Lecture Notes Are Based on a Series of Lectures Given by Myself from April to June 2009 in the School of Mathematics at Trinity College, Dublin

Christian S¨amann Notes by Anton Ilderton

August 24, 2009

1 Contents

1 Introduction 5 1.1 A SUSY toy model ...... 5 1.2 What is it good for? ...... 10

2 Spinors 12 2.1 The Poincar´egroup ...... 12 2.2 Spin and pin groups ...... 18 2.3 Summary ...... 22

3 The SUSY algebra 24 3.1 SUSY algebra on R1,3 ...... 24 3.2 Representations ...... 25 3.3 The Wess–Zumino model ...... 28

4 Superspace and superﬁelds 32 4.1 Reminder: Graßmann numbers...... 32 4.2 Flat superspace ...... 33 4.3 Superﬁelds ...... 35

5 SUSY–invariant actions from superfields 38 5.1 Actions from chiral sfields ...... 38 5.2 Actions with vector superfields ...... 42

6 SUSY quantum field theories 49 6.1 Abstract considerations ...... 49 6.2 Sfield quantisation of chiral sfields ...... 51 6.3 Quantisation of Super Yang–Mills theory ...... 58

7 Maximally SUSY Yang–Mills theories 63 7.1 Spinors in arbitrary dimensions ...... 63 7.2 Actions and constraint equations ...... 64 7.3 d = 4, N = 4 super Yang–Mills theory ...... 67

8 Seiberg-Witten Theory 72 8.1 The moduli space of pure N = 2 SYM theory ...... 72 8.2 Duality ...... 74 8.3 The exact eﬀective 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 field theories. SUSY was discovered in the early 1970’s and has attracted growing attention ever since, even though there is still no experimental evidence for its existence up to this day. There are essentially two reasons why high-energy physicists keep interested in supersymmetry: From a phenomenological point of view, the supersymmetric extension of the Standard Model provides very reasonable solutions to some of the remaining puzzles in particle physics. On the other hand, supersymmetric field theories have many intriguing features which often can be accessed analytically, making them ideal toy models for theorists. These lecture notes are based on a series of lectures given by myself from April to June 2009 in the School of Mathematics at Trinity College, Dublin. The material covered in these notes was presented during eleven lectures, each lasting 90 mins, which was a little ambitious, retrospectively. The aim of the lectures was to give a reasonable overview of this topic while being thorough enough to provide a graduate student with the necessary tools to do research involving supersymmetric field theories herself. I put some emphasis on motivating spinors, as I still remember struggling with the reasons for their existence when I first encountered them. Besides the standard material presented in almost any course on supersymmetry, I chose superfield quantisation, maximally supersymmetric Yang-Mills theories and Seiberg-Witten theory as additional topics for these lectures. I had just used supergraphs in a research project myself and their usefulness and simplicity was still fresh in my mind. The maximally supersymmetric Yang-Mills theories with their amazing properties were included because of the important role they play in string theory today. Finally, the section on Seiberg-Witten theory demonstrates why supersymmetric field theories are indeed beautiful toy models. If you should find any typos or mistakes in the text, please let us know by sending an email to [email protected] or [email protected]. The most recent version of 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 notebook which performs many of the tedious computations necessary for understanding these lectures automatically. I am very grateful to Anton Ilderton for doing such an amazing job with writing up these notes, correcting prefactors and providing typed-up solutions to all the steps left as exercises during the lectures. I would also like to thank all the people attending the lectures at Trinity for making them more interesting and lively by asking questions.

Christian S¨amann

3 General remarks

Our metric is (−, +, +, +), which diﬀers by a sign from the metric of most quantum ﬁeld theory textbooks, as e.g. Peskin–Schroeder. The remainder of our conventions are introduced 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 familiar with 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 diﬀerent 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 ﬁelds. 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 and supergravity: 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 one thousand 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 section we will discuss a simple model which will illustrate many of the important physical properties of SUSY theories which are commonly discussed in a field theory context. It will also serve to illustrate the fundamentals of many calculations we will later perform in field theory, in a simple and accessible setting. Much of the following discussion can be found in [9, 10] in more detail. §1 Definition. Our model is quantum mechanical. We have a hermitian Hamiltonian H and non–hermitian operators Q, Q† related through the anti–commutator 1 1 H = {Q, Q†} ≡ (QQ† + Q†Q) , (1) 2 2 where the operators obey the following “0 + 1 dimensional SUSY algebra”,

{Q, Q} = {Q†,Q†} = 0 (2) [Q, H] = [Q†,H] = 0 .

The Q and Q† are called supercharges and generate supersymmetry transformations. We always count real supercharges, so here we have two of them. It is a direct consequence 2 †2 of (2) that Q = Q = 0. ¡ ¢ §2 Properties. Consider a Hilbert space H, ⟨ | ⟩ carrying a representation of H, Q and Q†. From the algebra (2), it follows that H is positive deﬁnite, for, given any state | Ψ ⟩ ∈ H,

2⟨ Ψ |H| Ψ ⟩ = ⟨ Ψ |{Q, Q†}| Ψ ⟩ = ⟨ Ψ |QQ†| Ψ ⟩ + ⟨ Ψ |Q†Q| Ψ ⟩ (3) = ∥Q| Ψ ⟩∥2 + ∥Q†| Ψ ⟩∥2 ≥ 0 .

We see that states in a SUSY theory have non–negative energy; note that the minimum may or may not be obtained. To further examine the state space, we diagonalise H and consider the eigenstates | n ⟩ such that

H| n ⟩ = En| n ⟩. (4)

We must treat the two cases En = 0 and En > 0 separately. We√ begin with the√ case † † En = E > 0. We may here introduce the scaled operators a = Q/ 2E and a = Q / 2E which, within the space of states of energy E, obey the algebra

{a, a†} = 1 , {a, a} = {a†, a†} = 0 . (5)

This is a simple example of a Cliﬀord algebra, which we will study later on. We now construct 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 have a2 = a†2 = 0, from which it follows that the only eigenvalue of a is 0. Call the state with this eigenvalue | − ⟩. We can create no more states by acting on this with a, and we can create only one more by acting with a† (since a†2 = 0), which we call | + ⟩ ≡ a†| − ⟩. Hence we have a subsystem of two states obeying

a†| − ⟩ = | + ⟩ , a| + ⟩ = | − ⟩ , a| − ⟩ = a†| + ⟩ = 0 .

A simple 2d representation of the algebra (5) is given by the following matrices and vectors, Ã ! Ã ! Ã ! Ã ! 0 0 0 1 1 0 a = , a† = , | + ⟩ = , | − ⟩ = . (6) 1 0 0 0 0 1

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 transformed into each other by the action of the SUSY generators, and more generally we will see that Q| boson ⟩ = | fermion ⟩ ,Q| fermion ⟩ = | boson ⟩ , and similarly for the action of Q†. We also have an example of the SUSY property that states of non–zero energy are degenerate and appear in pairs; a possible non–zero spectrum of a SUSY theory is sketched in Fig. 1.

E − + − + − + − +

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

We now turn to the states of zero energy, H| 0 ⟩ = 0. Directly from equation (3), we must have 0 = ∥Q| Ψ ⟩∥2 + ∥Q†| Ψ ⟩∥2 , so that a vacuum state | 0 ⟩ exists if and only if

Q| 0 ⟩ = Q†| 0 ⟩ = 0 . (7)

6 §3 SUSY breaking. The vacuum should be unique in quantum mechanics and therefore invariant under supersymmetry transformations – this is just the statement that Q| 0 ⟩ = Q†| 0 ⟩ = 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 forms shown 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 rotational symmetry, but not SUSY. The quadratic potential with a non–zero minimum, on the other hand, breaks SUSY but not the rotational symmetry. We will return to these points in a moment, after looking at a representation of the SUSY algebra on a Hilbert space of wavefunctions.

x x

Figure 2: The ‘quadratic’ potential on the left breaks SUSY but not the rotational symmetry, whereas a minimum in the ‘Mexican hat’ potential breaks the rotational symmetries but not SUSY.

§4 SUSY quantum mechanics. To have a Hilbert space on which we can act with Q, Q† and H we choose a set of functions in L2(R) ⊗ C2 , Ã ! ψ (x) ψ : bosons, ψ = + + (8) ψ−(x) ψ− : fermions, following (6). We now deﬁne the SUSY generators, Ã ! Ã ! 0 1 ¡ ¢ 0 0 ¡ ¢ Q† = P + iW ′ ,Q = P − iW ′ , (9) 0 0 1 0 where W (x) is a real function, a dash denotes diﬀerentiation with respect to x and

P = −i~∂x.

Exercise 1. Show that the Hamiltonian of this system is 1 1¡ ¢ ~ H ≡ {Q†,Q} = P 2 + W ′2 1 − W ′′σ , (10) 2 2 2 3 where σ3 = 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 this theory. States annihilated by both Q and Q† take one of the forms Ã ! ψ ψ = + &(P − iW ′)ψ = 0 =⇒ ψ ∝ e−W/~ , 0 + + or (11) Ã ! 0 ′ W/~ ψ = &(P + iW )ψ− = 0 =⇒ ψ− ∝ e . ψ−

The choice of ground state is dictated by normalisability, since if W (±∞) = ∞ only ψ+ is normalisable, and if W (±∞) = −∞ only ψ− is normalisable. Note that if W (∞) = −W (−∞) neither of the states are normalisable, and so there can be no ground state wavefunction. The role played by these boundary conditions gives us an explicit realisation of SUSY breaking discussed above. Note that for the cases W (±∞) = ±∞, i.e. a ground state wavefunction exists, the derivative W ′ must be somewhere vanishing, and so the potential W ′2 in our theory has a zero, i.e. SUSY is unbroken. Conversely, if W ′2 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 more precise, one has to take into account non-perturbative contributions: although W (∞) = −W (−∞) still allows for points x with W ′(x) = 0, there has to be an even (counted with multiplicities) number of such points (consider, e.g., the potential W ′2(x) = x4). The true quantum vacuum is unique and will be a superposition of wave functions localised at these points. Here, however, non-perturbative corrections will lift W ′(x) to be strictly positive and thus break SUSY. This explains why we do not ﬁnd SUSY ground states for W (∞) = −W (−∞). §5 Witten index. Introduce the operator (−)F , which gives +1 on bosonic states and −1 on fermionic states [10]. For our states of non–zero energy, we have Ã ! 1 0 (−)F = 2aa† − 1 ↔ . (12) 0 −1

More generally, we would have states of arbitrary energy labelled as | +; j ⟩ and | −; j ⟩ where j is some collection of quantum numbers, and

(−)F | +; j ⟩ = | +; j ⟩ , (−)F | −; j ⟩ = −| −; j ⟩ .

Consider the operator X Tr(−)F ≡ ⟨ n |(−)F | n ⟩ , (13) n where n runs over the energy eigenvalues. This operator counts the number of bosonic states minus the number of fermionic states, or nB − nF . However, since states with

8 E=0 − E=0 E > 0 always occur in pairs, all this operator really counts is nB nF , i.e. the number 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 coeﬃcients in some potential, the volume 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 only boson–fermion pairs of states can move away from the vacuum E = 0, which clearly E=0 − E=0 cannot change the value of nB nF . The value of the index therefore tells us something about SUSY breaking:

Tr(−)F ≠ 0 =⇒ SUSY is unbroken , (14) since if the index is non–zero then there is at least one state in the vacuum sector. Note that if we ﬁnd that the index is zero, we cannot infer that SUSY is broken. As the index is analytic, it is easy to calculate. It may be regulated, for example, by h i Tr (−)F e−βH .

§6 The Witten index as an operator index. We now show that the Witten index is, indeed, an index in the sense defined in the mathematics literature, see e.g. [11]. We can do this quite generally. Given a SUSY theory, we split the Hilbert space into bosonic and fermionic sectors, H = HB ⊕ HF , on which the SUSY generators must take the block off–diagonal form Ã ! Ã ! † 0 M1 † 0 M2 Q = ,Q = † , (15) M2 0 M1 0 for some operators M1 and M2. The above forms follow from the statement that the SUSY generators transform bosons into fermions and vice versa – they are the generalisations of the matrices in (6). Now define the operator Qe = Q + Q†, which is hermitian and annihilates E = 0 states, Ã ! † e 0 M1 + M2 Q = † M2 + M1 0

E=0 † E=0 † We then have that nB = dim ker(M2 + M1 ), and nF = dim ker(M1 + M2 ). Hence, E=0 − E=0 † − † nB nF = dim ker(M2 + M ) dim ker(M1 + M ) 1 2 (16) ≡ † ind M2 + M1 , ¡ ¢ E=0 E=0 † † as claimed. We also have nB + nF = dim ker Q /imQ .

9 −1 −1 αEM αEM

−1 −1 αW αW

−1 −1 αS αS

logΛ/GeV logΛ/GeV

Figure 3: The strong, weak and electromagnetic couplings α−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 explicit representations, a function W , the superpotential. Here we collect some motivating reasons for studying SUSY. §1 Coleman–Mandula theorem. We may ask the question: can one extend spacetime symmetries non–trivially beyond the Poincar´egroup? The answer goes as follows. Assume G is the symmetry group of a theory with S–matrix S such that

• G contains the Poincar´egroup,

• all particles have positive energy, with ﬁnitely many particles of mass m < m0 for all m0,

• S–matrix elements ⟨ out |S| in ⟩ are analytic and non–trivial, ∼ then the Coleman–Mandula theorem tells us that G = Poincarégroup × internal symmetries. So the answer to the above question appears to be no. There was a hidden assumption in this theorem, however – that the Lie algebra of G was generated by commutators. As we saw above, SUSY algebras, however, include anticommutators and therefore provide a loophole to the Coleman–Mandula theorem. §2 Gauge coupling unification. The gauge group of the standard model is SU(3) × SU(2) × U(1). Various attempts have been made at constructing a grand unified theory, or GUT, which unifies the standard model at some high energy scale, within a single group, be it SU(5), SO(10), E6, etc. A unified theory would imply a unified coupling; however, the couplings in the standard run as shown in the left panel of Fig. 3, and do not appear to intersect. With SUSY, however, the situation is improved – the couplings very nearly unify at the order of 1016 GeV. §3 Hierarchy problem. The masses in the standard model are generated by the Higgs particle. Experimentally, we have ⟨H⟩ ≃ 174 GeV, but this is very sensitive to quantum corrections which can be estimated to be of the order 1032 GeV. As keeping the experimentally desired value of 174 GeV would require an unnatural amount of fine tuning, it seems that some protection mechanism is at work, and SUSY is a good candidate for

10 q ⟨ ⟩ ∼ − 2 this. Here, H mH , and we are thus looking at quantum corrections to the mass of the Higgs boson. Many Feynman diagrams contributing to mass corrections in SUSY theories cancel against other Feynman diagrams in which a particle loop is replaced by its superpartner loop, see Fig. 4 .

f Hb H + =0

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

§4 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 likely a number of different constituents of dark matter. One of the most important candidates besides neutrinos is the neutralino, the lightest particle of the minimal supersymmetric standard model (MSSM) yet to be found. The hope is, of course, that the LHC will find the 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 there are actually good theoretical reasons for discovering first the particles of the ordinary standard model, if one assumes a MSSM with broken SUSY. §5 Theorist’s reasons. SUSY theories are highly constrained by symmetries and are therefore ideal toy models. Local SUSY theories contain gravity, and perhaps even give good theories of quantum gravity. N = 8 supergravity might actually be finite, similarly to N = 4 super Yang-Mills theory [12]. String theories are also much nicer with SUSY included, as the tachyonic states of the bosonic string can be safely removed from the spectrum. In mathematics, it could be that mirror symmetric partners of rigid Calabi– Yau manifolds are Calabi–Yau supermanifolds, and therefore mirror symmetry might require 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 this topic in some detail in this part of the lectures, and things will be clearer for it later. The aim is to find all irreducible representations (‘irreps’) of the Poincarégroup, as all fields in physics live in such representations.

2.1 The Poincarégroup §1 Definition. The Poincarégroup is the group of isometries (maps preserving distance) on Minkowski space R1,3. It is a non–compact Lie group. Its generators are four translations, Pµ, and a total of six boosts and rotations, Mµν = −Mνµ. The Lie algebra relations are ¡ ¢ [P ,M ] = i η P − η P ≡ iη P + symm. , (17) ρ µν ¡µρ ν νρ µ ¢ µρ ν [Mµν,Mρσ] = −i ηµρMνσ + symm. . (18)

In the ﬁrst line we have introduced the operation “symm.” which takes account of the symmetries of the generators, i.e. it symmetrises or antisymmetrises as appropriate. For example, the left hand side of (17) is antisymmetric in µ and ν, hence “symm.” generates the second term on the right hand side of (17).

Exercise 2. Check you understand the deﬁnition of “symm.” by generating the remaining three terms on the right hand side of (18).

The Poincar´egroup is R1,3 o O(1, 3), the semi–direct product of the abelian group of 1,3 translations R generated by Pµ and the Lorentz group O(1, 3) generated by the Mµν. 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. §2 The Lorentz subgroup. Consider the vector representation of O(1, 3). That is, given an element x ∈ R1,3,   x0   x1 x =   , x2 x3 an element of the Lorentz group will be represented by a 4 × 4 matrix. There are four special elements of the group we would like to consider. They are:

12   1    1  1 =   The identity element, which takes any vector to itself. 4  1  1   −1    1  T =   The time reversal operator, which takes x0 → −x0,  1  but leaves spatial components unchanged. 1   1    −1  P =   The parity operator, which takes us to a mirror world  −1  xj → −xj, but leaves the time direction unchanged. −1

PT = −14 The combined parity–time transformation which takes xµ → −xµ.

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

↑ ∋ 1 ↓ ∋ T ↑ ∋ P ↓ ∋ PT L+ ,L+ ,L− ,L− , (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 called ↑ proper orthochronous. Those transformations with determinant +1, i.e. those in L+ and ↓ L−, connected to 14 or PT , form SO(1, 3). Note that the four components in (19) are disconnected. Pictorially, the Lorentz group breaks down into components as shown in Fig. 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 show there is no continuous path from SO(1, 3) to any element not in SO(1, 3), which realises the 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.

§3 Representations of the Poincar´eand Lorentz groups. Particles in QFT are 1,3 described by ﬁelds ϕρ on R which, if we perform a Lorentz transformation x → Lx, should transform under some representation of the Lorentz group, i.e.

−1 ϕρ(x) → ρ(L)ϕ(L x) , (20) where ρ(L) is some matrix representation of the Lorentz group. Under sequential Lorentz transformations x → L2L1x =: L3x we should have

→ −1 −1 −1 ϕρ(x) ρ(L2)ρ(L1)ϕ(L1 L2 x) = ρ(L3)ϕ(L3 x) , (21) and thus the fields ϕρ form representations of the Lorentz (and Poincaré)group. To label representations we need the Casimir operators, which commute with every generator of the group. Casimirs for the Poincarégroup are 1 P P µ and W W µ ,W µ := ϵµνρσP M . (22) µ µ 2 ν ρσ W µ 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 which P µ = (m, 0, 0, 0) so P 2 = −m2. It can then be shown that W 2 = −m2s(s + 1) where s ∈ Z/2 is the spin of the particle or ﬁeld.

2. Massless representations, m2 = 0. In this case there is no rest frame, but we

can always choose a frame such that Pµ = (E, 0, 0,E), in which case we ﬁnd 2 2 P = W = 0. However, it can be shown that Wµ = λPµ here, where λ is the helicity of the particle, λ ∈ Z/2, and this is used to label the representations.

Elementary particles should form (“sit in”) irreducible representations of the Poincaré group. Hence, the problem of writing down all possible fields in QFT is equivalent to finding all possible irreps of the Poincarégroup. §4 Remarks. We restrict ourselves in the sequel to finding all irreps of the Lorentz subgroup – the extension to the full Poincarégroup is trivial. Representations should be linear (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 such representations up to a phase, to representations up to a sign, and Bargmann showed later [15] that studying all unitary irreps of the Poincarégroup up to a sign corresponds to studying all irreps of the universal covering group. The universal covering group of a group X is a simply connected space Y with a map f : 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 encounter the double cover of a space, where Y is not necessarily simply connected and the map f is 2:1. We turn now to the universal cover of the Lorentz group, beginning with the proper orthochronous component. § ↑ ↑ ≡ 5 The universal cover of L+. We begin by showing that the universal cover of L+ + SO (1, 3) is SL(2, C). First deﬁne the four–vector of Pauli matrices σµ, Ã ! Ã ! Ã ! Ã ! 1 0 0 1 0 −i 1 0 σ = , σ = σ = σ = . (23) 0 0 1 1 1 0 2 i 0 3 0 −1

Using these matrices, we construct the carrier space on which our representation must act – this space ‘contains’ all of our original vectors xµ. The carrier space is deﬁned by Ã ! x0 + x3 x1 − ix2 X(x) := xµσ = . (24) µ x1 + ix2 x0 − x3

The inverse map, which extracts xµ from X, is1 1 ¡ ¢ xµ = Tr σ X . (25) 2 µ µ The vector norm is x xµ = − det X, where the minus sign is due to our metric conventions. The determinant of this matrix must therefore be preserved under Lorentz transformations L; the relevant representation is ρ(L) ∈ SL(2, C). We will give the explicit form of ρ(L) below, but we ﬁrst note that the action of L on X(x) is

L ◃ X(x) = ρ(L)X(x)ρ†(L) , (26) which clearly preserves the determinant of X since det(ρ(L)) = det(ρ†(L)) = 1. Now, → for a rotation by an angle θ around a unit direction n, and a boost with rapidity tanh α → in the direction n, the matrix ρ(L) is · ³ ´ ¸ 1 → → → ρ(L) = exp αn − iθn · σ ≡ B R . (27) 2 L L (Boosts along an axis commute with rotations about that same axis, hence the exponential 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 a boost. Note that the map is 2 : 1, since for rotations of angles θ and θ′ := θ + 2π

Rθ′ = −Rθ , (28) but the presence of both ρ(L) and ρ†(L) in (26) means that the sign is killed and these correspond to the same Lorentz transformation. We will see a peculiar consequence of this in the next section. 1Note the index positions, which appear unnatural. Once we have introduced theσ ¯ matrices, the µ − 1 µ reader will understand that what we should really write is x = 2 Tr(¯σ X).

15 Finally, we leave it as an exercise to show that SL(2, C) is simply connected – it is therefore exactly the group we need for classifying the irreps of SO+(1, 3). We now look for 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 elements M,N ∈ SL(2, C) can always be continuously deformed into each other (“are homotopically equivalent”).

∼ §6 Weyl spinors. In the fundamental representation, the vector (carrier) space is W = C2, the space of two–component complex Weyl spinors ψ, Ã ! ψ1 ψ = , ψj ∈ C. (29) ψ2

The action of the Lorentz group is

L ◃ ψ = ρ(L) ψ , (30) where ρ(L) are the matrices (27). Now, since there is only one ρ present, rotations of angle 2π do not leave spinors invariant – they pick up a minus sign from (28). Instead, it takes a rotation of angle 4π to bring a spinor back to itself. This shows us that spinors really 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 Lorentz invariant. To do this, we need a dual space3. We write the dual space as W ∨ ≃ C2 which consists of elementsχ ˜ on which the Lorentz group acts as Ã ! χ1 χ˜T = ∈ W ∨ , L ◃ χ˜ =χ ˜ ρ(L−1) . (31) χ2

To construct the scalar product (which is really a bilinear form as it is not positive deﬁnite), we need a map M : W → W ∨, which takes ψ → ψT M. The pairing

(ψ, ψ′) ≡ ψT Mψ′ (32) must be invariant under Lorentz transformations, i.e. ρT (L)Mρ(L) = M. The only possible M are Ã ! 0 ±1 M = . (33) ∓1 0

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 xµ – we construct a map to the µ ν 2 µ dual space using the metric, x → xµ = ηµν x , which furnishes us with the inner product x = x xµ = µ ν x ηµν x .

16 Exercise 5. Prove the result (33).

∨ α αβ ∨ We now deﬁne W as follows: for ψα ∈ W , ψ := ϵ ψβ ∈ W , where our choice of M is ϵ12 = 1, ϵ21 = −1, i.e. Ã ! 0 1 ϵαβ = . (34) −1 0 Then, α αβ βα (ψ, χ) ≡ ψ χα = ϵ ψβχα = −ϵ ψβχα . (35)

For the scalar product to be symmetric we ﬁnd ψβχα = −χαψβ, i.e. spinors must anticommute. We assume this from now on. We will also abbreviate the inner product as follows: α (ψ, χ) → ψ χα or ψχ (= χψ) .

§7 Complex conjugate Weyl spinors. We also deﬁne the complex conjugate representation, W : its elements4 ψ¯ are labelled by upper dotted indices, ψ¯α˙ and transform under Lorentz transformations as ρ†(L)ψ¯. The space dual to W¯ is deﬁned by ¯ ¯β˙ ψα˙ = ϵα˙ β˙ ψ . (36)

Here ϵ12 = −1, ϵ21 = +1, so Ã ! 0 −1 ϵ = ϵ = . (37) αβ α˙ β˙ 1 0

αβ α β˙γ˙ γ˙ Note that ϵ ϵβγ = δγ and ϵα˙ β˙ ϵ = δα˙ . §8 Other examples of spinor representations. We have just seen two examples of spinor representations; the Weyl spinors and their conjugates. These representations are labelled by their helicities,

1 W : ψα , ( , 0) 2 (38) α˙ ¯1 W : ψ , (0, 2 ) We also have the vector representation:

µ µ αα˙ 1 ¯1 Vector : x = σαα˙ x , ( 2 , 2 ) . (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 tensor products of the 2d Weyl spinor rep. of SL(2, C), m n¯ ϕ α˙ 1...α˙ n , ( , ) . (40) α1...αm 2 2 4The reader who is wary of this notation for complex conjugation, or more familiar with Dirac spinors ¯ † ¯ α† where we write Ψ = Ψ γ0, might like to think of the Weyl spinor ψβ˙ as ψ σ0αβ˙ , as this is exactly the same thing.

17 For example, the Yang–Mills ﬁeld strength Fµν ∼ [∇µ, ∇ν] may be written

˙ F µν → σ[µ σν] F ααβ˙ β . (41) αα˙ ββ˙

The antisymmetry in (µ, ν) implies that F ααβ˙ β˙ can be written

˙ ˙ ˙ F ααβ˙ β = ϵαβf α˙ β + ϵα˙ βf αβ . (42)

R4 After a Wick rotation to Euclidean space , if fα˙ β˙ = 0 then F is self-dual: F = ⋆F , which corresponds to a gauge field configuration known as an instanton, while if fαβ = 0 then F = − ⋆ F ; this is an anti–instanton. §9 Other signatures. In Euclidean signature the symmetry group is SO(4), and the corresponding double cover is SU(2) × SU(2). For the metric diag(+, +, −, −) with Kleinian 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 by factors of i such that the condition ∥x∥2 = − det X(x) is maintained. Note that for finding all unitary ray representations for other signatures and other dimensions, the double cover of the group of isometries is sufficient.

2.2 Spin and pin groups ↑ So far we have looked at L+, but we are still looking for the full universal cover of O(1, 3). We pursue this below. For a more detailed exposition of the material, see for example [16]. § 5 1 The Cliﬀord algebra. Consider an algebra generated by d objects γµ, for µ = 0, 1, . . . d − 1, { } − 1 γµ, γν = 2ηµν d , (43) along with the vector space V = Span(γµ). The sign on the right hand side of this equation is pure convention. Most books working with a mostly minus metric (WCM) use the opposite sign. The Cliﬀord algebra C(V ) is M C(V ) = V ∧n = C ⊕ V ⊕ V ∧ V ⊕ V ∧ V ∧ V ⊕ ... ⊕ V ∧d . (44) n≥0 C ∋ 1 ∋ ∧ ∋ ≡ Here, d, V γµ, V V γµν γ[µγν], etc, where we antisymmetrise over indices as symmetric combinations of the γµ can be reduced to expressions containing fewer γ’s using the anticommutator (43). The sum in (44) terminates for the same reason. The algebra decomposes as C(V ) = C+(V ) ⊕ C−(V ) , (45)

5We use a heavy script to denote the abstract elements of the Cliﬀord algebra. Later, we will use normal script γ for the usual gamma matrices.

18 + − where C (V ) contains products of even numbers of γµ while C (V ) contains products of odd numbers of the γµ. With the definition of the following operation τ (an anti- involution) on this space, ¡ ¢ ¡ ¢ τ − n γµ1 ... γµn = ( 1) γµn ... γµ1 , (46) we can now define the pin group. §2 The pin group. On the vector space V with isometry group SO(V ), the associated pin group is defined as

Pin(V ) := {Λ ∈ C(V ) | Λ.Λτ = 1 , Λ.V.Λτ ⊂ V } . (47)

The interesting point is that Pin(V ) carries a natural representation of the group SO(V ), where the action of an element L ∈ SO(V ) on C(V ) is given by the following expression with ΛL ∈ C(V ): τ −1 ν ΛLγµΛL = ΛLγµΛL = γνL µ . (48) Note that we could replace τ with the inverse because of the deﬁnition of the pin group.

§3 Examples. One can now easily ﬁnd a couple of explicit examples for ΛL. Consider, for example, the parity transformations P. The associated pin group element is given by τ ΛP = iγ0, which obeys ΛP ΛP = 1 as required using the fundamental anticommutator. We can now check that the action of ΛP reverses the sign of γµ for µ = 1, 2, 3, but leaves γ0 alone, hence − Pν iγ0γµ( iγ0) = γν µ , (49) and we have correctly identiﬁed the parity operator. The time reversal transformation T corresponds to ΛT = γ1γ2γ3, and its action reverses the sign of γµ only for µ = 0. We also want the continuous Lorentz transformations. These are µ ¶ 1 ↑ Λ = ± exp [γ , γ ]θµν ,L ∈ L . (50) L 8 µ ν +

≡ 1 Writing Σµν 4 [γµ, γν], one can show that Σµν satisfy the algebra of the generators of the Lorentz algebra Mµν. §4 The spin group. © ª Spin(V ) := Λ ∈ C+(V ) | Λ.Λτ = 1 , Λ.V.Λτ ⊂ V = Pin(V ) ∩ C+(V ) . (51)

Here, as above, C+(V ) contains only even numbers of γ’s. The relationship between the spin and pin groups is shown in Fig. 6. Spin(V ) and Pin(V ) are connected and form double covers of SO(V ) and O(V ),

1 → Z2 → Pin(V ) → O(V ) → 1 ,

1 → Z2 → Spin(V ) → SO(V ) → 1 ,

19 ±Λ1 ±ΛPT ±ΛP ±ΛT 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. §5 Representations of the Clifford algebra C(R1,3). We now move on to the representations of the Clifford algebra. This will lead us to all the fields we need to consider in Lorentz invariant theories. The minimal representations are four dimensional, i.e. Ã ! 0 σµ γµ = (52) σ¯µ 0 or, explicitly: (Ã ! Ã ! Ã ! Ã !) −1 0 0 1 0 −i 1 0 µ 0 0 −1 0 1 0 0 i 0 0 0 −1 γ = −1 0 , 0 −1 , 0 i , −1 0 , (53) 0 −1 0 −1 0 0 −i 0 0 0 1 0 µ µν µ where σ = η σν, see (23), andσ ¯ is defined by ˙ σ¯µ αα˙ := ϵα˙ βϵαβσµ . (54) ββ˙ Note that in (54), both σ andσ ¯ are given with their natural index positions. The form of the gamma matrices given here is the so-called Weyl representation. There is also the Dirac representation, commonly employed in QFT textbooks. We always use the Weyl representation in these lectures. The gamma matrices obey

µ ν µν {γ , γ } = −2η 14 . (55) † † − Their conjugates are γ0 = γ0 and γj = γj. The γ matrices act on a pair Ψ of Weyl spinors χ ∈ W and ψ¯ ∈ W , Ã ! χ Ψ = α (56) ψ¯α˙ † This pair is called a Dirac spinor. We also deﬁne γ5 := iγ0γ1γ2γ3, with γ5 = γ5, which is the matrix Ã ! 12 0 γ5 = , (57) 0 −12

20 in our Weyl representation. The Weyl spinors are eigenvectors of the projectors 1 1 ± P± := 2 ( 4 γ5).

Exercise 6. It is very useful to realise that, writing σµ ≡ (σ0, σ1, σ2, σ3), the barred sigma matrices are simply

σ¯µ = (σ0, −σ1, −σ2, −σ3) . Prove this result for yourself. This is the origin of the (baﬄing) choice of seemingly µ Lorentz–ignorant conventions “σ¯µ = −σ ”, which we (tacitly used earlier and) shall never speak of again.

§6 Parity and time reversal. Recalling §3, the parity and time reversal operators in terms of explicit matrices are given by

ΛP = ±iγ0 , ΛT = γ1γ2γ3 , (58) which can be checked by calculating −1 −1 −1 ν ΛP γ0ΛP = γ0 , and ΛP γjΛP = −γj , =⇒ ΛP γµΛP = γνP µ , (59) −1 −1 −1 ν ΛT γ0ΛT = −γ0 , and ΛT γjΛT = γj , =⇒ ΛT γµΛT = γνT µ . Letting these parity and time reversal operators act on the Dirac spinor, we ﬁnd Ã ! Ã ! ψ¯ −iψ¯ ΛP Ψ = ± i , ΛT Ψ = . χ iχ

Exercise 7. Show that (59) holds for the deﬁnitions of ΛP and ΛT given in (58).

§7 The Dirac equation and the Lagrangian. We now make contact to physics by giving the equation of motion for Dirac spinors. This is, ¡ ¢ µ iγ ∂µ − m Ψ = 0 , (60)

µ µ where γ ∂µ is C(V )-valued. The factor of γ yields the appropriate coupling of vectors (∂µ) to spinors (Ψ). The equations of motion have the following ‘Weyl decomposition’ in terms of Weyl spinors, µ iσ ∂µψ¯ − mχ = 0 , (61) µ iσ¯ ∂µχ − mψ¯ = 0 . A Lagrangian density which yields the Dirac equation is, ¡ ¢ µ L = Ψ¯ iγ ∂µ − m Ψ , (62) † where Ψ¯ ≡ Ψ γ0 . In terms of Weyl spinors it may be written µ µ L = iχ¯ σ¯ ∂µχ + iψ σ ∂µψ¯ − mψχ − mχ¯ψ¯ . (63)

21 §8 Charge conjugation. Along with the discrete symmetries P, T and PT , there is another discrete transformation. Consider the above equations of motion, but including minimal coupling to a hermitian U(1) gauge potential, ¡ ¢ µ iγ (∂µ − ieAµ) − m Ψ = 0 .

Note that if we take the complex conjugate, we obtain ¡ ¢ µ∗ ∗ − iγ (∂µ + ieAµ) − m Ψ = 0 , (64) where the sign of the coupling has changed. We introduce C which acts as follows,

µ ∗ −1 µ C ∗ ΛC(γ ) ΛC = −γ , Ψ ≡ ΛCΨ . (65)

The resulting equation of motion for ΨC is thus ¡ ¢ µ C iγ (∂µ + ieAµ) − m Ψ = 0 .

C C ∗ 1,3 We want (Ψ ) = Ψ, which implies ΛCΛC = 1. On R in the Weyl/chiral representation, we can choose ΛC = ±γ2. Physically, C changes particles into their antiparticles of the same mass but the opposite charge. §9 The CPT theorem. Both P and CP are violated in nature: the ﬁrst was observed in Cobalt 60 decays in 1956, the second in Kaon decays in 1964. Both appear in the weak nuclear force, and it is unknown if CP violation also exists for strong nuclear interactions. This violation is believed to be the reason for the existence of more matter than antimatter. In 1955, Pauli proved that if you have a quantum ﬁeld theory which is

↑ 1. invariant under L+, 2. causal and local,

3. has a Hamiltonian which is bounded below, then the quantum ﬁeld theory is invariant under the combined transformation CPT .

2.3 Summary Since all our ﬁelds will transform under irreps of O(1, 3), we spent some time constructing all possible irreps. Group theory told us that we needed the universal covering group ↑ C which, for L+ for example, is SL(2, ). We discussed the following representations (the examples column contains some objects we will meet later, for reference):

22 Carrier space Name Examples in this rep. ∼ W = C2 Weyl spinors ψα SUSY charges Qα, ϵα parameters. ∨ ∼ 2 W = C Dual Weyl spinors ψα SUSY charges Qα, ϵα parameters. ∼ 2 W = C Conjugate Weyl spinors ψ¯α˙ SUSY charges Q¯α˙ , ϵ¯α˙ parameters. ∨ ∼ W = C2 Conjugate, dual Weyl spinors ψ¯α˙ SUSY charges Q¯α˙ , ϵ¯α˙ parameters.

Spinors are anticommuting objects. Once we allow for the discrete transformations P, T and PT we need the double cover of SO(1, 3), which is Spin(1, 3), and the double cover of the full group O(1, 3) is the pin group. With the Dirac matrices, Ã ! 0 σµ γµ = , (66) σ¯µ 0 the basic representations are given by Dirac spinors Ψ which decompose into two Weyl spinors, Ã ! ψ Ψ = α . (67) χα˙

µ µ µ µ αα˙ Note that the natural index structure on σ andσ ¯ is σαα˙ andσ ¯ , so that the gamma matrices (66) act naturally on the Dirac spinor (67). §1 Hermitian conjugation. Before continuing it is worth cementing our conventions as regards complex, or hermitian, conjugation of Weyl spinors. The conjugate of a product of objects, be they vector or spinor, is

(AB . . . Z)† = Z† ...B†A†. (68)

There are no minus signs, and all indices remain in their allotted positions. It may be useful in calculation to explicitly put dots over spinor indices which become conjugated spinors. A conjugated spinor is ψ† ≡ ψ¯. 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 may feel6 should be more than compensated for by the ease of calculation these conventions provide. For example, we then have

(ψχ)† = χ†ψ† ≡ χ¯ψ¯ = ψ¯χ¯ and (ψσµχ¯)† = χσµψ¯ .

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 Poincaréalgebra we met earlier. A theorem due to Haag et al [17] says that the SUSY algebra we will present below is the only possible extension of the Poincarégroup consistent with the axioms of quantum field theory.

3.1 SUSY algebra on R1,3

§1 SUSY algebra. Along with the generators Pµ and Mµν which obey the algebra (17)–(18), we choose N ∈ N+, and then for i = 1 ... N we introduce the supersymmetry i ¯ charges Qα and Qiα˙ which obey { i j } { ¯ ¯ } Qα,Qβ = Qiα˙ , Qjβ˙ = 0 , (69) { i ¯ } i µ Qα, Qjα˙ = 2 δj σαα˙ Pµ , (70) where Q¯ is the complex conjugate of Q. Note the positions of the i indices labelling the diﬀerent supercharges, which will serve as a bookkeeping device for correct contractions later. Note also that the zero vector–component on the right hand side of (70) looks like the Hamiltonian in quantum mechanics, just as we had in our toy model, earlier. The commutators of the SUSY charges with the generators of the Poincar´egroup are

[P ,Qi ] = [P , Q¯ ] = 0 , (71) £ µ α¤ µ iα˙ i β i Mµν,Qα = i(σµν)α Qβ , (72) £ ¤ ¯α˙ α˙ ¯β˙ Mµν, Qi = i(¯σµν) β˙ Qi , (73) where we have introduced µ ¶ 1 (σµν) β ≡ σµ σ¯ν αβ˙ − σν σ¯µ αβ˙ , α 4 αα˙ αα˙ µ ¶ (74) 1 (¯σµν)α˙ ≡ σ¯µ αα˙ σν − σ¯ν αα˙ σµ . β˙ 4 αβ˙ αβ˙ If N > 1 we call this the N –extended SUSY algebra. The number of real supercharges in the game is 4N (in four dimensions) since each of the Qi is a two component complex Weyl spinor. §2 Theorem of Haag, Sohnius, Lopuszanski. Up to introducing ‘central charges’ Z[i,j] such that { i j } [i,j] Qα,Qβ = ϵαβZ , (75) where the Z are just complex numbers, the N –extended SUSY algebra is the only extension of the Poincar´egroup which is consistent with the axioms of relativistic quantum ﬁeld 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, §6 when discussing the massive representations of the SUSY algebra.

24 3.2 Representations §1 Casimir operators. For the massive representations, P 2 remains a Casimir operator. We also label representations by their superspin, C2, with

2 2 2 − 1 2 C = P W 4 (PW ) . (76) The eigenvalues of the superspin operator are −m4s(s+1). For the massless representa- − 1 αα˙ { i ¯ } tions, we introduce Lµ = Wµ 16 σ¯µ Qα, Qiα˙ , and then label representations by their superhelicity κ, where ¡ ¢ 1 Lµ = κ + 4 Pµ , (77) and κ ∈ Z/2 (compare with the relation Wµ = λPµ defining the helicity for the massless Poincarérepresentations). §2 Massless representations. As before, to construct the massless representations, we go to the frame Pµ = (E, 0, 0,E), and remain there in this paragraph. The key to understanding the massless reps lies in evaluating (70) in our chosen frame. We find Ã ! 0 0 σµ P = . (78) αα˙ µ 0 2E αα˙ i ¯ 7 Comparing with our toy model, we find that Q1 = Qi1˙ = 0 on such states , since for arbitrary ψ, ⟨ |{ ¯ }| ⟩ ∥ | ⟩∥2 ∥ ¯ | ⟩∥2 ψ Q1, Q1˙ ψ = Q1 ψ + Q1˙ ψ = 0 , { i ¯ } i N while Q2, Qj2˙ = 4Eδj, i.e. we have copies of our toy model Clifford algebra. Now, in this frame, the helicity operator is just J3 ≡ M12. Calculating its commutator with the SUSY charges we find 1 1 [J ,Qi ] = − Qi , [J , Q¯ ] = + Q¯ . (79) 3 2 2 2 3 i2˙ 2 i2˙ i ¯ These commutators simply say that Q2 lowers helicity by 1/2 while Qi2˙ raises helicity by 1/2, so that repeated application of Q or Q¯ changes the spin of a state from half–integral to 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 this representation. (This is just what we did in our toy model, starting from the state annihilated by Q.) Our lowest weight state will be | h ⟩, which is annihilated by all the i ¯ Q2. All we can do to construct other states is act with the Qi2˙ , but we can act with each at 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 N ¯ there are a total of 2 states (each of the Qi2˙ may be present once or not at all) of the form Q¯ ... Q¯ | h ⟩ . (80) i12˙ in2˙ The range of helicities of these states is h up to h + N /2, possessed by the state ¯ ¯ ··· ¯ | ⟩ Q12˙ Q22˙ QN 2˙ h . 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 Q¯i corresponds to the lattice distance to the corner corresponding to the state | h ⟩. We now give examples of the massless representations for various N . Representations are labelled by their mass and superspin or superhelicity. The states which live inside any such representation are collectively called a (super)multiplet. As described above, these states are related by the action of the SUSY charges, and are of equal mass. A multiplet may be seen as the set of all states which are needed to form a closed representation of the (SUSY) algebra. §3 Example 1: N = 1 multiplets. As in our toy model, we have a 21 = 2 state | ⟩ | 1 ⟩ ¯ | ⟩ system, comprising the lowest weight state h and h + 2 = Q2˙ h . For physical reasons we wish to avoid particles with helicities larger than 2 (the theories of which are thought not to be well deﬁned) and with helicities 2 and 3/2, which correspond to the graviton and gravitino (as we want to steer clear of gravity). Hence we have two possibilities,

State helicities Name

1 (1, 2 ) The vector multiplet (since helicity h = 1 corresponds to the photon, which is a vector).

1 ( 2 , 0) The chiral multiplet.

The 1-dimensional cube is just a line segment with the end points corresponding to the two diﬀerent states. Taking the CPT conjugates of these multiplets gives us multiplets − 1 − 1 − of helicity (0, 2 ) and ( 2 , 1). §4 Example 2: N = 2 multiplets. Starting from our lowest weight state | h ⟩, we can ¯ ¯ 1 now act with either Q12˙ or Q22˙ to generate two states of helicity h + 2 . We can also act with both operators to generate a state of helicity h + 1. As before, we wish to steer clear of spin larger than 1, which gives us two possible multiplets,

State helicities Name

1 1 N (1, 2 , 2 , 0) The = 2 vector multiplet.

1 − 1 ( 2 , 0, 0, 2 ) 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 − 1 − 1 − (0, 2 , 2 , 1), while the hypermultiplet is its own charge conjugate. §5 Example 3: N = 3 and N = 4 multiplets. The case N = 3 is usually not discussed seperately from the case N = 4 for the following reason: in principle, there × × 1 × × − 1 × 1 × × − 1 × − are two multiples: ( 1, 2 , 0, 2 ) and ( 2 , 0, 2 , 1), where the plain number in each product is the helicity while the circled number is the number of states with that helicity. In the cubic picture, the eight states correspond to the corners of the cube with lattice distances (0, 1, 2, 3). These two multiplets are each other’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 µ ¶ 1 1 × 1, × , × 0, × − , × −1 , (81) 2 2 We have a total of 1 + 4 + 6 + 4 + 1 = 16 states, which correspond to the 16 corners of a 4-dimensional cube with lattice distance (0, 1, 2, 3, 4). The number of states of course matches with our general form of 2N states, for N = 4. Note that this supermultiplet is its own charge conjugate. It gives the entire ﬁeld content of N = 4 super Yang–Mills theory, which we will meet again later in these lectures. §6 Massive representations. In these lectures we will mostly deal with massless representations (taking mass generation to be due to Higgsing), but for completeness we include this discussion of the massive representations. We consider the most general case by leaving in the central charges (75),

{ i j } [i,j] { ¯ ¯ } ¯ Qα,Qβ = ϵαβZ , Qiα˙ , Qjβ˙ = ϵα˙ β˙ Z[i,j] . (82)

We go to the rest frame of the particle, in which Pµ = (m, 0, 0, 0). In order to understand the representations we will write our commutators, in this frame, in such a way that the technology 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,   0 z1  0 0   −z1 0 2 2    [i,j]  2  Z =  0 z  . (83)  02 2 02   −z 0  .. 02 02 .

Next, introduce the following linear combinations of the supercharges8, ¡ ¢ ¡ ¢ i 1 2i−1 2i † i 1 2i−1 2i † a := √ Q + ϵαβ Q , b := √ Q − ϵαβ Q , (84) α 2 α β α 2 α β

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 are { i j †} − i aα, a = (2m z )δijδαβ , no sum, β (85) { i j †} i bα, bβ = (2m + z )δijδαβ , no sum. Now recall, from our toy model example, that positivity of the Hilbert space requires 2m ≥ |zi| (and so in particular central charges must vanish in the massless case). Assume that k of the zi saturate this bound, i.e. |zi| = 2m, i = 1 . . . k, for k ≤ N /2. Then one of the pair {ai, ai †} or {bi, bi †} vanishes, which implies either ai or bi must be put to zero. We therefore have a total of 2N −2k non–zero fermionic oscillators, and so 22N −2k states. To illustrate, if k = 0 and N = 1, we can construct the multiplets µ ¶ µ ¶ 1 1 1 1 , 0, 0, − , 1, , , 0 . 2 2 2 2 It follows that representations are constructed just as in the massless case, for k > 0, and those multiplets are called short or BPS. If k takes its maximal value of N /2 then we have an ultrashort multiplet. Note that only ultrashort multiplets can become massless, as it is only then that 22N −2k = 2N and so the number of states matches that in a massless multiplet.

3.3 The Wess–Zumino model We are now ready to look at our ﬁrst SUSY ﬁeld theory. We begin with a condensing of our notation by combining commutators and anti–commutators into a single bracket. §1 Grading and supercommutators. Introduce a parity to every object. This is a

Z2 grading such that everything ‘bosonic’ (i.e. with integral spin) carries a label 0, while everything ‘fermionic’ (i.e. with half–integral spin) carries the label 1. We denote the parity of an operator by writing a tilde over it, i.e.

e ej Pµ = 0 , Qα = 1 . Noting that our SUSY algebra comprised anticommutators between the odd objects and commutators otherwise, we introduce the supercommutator

˜ ˜ {[A, B]} := AB − (−)ABBA. (86)

This recovers all our previous commutators (e.g. {[Pµ,Pν]} = [Pµ,Pν]) and anticommutators (e.g. {[Q, Q¯]} = {Q, Q¯}). §2 Representation of the SUSY algebra on component fields. We are looking for some set of fields φ, ψ. . . 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 fields. µ On any field, we have Pµ = −i∂µ. Given an infinitesimal v , we can define the infinitesimal transformation of, say, a scalar φ as

µ µ δvφ = v Pµφ = −iv ∂µφ . (87)

28 Inﬁnitesimal supersymmetry transformations are accordingly generated by the combination ¡ ¢ δϵφ = ϵQ +ϵ ¯Q¯ φ , (88) and similarly on other ﬁelds. Here, we have introduced anticommuting parameters ϵα such that {[ϵα, ·]} = 0. It follows that (ϵα)2 = 0, i.e. that the ϵα are just Graßmann numbers. These parameters have parity +1, and are combined with the supercharges to form the bosonic objects in (88),

α α˙ ϵQ ≡ ϵ Qα , ϵ¯Q¯ ≡ ϵ¯α˙ Q¯ . (89)

Such contracted combinations of Weyl spinors and their conjugates will appear often from here on. For objects χ, ψ andχ, ¯ ψ¯ in W and W¯ respectively, contractions are always taken as follows,

α χ ψ := χ ψα , α˙ χ¯ ψ¯ :=χ ¯α˙ ψ¯ , (90) µ ¯ α µ ¯α˙ χ σ ψ := χ σαα˙ ψ . Introducing a second Graßmann number ξ, we can write our basic commutators as

[ϵQ, ξQ] = [¯ϵQ,¯ ξ¯Q¯] = 0 , (91) ¯¯ α µ ¯α˙ µ ¯ [ϵQ, ξQ] = 2 ϵ σαα˙ ξ Pµ =: 2 ϵσ ξPµ .

The action of Q and Q¯ remains somewhat abstract – in order to make things more explicit, let us act with two supersymmetry transformations, and take their commutator, i.e. ¡ ¢ ¯ δϵδξ − δξδϵ φ = [ϵQ, ξQ¯]φ − [ξQ, ϵ¯Q¯]φ ¡ ¢ (92) µ µ = −2i ϵσ ξ¯− ξσ ϵ¯ ∂µφ .

Now, for simplicity we deﬁne a spinor ﬁeld ψ through the transform of the scalar, √ √ δϵφ = 2 ϵψ , δϵφ¯ = 2ϵ ¯ψ¯ . (93)

From (92) it follows that, under a SUSY transform, the field ψ must transform into something proportional to ∂µφ, plus contributions from other fields which cancel from the commutators in (92). We find √ √ µ α˙ δϵψα = i 2σ ϵ¯ ∂µφ + 2ϵαF, √ αα˙ √ (94) ¯ − α µ ¯ δϵψα˙ = i 2ϵ σαα˙ ∂µφ¯ + 2¯ϵα˙ F. where the first term is fixed by the algebra, while we are free to include the second term, which introduces a third field F .

29 Exercise 9. The following identities are useful when dealing with component ﬁelds. Prove them. 1. ψχ = χψ , α α 2. ψ χα = −ψαχ , 3. χσµψ¯ = −ψ¯σ¯µχ , 4. (χσµψ¯)† = ψσµχ¯ .

A straightforward calculation now shows that (92) is satisﬁed. A more involved calculation shows that the commutator of two SUSY transformations acting on ψ requires √ √ µ αα˙ µ αα˙ δϵF = i 2ϵ ¯α˙ σ¯ ∂µψα , δϵF¯ = −i 2 ∂µψ¯α˙ σ¯ ϵα . (95)

It can be checked that F transforms correctly under two SUSY transformations, giving a rep. of (70) and there is then no need to introduce any further ﬁelds – the SUSY algebra closes on the multiplet (φ, ψ, F ), which is the entire ﬁeld content of the Wess–Zumino model.

Exercise 10. Show that (94) implies that ψ¯α˙ transforms as √ √ α˙ µ αα˙ α˙ δϵψ¯ = i 2¯σ ϵα∂µφ¯ + 2¯ϵ F,¯ and that the transformation of F¯ in (95) can be written √ ¯ α µ ¯α˙ δϵF = +i 2ϵ σαα˙ ∂µψ .

§3 Dimensions of fields. Before writing down an action for our fields, we briefly consider their mass dimension. In four dimensions, [φ] = [P ] = 1. It follows from the algebra {Q, Q¯} ∼ P that [Q] = [Q¯] = 1/2. We then find that [ψ] = 3/2 and [F ] = 2. Therefore, terms quadratic in F cannot contain derivatives, as F 2 already has mass dimension 4, and so F cannot have a kinetic term. F is therefore an auxiliary field, with a purely algebraic equation of motion, and can be integrated out. This means eliminating the field by using its equation of motion, or, alternatively, performing the Gaussian path integral over this field. The actual multiplet 1 N therefore contains only the fields (φ, ψ) with helicities (0, 2 ) – this is the = 1 chiral multiplet introduced earlier. §4 The Lagrangian. From the dimensional analysis above, the only Lorentz invariant terms we could include in a free action are

2 µ L0 =φ ¯ ∂ φ − iψ¯σ¯ ∂µψ + FF.¯ (96)

In principle, one can include further interaction terms. We will postpone the discussion of these terms for simplicity to Sect. 5.1. A small calculation reveals that

δϵL0 = 0 , (97)

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

R 4 Exercise 11. Show that the action d x L0 is invariant under SUSY transformation (up to boundary terms). The following identities are useful: ¡ ¢ σµσ¯ν + σνσ¯µ β = −2ηµνδ β , ¡ ¢α α σ¯µσν +σ ¯νσµ α˙ = −2ηµνδα˙ . β˙ β˙

Going through the above exercise (and remembering that this is the simplest SUSY ﬁeld theory we could consider), it becomes apparent that many identities need to be employed 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 without the kind of labour implied by the above. We turn to this below.

31 4 Superspace and superﬁelds

Introducing the idea of superspace will provide us with a natural and easy-to-use representation of the SUSY algebra. Note that the preﬁx ‘super’ always refers to the presence of the poorly–named ‘parity’ of all objects (the Z2–grading) introduced earlier. We begin with a reminder of Graßman numbers and their properties.

4.1 Reminder: Graßmann numbers. §1 Deﬁnition. Graßmann numbers are, say, n formal parameters generating the Graß- mann algebra Λn, i j i j θ , θ ∈ Λn : {θ , θ } = 0 , i 2 so that (θ ) = 0. It follows that any Z ∈ Λn must have the form

i i j i1 ··· in Z = z0 + ziθ + zijθ θ + . . . zi1...in θ θ , (for some complex–valued coeﬃcients z) since each generator can appear at most once. n The number of elements in the basis of Λn is clearly 2 . Elements of Λ are called supernumbers, which are just functions of n Graßmann numbers. The fact that (θi)2 = 0 implies that there is no inverse for θi. Their parity is θ˜i = 1. (Parities are combined as a˜˜b =a ˜ + ˜b mod 2.) The algebra can be split according to the parity of its elements,

Λ = Λ0 ⊕ Λ1 , where the first (second) component contains elements with only even (odd) powers of i θ ’s. Another decomposition is Λ = ΛB ⊕ ΛS, body and soul components, where the ∼ former contains only the c–number z0, so ΛB = C. §2 Differentiation and integration. A derivative should be a linear map annihilating constants and satisfying the Leibnitz rule. Consider Z = a + θb, where θ ≡ θj for some ∈ R C | j and a, b , , or Λ θj =0. We define the derivative with respect to θ by ∂Z = b . ∂θ

The super–Leibnitz rule is, for Z1 and Z2 two elements of Λ, ∂ ∂Z ∂Z Z Z = 1 Z + (−)Z1 Z 2 . ∂θ 1 2 ∂θ 2 f 1 ∂θ The integral should be a superlinear functional obeying Z Z ∂ ∂ 1. dθf = 0 , 2. dθ f = 0 . ∂θ ∂θ The ﬁrst condition states that the integral should be independent of the variable which has been integrated over, the second condition is the foundation of Stokes theorem and integration by parts. We are therefore led to demand Z Z dθ = 0 , dθ θ = 1 ,

32 which ﬁxes integration completely: It follows that, for Z = a + bθ with a and b independent of θ, Z ∂Z = b = dθ Z . ∂θ

4.2 Flat superspace §1 Split supermanifolds. Consider a manifold M together with a vector bundle E of rank k. Locally the total space is described by co–ordinates x1 . . . xn describing points on the base M, and co–ordinates v1 . . . vk on E, as shown in Fig. 7.

∼ E µ = Ck x1 Ck : v1 ...vk E µ x0

M R1,d−1 : xµ

Figure 7: The base manifold M, which is R1,3 for us, with a vector bundle of rank k ∼ over 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 Π to the co–ordinates on the fibers: so ΠE is locally described by co–ordinates x1 . . . xn, θ1 . . . θk, where the θi are Graßmann variables. The space ΠE is called a split supermanifold – these are the supermanifolds relevant to physics. The picture of our space is now as in Fig. 8: the fibers become infinitesimal directions off the base manifold M. This global split in the real directions (on M) and Graßmann directions is the origin of the name ‘split’ supermanifold. §2 Flat superspace. The space we will mostly be dealing with is R1,3|4 – this is ΠE ∼ with E the trivial bundle W × R1,3. Recalling that W = C2, the bundle W × R1,3 just describes flat Minkowski space, with, at every point, two new complex (or four real) directions in C2. Applying the parity changing operator, these new directions become infinitesimal, as they are parameterised by two complex Graßmann numbers. So, the

Figure 8: The split supermanifold ΠE, with the base manifold M (sometimes referred to as the body) and inﬁnitesimal directions θj oﬀ the base (sometimes referred to as the soul).

33 co–ordinates on our superspace are

(xµ, θi) ≡ (x0, x1, x2, x3, θ1, θ2) . (98)

Note that it is very common to see θ¯1, θ¯2 included in the list of co–ordinates – this is just notation emphasising that we have two complex directions. We briefly comment that for N –extended theories (N > 1) we would work on the superspace R1,3|4N = ΠEN , where EN = W N × R1,3. However, this is less useful because one can often only work with equations of motion, not with Lagrangians. We will return to these points later. §3 A caveat. In the following section we generate transformations on superspace by exponentiating the infinitesimal generators Pµ, ϵQ andϵ ¯Q¯. We will use this to construct a representation of the SUSY algebra which acts on ‘superfields’, which are just functions on superspace.

There is an important caveat. Up to now, we have written Pµ = −i∂µ when acting on fields. It is a (somewhat tiresome) quirk that the path we will re–tread below, and which much of literature also follows, leads to a representation in which Pµ = +i∂µ. We will explain why in due course, but the reader should be prepared for this change in notation. §4 Group of translations on flat superspace. We now exponentiate our generators (with parameters), defining the group element G(x, θ, θ¯) by £ ¤ µ G(x, θ, θ¯) = exp −ix Pµ − iθQ − iθ¯Q¯ . (99)

Using the BCH formula, which just says that composition of group elements gives other group elements, ¡ ¢ 1 exp A exp B = exp A + B + 2 [A, B] + ... we can show that two of these group elements compose as

G(0, ξ, ξ¯)G(xµ, θ, θ¯) = G(xµ + iθσµξ¯− iξσµθ,¯ θ + ξ, θ¯ + ξ¯) , (100) where the terms denoted by ellipses in the BCH formula vanish because of the nilpotency of θi.

Exercise 12. Derive (100).

From (100) and the explicit expression of G(0, ξ, ξ¯), we can directly read oﬀ the realization of Q and Q¯ in terms of diﬀerential operators (and this is the form in which they will act on functions of superspace co–ordinates):

∂ − µ ¯α˙ ∂ Qα = α iσαα˙ θ µ , ∂θ ∂x (101) ∂ ∂ Q¯ = − + iθασµ . α˙ ∂θ¯α˙ αα˙ ∂xµ

34 It follows that {Q, Q} = {Q,¯ Q¯} = 0 and { ¯ } µ µ Qα, Qα˙ = 2iσαα˙ ∂µ =: 2iσαα˙ Pµ . (!!) Here we see the result of considering ‘left actions’ of the group in (100); we must now identify Pµ = +i∂µ. Considering ‘right actions’ would have maintained our conventions, but sadly this is not the way things unfolded historically. We would in that case have been led to the representations Dα and D¯α˙ for the SUSY generators, deﬁned by ∂ µ ¯α˙ ∂ Dα = α + iσαα˙ θ µ , ∂θ ∂x (102) ∂ ∂ D¯ = − − iθασµ . α˙ ∂θ¯α˙ αα˙ ∂xµ The only diﬀerence between this and (101) is a change in sign in the second terms. These operators obey { ¯ } − µ Dα, Dα˙ = 2iσαα˙ ∂µ , (103) as we might have expected. They will still play an important role in what follows. We note that the Q and D operators anticommute completely – they do not talk to each other at all, {Q, D} = {Q,¯ D¯} = {Q, D¯} = {Q,¯ D} = 0 . (104)

4.3 Superfields We are now ready to consider the superfields – functions on superspace. §1 General superfields. Due to the nilpotency of the Graßmann numbers, the θ de- pendence of an arbitrary function, or superfield, on R1,3|4 can be written down explicitly. Writing θ2 ≡ θθ and θ4 ≡ θ2θ¯2, a general superfield has the form F (x, θ, θ¯) = f(x) + θϕ(x) + θ¯χ¯(x) + θ2 m(x) + θ¯2 n(x) (105) µ 2 2 4 + θσ θA¯ µ(x) + θ¯ θλ(x) + θ θψ¯ (x) + θ d(x) . Note that the product of two superfields is again a superfield. Superfields clearly form a representation of the supersymmetry algebra, where the action of the SUSY generators on superfields is given by ¯ α ¯α˙ δξF = (ξQ + ξQ¯)F ≡ (ξ Qα − ξ Q¯α˙ )F, with explicit formula resulting from inserting the expressions (101). Note that we have used one of the identities from Exercise 9 to write the SUSY generators in terms of Qα and Q¯α˙ so that (101) is easily applied. Performing the derivatives, and matching powers of θ on both sides, allows us to extract the transformation laws of the component fields. Our Q and Q¯ give us a linear rep of the SUSY algebra. For example, ¯ ¯ ¯ ¯ δξf(x) ≡ δξF ¯ = ξϕ(x) + ξχ¯(x) . θ=θ¯=0 Exercise 13. Confirm the above transformation of the component field f(x).

35 §2 Reducibility. The general superfield above has sixteen (complex) degrees of freedom. Recall, however, that the fields (φ, ψ, F ) of our Wess-Zumino model had only four (complex) degrees of freedom. In fact, the representation (105) of the SUSY algebra is highly reducible – there are a number of conditions we can impose on F which reduce the number of components. To ensure that the reduced components still form a representation of the SUSY algebra, the imposed conditions have to be invariant under supersymmetry transformations. One example is D¯α˙ F = 0, which is compatible with our SUSY representation since {Q, D¯} = {Q,¯ D¯} = 0, giving the so–called chiral superfield. Another condition is F = F †, giving the vector superfields. We will consider these two cases in more detail below. §3 Chiral and antichiral superfields. Since D and D¯ anticommute with the SUSY generators Q and Q¯, the constrained superfields Φ and Φ¯ obeying the chiral and antichiral conditions D¯Φ = 0 and DΦ¯ = 0 , (106) respectively, still form representations of the SUSY algebra: the conditions (106) are preserved under SUSY transformations.

From the definition of D¯, a function Φ obeying D¯α˙ Φ = 0 must depend on the combination yµ := xµ + iθσµθ¯ . (107) Note that as D¯ contains no factors of θ, our function Φ can also depend on θ, so we find Φ ≡ Φ(y, θ). yµ is called a chiral co–ordinate of chiral superspace. We can use it to write down the component field expansion of a chiral sfield9 Φ, √ α 2 Φ(y, θ) = φ(y) + 2θ ψα(y) + θ F (y) .

We have two complex scalars and one Weyl spinor – four complex degrees of freedom in total. This is precisely the ﬁeld content of the Wess–Zumino model.

Exercise 14. In order to obtain something we can calculate with, we have to Taylor expand our functions, in y, around x. Show that the chiral sfield then becomes: √ µ 1 2 2 2 i 2 µ 2 Φ(y, θ) = φ(x) + iθσ θ∂¯ µφ(x) + θ θ¯ ∂ φ(x) + 2θψ(x) − √ θ ∂µψ(x)σ θ¯+ θ F (x) . 4 2 You might find it useful to first prove that

1 ˙ 1 ˙ 1 θαθβ = − θ2ϵαβ, θ¯α˙ θ¯β = + θ¯2ϵα˙ β and θσµθ¯ θσνθ¯ = − θ2θ¯2ηµν 2 2 2

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

36 We can also compute the SUSY transformation rules of the component ﬁelds using chiral co–ordinates, i.e. we compute

α ¯ α˙ δξΦ = (ξ Qα + ξα˙ Q¯ )Φ , (108) after ﬁrst changing variables (x, θ, θ¯) to (y, θ, θ¯). The chain rule gives us

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ → , → + iσµ θ¯α˙ , → − iθασµ , ∂xµ ∂yµ ∂θα ∂θα αα˙ ∂yµ ∂θ¯α˙ ∂θ¯α˙ αα˙ ∂yµ from which it follows that Qα and Q¯α˙ in our ‘new’ co–ordinates are ∂ ∂ ∂ Q = , Q¯ = − + 2iθασµ . (109) α ∂θα α˙ ∂θ¯α˙ αα˙ ∂yµ

We now compute (108), finding √ √ µ ¯ 2 µ ¯α˙ α δξΦ = 2ξψ + 2θξF + 2iθσ ξ∂µφ + 2iθ σαα˙ ξ ∂µψ , from which we read off that √ δ φ = 2ξψ , ξ √ √ δ ψ = 2iσµ ξ¯α˙ ∂ φ + 2ξ F, ξ α √ αα˙ µ α µ ¯ δξF = −i 2(∂µψ)σ ξ , which are precisely the transformation laws of the Wess–Zumino model fields we found previously in (93), (94) and (95). Analogously, one can find the SUSY transformation laws for the components of antichiral superfields by complex conjugating the chiral coordinates to obtain antichiral coordinates on antichiral superspace. §4 Vector and complex linear superfields. Another constraint we can demand is that a superfield V should be real, i.e.

V¯ (x, θ, θ¯) = V (x, θ, θ¯) .

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

37 5 SUSY–invariant actions from superﬁelds

5.1 Actions from chiral sfields §1 Invariant actions. Our aim is now to construct actions for our sfields. Note that the SUSY transform (95) of the component field F is a total derivative. Hence, the following action will be SUSY invariant: Z Z S = d4x d2θ Any chiral sfield. (110)

The justification of this claim is that the integral over d2θ picks out only the F – component of the chiral sfield. The resulting integrand transforms into a total derivative under SUSY, and so the action will be SUSY invariant (neglecting boundary contributions in the usual way). It can also be shown that the combination Z Z d4x d2θd2θ¯ a general sfield (111) is also invariant under SUSY as the θ2θ¯2 term of a general superfield transforms as a total derivative. Such actions may appear rather uninteresting if we consider only a single sfield placed under the integral, but recall that products of sfields are again sfields and products of arbitrary chiral (antichiral) sfields are chiral (antichiral) sfields. This is because D¯ obeys the Leibnitz rule, so that D¯(ΦΦ′) = 0. In the following, we will build actions both from terms of form (111) and (110). §2 Free action. The free action should be quadratic in component fields and we are thus led to try a product of, say, a chiral field with its conjugate (an antichiral field), and projecting onto its θ2θ¯2-component as follows, Z Z Z 4 2 2 S0 = d x d θ d θ¯ ΦΦ¯ Z (112) 4 2 µ = d x FF¯ +φ∂ ¯ φ − iψ¯σ¯ ∂µψ , which is the integral of the Wess–Zumino Lagrangian we found in (96). Note that for evaluating the integral over Graßmann variables, one has to use the expansion of the chiral and antichiral superfields in x-space. We now have a method of constructing free actions for component fields. §3 The superpotential. We saw above that integrating ΦΦ¯ over d4θ gives rise to the free Wess–Zumino Lagrangian on Minkowski space R1,3. Alternatively, we could have written an action of the form (110) by picking out the ‘F ’–term of a polynomial W (Φ) of a chiral sfield Φ, Z Z 2 2 Lint = d θ W (Φ) + d θ¯ W¯ (Φ) .

38 The polynomial W is called the superpotential. The second term, above, is just the complex conjugate of the ﬁrst, included so that (what will be) our Lagrangian is manifestly real.

Exercise 15. Working in chiral co–ordinates, show that the θ2 component of ΦN (y, θ), for N ≥ 1, is given by

N N−1 1 N−2 α Φ | 2 = Nφ F − N(N − 1)φ ψ ψ . θ 2 α

It follows from the above exercise that for a polynomial superpotential W (Φ) = 2 a1Φ + a2Φ + ..., we can write ¯ ¯ X ¯ N−1 1 N−2 α W (Φ)¯ = aN Nφ F − aN N(N − 1)φ ψ ψα 2 2 θ µN=1 ¶ µ ¶ X 1 X = a NφN−1 F − a N(N − 1)φN−2 ψαψ (113) N 2 N α N=1 N=1 ∂W (φ) 1 ∂2W (φ) = F − ψαψ . ∂φ 2 ∂φ2 α The ﬁnal line gives us a compact way of writing the result. Note that although we worked in chiral co–ordinates, our action is an integral over d4y = d4x so that for the action we can simply take the above results and write the arguments of the component ﬁelds as x. Our action is Z Z Z Z S = d4x d4θ ΦΦ¯ + d2θ W (Φ) + d2θ¯ W¯ (Φ)¯ Z 4 2 µ = d x FF¯ +φ∂ ¯ φ − iψ¯σ¯ ∂µψ (114) ∂W (φ) ∂W¯ (φ ¯) 1 ∂2W (φ) 1 ∂2W¯ (φ ¯) + F + F¯ − ψψ − ψ¯ψ¯ . ∂φ ∂¯φ¯ 2 ∂φ2 2 ∂φ¯2

§4 Equations of motion for F . The action (112), generated by the combination ΦΦ,¯ is quadratic in the component ﬁelds, in particular in F . The superpotential of the previous paragraph contributed terms linear in F to the action. Combining our two actions as in (114), the resulting equations of motion for F are

∂W (φ) ∂W¯ (φ ¯) F¯ + = 0 ,F + = 0 , (115) ∂φ ∂φ¯ which are algebraic. Integrating out F , we generate a term

∂W (φ) W¯ (φ ¯) − ⊂ L , (116) ∂φ ∂φ¯ int

39 which is a nontrivial self–interaction of the scalar field φ. In four dimensions, the max- imum interaction power in a renormalisable scalar field theory is 4, which implies that the most general renormalisable Wess–Zumino model interaction is generated by the superpotential m λ W (Φ) = a Φ + Φ2 + Φ3 . (117) 1 2! 3! We have chosen a particular parameterisation of coefficients which will be used in the following section. Note that a1 can always be eliminated by a shift of the field Φ. §5 Mass terms. The second term of (117) generates a mass term for both the scalar fields φ and the spinors ψ. As should be expected, these component fields have the same mass. Explicitly, m ∂W (φ) W¯ (φ ¯) W (φ) = φ2 =⇒ − = −m2φφ¯ , 2 ∂φ ∂φ¯ (118) 1 ∂2W (φ) m =⇒ − ψαψ = − ψαψ . 2 ∂φ2 α 2 α Note that we are using Weyl spinors, not Dirac spinors. Hence, the kinetic term is not −mΨΨ,¯ as it is for four–spinors Ψ, but rather −mψψ/2 − mψ¯ψ/¯ 2. The factor of 1/2 is important for the masses of the spinor and the scalar to be equal. §6 SUSY transformations and F . Before integrating out the auxiliary field F , our SUSY transforms were linear in the component fields. After integrating out F , we have the replacement F → −∂W¯ (φ ¯)/∂φ in the SUSY transformations. As a result, the transformations become non–linear, and close only up to terms proportional to the equations of motion, i.e. the SUSY algebra closes only on–shell. In general, SUSY transformations can close up to isometries, gauge transformations and equations of motion. Often, one can even deduce the equations of motion from the SUSY transformations. §7 Generalisations. Consider now n chiral sfields Φj, j = 1 . . . n. We can write down the following invariant action, Z Z Z Z 4 4 j 2 1 n 2 d x d θ Φ¯ jΦ + d θ W (Φ ,..., Φ ) + d θ¯ W¯ (Φ¯ 1,..., Φ¯ n) .

The first term gives us the correct kinematic terms for the component fields. It is possible j k to insert an arbitrary hermitian matrix here, sending the first term to Φ¯ jM kΦ , but we can always perform a field transformation which removes such a matrix, and hence we omit it. Giving up renormalisability of the action allows us to construct supersymmetric nonlinear sigma–models. Recall that a sigma–model action is Z 1 S = d4x g (ϕ)∂ ϕi∂µϕj , (119) 4 ij µ where ϕ is a map from the worldsheet, in this case R1,3 to target space, a manifold Md with 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 theory nonlinear sigma–model in a ﬂat background. If we ‘superize’ the sigma–model, we arrive at an action where M is a complex manifold, Z Z S = d4x d4θ K(Φj, Φ¯ k) .

Expanding this action in components, one arrives at a function called the K¨ahlerpotential

K, giving rise to the Kählermetric gij, ∂2K(ϕk, ϕ¯l) = g (ϕ) . ∂ϕi∂ϕj ij The existence of the Kählerpotential turns M in a KählerManifold, which is just a complex manifold on which the metric gij comes from a potential. The action starts with the bosonic part (119), where gij is the Kählermetric. §8 Complex linear superfields. Now that we know how to construct actions from sfields, we can give an interesting example of duality in SUSY theories. We take a vector sfield V , and use it to define a chiral sfield Φ via

Φ ∼ D¯ 2V. (120)

(We could have started with a general superfield, but this setup will reappear later in gauge theory, so we might as well use it here.) This field is clearly chiral since D¯Φ ∼ D¯ 3V ≡ 0, from the anticommutation relation10 of D¯. There is a redundancy in our definition of Φ; we could shift the vector sfield by any Σ such that D¯ 2Σ = 0, and obtain the same chiral sfield Φ. The sfield Σ is called a complex linear sfield. Recall that, earlier, we found there were essentially only two massless representations of the SUSY algebra, given by the two multiplets of helicities 1 1 (0, 2 ) and ( 2 , 1). The former is the content of the chiral sfield, while the latter will turn out to be the content of the vector sfield. Our new sfield Σ is something else, and doesn’t seem to fit here. The reason is that Σ is dual to a chiral sfield, in the following sense.

Starting with arbitrary superfields Σ0 and Y , we can construct an invariant action from Z Z 4 4 2 2 d x d θ Σ¯ 0Σ0 + Y D¯ Σ0 + YD¯ Σ¯ 0 . R R Integrating out Y gives us the action d4x d4θ ΣΣ¯ in terms of a complex linear sfield 2 Σ. Alternatively, integrating over Σ0, and defining Ψ = iD¯ Y , we arrive at the action Z Z d4x d4θ ΨΨ¯ for the chiral sfield Ψ. Hence, Σ is dual to a chiral field.

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 ≡ D¯ 3 ≡ 0.

41 5.2 Actions with vector superfields Above, we formed actions containing kinetic (free) and potential (interacting) terms for chiral sfields, i.e. we formed actions for the particles contained in the chiral multiplet {| ⟩ | 1 ⟩} of states 0 , 2 . In fact, we gave the most general renormalisable action for these fields, and so we have done all that we can with them. We turn now to the inclusion of {| ⟩ | 1 ⟩} the vector sfield, i.e. the vector multiplet of states 1 , 2 and use them to construct SUSY gauge theories.

§1 Gauge theory reminder. We have a vector potential Aµ taking values in the Lie algebra g of a gauge group G, ≡ a a Aµ Aµτ , (121) a 11 a† a a where the τ are hermitian generators of the Lie algebra g, τ = +τ , while the Aµ † are real ﬁelds, so that Aµ = Aµ. The vector potential is part of the covariant derivative, which we write as ∇ since we have used up all the D’s already,

∇µ := ∂µ + igAµ . (122)

This gives rise to the field strength Fµν, 1 F = [∇ , ∇ ] = ∂ A − ∂ A + ig[A ,A ] . (123) µν ig µ ν µ ν ν µ µ ν Gauge transformations are given by functions U taking values in G. The gauge potential → ′ 1 −1∇ transforms according to Aµ Aµ = ig U µU while the corresponding field strength → ′ −1 transforms as Fµν Fµν = U FµνU. On infinitesimal level, gauge transformations are given by a function α taking values in g which leads to a change δAµ = ∇µα. §2 Abelian gauge invariance. The expansion of a real vector sfield, which lives on full superspace, is 1 1 V (x, θ,θ¯) = C(x) + iθχ(x) − iθ¯χ¯(x) + θ2M(x) + θ¯2M¯ (x) 2 2 ¡ i ¢ ¡ i ¢ − θσµθA¯ (x) + iθ2θ¯ λ¯ + σ¯µ∂ χ(x) − iθ¯2θ λ + σµ∂ χ¯(x) (124) µ 2 µ 2 µ 1 ¡ 1 ¢ + θ2θ¯2 D(x) + ¤C(x) . 2 2

With C,Aµ and D real, this sﬁeld manifestly obeys V¯ = V . What is not so clear is the reason for the mixing of the component ﬁelds 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 sﬁeld 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 to make. We will see shortly that it makes a subsequent calculation very simple. As with the chiral sﬁeld, there appears to be some redundancy in the expansion

(124). We expect the gauge field Aµ to correspond to the spin one state | 1 ⟩, and λ to | 1 ⟩ correspond to 2 , and we might expect an auxiliary field, from our previous discussions. In addition, though, we also have here the fields C, M and χ. In fact, these fields can be eliminated by introducing an abelian gauge symmetry, under which

V → V + Λ + Λ¯ , (125) where Λ is a chiral sﬁeld, D¯α˙ Λ = 0. But what does this have to do with gauge transformations in the usual U(1) sense? Using the expansion given in Exercise 14, we write √ √ 2 2 µ Λ + Λ¯ = φ +φ ¯ + 2θψ + 2θ¯ψ¯ + θ F + θ¯ F¯ + iθσ θ∂¯ µ(φ − φ¯)

i 2 µ i 2 µ 1 2 2 (126) + √ θ θ¯σ¯ ∂µψ + √ θ¯ θσ ∂µψ¯ + θ θ¯ ¤(φ +φ ¯) , 2 2 4 and comparing components, we see that under the transformation (125)

Aµ → Aµ − i∂µ(φ − φ¯) = Aµ + ∂µα , (127) which is just the usual transformation law of an abelian vector potential, upon writing α = 2ℑ(φ). Upon further inspection, we see that under gauge transformations, C → C + φ +φ ¯, so that we may choose Λ such that φ +φ ¯ = −C, and so C is gauged to zero. Similarly, we can choose F and ψ to gauge away M and χ from the vector sﬁeld.

Exercise 16. Write down the gauge choice of fields φ, ψ and F which removes the component fields C, χ and M from the vector sfield expansion (124).

We see now the reason for our choice of component fields in (124) – the expansion is such that the additional component fields can be gauged away nicely using (125). (Another reason for defining the gauge transformation in terms of chiral sfields will be met shortly, when we couple to matter.) The resulting expansion of our vector sfield reads as 1 V (x, θ, θ¯) = −θσµθA¯ (x) + iθ2θ¯λ¯(x) − iθ¯2θλ(x) + θ2θ¯2D(x) , (128) µ 2 which contains a vector Aµ, a Weyl spinor λ and an auxiliary field D. If we calculate the square of our vector sfield (128), we find 1 V 2 = − θ2θ¯2A Aµ =⇒ V 3 = 0 , (129) 2 µ which makes this a nice gauge to work in. We have gone to what is called Wess– Zumino (WZ) gauge, removing the redundant fields from our vector sfield expansion.

43 We have not specified every component of Λ, however; the remaining component is ℑ(φ), so that WZ gauge fixes everything except the usual gauge symmetry of the vector potential (127). So, we have seen that our supergauge transformations split up into ordinary gauge transformations and supertransformations, the latter of which are used to eliminate redundant sfield components. §3 In components – SUSY transformations. If we make a SUSY transformation ¯ then, using δξV = (ξQ + ξQ¯)V as usual, we find that the components of a vector field transform as ¯ δξC = iξχ − iξχ¯ , ¯¯ µ ¯ δξM = 2iξλ + 2∂µχσ ξ , (130)

δξAµ = ...

In particular, note that the field M, which was gauged away in WZ gauge, re–enters once we make a gauge transformation, due to the term ξ¯λ¯ appearing in (130). In other words, SUSY transformations break WZ gauge. However, we know that the fields which are being re–introduced can just be gauged back to zero, so there is always a compensating gauge transformation which brings us back to WZ gauge (called the de Wit–Freedman transformation [18]). We then say that the ‘SUSY transformations’ of our component fields are as follows, but by this we really mean a combined SUSY transformations and compensating gauge transformation: ¯ ¯ δξAµ = iξσµλ − iλσµξ , α α µνα β δξλ = iξ D + σ βξ Fµν , (131) µ ¯ µ ¯ δξD = −ξσ ∇µλ − ∇µλ σ ξ .

Note that, until now, all our SUSY transformations on component fields have been linear (provided the auxiliary fields are left in play). Here, though, the transformations are nonlinear due to the minimal coupling ∇µλ term (the Fµν term is also nonlinear in the non–abelian theory). It is the compensating gauge transformation which generates these nonlinearities. µν µ §4 Action. For an action we need to create terms of the form F Fµν and λ¯σ¯ ∇µλ; from (129), we see that we will need expressions which are quadratic in V in order to generate terms quadratic in A. In order to do so, we appeal to our previous experience with chiral sfields. We define g W ≡ W (V ) = − D¯ 2D V, (132) α α 4 α

3 noting that Wα is chiral since DW¯ ∼ D¯ ≡ 0. We already know how to construct actions from chiral sﬁelds, and so we try as an action Z Z Z Z 1 1 d4x d2θ W αW + d4x d2θ¯ W¯ W¯ α˙ . (133) 4g2 α 4g2 α˙

44 α Under gauge transformations V → V + Λ + Λ,¯ our W is, like Fµν, gauge invariant, and indeed the resulting action in component ﬁelds is Z 1 1 S = d4x D2 − F F µν − iλ¯σ¯µ∇ λ . (134) 2 4 µν µ

This is the free U(1) super Yang–Mills action, comprising the photon Aµ (a boson), the photino λ (a fermion) and the axillary ﬁeld D which appears as D2, rather than DD¯ , because it comes from a real superﬁeld.

Exercise 17. Show that Wα deﬁned in (132) is gauge invariant.

§5 Abelian matter couplings. To minimally couple matter fields to ordinary gauge theory, we replace ∂µ with ∇µ, which requires matter fields to take values in a representation of the gauge group. Typically we consider matter in the fundamental, antifundamental and adjoint reps, the last of which is trivial for U(1) theories since adjoint matter is neutral. In analogy with ordinary U(1) gauge invariance, we define the gauge transformations of chiral sfields Φ and Φ¯ in the fundamental and antifundamental reps as12

Φ → e−gΛ Φ , fundamental , ¯ (135) Φ¯ → Φ¯ e−gΛ , antifundamental .

Recall that the ‘ordinary’ gauge parameter α is the lowest component of ℑ(Λ), and therefore (135) yields indeed the well-known gauge transformations

φ → e−igαφ andφ ¯ → φe¯ igα . (136)

We see here another reason for taking the gauge transformations to be generated by chiral sfields Λ; it is only then that the chiral sfield Φ remains chiral under gauge transformations. Recall the free chiral sfield action (112), with kinetic term ΦΦ.¯ This is clearly not invariant under (135), picking up the exponent of −g(Λ + Λ).¯ Nevertheless, it is straightforward to write down a coupling which is gauge invariant, and which reduces to (112) when g → 0, Z Z d4x d4θ Φ¯ egV Φ . (137)

Recalling also that the chiral sfield contains the component fields φ, ψ and F , the effect of including the exponential term in (137) is to make the replacements

µ µ ∂µφ → ∇µφ , and λ¯ σ¯ ∂µλ → λ¯ σ¯ ∇µλ ,

12Note that Wess & Bagger switch from ‘V → V + Φ + Φ’¯ to ’V → V + iΛ − iΛ’.¯ We maintain our convention of (125).

45 This gives us kinetic terms for matter in the fundamental and antifundamental. What about potential terms? From (117), the superpotential terms involve only Φ or Φ,¯ and we cannot make these gauge invariant using insertions of the type in (137). If we want to include matter, we can do so provided that we have at least two chiral sﬁelds of charges gj, i.e., j −gj Λ j −gj Λ¯ Φ → e Φ , Φ¯ j → Φ¯ je . We can then write down superpotentials of the form

j i j i j k W (Φ ) = mijΦ Φ + λijkΦ Φ Φ , (138) which are gauge invariant provided the couplings mij and λijk vanish whenever the combinations of charges gi + gj and gi + gj + gk are non–zero. With the inclusion of the kinetic terms (137) for each of the chiral sﬁelds, our gauge–matter U(1) Lagrangian becomes Z Z Z 1 1 L = d2θ W αW + d2θ¯ W¯ W¯ α˙ + d4θ Φ¯ egj V Φj 4g2 α 4g2 α˙ j Z Z (139) 2 j 2 + d θ W (Φ ) + d θ¯ W¯ (Φ¯ j) .

The ﬁrst line contains the kinetic terms, while the second line contains the superpotential terms as in (138). §6 Non–abelian gauge invariance. We now wish to extend the above discussion to non–abelian theories. Note that with our conventions, vector ﬁelds V become V ≡ V aτ a with the reality condition V = V † being equivalent to V a = V¯ a (which is a nice consequence of choosing hermitian generators).

We begin by generalising the superfield Wα to a non–abelian sfield, and then con- − g ¯ 2 struct the action for the gluon and gluino. Previously, we had Wα = 4 D DαV . Noting that V was essentially an element of the algebra of our gauge group, we recall the rela- −tλ tλ λ tion λ = e ∂t e relating algebra (λ) and group (e ) elements and make the educated guess for the non–Abelian Wα 1 ¡ ¢ W = − D¯ 2 e−gV D egV . (140) α 4 α To identify how this sfield behaves under gauge transformations, we must extend the transformation law of the vector fields. Note that in (139), it is really the transformation rule for exp V , rather than V alone, which gives us a gauge invariant action. It is this transformation which we must generalise to the non–abelian theory, as follows ¯ ¯ egV → egΛegV egΛ , =⇒ e−gV → e−gΛe−gV e−gΛ . (141)

We leave it to the reader to conﬁrm, using (141), that the ﬁeld Wα transforms as −gΛ gΛ Wα → e Wα e , (142)

−1 which parallels the non–abelian ﬁeld strength transformation Fµν → U FµνU.

46 Exercise 18. Verify (142).

With this, all that needs to be done to generalise the action (139) is to add a trance over matrix indices, so that we arrive at Z Z Z Z 1 1 S = d4x d2θ TrW αW + d4x d2θ¯ TrW¯ α˙ W¯ . (143) 4g2 α 4g2 α˙ In components we find Z 1 1 S = Tr d4x D2 + F F µν − iλ¯σ¯µ∇ λ , (144) 2 4 µν µ which is, as it should be, precisely as in (139) but where all fields are non–abelian, hence the addition of the matrix trace. §7 Gauge transformations. As for abelian SUSY theories, it is possible to use superspace transformations to go to a WZ gauge. Note that unlike, say, Coulomb gauge in Yang–Mills theory, the WZ SUSY gauge is well defined, i.e. it does not suffer from a Gribov ambiguity – the gauge fixing here amounts to comparing and cancelling ‘coefficients’ (the component fields) of the finitely many allowed functions of θ and θ¯. This can clearly always be done exactly. Expanding (141) in the coupling, we see that the non–abelian vector sfield transforms as V → V + Λ + Λ¯ + ... to first order. The variation in V under an infinitesimal gauge transformation can be computed exactly, as there is a version of the BCH formula which is exact to first order in B. This is ¡ ¢ 2 exp(A) exp(B) = exp A + LA/2 B + coth(LA/2)B + O(B ) , L 1 where A/2(B) = 2 [A, B], and coth should be understood in terms of its series expansion. Applying this result to (141), we can calculate, for infinitesimal gauge transformations, µ ¶ ¯ ¯ V → V + LV/2 Λ − Λ + coth(LV/2)(Λ + Λ) . (145)

We will make some use of this formula in a later lecture. The SUSY transformations of the component fields (including the gauge transformation which brings us back to WZ gauge) are as in (131), but decorated with a colour index. §8 Gauge invariant matter couplings and interactions What about matter couplings and interactions? This depends crucially on the representation of the matter fields. If the matter fields are in the fundamental representation of the gauge group (i.e. Φ → e−gΛΦ), the minimally coupled action reads as Z Z S = d4x d4θ Φ¯egV Φ , (146)

47 and no superpotential terms are allowed, as they cannot be rendered gauge invariant. If, however, the matter ﬁelds are in the adjoint representation(i.e. Φ → e−gΛΦegΛ), the minimally coupled action is given by Z Z 4 4 −gV gV Sminimal = Tr d x d θ e Φ¯e Φ , (147) and arbitrary polynomials are admissible as superpotentials: Z Z X 4 2 n Sint = Tr d x d θ anΦ + h.c. . (148) n The overall trace renders these actions gauge invariant. Of course, renormalizability will again restrict the superpotential to polynomials of degree 3. We will give an example of a supersymmtric nonabelian gauge theory with superpotential couplings in Sect. 7.3, where we write down the action of N = 4 super Yang-Mills theory in the N = 1 superﬁeld formalism.

48 6 SUSY quantum ﬁeld theories

In this part of the lectures we quantise the classical SUSY theories we constructed above.

6.1 Abstract considerations We begin with a taster of some of the important consequences of SUSY at the quantum level. Using only the SUSY algebra, we will shortly derive, for example, a nonrenormalisation theorem, without having to write down Feynman rules. We begin by introducing another important symmetry of our SUSY theories. §1 R–symmetry. The Coleman–Mandula theorem tells us that internal bosonic symmetries cannot mix with Poincar´egenerators. There is, however, a loophole to this no-go theorem, which is that certain internal bosonic symmetries can mix with SUSY generators. These symmetries are called R-symmetries and are given by the automorphism group of the supersymmetry algebra. In the case N = 1, the automorphism group is just U(1) and it is generated by an operator R which satisﬁes the following commutation relations:

[R,Qα] = −Qα , [R, Q¯α˙ ] = +Q¯α˙ . (149)

§2 Associating R–charge. We will define the R–charge of a chiral sfield Φ to be the R–charge of its lowest component. If the lowest component φ of the sfield has R–charge r, then it follows from (149) that

[φ]R = r =⇒ [ψ]R = r − 1 , [F ]R = r − 2 , (150) and further that [θ]R is +1. It follows that [dθ]R = −1 and so, since the Lagrangian should be R–invariant, the superpotential W must have R–charge [W ]R = +2. §3 Nonperturbative nonrenormalisation theorem. Consider the Wess–Zumino model with superpotential W = mΦ2 + λΦ3, and assign the following R–charges,

Φ m λ

UR(1) 1 0 -1 . Note that, aside from R–symmetry, the action has an additional U(1) symmetry, the generator of which does not mix with any of the Poincar´eor SUSY generators, such that W is neutral and we assign

Φ m λ U(1) 1 -2 -3 .

(The assignment of the chiral ﬁeld charge is up to us; all the other assignments follow.) Now, we will assume that these symmetries of the classical action are also symmetries of the quantum eﬀective action. Generally, of course, symmetries need not survive

49 quantisation, they can be anomalous, but see [19] for reasons we expect them to be preserved in this case. We assume that the effective action is again given by a Wess–Zumino model, but we must allow the superpotential to contain arbitrary powers of the chiral sfield. Taking into account the preservation of our UR(1) and U(1) symmetries, we find the most general effective superpotential has the form X n 1−n n+2 Weff(Φ) = anλ m Φ . n≥0 For small λ and m we should be able to return to the free theory, and this forbids the m−1 terms produced by taking n > 1, so that our effective superpotential can only contain two terms, 2 3 Weff(Φ) → meffΦ + λeffΦ . Again, in the limit of small coupling, the coefficients of this superpotential should match up, so that meff = m, λeff = λ, and we find that the superpotential is not renormalised. §4 SUSY Ward–Takahashi identities. Consider the Green’s function

GWT := ⟨ Ω |φ1(x1) . . . φn(xn)| Ω ⟩ , where the φj are the first components of a set of chiral sfields Φ1 ... Φn. To proceed, we will need the component field results of the following exercise.

Exercise 19. The transformation laws of component ﬁelds were originally introduced in (93) and (94). Use these to extract the following: √ { ¯ } − µ ¯ Qα˙ , ψβ = i 2 σαβ˙ ∂µφ , [Qα˙ , φ] = 0 . (151)

Using the first of (151) we take a derivative of GWT with respect to the argument of φ1, √ − µ ∂ ⟨ |{ ¯ } | ⟩ i 2 σαβ˙ µ GWT = Ω Qα˙ , ψβ(x1) . . . φn(xn) Ω . ∂x1 The vacuum should be supersymmetric and therefore it is annihilated by Q¯, which leaves only one term in the anticommutator. Using the second result of (151), we can commute the Q¯ of this term through to the right, where it annihilates | Ω ⟩. Our derivative therefore kills the correlation function. Since the Pauli matrices are invertible, we can conclude ∂ µ GWT = 0 , ∂x1 and similarly the correlation function must be independent of all the xj. We can therefore choose to take the φj infinitely far apart and therefore, by the axioms of QFT, there should be no interactions between the various fields. Thus, the correlation function becomes a product of free one–point functions,

GWT = ⟨φ1⟩⟨φ2⟩ ... ⟨φn⟩ . (152)

50 §5 Holomorphy of correlation functions. For simplicity, we restrict our discussion here to a simple example: consider some set of chiral sﬁeld Φ1 ··· Φn with interaction Lagrangian Z Z 2 2 Lint = d θ λjΦj + d θ¯ λ¯jΦ¯ j .

The λj are some set of coupling constants. Deﬁne the correlation function G of only chiral ﬁelds (not their conjugates) by Z

iS0+iSint G = D(ﬁelds) φ1(x1) . . . φn(xn) e = ⟨φ1(x1) . . . φn(xn)⟩ .

Taking a derivative with respect to λ¯j gives us Z Z ∂ 4 2 ¯ ⟨¯ ⟩ ¯ G = d y d θ Φj(y)φ1(x1) . . . φn(xn) ∂λj Z 4 = d y ⟨F¯j(y)φ1(x1) . . . φn(xn)⟩ Z ∝ 4 ⟨{ ¯ ¯α˙ } ⟩ d y Qα˙ , ψj (y) φ1(x1) . . . φn(xn) = 0 , where the ﬁnal line follows from the same argument as employed above in §4– Q¯ can be commuted through to the left, and right, so that it annihilates the vacuum. The result is that the correlation function of only φ’s is holomorphic, in that it depends only on the λ, not λ¯. Similarly, correlation functions of theφ ¯ depend only on the λ¯.

6.2 Sfield quantisation of chiral sfields §1 Reminder of quantisation in QFT. Consider a scalar field theory given by an action functional S[φ] with some free and interacting split, S = S0 +Sint. The generating functional of correlation functions is defined by Z · Z ¸ Z[J] = Dφ exp iS + d4x Jφ , from which we construct correlation functions by taking derivatives with respect to J, ¯ n ¯ 1 δ Z[J] ¯ G(x1, . . . xn) = ··· ¯ . Z[0] δJ(x1) δJ(xn) J=0

The choice of split in free and interaction terms of the action is made such that S0 is quadratic in the ﬁelds, the reason being that Gaussian functional integrals are one of the few we can perform explicitly, whereas we cannot evaluate Z[J] for an interacting theory in a closed form. Instead we resort to perturbation theory in the interaction term, writing Z µ Z ¶ ¡ ¢ δ D 4 Z[J] = exp iSint[ δJ ] Z0[J] ,Z0[J] := φ exp iS0 + d xJφ ,

51 which, when expanded in the coupling, yields the usual Feynman diagrams. From the quadratic term of the action, the propagator is given by (ϵ − iA)−1 → iA−1, where the iϵ prescription has been employed to make the quadratic term regular, and therefore invertible. Alternatively, one can rotate to Euclidean space, which is the route we will take later, as we translate this quantisation prescription from fields to sfields. §2 Sfield quantisation. To quantise SUSY theories we will introduce a generating superfunctional, replacing fields by sfields, and then follow the same path as in field theory. We could choose to expand all our superfields in θ, working with the component fields which constitutes a ‘normal’ field theory. Defining the usual Feynman propagator ∆ by 1 △ = , (153) ¤ − m2 we would find the component field propagators

⟨ 0 |T φ(x)φ†(y)| 0 ⟩ = i△(x − y) , (154) ⟨ |T ¯ | ⟩ µ △ − 0 ψα(x)ψα˙ (y) 0 = σαα˙ ∂µ (x y) , and so on.

Exercise 20. Verify the propagators (154) using the explicit component ﬁeld form of the free action given by (112), with mass terms generated as in (118). Derive also the various fermion-fermion propagators.

§3 Superfield propagators. It is preferable to work with the sfields directly, rather than their components. In terms of the sfields, our action is Z Z Z µ Z ¶ m λ S = d4x d4θ ΦΦ¯ + d4x d2θ Φ2 + Φ3 + h.c. . (155) 2 3!

The immediate problem we face is that the quadratic terms are written partly as an integral over all of superspace, and partly as an integral over chiral superspace.

Exercise 21. For a chiral sﬁeld Φ, prove the identity

α˙ α D¯α˙ D¯ DαD Φ = 16¤Φ .

From the exercise above, we find that we can replace a chiral sfield Φ with D¯ 2D2Φ/16¤ as we please. Now, suppose we have any field f(x, θ, θ¯) = g(x)θ¯θ¯ + ..., where the ellipses denote all other terms in the superfield. We then have Z Z Z Z Z d4x d2θ¯ f(x, θ, θ¯) = d4x d2θ¯ g(x)θ¯θ¯ = d4x g(x) . (156)

52 We also have, acting with D¯ 2, Z Z Z 1 1 d4x − D¯ 2f(x, θ, θ¯) = d4x − D¯ 2g(x)θ¯θ¯ = d4x g(x) , (157) 4 4 which implies Z Z Z 1 d4x d2θ¯ ≡ − d4x D¯ 2 . (158) 4 Note that this result is only valid when the integration over xµ is included. This is needed to kill total derivative terms coming from the ∂/∂x terms in D¯ 2. We may therefore write those parts of the action (155) which are integrals over chiral superspace as integrations over full superspace, thus: Z Z Z ¯ 2 2 4 2 2 D D d x d θ ΦΦ = d θ Φ ¤ Φ Z Z 16 D2 4 2 ¯ 2 (159) = d x d θ D Φ ¤Φ since Φ is chiral, Z Z 16 D2 = − d4x d4θ Φ Φ , 4¤ and similarly for the conjugate term. Our quadratic action may therefore be written Z Z µ ¶ µ ¶ m 1 D2 m 1 D¯ 2 S = d4x d4θ ΦΦ¯ + Φ − Φ + Φ¯ − Φ¯ . (160) 0 2 4 ¤ 2 4 ¤

We can now extract the propagators of our theory from the free partition superfunction

Z0[J, J¯] completely analogously to the case of an ordinary QFT. For a detailed justiﬁca- tion that this indeed yields the same result, see [3]. Note that in the remainder of this section we work in Euclidean space. We have Z0[J, J¯], Z £ ¤ 2 Z0[J, J¯] = D Φ exp S0 + Ssource , (161) where the source terms are, in terms of a chiral sﬁeld J, Z Z Z Z ¡ ¢ 4 2 → 4 4 − 1 2 ¤ Ssource = d x d θ JΦ + h.c. d x d θ J 4 D / Φ + h.c. , (162) and we have used the same trick as above to write the source terms as an integral over all of superspace. Note that because the sources are chiral (i.e. constrained), taking derivatives with respect to them is a little subtle – one may show that, writing13 z = (x, θ, θ¯) as the co–ordinate on full superspace, δ J(z ) = − 1 D¯ 2 δ8(z − z ) , 2 4 z1 1 2 δJ(z1) (163) δ8(z) ≡ δ4(x)δ2(θ)δ2(θ¯) ≡ δ4(x) θθ θ¯θ¯ .

13 µ iα ¯α˙ Note that the index positions in the z will read as zi = (xi , θ , θi ).

53 The end result, however, is simple – if we simply diﬀerentiate our partition function with respect to the sources, it generates the correlation functions we expect – see the following exercise.

Exercise 22. Using the above results, verify that diﬀerentiation of the free partition superfunction (161), with respect to J, yields Φ as it should, i.e. Z ¡ ¢ δ 8 − 1 2 ¤ d z2 Φ(z2) 4 D / J(z2) = Φ(z1) . δJ(z1)

We are now ready to carry out the Gaussian integral and derive the propagators. It is most convenient to present the calculation in terms of the two component vector X, and the two–component source vector K, deﬁned by Ã ! Ã ! Φ 1 − m D¯ 2J/¯ ¤ X = ,K = 4 . ¯ − m 2 ¤ Φ 2 4 D J/

The reason for this is that we can immediately write Z0 in the form (where the bar is, as usual, just complex conjugation and we subsume integrals into the indices i and j), Z £ ¤ 2 Z0[J, J¯] = D X exp − X¯iAijXj + X¯jKj + K¯jXj , (164) and apply the standard Gaussian integral formula Z Y d2z ¡ ¢ ¡ ¢ k exp − z¯ A z +u ¯ z +z ¯ u = det(A)−1 exp u¯ A−1u . (165) 2πi k kl l i i i i i ij j k The quadratic form A and its inverse are Ã ! Ã ! 1 1 m D¯ 2/¤ 2 ¤ 1 − m D¯ 2/¤ A = − 4 ,A−1 = − 4 . m 2 ¤ ¤ − 2 − m 2 ¤ 2 4 D / 1 m 4 D / 1 Note that we have projected the inverse operator back into the space of chiral/anti–chiral sfields on which it acts, hence its simplicity. Plugging this into (165), we find that the free partition superfunction is the exponent of Z Z 1 1 − m D2 − m D¯ 2 d8z J¯ J + d8z J 4¤ J + J¯ 4¤ J.¯ (166) −¤ + m2 2 −¤ + m2 −¤ + m2 From this, we read off the propagators by direct differentiation with respect to J and J¯. In momentum space (for the xµ), we find 1 ⟨Φ(z )Φ(¯ z )⟩ = δ4(θ1 − θ2) , 1 2 p2 + m2 m D2(p, 1) ⟨Φ(z )Φ(z )⟩ = 4 δ4(θ1 − θ2) , (167) 1 2 p2(p2 + m2) m D¯ 2(p, 1) ⟨Φ(¯ z )Φ(¯ z )⟩ = 4 δ4(θ1 − θ2) , 1 2 p2(p2 + m2)

54 µ 2 where the shift from x → pµ sends ∂µ → −ipµ and so ¤ → −p and so the covariant SUSY derivatives read as ∂ D(p, i) ≡ + σµ θ¯α˙ p . (168) α ∂θiα αα˙ i µ

§4 Vertices. Now that we have our free partition superfunction, we can construct vertices just as in quantum field theory. The remaining terms in the superpotential W 8 8 are cubic in the sfields, hence we evaluate, employing the shorthand δ (z41) ≡ δ (z4 −z1), · ¸ Z 3 δ λ 4 2 4 δ Sint J(z1)J(z2)J(z3) = d x4d θ 3 J(z1)J(z2)J(z3) δJ Z 3! δJ (z4) £ ¤£ ¤£ ¤ = λ d4x d2θ4 − 1 D¯ 2 δ8(z ) − 1 D¯ 2 δ8(z ) − 1 D¯ 2 δ8(z ) (169) 4 4 z4 41 4 z4 42 4 z4 43 Z £ ¤£ ¤ = λ d8z δ8(z ) − 1 D¯ 2 δ8(z ) − 1 D¯ 2 δ8(z ) . 4 41 4 z4 42 4 z4 43 − 1 ¯ 2 To obtain the final line, we have pulled one factor of 4 D to the front (if it acts on any other terms we obtain a D¯ 3 = 0), and converted it into an integral over full superspace. Our interaction is an integral over all of superspace, at which two propagator legs come with a factor of −D¯ 2/4. Similarly, the Φ¯ 3 vertex is integrated over all of superspace and on which two of the three propagator legs have factors of −D2/4 acting on then. §5 Improved Feynman rules. Following [20], we give the following Feynman rules. We work in full superspace, but with xµ → pµ. 1. The propagators in momentum space are 1 ⟨ΦΦ¯⟩ = δ4(θ12) , p2 + m2 m/4 ⟨ΦΦ⟩ = D2(p, 1)δ4(θ12) , (170) p2(p2 + m2) m/4 ⟨Φ¯Φ¯⟩ = D¯ 2(p, 1)δ4(θ12) . p2(p2 + m2) 2. Vertices are derived from the superpotential, as we did above. For an n–point chiral vertex there are n − 1 factors −D¯ 2/4 acting on the attached propagators. For an n–point anti–chiral vertex there are n − 1 factors of −D2/4 acting on the attached propagators.

3. Integrate over loop momenta, external momenta (with overall momentum conser- vation) and d4θ at vertices.

4. Include the usual combinatorics and factors of −1 for ghost loops (see below). We will be interested in the eﬀective action, for which we have an additional Feynman rule. This is that we amputate external legs, multiply by sources Φ or Φ,¯ omitting a further factor of −D¯ 2/4 or −D2/4 respectively (and integrate over the source arguments). The presence of the sources allows us to perform some necessary integrations by parts.

55 2 k 2 D (−k,θ1) D¯ (k,θ2) p p θ1 θ2 Φ(¯ −p,θ1) Φ(p,θ2)

p + k

Figure 9: One loop correction to the eﬀective action.

§6 Propagator correction. As an example, we calculate a one–loop correction to the eﬀective action at m = 0, as shown in Fig. 9 and following e.g. in [20]. The following identities will prove useful.

Exercise 23. Show that ←− 4 1 2 4 1 2 δ (θ − θ )D α(−p, 1) = −Dα(−p, 2)δ (θ − θ ) , (171) and D¯ 2D2δ4(θ1 − θ2) = 16 . (172)

On the left is a Φ3 vertex, on the right a Φ¯ 3 vertex. From our Feynman rules, having amputated the external legs and attached the external ﬁeld, this diagram is given by Z Z λ2 d4p Γ(Φ, Φ)¯ = d4θ1d4θ2 Φ(¯ −p, θ1)Φ(p, θ2)× 2 (2π)4 Z µ ¶ d4k 1 £ ¤ 1 £ ←− ¤ δ4(θ12) − 1 D¯ 2(k, 2) δ4(θ12) − 1 D 2(−k, 1) . (2π)4 (p + k)2 4 k2 4

1 2 The initial factor of 2 is a symmetry factor. We now use (171), and integrate over θ , Z Z λ2 d4p Γ(Φ, Φ)¯ = d4θ1 Φ(¯ −p, θ1)Φ(p, θ1)× 2 (2π)4 Z ¯ 4 £ ¤ ¯ d k 1 1 2 ¯ 2 4 12 ¯ 4 2 2 16 D (k, 1)D (k, 1) δ (θ )¯ . (2π) k (p + k) θ1=θ2 Finally, apply (172) to ﬁnd Z Z λ2 d4p Γ(Φ, Φ)¯ = d4θ1 Φ(¯ −p, θ1)Φ(p, θ1) K(p) , 2 (2π)4 where Z d4k 1 K(p) := . (173) (2π)4 k2(p + k)2 This integral is divergent and requires regularisation. Here, a momentum cutoﬀ Λ is K log Λ equivalent to the usual dimensional regularisation and yields (p) = 16π2 .

56 §7 Remarks on regularisation. Unless care is taken to choose a regulator which preserves them, symmetries of the theory may be violated in perturbative calculations. In particular, one often wishes to use Ward identities, which can be violated by bad choices of regularisation. For SUSY theories, it is necessary to keep the number of bosons and fermions equal, which is diﬃcult in dimensional regularisation (DREG). This led Siegel to introduce dimensional reduction, or ‘DRED’ [21]. Here, one splits, for example, the gauge boson Aµ into (Aµˇ, ϕϵ) into a 4 − ϵ dimensional gauge potential, and an ϵ dimensional scalar. Although it is known to be correct up to two loops, this regularization has unfortunately problems of its own [22]. A detailed summary of the current situation is found in [23]. §8 Localisation theorem. The localisation theorem states that each term in the effective action is supported on a single d4θ integral. Consider an arbitrary diagram, and pick a loop. From the Feynman rules, this part of the diagram contains a string of δ’s like δ(θ12)δ(θ23) . . . δ(θn1) with D’s acting on them. Using partial integration, we can remove all D’s from δ(θn1), and perform the integral over δ(θn1). At the end, one is left with a term containing Z Z £ ¤ d4θ1 d4θ2 δ4(θ12) D . . . Dδ4(θ12) .

We can evaluate this by reducing it to a product of four D’s and then evaluate it – the 2 ¯ 2 4 12 only nonvanishing term is of the form D D δ (θ )|θ1=θ2 = 16. §9 Cancellation theorem. We brieﬂy return to chiral co–ordinates in order to demon- strate an important result. From (166), we can integrate over θ¯ and so obtain the Φ–Φ propagator in chiral superspace,

Z − Z 1 m D2 1 m d8z J 4¤ J = d4xd2θ J J, 2 −¤ + m2 2 −¤ + m2 m =⇒ ⟨ΦΦ⟩ = δ2(θ1 − θ2) . p2 + m2 Now consider a closed chiral loop. The Feynman diagram is proportional to Z Y d2θj δ2(θ1 − θ2)δ2(θ2 − θ3) . . . δ2(θn−1 − θn)δ2(θn − θ1) j Z Y = d2θj δ2(θ1 − θ2)δ2(θ1 − θ3) . . . δ2(θ1 − θn)δ2(θn − θ1) , j where we used δ(z1 − z2)f(z2) = δ(z1 − z2)f(z1). Consider the ﬁnal product of terms in the above. We have δ2(θ)δ2(θ) = (θθ)(θθ) ≡ 0 , since θ has only two components. Therefore our diagram vanishes. Similarly, every Feynman diagram with a closed chiral loop vanishes identically. If we were to work with

57 the component ﬁelds, the same result would eventually emerge as the mutual cancellation of the bosonic and fermionic loops contributing to the diagram – it is clearly much more direct to obtain this result at the level of superﬁelds. One consequence of this result is that all tadpole diagrams vanish.

6.3 Quantisation of Super Yang–Mills theory R § 4 2 α 1R Classical action. First, note that the expressions Tr d xd θ W Wα and 4 2 α˙ Tr d xd θ¯ W¯ α˙ W¯ are identical after performing a Wick rotation. We thus start from the action Z Z 1 S = Tr d4xd2θ W αW + Tr d4xd4θ Φ¯egV Φ + ... (174) YM 2g2 α where Wα is exactly as in (140) and the ellipses denote the potential terms for Φ. As before, we begin by identifying the propagator of the vector superﬁeld V . Recall that

Wα is a chiral superﬁeld, and so we begin by performing the tricks introduced in (159) to write the action as an integral over the whole of superspace, Z 1 S = Tr d4xd4θ − (e−gV DαegV )D¯ 2(e−gV D egV ) + Φ¯egV Φ + ... (175) YM 32g2 α To extract the propagator, we expand in powers of V and retain only the quadratic terms, Z 1 S = Tr d4xd4θ V DαD¯ 2D V + ... (176) YM 32 α The problem with this quadratic term is that, just as in ordinary Yang–Mills theory, the kernel appearing is a projection operator, and therefore not invertible. To see this, deﬁne ¡ ¢ 1 α 2 1 2 2 2 2 Π 1 = − D D¯ Dα , Π0 = D¯ D + D D¯ . (177) 2 8¤ 16¤ These are projection operators which also obey

Π0 + Π 1 = 1 =⇒ Π0Π 1 = Π 1 Π0 = 0 . (178) 2 2 2

Exercise 24. Useful practice with ‘D–algebra’: show that Π 1 and Π0 are projection 2 operators, and that Π 1 + Π0 = 1. 2

We have an obvious analogy with the ordinary transverse and longitudinal projectors

(respectively, Π 1 , which appears in the action, and Π0). What we need to get around 2 this problem is of course a gauge ﬁxing.

58 §2 Gauge fixing. Our aim is to employ the Faddeev–Popov technique to introduce a gauge fixing term into the action. This will allow us perform the integral over V and so construct a propagator, in complete analogy to what we do in ordinary Yang–Mills theory. From (176)–(178), we can see that the most natural way to do this would be to arrange for the gauge fixing term to take the form Z 1 S = − Tr d4xd4θ V Π ¤ V, (179) GF 4α 0 as then the quadratic total terms in the action would collectively become Z · ¸ 1 4 4 1 SYM + SGF = − Tr d xd θ V Π 1 + Π0 ¤ V, (180) 4 2 α which is analogous to the usual choice of covariant gauges in Yang–Mills theories, with the presence of both projectors allowing us to invert the quadratic term. The price we will have to pay for this, of course, is the introduction of ghosts. To see all this explicitly, let us derive (179). The functional integral quantising the V sfield theory is Z Z[V ] = DV eSYM[V ] .

The choice of a gauge fixing condition is not quite obvious. We have to choose a gauge variant function, which can be rendered zero by a gauge transformation. Therefore, it should have the same spin and superspin, cf. [4]. In any case, the following choice is clearly admissible and will lead to the desired result: 1 F [V ] = − D¯ 2V − f , (181) 4 This is an equation on a chiral superfield f, and therefore the gauge fixing condition appears as follows in the resolution of unity, which is inserted into the functional integral: Z ¡ ¢ ¡ ¢ Λ Λ 1 = ∆FP(V ) DΛDΛ¯ δ F [V ] δ F¯[V ] , (182) where the symbol V Λ denotes the transformation generated by the chiral superfield parameter Λ acting on V . We now invoke gauge invariance in the usual way, which allows us to pull DΛDΛ¯ out as a volume (which we move into a normalisation factor), and rename V Λ as V everywhere, leaving Z ¡ ¢ SYM[V ] Z[V ] → DV ∆FP(V ) δ (F [V ]) δ F¯[V ] e .

Substituting our choice of gauge ﬁxing condition and averaging over all choices of f with a Gaussian weight, we arrive at Z · Z ¸ Z 1 ¡ ¢ ¯ 4 4 ¯ SYM[V ] Z[V ] = DfDf exp − Tr d xd θff DV ∆FP(V ) δ (F [V ]) δ F¯[V ] e Z 2α

SYM+SGF = DV ∆FP(V ) e .

59 The integrations over f and f¯ are ﬁxed by the delta functions, yielding our gauge ﬁxing action, Z Z 1 1 1 S = − Tr d4xd4θ D¯ 2VD2V = − Tr d4xd4θ V Π ¤V, GF 2α 16 4α 0 which is precisely as in (179). It remains only to evaluate the Faddeev–Popov determinant, which we can write as Z Z · Z µ ¶ ¸ δF δF ∆−1(V ) = D(Λ, Λ)¯ D(Λ′, Λ¯ ′) exp Tr d4xd2θ Λ′ Λ + Λ¯ + h.c. . FP δΛ δΛ¯

This is evaluated using (145), and inverted by replacing the chiral sﬁelds Λ, Λ′ with chiral ghosts ic, ic′, Z Z £ ¤ ′ ′ ∆FP(V ) = DcDc¯ Dc Dc¯ exp Sghost , Z £ ¤ (183) 4 4 ′ ′ Sghost := Tr d xd θ (¯c − c )LgV/2 (¯c + c) + coth(LgV/2)(c − c¯) .

− 1 ¯ 2 − 1 2 ′ ′ Note that the 4 D and 4 D terms coming from F can be moved past c andc ¯ respectively, by chirality, and then replaced by d2θ¯ and d2θ, giving us the integral over full superspace in (183). §3 The kinetic terms. We now have our gauge ﬁxing and ghost actions. As a result, the kinetic terms for the vector sﬁeld V are, in total, just as in (180). A natural choice of the parameter α is given by working in SUSY Feynman gauge α = 1, in which case the kinetic terms reduce to Z 1 S + S ∼ − Tr d4xd4θ V ¤V. (184) YM GF 4 We can also expand the ghost action and obtain their kinetic terms, Z 4 4 ′ ′ Sghost = Tr d xd θ c¯ c + c c¯ + ... (185)

(Note that the terms c′c andc ¯′c¯, which are products of two chiral or two antichiral sﬁelds, vanish under the integral over full superspace.) §4 BRST invariance. The total action is invariant under the following BRST transformations: ¡ ¢ δgV = ξLgV/2 (c +c ¯) + coth[LgV/2](c − c¯) , g g (186) δc′ = ξD¯ 2D2V, δc¯′ = ξD2D¯ 2V, δc = −ξc2 , δc¯ = −ξc¯2 . α α

60 §5 Feynman rule summary.

− 2 4 1 − 2 1. The vector sfield propagator is p2 δ (θ θ ), while the ghost propagator is 1 4 1 − 2 + p2 δ (θ θ ). 2. Vertices are read off from the action, with the appropriate factors of D2 and D¯ 2. For example, the cubic terms in the vector sfield V are Z g S = ... + Tr d4xd4θ [V, (DαV )]D¯ 2D V + ..., (187) YM 16 α

from which we read oﬀ a three point vertex with factors of D and D¯ distributed as shown below in Fig. 10.

3. Include combinatorics and factors of (−) for each ghost loop.

4. For amputated diagrams, multiply by external superﬁelds (so that we can integrate by parts).

5. Integrate over all θ’s at vertices and over loop momenta.

Dα b f abc a

¯ 2 c D Dα

Figure 10: Cubic V vertex with factors of D and D¯.

§6 Example calculation. Consider the one–loop correction to the V propagator. There are three contributions, as three species of sﬁeld can circulate in the loop – V , the scalars Φ and Φ,¯ or the ghosts. We will now describe the calculation of the scalar contribution, shown in Fig. 11, which from our Feynman rules is Z Z d4pd4k 1 Γ = g2 d4θ1d4θ2 V (−p, θ1)V (p, θ2) scalar (2π)8 k2(p + k)2 ¡ ¢¡ ¢ (188) 1 ¯ 2 − − ¯ 2 − − 4 1 ¯ 2 ¯ 2 4 16 D ( p k, 2)D ( p k, 2)δ (θ12) 16 D (k, 2)D (k, 2)δ (θ12) , where the factors of D2 and D¯ 2 are those acting on the chiral propagator legs. We now integrate by those D’s which see p + k by parts, after which they act on the product £ ¤ 2 2 4 2 D¯ (k, 2)D¯ (k, 2)δ (θ12) V (p, θ ) , and will leave another δ4(θ12) exposed so that we can perform one of the θ integrals. They key to performing these integrations to observer that D(−p − k, 2), when we integrate by parts, splits up, suing the Leibnitz rule, into pieces like D(p, 2) acting on V (which

61 k 2 2 D (−k,θ1) D¯ (k,θ2) p p θ1 θ2 V (−p,θ1) V (p,θ2) 2 2 D (−p − k,θ1) D¯ (p + k,θ2) p + k

Figure 11: One–loop vector propagator – the scalar contribution. sees p, not k) and D(k, 2) acting on the D2D¯ 2δ (which sees k, not p). For three sfields X ≡ X(θ22), Y ≡ Y (k, θ2) and Z ≡ Z(p, θ2), we have Z £ ¤ 4 2 d θ Dα(−p − k, 2)X YZ Z µ ¶ 4 2 X˜ 2 Y˜ α 2 = d θ (−) X D YZ + 2(−) D YDαZ + YD Z , where D ≡ D(k, 2) if it sees Y and D ≡ D(p, 2) if it sees Z. Using this result, and some of our earlier D–algebra identities, we find that Z Z Z 2 4 d 2 1 αα˙ 1 2 2 g d p d k −k − k DαD¯α˙ + D D¯ Γ = d4θ V (−p, θ) 2 16 V (p, θ) . scalar 2 (2π)4 (2π)4 k2(p + k)2 One now goes to 4 − ϵ dimensions and uses the results Z Z 1 (2k + p) d4−ϵk = d4−ϵk µ = 0 , (189) (p + k)2 (p + k)2 to finally arrive at Z Z 2 4 g d p 4 2 Γscalar = d θ V (−p, θ) p Π 1 V (p, θ) . (190) 2 (2π)4 2 There are two further contributions, coming from the vector and ghost loops, which are, respectively, Z Z 4 ¡ ¢ 2 d p 4 − 2 − 5 1 Γvector = g d θ V ( p, θ) p 2 Π 1 + 2 Π0 V (p, θ) , (2π)4 2 Z Z (191) 4 ¡ ¢ 2 d p 4 − 2 − 1 − 1 Γghost = g d θ V ( p, θ) p 2 Π 1 2 Π0 V (p, θ) . (2π)4 2

Note that, adding the three contributions in (190) and (191), the Π0 contribution vanishes – if we were to expand in components, we would see that this is just the SUSY µ version of the usual Ward identity pµA = 0 on the amplitude A. §7 Localisation theorem. The localisation theorem for chiral superﬁelds (§8 in section 6.2) evidently also holds for SYM theory and the proof is precisely the same.

62 7 Maximally SUSY Yang–Mills theories

Maximally supersymmetric Yang-Mills theories are particularly interesting theories, as their maximal symmetry yields many unusual or simplifying features. In particular, they serve as eﬀective descriptions of D-branes in type II superstring theories. The results presented here are mainly due to [24, 25]. Further useful material on SUSY in higher dimensions is found in [8].

7.1 Spinors in arbitrary dimensions For the description of all maximally SUSY YM theories, we need to extend our understanding of spinors to arbitrary dimensions. That is, we want again the double covers of the Lorentz group: the pin and spin groups. §1 Cliﬀord algebra and representations. Consider the Cliﬀord algebra C(Rt,s), where t (s) gives the number of negative (positive) entries on the diagonal of the metric

ηµν, and we have the algebra © ª − 1 γµ, γν = 2gµν . (192)

(We write g instead of η as we will consider various diﬀerent signatures here.) As before, 1 the combination γµν = 4 [γµ, γν] yields the spinor representation of SO(t, s), and we have ¡ ¢ B µν µν L ψ = exp α γµν ψ , α some real parameters. (193) Of course, for explicit computations we need the γ–matrices in arbitrary dimensions. One particularly simple way of constructing them is as follows14. Starting in d ≥ 2 dimensions with Euclidean metric, i.e. t = 0, s = d, the γ–matrices are given by the tensor products

γ0 = iσ1 ⊗ σ0 ⊗ σ0 ⊗ σ0 ⊗ ...

γ1 = iσ2 ⊗ σ0 ⊗ σ0 ⊗ σ0 ⊗ ...

γ2 = iσ3 ⊗ σ1 ⊗ σ0 ⊗ σ0 ⊗ ...

γ3 = iσ3 ⊗ σ2 ⊗ σ0 ⊗ σ0 ⊗ ...

γ4 = iσ3 ⊗ σ3 ⊗ σ1 ⊗ σ0 ⊗ ...

γ5 = iσ3 ⊗ σ3 ⊗ σ2 ⊗ σ0 ⊗ ... . . which we hope is enough information for the pattern to be clear. If d is even, the representation is 2d/2 dimensional, whereas if d is odd, the representation is 2(d−1)/2 ≡ 2[d/2] dimensional, and this tells us when to stop tensoring. For example, in three

14Note that this does not yield the Weyl representation which we have employed thus far

63 dimensions, the gamma matrices are γ0 = σ1, γ1 = σ2, γ2 = σ3, which it is easily checked form a representation of (192) with Euclidean signature. Finally, in the case that t ≠ 0, we simply need to drop the factor of i in the ﬁrst t gamma matrices.

Exercise 25. Write down the γ matrices in d = 4 Euclidean space, using the above, and check that they obey the algebra (192). Convert the matrices you obtain into those for the algebra (55) in our usual signature (−, +, +, +).

§2 Irreducible spinors. A Dirac spinor in four dimensions splits into two Weyl spinors, P 1 1 ± which are eigenspinors of the projection operators ± = 2 ( γ5). To generalise this idea to other dimensions without confusing notation, we replace γ5 by γchiral. Then, in even dimensions, we can construct

n γchiral = i γ0γ1 . . . γd−1 , (194) 2 1 with n ﬁxed by the convention γchiral = , which obeys

{γchiral, γµ} = 0 , [γchiral, γµν] = 0 . (195)

This shows us that the representation is reducible into chiral and antichiral pieces, or d −1 ‘Weyl’ and ‘Weyl’ pieces, each of dimension 2 2 . A second condition one can impose on spinors is the ‘Majorana condition’, that is

C ∗ ψ = ψ = Cγ0ψ , (196) which is only consistent in certain dimensions (for more details, see [26, Appendix B]). For SO(1, d − 1), the possible types of spinors in various dimension are summarised in the following table d= 2 3 4 5 6 7 8 9 10 11 12 Majorana x x (x) (x) x x x (x) Weyl x x x x x x Majorana–Weyl x x Note the cases d = 4, 8, 12. While one can in principle impose a Majorana condition in these cases, this condition is incompatible here with the Weyl condition and thus there are no Majorana-Weyl spinors for d = 4, 8, 12. Of particular importance are d = 2 and d = 10, because of their relevance to string theory.

7.2 Actions and constraint equations §1 N = 1 super Yang–Mills. Consider R1,d−1, a connection ∇, corresponding cur- 15 vature Fµν and a spinor λ taking values in the algebra. The action of N = 1 SYM

15 We will use anti–hermitian generators in this section, so ∇µ = ∂µ + gAµ.

64 theory is Z 1 S = Tr ddx − F F µν + iλ¯Γµ∇ λ , (197) 4 µν µ where Γµ are the d–dimensional gamma matrices. A requirement of SUSY is of course that we have equal numbers of bosonic and fermionic degrees of freedom, so we should ﬁrst ask in what dimensions this action can be supersymmetric. A gauge potential in d dimensions has d − 2 on–shell real degrees of freedom, and we saw above that Dirac spinor has 2[d/2] on–shell, real degrees of freedom, and these two must match. Imposing a Majorana or Weyl condition on the spinor, though, reduces its degrees of freedom, each by a factor of one half. Starting with d = 3, the various possibilities for matching degrees of freedom are shown in the following table.

Aµ λD λM λW λMW d = 3 1 2 1 -- d = 4 2 4 2 2 - d = 6 4 8 - 4 - d = 10 8 32 16 16 8

Note that in d > 10 there are no solutions to our counting problem: d = 10 is the highest dimension in which the action (197) can be supersymmetric. §2 N = 1, d = 10 SYM. In d = 10, the action is just as above, and λ is a Majorana– Weyl spinor. Beginning with the 32–component Dirac spinor, we reduce the spinor into 1 1± chiral parts λ± deﬁned by 2 ( Γchiral) = 0, and then drop half of these, leaving us with a Weyl spinor with 16 degrees of freedom. Imposing the Majorana condition λ = CλT brings us down to the required eight degrees of freedom. The SUSY transformations of the ﬁelds are ¡ ¢ µν 1 δAµ = iα¯Γµλ , δλ = ΣµνF α Σµν := 4 [Γµ, Γν] . (198)

In the above, we have a single Majorana–Weyl spinor α parameterising the SUSY transformations. This is referred to as N = 1 SUSY – compare with our previous discussion of N = 1 in four dimensions, where a single Weyl spinor ϵ parameterised the SUSY transformations through the generator ϵQ +ϵ ¯Q¯. The number of supercharges, however, depends on the dimension. In d = 4, we had 4N real supercharges, and in d = 10 there are 16N real supercharges. Note from (198) that for gauge groups other than U(1) we have a nonlinear representation of the SUSY algebra, and no auxiliary field. The number of auxiliary fields needed to make the theory linear in d = 10 is actually infinite. §3 d = 10 constraint equations. It is possible to give a superspace description of N = 1 SYM in d = 10, if only classically. Our superspace is now R1,9|16, onto which we

65 introduce the superconnection

∇µ = ∂µ + gAµ , ∂ (199) ∇ = D + gA ,D = + iΓµ θβ∂ , α α α α ∂θα αβ µ in terms of sﬁelds Aµ and Aα such that

{∇ ∇ } ! µ ∇ α, β = 2Γαβ µ . (200)

In the above, Aµ is a bosonic sﬁeld while Aα is fermionic, with expansions

0 1 α 0 1 β Aµ = Aµ + Aµαθ + ...,Aα = Aα + Aαβθ + ..., up to terms with 16 θ’s. We also introduce the curvature and spinor sfield λ, 1 F = [∇ , ∇ ] , µν g µ ν (201) 1 λα = Γµαβ[∇ , ∇ ] , 10g µ β respectively. Using the Bianchi identities and Fierz identities for 10d Dirac matrices, and the fact that λ = CλT , one obtains a superspace ‘Weyl equation’ for λ and ‘equation of motion’ for Fµν, 1 Γµ ∇ λβ = 0 , ∇µF + Γ {λα, λβ} . (202) αβ µ µν 2 ναβ In fact, (200) is equivalent to (202). These are the equations of motion of 10d SYM, but living on the superspace R1,9|16. One can show that these equations are satisfied if and only if they are satisfied to zeroth order in θ [25]: 10d, N = 1 SYM on superspace R1,9|16 is equivalent to 10d, N = 1 SYM on R1,9. However, this is manifestly an on–shell correspondence as it holds at the level of the equations of motion, rather than the action, and so cannot be used to quantise in a superspace formulation. §4 Dimension reduction/Kaluza–Klein compactification. Consider compactifying one spatial dimension on a circle of radius R, so R1,d becomes R1,d−1 × S1. We decompose our co–ordinates xµ into xµ¯ on the extended dimensions and y on the circle. The limit R → 0, in which the compactified dimension shrinks to zero size, is called ‘dimensional reduction’. To understand what happens to spacetime fields under this action, we begin with the simplest case of a complex scalar with periodic boundary conditions on the S1, which has the Fourier expansion

X iny/R µ¯ µ¯ e ϕ(x , y) = ϕn(x ) √ . (203) n∈Z 2πR

66 The kinetic action term for this ﬁeld is Z Z Z 2 d µ † 2 d−1 µ¯ † ∂ 2 d x ϕ (¤ − m )ϕ = d x dy ϕ (¤ − + − m )ϕ d 1 ∂y2 Z µ ¶ X 2 d−1 µ¯ † µ¯ 2 n µ¯ = d x ϕ (x ) ¤ − − m − ϕ (x ) , n d 1 R2 n n∈Z

µ¯ 2 which describes the Fourier modes ϕn(x ) as having curvature dependent masses m + 2 2 n /R . As R → 0, the modes ϕn for n ≠ 0 become infinitely massive, i.e. it would cost an infinite amount of energy to excite such modes, and they therefore decouple from the theory in a spectral flow. All that remains is the zero mode ϕ0, with action Z d−1 µ¯ † ¤ − 2 d x ϕ0( d−1 m )ϕ0 .

More generally, consider compactiﬁcation on a torus Tq = S1 × ... × S1, q times, with each circle of radius R, so that spacetime becomes R1,d−1−q × Tq. Before we take the limit R → 0, we note that the Lorentz group splits accordingly,

SO(1, d − 1) → SO(1, d − 1 − q) × isometries on Tq.

What about fields coupled to gauge theory? A gauge connection ∇µ = ∂µ + Aµ acting on the zero mode of a field simply reduces to ∇µ¯ for µ =µ ¯. However, the components of the connection ∇i for i = 1 . . . q in the circular directions, acting on a zero mode is just Ai on the mode, since the zero modes are constants in the circular directions and the components Ai, i = 1 . . . q, become a collection of scalar fields after reduction. In the limit R → 0, the isometries on T q become the rotations on Rq and we have the splitting SO(1, d − 1) → SO(1, d − 1 − q) × SO(q) . A spinor field decomposes into direct sums of representations of Spin(1, d−1−q) because of the tensor product structure of the Clifford algebra and we have the splitting

Spin(1, d − 1) → Spin(1, d − 1 − q) × Spin(q) .

We will put this to use in the following section.

7.3 d = 4, N = 4 super Yang–Mills theory We reduce N = 1, d = 10 SYM to N = 4 SYM in d = 4. §1 From 10 to 4 dimensions. The Lorentz group and its double cover split according to

SO(1, 9) → SO(1, 3) × SO(6) , ∼ Spin(1, 9) → Spin(1, 3) × Spin(6) = Spin(1, 3) × SU(4) .

67 Field reduction: Aµ → Aµ¯, forµ ¯ = 0, 1, 2, 3, and scalars ϕy for y = 1 ... 6. The SO(6) becomes an internal symmetry mixing these six scalars, which sit in the fundamental rep, i.e. ϕy is a 6–dimensional vector. This is in fact an R-symmetry, as we will see below. To expose the SU(4) structure implied by the split of the double cover, we put the scalars into the 6 of SU(4). An antisymmetric 4 by 4 matrix has 4 × 3/2 = 6 components. Call this matrix φij. Now, take our six scalars and define the components 1 ¡ ¢ φi4 = √ ϕi+3 + iϕi+6 , (204) 2 where i is counted modulo 6. Antisymmetry then defines the elements φ4i. The rest of the matrix is filled out by the definition 1 φij = ϵijklφ = φ∗ . (205) 2 kl ij ij 1 ij We can go back to SO(6) from SO(4) using the ’t Hooft tensors ηy , ϕy = 2 ηy φij. That deals with the scalars – what about spinors? The Clifford algebra splits up as follows; Ã ! ∼ 0 ρij Γ → γ ⊗ 1 , Γ = γ = γ ⊗ , (206) µ µ¯ 8 y ij 5 ρij 0 where the γµ¯ are the ordinary 4 by 4 gamma matrices and the 4 by 4 ρ matrices are given by 1 (ρ ) = ϵ , (ρij) = ϵijmnϵ , (207) ij kl ijkl kl 2 mnkl and the chiral gamma matrix, in terms of our usual γ5, is

Γchiral = Γ0 ... Γ9 = γ5 ⊗ 18 . (208) Finally, the 10 dimensional charge conjugation matrix is related to the four dimensional matrix C by Ã ! 0 14 C10 = C ⊗ . (209) 14 0 Now, our 32 component spinor splits up as Ã ! Lχi 1 + γ 1 − γ λ = ,L = 5 ,R = 5 , (210) Rχ˜i 2 2 andχ ˜ = Cχ¯T . We have four left–handed and four right–handed Weyl spinors, which is exactly what we want. Why? Recall, from Sect. 3.2, §5, that the d = 4, N = 4 supermultiplet had the form i i fαβ χa φij χ¯a fα˙ β˙ 1 4 6 4 1 , where the second row counts the number of real components. We have therefore re- covered the correct content of N = 4 SYM in d = 4 from a dimensional reduction of N = 1 SYM in d = 10. The number of supercharges, 16, is the same before and after dimensional reduction.

68 §2 Action. The action is obtained from simply plugging all the reduced ﬁelds into the 10d action, Z 4 1 µν µ ij µ i S = Tr d x − F Fµν + ∇µφij∇ φ + iχσ¯ ∇µχ 4 (211) 2 ij kl i j ij + 2g [φij, φkl][φ , φ ] − gχ [χ , φij] − gχ¯i[¯χj, φ ] .

The action is invariant under the following SUSY transformations:

µ − jα µ α˙ − jα µ α˙ δA = iχ σαα˙ ϵ¯j iϵ σαα˙ χ¯j , 1 ¡ ¢ δχi = − F σµνβϵi + 4i D/ φij ϵ¯α˙ − 8g[φ ¯ , φki]ϵj , α 2 µν α β αα˙ j jk α (212) 1¡ ¢ 1 δφij = χiαϵj − χjαϵi + ϵijklϵ¯ χ¯α˙ . 2 α α 2 kα˙ l From these SUSY transformations, one sees that the group SU(4) acting on the indices i, j, ... indeed is an R-symmetry as it mixes nontrivially with supersymmetry transformations. §3 Superfield formulation, N = 1. The bosonic field content is 6 scalars, 1 vector, or in N = 1 language, 1 vector field and 3 chiral sfields. The actions takes the form Z Z 1 1 L = Tr d4θ Φ¯egV Φ + d2θ W αW + c.c. 4g2 4 α √ µ Z Z ¶ (213) 2 + i d2θ ϵ Φi[Φj, Φk] + d2θ¯ Φ¯ i[Φ¯ j, Φ¯ k] . 3! ijk

Although we cannot set up a manifestly N = 4 supersymmetric framework for performing computations in N = 4 SYM theory, we can at least use the manifestly N = 1 supersymmetric formalism. That is, we can use N = 1 supergraph techniques to simplify quantum computations, for example. §4 Reduction to other N SYM. Taking the N = 4 action (211) and restricting i to run from 1 to N yields other SYM theories, for N = 1 and 2. In these cases, one can split the supermultiplet into self dual and anti–self dual parts, i.e. ﬁelds with helicities > 0 and < 0 (the complex scalar ﬁeld for N = 2 is split into two reals, one belonging to each of the multiplets). These parts are each others’ CPT conjugate. In the reduced theories, these parts are closed under supersymmetry transformations, if the respective other part is put to zero, as is seen from the above supersymmetry transformations. µν (Fµνσ )αβ = fαβ has helicity +1 and this is an instanton (strictly speaking, only on CPT µν − Euclidean space). The conjugate (Fµνσ¯ )α˙ β˙ = fα˙ β˙ , with helicity 1, is an anti– instanton. One can extend this to N = 3, which is, at the level of the equations of motion, identical to N = 4, cf. the discussion in Sect. 3.2, §5. However, as we have seen, the N = 4 SYM multiplet is its own CPT conjugate and therefore irreducible.

69 §5 Constraint equations. Again, there are constraint equations living on R1,3|16 which imply the superﬁeld equations, and are equivalent to the ordinary ﬁeld equations.

{∇i ∇j } − ij {∇¯ ∇¯ } − α, β = 2ϵαβφ , iα˙ , jβ˙ = 2ϵα˙ β˙ φ¯ij , {∇i ∇ } − i µ ∇ α, jα˙ = 2δjσαα˙ µ

Further superﬁelds whose lowest components are further physical ﬁelds are given by 1 F = [∇ , ∇ ] µν g µ ν ∇ ∇ ββ˙ ∇i ∇ ββ˙ i [ iα, µ] = σµ ϵαβχ¯iβ , [ α˙ , µ] = σµ ϵα˙ β˙ χβ , ∇ i where the supergauge potential A in = ∂ + A has components A = (Aµ,Aα,Aiα˙ ). §6 Remarks.

• N = 4 SYM theory is perturbatively ﬁnite (the β–function vanishes) to all orders, see e.g. [27] and references therein. This is also believed to hold non–perturbatively.

• A stack of N Dp–branes in R1,9 is effectively described by N = 1, d = 10, U(N) SYM theory, reduced to R1,p, the world volume of the brane, via Kaluza–Klein ∼ reduction. Decomposing U(N) = U(1)×SU(N), the U(1) of the group corresponds to the center of mass motion of the branes, the SU(N) describes strings stretching from the ith to the jth brane, and the scalar fields describe transverse fluctuations of the branes. For example, a stack of D3–branes is described by N = 4 SYM in d = 4. The six scalar fields describe the fluctuations of the branes in the six dimensions transverse to the brane’s 3+1 dimensional world volume.

• N = 4 SYM is classically conformally invariant, i.e. all couplings are dimensionless, and, as the beta function vanishes, this is also true quantum mechanically (at least to all orders in perturbation theory). The 16 supercharges we had combine with 16 superconformal transformations, and in total the symmetry group becomes PSU(2, 2|4) (the SUSY extension of the conformal group on R1,3, SO(4, 2), for ∼ ∼ SU(2, 2) = SO(4, 2) and SO(6) = SU(4)):   SU(2, 2) 8 SUSY,    8 superconformal    , (214)  8 SUSY,  8 superconformal SU(4)

which is 8 by 8.

5 • The bosonic part of this is also the symmetry group of AdS5 × S . This is why 5 N = 4 SYM is connected to AdS5 × S – they have the same symmetries. IIB 5 string theory on AdS5 × S with background ﬁelds corresponding to N D3–branes is equivalent to the SYM theory on these branes (strongest form of the conjecture).

70 • This symmetry group is also the symmetry group of CP3|4 (recall that CPd has isometry group SU(d + 1), the signature and the P appearing in PSU(2, 2|4) is a mere technicality). This space is a CY supermanifold, so one can put topological strings on it. It is also a twistor space, so that we can use it in the description of a 4d theory [28].

• Other dimensional reductions are given by the chains 10d, N = 1 → 6d, N = 2 → 4d, N = 4 → 3d, N = 8 → 2d N = (8, 8) 6d, N = 1 → 4d, N = 2 → 3d, N = 4 → 2d, N = (4, 4) 4d, N = 1 → 3d, N = 2 → 2d, N = (2, 2)

71 8 Seiberg-Witten Theory

In this section, we present a sketch of the construction of the exact eﬀective action of N = 2 SYM theory as obtained by Seiberg and Witten [29]. For simplicity, we restrict ourselves to the minimal theory without additional ﬂavours and gauge group SU(2). For a more detailed presentation of this material, see e.g. [7, 30, 31, 32].

8.1 The moduli space of pure N = 2 SYM theory §1 N = 2 superspace. Interestingly, N = 2 SYM theory is one of the few examples of a theory with extended supersymmetry which admits a manifestly supersymmetric formulation in extended superspace16. Consider the superspace R1,3|8 with coordinates xµ, θα, ϑα. On this space, we introduce the chiral coordinates

y˜µ ≡ xµ + iθσµθ¯ + iϑσµϑ¯ (215) ≡ yµ + iϑσµϑ.¯

An N = 2 chiral superfield Ψ is defined by the constraints D¯iαΨ = 0 as well as iα j ¯ ¯ α˙ † D DαΨ = Diα˙ Dj Ψ , where in both expressions i, j denote either θ or ϑ. Such a field has the following expansion in terms of components: √ α α Ψ(˜y) = Φ(˜y, θ) + 2ϑ Wα(˜y, θ) + ϑ ϑαG(˜y, θ) , (216)

1,3|4 where Φ, Wα and G are chiral superfields on our previous superspace R described µ by the coordinates x , θ. Furthermore, Wα is the usual chiral field strength of a real superfield V . The chirality constraints on Ψ imply that Z 1 µ µ ¯ ¯ µ µ ¯ ¯ G(˜y, θ) = − d2θ¯ e−gV (˜y −iθσ θ,θ,θ)Φ†(˜yµ − iθσµθ,¯ θ) egV (˜y −iθσ θ,θ,θ) . (217) 2 The field content of our theory is thus that of an N = 1 chiral multiplet plus that of an N = 1 vector multiplet, which corresponds to an N = 2 vector supermultiplet. §2 Action. The action should be quadratic in the fields, so we try a quadratic ansatz for the Lagrangian Ψ2. The relevant component of this expression is

2 α Ψ |(θ)2(ϑ)2 = W Wα|(θ)2 + 2GΦ|(θ)2 , (218) and this is indeed the desired expression. The full action is · Z ¸ τ 1 S = ℑ d4xd2θd2ϑ Tr(Ψ2) . (219) 8π 2

Θ 4πi Here, we have introduced the complexiﬁed coupling constant τ = 2π + g2 . While g µν is the ordinary Yang-Mills coupling and corresponds to the term F ∧ ∗F or FµνF in

16See, e.g. [8] for more details on this formulation in N = 2 superspace.

72 µνσρ the action, Θ is responsible for the topological term F ∧ F or FµνFσρε . In N = 1 notation, the action becomes, noting that e−gV Φ†egV Φ is real, · Z µ Z Z ¶¸ τ S = ℑ Tr d4x d2θ W αW + d2θd2θ¯ e−gV Φ†egV Φ 8π α · Z Z ¸ Z Z (220) τ 1 = ℑ Tr d4x d2θ W αW + Tr d4x d2θd2θ¯ e−gV Φ†egV Φ . 8π α 2g2

For simplicity, we will restrict our discussion in the following to gauge group SU(2). §3 General theory. The most general SUSY action of Ψ is that given above, as renormalisability forbids higher terms. We will be interested in the Wilsonian eﬀective action of this theory, so we can neglect the condition of renormalisability. Then one can show that the most general action takes the form · Z ¸ 1 S = ℑ Tr d4xd2θd2ϑ F(Ψ) , (221) 8π where F, the so-called N = 2 prepotential, depends only on Ψ and not on Ψ†. Introduc- ing the derivatives

∂F(Φ) ∂2F(Φ) F (Φ) = and F (Φ) = , (222) a ∂Φa ab ∂Φa∂Φb a where a runs over the colour index, i.e. Φ ≡ Φ σa, we ﬁnd that · Z µZ Z ¶¸ 1 S = ℑ Tr d4x d2θ F (Φ)W αaW b + d2θd2θ¯ (e−gV Φ†egV )aF (Φ) . (223) 8π ab α a

F 1 2 Note that we recover the action (220) by putting (Φ) = 2 τTr(Φ ). §4 Flat directions and the moduli space. The potential of the complex scalar field 1 † 2 ϕ appearing in the chiral superfield Φ is V (ϕ) = 2 tr([ϕ , ϕ] ). This potential has flat directions, i.e. continues paths in ϕ-space along which V (ϕ) = 0. This is a typical feature of gauge theories coupled to matter. Note that we can always fix a gauge such 1 → − that ϕ = 2 aσ3. The remaining gauge transformations can still change a a, however, the quantity u = Tr(ϕ2) is indeed fully gauge invariant. For this reason, u is a good coordinate on the moduli space of the theory: the space of vacuum configurations modulo the action of gauge transformations. In the quantum case, we will have to use u = ⟨Trϕ2⟩ ⟨ ⟩ 1 and ϕ = 2 aσ3 instead. For a ≠ 0, the SU(2) gauge symmetry is broken down to U(1) as usual in the Higgs mechanism. For gauge group U(1), the action (223) simplifies to · Z Z Z Z ¸ 1 S = ℑ d4x d2θ F ′′(Φ)W αW + d4x d2θd2θ¯ Φ†egV F ′(Φ) . (224) 8π α

73 8.2 Duality §1 Electromagnetic duality. Dirac observed that the source free Maxwell equations,

µν ∇µF = 0 ⇔ ∇ ∗ F = 0 and ∇[µFρσ] = 0 ⇔ ∇F = 0 , (225)

˜ 1 σρ are symmetric under an exchange of Fµν and Fµν := 2 εµνσρF or, written in diﬀerential forms, F and ∗F . To preserve this symmetry when sources for the electric charge are present, one needs to introduce sources for the magnetic charge, as well. These are the so-called magnetic monopoles and they are the partners of electrons under the above exchange. One can show that there is a charge quantisation condition q = 2πn , n ∈ Z m qe (cf. e.g. [11]) where qe and qm are the magnetic and the electric charge, respectively. As the charges characterise the strength of the coupling, the EM duality, i.e. the exchange of F with ∗F , relates strongly coupled theories to weakly coupled ones and vice versa. By using such dualities, one can apply perturbation theory indirectly to strongly coupled theories. §2 Duality for N = 2 SYM. The duality transformation we are interested in is the Legendre transformation

′ ΦD = F (Φ) , FD(ΦD) = F(Φ) − ΦΦD , (226)

F ′ − which implies D(ΦD) = Φ. Let us examine the behaviour of the action (224) under this transformation. For the part containing the chiral superfield, we have Z Z Z Z ¡ ¢ 4 2 2 ¯ † gV ′ 4 2 2 ¯ ′ † gV ℑ d x d θd θ Φ e F (Φ) = ℑ d x d θd θ − F (ΦD) e ΦD Z Z (227) ℑ 4 2 2 ¯ † gV F ′ = d x d θd θ ΦDe D(ΦD) , where we have used ℑ(a¯b) = −ℑ(¯ab) and the fact that V is real. We see that this part of the action is invariant under the duality transformation. To see the invariance of the gauge kinetic terms under the duality, we have to do a little more work. Consider this part of the action under the path integral: Z Z Z i DV exp ℑ d4x d2θ F ′′(Φ)W αW . (228) 8π α It is possible to promote W α from a field strength to an arbitrary field, provided it satisfies the condition α ℑ(D Wα) = 0, (229) which is the analogue of the Bianchi identity which needs to be imposed if one trades a vector potential Aµ for the field strength Fµν, and is equivalent to the condition − g ¯ 2 Wα = 4 D DαV for some real superfield V .

74 We include the condition (229) by introducing the Lagrange multiplier superﬁeld VD, so that (228) becomes Z · Z Z ¸ Z Z i i DW DV exp ℑ d4x d2θ F ′′(Φ)W αW + d4x d2θd2θ¯ V ℑ(DαW ) . D 8π α 16π D α We now need the following result:

Exercise 26. Show thatZ Z 1 d2θd2θ¯ V DαW = − d2θ W αW . (230) D α g D α

Using this, we integrate over W and arrive at the expression Z Z µ ¶ i 1 DV exp ℑ d4xd2θ − W αW . (231) D 16π F ′′(Φ) D Dα F ′′ − F ′′ Finally, from (226), it can be checked that D(ΦD) = 1/ (Φ), and so the action (224) is indeed invariant under the duality transformation Φ → ΦD and V → VD. §3 Extended duality. We rewrite the action (224) as · Z ¸ Z 1 dΦ 1 † S = ℑ d4xd2θ D W αW + d4xd2θd2θ¯ egV (Φ†Φ − Φ Φ), (232) 8π dΦ α 4g2 D D where we have explicitly evaluated the imaginary part for the second term. The unusual coupling in front of the term containing the matter fields can be absorbed by a field rescaling. From this expression, it is easy to notice that the transformation Ã ! Ã !Ã ! Φ 1 b Φ D → D (233) Φ 0 1 Φ leavesR the second term in the action invariant, while the first changes by an amount − b 4 ˜µν − ∈ Z ∈ Z 4π d xFµνF = 2πbν, ν . Thus, if b , the path integral remains invariant. This is an additional duality transformation leaving the action (224) invariant.

8.3 The exact effective action §1 Form of the effective action. Here, we have to be a bit sketchy and quote a F 1 2 result from the literature: recall that classically, we had class(Ψ) = 2 τclassTr(Ψ ). At the quantum level, the effective action at 1-loop was calculating in [33]. Due to nonrenormalization arguments, there are no perturbative contributions beyond 1-loop. Its non-perturbative structure was also derived there, with the result being µ ¶ 2 X∞ 4k i 2 Ψ 2 Λ FQM (Ψ) = Ψ ln + Ψ Fk , (234) |2π {z Λ2} Ψ | k=1 {z } 1-loop exact result instanton corrections with the Fk to be determined. The perturbative part is obtained from U(1)R–symmetry considerations, as applied to the non-renormalisation theorem in Sect. 6.1 §3.

75 §2 Outline of the computation. In the following, we will determine the coeﬃcients

Fk in (234) analytically. This will be done by employing duality on quantum level. Thus, we have to switch from classical field configurations Φ ↔ ΦD to the expectation 2 values a(u) ↔ aD(u), where u = ⟨Tr(Φ )⟩ is again the coordinate on the quantum ′ ′ moduli space. Note that ΦD = F (Φ) goes over into aD(u) = F (a(u)). Thus, by finding a(u) and aD(u), we can solve the first relation for u and plug this into the second relation, which we can integrate to obtain F(a). This function then is the exact N = 2 prepotential for the effective theory.

The functions a(u) and aD(u) turn out to be multivalued functions living on multiple Riemann sheets. Following such a function on a continuous path in the complex plane, we will encounter transitions from one Riemann sheet to another, if we cross a branch cut. That is, the functions have non-trivial monodromies, and knowing these monodromies is sufficient for setting up a differential equation determining a(u) and aD(u) completely. §3 Monodromy at infinity. At large |a(u)|, the theory is asymptotically free: τ(a) = F ′′ ∼ 1 → → ∞ (a) g , and with the expression (234), we see indeed that g 0 as a . In ∼ 1 2 this limit, we have also u(a) 2 a . In this regime, we can neglect all non-perturbative contributions, and we have i a2 F(a) = a2 ln . (235) 2π Λ2 From our duality transformation, we have the following expression for the dual of a: µ ¶ ∂F ia a2 a (u) = = ln + 1 . (236) D ∂a π Λ2 Consider now following a circle with very large radius in the moduli space parameterised by u: u → e2πiu. This amounts to a → eπia, and17 µ ¶ i e2πia2 a → (−a) ln + 1 = −a + 2a (237) D π Λ2 D in the dual description. We thus find a monodromy of the form Ã ! Ã !Ã ! a −1 2 a D → D . (238) a 0 −1 a

§4 Further monodromies. Other monodromies require singularities, which appear from integrating out massless modes. As an ansatz (which can be justiﬁed a posteriori), we assume that this happens at exactly two points u = u0 and u = −u0. Consistency of the monodromies then requires that M−u0 Mu0 = M∞. Near the point Mu0 , one can show by considering the 1–loop β–function that i a = a + c (u − u ) ln(u − u ) , (239) 0 π 0 0 0 17The following line is written in the usual slightly sloppy way. Meant is the following: The logarithm of the product has to be split into a sum and taken, the factor e2πi cannot be simpliﬁed to 1 here.

76 from which we read oﬀ the monodromy Mu0 to be Ã ! Ã !Ã ! a 1 0 a D → D . (240) a −2 1 a

− Because of the consistency condition M−u0 Mu0 = M∞, we also have at u0: Ã ! Ã !Ã ! a −1 2 a D → D . (241) a 2 3 a

A more thorough analysis shows that at u0 and −u0 monopoles and dyons (combinations of monopoles and electrons) become massless. One can also show that this choice of singularities is correct and unique.

§5 The solution. We need to find functions aD(u) and a(u) with the given nontrivial monodromies around ±u0 and ∞. There is a nice map of this problem to differential equations: consider the Schrödingerequation µ ¶ d2 − + V (z) ψ(z) = 0 . (242) dz2

This diﬀerential equation has two linearly independent solutions satisfying the conditions Ã ! Ã ! ψ1 2πi ψ1 (z + e (z − zi)) = Mi (z) , (243) ψ2 ψ2 with singularities at zi and corresponding monodromies Mi. Choosing the potential − 1 1 V (z) = 4 (z+1)(z−1) , one obtains the solutions with the desired monodromies: µ ¶ u − 1 1 1 1 − u a (u) = iψ (u) = i F , , 2; , D 2 2 2 2 2 µ ¶ (244) √ 1 1 2 a(u) = −2iψ (u) = 2(u + 1)1/2F − , , 1; . 1 2 2 u + 1

If one now inverts the last relation to obtain u(a), plugs this into aD(u) and integrates the resulting function aD(a), one obtains F(a), which gives the exact result. §6 Final remarks. Although the presentation of the derivation of the Seiberg–Witten solution was rather sketchy, the reader is now hopefully convinced that SUSY theories, and in particular SUSY gauge theories have indeed very nice properties. Compactifying the complex plane by adding a point at ∞, multivalued functions turn out to live on a Riemann surface. In our case, the Riemann surface arising from compactifying the moduli space is the torus. It has become a successful strategy to look for string theoretic meaning of any geometry appearing in the discussion of ﬁeld theory. The torus here naturally appears in the D-brane engineering of the N = 2 SYM theory we started from.

77 Appendices

A Conventions and identities

The following details our various conventions and useful relations. All of the relations given here are introduced as and when they are needed in the text, and exercises are provided against which the reader may check his/her understanding of the material. Further useful identities are found in [1], appendices A and B.

Metric

µν Our metric tensor is ηµν = η = diag(−1, +1, +1, +1). As a result, the mass–shell 2 2 µ condition is p + m = 0. We write ¤ ≡ ∂ ∂µ.

Epsilon tensors

The epsilon tensors, which raise and lower spinor indices, are Ã ! 0 −1 ϵ = ϵ = , (1) αβ α˙ β˙ 1 0 Ã ! ˙ 0 1 ϵαβ = ϵα˙ β = . (2) −1 0

Spinor contractions

Spinors anticommute. Spinors in the Weyl representation are naturally contracted ‘in the ↘ direction’,

α αβ βα βα β ψχ ≡ ψ χα = ϵ ψβχα = −ϵ ψβχα = +ϵ χαψβ = χ ψβ = χψ . (3)

Spinors in the conjugate Weyl representation are naturally contracted in the ↗ direction,

¯ ≡ ¯ α˙ ¯β˙ α˙ − ¯β˙ α˙ α˙ ¯β˙ ¯β˙ ¯ ψχ¯ ψα˙ χ¯ = ϵα˙ β˙ ψ χ¯ = ϵβ˙α˙ ψ χ¯ = +ϵβ˙α˙ χ¯ ψ =χ ¯β˙ ψ =χ ¯ψ . (4)

Fixing the directions of these contractions means that the inner products are symmetric. The products of spinor components are proportional to the ϵ tensors,

1 ˙ θαθβ = − 1 θ2ϵαβ, θ¯α˙ θ¯β˙ = + θ¯2ϵα˙ β , (5) 2 2 1 θ θ = 1 θ2ϵ , θ¯ θ¯ = − θ¯2ϵ . (6) α β 2 αβ α˙ β˙ 2 α˙ β˙ We sometimes abbreviate θθ ≡ θ2, θ¯θ¯ = θ2 and θθ θ¯θ¯ = θ4. A useful result is

δ4(θ) ≡ θ2θ¯2 . (7)

78 Pauli matrices

The Pauli sigma matrices are deﬁned with lower Lorentz index, Ã ! Ã ! Ã ! Ã ! 1 0 0 1 0 −i 1 0 σ = , σ = σ = σ = . (8) 0 0 1 1 1 0 2 i 0 3 0 −1

Their natural spinor indices are σαα˙ . The barred sigma matrices, with their natural spinor indices are ˙ σ¯µ αα˙ := ϵα˙ βϵαβσµ . (9) ββ˙ Explicitly

σ¯µ = (σ0, −σ1, −σ2, −σ3) . (10)

Dirac matrices

We use the Weyl representation for Dirac matrices, Ã ! 0 σµ γµ = , (11) σ¯µ 0 which yields the following explicit form: (Ã ! Ã ! Ã ! Ã !) −1 0 0 1 0 −i 1 0 µ 0 0 −1 0 1 0 0 i 0 0 0 −1 γ = −1 0 , 0 −1 , 0 i , −1 0 . (12) 0 −1 0 −1 0 0 −i 0 0 0 1 0

The Dirac matrices obey the anticommutation relations (Cliﬀord algebra)

{γµ, γν} = −2ηµν . (13)

Relations between Pauli matrices

1 θσµθ¯ θσνθ¯ = − θ2θ¯2ηµν , (14) 2 µ ¶ 1 (σµν) β ≡ σµ σ¯ν αβ˙ − σν σ¯µ αβ˙ , α 4 αα˙ αα˙ µ ¶ (15) 1 (¯σµν)α˙ ≡ σ¯µ αα˙ σν − σ¯ν αα˙ σµ , β˙ 4 αβ˙ αβ˙ ¡ ¢ µ ν ν µ β − µν β σ σ¯ + σ σ¯ = 2η δα , ¡ ¢α (16) σ¯µσν +σ ¯νσµ α˙ = −2ηµνδα˙ . β˙ β˙ From the last relations, we easily read oﬀ that

µ ν µ ναα˙ − µν Tr(σ σ¯ ) = σαα˙ σ¯ = 2η (17)

79 D–algebra

In quantum computations using supergraphs, one has to reshuﬄe the supersymmetric covariant derivatives. In the text we needed the following formulæ

4 4 Dα(p, θ1)δ (θ1 − θ2) = −Dα(−p, θ2)δ (θ1 − θ2) , (18) ¯ ¯ 2 2 4 ¯ D¯ D δ (θ − ϑ)¯ = 16 . (19) θ=ϑ D¯ 2D2D¯ 2 = 16D¯ 2¤ ,D2D¯ 2D2 = 16D2¤ . (20)

2 2 2 2 α 2 D¯ D + D D¯ − 2D D¯ Dα = 16¤ . (21) For a chiral sﬁeld Φ: α˙ α D¯α˙ D¯ DαD Φ = 16¤Φ . (22) More helpful identities are summarized in [3], section 2.5.6 and [4], eqs. (6.3.28).

80 B Solutions to exercises

Solution 1. Directly from the given Q and Q† we have Ã ! 1 (P + iW ′)(P − iW ′) 0 H = 2 0 (P − iW ′)(P + iW ′) Ã ! 1 i 1 0 = (P 2 + W ′2)1 − [P,W ′] 2 2 0 −1 1 ~ = (P 2 + W ′2)1 − W ′′ σ . 2 2 3

Solution 2. The complete expression must be antisymmetric under interchange of both (µ, ν) and (σ, ρ). First antisymmetrise over (µ, ν), to generate one additional term (the second on the right hand side, below), then antisymmetrise this expression over (σ, ρ) to generate the ﬁnal two terms: ¡ ¢ [Mµν,Mρσ] = −i ηµρMνσ − ηνρMµσ − ηµσMνρ + ηνσMµρ .

Solution 3. First, recall that it follows from preservation of the norm that the determinant of any Lorentz transformation (matrix) is ±1, since

ρ σ T ηµν = L µgρσL ν ⇐⇒ 14 = L L.

Now, suppose we have a path L(s) : [0, 1] → O(1, 3) through O(1, 3), such that L(0) ∈ SO(1, 3) but L(1) ̸∈ SO(1, 3). Then we have

det L(0) = 1 , det L(1) = −1 .

However, det L(s) = ±1 ∀s so the path L(s) cannot be continuous. This establishes the disconnection of the component SO(1, 3). Within SO(1, 3) we ↑ ↓ have the two components L+ and L−. Again suppose we have a map L(s) such that, for example, the (0, 0) components obey L(0) = 1, L(0) = −1. Consider the vector xµ = (1, 0, 0, 0). Preservation of the norm of this vector reads ¢ 0 2 0 (L 0(s) = 1 =⇒ L 0(s) = ±1 , so again the map cannot be smooth.

Solution 4. We will ﬁrst show that SL(2, C) is path connected. Deﬁning arbitrary matrix powers via the exponential map, i.e. M t := exp(t log M), consider the matrix Z(t) = M tN 1−t. This is a 2 × 2 complex matrix such that Z(0) = N and Z(1) = M, with determinant det Z(t) = det(M)t det(N)1−t = 1 .

81 Therefore, Z(t) ∈ SL(2, C)∀t, and so SL(2, C) is path connected.

Now consider any two such paths Z1(t) and Z2(t). Using the same idea, for any ∈ s 1−s C s [0, 1], Z1(t)Z2 (t) is also in SL(2, ) and is a path between N and M, so the two paths Z and W are homotopic. Thus, SL(2, C) space is path connected and any two paths between given endpoints are homotopically equivalent. Together these conditions imply the space is simply connected. Solution 5. ρ(L) is a 2×2 matrix of determinant one, and is therefore invertible. Write the condition of Lorentz invariance as Mρ−1(L) = ρT (L) M. (23) Writing out the equations for the four components, we conclude first that (since ρ is essentially arbitrary) the diagonal elements of M vanish, and then that the off-diagonal elements obey M12 = −M21. The condition det M = 1 then implies M12 = ±1. Solution 6. Starting from (54), we can write ˙ σ¯ = −ϵα˙ βσT ϵβα , µ αα˙ µ ββ˙ where the minus sign comes from flipping the indices on the second ϵ, and we have flipped the indices on the Pauli matrix and written T for transpose. The reason is that all the indices are now in the right place to write this as a matrix equation, Ã ! Ã ! 0 1 0 1 σ¯ = − σT . µ −1 0 µ −1 0 The desired result follows from simply inserting the Pauli matrices and checking each of the four cases. ± −1 ∓ −1 3 Solution 7. Let’s begin with ΛP = iγ0, so that ΛP = iγ0. Then, ΛP γ0ΛP = γ0 = 2 γ0, since γ0 = 1. Similarly, −1 − 2 − ΛP γjΛP = γ0γjγ0 = γ0 γj = γj , and so ±γ0 gives us the parity operation as claimed. −1 Now, for ΛT = γ1γ2γ3, we find ΛT = −γ3γ2γ1. We then have −1 ΛT γ0ΛT = γ1γ2γ3γ3γ2γ1 γ0 , passing γ0 to the right, 3 = (−1) γ0 = −γ0 . 2 − The second line follows from γj = 1. Finally, taking γ3 to illustrate, we calculate −1 − 3 ΛT γ3ΛT = γ1γ2γ3 γ2γ1

= γ1γ2γ3γ2γ1

= γ1γ2γ2γ1 γ3 2 = (−1) γ3 = γ3 .

So, the posited ΛT does indeed generate time reversal.

82 Solution 8. The result of Exercise 6 comes in very useful here. Noting that all of the indices are in the right place to be contracted (i.e. we just have matrices), we ﬁnd

i [M ,Qi ] = (σ σ¯ − σ σ¯ ).Qi using the deﬁnition of J and the commutator (72) 12 α 4 1 2 2 1 3 i = − (σ σ − σ σ ).Qi from Exercise 6 4 1 2 2 1 2i2 = − ϵ σ .Qi 4 12k k 1 = (Qi − Qi ) 2 1 2 1 = − Qi . 2 2 (24)

The third and fourth line use the standard commutator of the Pauli matrices, while the i ﬁnal line uses Q1 = 0 on our states. Taking the hermitian conjugate of this result immediately gives the ‘raising’ commutator.

Solution 9.

α α β 1. By deﬁnition, ψχ ≡ ψ χα = ψ ϵαβχ , now transpose the epsilon tensor and swap β α the spinors, which introduces two minus signs. We then have ψχ = χ ϵβαψ = β χ ψβ = χψ.

2. Introducing epsilon tensors to raise and lower both indices, we have α αβ γ ψ χα ≡ ϵ ψβϵαγχ . Now transpose one of the epsilon’s at the cost of a minus α βα γ β γ α sign, ﬁnding ψ χα = −ϵ ϵαγψβχ = −δγ ψβχ = −ψαχ .

3. We drop the Lorentz index to simplify notation, as it plays no role here. χσψ¯ ≡ α α˙ χ σαα˙ ψ¯ . Now raise and lower the spinor indices with epsilon tensors,

¯ αβ α˙ β˙ ¯ χσψ = ϵ χβσαα˙ ϵ ψβ˙ , ¡ ¢ β˙α˙ βα = χβ ϵ ϵ σαα˙ χβ˙ , transposing ϵ indices to convert σ into σ¯ , ββ˙ ¯ = χβσ¯ ψβ˙ , − ¯ ββ˙ = ψβ˙ σ¯ χβ swapping the spinors , ≡ −ψ¯σχ¯ .

4. Directly, (χσψ¯)† = ψσ†χ† ≡ ψσχ¯.

83 β˙α˙ Solution 10. For the ﬁrst part, hit the transformation of ψ¯α˙ with ϵ , √ √ β˙ β˙α˙ α µ β˙ α αγ γα δεψ¯ = −i 2 ϵ ε σ ∂µφ¯ + 2ε ¯ F,¯ now write ε as εγϵ = −εγϵ √ αα˙ √ β˙α˙ γα µ β˙ = +i 2 ϵ εγϵ σ ∂µφ¯ + 2ε ¯ F,¯ note the index ordering, hence the sign, √ αα˙ √ µβγ˙ β˙ = +i 2σ ¯ εγ∂µφ¯ + 2ε ¯ F,¯ as required, using the deﬁnition of σ¯.

For the second part, we can compactly write the transformation of F¯ as √ √ µ µ δεF¯ = −i 2 ∂µψ¯ σ¯ ε = +i 2 εσ ∂µψ¯ , where the second inequality follow from the result of Exercise 9.3.

Solution 11. We will label the terms of the Lagrangian as follows,

2 µ = φ∂ φ , = −iψ¯σ¯ ∂µψ , = FF.¯

Making a SUSY transformation on each term, we ﬁnd (note, we may have integrated the right hand sides by parts) √ √ δ = 2ξ¯ψ∂¯ 2φ + 2ξψ∂2φ¯ ξ √ √ √ √ µ ν ν µ ν µ δ = + 2ξσ σ¯ ψ∂ν∂µφ¯ − i 2F¯ ξ¯σ¯ ∂νψ + 2ψ¯σ¯ σ ξ∂¯ µ∂νφ − i 2ψ¯σ¯ ξ∂µF , ξ √ √ ¯ µ ν ¯ δξ = i 2F¯ ξσ¯ ∂µψ + i 2F ξσ ∂νψ .

We will deal first with the terms underlined above (call those in ,“”). The first term in cancels with the first term of . Integrating by parts, what remains is we then have √ √ ¯ µ ν ¯ δξ + δξ = −i 2ψσ¯ ξ∂µF − i 2ξσ ψ∂νF.

The right hand side vanishes upon applying the identity ψσµχ¯ = −χ¯σψ¯ from Exercise 13. It remains to cancel the remaining terms of with . Using the symmetry of the derivatives ∂µ∂νφ, we can write 1 ¡ ¢ ξσµσ¯νψ∂ ∂ φ¯ = ξ σµσ¯ν + σνσ¯µ ψ∂ ∂ φ¯ ν µ 2 µ ν = −ξψ∂2φ¯ , where the second line uses the identity given in the Exercise. So, the remaining terms of contribute √ √ 2 ¯¯ 2 − 2ξψ ∂ φ¯ − 2ψξ∂ φ = −δξ , and so the remaining terms vanish. The action is therefore SUSY invariant.

84 Solution 12. Note that the generators (combined with their parameters) have parity 0, so we don’t need to worry about supercommutators. We identify

µ iA = ξQ + ξ¯Q¯ , iB = x Pµ + θQ + θ¯Q,¯ and calculate the commutator

2 µ µ [A, B] = (−i) (2ξσ θP¯ µ − 2θσ ξP¯ µ) , which clearly commutes with both A and B, meaning we can drop all higher order terms in the BCH formula. The result is ¡ ¡ ¢ ¢ µ µ µ exp A exp B = exp − i x + iθσ ξ¯− iξσ θ¯ Pµ − i(ξ + θ)Q − i(ξ¯+ θ¯)Q¯ , which is the desired result.

Solution 13. Recall that the SUSY transform is ∂ ∂ δ = ξαQ + ξ¯ Q¯α˙ = ξαQ − ξ¯α˙ Q¯ = ξα + ξ¯α˙ + ..., ξ α α˙ α α˙ ∂θα ∂θ¯α˙ where we omit terms which do not contribute to δξf(x). The only components of the ¯ superﬁeld we need consider are θϕ(x) and θχ¯(x) as, after acting with δξ, all other terms will die at θ = θ¯ = 0. We then have ¯ ¯ ∂ ˙ δ f ≡ δ F ¯ = ξαϕ +χ ¯ ξ¯α˙ θ¯β ξ ξ ¯ α β˙ ¯α˙ θ=θ¯=0 ∂θ = ξϕ +χ ¯ξ¯ ≡ ξϕ + ξ¯χ¯ .

α β − 1 2 αβ Solution 14. Let us prove the required identities ﬁrst. We start with θ θ = 2 θ ϵ . This clearly holds if the indices are equal, so we consider 1 1 1 − θθϵ12 = − (θ1θ + θ2θ ) = − (θ1(−1)θ2 + θ2(+1)θ1) = θ1θ2 . 2 2 1 2 2

The second case follows. A similarly uncivilised proof by exhaustion conﬁrms that θ¯α˙ θ¯β˙ = 1 ¯2 α˙ β˙ + 2 θ ϵ . With these results in hand, we can show that

˙ θσµθ¯ θσνθ¯ ≡ −θαθβσµ σν θ¯α˙ θ¯β moving a θ to the left αα˙ ββ˙ 1 ˙ = + θ2θ¯2ϵαβϵα˙ β σµ σν using the above results 4 αα˙ ββ˙ 1 = + θ2θ¯2 σµ σ¯ν αα˙ converting σ to σ¯ 4 αα˙ 1 = θ2θ¯2 Tr(σµσ¯ν) . 4 In arriving at the ﬁnal line, we have noted that the indices on the σ’s are in the right places for them to be treated as matrices. We now have to evaluate the trace.

85 If µ ≠ ν, the trace vanishes. If µ = ν = 0, we have Tr(σ0σ0) = 2, while if µ = ν = j we have Tr((σj)(−σj)) = −2. Hence, 1 θσµθ¯ θσνθ¯ = − θ2θ¯2ηµν . 2 The component expansion of the chiral superﬁeld now follows from applying Taylor’s theorem and applying these results. Explicitly, we ﬁrst expand around x,

µ 1 µ ν Φ(y, θ) = φ(x) + iθσ θ∂¯ µφ(x) − (θσ θ¯)(θσ θ¯)∂µ∂νφ(x) √ ¡ 2 ¢ α µ + 2θ ψα(x) + iθσ θ∂¯ µψα(x) + θθ F (x) .

All other terms vanish because they contain more than two θ’s. The third line is complete, the ﬁrst line can be simpliﬁed immediately using our identities. For the second line, we have

˙ θα θσµθ∂¯ ψ ≡ θαθβ σµ θ¯β∂ ψ µ ββ˙ µ α 1 ˙ = − θθ ϵαβσµ θ¯β∂ ψ using the above results 2 ββ˙ µ α 1 ˙ = − θθ ϵβα∂ ψ σµ θ¯β 2 µ α ββ˙ 1 ≡ − ∂ ψσµθ¯ . 2 µ In the third line, we have pushed a ψ to the left and transposed the ϵ tensor, at the cost of two minus signs. In the ﬁnal line, we have used ϵ to raise the index on ψ, and contracted. Putting this all together, we ﬁnally have 1 Φ(y, θ) = φ(x) + iθσµθ∂¯ φ(x) + θθ θ¯θ¯ ∂2φ(x) µ 4 √ i µ + 2θψ(x) − √ θθ ∂µψ(x)σ θ¯ 2 + θθ F (x) .

√ Solution 15. Write the chiral superﬁeld as follows, Φ = φ + { 2θψ + θ2F }, where the braces simply indicate that we will think of φ as term x and the rest as term y. Now, powers of y higher than two must vanish, since they will contain products of at least three θ’s, which all vanish by antisymmetry. Hence, contributions to θ2 terms can only come from xN−1× y and xN−2× y2. From the binomial expansion we then

86 have ¯ ¯ ¯ √ √ ¯ N ¯ N−1 2 1 N−2 2 2¯ Φ ¯ = Nφ ( 2θψ + θ F ) + N(N − 1)φ ( 2θψ + θ F ) ¯ 2 θ2 ¯ θ2 £ ¤¯ N−1 1 N−2 α β ¯ = Nφ θθ F + N(N − 1)φ 2θ ψαθ ψβ ¯ 2 θ2 1 £ ¤ = NφN−1F + N(N − 1)φN−2 ϵαβψ ψ 2 α β 1 = NφN−1F − N(N − 1)φN−2ψαψ . 2 α Solution 16. By inspection of (125) and (126), we ﬁnd M iχ φ +φ ¯ = −C,F = − , ψ = −√ . 2 2 Solution 17. Under the gauge transformation we have g W → W − D¯ 2D Λ , α α 4 α where the term in Λ¯ dies because Λ¯ is antichiral. The idea is now to kill the term in Λ, ¯ ¯ 2 by using the property that it is chiral. We extract one of the Dβ˙ from D , writing g ˙ W → W + D¯ β{D¯ ,D } Λ , α α 4 β˙ α where the minus sign comes from a choice of index ordering, and the anticommutator appears because the term like DD¯Λ is zero. We then use the algebra of the D’s to see that ig ˙ W → W − D¯ βσµ ∂ Λ α α 2 αβ˙ µ ig ˙ = W − σµ ∂ D¯ β Λ commuting a D¯ past ∂ α 2 αβ˙ µ µ = Wα since Λ is chiral.

Solution 18. Applying (141) to (140), · ¸ ¡ ¢ 1 2 −gΛ −gV −gΛ¯ gΛ¯ gV gΛ Wα → − D¯ e e e Dα e e e 4 · ¸ ¡ ¢ 1 −gΛ 2 −gV gV gΛ = e D¯ e Dα e e using DΛ¯ = D¯Λ = 0 4 · ¸ g ¡ = e−gΛD¯ 2 e−gV D egV )egΛ + D egΛ 4 α α 1 = e−gΛW egΛ − e−gΛD¯ 2D egΛ pushing the final egΛ to the right, first term α 4 α −gΛ gΛ = e Wαe . To obtain the final line, the second term is killed using the same argument as for Solu- tion 17.

87 Solution 19. From (93), we see that the SUSY variation of φ sees only ϵ, not ϵ¯, and so we conclude that φ is not transformed by Q¯,

[¯ϵQ,¯ φ] = 0 =⇒ [Q¯α˙ , φ] = 0 .

Similarly, from (94), only one term in δϵψ sees ϵ¯ and therefore Q¯, so √ √ ¯ µ α˙ ⇒ −{ ¯ } α˙ µ α˙ [¯ϵQ, ψβ] = i 2 σβα˙ ϵ¯ ∂µφ = Qα˙ , ψβ ϵ¯ = i 2 σβα˙ ϵ¯ ∂µφ , where we arrive at an anticommutator from passing ϵ¯ to the right, past ψ. These are the desired results. Solution 20. The scalar case is the easiest, so we begin with that. From (112) and (118), the relevant parts of i× the action are Z ¡ iS = − d4x φ¯ ϵ − i(¤ − m2))φ , from which we read oﬀ the propagator 1 i = ≡ i△ . ϵ − i(¤ − m2) ¤ − m2 + iϵ Now for the fermions. The relevant terms of the action are Z Z 1 iS = − d4x − ψ¯σ.∂ψ¯ − ψσ.∂ψ¯ + imψψ + imψ¯ψ¯ + d4x ηψ +η ¯ψ¯ 2 which we have made more symmetric by halving the kinetic term and integrating by parts. We also introduced source terms. Recall the fermionic Gaussian integral formula Z µ Z ¶ ¡ ¢ DΨDΨ¯ exp − Ψ¯ AΨ + ωΨ + Ψ¯¯ ω = Det(A) exp ωA−1ω¯ .

This result is most easily applied by converting everything to Dirac spinors, so that we deﬁne Ã ! Ã ! Ã ! ψ 1 η¯α˙ 1 −σ.∂¯ αβ˙ im δα˙ Ψ := α , ω¯ := A := β˙ . ψ¯α˙ 2 η 2 β − α im δα σ.∂αβ˙ Performing the Gaussian, we arrive at the exponential of Ã !Ã ! Z ³ ´ γ Z 1 σ.∂βγ˙ im δ η¯γ˙ im im ηβ η¯ β = d4x ησ.∂△η¯ + △ηη + △η¯η¯ . 2 β˙ β˙ βγ˙ η 2 2 im δγ˙ σ.∂¯ γ We can now read oﬀ the various propagators:

δ δ µ ⟨ψα(x)ψ¯ ˙ (y)⟩ ∼ − = σ ∂µ△(x − y) , β δηα(x) δη¯β˙ (y) αβ˙ δ δ ⟨ψ (x)ψγ(y)⟩ ∼ ϵγβ = im δγ △(x − y) . α δηα(x) δηβ(y) α

88 Solution 21. The idea is to (anti)commute the D¯’s around so that they hit Φ and are killed. Remembering that the anticommutator {D, D¯} commutes with both D and D¯, we jump straight in: D¯ 2D2Φ ≡ D¯ D¯ α˙ DβD Φ α˙ ¡ β ¢ α˙ β β α˙ = D¯α˙ {D¯ ,D } − D D¯ DβΦ α˙ β β α˙ = {D¯ ,D }{D¯α˙ ,Dβ} − D¯α˙ D {D¯ ,Dβ}Φ = {D¯ α˙ ,Dβ}{D¯ ,D } − {D¯ ,Dβ}{D¯ α˙ ,D }Φ ¡ α˙ β ¢α˙ β − µ αβ˙ ν ν γβ˙ µ = 4 σ¯ σβα˙ +σ ¯ σβγ˙ ∂µ∂νΦ µν = −4Tr 12(−2η )∂µ∂νΦ = 16¤Φ .

Solution 22. Z µ ¶ Z µ ¶ µ ¶ δ D2 D2 1 d8z Φ(z ) − J(z ) = d8z Φ(z ) − − D¯ 2 δ8(z − z ) δJ(z ) 2 2 4¤ 2 2 2 4¤ 4 1 2 1 Z µ ¶ 2 ¯ 2 8 D D 8 − = d z2 ¤ Φ(z2) δ (z1 z2) by parts Z 16 8 8 = d z2 Φ(z2) δ (z1 − z2) using the previous Exercise

= Φ(z1) . ←− 2 1 2 Solution 23. For (171), it is suﬃcient to show that δ (θ − θ )D α(p, 1) = 2 1 2 −Dα(−p, 2)δ (θ − θ ): ←− ¡ ¢←− 1 2 β 1 2 1 2 1 2 2 2 (θ − θ ) (θ − θ )β D α(p, 1) = (θ ) − 2θ θ + (θ ) D α(p, 1) ←− ¡ ¢ ∂ = (θ1)2 − 2θ1θ2 + (θ2)2 ∂θ1α 1 − 2 = 2(θα θα) ¡ ¢ − ∂ 1 2 − 1 2 2 2 = 2α (θ ) 2θ θ + (θ ) ∂θ ¡ ¢ 1 2 1 2 2 2 = −Dα(−p, 2) (θ ) − 2θ θ + (θ ) ←− 4 2 2 4 From this, it immediately follows that δ (θ12)D (−p, 1) = D (p, 2)δ (θ12). For (172) we use the explicit representation δ4(θ) ≡ (θθ)(θ¯θ¯), as so,

2 ¯ 2 4 1 − 2 ∂ ∂ ∂ ∂ 1 − 2 1 − 2 ¯ − ¯ ¯ − ¯ D (p, 1)D (p, 1)δ (θ θ ) = 1 1 (θ θ )(θ θ )(θ1 θ2)(θ1 θ2) . ∂θ ∂θ ∂θ¯1 ∂θ¯1 We clearly need only calculate ¡ ¢ ∂ ∂ β ∂ − β ∂ α θ θβ = θα θ ϵβα = 2 θα = 4 , ∂θα ∂θ ∂θα ∂θα from which it follows that D¯ 2D2δ4(θ) = 16.

89 Solution 24. We begin by showing that D¯ 2D2D¯ 2 = 16D¯ 2¤, by using the algebra of the supersymmetric derivaties and D3 = D¯ 3 = 0. We have

˙ ˙ ˙ D¯ 2DαD D¯ D¯ β = D¯ 2Dα(−2iσµ )∂ D¯ β + D¯ 2D D¯ DαD¯ β α β˙ αβ˙ µ α β˙ ˙ ˙ = D¯ 2(−2iσ¯νβα)(−2iσµ )∂ ∂ + D¯ 2(−2iσµ )(−2iσ¯νβα)∂ ∂ αβ˙ µ ν αβ˙ µ ν 2 µ ν 2 = D¯ (−8Tr(σ σ¯ ))∂µ∂ν = 16D¯ ¤ .

By complex conjugation, we also have D2D¯ 2D2 = 16D2¤. It immediately follows that 2 Π0 = Π0: 1 1 Π2 = (D¯ 2D2D¯ 2D2 + D2D¯ 2D2D¯ 2) = Π . 0 16¤ 16¤ 0 2 To see that Π 1 = Π 1 , note that we have 2 2

2 1 α ¯ 2 β ¯ 2 − 1 α ¯ 2 1 2 ¯ 2 β Π 1 = 2 2 D D DαD D Dβ = 2 2 D D ( 2 D ϵαβ)D D = Π 1 . 2 8 ¤ 8 ¤ 2

So, we have shown that Π0 and Π 1 are projectors. We now show that their sum is the 2 2 unit operator. We begin with Π 1 and push the D¯ to the right. We ﬁnd 2 ¡ ¢ 1 α α˙ α˙ Π 1 = − D D¯α˙ {D¯ ,Dα} − DαD¯ 2 8¤ 1 1 ¡ ¢ = − DαD¯ {D¯ α˙ ,D } + Dα {D¯ ,D } − D D¯ D¯ α˙ 8¤ α˙ α 8¤ α˙ α α α˙ i ˙ 1 = − Dασµ D¯ β∂ − D2D¯ 2 . 2¤ αβ˙ µ 8¤

The point of this operation is that we have generated one of the terms in Π0. Now, we could also have pushed D¯ 2 to the left, in which case we would have found

− i µ ¯ β˙ α − 1 2 ¯ 2 Π 1 = σ ˙ ∂µD D D D . 2 2¤ αβ 8¤

Adding our two expressions for Π 1 together, we ﬁnd 2

− − i µ { α ¯ β˙ } Π 1 = Π0 σ ˙ D , D ∂µ 2 4¤ αβ 1 = −Π − Tr(σµσ¯ν)∂ ∂ 0 2¤ µ ν 1 = −Π − Tr(σµσ¯ν + σνσ¯µ)∂ ∂ 0 4¤ µ ν 1 = −Π − 4¤ = −Π + 1 , 0 4¤ 0 as desired.

90 Solution 25. Remembering that σ0 ≡ 12, the tensor products give us Ã ! Ã ! Ã ! Ã ! 0 i12 0 12 σ1 0 σ2 0 γ0 = , γ1 = , γ2 = i , γ3 = i . i12 0 −12 0 0 −σ1 0 −σ2

It can now be checked explicitly that {γµ, γν} = −2δµν . If we go back to our usual signature, then the anticommutator of the matrices should be −2ηµν = 2diag(+, −, −, −). To obtain this, we simply drop the factor of i in γ0, so that Ã ! Ã ! Ã ! Ã ! 0 12 0 12 σ1 0 σ2 0 γ0 = , γ1 = , γ2 = i , γ3 = i . 12 0 −12 0 0 −σ1 0 −σ2

Solution 26. Z Z 2 2 ¯ α 2 2 ¯ α d θd θ VDD Wα = − d θd θ D VDWα by parts Z Z ¡ 2 1 2 α 4 = − d θ − D¯ D VDWα) since we use the result under d x Z µ 4 ¶ 2 1 2 α = − d θ − D¯ D VD Wα since Wα is chiral Z 4 1 = − d2θ W αW by deﬁnition. g D α

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