Quantum D-modules for toric nef complete intersections

TL;DR

This paper develops a framework for quantum D-modules associated with toric nef complete intersections, linking mirror symmetry, GKZ systems, and Gromov-Witten invariants to understand their algebraic and geometric structures.

Contribution

It introduces a new twisted quantum D-module for complete intersections and relates it to GKZ systems via mirror symmetry, providing explicit quotient descriptions.

Findings

The twisted quantum D-module is isomorphic to an ambient part of the quantum D-module of Z.

In the toric case, these quantum D-modules are cyclic.

The paper constructs the defining system of equations for the ambient quantum D-module as a quotient of the GKZ system.

Abstract

On a smooth projective variety with k ample line bundles, we denote by Z the complete intersection subvariety defined by generic sections. We define the twisted quantum D-module which is a vector bundle with a flat connection, a flat pairing and a natural integrable structure. An appropriate quotient of it is isomorphic to the ambient part of the quantum D-module of Z. When the variety is toric, these quantum D-modules are cyclic. The twisted quantum D-module can be presented via mirror symmetry by the GKZ system associated to the total space of the dual of the direct sum of these line bundles. A question is to know what is the system of equations that define the ambiant part of the quantum D-module of Z. We construct this system as a quotient ideal of the GKZ system. We also state and prove the non-equivariant twisted Gromov-Witten axioms in the appendix.