arXiv 2107.094952021-07-20hep-thgr-qcmath.DGmath.RA

超越洛伦兹对称性

Beyond Lorentzian Symmetry

Ross Grassie

备注：: Thesis completed in the School of Mathematics of the University of Edinburgh under the supervision of Jose Figueroa-O'Farrill

摘要：

这篇论文提出了一个框架，在这个框架中，我们将探索超越标准洛伦兹情形的运动学对称性。这个框架包括一个代数分类，几何分类，并推导所需的几何性质，以定义物理理论的分类时空几何。在充实这个框架的运动学，超运动学，超巴格曼对称性完成的工作构成了本论文的主体。为此，回顾了[3，4]中提出的空间维$D = 3$中运动学李代数的分类，以及[5]中空间各向同性齐次时空的分类。推导的几何性质，如非紧性的boosts，焊接形式和vielbeins，和不变的仿射连接的空间。我们继续分类的$\mathcal{N}=1$运动学李超代数在三维空间，发现43个同构类的李超代数。一旦确定了这些代数，我们就对相应的单连通齐次代数进行分类（4| 4）维超空间，并显示如何产生的27个齐次超空间可以通过几何极限彼此相关。最后，我们将注意力转向广义巴格曼超代数。在目前的工作中，这些将是$\mathcal{N}=1$和$\mathcal{N}=2$的Bargmann和Newton-Hooke代数的超扩展，以及中心扩展静态运动李代数，其中前三个都出现的变形。只专注于三个空间维度，我们发现$9$同构类的$\mathcal{N}=1$的情况下，我们确定超代数的$22$分支的$\mathcal{N}=2$的情况下。

英文摘要：

This thesis presents a framework in which to explore kinematical symmetries beyond the standard Lorentzian case. This framework consists of an algebraic classification, a geometric classification, and a derivation of the geometric properties required to define physical theories on the classified spacetime geometries. The work completed in substantiating this framework for kinematical, super-kinematical, and super-Bargmann symmetries constitutes the body of this thesis. To this end, the classification of kinematical Lie algebras in spatial dimension $D = 3$, as presented in [3, 4], is reviewed; as is the classification of spatially-isotropic homogeneous spacetimes of [5]. The derivation of geometric properties such as the non-compactness of boosts, soldering forms and vielbeins, and the space of invariant affine connections is then presented. We move on to classify the $\mathcal{N}=1$ kinematical Lie superalgebras in three spatial dimensions, finding 43 isomorphism classes of Lie superalgebras. Once these algebras are determined, we classify the corresponding simply-connected homogeneous (4|4)-dimensional superspaces and show how the resulting 27 homogeneous superspaces may be related to one another via geometric limits. Finally, we turn our attention to generalised Bargmann superalgebras. In the present work, these will be the $\mathcal{N}=1$ and $\mathcal{N}=2$ super-extensions of the Bargmann and Newton-Hooke algebras, as well as the centrally-extended static kinematical Lie algebra, of which the former three all arise as deformations. Focussing solely on three spatial dimensions, we find $9$ isomorphism classes in the $\mathcal{N}=1$ case, and we identify $22$ branches of superalgebras in the $\mathcal{N}=2$ case.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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