Beyond Lorentzian Symmetry Ross Grassie arXiv:2107.09495v1 [hep-th] 20 Jul 2021 Doctor of Philosophy University of Edinburgh 2021 Declaration I declare that this thesis was composed by myself and that the work contained therein is my own, except where explicitly stated otherwise in the text. No part of this thesis has been submitted here or elsewhere for any other degree or qualification. Chapter 3 of this thesis is based on a published paper [1] authored by Stefan Prohazka, my supervisor José Figueroa-O’Farrill, and myself. Additionally, Chapter 4 is based on a published paper [2] authored by my supervisor José Figueroa-O’Farrill and myself. Chapter 2 contains aspects from both of these papers. (Ross Grassie) iii In Memory of Winifred Whittle iv Acknowledgements I would like to start by thanking my supervisor José Figueroa-O’Farrill. His enthusiasm for mathematics and physics has certainly had a profound impact on my approach to research, teaching me not to be so serious all the time and remember that there is a lot of fun to be had in exploring new material. I would also like to thank my family and friends for their constant support through what has been a very difficult few years. In particular, I would like to thank my partner Katie Meyer for always believing in me and picking me up when I felt as though the world was falling apart. I don’t think my PhD would have been half as fun as it was without my office mates, Fred Tomlinson, Graeme Auld and Ollie Burke. Whether it was going to the pub perhaps a bit too early on Friday or chatting complete nonsense for a few hours in the office, they made the last couple of years infinitely more enjoyable. I think a special mention should also be reserved for Sam Stern and Manya Sahni. If it were not for them, I would never have made it through my undergraduate of masters degrees to even start my PhD! Finally, I would like to thank my gran, Winifred Whittle. I do not think anything I can write here can do justice to how much of a positive influence she has had on my life. It is primarily due to the work ethic she passed on to me that this thesis exists. v vi Lay Summary The goal for any physicist is to better understand the world around them. To complete this task, we need to construct models that estimate, characterise, or otherwise describe the par- ticular phenomenon we are interested in. This may be how an object moves through space, what happens to it over time, what happens when it is exposed to extreme temperatures or pressures, how it interacts with other things; whatever it may be, we want to describe its place in the world. The hope is that by using insightful and robust models, we will accurately pre- dict interesting aspects of future occurrences of the phenomenon. Equipped with the ability to produce these predictions, we may say we have crucial insight into how the world works. However, our understanding of these phenomena will be heavily influenced by the model we choose. Therefore, we need to make sure we are constructing models which are fit for purpose. Constructing a model for a particular phenomenon can be thought of as consisting of two parts. The first part captures the initial conditions of the system. Using the movement of a planet as an example, we would like to know its starting position and how fast it is moving. The second part contains what we may call the “laws of physics” or the “laws of nature”. These are the ideas we believe hold true irrespective of the initial conditions; for example, Newton’s gravitational laws. As the initial conditions will necessarily be different each time we conduct an experiment, the second part of our model gives us the desired predictive power to better understand the world around us. However, building this part of our models is challenging; therefore, we would like to recognise and exploit any inherent structure and coherence in these natural laws. Since mathematics is the chosen language of physicists, we want a mathematical principle that will capture this information for us. The current principle of choice, which guides us in producing our models of the world, is symmetry. In this thesis, we will classify symmetries and construct models of space and time that em- ploy these symmetries. Our hope is that future researchers will use these classifications to accurately describe and better understand the world we live in. vii viii Abstract This thesis presents a framework in which to explore kinematical symmetries beyond the stan- dard Lorentzian case. This framework consists of an algebraic classification, a geometric classi- fication, 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 kine- matical, 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 pre- sented 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 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 geo- metric limits. Finally, we turn our attention to generalised Bargmann superalgebras. In the present work, these will be the N = 1 and N = 2 super-extensions of the Bargmann and Newton-Hooke alge- bras, 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 N = 1 case, and we identify 22 branches of superalgebras in the N = 2 case. ix x Contents Acknowledgements v Lay Summary vii Abstract ix Contents xii 1 Introduction 1 1.1 ABriefHistoryofKinematicalSymmetry . ........ 1 1.2 BeyondLorentzianSymmetries . ...... 2 1.2.1 ClassesofKinematicalSpacetime . ...... 3 1.2.2 Supersymmetry ................................. 4 1.3 OutlineofThesis ................................. 5 2 Mathematical Preliminaries 7 2.1 Algebra......................................... 7 2.1.1 LieAlgebrasandLieSuperalgebras. ...... 7 2.1.2 KinematicalLieAlgebras . 8 2.1.3 AristotelianLieAlgebras. ..... 9 2.1.4 GeneralisedBargmannAlgebras . .... 9 2.1.5 KinematicalLieSuperalgebras . ..... 10 2.1.6 GeneralisedBargmannSuperalgebras . ...... 11 2.2 Geometry........................................ 12 2.2.1 LieGroupsandtheirLieAlgebras . .... 13 2.2.2 ExponentialCoordinatesonaLieGroup . ..... 14 2.2.3 CosetSpaces................................... 15 2.2.4 HomogeneousSpaces ............................. 16 2.2.5 KinematicalSpacetimes . 18 2.2.6 Lie Supergroups and Homogeneous Superspaces . ....... 19 2.3 GeometricProperties ............................. ..... 21 2.3.1 The Group Action and the Fundamental Vector Fields . ....... 21 2.3.2 InvariantConnections . 23 2.3.3 The Soldering Form and the Canonical Connection . ........ 24 2.3.4 InvariantTensors .............................. 25 3 Kinematical Spacetimes 27 3.1 Classification of Kinematical Lie Algebras . ........... 27 3.1.1 Classification of Aristotelian Lie Algebras . .......... 29 3.2 Classification of Kinematical Spacetimes . ........... 29 3.2.1 Admissibility................................. 30 3.2.2 Effectivity .................................... 31 3.2.3 GeometricRealisability. ..... 31 3.2.4 Classification................................. 31 3.3 LimitsBetweenSpacetimes. ...... 33 3.3.1 ContractionLimits ............................. 33 xi 3.3.2 Remaining Galilean Spacetimes . ..... 37 3.3.3 LimitoftheCarrollianLightCone . ..... 37 3.3.4 AristotelianLimits .. .. .. .. .. .. .. .. .. .. .. .. .. .. 37 3.3.5 ANon-ContractingLimit . 38 3.3.6 Summary..................................... 38 3.4 Geometric Properties of Kinematical Spacetimes . ............ 39 3.4.1 TheActionoftheRotations. 40 3.4.2 TheActionoftheBoosts. 41 3.4.3 Invariant Connections, Curvature, and Torsion for Reductive Spacetimes 43 3.4.4 Pseudo-Riemannian Spacetimes and their Limits . ......... 47 3.4.5 TorsionalGalileanSpacetimes . ...... 58 3.4.6 AristotelianSpacetimes . .... 67 3.4.7 Carrollian Light Cone (LC) .......................... 70 3.5 Conclusion ...................................... 74 4 Kinematical Superspaces 75 4.1 Classification of Kinematical Superalgebras . ............ 75 4.1.1 SetupfortheClassification . .... 75 4.1.2 Classification of Kinematical Lie Superalgebras . ........... 82 4.1.3 Summary..................................... 91 4.1.4 Classification of Aristotelian Lie Superalgebras . ............ 93 4.1.5 CentralExtensions ............................. 95 4.1.6 Automorphisms of Kinematical Lie Superalgebras . ......... 96 4.2 Classification of Kinematical Superspaces . ........... 107 4.2.1 AdmissibleSuperLiePairs . 109 4.2.2 EffectiveSuperLiePairs. 112 4.2.3 Aristotelian Homogeneous Superspaces. ........ 116 4.2.4 Summary..................................... 116 4.2.5 Low-RankInvariants . .. .. .. .. .. .. .. .. .. .. .. .. .. 118 4.3 LimitsBetweenSuperspaces . ...... 119 4.3.1 ContractionsoftheAdSSuperalgebra . ...... 120 4.3.2 RemainingGalileanSuperspaces