Topological Strings on Local Curves

Abstract

We review some perturbative and nonperturbative aspects of topological string theory on the Calabi-Yau manifolds X p = O(−p) ⊕ O(p − 2) → P 1 . These are exactly solvable models of topological string theory which exhibit a nontrivial yet simple phase structure, and have a phase transition in the universality class of pure two-dimensional gravity. They don't have conventional mirror description, but a mirror B model can be formulated in terms of recursion relations on a spectral curve typical of matrix model theory. This makes it possible to calculate nonperturbative, spacetime instanton effects in a reliable way, and in particular to characterize the large order behavior of string perturbation theory.

Lectures on non‐perturbative effects in large N gauge theories, matrix models and strings

“…However, the geometric or physical meaning of this trans-series is not yet known. The Hurwitz model that we have discussed can be generalized to a family of topological string theories on non-compact Calabi-Yau manifolds (the so-called "local curves") where nonperturbative effects can be also computed by using matrix model techniques, see [93,100,95].…”