Algebraic Structures of Mathematical Physics

项目负责人：Alexander a . voronov该项目的目标是解决拓扑场论（TFT）、辛场论（SFT）和Gromov-Wittentheory等快速发展领域的一些重要问题。项目的第一部分将涵盖高范畴理论、协同论和量子场论。该计划是将带有角的流形的协协置于适当的n范畴框架内，并将tft描述为n个函子，从n范畴的协协到n个向量空间的协协，以及表明物理模型，如规范（Wess-Zumino-Witten）， Yang-Mills, chen - simons， Seiberg-Wittentheories和sigma模型可以被描述为这样的高阶tft。该项目的第二部分包括将代数几何和辛方法结合在一起，以构造所谓的量子主方程的完整解。该方程描述了全纯曲线和相关代数结构的模空间的拓扑结构，提供了辛几何和代数几何中重要的不变量。该项目的第三部分旨在将Gromov-Witten理论提升到（Floer）链水平，并发展Gromov-Witten理论的组合版本，从而连接枚举代数几何，辛Floer理论和图同调的领域。最后，该项目的SFT部分将导致构造riemann曲面模空间的一个新的紧化，这将控制SFT中产生的代数运算和不变量。这种紧化将是与gromov - witten理论相关的delign - mumford紧化的sft模拟。该项目旨在发现和研究由数学物理，特别是弦理论，辛场论和Gromov-Witten理论提出或激发的拓扑中的新代数结构。另一个长期目标是在研究数学物理相关问题的几种数学文化之间建立一座桥梁。这些文化包括几何学家、代数拓扑学家、辛几何学家、代数几何学家和几何拓扑学家，仅举几例。代数结构是物理理论的基本结构作为它们之间的相关和关系（沃德恒等式）的数学转世。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，该项目在物理学的推动下产生了新的数学思想、新的代数、几何和拓扑。

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.