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I have edited and expanded wall crossing a little
One question to Zoran:
you have designed the entry to cover the notion in great generality. But most of the references that you already had, and now also all that I have added, concern wall crossing of BPS states. Eventually we need to do something to make the entry more systematic on this point. Should we split off an “wall crossing of BPS states”, maybe?
I prefer not to split. Most of existing references are anyway about BPS states, so if one thinks of wall crossing one usually thinks of this case. On the other hand, it is beneficial for that case, to attempt the meaning which is more general. It is like Poincare duality. It is mainly about manifolds. Though it is useful sometimes to think more generally. We did not make a special entry about Poincare duality for manifolds. Do you think we should ?
Another reason why I would not like to split is that wall crossing is sometimes about BPS states essentially but it is phrased in a different language.
Of course I agree that the ’general section’ is so far vague (I originally wrote the entry on wall crossing AND stability, and the text much reflects the second aspect), but I will need more thought to make it more direct and closer to the BPS case. So if you excuse me until mid August I’d repair that (I’d appreciate some reminder if it slips my mind).
I added a new preprint of Kontsevich and Soibleman from few days ago at wall crossing and Donaldson-Thomas invariant. The web page of Aarhus lectures cited as the main website at wall crossing in Aarhus. Can somebody correct the link ?
I remember seeing wall crossing before and not finding what I was expecting to find: intuitively, a codimension 1 subset of a space of functions or configurations (typically infinite-dimensional) which separates chambers = connected components of the subspace consisting of nondegenerate functions or configurations. These are studied in Morse-Cerf theory; see for example here. Is there a connection between this and the material discussed in the nLab article?
Well, walls are indeed codimension one, but the wall crossing research is not concerned with how the walls may look like in general geometric situations, but when they represent the abrupt/discontinuous change in parameters and provided the space has to do with solutions of something, in the cases the entry mainly describes, these are the space of stability conditions. Thus you have something what depends on stability conditions, like observables of QFT, and you look their dependence.
I’d think there should be room in the $n$Lab entry to mention the use of “wall crossing” as in the article that Todd is pointing to and to make the discussion of the various uses of the term more specific.
I am not sure. It may be a completely different concept and should be a separate entry (with a link) Cerf wall crossing. If I understand, Cerf crossing is a geometric path which crosses a codimension one wall in certain setup of Morse theory, so it is about a shape of something crossing the wall in the usual manifold. On the other hand, wall crossing in the sense of our entry is a phenomenon of crossing in the more complicated moduli stack of parameters. Wikipedia also calls it “phenomenon in string theory” here. I am however not an expert to judge. Having an entry about Cerf theory does not hurt, and having a link to there from wall crossing neither. But trying to explain that these are cases of a same concept may be overstretching and giving a wrong information. It would be nice if somebody can of course trace if it may be so.
Cerf wall crossing stub. Please substantiate.
Thanks. I have further cross-linked with wall crossing by adding disambiguation lines at the top of the entries.
I am yet not sure where to put the wall crossing in representation theory, so I will also separate it: wall crossing functors with redirect wall crossing in representation theory.
did some minimal editing and added further references at wall crossing.
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