Groupoids and the Geometry of Gauge Groups

9803593 brylinski J-L。Brylinski将继续研究数学物理中的几何结构，特别是规范理论、环空间和向量束的模空间。规范群在现代几何中具有重要的基础意义。圆上的规群是环群，二维规群与共形场理论密切相关。Brylinski将研究他为规范群构造的某些上同调类，这些上同调类是二次上同调类的自然推广，给出了环群的中心扩展。这些类是由chen - simons类和Beilinson的chernclass的侵进过程得到的。第一个新例子是一个闭曲面规范群中的三次生态同调类。brylinski已经证明，这些类服从推广共形场论的互易律。他将研究它们在多大程度上与高维场论相关。他将继续与P. Foth的工作，研究向量束模空间的Kaehler几何，并打算研究这些模空间的几何量子化。Brylinski还将继续他在行列式线束上的quillen度规的研究，这将导致在许多情况下用几何方法来计算它，而不是用解析方法。在这里，他将使用他发展的格布微分几何和他引入的德列涅上同调的一种变体，它包含了厄米度量。Brylinski将继续他对光滑三流形中结点空间的微分几何的研究，他之前展示的是aKaehler流形，更准确地说，是一组非模微分同态的协轨道的联合。一个特殊的研究对象将是关于泊松括号的各种泛函李代数的结构。j - l。布林斯基关注的是几何和数学物理之间的联系。这个界面在最近的过去稳步发展，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，Edward Witten（自然科学学院，高等研究院）在与chen - simons类相关的三维场论中解释了结点的Jones多项式。此外，Verlinde利用共形场论的思想，推导出了在黎曼曲面上的束的模空间上的线束的黎曼-罗奇数的公式。所有这些事实都表明需要一种新的几何类型，包括向量束或主束概念的推广。为此，Brylinski发展了一种群拟和格贝的微分几何理论，并将其应用于规范群、向量束的模空间、Quillenline束和结空间的研究。虽然他所使用的基本概念是抽象的，但它们往往可以导致具体的公式。例如，他得到了一些几何行列式线束上与Riemann曲面上的线束相连的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

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.***