The B-model topological recursion, holonomic systems, and the integrability

B 模型拓扑递归、完整系统和可积性

• 批准号: 1309298
• 负责人: Motohico Mulase
• 金额: $ 13.07万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-15 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1309298&HistoricalAwards=false
• 关键词: model; topological; recursion; holonomic; systems

The proposed project is aimed at discovering a mathematical relation between the "classical" topological invariants and the "quantum" ones of a given space, such as a knot complement, in terms of the analysis on Riemann surfaces. Within the last two years, many new insights have been established around the idea of mirror symmetry and "quantization" of topological invariants for a large class of geometric spaces. The central driving force of the recent development is the discovery, due mainly to physicists Eynard, Orantin, Marino, and others, of a recursive formula for quantum invariants in terms of integral transforms over Riemann surfaces. For example, this recursion formula computes both open and closed Gromov-Witten invariants for all genera of an arbitrary toric Calabi-Yau three-fold. Here what we call the classical invariants of the Calabi-Yau space determine a Riemann surface as its mirror dual. Then the Eynard-Orantin recursion formula computes the quantum invariants, i.e., the higher-genus Gromov-Witten invariants, of the original Calabi-Yau space. The mathematical structure of this miraculous procedure is best understood by the simplest example of the theory discovered by the PI, in collaboration with Dumitrescu and others. This example is based on the quest of mirror symmetric dual of the Catalan numbers, which has led the authors to an unexpectedly rich theory. The application of the mirror symmetry, and then of the quantization process of Eynard-Orantin, to the Catalan numbers, we obtain: (1) the virtual Poincare polynomials of the moduli spaces of smooth pointed curves; and (2) the intersection numbers of the tautological classes of the moduli spaces of stable pointed curves. For this particular example, and later for many other examples, the PI and his collaborators Bouchard, Shadrin, Sulkowski, and others, have discovered that the partition function (a special choice of the generating function of quantum invariants) satisfies the "quantum curve" equation, which is a Schrodinger equation. The idea of quantum curves is due to physicists Aganagic, Dijkgraaf, Klemm, Marino, Vafa, and others. For all examples that admit the quantum curve, it has also been verified that the partition function is a Baker-Akhiezer function of an integrable system of the KdV/KP type. Very surprisingly, a torsion condition of a Steinberg symbol in the second K-group in algebraic K-theory holds for all these examples. Establishing a mathematical understanding of the relation between this higher algebraic K-theory condition and the quantizability of a Riemann surface (in particular a knot A-polynomial), and the existence of a Baker-Akhiezer function behind the scene, in the context of quantum topological invariants, is the goal of the proposed project.Pure mathematical research is often inspired by radical ideas from theoretical physics. The mirror symmetry is an example of such ideas: there are two very different mathematical ways of describing the physical universe. Since the universe is unique, we must conclude that these two mathematical theories are equivalent. The "mirror symmetry" refers to the relation of these two theories. Further extending the idea of mirror symmetry more mathematically, one arrives at a naive, but also quite radical, question: what is the mirror symmetric partner of Catalan numbers? If we consider Catalan numbers as a "classical" object, then what are the "quantum" generalization of them? The PI and his collaborators have discovered an affirmative answer to these questions. To their surprise, the answer turns out to provide the simplest mathematical example of a powerful speculative theory due to theoretical physicists. The physics theory predicts a concrete and universal formula to calculate an infinite series of characteristic numbers (called invariants) of the possible universe. This is a rather involved theory, and mathematical proofs of the formulas are also complicated. Our simplest example illustrates what is happening in an elementary language, and helps understanding the general theory. Building on the concrete mathematical foundation the PI and his collaborators have established, the PI proposes to study newly proposed conjectures on classical and quantum knot invariants by physicists. The quantum generalization of Catalan numbers count certain topological graphs. It is interesting to note that these numbers also appear in biology as the free energy of complex molecules such as DNA, RNA, and proteins. For example, the quantum generalized Catalan numbers count the "secondary" structures of an RNA. The primary structure is the linear sequence of nucleotides. The secondary structure refers to the complication of its position due to knotted, tangled, and bridged, structures. Being a result of quantization, the generating function of these quantized Catalan numbers satisfies a Schrodinger equation. This also provides a simple example of another set of radical physics predictions related to quantization of surfaces. These predictions include new conjectures in knot theory. The proposed research project is aimed at establishing mathematically rigorous results, verifying speculative predictions from theoretical physics. The work is expected to have impact on several areas of pure mathematics. It is also expected to have an application in the study of secondary structures of complex molecules in biology. Through the construction and analysis of simple examples of the mysterious and miraculous theories with quite involved nature, the PI has been able to attract undergraduate students participating in these research topics. The proposed project contains an REU component to engage undergraduate students in the exciting research frontier.

拟议的项目旨在发现一个数学关系之间的“经典”拓扑不变量和“量子”的一个给定的空间，如一个结补，在黎曼曲面的分析。在过去的两年里，围绕着镜像对称和拓扑不变量的“量子化”的概念，为一大类几何空间建立了许多新的见解。最近发展的中心驱动力是发现，主要是由于物理学家Eynard，Orantin，Marino和其他人，量子不变量的递归公式在黎曼曲面上的积分变换。例如，这个递归公式计算任意复曲面卡-丘三重的所有属的开和闭Gromov-Witten不变量。这里我们称之为卡-丘空间的经典不变量决定了黎曼曲面是它的镜像对偶。然后Eynard-Orantin递归公式计算量子不变量，即，原卡-丘空间的高阶格罗莫夫-威滕不变量。这个神奇的过程的数学结构最好通过PI与Dumitrescu和其他人合作发现的最简单的理论例子来理解。这个例子是基于对Catalan数的镜像对称对偶的追求，这使作者获得了一个意想不到的丰富理论。应用镜像对称性，然后应用Eynard-Orantin的量子化过程，我们得到：（1）光滑尖曲线模空间的虚Poincare多项式;（2）稳定尖曲线模空间重言类的交数。对于这个特殊的例子，以及后来的许多其他例子，PI和他的合作者Bouchard，Shadrin，Sulkowski等人发现配分函数（量子不变量的生成函数的特殊选择）满足“量子曲线”方程，这是薛定谔方程。量子曲线的概念是由物理学家Aganagic，Dijkgraaf，Klemm，Marino，Vafa等人提出的。对于所有允许量子曲线的例子，也已经证实了配分函数是KdV/KP型可积系统的Baker-Akhiezer函数。非常令人惊讶的是，代数K-理论中第二个K-群中斯坦伯格符号的扭转条件对所有这些例子都成立。该项目的目标是在量子拓扑不变量的背景下，建立一个数学上的理解，来理解这种高级代数K理论条件和黎曼曲面（特别是纽结A多项式）的可量子化性之间的关系，以及背后的Baker-Akhiezer函数的存在性。纯数学研究通常受到理论物理学激进思想的启发。镜像对称就是这样一个例子：有两种非常不同的数学方法来描述物理宇宙。既然宇宙是唯一的，我们就必须得出结论，这两个数学理论是等价的。“镜像对称”指的是这两种理论的关系。进一步在数学上扩展镜像对称的概念，我们会得到一个天真但也相当激进的问题：加泰罗尼亚数的镜像对称伙伴是什么？如果我们把卡塔兰数看作一个“经典”对象，那么它们的“量子”推广是什么？PI和他的合作者已经发现了这些问题的肯定答案。令他们惊讶的是，这个问题的答案是理论物理学家提出的一个强有力的思辨理论的最简单的数学例子。物理学理论预言了一个具体的和普遍的公式来计算可能宇宙的无限系列的特征数（称为不变量）。这是一个相当复杂的理论，公式的数学证明也很复杂。我们最简单的例子说明了在基本语言中发生的事情，并有助于理解一般理论。在PI和他的合作者已经建立的具体数学基础上，PI建议研究物理学家新提出的经典和量子结不变量。Catalan数的量子推广计算某些拓扑图。值得注意的是，这些数字在生物学中也作为复杂分子（如DNA、RNA和蛋白质）的自由能出现。例如，量子广义卡塔兰数计算RNA的“二级”结构。一级结构是核苷酸的线性序列。二级结构是指由于打结、缠结和桥接的结构而使其位置复杂化。作为量子化的结果，这些量子化Catalan数的生成函数满足薛定谔方程。这也提供了另一组与表面量子化相关的激进物理学预测的简单例子。这些预言包括纽结理论的新发现。拟议的研究项目旨在建立数学上严格的结果，验证理论物理学的推测性预测。这项工作预计将对纯数学的几个领域产生影响。它也有望在生物学中复杂分子二级结构的研究中得到应用。通过构建和分析具有相当复杂性质的神秘和神奇的理论的简单例子，PI已经能够吸引本科生参与这些研究课题。拟议的项目包含一个REU组件，让本科生参与令人兴奋的研究前沿。