In this talk, I will describe a conjectural correspondence between certain Calabi-Yau manifolds, which arise as geometric backgrounds in topological string theory, and a new class of trace class operators in one-dimensional quantum mechanics. Using this correspondence, it is possible to find exact and explicit expressions for the Fredholm determinants of these operators in terms of topological string partition functions, i.e. in terms of enumerative invariants of the Calabi-Yau's. We obtain in this way an explicit connection between spectral theory and enumerative geometry, with fruitful implications for both fields. Using similar ideas, we also find exact quantization conditions for the relativistic Toda chain. The correspondence leads to a rigorous, non-perturbative definition of topological string theory on these backgrounds, which can be realized in terms of matrix models.