Quantum K-theory and other topics in enumerative geometry

量子 K 理论和计数几何中的其他主题

• 批准号: 0906148
• 负责人: Anders Buch
• 金额: $ 15.65万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-15 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0906148&HistoricalAwards=false
• 关键词: Quantum; theory; other; topics; enumerative

Schubert calculus is the study of the (singular) cohomology ring of a homogeneous space X, with an emphasis on multiplication of Schubert classes. Such products can be interpreted as the number of points in an intersection in general Schubert varieties, and therefore provide an essential tool in enumerative algebraic geometry. While this ring contains information about the components of an intersection of Schubert varieties, the K-theory ring also provides information about how these components intersect. The equivariant cohomology ring refines the ring structure by taking a group action into account. And the quantum cohomology ring encodes the Gromov-Witten invariants, which give the number of curves of fixed degree that meet general Schubert varieties. All these rings can be combined to one single equivariant quantum K-theory ring (to rule them all). Until recently, not much has been known about the structure of this ring, since the Gromov-Witten invariants used to define it have been hard to compute. The investigator will attempt to overcome this problem and uncover the ring structure in as many cases as possible. He will also seek answers to some important open questions concerning the orbit closures of quiver representations. These objects from representation theory generalize many interesting degeneracy loci, including Schubert varieties and determinantal varieties, and their classes generalize important polynomials from geometry and combinatorics, including Schur, Schubert, and Grothendieck polynomials. Studies of orbit closures for relatively simple quivers have in the past resulted in many beautiful formulas, constructions, and positivity results. The proposer hopes to generalize these results to more general quivers.The investigator will attack problems in the broad area of enumerative geometry, especially problems concerning quantum rings and quiver cycles. Enumerative geometry aims to provide the tools required to count the number of geometric objects of given type that satisfy specified conditions. A typical example is the fact that exactly two lines (possibly with complex coordinates) meet 4 randomly chosen fixed lines in 3-space. Enumerative problems are often approached by constructing a moduli space, which has one point for each geometric object of the given type, and then translating the specified conditions into polynomial equations on the moduli space. This transforms the problem into one of counting the solutions to such equations. Quiver cycles can be understood as a way to organize equations that arise in many important situations, and knowledge about geometric invariants of quiver cycles adds to their utility for counting solutions. Ideas from string theory in physics led to the definition of the quantum cohomology ring of a homogeneous space, which provides an efficient tool for counting the number of curves of given degree that meet fixed subvarieties in the space, at least when this number is finite. When the number of solution curves is infinite, the set of these curves form a space called a Gromov-Witten variety, and geometric invariants of this space can tell us much about the problem. For example, if the Gromov-Witten variety has non-zero Euler characteristic, then solution curves do exist. Quantum K-theory is a generalization of quantum cohomology that encodes the Euler characteristic of Gromov-Witten varieties. The investigator plans to study these topics with a combination of geometric and combinatorial methods.

舒伯特演算是研究齐次空间X的（奇异）上同调环，重点是舒伯特类的乘法。这样的乘积可以解释为在一般舒伯特簇中相交的点的数目，因此提供了枚举代数几何中的一个重要工具。虽然这个环包含舒伯特簇的交集的分量的信息，但K理论环也提供了这些分量如何相交的信息。等变上同调环通过考虑群作用来细化环的结构。量子上同调环编码了Gromov-Witten不变量，它给出了满足一般Schubert簇的固定次数曲线的个数。所有这些环可以组合成一个单一的等变量子K理论环（来统治它们）。 直到最近，人们对这个环的结构还知之甚少，因为用来定义它的Gromov-Witten不变量很难计算。研究人员将试图克服这个问题，并在尽可能多的情况下揭示环结构。 他还将寻求一些重要的悬而未决的问题的答案，有关轨道关闭的卫星表示。 这些来自表示论的对象概括了许多有趣的退化轨迹，包括舒伯特簇和行列式簇，它们的类概括了几何和组合学中的重要多项式，包括舒尔、舒伯特和格罗滕迪克多项式。 在过去，对相对简单的箭点的轨道闭合的研究已经产生了许多漂亮的公式、构造和正性结果。提出者希望将这些结果推广到更一般的箭图，研究者将在计数几何的广泛领域中解决问题，特别是关于量子环和量子环的问题。 枚举几何旨在提供计算满足指定条件的给定类型的几何对象的数量所需的工具。一个典型的例子是，在三维空间中，正好有两条直线（可能具有复坐标）与4条随机选择的固定直线相交。 枚举问题通常通过构造一个模空间来解决，该空间对给定类型的每个几何对象都有一个点，然后将指定的条件转换为模空间上的多项式方程。 这就把问题转化为计算这些方程的解的问题。箭壶圈可以被理解为一种组织在许多重要情况下出现的方程的方法，关于箭壶圈几何不变量的知识增加了它们在计算解时的效用。 物理学中弦理论的思想导致了齐次空间的量子上同调环的定义，它提供了一个有效的工具来计算空间中满足固定子簇的给定次数的曲线的数量，至少当这个数量是有限的时候。 当解曲线的数目是无限时，这些曲线的集合形成一个称为Gromov-Witten簇的空间，这个空间的几何不变量可以告诉我们很多关于这个问题的信息。 例如，如果Gromov-Witten簇具有非零的欧拉特征，则解曲线确实存在。 量子K理论是量子上同调的一个推广，它编码了Gromov-Witten簇的欧拉特征。 研究者计划结合几何和组合方法来研究这些主题。