Timeline for Meaning/origin of Seiberg-Witten equations/invariants
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 6, 2022 at 11:53 | history | edited | YCor | CC BY-SA 4.0 |
formatting
|
| S Apr 19, 2014 at 20:15 | history | suggested | Geom math | CC BY-SA 3.0 |
improved formatting
|
| Apr 19, 2014 at 20:02 | review | Suggested edits | |||
| S Apr 19, 2014 at 20:15 | |||||
| Feb 16, 2012 at 7:42 | vote | accept | Chris Gerig | ||
| Feb 15, 2012 at 14:07 | comment | added | Tim Perutz | @Chris: picking up Liviu's point, the idea that SW and Donaldson theories are interchangeable is not true, in either direction. What is true is that the primary invariants of these theories are equal. There are further invariants (notably, the homotopical SW invariants) which are only defined in one of the theories; and each sees geometry which is not visible in the other. Kronheimer-Mrowka's proofs of "Property P for knots", or Andrei Teleman's results on curves in Class VII surfaces, say, use instantons to see thing that monopoles don't. | |
| Feb 15, 2012 at 10:31 | comment | added | Liviu Nicolaescu | @Chris Gerig To paraphrase Yoggi Berra, in theory there is no difference between SW and YM. In practice there is. There are things that SW theory sees more clearly than YM. (The work of Taubes on symplectic $4$-manifolds, and contact $3$-manifolds is a good example.) As an analogy, try proving Kodaira's vanishing theorem using the sheaf theoretic description of cohomology but not the Hodge theoretic version. In this classical situation you have two theories describing the same object, but in concrete situations one theory may be more useful than the other. | |
| Feb 15, 2012 at 8:28 | comment | added | Alexander Chervov | @Chris " that no new information is obtained from SW Theory over Donaldson Theory (and vice versa)" YES (as far as I understand), at least idealogically it is surely like this, I am not sure if there are technical caveats. From the physical point of view it is in some sense the crucial belief going back to 60-ies that Yang-Mills-type theory can be equally described in "monopole" variables. This is crucial for the explainaition of one the main hep-th problem - "quark confiment". Something like "monopole strings tire the quarks and do not allow them to be free". | |
| Feb 15, 2012 at 8:22 | comment | added | Alexander Chervov | arxiv.org/abs/dg-ga/9507004 Localisation of the Donaldson's invariants along Seiberg-Witten classes; Victor Pidstrigach, Andrei Tyurin; This article is a first step in establishing a link between the Donaldson polynomials and Seiberg-Witten invariants of a smooth 4-manifold. Andrei Tyurin passed away several years ago (27.10.2002) was an influential Russian algebraic geometer, e.g. Tyurin parameters for vector bundles on curves widely used (e.g. S.P. Novikov, Kriechever arxiv.org/abs/math-ph/0308019 ) | |
| Feb 15, 2012 at 7:26 | comment | added | Deane Yang | Tim, thanks for correcting my rather imprecise comments. | |
| Feb 15, 2012 at 5:30 | comment | added | Chris Gerig | @Tim Perutz, so it is safe to say (and rigorously shown) that no new information is obtained from SW Theory over Donaldson Theory (and vice versa)? | |
| Feb 15, 2012 at 5:20 | answer | added | Chris Gerig | timeline score: 15 | |
| Feb 15, 2012 at 3:30 | comment | added | Tim Perutz | Deane, when you said Turaev, I think you meant Tyurin (and the someone else was Pidstrigatch). They proposed a certain "master theory" that has instanton and SW moduli spaces as boundary strata. Their proposal was carried out in a series of papers by Feehan and Leness. | |
| Feb 14, 2012 at 22:02 | comment | added | user17945 | To add to Paul Siegel's answer, you can download the relevant parts of Naber's book directly from his website pages.drexel.edu/~gln22. He also has a 17-part series of lectures on Seiberg-Witten Theory ("Lectures on the Witten Conjecture"). I haven't studied any of these, so I have no idea how useful they will be in answering your question. But they seem to at least partly address the motivation behind the Seiberg-Witten equations, and Naber tends to be a very clear expositor in general. | |
| Feb 14, 2012 at 16:55 | comment | added | Deane Yang | VERY VAGUE: My memory from decades ago is that Seiberg and Witten's original paper showed that there was some kind of quantum field theory, maybe even a topological quantum field theory, with a parameter, where one limit of the parameter gave the Donaldson invariants and the other limit the Seiberg-Witten invariants. This is what triggered the initial mathematical investigation of Seiberg-Witten invariants. And my impression is that Turaev, possibly with someone else, found a rigorous mathematical way to show the equivalence of the two types of invariants (maybe via some kind of cobordism?). | |
| Feb 14, 2012 at 15:49 | comment | added | Paul Siegel | A suggestion from one confused soul to another: you might try looking at Naber's books "Topology, Geometry and Gauge Fields" (there are two of them). The goal of those books is to explain the rudiments of Donaldson theory and Seiberg-Witten theory to someone who hasn't necessarily seen the definition of a manifold. The treatment of these topics is obviously not very deep, but it might be a good place to look for intuition and historical background. | |
| Feb 14, 2012 at 14:00 | answer | added | Liviu Nicolaescu | timeline score: 3 | |
| Feb 14, 2012 at 7:20 | history | asked | Chris Gerig | CC BY-SA 3.0 |