Moduli spaces, homotopy Frobenius algebras and mirror symmetry

摘要：黎曼曲面的模空间在数学和理论物理的许多领域，包括几何和代数拓扑、辛拓扑、弦理论和数论中起着核心作用。这个项目探索了黎曼曲面模空间的另一种表现形式，这可能不太为人所知。这是它们在同调代数中的表现，更准确地说，是在循环a无穷代数（一种“弗洛贝纽斯代数至同伦”）的研究中。这个项目背后的哲学是，关于各种模空间的同伦类型的一切都可以用循环a无穷代数的同伦理论来表达。这种哲学的一个精确表现（将在本项目中得到证明）是曲面的模空间作为环无穷代数的某些“同调运算”出现。这一理论结果将用于研究镜像对称猜想的一些具体的高格方面，镜像对称猜想在当前的代数和辛几何工作中起着突出的作用。特别地，本项目将尝试在一些例子中计算高格Gromov-Witten不变量的推测镜像伙伴。所有可能的二维形状的空间——被称为曲面的模空间——长期以来一直是许多数学领域的基本研究对象，从几何到数论。这个空间在假定的“万有理论”——弦理论中也起着重要的作用。这个项目关注的是建立二维几何（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来自弦理论的某些数学猜想。弦理论学家预测，弦理论的两种不同的简化模型——被称为A模型和B模型——在数学上是等价的。这一预测在过去15年中刺激了大量的数学工作。pi将处理B模型中迄今为止无法实现的一些计算。这些计算的结果将与A模型中已知的计算结果进行比较，希望能进一步验证弦理论的预测。

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.