String-Steilkurs 2008:Supersymmetry and Supergravity

M. Zagermann

29.09 – 2.10.2008Albert-Einstein-Institut in Golm

Letzte Aktualisierung und Verbesserung: October 4, 2008

Mitschrift der Vorlesungsreihe Supersymmetry and Supergravityvon Herrn M. Zagermann auf dem String-Steilkurs 2008

von Marco Schreck.

Dieser Mitschrieb erhebt keinen Anspruch auf Vollstandigkeit und Korrektheit.Kommentare, Vorschlage und konstruktive Kritik bitte an Marco.Schreck@gmx.de.

Contents

1 Introduction to Supersymmetry 51.1 Symmetries in Particle Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 The Poincare group P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 “Known” particle spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 The Coleman-Mandula theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 The Super-Poincare algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Representations of the super Poincare algebra 92.1 Case 1: N = 1 SUSY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Consequences for the standard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Case 2: N -extended SUSY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Globally supersymmetric field theories 133.1 Fields and symmetry operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 SUSY analogon of the above . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Introduction to Supergravity 174.1 Fermions in curved space-time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3

Chapter 1

Introduction to Supersymmetry

1.1 Symmetries in Particle Physics

1.1.1 The Poincare group P

The Poincare is defined as the isometry group of four-dimensional Minkowski space. Its Lie algebra is spannedby two types of generators, namely Pm, Mmn, whereas m, n = 0, 1, 2, 3. Pm are the translation generatorsand Mmn generate the Lorentz transformations. These generators can be written in a more familiar form. Theangular momentum operators Li ≡ εijkMjk (i = 1, 2, 3) generate the rotations and Ki ≡ M0i generate theboosts.

[Pm, Pn] = 0 , [Pm,Mnp] = iηmpPn − iηmnPp , ηmn = diag(−, +, +,+) , (1.1)

[Mmn,Mpq] = iηmpMnq − iηnpMmq − (p ↔ q) . (1.2)

The generators of the Poincare group transform themselves under the Lorentz group. Defining

J±i ≡ 12(Li ± iKi) , (1.3)

one finds that

[J±i , J±j ] = iεijkJk , [J+i , J−j ] = 0 . (1.4)

The irreducible representations can be labeled by two half-integers (j, j′). One of these corresponds to thespin quantum number J+

i and the other one to J−i . The representations split into two types. Pm and Mmn

themselves correspond to

Pm '(

12,12

). (1.5)

Mmn ' (1, 0)⊕ (0, 1) , with (1, 0) for J+i and (0, 1) for J−i . (1.6)

Remark: Irreducible representations (j, j′) with j+j′ = half-integer correspond to “double-valued” representa-tions of the Lorentz group SO(3,1). (However, they are single-valued representations of the universal coveringgroup SO(3, 1) ' SL(2,C) of the Lorentz group.) Analogy: quantum mechanics of angular momentum:SO(3,1) ↔ SO(3). (Spin 1/2: double-valued representation of SO(3), but they are single-valued representa-tions of SO(3) ' SU(2).) Hence, (j, j′) with j + j′ = half-integer correspond the fermionic quantities. Youmake the following observation: The known particle physics is invariant under

Lie(P)⊕ Lie(Gint) , (1.7)

with Gint being an “internal” symmetry group, which is given by

GSM = SU(3)× SU(2)×U(1) . (1.8)

The Lie algebra of the internal symmetry group is spanned by generators Ta, for which it holds that [Ta, Tb] =if c

ab Tc (a = 1, . . ., dim(Gint)). The dimension of the internal group is at least 12. “⊕” means direct sum, sowe have

[Ta, Pm] = [Ta,Mmn] = 0 , (1.9)

5

CHAPTER 1. INTRODUCTION TO SUPERSYMMETRY

so the two algebras do not talk to each other. They do not transform under the Lorentz group; hence,the generators live in the trivial representation (0, 0). Gint cannot change the mass (m2 = −PaP a) or thespin/helicity of a particle:

LiTa|s,m〉 = TaLi|s,m〉 = sTa|s,m〉 . (1.10)

Similarly this can be proofed for the mass.

1.2 “Known” particle spectrum

spin/helicity 2 3/2 1 1/2 0particles “graviton” ? photon leptons Higgs (?)

W±, Z quarksgluons

Gint can at most operate “vertically”.(

νe

e−

)

L

' 2 of SU(2) , (1.11)

dr

dg

db

R

' 3 of SU(3) , gluons ' 8 of SU(3) . (1.12)

Grand unified theories, as for example Gint = SO(10) ⊃ GSM:(

quarksleptons

)' 16 of SO(10) , (1.13)

(all gaugebosons

)⊂ 45 of SO(10) . (1.14)

There arises one question: Can one also “unify” the particles of different spin? Can you, in particular, unifythe graviton (s = 2) and the gauge bosons (s = 1)? The disappointing news is the Coleman-Mandula theorem(1967).

1.3 The Coleman-Mandula theorem

Under the (reasonable) assumptions that

• the S-matrix is nontrivial and analytic in the scattering angles

• and that there exist only a finite number of particles below a fixed mass,

symmetries, which are described by Lie algebras, must be of the following form:

Lie(P)⊕ Lie(Gint) . (1.15)

This implies that Gint can never change the spin! Are there perhaps symmetries that do not form Lie algebras?Then, we could circumvent the Coleman-Mandula theorem and connect particles with different spin. Supposethat you have some general symmetry generator that acts additively on multi-particle states. Then, it mustbe of the following form:

G =∑

i,j

∫d3p

∫d3q a†i (p)Kij(q, p)aj(q) = a† ∗K ∗ a , (1.16)

where a†i (p) are creation and aj(q) annihilation operators. Bosons have creation and annihilation operatorsthat satisfy a commutation relation of the form

[bi(p), b†j(q)] = δijδ(3)(p− q) . (1.17)

Fermions satisfy an anticommutation relation

{fi(p), f†j (q)} = δijδ(3)(p− q) . (1.18)

6

1.4. THE SUPER-POINCARE ALGEBRA

Hence, G can be split in B + F with

B = b† ∗Kbb ∗ b + f† ∗Kff ∗ f , (1.19)

which transforms bosons and fermions under themselves or

F = f† ∗Kfbb + b† ∗Kbf ∗ f , (1.20)

which transforms bosons to fermions and vice versa.

[B1, B2] = B3 , [F1, B2] = F3 , [F1, F2] = not bilinear in a†, a , (1.21)

so this is not a proper symmetry generator. But, it holds that

{F1, F2} = B3 . (1.22)

So, allowing for Bose-Fermi symmetries F , results in a “graded Lie algebra” (or “super Lie algebra”).Schematically, it holds that

[G1, G2} = G3 , [G1, G2} ≡ G1G2 − (−1)η1η2G2G1 , (1.23)

where η(B) = 0 and η(F ) = −1. Graded Jacobi identity:

(−1)ηaηc [[Ga, Gb}Gc}+ cyclic = 0 . (1.24)

Symmetries between bosons and fermions are called supersymmetries and are described by anti-commutingsymmetry generators F . These are called SUSY generators or “odd” or “fermionic”. The B generators areeither called “even” or “bosonic”. From now on we will write F → Q.

1.4 The Super-Poincare algebra

Three remarks:

• The supersymmetry generators Q change the spin = (j, j′). They cannot be the trivial representation(0,0), because then, the spin could not be changed.

• Q† ' (j′, j),

• {Q,Q†} ' (j + j′, j + j′)⊕ . . . (non-zero, bosonic).

The Coleman-Mandula theorem tells us that a state of the form (j + j′, j + j′) must be either an internalsymmetry generator with (0,0), or Pm with (1/2,1/2) or Mmn with (1,0)⊕(0,1). The first and the thirdpossibility go out, what stays is the second one.

Q '(

12, 0

)or

(0,

12

), {Q, Q†} ∼ Pm . (1.25)

Without loss of generality one can define Q 7→ (1/2, 0) and Q† 7→ (0, 1/2). We use the notation QA (A = {1, 2},labels (1/2,0)) and Q†

A(A = {1, 2}, labels (0,1/2)) and the 2× 2-matrices r(Mab)AB and r(Mab)AB .

[Mab, QA] = r(Mab)ABQB , [Mab, Q†A] = r(Mab)ABQ†

B. (1.26)

A further generalization is QA → QrA (with r = 1, . . ., N). N = 1 is denoted as “minimal” or “simple” SUSYV

and N > 1 as “N -extended” SUSY. Using similar arguments than above plus the graded Jacobi identity onefinds (Haag-ÃLopuszanski-Sohnius, 1975):

{QrA, (Q†

B)s} = 2δrs(σm)ABPm , σ0

AB=

(−1 00 −1

), σi

AB= Pauli matrices (i = 1, 2, 3) . (1.27)

With ηmn = diag(−, +, +,+) it holds that (σ0)AB = −(σ0)AB . Now, we are interested in {QrA, Qs

B}. From(

12, 0

)×

(12, 0

)= (0, 0)⊕ (1, 0) , (1.28)

7

CHAPTER 1. INTRODUCTION TO SUPERSYMMETRY

one can state that {QrA, Qs

B} must either contain Ta or Mmn. However, Mmn is excluded, because then theequations

[Pm, QrA] = [Pm, (Q†

A)s] = 0 , (1.29)

could not be fulfilled. It has to hold that

{QrA, Qs

B} = eABZrs , eAB =(

0 1−1 0

), (1.30)

where the Zrs is some linear combination of the Ta: Zrs = −Zsr = (ca)rsTa. These linear combinationscannot be arbitrary and must commute with everything else of the algebra (using the graded Jacobi identity).These are then called “central charges”. They, of course, commute with the generators of the Poincare group,since they are built up of internal symmetry generators. Anyway, they must form an Abelian subgroup of theinternal symmetry group. Note that for N = 1 SUSY the central charges have to vanish. It must be

[Mmn, QrA] = r(Mmn) B

A QrB , (1.31)

where r(Mmn) BA is the (1/2, 0)-representation matrix of Lorentz transformations.

[Ta, QrA] = (ta)r

sQsA , (1.32)

whereas the choice (ta)rs is possible. The Ta with (ta)r

s 6= 0 generate the “R-symmetry group”. R-symmetries are internal symmetries that rotate the Qr

A and therefore act nontrivially on supercharges. Ingeneral, the R-symmetry group is U(N)R, for N = 1 SUSY this is the U(1)R. The standard model gaugegroup has to commute with the supercharges.

Remarks

0) Adding also [P, P ], [P, M ], . . . this is called a super Poincare algebra.

1) Acting with Q and Q† on a state raises or lowers the spin/helicity by 1/2.

2) Acting repeatedly with QrA or (Q†

A)s creates new states of different spins/helicities: {|s〉, Q|s〉, Q2|s〉, . . .}.

At a certain point any of this products of Q will be zero (because they anti-commute), so the sequenceterminates after finitely many steps. The states so obtained form a so-called “supermultiplet” (rep-resentation of the SUSY algebra). They are “superpartners” of one another.

3) [P, Q] = 0 = [P, Q†] implies that [m2, Q] = 0 = [m2, Q†] (m2 = PmPm, m = 0, 1, 2, 3). Q and Q†

cannot change the mass; hence, the masses within a supermultiplet are all the same. However, this isnot observed in nature! So, SUSY can at best be a broken symmetry! The usual assumption is thatthe masses of the standard model superpartners are at least of O(100GeV).

8

Chapter 2

Representations of the super Poincarealgebra

Our focus will be on massless supermultiplets only because of

i) lack of time and, what is more important,

ii) SUSY is only a good approximation for E À ∆m ≈ msuperpartner ¿ mSM−particle. Hence, m ≈ 0 is agood approximation for large E.

iii) In the standard model the masses all vanish before the electroweak symmetry breaking.

Choose a Lorentz frame such that P 1 = P22 = 0, with Pm = (E, 0, 0, E). This implies that

{QrA, Q†

B

s} = 2δrs(σ0P0 + σ3P

3)AB = 2Eδrs

(2 00 0

)

AB

. (2.1)

From this, we can read off

{Qr1, Q

†1

s} = 4Eδrs , {Qr2, Q

†2

s} = 0 , (2.2)

and furthermore (no sum!):

0 = 〈φ|{Qr2, (Q

†2)

r}|φ〉 = ‖(Q†2)r|φ〉‖2 + ‖Qr2|φ〉‖2 ∀ |φ〉 ∈ H ⇒ Qr

2 = Q†2

r= 0 . (2.3)

So, we can forget about the Q2 and just consider the Q1.

{Qr1, Q

s1} = e11Z

rs = 0 . (2.4)

For massless supermultiplets the central charge has no affect on the anti-commutator. However, one can alsoshow (as an exercise) that Zrs = 0. One can construct massless supermultiplets from acting with Qr

1 or Q†1

r.

{Qr1, Q

s1†} = 4Eδrs , {Qr

1, Qs1} = 0 . (2.5)

Redefining qr := (4E)−12 Qr

1 we arrive at

{qr, (qs)†} = δrs , {qr, qs} = 0 . (2.6)

Remark: qr lowers helicity by 1/2 and (qr)† raises helicity by 1/2 (by calculating [qr1, Li]).

2.1 Case 1: N = 1 SUSY

Our algebra is {q, q†} = 1 and {q, q} = 0 (hence q2 = 0 = (q†)2). Start with state of maximal helicity |λmax〉.From that if follows

q†|λmax〉 = 0 , q|λmax〉 =:∣∣∣∣λmax − 1

2

⟩. (2.7)

9

CHAPTER 2. REPRESENTATIONS OF THE SUPER POINCARE ALGEBRA

By using the anti-commutator one ends up with⟨

λmax − 12

∣∣∣∣ λmax − 12

⟩= 〈λmax|q†q|λmax〉 = 〈λmax|λmax〉 6= 0 ⇒

∣∣∣∣λmax − 12

⟩6= 0 . (2.8)

Anyway, one obtains:

q

∣∣∣∣λmax − 12

⟩= q2|λmax〉 = 0 , (2.9)

and

q†∣∣∣∣λmax − 1

2

⟩= q†q|λmax〉 = |λmax〉 . (2.10)

So, we have a complete supermultiplet{|λmax〉,

∣∣∣∣λmax − 12

⟩}. (2.11)

It is irreducible, so every massless particle of helicity λ has exactly one superpartner of helicity λ± 1/2.

2.1.1 Consequences for the standard model

Assume [Lie(GSM), Q] = 0. (For SU(3), SU(2) this is automatically the case.) This implies that the super-partners are in the same [SU(3) × SU(2)× U(1)]-representation.

i) Quarks with λ = ±1/2 can be either have a superpartner with λ = 0 or λ = 1. The case λ = 1 wouldcorrespond to vector bosons in 3 of SU(3). That would be a non-unitary theory, so this possibility is ruledout. As a result of that it must hold that λpartner = 0 and the superparnters must be scalar colour-triplet(in the 3 of SU(3)) particles, so-called “squarks”.

ii) The leptons with λ = ±1/2 have superpartners with λpartner = 0 because of the same reason as abovefor the quarks. These are called “sleptons”.

iii.) Gauge bosons with λ = ±1 could either have λpartner = ±1/2 or λpartner = ±3/2. Particles withλ = ±3/2 cannot be coupled in a renormalizable way. So, this possibility is ruled out, if one claimsrenormalizability. Therefore, it must be λpartner = ±1/2 and these superpartners are called “gauginos”.These are fermions that live in the adjoint representation of the standard model gauge group GSM.

iv.) From λ = 0 ist follows that λ = ±1/2. The partners are called “Higgsinos”.

v.) For the graviton with λ = ±2 it follows that λpartner = ±3/2 or λpartner = ±5/2. Partners with spin ±5/2are not known how to couple to gravity, so that is out. λ = ±3/2 causes problems with renormalizabilityas mentioned above. But that is not important here, because gravity anyway is not renormalizable! Thesuperpartner of the graviton is called “gravitino”.

2.2 Case 2: N -extended SUSY

{qr, q†s} = δrs , {qr, qs} = 0 . (2.12)

From |λmax〉 one obtains N states of helicity λmax − 1/2

{q|λmax, q2|λmax, . . . , q

N |λmax} , (2.13)(N2

)states with helicity λmax − 1

{q1q2|λmax〉, q1q3|λmax〉, . . . , qN−1qN |λmax〉} , (2.14)

and one state with λmax −N/2 = λmin:

{q1q2 . . . qN |λmax〉 . (2.15)

10

2.2. CASE 2: N-EXTENDED SUSY

• N = 2:λ 2 3/2 1 1/2 0 -1/2 -1 -3/2 -2

1 2 11 2 1

1 2 1

• N = 4 (super Yang-Mills theory, N = 4 is the largest N which maintains renormalizability)

λ 2 3/2 1 1/2 0 -1/2 -1 -3/2 -21 4 6 4 1

1 4 6 4 1

• N = 8 (SUGRA N = 8 = Nmax):

λ 2 3/2 1 1/2 0 -1/2 -1 -3/2 -21 8 28 56 70 56 28 8 1

Perhaps, N = 8 SUGRA could be the theory of everything. It could be renormalizable or even finite,because different loop contributions cancel. However, it turns out, that N = 8 SUGRA has someproblems.

– The largest gauge group is SO(8) 6⊃ SU(3)× SU(2)×U(1).

– N ≥ 2 only has non-chiral gauge interactions. Hence, they cannot describe the electroweak sector.

To summarize, N = 1 SUSY seems the only realistic option.

2 3/2 1 1/2 01 1

1 1

It does not unify gravity with gauge interactions.

11

CHAPTER 2. REPRESENTATIONS OF THE SUPER POINCARE ALGEBRA

12

Chapter 3

Globally supersymmetric field theories

SUSY imposes strong constraints on

a) possible particle spectrum,

b) possible interactions.

These have a large impact on QFT and therefore we need supersymmetric field theories.

3.1 Fields and symmetry operators

Schematically it holds that |x〉 = φ(x)|0〉. Consider translations:

φ(x+a)|0〉 = |X+a〉 = exp(ipa)|x〉 = exp(ipa)φ(x)|0〉 = exp(ipa)φ(x) exp(−ipa) exp(ipa)|0〉 = exp(ipa)φ(x) exp(−ipa)|0〉 ,(3.1)

from which it follows that

exp(ipa)φ(x) exp(−ipa) = φ(x + a) . (3.2)

Infinitesimally, this means:

i) δaφ ≡ φ(x + a)− φ(x)|a 7→0 = iam[Pm, φ(x)] = am∂mφ(x)

ii) Pm × φ(x) := [Pm, φ(x)] = −i∂mφ(x) (action of Pm on φ(x))

That must be consistent with the expression

0 = [Pm, Pn]× φ = (−i)2[∂m, ∂n]φ = 0 . (3.3)

This is consistent with the representation of the symmetry algebra on the field φ(x).

3.2 SUSY analogon of the above

QA × (bosonic field) = [QA, bosonic field] = fermionic field , (3.4a)

QA × (fermionic field) = {QA, fermionic field] = bosonic field . (3.4b)

This must be consistent with the SUSY algebra

{QA, Q†A} × field = 2σm

AAPm × field = −2iσm

AA∂m(field) . (3.5)

This constraint restricts the way the fields can occur on the right-hand of Eq. (3.4). Often, Eq. (3.4) is writtenin the form (i) by introducing “infinitesimal parameters” εA, εA (A = 1, 2). Then, we find

δεφ = (εAQA + εAQ†A)× φ =

{εA[QA, φB ] + . . .

εA{QA, φF }+ εA{Q†A, φ} . (3.6)

13

CHAPTER 3. GLOBALLY SUPERSYMMETRIC FIELD THEORIES

Useful trick: Assume εA, εA to be anti-commuting: ε1ε2 = −ε2ε1 (Grassmann variables). One also requiresthat they anti-commute with fermionic operators, but commute with bosonic ones:

εAφF = −φF εA , εAφB = φBεA . (3.7)

As a result of that all brackets become commutators, if we write εA, εA inside the brackets:

εA{QA, φF } = [εAQA, φF ] , (3.8)

etc. Also:

[εAQA, εAQA] = −2εAσmAA

εAPm . (3.9)

[δε1 , δε2 ]φ = −2εA1 σm

AAεA2 Pm × φ︸ ︷︷ ︸

=−i∂mφ

. (3.10)

This gives us a representation of SUSY algebra on fields. Then, we have to require that dynamics is alsoinvariant under SUSY:

δεS[φ1, . . . , φn] = 0 . (3.11)

The is equivalent to the statement that the action is supersymmetric. There are two cases:

i.) εA = const.: global SUSY

ii.) εA = εA(x): local SUSY (supergravity)

To write down an example, we switch to four-component spinor notation. From now on we will use the∗ for Hermitian conjugate of a Hilbert space operator instead of the symbol †, because we would like to use †for Hermitian conjugation plus transposition:

(Q1

Q2

)†=

(Q1

Q2

)∗,ᵀ= (Q∗1, Q

∗2) . (3.12)

Combine QA and Q∗A

into a four-component spinor as follows:

Qα =(

e ·Q∗(2)Q(2)

)

α

, e =(

0 1−1 0

), (3.13)

with α = 1, 2, 3, 4. Define matrices (γm)αβ (m = 0, 1, 2, 3 and α = 1, 2, 3, 4):

γ0 :=(

0 i12

i12 0

), γj =

(0 −iσj

iσj 0

), (3.14)

with j = 1, 2, 3. That is the so-called Weyl representation, which is very convenient, if one talks abouttwo-component spinors. The γ matrices obey the Clifford algebra:

{γm, γn} = 2ηmn14 . (3.15)

We define

γ5 := iγ0γ1γ2γ3 =(12 00 −12

), (γ5)2 = 14 , {γ5, γm} = 0 . (3.16)

The Dirac conjugate is given by ψ ≡ −iψ†γ0. A four-component ψ with

ψ = ψᵀC , C =(

e 00 −e

), (3.17)

is called a “Majorana spinor”. C is the charge conjugation matrix. Eq. (3.17) is equivalent to ψ = ψc.Hence, these are spinors that describe particles which are their own anti-particles. Because of this constraint,Majorana spinors only have two independent complex components:

ψ =(

eχ∗

χ

), (3.18)

14

3.2. SUSY ANALOGON OF THE ABOVE

whereas χ is the two-component spinor containing the two independent degrees of freedom. A general Diracspinor, which is needed to describe particles that are not their own antiparticles (as for example the electron),can be written as

ψ =(

eχ∗

λ

), λ 6= χ . (3.19)

Hence, Qα is a Majorana spinor.

{Qα, Qβ} = −2iγmαβPm . (3.20)

In the following, all spinors are meant to be anti-commuting Majorana spinors. Let us consider the followingexample:

ψχ = χᵀCχ = ψαCαβχβ = −χβCαβψα = −χᵀCᵀψ = χᵀCψ = χψ . (3.21)

We introduce chiral projectors

PL = (14 + γ5) =(12 00 0

), PR =

12(14 − γ5) =

(0 00 12

). (3.22)

With these one can write

χL ≡ PLχ , χR ≡ PRχ , χR ≡ χPR = χL , χL ≡ χPL = χR . (3.23)

• Exercise 1: Verify γᵀm = −CγmC−1, γᵀ

5 = Cγ5C−1 and Cᵀ = −C.

• Exercise 2: Use this to show ψχ = χψ, ψγmχ = −χγmψ and ψγmγ5χ = χγmγ5ψ.

• Exercise 3: Show that PL + PR = 14, γmPL = PRγm, P 2L = PL, PRPL = 0, ψLχL = χLψL, ψLγ5χR =

−χRγmψL and finally ¢∂¢∂ = γm∂mγn∂nφ(x) = ¤φ(x)14.

With these preparations we are now ready to write down the simplest four-dimensional supersymmetric fieldtheory: the free massless Wess-Zumino model. This model contains one complex scalar φ(x), one Majoranafermion χ(x) and another complex scalar F (x).

δεφ = εLχL , δεφ∗ = εRχR , (3.24a)

δεχL =12(¢∂φ)εR +

12FεL , δεχR =

12(¢∂φ∗)εL +

12F ∗εR , (3.24b)

δεF = εR¢∂χL , δεF∗ = εL¢∂χR . (3.24c)

These satisfy:

[δε1 , δε2 ]

φχF

=

12ε2γ

mε1∂m

φχF

∼ ε2γ

mε1Pm ×

φχF

. (3.25)

Hence, (φ, χ, F ) represents the SUSY algebra! The Lagrangian of the Wess-Zumino model is given by

L = −(∂mφ)(∂mφ∗)− χ¢∂χ + FF ∗ . (3.26)

Claim: L is SUSY-invariant modulo a total derivative. We want to proof that and first write up a lemma:

χ¢∂χ = 2χL¢∂χR + total derivatives . (3.27)

The proof of this lemma is not so difficult. We use equations of the third exercise:

χ¢∂χ(1)= χ¢∂(PL + PR)χ

(3)= χ¢∂PLχL + χ¢∂PRχR = χR¢∂χL + χL¢∂χR =

(6)= −χL

←−¢∂ χR + χL

−→¢∂ χR

p.I.= 2χL¢∂χR .¤ (3.28)

15

CHAPTER 3. GLOBALLY SUPERSYMMETRIC FIELD THEORIES

Now we can proceed with the proof of our claim (with the supercurrent JmR = −χRγm(¢∂φ))

δε[−χ¢∂χ] Lemma= δε[−2χL¢∂χR] = −2δχL¢∂χR − 2χL¢∂(δχR)(6)+p.I.

= +2χR

←−¢∂ (δχL) + 2χL

←−¢∂ (δχR) =

= 2χR

←−¢∂

{12(¢∂φ)εR +

12FεL

}+ h.c. = χR

←−¢∂ (¢∂φ)εR + χR

←−¢∂ εLF + h.c. =

p.I.= −χR(¢∂¢∂φ)εR − χRγm(¢∂φ)(∂mεR) + χR

←−¢∂ εLF + h.c. =

(7),(5),(6)= −εRχR¤φ + Jm

R (∂mεR)− εL

−→¢∂ εRF + h.c. =: A + B + C + h.c. (3.29)

As a further exercise it can be shown that

(A + h.c.) = −δε[−(∂mφ)(∂mφ∗)] , (C + h.c.) = −δε(FF ∗) . (3.30)

The result is:

δεL = JmR (∂mεR) + Jm

L (∂mεL) = 0 , (3.31)

for global SUSY (constant ε). Let us now have a look at the equations of motion for F . They are F = 0,hence, we could set F = 0 everywhere and L|F=0 would be invariant under δφ|F=0, δχ|F=0. So, why have weintroduced the field F? [δε1 , δε2 ]φ would still be okey, but

[δε1 , δε2 ]χ =12(ε2γ

mε1)∂mχ + [. . .]¢∂χ , (3.32)

where the second term is bad. This second term vanishes, if the equations of motion are used. Eliminatingthe auxiliary field F the SUSY algebra only holds “on-shell” (i.e. after use of equations of motion.) With F ,it holds off-shell. For the physical content one does not need the off-shell formulation, but one cannot usesuperspace formulation. For N ≥ 2, off-shell formulations are often unknown.

16

Chapter 4

Introduction to Supergravity

There are three equivalent definitions:

i) These are locally supersymmetric field theories (ε = ε(x)).

ii) Supersymmetrizations of general relativity

iii) Only known field theories of interacting spin 3/2 fields.

In order to illustrate the equivalence of the three definitions we consider the locally supersymmetric Wess-Zumino model with the Lagrangian

LWZ = −(|∂φ|2 + χ¢∂χ) . (4.1)

The variation under local SUSY is given by

δεLWZ = JmR (∂mεR) + Jm

L (∂mεL) 6= 0 , (4.2)

with the supercurrents

JmR ≡ −χRγm¢∂φ , Jm

L = (JmR ) + h.c. . (4.3)

Consider the Lagrangian of free quantum electrodynamics: LQEDfree = −λ¢∂λ. Under U(1) transformations

λ 7→ exp(iΛ(x))λ the variation does not vanish:

δU(1)Lfree = Jmem∂mΛ(x) . (4.4)

To compensate this contribution one has to couple a vector field to the current and impose a certain transfor-mation property of this vector field:

Lint = −eJmemAm , δU(1)Am =

1e∂mΛ . (4.5)

We do the same in the Wess-Zumino model. Add

L′WZ = −κJmR ψm,R − κJm

L ψm,L , δεψm =1κ

∂mε , [κ] =1

mass, [κ](D) =

(1

mass

)D−22

. (4.6)

The field ψ has a vector index m and a spinor index α. ψm,α is the “gauge field of local SUSY”. However, itturns out that this is not enough.

δε[LWZ + L′WZ] = −κ(δJmR )ψm,R − κ(δJm

L )ψm,L , (4.7)

with

δJmR = δ(−χRγm¢∂φ) = εγmηmn

[|∂φ|2 + χR¢∂χL + . . .]

. (4.8)

One realizes that

ηmn[|∂φ|2 + χR¢∂χL + . . .

]= Tmn . (4.9)

17

CHAPTER 4. INTRODUCTION TO SUPERGRAVITY

Hence, the result is

δε(LWZ + L′WZ) ' κεγmψnTmn + . . . , (4.10)

and we have to add a second term

L′′WZ ∼ gmnTmn , δεgmn ∼ κεγ(mψn) . (4.11)

gmn is some tensor field, which turns out to be the space-time metric. This implies that local SUSY impliesthe coupling to gravity. (In the literature one sometimes finds another hand-waving argument. We look at theSUSY algebra:

{Q, Q} ∼ Pm . (4.12)

If one tries to make the left-hand side local, one also has to consider local translations, which is equivalent togeneral coordinate transformations (→ gravity).)

• ψm is the superpartner of the metric (“gravitino”).

• κ = 1/Mpl =√

8πG = (108 GeV)−1.

These considerations tell us that there should be a pure N = 1 supergravity theory. Just consider (gmn, ψm),no (φ, χ), . . .. We expect:

Lpure = Lkin[gmn] + Lkin[ψm] + Lint[gmn, ψm] , (4.13)

with

Lkin[gmn] = − 12κ2

√gR . (4.14)

Lkin[ψm] should be the spacetime covariant version of

LRavita−Schwinger = −12ψmγmn%∂nψp , γmnp = γ[mγnγp] . (4.15)

For spin-3/2 terms this is the only kinetic term which is suitable without having ghosts. We know that thevariations are given by

δεgmn ∼ κεγ(mψn) , δεψm ∼ 1κ

∂mε

∣∣∣∣cov.

+ . . . . (4.16)

However, we have to know, how fermions are to be described in a curved space-time, since Eq. (4.15) comesfrom a Minkowski-like description.

4.1 Fermions in curved space-time

In Minkowski space-time vectors are described by vector representations of SO(3,1) and spinors by double-values spinor representations of SO(3,1). In curved space, vectors are described by

V µ′ =∂xµ′

∂xνV ν ,

∂xµ′

∂xν∈ GL(4,R) ⊃ SO(3, 1) . (4.17)

However, general linear groups do not have spinor representations. Hence, we choose a local inertial frame ateach point. Instead of using tangent vector field ∂µ we will use orthogonal fields (vierbein formalism). That isunique up to local SO(3,1).

18

4.2. GENERALIZATIONS

em = eµm∂µ . (4.18)

Local SO(3,1) transformations act on the index m and general coordinate transformations on the index µ. Onethen writes

gµν = emµ en

ν ηmn , (4.19)

where emµ is the inverse of eµ

m. The Christoffel symbols are given by

Γ %µν (g) 7→ ω m

µ n(e) , (4.20)

where the right-hand side is called spin-connection. That implements local SO(3,1) covariance. Thecovariant derivative is now given by

∇µλ =(

∂µ +12ω mn

µ Σmn

)λ , Σmn ≡ 1

4[γm, γn] = r(Mmn) . (4.21)

Σmn is the generator for Lorentz transformations in the spinor representation. There is one subtlety for thegravitino. Its covariant derivative is given by

∇µψν = ∂µψν +12ω mn

µ Σmnψν − Γ %µν (g)ψ% . (4.22)

It is local Lorentz-invariant and also invariant under general coordinate transformations, which act on thevector-like index. ∇[µψν] implies Γ %

[µν] = 0. The curvature tensor is given by

R mµν n(e) = ∂µω m

ν n + ω nµ pω

pν m − (µ ↔ ν) . (4.23)

Furthermore, the curvature scalar is

R(e) = R mnµν eν

meµn = R[g] , (4.24)

and

γµ = emµ γm ,

√g =

√(det em

µ )2 = det(emµ = e . (4.25)

Now, we can write up the Lagrangian of pure supergravity:

Lpure SUGRA = −e

2M2

plR(e)− e

2ψµγµν%∇νψ% + L4−fermion , L4−fermion = O

(1

M2pl

ψψψψ

), (4.26)

δemµ =

12Mpl

εγmψµ ⇒ δgµν =1

Mplεγ(µγν) , (4.27)

δψµ = MPl∇µε + 3-fermion terms , 3-fermion terms = O(

1Mpl

ψψε

). (4.28)

The interaction described by the four-fermion-term is supressed by two powers of the Planck mass. Hence, itis not very important. SUSY invariance can be shown as follows:

δε

(−e

2ψµγµν%∇νψ%

)= −eψµγµν%∇[ν∇%]ε , ∇[ν∇%] =

18R mn

ν% γmnε ∼ Rεγψ ∼ Rδεenµ = −δSEH

δemµ

·δεemµ +. . . .

(4.29)

The whole calculation is done in Freedman, de Wit [Lectures, NATO, ∼ 1985]. Unfortunately, it containsmany typos. (The formulation via uperspace becomes very messy in SUGRA. That is why we are using theon-shell formulation here.)

4.2 Generalizations

i) Add a cosmological constant term Λ < 0 (AdS). A constant Λ > 0 (dS) is not possible in supergravity.For Λ > 0 one needs to add matter multiplets.

19

CHAPTER 4. INTRODUCTION TO SUPERGRAVITY

ii) Add matter multiplets.

The most general field content in N = 1, D = 4 supergravity is

[(emµ , ψµ) (SUGRA)]⊕ [nc chiral multiplets (φa, χa), a = 1, . . . , nc]

⊕ [nvvector multiplets (AIµ, λI), I = 1, . . . , nv] . (4.30)

The Lagrangian is uniquely specified by the following data:

i) Kahler potential: It is a function of the scalar fields: K(φa, (φa)∗) with (φa)∗ ≡ φa. That determinesthe kinetic terms of the chiral multiplets in a sense that it is of the form:

Lkin[φa, χa] = −gab(φ, φ∗)[∂µφa∂µφb + fermionic terms

]. (4.31)

gab(φ, φ∗) is a metric on a scalar manifold (space of all fields). This metric is given in terms of theKahler potential and is a nonlinear σ-model.

gab =∂

∂φa

∂

∂φb. (4.32)

Hence, Mscalar is a “Kahler manifold”. Kahler terms can destroy the renormalizability of the theory.However, this is not an issue here, because gravity is nevertheless not renormalizable.

ii) Superpotential W (φa) (holomorphic)This leads to self-interaction of chiral multiplets (φa, χa) described by the F-term potential:

VF = exp(

K

M2Pl

) [gabDWDbW

∗ − 3|W |2M2

pl

], DaW ≡ ∂aW +

(∂aK

M2pl

)W . (4.33)

The derivative D is called the Kahler covariant derivative. Decoupling SUGRA (equivalent withtaking the global SUSY limit) in the limit Mpl 7→ ∞ one obtains:

VF = gab∂aW∂bW∗ ≥ 0 . (4.34)

iii) gauge kinetic function fIJ(φa) (holomorphic)

Lkin[AIµ, λI ] = −1

4(Re(fIJ))F I

µνFµν, J + fermionic terms . (4.35)

One still has to define the gauge group G of the theory plus its action on (φa, χa). The gaugegroup has to act on the matter fields that sit in the chiral multiplets. Then, there is an additionalcontribution to the scalar potential, the so-called D-term potential:

VD =12(Re(fIJ))DIDJ , DI ∼ (∂aK)δgauge,Iφ

a . (4.36)

We realize VD ≥ 0 even in supergravity. This does not hold for the F-term potential.

iv) possible Fayet-Iliopoulos constants ξI for Abelian factors of G

DI 7→ (DI + ξI) , (4.37)

which only works for those ξI which correspond to Abelian vector fields.

The important task in string theory is to derive the above date from compactification data.

iii) Extended supergravity: N = 1 7→ N > 1.

In 4D, for N = 1 you find V = VF + VD with VD 6= 0 only for gauge interactions. VF is not relatedat all to gauge interactions. This splitting ends to exist in supergravity theories in the following sense.For N > 1, it holds that V 6= 0 if and only if there exist non-trivial gauge interactions (“gaugedsupergravity”). They are important for moduli stabilizations of AdS/CFT correspondence.

20

4.2. GENERALIZATIONS

iv) Replace D = 4 by something D 6= 4.

How do we get spinor representations in arbitrary space-time dimensions? Look at the Clifford algebra:

{γM , γN} = 2ηMN1D , (4.38)

with (M,N) = 0, . . . , D. The Clifford algebra induces spinor representations of the correspondingLorentz group. ΣMN ≡ 1/4[γM , γN ] is a spinor representation of SO(1,D− 1). But, ΣMN is, in general,reducible. Therefore, one has to impose constraints on the spinors:

i) chirality (Weyl condition):

PLψ = 0 , PL =12

(1+ “γ5”) , (4.39)

with the D-dimensional analogue of the γ5 matrix: γ5 ∼ γ0 . . . γD−1. This is only possible forD = 2n, because otherwise γ5 turns out to be proportional to the unit matrix.

ii) reality condition (Majorana):

ψ∗ = Bψ , γ∗M = ηBγMB−1 , (4.40)

with a constant η = ±1. What η is, depends on the dimension. The general proof can be foundin [Weinberg vol. 3, Ch. 32]. The Majorana condition should be self-consistent in the sense thatφ∗∗ = ψ and therefore, one needs B∗B = 1. However, it holds that B∗B = ε1, where ε depends onthe dimension D and η. The number of space-time components of spinors depends strongly on thedimension (look at the table). If scalars transform under the R-symmetry group, the R-symmetrygroup appears as a factor of the holonomy group of Mscalar, which is the group of all rotations onegets by parallel transporting. A large holonomy group strongly constraints a manifold. (In fivedimensions: N = 8: USP(8) ≡ SP(8, R) ∩U(8), 42 real scalars ∈ Mscalar ≡ E6(6)/USP(8))

21