Invariant Stability Conditions on Certain Calabi-Yau Threefolds

TL;DR

This paper constructs and analyzes stability conditions on certain Calabi-Yau threefolds, linking them to Donaldson-Thomas invariants and wall-crossing phenomena, with explicit descriptions and applications to local geometries.

Contribution

It introduces new stability conditions on derived categories of specific Calabi-Yau threefolds and relates them to DT invariants and wall-crossing, expanding understanding of their geometric and algebraic structures.

Findings

Explicit stability conditions on the derived category of the total space of the canonical bundle of P^1×P^1

Complete description of the associated Donaldson-Thomas invariants

Demonstration of analytic wall-crossing structures in these settings

Abstract

We apply results on inducing stability conditions to local Calabi-Yau threefolds and obtain applications to Donaldson-Thomas (DT) theory. A basic example is the total space of the canonical bundle of $Z=\mathbb{P}^1\times \mathbb{P}^1$. We use a result of Dell to construct stability conditions on the derived category of $X$ for which all stable objects can be explicitly described. We relate them to stability conditions on the resolved conifold $Y=\mathscr{O}_{\mathbb{P}^1}(-1)^{\oplus 2}$ in two ways: geometrically via the McKay correspondence, and algebraically via a quotienting operation on quivers with potential. These stability conditions were first discussed in the physics literature by Closset and del Zotto, and were constructed mathematically by Xiong by a different method. We obtain a complete description of the corresponding DT invariants, from which we can conclude that they define analytic wall-crossing structures in the sense of Kontsevich and Soibelman. In the last section we discuss several other examples of a similar flavour.