Real and integral structures in quantum cohomology I: toric orbifolds

TL;DR

This paper investigates real and integral structures in quantum cohomology of toric orbifolds, demonstrating their geometric properties and calculating natural integral structures via mirror symmetry, with implications for Ruan's conjecture.

Contribution

It establishes conditions under which real structures induce pure polarized tt^*-geometry and computes natural integral structures using mirror symmetry for toric orbifolds.

Findings

Real structures lead to pure polarized tt^*-geometry near the large radius limit.

Integral structures can be explicitly described using topological data and mirror symmetry.

Quantum parameters should specialize to roots of unity in Ruan's crepant resolution conjecture.

Abstract

We study real and integral structures in the space of solutions to the quantum differential equations. First we show that, under mild conditions, any real structure in orbifold quantum cohomology yields a pure and polarized tt^*-geometry near the large radius limit. Secondly, we use mirror symmetry to calculate the "most natural" integral structure in quantum cohomology of toric orbifolds. We show that the integral structure pulled back from the singularity B-model is described only in terms of topological data in the A-model; K-group and a characteristic class. Using integral structures, we give a natural explanation why the quantum parameter should specialize to a root of unity in Ruan's crepant resolution conjecture.