Groupoids and the Geometry of Gauge Groups
群形和规范群的几何
基本信息
- 批准号:9803593
- 负责人:
- 金额:$ 13.65万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Continuing Grant
- 财政年份:1998
- 资助国家:美国
- 起止时间:1998-08-01 至 2002-07-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
9803593Brylinski J-L. Brylinski will continue his work on the geometric structuresoccurring in mathematical physics, particularly in gauge theory, loopspaces, and moduli spaces of vector bundles. Gauge groups are offundamental importance in modern geometry. The gauge groups on acircle are loop groups, and gauge groups in dimension 2 are closelyrelated to conformal field theory. Brylinski will study certaincohomology classes he has constructed for gauge groups, which arenatural generalizations of the degree-two cohomology class giving thecentral extension of a loop group. These classes are obtained by atransgression process from the Chern-Simons classes and the Chernclasses of Beilinson. The first new example is that of a degree-threecohomology class in the gauge group of a closed surface. Brylinskihas shown that these classes obey reciprocity laws that generalizethose of conformal field theory. He will investigate to what extentthey are related to higher-dimensional field theories. He will studythe Kaehler geometry of moduli spaces of vector bundles, continuinghis work with P. Foth, and intends to study the geometric quantizationof these moduli spaces. Brylinski will also continue his work on theQuillen metric on determinant line bundles, which should lead tomethods to compute it geometrically, as opposed to analytically, inmany situations. Here he will use both the differential geometry ofgerbes that he developed and a variant of Deligne cohomology heintroduced, which incorporates hermitian metrics. Brylinski willcontinue his investigation of the differential geometry of the spaceof knots in a smooth three-manifold, which he previously showed is aKaehler manifold, more precisely a union of coadjoint orbits of thegroup of unimodular diffeomorphisms. A particular object of studywill be the structure of various Lie algebras of functionals withrespect to the Poisson bracket. The research of J.-L. Brylinski is concerned with the interfacebetween geometry and mathematical physics. This interface has beengrowing steadily in the recent past and has had profound consequencesfor various branches of geometry: algebraic, differential, and symplectic,among others. For instance, Edward Witten (School of Natural Sciences,Institute for Advanced Study) interpreted the Jones polynomialfor knots in terms of field theory in three dimensions connected tothe Chern-Simons class. Also, Verlinde derived a formula forRiemann-Roch numbers of a line bundle over a moduli space of bundlesover a Riemann surface, using ideas from conformal field theory. Allthese facts point to the need for new types of geometry that includegeneralizations of the concept of a vector bundle or a principalbundle. For this purpose, Brylinski has developed a theory of thedifferential geometry of groupoids and gerbes, which he is applying tothe study of gauge groups, moduli spaces of vector bundles, Quillenline bundles, and the space of knots. Although the basic concepts heemploys are of an abstract nature, they can often lead to concreteformulas. For instance, he has obtained a concrete description ofQuillen metrics on some geometric determinant line bundles connectedwith line bundles over Riemann surfaces. This research is expected toresult in a better understanding of the geometric underpinnings ofmathematical physics.***
小行星9803593 J-L Brylinski将继续他的工作的几何structureoccurring在数学物理,特别是在规范理论,loopspaces,和模空间的向量束。 规范群在现代几何学中具有重要的基础性。 圆上的规范群是圈群,二维规范群与共形场论密切相关。 Brylinski将研究某些上同调类,他已经为规范组,这是自然的推广度2上同调类给出了一个循环组的中心扩展。 这些类是由Chern-Simons类和Beilinson的Chernclasses类通过一个变换过程得到的。 第一个新的例子是一个度threocohomology类的规范组的封闭表面。 Brylinski表明,这些类服从互易定律,概括了那些共形场论。 他将调查到什么程度,他们是相关的高维场理论。 他将研究Kaehler几何的模空间的向量丛,继续他的工作与P. Foth,并打算研究几何quantizationof这些模空间。 Brylinski还将继续他的工作对quillen度量行列式线丛,这应该导致tomods计算它的几何,而不是分析,在许多情况下。 在这里,他将使用的微分几何ofgerbes,他开发和一个变种德利涅上同调hesintroduced,其中包括埃尔米特度量。 Brylinski将继续他的调查微分几何的空间结在一个顺利的三个流形,他以前表明是一个Kaehler流形,更确切地说是一个工会的coadjoint轨道的一组unimodulated crystalomorphisms。 一个特殊的研究对象将是结构的各种李代数的泛函相对于泊松括号。 J的研究- L. Brylinski关心的是几何和数学物理之间的接口。 这个接口在最近的过去一直在稳步增长,并对几何的各个分支产生了深远的影响:代数,微分和辛等。 例如,爱德华维滕(自然科学学院,高等研究所)解释了琼斯多项式的纽结在场论方面在三维连接到陈-西蒙斯类。 此外,Verlinde推导出一个公式forRiemann-Roch数的线丛超过模空间的soules超过黎曼曲面,使用的想法从共形场理论。 所有这些事实都表明需要新的几何类型,包括向量丛或主丛概念的推广。 为此,Brylinski制定了理论的微分几何的groupoid和gerbes,他是适用于研究规范群,模空间的向量丛,Quillenline丛,和空间的结。 虽然他所使用的基本概念是抽象的,但它们往往可以导出具体的公式。 例如,他在与黎曼曲面上的线丛相连的某些几何行列式线丛上得到了Quillen度量的具体描述。 这项研究有望使人们更好地理解数学物理的几何基础。*
项目成果
期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
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Jean-Luc Brylinski其他文献
Springer's Weyl group representations through characteristic classes of cone bundles
- DOI:
10.1007/bf01458071 - 发表时间:
1987-03-01 - 期刊:
- 影响因子:1.400
- 作者:
Walter Borho;Jean-Luc Brylinski;Robert MacPherson - 通讯作者:
Robert MacPherson
Jean-Luc Brylinski的其他文献
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{{ truncateString('Jean-Luc Brylinski', 18)}}的其他基金
Mathematical Sciences: Geometry of Loop Spaces and Groupoids
数学科学:循环空间和群形的几何
- 批准号:
9504522 - 财政年份:1995
- 资助金额:
$ 13.65万 - 项目类别:
Continuing Grant
Mathematical Sciences: Characteristic Classes, the Space of Knots and Groups of Diffeomorphisms
数学科学:特征类、结空间和微分同胚群
- 批准号:
9203517 - 财政年份:1992
- 资助金额:
$ 13.65万 - 项目类别:
Continuing Grant
Mathematical Sciences: Group Actions on Manifolds, Cyclic Homology and Elliptic Cohomology
数学科学:流形、循环同调和椭圆上同调的群作用
- 批准号:
8903248 - 财政年份:1989
- 资助金额:
$ 13.65万 - 项目类别:
Continuing Grant
Mathematical Sciences: Cyclic Homology and D-Modules
数学科学:循环同调和 D 模
- 批准号:
8815744 - 财政年份:1988
- 资助金额:
$ 13.65万 - 项目类别:
Standard Grant
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