Quantum 𝒟-modules for toric nef complete intersections

Abstract

Let [Formula: see text] be a smooth projective toric variety with [Formula: see text] ample line bundles. Let [Formula: see text] be the zero locus of [Formula: see text] generic sections. It is well known that the ambient quantum [Formula: see text]-module of [Formula: see text] is cyclic i.e. is defined by an ideal of differential operators. In this paper, we give an explicit construction of this ideal as a quotient ideal of a GKZ system associated to the toric data of [Formula: see text] and the line bundles. This description can be seen as a “left cancellation procedure”. We consider some examples where this description enables us to compute generators of this ideal, and thus to give a presentation of the ambient quantum [Formula: see text]-module.