String structures via higher gauge theory

基于高规范理论的弦结构

• 批准号: EP/I010610/1
• 负责人: Daniel Stevenson
• 金额: $ 5.99万
• 依托单位: University of Glasgow
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2010
• 资助国家: 英国
• 起止时间: 2010 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FI010610%2F1
• 关键词: String; structures; via; higher; gauge

The project can be described as `categorified geometry'. Generally, the objects studied in mathematics are sets with extra structure, for example given two elements of a set we might be able to add them and get a new element of the set. However, in a set we only have limited scope to compare elements: we have the notion of equality. The notion of category is a generalization of the notion of a set; we have objects, but we also have morphisms between them with which we can compare different objects. We can picture a set as a collection of dots and we can picture a category as a collection of dots together with arrows connecting different dots. For a category, the analog of the notion of an equation between elements of a set, is the notion of an isomorphism between objects. This is a much richer notion than the notion of equality, two objects can be isomorphic in many different ways. There is a process from which we can condense a category down to a set: we can form the set of isomorphism classes of objects in the category. So if we think of a category as a collection of dots with arrows joining them, then we remove the arrows so that we have just have the collection of dots but moreover we regard two dots as the same if there was an isomorphism connecting them. This process is sometimes called `decategorification'. For example we could take the category of finite sets and functions between them. Two sets are isomorphic if they have exactly the same number of elements. The decategorification of this category is the set of natural numbers. Decategorification is a process which loses information. Categorification is a process which attempts to recover this lost information, as a result this is generally quite difficult. Categorification has been implicit in much progress in modern mathematics. It is particularly important in algebraic topology, where one wants to distinguish between different spaces by assigning `invariants' to a space in the form of algebraic information. If two spaces have different invariants then the spaces must be different. The crudest invariant one could assign to a space is a number: for instance one could count the number of holes in a surface - a donut shape has one hole while a sphere has no holes. This crude invariant was quickly categorified, the number associated to the surface was recognized as the dimension of a certain vector space which could be assigned to the surface. A more sophisticated invariant would be the assignment of a category to the surface. In this project we will study categorified geometry. We will take the usual objects studied in differential geometry at a set theoretic level (for instance bundles and Lie groups) and define and study category theoretic versions of them. In fact, we will not just be interested in the notion of a mere category, but a higher dimensional generalizations of this notion in which we have not just morphisms between objects but 2-morphisms between morphisms, 3-morphisms between 2-morphisms and so on. These structures are playing an ever more prominent role in mathematics. We will lay some foundation stones for a theory of differential geometry in this higher dimensional context. We will use this theory to develop and illuminate some aspects of the notion of `string structure' which is important in homotopy theory and string theory.

该项目可以说是“分类几何”。一般来说，数学研究的对象是具有额外结构的集合，例如，给定一个集合的两个元素，我们可以将它们相加并得到集合的一个新元素。然而，在集合中，我们只有有限的范围来比较元素：我们有相等的概念。范畴的概念是集合概念的推广;我们有对象，但我们也有它们之间的态射，我们可以用它来比较不同的对象。我们可以把一个集合想象成一个点的集合，我们可以把一个类别想象成一个点的集合，还有连接不同点的箭头。对于一个范畴，集合元素之间的等式概念的类似物是对象之间的同构概念。这是一个比相等的概念更丰富的概念，两个对象可以以许多不同的方式同构。有一个过程可以使我们把一个范畴压缩成一个集合：我们可以形成范畴中对象的同构类的集合。因此，如果我们把一个范畴看作是由箭头连接的点的集合，那么我们去掉箭头，这样我们就得到了点的集合，而且如果有同构连接它们，我们就把两个点看作是相同的。这一过程有时被称为“去范畴化”。例如，我们可以把有限集合和它们之间的函数的范畴。如果两个集合的元素个数完全相同，则它们是同构的。这个范畴的非范畴化是自然数的集合。非范畴化是一个信息丢失的过程。分类是试图恢复这些丢失信息的过程，因此这通常是相当困难的。范畴化已经隐含在现代数学的许多进展中。它在代数拓扑学中特别重要，在代数拓扑学中，人们希望通过以代数信息的形式将“不变量”分配给空间来区分不同的空间。如果两个空间有不同的不变量，那么这两个空间一定是不同的。人们可以赋予空间的最粗糙的不变量是一个数字：例如，人们可以计算表面上的孔的数量-甜甜圈形状有一个孔，而球体没有孔。这个粗糙的不变量很快被归类，与曲面相关的数被认为是某个向量空间的维数，该向量空间可以分配给曲面。一个更复杂的不变量是给曲面指定一个类别。在这个项目中，我们将研究分类几何。我们将采取通常的对象研究微分几何在一组理论水平（例如丛和李群），并定义和研究范畴理论版本。事实上，我们感兴趣的不仅仅是一个范畴的概念，而是这个概念的高维推广，其中我们不仅有对象之间的态射，而且有态射之间的2-态射，2-态射之间的3-态射等等，这些结构在数学中扮演着越来越重要的角色。我们将在这个更高维度的背景下为微分几何理论奠定一些基石。我们将使用这个理论来发展和阐明“弦结构”概念的某些方面，这在同伦理论和弦理论中很重要。

项目成果

The Faddeev-Mickelsson-Shatashvili Anomaly and Lifting Bundle Gerbes

Faddeev-Mickelsson-Shatashvili 异常和升力束 Gerbes

• DOI: 10.1007/s00220-012-1608-7
• 发表时间: 2012
• 期刊: Communications in Mathematical Physics
• 影响因子: 2.4
• 作者: Hekmati P