From quantum curves to topological string partition functions. (English) Zbl 1522.81336
Summary: This paper describes the reconstruction of the topological string partition function for certain local Calabi-Yau (CY) manifolds from the quantum curve, an ordinary differential equation obtained by quantising their defining equations. Quantum curves are characterised as solutions to a Riemann-Hilbert problem. The isomonodromic tau-functions associated to these Riemann-Hilbert problems admit a family of natural normalisations labelled by the chambers in the extended Kähler moduli space of the local CY under consideration. The corresponding isomonodromic tau-functions admit a series expansion of generalised theta series type from which one can extract the topological string partition functions for each chamber.
MSC:
| 81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |
| 14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |
| 14H15 | Families, moduli of curves (analytic) |
| 35Q15 | Riemann-Hilbert problems in context of PDEs |
| 14D21 | Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) |
| 34M56 | Isomonodromic deformations for ordinary differential equations in the complex domain |
| 11F27 | Theta series; Weil representation; theta correspondences |
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