发表机构
North Carolina State University（北卡罗来纳州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究结合复代数几何与精确WKB方法，解决了高亏格势下P/NP关系是否存在的问题，推导了适用于任意亏格势的全阶量子P/NP关系，并针对特定非谐振荡器明确给出了结果。
AI 中文摘要
在亏格1势的量子力学中，拓扑平凡鞍点附近的微扰论与非微扰瞬子鞍点通过P/NP关系建立了构造性关联。这类关系是否适用于更高亏格势仍是未解决的问题。本研究结合复代数几何工具与精确WKB方法解决了该问题。经典能量守恒关系$p^2 = 2(E-V(q))$定义了亏格为g的黎曼面X，其周期通过Abel-Jacobi映射自然由雅可比簇$J(X) = \text{C}^g/\text{Λ}$组织，也独立由相关Picard-Fuchs方程的解基组织。我们证明，与亏格1情形不同，当$g\neq2$时，物理相关的WKB活跃周期通常通过$\text{SL}(2g,\text{Z})$中的线性变换与Picard-Fuchs基关联，而非辛群$\text{Sp}(2g,\text{Z})$中的变换；这种不匹配是导致明确的高亏格P/NP关系难以获得的障碍。我们证明，通过采用经适当变换的相交矩阵构建修正的黎曼双线性恒等式可解决该问题，而WKB活跃周期恰好满足该恒等式。最终得到适用于任意亏格势的全阶量子P/NP关系，我们针对亏格2（五次和六次）及亏格3（七次和八次）非谐振荡器明确推导了该关系。
英文摘要
In quantum mechanics of genus-1 potentials, perturbation theory around the topologically trivial saddle and non-perturbative instanton saddle are constructively related via the P/NP relation. Whether such a relation persists for higher-genus potentials has remained an open problem. In this work, we resolve it using tools from complex algebraic geometry combined with exact WKB. The classical energy conservation relation $p^2 = 2(E-V(q))$ defines a genus-g Riemann surface $X$, whose periods are naturally organized by its Jacobian variety $J(X) = \mathbb{C}^{\text{g}}/Λ$ via the Abel-Jacobi map, and independently by the solution basis of the associated Picard-Fuchs equations. We show that the physically relevant, WKB-active cycles are generally related to the Picard-Fuchs basis by a linear transformation that lies in $\text{SL}(2\text{g},\mathbb{Z})$ but not in the symplectic group $\text{Sp}(2\text{g},\mathbb{Z})$ once $\text{g}\geq2$, in contrast to the genus-1 case, where the two groups coincide. This mismatch is the obstruction that has made an explicit higher-genus P/NP relation elusive. We show that it can be resolved by passing to a modified Riemann bilinear identity, built from a suitably transformed intersection matrix, which the WKB-active periods do satisfy exactly. The result is an all-orders quantum P/NP relation valid for potentials of arbitrary genus, which we derive explicitly for genus-2 (quintic and sextic) and genus-3 (septic and octic) anharmonic oscillators.
Comments34 pages, 4 figures
