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How exactly do we construct the $T^2\times \mathbb{R}$ toric Calabi-Yau three-fold?

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I am trying to understand why and how the functions $r_{a}(z) = |z_1|^2 - |z_3|^2$, $r_{b}(z)=|z_2|^2 - |z_3|^2$ and $r_{c}(z)=\Im(z_1z_2z_3)$ "generate" the toric CY threefold $T^2 \times \mathbb{R}$ fibration. In these lecture notes in section D.1 the author says that the above three functions (Hamiltonians) generate a flow on $\mathbb{C}^3$ via the symplectic form $$ \omega = \sum_i dz_i \wedge d\bar{z}_i $$ and the Poisson bracket $$ \partial_uz_i = \{ r_u,z_i \}_{\omega} $$ for $u=a,b,c$.

I would like some more detailed and "dumbed down" explanation if possible. Additionally I would like some explanation on why the 2-torus is generated by $e^{iar_a + ibr_b}$ and what exactly do I see when I see the toric diagram! (I know there are some cycles that degenerate along the lines etc but I would like to have some connection with the above construction).

This post imported from StackExchange MathOverflow at 2015-10-21 15:13 (UTC), posted by SE-user user39726
asked Apr 6, 2015 in Theoretical Physics by user39726 (25 points) [ no revision ]
retagged Oct 21, 2015

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