Stability in Maximal Supergravity

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Stability in Maximal Supergravity

S. Bielleman, s1712136,

RuG

Supervisor : Dr. D. Roest

August 25, 2014

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Abstract

In this thesis, we look for a bound on the lightest scalar mass in maximal supergravity. The goal of 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 direction would prove that there are no stable de Sitter vacua in maximal supergravity. We give short introductions on stability, supersymmetry and supersymmetric field theories. Then, we study attempts 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 possible directions, inspired by sponteneous supersymmetry breaking and the sGoldstini, with which to project the scalar mass matrix. Unfortunately, we find a counter example (vacuum for which the projection gives a positive value) for all of the directions we consider. This does not imply that there exist stable de Sitter vacua but we are unable to prove that there aren’t any.

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Introduction

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

There are many examples of quantum field theories that have survived careful experimental verification. An example is Quantum electrodynamics which combines special relativity with classical electromagnetism. This theory is renormalizable which means that calculations will give finite answers at finite energies. It does not mean that the theory can be trusted up to arbitrary energies because at some energy the strong and weak interaction will become important. The most complete theory of the electroweak and strong interactions is the Standard Model which has also been tested to great extent. In fact, the full particle content of the Standard Model has been observed 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 curved background. The curvature of space time is due to the matter content and is given by the Einstein equations. It is possible to write general relativity as a field theory using the metric as the field. The action of the theory is the Einstein-Hilbert action, whose variation gives the classical Einstein equations. This quantum field theory of gravity is non renormalizable and higher order terms become increasingly important because the coupling constant G_N of the theory has mass dimension −2.

The unification of the Standard Model and general relativity is one of the main problems in theoretical physics today. Such a theory would have to contain the gauge group of the Standard Model, SU (3) × SU (2) × U (1) and a spin-2 particle that mediates the gravitational force. The construction 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”. Supersymmetry provides a loophole to the Coleman-Mandula theorem since the theorem assumes the symmetry to be based on a Lie algebra, supersymmetry is based on a graded Lie algebra. Supersymmetric extensions of the Standard Model give a solution to a number of theoretical problems including the hierarchy problem and it provides a candidate for dark matter, namely the lightest superparticle which could be stable. These extensions do not contain a spin-2 particle.

An interesting thing happens when the supersymmetry generators are made local or equivalently 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 called supergravity theories or simply SUGRA. Supergravity can be viewed as a field theory with local supersymmetry or equivalently as the supersymmetrized version of general relativity. Supergravity theories, in 4 space time dimensions are usually characterized by the number N which gives the number 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 extension of known theories and as the low energy limit of string theory. This suggests that there is some

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

Supergravity becomes more and more constraint as N increases, for N = 8 in 4 space time dimensions, we have maximal supergravity. The field content of maximal supergravity is fixed (if we demand that the maximal spin is 2). The only freedom that is left in this theory is given by the gauging of the internal symmetry group and the symmetry breaking patterns. One way of breaking the symmetry of a theory is by choosing a vacuum that breaks the gauge group/supersymmetry explicitly. Maximal supergravity has a wide variety of gauge groups and vacua and at present no full classification exists.

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

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

In this thesis we will focus on the possibility of stable de Sitter vacua in maximal supergravity in 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 are 70 scalars in the theory and an unknown number of possible gaugings. Our approach is inspired by 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 to different concepts that the reader is probably not familiar with. However, we encourage the reader to also read the literature on these subjects. In chapter 3 we describe stability analyses in N = 1, 2 supersymmetric theories. We will use these analyses as a guideline for the N = 8 case. In chapter 4 we introduce maximal supergravity. We will focus mainly on the coset structure and embedding tensor formalism. This will allow us to introduce the objects with which we do the calculations. We will also introduce a number of critical points that we will use to test our hypotheses . In chapter 5 we will discuss various attempts at finding a bound for the lightest scalar in maximal supergravity. All of the expressions and conventions that are too big to put in the main body of this text have been collected in the appendices. We refer to these appendices throughout the text. The final appendix contains short instructions on how to do the calculations of chapter 5. The calculations are not difficult but it is very convenient to have a short introduction. This appendix is added should the reader feel like he/she wants to check the calculations or try another but similar approach of his/her own.

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Chapter 2

Stability and supersymmetry

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

In this chapter we will introduce a couple of different topics that are either useful later on or interesting for a master student. The first section will briefly explain what we mean with stability of a vacuum. The next two sections are on the construction of the particle content in supersymmetric theories, a topic that is accessible and interesting. Finally, we will discuss the K¨ahler and superpotential which will be used in the discussion of the lower supersymmetric theories. The treatment of the topics is compact and we encourage the reader to read the related literature for a more complete explanation.

2.1 Stability

The main topic of this thesis is stability in maximal supergravity. Stability is important in more areas of physics then just supergravity, so it can be treated in a more general quantum field theoretic setting. In general, stability of quantum field theories is related to the masses of the particles in a given critical point, we will focus on the scalar masses. The rule of thumb for Minkowski 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 mass squared gives. The classical vacuum of the theory is defined by an critical point of the scalar potential. The dynamics of the theory are obtained by expanding the fields around this critical point. 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^−ip^µ^x^µ (2.2)

This solution will blow up whenever m² is negative and hence this implies that we can not trust perturbation theory around the vacuum at late times when m²is negative.

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

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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 µ²is positive then the potential has a global minimum in the origin and all is well. If µ² is negative then the Higgs potential has a global maximum in the origin and no global minimum at all. The origin has become an unstable vacuum and the Higgs field will start rolling down the potential towards −∞.

The situation becomes a bit more subtle when we consider (A)dS vacua. We have to account for the curved background so there will always be an interaction between the Ricci scalar and the 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 still holds and we have stability when all scalar masses are positive [16]. One has to be careful to give a 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 still have a stable Anti de Sitter vacuum when we have negative masses. Breitenlohner and Freeman showed 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 Sitter vacua. 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 supersymmetric theories. In non supersymmetric particle physics particles are usually defined to be irreducible representations of the Poincaré algebra. Since the Poincaré algebra is a subalgebra of the supersymmetry algebra, any irreducible representation of the supersymmetry algebra is a representation of the Poincaré algebra. In general this representation will be reducible in irreps of the Poincaré algebra. This means that a superparticle corresponds to a collection of particles that are related to each other via the odd generators of the supersymmetry algebra. The irreducible representation 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 one antisymmetric tensor Mµν: M0,i= Kiand Mij= ijkJk. The Lorentz algebra is fully determined by Mµν together with its commutator relations. The Poincar´e group is the Lorentz group aug- mented by the space time translation generators Pµ, the Poincar´e algebra is fully determined by Mµν, Pµ and their commutator relations. The only additional generators that are present in the supersymmetry algebra are the odd (or fermionic): Qⁱ_α, 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 8 odd generators (4 + 4). We denote the complex conjugates by (Qⁱ_α)^∗= Q_αi_˙ ¹. The supersymmetry

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

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algebra, besides the commutators between Pµ and Mµν reads:

[Pµ, Qⁱ_α] = 0 (2.6)

[P_µ, Q_αi_˙ ] = 0 (2.7)

[Mµν, Qⁱ_α] = i(σµν)αβQⁱ_β (2.8) [Mµν, Q_αi_˙ ] = i(σµν)α˙

β˙Qβi˙ (2.9)

{Qⁱ_α, Qβj˙ } = 2σ^µ

α ˙βPµδ_jⁱ (2.10)

{Qⁱ_α, Q^j_β} = αβZ^ij (2.11)

Where Z^ij are bosonic central generators that are related to the generators of the internal symmetry group of the theory at hand. Of particular interest for the following part of this section is the commutator between Qⁱ_αand 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 Qⁱ₂and Q1i˙ raise the spin of a particle by half unit and that Qⁱ₁ and Q2i˙ lower the spin by half unit.

Lets start constructing supermultiplets, the procedure we follow is described in [13]. The easiest case to consider is the massless case with no central charges, Z^ij = 0 (we will see later that the central charges are always zero in the massless case). We can go to the rest frame where Pµ= (E, 0, 0, E) such that the commutators become:

{Qⁱ_α, Qβj˙ } = 0 0 0 4E

δ_jⁱ (2.14)

{Qⁱ_α, Q^j_β} = 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 Qⁱ2˙ generators of the super algebra. We can redefine the generators in the following way:

ai = 1

√4EQⁱ₂ (2.19)

a^†_i = 1

√

4EQ2i˙ (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 a_i are the annihilation operators and a^†_i are the creation operators. We will interpret the

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generators of the superalgebra as operators from this point that work on a Hilbert space. We already mentioned that these operators raise/lower the helicity of a state by ¹₂. To construct a supermultiplet one can start with choosing a state that is annihilated by all the ai (a so-called Clifford vacuum). This state will carry an irrep of the Poincar´e algebra (ie. a particle) with a definite helicity |λ_max>. We can create other states by acting with a^†_i on the state with maximum helicity. In this way we create a tower of states:

|λ_max> (2.24)

a^†_i|λ_max> = |λ_max−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 λ_max−^k₂ is equal to ^N_k and that the total number of states in a supermultiplet is equal to 2^N. As an example consider the multiplet with N = 8 and λ_max = 2. This is the multiplet of maximal supergravity. We have 1 state with helicity 2, 8 states with helicity ³₂, 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 the states with positive helicity. As a second example consider the case N = 6 and λ_max = 2. We naively 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 and write:

[(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 supermultiplets can be constructed. They can be found for instance in [12]. There is another instance of CPT doubling that is group theoretical in nature. Let us just briefly show it in case of the N = 2 hypermultiplet. 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 helicity 0 states behave as a doublet while the fermionic states are singlets. If the multiplet were CPT self-conjugate the two scalars would have to be real (or else they will not be invariant). However two 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 the massless 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 the anticommutator:

{Qⁱ_α, Qβj˙ } = 2m 0

0 2m

δ_jⁱ (2.32)

{Qⁱ_α, Q^j_β} = 0 (2.33)

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

aⁱ_α = 1

√2mQⁱ_α (2.34)

a^†_αi_˙ = 1

√2mQ_αi_˙ (2.35)

We define a Clifford vacuum as a state that is annihilated by all aⁱ_α. From this we can start building supermultiplets starting with a state that is a Clifford vacuum. We assume that the Clifford vacuum is uniquely given (and always exists) for given N and given spin λ0. A difference with the massless case is that now we can create states with lower and higher spin than the Clifford 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 with Clifford vacuum |λ₀> then we can build the following states:

λ = λ₀ |λ₀> (2.36)

a^†_˙

1a^†_˙

2|λ0> (2.37)

λ = λ₀−1

2 a^†_˙

2|λ₀> (2.38)

λ = λ₀+1

2 a^†_˙

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 name long multiplet. We should in principle distinguish between the states |λ0 > and a^†_˙

1a^†_˙

2|λ0 > since they transform differently under parity. There are 2^2N 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:

Z^ij =







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 Qⁱ_αand Q_αi_˙ such that they form an oscillator algebra. It turns out that the correct way to do this is (we assume N even, the odd case is a trivial expansion

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of the algebra):

a^I_α = 1

√2

Q^2I−1_α + _αβQ_γ(2I)_˙ δ^γβ^˙

(2.44) a^†_αI_˙ = 1

√2

Q_α(2I−1)_˙ + _{α ˙}_˙_βδ^βγ^˙ Q_γ(2I)

(2.45) b^I_α = 1

√2

Q^2I−1_α − αβQ_γ(2I)_˙ δ^γβ^˙

(2.46) b^†_αI_˙ = 1

√2

Q_α(2I−1)_˙ − _{α ˙}_˙_βδ^βγ^˙ Q_γ(2I)

(2.47) Where the capital index I runs from 1 to N /2. These operators have the following algebra:

n

a^I_α, a^†_αJ_˙ o

= (2m − Z_I)δ_{α ˙}_αδ_J^I (2.48) n

b^I_α, b^†_αJ_˙ o

= (2m + ZI)δα ˙αδ_J^I (2.49) And all the other anti commutators equal to zero. From the algebra and positivity of the Hilbert space it follows that 2m ≥ |ZI| by a similar argument as before. This is why we did not need to consider the massless case with non-trivial central charges. As before creation operators with ˙2 lower the spin of a state by ¹₂ 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 the same number of creation operators as in the massive case without central charges and we get the same number of degrees of freedom as in that case. If all of the operators saturate the bound we have the same number of creation operators as in the massless case (which was fewer then the massive case without central charges since a few of the operators become trivial). Clearly, we get some interesting results when q of the operators saturate the bound. The resulting multiplet is called 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 freedom counting this is not necessary however. This is because when we have a short multiplet with q operators that satisfy the BPS bound then we have 2(N − q) non-trivial operators. This is the same number of operators as for a long multiplet of N − q supersymmetry. Since we know the states in those multiplets we are done if we properly account for CPT doubling. It turns out that all short super multiplets have to be doubled due to CPT invariance. This is because they all carry 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 is quite a general thing, we did not write an explicit Lagrangian for the theories. Nevertheless we can give some necessary conditions for local supersymmetry breaking based on the previous section.

When we partially break supersymmetry from N to N⁰, we might wonder if it is possible to reorder the particles from the massless N supermultiplet into massive (since we break a local symmetry) N⁰ supermultiplets. We will use N to denote the unbroken local supersymmetric theory and N⁰ to denote the broken local supersymmetric theory.

Since we assume that our supersymmetry N is locally supersymmetric it is a theory of gravity and hence contains a graviton. We will assume that the gravity multiplet is the only multiplet in N . Breaking to N⁰implies that we need a massless gravity multiplet of N⁰and N − N⁰ massive ³₂ multiplets of N⁰due to the super Higgs mechanism. The gravitini will absorb some spin-¹₂particle and the spin-1 particles will absorb a scalar in the super Higgs mechanism. The particles that remain after this have to be grouped in massless multiplets of N⁰.

The fact that short multiplets have to be doubled and that we do not have long multiplets for N > 3 excludes any breaking from N = 8, 6 to N⁰ = 5 and from N = 5 to N⁰ = 4. This is

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

Consider a N = 6 supergravity theory that breaks to a N⁰ = 3 supergravity theory then the decomposition 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 have two different massive gravitino multiplets in N⁰ = 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 gravitini and vectors simultaneously. Hence breaking from N = 6 to N⁰ = 3 is impossible in supergravity.

Another interesting example is breaking from N = 8 to N⁰ = 6. The massless multiplets in this 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 have at 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 been eaten by the 6 vectors and gravitino to gain a mass. Hence it is possible to break supersymmetry from N = 8 to N⁰= 6.

It should be noted that just because we can reorder the particles in massive multiplets does not mean that there exists a theory that allows for such a symmetry breaking. There could be a reason why a particular supersymmetry breaking is not allowed that does not show up at the particle level. If a theory allows for a supersymmetry breaking then it must also be possible to put the particle content in the correct multiplets. So at this level we are only able find necessary constraints 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 the supersymmetry algebra. We did not discuss the possibility of building a consistent field theory from the given algebra. There are at least two different ways of constructing a supersymmetry invariant Lagrangian given the field content. The first approach is to simply start with a free field Lagrangian, balance the degrees of freedom and ”guess” the supersymmetry transformations on the fields. It can be very tedious to check if the Lagrangian is invariant and, if not, what additional terms are needed to make the Lagrangian invariant. The second approach is the so called superspace formalism, in this formalism supersymmetry is manifestly present.

Superspace has two sets of coordinates, space time x^µ and fermionic θ^a which anticommute.

We can define superfields on the superspace similarly to fields in quantum field theory and define a representation of the supersymmetry algebra that acts on the superfields. In order to obtain irreducible 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

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representation of the algebra. The reader can find many examples of this in the literature. As an example 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, θ^a are the fermionic coordinates of the superspace and F is an auxiliary field which balances the degrees of freedom and has no kinetic term. The field is called chiral due to the constraint:

(−∂_a_˙ − i(σ^µ)_{a ˙}_aθ^a∂_µ)Φ = 0 (2.57) The field content corresponds to the well known Wess-Zumino theory. We see that the different fields are the coefficients of the expansion in terms of the fermionic coordinates. They can be extracted by differentiating the superfield and setting the fermionic coordinates to zero. We know that any theory that we build from this superfield will behave properly under supersymmetry because it is defined to be an irrep of the algebra.

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

L =

Z

d⁴θK(Φ, Φ^†) + Z

d²W (Φ) + Z

d²θW (Φ^†) (2.58)

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

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

One of the simplest supersymmetric theories of gravity is the supersymmetrized version of general relativity. The dynamical variable in GR is the metric which can be written locally using the vierbein, e^a_µ:

gµν(x) = e^a_µe^b_ν(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 field ω_µ^ab (also called the spin-connection). It might look as if the introduction of the spin connection gives too many degrees of freedom in the theory but it is possible to write the spin connection in terms 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 all objects. We have the supervierbein: E^M_Λ (x, θ), the superspin connection Ω^{M N}_Λ , superlocal Lorentz etc. 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:

E_µ^m(x, θ = 0) = e^m_µ (2.60)

E_µ^a(x, θ = 0) = φ^a_µ (2.61)

Ω^mn_µ (x, θ = 0) = ω^mn_µ (2.62)

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From which we see that in the bosonic part of superspace we have the vierbein, spin connection and newly introduced gravitino: φ^a_µ. The action for N = 1 supergravity in superspace is then given by the simplest invariant possible:

S = 1 2k²

Z

d⁴xd⁴θ sdet(E_Λ^M) (2.63)

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

If we want to couple matter to gravity in the superspace formalism we need to write invariant combinations of the supervierbein with the K¨ahler potential and superpotential. For chiral superfields the most general invariant action is given by:

S =

Z

d⁴xd⁴θE[K(Φ, Φ^†) + Φ^†e^VΦ] + Z

d⁴xd⁴θE [W (Φ) + T r(W²)] (2.64) Where E is the sdet of the supervierbein, V is a super gauge field and E is proportional to the second covariant derivative of E. We see that also for N = 1 supergravity coupled to chiral matter fields the freedom in the theory is contained in the K¨ahler potential and the superpotential.

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

The scalar manifold structure is also important in maximal supergravity and we will spend some time on the arguments leading to the coset structure of that theory. The actual analysis in maximal supergravity will rely on objects that are functions on the scalar manifold just as in the lower supersymmetric case. However, we can do the full analysis in the origin of the scalar manifold 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, 2 supersymmetric 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 of lecture notes when one starts learning about supersymmetry and supergravity. We have left out a tremendous amount of information and derivation in this section, do read the notes.

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Chapter 3

Stability in N = 1, 2

supersymmetric theories

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

3.1 Global supersymmetry

Let us first look at a study to find a bound on the lightest scalar in theories with global supersymmetry (theories without gravity), this section is a discussion of [10]. Lets start with explaining why a 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 the same mass. For example in a N = 1 supersymmetric theory this implies that particles in the same chiral multiplet have the same mass and that particles in the same vector multiplet have the same mass.

When only supersymmetry is broken there exists a physical Goldstone fermion which has zero mass due to Goldstone’s theorem. This Goldstone fermion is usually called the Goldstino. In general for N = n supersymmetry there are n Goldstini and, since we have N = 1 supersymmetry, a single Goldstino exists. The Goldstino has two scalar superpartners, the sGoldstini, that are massless in the supersymmetric limit because the Goldstino is massless in this limit. The masses of the sGoldstini are fully determined by splitting effects that depend on the amount of supersymmetry 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 degree of freedom. To balance the degrees of freedom in the vector multiplet, it absorbs the chiral multiplet of the would-be Goldstone. The newly formed multiplet contains one complex scalar, two two-component fermions and one three-component vector (we have seen this in chapter 2.3). The mass of the vector is determined by the Higgs mechanism and hence the mass of the physical scalar particle, the would-be Goldstone partner, is also determined by the Higgs mechanism. In general for each gauge vector, there is a single would-be Goldstone partner and a single massive mode due to the Higgs mechanism.

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

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

We will first expand on the discussion of supersymmetric field theories from section 2.4 and use [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 Vâ. The indices i, j, k, · · · label the chiral multiplets and their scalars φⁱand spinors ψⁱand the indices a, b, · · · label the vector multiplets and their spinors λâ and vector bosons Aâ_µ. These multiplets define a set of superfields which are used to specify the Lagriangian of the theory via a real Kähler potential K and a holomorphic superpotential W as before. In addition the gaugings introduces a holomorphic gauge kinetic function Hab(which is set constant at Hab= hab) and holomorphic Killing vectors X_aⁱ. These quantities are not independent but have to satisfy a number of relations to guarantee invariance of the Lagrangian.

Furthermore, we know that in general, the K¨ahler potential defines a metric g_ij and a Riemann tensor R_ijkl that determine the scalar geometry of the theory. The scalar geometry comes with a covariant derivate ∇i, we will write covariant derivatives as: Ki := ∇iK. Finally, we define the scalar dependent matrices: Qⁱ_aj = i∇jX_a^jwhich appear in the linear transformations of the fermion fields. In addition to the physical fields we also have a number of auxiliary fields to balance the off-shell degrees of freedom. These auxiliary fields are denoted by Fⁱ and D^a. They are auxiliary in the sense that they do not have a kinetic term in the Lagrangian and hence their equations of motion are purely algebraic. These fields are nevertheless important for our discussion since they play a crucial role as symmetry breaking order parameters.

We are trying to study the scalar mass spectrum of the above theory for arbitrary values of the Kähler potential and the superpotential and for arbitrary gaugings (specified by the Killing vectors). In order to do that, we must have a general expression for the scalar mass matrix. The scalar mass matrix is in general found by defining a vacuum and looking at small fluctuations around the vacuum solution. In general the vacuum is defined by constant values of the scalar fields φⁱ and vanishing values of the fermions ψⁱ, λâ and the vectors Aâ_µ such that the potential energy has a local minimum.

We can write the scalar potential for this theory as:

V =g_ijFⁱF^j+1

2HabD^aD^b (3.1)

We see that the value of the scalar potential is always non negative. This is always true for theories with global supersymmetry. In fact, it is well known that for theories with global supersymmetry the supersymmetric ground state has lowest energy V = 0 and any vacuum with V > 0 has broken supersymmetry. From our expression of the scalar potential we can see that this implies that either Fⁱ6= 0 or Dâ 6= 0 whenever supersymmetry is broken which underlines our earlier claim that Fⁱ and Dâ can be viewed as order parameters of supersymmetry breaking. We define the following vector: φÎ = (φⁱ, φⁱ), such that the Lagrangian for the scalar fields can be written as:

L =1

2g_IJ∂µφ^I∂^µφ^I−1

2m²_IJφ^Iφ^J (3.2)

And the scalar mass matrix has the following form:

m²_IJ= m²

ij m²_ij m²

ij m²

ij

!

(3.3) With entries:

m²_ij =g^kl∇iW_k∇_jW_l− R_ijklF^kF^l+ g²hâbX_aiX_bj+ gQ_aijDâ (3.4) m²_ij = − ∇i∇jWkF^k− g²hâbXaiXbj (3.5)

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From these expressions we see that the scalar masses are not only determined by the superpotential but also by the auxiliary fields and the Killing vectors. This reflects the discussion in the introduction where we said that some scalar particles have masses that are determined by splitting effects 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:

M_ab² =2g²g_ijX_(aⁱ X^j_b) (3.9)

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

We already mentioned that there exists a physical Goldstino fermion with vanishing mass whenever supersymmetry is broken. This fermion is given by the following combination of fields:

η ∝ F_iψⁱ+^√ⁱ

2D_aλ^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 symmetry is spontaneously broken whenever the vector bosons have a mass i.e. whenever: M_ab² 6= 0. In that case there exist non physical would-be Goldstone scalars σ_a ∝ X_aiφⁱ+ X_aiφⁱ. These scalars are absorbed by the gauge bosons through the Higgs mechanism and their scalar partners that become a part of the same multiplet are interesting because they have a mass that has to be equal to the vector boson mass.

Next we consider how to find the physical masses of the scalars. They correspond to the eigenvalues of the scalar mass matrix. In order to find the eigenvalues we define a basis by a set of vectors that is orthonormal with respect to the metric g_IJ. The elements of the scalar mass matrix in this basis will be the eigenvalues. The main result from linear algebra that will be used is that the eigenvalues of any principal submatrix of m²_IJ must always be larger then the minimal eigenvalue of the whole matrix. The larger the principal submatrix the better the bound on the lowest eigenvalue will be in general. A principal submatrix is constructed by projecting the scalar mass 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 one spanned by the directions specified by the sGoldstino and Goldstone partners because these particles have masses that are constrained by symmetry and cannot be freely tuned by tuning the superpotential as is the case for the other scalar masses. These directions are specified by Fⁱ and X_aⁱ. The sGoldstino direction is always orthogonal to the Goldstone directions. However, the Goldstones X_aⁱ are in general not orthogonal to each other. However, it is possible to do a rotation such that the rotated vectors X_aⁱ are orthogonal to each other. Following this discussion we define the following normalized vectors:

fⁱ=Fⁱ

F (3.10)

xⁱ_a=√ 2gX_aⁱ

Ma

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

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to project the scalar mass matrix on:

f_A^I =(fⁱ, 0) (3.12)

f_B^I =(0, fⁱ) (3.13)

x^I_a+= 1

√2(xⁱ_a, xⁱ_a) (3.14)

x^I_a−= i

√2(xⁱ_a, −xⁱ_a) (3.15)

The two directions f_AÎ and f_BÎ are related via a rotation to the real partners of the Goldstino. The direction xÎ_a+describes the unphysical would-be Goldstone modes and the direction xÎ_a−describes their partners. Hence the vectors f_AÎ, f_BÎ and xÎ_a− define a 2 + k dimensional subspace which contain the directions that are dangerous to stability. The projected mass matrix can be written as:

m²_αβ=





 m²

f f ∆ −√

2im^2∗_{f x}

b

∆^∗ m²

f f

√2im²_{f x}

√ b

2im²_{f x}_a −√

2im^2∗_{f x}_a 2m²_x_a_x_b





 (3.16)

Where we define ∆ = m²_ijfⁱf^j and m²

f f = m²

ijfⁱf^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 the entire complex plane. The other expressions have the following form:

m²_{f f} = − A_{f f}|F |² (3.17)

m²_{f x}

b= − Af x_b|F |² (3.18)

m²_x

ax_b=1

2M_ab² − Bxax_b|F |² (3.19) The explicit expressions can be found in [10]. From the above expressions it is clear that the sGoldstino masses are indeed proportional to |F |² and that the Goldstone partner masses are proportional to M_ab² − |F | as one would expect. In order to find an upper bound for the lowest scalar 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 bound is given by the sGoldstino mass. When the Goldstone partners have masses comparable to the sGoldstini masses the analysis is not so easy. It is for this reason that we will focus on the simple U (1) case where k = 1 and the matrix m²

αβ is 3 × 3. Calculating the smallest eigenvalue of this matrix one finds:

m²= max 1

2(m²_{f f}+ 2m²_xx) −1 2

q(m²

f f− 2m²_xx)²+ 8|m²_{f x}|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 |∆| and arg(∆) given that we want to maximize the eigenvalues which solves the first complication. The second complication is the implicit dependence of the eigenvalues on the vacuum solution. It is for 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 tune the superpotential in such a way that the bound is saturated.

In the simplest case of renormalizable gauge theories with a U (1) gauge symmetry, where the K¨ahler potential is quadratic and the Killing vector linear, it is possible to do the analysis explicitly. 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

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theory with a non-trivial K¨ahler potential but with a U (1) gauge symmetry. In this case we have a curved scalar manifold and the lightest scalar is in general identified with a linear combination of the sGoldstini and the Goldstone partners. To get an explicit expression one would have to do a case by case analysis to calculate the values of the expressions explicitly as well as to tune the superpotential 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 to supergravity theories. We give a discussion of [11]. In N = 1 supergravity with n chiral multiplets we have the gravity multiplet containing the graviton and the gravitino and the chiral multiplets containing complex scalar fields φⁱ and chiral fermions χⁱ, where i labels the multiplet. This theory is described by the holomorphic superpotential W (φ) and the K¨ahler potential K(φ, φ) as before. The scalar manifold in this case is K¨ahler-Hodge with a metric given by: g_ij = K_ij. In order to analyze the theory we will define a new potential:

L = e^K/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) with respect to the covariant derivative defined on the scalar manifold ∇. The holomorphic derivative of 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 = NⁱN_i− 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:

m²_3/2= 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 not consider the Goldstone partners. In fact it is not clear how much new information can be found in the Goldstone partners. It is known that the Goldstone partner directions and the sGoldstino directions are no longer orthogonal as was the case in the global case [10]. This will lead to a smaller 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 splittings due to supersymmetry breaking effects as was the case in globally supersymmetric theories. Since the article [11] does not consider broken gauge symmetry we will not consider the Goldstone partners in this chapter.

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

Niχⁱ 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 the projection of the scalar mass matrix on the sGoldstino direction:

m²_η =

m²_0ijNⁱN^j

|N |² (3.24)

= 3(Rη+2

3)m²_3/2+ RηV (3.25)

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

R_η = −R_ijpqNⁱN^jN^pN^q (N^kNk)²

(3.26) Where R_ijpq is the Riemann tensor of the scalar manifold. The sGoldstino mass gives an upper bound on the lowest eigenvalue of the full scalar mass matrix because it is the average of the two real sGoldstini. Stability requires m_η ≥ 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 define γ = _3m^V2

3/2

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 becomes bigger, which corresponds to an increasing cosmological constant. This bound gives an necessary restriction on the sectional curvature of the scalar manifold, but it can always be satisfied by tweaking the cosmological constant. We find that there can be stable de Sitter vacua in N = 1 supergravity with only chiral multiplets. This should come as no surprise since such vacua have been 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 sections but since there are more sGoldstini the problem gets a bit more complicated compared to the N = 1 supergravity case. Since we are working with N = 2 supergravity we will have a gravity multiplet containing the graviton gµν, two gravitini ψ_µ^A and a graviphoton Aµ. The index A is a SU (2) fundamental index. The hypermultiplets contain 2n complex fermions ζ^α, α is an Sp(2n) index, and 4n real scalars q^u. The SU (2) indices A, B are raised and lowered by the invariant tensor AB and a symmetric pair (AB) can be replaced with a triplet index x by using the Pauli matrices σ_B^xA:

N^x= iσ_A^xC_CBN^(AB) (3.29)

We can raise and lower the Sp(2n) indices using antisymmetric symplectic tensors Cαβ and C^αβ 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 scalar manifold is a quaternionic K¨ahler manifold of dimension 4n with holonomy group Sp(2n) × SU (2).

On the scalar manifold we have naturally a triplet of almost complex structures J^x that satisfy an SU (2) algebra. Associated with this triplet is a triplet of Hyperk¨ahler two forms Ω^x_uv that are identified with the field strength of the SU (2) part of the holonomy group. It should come as no surprise that the curvature of the SU (2) part is associated with the field strength of the SU (2) part and hence with Ω^x_uv.

R^AB_uv = −iΩ^x_uvσ^xAB (3.30)

We can define a vielbein on the scalar manifold U_u^αA that is related to the metric in the usual way:

huv = ABU_u^αAU_v^βB (3.31)

The inverse vielbein is given by U_αAû and it satisfies U_αAû U_v^αA= δ_vû. Using the vielbein we can give the curvature form associated with the Sp(2n) part of the scalar manifold:

R^αβ_uv = ABU_[u^γAU_v]^δB(−2δ_(γ^αδ_βδ)^β + Σ^αβ_γδ) (3.32)