Timeline for Characterisation of (integrable) connections on (trivial) principal bundle
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| Oct 16 at 1:03 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Jun 18 at 3:00 | comment | added | cheyne | If I understand your question this paper answered things for me a while back: arxiv.org/pdf/0705.0452 | |
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| Jun 4, 2021 at 9:08 | comment | added | user122276 | @PraphullaKoushik - you find a similar question (and answers) here: mathoverflow.net/questions/123942/… | |
| Jun 3, 2021 at 15:26 | comment | added | Joshua Mundinger | @PraphullaKoushik on the trivial principal bundle, a $\mathfrak g$-valued form $\omega$ corresponds to an integrable connection if and only if $\omega$ satisfies the Maurer-Cartan equation $d\omega + \frac{1}{2} [\omega,\omega] = 0$. | |
| S Jun 3, 2021 at 12:56 | history | edited | Praphulla Koushik |
Some added tags.
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| S Jun 3, 2021 at 12:56 | history | suggested | user122276 |
Some added tags.
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| Jun 3, 2021 at 12:13 | review | Suggested edits | |||
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| May 8, 2020 at 19:15 | history | edited | Praphulla Koushik | CC BY-SA 4.0 |
added 328 characters in body
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| May 8, 2020 at 18:49 | answer | added | Praphulla Koushik | timeline score: 0 | |
| May 7, 2020 at 7:06 | comment | added | Adittya Chaudhuri | @PraphullaKoushik Thanks for the notion of integrable connection. Ok I will try to give the answer :). | |
| May 7, 2020 at 5:21 | comment | added | Praphulla Koushik | In this site, at least for my questions, partial answers are welcome... You can add what ever you know.. Not a problem at all.. | |
| May 7, 2020 at 5:19 | comment | added | Praphulla Koushik | A distribution $\mathcal{H}\subseteq TP$ on a manifold $P$ is said to be an integrable distriibution if, for any two vector fields $X,Y:P\rightarrow TP$ that lies in $\mathcal{H}$, in the sense $X(p),y(p)\in \mathcal{H}_pP$ for each $p\in P$, then, the Lie bracket $[X,Y]$ lies in $\mathcal{H}$, in the above sense.. A connection on a principal bundle is given in terms of distribution.. If it is integrable, we call it integrable connection | |
| May 7, 2020 at 5:13 | comment | added | Adittya Chaudhuri | @PraphullaKoushik Can you please define what do you mean by an integrable connection on a principal $G$ bundle? | |
| May 7, 2020 at 5:11 | comment | added | Adittya Chaudhuri | @PraphullaKoushik To make it an answer my explanation should be complete. For example I did not talk about "integrable connections" at all(I do not have much idea about those right now.). So it's better if an expert(which I am definitely not) answer your question with full details so that we both can learn from it :). | |
| May 7, 2020 at 5:05 | comment | added | Adittya Chaudhuri | @PraphullaKoushik Path groupoid can also be treated as a diffeological space (see the definition of path groupoid here ncatlab.org/nlab/show/path+groupoid) | |
| May 7, 2020 at 5:04 | comment | added | Praphulla Koushik | :) See if you can make it as an answer by adding as many details as you can.. | |
| May 7, 2020 at 5:02 | comment | added | Adittya Chaudhuri | @PraphullaKoushik In Generalised smooth space (like diffeologocal space, Chen space..etc) quotient always exist which fails in smooth manifolds. (see arxiv.org/pdf/0807.1704.pdf)Though Path groupoid in general do not have standard manifold structure but can be considered as generalised space (According to jstor.org/stable/pdf/1970846.pdf?seq=1 set of piecewise differentiable paths has a Chen space structure.Now underlying set of Path groupoid is the quotient of set of path(here thin homotopy) ... hence quotient is also Chen space which is an example of gen smooth space.) | |
| May 7, 2020 at 4:35 | comment | added | Praphulla Koushik | @AdittyaChaudhuri I mean to share your understanding of smooth structure, not the link :P I have seen that before... Did not understand much... | |
| May 7, 2020 at 4:28 | comment | added | Adittya Chaudhuri | @PraphullaKoushik Sure! Please checkout this link ncatlab.org/nlab/show/smooth+structure+of+the+path+groupoid | |
| May 7, 2020 at 3:51 | comment | added | Praphulla Koushik | @AdittyaChaudhuri +1 for Theorem $1$ that gives a one-one correspondence between $\mathfrak{g}$-valued $1$-forms on $M$ and the set of connections on the trivial principal bundle $M\times G\rightarrow M$... I do not understand the other half of the theorem that says there is a one one correspondence between the above mentioned set and the set of "smooth" functors $\mathcal{P}_1(M)\rightarrow G$.. I never understood what is smooth structure on $\mathcal{P}_1(M)$.. Do you want to shre your thoughts on smoothness of these functors? | |
| May 6, 2020 at 21:17 | comment | added | Adittya Chaudhuri | Atleast for trivial Principal $G$ bundles a partial answer to your question is mentioned in the Theorem 1 in arxiv.org/pdf/1003.4485.pdf. Also for non trivial Principal $G$ bundles a partial answer can be found in the section "Idea" in ncatlab.org/nlab/show/connection+on+a+bundle where you may find a one-one correspondence(not sure about 1 direction) between connections on Principal $G$ bundles over $M$ and an appropriate subset of functors from Path groupoid of $M$ to the Atiyah Lie Groupoid of the principal $G$ bundle. | |
| May 4, 2020 at 15:46 | comment | added | Praphulla Koushik | @MikeMiller I have checked Kobayashi before asking you.. It was not there.. So, I had to bother you... I see the theorem.... I will read that... Frankly, I do not completely understand the second comment.. I will come back to second comment after reading that theorem 13.2.. Thanks for the reference... | |
| May 4, 2020 at 15:35 | comment | added | mme | Yes, universal cover. The stuff in the first comment is in most references on connections; I am surprised to not see it in Kobayashi-Nomizu vol 1, but see eg Taubes, "Differential geometry: [...]", Theorem 13.2 (here he does not separate the set of flat connections by the topological type of the bundle, but rather just lists off all flat connections on all principal bundles). I am not sure what it would mean to give a reference for my second comment, unless you mean the comment about $U(1)$, in which case try playing with UCT and the LES in homology coming from SES of the coefficient groups. | |
| May 4, 2020 at 15:04 | comment | added | Praphulla Koushik | @MikeMiller any reference regarding your second comment would be useful. | |
| May 4, 2020 at 3:47 | comment | added | Praphulla Koushik | What is $\tilde{M}$ here? Is it the universal cover of $M$? @MikeMiller Can you suggest some reference for “After quotienting this descends to a bijection from the space of flat conns to the subset of $Hom(\pi_1 M,G)$” | |
| May 3, 2020 at 17:56 | comment | added | mme | If your question is how to tell what the topological type of the bundle $\tilde M \times_{\pi_1 M} G$ is given the homomorphism $\rho: \pi_1 M \to G$, you will probably first want to know how to determine the isomorphism classes of principal G-bundles over M, and then to determine it by hand for your representation $\rho$... in particular it's not hard if $G = U(1)$! | |
| May 3, 2020 at 17:53 | comment | added | mme | Connections are affine over $\Omega^1(M;\mathfrak g)$, and in particular form an affine space. Integrable connections are the same thing as flat conns, which are the zeroes of the curvature map $F_A$. Such a map gives rise to a holonomy map $\pi_1 M \to G$. The space of all flat connections on $P$ carries the action of the automorphism group of $P$, and the space of homomorphisms to $G$, the action of conjugation. After quotienting this descends to a bijection from the space of flat conns to the subset of $\text{Hom}(\pi_1 M, G)$ so that $\tilde M \times_{\pi_1 M} G \cong P$ as G-bundles. | |
| May 3, 2020 at 16:54 | comment | added | Praphulla Koushik | Consider the trivial bundle $pr_1:M\times G\rightarrow M$. See that, for each $(m,g)\in M\times G$, the kernel of $(pr_1)_{*,(m,g)}=T_gG$. The assignment $(m,g)\mapsto T_mM$ then defines a connectionon on the bundle $M\times G\rightarrow M$. This is an integrable connection... | |
| May 3, 2020 at 16:54 | history | asked | Praphulla Koushik | CC BY-SA 4.0 |