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研究的中心对象。在过去的几十年里，出现了Calabi-Yau流形的微妙不变量，通常在数学和物理中有相似的描述。在数学中，不变量来源于与Calabi-Yau流形相关的各种模空间的几何；在物理学中，它们产生于与Calabi-Yau流形相关的各种量子场论。******这些不变量已经被证明具有惊人丰富的结构，并且与数学的其他分支有着惊人的联系，因此在过去的15年里成为了密集研究的对象。这种不变量的中心例子是所谓的Donaldson-Thomas不变量。从几何上讲，这些是对歧管上的轴的微妙“计数”，特别是，它们可以计算曲线在歧管内部和周围移动的方式。在物理上，这些不变量对应于一定弦理论中d膜状态的计数。粗略地说，计数告诉了相关量子理论的粒子谱。******在过去的几年里，Donaldson-Thomas理论和数论之间出现了惊人而深刻的联系。通过一些研究者的一系列计算和猜想，我们注意到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为一个函数，而Jacobi模形式是数论中出现的具有非凡对称性的函数，并且已经以各种形式研究了数百年。几何和数论之间的这种惊人联系似乎出现在一类特殊的Calabi-Yau流形中，即那些椭圆纤维的流形。******该项目的目标是通过检查特定的几何形状和开发计算配分函数的新技术来完善和深化我们对这种猜想现象的理解。我最近开发了一种新的计算工具，它对这类几何非常有效，我已经看到了雅可比形式如何从几何中出现的诱人暗示。充分理解这一现象将为雅可比模形式的古老主题和卡拉比-丘流形的几何和物理新主题带来新的启示。

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.