Counting curves with symmetry

枚举代数几何，又名。“曲线计数”，在过去的30年里一直是代数几何的重要组成部分。它指的是通过计算曲线(有时是更一般的物体)坐在空间内的方式的数量而获得的空间的微妙不变量。这个主题起源于理论物理(弦理论)的思想，在理论物理中，曲线可以被认为是弦的世界薄片。对于任何给定的空间，存在无限数量的曲线计数不变量，并且数字集合被编码成称为该空间的配分函数的单个函数。计算空间的配分函数(特别是对于一类称为紧Calabi-Yau三重空间的六维空间)，是代数几何和物理学的主要目标。这个项目的目标是丰富我们对具有额外对称性(包括某些被称为“派生”对称的“隐藏”对称)的空间的曲线计数不变量的理解。一般说来，人们可以计算在对称性下不变的曲线。这将导致新的曲线计数不变量，这反过来将使我们对原始不变量有更好的理解。初步研究表明，对于某些空间，与这些新的曲线计数理论相关的配分函数将以模形式给出-在数论中出现并已被研究了一百多年的非常函数。这在几何、物理和数论这三门完全不同的学科之间提供了强大的联系。本课题的主要目的之一是得到任意紧的Calabi-Yau三元系的配分函数的一个完整而明确的公式。这个项目的一个目标正是要做到这一点：找到一个非常特殊的Calabi-Yau三重流形，即Schoen流形的配分函数的明确的模形式公式。这个想法是，这个空间有大量的对称性，包括上面提到的许多“隐藏的”对称性。这一点，加上引入的新几何概念，应该允许我们给出这个配分函数的完整计算，并以模形式表示它。从来没有人能够计算紧凑的Calabi-Yau三重(弦理论中最相关的几何)的配分函数。拥有这样的函数将使我们深入了解相应的物理理论，从而在几何学和物理学中都将产生重大影响。

Enumerative algebraic geometry, a.k.a. "curve counting", has been a major part of algebraic geometry for the last 30 years. It refers to subtle invariants of spaces obtained by counting the number of ways that curves (or sometimes more general objects) can sit inside a space. The subject has its origin in ideas coming from theoretical physics (string theory) where the curves can be thought of as the world-sheets of strings. There is an infinite number of curve counting invariants for any given space, and collection of numbers is encoded into a single function called the partition function of the space. Computing the partition function of a space (especially for a certain kind of six dimensional space called a compact Calabi-Yau threefold), is a major goal of both algebraic geometry and physics. The goal of this project is to enrich our understanding of curve counting invariants for spaces which have extra symmetries (including certain "hidden" symmetries called "derived" symmetries). Broadly speaking one can count curves which are invariant under the symmetry. This will lead to new curve counting invariants, which will in turn give us greater understanding of the original invariants. Preliminary investigations suggest that for certain spaces, the partition functions associated to these new curve counting theories will be given by modular forms --- extraordinary functions that arise in number theory and have been studied for well over a hundred years. This provides a powerful link between three very different subjects: geometry, physics, and number theory. One of the holy grails in this subject is to obtain a complete and explicit formula for the partition function of any compact Calabi-Yau threefold. One aim of this project is to do exactly that: find an explicit formula for the partition function, in terms of modular forms, of a very special Calabi-Yau threefold, namely the Schoen manifold. The idea is that this space has an enormous number of symmetries, including many of the "hidden" symmetries alluded to above. This, along with new geometric ideas introduced, should allow us to give a complete computation of this partition function and express it in terms of modular forms. No one has every been able to compute the partition function of a compact Calabi-Yau threefold (the geometry most relevant in string theory). Having such a function would give us great insight into the corresponding physical theory and thus would have a significant impact in both geometry and physics.