# Drawing the toric diagram for $\mathcal{L}^{m} \rightarrow T^{2}$ geometries.

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 $$\mathbb{P}^{2}$$).

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!