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.

arXiv comment88+40 pages, one figure, quivers aplenty; v5: various misprints corrected

References in corpus (85)

• N=2 Extremal Black Holes
• D-branes, Quivers, and ALE Instantons
• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• A Strong Coupling Test of S-Duality
• Orbifold Resolution by D-Branes
• D-Brane Gauge Theories from Toric Singularities and Toric Duality
• Statistical Entropy of Four-Dimensional Extremal Black Holes
• Brane Dimers and Quiver Gauge Theories
• Split States, Entropy Enigmas, Holes and Halos
• The spectrum of BPS branes on a noncompact Calabi-Yau
• D-branes, Categories and N=1 Supersymmetry
• Fractional Branes and Boundary States in Orbifold Theories
• Toric Duality Is Seiberg Duality
• Witten Index and Wall Crossing
• Phase Structure of D-brane Gauge Theories and Toric Duality
• Building the Standard Model on a D3-brane
• Wall-Crossing from Boltzmann Black Hole Halos
• Symmetries of Toric Duality
• Mergers and Typical Black Hole Microstates
• A Farey Tail for Attractor Black Holes
• N=2 Quantum Field Theories and Their BPS Quivers
• E-Strings and N=4 Topological Yang-Mills Theories
• The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli
• Derived Categories and Zero-Brane Stability
• Toric bases for 6D F-theory models
• Split attractor flows and the spectrum of BPS D-branes on the Quintic
• Computation of Superpotentials for D-Branes
• Black hole bound states in AdS_3 x S^2
• The tropical vertex
• Brane Tilings and Reflexive Polygons
• Quiver Theories from D6-branes via Mirror Symmetry
• Dibaryons from Exceptional Collections
• BPS states and algebras from quivers
• A fixed point formula for the index of multi-centered N=2 black holes
• From Black Holes to Quivers
• Theta functions and Hodge numbers of moduli spaces of sheaves on rational surfaces
• Stability and duality in N=2 supergravity
• Vafa-Witten invariants for projective surfaces I: stable case
• Quiver Yangian from Crystal Melting
• On the Coulomb and Higgs branch formulae for multi-centered black holes and quiver invariants
• Superpotentials for Quiver Gauge Theories
• The M5-Brane Elliptic Genus: Modularity and BPS States
• On blowup formulae for the S-duality conjecture of Vafa and Witten
• Seiberg Duality is an Exceptional Mutation
• Exceptional Collections and del Pezzo Gauge Theories
• Vafa-Witten invariants for projective surfaces II: semistable case
• Euler characteristics of SU(2) instanton moduli spaces on rational elliptic surfaces
• D-branes on Calabi-Yau manifolds and helices
• Multiple D3-instantons and mock modular forms I
• D-Branes on Vanishing del Pezzo Surfaces
• Line Defects, Tropicalization, and Multi-Centered Quiver Quantum Mechanics
• Exceptional Sequences of Invertible Sheaves on Rational Surfaces
• Indefinite theta series and generalized error functions
• Equivariant K-theory and refined Vafa-Witten invariants
• On Correspondences Between Toric Singularities and (p,q)-webs
• Exact results for $\mathcal{N}=2$ supersymmetric gauge theories on compact toric manifolds and equivariant Donaldson invariants
• Euler characteristics of moduli spaces of torsion free sheaves on toric surfaces
• Attractor flow trees, BPS indices and quivers
• Virtual refinements of the Vafa-Witten formula
• Three-block exceptional collections over Del Pezzo surfaces
• S-duality and refined BPS indices
• Exceptional collections and D-branes probing toric singularities
• Monopole contributions to refined Vafa-Witten invariants
• N=4 Supersymmetric Yang-Mills Theory on a Kaehler Surface
• Multiple D3-instantons and mock modular forms II
• The Betti numbers of the moduli space of stable sheaves of rank 3 on P2
• On Calabi-Yau generalized complete intersections from Hirzebruch varieties and novel K3-fibrations
• The Coulomb Branch Formula for Quiver Moduli Spaces
• Invariants of moduli spaces of stable sheaves on ruled surfaces
• Examples of M5-Brane Elliptic Genera
• Two-dimensional crystal melting and D4-D2-D0 on toric Calabi-Yau singularities
• Vafa-Witten theory and iterated integrals of modular forms
• Duality and Mock Modularity
• Black holes and higher depth mock modular forms
• A 5d/2d/4d correspondence
• Generalized quiver mutations and single-centered indices
• Vafa-Witten invariants from modular anomaly
• The quiver approach to the BPS spectrum of a 4d N=2 gauge theory
• Intersection cohomology of moduli spaces of sheaves on surfaces
• Rank $N$ Vafa-Witten invariants, modularity and blow-up
• Discrete Integrable Systems, Supersymmetric Quantum Mechanics, and Framed BPS States - I
• Moduli of semistable sheaves as quiver moduli
• Topological correlators of $SU(2)$, $\mathcal{N}=2^{\ast }$ SYM on four-manifolds
• Generalized Donaldson-Thomas invariants on the local projective plane
• Moduli of Bridgeland semistable objects on ${\mathbb{P}}^2$

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• Exponential BPS graphs and D brane counting on toric Calabi-Yau threefolds: Part I
• 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