Moduli spaces, homotopy Frobenius algebras and mirror symmetry

模空间、同伦 Frobenius 代数和镜像对称

• 批准号: 0706945
• 负责人: Kevin Costello
• 金额: $ 19.01万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-09-01 至 2011-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0706945&HistoricalAwards=false
• 关键词: Moduli; spaces; homotopy; Frobenius; algebras

AbstractAward: DMS-0706945Principal Investigator: Kevin J. CostelloThe moduli spaces of Riemann surfaces play a central role in manyareas of mathematics and theoretical physics, including geometricand algebraic topology, symplectic topology, string theory, andnumber theory. This project explores another manifestation ofthe moduli spaces of Riemann surfaces, which is maybe not sowidely known. This is their appearance in homological algebra,and more precisely in the study of cyclic A-infinity algebras (akind of "Frobenius algebra up to homotopy"). The philosophyunderlying much of this project is that everything one can sayabout the homotopy types of the various moduli spaces can beexpressed in terms of the homotopy theory of cyclic A-infinityalgebras. One precise manifestation of this philosophy (whichwill be proved in this project) is that the moduli space ofsurfaces arises as certain "homology operations" for cyclicA-infinity algebras. This theoretical result will be used toinvestigate some concrete higher-genus aspects of the mirrorsymmetry conjecture, which plays a prominent role in current workon algebraic and symplectic geometry. In particular, thisproject will attempt to compute the conjectural mirror partner ofthe higher-genus Gromov-Witten invariants in some examples.The space of all possible two-dimensional shapes -- known as themoduli space of surfaces -- has long been a fundamental object ofstudy in many areas of mathematics, from geometry to numbertheory. This space also plays an important role in stringtheory, the putative "theory of everything". This project isconcerned with setting up a correspondence betweentwo-dimensional geometry (the moduli space of surfaces) and akind of abstract algebra. This correspondence will be used totest certain mathematical conjectures coming from string theory.String theorists have predicted that two different simplifiedmodels of string theory-- known as the A model and the B model --are mathematically equivalent. This prediction has stimulated agreat deal of mathematical work in the last 15 years. The PIwill tackle some computations in the B model which have beenheretofore out of reach. The results of these computations willthen be compared with known computations in the A model,hopefully leading to further verification of the predictions ofstring theory.

AbstractAward：DMS-0706945首席研究员：Kevin J. Costello黎曼曲面的模空间在数学和理论物理的许多领域中发挥着核心作用，包括几何和代数拓扑、辛拓扑、弦理论和数论。 这个项目探索了黎曼曲面模空间的另一种表现形式，这可能不是很清楚。 这是它们在同调代数中的出现，更确切地说是在循环A-无穷代数（一种“Frobenius代数直到同伦”）的研究中。 这个项目的哲学基础是，人们所能说的关于各种模空间的同伦类型的一切都可以用循环A-无限代数的同伦理论来表达。 这一哲学的一个精确表现（这将在本项目中得到证明）是，曲面的模空间作为循环A-无穷代数的某些“同调运算”而出现。 这一理论结果将被用来调查一些具体的高亏格方面的镜像对称猜想，这起着突出的作用，在目前的工作代数和辛几何。 特别是，这个项目将试图计算一些例子中的高格Gromov-Witten不变量的几何镜像伙伴。所有可能的二维形状的空间--被称为曲面的模空间--长期以来一直是从几何到数论的许多数学领域的基本研究对象。 这个空间在弦理论中也扮演着重要的角色，弦理论被认为是“万物理论”。 这个项目是关于建立二维几何（曲面的模空间）和一种抽象代数之间的对应关系。 这种对应关系将被用来检验弦理论的某些数学模型。弦理论家已经预言，弦理论的两种不同的简化模型--A模型和B模型--在数学上是等价的。 在过去的15年里，这个预言激发了大量的数学工作。 PI将解决B模型中的一些迄今为止无法实现的计算。 这些计算的结果将与已知的A模型的计算结果进行比较，希望能进一步验证弦理论的预言。