Timeline for Mirror symmetry for elliptic curves
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 10, 2010 at 21:24 | vote | accept | Mohammad Farajzadeh-Tehrani | ||
| Nov 10, 2010 at 18:24 | comment | added | Arend Bayer | Sorry Mohammed, you are right. What I meant is that the choice of complex structure and symplectic structure get interchanged. However, $\rho$ does not correspond to a choice of symplectic structure on the (fixed) differentiable manifold $E$, but a choice of symplectic structure on $\mathbb R^2$. I will correct my answer later. | |
| Nov 10, 2010 at 16:43 | comment | added | Mohammad Farajzadeh-Tehrani | If $\tau=a+ib$ and $\rho=c+id$ then as a symplectic manifold $E_{tau}^{rho}$ is isomorphic to $E_{i}^{b\rho}$ so has to correspond to complex tori $E_{b\rho}$ on the B-side and not $E_{\rho}$. | |
| Nov 10, 2010 at 16:38 | comment | added | Mohammad Farajzadeh-Tehrani | The maps $m_k$ of $A_{\infty}$ category are very sensitive to the choice of Lagrangian so if we want to have a real correspondence between derived category of coherent sheaves in B-side and Fukaya category in A-side we can not consider any arbitrary Lagrangian (with zero maslov class). But even forgetting this issue the correspondence $E_{\tau}^{\rho} \leftrightarrow E_{\rho}^{\tau}$ can not be true by the following reason: | |
| Nov 10, 2010 at 16:27 | comment | added | Tim Perutz | Mohammad, Kevin is correct. A special Lagrangian (if one exists, which is rarely known) gives a preferred representative within a Hamiltonian isotopy classes of Lagrangians. Hamiltonian-isotopic Lagrangians are quasi-isomorphic objects of the Fukaya category. | |
| Nov 10, 2010 at 15:10 | comment | added | Mohammad Farajzadeh-Tehrani | No in Fukaya category they work with special Lagrangians, for many reasons. set of all Lagrangians is very huge and is not suitable. | |
| Nov 10, 2010 at 5:22 | comment | added | Kevin H. Lin | @Mohammad: The special Lagrangians change as the complex structure changes, but the Lagrangians do not change. The Fukaya category has objects Lagrangians, not special Lagrangians. | |
| Nov 10, 2010 at 2:10 | comment | added | Mohammad Farajzadeh-Tehrani | Infact there is a lack of symmetry in the homological mirror symmetry which I can not digest: The A-side (Special Lagrangians) depends on both complex structure and symplectic structure but the B-side depends only on complex structure. I wish some body can explain it to me. | |
| Nov 10, 2010 at 2:08 | comment | added | Mohammad Farajzadeh-Tehrani | At the first glance this seems to be the answer, but I think it is not. Here is the reason: Consider $E_{\tau}^{\rho}$ in the A-side while $\rho$ is fixed and $\tau$ is moving. As $\tau$ moves the special Lagrangians deform and the map $m_2$ of $A_{\infty}$ structure change (due to the change of area of holomorphic triangles) so the A-side deforms. But from your claim the B-model is $E_{\rho}^{\tau}$ and so the complex structure is fixed and nothing changes. | |
| Nov 10, 2010 at 1:26 | history | answered | Arend Bayer | CC BY-SA 2.5 |