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The Gross-Siebert program is said to be an algebraic analog of the SYZ conjecture and they used toric degeneration to construct a mirror dual of Calabi-Yau varieties. It seems like the singular central fiber, as is mostly given by some combinatoric data, has some sort of "mirror dual" that's easily written down.

My question is, how far can one get with this program? Are the CY varieties arising this way "the" mirror dual of the original CY varieties or just one candidate of its mirror? What are the most recent progress on it?

Also, this seems very different from the homological mirror conjecture saying that the equivalence of the derived category on one and Fukaya category on another. Can anyone here connect the dots for me?

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  • $\begingroup$ Could you explain what you mean by "[a]lso, this seems very different from the homological mirror conjecture saying that the equivalence of the derived category on one and Fukaya category on another. Can anyone here connect the dots for me?" I might be able to say something helpful, but I'm not sure what you interpret as the Gross-Siebert conjecture. $\endgroup$
    – Daniel Pomerleano
    Commented Mar 1, 2012 at 19:10
  • $\begingroup$ I have a feeling roughly Gross-Siebert is doing the following: take a Calabi-Yau, find a toric degeneration for it, so we get a singular fiber, which is basically given by some combinatoric data. Apply the Legendre transformation on those data, we get another "singular fiber", and its smoothing shall be a mirror dual of the original Calabi-Yau. (That's my understanding. I'm not sure if there is a fully-fledged conjecture.) I don't see how the stuff from derived category and Lagrangian submanifolds fit in this picture. (Also, how does this connect hodge theory and period integral?) $\endgroup$
    – temp
    Commented Mar 2, 2012 at 1:40

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Maybe I should comment. The short answer is "Hopefully all the way", but there are some caveats. Our program indeed started out by the observation that from a physical reasoning mirror symmetry for Calabi-Yau varieties only works near degeneration limits. The reason is that while the topological B-model (the complex side) works for any Calabi-Yau variety, the topological A-model (the symplectic side) becomes unreliable for small Kähler classes. As one example of a mathematical manifestation of this there are Calabi-Yau manifolds with two different maximal degenerations leading to non-deformation equivalent mirrors, such as the Pfaffian Calabi-Yau (http://arxiv.org/abs/math/9801092). Another mathematical manifestation already mentioned by Scott Carnahan is the fact that the Fukaya category is only defined over the Novikov ring and at best converges for large symplectic forms. One might phantasize about a strict analogue of the complex moduli space on the mirror side, a stringy Kähler moduli space, but at present it is not clear what this should be. Going over to the homological point of view does not appear to help directly either, as the example of the Fukaya category shows.

The point I want to make is that if you want a statement staying within the realm of Calabi-Yau varieties and Fukaya categories I don't see a way around a perturbative formulation. There are nevertheless non-perturbative manifestations of mirror symmetry and powerful non-perturbative computational techniques. For example, the global structure of the complex moduli space has been repeatedly put to use with great effect, e.g. to solve the holomorphic anomaly equation. We have no means to see this input from the global geometry of the complex moduli space perturbatively, and this is probably what Mark meant in his answer in Michigan in 2008 (but see below for recent progress on this via theta functions). The one exception I am aware of is the recent Chiodo-Ruan-fantasy about "global mirror symmetry". While I haven't thought deeply about this, the examples are about hypersurfaces, and these can be studied as Landau-Ginzburg models by working with the homogeneous equations on affine space (LG/CY-correspondence). Landau-Ginzburg models fit into our program as well, see my paper with Michael Carl and Max Pumperla (on my webpage, yet to be polished), so hopefully this has an interpretation from our point of view as well.

As for the question on a relation to homological mirror symmetry, we are just about to finish a survey "Theta functions and mirror symmetry" that sketches how we imagine to prove homological mirror symmetry via tropical Morse trees. The point is that our construction comes with a canonical basis of sections ("theta functions") of powers of the ample line bundle. These are mirror dual to intersection points of a class of Lagrangian sections of an SYZ fibration, viewed in the degeneration limit. For the elliptic curve Mark has worked out the correspondence in great detail in Chapter 8 of the book "Dirichlet branes and mirror symmetry". The theta functions, by the way, often turn out to be algebraic and may help to probe deeper into the moduli space, but this is still to be understood.

Tropical curves should also be at the heart of the correspondence between Gromov-Witten invariants and periods that you were also wondering about. There are some loose ends here, but Mark's paper "Mirror symmetry for P^2 and tropical geometry" and our joint paper with Rahul Pandharipande should give an impression of the picture.

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  • $\begingroup$ Nice to see comments given by the original authors. It will take me some time to really understand all this, but I decided to click the check and recognize this and the answer to my question. Thanks a lot. $\endgroup$
    – temp
    Commented Mar 20, 2012 at 5:47
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Here is one comment that might help you. In the question session after his talk at the 2008 Michigan workshop, Gross expressed skepticism that his methods could see any information that was not visible on an infinitesimal neighborhood of the singular central fiber which you mention. He did not have a good suggestion for what sort of information might be available in the interior of the moduli space, and not on such an infinitesimal neighborhood.

MAJOR DISCLAIMERS: I am not an expert, this was 4 years ago, this was a spoken statement which was probably less carefully formulated than something meant to be recorded in writing. I am, for these reasons, nervous about posting it in public at all. But, if taken with all of these caveats in mind, it might be a good answer to "how far can one get with this program".

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    $\begingroup$ I am also not an expert, but I have heard that the Fukaya category of a CY is also difficult to define away from infinitesimal deformations of singular degenerations. (I'm afraid to speculate in the company of specialists, though...) $\endgroup$
    – S. Carnahan ♦
    Commented Mar 2, 2012 at 8:41
  • $\begingroup$ Scott, my understanding is that the Fukaya category is equally difficult to define for a rigid CY and for a degenerating one. (The coefficients are in a Novikov series ring.) But in the case of a maximal degeneration, one expects a homological mirror symmetry description of the category, and that the Novikov series actually converge. $\endgroup$
    – Tim Perutz
    Commented Mar 10, 2012 at 17:19
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For the question how SYZ is related to HMS, I have recently posted a paper on the arXiv (http://arxiv.org/abs/1201.6454) which explains this connection as a sort of Fourier-Mukai transform. There I gave a local explanation. Global situation requires more machinery, but it is a similar picture.

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