题目：Quantum Curves, Toric Calabi-Yau and Non-perturbative Topological String
时间：7月14日 周四 下午3:00-4:00
摘要: We establish the precise relation between the Nekrasov-Shatashvili (NS) quantization scheme and Grassi-Hatsuda-Marino conjecture for the mirror curve of arbitrary toric Calabi-Yau threefold. 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 enjoy a self S-duality with respect to Planck constant and can be derived from the Lockhart-Vafa partition function of non-perturbative topological string. Based on a recent observation on the correspondence between spectral theory and enumerative geometry, another quantization scheme was proposed by Grassi-Hatsuda-Marino, in which 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. This highly nontrivial coincidence between the two quantization schemes requires infinite constraints among the refined Gopakumar-Vafa invariants. We also find a set of novel identities among the topological string amplitudes which guarantee the equivalence.