Super Yang-Mills Theory Using Pure Spinors

Thesis for the Degree of Master of Science in Engineering Physics Super Yang-Mills Theory using Pure Spinors

Fredrik Eliasson

Fundamental Physics Chalmers University of Technology Goteborg,¨ Sweden 2006 Super Yang-Mills Theory using Pure Spinors FREDRIK ELIASSON

°c FREDRIK ELIASSON, 2006

Fundamental Physics Chalmers University of Technology SE-412 96 Goteborg¨ Sweden

Chalmers Reproservice Goteborg,¨ Sweden 2006 Super Yang-Mills Theory using Pure Spinors

Fredrik Eliasson

Department of Fundamental Physics Chalmers University of Technology SE-412 96 Goteborg,¨ Sweden

Abstract

The main purpose of this thesis is to show how to formulate super Yang-Mills theory in 10 space-time dimensions using the pure spinor method developed by Berkovits. For comparison we also introduce super Yang-Mills in the ordinary component form as well as the usual superspace formulation with constraints. Furthermore we show how the extra ﬁelds in the cohomology of the pure spinor approach can be explained by introducing the antiﬁeld formalism of Batalin-Vilkovisky for handling gauge theories.

iii Acknowledgements

I wish to thank my supervisor Bengt E.W. Nilsson.

iv Contents

1 Introduction 1

2 SYM and Bianchi identities 3

2.1 Ordinary YM ...... 3

2.2 Super Yang-Mills in component form ...... 4

2.2.1 The abelian case ...... 5

2.2.2 The non-abelian case ...... 8

2.3 Introducing superspace ...... 10

2.3.1 Introducing the supermanifold ...... 10

2.3.2 Recalling differential geometry and gauge theory . . . . 12

2.3.3 Back to superspace ...... 15

2.4 Bianchi identities and their solution ...... 19

2.4.1 The conventional constraint ...... 20

2.4.2 The dynamical constraint ...... 22

2.4.3 Solving the Bianchi identities ...... 22

2.5 Gauge and SUSY-transformations in superspace ...... 27

3 SYM using pure spinors 33

3.1 The Pure Spinor ...... 33

3.2 Q and its cohomology ...... 34

v 3.3 More Fields ...... 43

3.3.1 Level zero ...... 43

3.3.2 Level two ...... 45

3.3.3 Level three ...... 46

3.4 Extending to the non-abelian case ...... 46

4 BRST and antiﬁelds 49

4.1 Antiﬁelds and the master action ...... 49

4.1.1 Fadeev-Popov quantisation ...... 51

4.1.2 BRST-quantisation ...... 55

4.1.3 BV-quantisation ...... 59

4.2 Antiﬁelds for super Yang-Mills ...... 63

5 Conclusions 67

A Some conventions 69

B Spinors and γ-matrices in D=10 71

B.1 Spinors ...... 71

B.2 Fierzing ...... 74

B.3 Some γ-matrix identities ...... 76

C Solving the pure spinor constraint 79

vi Chapter 1

Introduction

The usual framework for describing the fundamental structure of matter and interactions in nature is that of quantum field theory (QFT). A specific QFT is given by specifying its action, a functional of the different fields of the theory, which can be used to calculate all measurable quantities of interest. The perhaps most interesting property of any action is the symmetries is possesses. By demanding that an action should satisfy certain symmetries we can severely limit the fields it can contain and the shape it can take. The most common example is that for a QFT to be compatible with the theory of special relativity we have to demand symmetry under global Lorentz transformation. By studying the algebra of the generators of these symmetry transformations we can find exactly what fields can be allowed to appear in the action.

Note that we said global Lorentz transformations. This means that we are considering a continuous family of transformations parametrised by one or more constants on space-time. It is then of course natural as a next step to consider transformations with parameters that are functions on space-time. These kind of symmetries are known as gauge symmetries. It turns out that the interactions we can observe in nature are described very well by gauge theories — theories that possess gauge symmetries — e.g. quantum electrodynamics, quantum chromodynamics and the standard model. All of these theories are Yang-Mills theories — a speciﬁc kind of gauge theory.

In the ordinary Standard Model there is a problem related to the Higgs particles mass — the hierarchy problem — which can be solved by introducing a new rather peculiar symmetry called supersymmetry (SUSY). The simplest modi- fication of the standard model that includes SUSY is the Minimal Supersym- metric Standard Model (MSSM) which also has the added benefit of coupling constant unification. The introduction of SUSY means enlarging the Poincare´ group by postulating a new symmetry transformation that relates fermions and bosons. At the moment there are no firm indications that such a symmetry actually exists in nature, but nevertheless it is an interesting subject to study. Aside from the problems mentioned above, the quest to unify gravity with

1 2 CHAPTER 1. INTRODUCTION quantum mechanics through for instance string theory has led to predictions of supersymmetry.

In this thesis we will study super Yang-Mills theory (SYM). This is simply the theory you arrive at when you try to make ordinary Yang-Mills theory supersymmetric. Specifically our aim is to show how SYM can be formulated using a relatively recently discovered method that involves what is called pure spinors. First we will briefly discuss the simplest formulation of SYM — that of simply writing down the action in therms of the involved fields. This we call the component formalism. We will then go on to describe the so called super-space formulation of SYM and then demonstrate how this is related to the new pure spinor formulation. Finally we will introduce some very general tools for the quantization of gauge theories, the Batalin-Vilkovisky formulation (BV), to explain some additional elements that appear in the pure spinor formulation as compared to the super-space one. Chapter 2

Super Yang-Mills in D=10 from constrained Bianchi identities

2.1 Ordinary YM

The most convenient way to formulate a Yang-Mills theory is to utilise the language of differential forms. The reason is that the gauge invariance of Yang- Mills theory then can be seen as being due to the nilpotency of the exterior derivative, d2 = 0, and thus becomes completely transparent.

µ If we introduce the gauge potential as a 1-form, A = Aµdx , and then simply let the ﬁeld strength be the 2-form, F = dA, we will immediately have gauge invariance under A → A + dΛ, where Λ is an arbitrary 0-form. This is true because then we have δF = d(dΛ) = d2Λ = 0.

Expanding the forms in their components one ﬁnds that this corresponds directly to the usual formulation of Maxwell’s electromagnetism. That is Fµν = ∂µ Aν − ∂ν Aµ and the transformation Aµ → Aµ + ∂µΛ.

Of course Maxwell’s theory is only a very particular type of Yang-Mills theory, namely the abelian one, but this formalism can also be extended to non-abelian theories. The exterior derivative then has to be extended to a covariant version D.

Because F = dA, for the abelian case, it’s obvious that F satisﬁes the identity dF = 0. This identity is known as the Bianchi identity. In fact as long as our spacetime has no topological subtleties, saying that F satisﬁes the Bianchi identity implies that it’s possible to construct F from a gauge potential the way we have done. In the non-abelian case there is also a Bianchi identity involving

3 4 CHAPTER 2. SYM AND BIANCHI IDENTITIES the covariant exterior derivative in a similar way. The equivalence between constructing F from a potential A and demanding that it satisﬁes the Bianchi identity will be of importance when we try to formulate super Yang-Mills theory in superspace.

2.2 Super Yang-Mills in component form

When the Poincare´ group is extended to the super-Poincare´ group we need to consider what representations the new group has. Since it consists of both fermionic and bosonic elements the representation space will have both a fermionic and a bosonic sector. Furthermore since the ordinary Poincare´ group is a bosonic subgroup both of these sectors should consist of representations of the Poincare´ group. A collection of fields living in such a representation of the super-Poincare´ group is called a supermultiplet. It consists of bosonic and fermionic fields that are mixed when transformed by the supersymmetry generators. The supersymmetry transformation maps bosons into fermions and vice versa. Because of this the degrees of freedom of the bosonic fields in the multiplet has to equal the degrees of freedom of the fermionic fields.

The simplest example of a supermultiplet is the Wess-Zumino multiplet in four dimensions. This multiplet contains simply a complex scalar, ϕ, and a Majo- rana spinor, Ψa. The index a in this case takes 4 different values and because of the Majorana condition this means that the spinor consists of four real components if we work in an appropriate basis. The complex scalar on the other hand can be regarded as being composed of two real components. The number of components of the fermionic and bosonic fields does obviously not match as we above claimed they must. The solution is to require that the fields are on-shell. The equation of motion for the scalar is the Klein-Gordon equation and reads p2ϕ = 0. On the mass shell p2 = 0 so ϕ is not restricted in any way 1 and thus we really have two independent¡ ¢ degrees of freedom . The spinor has µ b µ to satisfy the Dirac equation, γ a pµΨb = 0. Since the Dirac operator, pµγ , squares to zero on the mass shell2it follows that the dimensionality of its kernel is half the dimension of the γ-matrices. We can conclude that the Dirac equation halves the number of degrees of freedom in the spinor from four to two, thus matching the scalar. Note that this matching only occurs when considering on-shell fields. If we wish to work off-shell we must introduce extra auxiliary fields to absorb the difference in number of degrees of freedom.

The SUSY generators are denoted Qa. Their actions on the ﬁelds are not particularly complicated but since we do not really need them we will only give them in a schematic form:

Qaϕ ∼ Ψa (2.1) Ψ α Qa b ∼ (γ )ab ∂αϕ

1Chapter ten of [1] has as an elementary introduction to the counting of degrees of freedom. 2 µ ν µ ν µν 2 Since pµγ pν γ = 1/2pµ pν {γ , γ } = pµ pν η = p = 0. 2.2. SUPER YANG-MILLS IN COMPONENT FORM 5

D Dirac spinor Weyl spinor Maj. spinor Maj.-Weyl spinor Vector 3 4 (2) 2(1) 3 (1) 4 8 (4) 4(2) 4(2) 4 (2) 6 16 (8) 8(4) 6 (4) 10 64 (32) 32 (16) 32 (16) 16(8) 10 (8)

Table 2.1: The number of components of spinors in the dimensions where SYM i possible. The number in parentheses is the degrees of freedom when gauge invariance and equations of motion are imposed. The cases that can be used in SYM are boxed. Note that we count real components.

One can check that the the following anticommutation relation is satisﬁed by the SUSY generators: α {Qa, Qb} = 2 (γ )ab Pα (2.2) where Pα is the ordinary momentum operator that generates translations in space-time. The standard reference on supersymmetry is [2].

2.2.1 The abelian case

We now wish to construct a supersymmetric version of Yang-Mills theory. The fields of this theory must live in a supermultiplet and one of the members of this multiplet should be a vector corresponding to the gauge field in the ordinary theory. In D dimensions a vector has D − 2 degrees of freedom. The vector has D components and the equation of motion p2 Aµ − pµ p · A = 0 reduces to pµ p · A = 0 on the mass shell which implies that p · A = 0. This can be used to eliminate one of the components of Aµ in terms of the others, for instance A0 = pi Ai/p0. At the same time we have the gauge invariance δAµ = pµΛ which can be used to remove another component of Aµ, for instance A1 by taking Λ = −A1/p1. Thus the D − 2 degrees of freedom. Note that we will be considering only the on-shell case. It is only in certain specific dimension that it is possible to find a field in a spinorial representation with as many degrees of freedom. One such dimension is D = 10. In this case the vector will have 8 degrees of freedom. A Weyl spinor will have 16 complex components. Also introducing the Majorana condition (for D = 10 it is possible to impose both the Weyl and the Majorana conditions simultaneously) makes these real. Finally demanding that the spinor satisfies the Dirac equation gives the desired 8 on- shell degrees of freedom. Super Yang-Mills is also possible for D = 3, 4, 6. Table 2.1 shows the dimensionality of the spinors for these cases. Also see appendix B for more information about the spinors.

The action for our super Yang-Mills theory is Z £ 1 i ¤ S = d10x − F Fµν + χγρ∂ χ (2.3) 4 µν 2 ρ 6 CHAPTER 2. SYM AND BIANCHI IDENTITIES where F is the ordinary ﬁeld strength constructed out of the vector in the multiplet as Fµν = ∂µ Aν − ∂ν Aµ and χa is the Majorana-Weyl spinor with a taking values from 1 to 16. Since we are working with Weyl spinors the γ-matrices are really the 16x16 blocks of the ordinary 32x32 γ-matrices in D = 10 in the block off-diagonal Weyl representation. Once again we refer to appendix B for a further discussion. Varying the action yields the familiar equations of motion:

µ ∂ Fµν = 0 (2.4) ¡ µ¢ab γ ∂µχb = 0

Now consider the following supersymmetry transformation on the fields: ¡ ¢ (δε A)µ = − εγµχ i (2.5) (δ χ)a = − F (γµν ε)a ε 2 µν where ε is a constant Majorana-Weyl spinor parameter. A simple calculation gives the corresponding variation of the action: Z h 1 i ¡ ¢ i ¡ ¢i δS = d10x − F (δ F)µν + δ χ 6∂χ + χ 6∂δ χ = 2 µν ε 2 ε 2 ε Z h 1 ¡ ¢ 1 ¡ ¢i = d10x − F ∂µ(δ A)ν + F εγµν γρ∂ χ + ∂ F χγργµν ε = µν ε 4 µν ρ 4 ρ µν Z h ¡ ¢ 1 ¡ ¢ 1 ¡ ¢i = d10x F εγν ∂µχ + F εγµν γρ∂ χ − ∂ F εγµν γρχ = µν 4 µν ρ 4 ρ µν Z h ¡ ¢ 1 ¡ ¢ 1 ³ ¡ ¢´ = d10x F εγν ∂µχ + F εγµν γρ∂ χ − ∂ F εγµν γρχ + µν 4 µν ρ 4 ρ µν 1 ¡ ¢i + F εγµν γρ∂ χ = 4 µν ρ Z h ¡ ¢ 1 ¡ ¢ ³ í = d10x F εγν ∂µχ + F εγµν γρ∂ χ + ∂ ··· = µν 2 µν ρ ρ Z h ¡ ¢ 1 ¡ ¢ ¡ ¢ = d10x F εγν ∂µχ + F εγµνρ∂ χ + F εγ[µην]ρ∂ χ + µν 2 µν ρ µν ρ ³ í + ∂ρ ··· = Z h ¡ ¢ 1 ¡ ¢ ¡ ¢ ³ í = d10x F εγν ∂µχ + F εγµνρ∂ χ + F εγµ∂ν χ + ∂ ··· = µν 2 µν ρ µν ρ Z h1 ¡ ¢ ³ í = d10x F εγµνρ∂ χ + ∂ ··· = 2 µν ρ ρ Z h ³1 ¡ ¢´ 1 ¡ ¢ ³ í = d10x ∂ F εγµνρχ − ∂ F εγµνρχ + ∂ ··· = ρ 2 µν 2 ρ µν ρ Z ³ ´ 10 = d x∂ρ ··· where we have used that γµγνρ = γµνρ + 2ηµ[ν γρ], the symmetry properties of the γ-matrices and the spinors, and in the last step the Bianchi identity, ∂[µ Fνρ] = 0. As can be seen the variation consists of a boundary term. Thus the action is invariant under the SUSY transformation (2.5) if we assume, as is usually done, that the fields goes to zero as x → ∞. 2.2. SUPER YANG-MILLS IN COMPONENT FORM 7

To derive the algebra of the generators of our symmetry we will now calculate the effect of making two successive transformations on a ﬁeld. Denoting a transformation with parameter εi by δi we get

¡ ¢ ¡ ¢ i ¡ ¢ δ δ A = − δ ε γ χ = − ε γ δ χ = Fρσ ε γ γ ε = 2 1 µ 2 1 µ 1 µ 2 2 1 µ ρσ 2 i ¡ ¢ ¡ ¢ = Fρσ ε γ ε + iFρσ ε η γ ε = 2 1 µρσ 2 1 µ[ρ σ] 2 i ¡ ¢ ¡ ¢ = Fρσ ε γ ε + iF ε γσε 2 1 µρσ 2 µσ 1 2

If we now use the fact that γµρσ is antisymmetric while γσ is symmetric in combination with the anticommuting property of the εi we get: h i ¡ σ ¢ δ1, δ2 Aµ = − 2iFµσ ε1γ ε2 (2.6)

Let us now denote the generator of the SUSY transformation by Qa, that is a δ1 Aµ = ε1 Qa Aµ. We can then rewrite the commutator of transformations, [δ1, δ2], a b as an anticommutator of generators, ε1ε2{Qa, Qb}. From equation (2.6) we can thus deduce that n o ¡ ν ¢ ¡ ν ¢ ¡ ν ¢ Qa, Qb Aµ = − 2iFµν γ ab = −2i∂µ Aν γ ab + 2i∂ν Aµ γ ab = ³ ´ (2.7) ¡ ν ¢ ¡ ν ¢ = ∂µ −2iAν γ ab + 2i γ ab∂ν Aµ

The ﬁrst term in this expression is a gauge transformation of A, the second is proportional to the momentum operator, Pµ ∼ ∂µ, in its coordinate realisation. If we instead consider the gauge invariant quantity F we get: n o ³ ´ ³ ´ ¡ ρ¢ ¡ ρ¢ Qa, Qb Fµν =∂µ 2i γ ab∂ρ Aν − ∂ν 2i γ ab∂ρ Aµ ³ ´ (2.8) ¡ ρ¢ ¡ ρ¢ =2i γ ab∂ρ ∂µ Aν − ∂ν Aµ = 2i γ ab∂ρ Fµν

We recognise this as the desired form of the SUSY algebra. Let us now repeat this calculation but instead acting on the spinor:

i ³ ¡ ¢ ´ ¡ ¢¡ ¢ δ δ χa = − δ F γµν ε a = −i∂ δ A γµν ε a = 1 2 2 1 µν 2 µ 1 ν 2 ¡ ¢¡ ¢ i ¡ ¢h¡ ¢ ¡ ¢ i =i∂ ε γ χ γµν ε a = ε γ ∂ χ γµγν ε a − γν γµε a = µ 1 ν 2 2 1 ν µ 2 2 i h¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ i = − εbεc γ γµ a γν e − γ γν a γµ e ∂ χd 2 1 2 ν bd e c ν bd e c µ This leads to h i i h¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ δ , δ χa = − εbεc γ γµ a γν e − γ γν a γµ e + 1 2 2 1 2 | ν bd {z e }c | ν bd {z e }c I II ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ i + γ γµ a γν e − γ γν a γµ e ∂ χd | ν cd {z e }b | ν cd {z e }b µ III IV 8 CHAPTER 2. SYM AND BIANCHI IDENTITIES

Terms I and III can be rewritten as:

¡ ¢ ¡ µ¢a ¡ ν ¢e ¡ ¢ ¡ µ¢a ¡ ν ¢e I + III = γν bd γ e γ c + γν cd γ e γ b = ¡ µ¢a ¡ ν ¢e ¡ ¢ ¡ µ¢a ¡ ν ¢e ¡ ¢ =3 γ e γ (c γν bd) − γ e γ d γν bc = ¡ µ¢a e ¡ µ¢a ¡ ν ¢e ¡ ¢ =3 γ e Q cbd − γ e γ d γν bc ¡ ¢ ¡ ¢ where Q is deﬁned by Qe ν e . We can utilise the Clifford algebra, ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢cbd = γ (c γν bd) b c b c c γµ a γν b + γν a γµ b = 2ηµν δa, to rewrite terms II and IV as:

¡ ¢ ¡ ν ¢a ¡ µ¢e ¡ ¢ ¡ ν ¢a ¡ µ¢e II + IV = γν bd γ e γ c + γν cd γ e γ b = ¡ ¢ µν a ¡ ¢ ¡ µ¢a ¡ ν ¢e ¡ ¢ µν a =2 γν bdη δc − γν bd γ e γ c + 2 γν cdη δb− ¡ ¢ ¡ µ¢a ¡ ν ¢e − γν cd γ e γ b = ¡ µ¢ a ¡ µ¢ a ¡ µ¢a e ¡ µ¢a ¡ ν ¢e ¡ ¢ =2 γ bdδc + 2 γ cdδb − 3 γ e Q cbd + γ e γ d γν bc Combining everything we get h i h a i b c ¡ µ¢a e ¡ µ¢ a ¡ µ¢ a δ1, δ2 χ = − ε1ε2 6 γ e Q cbd − 2 γ bdδc − 2 γ cdδb− 2 i ¡ µ¢a ¡ ν ¢e ¡ ¢ d − 2 γ e γ d γν bc ∂µχ = h i b c ¡ µ¢a e ¡ µ¢ a ¡ µ¢ a = − ε1ε2 6 γ e Q cbd − 2 γ bdδc − 2 γ cdδb− 2 i µν a¡ ¢ ¡ ν ¢a ¡ µ¢e ¡ ¢ d − 4η δd γν bc + 2 γ e γ d γν bc ∂µχ = h i b c ¡ µ¢a e ¡ µ¢ a ¡ µ¢ a = − ε1ε2 6 γ e Q cbd − 2 γ bdδc − 2 γ cdδb− 2 i µν a¡ ¢ ¡ ν ¢a ¡ µ¢e ¡ ¢ d − 4η δd γν bc + 2 γ e γ d γν bc ∂µχ

To get to the desired form we will¡ ¢ now demand that the spinor χ is on shell, µ b e i.e. satisﬁes the Dirac equation γ ab∂µχ = 0, and recall that in D = 10 Q abc is identically zero (see appendix B). Only the penultimate term above survives to give: n o c ¡ µ¢ c Qa, Qb χ = 2i γ ab∂µχ (2.9) As is noted in the appendix Q vanish also for D = 3, 4, 6 so we will get the super Poincare algebra on-shell in those cases as well. Note however that ,the action (2.3) was shown to be invariant without using Q. This is an indication of the fact that this action actually is invariant under the transformations 2.5 independently of the dimension or what kind of spinor χ is. We didn’t show this as our derivation assumed that χ was a Majorana-Weyl spinor in ten dimensions, but it is possible to do. For further details see [3]. This fact will no longer be true in the non-abelian case.

2.2.2 The non-abelian case

Everything in section 2.2.1 can be generalised to the non-abelian case. Now the ﬁelds, both Aµ and χc, take their values in the gauge Lie algebra. By introducing a basis Ti for the lie algebra we could make this explicit by expanding the 2.2. SUPER YANG-MILLS IN COMPONENT FORM 9

i i ﬁelds like Aµ = AµT . However to keeps our formulas clearer we will refrain from this. Note that in contrast to for instance a fermion in QED here also the spinor is Lie-algebra valued. This is not really strange since it is not a fermion making up matter but rather the superpartner to the Lie-algebra valued gauge boson.

Furthermore, the ordinary derivative ∂µ will have to be replaced by the gauge covariant derivative ∇µ everywhere it appears. This includes when it is “hid- den” inside the deﬁnition of Fµν . The covariant derivative acts in the ordinary way on Aµ and χc. With our conventions, see appendix A, this is given by

∇µ Aν = ∂µ Aν − Aµ Aν

∇µχc = ∂µχc − [Aµ, χc]

The action becomes Z £ 1 i ¤ S = d10x tr − F Fµν + χγρ∇ χ (2.10) 4 µν 2 ρ where the trace over the algebra has to be introduced to render the action gauge invariant. It’s worth pointing out that unlike the abelian case we no longer have a free theory. The covariant derivative has introduced an interaction between the ﬁelds Aµ and χc. We could have included a new parameter in the deﬁnition of the covariant derivative giving the strength of this interaction.

The equations of motion for the ﬁelds are modiﬁed to become

¡ α¢ab γ ∇αχb = 0 i ¡ ¢ (2.11) ∇α F = γ ab{χ , χ } αβ 2 β a b where we see that the interaction between the ﬁelds lead to a current term in the second equation.

The supersymmetry variations will be exactly the same as before, given in equation (2.5). When verifying the invariance of the new action under these transformations one proceeds like in the abelian case. There is however the added complication that you have to remember to vary also the A-ﬁeld inside the covariant derivative. As long as it’s a derivative appearing inside F this will, due to cancellations, not give anything new, but because of the covariant derivative in the spinor part of the action a new term appears which means a that the transformation of the action is proportional to the object Qbcd that we deﬁned earlier. So in the non-abelian case the action is only invariant in those cases where Q vanish. As we have noted this happens in particular for D = 10.

Working out the algebra one encounters¡ ¢ no additional obstacles. On shell it is α as expected given by {Qa, Qb} = 2i γ ab∇α.

For details on the non-abelian case consult [3] and [4]. 10 CHAPTER 2. SYM AND BIANCHI IDENTITIES

2.3 Introducing superspace

As we saw in section 2.1 it is convenient to formulate Yang-Mills theory using the language of differential forms. It is now natural to ask whether an analogous construction can be made for super Yang-Mills. The answer is “sort of”. Below we will show how we can embed our fields in a super differential form over a supermanifold and then from it construct a field strength and eventually obtain the equations of motion. What we do will however not be to completely mimic the construction of ordinary Yang-Mills theory only re- placing manifold with a supermanifold. We will not gain any complete geo- metric understanding of SYM. The benefit of the superspace formulation to be presented is rather that the supersymmetry is made completely transparent. In this language supersymmetry transformations will be on an equal footing with change of coordinates in spacetime. In fact they will be certain coordinate transformations on the supermanifold and thus the use of objects that are inherently coordinate invariant, such as differential forms, will guarantee a supersymmetric theory. The supersymmetry generators will be represented by certain fermionic derivatives just as the momentum generators are represented by ∂µ. In contrast to the purely bosonic case we will not construct an action to derive the equations of motion. Instead these will arise due to a specific constraint being imposed on the field strength. Our treatment of super Yang-Mills in superspace is entirely based on [5].

2.3.1 Introducing the supermanifold

A supermanifold, M, is a topological space where each open set can be param- eterised by a set of coordinates ZM = (xµ, θm), where the xµ are commuting real numbers and the θm are anticommuting real Grassmann numbers. For the case we consider xµ is a vector with 10 components and θm has 16 components. This is of course to match the D = 10 vector and Majorana spinor. We will not be bothered with any global issues and proceed as if the entire manifold could be covered with a single chart. Just as for an ordinary manifold we can introduce the tangent bundle, TM. This is the union of the tangent spaces at all points of the manifold. The coordinate basis for the tangent bundle TM consists of the µ m derivatives ∂M = (∂µ, ∂m) = (∂/∂x , ∂/∂θ ). Note that the derivatives commute in the same way as the coordinates. The dual space to TM is the bundle of 1-forms, T∗ M or in other words the union of all cotangent spaces. The coordinate basis for this bundle is simply the dual basis to the coordinate basis of the tangent bundle. It is usually denoted dZm. It is now of course easy to construct higher forms by taking the usual alternating tensor product of one forms — the ∧-product. One has to be careful though when commuting different objects and take into account both their form degree and their Grassmann properties. For instance, if we let |M| = 1 when ZM is anticommuting and |M| = 0 when ZM is commuting we have dZM ∧ dZN = −(−1)|M||N|dZN ∧ dZM. To save some writing we will from now on drop the | · · · | in such expressions. You then have to remember that indices appearing over (−1) are not ordinary indices and should not be summed over. For forms of higher degree all this 2.3. INTRODUCING SUPERSPACE 11 generalises to (p) (q) (q) (p) ϕ ∧ ω = (−1)pq+|ω||ϕ| ω ∧ ϕ where the form degree is displayed above the forms and |ϕ| = 0 if ϕ is commuting etc. Note that in the future we will usually not write out the wedge V between forms. We will denote the space of k-forms by k M and the space of V all forms by ∗ M. Any k-form can be expanded in terms of the basis elements dZM1 ··· dZMk like this:

(k) 1 ω = dZMk ··· dZM1 ω k! M1···Mk Please note the order of the indices and the fact that we place the components after the forms. V A natural thing to do is to try to deﬁne an exterior derivative, d : k M → Vk+1 µ M. Usually it is given by d = dx ∂µ so a natural generalisation would be M dZ ∂M. This is indeed the deﬁnition we will adopt though with the added complication that it will act from the right. This is best explained by an example. On a p-form:

(p) 1 dω = dZMp ··· dZM1 dZN∂ ω p! N M1···Mp

Note that no extra signs appeared since the right action means that d never have to commute past the forms dZM, it acts directly on the form components standing on the right. On products of forms the rule is:

¡(p) (q)¢ (p) (q) (p) (q) d ω ∧ ϕ = ω ∧ dϕ + (−1)q dω ∧ ϕ

In the second term d have been commuted past the q-form, thus the extra sign. Also note that d is purely bosonic.

The characteristic property of d is that it satisﬁes d2 = 0. This is simply encod- ing the fact that partial derivatives commute/anticommute:

(p) ¡ 1 ¢ d2 ω =d2 dZMp ··· dZM1 ω = p! M1···Mp 1 ¡ ¢ = d dZMp ··· dZM1 dZN∂ ω = p! N M1···Mp 1 = dZMp ··· dZM1 dZNdZK ∂ ∂ ω = p! | {z } | K{z N} M1···Mp NK+1 K N NK (−1) dZ dZ (−1) ∂N ∂K =0 where the last step follows since the two braced quantities have opposite symmetries in the N and K indices.

Let us now assume that we have a metric on our manifold. At each point it is given by an inner product on the tangent space, g(p): Tp M × Tp M → R. M N We can express the metric in terms of the coordinate basis, g(V ∂M, W ∂N) = 12 CHAPTER 2. SYM AND BIANCHI IDENTITIES

M N V gMNW , where gMN = g(∂M, ∂N). Now using the metric we can always construct another basis for the tangent space that is orthonormal. Let us de- M note such a basis by EA = EA ∂M. We will assume that the signature of our metric is lorentzian.³ ´ Then the orthonormality is given by, g(EA, EB) = gAB = M N ηαβ 0 E gMN E = . We will use letters from the beginning of the alphabet, A B 0 δab like A, to denote non-coordinate bases. Letters from the middle of the alphabet, like M, will be reserved for coordinate bases. Note that the orthogonal basis is by no means unique. We can always let the Eα transform as a vector under the Lorentz group and the Ea as a Majorana-Weyl spinor. This will not affect the orthonormality. In fact we can make a local Lorentz transformation so that the M matrices EA depends on where on the manifold we are. We will assume that M EA (Z) depends smoothly on Z.

2.3.2 Recalling differential geometry and gauge theory

Before proceeding further with our supermanifold it might be wise to brieﬂy recall how you in general do differential geometry and gauge theory. This will help us keep our head clear when we try to do the same on the supermanifold. Let us proceed in steps:

1. First let set the stage. We start with a manifold, M. This will always be spacetime. On this manifold we introduce a vector bundle. This we think of simply as a union of vector spaces, one for each point of the manifold. A familiar example is the tangent bundle. A trivial vector bundle would be V × M, where V is a vector space and M the manifold, but the interesting cases are when the bundle is a so called twisted product between a vector space and the manifold. Basically this means that locally the bundle is a direct product but when you move to a neighbouring region the vector space above it will have been rotated by some element of a group G relative to the space over the ﬁrst region. Of course in reality the group element acts on the vector space through some given representation. The group G is called the structure group of the vector bundle and the bundle itself is called a G-bundle. In the case of the tangent bundle this group would be the Lorentz group if we decide to use a Minkowski metric.

2. The physical ﬁelds will be sections of vector bundles, that is functions from the manifold into the bundle — vector ﬁelds if you wish. We will need to be able to differentiate these sections. To do this there must be a way of comparing vectors that lives in vector spaces at two different points. This is where the connection enters. The connection is really a way to do differentiation on sections of bundles. Note that there could be many different ways to do this for a given vector bundle. Every connection Dv (doing differentiation in the direction of the vector v) can be writ- µ ten as Dv = v ∂µ + A(v) where A(v) is an endomorphism on the vector bundle. As physicists we usually call the endomorphism-valued 1-form A the connection. When working with G-bundles we won’t let A(v) be any old endomorphism. To make the connections “compatible” with the 2.3. INTRODUCING SUPERSPACE 13

structure group we have to require it to belong to the Lie algebra g of G.

3. The physical theories we want should be gauge invariant. This means that if we take a particular section of a G-bundle, say s, and for every point of the manifold act on it with an element of G, that is s → gs where g : M → G, then the new section gs should describe the same physics as the original section. What happens to the connections when we do gauge transformations? For a given connection D there exists another one D0 such that D0(gs) = g(Ds). This D0 is the gauge transformation of D. In terms of the vector potential A this is expressed as A → A0 = gAg−1 + gdg−1, when A is Lie-algebra valued this transformation rule makes A0 Lie-algebra valued too. The physics should be invariant under gauge transformations of the connection together with gauge transformations of the ﬁelds. Since Ds transform in the same way as s we call D a covariant derivative.

4. To do gauge invariant physics it’s convenient to have quantities that are invariant under gauge transformations. One such object that we can construct using only the connection is the curvature. It measures how taking covariant derivatives in different directions fails to commute. It will be a linear function£ of those¤ two directions, in fact a 2-form. To be more specific we have Dv, Dw s = F(v, w)s + D[v,w]s where s is a section of the vector bundle. F is the curvature 2-form. For a G-bundle it will be Lie algebra-valued just like A. The last term in the expression above is the torsion term. It appears since it might happen that first moving a small step in the direction of v on the manifold and then in the direction of w lands you in a different point than first moving along w and then along v. Above we implied that the curvature was gauge invariant. This is actually not true. It transform as F → gFg−1. But it’s easy to construct truly gauge invariant objects from it — you only have to take the trace. Furthermore products of F transform in the same way as F itself.

5. We can regard the connection as a covariant exterior derivative. Let E be the bundle we are working on. Then a section s of this bundle will be an E-valued 0-form. The covariant exterior derivative dD of this will be an E-valued 1-form. We define it as (dDs)(v) = Dv(s) for a vector field v. This then generalises in the natural way to higher forms. Us- 2 ing this language we can define the curvature by dDs = F ∧ s where s is an E-valued form. Now the torsion term only appears if we ex- a 2 a pand the left hand side in terms of a basis, say E : dDs = dD(E Das) = a a a b c c 1 a b c c dD(E )Das − E dD(Das) = E E (Da Dbs + Tab Dcs). T = 2 E E Tba = dD E is the torsion 2-form. Starting from a connection on E we can construct a corresponding connection on End(E) and use this to get an exterior derivative on End(E)-valued forms. This allows us to also do exterior differentiation on objects like A and F. We can write the exterior derivatives in terms of A as dDs = ds + A ∧ s when s is an E-valued form and as dD B = dB + [A, B] when B is an End(E)-valued form. From the first one of those you can deduce that F = dA + A ∧ A.

6. Since F is an End(E)-valued 2-form we can take the covariant exterior derivative of it: dD F = dF + [A, F]. By using the formula for F in step 5 it 14 CHAPTER 2. SYM AND BIANCHI IDENTITIES

immediately follows that dD F = 0. This is the Bianchi identity.

That’s all the basics. So let’s move on to the physics. There are two applications: general relativity and gauge theory. We will begin with the ﬁrst. When doing GR we are working on a base manifold, space-time, that is semi-riemannian. This means that we have a metric — an inner product on the tangent space. Using this metric we can construct an orthonormal basis for the tangent space a each point. This is exactly what we did for the supermanifold a little earlier. Such a choice of basis for the entire tangent bundle is called a frame and will be denoted by ea (the orthonormality means that g(ea, eb) = δab). There are of course many different orthonormal bases and the collection of all of them is called the frame bundle. Given a particular frame we can do local rotations, 0 Λb ea(x) → ea(x) = a(x)eb(x), to get another one. For a minkowskian metric the Λb rotations, the matrix a above, will belong to the Lorentz group. These local Lorentz rotations are the gauge transformations we can do on the frame bundle whose structure group thus is the Lorentz group.

Our ﬁelds will be sections of vector bundles associated to the frame bundle. Different representations of the Lorentz group gives different bundles, e.g. vector bundle and spinor bundle. We now have to answer how we are going to decide what connection to use on those bundles. First of all we should recall that the curvature of space-time is found by solving the Einstein equations. The curvature then gives the connection through the equation b b c b Ra = dωa + ωa ∧ ωc b where we have denoted the connection by ωa . It is a matrix of 1-forms belonging to the Lorentz algebra. To proceed we will have to make an assumption. It is that the connection is torsion free. Remarkably there is only a single connection compatible with this assumption for a given metric; the Levi-Civita connection. So Einstein’s equations gives curvature, which gives the connection which in turn gives a metric — you only need to solve the differential equations. The main use of the Levi-Civita connection is that we can formulate covariant equations of motions for our matter ﬁelds by using the covariant derivative constructed with it.

For the special case of ﬂat space the curvature vanishes. You can then show that there is a choice of orthonormal basis for tangent space so that the connection is zero for all point on the manifold. That is, we can always make a local Lorentz transformation on the frame bundle so that the connection potential w transforms to zero. This does not completely ﬁx the connection. You can still do global Lorentz rotations and the connection will remain zero. We can now make a suitable choice of coordinates on space-time so that the orthonormal basis equals the coordinate basis ∂µ for these coordinates. Note that it is only when the torsion is zero that this choice of coordinates is compatible with the global vanishing of the connection.

Next up is gauge theory. Since we want both gauge and Lorentz invariance our matter fields will be sections of the tensor product bundle of a Lorentz bundle and a G-bundle, where G is som Lie group. This means that our connection will be of the form dD = d + A + ω where A is g-valued and ω is Lorentz 2.3. INTRODUCING SUPERSPACE 15 algebra-valued as above. The curvature will split in two. One part, R, will be determined from ω and measure space-time curvature as before. The other part, which we call F, is determined from A and is simply the field strength of the gauge potential. The curvature part of the connection will be the Levi- Civita connection while A is given by solving the Yang-Mills equation for the field strength. The Yang-Mills equation follows from the action: Z SYM = tr(F ∧ ∗F) M where ∗ is the usual Hodge star operator. The equation of motion coming from this action is dD ∗ F = 0.

For further details of differential geometry and gauge theory [6] is recommended.

2.3.3 Back to superspace

We will now apply our knowledge of gauge theory summarised in the last section to our supermanifold. Of course the superisation will introduce some differences. Commutators are replaced by graded commutators, that is anticommutator on two fermionic objects, otherwise ordinary commutator, and things acting from the left are replaced by things acting from the right, specifically the exterior derivative and gauge transformations. We have already introduced the supermanifold and showed how an orthonormal basis of tangent space can be locally Lorentz rotated. From now on we will restrict ourselves to ﬂat superspace. If we had worked in eleven dimensions curved superspace would have allowed us to derive supergravity, but for now we are only interested in super Yang-Mills. Flat superspace means that the curvature R is zero. We can then choose a basis for our tangent space so that the connection ω is zero everywhere and as above there is a corresponding choice of coordinates of the supermanifold so that the tangent basis is the coordinate basis. These coordinates will be called ZM.

We will however not use this basis for the tangent bundle but instead use the one spanned by the vectors DA given by:

Dα = ∂α (2.12) ¡ µ¢ b Da = ∂a − i γ abθ ∂µ

M Expressed in terms of the coordinate basis this takes the form DA = EA ∂M M where the matrix EA is given by

µ µ ¶ M δα 0 EA = µ m −i(γ θ)a δa

We can introduce a corresponding 1-form basis which we denote by EA. It is A A M A given in terms of the coordinate basis by E = EMdZ where the matrix EM is 16 CHAPTER 2. SYM AND BIANCHI IDENTITIES

M the inverse of EA . It is easily calculated to be µ ¶ α A δµ 0 EM = α a i(γ θ)m δm

We will now explain why this basis is the preferable one. As we mentioned in the beginning of section 2.3 the purpose of superspace is to realise SUSY transformations as coordinate transformations. The fields we are interested in are functions of the coordinates of superspace: F = F(Z). When doing the SUSY a transformation they should change like δF = ε Qa F where the generator¡ ¢ of µ the transformation should satisfy the defining property {Qa, Qb} = 2 γ ab Pµ. Now this should be realised as coordinate transformations. For an arbitrary, small change of coordinates ZM → Z0M = ZM + ξ M(Z) we can easily compute the induced change in the field. We should have

F0(Z0) = F(Z) for corresponding Z and Z0. Taylor expanding gives

0 0 0 M 0 0 0 F (Z ) = F(Z ) − ξ (Z )∂M F(Z ) were we used that ZM ≈ Z0M − ξ M(Z0). If such a change of coordinates really should give a SUSY transformation we would then have to choose ξ so that a M ε Qa = −ξ ∂M, or equivalently that the action of the differential operator Qa a M M is ε Qa Z = −ξ . Notice that Q generates the coordinate transformation apart M a M from a sign, that is δZ = ε Qa Z . The sign is the same one that appears when changing from active to passive transformations, in fact that’s exactly what we are doing. So what we have to do is to find a differential operator that satisfies the defining commutation relation and then minus this operator will give the coordinate transformations that induces SUSY transformations of the fields. ¡ ¢ α On the right hand side of the {Qa, Qb} = 2 γ ab Pα the momentum operator Pα appears. We will work with the definition where Pµ = +i∂µ when acting on the fields. It’s now easy to see that the defining¡ ¢ commutation relation is µ b fulfilled by the differential operator Qa = ∂a + i γ abθ ∂µ. Explicitly we have

¡ µ ¢ ¡ µ ¢ {Qa, Q } = {∂a, ∂ } +{∂a, i γ θ ∂µ} + {i γ θ , ∂a}− b | {z b } b a =0 ¡ µ ¢ ¡ ν ¢ £ ¤ ¡ µ¢ ¡ µ¢ − γ θ γ θ ∂µ, ∂ν = i γ ∂µ + i γ ∂µ = a b | {z } ab ab =0 ¡ µ¢ = 2i γ ab∂µ

M 0M M a So to recap, if we do a coordinate transformation Z → Z = Z − ε Qa Z with the Q just defined the induced transformation on the field F(Z) → F0(Z) = a F(Z) + ε Qa F(Z) will be precisely a SUSY transformation. However, there is a complication that we haven’t mentioned yet. The fields we are interested in are not simply scalar fields on superspace, but rather things like vector fields, 1- forms etc. We expect the spinor and vector field that we had in the component formulation of super Yang-Mills to sit in the components AM(Z) of a 1-form 2.3. INTRODUCING SUPERSPACE 17 and the change of those components is not so simple under coordinate transformations as for the scalar F(Z) above. As an example let us consider a tangent M vector to superspace, V = V(Z) ∂M. For an arbitrary coordinate transforma- M 0M M 0 ∂ZN tion Z → Z = f (Z) we have ∂M → ∂M = ∂Z0M ∂N. Using the transformed vector basis and the transformed coordinates the expansion of our vector looks 0M 0 0 like V = V (Z )∂M. Since the vectorfield itself is independent of both basis and coordinates we should have, for corresponding Z and Z0,

M 0N 0 0 V (Z)∂M = V (Z )∂N ⇔ ∂ZM VM(Z)∂ = V0N(Z0) ∂ M ∂Z0N M ⇔ ∂Z0N ∂Z0N V0N(Z0) = VM(Z) = VM( f −1(Z)) ∂ZM ∂ZM

If we now assume that the transformation is small as we did earlier, f M(Z) = ZM + ξ M(Z), we can Taylor expand and get

0N N ¡ N¢ M M N V (Z) = V (Z) + ∂Mξ V (Z) − ξ ∂MV (Z) | {z } | {z } rotation term transport term

Notice that there are two terms here. One of them, the transport term, is solely due to the changing coordinates. The other, the rotation term, appears since the basis also changes. It is the first of these that would generate the desired SUSY transformation of the field under coordinate transformations generated by −Qa defined above. The rotation term however destroys this and means that we also get another undesirable term.

It is now that the new basis, introduced in (2.12), comes to the rescue. The Da only differs by a sign from the SUSY generators Qa and it is exactly this that makes them satisfy {Da, Qa} = 0. Explicitly

¡ µ ¢ ¡ µ ¢ {Da, Q } = {∂a, ∂ } +{∂a, i γ θ ∂µ} − {i γ θ , ∂a}+ b | {z b } b a =0 ¡ µ ¢ ¡ ν ¢ £ ¤ ¡ µ¢ ¡ µ¢ + γ θ γ θ ∂µ, ∂ν = i γ ∂µ − i γ ∂µ = 0 a b | {z } ab ab =0

We also trivially have [Dα, Qb] = 0. We now like to see how this basis transform when we do the SUSY coordinate transformation. The action of DA on M M the coordinates is given by the matrix we introduced earlier: DA Z = EA (Z). 18 CHAPTER 2. SYM AND BIANCHI IDENTITIES

Under the transformation this goes to

0 M a M M 0 DA(Z − ε Qa Z ) = EA (Z ) ⇔ M a M M a M (DA + δDA)(Z − ε Qa Z ) = EA (Z) − ε Qa EA (Z) ⇔ M aA a M a M δDA Z − (−1) ε DA Qa Z = ε Qa EA (Z) ⇔ δD ZM − (−1)aAεa[D , Q }ZM − εa Q D ZM = εa Q EM(Z) A A a a | A{z } a A M =EA ⇔ M a M a δDA Z = ε [Qa, DA}Z ⇔ δDA = ε [Qa, DA} = 0 In short our particular basis of tangent vectors does not change when doing coordinate transformations with Qa. This ensures that the rotation term mentioned above is eliminated and thus the components of vectors expressed in this basis really are SUSY-transformed under those changes of coordinates. All of this works the same way for 1-forms and the dual basis, EA.

We are now ready to introduce gauge theory on superspace. Just as described in the previous section we should then start with a G-bundle where the gauge group G is some Lie group. Note that this will be simply an ordinary group — no superisation. What we really are interested in a connection on this bundle. As mentioned any connection is specified by giving a g-valued 1-form, say A, where g is the Lie algebra of G. A can of course be expanded in the EA-basis and the components will then be SUSY-fields. By adding A to the exterior derivative we get a covariant exterior derivative. It will be denoted by D and is given by D = d + ∧A (note right action) on G-valued sections. We will only be dealing with g-valued fields where the action is D = d + [ , A]. Writing out A A A this in terms of our basis it takes the shape D = E DA = E DA + ∧E AA for the first case. The next step is to introduce the field strength 2-form for A: 1 A B F = 2 E E FBA. It is easy to derive the equation for F in terms of A following step 5 in the previous section (here s is group valued):

2 2 s ∧ F = D s = D(ds + s ∧ A) = |{z}d s +ds ∧ A + d(s ∧ A) + s ∧ A ∧ A = =0 = ds ∧ A + s ∧ dA − ds ∧ A + s ∧ A ∧ A = s ∧ (dA + A ∧ A) (note how F and A acts from the right). Expanding in the orthonormal basis gives: 1 EB EA F = d(EC A ) + (−1)AB EB EA A A = 2 AB C B A C A C AB B A = E E DA AC + d(E )AC + (−1) E E AB AA

C 1 A B C Here a torsion term appears due to our choice of basis. It is dE = 2 E E TBA. So the components of the ﬁeld strength are

AB C FAB = 2D[A AB} + 2(−1) A[B AA} + TAB AC (2.13) 2.4. BIANCHI IDENTITIES AND THEIR SOLUTION 19

The components of the torsion are easily calculated using

C C M C M C D M A C T = d(E ) = d(dZ EM) = dZ dEM = E ED E DA EM = A(D+M) D A M C = (−1) E E ED DA EM 1 C B(A+M) M C which gives 2 TBA = (−1) E[ADB} EM. Plugging in the values of the matri- M C ces EA and EM given earlier it turns out that all components of the torsion is zero except for α ¡ α¢ Tab = 2i γ ab (2.14) Using this value for the torsion in equation (2.13) we can write out the different superﬁeld components. They will be of use later:

Fαβ = ∂α Aβ − ∂β Aα − [Aα, Aβ] (2.15) ¡ σ¢ Fab = Da Ab + Db Aa − {Aa, Ab} + 2i γ ab Aσ (2.16)

Fαb = ∂α Ab − Db Aα − [Aα, Ab] (2.17)

By expanding D2s in terms of the basis elements we can derive an important relation for the covariant derivatives. When s is a group-valued 0-form (bosonic) we have 2 A A B A D s = D(E DAs) = E E DBDAs + d(E )DAs which leads to £ C FAB = DB,DA} + TBADC

Of course this field strength satisfy the Bianchi identity, D F = 0. As in the equation for the field strength we will find that torsion terms appears when expanding in the orthonormal basis: 1 1 1 D F = D( EA EB F ) = EA EB ECD F + EA EB ECTD F − 2 BA 2 C BA 4 CB DA 1 1 − EC EDTA EB F = EA EB EC(D F + TD F ) 4 DC BA 2 C BA CB DA So the Bianchi identity takes the form

D D[A FBC} + T[AB F|D|C} = 0 (2.18) This equation will the main topic of the whole next section.

2.4 Bianchi identities and their solution

Writing out the components of the Bianchi¡ identity¢ (2.18), recalling that the α α only non-zero torsion component is Tab = 2i γ ab, we get

D[α Fβγ] = 0 (2.19)

2D[α Fβ]c + Dc Fαβ = 0 (2.20) δ Dα Fbc + 2D(b Fc)α + 2iγbc Fδα = 0 (2.21) δ D(a Fbc) + 2iγ(ab F|δ|c) = 0 (2.22) 20 CHAPTER 2. SYM AND BIANCHI IDENTITIES

Of course these are identities so if F is constructed from A in the prescribed way they will be trivially satisfied. But on the other hand, just as is the case for ordinary YM, the Bianchi identities are equivalent to being able to write F in terms of A. So instead of working with the potential A we might as well forget all about it and consider the field strength, satisfying the Bianchi identities, as our fundamental field. This is what we are going to do now. The reason we are doing it this way is that then we do not have to worry about the gauge invariance when looking for the physical degrees of freedom. What we want to do is basically to expand F in a power series in θ and see what physical field appear at each level. Those fields will belong to representations of SO(9,1) and depend only on the x-coordinate.

In ordinary Yang-Mills theory we eliminate the unphysical degrees of freedom by imposing the Bianchi identity and the equation of motion on the field strength. Normally we would introduce an action to derive the equations of motion but here we will proceed in a completely different way. We will by hand impose another set of constraints on the field strength. Then we will show that those constraints together with the Bianchi identity implies that F contains the relevant super Yang-Mills fields and that they satisfy the correct equations of motion. This is very similar to what happens with self-dual field strengths for ordinary Yang-Mills theory in four dimensions. In this case taking the Hodge dual of a 2-form returns another 2-form so it is possible to consider fields that are self dual: F = ∗F. Since the Bianchi identity is dF = 0 such fields automatically solves d ∗ F = 0 which is the equation of motion.

Of course imposing some ad-hoc constraints with the only motivation that “it works” feels slightly awkward. Certainly by looking at the field content of the θ-expansion of the different components of F one will find a large number of fields that are irrelevant for super Yang-Mills and to end up with this theory they need to be eliminated by some additional mechanism, only imposing the ordinary equation of motion is not enough.

2.4.1 The conventional constraint

Let us start with Fab. Since it has two spinor indices we can expand it in terms of the γ-matrices. Furthermore it is symmetric and both indices have the same chirality so we only need to use γ(1) and γ(5),

i F = iγα F˜ + γα1···α5 F˜ (2.23) ab ab α 5! α1···α5

We can now proceed to expand the field in the first term in powers of θ: F˜α(x, θ) = F˜(0)(x) + θb F˜(1)(x) + ··· . The second θ-level can be decomposed in irreps as α ¡ α¢b ¡ ¢ ˜(1) c 0 ˆ ˆ α b ˆ Fαb (x) = γα b Fc(x) + Fαb. Here F is γ-traceless, that is γ a Fαb = 0, but the 0 interesting part is the spinor Fc. After all the theory we are looking for should contain a spinor field. Now let us expand Fαb in the same way ¡ ¢ ˜ α c ˜ Fαb = Fαb + γ b Fc (2.24) 2.4. BIANCHI IDENTITIES AND THEIR SOLUTION 21

˜ ˜ where Fαb is γ-traceless. Of course the zeroth θ-level of Fc(x, θ) is a spinor ﬁeld. This means that we have found two independent spinor ﬁelds, so there is at least one too many.

It is perhaps worth pointing out that both of these fields have a physical dimension. If we base our calculus of dimensions on the case of four dimensions where Fαβ has dimension −2 we can deduce that the 2-form F should be dimen- sionless and thus that [Fab] = −1 and [Fαb] = −3/2 by using that the dimension of θ and of dθ is 1/2. It then follows that the spinors defined above have di- 0 ˜ mension [Fc] = −3/2 and [Fc(x, θ = 0)] = −3/2. Going back to the action in component form (2.3) we see that −3/2 is indeed the physical dimension for the spinor field.

The simplest way to eliminate one of the spinor ﬁelds is to put the so called conventional constraint on our ﬁelds

¡ β¢ab γ Fab = 0 (2.25)

This is equivalent to F˜α = 0. So we get rid of the spinor contained at the ﬁrst 0 θ-level of this ﬁeld, Fc.

If we for a moment go back to Fs deﬁnition in terms of A ,see equation (2.16), we notice that the conventional constraint implies the following relation between the components of the connection

i ¡ ¢ i ¡ ¢ A = γ abD A − γ ab A A σ 16 σ a b 16 σ a b

Recall that the connection was defined to make a derivative that transform in a covariant way under gauge transformations possible. That is (DAV) → (DAV)g when V → Vg where g is an element of the gauge group in question. It is obvious that given a specific connection we can construct a new connection by adding a quantity that is covariant, f → g−1 f g, to the old one. Examples of quantities that transform in this way are of course the components of the field strength. Let us now assume that AA = (Aα, Aa) is som¡ arbitrary¢ connection. 0 ab From it we construct a new connection AA = (Aα + i/32 γα Fab, Aa) using the just mentioned method. Plugging in the expression for Fab in terms of Aα and Aa you can see that the new connection¡ ¢ only depends¡ ¢ on the components Aa 0 ab ab of the old connection: AA = (i/16 γα Da Ab − i/16 γα Aa Ab, Aa). When we 0 calculate the field strength for the new connection AA we find that it automatically satisfies the conventional constraint:

¡ ¢ab 0 ¡ ¢ab 0 ¡ ¢ab 0 0 ¡ ¢ab¡ α¢ 0 γσ Fab = 2 γσ Da Ab − 2 γσ Aa Ab + 2i γσ γ ab Aα = ¡ ¢ab ¡ ¢ab 0 = 2 γσ Da Ab − 2 γσ Aa Ab + 32iAσ = 0

The fact that we always can do such a redeﬁnition of the ﬁeld Aα is why this constraint is called conventional. 22 CHAPTER 2. SYM AND BIANCHI IDENTITIES

2.4.2 The dynamical constraint

Imposing only the conventional constraint on the ﬁeld strength certainly yields interesting results [7, 8], but to get plain and simple super Yang-Mills more is needed – the dynamical constraint: ¡ ¢ ρ1ρ2ρ3ρ4ρ5 ab γ Fab = 0 (2.26) ˜ This puts Fρaρ2ρ3ρ4ρ5 to zero and together with the conventional constraint elim- inates all of Fab. It is often usefull to instead consider the constraint in the form (5) cd (5) γab γ(5) Fcd = 0. The difference is subtle but in this form the self-duality of γ (as explained in appendix B) can be used to make simpliﬁcations in some calculations. We will refer to this form of the constraint as the “weak” one.

There are ways to explain this constraint as being due to “integrability on light- like lines”, see [9], but we will not discuss this further in this thesis.

2.4.3 Solving the Bianchi identities

We will now try to ﬁnd out exactly what our constraints (2.25) and (2.26), together giving

Fab = 0 (2.27) implies for the ﬁeldstrength components when combined with the Bianchi identities. Plugging in Fab = 0 in (2.19) to (2.22) yields

D[α Fβγ] = 0

2D[α Fβ]c + Dc Fαβ = 0 δ (2.28) 2D(b Fc)α + 2iγbc Fδα = 0 δ γ (ab F|δ|c) = 0

The fourth identity

Let us ﬁrst concentrate on the last one¡ of these.¢ We have already noted that we ˜ c ˜ can do the decomposition Fαb = Fαb + γα b Fc, which when combined with the last line of (2.28) gives ¡ ¢ ¡ ¢ ¡ ¢ δ ˜ δ d ˜ γ (ab F|δ|c) + γ (ab γ|δ| c) Fd = 0 (2.29)

d In the second term we recognise the combination of γ-matrices denoted Qabc in section 2.2.1. We also recall that, as shown in appendix B, Q is identically zero 3 in D = 10. Thus the last Bianchi identity really says that F˜δc = 0 or equivalently ¡ ¢ c ˜ Fαb = γα b Fc (2.30) ¡ ¢ 3You can see that this is true by contracting equation (2.29) with γβ ab. You also have to use that F˜ is γ-traceless. 2.4. BIANCHI IDENTITIES AND THEIR SOLUTION 23

The third identity

Let us now turn to the third equation in (2.28). Recalling Fbα = −Fαb and using (2.30) we get ¡ ¢ ¡ ¢ d ˜ δ −2D(b γ|α| c) Fd + 2i γ bc Fδα = 0 (2.31) The symmetry in (bc) means that those two indices can be expanded in terms of γ(1) and γ(5). Since these are linearly independent the coefﬁcient of each has to be zero. The coefﬁcients are proportional to the contraction with the respective γ-matrix. For γ(1) we get

¡ ¢bd −2 γβ γα Db F˜d + 32iFβα = 0 ⇔ bd ¡ ¢bd ηβαC Db F˜d + γβα Db F˜d − 16iFβα = 0

Here the middle and last term are antisymmetric in β and α while the ﬁrst term is symmetric. They thus have to be zero separately. The symmetric part yields

bd C Db F˜d = 0 (2.32) and the antisymmetric part gives

i ¡ ¢ F = − γ abD F˜ (2.33) αβ 16 αβ a b

We also have to take the contraction of equation (2.31) with γ(5). Because of the orthogonality of the different γ-matrices with respect to the trace only the ﬁrst term in (2.31) survives to give ¡ ¢ ρ1ρ2ρ3ρ4ρ5 bd γ γα Db F˜d = 0 (2.34)

To proceed any further we will once again have to do an expansion in terms of γ-matrices, though this time we will have to be slightly careful. If we assume that the index on the basis form Ea is an anti-Weyl index then obviously the index on the component Da has to be Weyl. Similarly both indices on Fab should be Weyl, but when doing the expansion in equation (2.30) F˜c must have an anti- (1) Weyl index since γ has two indices of the same type. The two indices in Db F˜d thus have opposite chirality (but no deﬁnite symmetry) so we can expand them as ¡ ¢ ¡ ¢ F˜ = C Λ(0) + γαβ Λ(2) + γσ1σ2σ3σ4 Λ(4) Db d bd bd αβ bd σ1σ2σ3σ4 Now we note that equation (2.32) that we just derived from the third Bianchi identity puts Λ(0) to zero. Similarly equation (2.33) implies that

i F = − tr(γ γαβ )Λ(2) = 2iΛ(2) στ 16 στ αβ στ Using this expansion in equation (2.34) we ﬁnd

i − tr(γρ1···ρ5 γ γ )Fβδ − tr(γρ1···ρ5 γ γ )Λ(4)σ1···σ4 = 0 2 α βδ α σ1···σ4 24 CHAPTER 2. SYM AND BIANCHI IDENTITIES where we for convenience raised and lowered some indices. Utilizing the identity γαγβδ = γαβδ + ηα[β γδ] we see that the first term vanish since it consists of traces of products of γ-matrices with different number of indices. For the second term we have the analogous identity γαγσ1···σ4 = γασ1···σ4 + 4ηα[σ1 γσ2σ3σ4]. Here the first γ has five indices so it survives when traced together with the γρ1···ρ5 giving 16 · 5!δρ1···ρ5 Λ(4)σ1···σ4 = 0 ασ1···σ4

By letting ρ1 = α being equal to different values we see that this is the same as Λ(4) saying that σ1···σ4 = 0.

Finally we can conclude that the third Bianchi identity is equivalent to

i ¡ ¢ D F˜ = − γαβ F (2.35) a b 2 ab αβ

The second identity

Using the constraint (2.30) in the second Bianchi identity in (2.28) gives us ¡ ¢ d ˜ Dc Fαβ + 2 γ[β |c| Dα] Fd = 0 (2.36) To see what use this equation has let us study the field content of our superfields. As we have seen the only superfields we have to care about are Fαβ and F˜c. Fαβ contains at the zeroth θ-level a 2-form field. This is of course the ordinary x-space field strength. We have no control over the higher θ-levels at this point. The θ = 0 component of F˜c is of course the ordinary spinor in super Yang-Mills. The fields at the first θ-level of F˜c is equal to the fields at the zeroth b θ-level of Db F˜c since the θ independent part of Db reduces the power of θ by one. Looking at equation (2.35) we can thus conclude that at the first θ-level of F˜c we only find the physical 2-form and no new independent fields. We also see that all the higher θ-levels of F˜c are given by the higher levels of Fαβ . The fields at the second θ-level of F˜c will for instance be those at the first θ-level of Fαβ . But we can also study the second θ-level of F˜c by looking at the zeroth θ-level of DaDb F˜c. n o i ¡ ¢ D D F˜ = (2.35) = − γαβ D F a b c 2 bc a αβ Now we can use (2.36) to rewrite this as

¡ αβ ¢ ¡ ¢ d DaDb F˜c = i γ bc γβ a Dα F˜d (2.37)

So at the second level of F˜c there are no new fields at all, only a derivative of the physical spinor. Continuing taking more spinorial derivatives we would, at the higher levels, alternatingly find space-derivatives of the spinor field or the 2-form field. That is, the only independent fields contained in F˜c are the 2-form and the spinor of super Yang-Mills. And as noted above there can be nothing more at the higher levels of Fαβ either. We will denote the zeroth θ level of the two superfields by fαβ and χc respectively. The structure of the rest of the superfields is then as depicted in table 2.2. 2.4. BIANCHI IDENTITIES AND THEIR SOLUTION 25

θ0 θ1 θ2 θ3 ...

Fαβ fαβ ∇αχc ∇γ fαβ ... F˜c χc fαβ ∇αχc ∇γ fαβ ...

Table 2.2: The ﬁeld content of the superﬁelds at different θ-levels, when the constraints and three of the Bianchi identities are imposed.

The ﬁrst identity

The first equation in (2.28) has as its θ = 0 component the ordinary Bianchi identity for the physical 2-form field. This is of course very good, since to be physical fαβ should satisfy a Bianchi identity. It is not clear, however, if the higher θ-levels imply something more. To find out we begin by noting that this Bianchi identity is equivalent to

Dα Fβγ + Dβ Fγα + Dγ Fαβ = 0 When applying one fermionic derivative we get terms like

DcDα Fβγ = {Dc,Dα}Fβγ + DαDc Fβγ If we recall that F is an endomorphism-valued 2-form and that A in the covariant derivative thus acts on both sides we see that we get n o ³ ´ ¡ ¢ d {Dc,Dα}Fβγ = Fβγ Fcα − Fcα Fβγ = (2.30) = − γα c Fβγ F˜d − F˜d Fβγ

For the second term we ﬁrst use equation (2.36) to get ¡ ¢ ¡ ¢ ¡ ¢ d ˜ d ˜ d ˜ DαDc Fβγ = −2 γ[γ c D|α|Dβ] Fd = − γγ c DαDβ Fd + γβ c DαDγ Fd When adding up the three permutations of the indices in the Bianchi identity we will get 12 terms that can be collected in three groups of four like

Dα Fβγ + permutations = ³ ´ ¡ ¢ d = − γα c Fβγ F˜d − F˜d Fβγ + DβDγ F˜d − DγDβ F˜d + ··· = ³ ´ ¡ ¢ d £ ¤ = − γα c Fβγ F˜d − F˜d Fβγ + Dβ ,Dγ F˜d + ··· = ³ ´ ¡ ¢ d = − γα c Fβγ F˜d − F˜d Fβγ + F˜d Fβγ − Fβγ F˜d + ··· = 0

This means that the first θ-level of the first Bianchi identity is automatically satisfied due to the other Bianchi identities. It does not gives any new information about the fields. Taking further fermionic derivatives we would find that even higher levels are also trivially true. Thus the only consequence of D[α Fβγ] = 0 is that the physical 2-form fβγ has to satisfy the ordinary Bianchi identity.

The equations of motion

So far, we have shown that by starting with a super field strength and imposing the conventional and the dynamical constraint as well as the super Bianchi 26 CHAPTER 2. SYM AND BIANCHI IDENTITIES identity, we end up with a theory containing only a 2-form field satisfying the normal Bianchi identity and a spinor field. These are just the building blocks needed for super Yang-Mills. The only step left is to write down the equations of motion. What is remarkable of the superspace formulation of SYM is that we have already done it! As we will now show the relevant equations of motion has already been imposed, although not explicitly, by the constraints and the super Bianchi identity.

The crucial step is to notice that in the same way as the SUSY generators Q satisfy¡ their¢ commutation relation, the covariant derivatives satisfy {Da,Db} = γ −2i γ abDγ . We recognise in the right hand side the Dirac operator which means that we have

¡ ¢ i n o γα bD F˜ = {D ,D }F˜b = iCbcD D F˜ = (2.37) = a α b 2 a b (a b) c ¡ ¢ ¡ ¢ 1 ¡ ¢ = −Cbc γαβ γ dD F˜ = tr(γαβ ) γ dD F˜ + c(b β a) α d 2 | {z } β a α d =0

1¡ αβ ¢ d 9¡ α¢ d + γ γβ a Dα F˜ = γ a Dα F˜ 2| {z } d 2 d =9γα As the right hand side is proportional to the left hand side (and the constant of proportionality is not 1) F˜c must satisfy the Dirac equation. This of course also applies to its lowest component, the spinor χc:

¡ α¢ c γ a ∇αχc = 0 (2.38)

At the second level of F˜c we ﬁnd fαβ . To ﬁnd out what the Dirac equation means for fαβ we apply one fermionic derivative

³ ´ ¡ α¢ b ¡ α¢ b£ ¤ ¡ α¢ b 0 = Dc γ a Dα F˜ = γ a Dc,Dα F˜ + γ a DαDc F˜ b | {z }b | {z }b I II

In term (I) we have to remember that [Dc,Dα]F˜b = F˜b Fcα + Fcα F˜b (plus in front ˜ of¡ the¢ second term since both Fb and Fcα are fermionic). Recalling that Fcα = d − γα c F˜d we get

¡ α¢ b¡ ¢ d I = − γ a γα c {F˜b, F˜d}

Using equation (2.35) the second term becomes

i ¡ ¢ ¡ ¢ i ¡ ¢ II = − γ b γβγ D α F = − γ γβγ D α F 2 α a cb βγ 2 α ac βγ 2.5. GAUGE AND SUSY-TRANSFORMATIONS IN SUPERSPACE 27

¡ ¢ ac Upon combining the two terms and contracting with γσ you get

¡ ¢ac¡ ¢ γσ I + II = 0 ⇔

¡ α ¢bd i βγ α γ γσγα {F˜ , F˜ } + tr(γσγαγ )D Fβγ = 0 | {z } b d 2 =−8γσ ⇔ ¡ ¢ i i −8 γ bd{F˜ , F˜ } + tr(γ γβγ )D α F + tr(η γβγ )D α F = 0 σ b d 2 | σα{z } βγ 2 | σα{z } βγ βγ =0 =2·16δσα ⇔ i ¡ ¢ D α F = γ ab{F˜ , F˜ } (2.39) ασ 2 σ a b

This is the Yang-Mills equation for Fαβ with a current constructed from the spinor F˜c. The same equation is also satisﬁed by the physical ﬁelds fαβ and χc.

i ¡ ¢ ∇α f = γ ab{χ , χ } ασ 2 σ a b

This is of course an exact match with the equation given when studying the component formulation.

2.5 Gauge and SUSY-transformations in superspace

We would now like to show that the supersymmetry transformations on the superfields generate exactly the same transformations for the component fields as those given in chapter 2.2. To this end we will first have to reintroduce the potential AA and look at its θ-expansion.

Let us begin with recalling that the conventional constraint relates the vector and spinor part of AA through

i ¡ ¢ ¡ ¢ A = γ ab D A − A A (2.40) α 16 α a b a b

This mean that we can restrict our attention to Aa since all component fields will be contained in it. In particular the zeroth θ-level of Aα appears at the first level of Aa. This field is of course the ordinary gauge potential that gives the field strength fαβ . We will call it aα.

When working with the ﬁeld strength we had to consider the Bianchi identities. For the potential we instead have the gauge invariance. It is given by

δAA = DAΛ 28 CHAPTER 2. SYM AND BIANCHI IDENTITIES where the gauge parameter Λ is a Lie-algebra valued bosonic superﬁeld. For the spinor component this gives ¡ ¢ Λ α Λ Λ δAa = ∂a − i θγ a∂α + [ , Aa]

(0) b (1) b1 b2 (2) If we expand Aa in powers of θ like Aa = Aa + θ A + θ θ A + ··· and ab ab1b2 (1) (2) similarly for Λ, Λ = Λ(0) + θbΛ + θb1 θb2 Λ + ··· , we ﬁnd that the gauge vari- b b1b2 ation of the different components becomes

(0) Λ(1) Λ(0) (0) δAa = a + [ , Aa ] ¡ ¢ (1) Λ(2) α Λ(0) Λ(1) (0) Λ(0) (1) δAab = 2 ab − i γ ab∂α + [ b , Aa ] + [ , Aab ] (2) (3) ¡ α¢ (1) (2) (0) (1) (1) (0) (2) δA = 3Λ − i γ ∂αΛ + [Λ , A ] − [Λ , A ] + [Λ , A ] ab1b2 b1b2a ab1 b2 b1b2 a b1 ab2 ab1b2 ··· (2.41)

(1) (0) (0) (0) It is now apparent that by choosing Λ = −Aa − [Λ , Aa ] we can completely (0) (0) gauge Aa to zero. This is a good thing since Aa is a spinor field with the (0) unphysical dimension [Aa ] = −1/2. From now on we will assume that we are (1) in this specific gauge. The next component is Aab which can be expanded in terms of a γ(1), a γ(3) and a γ(5)-matrix since the spinor indices have the same chirality and no specific symmetry. Due to the anticommuting property of the Λ(2) (3) θ, ab will be proportional to only γ . We thus see from (2.41) that by choosing Λ(2) (3) (1) ab properly the γ -part of Aab can be gauged away.

Thus far we have not mentioned the dynamical constraint. It is easy to see that for the potential it implies

¡ ¢ ¡ (5)¢ab ¡ ¢ ¡ (5)¢ab ¡ α¢ γ(5) cd γ Fab = 0 ⇔ γ(5) cd γ (∂a Ab − i θγ a∂α Ab − Aa Ab) = 0 Note that we here are using the “weaker” form with the extra γ(5)-matrix. This is essential for the following calculations to come out right. Each θ-level of the equation has to vanish separately. Since A(0) is zero the lowest level (the θ0- ¡ ¢ a (5) ab (1) (5) (1) level) says that γ Aba = 0. This will eliminate the γ -part of Aab leaving only the vector part proportional to γ(1).

From equation (2.41) we see that we still haven’t utilised the terms containing Λ(0) to do gauge transformations with (the term with Λ(1) doesn’t enter since ¡ a ¢ (0) (1) (1) α Aa is zero). If we write what is left of Aab as Aab = −i γ abaα we see that this new vector ﬁeld gauge transform as

(0) (0) δaα = ∂αΛ + [Λ , aα]

This is immediately recognised as the gauge transformation of a gauge potential. Furthermore we see from the conventional constraint (2.40) that in our (0) gauge Aα = aα.

Proceeding with the higher θ-levels of the dynamical constraint and the gauge transformation we would ﬁnd that we can choose a gauge such that the spinor 2.5. GAUGE AND SUSY-TRANSFORMATIONS IN SUPERSPACE 29 gauge potential takes the form ¡ ¢ i ¡ ¢¡ ¢ α σ1σ2σ3 Aa = −i θγ aaα − θγ θ γσ1σ2σ3 χ a+ 36 (2.42) 1 ¡ ¢ ¡ ¢¡ ¢ + γσ1σ2σ3σ4σ5 θ θγ θ ∂ a + a a + ··· 24 a σ1σ2σ3 σ4 σ5 σ5 σ4 It is the coefﬁcient of the third term that requires the “weaker” form of the dynamical constraint. This is because we have to use the duality of the γs to convert a γ(7) to a γ(3) when calculating it. Doing this we get an ε-tensor that can then be absorbed in the extra γ(5)-factor due to its self-duality. The expression above leads to the following vector part: ¡ ¢ Aα = aα − θγαχ + ···

We will now go back to the field strength. We have already indicated the structure of the θ-expansion in table 2.2. Now we would like to have this expansion in its exact form with the correct coefficients etc. To get this you have to use the equations for the fermionic covariant derivatives acting on F˜c derived from the Bianchi identities in the last section. When taking θ = 0 in these expression we will get equations for the different levels of F˜c. Since Da contains Aa we will have use for the expansion in equation (2.42). The first θ-level is easy to get using equation (2.35) while the second one requires more involved γ-matrix manipulations. First we have to put θ = 0 in equation (2.37), taking care not to forget any of the terms on the left hand side: ¡ ¢ ¡ ¢ ¡ ¢ β ˜(2) αβ d −i γ ab∇β χc + 2Fcba = i γ bc γβ a ∇αχd

˜ ˜ b a ˜(2) where the second level of Fc is given by Fc = ... + θ θ Fcba + .... Now the trick is to expand¡ the¢ a¡and b indices¢ ¡ on the¢ right hand side in terms of the three σ σ1···σ3 σ1···σ5 (5) γ-matrices γ ab, γ ab and γ ab. When contracting with the γ we will get two terms containing a γ(6) and a γ(4) respectively. Using the duality (6) (4) from appendix B we can then rewrite the γ as ε(10)γ . The Levi-Civita ε can (5) then be gotten rid¡ of using¢ ¡ the self-duality¢ of the γ so that we have two terms σ1···σ5 d proportional to γ ab γσ1···σ4 c ∇σ5 χd. Doing the calculation carefully one luckily ﬁnds that these two terms cancel.

(1) d Proceeding¡ ¢ with the γ -term we get one term with just a δc and one term with d a γασ c . The second of these can be rewritten as ηασ − γσγα. Now we see that when contracting this with ∇αχd the last of those terms will yield the Dirac operator and since χ satisfy the Dirac equation it¡ disappears.¢ The result is that (1) α the γ -term on the right hand side becomes −i γ ab∇αχc exactly matching (1) ˜(2) (3) the γ -term on the left. This means that Fcba is equal to the γ part of the right side. This is not surprising since the contractions with the θs means that a and b are antisymmetric. Explicitly we have 5i ¡ ¢ ¡ ¢ i ¡ ¢ ¡ ¢ F˜(2) = γ γσ1σ2 d∇αχ + γσ1σ2σ3 γ d∇αχ cba 64 ασ1σ2 ab c d 32 ab ασ1σ2σ3 c d Here we can make a simpliﬁcation by using the Dirac equation in the same (1) α way as when calculating the γ -term. When γ ∇αχ = 0 one can show that 30 CHAPTER 2. SYM AND BIANCHI IDENTITIES

α the relation γασ1σ2σ3 ∇ χ = 3γ[σ2σ3 ∇σ1]χ is true. This means that the two terms ˜(2) ˜ in Fcba combine to one. The full expression for Fc to second order is now given by i ¡ ¢ i ¡ ¢¡ ¢ F˜ = χ − θγαβ f − θγσ1σ2σ3 θ γ ∇ χ + ··· (2.43) c c 2 c αβ 8 σ2σ3 σ1 c

Having the exact form of F˜c up to the second level allows us to calculate Fαβ to the first level using equation (2.33). For the zeroth level we of course only get fαβ while the first level will have a contribution from both the zeroth and second level of F˜c. From the zeroth level when the θ-part of Da acts on χb as (1) well as from the anticommutator between χb and Aac and from the second level due to the ∂a in Da. Specifically these contributions are i ¡ ¢ h i ¡ ¢ ¡ ¢ ab σ1σ1σ3 Fαβ = · · · − γαβ − γ θ a γσ2σ3 ∇σ1 χ b− 16 4 i ¡ σ¢ ¡ σ¢ − i θγ a∂σχb − {χb, −i θγ aaσ} + ··· = 1 ¡ ¢ 1 ¡ ¢ = ··· + θγσ1σ2σ3 γ γ ∇ χ − θγσγ ∇ χ + ··· 64 αβ σ2σ3 σ1 16 αβ σ

Rewriting the products of the γ-matrices while utilizing that χc satisﬁes the Dirac equation like above we ﬁnd that ¡ ¢ Fαβ = fαβ − 2 θγ[β∇α]χ + ··· (2.44)

We are now able to consider the supersymmetry transformations. As was ex- a plained these are given by the operator Qa. Acting with ε Qa on a field would give the SUSY transformation with parameter εa of the field. By looking at the different θ-levels you can then deduce the transformation properties of the component field. Let us do this first for F˜c given in (2.43):

¡ ¢ ¡ a ¡ α ¢ ¢ δε F˜c = εQ F˜c = ε ∂a + i εγ θ ∂α F˜c = i ¡ ¢ ¡ ¢ i ¡ ¢¡ ¢ = − εγαβ f + i εγαθ ∂ χ − εγσ1σ2σ3 θ γ ∇ χ + ··· 2 c αβ α c 4 σ2σ3 σ1 c

i αβ From this we immediately get that δεχc = − 2 (εγ )c fαβ which is exactly the same transformation as the one deﬁned in equation (2.5). To get the variation (2) of fαβ we could remove the θ and project with a γ -matrix but it is simpler to instead consider the variation of Fαβ . This gives ¡ ¢ δε Fαβ = −2 εγ[β∇α]χ + ···

We don’t actually need¡ to go further¢ than this ﬁrst term because it directly implies that δε fαβ = −2 εγ[β∇α]χ which is the desired transformation from equation (2.5). The same transformation would have appeared if we had pro- ceeded with the next θ-level of δε F˜c above, but only after slightly harder work.

If we had wished we could have applied the SUSY transformation directly to the expansions of the gauge ﬁelds AA that we derived earlier. The problem with this though is that it does yield the SUSY transformation of the component 2.5. GAUGE AND SUSY-TRANSFORMATIONS IN SUPERSPACE 31

ﬁelds only up to a gauge transformation. To get the same result as above we should combine a SUSY transformation of AA with a gauge transformation of AA. This is of course messier than working directly with the gauge covariant ﬁeld strengths. 32 CHAPTER 2. SYM AND BIANCHI IDENTITIES Chapter 3

Super Yang-Mills in D=10 using pure spinors

In this chapter we will show how to formulate super Yang-Mills in yet another way. Here the equations of motion and gauge variations will arise as a consequence of demanding that the physical fields belong to the cohomology of a certain operator, denoted Q. This operator will be constructed using a certain kind of spinors called pure spinors that we will define shortly. The pure spinor approach to SYM was introduced by Berkovits [10, 11, 12]. The original purpose was to covariantly quantise the Green-Schwarz superstring, but it is also applicable to our case. Besides giving us another way to formulate SYM we will also see that there is something more to this approach – we will find additional fields.

3.1 The Pure Spinor

To do super Yang-Mills the Berkovits way we have to work with the so called pure spinor. A pure spinor, λc, is in D = 10 a Weyl spinor satisfying the condition ¡ ¢ λγσλ = 0 (3.1)

For this to make sense λc has to be a bosonic parameter rather than fermionic since otherwise the pure spinor constraint would be trivially true.

Note that because the constraint is completely covariant this deﬁnes, in a sense, a representation of the Lorentz group – making a Lorentz transformation of a pure spinor will yield another pure spinor — the pure spinors does however not form a module. It can be shown that the pure spinor constraint reduces the number of independent components to 11. To show this it is simplest to break the manifest Lorentz symmetry and choose another basis for the γ-matrices.

33 34 CHAPTER 3. SYM USING PURE SPINORS

We will however not have any need for this result so this calculation has been relegated to appendix C.

The pure spinor was ﬁrst introduced by Cartan in a slightly different form and for other purposes [13]. There are earlier attempts to utilize pure spinors when trying to formulate a theory of SYM that closes off-shell [14].

3.2 Q and its cohomology

Let us deﬁne the operator Q by

a a α Q = λ Da = λ (∂a − i (γ θ)a∂α) (3.2) where λ is a pure spinor. This Q should not be confused with the SUSY generator Qa. It is easy to show that Q is nilpotent when acting on an arbitrary ﬁeld:

2 ¡ a α ¢¡ b β ¢ Q Ψ = λ ∂a − i(λγ θ)∂α λ ∂b − i(λγ θ)∂β Ψ = a b ¡ β ¢ a¡ β ¢ b¡ α ¢ = λ λ ∂a∂ Ψ − i λγ λ ∂βΨ + iλ λγ θ ∂β ∂aΨ − iλ λγ θ ∂α∂ Ψ − | {z b } | {z } | {z } | {z b } I II III IV ¡ α ¢¡ β ¢ − i λγ θ λγ θ ∂α∂βΨ = 0 | {z } V Here I is zero because the product of the two pure spinors is symmetric while the two derivatives are anticommuting, II is zero because of the pure spinor constraint, III and IV cancel each other and V vanish because of the opposite symmetry in the α and β indices on the γ-matrices compared to the derivatives.

One of the things that are interesting to calculate for a nilpotent operator is its cohomology. For a nilpotent operator acting on a vector space, Q : Λ → Λ, the cohomology is the quotient vector space

Ker(Q) H(Q) = Im(Q) or in words the Q-closed elements that are not Q-exact. In our case the space Λ is the space of functions of the variables xα, θa and λa. More speciﬁcally we will restrict ourselves to functions that can be expanded in a power series in positive powers of θa and λa. Because of the antisymmetry the highest power of θ is 16. A general function in the space Λ can thus be expanded as

A(x, θ, λ) = A(0,0) + ··· +θb1 ··· θb16 A(0,16) + b1···b16 + λa1 A(1,0) + ··· +λa1 θb1 ··· θb16 A(1,16) + a1 a1b1···b16 + λa1 λa2 A(2,0) + ··· +λa1 λa2 θb1 ··· θb16 A(2,16) + a1a2 a1a2b1···b16 . . 3.2. Q AND ITS COHOMOLOGY 35

Notice that this power expansion introduces a natural bigrading of the vector (i,j) (i,j) space: Λ = ⊕i,jΛ , where Λ is the subspace of functions with θ-power j and λ-power i. The operator Q has one term that increases the θ-level by one and one term that decreases it. We will denote the two terms with Q+ and Q− respectively. Both of those adds one to the λ-level. Thus the total Q acts on the subspaces like Q : Λ(i,j) → Λ(i+1,j−1) ⊕ Λ(i+1,j+1).

Since Q does not mix the different λ-levels we can consider the cohomology on each subspace of a speciﬁc λ-power separately. If we deﬁne the subspace L Λ(i) 16 Λ(i,j) Λ(i) = j=1 and let Q(i) be the restriction of Q to the cohomology (i) for a given power i of λ is given by H (Q) = H(Q(i)) = Ker(Q(i))/Im(Qi−1). The complete cohomology can be reconstructed from the subcohomologies: L (i) H(Q) = i H (Q).

We will now start our calculation of the cohomology of Q by considering only the ﬁrst λ-level, H(1)(Q). In this case the space of ﬁelds is given by functions on a the form λ Aa(x, θ) where Aa can be expanded like

(0) b1 (1) b1 b16 (16) Aa(x, θ) = a (x) + θ a (x) + ··· + θ ··· θ a (x) a ab1 ab1···b16 If a particular field configuration is to belong to the cohomology it must sat- a c a c a isfy the equation 0 = Qλ Aa = λ Dcλ Aa = λ λ Dc Aa where the last equality follows since Dc contains no derivatives with respect to λa. Now the two indices on the λs can be expanded in γ-matrices. Since the pure spinors are commuting the only possible terms are the one proportional to γ(1) and the one proportional to γ(5). However due to the pure spinor constraint the first one of these is killed. We thus have 1 ¡ ¢ ¡ ¢ λcλa = γ ca λγσ1···σ5 λ 16 · 5! σ1···σ5 ¡ ¢ (5) ab and the equation for the Q-closed elements is equivalent to γ Da Ab = 0. This equation will be referred to as the equation of motion. We must of course not forget that in the cohomology we have to identify fields only differing by something Q-exact. We can interpret this as the following gauge-invariance: a a (0) λ δAa = QΩ = λ DaΩ. Here Ω ∈ Λ and can be expanded like

Ω(x, θ) = ω(0)(x) + θb1 ω(1)(x) + ··· + θb1 ··· θb16 ω(16) (x) b1 b1···b16 To proceed we will look at what our two equations means level by level in θ. The lowest level of the gauge invariance says that

(0) (1) δaa = ωa

(1) (0) It is apparent that by choosing ωa appropriately we can always gauge aa to zero and the superﬁelds in the cohomology will thus not contain this component. Moving on to the next level we have ³ ´ a b (1) a b (2) ¡ σ¢ (0) λ θ δaab = λ θ 2ωab − i γ ab∂σω

Now it’s time to do some γ-expansion. There is no symmetry between a and b (1) (1) (3) (5) (2) in aba so the expansion will contains term proportional to γ , γ and γ . ωab 36 CHAPTER 3. SYM USING PURE SPINORS on the other hand is completely antisymmetric so here the only term is the one (3) (2) (3) (1) with γ . Now we see that we can use ωab to gauge away the γ -part of aba . If we now turn to the zeroth level of the equation of motion we see that this says that ¡ (5)¢ab (1) γ aba = 0 This obviously tells us that also the γ(5)-term is zero. We conclude that a(1) must ¡ ¢ ab (1) α take the form aab = −i γ abaα where the constant is chosen with the beneﬁt of hindsight. Furthermore the term containing ω(0) in the gauge-variance means (0) that δaα = ∂αω . This is certainly starting to look familiar, but there is more to come!

Moving up to the second θ-level the gauge invariance is ³ ¡ ¢ ´ a b1 b2 (2) a b1 b2 (3) σ (1) λ θ θ δa = λ θ θ 3ω − i γ ∂σω (3.3) ab1b2 ab1b2 ab1 b2

We will have to expand a(2) in irreps. Of course due to the antisymmetry the ab1b2 ¡ ¢ (3) (2) σ σ σ b indices must be sitting on a γ , a = γ 1 2 3 kaσ σ σ , leaving only the ab1b2 b1b2 1 2 3 expansion of the 3-form- spinor k . Earlier we have used that a vector- aσ1σ2σ3 ¡ ¢ b spinor can be expanded like Vaα = V˜ aα + γα abs , where the tilde denotes γ-tracelessness and sb is a cospinor. This type of expansion can be extended to other tensor-spinors. A given n-form-spinor can be expanded in one γ- tracelessness part and terms with γ(1) up to γ(n)-matrices. These matrices will be multiplied with appropriate γ-traceless tensor-spinors with enough vector indices to give a total of n in each term. To keep the correct symmetry all vector indices must be antisymmetrised. As an example consider¡ a 2-form-¢ spinor V . To begin with we can do the expansion V = V˜ + γ T + ¡ ¢ αβa αβa αβa [α β] a b b γαβ S a, where Tβ is a vector-cospinor and S is a spinor. Using the decomposition of a vector-cospinor into a traceless part¡ and a¢ spinor¡ part¢ that can be b ˜ ˜ absorbed into the S term we get Vαβa = Vαβa + γ[αTβ] a + γαβ S a. It is easy to convince yourself that if you try to include terms with higher γ-matrices than γ(2) they can always be rewritten as a sum of terms like the ones we already have. The case of a 3-form-spinor is now straightforward: ¡ ¢ ¡ ¢ ¡ ¢ k = k˜ + γ s˜1 + γ s˜2 + γ s aσ1σ2σ3 aσ1σ2σ3 [σ1 σ2σ3] a [σ1σ2 σ3] a σ1σ2σ3 a Here k˜ is a 3-form-spinor, s˜1 is a 2-form-cospinor, s˜2 is a vector-spinor and s is a cospinor. The ﬁrst three all have vanishing γ-trace.

Returning to the gauge variation we have ω(3) on the right hand side with ab1b2 all indices antisymmetric. Of course we can still do the same expansion as for a(2) in the b indices ab1b2 ¡ ¢ h ¡ ¢ ¡ ¢ ¡ ¢ i (3) σ1σ2σ3 1 2 ω = γ m˜ aσ σ σ + γ r˜ + γ r˜ + γσ σ σ r ab1b2 b1b2 1 2 3 [σ1 σ2σ3] a [σ1σ2 σ3] a 1 2 3 a

Since each term belongs to a separate irrep the additional antisymmetry of the a index has to be present¡ in¢ each one of the terms by itself. To investigate ab1 this we project with γτ1τ2τ3 . When doing this we get several products of γ-matrices, one for each term, that have to be calculated. For doing this the 3.2. Q AND ITS COHOMOLOGY 37

Mathematica package GAMMA [15] is recommended. This will lead to the following identities for the different terms:

¡ ¢ 1¡ ¢ σ1σ2σ3 τ1τ2τ3 γ b1b2 m˜ aσ1σ2σ3 = γ b1am˜ b2τ1τ2τ3 ¡ ¢ ¡ ¢ 2 ¡ ¢ ¡ ¢ σ1σ2σ3 1 τ1τ2τ3 1 γ b b γ[σ r˜ = − γ b a γ[τ r˜ 1 2 1 σ2σ3] a 1 1 τ2τ3 b2 ¡ ¢ ¡ ¢ 1¡ ¢ ¡ ¢ σ1σ2σ3 2 τ1τ2τ3 2 γ b1b2 γ[σ1σ2 r˜σ ] = γ b1a γ[τ1τ2 r˜τ ] 3 a 2 3 b2 ¡ ¢ ¡ ¢ 1¡ ¢ ¡ ¢ σ1σ2σ3 τ1τ2τ3 γ b b γσ σ σ r = γ b a γτ τ τ r 1 2 1 2 3 a 2 1 1 2 3 b2 Of these only the second has the desired antisymmetry. Thus the traceless 2- form-spinor is the only field in the expansion that isn’t necessarily zero: ω(3) = ¡ ¢ ¡ ¢ ab1b2 γσ1σ2σ3 γ r˜1 . b1b2 [σ1 σ1σ2] a After this long exercise in γ-matrix expansions we can finally return to the gauge variation. Comparing the expansions on both sides it is obviously always possible to choose r˜1 so that the corresponding term in a(2) is gauged aσ1σ2 ab1b2 away. I.e. s˜1 = 0. To get information on the other irreps in a(2) we will now aσ1σ2 ab1b2 look at the equation of motion at the first θ-level. It says that ¡ ¢ ³ ¡ ¢ ´ (5) a1a2 b1 (2) α (0) γ θ 2a − i γ ∂αa = 0 (3.4) a2a1b1 a1b1 a2

(0) Of course as we already have pointed out aa2 can be gauged to zero so we can disregard the second term. When contracting the γ(5) with the expansion of a(2) we get the following different terms a2a1b1 ¡ ¢ ¡ ¢ σ1σ2σ3 τ1τ2τ3τ4τ5 ˜ [τ1τ2 ˜τ3τ4τ5] γ γ kσ1σ2σ3 b = −480 γ k b ¡ ¢2 ¡ ¢ 2 σ1σ2σ3 τ1τ2τ3τ4τ5 2 2 γ γ γσ1σ2 s˜σ = 80 γ[τ1τ2τ3τ4 s˜τ ] ¡ 3¢b2 5 b2 σ1σ2σ3 τ1τ2τ3τ4τ5 γ γ γσ σ σ s = 0 1 2 3 b2 that we have calculated using GAMMA. These equations means that due to ˜ 2 the Q-closedness kaσ1σ2σ3 and s˜aσ both have to be zero. They cannot cancel each other since they belong to different irreps. All that remains at the second θ- ¡ ¢ ¡ ¢ (2) i σ σ σ level of Aa is thus a = − γ 1 2 3 γσ σ σ χ where we have rescaled ab1b2 36 b1b2 1 2 3 a the cospinor in a convenient way.

The reader will by now have realised that the field Aa belonging to the Q- cohomology contains the same component fields as the gauge field Aa we got from the constrained superspace in the last chapter. To investigate this further we would now like to move on to the higher θ-levels, but the expansion of fields with more and more spinor indices would become prohibitively cum- bersome. There would be too many terms living in different irreps making us loose track of what we’re doing. Fortunately we can simplify the expansion process by using the computer program LiE [16]. The equations we are deal- a1···am ing with typically have the following spinor index structure A Ba1···am = 0 (there might be uncontracted spinor indices that are not written out). The object B lives in the space Sm and A lives in Cm where S and C are the vectorspaces 38 CHAPTER 3. SYM USING PURE SPINORS

Dynkin label Irreducible representation of SO(1,9) (10000) vector (00010) spinor (00001) cospinor (01000) 2-form (00100) 3-form (00011) 4-form (00020)⊕(00002) 5-form (10010) γ-traceless vectorspinor (01010) γ-traceless 2-formspinor

Table 3.1: Some irreducible representations of the ten-dimensional Lorentz group and their corresponding Dynkin label. of the spinor and cospinor representations of the Lorentz group. These product spaces can be decomposed into irreducible representation spaces, i.e. Sm = R1 + R2 + ··· and Cm = T1 + T2 + ··· . In the same way that the cospinor is dual to the spinor the subspace Ti will be dual to Ri. A contraction, or inner product, between something in Ri and T j will only be non-zero for i = j.

The different symmetries among the indices on A and B will restrict those multispinors to only have components living in some of the irreducible subspaces Ri and Ti. For instance if B has a component in the irrep R5 (whatever it is) but A has no component in the dual irrep T5 the equation will have no implications for this part of B or A. The LiE program is useful because it allows us to systematically list the different irreps that are contained in a multispinor with a given symmetry. It gives the irreps in terms of their Dynkin labels. A list of some common irreps and their label is found in table 3.1. Using this method will only investigate what irreps might possibly enter. We would still have to do some computations to ﬁnd the exact coefﬁcients.

As an example let us go back and look at the calculation of a(2) again. The ab1b2 relevant equation of motion was given in equation (3.4). Using LiE the ﬁelds that appear have the following representation content (we have reintroduced the pure spinors missing in (3.4))

λa1 λa2 θb (00003) (00101) (10001) a(2) ((00110)h(hh(h( (10010) (00001) (01001) ¡ ¢ a1a2b α (0) γ a1b∂αaa2 (10010) (00001)

We see that there are two irreps where the equation of motion has implications for a(2) . One of them (the second column) kills the 3-formspinor part. The a1a2b other (the third column) says that the vectorspinor must be proportional to a (0) derivative of aa2 . The gauge variation was given in equation (3.3). The partici- pating ﬁelds contains the following irreps 3.2. Q AND ITS COHOMOLOGY 39

λaθb1 θb2 (00010) (00101) (01010) (10001) a(2) (00001) (01001) (10010) ab1b2 ω(3) (01001) ¡ ¢ ab1b2 σ (1) γ ∂σω (00001) (10010) ab1 b2

Here we have only included the reps in a(2) that are still left after the equation ab1b2 of motion have been applied. There is one match in the third column which tells us that the traceless 2-formspinor can be gauged away. Furthermore we see from the last column that the vectorspinor, which above was shown to be (0) proportional to a derivative of aa , can be gauge transformed with a derivative (1) (0) of ωa . This is not surprising since we already know that aa is put to zero by (1) (1) (0) choosing ωa properly. Thus choosing ωa so that aa vanish will also remove the vectorspinor in a(2) . The only remaining part of a(2) is then the cospinor. ab1b2 ab1b2 (1) (1) In the gauge variation the term with ωa also has a cospinor part but since ω (0) has already been used to put aa to zero there is no freedom left to do anything more and so we still have all degrees of freedom left in the cospinor. The result we have reached is the same as what we got earlier by doing the full explicit expansion.

Now that we have introduced this more schematic method for expansion of multispinors we are able to proceed to the second θ-level of the equation of motion and the third θ-level of the gauge variation without the complication of doing an explicit expansion. First the gauge variation: ³ ¡ ¢ ´ a b1 b2 b3 (3) (4) α (2) λ θ θ θ δa − 4ω + i γ ∂αω = 0 ab1b2b3 ab1b2b3 ab1 b2b3 LiE tells us that the relevant irreps are

λaθb1 θb2 θb3 (00011) (01000) (01011) (10100) (02000) (10020) a(3) (00011) (01000) (01011) (10100) (02000) (10002) ab1b2b3 ω(4) (02000) (10002) ¡ ¢ab1b2b3b4 α (2) γ ∂αω (00011) (01000) (10100) ab1 b2b3

The two rightmost irreps contained in a(3) in this table can be gauge to zero. ab1b2b3 (2) (2) Three of the four others gaugetransform with a derivative on ω b2b3 . Since ω b2b3 (1) was used to gauge parts of aab to zero there is no freedom left in this ﬁeld to do further transformations. On to the equation of motion: ³ ¡ ¢ ´ a1 a2 b1 b2 (3) α (1) λ λ θ θ 3a − i γ ∂αa = 0 a2a1b1b2 a1b1 a2b2 Now the table of irreps looks like this

λa1 λa2 θ θ (00011) (00102) (01000) (01011) (10002) (10100) b1 b2 a(3) ((00011)hh(h( (01000) ((01011)hh(h( ((10100)hh(h( ¡a2a2¢b1b2 α (1) γ a b ∂αa (01000) (00000) (20000) 1 1 a2b2 40 CHAPTER 3. SYM USING PURE SPINORS

This immediately yields that three of the four remaining parts of a(3) must ab1b2b3 be zero. The only nontrivial part is thus a 2-form which should be given by (1) some expression involving ∂αa . To deduce the exact expression we must a1b2 ﬁnd a way to combine γ-matrices and the 2-form into an expression with four spinor indices with the same symmetry as a(3) . This is actually easy. The ab1b2b3 last two b indices are antisymmetric so they can be put on a γ(3)-matrix. Now place the other two indices on a γ(5)-matrix and then contract the vector indices on the γ(3) with those on the γ(5). This leaves two antisymmetric vector (5) indices on¡ the γ ¢ that¡ can be¢ contracted with the 2-form. The ansatz is thus (3) σ ···σ a = γ 1 5 γσ σ σ pσ σ . Here we had to put in the explicit an- ab1b2b3 a[b1 1 2 3 b2b3] 4 5 tisymmetrisation since we don’t write out the θs. You could be worried that this might¡ lead¢ to¡ the¢ identical vanishing of our expression. Compare for in- σ stance to γ a(b γσ cd) which we know is zero. However, as can be veriﬁed by (3) contracting for instance the b1 and b2 index with a γ , this is not the case.

Of course the ansatz we have written down is not the only possibility with the ¡desired¢ ¡ symmetry.¢ Specifically¡ there¢ are¡ two other¢ ways to write such terms: σ1σ2σ3 σ1σ2σ3 σ4 γσ1 a[b1 γ b2b3] pσ2σ3 and γ a[b1 γσ1σ2 b1b2] pσ3σ4 . However since we know a priori from the LiE-expansion that there should be only one 2-form in a(3) all the different ways to write this out must be related to each other. The exact relations are possible to work out by Fierz expanding in the different pairs of indices. Doing this you find that the different terms satisfy ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ γσ1···σ5 γ p = −6 γ γσ2σ3 p = a[b1 σ1σ2σ3 b2b3] σ4σ5 σ1 a[b1 ¡ b2b3] ¢σ3σ4 ¡ ¢ σ1σ2σ3 σ4 = −3 γ a[b1 γσ1σ2 b2b3] pσ3σ4 It is thus OK for us to decide to work with only one of the forms. ¡ ¢ (1) α If we use, as was shown earlier, that aab = −i γ abaα the 2-form part of the equation of motion says that ¡ ¢¡ ¢ ¡ ¢¡ ¢ σ1···σ5 σ1···σ5 λγ λ θγσ1σ2σ3 θ pσ4σ5 − 2 λγ θ λγσ1σ2σ3 θ pσ4σ5 − ¡ α ¢¡ β ¢ − λγ θ λγ θ ∂αaβ = 0 To get something out of this equation we have to rewrite each term in a canonical way. We do this by a Fierz expansion of the indices the two λs in the second and third terms are contracted with. Since the pair of λs can only sit on a γ(5) due to their pureness we get ¡ ¢¡ ¢¡ 2880 40 ¢ λγσ1···σ5 λ θγ θ p − p − ∂ a = 0 σ1σ2σ3 σ4σ5 16 · 5! σ4σ5 16 · 5! σ4 σ5 To arrive at this result the duality between γ(7) and γ(3) as well as the self- duality of γ(5) had to be used (see appendix B). We see that the 2-form we are looking for is really given by the field strength constructed from the vec- 1 tor potential: pαβ = 14 ∂[αaβ]. This still matches the spinor potential from the superspace construction, though only if we assume an abelian gauge group.

There might not appear to be any beneﬁts of continuing the calculation of the cohomology to higher orders, but there is! Already at the next level some interesting things will start to happen so we push on. We proceed exactly in the 3.2. Q AND ITS COHOMOLOGY 41 same way by writing down the irreps of the next equation of motion and gauge variation. Since the number of representations are growing they are displayed in the separate tables 3.2 and 3.3.

Starting with the gauge variation we notice that a(4) has multiplicity two in ab1···b4 one representation. One of the ﬁelds in this representation can be gauged away. Furthermore the representation (00003) can be removed. Next the equation of motion kills all but one of the remaining terms in a(4) . This is the traceless ab1···b4 vector-cospinor which is put proportional to a derivative on a(2) — earlier a2b2b3 shown to contain only the cospinor χc. Recall that we can do the decomposition 1 ¡ ¢ ∂ χc = γ γσ∂ χ c + X˜ c (3.5) α 10 α σ α into a spinor part and, what we need here, a γ-traceless vector-cospinor. Now there are several different ways to write an expression with γ-matrices contracted with each other and a vector-cospinor with spinor indices matching the ones on a(4). From table 3.3 we know that there is only one such component in a(4) so just as for a(3) the different ways of writing such an expression should be related to each other. In this case there are more possibilities than for a(3), some involving γ(n) with n greater than 5. Those terms can trivially be rewritten with γ(n)s where n only is less or equal to 5 using the duality between γ(n) and γ(10−n). This leaves three different possibilities. For these you ﬁnd, by Fierz expansion, that two actually vanish identically. We thus have to use the remaining term to make the ansatz ¡ ¢ ¡ ¢ ¡ ¢ (4) σ1σ2σ3 σ4σ5σ6 a ∼ γ γ γσ ···σ X˜ σ ab1···b4 [b1b2 b3b4] 1 5 6 a

To deduce the exact coefﬁcient we would have to put this ansatz into the equation of motion and solve it. Unfortunately this requires some quite involved γ-matrix calculations so we refrain from doing it.

Actually the most interesting part of the equation of motion at the third θ-level is not that it gives the expression for a(4). If we look at the ﬁrst column of table 3.3 we see that there is one term, the spinor part, that does not involve a(4) at all but only gives implications for a(2). More explicitly this part of the equation tells us that the spinor part in the decomposition (3.5) above is zero. That is

¡ σ ¢ γ ∂σχ a = 0

˜ c or in words — χ satisﬁes the Dirac equation. Of course this also means that Xα (4) c that was shown to appear in a above must simply be equal to ∂αχ .

Now it is not hard to guess what happens at the next level. There we expect to ﬁnd the equation of motion for the gauge potential. That this is actually the case is easy to see from table 3.5 and 3.4. The pattern is the same as in the previous case. In a suitable gauge a(5) will consist of only a single irrep (3) proportional to a derivative on a which we know to be fβδ. At the same time the other irreps in the derivative term, a 3-form and a vector, must be zero. So 42 CHAPTER 3. SYM USING PURE SPINORS al .:Teirdcberpeetto otn fteeuto fmto ttethird the at motion of equation the of content ﬁelds. representation eliminate irreducible The 3.3: Table ¡ λ γ a α 1 ¢ a λ a (4) a a 1 1 2 b a θ 2 1 b ∂ b 1 1 α b al .:Teirdcberpeetto otn fteguevraina h fourth the at variation gauge the of content representation irreducible The 3.2: Table θ 2 a b b (2) a 3 2 2 ¡ θ b γ 2 b λ b 3 α 3 a ¢ θ ω a ab b (4) ab ab (5) 000 (10001) (11010) (10101) (10021) (10010) (02001) (01012) 2x(01001) (00010) (00110) (00012) (00001) 1 1 ∂ 1 · · · 1 ··· ··· α b b ω 4 θ 4 b (3) b 2 4 b 3 b 4 003 (11001) 2x (10012) 2x(11010) (10001) (10021) (02010) (10010) (01010) (02001) (00003) (00101) (01001) ( (00003) h (00110) (00030) ( h h ( h ( ( (00101) h ( h 011 000 101 (11001) (10001) (01010) (00101) h ( h ( ( (01010) h ( h h ( h ( 000 (10001) ( (02010) h ( h h ( h ( θ lvl h ag aito a enue to used been has variation gauge The -level. ( (10012) h ( h h ( h ( ( (11001) h θ ( h -level. h ( h ( ( (11001) h ( h h ( h ( 3.3. MORE FIELDS 43 in the cohomology we must have

∂[α fβδ] = 0 α ∂ fαβ = 0

The first equation is the Bianchi identity, which is trivially satisfied, and the second is the expected equation of motion for the abelian Yang-Mills field strength.

It is now clear that the cohomology of the Berkovits Q-operator at the first λ-level is completely identical to the constrained superspace field Aa defined earlier and thus provides yet another description of super Yang-Mills. This can actually be seen very easily without the longwinded explicit calculation we’ve a just gone¡ through.¢ Start with λ Aa and use this field to define a new vector field i ab Aα = 16 γα Da Ab. This definition matches the relation derived between Aa and Aα in the superspace formulation. If we now use the two fields to define field strengths just as in equation (2.17) (but without the non-abelian terms), the a conventional constraint will of course be satisfied. What is more, λ Aa being Q-closed implies that the field strength also satisfies the¡ dynamical¢ ¡ ¢ constraint. (5) ab This follows since the Q-closedness¡ is¢ equivalent¡ ¢ to γ(5) cd γ Da Ab = 0 (5) ab and this is of course nothing but γ(5) cd γ Fab = 0 which is the “weaker” form of the dynamical constraint as given in equation (2.26).

3.3 More Fields

We have now shown that the cohomology of Q at the first λ-level gives us on- shell super Yang-Mills theory. But what about the other levels? It would be unnatural to restrict to only one specific level, so we will now investigate the field content of the other levels using the same technique as for level one.

3.3.1 Level zero

The zeroth level, where the general field take the form C(x, θ) with no λ-dependence, is particularly simple since there is no gauge transformation to take into account. In other words, such a field can not be Q-exact. We expand the field as follows C(x, θ) = c(0)(x) + θa1 c(1)(x) + ... + θa1 . . . θa16 c(16) (x) a1 a1...a16 It is immediately clear that the equation of motion at the zeroth level forces c(1) to be zero. The equation at the first level is

a b¡ (2) ¡ α¢ (0)¢ λ θ 2cab − i γ ab∂αc = 0

Because of the antisymmetry in the indices of c(2) the ﬁrst term contains only a 3-form while the second obviously is a vector. This means that both has to be (2) (0) zero separately, that is c = 0 and ∂αc = 0. 44 CHAPTER 3. SYM USING PURE SPINORS account. al .:Teirdcberpeetto otn fteeuto fmto ttefourth the at motion of equation the of content representation irreducible The 3.5: Table ¡ γ λ α a ¢ θ ω a ab b (5) ab ab (6) 1 1 ∂ 1 · · · 1 ... ··· α b b ω 5 5 θ b (4) b 2 5 ··· b 5 002 002 010 101 100 100 (20002) (12000) (11000) (10011) (01100) (20100) (01002) (20020) (12000) (11011) (11000) (10011) (00002) 2x (01100) (00013) 2x(01020) (00002) (00031) (00020) ¡ γ λ α a ¢ 1 a al .:Teirdcberpeetto otn fteguevraina h ﬁfth the at variation gauge the of content representation irreducible The 3.4: Table a 1 λ (5) a b 1 a 1 2 a 2 ∂ θ b α b 1 1 a ··· (3) a · · · 2 b b 4 θ 2 b b 4 3 b 4 002 000 001 010 011 002 000 010 002 100 x101 102 100 x101 100 (20020) (12000) 2x(11011) (11000) (10022) 2x(10011) (10000) (02002) (01100) (01020) (01002) (00111) (00100) (00031) (00020) (00002) ( (00002) h 002 (20100) (20002) (12000) (11011) (11000) (10011) (01002) (01100) ( (01002) h h ( ( h h ( ( (00013) h h ( h ( 010 100 (11000) (10000) (00100) ( (01002) h h ( ( (01100) h h ( 101 (11000) ( (10011) h h ( θ lvl h ag aito a entkninto taken been has variation gauge The -level. ( (11011) h ( (20100) h h ( ( h θ -level. h ( ( (12000) h h ( h ( ( (20002) h h ( 3.3. MORE FIELDS 45

Now since c(1) = 0 and c(2) = 0 all terms c(k) with k ≥ 3 must be zero too. This can be¡ seen as follows¢ by induction. The equations of motion take the shape λθ(n−1) c(n) − i∂c(n−2) = 0. When c(n−2) vanish this reduces to λθ(n−1)c(n) = 0. This equation actually kills all irreps in a(n) since all the irreps in the product of n antisymmetric spinors are contained in the set of irreps you get when replac- ing one of the antisymmetric spinors with one without any speciﬁc symmetry properties (λ in our case). So, since we have already shown that c(1) is required to be zero induction gives that all terms with odd powers of θs vanish — the same thing happens with c(2) and the even terms.

3.3.2 Level two

We will denote the field at level two with A∗ (x, θ) and the expansion coeffi- a1a2 cients will be called a∗(n) (x). In contrast to the level zero case here we will a1a2b1···bn once again have to take into account the gauge invariance which leads to some differences. It is for instance easy to show, using LiE calculations, that the first two θ-levels in the cohomology are zero. This does however not mean that all higher terms also vanish as was the case for level zero. The reason for this is ¡ ∗(n) that now the equations of motion take the shape λa1 λa2 λa3 θb2 ··· θbn a + ¢ a1a2a3b2···bn ··· = 0 while the coefficient a∗(n)s field content is governed by the fact that it is contracted with the spinor product λa1 λa2 θb1 ··· θbn . When one of the θs was traded for a λ in the level zero case we could be certain that none of the irreps were eliminated from the equation of motion. In this case however the fact that we already have a pure spinor contracted to the field means that there will be more than one λ appearing in the equation and thus the pureness can come into play. This means that in some cases representations appearing in the field a∗(n) will be killed by the spinor product λ ··· θ and thus not be restricted by the equation of motion. It is of course still possible that they can be gauged away.

A careful (but trivial) analysis with LiE shows that in a suitably chosen gauge the ﬁeld content of A∗ is given by a1a2 ¡ ¢¡ ¢¡ ¢ a1 a2 ∗ σ1...σ5 ∗ λ λ A = λγ λ θγσ σ σ θ θγσ σ χ + a1a2 1 2 3 ¡ 4 5 ¢¡ ¢¡ ¢ σ1σ2σ3 σ4σ5α ∗ + λγσ1···σ5 λ θγ θ θγ θ aα + ···

∗ ∗ that is one spinor χc at the third θ-level and one vector aα at the fourth θ-level. The higher level terms contains, as usual, derivatives on these fields but no other new fields. Both fields can be gauge transformed in the following way

∗ β δaα = ∂ ∂[β sα] ∗ ¡ α¢ a δχc = γ ca∂ακ and still remain in the cohomology. However only the vector is required to α ∗ satisfy an equation of motion: ∂ aα = 0. Note that in contrast to the physical vector and spinor at the first λ-level the ones we have found here have the “wrong” statistics. The spinor field is commuting while the vector field is anticommuting. 46 CHAPTER 3. SYM USING PURE SPINORS

Field Equation Transformation Fermion/Boson Level α β α a ¡ ∂¢ ∂[β aα] ∂ ω B λθ a α a b 2 χ γ b∂αχ = 0 0 F λθ c ∂αc = 0 0 F 0 ∗ α ∗ β 2 4 aα ∂ aα = 0 ¡ ∂ ¢∂[β sα] F λ θ ∗ α b 2 3 χa 0 γ ab∂ακ B λ θ ∗ α 3 5 c 0 ∂ tα B λ θ

Table 3.6: All the different ﬁelds contained in the abelian pure spinor cohomology and their properties.

3.3.3 Level three

We denote the field at the third λ-level λa1 λa2 λa3 C∗ (x, θ) with expansion co- a1a2a3 efficients c∗(n) . Using the now well described methodology you easily a1a2a3b1···bn find, using LiE, that the only new field is a scalar appearing at the fifth θ-level. It can be written out explicitly in the following way ¡ ¢¡ ¢¡ ¢¡ ¢ λa1 λa2 λa3 C∗ = λγσ1 θ λγσ2 θ λγσ3 θ θγ θ c∗(x) a1a2a3 σ1σ2σ3

There will be no higher order terms with derivatives on c∗ and furthermore there will be no equation of motion for it. We do however have a gauge varia- ∗ α tion: δc = ∂ tα.

So in summary, we have found that in addition to the super Yang-Mills multiplet the Q-cohomology contains four other fields: Two scalars c and c∗, one ∗ anticommuting and one commuting, one anticommuting vectorfield aα and ∗ one commuting spinor field χc . The equations the fields have to satisfy are summarised in table 3.6. The extra fields might at first appear puzzling, but we will later explain why their appearance is actually not at all a bad thing.

3.4 Extending to the non-abelian case

The cohomology of the operator Q defined above only yields the abelian version of super Yang-Mills. To extend to the more general case we need to introduce a covariant version of Q. Since a field Ψ in the cohomology contains the super-connection Aa a natural guess would be to use Ψ itself a a sort of connection. For an arbitrary Lie-algebra valued field B that gauge transform as δB = [Λ, B] one can easily show that the following is the correct definition for a covariant version of Q QΨ B = QB − [Ψ, B}

For this to transform as it should, δ(QΨ B) = [Λ, QΨ B], the connection should have the gauge variation

δΨ = QΛ − [Ψ, Λ] = QΨΛ 3.4. EXTENDING TO THE NON-ABELIAN CASE 47

The equation of motion for Ψ should similarly be deﬁned as

QΨΨ = QΨ − ΨΨ = 0

Note that now all the ﬁelds should take values in the Lie algebra corresponding to the non-abelian gauge group.

Is the new QΨ still nilpotent? The answer is yes, but only on shell. Note that 0 0 for a Ψ on the form Ψ = Ψ + δΨ = Ψ + QΨΛ we have

0 0 0 0 QΨ0 Ψ = QΨ − Ψ Ψ = QΨ + Q(QΨΛ) − (Ψ + QΨΛ)(Ψ + QΨΛ) ≈ ≈ QΨ − Q[Ψ, Λ] − ΨΨ − {Ψ, QΛ} + {Ψ, [Ψ, Λ]} = = QΨΨ − [QΨ, Λ] − [Λ, ΨΨ] = QΨΨ + [Λ, QΨΨ] to the first order in Λ. This means that if Ψ belongs to the kernel of the covariant Q so does a Ψ0 in the form above trivially. It is thus meaningful to consider the cohomology as before. A careful analysis shows that the independent field content doesn’t change. However the equations of motions and the gauge variations are modified and turns out to give

i ¡ ¢ ∇ f αβ = γ ab{χ , χ } α 2 β a b δaα = ∇αω ¡ α¢ c γ a ∇αχc = 0

δχc = [ω, χc] . .

We see that the physical (in the sense that we now what they are, not that they are gauge invariant) ﬁelds aα and χc now satisfy the equations of non-abelian super Yang-Mills. So the modiﬁcation we did to Q appears to be the correct one. 48 CHAPTER 3. SYM USING PURE SPINORS Chapter 4

BRST and antiﬁelds

4.1 Antiﬁelds and the master action

So far we have been dealing with the classical super Yang-Mills theory, but we do of course know that for any theory to be applicable to the real world it has to be quantised. However, gauge theories are generically problematic in this regard due to the fact that infinitely many field configurations (those related by gauge transformations) describe exactly the same physical situation. This means that in the partition function Z Z = DϕeiS[ϕ] you will be summing over the same integrand infinitely many times which leads to a divergent result. For the time being we will be working with some generic gauge theory with the all the fields collectively denoted ϕ. It is frequently stated that this resulting infinity means that simply doing the naive path integral is wrong for gauge theories. This is however not really the problem. You have to remember that what you really would like to do is to compute matrix elements of some operator O[ϕ] and the formula for doing this is of the following form R DϕO[ϕ]eiS[ϕ] hα|O |βi = R DϕeiS[ϕ] If O is a gauge invariant operator, as it should be to be observable, the same divergence will be present in both the nominator and denominator of this expression and thus cancel each other. So the matrix elements could possibly still be finite and meaningful. The real problem with the gauge invariance has rather to do with the perturbation expansion. A general action takes the form S = S0 + Si where S0 is the part of the action that is quadratic in the fields and governs their dynamics. The term Si is the interaction part. When in- serted in the path integral the part due to the interaction terms can be Tay- lor expanded (assuming a small coupling) meaning that we in reality only

49 50 CHAPTER 4. BRST AND ANTIFIELDS have to be able to calculate correlation functions for the free theory which is given by the quadratic part of the action. Now note that by introducing deltaRR functions and doing partial integrations we can always write this part as S0 = dxdyϕ(x)K(x, y)ϕ(y) for some K. The gauge invariance means that the complete action satisﬁes Z δS dx δϕ(x) = 0 δϕ(x)

Rwhere δϕ(x) is the transformation of the ﬁeld ϕ(x). We will write this as δϕ(x) = dyR[ϕ](x, y)α(y) with α(y) as the parameter of the gauge transformation. Note that this form of writing the variation can accommodate derivatives on α since they by partial integration can be included in R. Now if we take a functional derivative with respect to ϕ(y) on the expression for the variation of the action above we ﬁnd Z ZZ δ2S δS δR[ϕ](x, z) dx δϕ(x) + dxdz α(z) = 0 δϕ(y)δϕ(x) δϕ(x) δϕ(y)

If we recall that δS/δϕ(x) actually gives the equation of motion and thus is zero for ﬁelds that are on shell we see that if we evaluate the above equation for such a ﬁeld — let’s call it ϕ0 — we get Z ¯ δ2 S ¯ dx ¯ δϕ0(x) = 0 δϕ(y)δϕ(x) ϕ=ϕ0

Unless the action contains terms linear in the ﬁeld, identically equal to zero will be a solution to the equations of motion and for this particular choice of ϕ0 the second derivative on S in the integrand above reduces to the quadratic part of the action, exactly what we denoted K earlier. So this small calculation thus shows that K has zero modes Z dxK(y, x)V(x) = 0 R where the vector with zero eigenvalue, V(x), is given by dzR[0](x, z)α(z). The point of all this is that it means that K lacks an inverse. This can easily be seen as follows. Recall that the deﬁnition of the inverse is Z dzK−1(x, z)K(z, y) = δ(x − y)

If we let the left hand side act on the (nonzero) vector V(y) we get Z dzdyK−1(x, z) K(z, y)V(y) = 0 | {z } =0 The right hand side would of course yield V(x) thus implying that V = 0 which is a contradiction. So in this case K−1 must be nonexistent. As is well known the inverse of K is the propagator and it is essential when computing the correlation functions. Thus we see that the existence of the gauge invariance prohibits us from doing perturbation theory using the ordinary Feynman dia- grams. There is however a small problem with the argument above. We could 4.1. ANTIFIELDS AND THE MASTER ACTION 51 equally well restrict α(x) to be a constant. We would then be dealing with a global symmetry but still find that the quadratic part of the action has zero modes. But we know from experience that global symmetries does not stop us from finding propagators so obviously there has to be something more to it than we have said so far. The answer lies in the fact that the amount of zero modes is far greater in the case of a local symmetry. There is one for every function α(x) whereas in the global case there is only one for every constant α. Note that even if an operator is zero on a subspace it still has an inverse when restricted to the rest of the space. In the case of the global symmetry the space of zero modes is so small that it does not matter in the path integral — it has measure zero so to speak. Whereas for the gauge case there are actually field degrees of freedom with no propagator.

4.1.1 Fadeev-Popov quantisation

It is apparent that we can’t do perturbation theory with an action that is gauge invariant and to circumvent this problem we have to come up with an equivalent action, in the sense that it yields the same physics, but without the gauge invariance. The standard way to do this is by the Fadeev-Popov procedure. This procedure is discussed, more or less clear, in almost any book on quantum ﬁeld theory. One of the better is [17], even though the treatment there differs somewhat from ours.

The essence of this method is that we really should only need to sum once over each physically distinct field configuration. When doing the naive path integral we integrate over the whole space of field configurations illustrated in figure 4.1. Starting from a given field the gauge slice or gauge orbit this field belongs to is constructed by performing all possible gauge transformations on the field. All fields in the slice describe the same physics. By choosing a suitable function F(ϕ) of the fields and demanding that F(ϕ) = 0 we can pick out one field from every slice. F(ϕ) = 0 is thus a gauge fixing condition. Now it should be possible to rewrite the path integral so that the integration over the field ϕ(x) is performed by integration on one hand along the gauge fixation curve and on the other hand along the gauge slices, i.e. over α(x). To perform this “change of variables” we first note the following useful identity Z δF(ϕ ) Dαδ(F(ϕ )) det α = 1 α δα where ϕα is the gauge transformation of ϕ with gauge parameter α. This identity is a straightforward generalisation of the analogous version with an ordinary integral. For a gauge fixing condition, F, that is linear in the fields the Jacobian in the determinant will be independent of the field α sinceR we have assumed that the variation itself is linear in α (recall that δϕ = R[ϕ]α). We thus introduce the following notation F (ϕ) = δF(ϕα)/δα. Inserting this factor of one in the path integral yields Z Z iS[ϕ] Z = Dα Dϕδ(F(ϕα)) det F [ϕ]e 52 CHAPTER 4. BRST AND ANTIFIELDS

gauge slices

(x) F (ϕ) = 0

PSfrag replacements

ϕ(x)

Figure 4.1: The space of field configurations with gauge slices with the fields connected by gauge transformations and a gauge fixing function F( f ield) = 0.

Now we can utilise the gauge invariance of the action to rewrite this as Z Z iS[ϕα] Z = Dα Dϕδ(F(ϕα)) det F [ϕ]e

0 As the next step note that if we perform a change of variables ϕ → ϕ = ϕα the Jacobian of this transformation will be δϕα/δϕ. Fortunately for ordinary Yang- Mills theory the transformation is only a shift and a rotation and it is easy to show that the the determinant of the Jacobian simply is unity. Furthermore the ﬁeld ϕ can be written as the corresponding inverse gauge transformation of the ﬁeld ϕ0. We will denote the gauge parameter that generates this transformation −1 0 by α so that we have ϕ = ϕα−1 in the notation introduced above. Expressing the partition function using the new variable we thus get Z Z 0 0 0 iS[ϕ0] Z = Dα Dϕ det F [ϕα−1 ]δ(F(ϕ ))e

To simplify the Jacobian in the determinant we will have to do two successive gauge transformations. To do this we use that due to the group structure two successive gauge transformations with parameters τ and λ are identical to one single gauge transformation with a parameter Λ = Λ(τ , λ). We get 0 Z 0 Λ −1 0 δF((ϕ a−1 )λ) δF(ϕΛ(α−1,λ)) δF(ϕΛ) δ (α , λ) F (ϕ −1 ) = = = α δλ δλ δΛ δλ where we used the chain rule of functional differentiation in the last step. The ﬁrst factor we recognise as F (ϕ0) and it is, as noted earlier, independent of Λ and thus independent of both α−1 and λ. Since the left hand side is independent of λ this means that the second factor also can’t depend on λ. We 4.1. ANTIFIELDS AND THE MASTER ACTION 53 can however not say anything about its dependence on α−1. When taking the determinant of this we get a product of two determinants, and our partition function takes the following form

³Z −1 ´ Z δΛ(α , λ) 0 Z = Dα det Dϕ0δ(F(ϕ0) det F (ϕ0)eiS[ϕ ] δλ Note how the only α-dependence left is in δΛ/δλ which does not depend on the ﬁeld ϕ0 so that this factor could be brought outside of the ϕ0 integral and in this way giving a simple constant. Of course this constant is really inﬁnite since it corresponds to the integration along the gauge orbits. In matrix elements calculations it will cancel as we have already explained. This however means that we get exactly the same physics by simply neglecting this extra factor and taking the remaining functional integral as our fundamental partition function. This is what we will do know. What is left is Z Z = Dϕδ(F(ϕ)) det F (ϕ)eiS[ϕ]

Here the delta function stops the gauge invariance of the action from being a problem. We are in reality only integrating along the gauge-ﬁxed surface in ﬁeld space.

When doing perturbation theory the form of the path integral we have arrived at is not particularly convenient so we will have to rewrite it. One good thing about the formula above is that it does not depend on the specific gauge fixing function you choose. This means that if we take F(ϕ) = f (ϕ) − ω, where f is some function and ω is a field, thenR Z will not depend on ω. If we then come up with a function ρ(ω) so that Dωρ(ω) = 1 this weight function can then be used to integrate over Z without changing its value. I.e., we have Z Z Z Z = Dωρ(ω)Z = Dϕ Dωρ(ω)δ( f (ϕ) − ω) det F (ϕ)eiS[ϕ] = Z = Dϕρ( f (ϕ)) det F (ϕ)eiS[ϕ]

Now since we only know how to do functional integrals over gaussian inte- i R 2 grands it is convenient to choose ρ in such a way: ρ(ω) = exp(− 2ξ dxω ). Here ξ is an arbitrary constant. Strictly speaking we should include a normalisation constant to ensure that ρ integrates to 1, but as constants cancels anyway when calculating physical quantities we can ignore it. This exponential factor can then be incorporated as a new term in the action. As the next step we rewrite the determinantR asRR a functional integral over two new fermionic fields. Recall that DbDc exp(i bMc) is proportional to det M when c and b are fermionic. The c-field is called a ghost and b goes under the name antighost. The modified Fadeev-Popov action we end up with is Z Z Z 1 δ f (ϕ (y) S = S − dx f 2(ϕ(x)) + dy dxb(y) λ c(x) (4.1) FP 2ξ δλ(x) We should note that there is another way to include the delta function in the action, using its fourier expansion: Z δ(F(ϕ)) ∼ DheihF(ϕ) 54 CHAPTER 4. BRST AND ANTIFIELDS

To utilise this a new auxiliary ﬁeld h would have to be introduced and path integrated over. In fact this way of handling the delta function can be combined with the one that yielded the f 2-term by using the formula Z ih( ξ h− f ) − i f 2 Dhe 2 ∼ e 2ξ which can be derived by completing the square in the exponent. In this way the action in equation (4.1) can be rewritten as Z ZZ ξ δ f (ϕ (y) S = S + dx( h(x) − f (x))h(x) + dxdyb(y) λ c(x) (4.2) FP 2 δλ(x)

As an example let us apply the Fadeev-Popov method to the case of SYM. The gauge transformations are as we know given by δAµ = ∂µα − [Aµ, α] and δχc = −[χc, α]. We can rewrite this in the form above in the following way if i i we expand the ﬁeld in terms of the generators of the Lie algebra, Aµ = AµT , χc = χciTi and α = αiTi, and use the structure constants of the Lie algebra, [Ti, T j] = f i jkTk, Z i 10 i j kji k j δAµ(x) = d y(δ ∂µ − f Aµ(x))δ(x − y)α (y) Z δχci(x) = − d10 y f kjiχckδ(x − y)α j(y)

i µ i The conventional gauge ﬁxing condition is to choose f (A) = ∂ Aµ. This means that Z i µ i 10 i j 2 kji µ k kji k µ j f (Aα) = ∂ Aµ + d y(δ ∂ − f ∂ Aµ(x) − f Aµ(x)∂ )δ(x − y)α (y)

Taking the functional derivative we get

δ f i(A (x)) α = (δi j∂2 − f kji∂µ Ak (x) − f kji Ak (x)∂µ)δ(x − y) δα j(y) µ µ so that the ghost term in the action becomes Z Z 10 10 i i j 2 kji µ k kji k µ j d y d xb (x)(δ ∂ − f ∂ Aµ(x) − f Aµ(x)∂ )δ(x − y)c (y) = Z Z 10 i µ i 10 µ = d xb ∂ ∇µc = tr d xb∂ ∇µc

Note that the ghost field c has to belong to the same representation as the gauge parameter but with the opposite statistics. The antighost should be a field that is dual to the gauge fixing condition f (a vector if f is a vector and a cospinor if f is a spinor etc), again with the opposite Grassmann parity. The complete Fadeev-Popov action for SYM becomes Z h 1 1 i i S = tr d10x − F Fµν − ∂µ A ∂ν A + χγρ∇ χ + b∂µ∇ c FP 4 µν 2ξ µ ν 2 ρ µ using the form in equation (4.1). The gauge transformation of the ghost fields is given directly by the fact that they are Lie algebra valued. There is no longer 4.1. ANTIFIELDS AND THE MASTER ACTION 55 any gauge invariance though, because of the gauge fixing term. One negative thing with the Fadeev-Popov action is that it is no longer invariant under the supersymmetry transformations. This is easy to see by computing the variation of the added terms. The situation here is comparable to what happens when you choose a non-covariant gauge fixing — then manifest Lorentz symmetry is lost. Here the problem is that when choosing a gauge the way we do supersymmetry goes away. The c-ghost introduced here is the same one as we earlier found at the lowest θ and λ-level when doing the pure spinor expansion. This will become apparent in a short while.

4.1.2 BRST-quantisation

One drawback of the Fadeev-Popov method is that it does not yield the right result in all cases. It does work for abelian and non-abelian Yang-Mills theory so it covers the cases we are interested in. There are however more general types of gauge theories where something else is necessary. In particular this is the case for

1. reducible theories (there is gauge invariance in the gauge parameter) 2. open gauge algebras (i.e. it is only an algebra on-shell)

In addition to these limitations not all gauges will yield a correct result with the Fadeev-Popov procedure. To generalise the Fadeev-Popov result so that it is valid for other cases the key point is to observe that the Fadeev-Popov action has a new global symmetry that didn’t exist in the original action. This new symmetry is the BRST-symmetry (named after Becchi, Rouet, Stora and Tyutin). It involves a fermionic transformation of the ﬁelds — that is the parameter of the transformation is a fermionic constant which we will denote by θ. It is convenient to introduce the BRST-operator s in the following way

δBRST A = (sA)θ Note that this deﬁnition means that s will be acting from the right: s(AB) = A(sB) + (−1)B(sA)B This turns out to be convenient as it avoids some extra signs later on.

For the case of SYM s is given by

sAµ = ∇µc sχc = {χc, c} on the fields. This is just a gauge transformation with the ghost as gauge parameter. On the ghosts on the other hand we have 1 sc = − {c, c} 2 sb = −h 56 CHAPTER 4. BRST AND ANTIFIELDS and on the auxiliary field sh = 0 The BRST-transformation has the peculiar property that it is nilpotent, s2 = 0. This is easy to verify from the definitions above with frequent use of the Jacobi identity. The gauge fixed action is of the form SFP = S + Sextra where S is the original action that has no dependence on the ghost or the auxiliary field (it is easiest to work with the form of the action in equation (4.2)). All fields of this kind appear in the extra terms in Sextra. Since the BRST-transformation of the original fields is just a gauge transformation (albeit with a peculiar parameter) it is trivial that S is invariant under s, sS = 0. To show that the same is true for Sextra we first note that we can rewrite the integrand of this part of the action as ξ ξ ( h − ∂µ A )h + b∂µ∇ c = s(b∂µ A − bh) 2 µ µ µ 2

Now the nilpotency immediately tells us that also Sextra and thus the entire Fadeev-Popov action is BRST-invariant.

To describe the BRST-symmetry in some more generality we will ﬁrst introduce a compact notation that in this case is quite useful. In this so called De Witt notation we will let integration over space-time be implied by summation µ R µ over indices. As an example A Bµ should be taken to mean dxA (x)Bµ(x) in ordinary notation. In this way a general gauge transformation can be written

i i A δαϕ = R[ϕ]Aα ϕi denotes all the fields in the theory and αA the gauge parameters. Gauge i i invariance of the action is now compactly written as δR S/δϕ R[ϕ]A = 0. Here k we introduced the right functional derivative δR/δϕ . It simply means that the field ϕk is “removed” towards the right when taking the derivative. It is important to keep track of this when we are dealing with a mixture of fermionic and bosonic quantities. There is of course a corresponding definition of a left derivative. To write down the BRST-transformation of the ordinary fields we have to introduce ghost fields, cA — one for every gauge parameter — and then we simply define i i A sϕ = R[ϕ]Ac Now the question is how s should act on the ghosts. The interesting property of the BRST-transformation for the Yang-Mills case really was that it was nilpotent so we will demand the same for the general case:

δ Ri 0 = s2ϕi = s(R[ϕ]i cA) = Ri scA + (−1)A+1 R A (sϕk)cA = A A δϕk δ Ri = Ri scA + (−1)A+1 R A Rk cBcA A δϕk B ⇔ 1³δ Ri δ Ri ´ Ri scA = (−1)A R A Rk + (−1)(A+1)(B+1) R B Rk cBcA A 2 δϕk B δϕk A Here it is perhaps worth pointing out that when an index such as A or B appears over −1 we mean the Grassmann parity of the gauge parameter αA. This 4.1. ANTIFIELDS AND THE MASTER ACTION 57 is why we have to write A + 1 etc. to get the parity of the ghost cA. To fulﬁll this equation we should take scA to be given by

1 scA = TA [ϕ]cCcB 2 BC where T is deﬁned by

δ Ri δ Ri R B Rk + (−1)(B+1)(C+1) R C Rk = Ri TA + ··· δϕk C δϕk B A BC

This is the commutation relation of a general gauge algebra. You would find this expression if you tried to compute the commutator of two gauge transfor- k mations: [δα1 , δα2 ]ϕ . For the case of a closed algebra there are no more terms on the right hand side, but for open algebras you also have a term proportional to the equation of motion. For instance for Yang-Mills theory this would be equivalent to the Lie algebra commutation relation and T would really be given by the structure constants of the Lie algebra. The definition of scA we have presented does actually only work for the case of closed algebras. Open algebras can be handled but requires extra complications so we will restrict ourselves to the closed case. With the definition of s we have given, you will also have nilpotency when acting on c. To show this you only need to use that T satisfy the analogous thing of a Jacobi identity. This far we have been able to introduce a nilpotent BRST-operator, but we have seen no sign of the antighost and the auxiliary field. The BRST-transformation of the fields and the ghosts only includes these two types of fields themselves so there is no apparent need for anything else. However we can alway introduce so called trivial pairs of fields if we let s act on them in the following way

sA = B sB = 0

In this way the fact that s2 = 0 on A and B is trivial. By comparing with the example of Yang-Mill theory that we presented above we see that the antighost bA and the auxiliary ﬁeld hB are of precisely this kind

sbA = hA shA = 0

Exactly why we need this trivial pair will be demonstrated shortly. First we will go on to the next step after s has been completely defined — that of constructing a BRST-invariant action. The new action we wish to have should of course have some resemblance to the original action we start from. It is for instance reasonable to demand that if we put all new fields to zero the modified action should reduce to the old one. This can be achieved by the following split

Smod = S + Snew

Exactly as for Yang-Mills the gauge invariance of the ordinary action S implies that it is also BRST-invariant. Now the question is how to construct a functional Snew of all the fields that is BRST-invariant. For the case of Yang-Mills this 58 CHAPTER 4. BRST AND ANTIFIELDS term was of the form s(something). This is in fact the general solution. It can be shown that every BRST-invariant action can be written on the form gauge invariant functional of only the ordinary field plus s on functional of all the fields, see [17] for a proof. So Snew has to be of the form sΨ if it is to satisfy sSnew = 0. Since s is fermionic the same has to be true for the functional Ψ to make sure that the action is bosonic. To further investigate the properties of Ψ is is useful to introduce a new grading, the ghost number. Ordinary fields are no ghosts so we will assign them ghost number zero, gh(ϕ) = 0, and since the ghosts are ghosts they have to have ghost number one, gh(c) = 1. By looking at the Fadeev-Popov action for Yang-Mills and assuming that it is purely of ghost number zero we see that gh(h) has to be zero (because of the h2-term) and gh(b) = −1 (because of the b∂∂c-term). Now make note of the fact that the operator s always increases ghost number by one unit. This means that the fermion Ψ has to have ghost number equal to -1 if we want gh(Snew) to vanish. It is now apparent why we had to introduce the trivial pair of the antighost and the auxiliary field — otherwise it would have been impossible to construct a functional with ghost number -1. It could not have been done with only ϕ and c since they both have non-negative ghost numbers.

The BRST-procedure [18] really only says that we should construct an action that is invariant under BRST-symmetry (and other globals symmetries of the original action) and then use this action for all computations. For Yang-Mills we can see that choosing the fermion Ψ in an appropriate way we will always give us back the FP-action and so in this case the BRST-method obviously gives a correct gauge fixed action for the gauge theory we started from. But what about other choices for Ψ? Do we have to put restrictions on the way to choose Ψ? The answer is that under certain conditions to be explained shortly the choice of Ψ does not matter as amplitudes computed with the BRST-action will be completely independent of the choice of Ψ. In fact we can consider different Ψs to correspond to different ways of fixing the gauge. Some Ψs give the same result as the Fadeev-Popov procedure, but others do not so we do have a greater generality here. For the independence on Ψ to come true we have to restrict the set of operators we are allowed to calculate matrix elements for. Recall that with the ordinary gauge invariant action we are only interested in gauge invariant operators since they are the only ones that are measurable. Now that we in addition to the ordinary fields also have an assortment of ghost etc to construct operators with there is obviously need for a generalisation. The key is to define observables to be operators with ghost number zero that satisfy sO = 0. Indeed if O only depends on ϕ this reduces to demanding gauge invariance. We also have to define which of the states in the now enlarged Hilbert space are physical. To do this we assume that it is possible to introduce a generator for the BRST-transformation: sO = [Q,O}. Then physical state vectors are defined by Q |Vi = 0. It is now easy to show that it is precisely for matrix elements of physical operators between physical states that Ψ can be chosen at will. Indeed upon a change Ψ → Ψ + δΨ the matrix element hα|O |βi 4.1. ANTIFIELDS AND THE MASTER ACTION 59 changes by Z Z i hα|Oi sΨ |βi = i hα|O{Q, Ψ} |βi = Z Z = i hα|OQΨ |βi + i hα|OΨ Q |βi = | {z } =0 Z Z = i hα| [O, Q} Ψ |βi ± i hα| QOΨ |βi = 0 | {z } | {z } =0 =0 where both the O and the states are assumed to be physical. Operators on the form sP trivially satisfy the physicality condition but their matrix elements vanish for physical states: hα| sP |βi = hα| QP |βi ± hα| P Qκ |βi = 0. Further- more it follows from s2 = 0 that Q2 = 0 so that states on the form Q |γi also are trivially physical. Also in this case matrix elements vanish: hα|OQ |γi = hα| [Q,O} |γi ± hα| QO |γi = 0. Physical observables and physical states could thus really be said to live in the cohomology of s and Q respectively.

4.1.3 BV-quantisation

For closed algebras the application of the BRST-procedure is very straightforward as demonstrated above. But for open algebras things are more complicated. By following the antiﬁeld procedure of Batalin and Vilkovisky we will reach exactly the same goal of a BRST invariant action, but in a more systematic way. In short this method consists of three steps

1. Introduce antiﬁelds.

2. Solve the master equation with boundary conditions to give the master action.

3. Gauge ﬁx the master action.

Of course we will have to deﬁne what each step means in some more detail but the end result is that the gauge ﬁxed master action is the BRST-invariant action we are looking for. A nice introduction to the BV-formalism is the review [19], the short paper [20] and chapter 15 of [17]. There is also the comprehensive and mathematical book [21].

Let us start by defining the fields we are working with. First of all we have to introduce the ghost fields in the same way as earlier. To be able to refer to both the ordinary fields and the ghosts in a compact way we introduce the general field ΦI = (ϕi, cA). Now we introduce for both the ordinary field ϕ and ∗ ∗ the ghost c a corresponding antifield. They will denoted by a ∗: ϕi and cA, or Φ∗ Φ∗ in short I . The antifield I will be defined to have the opposite Grassmann ΦI Φ∗ ΦI parity of and its ghost number will be gh( I ) = −gh( ) − 1. The antifield of the ghost should not be confused with the antighost as they are not the same. 60 CHAPTER 4. BRST AND ANTIFIELDS

We will now go on to define the master equation and to do this we first have to introduce the antibracket. This is a bilinear bracket that acts on functionals of the fields and antifield in much the same way as the Poisson bracket acts on functionals of fields and their conjugate momenta. The definition is

δ A δ B δ A δ B (A, B) = R L − R L (4.3) ΦI Φ∗ Φ∗ ΦI δ δ I δ I δ

i i Note that just like the Poisson bracket satisfy {ϕ , πj}PB ∼ δj the antibracket ΦI Φ∗ I satisfy ( , J ) = δJ so in some sense the fields and antifields can be said to be conjugate to each other. The symmetry properties of the antibracket are however not the same as for the Poisson bracket. The most noteworthy thing is the following (A, B) = −(−1)(A+1)(B+1)(B, A) So for two bosonic functionals the bracket is symmetric while in all other cases it is antisymmetric. Note from the definition that the bracket is fermionic and always increases the ghost number by one unit.

We are now ready to introduce the (classical) master equation:

(S, S) = 0 (4.4)

If we assume that the functional S is bosonic we can use the definition of the bracket and the relation between left and right functional derivatives1 to rewrite this as δ S δ S L L = 0 ΦI Φ∗ δ δ I The master action, S, will be required to be a solution of this equation with zero ghost number and as the name implies it will have something to do with the action. To make contact with the gauge theory of our choice we must require that if we put all the antifields to zero we get back the usual action, S(Φ, Φ∗ = 0) = S(Φ). We will actually need to impose more “boundary” conditions since otherwise the action S itself would be a solution due to it not depending on any antifields and this solution would obviously not be very interesting since it does not provide anything new. We will not dwell on this extra condition, but one can show that in S there are gauge invariances and what the condition says is roughly that the number of these has to equal them number of antifields. A solution satisfying these condition is called a proper solution. We can in very much the same way as for the BRST procedure introduce new variables as trivial pairs if we to the master action add terms of the following kind S0 + A ∗ h bA. It is trivial that such terms does not alter the fact that (S, S) = 0. In this way antighosts, auxiliary fields and their corresponding antifields can be ΦI Φ∗ added to the set of available fields, and I .

A remarkable feature of the master action is that it encodes all information about the gauge structure of the theory. This can be seen by expanding S in terms of the number of antiﬁeld: S = S + S1 + S2 + ··· . The master equation

1 δL A (A−I)I δR A When acting on objects that are not algebra valued or similarly we have I = (−1) ΦI δΦ δ 4.1. ANTIFIELDS AND THE MASTER ACTION 61 has to be satisﬁed at each antiﬁeld level separately and this gives the following sequence of equations

δ S δ S level -1: 0 = L L = 0 ΦI Φ∗ δ δ I δ S δ S δ S δ S level 0: 0 = L L 1 = L L 1 ΦI Φ∗ i ∗ δ δ I δϕ δϕi δ S δ S δ S δ S level 1: 0 = L L 2 + L 1 L 1 ΦI Φ∗ ΦI Φ∗ δ δ I δ δ I . .

The first equation is of course trivially true as S contains no antifields. The next ∗ i A equation is fulfilled by S1 = ϕi R[ϕ]Ac . It is possible to show that S1 must contain this term for the solution to be proper. We can see that c must appear ∗ since the master action should have ghost number zero and gh(ϕi ) = −1 is here compensated by gh(c) = 1. There can be a further term in S1 that contain the antifield of the ghost (ghost number -2) combined with two ghost fields to bring the ghost number to zero. This term will enter in the third equation, part of which really is nothing but the commutation relation of the gauge algebra in disguise. For a closed algebra one would find that the second term of S1 ∗ A B C just mentioned would have to look like cATBCc c where T is the structure “constant” of the algebra. In the closed case all the higher terms of the master action are zero and the rest of the equations would only be manifestations of different consistency conditions of the algebra like the Jacobi identity. For this case the master action looks like

∗ i A ∗ A B C ∗ S = S + ϕi RAc + cATBCc c + bAhA (4.5) with trivial pairs included.

The antibracket itself also satisfy a Jacobi identity:

(A, (B, C)) + (−1)(B+1)(A+1)(B, (C, A)) + (−1)(C+1)(B+1)(C, (A, B)) = 0

Due to this we can always, given one solution to the master equation, construct another one of the form S + (δF, S) where δF has to be a fermionic inﬁnitesimal functional of ghostnumber = -1. In direct comparison with the poisson bracket this could be called a canonical transformation as it does not alter the canonical ΦI Φ∗ I Φ Φ equation ( , J ) = δJ to change the variables by δ = (δF, ) and analogously for the antiﬁelds (this is easy to check). We can see that this transformation gives a new solution to the master equation by noting that the Jacobi identity implies ((A, S), S) = ±1/2(A, (S, S)) = 0 for any A due to S solving the master equation. Using this gives us

(S + (δF, S), S + (δF, S)) = (S, S) −2 ((δF, S), S) = 0 | {z } | {z } =0 =0 One particular choice of the δF is to take it to be a fermion that only depends on the fields and not the antifields: δF[Φ, Φ∗] = εΨ[ϕ]. Note that this forces one to introduce a trivial pair with an antighost and an auxiliary field since 62 CHAPTER 4. BRST AND ANTIFIELDS otherwise there is no way to get the desired ghost number of minus one. Us- ing the definition of the antibracket shows us that the corresponding change Ψ Ψ Ψ δR δLS δL δRS in S is δS = (ε , S) = ε ΦI Φ∗ = ε ΦI Φ∗ . We can interpret this as an in- δ δ I δ δ I Ψ 0 Φ Φ∗ Φ Φ∗ δL finitesimal change of variables: S → S [ , ] = S[ , + ε δΦ ]. The infinitesimal transformation can be integrated to yield the finite field transformation Ψ Φ∗ Φ0∗ Φ∗ δL I → I = I + δΦI . The utility of this particular transformation will become clearer in a short while.

There is another very important consequence of the fact that (A, (S, S)) = 0 and that is that because of this we can utilise the bracket to define an operator that is guaranteed to be nilpotent. This operator, the generalised BRST-operator, is given by sO = (O, S). Applying this to the fields we find δ S sΦI = (ΦI , ) = L S Φ∗ δ I In this sense the antifields can be said to be the generators of the BRST-transformation. This is certainly the case when the master action is linear in the antifields and closed gauge algebras is a particular example. It is easy to see from the action in (4.5) that here the generalised BRST-transformation of the fields exactly matches the one defined in the previous part. The antifields transform as δ S sΦ∗ = (Φ∗, S) = − L I I δΦI It is quite remarkable that in the BV-formulation one single object encodes all information about both the gauge structure and the BRST-transformation for a gauge theory.

As we noted there exists gauge symmetries in the master action so it is, as it stands, unsuitable to use as an action for path integral calculation. It is indeed very easy to demonstrate the existence of these gauge invariances — all you have to do is to take a derivative of the master equation δ ³ δ S δ S ´ ³ δ δ S ´ δ S δ S ³ δ δ S ´ 0 = L L L = L L L + (−1)JI L L L = ΦI ΦJ Φ∗ ΦI ΦJ Φ∗ ΦJ ΦI Φ∗ δ δ δ J δ δ δ J δ δ δ J ³ δ δ S ´ δ S ³ δ δ S ´ δ S = L L L + L L L ΦI ΦJ Φ∗ ΦI Φ∗ ΦJ δ δ δ J δ δ J δ This shows that the variation of the action is zero for the transformations

J I δL δLS δΛΦ = Λ δΦI δΦ∗ J (4.6) ∗ I δL δLS δΛΦ = Λ J δΦI δΦJ By taking a derivative with respect to an antiﬁeld instead we also ﬁnd the following gauge invariances δ δ S δ0 ΦJ = (−1)JΩ L L Ω I δΦ∗ δΦ∗ I J (4.7) δ δ S δ0 Φ∗ = Ω L L Ω J I Φ∗ ΦJ δ I δ 4.2. ANTIFIELDS FOR SUPER YANG-MILLS 63

So in total we have found 2N gauge symmetries if there are N fields, but if we are to have a proper solution all of those can’t be independent. You can show that the symmetries are reducible so that in total we only have half of the degrees of freedom remaining in the gauge parameters. You can now imagine that those gauge symmetries can be used to somehow gauge fix all the antifields. What you do at this stage is to, rather ad hoc, adopt the gauge condition Φ0∗ Φ0∗ I = 0 where is the antifield transformed with the canonical transformation used earlier. You simply replace the antifields everywhere they appear Ψ Φ∗ δL with I = δΦI . Doing this you arrive at the so-called gauge fixed master action Ψ Φ δL SΨ = S[ , δΦ ]. The gauge fixed action is then invariant under the gauge fixed I I BRST-transformation sΦ = (Φ , S)| ∗ (for the closed case this transfor- Φ =δLΨ/δΦ mation is the same as it was before due to the linearity of the antifields). The good thing about the gauge fixed action is that a path integral where this replacement has been done will not depend on the particular choice of Ψ and indeed what one arrives at is the same as when following the BRST-procedure. The Ψ here is the same as the gauge fixing fermion in the BRST method.

We have ignored some details in our discussions and will continue to do so as they are irrelevant to the case we will be considering. These details are related to the fact that a matrix elements calculated with SΨ may actually not be independent of Ψ due to the measure of the path integral not being BRST- invariant. To take care of this we are forced to introduce a modiﬁcation of the master equation called the quantum master equation and instead of S consider quantum master actions that are solutions to this new equation. However, as we said, this is not relevant for the case at hand.

4.2 Antiﬁelds for super Yang-Mills

We will now put the Batalin-Vilkovisky machinery to work on super Yang- Mills theory. Our fields, including the ghost, are ΦI = (aµ, χb, c). This leads Φ∗ ∗ ∗ ∗ us to introduce the antifields I = (aµ, χb , c ). It doesn’t take a genius to notice that the fields we have at this point exactly match what we found when considering the pure spinor cohomology earlier on (of course the suggestive notation we used at that point helps). One check you can do is to notice that the Grassmann parities of the antifields and the similarly looking fields in the pure spinor case are the same. Moving on we can directly from equation (4.5) write down the master action for SYM (we are going back to the normal notation with explicit integrals now)

Z ³ 1 i ´ S = tr d10x − f αβ f + (χγα∇ χ) + ia∗ ∇αc − iχ∗{c, χa} − ic∗cc (4.8) 4 αβ 2 α α a 64 CHAPTER 4. BRST AND ANTIFIELDS

Varying this action gives the following equations of motion

i ¡ ¢ δa : ∇α f − γ ab{χ , χ } − i{a∗ , c} = 0 αβ 2 β a b β ¡ α¢ b ∗ δχ : γ a ∇αχb + [χb , c] = 0 α ∗ a ∗ ∗ δc : ∇ aα − [χ , χa ] − [c, c ] = 0 ∗ δa : ∇αc = 0 δχ∗ : {c, χa} = 0 δc∗ : cc = 0

We only ever computed the pure spinor cohomology in full for the abelian case so to compare these equations we would have to remove all commutators. Do- ing this you see that all equations reduces to what we had for the cohomology. In particular we can now explain the, at that point rather mysterious, equations α ∗ ∂ aα = 0 and ∂αc = 0 as the equations of motions for the vector antiﬁeld and the ghost. For the non-abelian case some of the equations above may appear rather peculiar. The last equation, with Lie algebra indices written out, says that f i jkcicj = or in other words that the structure constant should be antisymmetric in the last two indices, but this is of course true by deﬁnition.

In the pure spinor approach we also found gauge transformations of the ﬁelds. The same variations should now preferably be symmetries of the master action. We did show that all the gauge symmetries of S are given by equations (4.6) and (4.7). So the question is how to chose Λ or Ω to yield the desired result. If we ﬁrst consider equation (4.6) and take Λ to only have a component in the c-direction, denoted by λ, we can calculate the variation of for instance aα as follows: Z Z α 10 i δL δLS 10 i δL α δλa (x) = d yλ (y) i ∗ = i d yλ (y) i ∇ c(x) = δc (y) δaα(x) δc (y) Z δ ¡ ´ = i d10 yλi(y) L ∂αck(x)Tk + akα(x)cl(x) f klmTm = δci(y) Z ¡ ¢ = i d10 yλi(y) −Ti∂αδ(y − x) + akα(y) f kimTmδ(y − x) = i∇αλ(x)

Repeating the procedure for the other fields you find that they transform as δ(field) = i[λ, field}. So this transformation is really the original gauge symmetry. Removing the commutators we of course get the variation found for a in the abelian pure spinor cohomology. More interesting is to let Ω in equation ∗ (4.7) have only the component vα in the aα-direction. Doing all the calculations you see that this makes the fields transform as:

∗ δaα = i[vα, c] ∗ α δc = −i∇ vα

This reduce, in the abelian case, to the variation of c∗ we found using pure 4.2. ANTIFIELDS FOR SUPER YANG-MILLS 65 spinors. Taking Λ to only have the component tα in the direction of aα we get

δaα = i{c, tα} ∗ β α δaα = 2∇ ∇[β tα] − 2∇[β∇α]t ∗ ¡ α¢ b δχa = i γ a {χb, tα} ∗ ∗ α δc = −i{aα, t } In the abelian limit the second of those gives us the same variation of a∗ as in the pure spinor case. Finally taking Λ to be τ a in the direction of χa we arrive at

δχa = −i[c, τa] ∗ ¡ ¢ab δaα = i γα [χa, τb] ∗ ¡ α¢ b δχa = i γ ab∇ατ ∗ ∗ a δc = −i[χa , τ ] Here we ﬁnd the variation of χ∗ that matches what we had in the abelian cohomology. The full non-abelian transformations should match what you would ﬁnd calculating the non-abelian pure spinor cohomology in its full glory. That would of course be considerably harder than following the simple and auto- matic procedure we have here.

The generalised BRST-transformations of the ﬁelds are easy to write down now that we have the master action. We have already mentioned that sΦI = Φ∗ Φ∗ ΦI δLS/δ I and s I = −δLS/δ . This yields saα = i∇αc ∗ β ¡ ¢ ∗ saα = ∇ fβα − i χγαχ − i{c, aα} sχa = −i{c, χa} ∗ ¡ α¢ b ∗ sχa = −i γ a ∇αχb − i[χa , c] sc = −icc ∗ α ∗ ∗ a ∗ sc = −i∇ aα + i[χa , χ ] + i[c , c] If we for the moment restrict to the abelian case

saα = i∂αc ∗ β saα = ∂ ∂[β aα] ∗ ¡ α¢ b sχa = −i γ a ∂αχb ∗ α ∗ sc = −i∂ aα with zero on the rest of the ﬁelds, we can make the observation that this is exactly the same transformation that would follow from sΨ(x, θ, λ) = QΨ(x, θ, λ). Where Ψ is a superﬁeld in the pure spinor cohomology and Q it the operator we introduced in chapter 3.2. We can now understand that the reason for Q being nilpotent was that it is really the BRST-operator of SYM. In addition the pure spinor cohomology can now in a sense be regarded as the BRST- cohomology. 66 CHAPTER 4. BRST AND ANTIFIELDS Chapter 5

Conclusions

We have in this thesis showed how super Yang-Mills theory can be formulated using the concept of pure spinors and how this formulation has the nice property of automatically also including ghosts and antiﬁelds in the theory. We have however not said anything about how this new formulation might be useful. In the introduction to chapter 3 we brieﬂy mentioned that the pure spinor approach was introduced in conjunction with the Green-Schwarz superstring. In fact if we quantise the superparticle, which really is nothing but the zero mode of the GS superstring, we will as a result get a wavefunction that satisfy all the equations of SYM. This quantisation has however always been plagued with the problem that it is not known how to do it covariantly while it can easily be done in the light-cone. Berkovits showed that the superparticle can be reformulated by including pure spinors in the action and in an ad-hoc way imposing the Q = 0 constraint. This does not really solve the problem of covariant quantisation as manifest Lorentz invariance appears to have to be broken to show the equality of the new and old formulation. However, when extended to the superstring, the pure spinor formulation turns out to be useful when calculating scattering amplitudes.

Another interesting subject that we have overlooked is that of Berkovits’s action. We wrote down the BV-action for SYM in equation (4.8) directly in terms of the component ﬁelds. It would of course be interesting if one could some- a how rewrite this in terms of the superﬁeld Ψ = C(x, θ) + λ Aa(x, θ) + ··· itself. Berkovits has suggested a way of doing this that involves a very peculiar measure. This measure is based on the fact that there exist a scalar in a product of 5 θs and 3 λs.

Additional we have not had time to consider the relationship between the pure spinor formalism and the spinorial cohomology introduced in [7, 8]. There the constraints of the superspace formulation of SYM are relaxed and a set of ﬁelds resembling the one in Berkovits cohomology is found. It would also have been interesting to consider what happens if you allow ﬁelds that are not only restricted to positive powers of the pure spinor λ.

67 68 CHAPTER 5. CONCLUSIONS Appendix A

Some conventions

We use a metric with mostly minus signs: ηµν = diag(1, −1, ··· , −1) where the zero-direction corresponds to time.

We use small greek letters to denoted vector indices. Sometimes we will distinguish curved indices from flat ones by using letters from the beginning of the alphabet to denote the flat ones and from the middle of the alphabet to denote the curved ones. Small latin letters will denote spinors in the same way. Large latin letters will be used for superspace indices. In the chapter about antifields latin letters will also be used to collectively denoted several indices as well as Lie algebra indices.

Square brackets enclosing indices means that they are antisymmetrised while curly brackets denotes symmetrisation. One squared bracket and one curly bracket denotes graded symmetrisation. All symmetrisations and antisym- metrisations are accompanied by a factor 1/n!, where n is the number of indices enclosed.

We can let a group element act on a vector space through a representation in two ways — from the left and from the right. In addition we can define the covariant derivative on a field living in this vector space in two ways. The combination we choose here dictates the transformation properties of the connection. In table A.1 we summarises the four possibilities. We are usually interested in infinitesimal transformations and for this purpose we expand the group element in terms of a corresponding lie algebra element: g = e±ω.

∇α = ∂α + Aα ∇α = ∂α − Aα 0 0 −1 −1 0 −1 −1 V = gV Aα = gAαg − ∂αgg Aα = gAαg + ∂αgg 0 0 −1 −1 0 −1 −1 V = Vg Aα = g Aαg − g ∂αg Aα = g Aαg + g ∂αg Table A.1: Different possibilities for the gauge transformation of the connection A. g acts on the vector space V lives in through some representation of the gauge group.

69 70 APPENDIX A. SOME CONVENTIONS

Once again there are two choices. On ordinary fields we will let the group element act from the left and the covariant derivative with a minus which corresponds to the upper right entry in table A.1. With this convention the in- 0 finitesimal transformation of A becomes Aα = A ± [ω, Aα] ± ∂αω. We want the derivative to have a positive sign so we have to chose g = e+ω and thus get δAα = ∂αω + [ω, Aα]. The transformation of an endomorphism on the representation vector space must with our convention be F0 = gFg−1 and the covariant derivative ∇α F = ∂α F − [Aα, F]. We get δAα = ∇αω as expected. The field strength is with our definitions Fαβ = ∂α Aβ − δβ Aα − [Aα, Aβ].

When introducing the exterior derivative on superspace we will instead use a right action and a covariant derivative with a plus sign, but when looking at the component level one will ﬁnd that it all reduces to the above deﬁnition.

α Our superspace derivative is Da = δa − i(γ θ)a∂α. Appendix B

Spinors and γ-matrices in D=10

B.1 Spinors

When doing relativistic physics we are interested in representations of the Lorentz algebra. The homogeneous Lorentz algebra is deﬁned by the commutation relations :

[Jµν , Jρσ] = i(ηνρ Jµσ − ηµρ Jνσ − ηνσ Jµρ + ηµσ Jνρ) (B.1)

The simplest (non-trivial) representation is the ordinary deﬁning vector representation.

The spinor representations can be constructed by ﬁnding matrices Γµ that satisfy a Clifford algebra: {Γµ, Γν } = 2ηµν (B.2)

It is then easy to show that generators that satisfy (B.1) can be deﬁned as µν i Γµ Γν J = 4 [ , ]. One can show that for a given dimension there is only one representation of the Clifford algebra and it can be chosen to be unitary. That there is only one representation means that if we can ﬁnd two sets of (nontrivial) matrices that satisfy (B.2) they must be related by a similarity transformation: Γ0 = MΓM−1. That the representation can be chosen to be unitary means that a given set of Γ-matrices can be similarity transformed so that they satisfy Γµ(Γµ)† = 1 (no summation). We will from now on assume that we have made this choice.

By taking the transpose, complex conjugation and hermitian conjugation of equation (B.2) we ﬁnd that if Γµ gives a Clifford algebra so does (Γµ)T, (Γµ)∗ and (Γµ)†. Because of the uniqueness of the representation as stated above we

71 72 APPENDIX B. SPINORS AND γ-MATRICES IN D=10 can then ﬁnd matrices so that

(Γµ)† = AΓµ A−1 (Γµ)∗ = BΓµ B−1 (B.3) (Γµ)T = CΓµC−1

The similarity transformations A, B and C are not unrelated. By using that † = ∗T = T∗, T = †∗ = ∗† and ∗ = †T = T† one can derive the following six identities:

A ∼ C∗ BA ∼ C† BA ∼ B†C (B.4) C ∼ BT AB ∼ A∗CC ∼ A∗ B (B.5)

As an example we have AΓµ A−1 = (Γµ)† = (Γµ)∗T = (BΓµ B−1)T = (B−1)TCΓµ((B−1)TC)−1. From this we can deduce that A ∼ (B−1)TC which is equivalent to the ﬁrst relation in (B.5). It follows from the equations above that C∗ ∼ C† ⇔ C ∼ CT (upper left and upper middle), B∗ B ∼ 1 (upper left and lower right), A∗ A∗ ∼ 1 ⇔ AA ∼ 1 (lower middle and lower right) and that A ∼ A† (upper middle and lower left together with B∗ B ∼ 1).

The existence of the similarity transformations in (B.3) enables us to create real Lorentz invariant scalars. The transformation properties of a spinor s in the space S that the Γ-matrices act on is such that it transform under Lorentz trans- µν formations as s → (1 + iωµν J )s. Here ω are the inﬁnitesimal parameters of the transformation. Now consider instead t¯ = t† A. This object must transform as

¡ −i ¢ t¯ → t†(1 + iω Jµν )† A = t† 1 − iω [(Γν )†, (Γµ)†] A = µν µν 4 † ¡ µν ¢ ¡ µν ¢ = t A 1 − iωµν J = t¯ 1 − iωµν J where we used (B.3) to move A past the Γ-matrices in J. It is now obvious ¯ that the bispinor ts is invariant under Lorentz transformations¡ ¢ (at least to the Γµ νρ νρ µ Γσ ﬁrst order in ω). It is possible to show that [ , J ] = Jvector σ . Because of this we have that t¯Γµs and more generally t¯Γµ1 ··· Γµn s transform as Lorentz tensors.

The Grassmann algebra tells us that Γ0Γ0 = 1 and ΓiΓi = −1 (no summation) which together with the unitarity condition Γµ(Γµ)† = 1 (no summation) means that (Γ0)−1 = (Γ0)† = Γ0 and (Γi)−1 = (Γi)† = −Γi. Because of this we can take the similarity transformation matrix A to be equal to Γ0 (note that this is compatible with the equations AA ∼ 1 and A ∼ A† we had earlier). From now on we will assume that this choice has been made. Since A now is a Γ-matrix itself we can apply (B.3) on it so that we have A∗ = BAB−1 and AT = CAC−1. This together with the relations in (B.4) and (B.5) allows us to deduce that B† B ∼ 1 (lower right and upper right). As the proportionality constant has to be real and positive we might just as well normalise B so that it is one — that is B† B = 1.

We can now show that if we take t and s to be the same spinor in the com- binations above and only allow antisymmetrised products of Γ-matrices they B.1. SPINORS 73

are either purely real or imaginary. This can be seen as follows: (s¯Γµ1···µn s)† = (1/n!)s†(Γ[µn )† ··· (Γµ1])† A†s = (1/n!)s† AΓ[µn ··· Γµ1]s = (−1)n(n−1)/2s¯Γµ1···µn s. Note that this formula also works for the case n = 0. It is now easy to construct real bispinor terms to use in the action by multiplying with an i when appropriate.

The representation of the Clifford algebra will be 2d/2-dimensional when the number of spacetime dimensions d is even and 2(d−1)/2-dimensional when it is odd. In particular this means that in a 10-dimensional spacetime the spinor representation constructed with the Γ-matrices will be 32-dimensional. This spinor representation is however not always irreducible. Consider the matrix Γ¯ = Γ0 ··· Γd. The square of this matrix is proportional to the unit matrix: Γ¯ Γ¯ = (−1)(d−1)+d(d−1)/2 (the ﬁrst term in the exponent is due to ΓiΓi = −1 the other term comes from the anticommutations). Particularly in d = 10 we get a plus sign on the right. Had we been working in a dimension that gave a minus sign we would have to multiply Γ¯ with an i in what follows. The matrix Γ¯ can now be used to construct a projector. Any spinor can be decomposed as 1 Γ¯ 1 Γ¯ + 1 Γ¯ − s = 2 (1 + )s + 2 (1 − )s. The projection operators are P = 2 (1 + ) and P = 1 Γ¯ ± 2 ± + − − + 2 (1 − ). They satisfy (P ) = P and P P = P P = 0. In this way the space of spinors S split into two subspaces S+ and S− each having half the dimension of S. So in 10 spacetime dimension dim S± = 16. Note that this construction only is meaningful for even dimensions. I turns out that for d odd Γ¯ is actually proportional to the identity itself. It is easy to show that [P±, Jµν ] = 0 which means that J can be made block-diagonal by a suitable change of basis and thus the original spinor representation is reducible. The two smaller representations are called Weyl spinors¡ ¢ and anti-Weyl spinors respectively. We will chose our s+ Γ basis so that s = s− . The -matrices themselves are not necessarily block diagonal in such a basis. We have ΓΓ¯ µ = (−1)d−1ΓµΓ¯ . In even dimensions the minus sign means that P+Γµs+ = Γµ P−s+ = 0 and P−Γµs+ = Γµ P+s+ = Γµs+ so here a Γ-matrix changes a Weyl spinor to an anti-Weyl and vice-versa. This means that the Γ-matrices are block off-diagonal in our basis. We will denote ¡ µ ¢ Γµ 0γ ˜ Γ0 the off-diagonal blocks with a small γ: = γµ 0 Since A = the product AΓµ in t¯Γµs = t† AΓµs will be block diagonal and connect Weyl spinors with Weyl spinors and anti-Weyl spinors with anti-Weyl spinors. Writing out the details we have t¯Γµs = (t+)†γ˜ 0γµs+ + (t−)†γ0γ˜ µs−. From now on we will not write out neither the γ0, the γ˜ 0 nor the tilde on γµ. We will rather use the one symbol γµ to denote the appropriate thing according to the context it appears in. When between two Weyl spinors it should be assumed to really mean γ˜ 0γµ etc. For instance the Grassmann like commutation relation {γµ, γν } = 2ηµν should be interpreted as γ˜ µγν + γ˜ ν γµ = 2ηµν or γµγ˜ ν + γν γ˜ µ = 2ηµν .

In some cases the Weyl and anti-Weyl representations are further reducible. Re- call that C was a similarity transformation matrix that satisfied (Γµ)T = CΓµC−1. In ten spacetime dimensions it is possible to transform the Γ-matrices so that C also can be taken to be Γ0. This is compatible with equations (B.4) and (B.5). It is then possible to consider the spinors that are such that s¯ = s† A = sTC our in other word spinors with real components: s∗ = s. These spinors form a subspace of all the spinors that is invariant under Lorentz rotations, they are known as Majorana spinors. This way of choosing the Γ-matrices is known as the real basis since it turns out that also the matrices themselves are real. To 74 APPENDIX B. SPINORS AND γ-MATRICES IN D=10 see this note that the first equation in (B.5) tells us that B is just proportional to unity now that C and A are equal, thus (Γµ)∗ = BΓµ B−1 = Γµ. With Majorana spinors we can do so called Majorana flips. Since t¯Γµ1···µn s is a scalar (in spinor space), we should get back the same thing when we take the transpose of it:

¡ ¢T t¯Γµ1···µn s = t¯Γµ1···µn s = −sT(Γµ1···µn )TCTt = −(−1)n(n−1)/2s¯Γµ1···µn t (B.6) Note how s and t have changed places. Here we did use that CT = (Γ0)T = Γ0Γ0(Γ0)−1 = Γ0. A tricky point is the minus that appears after the second equality. It only appears if we assume that s and t are anticommuting spinors. In the case of d = 10 it is also possible to carry over this construction to the Weyl spinors so that we get Majorana-Weyl spinors. These are used exten- sively throughout this thesis. When dealing with Majorana-Weyl spinors we will not write out the T for transpose in product of spinors. So sγµt should be read as sTγ˜ 0γµt if s and t both are Majorana-Weyl (not anti-Weyl) for instance.

The matrix C acts like an inner product on the space of spinors, C : S ⊗ S → R, where s ⊗ t is mapped on sTCt. We can then identify the space of spinors with its dual, S ' S∗, in the standard way — the dual of s can be identiﬁed with the s∗ that satisfy s∗(t) = C(s, t). It is easy to see that since sTCP+t = (P−s)TCt the dual of the Weyl spinors will actually be equal to the anti-Weyl spinors. In this thesis we have not used different indices to distinguish between Weyl and anti-Weyl spinors but this is frequently done. If we for a moment adopt such a + − convention and let ea be a basis for S and ei a basis for S one ﬁnds that the T a matrix Cia = (ei) Cea can be used to raise and lower indices. That is if s are the a components of a Weyl spinor then Cias will give the components of the dual of a anti-Weyl spinor which can then be contracted with the components of a i i a anti-Weyl spinor, for instance t , like this: t Cias . But as noted we will not use different indices in this way.

For further information about spinors and representations of algebras consult [22].

B.2 Fierzing

The matrices Cγµ1···µn for n = 0 ··· 10 provides a basis for all 16x16 matrices. They are orthogonal with respect to the inner-product given by taking the trace of a product of two of them. Their properties are summarised in table B.1. The properties given there are easily calculated using the relations given in the previous section. Note that if you sum up the total number of available matrices in table B.1 you get 1024 which is not equal to the number of components of a 16x16 matrix which is 256. The number 1024 is rather the number of components in a 32x32 matrix. To arrive at the correct number we have to note that since γµ1···µ10 = εµ1···µ10 γ¯ and since γ¯ is 1 (−1) when acting on (anti-)Weyl spinors we have a relation between the matrices γ(n) and the matrices γ(10−n). Let us assume that we are acting on Weyl spinors. Then we have

µ1···µ10 µ1···µ10 (10−n−1)(10−n)/2 µ1···µn ε γγ¯ µn+1···µ10 = γ γµn+1···µ10 = (10 − n)!(−1) γ B.2. FIERZING 75

n Symmetry Weyl properties Number 0 S W-A 1 1 S W-W 10 2 A W-A 45 3 A W-W 120 4 S W-A 210 5 S W-W 252 6 A W-A 210 7 A W-W 120 8 S W-A 45 9 S W-W 10 10 A W-A 1

Table B.1: Properties of γ-matrices (really Cγ). S and A denotes symmetric respectively antisymmetric. W-A means that the matrices connects Weyl spinors with anti-Weyl spinors or vice-versa. Analogously W-W connects Weyl spinors with Weyl spinors and anti-Weyl spinors with anti-Weyl spinors. The last column gives the number of independent matrices of the given type.

When acting with this equation on Weyl spinors the γ¯ on the left will contribute a sign factor (−1)10−n so that we get

1 γµ1···µn = −(−1)n(n−1)/2 εµ1···µnµn+1···µ10 γ (B.7) (10 − n)! µn+1···µ10

When acting on anti-Weyl spinors we would get an extra factor of −1. Equation (B.7) means that we don’t need the γ(n) for n > 5 to get a complete basis. Fur- thermore we will only need half of the γ(5) since they are self-dual according to (B.7). If we now sum the number of matrices that are marked with W-W in the table and take this new information in account we ﬁnd 10 + 120 + 252/2 = 256 which is the expected answer. The same thing is true for the matrices marked with W-A.

We can now expand any bispinor in terms of γ-matrices. This is what is known as a Fierz expansion. Since a simple γ-matrix takes a Weyl spinor to an anti- Weyl we have γµ ∈ Hom(S+, S−) ' S− ⊗ (S+)∗. Now because of the isomor- phism between (S+)∗ and S− given by C we can rewrite this as Cγµ ∈ (S+)∗ ⊗ (S+)∗. Of course it does work in the same way for (S−)∗ ⊗ (S−)∗, S+ ⊗ S+, and S− ⊗ S−. This is the reason why the matrices Cγ ··· can be used to expand bispinors. Written out with different Weyl and anti-Weyl indices what we are µ i saying here is that while the normal γ-matrix has the index structure (γ ) a µ once we multiply by C we get (Cγ )ba which lives in the same space, and can be used to expand, an object like sbta.

For an object with two Weyl or two anti-Weyl indices the expansion is given by ¡ ¢ ¡ ¢ ¡ ¢ S = k1 γµ + k3 γµ1µ2µ3 + k5 γµ1···µ5 (B.8) ab µ ab µ1µ2µ3 ab µ1···µ5 ab

If on the other hand the indices are of opposite type, i.e. one Weyl and one 76 APPENDIX B. SPINORS AND γ-MATRICES IN D=10 anti-Weyl, the expansion looks like ¡ ¢ ¡ ¢ S = k0C + k2 γµ1µ2 + k4 γµ1···µ4 (B.9) ab ab µ1µ2 ab µ1···µ4 ab To ﬁnd the values of the different coefﬁcients k one should multiply by the different γ(m) and take the trace. By using the orthogonality condition

tr γµ1···µn γ = δmn(−1)n(n−1)/216n!δµ1···µn ν1···νm ν1···νn one ﬁnds that the coefﬁcients are 1 k0 = tr CS 16 1 k1 = tr γ S µ 16 µ 2 1 k = − tr γµ µ S µ1µ2 16 · 2! 1 2 3 1 k = − tr γµ ···µ S µ1···µ3 16 · 3! 1 3 4 1 k = tr γµ ···µ S µ1···µ4 16 · 4! 1 4 5 1 k = tr γµ ···µ S µ1···µ5 16 · 5! 1 5

If the bispinor has a deﬁnite symmetry, that is symmetric or antisymmetric, the expansion will be restricted to those γ-matrices with the corresponding symmetry.

B.3 Some γ-matrix identities

It follows directly from the Clifford algebra that

γµγν = γµν + ηµν (B.10)

This can easily be generalised to other cases with products between γ-matrices. The next simplest case is

γµγνρ = γµνρ + 2ηµ[ν γρ] (B.11)

α One frequently encounters products of γ-matrices of the form γ γρ1···ρn γα where n of course can’t be greater than the dimension of spacetime D. These products can be simplified in the following general way. When summing over the repeated index α it will be different from all the indices ρ1 to ρn in D − n cases (all the ρi must be different due to the antisymmetry). In those terms the first n γ-matrix can be commuted past all the γri picking up a factor of (−1) . When multiplied with the last γ-matrix it just gives the identity. In the remaining n cases the α-index will match one of the ρi. In this case we can commute the first B.3. SOME γ-MATRIX IDENTITIES 77

α γ until it is next to the matching γρi and then continue to commute this γρi until it reaches the second γα and combines with this to give a factor 1. Along the way we pick up a factor (−1)n−1. Thus the complete product can be rewritten as

α n n γ γρ1···ρn γα = (−1) (D − n − n)γρ1···ρn = (−1) (D − 2n)γρ1···ρn (B.12)

Alternatively, we can utilise successive expansions like in (B.10) and (B.11) to derive this.

The following identity is often very useful

¡ µ¢ ¡ ¢ γ a(b γµ cd) = 0 (B.13)

It can be shown by a Fierz expansion in the indices c and d. Because of the explicit symmetry the expansion will only contain a γ(1)-term and a γ(5)-term. If we denote the left hand side by Qabcd we have 1 ¡ ¢ ¡ ¢ 1 ¡ ¢ ¡ ¢ Q = γρ γ e f Q + γρ1ρ2ρ3ρ4ρ5 γ e f Q abcd 16 cd ρ abe f 5!16 cd ρ1ρ2ρ3ρ4ρ5 abe f We will now calculate each of the two terms and show that they both vanish. In the ﬁrst term we have: ¡ ¢ 2¡ ¢ ¡ ¢ ¡ ¢ 1¡ ¢ ¡ ¢ ¡ ¢ γ e f Q = γ e f γµ γ + γ e f γµ γ = ρ abe f 3 ρ ae µ b f 3 ρ ab µ e f 2¡ ¢ 1 ¡ ¢¡ ¢ = γµγ γ + tr γ γ γµ = 3 ρ µ ab 3 ρ µ ab 16¡ ¢ 16 ¡ ¢ = − γ + η γµ = 0 3 ρ ab 3 ρµ ab where we have used equation (B.12). The second term works out similarly:

¡ ¢e f 2¡ µ ¢ 1 ¡ ¢ ¡ µ¢ γρ ρ ρ ρ ρ Q = γ γρ ρ ρ ρ ρ γµ + tr γρ ρ ρ ρ ρ γµ γ = 1 2 3 4 5 abe f 3 1 2 3 4 5 ab 3 | 1 2{z3 4 5 } ab =0 2¡ ¢¡ ¢ = − 5 − 5 γ = 0 3 ρ1···ρ5 ab where we used equation (B.12) and the trace-orthogonality of the γ-matrices. We have thus shown that Qabcd vanishes in D = 10. This is also true for the dimensions D = 3, 4, 6 [3, p. 301]. 78 APPENDIX B. SPINORS AND γ-MATRICES IN D=10 Appendix C

Solving the pure spinor constraint

To solve the pure spinor constraint

λγµλ = 0 (C.1) it is convenient to break the explicit Lorentz covariance. First note that we might just as well replace the matrices γi with iγi — this doesn’t alter the content of the constraint. This replacement is equivalent with having the γ- matrices satisfy the commutation relation

{γµ, γν } = 2δµν instead of the ordinary one. Equation (C.1) with the modiﬁed γs is equivalent to demanding that + λγI λ = 0 (C.2) and − λγI λ = 0 (C.3) ± where the ten linearly independent matrices γI are deﬁned by 1 γ± = √ (γ2(I−1) ± iγ2I−1) (C.4) I 2 for I = 1 ··· 5. It is easy to see that these new matrices have the following commutation relations

+ + {γI , γJ } = 0 − − {γI , γJ } = 0 + − {γI , γJ } = δIJ

− Since all the γI square to zero they must have nullvectors and furthermore since they all anticommute we can ﬁnd a common nullvector which we will

79 80 APPENDIX C. SOLVING THE PURE SPINOR CONSTRAINT denote λ−−−−−. We will now introduce a new set of spinors by acting with + −−−−− +−−−− different numbers of γI -matrices on λ . For instance λ will be + −−−−− ++−−− + + −−−−− deﬁned by γ1 λ , λ by γ2 γ1 λ etc. Note the order of the matrices in the last example — we will always put the lowest index to the right. + The notation is such that acting with a γI on a spinor where the Ith index is a plus will annihilate the spinor whereas the index will be turned into a plus if it is a minus, perhaps accompanied with a sign to accommodate for the ordering − of the matrices. Similarly a γI kills all spinors with Ith index being a minus and turns a plus index into a minus index.

All the different λ±±±±± are easily seen to be linearly independent. As an +−+−− +++−− −−−++ + example assume that λ = k1λ + k2λ , then by acting with γ1 +−−++ you ﬁnd k2λ = 0 or in other words k2 = 0. Next, acting with for instance − +−+−− γ2 yields k1 = 0 which in turn gives the contradiction λ = 0. One can easily convince oneself that it works out in the same way for the general case. Obviously there are 25 = 32 different spinors so we can conclude that they must form a basis for both Weyl and anti-Weyl spinors. To only get one or the other we have to restrict to the spinors constructed by using either an even or an odd number of γ-matrices. Indeed, if we assume that λ−−−−− is Weyl we see +−−−− + −−−−− that λ = γ1 λ is anti-Weyl since, as we have demonstrated earlier, the γ-matrices turns Weyl spinors into anti-Weyl spinors. On the other hand, acting with two matrices we should really reintroduce the tilde on one of the ++−−− + + −−−−− matrices giving λ = γ˜2 γ1 λ . Here the ﬁrst matrix turns the Weyl spinor into an anti-Weyl and the second turns this anti-Weyl spinor back into a Weyl one.

We thus get a basis for the space of Weyl spinors by considering the 16 different spinors given by

λ− = λ−−−−− 1 spinor IJ + + − JI λ = γJ γI λ = −λ 10 spinors 1 λI = ε γ+γ+γ+γ+ λ− 5 spinors 4! IJKLM J K L M We now wish to say something about the normalisation of λ−, but note that − a − − − +++++ a − + f − − ba − (λ ) (λ )a = (γ1 ··· γ5 λ ) (λ )a = (λ ) (γ5 ) f e ··· (γ1 ) (λ )a = 0. Indeed if we try to contract a spinor with any spinor that has the same sign in at least one of the indices we will in the same way get zero. Thus the only possibility is to contract λ− with λ+ = λ+++++ and so we chose our normalisation so that

+ a − − + + + + + − 1 = (λ ) (λ )a = λ γ1 γ2 γ3 γ4 γ5 λ Using this it is easy to derive the following useful relations:

− + I I λ γP λ = δp KL + IJ λ γP λ = εKLPJI L − IJ L λ γK λ = 2δ[I δJ]K All other contractions among spinors in the Weyl basis vanish. We can now use that a general Weyl spinor can be written in term of the basis spinors as 81

− IJ I λ = u−λ + uIJ λ + uI λ to solve the pure spinor constraint. Equation (C.2) becomes

+ − + J JK + LM J + − 0 = λγI λ = u−uJ λ γI λ + uJKuLMλ γI λ + uJ u−λ γI λ = J J = u−uJ δI + uJKuLMεJKILM + uJ u−δI = 2uI u− + εIJKLMuJKuLM and equation (C.3) becomes

− JK − L J − KL 0 = λγI λ = uJKuLλ γI λ + uJ uKLλ γI λ = L J = 2uJKuLδ[J δK]I + 2uJ uKLδ[KδL]I = 2uLI uL + 2uJ uJI = 4uJ uJI

1 The ﬁrst equation is, when u− 6= 0, solved by uI = − 2 εIJKLMuJKuLM. This solution does in fact also satisfy the second equation. We have

uJ uJI ∼ εJKLMNuKLuMNuJI = εJKLMNu[KLuMNuJ]I and the particular antisymmetrisation over the uIJ that appear here is actually identically zero. To see this it is simplest to expand the antisymmetrisation and note that the value of the index I has to match one of the ﬁve indices inside the brackets. This leads to some terms dropping out while the ones that are left cancel.

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