Timeline for what is $\mathrm{Bun}(G)$?
Current License: CC BY-SA 3.0
21 events
| when toggle format | what | by | license | comment | |
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| Apr 28, 2017 at 15:26 | comment | added | nfdc23 | Strictly speaking, if $V$ is a rank-$n$ vector bundle on a scheme $X$ then it is the sheaf ${\rm{Isom}}(V,O^n)$ (assigning to open $U\subset X$ the set of $O_U$-linear isomorphisms $V|_U\simeq O_U^n$) that is the ${\rm{GL}}_n$-bundle, through the post-composition action of ${\rm{GL}}_n(O_U)$. This defines an equivalence of categories between rank-$n$ vector bundles and GL$_n$-bundles for the Zariski topology on $X$. Likewise, rank-$n$ non-degenerate quadratic spaces are often conflated with ${\rm{O}}_n$-bundles but for this the etale and not Zariski topology is crucial for interesting cases. | |
| Apr 28, 2017 at 15:20 | comment | added | Will Sawin | @johnmangual Yes, the relationship between the original number theory Langlands and the physics Langlands is explained in the second reference I gave you. For examples, look to examples of line bundles and vector bundles in algebraic geometry. Line bundles are $GL_1$-bundles and vector bundles "are" $GL_n$-bundles. The simplest example is the tangent bundle of an algebraic curve. The key point is that they do not admit a connection, but do admit a holomorphic structure. | |
| Apr 28, 2017 at 15:01 | vote | accept | john mangual | ||
| Apr 28, 2017 at 14:54 | comment | added | john mangual | @WillSawin Questions of that begin "What is... ?" are often not useful at this level because we can't say what is an acceptable answer. My point is just that $Bun(G)$ is everywhere -- I would still just like to see one $G$-bundle (even over a scheme); at least in Physics the object you're solving for in Yang-Mills theory is a connection which defines a $G$-bundle. These moduli space problems have been a source of terminal confusion. The language is just impenetrable. | |
| Apr 28, 2017 at 14:51 | comment | added | john mangual | @WillSawin it's very interesting you said math and number theory langlands. There is a physics Langlands [1] , an equally all-encompassing program using supersymmetric gauge theory. These days, having left school, I've been noticing the original Langlands from number theory; it start off about automorphic forms, but ultimately becomes a deep discussion of $p$-adic Lie groups. I have no idea really | |
| Apr 27, 2017 at 16:31 | comment | added | Qfwfq | @dhy: this MO question mathoverflow.net/questions/235064/… persuaded me that even a "drawing" of $\mathrm{Bun}_n(\mathbb{P}^1)$ would be quite messy... | |
| Apr 27, 2017 at 15:18 | answer | added | nfdc23 | timeline score: 42 | |
| Apr 27, 2017 at 14:18 | answer | added | Qfwfq | timeline score: 5 | |
| Apr 27, 2017 at 6:04 | comment | added | dhy | On a somewhat different note, I've always been curious if anyone has drawn pictures of $Bun_G$. | |
| Apr 27, 2017 at 5:39 | answer | added | S. Carnahan♦ | timeline score: 12 | |
| Apr 26, 2017 at 21:36 | comment | added | Will Sawin | @johnmangual I think you might want to reinterpret your question as "Why is the space of $G$-bundles, which I am familiar with from physics, important in number theory at all?" Once you understand that it will be easier to think about which particular spaces of $G$-bundles are relevant, and how they should be viewed. Frenkel attempts to answer this in arxiv.org/pdf/hep-th/0512172.pdf Specifically Lemma 2 of section 2 is where the moduli space is introduced. | |
| Apr 26, 2017 at 21:28 | comment | added | Will Sawin | @sdf Is your issue really of the form "What is $\textrm{Bun}_G$?", or do you seek to understand specific properties of $\textrm{Bun}_G$ with a view towards a particular set of applications? Because those seem like very different questions to me. | |
| Apr 26, 2017 at 19:28 | comment | added | john mangual | @მამუკაჯიბლაძე the topology keeps changing. Yes. I appreciate it. | |
| Apr 26, 2017 at 19:25 | comment | added | მამუკა ჯიბლაძე | For me personally it was helpful to view this $\mathrm{Bun}_G$ as an algebro-geometric counterpart/model of the classifying space of $G$ that is so important in homotopy theory. So these are various forms/analogs of the Grassmanian. In some cases I've seen (e. g. in Beilinson-Drinfeld or Ben Zwi-Frenkel) it really is a(n affine) Grassmanian and can be realized as an ind-scheme. For $G=GL_n$ one, probably oldest and simplest possibility is to consider it as something "almost equal to" $GL_{n+N}/(GL_n\times GL_N)$ or something similar, for "very large" $N$. | |
| Apr 26, 2017 at 19:05 | comment | added | Stiofán Fordham | I think this question is not for the site since it does not fall within the remit. But I'd like to see it stay because I do not know, and would like to see a list of good sources for learning more about it (I have looked previously and had the same problem, namely that the literature is enormous). | |
| Apr 26, 2017 at 18:39 | comment | added | john mangual | @WillSawin I have no idea what I'm doing. I had been watching some recent IHES video lectures of Peter Scholze, which went way over my head. There it was phrased in generality. But I have seen these kind of moduli problems before from mathematical physics, where $X$ is a Riemann surface, $G$ is just a Lie group, and $k = \mathbb{C}$. I'm afraid to admit there is enormous literature on the subject. | |
| Apr 26, 2017 at 18:29 | comment | added | Will Sawin | Why are you interested in this topic? As you have noticed, there is a lot to say about this. It might help to understand what perspective you are coming from. For instance you might only need the case where the base field is $\mathbb C$, or you might only need $G=GL_n$, where one can work with vector bundles, which are conceptually and practically simpler. | |
| Apr 26, 2017 at 18:12 | comment | added | Will Sawin | Have you looked here? users.ictp.it/~pub_off/lectures/lns001/Sorger/Sorger.pdf | |
| Apr 26, 2017 at 18:06 | comment | added | Daniel Barter | As a special case, consider the bundle $\mathbb{R} \times G$. The automorphism group of this thing is the loop group ${\rm maps}(\mathbb{R},G)$ | |
| Apr 26, 2017 at 18:03 | comment | added | Daniel Barter | If you are working over a space $X$, then the points should be principal $G$-bundles over $X$ and the 1-cells should be isomorphisms of principal $G$-bundles. The way to present this object very much depends on which field you are working in / how you want to use it. | |
| Apr 26, 2017 at 17:45 | history | asked | john mangual | CC BY-SA 3.0 |