Abstract

Unlike in cases with higher supersymmetry, the spectrum of BPS states in type II string theory compactified on a Calabi-Yau threefold $X$ with generic $SU(3)$ holonomy remains partially understood. Mathematically, the BPS indices coincide with the generalized Donaldson-Thomas invariants associated to the derived category of coherent sheaves, but they are rarely known explicitly. String dualities indicate that suitable generating series of rank 0 DT indices counting D4-D2-D0 bound states  should transform as vector-valued mock modular forms, in a very precise sense. I will spell out and test these predictions in the context of non-compact threefolds of the form $X=K_S$ where $S$ is a Fano surface, and for compact Calabi-Yau threefolds with $b_2=1$ such as the quintic. Based on earlier work with Sergey Alexandrov, Sibasish Banerjee, Nava Gaddam, Jan Manschot, and work in progress with Sergey Alexandrov, Soheyla Feyzbakhsh, Albrecht Klemm and Thorsten Schimannek.