Groupoids and the Geometry of Gauge Groups

9803593布赖林斯基J-L。Brylinski将继续他在数学物理中出现的几何结构的工作，特别是在规范理论、回路空间和向量丛的模空间中。规范群在现代几何学中占有极其重要的地位。圆周上的规范群是环群，而2维的规范群与保形场理论密切相关。Brylinski将研究他为规范群构造的某些上同调类，这些上同调类不是给出环群中心扩张的二次上同调类的自然推广。这些类是由贝林森的Chern-Simons类和Chern类通过递进过程得到的。第一个新的例子是闭曲面规范群中的三次上同调类。布赖林斯基已经证明，这些类遵循共形场理论的广义互易定律。他将调查它们在多大程度上与更高维的场论有关。他将研究向量丛的模空间的Kaehler几何，继续他与P.Foth的工作，并打算研究这些模空间的几何量化。Brylinski还将继续他关于行列式线丛的Quillen度量的工作，这将导致在许多情况下用几何而不是解析的方法来计算它。在这里，他将使用他开发的Gerbe的微分几何和他引入的Deligne上同调的变体，该变体结合了厄米特度量。Brylinski将继续他对光滑三维流形中纽结空间的微分几何的研究，他以前证明了它是Kaehler流形，更准确地说是么模微分同构群的余伴轨道的并。一个特殊的研究对象将是关于Poisson括号的各种泛函李代数的结构。布赖林斯基的研究涉及几何学和数学物理之间的接口。这种界面在最近稳步发展，并对几何学的各个分支产生了深远的影响：代数、微分和辛等。例如，Edward Witten(高等研究院自然科学学院)根据与Chern-Simons类有关的三维场论解释了纽结的琼斯多项式。此外，Verlinde利用共形场论的思想，导出了黎曼曲面上的线丛的模空间上的线丛的Riemann-Roch数的公式。所有这些事实表明，需要新类型的几何，包括向量丛或主丛的概念的推广。为此目的，Brylinski发展了群胚和Gerbe的微分几何理论，并将其应用于规范群、向量丛的模空间、Quillenline丛和纽结空间的研究。虽然他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，他得到了黎曼曲面上与线丛相连的一些几何行列式线丛上的Quillen度量的具体描述。这项研究有望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.***