The mock-modularity conjecture for BPS invariants of type II limits

Consider a type II$_{b}$ limit associated with a Tyurin degeneration of the form $V_0=V_1\cup_Z V_2$ and a special Lagrangian 3-cycle $\Gamma_0$ dual to an element of $H^3(V,\mathbb{Z})\cap\operatorname{Gr}_2(\Delta)$. Let $C_0=\partial_*\iota_*\Gamma_0\in\Lambda_{\rm trans}$ denote the curve on $Z$ over which the asymptotic form of $\Gamma_0$ is fibered near the Tyurin degeneration. BPS mock-modularity conjecture. There exists a meromorphic mock-modular form, generally with respect to a subgroup $G\subseteq SL(2,\mathbb{Z})$, of the form

$$\theta(q)=\sum_{n\in\mathcal I}c(n)q^n,$$

where $\mathcal I$ is a suitable subset of $\mathbb Q$, whose coefficients count the four-dimensional $\mathcal N=2$ BPS invariants associated with $\Gamma_0$ through

$$\Omega_{\rm BPS}(\Gamma_0)=c\left(\frac{1}{2}C_0\mathbin{\cdot_Z}C_0\right).$$

If $\operatorname{Gr}_3=\emptyset$, then $\theta(q)$ is modular. The conjecture relates BPS state counting to modular or mock-modular forms; its general status is not specified in the source.