Talk:Mirror symmetry conjecture

About page

This page is suppose to contain a discussion of mirror symmetry for a mathematical audience. Because of this, can someone create a redirect at the top giving the disambiguation for the non-technical article Mirror symmetry.

I recently changed this back to a disambiguation page. Please add your comment to https://en.wikipedia.org/wiki/Wikipedia_talk:WikiProject_Mathematics Wundzer (talk) 21:56, 19 August 2020 (UTC)

Todo

Original construction

Add section giving overview of original construction. This should contain

• variation of hodge structures in the second

Yukawaw coupling

Look in Cox, Katz book for the following:

• discuss equivalent SCFT's
• discuss Yukawa couplings
• give power series from second Yukawa coupling
• also give conjectures from this, such as the integrality of the $n_{d}$

Ideas from string theory

• include moduli spaces from Auroux's lectures and create subsections mentioning them as the original moduli spaces string theorists considered.

Statement

• There should be a section discussing an overview of quantum cohomology
• Given relation between quantum cohomology and VHS integrals
• GW-invariants + associativity of quantum cohomology give big results like counting plane curves intersecting points.

(the above was added by Wundzer, who left it unsigned)

Please try to make the current page more readable before adding extra material. Gumshoe2 (talk) 02:29, 15 July 2020 (UTC)

Purpose

What is meant to be on this page that could not be more easily absorbed into the pages

• Gromov-Witten invariant
• Quantum cohomology
• Homological mirror symmetry
• SYZ conjecture
• Type II string theory

and which could be synthesized together in a section on Mirror symmetry (string theory)? Gumshoe2 (talk) 18:35, 14 July 2020 (UTC)

The page is meant to act similar to a survey article for the concepts related to mirror symmetry. Furthermore, the page Mirror symmetry (string theory) reads much more like a popular science article, and I think combining the two will attempt to speak towards two audiences at once. Wundzer (talk) 15:33, 17 August 2020 (UTC)

I can see the merit of a survey article aimed at mathematicians, which seems to be what you mean. I'm not sure it warrants a separate article, but I will leave to others to decide. A more appropriate title would probably be "Mirror symmetry (mathematics)". However, I think the current content of the article must be rewritten to be more accessible, even to other mathematicians. Maybe you could start by clarifying (here on the talk page) what level it is meant for and what the intended prerequisites are. Gumshoe2 (talk) 16:20, 17 August 2020 (UTC)

I think the name change is warranted. Can you make that change? I'm not sure a rewrite is needed, just more material. Mirror symmetry touches upon a lot of advanced mathematics, such as variations of mixed Hodge structures, A-infinity algebras/categories, Gromov-witten theory, and so forth. Wundzer (talk) 02:42, 18 August 2020 (UTC)

Unfortunately I have no idea how to change a title. If you don’t have a clear sense of who you’re writing for, I think the result will be a very bad article. If you’re writing for someone who already understands variation of mixed Hodge structure, A infinity categories, and Gromov-Witten theory, the audience will be extremely small, and likely would already be familiar with much of the material. I think it’s an interesting article topic which I suspect could at least reach readers with a year of graduate-level algebraic geometry and differential geometry. I suspect the prerequisites for a good article could even be much less than that. As it stands, most readers (even the mathematicians) will not even be aware of what they would have to learn in order to understand the article; I think that’s not good. Gumshoe2 (talk) 04:18, 18 August 2020 (UTC)

For how to rename an article, see HELP:MOVE.  --Lambiam 18:38, 20 August 2020 (UTC)

That is not going to work, since there is already a page Mirror symmetry (mathematics). See further atthe bottom of this section.  --Lambiam 18:54, 20 August 2020 (UTC)

I'm writing for a general algebraic geometry audience, but by that, it generally means having a decent level of maturity. While all these subjects cannot be introduced in this article, I think linking to them and providing relevant examples in those articles certainly makes this article more approachable. If I were picking an audience, it would be people looking for a top-down introduction to mathematical mirror symmetry who are looking for a view of the landscape. In addition, there should be ample references to literature so they can go ahead and deep-dive into whatever piques their interest. I think I'm doing a decent job making citations for the relevant material, but please correct me about any gaps you find - I'm more than happy to fill them either in this article, or in others. Wundzer (talk) 04:49, 18 August 2020 (UTC)

Here's an example: it seems like the use of $\Gamma (\mathbb {P} ^{4},{\mathcal {O}}_{\mathbb {P} ^{4}}(5))$ is unnecessary, as it can be explained purely in terms of homogeneous polynomials in five variables. I'm sure this isn't how it goes for practitioners of the field, but I think it'd be more appropriate for an entry such as this. Also, it isn't said what $\mathbb {G} _{m}$ is, or what $U_{\text{smooth}}$ is; these seem like major gaps. Could it be explained purely in the case of the complex field? If so, it probably should be. Is X assumed to be smooth? Gumshoe2 (talk) 17:31, 18 August 2020 (UTC)

For instance the opening of the Complex Moduli section could say

A "quintic threefold" in complex projective space ℂℙ⁴ refers to a subset ${\big \{}[z_{0},\ldots ,z_{4}]\in \mathbb {CP} ^{4}:p(z_{0},\ldots ,z_{4})=0{\big \}},$ where p is a degree-five homogeneous polynomial of of five complex variables. The space of such polynomials forms a complex vector space of dimension 126.

followed by a few sentences about group actions on the vector space. I believe this would be much more accessible. Gumshoe2 (talk) 17:34, 18 August 2020 (UTC)

(re: Gumshoe2) I disagree that the vector space of global sections should be removed since understanding where the CY manifold comes from is useful during the construction of the associated Kontsevich moduli space ${\mathcal {M}}_{g,n}(X,\beta )$. I can fix the other issues where the algebraic group and $U_{\text{smooth}}$ are left undefined/un-referenced. Also I can add the smoothness hypothesis to the CY manifold. I think the "quintic threefold" text should refer to the quintic threefold page, so a more down-to-earth description can be given there. For the group actions, maybe that can reference the GIT page, or a page related to algebraic group actions. That way those issues can be brushed aside from this page, making it more distracting to read. Wundzer (talk) 17:44, 18 August 2020 (UTC)

As it exists in the article at the moment, it seems that the space of sections is not part of the exposition of the Kontsevich moduli space. At any rate, I'm just saying that the material should be as down to earth as possible, since wikipedia is an encyclopedia and not lecture notes. If that can be done by linking to other pages, then I guess that's ok. As it stands, the quintic threefold page also seems useless unless the reader already knows the line bundle O(d). Gumshoe2 (talk) 17:53, 18 August 2020 (UTC)

Great! I made some changes to quintic threefold. I can try and continue going down to the bottom so someone could hypothetically go from this article, keep going down the layers of abstraction, and understand how the tools come together for the construction in this article. Please keep up your concerns! Wundzer (talk) 18:03, 18 August 2020 (UTC)

It seems like there are some new problems there, like an apparent definition of a CY manifold which is obviously incorrect. I'm still trying to understand why the use of $\Gamma (\mathbb {P} ^{4},{\mathcal {O}}_{\mathbb {P} ^{4}}(5))$ is necessary in this article. Gumshoe2 (talk) 18:16, 18 August 2020 (UTC)

Let me see if I understand the complex moduli section, so that I can try to help improve it. (I'm not an algebraic geometer.) Is the following correct? (I'm not proposing it as article text, just a personal summary.)

A degree-five (complex) homogeneous polynomial in five complex variables defines a quintic threefold. Due to its projective structure, a smooth quintic threefold is automatically a Kähler manifold whose first Chern class vanishes, and its Hodge diamond can be determined from the Lefschetz hyperplane theorem.

The complex vector space of degree-five complex homogeneous polynomial in five complex variables is 126-dimensional. There is a natural action of a 25-dimensional Lie group on this vector space, and there is a natural correspondence between the orbits and the equivalence classes of (smooth?) quintic threefolds (what exactly is the equivalence relation?). According to geometric invariant theory, the space of orbits naturally has the structure of a generically smooth scheme of dimension 101. This equals the Hodge number h^2,1 of a smooth quintic threefold. The Tian-Todorov theorem says that any infinitesimal deformation of Calabi-Yau structure comes from a one-parameter family of Calabi-Yau structures. This shows that the smooth part of the above quotient space gives a smooth parametrization of the Calabi-Yau moduli of quintic threefolds. Gumshoe2 (talk) 19:03, 18 August 2020 (UTC)

You got everything except for the BTT theorem correct. BTT just means any deformation of a calabi-yau manifold is unobstructed, so the cohomology groups $H^{1}(X,T_{X})$ are the tangent space of a hypothetical moduli space of Calabi-Yau manifolds. Wundzer (talk) 20:22, 18 August 2020 (UTC)

I must have some misunderstanding. I thought that unobstructedness means that for any element of H¹(X, TX), there is a one-parameter family of Calabi-Yau metrics (as a mapping of a complex manifold to a small one-dimensional complex disk) which induces that element. I thought that was the meaning of the phrase "the cohomology groups $H^{1}(X,T_{X})$ are the tangent space of a hypothetical moduli space of Calabi-Yau manifolds". Is that incorrect? Gumshoe2 (talk) 20:56, 18 August 2020 (UTC)

No, by unobstructedness I mean it as a deformation theoretic term for deforming the complex structure on a Calabi-Yau manifold. This means equipping the underlying real manifold with a different, non-isomorphic, complex structure. Furthermore, the Calabi-Yau condition just means (in the most light-weight sense) that the canonical bundle is trivial. Some authors require a compactness, or projective condition, but that leaves out affine Calabi-Yau manifolds, which can be interesting objects in of themselves. The one-parameter family considered is the Dwork family. It could very well be a deformation of the CY manifold could be equivalent to a deformation of a CY metric, but I am unfamiliar with the differential geometry. I think the algebraic notion of Calabi-Yau was popularized because the Kahler structure can be pulled back from the class $H\in H^{2}(\mathbb {P} ^{4};\mathbb {Z} )$, which is a one-dimensional module. Explicitly representing the metric is in general an open problem from what I recall. E.g. see this article: https://arxiv.org/abs/1503.02899 .Wundzer (talk) 21:08, 18 August 2020 (UTC)

Thanks, I think I understand now. The existence and uniqueness theorem for CY metrics means that deformation of complex structure gives a deformation of CY metric. So there’s nothing to be gained by thinking on a primary level about deforming the CY metric. One last question: how many singular points are there on the scheme quotient above? Probably the smooth part can be described explicitly? Gumshoe2 (talk) 21:21, 18 August 2020 (UTC)

Awesome! For the singularities, I should have written blowing up the relevant singularities introduces 100 new divisors - that needs to be fixed. Checkout this link starting in the middle of page 2 Wundzer (talk) 21:56, 18 August 2020 (UTC)

• @Wundzer and Gumshoe2: For the earlier question about moving the page, use the following template at the bottom of this talk page:

{{subst:requested move|NewName|reason=Place here your rationale for the proposed page name change, ideally referring to applicable naming convention policies and guidelines, and providing evidence in support where appropriate. If your reasoning includes search engine results, please prioritize searches limited to reliable sources (e.g. books, news, scholarly papers) over other web results. You don't need to add your signature at the end, as this template will do so automatically.}}

with further instructions available at Wikipedia:Requested moves#Requesting a single page move. — MarkH₂₁^talk 20:46, 18 August 2020 (UTC)

There is already a page with the title Mirror symmetry (mathematics). It redirects to the disambiguation page Mirror symmetry (disambiguation), which lists, among other things:

• Mirror symmetry (string theory), a relation between two Calabi–Yau manifolds in string theory
• Homological mirror symmetry, a mathematical conjecture about Calabi–Yau manifolds made by Maxim Kontsevich

The new page title should be sufficiently distinct from these (as should its contents). If no page with the new title exists, the instructions of HELP:MOVE should suffice.  --Lambiam 18:54, 20 August 2020 (UTC)

What is the conjecture

A conjecture is a mathematical statement that is suspected to be true but for which no proof is known. The article does not explicitly state what the statement is here that is conjectured to hold. I think this is an omission.  --Lambiam 11:29, 1 August 2020 (UTC)

I will fix this soon. Look at page 3-4 of https://ocw.mit.edu/courses/mathematics/18-969-topics-in-geometry-mirror-symmetry-spring-2009/lecture-notes/MIT18_969s09_lec01.pdf . I wanted to get all of the preliminary material on the page before writing out the conjectural relationships between mirror manifolds. You can also look at https://ocw.mit.edu/courses/mathematics/18-969-topics-in-geometry-mirror-symmetry-spring-2009/lecture-notes/MIT18_969s09_lec07.pdf for a more precise statement. I'm trying to get the Hodge structure pages setup before finishing this bit. It requires knowledge of a lot of Hodge theory in order to properly understand. Wundzer (talk) 15:39, 17 August 2020 (UTC)

Here, I added some of the basic material + conjecture. I will try and expand it tomorrow, or later on this week. Please let me know if you have any suggestions for improvement! Wundzer (talk) 04:14, 18 August 2020 (UTC)

Thanks a lot. Please take your time. It is more important that it is correct (and understandable) than that it is complete.  --Lambiam 18:32, 20 August 2020 (UTC)