TITLE: The use of Hall algebras in physics
QUESTION [12 upvotes]: I once read a statement (not memorized precisely) that a certain physics quantity between two states of charge $d_1$ and $d_2$ respectively could be computed by running over the states of charge $d_1+d_2$ which is the extension of the original two states. Therefore we need to consider some Hall algebras on a moduli space.
I couldn't find that literature any more, so I am not sure that this statement is correct. Could anyone help me to make clear this sort of things? Thanks a lot!
My questions are:
1) What is the basic physics setting of this story?
2) Why is this "extension" important?
3) If this is not correct, what is the correct statement/why do physicists care about Hall algebras?

REPLY [4 votes]: Let me add a small comment to Jeff Harvey's nice answer. Some years after the original BPS-algebra papers of Harvey-Moore, there was a paper by Fiol-Marino, hep-th/0006189, which analyzed the algebra of BPS-states for local compactifications where the BPS-states can be described by quivers. They gave a rather explicit realization of Harvey and Moore's "correspondence conjecture", and they also noted that the algebra of BPS-states is essentially that of a Ringel-Hall algebra.