Cayley submanifolds in Spin(7)-manifolds

1955年，Marcel Berger对流形M(满足一些基本条件)上黎曼度量g的可能完整群进行了分类，给出了可能完整群的列表SO(N)，U(M)，Su(M)，Sp(M)，G2和Spin(7)。如果完整群不是so(N)，则g与M上的附加几何结构相容--这使得它特别而有趣。具有完整度规SU(M)、Sp(M)、G2和自旋(7)的度规是Ricci平坦的，在物理学中特别是在弦理论和M-理论中是重要的。G2(7维)和自旋(7)(8维)称为例外完整群。Joyce教授在1993-2005年构建了具有完整G2和自旋(7)的紧致流形的第一个例子。G2-流形在M-理论中尤其重要，因为它是构建宇宙的基础。刻度几何是黎曼完整群的自然伴生。对于任何特殊的完整群，它定义了特殊的极小子流形的有趣类，称为定标子流形。在Calabi-Yau流形(具有完整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-折叠和联合4-折叠。在自旋(7)流形中有Cayley 4折叠。定标子流形在弦理论和M理论中是重要的，作为膜的经典几何基础。1995年Syz猜想用特殊的拉格朗日子流形，包括奇异纤维，用Calabi-Yau流形的对偶纤丝来解释镜面对称性。从那时起，利用包括奇异纤维在内的定标子流形来构造具有特殊完整的紧致流形的纤颤的例子一直是一个重要的公开问题。有趣的可能性是由特殊拉格朗日量纤维的Calabi-Yau流形，由共结合4-折叠纤维的G2流形，以及由Cayley 4-折叠纤维的自旋(7)流形。我们建议研究紧致和自旋(7)流形中的Cayley 4-折叠，包括奇异的Cayley 4-折叠(可能具有孤立的锥形奇点)。我们应该看看他们的变形理论，以及奇点的解决方法。一个长期目标，可能在PHD中实现，也可能无法实现，将通过紧凑的Cayley 4折叠(包括奇异纤维)构建具有纤颤的紧凑自旋(7)流形的例子。在我们到达那里之前，我们需要开发技术来处理Cayley四重数族，包括奇点，以及例子中的族的构造。有几种不同强度的“纤维”概念：a)最强的是每个点恰好有一根纤维通过。B)较弱的概念是，具有由紧凑的4-流形参数化的Cayley 4-折叠的族，使得一根纤维穿过用符号计数的每个点。c)人们还可以要求包括流形的体积的99%的开集，其中由Cayley 4-折叠对开集进行纤化；或者可能要求流形、开集和纤化的族，使得在纤化中的体积比例(例如99%)在极限中趋于100%。用胶合来研究奇点Cayley 4折叠和奇点的分辨率可能会产生b)型纤维(如果我们努力工作的话)。我们还不清楚这种颤动是否也会满足a)。我们可以研究的一个问题是，在给定的奇异模型下，a型纤颤是否在小扰动下是稳定的，因为这可能有助于确保a)在例子中成立。如果Y是G2流形，则X=Y x S1是(退化的)自旋(7)流形。由共结合的4折对Y的纤化产生由Cayley 4折对X的纤化。相反地，X的Cayley 4-折叠的S1-不变纤维下降为Y的共结合4-折叠的纤维。这个项目属于EPSRC的几何和拓扑学领域。

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.