Operads and Homotopy Algebra

运算和同伦代数

• 批准号: 9971434
• 负责人: Alexander Voronov
• 金额: $ 4.61万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-08-15 至 2002-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971434&HistoricalAwards=false
• 关键词: Operads; Homotopy; Algebra

9971434Voronov There are two objectives of this project. One is to work on andeventually prove the following conjecture of Deligne: There exists the(natural) structure of an algebra over a chain operad of the littledisks operad on the Hochschild complex of any associative algebra.The other objective is to study holomorphic line bundles over themoduli space of algebraic curves with a holomorphic disk and prove theanalogue of Borel-Weil-Bott Theorem for this space, considered ashomogeneous space for the Virasoro algebra. The significance of theproject on Deligne's Conjecture consists in studying presumably deepconnections between one complex variable and associative algebras.The proof of Deligne's Conjecture will also further simplify the proofof the existence of a deformation quantization of a Poisson manifold,originally proved by Kontsevich and simplified by Tamarkin. Thecompletion of the Borel-Weil-Bott part of the project will establishimportant relationship between the geometry of the moduli space andthe representation theory of the Virasoro algebra. The structure of an algebra over a chain operad indicates, usuallyexplicitly, the presence of a ``homotopy something'' algebra, analgebra with certain identities, such as associativity, satisfied inonly an approximate way known as ``up to homotopy.'' Such structureshave been used by Stasheff and May in their study of loop spaces,Beilinson and Ginzburg and Hinich and Schechtman in the study ofdeformation theory of algebraic varieties and vector bundles overthem, and by Kontsevich toward knot invariants. Witten and Zwiebachhave effectively used the homotopy Lie structure in string theory.Deligne's Conjecture may be reformulated as the existence of a certainstring theory associated to every associative algebra. Thus, theproof of the conjecture will create ground for considerable progressin studying applications of algebra to geometry and theoreticalphysics. The significance of the moduli space part of the project isin extending the fundamental relationship between the geometry of ahomogeneous space and the representation theory of a semisimple Liegroup to the case of the moduli space of Riemann surfaces and theVirasoro algebra, respectively. As in the classical case, thecompletion of this project would contribute mutually to the twosubjects: the geometry of the moduli space and the representationtheory of the Virasoro algebra. Both subjects are much morecomplicated and less studied than those in the classical situation.The project aims to advance the general understanding of each. Ingeneral, the study of the geometry and topology of moduli spaces hasrecently become very important, as moduli spaces play now asignificant role in four-dimensional topology (Donaldson andSeiberg-Witten invariants), knot theory (Vassiliev invariants),higher-dimensional topology (Gromov-Witten invariants), anddeformation quantization (Kontsevich's quantization of Poissonmanifolds). Moreover, although moduli spaces have been classicalobjects of algebraic geometry (since the seminal work of P. Deligneand D. Mumford), their topology has not been understood yet and haspresented a challenge for algebraic geometers for the last twentyyears. Thus unraveling the topology of moduli spaces seems to be veryimportant for progress in both topology and geometry.***

小行星9971434 该项目有两个目标。 一是研究并最终证明了Deligne的猜想：结合代数的Hochschild复形上的小圆盘算子的链算子上的代数存在（自然）结构.另一个目的是研究具有全纯圆盘的代数曲线的模空间上的全纯线丛，证明Borel-Weil-Bott定理在此空间上的类似定理，被认为是Virasoro代数的非齐性空间。 Deligne猜想项目的意义在于研究一个复变元与结合代数之间可能存在的深层联系，Deligne猜想的证明也将进一步简化Poisson流形的变形量子化存在性的证明，该证明最初由Kontsevich证明，后来由Tamarkin简化。 该项目Borel-Weil-Bott部分的完成将建立模空间几何与Virasoro代数表示理论之间的重要关系。 链操作上的代数的结构通常明确地表明存在一个“同伦的东西”代数，具有某些恒等式的代数，例如结合性，只满足一种近似的方式，称为“上至同伦”。这种结构已被使用的Stasheff和可能在他们的研究循环空间，贝林森和金兹伯格和Hinich和Schechtman在研究变形理论的代数簇和向量丛在他们身上，并由Kontsevich对结不变量。 维滕和兹维巴赫在弦理论中有效地利用了同伦李结构，德利涅猜想可以重新表述为存在一个与每个结合代数相联系的弦理论。 因此，猜想的证明将为研究代数在几何和理论物理中的应用创造相当大的进展。 该项目模空间部分的意义在于将齐次空间几何与半单李群表示理论之间的基本关系分别扩展到Riemann曲面模空间和Virasoro代数的情况。 在经典的情况下，这个项目的完成将有助于两个主题：模空间的几何和Virasoro代数的表示理论。 这两个主题都比经典情况下的主题复杂得多，研究得也少。该项目旨在促进对每一个主题的普遍理解。 一般来说，模空间的几何和拓扑的研究最近变得非常重要，因为模空间在四维拓扑（唐纳森和塞伯格-威滕不变量）、纽结理论（瓦西里耶夫不变量）、高维拓扑（格罗莫夫-威滕不变量）和变形量子化（孔采维奇的泊松流形量子化）中扮演着重要的角色。 此外，尽管模空间一直是代数几何的经典对象（自从P. Deligne和D. Mumford），他们的拓扑结构还没有被理解，并提出了一个挑战，代数几何学家在过去的20年。 因此，揭开模空间的拓扑似乎对拓扑学和几何学的发展都非常重要。