Timeline for H^d[U(1)^n,U(1)] of the Borel cohomology and Chern-Simons theory
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23 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| May 27, 2013 at 4:17 | comment | added | Mariano Suárez-Álvarez | By the way, there is no reason for you to apologize for being a physicist! :-) | |
| May 13, 2013 at 7:24 | comment | added | Konrad Waldorf | Thank you for clarifying which cohomology you meant. So by $H^n(U(1),U(1))$ you mean $H^{n+1}(BU(1),\mathbb{Z})$, the singular cohomology of the topological space $BU(1)$. $BU(1)$ is a $K(\mathbb{Z},2)$, whose cohomology is $\mathbb{Z}$ in even, and $0$ in odd degrees. Now you can compute all your cohomology groups using the Künneth formula. The expression $H^4(BU(1),\mathbb{Z})=\mathbb{Z}^{n+\frac{1}{2}n(n-1)}$ that you provided somewhere seems to be correct. | |
| May 13, 2013 at 2:19 | history | edited | wonderich | CC BY-SA 3.0 |
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| May 12, 2013 at 1:37 | comment | added | Mariano Suárez-Álvarez | I for one still don't know what cohomology you are talking aboout —Konrad asked you to be specific about this a few weeks ago but you haven't answered that afaict. Without knowing that, it is simply impossible to even make sense of what you are asking! For example, the answer by Henr.L below seems to have decided that you are taking about singular cohomology of the space $U(1)^n$, which I think is quite surely no the case... | |
| Apr 28, 2013 at 1:00 | history | edited | wonderich | CC BY-SA 3.0 |
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| Apr 27, 2013 at 23:47 | comment | added | YangMills | @Idear: why do you keep capitalizing PHYSICS? | |
| Apr 27, 2013 at 22:39 | answer | added | wonderich | timeline score: 3 | |
| Apr 24, 2013 at 0:29 | comment | added | Steven Landsburg | Where are you getting the "known fact" that $H^3(U(1),U(1)={\mathbb Z}$? | |
| Apr 23, 2013 at 12:10 | comment | added | Konrad Waldorf | Can you say more precisely which cohomology you consider? You say "Moore cohomology". Do you mean Borel-Moore cohomology, which is ordinary cohomology for compact spaces as we have here? Or do you mean the equivariant Moore cohomology, in which case you have to specify an action of a group on your spaces? Or do you mean a version of group cohomology? If so, which one (algebraic group cohomology, continuous group cohomology, smooth group cohomology)? | |
| Apr 22, 2013 at 23:47 | history | edited | wonderich | CC BY-SA 3.0 |
change the title slightly to draw people's attention.
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| Apr 22, 2013 at 5:05 | answer | added | Henry.L | timeline score: 1 | |
| Apr 22, 2013 at 3:53 | comment | added | wonderich | Thank you Mariano. Again, do someone please know the answers of: $ \begin{cases} H^3[U(1),U(1)]=Z, \newline H^3[U(1)\times U(1),U(1)]=? \newline H^3[U(1)\times U(1)\times U(1),U(1)]=? \newline \end{cases} $ | |
| Apr 22, 2013 at 3:48 | comment | added | Mariano Suárez-Álvarez | «Richard Feynmann might have done this» is not exactly a great justification for anything, really... Euler also got away with lots of things, but, well, he was Euler. You seem to be operating under the intuition that ℤn somehow converges to ℤ (and that the various functors you are trying to compute are compatible with this limit) But that is an incorrect intuition, at least here. | |
| Apr 22, 2013 at 3:43 | comment | added | Mariano Suárez-Álvarez | That $Tor_1^{\mathbb Z}(\mathbb Z,\mathrm{anything})=0$ does tell you something non-trivial: that $\mathbb Z$ is a flat $\mathbb Z$-module, and this is so simply because it is free. | |
| Apr 22, 2013 at 3:07 | history | edited | wonderich | CC BY-SA 3.0 |
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| Apr 22, 2013 at 2:10 | comment | added | wonderich | For people who are interested in PHYSICS motivation for my problem, please take a look on: "Symmetry protected topological orders and the group cohomology of their symmetry group" arxiv.org/abs/1106.4772v6 or the Science paper: Science 338, 1604 (2012) sciencemag.org/content/338/6114/1604.abstract The formulas I used are listed in 1106.4772v6 paper. Again, my intuition for my guessed answer of (Q1) is from: Chern-Simons theory. For example, $N \times N$ K matrix multiplet Chern-Simons theory $K_{ij} a_i \wedge d a_j$ with $U(1)^N$ gauge group. | |
| Apr 22, 2013 at 2:04 | comment | added | wonderich | Thank you Steven for the comment. The inconsistency also happens to the known fact $H^3[U(1),U(1)]=Z$. So what I was driving to is "whether there is some extra constraint on universal coefficient theorem" applying to continuous group $U(1)$? Or something else was wrong. Although my (Q1) is just a guessed answer, I somehow feel it still makes some sense. The understanding of my guess is based on K matrix multiplet Chern-Simons theory K_{ij} a_i \wedge d a_j, where the classification should lead to the same answer as (Q1). For people who are interested in PHYSICS motivation for my problem(contd) | |
| Apr 22, 2013 at 1:57 | history | edited | wonderich | CC BY-SA 3.0 |
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| Apr 22, 2013 at 1:19 | comment | added | Steven Landsburg | So you are making guesses that you say you can prove are wrong (by invoking the universal coefficient theorem) and then asking whether your guesses are right? | |
| Apr 22, 2013 at 0:20 | history | edited | wonderich | CC BY-SA 3.0 |
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| Apr 22, 2013 at 0:12 | history | edited | wonderich | CC BY-SA 3.0 |
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| Apr 22, 2013 at 0:05 | history | asked | wonderich | CC BY-SA 3.0 |