Topological recursion, the Laplace transform, and integrable systems

项目编号：dms -1104734项目负责人：Motohico mulas该项目旨在确定最初在随机矩阵理论和统计物理中发现的eynard - orantin递归公式的几何结构。该项目特别强调的是开发作为镜像映射的拉普拉斯变换的观点。该项目的目标是解决由弦理论物理学家Marino和Bouchard-Klemm-Marino-Pasquetti提出的代数几何中的建模猜想。在黎曼曲面复分析的基础上，给出了一个具体的、通用的递推公式，用于计算任意环面calabi - yau 3-fold的开放和闭合Gromov-Witten不变量。Bouchard和Marino提出了一个关于hurwitz数的猜想，作为重构猜想的极限情况。他们定义了一系列简单Hurwitz数的生成函数，并推测这些量满足Eynard-Orantinrecursion公式。布沙尔-马里诺猜想在2009- 2010年由皮耶和他的合作者解决。自PI的论文发表以来，目前的理解如下。环形Calabi-Yau三折的Gromov-Witten理论是拓扑弦理论a模型侧的一个组合计数问题。这个计数问题的解的生成函数满足一个组合方程，被周称为切割连接方程。现在求这个函数的拉普拉斯变换。结果是一个定义在黎曼曲面积上的对称亚纯函数。这个黎曼曲面被认为是环形Calabi-Yau 3-fold的镜像曲线。组合方程的拉普拉斯变换（推测）成为b模型侧的eynard - orantin递归。PI对这幅图的贡献在于它确定了拉普拉斯变换作为镜像映射的作用。这种类型的拉普拉斯变换在PI最近的论文中得到了进一步的研究，并被其他研究人员用于解决许多相关问题。纯数学研究是关于发现的强烈刺激。这种兴奋激励着理工科的年轻学生。PI一直在与本科生和研究生合作，让他们参与到研究的兴奋之中。这些学生参与了数学发现的真正兴奋，而不是窥视研究经验。这些持久的影响激发了学生们成为研究数学家，并且他们还在继续产生新的结果。代数几何、辛几何、组合学、随机矩阵理论、拓扑学、可积系统和弦理论在格罗莫夫-威滕理论中的关系在20世纪90年代初就很明显了。在每个领域都取得了卓有成效的发展之后，我们又回到了一个新的互动水平。拟议的项目旨在理解一种新的观点，而不是推动现有的问题或猜想。它有望为数学科学中这些完全不同的思想/领域带来进一步的交叉施肥。在过去的两年半里，pi组织了几次专门针对这一主题的研讨会，许多年轻的研究人员参加了这些研讨会。他将继续这样做，以便进一步传播。

AbstractAward: DMS-1104734Principal Investigator: Motohico MulaseThe project is aimed at identifying the geometric structure of theEynard-Orantin recursion formula that was originally discovered inrandom matrix theory and statistical physics. A particular emphasis ofthe project is placed on developing a point of view of the Laplacetransform as the mirror map. The goal of the project is to solve theRemodeling Conjecture in algebraic geometry due to string theoryphysicists Marino and Bouchard-Klemm-Marino-Pasquetti. The RemodelingConjecture presents a concrete and universal recursion formula, basedon the complex analysis of a Riemann surface, that computes bothclosed and open Gromov-Witten invariants of an arbitrary toricCalabi-Yau 3-fold. Bouchard and Marino proposed a conjecture onHurwitz numbers as a limit case of the Remodeling Cojecture. Theydefined a sequence of generating functions of simple Hurwitz numbers,and conjectured that these quantities satisfy the Eynard-Orantinrecursion formula. The Bouchard-Marino conjecture was solved by the PIand his collaborators in 2009-10. The current understanding that hasemerged since the publication of the PI's papers is thefollowing. The Gromov-Witten theory of a toric Calabi-Yau 3-fold is acombinatorial counting problem on the A-model side of a topologicalstring theory. The generating function of the solution to thiscounting problem satisfies a combinatorial equation, called the cut-and-join equation by Zhou. Now take the Laplace transform of thisfunction. The result is a symmetric meromorphic function defined onthe product of a Riemann surface. This Riemann surface is identifiedas the mirror curve of the toric Calabi-Yau 3-fold. The Laplacetransform of the combinatorial equation becomes (conjecturally) theEynard-Orantin recursion on the B-model side. The PI'scontribution to this general picture is the identification of the roleof the Laplace transform as the mirror map. This type of the Laplacetransform was further investigated in the PI's recent papers, andhas been utilized by other researchers in solving many relatedproblems.Pure mathematical research is about a sharp excitement ofdiscovery. This excitement energizes young students in scienceand engineering. The PI has been collaborating with bothundergraduate and graduate students, engaging them into the heartof the research excitement. Instead of peeking into researchexperience, these students have participated in the realexcitement of mathematical discovery. These lasting impacts haveinspired the students to become research mathematicians, and theyare continuing to produce new results. The relation betweenalgebraic geometry, symplectic geometry, combinatorics, randommatrix theory, topology, integrable systems, and string theory inthe Gromov-Witten theory was apparent in the early 1990s. After along time of extremely fruitful developments in each individualarea, we are back to a new level of interaction onceagain. Instead of pushing an existing problem or conjecture, theproposed project is aimed at understanding a new point ofview. It is expected to bring a further cross-fertilization ofthese quite different ideas/areas in mathematical sciences. ThePI has organized several workshops specifically aimed at thissubject in the last two and a half years, which were attended bymany young researchers. He will continue to do so for furtherdissemination.