Stability in Maximal Supergravity

S. Bielleman,s1712136,

RuG

Supervisor : Dr. D. Roest

August 25, 2014

Abstract

In this thesis, we look for a bound on the lightest scalar mass in maximal supergravity. The goalof this thesis is to find a direction on the scalar manifold of maximal supergravity, E7(7)/SU(8),that always has a negative mass when the scalar potential has a positive value. Such a directionwould prove that there are no stable de Sitter vacua in maximal supergravity. We give shortintroductions on stability, supersymmetry and supersymmetric field theories. Then, we studyattempts to find a bound on the lightest scalar mass in N = 1, 2 (local) supersymmetric theories,[10] and [11]. After this discussion, we move to maximal supergravity and discuss several possibledirections, inspired by sponteneous supersymmetry breaking and the sGoldstini, with which toproject the scalar mass matrix. Unfortunately, we find a counter example (vacuum for which theprojection gives a positive value) for all of the directions we consider. This does not imply thatthere exist stable de Sitter vacua but we are unable to prove that there aren’t any.

Contents

1 Introduction 3

2 Stability and supersymmetry 52.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Supersymmetry algebra and super multiplets . . . . . . . . . . . . . . . . . . . . . 62.3 Local supersymmetry breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Supersymmetric field theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Stability in N = 1, 2 supersymmetric theories 143.1 Global supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 N = 1 supergravity with chiral multiplets . . . . . . . . . . . . . . . . . . . . . . . 183.3 N = 2 supergravity with only hypermultiplets . . . . . . . . . . . . . . . . . . . . . 19

4 Maximal supergravity 234.1 SO(8) maximal supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2 The embedding tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.3 Going to the origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.4 Symmetry breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.5 Vacua of maximal supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Stability in maximal supergravity 355.1 The scalar mass matrix and the sGoldstino mass matrix . . . . . . . . . . . . . . . 355.2 Comparing N = 8 with N = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.3 Singlet projections of the scalar mass matrix . . . . . . . . . . . . . . . . . . . . . 41

5.3.1 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.3.2 Analytical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.4 Eigenvalue equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.5 Discussion of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6 Conclusion 47

7 Acknowledgements 51

A Conventions 52A.1 Identities for 70± . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52A.2 Quadratic variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53A.3 Quartic singlets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55A.4 Quartic non singlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

B Scalar mass matrix in N = 2 supergravity projected on NuxP

x 59

1

C Scalar mass matrix in N = 8 supergravity 61C.1 Scalar mass matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61C.2 sGoldstino mass matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62C.3 Mass matrix * sCrucchino/sCrostino . . . . . . . . . . . . . . . . . . . . . . . . . . 65

D Singlet projections, numerical results 66

E How to contract (anti)-symmetric indices 67

2

Chapter 1

Introduction

Quantum field theories are among the most successful theories in physics. They are importantboth in high energy physics and in condensed matter physics, developments in one of these areasoften lead to developments in the other. The process of symmetry breaking is an example of aconcept that was developed mutually in high energy physics and condensed matter physics.

There are many examples of quantum field theories that have survived careful experimentalverification. An example is Quantum electrodynamics which combines special relativity withclassical electromagnetism. This theory is renormalizable which means that calculations will givefinite answers at finite energies. It does not mean that the theory can be trusted up to arbitraryenergies because at some energy the strong and weak interaction will become important. Themost complete theory of the electroweak and strong interactions is the Standard Model which hasalso been tested to great extent. In fact, the full particle content of the Standard Model has beenobserved with the discovery of the Higgs particle last year [23].

On the other hand, one of the most well known non quantum field theories is general relativity.It describes gravity as a phenomenon that emerges when free particles follow geodesics on a curvedbackground. The curvature of space time is due to the matter content and is given by the Einsteinequations. It is possible to write general relativity as a field theory using the metric as thefield. The action of the theory is the Einstein-Hilbert action, whose variation gives the classicalEinstein equations. This quantum field theory of gravity is non renormalizable and higher orderterms become increasingly important because the coupling constant GN of the theory has massdimension −2.

The unification of the Standard Model and general relativity is one of the main problems intheoretical physics today. Such a theory would have to contain the gauge group of the StandardModel, SU(3) × SU(2) × U(1) and a spin-2 particle that mediates the gravitational force. Theconstruction of such a theory is hindered by the Coleman-Mandula theorem which states that“space time and internal symmetries can only be combined in a trivial way”. Supersymmetryprovides a loophole to the Coleman-Mandula theorem since the theorem assumes the symmetryto be based on a Lie algebra, supersymmetry is based on a graded Lie algebra. Supersymmetricextensions of the Standard Model give a solution to a number of theoretical problems including thehierarchy problem and it provides a candidate for dark matter, namely the lightest superparticlewhich could be stable. These extensions do not contain a spin-2 particle.

An interesting thing happens when the supersymmetry generators are made local or equiva-lently when we gauge supersymmetry. The theory then requires a spin-2 particle to be consistent,a local supersymmetric theory is a theory of gravity. Theories with local supersymmetry are calledsupergravity theories or simply SUGRA. Supergravity can be viewed as a field theory with localsupersymmetry or equivalently as the supersymmetrized version of general relativity. Supergravitytheories, in 4 space time dimensions are usually characterized by the number N which gives thenumber of supersymmetric generators, the particle content and the internal gauge group.

Supergravity has a special place in high energy physics since it can be viewed as an extensionof known theories and as the low energy limit of string theory. This suggests that there is some

3

known relation between string theory and the Standard Model but this is not true. Even thoughone can obtain a supergravity theory from different directions the exact relation between thosetheories is unclear. This remains a field of active research.

Supergravity becomes more and more constraint as N increases, for N = 8 in 4 space timedimensions, we have maximal supergravity. The field content of maximal supergravity is fixed (ifwe demand that the maximal spin is 2). The only freedom that is left in this theory is given by thegauging of the internal symmetry group and the symmetry breaking patterns. One way of breakingthe symmetry of a theory is by choosing a vacuum that breaks the gauge group/supersymmetryexplicitly. Maximal supergravity has a wide variety of gauge groups and vacua and at present nofull classification exists.

Of particular interest are vacua that have a positive cosmological constant, space time is deSitter for these theories. Both the era of inflation and the late time expansion of our universe arede Sitter (like) spaces. For this reason alone it is important to understand the place of de Sittervacua in high energy theories.

It is an interesting observation that it becomes more and more difficult to construct stable deSitter vacua as the number of supersymmetric generators is increased. For N = 1 supergravitiesa number of stable de Sitter vacua have been found. There exists a no-go theorem for N = 2supergravity with certain matter content but there exist stable de Sitter vacua for certain mattercontent. For N = 4 there are bounds on the mass of the lightest scalar depending on the waysupersymmetry is broken. For N = 8 supergravity there are stringent conditions on certainscalars. No stable de Sitter vacua have been found for N = 4, 8 supergravities [1] raising thequestion whether this is possible at all.

In this thesis we will focus on the possibility of stable de Sitter vacua in maximal supergravityin 4 dimensions. We will try to find eigenvalues of the scalar mass matrix that are always negative,independent of the gauging and critical point. This is quite an ambitious goal because there are70 scalars in the theory and an unknown number of possible gaugings. Our approach is inspiredby work done in N = 1, 2 (local) supersymmetric theories.

The structure of this work is as follows. In chapter 2 we will give a short introduction todifferent concepts that the reader is probably not familiar with. However, we encourage thereader to also read the literature on these subjects. In chapter 3 we describe stability analysesin N = 1, 2 supersymmetric theories. We will use these analyses as a guideline for the N = 8case. In chapter 4 we introduce maximal supergravity. We will focus mainly on the coset structureand embedding tensor formalism. This will allow us to introduce the objects with which we dothe calculations. We will also introduce a number of critical points that we will use to test ourhypotheses . In chapter 5 we will discuss various attempts at finding a bound for the lightest scalarin maximal supergravity. All of the expressions and conventions that are too big to put in the mainbody of this text have been collected in the appendices. We refer to these appendices throughoutthe text. The final appendix contains short instructions on how to do the calculations of chapter5. The calculations are not difficult but it is very convenient to have a short introduction. Thisappendix is added should the reader feel like he/she wants to check the calculations or try anotherbut similar approach of his/her own.

4

Chapter 2

Stability and supersymmetry

Supersymmetry has been studied intensely in the past 40 years for a variety of reasons: supersym-metry is the only way to combine space time and internal symmetries in a non trivial way (thisis the already mentioned Coleman-Mandula theorem), supersymmetric theories have in generalless problems with divergences then non supersymmetric theories, local supersymmetric theoriesare theories of gravity, supersymmetry is needed to introduce fermions in string theory, · · · . Wecan not and will not give a complete overview of supersymmetry in this chapter (or even in thisthesis).

In this chapter we will introduce a couple of different topics that are either useful later onor interesting for a master student. The first section will briefly explain what we mean withstability of a vacuum. The next two sections are on the construction of the particle contentin supersymmetric theories, a topic that is accessible and interesting. Finally, we will discussthe Kahler and superpotential which will be used in the discussion of the lower supersymmetrictheories. The treatment of the topics is compact and we encourage the reader to read the relatedliterature for a more complete explanation.

2.1 Stability

The main topic of this thesis is stability in maximal supergravity. Stability is important in moreareas of physics then just supergravity, so it can be treated in a more general quantum fieldtheoretic setting. In general, stability of quantum field theories is related to the masses of theparticles in a given critical point, we will focus on the scalar masses. The rule of thumb forMinkowski and de Sitter vacua is that we do not want any tachyons in the scalar spectrum, i.e.we want all the eigenvalues of the scalar mass matrix to be non negative.

We consider the Klein-Gordon equation to illustrate some of the problems a negative masssquared gives. The classical vacuum of the theory is defined by an critical point of the scalarpotential. The dynamics of the theory are obtained by expanding the fields around this criticalpoint. The easiest way to see the problems with negative square masses is in a Minkowski vacuum.The equation governing the scalar dynamics is the Klein Gordon equation:

∂µ∂µφ+m2φ = 0 (2.1)

Which has solutions:

φ ∝ e−ipµxµ

(2.2)

This solution will blow up whenever m2 is negative and hence this implies that we can not trustperturbation theory around the vacuum at late times when m2 is negative.

As another example consider the potential of the only scalar particle in the standard model,

5

the Higgs:

V = µ2η2 + λνη3 +1

4λ4η (2.3)

ν2 =µ2

λ(2.4)

The mass of the Higgs field is given by µ. If µ2 is positive then the potential has a global minimumin the origin and all is well. If µ2 is negative then the Higgs potential has a global maximum inthe origin and no global minimum at all. The origin has become an unstable vacuum and theHiggs field will start rolling down the potential towards −∞.

The situation becomes a bit more subtle when we consider (A)dS vacua. We have to accountfor the curved background so there will always be an interaction between the Ricci scalar andthe scalar field in the Lagrangian, we also lose the simple form of the Klein Gordon equation.Fortunately, it turns out that for de Sitter vacua our naive view from Minkowski physics stillholds and we have stability when all scalar masses are positive [16]. One has to be careful to givea proper definition of mass in this case, we will assume that we did.

Stability is even more interesting when we consider Anti de Sitter spaces because we can stillhave a stable Anti de Sitter vacuum when we have negative masses. Breitenlohner and Freemanshowed that in the case of an Anti de Sitter vacuum all scalar masses have to satisfy:

m2 ≥ − 3

4V (2.5)

Which is called the BF bound [17]. In this thesis we will only be interested in stability of de Sittervacua. We will consider our vacuum stable when all scalar masses are non negative.

2.2 Supersymmetry algebra and super multiplets

In this section we will use the superalgebra to construct the particle content of different super-symmetric theories. In non supersymmetric particle physics particles are usually defined to beirreducible representations of the Poincare algebra. Since the Poincare algebra is a subalgebraof the supersymmetry algebra, any irreducible representation of the supersymmetry algebra is arepresentation of the Poincare algebra. In general this representation will be reducible in irreps ofthe Poincare algebra. This means that a superparticle corresponds to a collection of particles thatare related to each other via the odd generators of the supersymmetry algebra. The irreduciblerepresentation of the supersymmetry algebra is usually called a supermultiplet.

The Lorentz group is generated by the rotations Ji and boosts Ki that can be put in oneantisymmetric tensor Mµν : M0,i = Ki and Mij = εijkJk. The Lorentz algebra is fully determinedby Mµν together with its commutator relations. The Poincare group is the Lorentz group aug-mented by the space time translation generators Pµ, the Poincare algebra is fully determined byMµν , Pµ and their commutator relations. The only additional generators that are present in thesupersymmetry algebra are the odd (or fermionic): Qiα, α = 1, 2 and i = 1, · · · ,N . α is a SU(2)index and i labels the generators. For instance for N = 4 supersymmetry there are a total of 8odd generators (4 + 4). We denote the complex conjugates by (Qiα)∗ = Qαi

1. The supersymmetry

1The notation with the dotted index is called the Van der Waerden notation. The dots have to do with thechirality of the spinor. It plays no role in this thesis other then to use the correct notation in this section and thereader should not worry too much about it.

6

algebra, besides the commutators between Pµ and Mµν reads:

[Pµ, Qiα] = 0 (2.6)

[Pµ, Qαi] = 0 (2.7)

[Mµν , Qiα] = i(σµν)α

βQiβ (2.8)

[Mµν , Qαi] = i(σµν)αβQβi (2.9)

Qiα, Qβj = 2σµαβPµδ

ij (2.10)

Qiα, Qjβ = εαβZ

ij (2.11)

Where Zij are bosonic central generators that are related to the generators of the internal sym-metry group of the theory at hand. Of particular interest for the following part of this section isthe commutator between Qiα and M12 = J3. This commutator can be written as:

[J3, Qi1] = − 1

2Qi1 (2.12)

[J3, Qi2] =

1

2Qi2 (2.13)

And taking the complex conjugate we see that Qi2 and Q1i raise the spin of a particle by half unitand that Qi1 and Q2i lower the spin by half unit.

Lets start constructing supermultiplets, the procedure we follow is described in [13]. Theeasiest case to consider is the massless case with no central charges, Zij = 0 (we will see laterthat the central charges are always zero in the massless case). We can go to the rest frame wherePµ = (E, 0, 0, E) such that the commutators become:

Qiα, Qβj =

(0 00 4E

)δij (2.14)

Qiα, Qjβ = 0 (2.15)

The first commutator implies that:

Qi1, Q1j = 0 (2.16)

⇒ 0 = < φ|Qi1, Q1j|φ > (2.17)

= ||Qi1|φ > ||2 + ||Q1i|φ > ||2 (2.18)

And by the positive definiteness of the Hilbert space this implies: Qi1 = Q1i = 0. We are left with

N Qi2 and N Qi

2 generators of the super algebra. We can redefine the generators in the followingway:

ai =1√4E

Qi2 (2.19)

a†i =1√4E

Q2i (2.20)

Such that they satisfy the following anticommutator relations:

ai, a†j = δij (2.21)

ai, aj = 0 (2.22)

a†i , a†j = 0 (2.23)

These are exactly the anticommutation relations of a set of creation and annihilation operators,where ai are the annihilation operators and a†i are the creation operators. We will interpret the

7

generators of the superalgebra as operators from this point that work on a Hilbert space. Wealready mentioned that these operators raise/lower the helicity of a state by 1

2 . To construct asupermultiplet one can start with choosing a state that is annihilated by all the ai (a so-calledClifford vacuum). This state will carry an irrep of the Poincare algebra (ie. a particle) with a

definite helicity |λmax >. We can create other states by acting with a†i on the state with maximumhelicity. In this way we create a tower of states:

|λmax > (2.24)

a†i |λmax > = |λmax −1

2>i (2.25)

a†ia†j |λmax > = |λmax − 1 >ij (2.26)

It is not hard to see that the total number of states at helicity level λmax − k2 is equal to

(Nk

)and

that the total number of states in a supermultiplet is equal to 2N . As an example consider themultiplet with N = 8 and λmax = 2. This is the multiplet of maximal supergravity. We have 1state with helicity 2, 8 states with helicity 3

2 , etc. This is written as:

[(2), 8(3

2), 28(1), 56(

1

2), 70(0)] (2.27)

The number in parenthesis represents the helicity of the state and the number in front the multi-plicity. We do not write the states with negative helicity, they have the same multiplicities as thestates with positive helicity. As a second example consider the case N = 6 and λmax = 2. Wenaively get the following multiplet:

[(2), 6(3

2), 15(1), 20(

1

2), 15(0), 6(−1

2), (−1)] (2.28)

However this multiplet is not CPT invariant since the helicity λ goes to −λ under a CPT trans-formation. In order to remedy this we include the conjugate multiplet with opposite helicities andwrite:

[(2), 6(3

2), 16(1), 26(

1

2), 30(0)] (2.29)

Which is CPT invariant. This is called CPT doubling. In this way all the massless supermultipletscan be constructed. They can be found for instance in [12]. There is another instance of CPTdoubling that is group theoretical in nature. Let us just briefly show it in case of the N = 2hypermultiplet. Going by the above we construct:

[(1

2), 2(0)] (2.30)

Under SU(2) R-symmetry (rotation of the superalgebra generators into each other) the helicity0 states behave as a doublet while the fermionic states are singlets. If the multiplet were CPTself-conjugate the two scalars would have to be real (or else they will not be invariant). Howevertwo real states can not form a SU(2) doublet and hence we need to use CPT doubling in this case.So we get the hypermultiplet:

[2(1

2), 4(0)] (2.31)

The only cases where we do not need CPT doubling is when N is a multiple of 4 and λmax = N/4[13].

The next step is to construct massive supermultiplets. The approach is the same as in themassless case but there are a few differences. We can once again go to the rest frame Pµ =(m, 0, 0, 0) of a state with mass m. If we assume that the central charges are still zero then

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we get the full set of 2N creation and 2N annihilation operators. This can be seen from theanticommutator:

Qiα, Qβj =

(2m 00 2m

)δij (2.32)

Qiα, Qjβ = 0 (2.33)

From which we deduce that none of the operators Qiα have to be set to zero. A massive super-multiplet with no central charges is generically called a long multiplet. It is again trivial to defineoperators that satisfy the usual oscillator algebra:

aiα =1√2m

Qiα (2.34)

a†αi =1√2m

Qαi (2.35)

We define a Clifford vacuum as a state that is annihilated by all aiα. From this we can startbuilding supermultiplets starting with a state that is a Clifford vacuum. We assume that theClifford vacuum is uniquely given (and always exists) for given N and given spin λ0. A differencewith the massless case is that now we can create states with lower and higher spin than theClifford vacuum hence it is no longer true that the spin of the Clifford vacuum is the maximal(or equivalently the minimal) spin in the multiplet. Consider as an example the case N = 1 withClifford vacuum |λ0 > then we can build the following states:

λ = λ0 |λ0 > (2.36)

a†1a†

2|λ0 > (2.37)

λ = λ0 −1

2a†

2|λ0 > (2.38)

λ = λ0 +1

2a†

1|λ0 > (2.39)

From which we can read of several supermultiplets (keeping in mind CPT doubling), for instance:

λ0 = 0 [(1

2), 2(0)] (2.40)

λ0 =1

2[(1), 2(

1

2), (0)] (2.41)

λ0 = 1 [(3

2), 2(1), (

1

2)] (2.42)

We see that these multiplets are usually longer than their massless counterparts hence the namelong multiplet. We should in principle distinguish between the states |λ0 > and a†

1a†

2|λ0 > since

they transform differently under parity. There are 22N states in a supermultiplet.The only case left to consider is the massive theory with non-trivial central charges. We can

always choose a basis such that the central charges go in block diagonal form:

Zij =

0 z1 0 0 · · ·−z1 0 0 0 · · ·

0 0 0 z2 · · ·0 0 −z2 0 · · ·...

......

.... . .

(2.43)

We will once again define operators out of Qiα and Qαi such that they form an oscillator algebra. Itturns out that the correct way to do this is (we assume N even, the odd case is a trivial expansion

9

of the algebra):

aIα =1√2

(Q2I−1α + εαβQγ(2I)δ

γβ)

(2.44)

a†αI =1√2

(Qα(2I−1) + εαβδ

βγQγ(2I)

)(2.45)

bIα =1√2

(Q2I−1α − εαβQγ(2I)δ

γβ)

(2.46)

b†αI =1√2

(Qα(2I−1) − εαβδ

βγQγ(2I)

)(2.47)

Where the capital index I runs from 1 to N/2. These operators have the following algebra:aIα, a

†αJ

= (2m− ZI)δααδIJ (2.48)

bIα, b†αJ

= (2m+ ZI)δααδ

IJ (2.49)

And all the other anti commutators equal to zero. From the algebra and positivity of the Hilbertspace it follows that 2m ≥ |ZI | by a similar argument as before. This is why we did not need toconsider the massless case with non-trivial central charges. As before creation operators with 2lower the spin of a state by 1

2 and creation operators with 1 raise the spin of a state by 12 .

Now an important question regards the number of operators that saturate the bound 2m = |Zi|(this is called the BPS bound). If none of the operators satisfy this equality then we have thesame number of creation operators as in the massive case without central charges and we get thesame number of degrees of freedom as in that case. If all of the operators saturate the bound wehave the same number of creation operators as in the massless case (which was fewer then themassive case without central charges since a few of the operators become trivial). Clearly, we getsome interesting results when q of the operators saturate the bound. The resulting multiplet iscalled short or q/N BPS. In principle we need to do the same as before, namely choose a Clifford

vacuum and construct the states using the non-trivial a†αI and b†αI . For a simple degree of freedomcounting this is not necessary however. This is because when we have a short multiplet with qoperators that satisfy the BPS bound then we have 2(N − q) non-trivial operators. This is thesame number of operators as for a long multiplet of N − q supersymmetry. Since we know thestates in those multiplets we are done if we properly account for CPT doubling. It turns out thatall short super multiplets have to be doubled due to CPT invariance. This is because they allcarry a BPS charge. The correct multiplets are all recorded in [12].

2.3 Local supersymmetry breaking

So far we have talked about the construction of supermultiplets from the superalgebra. This isquite a general thing, we did not write an explicit Lagrangian for the theories. Nevertheless we cangive some necessary conditions for local supersymmetry breaking based on the previous section.When we partially break supersymmetry from N to N ′, we might wonder if it is possible to reorderthe particles from the massless N supermultiplet into massive (since we break a local symmetry)N ′ supermultiplets. We will use N to denote the unbroken local supersymmetric theory and N ′to denote the broken local supersymmetric theory.

Since we assume that our supersymmetry N is locally supersymmetric it is a theory of gravityand hence contains a graviton. We will assume that the gravity multiplet is the only multiplet inN . Breaking to N ′ implies that we need a massless gravity multiplet of N ′ and N −N ′ massive 3

2multiplets of N ′ due to the super Higgs mechanism. The gravitini will absorb some spin- 1

2 particleand the spin-1 particles will absorb a scalar in the super Higgs mechanism. The particles thatremain after this have to be grouped in massless multiplets of N ′.

The fact that short multiplets have to be doubled and that we do not have long multipletsfor N > 3 excludes any breaking from N = 8, 6 to N ′ = 5 and from N = 5 to N ′ = 4. This is

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the first example of a restriction on supersymmetry breaking due to the particle content of thetheories. We shall give a few more examples:

Consider a N = 6 supergravity theory that breaks to a N ′ = 3 supergravity theory then thedecomposition in massless multiplets is:

[(2), 6(3

2), 16(1), 26(

1

2), 30(0)] → [(2), 3(

3

2), 3(1), (

1

2)] (2.50)

+ 3[(3

2), 3(1), 3(

1

2), 2(0)] + 4[(1), 4(

1

2), 6(0)] (2.51)

The 3 gravitino multiplets have to become massive due to the super Higgs mechanism. We havetwo different massive gravitino multiplets in N ′ = 3:

[(3

2), 6(1), 14(

1

2), 14(0)] 2[(

3

2), 4(1), 6(

1

2), 4(0)] (2.52)

It is easy to see that no combination of these two multiplets can match the number of gravitiniand vectors simultaneously. Hence breaking from N = 6 to N ′ = 3 is impossible in supergravity.

Another interesting example is breaking from N = 8 to N ′ = 6. The massless multiplets inthis case read:

[(2), 8(3

2), 28(1), 56(

1

2), 70(0)] → [(2), 6(

3

2), 16(1), 26(

1

2), 30(0)] (2.53)

+ 2[(3

2), 6(1), 15(

1

2), 20(0)] (2.54)

The gravitini multiplets will have to become massive again. The massive multiplets that we haveat our disposal are:

2[(3

2), 6(1), 14(

1

2), 14(0)] (2.55)

We see that this matches the gravitino and vector content and that a spinor and 6 scalars have beeneaten by the 6 vectors and gravitino to gain a mass. Hence it is possible to break supersymmetryfrom N = 8 to N ′ = 6.

It should be noted that just because we can reorder the particles in massive multiplets doesnot mean that there exists a theory that allows for such a symmetry breaking. There could bea reason why a particular supersymmetry breaking is not allowed that does not show up at theparticle level. If a theory allows for a supersymmetry breaking then it must also be possible toput the particle content in the correct multiplets. So at this level we are only able find necessaryconstraints for symmetry breaking, not sufficient ones.

2.4 Supersymmetric field theories

In the previous sections we have built the particle content of supersymmetric theories using thesupersymmetry algebra. We did not discuss the possibility of building a consistent field theoryfrom the given algebra. There are at least two different ways of constructing a supersymmetryinvariant Lagrangian given the field content. The first approach is to simply start with a freefield Lagrangian, balance the degrees of freedom and ”guess” the supersymmetry transformationson the fields. It can be very tedious to check if the Lagrangian is invariant and, if not, whatadditional terms are needed to make the Lagrangian invariant. The second approach is the socalled superspace formalism, in this formalism supersymmetry is manifestly present.

Superspace has two sets of coordinates, space time xµ and fermionic θa which anticommute.We can define superfields on the superspace similarly to fields in quantum field theory and define arepresentation of the supersymmetry algebra that acts on the superfields. In order to obtain irre-ducible representations of the supersymmetry algebra we must put constraints on the superfields.There are many choices here that depend on the nature of the superfield (scalar, spinor, · · · ) and

11

representation of the algebra. The reader can find many examples of this in the literature. As anexample consider the chiral superfield:

Φ(y, θ) = ψ(y) +√

2θφ(y) + θθF (y) (2.56)

Where ψ is a scalar field, φ is a spinor field, θa are the fermionic coordinates of the superspaceand F is an auxiliary field which balances the degrees of freedom and has no kinetic term. Thefield is called chiral due to the constraint:

(−∂a − i(σµ)aaθa∂µ)Φ = 0 (2.57)

The field content corresponds to the well known Wess-Zumino theory. We see that the differentfields are the coefficients of the expansion in terms of the fermionic coordinates. They can beextracted by differentiating the superfield and setting the fermionic coordinates to zero. We knowthat any theory that we build from this superfield will behave properly under supersymmetrybecause it is defined to be an irrep of the algebra.

The most general Lagrangian that we can write for the chiral superfield is:

L =

∫d4θK(Φ,Φ†) +

∫d2W (Φ) +

∫d2θW (Φ†) (2.58)

K is called the Kahler potential and W is called the superpotential. We integrate with respect tothe fermionic coordinates to extract the different coefficients in the θ expansion of the potentials,integration is the same as differentiation for fermionic variables. The Kahler potential gives thekinetic terms and the superpotential gives the interaction terms of the theory. The potentials are inprinciple arbitrary functions of their arguments but there can be additional constraints dependingon physical requirements (for instance, if we want a renormalizable theory). The specific theorythat one wishes to study is specified by the field content (superfields+constraints), the Kahlerpotential and the superpotential. If we add different multiplets to the theory then we will havedifferent Kahler potentials and superpotentials for each multiplet.

Supergravity is nothing more then a local supersymmetric field theory. It is a surprising factthat if one makes the supersymmetry generators depend on the local coordinates then one hasto introduce a spin-2 particle which acts as the graviton. It is possible to write a superspaceformalism for low N supergravity but it is unknown if there exists a general superspace formalismof maximal supergravity.

One of the simplest supersymmetric theories of gravity is the supersymmetrized version ofgeneral relativity. The dynamical variable in GR is the metric which can be written locally usingthe vierbein, eaµ:

gµν(x) = eaµebν(x)ηab (2.59)

Where the Latin indices a, b are so called flat indices and the Greek indices µ, ν are curved indices.The flat indices can be acted upon by a local Lorentz group, which comes with its own gauge fieldωabµ (also called the spin-connection). It might look as if the introduction of the spin connectiongives too many degrees of freedom in the theory but it is possible to write the spin connection interms of the vierbein balancing the degrees of freedom.

In order to move from gravity to supergravity all one has to do is write super- in front of allobjects. We have the supervierbein: EMΛ (x, θ), the superspin connection ΩMN

Λ , superlocal Lorentzetc. The indices are now capital to remind us that they have a bosonic and a fermionic component.We can gauge fix the different components of the different superfields conveniently as:

Emµ (x, θ = 0) = emµ (2.60)

Eaµ(x, θ = 0) = φaµ (2.61)

Ωmnµ (x, θ = 0) = ωmnµ (2.62)

12

From which we see that in the bosonic part of superspace we have the vierbein, spin connectionand newly introduced gravitino: φaµ. The action for N = 1 supergravity in superspace is thengiven by the simplest invariant possible:

S =1

2k2

∫d4xd4θ sdet(EMΛ ) (2.63)

Where sdet is the superdeterminant which is just the generalization of the normal determinant onsuperspace.

If we want to couple matter to gravity in the superspace formalism we need to write invari-ant combinations of the supervierbein with the Kahler potential and superpotential. For chiralsuperfields the most general invariant action is given by:

S =

∫d4xd4θE[K(Φ,Φ†) + Φ†eV Φ] +

∫d4xd4θE [W (Φ) + Tr(W 2)] (2.64)

Where E is the sdet of the supervierbein, V is a super gauge field and E is proportional to thesecond covariant derivative of E. We see that also for N = 1 supergravity coupled to chiral matterfields the freedom in the theory is contained in the Kahler potential and the superpotential.

A useful feature of supersymmetric theories is that the scalar fields can be viewed as coordinatesof a manifold of a type that is generally fixed by the theory. This allows us to talk about thescalar sector of the theory in a geometric language. This geometrical language will be apparentin the stability analysis of N = 2 supergravity. The interesting objects in that case are directlydefined as vectors on the scalar manifold. This should come as no surprise since paths and hencedirections can be seen as the scalars of the theory. We are interested in special scalars and whichcorrespond to special directions on the scalar manifold.

The scalar manifold structure is also important in maximal supergravity and we will spendsome time on the arguments leading to the coset structure of that theory. The actual analysisin maximal supergravity will rely on objects that are functions on the scalar manifold just as inthe lower supersymmetric case. However, we can do the full analysis in the origin of the scalarmanifold so the scalar dependence of the objects drops out in this case.

We already mentioned that the superspace formalism of maximal supergravity is unknown.We will devote a chapter of this thesis to introduce the relevant objects in maximal supergravity.This section was merely used to introduce the different potentials that play a role in N = 1, 2supersymmetric theories. They will be the starting point of our analysis in the next chapter.

Most of the information in this section comes from [3] which is a very nice and complete set oflecture notes when one starts learning about supersymmetry and supergravity. We have left outa tremendous amount of information and derivation in this section, do read the notes.

13

Chapter 3

Stability in N = 1, 2supersymmetric theories

Our approach of the analysis of stability in maximal supergravity is inspired by the discussion ofstability in N = 1 and N = 2 supergravity theories and by the discussion of stability in globallysupersymmetric theories. In this chapter we will give an overview of the relevant results in theliterature (references [10] and [11]). We will also add a few results of our own in the case of N = 2supergravity.

3.1 Global supersymmetry

Let us first look at a study to find a bound on the lightest scalar in theories with global supersym-metry (theories without gravity), this section is a discussion of [10]. Lets start with explaining whya bound on the lightest scalar mass exists for a theory with broken gauge and/or supersymmetry.It is well known that when symmetry is preserved particles in the same multiplet must have thesame mass. For example in a N = 1 supersymmetric theory this implies that particles in the samechiral multiplet have the same mass and that particles in the same vector multiplet have the samemass.

When only supersymmetry is broken there exists a physical Goldstone fermion which has zeromass due to Goldstone’s theorem. This Goldstone fermion is usually called the Goldstino. Ingeneral for N = n supersymmetry there are n Goldstini and, since we have N = 1 supersymme-try, a single Goldstino exists. The Goldstino has two scalar superpartners, the sGoldstini, thatare massless in the supersymmetric limit because the Goldstino is massless in this limit. Themasses of the sGoldstini are fully determined by splitting effects that depend on the amount ofsupersymmetry breaking.

Breaking only gauge symmetry puts a constraint on the lightest scalar via the Higgs mechanism.This can be understood in terms of multiplets. The gauge vectors are part of vector multiplets.Breaking gauge symmetry implies that such vectors absorb a scalar particle, the would-be Gold-stone, to obtain a mass. This, in turn, implies that the vector particle gets an additional degreeof freedom. To balance the degrees of freedom in the vector multiplet, it absorbs the chiral mul-tiplet of the would-be Goldstone. The newly formed multiplet contains one complex scalar, twotwo-component fermions and one three-component vector (we have seen this in chapter 2.3). Themass of the vector is determined by the Higgs mechanism and hence the mass of the physicalscalar particle, the would-be Goldstone partner, is also determined by the Higgs mechanism. Ingeneral for each gauge vector, there is a single would-be Goldstone partner and a single massivemode due to the Higgs mechanism.

When both gauge symmetry and supersymmetry are broken the spectrum becomes a bit morecomplicated. For the chiral multiplets that are not absorbed by a vector multiplet the result isthe same as when only supersymmetry is broken. The masses of the would-be Goldstone partners

14

split from the masses of the vector particles. For a theory with N = 1 with n scalars and kgauge vectors the scalar spectrum consists of two sGoldstini with masses that are fully determinedby splitting effects, k would-be Goldstone partners with masses that deviate from the vectormasses via splitting effects and n− k − 2 scalars with arbitrary masses that can be tuned via thesuperpotential.

We will first expand on the discussion of supersymmetric field theories from section 2.4 anduse [10] to introduce a few additional concepts before we make the above discussion more precise.In the end we will analyze a simple theory (U(1) gauge group) as an example.

We consider a generic N = 1 theory with n chiral multiplets Φi and k vector multiplets V a.The indices i, j, k, · · · label the chiral multiplets and their scalars φi and spinors ψi and the indicesa, b, · · · label the vector multiplets and their spinors λa and vector bosons Aaµ. These multipletsdefine a set of superfields which are used to specify the Lagriangian of the theory via a real Kahlerpotential K and a holomorphic superpotential W as before. In addition the gaugings introducesa holomorphic gauge kinetic function Hab (which is set constant at Hab = hab) and holomorphicKilling vectors Xi

a. These quantities are not independent but have to satisfy a number of relationsto guarantee invariance of the Lagrangian.

Furthermore, we know that in general, the Kahler potential defines a metric gij and a Riemanntensor Rijkl that determine the scalar geometry of the theory. The scalar geometry comes with acovariant derivate ∇i, we will write covariant derivatives as: Ki := ∇iK. Finally, we define thescalar dependent matrices: Qiaj = i∇jXj

a which appear in the linear transformations of the fermionfields. In addition to the physical fields we also have a number of auxiliary fields to balance theoff-shell degrees of freedom. These auxiliary fields are denoted by F i and Da. They are auxiliaryin the sense that they do not have a kinetic term in the Lagrangian and hence their equations ofmotion are purely algebraic. These fields are nevertheless important for our discussion since theyplay a crucial role as symmetry breaking order parameters.

We are trying to study the scalar mass spectrum of the above theory for arbitrary values ofthe Kahler potential and the superpotential and for arbitrary gaugings (specified by the Killingvectors). In order to do that, we must have a general expression for the scalar mass matrix. Thescalar mass matrix is in general found by defining a vacuum and looking at small fluctuationsaround the vacuum solution. In general the vacuum is defined by constant values of the scalarfields φi and vanishing values of the fermions ψi, λa and the vectors Aaµ such that the potentialenergy has a local minimum.

We can write the scalar potential for this theory as:

V =gijFiF

j+

1

2HabD

aDb (3.1)

We see that the value of the scalar potential is always non negative. This is always true for theorieswith global supersymmetry. In fact, it is well known that for theories with global supersymmetrythe supersymmetric ground state has lowest energy V = 0 and any vacuum with V > 0 has brokensupersymmetry. From our expression of the scalar potential we can see that this implies that eitherF i 6= 0 or Da 6= 0 whenever supersymmetry is broken which underlines our earlier claim that F i

and Da can be viewed as order parameters of supersymmetry breaking. We define the following

vector: φI = (φi, φi), such that the Lagrangian for the scalar fields can be written as:

L =1

2gIJ∂µφ

I∂µφI − 1

2m2IJφIφJ (3.2)

And the scalar mass matrix has the following form:

m2IJ

=

(m2ij

m2ij

m2ij

m2ij

)(3.3)

With entries:

m2ij

=gkl∇iWk∇jW l −RijklFkF

l+ g2habXaiXbj + gQaijD

a (3.4)

m2ij =−∇i∇jWkF

k − g2habXaiXbj (3.5)

15

From these expressions we see that the scalar masses are not only determined by the superpo-tential but also by the auxiliary fields and the Killing vectors. This reflects the discussion in theintroduction where we said that some scalar particles have masses that are determined by splittingeffects due to symmetry breaking. The fermion masses are found to be:

µij =∇iWj (3.6)

µab =0 (3.7)

µia =√

2Xai (3.8)

And the vector boson masses read:

M2ab =2g2gijX

i(aX

j

b) (3.9)

The masses of the vector bosons are determined by the Killing vectors. The vector masses are zerowhenever gauge symmetry is preserved and they are different from zero whenever gauge symmetryis broken. This implies that the Killing vectors can be viewed as an order parameter of gaugesymmetry breaking.

We already mentioned that there exists a physical Goldstino fermion with vanishing masswhenever supersymmetry is broken. This fermion is given by the following combination of fields:η ∝ F iψ

i + i√2Daλ

a. The fact that this expression is proportional to the auxiliary fields once

again shows that they are intimately related to supersymmetry breaking. Local gauge symmetryis spontaneously broken whenever the vector bosons have a mass i.e. whenever: M2

ab 6= 0. In

that case there exist non physical would-be Goldstone scalars σa ∝ Xaiφi +Xaiφ

i. These scalars

are absorbed by the gauge bosons through the Higgs mechanism and their scalar partners thatbecome a part of the same multiplet are interesting because they have a mass that has to be equalto the vector boson mass.

Next we consider how to find the physical masses of the scalars. They correspond to theeigenvalues of the scalar mass matrix. In order to find the eigenvalues we define a basis by a setof vectors that is orthonormal with respect to the metric gIJ . The elements of the scalar massmatrix in this basis will be the eigenvalues. The main result from linear algebra that will be usedis that the eigenvalues of any principal submatrix of m2

IJmust always be larger then the minimal

eigenvalue of the whole matrix. The larger the principal submatrix the better the bound on thelowest eigenvalue will be in general. A principal submatrix is constructed by projecting the scalarmass matrix on some subspace by using a subset of the basis vectors.

From the discussion above, we know that an interesting submatrix to consider is the onespanned by the directions specified by the sGoldstino and Goldstone partners because these par-ticles have masses that are constrained by symmetry and cannot be freely tuned by tuning thesuperpotential as is the case for the other scalar masses. These directions are specified by F i

and Xia. The sGoldstino direction is always orthogonal to the Goldstone directions. However, the

Goldstones Xia are in general not orthogonal to each other. However, it is possible to do a rotation

such that the rotated vectors Xia are orthogonal to each other. Following this discussion we define

the following normalized vectors:

f i =F i

F(3.10)

xia =√

2gXia

Ma(3.11)

With these normalized vectors we can write the following real orthonormal basis that will be used

16

to project the scalar mass matrix on:

f IA =(f i, 0) (3.12)

f IB =(0, fi) (3.13)

xIa+ =1√2

(xia, xia) (3.14)

xIa− =i√2

(xia,−xia) (3.15)

The two directions f IA and f IB are related via a rotation to the real partners of the Goldstino. Thedirection xIa+ describes the unphysical would-be Goldstone modes and the direction xIa− describestheir partners. Hence the vectors f IA, f IB and xIa− define a 2 + k dimensional subspace whichcontain the directions that are dangerous to stability. The projected mass matrix can be writtenas:

m2αβ

=

m2ff

∆ −√

2im2∗fxb

∆∗ m2ff

√2im2

fxb√2im2

fxa−√

2im2∗fxa

2m2xaxb

(3.16)

Where we define ∆ = m2ijf

if j and m2ff

= m2ijf if

j. These entries can in principle be calculated

explicitly. It turns out that all of the entries take a nice form after some manipulation except for∆. In fact, ∆ depends on the third derivatives of the superpotential and hence can vary over theentire complex plane. The other expressions have the following form:

m2ff

=−Aff |F |2 (3.17)

m2fxb

=−Afxb |F |2 (3.18)

m2xaxb

=1

2M2ab −Bxaxb |F |2 (3.19)

The explicit expressions can be found in [10]. From the above expressions it is clear that thesGoldstino masses are indeed proportional to |F |2 and that the Goldstone partner masses areproportional to M2

ab − |F | as one would expect. In order to find an upper bound for the lowestscalar mass we have to find an upper bound for the smallest eigenvalue of the matrix m2

αβ. In the

case where the Goldstone partners are very heavy compared to the sGoldstini the upper boundis given by the sGoldstino mass. When the Goldstone partners have masses comparable to thesGoldstini masses the analysis is not so easy. It is for this reason that we will focus on the simpleU(1) case where k = 1 and the matrix m2

αβis 3 × 3. Calculating the smallest eigenvalue of this

matrix one finds:

m2 = max[12

(m2ff

+ 2m2xx)− 1

2

√(m2

ff− 2m2

xx)2 + 8|m2fx|2

](3.20)

There are two complications in calculating this bound. The first comes from the contribution of∆. It is possible to use some general considerations from linear algebra to determine |∆| andarg(∆) given that we want to maximize the eigenvalues which solves the first complication. Thesecond complication is the implicit dependence of the eigenvalues on the vacuum solution. It isfor this reason that the above bound contains a max

[ ]. This means that the superpotential is

tuned such that the eigenvalue is maximal. It is possible to show that it is always possible to tunethe superpotential in such a way that the bound is saturated.

In the simplest case of renormalizable gauge theories with a U(1) gauge symmetry, wherethe Kahler potential is quadratic and the Killing vector linear, it is possible to do the analysisexplicitly. In this case we have a flat scalar manifold and a linear relation between Ki and Xi.The lightest scalar field is one of the sGoldstini and it has a positive square mass. Next, consider a

17

theory with a non-trivial Kahler potential but with a U(1) gauge symmetry. In this case we havea curved scalar manifold and the lightest scalar is in general identified with a linear combinationof the sGoldstini and the Goldstone partners. To get an explicit expression one would have to doa case by case analysis to calculate the values of the expressions explicitly as well as to tune thesuperpotential explicitly to saturate 3.20.

3.2 N = 1 supergravity with chiral multiplets

After this overview of a way to describe stability in globally supersymmetric theories we move tosupergravity theories. We give a discussion of [11]. In N = 1 supergravity with n chiral multipletswe have the gravity multiplet containing the graviton and the gravitino and the chiral multipletscontaining complex scalar fields φi and chiral fermions χi, where i labels the multiplet. Thistheory is described by the holomorphic superpotential W (φ) and the Kahler potential K(φ, φ) asbefore. The scalar manifold in this case is Kahler-Hodge with a metric given by: gij = Kij . Inorder to analyze the theory we will define a new potential:

L = eK/2W (3.21)

That will make the expressions for the scalar potential and sGoldstino resemble the N = 2 case,which is richer and therefore more interesting. This potential is anti holomorphic (∇iL = 0) withrespect to the covariant derivative defined on the scalar manifold ∇. The holomorphic derivativeof L is related to the order parameter of supersymmetry breaking:

∇iL = Ni (3.22)

Using these quantities the scalar potential takes the following form:

V = NiNi − 3|L| (3.23)

The scalar mass matrix is defined by the second covariant derivative of the scalar potential:∇i∇jV = m2

0ij and the mass of the gravitino is given by the scale of supersymmetric AdS:

m23/2 = L.

The study of stability in globally supersymmetric theories relied on the sGoldstino and Gold-stone partner directions. In this section we only consider the sGoldstino directions and we will notconsider the Goldstone partners. In fact it is not clear how much new information can be foundin the Goldstone partners. It is known that the Goldstone partner directions and the sGoldstinodirections are no longer orthogonal as was the case in the global case [10]. This will lead to asmaller subspace of directions then when the directions are orthogonal.

All multiplets in the theory can be made arbitrarily massive by tuning the superpotential W .The only exception to this is the Goldstino multiplet that is only allowed to have mass splittingsdue to supersymmetry breaking effects as was the case in globally supersymmetric theories. Sincethe article [11] does not consider broken gauge symmetry we will not consider the Goldstonepartners in this chapter.

Supersymmetry is broken whenever the fermion shifts are nonzero, Ni 6= 0, making themsupersymmetry order breaking parameters. The associated Goldstino direction is then given by:Niχ

i and the sGoldstino is given by: η = Niφi. The sGoldstino is a complex field and hence

carries two degrees of freedom. The mass associated to the sGoldstino direction is given by theprojection of the scalar mass matrix on the sGoldstino direction:

m2η =

m20ijNiN j

|N |2(3.24)

= 3(Rη +2

3)m2

3/2 +RηV (3.25)

18

Where Rη is the normalized holomorphic sectional curvature along the sGoldstino direction:

Rη = −RijpqN

iN jN

pNq

(NkNk)2

(3.26)

Where Rijpq is the Riemann tensor of the scalar manifold. The sGoldstino mass gives an upperbound on the lowest eigenvalue of the full scalar mass matrix because it is the average of thetwo real sGoldstini. Stability requires mη ≥ 0 for Minkowski and de Sitter vacua and m2

η ≥ 34V .

From this we can extract a bound on Rη and on the scalar geometry of the theory. If we defineγ = V

3m23/2

then the bound becomes:

R ≥ −2

3

1

1 + γ(3.27)

For Minkowski and de Sitter vacua and:

R ≥ −2

3

1− 98γ

1 + γ(3.28)

for Anti de Sitter vacua. It is easy to see that the bound becomes more restrictive as γ becomesbigger, which corresponds to an increasing cosmological constant. This bound gives an necessaryrestriction on the sectional curvature of the scalar manifold, but it can always be satisfied bytweaking the cosmological constant. We find that there can be stable de Sitter vacua in N = 1supergravity with only chiral multiplets. This should come as no surprise since such vacua havebeen found [18].

3.3 N = 2 supergravity with only hypermultiplets

Next we consider N = 2 supergravity. The approach will be the same as in the previous sectionsbut since there are more sGoldstini the problem gets a bit more complicated compared to theN = 1 supergravity case. Since we are working with N = 2 supergravity we will have a gravitymultiplet containing the graviton gµν , two gravitini ψAµ and a graviphoton Aµ. The index A is aSU(2) fundamental index. The hypermultiplets contain 2n complex fermions ζα, α is an Sp(2n)index, and 4n real scalars qu. The SU(2) indices A,B are raised and lowered by the invarianttensor εAB and a symmetric pair (AB) can be replaced with a triplet index x by using the Paulimatrices σxAB :

Nx = iσxCA εCBN(AB) (3.29)

We can raise and lower the Sp(2n) indices using antisymmetric symplectic tensors Cαβ and Cαβ

such that a tensor with a raised/lowered index transforms in the right way under Sp(2n).This theory has a nice description in terms of geometric quantities and we will first take some

time to properly define all the necessary objects. It follows from the field content that the scalarmanifold is a quaternionic Kahler manifold of dimension 4n with holonomy group Sp(2n)×SU(2).On the scalar manifold we have naturally a triplet of almost complex structures Jx that satisfyan SU(2) algebra. Associated with this triplet is a triplet of Hyperkahler two forms Ωxuv that areidentified with the field strength of the SU(2) part of the holonomy group. It should come as nosurprise that the curvature of the SU(2) part is associated with the field strength of the SU(2)part and hence with Ωxuv.

RABuv = −iΩxuvσxAB (3.30)

We can define a vielbein on the scalar manifold UαAu that is related to the metric in the usualway:

huv = εABUαAu UβBv (3.31)

The inverse vielbein is given by UuαA and it satisfies UuαAUαAv = δuv . Using the vielbein we can give

the curvature form associated with the Sp(2n) part of the scalar manifold:

Rαβuv = εABUγA[u U δBv] (−2δα(γδ

ββδ) + Σαβγδ ) (3.32)

19

The tensor Σαβγδ is fully symmetric but otherwise arbitrary. It represents the only freedom inthe curvature of the scalar manifold. This is not surprising since it is associated, via the groupSp(2n), with the only freedom we allow in the theory, namely the number of hypermultiplets n.The full Riemann tensor of the scalar manifold is given by:

RαAβBuv = RABuv Cαβ +Rαβuv ε

AB (3.33)

We gauge the theory using a Killing vector ku of the metric huv. in other words we gauge anisometry of huv with the graviphoton. Associated with the Killing vector is a triplet of Killingpotentials, P x defined by:

∇uP x = 2Ωxuvkv (3.34)

It turns out that the fermion shifts, and hence the order parameters of SUSY breaking, are relatedto the Killing vectors:

NAα = 2UAuαk

u (3.35)

A comparison with the N = 1 case is in order. We can identify the Killing potentials P x with thepotential L and the fermion shifts NA

α with Ni. So we see that even though a lot of details aredifferent we can still make cross relations between the different cases.

In order to study stability we first need the potential:

V = NαAN

Aα − 3P xP x (3.36)

Notice the resemblance to the potential in the N = 1 theory 3.23. A critical point is defined asa point where ∇uV = 0 and the scalar masses are given by the second covariant derivative of thepotential:

m20uv =

1

2∇u∇uV (3.37)

The mass of the graviphoton is determined by the Killing vector:

m21 = 4kuku (3.38)

This seems to imply a relation between the Killing vector and the Goldstone direction (sincethe Goldstone particle gives mass to the vector bosons), this relation was given explicitly in theglobal case. Furthermore, the Killing vector is also related to the scale of SUSY breaking viaNAα = 2UAuαk

u which is reflected in the statement that the Goldstone partner directions and thesGoldstino directions are no longer orthogonal. The final interesting mass is the gravitino masssince this is related to the supersymmetric AdS scale of the theory:

m23/2 = P xP x (3.39)

Next we define the sGoldstino directions. The Goldstino fermions are related to SUSY breakingin the vacuum and the corresponding directions are given by the order parameter of SUSY breakingNAα . The two independent Goldstinos transform as a doublet under SU(2) and they have four

real sGoldstino partners given by: ηAB = NABu qu (qu are the scalar fields). The quantity NAB

u

transforms as a tensor product of two SU(2) doublets and hence contains a singlet and a tripletof SU(2). The sGoldstino directions can be computed by acting with the inverse vielbein on NB

α ,one finds:

NABu = Nuε

AB + iNxuσ

xAB (3.40)

The singlet corresponds to the Killing vector ku and hence is an unphysical gauge direction, thetriplet gives the physical sGoldstino directions and can be written as Nx

u qu. We say that the

physical sGoldstino directions are given by Nxu .

Now we have a variety of approaches that we can use to study stability of non SUSY vacua. Sofar we have only considered projecting the mass matrix on the sGoldstino (and Goldstone partner)directions. Another approach could be to trace the whole mass matrix or to find an interesting

20

eigenvector of the whole mass matrix. We will come back to these possibilities later, first we willexplore what we can conclude from projecting the mass matrix in the sGoldstino directions.

By projecting the full scalar mass matrix on the sGoldstino directions we find:

m2xyη =

m20uvN

uxNvy

NwNw(3.41)

This tensor is contained in the symmetric tensor product of the triplet with itself: (3⊗3)s = 1⊕5.It can be shown using a straightforward calculation that the sGoldstino mass matrix is equal to:

m2xyη = −(Rxyη + 3δxy)(V + 3m2

3/2) + 4(δxy − πxy)m23/2 (3.42)

This mass matrix already has a number of interesting features. The projector:

πxy =P xP y

P zP z(3.43)

specifies a direction on the scalar manifold and it has as an eigenvector P x. The combinationδxy − πxy, clearly identifies directions orthogonal to P x. The tensor Rxyη holds all the freedom inthe curvature and it can be written as:

Rxyη = −2δxy − ΣαβγδNαANβANγANδA

(N εENεE)2(3.44)

The tensor Σ is traceless and hence is contained in the 5 and the delta is a singlet of SU(2), hencethe 1 + 5 structure can be seen clearly here. The geometric interpretation of Rxyη is that it isrelated to the sectional curvature of the sGoldstino directions.

Metastability of the vacuum implies that the eigenvalues of m2xyη are positive or above the BF

bound. The freedom of choosing Σ makes it difficult to find a general expression for the eigenvaluesof this matrix, but the fact that the Σ lives in the 5 (contractions with the δ vanish) suggeststhat some general (n independent) information can be found in the singlet. Indeed, the singletcorresponds to the trace of the mass matrix. The trace of a matrix is the sum of its eigenvaluesand hence it should always be positive (or above the BF bound) as a necessary condition. Informula form the above discussion implies for Rxyη that:

Rxyη δxy = −2− ΣαβγδN

αANβANγANδA

(N εENεE)2δxy (3.45)

= −2 (3.46)

And hence the trace of the complete sGoldstino mass matrix is:

m2η = −1

3(1 + 9γ)m2

3/2 (3.47)

Where γ = V3m2

3/2

. From the above formula it is obvious that there is no positive value for γ for

which m2η is positive and hence we can conclude that there are no stable de Sitter vacua in this

theory.We might wonder if there is any interesting information in the 5. To determine the 5 we have

to look at the traceless part of m2xyη :

5 = m2xy −Am2ηδxy (3.48)

The coefficient A is determined from the tracelessness requirement: m2xyη δxy − 3Am2

η = 0, fromwhich we find A = 1. A short calculation shows that:

5 = −(2δxy −Rxyη )(V + 3m23/2) + 4(

1

3δxy − πxy)m2

3/2 (3.49)

21

The first term is clearly proportional to Σ and the second term is the traceless part of πxy. It ishowever unclear what the general interpretation of this irrep could be. That is, there is not such aclear relation between the 5 and the eigenvalues of the sGoldstino mass matrix as there is betweenthe singlet and the eigenvalues of the sGoldstino mass matrix.

An illuminating special case occurs when Σ = 0. In that case we can derive a stronger boundthen given in 3.47. There are several equivalent ways of deriving this bound. Let us start bytaking the sGoldstino mass matrix 3.42 and setting Σ = 0:

m2xyη = −δxy(V + 3m2

3/2) + 4(δxy − πxy)m23/2 (3.50)

= (δxy − πxy)(m23/2 − V )− (V + 3m2

3/2)πxy (3.51)

This clearly shows that there is a direction given by P x for which the projection of the massmatrix is always negative:

m2xyη P x = −(V + 3m2

3/2)P y (3.52)

Rewriting the eigenvalue of this equation:

−3(1 + γ)m23/2 (3.53)

Which gives a stronger bound on stability then 3.47. A slightly different way to derive the sameresult is to look at the irreps of the sGoldstino mass matrix. The singlet remains the same whenΣ = 0 but the 5 changes:

5 = 4(1

3δxy − πxy)m2

3/2 (3.54)

=4

3(δxy − πxy)m2

3/2 −8

3πxym2

3/2 (3.55)

Projecting the 5 on the direction P x we find as eigenvalue − 83m

23/2. This does not give a bound

on the eigenvalues of the sGoldstino mass matrix, for this we should include the information thatwe got from the singlet. We can find an eigenvalue of the sGoldstino mass matrix when we addthe singlet to an eigenvalue of the 5:

−8

3m2

3/2 −1

3(1 + 9γ)m2

3/2 = −3(1 + γ)m23/2 (3.56)

Which is the same bound as we found before. In this case it is slightly more complicated to firstextract the irreps from the original sGoldstino mass matrix and then look at their eigenvalues thanto look directly at eigenvalues of the sGoldstino mass matrix. However, for more complicated massmatrices it could be worth the effort to extract the irreps. This is because it may be impossibleto find an eigenvector of the whole mass matrix while it is possible to find eigenvectors for theindividual irreps. We could then find a bound for the special case where all the irreps for which wedo not have an eigenvector are zero. This would be like proving that de Sitter vacua are unstablefor a subset of all cases.

Instead of looking for eigenvectors of the sGoldstino mass matrix it might be interesting tolook for eigenvectors of the full scalar mass matrix. The full scalar mass matrix is given by:

m20uv = 4∇ukw∇vkw − 4Rusvtk

skt − 3∇uP y∇vP y − 3P y∇(u∇v)Py (3.57)

One natural candidate to be an eigenvector is the sGoldstino direction NuxP x. The result ofcontracting the scalar mass matrix with this direction is:

m20uvN

vxP

x = (−45

8P yP y − 7

6ktkt)N

xuP

x − 2Σusvtkskt∇vP xP x (3.58)

The derivation of this statement is done in Appendix B. It is clear that the sGoldstino is aneigenvector if Σ = 0. It might be very interesting to see if we can do analogous calculations inN = 8 supergravity.

22

Chapter 4

Maximal supergravity

In this chapter we introduce N = 8 supergravity in 4 space time dimensions. We will start bydiscussing the SO(8) invariant theory. Then we will move on to more general gauged maximalsupergravity and introduce the embedding tensor. Finally we will briefly discuss the going to theorigin approach and give several explicit examples of vacua.

Maximal supergravity is a theory with a unique field content which consists of 1 graviton, 8gravitino, 28 vectors, 56 Majorana spinors and 70 scalars. It was shown before that this is aconsistent multiplet for a theory with N = 8 and with a maximal spin 2 particle. In the ungaugedversion the theory is unique and there is no room for interaction terms or a scalar potential.Gauging the theory introduces interactions between the different particles, the gauge interactions,where a subset of the vector particles become gauge bosons. Gauging the theory also introduces ascalar potential. The first attempts at gauging maximal supergravity consistent of choosing somegroup as gauge group and all the couplings as well as the scalar potential had to be derived in acase by case fashion and checked for consistency with supersymmetry.

The embedding tensor parametrizes the gauge group in a more abstract way. As long as theembedding tensor components satisfy a number of constraints the gauging will give a consistenttheory. Once the theory is gauged using the embedding tensor, we find that the scalar potentialis fully determined by two irreducible tensors A1 and A2, that naturally appear in the derivation.Analyzing the vacuum structure for a given gauging makes use of the going to the origin approachwhich greatly simplifies the analysis.

Another important feature of maximal supergravity is the duality invariance of the theory. Thetheory allows for the definition of dual vector fields that, after a choice of basis, can also be usedas gauge vectors. This duality property is central to a lot of research in maximal supergravity.For instance, the vacuum structure depends on the choice of duality frame as well as on the gaugegroup. We will only shortly mention the duality invariance and not spend too much time on it.

Both the embedding tensor as well as the going to the origin approach are tools that werederived in the past decade. Without these tools the analysis of stability in maximal supergravitywould be far more difficult if not impossible. Even so, making general statements on stability thatare independent of a specific gauging is still quite hard.

4.1 SO(8) maximal supergravity

The first gauged version of maximal supergravity that was fully explored was SO(8) supergravityby de Wit and Nicolai in the early 80s. We will also start our overview of maximal supergravitywith SO(8) supergravity. In particular, we will introduce the E7(7)/SU(8) coset structure of thetheory and the so-called T -tensor in the SO(8) setting. We will not give the full Lagrangian norwill we discuss its explicit supersymmetry transformations nor will we go too deep into the grouptheoretic arguments that are used in [4]. If the reader is interested in a more rigorous derivation,we are happy to direct your attention to that paper.

23

The reason to start with the coset structure of the theory is that it plays a central role in theconstruction of maximal supergravity. In particular, the duality group E7(7) that comes from thecoset structure plays a crucial role in the background. The SO(8) theory will allow us to introducetwo SU(8) covariant tensors A1 and A2 that fully determine the scalar potential, the interactionsand the particle masses for maximal supergravity (for any gauging, as it turns out).

The fermions of maximal supergravity transform internally in the 8 (gravitini) and 56 (spinors)representations of SU(8). The graviton, being a single state, has trivial internal transformationproperties. The symmetry properties of the scalars are off course determined by their cosetstructure, which is E7(7)/SU(8). Maximal supergravity contains 28 vector fields Aµ

IJ (capitalindices are SO(8)) that enter the Lagrangian via their field strengths Fµν

IJ = 2∂[µAν]IJ . The

vectors are invariant under a global SO(8) symmetry group. This is the largest invariance groupthat they allow since there are only 28 vectors and hence the maximal dimension of the adjoin repof the gauge group is 28. Even though the vector potentials have SO(8) as their largest invariancegroup it is possible to extend this to a bigger invariance group when we include the (magnetic)duals of the vectors. The concept of introducing dual objects to extend the symmetry is called(generalized) duality invariance. The arguments leading up to the E7(7)/SU(8) coset structureare based on this generalized duality invariance.

The vector field equations of motion of the ungauged theory are given by (obtained by varyingthe Lagrangian with respect to the field strengths):

∂µ[e(G+µνIJ +G−µνIJ)] = 0 (4.1)

Where:

G+µνIJ = − 4

e

δL∂F+

µνIJ(4.2)

is the dual vector of F+µνIJ , F+

µνIJ the self dual part1 of the field strength and e is the de-terminant of the inverse vierbein. G−µνIJ is defined similarly. As usual, we denote a self dualtensor with a + and an anti self dual tensor with a −. In addition to the field equations, the fieldstrengths have to satisfy the Bianchi identity which gives an independent condition on the fieldstrengths. The Bianchi identity is given by:

∂µ[e(F+µνIJ − F−µνIJ)] = 0 (4.3)

Using the dual vector field strengths, the equation of motion and the Bianchi identity it is possibleto enhance the symmetry of the vectors. In order to do this we define two linear combinations ofthe field strengths and the dual field strengths F1,2 = G+±F+ and put them in one 56-dimensionalvector H+ = (F+

1 , F+2 )2. We can now write the equations of motion and the Bianchi identity in

a single equation:

∂µ[eH+ + eωH+∗] = 0 (4.4)

Where ω is the 56× 56 matrix:

ω =

(0 II 0

)(4.5)

Generalized duality transformations are complex rotations of the vector H+ that leave theabove equation invariant and the duality group of the theory is the group of generalized dual-ity transformations. The explicit ω dependence of the above equation implies that that such atransformation group has to be a subgroup of the symplectic group Sp(56). In order to get moreconstraints on the form of the duality group we need to look at additional constraints on H. These

1self dual means: FµνIJ = εIJKLFKLµν and anti self dual mean: FµνIJ = −εIJKLFKLµν

2we are a bit sloppy with the indices here

24

come from constraints on the vectors F1 and F2, they are not independent. Instead, they have tosatisfy a constraint relation that contains a matrix which is depended on the scalar fields:

1

2(1− Ω)VH+ =

(0O+µν

)(4.6)

Where V is a scalar dependent 56 × 56 matrix (it is in fact a representative of the coset),Ω = diag(I,−I) and O+

µν is a bilinear in the fermion fields that appears in the Lagrangian incombination with the vector field strengths. If we do not include the matrix V then the maximalsymmetry group of H that leaves the above equation invariant is SU(8), including V allows us toenlarge that symmetry group even further.

The true goal so far has not been to derive the duality group of the ungauged theory butrather it has been to find the coset structure. This coset is in principle a coset group manifoldand V is an element of the coset. If we know what transformations leave V invariant and whichtransformations transform V into another element of the coset then we have found the coset.

If we demand that equation 4.6 is invariant under duality transformations of the vector H (H →EH E ∈duality group) then we have to let V transform under inverse duality transformations fromthe right. In addition to this V has to transform in the same way as O+

µν from the left. In formulaV transforms as:

V → UVE−1 (4.7)

O+µν is a function of the fermions which belong to representations of SU(8) we postulate that O+

µν

also transforms under SU(8) such that U ∈ SU(8). Not all the fields of V will correspond tophysical degrees of freedom. There are 70 physical scalars in the theory and in principle we canuse the SU(8) gauge freedom to gauge another 63 degrees of freedom away from V which impliesthat V has 133 degrees of freedom which is precisely the number of generators of E7. This countingargument was first used to suggest that E ∈ E7 by Cremmer and Julia in [9] and repeated in [4]which is where we found it.

The above argument is not sufficient to determine that the group of duality transformationsis indeed E7 (or rather E7(7)) but it is a lot more tractable then the full derivation that can befound in [4] and related literature. If we assume that the group of duality transformations is E7(7)

than that indeed implies that V depends on 133 scalar fields. Equation 4.7 then implies thatthese scalar fields fall in classes that are gauge equivalent with respect to SU(8) which in turnimplies that the coset manifold is E7(7)/SU(8). The coset representative V can be used to givea parametrization of the coset manifold and hence can be used to parametrize the scalar fieldsof the theory. For example the kinetic term for the scalar fields appearing in the Lagrangian isproportional to:

Tr(DµVDµV−1) (4.8)

Where Dµ is a covariant derivative. Instead of talking about scalar fields φijkl we can also talkabout V.

We conclude that the ungauged Lagrangian is E7×SU(8) invariant on-shell from which it is inprinciple possible to derive all the transformation rules of the fields and write down the completeungauged supersymmetry invariant Lagrangian of the theory. Checking that this Lagrangian isconsistent with SUSY transformations is quite laborious work and we will not discuss it here (weonce again refer to [4]). The ungauged theory is not that interesting from our perspective as itlacks mass terms and a scalar potential. Instead we will turn our attention to gauging the theory.In the remaining part of this section we will briefly consider the SO(8)× SU(8) invariant theory.In particular we will look at the so-called T -tensor since it is this tensor that plays a central rolein the discussions to come.

We already remarked that the vector potentials fit into the adjoint representation of SO(8)and one could try to make this symmetry a local symmetry in the same way as is done for instancein the Standard Model. Gauging the theory violates the duality invariance of the field equationsand E7(7) will no longer be a symmetry of the equations of motion and the Bianchi identity.

25

In order to gauge the theory we have to make the field strengths and derivatives covariantwith respect to SO(8). Varying the Lagrangian under a symmetry transformation can in principleintroduce additional terms. We have to introduce additional terms in the Lagrangian to ensurethat the Lagrangian is invariant under the gauge group. These additional terms are interactionterms, mass terms and the scalar potential. All of the additional terms turn out to be proportionalto certain combinations of the scalar fields that neatly fit in a tensor we call the T -tensor. Actuallydoing the calculation to check for invariance of the Lagrangian is off course messy work and luckilyit has been done for us.

Instead of doing the calculation let us define the T -tensor directly in terms of components ofthe coset representative. We write the coset representative as:

V =

(uij

IJ vijKLvklIJ uklKL

)(4.9)

Where the capital indices refer to E7 and the little indices refer to SU(8) and u and v are 28× 28matrices. We define the T -tensor in terms of u and v:

T jkli = (uklKL + vklIJ)(uimJKujmKI − vimJKvjmKI) (4.10)

This tensor is SU(8) covariant and SO(8) invariant. The T -tensor has to satisfy a number ofidentities that follow from the coset structure of the theory. It is interesting to know that theseidentities play an important role in proving the consistency of the SO(8) gauging. The cosetstructure of the theory is also the reason that the T -tensor contains two irreducible components:

Aij1 =4

21T ikjk (4.11)

Ajkl2i = − 4

3T

[jkl]i (4.12)

Which naturally live in the 36 and 420 representation of SU(8). A1 is symmetric in its indicesand A2 is traceless and fully antisymmetric in its 3 upper indices:

Aij = A(ij)1 (4.13)

Aijki = 0 (4.14)

Ajkli = A[jkl]i (4.15)

The mass terms, interaction terms and the scalar potential that are due to the gauging withrespect to SO(8) all depend on the tensors Aij1 and Ajkl2i . In particular the scalar potential lookslike:

V = − 3

4|A1|2 +

1

24|A2|2 (4.16)

We should not forget that the T tensor is defined in components of the coset representative V andhence that it depends on the scalars T = T (φ). This implies off course that also A1 and A2 arefunctions of the scalar fields and hence that the scalar potential is dependent on the scalar fields.Which in a sense is saying that the form of the potential is dependent on the vev of the theory.

If we write the mass like terms that arise in the Lagrangian:

Lmass =g√2Aijψ

i

µγµνψjν +

g

6Ajkli ψ

i

µγµχjkl +

g√

2

144εijkpqr[lmAn]

pqrχijkχlmn (4.17)

We see that A1 sets the masses of the gravitini, that A2 sets the masses of the spinors and A2

gives rise to a bilinear term in the gravitini and spinors. The term that is bilinear in the gravitiniand the spinors breaks supersymmetry of the critical points of the scalar potential and hencesupersymmetry is broken whenever A2 6= 0 in a critical point. The tensor A2 plays the role of

26

order parameter of supersymmetry breaking. Setting A2 = 0 ensures that we have a maximallysupersymmetric theory and looking at the scalar potential we see that this implies that space timeis anti de Sitter with a scale given by A1.

Even though we did not do the calculations it should be obvious that it is hard work to picka gauge group (for instance a subgroup of SO(8) like SO(4, 4)) and check whether it is possibleto write Lagrangian that is invariant under this gauge group and check for consistency withsupersymmetry. It is for this reason that the embedding tensor is such a nice tool. It allows oneto construct a consistent theory without having to construct additional terms for the Lagrangianand checking its consistency. The T -tensor components A1 and A2 appear rather naturally in thiscontext as we shall see.

4.2 The embedding tensor

In the previous section we described the coset structure of maximal supergravity using the gener-alized duality invariance of the theory. We closed the last section with the remark that it is hardwork to write down a consistent theory given a gauge group and that the embedding tensor is away to avoid doing the calculations. We will introduce the embedding tensor in this section byclosely following the article by de Wit, Sambtleben en Trigiante [5].

The gauge group of the theory is always a subgroup of the duality group E7(7) but the gaugevectors for different gaugings do not necessarily have to be a subset of the same 28 electric vectorsthat appear in the ungauged Lagrangian. This is because the choice of electric and magneticvectors is a choice of basis and depends on the embedding of E7(7) in the larger group Sp(56).In the previous section we used all of the 28 electric vectors to gauge the theory since they couldeasily be put in the adjoint representation of SO(8). It is possible to use the 28 magnetic vectorsto gauge the theory with SO(8). Surprisingly this gives a different vacuum structure [2].

The most general approach would be to gauge any combination (maximal 28 vectors) of theelectric and magnetic vector fields. In order to do this we consider the 56 vector fields AMµ asa single vector that transforms in the 56 dimensional representation of E7(7), M runs from 1 to56, that splits into 28 + 28 upon a choice of duality frame. The 56 dimensional representationhas generators (tα)NM , where α is the fundamental index of E7(7) and hence runs from 1 to 133.The gauge group is contained in the duality group since the duality group contains all possibletransformations of the vectors. This implies that the generators of the gauge group can be writtenin terms of the generators of the duality group. The embedding of the gauge group in E7(7) isencoded in the embedding tensor that selects which of the generators of E7(7) are also generators,XM , of the gauge group.

XM = ΘαM tα (4.18)

In this way we can select an arbitrary combination of electric and magnetic vectors that will beused to gauge the theory, consistency will ultimately require that the maximal number of gaugevectors is 28. From the above equation it is clear that the embedding tensor lives in the 56× 133representation of E7(7). We define the generalized structure constants of the gauge group in thefollowing way (inspired by a general gauge transformation on the vector fields):

(XM )NP = ΘM

α(tα)NP (4.19)

Which has to satisfy the following relation:

[XM , XN ] = −XPMNXP (4.20)

This signals a problem because XPMN is not antisymmetric in MN while the left hand side of

the equation is. This problem is solved by requiring that the gauge group generators close into aLie algebra. This leads to the so called quadratic constraints that ensure that the gauge groupis a proper subgroup embedded in the duality group E7(7). The constraints are called quadratic

27

because they are quadratic in the embedding tensor components. If we use the SP (56) invariantmatrix ΩMN then the quadratic constraints take the following form (assuming that the embeddingtensor lives in the 912):

ΘMαΘN

βΩMN = 0 (4.21)

In general, the embedding tensor is contained in the product 56 × 133 = 56 + 912 + 6480representation of E7(7). However, supersymmetry restricts the embedding tensor to the 912representation [15]. This is equivalent to the following contractions:

tαMNΘN

α = 0 (4.22)

(tβtα)NMΘβ

N = − 1

2ΘαM (4.23)

Where we raise the fundamental E7(7) index α using the Killing metric. These equations areknown as the linear constraints on the embedding tensor (LC). Like the quadratic constraints,they are necessary for a consistent theory. We can use the Sp(56) matrix ΩMN to lower the indexof XMN

P . The constraints on the embedding tensor lead to the the following properties of XMNP :

XM [NP ] = 0 (4.24)

X(MNP ) = 0 (4.25)

XMNN = XMN

M = 0 (4.26)

As always in maximal supergravity, gauging the theory leads to mass terms and a scalarpotential. In the previous section we defined the T -tensor which had a natural place in the SO(8)theory because its irreducible components appeared in the mass terms and scalar potential of thetheory. The T -tensor was defined in terms of the coset representative V ∈ E7(7). It seems logicalto expect a relation between the embedding tensor and some generalization of the T -tensor.

We will use a coset representative V ∈ E7(7) to elevate the embedding tensor to a SU(8)covariant, field dependent tensor that will naturally take the role the T -tensor had in the SO(8)theory. Naturally, this tensor is called the T -tensor:

TMNP [Θ, φ] = V−1

MMV−1

NNVP PXMN

P (4.27)

The T -tensor as defined above has underlined local SU(8) indices. In order to make contact withthe fields in the theory that all carry small Latin indices we have to go from underlined indicesM to small Latin indices i, j. This can be accomplished by choosing a suitable basis in which thecoset representative can be written as:

VMN = (VM

ij , VMkl) (4.28)

Where the indices i, j, · · · are antisymmetric SU(8) indices. We can decompose the T -tensor inthis basis as:

TMNP = (TijN

P , T klNP ) (4.29)

Where the TijNP are:

Tij =

(− 2

3δ[k[pT q]l]ij

124εklrstuvwT

tuvwij

Tmnpqij23δ[r

[mTs]n]ij

)(4.30)

Factors are chosen to agree with previous conventions in the literature. This tensor depends onthe scalar fields of the theory via the coset Representative and on the gauging via the embeddingtensor. The coset geometry of the theory and the requirements for a consistent gauging putconstraints on the T tensor.

28

We look at constraints of the T -tensor due to the constraints on the embedding tensor. Theembedding tensor is constrained to live in the 912 of E7(7) this implies that the T tensor lives inthe 912 of E7(7) as well. The embedding tensor is a SU(8) covariant object so the 912 of E7(7)

branches in irreps of SU(8), which happens in the following way:

912→ 36 + 420 + cc (4.31)

The branching of 912 into irreps of SU(8) implies that the explicit expression of the T tensor canbe written as a linear combination of objects living in the 36 and 420.

Looking at the components in the T tensor we infer that there must be a proportionality

relation between T klmnij and δ[i[kT

lmn]j] . The following proportionality relation is found:

T klmnij = − 4

3δ[i

[kTlmn]j] (4.32)

And the decomposition of Tijkl into components is given by:

Tijkl = − 3

4A2i

jkl − 3

2A1

j[kδl]i (4.33)

Where A2 is fully antisymmetric in its upper indices and traceless and A1 is fully symmetric it itsindices. It is easy to see that A1 lives in the 36 and A2 lives in the 420 of SU(8). They are thesame tensors that were used in the SO(8) theory. As before, its these components of the T -tensorthat appear in the mass terms and scalar potential of the Lagrangian.

A consistent gauging was characterized by a number of constraints quadratic in the embeddingtensor. The T tensor is linear in the embedding tensor hence there should exist a number ofconstraints on the square of the T tensor. These constraints then translate to quadratic constraintson A1 and A2. The constraints that have to be satisfied for a consistent gauging of the theory are[1]:

9ArstmArsti −AirstAmrst − δmi |A2|2 = 0 (4.34)

3ArstmArsti −AirstAmrst + 12AirA

mr − 1

4δmi |A2|2 −

3

2δmi |A1|2 = 0 (4.35)

Aijv[mAvnpq] +Ajvδ

i[mA

vnpq] −Aj[mAinpq]

+1

4!εmnpqrtstu(Aj

ivrAvstu +AivδrjAv

stu −AirAjstu) = 0 (4.36)

AirsmAnjrs −AjrsnAmirs + 4A(m

ijrAn)r − 4A(i

mnrAj)r

−1

8δni (Ar

stmArstj −ArrstAmrst) +1

8δmj (Ar

stnArsti −AirstAnrst) = 0 (4.37)

ArmnpArijk − 9A[i

r[mnAp]jk]r − 9δ[m[i Aj

rs,nAp]k]rs

−9δ[mn[ij Ar

p,stArk]st + δmnpijk |A2|2 = 0 (4.38)

These constraints live in the 63, 63, 70− + 378 + 3584, 945 + 945 and 2352 irreps of SU(8),respectively. We can extract the pure 70− and 378 irreps by contracting the third equation withdeltas. We find:

Ar [ijkAl]r −3

4Ars[ijA

skl]r =

1

4!εijklmnpq(Ar

mnpAqr +3

4Ar

smnAsrpq) (4.39)

3

4ArijkAlr +

3

4Arl[ijAk]r =

1

4!εmnpqrijkAl

qrsAsmnp +

3

4

1

4!εijklmnpqAr

spqAsrmn (4.40)

Note that the expression that lives in the 70− is a self duality condition. The above constraintsare also often called the quadratic constraints, since they are quadratic in A1 and A2 and from nowon in this thesis when we talk about QC we mean the above equations and not the correspondingquadratic constraints on the embedding tensor.

29

To summarize, consistent gauging are characterized by an embedding tensor that satisfiesthe two types (linear and quadratic) constraints. Mass terms that are induced by the gaugingare conveniently expressed in terms of irreducible components of the T tensor. The linear andquadratic constraints on the embedding tensor lead to the corresponding constraints on the T -tensor. Which in turn puts constraints on the tensors A1 and A2. The quadratic constraintsimply that we can not have arbitrary scalar masses. Ultimately, we need to use the QC to finda bound on the lightest scalar mass in the theory since the constraints are independent of thespecific gauging that is used. The QC represent one of the few general handles that exist for thetheory and it is certainly one of the most direct when one talks about the scalar masses.

4.3 Going to the origin

A vacuum of the theory is specified by a point on the coset manifold for which the scalar potentialhas an extremal point. It is in general not enough to specify a consistent gauging when one isdiscussing stability of a theory because stability only makes sense when one specifies a vacuumaround which the theory is expanded. In this section we will briefly discuss how we ensure thatwe work around a vacuum of the theory.

A critical procedure for our analysis of stability in maximal supergravity is the so-called goingto the origin procedure. The idea of this procedure was outlined in [6] and [7] and applied tomaximal supergravity in [8]. We already know that all the possible gaugings of the theory areencoded in the embedding tensor Θα

M which has to satisfy a number of constraints in order forthe gauging to be consistent. Given a gauging, finding a vacuum amounts to finding an extremalpoint of the scalar potential specified by that gauging. This is in general a non trivial task becausein general the scalar potential depends non-linearly on the 70 scalar fields and finding a solutionto the equation ∂φV = 0 becomes a very complicated non linear task. The going to the originapproach reduces the problem of finding extrema of the scalar potential to solving some additionalconstraints on the embedding tensor components.

The idea is pretty simple. The scalar manifold of maximal supergravity is the coset spaceE7(7)/SU(8). This is a homogeneous space which implies that any point on the scalar manifoldcan be mapped to any other point via an E7(7) transformation. The scalar potential is invariantunder a duality transformation that acts simultaneously on the coset representative and on theembedding tensor. These two statements suggest that instead of finding critical points of thescalar potential by scanning over all the points of the scalar manifold, we can solve the criticalpoint conditions in the origin and vary the values of the embedding tensor to scan over all thepossible vacua. We will use the rest of this section to make this statement more precise.

The scalar potential is proportional to:

V (φ) ∝ (XMNRXPQ

SMMPMNQMRS + 7XMNQXPQ

NMMP ) (4.41)

Where M is a SU(8) invariant matrix determined by the coset representative VTV. It is clearthat the scalar potential depends on the scalar fields. The structure constants XMN

P depend onthe embedding tensor (thus on the gauging). As we said before it is important that we are able toredefine the scalar potential such that its value at any point is mapped to the origin. We can mapa point on the scalar manifold to the origin using an E7(7) transformation. The scalar potentialdepends on the scalar fields via the coset representative so we will have to check the effect of anE7(7) transformation on the coset representative. The action of an element E ∈ E7(7) a cosetrepresentative V(φ) will in general amount to changing the coset of which V is a representative.Which is the same as evaluating V for a different value of the scalar fields times a field dependentSU(8) transformation:

EV(φ) = V(φ′)h(φ, φ′) (4.42)

Since the scalar potential depends only on the SU(8) invariant matrix M we see that the scalarpotential is invariant under h. Since we use an E7(7) transformation to map the scalar potential

30

V (φ) to V (0) and we want the scalar potential to be invariant under this transformation we needto somehow compensate for E. The crucial observation is that the scalar potential in fact onlydepends on the combination V−1Θ and hence that it is invariant under an E7(7) transformationthat acts both on the coset representative V as well as on the embedding tensor Θ. We seethat evaluating the scalar potential at a different point of the scalar manifold is equivalent toevaluating the scalar potential with a modified embedding tensor. This in turn implies that wecan calculate the scalar potential and its derivatives in a single point (for instance φ = 0) and varythe embedding tensor to search for vacua. The scalar potential at the origin takes the followingform in terms of the irreducible components of the T-tensor:

V (0) = − 3

4|A1(0)|2 +

1

24|A2(0)|2 (4.43)

Which is a quadratic function in the embedding tensor components. Minimizing this potentialgives an additional quadratic condition on the embedding tensor which should be satisfied alongwith the quadratic constraints defining a consistent gauging. A solution to this set of constraintsdefines at the same time the gauge group, the value of the scalar potential and the masses ofthe scalar particles. The requirement that the scalar potential has a critical point in the origintranslates to:

Ar [ijkAl]r +3

4Ars[ijA

skl]r = − 1

4!εijklmnpq(Ar

mnpAqr +3

4Ar

smnAsrpq) (4.44)

This equation has to be satisfied together with the quadratic constraints and hence forms anadditional constraint on A1 and A2. Note that it follows from this equation that the self dualpart of the left hand side is equal to zero. In other words, this combination of A1 and A2 isanti self dual. This equation can also be obtained from the field equations for the scalars of thetheory and hence lives in the 70+ [1]. Note that equation 4.44 seems to be related to equationA.7. The adding or subtracting of A2A2 seems to select the self dual or anti self dual part of thecombination A1A2. The importance of the fact that these combinations are self dual/anti self dualwill be explored later on in this thesis. By requiring equation 4.44 in combination with the QCwe ensure that we scan over all the possible vacua of the theory with all the possible gaugings.

4.4 Symmetry breaking

So far we have discussed stability in N = 1, 2 supersymmetric theories and we have given a generalbackground in maximal supergravity (at least the parts we need). Before doing an analysis ofstability in maximal supergravity we feel that it is convenient to discuss a number of vacua thatwe will use to test our hypotheses. Before discussing vacua of a quantum field theory it is naturalto discuss symmetry breaking, which is what we will do in this section.

There are several different ways in which we can break a symmetry of a theory; we can addexplicit terms to the Lagrangian such that the Lagrangian is no longer symmetric, a symme-try can be broken by quantum effects which arise dynamically or the symmetry can be brokenspontaneously. We will only consider the last case, where the Lagrangian is (super)symmetricbut the vacuum breaks the symmetry group down to a subgroup. In general, this gives rise toGoldstone and Higgslike particles via the Goldstone theorem and Higgs mechanism. There is anenormous amount of literature written on the subject of symmetry breaking because of the phys-ical implications the breaking of a symmetry can have. In supersymmetric theories the breakingof supersymmetry is important because of simple observations, no superparticles have ever beendetected so our universe clearly is not supersymmetric. If supersymmetry is a symmetry of naturethan it has to be broken.

Despite the vast literature and importance of symmetry breaking, we will take on a pragmaticview. Rather then going over the different ways to break the symmetry of a theory, we will brieflyshow why:

V rs[ijkl] = δr[iAsjkl] (4.45)

31

are related to supersymmetry breaking. From the index structure it is clear that this combinationdefines 64 directions on the scalar manifold. In fact, these are the 64 sGoldstini, the partners ofthe 8 Goldstinos that can arise when breaking supersymmetry. The 8 Goldstini are defined by:

ηi = Aijklχjkl (4.46)

Which give mass contributions to the gravitini [5]:

Lfermionmass = eg(1

2

√2Aijψ

i

µγµνψjν +

1

6Ajkli ψ

i

µγµχjkl + terms proportional to χ2) (4.47)

Which can be put in canonical form by a transformation ψi → ψi + ηi. The ηi hence representthe degrees of freedom that have been transformed from the spin 1

2 sector to the spin 32 sector.

These directions then correspond to the would-be Goldstones as a result of the Higgs theorem.The super partners of ηi are obtained by a supersymmetry transformation and are found to beV rs[ijkl]φ

ijkl. This gives these directions the same status as the sGoldstini used in chapter 3, we willuse them analogously in chapter 5.

We also note that by far the easiest way to see that V rs[ijkl] defines special directions whensupersymmetry is broken is to notice that A2 is the supersymmetry order parameter. We can usethe tensor A1 to select one direction from among the 64 contained in V rs[ijkl]. This direction willbe called the Scrucchino in the next chapter.

Finally we will also be interested in the combination:

Ars[ijAskl]r (4.48)

This combination singles out the (anti) self dual part of the Scrucchino as a result of the quadraticconstraints and critical point condition. They also arise in the mass terms of the spin 1

2 sectorwhich means they are also related to symmetry breaking.

4.5 Vacua of maximal supergravity

In this section we will briefly discuss a few different vacua of maximal supergravity. We will usethese vacua to test several of our hypotheses in chapter 6 which is why it is useful to introducethem here. If we set A2 = 0 in the SO(8) theory then we obtain the unique N = 8 critical point ofthe theory [1]. The scalar masses necessarily coincide in this point and are all equal to m2 = − 2

3V .They satisfy the BF bound and hence the vacuum is stable. It is possible to show that the onlypossible gauging with A2 = 0 and hence with N = 8 preserved is the SO(8) gauging. We alsonote that a N = 8 Minkowski vacuum implies A1 = A2 = 0 and is therefore trivial.

It was long believed that gauging the theory with SO(8) uniquely determined the theory, up tothe choice of vacuum. Recently, it was shown that this is not the case. Instead, the SO(8) gaugegives a one parameter family of different theories, see for instance [19]. The embedding tensorshould be a singlet of the gauge group (otherwise, the charges of the theory would depend on thegauge). This implies that the embedding tensor lives in a singlet that is left after the branchingof 912 of E7 under the gauge group. When we consider SO(8) as gauge group then the followingirreps appear:

912 = 2× (1 + 35v + 35s + 35c + 350) (4.49)

We see that there are two singlets in which the embedding tensor can live. In principle theembedding tensor can live in any linear combination of the two and this gives rise to the oneparameter family where the parameter interpolates between the two singlets. The parameter isusually denoted by ω. Physically the reason for the family of gaugings is the choice of gaugevectors. We are free to choose any combination of the electric and magnetic gauge vectors whichgives rise to different theories.

In general a critical point of the SO(8) gauged theory will only preserve a subgroup of SO(8),breaking the full gauge symmetry and giving rise to additional contributions to the mass terms.

32

Before the discovery of the new (ω independent) maximal supergravities already a number ofcritical points with either SU(3) SO(4) invariance were classified, [21] and [22]. The effect ofturning on ω on the critical points of the theory has only recently been discussed in the literature.Chapter 10 of [1] gives a discussion on the ω dependence of the critical points that are eitherSU(3) or SO(4) invariant. It turns out that in some cases the scalar masses are depended on ωand in some cases they are not. In particular the SO(4) invariant sector has a very interestingorbit with critical points that range from Anti de Sitter to de Sitter space times. In addition tothe sign of the cosmological constant, also the residual symmetry group changes over the orbit. It

0 Π

6Π

4Π

2

0

0Π

6Π

4Π

2

0

Θ

VHΘL

0 Π

6Π

4Π

2

-5-3

0

35

10

15

0Π

6Π

4Π

2

-5-3

0

35

10

15

Θ

m2 L

2

Figure 4.1: Top: Scalar potential interpolating between the AdS and dS critical points. Bottom:Behavior of the scalar masses as a function of θ. Taken from [14]

is interesting to note from figure 4.1 that the scalar masses are almost continuous when one crossesover the Minkowski point. In addition, it is clear from the picture that the de Sitter critical pointsare always unstable but the closer we approach Minkowski the closer the tachyonic directions tendto zero.

Maximal supergravity can also be viewed as the result of a compactification of certain higherdimensional string theories. In particular, a number of critical points that arise from geometriccompactifications of type IIA string theory where described in [7]. These points where describedin a N = 4 setting but they can be embedded in a N = 8 background. From our viewpoint thesepoints are interesting because they do not belong to the original SO(8) gauged theory so theycould contain features missing in the SO(8) gauged theory. There are 4 points that are obtainedin this way.

The different critical points with their most tachyonic direction and their cosmological constantare given in table 4.1, for a more exhaustive discussion on the construction of the vacua we referthe reader to the references ([1], [8], [14], · · · ) and references in those works. In this chapter we

33

Critical point |V | Min(m2 ) / |V | StableSO(8) AdS − 2

3

√

SO(4, 4) dS −2 ×SO(3, 5) dS −6 ×

SO(4), θ = π/4 Minkowski 0√

SO(4), θ = π/2 dS −3 +√

3−√

6(4−√

3) ×Geo IIa 1 AdS − 2

3

√

Geo IIa 2 AdS − 45 ×

Geo IIa 3 AdS 0√

Geo IIa 4 AdS − 43 ×

Table 4.1: Critical points of maximal supergravity

have introduced maximal supergravity, first in the framework of the SO(8) gauging and second inthe framework of the embedding tensor. Our discussion culminated in the quadratic constraintsand the critical point condition (they can be found in this chapter or in appendix A.2). Togetherwith the scalar mass matrix these equations are everything we need to study stability using ourmethod of projections. We will use numerical tests to exclude possible master relations using thevacua that are introduced in this section.

34

Chapter 5

Stability in maximal supergravity

In this section we will describe attempts to study stability in N = 8 supergravity in 4 space timedimensions. In particular we are interested in an upper bound on the lightest scalar mass whenthe value of the scalar potential is positive (when we have a de Sitter vacuum). We have studiedstability before in this thesis in the framework of N = 1, 2 theories. Analogous to the lowersupersymmetric case, we are looking for tachyonic directions on the scalar manifold. Finding adirection that is always negative constitutes a proof of instability because there always exists astate in the theory with a smaller or equal mass than any direction on the scalar manifold.

The scalar masses correspond to the eigenvalues of the scalar mass matrix. We can either tryto find an eigenvector of the scalar mass matrix or (as was done in the globally supersymmetriccase) find an eigenvector of a smaller matrix that is a projection of the full scalar mass matrix.Another approach is to find a singlet projection of the scalar mass matrix, this will also give anupper bound on the lowest eigenvalue of the scalar mass matrix. The singlet projection is offcourse a special case of the projection onto a smaller matrix since it is a projection on a 1 × 1matrix.

We start with the easiest possible projections of the mass matrix and with the results generatedin [1]. After that we will discuss various extensions of the method using numerical examples andanalytical calculations. There is a small summary at the end of this chapter with an overview ofthe results.

5.1 The scalar mass matrix and the sGoldstino mass matrix

The scalar mass matrix in the origin is given in terms of the embedding tensor components by (upto quadratic constraints):

m2ijkl

vstu = 4δijklvstuV + 20δ[ijk

[vstAu]pAu]p + 6δ[ij

[vsAkpq,tA

u]l]pq

− 2

3δ[i

[vApstu]Apjkl] −

2

3A[i

[vstAjkl]u] (5.1)

This expression is Hermitean in the sense that:

(m2ijkl

vstu)∗ = m2vstu

ijkl (5.2)

Group theoretically it contains the following SU(8) irreps:

70× 70 = 1 + 720 + 1764 (5.3)

The explicit decomposition of the mass matrix in terms of its irreps is recorded in Appendix C.The easiest bound that one can consider is taking the trace of the mass matrix since the trace

of a matrix is equal to the sum of its eigenvalues. This bound is extremely crude and one should

35

not expect to find it to be to restrictive. The trace of the above matrix is given by:

m2 =1

70m2ijkl

vstuδvstuijkl (5.4)

= − 1

2|A1|2 +

1

20|A2|2 (5.5)

=6

5V +

2

5|A1|2 (5.6)

where we have used equation 4.43 for the scalar potential. We see that the trace is always positivefor de Sitter vacua.

We know that A1 represents the scale of the supersymmetric AdS vacuum and that A2 canbe viewed as an order parameter of supersymmetry breaking. When the vacuum is fully super-symmetric, A2 is zero and the trace is given by: − 1

2 |A1|2. In fact, all of the scalar masses areequal to m2 = − 1

2 |A1|2 because the vacuum is supersymmetric all scalars have to be in the samemultiplet and hence have the same mass. This is fully analogous to the N = 1, 2 cases. Whensupersymmetry is broken |A2|2 6= 0 and the trace of the scalar mass matrix can in principle betuned to any value. So indeed, the trace of the full mass matrix does not give a usable bound onthe mass of the lightest scalar. Note that using the delta to trace the mass matrix is equivalentto projecting the mass matrix on 2nd order singlets. The most general 2nd order singlet is indeedgiven by:

a|A1|2 + b|A2|2 (5.7)

Since this is the only quadratic combination of A1 and A2 that has no free indices. Taking thetrace amounts to fixing the vector (a, b). This is in general the case, when we project the massmatrix we will always have to consider all possible singlet combinations of A1 and A2 at the rightorder. Projecting in this case means calculating the coefficients in front of the singlets. Thequadratic constraints and the critical point (vacuum) condition give various relations between thesinglets that we have to take into account. We could also try to find an eigenvector of the massmatrix directly. We will come back to that later in this chapter.

Another approach is inspired by the discussion on stability in N = 1, 2 supergravity. Thescalar mass matrix is projected on specific directions determined by the sGoldstinos, which arestates that are singled out by supersymmetry breaking. This projected matrix is in general smallerthen the full scalar mass matrix, but the trace of this smaller matrix still gives a bound on thesmallest scalar mass.1 Taking the trace of a submatrix is the approach applied to the N = 8 casein [1].

Since we have N = 8 supergravity we can in principle break 8 symmetries and hence thereare 8 Goldstinos, which implies that there are 64 sGoldstinos. The directions are defined as (seechapter 4):

V rsijkl = δr[iAsjkl] (5.8)

This 64-dimensional representation splits up in the 36 symmetric and 28 antisymmetric irreps. Itturns out that the antisymmetric sGoldstino directions are gauge directions that can be gaugedaway and hence are unphysical. We therefore only have to focus on the symmetric directions:

V (rs)ijkl = δ

(r[i A

s)jkl] (5.9)

We can use these directions to project the scalar mass matrix on the so called sGoldstino massmatrix. Afterwards we are left with two options, either project the sGoldstino mass matrix on asinglet or try to find an eigenvector of the sGoldstino mass matrix. The reason for looking in thesedirections is basically the same as for the globally supersymmetric case and for the N = 1, 2 cases,these directions are singled out by supersymmetry breaking. Hence they are thought to representdirections on the scalar manifold that are relatively stable with respect to tuning of the theory.

1This is an elementary linear algebra exercise.

36

An important difference (computationally) with taking the trace of the full scalar mass matrixcompared to taking the trace of the projected sGoldstino mass matrix is that the sGoldstino massmatrix is 4th order instead of 2nd order in the embedding tensor components. This will makethe calculations more difficult. It is possible to define all the 4th order singlets, they are recordedin appendix A of this thesis. There are 15 real singlets xi, i = 1, · · · , 15, and 7 complex singletszj , j = 1, · · · , 7. Which means a total increase of 27(= 15 + 14 − 2) in the number of singletscompared to 2nd order.

These singlets are not independent. After all, the tensors A1 and A2 are not independent. The4th order singlets have to satisfy quartic constraints which are recorded in appendix A. Noticethat the number of quartic constraints is bigger than the number of quadratic constraints. This isa result of the different ways to contract the quadratic constraints and uplift them to 4th order.This growth in the number of constraints (and number of singlets) will be even bigger when onegoes to 6th order. Making sure that one has all possible constraints and singlets becomes anincreasingly difficult task.

After projecting the mass matrix and taking the trace, we find the sGoldstino square mass(trace of the sGoldstino mass matrix):

M2sG = V (rs)

ijklVmnpq

rsm2mnpq

ijkl (5.10)

=1

32(x1 − 12x10 + 672x4) (5.11)

This expression has a non definite sign due to the relative minus sign and the fact that x4 can beeither positive or negative (x1 and x10 are positive by definition). In order to find constraints onthe value of the sGoldstino square mass we need to find additional constraints on the 4th ordersinglets (we have already used the quadratic constraints to put the projection in this simple form).There are several sources of the additional constraints. The first set of constraints comes fromthe definition of the 4th order singlets, some of them are positive by definition. The second kindcomes from constructing non negative combinations at 4th order by taking the square of a 2ndorder expression (which lives in some irrep). The easiest example of this kind of constraint isillustrated by taking the square of:

AirAim − 1

8δmr |A1|2 (5.12)

From which it follows that |α|2x4 > 0 for some parameter α. The third set of constraints comesfrom identities in linear algebra related to hermitian matrices and the 4th set of constraints followfrom the geometry of the scalar manifold. A list of constraints can be found in [1]. It is notguaranteed that this list is complete, which illustrates how difficult it is to make sure that one haswritten down all possibilities. The allowed region for the sGoldstino square mass is indicated infigure 5.1. There are two regions plotted in the figure: the off shell region and the on shell region.The off shell region only incorporates the quadratic constraints and the on shell window whichalso incorporates the field equations living in the 70+. The additional constraint for the on shellwindow is:

Ar[ijkAl]r +3

4Ars[ijA

skl]r = 0 (5.13)

The points correspond to different gaugings. A more thorough discussion can be found in [1].From the figure it is obvious that the sGoldstino square mass is not always negative for positivevalues of the cosmological constant. Off course we can try to find additional constraints thatensure that the sGoldstino square mass is always non positive. We consider a few different cases:

A1 = 0

For all cases where we have A1 = 0 it was shown in [1] that the sGoldstino square mass is always0. In this case the sGoldstino square mass reduces to:

M2sG =

1

32(x1 − 12x10) (5.14)

37

SO(4,4)

SO(4)SU(4)

_

SO(5,3)

Figure 5.1: Allowed values of the sGoldstino square mass in units of 34 |A1|2 (horizontally) versus

the value of the scalar potential (vertically). Taken with permission from [1]

38

The constraint coming from the square of the 720 irrep gives M2sG ≥ 0 and the constraint coming

from the square of the 1764 irrep gives M2sG ≤ 0 implying that the sGoldstino square mass is

always zero.

Aij full trace

The case A1 full trace is equivalent to saying that x4 = 0:

AirAim − 1

8δmr |A1|2 = 0 (5.15)

There are two main reasons to try this choice. The first reason is group theoretical, there arethree different quadratic combinations of A1 and A2 that live in the 63 and two of the three areset to zero by the quadratic constraints. Thus, this is the simplest possible additional constraintthat we have. The second reason is that this choice has a potentially easy physical interpretationin terms of the gravitini masses. Unfortunately, we find that the sGoldstino square mass is alwaysnon negative in this case, M2

sG ≥ 0. This follows from the constraint coming from the square ofthe 2352.

Armnp full trace

In this case we have the additional constraint:

Amnpr Arijk =9

4δ

[m[i A

np]sr Arjk]s −

9

10δ[ij

[mnAp]str Ark]st +1

20δijk

mnp|A2|2 (5.16)

Which corresponds to setting the quartic constraint coming from the square of the 2352 equal to0. This choice is motivated by the previous case x4 = 0. In this case it is not difficult to showthat the sGoldstino square mass is non positive, M2

sG ≤ 0, however there is no clear physicalinterpretation. We conclude that for all gaugings where A2 is full trace there is no meta stable deSitter vacuum.

An interesting observation that can be made from the above examples is that the sGoldstinosquare mass seems to have the structure:

M2sG = − (63)2 + (2352)2 (5.17)

This might be just a coincidence due to the quadratic constraints.We conclude that the sGoldstino square mass can have either sign. Furthermore, even though

we can find an additional constraint such that the sGoldstino square mass is always negative,the physical interpretation of the constraint itself is not very clear. This is mainly because theconstraint restricts the tensor A2 which shows up in a lot of places in the Lagrangian. Also, theinterpretation is harder because the constraint lives in such a high dimensional irrep. We do notethat the spin 1

2 fields of the theory carry the right number of indices (3 antisymmetric) so thatif a physical interpretation exists it should be related to the spinors. In addition, making A2 fulltrace could be related to certain supersymmetry breaking patterns since A2 is the supersymmetricorder breaking parameter.

In addition to the analytical treatment of the above constraints, we have also used Mathematicato sketch similar plots as figure 5.1 to see if there is some additional constraint that we could use.As possible constraints we used the list given in [1] but instead of (irrep)2 ≥ 0 we set (irrep)2 = 0for each constraint in turn. The qualitative results are given in table 5.1. We found in this waythat the only additional constraint that ensures that we have a negative sGoldstino square masslives in the 3584 which has an even higher dimension than the 2352 constraint we found earlier.We have been unable to derive this result by hand. We should note that a negative sGoldstinosquare mass proves that the vacuum is unstable, the opposite statement is not true. A positivesGoldstino square mass does not prove that the vacua are stable, there could still be tachyonicdirections that are not captured by the sGoldstino square mass.

39

Constraint Off shell On shell commentA1 = 0 M2

sG = 0 M2sG = 0

x4 = 0 M2sG ≥ 0 M2

sG ≥ 0720 = 0 M2

sG indef sign M2sG indef sign for any combination in the 720

945 = 0 M2sG indef sign M2

sG indef sign No visible effect on the allowed region1232 = 0 no plots produced1764 = 0 M2

sG indef sign M2sG indef sign

2352 = 0 M2sG ≤ 0 M2

sG ≤ 03584 = 0 M2

sG ≤ 0 M2sG ≤ 0

Table 5.1: All possible constraints with a qualitative description of the form of the sGoldstinosquare mass as given by Mathematica

We end this section with a few comments on the the sGoldstino mass matrix:

m2sG

(mn)(ij) = V (mn)

rstuVklpq

(ij)m2klpq

rstu (5.18)

Which contains the following irreps:

36× 36 = 1 + 63 + 1232 (5.19)

Where the singlet corresponds to the sGoldstino square mass. The explicit expression and de-composition in terms of the different irreps of the sGoldstino mass matrix can be found in theappendix. We have studied the trace of this matrix in some detail in this section. However, wecould also try to find an eigenvector of the sGoldstino mass matrix and use its eigenvalue as abound on the lowest mass. We will come back to this later in this chapter.

5.2 Comparing N = 8 with N = 2

It is illustrative to compare the N = 8 case with our discussion on stability in N = 2 supergravitywith only hypermultiplets. This was based on work by Scrucca et al. First a few differencesbetween the two theories. The particle content for N = 8 is fixed by supersymmetry, for N = 2this is not the case. The scalar manifold for N = 8 is fixed, for N = 2 this is not the case.The number of sGoldstini is a lot higher for N = 8 supergravity compared to N = 2. Despitethese (and other) differences, it is still easy to see similarities in the two theories when we discussstability. This is because the same symmetry principles are at work and hence similar objectsarise in similar places (supersymmetric order breaking parameters, AdS scale, · · · ).

If we look at the roles of A1 (mass of gravitini/AdS scale) and A2 (Supersymmetry orderbreaking parameter) we have the following relations between the objects in the theories:

P x ↔ Aij (5.20)

NAα ↔ Aixyz (5.21)

This is best seen by comparing the two scalar potentials (equations 3.36 and 4.43). Let us comparethe approach used in chapter 3 with the one used in the previous section.

The physical sGoldstini in the case of N = 2 supergravity are given by:

ηsG = Nxu (5.22)

Where Nxu is the symmetric part of UαBu NA

α . Compared to the physical sGoldstini in the N = 8(equation 5.9) case there is little difference. The sGoldstino mass matrix is defined as the scalarmass matrix contracted with two sGoldstino vectors in both cases. There is a difference in the

40

expression for the sGoldstino mass matrix, namely the different terms in the N = 2 case all havea clear interpretation in terms of the scalar geometry. This is not the case for N = 8.

The trace of the sGoldstino mass matrix is enough to prove that there are no stable de Sittervacua in the N = 2 theory with only hypermultiplets. We have already discussed that this is notthe case for maximal supergravity. We could derive a stronger bound on the lightest scalar whenwe only considered a subset of all scalar geometries, namely those for which Σ = 0. We found thatP x is an eigenvector of the sGoldstino mass matrix and (naturally) that Nu

xPx is an eigenvector

of the scalar mass matrix when Σ = 0. The eigenvalue belonging to this eigenvector was alwaysnegative.

Given the failure of using the sGoldstino square mass in the N = 8 theory, the fact that NuxP

x

is an eigenvector for certain scalar geometries and the fact that the scalar geometry in the N = 8case is fixed leads us to introduce A1 as a tool in projecting the scalar mass matrix in maximalsupergravity.

5.3 Singlet projections of the scalar mass matrix

In the first section of this chapter we have used the trace of the sGoldstino mass matrix to get abound on its lowest eigenvalue. We found an allowed area where the sGoldstino square mass ispositive and hence we have an inconclusive bound on the lightest scalar. In this section we willdiscuss other possible singlet projections of the scalar mass matrix. The simplest such weight isthe tensor A1. We briefly discussed the analogous treatment in N = 2 theories in the previoussection. In that case the weight was P x. We define the combination Ap[rA

pstu], it is clear that if we

contract the scalar mass matrix with this combination we use the tensor A1 to weigh the differentdirections2. We will call this 2nd order tensor the sCrucchino. It is thought that this combinationwill project the scalar mass matrix closer to the unstable directions.

The sCrucchino lives in the 70 of SU(8) which is not an irrep. The 70 can be decomposed in aself dual and anti self dual representation 70 = 70+ + 70− which means that the sCrucchino canbe decomposed into two objects living in the two irreps. It follows from the equations of motionand the critical point condition that:

[Ar[ijk]Al]r −3

4Ars[ijA

skl]r]ASD = 0 (5.23)

[Ar[ijk]Al]r +3

4Ars[ijA

skl]r]SD = 0 (5.24)

Hence that the first equation is the self dual part of the sCrucchino and the second equation isthe anti self dual part of the sCrucchino. These combinations are called the sCrostinoSD/ASD

3.In this section we will not always write the indices explicitly, we set the following definitions:

sCruc = Ar[iArjkl] (5.25)

sCrSD = Ar[ijkAl]r −3

4Ars[ijA

skl]r (5.26)

sCrASD = Ar[ijkAl]r +3

4Ars[ijA

skl]r (5.27)

sCrASD + sCrSD = − 2sCruc (5.28)

Notice the minus sign when summing the sCrostini, this is due to the order of the indices andtheir antisymmetrization. If we project the mass matrix with the sCrucchino or sCrostino thenwe are left with a 6order singlet. If we try to find an eigenvalue equation using the sCrucchinoor sCrostino than we find a 4th order object living in the 70. Doing the calculations by handand computing all the 4th order and 6th order constraints becomes an increasingly difficult task.Instead of diving into the calculations we first run a number of simulations where we calculate theprojected mass matrix for different gaugings and different vacua.

2note that: Ap[rApstu]

= Apqδq[rApstu]

3named after some Italian snack

41

5.3.1 Numerical analysis

In this section we will give an overview of all the numerical calculations that were performed inlight of this thesis. Only at the end will we comment on the significance of the results. If wepick a gauging for which the sCrASD = 0 then we have sCrSD ∝ sCruc and vice versa. In thosecases we can use the gauging as an example for both the sCrucchino projection and the sCrostinoprojection since they are equivalent.

sCruc m2 sCruc

The least general equation that was checked in this case was for two gaugings in which sCrASD = 0.The relation itself is:

sCrSD m2L2 sCrSD = λ|sCrSD|2 (5.29)

Where L2 = 3V and λ is some general number that has to be the same for all gaugings. This is

the same as the sCrucchino projection because the anti self dual part of the sCrostino is zero. ForSO(3, 5) λ = −6 and for SO(4) at θ = π

2 , λ =√

3 − 5. Both these vacua are de Sitter but thecoefficient is different (while still negative) excluding a general relation of this form.

The second relation that one can think of has the form:

sCruc m2 sCruc = λ(A1, A2)|sCruc|2 (5.30)

Where the singlet λ is now dependent on the tensors A1 and A2 and has the general form:

λ = a|A1|2 + b|A2|2 (5.31)

Which is the most general second order singlet. The coefficients a and b were determined with theSO(3, 5) and SO(4), θ = 0 gaugings and checked with the SO(4), θ = π

2 gaugings (again we havechecked with a self dual sCrostino in gaugings where the anti self dual part is zero). For thesegaugings the relation does not work.

At this point there are two more relations that one can write down:

sCruc m2 sCruc = (a|A1|2 + b|A2|2)×× (Ax1 +Bx2 + Cx3 +Dx4 + Ex10 +Kz1 + Lz2 +Mz4) (5.32)

sCruc m2 sCruc = f(Most general 6th order singlets) (5.33)

The first relation is the most general decomposition of a 6th order singlet into a product of asecond order and a 4th order singlet and the second relation is the most general 6th order singletthat does not have to be decomposable in lower order singlets. If either of these relations wereto hold for all gaugings it would still be very interesting and it might still be possible to extracta meaningful bound. However, we did not check these relations numerically and we went to thesCrostino projections.

We have also checked if any of the irreps of the (sGoldstino) mass matrix has as eigenvectorthe sCrucchino or A1. If this is true then we could set the other irrep to zero as an additionalconstraint and work on the eigenvalue problem. Non of the irreps (other then the trivial ones) hasthe sCrucchino or A1 as eigenvector.

sCrSD/ASD m2 sCrSD/ASD

The most simple relations of the form:

sCr m2 sCr = λ(A1, A2)|sCr|2 (5.34)

Were already excluded for a self dual sCrostino. We have computed no examples of this relationwith a purely anti self dual sCrostino. The above relation was also tested for points 2, 3 and 4 of

42

the geometric type IIA compactification for both the self dual and anti self dual sCrostino. Forthese gaugings the sCrostino is a mix of a self dual and anti self dual part. However, the relationfails for these gaugings as well.

In principle we could write down relations with a general second times 4th order singlet or ageneral 6th order singlet on the right hand side just as for the sCrucchino, we have not tested thisnumerically. The reason for this is the SO(4) orbit. We have found that the projection:

sCrSD m2 sCrSD (5.35)

becomes increasingly positive when θ → π4 from the right hand side, i.e. when we approach the

Minkowski vacuum from the de Sitter vacuum. The reason is that while the sCrostino picks up atachyonic direction (which becomes massless in the Minkowski vacuum) it also picks up a directionthat becomes very massive in the Minkowski vacuum. The change in mass is continuous and hencethe sCrostino will become positive before the Minkowski point. We did not check this relation forthe anti self dual sCrostino but we do not doubt that something similar will happen. This vacuumputs an end to our hopes of finding a general master relation using the sCrostino.

5.3.2 Analytical analysis

We know from the numerical analysis that there is no master relation of the form:

sCr m2 sCr < 0 (5.36)

For de Sitter vacua. There are two possible ways to proceed from here. Either we find someadditional 6th order constraint such that the master relation does hold or we try to make a plotwith an allowed region as was done for the sGoldstino square mass (figure 5.1). Both approacheshave their own difficulties but in general the problem is the number of 6th order singlets and thenumber of 6th order constraints that can be constructed.

We have not made an exhaustive list of all 6th order singlets but here is an attempt to getto the right order of magnitude. First the simplest singlets, there are 22 4th order singlets whichmeans that there are 44 6th order singlets of the form |Ai|2 · 4th order singlet. Then there are anumber which are squares of (we indicate the number of free indices):

2u0d ArstpAstqr Apxyq

2u0d ArpAqxArq

3u1d Astpr ArqAsx

4u2d Astmr ArijkAix

4u2d Astmr ApijkAir

6u0d Astmr Aijkp Arp

6u0d Astpr Aijkp Arq

7u1d Astpr Aijkp Arpqx

7u1d Astpr Aijkp Arxyi

Giving 13 additional singlets from these cubic relations alone (these are not all cubics) leading toa minimum of 60 6th order singlets. The result is that even if we had a complete classification ofsinglets an analytical approach would be to difficult.

The number of constraints coming from the quadratic constraints on 2nd order (there are 7at that order) is already 17 on 4th order and will also increase significantly on 6th order. Theupside here is that we know all the quadratic constraints and that it is simpler to construct anexhaustive list of 6th order constraints from this set than it is to make an exhaustive list of all6th order singlets.

We can also write down a number of boundary conditions much like was done at 4th order forthe sGoldstino square mass. These would consist of the 4th order ones supplemented with the

43

square of the traceless part of all the 3rd order non singlets such as those listed above. Once all ofthis is obtained, we can put all this information in a computer and produce a plot like figure 5.1.

When this is done, then we can try to put additional bound to zero as was done in thesGoldstino square mass case. There is an additional difficulty here because while it is easy tomathematically set some bound to zero, there would have to be a physical interpretation of thisadditional constraint. This was already a challenge in the sGoldstino analysis and will not becomeeasier in this case because all the physical quantities (masses, interactions etc) are 2nd order inthe embedding tensor components.

5.4 Eigenvalue equations

Another approach to the problem would be to find an eigenvector of the (projected) mass matrix.This could be A1 in the case of the sGoldstino mass matrix or the sCrucchino or sCrostino in caseof the full mass matrix. In the previous section we have already excluded a lot of possibilities inthis direction since we know that:

x m2 x 6= λ|x|2 (5.37)

Where x is either the sCrucchino or the sCrostino. If x is an eigenvector of the mass matrix thenthe above relation would have to be an equality, so x is not an eigenvector of the mass matrix.

We can again try to find additional constraints such that x is an eigenvector of the massmatrix. The upside of these calculations is that they are 4th order, the downside is that theyare non singlet. The fact that the expressions are non singlet is not that big of a problem for afirst calculation because the quadratic constraints are non singlet. It is relatively easy to pick a4th order term and find a quadratic constraint times second order tensor to rewrite the 4th orderexpression. Consider as an example the δ1 term in m2Scruc (the full expression is given in theappendix):

δv[rAxyza Aastu][3A

pvxyAzp −ApxyzAvp] (5.38)

The first two A2 are just two A2 tensors with a single contracted index and hence lives in the 945.There is also a quadratic constraint that lives in that irrep so we could use that to rewrite theexpression. Rewriting these expressions the goal is to get all the free indices in the vector x suchthat we are left with a singlet times x. This is quite difficult most of the times. Even the secondterm in the above expression which looks almost right has the internal indices a and p wrong.

We know that we would have to introduce additional constraints at some point in order tomake the eigenvalue equation work. A good candidate to try first is to set x4 = 0. This is becauseof two reasons, first it has a strong (but wrong) effect on MsG and second it is a simple constraintwith a lot of impact on the expressions.

Setting x4 equal to zero we find that the δ4 and δ3 terms get in the right form for m2Scrucand (even better) that the δ1 and δ0 terms cancel. We are then left with the δ2 term which is ingeneral the most difficult term to deal with because the free indices are the most irregular here:

δ[rsvxAabyt Azu]ab[A

pvxyAzp −ApvxzAyp + 2ApvyzAxp] (5.39)

which has free indices on 3 and even on 4 tensors. It turns out that we can set this combinationto zero if we set the 720 = 0 in combination with x4 = 0. We are then left with:

m2 sCruc = (−17

4|A1|2 +

5

84|A2|2) sCruc (5.40)

Which has indefinite sign for de Sitter vacua.We have tried using the sCrostino instead of the sCrucchino and we have tried to find other

constraints to deal with the different terms. The result is always the same, the δ4 and δ3 termsgive some mostly positive contribution, the δ0 and δ1 terms cancel and the δ2 term is always themost problematic. It is possible that when we contract the δ2 term from the left with anothersCrucchino that the problematic terms simplify in some way. As discussed in the previous section,we have not explored this possibility.

44

5.5 Discussion of the results

In this chapter we have looked for a bound on the lightest scalar particle in maximal supergravity,analogous to the analysis done in the N = 1, 2 supergravity theories. The simplest approach,taking the trace of the sGoldstino mass matrix, was explored by A. Borghese and ultimately gavean allowed region in the (MsG, V ) plane. This analysis shows that if stable de Sitter vacua exist inmaximal supergravity it is probably difficult to find them. Adding additional constraints A1 = 0,2352 = 0 or 3584 = 0 ensures that the sGoldstino square mass is non positive. Any stable deSitter vacuum needs to have a non zero A1 = 0, 2352 = 0 and 3584 = 0. Adding the constraintx4 = 0 ensures that the sGoldstino square mass is non negative and hence this does not proveanything.

We have used numerical analysis to examine using the sCrucchino or sCrostino to project themass matrix. We have found that any relation of the form:

x m2 x = λ|x|2 (5.41)

does not work in general. In addition, we have found a stable sCrostino in the SO(4) orbit whichimplies that we can not find a general master relation which is always negative using the sCrostino.

We have then looked at the possibility to examine the 6th order expressions similarly to thesGoldstino square mass with the goal of constructing an allowed region in the (MScruc, V ) plane.We have not made any progress in this direction due to the number of constraints and singletsinvolved.

Finally we have looked at the eigenvalue equation:

m2 x = λx (5.42)

We know that this relation can not hold in general, but maybe we could find some nice constraintsfor these equations. The difficult term in this direction is always the term arising from the δ2 termin the mass matrix. Finding a proper constraint for this term turned out to be like shooting inthe dark.

We know that our approach does not work, it is interesting to know why it does not work.Unfortunately, this question is difficult to answer. When considering the sGoldstino square masswe take the trace over a 36 × 36 matrix and hence sum over 36 eigenvalues. In that case onecan argue that the negative eigenvalues are compensated by the positive eigenvalues such that thetrace is positive. Indeed, this seems to be the case.

The failure of the sCrucchino and sCrostino is more difficult to explain. We are looking inone direction, specified by A1, so there is no averaging over all the eigenvalues. We know also inthose cases that the masses can be positive for de Sitter vacua. The most information that wehave is contained in the behavior of the sCrostino in the SO(4) orbit. In the basis used in [14]the sCrostino selects an tachyonic and non tachyonic direction such that the sum of the two ispositive.

If we consider all de Sitter vacua that are linked to Minkowski vacua through some continuousorbit then the behavior of the SO(4) orbit could well be generic. There are a few scalar masses thatbecome increasingly positive which makes the sCrucchino mass positive close to the Minkowskipoint. In this case the only vacua that yield to a sCrucchino analysis are those that are connectedto a Minkowski vacuum with all directions sufficiently (normalized mass equal to zero in theMinkowski limit) flat and the de Sitter vacua that are not connected to a Minkowski vacuum(either because they are not part of an orbit or because the orbit consists solely of de Sittervacua). In addition, we suspect that the sCrucchino mass will tend zero or a negative value whenwe move far away from Minkowski.

There could be a simple explanation why our approach does not work; there exist stable deSitter vacua in maximal supergravity. In this case one certainly can not expect a master relationto hold. Probably the easiest way to show that there exist stable de Sitter vacua is to constructan explicit example. We have not made an attempt in that direction but others have.4 Following

4See for instance the thesis by Remko Klein.

45

the above reasoning we conclude that stable de Sitter vacua (if they exist) are either close (inan orbit) to some Minkowski vacuum with strictly positive directions or are not part of an orbitat all. In fact if we consider a Minkowski vacuum with strictly positive directions then the deSitter vacua that are close to it, would have to be stable due to continuity of the scalar masses.Because of this it is interesting to investigate Minkowski vacua with strictly positive directions.A Minkowski vacuum with one or more flat directions will probably be connected to unstable deSitter vacua since the flat directions can be pushed away from zero due to splitting effects.

46

Chapter 6

Conclusion

In this work we have tried to prove that there are no stable de Sitter vacua in maximal supergravity.Our approach was based on ideas used in N = 1, 2 supersymmetric theories. Certain scalardirections, that are related to supersymmetry breaking, have masses that are fully determined bysplitting effects and hence can not be tuned to arbitrary values. In the lower supersymmetric casesthe scalar mass matrix is projected on these directions and the trace of the projected matrix givesa bound on the lowest scalar mass.

We have defined similar directions in maximal supergravity by taking specific combinations ofthe embedding tensor components. The easiest projection is to sandwich the scalar mass matrixwith 2 A2 and taking the trace. This was a 4th order calculation and the results are contained infigure 5.1. There is clearly an allowed region in the (MsG, V ) plane with positive scalar potentialand sGoldstino square mass. We have tried to find additional constraints, both numerically asanalytically, such that the sGoldstino square mass is always non positive. The sGoldstino squaremass is always zero when A1 is zero, non negative when A1 is full trace and non positive whenA2 is full trace (the theory is stable when A2 is zero since then we have the fully supersymmetriccase).

Moving from 4th order to 6th order we can use the sCrucchino and the sCrostino to projectthe mass matrix. We hoped that the additional weight provided by A1 in the sCrucchino caseor special properties of the sCrostino (which is the (anti) self dual part of the sCrucchino) wouldsingle out the tachyonic directions. We can conclude after a number of numerical examples thatneither the sCrostino nor the sCrucchino is an eigenvector of the mass matrix in general and evenworse that there are examples where the mass matrix is projected on a positive number when weuse the sCrostino.

This implies that the best we can hope for is a similar result as in figure 5.1 with a (MScrucc/Scros, V )plane. It is unclear if this analysis will give a significant improvement over the constraint areaobtained for the sGoldstino square mass. If it does it could show a number of things. Ei-ther the allowed area is a lot smaller for positive values of the mass matrix in which case thesCrucchino/sCrostino analysis really is closer to the tachyonic directions than the sGoldstino di-rections, or the allowed area is a lot smaller for negative values of the mass matrix in which casethe sCrucchino/sCrostino is further away from the tachyonic directions, or the allowed regionbecomes smaller in both the positive and negative putting more and more constraints on stablevacua. Only the first and third possibility can really count as an improvement over the sGoldstinoanalysis.

We have also tried a variety of additional constraints to see if we can find some that make thesCrucchino/sCrostino mass matrix or eigenvalue equation non positive. We have only been ableto find a set of constraints such that the projected matrix is non negative. The main difficulty isthat we do not have an exhaustive list of 6th order singlets and constraint equations, such a listwould be needed if one wants to construct a (MScrucc/Scros, V ) plane.

Given the fact that the sCrostino projection will not give a conclusive prove of instability inmaximal supergravity and the many hours it would take to construct all 6th order constraints

47

and singlets we think that this is not the right approach for this problem. One could try to go tohigher order equations to find a conclusive prove. For instance at 8th order one could sandwichthe mass matrix with two 3th order non singlets and do a similar analysis as was done at 4th and6th order. There are a number of problems with this, first calculations would become more timeconsuming once again and second (and we feel more importantly) the physical interpretation of a3th order object is far from clear since theoretically all objects are 2nd order. Mathematically itcould work however.

If we abandon this line of thought then we are not entirely sure what would be a good alternativeto our approach. It might be fruitful to try and construct large number of de Sitter vacua. Recentlythere where some advantages in the construction of de Sitter vacua in N = 1 supergravity theories[18]. Kallosh et al found a procedure to deform Minkowski vacua to stable de Sitter vacua for aspecial class of theories. In our case, Minkowski would imply that:

|A1|2 ∝ |A|2|2 (6.1)

And deforming to de Sitter means increasing the modulus of A2 slightly. Kallosh uses a numberof superfields, one of which is of no scale type and has the greatest symmetry order parametervalue. THey set up a hierarchy of symmetry breaking such that symmetry breaking in the no scalemodulus is big and in the other moduli symmetry breaking is comparable to a parameter ε. Theno scale modulus T is used to balance the negative definite contribution in the scalar potentialsuch that we are in a de Sitter vacuum. It is shown (under the condition of having one no scalemodulus and a hierarchy in symmetry breaking) that one can always find values for ε and thecosmological constant (> 0) such that the scalar mass matrix is positive definite.

The approach used in [18] relies on the superspace formalism of N = 1 supergravity. If onewants to translate the approach to maximal supergravity then one runs in a number of problems.First, there is no superspace formalism, but it might be possible to translate to the embeddingtensor formalism. Second, there is one order parameter of symmetry breaking, A2, and it is unclearhow to construct a hierarchy of symmetry breaking in this case (if that is possible at all). We give[20] as a possible reference on no scale structure in extended supergravity.

Instead of trying to generalize the no scale result from lower supersymmetric theories onecan also try to construct several orbits such as the SO(4) orbit. The least unstable de Sittervacua lie close to Minkowski in that orbit, we have already expressed our believe that if stable deSitter exist then they probably have a low cosmological constant. It would be interesting to havemore examples of the behaviour of the scalar masses near a Minkowski point to see if the SO(4)behaviour holds for other gaugings as well. It is also known from figure 5.1 that the sGoldstinosquare mass favors stable de Sitter vacua with small cosmological constants.

We have shown that a simple projection of the scalar mass matrix does not give a meaningfulbound. This does not mean that there are stable de Sitter vacua in maximal supergravity. Wedo believe that a proof of instability will not be based on projections of the scalar mass matrix.However, we are also unsure of another approach that shows the same potential as the one we usedin this thesis. It is also possible that stable de Sitter vacua exist. In that case we expect them tohave a small cosmological constant and a possible way to construct them is to deform Minkowskivacua in a similar way as in [18].

48

Bibliography

[1] A. Borghese, Cosmological and holographic applications of supergravity, 2013

[2] A. Borghese, A. Guarino, D. Roest, The many surprises of maximal supergravity 2013, [hep-th]1307.6919v1

[3] H. Nastase, Introduction to supergravity 2012, [hep-th] 1112.3502v2

[4] B. De Wit, H. Nicolai, N = 8 Supergravity 1982, Nuclear physics B208, 323-364

[5] B. de Wit, H. Samtleben, M. Trigiante, The maximal D=4 supergravities, 2007, [hep-th]0705.2101v2

[6] G. Inverso, de Sitter vacua in N = 8 supergravity, Master thesis 2009/2010, Padova Uni.

[7] G. Dibitetto, A. Guarino, D. Roest, Charting the landscape of N = 4 flux compactifications,JHEP 1103 (2011), arxiv: 1102.0239 [hep-th]

[8] G. Dall’Agata, G. Inverso, On the vacua of N = 8 gauged supergravity in 4 dimensions, 2012,[hep-th] 1112.3345v2

[9] E. Cremmer, B. Julia, The SO(8) supergravity Phys. Lett 80B (1978) 48;

[10] L. Brizi, C.A. Scrucca, The lightest scalar in theories with broken supersymmetry, 2011, [hep-th] 1107.1596

[11] M. Gomez-Reino, J. Louis, C. A. Scrucca, No metastable de Sitter vacua in N = 2 supergravitywith only hypermultiplets, 2008, [hep-th] 0812.0884

[12] L. Andrianopoli, R. D’Auria, S. Ferrara, M.A. Lledo, On the super higgs effect in extendedsupergravity 2002, [hep-th] 0202116v2

[13] M. Bertolini, Lectures on supersymmetry Sissa, Trieste, Italy

[14] A. Borghese, A. Guarino, D. Roest, Triality, Periodicity and Stability of SO(8) Gauged Su-pergravity 2013, [hep-th] 1302.6057v2

[15] B. de Wit, H. Samtleben, M. Trigiante, On lagrangians and gaugings of maximal supergravi-ties, Nucl. Phys. B655 (2003) [hep-th] 0212239

[16] T. Garidi, What is mass in de Sitterian physics? [hep-th] 0309104

[17] P. Breitenlohner, D.Z. Freeman, Stability in Gauged Extended Supergravity Annals Phys, 144,(1982) 249

[18] R. Kallosh, A. Linde, B. Vercnocke, T. Wrase, Analytic classes of metastable de Sitter vacua[hep-th] 1406.4866v1

[19] G. Dall’Agata, G. Inverso, M. Trigiante, Evidence for a family of SO(8) gauged supergravitytheories [hep-th] 1209.0760

49

[20] L. Andrianopoli, R. D’Auria, S. Ferrara, M.A. Lledo, Gauged extended supergravity withoutcosmological constant: no-scale structure and supersymmetry breaking, [hep-th] 0212141v1

[21] N. Warner, Some new extrema of the scalar potential of gauged N = 8 gauged supergravitytheories, Phys. Lett. B128 (1983) 169

[22] N. Warner, Some properties of the scalar potential in gauged supergravity theories, Nucl. Phys.B231 (1984) 250

[23] CMS Collaberation, Observation of a new boson at a mass of 125 GeV with the CMS exper-iment at LHC [hep-ex] 1207.7235

50

Chapter 7

Acknowledgements

This thesis could not have been written without the help of a few people. First, I would like tothank Andrea for helping me understand the theory and the problem. Unfortunately, I was unableto significantly improve on his work. Secondly, I would also like to thank Adolfo for that week inNovember when we had a lot of discussions and for the many hours of computations that he didfor us. All of the numerical results in this thesis are due to him. Thanks for all the help, Andreaand Adolfo.

I should not forget my fellow master students: Ricardo, Remko, Jan en Job. We helped eachother to stay motivated during the difficult times and we also had some fun hallway conversations.Good luck in the future, guys. I also have to thank Emma for reading most of my thesis andchecking it for errors. This was extra difficult since she has not done any physics in the last 8years.

Finally, I want to thank Diederik for supervising my project. We usually had a discussion inour weekly meetings and they always gave me some new things to think about. So also to you,Diederik, I give you my thanks.

51

Appendix A

Conventions

A.1 Identities for 70±

Indices [ijkl] belong to the complex 70 representation of SU(8). That’s not an irrep. Indeed, dueto the presence of the ε symbol it can be split in two different irreps which we call 70+ and 70−.Explicitly we have:

ϕijkl ≡(ϕijkl

)∗, ϕijkl± ≡ 1

2

(ϕijkl ± 1

4! εijklmnpqϕmnpq

)(A.1)

Whenever the set of indices ijkl appears on a physical scalar field φ we implicitly assume thatthe indices must be taken to be antisymmetric and that the field is self dual. This is not the casewhen we deal with embedding tensor components. In there any antisymmetrization is indicatedexplicitly and in principle both self-dual and anti self-dual components of the 70 representationare present. Clearly, from the very definition:

14! εijklmnpq ϕ

mnpq± = ±ϕ± ijkl = ±

(ϕijkl±

)∗From this expression we can obtain the following results for the contraction of two scalar fields:

ϕijklϕijkl =∣∣ϕijkl∣∣2 = ϕijkl+ ϕ+ ijkl + ϕijkl− ϕ− ijkl + ϕijkl+ ϕ− ijkl + ϕijkl− ϕ+ ijkl

=∣∣∣ϕijkl+

∣∣∣2 +∣∣∣ϕijkl− ∣∣∣2

Whenever the two fields are contracted on a smaller number indices we could extract the followingrelations:

ϕmrst± ϕ± irst = 18 δ

mi

∣∣ϕrstu± ∣∣2 ,ϕmnpr± ϕ± ijkr = − 1

16 δmnpijk

∣∣ϕrstu± ∣∣2 + 94 δ

[m[i ϕ

np]rs± ϕ± jk]rs

52

A.2 Quadratic variables

We define the following quadratic contractions of the fluxes:

Rmi ≡ Amr Air , Smi ≡ AirstAmrst , Smi ≡ ArstmArsti ,

Tmnij ≡ AirsmAnjrs , Tmnij ≡ ArsmnArsij , T[ijkl] ≡ Ars[ij Askl]r ,

Vmnpijk ≡ Ai

rmnApjkr , Vmnpijk ≡ Ar

mnpArijk , Xmnpqijkl ≡ Ai

mnpAqjkl ,

U(rs)ijkl ≡ δ

(r[i A

s)jkl] , V[ijkl] ≡ Ar [ijk Al]r , Sijkl ≡ Ar [ijk Al]r − 3

4 Ars[ij A

sr|kl]

These combinations do not belong to specific representations of SU(8). Instead they are a combi-nation of different irreps. If we want to write specific combinations belonging to SU(8) irreps wehave to subtract traces. By doing that we obtain the following (The square of these expressionsis what leads to the non negative singlet bounds on the sGoldstino square mass in chapter 6):

ArmstArist

63

= ArmstArist − 1

8 δmi |A2|2

AirstAmrst

63

= AirstAmrst − 1

8 δmi |A2|2

AmrAir63

= AmrAir − 18 δ

mi |A2|2

ArmnsArijs

720

= ArmnsArijs − 2

3 δ[m[i Ar

n]stArj]st + 121 δ

mnij |A2|2

A[irs[mAn]

j]rs

720

= A[irs[mAn]

j]rs + 16 δ

[m[i Ar

n]stArj]st + 16 δ

[m[i Aj]

rstAn]rst − 1

42 δmnij |A2|2

ArmnpArijk

2352

= ArmnpArijk − 9

4 δ[m[i Ar

np]sArjk]s + 910 δ

[mn[ij Ar

p]stArk]st − 120 δ

mnpijk |A2|2

A[ir[mnAp]jk]r

2352

= A[ir[mnAp]jk]r − 1

4 δ[m[i Ar

np]sArjk]s + δ[m[i Aj

rs|nAp]k]rs+

+ 15 δ

[mn[ij Ar

p]stArk]st + 110 δ

[mn[ij Ak]

rstAp]rst − 160 δ

mnpijk |A2|2

A[i[mnpAq]jkl]

1764

= A[i[mnpAq]jkl] + 1

2 δ[m[i Ar

npq]Arjkl] + 92 δ

[m[i Aj

r|npAq]kl]r+

− 32 δ

[mn[ij Ar

pq]sArkl]s + 3 δ[mn[ij Ak

rs|pAq]l]rs+

+ 34 δ

[mnp[ijk Ar

q]stArl]st + 14 δ

[mnp[ijk Al]

rstAq]rst − 120 δ

mnpqijkl |A2|2

53

The quadratic constraints on due to the embedding tensor are repeated for completeness:

9ArstmArsti −AirstAmrst − δmi |A2|2 = 0 (A.2)

3ArstmArsti −AirstAmrst + 12AirA

mr − 1

4δmi |A2|2 −

3

2δmi |A1|2 = 0 (A.3)

Aijv[mAvnpq] +Ajvδ

i[mA

vnpq] −Aj[mAinpq]

+1

4!εmnpqrtstu(Aj

ivrAvstu +AivδrjAv

stu −AirAjstu) = 0 (A.4)

AirsmAnjrs −AjrsnAmirs + 4A(m

ijrAn)r − 4A(i

mnrAj)r

−1

8δni (Ar

stmArstj −ArrstAmrst) +1

8δmj (Ar

stnArsti −AirstAnrst) = 0 (A.5)

ArmnpArijk − 9A[i

r[mnAp]jk]r − 9δ[m[i Aj

rs,nAp]k]rs

−9δ[mn[ij Ar

p,stArk]st + δmnpijk |A2|2 = 0 (A.6)

These constraints live in the 63, 63, 70− + 378 + 3584, 945 + 945 and 2352 irreps of SU(8),respectively. We can extract the pure 70− and 378 irreps by contracting the third equation withdeltas. We find:

Ar [ijkAl]r −3

4Ars[ijA

skl]r =

1

4!εijklmnpq(Ar

mnpAqr +3

4Ar

smnAsrpq) (A.7)

3

4ArijkAlr +

3

4Arl[ijAk]r =

1

4!εmnpqrijkAl

qrsAsmnp +

3

4

1

4!εijklmnpqAr

spqAsrmn (A.8)

The requirement that the scalar potential has a critical point in the origin translates to:

Ar [ijkAl]r +3

4Ars[ijA

skl]r = − 1

4!εijklmnpq(Ar

mnpAqr +3

4Ar

smnAsrpq) (A.9)

54

A.3 Quartic singlets

Quartic variables can be constructed using the following real contractions

x1 = |A2|2 |A2|2 , (A.10)

x2 = |A2|2 |A1|2 ,x3 = |A1|2 |A1|2 ,x4 = AmrAir A

isAms − 18 |A1|2 |A1|2 = (Rm

i − 18 δ

mi |A1|2)(Ri

m − 18 δ

im |A1|2) = |trl R|2 ,

x5 = ArstmArstiAu

vziAuvzm = Smi Sim = S S ,

x6 = ArstmArstiAm

uvzAiuvz = Smi Sim = S S ,

x7 = AirstAmrstAm

uvzAiuvz = Smi Sim = S S ,

x8 = ArstmArstiA

iuAmu = Smi Rim = S R ,

x9 = AirstAmrstA

iuAmu = Smi Rim = S R ,

x10 = ArsmnArsij At

uijAtumn = Tmnij Tijmn = T T ,

x11 = ArsmnArsij Am

tuiAjntu = Tmnij Tijmn = T T ,

x12 = AirsmAnjrsAm

tuiAjntu = Tmnij Tijmn = T T ,

x13 = AirsmAnjtuAm

tujAintu = Tmnij Tjimn = T TT ,

x14 = AirsmAnjrsA

ijAmn = Tmnij AmnAij = TA1A1 ,

x15 = Ars[ijAs

r|kl]AtuijAutkl = T[ijkl] Tijkl .

And complex contractions

z1 = 14! εijklmnpq Ar

ijkAlr AsmnpAqs , z2 = 1

4! εijklmnpq ArsijAs

rklAtumnAu

tpq ,

z3 = Ars[ijAsr|kl]At

ijkAlt , z4 = 14! εijklmnpq Ar

sijAsrklAt

mnpAqt ,

z5 = 14! εijkmnprsAt

ijk AvzAvmnpAz

rst , z6 = ArsmnArsij A

imntA

jt ,

z7 = 14! εijkmnprsAv

zrtAzvsuAt

ijkAumnp

55

These variables are not independent but they are constraint due to the quadratic constraints. Thelist of constraints derived in [1] is:

0 = x5 −1

8x1 − 4x4 (A.11)

0 = x6 −1

8x1 − 36x4 (A.12)

0 = x7 −1

8x1 − 324x4 (A.13)

0 = x8 −1

8x2 − 2x4 (A.14)

0 = x9 −1

8x2 − 18x4 (A.15)

0 =1

2x9 −

3

2x14 − (z1 + z1)− 3

4(z3 + z3) +

3

4(z4 + z4) (A.16)

0 =3

4(z3 + z3)− 3

4(z4 + z4) +

9

16(z2 + z2)− 9

8x15 (A.17)

0 =1

2x9 −

3

2x14 + (z1 + z1) +

3

4(z3 + z3) +

3

4(z4 + z4) (A.18)

0 =3

4(z3 + z3) +

3

4(z4 + z4) +

9

16(z2 + z2) +

9

8x15 (A.19)

0 = x9 + x14 −4

3z5 − z4 (A.20)

0 =4

3z5 + z4 −

1

6x6 − x11 −

1

2x13 +

3

4x15 (A.21)

0 = x10 − x12 − 4z6 + 4z3 −1

8x5 +

1

4x6 −

1

8x7 (A.22)

0 = 2z6 − 2z3 − 4x8 − 4x14 (A.23)

0 = x11 + x13 − 32x4 (A.24)

0 =1

3x1 − 3x5 − x6 + x10 − 5x11 − x12 − x13 + 6x15 (A.25)

0 = − 3

4z6 −

3

4x8 −

1

4x9 +

1

4x2

3

2z4 + z5 − z1 (A.26)

0 = − z7 +3

2z4 +

1

4x6 +

3

4x11 −

3

4z3 −

3

4z6 (A.27)

56

A.4 Quartic non singlet

These were all the quartic variables in the singlet representation of SU(8). It is also possible toclassify quartic variables in non-singlet representations.

T(kl)(ij) V

1232

=(T

(kl)(ij) −

110 δ

(k(i S

l)j) −

110 δ

(k(i S

l)j) + 1

90 δ(kl)(ij) |A2|2

)V ,

Vq2(kl)(ij)q1

Rq2q2

1232

=(Vq2(kl)(ij)q1

+ 110 δ

(k(i T

|l)q2j)q1

+ 110 δ

(k(i T

q2|l)j)q1− 1

90 δ(kl)(ij) Sq2q1

)Rq1q2 ,

Tq(k(ij) Rl)

q + T(kl)(i|q Rq

j)

1232

= Tq(k(ij) Rl)

q + T(kl)(i|q Rq

j) −110 δ

(k(i

(Tl)q2j)q1

+ 2 Tq2|l)j)q1

+ Tq2|l)q1|j)

)Rq1q2+

− 110 δ

(k(i Sqj) Rl)

q − 110 δ

(k(i Sl)q Rq

j) + 145 δ

(kl)(ij) Sq2q1 Rq1

q2 ,S

(k(i R

l)j)

1232

= S(k(i R

l)j) −

110 δ

(k(i Sqj) Rl)

q − 110 δ

(k(i Sl)q Rq

j) −110 δ

(k(i S

l)j) |A1|2+

− 110 δ

(k(i R

l)j) |A2|2 + 1

90 δ(kl)(ij) Sq2q1 Rq1

q2 + 190 δ

(kl)(ij) |A1|2 |A2|2 ,

Xp2q2(kl)(ij)p1q1

Tp1q1p2q2

1232

=(Xp2q2(kl)(ij)p1q1

− 110 δ

(k(i V

l)p2q2j)p1q1

− 110 δ

(k(i V

p2q2|l)j)p1q1

+ 190δ

(kl)(ij) Tp2q2p1q1

)Tp1q1p2q2 ,

Vq2(kl)(i|p1q1

(Tp1q1j)q2

− Tp1q1q2|j))+

−Vp2q2(k(ij)q1

(Tl)q1p2q2 − Tq1|l)p2q2

)1232

= Vq2(kl)(i|p1q1

(Tp1q1j)q2

− Tp1q1q2|j))− V

p2q2(k(ij)q1

(Tl)q1p2q2 − Tq1|l)p2q2

)+

+ 110 δ

(k(i Tl)q2p1q1

(Tp1q1j)q2

− Tp1q1q2|j))

+ 110 δ

(k(i Tp2q2j)q1

(Tl)q1p2q2 − Tq1|l)p2q2

)+

+ 15 δ

(k(i

(Vp1q1|l)p2|j)q2 + V

p1q1|l)|j)p2q2

)T[p2q2]p1q1 −

15 δ

(k(i

(Vq1p1|l)j)p2q2

+ Vq1|l)p1|j)p2q2

)Tp2q2[p1q1]

− 245 δ

(kl)(ij) Tp2q2p1q1 Tp1q1p2q2 ,

Tq2(k(i|q1

(Tl)q1j)q2− T

l)q1q2|j)+

−Tq1|l)j)q2

+ Tq1|l)q2|j)

)1232

= Tq2(k(i|q1

(Tl)q1j)q2− T

l)q1q2|j) − T

q1|l)j)q2

+ Tq1|l)q2|j)

)+ 1

10 δ(k(i T

q1|l)j)q2

(Sq2q1 + Sq2q1

)− 1

10 δ(k(i

(Tl)q1j)q2− T

l)q1q2|j) − T

q1|l)j)q2

+ Tq1|l)q2|j)

)Sq2q1+

− 15 δ

(k(i Tq2|l)p1q1

(T

[p1q1]j)q2

− T[p1q1]q2|j)

)+ 1

5 δ(k(i Tp2q2j)q1

(Tl)q1[p2q2] − T

q1|l)[p2q2]

)+

− 190 δ

(kl)(ij) Sq2q1

(Sq1q2 + Sq1q2

)− 2

45 δ(kl)(ij) Tp2q2p1q1 T

[p1q1][p2q2] ,

Tp2q2(ij) T(kl)q2p2

1232

= Tp2q2(ij) T(kl)q2p2 + 1

90 δ(kl)(ij)

(Tp2q2p1q1 Tp1q1q2p2 + Tp2q2p1q1 Tq1p1q2p2

)+

+ 110 δ

(k(i

(Tp2q2j)q1

Tl)q1q2p2 + Tp2q2j)q1Tq1|l)q2p2 + Tp2q2q1|j) Tl)q1q2p2 + Tp2q2q1|j) Tq1|l)q2p2

)

Sq(i T(kl)q|j) + S(k

q Tl)q(ij)

1232

= Sq(i T(kl)q|j) + S(k

q Tl)q(ij) −

110 δ

(k(i Sq2q1

(Tl)q1j)q2

+ 2 Tl)q1q2|j) + T

q1|l)q2|j)

)+

− 15δ

(k(i Sqj) Sl)q + 1

45 δ(kl)(ij) Sq2q1 Sq1q2 ,

S(k(i S

l)j)

1232

= S(k(i S

l)j) −

15 δ

(k(i

(Sqj) Sl)q + S

l)j) |A2|2

)+ 1

90 δ(kl)(ij)

[Sq2q1 Sq1q2 +

(|A2|2

)2],

Tp2q2(ij) T(kl)p2q2

1232

= Tp2q2(ij) T(kl)p2q2 + 1

90 δ(kl)(ij)

(Tp2q2p1q1 Tp1q1p2q2 + Tp2q2p1q1 Tq1p1p2q2

)+

110 δ

(k(i

(Tp2q2j)q1

Tl)q1p2q2 + Tp2q2j)q1Tq1|l)p2q2 + Tp2q2q1|j) Tl)q1p2q2 + Tp2q2q1|j) Tq1|l)p2q2

)

Sq(i T(kl)j)q + S(k

q Tq|l)(ij)

1232

= Sq(i T(kl)j)q + S(k

q Tq|l)(ij) −

110 δ

(k(i Sq2q1

(Tl)q1j)q2

+ 2 Tq1|l)j)q2

+ Tq1|l)q2|j)

)+

− 110 δ

(k(i

(Sl)q Sqj) + Sqj) Sl)q

)+ 1

45 δ(kl)(ij) Sq2q1 Sq1q2 . (A.28)

57

while those in the 63 are always of the formδ

(k(i Aj)

npq Al)npq63

= δ(k(i

(Aj)

npq Al)npq − 18 δ

l)j)|A2|2

)f(|A1|2, |A2|2) ,

δ(k(i Aj)

npq1 Al)npq263

= δ(k(i

(Aj)

npq1 Al)npq2 − 18 δ

l)j)Am

npq1 Amnpq2)f(Sq2q1 , Sq2q1 , Rq2

q1) ,δ

(k(i Aj)

np1q1 Al)np2q263

= δ(k(i

(Aj)

np1q1 Al)np2q2 − 18 δ

l)j)Am

np1q1 Amnp2q2)f(Tp2q2p1q1 , Tp2q2p1q1) ,

...

In other words one always has to subtract 1/8 times the trace.

58

Appendix B

Scalar mass matrix in N = 2supergravity projected on Nu

xPx

The sGoldstino is an eigenvector of the scalar mass matrix if Σ = 0. We will first calculate thecontraction of the scalar mass matrix with the sGoldstino: NvxP x = − 1

2∇vP xP x. The scalar

mass matrix is given by:

m20uv = 4∇ukw∇vkw − 4Rusvtk

skt − 3∇uP y∇vP y − 3P y∇(u∇v)Py (B.1)

To keep the expressions easy to handle we will do the calculation term by term. The centralidentity needed for the first term is given in [11]:

∇ukw∇uP x = 3P xkw +1

2εxyzP y∇wP z (B.2)

Such that:∇ukw∇uP xP x = 3P xP xkw (B.3)

Because of the antisymmetric property of the ε-tensor. Doing the calculation for the first term:

−2∇ukw∇vkw∇vP xP x = −2∇ukw∇vkw∇vP xP x (B.4)

= −6kw∇ukwP xP x (B.5)

= −9

2P yP y∇uP xP x (B.6)

In the last line we have used the fact that we are in a stationary point. For the second term weneed the Riemann tensor:

Rusvt = −hu[vhst] − ΩyusΩyvt − Ωyu[vΩ

yst] − Σusvt (B.7)

59

We will have to use: ∇uP x = 2Ωxuvkv. The calculation for the second term becomes:

2Rusvtkskt∇vP xP x = −2

3ktkt∇uP xP x

−2∇uP yNyvN

vxP x

−2

3Ωystk

sktΩyuv∇vP xP x

−2Σusvtkskt∇vP xP x (B.8)

= −2

3ktkt∇uP xP x

−2ktkt∇uP xP x

−2Σusvtkskt∇vP xP x (B.9)

= −8

3ktkt∇uP xP x

−2Σusvtkskt∇vP xP x (B.10)

Where we have used that NxuNyu = kukuδ

xy and 0 = NuNxu = Ωxsuk

sku. The third term is theeasiest and uses again the orthogonality of the sGoldstino directions.

3

2∇uP y∇vP y∇vP xP x =

3

2ktkt∇uP xP x (B.11)

For the last term we need:ΩxuwΩywv = −huvδxy − εxyzΩzuv (B.12)

From which it follows that:

3

2P y∇(u∇v)P

y∇vP xP x = −9

8P yP y∇uP xP x (B.13)

The full mass matrix contracted with the sGoldstino then becomes:

m20uvN

vxP

x = (−45

8P yP y − 7

6ktkt)∇uP xP x − 2Σusvtk

skt∇vP xP x (B.14)

From which it is clear that the sGoldstino is an eigenvector if Σ = 0.

60

Appendix C

Scalar mass matrix in N = 8supergravity

C.1 Scalar mass matrix

The scalar mass matrix is composed of three irreps(70× 70

)sym

= 1 + 720 + 1764 . (C.1)

In terms of the embedding tensor components the single irreps are given (modulo quadratic con-straints) by the following expressions

1 −→ δm1n1p1q1m2n2p2q2

(− 1

2 |A1|2 + 120 |A2|2

). (C.2)

720 −→ − δ[m1n1

[m2n2 Arp2q2]sArp1q1]s + 5 δ[m1n1

[m2n2 Ap1rs|p2 Aq2]

q1]rs +

+ 32 δ[m1n1p1

[m2n2p2 Arq2]stArq1]st + 5

6 δ[m1n1p1[m2n2p2 Aq1]

rstAq2]rst +

− 16 δm1n1p1q1

m2n2p2q2 |A2|2 . (C.3)

1764 −→ − 23 A[m1

[m2n2p2 Aq2]n1p1q1] − 2

3 δ[m1

[m2 Arn2p2q2]Arn1p1q1] +

+ δ[m1n1

[m2n2 Arp2q2]sArp1q1]s + δ[m1n1

[m2n2 Ap1rs|p2 Aq2]

q1]rs +

+ 52 δ[m1n1p1

[m2n2p2 Arq2]stArq1]st − 1

6 δ[m1n1p1[m2n2p2 Aq1]

rstAq2]rst +

− 310 δm1n1p1q1

m2n2p2q2 |A2|2 . (C.4)

Using the quadratic constraints we can rewrite this to:

720 −→ 5δ[m1n1

[m2n2 Ap1rs|p2 Aq2]

q1]rs − δ[m1n1

[m2n2 Arp2q2]sArp1q1]s

+ δ[m1n1p1[m2n2p2 Aq1]

rstAq2]rst (C.5)

1764 −→ − 23 A[m1

[m2n2p2 Aq2]n1p1q1] − 2

3 δ[m1

[m2 Arn2p2q2]Arn1p1q1] +

+ δ[m1n1

[m2n2 Arp2q2]sArp1q1]s + δ[m1n1

[m2n2 Ap1rs|p2 Aq2]

q1]rs +

+ 19 δ[m1n1p1

[m2n2p2 Aq1]rstAq2]

rst − 145 δm1n1p1q1

m2n2p2q2 |A2|2 . (C.6)

61

C.2 sGoldstino mass matrix

The sGoldstino mass matrix is composed of three irreps(36× 36

)sym

= 1 + 63 + 1232 . (C.7)

These irreps take the following form in terms of the quadratic variables (coefficients such that theysum to the full mass matrix)

1 −→ δ(ij)(mn)

[ 17

216|A2|2|A2|2 −

1

12TT +

1

3T T +

1

12T TT − 13

36|A2|2|A1|2

− 1

24SS − 5

72SS − 1

108SS +

25

12SR+

5

36SR]. (C.8)

(= 29m

2sGδ

(nm)(ij) )

63 −→ δ(i(m[3

8Vj)xy

tu,n)Ttuxy +

3

20Vxy,j)

tu,n)Ttu[xy] +

3

20Vj)xy

n)tuT[tu]xy

− 3

40Vj)xy

n)tuTtuxy +

3

40Tj)x

tuTtux,n) − 3

40Tj)x

tuTtun)x

− 3

40Txy

n)tTj)txy +

3

40Txy

n)tTt,j)xy +

3

80Tx,j)

tuTtun)x

− 3

80Tx,j)

tuTutn)x +

9

80Tj)x

tuTtux,n) − 9

80Tj)x

tuTutx,n)

+3

80StxTj)t

n)x − 3

80StxTt,j)

n)x +3

80StxTt,j)

x,n)

− 3

40StxTj)t

x,n) − 1

16StxTj)t

x,n) +1

40StxTt,j)

n)x

+9

4RtxTj)t

x,n) +3

4RtxTj)t

x,n)

− 3

8RtxTt,j)

x,n) − 1

80Sn)x S

xj) −

1

80Sn)x S

xj)

− 3

8RtxTj)t

n)x − 1

120Sn)x S

xj)

+9

40|A1|2Sn)

j) +1

8|A2|2Rn)

j) −3

8Rn)x S

xj)

− 3

8Rxj)S

n)x −

2

5|A1|2Sn)

j) +1

8Rxj)S

n)x

+1

8Rn)x S

xj) −

1

80|A2|2Sn)

j) +3

80|A2|2Sn)

j)

]+ δ

(mn)(ij)

[ 1

160|A1|2|A2|2 −

1

480SS − 3

32SR− 1

320|A2|4

+3

160TT − 3

160T TT +

3

320SS +

1

64SS − 1

32RS]. (C.9)

(= 25 (m2

sG(m(i δ

n)j) −

18m

2sGδ

(nm)(ij) ))

62

1232 −→ − 3

4Vxy(i

tu(mVj)tun)xy − 3

4V(ij)x

tu(mTtun)x +

3

4V(ij)x

tu(mTtux,n)

+3

4V(i,xy

t(mn)Tj)txy − 3

4V(i,xy

t(mn)Tt,j)xy +

15

2V(ij)x

t(mn)Rxt

+3

8T(ij)

tuTtu(mn) − 3

8T(ij)

tuTut(mn) +

3

8T(i,x

t(mTj)tn)x

− 3

8T(i,x

t(mTj)tx,n) − 3

8T(i,x

t(mTt,j)n)x +

3

8T(i,x

t(mTt,j)x,n)

− 1

8St(iTj)t

(mn) +1

8St(iTt,j)

(mn) − 1

8S

(mt T(ij)

t,n)

+1

8S

(mt T(ij)

n)t +9

4|A1|2T(ij)

(mn) − 1

8|A2|2T(ij)

(mn)

+1

24S

(m(i S

n)j) −

15

4T(ij)

t(mRn)t −

15

4T(i,x

(mn)Rxj)

+5

4S

(m(i R

n)j)

− δ(i(m[− 3

8Vj)xy

tu,n)Ttuxy +

3

20Vxy,j)

tu,n)Ttu[xy] +

3

20Vj)xy

n)tuT[tu]xy

− 3

40Vj)xy

n)tuTtuxy +

3

40Tj)x

tuTtux,n) − 3

40Tj)x

tuTtun)x

− 3

40Txy

n)tTj)txy +

3

40Txy

n)tTt,j)xy +

3

80Tx,j)

tuTtun)x

− 3

80Tx,j)

tuTutn)x +

9

80Tj)x

tuTtux,n) − 9

80Tj)x

tuTutx,n)

+3

80StxTj)t

n)x − 3

80StxTt,j)

n)x +3

80StxTt,j)

x,n)

− 3

40StxTj)t

x,n) − 1

16StxTj)t

x,n) +1

40StxTt,j)

n)x

+9

4RtxTj)t

x,n) +3

4RtxTj)t

x,n)

− 33

8RtxTt,j)

x,n) − 1

80Sn)x S

xj) −

1

80Sn)x S

xj)

− 3

8RtxTj)t

n)x +1

30Sn)x S

xj)

+9

40|A1|2Sn)

j) +1

8|A2|2Rn)

j) −3

8Rn)x S

xj)

− 3

8Rxj)S

n)x +

7

20|A1|2Sn)

j) +1

8Rxj)S

n)x

+1

8Rn)x S

xj) −

1

80|A2|2Sn)

j) −1

240|A2|2Sn)

j)

]− δ(ij)(mn)

[ 653

8640|A2|2|A2|2 −

31

480TT +

1

3T T +

31

480T TT − 511

1440|A2|2|A1|2

− 31

960SS − 31

576SS − 49

4320SS +

191

96SR+

31

288SR]

(C.10)

(= m2sG

(nm)(ij) − 63− 1)

63

Combining the irreps we find the following sGoldstino mass matrix:

m2sG(ij)

(mn) −→ − 3

4Vxy(i

tu(mVj)tun)xy − 3

4V(ij)x

tu(mTtun)x +

3

4V(ij)x

tu(mTtux,n)

+3

4V(i,xy

t(mn)Tj)txy − 3

4V(i,xy

t(mn)Tt,j)xy +

15

2V(ij)x

t(mn)Rxt

+3

8T(ij)

tuTtu(mn) − 3

8T(ij)

tuTut(mn) +

3

8T(i,x

t(mTj)tn)x

− 3

8T(i,x

t(mTj)tx,n) − 3

8T(i,x

t(mTt,j)n)x +

3

8T(i,x

t(mTt,j)x,n)

− 1

8St(iTj)t

(mn) +1

8St(iTt,j)

(mn) − 1

8S

(mt T(ij)

t,n)

+1

8S

(mt T(ij)

n)t +9

4|A1|2T(ij)

(mn) − 1

8|A2|2T(ij)

(mn)

+1

24S

(m(i S

n)j) −

15

4T(ij)

t(mRn)t −

15

4T(i,x

(mn)Rxj)

+5

4S

(m(i R

n)j)

+ δ(m(i

[3

4Vj)xy

tu,n)Ttuxy − 1

24Stj)S

n)t +

15

4Tj)x

t,n)Rxt

+1

24|A2|2Sn)

j) −3

4|A1|2Sn)

j)

](C.11)

64

C.3 Mass matrix * sCrucchino/sCrostino

We write m2rstu

vxyzAp[vxyAz]p, this is the mass matrix times the sCrucchino. We have kept the

various deltas in the expressions to give more insight in where the different terms come from.Removing the deltas is trivial if one takes care of the signs that might appear.

m2rstu

vxyzAp[vxyAz]p = 4V δrstuvxyzApvxyAzp

+ 5δ[rstvxyAu]aA

za[ApvxyAzp − 3ApvxzAyp]

+3

2δ[rs

vxAabyt Azu]ab[ApvxyAzp −ApvxzAyp + 2ApvyzAxp]

− 1

6δv[rA

xyza Aastu][3A

pvxyAzp −ApxyzAvp]

− 1

6Avxy[r Azstu][A

pvxyAzp − 3ApvxzAyp] (C.12)

In addition we also write − 34m

2rstu

vxyzApq[vxAqyz]p so that one can sum the two expressions to get

the mass matrix multiplied with the (anti) self dual scrostino. We have kept the deltas in thisexpression as well.

−3

4m2rstu

vxyzApq[vxAqyz]p = − 3V δrstu

vxyzApqvxAqyzp

− 15δ[rstvxyAu]aA

zaApqvxAqyzp

− 3

2δ[rs

vxAabyt Azu]ab[AqpvxA

pyzq − 2AqpvyA

pxzq]

+1

2δv[rA

astu]A

xyza ApqvxA

qyzp

+1

2Avxy[r Azstu]A

pqvxA

qyzp (C.13)

65

Appendix D

Singlet projections, numericalresults

Gauging sCrucchino sCrostino Vacuum CommentSO(3,5) − 45

64 −6 dS sCrucchino is dimensionfulSO(4,4) 0 0 Minkowski A1 = 0

SO(4) θ = π2

√3− 5 dS sCrASD = 0

SO(4) θ = 26π100 >> 0 dS

SO(4) θ = π4

965 Minkowski

SO(4) θ = 0√

3 + 5 AdS sCrASD = 0Geo IIa point 2 1468

85 AdSGeo IIa point 3 27

1124817 AdS sCrucchino is dimensionful

Geo IIa point 4 8921

372 AdS sCrucchino is dimensionful

Table D.1: The sCrucchino column gives the value of x for A1A2m2A2A1 = x|A1|2 and the

sCrostino column gives the value of x for sCrSDm2sCrSD = x|sCrSD|2, these numbers may be

dimensionful. The SO(4) is a reference to the SO(4) orbit discussed in [14]. We only give theprojections for which we know the value of x.

66

Appendix E

How to contract (anti)-symmetricindices

Contracting (anti)-symmetric indices comes down to counting. There are a few (inductive) rulesthat one has to follow but the basic idea is to write down all inequivalent permutations with thecorrect normalization. The best way to show how to contract indices is to give a few examples.We will first show what do we mean with the symbol antisymmetric and symmetric symbols []and () before starting to contract. For two indices we mean:

A(ij) =1

2(Aij +Aji) (E.1)

A[ij] =1

2(Aij −Aji) (E.2)

and for three:

A(ijk) =1

3(Ai(jk) +A(j|i|k) +A(jk)i) (E.3)

=1

6(Aijk +Aikj +Ajik +Akij +Ajki +Akji) (E.4)

A[ijk] =1

3(Ai[jk] −A[j|i|k] +A[jk]i) (E.5)

=1

6(Aijk −Aikj −Ajik +Akij +Ajki −Akji) (E.6)

From which we see that the number in front is always 1 over the number of tensors betweenbrackets (an excellent check you should always do to look for mistakes is to see if this is the case)and that the minus sign in the antisymmetric case is equal to (−1)number of permutations.

If the tensor Aij is symmetric in its indices then A(ij) = Aij and A[ij] = 0 and vice versafor antisymmetric indices. This also holds for arbitrary number of indices. If the tensor Aijk issymmetric in the last two indices Aijk = Aikj then:

A(ijk) =1

3(Aijk +Ajik +Akij) (E.7)

And once again similarly for more and antisymmetric indices.Let us start contracting indices. First, (anti)-symmetrization carries over under contracted

indices:

A(ijk)A(ijk) = A(ijk)A

ijk = AijkA(ijk) (E.8)

This naturally implies that A(ij)A[ij] = 0. Contracting indices then means that we simply write

down all the possibilities. For instance:

Ai[jAk]lBjk =

1

2(AijAklB

jk −AikAjlBjk) (E.9)

67

Then there is the generalized delta function: δj1···jni1···in which is always a fully antisymmetriccollection of single deltas. For instance:

δklij =1

4(δki δ

lj − δkj δli + δljδ

ki − δliδkj ) (E.10)

=1

2(δki δ

lj − δkj δli) (E.11)

From which it follows that (for i = 1, · · · 8):

δikij =7

2δkj (E.12)

We are now ready to tackle any contraction. We will now use the tensors A1 and A2 (using their(anti)-symmetric and tracelessness properties) for a few more examples:

δk[iAxyzj] Akl =

1

2(Axyzj Ail −Axyzi Ajl) (E.13)

= Axyz[j Ai]l (E.14)

δvxy[rstAu]aAzaAp[vxyAz]p =

1

4δvxy[rstAu]aA

za(ApvxyAzp − 3ApzvxAyp) (E.15)

Avxy[r Azstu]Apq[vxA

qyz]p =

1

4Avxy[r Azstu](2A

pqvxA

qyzp − 2ApqzvA

qxyp) (E.16)

=1

4Avxy[r Azstu](2A

pqvxA

qyzp + 2ApvzqA

qxyp) (E.17)

= Avxy[r Azstu]ApqvxA

qyzp (E.18)

In the last example I have relabeled some of the contracted indices.

68