Algebraic Structures of Mathematical Physics

摘要奖：DMS-0805785 主要研究者：亚历山大A. Voronov该项目的目标是解决拓扑场论（TFT），辛场论（SFT）和Gromov-Witten理论等快速发展领域的一些重要问题。该项目的第一部分将跨越从更高的范畴理论，以cobordisms和量子场论。该计划是放置cobordisms的流形与角在一个适当的n-范畴的框架和描述TFT作为n-functorsfrom的n-范畴的cobordisms的n-向量空间，以及显示，物理模型，如规范（Wess-Zumino-维滕），杨-米尔斯，陈-西蒙斯，塞伯格-威腾理论，和西格玛模型可以描述为这样的higherTFT。该项目的第二部分包括bringingtogether代数几何和辛的方法来构造一个完整的解决方案，所谓的量子主方程在Gromov-Witten理论。这个方程描述了全纯曲线的模空间的拓扑和相关的代数结构，在辛几何和代数几何中提供了重要的不变量。第三部分的工作是将Gromov-Witten理论提升到（Floer）链的层次，发展Gromov-Witten理论的组合形式，从而在枚举代数几何、辛Floer理论和图同调等领域建立一个桥梁。最后，SFT部分的项目将导致建立一个新的紧化的模空间的黎曼曲面，这将支配代数运算和不变量中出现的SFT。 这个紧化将是与Gromov-Witten理论相关的Deligne-Mumford紧化的一个SF Tansomer.The项目旨在发现和研究由物理学，特别是弦理论，辛场论和Gromov-Witten理论提出或激发的拓扑学中的新代数结构。 另一个长期目标是在几种数学文化之间建立桥梁，解决与数学物理有关的问题。这些文化包括gebraists，代数拓扑学家，辛几何学家，代数几何学家，几何拓扑学家，仅举几例。代数结构是物理学理论的基本结构，如算符和它们之间的关系（沃德恒等式）的数学再现。理解这种结构对于理解物理理论至关重要。从数学的角度来看，该项目导致了新的数学思想，新的代数，几何和拓扑，物理的动机。

AbstractAward: DMS-0805785 Principal Investigator: Alexander A. VoronovThe goal of the project is to solve a number of importantproblems in the rapidly developing fields of Topological FieldTheory (TFT), Symplectic Field Theory (SFT), and Gromov-Wittentheory. The first part of the project will span from highercategory theory, to cobordisms and to quantum field theories. Theplan is to place cobordisms of manifolds with corners within anappropriate n-category framework and describe TFTs as n-functorsfrom the n-category of cobordisms to that of n-vector spaces, aswell as show that physical models, such as gauge(Wess-Zumino-Witten), Yang-Mills, Chern-Simons, Seiberg-Wittentheories, and sigma-model may be described as such higherTFTs. The second part of the project consists in bringingtogether algebraic geometric and symplectic methods to constructa full solution to the so-called Quantum Master Equation inGromov-Witten theory. This equation describes the topology of themoduli spaces of holomorphic curves and relevant algebraicstructures, providing important invariants in symplectic andalgebraic geometry. The third part of the project aims at liftingGromov-Witten theory to the (Floer) chain level and developing acombinatorial version of Gromov-Witten theory, thus bridging theareas of enumerative algebraic geometry, symplectic Floer theory,and graph homology. The last, SFT part of the project will resultin constructing a new compactification of the moduli space ofRiemann surfaces, which would govern the algebraic operations andinvariants arising in SFT. This compactification will be an SFTanalogue of the Deligne-Mumford compactification relevant toGromov-Witten theory.The project aims at discovering and studying new algebraicstructures in topology suggested or motivated by mathematicalphysics, in particular, string theory, Symplectic Field Theory,and Gromov-Witten theory. Another long-term goal is to build abridge between several mathematical cultures working on problemsrelated to mathematical physics. These cultures includealgebraists, algebraic topologists, symplectic geometers,algebraic geometers, and geometric topologists, to name afew. The algebraic structures is a mathematical reincarnation ofsuch fundamental structures of physical theories as correlatorsand relations between them (Ward identities). Understanding thisstructure is crucial for understanding the physical theory. Fromthe point of view of mathematics, the project leads to newmathematical ideas, new algebra, geometry, and topology,motivated by physics.