Singular spaces of special and exceptional holonomy.

特殊而独特的独特空间。

• 批准号: EP/L001527/1
• 负责人: Mark Haskins
• 金额: $ 32.27万
• 依托单位: Imperial College London
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2013
• 资助国家: 英国
• 起止时间: 2013 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FL001527%2F1
• 关键词: Singular; spaces; special; exceptional; holonomy.

M theory is an 11-dimensional theory in theoretical physics; it is thought a promising candidate for a consistent quantum theory of gravity, i.e. a theory that unites quantum mechanics with Einstein's theory of General Relativity in a consistent way. To obtain 4-dimensional universes like ours from an 11-dimensional theory one postulates that the ``extra 7 dimensions'' are small. This process is called compactification and the small 7-dimensional space is called the compactifying space. At low enough energy scales the physics is then 4-dimensional and the properties of the 4-dimensional physics is determined by the features of the compactifying space. M theory compactifications on special 7-dimensional spaces--called manifolds with G2-holonomy--attracted particular attention because of their potential to incorporate theories that can accurately describe all the currently known fundamental particles in nature into a unified theory containing gravity. These same G2 holonomy spaces had already been studied for many years by mathematicians studying geometry. Geometers knew that if they could find such spaces then they would have very special geometric properties involving their curvature. Geometers call these spaces Ricci-flat because some part of their curvature, their so-called Ricci curvature vanishes. Ricci curvature also plays an important role in the basic equations in General Relativity. For this reason mathematicians regard Ricci-flat spaces as very special; they are like analogues of the spaces in General Relativity one would see if no matter were present. However Ricci-flat spaces that were not already totally flat proved very hard to find and none were known until 1978. Manifolds with G2 holonomy (also called G2-manifolds) proved even harder to find; it wasn't until the mid 1990s that Joyce found a way to produce G2-manifolds of finite extent. Finding such G2-manifolds was considered a major achievement and involved solving systems of so-called nonlinear partial differential equations in a rather indirect way---by first solving different and very slightly easier equations, and then proving that an appropriate small adjustment of this solution would solve the original system of equations. However, when theoretical physicists started studying the physical properties of M theory compactifications on the G2-manifolds found by Joyce, they realised there was a problem. The physical theories they got out turned out not to be compatible with the basic known facts about the fundamental particles. Later other theoretical physicists re-examined these problems in M theory and realised a way out of their dilemma. If the 7-dimensional compactifying space still had the special Ricci-curvature property described, but in addition had some very special points that look different from surrounding points and at which the full curvature can be infinite, then the physicists could get more complicated and realistic theories. Geometers called these special points, singularities, because of the curvature being infinite there. If M theorists supposed that singular G2 holonomy spaces with very special kinds of singularities existed, then they found that they got out physics compatible with the basic known facts about the fundamental particles. The only problem was that mathematicians could no longer demonstrate that such singular G2 holonomy spaces exist. The method Joyce had pioneered broke down in the presence of the singularities that the physicists needed to get out realistic physics. Even today geometers have still not be able to demonstrate that the singular G2 spaces needed by M theorists exist. This proposal sets out to develop the new mathematics needed to find these kinds of singular G2 spaces (and other singular spaces with similar curvature properties) as part of a collaborative project involving both geometers and M theorists.

M理论是理论物理中的11维理论；它被认为是一致量子引力理论的有希望的候选者，即以一致的方式将量子力学和爱因斯坦的广义相对论统一起来的理论。为了从11维理论中获得像我们这样的4维宇宙，人们假定“额外的7维”是小的。这个过程被称为紧凑化，而小的7维空间被称为紧凑化空间。在足够低的能量尺度下，物理是4维的，而4维物理的性质是由紧致空间的特征决定的。在特殊的7维空间上的M理论压缩--称为具有G2-完整的流形--引起了特别的关注，因为它们有可能将能够准确描述自然界中所有目前已知的基本粒子的理论整合到一个包含重力的统一理论中。这些相同的G2完整空间已经被研究几何的数学家研究了很多年。几何学家知道，如果他们能找到这样的空间，那么他们就会拥有与曲率有关的非常特殊的几何性质。几何学家称这些空间为Ricci平坦空间，因为它们的曲率的一部分，即所谓的Ricci曲率消失了。在广义相对论的基本方程中，Ricci曲率也起着重要作用。出于这个原因，数学家认为利玛窦平坦空间是非常特殊的；它们就像是广义相对论中的空间的类比，如果没有物质存在，人们就会看到。然而，事实证明，还不是完全平坦的利玛窦-平坦空间很难找到，直到1978年才为人所知。事实证明，具有G2完整流形(也称为G2-流形)的流形更难找到；直到20世纪90年代中期，Joyce才找到了产生有限范围的G2-流形的方法。找到这样的G2流形被认为是一项重大的成就，涉及到以一种相当间接的方式求解所谓的非线性偏微分方程组-首先求解不同的、非常简单的方程，然后证明对该解进行适当的小调整就可以求解原始的方程组。然而，当理论物理学家开始研究乔伊斯发现的G2流形上M理论紧致的物理性质时，他们意识到存在一个问题。事实证明，他们得出的物理理论与基本粒子的基本已知事实并不相容。后来，其他理论物理学家重新研究了M理论中的这些问题，并意识到了一种摆脱困境的方法。如果7维紧空间仍然具有所描述的特殊的Ricci-曲率性质，但又有一些看起来与周围点不同的非常特殊的点，并且在这些点上的全曲率可以是无穷的，那么物理学家就可以得到更复杂和更现实的理论。几何学家称这些特殊的点为奇点，因为那里的曲率是无限的。如果M理论家假设存在具有非常特殊奇点的奇异G2完整空间，那么他们发现他们得到了与基本粒子的基本已知事实相一致的物理学结论。唯一的问题是，数学家再也无法证明这种奇异的G2完整空间是存在的。乔伊斯开创的方法在出现奇点的情况下失败了，物理学家们需要这些奇点来研究现实物理学。即使在今天，几何学家仍然不能证明M理论家所需的奇异G2空间的存在。作为一个涉及几何学家和M理论家的合作项目的一部分，这项提议着手开发寻找这些类型的奇异G2空间(以及其他具有类似曲率性质的奇异空间)所需的新数学。

项目成果

New G2 holonomy cones and exotic nearly Kaehler structures on the 6-sphere and the product of a pair of 3-spheres

新的 G2 完整锥体和 6 球体上的奇异近凯勒结构以及一对 3 球体的乘积

• DOI: 10.48550/arxiv.1501.07838
• 发表时间: 2015
• 作者: Foscolo L

ALF gravitational instantons and collapsing Ricci-flat metrics on the $K3$ surface

$K3$ 表面上的 ALF 引力瞬子和塌陷 Ricci 平坦度量

• DOI: 10.4310/jdg/1557281007
• 发表时间: 2019
• 期刊: Journal of Differential Geometry
• 影响因子: 2.5
• 作者: Foscolo L

ALF gravitational instantons and collapsing Ricci-flat metrics on the K3 surface

K3 表面上的 ALF 引力瞬子和塌陷 Ricci 平坦度量

• DOI: 10.48550/arxiv.1603.06315
• 发表时间: 2016
• 作者: Foscolo L

Asymptotically cylindrical Calabi-Yau 3-folds from weak Fano 3-folds

来自弱 Fano 3 倍的渐近圆柱 Calabi-Yau 3 倍

• DOI: 10.2140/gt.2013.17.1955
• 发表时间: 2013
• 期刊: Geometry & Topology
• 影响因子: 2
• 作者: Corti A

Asymptotically conical Calabi-Yau metrics on quasi-projective varieties

准射影簇的渐近圆锥形 Calabi-Yau 度量

• DOI: 10.48550/arxiv.1301.5312
• 发表时间: 2013
• 作者: Conlon R