There is a version of mirror symmetry, called "local mirror symmetry", for certain non-compact Calabi-Yaus, for example the total space of the canonical bundle of P^2 (exercise: show this is CY). The mirror (or rather one possible mirror) of this non-compact Calabi-Yau is an affine elliptic curve in (C^*)^2. I don't think that there is as yet a version of mirror symmetry for more general non-compact CYs, though I don't know too much about this story. In all of the papers that I've looked at on this stuff at least, the only non-compact CYs that have been considered in mirror symmetry so far are total spaces of vector bundles over compact (probably Fano) things.
I guess we should probably get some sort of "Hodge diamond symmetry" in local mirror symmetry, but the story becomes more complicated. One immediate thing to notice is that, at least in the example I've given, the dimensions of the manifolds aren't the same! So things will have to be modified.
The Hodge diamond symmetry in mirror symmetry for compact Calabi-Yaus should really be thought of as coming from a correspondence between certain deformations of a Calabi-Yau and certain deformations of its mirror. In the non-compact case, the deformations that we should consider will be somewhat different from the deformations that we should consider in the compact case.
A somewhat recent point of view, due to Kontsevich, is that this correspondence between deformations can be gotten from homological mirror symmetry. In homological mirror symmetry for compact Calabi-Yau manifolds, we consider a derived category of coherent sheaves on one side and a Fukaya category on the other side. Then we should have an equivalence of categories, plus an equivalence of a certain structure on their Hochschild cohomologies -- in particular their Hochschild cohomologies should be equivalent at least as vector spaces. These Hochschild cohomologies should be thought as the appropriate deformation spaces (or maybe rather the tangent spaces to the appropriate deformation spaces?), and an appropriate identification of the Hochschild cohomologies (plus the extra structure that I mentioned) should give in particular the Hodge diamond symmetry. There should also be homological mirror symmetry for non-compact Calabi-Yau manifolds, but we must define the analogues of derived category and Fukaya category in this situation appropriately. Then there should be an analogous story on the Hochschild cohomologies of the categories, and an appropriate analogue of the Hodge diamond symmetry. See the paper "Hodge theoretic aspects of mirror symmetry" by Katzarkov-Kontsevich-Pantev for more details.
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?
Best Answer
Since B-fields are being discussed in the other question, i'll try to answer your question on the complexified Kahler moduli. The set of Kahler classes of a Kahler manifold is an open cone in $H^{1,1}(X, C) \cap H^2(X, R)$. There are results on the structure of this cone which roughly say that it is rational polyhedral away from some set (I think!). In the construction of the complexified Kahler cone you have to take a "framing" of this cone, in the end you get something isomorphic to $(\Delta^{\ast})^{h^{1,1}}$, where $\Delta^*$ is the punctured disc in $C$. This matches the moduli of complex structures of the mirror, near a large complex structure limit point. These things are discussed in the book "Calabi-Yau manifolds and related geometries", by Gross, Huybrechts and Joyce.