Introduction to Supersymmetry

Christian Samann

Notes by Anton Ilderton

August 24, 2009

1

Contents

1 Introduction 51.1 A SUSY toy model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 What is it good for? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Spinors 122.1 The Poincare group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Spin and pin groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 The SUSY algebra 243.1 SUSY algebra on R1;3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3 The Wess{Zumino model . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 Superspace and superelds 324.1 Reminder: Gramann numbers. . . . . . . . . . . . . . . . . . . . . . . . . 324.2 Flat superspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.3 Superelds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5 SUSY{invariant actions from superelds 385.1 Actions from chiral selds . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.2 Actions with vector superelds . . . . . . . . . . . . . . . . . . . . . . . . 42

6 SUSY quantum eld theories 496.1 Abstract considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.2 Seld quantisation of chiral selds . . . . . . . . . . . . . . . . . . . . . . 516.3 Quantisation of Super Yang{Mills theory . . . . . . . . . . . . . . . . . . 58

7 Maximally SUSY Yang{Mills theories 637.1 Spinors in arbitrary dimensions . . . . . . . . . . . . . . . . . . . . . . . . 637.2 Actions and constraint equations . . . . . . . . . . . . . . . . . . . . . . . 647.3 d = 4, N = 4 super Yang{Mills theory . . . . . . . . . . . . . . . . . . . . 67

8 Seiberg-Witten Theory 728.1 The moduli space of pure N = 2 SYM theory . . . . . . . . . . . . . . . . 728.2 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 748.3 The exact eective action . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

A Conventions and identities 78

B Solutions to exercises 81

2

Preface

Supersymmetry, or SUSY for short, is an extension of the classical symmetries of eldtheories. SUSY was discovered in the early 1970's and has attracted growing attentionever since, even though there is still no experimental evidence for its existence up tothis day. There are essentially two reasons why high-energy physicists keep interested insupersymmetry: From a phenomenological point of view, the supersymmetric extensionof the Standard Model provides very reasonable solutions to some of the remainingpuzzles in particle physics. On the other hand, supersymmetric eld theories have manyintriguing features which often can be accessed analytically, making them ideal toymodels for theorists.

These lecture notes are based on a series of lectures given by myself from April toJune 2009 in the School of Mathematics at Trinity College, Dublin. The material coveredin these notes was presented during eleven lectures, each lasting 90 mins, which was alittle ambitious, retrospectively.

The aim of the lectures was to give a reasonable overview of this topic while beingthorough enough to provide a graduate student with the necessary tools to do researchinvolving supersymmetric eld theories herself. I put some emphasis on motivatingspinors, as I still remember struggling with the reasons for their existence when I rstencountered them. Besides the standard material presented in almost any course onsupersymmetry, I chose supereld quantisation, maximally supersymmetric Yang-Millstheories and Seiberg-Witten theory as additional topics for these lectures. I had justused supergraphs in a research project myself and their usefulness and simplicity wasstill fresh in my mind. The maximally supersymmetric Yang-Mills theories with theiramazing properties were included because of the important role they play in string theorytoday. Finally, the section on Seiberg-Witten theory demonstrates why supersymmetriceld theories are indeed beautiful toy models.

If you should nd any typos or mistakes in the text, please let us know by sendingan email to antoni@maths.tcd.ie or saemann@maths.tcd.ie. The most recent versionof these lecture notes can be found on the lecture series homepage,

http://www.christiansaemann.de/supersymmetry.html .

On this webpage, there is also a (still preliminary) version of a Mathematica notebookwhich performs many of the tedious computations necessary for understanding theselectures automatically.

I am very grateful to Anton Ilderton for doing such an amazing job with writing upthese notes, correcting prefactors and providing typed-up solutions to all the steps leftas exercises during the lectures. I would also like to thank all the people attending thelectures at Trinity for making them more interesting and lively by asking questions.

Christian Samann

3

General remarks

Our metric is (;+;+;+), which diers by a sign from the metric of most quantumeld theory textbooks, as e.g. Peskin{Schroeder. The remainder of our conventions areintroduced in the text, as they are needed. A summary is included in the appendices.Numerous exercises are included, the intent of which is mainly to make the user familiarwith the kind of methods, tricks and identities needed to perform SUSY calculations.Full solutions can be found in the appendices.

Recommended textbooks and lecture notes (with metric conventions)

Recall that the two dierent choices for Minkowski metric, the mostly plus (;+;+;+)and the mostly minus (+;;;), are often referred to as the East Coast Metric (ECM)and the West Coast Metric (WCM), respectively.

[1] J. Wess and J. Bagger, \Supersymmetry and supergravity", Princeton, USA: Univ.Pr. (1992) 259 p.ECM, the essentials.

[2] S. Weinberg, \The quantum theory of elds. Vol. 3: Supersymmetry", Cambridge,UK: Univ. Pr. (2000) 419 p.ECM, uses Dirac spinors, more physics.

[3] I. L. Buchbinder and S. M. Kuzenko, \Ideas and methods of supersymmetry andsupergravity: Or a walk through superspace", Bristol, UK: IOP (1998) 656 p.ECM, supermathematics, NR theorems, super Feynman rules.

[4] S. J. Gates, Marcus T. Grisaru, M. Rocek and W. Siegel, \Superspace, or onethousand and one lessons in supersymmetry," Front. Phys. 58 (1983) 1 [hep-th/0108200].ECM, many useful things, in particular super Feynman rules.

[5] A. Van Proeyen, \Tools for supersymmetry", arXiv:hep-th/9910030.ECM, more algebraic.

[6] S. P. Martin, \A Supersymmetry Primer", arXiv:hep-ph/9709356.ECM, particle physics, MSSM.

[7] A. Bilal, \Introduction to supersymmetry", arXiv:hep-th/0101055.WCM, for particle physics conventions.

[8] J. D. Lykken, \Introduction to supersymmetry", arXiv:hep-th/9612114.WCM, useful as a reference.

Cover: an otter, and it's supersymmetric partner, the sotter.

4

1 Introduction

1.1 A SUSY toy model

At the beginning of the 1970's, people started looking at SUSY toy models. In this sec-tion we will discuss a simple model which will illustrate many of the important physicalproperties of SUSY theories which are commonly discussed in a eld theory context. Itwill also serve to illustrate the fundamentals of many calculations we will later performin eld theory, in a simple and accessible setting. Much of the following discussion canbe found in [9, 10] in more detail.x1 Denition. Our model is quantum mechanical. We have a hermitian HamiltonianH and non{hermitian operators Q, Qy related through the anti{commutator

H =12fQ;Qyg 1

2(QQy +QyQ) ; (1)

where the operators obey the following \0 + 1 dimensional SUSY algebra",

fQ;Qg = fQy; Qyg = 0[Q;H] = [Qy;H] = 0 :

(2)

The Q and Qy are called supercharges and generate supersymmetry transformations. Wealways count real supercharges, so here we have two of them. It is a direct consequenceof (2) that Q2 = Qy2 = 0.x2 Properties. Consider a Hilbert space H; h j i carrying a representation of H, Qand Qy. From the algebra (2), it follows that H is positive denite, for, given any statej i 2 H,

2h jHj i = h jfQ;Qygj i= h jQQyj i+ h jQyQj i= kQj ik2 + kQyj ik2 0 :

(3)

We see that states in a SUSY theory have non{negative energy; note that the minimummay or may not be obtained. To further examine the state space, we diagonalise H andconsider the eigenstates jn i such that

Hjn i = Enjn i: (4)

We must treat the two cases En = 0 and En > 0 separately. We begin with the caseEn = E > 0. We may here introduce the scaled operators a = Q=

p2E and ay = Qy=

p2E

which, within the space of states of energy E, obey the algebra

fa; ayg = 1 ; fa; ag = fay; ayg = 0 : (5)

This is a simple example of a Cliord algebra, which we will study later on. We nowconstruct all the states with energy E, in analogy to the construction of harmonic

5

oscillator states via creation operators acting on the vacuum. Again, we clearly havea2 = ay2 = 0, from which it follows that the only eigenvalue of a is 0. Call the statewith this eigenvalue j i. We can create no more states by acting on this with a, and wecan create only one more by acting with ay (since ay2 = 0), which we call j+ i ayj i.Hence we have a subsystem of two states obeying

ayj i = j+ i ; aj+ i = j i ; aj i = ayj+ i = 0 :

A simple 2d representation of the algebra (5) is given by the following matrices andvectors,

a =

0 01 0

!; ay =

0 10 0

!; j+ i =

10

!; j i =

01

!: (6)

We see here a basic example of the existence of two types of states in SUSY theories;`+' states and `' states which will later be bosons and fermions. They are transformedinto each other by the action of the SUSY generators, and more generally we will seethat

Qjboson i = j fermion i ; Qj fermion i = jboson i ;and similarly for the action of Qy. We also have an example of the SUSY propertythat states of non{zero energy are degenerate and appear in pairs; a possible non{zerospectrum of a SUSY theory is sketched in Fig. 1.

E

+

+

+ +

Figure 1: Schematic of the non{zero energy spectrum in SUSY theories: states appearin pairs of equal energy.

We now turn to the states of zero energy, Hj 0 i = 0: Directly from equation (3), wemust have

0 = kQj ik2 + kQyj ik2 ;so that a vacuum state j 0 i exists if and only if

Qj 0 i = Qyj 0 i = 0 : (7)

6

x3 SUSY breaking. The vacuum should be unique in quantum mechanics and there-fore invariant under supersymmetry transformations { this is just the statement thatQj 0 i = Qyj 0 i = 0. If, however, there is no such state we say that SUSY is sponta-neously broken. Consider introducing a potential V to our system, of one of the formsshown in Fig. 2. Note that the `Mexican hat' potential possesses a rotational symmetry,but any particular minimum of the potential x0 with V (x0) = Vmin breaks the rotationalsymmetry, but not SUSY. The quadratic potential with a non{zero minimum, on theother hand, breaks SUSY but not the rotational symmetry. We will return to thesepoints in a moment, after looking at a representation of the SUSY algebra on a Hilbertspace of wavefunctions.

x x

Figure 2: The `quadratic' potential on the left breaks SUSY but not the rotationalsymmetry, whereas a minimum in the `Mexican hat' potential breaks the rotationalsymmetries but not SUSY.

x4 SUSY quantum mechanics. To have a Hilbert space on which we can act withQ, Qy and H we choose a set of functions in L2(R) C2 ,

=

+(x)(x)

!+ : bosons, : fermions,

(8)

following (6). We now dene the SUSY generators,

Qy =

0 10 0

!P + iW 0

; Q =

0 01 0

!P iW 0 ; (9)

where W (x) is a real function, a dash denotes dierentiation with respect to x andP = i~@x.

Exercise 1. Show that the Hamiltonian of this system is

H 12fQy; Qg = 1

2P 2 +W 02

1 ~

2W 003 ; (10)

where 3 = diag(1;1) is the usual Pauli matrix.

7

The three terms in the above Hamiltonian describe, respectively, kinetic energy,potential energy and a magnetic interaction. Let us look for the vacuum state of thistheory. States annihilated by both Q and Qy take one of the forms

=

+0

!& (P iW 0)+ = 0 =) + / eW=~ ;

or

=

0

!& (P + iW 0) = 0 =) / eW=~ :

(11)

The choice of ground state is dictated by normalisability, since if W (1) =1 only +is normalisable, and if W (1) = 1 only is normalisable. Note that if W (1) =W (1) neither of the states are normalisable, and so there can be no ground statewavefunction.

The role played by these boundary conditions gives us an explicit realisation ofSUSY breaking discussed above. Note that for the cases W (1) = 1, i.e. a groundstate wavefunction exists, the derivative W 0 must be somewhere vanishing, and so thepotential W 02 in our theory has a zero, i.e. SUSY is unbroken.

Conversely, ifW 02 is non vanishing, then the potential is strictly positive. In this case,there is no normalisable ground state wavefunction, and so SUSY is broken. To be moreprecise, one has to take into account non-perturbative contributions: although W (1) =W (1) still allows for points x withW 0(x) = 0, there has to be an even (counted withmultiplicities) number of such points (consider, e.g., the potential W 02(x) = x4). Thetrue quantum vacuum is unique and will be a superposition of wave functions localisedat these points. Here, however, non-perturbative corrections will liftW 0(x) to be strictlypositive and thus break SUSY. This explains why we do not nd SUSY ground statesfor W (1) = W (1).x5 Witten index. Introduce the operator ()F , which gives +1 on bosonic states and1 on fermionic states [10]. For our states of non{zero energy, we have

()F = 2aay 1$1 00 1

!: (12)

More generally, we would have states of arbitrary energy labelled as j+; j i and j ; j iwhere j is some collection of quantum numbers, and

()F j+; j i = j+; j i ; ()F j ; j i = j; j i :

Consider the operatorTr()F

Xn

hn j()F jn i ; (13)

where n runs over the energy eigenvalues. This operator counts the number of bosonicstates minus the number of fermionic states, or nB nF . However, since states with

8

E > 0 always occur in pairs, all this operator really counts is nE=0B nE=0F , i.e. thenumber of bosonic states minus the number of fermionic states in the vacuum sector.This is the Witten index.

Varying the parameters of the theory, such as the coecients in some potential, thevolume of the system, etc, and assuming analyticity of the states in these parameters(there are good reasons for this [10]), the index never changes { the reason is that onlyboson{fermion pairs of states can move away from the vacuum E = 0, which clearlycannot change the value of nE=0B nE=0F . The value of the index therefore tells ussomething about SUSY breaking:

Tr()F 6= 0 =) SUSY is unbroken ; (14)

since if the index is non{zero then there is at least one state in the vacuum sector. Notethat if we nd that the index is zero, we cannot infer that SUSY is broken. As the indexis analytic, it is easy to calculate. It may be regulated, for example, by

Trh()FeH

i:

x6 The Witten index as an operator index. We now show that the Witten indexis, indeed, an index in the sense dened in the mathematics literature, see e.g. [11].We can do this quite generally. Given a SUSY theory, we split the Hilbert space intobosonic and fermionic sectors, H = HB HF , on which the SUSY generators must takethe block o{diagonal form

Q =

0 M1M2 0

!; Qy =

0 M y2M y1 0

!; (15)

for some operators M1 and M2. The above forms follow from the statement that theSUSY generators transform bosons into fermions and vice versa { they are the generali-sations of the matrices in (6). Now dene the operator eQ = Q+Qy, which is hermitianand annihilates E = 0 states,

eQ = 0 M1 +M y2M2 +M

y1 0

!

We then have that nE=0B = dimker(M2 +My1), and n

E=0F = dimker(M1 +M

y2). Hence,

nE=0B nE=0F = dimker(M2 +M y1) dimker(M1 +M y2) ind M2 +M y1 ;

(16)

as claimed. We also have nE=0B + nE=0F = dim

kerQy=imQy

.

9

1S

1W

1EM

1S

1W

1EM

log /GeV log/GeV

Figure 3: The strong, weak and electromagnetic couplings 1 of the standard model,with and without SUSY.

1.2 What is it good for?

We have seen that SUSY theories are constructed from an algebra and, for explicitrepresentations, a function W , the superpotential. Here we collect some motivatingreasons for studying SUSY.x1 Coleman{Mandula theorem. We may ask the question: can one extend space-time symmetries non{trivially beyond the Poincare group? The answer goes as follows.Assume G is the symmetry group of a theory with S{matrix S such that

G contains the Poincare group,

all particles have positive energy, with nitely many particles of mass m < m0 forall m0,

S{matrix elements h out jSj in i are analytic and non{trivial,

then the Coleman{Mandula theorem tells us that G = Poincare group internal sym-metries. So the answer to the above question appears to be no. There was a hiddenassumption in this theorem, however { that the Lie algebra of G was generated by com-mutators. As we saw above, SUSY algebras, however, include anticommutators andtherefore provide a loophole to the Coleman{Mandula theorem.x2 Gauge coupling unication. The gauge group of the standard model is SU(3)SU(2)U(1). Various attempts have been made at constructing a grand unied theory,or GUT, which unies the standard model at some high energy scale, within a singlegroup, be it SU(5), SO(10), E6, etc. A unied theory would imply a unied coupling;however, the couplings in the standard run as shown in the left panel of Fig. 3, and donot appear to intersect. With SUSY, however, the situation is improved { the couplingsvery nearly unify at the order of 1016 GeV.x3 Hierarchy problem. The masses in the standard model are generated by the Higgsparticle. Experimentally, we have hHi ' 174 GeV, but this is very sensitive to quantumcorrections which can be estimated to be of the order 1032 GeV. As keeping the exper-imentally desired value of 174 GeV would require an unnatural amount of ne tuning,it seems that some protection mechanism is at work, and SUSY is a good candidate for

10

this. Here, hHi qm2H , and we are thus looking at quantum corrections to the mass

of the Higgs boson. Many Feynman diagrams contributing to mass corrections in SUSYtheories cancel against other Feynman diagrams in which a particle loop is replaced byits superpartner loop, see Fig. 4 .

b+

fH H

= 0

Figure 4: In SUSY theories, contributions from bosonic (b) and fermionic (f) superpart-ners often cancel exactly.

x4 Dark matter. The energy content of the universe is roughly 4% ordinary matter,22% dark matter and 74% dark energy (or cosmological constant). There are most likelya number of dierent constituents of dark matter. One of the most important candidatesbesides neutrinos is the neutralino, the lightest particle of the minimal supersymmetricstandard model (MSSM) yet to be found. The hope is, of course, that the LHC will ndthe neutralino and heavier SUSY particles. In fact { and add your own pinch of salt here {we have already found half of the SUSY spectrum: that of the ordinary standard model.Although this is clearly no evidence for SUSY whatsoever, it is reassuring that thereare actually good theoretical reasons for discovering rst the particles of the ordinarystandard model, if one assumes a MSSM with broken SUSY.x5 Theorist's reasons. SUSY theories are highly constrained by symmetries and aretherefore ideal toy models. Local SUSY theories contain gravity, and perhaps even givegood theories of quantum gravity. N = 8 supergravity might actually be nite, similarlyto N = 4 super Yang-Mills theory [12]. String theories are also much nicer with SUSYincluded, as the tachyonic states of the bosonic string can be safely removed from thespectrum. In mathematics, it could be that mirror symmetric partners of rigid Calabi{Yau manifolds are Calabi{Yau supermanifolds, and therefore mirror symmetry mightrequire us to introduce a notion of supersymmetry [13].

11

2 Spinors

To understand SUSY QFT, we need a good understanding of spinors. We will cover thistopic in some detail in this part of the lectures, and things will be clearer for it later.The aim is to nd all irreducible representations (`irreps') of the Poincare group, as allelds in physics live in such representations.

2.1 The Poincare group

x1 Denition. The Poincare group is the group of isometries (maps preserving dis-tance) on Minkowski space R1;3. It is a non{compact Lie group. Its generators are fourtranslations, P, and a total of six boosts and rotations, M = M. The Lie algebrarelations are

[P;M ] = iP P

iP + symm. ; (17)[M ;M] = i

M + symm.

: (18)

In the rst line we have introduced the operation \symm." which takes account of thesymmetries of the generators, i.e. it symmetrises or antisymmetrises as appropriate. Forexample, the left hand side of (17) is antisymmetric in and , hence \symm." generatesthe second term on the right hand side of (17).

Exercise 2. Check you understand the denition of \symm." by generating the remain-ing three terms on the right hand side of (18).

The Poincare group is R1;3oO(1; 3), the semi{direct product of the abelian group oftranslations R1;3 generated by P and the Lorentz group O(1; 3) generated by the M .It is not a direct product because translations and boosts do not in general commute.We now look in more detail at the Lorentz subgroup.x2 The Lorentz subgroup. Consider the vector representation of O(1; 3). That is,given an element x 2 R1;3,

x =

0BBB@x0

x1

x2

x3

1CCCA ;an element of the Lorentz group will be represented by a 4 4 matrix. There are fourspecial elements of the group we would like to consider. They are:

12

14 =

0BBB@1

11

1

1CCCA The identity element, which takes any vector to itself.

T =

0BBB@1

11

1

1CCCA The time reversal operator, which takes x0 ! x0,but leaves spatial components unchanged.

P =

0BBB@11

11

1CCCA The parity operator, which takes us to a mirror worldxj ! xj , but leaves the time direction unchanged.

PT = 14 The combined parity{time transformation which takesx ! x.

The reason for introducing these special elements is that the Lorentz group splits intofour components, each of which is continuously connected to one of the four elementsabove. These components are written

L"+ 3 1 ; L#+ 3 T ; L" 3 P ; L# 3 PT ; (19)

with " denoting those transformations which preserve the sign of the time direction,while + denotes those which preserve the sign of the spatial components. Those trans-formations in L"+, i.e. those continuously connected to the identity element, are calledproper orthochronous. Those transformations with determinant +1, i.e. those in L"+ andL#, connected to 14 or PT , form SO(1; 3). Note that the four components in (19) aredisconnected. Pictorially, the Lorentz group breaks down into components as shown inFig. 5.

L+ L

L

+ L

SO(1, 3)

Figure 5: The disconnected components of the Lorentz group O(1; 3).

Exercise 3. Verify the decomposition of the Lorentz group shown in Fig. 5. First showthere is no continuous path from SO(1; 3) to any element not in SO(1; 3), which realisesthe rectangular boundary shown. Next, show there is no continuous path between ele-

13

ments with (0; 0) matrix-component +1 and 1, so realising the circular boundaries.

x3 Representations of the Poincare and Lorentz groups. Particles in QFT aredescribed by elds on R1;3 which, if we perform a Lorentz transformation x ! Lx,should transform under some representation of the Lorentz group, i.e.

(x)! (L)(L1x) ; (20)

where (L) is some matrix representation of the Lorentz group. Under sequential Lorentztransformations x! L2L1x =: L3x we should have

(x)! (L2)(L1)(L11 L12 x) = (L3)(L13 x) ; (21)

and thus the elds form representations of the Lorentz (and Poincare) group. To labelrepresentations we need the Casimir operators, which commute with every generator ofthe group. Casimirs for the Poincare group are

PP and WW ; W :=

12PM : (22)

W is called the \Pauli{Lubanski vector". We label representations by the values of P 2

and W 2. There are two cases to consider:

1. Massive representations, m2 > 0. In this case there exists a rest frame in whichP = (m; 0; 0; 0) so P 2 = m2. It can then be shown that W 2 = m2s(s + 1)where s 2 Z=2 is the spin of the particle or eld.

2. Massless representations, m2 = 0. In this case there is no rest frame, but wecan always choose a frame such that P = (E; 0; 0; E), in which case we ndP 2 = W 2 = 0. However, it can be shown that W = P here, where is thehelicity of the particle, 2 Z=2, and this is used to label the representations.

Elementary particles should form (\sit in") irreducible representations of the Poincaregroup. Hence, the problem of writing down all possible elds in QFT is equivalent tonding all possible irreps of the Poincare group.x4 Remarks. We restrict ourselves in the sequel to nding all irreps of the Lorentzsubgroup { the extension to the full Poincare group is trivial. Representations should belinear (i.e. given by linear transformations) and QFT requires unitary representations(up to a phase { recall that to obtain physical information from QFT, we calculate mod{squared amplitudes, which remove phases). Wigner showed [14] that one can reduce suchrepresentations up to a phase, to representations up to a sign, and Bargmann showedlater [15] that studying all unitary irreps of the Poincare group up to a sign correspondsto studying all irreps of the universal covering group.

The universal covering group of a group X is a simply connected space Y with a mapf : Y ! X which is locally homeomorphic (locally it looks like the original group) and

14

surjective (i.e. it contains every element of the original group). We will also encounterthe double cover of a space, where Y is not necessarily simply connected and the mapf is 2:1. We turn now to the universal cover of the Lorentz group, beginning with theproper orthochronous component.x5 The universal cover of L"+. We begin by showing that the universal cover of L"+ SO+(1; 3) is SL(2;C). First dene the four{vector of Pauli matrices ,

0 =

1 00 1

!; 1 =

0 11 0

!2 =

0 ii 0

!3 =

1 00 1

!: (23)

Using these matrices, we construct the carrier space on which our representation mustact { this space `contains' all of our original vectors x. The carrier space is dened by

X(x) := x =

x0 + x3 x1 ix2x1 + ix2 x0 x3

!: (24)

The inverse map, which extracts x from X, is1

x =12TrX

: (25)

The vector norm is xx = detX, where the minus sign is due to our metric con-ventions. The determinant of this matrix must therefore be preserved under Lorentztransformations L; the relevant representation is (L) 2 SL(2;C). We will give theexplicit form of (L) below, but we rst note that the action of L on X(x) is

L . X(x) = (L)X(x)y(L) ; (26)

which clearly preserves the determinant of X since det((L)) = det(y(L)) = 1. Now,for a rotation by an angle around a unit direction

!n , and a boost with rapidity tanh

in the direction!n , the matrix (L) is

(L) = exp12

!n i!n

! BLRL : (27)

(Boosts along an axis commute with rotations about that same axis, hence the expo-nential factorises.) This representation provides us with a map SL(2;C) ! SO+(1; 3),i.e. it tells us how to take an element of SL(2;C) and transform it into a rotation and aboost. Note that the map is 2 : 1, since for rotations of angles and 0 := + 2

R0 = R ; (28)

but the presence of both (L) and y(L) in (26) means that the sign is killed and thesecorrespond to the same Lorentz transformation. We will see a peculiar consequence ofthis in the next section.

1Note the index positions, which appear unnatural. Once we have introduced the matrices, the

reader will understand that what we should really write is x = 12Tr(X).

15

Finally, we leave it as an exercise to show that SL(2;C) is simply connected { it istherefore exactly the group we need for classifying the irreps of SO+(1; 3). We now lookfor all irreps of SL(2;C) and for this, we have to study the fundamental representation.

Exercise 4. Show that SL(2;C) is simply connected, i.e. two paths connecting elementsM;N 2 SL(2;C) can always be continuously deformed into each other (\are homotopi-cally equivalent").

x6 Weyl spinors. In the fundamental representation, the vector (carrier) space isW =C2, the space of two{component complex Weyl spinors ,

=

12

!; j 2 C: (29)

The action of the Lorentz group is

L . = (L) ; (30)

where (L) are the matrices (27). Now, since there is only one present, rotations ofangle 2 do not leave spinors invariant { they pick up a minus sign from (28). Instead, ittakes a rotation of angle 4 to bring a spinor back to itself. This shows us that spinorsreally are something new, and unrelated to vectors which have no such property2.

To begin to make contact with QFT, we need a scalar product which is Lorentzinvariant. To do this, we need a dual space3. We write the dual space as W_ ' C2which consists of elements ~ on which the Lorentz group acts as

~T =

1

2

!2W_ ; L . ~ = ~(L1) : (31)

To construct the scalar product (which is really a bilinear form as it is not positivedenite), we need a map M :W !W_, which takes ! TM . The pairing

(;0) TM0 (32)

must be invariant under Lorentz transformations, i.e. T (L)M(L) = M . The onlypossible M are

M =

0 11 0

!: (33)

2For entertaining illustrations of this fact, look for Dirac's belt trick, Feynman's plate trick and the

game Tangloids.3Compare with the example of constructing the scalar product on x { we construct a map to the

dual space using the metric, x ! x = x , which furnishes us with the inner product x2 = xx =xx

.

16

Exercise 5. Prove the result (33).

We now dene W_ as follows: for 2W , := 2W_, where our choice ofM is 12 = 1, 21 = 1, i.e.

=

0 11 0

!: (34)

Then,(; ) = = : (35)

For the scalar product to be symmetric we nd = , i.e. spinors mustanticommute. We assume this from now on. We will also abbreviate the inner productas follows:

(; )! or (= ) :

x7 Complex conjugate Weyl spinors. We also dene the complex conjugate repre-sentation, W : its elements4 are labelled by upper dotted indices, _ and transformunder Lorentz transformations as y(L) . The space dual to W is dened by

_ = _ _ _: (36)

Here 12 = 1, 21 = +1, so

= _ _ =

0 11 0

!: (37)

Note that = and _ __ _ = __.

x8 Other examples of spinor representations. We have just seen two examples ofspinor representations; the Weyl spinors and their conjugates. These representations arelabelled by their helicities,

W : ; (12 ; 0)

W : _ ; (0; 12)(38)

We also have the vector representation:

Vector : x = _x _ ; (12 ;

12) : (39)

The above furnishes us with a map from bispinors, on the left, to vectors on the right.More generally, arbitrary representations of SO+(1; 3) can be constructed from tensorproducts of the 2d Weyl spinor rep. of SL(2;C),

1:::m_1::: _n ; (

m

2;n2) : (40)

4The reader who is wary of this notation for complex conjugation, or more familiar with Dirac spinors

where we write = y0, might like to think of the Weyl spinor _ as y0 _ , as this is exactly thesame thing.

17

For example, the Yang{Mills eld strength F [r;r ] may be written

F ! [ _ ] _ F _ _ : (41)

The antisymmetry in (; ) implies that F _ _ can be written

F __ = f _ _ + _ _f : (42)

After a Wick rotation to Euclidean space R4, if f _ _ = 0 then F is self-dual: F = ?F ,which corresponds to a gauge eld conguration known as an instanton, while if f = 0then F = ? F ; this is an anti{instanton.x9 Other signatures. In Euclidean signature the symmetry group is SO(4), and thecorresponding double cover is SU(2) SU(2). For the metric diag(+;+;;) withKleinian signature and symmetry group SO(2; 2) the double cover is SL(2;R)SL(2;R).To adapt the above to these metrics, it is a matter of adjusting the sigma matrices byfactors of i such that the condition kxk2 = detX(x) is maintained. Note that fornding all unitary ray representations for other signatures and other dimensions, thedouble cover of the group of isometries is sucient.

2.2 Spin and pin groups

So far we have looked at L"+, but we are still looking for the full universal cover ofO(1; 3). We pursue this below. For a more detailed exposition of the material, see forexample [16].x1 The Cliord algebra. Consider an algebra generated by d objects5 , for =0; 1; : : : d 1,

f;g = 21d ; (43)along with the vector space V = Span(). The sign on the right hand side of thisequation is pure convention. Most books working with a mostly minus metric (WCM)use the opposite sign. The Cliord algebra C(V ) is

C(V ) =Mn0

V ^n = C V V ^ V V ^ V ^ V : : : V ^d : (44)

Here, C 3 1d, V 3 , V ^V 3 [], etc, where we antisymmetrise over indicesas symmetric combinations of the can be reduced to expressions containing fewer 'susing the anticommutator (43). The sum in (44) terminates for the same reason. Thealgebra decomposes as

C(V ) = C+(V ) C(V ) ; (45)5We use a heavy script to denote the abstract elements of the Cliord algebra. Later, we will use

normal script for the usual gamma matrices.

18

where C+(V ) contains products of even numbers of while C(V ) contains productsof odd numbers of the . With the denition of the following operation (an anti-involution) on this space,

1 : : :n = (1)nn : : :1 ; (46)

we can now dene the pin group.x2 The pin group. On the vector space V with isometry group SO(V ), the associatedpin group is dened as

Pin(V ) := f 2 C(V ) j: = 1 ; :V: V g : (47)

The interesting point is that Pin(V ) carries a natural representation of the group SO(V ),where the action of an element L 2 SO(V ) on C(V ) is given by the following expressionwith L 2 C(V ):

LL = L

1L = L

: (48)

Note that we could replace with the inverse because of the denition of the pin group.x3 Examples. One can now easily nd a couple of explicit examples for L. Consider,for example, the parity transformations P. The associated pin group element is given byP = i0, which obeys PP = 1 as required using the fundamental anticommutator.We can now check that the action of P reverses the sign of for = 1; 2; 3, but leaves0 alone, hence

i0(i0) = P ; (49)and we have correctly identied the parity operator. The time reversal transformationT corresponds to T = 123, and its action reverses the sign of only for = 0.We also want the continuous Lorentz transformations. These are

L = exp18[; ]

; L 2 L"+ : (50)

Writing 14 [; ], one can show that satisfy the algebra of the generators ofthe Lorentz algebra M .x4 The spin group.

Spin(V ) := 2 C+(V ) j: = 1 ; :V: V = Pin(V ) \ C+(V ) : (51)

Here, as above, C+(V ) contains only even numbers of 's. The relationship betweenthe spin and pin groups is shown in Fig. 6. Spin(V ) and Pin(V ) are connected and formdouble covers of SO(V ) and O(V ),

1! Z2 ! Pin(V )! O(V )! 1 ;1! Z2 ! Spin(V )! SO(V )! 1 ;

19

1 PT P T

Spin(V )

Pin(V )

Figure 6: The pin and spin groups.

where V is an arbitrary Rp;q. Some explicit examples of spin groups are

Spin(4) = SU(2) SU(2) ;Spin(3; 1) = SL(2;C) ;Spin(2; 2) = SL(2;R) SL(2;R) :

We will come back to spinors in arbitrary dimensions in section 7.1.x5 Representations of the Cliord algebra C(R1;3). We now move on to the rep-resentations of the Cliord algebra. This will lead us to all the elds we need to considerin Lorentz invariant theories. The minimal representations are four dimensional, i.e.

=

0

0

!(52)

or, explicitly:

=

(0 1 00 1

1 00 1 0

!;

0 0 11 0

0 11 0 0

!;

0 0 ii 00 ii 0 0

!;

0 1 00 1

1 00 1 0

!); (53)

where = , see (23), and is dened by

_ := _ _ _

: (54)

Note that in (54), both and are given with their natural index positions. The formof the gamma matrices given here is the so-called Weyl representation. There is also theDirac representation, commonly employed in QFT textbooks. We always use the Weylrepresentation in these lectures. The gamma matrices obey

f; g = 214 : (55)Their conjugates are y0 = 0 and

yj = j . The matrices act on a pair of Weyl

spinors 2W and 2W , =

_

!(56)

This pair is called a Dirac spinor. We also dene 5 := i0123, with y5 = 5, which

is the matrix

5 =

12 00 12

!; (57)

20

in our Weyl representation. The Weyl spinors are eigenvectors of the projectorsP := 12(14 5).

Exercise 6. It is very useful to realise that, writing (0; 1; 2; 3), the barredsigma matrices are simply

= (0;1;2;3) :Prove this result for yourself. This is the origin of the (baing) choice of seeminglyLorentz{ignorant conventions \ = ", which we (tacitly used earlier and) shallnever speak of again.

x6 Parity and time reversal. Recalling x3, the parity and time reversal operators interms of explicit matrices are given by

P = i0 ; T = 123 ; (58)which can be checked by calculating

P01P = 0 ; and Pj1P = j ; =) P1P = P ;

T 01T = 0 ; and T j1T = j ; =) T 1T = T :(59)

Letting these parity and time reversal operators act on the Dirac spinor, we nd

P = i

!; T =

i i

!:

Exercise 7. Show that (59) holds for the denitions of P and T given in (58).

x7 The Dirac equation and the Lagrangian. We now make contact to physics bygiving the equation of motion for Dirac spinors. This is,

i@ m = 0 ; (60)

where @ is C(V )-valued. The factor of yields the appropriate coupling of vectors(@) to spinors (). The equations of motion have the following `Weyl decomposition'in terms of Weyl spinors,

i@ m = 0 ;i@m = 0 :

(61)

A Lagrangian density which yields the Dirac equation is,

L = i@ m ; (62)where y0 . In terms of Weyl spinors it may be written

L = i @+ i @ mm : (63)

21

x8 Charge conjugation. Along with the discrete symmetries P, T and PT , there isanother discrete transformation. Consider the above equations of motion, but includingminimal coupling to a hermitian U(1) gauge potential,

i(@ ieA)m = 0 :

Note that if we take the complex conjugate, we obtain i(@ + ieA)m = 0 ; (64)where the sign of the coupling has changed. We introduce C which acts as follows,

C()1C = ; C C : (65)

The resulting equation of motion for C is thusi(@ + ieA)m

C = 0 :

We want (C)C = , which implies CC = 1. On R1;3 in the Weyl/chiral representa-tion, we can choose C = 2. Physically, C changes particles into their antiparticles ofthe same mass but the opposite charge.x9 The CPT theorem. Both P and CP are violated in nature: the rst was observedin Cobalt 60 decays in 1956, the second in Kaon decays in 1964. Both appear in theweak nuclear force, and it is unknown if CP violation also exists for strong nuclearinteractions. This violation is believed to be the reason for the existence of more matterthan antimatter.

In 1955, Pauli proved that if you have a quantum eld theory which is

1. invariant under L"+,

2. causal and local,

3. has a Hamiltonian which is bounded below,

then the quantum eld theory is invariant under the combined transformation CPT .

2.3 Summary

Since all our elds will transform under irreps of O(1; 3), we spent some time constructingall possible irreps. Group theory told us that we needed the universal covering groupwhich, for L"+ for example, is SL(2;C). We discussed the following representations (theexamples column contains some objects we will meet later, for reference):

22

Carrier space Name Examples in this rep.W = C2 Weyl spinors SUSY charges Q,

parameters.W_ = C2 Dual Weyl spinors SUSY charges Q,

parameters.W = C2 Conjugate Weyl spinors _ SUSY charges Q _,

_ parameters.W

_ = C2 Conjugate, dual Weyl spinors _ SUSY charges Q _, _ parameters.

Spinors are anticommuting objects. Once we allow for the discrete transformations P, Tand PT we need the double cover of SO(1; 3), which is Spin(1; 3), and the double coverof the full group O(1; 3) is the pin group. With the Dirac matrices,

=

0

0

!; (66)

the basic representations are given by Dirac spinors which decompose into two Weylspinors,

=

_

!: (67)

Note that the natural index structure on and is _ and _, so that the gamma

matrices (66) act naturally on the Dirac spinor (67).x1 Hermitian conjugation. Before continuing it is worth cementing our conventionsas regards complex, or hermitian, conjugation of Weyl spinors. The conjugate of aproduct of objects, be they vector or spinor, is

(AB : : : Z)y = Zy : : : ByAy: (68)

There are no minus signs, and all indices remain in their allotted positions. It may beuseful in calculation to explicitly put dots over spinor indices which become conjugatedspinors. A conjugated spinor is y .

A consequence of these conventions is that the Pauli matrices behave as just that {matrices, rather than bispinors. Any lack of mathematical satisfaction the reader mayfeel6 should be more than compensated for by the ease of calculation these conventionsprovide. For example, we then have

()y = yy = and ( )y = :

6as, e.g., that the { naive { Koszul sign rule is not obeyed

23

3 The SUSY algebra

The SUSY algebra is an extension of the Poincare algebra we met earlier. A theoremdue to Haag et al [17] says that the SUSY algebra we will present below is the onlypossible extension of the Poincare group consistent with the axioms of quantum eldtheory.

3.1 SUSY algebra on R1;3

x1 SUSY algebra. Along with the generators P and M which obey the algebra(17){(18), we choose N 2 N+, and then for i = 1 : : :N we introduce the supersymmetrycharges Qi and Qi _ which obey

fQi; Qjg = f Qi _; Qj _g = 0 ; (69)fQi; Qj _g = 2 ij _P ; (70)

where Q is the complex conjugate of Q. Note the positions of the i indices labelling thedierent supercharges, which will serve as a bookkeeping device for correct contractionslater. Note also that the zero vector{component on the right hand side of (70) looks likethe Hamiltonian in quantum mechanics, just as we had in our toy model, earlier. Thecommutators of the SUSY charges with the generators of the Poincare group are

[P; Qi] = [P; Qi _] = 0 ; (71)M ; Q

i

= i()

Qi ; (72)M ; Q _i

= i() _ _ Q

_

i ; (73)

where we have introduced

() 14

_

_ _ _;

() _ _ 14

_

_ _

_

:

(74)

If N > 1 we call this the N{extended SUSY algebra. The number of real superchargesin the game is 4N (in four dimensions) since each of the Qi is a two component complexWeyl spinor.x2 Theorem of Haag, Sohnius, Lopuszanski. Up to introducing `central charges'Z [i;j] such that

fQi; Qjg = Z [i;j] ; (75)where the Z are just complex numbers, the N{extended SUSY algebra is the only exten-sion of the Poincare group which is consistent with the axioms of relativistic quantumeld theory [17]. We will set the central charges to zero in the majority of these lectures{ for completeness, we retain them only in Sect. 3.2, x6 when discussing the massiverepresentations of the SUSY algebra.

24

3.2 Representations

x1 Casimir operators. For the massive representations, P 2 remains a Casimir oper-ator. We also label representations by their superspin, C2, with

C2 = P 2W 2 14(PW )2 : (76)

The eigenvalues of the superspin operator are m4s(s+1). For the massless representa-tions, we introduce L =W 116 _ fQi; Qi _g, and then label representations by theirsuperhelicity , where

L =+ 14

P ; (77)

and 2 Z=2 (compare with the relation W = P dening the helicity for the masslessPoincare representations).x2 Massless representations. As before, to construct the massless representations,we go to the frame P = (E; 0; 0; E), and remain there in this paragraph. The key tounderstanding the massless reps lies in evaluating (70) in our chosen frame. We nd

_P =

0 00 2E

! _

: (78)

Comparing with our toy model, we nd that Qi1 = Qi _1 = 0 on such states7, since for

arbitrary ,h jfQ1; Q _1gj i = kQ1j ik2 + k Q _1j ik2 = 0 ;

while fQi2; Qj _2g = 4Eij , i.e. we have N copies of our toy model Cliord algebra. Now,in this frame, the helicity operator is just J3 M12. Calculating its commutator withthe SUSY charges we nd

[J3; Qi2] = 12Qi2 ; [J3; Qi _2] = +

12Qi _2 : (79)

These commutators simply say that Qi2 lowers helicity by 1=2 while Qi _2 raises helicity by1=2, so that repeated application of Q or Q changes the spin of a state from half{integralto integral and back, i.e. it turns bosons into fermions into bosons, etc.

Exercise 8. Derive the `raising' and `lowering' commutators (79).

We now choose a lowest weight state from which to construct all the states in thisrepresentation. (This is just what we did in our toy model, starting from the stateannihilated by Q.) Our lowest weight state will be jh i, which is annihilated by all theQi2. All we can do to construct other states is act with the Qi _2, but we can act with eachat most once, since they anticommute amongst themselves and square to zero. Hence,

7Note that the term state referes here { slightly sloppily { to an element of the module which serves

as the carrier space of the SUSY representation under consideration.

25

there are a total of 2N states (each of the Qi _2 may be present once or not at all) of theform

Qi1 _2 : : :Qin _2jh i : (80)

The range of helicities of these states is h up to h + N=2, possessed by the stateQ1_2 Q2_2 QN _2jh i. If it helps, you may like to imagine all such states as being ar-ranged into corners of an N -dimensional cube, where the number of Qi corresponds tothe lattice distance to the corner corresponding to the state jh i.

We now give examples of the massless representations for variousN . Representationsare labelled by their mass and superspin or superhelicity. The states which live inside anysuch representation are collectively called a (super)multiplet. As described above, thesestates are related by the action of the SUSY charges, and are of equal mass. A multipletmay be seen as the set of all states which are needed to form a closed representation ofthe (SUSY) algebra.x3 Example 1: N = 1 multiplets. As in our toy model, we have a 21 = 2 statesystem, comprising the lowest weight state jh i and jh + 12 i = Q _2jh i. For physicalreasons we wish to avoid particles with helicities larger than 2 (the theories of whichare thought not to be well dened) and with helicities 2 and 3=2, which correspond tothe graviton and gravitino (as we want to steer clear of gravity). Hence we have twopossibilities,

State helicities Name

(1; 12) The vector multiplet (since helicity h = 1 corresponds to thephoton, which is a vector).

(12 ; 0) The chiral multiplet.

The 1-dimensional cube is just a line segment with the end points corresponding to thetwo dierent states. Taking the CPT conjugates of these multiplets gives us multipletsof helicity (0;12) and (12 ;1).x4 Example 2: N = 2 multiplets. Starting from our lowest weight state jh i, we cannow act with either Q1_2 or Q2_2 to generate two states of helicity h+

12 . We can also act

with both operators to generate a state of helicity h + 1. As before, we wish to steerclear of spin larger than 1, which gives us two possible multiplets,

State helicities Name

(1; 12 ;12 ; 0) The N = 2 vector multiplet.

(12 ; 0; 0;12) The hypermultiplet.

26

The 2-dimensional cube is a square, and the corners follow the pattern boson-fermion-boson-fermion. The charge conjugate of the vector multiplet has helicities(0;12 ;12 ;1), while the hypermultiplet is its own charge conjugate.x5 Example 3: N = 3 and N = 4 multiplets. The case N = 3 is usually notdiscussed seperately from the case N = 4 for the following reason: in principle, thereare two multiples: ( 1; 12 ; 0;12) and ( 12 ; 0;12 ;1),where the plain number in each product is the helicity while the circled number is thenumber of states with that helicity. In the cubic picture, the eight states correspond tothe corners of the cube with lattice distances (0; 1; 2; 3). These two multiplets are eachother's CPT conjugate and in a physical theory, we expect both to be present. Then,however, they combine to form the N = 4 supermultiplet, which reads as

1; 12; 0; 1

2; 1

; (81)

We have a total of 1 + 4 + 6 + 4 + 1 = 16 states, which correspond to the 16 corners ofa 4-dimensional cube with lattice distance (0; 1; 2; 3; 4). The number of states of coursematches with our general form of 2N states, for N = 4. Note that this supermultipletis its own charge conjugate. It gives the entire eld content of N = 4 super Yang{Millstheory, which we will meet again later in these lectures.x6 Massive representations. In these lectures we will mostly deal with masslessrepresentations (taking mass generation to be due to Higgsing), but for completenesswe include this discussion of the massive representations. We consider the most generalcase by leaving in the central charges (75),

fQi; Qjg = Z [i;j] ; f Qi _; Qj _g = _ _ Z[i;j] : (82)

We go to the rest frame of the particle, in which P = (m; 0; 0; 0). In order to understandthe representations we will write our commutators, in this frame, in such a way that thetechnology we applied to the massless representations can also be employed here. First,we use U(N ) rotations to bring the central charges to the following form,

Z [i;j] =

0BBBBBB@0 z1

z1 0 02 02

020 z2

z2 0 02

02 02. . .

1CCCCCCA : (83)

Next, introduce the following linear combinations of the supercharges8,

ai :=1p2

Q2i1 + Q

2i y

; bi :=

1p2

Q2i1 Q2i y

; (84)

8The strange contractions between indices will only appear in this section, in this frame (Lorentz

invariance is broken).

27

for r = 1 : : :N=2. The only non-vanishing commutators arefai; aj y g = (2m zi)ij ; no sum,fbi; bj y g = (2m+ zi)ij ; no sum.

(85)

Now recall, from our toy model example, that positivity of the Hilbert space requires2m jzij (and so in particular central charges must vanish in the massless case). Assumethat k of the zi saturate this bound, i.e. jzij = 2m, i = 1 : : : k, for k N=2. Then oneof the pair fai; ai yg or fbi; bi yg vanishes, which implies either ai or bi must be put tozero. We therefore have a total of 2N 2k non{zero fermionic oscillators, and so 22N2kstates. To illustrate, if k = 0 and N = 1, we can construct the multiplets

12; 0; 0;1

2

;

1;12;12; 0

:

It follows that representations are constructed just as in the massless case, for k > 0, andthose multiplets are called short or BPS. If k takes its maximal value of N=2 then wehave an ultrashort multiplet. Note that only ultrashort multiplets can become massless,as it is only then that 22N2k = 2N and so the number of states matches that in amassless multiplet.

3.3 The Wess{Zumino model

We are now ready to look at our rst SUSY eld theory. We begin with a condensing ofour notation by combining commutators and anti{commutators into a single bracket.x1 Grading and supercommutators. Introduce a parity to every object. This is aZ2 grading such that everything `bosonic' (i.e. with integral spin) carries a label 0, whileeverything `fermionic' (i.e. with half{integral spin) carries the label 1. We denote theparity of an operator by writing a tilde over it, i.e.eP = 0 ; eQj = 1 :Noting that our SUSY algebra comprised anticommutators between the odd objects andcommutators otherwise, we introduce the supercommutator

f[A;B]g := AB () ~A ~BBA : (86)This recovers all our previous commutators (e.g. f[P; P ]g = [P; P ]) and anticommu-tators (e.g. f[Q; Q]g = fQ; Qg).x2 Representation of the SUSY algebra on component elds. We are lookingfor some set of elds ', . . . etc, on which the SUSY algebra closes. As we look for this,we will build a concrete representation of the SUSY generators acting on various elds.

On any eld, we have P = i@. Given an innitesimal v, we can dene theinnitesimal transformation of, say, a scalar ' as

v' = vP' = iv@' : (87)

28

Innitesimal supersymmetry transformations are accordingly generated by the combi-nation

' =Q+ Q

' ; (88)

and similarly on other elds. Here, we have introduced anticommuting parameters

such that f[; ]g = 0. It follows that ()2 = 0, i.e. that the are just Gramannnumbers. These parameters have parity +1, and are combined with the supercharges toform the bosonic objects in (88),

Q Q ; Q _ Q _ : (89)

Such contracted combinations of Weyl spinors and their conjugates will appear oftenfrom here on. For objects ; and ; in W and W respectively, contractions arealways taken as follows,

:= ;

:= _ _ ;

:= _ _ :

(90)

Introducing a second Gramann number , we can write our basic commutators as

[Q; Q] = [ Q; Q] = 0 ;

[Q; Q] = 2 __P =: 2 P :

(91)

The action of Q and Q remains somewhat abstract { in order to make things moreexplicit, let us act with two supersymmetry transformations, and take their commutator,i.e.

' = [Q; Q]' [Q; Q]'= 2i @' : (92)

Now, for simplicity we dene a spinor eld through the transform of the scalar,

' =p2 ; ' =

p2 : (93)

From (92) it follows that, under a SUSY transform, the eld must transform intosomething proportional to @', plus contributions from other elds which cancel fromthe commutators in (92). We nd

= ip2 _

_@'+p2F ;

_ = ip2 _@ '+

p2 _ F :

(94)

where the rst term is xed by the algebra, while we are free to include the second term,which introduces a third eld F .

29

Exercise 9. The following identities are useful when dealing with component elds.Prove them.

1: = ;

2: = ;3: = ;4: ( )y = :

A straightforward calculation now shows that (92) is satised. A more involvedcalculation shows that the commutator of two SUSY transformations acting on requires

F = ip2 _ _@ ; F = i

p2 @ _ _ : (95)

It can be checked that F transforms correctly under two SUSY transformations, giving arep. of (70) and there is then no need to introduce any further elds { the SUSY algebracloses on the multiplet (';; F ), which is the entire eld content of the Wess{Zuminomodel.

Exercise 10. Show that (94) implies that _ transforms as

_ = ip2 _@ '+

p2 _ F ;

and that the transformation of F in (95) can be written

F = +ip2 _@

_ :

x3 Dimensions of elds. Before writing down an action for our elds, we brieyconsider their mass dimension. In four dimensions, ['] = [P ] = 1. It follows from thealgebra fQ; Qg P that [Q] = [ Q] = 1=2. We then nd that [] = 3=2 and [F ] = 2.Therefore, terms quadratic in F cannot contain derivatives, as F 2 already has massdimension 4, and so F cannot have a kinetic term.

F is therefore an auxiliary eld, with a purely algebraic equation of motion, and canbe integrated out. This means eliminating the eld by using its equation of motion, or,alternatively, performing the Gaussian path integral over this eld. The actual multiplettherefore contains only the elds (';) with helicities (0; 12) { this is the N = 1 chiralmultiplet introduced earlier.x4 The Lagrangian. From the dimensional analysis above, the only Lorentz invariantterms we could include in a free action are

L0 = '@2' i @ + FF : (96)In principle, one can include further interaction terms. We will postpone the discussionof these terms for simplicity to Sect. 5.1. A small calculation reveals that

L0 = 0 ; (97)

30

up to total derivatives, so that the action is invariant under SUSY transforms. From thetransformation laws (93){(95), we see that our representation of the SUSY algebra islinear. However, if we were to integrate out F , then SUSY would be realised nonlinearly,and the algebra would only close on{shell.

Exercise 11. Show that the actionRd4xL0 is invariant under SUSY transformation

(up to boundary terms). The following identities are useful: +

= 2 ; +

__ = 2 __ :

Going through the above exercise (and remembering that this is the simplest SUSYeld theory we could consider), it becomes apparent that many identities need to beemployed in SUSY calculations, and that there is great opportunity for making, say,sign errors. We would like to adopt a formalism which allows us to calculate withoutthe kind of labour implied by the above. We turn to this below.

31

4 Superspace and superelds

Introducing the idea of superspace will provide us with a natural and easy-to-use repre-sentation of the SUSY algebra. Note that the prex `super' always refers to the presenceof the poorly{named `parity' of all objects (the Z2{grading) introduced earlier. We beginwith a reminder of Graman numbers and their properties.

4.1 Reminder: Gramann numbers.

x1 Denition. Gramann numbers are, say, n formal parameters generating the Gra-mann algebra n,

i; j 2 n : fi; jg = 0 ;so that (i)2 = 0. It follows that any Z 2 n must have the form

Z = z0 + zii + zijij + : : : zi1:::ini1 in ;

(for some complex{valued coecients z) since each generator can appear at most once.The number of elements in the basis of n is clearly 2n. Elements of are calledsupernumbers, which are just functions of n Gramann numbers. The fact that (i)2 = 0implies that there is no inverse for i. Their parity is ~i = 1. (Parities are combined as~a~b = ~a+~b mod 2.) The algebra can be split according to the parity of its elements,

= 0 1 ;where the rst (second) component contains elements with only even (odd) powers ofi's. Another decomposition is = B S , body and soul components, where theformer contains only the c{number z0, so B = C.x2 Dierentiation and integration. A derivative should be a linear map annihilatingconstants and satisfying the Leibnitz rule. Consider Z = a+ b, where j for somej and a; b 2 R;C, or jj=0. We dene the derivative with respect to by

@Z

@= b :

The super{Leibnitz rule is, for Z1 and Z2 two elements of ,

@

@Z1Z2 =

@Z1@

Z2 + ()fZ1Z1@Z2@

:

The integral should be a superlinear functional obeying

1:@

@

Zdf = 0 ; 2:

Zd

@

@f = 0 :

The rst condition states that the integral should be independent of the variable whichhas been integrated over, the second condition is the foundation of Stokes theorem andintegration by parts. We are therefore led to demandZ

d = 0 ;Zd = 1 ;

32

which xes integration completely: It follows that, for Z = a+ b with a and b indepen-dent of ,

@Z

@= b =

Zd Z :

4.2 Flat superspace

x1 Split supermanifolds. Consider a manifoldM together with a vector bundle E ofrank k. Locally the total space is described by co{ordinates x1 : : : xn describing pointson the base M, and co{ordinates v1 : : : vk on E, as shown in Fig. 7.

R1,d1 : x

Ck : v1 . . . vk

M

Ex1

= C

k

Ex0

Figure 7: The base manifold M, which is R1;3 for us, with a vector bundle of rank kover it. For us, this is the trivial bundle W R1;3 = C2 R1;3.

Now, we apply the parity changing (in the Z2{sense) operator to the co{ordinateson the bers: so E is locally described by co{ordinates x1 : : : xn; 1 : : : k, where thei are Gramann variables. The space E is called a split supermanifold { these arethe supermanifolds relevant to physics. The picture of our space is now as in Fig. 8:the bers become innitesimal directions o the base manifold M. This global split inthe real directions (on M) and Gramann directions is the origin of the name `split'supermanifold.x2 Flat superspace. The space we will mostly be dealing with is R1;3j4 { this is Ewith E the trivial bundle W R1;3. Recalling that W = C2, the bundle W R1;3 justdescribes at Minkowski space, with, at every point, two new complex (or four real)directions in C2. Applying the parity changing operator, these new directions becomeinnitesimal, as they are parameterised by two complex Gramann numbers. So, the

M

Figure 8: The split supermanifold E, with the base manifold M (sometimes referredto as the body) and innitesimal directions j o the base (sometimes referred to as thesoul).

33

co{ordinates on our superspace are

(x; i) (x0; x1; x2; x3; 1; 2) : (98)

Note that it is very common to see 1; 2 included in the list of co{ordinates { this is justnotation emphasising that we have two complex directions. We briey comment that forN{extended theories (N > 1) we would work on the superspace R1;3j4N = EN , whereEN = WN R1;3. However, this is less useful because one can often only work withequations of motion, not with Lagrangians. We will return to these points later.x3 A caveat. In the following section we generate transformations on superspace byexponentiating the innitesimal generators P, Q and Q. We will use this to constructa representation of the SUSY algebra which acts on `superelds', which are just functionson superspace.

There is an important caveat. Up to now, we have written P = i@ when actingon elds. It is a (somewhat tiresome) quirk that the path we will re{tread below, andwhich much of literature also follows, leads to a representation in which P = +i@. Wewill explain why in due course, but the reader should be prepared for this change innotation.x4 Group of translations on at superspace. We now exponentiate our generators(with parameters), dening the group element G(x; ; ) by

G(x; ; ) = expixP iQ i Q : (99)

Using the BCH formula, which just says that composition of group elements gives othergroup elements,

expA expB = expA+B + 12 [A;B] + : : :

we can show that two of these group elements compose as

G(0; ; )G(x; ; ) = G(x + i i; + ; + ) ; (100)

where the terms denoted by ellipses in the BCH formula vanish because of the nilpotencyof i.

Exercise 12. Derive (100).

From (100) and the explicit expression of G(0; ; ), we can directly read o therealization of Q and Q in terms of dierential operators (and this is the form in whichthey will act on functions of superspace co{ordinates):

Q =@

@ i _ _

@

@x;

Q _ = @@ _

+ i _@

@x:

(101)

34

It follows that fQ;Qg = f Q; Qg = 0 andfQ; Q _g = 2i _@ =: 2i _P : (!!)

Here we see the result of considering `left actions' of the group in (100); we must nowidentify P = +i@. Considering `right actions' would have maintained our conventions,but sadly this is not the way things unfolded historically. We would in that case havebeen led to the representations D and D _ for the SUSY generators, dened by

D =@

@+ i _

_ @

@x;

D _ = @@ _

i _@

@x:

(102)

The only dierence between this and (101) is a change in sign in the second terms. Theseoperators obey

fD; D _g = 2i _@ ; (103)as we might have expected. They will still play an important role in what follows. Wenote that the Q and D operators anticommute completely { they do not talk to eachother at all,

fQ;Dg = f Q; Dg = fQ; Dg = f Q;Dg = 0 : (104)

4.3 Superelds

We are now ready to consider the superelds { functions on superspace.x1 General superelds. Due to the nilpotency of the Gramann numbers, the de-pendence of an arbitrary function, or supereld, on R1;3j4 can be written down explicitly.Writing 2 and 4 22, a general supereld has the form

F (x; ; ) = f(x) + (x) + (x) + 2m(x) + 2 n(x)

+ A(x) + 2 (x) + 2(x) + 4d(x) :(105)

Note that the product of two superelds is again a supereld. Superelds clearly form arepresentation of the supersymmetry algebra, where the action of the SUSY generatorson superelds is given by

F = (Q+ Q)F (Q _ Q _)F ;with explicit formula resulting from inserting the expressions (101). Note that we haveused one of the identities from Exercise 9 to write the SUSY generators in terms of Qand Q _ so that (101) is easily applied. Performing the derivatives, and matching powersof on both sides, allows us to extract the transformation laws of the component elds.Our Q and Q give us a linear rep of the SUSY algebra. For example,

f(x) F==0

= (x) + (x) :

Exercise 13. Conrm the above transformation of the component eld f(x).

35

x2 Reducibility. The general supereld above has sixteen (complex) degrees of free-dom. Recall, however, that the elds (', , F ) of our Wess-Zumino model had onlyfour (complex) degrees of freedom. In fact, the representation (105) of the SUSY alge-bra is highly reducible { there are a number of conditions we can impose on F whichreduce the number of components. To ensure that the reduced components still form arepresentation of the SUSY algebra, the imposed conditions have to be invariant undersupersymmetry transformations. One example is D _F = 0, which is compatible withour SUSY representation since fQ; Dg = f Q; Dg = 0, giving the so{called chiral super-eld. Another condition is F = F y, giving the vector superelds. We will consider thesetwo cases in more detail below.x3 Chiral and antichiral superelds. Since D and D anticommute with the SUSYgeneratorsQ and Q, the constrained superelds and obeying the chiral and antichiralconditions

D = 0 and D = 0 ; (106)

respectively, still form representations of the SUSY algebra: the conditions (106) arepreserved under SUSY transformations.

From the denition of D, a function obeying D _ = 0 must depend on thecombination

y := x + i : (107)

Note that as D contains no factors of , our function can also depend on , so we nd (y; ). y is called a chiral co{ordinate of chiral superspace. We can use it to writedown the component eld expansion of a chiral seld9 ,

(y; ) = '(y) +p2(y) + 2F (y) :

We have two complex scalars and one Weyl spinor { four complex degrees of freedom intotal. This is precisely the eld content of the Wess{Zumino model.

Exercise 14. In order to obtain something we can calculate with, we have to Taylorexpand our functions, in y, around x. Show that the chiral seld then becomes:

(y; ) = '(x) + i@'(x) +1422@2'(x) +

p2(x) ip

22 @(x)+ 2F (x) :

You might nd it useful to rst prove that

= 122 ; _ _ = +

122 _ _ and = 1

222

9We will abbreviate supereld to seld from here on. Some other supernouns may also become snouns.

36

We can also compute the SUSY transformation rules of the component elds usingchiral co{ordinates, i.e. we compute

= (Q + _ Q _) ; (108)

after rst changing variables (x; ; ) to (y; ; ). The chain rule gives us

@

@x! @

@y;

@

@! @

@+ i _

_ @

@y;

@

@ _! @

@ _ i _

@

@y;

from which it follows that Q and Q _ in our `new' co{ordinates are

Q =@

@; Q _ = @

@ _+ 2i _

@

@y: (109)

We now compute (108), nding

=p2 + 2F + 2i@'+

p2i2 _

_@ ;

from which we read o that

' =p2 ;

=p2i _

_@'+p2F ;

F = ip2(@) ;

which are precisely the transformation laws of the Wess{Zumino model elds we foundpreviously in (93), (94) and (95).

Analogously, one can nd the SUSY transformation laws for the components ofantichiral superelds by complex conjugating the chiral coordinates to obtain antichiralcoordinates on antichiral superspace.x4 Vector and complex linear superelds. Another constraint we can demand isthat a supereld V should be real, i.e.

V (x; ; ) = V (x; ; ) :

This is called a vector supereld, which we will later employ in the construction of Super-Yang-Mills (SYM) theories. There is also something called a complex linear supereld,and we will meet these in the course of the next section.

37

5 SUSY{invariant actions from superelds

5.1 Actions from chiral selds

x1 Invariant actions. Our aim is now to construct actions for our selds. Note thatthe SUSY transform (95) of the component eld F is a total derivative. Hence, thefollowing action will be SUSY invariant:

S =Zd4x

Zd2 Any chiral seld: (110)

The justication of this claim is that the integral over d2 picks out only the F{component of the chiral seld. The resulting integrand transforms into a total derivativeunder SUSY, and so the action will be SUSY invariant (neglecting boundary contribu-tions in the usual way). It can also be shown that the combinationZ

d4xZd2d2 a general seld (111)

is also invariant under SUSY as the 22 term of a general supereld transforms as atotal derivative.

Such actions may appear rather uninteresting if we consider only a single seld placedunder the integral, but recall that products of selds are again selds and products ofarbitrary chiral (antichiral) selds are chiral (antichiral) selds. This is because D obeysthe Leibnitz rule, so that D(0) = 0. In the following, we will build actions both fromterms of form (111) and (110).x2 Free action. The free action should be quadratic in component elds and we arethus led to try a product of, say, a chiral eld with its conjugate (an antichiral eld),and projecting onto its 22-component as follows,

S0 =Zd4x

Zd2

Zd2

=Zd4x FF + '@2' i @ ;

(112)

which is the integral of the Wess{Zumino Lagrangian we found in (96). Note that forevaluating the integral over Gramann variables, one has to use the expansion of thechiral and antichiral superelds in x-space. We now have a method of constructing freeactions for component elds.x3 The superpotential. We saw above that integrating over d4 gives rise to thefree Wess{Zumino Lagrangian on Minkowski space R1;3. Alternatively, we could havewritten an action of the form (110) by picking out the `F '{term of a polynomial W ()of a chiral seld ,

Lint =Zd2 W () +

Zd2 W () :

38

The polynomialW is called the superpotential. The second term, above, is just the com-plex conjugate of the rst, included so that (what will be) our Lagrangian is manifestlyreal.

Exercise 15. Working in chiral co{ordinates, show that the 2 component of N (y; ),for N 1, is given by

N j2 = N'N1F 12N(N 1)'N2 :

It follows from the above exercise that for a polynomial superpotential W () =a1+ a22 + : : :, we can write

W ()2=XN=1

aNN'N1F 1

2aNN(N 1)'N2

=X

N=1

aNN'N1

F 1

2

XN=1

aNN(N 1)'N2

=@W (')@'

F 12@2W (')@'2

:

(113)

The nal line gives us a compact way of writing the result. Note that although weworked in chiral co{ordinates, our action is an integral over d4y = d4x so that for theaction we can simply take the above results and write the arguments of the componentelds as x. Our action is

S =Zd4x

Zd4 +

Zd2 W () +

Zd2 W ()

=Zd4x FF + '@2' i @

+@W (')@'

F +@ W ( ')@ '

F 12@2W (')@'2

12@2 W ( ')@ '2

:

(114)

x4 Equations of motion for F . The action (112), generated by the combination, is quadratic in the component elds, in particular in F . The superpotential of theprevious paragraph contributed terms linear in F to the action. Combining our twoactions as in (114), the resulting equations of motion for F are

F +@W (')@'

= 0 ; F +@ W ( ')@ '

= 0 ; (115)

which are algebraic. Integrating out F , we generate a term

@W (')@'

W ( ')@ '

Lint ; (116)

39

which is a nontrivial self{interaction of the scalar eld '. In four dimensions, the max-imum interaction power in a renormalisable scalar eld theory is 4, which implies thatthe most general renormalisable Wess{Zumino model interaction is generated by thesuperpotential

W () = a1+m

2!2 +

3!3 : (117)

We have chosen a particular parameterisation of coecients which will be used in thefollowing section. Note that a1 can always be eliminated by a shift of the eld .x5 Mass terms. The second term of (117) generates a mass term for both the scalarelds ' and the spinors . As should be expected, these component elds have the samemass. Explicitly,

W (') =m

2'2 =) @W (')

@'

W ( ')@ '

= m2 '' ;

=) 12@2W (')@'2

= m2 :

(118)

Note that we are using Weyl spinors, not Dirac spinors. Hence, the kinetic term is notm, as it is for four{spinors , but rather m=2m =2. The factor of 1=2 isimportant for the masses of the spinor and the scalar to be equal.x6 SUSY transformations and F . Before integrating out the auxiliary eld F , ourSUSY transforms were linear in the component elds. After integrating out F , we havethe replacement F ! @ W ( ')=@' in the SUSY transformations. As a result, the trans-formations become non{linear, and close only up to terms proportional to the equationsof motion, i.e. the SUSY algebra closes only on{shell. In general, SUSY transformationscan close up to isometries, gauge transformations and equations of motion. Often, onecan even deduce the equations of motion from the SUSY transformations.x7 Generalisations. Consider now n chiral selds j , j = 1 : : : n. We can write downthe following invariant action,Z

d4xZd4 jj +

Zd2 W (1; : : : ;n) +

Zd2 W (1; : : : ; n) :

The rst term gives us the correct kinematic terms for the component elds. It is possibleto insert an arbitrary hermitian matrix here, sending the rst term to jM jkk, butwe can always perform a eld transformation which removes such a matrix, and hencewe omit it.

Giving up renormalisability of the action allows us to construct supersymmetric non-linear sigma{models. Recall that a sigma{model action is

S =Zd4x

14gij()@i@j ; (119)

where is a map from the worldsheet, in this case R1;3 to target space, a manifold Mdwith Riemannian metric gjk. The reader is perhaps more familiar with the case in which

40

the worldsheet is R1;1, and target space is M = R1;9, which is just the string theorynonlinear sigma{model in a at background.

If we `superize' the sigma{model, we arrive at an action where M is a complexmanifold,

S =Zd4x

Zd4 K(j ; k) :

Expanding this action in components, one arrives at a function called theKahler potentialK, giving rise to the Kahler metric gij ,

@2K(k; l)@i@j

= gij() :

The existence of the Kahler potential turns M in a Kahler Manifold, which is just acomplex manifold on which the metric gij comes from a potential. The action startswith the bosonic part (119), where gij is the Kahler metric.x8 Complex linear superelds. Now that we know how to construct actions fromselds, we can give an interesting example of duality in SUSY theories. We take a vectorseld V , and use it to dene a chiral seld via

D2V : (120)(We could have started with a general supereld, but this setup will reappear laterin gauge theory, so we might as well use it here.) This eld is clearly chiral sinceD D3V 0, from the anticommutation relation10 of D.

There is a redundancy in our denition of ; we could shift the vector seld byany such that D2 = 0, and obtain the same chiral seld . The seld is calleda complex linear seld. Recall that, earlier, we found there were essentially only twomassless representations of the SUSY algebra, given by the two multiplets of helicities(0; 12) and (

12 ; 1). The former is the content of the chiral seld, while the latter will turn

out to be the content of the vector seld. Our new seld is something else, and doesn'tseem to t here. The reason is that is dual to a chiral seld, in the following sense.Starting with arbitrary superelds 0 and Y , we can construct an invariant action fromZ

d4xZd4 00 + Y D20 + Y D2 0 :

Integrating out Y gives us the actionRd4x

Rd4 in terms of a complex linear seld

. Alternatively, integrating over 0, and dening = i D2Y , we arrive at the actionZd4x

Zd4

for the chiral seld . Hence, is dual to a chiral eld.10Note that these arguments are typical and common in SUSY, but the reader should beware that

they depend on the dimension of the theory under consideration. For us, working in 3 + 1 dimensions,

Weyl spinors are two component objects and so the product of any three such components is identically

zero, i.e. D3 D3 0.

41

5.2 Actions with vector superelds

Above, we formed actions containing kinetic (free) and potential (interacting) terms forchiral selds, i.e. we formed actions for the particles contained in the chiral multipletof states fj 0 i; j 12 ig. In fact, we gave the most general renormalisable action for theseelds, and so we have done all that we can with them. We turn now to the inclusion ofthe vector seld, i.e. the vector multiplet of states fj 1 i; j 12 ig and use them to constructSUSY gauge theories.x1 Gauge theory reminder. We have a vector potential A taking values in the Liealgebra g of a gauge group G,

A Aaa ; (121)where the a are hermitian generators11 of the Lie algebra g, ay = +a, while the Aaare real elds, so that Ay = A. The vector potential is part of the covariant derivative,which we write as r since we have used up all the D's already,

r := @ + igA : (122)

This gives rise to the eld strength F ,

F =1ig[r;r ] = @A @A + ig[A; A ] : (123)

Gauge transformations are given by functions U taking values in G. The gauge potentialtransforms according to A ! A0 = 1igU1rU while the corresponding eld strengthtransforms as F ! F 0 = U1FU . On innitesimal level, gauge transformations aregiven by a function taking values in g which leads to a change A = r.x2 Abelian gauge invariance. The expansion of a real vector seld, which lives onfull superspace, is

V (x; ;) = C(x) + i(x) i(x) + 122M(x) +

122 M(x)

A(x) + i2 + i

2@(x)

i2+ i2@ (x)

+1222

D(x) +

12C(x)

:

(124)

With C;A and D real, this seld manifestly obeys V = V . What is not so clear isthe reason for the mixing of the component elds in the latter terms. For the moment,

11Our choice of the physicists' convention of hermitian Lie algebra generators over the Mathematicians'

convention of anti-hermitian generators has a number of reasons. Most importantly, our conventions will

agree with the literature we rely on in the following and thus make it easier to read and compare the

original papers. Furthermore, it will render a (real) vector seld hermitian instead of anti-hermitian. In

one's everyday research, one should certainly use anti-hermitian generators, and translating the formulas

given here to the other convention is an important exercise everyone should go through once.

42

the reader should content themselves with this clearly being a choice we are allowed tomake. We will see shortly that it makes a subsequent calculation very simple.

As with the chiral seld, there appears to be some redundancy in the expansion(124). We expect the gauge eld A to correspond to the spin one state j 1 i, and tocorrespond to j 12 i, and we might expect an auxiliary eld, from our previous discussions.In addition, though, we also have here the elds C, M and . In fact, these elds canbe eliminated by introducing an abelian gauge symmetry, under which

V ! V + + ; (125)

where is a chiral seld, D _ = 0. But what does this have to do with gauge transfor-mations in the usual U(1) sense? Using the expansion given in Exercise 14, we write

+ = '+ '+p2 +

p2 + 2F + 2 F + i@(' ')

+ip22@ +

ip22 @ +

1422('+ ') ;

(126)

and comparing components, we see that under the transformation (125)

A ! A i@(' ') = A + @ ; (127)

which is just the usual transformation law of an abelian vector potential, upon writ-ing = 2=('). Upon further inspection, we see that under gauge transformations,C ! C +'+ ', so that we may choose such that '+ ' = C, and so C is gauged tozero. Similarly, we can choose F and to gauge away M and from the vector seld.

Exercise 16. Write down the gauge choice of elds ', and F which removes thecomponent elds C, and M from the vector seld expansion (124).

We see now the reason for our choice of component elds in (124) { the expansionis such that the additional component elds can be gauged away nicely using (125).(Another reason for dening the gauge transformation in terms of chiral selds will bemet shortly, when we couple to matter.) The resulting expansion of our vector seldreads as

V (x; ; ) = A(x) + i2(x) i2(x) + 1222D(x) ; (128)

which contains a vector A, a Weyl spinor and an auxiliary eld D. If we calculatethe square of our vector seld (128), we nd

V 2 = 1222AA =) V 3 = 0 ; (129)

which makes this a nice gauge to work in. We have gone to what is called Wess{Zumino (WZ) gauge, removing the redundant elds from our vector seld expansion.

43

We have not specied every component of , however; the remaining component is=('), so that WZ gauge xes everything except the usual gauge symmetry of the vectorpotential (127). So, we have seen that our supergauge transformations split up intoordinary gauge transformations and supertransformations, the latter of which are usedto eliminate redundant seld components.x3 In components { SUSY transformations. If we make a SUSY transformationthen, using V = (Q + Q)V as usual, we nd that the components of a vector eldtransform as

C = i i ;M = 2i + 2@ ;

A = : : :

(130)

In particular, note that the eldM , which was gauged away in WZ gauge, re{enters oncewe make a gauge transformation, due to the term appearing in (130). In other words,SUSY transformations break WZ gauge. However, we know that the elds which arebeing re{introduced can just be gauged back to zero, so there is always a compensatinggauge transformation which brings us back to WZ gauge (called the de Wit{Freedmantransformation [18]). We then say that the `SUSY transformations' of our componentelds are as follows, but by this we really mean a combined SUSY transformations andcompensating gauge transformation:

A = i i ;

= iD + F ;

D = r r :(131)

Note that, until now, all our SUSY transformations on component elds have been linear(provided the auxiliary elds are left in play). Here, though, the transformations arenonlinear due to the minimal coupling r term (the F term is also nonlinear in thenon{abelian theory). It is the compensating gauge transformation which generates thesenonlinearities.x4 Action. For an action we need to create terms of the form FF and r;from (129), we see that we will need expressions which are quadratic in V in order togenerate terms quadratic in A. In order to do so, we appeal to our previous experiencewith chiral selds. We dene

W W(V ) = g4D2DV ; (132)

noting thatW is chiral since DW D3 0. We already know how to construct actionsfrom chiral selds, and so we try as an action

14g2

Zd4x

Zd2 WW +

14g2

Zd4x

Zd2 W _ W _ : (133)

44

Under gauge transformations V ! V ++ , our W is, like F , gauge invariant, andindeed the resulting action in component elds is

S =Zd4x

12D2 1

4FF

ir : (134)

This is the free U(1) super Yang{Mills action, comprising the photon A (a boson), thephotino (a fermion) and the axillary eld D which appears as D2, rather than DD,because it comes from a real supereld.

Exercise 17. Show that W dened in (132) is gauge invariant.

x5 Abelian matter couplings. To minimally couple matter elds to ordinary gaugetheory, we replace @ with r, which requires matter elds to take values in a repre-sentation of the gauge group. Typically we consider matter in the fundamental, anti-fundamental and adjoint reps, the last of which is trivial for U(1) theories since adjointmatter is neutral. In analogy with ordinary U(1) gauge invariance, we dene the gaugetransformations of chiral selds and in the fundamental and antifundamental repsas12

! eg ; fundamental ;! eg ; antifundamental :

(135)

Recall that the `ordinary' gauge parameter is the lowest component of =(), andtherefore (135) yields indeed the well-known gauge transformations

'! eig' and '! 'eig : (136)

We see here another reason for taking the gauge transformations to be generated bychiral selds ; it is only then that the chiral seld remains chiral under gaugetransformations.

Recall the free chiral seld action (112), with kinetic term . This is clearlynot invariant under (135), picking up the exponent of g( + ). Nevertheless, it isstraightforward to write down a coupling which is gauge invariant, and which reducesto (112) when g ! 0, Z

d4xZd4 egV : (137)

Recalling also that the chiral seld contains the component elds ', and F , the eectof including the exponential term in (137) is to make the replacements

@'! r' ; and @! r ;12Note that Wess & Bagger switch from `V ! V + + ' to 'V ! V + i i'. We maintain our

convention of (125).

45

This gives us kinetic terms for matter in the fundamental and antifundamental. Whatabout potential terms? From (117), the superpotential terms involve only or , andwe cannot make these gauge invariant using insertions of the type in (137). If we want toinclude matter, we can do so provided that we have at least two chiral selds of chargesgj , i.e.,

j ! egjj ; j ! jegj :We can then write down superpotentials of the form

W (j) = mijij + ijkijk ; (138)

which are gauge invariant provided the couplings mij and ijk vanish whenever thecombinations of charges gi + gj and gi + gj + gk are non{zero. With the inclusion ofthe kinetic terms (137) for each of the chiral selds, our gauge{matter U(1) Lagrangianbecomes

L = 14g2

Zd2 WW +

14g2

Zd2 W _ W _ +

Zd4 j egjV j

+Zd2 W (j) +

Zd2 W (j) :

(139)

The rst line contains the kinetic terms, while the second line contains the superpotentialterms as in (138).x6 Non{abelian gauge invariance. We now wish to extend the above discussion tonon{abelian theories. Note that with our conventions, vector elds V become V V aa with the reality condition V = V y being equivalent to V a = V a (which is a niceconsequence of choosing hermitian generators).

We begin by generalising the supereld W to a non{abelian seld, and then con-struct the action for the gluon and gluino. Previously, we hadW = g4 D2DV . Notingthat V was essentially an element of the algebra of our gauge group, we recall the rela-tion = et@t et relating algebra () and group (e) elements and make the educatedguess for the non{Abelian W

W = 14D2egVD egV

: (140)

To identify how this seld behaves under gauge transformations, we must extend thetransformation law of the vector elds. Note that in (139), it is really the transformationrule for expV , rather than V alone, which gives us a gauge invariant action. It is thistransformation which we must general