Uppsala University

Supergravity and Kaluza-Klein dimensional reductions

Author: Supervisor: Roberto Goranci Giuseppe Dibitetto

May 13, 2015

Abstract This is a 15 credit project in basic supergravity. We start with the supersym- metry algebra to formulate the relation between fermions and bosons. This project also contains Kaluza-Klein dimensional reduction on a circle. We then continue with supergravity theory where we show that it is invariant under supersymmetry transformations, it contains both D = 4, N = 1 and D = 11, N = 1 supergravity theories. We also do a toroidal compactification of the eleven-dimensional super- gravity. I Contents

1 Introduction 1

2 Supersymmetry algebra 3 2.1 Poincaré symmetry ...... 3 2.2 Representation of spinors ...... 4 2.3 Generators of SL(2, C)...... 5 2.4 Super-Poincaré algebra ...... 5

3 Clifford algebra 7 3.1 The generating γ-matrices ...... 7 3.2 γ-matrix manipulation ...... 8 3.3 Symmetries of γ-matrices ...... 8 3.4 Fierz rearrangement ...... 9 3.5 Majorana spinors ...... 10

4 Differential geometry 11 4.1 Metric on manifold ...... 11 4.2 Cartan formalism ...... 11 4.3 Connections and covariant derivatives ...... 12 4.4 Einstein-Hilbert action in frame and curvature forms ...... 17

5 Kaluza-Klein theory 18 5.1 Kaluza-Klein reduction on S1 ...... 18

6 Free Rarita-Schwinger equation 21 6.1 Dimensional reduction to the massless Rarita-Schwinger field ...... 22

7 The first and second order formulation of general relativity 23 7.1 Second order formalism for gravity and fermions ...... 23 7.2 First order formalism for gravity and fermions ...... 25

8 4D Supergravity 27 8.1 Supergravity in second order formalism ...... 27 8.2 Supergravity in first order formalism ...... 29 8.3 1.5 order formalism ...... 29 8.4 Local supersymmetry of N = 1, D = 4 supergravity ...... 30

9 Toroidal compactification 33 9.1 D=11 supergravity ...... 33 9.2 11-Dimensional toroidal compactification ...... 38

10 Conclusion 43

A Riemann tensor for Einstein-Hilbert action in D dimensions 45

B Equations of motion for Einstein-Hilbert action in D dimensions 47

C Covariant derivative of γν 48

D Variation of the spin connection 48

II E Chern-Simons dimensional reduction 49

III IV 1 Introduction

The aim of this thesis is to study supergravity from its four-dimensional formulation to its eleven-dimensional formulation. We also consider compactifications of higher dimensional supergravity theories which results in extended supergravity theories in four-dimensions. Supergravity emerges from combining supersymmetry and general relativity. General relativity is a theory that describes gravity on large scales. The predictions of general relativity has been confirmed by experiments such as gravitational lensing and time dilation just to mention a few. However general relativity fails to describe gravity on smaller scales where the energies are much higher than current energy scales tested at the LHC. The Standard model took shape in the late 1960’s and early 1970’s and was con- firmed by the existence of quarks. The Standard model is a theory that describes the electromagnetic the weak and strong nuclear interactions. The interactions between these forces are described by the Lie group SU(3)⊗SU(2)⊗U(1) where SU(3) describes the strong interactions and SU(2) ⊗ U(1) describes the electromagnetic and weak in- teractions. Each interaction would have its own force carrier. The Weinberg-Salam model of electroweak interactions predicted four massless bosons, however a process of symmetry breaking gave rise to the mass of three of these particles, the W ± and Z0. These particles are carriers of the weak force, the last particle that remains massless is the photon which is the force carrier for the electromagnetic force. We know that there are four forces in nature and the Standard model neglects grav- itational force because its coupling constant is much smaller compared to the coupling constants of the three other forces. The Standard model does not have a symmetry that relates matter with the forces of nature. This led to the development of supersymmetry in the early 1970’s. Supersymmetry is an extended Poincaré algebra that relates bosons with fermions. In other words since the fermions have a mass and all the force carriers are bosons, we now have a relation between matter and forces. However unbroken supersymmetry requires an equal mass for the boson-fermions pairs and this is not something we have observed in experiments. So if supersymmetry exist in nature it must appear as a broken symmetry. Each particle in nature must have a superpartner with a different spin. For example the gravitino which is a spin-3/2 particle has a superpartner called the graviton with spin-2. Graviton in theory is the particle that describes the gravitational force. We now see why supersymmetry is an important tool in the development of a theory that describes the three forces in the Standard model and gravity force. With supersymmetry one could now develop a theory that unifies all the forces in nature. One theory in particular that describes gravity in the framework of supersymmetry is called supergravity. Supergravity was developed in 1976 by D.Z Freedman, Sergio Ferrara and Peter Van Nieuwenhuizen. They found an invariant Lagrangian describing the gravitino field which coincides with the gravitational gauge symmetry. In 1978 E.Cremmer, B.Julia and J.Scherk discovered the Lagrangian for eleven dimensional supergravity. The eleven dimensional theory is a crucial discovery which is the largest structure for a consistent su- pergravity theory. Eleven dimensions is the largest structure where one has a consistent theory containing a graviton, however there exist theories that contain higher dimen- sions which have interactions containing spin 5/2 particles. From the eleven-dimensional theory one can obtain lower-dimensional theories by doing dimensional reductions, one then finds ten-dimensional theories known as Type IIA and Type IIB and these the- ories are related to superstring theories with the same name. Supergravity appears as low-energy limit of superstring theory. The maximally extended supergravity theories

1 start from the eleven-dimensional Lagrangian, one obtains maximally extended theo- ries by doing dimensional reductions, for example Type IIA is a maximally extended supergravity theory since it contains two supercharges in ten dimensions. The outline of this thesis goes as follows. In section 2 we begin with constructing Supersymmetry algebra and its representations. In section 3 we discuss Clifford algebra which plays an important role in Supergravity because it contains spinors. In section 4 we go trough some differential geometry, in particular the Cartan formalism and differential forms. In section 5 we consider the (D + 1) dimensional Einstein-Hilbert action and perform a dimensional reduction on a circle this is known as the Kaluza-Klein reduction. We obtain a D-dimensional Lagrangian that contains gravity, electromagnetic fields and scalar fields. In section 6 we go through the Rarita-Schwinger equation and discuss degree’s of freedom for the gravitino field. Section 7 contains first and second order formulations of general relativity. In section 8 we consider the four-dimensional Supergravity theory and show the invariance of its Lagrangian. Section 9 contains eleven-dimensional Su- pergravity theory where we construct a Lagrangian that is invariant. We also perform dimensional reductions on the eleven-dimensional theory using Kaluza-Klein method to obtain lower dimensional theories.

2 2 Supersymmetry algebra

Supersymmetry is a symmetry that relates two classes of particles, fermions and bosons. Fermions have half-integer spin meanwhile bosons have integer spin. When one con- structs a supersymmetry the fermions obtains integer spin and the bosons obtain half- integer spin, each particles has its own superpartner. So for example a graviton has spin-2 meanwhile its superpartner gravitino has spin-3/2. The key references in this section will be [1], [2] and [3].

2.1 Poincaré symmetry The Poincaré group ISO(3, 1) is a extension of the Lorentz group. The Poincaré group is a 10-dimensional group which contains four translations, three rotations and three boosts. The Poincaré group corresponds to basic symmetries of special relativity, it acts on the spacetime coordinate xµ as

µ ′µ µ ν µ x → x = Λ νx + a . (2.1) Lorentz transformation leaves the metric invariant

ΛT ηΛ = η

Λ is connected to the proper orthochronous group SO(3, 1) and the metric is given as η = (−1, 1, 1, 1). The generators of the Poincaré group are M µν, P σ with the following algebras

[P µ,P ν] = 0 (2.2) [M µν,P σ] = i(P µηνσ − P νηµσ) (2.3) [M µν,M ρσ] = i(M µσηνρ + M νρηµσ − M µρηνσ − M νσηµρ) (2.4) where P is the generator of translations and M is the generator of Lorentz transforma- tions. The corresponding Lorentz group SO(3, 1) is homeomorphism to SU(2), where the following SU(2) generators correspond to Ji of rotations and Ki of Lorentz boosts defined as 1 J = ε M ,K = M i 2 ijk jk i 0i and their linear combinations 1 1 A = (J + iK ) ,B = (J − iK ) . i 2 i i i 2 i i The linear combinations satisfy SU(2) commutation relations such as

[Ji,Jj] = iεijkJk (2.5)

[Ji,Kj] = iεijkKk (2.6)

[Ki,Kj] = −iεijkKk (2.7)

[Ai,Aj] = 0 (2.8)

[Bi,Bj] = 0 (2.9)

[Ai,Bj] = 0 . (2.10) ∼ We can interpret J⃗ = A⃗ + B⃗ as physical spin. There is a homeomorphism SO(3, 1) = SL(2, C) which gives rise to spinor representations. We write down a 2 × 2 hermitian matrix that can be parametrized as ( ) x + x x − ix ⃗x = 0 3 1 2 . (2.11) x1 + ix2 x0 − x3

3 1 1 This will lead to a explicit description of the ( 2 , 0) and (0, 2 ) representation and this is the central treatment for fermions in quantum field theory. 1

2 µ ν The determinant of equation (2.11) gives us ⃗x = −x ηµνx which is negative of the Minkowski norm of the four vector xµ. This gives rise to a relation between the linear space of the hermitian matrix and four-dimensional Minkowski space. The transforma- tion xµ → xµΛ under SO(3, 1) leaves the square of the four vector invariant, but what happens to the determinant if we consider a linear mapping 2.

Let N be a matrix of SL(2, C) and consider the linear mapping

⃗x → N⃗xN † the four-vectors are linearly related i.e xµ → xµΛ and from the 2 × 2 matrix we can obtain the explicit form of the matrix Λ. The linear transformation gives us that the determinant is preserved, and the Minkowski norm is invariant. This means that the ∼ matrix Λ must be a Lorentz transformation. Since SO(3, 1) = SU(2) ⊗ SU(2) is valid then the topology of SL(2, C) is connected due to the fact that Λ is a Lorentz trans- ∼ formation of SO(3, 1) and the relation SO(3, 1) = SL(2, C). The spinor representation will play a important role in the future chapters.

2.2 Representation of spinors Let us consider the spinor of representations of SL(2, C). The corresponding spinors are 1 called Weyl spinors. The ( 2 , 0) is given by the irreducible representation of the chirality ≡ 1 2L (ψα, α = 1, 2) and the spinor representation (0, 2 ) is given by 2R = (χα˙ , α˙ = 1, 2). The first representation gives us the following Weyl spinor

′ β ψα = Nα ψβ, α, β = 1, 2 (2.12) this is the left-handed Weyl spinor. The other representation is given by

′ ∗β˙ ˙ χ¯α˙ = Nα˙ χ¯β˙ , α,˙ β = 1, 2 (2.13) this is the right-hand Weyl spinor. These spinors are the representation of the basic representation of SL(2, C) and the Lorentz group. From the previous chapter we can now see the relation between the Lorentz group and SL(2, C). Introducing another spinor εαβ which will be a contravariant representation of SL(2, C) defined as ( ) ˙ 0 1 εαβ = εα˙ β = = −ε = −ε (2.14) −1 0 αβ α˙ β˙ hence we see that the spinor ε raises and lowers indices. Our representations of the Weyl spinors then becomes α αβ α˙ α˙ β˙ ψ = ε ψβ, χ¯ = ε χ¯β˙ . (2.15) When we have mixed indices of both SO(3, 1) and SL(2, C) the transformations of the components xµ should look the same as we have seen in section 2.1. We saw that the determinant is invariant under transformation of SO(3, 1) or a transformation via the µ matrix ⃗x = xµσ , hence µ µ → β ν ∗γ˙ ν µ (x σ )αα˙ Nα (xνσ )βγ˙ Nα˙ = Λµ xνσ 1The reason why we can find spinor representations is that we have written our 2 × 2 matrix with µ Pauli matrices such that ⃗x = xµσ 2Since we know that we have a relation between Minkowski space and a linear space of the hermitian matrix, it is valid to consider a linear mapping of the hermitian matrix

4 the right transformation rule is then given by

µ β ν −1 µ ∗γ˙ (σ )αα˙ = Nα (σ )βγ˙ (Λ ) νNα˙ we can also introduce the matricesσ ¯µ defined as

ν αα˙ αβ α˙ α˙ µ (¯σ ) = ε ε (σ )ββ˙

2.3 Generators of SL(2, C) We can define the tensors σµν,σ ¯µν as antisymmetric products of σ matrices i (σµν) β = (σµσ¯ν − σνσ¯µ) β (2.16) α 4 α ˙ i (¯σµν) β = (¯σµσν − σ¯νσµ)α˙ (2.17) α˙ 4 β˙ which satisfy the Lorentz algebra

[σµν, σλρ] = i(ηµρσνλ + ηνλσµρ − ηµλσνρ − ηνρσµλ) (2.18)

The finite Lorentz transformation is then represented as ( ) i µν β ψα → exp − ωµνσ ψβ (2.19) 2 α which is the left handed spinor, the right hand spinor is defined as ( ) α˙ i ˙ χ¯α˙ → exp − ω σ¯µν χ¯β (2.20) 2 µν β˙ We can obtain some useful identities concerning σµ and σµν, defined as 1 σµν = εµνρσσ (2.21) 2i ρσ 1 σ¯µν = − iεµνρσσ¯ (2.22) 2 ρσ these identities are known as the self duality and anti self duality. These identities are important when we discuss Fierz identities which rewrite bilinear of the product of two spinors as a linear combination of products of the bilinear of the individual spinors. We will use Fierz identities when we consider Clifford algebra.

2.4 Super-Poincaré algebra In this section we will consider the Super-Poincaré group which is an extension of the A Poincaré group but with a new generator known as the spinor supercharge Qα , where α is the spacetime spinor index and A = 1,..., N labels the supercharges. A superalgebra contains two classes of elements , even and odd. Let us introduce the concept of Graded algebras.

Let Oa be a operator of the Lie algebra then

ηaηb e OaOb − (−1) ObOa = iC abOe (2.23) where ηa takes the value { 0 : Oa bosonic generator ηa = 1 : Oa fermionic generator

5 This structure relations include both commutators and anti commutators in the pattern [B,B] = B,[B,F ] = F and {F,F } = B. We know the commutation relations in the Poincaré algebra, we now need to find the commutation relations between the super- charge and the Poincaré generators.

The following commutation relations with the supercharge generator are defined as

µν µν β [Qα,M ] = (σ )α Qβ (2.24) µ [Qα,P ] = 0 (2.25)

{Qα,Qβ} = 0 (2.26) { ¯ } µ Qα, Qβ˙ = 2(σ )αβ˙ Pµ (2.27) When N = 1 then we have a simple SUSY, when N > 1 then we have a extended SUSY. Let us consider N = 1 and the commutator (2.27). We consider the N = 1 3 SUSY representation, in any on-shell supermultiplet the number nB of bosons should 4 be equal to the number nF of fermions . Proof. Consider the fermion operator (−1)F = (−1)F defined via (−)F |B⟩ = |B⟩, (−)F |F ⟩ = −|F ⟩

F The new operator (−) anticommutes with Qα, since F F F F (−) Qα|F ⟩ = (−) |B⟩ = |B⟩ = Qα|F ⟩ = −Qα(−) |F ⟩ → {(−) ,Qα} = 0 (*) Next, consider the trace { } { } F F F Tr (−) {Qα, Q¯ ˙ } = Tr (−) QαQ¯ ˙ + (−) Q¯ ˙ Qα β { β β } − − F ¯ − F ¯ = Tr Qα( ) Qβ˙ + Qα( ) Qβ˙ = 0 { ¯ } µ It can also be evaluated using Qα, Qβ˙ = 2(σ )αβ˙ Pµ { { }} { } − F − F µ µ { − F } Tr ( ) Qα,Qβ˙ = Tr ( ) 2(σ )αβ˙ Pµ = 2(σ )αβ˙ pµ Tr ( ) = 0 µ where Pµ is replaced by the eigenvalues p for the specific state. The conclusion is ∑ ∑ 0 = Tr{(−)F } = ⟨B|(−)F |B⟩ + ⟨F |(−)F |F ⟩ bosons fermions ∑ ∑ = ⟨B|B⟩ − ⟨F |F ⟩ = nB − nF bosons fermions

Each supermultiplet contains both fermion and boson states which are known as the superpartners of each other. This proof also shows that there are equal number of bosonic and fermionic degrees of freedom. This statement is only valid when the Super- Poincaré algebra holds. It is also valid when we have auxiliary fields that closes the algebra off-shell i.e when off-shell degree’s of freedom disappear on-shell. If we consider the massless representation the eigenvalues are pµ = (E, 0, 0,E), the algebra is ( ) 1 0 {Q , Q¯ } = 2(σµ) P = 4E α β˙ αβ˙ µ 0 0 αβ˙

3a supermultiplet is a representation of a SUSY algebra 4However there exist some examples where one also has off-shell supermultiplet that have equal amount of bosonic and fermionic degree’s of freedom. Off-shell equality holds for some extended SUSY and higher dimensional theories.

6 letting α take the value of 1 or 2 gives us the following commutation relations

{ ¯ } Q2, Q2˙ = 0 (2.28) { ¯ } Q1, Q1˙ = 4E. (2.29)

We can define the creation- and annihilation operators a and a† as

Q Q¯ a = √1 , a† = √1˙ (2.30) 2 E 2 E

1 we can see that the annihilation operator is in the representation (0, 2 ) and has the − 1 helicity λ = 1/2 and the creation operator is in the representation ( 2 , 0) and has the helicity λ = 1/2. We can build the representation by using a vacuum state of minimum helicity λ, lets call it |Ω⟩, if we let the annihilation operator act on this state we get zero. Letting the creation operator act on the vacuum state we obtain that the whole multiplet consisting of

|Ω⟩ = |pµ, λ⟩, a†|Ω⟩ = |pµ, λ + 1/2⟩ so for example we have that the graviton has helicity λ = 2 and the superpartner gravitino has the helicity λ = 3/2.

3 Clifford algebra

In section 1.2 we only considered Weyl spinors. We will now introduce Clifford algebra which uses gamma matrices. The Clifford algebra satisfies the anticommutation relation

γµγν + γνγµ = 2ηµν1 (3.1) these matrices are the generating elements of the Clifford algebra. The key references of this section are [2] and [4].

3.1 The generating γ-matrices Let is discuss the Clifford algebra associated with the Lorentz group in D dimensions. We can construct a Euclidean γ-matrices which satisfy (3.1) with Minkowski metric ηµν. The representation of the Clifford algebra can be written in terms of σ matrices which are hermitian with square equal to 1 and cyclic

γ1 = σ1 ⊗ 1 ⊗ 1 ⊗ ...

γ2 = σ2 ⊗ 1 ⊗ 1 ⊗ ...

γ3 = σ3 ⊗ σ1 ⊗ 1 ⊗ ...

γ4 = σ3 ⊗ σ2 ⊗ 1 ⊗ ...

γ5 = σ3 ⊗ σ3 ⊗ σ1 ⊗ ...... = ... there are two representations of this algebra mainly an even representation and an odd representation. Let us assume that D = 2m is even then the dimension of the representation is 2D/2, this means that we need m factors to construct the γ-matrices. For odd representations we have D = 2m + 1 which gives us the same dimension of the representation. This is due to the fact that we need γ2m+1 matrices to construct the algebra but since we only keep the factor of m and delete a σ1 matrices which does not increase the dimension. So the general construction of the algebra gives a

7 representation of the dimension 2D/2. We can construct the Lorentzian γ-matrices by picking any matrix from the Euclidean construction and multiplying it by i and label it γ0 for the time-like direction. The hermiticity property of the Lorentzian γ are defined as µ† µ γ = γ0γ γ0 . (3.2) To preserve the Clifford algebra we introduce the definition of conjugacy

γ′µ = U −1γµU. (3.3)

We only consider hermitian representation in which (3.2) holds, then the matrix U has to be unitary. Given two equivalent representations the transformation matrix U is unique.

3.2 γ-matrix manipulation We need to define some γ-matrix manipulation in order to later use it for fermion spin calculations. Clifford algebra are needed to explore the physical properties of thse fields. These manipulations are valid in odd and even dimensions D. Consider the index contraction such as µν µ γ γν = (D − 1)γ . (3.4) In general we obtain

(D − r)! γµ1...µrν1...νs γ = γµ1...µr . (3.5) νs...ν1 (D − r − s)!

The general order reversal symmetry is defined as

− γν1...νr = (−)r(r 1)/2γνr...ν1 (3.6) the sign factor (−)r(r−1)/2 is negative for r = 2, 3 mod 4. Another useful property is the contraction which is defined as

µ1µ2 ν1...νD − µ2µ1ν3...νD γ γν1...νD ε = D(D 1)ε γν3...νD (3.7)

γ-matrix without index contraction in the simplest case is defined as

γµγν = γµν + ηµν . (3.8)

This follows directly from the definitions: the antisymmetric part of the product is defined to be γµν and the symmetric part is ηµν. In general one writes the totally antisymmetric Clifford matrix that contains all the indices and then add terms of possible index pairings. Another example is

µνρ µνρ [µν ρ] [µ ν ρ] γ γστ = γ στ + 6γ [τ δ σ] + 6γ δ [τ δ σ] (3.9)

3.3 Symmetries of γ-matrices The Clifford algebra of 2m ×2m matrices for both even and odd representations, one can distinguish the antisymmetric and the symmetric matrices with a symmetry property called charge conjugation matrix. There exist a unitary matrix C such that each matrix CγA is either antisymmetric or symmetric. Symmetry only depends on the rank r of the matrix γA, so we can write

(r) T (r) (Cγ ) = −trCγ , tr = ±1 (3.10)

8 where γA is the basis of the Clifford algebra defined as

γA = 1, γµ, γµ1µ2 , . . . , γµ1...µD .

For rank r = 0 and 1 we obtain

T µT µ −1 C = −t0C, γ = t0t1Cγ C (3.11)

Given two possibilities in the representation for even dimensions we can construct the Clifford algebra with Pauli matrices

C+ = σ1 ⊗ σ2 ⊗ σ1 ⊗ σ2 ⊗ . . . , t0t1 = 1 (3.12)

C− = σ2 ⊗ σ1 ⊗ σ2 ⊗ σ1 ⊗ . . . , t0t1 = −1 (3.13) for odd dimensions only one of the two can be used. To preserve the algebra the charge conjugation matrix transforms as

C′ = U T CU. (3.14)

The charge conjugation needs to be unitary C† = C−1 in any representation.

3.4 Fierz rearrangement In this section we study the importance of the completeness of the Clifford algebra basis γA. Fierz rearrangement properties are frequently used in supergravity, these properties involve changing the pairing of spinors in product of spinor bilinears.

Let us derive the basic Fierz identity. Using spinor indices defined as

µβ νδ β δ MαµM MγνM = δα δγ . (3.15)

We can expand the matrix M in the complete basis γA using our spinor indices we obtain 1 ∑ δ βδ δ = (m ) δ(γA) β (3.16) α γ 2m A α γ A −m δ β γ the coefficients are mA = 2 δα δγ (γA)β . Therefore we obtain the basic rearrange- ment lemma 1 ∑ δ βδ δ = (γ ) δ(γA) β . (3.17) α γ 2m A α γ A The Fierz rearrangement is valid for any set of four anticommuting spinor fields. The basic Fierz identity (3.17) gives us 1 ∑ (λ¯ λ )(λ¯ λ ) = − (λ¯ γAλ )(λ¯ γ λ ) (3.18) 1 2 3 4 2m 1 4 3 A 2 A Useful Weyl spinor identities, which involves manipulating σ matrices interacting with two spinors, the symmetry properties are gives as

ψσµχ¯ = −χ¯σ¯µψ (3.19) ψσµσ¯νχ = χσνσ¯µψ (3.20) ψσµνχ = −χσµνψ . (3.21)

These manipulations will be useful later on.

9 Let us define the Majorana conjugate of any spinor λ using its transpose and the charge conjugation matrix λ¯ = λT C (3.22) This equation is useful in SUSY and supergravity in which symmetry properties of γ- matrices and of spinor bilinears are important and these properties are determined by C. Using the definition of Majorana conjugate and equation (3.10) we obtain ¯ λγµ1...µr χ = trχγ¯ µ1...µr λ (3.23) the minus sign obtained by changing the Grassmann valued spinor components. The symmetry property is valid for Dirac spinors, but its main application is with Majorana spinors. Therefore we call it Majorana flip relations.

3.5 Majorana spinors We consider the reality constraint and write down the following equations

ψ = ψC = B−1ψ∗, i.e ψ∗ = Bψ (3.24) the constraint is compatible with Lorentz symmetry, the equation (3.24) is the defining condition for Majorana spinors. In order to have Majorana spinors we consider when T D = 4, using that C = −t1C and equation (3.12). These equations and the condition indeed satisfies the equation (3.24). There are representations of γ-matrices that are real and may be called really real representations. Here is a real representation for D = 4 ( ) 0 1 γ = = iσ ⊗ 1 (3.25) 0 −1 0 2 ( ) 1 0 γ = = σ ⊗ 1 (3.26) 1 0 −1 3 ( ) 0 −iσ2 γ2 = = σ1 ⊗ σ1 (3.27) iσ2 0 ( ) 0 σ3 γ3 = = σ1 ⊗ σ3 (3.28) σ3 0 . (3.29)

The physics of Majorana spinors is the same, in any Clifford algebra the complex conju- gate can be replaced with the charge conjugation. For example the complex conjugate ofχγ ¯ µ1...µr ψ where χ and ψ are Majorana, is computed in the following way

∗ C C (¯χγµ1...µr ψ) = (¯χγµ1...µr ψ) =χ ¯(γµ1...µr ) ψ =χγ ¯ µ1...µr ψ where we have used ψC = ψ andχ ¯C =χ ¯.

10 4 Differential geometry

In this section we will discuss differential geometry, where we will formulate the Cartan formalism also known as vielbein, spin connection and p-forms mainly. In supergravity fermions couple to gravity. The key references in this section will be [4] and [5].

4.1 Metric on manifold A metric or inner product on a real vector space V is a bilinear map from V ⊗V → R.The inner product of two vectors u and v ∈ V must satisfy the following properties:

• bilinearity, (u, c1v1 + c2v2) = c1(u, vi) + c2(u, v2) and (c1v1 + c2v2, u) = c1(v1, u) + c2(v2, u) • non-degeneracy, if (u, v) = 0 ∀v ∈ V then u = 0

• symmetry (u, v) = (v, u). The metric on a manifold is smooth assignment of an inner product map on each 5 Tp(M) ⊗ Tp(M) → R . In local coordinates the metric is specified by a covariant rank-2 symmetric tensor field gµν and the inner product of two contravariant vectors µ µ µ µ U (x) and V (x) is gµνU (x)V (x) which is a scalar field. We can summarize the properties of a metric by the line element

2 µ ν ds = gµνdx dx . (4.1)

µν non-degeneracy means that det(gµν) ≠ 0 so the inverse metric g exist as a rank-2 symmetric contravariant tensor which satisfy

µρ ρµ µ g gρν = gνρg = δν . (4.2)

ν µ We can use this relation to raise and lower indices e.g Vµ(x) = gµνV (x) and ω (x) = µν g ων(x). In gravity the metric has the following signature − + + + ... +. The metric −1 a a tensor gµν may be diagonalized by an orthogonal transformation (O )µ = O µ, and

a b gµν = O µDabO ν (4.3)

a 0 1 D−1 with positive eigenvalues λ in Dab = diag(−λ , λ , . . . , λ )

4.2 Cartan formalism Let us define an important auxiliary quantity √ a a a e µ(x) = λ (x)O µ(x) . (4.4)

In four dimensions this quantity is known as the tetrad or vierbien. In general dimensions it is called vielbein but when we discuss gravity we prefer the term frame field. We can write the metric as a b gµν = e µηabe ν (4.5) where ηab = diag(−1, 1,..., 1) is the metric of flat D-dimensional Minkowski space- a time. Given a x-dependent matrix Λ b(x) which leaves ηab invariant, which allows us to construct the equation (3.5) with a Lorentz transformation i.e

′a −1a b eµ (x) = Λ b(x)eµ(x) . (4.6)

5i.e the tangent bundle of a smooth manifold M assigns to each fixed point of M a vector space called the tangent space and each tangent space can be equipped with an inner product.

11 all frame fields related by a local Lorentz transformation are viewed as equivalent. Local Lorentz transformations in curved spacetime differ from global Lorentz transformation a if Minkowski space. We require that the frame field e µ transform as a covariant vector under coordinate transformations ρ ′a ′ ∂x a e (x ) = ′ e (x) . (4.7) µ ∂x µ ρ a µ a µ a µ a The vielbein e µ has an inverse frame field e a which satisfy e µeb = δb and ea eν = µ δν . The vielbein transforms under a local transformation, hence µ µν b µ ν e a = g ηabe ν, e agµνe b = ηab . (4.8)

This term shows that the inverse frame field can be used relate a general metric to the µ Minkowski metric. The second relation of (4.8) indicates that e a form an orthonormal µ set of Tp(M). Since it is non-degeneracy the det e a ≠ 0 we have a basis of each tangent space. Any contravariant vector field has a unique expansion in the new basis, i.e µ a µ a µ a a V = V e a with V = V e µ. The V are the frame components of the original vector ′a −1a b field V µ. The vector fields transform under a Lorentz transformation, i.e V = Λ bV . µ a We can then use e a and e µ to transform vector and tensor fields back and forth between a coordinate basis with indices µ, ν, . . . and a local Lorentz basis with indices a, b, . . . in the metric ηab. We can do an exercise to show how all these transformations work:

µ µ a b µ ν a b a U Vµ = gµνU Vν = e µηabe νU V = ηabU V = U Va

ν where the second term is obtain from Vµ = gµνV , the third term we use equation (3.8) and the last term we use the same relation as the second term. We have now constructed a new local invariance, we have enlarge the symmetry of GR to be general coordinate transformations and Lorentz transformations. We need frame fields to describe fermions in general relativity.

a p We can also use the frame field e µ to define the new basis Λ (M) of differential forms. The local Lorentz basis of 1-form is

a a µ e ≡ e µdx . (4.9)

For 2-forms the basis consist of the wedge product ea ∧ eb and so on. Local frames are useful when we consider fermions coupled to gravity, because spinors transform under Lorentz transformations.

4.3 Connections and covariant derivatives Contravariant derivative on a manifold is a rule to differentiate a tensor of type (p, q) ρ producing a tensor (p, q + 1). We need to introduce the affine connection Γ µν. On vector fields the covariant derivative is defined as

ρ ρ ρ ν ∇µV = ∂µV + Γ µνV (4.10) ρ ∇νVν = ∂νVν − Γ µνVµ . (4.11)

a In supergravity we will use the frame field e µ. Gravitational theories with fermions is ab ab µ ab antisymmetric in Lorentz indices ω = ωµ dx . The components ωµ are called the spin connections because they describe the spinors on the manifold.

Given the 1-forms ea we examine the 2-form 1 dea = (∂ ea − ∂ ea )dxµ ∧ dxµ (4.12) 2 µ ν ν µ

12 the antisymmetric components transforms as (0, 2) tensor under coordinate transforma- tion. Using the Lorentz transformation (4.6) we obtain the following equation

′a −1a b −1a b de = Λ bde + dΛ b ∧ e (4.13) the second term spoils the vector transformation so we need to add an extra term that −1a absorbs the term involving dΛ b. We introduce the spin connection which is a 2-form, giving us a a b a de + ω b ∧ e ≡ T . (4.14) a If ω b is defined to transform under Lorentz transformation as

′a −1a c −1a c d ω b = Λ cdΛ b + Λ cω dΛ b (4.15)

a ′a −1a b then T transforms as a vector, T = Λ bT . We know that any tensor transforms as a vector i.e µ′ ν a′µ′ a′ ∂x b ∂x aµ T ′ ′ = Λ Λ ′ ′ T b ν a ∂xµ b ∂xν bν ′a −1a b so when we do a Lorentz transformation in the frame field i.e e µ → Λ be µ we have to add an extra term when taking the covariant derivative of the transformation. Therefore when performing a covariant derivative on the frame field we obtain a (0, 2) tensor and a Lorentz vector which coincides with the connection transformation. The 2-form T a is called the torsion 2-form of the connection and (4.14) is called the first Cartan structure equation.

Spinor fields Ψ(x) are crucial for supergravity theories, we describe the spinor fields through their local frame components. The local Lorentz transformation rule ( ) 1 ψ′ = exp − λabγ ψ (4.16) 4 ab determines the covariant derivative ( ) 1 D ψ = ∂ + ω γab ψ (4.17) µ µ 4 µab this will be very useful when we do calculations in supergravity.

Let us transform Lorentz covariant of the vector and tensor frame fields to the coordi- nate basis where they become covariant derivatives with respect to general coordinate transformations. We can write out the spin connection in terms of the affine connection ρ ∇ ν ν a Γµν. The quantity µV = e aDµV is the transform to coordinate basis of a frame field and covariant vector field. We can show that this quantity can take the form of equation (4.10) and (4.11)

ρ ρ a ∇µV = e aDµV ρ a ν = e aDµ(e νV ) (4.18) ρ ρ a a b ν = ∂µV + e a(∂µe ν + ωµ be ν)V

ν ν a ν ρ a ρ where we have used that DµV = ∂µV + ωµ bV and e ae ν = δν . We can show that a ∇µVν = e νDµVa in the same way

a a ρ ∇µVν = e νDµVa = e νDµ(e aVρ) a ρ a ρ = e νDµe aVρ + e νe aDµVρ (4.19) ρ a a b = ∂µVν + e a(∂µe ν + ωµ be ν)Vρ

13 These two examples can be written down in a general Lorentz tensor of type (p, q). The transformation to coordinate basis is given by

a ...a ∇ T ρ1...ρp ≡ eρ1 . . . eρp eb1 . . . ebq D T 1 p (4.20) µ ν1...νq a1 ap ν1 νq µ b1...bq define the tensor of type (p, q + 1) with the properties of the covariant derivative. The affine connection relates to the spin connection by

ρ ρ a a b Γµν = e a(∂µe ν + ωµ be ν) (4.21) we can rewrite this as a a b − a ρ ∂µe ν + ωµ be ν e ρΓµν = 0 . (4.22) This property is called vielbein postulate. When we considered a connection without spin connection we had an restriction mainly the that the connection is metric compatible i.e ∇gµν = 0. We have a similar restriction on the connection above, ∇ − σ − σ µgνρ = ∂µgνρ Γµνgσρ Γµρgνσ = 0 (4.23)

ρ the metric postulate means that the tensor is covariantly constant. The Γµν is the Christoffel symbols defined as 1 Γρ = gρσ(∂ g + ∂ g − ∂ g ) (4.24) µν 2 µ σν ν µσ σ µν this is the torsion-free connection. When the torsion is present the affine connection is not symmetric rather ρ − ρ ρ Γµν Γνµ = Tµν (4.25) Using equation (4.23) we can calculate the nonvanishing torsion, we start with

0 = ∇agbc + ∇bgca − ∇cgab (4.26) using the metric compatibility property we obtain the following equation

d d d d d d ∂agbc + ∂bgca − ∂cgab = Γa bgdc + Γa cgbd + Γb agcd + Γb cgda − Γc agdb − Γc bgad d d d = 2Γ(a b)gdc + 2Γ(a c)gbd + 2Γ(b c)gda d d d = 2Γ(a b)gdc + T[a c]gbd + T[b b]gda (4.27) this connection is not unique. If the torsion vanishes we get back the affine connection.

We can now consider curvature, recall that we defined the curvature as

a a a a e a e R bcd = ∂cΓd b − ∂dΓc b + Γc eΓd b − Γd eΓc b (4.28) we also have another connection which will have curvature too

µ µ µ σ µ σ e µ µ σ (∇c∇d − ∇d∇c)V = ∂c∂dV + (∂cωd σ)V + ωd σ∂cV − Γc d(∂eV + ωe σV ) µ λ λ σ + ωc λ(∂dV + ωd σV ) − (c ↔ d) (4.29) we would like this to be the spin curvature and a torsion term. We look at the terms involving V and no derivatives of V , and identify

µ µ µ µ λ µ λ Rcd σ = ∂cωd σ − ∂dωc σ + ωc λωd σ − ωd λωc σ (4.30)

14 as the curvature of the spin connection. The remaining terms are

e e µ µ σ e µ (−Γc d + Γd c)(∂eV + ωe σV ) = −Tc d∇eV (4.31)

Because of the antisymmetry in cd, it is also a (0, 2) tensor under general coordinate transformation, called the curvature tensor. Thus we can define the curvature 2-form 1 Rµ = Rµ dxa ∧ dxb (4.32) σ 2 σab using equation (4.30) we then see that the curvature 2-form is related to connection 1-form by µ µ ν µ dω σ + ω ν ∧ ω σ = R σ (4.33) this is known as the second Cartan structure equation. We can derive the Bianchi identities for the curvature tensor, but first we will introduce the basis of 1-forms

µ µ a E = ea dx . (4.34)

We can define the torsion 2-form 1 T λ = e λ T a dxb ∧ dxc . (4.35) a 2 b c We can use these p-forms to obtain the Cartan first and second structure equations in a much faster way. We consider the first structure equation where we will use the equation (4.34), rewriting the first structure equation and doing the calculation yields us

µ µ ν µ a µ a ν b dE + ω ν ∧ E = d(ea dx ) + ωa νdx ∧ eb dx µ b a µ ν a b = (∂bea )dx ∧ dx + ωa νeb dx ∧ dx 1 = (∂ e µ − ∂ e µ + ω µ e ν − ω µ e ν)dxa ∧ dxb 2 a b b a a ν b b ν a 1 = (Γ c e µ − ω µ e σ − Γ c e µ + ω µ e σ + ω µ e ν − ω µ e ν)dxa ∧ dxb 2 a b c a σ b b a c b σ a a ν b b ν a 1 = T c e µdxa ∧ dxb = T µ 2 a b c c − σ this is Cartan’s first structure equation, where we have used ∂beaν = Γb aecν ωbνσea and contracted the remaining terms which disappears when doing so. We can do the same with the curvature µ µ ∧ ν µ a µ a ∧ ν dω σ + ω ν ω σ = d(ω σadx ) + ω νadx ω σ µ a µ a ∧ ν b = dω σadx + ω νadx ω σbdx 1 µ µ (4.36) = (∂ ωµ − ∂ ω + ωµ ων − ω ων )dxa ∧ dxb 2 b σa a σb νa σb νb σa 1 = Rµ dxa ∧ dxb = Rµ 2 σab σ We can now find the Bianchi identities for Cartan’s first and second structure equation. Taking d of Cartan’s first structure equation gives us

µ µ µ ν dT = d(dE + ω ν ∧ E ) µ µ µ ν = dω ν ∧ E − ω ν ∧ dE µ µ ρ ν µ ν ν σ = (R σ − ω ρ ∧ ω σ) ∧ E − ω ν ∧ (T − ω σ ∧ E ) (4.37) µ ν µ ρ ν µ ν µ ν σ = R σ ∧ E − ω ρ ∧ ω σ ∧ E − ω ν ∧ T + ω ν ∧ ω σ ∧ E µ ν µ µ = R σ ∧ E − ω ν ∧ T

15 and taking the d of Cartan’s second structure equation gives us

λ λ λ ρ dR µ = d(dω µ + ω ρ ∧ ω σ) λ ρ λ ρ = dω ρ ∧ ω σ − ω ρ ∧ dω σ (4.38) µ µ ρ ρ λ ρ ρ δ = (R σ − ω ρ ∧ ω σ) ∧ ω σ − ω ρ ∧ (R σ − ω δ ∧ ω σ) µ ρ λ ρ = R σ ∧ ω σ − ω ρ ∧ R σ .

a a b Using the following relation Rµνρ = Rµνb e ρ, we obtain a a − a − a − a Rµνρ + Rνρµa + Rρµν = DµTνρ DνTρµ DρTµν (4.39) ab ab ab DµRνρ + DνRρµ + DρRµν = 0 . (4.40)

The derivatives Dµ in the equation above are Lorentz covariant derivatives and contain only the spin connection. The first relation is the first Bianchi identity for the curvature tensor, but we include a torsion tensor. The second relation is known as the Bianchi identity for for the curvature. The commutator of Lorentz covariant derivatives leads to Ricci identities. We define them as 1 [D ,D ]Φ = R M abΦ (4.41) µ ν 2 µνab a a b [Dµ,Dν]V = Rµν bV (4.42) 1 [D ,D ]Ψ = R γabΨ (4.43) µ ν 4 µνab where Ψ is the spinor fields, Φ is a field transforming in a representation of proper Lorentz group with the generator M ab.

16 4.4 Einstein-Hilbert action in frame and curvature forms Let us show that the Einstein-Hilbert action for GR can be written as the integral of a volume form involving the frame and curvature. ∫ 1 ε ec1 ∧ ... ∧ ecD−2 ∧ Rab (D − 2)! abc1...cD−2 ∫ 1 1 = ε ec1 ∧ ... ∧ ecD−2 ∧ Rab ec ∧ ed (D − 2)! abc1...cD−2 2 cd ∫ 1 = ε ec1 ∧ ... ∧ ecD−2 ∧ ec ∧ edRab 2(D − 2)! abc1...cD−2 cd ∫ (4.44) −1 = ε εc1...cD−2cdRab dV 2(D − 2)! c1...cD−2ab cd ∫ −2 cdc1 cD−2 ab = (D − 2)!δ . . . δ R cddV 2(D − 2)! abc1 cD−2 ∫ ∫ √ c d ab D = − δa δb R cddV = − d x − det gR where we have used the equation (4.32) and volume form defined as √ dV = dDx − det g (4.45) and ea1 ∧ ... ∧ eaq ∧ eb1 ∧ ... ∧ ebp = −εa1...aqb1...bp dV . (4.46) This form of the action reveals how the connection with torsion is realised in the physical setting of gravity coupled to fermions.

17 5 Kaluza-Klein theory

In this section we will consider Kaluza-Klein theory which will be very useful when we do dimensional reductions. In this chapter we will go through dimensional reduction of Einstein-Hilbert action (EH) in D + 1 dimensions. The dimensional reduction of E-H action will yield us a D-dimensional Einstein field equations, the Maxwell equations for electromagnetic fields and an wave equation for scalar fields. The key references in this section are [1] and [6]

5.1 Kaluza-Klein reduction on S1 Let us assume we start from Einstein gravity in (D + 1) dimensions described by the Einstein-Hilbert Lagrangian √ L = −gˆRˆ (5.1) where Rˆ is the Ricci scalar andg ˆ denotes the determinant of the metric tensor. We now want to reduce the action to D dimensions by doing so we need to compactify one of the coordinates on a circle S1 of radius L. We can expand all the components of D + 1-dimensional metric as a Fourier series ∑ (n) inz/L gˆMN = gNM (x)e (5.2) n If we do this then we obtain infinite numbers of fields in D dimensions by labelled the Fourier mode number n. So if we consider a scalar field in 5 dimensions defined by the action ∫ 5 M ˆ ˆ S5D = d x∂ ϕ∂M ϕ (5.3) we set the extra dimensions x4 = z defining a circle with the radius L with z = z + 2πL, 1 the spacetime is now M4 × S . Using the Fourier expansion where we expand around ϕˆ, the coefficients describe infinite amount of D-dimensional scalar fields, which satisfy the equation of motion n2 2ϕ − ϕ = 0 . (5.4) n L2 n This means that we have infinite amount of Klein-Gordon equations for massive D- 2 n2 dimensional fields and that each ϕn is a D-dimensional particle with mass mn = L2 . We assumed that the circle is very small, which tells us that the particles should be around the Planck mass. However since the modes are way beyond intergalactic scales, we can not observed these types of particles and we can thus neglect them.

Going back to the EH action, the Kaluza-Klein ansatz will beg ˆMN (x, z) where the metric tensor will not depend on the z coordinate. The index M runs over (D + 1) values of the higher dimension which splits into D values of lower dimension, i.e takes the value associated with the compactification. Let us define the (D + 1)-dimensional metric in terms of D-dimensional fields gµν, Aµ and ϕ

2 2αϕ 2 2βϕ 2 ds˜D+1 = e dsD + e (dz + A) . (5.5)

This ansatz is independent of z thus we have express the components of the higher dimensional metricg ˆMN which are given by the lower dimensional fields ( ) e2αϕg + e2βϕA A e2βϕA gˆ = µν µ ν µ (5.6) MN 0 e2βϕ

18 To calculate the Ricci tensor for the Einstein-Hilbert Lagrangian we must choose a vielbein basis defined as

eˆa = eαϕea, eˆz = eβϕ(dz + A) . (5.7)

We can now calculate the one-forms, where we set the torsion tensor to zero.

deˆa = αdϕeαϕ ∧ ea − eαϕdea (5.8) −αϕ b a a b = α∂bϕe eˆ ∧ eˆ − ωˆ b ∧ eˆ and the other one-form can be calculated in the same way

deˆz = βdϕ ∧ eβϕ(dz + A) + eβϕdA 1 (5.9) = βe−αϕ∂ ϕeˆa ∧ eˆz + e(β−2α)ϕF eˆa ∧ eˆb a 2 ab where we have used that F = dA defined as 1 1 F = F ea ∧ eb = (∂ A − ∂ A )ea ∧ eb . (5.10) 2 ab 2 a b b a We can now obtain the spin connection by using Cartan’s first structure equation (3.14), 1 ωˆab = ωab + αe−αϕ(∂bϕeˆa − ∂aϕeˆb) − F abe(β−2α)ϕeˆz (5.11) 2 1 ωˆaz = −βe−αϕ∂aϕeˆz − F a e(β−2α)ϕeˆb . (5.12) 2 b Now that we have found the spin connection we can obtain the two-form by using Car- tan’s second structure equation. Before we obtain the curvature, we want to choose the values√ of α and β such that the dimensionally-reduced Lagrangian is defined as L − L = gR√. If we do not fix the constants our Lagrangian takes the form = e(β−(D−2)α)ϕ −gR, we now see that we have to set β = −(D − 2)α to achieve the Lagrangian we defined at the beginning. This is called the Einstein frame and it comes from our metric ansatz where we have Weyl rescaling e2αϕ and e2βϕ which results in modified Ricci tensors. We also want to ensure that the scalar field ϕ acquires a kinetic 1 2 term with a canonical normalisation 2 (∂ϕ) in the Lagrangian. Therefor we choose the constant to be 1 α2 = , β = −(D − 2)α (5.13) 2(D − 1)(D − 2) The Ricci tensor are ( ) 1 1 Rˆ = e−2αϕ R − ∂ ϕ∂ ϕ − αη 2ϕ − e−2DαϕF cF (5.14) ab ab 2 a b ab 2 a bc ( ) 1 Rˆ = e(D−3)αϕ∇b e−2(D−1)αϕF (5.15) az 2 ab 1 Rˆ = (D − 2)αe−2αϕ2ϕ + e−2DαϕF 2 . (5.16) zz 4

AB ab We can obtain the Ricci scalar Rˆ = η RˆAB = η Rˆab + Rˆzz, ( ( ) ) 1 1 Rˆ = ηab e−2αϕ R − ∂ ϕ∂ ϕ − αη 2 − e−2DαϕF cF ab 2 a b ab 2 a bc 1 + (D − 2)αe−2αϕ2ϕ + e−2DαϕF 2 (5.17) ( 4 ) 1 1 = e−2αϕ R − (∇ϕ)2 + (D − 3)α2ϕ − e−2DαϕF 2 . 2 2

19 We can now determine the determinant of the metric tensor, √ √ √ −gˆ = e(β+Dα)ϕ −g = e2αϕ −g (5.18) the Einstein-Hilbert Lagrangian now takes the form ( ) √ 1 1 L = −g R − (∇ϕ)2 − e−2(D−1)αϕF 2 . (5.19) 2 4 Doing the integral over the z coordinate now yields us the following action ∫ √ 1 D+1 SEH = d −gˆRˆ 16πGD+1 N ( ) ∫ √ (5.20) 2πL D − − 1 ∇ 2 − 1 −2(D−1)αϕ 2 = D+1 d x g R ( ϕ) e F . 16πGN 2 4 the scalar field ϕ is called a dilaton. We can work out the field equations for this Lagrangian that tells us more about the fields. If one were to set the scalar field to zero we would obtain the Einstein-Maxwell Lagrangian in D dimensions, but this is not allowed due to the equations of motion. ( ) ( ) − 1 1 − 1 2 1 −2(D−1)αϕ 2 − 1 2 Rµν Rgµν = ∂µϕ∂νϕ (∂ϕ) gµν + e Fµν F gµν ( 2 ) 2 2 2 4 µ −2(D−1)αϕ ∇ e Fµν = 0 1 2ϕ = − (D − 1)αe−2(D−1)αϕF 2 . 2 (5.21) the reason we have dropped the 2ϕ term in the Lagrangian is because it is a total derivative in L which doesn’t contribute to the field equations. From the last equation in (5.21) we see that we cannot set ϕ = 0 because it involves a F 2 term on the right-hand side. This means that the lower dimensional fields prevent the truncation of the scalar [6]. We neglected the massive modes in our Fourier expansion but assume we kept the massive modes, would we then be able to set the equation of motion of the massive field to zero. When we do a dimensional reduction on a circle our fields must transform covariantly under lower-dimensional U(1) gauge symmetry and under diffeomorphism of the lower-dimensional spacetime. Thus any mixture of massless modes with towers of massive modes will be avoided. Lower-dimensional solutions must be solutions to higher- dimensional theories which guarantees consistent truncation. So in other words when the Kaluza-Klein ansatz satisfies the equations of motion we have consistent truncation.

20 6 Free Rarita-Schwinger equation

Let us now introduce a free spin-3/2 field. We only consider fields that don’t interact and we consider them separately. In particular we consider Ψµ(x) as a free field, the gauge transformation is Ψµ(x) → Ψµ(x) + ∂µεx . (6.1) [D/2] We will also assume that Ψµ and ε are complex spinors with 2 spinor components for spacetime dimension D. The gauge field Ψµ(x) should have anti-symmetric derivative of the gauge potential ∂µΨν − ∂νΨµ which is invariant under gauge transformation. We now consider a action that is Lorentz invariant, first order in derivatives and invariant under gauge transformations (6.1) and conjugate transformation of Ψ¯ µ. We also demand that the action is hermitian so that the equations of motion for Ψ¯ µ is Dirac conjugate of Ψµ. We obtain the following action ∫ D µνρ S = − d xΨ¯ µγ ∂νΨρ (6.2) we see that the action contains a third rank of Clifford algebra γµνρ which satisfies all these properties. Using the variational principle on the action we obtain the Euler- Lagrange equation given as µνρ γ ∂νΨρ = 0 . (6.3) Let us consider the on-shell degree’s of freedom for the Rarita-Schwinger field. We need to fix the gauge so we impose the constraint

i γ Ψi = 0 . (6.4)

We rewrite the components of (6.3) as ν → 0 and ν → i which gives us

ij γ ∂iΨj = 0 (6.5) 0 ij ijk −γ γ (∂0Ψj − ∂jΨ0) + γ ∂jΨk = 0 .

Using γij = γiγj − δij and the gauge condition we obtain

i ∂ Ψi = 0 . (6.6)

Using the same gamma-matrix manipulation on the second equation in (6.5) and multi- plying it with γi as well as using the gauge condition, we obtain that the two first terms are zero due to Ψ0 = 0. The second equation then takes the form

0 j µ γ ∂0Ψ0 + γ ∂jΨi = γ ∂µΨi = 0 . (6.7)

We have now shown that the components of Ψi satisfy a Dirac equation. Hence the classical degree’s of freedom after imposing the gauge condition and (6.6) is given by (D − 3)2[D/2]. The spinor field Ψ is a representation of SO(D − 2) which would result in on-shell degree’s of freedom given by (D − 2)[D/2]−1. However when one subtracts the γ-trace of the vector-spinor representation, we end up with a irreducible representation that contains (D − 3)[D/2] components.

µνρ νρ We can rewrite the equation (6.3) using the γ-matrix relation γµγ = (D − 2)γ , νρ which imples that the equation γ ∂νΨρ = 0 is zero in spacetime dimension D > 2. We also use that γµνρ = γµγνρ − 2ηµ[γρ]. Using this we obtain the following equation

µ γ (∂µΨν − ∂νΨµ) = 0 . (6.8)

21 This equation is equivalent to the equation of motion above by applying γν and obtain νρ γ ∂µΨρ = 0. We can also apply ∂ρ to obtain the following equation of motion defined as ∂/(∂ρΨν − ∂νΨρ) = 0 . (6.9) If we now take SUSY into account we know that the spin-3/2 particle should have a superpartner. If the spin-3/2 particle from the Rarita-Schwinger equation represents a gravitino then the superpartner should be a graviton. The graviton is expected to be massless, but the gravitino is expected to have mass.

6.1 Dimensional reduction to the massless Rarita-Schwinger field Let us apply dimensional reduction the Rarita-Schwinger field in D + 1 dimensions with D = 2m. We assume that the field Ψµ(x, y) is periodic in y so that the Fourier series involves modes with half-integer k. This yields us only massive modes since k ≠ 0 occurs. Let us derive the wave equation of a massive gravitino in D-dimensional Minkowski space. We impose a gauge condition ΨDk(x) = 0 thus eliminating the field component D ΨD(x, y). Writing out the wave equation using γ = γ∗ where µ = D and µ ≠ D − 1: νρ γ ∂νΨρk = 0 (6.10)

µνρ k µρ (γ ∂ − i γ∗γ )Ψ = 0 . (6.11) ν L ρk We can obtain the equation of motion for a massive particle from (6.11) by applying the chiral transformation Ψ = e−iπγ∗/4Ψ′. We can rewrite the chiral transformation as a phase factor (cos(2β) + iγ∗ sin(2β))Ψ then we see that the only contribution is the sinus term and the Fourier series mode eiky/LΨ gives rise to the massive modes since k ≠ 0, therefore our equation of motion takes the form

µνρ µρ (γ ∂ν − mγ )Ψρ = 0 (6.12) where m = k/L which is the mass of the vector fields. One can further investigate the µ constraints on the equation of motion by finding γ Ψν = 0 and letting µ = 0 which yields us the following constraint

ij j (γ ∂i − mγ )Ψj = 0 . (6.13) Gathering the information about the constraints of the equation of motion gives us information about degrees of freedom.

µ γ Ψµ = 0 (6.14) ij j (γ ∂i − mγ )Ψj = 0 (6.15)

(∂/ + m)Ψµ = 0 . (6.16)

The initial data are the values t = 0 of the Ψµ restricted by the fist two equations above. D/2 D/2 The complex scalar field Ψµ with D × 2 degrees of freedom contains (D − 2) × 2 1 − × D/2 independent classical degrees of freedom and thus 2 (D 2) 2 on-shell physical states [4].

We can obtain a more general action defined as ∫ D µνρ µρ ′ µρ S = − d xΨ¯ µ(γ ∂ν − mγ − m η )Ψρ (6.17) thus from the dimensional reduction we obtained that the mass term for massive grav- µν itino should be mΨ¯ µγ Ψν and the extra term in the action is a Lorentz invariant term which has the dimension of mass.

22 7 The first and second order formulation of general relativity

In this chapter we will discuss the first and second order formulation of GR, which will be important when we consider supergravity for N = 1. The second order formulation in which the metric tensor or the frame field describes gravity. If fermions are involved we must use the frame field. a In first order a.k.a Palatini formalism one starts with an action in which e µ and ωµab are independent variables and the Euler-Lagrange equations are first order in derivatives. When we couple gravity to spinor fields the ωµab field equation contains terms bilinear in the spinors and the solution is ωµab = ωµab(e)+Kµab with a contorsion tensor determined as a bilinear expression in the spinor fields.

7.1 Second order formalism for gravity and fermions We consider the massless Dirac field Ψ(x), the action is defined as ∫ ( ) 1 1 1 ←− S = S + S = dDxe eµ eν R ab(ω) − Ψ¯ γµ∇ Ψ + Ψ¯ ∇ γµΨ . (7.1) 2 1/2 2κ2 a b µν 2 µ 2 µ We can obtain the relation

δS2(e, ω) ω=ω(e) = 0 (7.2) δωµab by using that the variation of the curvature is given by

ab ab ab δRµν = Dµδων − Dνδωµ (7.3)

ab a a thus making δRµν ∝ Dµe ν − Dνe µ. Using Cartan’s first structure equation we have that the spin connection can be expressed as a unique torsion free spin connection defined as ab ν[a b] ν[a b]σ c ωµ (e) = 2e ∂[µeν] − e e eµc∂νeσ . (7.4) The covariant derivatives defined from previous chapters: 1 ∇ Ψ = D Ψ = (∂ + ω abγ )Ψ (7.5) µ µ µ 4 µ ab ←− ←− ←− 1 Ψ¯ ∇ = Ψ¯ D = Ψ(¯ ∂ − ω abγ ) . (7.6) µ µ µ 4 µ ab Frame fields are used to transform frame vector indices to a coordinate basis, e.g µ µ a µν µ γ = e aγ = g γν. Thus γ transforms as a covariant vector under coordinate trans- formation. The covariant derivative of γµ is therefore 1 ∇ γ = ∂ γ + ω ab[γ , γ ] − Γρ γ . (7.7) µ ν µ ν 4 µ ab ν µν ρ The spin connection appears with the commutator as required for a spinor. We can obtain the following result

∇ a b − ρ µγν = γ (∂µeav + ωµabe ν Γµνeaρ) = 0 (7.8) which tells us that covariant derivatives commute with multiplication by γ-matrices. One obtains this by using the vielbein postulate (4.22). This relation holds for any affine connection the complete derivation can be seen in the appendix appendix C. For example the Dirac field ∇µ(γνΨ) = γν∇µΨ, and from the (7.1) one can find the equation of motion for the massless covariant Dirac equation

µ γ ∇µΨ = 0 . (7.9)

23 Let us now take the variation of the action defined in (7.1) with respect to eaµ, the goal is to find the Einstein field equations for a conserved symmetric fermionic stress tensor Tµν. The variation of the action is ∫ ( ( ) 1 1 δS = dDxe ebνR − e R δeaµ κ2 µνab 2 aµ ) (7.10) 1 ←→ 1 − Ψ¯ γa ∇ Ψδeaµ − Ψ¯ {γµ, γab}Ψδω 2 µ 8 µab we have used (7.2), and we have dropped the terms that are satisfied by the equation of motion (7.9) in the fermionic part of the Lagrangian. Let us now consider the variation of the spin connection in the last part of the action above, when we integrate by parts we will obtain a stress tensor. The variation of the spin connection is defined as b a ab a a − a a a a e ρe νδωµ = D[µδν]e ρ D[νδρ]e µ + (D[ρδµ]e ν . (7.11) µ cab The anti commutator is equal to a third rank Clifford matrix ec γ . Using the variation of the spin connection above we see that it is totally anti symmetric in µ, ν and ρ, thus we are able to write the last term 1 1 − Ψ¯ γµνρΨea eb δω = − Ψ¯ γµνρΨeb ∇ δe . (7.12) 4 ν ρ µab 4 ρ µ νb 1 { a b} One obtains this relation by doing some γ-matrix manipulations using 2 γ , γ and inserting this into the anti-commutation relation above, after doing some calculations 1 ¯ [a b]µ we obtain that this is 4 Ψγ η Ψδωµab using that the gamma matrices are constant µ µ a matrices i.e γ = e aγ and using (7.11) one obtains the result in (7.12). Integrating by parts, and using the Dirac equation (7.9) and γ-matrix manipulations we obtain 1 ←→ ←→ 1 ←→ ←→ − e δaΨ(¯ γν ∇ ρ − γρ ∇ ν)Ψ = Ψ(¯ γ ∇ − γ ∇ eρ )Ψδeaµ (7.13) 4 aρ ν 4 a µ µ ρ a Thus we arrived at the final expression defined as ∫ ( ( ) ( ) ) 1 1 1 ←→ ←→ δS = dDx ebνR − e R δeaµ − Ψ¯ γ ∇ − γ eρ ∇ Ψδeaµ . κ2 µνab 2 aµ 4 a µ µ a ρ (7.14) since δeaµ is arbitrary we can set it to zero thus we obtain the Einstein equation of the form ( ) 1 κ2 ←→ ←→ R − g R = Ψ¯ γ ∇ + γ ∇ Ψ . (7.15) µν 2 µν 4 µ ν ν µ The Einstein equations are consistent only if the matter stress tensor is covariantly conserved Tµν = 0, and symmetric Tµν = Tνµ. This follows from the requirements that the matter action is invariant under general coordinate transformation and local Lorentz transformations. Let us show that the stress tensor is Lorentz invariant, we consider the action ∫ S = dDxL(e, ω(e), ψ) (7.16) where ψ is the spinor field, thje independent fields transform as 1 1 δe = −λ be , δΨ = − λ γabΨ, δψ = − λ γabψ . (7.17) aµ a bµ 4 ab µ 4 ab µ

The ψµ field is the gravitino field that appears in the Rarita-Schwinger equation and λab is the infinitesimal Lorentz transformation. We assume that the action is invariant so the variation is given by ∫ ( ) D δS δS δS = d x δeaµ + δψ = 0 (7.18) δeaµ δψ

24 From the first term we obtain 1 δS T aµ = (7.19) e δaµ Thus if the equation of motion (7.9) is satisfied then the equation above becomes ∫ D µν a b δS = d xeT e νeµλab(x) = 0 (7.20) since λab(x) is antisymmetric the stress tensor must be symmetric. Symmetry is guar- anteed only if the fermion equation of motion is satisfied.

7.2 First order formalism for gravity and fermions

The field equation for ωµab contains terms bilinear in the spinors giving a connection with torsion ωµab = ωµab(g) + Kµab. We will use the action (7.1) and the covariant derivative (7.6) for the first order action. In first order formalism the frame field and the spin ab connection ωµab are independent variables and that the curvature tensor Rµν can be constructed from this. The field equation for the spin connection gives a connection with the torsion ωµab = ωµab(e) + Kµab(ϕ) (7.21) The variation of the gravitational action is ∫ ( ) 1 δS = dDx eeµ eν D δω ab − D δω ab (7.22) 2 2κ2 a b µ ν ν µ ∫ √ D µ ν ab = d x −ge ae bDµδων . (7.23)

The Lorentz covariant derivatives can be integrated√ by parts. We have to include√ the − µ ν − action√ of Dµ on each of the three factors in ge ae b. Using the relation ∂µ g = − ρ g Γρµ(g), we obtain ∫ √ ( ) − D − ρ µ µ ν µ ν ab δS2 = d x g (Γρµ(g)e a + Dµe a)e b + eb Dµe b δων (7.24) the firs two terms can be written as ρ − ρ µ ∇ µ (Γρµ(g) Γρµ)e a + µe a . (7.25)

µ The last term is a total covariant derivative of e a which vanishes as a consequence of ρ ρ − ρ the vielbein postulate. Using that Γµν = Γµν(g) Kµν and the contorsion tensor 1 K = − (T − T + T ) (7.26) [µ[νρ] 2 [µν]ρ [νρ]µ [ρµ]ν thus we can rewrite the equation (7.25) as ρ − ρ µ ρ (Γρµ(g) Γρµ)e a = Taρ (7.27) The last term in the action can be written as

ν ρ c ν Dµe b = −e b(Dµe ρ)e c (7.28)

The integrand of the second term is then

− µ ρ c ν ab −1 µ ρ c − c ν ab e ae bDµe ρe cδων = e ae b(Dµe ρ Dρeµ)e cδων 2 (7.29) 1 = − T νδω ab 2 ab ν

25 this follows from equation (4.12). Combining all the information above we arrive at the final expression of the variation of S2 defined as ∫ 1 √ δS = dDx −g (T ν − T ρeν + T ρeν ) δω ab . (7.30) 2 2 ab aρ b bρ a ν Let us look at the connection variation of the spinor action using equation (7.6), we obtain ∫ 1 δS = − dDx e Ψ¯ γν Ψδω ab . (7.31) 1/2 4 ab ν

ab We can now find a equation for the torsion tensor, by considering δων to be arbitrary and thus we obtain 1 T ν − T ρeν + T ρeν = κ2Ψ¯ γ νΨ . (7.32) ab aρ b bρ a 2 ab We have seen that the spin connection in first order formalism has a connection with the torsion tensor i.e ωµab = ωµab(e) + Kµab(ϕ), thus we see that the spin connection ωµab is a function of the other fields. The contorsion tensor appears when we couple gravity to fermions

26 8 4D Supergravity

We have now arrived to supergravity theory, which is a locally invariant supersymmetry theory. This means that the action is invariant for SUSY transformations where ε(x) is a function of the spacetime coordinates. A supergravity theory is nonlinear which is a interacting field theory that contains gauge and gravity multiplets and other multiplets a describing scalar fields. The gauge multiplet consist of a frame field e µ(x) describing N i N the graviton and of vector-spinor fields Ψµ(x) where i = 1,..., . In this section we will focus on D = 4 with N = 1.

8.1 Supergravity in second order formalism We will not start with the second order formalism of supergravity. We have the Einstein- Hilbert Lagrangian which should be interpret as the kinetic term for gravity and we should add a kinetic term for the spin 3/2 field which is known as the Rarita-Schwinger Lagrangian. The action should be of form S = S2 + S3/2, hence the supergravity action is defined as ∫ ( ) 1 S = dDx e eaµebνR − e ψ¯ γµνρD ψ (8.1) SG 2κ2 µνab µ ν ρ a where e stands for the determinant of e µ, the gravitino covariant derivative is given by 1 D ψ = ∂ ψ + ω γabψ . (8.2) ν ρ ν ρ 4 νab ρ The supersymmetry transformations laws of the vielbein and the gravitino are given by 1 δea = εγ¯ aψ (8.3) µ 2 µ δψµ = Dµε(x) . (8.4)

The gravitino is the gauge field of local supersymmetry, and one can interpret (8.3) as a bosonic vielbein that transforms into its superpartner. We have to check that the transformations satisfy the supersymmetry algebra. The commutator of two supersym- metry transformations should give rise to a suitable combination of local supersymmetry transformations. Thus the commutator of two infinitesimal supersymmetry transforma- tions yields space-time dependent vector fieldεγ ¯ µε. Consider the commutator of two supersymmetry transformations on the vielbein is

a a [δε1 , δε2 ]e µ = 2δ[ε1 δε2]e µ a − a = δε2 (¯ε1γ ψµ) δε1 (ε ¯2γ ψµ) a a (8.5) =ε ¯1γ Dµε2 − ε¯2γ Dµε1 a = Dµ(ε ¯2γ ε1) .

µ a We define the space-time vector field ξ =ε ¯2γ ε1, which yields us

ν a a ν [δε1 , δε2 ] = ξ Dµeν + eν ∂µξ (8.6)

We have used that Dµ is the covariant derivative which acts as a space-time derivative on ξν. For now we set our torsion tensor to zero thus we have a relation from the vielbein postulate that allows us to write a D[µeν] = 0 . (8.7) Considering this property and the equation (8.6), we can write the covariant derivative as a ν a a ν ν a b [δε1 , δε2 ]eµ = ξ ∂νeµ + eν ∂µξ + ξ ων beµ (8.8)

27 the first and second term are infinitesimal action of a diffeomorpishm on the frame field a a and the last term as a infinitesimal internal Lorentz transformation δΛeµ with Λ b = ν a ξ ων b. We have shown that the commutator of two supersymmetry transformations can be written as a [δε1 , δε2 ]eµ = δξ + δΛ (8.9) which is a combination of local symmetries: diffeomorphisms and Lorentz transforma- tions. So the space-time dependent vector field ξµ is a element of the group of local diffeomorphism on space-time. We have to treat the space-time metric as a dynamical object because it is coordinate invariant.

Let us go back to our action we want to show that the action (8.1) is invariant under supersymmetry transformations. We start with the gravitational part of the action ∫ 1 δS = dDx e eaµebνR (8.10) 2 2κ2 µνab we will follow the same procedure as we did in section (7.1), where we take the curvature tensor to depend on the spin connection ω, hence Rµνab(ω). Thus the variation of the gravitational action under (8.3) is ∫ ( ) 1 δS = dDx (2e(δeaµ)ebν + (δe)eaµebν)R + eeaµebνδR 2 2κ2 µνab µνab ∫ ( ) (8.11) 1 1 = dDx e R − g R (−εγ¯ µψµ) . 2κ2 µν 2 µν

Due to symmetry Rµνab = Rνµba gives the factor of 2 in the first line, the second term vanishes because of the variation on the action where we set the variations on the boundary to zero. Let us now consider the gravitino variation. We want to partial integrate so we vary δψ¯µ and multiply with 2. We then obtain ∫ 1 ←− δS = − dDx e ε¯D γµνρD ψ 3/2 2κ2 µ ν ρ ∫ ∫ (8.12) 1 1 = ddx e εγ¯ µνρD D ψ = dDx e εγ¯ µνρR γabψ κ2 µ ν ρ 8κ2 µνab ρ we integrated by parts and used (7.8) to obtain the second line and we replaced the ∇µ by Dµ because of antisymmetry and no absence of torsion. The last expression can be obtain by using the Ricci identity (4.43). We have to evaluate the γµνργab. We contract the Riemann tensor with the gamma matrices and using the relation (3.9), we obtain

µνρ ab µνρab [ρ µν]b [µ ρν] Rµνabγ γ = γ Rµνab + 6Rµν bγ + 6γ Rµν µνρab ρ µνb µ νρb = γ Rµνab + 2Rµν bγ + 4Rµν bγ (8.13) µ ρν ρ νµ + 4γ Rµν + 2γ Rµν .

The first and second term vanishes from the Bianchi identity (4.39) with a zero torsion and the third term vanishes because of symmetry clash between the symmetric Ricci µ νρb tensor Rνb = Rµν b and the anti symmetry of γ . We have now obtain the variation on the gravitino field, given as ∫ ( ) 1 1 δS = dDx e R − g R (¯εγµψν) (8.14) 3/2 2κ2 µν 2 µν

We see that δS2 + δS3/2 = 0 and thus confirms local supersymmetry to linear order in ψµ for any spacetime dimension.

28 8.2 Supergravity in first order formalism In the previous section we only considered linear order terms but it becomes more complicated when we go beyond linear order, due to complication in establishing local supersymmetry for supergravity theory. Let us now consider the spin connection ωµab as an independent variable. Considering the action (8.1), we will use the connection variation (7.30) of S2. We are left with the spin variation on the spin-3/2 field, we obtain the following variation ∫ ( ) 1 δS = − dDx e ψ¯ γµνργ ψ δω ab . (8.15) 3/2 8κ2 µ ab ρ ν The rank 3 terms from (3.9) vanish because of antisymmetry in the gravitino indices µ, ρ, therefore we obtain ( ) ¯ µνρ ¯ µνρ [µ ν ρ] ψµγ γabψρ = ψµ γ ab + 6γ e [be a] ψρ . (8.16)

Using the equation (7.30) we can solve S2 + S3/2 = 0, we obtain the following solution to the torsion 1 1 T ν = ψ¯ γνψ + ψ¯ γµνρ ψ . (8.17) ab 2 a b 4 µ ab ρ The first formalism determines a connection with torsion in supergravity. The fifth rank term in the torsion vanishes in D = 4, and we obtain the same torsion as in section (7.2). We will see that writing the spin connection as ω = ω(e) + K will leave the supergravity action invariant under the transformation rules (8.3) and (8.4).

We can substitute the torsion above in the action (8.1) to obtain the second order action of supergravity. ∫ ( ) 1 S = d4x e R(e) − ψ¯ γµνρD ψ + L (8.18) 2κ2 µ ν ρ SG,torsion where ( ) 1 L = − (ψ¯ργµψν)(ψ¯ γ ψ + 2ψ¯ γ ψ ) − 4(ψ¯ γ · ψ)(ψγ¯ · ψ) (8.19) SG,torsion 16 ρ µ ν ρ ν µ µ

8.3 1.5 order formalism The second order action (8.18) is invariant under supersymmetry transformations, with the connection

ωµab = ωµab(e) + Kµab , (8.20) 1 K = − (ψ¯ γ ψ − ψ¯ γ ψ + ψ¯ γ ψ ) (8.21) µνρ 4 µ ρ ν ν µ ρ ρ ν µ that includes the gravitino torsion. For higher order terms using first and second or- der formalism becomes very complicated to solve, because of the variations on different orders in the gravitino field. In the first order the spin connection is an independent variable this approach becomes complicated when we couple matter multiplets to su- pergravity. The simplest approach to supergravity is between the first and second order called the 1.5 formalism, we can apply this formalism in any dimensions. We are working in the second order formalism since there are only two independent µ fields ψµ and e a. Let us consider a functional of three variables S[e, ω, ψ], we use the chain rule to calculate its variation in the second order formalism ∫ ( ) δS δS δS δS[e, ω(e) + K, ψ] = dDx δe + δ(ω(e) + K) + δψ . (8.22) δe δω δψ

29 We can neglect all the δω variations, because in 1.5 order formalism we don’t need to calculate the algebraic field equation δS/δω = 0. We can summarize this

• Use the first order form of the action S[e, ω, ψ] and the transformation rules δe and δψ with an unspecified connection

• Ignore the connection variation and calculate ∫ ( ) δS δS δS = dDx δe + δψ (8.23) δe δψ

8.4 Local supersymmetry of N = 1, D = 4 supergravity We will now use the 1.5 order formalism to show that supergravity in D = 4 and N = 1 is invariant under the transformations rules (8.3) and (8.20) and (8.21). To simplify the gravitino action we introduce the highest rank Clifford element γ∗ = iγ0γ1γ2γ3. We can express the third rank Clifford matrices as

abc abcd µνρ µνρσ γ = −iε γ∗γd, γ = −iε γ∗γσ . (8.24)

The first relation holds for local frames and the second is needed for the coordinate basis. We can now rewrite the gravitino action using these relations and we obtain ∫ i 4 µνρσ S = d xε ψ¯ γ∗γ D ψ . (8.25) 3/2 2κ2 µ σ ν ρ

The advantage of this form is that the frame field variation is needed only in γσ instead of eγµνρ. Since we are working in 1.5 order formalism we can ignore the variation δω of Rµνρσ, so we consider the following terms

δS = δS2 + δS3/2,e + δS3/2,ψ + δS3/2,ψ¯ (8.26) where the first term is the variation of the gravity and the second term is the variation ¯ of frame field in S3/2, meanwhile the third and forth term are variations of ψ and ψ. The variation of the gravity was obtain from (8.11) ∫ ( ) 1 1 δS = dDx e R (ω) − g R (−εγ¯ µψµ) . (8.27) 2 2κ2 µν 2 µν and the second term in (8.26) is obtained by the variation of the frame field which is obtained in the same fashion as the second order formalism ∫ i 4 µνρσ a δS = d xε (¯εγ ψ )(ψγ¯ ∗γ D ψ ) . (8.28) 3/2,e 4κ2 σ a ν ρ

The variation of the S3/2,ψ is given by ∫ i 4 µνρσ δS = d xε ψ¯ γ∗γ D D ε 3/2,ψ 2κ2 µ σ ν ρ ∫ (8.29) i 4 µνρσ ab = d xψ¯ ε γ∗γ γ R (ω)ε 16κ2 µ σ νρab the last line is obtained from the equation (4.43), and we were allowed to shift the derivative Dν due to partial integration. Let us now write down the variation of S3/2,ψ¯, where we will use the relation (3.23) with t3 = 1. We obtain ∫ ←− i 4 µνρσ δS = d xε ψ¯ D γ∗γ D ε (8.30) 3/2,ψ¯ 2κ2 ρ ν σ µ

30 we can now use the covariant derivative given in previous chapters ←− 1 ψ¯ D = ∂ ψ¯ − ψ¯ ω γab (8.31) ρ ν ν ρ 4 ρ νab if we then partial integrate we can use the same property as above which gives us the right to shift derivatives and we now obtain the following variation ∫ − i 4 µνρσ ¯ ∗ δS3/2,ψ¯ = 2 d xε ψργ ((Dνγσ)Dµε + γσDνDµε) 2κ ∫ ( ) − (8.32) i 4 µνρσ 1 a 1 ab = d xε ψ¯ γ∗ T γ D ε − γ γ R (ω)ε . 2κ2 ρ 2 νσ a µ 8 σ µνab The torsion term comes from when we add the Christoffel symbols and use antisymmetry in νσ, this relation comes from (4.25). We see that the Riemann term R(ω) is equal in both variations on the gravitino fields thus we can add them and we obtain the following variation ∫ ( ) − i 4 µνρσ 1 a 1 ab δS + δS ¯ = d xε ψ¯ γ∗ T γ D ε − γ γ R (ω)ε (8.33) 3/2,ψ 3/2,ψ 2κ2 ρ 2 νσ a µ 4 σ ab µν Using the following γ-matrix manipulation

d c γσγab = γσab + 2eσ[aγb] = ieσεabcdγ∗γ + 2eσ[aγb] (8.34) where we use the relation from (8.24), using this relation and plugging it back into c c the action (8.33) and keeping in mind that ψ¯µγ ε = −εγ¯ ψµ. We consider the parts separated, let us start with the first part and we see that we have a contraction of two ρ ...ρ µ1...µnρ1...ρp − 1 p Levi-Civita symbols, using εµ1...µnν1...νp ε = p!n!δν1...νp . The contraction of two Levi-Civita with the Riemann tensor R(ω) becomes ( ) εµνρσε ed R ab = −2e eµ eν eρ + eµeν eρ + eµeν eρ R ab(ω) abcd σ νρ ( a b c b c )a c a b νρ 1 (8.35) = 4e R µ(ω) − eµR(ω) . c 2 c

Using this and plugging it back in to the action (8.27), with the majorana spinor con- dition above we see that these terms cancel out. We see that we have obtained the same result as in the second order formalism of supergravity. Which tells us that we have a linear local supersymmetry. The second term in (8.34) to the integrand in (8.32) involves the factor µνρσ µνρσ ε Rνρσb(ω) = −ε DνTρσb (8.36) where we have used the Bianchi identity (4.39), where the derivative Dν is a Lorentz covariant derivative and contain only the spin connection acting on the index b. We are left with ∫ − i 4 µνρσ a a δS + δS + δS ¯ = d xε ψ¯ γ∗γ (T D ε + (D T )ε) (8.37) 2 3/2,ψ 3/2,ψ 4κ2 µ a ρσ ν ν ρσ

We now have the variation of the frame field for the S3/2 action, ∫ i 4 µνρσ a δS = d xε (¯εγ ψ )(ψγ¯ ∗γ D ψ ) . (8.38) 3/2,e 4κ2 σ a ν ρ We need to reorder the spinors using Fierz rearrangement, using 1 ∑ (γµ) β(γ ) δ = v (Γ ) β(ΓA) β (8.39) α µ γ 2m A A α γ A

31 r the expansion coefficient vA = (−) A (D−2rA) where rA is the tensor rank of the Clifford basis element ΓA, thus for even dimensions D = 2m we have that 0 ≤ rA ≤ D. Applying this to the integrand above we obtain ∑ a 1 r A (¯εγ ψ )(ψγ¯ ∗γ D ψ ) = − (−) A (4 − 2r )(¯εΓ γ∗D ψ )(ψ¯ Γ ψ ) σ a ν ρ 4 A A ν ρ µ σ A 1 a (8.40) = (¯εγ γ∗D ψ )(ψ¯ γ ψ ) 2 a ν ρ µ σ a = (¯εγ∗γaDνψρ)Tµν .

Due to antisymmetry in µσ of the action S3/2,e with the contraction of the Levi-Civita symbols, allows us to write the last line using the D = 4 torsion tensor defined as (8.17). We insert this into (8.38) and reorder the (¯ε . . . Dψ) bilinear and exchange the indices µρ to obtain ∫ − ←− i 4 µνρσ a δS = d xε T ψ¯ D γ∗γ ε . (8.41) 3/2,e 4κ2 ρσ µ ν a We have now expressed all the variations in terms of the torsion tensor and adding them together we obtain ∫ − ( ←− ) i 4 µνρσ a b ¯ ∗ ¯ ∗ ¯ ∗ δS = 2 d xε Tρσ (ψµγ γaDνε + ψµ D νγ γbε) + DνTρσ ψµγ γbε 4κ ∫ − ( ) (8.42) i 4 µνρσ b = d xε ∂ T ψ¯ γ∗γ ε = 0 . 4κ2 ν ρσ µ b We have shown that the action is invariant up to a total derivative. We also see that the spin connection among the three terms in the second line cancel, thus we have showed that the N = 1, D = 4 supergravity theory is locally supersymmetric.

32 9 Toroidal compactification

We have seen Kaluza-Klein reductions on a circle, in this chapter we will consider the toroidal compactification of the eleven-dimensional supergravity Lagrangian. We will reduce the 11D supergravity Lagrangian down to 4D. Performing dimensional reductions will yield us e.g Type IIA and Type IIB theories which are low-energy theories of superstring theory with the same name. The main references in this chapter will be [6], [10] and [11] .

9.1 D=11 supergravity We will now consider supergravity in 11 dimensions, before we construct the Lagrangian and the transformation rules we must first investigate the field content for D = 11 supergravity. a The field content of D = 11 theory, consists of elfbein e µ, a Majorana spin 3/2 ψµ and a antisymmetric gauge tensor with three indices Aµνρ, this was first formulated by [8]. To obtain the set of fields we simply calculate the number of states. The formulas for counting the states are defined as ( ) d(d − 3) (d − 3)2(d/2) d − 2 ea = , ψ = ,A = . (9.1) µ 2 µ 2 µ1,...µp p

We see that we obtain 44 degrees of freedom for the vierbein and we have 84 degrees of freedom. This comes from the fact that we need equal amounts of boson and fermion states. We know that doing a dimensional reduction on T 7 generates the D = 4, N = 8 supergravity theory, which contains one graviton gµν, seven vectors gµi and gij contains 28 scalars. The Majorana field decomposes into 8 spin 3/2 fields, and 56 1/2 fields. The N = 8 theory contains 28 vectors and the missing 21 are supplied by the 3-form component Aµij. The form components Aijk contain 35 scalars and the components Aµνi give additional 7 scalars. This comes from the fact that antisymmetric tensor fields in four dimensions are equivalent to scalars or vector fields by a symmetry known as duality. Thus we have 70 scalars, combining all the fields we end up with the same degrees of freedom as D = 11 N = 1 supergravity theory. We can now begin to construct the 11-dimensional Lagrangian and its transformation rules. We begin with the 3-form potential Aµνρ this field must be coordinate invariant and local Lorentz invariant, it must also transform under gauge transformation involving a parameter θνρ. The theory involves a gauge invariant 4-form field strength Fµνρσ. The equations are

δAµνρ = ∂µθνρ + ∂νθρµ + ∂ρθµν (9.2)

Fµνρσ = ∂µAνρσ − ∂νAρσµ + ∂ρAσµν − ∂σAµνρ (9.3)

∂[τ Fµνρσ] = 0 (9.4) the last equations follows from the Bianchi identity, this follows from the fact that letting a covariant derivative act on F = dA gives us zero when written in differential form. We will also use Majorana flip relation, we have the same properties as in four dimensions

µ1...µr µ1...µr χγ¯ λ = trλγ¯ χ, t0 = t3 = 1, t1 = t2 = −1, tr+4 = tr . (9.5)

We know that the 11-dimensional supergravitry theory must have graviton and gravitino terms like we had in the D = 4 theory, we also have to include the 3-form potential

33 Aµνρ plus several terms which we need to find by evaluating the gravitino field and the potential. We begin with the action defined as ∫ ( ) 1 1 S = d11x e eaµebνR − ψ¯ γµνρD ψ − F µνρσF + ... (9.6) 2κ2 µνab µ ν ρ 24 µνρσ where ... are the terms we need to find. Let us evaluate the gravitino and 3-form poten- tial now using the second order formalism where we have a torsion-free spin connection ωµab(e). The transformation rules are given by 1 δea = εγ¯ aψ (9.7) µ 2 µ αβγδ βγδ α δψµ = Dµε + (aγ µ + bγ δµ )Fαβγδε (9.8) 1 δA = − cε¯(γ ψ + γ ψ + γ ψ ) . (9.9) µνρ 3 µν ρ νρ µ ρµ ν We have expressed the transformations rules that contain numerical constants a, b, c, which we have to determine. The action for a free theory where we don’t have any interactions, is given by ∫ ( ) 1 1 S = d11x −ψ¯ γµνρ∂ ψ − F µνρσF (9.10) 0 2 µ ν ρ 24 µνρσ Let us now take the variation of this action, which is needed to determine the numerical constants. We vary the action w.r.t ψ¯µ and Aµνρ ∫ ( ) 1 δS = d11x ε¯ (aγαβγδ − bγβγδδα)F γµνρ∂ ψ − cγ ψ ∂F µνρσ 0 µ µ αβγδ ν ρ 6 νρ σ ∫ ( ) (9.11) 1 = d11x ε¯ (−aγαβγδ + bγβγδδα)∂ F γµνρψ − cγ ψ ∂ F µ µ ν αβγδ ρ 6 νρ σ µ µνρσ where we have used

¯ αβγδ − βγδ α µνρ δψµ =ε ¯(aγ µ bγ δµ )Fαβγδγ . (9.12)

The factor of 2 disappears because we vary the fields twice hence we can combine them and only vary one field, the variation of the field strength tensor was obtained by using (9.9). We now see that we need to reduce the products of the γ matrices, to sums over rank 6, rank 4 and rank 2 elements of ΓA of the Clifford algebra. We start with the first term in (9.11)

αβγδ µνρ αβγδνρ αβγ[ν ρ] γ µγ Fαβγδ = (D − 6)γ Fαβγδ + 8(D − 5)γ F αβγ (9.13) αβ νρ − 12(D − 4)γ Fαβ we can obtain this by using γ-matrix manipulations. In the first term we see that µ can run over all D values except αβγδ and νρ which gives us (D − 6) factor in front of the first term. We can use the same logic for the other terms. In the second term we see that ν can run over all D values except the five indices. We can also choose four indices from the first factor and two indices from the second term which gives us the factor 8. We also have one contraction from the δ-index thus combining all this we obtain the second term. The third term contains a contraction from the γ-index and the same logic applied we obtain the last term. Let us now take the second term

βγδ µνρ νραβγδ αβγ[ν ρ] αβ νρ γ γ Fµβγδ = −γ Fαβγδ − 6γ F αβγ + 6γ Fα . (9.14)

34 This relation holds directly from the definition of the Clifford algebra (3.1), we have seen this relation in (3.8) and (3.9). We first write the totally antisymmetric Clifford matrix that contains all the indices and then add possible pairings. We also use the same logic as above hence the constants in front of the second and third term. We can now use these relations above in the variational action (9.11) to solve the numerical constants. By doing so we see that the rank 6 elements disappear due to the Bianchi identity (9.4) we are left with the rank 4 elements and the rank 2 elements of the Clifford algebra. We can summarize this as

8a(D − 5) + 6b = 0 (9.15) 1 12a(D − 4) + 6b = c (9.16) 6 c − we solve for D = 11 and we obtain that a = 216 and b = 8a. The transformation rules are then given as ( ) c δψ = ∂ ε + γαβγδ − 8γβγδδα F ε (9.17) µ µ 216 µ µ αβγδ δAµνρ = −cεγ¯ [µνψρ] . (9.18)

We now have to solve the c constant, we do so by examining the commutator of two SUSY transformations and require that they agree with the local supergravity. Let us write down the SUSY transformation for the gauge potential, defined as ( ) 1 [δ , δ ]A = − c2ε¯ γ γαβγδ − 8γβγδδα ε F − (1 ↔ 2) . (9.19) 1 2 µνρ 216 2 [µν ρ] ρ] 1 αβγδ We know that each contraction in these terms contains a pair of indices so the contri- butions to the Clifford algebra will be of rank 1, 3, 5 , 7. We can do the same γ-matrix manipulations as above, but we can simplify this by observing that the parameters ε1 and ε2 are antisymmetric and we only obtain contributions from the rank 1 and rank 5 terms. We also see that the first term does not have any rank 1 terms because we would need to contract three pairs of indices to obtain a non-vanishing rank 1 term, and the antisymmetrization do not allows this. We can see this by doing the γ-matrix manipulation where the contraction is between the antisymmetric indices hence why we are not allowed to do this. We can see that the rank 3 terms vanish when we consider the antisymmetry of the parameters ε1 and ε2, using the relations in (3.11) and a very useful table in [2]

d( mod 8) S A ε η 0 0,3 2,1 -1 +1 0,1 2,3 -1 -1 1 0,1 2,3 -1 -1 2 1,0 3,2 -1 -1 1,2 3,0 +1 +1 3 1,2 0,3 +1 +1

Table 1: The S and A stands for symmetric and antisymmetri, for a detailed description of the table we refer to [2]

note that ε = ±t0 and η = ±t0t1. We can see that when we do the calculation for rank 3 elements they all cancel with each other. The second term is already given in (3.9). We know from the definition that the σ SUSY algebra contain a spacetime translation which involves a rank 1 bilinearε ¯1γ ε2

35 from the second term in (9.19). We then obtain the commutator 4 [δ , δ ]A = − c2ε¯ γσε F (9.20) 1 2 µνρ 9 1 2 σµνρ SUSY requires that the rank 5 elements vanish in the term (9.19) which they do. Let us interpret the SUSY commutation relation by looking at the field strength defined in (9.3). The first term ∂σAµνρ is the space-time translation from the SUSY algebra, the remaining terms just add up to the 3-form potential. The gauge field is σ proportional gauge transformation θµν = −ε¯1γ ε2Aµνν. These gauge transformations can be found when one considers the super Yang-Mills theory and also in local super- symmetry for supergravity theories. 2 We can now find the c constant,√ from above we see that c = 9/8. We choose the positive root and obtain c = 3/2 2. We can find the supercurrent in (9.11) by considering the spinor parameter ε de- pending on xµ, taking the variation of the action and performing partial integration. We find that the variation contains a term that is proportional to Dνε whose coefficients are the supercurrent √ 2 J ν = (γαβγδνρF + 12γαβF νρ)ψ . (9.21) 96 αβγδ αβ ρ This is a new term to the 11-dimensional supergravity. We can now extend the results a from the free system to a interacting system. We consider a general frame field e µ(x) and consider the general ε(x), we then have the transformation rules

a 1 a δe µ = εγ¯ ψµ (9.22) 2 √ 2 αβγδνρ αβ νρ δψµ = Dµε + (γ Fαβγδ + 12γ Fαβ )ψρ (9.23) √ 96 3 2 δA = − εγ¯ ψ (9.24) µνρ 4 [µν ρ] and the action ∫ ( 1 1 S = d11x e eaµebνR − ψ¯ γµµρD ψ − F µνρσF 2κ2 µνab µ ν ρ 24 µνρσ √ ) (9.25) 2 − ψ¯ (γαβγδνρF + 12γαβF νρ)ψ + ... . 96 ν αβγδ αβ ρ

We still need to figure out the rest of the Lagrangian. We have seen that the first and second term cancel in previous section and thatεF ¯ αβγδψρ also cancel. We are now left with the variation of the field strengthεF ¯ 2ψ and the variation of ψF¯ ψ term. Writing the variational terms explicitly, ( ) 1 1 δL = 4¯εγµψν − gµνεγ¯ · ψ F ρστ F FF 48 2 µ νρστ ( ) 1 α′β′γ′δ′ β′γ′δ′ α′ (9.26) δL = ε¯ γ + 8γ δ F ′ ′ ′ ′ ψF¯ ψ 96 × 144 ν ν α β γ δ αβγδνρ αβ ν × (γ Fαβγδ + 12γ Fαβ )ψρ .

We have now used the 1.5 order formalism and taking the linear parts in the δS, the terms of the formεψ ¯ in the variation vanish as it did in D = 4 supergravity. The products of γ-matrices contain rank 9, 7, ,5 ,3, 1 terms. For a detailed description on how to solve these γ-matrices we refer to [9]. The rank 1 terms in (9.26) cancel between

36 each other and several rank 3, 5 and 7 terms cancel withing the term ψF¯ ψ. We have the rank 9 terms, which we can obtain from the γ-matrix products

α′β′γ′δ′ αβγδνρ α′β′γ′δ′αβγδ α′β′γ′δ′αβγδρ γ νγ = (D − 9)γ + ... = 2γ + ... (9.27) ′ ′ ′ ′ ′ ′ ′ ′ γβ γ δ γαβγδα ρ = −γα β γ δ αβγδρ + ... (9.28) where the dots are the lower ranked terms which we ignore. Combining these results we are left with

1 α′β′γ′δ′αβγδρ δL + δL = − εγ¯ F ′ ′ ′ ′ F (9.29) FF ψF¯ ψ 16 × 144 α β γ δ αβγδ this is the term that survives from the variational terms. In order to make the Lagrangian invariant the terms must cancel, hence we need Clifford algebra in odd spacetime di- mensions D = 2m + 1. For D = 11 the generating matrices are 32 × 32, using

µ 0 1 (2m−1) 2m γ± = (γ , γ , . . . , γ , γ ) = ±γ∗ (9.30) we obtain that the rank 9 Clifford element is related to rank 2 by this equation. Thus the Clifford algebra is given as

′ ′ ′ ′ 1 ′ ′ ′ ′ γα β γ δ αβγδρ = − ϵα β γ δ αβγδρµνγ . (9.31) 2e νµ Rewriting the equation (9.29) we get ( ) 1 α′β′γ′δ′αβγδρµν e δL + δL ¯ = ϵ εγ¯ ψ F ′ ′ ′ ′ F FF ψF ψ 32 × 144 νµ ρ α β γ δ αβγδ (9.32) 4 α′β′γ′δ′αβγδρµν = √ ϵ (δAµνρ)Fα′β′γ′δ′ Fαβγδ 3 2 × 32 × 144 We are left with the following variation, called the Chern-Simons term for the 3-form potential ∫

SC−S = A3 ∧ F4 ∧ F4 (9.33) there is a U(1) gauge symmetry where the gravitino is charged and for which the 3-form potential is the gauge field. We also note that F4 = dA3 is the U(1) field strength 4-form. The Chern-Simons term is gauge invariant up to a total derivative. The action transform as ∫ ∫ ∫

A3 ∧ F4 ∧ F4 = A3 ∧ F4 ∧ F4 + dΛ2 ∧ F4 ∧ F4 ∫ ∫ (9.34) = A3 ∧ F4 ∧ F4 + d(Λ2 ∧ F4 ∧ F4) the last term vanishes due to the Bianchi identity dF4 = 0, and we have used the gauge transformation A3 → A3 + dΛ2. This variation does not produce any further variations since there are no frame fields in the expression. A typical property of the Chern-Simons action always guarantees gauge invariance. We have now obtained all the transformations rules. We can thus write the 11- dimensional Lagrangian as ∫ ( ( ) 1 ω +ω ˆ 1 S = d11x e eaµebνR − ψ¯ γµµρD ψ − F µνρσF 2κ2 µνab µ ν 2 ρ 24 µνρσ √ 2 ¯ αβγδνρ αβ γν δρ − ψν(γ + 12γ g g )ψρ(Fαβγδ + Fˆαβγδ) (9.35) 192√ ) 2 2 α′β′γ′δ′αβγδρµν − ϵ F ′ ′ ′ ′ F A . (144)2 α β γ δ αβγδ µνρ

37 We replaced the spin-connection field by ω with (ω +ω ˆ)/2 in the covariant derivative of the gravitino kinetic term and we also replaced the field-strength with (Fαβγδ + Fˆαβγδ)/2, these substitutions ensure that the field equations are supercovariant. The hatted connection and field strength are given by

ωµab = ωµab(e) + Kµab , (9.36) 1 ωˆ = ω (e) − (ψ¯ γ ψ − ψ¯ γ ψ + ψ¯ γ ψ ) , (9.37) µab µab 4 µ b a a µ b b a µ 1 1 K = − (ψ¯ γ ψ − ψ¯ γ ψ + ψ¯ γ ψ ) + ψ¯ γνρ ψ , (9.38) µab 4 µ b a a µ b b a µ 8 ν µab ρ 3√ Fˆ = 4∂ A + 2ψ¯ γ ψ . (9.39) µνρσ [µ νρσ] 2 [µ νρ σ] The action is invariant under the transformations defined in (9.22 − 9.24)

9.2 11-Dimensional toroidal compactification Before we start with the dimensional reduction on the eleven-dimensional supergravity Lagrangian we first have to investigate the reductions of antisymmetric tensor field strengths. We will consider the reduction from (D + 1) to D dimensions, we assume that we have an n-index field strength in higher dimension which we denote Fˆn. Let us rewrite the field strength in terms of the gauge potential Aˆn−1 so that Fˆn = dAˆn−1. Thus after reduction we obtain two dimensional reduced potentials, namely (n−1) which lies in the D-dimensional spacetime and the (n − 2) which lies is in the S1 direction. We can express this as Aˆn−1(x, z) = An−1(x) + An−2(x) ∧ dz (9.40) the field strength can be expressed as

Fˆn = Fn + Fn−1 ∧ (dz + A1) (9.41) where A1 is the metric reduction. Thus we obtain that the D-dimensional field strength are given by Fn = dAn−1 − dAn−2 ∧ A1,Fn−1 = dAn−2 (9.42) we can then conclude that the field strengths after dimensional reduction will mean different things. For each reduction we do we obtain one field strength of same rank as the one we reduce from and the other field strength has (n − 1) rank. The m-th reduction for the gauge-potential can now be written as

a 1 a b Aˆ − = A − + A − ∧ dz + A − ∧ dz ∧ dz n 1 n 1 (n 2)a 2! (n 3)ab m,n∑ (9.43) 1 a a = A − − ∧ dz 1 ∧ ... ∧ dz p p! (n 1 p)a1...ap p=0 and the field strength can be written as m,n∑ 1 a a F = F − ∧ dz 1 ∧ ... ∧ dz p . n (n p)a1...ap (9.44) p=0 p! We have now established the dimensional reduction of field strengths and gauge-potentials. Let us now begin with finding the D-dimension Lagrangian for the toroidal compact- ifiation, we consider the (D + 1)-dimensional Lagrangian of the form where we follow the notations used in [11]

1 1 ˆ L =eR ˆ − eˆ(∂ϕ)2 − eeˆ aˆϕFˆ2 (9.45) 2 2n! n

38 doing a dimensional reduction on this Lagrangian gives us the following Lagrangian 1 1 L = eR − e(∂ϕ)2 − ee−2(D−1)αϕF 2 2 4 1 1 (9.46) − ee−2(n−1)αϕ+ˆaϕF 2 − ee2(D−n)αϕ+ˆaϕF 2 2n! n 2(n − 1)! n−1 where we have the 2-form defined as F = dA, so for each time we reduce the Lagrangian we obtain a new two-form field strength from the metric, and one dilaton scalar so there will be (D − 11) dilaton scalars. The dilaton factor of the field strength in (9.45) takes ⃗ aˆD+1·ϕD+1 ⃗ the general form of e where ϕD+1 = (ϕ1, ϕ2, . . . , ϕ10−D). We can see that there is no dilaton factor in the eleven-dimensional supergravity Lagrangian so we seta ˆ11 = 0, we can obtain the lower dimensional vectors ⃗a by the following recursion formula  ( √ )  −(n − 1), for Fˆ → Fˆ 2 n n ⃗aD = ⃗aD+1, x with x = (D − n), for Fˆ → F − . (D − 1)(D − 2)  n n 1 −(D − 1), for F (9.47) So for each reduction we obtain n-form field strength from the n-form in (D + 1) dimen- sion, an (n − 1)-form from the n-form field strength and the 2-form from the metric. We note that the 2-form appears in the D dimensional Lagrangian for the first time since there is no dilaton vectors in the eleven-dimensional supergravity theory we set ⃗aD+1 = 0, from this formula we can obtain all the factors for the dilaton vectors that appear when we do a dimensional reduction. One can easily calculate the prefactors using this formula. Let us denote the dilaton vector for the 4-form as FMNPQ, the 3-forms as FMNP i and the 2-forms as FMNij 1-forms as FMijk by ⃗a,⃗ai,⃗aij,⃗aijk where i labels the internal (11- F F j D) indices in D dimensions. We also obtain 2-forms MN i and 1-forms Mi with i < j which comes from the dimensional reduction of the vielbein. We denote the vielbein dilaton scalars with ⃗bi and ⃗bij, we summarize this with this table:

FMNPQ vielbein 4-form: ⃗a = −g 3-forms: ⃗ai = f⃗i − ⃗g 2-forms: ⃗aij = f⃗i + f⃗j − ⃗g ⃗bi = −f⃗i ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ 1-forms: ⃗aijk = fi + fj + fk − ⃗g bij = −fi + fj Table 2: Dilaton vector

where the vectors ⃗g and f⃗ have (11 − D) components in D dimensions and are given by

⃗g = 3(s1, s2, . . . , s11−D) (9.48) ⃗ − fi = (0|, 0{z..., 0}, (10 1)si, si+1, . . . , s11−D) (9.49) i−1 √ where si = 2/((10 − i)(9 − i)). We can now obtain the Lagrangian for the toroidal compactification of the eleven-dimensional supergravity theory. We write down the compact form of the elevel-dimensional Lagrangian, given as 1 1 L = eR − eFˆ2 + Fˆ ∧ Fˆ ∧ Aˆ (9.50) 48 4 6 4 4 the Chern-Simons term needs to be treated separately, which we will come back to. We can apply the rules above to the kinetic term where we consider a modification to the

39 equation (9.44), where we now include the dilaton vectors and order them as following i < j < k, which follows directly from the Table 2. The field strength dimensional reduction equation takes the general form

n∑−1 ∑ 1 2 1 2 (− n−1 ⃗g+f⃗ +...+f⃗ )ϕ⃗ F = (F − ) e 3 a1 ap (9.51) n! n (n − p)! (n p)a1...ap p=0 a1<...

The complete Lagrangian for the toroidal compactification of the eleven-dimensional supergravity theory is gives as ∑ ∑ 1 1 ·⃗ 1 ·⃗ 1 ·⃗ L = eR − e(∂ϕ⃗)2 − ee⃗a ϕF 2 − e⃗ai ϕ(F i)2 − e⃗aij ϕ(F ij)2 2 48 4 12 3 4 2 i i

We have now written out the toroidal compactified Lagrangian, let us now consider the Chern-Simons term. We begin with writing out the 3-form gauge potential amd the 4-form field strengths associated with the Chern-Simons term 1 1 Fˆ = F˜ + F˜i ∧ dzi − F˜ij ∧ dzi ∧ dzj − F˜ijk ∧ dzi ∧ dzj ∧ dzk (9.55) 4 4 3 2 2 6 1 1 1 Fˆ = F˜ + F˜l ∧ dzl − F˜lm ∧ dzl ∧ dzm − F˜lmn ∧ dzl ∧ dzm ∧ dzn (9.56) 4 4 3 2 2 6 1 1 1 Aˆ = A + Ap ∧ dzp − Apq ∧ dzp ∧ dzq − Apqr ∧ dzp ∧ dzq ∧ dzr . (9.57) 3 3 2 2 1 6 0

Note that we have Fˆ4 ∧Fˆ4 ∧Aˆ3, thus giving us 64 terms in total. The complete expression is written out in the appendix. We also note that the amount of indices the terms have will tell us what dimension the Lagrangian will have after the compactification on a torus. Before we go on let us define a important property of differential forms, namely

d(ωp ∧ ωq) = dωp ∧ ωq + (−1)pωp ∧ dωq (9.58)

When we do our calculations we set the right-hand side to zero this is due to having a eleven-dimensional theory. When we let a differential operator act on a form we increase its degree by one, which would result in a having p-forms of higher degree than our theory where our Lagrangian is written out as an 11-form. From a more technical point of view we see that letting a differential operator act on Fˆ4 ∧ Fˆ4 ∧ Aˆ3 will result in having Bianchi identities dFˆ4 = 0, hence why we set the right-hand side to zero. Let us begin with the simplest case where we have terms with 1 index, we write out the Lagrangian as ∫ ( ) 1 Ln=1 = F˜ ∧ F˜ ∧ Ai ∧ dzi + F˜ ∧ F˜i ∧ dzi ∧ A + F˜i ∧ dzi ∧ F˜ ∧ A . (9.59) FFA 6 4 4 2 4 3 3 3 4 3

40 The next step is to partial integrate the terms. Let us consider the second term where we now perform a partial integration and use the differential form property above ∫ ( ) ∫ ( ) ˜ ∧ ˜i ∧ i ∧ i ∧ ˜ ∧ ∧ i F4 F3 dz A3 = dA2 F4 A3 dz ∫ ( ) i ∧ ˜ ∧ ∧ i = A2 d(F4 A3) dz ∫ ( ) i ∧ ˜ ∧ ˜ ∧ ∧ i = A2 (dF4 A3 + F4 dA3) dz (9.60) ∫ ( ) i ∧ ˜ ∧ ˜ ∧ i = A2 F4 F4 dz ∫ ( ) i = F˜4 ∧ F˜4 ∧ A2 ε where we have used the differential form property in the first line, in the third line we see that the first term we obtain is zero due to the Bianchi identity. We are now left with something that has form as the first term. Applying the same procedure to the last term should give us the same terms as above, one could easily see from the similarities of the terms 6. We can now gather all the terms and obtain the following Lagrangian ∫ ( ) 1 Ln=1 = F˜ ∧ F˜ ∧ A + F˜ ∧ F˜ ∧ A + F˜ ∧ F˜ ∧ A εi FFA 6 4 4 2 4 4 2 4 4 2 ∫ ∫ (9.61) 1 1 = 3(F˜ ∧ F˜ ∧ A )εi = (F˜ ∧ F˜ ∧ A )εi . 6 4 4 2 2 4 4 2 We have now found the Chern-Simons term for D=10 Lagrangian. Note that we now have a Lagrangian of 10-form, we can thus conclude that each time we do a dimensional reduction we integrate over the coordinate in the toroidal direction which results in having one dimension less. We refer to the appendix for the rest of the dimensional re- ductions. Let us write down all the Lagrangian one obtains after dimensional reduction: 1 D = 10 : F˜4 ∧ F˜4 ∧ A2 , 2( ) 1 ˜ ∧ ˜ ∧ − 1 ˜i ∧ ˜j ∧ ij D = 9 : F4 F4 Aij1 F3 F3 A3 ε , (4 2 ) 1 1 1 D = 8 : F˜ ∧ F˜ ∧ Aijk − F˜i ∧ F˜j ∧ Ak + F˜i ∧ F˜jk ∧ A εijk , 12 4 4 0 6 3 3 2 2 3 2 3 ( ) 1 1 1 D = 7 : F˜ ∧ F i ∧ Ajkl − F˜i ∧ F˜j ∧ Akl + F˜ij ∧ F˜kl ∧ A εijkl , 6 4 3 0 4 3 3 2 8 2 2 3 ( ) 1 ˜ ∧ ˜ij ∧ klm − 1 ˜i ∧ ˜j ∧ klm 1 ˜ij ∧ ˜kl ∧ m ijklm D = 6 : F4 F2 A F3 F3 A0 + F2 F2 A2 ε , (12 12 8 ) 1 ˜i ∧ ˜jk ∧ lmn 1 ˜ij ∧ ˜kl ∧ mn − 1 ˜ijk ∧ ˜lmn ∧ ijklmn D = 5 : F3 F2 A0 + F2 F2 A1 F1 F1 A3 ε , (12 48 )72 1 1 D = 4 : F˜ij ∧ F˜kl ∧ Amnp − F˜ijk ∧ F˜lmn ∧ Ap εijklmnp , 48 2 2 0 72 1 1 2 1 D = 3 : − F˜ijk ∧ F˜lmn ∧ Apqεijklmnpq , 144 1 1 1 1 D = 2 : − F˜ijk ∧ F˜lmn ∧ Apqrεijklmnpqr . 1296 1 1 0 These are all the reduced Lagrangian we can obtain from the D = 11 Chern-Simons term. 6 ˜ ˜i One could choose to partial integrate the first term and obtain something similar to F4 ∧ F3 ∧ A3 but we chose to partial integrate the other two terms to obtain the maximally ranked field-strengths in our Lagrangian.

41 These reductions are from the unmodified field strengths, i.e the field strength re- duction (9.55 − 9.57) are not expressed in the vielbein basis. However if one were to consider dualisations as [10] one would need modified field strengths which are expressed in vielbein basis. Let us re express the toroidal coordinates dzi in terms of

i i Ai Aij j h = dz + 1 + 0 dz . (9.62)

We obtain this from the dimensional reduction of the metric ansatz ∑ 1 2 ⃗g·ϕ⃗ 2 2⃗γi·ϕ⃗ i 2 ds11 = e 3 dsD + e (h ) (9.63) i

1 − 1 ⃗ where γi = 6⃗g 2 fi. We have considered the field strengths in the Chern-Simons term to be associated with the gauge potentials e.g F4 = dA3 and so one, in general one would also obtain non-linear Kaluza-Klein modifications. If we express the field strength reductions in terms of (9.62), we obtain the following modified field strengths 1 1 F = F˜ − γijF˜i ∧ Aj − γikγjlF˜ij ∧ Ak ∧ Al + γilγjmγknF˜ijk ∧ Al ∧ Am ∧ An 4 3 1 2 2 1 1 6 1 1 1 1 1 F i = γijF˜j − γijγklF˜jk ∧ Al − γijγkmγlnF˜jkl ∧ Am ∧ An 3 3 2 1 2 1 1 1 ij ki lj ˜kl − ki lj mn ˜klm ∧ An F2 = γ γ F2 γ γ γ F1 1 ijk li mj nk ˜klm F1 = γ γ γ F1 (9.64) these modified field strengths appear in the kinetic term of the Lagrangian (9.54). The modified field strengths that come from the vielbein are given by

F i = F i − γjkF ij ∧ Ak 2 2 1 1 (9.65) F ij kjF ik 1 = γ

Aij Notice that we do not have any 0-form potentials, this is because 0 are defined only Aij ≥ for i < j, in other words 0 = 0 for i j. This can be seen when we to dimensional reduction from the Kaluza-Klein vielbein ansatz by taking the i’th step of the reduction process. We obtain the following relation

i il l − Al dz = γ (h 1) (9.66) one then sees that the 0-form potentials are eliminated7, notice that γij = 0 when i > j and γij = 1 for i = j. If one were to consider brane’s we would need to have these modified Chern-Simons terms in the kinetic term. For example in [11] the authors attempt to find p-brane solutions in maximal su- pergravity, where they find the equations of motion of the supergravity Lagrangian and impose additional constraints on the Chern-Simons modifications of the field strengths. These constraints gives rise to which field strengths can be dualised in certain dimen- sions.

7A detailed mathematical description can be found in [11]

42 10 Conclusion

In this thesis we have seen how Supergravity theories are invariant under the local SUSY transformations where the spinor parameters ε(x) are arbitrary functions of spacetime coordinates. Showing that the Lagrangians are invariant tells us that Supergravity is consistent as a classical theory of graviton and gravitino. In section 5 we saw how one obtained a theory containing gravity, electromagnetic fields and scalar fields from a (D + 1)-dimensional Einstein-Hilbert action by using Kaluza-Klein theory. This method turns out to be very important when considering Supergravity theories. When we performed a dimensional reductions of the eleven- dimensional Supergravity theory on a 7-torus, we obtained the degrees of freedom of D = 4, N = 8 theory. The theory contains one graviton gµν, seven vectors gµi and 28 scalars gij. The Majorana field is decomposed into eight gravitinos and 56 spin-1/2 fields this is obtained from calculating the states for the Rarita-Schwinger field. We know from Supersymmetry that a theory should contain equal amount of bosonic and fermionic states on-shell degrees of freedom, so we are missing bosons to account for the fermions. These bosons are obtained from the 3-form gauge potential Aµij which supplies 21 scalars. The form components Aijk contain 35 scalars and the components Aµνi give additional seven scalars. This completes the field content of the D = 4, N = 8 theory. We also saw that we obtained ten-dimensional Supergravity theories, Type IIA and Type IIB which are low-energy limits of superstring theories of the same name.

Type IIB and and SO(6)-gauged D = 5 supergravities have important applications in AdS/CFT-correspondence. The modified field strengths appear in the dimensional re- duction of the Chern-Simons term in the eleven-dimensional supergravity. The modified field strengths are needed to guarantee gauge invariance in the kinetic term. When one considers external sources such as brane worlds one needs the modified field strengths. With the modified field strengths one can consider dualisations of higher dimensional theories to lower dimensional theories. The eleven-dimensional Supergravity theory com- bined with M2 and M5-brane solutions is the basis for M-theory, which is the theory that contains all the various string theories.

Acknowledgments

I would like to thank Giuseppe for his guidance and the useful discussions during our meetings. I would also like to thank Johan Asplund and Daniel Neiss for taking their time to go through some of the calculations.

References

[1] F.Quevedo, S.Krippendorf, O.Schlotter Cambrigde lectures on supersym- metry and extra dimensions arXiv:1011.1491

[2] A. Van Proeyen Tools for supersymmetry arXiv:hep-th/9910030v6

[3] P.Binétruy Supersymmetry Oxford Graduate texts

[4] D.Z.Freedman, A. Van Proeyen Supergravity Cambridge university press (2012)

[5] M.Perry Applications of differential geometry to physics (2009)

43 [6] C.N Pope Kaluza-Klein theory people.physics.tamu.edu/pope/

[7] H. Samtleben Introduction to Supergravity http://www.itp.uni- hannover.de/saalburg/Lectures/samtleben.pdf

[8] E. Cremmer, B. Julia, J. Scherk Supergravity theory in 11 dimensions Phys. Lett. B76 1978

[9] S. Naito, K. Osada, T. Fukui Fierz identities and invariance of 11- dimensional supergravity action Phys. Rev D 1986

[10] E. Cremmer, B.Julia, H.Lü, C.N Pope Dualisation of Dualities arXiv:hep- th/9710119v2

[11] H.Lü, C.N Pope p-brane Solitons in Maximal Supergravities arXiv:hep- th/951201v1

44 A Riemann tensor for Einstein-Hilbert action in D dimensions

The derivatives of the ON-Basis are defined as

i −βϕ j i i j dE = βe ∂jE ∧ E − ωˆ j ∧ E (A.1) 1 dE5 = αe−βϕ∂ ϕEj ∧ E5 + e(α−2β)ϕF Ei ∧ Ej . (A.2) j 2 ij The spin connection can be obtain by using Cartan’s first structure equation and we obtain 1 ω5 = α∂ ϕe−βϕE5 + e(α−2β)ϕF Ej (A.3) i i 2 ij 1 ωi =ω ˆi − βe−βϕ(∂ ϕE − ∂ ϕEi) − e(α−2β)ϕF i E5 (A.4) j j i j j 2 j Cartan’s second structure equation is defined as

µ µ µ ν R σ = dω σ + ω ν ∧ ω σ (A.5)

5 µ Let us now find the Riemann tensor R i, we start with the first component dω σ 1 dω5 = d(α∂ ϕe−2βϕE5 + e(α−2β)ϕF Ej) i i 2 ij −2βϕ j 5 −2βϕ 5 −βϕ 5 = α∂j∂iϕe E ∧ E − αβ∂iϕe dϕE + α∂iϕe dE 1 + (dF e(α−2β)ϕEj + (α − 2β)F dϕe(α−2β)ϕEj + e(α−2β)ϕF dEj 2 ij ij ij −2βϕ j 5 −2βϕ j 5 = α∂j∂iϕe E ∧ E − αβ∂iϕe ∂jϕE ∧ E 1 + α∂ ϕe−βϕ(αe−βϕ∂ ϕEj ∧ E5 + e(α−2β)ϕF Ei ∧ Ej) i j 2 ij 1 1 + ∂ ϕF e(α−3β)ϕEk ∧ Ej + (α − 2β)F ∂ ϕe(α−3β)ϕEk ∧ Ej 2 k ij 2 ij k 1 + e(α−2β)ϕF (βe−βϕ∂ Ek ∧ Ej − ωˆj ∧ Ek) 2 ij k k −2βϕ j 5 −2βϕ j 5 = α∂j∂iϕe E ∧ E − αβ∂iϕe ∂jϕE ∧ E 1 + α2∂ ϕ∂ ϕe−2βϕEj ∧ E5 + α∂ ϕF e(α−3β)ϕEi ∧ Ej i j 2 i ij 1 1 + ∂ F e(α−3β)ϕEk ∧ Ej + (α − 2β)F ∂ ϕe(α−3β)ϕEk ∧ Ej 2 k ij 2 ij k 1 + β∂ ϕF e(α−3β)ϕEk ∧ Ej − e(α−2β)ϕF ωˆj ∧ Ek 2 k ij ij k Now we consider the second part of the equation above and we obtain

5 j −βϕ 5 j j −2βϕ 5 j ω j ∧ ω i = α∂jϕe E ∧ ωˆ i − αβ∂jϕ∂ ϕe E ∧ E 1 + αβ∂ ϕ∂ ϕe−2βϕE5 ∧ Ej + e(α−2β)ϕF Ek ∧ ωˆj j i 2 kj i 1 1 + βe(α−3β)ϕ∂ ϕF Ek ∧ Ej − e2(α−2β)ϕF k F k Ej ∧ E5 2 k ij 4 j i 1 + βη F ∂lϕe(α−3β)ϕEk ∧ Ej 2 i[j k]l

45 combining all the equations now we obtain the following Riemann tensor

z −2βϕ j 5 −βϕ j 5 2 −2βϕ j 5 R i = α∂j∂iϕe E ∧ E − αβ∂iϕe ∂jϕE ∧ E + α ∂iϕ∂jϕe E ∧ E 1 1 + α∂ ϕF e(α−3β)ϕEi ∧ Ej + βη F ∂lϕe(α−3β)ϕEk ∧ Ej 2 i ij 2 i[j k]l 1 1 + ∂ F e(α−3β)ϕEk ∧ Ej + (α − 2β)F ∂ ϕe(α−3β)ϕEk ∧ Ej 2 k ij 2 ij k 1 + β∂ ϕF e(α−3β)ϕEk ∧ Ej − e(α−2β)ϕF ωˆj ∧ Ek 2 k ij ij k −βϕ 5 j j −2βϕ 5 i + α∂jϕe E ∧ ωˆ i − αβ∂jϕ∂ ϕe E ∧ E 1 − αβ∂ ϕ∂ ϕe−2βϕE5 ∧ Ej + e(α−2β)ϕF Ek ∧ ωˆj j i 2 kj i 1 1 + βe(α−3β)ϕ∂ ϕF Ek ∧ Ej − e2(α−2β)ϕF j F j Ej ∧ E5 2 k ij 4 i i simplify this by gather all the wedge products under same brackets we obtain ( ) 1 5 −2βϕ k 2(α−2β)ϕ ki j 5 R i = e α(α − 2β)∂iϕ∂jϕ + α∂j∂iϕ + ηijαβ∂kϕ∂ ϕ − e FkjF E ∧ E ( 4 ) 1 1 1 1 + e(α−3β)ϕ (α − β)∂ ϕF − (α − β)∂ ϕF − ∂ F + βη F ∂lϕ Ek ∧ Ej 2 i kj 2 [k j]i 2 [k j]i 2 i[j k]l ( ) 1 − e(α−3β)ϕ F ωˆj ∧ Ek + F ωˆj ∧ Ek + α∂ ϕe−βϕE5 ∧ ωˆj 2 ij k jk i j i

i We have another curvature tensor to calculate namely R j we follow the same procedure as above i i i ν R j = dω j + ω ν ∧ ω j We start with the first component which yields us

i −2βϕ k −2βϕ k i 2 −2βϕ i k dω j =βe ∂i∂kϕE ∧ Ej − βe ∂kj∂kϕE ∧ E + β e ∂ ∂kϕE ∧ Ej −βϕ i j k 2 −2βϕ j i −βϕ i j − βe ∂ ϕωˆ k ∧ E − β e ∂jϕ∂jϕE ∧ E + βe ∂jϕωˆ j ∧ E 1 − β2e−2βϕ∂ ϕ∂ ϕEk ∧ E + β2e−2βϕ∂ ϕ∂ ϕEk ∧ Ei + ∂ ϕF i e(α−2β)ϕEk ∧ E5 i k j j k 2 k j 1 1 1 − (α − 2β)∂ F i e(α−3β)ϕEk ∧ E5 + e(α−3β)ϕα∂ ϕF i Ej ∧ E5 − F k F e2(α−2β)ϕEi ∧ Ej 2 k j 2 j j 4 j kj the second term i ν i 5 i k ω ν ∧ ω j = ω 5 ∧ ω j + ω k ∧ ω j Which yields us the following expression without going through the entire calculation 1 1 ωi ∧ ων = α∂ ϕF e(α−3β)ϕE5 ∧ Ek + F α∂ ϕe(α−3β)ϕEj ∧ E5 ν j 2 i jk 2 ij j 1 + F F e2(α−2β)ϕEj ∧ Ek +ω ˆi ∧ ωˆk − β∂kϕe−βϕωˆi ∧ E 4 ij jk k j k j −βϕ i k 2 i k −2βϕ 2 i −2βϕ k k + β∂jϕe ωˆ k ∧ E + β ∂ ϕ∂ ϕe Ek ∧ Ej − β ∂ ϕ∂jϕe E ∧ E 2 k −2βϕ i 2 −2βϕ i k − β ∂kϕ∂ ϕe E ∧ Ej + β ∂kϕ∂jϕe E ∧ E 1 1 + β∂iϕF k e(α−3β)ϕE ∧ E5 − β∂ ϕF k e(α−3β)ϕEi ∧ E5 2 j k 2 k j 1 1 1 − F i e(α−2β)ϕE5 ∧ ωˆk + β∂kϕF i e(α−3β)ϕE5 ∧ E − β∂ ϕF i E5 ∧ Ek 2 k j 2 k j 2 j k

46 Gathering all the terms we obtain the following Riemann tensor ( 1 1 Ri =ri + Ez ∧ Eke(α−3β)ϕ (α − β)F i ∂ ϕ + ∂ F i − (α − β)(∂iϕF − F i ∂ ϕ) j j j k 2 k j 2 jk k j ) ( ( ) 1 ( ) + β ∂ ϕF n δi + F i ∂lϕη + Ek ∧ Ele−2βϕ β ∂ ∂ ϕδi − ∂ ∂iϕη 2 n j k l jk j [k l] [k l]j ( ) ( ) 1 2(α−2β)ϕ i i 2 i p i i − e F jFkl + F [k|Fj|l] + β ∂ ϕ∂[kϕηl]j − ∂pϕ∂ ϕδ [kηl]j − ∂jϕ∂[kϕδ l] 4 ( ) ( ) 1 + βe−βϕ ∂ ϕEi ∧ ωˆk − ∂kϕωˆi ∧ E − βe(α−2β)ϕ + F k ωˆi ∧ E5 − F i ωˆk ∧ E5 k j k j 2 j k k j B Equations of motion for Einstein-Hilbert action in D dimen- sions

The derivation for the equations of motion for Einstein-Hilbert action in D dimensions where the action is given as ∫ ( ) √ 1 1 S = κ−1 dDx −g R − (∇ϕ)2 − e−2(D−1)αϕF 2 (B.1) EH 2 4 we vary this action with respect to the metric gµν ∫ ( ) √ 1√ 1√ δS = κ−1 dDxδ −gR − −g(∇ϕ)2 − −ge−2(D−1)αϕF 2 EH 2 4 ( ∫ √ √ √ −1 D µν µν = κ d x −gδRµνg + −gRµνδg + Rδ −g 1√ 1√ − −g(∂ δϕ∂νϕ + ∂ ϕ∂νδϕ) − −g(∇ϕ)2δgµν 2 µ µ 2 1 √ 1√ − (∂ϕ)2δ −g − −ggµνe−2(D−1)αϕ(δF F µν + F δF µν) 2 4 µν µν √ √ − 1 − 2 −2(D−1)αϕ µν 1 2 −2(D−1)αϕ − gFµνe δg + F e δ g 4 4 ) 1√ + −g(D − 1)αδϕe−2(D−1)αϕF 2 2

ν ν µν µν we set ∂µδϕ∂ ϕ = ∂µϕ∂ δϕ and δFµνF = FµνδF and all the boundary terms are set to zero. Using the relation √ 1√ δ −g = − −gg δgµν (B.2) 2 µν and integrating by parts we obtain

47 ∫ ( √ 1√ √ δS = κ−1 dDx −gR δgµν − −gRg δgµν − −g∂ δϕ∂νϕ EH µν 2 µν µ 1√ 1√ − −g∂ ϕ∂νϕδgµν + −g(∂ϕ)2g δgµν 2 µ 4 µν √ √ − 1 − µν ∇µ −2(D−1)αϕ − 1 − 2 −2(D−1)αϕ µν gg Fµνδ Aνe gFµνe δg 2 4 ) 1√ 1√ − −gg F 2e−2(D−1)αϕδgµν + −g(D − 1)αδϕe−2(D−1)αϕF 2 8 µν 2 ∫ ( ( √ 1 1 1 = κ−1 dDx −g δgµν R − Rg − (∂ ϕ∂νϕ + (∂ϕ)2g µν 2 µν 2 µ 2 µν ) 1 1 − e−2(D−1)αϕ(F 2 − F 2 g ) 4 µν 4 µν µν ( ) µ −2(D−1)αϕ − δAν∇ Fµνe ( ) 1 − δϕ ∂µ∂νϕ − (D − 1)αe−2(D−1)αϕF 2 . 2 The equations of motion can be read of as ( ) ( ) − 1 1 − 1 2 1 −2(D−1)αϕ 2 − 1 2 Rµν Rgµν = ∂µϕ∂νϕ (∂ϕ) gµν + e Fµν F gµν ( 2 ) 2 2 2 4 µ −2(D−1)αϕ ∇ e Fµν = 0 1 2ϕ = − (D − 1)αe−2(D−1)αϕF 2 2 (B.3)

C Covariant derivative of γν

1 ∇ γ = ∂ γ + ω ab[γ , γ ] − Γρ γ µ ν µ ν 4 µ ab ν µν ρ 1 = ∂ γ + ω ab[γ γ , γ ] − Γρ γ µ ν 4 µ [a b] ν µν ρ 1 = ∂ γ + ω ab([γ γ , γ ] − (a ↔ b)) − Γρ γ µ ν 8 µ a b ν µν ρ 1 = ∂ γ + ω ab(γ γ γ − γ γ γ ) − (a ↔ b)) − Γρ γ µ ν 8 µ a b ν ν a b µν ρ 1 (C.1) = ∂ γ + ω ab(γ γ γ − γ γ γ − 2η γb − (a ↔ b)) − Γρ γ µ ν 8 µ a b ν a ν b aν µν ρ 1 = ∂ γ + ω ab(γ γ γ − γ γ γ − 2η γb + 2η γa − (a ↔ b)) − Γρ γ µ ν 8 µ a b ν a b ν aν bν µν ρ 1 = ∂ γ + ω ab(2γaη − 2γbη − (a ↔ b)) − Γρ γ µ ν 8 µ bν aν µν ρ ab [a b]ν − ρ = ∂µγν + ωµ γ η Γµνγρ a ab b − ρ = γ (∂µeaν + ωµ e ν Γµνeaρ) = 0

D Variation of the spin connection

We are interested in finding the variation of the spin connection which will be useful when doing calculations in the second order formalism of gravity and fermions. We start by varying the first Cartan structure equation a a b a b dδe + ω b ∧ δe + δω b ∧ e = 0

48 c Let us multiply with ηace ρ and do a cyclic permutation of µ, ν and ρ with the sign + − + ( ) a c ab c ηac D[µδν]e ρ + δω[µ eν]e ρ ( ) − a c ab c ηac D[νδρ]eµ + δω[ν eρ]eµ ( ) a c ab c ηac D[ρδµ]eν + δω[ρ eµ]eν = 0 writing out the explicit terms of the spin connection we obtain ( ) a c − 1 ab c 1 ab c ηac D[µδν]e ρ δωµ eνbe ρ + δωµ eµbe ρ ( 2 2 ) − a c − 1 ab c 1 ab c ηac D[νδρ]eµ δων eρbeµ + δωρ eνbeµ ( 2 2 ) 1 1 η D δa ec − δω abe ec + δω abe ec = 0 ac [ρ µ] ν 2 ρ µb ν 2 µ ρb ν we see that certain terms vanish and we have contracted the metric so all the c are replaced with a and we obtain b a ab a a − a a a a e ρe νδωµ = D[µδν]e ρ D[νδρ]e µ + D[ρδµ]e ν (D.1)

E Chern-Simons dimensional reduction ( ) ˆ ∧ ˆ ∧ ˆ ˜ ˜i ∧ i − 1 ˜ij ∧ i ∧ j − 1 ˜ijk ∧ i ∧ j ∧ k F4 F4 A3 = F4 + F3 dz F2 dz dz F1 dz dz dz ( 2 6 ) ∧ ˜ ˜l ∧ l − 1 ˜lm ∧ l ∧ m − 1 ˜lmn ∧ l ∧ m ∧ n F4 + F3 dz F2 dz dz F1 dz dz dz ( 2 6 ) 1 1 ∧ A + Ai ∧ dzi − Aij ∧ dzi ∧ dzj − Aijk ∧ dzi ∧ dzj ∧ dzk 3 2 2 1 6 0 (E.1) In order to obtain the dimensional reductions from eleven-dimensional supergravity to four-dimensional supergravity, we need to expand the expression above and we obtain a very messy expression. In order to avoid having a expression that is to long we will break it into 4 parts, for the simplicity. 1 Fˆ ∧ Fˆ ∧ Aˆ = F˜ ∧ F˜ ∧ A + F˜ ∧ F˜ ∧ Ai ∧ dzi − F˜ ∧ F˜ ∧ Aij ∧ dzi ∧ dzj 4 4 3 4 4 3 4 4 2 2 4 4 1 1 − F˜ ∧ F˜ ∧ Aijk ∧ dzi ∧ dzj ∧ dzk + F˜ ∧ F˜l ∧ dzl ∧ A 6 4 4 0 4 3 3 1 + F˜ ∧ F˜l ∧ dzl ∧ Ai ∧ dzi − F˜ ∧ F˜l ∧ dzl ∧ Aij ∧ dzi ∧ dzj 4 3 2 2 4 3 1 1 1 − F˜ ∧ F˜l ∧ dzl ∧ Aijk ∧ dzi ∧ dzj ∧ dzk − F˜ ∧ F˜lm ∧ dzl ∧ dzm ∧ A 6 4 3 0 2 4 2 3 1 1 − F˜ ∧ F˜lm ∧ dzl ∧ dzm ∧ Ai ∧ dzi + F˜ ∧ F˜lm ∧ dzl ∧ dzm ∧ Aij ∧ dzi ∧ dzj 2 4 2 2 4 4 2 1 1 + F˜ ∧ F˜lm ∧ dzl ∧ dzm ∧ Aijk ∧ dzi ∧ dzj ∧ dzk 12 4 2 0 1 − F˜ ∧ F˜lmn ∧ dzl ∧ dzm ∧ dzn ∧ A 6 4 1 3 1 − F˜ ∧ F˜lmn ∧ dzl ∧ dzm ∧ dzn ∧ Ai ∧ dzi 6 4 1 2 1 + F˜ ∧ F˜lmn ∧ dzl ∧ dzm ∧ dzn ∧ Aij ∧ dzi ∧ dzj 12 4 1 1 1 + F˜ ∧ F˜lmn ∧ dzl ∧ dzm ∧ dzn ∧ Aijk ∧ dzi ∧ dzj ∧ dzk 36 4 1 0

49 this is the expression for the first extensions that contains 16 terms, we see that we should in total have 64 terms in total. The next extensions are expressed as

1 Fˆ ∧ Fˆ ∧ A = F˜i ∧ dzi ∧ F˜ ∧ A + F˜i ∧ dzi ∧ F˜ ∧ Ai ∧ dzi − F˜i ∧ dzi ∧ F˜ ∧ Aij ∧ dzi ∧ dzj 4 4 3 3 4 3 3 4 2 2 3 4 1 1 − F˜i ∧ dzi ∧ F˜ ∧ Aijk ∧ dzi ∧ dzj ∧ dzk + F˜i ∧ dzi ∧ F˜l ∧ dzl ∧ A 6 3 4 0 3 3 3 1 + F˜i ∧ dzi ∧ F˜l ∧ dzl ∧ Ai ∧ dzi − F˜i ∧ dzi ∧ F˜l ∧ dzl ∧ Aij ∧ dzi ∧ dzj 3 3 2 2 3 3 1 1 1 − F˜i ∧ dzi ∧ F˜l ∧ dzl ∧ Aijk ∧ dzi ∧ dzj ∧ dzk − F˜i ∧ dzi ∧ F˜lm ∧ dzl ∧ dzm ∧ A 6 3 3 0 2 3 2 3 1 1 − F˜i ∧ dzi ∧ F˜lm ∧ dzl ∧ dzm ∧ Ai ∧ dzi + F˜i ∧ dzi ∧ F˜lm ∧ dzl ∧ dzm ∧ Aij ∧ dzi ∧ dzj 2 3 2 2 4 3 2 1 1 + F˜i ∧ dzi ∧ F˜lm ∧ dzl ∧ dzm ∧ Aijk ∧ dzi ∧ dzj ∧ dzk 12 3 2 0 1 1 − F˜i ∧ dzi ∧ F˜lmn ∧ dzl ∧ dzm ∧ dzn ∧ A − F˜i ∧ dzi ∧ F˜lmn ∧ dzl ∧ dzm ∧ dzn ∧ Ai ∧ dzi 6 3 1 3 6 3 1 2 1 + F˜i ∧ dzi ∧ F˜lmn ∧ dzl ∧ dzm ∧ dzn ∧ Aij ∧ dzi ∧ dzj 12 3 1 1 1 + F˜i ∧ dzi ∧ F˜lmn ∧ dzl ∧ dzm ∧ dzn ∧ Aijk ∧ dzi ∧ dzj ∧ dzk 36 3 1 0

1 1 Fˆ ∧ Fˆ ∧ A = F˜ij ∧ dzi ∧ dzj ∧ F˜ ∧ A − F˜ij ∧ dzi ∧ dzj ∧ F˜ ∧ Ai ∧ dzi 4 4 3 2 2 4 3 2 2 4 2 1 + F˜ij ∧ dzi ∧ dzj ∧ F˜ ∧ Aij ∧ dzi ∧ dzj 4 2 4 1 1 1 + F˜ij ∧ dzi ∧ dzj ∧ F˜ ∧ Aijk ∧ dzi ∧ dzj ∧ dzk − F˜ij ∧ dzi ∧ dzj ∧ F˜l ∧ dzl ∧ A 12 2 4 0 2 2 3 3 1 1 − F˜ij ∧ dzi ∧ dzj ∧ F˜l ∧ dzl ∧ Ai ∧ dzi + F˜ij ∧ dzi ∧ dzj ∧ F˜l ∧ dzl ∧ Aij ∧ dzi ∧ dzj 2 2 3 2 4 2 3 1 1 + F˜ij ∧ dzi ∧ dzj ∧ F˜l ∧ dzl ∧ Aijk ∧ dzi ∧ dzj ∧ dzk 12 2 3 0 1 1 + F˜ij ∧ dzi ∧ dzj ∧ F˜lm ∧ dzl ∧ dzm ∧ A + F˜ij ∧ dzi ∧ dzj ∧ F˜lm ∧ dzl ∧ dzm ∧ Ai ∧ dzi 4 2 3 3 4 2 3 2 1 − F˜ij ∧ dzi ∧ dzj ∧ F˜lm ∧ dzl ∧ dzm ∧ Aij ∧ dzi ∧ dzj 8 2 3 1 1 − F˜ij ∧ dzi ∧ dzj ∧ F˜lm ∧ dzl ∧ dzm ∧ Aijk ∧ dzi ∧ dzj ∧ dzk 24 2 3 0 1 − F˜ij ∧ dzi ∧ dzj ∧ F˜lmn ∧ dzl ∧ dzm ∧ dzn ∧ A 12 1 3 1 − F˜ij ∧ dzi ∧ dzj ∧ F˜lmn ∧ dzl ∧ dzm ∧ dzn ∧ Ai ∧ dzi 12 1 2 1 − F˜ij ∧ dzi ∧ dzj ∧ F˜lmn ∧ dzl ∧ dzm ∧ dzn ∧ Aij ∧ dzi ∧ dzj 24 1 1 1 − F˜ij ∧ dzi ∧ dzj ∧ F˜lmn ∧ dzl ∧ dzm ∧ dzn ∧ Aijk ∧ dzi ∧ dzj ∧ dzk 72 1 0

50 1 1 Fˆ ∧ Fˆ ∧ A = − F˜ijk ∧ dzi ∧ dzj ∧ dzk ∧ F˜ ∧ A − F˜ijk ∧ dzi ∧ dzj ∧ dzk ∧ F˜ ∧ Ai ∧ dzi 4 4 3 6 1 4 3 6 1 4 2 1 + F˜ijk ∧ dzi ∧ dzj ∧ dzk ∧ F˜ ∧ Aij ∧ dzi ∧ dzj 12 1 4 1 1 + F˜ijk ∧ dzi ∧ dzj ∧ dzk ∧ F˜ ∧ Aijk ∧ dzi ∧ dzj ∧ dzk 36 1 4 0 1 1 − F˜ijk ∧ dzi ∧ dzj ∧ dzk ∧ F˜l ∧ dzl ∧ A − F˜ijk ∧ dzi ∧ dzj ∧ dzk ∧ F˜l ∧ dzl ∧ Ai ∧ dzi 6 1 3 3 6 1 3 2 1 + F˜ijk ∧ dzi ∧ dzj ∧ dzk ∧ F˜l ∧ dzl ∧ Aij ∧ dzi ∧ dzj 12 1 3 1 1 + F˜ijk ∧ dzi ∧ dzj ∧ dzk ∧ F˜l ∧ dzl ∧ Aijk ∧ dzi ∧ dzj ∧ dzk 36 1 3 0 1 + F˜ijk ∧ dzi ∧ dzj ∧ dzk ∧ F˜lm ∧ dzl ∧ dzm ∧ A 12 1 3 3 1 + F˜ijk ∧ dzi ∧ dzj ∧ dzk ∧ F˜lm ∧ dzl ∧ dzm ∧ Ai ∧ dzi 12 1 3 2 1 − F˜ijk ∧ dzi ∧ dzj ∧ dzk ∧ F˜lm ∧ dzl ∧ dzm ∧ Aij ∧ dzi ∧ dzj 24 1 3 1 1 − F˜ijk ∧ dzi ∧ dzj ∧ dzk ∧ F˜lm ∧ dzl ∧ dzm ∧ Aijk ∧ dzi ∧ dzj ∧ dzk 72 1 3 0 1 + F˜ijk ∧ dzi ∧ dzj ∧ dzk ∧ F˜lmn ∧ dzl ∧ dzm ∧ dzn ∧ A 36 1 3 3 1 + F˜ijk ∧ dzi ∧ dzj ∧ dzk ∧ F˜lmn ∧ dzl ∧ dzm ∧ dzn ∧ Ai ∧ dzi 36 1 3 2 1 − F˜ijk ∧ dzi ∧ dzj ∧ dzk ∧ F˜lmn ∧ dzl ∧ dzm ∧ dzn ∧ Aij ∧ dzi ∧ dzj 72 1 3 1 1 − F˜ijk ∧ dzi ∧ dzj ∧ dzk ∧ F˜lmn ∧ dzl ∧ dzm ∧ dzn ∧ Aijk ∧ dzi ∧ dzj ∧ dzk . 216 1 3 0

Continue with terms with 2 indices ∫ ( 1 Ln=2 = − F˜ ∧ F˜ ∧ Aij ∧ dzi ∧ dzj + F˜ ∧ F˜ ∧ dzi ∧ Aj ∧ dzj FFA 2 4 4 1 4 3 2 − 1 ˜ ∧ ˜ij ∧ i ∧ j ∧ ˜i ∧ i ∧ ˜ ∧ j ∧ j F4 F2 dz dz A3 + F3 dz F4 A2 dz (E.2) 2 ) 1 + F˜i ∧ dzi ∧ F˜j ∧ dzj ∧ A − F˜ij ∧ dzi ∧ dzj ∧ F˜ ∧ A 3 3 3 2 2 4 3

rearranging the terms using the property of changing sign each time we run a p-form through another q-form, we obtain ∫ ( 1 Ln=2 = − F˜ ∧ F˜ ∧ Aij ∧ dzi ∧ dzj − F˜ ∧ F˜i ∧ Aj ∧ dzi ∧ dzj FFA 2 4 4 1 4 3 2 − 1 ˜ ∧ ˜ij ∧ ∧ i ∧ j − ˜ ∧ ˜i ∧ j ∧ i ∧ j F4 F2 A3 dz dz F4 F3 A2 dz dz (E.3) 2 ) 1 − F˜i ∧ F˜j ∧ A ∧ dzi ∧ dzj + F˜ij ∧ F˜ ∧ A ∧ dzi ∧ dzj 3 3 3 2 2 4 3

Note that the factors in front of the various terms can help us see what type of term we should obtain after integration. Consider the first term we see that we have 4 ∧ 4 ∧ 1 i.e we have four “leg” in the first and second field strength and one “leg” in the gauge potential. So we should perform partial integration on the terms with the same factor

51 in front but different setup of legs in each term 8.

− ˜ ∧ ˜i ∧ j ∧ i ∧ j − ∧ i ∧ ˜j ∧ i ∧ i ∧ j F4 F3 A2 dz dz = dA3 A2 F3 A2 dz dz − ∧ ˜i ∧ j ∧ i ∧ j = A3 d(F3 A2) dz dz − ∧ ˜i ∧ ˜j ∧ i ∧ j = A3 F3 F3 dz dz − ˜i ∧ ˜j ∧ ∧ i ∧ j = F3 F3 A3 dz dz we see that after partial integration we obtain a term that has the same form as the fifth term in the integral, doing a partial integration on the term that has a 1/2 factor in front gives us 1 1 − F˜ ∧ F˜ij ∧ A ∧ dzi ∧ dzj = − F˜ ∧ dAij ∧ A ∧ dzi ∧ dzj 2 4 2 3 2 4 1 3 1 = + Aij ∧ d(F˜ ∧ A ) ∧ dzi ∧ dzj 2 1 4 3 1 = − Aij ∧ F˜ ∧ F˜ ∧ dzi ∧ dzj 2 1 4 4 1 = − F˜ ∧ F˜ ∧ Aij ∧ dzi ∧ dzj . 2 4 4 1 We have now performed partial integration on two terms with different setup of legs, the same procedure applies to the other terms and we obtain

∫ ( ) 1 3 Ln=2 = − F˜ ∧ F˜ ∧ Aij − 3F˜i ∧ F˜j ∧ A dzi ∧ dzj FFA 6 2 4 4 1 3 3 3 ∫ ( ) (E.4) 1 1 = − F˜ ∧ F˜ ∧ Aij − F˜i ∧ F˜j ∧ A εij 4 4 4 1 2 3 3 3 thus we have found the D = 9 C-S term. We can continue with this procedure and obtain dimensional reductions all the way down to D = 2. Let us continue with 3 indices which should yield us a D = 8 C-S term. Gathering all the terms now with 3 indices gives us the following Lagrangian ∫ ( 1 1 1 Ln=3 = − F˜ ∧ F˜ ∧ Aijk ∧ dzi ∧ dzj ∧ dzk − F˜ ∧ F i ∧ dzi ∧ Ajk ∧ dzj ∧ dzk FFA 6 6 4 4 0 2 4 3 1 1 1 − F˜ ∧ F˜ij ∧ dzi ∧ dzj ∧ Ak ∧ dzk − F˜ ∧ F˜ijk ∧ dzi ∧ dzj ∧ dzk ∧ A 2 4 3 2 6 4 1 3 1 − F˜i ∧ dzi ∧ F˜ ∧ Aij ∧ dzi ∧ dzj + F˜i ∧ dzi ∧ F˜i ∧ dzj ∧ Ak ∧ dzk 2 3 4 1 3 3 2 − 1 ˜i ∧ i ∧ ˜jk ∧ j ∧ k ∧ − 1 ˜ij ∧ i ∧ j ∧ ˜ ∧ k ∧ k F3 dz F2 dz dz A3 F2 dz dz F4 A2 dz 2 2 ) 1 1 − F˜ij ∧ dzi ∧ dzj ∧ F˜k ∧ dzk ∧ A − F˜ijk ∧ dzi ∧ dzj ∧ dzk ∧ F˜ ∧ A 2 2 3 3 6 1 4 3 (E.5)

We start by noticing that we have three terms with 1/6-term in front, these terms should yield us the same expression. Let us assume we want something of the form like the

8Hope this is understandable, if we think in this way we can save some time on the calculations

52 first term i.e 4 ∧ 4 ∧ 0 ∧ 1 ∧ 1 ∧ 1. The second term with the same factor is expressed as 1 1 − F˜ ∧ F˜ijk ∧ dzj ∧ dzj ∧ dzk ∧ A = F˜ ∧ F˜ijk ∧ A ∧ dzi ∧ dzj ∧ dzk 6 4 1 3 6 4 1 3 1 = F˜ ∧ dAijk ∧ A ∧ dzi ∧ dzj ∧ dzk 6 4 0 3 1 = − Aijk ∧ d(F˜ ∧ A ) ∧ dzi ∧ dzj ∧ dzk 6 0 4 3 1 = − Aijk ∧ F˜ ∧ F˜ ∧ dzi ∧ dzi ∧ dzk 6 0 4 4 1 = − F˜ ∧ F˜ ∧ Aijk ∧ dzi ∧ dzj ∧ dzk 6 4 4 0 we see that this term has the same form as the first term mainly 4 ∧ 4 ∧ 0 ∧ 1 ∧ 1 ∧ 1, we also obtain the same expression for the last term with the factor 1/6 one can also show this but we see that the term resembles the term above. Let us now look on the terms with the factors 1/2 and see what expression we obtain after partial integration 1 1 − F˜ ∧ F˜i ∧ dzi ∧ Ajk ∧ dzj ∧ dzk = F˜ ∧ F˜i ∧ Ajk ∧ dzi ∧ dzj ∧ dzk 2 4 3 1 2 4 3 1 1 = dA ∧ F˜i ∧ Ajk ∧ dzi ∧ dzj ∧ dzk 2 3 3 1 1 = A ∧ d(F˜i ∧ Ajk) ∧ dzi ∧ dzj ∧ dzk 2 3 3 1 1 = A ∧ F˜i ∧ F˜jk ∧ dzi ∧ dzj ∧ dzk 2 3 3 2 1 = F˜i ∧ F˜jk ∧ A ∧ dzi ∧ dzj ∧ dzk . 2 3 2 3 we can see that we obtain the same expression as the other 1/2 factor terms. Gathering all of them together we obtain the following integral ∫ ( ) 1 1 Ln=3 = − F˜ ∧ F˜ ∧ Aijk − F˜i ∧ F˜j ∧ Ak + 3F˜i ∧ F˜jk ∧ A εijk FFA 6 2 4 4 0 3 3 2 3 2 3 ∫ ( ) (E.6) 1 1 1 = − F˜ ∧ F˜ ∧ Aijk − F˜i ∧ F˜j ∧ Ak + F˜i ∧ F˜jk ∧ A εijk . 12 4 4 0 6 3 3 2 2 3 2 3 We thus see that we have 8-forms which yields us a 8-dimensional Chern-Simons term.

53 Let us continue now with terms that have four index. The following integral becomes ∫ ( 1 Ln=4 = − F˜ ∧ F˜i ∧ dzi ∧ Ajkl ∧ dzj ∧ dzk ∧ dzl FFA 6 4 3 0 1 + F˜ ∧ F˜ij ∧ dzi ∧ dzj ∧ Akl ∧ dzk ∧ dzl 4 4 2 1 1 − F˜ ∧ F˜ijk ∧ dzi ∧ dzj ∧ dzk ∧ Al ∧ dzl 6 4 1 2 1 − F˜i ∧ dzi ∧ F˜ ∧ Ajkl ∧ dzj ∧ dzk ∧ dzl 6 3 4 0 1 − F˜i ∧ dzi ∧ F˜j ∧ dzj ∧ Akl ∧ dzl ∧ dzk 2 3 3 1 − 1 ˜i ∧ i ∧ ˜jk ∧ j ∧ k ∧ l ∧ l F3 dz F2 dz dz A2 dz 2 (E.7) 1 − F˜i ∧ dzi ∧ F˜jkl ∧ dzj ∧ dzk ∧ dzl ∧ A 6 3 1 3 1 + F˜ij ∧ dzi ∧ dzj ∧ F˜ ∧ Akl ∧ dzk ∧ dzl 4 2 4 1 1 − F˜ij ∧ dzi ∧ dzj ∧ F˜k ∧ dzk ∧ Al ∧ dzl 2 2 3 2 1 + F˜ij ∧ dzi ∧ dzj ∧ F˜kl ∧ dzk ∧ dzl ∧ A 4 2 2 3 − 1 ˜ijk ∧ i ∧ j ∧ k ∧ ˜ ∧ l ∧ l F1 dz dz dz F4 A2 dz 6 ) 1 − F˜ijk ∧ dzi ∧ dzj ∧ dzk ∧ F˜l ∧ dzl ∧ A . 6 1 3 3

We see that the terms with the factor 1/6 should resemble each other and so on. Like always we choose a term and perform a partial integration. 1 1 F˜ ∧ F˜ijk ∧ dzi ∧ dzj ∧ dzk ∧ Al ∧ dzl = − F˜ ∧ F˜ijk ∧ Al ∧ dzi ∧ dzj ∧ dzk ∧ dzl 6 4 1 2 6 4 1 2 1 = dAijk ∧ F˜ ∧ Al ∧ dzi ∧ dzj ∧ dzk ∧ dzl 6 0 4 2 1 = Aijk ∧ d(F˜ ∧ Al ) ∧ dzi ∧ dzj ∧ dzk ∧ dzl 6 0 4 2 1 = Aijk ∧ F˜ ∧ F l ∧ dzi ∧ dzj ∧ dzk ∧ dzl 6 0 4 3 1 = F˜ ∧ F i ∧ Ajkl ∧ dzi ∧ dzj ∧ dzk ∧ dzl . 6 4 3 0 We thus see that this configuration becomes the first term in our integral above, after partial integration.

54 We continue with the 1/2 factor terms and see what we obtain after partial integra- tion 1 1 − F˜i ∧ dzi ∧ F˜jk ∧ dzj ∧ dzk ∧ Al ∧ dzl = − F˜i ∧ F˜jk ∧ Al ∧ dzi ∧ dzj ∧ dzk ∧ dzl 2 3 2 2 2 3 2 2 1 = dAij ∧ F˜k ∧ Al ∧ dzi ∧ dzj ∧ dzk ∧ dzl 2 1 3 2 1 = Aij ∧ d(F˜k ∧ Al ) ∧ dzi ∧ dzj ∧ dzk ∧ dzl 2 1 3 2 1 = − Aij ∧ F˜k ∧ F l ∧ dzi ∧ dzj ∧ dzk ∧ dzl 2 1 3 3 1 = − F˜i ∧ F˜j ∧ Akl ∧ dzi ∧ dzj ∧ dzk ∧ dzl . 2 3 3 1 The terms left now are the terms with the factor 1/4, doing a partial integration on for example the term 1 1 F˜ ∧ F˜jk ∧ dzi ∧ dzj ∧ Akl ∧ dzk ∧ dzl = dA ∧ F˜ij ∧ Akl ∧ dzi ∧ dzj ∧ dzk ∧ dzl 4 4 2 4 3 2 1 1 = − A ∧ d(F˜ij ∧ Akl) ∧ dzi ∧ dzj ∧ dzk ∧ dzl 4 3 2 1 1 = A ∧ F˜ij ∧ F˜kl ∧ dzi ∧ dzj ∧ dzk ∧ dzl 4 3 2 2 1 = F˜ij ∧ F˜kl ∧ A ∧ dzi ∧ dzj ∧ dzk ∧ dzl . 4 2 2 3 We see that this term is the same as the other 1/4 factor terms, we can now add them together and obtain the following integral ∫ ( ) 1 1 1 Ln=4 = F˜ ∧ F i ∧ Ajkl − F˜i ∧ F˜j ∧ Akl + F˜ij ∧ F˜kl ∧ A εijkl (E.8) FFA 6 4 3 0 4 3 3 2 8 2 2 3

55 Let us continue with 5 indices now, collecting all the term with 5 indices gives us the following integral ∫ ( 1 1 Ln=5 = + F˜ ∧ F˜ij ∧ dzi ∧ dzj ∧ Aklm ∧ dzk ∧ dzl ∧ dzm FFA 6 12 4 2 0 1 + F˜ ∧ F˜ijk ∧ dzi ∧ dzj ∧ dzk ∧ Alm ∧ dzl ∧ dzm 12 4 1 1 1 − F˜i ∧ dzi ∧ F˜j ∧ dzj ∧ Aklm ∧ dzk ∧ dzl ∧ dzm 6 3 3 0 1 + F˜i ∧ dzi ∧ F˜jk ∧ dzj ∧ dzk ∧ Alm ∧ dzl ∧ dzm 4 3 2 1 1 − F˜i ∧ dzi ∧ F˜jkl ∧ dzj ∧ dzk ∧ dzl ∧ Am ∧ dzm 6 3 1 2 1 + F˜ij ∧ dzi ∧ dzj ∧ F˜ ∧ Aklm ∧ dzk ∧ dzl ∧ dzm 12 2 4 0 1 + F˜ij ∧ dzi ∧ dzj ∧ F˜kl ∧ dzk ∧ dzl ∧ Am ∧ dzm 4 2 2 2 1 − F˜ij ∧ dzi ∧ dzj ∧ F˜klm ∧ dzk ∧ dzl ∧ dzm ∧ A 12 2 1 3 1 + F˜ijk ∧ dzi ∧ dzj ∧ dzk ∧ F˜ ∧ Alm ∧ dzl ∧ dzm 12 1 4 1 − 1 ˜ijk ∧ i ∧ j ∧ k ∧ ˜l ∧ l ∧ m ∧ m F1 dz dz dz F3 dz A2 dz 6 ) 1 + F˜ijk ∧ dzi ∧ dzj ∧ dzk ∧ F˜lm ∧ dzl ∧ dzm ∧ A 12 1 2 3

Let us calculate the partial integration for one of the 1/12 terms and see what we obtain 1 1 F˜ ∧ F˜ijk ∧ Alm ∧ dzi ∧ dzj ∧ dzk ∧ dzl ∧ dzm = − dAijk ∧ F˜ ∧ Alm ∧ dzijklm 12 4 1 1 12 0 4 1 1 = Aijk ∧ d(F˜ ∧ Alm) ∧ dzijklm 12 0 4 1 1 = Aijk ∧ F˜ ∧ F˜lm ∧ dzijklm 12 0 4 2 1 = F˜ ∧ F˜ij ∧ Aklm ∧ dzijklm 12 4 2 we introduced a short-hand notation to save space i.e, dzi∧dzj ∧dzk∧dzl∧dzm = dzijklm. We continue with the factor 1/6 term and obtain the following 1 1 − F˜ijk ∧ dzi ∧ dzj ∧ dzk ∧ F˜l ∧ dzl ∧ Am ∧ dzm = F˜ijk ∧ F˜l ∧ Am ∧ dzijklm 6 1 3 2 6 1 3 2 1 = dAijk ∧ F˜l ∧ Am ∧ dzijklm 6 0 3 2 1 = − Aijk ∧ d(F˜l ∧ Am) ∧ dzijklm 6 0 3 2 1 = − Aijk ∧ F˜l ∧ F˜m ∧ dzijklm 6 0 3 3 1 = − F˜i ∧ F˜j ∧ Aklm ∧ dzijklm 6 3 3 0

56 and finally the last term gives us 1 1 F˜i ∧ dzi ∧ F˜jk ∧ dzj ∧ dzk ∧ Alm ∧ dzl ∧ dzm = F˜i ∧ F˜jk ∧ Alm ∧ dzijklm 4 3 2 1 4 3 2 1 1 = dAi ∧ F˜jk ∧ Alm ∧ dzijklm 4 2 2 1 1 = Ai ∧ d(F˜jk ∧ Alm) ∧ dzjiklm 4 2 2 1 1 = Ai ∧ F˜jk ∧ F˜lm ∧ dzijklm 4 2 2 2 1 = F˜ij ∧ F˜kl ∧ Am ∧ dzijklm 4 2 2 2 we have now done all the partial integrations we need, collecting the results gives us ∫ ( ) 1 1 1 3 Ln=5 = F˜ ∧ F˜ij ∧ Aklm − F˜i ∧ F˜j ∧ Aklm + F˜ij ∧ F˜kl ∧ Am εijklm FFA 6 2 4 2 2 3 3 0 4 2 2 2 ∫ ( ) (E.9) 1 1 1 = F˜ ∧ F˜ij ∧ Aklm − F˜i ∧ F˜j ∧ Aklm + F˜ij ∧ F˜kl ∧ Am εijklm 12 4 2 12 3 3 0 8 2 2 2

57