Vafa-Witten invariants from exceptional collections

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 $\gamma$ and polarization $J$ on $S$ vs. the dimension vector $\vec{N}$ and stability parameters $\vec{\zeta }$ on the quiver side. For all examples that we consider, which include all del Pezzo and Hirzebruch surfaces, we find that the BPS indices ${\Omega }_{\star }(\gamma )$ 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.

88+40 pages, one figure, quivers aplenty; v5: various misprints corrected

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Cited by in corpus (16)

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• On the BPS spectrum of 5d SU(2) super-Yang-Mills
• 5D BPS Quivers and KK Towers
• Vafa-Witten invariants from modular anomaly
• Rank $N$ Vafa-Witten invariants, modularity and blow-up
• Gauge theories on compact toric manifolds
• BPS Dendroscopy on Local $P^2$
• The $U$-plane of rank-one 4d $\mathcal{N}=2$ KK theories
• Mock modularity at work, or black holes in a forest
• On the existence of scaling multi-centered black holes
• Quivers and curves in higher dimension
• Tame algebras have dense $\mathbf{g}$-vector fans
• Attractor invariants, brane tilings and crystals
• GW/DT invariants and 5D BPS indices for strips from topological recursion
• Topological string amplitudes and Seiberg-Witten prepotentials from the counting of dimers in transverse flux
• Seiberg-Witten geometry, modular rational elliptic surfaces and BPS quivers