The gauge/string correspondence postulates an equivalence between certain string theories and gauge theories at large N. Topological strings are simplified versions of string theory, and one could suspect that they have even simpler dual descriptions. In this talk I present a conjectural correspondence between topological string theory on toric Calabi-Yau manifolds, and the spectral properties of quantum-mechanical operators on the real line, in the spirit of large N dualities. The operators turn out to be quantum versions of the algebraic curves which define the mirror manifolds to the Calabi-Yau manifolds.
This conjecture provides a rigorous, non-perturbative definition of topological string theory on these backgrounds. In addition, it gives a precise and explicit prediction for the Fredholm determinants and spectra of the corresponding operators, providing in this way a new class of exactly solvable problems in spectral theory.