The following is an archived discussion of a featured article nomination. Please do not modify it. Subsequent comments should be made on the article's talk page or in Wikipedia talk:Featured article candidates. No further edits should be made to this page.

The article was promoted by GrahamColm 23:51, 13 April 2014 (UTC) [1].[reply]

---

Mirror symmetry (string theory)[edit]

Mirror symmetry (string theory) (edit | talk | history | links | watch | logs)

• Featured article candidates/Mirror symmetry (string theory)/archive1

Toolbox
Analysis Citation bot External links

Nominator(s): Polytope24 (talk) 04:48, 7 March 2014 (UTC)[reply]

Mirror symmetry is an interesting example of an idea in theoretical physics that has had a significant impact on pure mathematics. This article was promoted to GA status back in September, and I've done a lot of work since then to bring it to FA status. Polytope24 (talk) 04:48, 7 March 2014 (UTC)[reply]

• Support, in light of the work done on the article since its nomination. Ozob (talk) 02:54, 20 March 2014 (UTC) Comment. While I enjoyed reading this article, I'm a professional mathematician, and sentences like, "Then the area element of the torus is ${\displaystyle R_{1}R_{2}\,dx\,dy}$ which integrates to ${\displaystyle R_{1}R_{2))$ on the unit square" are perfectly clear to me. That's not so for the average reader. The average reader finds mathematics at best unintuitive; at worst they find it terrifying, even if they think that physics is fascinating. In particular, most laymen are intimidated by equations. Stephen Hawking, when he wrote A Brief History of Time, was told by his editor that every equation halves the number of people who will read the book. There was a nice article by Steven Strogatz in the Notices of the AMS recently on writing mathematics for the general public which discussed this same issue in depth. So while I think that the article is very good for someone with some technical background, I am not sure that others will find it so appealing. Ozob (talk) 23:29, 9 March 2014 (UTC)[reply]

Too, too, true. Even if, as I expect is the case here, the topic is never going to be understood by those without some advanced maths, there should at the least be a summary in the lead that bends over backwards to explain in lay terms roughly where the topic fits in, and what those who do understand it use it for, or in. I can see an effort has been made here, but it does not really work. The lead seems generally too short - Enumerative geometry (which seems a bit simpler to grasp as a concept) gets a long section below, but is not mentioned. Johnbod (talk) 00:51, 10 March 2014 (UTC)[reply]

Thank you both for your comments. I have modified the lead slightly, following Johnbod's suggestion, to explain the idea of enumerative geometry. Please let me know if this is an improvement.

About the level of mathematics, I'm not sure what I can do exactly. The current article does require a little knowledge of complex numbers and calculus, but this is already way, way less background than you would really need to understand mirror symmetry. A proper treatment of mirror symmetry would require very advanced ideas from complex algebraic geometry, symplectic geometry, and homological algebra, not to mention very advanced knowledge of quantum field theory and string theory.

Somehow I would like to include a precise discussion in the article without assuming all this background. The approach taken here seemed appropriate because it comes from another encyclopedia entry by Eric Zaslow in "The Princeton Companion to Mathematics". I welcome any specific ideas about how I might make the mathematical parts of this article more accessible. Polytope24 (talk) 06:15, 10 March 2014 (UTC)[reply]

Well, let me suggest some changes. Consider the Overview section. It starts off with a subsection called Idea, and this section attempts to explain where mirror symmetry might come from. The Idea subsection can be caricatured as:

1. Duality is a thing relating two theories
2. String theory requires extra dimensions
3. Extra dimensions are eliminated by compactification
4. There are two Calabi–Yau manifolds
5. Mirror symmetry is interesting to mathematicians

This is oversimplified, and not exactly what the article says, but it's what I think a lay reader will get out of it. Imagine that someone who has no prior knowledge of mirror symmetry comes to this article and reads this subsection. They are looking for The Story of Mirror Symmetry. They stop near the beginning of the next section when they hit the first equation in the article, ${\displaystyle \omega _{1}/\omega _{2))$ (technically it's not an equation because there's no equals sign, but those weird w's are intimidating nonetheless). The story that they get is the one I outlined above, and—to be honest—it's going to be opaque to them. It starts off with abstract nonsense (duality), continues with five-dimensional unicorns (extra dimensions), then a meaningless long word (compactification), something named after some Calabi-Yau fellow, and then bland assertions that mathematicians think it's important. I think the article presupposes background in mathematics or physics that the general reader will not have.

I would suggest greatly expanding this section, roughly along the following lines (I apologize for any errors; I'm a mathematician, not a physicist):

1. String theory is a model of the physical world where elementary particles like protons and electrons are modeled by tiny one-dimensional strings.
2. String theory in four dimensions is inconsistent, but string theory is also the best available model for quantum gravity.
3. If we add extra dimensions to string theory, then it's not obviously inconsistent.
4. In order to make these extra dimensions accord with both everyday experience and with existing experimental verification of general relativity and quantum physics, they must not show up in any experiment we can do right now.
5. One way of doing this is if the extra dimensions close up on themselves. If they close up on themselves and are too small for our equipment to detect, then they will be consistent with prior experiments.
6. Not all ways of having these extra dimensions close up on themselves have the necessary physical properties. The extra dimensions must be shaped like something called a Calabi–Yau manifold (named after Eugenio Calabi and Shing-Tung Yau).
7. Even if string theory with the aforementioned extra dimensions is a correct model for the physical world, nobody knows which Calabi–Yau manifold is the right model for the physical world. This requires experimental observations that we cannot currently do.
8. Furthermore, there is more than one way to create a string theory from a choice of Calabi–Yau manifold. Two ways of doing so are called the A-model and the B-model.
9. However, it turns out that the choice of an A-model or a B-model is irrelevant. An A-model on a Calabi–Yau manifold is the same as a B-model on a different but related Calabi–Yau manifold. This related manifold is called the mirror of the original Calabi–Yau manifold. Mirror symmetry is the study of how A-models on a Calabi–Yau manifold relate to B-models on its mirror.
10. Mirror symmetry is a type of duality, meaning that the mirror of a mirror is what you started with. There are many kinds of dualities in physics.
11. Mirror symmetry also has mathematical consequences such as enumeration of rational curves.
12. In physics, mirror symmetry is justified on physical grounds. Mathematicians want to be able to prove the existence of mirror symmetry using pure logic, with no appeals to physical necessity. In some but not all cases this has been done.

Now, that's much, much longer; it's a lot to write, and a lot to read. But it can be done without any mathematics whatsoever, and it can be done so that the reader understands The Story of Mirror Symmetry. In addition, a reader who has the background to understand some or all of the mathematical details will be much better equipped for the rest of the article. That said, I think that this treatment applies to many of the later sections: They can be written using less formal mathematics and more prose and pictures. For example, complex structures on a torus can be described as measurements of the sizes of the torus's holes (since that's most of what the period lattice is capturing). In some ways the symplectic geometry section already does this, since it talks about area, but the discussion is in terms of a volume form ${\displaystyle \rho \,dx\,dy}$, and that kind of language foreign to many potential readers. The more that the article relies on prose, the more accessible it becomes.

I think a good example in this regard is homotopy groups of spheres, which explains a very advanced mathematical topic to a general audience. Eventually, the article becomes more technical and less accessible, but for a while at the beginning the reader needs very little background. Since you're aiming for mirror symmetry (string theory) to be an FA you'll have to aim even higher and correspondingly make the descriptions even simpler. Another excellent article is group (mathematics); it's written at a level where anyone interested in recreational mathematics should be able to follow (and the more technical stuff is in group theory). I don't pay much attention to physics articles, so I don't know whether there are any good examples like this among the physics articles; perhaps yours will be the first! Ozob (talk) 03:47, 11 March 2014 (UTC)[reply]

Thank you for these helpful suggestions, Ozob. I just finished making some major revisions to the article. I have significantly expanded the Overview section following your suggestions, and I have also collected the mathematical portions of the article in a separate section that readers can skip if they do not have the background. I don't really see how I can remove any of the math while still explaining what I wanted to explain (the relationship between complex and symplectic structures of mirror manifolds), but I made a serious effort to simplify the exposition and make the mathematics less dense. Please let me know if there's anything else I can do. Polytope24 (talk) 06:55, 12 March 2014 (UTC)[reply]

It works a lot better for me, but I will wait for expert comments. As I said above, I never expected to be able to understand most of the article, but the lead and overview now seem to explain pretty clearly where it "fits in" (to quote myself above). Thanks for the prompt and extensive rewriting. Johnbod (talk) 12:42, 12 March 2014 (UTC)[reply]

Thanks. I'll wait for Ozob to chime in here, but I just wanted to say that if the mathematical sections of the article are still a problem, I have an idea about how to remove most of this material and expand other sections of the article, possibly making it more accessible. Polytope24 (talk) 02:55, 13 March 2014 (UTC)[reply]

I like what you've done with the introduction, and what I want to say is: More! I think you could give the entire rest of the article the same treatment. The geometry of tori, for example, is very visual, and I believe that you can give a treatment that is accessible to everyone. And similarly for symplectic manifolds because they're so closely tied to classical mechanics. I think the article's progressing very well, but it's still not done. Ozob (talk) 03:49, 13 March 2014 (UTC)[reply]

Okay, I made a bunch of changes to the article, and I think it's much more accessible. The content is basically the same, but I've removed all the technical mathematics and described these concepts more intuitively in other sections of the article. Please let me know what you think. Thank you. Polytope24 (talk) 01:48, 14 March 2014 (UTC)[reply]

Very good! The article is much, much more accessible. I have a few more comments, though:

1. The overall layout feels odd. The history seems like it should be closer to the start (often a historical introduction to a subject reads well), and the applications seem like they should be further down.
2. Maybe I missed it, but I don't think the article says clearly anywhere that constructing mirrors is still conjectural. My understanding of the state of affairs is that it's much better understood than it used to be (in particular I think Mark Gross has some nice work on this – by the way, the article links to the wrong Mark Gross), but there are still cases where it's somewhat mysterious.
3. Also, now the article has no equations whatsoever. While I think the article is better now than it was before, it might be that it's gone too far towards accessibility and now does not have a full and complete description of the subject. Take a look at Emmy Noether, for example: It gives the reader a look not only to Noether herself but also at her work. Some parts of it are inaccessible without some background, but those parts are relatively few, and they often come together with simply descriptions.

Despite my suggestions, I'm running out of feedback for the article. I think it's now at the point where it would be good for some non-technical people to look at it. Ozob (talk) 03:50, 14 March 2014 (UTC)[reply]

Thank you so much for your comments, Ozob. I think the article has improved significantly as a result of your input. I've made some changes to the to address your points 1 and 2. The history section is now the second section rather than the last.

I'm not sure what to do about your third point. Mirror symmetry is not a subject where you have one big central equation, and my previous attempts to include math in the article involved too much handwaving and circumlocution. I considered adding more information about Hodge numbers of mirror manifolds, but this topic is only tangentially related to the actual mirror duality, and I would need a lot of space to explain it in a reasonably accessible way.

Actually, I think the article looks much better now that I've changed the order of the sections. The previous version had the applications section right after the overview, and this made it seem like something was missing from the article. Please let me know what you think, and tell me if you have any specific ideas about what sort of technical details I should be including in the article. Thanks. Polytope24 (talk) 03:22, 15 March 2014 (UTC)[reply]

Well, since I'm not an expert in mirror symmetry, I don't know what kind of technical information is important to include and what isn't. My feeling about the current version is that it stops just shy of giving any details because those details are hard to make accessible. For example, I don't know how one explains derived categories to non-specialists. But if the article is really going to try to explain homological mirror symmetry, then it needs to have some kind of description of what the derived category is, what the derived categories of coherent sheaves and of the Fukaya category look like, why these are expected to provide interesting information, and how they're expected to be related. Right now, the article starts down this road, then quickly stops. There is much more to say about homological mirror symmetry, and the reader can see that, but the article avoids it. Until the article really wrestles with details and makes the details accessible (if not to a complete layperson, then to someone with very minimal training in physics and mathematics), then the article isn't complete. This is a very hard thing to do, but the article isn't FA quality without it. Ozob (talk) 04:51, 16 March 2014 (UTC)[reply]

Personally I'm perfectly happy if a maths article becomes completely incomprehensible to me in the lower sections, so long as I feel the broad topic has been explained in terms a lay person can just about understand at the start. There should be enough mathematical detail to keep mathemeticians, who are always likely to be the main readership, happy, but it doesn't need to be (and probably can never be) comprehensible to the lay reader throughout - I for one am content to drop out after a certain point. Johnbod (talk) 05:09, 16 March 2014 (UTC)[reply]

I may be overstating the need for accessibility in later sections of the article. I don't think we can seriously expect the article to make derived categories accessible to the layperson (I remember struggling with them myself). But the article shouldn't omit these topics entirely. Ozob (talk) 16:31, 16 March 2014 (UTC)[reply]

Okay, I expanded the section on homological mirror symmetry. It's should still be accessible to a broad audience, but now it gives a more detailed description of the two categories. Polytope24 (talk) 00:04, 17 March 2014 (UTC)[reply]

Much better! I object, however, to the last sentence: There are well-known links between symplectic and complex geometry. These are best explained by Unitary_group#2-out-of-3_property.

Looking over the article again, I realized that I don't really know what mirror symmetry is good for in physics. I get the impression from the article that it helps in doing calculations; why? Why should looking at a mirror manifold be any easier than looking at the original manifold? Ozob (talk) 03:32, 17 March 2014 (UTC)[reply]

I made some changes to address your comments. Polytope24 (talk) 04:06, 18 March 2014 (UTC)[reply]

I'm getting the impression that mirror symmetry appears in physics as a kind of technical tool; is there physical intuition for why it should sometimes be easier (or harder) to work on the mirror manifold? Ozob (talk) 14:14, 18 March 2014 (UTC)[reply]

It's a useful technical tool in physics because it simplifies things mathematically. I'm not sure if there is a more intuitive reason for this.

I think you're assuming that the change of manifolds is what simplifies the calculation when in fact it's the change of theories. Let me know if there's a way to clarify this in the article. Polytope24 (talk) 15:30, 18 March 2014 (UTC)[reply]

You're right, I am. I guess I don't really understand the difference between type IIA and type IIB string theories and therefore I don't understand why switching theories would be significant. The article doesn't currently explain how either of these theories work, so we can't expect the reader to understand why mirror symmetry would be physically important. I guess it would help if the article tried to address this as part of the overview section. Then, later in the article, it could sketch how the correspondence between theories on mirror pairs works. That would not only avoid readers getting misconceptions (as I did) but also make the article more comprehensive. Ozob (talk) 01:07, 19 March 2014 (UTC)[reply]

Okay, I made some changes to clarify these points. Polytope24 (talk) 19:45, 19 March 2014 (UTC)[reply]

Ah, OK! This gives me a much better sense of why mirror symmetry would be useful and physically relevant.

At this point, I'm out of criticism for the article; I don't know how to make it better. I'll mark myself as supporting this article's nomination. Ozob (talk) 02:54, 20 March 2014 (UTC)[reply]

Thanks for all your help! I'm really happy with the improvements we've made! Polytope24 (talk) 03:02, 20 March 2014 (UTC)[reply]

• Support Now ready. Near to support After the responses to my comments above, and Ozub's change to support, I'm very near to support. I think the comments just below are sensible, and I also wonder if the 6 "see also"s can easily be reduced by mentioning some in the text (I hope none are already mentioned). Johnbod (talk) 18:08, 27 March 2014 (UTC)[reply]

Hi Johnbod. The topics linked in the "See also" section are actually not part of mirror symmetry per se. What I've tried to do here is collect links to a handful of other examples of ideas in modern theoretical physics that have influenced the development of mathematics. These would presumably be of interest to readers interested in mirror symmetry. Please let me know if this is an appropriate use of the "See also" section. Polytope24 (talk) 20:26, 27 March 2014 (UTC)[reply]

@Polytope24: My understanding of ALSO is that it probably isn't an appropriate use: "The links in the "See also" section should be relevant, should reflect the links that would be present in a comprehensive article on the topic ..." Of course, a featured article is supposed to be comprehensive. If you do keep some links: "Editors should provide a brief annotation when a link's relevance is not immediately apparent, when the meaning of the term may not be generally known, or when the term is ambiguous." RockMagnetist (talk) 20:59, 27 March 2014 (UTC)[reply]

Yes, FAs, at least when passed, tend to have 3 SAs or fewer. It might be better to string a sentence together about "ideas in modern theoretical physics that have influenced the development of mathematics" with them as examples. Johnbod (talk) 21:17, 27 March 2014 (UTC)[reply]

Thanks for the explanation. I went ahead and removed several of the links. I left only the ones that are rather closely related to mirror symmetry but not closely enough to include in a survey like this article. Polytope24 (talk) 02:09, 28 March 2014 (UTC)[reply]

Thanks for your help, Johnbod! Polytope24 (talk) 02:49, 30 March 2014 (UTC)[reply]

• Comments - I am a physicist, but I have had little exposure to string theory, so my comments will be mainly about readability. Overall, this is a well-written article, and I congratulate Polytope24 for doing a great job of making it accessible. However, there are some stylistic problems of the sort that tend to creep into mathematical writing. Some I fixed myself, but there are a few that I didn't feel BOLD enough to fix. For now, I have only looked at the Overview:

• Consider making the first section heading String theory instead of Overview.
• Per Manual of Style, the section headings generally should not repeat the article title. I'm wondering whether Homological mirror symmetry could be replaced by Homology.
• 'the manifolds are said to be "mirror" to one another' is ugly. Would mathematicians object if it were replaced by "the manifolds are said to mirror one another"?
• The garden hose analogy is a bit puzzling, since the hose is actually three-dimensional. Is the ant really "inside" or is it restricted to the surface?
• The paragraph on duality: It's jarring to start discussing "physical systems" and switch to "theories", unless you explain why they are the same thing. Also, there is some flab here. How about replacing

If two theories are related by a duality, it means that one theory can be transformed so that it ends up looking just like the other theory. The two theories are then said to be dual to one another under the transformation.

by

If one theory can be transformed so it looks just like another theory, the two are said to be dual under that transformation.

And duality needs disambiguating.

• "In the late 1980's, it was noticed" needs to be replaced by something more definite (i.e., the names of the noticers).
• "In the context of topological string theory, mirror symmetry states that two theories, the A-model and B-model, are equivalent in a certain precise sense." This doesn't seem to add any new information to the preceding text.

RockMagnetist (talk) 18:18, 27 March 2014 (UTC)[reply]

• In History, three more instances of "it was noticed". RockMagnetist (talk) 18:34, 27 March 2014 (UTC)[reply]

Thank you so much for the review. I'll try to revise the article later today. Polytope24 (talk) 20:18, 27 March 2014 (UTC)[reply]

You're welcome. The illustrations are gorgeous, by the way. I moved a couple to the right because I think they look better there, but I don't mind if you move them back. RockMagnetist (talk) 21:04, 27 March 2014 (UTC)[reply]

Okay, I've made a bunch of changes to the article. Here's a summary of what I did to address your points:

1. I changed the headings in the overview section to include a reference to string theory.

2. I don't see any way of abbreviating the term homological mirror symmetry. I hope that's okay. This is a phrase that appears in all the literature on the subject. Also, the term "homology" is only very indirectly related to homological mirror symmetry.

3. Changed "mirror" to mirror.

4. Fixed garden hose analogy.

5. Revised the paragraph on duality following your suggestion.

6. Credited Cumrun Vafa and others with the discovery of mirror symmetry.

7. Tweaked the sentence on topological string theory to emphasize that I'm introducing two new terms here: the "A-model" and "B-model".

Polytope24 (talk) 02:08, 28 March 2014 (UTC)[reply]

Most of these changes look good. However, my third point had to do with grammar, not typography - "to mirror" would be better than "to be mirror to". It's something I have noticed about the use of mathematical terminology: often, a definition that makes grammatical sense in its original context is grafted without change into a sentence where it is not good grammar. Also, for point 7 it might help to have a link to the section in which they are described. RockMagnetist (talk) 05:09, 28 March 2014 (UTC)[reply]

I added the links you requested and changed the terminology slightly. It would be a little odd and unconventional to say the manifolds mirror one another, so instead I wrote that they are mirror manifolds. Polytope24 (talk) 05:56, 28 March 2014 (UTC)[reply]

• Comments on Homological mirror symmetry:

• Is it correct to refer to a Calabi-Yau manifold as "a Calabi-Yau"?
• I'm wondering whether the description of categories is complete. Is the morphism a structure-preserving mapping between branes? If so, what structure is preserved? Is the "state" of an open string entirely determined by the endpoints, or is it something more?

RockMagnetist (talk) 22:09, 27 March 2014 (UTC)[reply]

People often refer to Calabi-Yau manifolds as Calabi-Yaus, but I can change this in the article if you like. As for the morphisms in the derived category, I have worded the explanation such a way that it is technically accurate without going into too much detail. The morphisms in this category are indeed structure-preserving maps; they are obtained from morphisms of sheaves. The set of all morphisms between two branes has the structure of a vector space, which physicists regard as the state space for strings stretched between the two branes. If you like, I can explain this in the article, but I think it's probably best to leave it as it is. Polytope24 (talk) 02:19, 28 March 2014 (UTC)[reply]

I'll defer to your judgement whether to say "Calabi-Yau manifold" or "Calabi-Yaus". I just thought I'd check on that. RockMagnetist (talk) 05:41, 28 March 2014 (UTC)[reply]

As for your detailed explanation of morphisms - I think you should put it in. You're near the end of the article, and this is by far the most difficult section (in fact, it probably should be the last section). It's difficult because it is loaded with abstract concepts, and I'm sure a reader who made it this far would appreciate anything that you could do to make it more concrete. RockMagnetist (talk) 06:15, 28 March 2014 (UTC)[reply]

• Comments on Strominger-Yau-Zaslow conjecture:

• I removed the figure that just pictures a torus, because (though pretty) it is redundant and forces you to specify which picture of a torus you're referring to in the text.
• "says how these circles are arranged" - "determines" maybe? Also, aren't there an infinite number of choices for B? Later you mention an "auxiliary space", so a bridge to this concept would help.
• In the second last paragraph, does applying T-duality amount to mapping the red and pink circles onto each other? If not, is there some other way of picturing the dual torus?

RockMagnetist (talk) 05:41, 28 March 2014 (UTC)[reply]

I tweaked the language in this last section to hopefully address your concerns. The point about T-duality is that all of the longitudinal circles are inverted, giving a new torus which is "fatter" or "skinnier" than the original. I hope my revision makes this clear. Polytope24 (talk) 06:08, 28 March 2014 (UTC)[reply]

• General comment - I think this will be my last comment. This article is looking really good, and I'm ready to support it. However, there are still several places where you use the words "some", "many" or "X are certain Y", and these are often vague. If you can replace any of them with more concrete statements, the prose would be more crisp (as in User:Tony1/How to improve your writing). RockMagnetist (talk) 06:20, 28 March 2014 (UTC)[reply]

Fixed. Polytope24 (talk) 19:31, 28 March 2014 (UTC)[reply]

• Support - I think it meets all the featured article criteria. I don't have the expertise to say whether it is comprehensive, but there are no obvious gaps, and Polytyope24 has done an excellent job making it comprehensible to a non-expert. RockMagnetist (talk) 21:28, 28 March 2014 (UTC)[reply]

Thank you for all your help, RockMagnetist. Polytope24 (talk) 07:27, 29 March 2014 (UTC)[reply]

• Comments Sorry, Polytope24, for being late to the review process. All your hard work with Ozob and RockMagnetist has paid off, as the article is very readable given the mathematical sophistication of the topic. From a content point of view I think I am very near full support for FA status. Just a couple of comments.

1. In the context of comprehensiveness, in Strings and compactification section you mention superstrings, but nowhere else in the article. The rest of the article talks about Calabi-Yau manifolds as seemingly ordinary varieties with e.g., their defining polynomials containing no Grassmann variables. For superstrings, are there super-Calabi-Yau manifolds? Do these supermanifolds have supermirror symmetry? It would be super if you could clarify this.
2. In the Theoretical physics section, it is said, "mirror symmetry can be used to understand properties of gauge theories (a class of highly symmetric physical theories appearing, for example, in the standard model of particle physics). Such theories arise from strings propagating on a nearly singular background, and mirror symmetry is a useful tool for doing computations in these theories.[45]" But reference 45 doesn't really support this assertion. In chapter 36 of the Hori et.al tome, there is no mention of mirror symmetry being applicable to the standard model. It only discusses gauge theories, like N = 2 supersymmetric SU(2) gauge theory in four dimensions and large-N Chern-Simons theory. These are quite specialized gauge theories with special symmetries. As far as I know, the geometric engineering approach hasn't mapped the standard model in an analogous fashion. Hence it isn't clear that the standard model has a mirror symmetry.

--Mark viking (talk) 22:45, 28 March 2014 (UTC)[reply]

Hi Mark viking,

1. The part of mirror symmetry that I'm most familiar with is homological mirror symmetry, and there you certainly don't need any supermanifolds. Everything is formulated in terms of an ordinary Calabi-Yau variety. The book by Hori et al. discusses the superspace formalism for supersymmmetric field theories in Chapter 12, but there it's the worldsheet that's being treated as a supermanifold, not the target space. I don't know if there's some formulation of mirror symmetry where the Calabi-Yau manifolds are promoted to supermanifolds.

2. My reference to the standard model was only supposed to provide a context for the discussion of gauge theory. I have modified the language in this section to emphasize that the gauge theories studied using mirror symmetry are not part of the standard model.

Polytope24 (talk) 07:26, 29 March 2014 (UTC)[reply]

Thanks, Polytope24, you have addressed both my issues and new wording concerning the standard model looks good to me. The article has my support for FA status. --Mark viking (talk) 21:40, 29 March 2014 (UTC)[reply]

• Support - I think this article meets all the featured article criteria for content: the prose is clear and concepts well-explained, the the coverage of the topic is comprehensive, the inline citations look to support the assertions (my quibble regarding the standard model above was a matter of wording, not a bad reference) and a spot check of some of the references showed that they all checked out. I'm not an expert on MOS compliance in FA, etc, but in regards to the content, the article is FA class. Well done, Polytope24! --Mark viking (talk) 21:40, 29 March 2014 (UTC)[reply]

Thanks for your help, Mark viking! Polytope24 (talk) 02:49, 30 March 2014 (UTC)[reply]

• Comment: It looks as though the article has had quite a bit of attention since your crie de coeur on the FAC talkpage, which is how I noticed it. I can't comment sensibly on any of the article's substantial content, which is way outside my range of comprehension. However, I did notice that in the relatively short lead the words "mirror symmetry" occur ten times, which is over-repetitive and could, I am sure, be reduced by a few textual manoeuvres. Brianboulton (talk) 12:02, 30 March 2014 (UTC)[reply]

Thanks for the suggestion. Fixed. Polytope24 (talk) 16:07, 30 March 2014 (UTC)[reply]

• Comments First, given the complicated technical nature of the subject, this article does an excellent job in providing a relatively accessible account. While reading I noticed a few things:
  • As currently phrased the tone of some of the remarks about lack of mathematical rigour in the physical formulation of mirror symmetry can be misread as being a little judgemental. (I am sure this is not intended, but different people may read things in different ways). Take for example the line in the lede: " Although the original approach to mirror symmetry was based on nonrigorous ideas from theoretical physics, mathematicians have gone on to rigorously prove some of its mathematical predictions". This could be phrased more neutrally as (for example) "Since the original formulation of mirror symmetry based on physical intuition, rigorous mathematical proofs of some of it predictions have been formulated." (OK, maybe not the smoothest suggestion.) This avoids a few unwanted suggestions, such as a dichotomy between physicists and mathematicians (even if that does exist to some extent. Similar phrasing occurs again in the "enumerative geometry" section.
  • "A Calabi–Yau manifold is a special (typically six-dimensional) space..." If I recall correctly, Calabi-Yau manifolds can have any (even) number of dimensions. The word "typical" seems out of place. (The thing you seem to want to stress is that the six dimensional case is of most physical interest.)
  • "In a paper from 1985, Philip Candelas, Gary Horowitz, Andrew Strominger, and Edward Witten showed that by compactifying string theory on a Calabi–Yau manifold, one obtains a theory roughly similar to the standard model of particle physics" It has been a while for me, but isn't Calabi-Yau compactition required in order to retain some degree of supersymmetry? (Which is necessary for some popular models of beyond the standard model physics, but not for the standard model per se.) Please double check that this phrasing is accurate.
  • "Such dualities arise frequently in modern physics, especially in string theory." Is "frequently" the best adverb here? You could take the position that dualities are actually quite rare. However, they are very useful when they exist and therefor very important.

TR 16:40, 3 April 2014 (UTC)[reply]

• One more comment, some of the papers referenced are missing DOIs, while others do include them. This should be at least consistent. (As annoying as the task of tracking them down an adding them can be.)TR 16:56, 3 April 2014 (UTC)[reply]

Thanks for your comments. I made the following changes to the article:

1. I certainly didn't want the discussion of mathematical rigor to sound like an attack on physicists. I have changed the wording to emphasize that this has to do with methodological differences between the two subjects.

2. It is possible to talk about Calabi-Yau manifolds in dimensions other than 6, but the six-dimensional ones are the most interesting even for pure mathematicians. I changed the wording to avoid confusing the reader.

3. The sentence about Calabi-Yau compactification is true as stated, and it's also related to supersymmetry. I added a few words mentioning this fact in the article.

4. The word "frequently" is perhaps not the best choice, but I would say that dualities are really quite ubiquitous in theoretical physics. I changed the wording in this sentence.

5. I'm working on tracking down the DOIs of these articles.

Polytope24 (talk) 19:55, 3 April 2014 (UTC)[reply]

Thanks for the edits.

1&2 are indeed an improvement.

The change you made for 3 has a small problem. As it reads now, it suggests that the standard model has supersymmetry.

And for 4. I would call "ubiquitous" somewhat of an over statement for someting for which the occurences outside string theory can be counted on one hand. It would be a lot safer to state that: "Such dualities play an important role in modern physics, especially string theory." This is also more informative.

TR 22:28, 3 April 2014 (UTC)[reply]

I just fixed #3, and I'll take care of the DOIs as soon as I get the chance. As for #4, I disagree that "ubiquitous" is an overstatement. In addition to the dualities relating different versions of string theory, you have lots of dualities in field theory, like Montonen-Olive duality, Seiberg duality, the relationship between Chern-Simons theory and the Wess-Zumino-Witten model, the AGT correspondence, 3D-3D correspondence, 3D mirror symmetry, and various generalizations of the AdS/CFT correspondence. In many cases, these dualities don't just relate individual theories, but entire families of theories. It is true, of course, that all of these theories are related to string theory. If you like, I could emphasize this more in the article, or I could list some of these dualities to justify my use of the term ubiquitous. Let me know what you think. Polytope24 (talk) 23:17, 3 April 2014 (UTC)[reply]

_{Remember that there is lots of modern physics that is unrelated to string theory.TR 22:04, 7 April 2014 (UTC)}[reply]

_{Yes, of course you're right! Polytope24 (talk) 00:51, 8 April 2014 (UTC)}[reply]

Okay, I added the DOIs to the references. I also went ahead and followed your advice on #4. Polytope24 (talk) 06:11, 4 April 2014 (UTC)[reply]

I think that covers my comments. (Just browsing through the article, I noticed that some (all?) figures are missing WP:ALTTEXT.)

Fixed. Polytope24 (talk) 00:51, 8 April 2014 (UTC)[reply]

• Concern In general the article is very good. I think at this point the only concern is comprehensiveness. The biggest problem with this is that most reviewers (myself included) are not in a good position to judge whether the article is comprehensive. However, I am a bit concerned that the article is missing the state-of-the-art. Reading the article, a reader may be left with the suggestion that nothing has happened in the field since 2000. Maybe this is indeed the case, but that seems unlikely to me for a field that seems rather active.TR 10:28, 8 April 2014 (UTC)[reply]

Thanks for your comment, TR. You make a fair point. I just added a paragraph to the end of the history section mentioning some of the recent developments in mirror symmetry. Polytope24 (talk) 04:00, 9 April 2014 (UTC)[reply]

• Closing note: This candidate has been promoted, but there may be a delay in bot processing of the close. Please see WP:FAC/ar, and leave the ((featured article candidates)) template in place on the talk page until the bot goes through. Graham Colm (talk) 23:29, 13 April 2014 (UTC)[reply]

The above discussion is preserved as an archive. Please do not modify it. No further edits should be made to this page.