Abstract loop equations, topological recursion and new applications

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2015-01-01 DOI:10.4310/CNTP.2015.V9.N1.A2

G. Borot, B. Eynard, N. Orantin

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引用次数: 57

Abstract

loop equations, topological recursion and applications Gaetan Borot, Bertrand Eynard, Nicolas Orantin Abstract We formulate a notion of ”abstract loop equations”, and show that their solution is provided by a topological recursion under some assumptions, in particular the result takes a universal form. The Schwinger-Dyson equation of the one and two hermitian matrix models, and of the Opnq model appear as special cases. We study applications to repulsive particles systems, and explain how our notion of loop equations are related to Virasoro constraints. Then, as a special case, we study in detail applications to enumeration problems in a general class of non-intersecting loop models on the random lattice of all topologies, to SUpNq Chern-Simons invariants of torus knots in the large N expansion. We also mention an application to Liouville theory on surfaces of positive genus.We formulate a notion of ”abstract loop equations”, and show that their solution is provided by a topological recursion under some assumptions, in particular the result takes a universal form. The Schwinger-Dyson equation of the one and two hermitian matrix models, and of the Opnq model appear as special cases. We study applications to repulsive particles systems, and explain how our notion of loop equations are related to Virasoro constraints. Then, as a special case, we study in detail applications to enumeration problems in a general class of non-intersecting loop models on the random lattice of all topologies, to SUpNq Chern-Simons invariants of torus knots in the large N expansion. We also mention an application to Liouville theory on surfaces of positive genus.

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抽象环方程、拓扑递归及其新应用

摘要本文提出了“抽象环方程”的概念，并在一定的假设条件下证明了它们的解是由拓扑递推提供的，特别是结果具有全称形式。一厄米矩阵模型和二厄米矩阵模型的Schwinger-Dyson方程以及Opnq模型的Schwinger-Dyson方程作为特例出现。我们研究了排斥粒子系统的应用，并解释了我们的循环方程的概念是如何与Virasoro约束相关的。然后，作为一个特例，我们详细研究了在所有拓扑的随机格上的一般不相交环模型的枚举问题中的应用，即环面结在大N展开中的SUpNq chen - simons不变量。我们还提到了Liouville理论在正属曲面上的一个应用。我们提出了“抽象循环方程”的概念，并在一定的假设条件下，证明了它们的解是由拓扑递推提供的，特别是结果具有全称形式。一厄米矩阵模型和二厄米矩阵模型的Schwinger-Dyson方程以及Opnq模型的Schwinger-Dyson方程作为特例出现。我们研究了排斥粒子系统的应用，并解释了我们的循环方程的概念是如何与Virasoro约束相关的。然后，作为一个特例，我们详细研究了在所有拓扑的随机格上的一般不相交环模型的枚举问题中的应用，即环面结在大N展开中的SUpNq chen - simons不变量。我们还提到了Liouville理论在正属曲面上的一个应用。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.