the Poincaré series $S_\eps(x,y)=\sum_{\gt\in\Gt}j(\gt,x)/(γx-y)\,(dx\,dy)^{1/2}$ over a lift $\Gt\subset\mathrm{SL}_2(\C)$ of $Γ$. The $\Ng:=2^{g}$ lifts give $\Ng$ trivial $α$-holonomy even spin structures $\Lg$. The remaining $2^{2g}-\Ng$ 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 $\half$ (any genus $g\ge 1$) the following: absolute and locally uniform convergence on $Ω_0\timesΩ_0$, where $Ω_0$ is the full region of discontinuity including the point $\infty$ (treated in homogeneous coordinates); invariance without Eichler-type defect; skew-symmetry; a simple pole on the diagonal as the only singularity; $H^0(\Sg,\Lg)=0$; and hence identification of $S_\eps$ with the Szegő kernel of $\Lg$. 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.