A Mathematical Study of Superspace in Fourdimensional, Unextended Supersymmetry

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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 constructions differ in methods, elucidating different important aspects of superspace. Rogers focuses on the underlying structure through the study of supermanifolds, and Buchbinder-Kuzenko the direct correlation with the Poincarésuperalgebra, 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 G∞ and H∞ 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

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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

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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 supersymmetry, 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 different 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 algebraic 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 approach inspired by Akulov-Volkov [1] that discovered the concept of superspace 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 Poincarégroup, 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 Poincarégroup. Both this and the following section will stick to the notation of Lindström [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 ||xµ|| = x0 − x1 − x2 − x3 (2.1)

4 4 In other words, M = R with metric tensor  1 0 0 0   0 −1 0 0  ηab =   (2.2)  0 0 −1 0  0 0 0 −1

4 Some authors refer to M as R(1, 3), which is a notation that will be expanded 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.

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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 Poincarégroup 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 Poincarégroup, but readers should be wary that many authors refer to the unrestricted Poincarégroup 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 Poincaréalgebra, 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.

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2.4 An Algebraic Minkowski Space

The process leading up to P can be effectivly reversed, resulting in a parametrisation of Π (and as a direct consequence M4) in terms of the Poincaré generators Pa and Mab. The exponentiation function relates an element of a Lie algebra to its corresponding Lie group, recreating the group operation with the Baker-Campbell- Hausdorff formula:

1 1 1 exp(u) exp(v) = exp u + v + [u, v] + [u, [u, v]] + [v, [v, u]] + ... 2 12 12 (2.6) where the dots symbolizes commutators of third degree and higher. Applying the exponentiation function to P results in the following expression for elements of Π, parametrized by the 4-vector xa and the 4-by-4-matrix ωab

4 4,4 a bc a bc g :(R , M ) → Π g(x , ω ) = exp x Pa + ω Mbc (2.7)

M4 is found by isolating the subgroup of translations∗ as the the coset space of Π with respect to SO+(1, 3).

M4 ' Π/SO+(1, 3) = {g|g ∈ Π} g = gh|h ∈ SO+(1, 3) (2.8) with elements parametrized as

4 4 a a g : R → M g(x ) = exp (x Pa) (2.9)

3 Supersymmetry and Superspace

It would be reasonable to ask whether the Poincaréalgebra can be expanded by introducing some new kind of undiscovered symmetries. It was proven by Haag, Lopszanski and Sohnius [9] that it could, although the only way that did not directly contradict modern physics adds an odd part to the Poincaréalgebra, transforming it into a Lie superalgebra whose even part is the regular Poincaréalgebra.

∗Identifying a point in M4 as the translation taking the origin to that point.

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This expansion has many deep implications. Odd symmetries transform fermionic quantum fields into bosonic, and vice versa, meaning there is a kind of symmetry that connects bosons (force-carrier particles; e.g. photons, Higgs bosons) and fermions (matter particles; e.g. protons, electrons). Since no currently known particles exist that fit the bill as supersymmetric copies, supersymmetry necessitates a whole new (as of now undiscovered) class of particles, bosons corresponding to known fermions and vice versa. Supersymmetry has also given rise to a lot of new and interesting mathematics (some of which will be touched on in this paper), and is a necessarry component of, among other theories, string theory. The section will be prefaced by the definition of one of the fundamental structures of supermathematics: Lie superalgebras.∗ This structure will later be used to define the supersymmetric analog of the Poincaréalgebra.

3.1 Lie Superalgebras

Deﬁnition 3.1. A real (complex) super vector space is a real (complex) vector space A, with two linear subspaces 0A and 1A such that

A = 0A ⊕ 1A (3.1) where indices should be read modulo 2.† Deﬁnition 3.2. A real (complex) superalgebra is a real (complex) super vector space A, with a bilinear operation A × A → A such that

iA jA ⊆ i+jA (3.2) where indices should be read modulo 2.

In other words, the product of two superalgebra elements both belonging to either 0A or 1A lies in 0A while the product of two elements where one lies in 0A and one in 1A lies in 1A. 0A is called the even part of A, and it’s elements even. Analogously, 1A is called the odd part of A, and it’s elements odd. The parity‡ of even or odd elements is denoted |a| and is equal to 0 and 1 respectively. The even and odd elements of A are collectively called pure.§

∗There is some ambiguity in the scientiﬁc community about the term ”Lie superalgebra” versus ”super Lie algebra”. I have chosen the former, and will stick with it and its analogs throughout this paper, but note that the terms most often refer to the same object. Exeptions are cases where authors keen on confusing their readers use them both for diﬀerent structures, see [4]. † 2 This partition is an example of a Z grading. Some, especially mathematical, authors 2 refers to super vector spaces as Z graded vector spaces. ‡Called the degree by many authors. §Some authors refers to elements of 0A ∪ 1A as homogenous

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Deﬁnition 3.3. A Lie superalgebra is a superalgebra where the bilinear ∗ product [·, ·]S (called the superbracket) obeys the following two restrictions when acting on pure elements:

1. Super anti-commutativity:

|x||y| [x, y]S = −(−1) [y, x]S (3.3)

2. The super Jacobi identity:

|z||x| |y||z| |x||y| (−1) [x, [y, z]S]S + (−1) [z, [x, y]S]S + (−1) [y, [z, x]S]S = 0 (3.4)

Just like a regular Lie algebra can be defined by providing a basis (whose elements are called the Lie algebra ”generators”) and commutations relations between the basis vectors, it is sufficient to define a Lie superalgebra to supply even and odd generators, and their supercommutation relations. This will be done in the following section when defining the Poincarésuper- algebra. One can impose a Lie superalgebra structure on a superalgebra by defining the superbracket as the supercommutator, defined as:

|x| |y| [x, y]S = xy − (−1) yx (3.5)

Or in other words:

[x, y] = xy − yx if x or y is even [x, y] = (3.6) S {x, y} = xy + yx if x and y are odd

If the parity of the elements being acted upon is known it is convenient to denote the supercommutator of two elements x and y as either a commutator [x, y] = xy − yx or the anticommutator {x, y} = xy + yx, depending on its function at the moment. The Lie superalgebras being considered in this paper can all be constructed in the way of equation (3.5), and the superbrackets will all be denoted as commutators and anticommutators.

∗Authors on supermathematics differ wildly on the point of how to denote a superbracket. Popular choices include simple brackets [·, ·] and the assymetric [·, ·}. It is the firm opinion of this humble author that both of these have significant drawbacks, and that a bracket with a subscript S for ”super” is the proper choice of notation.

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3.2 The Poincar´eSuperalgebra

According to the Coleman-Mandula theorem [5], as expanded by Haag, Lop- szanski and Sohnius [9], the only way to expand the Poincaréalgebra while respecting the axioms of a relativistic quantum field theory is to transform it into a superalgebra by adding an odd subalgebra generated by four real anticommuting parameters that combine into a Majorana spinor, or equivalently into two Weyl spinors.∗ I will present the Poincarésuperalgebra, SP, in both Majorana and Weyl notation. The Majorana notation was the one used by Salam and Strathdee originally, and will be used heavily in section 6. The Weyl notation is presented here mostly for completeness, it is covenient to use in many calcula- tions that lie beyond the scope of this paper. Below, σ represents the dirac matrices: 1 0 0 1 0 −i 1 0 σ = , , , (3.7a) a 0 1 1 0 i 0 0 −1

σ˜a = (σ0, −σ1, −σ2, −σ3) (3.7b)

1 (σ )β = − (σ σ˜ − σ σ˜ )β (3.7c) ab α 4 a b b a α

˙ 1 ˙ (˜σ )β = − (˜σ σ − σ˜ σ )β (3.7d) ab α˙ 4 a b b a α˙ Definition 3.4. The Poincarésuperalgebra, SP, is defined in Majorana notation as follows: (a, b, c, d = 0,..., 3 α, β = 0, 1)

• Even generators: Pa,Mab

• Odd generators: Qα, Q¯α˙ • Super commutation relations:

[Pa,Pb] = 0 (3.8a)

−i[Mab,Pc] = ηacPb + ηbcPa (3.8b)

−i[Mab,Mcd]= ηacPbd + ηbdPac − ηadPbc − ηbcPad (3.8c)

[Pa,Qα] = 0 (3.8d) α i[Mab,Qα] = (σab)β Qβ (3.8e) ¯ ¯ {Qα,Qβ} = {Qα˙ , Qβ˙ } = 0 (3.8f) ¯ a {Qα, Qα˙ } = 2 (σ )αα˙ Pa (3.8g) ∗This equivalence is speciﬁc to fourdimensional supersymmetry.

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3.3 Superspace

Abdus Salam and J. Strathdee introduced the concept of superspace in their groundbreaking paper [12]. By mirroring the process from section 2.4 of constructing Minkowski space from the Poincaréalgebra, they constructed an analog of Minkowski space with some added structure corresponding to the introduction of supersymmetries into our model of the universe. Below is a quick review of the concept of superspace following Salam and Strathdee, a construction that the rest of this paper is dedicated to the correction of. Consider the exponential map applied to SP, resulting in a supersymmetric analog to the Poincarégroup Π, named the Poincarésupergroup SΠ parametrized by commuting parameters x and ω, and anticommuting parameters θ as follows:

µ α ab exp x Pµ + θ Qα + ω Mab (3.9)

Just as in the case with Minkowski space, superspace is found by qouting out the restricted Lorentz group SO+(1, 3).

Superspace = SΠ/SO+(1, 3) (3.10)

With points parametrized as∗

µ α α˙ g(x, θ, θ¯) = exp (x Pµ) exp θ Qα + θ¯ Q¯α˙ (3.11)

This construction seem straighforward, but exponentiation of the Poincar´e superalgebra is not well deﬁned. The next section will discuss supermanifolds in some depth resulting in the Rogers construction of superspace in section 5. Section 6 will bring the Buchbinder-Kuzenko construction.

4 Supermanifolds and Graded Manifolds

The Rogers construction of superspace in section 5 necessitates a discussion on supermanifolds. In the ﬁrst few sections I present the two primary structures used as supermanifolds, the supermanifolds deﬁned with sets and atlases (popularized by DeWitt [6]) and the graded manifolds of Berezin-Leites [3] and Kostant

∗ Note that since [Qα,Pa] = 0 the Baker-Campbell-Hausdorﬀ formula contributes only trivially and I am able to easily separate the exponentials.

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[7] using sheafs of continous functions. Section 4.5 will present the direct correlation between the two. This discussion will culminate in section 5.1 with the deﬁnition of Lie supergroups, the global objects that corresponds to Lie superalgebras in the way that Lie groups relate to Lie algebras.

m,n 4.1 Flat superspace RS

4.1.1 Real Grassman Numbers

Deﬁnition 4.1. The real (complex) Grassmann algebra, RS (CS), is the real ∗ (complex) superalgebra generated by the set {1, v1, v2, ···} with relations

1vi = vi = vi1 (4.1a)

vivj + vjvi = 0 (4.1b)

A convenient basis for the Grassmann algebra is the set of non-redundant† ‡ products of generators vi whose factors are ordered by increasing i.

{1, v1, v2, ··· , v1v2, v1v3, ··· , v3v5v8, ···} (4.2)

To formulate this properly, let Mn be the set of non-redundant selections of positive integers less then or equal to n ordered by value.   m = 0, 1, . . . , n   Mn = Γ = (λ1, . . . , λm) λi = 1, 2, . . . , n (4.3)

 i < j ⇒ λi < λj 

Also note that the empty set is an allowed selection: m = 0 ⇒ Γ = ∅.

Let M∞ denote the set where the values of m and λi are allowed to be arbitrarily large.

  m = 0, 1,...   M∞ = Γ = (λ1, . . . , λm) λi = 1, 2,... (4.4)

 i < j ⇒ λi < λj 

∗ RS can be equivalently defined as the exterior algebra or as the Clifford algebra of the trivial quadratic form Q(v) = 0. This definition however, suits our needs best. †We can demand non-redundant products (products where no factor occurs more than 2 once) since vi = 0 by (4.1b) ‡We can demand ordered products since because of (4.1b) permutating the objects of a product of generators only has the effect of flipping the sign by the sign of the permutation. v1 ··· vm = sgn(σ)vσ(1) ··· vσ(m)

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The elements from (4.2) can now be written explicitly with the help of multi-indices Γ from the set M∞.

A general element of RS is written as X a = aΓvΓ aΓ ∈ R Γ∈M∞ Γ ∈ M∞ vΓ = vλ1 ··· vλm (4.5) Γ = ∅ ⇒ vΓ = 1

The even and odd elements of RS are deﬁned as follows:

• An element of RS (CS) is even if and only if

aΓ = 0 if m is odd (4.6a)

• An element of RS (CS) is odd if and only if

aΓ = 0 if m is even (or zero) (4.6b)

It is easily verfiable that even elements commute with all elements of RS and that odd elements anti-commute among themselves. At some points in the paper it will be useful to consider the finitedimensional Grassman algebra, denoted as RS[L]. it is generated by L anticommuting generators, with basis vΓ, Γ ∈ ML. Physical considerations beyond the scope of this paper unearths unrealistic consequences of using any finite-dimensional Grassmann algebra for the construction of superspace, but RS[L] is still useful in many definitions and proofs.

Deﬁnition 4.2. Flat (m, n) superspace is the set

m copies n copies z0 }| 0 { z1 }| 1 { m,n RS × · · · × RS × RS × · · · × RS ≡ RS (4.7)

A point in ﬂat superspace is speciﬁed by giving m even and n odd Grass- m,n man numbers. We denote the coordinates of a point in RS as (a; b) = 1 m 1 n i 0 i 1 (a , ··· , a ; b , ··· , b ) with a ∈ RS and b ∈ RS. At times it will be convenient to use the combined coordinate (c) = (c1, ··· , cm+n).

( i 0 i a ∈ RS (i = 1, ··· , m) c = i 1 (4.8) b ∈ RS (i = m + 1, ··· , m + n)

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4.1.2 The DeWitt Topology

m,n Several topologies have been considered on RS , most notable is the DeWitt topology, as described in [6]. m,n At its core it reduces the concept of an open subset of RS to wether or m m not the subset projected unto R is an open subset of R . The terms body and soul (after DeWitt) will prove helpful.

Deﬁnition 4.3. • The body of a Grassman number a is the real number (a) = a∅, the coeﬃcient coupled with the identity element v∅ = 1 in the basis expansion (4.5). Grassmann numbers a where (a) = 0 are termed bodyless.

• The soul of a Grassman number a is the Grassman number σ(a) = a − (a). Grassmann numbers a where σ(a) = 0 are termed soulless.

m,n i • The body of an element (a; b) of RS is the point m,n(a; b) = (a ) m in R with coordinates equal to the body of the corresponding even m,n m,n m coordinates in RS . The mapping m,n : RS → R is called the body projection.

Note that since all odd elements of RS by deﬁnition has a∅ = 0, all odd m,n points in RS are bodyless. m,n m,n Deﬁnition 4.4 (The DeWitt Topology on RS ). A subset of RS is called open in the DeWitt topology if and only if it has the property

−1 U = m,n(V ) = {a | m,n(a) ∈ V } (4.9)

In other words, all open subsets in the DeWitt topology consists of all points m,n m in RS that maps into a certain open subset of R under the projection ∗ map m,n. This topology is not Hausdorﬀ , but does capture the essential properties of the Grassman algebra.

4.1.3 The Rogers Vector Space Topology

Another possible topology on ﬂat superspace was proposed by Rogers in 1980 [10]. Given the basis expansion given in (4.5) and viewing RS as a vectorspace we can deﬁne a natural norm as follows: X X a = aΓvΓ ⇒ ||a|| = aΓ (4.10)

Γ∈M∞ Γ∈M∞

∗These types of topologies are sometimes refered to as projectively Hausdorﬀ.

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Rogers goes on to restrict the Grassman algebra to elements whose norm from (4.10) is a convergent sum. Unlike the (coarser) DeWitt topology, the natural topology induced by the norm in (4.10) is Hausdorff, and allows for non-trivial structure in the odd part of supermanifolds.∗ However, DeWitt remarked in [6] (and Rogers later agreed in [11]) that the vector-space topology above ignores much of the algebraic structure of m,n flat superspace, and is thus less suited as a topological structure on RS . It is generally understood in modern literature that the DeWitt topology (though coarser and non-hausdorff) provides the most useful structure for further studies of supermanifolds.†

m,n 4.2 Smooth Functions on RS

Supermanifolds can be defined at this point as sets locally isomorphic to m,n n RS (in the same way that manifolds are locally isomorphic to R ) but for supermanifolds to be physically relevant, it is necessary to impose a m,n ”smoothness” restriction on functions of RS . There are several different widely used smoothness conditions and they de- m,n pend heavily on the topology on RS . This section includes the DeWitt “differentiability” concept and Rogers’ G∞‡ and H∞ classes of superfunc- tions, all using the DeWitt topology.

4.2.1 DeWitt Diﬀerentiable Functions

Deﬁnition 4.5. A function of even supernumbers is DeWitt diﬀerentiable if it is of the form

0 f : RS → CS ∞ X 1 f(x) = f (n)((x)) (x − (x))n (4.11) n n=0

Recall that : RS → R is the projection of a Grassman number onto its body, as deﬁned in section 4.1.2.

∗In [10] Rogers gives an example of a two-dimensional torus given the structure of a (1, 1) dimensional supermanifold. †Concrete examples can be found in DeWitt [6] and Rogers [11] ‡In the sense described in [11]. In [10] Rogers deﬁned another class of functions as G∞ using the Rogers vector space topology.

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Deﬁnition 4.6. A function of odd supernumbers is DeWitt diﬀerentiable if it is of the form

1 f : RS → CS

f(x) = ax + b a, b ∈ CS (4.12) Definition 4.7. A function of flat superspace is DeWitt differentiable if the coordinates of its image points are DeWitt differentiable functions of the domain point coordinates.

4.2.2 Rogers G∞ and H∞ functions

Superﬁelds can always be expressed in the form∗

X Γ Γ λ1 λm f(a; b) = fΓ(a)b b ≡ b ··· b (4.13) Γ

Where, as in section 4.1.1, each Γ = (λ1, ··· , λm) is a non-empty, non- redundant selection of natural numbers ordered by increasing values. The sum is taken over all such tuples with natural numbers less than m. The dependence on b is completely determined by the above expression, so the problem of classifying smooth superﬁelds can be reduced to putting suitable restrictions on the functions f.

Recall the ﬁnitedimensional Grassman algebra, RS[L], from section 4.1.1. It plays a fundamental part in the deﬁnition of G∞ functions.

∞ Definition 4.8 (C RS-valued functions). Let V be an open subset of R and PL : RS → RS[L] the projection map that maps generators vi to zero when i > L. ∞ ∞ f : V → RS is said to be C if PL ◦ f : V → RS[L] is C (with RS[L] seen as a finitedimensional vectorspace) for all positive integers L. The set of all ∞ such functions is called C (V, RS). Definition 4.9 (Grassman Analytical Continuation). For every function f ∞ ˆ −1 in C (V, RS) the Grassman analytical continuation f :(m,0(V ) → RS is

fˆ(a; 0) ≡

∞ 1 ∂ i1 ∂ im X 1 i1 m im 1 ··· m f(m,n(a; 0))×σ(x ) ··· σ(x ) i1! . . . im! ∂a ∂a i1=0,··· ,im=0 (4.14)

∗This can be seen by considering the taylor expansion in terms of b. More details in Rogers [11].

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Recall from section 4.1.2 that σ(x), x ∈ RS denotes the soul of the Grass- mann number, i.e. σ(x) = x − (x).

∞ m,n Deﬁnition 4.10 (G functions). Let U be an open subset of RS . ∞ The function f : U → RS is said to be G if and only if there exists a ∞ collection of functions fΓ : m,n(U) → RS in C (m,n(U), RS) satisfying:

∞ fΓ ∈ C (m,n(U), RS)Γ ∈ Mn

X ˆ Γ f(a; b) = fΓ(a)b (4.15)

Γ∈Mn

The set of such functions is denoted G∞(U).

With only a slight adjustment to the demand of the fΓ family of functions, one arrives at the H∞ class of functions. These functions are the key to the correspondence between supermanifolds and graded manifolds.

∞ m,n Deﬁnition 4.11 (H functions). Let U be an open subset of RS . ∞ The function f : U → RS is said to be H if and only if there exists a ∞ collection of functions fGamma : m,n(U) → R in C (m,n(U)) satisfying:

∞ fΓ ∈ C (m,n(U), RS)Γ ∈ Mn

X ˆ Γ f(a; b) = fΓ(a)b (4.16)

Γ∈Mn

The set of such functions is denoted H∞(U). ∞ ∞ The distinction between G and H functions lies in the functions fΓ that ∞ ∞ takes values in RS in a G function and in R in a H function. It follows that H∞(U) ⊂ G∞(U), or in other words that every G∞ function can be given the structure of a H∞ ditto.

4.3 Supermanifolds

With a suitable “smoothness restriction” it is now possible to deﬁne smooth supermanifolds, directly analogous to the standard deﬁnition of C∞ manifolds.

Deﬁnition 4.12 (Chart). Let M be a set. An (m, n) chart on M is a pair m,n (U, φ) such that U ⊂ M is open (in some topology) and φ : U → RS is bijective.

Deﬁnition 4.13 (Complete Atlas). An (m, n) smooth atlas on M is a collection {(Ua, φa)|a ∈ A} (A is some index-set) of (m, n) charts on M such that

16 Uppsala Univeritet Eric Friden´ [email protected]

S 1. M = a∈A Ua 2. For each (a, b) ∈ A × A, the function

−1 φb ◦ φa : φa(Ua ∩ Ub) −→ φb(Ua ∩ Ub) (4.17)

is smooth.

The atlas is called complete if it is not contained in any other smooth atlas.

Deﬁnition 4.14 (Smooth supermanifold). An (m, n) smooth supermanifold is a set M with an (m, n) smooth complete atlas.

Set-atlas supermanifolds can be classified by the type of flat superspace they use as “model space”, it’s topology and the family of superfields considered “smooth”. For example:

m,n ∞ • A RS , DeWitt,G supermanifold is a set with a complete atlas to m,n ∞ RS with transition functions in G . It is the standard supermanifold described by Rogers in [11]. m,n • A RS , DeWitt, Diﬀ supermanifold has the same structure as above, except the transition functions are required to be DeWitt diﬀeren- tiable.

m,n • A RS[L], FDVS, F supermanifold uses the ﬁnitedimensional Grass- m,n man algebra and the standard topology regarding RS[L] as a ﬁnite- dimensional vector space.

This notation will later on be refered to as the Rogers triplet notation.

4.3.1 The Body of a Supermanifold

m,n ∞ Deﬁnition 4.15 (Body of a RS , DeWitt,G supermanifold). Let M m,n ∞ be a RS , DeWitt,G supermanifold with atlas {(Ua, φa)|a ∈ A}and let ∼ be the relation that calls two elements p, q ∈ M equivalent if and only if there exists a ∈ A such that

p, q ∈ Ua m,n(φa(p)) ∼ m,n(φa(q)) (4.18)

Then, these two statements are true:

• The relation ∼ is an equivalence relation.

17 Uppsala Univeritet Eric Friden´ [email protected]

∞ • The space M[∅] = M/ ∼ is an m-dimensional C manifold with atlas (U[∅]a, φ[∅]a)|a ∈ A where

– U[∅]a = [p]|p ∈ Ua

– φ[∅]a = m,n ◦ φa

M[∅] is called the body of M, and the canonical projection of the quotient is denoted : M → M[∅].

4.4 Graded Manifolds

The graded manifolds presented independently by Berezin and Leites [3] in 1976 and Kostant [7] in 1977 represents a different approach to supermanifolds. Using a sheaf of superalgebras of functions on a regular manifold, the definition of graded manifolds avoids the algebraically complex Grassmann algebra, a small victory since the Grassmann numbers are needed for the correspondence between Lie supergroups and Lie superalgebras regardless. This section starts out by defining sheafs, the central concept in the definition of graded manifolds in the following section.

4.4.1 Sheafs

I will for the sake of brevity say ”presheaf/sheaf” where I technically mean ”presheaf/sheaf of superalgebras”. For a more general deﬁnition, replace the superalgebras F (U) in the deﬁnition below with objects from an arbitrary category.∗

Deﬁnition 4.16 (Presheaf). Let X be a topological space. A collection of superalgebras {F (U)|U open subset of X} is called a presheaf over X if the following demands are met: (U, V and W are open subsets of X)

1. The restriction map is a homomorphism of superalgebras deﬁned through

U ⊂ V ⇒ resV,U : F (V ) → F (U) (4.19)

2. Given U ⊂ V ⊂ W

resV,W ◦ resU,V = resU,W (4.20)

∗Readers with a grasp of category theory can skip the rather longwinded deﬁnition and simply note that a presheaf is a contravariant functor between the category of open subsets of a topological space where morphisms are inclusions to an arbitrary category, in this case the category of superalgebras.

18 Uppsala Univeritet Eric Friden´ [email protected]

A presheaf is called a sheaf if it obeys the following rules given an open subset U of X and an open covering of U, {Ua|a ∈ A} for some index set A.

1. Given f, g in F (U), if resU,Ua f = resU,Ua g for all a ∈ A then f = g.

2. Given fa in F (Ua) for each a ∈ A such that resUa,Ua∩Uµ fa = resUµ,Ua∩Uµ fµ,

there exists f ∈ F (U) such that fa = resU,Ua f.

4.4.2 Graded Manifolds

Given a regular manifold M, the sheaf C∞ = {C∞(U)|U open in M} deﬁnes C∞(U) as the algebra of smooth functions on U. This sheaf is central in the deﬁnition of graded manifolds.

Deﬁnition 4.17. A smooth real graded manifold of dimension (m, n) is a pair (M,F ), where M is an m-dimensional smooth manifold and F a sheaf of superalgebras over M, that obeys the following rules:

1. There exists an open cover {Ua|a ∈ A} such that for all a ∈ A

∞ F (Ua) = C (Ua) ⊗ RS[n−1] (4.21)

2. If N is the sheaf of nilpotents in F , then (M,F/N) ' (M,C∞)

4.5 The Correspondence Between Graded Manifolds and Su- permanifolds

The connection between graded manifolds and supermanifolds was ﬁrst es- tablished by Batchelor in [2]. I will again follow the notation of Rogers. m,n ∞ Given a RS , DeWitt,H supermanifold M, deﬁne A as the sheaf of superalgebras that maps an open subset V ∈ M[∅] to the superalgebra of H∞ functions on the set of points in M whose bodies lies in V .

V ⊂ M[∅]

A(V ) = H∞ −1(V ) (4.22)

The set (m,n(M),A) can now be shown to be a graded manifold as deﬁned in section 4.4.2

19 Uppsala Univeritet Eric Friden´ [email protected]

5 The Rogers Construction

5.1 Lie Supergroups

A Lie supergroup is a group whose underlying set is a smooth supermanifold.∗ Defining Lie supergroups, their corresponding Lie superalgebras and the exponential map between the two is done in more or less direct analogy to regular Lie group theory. I will bypass most technicalities and simply present the results, Rogers [11] offers a more thorough walk through Lie supergroup theory. Definition 5.1 (Lie supergroup). An (m, n)-dimensional super Lie group is a group whose underlying set is an (m, n)-dimensional smooth supermanifold, where the group operation is such that f(g, h) = gh−1 is “smooth” in ∞ m,n ∞ the same sense as the supermanifold (e.g. G for a RS , DeWitt,G supermanifold).

The Lie algebra corresponding to a certain Lie group is classically found as the set of left-invariant vectorﬁelds on the group, and this method can be neatly transferred to the ”super-case”. In all but a few esoteric cases the set of all left invariant vector ﬁelds on a given Lie supergroup G is isomorphic to u = RS ⊗ g called the super Lie module of G, where g is a Lie superalgebra refered to as the Lie superalgebra of the Lie supergroup G.

5.2 The Poincar´eSupergroup and Rogers Superspace

Instead of constructing SΠ from exponentiation as in section 3.3 we will define it constructively and note that its corresponding Lie superalgebra is the Poincarésuperalgebra from definition 3.4. Definition 5.2. The (m, n) - dimensional super translation group (T m,n) m,n† is the supermanifold RS given the structure of an supergroup by the operation ◦ defined below. (x, θ) ◦ (y, ϕ) = (z, ψ) (5.1a) n i i i X α β i z = x + y + θ ϕ Cγ αβ (5.1b) α,β=1 ψi = θi + ϕi (5.1c)

∗Some early attempts at creating a super-analog to Lie groups was made by Berezin and Kac in [3] and Kostant (using graded manifolds) in [7]. Neither the formal Lie supergroups of Berezin and Kac nor the graded Lie Groups of Kostant were infact groups. Rogers [11] notes that in many ways they relate to the super Lie groups considered in this paper in the same way that the formal Lie groups of Bochner relates to regular Lie groups. † m,n It is simple to show that RS is itself a supermanifold

20 Uppsala Univeritet Eric Friden´ [email protected]

Deﬁnition 5.3. The Poincar´esupergroup SΠ is the semi direct product of the (4, 4) - dimensional super translational group and spin(1, 3), the double cover of SO+(1, 3).

4,4 SΠ = T o spin(1, 3) (5.2)

The Lie superalgebra corresponding to SΠ can now be shown to be the super Poincar´ealgebra, and since∗

4,4 RS = SΠ/spin(1, 3) (5.3)

4,4 the conclusion can be drawn that RS fulﬁlls the role of superspace, as laid out by Salam-Strathdee.

5.3 A Note on Exponentiation

This subsection will leave Rogers behind to make a short note on parametris- ing the elements of superspace in terms of Poincar´esuperalgebra generators. This section is here for comparative purposes, and is not up to snuﬀ when it comes to mathematical rigor. The subject of exponentiation and Lie superalgebras will be treated with more respect in the next section regarding the Buchbinder-Kuzenko construction of superspace. Let an element (x, θ) of T 4,4 be denoted as

a α exp (x Pa + θ Qα) (5.4) where Pa and Qα are Poincarésuperalgebra generators in Majorana notation as defined in definition 3.4. The multiplication defined in definition 5.2 now corresponds with the Baker-Campbell-Hausdorff formula from (2.6). The 4,4 end result is a parametrisation of the space RS in terms of exponentials of Poincarésuperalgebra generators, or in other words: superspace.

6 The Buchbinder-Kuzenko Construction

The second construction of superspace in this paper will be the one given by Buchbinder and Kuzenko in [4]. It follows the tradition of Akulov and Volkov, two soviet physicists that published a paper [1] containing the

∗ 4,4 4,4 Recall that T is as a set equal to RS , and that since spin(1, 3) is not a normal subgroup we lose the group structure in the quotient.

21 Uppsala Univeritet Eric Friden´ [email protected] concept of superspace at the same time as (and independent of) Salam- Strathdee. It is a more concrete and less general approach, giving speciﬁc matrix representations and explicitly giving a parametrisation of superspace in terms of exponents of Poincar´esuperalgebra generators.

6.1 Matrix Realisation of The Poincar´eSuperalgebra

One of the important characteristics of the Buchbinder-Kuzenko construction is the matrix representations given of the relevant structures. Below are matrix representations of the generators of the Poincar´esuperalgebra as complex supermatrices, matrices on the form

  n ν Am Bm   n ν 0 ν n 1   Am,Dµ ∈ CS Bm,Cµ ∈ CS (6.1)  n ν  Cµ Dµ

    1 0 0 02 − σa −iσab 02  4 0   0      Pa =  0  Mab =  0  (6.2a)  02 02   02 −iσ˜ab   0   0  0 0 0 0 0 0 0 0 0 0

 0   1   −1   0   04   04  Q0 =  0  Q1 =  0  (6.2b)      0   0  0 0 0 0 0 0 0 0 0 0

 0   0   0   0  ¯  04  ¯  04  Q˙ =  0  Q˙ =  0  (6.2c) 0   1    0   0  0 0 −1 0 0 0 0 0 1 0

Note that σa,σ ã and σab represents the Dirac matrices (and related sets of matrices) defined in section 3.2. These matrices satisfy the commutation relations defined in section 3.4 given the multiplication of supermatrices, a supersymmetric variant of regular matrix multiplication.

22 Uppsala Univeritet Eric Friden´ [email protected]

6.2 Lie Supergroups

6.2.1 Grassmann Shells of Lie Superalgebras

Recall from section 3.1 the deﬁnition of a Lie superalgebra.

Deﬁnition 6.1. (Grassmann Shell) Given a complex Lie superalgebra A with basis

0 1 eI = {ei, eα} ei ∈ A eα ∈ A (6.3) the Grassmann Shell of A, denoted A(CS), is the set of elements

I I u = z eI z ∈ RS (6.4)

I with operations: (ui = zi eI )

• Addition

I I u1 + u2 = z1 + z2 eI (6.5a)

• Left Multiplication by Supernumbers

I α · u = αz eI (6.5b)

• Right Multiplication by Supernumbers

|eI ||α| I u · α = (−1) αz eI (6.5c)

Here α is a pure element of CS, and |x| denotes the parity of elements in their respective superalgebras) The Grassmann shell of a Lie superalgebra is divided into even and odd parts as follows:

0 1 u ∈ A(CS) ⇒ u = u + u (6.6a)

0 I i α i 0 α 1 u = ξ eI = ξ ei + ξ eα ξ ∈ CS ξ ∈ CS (6.6b) 1 I i α i 1 α 0 u = η eI = η ei + η eα η ∈ CS η ∈ CS (6.6c)

The exact structure of the Grassmann shell is uncomplicated, but takes up a lot of real estate and is largely unnecessary for this paper. For a more in depth look, see [4].

23 Uppsala Univeritet Eric Friden´ [email protected]

0 ∗ The structure of interest is the subalgebra A(CS)R of the even and real elements of the Grassmann shell A(CS), with multiplication given by

J I [u, v] = ζ ξ [eI , eL]

0 I J u, v ∈ A(CS)R u = ξ eI v = ζ eJ (6.7) having the properties of a Lie algebra bracket:

[u, v] = −[v, u] (6.8a) [u, [v, w]] + [v, [w, u]] + [w, [u, v]] = 0 (6.8b)

[αu + βv, w] = α[u, w] + β[v, w] α, β ∈ RS (6.8c)

6.2.2 Exponentiation

The even and real subalgebra of a Grassmann shell over a Lie superalgebra has an interesting property: it contains the information of the Lie superalgebra, but does so within a regular Lie algebra structure. Consecuently, the Baker-Campbell-Hausdorﬀ formula is applicable and exponentiation is possible. Buchbinder and Kuzenko formally associates every element

0 I u ∈ A(CS)R u = ξ eI (6.9) with the symbol

g(ξI ) = exp u (6.10) and makes it a group by applying the Baker-Campbell-Hausdorﬀ formula.

1 1 1 exp(u) exp(v) = exp u + v + [u, v] + [u, [u, v]] + [v, [v, u]] + ... 2 12 12 (6.11)

∗Invoking reality here means introducing an operation of complex conjugation such 0 that the subset of real (invariant under conjugation) elements of A(CS ) is a subalgebra.

24 Uppsala Univeritet Eric Friden´ [email protected]

6.3 Exponentiating the Poincar´eSuperalgebra

A general element of the Poincar´esuperalgebra is on the form:

X ∈ SP

1 √ X = i −baP + ωabM + i(καQ + κα˙ Q¯ ) a 2 ab α α˙ a ab ba α α˙ b , ω = −ω ∈ R κ , κ = κα ∈ C (6.12)

Its complex shell SP(C) consists of elements on the form:

X ∈ SP(C) 1 X = xaP + xabM + xαQ + xα˙ Q¯ ) a 2 ab α α˙ a ab ba α α˙ x , x = −x , x , x ∈ C (6.13)

The Grassmann shell SP(CS) consists of elements on the form:

1 X = ξaP + ξabM + ξαQ + ξα˙ Q¯ ) a 2 ab α α˙

a ab ba α α˙ ξ , ξ = −ξ , ξ , ξ ∈ CS (6.14)

Introducing a conjugation operator according to:

∗ (Pa)∗ ≡ −Pa (Mab)∗ ≡ −Mab (Qα) ≡ −Q¯α˙ (6.15)

0 results in a subalgebra SP(CS)R of real, even elements on the form

1 X = i −baP + ωabM + αQ + α˙ Q¯ a 2 ab α α˙

a ab ba 0 α α˙ α ∗ 1 b , ω = −ω ∈ RS , = ( ) ∈ CS (6.16)

As in section 6.2.2, the even, real subalgebra of a grassmann shell can be exponentiated. The result will be the Poincar´esupergroup, SΠ, of elements on the form

25 Uppsala Univeritet Eric Friden´ [email protected]

g ∈ SΠ g = g(b, , , ω)

1 g(b, , , ω) = exp i −baP + ωabM + αQ + α˙ Q¯ (6.17) a 2 ab α α˙

Elements on the form

a a 0 exp (−ib Pa) b ∈ RS (6.18a) are called translations, and the subset of elements with soulless parameters ba is the translation subgroup of the Poincar´egroup. Elements on the form 1 exp ωabM ωab ∈ 0 (6.18b) 2 ab RS are called Lorentz tranformations, and the subset of elements with soulless parameters ωab is the restricted Lorentz group SO+(1, 3). Elements of the + 0 form (6.18b) form a subgroup of SΠ denoted SO (1, 3| RS) read as the 0 ”Lorentz group over RS”. Elements on the form

α α˙ ¯ α α˙ α ∗ 1 exp i( Qα + Qα˙ ) , = ( ) ∈ CS (6.18c) are called ”supersymmetry transformations”. Elements of the form (6.18a) and (6.18c) together form a subgroup of SΠ called the group of ”supertrans- lations”.

6.4 Buchbinder-Kuzenko’s Superspace

Because of the Baker-Campbell-Hausdorﬀ formula, and since [M,P ] ∼ P and [M,Q] ∼ Q, every element of SΠ can be written as

g(b, , , ω) = g(x, θ, θ, 0)g(0, 0, 0, ω)

a ab 0 α α˙ α ∗ x , ω ∈ RS θ , θ = (θ ) (6.19)

+ 0 Equation (6.19) makes it easy to construct the coset space SΠ/SO (1, 3| RS), the set of equivalence classes

+ 0 g = gh, h ∈ SO (1, 3| RS) (6.20)

26 Uppsala Univeritet Eric Friden´ [email protected] as the set of elements of SΠ on the form

a α α˙ exp i −b Pa + Qα + Q¯α˙ (6.21)

Being parametrised by four even Grassmann numbers and a Weyl spinor (that in four dimensions is equivalent to a Majorana spinor represented by four odd Grassmann numbers) the subgroup of elements on the form (6.21) m,n is as a set equal to RS , and forms Buchbinder-Kuzenkos superspace.

7 Conclusions and Comparisons

7.1 Superspaces

This paper has presented three separate constructions of superspace. The first of which being the original Salam-Strathdee construction, as represented by Lindstrom [8]. A functional though mathematically unrigorous method that primarily serves the function of ilucidating the concept and primary functions of superspace. The other two, termed in this paper as the Rogers and Buchbinder-Kuzenko constructions, both represent ways of making that original method mathematically sound. The Rogers construction as seen in [11], acting as a representative of the works of (among others) DeWitt and Leites, is made in a roundabout fash- ion, putting focus on the underlying structure and the concept of supermanifolds. Rogers explicitly defines the 4,4-dimensional Lie supergroup as 4,4 R with an explicit group structure, and constructs the Poincarésuper- group as the semidirect product between this supergroup and the double cover of SO(1, 3), SL(2, C) (see section 5 for details). The construction is undoubtedly correct, but lacks a direct parametrisation in terms of Poincaré superalgebra elements. Buchbinder and Kuzenko [4] used their matrix-based approach to construct the Poincarésupergroup in a way that more closely resembles the construction of Minkowski space seen in 2.4. By carefully defining what scalars to use in the Poincarésuperalgebra, they define a structure containing the relevant information where the exponentiation mapping is applicable. Using the Baker-Campbell-Hausdorff formula the resulting structure can be shown to be isomorphic to that defined by Rogers. But little is learned about the underlying structure of superspace without the Rogers structure connecting it to supermanifold theory.

27 Uppsala Univeritet Eric Friden´ [email protected]

7.2 Supermanifolds and Graded Manifolds

Since the structure of superspace is closely connected to the study of supermanifolds (or graded manifolds), a section is given discussing the different ways in which to define a supersymmetric analog of the manifold. These structures come in two basic varieties: The supermanifolds of De- Witt [6] and Rogers and the graded manifolds of Berezin-Leites [3] and Kostant [7] (the former sets locally isomorphic to a model superspace and the latter regular manifolds equipped with a sheaf of superalgebras. Though on the surface radically different, the work of Marjorie Batchelor [2] connects the two, showing that every graded manifold can be identified with m,n ∞ a RS , DeW itt, H supermanifold (in the Rogers triplet notation from section 4.3). The Rogers triplet notation is a good comparative tool when studying supermanifolds as it isolates the three disinguishing factors of modern supermanifolds. The first post in the triplet is the choice of underlying Grassmann algebra. The distinction worth making here is primarily between the finite dimensional Grassmann algebra RS[L] and its infinitedimensional counterpart RS. (see section 4.1.1 for details). As is noted in section 4.1.1, theories based on RS[L] becomes physically unrealistic on levels deeper then this paper. The second post in the Rogers triplet notation is the choice of topology to m,n apply to RS . The DeWitt topology (see section 4.1.2 for more details) is widely regarded by modern authors to be the natural choice. Rogers presented an alternative topology in [10] that unlike the DeWitt topology is Hausdorff and allows for non-trivial structure in the odd part of a supermanifold. Unlike the first two (where a general consensus has been reached) the third post in the Rogers triplet notation, the smoothness restriction of functions on the supermanifold, is still a point of some contention. In this paper, three of these are presented: Rogers G∞ functions, DeWitts differentiable functions and Batchelors smooth functions (called H∞ in this paper, after Rogers).

References

[1] V. P. Akulov and D. V. Volkov. Goldstone Fields with Spin 1/2. Teor. Mat. Fiz., 18:28–35, 1974.

[2] M. Batchelor. Two Approaches to Supermanifolds. Transactions of the American Mathematical Society, 258:257–270, 1980.

28 Uppsala Univeritet Eric Friden´ [email protected]

[3] F. Berezin and D. Leites. Supermanifolds. Soviet Maths Doklady, 16:1218–1222, 1976.

[4] I. L. Buchbinder and S. M. Kuzenko. Ideas and Methods of Supersym- metry and Supergravity: Or a Walk Through Superspace. IOP Publish- ing Ltd, 1998.

[5] S. R. Coleman and J. Mandula. All Possible Symmetries of the S Matrix. Phys. Rev., 159:1251–1256, 1967.

[6] Bryce DeWitt. Supermanifolds. Press Syndicate of the University of Cambridge, 1984.

[7] B. Kostant. Graded Manifolds, Graded Lie Theory and Prequantiza- tion. In Diﬀerential Geometrical Methods in Mathematical Physics : Proceedings of the Symposium Held at the University of Bonn, July 1-4, 1975, volume 570, pages 177–306. Springer, 1977.

[8] Ulf Lindstr¨om.Supersymmetry, a Biased Review. http://arxiv.org/ abs/hep-th/0204016, 2002. Lectures given at the 22nd Winter School Geometry and Physics, Srni, Czech Republic, 12–19 January, 2010.

[9] J. T. Lopuszanski R. Haag and M. Sohnius. All Possible Generators of Supersymmetries of the S Matrix. Nucl. Phys. B, 88:257–274, 1975.

[10] Alice Rogers. A Global Theory of Supermanifolds. J. Math. Phys., 21:1352, 1980.

[11] Alice Rogers. Supermanifolds: Theory and Application. World Scien- tiﬁc, 2007.

[12] Abdus Salam and J. Strathdee. Super-gauge Transformations. Nucl. Phys. B, 76:477–482, 1974.