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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�
�
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�
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 � � � � � � � � � � � � � � � � � � � � � � � ��
�
� 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 � � � � � � � � � � � � � � � � � � �
CONTENTS �
E K�ahler Geometry and Chiral Fields ���
E�� Connection and Covariant Derivative � � � � � � � � � � � � � � ���E�� The K�ahler Potential � � � � � � � � � � � � � � � � � � � � � � � ���E�� Curvature � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��E�� Chiral Models � � � � � � � � � � � � � � � � � � � � � � � � � � � ��
� CONTENTS
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
�
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�
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
� 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
���� 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� � � �
�� 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
���� 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
�� 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 � ��
���� 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
�� 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 ����
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� ��� �
��
� 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�
���� 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�
�
� 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 �
���� 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 ����
�� CHAPTER �� THE WESS�ZUMINO MODEL
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��� �� �� ��� ��
�
��
�� 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 �
���� 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
���
�� 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�
���� 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�
����
� 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 ����
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� � �����
�
� 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�
�����
���� 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��
����������
�� 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�
�����������
���� 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�����������
�� 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�
���� 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�
�� 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�
����
���� 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 ����
� CHAPTER �� THE WESS�ZUMINO MODEL IN SUPERSPACE
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� �
�
� 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
���� 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�
��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��
��
���� 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�
��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�
���� 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�
��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
���� 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�
�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 �����
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 �� �
�
� 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 ��
��
�������� ��
�
��� 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
�� 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��
��� 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�
�
�� 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 ����
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
��
�� 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�� ��
���� 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 ����
� 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
���� 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 � ����
� 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 ����
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
��
�� 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�� ��� �
�
���� 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
�� 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��
���� 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 � �
�� 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 �� �
���� 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��
� 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
���� 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
����
� 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�
���� 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 ����
�� CHAPTER �� THE N�� BIANCHI IDENTITIES
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 � �� ���
��
�� 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 � ����
� ��� 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 �
�� 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 � �� �� �
� ��� 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
� 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 �
� ��� 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
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 �� ���
� ��� 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 �
� 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
�
� ��� 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 �
� 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 �� ��
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��
�
� 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�
����� 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
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 �
����� 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 � �
������
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��
����� 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 �����
� CHAPTER ��� THE N�� YANG�MILLS THEORY
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���
�
� 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�� �
� � ��
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��
� 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����
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�
APPENDIX A� CONVENTIONS
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 ���
�
� 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 ���
�
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 ���
������� � ��
�����
������ ������ � �����
� � 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�� �
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����
� � APPENDIX B� SOME USEFUL FORMULAS
Proof� multiply by �� from the left���
��
��� �
�
��
��� �
���� �� � ����
���� �
��� � ����
Consistent�
D����������� � �� �B����
�The de�nition of D� is given in ����Proof�
D������������
� �
��
������ � �
��
��� �
� ���� ��� �
�
� ��
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 �
� �
� � 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���
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�
� 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��
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����
�� APPENDIX C� CONVENTIONS IN YANG�MILLS THEORY
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�
���
���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���
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��
���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
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 ����
���APPENDIXD� WESS�ZUMINOMODEL IN MAJORANA NOTATION
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��
���
�� 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
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 � �
�� 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�� �
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�
��
��� 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����
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
��� 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
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 �
��� 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����
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
�� 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 ����
Bibliography
��� Y� A� Goldfand and E� S� Likhtman �� JETP Letters ����� ���
��� S� Coleman and J� Mandula �� Physics Review ����� ����
��� S� Ferrara An Overview on Broken Supergravity Models LNF �� � ��
��� J� Wess and J� Bagger Supersymmetry and Supergravity Princeton Uni�versity Press ��
��� P� West Introduction to Supersymmetry and Supergravity World Scien�ti�c Publishing ��
��� M� Nakahara Geometry� Topology and Physics IOP Publishing �
��� B� E� W� Nilsson Lecture Notes ��
�� B� E� W� Nilsson Private Notes
�� P� Sundell Private Discussions
�� � R� Grimm� M� Sohnius and J� Wess Extended Supersymmetry and GaugeTheories B��� Nuclear Physics ���� ���
���� H� Osborn Topological Charges for N�� Supersymmetric Gauge The�
ories and Monopoles of Spin � ��B Physical Letters No �� � �������
���� L� Brink� H� Schwarz and J� Scherk Supersymmetric Yang�Mills Theo�
ries B��� Nuclear Physics ����� �
���� J� O� Winnberg Super�elds as an Extension of the Spin Representa�
tion of the Orthogonal Group �� Journal of Mathematical Physics No� ����� ���
��