Attractor-invariant formula under quiver mutation

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Let (Q,W)(Q,W) and (Q,W)(Q',W') be quivers with potential related by mutation at a vertex kk, and let d,dd,d' and stability weights θ,θ\theta,\theta' be related by the mutation formulas in the source. Let Ωθ(d,y)\Omega_\theta(d,y) and Ωθ(d,y)\Omega'_{\theta'}(d',y) be the corresponding DT invariants. Suppose that dd is not collinear to eke_k, equivalently that dd' is not collinear to eke'_k. Attractor-invariant mutation conjecture. If dNQ0d\in\mathbb{N}^{Q_0} and dNQ0d'\in\mathbb{N}^{Q'_0}, then Ωθ(d,y)=Ωθ(d,y)\Omega_\theta(d,y)=\Omega'_{\theta'}(d',y); if dNQ0d\in\mathbb{N}^{Q_0} but dNQ0d'\notin\mathbb{N}^{Q'_0}, then Ωθ(d,y)=0\Omega_\theta(d,y)=0; and if dNQ0d'\in\mathbb{N}^{Q'_0} but dNQ0d\notin\mathbb{N}^{Q_0}, then Ωθ(d,y)=0\Omega'_{\theta'}(d',y)=0. In particular, the attractor invariants coincide or vanish under the same conditions. This expresses the expected mutation, or Seiberg-duality, invariance of DT invariants away from the mutated basis direction; the source gives no resolution status.

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Primary source

Sergey Mozgovoy and Boris Pioline, “Attractor invariants, brane tilings and crystals”, arXiv:2012.14358 (2022).

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