SFB 1624: Higher structures, moduli spaces and integrability

这个CRC是数学家和物理学家的合资企业。它在这两个领域都有同样强烈的动机。这种互动对双方都非常有益。数学为物理理论提供概念，并为推导预测提供工具。物理学揭示了数学不同部分之间的几种深刻而令人惊讶的关系。关于我们宇宙的起源和物质的基本组成的许多重要问题仍然悬而未决。虽然我们确实有非常有希望的基础理论候选者，但从它们中推导出具体物理问题的预测存在巨大问题。需要新的数学来克服这些障碍。其中最主要的三个困难是：第一，通常不清楚哪一个量或数学对象最适合展示一个理论的物理内容。量子场论和弦论可以表述为场的理论，即在空间和时间内变化的量。然而，不同的场组态可以描述相同的物理。第二，需要找到具有直接物理相关性的称为缺陷的场的平均值。缺陷取决于在时空中执行平均的区域的选择。需要描述与不同几何形状的区域相关联的缺陷之间的关系。第三，求解定义基本理论的方程可能非常困难。构造和分析解决方案是一个巨大的数学挑战。现代数学发展工具解决这些问题。高级结构是数学概念，它可以描述与不同维度区域相关联的数学对象之间的关系层次。模空间是辅助几何空间，其点与可以被认为是等价的场配置的集合相关联。可积性是数学物理方程可以展示的一个附加特征，它允许人们找到精确的解，从中我们可以学到很多东西。我们的CRC将联合收割机研究更高的结构，模空间和可积性在一个全新的方式。它将为这些主题的结果的数学合成铺平道路，并发现数学不同部分之间的新关系。这种类型的相互作用已经导致了几个数学突破，如镜像对称。这种新的数学也将使我们能够发展强大的技术来精确地解决量子场论和弦理论的范例。这样，我们将克服长期以来阻碍在这些方面取得进展的障碍。在纯数学和理论物理的相关领域的专业知识的理想融合，再加上这些学科之间的互动非常成功的传统，使汉堡成为这条研究线的理想场所。

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.