In type II strings compactified on a Calabi-Yau threefold $X$, the Donaldson-Thomas (DT) invariants $\Omega(\gamma,z)$ counting BPS black holes have an intricate dependence on the moduli $z$, due to wall-crossing phenomena. When $X$ is a toric threefold, these indices are related to the DT invariants of a quiver with potential with superpotential, encoded by a brane tiling. I will present a conjecture for the DT invariants for all dimension vectors $d$ in a certain chamber $z_*(d)$ known as the attractor (or self-stability) chamber. In short, "DT invariants all vanish, except when they are known not to." In combination with the attractor flow tree formulae, this conjecture provides an algorithmic way of computing the DT invariants $\Omega(\gamma,z)$ for any $\gamma,z$. The conjecture is supported by a large number of checks, including a successful comparison with the Vafa-Witten invariants of a Fano surface $S$ when $X$ is the total space of the canonical bundle $K_S$, and with the counting of molten crystals for framed DT invariants in the non-commutative chamber. Based on works with G. Beaujard, J. Manschot and S. Mozgovoy, arXiv:2004.14466 and 2012.14358 Join Zoom Meeting https://zoom.us/j/94029175955?pwd=b1hvVnVxbFZjTmM5bkxaWi93VkpzUT09 Meeting ID: 940 2917 5955 Passcode: 086150

BPS states in N=2 gauge theories or string vacua are generically stable but liable to decay across certain codimension-one loci in moduli space. This process is easily understood by viewing BPS states as a bound state of more elementary BPS constituents, described classically by multi-centered solutions of the low energy effective action. The semi-classical quantization of the space of such solutions agrees with the wall-crossing formulae derived in the mathematical literature on BPS invariants, providing a physically elementary justification of the latter. Using this intuition, it is possible to express the BPS index, at any point in moduli space, in terms of indices associated to elementary (or single-centered) constituents. If time permits, I will present evidence for this idea in the case of BPS states described by quiver representations.

Recently, Ooguri, Strominger and Vafa have proposed to identify the macroscopic Bekenstein-Hawking-Wald entropy of extremal charged Black Holes in type II string compactified on a Calabi-Yau threefold, with the all order microscopic entropy in a specific statistical ensemble. We present a test of this conjecture in the case of 'small' black holes (which have zero Bekenstein-Hawking entropy at tree level, but obtain a non-zero entropy after quantum corrections) using heterotic / type II duality. We find a remarkable agreement to all orders in a class of N=2 and N=4 compactifications. Nevertheless, the OSV proposal misses non-perturbative corrections which are computable on the heterotic side. We comment on possible improvements of this proposal.