# Two equivalent definitions of fibration structure on toric CY $3$-fold

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.)

• 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. – leastaction Mar 30 at 15:22