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The aim of this question is to understand SYZ conjecture ("Mirror symmetry is T-Duality"). I don't expect a full and quick answer but to find a better picture from answers and comments.

The whole idea is to construct the Mirror C.Y $Y$ from $X$ intrinsically as follows. One considers the moduli of special Lagrangian tori with a flat $U(1)$ bundle on it in $X$.

Then we put a metric on this moduli (plus corrections coming from J-holomorphic disks) and expect that this moduli and the metric given on it is the mirror C.Y we were looking for.

Here are the things I can not understand:

---What is the metric given in the paper "Mirror symmetry is T-Duality"? Where does it come from? (I can not understand the formulation of metric there).

and more importantly --How do we deform the metric using J-holomorphic disks(instantons)?

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I would appreciate if some body give me an expository reference for the discussion of metric given there. –  Mohammad F. Tehrani Jun 30 '10 at 2:38
    
Can you be more precise about which metric you're asking about?, e.g. which page of the paper? –  Kevin H. Lin Jun 30 '10 at 2:39
    
The best mathematical exposition of SYZ that I've seen is this paper of Denis Auroux: arxiv.org/pdf/0706.3207v2 . Although this paper does not focus on the Calabi-Yau case (which is probably the case that SYZ consider in their paper?), I think the basic ideas of the constructions are still the same. –  Kevin H. Lin Jun 30 '10 at 2:43
    
same paper I mentioned above, " mirror symmetry is T-Duality" all the paper is about this moduli and the metric on it. this is the paper where SYZ comes from. –  Mohammad F. Tehrani Jun 30 '10 at 2:54
    
Oh, I'm sorry, I read your question too quickly. Take a look at section 2 of the Auroux paper I linked to -- there he explains how one can obtain a Kaehler structure (and so in particular a metric) on the moduli space of special Lagrangians with flat U(1) connections. –  Kevin H. Lin Jun 30 '10 at 3:11
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3 Answers 3

up vote 12 down vote accepted

Hi-

Just saw this thread. Maybe I should comment. The conjecture can be viewed from the perspective of various categories: geometric, symplectic, topological. Since the argument is physical, it was written in the most structured (geometric) context -- but it has realizations in the other categories too.

Geometric: this is the most difficult and vague, mathematically, since the geometric counterpart of even a conformal field theory is approximate in nature. For example, a SUSY sigma model with target a compact complex manifold X is believed to lie in the universality class of a conformal field theory when X is CY, but the CY metric does not give a conformal field theory on the nose -- only to one loop. Likewise, the arguments about creating a boundary conformal field theory using minimal (CFT) + Lagrangian (SUSY) are only valid to one loop, as well. To understand how the corrections are organized, we should compare to (closed) GW theory, where "corrections" to the classical cohomology ring come from worldsheet instantons -- holomorphic maps contributing to the computation by a weighting equal to the exponentiated action (symplectic area). The "count" of such maps is equivalent by supersymmetry to an algebraic problem. No known quantity (either spacetime metric or Kahler potential or aspect of the complex structure) is so protected in the open case, with boundary. That's why the precise form of the instanton corrections is unknown, and why traction in the geometric lines has been made in cases "without corrections" (see the work of Leung, e.g.). Nevertheless, the corrections should take the form of some instanton sum, with known weights. The sums seem to correspond to flow trees of Kontsevich-Soibelman/ Moore-Nietzke-Gaiotto/Gross-Siebert, but I'm already running out of time.

Topological: Mark Gross has proven that the dual torus fibration compactifies to produce the mirror manifold.

Symplectic: Wei Dong Ruan has several preprints which address dual Lagrangian torus fibrations, which come to the same conclusion as Gross (above). I don't know much more than that.

Also-

Auroux's treatment discusses the dual Lagrangian torus fibration (even dual slag, properly understood) for toric Fano manifolds, and produces the mirror Landau-Ginzburg theory (with superpotential) from this.

With Fang-Liu-Treumann, we have used T-dual fibrations for the same fibration to map holomorphic sheaves to Lagrangian submanifolds, proving an equivariant version of homological mirror symmetry for toric varieties. (There are many other papers with similar results by Seidel, Abouzaid, Ueda, Yamazaki, Bondal, Auroux, Katzarkov, Orlov -- sorry for the biased view!)

Reversing the roles of A- and B-models, Chan-Leung relate quantum cohomology of a toric Fano to the Jacobian ring of the mirror superpotential via T-duality.

Help or hindrance?

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Thanks for answer The thing I would like to know most is this: How J-holomorphic disks with boundary on Lagrangians enter into the story? How do we do quantum corrections to metric defined on the moduli of special lagrangians using moduli of open disks with two marked points ( as claimed in "Mirror symmetry is T-Duality" )? why the correct kahler metric on this moduli (which is going to give the mirror CY) should incorporate quantum corrections coming form open disks? I would appreciate if you lead me to more expository references on this issue. –  Mohammad F. Tehrani Jul 16 '10 at 23:51
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Here are some papers on SYZ worth reading:

Hitchin's paper was written shorly after Mirror Symmetry is T-duality and it is a matematical explanation of the paper. Essentially Maclean proved that the moduli space of sL submanifolds is unobstructed and its tangent space is the space of harmonic 1-forms on the sL submanifold. A natural metric which you can put on the moduli space is the $L^2$ metric on harmonic forms. When the sL submanifold is a torus, the moduli space also has an "affine structure". This was already known from integrable systems, they are called action coordinates. They are affine because they are defined up to affine transformations (with linear part having integral coefficients). Hitchin shows that with respect to these coordinates the metric can be expressed as the Hessian of a function. Hitchin also shows that the moduli space has two affine strutures (this is because of the "special" condition). The two affine structures are related by Legendre transform using the Hessian (i.e.the metric). So one could say that mirror symmetry is "Legendre transform".

Things have developed a lot since Hitchin's paper, and M. Gross surveys these developements. How to do "quantum corrections" to the metric is a major open problem, there are many approaches. They seem all quite difficult.... Auroux in the paper mentioned above deals with it. I heard a talk of Fukaya where he wants do do it with Floer homology for families, but I do not know much about it. Then there is the program of Kontsevich and Soibelmann, using rigid analytic geometry and the Gross-Siebert program. It seems that quantum corrections could be understood in terms of "tropical geometry" in the moduli space of SL tori (an "affine manifold with sigularities"). In a recent paper of M. Gross ("Mirror symmetry for $\mathbb{P}^{2}$ and tropical geometry"), he explains how "period calculations" can be understood in terms of tropical geometry (at least for $\mathbb{P}^{2}$). Here you can find a link to a book of M. Gross where he explains the connection between tropical geometry and mirror symmetry.

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And here's the link to "Mirror symmetry for $\mathbb{P}^2$ and tropical geometry" arxiv.org/abs/0903.1378 –  j.c. Jun 30 '10 at 12:14
    
so you mean, it is still unknown that how we should modify the metric using holomorphic disks ?! Is there any idea that how should this modification look like ! –  Mohammad F. Tehrani Jun 30 '10 at 15:05
    
the most difficult part is to understand how to modify the complex structure on the moduli space. As I said, Kontsevich-Soibelman and Gross-Siebert make a lot of progress on this using tropical geometry. I think the role of holomorphic disks is still unclear, although Auroux, Fukaya (and maybe others) are working on it. –  Diego Matessi Jul 1 '10 at 6:53
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You might also want to check out

Riemannian Holonomy Groups and Calibrated Geometry by Dominic Joyce.

In Chapter 9, he gives a nice introduction to SYZ that is very accessible. He also points out reasons why SYZ as originally formulated (even though it does not have a precise formulation) could not be true. He proposes modified versions of SYZ that he believes are likely to be closer to any eventual true statement. (I have not looked at Auroux's recent work, so I can't comment on that.)

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