Algebraic Structures of Mathematical Physics

摘要奖：DMS-0805785首席研究员：Alexander A.Voronov该项目的目标是解决拓扑场论、辛场论和格罗莫夫-维滕理论等迅速发展的领域中的一些重要问题。该项目的第一部分将从高级范畴理论，到密码学，再到量子场论。我们的计划是将具有角点的流形的协边集置于适当的n-范畴框架内，并将TFT描述为从协边集的n-范畴到n-向量空间的n-函数式，并表明物理模型，如Gauge(Wess-Zumino-Witten)、Yang-Mills、Chern-Simons、Seiberg-Witten理论和Sigma-模型可以被描述为这样的更高的TFT。该项目的第二部分是将代数、几何和辛方法结合起来，以构造Gromov-Witten理论中所谓的量子主方程的完整解。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。该项目的第三部分旨在将Gromov-Witten理论提升到(Floer)链级别，并开发Gromov-Witten理论的组合版本，从而连接列举代数几何、辛Floer理论和图同调领域。最后，该项目的SFT部分将导致构造Riemann曲面的模空间的一种新的紧化，它将支配SFT中产生的代数运算和不变量。这个紧化将是与Gromov-Witten理论相关的Deligne-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、辛场理论和Gromov-Witten理论所建议或激发的拓扑学中的新的代数结构。另一个长期目标是在致力于数学物理相关问题的几种数学文化之间建立桥梁。这些文化包括地质学家、代数拓扑学家、辛几何学家、代数几何学家和几何拓扑学家，不一而足。代数结构是物理理论的基本结构，如关联器和它们之间的关系(Ward恒等式)的数学转世。理解这一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、几何和拓扑学。

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.