Studies Of Quantum Spectral Problems In Topological String Theory

In recent years, the researches in topological string theory on local Calabi-Yau manifold revealed that there exist some non-perturbative effects for many physical quantities. The formulations of these non-perturbative effects can be determined, in some extent, by asking the singularities coming from non-perturbative contribution can-cel the singularities coming from the perturbative contributions. In this paper, we study these non-perturbative effects from the aspect of spectral problems of some local Calabi-Yau manifolds.In chapter1, we introduce the spectral problem that we are going to talk about in this paper, based on the results given in the literatures. Through the mirror curve of local Calabi-Yau manifold, we can construct a quantum operator and then result in a well-defined spectral problem. This will be demonstrated in chapter2.In chapter3, we talk about the spectral problem corresponding to local P2model. After solving the quantum Hamiltonian, we first consider the perturbative case of small Plank constant h. Using Bohr-Sommerfeld quantization condition and time-independent perturbation theory respectively, we compute the perturbative energy spectral and yield same results. The periods of this model can be computed perturbatively by the re-fined topological string amplitude in the Nekrasov-Shatashvili limit, which further give the exact perturbative phase volume. However, there exist singularities in this vol-ume, which become significant when the Plank constant is large. In order to cancel these singularities, we introduce some non-perturbative contribution and then obtain a well-defined total phase volume, which further deduce the energy spectral by Bohr-Sommerfeld quantization condition. We take different values of n, compare the results of energy spectral from this method with numerical results, and find that it is not enough by only asking the singularities coming form different parts cancel each other. There-fore, instead, we bring some higher order non-perturbative corrections in the phase vol-ume. These higher order corrections will not contribute new singularities, but can give revised energy spectral which match the numerical results better. We also talk about the case when hâ†'oo, and give the asymptotical results.In chapter4, we continue to study the energy spectral of local P1×P1model. We first consider the energy spectral for the most symmetric case of local P1×P1model, using the same method as local P2model. We find there also exist some higher order non-perturbative corrections in the phase volume and we give part formulations of these corrections. Then we consider ABJM matrix model. This model can be related to the refined topological string on local P1x P1model by a coordinate transformation. Thus we can calculate the energy spectral of this model similar as before. These spectral problem has already been studied in the literatures. Based on the non-perturbative con-tribution given already, we find some new higher order non-perturbative corrections, the formulation of which is very similar to the most symmetric case of local P1×P1model.In chapter5, we talk about the energy spectral of local F1model, where there are also some corrections to the non-perturbative contribution of phase volume. However, the corrections are very small and appear until to the8th order, which thus would be hard to notice if we do not know there may be higher order non-perturbative corrections in this kind of models.Finally, in chapter6, we will give a summary and some related outlook.