Categorical wall-crossing for higher-framed DT/PT quivers

Abstract

We study a framed three-loop quiver with $a+r$ incoming arrows and $a$ outgoing arrows. For a generic window parameter, we construct a semiorthogonal decomposition of the bounded derived category of the positive GIT quotient into products of unframed window categories and categories of the negative quotient. The arrow imbalance $r$ determines an interval of admissible slopes of length $r$. The proof separates a lower-convex-hull decomposition of dominant weights, Koszul induction, full faithfulness and semiorthogonality of Hall functors, and the two window equivalences. We extend the decomposition to trace potentials satisfying block additivity. Applications include a precisely specified open chart of the framed conifold quiver, an algebraically defined extended ADHM critical model, and the critical presentation of the Quot scheme of points on ${\mathbb{A}}^3$. We do not identify the extended ADHM model with moduli of higher-rank pairs having a generically rank-one reduced-curve sheaf. A formal MacMahon identity is proved separately, and the localized categorical rank formula is deduced using the localized strict-slope product formula of Pădurariu--Toda.

arXiv comment20 pages. v2: Presentation shortened. v3: Delete the invalid geometric identifications in Section 2 of v.2. Comments are welcome