来自q矩阵模型的拉格朗日簇
Lagrangian varieties from $q$-matrix models
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- MIPT（莫斯科物理技术学院）
- NRC ”Kurchatov Institute”（库尔恰托夫国家研究中心）
- Uppsala University（乌普萨拉大学）
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中文总结 AI 辅助
本文研究三种q变形矩阵模型，将其逆特征多项式的关联函数解释为拉格朗日子簇的量子化，为分析矩阵模型关联函数的半经典大N展开提供几何框架，揭示其与开拓扑弦几何的关联。
中文摘要 AI 辅助
我们研究三种特定的q变形矩阵模型，即陈-西蒙斯（Chern-Simons）矩阵模型、q-拉盖尔（q-Laguerre）矩阵模型和q-高斯（q-Gaussian）矩阵模型，这些模型可利用超可积性性质显式求解。我们证明，逆特征多项式的单点和两点函数可分别解释为(ℂ*)²和(ℂ*)⁴中拉格朗日子簇的量子化。虽然陈-西蒙斯矩阵模型的这类几何图像已被预期，但我们关于q-拉盖尔和q-高斯模型的结果是新的，且呈现出额外特征，特别是特定的反辛双有理对合在构造中起关键作用。这种重新表述为分析矩阵模型关联函数的半经典大N展开提供了几何框架，该图像暗示了开拓扑弦背后的几何，尽管两种构造并不相同，二者间的确切关系仍有待理解。
英文摘要
We consider three specific $q$-deformed matrix models (the Chern-Simons, $q$-Laguerre, and $q$-Gaussian matrix models) which can be solved explicitly using the property of superintegrability. We show that one- and two-point functions of inverse characteristic polynomials can be interpreted as quantizations of Lagrangian subvarieties in $(\mathbb{C}^*)^2$ and $(\mathbb{C}^*)^4$, respectively. While such a geometric picture is expected for the Chern-Simons matrix model, our results for the $q$-Laguerre and $q$-Gaussian models are new and exhibit additional features. In particular, specific anti-symplectic birational involutions play an essential role in the construction. This reformulation provides a geometric framework for analyzing the semiclassical, large-$N$ expansion of matrix-model correlators. This picture is suggestive of the geometry underlying open topological strings, although the two constructions are not identical and the precise relation between them remains to be understood.
