FRG: Collaborative Research: How the Algebraic Topology of Closed Manifold Relates to Strings and 2D Quantum Field Theory

二维量子场理论和弦理论在数学上非常有趣，但仍然没有精确定义。二维场理论是与几何表面相关的代数和分析结构。弦理论是二维场理论在所有表面几何上的平均或积分。在这个项目中，我们研究各种二维场论，它们的积分和它们共同的基本结构。例如，我们将研究具有与模形式相关的非平凡内部结构的2D理论的Atiyah-Segal-维滕概念的超对称环。这里的重点是构造例子，并将这些思想与拓扑模形式上同调理论联系起来。我们将探索一个新的不确定性分类版本的二维场理论。有许多自然的例子来研究和验证二维理论与三维拓扑量子场论的关系。我们打算澄清“弦拓扑”二维场理论与任何“目标流形”。这些理论都有一个主方程解dX + X*X = 0，这是形成弦理论积分的类似物所必需的。代数形式主义与辛拓扑的J全纯曲线共鸣。更一般地说，我们将研究主方程的解的概念意义，对偶结构的变形，以及与多线性函数或称为双线性算子的运算相关的形式主义。科学和数学的不同部分交织在一起，形成了一幅丰富的挂毯，上面很好地说明了这一现象。该项目的研究将影响到大学中广泛的年轻研究人员以及博士生和本科生。其他影响最终可能包括技术的改进。科学家的实际应用有时是偶然的，就像青霉素和合成橡胶一样。然而，有时，就像晶体管和微电子学的发现一样，它们依赖于对量子理论等学科的深刻理解。对二维量子场论的更深入理解的一个非常投机的应用，这个项目的主题，可能是量子计算机的物理实现。量子计算机理论上可以解决非量子计算系统理论上无法解决的问题。其中一个问题是破解金融系统和政府机构使用的安全系统的因子分解部分的理论可能性。根据Michael Freedman的说法，实现量子计算机的技术困难可以使用上面提到的三维拓扑理论来重新设计。此外，这些三维理论与二维理论的关系表明了一个解决技术难题的方向：在二维中发生的实验物理，凝聚态研究的非常活跃的领域。现在有机会将项目参与者的这些不同观点和能量组织成一个连贯的运动，以照亮该地区。二维场论、弦论和对偶结构变形的思想圈很可能成为20世纪数学的重要组织中心，就像拓扑学在20世纪的作用一样。

Two dimensional quantum field theory and string theory are mathematically very interesting but still not precisely defined. Two dimensional field theories are algebraic and analytic structures associated to geometric surfaces. String theories are averages or integrals of two dimensional field theories over all surface geometries. In this project we study various two dimensional field theories , their integrals and their common underlying structure. For example we will study a supersymmetric refinement of the Atiyah-Segal-Witten notion of 2D field theories with a nontrivial internal stucture related to modular forms. The point here is to construct examples and relate these ideas to the topological modular form cohomology theory. We will explore a new infinity categorical version of 2D field theory. There are many natural examples to investigate and categorifications of two dimensional theories relate to three dimensional topological quantum field theories. We intend to clarify the "string topology" two dimensional field theories associated to any "target manifold" . These theories feature a master equation solution, dX + X*X = 0, required to form the analogue of the string theory integral. The algebraic formalism resonates with the J holomorphic curves of symplectic topology. More generally, we will research the conceptual meaning of solutions to master equations, deformations of structures with duality, and the formalism related to multilinear functions or operations called correlators. The different parts of science and mathematics are woven together in a rich tapestry and this phenomenon is well illustrated by the above. The studies of this project will impact a wide range of younger researchers in the universities as well as PhD students and undergraduates. Other impacts might eventually include improved technology. Practical applications by scientists are sometimes accidental like penicillin and synthetic rubber. Sometimes however, like the discovery of transistors and microelectronics, they depend on a deep understanding of subjects like quantum theory. A very speculative application of a deeper understanding of two dimensional quantum field theory, the subject of this project, could be to the physical realization of quantum computers. One knows quantum computers can theoretically solve problems not known to be solved theoretically by non quantum computing systems. One such problem is the theoretical possibility of cracking the factorization part of the security systems used by financial systems and government agencies. The technical difficulties to realizing quantum computers can be recast according to Michael Freedman, using the three dimensional topological theories alluded to above. Furthermore the relationship of these three dimensional theories with two dimensional theories suggests a direction to look to solve the technical difficulties: the experimental physics that takes place in two dimensions, the very active area of condensed matter research. There is an opportunity just now to organize these different perspectives and energies of the participants of the project into a coherent campaign to illuminate the area. The circle of ideas from two dimensional field theory, string theory, and deformations of structures with duality may very well become an important organizing center for twenty first century mathematics in the way that topology influenced the twentieth century.