Invariant Stability Conditions on Certain Calabi-Yau Threefolds

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.