Motivic invariants and categorification

动机不变量和分类

基本信息

批准号：
EP/I033343/1
负责人：
Dominic Joyce
金额：
$ 236.96万
依托单位：
University of Oxford
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2011
资助国家：
英国
起止时间：
2011 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FI033343%2F1
关键词：
Motivic invariants categorification

项目摘要

The proposal aims to discover new structures in geometry, and algebra, and string theory in theoretical physics. Beginning with some classical situation which is already well understood, we aim to generalize it in two directions: we can make the classical situation motivic , or we can categorify it.These are technical words, so an analogy may help. The thing we already understand, the classical mathematics, is like a 2-dimensional shadow on the wall, cast by some 3-dimensional object. Our goals are analogous to understanding this 3-dimensional object, exploring the implications of the extra third dimension, and then seeing what new things we can find out about the shadow by viewing it as the projection of a more complex 3-dimensional object.In both mathematics and physics, there are good notions of the dimension of a mathematical structure - for instance, in physics an n-dimensional field theory is a quantum theory which quantizes maps from n-dimensional objects into some space-time. Oversimplifying rather, classical quantum theory regards particles as points (0-dimensional objects) moving in space-time, so is a 0-dimensional field theory. String theory regards particles as 1-dimensional loops of string moving in space-time, so is a 1-dimensional field theory; more recent developments in physics (M-theory) consider higher dimensional membranes moving in space-time.The idea of categorification is to replace n-dimensional mathematical structures by (n+1)-dimensional structures in a problem, in some systematic way, so that you get the original n-dimensional structure back again when you reduce dimension by one - like passing from a 2-dimensional shadow, to the 3-dimensional object that casts it.In geometry, an invariant is usually a number which counts some class of objects. But because the classes of objects we want to count are usually infinite, this counting has to be done in a complicated way. If you count the objects in just the right way, your invariant may turn out to have some special properties - for instance, it may be unchanged when you deform the underlying space. This kind of thing makes mathematicians excited, as it suggests the invariant is measuring some deeper underlying structure, and we want to know what this is. For example, mirror symmetry is a circle of conjectures coming from physics, which are slowly being proved. One central claim is a surprising equality of invariants: invariants counting curves in a space X should be equal to invariants counting something else on a different space Y, because the quantum theories of X and Y are related. On the face of it, this is as bizarre as saying that quantum theory requires the numbers of giraffes in the Gambia, and of zebras in Zambia, to be the same.An invariant is something which counts the points in a space. It could be a number (integer), or something more general. An invariant of spaces is motivic if, when you cut the space into two pieces, the invariant is the sum of the invariants of the pieces. The most basic is the Euler characteristic , but there are also many other more complicated motivic invariants.Some of the invariants studied in geometry (for instance, Donaldson-Thomas invariants of Calabi-Yau 3-folds, which appear in string theory) use Euler characteristics to do the actual counting. One can try to define a new invariant which counts the same things, but using some other motivic invariant instead of Euler characteristics. This is what we mean by a motivic generalization. The new invariants should be richer, with more structure and information. They may also make new things possible.As one application, we hope to help physicists understand a bit more about what string theory actually is. String theory (in its final form) may be the mathematics underlying the universe, and has been a fertile source of new mathematics for decades, but much of it is still a mystery.

该提案旨在发现几何形状和代数的新结构以及理论物理学中的弦理论。从已经被充分理解的经典状况开始，我们旨在将其概括为两个方向：我们可以使经典的情况动机，或者我们可以对其进行分类。这些都是技术词，因此类比可能会有所帮助。我们已经理解的是古典数学，就像墙上的二维阴影一样，由一些三维对象铸造。 Our goals are analogous to understanding this 3-dimensional object, exploring the implications of the extra third dimension, and then seeing what new things we can find out about the shadow by viewing it as the projection of a more complex 3-dimensional object.In both mathematics and physics, there are good notions of the dimension of a mathematical structure - for instance, in physics an n-dimensional field theory is a quantum theory which quantizes maps from n-dimensional objects进入一些时空。过度简化的，经典的量子理论将粒子视为在时空移动的点（0维对象），因此0维场理论也是如此。弦理论将粒子视为在时空中移动的弦的一维循环，因此1维场理论也是如此。 more recent developments in physics (M-theory) consider higher dimensional membranes moving in space-time.The idea of categorification is to replace n-dimensional mathematical structures by (n+1)-dimensional structures in a problem, in some systematic way, so that you get the original n-dimensional structure back again when you reduce dimension by one - like passing from a 2-dimensional shadow, to the 3-dimensional object that casts it.In几何形状，一个不变的通常是一个数字，它计算某些对象。但是，由于我们要计算的对象类别通常是无限的，因此必须以复杂的方式完成此计数。如果您以正确的方式计算对象，则您的不变可能会具有一些特殊的属性 - 例如，当您变形基础空间时，它可能会不变。这种事情使数学家感到兴奋，因为它表明不变的是测量了一些更深的基础结构，我们想知道这是什么。例如，镜像对称是来自物理学的一个猜想圈，它们正在缓慢地证明。一个中心主张是不变的令人惊讶的平等：不变的空间x中的曲线应等于不变的曲线，因为x和y的量子理论是相关的，因为在其他空间y上计算其他东西。从表面上看，这与说量子理论需要冈比亚的长颈鹿数量和赞比亚的斑马一样奇怪。它可能是一个数字（整数），也可能是更通用的东西。如果将空间切成两个零件时，不变的空间是动机，那么不变的是碎片的不变式的总和。最基本的是Euler的特征，但是还有许多其他更复杂的动机不变性。在几何形状研究的不变性人物（例如，Calabi-yau的Donaldson-Thomas不变性3倍，以弦乐理论出现）使用Euler特征来完成实际计数。人们可以尝试定义一个新的不变式，该新不变式计算相同的事物，但使用其他一些动机不变而不是Euler特征。这就是动机概括的意思。新不变的人应该更丰富，并具有更多的结构和信息。它们也可能使新事物成为可能。作为一个应用程序，我们希望帮助物理学家更多地了解弦理论的实际是什么。弦理论（以最终形式）可能是宇宙的基础数学，数十年来一直是新数学的肥沃来源，但其中大部分仍然是一个谜。