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 Seps(x,y)=∑gt∈Gtj(gt,x)/(γx−y) (dx dy)1/2 over a lift Gt⊂SL2(C) of Γ. The Ng:=2g lifts give Ng trivial α-holonomy even spin structures Lg. The remaining 22g−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≥ 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(Sg,Lg)=0; and hence identification of Seps 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.
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BibTeX
@misc{zuevsky2026szegő,
title = {{Szegő quasiforms: a Schottky-scheme construction for vertex operator superalgebras}},
author = {A. Zuevsky},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06446},
eprint = {2611.06446},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06446},
note = {CC0},
}
RIS
TY - UNPB
TI - Szegő quasiforms: a Schottky-scheme construction for vertex operator superalgebras
AU - A. Zuevsky
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06446
UR - https://arxiv.org/abs/2611.06446
AB - 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)=∑_gt∈Gtj(gt,x)/(γx−y) (dx dy)^1/2 over a lift Gt⊂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≥ 1) the following: absolute and locally uniform convergence on Ω_0×Ω_0, where Ω_0 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; 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.
ER -