Abstract: |
In type II strings compactified on a Calabi-Yau threefold $X$, the Donaldson-Thomas (DT) invariants counting BPS black holes have an intricate dependence on the moduli, due to wall-crossing phenomena. When $X$ is toric, 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 value of the DT invariants for any dimension vector in a certain chamber known as the attractor (or self-stability) chamber. In short, "attractor invariants always vanish, except when they are known not to". In combination with the attractor flow tree formulae, which are now rigorously established, this conjecture provides an algorithmic way of computing the DT invariants for any dimension vector and stability parameters. The conjecture passes 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 |