## – Europe/Lisbon — Online

Sergey Mozgovoy, Trinity College Dublin

Attractor invariants, brane tilings and crystals

Given a CY3-fold $X$, we define its refined DT invariants $\Omega_Z(d)$ by counting objects in the derived category $D^b(X)$, semistable with respect to a stability condition $Z$ and having Chern character $d$. Attractor invariants $\Omega_*(d)$ correspond to a special stability condition that depends on the Chern character $d$. They usually have a particularly simple form. If known, attractor invariants can be used to determine DT invariants for all other stability conditions using wall-crossing formulas or flow tree formulas. A wide class of non-compact toric CY3-folds is encoded by combinatorial data called brane tilings or by associated quivers with potentials. In this setting the derived category of $X$ can be substituted by the derived category of a quiver with potential and the counting problems can be reduced to representation theoretic problems and then solved under suitable conditions. I will survey known results about DT invariants and some new conjectures about attractor invariants in this setting. I will also explain how these formulas in the unrefined limit correspond to the counting of molten crystals associated with brane tilings.