Talk:Calabi–Yau manifold

Confused with the sentence...
"These conditions imply that the first integral Chern class c_1(M) of M vanishes. Nevertheless, the converse is not true. The simplest examples where this happens are hyperelliptic surfaces, finite quotients of a complex torus of complex dimension 2, which have vanishing first integral Chern class but non-trivial canonical bundle."

Should it really be written "integral Chern class" and not "real Chern class"? It seems to me that the conditions above the sentence should be equivalent to the vanishing of the INTEGRAL first Chern class since in that case c_1(canonical bundle)=-c_1(M)=0 and since line bundles are classified by INTEGRAL c_1 the canonical line bundle would be trivial. Do hyperelliptic surfaces really have vanishing integral first Chern class?
 * I think it is correct. Integral first Chern classes classify smooth line bundles. A holomorphic line bundle can have vanishing first Chern class (i.e. it is smoothly trivial) without being holomorphically trivial. For example on an elliptic curve, every line bundle of degree zero has vanishing first Chern class.JonasEhrhard (talk) 08:45, 25 April 2024 (UTC)

Pop culture
It's really starting to bug me how even a serious page about a difficult mathematical topic has a "... in popular culture" section. These sections add nothing. I don't care if Calabi-Yau manifolds were mentioned in a particular episode of care bears, or in a particular track off of an obscure icelandic death metal band. What could this possibly add to my understanding of Calabi-Yau manifolds? They are, at best, arbitrary factoids introduced by followers of particular fandoms, with no business being discussed in a serious scientific topic. —Preceding unsigned comment added by 131.142.152.101 (talk) 17:34, 23 July 2008 (UTC)


 * I see some cause to challenge most of these on notability grounds since most seem merely to use the term as the buzzword of the day. But I see the "in popular culture" section fulfilling a vital role in the discussion of a topic. Namely it is important to illuminate any significant influences that a concept has on the rest of society. For example, relativity and quantum mechanics both have a profound effect on popular culture. -- KarlHallowell (talk) 16:17, 25 July 2008 (UTC)


 * The section has been removed now... I'm not a great wikipedia expert, so maybe this information is out there but unfindable to me, but was there some reason given? Otherwise I agree with KarlHallowell on the indicative nature of a "In Popular Culture" section. For equality's sake, was the "In Popular Culture"-like section also removed from the for instance Wormhole page? Hmm... no it turns out. In fact the section has its *own* page. Just wondering here. -- Zsejk (talk) 15:07, 11 February 2009 (UTC)

In simple words
Here's my rather superficial understanding or a CY manifold. I am sorry if any of this "simplistic" definition is wrong.

«A Calabi-Yau manifold might look like a folded lasagna pasta in higher dimensions. It is a complicated surface and, when projected in 3D or in 2D, it would appear to twist and swirl and even cross itself. It has a smooth surface (without angles, singularities or holes) but it may or may not have an inside volume (compare a balloon to a coffee cup, for instance).» 207.134.187.165 (talk) 13:50, 16 April 2008 (UTC)

Too technical?
Can someone explain this in a manner that gives examples of how it relates to our world, or give analogies to so the general public can better understand it?
 * I highly doubt it.


 * You might refer to the research project of one, Neal Wadha, who's discovery of an exception to Yau's conjecture won him recognition as a 2005 Intel Science Talent Search Competition Finalist.

You may want to read one of Brian Greene's books. And howabout someone fire up Mathematica and do a projection of the manifold into 3-space so that we can get a nice pretty picture for the page?


 * I read Brian Greene's "Elegant Universe" and I have a fague understanding of the simplest and most general ideas of Calabi Yau shapes, but I do not understand why if (as he states) 11 dimensions (10 space and 1 time) seem to make the equations balance and we live in a 3 expaned dimensional world, that Calabi Yau shapes are persued. 3 dimensions experienced + 6 curled up Calabi Yau dimensions + 1 time dimension only makes 10 dimensions. Is there another type of manifold that has 7 dimensions? Is the remaining dimension expanded? Specifically, if there are tens of thousands of variations on Calabi Yau shapes, how many viable variations exist for 7 dimensional manifolds? -- Alex R April 23, 2006


 * The seven dimensional manifold you are looking for is a G2 manifold. That is, a manifold whose holonomy group is the exceptional Lie group G2.  Just as string theory in 10 dimensions can be thought of as a limit of M-theory in 11 dimensions, there is a way to turn Calabi-Yau threefolds into G2 manifolds. -lethe talk [ +] 22:40, 23 April 2006 (UTC)


 * If 7 curled up it uses the extra one that time does not use because time is a singualrity and not the fourth dimension. [This sentence was improperly inserted into Alex R's remarks, above, by 206.248.255.66 in August 2011. —Tamfang (talk) 21:53, 24 December 2011 (UTC)]

Please sign your posts people. A projection of a CY to 3-space would look like a fairly meaningless blob. Imagine trying to project a sphere or other surface onto a line. And that's only going down one dimension. -- Fropuff 16:24, 24 January 2006 (UTC)

i added the tag to this article because i have no idea what it's talking about. --Someones life 22:26, 4 February 2006 (UTC)
 * And I, in turn, removed it. -lethe talk [ +] 00:29, 5 February 2006 (UTC)
 * Could you please explain why? This article seems ridiculously inaccessible to a general audience. Perhaps you would like to give reasons why you think this article is unavoidably technical, or maybe add a list of prerequisites for understanding this article. If, on the other hand you think this article is already accessible to a general audience, as someone with more than a high school education and understands nothing about this article, i would have to dissagree. --Someones life 19:41, 5 February 2006 (UTC)
 * The link you post, Make technical articles accessible states that "you should put an explanation on the talk page with comments or suggestions for improvement". You didn't do that, so the presence of a technical tag was completely useless.  People putting technical tags on templates without any effort at all to improve the article is a pet peeve of mine.  I happen to think that this article can't really be made much more accessible without embedding it in a one semester course.  However, if you had specific problems, it might be possible to address them.  Instead, all you said is you had no idea what the article is talking about. Well, doesn't help anything.  -lethe talk [ +] 01:14, 6 February 2006 (UTC)
 * Ok, I can see where you're coming from, but how can i even try to improve this article when i barely even know what it's about?... something about multiple dimensions and string theory is all i can extract. I'll try to be more specific about what i think could be improved on by someone who knows what this is all about. There are many technical terms that are used bluntly as though one is expected to be familliar with them, whithout saying what they mean or how they apply to whatever a Calabi-Yau manifold is. Basically I think the article should try to have more of an explanation of what it is, rather than definition and examples. If, as you say, you think it would take far too long to sufficiently explain what this thing is about, why not take my suggestion of applying the prerequisites template so that people who want to learn at least know where to start? --Someones life 05:03, 6 February 2006 (UTC)
 * Well, since I got no response, and i still think the article needs to be fixed, and i've fully explained why, i'm putting the technical tag back on.--Someones life 16:56, 8 February 2006 (UTC)
 * So a Calabi-Yau is a special kind of Kähler manifold. Would you be satisfied if we explained what a Kähler manifold is?  In my opinion, a layman's description of a Kähler manifold cannot be done in a few words, and we already have the link there.  I might be willing to write a paragraph attempting at a layman's description, but right now, the article is only 3 paragraphs long; the layman's description would account for a full 25% of the article, which I'm not comfortable with.  Moreover, I'm fairly convinced that no one would find it useful.  Not the layman, not the expert.  It would look something like this:


 * A Calabi-Yau manifold is a particular type of Kähler manifold, which is a manifold which carries a complex structure and a Hermitian structure which are compatible in each other in a technical way. For a Kähler manifold to be Calabi-Yau manifold, its curvature form must be an exact differential, which also implies that the trace of its curvature form vanishes, according to a famous result by Calabi and Yau.


 * I see problems with this attempt. For the layman: if I don't know what a manifold, complex structure, Hermitian structure, exact differential, trace or curvature are, then this is still completely inaccessible.  For person who has the prerequisites, and is now trying to learn about Calabi-Yaus, these descriptions are far too short to be of use, they're too imprecise.  In short, I can't imagine saying anything that would make this article more approachable without simply copying and pasting definitions of all the hyperlinked words.  Last week I think I really helped make the intro for Hilbert space more accessible, because a Hilbert space is really just an infinite dimensional version of the inner product vector space that we all learned in high school.  But Calabi-Yaus don't (as far as I know) admit such a description.  I'm going to leave the technical notice though.  While I personally don't think much can be done, and therefore don't think it's appropriate, maybe someone else will disagree.  -lethe talk [ +] 18:31, 8 February 2006 (UTC)


 * I agree with Lethe that this article is unavoidable technical; there just isn't a good way to explain the concept to someone unfamiliar with manifolds, let alone Kähler manifolds. The problem is that layman do hear about the concept because of its connection to string theory. No one complains that the article on Kähler manifolds is too technical because no layman ever looks at it. We could try an intro along the lines of
 * A Calabi-Yau manifold is a special type of manifold that shows up in certain branches of mathematics such as algebraic geometry, as well as in theoretical physics. Particularly in superstring theory, the extra dimensions of spacetime are sometimes conjectured to take the form a 6-dimensional Calabi-Yau manifold. The precise definition of a Calabi-Yau manifold, given below, is somewhat technical.
 * Not that that really explains anything, but it might ward off complaints. -- Fropuff 20:57, 8 February 2006 (UTC)


 * I agree with you, Lethe, that your attempt would still not help the layman, i didn't understand it. You're right about the fact that all the hyperlinked words would need to be defined for the reader in order for them to understand this article, and I have no problem with that in itself, thats how wikipedia works. The problem i have is that all those links go to articles which are equally unintelligeable to the average user. I think Fropuff's suggestion is exactly the kind of thing that would help, especially because the article currently links only to Kähler manifold, which, in turn, links to Complex manifold, which still doesn't make much sense to me, but links to manifold, as Fropuff did, which finally gave me some inkling of what a Calabi-Yau manifold is. I just dont think its fair for people to have to navigate a maze of links just to figure out what an article is about. --Someones life 17:58, 9 February 2006 (UTC)


 * Actually I think Fropuff's version is pretty good for an intro. Although I really dislike the vacuous "The precise definition of a Calabi-Yau manifold, given below, is somewhat technical." Of course it is given below. Technicality is relative/POV. A prerequisite for this is that manifold itself remains general and non-technical enough and doesn't effectively become topological manifold. --MarSch 11:31, 24 April 2006 (UTC)

since the article was first tagged, edits have added an introductory paragraph, a diagram, more details in the definition, additional examples, and more links. is it time to remove the "too technical" tag? Lunch 03:50, 24 September 2006 (UTC)

It's difficult to believe that "In mathematics, Calabi–Yau manifolds are complex manifolds that are higher-dimensional analogues of K3 surfaces. They are sometimes defined as compact Kähler manifolds whose canonical bundle is trivial..." is considered a satisfactory general introduction to this subject. I suspect it might improve if the technically competent contributors started by giving a brief explanation as to how a Calabi Yau manifold helps mathematicians and cosmologists, rather than start off with a technically-dense definition. When reading about something you don't understand, every element of the explanation which itself is not understood causes a mental 'trip'; after enough trips you fall over. In the section I quoted I count five trips - and I already had a vague idea of what they are!87.194.231.27 (talk) 13:03, 20 July 2011 (UTC)

Taking Fropuff's suggestion, which I think is very useful, I have edited the introduction to the article. --Volons (talk) 11:11, 30 October 2011 (UTC)

The compactness requirement
A Calabi-Yau (C-Y) manifold isn't required to be compact (it's probably better to say that compactness is an optional part of the definition). Otherwise, it makes little sense to discuss compactification of C-Y manifolds as is done in the "Applications in String Theory" section. If the manifold is compact, then it is its compactification. -- KarlHallowell 20:04, 15 February 2006 (UTC)


 * Look at this preprint, Topological Strings and Their Physical Applications, by Andrew Neitzke and Cumrun Vafa (see pp. 3-4). They replace the conditions of compactness and the vanishing of the first Chern class with the condition that the manifold admits a global, nonvanishing, holomorphic n-form. It coincides with with our definition when the manifold is compact. -- KarlHallowell 20:21, 15 February 2006 (UTC)


 * Still seems to be a problem, but perhaps not a significant one? In string theory, Calabi-Yau manifolds aren't necessarily compact (I see the compactification comment still is there), but I gather they are in mathematics proper. -- KarlHallowell 02:22, 10 March 2007 (UTC)

In fact, smoothness (implied by the term "manifold") is not necessary for applications in higher-dimensional theories such as string theory. As best known, certain mild singularities are permissible but all the precise bounds on the severity of singularizations is not. From the "pure mathematics" side, the inclusion of spaces with such mild singularities is also desirable, since then the whole collection of Calabi-Yau n-folds, for any particular dimension (= n), admits a more universal approach of analysis. For Calabi-Yau 3-folds, Miles Reid conjectured [The Moduli Space of 3-Folds with K=0 may Nevertheless be Ireducible. Math. Ann. 278 (1987)329] that the (possibly infinitely many topological types of) Calabi-Yau 3-folds could nevertheless be deformed one into another, while passing only through certain mild singularizations.Tristan 13:08, 18 March 2007 (UTC)

Historical Motivation
It think this entry would be enriched if the reason that led Calabi to define Calabi-Yau manifolds from his study of Kahler structures is added.


 * I know little of the historical motivation for these manifolds. Do you have any sources you can recommend? -- KarlHallowell 19:29, 3 August 2006 (UTC)

I don't know any reference for this; I actualy visited this entry with hopes to find it out.

Perhaps they were introduced as a class of manifolds that generalize K3 surfaces? Since so much is known about K3 surfaces (which are compact surfaces with trivial canonical line bundle and trivial first Betti number) it seems a good idea to study a generalization of them to the study of (algebraic or at least Kahler) compact n-folds with trivial canonical line bundles (Betti = 0 only means that the space is not a product of some simpler spaces and there are some definitions of Calabi-Yau manifolds that assume that this number is zero). In any case, the two definitions are so much alike that it's hard not to assume that Calabi-Yau manifolds should actually be called K3 n-folds! =)
 * Shing-Tung Yau and Steve Nadis; The Shape of Inner Space, Basic Books, 2010. The book is as much about the motivations and historical development as the manifolds themselves. &mdash; Cheers, Steelpillow (Talk) 14:55, 18 November 2015 (UTC)
 * Calabi conjectured, as a matter of pure geometry, that certain manifolds manifesting a Ricci curvature could be given a metric having zero curvature, i.e. their geometry would be Ricci-flat. Yau published a refutation, which was widely accepted at the time, but Calabi raised questions to Yau, who subsequently came to realise that the conjecture was after all true. His proof of the Calabi conjecture was therefore also the proof that Calabi-Yau manifolds existed. Ricci flatness is therefore a foundational defining cgharacteristic of these manifolds. I'll have to revisit the book to check exactly which class of surfaces the proof starts from, certainly something to do with Kahler manifolds. Calabi and Yau were intuitively aware of the implications to physics, but it took some time for physicists to discover their work. &mdash; Cheers, Steelpillow (Talk) 15:12, 18 November 2015 (UTC)

Formal definition
As currently stated, the last sentence in the paragraph defining Calabi-Yau manifold directly contradicts the supposedly equivalent definition by the existence of a holomorphic n-form in the second paragraph. Arcfrk 05:35, 26 March 2007 (UTC)

Free statement
The following statement in the leading paragraph: "Physical insights about Calabi-Yau manifolds, especially mirror symmetry, led to tremendous progress in pure mathematics." isn't developed in the rest of the article (or at least I cannot extract it from the jungle). What were those insights? Why did they have such an impact? I find it very interesting and would be pleased if it is explained. --euyyn 17:23, 22 May 2007 (UTC)

Phonetic
Does anyone know what the proper way to say Calabi-Yau is. Could a phonetic spelling be included? 74.12.128.198 20:24, 23 July 2007 (UTC)
 * Eugenio Calabi's name is pronounced with the vowel patter of "wasabi", with the stress on the second syllable; Shing-Tung Yau's name is pronounced to rhyme with "how". --Tristan 03:54, 5 July 2008 (UTC)

WikiProject class rating
This article was automatically assessed because at least one WikiProject had rated the article as start, and the rating on other projects was brought up to start class. BetacommandBot 09:45, 10 November 2007 (UTC)

SU(m) structure group
Could someone point to a reference that shows that the reducibility of the structure group is equivalent to the other statements in the definition (in particular to the one that defines a calabi yau as having a trivial canoninical bundle). Or should this be obvious? —Preceding unsigned comment added by 145.97.232.199 (talk) 19:35, 16 December 2008 (UTC)

merger proposal
I agree with merging Calabi–Yau four-fold with calabi-yau manifold. Jakob.scholbach (talk) 12:33, 18 December 2008 (UTC)

Dimensions
According to Shing-Tung Yau himself, along with Steve Nadis, in The Shape of Inner Space, Basic Books, 2010, Page 176, the humble 2-torus is the simplest Calabi-Yau manifold and the K3 surfaces are 4 (real) dimensional examples. I think this needs adding to the article and the lead modifying accordingly. But I'd like to clarify one thing first. The article section on different definitions is interminable and I am sure that many of these are simply relative to the topic of the paper concerned and are not meant to be global definitions. But do any global definitions refute the idea that Calabi-Yau manifolds exist in all even numbers of dimensions? &mdash; Cheers, Steelpillow (Talk) 14:51, 18 November 2015 (UTC)

Assessment comment
Substituted at 10:41, 29 April 2016 (UTC)

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