Cayley submanifolds in Spin(7)-manifolds

Spin(7) 流形中的 Cayley 子流形

• 批准号: 2422851
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2422851
• 关键词: Cayley; submanifolds; Spin; manifolds

In 1955, Marcel Berger classified the possible holonomy groups of Riemannian metrics g on a manifold M (satisfying some basic conditions, giving the list SO(n), U(m), SU(m), Sp(m), G2 and Spin(7) of possible holonomy groups. If the holonomy group is not SO(n) then g is compatible with additional geometric structures on M -- this makes it special and interesting. Metrics with holonomy SU(m), Sp(m), G2 and Spin(7) are Ricci-flat, and are important in Physics, especially in String Theory and M-Theory. G2 (in 7 dimensions) and Spin(7) (in 8 dimensions) are called the exceptional holonomy groups. Professor Joyce constructed the first examples of compact manifolds with holonomy G2 and Spin(7) in 1993-5. G2-manifolds are especially important in M-Theory, as ingredients to build the universe from. Calibrated geometry is a natural companion subject to Riemannian holonomy groups. For any special holonomy group, it defines intersting classes of special minimal submanifolds called calibrated submanifolds. In Calabi-Yau manifolds (with holonomy SU(m)) these are special Lagrangian submanifolds (SL m-folds). In G2 manifolds there are associative 3-folds and coassociative 4-folds. In Spin(7) manifolds there are Cayley 4-folds. Calibrated submanifolds are important in String Theory and M Theory, as the classical geometry underlying 'branes'. The SYZ Conjecture in 1995 explained Mirror Symmetry in terms of dual fibrations of a Calabi-Yau manifold by special Lagrangian submanifolds, including singular fibres. Ever since then, it has been an important open question to construct examples of fibrations of compact manifolds with special holonomy by calibrated submanifolds, including singular fibres. The interesting possibilities are Calabi-Yau manifolds fibred by special Lagrangians, G2 manifolds fibred by coassociative 4-folds, and Spin(7) manifolds fibred by Cayley 4-folds. We propose to study Cayley 4-folds in compact and Spin(7) manifolds, including singular Cayley 4-folds (probably with 'isolated conical singularities'). We should look at their deformation theory, and resolution of singularities. A long term goal, which may or may not be achieved in the PhD, would be to construct examples of compact Spin(7) manifolds with fibrations by compact Cayley 4-folds, including singular fibres. Before we get there, we need to develop technology to deal with families of Cayley 4-folds, including singularities, and construction of families in examples. There are several notions of 'fibration', of varying strength: a) The strongest is that exactly one fibre passes though each point. b) A weaker notion is that one has a family of Cayley 4-folds parametrized by a compact 4-manifold, such that one fibre passes through each point counted with signs.c) One could also ask for an open set comprising 99% of the volume of the manifold, with a fibration of the open set by Cayley 4-folds; or maybe for a family of manifolds, open sets and fibrations such that the proportion of volume in the fibration (e.g. 99%) tends to 100% in a limit. Studying singular Cayley 4-folds and resolutions of singularities using gluing may yield a fibration of type b) (if we work hard). It is not yet clear to us whether the fibration will also satisfy a). One question we could investigate is whether fibrations of type a), with given singular models, are stable under small perturbations, as this would probably help ensure a) holds in examples. If Y is a G2 manifold then X = Y x S1 is a (degenerate) Spin(7) manifold. A fibration of Y by coassociative 4-folds yields a fibration of X by Cayley 4-folds. Conversely, an S1-invariant fibration of X by Cayley 4-folds descends to a fibration of Y by coassociative 4-folds. This project falls within the EPSRC's Geometry and Topology area.

1955年，Marcel Berger对流形M（满足一些基本条件）上的黎曼度量g的可能的完整群进行了分类，给出了可能的完整群的列表SO（n），U（m），SU（m），Sp（m），G2和Spin（7）。如果完整群不是SO（n），那么g与M上的其他几何结构相容--这使得它特别而有趣。SU（m），Sp（m），G2和Spin（7）都是Ricci平坦的，在物理学，特别是弦论和M理论中有重要的意义。G2（7维）和Spin（7）（8维）被称为例外完整群。Joyce教授在1993-5年用holonomy G2和Spin（7）构造了紧致流形的第一个例子。G2-流形在M理论中特别重要，作为构建宇宙的成分。校准几何是黎曼完整群的自然伴侣。对于任何特殊的完整群，它定义了一类有趣的特殊极小子流形，称为校准子流形。在卡拉比-丘流形（具有完整性SU（m））中，这些是特殊的拉格朗日子流形（SL m-折叠）。在G2流形中有结合3-折叠和余结合4-折叠。在Spin（7）流形中有Cayley 4-folds。校准子流形在弦论和M理论中很重要，因为它是经典几何学中的“膜”。1995年的SYZ猜想解释了镜像对称性，即卡-丘流形通过特殊的拉格朗日子流形（包括奇异纤维）的对偶纤维化。从那时起，利用标定子流形（包括奇异纤维）构造具有特殊完整性的紧致流形的纤维化的例子就成为一个重要的公开问题。有趣的可能性是Calabi-Yau流形被特殊的拉格朗日量所包围，G2流形被余结合4-folds所包围，以及Spin（7）流形被Cayley 4-folds所包围。我们打算研究紧流形和Spin（7）流形中的Cayley 4-folds，包括奇异Cayley 4-folds（可能具有“孤立的圆锥奇点”）。我们应该看看他们的变形理论，和奇异性的解决方案。一个长期的目标，这可能会或可能不会实现在博士学位，将构造紧凑的自旋（7）流形的例子与纤维化紧凑凯莱4倍，包括奇异纤维。在我们到达那里之前，我们需要开发技术来处理Cayley 4-folds族，包括奇点，以及示例中的族的构造。有几种不同强度的“纤维化”概念：a）最强的是恰好有一根纤维穿过每个点。B）一个较弱的概念是，我们有一个由紧致4-流形参数化的凯莱4-折叠族，使得一个纤维穿过用符号计数的每个点。c）我们也可以要求一个包括流形体积的99%的开集，开集由凯莱4-折叠纤维化;或者可能对于一系列歧管、开集和纤维化，使得纤维化中的体积比例（例如99%）在极限内趋于100%。研究奇异的凯莱4-折叠和奇异性的解决方案，使用胶合可能会产生类型B）的纤维化（如果我们努力工作）。我们还不清楚纤维化是否也满足a）。我们可以研究的一个问题是，对于给定的奇异模型，a）型纤维化在小扰动下是否稳定，因为这可能有助于确保a）在例子中成立。如果Y是一个G2流形，那么X = Y x S1是一个（退化的）Spin（7）流形。Y通过共联想4重的纤维化产生X通过凯莱4重的纤维化。相反，X的凯莱4-折叠的S1-不变纤维化下降到Y的共结合4-折叠的纤维化。 该项目属于EPSRC的几何和拓扑区域的福尔斯。