Timeline for Usefulness of using TQFTs
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
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| Mar 29, 2022 at 7:04 | comment | added | David Roberts♦ | The link in Ian's comment is broken, here's a replacement: arxiv.org/abs/math/0503054 | |
| S Dec 4, 2013 at 9:40 | history | suggested | Gejza Jenča | CC BY-SA 3.0 |
corrected broken equation
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| Dec 4, 2013 at 8:58 | review | Suggested edits | |||
| S Dec 4, 2013 at 9:40 | |||||
| Oct 7, 2011 at 10:17 | comment | added | Kelly Davis | @Agol Unfortunate, I didn't hear about Freedman et al. arXiv:math/0503054v4 [math.GT] until after I posted the preprint. I just updated my article with a reference to Freedman et al.'s and also to a few results Danny Calegari tipped me off to from the related research program of Kreck, Teichner, Calegari, Freedman, and Walker. It should go out with the next posting. | |
| Jul 1, 2011 at 15:50 | comment | added | Ian Agol | @Kelly: the main theorem of your article (Theorem 4.1) seems to have been proven by Freedman et. al.: front.math.ucdavis.edu/0503.5054 (Theorem 4.2) | |
| Jun 14, 2011 at 17:07 | comment | added | Kim Morrison | @Kelly has just posted an article following up on this last comment: arxiv.org/abs/1106.2358 | |
| May 9, 2011 at 6:24 | comment | added | Kelly Davis | @Dylan In 4-dimensions no axiomatic TQFT can detect changes in smooth structure. (One can prove this using the TQFT axioms and addenda B,C, and D of math/9712231.) However, in 4-dimensions Witten-Donaldson theory does detect changes in smooth structure. My guess is that the the axioms of TQFT are sick, and one just happens to see this explicitly in 4-dimensions. | |
| Apr 5, 2011 at 20:19 | comment | added | Dylan Thurston | @Tim: Thanks for explaining the confusion. I guess the length of the second part of my answer is an argument against slavishly following Atiyah-Segal... | |
| Apr 5, 2011 at 19:48 | comment | added | Kelly Davis | There are more things in heaven and earth, Dylan, Than are dreamt of in the Atiyah-Segal axioms. :-) | |
| Apr 5, 2011 at 18:16 | vote | accept | ISH | ||
| Apr 5, 2011 at 15:04 | comment | added | Tim Perutz | @Kelly: there's a terminological issue, I think. Regarded as a QFT, Chern-Simons theory (for example) is considered by Witten to be a TQFT just because the action does not involve the metric. However, it does involve a hard-to-interpret path integral. Mathematicians usually prefer the Atiyah-Segal axioms for TQFT, and that's what Dylan is referring to. | |
| Apr 5, 2011 at 11:57 | history | edited | Dylan Thurston | CC BY-SA 2.5 |
added 1598 characters in body
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| Apr 5, 2011 at 11:23 | comment | added | Dylan Thurston | @Kelly It's not a standard TQFT, if only because the invariants (both Donaldson and SW) are only defined for 4-manifolds with $b_2^+ \ge 2$. I also don't know what vector spaces to associate to a general 3-manifold for the Donaldson invariants. The theory would be call "instanton Floer homology", but, as I understand it, there are severe technical issues with reducibles in general. If you know how to define such a TQFT related to Donaldson theory, I'm very interested to hear about it. | |
| Apr 5, 2011 at 7:10 | comment | added | Kelly Davis | @Dylan First, Donaldson's invariants are interpretable as a TQFT, ala Witten. Also, you can interpret Donaldson's invariants as classical homology/cohomology invariants on the moduli space of connections. | |
| Apr 5, 2011 at 5:32 | comment | added | Greg Friedman | Is it possible to say a few quick words about how Donaldson/Seiberg-Witten invariants are related to TQFTs? | |
| Apr 5, 2011 at 1:08 | history | answered | Dylan Thurston | CC BY-SA 2.5 |