Gopakumar and Vafa demonstrated in an important paper (1999) that there is a closed string theory leading to the resummations (2.179) and (2.181). The intuition behind the result of Gopakumar and Vafa is that open/closed string dualities are related to geometric transitions in the background geometry. Since Chern-Simons theory is an open topological string on the deformed conifold geometry with N topological D-branes wrapping the three-sphere, it is natural to conjecture that at large N the D-branes induce a conifold transition in the background geometry. This yields the resolved conifold and no D-branes. But in the absence of D-branes that enforce boundary conditions, a theory of closed topological strings remains. Following this reasoning, Gopakumar and Vafa conjectured that Chern-Simons theory on S3 is equivalent to closed topological string theory on the resolved conifold. This chapter analyzes geometric transitions for Chern-Simons theory and type-A topological strings as well as matrix models and type-B topological strings.
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