What is BPS state? I am preparing a presentation about black holes thermodynamics in string theory. Both in polchinski and in many famous articles there is a discussion about BPS and I am not able to understand why it is related.
I know what is the definition of BPS state but why is it relevant for the discussion about black $p$-branes?
 A: Given a system, if some of the supersymmetries (but not necessarily all of them are) conserved then it can be shown that the mass (or energy) of the states is bounded below by the charge associated with the conserved supersymmetry - this is known as the BPS bound.
BPS states is a state which has the minimal possible mass according to this bound.
Dp branes are essentially such states with the Ramond-Ramond charge as the associated charge.
The important thing for the discussion about black p branes is that for BPS states there is a non renormalization theorem which state that we can vary the string coupling parameter g without changing the total mass. This is useful because it allows us to work both in the regime where g is small (hence the low energy effective theory is valid) and in the regime where g is large (hence the strings sitting on the D brane picture is valid).
Then we can do the relevant calculation in the regime in which it is easier and carry the solution to the other regime.
This is very useful for example, in calculations of entropy: one can work in the low energy effective theory, calculate the surface area associated with the black p brane and then derive the Bekenstein-Hawking entropy.
Then, one can work with the D brane picture at small coupling and actually count the number of states and derive the entropy from it.
The two calculation then should agree because of the non renormalization theorem described above, and we can use it to check weather string theory is consistent with Bekenstein-Hawking entropy.
