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I asked the same question in mo. I think maybe here there are more physics guys to help me.

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?

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?

PS: I think these questions are also related to the representations of particles, and the decomposition of particles.

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The MathOverflow answer really is good: Harvey & Moore's papers are pretty much required reading in this subject.

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