Topological Strings And Quantum Mechanics

Non-perturbative phenomena are ubiquitous in quantum theories.However,we do not have an effective method studying it in general situations.Based on topological strings and quantum mechanics correspondence,we suggest exact quantization con-ditions of a class of relativistic quantum integrable systems.We name this method of quantization as Nekrasov-Shatashvili(NS)quantization scheme.Another scheme of quantization consider the spectrum of operators constructed from the quantum inte-grable systems.The quantization conditions turn out to be vanishing of the spectrum determinants of these operators and the solution is a set of divisors.This kind of method of quantization was first established by Grassi-Hatsuda-Marino(GHM),we name it as GHM quantization scheme.These two quantization schemes are quite different.By in-troducing a set of ri fields,we demonstrate that there are several spectral determinants differ by phases ri,the intersections of these divisors correspond to different spectral determinants coincide with NS quantization conditions.We argue that this statement is guaranteed by a set of identities,which are actually a special case of blowup equations.We then study blowup equations for details and we show that the blowup equations must be modular invariant.This property predicts all the ri fields and solve all BPS invariants of local Calabi-Yau threefolds from perturbative information of topological strings,at least for all the models we have checked in this thesis and previous papers.We also find a certain form of blowup equations exist at generic loci of the moduli space.