Topological recursion in enumerative geometry and random matrices
Topological recursion in enumerative geometry and random matrices
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DOI：
10.1088/1751-8113/42/29/293001
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发表时间：
2009-07
期刊：
影响因子：
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通讯作者：
B. Eynard;N. Orantin
中科院分区：
文献类型：
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作者：
B. Eynard;N. Orantin
我们回顾了最近引入的辛不变量方法在所谓的拓扑展开中求解矩阵模型的环方程，并进一步推广到矩阵模型之外。对于任何给定的谱曲线，定义一个微分形式序列和一个称为辛不变量的复数序列Fg。我们回顾了Fg的定义，并解释了它们的主要性质，特别是辛不变性、可积性、模性，以及它们的极限和变形。然后，我们给出了几个应用实例，特别是矩阵模型，离散曲面（映射）的枚举，代数几何和拓扑弦，以及非相交布朗运动。
We review the method of symplectic invariants recently introduced to solve matrix models' loop equations in the so-called topological expansion, and further extended beyond the context of matrix models. For any given spectral curve, one defines a sequence of differential forms and a sequence of complex numbers Fg called symplectic invariants. We recall the definition of Fg's and we explain their main properties, in particular symplectic invariance, integrability, modularity, as well as their limits and their deformations. Then, we give several examples of applications, in particular matrix models, enumeration of discrete surfaces (maps), algebraic geometry and topological strings, and non-intersecting Brownian motions.