In a lot of physics papers I hear talk of "wrapping a stack of D-branes on a supersymmetric cycle in a Calabi-Yau threefold $X$" and I'm wondering if I understand this correctly from a mathematical perspective.

Let's assume we want to use D-branes to engineer BPS *particles* in four dimensions. Therefore, our D$p$-branes will have $p$ real spatial dimensions in $X$, and only the worldline supported in four dimensions. Okay, in Type IIA we therefore have D6, D4, D2, and D0 branes. The mass of the particle should be proportional to the volume of the brane. Therefore, if we want BPS particles, we want *minimal mass* and hence, minimal volume. We therefore want our branes wrapping calibrated submanifolds (these minimize volume within their homology class). Since $X$ is Calabi-Yau, the calibrated submanifolds are either holomorphic submanifolds or special Lagrangian submanifolds.

It follows that using Type IIA, we can get BPS particles with D-branes wrapping holomorphic cycles in $X$ and using Type IIB we get BPS particles with D3-branes wrapping special Lagrangians. (Please correct me if any of this is mistaken!)

**(In the Type IIA case) are supersymmetric cycles simply holomorphic cycles? Meaning we necessarily get BPS particles?**

The reason I'm confused is because on page 31 of (https://arxiv.org/abs/math/0412328) they write "...in order to give a supersymmetric configuration the mass $M$ of a D-brane and its central charge have to satisfy the inequality $M \geq |Z(Q)|$."

So are supersymmetric cycles when that bound is *saturated* or when that bound is satisfied? Is the idea that supersymmetric cycles are those homologous to a holomorphic submanifold? So wrapping a brane on the holomorphic submanifold itself gives you a BPS state whereas wrapping it on a cycle homologous will just give you a supersymmetric state?