Modularity of quantum invariants of Calabi-Yau threefolds

背景：Calabi-Yau流形是代数几何数学和弦理论物理学的中心研究对象。在过去的几十年里，Calabi-Yau流形的微妙不变量已经出现，通常在数学和物理中有类似的描述。在数学中，不变量诞生于与Calabi-Yau流形相关的各种模空间的几何；在物理学中，它们源于与Calabi-Yau流形相关的各种量子场理论。*这些不变量被证明具有惊人的丰富结构，并与数学的其他分支有惊人的联系，因此在过去的15年里成为了密集研究的对象。这种不变量的中心实例是所谓的Donaldson-Thomas不变量。从几何上讲，这些都是歧管上滑轮的微妙“计数”，特别是它们可以计算曲线位于歧管内部和围绕歧管移动的方式。在物理上，这些不变量对应于一定弦理论中的D-膜态的计数。粗略地说，这些计数说明了相关量子理论的粒子光谱。*在过去的几年里，唐纳森-托马斯理论和数论之间出现了令人惊讶的深层次联系。通过几位研究者的一系列计算和猜想，人们注意到Donaldson-Thomas配分函数通常是由Jacobi模形式给出的。Calabi-Yau流形的Donaldson-Thomas配分函数将所有这些几何不变量编码成一个函数，而Jacobi模形式是出现在数论中的具有特殊对称性的函数，数百年来人们以各种形式对其进行了研究。几何和数论之间的这种惊人的联系似乎发生在一类特殊的Calabi-Yau流形上，即那些椭圆纤维流形。*这个项目的目标是通过研究特定的几何结构和开发计算配分函数的新技术来完善和加深我们对这一猜想现象的理解。最近开发了一种新的计算工具，对这些类型的几何非常有效，我看到了一些诱人的迹象，说明雅可比形式是如何从几何中出现的。充分理解这一现象将为雅可比模形式这一古老的主题和Calabi-Yau流形的几何和物理这一较新的主题带来新的启示。

Background: Calabi-Yau manifolds are central objects of research in both the mathematics of algebraic geometry and the physics of string theory. In the last few decades, subtle invariants of Calabi-Yau manifolds have arisen, often having parallel descriptions in math and physics. In math, the invariants are born from the geometry of various moduli spaces associated to the Calabi-Yau manifold; in physics, they arise out of various quantum field theories associated to the Calabi-Yau manifold.******These invariants have turned out to have amazingly rich structure and surprisingly many connections to other branches of mathematics and have consequently become the objects of intense study in the last 15 years. The central instance of such invariants are the so-called Donaldson-Thomas invariants. Geometrically, these are subtle "counts" of sheaves on the manifold, in particular, they can count the ways that curves can sit inside and move around the manifold. Physically, these invariants correspond to counts of D-brane states in a certainly string theory. Roughly speaking, the counts tell about the particle spectrum of the associated quantum theory. ******In the last few years, a surprising and deep connection between Donaldson-Thomas theory and number theory has emerged. Through a series of computations and conjectures of several researchers, it has been noticed that the Donaldson-Thomas partition function is often given by a Jacobi modular form. The Donaldson-Thomas partition function of a Calabi-Yau manifold encodes all these geometric invariants into a single function, whereas Jacobi modular forms are functions with extraordinary symmetries which arise in number theory and have been studied in various forms for hundreds of years. This amazing connection between geometry and number theory appears to occur for a special class of Calabi-Yau manifolds, namely those which are elliptically fibered. ******The goal of this project is to refine and deepen our understanding of this conjectural phenomenon both by examining specific geometries and by developing new technology for computing partition functions. Having recently developed a new computational tool which is very effective for these sorts of geometries, I've seen tantalizing hints of how Jacobi forms emerge from the geometry. Fully understanding this phenomenon will shed new light on both the venerable subject of Jacobi modular forms and the newer subject of the geometry and physics of Calabi-Yau manifolds.