dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ......

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Transcript of dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ......

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Abstract

Supersymmetry has been intensively studied by physicists in the lasttwenty years� This thesis is a brief introduction to the fundamental conceptsof supersymmetry � In the �rst chapters the Wess�Zumino model and theN�� Yang�Mills theory will be derived� The second part deals with thedi�erential geometrical approach to supersymmetric gauge theories� In thelast chapter� the N�� super Yang�Mills Lagrangian will be derived throughdimensional reduction�

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Acknowledgments

I would like to thank my supervisor Bengt E� W� Nilsson for his contin�uous support and many hours of discussions during the time I have workedon this thesis� I would also like to thank Per Sundell for many enlighten�ing monologues� Finally Helena Karlsson has been very helpful� reading themanuscript and correcting my worst grammatical errors�

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Contents

� Introduction �

� The Supersymmetry Algebra �

��� Irreducible Representations of the Supersymmetry Algebra � ��� The Massive Case � � � � � � � � � � � � � � � � � � � � � � � �

��� Central Charges � � � � � � � � � � � � � � � � � � � � � � � � � ����� The massless Case � � � � � � � � � � � � � � � � � � � � � � � � ��

� The Wess�Zumino Model ��

��� Supersymmetry Transformations of Component Fields � � � � ��

� Superspace and Super�elds ��

��� Supersymmetry Transformations and Covariant Derivatives � ����� Super�elds � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

��� Integration and Functional Derivatives � � � � � � � � � � � � � ��

� The Wess�Zumino Model in Superspace ��

��� Supersymmetry Transformations � � � � � � � � � � � � � � � � �

��� The Wess�Zumino Lagrangian � � � � � � � � � � � � � � � � � � � ��� Equations of Motion � � � � � � � � � � � � � � � � � � � � � � � ��

� The N� Yang�Mills Theory the Abelian Case ��

��� The Vector Field � � � � � � � � � � � � � � � � � � � � � � � � � ��

��� The Chiral Field Strength � � � � � � � � � � � � � � � � � � � � �

��� Supersymmetry Transformations � � � � � � � � � � � � � � � � ����� The N�� Yang�Mills Lagrangian � � � � � � � � � � � � � � � � ��

� Abelian and non�Abelian Interactions ��

��� Abelian Interactions � � � � � � � � � � � � � � � � � � � � � � � ��

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� CONTENTS

��� Non�Abelian Interactions � � � � � � � � � � � � � � � � � � � � �

� Di�erential Geometry in Superspace ��

�� Di�erential Forms � � � � � � � � � � � � � � � � � � � � � � � � ��

�� The Covariant Derivative and the Field Strength � � � � � � � ��

�� The Bianchi Identities � � � � � � � � � � � � � � � � � � � � � � ��

The N� Bianchi Identities �

�� Constraints � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

�� Solving the Bianchi Identities � � � � � � � � � � � � � � � � � � ��

�� The Lagrangian for the non�Abelian N�� Yang�Mills Theory ��

�� The N� Yang�Mills Theory ��

� �� The Bianchi Identities � � � � � � � � � � � � � � � � � � � � � � ��

� �� Yang�Mills Theories for General N � � � � � � � � � � � � � � � �

� �� Reduction � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

�� The N� Yang�Mills Theory ��

���� The � �dimensional Lagrangian � � � � � � � � � � � � � � � � � �

���� Dimensional Reduction � � � � � � � � � � � � � � � � � � � � � � �

A Conventions �

A�� Majorana Spinors � � � � � � � � � � � � � � � � � � � � � � � � � �

A�� Weyl Spinors � � � � � � � � � � � � � � � � � � � � � � � � � � � �

B Some useful Formulas �

B�� Sigma Matrices � � � � � � � � � � � � � � � � � � � � � � � � � � �

B�� Spinor Algebra � � � � � � � � � � � � � � � � � � � � � � � � � �

B�� Derivatives in Superspace � � � � � � � � � � � � � � � � � � � � � �

C Conventions in Yang�Mills Theory ���

C�� Raising and Lowering Indices � � � � � � � � � � � � � � � � � � � �

C�� Yang�Mills Derivatives � � � � � � � � � � � � � � � � � � � � � � � �

C�� The Field Strength � � � � � � � � � � � � � � � � � � � � � � � � � �

C�� Reality Conditions � � � � � � � � � � � � � � � � � � � � � � � � � �

D Wess�Zumino Model in Majorana Notation ��

D�� The Wess�Zumino Lagrangian � � � � � � � � � � � � � � � � � � �

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CONTENTS �

E K�ahler Geometry and Chiral Fields ���

E�� Connection and Covariant Derivative � � � � � � � � � � � � � � ���E�� The K�ahler Potential � � � � � � � � � � � � � � � � � � � � � � � ���E�� Curvature � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��E�� Chiral Models � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

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� CONTENTS

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Chapter �

Introduction

Theoretical physicists are searching for simple and universal principles ofNature� Many of these principles� or laws of Nature� can be expressed assymmetries� An example from classical mechanics is invariance under trans�lation in time� which gives the principle of conserved energy� Another exam�ple is local gauge invariance� which is responsible for most of the interactionsin the standard model�

Supersymmetry is a global symmetry� which relates states of di�erentstatistics to each other� This property is very important in the renormal�ization of the supersymmetric standard model� because some in�nities willcancel due to the equal number of fermionic and bosonic states� Furthermore�many physicists hope that supersymmetric theories will provide a consistenttheory of gravitation and quantum mechanics� and at the same time unifygravity with all other forces in Nature� The most promising candidate issuperstring theory�

To this day� there is no �rm evidence that supersymmetry is realizedin Nature� Neither is there any completely compelling reason that super�symmetry is required to solve the paradoxes in modern elementary particlephysics� However it is possible that �shadows� of supersymmetry will helpus �or even be required� to explain new phenomena� found in particle accel�erators in the near future�

Supersymmetry was �rst discovered more than twenty years ago by Gol�fand and Likhtman ���� However it only became widely known when a fourdimensional theory was constructed� This theory is called the Wess�Zuminotheory� and it will be derived in chapter � �and again in chapter �� this

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CHAPTER �� INTRODUCTION

time in superspace�� However the natural way to begin a thesis about su�persymmetry is with a discussion on the supersymmetry algebra and itsrepresentations� This is done in chapter �� In chapter � we will derive thesimplest supersymmetric gauge theory� the N�� Yang�Mills theory�

Most of the second part of the thesis �chapters � � � will deal withdi�erential geometry in superspace� We will use the geometrical formulationof gauge theories� to derive the N�� and N�� supersymmetric Yang�Millstheories� The N�� Yang�Mills theory will be derived in the last chapter�This will not be done with a geometrical approach� Instead we will usedimensional reduction of an N�� supersymmetric gauge theory in ten space�time dimensions�

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Chapter �

The Supersymmetry Algebra

It is well known �at least to physicists� that the underlying space�time sym�metry group of the S�matrix is the Poincar�e group�

�Pm� Pn� �

�Pm� Jnp� � ��mnPp � �mpPn�

�Jmn� Jpq� � ���mpJnq � �nqJmp � �mqJnp � �npJmq�

where �mn is the Minkowski matric with signature ��� �� �� ���

Much e�ort has been made trying to add an internal symmetry group�which mix with the Poincar�e group in an nontrivial way� However Colemanand Mandula ��� showed that this is impossible in the context of an ordinaryLie algebra� This no�go theorem was proven under very general assumptions�the S�matrix should for example be consistent with local relativistic quantum�eld theory�To �nd a way around the no�go theorem we must relax the assumptions�One way of doing this is to except anticommutators as well as commutators

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� CHAPTER �� THE SUPERSYMMETRY ALGEBRA

of group generators� In this way we get the supersymmetry algebra

�Qi��

�Q ��j�

� �m� ��Pm�ijn

Qi�� Q

j�

o� ���Z

ij

n�Q ��i� �Q ��j

o� � �� ��Z

�ij�

Pm� Qi�

�� �

Pm� �Q ��i�

�Pm� Pn� �

�����

where� � �� � �� � �� �m � �� � � � � � i � �� � � � � N

where Qi� and �Q ��j are the supersymmetry generators�

The Greece indices are Weyl spinor indices �see Appendix A�� The Latin in�dices from the middle of the alphabet are Lorenz indices� If we just considerthese indices �put N � ��� we get the simplest N�� SUSY algebra� Thisis the algebra we are going to discuss further in chapters � � � However� ���� represent a more general SUSY algebra� the so called N extendedsupersymmetry algebra �N � ���The central charges� Zij � commute with all other operators in the algebra�hence they belong to the center of the algebra and are bosonic�� Further�more they are antisymmetric �Zij � �Zji� so they never enter in the N��case� However we are going to discuss central charges later in section ��� ofthis chapter�There are of course more relations in the supersymmetry algebra� but wewill only need the ones given in � �����

��� Irreducible Representations of the Supersym�

metry Algebra

Supersymmetry is a set of symmetry operations which transforms particlesof di�erent statistics into each other� We shall now construct irreduciblerepresentations of the supersymmetry algebra� This is done by operatingwith fermionic supersymmetry operators on a Cli�ord vacuum state� The

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���� THE MASSIVE CASE ��

energy�momentum four vector Pm commutes with the supersymmetry oper�ators Q�� �Q ��� This means that the mass operator P � is a Casimir operatorand therefore all particle states of an irreducible representation has the samemass� When we construct our representations we shall consider the two casesM � constant �� and M � �

Before doing this let�s prove that every representation of a supersymme�try algebra contains equal number of fermionic and bosonic states�

Proof� Introduce the fermionic number operator Nf � such that ���Nf haseigenvalue �� on a fermionic state and �� on a bosonic state� We see that

���NfQ� � �Q����Nf

because Q� is a fermionic operator� This combined with the fact that thetrace is cyclic Tr�XY � � Tr�Y X� gives

Tr����Nf

nQi��

�Q ��j

o�� Tr

����NfQi

��Q ��j � ���Nf �Q ��jQ

i�

� Tr��Qi

����Nf �Q ��j �Qi����Nf �Q ��j

� �����

On the other hand we have� from the supersymmetry algebra�

Tr����Nf

nQi��

�Q ��j

o�� Tr

����Nf�m

� ��Pm�

ij

� �m� ���ijTr

����NfPm

������

Combining � ���� and � ���� gives

Tr����Nf

�� �����

for �xed nonzero momentum�

��� The Massive Case

We can now start to construct representations of the supersymmetry algebra�corresponding to massive one�particle states� First we use the Poincar�e�algebra to boost the system into the rest frame

P � � �M� � P � �M� � � �

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�� CHAPTER �� THE SUPERSYMMETRY ALGEBRA

�Note� space�like metric��The supersymmetry algebra then takes the form

nQi�� �Q ��j

o� M��� ���

ij

nQi�� Q

j�

o�

n�Q ��i� �Q ��j

o�

After a convenient rescaling of the supersymmetry operators we get

nai�� �a ��j

o� ��� ���

ij

nai�� a

j�

o�

n�a ��i� �a ��j

o�

where

ai� ��pM

Qi�

�a ��i ��pM

�Q ��i

The representations of this algebra is well known� The new operators ai� and�a ��i can be seen as annihilation and creation operators� The representationis then constructed from a Cli�ord vacuum �� de�ned by

ai�� �

Higher spin states are produced by acting with creation operators on thevacuum

��n�������ni����in �

�pn�

�ai���

�y � � ��ain�n�y�Each index pair ��i� Ai� can take �N di�erent values� The creation operatorsanticommutes� so the order of the index pairs can be rearranged �picking up

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���� CENTRAL CHARGES ��

signs�� For a �xed n we then get

��Nn

�di�erent states� The total dimension

of the representation becomes

d ��NXn��

��Nn

�� �� � ���N

� ��N

Here we can again check that there is equal number of bosonic and fermionicstates in the representation� If we act with ���Nf on each state� we get

�NXn��

��Nn

�����n � ��� ���N

The highest spin state is a singlet with spin S � N���

We can now list a table over the massive representations of the N�extendedsupersymmetry algebra

NnJ � ��� � ���

� � �� � �

� � �� � �

� � � �� � �� � �

� � �� � � �

� � �� � ��

�����

��� Central Charges

We will now look at a massive representation including central charges� Forconvenience we restrict ourselves to the N�� case� We start by making aunitary transformation on Zij �

Zij � U ikU

jlZ

kl

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�� CHAPTER �� THE SUPERSYMMETRY ALGEBRA

such that

Z�ij � Z ��

���

where Z is diagonal N���dimensional matrix with real eigenvalues� This isalways possible because Zij is antisymmetric� The N � � supersymmetryalgebra now becomes n

Q�i��

�Q��j

o� � i

j ����M

nQ�i�� Q

�j�

o� ����

ijZ

n�Q�

��i� �Q���j

o� � �� ���ijZ

Now we de�ne new operators

a� ��p�

�Q�� � ��� �Q��

b� ��p�

�Q�� � ��� �Q��

Note that we don�t have to worry about the Lorenz invariance anymorebecause we have chosen to be in the rest frame� The anticommutators of thenew operators are zero in most cases

fa�� b�g � fa�� a�g � fb�� b�g � fb�� b�g � n�a ����b ��

o�

n�a ��� �a ��

o�

n�b ����b ��

o�

n�b ����b ��

o� n

a���b ��

o�

The only nonzero ones are

fa�� �a�g � �M � Z� ����b���b�

�� �M � Z� ���

If the central charge is equal to the mass �Z � M�� the only interesting oper�ator is a� �and its hermitian conjungate �a��� In that case the representationis exactly the same as for N � ��

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���� THE MASSLESS CASE ��

��� The massless Case

We will now look at the massless case� where we can choose P � ��E� � � E��The supersymmetry algebra becomes

nQi��

�Q ��j

o�

�E

���

�ij

nQi�� Q

j�

o�

n�Q ��i� �Q ��j

o�

We see that we can put Qi� � � After rescaling� we get

nai� �aj

o� �ij

nai� aj

o�

f�ai� �ajg �

were

ai ��pEQi

�aj ��pE

�Q ��j

Just as before� we choose ai to be annihilation operator

ai�� �

where � is the lowest �vacuum� helicity� The states are build up as

��n���n

��i������in �

�pn�

�ain � � � �ai���

This state has helicity �� n� and is

�Nn

��times degenerated� The represen�

tation has dimension d � �N � This is derived in the same way as for the

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�� CHAPTER �� THE SUPERSYMMETRY ALGEBRA

massive case�

If we require a given mass less representation to be TCP selfconjugate�we must add the representation produced from ��� to the representationabove �except for the case when N��� this representation is automaticallyTCP selfconjugate��If this is done� we get the following table

Nn� � ���

� � �� �

� � � �� �

� � � �� �� �� �

�����

A more detailed discussion on the subjects covered in this chapter isgiven in ���� ��� and ����

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Chapter �

The Wess�Zumino Model

In this chapter we will construct the simplest supersymmetric �eld theory�the Wess�Zumino theory �this we will be done in Weyl notation and thesame results� but in Majorana notation� is given in Appendix D�� To dothis we start with a complex scalar �eld in ��dimensional spacetime� and itssupersymmetry partner� The partner will be an anticommuting spinor �eld�We examine how the �elds transform under supersymmetry transformations�The algebra has to close� which gives restrictions on the spinor �eld andthe transformations �see below�� We will also see that o��shell� we needto introduce a new bosonic �eld� This must be done in order to close thealgebra� but we can also see that the fermion�boson symmetry requires thisextra �eld�

��� Supersymmetry Transformations of Component

Fields

We introduce Grassmann parameters ���� �� ���� These anticommuting para�meters satisfy the following relations

n��� ��

o�

f��� Q�g �

���

�Pm� ��� �

��

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� CHAPTER �� THE WESS�ZUMINO MODEL

We can now express the supersymmetry algebra entirely in terms of com�mutators �

�Q� �� �Q�

� ��m��Pm

��Q� �Q� �

���

�Pm� ��� �

�����

where the Weyl indices are contracted in the natural way ��Q � ��Q��� seeAppendix B�In�nitesimal supersymmetry transformations are de�ned as

�� ���Q � �� �Q

�����

These operators can now act on the component �elds in our supermultiplet�A� �� ���� where A is a complex scalar bosonic �eld� �� is a fermionic spinor�eld etc� The supersymmetry transformations can now be written as

��A ���Q� �� �Q

A

��� ���Q� �� �Q

���

This transformation maps tensor �elds into spinor �eld and the other wayaround� From the algebra we see that the mass dimension of Q is ����see chapter ��� Bearing these observations in mind we can write down thetransformation of A

��A � ��

and take that as a de�nition of the spinor �eld ��� In the same way we canwrite

���� � a ��m���� mA� ��F

�where a � C�� This de�nes the �eld F � Finally we get

��F � b����mm�

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���� SUPERSYMMETRY TRANSFORMATIONS OF COMPONENT FIELDS�

�where b � C��The calculations below will show that it is necessary to introduce the auxil�iary �eld F � otherwise the algebra cannot close� However if we restrict our�elds by their equations of motion ��go on�shell��� it is consistent to putF � �

As was said in the beginning of this chapter� we want the algebra toclose� This gives us restrictions from which we can determine the constantsa and b� We start by calculating multiple supersymmetry transformationson A

������A � a���m���mA� ����F �����

On the other hand we have

���� � ��� �A ����Q� ��� �Q� ��Q � ��� �Q

�A

�����Q� ��Q� �

���Q� ��� �Q

������ �Q� ��Q

������ �Q� ��� �Q

�A

� ����m���Pm � ���

m���Pm�A � �Pm � im�

� i ����m��� � ���

m����mA �����

Combining � ���� and � ���� gives

a � �i

A similar calculation gives us b

�������� � a ��m����� m ����A� � b ����� ����F �

� �i ��m����� ��m�� b ����� �����mm�

If we go on�shell we see that the equation of motion for �� ��m� ��m�� �� � �

will make the second term vanish� Hence we can put F � on�shell and thealgebra will close anyway� Rewrite the �rst term in the last expression as

�i ��m����� ��m� � �i�m� ������� �

��m�� � �i�m� ��

����m��

��� ��� � � B���

� � i

��m� ��

����

n� ��

�����

��m�

�� ��n�

Page 20: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

� CHAPTER �� THE WESS�ZUMINO MODEL

� � i

�����

n����m� ����

���n m�� � � B��

�i

�����

n�������mn �

�� � �n� ����

m ����m��

� i ����n���m�� �

i

�����

n����n� ����m ���m�� � � B��

� i ����n���m�� � i ����� �����

mm�

Hence

�������� � i ����m���m�� � i ����� �����

mm�� b ����� �����mm�

On the other hand � ���� holds even if the commutator acts on a spinor �eld�and therefore we get

b � �iNow we know how all the component �elds in the Wess�Zumino model trans�forms under supersymmetry transformations

��A � ��

���� � �i ��m���� mA � ��F

��F � �i����mm������

Notice that F transforms into a total derivative� This is always the case forthe component of highest dimension in a supermultiplet�

What about the statement made in chapter �� saying that there shouldbe equal number of bosonic and fermionic degrees of freedom Well if welook at the �elds in the representation� this is indeed true o� shell

Field degrees of freedom �real�A � bosonic� � fermionicF � bosonic

On�shell we have the equation of motion

�A �

Page 21: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

���� SUPERSYMMETRY TRANSFORMATIONS OF COMPONENT FIELDS��

�m� ��m�� �� �

F �

The degrees of freedom then read

Field degrees of freedom �real�A � bosonic� � fermionicF bosonic

So we are relieved���

Even though this method of getting the supersymmetry transformationsis totally general� it is not so convenient �especially not for extended super�symmetry models�� In many extended supersymmetry models all o��shellcomponent �elds are not known �for example the N � � Yang�Mills theory��In these cases we have to go on�shell in order to close the supersymmetryalgebra �see chapter ���� In the next chapter we will introduce the conceptof superspace and super�elds� When we work in superspace we don�t have toworry about the algebra to close and how to �nd an invariant Lagrangian�because the super�elds transform into themselves� This is of course verynice and will save us much work�

A more detailed discussion on the subjects covered in this chapter isgiven in ��� and ����

Page 22: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

�� CHAPTER �� THE WESS�ZUMINO MODEL

Page 23: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

Chapter �

Superspace and Super�elds

It is now time to introduce a very convenient way of calculating with repre�sentations of supersymmetry � The key to the �trick� is to observe that thesupersymmetry algebra can be viewed as a Lie algebra with anticommutingparameters �just as in chapter ��� If we only add the space spanned bythese anticommuting parameters to our ordinary Minkowski space� we get asuperspace� Element of the superspace are labeled by z �

�x� �� ��

We are now ready to de�ne group elements corresponding to supersym�metry transformations �now acting as passive transformations on the para�meter space�

G�x� �� ��

� ei��x

mPm��Q��� �Q� �����

We can easily multiply two group elements� For example �using Baker�Hausdor��s formula�

G �y� �� ���G�x� �� ��

� ei��y

mPm��Q��� �Q�ei��xmPm��Q��� �Q�

� ei��ymPm��Q��� �Q�xmPm��Q��� �Q� �

���ymPm��Q��� �Q��xmPm��Q��� �Q��

� � ��� � ei���ym�xm�Pm������Q�������� �Q� i��m ��Pm� i

��m��Pm�

� G

y � x� i

���m�� �

i

���m��� �� �� ��� ��

��

Page 24: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

�� CHAPTER �� SUPERSPACE AND SUPERFIELDS

��� Supersymmetry Transformations and Covari�

ant Derivatives

We can get an explicit expression for the supersymmetry operators� as lineardi�erential operators in superspace

Q� � ��

� i��

m� ��

�� ��m

�Q �� � � ��

� i��

��m� ��m�����

These operators do indeed satisfy the supersymmetry algebra�Q�� �Q ��

�� ��m� ��Pm �Pm� Q�� �

fQ�� Q�g � �Pm� �Q ��

�� � �Q ��� �Q�

��

Proof�

�Q�� �Q ��

�� Q�

�Q �� � �Q ��Q�

���

i

��m� ��

����m

��

�� �� �i

����m� ��m

��

�� �� �i

����m� ��m

���

i

��m� ��

����m

� �

��

�� ��� i

��m� ��m �

i

����m� ��

��m �

i

��m� ��

����

�� ��m ��

��m� ��

�������n� ��mn �

�� ��

��� i

��m� ��m �

i

��m� ��

����

�� ��m �

� i

����m� ��

��m �

����m� ���

n

� ������mn

�� ��

��� i�m� ��m �

����n� ���

m

� ������mn �

Page 25: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

���� SUPERFIELDS ��

�� ��

���

����m� ���

n

� ������mn

� ��m� ��Pm

The other calculations are similar�

The sign change in the supersymmetry algebra� stems from the fact thatwe have chosen �for convenience� operators in superspace to act from theleft� In other words the transformations are no longer �active� operators�operating on component �elds� but passive �changing the parameter space��If we instead would have chosen �right action�� the supersymmetry operatorswould have been

D� � �� �

i��

m

� ������m

�D �� � � �� �� � i

����m� ��m

�����

Now these operators are introduced as covariant derivatives �covariant withrespect to the supersymmetry�� satisfying the following relations

�D�� �D ��

�� �m� ��Pm

nD ��� �Q ��

o�

fD�� D�g � � �D ��� Q�

��

��D ��� �D�

��

n�D ��� �Q ��

o�

fD�� Q�g �

�����

At this point a comment on dimensions is in order� To simplify our formulaswe have used �h � � and c � �� This means that the length has the dimensionof inverse mass� Then � ���� and � ���� gives �Q� � �D� � ���� ��� �

���� ����� in powers of the mass� This is consistent with � �����

��� Super�elds

Super�elds are functions of superspace� and are expressed in terms of theirpower series expansion in � and ��

F�x� �� ��

� a� ���� � �� �� ��� �� � ����b�� � �� ���� ��

�b� �����

����� ��W

���

Page 26: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

�� CHAPTER �� SUPERSPACE AND SUPERFIELDS

��

����� �� �� �

���� �

����� ���� ��

� �� ��� �

����� �� ���� ��C

�� ���� �����

where a� ��� ��� ��� b��� �b� ����� W

��� ��� are complex functions of x� All higher

terms of � and �� vanishes� because they are Grassmann variables�If we use B�� and B�� we can write � ���� as

F�x� �� ��

� a� ���� � �� �� ��� �� �

���b �

�����b� �

���m ��vm �

����� �� �

�� ��

������� � �

����� ��C �����

The supersymmetry transformations are now de�ned as

��F � ��a� ������ � �� ���� ������ � � �

��Q� �� �Q

F �����

Note that �� commutes with � and ���From this expression� we can �nd the explicit supersymmetry transforma�tions for the component �elds �more about this in the following chapters��

It is easy to see that both linear combinations and products of super��elds are again super�elds� Thus super�elds form a linear representation ofthe supersymmetry algebra� However these representations are in generalreducible� This is understandable because in superspace we have introducedcovariant derivatives in the algebra� These operators where not presentwhen we stated the supersymmetry algebra in chapter �� So the super�eldis an irreducible representation of the algebra containing both the opera�tors in the supersymmetry algebra and the covariant derivatives� To get anirreducible representation of the supersymmetry algebra� we must imposecovariant constraints on the super�elds� Examples of such constraints are

chirality� �DF �

and reality

�F y � F

�� Note that it is important not to re�

strict the x dependence of the component �elds� For example DF � �DF � gives F � �� where � is a constant�We shall examine the chirality and reality constraints in more detail soonenough�

Page 27: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

���� INTEGRATION AND FUNCTIONAL DERIVATIVES ��

��� Integration and Functional Derivatives

We must also de�ne integration in superspace� otherwise it is impossible towork with a Lagrangian �even though we will soon see that integration andderivation over Grassmann variables is basicly the same thing in superspace��Let us de�ne Z

�d��� �� � � ��

Z �d�� �� �� �� � � ��

��

Zd� �

Note that this is the only de�nition invariant under the shift � � � � ��Therefore integration and derivation is identical

Z�d��� �

��

The normalization is �the volume element dz � dxd��d��� �Zd�� �� � �

Zd��� ��� � �

It is now easy to verify the following relationsZdx �d��� � �

Zdx D�j������

Zdx �d��� � � B��� � ��

Zdx D�

���������Z

dz ��

��

Zdx D� �D�

���������

����

Finally we de�ne functional derivation in superspace

�F �z�

�F �z��� �

�z � z�

����

Page 28: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

� CHAPTER �� SUPERSPACE AND SUPERFIELDS

valid for unconstrained super�elds F �z��In the next chapter we will use this de�nition �or actually the chiral form ofit�� to get equations of motion from the Lagrangian�

A more detailed discussion on the subjects covered in is chapter is givenin ��� and ����

Page 29: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

Chapter �

The Wess�Zumino Model in

Superspace

In this chapter we will examine what the chirality condition on a super�eldmeans explicitly� Let us start by solving the chirality condition�

�D ��� � �����

We could of course solve this equation by letting �D act on a general su�per�eld� given in � ��� �� Then we would get nine equations �one for eachorder in � and ���� but only three are �non�trivial�� However we will solvethe chirality condition with a method which will be used frequently in therest of this thesis� Start by transforming all �elds and operators by thes�transformation�

s � ei��m ��m

We get

�� � s�

�D��� � s �D ��s

�� � �

�� ��

�D ��� � � �D����

� � s �D ��s��s� � �����

Page 30: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

� CHAPTER �� THE WESS�ZUMINO MODEL IN SUPERSPACE

Therefore �� � �� �x� �� and can be expanded as

�� � A� ��� ����F �����

where A� � and F are complex �elds� depending on x� The super�eld �� issaid to be in the chiral representation�It is now straightforward to transform �� back to our normal coordinatesystem

� �z� � A� ��� ����F � i

���m ��mA� i

���m ��m�� �

���������A

�����Proof�

� �z� � s���� � e�i��m ��m

A � ���

���F

�� i

���m��m � �

��m ����n��mn

�A� ���

���F

� � B� � A� ����

���F � i

���m��mA

� i

����m ��m��

���������A

We can now de�ne �consistent with the expansion�

� �z�j������ � A �x�

D�� �z�j������ � �� �x�

���D

�� �z����������

� F �x�

�����

Page 31: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

���� SUPERSYMMETRY TRANSFORMATIONS ��

��� Supersymmetry Transformations

It is now easy to calculate the supersymmetry transformations

��A � ��

���� � ��F � i��m ��

�mA

��F � �i����mm������

�compare with ����

Proof�

��� �z� � � ��� ���Q� �� �Q

� �z�

Note that Qj������ � Dj������� Hence

��A ���Q� �� �Q

��������� �

� �D� � �� ��D��j������

� ����

���� � ��D��j������ ���D � �� �D

D��

��������

� ��D�D�����������

� �� ���D

��D�����������

���D� �

� ��

������D

��

����������

� ����n�D ��� D�

o����������

� ��F � i�����m

� ��m�

���������

� ��F � i�m� ��

����mA

��F ���D � �� �D

��

�D��

����������

�hD�� �

i

� ��

��� �� �D ��D�D��

����������

Page 32: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

�� CHAPTER �� THE WESS�ZUMINO MODEL IN SUPERSPACE

��

��� �� � �D ��� D

��D��

����������

� �

��� ��D� �D ��D��

����������

�i

��� ���m�

��mD��

����������

� �

��� ��D� � �D ��� D�

��

����������

� � i

��m ��m�� i

��m ��m�

� �i�m ��m�

We see that this representation is exactly the same as the Wess�Zuminomodel we calculated in chapter ��

��� The Wess�Zumino Lagrangian

The most general action �in the free Wess�Zumino model�� invariant underthese supersymmetry transformations is

S� � �Rdz��� �

Rdx

�A�A� � FF � � i��mm��

�����

Proof�

S� � �

Zdz��� �

ZdxD� �D����

� �

Zdx���

����������

���j������ �

A� ���

���F � i

���m��mA

� i

����m ��m��

���������A

��

�A� � �����

�����F � �

i

���m��mA

��

i

������m�m���

���������A�

�����������

Page 33: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

���� THE WESS�ZUMINO LAGRANGIAN ��

��

��������A�A� �

i

��������m�m���

����� ��FF � �

���m ��mA��

n ��nA� �

i

����m ��m������

��������A�

�A

� � int� by parts� B� B���

��

�A�A� � i��mm��� FF �

This was the straight forward way of calculating the Lagrangian� There ishowever �at least� one other method� where we only use the supersymmetryalgebra together with ���

S� � �Zd��� �

Zd �D�D����

���������

� �� is chiral� ��

Zd �D�

���D��

����������

��

Zd��

�D���� �

D���� �

��D ���

� �D ��D�����������

�� �D�D�����������

�What do the terms mean We use the supersymmetry algebra and ���

�D������������

� ��F �

D�����������

� ��F

� �D ������������ � ��� ��

�D ��D�����������

�n�D ��� D�

oD��

���������

� D� �D ��D�����������

�n�D ��� D�

oD��

���������

� D�n�D ��� D�

o����������

� �n�D ��� D�

oD��

���������

� �i��m ���mD�����������

Page 34: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

�� CHAPTER �� THE WESS�ZUMINO MODEL IN SUPERSPACE

� �i��m ���m��

��j������ � AA�

�D�D�����������

� �D ��

n�D ��� D�

oD��

���������

� �D ��D� �D ��D��

���������

�n�D ��� D�

o� �D ��� D�

�����������

� � �D ��� D��n �D ��� D�

o����������

� �i��m ���i�n� ��mn����������

� � B��

� ��A

So if we put these expressions back into the Lagrangian� we get

S� �Zdx

�A�A� � FF � � i��mm��

just as expected� Note that we did not really have to use the power expansionof the chiral super�eld� In Appendix D it is checked that this Lagrangian isinvariant under supersymmetry transformations�

This was the Lagrangian for the free theory� we may also add couplingterms

S� � ��Rdx

�d��� � hc

� �

Rdx �F � hc�

Sm � mRdx

�d���� � hc

� m

Rdx

�AF � �

��� � hc

�Sg � �g

Rdx

�d���� � hc

� g

Rdx

�A�F � ��A� hc

����

Proof�

S� � ��

Zdx

�d��� � hc

�� ��

Zdx

�D��

���������

� hc�

� ��

Zd ��j�� � hc�

� �

Zdx �F � hc�

Page 35: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

���� EQUATIONS OF MOTION ��

Sm � m

Zdx

�d���� � hc

�� m

Zdx

��������

� hc�

� m

Zdx

�A� ���

���F � � � �

���������

� hc

� m

Zdx

AF � �

��� � hc

Sg ��g

��

Zdx

�d���� � hc

��g

Zd��������

� hc�

�g

Zdx

�A� ���

���F � � � �

���������

� hc

�g

Zdx

A� ���

���F � � � �

�A� � ���A� ��FA � �

�����

�������

� hc

�g

Zdx

�FA� � ��A� hc

�The total Lagrangian then becomes

Stot � S� � S� � Sm � Sg

�Rdx

�A�A� � FF � � i��mm��

� �

Rdx �F � hc��

mRdx

�AF � �

��� � hc

�� g

Rdx

�A�F � ��A� hc

����

��� Equations of Motion

In this section we will derive the equations of motion for the component�eld� �rst for the free model and then we consider also the coupling terms�When we act with a functional derivative on a chiral �eld� it will look like�see ���

�� �z�

�� �z��� �

�x� x�

���� � ��

� ��

��D��

�z � z�

Page 36: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

�� CHAPTER �� THE WESS�ZUMINO MODEL IN SUPERSPACE

The validity of this relation is most easily seen if we think of � as being inthe chiral representation �see �����We start by varying the free Lagrangian

�S���� �z�

� �Zdz�D��

�z � z�

��z��

� D�� �z� �

We use ��� to show that

D�����������

� � F �x� �

�D ��D�����������

� � �mm�� �x� �

�D�D�����������

� � �A �x� �

So the equations of motion in the free model are

F �x� �

�mm�� �x� �

�A �x� �

���� �

Before we are able to vary the total Lagrangian � ���� we must rewrite thecoupling terms in full superspace� This is most easily done by observing theidentity

��

�D�D�

�� � � � is chiral ������

which follows from the supersymmetry algebra�So we can rewrite the coupling terms as

S� � ��

Zdz

�D�

���

�D�

����

Sm � m

Zdz

��D�

�� ���D�

���

Sg ��g

��

Zdz

���D�

��� ���

�D�

����

Page 37: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

���� EQUATIONS OF MOTION ��

If we include these terms� �S� �

becomes

D�����

D� �D�

��

�m

D� �D�

��� �

g

���D� �D�

��� �

If we use ����� we can rewrite this as �remember that a constant is a specialcase of a chiral �eld�

D��� ��� �m�� � g��� � ������

This is the equation of motion for the coupled theory�

A more detailed discussion on the subjects covered in is chapter is givenin ��� and ����

Page 38: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

� CHAPTER �� THE WESS�ZUMINO MODEL IN SUPERSPACE

Page 39: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

Chapter �

The N�� Yang�Mills Theory�

the Abelian Case

The Wess�Zumino model has a scalar bosonic �eld as its lowest dimensionalcomponent �eld� The next representation we will look at has a �!��spinfermionic �eld as its lowest dimensional �eld� The simplest way to do thisis again to put covariant constraints on a general super�eld �but this timewe will also use a gauge choice�� The constraint will be reality� We will seethat super�elds satisfying this constraint will also be invariant under a localgauge transformation� This explains the name Yang�Mills theory�

��� The Vector Field

We are now going to examine the second covariant constraint mentioned inchapter �� the reality constraint�

V � V y �����

If a super�eld satis�es ��� it is called a vector �eld� The power series expan�sion of V reads�

V � C�x� � i� �x�� i�� � �x� �i

��� �M�x� � iN�x���

i

����� �M�x�� iN�x��� ���m��vm�x� �

Page 40: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

� CHAPTER �� THE N�� YANG�MILLS THEORY THE ABELIAN CASE

i����

p����x�� i

���mm �x�

�� i�����

p���x�� i

��mm � �x�

��

�������

D�x� �

�C�x�

where C� M � N � vm� D are all real�Why the coe"cient are chosen in this special way will soon be obvious� Theexpansion of V is not unique� it will still be real after the following super�eldgauge transformation

V � V � �� �y � is chiral �����

The power series expansion of the gauge shift is �see ����

�� �y � A�Ay � �� � �� �� ��

���F �

�����F � � i

���m��m

�A�Ay

�� i

�������mm� �

i

�������mm �� �

���������

�A �Ay

�Reading o�� term by term� we get

C � C � A�Ay

� � i�

M � iN � M � iN � iF

vm � vm �i

�m

�A� Ay

� � �

D � D

We see now that we can choose C� � M and N to be equal to zero� Thischoice is called the Wess�Zumino gauge� Then we get

V jWZ � ����m ��vm � ip�������� i

p��������

����� ��D

Page 41: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

���� THE CHIRAL FIELD STRENGTH ��

V ����WZ

� ������ ��vmvm

V ����WZ

� �����

The vector �eld V is a supersymmetric generalization of the Yang�Millspotential�

��� The Chiral Field Strength

Now we want to de�ne a gauge invariant chiral �eld strength W��

W� � ���D�D�V �����

The �eld strength is invariant under Wess�Zumino gauge transformations�

Proof�

W� � ��

��D�D�

�V � �� �y

� W� � �

��D�D��� �

��D�D��

y

� �� is chiral� � W� � �

��D ��

n�D ��� D�

o�

� W�

We will now calculate W� explicitly� Again we will save time by usingthe s�transform �see ����� in fact we will express W� in the transformedstate� The correction terms which occurs when transforming back to theordinary coordinate system� are of higher order in �� �� and will not e�ectour calculations�

W� � i� ��

mn��� Fmn � �p�����mm��

�� ip��� � ��D �����

where Fmn � mvn � nvm�

Page 42: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

��CHAPTER �� THE N�� YANG�MILLS THEORY THE ABELIAN CASE

Proof�

W �� � ��

��D��D�

�V�

�D��� � �

�� ��

D�� �

��� i�m

� ������m

V � � sV � � ���

� �

i

���m��m �

���������

��

����m��vm � i

p�������� i

p��������

����� ��D

� ����m��vm � ip�������� i

p��������

��

�������D �

i

����� ��mvm

D��V

� �

��� i�m

� ������m

��

����m��vm � i

p�������� i

p��������

����� ��D �

i

�������mvm

�Keep only terms of order ����

�� �i

p������� � ������D � i������

mvm �

�i�m� ��

������n ��mvn �

p��m

� ����������m��

� �ip������� � ������D � i������mvm �

i���� ��m��n��� mvn ��p��������m

� ��m��

��

Page 43: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

���� SUPERSYMMETRY TRANSFORMATIONS ��

Hence

W �� � ��

��D��D�

�V�

� �ip��� � ��D � i��mvm � i ��m��n��� mvn �

�p����m

� ��m��

��

The third and fourth terms give

i�m� ����n ��� �� ��mvn � i��

mvm � � B��

� i���mn� �

� � �mn� b�

���mvn � i��

mvm

� � i

���mn��� Fmn

Hence

W �� �

i

���mn��� Fmn � �p

�����mm��

�� ip��� � ��D

We see thatW� contains only the gauge invariant component �elds D� ��and Fmn� Furthermore the �eld strength is chiral and satis�es the condition

D�W�j������ � �D ���W ��

�������� �����

which only tells us that D is real�

��� Supersymmetry Transformations

Now we want to �nd the supersymmetry operators for the gauge invariantcomponent �elds� We use the same method as in chapter �

���� � ip���D � i

�p���mn��� Fmn

��D � �p�

���mm��� �� ��mm�

��Fmn � i

p����mn���� i

p������mn��

�����

Proof�

Page 44: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

��CHAPTER �� THE N�� YANG�MILLS THEORY THE ABELIAN CASE

��W�j������ ���Q� �� �Q

W�

�������� �

��D � �� �D

W�

��������

����D�

��ip��� � ��D �

i

���mn��� Fmn � � � �

�����������

� ��D �i

���mn��� Fmn

On the other hand

��W�j������ � �ip�����

Hence

���� �ip���D � i

�p���mn��� Fmn

To get the supersymmetry transformations for D of Fmn� we must use dif�ferent symmetries of the indexes�

Start by contracting D�W� with ���� This gives

���D�W�

���������

� �D�W�j������ � � B��

� ��D

and

���D � ��D�W�j������ �

���D� � �� �� �D ��

��D�W�

����������

��

������D

�W�

����������

� �� �� � �D ��� D�

�W�

���������

�p����m� ��m

�� �� � i�� ���m� ��m�ip���

�p���mm���

p�����mm�

Then we have the supersymmetry transformation of D

��D ��p�

���mm��� ����mm�

Page 45: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

���� SUPERSYMMETRY TRANSFORMATIONS ��

If we take the symmetric part instead� we get

D��W��

���������

� �����D �i

���mn�����Fmn � � B��

�i

���mn��� Fmn

This gives

i

���mn��� ��Fmn �

��D � �� �D

D��W��

���������

��

���D�����W��

����������

� �� ��n�D ��� D��

oW��

���������

� �p�����

m

�� ��m��

�� �p��� ����

u ���� m���

Thus we have the equation

i

���mn��� ��Fmn � �

p��������

m

�� ��m��

�� �p��� ����

m �������jmj���

The easiest way to solve this equation is to make the anzats

��Fmn � x���mn���� y�����mn�

Put the anzats into the equation

LHS �i

���mn�����

�x���mn���� y�����mn�

�� � B�� B��

�i

���x���m��j�j����m�� �� � �y�� ����

m ���� jmj���

� ix����m�� ��m

�� �� � iy� ����m ���� jmj���

If we compare LHS with RHS� we see that x �p�i and y �

p�i� Hence

��Fmn �p�i���mn����

p�i�����mn��

This completes the proof�

Page 46: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

��CHAPTER �� THE N�� YANG�MILLS THEORY THE ABELIAN CASE

��� The N� Yang�Mills Lagrangian

Since W� is chiral� the ���component of W�W� is a Lagrangian density

S� � �

Rdx

�d��W�W� � hc

Rdx

���D

� � �F

mnFmn � i��mm��� ����

Proof� Zdx

�d��W�W� � hc

��

Zdx �W�W�j�� � hc�

Thus we have to calculate W�W�� keeping only terms ���terms

W�W�j�� �

�p��� � ��D �

i

���mn��� Fmn � �p

�����mm��

� � � �� ��p��� � ��D �

i

���mn��� Fmn � �p

�����mm��

�� � ������

��

� i��mm���D� �i

�D�� ��mn� �

� ��Fmn

������

i

�D ��mn��� ����

������

� �

���mn��� �� ��pq� �

� ��FmnFpq

������

i��mm��

� �i��mm���D� � �

��mn� �

� ��pq� �� FmnFpq

The last term gives

��mn� �

� ��pq� �� FmnFpq �

�� �� FmnFmn � i

� �� �mnpqFmnFpq

��

�FmnFmn � i

��mnpqFmnFpq

Hence

W�W�j�� � D� � �i��mm��� �

�FmnFmn �

i

��mnpqFmnFpq

Page 47: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

���� THE N�� YANG�MILLS LAGRANGIAN ��

The last term drops out of the Lagrangian because it is a total derivative�

S� ��

Zdx �W�W� � hc� � � D and Fmn are real�

��

Zdx �W�W�� �

Zdx

�D� � i��mm��� �

�FmnFmn

The last thing we will do in this chapter is to check the invariance of theLagrangian under the supersymmetry transformations ����

Proof�

�L � D�D� �

�Fmn�Fmn � i���mm��� i��mm���

��p�D���mm��� ����mm�

ip�Fmn�

���mn���� �����mn��

��

ip�

i�D � �

���mn��Fmn

��uu��� ip

���nu

i��D �

���mn ��

Fmn

The�D� ��

�terms gives

�p�D��mm��� �p

�D��mm�� �

The �D� ���terms gives

� �p�D����mm��

�p���mm

���D � �Int� by parts�

The�Fmn � ��

�terms gives

ip�Fmn��mn����

i

�p���mn��Fmn�

uu�� �

ip�Fmn�mn��� i

�p���mn��Fmn�

uu�� �

h�mn�u � �mnu � ��u�m�n�� �uFmn� �

i�

Page 48: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

�CHAPTER �� THE N�� YANG�MILLS THEORY THE ABELIAN CASE

ip�Fmn�mn��� ip

�Fmn�mn�� �

The �Fmn� ���terms vanish in the same way�

So the Lagrangian is invariant�

A more detailed discussion on the subjects covered in this chapter is givenin ���� ��� and �����

Page 49: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

Chapter �

Abelian and non�Abelian

Interactions

In this chapter we will calculate the Lagrangian for a theory with inter�action between the Wess�Zumino theory and the N�� Yang�Mills theory�We start with the Abelian �U���� case� and then generalize to non�Abelianinteractions�

�� Abelian Interactions

The chiral super�eld �i has a supercharge ti

��i � e�i�ti�i

where � is a real constant�Then ��i is chiral too� because �D� � � Furthermore the ordinary Wess�Zumino Lagrangian �see chapter �� is invariant under this global gaugetransformation

Sglobal � �

Zdz�i�

yi �mij

Zdx

�d���i�j � hc

�� �����

�gijk��

Zd�d���i�j�k � hc

������

where mij � if ti � tj �� and gijk � if ti � tj � tk �� �

Page 50: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

� CHAPTER � ABELIAN AND NON�ABELIAN INTERACTIONS

Now we would like to make the U����invariance local� The gauge trans�formation gets a bit more complicated in this case

��i � e�ti��z��i �z�

����yi � e�ti�y�z��yi �z������

The chirality of � requires # to be chiral as well� The kinetic term ��i�yi �

in the Lagrangian is not invariant under this gauge transformation �because# �� #y�� If we remember the gauge transformation of the vector �eld � �����we might get an idea� The kinetic term transforms as

��i���iy

� �i�yie

�ti��y���

So if we put a vector �eld in the exponential� the kinetic term transformscorrectly

��ietiV

� ���iy

� �ietiV �yi

The total Lagrangian then becomes

SAtot � �

Rdx

�d��W�W� � hc

� �

Rdz�ie

tiV �yi �

mij

Rdx

�d���i�j � hc

gijk�

Rdx

�d���i�j�k � hc

�����

The only calculation that has not already been done �in chapter � and ��� isthe new kinetic term� So �let�s go to work��

�yetV ����������

Ay � �� �� �

�����F y �

i

���m��mA

y� i

�������m�m �� �

���������Ay

��

� � t

����m��vm � i

p�������� i

p������� �

�������D

�� t����� ��vmv

m

��

A� �� �

���F � i

���m��mA� i

�������mm� �

���������A

������������

��

��Ay�A� i

��� �����m ��m�

����������

��

�F yF �

���m����n ��mA

ynA����������

� i

�������m�

�m ��

��

�������� ��

Page 51: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

��� ABELIAN INTERACTIONS ��

��A�Ay � itAy��m��vm��

n��nA������� ��

� ip�tAy��������

���������

t

�AyAD � �t�� ����m ��vm��

������ �� � i

p�t�� ��������A

���������

it��m ��mAy��n��vnA

������� ��

� t�AyAvmvm

��

�Ay�A�

�F yF �

i

���mm �� � it

�AyvmmA�

itp�Ay�� �

t

�AyAD �

t

���m ��vm �

itp�A �����

it

�AvmmA

y � t�AyAvmvm

Then the total Lagrangian looks like

SAtot �

Rdx

���D

� � i��mm��� �F

mnFmn�

A�i�y�A�i� � F �i�yF �i� � i��i��mm ���i� � �it�i�A

�i�yvmmA�i��

�it�i�A�i�vmmA

�i�y � �p�it�i�A

�i�y���i� � �p�it�i�A

�i� ���i���� �t�i�A�i�yA�i�D�

t�i���i��m ���i�vm � �t��i�A

�i�yA�i�vmvm �mij

�AiFj � �

��i�j � hc��

gijk� �FiAjAk � �i�jAk � hc�

�����

This is the most general coupled Lagrangian� containing the following �elds

Yang�Mills

����

Fmn Vector multiplet�� Fermionic spinorD Real bosonic �eld �auxiliary�

Wess�Zumino

����

A Complex bosonic �eldF Complex bosonic �eld �auxiliary��� Fermionic spinor

Page 52: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

�� CHAPTER � ABELIAN AND NON�ABELIAN INTERACTIONS

�� Non�Abelian Interactions

The non�Abelian case of gauge interaction is a straitforward generalizationof the Abelian case �note the order�

�� � e���

����y � �ye��y�����

where # is a matrix

# ji � �T a� j

i #a

The generators of the gauge group �T a� are hermitian in the representationde�ned by �� and they are normalized in the adjoint representation

Tr�T aT b

�� k�ab

We must now check that our expressions still are gauge invariant� Let�s startwith the kinetic term

���yeV

�� � �yeV �

where V is in the adjoint representation of the Lie�group �V � V aT a��

Proof� we want to prove that

eV�

� e�y

eV e�

The Baker�Hausdor� formula

eAeB � eA�LA���B�coth�LA��B��

gives

V � � V � #� #y � � � �

This is the gauge transformation that gave the Wess�Zumino gauge �to �rstorder��

Page 53: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

��� NON�ABELIAN INTERACTIONS ��

The expansion involves only commutators of the group generators �thisfollows from the Baker�Hausdor� formula�� so the transformation does notdepend on the representation� Then the kinetic term is really invariant�

What about the �eld strength In the non�Abelian case it is de�ned as

W� � ���D�e�VD�e

V �����

which is a generalization of the Abelian case� Now the �eld strength trans�forms like

W �� � e��W�e

� ����

Proof�

eV�

� e�y

eV e� �heV

e�V�

� �i�

e�V�

� e��e�V e��y �

e�V�

D�eV �

� e��e�V e��y

D�e�y

eV e� � �chain rule� # is chiral�

� e���e�VD�e

V �D�

�e�

�Compare with A� gAg�� � gdg�� in the next chapter��Then we have

�D ��

�e�VD�e

V�

� �# is chiral�

� e����D ��e

�VD�eV�e� � e�� �D ��D�e

Finally

W �� � e��W�e

� � e�� �D ���D

��D�e�

However� the last term vanishes

e�� �D ���D

��D�e� � e�� �D ��

n�D

�� � D�

oe�

Page 54: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

�� CHAPTER � ABELIAN AND NON�ABELIAN INTERACTIONS

This completes the proof�

The fact that the trace is cyclic then guarantees that

Tr �W�W��

is a gauge invariant expression� The Lagrangian is then given by

Sn�Atot � �

��kg

Rdx

�d��Tr �W�W�� � hc

� �

Rdz ��i

�eV� j

i�j�R

dx�d��mij�i�j � hc

� �

Rdx

�d��gijk�i�j�k � hc

����

where mij and gijk must be symmetric and invariant under gauge transfor�mations� The normalization of the gauge �eld kinetic term is chosen to scaleback to our ordinary normalization if V � �gV �

In chapter we will see how we can derive the non�Abelian Yang�Millstheory with a geometric approach� We will also calculate an explicit expres�sion for the Lagrangian�

A more detailed discussion on the subjects covered in is chapter is givenin ��� and ����

Page 55: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

Chapter �

Di�erential Geometry in

Superspace

In this chapter we will work out the formalism of di�erential geometry insuperspace� in terms of di�erential forms� With this formalism� the theorywill be manifestly covariant under coordinate transformations� Covariantderivatives will also be introduced� to make the theory gauge invariant� Fi�nally we will derive the Bianchi identities in superspace� which will be solvedin the next two chapters�

��� Di�erential Forms

We will start by de�ning �ordinary� forms and then generalize to superspace�

A di�erential form of order r is a totally antisymmetric tensor of type� � r��

The Wedge product �� of r one forms is de�ned as

dx�� dx�� � � � dx�r �XP�Sr

sign �P �dx�� � dx�� � � � � � dx�r

where the sum means that we antisymmetriz the tensor product�

An r�form ��� can now be expanded in the vector space of r�forms �ac�tually the dual space to the ordinary vector space� spanned by the local

��

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�� CHAPTER �� DIFFERENTIAL GEOMETRY IN SUPERSPACE

coordinate axis�� at a point in the manifold

� ��

r�E�� � � � E�r��r �����

��

r�dx�� � � � dx�r��r �����

We now de�ne exterior product of an r�form ��� and a q�form ��� as

� � � dx�� � � � dx�r��r ����� dx�� � � � dx�q��q������ ��q �������r �����dx

�� � � � dx�r dx�� � � � dx�q

The exterior product maps an r�form and a q�form into an �r � q��form�Furthermore it is linear and associativ but commutes as

� � � ���rq � �We have to make one more important de�nition� the exterior derivative

d� � dx�r � � � dx�� dx� ��r �����x�

which maps r�forms into �r� ���forms �compare with the exterior product��The exterior derivative follows the following rules

d �� � �� � d� � d�

d �� �� � � d� � ���q d� �

dd� � ���r ����� �x� is smooth�

It is now easy to generalize the de�nitions and results above to superspace�The main di�erence is that elements in superspace generally do not commute

zAzB � ���f�A�f�B� zBzA

where A �and B� is any superspace index �m� �� ��� and f is a function whichtakes the value if the argument is a vector index and � if the argument isa spinor index�

The base in the vector space of ��forms is now de�ned as

EA ��Em� E�� �E ��

��dxm� d��� d�� ��

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���� THE COVARIANT DERIVATIVE AND THE FIELD STRENGTH��

The Wedge product between two ��forms commutes as

dzA dzB � � ���f�A�f�B�dzB dzA

A di�erential r�form is given by

� ��

r�EA� � � � EArWAr ���A�

��

r�dzA� � � � dzArWAr ���A�

Note that there is always an even number of indices between the two beingsummed over� The exterior product and the exterior derivatives obey thesame rules in superspace as before� From now on we shall drop the symbol for exterior multiplication �we know no other way to multiply forms�� Theexterior derivative is expanded as

d � EAA � dxm

xm� d��

��� d�� ��

�� ��

Theories which are described in terms of exterior derivatives and di�erentialforms are manifestly covariant under general coordinate transformations�

��� The Covariant Derivative and the Field Strength

Gauge theories are not only covariant under general coordinate transfor�mations� they are also covariant under a local structure group� This is acompact Lie group for Yang�Mills theories� In general a representation ofthe group is spanned by di�erential forms

�� � ��g

�g � ei�aTa � where T a are generators of the Lie group�� where the bar mark

that �� is in the fundamental representation of the Lie group �which meansthat �� is a row�vector��Objects that transform lineary under a representation of the structure groupare called tensors� The exterior derivative does not transform tensors intotensors

d�� � d���g � ��dg � d��g

As usual we must introduce a connection to compensate for the inhomoge�neous term ��dg� The connection must be a Lie algebra valued ��form

A � dzAAaAT

a ����

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� CHAPTER �� DIFFERENTIAL GEOMETRY IN SUPERSPACE

which transforms as

A � g��Ag � g��dg ����

under gauge transformations� �Note that A is in the adjoint representationof the Lie group� so A is a matrix��

Now we can de�ne the covariant derivative of an r�form in the fundamentalrepresentation

D��� d�� � ��A ����

This expression is covariant under gauge transformations

D�� � d���g� ���g �g��Ag � g��dg

� ��dg � d��g � ��Ag � ��dg

� d��g � ��Ag

��D��

g

The covariant derivative of an r�form in the adjoint representation is

D�� d���A� ���rA� ����

Proof�

d���A� ���r A� � d�g���g

�� g���g

�g��Ag � g��dg

��

���r�g��Ag � g��dg

��g���g

� g���dg� g��d�g � ���r dg���g � g���Ag �

g���dg� ���r g��A�g � ���r g��dgg���g

� g��d�g� g���Ag � ���r g��A�g

� g��D�g

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���� THE BIANCHI IDENTITIES �

This proves that the covariant derivative transforms correctly�

There is one tensor which can be constructed from the connection and itsderivatives� the �eld strength

F � dA� A� ����

The �eld strength tensor is a Lie�algebra valued two form

F ��

�EAEBF a

BATa ����

�The symmetry of the indices are FAB � � ���f�A�f�B�FBA��It transforms as

F � g��Fg

Proof�

F � dA�A�

� d�g��Ag � g��dg

���g��Ag � g��dg

��g��Ag � g��dg

� g��Adg � g��dAg � dg��Ag � dg��dg � g��A�g �

g��Adg � g��dgg��Ag � g��dgg��dg

�h � d

�g��g

�� dg��g � g��dg

i

� g��dAg � dg��Ag � dg��dg � g��A�g � dg��Ag � dg��dg

� g��Fg

��� The Bianchi Identities

The �eld strength is the only covariant tensor that can be constructed fromthe connection and its derivatives� Higher derivatives leads to identities�because dd � �� These identities are called the Bianchi identities

DD� � �F

DF � ����

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� CHAPTER �� DIFFERENTIAL GEOMETRY IN SUPERSPACE

Proof� the following algebraic calculation shows the �rst identity$

dD� � d �d���A� � dd���dA� d�A

� �dA� d�A � ��F �A�

�� �D�� �A�A

� �F �D�A

The other one is shown in a similar way

dF � AdA� dAA � A�F � A�

���F � A�

�A � AF � FA

Hence �remember F is in the adjoint representation�

DF � dF � FA �AF � �AF � FA� � FA �AF �

This completes the proof�

It is also useful to write out the explicit expression for the Bianchi iden�tities

DBDA� ��

�F aBA�T

a

D�CFBAg � ���

In the next chapter we will solve this Bianchi identities after we have put onsome constraints on the �eld strength� The solution will turn out to be theN�� Yang�Mills theory� just as we hoped���

A more detailed discussion on the subjects covered in is chapter is givenin ��� and ����

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Chapter �

The N�� Bianchi Identities

We will now use the di�erential geometry that we developed in chapter �to derive the N�� Yang�Mills theory� The idea is to put constraints onthe components of the �eld strength� Then the Bianchi identities becomesequations rather than identities� Deriving the Yang�Mills theory is thenequivalent to solving these equations�

�� Constraints

In chapter we used dzA as a basis for our di�erential forms� This basis ishowever not the most convenient one to use in explicit calculations� That iswhy we introduced covariant derivatives in chapter �� From now on we shalluse this more natural basis of covariant derivatives

Dm �

xm

D� �

��� i

��m� ��

����m

�D �� � �

�� �� �i

����m� ��m ����

Satisfying the conditions in ���The exterior derivative can now be expanded in this base

DA� � EAA�A

The reader who is familiar with vielbeins will recognize EAA� as vielbein �elds

�which are coe"cient functions of the vielbein forms�� The convention we

��

Page 62: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

�� CHAPTER �� THE N�� BIANCHI IDENTITIES

use is rather unusual� to get standard conventions we should put unprimedindices as curved indices �usually written as M��

The matrix EAA� can be read o� from ��

DA� � EAA�A �

�BBBB�

� mm�

� i��

m

�� ������ � �

��

i��

��m� ��� �� ���

��

�CCCCA

�BBBBB�

m

��

�� ��

�CCCCCA

The inverse is given by

EA�

A �

�BBBB�

� m�

m

i��

m�

� ������ � ��

i� ��

m� ����� �� �����

�CCCCA

This is very nice� but nothing comes for free� We have to pay a price forhaving covariant derivatives in the basis� and that is the covariant derivativeof the basis functions do not vanish� We have a torsion

DEA � �Flat space� � dEA � TA �

�ECEBT A

BC

The only non�vanishing components of the torsion is

Tm� �� � �i�m� ��

Tm���

� �i��m�������

Proof�

EA�

��Em�

� E��

� �E ���

���dxm� d��� d�� ��

EA�

A

dxm

�i

�d���m

� ���� �� �

i

�d�� ����

m� ������ d���

��d�� ���

gives

TA�

� dEA�

� �Remember right action� �

� i

�d���m

� ���� �� � i

�d�� ����

m� ���d��� �

���id���m�

� ��d�� ��� �

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���� CONSTRAINTS ��

Then the Bianchi identity �� can be written as

P�ABC�

�DAFBC � TD

ABFDC

�� ����

where �ABC� means that we take a cyclic sum� respecting the symmetriesof the ��form base�Proof�

DF ��

�D�EAEBFBA

��

�EAEBDFBA � �

�EADEBFBA �

�DEAEBFBA

��

�EAEBECDCFBA � �

�EAECEDTB

DCFBA ��

�ECEDTA

DCEBFBA

��

�EAEBEC

�DCFBA � TD

ABFDC

So

DF � �

EAEBEC�DCFBA � TD

ABFDC

��

Which component �elds are involved in the di�erent components of the �eldstrength

The components of the �eld strength is given in terms of the connection�see Appendix C�

FAB � DAAB � ���f�A�f�B�DBAA � i �AA� ABg� TCABAC

so it is better to ask for the component �elds in the connection �A�� Inchapter � we derived the representation of a vector �eld

V � � � � � ��m��Am � i������� i��������

�������D

which is invariant under the gauge transformation

V � V �#� #y

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�� CHAPTER �� THE N�� BIANCHI IDENTITIES

This gives

�D ��V � �D ��V � �D ��#y

D�V � D�V �D�#

D��D ��V � D�

�D ��V �D��D ��#

y � D��D ��V � i�m

� ��Dm#y

In the superspace form of the Yang�Mills theory� we see that it is consistentwith the expansion and the gauge shift to put

A �� �D ��V

A� D�V

A� �� � �m� ��Am D�

�D ��V

����

We see that A� involves all the component �elds needed to represent theYang�Mills theory �o��shell��

Now we want to transform the Bianchi identities into interesting equations�We do this by putting constraints on the �eld strength� These constraintsmust be restrictive enough to make the theory interesting� but if we �go toofar� we will end up with all the components of the �eld strength equal tozero� It turns out that

F�� � F �� �� � F� �� � ����

is the right choice� A more detailed discussion on the �eld strength �compo�nents� reality etc�� is given in Appendix C�Note that h

�D ��� �D ��

o� � �TC

�� ��DC�� iF �� ��� � iF �� ���

and if � is chiral

F �� �� �

Therefore F�� � F �� �� � is called representation preserving constraints�

The F� �� � constraint gives us Am in terms of A�� A �� and derivatives of

these two� This is natural because all the component �elds in the Yang�Millstheory is contained in A��

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���� SOLVING THE BIANCHI IDENTITIES ��

�� Solving the Bianchi Identities

If we put in the constraints �� in the Bianchi identities ��� we get the fol�lowing equations �listing only the equations which are not dependent troughhermitian conjugation�

D�mFnp� � ����

�D�mFn�� � D�Fmn � ����

DmF�� � �D��F��m � �

D��F��m � ���

DmF� �� �D�F ��m � �D ��F�m � i�n� ��Fnm � �

D�F ��m � �D ��F�m � i�n� ��Fnm ���

D��F��� � �

� ��� �

�D��F�� �� � �D ��F�� � iFm���m�� �� � �

�m� ��Fm� � �m� ��Fm� �����

�����

We are now going to solve these equations� Start by splitting Fm� intoirreducible parts

Fm� � �m� ��Fm� � F

�� ��

F�� �� � F���� �� � F���� ��

This leads us to make the following anzats

F�m � G�m ��

��m� ��

�W�� �����

where ��m���G�m � �

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�� CHAPTER �� THE N�� BIANCHI IDENTITIES

If we put this ansatz into ���� we get

�m� ��

G�m �

��m� ��

�W���� �m� ��

G�m �

��m� ��

�W���

� �

�m� ��G�m � �m� ��G�m ��

��m� ���m� �� � �m� ���m� ��

��W

�� � � � B��

�m� ��G�m � �m� ��G�m �

So we get no restriction on W�� � Multiply with ��n �� from the left

G�m � �����

This gives �see ����

F�m ��

��m� ��

�W�� �����

�F ��m ��

��m� ��W

� �����

Now we use ��� in �

D��F ��m � �D ��F�m � i�n� ��Fnm � � ����

�m� ��D�W� � �m� ��

�D ���W �� � �i�n� ��Fnm

Multiply by �� ���p � this gives

��m��p��

�D�W

� � ���p�m� �� ���D ��

�W �� � �iTr ��n��p�Fnm �� ��iFpm�

Now we split this equation in one symmetric and one antisymmetric part�

Start with �mp�

���m��p�

� �

�D�W

� �����p�m�

� ��

���D ��

�W �� � �

��m��p � �p��m� ��D�W

� � ���m�p � ��p�m� �� ���D ��

�W �� � �

D�W� � �D ��

�W �� �

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���� SOLVING THE BIANCHI IDENTITIES ��

where we used B�� in the last step� Thus we have

D�W� � �D ���W �� �����

The antisymmetric part gives �use B���

Fmn � � i

����mnD�W� � hc ����

The next identity we are going to use is �

D��F��m � � D���jmj�� �� �W �� � �

D��m� ���W �� �D��m� ��

�W �� �

Multiplying by ��m��� gives �see B���

��D��W

�� � �D��W

�� �

Thus we have

D��W �� � ����

The other Bianchi identities give nothing new� and we can now summarizethe results in this section

F�� � F� �� � F �� �� � �

F�m � ���m� ��

�W��

Fmn � � i�

��mnD�W� � hc

D�W� � �D ���W ��

D��W �� �

��� �

Note that this is exactly the same equations that we had for the �eld strengthW� in chapter � �but now we are dealing with non�Abelian Yang�Mills gaugetheories��

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� CHAPTER �� THE N�� BIANCHI IDENTITIES

�� The Lagrangian for the non�Abelian N� Yang�

Mills Theory

Now� when we have solved the Bianchi identities� we want to get the La�grangian� The method we are going to use is the same as we used in sec�tion ���� We start out with the following identities

W�j������� �p�i��

D�W� j������ �i

���mn��� Fmn � �

����D�W�

D�D�W� j������ �p�����

m� ��Dm

�� �� �p�����

m� ��Dm

���� �����

Proof� The �rst identity is actually a de�nition and can not be derived�However the second one follows from the Bianchi identities

D�W� j������ ��D��W�� �D��W��

����������

Start with the antisymmetric part

D��W�� � k���D�W� �hmultiply by ���

i�

D�W� � k ����D�W� �

k � ��

Then we use �

D�F ��m � �D ��F�m � i�n� ��Fnm � � � ��

��m� ��D�W� � �m� ��

�D ���W �� � �i�n� ��Fnm

Multiply by ��m ��

�� � D�W

� � �� �� ���

��D ��

�W �� � �i ��n��m� � Fnm

Multiply by �� and symmetrizise � and �

�D��W� � �i ��nm�� Fnm �

D��W� �i

���nm�� Fnm

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���� THE LAGRANGIAN FOR THENON�ABELIAN N�� YANG�MILLS THEORY�

Hence

D�W�j������ �i

���mn��� Fmn � �

����D�W�

So the second identity in ��� is all right� Let�s check the third

D�D�W� j������ � D�D��W��

���������

� D�D��W��

���������

Start with the anitsymmetric part

D�D�W� j������ � ��

����D�DW

����������

� ��

����D�

�D ��W �

����������

� � i

�����

m� �Dm

�W �����������

��p�����

m� �Dm

�� �

The symmetric part gives

D�D��W��

���������

�i

���mn��� D�Fmn

����������

� � ��

� �i ��mn��� D�mFn��

���������

� � � �

� � i

���mn��� Dm�n� ��

�W������������

��p���mn��� �n� ��Dm

���� � � B��

��p�

��m� ����

n ��� � �mn���

��n� ��Dm

���� � � B��

��p�

����m

� ����� � �m

� �����

�Dm

����

� �p�����

m

� ��Dm

���� �

�p�����

m

� ��Dm

����

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� CHAPTER �� THE N�� BIANCHI IDENTITIES

Hence

D�D�W� j������ �p�����

m

� ��Dm

���� �

p�����

m

� ��Dm

����

Now we will use ��� to get an explicit expression for the Lagrangian� Wewill use the cyclic properties of the trace to convert superspace covariantderivatives into Yang�Mills covariant derivatives

L �

Zd��Tr �W�W�� � ��

�D�D�Tr �W

�W��

����������

� ��

�Tr�D�D�W

�W�

So we have

��

�Tr�D�D�W

�W�

�����������

� ��

�Tr��D�W�

�W� � D�W�D�W�

�����������

��

�Tr��i�m�

��

�Dm

�� ������

i

���mn��� Fmn � �

����D�W�

��

i

���mn��� Fmn � �

����D�W�

��

��

�Tr

�i���m� ��Dm

�� �� � �

���mn� ��pq�FmnFpq�

�������D�W�DW � i

����D�W� ��

mn��� Fmn �

i

���mn��� Fmn���D�W�

The third and the fourth term vanishes because of B��� If we also use that

Tr ��mn�pq�FmnFpq � �FmnFmn � �i�mnpqFmnFpq

we getZd��Tr �W�W�� �

�Tr��i��mDm

��� FmnFmn�

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���� THE LAGRANGIAN FOR THENON�ABELIAN N�� YANG�MILLS THEORY��

i

��mnpqFmnFpq �

�D�W�D�W�

� �D�W�� �D�

��

�Tr

�D� � FmnFmn � �i��mDm

���i

��mnpqFmnFpq

�Note that the last term is a total derivative��

So the Lagrangian will for the non�Abelian N�� Yang�Mills theory is

S � �

Rdx

�d��Tr �W�W�� � hc

RdxTr

���D

� � �F

mnFmn � i��mDm��� �����

We recognize this expression from chapter �� note however the covariantderivative and the trace over the gauge indexes�

A more detailed discussion on the subjects covered in is chapter is givenin ����

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�� CHAPTER �� THE N�� BIANCHI IDENTITIES

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Chapter �

The N�� Yang�Mills Theory

The method we used to derive the N�� Yang�Mills theory in the last chapterwas based on a geometrical approach� This is a very useful way of looking atgauge theories� because the equations get to be manifestly covariant undergauge transformations� In this chapter we will use the same method to derivethe the N�� Yang�Mills theory� which is an extended supersymmetric gaugetheory�

���� The Bianchi Identities

The superspace element is now de�ned as

z ��xm� ��i� �� ��i

�where i � �� � is a SU ����index �see chapter ���

Then we have nDi�� �D ��j

o� i� i

j �m� ��Dm � i� i

j D� �� �� ���

�The index structure and the covariant derivatives are discussed in detail inAppendix C�� The component �elds get the following index structure �wetake FAB as an example�

FAB � Fmn� Fim�� Fm ��i� F

ij��� F

i� ��j � F �� ��ij

The Bianchi identities are now given by

DAFBC � TDABFDC � cyclic terms � �� ���

��

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�� CHAPTER � � THE N�� YANG�MILLS THEORY

Explicitly we get

Di�F

jk�� �Dj

�Fki�� �Dk

�Fij�� � �� ���

�D ��iF �� ��jk � �D ��jF �� ��ki � �D ��kF �� ��jk � �� ���

Di�F

j� ��k �Dj

�Fi���k

�D ��kFij���

i�m� ��Fim��

jk � i��m���F

jm��

ik � �� ���

DmFij��Di

�Fj�m � Dj

�Fim� � �� ���

DmF �� ��ij�D ��iF ��mj � �D ��jFm ��i � �� ���

DmFi

� ��j�Di

�F ��mj � �D ��jFim� � i�n

� ��Fnm�

ij � �� ��

DmFin� �DnF

i�m � Di

�Fmn � �� ��

DmFn ��i �DnF ��mi � �D ��iFmn � �� �� �

DmFnp �DnFpm � DpFmn � �� ����

Di�F �� ��jk � �D ��jF

i���k � �D ��kF

i

� ��j�

i�m� ��Fm ��k�

ij � i��m���Fm ��j�

ik � �� ����

If we use the reality conditions of the �eld strength �see Appendix C�� weget

� � �y � � � � �

� � ��y � � � ��

� � ��y � � � ��

� � ��y � � � ���

So we choose as a set of independent Bianchi identities � ��� � ��� � ��� � �� � �and � ����

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� ��� THE BIANCHI IDENTITIES ��

Now we would like to use the same method as we used in the N�� case�so we must �nd convenient covariant constraints� We start by trying thesame as in the last chapter

F ij�� � F i

� ��j�

However this turnes out to be to restrictive� as we will soon see� Equa�tion � �� gives

�m� ��Fim��

jk � ��m���F

jm��

ik � �

�m� ��Fim��

jk � �m� ��F

jm��

ik � �

F i�����

jk � F j

�����i

k �

Just as in the N�� case we split F i���� into two parts

F i����

� Gi

������ �H i������

Start with the symmetric part

Gi�������

jk � Gj

�������i

k � � �put i � k��

Gi������ �

The antisymmetric part gives

�H i�������

jk �Hj

�������i

k � � �put i � j��

�Hj

�� ���� � �Hj

������ � �

Hj

������ �

Then we have

Fm� � � � � ��

Fmn �

Page 76: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

�� CHAPTER � � THE N�� YANG�MILLS THEORY

We see that all the components of the �eld strength are zero� which impliesthat our covariant constraints were to restrictive� How can we relax theconstraints so we can get interesting information The right way to handlethis problem is to keep the �rst constraint F i

� ��j� and use the index

symmetries of the �eld strength

Fij�� � F

ji�� � F

�ij����� � F

�ij�����

We put

F ij�� � F ij

�� � �

F�ij����� �

This leads us to choose the following constraints on the �eld strength

F i� ��j

F ij�� � F

�ij����� �� ����

�see Appendix C�

Then � ��� can be written as

F i

� ��j�

F ij�� � �ij��� �W

�� ����

If we put this into the Bianchi identities � � �� � � ����� we get

�jk���Di��W � �ki���Dj

�u�W � �ij���Dk

��W � �� ����

�ij��� �D ��k�W � iF i

�����j

k � iF j�����

ik � �� ����

Di�F

j���� � Dj

�Fi���� � �ij���D� ��

�W � �� ����

�Di�F� �� ��j � �D ��jF

i���� � iF

� ��� ���i

j � �� ���

D� ��Fi

� ���� D

� ��Fi� ��� �Di

�F� ��� �� � �� ���

D� ��F� ��� �� � cyclic terms � �� �� �

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� ��� THE BIANCHI IDENTITIES ��

Where we have converted the free Lorenz indices to spinor indices �see Ap�pendix A� m� � ��� n� � ��� p� � ���

We will now solve these equations� just as in the last chapter� Start bymultiplying � ��� with �jk�

��

�Di��W � � i

j ��

� Dj��W � � i

j ��

� Dk��W � �

�Di��W �Di

��W � Di

��W �

Hence

Di��W � �� ����

We also have

�D ��iW � �� ����

That � ��� follows from � ��� is checked by the following calculation �seereality conditions in Appendix C�

F �� ��ij ��F ij��

�y���ij��� �W

�y� �ij� �� ��W

Now let�s split F i���� in the same manner as in chapter

F i���� � Gi

������ � ��� �H i��

We put this into � ���

�ij��� �D ��k�W � i� j

k

�Gi

������ � ��� �H i��

��

i� ik

�Gj

������ � ��� �Hj��

Multiplying by �ij gives

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� CHAPTER � � THE N�� YANG�MILLS THEORY

���� �D ��k�W � i�ik

�Gi

������ � ��� �H i��

��

i�kj�Gj

������ � ��� �Hj��

��

���� �D ��k�W � �i��� �H ��k

Thus we have

�H ��k � i �D ��k�W �� ����

To solve � ��� for G� we put j � k

��� �Di���W � �i

�Gi

������ � ��� �H i��

��

i�Gi

������ � ��� �H i��

��

��� �Di���W � �iGi

������

This gives

Gi������ � �� ����

Then � ��� and � ��� gives us

F i���� � i����

ij �D ��j�W �� ����

This also gives

F� �� ��i � �i� �� ���ikDk�W �� ����

because F� �� ��i � �

�F i����

�y�

Then � ��� gives

Di�

�i����

jk �D ��k�W�� Dj

�i����

ik �D ��k�W�

� �ij���D� ���W

Multiplying by �ij��� yields

�iDk��D ��k

�W � iDk��D ��k

�W � �D� ���W �

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� ��� THE BIANCHI IDENTITIES �

�iDk��D ��k

�W � �D� ���W � � � �� � ����

�inDk� �

�D ��k

o�W � �D� ��

�W �

D� ���W � D� ��

�W

So � ��� just gives consistency�

Let�s try � ��

�Di�

��i� �� ���jkDk

�W�� �D ��j

�i����

ik �D ��k�W�

� iF� ��� ���i

j

Now we make the following ansatz for F� ��� ��

F� ��� �� � �Fmn � �Fnm� � ��� �M� �� ��� � � �� ��M����

Then we get

i� �� ���jkDi�Dk

�W � i����ik �D ��j

�D ��k�W � i��� �M� �� ����

ij � i� �� ��M�����

ij

Put i � j

� �� ��D�kDk�W � ��� �Dk

���D ��k

�W � ���� �M� �� ��� � �� �� ��M����

Because of the symmetry the equation splits into two parts� we get

�M� �� ��� ��

��Dk���D ��k

�W

M���� ��

�D�kDk

�W

Now we have� from the anzats

F� ��� �� � �Fmn � �Fnm� � ��� �M� �� ��� � � �� ��M����

but on the other hand we have �see Appendix A�

F� ��� �� � �m� ���

n

� ��Fmn

��

��m� ���

n

� ��� �n� ���

m

� ��

�F�n

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CHAPTER � � THE N�� YANG�MILLS THEORY

These two relations gives

�Fmn � �Fnm� � ��� �M� �� ��� � � �� ��M���� ��

��m� ���

n� ��� �n� ���

m� ��

�Fmn

If we multiply this relation by ���� we get

� �M� �� ��� ��

��m� ���

n���Fmn � �

��n� ���

m���Fmn

Hence

�M� �� ��� � ���mn� �� �� Fmn �� ����

Now we want to get the inverse of � ���� Use B�� to write

Fmn ��

��� ��m ��

��n F� ��� �� �

��� ���m �

� ��n� F� ��� ��

��

��� ���m ��

��n�

���� �M� �� ��� � � �� ��M����

��

����mn�

�� �� �M� �� ��� ��

���mn�

��M�mn�

��

����mn�

�� �� �Dk���D ��k

�W ��

���mn�

�� D�kDk�W

So we have

Fmn ��

����mn�

�� �� �Dk���D ��k

�W ��

���mn�

�� D�kDk�W �� ���

However there is more information in � ��� If we multiply it with a tracelesssigma matrix ��ji �� we get

iDi��

ji �jkDk

�W� �� �� � i �D ��j�j

i �ik �D ��k�W��� � �

Di��ikDk

�W� �� �� � �D ��j�kj �D ��k

�W��� � �

� �� ������D�DW � �D� �D �W

This gives

D�DW � �D� �D �W �� ���

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� ��� YANG�MILLS THEORIES FOR GENERAL N �

The other Bianchi identities don�t give anything new� so we can summarizeour results � � ��� � � ��� as follows�

The constraints

F i� ��j

F ij�� � �ij��� �W

gives

Di��W �

�D ��iW �

F i���� � i����

ik �D ��k�W

F� �� ��i � �i� �� ���ikDk�W

F� ��� �� � �����

�Dk���D ��kW � �

�� �� ��D�kDk�W

D�DW � �D� �D �W

�� �� �

���� Yang�Mills Theories for General N

When we derived the N�� Yang�Mills theory� we had some problems gener�alizing the constraints we had in the N�� case� Our �rst suggestion turnedout to be too restrictive� In this section we will look at this calculation forgeneral N and see that the N�� case is di�erent from the others�We start out with the same constraints as in the N�� case

F ij�� � F i

� ��j�

where i and j are SU �N��indexes�

Just as before � �� gives

�m� ��Fim��

jk � ��m���F

jm��

ik � �

�m� ��Fim��

jk � �m� ��F

jm��

ik � �

F i�����

jk � F j

�����i

k �

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� CHAPTER � � THE N�� YANG�MILLS THEORY

Anzats

F i����

� Gi

������ �H i������

The symmetric part gives

Gi�������

jk � Gj

�������i

k � � �put i � k��

�N � ��Gi������ �

The antisymmetric part gives

�H i�������

jk �Hj

�������i

k � � �put i � j��

�Hj

������ �NHj

������ � �

�N � ��Hj

������ �

Now we see that if N � � there will be no restriction on Hj

������� but forN � � all the components of the �eld strength must be zero� That is whythe constraints can be more restrictive in the N���

���� Reduction

If we look at table ��� we could suspect that the representation N�� Yang�Mills theory is the direct sum of the N�� Yang�Mills theory and the Wess�Zumino theory� This is also what we could expect because of the absence ofcentral charges� which mix the supersymmetries� In this section we will doa calculation which shows that this is indeed the case�

The Lagrangian in the N�� case is

L � d�Tr �WW � � hc �� ����

where d� � d����� d�

���� d�

���� d�

���� �the SU ��� indexes are inside parentheses��

We transform the i � � part of the volume element to derivatives

L � d�Tr �WW � � ��

�d�����D���D���Tr �WW �

� ��

�d�����Tr

�D���D���WW

� ��

�d�����Tr

�WD���D���W �D���WD���W

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� ��� REDUCTION �

What do the two terms mean

We start with the �rst one� The super�eld W is chiral � �D����� W � �� so

if we integrate over the �����parameters it should be equal to � in chapter ��

The second part of the �rst term gives

D���D���W � �D��� �D��� �W �� ����

Proof� The last identity in � �� is

D�DW � �D� �D �W

where � is traceless�If we use �� and �� in this identity� we get

D����D���� W �D����D���

� W � � �D�����

�D��� �� �W � �D�����

�D��� �� �W

and

D����D���� W � D����D���

� W � �D�����

�D��� �� �W � �D�����

�D��� �� �W

Subtracting these identities gives � ����

So we have

Tr�WD���D���W

�� Tr

�W �D��� �D��� �W

� ��d������Tr�W �W

This gives the Wess�Zumino kinetic term in the Lagrangian � ����

The second term in the Lagrangian is D���WD���W � Now D���� W is nothing

but the �eld strength �W�� in the N�� Yang�Mills theory�

Proof� Wemust prove thatD���� W is chiral and thatD����D���

� W � �D��� ���D �����

�W

�corresponding to the identity D�W� � �D ��W���� The chirality condition is

trivial� because

�D��� ��D���� W � � C�� � i� �

� D� ��W �

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� CHAPTER � � THE N�� YANG�MILLS THEORY

The reality condition is a little more tricky

D����D���� W � � C�� � ���D�� �

i

�����

�� � �W�W�

�D��� ���D �����

�W � � C�� � � ����

�D���W �

i

�� ���� ���

�W� �W

�then � �� gives

D����D���� W � �D��� ��

�D �����

�W

So the N�� Lagrangian � ��� can be written as

L � �d�����Tr �W�W�� � d�����d������Tr �����

when the ���� dependence has been integrated away�

A more detailed discussion on the subjects covered in this chapter is givenin ��� and �� ��

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Chapter ��

The N�� Yang�Mills Theory

In this �nal chapter we will derive the Lagrangian and the supersymmetrytransformations of the N�� Yang�Mills theory� The N�� case is special inmany ways �the most important is the PCT�selfduality�� We will not dis�cuss these matters here� because our goal is only to get an explicit expressionfor the Lagrangian on�shell �the o��shell representation is not known�� Themethod we will use is dimensional reduction� We will start from a supersym�metric theory in � �dimensional space�time and then preform a dimensionalreduction down to an extended �N��� supersymmetric theory in � space�time dimensions�

���� The ���dimensional Lagrangian

The Lagrangian in � �dimensions is given by �� � diag ��� � � � � � ���

L � ��F

�a�ABF�a�AB � i

����a�%ADA�

�a� ������

where

FAB � AAB � BAA � �AA� AB�

DA� � A�� ���AA�

and A � � � � � � is a � �dimensional index� The %�matrices are �� �� Diracmatrices and the spinors are majorana �see Appendix A��

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� CHAPTER ��� THE N�� YANG�MILLS THEORY

The supersymmetry transformations are

�� � ��%

ABFAB�

�AA � i��%A� � �i��%A�������

The Lagrangian is invariant under the supersymmetry transformations�Proof�

�L � ��

�F �a�AB�F

�a�AB �

i

�����a�%ADA�

�a� �i

����a�%ADA��

�a�

We start by calculating

�F �a�AB � A��A

�a�B

��h�A

�a�A � A

�a�B

i� �A� B�

� i��%BA��a� � i

h��%A�

�a�� A�a�B

i� �A� B�

� i��%BDA��a� � �A� B�

and

����a� � ����a�y%�

��

�%ABF

�a�AB

�y%�

��

��y%yAB%�F

�a�AB �

��%��y

%A%� � %A�

��

���%ABF

�a�AB

This gives

�L � �i��%BDA��a�F�a�AB �

i

���%ABF

�a�AB%

CDC��a� �

i

����a�%ADA

�%BCFBC�

The two last terms cancels the �rst because

��%AB%CF�a�ABDC�

�a� � ���%ABC � ��C�A%B�

�F�a�ABDC�

�a�

� ���%BF�a�ABDA��a�

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����� DIMENSIONAL REDUCTION �

where we have used that D�AFBC� � � The last term gives the same

���a�%A%BCDF �a�BC� � ����a�%BDAF

�a�AB� � �%BF

�a�ABDA���a��

� ���%BF�a�ABDA��a�

So the Lagrangian is really invariant under the supersymmetry transforma�tions �����

���� Dimensional Reduction

In this section we will do the actual reduction� We do this by assuming thatthere is no the dependence on x� � � � � x�� We also split our matrices� startingwith the %�matrices

%m � �m � �� ��

%��i � �� �i � ��

%��j � �� � �j � ��

%�� � �� � � ��

where �i� �j �i� j � �� �� �� are � � real antisymmetric matrices �repre�senting SO ��� and acting on the internal supersymmetry indices�� satisfyingthe algebra n

�i� �jo

�n�i� �j

o� ���ij

h�i� �j

i� ���ijk�k

h�i� �j

i� ���ijk�k

h�i� �j

i�

The gauge �elds in the extra dimensions now become scalars

A��i� Ai

A��j� Bj

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CHAPTER ��� THE N�� YANG�MILLS THEORY

and the spinor �eld becomes a set of � Majorana �elds on which �i� �j act�actually on the supersymmetry index�

� � � � �p�

���i

�� � �� � �p�����i�

�� � �TCThe result Lagrangian for the � dimensional theory is

L � ��F

�a�mnF�a�mn � �

�DmA�a�iDmA�a�i � �

�DmB�a�jDmB�a�j �

i����a��mDm�

�a� � �����a�

h�iA

�a�i � i�jB

�a�j ��� ��a�

i� V �A�B�

������where

V �A�B� ��

hA�a�i� � A�a�i�

i hA�a�i�� A

�a�i�

i�

hB�a�j� � B�a�j�

i hB

�a�j�� B

�a�j�

i�

hA�a�i� B�a�j

i hA�a�i � B

�a�j

i

Proof� We start by splitting FABFAB in three terms representing A � � �� �� �� A � �� �� � and A � �� � � The �rst one gives

FABFAB FmnFmn � FmiFmi � FmjFmj

� FmnFmn ��mAi � iAm �

hAm� Ai

i��mAi � iAm � �Am� Ai�� �

�mBj � jBm �

hAm� Bj

i��mBj � jBm � �Am� Bj ��

� �iAm � � � FmnFmn �DmAiDmAi �DmBjDmBj

�DmAi means mAi � �Am� Ai���The second term gives

FABFAB F imFim � F i�i�Fi�i� � F ijFij

� DmAiDmAi �hAi� � Ai�

i�Ai� � Ai� � �

hAi� Bj

i�Ai� Bj �

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����� DIMENSIONAL REDUCTION

and the third

FABFAB F jmFjm � F ijFij � F j�j�Fj�j�

� DmBjDmBj �hAi� Bj

i�Ai� Bj � �

hBj� � Bj�

i�Bj� � Bj� �

So

FABFAB � FmnFmn � �DmAiDmAi � �DmBjDmBj � V �A�B�

The spinor part is given by

��%ADA� ���� � �p

�����i�

�%ADA

�� � �p

���i

��

� ��

��� � ����i� ��m � �� ��Dm � �� �i � ��Di�

�� � �j � ��Dj

�� �

���i

��

� �����mDm� ����i���

���i

�� �

����iDi� ����i���

���i

��

�����jDj� ����i����

��i

� ���mDm� � ���i�iAi � ���jBj � �

�The supersymmetry operators become

�Am � i���m�

�Ai � ���i�

�Bj � i�����j�

�� � ���

mnFmn � i�m��iDmAi � i���jBj

��

���

i�i�i��i� �Ai� � Ai� � � � ���

j�j�j��j� �Bj� � Bj� � ��

i�i�j �Ai� Bj � �

������

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CHAPTER ��� THE N�� YANG�MILLS THEORY

Proof�

�AA � i��%A��

�AA � i

�� � �p

�����i�

�%A

�p�

�� �

���i

��

�i

���� � ����i��

��m � �� �� � � � �i � ���

��� �j � ����

� ��

��i

���

�Am �i

����m�

�����i���

���i

��

� i���m�

�Ai �i

����i�

�����i���

���i

��

� ���i�

�Bj �i

������j�

�����i���

���i

��

� i�����j�

and

�� ��

�%ABFAB� �

�%A%BFAB�

��

��m � �� �� � �� �i� � �� � �� � �j� � ��

��

��n � �� �� � �� �i� � �� � �� � �j� � ��

��

�Fmn � Fmi� � Fmj� � Fi�n � Fi�i��

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����� DIMENSIONAL REDUCTION �

Fi�j� � Fj�n � Fj�i� � Fj�j�� ��� � �p

���i

��

��

��mnFmn��

��� � ��m�iDmAi����� � ��m���jDmBj��

����

�i��i� �Ai� � Ai� � ����� � �j��j� �Bj� � Bj� � ��

��� �

����i�j �Ai� Bj � �����

� �p�

���i

��

�� ��

��mnFmn � i�m

��iDmAi � i���jBj

���

��i�i�i��i� �Ai� � Ai� � � �

��j�j�j��j� �Bj� � Bj� � � �

i�i�j �Ai� Bj � �

In a similar way to this we could do dimensional reduction of a N�� supergauge theory in six dimensions� down to the N�� Yang�Mills theory �see������

A more detailed discussion on the subjects covered in is chapter is givenin ���� and �����

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� CHAPTER ��� THE N�� YANG�MILLS THEORY

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Appendix A

Conventions

In this appendix we will go through the conventions that are used in thisthesis� We will start with the conventions of Dirac �or more precisely Majo�rana� spinors and then go to Weyl notation�Note� The metric is space�like � � ����������

A�� Majorana Spinors

In this section we will look at Dirac spinors which are restricted by the Ma�jorana condition� We will also see how this leads to a natural relation to theVan der Waerden �i�e� two spinor� notation�

The ��matrices in four dimensions are de�ned as

�m �

� �m

��m

where

�m ���� �i

���m �

�����i

��m � � �� �� � i � �� �� ��Then the ��matrices satis�es the following algebra

f�m� �ng � ���mn �A���

and we have the following symmetry decomposition

�m�n � �mn � �mn �A���

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� APPENDIX A� CONVENTIONS

We de�ne �� as

�� � i�������� �A���

The charge conjugation matrix C is de�ned by

C���mC � � ��m�T

If we multiply this relation with C from the left we see that CT � �C� Thealgebra A�� and the fact that we have two symmetrical ��� and ��� and twoantisymmetrical ��� and ��� ��matrices� tells us that we can express thecharge conjugation matrix as a product of two ��matrices� We choose topick the two symmetrical matrices

C � i���� �

�� ��

� �CT � �C

��A���

�If we would have chosen the antisymmetrical ��matrices� the charge conju�gation matrix would have been C � diag ��� ��� This would have given otherrules for raising and lowering the Weyl spinor indices� see below��It is now easy to obtain the following symmetry table

n���n�C

�ab

symmetric��� or antisymmetric���

Cab �� ��mC�ab �

� ��mnC�ab �

� ��mnpC�ab ��

���C

ab

�A���

We split a Dirac spinor into one left handed and one right handed part�i�e� two Weyl spinors�

� �

��

��y �

����

The Dirac conjugation is de�ned in the usual way

��� �y�� �

� � ��

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A��� MAJORANA SPINORS �

We now de�ne the Majorana conjugate

�c � ��C �

� ������

Now the Majorana condition reads

� ��CT � � �A���

�Note that this gives also �� � �TC�� and � � �C ����What does the Majorana condition mean in terms of the upper and lowerparts of the Dirac spinor We use the split of � into � and

� ��CT � � ��

�����

��

���

Hence

� � � �A���

���� � �

Let us look at the index structure of this equations� �rst

�a�

���

� ��

� ��y

� � ��

� ��y � � ��

The right handed and the left handed parts ��� and �� of the Majoranaspinor are Weyl spinors�

Then A� can be written as

���� � ��

��� ����� �� � � �� �A��

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� APPENDIX A� CONVENTIONS

Now we de�ne raising and lowering of indices by

���� � �

��� � � � �A��

Then we have �from A��

� � ��

� �� � �� �� �A�� �

Hence� the Majorana spinor can be written as

� �

����� ��

��A����

This gives a relation between the Majorana notation and the Weyl notation�

Raising and lowering the indices must be done in an consistent way� thisgives

�� � ����� � �������� � ���������

��

������ � � ��

Note that this also gives ��� � ����

In this section we showed how to go from Majorana to Weyl notationin a consistent way� In the next section we are going to develop the Weylnotation from the Lorentz group�

A�� Weyl Spinors

We want to describe Lorentz transformations in � component �Weyl�� spinornotation� Let M � SL ���C�� then M represents Lorentz transformations

��� � M �

� �� ����� � M� ��

���� ��

��� � M���� �� ��� �� � �M���� ��

������

�A����

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A��� WEYL SPINORS �

�This gives ����� � �� ���� Now we want to rewrite Lorentz indices as Weyl

spinor indices �SL ���C��� This is done through Pauli�s sigma matrices

Pm � �m� ��Pm �

�P� � P� P� � iP�

P� � iP� P� � P�

��A����

Transformations is now performed as

P � � MPMy �A����

�This actually gives the index structure of the ��matrices��The determinant of P is invariant under Lorentz transformations

det��mP �

m

� �detM � �� � det ��mPm� �

� �P� � P�� �P� � P��� �P� � iP�� �P� � iP��

� PmPm

We have more Lorentz invariant expressions

���� � M���� ��M �

� �� � � �� ���� � ����

�� ���� �� � � � �� �� ��

����

���m� ��m�� �� � ���m

� ��m��

��

The ����tensor is also invariant because detM � ��Finally we de�ne raising and lowering of Weyl spinor indices as �compareto A��

�� � �����

�� � ������A����

In Appendix B we will calculate some useful formulas in Weyl spinor nota�tion�

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APPENDIX A� CONVENTIONS

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Appendix B

Some useful Formulas

B�� Sigma Matrices

The sigma matrices are de�ned as

�m ����� ��� ��� ��

��m ����������������

�where

�� �

�� �

��� �

� �ii

�� �

� ��

��� �

�� ��

The relation between �m and ��m is

��m ��� � � ��������m

� ��

The sigma matrices satisfy the usual algebra

f�m� �ng � ��mn

We now de�ne

��mn� ��

��n� ����

m ��� � �m� ����n ���

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� APPENDIX B� SOME USEFUL FORMULAS

We can now derive some useful formulas for sigma matrices

�m� ��

��n��� � ��mn� �

� � �mn� �� �B���

Proof�

��mn� �� �

��m� ����

n ��� � �n� ����n ���

� �m� ����n ��� � �

��m� ����

n ��� � �n� ����m ���

� �m� ����n ��� � �mn� �

��m��n � �n��m� �� � ���mn� �

� �B���

Proof� follows immediately from B���

�m� �������m � ��� �

� ����� �B���

Proof� Multiply the equality by � ����� this gives

�m� ��

�����m � ��� �

� � ��������� � � B��

�� mm � �

� � ��� ��

Consistent�

��mn� �� � �B���

Proof�

��mn� �� �

��n� ����

m ��� � �m� ����n ���

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B��� SPINOR ALGEBRA � �

��

��n� ����

m ��� � ����� �n� ����m ���

��mn� �� ��� � ��mn� �

� ��� �B���

Proof�

�m� ����n ������ � �n� ����

m ���

� ��n ��� �m� ��

� ��m� ����n ������

B�� Spinor Algebra

� � � �B���

Proof�

�� � � � ��� � � ��������� � �� �� ��

� ���

���� � ����

������� ��

������ �B���

Proof� multiply by ���

������� � ��

�����

������ ������ � �����

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� � APPENDIX B� SOME USEFUL FORMULAS

Note that

���� � ��

������

�� ������ �

���� ����

�� ���� �� ��

��n�� ����

follows immediately

��m ����n�� � �����

�� ���mn �B��

Proof�

���m� ���� �����n

� ������ � ����� �� ����

���m� ���n� ��

��

����� ��

������ ���m� ���n

� ��

��

��������m� ����

n ��� � � B��

� ��

����� ���mn

���� �� � ����

m

� ����m �� �B��

Proof� multiply by �����n

�������n

�� �� � ��

������n �m

� ����n �� � � B���

��n �� � ��

��� m

n ��m��

Consistent�

���� ���� � ��� ���� ���� �B�� �

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B��� DERIVATIVES IN SUPERSPACE � �

Proof�

�������� � ��������������

� ���������������

� ��

������������� ���� ����

� ��

����� ����

�� � �� �� � ����

n �� � �n� �� �B����

Proof�

�� � �� �� � �� �� �

���� ��

� �� �� � � B��

� ��

��n� ����

���n ��

� �� ��

��

����n

� ������ ��n� ��

B�� Derivatives in Superspace

We de�ne derivatives of the Grassmann variables in superspace

���� � � �

�� ���� �� � � ��

��

Then we can calculate the following relations

��� ��

� � ��

�B����

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� � APPENDIX B� SOME USEFUL FORMULAS

Proof� multiply by �� from the left���

��

��� �

��

��� �

���� �� � ����

���� �

��� � ����

Consistent�

D����������� � �� �B����

�The de�nition of D� is given in ����Proof�

D������������

� �

��

������ � �

��

��� �

� ���� ��� �

� ��

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Appendix C

Conventions in Yang�Mills

Theory

In this Appendix we will work out conventions needed mostly in chapter and � �

C�� Raising and Lowering Indices

We have already seen that raising and lowering spinor indices� goes like

�� � ����� �� � �����

and the complex conjugate

����� � �� ������

��� � ��

��

Now we want to know how these things work for the SU����indices in theN�� Yang�Mills theory� The raising and lowering are de�ned in the sameway as for the spinor indices

V i � �ijVj

Vi � V j�ji �C���

With these de�nition we can check that the only antisymmetric �� �� tensorin SU ��� is �ij �

� �

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� � APPENDIX C� CONVENTIONS IN YANG�MILLS THEORY

Proof�

�ij � �ik��� j

k�

� ���

��� �

��

� ���

���ij

� �ik��� j

k�

� ���

�� ��

��

�� ��

���ij

� �ik��� j

k�

� ���

�� �ii

��

�i i

���ij

� �ik��� j

k�

� ���

��� ��

��

� ����

We see that �ij� �ij �

However the SU����indices must be in their �natural� places� so complexconjugation is de�ned as

�V i��

� �Vi �C���

Consistency then requires

�Vi�� � � �V i � �ij �Vj

By reality we mean

�Lij � Lkl�ki�lj

C�� Yang�Mills Derivatives

The N�� Yang�Mills derivatives satisfy

�DA�DBg � �TCABDC � iFAB �C���

The constraints and the torsion now give �see chapter �

fD��D�g �

nD�� �D ��

o� iD

� �� �C���

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C��� THE FIELD STRENGTH � �

The N�� Yang�Mills derivatives satisfy

nDi��

�D ��j

o� i� i

j D� ��nDi��Dj

o� i����

ij �W�C���

and multiplication is de�ned as

Di�Dj

�� ���Dij � �ijD�� �

�i����

ij �W �C���

C�� The Field Strength

The �eld strength is de�ned as �see chapter �

F � dA� AA

The explicit expressions for the components in F is

FAB � DAAB � ���f�A�f�B�DBAA � �AA� ABg� T CAB AC �C���

Proof�

F � d�EBAB

�� EAAAE

BAB

� EBdAB � dEBAB � ���f�A�f�B�EBEAAAAB

� EBEADAAB ��

�EBEAT C

AB AC � ���f�A�f�B�EBEAAAAB

��

�EBEA

�DAAB � ���f�A�f�B�DBAA � �AA� ABg� T C

AB

But on the other hand

F ��

�EBEAFAB

So the statement holds�

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� APPENDIX C� CONVENTIONS IN YANG�MILLS THEORY

This gives us the explicit expressions �see ���

Fmn � mAn � nAm � �Am� An�

Fm� � mA� �D�Am � �Am� A��

Fm �� � mA �� � �D ��Am � �Am� A ���

F�� � D�A� �D�A� � fA�� A�g

F �� �� � �D ��A �� � �D ��A �� �nA ��� A ��

o

F� �� � D�A �� � �D ��A� �

nA�� A ��

o� i�m

� ��Am �C��

This is the most general solution to the Bianchi identities without con�straints� Note that F

� �� �see ��� gives us Am in terms of spinor componentsof A and their derivatives�

C�� Reality Conditions

In the Yang�Mills theory it is non�trivial to make the connection and the�eld strength real� We start with the N�� case� when we have no internalsymmetry� The connection can be expanded as

A � EmAm �A�A� � �E ��A��

This gives �remember �E��y � � �E ���

Ay � EmAym � �E �� �A �� �E�

�A�

� EmAm � �E ���A �� � E� �A�

In chapter we de�ned our generators of the Lie group �T a� to be antihermitian� this gives

Ay � �AaT a�y � � �Aa�y T a

If we want Aa to be real� we get

Ay � �A �C��

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C��� REALITY CONDITIONS �

This implies ��������

Aym � �Am

A �� � � �A ��

A� � � �A�

����������

�Aam�y � Aa

m

�Aa���y � Aa

��

�Aa��

y � Aa�

�C�� �

so we have

A � dxmAm � d��A� � d�� �� �A �� �C����

The reality condition on the connection also gives �see C��

Fmn � � �Fmn F�� � �F��

Fm� � � �Fm� F �� �� � �F �� ��

Fm �� � � �Fm �� F� �� � �F� ��

�C����

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�� APPENDIX C� CONVENTIONS IN YANG�MILLS THEORY

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Appendix D

Wess�Zumino Model in

Majorana Notation

SUSY�algebra�tex ���� TeX !home!tfe!martin!SUSY�algebra�tex SUSY�algebra�tex ���� TeX !home!tfe!martin!SUSY�algebra�tex

In the main chapters of this thesis all the spinor formalism are expressedin terms of Weyl notation� In this Appendix however� the Wess�Zuminomodel is given in Majorana notation� In the end the relation between thetwo representations are stated�

In chapter � we started by investigating how the supersymmetry op�erators transformed the component �elds� this way we calculated explicitexpressions for the transformations� In chapter � we derived the Lagrangianand the equations of motion� Here we will not do this calculations again�because it is similar to the one in Weyl notation� However we will showthat the Lagrangian is invariant under the supersymmetry transformation�In this way this Appendix �lls in a �missing blank� from the main chapters�

D�� The Wess�Zumino Lagrangian

We start with two real scalar �elds �A and B� and a supersymmetric partner��a� to this �elds� The spinor �eld ��a� is equal to its own charge conjugate��ca�� this is usually called the Majorana condition

� � ��c�T

where �c � ��C� More details are given in Appendix A�

���

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���APPENDIXD� WESS�ZUMINOMODEL IN MAJORANA NOTATION

All the �elds are massless� so the equations of motion are given by

�A � �B �

�mm�� �

The Lagrangian for our non�interacting �elds is given by

Son�shell� �

Zdx

�A�A�

�B�B �

i

����mm�

The supersymmetry transformations of the component �elds are ��� � i���������

��A � ��a�a

��B � ��a���� b

a�b

���a � �i ��m� ba m

�A� ��B

� c

b�c

However this Lagrangian and the supersymmetry transformations are onlyvalid on�shell �see chapter ��� If we don�t want to close the algebra with theequations of motion� we have to introduce auxiliary �elds �F and G�� BothF and G are real �having one degree of freedom each� just as we expected�and enters the Lagrangian as quadratic terms

So��shell� �

Rdx

���A�A� �

�B�B � i����mm�� �

�F� � �

�G��

�D���

The o��shell supersymmetry transformations are given by

��A � ��a�a

��B � ��a���� b

a�b

���a ��F � ��G

� b

a�b � i ��m� b

a m�A � ��B

� c

b�c

��F � �i��a ��m� ba m�b

��G � �i��a����m

� b

am�b �D���

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D��� THE WESS�ZUMINO LAGRANGIAN ���

The Lagrangian is invariant under these supersymmetry transformations�

Proof�

�Lo��shell� � �A�A ��B�B �

i

����a

��m� b

a m�b �

i

���a ��m� b

a m ���b� � F�F �G�G

We need ���a� The Dirac conjugation is de�ned in the usual way

��a ���y��

�aThis gives

����a �

�i ��m� b

d m�A� ��B

� c

b�c �

�F � ��G

� b

d�b

�y ����da

i�ycm

�A� ��yB

�cb��m�bd � �yb

�F � ��yG

�bd

�����da

� i��m�A� ��B

� b

c��m� a

b � ��b�F � ��G

� a

b

Hence

�Lo��shell� � ����A� ������B � iF ���mm�� iG�����mm��

i

�i��m

�A � ��B

��m�n � ��

�F � ��G

��n�n��

i

����mm

��i�nn

�A � ��B

���

�F � ��G

���

� ����A� ������B � iF ���mm�� iG�����mm���

���m

�A � ��B

��m�nn��

i

����F � ��G

��mm��

����mm�

nn�A� ��B

���

i

����mm

�F � ��G

��

� � A��

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���APPENDIXD� WESS�ZUMINOMODEL IN MAJORANA NOTATION

��

�����A�

�����B � i

�F ���mm�� i

�G�����mm��

����mm�

nm�A� ��B

�� �

i

����mm

�F � ��G

��

To get cancelation between these terms� we must use the majorana condition�see A��� A�� and A���� Then the ��th term becomes

����mm�

nm�A� ��B

�� � ��

����

�A� ��B

��

� ��

��� ��C����A� �

����� ��C����B

��

���C���A�

�����C���B

� ��

�����A� �

�������B

The ��th term becomes

i

����mm

�F � ��G

�� � � i

�Fm���m�� i

�Gm���m���

� � i

�Fm�� ���mC���� i

�Gm��

���m��C��

� � i

�Fm

����mC��

�i

�Gm

����m��C��

�i

�Fm ����m��� i

�Gm

����m���

�i

�F ���mm��

i

�G�����mm�

Hence

�So��shell� �

We could of course continue developing the Wess�Zumino theory in ma�jorana notation �calculating Lagrangian for interactions etc��� but this wouldgive nothing new compared to what has already been done in the main chap�ters� Instead we will end this Appendix by giving the relation between the

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D��� THE WESS�ZUMINO LAGRANGIAN ���

Wess�Zumino model in majorana and Weyl notations

Weyl notation Majorana notation

A � �p��A� iB�

F � �p��F � iG�

����

�� �

Then the action goes from Weyl to majorana like

A��A� i��mm��� F �F � �

��A� iB�� �A� iB� �

i

����mm��

�p��F � iG�

�p��F � iG�

��

�A�A�

�B�B �

i

����mm��

�F � �

�G�

The supersymmetry transformations are �translated� in the same way �notehowever that the supersymmetry transformations must be scaled with

p��

because the righthand side in the supersymmetry algebra is scaled with a���

A more detailed discussion on the subjects covered in is chapter is givenin ��� and ����

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���APPENDIXD� WESS�ZUMINOMODEL IN MAJORANA NOTATION

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Appendix E

K�ahler Geometry and Chiral

Fields

In the �rst part of this Appendix we will examine some of the propertiesof K�ahler geometry� which is a special type of complex analytic Riemannmanifold� In the second part we will use this formalism to express chiralmodels in a more general way than in chapter ��

E�� Connection and Covariant Derivative

The parametrisation of the Riemann manifold is given by

ai � ai�a�

a�i � a�i�a��

�E���

Di�erentials and derivatives transforms like

da�i � a�iaj

daj da��i � a��ia�j

a�i

� aj

a�iaj

a��i

� a�ja��i

a�j

�E���

Covariant vector �elds transforms like

V �i

�a�� a��

aj

a�iVj �a� a

��

V �i �a�� a�� �a�i

ajV j �a� a��

���

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�� APPENDIX E� K �AHLER GEOMETRY AND CHIRAL FIELDS

V �i��a�� a��

a�j

a��iVj� �a� a

��

V �i� �a�� a�� �a��i

a�jV j� �a� a��

Further we require that the K�ahler manifold should have a Hermitian metric�which is �nite and invertible�� Raising and lowering indexes then go like

Vi � gij�Vj� Vj� � gij�V

i

���

The covariant derivative must respect the analytical structure �see E�� and E���of the transformations� so the connection satis�es %k

ij � %k�

i�j � ��no mix�ing��

riVj �Vjai

� %kijVk

ri�Vj �Vja�i

� %ki�jVk

If the transformation rules for the connection are choosen as

%�kij �

al

a�iam

a�ja�k

an%nlm �

�an

a�ia�ja�k

an

%�ki�j �

a�l

a��iam

a�ja�k

an%nl�m �E���

Then the covariant derivatives transform correctly�Proof�

riVj �Vjai

� %kijVk

� V �j

a�i� %

�kijV

�k

a�i

�al

a�jVl

�� %

�kij

al

a�kVl

��al

a�ia�jVl �

al

a�jVl

a�i� %

�kij

al

a�kVl

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E��� THE K�AHLER POTENTIAL ��

��al

a�ia�jVl �

al

a�jak

a�iVl

ak�

�ap

a�iam

a�ja�k

an%npm �

�an

a�ia�ja�k

an

�al

a�kVl

� r�iV

�j �

�al

a�ia�jVl � �an

a�ia�ja�k

anal

a�kVl

� r�iV

�j �

�al

a�ia�jVl � �an

a�ia�j� ln Vl

� r�iV

�j �

� OK tensor

The �rst transformation rule � E�� � tells us that it is consistent to put thetorsion to zero

%kij � %kji �E���

The other one � E��� gives us freedom to put

%ki�j � � rj�Vi �Vi

a�j

The only nonzero components of the connection is then

%kij and�%kij

��� %k

i�j�

E�� The K�ahler Potential

The connection should be compatible with the Hermitian metric� this gives

rkgij� � rk�gij� �

This gives us a way to express the connection in terms of the metric

rkgij� � �gij�

ak� %lkiglj� � %lkj�%il � �

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�� APPENDIX E� K �AHLER GEOMETRY AND CHIRAL FIELDS

%lkiglj� �gij�

ak�

hgij�g

lj� � � li

i

� nl %lki � gnj

� gij�

ak

Hence

%nki � gnj� gij�

ak�E���

From E�� we get

gnl� gjl�

ai� gnl

� gil�

aj

and hence

gjl�

ai�

gil�

aj�E���

The same is true for the conjugated derivative

gjl�

a�i�

gil�

a�j�E���

The relations E�� and E�� tells us that we can write gij� as derivatives of apotential

gij� � ai

a�j

K �a� a�� �E��

K �a� a�� is called the K�ahler potential� We will soon see why it is so usefulin chiral models� The metric is invariant under the K�ahler transformation

K �a� a�� � K �a� a�� � F �a� � F �a�� �E��

E�� Curvature

We de�ne the curvature as

fri�rjgVk � Rlijk

fri�rj�gVk � Rlij�k �E�� �

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E��� CURVATURE ���

We can �nd more compact expressions for the curvature

Rlijk �

Rlij�k �

%pika�j

�E����

Proof� to prove that Rlijk � we only have to show that rirjVk is sym�

metric in i and j� In the calculation below we will only keep terms which arenot obviously symmetric in i and j� In the end� as we will see� there will beno terms left���

rirjVl � ri

Vl

aj� %kjlVk

ai

Vl

aj

��

ai

�%kjlVk

��

%pij

Vlap

� %kplVk

�� %pil

Vpaj

� %kjpVk

��Vlaiaj

� %kjlai

Vk � %kjlVkai

%pij �� � ��� %pil

Vp

aj� %kjpVk

� �%kjl

aiVk �

%kjl

Vkai

� %pilVpaj

�� %pil%

kjpVk

� �%kjl

aiVk � %pil%

kjpVk � � E�� �

� �

ai

gkl

� gll�

aj

�Vk � %pil%

kjpVk

� �gkl�

aigll�

aj� gpl

� gll�

aigkn

� gpn�

ajVk

The second term can be rewritten as

gpl� gll�

aigkn

� gpn�

ajVk �

�gpl

� gpn�

aj� �g

pl�

ajgpn�

��

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��� APPENDIX E� K �AHLER GEOMETRY AND CHIRAL FIELDS

� �gkn� gll�ai

gpn�gpl

aj

� �� kp

gll�

aigpl

aj

� �gll�ai

gkl�

aj

Hence

rirjVl � �gkl�

aigll�

aj� gpl

� gll�

aigkn

� gpn�

ajVk

� �gkl�

aigll�

aj� gll�

aigkl

aj

The second part of the proof is also strait forward calculations

�ri�rj� �Vk � rirj�Vk � rj�riVk

� ri

Vk

a�j

�� rj�

Vk

ai� %pikVp

��Vk

aia�j� %pik

Vp

a�j�

�Vkaia�j

� %pikVpa�j

� %pika�j

Vp

�%pika�j

Vp

E�� Chiral Models

We shall now examine the relation between chiral models with couplingterms and K�ahler geometry� The most general Lagrangian which could beconstructed from chiral �elds are

S �RdzK

��i� �yj

��Rdx

�d��P

��i

� hc �E����

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E��� CHIRAL MODELS ���

The super�elds K and P can be expanded in potential series of chiral super�elds

K��� �y

��

Xci����iN �j����jM�

i� � � ��iN�yj� � � ��yjM

P ��� �X

gi����iN�i� � � ��iN

We can now generalize the expressions in chapter �

P ��� � P �A� � ��iP �A�

Ai�

���

�F i P �A�

Ai� �

��i�j

�P �A�

AiAj

Note that all the �elds are in the �transformed� state �see chapter ���We have also

P y ��y� � P � �A�� � ����iP � �A��A�i �

�����

�F �iP

� �A��A�i � �

���i��j

�P � �A��A�iA�j

This nice� but we would like to get the explicit expression for the Lagrangian�Before we get that� we have to calculate P and K� Put

KMN � �i� � � ��iN�yj� � � ��yjM

Then we can express KMN in terms of the potential and perform the inverss�transform to get back to our ordinary coordinate system

KMN � s��P ��� sy��P y ��y�We break this expression �in half�� to make the calculation a bit easier

s��P ��� �

�� i

���m��m �

������ ���

��

�P �A� � ��i

P

Ai�

���

�F i P

Ai� �

��i�j

�P

Aiaj

��

� P � ��iP

Ai�

���

�F i P

Ai� �

��i�j

�P

Aiaj

��

i

���m��mP � i

���m��m

��i

P

Ai

��

������ ���P

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��� APPENDIX E� K �AHLER GEOMETRY AND CHIRAL FIELDS

In the same way we get

sy��P y ��y� � P y � ����iP y

Ayi ��

�����

�F yi P

Ayi ��

���i��j

�P y

AyiAyj

��

i

���m��mP

y �i

���m��m

�����i

P y

Ayi

��

������ ���P y

Now we multiply these two expressions� keeping only ���� ���terms

s��P ��� sy��P y ��y�������� ��

� ��iP

Ai

i

���m ��m

�����i

� P y

Ayi�

������������

�F i P

Ai� �

��i�j

�P

AiAj

��

�F yi� P y

Ayi� ��

���i

���j� �P y

Ayi�Ayj�

��

i

���m ��m

��i

P

Ai

�����i

� P y

Ayi�

�����������

���m ��mP��

n ��nPy����������

��

��P y�P �

��P�P y

We simplify term by term� starting with the last two terms

��P�P y �

��P y�P � ��

mPmP

y

�hK�MN

� Ai� � � �AinAyj� � � �AyjM

i

� ��

�K�MN

AkAylmAkmA

yl

The ��th term

���m��mP��

n ��nPy����������

� � B� � ��

mPmP

y

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E��� CHIRAL MODELS ���

The ��st term

��iP

Ai

i

���m��m

�����i

� P y

Ayi�

���������� ��

�i

����i��

��n� ��

���� P

Aim

��� ����

i� �� P y

Ayi�

������������

� � i

����� ��

�� �� ���i��

m

� ��

P

Ai�

�P y

Ayi� m��i

� �� � ��i� ��m

�P y

Ayi�

�������������

�i

�i��m

� ��

P

Ai

�P y

Ayi�m��i

� �� � ��i� ��m

�P y

Ayi�

��

The ��rd term looks pretty much the same

� i

���m��m

��i

P

Ai

�����i

� P y

Ai�

�����������

� � i

����m� ��

�� ��m

���i�

P

Ai

��� ��

��i� �� P y

Ayi�

��������� ��

� � i

�m� ��m

�i�

P

Ai

���i

� �� P y

Ayi�

�i

�i��m� ��

P

Ai

�P y

Ayi�m��i

� ��i�

m

�P y

Ayi�

��

So the ��st and the ��rd terms add up to

��st � ��th term �i

�K�MN

AiAyj�i�mm��j �

i

�K�MN

AiAyjAyk�i�m��jmA

yk

Finally we get

KMN j������ � s��P ��� sy��P y ��y����������

��

�F iF yj �K�

MN

AiAyj ��

F i��k��l

�K�MN

AiAykAyl �

F yi�k�l

�K�MN

AyiAkAl�

���i�j��k��l

K�MN

AiAjAykAyl �

��

�mAkmA

yl �K�

MN

AkAyl �i

��i�mm��j

�K�MN

AiAyj �

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��� APPENDIX E� K �AHLER GEOMETRY AND CHIRAL FIELDS

i

��i�m��jmA

yk �K�MN

AiAyjAyk �E����

If we replace K�MN with K�� we get the expression for the total polynomial

K�

Now it�s time to use the formalism that we developed in the beginningof this Appendix� this will give us much more compact expressions� Startby putting K� � K �a� a��� then we have �see E��

gik� ��K �a� a��aia�k

Equation E�� gives �inverted�

gnl�%nki �

gil�

ak�E����

Together these two relations gives

�K �a� a��aia�lak

� gnl�%nki �E����

Complex conjugation gives

�K �a� a��a�iala�k

� gln�%n�

k�i� �E����

If we use this in E���� we get

Kj������ ��

�F iF yjgij� � �

F i��k��lgin�%

n�

l�k� ��

F yi�k�lgni�%nkl �

���i�j��k��lgik��j�l� � �

�mAkmA

ylgkl� �

i

��i�mm��jgij� �

i

��i�m��jmA

ykgin�%n�

j�k�

The total Lagrangian E��� now looks like

L � ��gij�F

iF yj � ���gik��j�l��

i�j��k��l�

F i��gin�%

n�

l�k���l��k � P

Ai

��

F �i��gni�%

nkl�

l�k � P �A�i

�� �

�gkl�mAkmA

yl � i�gij��

i�n �Dn��j�

��

�PaiAj �

i�j � ��

�P �A�iA�j ��

i��j

�E����

Page 127: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

E��� CHIRAL MODELS ���

where �Dm��i � m��i � %i

j�k�mA�j��k is a covariant space�time derivative�

Note that if we put

K � A�iAi

P � �iAi �

�mijA

iAj ��

�gijkA

iAjAk

we get the ordinary expression for the Wess�Zumino Lagrangian �see ����

The auxiliary �elds can be eliminated by their Euler equation

S

F i�

S

F �i �

or explicit

�gij�F

�j � �

gin�%

n�

l�k���l��k �

P

Ai�

�gij�F

i � �

gi�n%

nkl�

l�k �P �

A�i �

When we put these expressions into the Lagrangian� the �rst and the thirdterm cancels� The fourth term gives

�F �i�

gni�%

nkl�

k�l � P �

A�i

�� ��

�%i

l�k���l��k � �gii

� P

Ai

��

�gni�%

nkl�

k�l � �P �

A�i

� � �

��gni�%

i�

l�k�%nkl�

k�l��k��l �

�%i

l�k���l��k

P �

A�i �

�%ikl�

k�l � �gii� P

Ai

P �

A�i

Page 128: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

�� APPENDIX E� K �AHLER GEOMETRY AND CHIRAL FIELDS

Remember the following identities �see E����

Rij�kl� � gml�%mika�j

��gkl�

aia�j� gmn�

gml�

a�j

�gkn�

ai

DiP �P

Ai

DiDjP ��P

Aij� %kij

P

Ak

Then we can write the total Lagrangian without the auxiliary �elds

L �Rd��d���K

��i� �yj

���R

d��P��i� hc

� �

��Rik�jl��i�j��k��l � �

gkl�mAkmA

yl�

igij��

i�m �Dm��j � �

�DiDjP�i�j�

��Di�Dj�P ��i��j � �gij

DiP �Dj�P

�E���

This Lagrangian describes the most general supersymmetric coupling of chi�ral multiplets� Each term in the Lagrangian has a natural interpretation inK�ahler geometry� The scalar �elds are coordinates of the K�ahler manifoldand the fermions are tensors in the tangent space�

Note that the Lagrangian is manifestly invariant under the K�ahler trans�form

K �a� a�� � K �a� a�� � F �a� � F �a��

A more detailed discussion on the subjects covered in is chapter is givenin ��� and ����

Page 129: dimensional reduction - Chalmersfy.chalmers.se/~tfebn/MartinNmasterthesis.pdf · Sigma Matrices ... p ersymmetry is with a discussion on the sup ersymmetry algebra and its ... ersymmetry

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��� S� Coleman and J� Mandula �� Physics Review ����� ����

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��