Twisted compactification of N=2 5D SCFTs to three and two dimensions from F(4) gauged supergravity

We study supersymmetric $AdS_4\times \Sigma_2$ and $AdS_3\times \Sigma_3$ solutions in half-maximal gauged supergravity in six dimensions with $SU(2)_R\times SU(2)$ gauge group. The gauged supergravity is obtained by coupling three vector multiplets to the pure $F(4)$ gauged supergravity. The $SU(2)_R$ R-symmetry together with the $SO(3)\sim SU(2)$ symmetry of the vector multiplets are gauged. The resulting gauged supergravity admits supersymmetric $AdS_6$ critical points with $SO(4)\sim SU(2)\times SU(2)$ and $SO(3)\sim SU(2)_{\textrm{diag}}$ symmetries. The former corresponds to five-dimensional $N=2$ superconformal field theories (SCFTs) with $E_1\sim SU(2)$ symmetry. We find new classes of supersymmetric $AdS_4\times \Sigma_2$ and $AdS_3\times \Sigma_3$ solutions with $\Sigma_{2,3}$ being $S^{2,3}$ and $H^{2,3}$. These solutions describe SCFTs in three and two dimensions obtained from twisted compactifications of the aforementioned five-dimensional SCFTs with different numbers of unbroken supersymmetry and various types of global symmetries.


Introduction
Field theories in six and five dimensions have been shown to posses non-trivial conformal fixed points [1,2]. However, higher dimensional superconformal field theories (SCFTs) are not well understood as their lower dimensional analogues. The study of five-dimensional SCFTs using the AdS/CFT correspondence [3] has attracted a lot of attention both from ten and six-dimensional point of views, see for example [4,5,6,7,8,9]. And recently, the investigation of supersymmetric AdS 6 solutions has been carried out systematically in [10,11,12].
An approach to understand higher dimensional field theories is to make some compactification of these theories to lower dimensions. The resulting lower dimensional field theories preserving some supersymmetry are usually obtained by twisted compactifications, and the holographic study via the AdS/CFT correspondence is still applicable at least in the large N limit [13]. From string/M theory point of view, these twisted field theories can be interpreted as wrapped branes on certain curved manifolds. In many cases, there is a description in terms of lower dimensional gauged supergravities. In particular, for the present case of five-dimensional SCFTs, the effective supergravity theory is the N = (1, 1) F (4) gauged supergravity and its matter-coupled version [14].
In this work, we will explore some aspects of twisted compactifications of fivedimensional SCFTs within the framework of half-maximal gauged supergravity in six dimensions coupled to matter multiplets [15,16]. A similar study in the pure F (4) gauged supergravity [17] have been carried out in [18] in which some AdS 4 × Σ 2 and AdS 3 × Σ 3 solutions have been identified along with their possible dual field theories. We will further investigate solutions of this type in the matter-coupled F (4) gauged supergravity. This could presumably give rise to more general solutions than those given in [18]. The result would also provide new solutions describing IR fixed points of the RG flows from SCFTs in five dimensions to three and two-dimensional SCFTs with different numbers of supersymmetry.
As a starting point, we add three vector multiplets to the F (4) gauged supergravity resulting in an SU(2) R × SU(2) ∼ SO(3) R × SO(3) gauge group with the first factor being the R-symmetry group. AdS 6 vacua of this theory including possible holographic RG flows between the dual SCFTs and RG flows to non-conformal field theories have already been studied in [8] and [9]. From the result in [8], there are two supersymmetric AdS 6 critial points. Both of them preserve the full sixteen supercharges, but one of them, with non-vanishing scalar fields, break the full SU(2) R × SU(2) symmetry to its diagonal subgruop. These two critical points are dual to certain N = 2 SCFTs in five dimensions by the usual AdS/CFT correspondence.
We then proceed by looking for possible AdS 4 ×Σ 2 and AdS 3 ×Σ 3 solutions for Σ 2,3 being S 2,3 or H 2,3 with different residual symmetries. The resulting solutions would be dual to SCFTs in three and two dimensions obtained from twisted compactifications of the above mentioned five-dimensional SCFTs. These will give new AdS 4 and AdS 3 solutions from six-dimensional gauged supergravity and provide appropriate gravity backgrounds in the holographic study of gauge theories in five and lower dimensions. The paper is organized as follow. We give a brief review of the F (4) gauged supergravity coupled to three vector multiplets in section 2. Possible supersymmetric AdS 4 and AdS 3 solutions are given in section 3 and 4, respectively. In section 5, we give some conclusions and comments about the results. We also include an appendix describing supersymmetric AdS 6 critical points previously found in [8] as well as an analytic RG flow between them.

Matter coupled
In this paper, we are interested in N = (1, 1) gauged supergravity with SU(2) × SU(2) gauge group. This gauged supergravity can be obtained by coupling three vector multiplets to the pure F (4) gauged supergravity constructed in [17]. The full construction by using the superspace approach can be found in [15,16]. Apart from different metric signature (− + + + ++), we will mostly follow the notations and conventions given in [15] and [16]. The matter coupled N = (1, 1) gauged supersymmetry consists of the supergravity multiplet given by e a µ , ψ A µ , A α µ , B µν , χ A , σ and three vector multiplets with the field content In the above expressions, ψ A µ , χ A and λ A denote the gravitini, the spin-1 2 fields and the gauginos, respectively. All spinor fields χ A , ψ A µ and λ A as well as the supersymmetry parameter ǫ A are eight-component pseudo-Majorana spinors with indices A, B = 1, 2 referring to the fundamental representation of the SU(2) R ∼ USp(2) R R-symmetry. Space-time and tangent space indices are denoted respectively by µ, ν = 0, . . . , 5 and a, b = 0, . . . , 5. e a µ and σ are the graviton and the dilaton. A α µ , α = 0, 1, 2, 3, are four vector fields in the gravity multiplet. Three of these vector fields will be used to gauge the SU(2) R R-symmetry. The index I = 1, 2, 3 labels the three vector multiplets, and finally B µν is the two-form field which admits a mass term.
with the following identification The structure constants of the full SU(2) R × SU(2) gauge group f Λ ΠΣ will be split into ǫ rst and C IJK = ǫ IJK for the two factors SU(2) R and SU(2), respectively. There are accordingly two coupling constants denoted by g 1 and g 2 .
In order to parametrize scalar fields described by the SO(4, 3)/SO(4) × SO(3) coset, we introduce basis elements of 7 × 7 matrices by (e ΛΣ ) ΓΠ = δ ΛΓ δ ΣΠ , Λ, Σ, Γ, Π = 0, . . . , 6 . In this paper, we are not interested in solutions with non-zero two-form field. We therefore set B µν = 0 from now on. The bosonic Lagrangian involving only the metric, vectors and scalar fields is given by [16] where e = √ −g. We have written the scalar kinetic term in term of P Iα µ = P Iα i ∂ µ φ i , i = 1, . . . , 12. The explicit form of the scalar potential is given by where N 00 is the 00 component of the scalar matrix N ΛΣ defined by Various quantities appearing in the scalar potential and the supersymmetry transformations given below are defined as follow (2.10) Finally, we need supersymmetry transformations of χ A , λ I A and ψ A µ to find supersymmetric bosonic solutions. These transformation rules with vanishing B µν field are given by where σ tC B are usual Pauli matrices, and ǫ AB = −ǫ BA . In our convention, the spacetime gamma matrices γ a satisfy {γ a , γ b } = 2η ab , η ab = diag(−1, 1, 1, 1, 1, 1), (2.14) and γ 7 = γ 0 γ 1 γ 2 γ 3 γ 4 γ 5 with γ 2 7 = 1. The covariant derivative of ǫ A is given by It should be noted that due to some difference in conventions, the above supersymmetry transformations do not coincide with those of the pure F (4) gauged supergravity given in [17] when all of the fields in the vector multiplets are set to zero. However, it can be verified that the transformation rules in [17] are recovered by using the identifications The SU(2) R ×SU(2) gauged supergravity admits maximally supersymmetric AdS 6 critical points when m = 0. One of them is the trivial critical point at which all scalars vanish after setting g 1 = 3m. This critical point preserves the full SU(2) R × SU(2) symmetry and should be dual to the five-dimensional SCFT with global symmetry E 1 ∼ SU (2). Furthermore, at the vacuum, the U(1) gauge field A 0 will be eaten by the two-form field resulting in a massive B µν field. Another supersymmetric AdS 6 critical point preserves only the diagonal subgroup SU(2) diag ⊂ SU(2) R × SU(2). This critical point has been mistakenly identified as a stable non-supersymmetric AdS 6 in [8], see also the associated erratum.
Actually, the non-trivial supersymmetric critical point can also be seen from the BPS equations studied in [9], but that paper mainly considers RG flows from fivedimensional SCFTs corresponding to the trivial AdS 6 critical point to non-conformal field theories in the IR. We give the analysis of these two supersymmetric AdS 6 critical points in the appendix together with an analytic RG flow between them. This flow solution have already been studied numerically in [8]. The critical points and the flow solution are similar to the corresponding solutions in the half-maximal gauged supergravity with SO(4) gauge group in seven dimensions studied in [19].

AdS 4 critical points
In this section, we consider solutions of the form AdS 4 × S 2 or AdS 4 × H 2 with S 2 and H 2 being a two-sphere and a two-dimensional hyperbolic space, respectively. The metric takes the form of for the S 2 case and for the H 2 case. In both cases, the warp factors F and G are functions only of r.
The non-vanishing spin connections of the above metrics are given respectively by and where ′ denotes the r-derivative.
To find supersymmetric solutions of the form AdS 4 ×Σ 2 with SO(2)×SO(2) symmetry, we turn on SO(2)×SO(2) gauge fields such that the spin connection along Σ is canceled.
In the present case, there are six gauge fields (A r , A I ) corresponding to SU(2) R ×SU(2) gauge group. We will turn on the following SO(2) × SO(2) gauge fields A 3 = a cos θdφ and for the S 2 case and A 3 = a y dx and for the H 2 case. To avoid confusion, we have given the gauge fields using the index Λ = 0, 1, . . . , 6. A 3 will appear in the covariant derivative of ǫ A since it is part of the SU(2) R gauge fields. We choose this particular form of the gauge field to cancel the spin connection on Σ 2 . Accordingly, the Killing spinors corresponding to unbroken supersymmetry will be constant spinors on Σ 2 provided that we impose the twist condition and a set of projection conditions given below. There are two scalars which are singlet under SO(2) × SO(2) generated by J 12 1 and J 12 2 . The SO(4, 3)/SO(4) × SO(3) coset representative can be written in terms of these scalars as Imposing the projection conditions for the S 2 case or for the H 2 case, we find that consistency of the BPS equations from δψ Aµ , for µ = 0, 1, 2, requires φ 1 = 0. Setting φ 1 = 0, we obtain the following BPS equations where λ = 1 and λ = −1 for S 2 and H 2 cases, respectively. We look for fixed point solutions satisfying G ′ = σ ′ = φ ′ 2 = 0 and F ∼ r. The γr projector is not necessary for constant scalars since γr only appears with the rderivative. The BPS equations are automatically satisfied by the fixed point solutions without imposing the γr projector. Furthermore, with φ 1 = 0, the γ 7 projection is not needed. Therefore, the AdS 4 fixed points will preserve half of the original supersymmetry corresponding to eight supercharges or N = 2 superconformal symmetry in three dimensions.
The explicit form of AdS 4 critical point is given by In the above equations, we have given a solution in the S 2 case for definiteness. A similar solution in the H 2 case can be obtained by replacing (a, b) by (−a, −b) in the above solution. For a < 0, the solution is valid provided that b < a or b > −a. When a > 0, we have a real solution for b < a or b > a. It can be checked that there exist both AdS 4 × S 2 and AdS 4 × H 2 fixed points.

AdS 3 critical points
We now look for a gravity dual of five-dimensional SCFTs compactified on a threemanifold Σ 3 which can be S 3 or H 3 . The IR effective theories would be two-dimensional field theories. We particularly look for the gravity solutions corresponding to conformal field theories in the IR, so the gravity solutions will take the form of AdS 3 × Σ 3 .
The metrics and the associated spin connections for each case are given by ds 2 7 = e 2F dx 2 1,2 + dr 2 + e 2G dψ 2 + sin 2 ψ(dθ 2 + sin 2 θdφ 2 ) (4.1) for Σ 3 = S 3 with the spin connections and ds 2 7 = e 2F dx 2 1,2 + dr 2 + e 2G y 2 (dx 2 + dy 2 + dz 2 ) (4.3) for Σ 3 = H 3 with the spin connections given by while, for the H 3 case, they are given by In both cases, the SO(3) gauge fields are related to the SU(2) R gauge fields by The two sets of gauge fields implement the SO(3) diag gauge fields. Furthermore, the twist condition implies a = b = c and g 1 a = 1. Using the following projectors we obtain the BPS equations (1 − e 4φ )], (4.10) (4.11) where λ = ±1 for S 3 and H 3 as in the previous cases. Note also that, in both cases, the last projector is not independent from the second and the third ones. Therefore, fixed point solutions will preserve four supercharges or equivalently N = (1, 1) superconformal symmetry in two dimensions. The analysis of unbroken supersymmetry can be done in a similar manner to that given in [18].
We begin with a simple fixed point solution with g 2 = g 1 . In this case, only AdS 3 × H 3 solution exists and is given by (4.14) For g 2 = g 1 , we find two classes of solutions. In the first class, only AdS 3 × H 3 is possible and given by where we have chosen g 1 > 0 and g 2 < −g 1 . For positive g 2 with g 2 > g 1 and g 1 > 0, we find another (4.16) The second class of solutions is given by 2 (e 4φ − 1) λa where φ is a solution to the following equation The explicit form of φ can be written, but we refrain from giving such a complicated expression. We will however give some examples of the solutions. Using the relation g 1 = 3m and choosing g 2 = 1 2 g 1 , we find an It is also possible to obtain an AdS 3 × H 3 solution for g 2 = 1 2 g 1 . This solution is given by  (2) singlet. This corresponds to the non-compact generator Y 03 . We then parametrize the coset representative as (4.22) The SO(3) R gauge fields are the same as in the previous case while the SO(2) gauge field will be chosen to be for Σ 3 = S 3 and Σ 3 = H 3 , respectively. Apart from the projectors in (4.8) and (4.9), in this case, we need to impose an additional projector involving γ 7 namely The critical points would then preserve only half of the supersymmetry in the previous case. This corresponds to N = (1, 0) superconformal symmetry in two dimensions. With all these and λ = ±1 for S 3 and H 3 , respectively, we find the following BPS (4.28) When Φ = 0 and b = 0, the above equations reduce to the pure F (4) gauged supergravity which admits only AdS 3 × H 3 solutions in agreement with [18]. We find an AdS 3 × Σ 3 fixed point given by .   [20].

Conclusions
All of these solutions correspond to IR fixed points of five-dimensional SCFTs with global symmetry SU(2) in lower dimensional space-time. There should be RG flows describing twisted compactifications of these SCFTs on 2 or 3-manifolds giving rise to these AdS 4 and AdS 3 geometries in the IR. We have not been able to find analytic solutions for these flows, but numerical solutions can be obtained as in other cases, see for example [20]. The results obtained in this paper are hopefully useful in the holographic study of five-dimensional SCFTs and their compactifications as well as the classification of vacua of the half-maximal gauged supergravity in six dimensions.
It would be interesting to find a possible embedding of these solutions in higher dimensions in particular in massive type IIA supergravity similar to the embedding of pure F (4) gauged supergravity [23] or in type IIB supergravity as in [11] and [24]. This could give an interpretation to these solutions in terms of wrapped D4-branes. However, since there is only one class of known AdS 6 solutions, as shown in [10], embedding the AdS 6 solutions with different SU(2) gauge coupling constants (if possible) might not be straightforward in massive type IIA theory.
It is also interesting to find dual field theories to the AdS 4 and AdS 3 critical points identified here. Another investigation would be to study other types of gauge groups such as non-compact gauge groups to the matter-coupled F (4) gauged supergravity and classify all possible gauge groups that admit supersymmetric AdS 6 vacuum similar to the recent analysis in seven dimensions [25,26]. Finally, gravity solutions with a non-vanishing two-form field could be of interest. A simple AdS 3 × R 3 solution with only the two-form and the dilaton turned on has been studied in [18]. It might be interesting to study this type of solutions and a more general twist involving B µν field within the framework of the matter coupled gauged supergravity. We leave these issues for future works.

Acknowledgments
This work is supported by Chulalongkorn University through Ratchadapisek Sompoch Endowment Fund under grant Sci-Super 2014-001. The author would like to give a special thank to Block 7 Company Limited where some parts of this work have been done.

A. Supersymmetric AdS 6 critical points and holographic RG flows
In this appendix, we review a description of supersymmetric AdS 6 critical points with SU(2)×SU(2) and SU(2) diag symmetries. We consider only the SU(2) diag singlet scalar corresponding to the non-compact generator Y s defined by The SO(4, 3)/SO(4) × SO(3) coset representative is accordingly parametrized by The scalar potential can be computed to be V = 1 16 e 2σ (g 2 1 + g 2 2 )[cosh(6φ) − 9 cosh(2φ)] + 8(g 2 2 − g 2 1 ) + 8g 1 g 2 sinh 3 (2φ) +e −6σ m 2 − 4e −2σ m(g 1 cosh 3 φ − g 2 sinh 3 φ). (A.3) We are mainly interested in supersymmetric critical points. Therefore, we set up the BPS equations from supersymmetry transformations of fermions given in (2.11), (2.12) and (2.13) with all but the metric and scalars σ and φ vanishing. The sixdimensional matric is taken to be the standard domain wall where the r-derivative is denoted by ′ . From these equations, it is clearly seen that there are two supersymmetric critical points namely φ = 0, σ = 1 4 ln 3m g 1 , V 0 = −20m 2 g 1 3m This critical point is valid for g 2 < −g 1 when g 1 > 0. For g 1 < 0, we need to take g 2 < g 1 . The full scalar mass spectrum at these two critical points can be found in [8].
(A. 12) In order to make this solution interpolate between the two critical points (A.8) and (A.9), we choose the constant C 1 to be which gives the solution for σ σ = 1 4 ln 3me −φ [(1 − e 2φ )g 1 + (1 + e 2φ )g 2 ] 2g 1 g 2 . (A.14) By the same procedure, we find the solution for A(r) up to an additive integration constant that can be absorbed by rescaling the coordinates in dx It should be noted that the critical points and the flow solution have a similar structure to those in seven dimensions [19].