Studies in Gromov-Witten Theory

格罗莫夫-维滕理论研究

• 批准号: 0702871
• 负责人: Ionut Ciocan-Fontanine
• 金额: $ 19.06万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-06-15 至 2011-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0702871&HistoricalAwards=false
• 关键词: Studies; Gromov; Witten; Theory

The first part of the project concerns a relationship, discovered by Ciocan-Fontanine and his collaborators, between the genus zero Gromov-Witten theories of nonsingular projective quotients of a projective manifold X by a complex reductive Lie group G, and by a maximal torus T in G. This relationship can be viewed as a kind of highly nontrivial functoriality in Gromov-Witten theory. Such instances of functorial behavior are quite rare and therefore very interesting. Ciocan-Fontanine proposes to significantly broaden the study of this abelian/nonabelian correspondence in several different directions. Specifically, the correspondence will be extended to higher genus Gromov-Witten invariants, and a version of the correspondence in terms of bounded derived categories of coherent sheaves will be explored. Further, some new and nontrivial examples will be studied. A second project deals with several applications of the derived moduli spaces introduced several years ago by Ciocan-Fontanine and Kapranov to the study of Gromov-Witten and Donaldson-Thomas invariants of some classes of projective varieties. For example, a construction of derived versions of the moduli of sheaves appearing in DT-theory is proposed. In the crucial rank one case, it is expected that a variant of the construction will lead to a very concrete expression of the virtual fundamental classes in terms of known complexes of bundles on simple varieties (products of Grassmannians). Using this expression, Ciocan-Fontanine will investigate the DT-theory of hypersurfaces in projective 4-space, about which little is known at the present time.This is research in the field of algebraic geometry, a highly developed branch of modern mathematics. In recent years, the methods and ideas of algebraic geometry, especially the study of moduli spaces, have been employed in string theory, a very active part of theoretical physics. Developments in string theory have sparked a fruitful interaction between the two communities of researchers and have led to the discovery and study of many unexpected new phenomena. The theories of Gromov-Witten and Donaldson-Thomas invariants (and the exciting conjectural relations between them) are particularly striking examples.

该项目的第一部分涉及的关系，发现了Ciocan-Fontanine和他的合作者之间的亏格为零的Gromov-Witten理论的非奇异投射流形X的一个复杂的还原李群G，并通过一个极大环面T在G。这种关系可以看作是Gromov-Witten理论中的一种高度非平凡的函数性。这种函子行为的实例非常罕见，因此非常有趣。Ciocan-Fontanine建议在几个不同的方向上显着扩大对这种阿贝尔/非阿贝尔对应的研究。具体而言，对应关系将被扩展到更高的属Gromov-Witten不变量，并将探讨在有界的相干层的派生类别的对应关系的版本。此外，将研究一些新的和非平凡的例子。第二个项目涉及几个应用程序的衍生模空间介绍了几年前由Ciocan Fontanine和Kapranov研究Gromov-Witten和唐纳森-托马斯不变量的某些类别的投影品种。例如，提出了DT理论中出现的层模的导出版本的构造。在关键的秩一的情况下，预计一个变体的建设将导致一个非常具体的表达虚拟的基本类在已知的复杂的丛简单品种（产品的格拉斯曼）。利用这个表达式，Ciocan-Fontanine将研究投射4-空间中的DT-超曲面理论，这是现代数学高度发展的分支-代数几何领域的研究。近年来，代数几何的方法和思想，特别是模空间的研究，已被应用于弦理论，这是理论物理中非常活跃的部分。弦理论的发展激发了两个研究群体之间富有成效的互动，并导致了许多意想不到的新现象的发现和研究。Gromov-Witten和Donaldson-Thomas不变量的理论（以及它们之间令人兴奋的几何关系）是特别引人注目的例子。