论文标题
产品歧管是一般线性对称性的实现
Product manifolds as realisations of general linear symmetries
论文作者
论文摘要
本文考虑了几何形状，对称性和基本相互作用 - 重力与仪表场介导的相互作用之间的关系。我们探索了a）具有仪表相互作用的必要对称性以及四维重力的产品空间，b）将其降低到其平面限制中的N维各大各向同性宇宙。关键技术是在坐标系更改下查看对称级别两种张量的操作员形式的轨道。含有对角矩阵的轨道被认为对应于产物歧管。分解的宇宙的GL（N，R）对称性在这种乘积时空上非线性作用。 我们探讨了所得的kaluza-klein理论，其中内部对称性在额外维度的空间上间接起作用，并给出了两个示例：一个六维模型，其中规格对称性为u（1）和一个七维模型，其中su（2）。我们确定可以放置在任何级别两个对称张量上的约束，以获得此类空间：多项式不变性之间的关系。其特征值的多重性决定了因子空间的维度，从而确定了量规的对称性。如果所讨论的张量是RICCI张量，则除二维因子空间外，所有因子空间都是爱因斯坦歧管。这种情况代表了Kaluza-Klein理论的经典真空
This paper considers the relationship between geometry, symmetry and fundamental interactions -- gravity and those mediated by gauge fields. We explore product spacetimes which a) have the necessary symmetries for gauge interactions and four-dimensional gravity and b) reduce to an N-dimensional isotropic universe in their flat space limit. The key technique is looking at orbits of the operator form of symmetric rank-two tensors under changes of coordinate system. Orbits containing diagonal matrices are seen to correspond to product manifolds. The GL(N, R) symmetry of the decompactified universe acts non-linearly on such a product spacetime. We explore the resulting Kaluza-Klein theories, in which the internal symmetries act indirectly on space of the extra dimensions, and give two examples: a six-dimensional model in which the gauge symmetry is U (1) and a seven-dimensional model in which it is SU (2). We identify constraints that can be placed on any rank-two symmetric tensor to obtain such spacetimes: relationships between polynomial invariants. The multiplicities of its eigenvalues determine the dimensionalities of the factor spaces and hence the gauge symmetries. If the tensor in question is the Ricci tensor, other than two-dimensional factor spaces all the factor spaces are Einstein manifolds. This situation represents the classical vacuum of the Kaluza-Klein theory
