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 Göteborg, Sweden 2006

Super Yang-Mills Theory using Pure Spinors FREDRIK ELIASSON c FREDRIK ELIASSON, 2006 Fundamental Physics Chalmers University of Technology SE-412 96 Göteborg Sweden Chalmers Reproservice Göteborg, Sweden 2006

Super Yang-Mills Theory using Pure Spinors Fredrik Eliasson Department of Fundamental Physics Chalmers University of Technology SE-412 96 Göteborg, 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 modification of the standard model that includes SUSY is the Minimal Supersymmetric Standard Model (MSSM) which also has the added benefit of coupling constant unification. The introduction of SUSY means enlarging the Poincaré 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, d 2 = 0, and thus becomes completely transparent. If we introduce the gauge potential as a 1-form, A = A µ dx µ, 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) = d 2 Λ = 0. Expanding the forms in their components one finds that this corresponds directly to the usual formulation of Maxwell s electromagnetism. That is F µν = µ A ν ν A µ and the transformation A µ A µ + µ Λ. 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 Poincaré group is extended to the super-poincaré 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 Poincaré group is a bosonic subgroup both of these sectors should consist of representations of the Poincaré group. A collection of fields living in such a representation of the super-poincaré 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 Majorana spinor, Ψ a. 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 p 2 ϕ = 0. On the mass shell p 2 = 0 so ϕ is not restricted in any way and thus we really have two independent degrees of freedom 1. The spinor has to satisfy the Dirac equation, ( γ µ) a b p µ Ψ b = 0. Since the Dirac operator, p µ γ µ, squares to zero on the mass shell 2 it 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 Q a. 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: Q a ϕ Ψ a (2.1) Q a Ψ b (γ α ) ab α ϕ 1 Chapter ten of [1] has as an elementary introduction to the counting of degrees of freedom. 2 Since p µ γ µ p ν γ ν = 1/2p µ p ν {γ µ, γ ν } = p µ p ν η µν = p 2 = 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: {Q a, Q b } = 2 (γ α ) ab P α (2.2) where P α 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 p 2 A µ p µ p A = 0 reduces to p µ p A = 0 on the mass shell which implies that p A = 0. This can be used to eliminate one of the components of A µ in terms of the others, for instance A 0 = p i A i /p 0. At the same time we have the gauge invariance δa µ = p µ Λ which can be used to remove another component of A µ, for instance A 1 by taking Λ = A 1 /p 1. 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 onshell 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 S = d 10 x [ 1 4 F µν F µν + i 2 χγρ ρ χ ] (2.3)

6 CHAPTER 2. SYM AND BIANCHI IDENTITIES where F is the ordinary field strength constructed out of the vector in the multiplet as F µν = µ A ν ν A µ and χ a 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 F µν = 0 ( γ µ ) ab µ χ b = 0 (2.4) Now consider the following supersymmetry transformation on the fields: (δ ε A) µ = ( εγ µ χ ) (δ ε χ) a = i 2 F µν (γ µν ε) a (2.5) where ε is a constant Majorana-Weyl spinor parameter. A simple calculation gives the corresponding variation of the action: Z [ δs = d 10 x 1 2 F µν(δ ε F) µν + i ( δε χ χ ) + i ( χ δε χ )] = 2 2 Z [ = d 10 x F µν µ (δ ε A) ν + 1 4 F µν( εγ µν γ ρ ρ χ ) + 1 4 ( ρf µν χγ ρ γ µν ε )] = Z [ = d 10 ( x F µν εγ ν µ χ ) + 1 4 F µν( εγ µν γ ρ ρ χ ) 1 4 ( ρf µν εγ µν γ ρ χ )] = Z [ = d 10 ( x F µν εγ ν µ χ ) + 1 4 F µν( εγ µν γ ρ ρ χ ) 1 ( 4 ( ρ F µν εγ µν γ ρ χ )) + + 1 4 F µν( εγ µν γ ρ ρ χ )] = Z [ = d 10 ( x F µν εγ ν µ χ ) + 1 2 F ( µν εγ µν γ ρ ρ χ ) ( )] + ρ = Z [ = d 10 ( x F µν εγ ν µ χ ) + 1 2 F µν( εγ µνρ ρ χ ) ( + F µν εγ [µ η ν]ρ ρ χ ) + )] + ρ ( = Z [ = d 10 ( x F µν εγ ν µ χ ) + 1 2 F ( µν εγ µνρ ρ χ ) ( + F µν εγ µ ν χ ) ( )] + ρ = Z 1 = d x[ 10 2 F ( µν εγ µνρ ρ χ ) ( )] + ρ = Z [ ( 1 = d 10 x ρ 2 F ( µν εγ µνρ χ )) 1 2 ( ρf µν εγ µνρ χ ) ( )] + ρ = Z ) = d 10 x ρ ( where we have used that γ µ γ νρ = γ µνρ + 2η µ[ν γ ρ], the symmetry properties of the γ-matrices and the spinors, and in the last step the Bianchi identity, [µ F νρ] = 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 ε i by δ i we get δ 2 δ 1 A µ = δ 2 ( ε1 γ µ χ ) = ( ε 1 γ µ δ 2 χ ) = i 2 Fρσ( ε 1 γ µ γ ρσ ε 2 ) = = i 2 Fρσ( ε 1 γ µρσ ε 2 ) + if ρσ ( ε 1 η µ[ρ γ σ] ε 2 ) = = i 2 Fρσ( ε 1 γ µρσ ε 2 ) + ifµσ ( ε1 γ σ ε 2 ) If we now use the fact that γ µρσ is antisymmetric while γ σ is symmetric in combination with the anticommuting property of the ε i we get: [δ 1, δ 2 ]A µ = 2iF µσ ( ε1 γ σ ε 2 ) (2.6) Let us now denote the generator of the SUSY transformation by Q a, that is δ 1 A µ = ε1 a Q aa µ. We can then rewrite the commutator of transformations, [δ 1, δ 2 ], as an anticommutator of generators, ε1 aεb 2 {Q a, Q b }. From equation (2.6) we can thus deduce that } ( {Q a, Q b A µ = 2iF µν γ ν ) = 2i ( ab µa ν γ ν ) + 2i ( ab ν A µ γ ν ) = ab ( = µ ( 2iA ν γ ν ) ) + 2i ( γ ν) (2.7) ab ab ν A µ The first term in this expression is a gauge transformation of A, the second is proportional to the momentum operator, P µ µ, in its coordinate realisation. If we instead consider the gauge invariant quantity F we get: {Q a, Q b }F µν = µ (2i ( γ ρ) ab ρa ν ) ν (2i ( γ ρ) ab ρa µ ) =2i ( γ ρ) ab ρ ( µ A ν ν A µ ) = 2i ( γ ρ) ab ρf µν (2.8) We recognise this as the desired form of the SUSY algebra. Let us now repeat this calculation but instead acting on the spinor: δ 1 δ 2 χ a = i 2 δ 1 This leads to (F µν ( γ µν ε 2 ) a ) = i µ ( δ1 A ν )( γ µν ε 2 ) a = ( =i µ ε1 γ ν χ )( γ µν ) a ε 2 = i 2 = i [ (γν 2 εb 1ε2 c ) bd ( ε1 γ ν µ χ )[ ( γ µ γ ν ε 2 ) a ( γ ν γ µ ε 2 ) a ] = ( γ µ ) ( a e γ ν ) e c ( ) ( γ ν bd γ ν ) ( a e γ µ ) e c ] µ χ d ] [δ 1, δ 2 χ a = i [ (γν 2 εb 1ε2 c ) bd( γ µ ) ( a e γ ν ) e }{{} c ( )bd( γ ν γ ν ) ( a e γ µ ) e c + }{{} I II + ( )cd( γ ν γ µ ) ( a e γ ν ) e }{{} b ( )cd( γ ν γ ν ) ( a e γ µ ) ] e }{{} b µ χ d III IV

8 CHAPTER 2. SYM AND BIANCHI IDENTITIES Terms I and III can be rewritten as: I + III = ( ( γ ν )bd γ µ ) ( a e γ ν ) e c + ( ) γ ν cd( γ µ ) ( a e γ ν ) e b = =3 ( γ µ) ( a e γ ν ) ( ) e (c γν bd) ( γ µ) ( a e γ ν ) ( e d γν )bc = =3 ( γ µ) a e Q e cbd ( γ µ) ( a e γ ν ) ( e d γν )bc where Q is defined by Q e cbd = ( γ ν) ( e (c γν )bd). We can utilise the Clifford algebra, ( γµ )a b( γ ν )b c + ( γ ν )a b( ) γ µ b c = 2η µν δa, c to rewrite terms II and IV as: ( γ ν ) ( a e γ µ ) e c + ( ) γ ν II + IV = ( γ ν )bd cd( γ ν ) a e ( γ µ ) e b = =2 ( ) γ ν bdη µν δc a ( ( γ ν )bd γ µ ) ( a e γ ν ) e c + 2 ( ) γ ν cdη µν δb a ( )cd( γ ν γ µ ) ( a e γ ν ) e b = =2 ( γ µ) bdδ a c + 2 ( γ µ) cdδ a b 3( γ µ) a e Q e cbd + ( γ µ) a e ( γ ν ) e d ( γν )bc Combining everything we get ] [δ 1, δ 2 χ a = i [ 2 εb 1ε2 c 6 ( γ µ) a e Q e cbd 2 ( γ µ) bdδc a 2 ( γ µ) cdδb a 2 ( γ µ) ( a e γ ν ) ( ] e d γν )bc µ χ d = = i [ 2 εb 1ε2 c 6 ( γ µ) a e Q e cbd 2 ( γ µ) bdδc a 2 ( γ µ) cdδb a 4η µν δd a ( γν )bc + 2 ( γ ν) ( a e γ µ ) ( ) ] e d γν bc µ χ d = = i [ 2 εb 1ε2 c 6 ( γ µ) a e Q e cbd 2 ( γ µ) bdδc a 2 ( γ µ) cdδb a 4η µν δd a ( γν )bc + 2 ( γ ν) ( a e γ µ ) ( ) ] e d γν bc µ χ d To get to the desired form we will now demand that the spinor χ is on shell, i.e. satisfies the Dirac equation ( γ µ) ab µ χ b = 0, and recall that in D = 10 Q e abc is identically zero (see appendix B). Only the penultimate term above survives to give: {Q a, Q b } χ c = 2i ( γ µ) ab µ χ c (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 A µ and χ c, take their values in the gauge Lie algebra. By introducing a basis T i for the lie algebra we could make this explicit by expanding the

2.2. SUPER YANG-MILLS IN COMPONENT FORM 9 fields like A µ = A i µt i. 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 hidden inside the definition of F µν. The covariant derivative acts in the ordinary way on A µ and χ c. With our conventions, see appendix A, this is given by µ A ν = µ A ν A µ A ν µ χ c = µ χ c [A µ, χ c ] The action becomes Z S = d 10 x tr [ 1 4 F µν F µν + i 2 χγρ ρ χ ] (2.10) 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 A µ and χ c. 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 ( γ α ) ab α χ b = 0 α F αβ = i 2( γβ ) ab {χ a, χ b } (2.11) 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 that the transformation of the action is proportional to the object Q a bcd 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 as expected given by {Q a, Q b } = 2i ( γ α) 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 replacing manifold with a supermanifold. We will not gain any complete geometric 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 parameterised by a set of coordinates Z M = (x µ, θ m ), where the x µ are commuting real numbers and the θ m are anticommuting real Grassmann numbers. For the case we consider x µ is a vector with 10 components and θ m 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, T M. This is the union of the tangent spaces at all points of the manifold. The coordinate basis for the tangent bundle T M consists of the derivatives M = ( µ, m ) = ( / x µ, / θ m ). Note that the derivatives commute in the same way as the coordinates. The dual space to T M 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 dz m. 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 Z M is anticommuting and M = 0 when Z M is commuting we have dz M dz N = ( 1) M N dz N dz M. 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) ϕ (q) pq+ ω ϕ (q) ω = ( 1) ω (p) 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 between forms. We will denote the space of k-forms by Vk M and the space of all forms by V M. Any k-form can be expanded in terms of the basis elements dz M 1 dz M k like this: (k) ω = 1 k! dzm k dzm 1 ω M1 M k Please note the order of the indices and the fact that we place the components after the forms. A natural thing to do is to try to define an exterior derivative, d : Vk M V k+1 M. Usually it is given by d = dx µ µ so a natural generalisation would be dz M 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: d (p) ω = 1 p! dzmp dz M 1 dz N N ω M1 M p Note that no extra signs appeared since the right action means that d never have to commute past the forms dz M, it acts directly on the form components standing on the right. On products of forms the rule is: d ((p) ω (q) ϕ ) = (p) ω d (q) ϕ + ( 1) q d (p) ω (q) 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 d 2 = 0. This is simply encoding the fact that partial derivatives commute/anticommute: d 2(p) ω =d 2( 1 p! dzmp dz M ) 1 ω M1 M p = = 1 p! d( dz Mp dz M 1 dz N N ω M1 M p ) = = 1 p! dzmp dz M 1 dz N dz K }{{} =0 ( 1) NK+1 dz K dz N K N ω }{{} = M1 Mp ( 1) NK N K 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) : T p M T p M R. We can express the metric in terms of the coordinate basis, g(v M M, W N N ) =

12 CHAPTER 2. SYM AND BIANCHI IDENTITIES V M g MN W N, where g MN = g( M, N ). Now using the metric we can always construct another basis for the tangent space that is orthonormal. Let us denote such a basis by E A = E M A M. We will assume that the signature of our metric is lorentzian. ( Then the orthonormality is given by, g(e A, E B ) = g AB = E M A g MNE N B = ηαβ 0 0 δ ). We will use letters from the beginning of the alphabet, ab 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 E α transform as a vector under the Lorentz group and the E a as a Majorana-Weyl spinor. This will not affect the orthonormality. In fact we can make a local Lorentz transformation so that the matrices E M A depends on where on the manifold we are. We will assume that (Z) depends smoothly on Z. E M A 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 D v (doing differentiation in the direction of the vector v) can be written as D v = v µ 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 D such that D (gs) = g(ds). This D is the gauge transformation of D. In terms of the vector potential A this is expressed as A A = gag 1 + gdg 1, when A is Lie-algebra valued this transformation rule makes A 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 [ D v, D w ] 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 d D of this will be an E-valued 1-form. We define it as (d D s)(v) = D v (s) for a vector field v. This then generalises in the natural way to higher forms. Using this language we can define the curvature by d 2 Ds = F s where s is an E-valued form. Now the torsion term only appears if we expand the left hand side in terms of a basis, say E a : d 2 Ds = d D (E a D a s) = d D (E a )D a s E a d D (D a s) = E a E b (D a D b s + Tab c D cs). T c = 1 2 Ea E b Tba c = d DE c 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 d D s = ds + A s when s is an E-valued form and as d D 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: d D F = df + [A, F]. By using the formula for F in step 5 it

14 CHAPTER 2. SYM AND BIANCHI IDENTITIES immediately follows that d D 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 e a (the orthonormality means that g(e a, e b ) = δ ab ). 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, e a (x) e a(x) = Λ b a(x)e b (x), to get another one. For a minkowskian metric the rotations, the matrix Λ b 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 R b a = dω b a + ω c a ω b c where we have denoted the connection by ω b a. 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 d D = 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 S YM = tr(f F) M where is the usual Hodge star operator. The equation of motion coming from this action is d D 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 Z M. We will however not use this basis for the tangent bundle but instead use the one spanned by the vectors D A given by: D α = α D a = a i ( γ µ) abθ b µ (2.12) Expressed in terms of the coordinate basis this takes the form D A = E M A M where the matrix E M A is given by ( δ E M µ ) A = α 0 i(γ µ θ) a δa m We can introduce a corresponding 1-form basis which we denote by E A. It is given in terms of the coordinate basis by E A = E A M dzm where the matrix E A M is

16 CHAPTER 2. SYM AND BIANCHI IDENTITIES the inverse of E M A. It is easily calculated to be ( ) δ EM A α = µ 0 i(γ α θ) m δm a 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 transformation they should change like δf = ε a Q a F where the generator of the transformation should satisfy the defining property {Q a, Q b } = 2 ( γ µ) abp µ. Now this should be realised as coordinate transformations. For an arbitrary, small change of coordinates Z M Z M = Z M + ξ M (Z) we can easily compute the induced change in the field. We should have F (Z ) = F(Z) for corresponding Z and Z. Taylor expanding gives F (Z ) = F(Z ) ξ M (Z ) MF(Z ) were we used that Z M Z M ξ M (Z ). If such a change of coordinates really should give a SUSY transformation we would then have to choose ξ so that ε a Q a = ξ M M, or equivalently that the action of the differential operator Q a is ε a Q a Z M = ξ M. Notice that Q generates the coordinate transformation apart from a sign, that is δz M = ε a Q a Z M. 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. On the right hand side of the {Q a, Q b } = 2 ( γ α) abp α the momentum operator P α appears. We will work with the definition where P µ = +i µ when acting on the fields. It s now easy to see that the defining commutation relation is fulfilled by the differential operator Q a = a + i ( γ µ) abθ b µ. Explicitly we have {Q a, Q b } = { a, b } +{ a, i ( γ µ θ ) }{{} b µ} + {i ( γ µ θ ), a a} =0 ( γ µ θ ) a( γ ν θ ) [ ] µ, b ν = i ( γ µ) ab µ + i ( γ µ) ab µ = }{{} =0 = 2i ( γ µ) ab µ So to recap, if we do a coordinate transformation Z M Z M = Z M ε a Q a Z with the Q just defined the induced transformation on the field F(Z) F (Z) = F(Z) + ε a Q a 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 A M (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 vector to superspace, V = V(Z) M M. For an arbitrary coordinate transformation Z M Z M = f M (Z) we have M M = ZN Z M N. Using the transformed vector basis and the transformed coordinates the expansion of our vector looks like V = V M (Z ) M. Since the vectorfield itself is independent of both basis and coordinates we should have, for corresponding Z and Z, V M (Z) M = V N (Z ) N V M (Z) M = V N (Z ) ZM Z N M V N (Z ) = V M (Z) Z N Z = M VM ( f 1 (Z)) Z N Z M If we now assume that the transformation is small as we did earlier, f M (Z) = Z M + ξ M (Z), we can Taylor expand and get V N (Z) = V N (Z) + ( M ξ N) V M (Z) ξ M M V N (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 Q a 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 D a only differs by a sign from the SUSY generators Q a and it is exactly this that makes them satisfy {D a, Q a } = 0. Explicitly {D a, Q b } = { a, b } +{ a, i ( γ µ θ ) }{{} b µ} {i ( γ µ θ ), a a}+ =0 + ( γ µ θ ) a( γ ν θ ) [ ] µ, b ν = i ( γ µ) ab µ i ( γ µ) ab µ = 0 }{{} =0 We also trivially have [D α, Q b ] = 0. We now like to see how this basis transform when we do the SUSY coordinate transformation. The action of D A on the coordinates is given by the matrix we introduced earlier: D A Z M = E M A (Z).

18 CHAPTER 2. SYM AND BIANCHI IDENTITIES Under the transformation this goes to D A(Z M ε a Q a Z M ) = E M A (Z ) (D A + δd A )(Z M ε a Q a Z M ) = E M A (Z) εa Q a E M A (Z) δd A Z M ( 1) aa ε a D A Q a Z M = ε a Q a E M A (Z) δd A Z M ( 1) aa ε a [D A, Q a }Z M ε a Q a D A Z }{{ M = ε } a Q a E M A (Z) =E M A δd A Z M = ε a [Q a, D A }Z M δd A = ε a [Q a, D A } = 0 In short our particular basis of tangent vectors does not change when doing coordinate transformations with Q a. 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, E A. 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 E A -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 this in terms of our basis it takes the shape D = E A D A = E A D A + E A A A for the first case. The next step is to introduce the field strength 2-form for A: F = 1 2 EA E B F BA. 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): s F = D 2 s = D(ds + s A) = }{{} d 2 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 2 EB E A F AB = d(e C A C ) + ( 1) AB E B E A A B A A = = E C E A D A A C + d(e C )A C + ( 1) AB E B E A A B A A Here a torsion term appears due to our choice of basis. It is de C = 1 2 EA E B T C BA. So the components of the field strength are F AB = 2D [A A B} + 2( 1) AB A [B A A} + T C AB A C (2.13)

2.4. BIANCHI IDENTITIES AND THEIR SOLUTION 19 The components of the torsion are easily calculated using T C = d(e C ) = d(dz M E C M) = dz M de C M = E D E M D E A D A E C M = = ( 1) A(D+M) E D E A E M D D A E C M which gives 1 2 TC BA = ( 1)B(A+M) E[A M D B}E C M. Plugging in the values of the matrices E M A and EC M given earlier it turns out that all components of the torsion is zero except for T α ab = 2i ( γ α) 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: F αβ = α A β β A α [A α, A β ] (2.15) F ab = D a A b + D b A a {A a, A b } + 2i ( γ σ) aba σ (2.16) F αb = α A b D b A α [A α, A b ] (2.17) By expanding D 2 s 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 D 2 s = D(E A D A s) = E A E B D B D A s + d(e A )D A s which leads to F AB = [ D B,D A } + T C BAD C 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: D F = D( 1 2 EA E B F BA ) = 1 2 EA E B E C D C F BA + 1 4 EA E B E C T D CBF DA 1 4 EC E D T A DCE B F BA = 1 2 EA E B E C (D C F BA + T D CBF DA ) So the Bianchi identity takes the form D [A F BC} + T D [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 only non-zero torsion component is T α ab = 2i( γ α) ab, we get D [α F βγ] = 0 (2.19) 2D [α F β]c + D c F αβ = 0 (2.20) D α F bc + 2D (b F c)α + 2iγ δ bc F δα = 0 (2.21) D (a F bc) + 2iγ δ (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 F ab. 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), F ab = iγ α ab F α + i 5! γα 1 α 5 F α1 α 5 (2.23) We can now proceed to expand the field in the first term in powers of θ: F α (x, θ) = F α (0) (x) + θ b F (1) αb (x) +. The second θ-level can be decomposed in irreps as F (1) αb (x) = ( γ α )b c F c(x) + ˆF αb. Here ˆF is γ-traceless, that is ( γ α) a b ˆF αb = 0, but the interesting part is the spinor F c. After all the theory we are looking for should contain a spinor field. Now let us expand F αb in the same way F αb = F αb + ( γ α) b c F c (2.24)