Chern-Simons AdS_5 supergravity in a Randall-Sundrum background

Chern-Simons AdS supergravity theories are gauge theories for the super-AdS group. These theories possess a fermionic symmetry which differs from standard supersymmetry. In this paper, we study five-dimensional Chern-Simons AdS supergravity in a Randall-Sundrum scenario with two Minkowski 3-branes. After making modifications to the D = 5 Chern-Simons AdS supergravity action and fermionic symmetry transformations, we obtain a Z_2-invariant total action S = \tilde{S}_bulk + S_brane and fermionic transformations \tilde{\delta}_\epsilon. While \tilde{\delta}_\epsilon \tilde{S}_bulk = 0, the fermionic symmetry is broken by S_brane. Our total action reduces to the original Randall-Sundrum model when \tilde{S}_bulk is restricted to its gravitational sector. We solve the Killing spinor equations for a bosonic configuration with vanishing su(N) and u(1) gauge fields.


Introduction
Chern-Simons AdS supergravity [1,2,3] theories can be constructed only in odd spacetime dimensions. As the name implies, they are gauge theories for supersymmetric extensions of the AdS group. 1 They have a fiber bundle structure and hence are potentially renormalizable [2]. The dynamical fields form a single adS superalgebra-valued connection and hence the supersymmetry algebra closes automatically off-shell without requiring auxiliary fields [4]. The Lagrangian in dimension D = 2n − 1 is a Chern-Simons (2n − 1)-form for the super-adS connection and is a polynomial of order n in the corresponding curvature. Unlike standard supergravity theories, there can be a mismatch between the number of bosonic and fermionic degrees of freedom. 2 For this reason, the 'supersymmetry' of Chern-Simons AdS supergravity theories is perhaps better referred to as a fermionic symmetry. D = 11, N = 1 Chern-Simons AdS supergravity may correspond to an off-shell supergravity limit of M-theory [2,3]. It has expected features of M-theory which are not shared by D = 11 Cremmer-Julia-Scherk (CJS) supergravity [5]. These features include an osp(32|1) superalgebra [6] and higher powers of curvature [7]. Hořava-Witten theory [8] is obtained from CJS supergravity by compactifying on an S 1 /Z 2 orbifold and requiring gauge and gravitational anomalies to cancel. This theory gives the low energy, strongly coupled limit of the heterotic E 8 × E 8 string theory. In light of the above discussion, it would be interesting to reformulate Hořava-Witten theory with D = 11, N = 1 Chern-Simons AdS supergravity.
Reformulating Hořava-Witten theory as described above may prove to be difficult. It is simpler to compactify the five-dimensional version of Chern-Simons AdS supergravity on an S 1 /Z 2 orbifold and ignore anomaly cancellation issues. Canonical sectors of D = 5 Chern-Simons AdS supergravity have been investigated in locally AdS 5 backgrounds possessing a spatial boundary with topology S 1 ×S 1 ×S 1 located at infinity [9]. In this paper, as a preamble to reformulating Hořava-Witten theory, we will study D = 5 Chern-Simons AdS supergravity in a Randall-Sundrum background with two Minkowski 1 The AdS group in dimension D ≥ 2 is SO(D − 1, 2). The corresponding super-AdS groups are given in [3]. For D = 5 and D = 11, the super-AdS groups are respectively SU (2, 2|N ) and OSp(32|N ).
2 For example, in D = 5 Chern-Simons AdS supergravity [1], the number of bosonic degrees of freedom (N 2 + 15) is equal to the number of fermionic degrees of freedom (8N ) only for N = 3 and N = 5. [10]. We choose coordinates x µ = (xμ, x 5 ) to parameterize the five-dimensional spacetime manifold. 3 In terms of these coordinates, the background metric takes the form

3-branes
is the warp factor, and ℓ is the AdS 5 curvature radius. The coordinate x 5 parameterizes an S 1 /Z 2 orbifold, where the circle S 1 has radius ρ and Z 2 acts as x 5 → −x 5 . We choose the range −πρ ≤ x 5 ≤ πρ with the endpoints identified as x 5 ≃ x 5 +2πρ. The Minkowski 3-branes are located at the Z 2 fixed points x 5 = 0 and x 5 = πρ. These 3-branes have corresponding tensions T (0) and T (πρ) and may support (3 + 1)-dimensional field theories.
This paper is organized as follows: In Section 2, we construct a Z 2invariant bulk theory. This bulk theory is obtained by making modifications to the D = 5 Chern-Simons AdS supergravity action and fermionic symmetry transformations which allow consistent orbifold conditions to be imposed. The variation of the resulting bulk action S bulk under the resulting fermionic transformations δ ǫ vanishes everywhere except at the Z 2 fixed points. We calculate δ ǫ S bulk in Section 3. In Section 4, we modify S bulk and δ ǫ to obtain a modified Z 2 -invariant bulk theory. The modified bulk actionS bulk is invariant under the modified fermionic transformationsδ ǫ . In Section 5, we complete our model by adding the brane action S brane . We show in Section 6 that our total action S =S bulk + S brane (1.2) reduces to the original Randall-Sundrum model [10] whenS bulk is restricted to its gravitational sector. In Section 7, we solve the Killing spinor equations for a purely bosonic configuration with vanishing su(N) and u(1) gauge fields. Our concluding remarks are given in Section 8. Finally, in the Appendix, we work out the fünfbein, spin connection, curvature 2-form components, Ricci tensor, and Ricci scalar for our metric (1.1). 3 We use indices µ, ν, . . . = 0, 1, 2, 3, 5 for local spacetime and a, b, . . . =0,1,2,3,5 for tangent spacetime. The corresponding metrics, g µν and η ab = diag(−1, 1, 1, 1, 1) ab , are related by g µν = e µ a e ν b η ab , where e µ a is the fünfbein. Barred indicesμ,ν, . . . = 0, 1, 2, 3, andā,b, . . . =0,1,2,3 denote the four-dimensional counterparts of µ, ν, . . . and a, b, . . ., respectively.

Z -invariant bulk theory
In this section, we construct a Z 2 -invariant bulk theory. The bulk theory is obtained by making modifications to the D = 5 Chern-Simons AdS supergravity [1] action and fermionic symmetry transformations which allow consistent orbifold conditions to be imposed.
The field content of D = 5 Chern-Simons AdS supergravity is the fünfbein e µ a , the spin connection ω µ ab , the su(N) gauge connection A µ = A i µ τ i , the u(1) gauge connection B µ , and N complex gravitini ψ µr which transform as Dirac spinors in a vector representation of su(N). 4 These fields form a connection for the adS superalgebra su(2, 2|N). The action and fermionic symmetry transformations are given in [9] in terms of the AdS 5 curvature radius ℓ. The only free parameter in the action is a dimensionless constant k. To allow consistent Z 2 orbifold conditions to be imposed, we make the following modifications: 1. Rescale the su(N) and u(1) gauge connections: 2. Replace g A , g B , ℓ −1 , and k by the Z 2 -odd expressions 5 In this manner, we obtain the bulk action and the transformations In these expressions, Γ a are the Dirac matrices 6 Note that the results in the Appendix can be used to show that the torsion vanishes for our metric. We impose the following orbifold conditions: 6 We choose a chiral basis for the 4 × 4 Dirac matrices where ε abcde is the Levi-Civita tensor and ε01235 = 1.

1.
Periodicity on S 1 . The fields and the fermionic parameters ǫ r , denoted generically by φ, are required to be periodic on the circle S 1 . That is, 2. Z 2 parity assignments. The bosonic field components That is, the Φ components are Z 2 -even and the Θ components are Z 2 -odd. For the gravitini, we require Γ˙5 ψμ r (xμ, Finally, the fermionic parameters ǫ r are required to satisfy Γ˙5 ǫ r (xμ, x 5 ) = + ǫ r (x µ , −x 5 ). (2.9) These conditions imply that the Z 2 -odd quantities vanish at the orbifold fixed points. It is straightforward to check that S bulk is Z 2 -even and that the transformations (2.3) are consistent with the Z 2 parity assignments.

Calculation of δ ǫ S bulk
The D = 5 Chern-Simons AdS supergravity action is invariant (up to a boundary term) under its fermionic symmetry transformations. In Section 2, we modified this action and its fermionic transformations to obtain a Z 2 -invariant bulk theory. Due to the signum functions introduced by the modifications, δ ǫ S bulk contains terms which have no counterpart in the unmodified theory. More specifically, the extra terms arise from ∂ 5 acting on the signum functions to yield delta functions. Such 'delta function' contributions to δ ǫ S bulk can potentially spoil the fermionic symmetry only at the Z 2 fixed points. Thus, S bulk is invariant under its fermionic transformations everywhere except perhaps at the Z 2 fixed points. In this section, we will calculate δ ǫ S bulk . For our metric and Z 2 parity assignments, the uncancelled variation δ ǫ S bulk arises from the variation of the 4-Fermi terms. The 4-Fermi terms are Let us now compute δ ǫ S bulk by applying δ ǫ to (3.1) and dropping all terms which contribute no delta functions. For our metric and Z 2 parity assignments, we can drop all but 1. The ∂ µ part of ∇ µ .

Modified Z 2 -invariant bulk theory
The result (3.3) for δ ǫ S bulk demonstrates that S bulk is not invariant under the fermionic transformations δ ǫ . In this section, we will modify S bulk and δ ǫ by replacing the adS 5 We will show that the modified bulk actioñ is invariant under the modified transformations That is, we will show that vanishes. It is straightforward to check thatS bulk is Z 2 -invariant and the transformationsδ ǫ are consistent with our Z 2 parity assignments.

Brane action
To complete our model, we add the brane action S brane . In the absence of particle excitations, the brane action consists of brane tensions. That is, where e (4) ≡ det(eμā). As discussed in Section 2, Z 2 -odd quantities vanish at the Z 2 fixed points. Thus, it is clear that S brane is Z 2 -even. Further discussion of 3-brane actions can be found in [11].

Connection with original RS model
In this section, we will show that our total action S =S bulk + S brane reduces to the original Randall-Sundrum model [10] whenS bulk is restricted to its gravitational sector. The gravitational sector ofS bulk is S grav , given by the first equation of (2.2). S grav consists of three terms: 1. The 'Gauss-Bonnet' term 1 8 Kε abcde R ab R cd e e /L. 2. The 'Einstein-Hilbert' term 1 8 · 2 3 Kε abcde R ab e c e d e e /L 3 . 3. The 'cosmological constant' term 1 8 · 1 5 Kε abcde e a e b e c e d e e /L 5 . For our metric, the first term can be expressed as a linear combination of the other two. Summing the three terms yields an effective Einstein-Hilbert term and an effective cosmological constant term. To demonstrate this explicitly, let us evaluate S grav for our metric. Using the results in the Appendix, we obtain where e ≡ det(e µ a ). Thus, where M is the five-dimensional gravitational mass scale 7 , Λ is the bulk cosmological constant, and Combining the result (6.2) with (5.1) , we obtain the action of the original Randall-Sundrum model. It is shown in [10] that the five-dimensional vacuum Einstein's equations for this system, 4) are solved by our metric provided that the relations are satisfied.

Killing spinors
In this section, we will solve the Killing spinor equations for a purely bosonic configuration with vanishing su(N) and u(1) gauge fields. In this case, the Killing spinor equations reduce to To solve these equations, split ǫ r into Z 2 -even (ǫ + r ) and Z 2 -odd (ǫ − r ) pieces: where ǫ ± r ≡ 1 2 (ǫ r ± Γ˙5ǫ r ) = ±Γ˙5ǫ ± r . We obtain the following system of equations: These equations are solved by where χ +(0) r and χ −(0) r are constant (projected) spinors. 8 Thus, our solution for the Killing spinors is
Thus, the fermionic symmetry is broken by S brane . Nevertheless, the Killing spinors of Section 7 are globally defined. 8 It is straightforward to check that (7.5) satisfies the first, second, and fourth equations of (7.4). There is, however, a subtlety in checking that (7.5) satisfies the third equation of (7.4). Unlike ǫ − r , ǫ + r is a smooth function of x 5 . Thus, the second term on the right side of the third equation of (7.4) contributes nothing.
Our model reduces to the original Randall-Sundrum model [10] wheñ S bulk is restricted to its gravitational sector. The original Randall-Sundrum model involves the fine-tuning relations Randall-Sundrum scenarios constructed from standard D = 5 supergravity theories yield these relations (up to an overall normalization factor) as a consequence of local supersymmetry (some examples are given in [12]). In our case, the relation Λ = −24M 3 /ℓ 2 follows from our metric choice. We do not obtain the relations T (0) = −T (πρ) = 24M 3 /ℓ as a consequence of local fermionic symmetry. These are fine-tuning relations in our model.
For the metric (1.1), the nonzero quantities here are and those related to (A.3) by R µν ab = −R νµ ab = −R µν ba . The prime symbol ′ denotes partial differentiation with respect to x 5 .