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?



The use of Hall algebras in physics
In supersymmetric field theories and string theories there are special states called BPS
states which are annihilated by some of the supercharges and whose mass is determined
in terms of their charges by the supersymmetry algebra. The study of these states
and how they behave as various moduli are varied has been one of the main tools physicists
have used to find evidence for various kinds of dualities, particularly S-duality which relates
weakly coupled theories to strongly coupled theories.


One particularly rich example of an S-duality involves a duality between the heterotic string
on $K3 \times E$ with a particular choice of $E\_8 \times E\_8$ gauge bundle where $E$ is an elliptic curve and Type II string theory on Calabi-Yau manifolds
which have the form of K3 surfaces fibered over rational curves. In the first paper mentioned by
Alexander Chervov, Greg Moore and I computed certain one-loop integrals on the heterotic string
side of this story and found two interesting facts. First, that these integrals were determined purely
by the spectrum of BPS states, and second that the answers involved denominator formulae for
Generalized Kac Moody algebras of the type studied previously by Borcherds. Given this
fact it was natural to think that there was an algebraic structure that one could define on the BPS
states that would ``explain'' why were getting the denominator formula for a GKM algebra. This was the physics motivation for the introduction of the algebra of BPS states. However we did not achieve
our original goal in that we were not able to find a direct connection between the BPS algebra
and the GKM denominator formulae. In spite of this failure the idea that there should be an
algebraic structure on the space of BPS states seems to have some merit. 


If you want to look at more recent developments you might have a look at arXiv:1102.1821
which finds a more direct relation between a particular Borcherds algebra and one-loop
integrals. On the mathematical side there is arXiv:1006.2706 by Kontsevich and Soibelman
where Hall algebras appear. They mention the idea of an algebra of BPS states as motivation
for their construction.  

I must confess though that I do not have the level of mathematical sophistication needed
to understand this paper and unfortunately my colleague Greg Moore, who does, is not on MO.


