The combinatorics of supertorus sheaf cohomology

Abstract

Affine superspace $\mathbb{C}^{1 \mid n}$ has a single bosonic coordinate $z$ and $n$ fermionic coordinates $\theta_1, \dots, \theta_n$. Let $M$ be the supertorus obtained by quotienting $\mathbb{C}^{1 \mid n}$ by the abelian group generated by the maps $S: (z,\theta_1, \dots, \theta_n) \mapsto (z + 1, \theta_1, \dots, \theta_n)$ and $T: (z, \theta_1, \dots, \theta_n) \mapsto (z + t, \theta_1 + \alpha_1, \dots, \theta_n + \alpha_n)$ where $t \in \mathbb{C}$ has positive imaginary part and $\alpha_1, \dots, \alpha_n$ are independent fermionic parameters. We compute the zeroth and first cohomology groups of the structure sheaf $\mathcal{O}$ of $M$ as doubly graded $\mathfrak{S}_n$-modules, exhibiting an instance of Serre duality between these groups. We use skein relations and noncrossing matchings to give a combinatorial presentation of $H^0(M,\mathcal{O})$ in terms of generators and relations.