SFB 1624: Higher structures, moduli spaces and integrability

SFB 1624：更高的结构、模块化空间和可集成性

• 批准号: 506632645
• 金额: --
• 依托单位: Technische Universität München (TUM)
• 依托单位国家: 德国
• 项目类别: Collaborative Research Centres
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/506632645?language=en
• 关键词: SFB; 1624; Higher; structures; moduli

This CRC is a joint venture of mathematicians and physicists. It has equally strong motivations from both fields. The interaction is highly beneficial for both sides. Mathematics provides concepts for physical theories, and instruments for deriving the predictions. Physics suggests several profound and surprising relations between different parts of mathematics. Many important questions about the origin of our universe, and about the basic constituents of matter are still wide open. While we do have very promising candidates for fundamental theories, there are huge problems in deriving predictions for concrete physical questions from them. New mathematics is needed to overcome these obstacles. Three of the most central difficulties are: First, it is often not clear which quantity or mathematical object is best-suited to exhibit the physical content of a theory. Quantum field theory and string theory can be formulated as theories of fields, quantities varying within space and time. However, different field configurations can describe the same physics. Second, one needs to find averages over the fields called defects having direct physical relevance. Defects depend on the choice of the region in space-time over which the average is performed. One needs to describe the relations among defects associated to regions of varying geometric shapes. Third, solving the equations defining fundamental theories can be very hard. Constructing and analysing solutions is an enormous mathematical challenge. Modern mathematics develops instruments addressing these issues. Higher structures are mathematical concepts which can describe hierarchies of relations among mathematical objects associated to regions of varying dimensions. Moduli spaces are auxiliary geometric spaces having points associated to the sets of field configurations which can be considered equivalent. Integrability is an additional feature that the equations of mathematical physics can exhibit, allowing one to find exact solutions from which we can learn a lot. Our CRC will combine research on higher structures, moduli spaces and integrability in a completely new way. It will pave the way towards a mathematical synthesis of results on these topics and discover new relations between different parts of mathematics. This type of interaction has already led to several mathematical breakthroughs like mirror symmetry. This new mathematics will also allow us to develop powerful techniques to solve paradigmatic examples of quantum field theories and string theories exactly. In this way we will overcome obstacles which have hampered progress in these directions for a long time. An ideal blend of expertise in the relevant areas of pure mathematics and theoretical physics, combined with a very successful tradition of interactions between these disciplines, make Hamburg a perfect place for this line of research.

该CRC是数学家和物理学家的合资企业。这两个领域都具有强大的动机。这种相互作用对双方都非常有益。数学提供了物理理论的概念，以及用于得出预测的工具。物理学提出了数学不同部分之间的几个深刻而令人惊讶的关系。关于我们宇宙起源以及物质基本成分的许多重要问题仍然是敞开的。虽然我们确实有非常有前途的基本理论候选人，但从他们的物理问题中提出预测时存在巨大问题。需要新的数学来克服这些障碍。最中心的三个困难是：首先，通常不清楚哪种数量或数学对象是最适合展示理论的物理内容的。量子场理论和弦理论可以作为场理论进行表述，数量在空间和时间内变化。但是，不同的现场配置可以描述相同的物理。其次，需要在称为缺陷的字段上找到具有直接物理相关性的缺陷。缺陷取决于在执行平均值的时空选择区域的选择。人们需要描述与不同几何形状区域相关的缺陷之间的关系。第三，解决定义基本理论的方程式可能非常困难。构建和分析解决方案是一个巨大的数学挑战。现代数学开发了解决这些问题的工具。较高的结构是数学概念，可以描述与不同维度区域相关的数学对象之间关系的层次结构。模量空间是辅助几何空间，其点与一组现场配置相关，可以认为是等效的。集成性是数学物理方程可以展示的附加功能，使人们可以找到可以从中学到很多东西的确切解决方案。我们的CRC将以一种全新的方式结合对更高结构，模量空间和集成性的研究。它将为这些主题的结果数学综合铺平道路，并发现数学不同部分之间的新关系。这种类型的互动已经导致了几个数学突破，例如镜像对称性。这种新的数学还将使我们能够开发出强大的技术来解决量子场理论和弦理论的范式示例。通过这种方式，我们将克服很长一段时间以来在这些方向上阻碍进步的障碍。在纯数学和理论物理学相关领域的专业知识的理想融合，再加上这些学科之间非常成功的互动传统，使汉堡成为这一研究的理想场所。