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  How exactly do we construct the T2×R toric Calabi-Yau three-fold?

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I am trying to understand why and how the functions ra(z)=|z1|2|z3|2, rb(z)=|z2|2|z3|2 and rc(z)=(z1z2z3) "generate" the toric CY threefold T2×R fibration. In these lecture notes in section D.1 the author says that the above three functions (Hamiltonians) generate a flow on C3 via the symplectic form ω=idzidˉzi

and the Poisson bracket uzi={ru,zi}ω
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 eiara+ibrb 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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