Beyond Lorentzian symmetry

Ross Grassie
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引用次数: 3

Abstract

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.
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超越洛伦兹对称
本文提出了一个框架,在其中探索超越标准洛伦兹的情况下的运动对称性。该框架由代数分类、几何分类和在分类时空几何上定义物理理论所需的几何性质的推导组成。在证实运动学、超运动学和超巴格曼对称框架方面所完成的工作构成了本文的主体。为此,回顾了[3,4]中提出的空间维$D = 3$的运动学李代数的分类;b[5]的空间各向同性均匀时空分类也是如此。在此基础上推导了凸形、焊接形和凸形的非紧性以及不变仿射连接的空间等几何性质。我们继续在三维空间中对$\math {N}=1$运动学李超代数进行分类,发现了43个李超代数的同构类。一旦这些代数被确定,我们分类相应的单连通齐次(4|4)维超空间,并展示了所得的27个齐次超空间如何通过几何极限相互关联。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,这些将是巴格曼和牛顿-胡克代数的$\mathcal{N}=1$和$\mathcal{N}=2$的超扩展,以及中心扩展的静态运动学李代数,其中前三种都是变形产生的。仅关注三维空间,我们在$\mathcal{N}=1$的情况下发现$9$同构类,并在$\mathcal{N}=2$的情况下识别了$22$超代数分支。
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