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)?
Best Answer
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?