For the purposes of this question, I imagine I want to restrict attention to M-theory compactified on $X \times S^{1}$, Type IIA or IIB compactified on $X$, or the A/B topological sigma models on with target space $X$. Here, $X$ is a smooth, projective Calabi-Yau threefold. I come from a mathematical background and only have a mediocre, basic understanding of string theory and supersymmetric physics, so I hope my questions below are at least well-posed.

I'm assuming that BPS D-branes have the minimal mass for fixed charge perhaps. In other words, they saturate the bound coming from the central charge. I also think that the mass of a D-brane is proportional to its volume. Now, we have the following result at the interface of Riemannian and Kahler/algebraic geometry:

Let $(X, \omega)$ be a compact Kahler manifold. If $Y$ is an $n$-dimensional closed Kahler submanifold of $X$, then $Y$ has minimal volume among all real submanifolds in the same homology class. Moreover, the volume of $Y$ is given by

$$\text{vol}(Y) = \frac{1}{n!}\int_{Y}\omega^{n}.$$

My questions are the following:

1. This result I quote seems to indicate to me that a BPS (i.e. volume minimizing) D-brane ought to neccesarily wrap a holomorphic cycle in $X$. Is this true? Is it true that a BPS D-brane cannot be supported along the spatial directions of the non-compact spacetime?

2. Probably a silly question, but does a BPS D-brane give rise to a BPS particle in 4d?

3. My next question pertains to a more specific formula for the mass of a D-brane. I have seen that the mass $M$ of a D-brane wrapping a submanifold $Y \subseteq X$ is given by (possibly up to constants) $$M = \frac{1}{\lambda} \text{vol}(Y),$$ where $\lambda$ is the string coupling constant. To what extent is this formula true, if at all? Does this string coupling constant exist in all the theories I mentioned at the beginning?

4. Finally, I've heard you can engineer a black hole in 4d with a very massive D-brane. From the above formula, you can also achieve large mass by taking $\lambda \to 0$. Is this correct? If so, what is the reason or interpretation?

• For your first question : it is very common in the daily practice of string theory to have D-branes supported along the spatial directions of non-compact space-time. In fact this the simplest way to construct a 4d theory : consider the world volume theory on some D3 branes spanning $\mathbb{R}^{1,3}$ inside $\mathbb{R}^{1,9}$ in type IIB. – Antoine May 16 '18 at 10:24
• However if your 10d space-time is $\mathbb{R}^{1,3} \times X_6$ then you can e.g. consider a D5 wrapping $\mathbb{R}^{1,3} \times \Sigma_2$ and the $\Sigma_2$ part will be a holomorphic cycle in $X_6$. – Antoine May 16 '18 at 10:33
• Right, I understand that, but I guess the crux of my question is whether what you describe above can be a BPS D-brane? Like, is what you describe volume-minimizing? By that mathematical result I quote, I'm wondering if BPS branes necessarily live only along the time direction in $\mathbb{R}^{1,3}$ and then wrap a holomorphic cycle in $X$? – Benighted May 16 '18 at 16:42
• Yes, these are BPS D-branes. The volume is infinite, so volume-minimizing is meaningless in that situation, however it makes sense locally (therefore, the branes span a hyperplane, with 0 curvature). – Antoine May 16 '18 at 18:29
• One way to see "physically" that infinite branes are BPS is to compute the forces felt by two such parallel branes. They attract each other via gravitational force, and repel each other via some kind of electrostatic repulsion. But the BPS property guarantees that these two forces precisely cancel (this is manifest in the equality between the charge density and the tension of the brane), so the configuration is stable. – Antoine May 16 '18 at 18:34