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Description
Four-dimensional string vacua and quantum field theories with $N=2$ supersymmetry exhibit a rich spectrum of supersymmetric bound states, which depends sensitively on the values of the moduli fields or gauge theory parameters. In the context of type II string theories compactified on a Calabi-Yau threefold, the index counting these states coincides with the Donaldon-Thomas invariants associated to the derived category of coherent sheaves in type IIA, or the Fukaya category in type IIB. The wall-crossing formula in principle determines the indices in any chamber, provided they are known in a particular chamber. More generally, the attractor flow tree formula (or equivalently, the scattering diagram in the space of stability conditions) determines them in terms of the so-called attractor indices.
For toric CY3 singularities, the category of BPS states is isomorphic to the derived category of quiver representations, and the attractor indices can be determined exactly, giving a complete characterization of the spectrum of BPS states in any chamber. For compact CY3 manifolds such as the quintic, the attractor invariants cannot be determined in general, but certain vanishing results allow to determine the so-called rank 0 Donaldson-Thomas invariants counting D4-D2-D0 brane bound states, in terms of the rank 1 DT invariants, in turn related to Gopakumar-Vafa invariants. These methods in particular provide precision tests of the (mock) modularity properties of generating series of D4-D2-D0 indices (or, in the local case, Vafa-Witten invariants) predicted by S-duality.