Topological recursion, the Laplace transform, and integrable systems

AbstractAward：DMS-1104734首席研究员：Motohico Mulase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。该项目的一个特别强调的是放在发展的拉普拉斯变换作为镜像映射的观点。该项目的目标是解决由于弦理论物理学家马里诺和Bouchard-Klemm-Marino-Pasquetti在代数几何中的重塑猜想。重塑猜想给出了一个具体而通用的递推公式，它基于Riemann曲面的复分析，计算任意复曲面Calabi-Yau 3-fold的封闭和开放Gromov-Witten不变量。 Bouchard和Marino提出了一个关于Hurwitz数的猜想，作为Remodeling猜想的一个极限情形.他们定义了简单Hurwitz数的生成函数序列，并证明了这些量满足Eynard-Orant递推公式。Bouchard-Marino猜想在2009-10年由PI和他的合作者解决。自PI的论文发表以来，目前的理解如下。复曲面Calabi-Yau 3-fold的Gromov-Witten理论是拓扑弦理论A-模型侧的组合计数问题。这个计数问题的解的母函数满足一个组合方程，周称之为割接方程。 现在对这个函数进行拉普拉斯变换。其结果是一个对称亚纯函数定义在黎曼曲面的乘积上。 这个Riemann曲面被确定为复曲面Calabi-Yau 3-fold的镜像曲线。组合方程的Laplace变换在B-模型侧变成Eynard-Orantin递归。 PI对这一总体图景的贡献是将拉普拉斯变换的作用确定为镜像映射。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兴奋。这种兴奋使科学和工程专业的年轻学生充满活力。PI一直在与本科生和研究生合作，让他们参与到研究兴奋的核心中。这些学生没有窥视研究经验，而是参与了真正令人兴奋的数学发现。这些持久的影响激发了学生成为研究数学家，他们继续产生新的结果。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。经过沿着时间在各个领域的卓有成效的发展，我们又一次回到了一个新的互动水平。而不是推动一个现有的问题或猜想，拟议的项目旨在了解一个新的观点。预计将带来进一步的交叉施肥这些相当不同的想法/领域的数学科学。 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.