Timeline for Wall Crossing in Physics and Mathematics
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
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| Jan 4, 2011 at 16:37 | comment | added | Tim Perutz | @David. The earliest usage that I'm aware of for walls and wall-crossing (perhaps the original coinage?) is Donaldson's 1987 paper "Irrationality and the h-cobordism conjecture", which proves a wall-crossing formula for an instanton invariant of 4-manifolds with $b^+=1$. The walls are period points for the abelian instantons which one encounters in generic 1-parameter families of conformal structures. | |
| Jan 4, 2011 at 8:41 | comment | added | domenico fiorenza | Thanks Andy, thanks David. I've now included your comments in my answer above. | |
| Jan 4, 2011 at 8:40 | history | edited | domenico fiorenza | CC BY-SA 2.5 |
included comments by Andy Neitzke and David Ben-Zvi
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| Jan 4, 2011 at 2:51 | comment | added | David Ben-Zvi | Also might be worth noting that wall crossing transition of moduli spaces by flops goes back at least to Thaddeus' influential work on the Verlinde formula, studying the dependence of geometric invariant theory quotients on the choice of stability (linearization of the action), and similar wall crossings were important eg in Donaldson theory. | |
| Jan 4, 2011 at 2:49 | comment | added | David Ben-Zvi | Nice answer. I realize you're being vague, but I don't think it's fair to say the space $M_C$ makes no sense without stability -- it's a perfectly nice stack for $C$ abelian (or higher stack in the derived setting), just not a scheme.. | |
| Jan 3, 2011 at 23:18 | comment | added | Andy Neitzke | It's not quite true that "the physicist's ill-defined quantity ... was defined before stability conditions." Even in the physical description of these quantities, a parameter certainly enters, which plays the role of the stability condition. What is true is that there is some other physically defined quantity $F$ that depends continuously on the parameter, and whose relation to $I$ is understood; this leads to the desired relation between $I_P$ and $I_Q$. | |
| Jan 3, 2011 at 21:16 | comment | added | domenico fiorenza | @José. Thanks! I've now followed your advice and split the answer into paragraphs. I've also fixed the $\mathcal{P}$ vs. $\mathcal{M}_P$ typo. | |
| Jan 3, 2011 at 21:15 | history | edited | domenico fiorenza | CC BY-SA 2.5 |
fixed typo, improved formatting
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| Jan 3, 2011 at 20:50 | comment | added | José Figueroa-O'Farrill | +1. Nice answer. Would it be too much to ask to perhaps split this into paragraphs to improve readability? Also you denote the fibre over $P \in \Sigma$ by $\mathcal{P}$, but don't you mean $\mathcal{M}_P$? | |
| Jan 3, 2011 at 20:15 | history | answered | domenico fiorenza | CC BY-SA 2.5 |