What's dual torus and mirror manifold? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T21:13:13Z http://mathoverflow.net/feeds/question/45881 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/45881/whats-dual-torus-and-mirror-manifold What's dual torus and mirror manifold? Johnny Ho 2010-11-13T00:25:11Z 2012-11-28T11:34:02Z <p>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?</p> <p>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. </p> http://mathoverflow.net/questions/45881/whats-dual-torus-and-mirror-manifold/45895#45895 Answer by Sasha Kirillov for What's dual torus and mirror manifold? Sasha Kirillov 2010-11-13T04:48:56Z 2010-11-13T04:48:56Z <p>The usual answer is that the dual lattice is <code>$\check{\Gamma}=\{f\in V^* | f(\gamma)\in \mathbb{Z}\ \forall \gamma \in \Gamma\}$</code>. It is defined for any lattice $\Gamma\subset V$ - no extra information needed. </p> http://mathoverflow.net/questions/45881/whats-dual-torus-and-mirror-manifold/45913#45913 Answer by Charles Matthews for What's dual torus and mirror manifold? Charles Matthews 2010-11-13T12:16:00Z 2010-11-13T12:16:00Z <p>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 <em>are</em> 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 <em>that</em>, 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. </p> http://mathoverflow.net/questions/45881/whats-dual-torus-and-mirror-manifold/114723#114723 Answer by Jay for What's dual torus and mirror manifold? Jay 2012-11-28T04:04:34Z 2012-11-28T04:04:34Z <p>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.</p> <p>Here is a good reference: <a href="http://www.math.cuhk.edu.hk/~kwchan/HMS_An.pdf" rel="nofollow">http://www.math.cuhk.edu.hk/~kwchan/HMS_An.pdf</a></p> <p>you can find a quick review of the construction.</p> http://mathoverflow.net/questions/45881/whats-dual-torus-and-mirror-manifold/114760#114760 Answer by Rhys Davies for What's dual torus and mirror manifold? Rhys Davies 2012-11-28T11:34:02Z 2012-11-28T11:34:02Z <p>Although it's true that the mirror of a torus is again a torus, it is <em>not</em> 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.</p>