Modularity of quantum invariants of Calabi-Yau threefolds

背景：卡-丘流形是代数几何数学和弦论物理的中心研究对象。在过去的几十年里，卡-丘流形的微妙不变量已经出现，通常在数学和物理学中有平行的描述。在数学中，不变量产生于与卡-丘流形相关的各种模空间的几何学;在物理学中，它们产生于与卡-丘流形相关的各种量子场论。 这些不变量已经被证明具有令人惊讶的丰富的结构和令人惊讶的许多连接到其他数学分支，并因此成为对象的紧张研究，在过去的15年。这种不变量的中心实例是所谓的唐纳森-托马斯不变量。几何上，这些是流形上层的微妙“计数”，特别是，它们可以计算曲线可以在流形内部和周围移动的方式。物理上，这些不变量对应于一个确定的弦理论中D-膜态的计数。粗略地说，这些计数说明了关联量子理论的粒子谱。 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刻联系。通过一些研究者的一系列计算和计算，人们注意到Donaldson-Thomas配分函数通常是由Jacobi模形式给出的。卡拉比-丘流形的唐纳森-托马斯配分函数将所有这些几何不变量编码成一个函数，而雅可比模形式是具有非凡对称性的函数，这些对称性出现在数论中，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发生在一类特殊的卡-丘流形上，即那些椭圆纤维。 该项目的目标是通过研究特定的几何形状和开发计算配分函数的新技术来完善和加深我们对这种几何现象的理解。最近开发了一种新的计算工具，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线索。充分理解这一现象将为雅可比模形式这一古老的课题和卡-丘流形的几何和物理这一较新的课题带来新的启发。

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.