# The use of Hall algebras in physics

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?

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–  Alexander Chervov Nov 9 '12 at 12:54

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.

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Thank you very much! Additional question: what motivated you to use that correspondence conjecture to define the product of BPS state? –  Xiao Xinli Nov 11 '12 at 3:19
I can't offer much beyond what we say in the paper. The main motivations were 1) Nakajima's (and others) work on instantons on ALE spaces and Kac-Moody algebras which used a similar construction 2) comparison to the algebra of BPS states on the heterotic side which could be computed using conformal field theory techniques (or VOA's if you prefer) and 3) some physical intuition related to the behavior of bound states in quantum mechanics. N=2 string duality is not something that is yet understood in a mathematically precise way, so I'm not sure how helpful 2) is to mathematicians. –  Jeff Harvey Nov 11 '12 at 15:01

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.

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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.

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Thank you very much! However, what I expected is something more physics. Those two papers are two mathematics. –  Xiao Xinli Nov 10 '12 at 4:23