精炼拓扑递归再探——性质与猜想
Refined topological recursion revisited -- properties and conjectures
AI总结：
本文针对任意超椭圆曲线定义了超椭圆精炼谱曲线与拓扑递归，推广并改进现有表述，揭示其几何结构及部分猜想性质，还建立Nekrasov-Shatashivili极限下的新递归并证明对应量子曲线存在。
摘要：
对于任意（可能奇异的）超椭圆曲线，我们给出了超椭圆精炼谱曲线和超椭圆精炼拓扑递归的定义，推广了Kidwai与作者针对零亏格特殊曲线类的表述，也改进了Chekhov和Eynard的方案。在此过程中，我们揭示了超椭圆精炼拓扑递归背后的基本几何结构并研究其性质——由于计算困难，部分性质仍为猜想。此外，我们建立了在所谓Nekrasov-Shatashivili极限下有效的新递归，并证明了对应量子曲线的存在性。
英文摘要：
For any (possibly singular) hyperelliptic curve, we give the definition of a hyperelliptic refined spectral curve and the hyperelliptic refined topological recursion, generalising the formulation for a special class of genus-zero curves by Kidwai and the author, and also improving the proposal by Chekhov and Eynard. Along the way, we uncover a fundamental geometric structure underlying the hyperelliptic refined topological recursion and investigate its properties -- parts of which remain conjectural due to computational difficulties. Moreover, we establish a new recursion valid in the so-called Nekrasov-Shatashivili limit and prove existence of the corresponding quantum curve.
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