The combinatorics of supertorus sheaf cohomology

Abstract

Affine superspace C1|n has a single bosonic coordinate z and n fermionic coordinates θ1,…,θn. Let M be the supertorus obtained by quotienting C1|n by the abelian group generated by the maps S:(z,θ1,…,θn)↦(z+1,θ1,…,θn) and T:(z,θ1,…,θn)↦(z+t,θ1+α1,…,θn+αn) where t∈C has positive imaginary part and α1,…,αn are independent fermionic parameters. We compute the zeroth and first cohomology groups of the structure sheaf O of M as doubly graded Sn-modules, exhibiting an instance of Serre duality between these groups. We use skein relations and noncrossing matchings to give a combinatorial presentation of H0(M,O) in terms of generators and relations.