Geometric structures and twisted supersymmetry

从牛顿时代开始，数学和理论物理的发展就有着密切的联系。现代最大的挑战之一是制定一个统一的理论，结合我们目前对宇宙的理解的两大支柱：粒子物理学的标准模型和爱因斯坦的广义相对论。尽管他们取得了非凡的成功，但这两种理论在根本上是不相容的，似乎要解决这个物理问题，又一次需要数学上的巨大进步。超弦理论是一个很有希望成为统一理论的候选者，但它只存在于十维时空中。解释我们通过弦理论观察到的四维世界的一种方法是，把额外的六个维度看作是一个微小的紧密的几何空间（就像一个球体的表面，但有更多的维度），它的存在只能由能量如此之高的粒子直接探测到，而这些粒子在当前的实验中是无法产生的。在这个结构中，这个紧凑空间的形状决定了我们在实验中看到的四维物理定律。此外，理解这些空间不仅对弦理论中的物理模型很重要——它们也是纯数学研究中深刻问题的主题。该项目旨在通过一种新的物理数学方法，对它们的几何性质进行新的理解。在最简单的情况下，人们需要的空间是被称为特殊完整流形的物体，数学家们已经研究了几十年。在这门课上，许多研究人员都把注意力集中在称为Calabi-Yau流形的空间上，因为它们在物理上很有前途，而且相对容易构建。然而，存在更一般的解决方案，具有称为通量的附加物理场，并且具有物理上理想的特征。最近，一个新的几何数学概念，“广义几何”，已经发展起来，它自然地包括了这些通量，这为更一般的解提供了一个优雅的描述。这只是物理学导致新数学的众多例子之一，这两个学科的混合经常导致双方取得惊人的进步。一个将弦理论和几何更紧密地联系在一起的重要观察是，在Calabi-Yau流形上存在简化的弦理论，称为拓扑弦理论，其物理性质更易于处理，并直接编码几何的有趣数学特征。特别是，它们编码的量在几何的光滑变形下是不变的。这些不变量被认为是空间的关键数学属性，例如，用于确定这些空间的构造示例是否真的不同，否则很难确定。相反，不变量编码了拓扑弦的物理信息，这反过来又可以为完整弦理论的计算提供精确的答案。然而，所有这些都只在Calabi-Yau流形的情况下被理解。有迹象表明，这样的理论存在于更一般的情况下，而广义几何是接近它们的构造的自然方式。与此特别相关的一个场景是另一种特殊的完整流形：称为G2流形的七维空间。这些已经成为几何社区的一个特别关注的焦点，即使它们仍然具有零通量，广义几何已经被视为为这些空间提供了一种优雅的“物理”方法。该项目旨在在更广泛的背景下构建拓扑弦理论的类似物，并利用这些来发现新的不变量，这将成为对这些空间的数学和物理理解的关键对象。

Since the era of Newton, there has been a close relationship between developments in mathematics and theoretical physics. One of the greatest challenges in the modern age is to formulate a unified theory combining the two pillars of our current understanding of the universe: the Standard Model of particle physics and Einstein's General Relativity. Despite their extraordinary successes, these two theories are fundamentally incompatible and it seems that, once again, a resolution of this physical problem will require great advances in mathematics.Superstring theory is a promising candidate for a unified theory, but it exists only in ten-dimensional spacetime. One way to explain the four-dimensional world that we observe via string theory is to view the extra six dimensions as a tiny compact geometric space (like the surface of a sphere, but with more dimensions) whose presence can only be directly detected by particles of such high energy that they are not created in current experiments. In this construction, the shape of this compact space determines the four-dimensional laws of physics we can see around us in experiments. Further, understanding these spaces is not only important for physical models in string theory - they are also the subject of deep questions in pure mathematics research.This project aims to develop new understanding of their geometric properties, via a new physical approach to the mathematics. In the simplest scenarios, the spaces which one requires turn out to be objects, called special holonomy manifolds, that have been studied by mathematicians for decades. Many researchers have focused on spaces in this class called Calabi-Yau manifolds, as they are physically promising and relatively easy to construct. However, more general solutions exist, with additional physical fields called fluxes, and these have physically desirable features. Recently, a new mathematical notion of geometry, "generalised geometry", has been developed which includes these fluxes naturally, and this provides an elegant description of the more general solutions. This is just one of many examples of physics leading to new mathematics and the mixing of these two disciplines has often led to astounding progress on both sides.An important observation, which linked string theory and geometry even more closely, was that on Calabi-Yau manifolds there exist simplified string theories, called topological string theories, whose physics are more tractable and directly encode interesting mathematical features of the geometries. In particular, they encode quantities which are invariant under smooth deformations of the geometry. These invariants are regarded as key mathematical properties of the spaces and are used, for example, to determine if constructed examples of such spaces are really different or not, which can be very hard to ascertain otherwise. Conversely, the invariants encode information about the physics of topological strings, which in turn can provide exact answers to calculations in the full string theory.However, all of this has only been understood in the cases of Calabi-Yau manifolds. There are signs that such theories exist in more general situations, and that generalised geometry is the natural way to approach their constructions. One scenario where this is particularly relevant is actually another type of special holonomy manifold: seven-dimensional spaces called G2 manifolds. These have become a particular focus of the geometry community, and even though they still have zero flux, generalised geometry has been seen to give an elegant "physical" approach to these spaces.This project aims to construct analogues of topological string theories in this wider context and to use these to discover new invariants, which will become key objects in both the mathematical and physical understanding of these spaces.