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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 ordi- nary component form as well as the usual superspace formulation with con- straints. 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 per- haps 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 ex- ample 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 con- sidering 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 inter- actions 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 super- symmetric. 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 com- ponent 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 ex- plain 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 di- rectly 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 iden- tity 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 the- ory 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 fer- mionic 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 com- ponents 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 ker- nel is half the dimension of the γ-matrices.
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