Self-dual tensors and partial supersymmetry breaking in five dimensions

We study spontaneous supersymmetry breaking of five-dimensional supergravity theories from sixteen to eight supercharges in Minkowski vacua. This N=4→N=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=4\to \mathcal{N}=2 $$\end{document} breaking is induced by Abelian gaugings that require the introduction of self-dual tensor fields accompanying some of the vectors in the gravity and vector multiplets. These tensor fields have first-order kinetic terms and become massive by a Stückelberg-like mechanism. We identify the general class of N=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=2 $$\end{document} vacua and show how the N=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=4 $$\end{document} spectrum splits into massless and massive N=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=2 $$\end{document} multiplets. In particular, we find a massive gravitino multiplet, with two complex massive tensors, and a number of massive tensor multiplets and hypermultiplets. We determine the resulting N=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=2 $$\end{document} effective action for the massless multiplets obtained by integrating out massive fields. We show that the metric and Chern-Simons terms of the vectors are corrected at one-loop by massive tensors as well as spin-1/2 and spin-3/2 fermions. These contributions are independent of the supersymmetry-breaking scale and thus have to be included at arbitrarily low energies.


Introduction
A systematic classification of supersymmetric vacua of supergravity theories in various dimensions has been a challenge since the first constructions of such theories. Supergravity theories with non-minimal supersymmetry can often admit Minkowski or anti-de Sitter ground states that preserve only a partial amount of supersymmetry. Finding such solutions is typically more involved than determining the fully supersymmetric solutions. For supergravity theories formulated in even space-time dimensions various breaking patterns have been investigated in detail. For example, the N = 2 to N = 1 breaking in fourdimensional supergravity theories has been investigated already in [1][2][3][4][5][6][7][8]. Recently, there has been a renewed interest in this direction [9][10][11], which was partially triggered by the application to flux compactifications of string theory [12]. The general analysis of [10] heavily employs the powerful techniques provided by the embedding tensor formalism [13,14].

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The general study of partial supersymmetry breaking in odd-dimensional theories has attracted much less attention. Such theories, however, possess the interesting new possibility that the dynamics of some fields can arise from Chern-Simons-type couplings that are topological in nature. As was pointed out already for three-dimensional supergravity theories [15] such couplings can allow for special supersymmetry breaking patterns. In this work we show that in five-dimensional supergravity theories with sixteen supercharges, denoted as N = 4, Chern-Simons-type couplings for two-from tensor fields can yield interesting new supersymmetry breaking patterns to vacua preserving eight supercharges, denoted as N = 2. Such tensor fields can have first-order kinetic terms and become massive by a Stückelberg-like mechanism in which they eat a dynamical vector field [16][17][18]. The degrees of freedom of such tensors are counted by realizing that they have zero degrees of freedom before eating the vector, but admit three degrees of freedom as massive fields. Hence they should be distinguished from tensors with standard kinetic and mass terms. They have been named self-dual tensors in [16]. In fact, such five-dimensional self-dual tensors arise, for example, as massive Kaluza-Klein modes of a six-dimensional self-dual tensor compactified on a circle [16,19]. The mechanism rendering the tensor fields massive by eating a vector will be called tensorial Higgs mechanism in the following.
In general, the couplings of the five-dimensional self-dual tensors to the vector fields and the form of the first order kinetic terms are encoded by a constant anti-symmetric matrix ξ M N , known as the embedding tensor [17,18]. A non-trivial ξ M N also induces vector gaugings and a scalar potential. We analyze the conditions on ξ M N that yield partial supersymmetry breaking to an N = 2 Minkowski vacuum. The massless and massive N = 2 spectrum comprising fluctuations around this vacuum are then determined systematically. We particularly stress the appearance of massive tensor fields and massive spin-1/2 and spin-3/2 fermions. This allows us to derive the key features of the effective N = 2 supergravity theory arising for the massless fluctuations around the ground state.
The N = 2 effective action for the massless fields comprises of two parts. Firstly, there are the classical couplings inherited from the underlying N = 4 theory. They are determined by truncating the original theory to the appropriately combined massless modes. At energy scales far below the supersymmetry breaking scale one might have expected that this determined already the complete N = 2 theory. However, as we show in detail in this work, the massive tensor, spin-1/2 and spin-3/2 modes have to actually be integrated out and induce non-trivial corrections. In fact, using the results of [20], one infers that if these massive fields are charged under some vector field, they induce non-trivial one-loop corrections to the Chern-Simons terms for the vector. One-loop corrections to the Chern-Simons terms due to massive charged spin-1/2 fermions have been considered in [21,22], but we stress here that in the N = 4 to N = 2 breaking both massive tensors and gravitini alter the result crucially. All these one-loop corrections are independent of the mass scale of these fields and therefore have to be taken into account in a consistent effective theory at scales well below the supersymmetry breaking scale.
Five-dimensional partial supersymmetry breaking can also occur in string compactifications. For example, compactifications of Type IIB string theory on a certain class of five-dimensional compact spaces can yield five-dimensional N = 4 theories that admit JHEP03(2015)008 half-supersymmetric vacua as discussed, for example, in [23][24][25][26][27][28][29][30]. The generic analysis of these effective theories yields the existence also of AdS vacua that are of particular interest for applications of the AdS/CFT duality. Our study of Minkowski vacua should be viewed as a crucial step in obtaining a full classification including also AdS vacua. We will see that already in the Minkowski vacua, arising due to the existence of tensor fields, we can highlight interesting features of partial supersymmetry breaking in five dimensions. In particular, we show that the N = 2 effective supergravity theory for the massless fluctuations around the N = 2 vacuum depends on the massive states of the underlying N = 4 theory in an intriguing way.
The paper is organized as follows. In section 2 we recall some general facts about N = 4 supergravity theories. For an Abelian configuration with a non-trivial embedding tensor ξ M N we determine the conditions for the scalar potential to admit supersymmetric vacua. Supersymmetry breaking from N = 4 to N = 2 is studied in detail in section 3. We give a detailed account of the tensorial Higgs mechanism and the super-Higgs mechanism. In section 4 we then determine the N = 2 spectrum and one-loop effective action. We argue that all massive multiplets induce one-loop corrections to the vector couplings of the theory that are independent of the supersymmetry breaking scale. In two supplementary appendices we summarize our conventions and give details on the computation of fermion and scalar masses.

Gauged N = 4 supergravity in five dimensions
In this section we summarize some important facts about N = 4 supergravity theories. We first discuss the spectrum and action in subsection 2.1. This will also allow us to state the conventions used throughout the paper. Further conventions and useful identities are supplemented in appendix A. In subsection 2.2 we study the vacua of an N = 4 theory characterized by the components ξ M N of the embedding tensor and comment on the possible amounts of supersymmetries preserved in the vacuum.

Generalities
Let us start reviewing the general properties of N = 4 gauged supergravity in five dimensions following [17,18]. 1 We first focus on ungauged Maxwell-Einstein supergravity that describes the coupling of n vector multiplets and a gravity multiplet. Note that in the ungauged theory one can replace the vector multiplets by dual tensor multiplets. The gravity multiplet consists of the fields with the metric g µν , four spin-3/2 gravitini ψ i µ , six vectors A ij µ , A 0 µ , four spin-1/2 fermions χ i and one real scalar σ. The indices i, j = 1, . . . , 4 run over the fundamental representation JHEP03(2015)008 of the R-symmetry group USp(4). We denote the symplectic form of USp(4) by Ω. It has the following properties Indices of USp(4) are raised and lowered according to 3) The 5 representation of USp(4) is denoted by the double index ij with the following properties Note that USp (4) is the spin group of SO(5). We will therefore often use the local isomorphism SO(5) ∼ = USp(4) to rewrite representations. We denote the indices of the fundamental representation of SO(5) by m, n = 1, . . . , 5. They are raised and lowered with the Kronecker delta δ mn . We stress that all massless fermions in this work are symplectic Majorana spinors. Further conventions and useful relations can be found in appendix A. Moreover we will conveniently use the definition where σ is the real scalar in the gravity multiplet (2.1). In addition to the gravity multiplet we have n vector multiplets, labeled by the indices a, b = 1, . . . , n, which are raised and lowered by the Kronecker delta δ ab . The field content is The A a µ denote vectors, λ ia spin-1/2 fermions and the φ ija are scalars in the 5 of USp(4). Collecting all scalars of the theory they span the coset manifold where the coset M 5,n is parametrized by the scalars φ ija in the vector multiplets, while the SO(1, 1) factor is described by the scalar σ in the gravity multiplet. Thus the global symmetry group of the theory is SO(5, n) × SO(1, 1). We also introduce SO(5, n) indices M, N = 1, . . . 5 + n, which are raised and lowered with the SO(5, n) metric (η M N ) = diag(−1, −1, −1, −1, −1, +1, . . . , +1). The generators t M N of SO(5, n) and t 0 of SO(1, 1) are given by 2 To couple the vector multiplets to the gravity multiplet we note that the vectors of both kind of multiplets transform as a singlet A 0 and the fundamental representation A M of SO(5, n): with SO(1, 1) charges −1 and 1/2 for A 0 and A M , respectively.
2 All antisymmetrizations in this paper include a factor of 1/n! .
where V ij M transforms in the 5 of USp(4) and Γ m ij are the components of the gammamatrix Γ m of SO (5).
There are now different possibilities to gauge some of the global symmetries. The various gaugings can be described in terms of the embedding tensors f M N P , ξ M N and ξ M that are totally antisymmetric in their indices. They specify a covariant derivative 3 (2.14) We note that in the ungauged theory the embedding tensors are spurionic objects transforming under the global symmetry group. As soon as we fix a value for the tensor components, the global symmetry group is broken down to a subgroup. In this paper we will focus solely on the gauging ξ M N , since the calculations simplify considerably. Furthermore, we will find that a non-vanishing ξ M N is essential in the tensorial Higgs mechanism and interesting parts of the structure arising with general gaugings are already present in the case of f M N P = ξ M = 0. Therefore, we will from now on set such that the covariant derivative (2.14) simplifies to We note that in the case of f M N P = ξ M = 0 there are no further constraints on ξ M N except of antisymmetry. It is important to remark that such a nontrivial gauging has the effect, JHEP03(2015)008 that we are forced to dualize some of the vector fields A M µ into two-forms B µν M . Therefore, in order to write down the most general gauged supergravity with f M N P = ξ M = 0, we have to consider an action where both A M µ and B µν M are present. In this formulation the tensor fields B µν M carry no on-shell degrees of freedom, they can, however, become massive by eating a dynamical vector and acquire three degrees of freedom. We will discuss this further in subsection 3.2.
The bosonic Lagrangian of this N = 4 supergravity theory reads [17,18] In this expression R denotes the Ricci scalar and we define The vectors are subject to gauge transformations with scalar parameters Λ 0 , Λ M , while the tensors transform under standard two-form gauge transformations with one-form parameters Ξ µ M . The variation of a vector under these transformations will play a prominent role in the work of this paper, since it allows to implement the tensorial Higgs mechanism. It reads for our choice of gaugings Next turn to the fermionic Lagrangian. For the purpose of this work it will be sufficient to only recall the kinetic terms, the mass terms as well as further terms quadratic in the fields. A more complete discussion can be found in [17,18]. The quadratic fermionic terms of interest read

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The first line in (2.21) are the kinetic terms of the fermions, while the remaining two lines summarize their mass terms. The covariant derivatives are given by where the dots indicate couplings to scalars, which are of no importance in this work. In order to simplify expressions like (2.21) we introduce some convenient notation. We denote contractions of the embedding tensor with the coset representatives by Thus these quantities are field-dependent and only become constant in the vacuum. Note that in the definition of ξ mn and ξ am the positions of the SO(5, n)-indices M, N are crucial because of (2.10). The local isomorphism of SO (5) and USp (4) where Γ ij mn are the components of the gamma-matrix product Γ mn = Γ [m Γ n] .

Vacuum conditions and supersymmetry breaking
In this subsection we formulate the conditions for finding vacua of the gauged N = 4 supergravity theory introduced above. We restrict our considerations to the case for which ξ M N is the only non-vanishing embedding tensor such that (2.15) holds. The scalar potential V found in (2.17) is then given by [17,18] Inserting the explicit expressions (2.11) and (2.12) of η M N and M M N in terms of the coset representatives gives where we have inserted (2.24). Using the fact that a and m indices are raised by the Kronecker delta, this implies that the scalar potential is a sum of positive semi-definite terms.
Determining the minima of this potential is now trivial. The derivative with respect to Σ yields

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Since the left-hand-side of this equation is a sum of non-negative quadratic terms, the solution simply reads 4 ξ am ! = 0 ∀a, m . (2.29) The potential at this point is The remaining derivatives with respect to the scalars in the vector multiplets are trivially vanishing since the potential is quadratic in the ξ am . In summary, for vanishing embedding tensors f M N P , ξ M the vacua are characterized by the condition ξ am = 0 for all a, m. Due to (2.30) all such vacua are necessarily Minkowskian. Furthermore, we note that due to the vacuum condition the fermion shift matrix A aij 2 given in (2.25) trivially vanishes.
Let us next discuss the amount of supersymmetry preserved by these vacua. It can be determined by analyzing the variations of the fermions under supersymmetry transformations, which can be found in [17]. Evaluated in the vacuum (2.29) they read is the supersymmetry parameter. The automatic vanishing of δλ a i is a direct consequence of the fact that A aij 2 = 0 in the vacuum. It is convenient to decompose the supersymmetry parameter ǫ i into a Killing spinor η and a spacetime independent Each conserved supersymmetry corresponds to a zero eigenvalue of the matrix A j 1 i . Therefore we now need to face the task of finding the general form of the eigenvalues of A j 1 i . The eigenvalues of A j 1 i were already determined in [31] and we will recall their analysis in the following. To begin with one considers the decomposition of the N = 4 R-symmetry group USp(4) under the choice of a special direction. Recall that there is the local isomorphism USp(4) ∼ = SO(5). One then defines an SO(5) vector which specifies a preferred direction in R 5 and therefore encodes the breaking Here ε mnpqr denotes the usual five-dimensional Levi-Civita symbol with ε 12345 = 1. One also defines a unit vector forX = 0

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Via this construction, the four eigenvalues of the hermitian matrix iA j 1 i are given by ±a 1± , where [31] The index of the eigenvalues indicates the corresponding SU(2) ± projected subspace. Recall that zero eigenvalues of A j 1 i are in one-to-one correspondence with conserved supersymmetries. From (2.37) we see that there are three different possibilities for the amount of supersymmetry in the vacuum: 5 supersymmetry condition In this section we study the supersymmetry breaking from N = 4 to N = 2. We first comment further on the N = 2 vacuum conditions in subsection 3.1. This is followed by a discussion of the spectrum parameterizing fluctuations around this vacuum. More precisely, we focus on the fields that become massive by a Higgs mechanism. In subsection 3.2 we show how some of the tensors become massive by eating vector degrees of freedom, a mechanism that we term tensorial Higgs mechanism. The Higgs mechanism rendering half of the gravitinos massive is summarized in subsection 3.3. Let us stress that both massive tensors and massive gravitinos will play a distinctive role in the evaluation of the one-loop effective action presented in section 4.

Solution to the N = 2 condition
Since we are focusing on N = 2 vacua in this work, we recall the corresponding supersymmetry condition from subsection 2.2. It takes the form where ξ mn is field dependent since it arises from the constant ξ M N via (2.24). In this case the eigenvalues a 1± of the gravitino mass matrix given in (2.37) take the form This implies that the unbroken R-symmetry is SU(2) + in (2.34). 5 Note that ξ mn ξmn is quadratic in each summand.

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In the following we aim to find the general solution to (3.1). In order to do that we recall from [31] that ξ mn only acts on the orthogonal complement to X m , i.e. it satisfies the vanishing condition We use the local SO(5) symmetry in order to rotate where the index0 refers to the direction of X m . Then (3.3) ensures that ξ0 m = 0. Note that we still retain the freedom of a reflection along the X m direction as well as SO(4) ⊂ SO (5) rotations orthogonal to X m . We continue with the latter in order to bring the skewsymmetric matrix ξmn into block-diagonal form [32] ( where γ,γ ∈ R and ε denotes the usual two-dimensional epsilon tensor. The values ±iγ , ±iγ are the imaginary eigenvalues of ξ mn . Inserting this expression for ξ mn into (3.1) yields the condition γ Therefore the N = 2 supersymmetry conditions can be summarized as Note that we can employ the SO(5) symmetry further to simplify the analysis. In fact, one can always choose γ > 0 andγ = γ , are rotations that leave the ξ0 m components invariant. The transformation (3.9) changes the absolute signs of γ andγ, while the transformations in (3.10) change the relative sign between γ andγ. Note that (3.9) leaves the direction of X m invariant, while the transformations (3.10) reflect along it.

The tensorial Higgs mechanism
In this subsection we introduce the tensorial Higgs mechanism that is key to implementing the N = 4 to N = 2 supergravity breaking. In order to do that we recall that the tensors appear in the combination (2.18) as In order to perform the Higgs mechanism we apply the gauge transformations (2.20) for the perturbations around the N = 2 vacuum. We will show explicitly which vector degrees of freedom get eaten by the tensors to render them massive.

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Since the gauge transformation (2.20) of A 0 is not dependent on the gauge parameters Ξ µ N of the tensors, A 0 will stay massless in the N = 2 vacuum. In contrast, some of the A M µ can be absorbed by the tensors. To make this explicit, we introduce Note that the position of the indices M, N is crucial in these definitions. Similarly, we can introduce the gauge parameters (Λ m , Λ a ) and (Ξ µ n , Ξ µ a ) by setting In this rotated basis the gauge transformations (2.20) read where we have inserted the vacuum condition ξ am = 0.
In the next step we apply the SO(5) × SO(n) symmetry to rotate ξ mn and ξ ab into a convenient basis. For ξ mn we have done this already in (3.5) and introduced the constant γ. Similarly, we now split a → (ã,â) ,â = 1, . . . , rank ξ ab . (3.16) Since ξ ab is anti-symmetric one notes that rank(ξ ab ) is even and we can define Using the SO(n) rotations one can then choose a basis such that ξ ab takes the form where ε is the two-dimensional epsilon tensor. In this expression the non-zero real eigenvalues γǎ,ǎ = 1, . . . , n T are parameterizing the non-trivial VEV of ξ ab . Together (3.5) and (3.18) provide a diagonalization of ξ M N after contraction with the VEVs V . Recall, however, that the eigenvalues ±iγ, ±iγǎ are identical to the eigenvalues of ξ M N , since the contraction with V corresponds to a similarity transformation.
Using the explicit expressions (3.5) and (3.18) in the gauge transformations (3.14) and (3.15) we first note that δA0 µ and δAã µ simply reduce to

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These are the standard gauge transformations for U(1) gauge fields. We conclude that the vector fields A 0 , A0, Aã remain massless and propagate in the N = 2 effective theory describing the massless fluctuations around the vacuum. Turning to the massive degrees of freedom, we can pick the gauge parameters Ξ µn , Ξ µb such that By definition of ξmn and ξâb in (3.5) and (3.18) both matrices have full rank and can be inverted to solve for Ξ µn , Ξ µb . This implies that the vectors Am µ and Aâ µ are pure gauge and can be absorbed by the tensors (B µνm , B µνâ ) to render them massive.
In computing the N = 2 effective action for the massless degrees of freedom it will be crucial to include one-loop effects of massive tensors. Therefore it is necessary to evaluate the action of the massive tensors (B µνm , B µνâ ) after the gauge-fixing (3.20). Inserting the definitions (2.24), (3.12), and the split form of ξ mn and ξ ab given in the first equalities of (3.5), (3.18) into the general N = 4 action (2.17) we extract all terms depending on the tensors. Explicitly, after gauging away the vectors Am µ , Aâ µ , they read The first two lines in this expression are the kinetic terms of (B µνm , B µνâ ), while the last line summarizes their mass terms. Since ξmn and ξâb have maximal rank, indeed all such tensors are massive. The Lagrangian (3.21) can be further simplified by going to the basis in which ξmn, ξâb are parametrized by the eigenvalues γ, γǎ as in (3.5), (3.18). The appearance of the twodimensional epsilon tensor in these expressions makes it natural to define the complex tensors

22)
Bǎ := B 2ǎ−1 + iB 2ǎ ,ǎ = 1, . . . , n T (use Bâ) with n T defined in (3.17). One can show that the α index corresponds to the fundamental representation of the N = 2 R-symmetry group SU (2). Here and in the following we will use boldface symbols to denote complex fields. Inserting these definitions together with (3.5) and (3.18) into (3.21) we find 6  In the last step we want to rescale the complex tensors in order to bring the action into the standard form This Lagrangian was used in the one-loop computations of [19,20]. This results in the identifications This concludes our discussion of the massive tensors. We found that evaluated around the N = 2 vacuum there are n T + 2 complex massive tensors (B α , Bǎ) with standard action (3.24) and characteristic data (3.27). For convenience we summarize the split of the fields induced by ξ M N in table 1.

The super-Higgs mechanism
In the N = 2 broken phase of a N = 4 theory a gravitino mass term has to be generated for half of the gravitino degrees of freedom. This mass arises in the sector of the broken SU(2) subgroup of the N = 4 R-symmetry group USp(4). In fact, two gravitini eat up two spin-1/2 goldstini from the gravity multiplet and become massive. In this super-Higgs mechanism the massive gravitini acquire four extra degrees of freedom. The appropriate description of the massive fields is in terms of a single Dirac spin-3/2 fermion ψ µ without JHEP03(2015)008 a symplectic Majorana condition. The massive gravitino combines with the two massive complex tensors B α of the gravity multiplet into a massive N = 2 gravitino multiplet (ψ µ , B α ). The construction of such a half-BPS multiplet has been discussed in [33]. In the following we will briefly discuss the super-Higgs mechanism and determine the mass and U(1) charge of the gravitino multiplet. Let us first consider the four N = 4 symplectic Majorana gravitini ψ i µ and the spin-1/2 fermions in the gravity multiplet χ i . These split under the breaking into ψ α µ , ψα µ and χ α , χα, respectively. As noted in the last subsection, the index α = 1, 2 refers to the fundamental representation of the N = 2 R-symmetry group SU(2) + , whilė α = 1, 2 corresponds to the broken SU(2) − part. Both indices are raised and lowered with the epsilon tensor analogous to (2.2) and (2.3). As one can see in (2.21) using the N = 2 vacuum conditions (2.29) and (3.1) all fermion bilinears involving ψ α µ and χ α vanish in the vacuum, leaving only the kinetic terms for these fields. Thus we find two massless N = 2 spin-3/2 symplectic Majorana fermions ψ α µ and two massless spin-1/2 symplectic Majorana fermions χ α . The reduced symplectic Majorana condition reads and similarly for ψ α µ , where C denotes the charge conjugation matrix and ǫ αβ is the twodimensional Levi-Civita symbol with ǫ 12 = +1. We note that throughout this paper all massless fermionic N = 2 fields are taken to be symplectic Majorana.
We proceed with the investigation of the remaining fields ψα µ and χα. The terms in the Lagrangian involving these fields are (ignoring fluctuations of scalars) where we used that due to (3.2) the two eigenvalues ±a 1− of A 1 β α are given by The χα actually are the goldstini, that render the ψα µ massive, and can be removed from the action by a shift of the gravitini analog to the one performed in [15,34]. We thus obtain

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It is now convenient to merge the two symplectic Majorana fermions into a single unconstrained Dirac spinor 7 The action then reads with D µ ψ ν = ∂ µ ψ ν − iγA 0 µ ψ ν . To conclude this section we compare the action (3.34) with the standard form where D ν ψ ρ = ∂ ν ψ ρ − iq ψ A ν ψ ρ and c ψ depends on the representation under the massive little group SO(4) as Comparing (3.34) with (3.35) we conclude that ψ is in the 1 2 , 1 representation of SO(4) and carries mass and A 0 µ -charge These data will be crucial in evaluating the one-loop corrections induced by the massive gravitino multiplet in the next section. The massive Dirac gravitino ψ combines with the massive tensors B α of the last section into a massive gravitino multiplet.

Mass spectrum and effective action
In this section we identify the complete spectrum parameterizing the fluctuations around the N = 2 vacuum. We determine the masses and U(1) charges of all fields and show how they reassemble in N = 2 multiplets in subsection 4.1. Furthermore, we derive the low-energy effective action of the massless modes with particular focus on the data determining the N = 2 vector sector. The classical truncation from N = 4 to N = 2 is discussed in subsection 4.2. The crucial inclusion of one-loop quantum corrections due to integrating out massive fermions and tensors is discussed in subsection 4.3. These induce extra contributions to the metric and Chern-Simons terms that are independent of the scale of supersymmetry breaking.

The N = 2 spectrum
The N = 2 spectrum and its properties can be determined by evaluating the N = 4 action in the vicinity of the N = 2 vacuum. To read off the masses and charges all kinetic terms and mass terms have to be brought into canonical form after spontaneous symmetry breaking. This diagonalization procedure is rather lengthy and therefore partially deferred to appendix B in detail. In the following we highlight some of the basic steps and summarize the results.
The key ingredients in the mass generation are the gaugings ξ M N . Recall that in the background we rotated ξ M N to ξ mn , ξ ab and found the components where ξmn and ξâb have maximal rank. This yielded the natural index split Here the splits ofm into [α1], [α2] arises due to the block-diagonalization in (3.5) with the first index α labeling the two blocks and the second index labeling the two entries of each block. The split ofâ into [ǎ1], [ǎ2] arises due to the blocks in (3.18).
In order to extract the massless and massive scalar spectrum it is convenient to reparameterize the element V of the coset space M 5,n introduced in (2.7). Denoting by V the value of V in the N = 2 vacuum as in (3.12), we write (4.5) To treat these more compactly we introduce, just as for tensors in (3.22), the complex scalars Note that in this way all φ ma of the split (4.5) except φ0ã are combined into complex scalars.

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Similarly, we proceed for the split of the N = 4 fermions λ a i . Note that as for the gravitino in subsection 3.3 one splits i → (α,α). Together with the index split of a given in (4.3) one has It turns out to be convenient to combine all λ a i except λã α into complex Dirac fermions α=1 , To justify the use of (4.8) we stress that the appearance of all spin-1/2 fermions can be expressed in terms of the unconstrained Dirac spinors λǎ α , λã, λǎ 1,2 . Concerning (4.8a), the other linear combination 1 is related to λǎ α by the symplectic Majorana condition. In (4.8b), by the same reasoning, the linear combinations with λ ȧ α=2 are related to those involving λ ȧ α=1 . All degrees of freedom of the massive spin-1/2 fermions are therefore captured by the spinors (4.8), dropping the symplectic Majorana condition.
We are now in the position to summarize the spectrum. From the N = 4 gravity multiplet the metric g µν , two gravitini ψ α µ , two spin-1/2 fermions χ α , two vectors A 0 , A0, and one scalar Σ remain massless. These fields group into the N = 2 gravity multiplet g µν , A0, ψ α µ and one N = 2 vector multiplet A 0 , Σ, χ α . Note that the vector multiplet A 0 , Σ, χ α is special, since the massive states, such as the tensors and gravitini discussed in subsection 3.2 and subsection 3.3, carry A 0 charge. In order to later derive the quantum effective action for the A 0 vector multiplet we need to determine the A 0 -charge of all massive states. Additional We stress that the identifications in (4.8b) are analogous to the definition of the massive gravitino (3.33). n − 2n T vector multiplets Aã, φ0ã, λã α remain massless. We have already discussed the massless vectors Aã in subsection 3.2. Inserting (4.4) into the N = 4 action we check in appendix B that the φ0ã and λã α are indeed massless scalars.

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is satisfied. We denote the number of such massless hypermultiplets by n H , and name them h Λ 1,2,3,4 , λ Λ α , with Λ = 1, . . . , n H . Due to the fact that the hypermultiplets appear in pairs, the existence of a massless hypermultiplet implies the existence of a massive hypermultiplet with mass √ 2Σ 2 γ. Furthermore, one can check that one can consistently choose all γǎ > 0 without changing the effective theory. In summary, one has 2n T − n H massive hypermultiplets with mass (4.10) out of the set φ αǎ 1,2 , λǎ 1,2 . Together with the φ αã , λã one finds a total of n − n H massive hypermultiplets.
To complete the summary of the spectrum recall that in subsection 3.2 and subsection 3.3 we have already identified and analyzed the N = 2 massive gravitino multiplet comprising a massive Dirac gravitino ψ µ and two complex massive tensors B α . Furthermore, we found n T complex massive tensors Bǎ. The latter combine with Dirac fermions λǎ α into n T complex massive tensor multiplets.
To conclude we list in table 2 the decompositions of the N = 4 fields in terms of N = 2 fields along with their masses and charges. The reorganization into N = 2 multiplets can be found in table 3.

General N = 2 action and classical matching
We are now in the position to derive the classical N = 2 effective action for the massless modes. In order to do that we simply truncate the N = 4 action to the massless sector. The discussion of the quantum corrections can be found in the next subsection.
To begin with we recall the canonical form of a general N = 2 ungauged supergravity theory. The dynamics of the gravity-vector sector is entirely specified in terms of a cubic potential where M I , I = 1, . . . , n − 2n T + 2 are very special real coordinates and k IJK is a symmetric tensor. The M I naturally combine with the vectors A I of the theory. However, since the vector in the gravity multiplet is not accompanied by a scalar degree of freedom, the M I have to satisfy one constraint. In fact, the N = 2 scalar field space is identified with the hypersurface N ! = 1 . The gauge coupling function and the metric are then obtained as . (4.14) The bosonic two-derivative Lagrangian is then given by Here we included the kinetic term for the hypermultiplet scalars h u with metric H uv . The canonical Lagrangian (4.15) has to be compared with the truncated N = 4 theory. In our set-up we found the vectors (A I ) = A 0 , A0, Aã , which sets the index range for I. JHEP03(2015)008 N = 4 N = 2 mass c field , SO(4) rep charge fields fields for massive fields under A 0 The massless scalars in the effective theory (except for Σ) are most conveniently described by SO(5, n)-rotated elements of the coset spacê  In fact, it turns out that all couplings involving the elements (4.17) can be expressed as functions ofVã 0 alone. In order to do that, one uses the relationŝ The elementVã 0 itself can be expanded aŝ after truncating all massive modes. This implies, in particular, thatVã 0 has no dependence on φmâ. Therefore, the effective action of the scalars in the N = 2 vector multiplets decouples from the potentially massless scalars in the hypermultiplets as expected from N = 2 supersymmetry. The reduced action then takes the simple form

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where H ab ΛΣ is the metric of the quaternionic manifold parametrized by the scalars in the massless hypermultiplets that we, however, do not discuss any further in this paper. Therefore, by comparison of (4.20) with (4.15) we find the identifications 21) and the real potential This result specifies the constant tensors k IJK at the classical level. It is interesting to realize that the constraint N ! = 1 translates with the identifications (4.21) into the condition (2.11) for the elements of the coset space. We conclude that the very special real manifold is the coset space which is the subspace of (2.7) spanned by the massless scalars in the vector multiplets.

One-loop effects and Chern-Simons terms
In this section we determine the one-loop corrections to the gravity-vector sector of the N = 2 theory specified in subsection 4.2. We focus on this sector, since the corrections due to integrating out massive fields are independent of the supersymmetry breaking scale and masses of the fields running in the loop. Let us stress that due to the preserved N = 2 supersymmetry and the fact that the Chern-Simons terms can only receive constant corrections, the integrating out process can only perturbatively correct the gravity-vector sector at the one-loop level.
To obtain the one-loop corrected N an analysis of the Chern-Simons terms is sufficient. The explicit loop computations were performed in [19,20] and we can simply apply these results to our set-up. The studied Chern-Simons terms are of the form Note that in [20] also the one-loop corrections to the gauge-gravitational Chern-Simons term, the second term in (4.24), were computed and we include the result for completeness. The one-loop corrections arise from integrating out massive fields that are charged under some gauge fields A I . We denote the charges of massive gravitini, tensors, and spin-1/2 fermions collectively by q I and denote by c ψ , c B and c λ the ±1 choice of representation under the massive little group SO(4  Table 4. One-loop corrections due to integrating out massive spin-1/2 fermions, tensors, and spin-3/2 fermions.
Using table 4 we find that the massive states summarized in table 2 induce by the one-loop couplings Furthermore, we find the gravitational one-loop Chern-Simons terms The existence of new Chern-Simons couplings implies that the effective theory still sees remnants of the underlying N = 4 theory at arbitrarily low energy scales. In fact, the mass of the fields listed in table 3 can be made arbitrarily large by choosing the VEV of the modulus Σ. The constants γ and γǎ appearing in (4.26) and (4.27) are the imaginary parts of the eigenvalues of ξ M N and therefore independent of the VEVs of the fields.

Conclusions
In this work we have studied partial supersymmetry breaking in N = 4 supergravity theories. We focused on a certain subclass of gauged supergravity theories that possess tensor fields with first order kinetic terms. These are characterized by a non-trivial embedding tensor ξ M N . The presence of these couplings also ensures that they can become massive by a tensorial Higgs mechanism and are gauged by a distinctive U(1) vector field A 0 . The Higgs mechanism allows that the tensors acquire three degrees of freedom by absorbing a dynamical vector. Clearly, this is just a special case of the mechanisms that are employed to construct general supergravity theories using tensor hierarchies [12,35]. We show that the presence of these couplings permits the existence of simple N = 2 supersymmetric Minkowski vacua.
The fluctuations around the N = 2 supersymmetric Minkowski vacua are split into massive and massless N = 2 multiplets. In fact, starting with the N = 4 gravity multiplet and n vector multiplets, we showed that the massless N = 2 spectrum consists of the gravity multiplet, n − 2n T + 1 vector multiplets, and n H hypermultiplets. In addition one finds a massive spin-3/2 multiplet, n T complex massive tensor multiplets, and n − n H massive hypermultiplets. The massive spin-3/2 multiplet contains in addition to the gravitino degrees of freedom also two complex massive tensors. The number n T is given by 2n T = rank(ξ M N ) − 4, where the 4 corresponds to the massive tensors in the spin-3/2 multiplet.

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The degeneracy n H is given by the number of hypermultiplets that satisfy the masslessness condition (4.11).
We also determined the N = 2 low-energy effective action for the massless modes particularly focusing on the vector multiplets. In order to do that we first extracted the classical N = 2 couplings obtained by dropping all massive fields. We argued, however, that it is crucial to evaluate the actions for the massive tensors, and massive spin-3/2 and spin-1/2 fermions that are charged under the vector field A 0 . This is due to the fact, that these massive fields induce corrections to the kinetic terms and Chern-Simons terms of A 0 at one-loop. Crucially, these corrections are independent of the supersymmetry breaking scale and thus have to be included despite the fact that they are of loop order. In other words, from the Chern-Simons terms of the effective theory, one can extract information about the underlying N = 4 theory. This is of similar spirit as the discussion in [36], where five-dimensional one-loop Chern-Simons terms were used to extract information about an underling six-dimensional theory.
Let us briefly discuss possible applications of our results and comment of further extensions. Five-dimensional theories with N = 4 supersymmetry can arise, for example, in compactifications of M-theory on K3 × T 2 , or Type IIB supergravity on K3 × S 1 . In order to obtain a gauged N = 4 theory the T 2 or S 1 reduction can, for example, be made non-trivial by demanding Scherk-Schwarz boundary conditions. In fact, in such reductions one expects that certain states are charged under the Kaluza-Klein vector that is identified with our A 0 . In particular, it would be interesting to carry out a Scherk-Schwarz reduction of a six-dimensional (2, 0) theory on a circle and determine its N = 2 vacua.
More involved five-dimensional gauged N = 4 supergravity theories arise from Mtheory on a general SU(2) structure manifold, or from Type IIB on a squashed Sasaki-Einstein manifold [23][24][25][26][27][28][29][30]. These theories in general also admit non-Abelian gaugings and allow for non-trivial Anti-de Sitter vacua. In a future project we hope to extend our analysis to include these more involved situations. We expect that similar to our discussion for the Minkowski vacua also one-loop effects will be essential when evaluating the effective theory. Note that one can use Γ mn := Γ [m Γ n] in order to switch between antisymmetric tensors T mn and symmetric tensors T ij in the following way

B Derivation of the mass terms and couplings
In this section we explicitly derive the masses for the spin-1/2 fermions and scalars induced by the supergravity breaking. We also evaluate the charges of the spin-1/2 fermions under the Abelian gauge field A 0 . We shortly note that the Lagrangian of a massive spin-1/2 Dirac spinor is given by The c 1/2 in (B.1) refers to the two inequivalent spinor representations of the massive little group, in detail q λǎ 1 = γǎ − γ , q λǎ 2 = −γǎ − γ . (B.12)

B.2 Scalar masses
Lastly we investigate the scalar degrees of freedom in the vacuum (except of Σ). In order to derive the scalar masses we insert the expansion (4.4) into the scalar potential (2.26) To read off the mass terms of the scalars, we focus on the terms quadratic in φ ma e −1 L φ, mass = − 1 16 Σ 4 φ ma φ nb (8ξ mn ξ ab + 4δ mn ξ ac ξ c b + 4δ ab ξ mp ξ p n ) . (B.14) According to the index split (4.3) the scalar fields arrange in four different groups: 1. φ0ã. The mass terms for these fields vanish: e −1 L φ, mass = 0 . (B.15) Thus we find n − 2n T massless real scalar fields φ0ã.