What's dual torus and mirror manifold?

Question

I guess this is a well known fact/definition for many people. It is mentioned in many places that if $\Gamma$ is a lattice of a vector space(vector bundle/affine bundle) $V$, then there is a dual lattice $\check{\Gamma}$ in $V^* $ and the torus $V/\Gamma$ has a dual torus $V^*/\check{\Gamma}$. What does this mean? When is it meaningful? I want to know the answer because I hope that we can get an easy topological description of ("toy model" of) mirror manifold (I have to admit that I am not sure if I really understand what's "mirror manifold"). For example, if $V$ is a vector bundle and $\Gamma$ is a lattice in $V$ with some extra structure or information, how can we find a torus bundle dual to $V/\Gamma$? What kind of extra structure or information we need? If $V$ is described by local chart and transitive groups {$U_\alpha,\mu_{\alpha \beta}$}, can we find a dual set {$U^* _\alpha,\mu^* _{\alpha \beta}$} to describe $V^* $? Is it unique or canonical?

The other question is about the canonical complex structure of tangent bundle. Of course, it relate to the above question. Again, like the symplectic structure on cotangent bundle, I guess it is well known, but I really can't find any reference. One source I found is a slide of Mark Gross (http://math.mit.edu/~auroux/frg/mit08-notes/M.%20Gross%20-%20Slides%20-%20From%20affine%20manifolds%20to%20complex%20manifolds.pdf). I am not sure if this is the standard one we understand. The complex structure looks much less natural then the canonical symplectic structure.

Answer 1

The usual answer is that the dual lattice is $\check{\Gamma}=\{f\in V^* | f(\gamma)\in \mathbb{Z}\ \forall \gamma \in \Gamma\}$. It is defined for any lattice $\Gamma\subset V$ - no extra information needed.

Answer 2

As said before, there is an answer in terms of the dual vector space, and this definition is purely algebraic. But there is a second answer, in terms of Pontryagin duality. These answers are not trivially the same, as one sees for the circle. They are the same, basically because we know the Fourier transform does make sense modulo some normalisation, taking functions on the real line to functions on the real line (rather than on the dual space of the real line). This gives you some idea what to expect for lattices in general: in passing to the quotient by a lattice, namely a torus, and then to the Pontryagin dual of that, you have another lattice lying naturally inside another vector space. If you chase down all the normalising factors explicitly, or hide them in notation, the algebraic and Fourier-analytic approaches are going to run out the same.

Answer 3

I think you are asking about the complex structure on tangent bundle of integral affine manifold, there we can cook up a simple one, as we did for symplectic structure on cotangent bundle.

Here is a good reference: http://www.math.cuhk.edu.hk/~kwchan/HMS_An.pdf

you can find a quick review of the construction.

Answer 4

Although it's true that the mirror of a torus is again a torus, it is not simply the dual torus. Roughly speaking, mirror symmetry 'exchanges the complex structure and the (complexified) Kähler form'. The lattice $\Gamma$ gives you the complex structure of the torus, but tells you nothing about the complex structure of its mirror.