This chapter discusses a particular class of Calabi-Yau geometries characterized by being non-compact, focusing on non-compact toric Calabi-Yau threefolds. These are threefolds that have the structure of a fibration with torus fibres. The manifolds have the structure of a fibration of IR3 by T2 x IR. It turns out that the geometry of these threefolds can be packaged in a two-dimensional graph that encodes the information about the degeneration locus of the fibration. These graphs are called the toric diagrams of the corresponding Calabi-Yau manifolds. A general introduction to the construction of non-compact Calabi-Yau geometries is presented, and the toric approach is discussed. Examples of closed string amplitudes are given.
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