Naturalness conjecture for the D-brane support morphism

Let $X$ be the Calabi–Yau threefold considered above, let $\beta\in H_2(X,\mathbb Z)$, and let

$$\pi_\beta:\widetilde M_\beta(X)\longrightarrow S_\beta(X)$$

be the surjective morphism from the normalization of the sheaf moduli space to the normalization of the image parametrizing support curves of homology class $\beta$. Naturalness conjecture for the D-brane support morphism. The morphism $\pi_\beta:\widetilde M_\beta(X)\longrightarrow S_\beta(X)$ is the natural morphism from the D-brane moduli space $\widetilde M_\beta(X)$ to the moduli space $S_\beta(X)$ of support curves whose homology class is $\beta$. This proposal identifies the sheaf moduli space with the natural moduli of D-branes wrapping support curves and is intended to provide the geometric basis for defining BPS invariants.