I'm having some problems to understand the geometry involved in the section 3.1 of the paper Two Dimensional Yang-Mills, Black Holes and Topological Strings.

Concretely, I was wondering to know if it is possible to visualize the geometry of the $X = \mathcal{L}^{-m} \oplus \mathcal{L}^{m} \rightarrow T^{2}$ Calabi-Yau threefold by means of toric diagrams, in other words, how should I draw the image of the moment map of $X$ in "the physics way" $(\ast)$. Here $T^{2}$ is a $2$-torus, $\mathcal{L}^{m}$ is the line bundle characterized by the fact that a holomorphic section of $\mathcal{L}^{m} \rightarrow T^{2}$ has a divisor of degree $m$ on $T^{2}$ and $\mathcal{L}^{-m}$ is the inverse bundle of $\mathcal{L}^{m}$.

$(\ast)$ By "the physics way" of draw toric diagrams I mean by considering non-compact toric Calabi-Yau manifold $X$ as $T^{2} \times \mathbb{R}$ fibrations over the image of the moment map of $X$ (see CY 3-folds are $T^2 \times \mathbb{R}$ fibrations over the base $\mathbb{R}^3$. What does it mean?). Below to the left I give the example of the toric diagram of $\mathbb{C}^{3}$ viewed as a $T^{2} \times \mathbb{R}$-fibration over $\mathbb{R}^{3}_{\geq 0}$; $D_{1}$,$D_{2}$ $D_{3}$ are the $2$-dimensional cones of $\mathbb{R}^{3}_{\geq 0}$. This example is discussed in detail in Topological strings and their physical applications (example 3.1, page 16). Below to the right represents the toric diagram of the cotangent bundle to a $\mathbb{P}^{2}$ embedded on a threefold (see Branes, Black Holes and Topological Strings on Toric Calabi-Yau Manifolds, section 3, page 8). Here $D_{0},D_{1}$,$D_{2}$ $D_{3}$ are the non-trivial divisors of the geometry where $D_{0}$ is a $\mathbb{P}^{2}$ and the remaining $\mathcal{O}(-p) \rightarrow \mathbb{P}^{1}$ bundles (the relevant $\mathbb{P}^{1}$ are the edges of $\mathbb{P}^{2}$).

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My problem that I can is that I can't draw the toric diagram of $\mathcal{L}^{-m} \oplus \mathcal{L}^{m} \rightarrow T^{2}$. My faliure goes back to the fact that I have no clue on how to draw a fibration over a codimension 2 cycle ($T^{2}$). I'm unable to draw the toric diagram of $\mathcal{L}^{m} \rightarrow T^{2}$ for example.

Any hint, comment or reference is welcomed!



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