The papers by J. Harvey and G. Moore seems relevant to subj.

http://arxiv.org/abs/hep-th/9609017 On the algebras of BPS states

http://arxiv.org/abs/hep-th/9510182 Algebras, BPS States, and String

See for example pages 24-25 section 7.2 of the second paper.
The "correspondence conjecture" (page 25 top) seems to be exactly the fact that physically defined BPS-state algebra for the particular situation of the IIA string on K3 is a Hall algebra related to the sheaves on K3.

conjecture, motivated by the work of
Nakajima, and of Ginzburg et. al.,
[56][57], is the following. Suppose
first that the three vectors Qi in
(7.2) represent positive BPS states.
Recall that the charges are Chern
characters of sheaves. There is only
one natural way that the three sheaves
E1, E2, E3 can be related and satisfy
(7.2). They must fit into an exact
sequence: 0 → E1 → E3 → E2 → 0 (7.3)
or 0 → E2 → E3 → E1 → 0 (7.4) The
ambiguity between (7.3) and (7.4) is
resolved by the requirement that E3 be
semistable: since Chern characters are
additive the inequality (5.2) cannot
hold for both F = E1 and F = E2. 23 We
define the correspondence region to be
the subset of M(Q1) ×M(Q2) ×M(Q3)
defined by the set of triples:
C+++(Q1,Q2;Q3) = {(E1, E2, E3) : 0 →
E1 → E3 → E2 → 0}

The correspondence conjecture given after that claims that the left hand-side product (physical definition of BPS-state algebra) equals to mathematical definition
which is Hall algebra (actually for its analogue for cohomologies, i.e. for Chern characters of sheaves, while true Hall algebra would be for sheaves themselves).

PS

Seems one of the authors is sometimes on MO, may be notify him to get expert's answer.