Local Calabi-Yau,Quantum Curves And Refined Topological Strings

Topological string theory on Calabi-Yau manifolds is one of the most beautiful and productive fields in mathematical physics.In mathematics,the mirror symmetry in topological string theories connects the symplectic geometry and complex geometry on Calabi-Yau manifolds.This thesis studies several problems in topologcial string theory.The first part confirms the n=3 case of Haghighat-Lockhart-Vafa conjecture which states n pairs of E-strings can be recombined into n heterotic strings.In mathematics,this concerns the properties of local half K3 Calabi-Yau manifold and some highly-nontrivial identities among E8 Weyl-invariant Jacobi forms.The second part studies the exact quantization conditions of the mirror curves of local Calabi-Yau manifolds.We establish the equivalent conditions between the exact Nekrasov-Shatashivili quati-zation and the Grassi-Hatsuda-Marino conjecture.For a mirror curve of genus g,the NS quantization scheme leads to g quantization conditions for the corresponding integrable system.The exact NS quantization conditions can be derived from the Lockhart-Vafa partition function of non-perturbative topological string.On the other hand,Grassi-Hatsuda-Marino conjecture there is a single quantization condition and the spectra are encoded in the vanishing of a quantum Riemann theta function.We demonstrate that there actually exist at least g nonequivalent quantum Riemann theta functions and the intersections of their theta divisors coincide with the spectra determined by the exact NS quantization conditions.The equivalence between these two quantization schemes is highly nontrivial and gives infinite constraints among the refined Gopakumar-Vafa invariants.