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I've been trying to reconcile two seemingly different definitions of what a toric space is, specifically a toric Calabi-Yau 3-fold. The first definition is from the paper Branes and Toric Geometry, by Leung and Vafa (arXiv:hep-th/9711013) who say (on page 12, section 2.1):

We are interested in manifolds admitting a $T^n$ action, with an $n$-dimensional base.

This seems to suggest that a toric Calabi-Yau $3$-fold (generalizing the construction in the paper) is a $T^3$ fibration over a real $3$-dimensional base.

However, in the thesis Crystal Melting and Wall Crossing Phenomena, by Yamazaki (arXiv:1002.1709 [hep-th]), the author explicitly says (on page 16, in section 3.2):

A toric Calabi-Yau threefold $X_\Delta$ is a $T^2 \times \mathbb{R}$ fibration over $\mathbb{R}^3$, where the fibers are special Lagrangian submanifolds.

Now I understand that the phrase "special Lagrangian submanifold" implies the existence in $X_{\Delta}$ of a special Lagrangian submanifold. But I would like to know the sense in which these two seemingly different definitions are equivalent. Any references to a further discussion about toric spaces would is very welcome! (I am a physicist learning about the underlying mathematical structures. I would not like to shy away from the formal definitions, so math references are particularly welcome.)

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  • $\begingroup$ The answer is in section 4.3 of Vincent Bouchard's lecture notes at arxiv.org/abs/hep-th/0702063. I would like to see the Lagrangian submanifold structure more explicitly but I guess I just need to work it out. $\endgroup$ – leastaction Mar 30 at 15:22

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