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.

13 pages