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A Mathematical Study of Superspace in Fourdimensional, Unextended Supersymmetry

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A Mathematical Study of Superspace in Fourdimensional, Unextended Supersymmetry It's Pretty Super! - A Mathematical Study of Superspace in Fourdimensional, Unextended Supersymmetry Eric Friden´ June 12, 2012 Abstract Superspace is a fundamental tool in the study of supersymmetry, one that while often used is seldom defined with a proper amount of mathematical rigor. This paper examines superspace and presents three different constructions of it, the original by Abdus Salam and J. Strathdee, as well as two modern methods by Alice Rogers and Buchbinder-Kuzenko. Though the structures arrived at are the same the two modern construc- tions differ in methods, elucidating different important aspects of super- space. Rogers focuses on the underlying structure through the study of supermanifolds, and Buchbinder-Kuzenko the direct correlation with the Poincar´esuperalgebra, and the parametrisation in terms of exponents. Thanks Thanks to my supervisor Ulf Lindstr¨omfor giving me the opportunity to dive deep into a scary ocean of unknown and exciting mathematics. It has been a wild ride. Hugs and high-fives to all my friends and acquaintances from the student organisation Moebius; for being a neverending spring of clever, clever people doing stupid, stupid things. Thanks to Valentina Chapovalova, who has been my closest friend, my role model, my anchor and my muse for all of my adult life. And lastly, a sign of love and respect to my parents and my extended family. Your support is the foundation upon which I have built my life. Thank you all. Uppsala Univeritet Eric Friden´ [email protected] Contents 1 Introduction 3 2 Minkowski Space - the Non-Super Case 4 2.1 Minkowski space . .4 2.2 The Poincar´eGroup . .5 2.3 The Poincar´eAlgebra . .5 2.4 An Algebraic Minkowski Space . .6 3 Supersymmetry and Superspace 6 3.1 Lie Superalgebras . .7 3.2 The Poincar´eSuperalgebra . .9 3.3 Superspace . 10 4 Supermanifolds and Graded Manifolds 10 m;n 4.1 Flat superspace RS ....................... 11 4.1.1 Real Grassman Numbers . 11 4.1.2 The DeWitt Topology . 13 4.1.3 The Rogers Vector Space Topology . 13 m;n 4.2 Smooth Functions on RS ................... 14 4.2.1 DeWitt Differentiable Functions . 14 4.2.2 Rogers G1 and H1 functions . 15 4.3 Supermanifolds . 16 4.3.1 The Body of a Supermanifold . 17 4.4 Graded Manifolds . 18 4.4.1 Sheafs . 18 4.4.2 Graded Manifolds . 19 4.5 The Correspondence Between Graded Manifolds and Super- manifolds . 19 5 The Rogers Construction 20 5.1 Lie Supergroups . 20 5.2 The Poincar´eSupergroup and Rogers Superspace . 20 5.3 A Note on Exponentiation . 21 1 Uppsala Univeritet Eric Friden´ [email protected] 6 The Buchbinder-Kuzenko Construction 21 6.1 Matrix Realisation of The Poincar´eSuperalgebra . 22 6.2 Lie Supergroups . 23 6.2.1 Grassmann Shells of Lie Superalgebras . 23 6.2.2 Exponentiation . 24 6.3 Exponentiating the Poincar´eSuperalgebra . 25 6.4 Buchbinder-Kuzenko's Superspace . 26 7 Conclusions and Comparisons 27 7.1 Superspaces . 27 7.2 Supermanifolds and Graded Manifolds . 28 2 Uppsala Univeritet Eric Friden´ [email protected] 1 Introduction Superspace is the fundamental tool of supersymmetry. It is, intuitively, the set of points (t; x; y; z; θ1; θ2; θ3; θ4) that expands the regular set of points in space-time with four anticommuting parameters, making it the natural "staging ground" for supersymmetry. Essential for studies in supersymme- try, superspace is also highly relevant in many parts of modern physics, notably as a necessary part of string theory. The purpose of this paper is to present superspace in a few of its differ- ent guises, give an overview of the mathematics that makes superspace a rigorously defined mathematical object and lastly to discuss the different superspace constructions, how they relate to each other and what their in- dividual strengths and weaknesses are. The paper starts out with a section on the algebraic representation of regular non-super Minkowski space. Hopefully, this will help readers unfamiliar with this topic to come to grips with some of the concepts of the later sections since both the methods and the motivations behind supersymmetry (and specifically superspace) are expansions of this method. Readers familiar with the subject matter should probably still give the section a good skimming since it introduces some notation used in later sections. The next section will introduce supersymmetry, how it relates to the alge- braic Minkowski space and give a first explanation of superspace from the perspective first introduced by Salam-Strathdee [12]. After the first dip into supersymmetry it is time to take a dive into the inviting waters of supermathematics. The next section is spent defining and investigating different kinds of supermanifolds before using this knowledge to put the notion of superspace on a secure mathematical footing; following in the footsteps of Alice Rogers [11]. The next section will present an alternative construction of superspace as presented by Buchbinder and Kuzenko [4], using an entirely different ap- proach inspired by Akulov-Volkov [1] that discovered the concept of super- space separately and concurrently with Salam-Strathdee. The paper closes with a comparative study of the benefits and drawbacks of the different superspaces discussed, the Salam-Strathdee original and the two approaches to modernisation as presented by Rogers and Buchbinder- Kuzenko, as well as a discussion of the different notions of supermanifolds and how they influence the properties of superspace. Expected preknowledge includes a base understanding of Lie group and Lie algebra theory, manifold theory and special relativity. Please note that much of the material in this paper is given little more than a glance, and I fully expect some readers (especially mathematically inclined 3 Uppsala Univeritet Eric Friden´ [email protected] ones) to find the occasional lack of proper motivations infuriating. For the sake of these readers I have been sure to include ample referenses to sources that (for the most part) present more rigorous discussions. 2 Minkowski Space - the Non-Super Case It is common practice in the study of mathematical objects to put focus on the different types of transformations of said object, rather than on the object itself. Of special interest are symmetries, transformations that leave certain values, invariants, unchanged. In our case, the object in question is space and time, represented by the four dimensional Minkowski space, and the invariants the laws of special relativity succinctly condensed into a metric (2.2). By collecting these symmetries into a Lie group, the Poincar´egroup, one can unearth a lot of deep information regarding the structure of Minkowski space. This section starts by defining Minkowski space and moves in to the realm of Lie groups and Lie algebras, towards the end goal of reconstructing Minkowski space as a coset space of the Poincar´egroup. Both this and the following section will stick to the notation of Lindstr¨om [8], with only a few small changes for the sake of clarity and consistency. 2.1 Minkowski space Minkowsi space (denoted M4) is defined as the set of 4-tuples of real numbers ∗ xµ = (x0; x1; x2; x3) equipped with the metric: 2 2 2 2 jjxµjj = x0 − x1 − x2 − x3 (2.1) 4 4 In other words, M = R with metric tensor 0 1 0 0 0 1 B 0 −1 0 0 C ηab = B C (2.2) @ 0 0 −1 0 A 0 0 0 −1 4 Some authors refer to M as R(1; 3), which is a notation that will be ex- panded on in later sections. ∗This tuple is sometimes denoted (ct; x; y; z), betraying that the first coordinate should be interpreted as time (multiplied with the speed of light c to put it in length units), while the others represent three dimensional space. 4 Uppsala Univeritet Eric Friden´ [email protected] 2.2 The Poincar´eGroup Definition 2.1. The restricted Poincar´eGroup, Π, is the 10-dimensional Lie group of all metric-preserving maps (isometries) on M4 on the form 0a a b b T x = Λb x + y Λ ηΛ = η (2.3) With the added restrictions that they: 1. do not reverse the direction of time 0 Λ0 ≥ 1 (2.4a) 2. do not reverse spatial orientation det Λ = 1 (2.4b) Π consists of two kinds of transformations, translations, yb, (forming a 4- ∗ a dimensional abelian subgroup) and Lorentz tranformations, Λb , forming the restricted Lorentz group SO+(1; 3). The unrestricted Poincar´egroup is the the isometry group on M4, without the restrictions of (2.4a) and (2.4b), and of generally less physical relevance. This paper will drop the "restricted" label and refer to Π simply as the Poincar´egroup, but readers should be wary that many authors refer to the unrestricted Poincar´egroup by that name. 2.3 The Poincar´eAlgebra The Lie algebra of left-invariant vectorfields on the universal cover of Π, its corresponding Lie algebra, can be conveniently defined using generators and commutation relations. Definition 2.2. The Poincar´ealgebra, P, is the 10-dimensional Lie algebra with generators Pa and Mab obeying the following commutation relations: (a; b = 0;:::; 3) [Pa;Pb] = 0 (2.5a) −i[Mab;Pc] = ηacPb − ηbcPa (2.5b) −i[Mab;Mcd] = ηacMbd − ηadMbc − ηbcMad + ηbdMac (2.5c) Here, Pa is the generator of translations, Mab is the generator of Lorentz transformations and ηab is the Minkowski metric tensor (2.2). ∗Boosts, transformation between frames of reference travelling at non-zero speed rela- tive to eachother and spatial rotations. 5 Uppsala Univeritet Eric Friden´ [email protected] 2.4 An Algebraic Minkowski Space The process leading up to P can be effectivly reversed, resulting in a parametri- sation of Π (and as a direct consequence M4) in terms of the Poincar´e generators Pa and Mab.
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