用于(2,p)极小刘维尔引力的显式四角字典
An explicit four-corner dictionary for (2,p) minimal Liouville gravity
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中文总结 AI 辅助
本文明确了(2,p)极小刘维尔引力四种代数描述间的对应关系，验证其三点比率与共形场论一致，实现谱曲线描述的x↔y互换并推导了零亏格下的变换框架。
中文摘要 AI 辅助
(2,p)极小刘维尔引力存在四种通过对偶性相互关联的代数描述。在弗罗贝尼乌斯流形（FM）侧，该理论可基于Q(y)=y²+u₁的A₁流形（y侧），或基于变量x的A_{p-1}流形（x侧）构建；在谱曲线/拓扑递归（SC）侧，对应的选择为切比雪夫曲线及其x↔y互换。FM-y、SC-y需要KdV时间与刘维尔耦合之间的共振变换，另外两种则不需要。我们在无散tau-函数层面明确了四种方法间的对应关系。所有四种表述中，与归一化无关的三点比率均与共形场论一致，匹配全部振幅可确定每个插入项的单个因子，该因子可再现带符号的Verlinde矩阵。联系两种谱曲线描述的x↔y互换（近期由Dekinga、Shadrin和Verlinde在一般情形下确立），在零亏格时表现为带势Φᵖ/p的广义Kontsevich积分变换的树级框架变换，推导在(2,5)至(2,11)模型中得到验证。
英文摘要
(2,p) minimal Liouville gravity admits four algebraic descriptions interrelated by dualities. On the Frobenius manifold (FM) side the theory is built either on the $A_{1}$ manifold of $Q(y)=y^{2}+u_{1}$ ($y$-side) or on the $A_{p-1}$ manifold in the variable $x$ ($x$-side). On the spectral curve / topological recursion (SC) side the analogous choice is the Chebyshev curve and its $x\leftrightarrow y$ swap. FM-$y$, SC-$y$ require a resonance transformation between the KdV times and the Liouville couplings, while the other two do not. We work out the explicit correspondence between the four approaches at the level of the dispersionless tau-function. The normalisation-independent three-point ratios agree with conformal field theory in all four formulations, and matching the full amplitudes fixes a single per-insertion factor reproducing the signed Verlinde matrices. We use the Kharchev--Marshakov integral transform of the basis functions, whose determinant gives an explicit relation between the corresponding tau-functions and realises the $x\!\leftrightarrow\! y$ swap at genus zero. Once the independently fixed linear SC-$x$ normalisation is imposed, the Laurent deformation in the second coordinate gives an alternative genus-zero derivation of the known compact resonance transformation for the whole (2,p) series, including all mixed coefficients.
发表机构
- Center for Geometry and Physics, Institute for Basic Science (IBS)（基础科学研究所几何与物理中心）
- Department of Physics, Ariel University（阿里埃尔大学物理系）
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