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

该项目旨在通过对黎曼曲面的分析，发现“经典”拓扑不变量与给定空间（如结补）的“量子”拓扑不变量之间的数学关系。在过去的两年中，许多新的见解已经围绕镜像对称和“量子化”拓扑不变量的思想建立了一大类几何空间。最近发展的核心驱动力是物理学家Eynard、Orantin、Marino等人的发现，他们用黎曼曲面上的积分变换来表示量子不变量的递归公式。例如，这个递归公式计算任意一个环形Calabi-Yau三倍的所有属的开和闭Gromov-Witten不变量。这里我们称之为Calabi-Yau空间的经典不变量决定了一个黎曼曲面作为它的镜像对偶。然后用Eynard-Orantin递推公式计算原Calabi-Yau空间的量子不变量，即高格Gromov-Witten不变量。PI与Dumitrescu等人合作发现了一个最简单的理论例子，可以最好地理解这个神奇过程的数学结构。这个例子是基于对加泰罗尼亚数的镜像对称对偶的探索，这使作者得出了一个意想不到的丰富理论。将镜像对称，进而将Eynard-Orantin的量化过程应用于Catalan数，得到：(1)光滑尖曲线模空间的虚庞加莱多项式；(2)稳定点曲线模空间的同义类的交点数。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（量子不变量生成函数的特殊选择）满足“量子曲线”方程，这是一个薛定谔方程。量子曲线的概念是由物理学家Aganagic， Dijkgraaf, Klemm, Marino， Vafa等人提出的。对于所有承认量子曲线的例子，也验证了配分函数是KdV/KP型可积系统的Baker-Akhiezer函数。令人惊讶的是，代数k理论中第二个k群中Steinberg符号的一个扭转条件对所有这些例子都成立。在量子拓扑不变量的背景下，建立对这种高代数k理论条件与黎曼曲面（特别是结a多项式）的量子化性之间关系的数学理解，以及幕后Baker-Akhiezer函数的存在性，是提议项目的目标。纯数学研究常常受到理论物理学的激进思想的启发。镜像对称就是这样一个例子：有两种截然不同的数学方法来描述物理宇宙。由于宇宙是唯一的，我们必须得出结论，这两种数学理论是等价的。“镜像对称”是指这两种理论的关系。将镜像对称的思想进一步扩展到数学上，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？如果我们把加泰罗尼亚数看作一个“经典”对象，那么它们的“量子”概括是什么？私家侦探和他的合作者已经找到了这些问题的肯定答案。令他们惊讶的是，这个答案为理论物理学家提供了一个最简单的数学例子，证明了一个强大的推测理论。物理学理论预测了一个具体的和通用的公式来计算无限系列的特征数（称为不变量）的可能的宇宙。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初级语言中发生的事情，并有助于理解一般理论。在PI和他的合作者建立的具体数学基础上，PI提议研究物理学家提出的关于经典和量子结不变量的新猜想。加泰罗尼亚数的量子推广计算了某些拓扑图。有趣的是，这些数字在生物学中也以复杂分子（如DNA、RNA和蛋白质）的自由能形式出现。例如，量子广义加泰罗尼亚数计算RNA的“二级”结构。初级结构是核苷酸的线性序列。二级结构是指由于结、缠结和桥接结构而使其位置复杂化。作为量子化的结果，这些量子化加泰罗尼亚数的生成函数满足薛定谔方程。这也为另一组与表面量子化有关的基本物理预测提供了一个简单的例子。这些预测包括结理论中的新猜想。拟议的研究项目旨在建立数学上严谨的结果，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。在生物学中复杂分子二级结构的研究中也有一定的应用前景。通过对具有相当复杂性质的神秘和神奇理论的简单实例的构建和分析，PI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，以吸引本科生在令人兴奋的研究前沿。

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.