Seiberg-Witten Theory

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Seiberg–Witten Theory

Derivation of a Low Energy Eﬀective Lagrangian

Sven Visser

Instituut-Lorentz

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[Source: http : / / www . physics . leidenuniv . nl / images / institute / education / master / diverse_files/Master_research_project.pdf]

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Acknowledgement

The author wants to thank his supervisor K. Schalm for letting the project last two years. Also S. C. F. van Opheusden is thanked for enlightening discussions and references to mathematical documents. B Najian is thanked for introducing the author to the document [1], which was useful in some proofs. Lastly also E. van Nieuwenburg, F. M. J. G. Coppens, and B. C. van Zuiden for some discussions that are of more entertaining nature.

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Introduction

Abstract A low energy eﬀective Lagrangian will be derived from a microscopic Lagrangian, that is a variation onYang–Mills–Higgs theory. Themicroscopic Lagrangian is a so-called N=2 massless supersymmetric Lagrangian that can be written in Dirac spinor notation as:

ℒ_𝒢aux(𝛸, 𝐴, 𝜑) = 1 𝑔_cl2 tr𝔤 −

1 4𝐹𝜇𝜈∘ 𝐹

𝜇𝜈_{+ 𝑔}2 cl

𝜃_cl 64𝜋2𝜀

𝜇𝜈𝜌𝜎_𝐹

𝜇𝜈∘ 𝐹𝜌𝜎+ D𝜇(𝜑†) ∘ D𝜇(𝜑)

− i ̄𝛸𝛼∘ (𝛾𝜇)_𝛼𝛽D𝜇(𝛸𝛽)

+ √2 ̄𝛸𝛼∘ (id_𝕊𝑋)_𝛼𝛽ad(Im 𝜑) + (−i𝛾5)_𝛼𝛽ad(Re 𝜑) (𝛸_𝛽)

−1 2[𝜑

†_{, 𝜑] ∘ [𝜑}†_{, 𝜑] ;}

where 𝛸 is a Dirac spinor, 𝐴 is the vectorpotential such that D𝜇 = ∂𝜇 − iad(𝐴𝜇) is the gauge

covariant derivative and 𝐹 is the correspondingfield tensor, and 𝜑 is theHiggs field that is a complex scalar field. The gauge transformationsin this Lagrangian are based onSU(2), which is non-Abelian. Furthermore𝑔_cl and𝜃_clarereal constants.

The effective Lagrangian is found indirectly in several steps. The first step is the Higgs perturbationwith the Higgs field𝜑, keeping the following Higgs potential zero:

1 2𝑔2_cltr𝔤 [𝜑

†_{, 𝜑] ∘ [𝜑}†_{, 𝜑] = 0.}

So the eﬀective Lagrangian, which only describes the remaining massless ﬁelds, is still N=2 massless supersymmetric, and hasgauge transformations based onU(1), which is Abelian.

Then the only remainingdegree of freedomin the effective Lagrangian is a branch cut of a multivalued function 𝑓. Such a function 𝑓 is fixed, up toSL(2, ℤ), using: quantum field theoretic perturbation theory, and a family ofcubic curvesas manifolds together with somecomplex analysis on holomorphic functions; thereby fixing the effective Lagrangian.

Chapter overview This document is meant to introduce and prove the so-called Seiberg– Witten theorem (Theorem 3.48), which is mostly contained in Part II.

In Part I, somemathematical concepts and notations are defined. In particular, chapter 1 contains definitions that are related to: manifolds, including spacetime, fiber bundles, and Lie groups. Then chapter 2 describes: spinor bundles, and superspaces; as well as some general concepts used in Lagrangians and quantum theoretic correlation functions.

Note that Part I doesnot contain all of the used definitions, instead one can use theindices on page 101 and page 104 to find mostdefinitions.

The last part, Part II, contains an introductory chapter (chapter 3), which ﬁrst describes the concept of a N=2 massless supersymmetric Lagrangian, and the microscopic Lagrangian

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before ﬁnding the eﬀective Lagrangian up to𝑓. Only then theSeiberg–Witten theoremis stated (Theorem 3.48), of which itsproof relates it to the chapters in Part II.

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Part I

Mathematical Preliminaries

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Chapter 1

Notation

Some definitions will follow in this chapter. These are related todifferentiable manifolds (Defin-ition 1.1), fiber bundles (Definition 1.3), and Lie algebras (Definition 1.10) which are mostly formalities.

The most important results here are the definition of the spacetime manifold 𝑋 (Defini-tion 1.2), and of thesectionsof a bundle (Definition 1.4), as well as the the following specificLie algebras: the special unitary Lie algebra𝔰𝔲(2) (section 1.2.2), and the unitary Lie algebra 𝔲(1)

(section 1.2.2).

One could be able to only skim this chapter, except for subsection 1.2.2 which is about speciﬁc Lie groups used in the remainder.

Einstein summation convention Einstein summation convention will be used in this docu-ment. This means that implicitly summations are added to products which use the same symbol for two indices, for example: 𝜓𝛼𝜒_𝛼 should actually read∑_𝛼∈𝐴𝜓𝛼𝜒_𝛼, where the set𝐴 is derived from the type the index𝛼has to be of, which is implied by the type of𝜓 and𝜒.

1.1 Manifold

First a definition of adifferentiable manifold will be given. That is followed by the definition of the used spacetime manifold 𝑋 in Definition 1.2, where some symbols are introduced that are more thoroughly defined in subsection 1.1.1.

Definition 1.1 (Differentiable manifold) A differentiable manifold of dimension 𝑛, for ex-ample 𝑋, is a topological space1 that has a smooth atlas [2]. A smooth atlas is a set of pairs called charts: {(𝑂_i, 𝜑_i)}_i2; that satisfies:

• The set{𝑂_i}_i is an open coverof 𝑋: each𝑂_i is an open set, and⋃_i𝑂_i= 𝑋3 is satisﬁed.

1_{It is a set of which a set of subsets is deﬁned, which is called its}_topology_{or the set of its}_{open sets}_{. Using this, a}

function between two topological spaces is calledcontinuousif and only if its inverse maps open sets to open sets.

2_A_set_{is completely deﬁned by its elements. Use}_{𝑎 ∈ 𝐴}_{to describe that}_𝑎_{is an}_element_{of the set}_𝐴_{. From any two}

sets𝐴and𝐵, deﬁne𝐴 × 𝐵to be the set containing every pair of the form(𝑎, 𝑏)where𝑎 ∈ 𝐴and𝑏 ∈ 𝐵. Furthermore also deﬁne{𝑎_i}_ito be the set with elements of the form𝑎_iwhere eachiis an element in an implicit set.

3_{From any two sets}

𝐴and𝐵, deﬁne theunionandintersectionrespectively as:𝑎 ∈ 𝐴 ∪ 𝐵if and only if𝑎 ∈ 𝐴or𝑎 ∈ 𝐵, and𝑏 ∈ 𝐴 ∩ 𝐵if and only if𝑏 ∈ 𝐴and𝑏 ∈ 𝐵. Then𝐴is a non-strictsubsetof𝐵,𝐴 ⊆ 𝐵, if and only if𝐴 ∩ 𝐵 = 𝐵. For any{𝑂_i}_i, the set⋃_i𝑂_icontains only the elements that are element of any𝐴 ∈ {𝑂_i}_i.

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• Each 𝜑_i: 𝑂_i → ℝ𝑛 is an one-to-one map such that the restriction 𝜑_i: 𝑂_i → 𝜑_i(𝑂_i) is an isomorphism in the category of topological spaces: both𝜑_i: 𝑂_i→ 𝜑_i(𝑂_i)and𝜑−1_i : 𝜑_i(𝑂_i) → 𝑂_i

are continuous maps.

• Additionally the smoothness is described in the following property: for all(𝑂_i, 𝜑_i), (𝑂_j, 𝜑_j) ∈ {(𝑂_i, 𝜑_i)}_i, the map𝜑_j∘ 𝜑−1_i : 𝜑_i(𝑂_i∩ 𝑂_j) → 𝜑_j(𝑂_i∩ 𝑂_j)4 is a diﬀeomorphism, in other words:

𝜑_j∘ 𝜑−1_i ∈ C∞(𝜑_i(𝑂_i∩ 𝑂_j), 𝜑_j(𝑂_i∩ 𝑂_j))5.

Deﬁnition 1.2 (Spacetime manifold,𝑋) Deﬁne𝑋 to be the spacetime manifold, as a4 di-mensional smooth Riemannian manifold described by the following atlas:

{(𝑋, 𝜑: 𝑋 → ℝ4)},

which is a set of a single chart, such thatT𝑋 = (𝑋×ℝ4, 𝜂)as a vectorspace; and the nondegenerate inner product 𝜂 ∈ Γ Hom(T𝑋 ⊗ T𝑋, ℝ) = Γ(T 𝑋⊗2) such that one can decompose T𝑋 as T𝑋 = ℝ1⊕ ℝ3:

𝜂(∂_𝜇, ∂_𝜇) = 𝜂𝜈𝜌(∂_𝜇)_𝜈(∂_𝜈)_𝜌= 𝜂_𝜇𝜇 = +1 |𝜇 = 0 −1 |𝜇 ∈ {1, 2, 3},

whereT𝑋 = span_ℝ{∂_𝜇|𝜇 ∈ {0, 1, 2, 3}}.

1.1.1 Fiber Bundle

First a definition of afiber bundleis given in Definition 1.3, after which the definition of asection of a bundle is given (Definition 1.4) which is is a variation of the concept of a ‘bundle-valued field on a manifold’6. The most important other concept described in this section is thetangent bundle, which is defined in Definition 1.6.

Deﬁnition 1.3 (Fiber bundle,𝜋) Aﬁber bundle𝒳is completely described by the tuple(𝒳, 𝑋, 𝜋, 𝐹), which contains the following types:

• 𝑋 is theunderlying manifold,

• 𝐹 is a topological space, for example a ﬁnite dimensional vectorspace with topology that is compatible with its inner product, and is called the prototypical ﬁber,

• 𝜋: 𝒳 → 𝑋 is theprojection map;

where some additional restrictions are present: there exists a local trivialisation{(𝑂_i, 𝜑_i)}_i that satisﬁes the following:

• the set{𝑂_i}_i is an open cover of 𝑋,

• for eachi,𝜑_i: 𝜋−1(𝑂_i) → 𝑂_i× 𝐹 is a homeomorphism, in other words an isomorphism in the category of topological spaces: both𝜑_i and the appropriate𝜑−1_i are continuous;

4_{For any function or mapping}_{𝑓: 𝐴 → 𝐵}_{, deﬁne}_↦_{as the following relation: for any}_{𝑥 ∈ 𝐴}_,_{𝑥 ↦ 𝑓(𝑥)}_{or in other words}

𝑓 maps𝑥to𝑓(𝑥). For any pair of functions𝑓: 𝐴 → 𝐵and𝑔: 𝐵 → 𝐶, thefunction compositioncan be used to make the function𝑔 ∘ 𝑓: 𝐴 → 𝐶such that(𝑔 ∘ 𝑓)(𝑥) = 𝑔(𝑓(𝑥)).

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The symbol C∞(𝐴, 𝐵)denotes the set of functions of the form 𝐴 → 𝐵 that are inﬁnitely diﬀerentiable. Inthis

document, if any of those sets is overℂ, then complex diﬀerentiability will be used which is equivalent to holomorphy.

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1.1. MANIFOLD 5

• the following diagramcommutes for eachi, where𝜋 : 𝑂_i× 𝐹 → 𝑂_i such that(𝑥, 𝑣) ↦ 𝑥:

.. ..

𝜋−1(𝑂_i) 𝑂_i× 𝐹..

..

𝑂_i ..

.

𝜋 . 𝜑i

.

𝜋

;

so𝜋 = 𝜋 ∘ 𝜑_i7.

Hence one can consider the respective ﬁber bundle as a set of spaces of the form 𝑂_i× 𝐹, which are pairwise knitted together on intersections such that each appropriate isomorphism

𝜑−1_i ∘ 𝜑_j: 𝜋−1(𝑂_i∩ 𝑂_j) → 𝜋−1(𝑂_i∩ 𝑂_j)is a restriction of the identity morphism id_𝒳. Also note that any ﬁber bundle is a manifold as well.

Note that using the previous notation, for each point on 𝑋 the set {𝜑_j∘ 𝜑−1_i : (𝑂_j∩ 𝑂_i) × 𝐹 → (𝑂_j∩ 𝑂_i) × 𝐹}_i,j constitutes a group with function composition, ‘∘’, as group structure. Hence one calls any ﬁber bundle with such a group equal to 𝐺_𝒳 a𝐺_𝒳-bundle, and the group will be called itsstructure group.

Definition 1.4 (Sections of a bundle) Define the set of smooth sections of any fiber bundle

𝒳 with any underlying manifold 𝑋, to be the set of smooth functions 𝑓: 𝑋 → 𝒳 that commute with the ﬁber bundle’s projection𝜋: 𝜋 ∘ 𝑓 = id_𝑋.

For any ﬁber bundle𝒳, deﬁne Γ 𝒳 to be the set of smooth sections.

One can easily see that any function like𝑓 : 𝑋 → 𝐹 can be rewritten to a section: 𝑓 ∈ Γ(𝑋 × 𝐹)

such that𝑓(𝑥) = (𝑥, 𝑓 (𝑥)); where one should note that thetrivial bundle𝑋 × 𝐹 is used.

Definition 1.5 (Hom(⋅, ⋅) fiber bundle) For any pair of vectorspace fiber bundles over the same manifold, use 𝐵₁ and 𝐵₂ as bundles over 𝑋, one defines the fiber bundle Hom(𝐵₁, 𝐵₂) over the manifold𝑋 as that containing fiber-wiselinear functions of the form𝑓: 𝐵₁|_𝑝→ 𝐵₂|_𝑝.

Note that creating fiber bundles from other fiber bundles is not always fiber-wise, for example a so-called pullback bundlecan be defined from a smooth function and an original fiber bundle, where the pullback bundle’s projection is 𝑓 ∘ 𝜋.

Definition 1.6 (Tangent bundle,T𝑋, T 𝑋) For any smooth manifold,𝑋, one can define the tangent bundle T𝑋 through: its prototypical fiber is the vectorspace of derivations, for example

span_ℝ{∂_𝜇}_𝜇, while these ﬁbers are knitted together such that those derivations locally describe directional derivatives on𝑋.

Similarly asT𝑋, one can define thecotangent bundleT 𝑋 for each smooth manifold𝑋. That bundle is defined such that each of its fibers is the dual vectorspaceof the corresponding fiber in

T𝑋, for example for each direction 𝜇 and𝑝 ∈ 𝑋: deﬁne for each ﬁber d𝑥𝜇|_𝑝 ∈ Hom(T𝑋, ℝ)|_𝑝 = T 𝑋|_𝑝= T 𝑋 ∩ 𝜋−1(𝑝)8 such that:

d𝑥𝜇|_𝑝(∂_𝜈|_𝑝) = 1 |𝜇 = 𝜈 0 |𝜇 ≠ 𝜈.

Vector ﬁeldsare actually sections of the tangent bundle: Γ T𝑋. For more detailed deﬁnitions, see for example [2].

7_{Similarly any other diagram that commutes, also known as a}_{commutative diagram}_{, implies all of its logical equations.} 8_{In other words: deﬁne the function}_d𝑥𝜇_|

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Deﬁnition 1.7 (Dual of a vector,⋅♭) For any vector 𝑉 in any tangent bundle T𝑋 with an inner product, deﬁne𝑉♭∈ T 𝑋to be the unique vector such that for any𝑊 ∈ T𝑋: 𝑉♭(𝑊) = 𝑉𝛼𝑊_𝛼.

Definition 1.8 (Differential of a scalar field,d⋅) For any smooth function 𝑓 ∈ Γ(𝑋 × ℝ) = C∞(𝑋, ℝ), define d𝑓 ∈ Γ T 𝑋 as: d𝑓(𝑋)|_𝑝= 𝑋(𝑓)|_𝑝= 𝑋|_𝑝(𝑓) for any𝑋 ∈ Γ T𝑋 and any 𝑝 ∈ 𝑋.

Note that this notation need not be completely compatible with the notation ‘d𝑥𝜇’, as defined in Definition 1.6, since such hypothetical𝑥𝜇 ∈ Γ(𝑋 × ℝ)need not exist globally on 𝑋. Instead these are more like functions, or can be considered locally in which one considers small enough neighbourhoods of each point on the manifold. Nevertheless, since Definition 1.2 defines the spacetime manifold to be flat, these functions𝑥𝜇 ∈ Γ(𝑋×ℝ)do exist on the spacetime manifold.

1.2 Lie Theory

In this section, ﬁrst the formal concept of a Lie group and Lie algebra will be given. Lastly two speciﬁc Lie groups will be discussed, which have as respective Lie algebra respectively𝔤_eff and

𝔤. Those speciﬁc algebras will be used in the deﬁnitions of two spaces of gauge transformations for Lagrangians in chapter 3.

Deﬁnition 1.9 (Lie group) A Lie group, 𝐺, is deﬁned as a smooth manifold that is also a group, such that the following is true [4, p. 226]:

• The group action⋅ ∘ ⋅: 𝐺 × 𝐺 → 𝐺 is a smooth function.

• The group’s inverse mapping⋅−1: 𝐺 → 𝐺 is also a smooth function.

1.2.1 Lie Algebra

The definition of a Lie algebra will be given in Definition 1.10. Also the concept of a linear representationof a Lie algebra is described, as well as such a specific representation, called the adjoint representation (Definition 1.12) which represents any element of the chosen Lie algebra by a linear mapping onto the same Lie algebra.

Additionally two nonstandard infix operators are defined, which are nothing more than ex-plicit forms offunction applications: ⋅ ⋅and⋅ ⋅. From these, the last form,⋅ ⋅, is not as generally defined as the first form, but is only meant for adjoint representations.

Definition 1.10 (Lie algebra) A Lie algebra, 𝔤, is a vectorspace, over some field 𝔽, with a non-associative product,[⋅, ⋅]: 𝔤 ⊗ 𝔤 → 𝔤calledcommutator, that satisfies the following axioms9:

∀𝑎, 𝑏, 𝑐 ∈ 𝔤 ∀𝜆 ∈ 𝔽:

[𝑎, 𝑏] = −[𝑏, 𝑎],

∧ [𝑎, 𝑏 + 𝜆𝑐] = [𝑎, 𝑏] + 𝜆[𝑎, 𝑐],

∧ 0 = [𝑎, [𝑏, 𝑐]] + [𝑐, [𝑎, 𝑏]] + [𝑏, [𝑐, 𝑎]]. (1.1)

9_{Define some}_logic_{symbols: for any}_𝑎_and_𝑏_{that are either true or false, define}_{𝑎 ∨ 𝑏}_{to be}_𝑎_or_𝑏_{, and define}_{𝑎 ∧ 𝑏}_to

be𝑎and𝑏. Then also define∀𝑎 ∈ 𝐴: 𝑓(𝑎)as: for any𝑎 ∈ 𝐴,𝑓(𝑎)is true; and define∃𝑎 ∈ 𝐴: 𝑓(𝑎)as: there exists an𝑎 ∈ 𝐴 such that𝑓(𝑎)is true. Similarly define the implication𝑎 ⇒ 𝑏as(¬𝑎) ∨ 𝑏, where¬𝑎is only true if𝑎is not, and define the equivalence𝑎 ⇔ 𝑏as(𝑎 ⇒ 𝑏) ∧ (𝑏 ⇒ 𝑎).

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1.2. LIE THEORY 7

Any ﬁnite dimensional Lie group, has an associated Lie algebra which10 _{is then the ﬁber}

above the identity element of the tangent bundle that is associated with the group’s manifold [4, p. 230]. Note that such a mapping from the set of Lie groups to the respective Lie algebra is not invertible11_.

Definition 1.11 (Linear representation (𝑉, 𝜌)of any Lie algebra 𝔤, [⋅, ⋅]) For any Lie al-gebra,𝔤over the field𝔽, define the tuple(𝑉, 𝜌)to be a linear representation of𝔤, if it satisfies the following:

• 𝑉 is a vectorspace over the ﬁeld 𝔽, which is the same𝔽 as noted before.

• 𝜌: 𝔤 → L(𝑉, 𝑉)is a linear mapping of 𝔤to the set of linear operators from𝑉 to itself12.

• 𝜌satisﬁes: ∀𝑎, 𝑏 ∈ 𝔤: 𝜌([𝑎, 𝑏]) = [𝜌(𝑎), 𝜌(𝑏)]; where[⋅, ⋅]: L(𝑉, 𝑉) ⊗ L(𝑉, 𝑉) → L(𝑉, 𝑉)is deﬁned as

[𝑎, 𝑏] = 𝑎 ∘ 𝑏 − 𝑏 ∘ 𝑎for any𝑎, 𝑏 ∈ L(𝑉, 𝑉).

Sometimes inthis document,⋅ ⋅: L(𝑉, 𝑉) × 𝑉 → 𝑉 is used, which is deﬁned as explicit function application: 𝑀 𝑎 = 𝑀(𝑎).

Definition 1.12 (Adjoint representation (𝔤, ad) of any Lie algebra 𝔤,⋅ ⋅) For any Lie al-gebra, 𝔤, one defines theadjoint representationto be the linear representation(𝔤, ad), whereadis defined through ∀𝑎, 𝑏 ∈ 𝔤: ad(𝑎)(𝑏) = [𝑎, 𝑏]. Note that (𝔤, ad)is a representation, since Equation 1.1 impliesad([𝑎, 𝑏]) = [ad(𝑎), ad(𝑏)].

For this representation, the ‘explicit reverse function application’⋅ ⋅: 𝔤 × ad𝔤 → 𝔤will be used, which is deﬁned as: 𝑎 ad𝑏 = [𝑎, 𝑏] = −ad𝑏 (𝑎); and isnonstandardnotation.

1.2.2 Speciﬁc Groups

Two speciﬁc Lie groups and algebras will be discussed here, these are: U(1), andSU(2). Both of these will be used later in spaces of gauge transformations.

Note that these Lie groups and algebras are just manifolds overℝ as ℂ ≃ ℝ2, even though they are subsets ofL(ℂn, ℂn).

Unitary Lie group, U(1)

The description of an unitary Lie group and its Lie algebra follows, including the surjective mapping from𝔤_eff toU(1): 𝑇 ↦ e−i𝑇.

Note that𝔤_eff will be used in Part II, as part of a space of gauge transformations.

Deﬁnition 1.13 (Unitary Lie group, U(1)) Deﬁne the Unitary Lie groupU(1)as a subset of invertible linear operators fromℂ1 to itself [4, p. 227]:

U(1) = 𝑀 ∈ L(ℂ1, ℂ1) 𝑀†∘ 𝑀 = id ;

where the group action is function composition.

Note that composing two functions fromU(1)does give a function in U(1).

10

Other deﬁnitions of such a Lie algebra is the algebra of so-called left-invariant sections of the tangent bundle that is associated with the group’s manifold. These are equivalent, also as topological spaces.

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For example any Lie group,𝐺, has the same Lie algebra as any double cover of𝐺: (ℤ/2ℤ) × 𝐺with group action (𝑎, 𝑓) ∘ (𝑏, 𝑔) = (𝑎 + 𝑏, 𝑓 ∘ 𝑔), andℤ/2ℤis the additive group of integers modulo2.

12_{By deﬁnition}

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Theorem 1.14 (Unitary Lie algebra, 𝔲(1)) The Lie algebra corresponding withU(1) (Deﬁn-ition 1.13), is:

𝔲(1) = 𝑇 ∈ L(ℂ1, ℂ1) 𝑇†= −𝑇 ,

with the addition of[𝑎, 𝑏] = 𝑎 ∘ 𝑏 − 𝑏 ∘ 𝑎 = 0; which is obtained fromU(1) = e𝔲(1).

Note that 𝔲(1) is isomorphic to another Lie algebra, deﬁne it 𝔤_eff, that consists only of Hermitian operators:

(𝔲(1), [⋅, ⋅]) ≃ (𝔤_eff, i[i−1⋅, i−1⋅]) = (ℝ, 0);

whereU(1) = e−i𝔤eff with complex product as group action.

Special unitary Lie group, SU(2)

The description of a special unitary Lie group and its Lie algebra follow, including the surjective mapping from𝔤toSU(2)/{− id, + id}: 𝑇 ↦ {−e−i𝑇, +e+i𝑇}. The Lie algebra𝔤will also be endowed with an inner product, making it aHilbert space.

The algebra𝔤will be used in Part II, as part of a space of gauge transformations.

Deﬁnition 1.15 (Special Unitary Lie group, SU(2)) Deﬁne the so-calledspecial unitary Lie group,SU(2), to be [4, p. 227]:

SU(2) = 𝑀 ∈ L(ℂ2, ℂ2) det(𝑀) = 1, 𝑀†∘ 𝑀 = 1 ;

where the group action is function composition.

Theorem 1.16 (Special unitary Lie algebra, 𝔰𝔲(2)) Deﬁne𝔰𝔲(2) as the Lie algebra corres-ponding with theSU(2)Lie group:

𝔰𝔲(2) = 𝑇 ∈ L(ℂ2, ℂ2) 𝑇†= −𝑇, tr_𝔤𝑇 = 0 ;

whereSU(2) = ±e𝔰𝔲(2)13.

Note that 𝔰𝔲(2) is isomorphic to another Lie algebra, deﬁne it 𝔤14, that consists only of Hermitian operators: thePauli sigma matrices[3], as follows:

(𝔰𝔲(2), [⋅, ⋅]) ≃ (𝔤, i[⋅, ⋅]) = (span_ℝ{𝜎1, 𝜎2, 𝜎3}, i[⋅, ⋅])

where [⋅, ⋅] is the ordinary commutator of L(ℂ2, ℂ2), over which 𝔤 is not closed. Also note

SU(2) = ±e−i𝔤.

Each 𝜎i is aPauli sigma matrix [3]:

(𝜎1, 𝜎2, 𝜎3) = ( 0 +1 +1 0 ,

0 +i −i 0 ,

+1 0 0 −1 );

which satisﬁes𝜎i𝜎j= 𝛿ij+ i𝜀ijk𝜎k, so i[𝜎i, 𝜎j] = −2𝜀ijk𝜎k.

Corollary 1.17 (𝔰𝔲(2) is simple Lie algebra) The Lie algebra𝔰𝔲(2)is asimple Lie algebra, and hence it satisﬁes the following. Any element of𝔰𝔲(2) can be obtained from the commutation of an appropriate pair of elements of 𝔰𝔲(2): [𝔰𝔲(2), 𝔰𝔲(2)] = 𝔰𝔲(2).

13

Note that− id ∈ SU(2)cannot be obtained from anye𝔰𝔲(2).

14

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1.2. LIE THEORY 9

Proof The proof of the algebra being a simple Lie algebra, exists as is noted by [3, p. 496]. One can also prove [𝔰𝔲(2), 𝔰𝔲(2)] = 𝔰𝔲(2), by comparing this commutator with the exterior

vector product, which has a similar corollary. ∎

Theorem 1.18 (Inner product space from 𝔤) The Lie algebra 𝔤 can be endowed with the inner product: tr_𝔤(⋅ ∘ ⋅); which has{√2−1⋅ 𝜎i}_i as an orthonormal basis. Later in this document,

𝑇a will be used as any orthonormal basisvector.

Note that for𝔰𝔲(2),tr_𝔤(ad(⋅) ∘ ad(⋅)) is negative deﬁnite.

Proof The existence of such a proof is noted in [3, p. 498]. ∎

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Chapter 2

Superspace

This chapter contains mostly only lesser known deﬁnitions1, most of which relate to concepts used to write theN=2 supersymmetric Lagrangian, that will be given in chapter 3.

The ﬁrst sections, section 2.1 to section 2.3, deﬁne: theWeyl spinor bundles𝕊_L𝑋 and 𝕊_R𝑋

(Definition 2.5), a special set ofdisjoint symbols𝒫 which is related to the Lagrangian (Defini-tion 2.8), then asupercommutative algebra (Definition 2.1) which contains products of anticom-muting spinor components 𝕊𝒫_L𝑋 ⊕ 𝕊𝒫_R̄𝑋 similar to those in quantum field theory’s correlators and is named 𝑂_𝑋0 (Definition 2.11), which has a special subset 𝑂0 Ξ_𝑋 (Definition 2.14) related to the function Ξ: 𝑂0 Ξ_𝑋 → Γ(𝑋 × ℂ). Then in section 2.3, a set like 𝒫 is defined and called ϴ

(Definition 2.16), which is then used to extend 𝑂_𝑋0 to 𝑂1_𝑋 by adding anticommuting vectors from 𝕊ϴ_L𝑋 ⊕ 𝕊ϴ_R̄𝑋 (Definition 2.17), which has special subsets that are used in the Lagrangian containing the so-called superfields 𝑂1𝛷_𝑋 and 𝑂1𝑊_𝑋L (Definition 2.18, and Definition 2.19). Also note that in Definition 2.29 and Definition 2.30mappingsfromnon-supersymmetric functionsto supersymmetric onesare described, which are also used significantly in chapter 3.

In section 2.4: the previously describedsuperalgebra𝑂1_𝑋 is combined with anyLie algebra us-ing thetensor product(Definition 2.32). After that, the relationship between theLagrangianand the previously described spaces is considered (Definition 2.33). Lastly the correlation function is defined as in quantum field theory (Definition 2.34).

Deﬁnition 2.1 (Supercommutative algebra) Asupercommutative algebra, for example𝑉, is an algebra2 that can be decomposed as: 𝑉 = 𝑉0⊕ 𝑉1; and has a so-called supercommutative product, that satisﬁes for any𝑣, 𝑤 ∈ 𝑉0∪ 𝑉1:

𝑣 ⋅ 𝑤 = (−1)deg(𝑣) deg(𝑤)𝑤 ⋅ 𝑣;

where the function deg: (𝑉0∪ 𝑉1) ⧵ {0} → {0, 1} ⊆ ℤ3 has been used, that is deﬁned as:

deg(𝑣) = 0 |𝑣 ∈ 𝑉

0

1 |𝑣 ∈ 𝑉1.

1

Note that the interpretation of the concepts used here is not always the conventional ones. Inthisdocument, [5] is partially followed, it is augmented with some additional definitions making some concepts more rigorous those in others. Furthermore any symbols representing spaces or algebras in this chapter are nonstandard. A significantly different interpretation is given in [6, 7].

2_An_algebra _{is a vectorspace with a product between vectors. This product satisﬁes} _{(𝑢 ⋅ 𝑣) ⋅ 𝑤 = 𝑢 ⋅ (𝑣 ⋅ 𝑤)} _and

(𝜆𝑢) ⋅ (𝜎𝑣) = (𝜆 ⋅ 𝜎)(𝑢 ⋅ 𝑣), and distributes over the vector addition: (𝑢 + 𝑣) ⋅ 𝑤 = 𝑢 ⋅ 𝑤 + 𝑣 ⋅ 𝑤.

3_{The set}

ℤis the set ofintegers, including negative integers. The setℕis the set of natural numbers, which only contains non-negative integers. The set of real numbers isℝ, and the set of complex numbers isℂ.

Note that for any pair of sets𝐴and𝐵, deﬁne the set𝐴 ⧵ 𝐵to contain only the elements that are in𝐴but not in𝐵.

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Later the concept of a supercommutative algebra being generated from a ‘commuting’ vec-torspace and an ‘anticommuting’ vecvec-torspace, is being used. This is meant to denote the most unrestrictive supercommutative algebra possible that contains sums of products from those vector-spaces, while also deﬁning theirdegrees: the commuting vectorspace is a subset ofdeg−1(0), while the anticommuting vectorspace is a subset ofdeg−1(1). Note that the anticommuting vectorspaces is not an algebra.

Also note that the category theory one should use is not quite an ordinary one: when mul-tilinear maps are considered, it is natural to use the supercommutative tensor productinstead [5].

2.1 Spinor Bundle

The purpose of this section is to define the concept of avectorspace bundleover𝑋 that contains ‘Weyl spinor components’whilenot being the trivial fiber bundle𝑋 × ℂ. This is similar to that being used in quantum field theory [3], but a more abstract approach is being used here.

Theset of Weyl spinorsthat can be used in those ‘Weyl spinor components’ will be encoded in the set 𝒫 (Definition 2.8), which contains symbols that are mostly being used like spinor fields. The actualunderlying Weyl spinor fieldcan be obtained using the function⋅s: 𝒫 → Γ 𝕊L𝑋

(Deﬁnition 2.8), or one can use Dirac conjugationbefore applying it to get a spinor withright chirality: ̄⋅: 𝒫 →𝒫̄and⋅s:𝒫 → Γ 𝕊̄ R𝑋.

Now the trick in the deﬁnition of the vectorspace of‘Weyl spinor components’ is to use the space 𝕊𝒫_L𝑋 = span_ℂ(𝒫) ⊗ 𝕊_L𝑋 for left chirality, and 𝕊𝒫_R̄𝑋 = span_ℂ( ̄𝒫 ) ⊗ 𝕊_R𝑋 for right chirality (Deﬁnition 2.9). Theaccompanying notationof a vector in𝕊𝒫_L𝑋and𝕊𝒫_R̄𝑋is slightly altered from the expected tensor product, to be more like a ‘spinor component’ (Remark 2.10): 𝜓𝛼 ∈ 𝕊𝒫_L𝑋

and𝜓̄_𝛼_̇ ∈ 𝕊𝒫_R̄𝑋 instead of respectively 𝜓 ⊗ 𝛼 and 𝜓 ⊗ ̇̄ 𝛼for any𝜓 ∈ 𝒫 and𝛼 ∈ 𝕊_L𝑋 as well as any𝛼 ∈ 𝕊̇ _R𝑋.

But first some definitions on thespinor bundle and Clifford bundle to be able to define the Weyl spinor bundles𝕊_L𝑋 and𝕊_R𝑋 (Definition 2.5).

Definition 2.2 (Spinor bundle over spacetime,𝕊𝑋) Define 𝕊𝑋 to be a fiber bundle over spacetime𝑋, such that among other restrictions4 _{the following are satisfied [3]:}

• its prototypical ﬁber is a four dimensional complex vectorspace,

• there is some additional structure relating it to a representation of the Cliﬀord bundle, which can be described through the use of sums of products of Dirac gamma matrices: there is a linear map T 𝑋 → Hom(𝕊𝑋, 𝕊𝑋) such that d𝑥𝜇 ↦ 𝛾𝜇 is true for each 𝜇, and

𝛾𝜇𝛾𝜇= (d𝑥𝜇)𝜈(d𝑥𝜇)_𝜈= 𝜂𝜇𝜇5.

Definition 2.3 (Basis of the Clifford bundle, 𝛾𝜇) Define each constant section𝛾𝜇∈ L(ℂ4, ℂ4) ⊆ Hom(𝕊𝑋, 𝕊𝑋)as in [1, 3, 8], which use the same 𝜂as used here:

∀i ∈ {0, 1, 2, 3}: 𝛾i= 0 𝜎

i

̄

𝜎i 0 ;

where each 𝜎i has been deﬁned fori ∈ {1, 2, 3}in Theorem 1.16, while𝜎0= id_ℂ4. The value of 𝜎̄i

is identical to𝜎0 fori = 0, and otherwise it is equal to−𝜎i, since then 𝛾𝜇𝛾𝜈+ 𝛾𝜈𝛾𝜇= 2𝜂𝜇𝜈.

4

This is an incomplete deﬁnition, which should not hurt the use of it, since it is only considered over spacetime as in Deﬁnition 1.2 and [3]’s is similarly incomplete.

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2.1. SPINOR BUNDLE 13

Note that here, a ‘basis’ of an algebra is diﬀerent from the basis of the vectorspace of the algebra: products of𝛾𝜇 are the basis of the vectorspace of the Cliﬀord bundle, which is an algebra.

Definition 2.4 (Top form in Clifford bundle,𝛾5) Define𝛾5= i𝛾0𝛾1𝛾2𝛾3, to be the preferred top form, as in [3]. This will be used to differentiate between the chiralities of each spinor in

𝕊𝑋:

𝛾5= − id𝕊L𝑋 0

0 + id_𝕊_R_𝑋 ;

where for all intents and purposes,𝕊_L𝑋 = 𝑋 × ℂ2 and𝕊_R𝑋 = 𝑋 × ℂ2 are trivial ﬁber bundles.

As one can note from the previous one can decompose 𝕊𝑋 into two subbundles, 𝕊_L𝑋 and

𝕊_R𝑋. This is done in the next deﬁnition.

Definition 2.5 (Subbundles of 𝕊𝑋,𝕊_L𝑋,𝕊_R𝑋, P_𝕊_L_𝑋, P_𝕊_R_𝑋) Since 𝛾5 is assumed to be glob-ally defined, define thesubbundles6_𝕊

L𝑋and𝕊R𝑋of𝕊𝑋, such that𝕊L𝑋 = P𝕊L𝑋(𝕊𝑋) =

1

2(id𝕊𝑋−𝛾5)(𝕊𝑋)

and 𝕊_R𝑋 = P_𝕊_R_𝑋𝕊𝑋 = ₂1(id_𝕊𝑋+𝛾5)(𝕊𝑋), so they are the eigenspaces of 𝛾5 for respectively the ei-genvalue −1, and+1.

Hence the spinor bundles can be related through a (ﬁber-wise) direct sum of vector bundles:

𝕊𝑋 = 𝕊_L𝑋 ⊕ 𝕊_R𝑋.

Now consider the action ofDirac conjugation on those subbundles.

Deﬁnition 2.6 (Dirac conjugation, )̄⋅ Split Dirac conjugation, which is originally ̄⋅: 𝕊𝑋 → 𝕊𝑋 such that𝛸 ↦ 𝛸†𝛾0 as in [3], to a polymorphic functionas a combination of the functions:

̄⋅: 𝕊_L𝑋 → 𝕊_R𝑋 and ̄⋅: 𝕊_R𝑋 → 𝕊_L𝑋; such that for any 𝛸 ∈ 𝕊𝑋 = 𝕊_L𝑋 ⊕ 𝕊_R𝑋: use 𝜓 ∈ 𝕊_L𝑋 and

𝜆 ∈ 𝕊_R𝑋 as in 𝛸 = 𝜓 ⊕ 𝜆, then:

𝛸 = 𝜓 𝜆 ,

̄

𝛸 = 𝜓 𝜆

† 0 𝜎0

̄

𝜎0 0 = 𝜆

†_𝜎_̄0 _𝜓†_𝜎0 _,

= 𝜆̄ 𝜓 .̄

So in other words the Dirac conjugation is𝜓 = 𝜓̄ † by the deﬁnition of 𝜎0.

Note that this deﬁnition cannot be used on symbols that are not spinors, for example 𝜎̄𝜇 is not from𝜎𝜇, also later ̄ðdoes not come fromð.

Deﬁnition 2.7 (Antisymmetric combination of two 𝛾 matrices, 𝜎𝜇𝜈,𝜎̄𝜇𝜈) Deﬁne𝜎𝜇𝜈and

̄

𝜎𝜇𝜈 as proportional to𝛾[𝜇,𝜈], and use the convention as in [1, p. 100]7:

i 4(𝛾

𝜇_𝛾𝜈_{− 𝛾}𝜈_𝛾𝜇_{) =} i

4

𝜎𝜇𝜎̄𝜈− 𝜎𝜈𝜎̄𝜇

̄

𝜎𝜇𝜎𝜈− ̄𝜎𝜈𝜎𝜇 , = 𝜎𝜇𝜈

̄

𝜎𝜇𝜈 .

Hence 𝜎𝜇𝜈 ∈ Hom(𝕊_L𝑋, 𝕊_L𝑋)and𝜎̄𝜇𝜈∈ Hom(𝕊_R𝑋, 𝕊_R𝑋).

6

A subbundle is a fiber bundle that is constructed from a previously defined fiber bundle, in such a way that it still has the same underlying manifold, but each fiber is a subset.

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Similarly one can derive𝜎𝜇𝜎̄𝜈+ 𝜎𝜈𝜎̄𝜇 = 𝜎𝜇𝜎̄𝜈+ 𝜎𝜈𝜎̄𝜇= 2𝜂𝜇𝜈 up to identity mapping.

Now there is enough deﬁned to introduce the set of symbols𝒫, that will be used to construct the vectorspace ofWeyl spinor component8_.

Definition 2.8 (Disjoint set of symbols denoting used spinors,𝒫,⋅s) Define for each Lag-rangian, 𝒫 as a set of disjoint symbols for which each symbol needs to correspond to a Weyl spinor fieldthat is an element ofΓ 𝕊L𝑋. Then define⋅s to be the mapping describing the earlier

mentioned correspondence: ⋅s: 𝒫 → Γ 𝕊L𝑋. So for example for𝒫 = {𝜓a, 𝜒a, 𝜆a|a ∈ [1, dim(𝔤)] ∩ ℕ}9,

of which each symbol needs to correspond to an element ofΓ 𝕊L𝑋, so: 𝜓as∈ Γ 𝕊L𝑋.

Extend this construction using Dirac conjugation . So deﬁne the set̄⋅ _𝒫̄ _{such that} _{̄⋅: 𝒫 →}_𝒫̄_.

Also deﬁne it such that ̄⋅: 𝕊_L𝑋 → 𝕊_R𝑋 and ⋅s:𝒫 → Γ 𝕊̄ R𝑋 commute, so 𝜓̄s = ( ̄𝜓)s = (𝜓s). Also

extend this to all similar notation, like for example ̄⋅: 𝕊_R𝑋 → 𝕊_L𝑋.

Note that in chapter 3, the set𝒫 is actually defined for each Lagrangian as the Weyl spinors in the Lagrangian’s domain, which will be called Lagrangian’sconfiguration spacein Definition 2.33. Hence the concept of gauge transformations might work out a bit unexpected.

Now follows the actual deﬁnition of theWeyl spinor component bundles.

Deﬁnition 2.9 (𝒫-spinor component bundles, 𝕊𝒫_L𝑋, 𝕊𝒫_R̄𝑋) Deﬁne the Weyl spinor com-ponent bundles𝕊𝒫_L𝑋 and𝕊𝒫_R̄𝑋, as: 𝕊𝒫_L𝑋 = span_ℂ(𝒫) ⊗ 𝕊_L𝑋, and𝕊𝒫_R̄𝑋 = span_ℂ( ̄𝒫 ) ⊗ 𝕊_R𝑋.

Then also extend ⋅s to those for these bundles: ⋅s: 𝕊𝒫_L𝑋 → Γ(𝑋 × ℂ) and ⋅s: 𝕊𝒫_R̄𝑋 → Γ(𝑋 × ℂ), such that𝜓 ⊗ 𝛼 ↦ (𝜓s)𝛼. In addition use the notation𝜓𝛼instead of𝜓 ⊗ 𝛼 for each vector in𝕊𝒫_L𝑋

and𝕊𝒫_R̄𝑋.

Note that the dimension of𝕊𝒫_L𝑋is the same as𝕊_R𝒫̄𝑋’s, and is equal todim(span_ℂ(𝒫)) ⋅ dim(𝕊_L𝑋) = dim(span_ℂ(𝒫)) ⋅ dim(𝕊_R𝑋) = 2 ⋅ card(𝒫).

Lastly some remarks on the notation.

Remark 2.10 (Spinor index notation) In equations, use Greek indices for 𝕊_L𝑋, like 𝜓_𝛽, and also for𝕊𝑋-directions when noted as such. Similarly usedotted Greek indicesfor𝕊_R𝑋, like

̄

𝜆𝛽̇. Expand the deﬁnition of the indices further to those of 𝒫 and _𝒫̄ _through _⋅s_{, but these are}

not scalars like those of 𝕊𝑋’s, instead these are respectively the vectorspaces𝕊𝒫_L𝑋 and 𝕊𝒫_R̄𝑋 in Deﬁnition 2.9.

In summary one can describe the spinor related notation using the following:

• Any 𝛸 ∈ Γ 𝕊𝑋 can be split into its left- and right-chiral Weyl spinors,𝜓s= P_𝕊

L𝑋𝛸 ∈ 𝕊L𝑋

and _𝜆s̄ _{= P}

𝕊R𝑋𝛸 ∈ 𝕊R𝑋, and are combined using 𝛸𝛽 =

𝜓_𝛽s

̄

𝜆𝛽 ṡ , when used as substitutions

with Einstein summation convention.

• The Dirac conjugation of 𝛸,_{𝛸 ∈ 𝕊𝑋}̄ _{, is similarly split in its Weyl components}_𝜓_̄s_{∈ 𝕊} R𝑋

and𝜆s∈ 𝕊_L𝑋, where𝜓, 𝜆 ∈ 𝒫 and𝜓, ̄̄ 𝜆 ∈𝒫̄.

• Then for each 𝕊_L𝑋-direction 𝛼 and 𝕊_R𝑋-direction _𝛽_:̇ _𝜓𝛼 _{∈ 𝕊}𝒫

L𝑋 and 𝜆̄

̇

𝛽 _{∈ 𝕊}𝒫̄

R𝑋, while

𝜓𝛼 s∈ Γ(𝑋 × ℂ)and_𝜆𝛽 s̄ ̇ _{∈ Γ(𝑋 × ℂ)}_.

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2.2. N=0 SUPERSPACE: ANTICOMMUTING SPINOR 15

• The polymorphic function ⋅s deﬁned up to now, can be described through the following commutative diagram:

.. ..

𝕊𝒫_L𝑋 𝒫.. _𝒫..̄ _𝕊𝒫_..̄ R𝑋

..

Γ(𝑋 × ℂ) Γ 𝕊..L𝑋 Γ 𝕊..R𝑋 Γ(𝑋 × ℂ)..

.

⋅s .

⋅𝛼

.

⋅s .

̄⋅

. _⋅s

.

⋅𝛽̇

.

⋅s

.

⋅𝛼

.

̄⋅

.

⋅𝛽̇

.

• The preferred indices for𝜎𝜇 and𝜎̄𝜇, can be derived from𝛾𝜇:

𝛾_𝛼𝛽𝜇𝛸_𝛽 = 0 𝜎

𝜇 𝛼 ̇𝛽

̄

𝜎𝜇 ̇𝛼𝛽 0

𝜓_𝛽s

̄

𝜆𝛽 ṡ ,

= (𝜎

𝜇_𝜆)̄

𝛼s

( ̄𝜎𝜇𝜓)𝛼 ṡ ;

where the indices in the last part are implicit, so eﬀectively extend 𝜎 and 𝜎̄ to: 𝜎𝜇 ∈ Hom(𝕊𝒫_L𝑋, 𝕊𝒫_R̄𝑋) and 𝜎̄𝜇 ∈ Hom(𝕊_R𝒫̄𝑋, 𝕊𝒫_L𝑋). Note that still for each 𝛼, ̇𝛼 indices, 𝜎𝜇_{𝛼 ̇}_𝛼∈ ℂ

and𝜎̄𝜇 ̇𝛼𝛼∈ ℂ.

• Similarly𝜎𝜇𝜈 ∈ L(𝕊L𝑋, 𝕊L𝑋), and𝜎̄𝜇𝜈∈ L(𝕊R𝑋, 𝕊R𝑋)have preferred indices as: (𝜎𝜇𝜈)𝛼𝛽𝜓𝛽=

(𝜎𝜇𝜈𝜓)_𝛼 and( ̄𝜎𝜇𝜈)𝛼̇_̇

𝛽𝜆̄

̇

𝛽 _{= ( ̄}_𝜎𝜇𝜈_𝜆)_̄𝛼̇_.

2.2 N=0 Superspace: Anticommuting Spinor

The main result in this section is thesupercommutative algebra 𝑂_𝑋0 that is generated from the Weyl spinor components (Deﬁnition 2.11). That algebra𝑂_𝑋0 is then endowed with an extension ofDirac conjugation (Deﬁnition 2.12).

Then the spinorial inner product is considered to be extended to 𝑂0_𝑋, which causes the introduction of the𝜀-tensor (Deﬁnition 2.13), and the subset𝑂_𝑋0 Ξ⊆ 𝑂0_𝑋∩ deg−1(0)on which one can consider extending the spinorial bilinearinner product 𝕊𝑋 ⊗ 𝕊𝑋 → ℂto, giving the function

Ξ: 𝑂0 Ξ_𝑋 → Γ(𝑋 × ℂ) (Deﬁnition 2.14) which satisﬁesΞ(𝜓𝛼𝜒_𝛼) = 𝜓𝛼 s𝜒_𝛼s andΞ( ̄𝜓_𝛼_̇𝜒̄𝛼̇) = ( ̄𝜓_𝛼_̇s𝜒̄𝛼 ṡ )for any𝜓, 𝜒 ∈ 𝒫.

Note that because of the anticommuting nature of Weyl spinor components in 𝑂0_𝑋, Ξ has

to have some unintuitive behaviour, since it cannot always distribute over𝑂_𝑋0’s product, this is considered in Remark 2.15.

Definition 2.11 (N=0 superspace,𝑂0_𝑋) Given (𝒫, ⋅s) (Definition 2.8), define 𝑂_𝑋0 to be the supercommutative algebra (Definition 2.1) that is generated by:

• commutativescalars ﬁeldsΓ(𝑋 × ℂ), and

• anticommutativespinor components𝕊𝒫_L𝑋 ⊕ 𝕊𝒫_R̄𝑋, which has been deﬁned in Deﬁnition 2.9.

Deﬁnition 2.12 (Extension of Dirac conjugation, )̄⋅ Extend the deﬁnition of Dirac con-jugationto ̄⋅: 𝑂0_𝑋 → 𝑂_𝑋0, such that it is similar to Hermitian conjugation ⋅†: it reverses factors, and is compatible withcomplex conjugation ⋅∗. So for all𝑎 ∈ Γ(𝑋 × ℂ),𝜓, 𝜆 ∈ 𝒫,𝛼, ̇𝛽-directions, and all 𝐴, 𝐵 ∈ 𝑂_𝑋0, the following are true: 𝑎 = 𝑎̄ ∗,(𝜓_𝛼_̇) = ( ̄𝜓)_𝛼_̇,𝐴𝐵 = ̄𝐵 ⋅ ̄𝐴, and𝐴 + 𝐵 =𝐴 + ̄̄ 𝐵.

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Definition 2.13 (Emulation of the spinor inner product in 𝑂_𝑋0, 𝜀𝛼𝛽 𝜀𝛼 ̇̇𝛽 (𝜓𝜒) ( ̄𝜓 ̄𝜒)) Since spinor fields𝕊𝒫_L𝑋⊕𝕊𝒫_R̄𝑋are anticommuting in the supercommutative algebra𝑂0_𝑋, one needs some antisymmetric tensors, which will be called𝜀with the following variations: 𝜀𝛼𝛽,𝜀_{𝛼 ̇}_̇_𝛽, and the re-spective inverses with the indices at a different height: 𝜀_𝛼𝛽 and𝜀𝛼 ̇̇𝛽.

Deﬁne𝜀 to be, where indices1, 2 and ̇1, ̇2 are used for the spinor-directions:

0 +1 −1 0 =

𝜀11 𝜀12 𝜀21 𝜀22 =

𝜀̇1 ̇1 𝜀̇1 ̇2 𝜀̇2 ̇1 𝜀̇2 ̇2 , 0 +1

−1 0

−1

= 0 −1 +1 0 =

𝜀₁₁ 𝜀₁₂ 𝜀₂₁ 𝜀₂₂ =

𝜀_{̇1 ̇1} 𝜀_{̇1 ̇2} 𝜀̇2 ̇1 𝜀̇2 ̇2 .

Using this notation, deﬁne the changes of height of indices on 𝕊𝒫_L𝑋, as: 𝜓𝛽 = 𝜀𝛽𝛼𝜓_𝛼 and

𝜓_𝛽 = 𝜀_𝛽𝛼𝜓𝛼; and on 𝕊_R𝒫̄𝑋 as: 𝜓̄𝛽̇= 𝜀𝛽 ̇𝛼̇𝜓̄𝛼̇ and𝜓̄𝛽̇= 𝜀𝛽 ̇𝛼̇ 𝜓̄𝛽̇[1, 8].

Using this tensor, one has the following in 𝑂0_𝑋: 𝜓𝛼𝜒_𝛼= 𝜒𝛼𝜓_𝛼, and 𝜓̄_𝛼_̇𝜒̄𝛼̇ = ̄𝜒_𝛼_̇𝜓̄𝛼̇. Note that these pairings, such as 𝜓𝛼𝜒_𝛼, arenot elements ofΓ(𝑋 × ℂ) but are still sums of products of two spinor components: 𝜓𝛼𝜒_𝛼= ∑_𝛼,𝛽𝜀𝛼𝛽𝜓𝛽𝜒_𝛼.

Later implicit pairings are being used, such as: (𝜓𝜒) = (𝜓𝛼𝜒_𝛼)and( ̄𝜓 ̄𝜒) = ( ̄𝜓_𝛼_̇𝜒̄𝛼̇); where one has to note the heights of the indices. These implicit pairings are compatible with the implicit indices used for tensors like𝜎: 𝜎𝜇_𝛼𝛼_̇ 𝜓𝛼= (𝜎𝜇𝜓)_𝛼_̇10.

The function ⋅s cannot easily be extended to a function from𝑂0_𝑋 toΓ(𝑋 × ℂ), so introduce a

subset𝑂0 Ξ_𝑋 ⊆ 𝑂_𝑋0 of terms and a functionΞ: 𝑂0 Ξ_𝑋 → Γ(𝑋×ℂ)which should effectively ‘extend’⋅sas far as possible. So now one should defineΞand𝑂_𝑋0 Ξmore formally, this is done in the following definitions11.

Definition 2.14 (Scalar sections from a subset 𝑂_𝑋0 Ξ⊆ 𝑂0_𝑋, Ξ, 𝑂0 Ξ_𝑋 ) Define the subset𝑂0 Ξ_𝑋 ⊆ 𝑂0_𝑋 as a subset of the commutative part of 𝑂_𝑋0, so𝑂0 Ξ_𝑋 ⊆ deg−1(0), and defineΞ: 𝑂_𝑋0 Ξ→ Γ(𝑋 × ℂ)

to at least satisfy the following:

• For any 𝜓, 𝜒 ∈ 𝒫, deﬁneΞ((𝜓𝜒)) = (𝜓s𝜒s) = 𝜓𝛼 s𝜒_𝛼s and similarly Ξ(( ̄𝜓 ̄𝜒)) = ̄𝜓_𝛼_̇s𝜒̄𝛼 ṡ .

• Because of the antisymmetry of 𝕊𝒫_L𝑋, one has𝜓1𝜓1= 0, anddim_ℂ(𝕊_L𝑋) = 2, deﬁne:

∀𝜓, 𝜒 ∈ 𝒫 ∀𝐴 ∈ 𝑂0 Ξ_𝑋 :

Ξ((𝜓𝜓) ⋅ 𝐴) = Ξ((𝜓𝜓)) ⋅ Ξ(𝐴|𝜓=0),

∧ 𝐴 = 𝐴|_𝜓,𝜒=0 ⇒ Ξ((𝜓𝜒) ⋅ 𝐴) = Ξ((𝜓𝜒)) ⋅ Ξ(𝐴); (2.1)

where the substitution mappings are such that𝜓̄_𝛼_̇|_𝜓=0= ̄𝜓_𝛼_̇ need not be0, but𝜓𝛼|_𝜓=0= 0is true for any𝛼. The similar restriction should be true for𝕊𝒫_R̄𝑋 instead of𝕊𝒫_L𝑋.

• StillΞ has to be afunction, so it should satisfy: ∀𝐴, 𝐵 ∈ 𝑂0 Ξ_𝑋 : 𝐴 = 𝐵 ⇒ Ξ(𝐴) = Ξ(𝐵).

• The functionΞhas to be linear: for any𝐴, 𝐵 ∈ 𝑂0 Ξ_𝑋 and any𝜆 ∈ Γ(𝑋×ℂ), deﬁneΞ(𝐴+𝜆𝐵) = Ξ(𝐴) + 𝜆 Ξ(𝐵).

• Hence 𝑂0 Ξ_𝑋 is a complex vectorspace, and not completely closed under 𝑂0_𝑋’s vector product.

10

Applying this to the𝜀-tensor would give:(𝜀𝜓)_𝛼= (𝜀_𝛼𝛽𝜓𝛽), which is equal to𝜓_𝛼 so the notation makes that implicit.

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2.3. N=1 SUPERSPACE 17

Note that Ξand𝑂0 Ξ_𝑋 are relatively badly deﬁned.

Lastly some remarks on the previous deﬁnition.

Remark 2.15 (Odd behaviour of Ξ) Since the spinor ﬁelds in𝑂0_𝑋 are anticommuting,Ξand

𝑂0 Ξ_𝑋 both have to have some odd behaviour relating to Equation 2.1. Most of these are related to the existence of 𝐴, 𝐵 ∈ 𝑂0 Ξ_𝑋 such that Ξ(𝐴 ⋅ 𝐵) ≠ Ξ(𝐴) ⋅ Ξ(𝐵) for some(𝒫, ⋅s), like:

• For example, for any𝜓, 𝜒 ∈ 𝒫: (𝜓𝜓)(𝜓𝜒) = 0; soΞ((𝜓𝜓)(𝜓𝜒)) = Ξ(0) = 0, which need not be equal toΞ((𝜓𝜓)) Ξ((𝜓𝜒)).

• Note the following identities that can be derived using a two dimensional basis: (𝜆𝜓)(𝜆𝜒) = −1₂(𝜆𝜆)(𝜓𝜒) and ( ̄𝜆 ̄𝜓)( ̄𝜆 ̄𝜒) = −1₂( ̄𝜆 ̄𝜃)( ̄𝜓 ̄𝜒) [3, 8]. Hence one has for any pairwise unequal

𝜆, 𝜓, 𝜒 ∈ 𝒫: Ξ((𝜆𝜓)(𝜆𝜒)) = −1₂Ξ((𝜆𝜆)) Ξ((𝜓𝜒)); while this is not true under the substitution 𝜒 ↦ 𝜆, since then it should be0 as given in the previous item.

In actual equations and Lagrangians, the role of Ξand by extension 𝑂0 Ξ_𝑋 , will be diminished by applying Ξwithout proof to elements of 𝑂0_𝑋.

2.3 N=1 Superspace

In this section the𝒫-like setϴ = {𝜃}is introduced (Deﬁnition 2.16), which is then used to extend the supercommutative algebra𝑂0_𝑋 to𝑂1_𝑋 (Deﬁnition 2.17). But this extension actually has more in common with the ordinaryexterior algebra as far as its functions/structure is concerned.

From that algebra 𝑂1_𝑋, two subsets are defined 𝑂1𝛷_𝑋 and 𝑂_𝑋L1𝑊 (Definition 2.18 and Defini-tion 2.19), which will later be used to construct the domains of each Lagrangian. Then some superderivationson𝑂_𝑋1 are defined only from ϴ, and not from any spinors in𝒫, and are called:

ð and ̄ð12 (Definition 2.23), D and D̄ (Definition 2.24). After which also some products similar to exterior algebra’stop form are introduced: (𝜃𝜃), ( ̄𝜃 ̄𝜃), and (𝜃𝜃)( ̄𝜃 ̄𝜃) (Definition 2.26); and the respective Berezin integrations,∫d2𝜃 and∫d2𝜃̄, are considered (Definition 2.27).

After that thesupercommutative algebra describingN=2 superspaceis constructed 𝑂_𝑋2 (Re-mark 2.31), but will only be used once in chapter 3.

Definition 2.16 (𝒫-like set,ϴ = {𝜃} and its bundles,𝕊ϴ_L𝑋,𝕊ϴ_R̄𝑋) Similar to a chosen 𝒫, define the setϴ = {𝜃}, with a spinor-like symbol, butwithoutextending⋅s. Then define𝕊ϴ_L𝑋 and

𝕊ϴ_R̄𝑋 respectively like 𝕊𝒫_L𝑋 and𝕊𝒫_R̄𝑋 as in Deﬁnition 2.9, but usingϴ instead of𝒫.

Definition 2.17 (N=1 superspace,𝑂1_𝑋) Given (𝒫, ⋅s) (Definition 2.8), define 𝑂1_𝑋 as the su-percommutative algebra (Definition 2.1) that is generated by those vectorspaces that generate

𝑂0_𝑋 (Deﬁnition 2.11) in addition to𝕊ϴ_L𝑋 ⊕ 𝕊ϴ_R̄𝑋:

• like𝑂0_𝑋,commutative scalars ﬁeldsΓ(𝑋 × ℂ), and

• also like𝑂0_𝑋,anticommutativespinor components𝕊𝒫_L𝑋 ⊕ 𝕊𝒫_R̄𝑋, and ﬁnally

• anticommutativespinor-like components 𝕊ϴ_L𝑋 ⊕ 𝕊ϴ_R̄𝑋 (Deﬁnition 2.16).

Note that hence one can consider 𝑂0_𝑋 a subalgebra of 𝑂_𝑋1, so𝑂0_𝑋⊆ 𝑂1_𝑋.

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Now the definitions of the subsets of 𝑂1_𝑋 that are called the sets of superfields, first the scalar-like𝑂1𝛷_𝑋 and second the left chiral Weyl spinor-like𝑂1𝑊_𝑋L.

Definition 2.18 (Scalar-like superfield, 𝑂1𝛷_𝑋 ) Define the set of scalar-like superfields 𝑂1𝛷_𝑋

to be the commutative subset of 𝑂1_𝑋∩ deg−1(0) that only contains sums of products, where each term contains at most one ﬁeld in 𝕊𝒫_L𝑋 ⊕ 𝕊𝒫_R̄𝑋. So any𝛷 ∈ 𝑂1𝛷_𝑋 can be described using some

𝜑, 𝐹₁, 𝐹₂, 𝐺 ∈ Γ(𝑋 × ℂ),𝐴 ∈ Γ(ℂ ⊗ T 𝑋), and𝜓₁, 𝜓₂, 𝜆₁, 𝜆₂ ∈ 𝒫 such that:

𝛷 = 𝜑 + √2(𝜃𝜓₁) + (𝜃𝜃)𝐹₁+ √2( ̄𝜃 ̄𝜓₂) + ( ̄𝜃 ̄𝜃)𝐹₂− (𝜃𝜎𝜇𝜃)𝐴̄ _𝜇+ (𝜃𝜃)( ̄𝜃 ̄𝜆₁) + ( ̄𝜃 ̄𝜃)(𝜃𝜆₂) + ( ̄𝜃 ̄𝜃)(𝜃𝜃)𝐺.

Note that dim(T 𝑋) = 4and dim(Hom(𝕊_L𝑋, 𝕊_R𝑋)) = dim(𝕊_L𝑋) ⋅ dim(𝕊_R𝑋) = 2 ⋅ 2 = 4, so there exists a ﬁber-wise linear bijection betweenΓ(𝕊L𝑋 ⊗ 𝕊R𝑋)andΓ(ℂ ⊗ T 𝑋) such that 𝑇𝛼 ̇𝛼= 𝜎𝜇𝛼 ̇𝛼𝐴𝜇.

Definition 2.19 (Left chiral spinor-like superfield, 𝑂1𝑊_𝑋L) Define the set of left chirality spinor-like superfields𝑂1𝑊_𝑋L to be the anticommutative subset of(𝑂1_𝑋∩ deg−1(1)) ⊗ 𝕊_L𝑋13 that only contains sums of products, where each term only contains at most one field in 𝕊𝒫_L𝑋 ⊕ 𝕊𝒫_R̄𝑋. So any 𝑊 ∈ 𝑂1𝑊_𝑋L can be described using some 𝐹, 𝐺 ∈ Γ(𝑋 × ℂ), 𝐴₁, 𝐴₂ ∈ Γ(ℂ ⊗ T 𝑋), and

𝜓₁, 𝜓₂, 𝜆₁, 𝜆₂, 𝜒 ∈ 𝒫 such that:

𝑊_𝛼= 𝜓_1𝛼+ 𝜃_𝛼𝐹 + (𝜃𝜃)𝜓_2𝛼+ ( ̄𝜃𝜎𝜇)_𝛼𝐴_1𝜇+ 𝜃_𝛼( ̄𝜃 ̄𝜒) + (𝜃𝜃)( ̄𝜃𝜎𝜇)_𝛼𝐴_2𝜇+ ( ̄𝜃 ̄𝜃)𝜆_1𝛼+ ( ̄𝜃 ̄𝜃)𝜃_𝛼𝐺 + (𝜃𝜃)( ̄𝜃 ̄𝜃)𝜆_2𝛼;

but that is not the only expansion, one could for example replace𝜃_𝛼( ̄𝜃 ̄𝜒)by (𝜃𝑇_𝜇)( ̄𝜃𝜎𝜇)_𝛼 for some

𝑇 ∈ Γ T 𝑋 ⊗ 𝕊𝒫_L𝑋 under the appropriate changes.

The following is a deﬁnition describing the projection map P𝜃 ̄₀₀𝜃: 𝑂1_𝑋 → 𝑂0_𝑋 which is rather simple.

Deﬁnition 2.20 (Projection to 𝜃-less space, P𝜃 ̄₀₀𝜃) DeﬁneP₀₀𝜃 ̄𝜃: 𝑂1_𝑋 → 𝑂_𝑋0 to be the projection mapthat maps all𝜃s and_𝜃_{s to the null vector, and hence to the subalgebra}̄ _𝑂0

𝑋 ⊆ 𝑂1𝑋. As far as

deﬁned notation, one can consider thisP𝜃 ̄₀₀𝜃(𝑎) = 𝑎|_{𝜃, ̄}_𝜃=0.

So the function is deﬁned through the following equations: it is linear P𝜃 ̄₀₀𝜃(𝐴 + 𝜆𝐵) = P𝜃 ̄₀₀𝜃(𝐴) + 𝜆 P𝜃 ̄₀₀𝜃(𝐵), and it distributes over the algebra’s product P𝜃 ̄₀₀𝜃(𝐴 ⋅ 𝐵) = P𝜃 ̄₀₀𝜃(𝐴) ⋅ P𝜃 ̄₀₀𝜃(𝐵).

Note that P𝜃 ̄₀₀𝜃 need not bedeg-invariant, depending on the number of factors of𝜃s, and_𝜃_s.̄

Note that using this deﬁnition one can deriveP𝜃 ̄₀₀𝜃(𝑂_𝑋1𝛷) = Γ(𝑋 × ℂ)and P₀₀𝜃 ̄𝜃(𝑂1𝑊_𝑋L) = 𝕊𝒫_L𝑋, which relates to the naming that has been given to those spaces: respectivelyscalar-likeandleft chiral spinor-like.

The following deﬁnition is used in order to describe the superderivations that are considered later.

Deﬁnition 2.21 (Supercommutator, [⋅, ⋅], ∂_𝜇) Deﬁne the supercommutator to be an exten-sion of the original commutator for functions, but then relating to supercommutativity which is encoded indeg, so each function described this way has an associateddeg:

[𝑓, 𝑔] = 𝑓 ∘ 𝑔 − (−1)deg(𝑓) deg(𝑔)𝑓 ∘ 𝑔.

13_{The notation used for any}_{𝑊 ∈ 𝑂}1𝑊

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2.3. N=1 SUPERSPACE 19

In this way the ordinary derivatives∂_𝜇∈ Γ T𝑋can be described as: deg(∂_𝜇) = 0, and using the product rule:

[∂_𝜇, (𝑏 ⋅ id_𝑂1

𝑋)](𝑐) = ∂𝜇(𝑏 ⋅ 𝑐) − 𝑏 ⋅ ∂𝜇(𝑐),

= (∂_𝜇(𝑏) ⋅ id_𝑂1 𝑋)(𝑐).

As an aside remember that both𝜃 ∈ ϴ and_{𝜃 ∈ ̄}̄ _ϴare constant, so: ∂_𝜇(𝜃𝛼) = 0.

Corollary 2.22 (Supercommutator for functions in𝑂1_𝑋⋅ id_𝑂1

𝑋) In particular, if one

con-siders linear mappings of the form: 𝑎 ⋅ id_𝑂1

𝑋 where 𝑎 ∈ 𝑂

1

𝑋 and deg(𝑎 ⋅ id𝑂1_𝑋) = deg(𝑎); then the

supercommutator is trivially0:

[𝑎 ⋅ id_𝑂1

𝑋, 𝑏 ⋅ id𝑂1𝑋](𝑐) = (𝑎 ⋅ 𝑏 − (−1)

deg(𝑎) deg(𝑏)_{𝑏 ⋅ 𝑎) ⋅ 𝑐,}

= (𝑎 ⋅ 𝑏 − 𝑎 ⋅ 𝑏) ⋅ 𝑐 = 0.

Now the ﬁrst and simplest anticommutative superderivation.

Deﬁnition 2.23 (Superderivation to𝜃,ð, ̄ð) Deﬁne derivation-like structures on 𝑂_𝑋1 that respectively reduces the number of factors of 𝜃s or _𝜃̄s, like respectivelyð𝛼=∂𝜃∂𝛼 or ̄ð𝛼̇=_{∂ ̄}_𝜃∂𝛼̇.

One can describe the use of ð and ̄ð through the concept of the supercommutator deﬁned

previously, using _deg(ð_𝛼) = deg( ̄ð_𝛼) = 1: for any𝑎, 𝑏 ∈ 𝑂1_𝑋

[ð𝛼, 𝑎 ⋅ id𝑂1_𝑋](𝑏) = (ð𝛼(𝑎) ⋅ id𝑂1_𝑋)(𝑏)

and similarly for ̄ð; and the deﬁnition:

ð𝛼(𝜃𝛽) = 𝛿𝛽𝛼=

1 |𝛼 = 𝛽

0 |𝛼 ≠ 𝛽, ð𝛼( ̄𝜃

̇

𝛽_{) = 0,}

̄ð𝛼̇( ̄𝜃𝛽̇) = 𝛿

̇

𝛽

̇

𝛼, ̄ð𝛼̇(𝜃𝛽̇) = 𝛿

̇

𝛽

̇

𝛼.

When multiplyingð and ̄ðwith 𝜀(Deﬁnition 2.13), one can derive:

ð𝛼(𝜃_𝛽) = −𝛿𝛼_𝛽, ̄ð𝛼̇( ̄𝜃𝛽̇) = −𝛿_𝛽𝛼̇̇.

The set of superderivations is a vectorspace, so the following is one too. This superderivation is used in particular since it is related with the so-called chiral superﬁelds(Deﬁnition 2.28).

Deﬁnition 2.24 (Supercovariant derivative,D,D̄) Deﬁne D and D̄, which are named su-percovariant derivatives, as:

D𝛼= + ð𝛼+i(𝜎𝜇𝜃)̄𝛼⋅ ∂𝜇, D̄𝛼̇= − ð𝛼̇−i(𝜃𝜎𝜇)𝛼̇⋅ ∂𝜇.

Hence deg(D𝛼) = 1anddeg( ̄D𝛼̇) = 1.

Now consider theHermitian conjugation⋅† of the superderivations.

Corollary 2.25 (Hermitian conjugation of superderivations,⋅†) Thesuperderivationson

𝑂_𝑋1: which are among others ∂_𝜇,ð𝛼, 𝛼̄ð ̄, D𝛼, and D̄𝛼̇; can be related to their Hermitian

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combination with: (𝑎 ⋅ id_𝑂1 𝑋)

† _{= ̄}_{𝑎 ⋅ id}

𝑂1_𝑋, since (𝐴 ∘ 𝐵)†= 𝐵†∘ 𝐴†, as well as deg(𝐴†) = deg(𝐴) and

deg(𝑎 ⋅ id_𝑂1

𝑋) = deg(𝑎); as follows:

∂_m(𝑎n) ⋅ id_𝑂1

𝑋 = [∂m, 𝑎

n_{⋅ id} 𝑂1

𝑋],

∂_m(𝑎n_{) ⋅ id}

𝑂1_𝑋 = [ ̄𝑎̇n⋅ id𝑂_𝑋1, ∂†ṁ],

= −(−1)deg(∂) deg(𝑎)∂†_m_̇( ̄𝑎̇n) ⋅ id_𝑂1 𝑋.

where ̇n and ṁ are indices that occur under the substitutions, where: 𝜇 = 𝜇̇ for T𝑋 and T 𝑋, which is not true for Weyl spinor indices like𝛼̇ from 𝕊_L𝑋 and𝕊_R𝑋, but𝛼 = 𝛼̈ .

For the speciﬁc superderivations, one ﬁnds:

∂_𝜇†= −∂_𝜇, ð𝛼†= + ̄ð𝛼̇, D𝛼†= − ̄D𝛼̇;

and similar equations for ̄ð† _and_D_̄†_.

Now the deﬁnition of the preferredtop form-like products in𝑂1_𝑋.

Deﬁnition 2.26 (Top forms of left- and right-chiral𝑂1_𝑋-subspaces, (𝜃𝜃),( ̄𝜃 ̄𝜃)) Deﬁne those top formsto be(𝜃𝜃) and( ̄𝜃 ̄𝜃). The combination (𝜃𝜃)( ̄𝜃 ̄𝜃)is also being used.

These are called top forms here, since dim(𝕊ϴ_L𝑋) = dim(𝕊ϴ_R̄𝑋) = 2, and hence the exterior algebras 𝛬𝕊ϴ_L𝑋 and 𝛬𝕊ϴ_R̄𝑋 have those products 𝜃𝛼∧ 𝜃𝛽 = −1₂𝜀𝛼𝛽𝜃𝛼∧ 𝜃_𝛼 ∈ 𝛬dim(𝕊ϴL𝑋)_𝕊ϴ

L𝑋 and

̄

𝜃𝛼̇𝜃̄𝛽̇= +1₂𝜀𝛼 ̇̇𝛽𝜃_𝛼̄_̇∧ ̄𝜃𝛼̇ ∈ 𝛬dim(𝕊ϴR̄𝑋)_𝕊ϴ̄

R𝑋.

A Berezin integrationis a map from𝑂1_𝑋 that removes any occurrences of 𝜃 or of_𝜃̄_{. This is}

used to force the Lagrangian to be supersymmetric, which means that it is invariant under some set of functions [1].

Deﬁnition 2.27 (Berezin integration,∫d2𝜃, ∫d2𝜃̄) Deﬁne theBerezin integrations,∫d2𝜃 and

∫d2𝜃̄, as functions in Hom(𝑂1_𝑋, 𝑂_𝑋1) such that each Berezin integration returns the coeﬃcient of the respective top form.

Since compositions ofð, ̄ð, and P𝜃 ̄₀₀𝜃 can ﬁlter out any of those coeﬃcients of 1,𝜃𝛼,_𝜃𝛼̄̇ _{and so}

on, the Berezin integrations can be deﬁned as:

d2𝜃 = −1 4(ð

𝛽

∘ ð𝛽), d2𝜃 = −̄

1 4( ̄ð𝛽̇∘ ̄ð

̇

𝛽_),

d2𝜃 (𝜃𝜃) = 1, d2𝜃 ( ̄̄ 𝜃 ̄𝜃) = 1.

Hence the Berezin integration∫d2𝜃d2𝜃̄ can be restricted as∫d2𝜃d2𝜃 : 𝑂̄ 1_𝑋→ 𝑂0_𝑋.

Deﬁne the set of scalar-like and left chiral spinor-likechiral superﬁelds in the following.

Definition 2.28 (Chiral superfield,( ̄D_⋅)−1(0) ∩ 𝑂1𝛷_𝑋 ,( ̄D_⋅)−1(0) ∩ 𝑂1𝑊_𝑋L) The most used super-fields, elements of 𝑂1𝛷_𝑋 and 𝑂1𝑊_𝑋L, in Part II are those also in _D̄−1

⋅ (0), so in each direction

the application of the respective_D̄ _{to such a ﬁeld will give}₀_{. These superﬁelds are called} _chiral

superﬁelds: ( ̄D_⋅)−1(0) ∩ 𝑂1𝛷_𝑋 and( ̄D_⋅)−1(0) ∩ 𝑂_𝑋L1𝑊.

Additionally the termchiralwill be more generally used to denote elements of( ̄D_⋅)−1(0)without the need of being a superﬁeld.

Seiberg-Witten Theory