Vafa-Witten invariants from exceptional collections
Guillaume Beaujard, Jan Manschot, Boris Pioline
TL;DR
This paper develops a general, quiver-based method to compute Vafa-Witten invariants for rational surfaces by leveraging full, strong exceptional collections to translate D-brane data into Beilinson-type quivers. A key finding is the attractor index vanishing for most dimension vectors on del Pezzo (and plausibly all rational) surfaces, which, together with flow-tree and Coulomb-branch formalisms, enables efficient computation of $c_{oldsymbol{ u},J}$ across chambers. The authors demonstrate precise agreement with VW invariants obtained through blow-up and wall-crossing techniques on $ ext{P}^2$, Hirzebruch, and del Pezzo surfaces, validating the quiver approach. They further speculate on mock modular properties of generating functions in this framework and conjecture vanishing of single-centered indices except for simple/D0 cases, highlighting a deeper structure behind BPS spectra in local Calabi–Yau geometries.
Abstract
Supersymmetric D-branes supported on the complex two-dimensional base $S$ of the local Calabi-Yau threefold $K_S$ are described by semi-stable coherent sheaves on $S$. Under suitable conditions, the BPS indices counting these objects (known as generalized Donaldson-Thomas invariants) coincide with the Vafa-Witten invariants of $S$ (which encode the Betti numbers of the moduli space of semi-stable sheaves). For surfaces which admit a strong collection of exceptional sheaves, we develop a general method for computing these invariants by exploiting the isomorphism between the derived category of coherent sheaves and the derived category of representations of a suitable quiver with potential $(Q,W)$ constructed from the exceptional collection. We spell out the dictionary between the Chern class $γ$ and polarization $J$ on $S$ vs. the dimension vector $\vec N$ and stability parameters $\vecζ$ on the quiver side. For all examples that we consider, which include all del Pezzo and Hirzebruch surfaces, we find that the BPS indices $Ω_\star(γ)$ at the attractor point (or self-stability condition) vanish, except for dimension vectors corresponding to simple representations and pure D0-branes. This opens up the possibility to compute the BPS indices in any chamber using either the flow tree or the Coulomb branch formula. In all cases we find precise agreement with independent computations of Vafa-Witten invariants based on wall-crossing and blow-up formulae. This agreement suggests that i) generating functions of DT invariants for a large class of quivers coming from strong exceptional collections are mock modular functions of higher depth and ii) non-trivial single-centered black holes and scaling solutions do not exist quantum mechanically in such local Calabi-Yau geometries.