N=2 SO(4) 7D gauged supergravity with topological mass term from 11 dimensions

We construct a consistent reduction ansatz of eleven-dimensional supergravity to $N=2$ $SO(4)$ seven-dimensional gauged supergravity with topological mass term for the three-form field. The ansatz is obtained from a truncation of the $S^4$ reduction giving rise to the maximal $N=4$ $SO(5)$ gauged supergravity. Therefore, the consistency is guaranteed by the consistency of the $S^4$ reduction. Unlike the gauged supergravity without topological mass having a half-supersymmetric domain wall vacuum, the resulting 7D gauged supergravity theory admits a maximally supersymmetric $AdS_7$ critical point. This corresponds to $N=(1,0)$ superconformal field theory in six dimensions. We also study RG flows from this $N=(1,0)$ SCFT to non-conformal $N=(1,0)$ Super Yang-Mills theories in the seven-dimensional framework and use the reduction ansatz to uplift this RG flow to eleven dimensions.


Introduction
Gauged supergravities in various dimensions play an important role in both string compactifications and in the AdS/CFT correspondence. In some cases, a consistent truncation can be made in such a way that a lower dimensional gauged supergravity is obtained via a dimensional reduction of a (gauged) supergravity in higher dimensions on spheres [1]. Embedding lower dimensional gauged supergravities is now of considerable interest since this provides a method to uplift lower dimensional solutions to string/M theory.
It is known that sphere reductions of 10 or 11 dimensional supergravities give rise to gauged supergravity in lower dimensions. Well-known examples of these consistent sphere reductions include S 7 and S 4 reductions of eleven-dimensional supergravity and S 5 reduction of type IIB theory giving rise to SO (8), SO(5) and SO(6) gauged supergravities in four, seven and five dimensions, respectively [2,3,4]. According to the AdS/CFT correspondence [5], seven-dimensional gauged supergravity is useful in the study of N = (2, 0) and N = (1, 0) field theories in six dimensions [6,7,8,9,10]. The latter describe the dynamics of M5-branes worldvolume in M-theory and are less-known on the field theory side. Therefore, seven-dimensional gauged supergravity is expected to give some insight to six-dimensional field theories via gauge/gravity correspondence.
In this paper, we are interested in obtaining N = 2 seven-dimensional gauged supergravity with SO(4) gauged group and topological mass term. In seven dimensions, the theory is obtained by coupling three vector multiplets to the pure SU(2) gauged supergravity constructed in [11]. This matter-coupled theory has been constructed in [12] and [13]. The SO(4) gauged supergravity has also been constructed in [14] by truncating the maximal N = 4 SO(5) gauged supergravity. All of these constructions have not included the topological mass term for the three-form field, and the resulting theory does not admit AdS 7 vacuum solutions. It has been shown in [15] that the topological mass term is possible. The massive gauged theory has been explored in [16] in which new AdS 7 vacua and the corresponding RG flow interpolating between these vacua have been given.
To give an interpretation to this solution in the string/M theory context, it is necessary to embed this solution to 10 or 11 dimensions. The reduction ansatz of elevendimensional supergravity giving rise to pure SU(2) gauged supergravity has been given in [17]. The SO(4) gauged theory without topological mass term from a dimensional reduction of eleven-and ten-dimensional supergravity has been given in [18] using the result of [19]. This result is clearly not sufficient to uplift the solution in [16]. The dimensionally reduced theory needs to include the topological mass term in order to admit AdS 7 vacua. We will give an extension to the result of [17,18] by constructing SO(4) gauged theory including topological mass term from a truncation of S 4 reduction of eleven dimensional supergravity. This provides an ansatz to uplift the 7-dimensional solutions of massive N = 2 SO(4) gauged supergravity to eleven dimensions.
The paper is organized as follow. In section 2, we give relevant formulae for N = 2 SO(4) gauged supergravity in seven dimensions. The embedding of this theory in eleven dimensions is obtained via a consistent truncation of the S 4 reduction of elevendimensional supergravity in section 3. We then use the resulting ansatz to uplift RG flow solutions from the maximally supersymmetric AdS 7 vacuum with SO(4) symmetry to non-conformal SYM in section 4. We end the paper by giving some conclusions and comments in section 5.

SO(4) N = 2 gauged supergravity in seven dimensions
In this section, we give a description of SO(4) N = 2 gauged supergravity in seven dimensions with topological mass term. All of the notations are the same as those in [15] to which the reader is referred for further details.
The SO(4) gauged theory is obtained by coupling three vector multiplets to the N = 2 supergravity multiplet. The field contents are given respectively by where an index r = 1, 2, 3 labels the three vector multiplets. Curved and flat spacetime indices are denoted by µ, ν, . . . and a, b, . . ., respectively. B µν and σ are a two-form and the dilaton fields. The two-form field will be dualized to a three-form field C µνρ . Indices i, j = 1, 2, 3 label triplets of SU(2) R . The 9 scalars φ ir are parametrized by SO ( (2.2) whose inverse is given by L −1 = (L I i , L I r ) where L I i = η IJ L Ji and L I r = η IJ L Jr . Indices i, j and r, s are raised and lowered by δ ij and δ rs , respectively while the full SO(3, 3) indices I, J are raised and lowered by η IJ = diag(− − − + ++).
The SO(4) ∼ SU(2) × SU(2) gauging is implemented by promoting the SU(2) × SU(2) ∼ SO(3) × SO(3) ⊂ SO(3, 3) to a gauge symmetry. The structure constants for the SU(2) × SU(2) gauge group, which will appear in various quantities, are given by To obtain SO(4) gauge group, we will later set g 2 = g 1 . The bosonic Lagrangian can be written in a form language as where the scalar potential is given by The constant h describes the topological mass term for the three-form C (3) with H (4) = dC (3) . The quantities appearing in the above Lagrangian are defined by The Chern-Simons three-form satisfying dω (3) = F I (2) ∧ F I (2) is given by It is also useful to give the corresponding field equations The Yang-Mills equation (2.10) can be written in terms of C ir and C irs by using the relation In obtaining the scalar equation (2.11), we have used the projections in the variations of scalars as in [12] which lead to (2.14) We finally give supersymmetry transformations for fermions with all fermionic fields vanishing. These are given by where SU(2) R doublet indices A, B, . . . on spinors are suppressed. σ i are the usual Pauli matrices.

Seven dimensional N = 2 gauged supergravity from eleven dimensions
We now construct a reduction ansatz for embedding SO(4) N = 2 gauged supergravity mentioned in the previous section in eleven dimensions. The ansatz will be obtained from a consistent truncation of the S 4 reduction of eleven-dimensional supergravity giving rise to the maximal N = 4 SO(5) gauged supergravity in seven dimensions.
To obtain the topological mass term, we will impose the so-called odd-dimensional self-duality as in [17].

N = 4 SO(5) gauged supergravity from seven dimensions
To set up the notations and make the paper self-contained, we briefly repeat the S 4 reduction of eleven-dimensional supergravity [3,20]. We will work in the notations of [19] and deal mainly with bosonic fields. The field content of eleven-dimensional supergravity consists of the gravitonĝ M N , gravitinoψ M and a four-form fieldF (4) . Eleven-dimensional space-time indices are denoted by M, N = 0, 1, . . . , 10. The S 4 reduction is characterized by the following ansatz where the quantities appearing in the above equations are defined by The symmetric matrix T ij , i, j = 1, . . . , 5 with unit determinant parametrize the SL(5, R)/SO(5) coset manifold. The bosonic field content of N = 4 gauged supergravity is given by the metric g µν , (1) gauging the SO(5) gauge group, five three-form fields S i (3) and four-teen scalars T ij . The corresponding field equations are given by All of these equation can be obtained from the Lagrangian where Ω (7) is the Chern-Simens three-form whose explicit form can be found in [22]. The scalar potential for T ij is given by We have not given Einstein equation since we will not consider Einstein equation in this paper. The consistency of the full truncation, including the Einstein equation, to N = 2 SO(4) gauged supergravity is guaranteed from the consistency of the S 4 reduction.
For completeness, we also repeat supersymmetry transformations of fermionic fields ψ µ and λˆi. Indicesî,ĵ = 1, . . . , 5 are vector indices of the composite SO(5) c symmetry. Additionally, both ψ µ and λˆi transform as a spinor under SO(5) c with the condition Γˆiλˆi = 0, but we have omitted the SO(5) c spinor indices to make the following expressions more compact. The SO(5) c gamma matrices will be denoted by Γˆi. The associated supersymmetry transformations are given by [22] where with Πˆi i being the SL(5, R)/SO(5) coset representative.

SO(4) N = 2 gauged supergravity from S 4 reduction
We now truncate the N = 4 gauged supergravity to N = 2 theory with topological mass term for the three-form field and SO(4) gauge group. In this process, the gauge group SO (5) is broken to SO(4). We will split the index i as (α, 5) with α = 1, . . . , 4. Furthermore, we will set T 5α , S α and F 5α to zero. The S 4 coordinates µ i will be chosen to be µ i = (cos ξµ α , sin ξ) in which µ α satisfy µ α µ α = 1. Similar to µ i , µ α are coordinates on S 3 . The scalar truncation is given by T ij = (T αβ , T 55 ) = (XT αβ , X −4 ) withT αβ being unimodular. The scalar field X will be related to the N = 2 dilaton. With these truncations, the three-form field equations (3.4) and (3.5) become We have used ǫ 5αβγδ = ǫ αβγδ . From (3.14), we see that the four-form X −4 * S 5 (3) is closed. We will denote it by To satisfy equation (3.15), we impose the odd-dimensional self-duality condition 2) , is the Chern-Simons term given by Equations for S α (3) are trivially satisfied. For the Yang-Mills equations, it can be verified that setting F 5α (2) = 0 satisfies their field equations. For F αβ (2) , we find where we have used the odd-dimensional self-duality condition. We then consider scalar equations. Equations for T 5α are trivially satisfied while the T 55 equation gives rise to the dilaton eqiation We can now use the X equation (3.22) and end up with With all of the above truncations, we find the following ansatz for the metric and the four-form field All of the above equations reduce to the pure N = 2 gauged supergravity with SU(2) gauge group forT αβ = δ αβ after using various relations given in [21]. Note that forT αβ = δ αβ , equation (3.24) gives * F αγ (3.28) which means that the SO(4) gauge fields A αβ (1) must be truncated to those of SU(2) satisfying F αβ (2) = ± 1 2 ǫ αβγδ F γδ (2) . This is expected since there are only three vector fields in the pure gauged supergravity which only admit SU(2) gauging.
The above equations can be obtained from the Lagrangian where the scalar potential is given by (3.30) ForT αβ = δ αβ , we findT αα =T αβTαβ = 4. The above potential becomes which is exactly the same as that given in [17] up to a redefinition of the coupling constant g. We can also check another truncation namely to U(1) × U(1) gauged supergravity. To preserve SO(2) × SO(2) symmetry, we take the scalar matrix to bẽ which takes the same form as that given in [23]. Finally, it should be remarked that the three-form field equation coming from the Lagrangian (3.29) needs to be supplemented with the odd-dimensional self-duality condition as in the pure SU(2) gauged supergravity discussed in [17]. The nine scalars, parametrized byT αβ , in the dimensionally reduced theory are encoded in the SL(4, R)/SO(4) coset manifold. Therefore, in order to compare the result with gauged N = 2 SO(4) supergravity given in the previous section, we need to use the relation between SL(4, R)/SO(4) and SO(3, 3)/SO(3)×SO(3) coset manifolds. This is given in [15]. For the details of this mapping, the reader is referred to [15]. We will only give the SO(3, 3)/SO(3) × SO(3) coset representative L A I = (L i I , L r I ) and that of SL(4, R)/SO(4), V α R with R = 1, . . . , 4, where Γ I and η A are chirally projected SO(3, 3) gamma matrices. It can be shown that the scalar potential can be written as This form is similar to the potential (3.30) ifT αβ is identified with T αβ . Note that T αβ and C, C ir contain the gauge coupling g 1 and g 2 . In order to compare the Lagrangian of the two theories, we need to multiply the Lagrangian (2.4) by two and separate the coupling constants g 1 and g 2 from the structure constants f IJK = (g 1 ǫ ijk , g 2 ǫ rst ). With these, the two scalar potentials are exactly the same if we identify We also need to redefine the following fields in the Lagrangian (2.4): (3.37) By using (3.34), it can also be checked that The field equations from the two theories also match. We now move to supersymmetry transformations of fermions. The maximal N = 4 theory contains the gravitini ψ µ and the spin-1 2 fields λˆi. The latter is decomposed into (λ R , λ 5 ). The SO(5) c Γˆi gamma matrices are accordingly decomposed as Γˆi = (Γ R , Γ 5 ). Γ 5 = Γ 1 Γ 2 Γ 3 Γ 4 acts as the chirality matrix of SO(4). Following [18], we make the truncation ǫ ± satisfy Γ 5 ǫ ± = ±ǫ ± with ǫ = ǫ + + ǫ − . We will now drop ± superscript from ǫ, λ and ψ µ . In accordance with the bosonic truncation T ij = (T αβ , T 55 ) = (XT αβ , X −4 ), we truncate the SL(5, R) coset representative as Πî i = (Π R α , Π5 5 ). With the identification Π R α = X − 1 2 V R α and Π5 5 = X 2 , we can writeT αβ in term of SL(4, R) coset representative V R α as We then find that equations (3.11) and (3.12) become The constraint Γˆiλˆi = 0 imposes the condition λ + 5 = −Γ R λ − R . Therefore, the independent fields will be ψ µ and λ R . This is the reason for excluding δλ 5 in the above equations. We then identify Γ R λ R with χ andλ R = λ R − 1 4 Γ R Γ S λ S with λ r in (2.17). Note thatλ R has only three independent components due to the condition Γ Rλ R = 0. With these and the odd-dimensional self-duality, we end up with, after some gamma matrix algebra, In the above equations, we have used the following definitions Notice that with our convention for Γ 5 ǫ = ǫ, Γ RS is anti-self dual. The field strength F RS (2) appearing in (3.43) and (3.44) must be accordingly anti-self dual. This should be identified with the SU(2) field strength F i (2) in (2.15) and (2.16). On the other hand, the self dual part of F RS (2) appears in (3.45) and should be identified with F r (2) in (2.17). Using the relation C = − 3 2 √ 2 g 1T and identifying F RS Γ RS = −2 √ 2iF i σ i , we can see that equations (3.43) and (3.44) match with equations (2.15) and (2.16) after using the relation g 1 = −2g and gamma matrix identities such as γ µ γ νρ = γ νρ Note that in order to match the gravitino variation, we need to multiply (3.43) by two. Comparing (2.17) and (3.45) is more complicated. The SO(4) gamma matrices Γ R need to be expressed in terms of

Embedding seven-dimensional RG flow to eleven dimensions
In this section, we will use the reduction ansatz obtained in the previous section to uplift some seven-dimensional solutions. The dimensional reduction gives rise to the condition g 2 = g 1 . This makes the supersymmetric AdS 7 critical point with SO(3) diag symmetry found in [16] disappears. Accordingly, the flow solution given in [16] cannot be uplifted to eleven dimensions with the present reduction ansatz. However, to give examples of the uplifted solutions, we will study other solutions in the case of g 2 = g 1 .

Uplifting AdS 7 solutions
We now further truncate the nine scalars given byT αβ to one scalar invariant under SO(3) diag ⊂ SO(3) × SO(3) ∼ SO(4). This scalar sector has already been studied in [16]. We will give more solutions in this section. Under SO(3) diag , the nine scalars transform as 1 + 3 + 5. There is only one singlet. It can be checked that the SO(3) diag singlet correspond to T αβ can be written more compactly asT αβ = (δ ab e φ , e −3φ ) for a, b = 1, 2, 3. By using (3.34) and the explicit form of Γ I and η A given in [15], it is easy to verify that this V precisely gives the SO(3, 3)/SO(3) × SO(3) coset representative L used in [16]. Using this and the relation X = e − σ 2 , we find the scalar potential This potential admits two AdS 7 critical points given by where we have used g = 8h or equivalently g 1 = −16h as given in [16]. By using the BPS equations given in [16], which are repeated below, we see that the second critical point is non-supersymmetric. Scalar masses at this critical point can be computed to be where the AdS 7 radius is given by L = − 15 V 0 . The three massless scalars are the expected Goldstone bosons corresponding to the symmetry breaking of SO(4) to SO(3). One of the 1 and 5 scalars have masses below the BF bound m 2 L 2 = −9, so this critical point is unstable.
The first critical point is the trivial point preserving all supersymmetries and the full SO(4) gauge symmetry. The scalar masses can be found in [16]. We will now uplift this AdS 7 vacuum to eleven dimensions. We begin with the coordinates µ α = (cos ψμ a , sin ψ) in whichμ aμa = 1. Since σ = φ = 0, we then find ∆ = 1 and where dΩ 2 2 is the metric on the two-sphere. The eleven dimensional geometry is given by AdS 7 × S 4 . Turning on the dilaton σ would deform the four-sphere but leave the S 3 inside invariant. If φ, σ = 0, the metric would be further deformed in such a way that the S 2 part described by dΩ 2 2 is invariant. The unbroken symmetry in this case is the SO(3) isometry of this S 2 identified with the unbroken SO(3) diag . The SO(3) critical point is however unstable. Therefore, we will not consider AdS 7 solution with SO(3) symmetry.

Uplifting RG flows to non-conformal SO(3) Super Yang-Mills
To give more examples, we will study RG flow solutions to non-conformal Super Yang-Mills theories in the IR. We will work in the theory of section 2. With g 2 = g 1 and the standard domain wall metric ansatz ds 2 7 = e A(r) dx 2 1,5 + dr 2 , the BPS equations taken from [16] become (4.9) in which d dr is denoted by ′ . After changing to the new coordinater given by dr dr = e − σ 2 , we find the solution The solution interpolates between an AdS 7 in the UV,r ∼ r → ∞, and a domain wall in the IR, 4hr →C, for a constantC. At the UV, the solution becomes The eleven-dimensional metric is given by (4.5).
In the IR, we find that φ blows up as φ ∼ − ln(4hr −C) (4.14) for a constantC. The behaviour of σ depends on the value of the integration constant C 2 . For C 2 = 0, we find where we have used the relation betweenr and r in the IR limit with C being another integration constant. The seven-dimensional metric is given by Both cases give V → −∞, so the solution is physical by the criterion of [24]. We now look at the eleven-dimensional geometry. For C 2 = 0 and C 2 = 0, the eleven dimensional metric is given respectively by where 14 3 hρ 6 7 = 4hr − C. As expected, when turning on φ and σ, the warped factors involve coordinates (ξ, ψ). The S 4 is then deformed leaving the S 2 intact. If only σ = 0, the S 3 part of the internal metric would be invariant as pointed in [17]. The deformation with only φ = 0 is not possible since the BPS equation for σ would imply φ = 0 as pointed out in [16].

Conclusions
In this paper, we have constructed N = 2 SO(4) gauged supergravity in seven dimensions with topological mass term. The resulting theory admit AdS 7 vacua and could be useful in the context of the AdS/CFT correspondence. The resulting reduction ansatz has been found by truncating the S 4 reduction leading to N = 4 SO(5) gauged supergravity and can be used to uplift seven-dimensional solutions to eleven dimensions. We have also constructed new seven-dimensional RG flow solutions and uplifted the resulting solutions to eleven dimensions. The flows can be interpreted as deformations of the UV N = (1, 0) SCFT in six dimensions with SO(4) symmetry to non-conformal SYM with SO(3) diag symmetry. These deformations are driven by vacuum expectation values of dimension 4 operators. Additionally, the result of this paper can be used to uplift flows to SO(2) non-conformal gauge theories studied in [16] for g 2 = g 1 .
However, the RG flow between two supersymmetric AdS 7 critical points recently found in [16] cannot be uplifted by using the reduction ansatz constructed here. It would be interesting to find an embedding of this solution in 10 or 11 dimensions. It is also interesting to extend the reduction ansatz given here to non-compact gauge groups SO(3, 1) and SO (2,2). The internal manifold should involve hyperbolic spaces H 3,1 and H 2,2 , respectively. Other possible non-compact gauge groups are SL(3, R), SO(2, 1) and SO(2, 2) × SO(2, 1). It would be very interesting to find higher dimensional origins for these gauge groups as well. Finally, more insight to six-dimensional gauge theories might be gained from studying these seven-dimensional gauged supergravities via AdS 7 /CFT 6 correspondence. We hope to come back to these issues in future works.