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利。数学为物理理论提供了概念，并为推导预测提供了工具。物理学表明，数学的不同部分之间存在着一些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成分的许多重要问题仍然悬而未决。虽然我们确实有非常有前途的基础理论候选者，但从这些基础理论中得出具体物理问题的预测存在巨大的问题。克服这些障碍需要新的数学方法。最核心的三个困难是：第一，通常不清楚哪个数量或数学对象最适合展示理论的物理内容。量子场论和弦理论可以被表述为场论，即量在空间和时间内变化的理论。然而，不同的场结构可以描述相同的物理。其次，人们需要找出被称为缺陷的领域的平均值，这些缺陷具有直接的物理相关性。缺陷取决于在时空中进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式可能非常困难。构造和分析解决方案是一个巨大的数学挑战。现代数学发展了解决这些问题的工具。更高的结构是数学概念，可以描述与不同维度的区域相关联的数学对象之间的关系层次结构。模空间是具有与可被认为是等价的场配置集相关联的点的辅助几何空间。可积性是数学物理方程可以表现出的另一个特征，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将以一种全新的方式结合对更高结构、模空间和可积性的研究。它将为数学综合这些主题的结果铺平道路，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，比如镜像对称。这种新的数学也将允许我们开发强大的技术来精确地解决量子场论和弦理论的范例。通过这种方式，我们将克服长期以来阻碍在这些方向取得进展的障碍。纯数学和理论物理相关领域的专业知识的完美融合，再加上这些学科之间相互作用的非常成功的传统，使汉堡成为这一研究领域的完美地点。

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.