Two-dimensional SCFTs from matter-coupled 7D N=2 gauged supergravity

We study supersymmetric $AdS_3\times M^4$ solutions of $N=2$ gauged supergravity in seven dimensions coupled to three vector multiplets with $SO(4)\sim SO(3)\times SO(3)$ gauge group and $M^4$ being a four-manifold with constant curvature. The gauged supergravity admits two supersymmetric $AdS_7$ critical points with $SO(4)$ and $SO(3)$ symmetries corresponding to $N=(1,0)$ superconformal field theories (SCFTs) in six dimensions. For $M^4=\Sigma^2\times\Sigma^2$ with $\Sigma^2$ being a Riemann surface, we obtain a large class of supersymmetric $AdS_3\times \Sigma^2\times \Sigma^2$ solutions preserving four supercharges and $SO(2)\times SO(2)$ symmetry for one of the $\Sigma^2$ being a hyperbolic space $H^2$, and the solutions are dual to $N=(2,0)$ SCFTs in two dimensions. For a smaller symmetry $SO(2)$, only $AdS_3\times H^2\times H^2$ solutions exist. Some of these are also solutions of pure $N=2$ gauged supergravity with $SU(2)\sim SO(3)$ gauge group. We numerically study domain walls interpolating between the two supersymmetric $AdS_7$ vacua and these geometries. The solutions describe holographic RG flows across dimensions from $N=(1,0)$ SCFTs in six dimensions to $N=(2,0)$ two-dimensional SCFTs in the IR. Similar solutions for $M^4$ being a Kahler four-cycle with negative curvature are also given. In addition, unlike $M^4=\Sigma^2\times \Sigma^2$ case, it is possible to twist by $SO(3)_{\textrm{diag}}$ gauge fields resulting in two-dimensional $N=(1,0)$ SCFTs. Some of the solutions can be uplifted to eleven dimensions and provide a new class of $AdS_3\times M^4\times S^4$ solutions in M-theory.


Introduction
One of the most interesting implications of the AdS/CFT correspondence [1] is the study of holographic RG flows. These solutions take the form of a domain wall interpolating between AdS vacua and holographically describe deformations of a conformal field theory (CFT) in the UV to another CFT in the IR or in some cases to a non-conformal field theory dual to a singular geometry, see [2,3,4] for example. Of particular interest are RG flows across dimensions in which a higher dimensional CFT flows to a lower dimensional CFT. This type of RG flows allows us to investigate the structure and dynamics of less known CFTs in higher, especially five and six, dimensions using the well-understood lower dimensional CFTs. In this paper, we will consider this type of RG flows in six-dimensional CFTs to two dimensions. Furthermore, the study along this direction is much more fruitful and controllable in the presence of supersymmetry. We are then mainly interested in RG flows within superconformal field theories (SCFTs).
Supersymmetric solutions of gauged supergravities play an important role in studying the aforementioned RG flows. In general, RG flows across dimensions from a d-dimensional SCFT to a (d − n)-dimensional SCFT are obtained by twisted compactification of the former on an n-dimensional manifold M n . The twist is needed for the compactification to preserve some amount of supersymmetry. This is achieved by turning on some gauge fields to cancel the spin connection on M n . In the supergravity dual, these RG flows are described by domain walls interpolating between an AdS d+1 vacuum to an AdS d+1−n × M n geometry. Solutions of this type have been studied in various dimensions, see [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26] for an incomplete list.
In this paper, we are interested in supersymmetric AdS 3 × M 4 solutions of N = 2 gauged supergravity in seven dimensions with SO(4) ∼ SO(3) × SO(3) gauge group. This gauged supergravity is obtained by coupling three vector multiplets to pure N = 2 gauged supergravity with SU(2) gauge group constructed in [27,28]. The matter-coupled gauged supergravity has been constructed in [29,30,31] with an extension to include a topological mass term for the threeform field, dual to the two-form in the N = 2 supergravity multiplet, given in [32]. This massive gauged supergravity admits supersymmetric AdS 7 vacua which has been extensively studied in [33,34,35]. These vacua are dual to N = (1, 0) SCFTs in six dimensions, and a number of RG flows of various types have already been studied [18,33,36]. However, holographic RG flows from N = (1, 0) six-dimensional SCFTs to two-dimensional SCFTs in the framework of mattercoupled N = 2 gauged supergravity have not appeared so far. To fill this gap, we will give a large class of AdS 3 × M 4 fixed points and the corresponding RG flows across dimensions within six-dimensional N = (1, 0) SCFTs.
We will consider a four-manifold M 4 with constant curvature of two types, a product of two Riemann surfaces Σ 2 × Σ 2 and a Kahler four-cycle M 4 k . In the first case, the twists can be performed by using SO(2) R ⊂ SO(3) R with SO(3) R being the R-symmetry. We will look for solutions with SO(2) × SO (2), SO(2) diag and SO(2) R symmetries. In the second case, M 4 k has a U(2) ∼ SU(2) × U(1) spin connection. Therefore, we can perform the twists by turning on either SO(2) R ⊂ SO(3) R or the full SO(3) R to cancel the U(1) or the SU(2) parts of the spin connection, respectively. It should also be noted that a twist by cancelling the full U(2) spin connection is not possible since the R-symmetry of N = 2 gauged supergravity is not large enough.
In general, the two SO(3) ∼ SU(2) factors in the SO(4) gauge group can have different coupling constants. However, for a particular case of equal SU (2) coupling constants, the resulting gauged supergravity can be embedded in elevendimensional supergravity via a truncation on S 4 [37]. The seven-dimensional solutions can accordingly be uplifted to eleven dimensions giving rise to new AdS 3 × M 4 × S 4 solutions of eleven-dimensional supergravity. Therefore, these solutions provide a number of new two-dimensional SCFTs with known M-theory dual. We also consider the uplifted solutions in this case.
The paper is organized as follow. In section 2, we give a short review of the matter coupled N = 2 seven-dimensional gauged supergravity and suppersymmetric AdS 7 vacua. In sections 3 and 4, we look for supersymmetric AdS 3 × Σ 2 × Σ 2 and AdS 3 × M 4 k solutions and numerically study interpolating solutions between these geometries and the AdS 7 fixed points. We finally give some conclusions and comments in section 5. Relevant formulae for the truncation of eleven-dimensional supergravity on S 4 giving rise to N = 2 gauged supergravity with SO(4) gauge group are reviewed in the appendix.
2 Seven-dimensional N = 2, SO(4) gauged supergravity and supersymmetric AdS 7 vacua We firstly review N = 2 gauged supergravity in seven dimensions coupled to three vector multiplets with SO(4) gauge group. Only relevant formulae involving bosonic Lagrangian and supersymmetry transformations of fermions will be presented. The detailed construction of general N = 2 seven-dimensional gauged supergravity can be found in [32], see also [38] for gaugings in the embedding tensor formalism.
2.1 Seven-dimensional N = 2, SO(4) gauged supergravity The seven-dimensional N = 2, SO(4) gauged supergravity is obtained by coupling the minimal N = 2 supergravity to three vector multiplets. The supergravity multiplet consists of the graviton eμ µ , two gravitini ψ a µ , three vectors A i µ , two spin-1 2 fields χ a , a two-form field B µν and the dilaton σ. Each vector multiplet contains a vector field A µ , two gaugini λ a , and three scalars φ i . We will use the convention that curved and flat space-time indices are denoted by µ, ν andμ, ν respectively. Indices i, j = 1, 2, 3 and a, b = 1, 2 label triplet and doublet of SO(3) R ∼ SU(2) R R-symmetry with the latter being suppressed throughout this work. The three vector multiplets will be labeled by indices r, s = 1, 2, 3 which in turn describe the triplet of the matter symmetry SO(3) under which the three vector multiplets transform.
From both supergravity and vector multiplets, there are in total six vector fields denoted collectively by A I = (A i , A r ). Indices I, J, . . . = 1, 2, . . . , 6 describe fundamental representation of the global symmetry SO (3,3) and are lowered and raised by the SO(3, 3) invariant tensor η IJ = diag(−1, −1, −1, 1, 1, 1) and its inverse η IJ . The two-form field will be dualized to a three-form C µνρ , which admits a topological mass term required by the existence of AdS 7 vacua.
The nine scalar fields φ ir parametrize SO(3, 3)/SO(3)×SO(3) coset manifold. They can be described by the coset representative with an index A = (i, r) corresponding to representations of the compact SO(3)× SO(3) local symmetry. The inverse of L I A will be denoted by with the relation L j Being an element of SO (3,3), the coset representative also satisfies the relation The bosonic Lagrangian of the N = 2, SO(4) gauged supergravity in form language can be written as The constant h describes the topological mass term for the three-form C (3) with the field strength H (4) = dC (3) . The gauge field strength is defined by The definition of the SO(4) structure constants f IJ K includes the gauge coupling constants where g 1 and g 2 are coupling constants of SO(3) R and SO (3), respectively. The scalar matrix a IJ appearing in the kinetic term of vector fields is given in term of the coset representative as follow The Chern-Simons three-form satisfying The scalar potential is given by where C-functions, or fermion-shift matrices, are defined as It should also be noted that indices i, j and r, s are raised and lowered by δ ij and δ rs , respectively. Finally, the scalar kinetic term is defined in term of the vielbein on the SO(3, 3)/SO(3) × SO(3) coset as To find supersymmetric solutions, we need supersymmetry transformations of fermionic fields ψ µ , χ and λ r . With all fermionic fields vanishing, these transformations read where σ i are the usual Pauli matrices. The dressed field strengths F i and F r are defined by the relations and The covariant derivative of the supersymmetry parameter ǫ is given by where Q i µ is defined in term of the composite connection Q ij µ as For convenience, we also give the full bosonic field equations derived from the Lagrangian given in (5) d(e −2σ * H (4) ) + 8hH (4)

Supersymmetric AdS 7 critical points
We now give a brief review of supersymmetric AdS 7 vacua found in [33]. There are two supersymmetric N = 2 AdS 7 critical points with SO(4) ∼ SO we can write non-compact generators of SO(3, 3) as Among the nine scalars from SO(3, 3)/SO(3)×SO(3), there is one SO(3) diag singlet corresponding to the non-compact generator The coset representative is then given by The scalar potential for the dilaton σ and the SO(3) diag singlet scalar φ is readily computed to be This potential admits two supersymmetric AdS 7 critical points I : Critical points I and II have SO(4) and SO(3) diag symmetries, respectively. We have also chosen g 1 = 16h to bring the SO(4) critical point to the value σ = 0. The cosmological constant is denoted by V 0 . According to the AdS/CFT correspondence, these critical points correspond to N = (1, 0) SCFTs in six dimensions with SO(4) and SO(3) symmetries, respectively. A holographic RG flow interpolating between these two critical points has already been studied in [33], see also [39] for more general solutions. In subsequent sections, we will find supersymmetric AdS 3 × M 4 solutions to this N = 2 SO(4) gauged supergravity and RG flow solutions from the above AdS 7 vacua to these geometries in the IR.
3 Supersymmetric AdS 3 × Σ 2 × Σ 2 solutions and RG flows In this section, we look for supersymmetric solutions of the form AdS 3 ×Σ 2 k 1 ×Σ 2 k 2 with Σ 2 k i for i = 1, 2 being two-dimensional Riemann surfaces. Constants k i describe the curvature of Σ 2 k i with values k i = 1, 0, −1 corresponding to a twodimensional sphere S 2 , a flat space R 2 or a hyperbolic space H 2 , respectively.
We will choose the ansatz for the seven-dimensional metric of the form in which dx 2 1,1 = η αβ dx α dx β , α, β = 0, 1 is the flat metric on the two-dimensional spacetime. The explicit form of the metric on Σ 2 k i can be written as The functions f k i (θ i ) are defined as By using an obvious choice of vielbein we can compute the following non-vanishing components of the spin connection Throughout the paper, we will use primes to denote derivatives of a function with respect to its argument for example solutions which admit nonvanishing Killing spinors, we perform a twist by turning on gauge fields along In the following discussions, we will consider various possible twists with different unbroken symmetries.

AdS 3 vacua with SO(2) × SO(2) symmetry
We first consider solutions with SO(2) × SO(2) symmetry. To perform the twist, we turn on the following SO(2) × SO(2) gauge fields on Σ 2 where p ij are constants magnetic charges. There is one SO(2) × SO(2) singlet scalar from SO(3, 3)/SO(3) × SO(3) coset corresponding to the non-compact generator Y 33 . We then parametrize the coset representative by with φ depending only on the radial coordinate r. By computing the composite connection Q ij µ along Σ 2 k 1 × Σ 2 k 2 , we can cancel the spin connections by imposing the following twist conditions together with the projection conditions Note that only the gauge field A 3 (1) enters the twist procedure since A 3 (1) is the gauge field of SO(2) R ⊂ SO(3) R under which the gravitini and supersymmetry parameters are charged.
From the gauge fields given in (39) and (40), we can straightforwardly compute the corresponding two-form field strengths It should also be noted that these field strengths give non-vanishing F I term. This term is present in the field equation of the three-form fied C (3) as can be seen from equation (22). Therefore, we need to turn on the three-form field with the corresponding four-form field strength given by This is very similar to the solutions of maximal SO(5) gauged supergravity considered in [8].
By imposing an additional projector required by δχ = 0 and δλ r = 0 conditions, we find the following BPS equations It can be verified that these BPS equations satisfy all the field equations. At large r, we have U ∼ V ∼ W ∼ r and φ ∼ σ ′ ∼ e − 4r L with the AdS 7 radius given by L = 1 4h , and the terms involving gauge fields and the three-form field are highly suppressed. We find the SO(4) AdS 7 fixed point from these BPS equations in this limit. The solutions are then symptotically locally AdS 7 as r → ∞.
We now look for supersymmetric AdS 3 solutions satisfying V ′ = W ′ = σ ′ = φ ′ = 0 and U ′ = 1 L AdS 3 in the limit r → −∞. We find a class of AdS 3 fixed point solutions where Note that the coupling constant g 2 does not appear in the above equations, so the solutions can be uplifted to eleven dimensions by setting g 2 = g 1 .
To obtain real solutions, we require that e 2V > 0, e 2W > 0, e σ > 0, and e φ > 0. It turns out that AdS 3 solutions are possible only for one of the two k i is equal to −1 with the seven-dimensional spacetime given by AdS 3 × H 2 × H 2 , AdS 3 ×H 2 ×R 2 and AdS 3 ×H 2 ×S 2 . Since the charges p 11 and p 12 are fixed by the twist conditions (42), there are only two parameters p 21 and p 22 characterizing the solutions. For g 1 = 16h and h = 1, regions in the parameter space (p 21 , p 22 ) for good AdS 3 vacua to exist are shown in figure 1. Note that these regions are precisely the same as supersymmetric AdS 3 × Σ 2 × Σ 2 solutions of maximal seven-dimensional SO(5) gauged supergravity in [8]. These AdS 3 fixed points preserve four supercharges due to the two projectors in (43) and correspond to N = (2, 0) SCFTs in two dimensions with SO(2) × SO(2) symmetry. On the other hand, the entire RG flow solutions interpolating between the AdS 7 fixed point and these AdS 3 geometries preserve only two supercharges due to an extra projector in (47). Examples of these RG flows from the AdS 7 fixed point to AdS 3 ×H 2 ×H 2 , AdS 3 ×H 2 ×R 2 and AdS 3 ×H 2 ×S 2 with h = 1 and different values of p 21 and p 22 are shown in figures 2, 3 and 4, respectively.
From the metric, we see that the SO(2) × SO(2) symmetry corresponds to the isometry along the α and β directions.   (3) symmetry. In this case, there are three SO(2) diag singlets from the nine scalars in SO(3, 3)/SO(3) × SO(3) coset. These correspond to non-compact gen-eratorsŶ

AdS
The coset representative takes the form of The ansatz for SO(2) diag gauge fields is obtained from that of SO(2) × SO (2) given in (39) and (40) by setting g 2 A 6 = g 1 A 3 or, equivalently, We will also simplify the notation by redefining the charges p 1 = p 11 and p 2 = p 12 .
In this case, the four-form field strength is given by Using the projection conditions (43) and (47), we obtain the corresponding BPS equations. It turns out that compatibility between these BPS equations and field equations requires either φ 1 = 0 or φ 3 = 0. Furthermore, setting φ 3 = 0 gives the same BPS equations as setting φ 1 = 0 with φ 3 and φ 1 interchanged. We will then consider only the φ 3 = 0 case with the following BPS equations In this case, solutions to the BPS equations are asymptotic to the two supersymmetric AdS 7 vacua with SO(4) and SO(3) diag symmetries at large r. Furthermore, unlike the previous case, all charge parameters are fixed by the twist conditions, and there exist only AdS 3 × H 2 × H 2 solutions. We now look for AdS 3 fixed points. The solutions also preserve four supercharges and correspond to N = (2, 0) SCFTs in two dimensions as in the previous case. We begin with a class of AdS 3 fixed points for φ 1 = 0 with g 2 > 3g 1 or g 2 < −3g 1 for AdS 3 vacua to exist. An example of RG flows from the SO(4) AdS 7 critical point to this AdS 3 × H 2 × H 2 fixed point for g 2 = 4g 1 and h = 1 is shown in figure 5 with φ 1 set to zero along the flow. Another class of AdS 3 × H 2 × H 2 solutions with φ 1 = 0 is given by     correspond to non-compact generators Y 31 , Y 32 and Y 33 . Therefore, the coset representative can be written as To perform the twist, we take the following ansatz for the SO(2) R gauge field The four-form field strength in this case is given by We can now repeat the same procedure as in the previous two cases to find the corresponding BPS equations. In this case, it turns out that compatibility between the BPS equations and second-order field equations allows only one of the φ i , i = 1, 2, 3, to be non-vanishing. We have verified that any of the φ i leads to the same set of BPS equations. We will choose φ 1 = φ 2 = 0 and φ 3 = 0 for definiteness. With this choice, the BPS equations are given by For these equations, there exist AdS 3 fixed points only for k 1 = k 2 = −1. The resulting AdS 3 × H 2 × H 2 solution is given by This solution again preserves four supercharges and corresponds to N = (2, 0) SCFT in two dimensions. An example of RG flow solutions from N = (1, 0) six-dimensional SCFT to this fixed point for h = 1 and φ 3 = 0 is shown in figure  9. Note that the AdS 3 fixed point and the RG flow are also solutions of pure N = 2 gauged supergravity with SU(2) gauge group. As in the case of AdS 3 solutions with SO(2)×SO(2) symmetry, the above solutions can be uplifted to eleven dimensions by setting g 2 = g 1 . The elevendimensional metric can be obtained from (61) by setting φ = 0 and A 6 (1) = 0, or  equivalently A 12 = A 34 ≡ A 3 . The result is given by with ∆ = e 2σ sin 2 ξ + e − σ 2 cos 2 ξ . (89)

Supersymmetric AdS 3 × M k solutions and RG flows
In this section, we repeat the same analysis for M 4 being a Kahler four-cycle and look for solutions of the form AdS 3 × M 4 k . For the constant k = 1, 0, −1, the Kahler four-cycle is given by a two-dimensional complex space CP 2 , a fourdimensional flat space R 4 , or a two-dimensional complex hyperbolic space CH 2 , respectively. The Kahler four-cycle has U(2) ∼ SU(2) × U(1) spin connection. We can perform a twist by using either SO(2) R ∼ U(1) R or SO(3) R ∼ SU(2) R gauge fields to cancel the U(1) or SU(2) parts of the spin connection.

AdS 3 vacua with SO(2) × SO(2) symmetry
We begin with AdS 3 vacua with SO(2) × SO(2) symmetry and take the following ansatz for the seven-dimensional metric The metric on the Kahler four-cycle M 4 k is given by with ϕ ∈ [0, π 2 ] and the function f k (ϕ) defined by τ i , i = 1, 2, 3, are SU(2) left-invariant one-forms satisfying dτ i = 1 2 ε ijk τ j ∧ τ k . Their explicit form is given by τ 1 = − sin χdθ + cos χ sin θdψ, τ 2 = cos χdθ + sin χ sin θdψ, τ 3 = dχ + cos θdψ . (93) The ranges of the coordinates are θ ∈ [0, π], ψ ∈ [0, 2π], and χ ∈ [0, 4π]. By choosing the following choice of vielbein we find non-vanishing components of the spin connection We can now perform the twist by turning on SO(2) × SO(2) gauge fields with the following ansatz The associated two-form field strengths are given by where J (2) is the Kahler structure defined by To implement the twist, we impose the following projectors on the Killing spinors together with the twist condition As in the previous cases, we need to turn on the three-form field with the field strength With all these and the γ r projector (47), we can derive the following BPS equations with φ being the SO(2) × SO(2) singlet scalar in (41). The BPS equations admit an AdS 3 × CH 2 fixed point given by The AdS 3 solution preserves four supercharges and exists for with g 1 = 16h, k = −1, and h > 0. The AdS 3 × CH 2 fixed point is dual to an N = (2, 0) two-dimensional SCFT. As in the Σ 2 × Σ 2 case, the AdS 3 × CH 2 fixed point and the associated RG flows can be uplifted to eleven dimensions by setting g 2 = g 1 . The elevendimensional metric can be obtained from (61) by replacing e 2V ds 2 by e 2V ds 2 M 4 k and using the gauge fields in (96). We will not repeat it here.

AdS 3 vacua with SO(2) diag symmetry
We next consider solutions with smaller residual symmetry SO(2) diag ⊂ SO(2) × SO(2) by imposing the condition g 2 p 2 = g 1 p 1 . There are three SO(2) diag singlet scalars with the coset representative given by (64). As in the previous section, compatibility between BPS equations and field equations requires φ 1 = 0 or φ 3 = 0, and these two cases are equivalent. We will consider the case of φ 3 = 0 with the following BPS equations There exist two classes of AdS 3 × CH 2 fixed points preserving four supercharges and corresponding to N = (2, 0) SCFTs in two dimensions with SO(2) diag symmetry. With k = −1, the first class of AdS 3 × CH 2 fixed points is given by with g 2 > 3g 1 or g 2 < −3g 1 for AdS 3 vacua to exist. An RG flow solution from the SO(4) AdS 7 critical point to AdS 3 × CH 2 fixed point for φ 1 = 0, g 2 = 4g 1 and h = 1 is shown in figure 11. Another class of AdS 3 × CH 2 fixed points is given by σ = 2 5 ln g 1 g 2 12h (g 2 + g 1 )(g 2 − g 1 ) , To obtain good AdS 3 vacua, we require that g 2 > g 1 . Various RG flows from N = (1, 0) six-dimensional SCFTs with SO(4) and SO(3) symmetries to these fixed points for g 2 = 4g 1 and h = 1 are shown in figures 12, 13 and 14.  As in the case of M 4 = Σ 2 × Σ 2 , all of these AdS 3 fixed points and RG flows cannot be uplifted to eleven dimensions using the truncation given in [37].

AdS 3 vacua with SO(2) R symmetry
By setting p 2 = 0 in the SO(2) × SO(2) case, we obtain solutions with SO(2) R ⊂ SO(3) R symmetry. As in the previous case, the three SO(2) R singlet scalars need to vanish in order for AdS 3 fixed points to exist. We will accordingly set all vector multiplet scalars to zero for brevity. The resulting BPS equations are given by After imposing the twist condition (100), we obtain an AdS 3 solution for k = −1 given by L AdS 3 = 8 3hg 4 1 1 5 .
(118) An RG flow from SO(4) AdS 7 to this fixed point for h = 1 is shown in figure 15.

AdS 3 vacua with SO(3) diag symmetry
For Kahler four-cycles with SU(2)×U(1) spin connection, we can also perform the twist by identifying SO(3) ∼ SU(2) ⊂ SU(2) × U(1) with the gauge symmetry In this case, we will use the metric on M 4 k in the form ds 2 with τ i being the SU(2) left-invariant one-forms given in (93) and f k (ϕ) defined in (36).
With the seven-dimensional vielbein we can compute the following non-vanishing components of the spin connection We then turn on the SO(3) diag gauge fields as follow with the two-form field strengths given by As in the previous cases, we also need a non-vanishing four-form field strength together with the twist condition and the following projectors It should be noted that the second condition in (128) consists of only two independent projectors since γ13 projector can be obtained from the product of those coming from γ12 and γ23. Therefore, the resulting AdS 3 fixed points preserve two supercharges corresponding to N = (1, 0) superconformal symmetry in two dimensions.
With all these and the coset representative for the SO(3) diag singlet scalar in (30), we find the following BPS equations We now look for AdS 3 fixed points for the case of g 2 = g 1 that can be embedded in eleven dimensions. Setting g 2 = g 1 in the above equations, we find the following AdS 3 × CH 2 fixed point . ( An RG flow interpolating between the SO(4) AdS 7 vacuum and this AdS 3 ×CH 2 fixed point is shown in figure 16.

Conclusions
We have studied supersymmetric AdS 3 ×M 4 solutions of N = 2 seven-dimensional gauged supergravity with SO(4) ∼ SU(2) × SU(2) gauge group. For M 4 being a product of two Riemann surfaces, we have found a large class of AdS 3 × H 2 × Σ 2 solutions with SO(2) × SO(2) symmetry for Σ 2 = S 2 , R 2 , H 2 similar to the corresponding solutions in maximal SO(5) gauged supergravity studied in [8]. Fur-thermore, there exist a number of AdS 3 × H 2 × H 2 solutions with SO(2) diag and SO(2) R symmetries. In the latter case, all scalars from vector multiplets vanish, so the AdS 3 × H 2 × H 2 solution can be interpreted as a solution of pure N = 2 gauged supergravity with SU(2) gauge group. We have also numerically given various holographic RG flows from supersymmetric AdS 7 vacua with SO(4) and SO (3) (2), SO(2) diag and SO(2) R symmetries. The solutions preserve four supercharges and correspond to N = (2, 0) two-dimensional SCFTs. For a twist along the SU(2) ∼ SO(3) part, we have performed the twist by turning on the SO(3) diag gauge fields. Unlike the previous cases, the AdS 3 fixed points in this case preserve only two supercharges. The solutions are accordingly dual to N = (1, 0) two-dimensional SCFTs. We have studied RG flows from supersymmetric AdS 7 vacua to these geometries as well.
All of these solutions provide a large class of AdS 3 ×M 4 solutions and RG flows across dimensions from six-dimensional SCFTs to two-dimensional SCFTs. The solutions might be useful in the holographic study of supersymmetric deformations of N = (1, 0) SCFTs in six dimensions to two dimensions. For equal SU(2) gauge coupling constants, the SO(4) gauged supergravity can be embedded in eleven-dimensional supergravity. We have also given the uplifted elevendimensional metric. These solutions with a clear M-theory origin should be of particular interest in the study of wrapped M5-branes on four-manifolds. It would be interesting to look for the embedding of other solutions in string/M-theory. Similar solutions in N = 2 gauged supergravity with other gauge groups also deserve further study. Finally, it should be noted that the RG flows across dimensions given here can be interpreted as supersymmetric black strings in asymptotically AdS 7 space. Our solutions should be useful in the study of black string entropy using twisted indices of N = (1, 0) SCFTs along the line of [40].
A Truncation ansatz of eleven-dimensional supergravity on S 4 In this appendix, we review relevant formulae for embedding solutions of N = 2 seven-dimensional gauged supergravity in eleven-dimensional supergravity. Since the AdS 3 × M 4 solutions involve all types of seven-dimensional fields namely scalar, vector and three-form fields, the eleven-dimensional four-form field strength is very complicated. Accordingly, we omit an explicit form of the four-form in each case for brevity. It can however be computed by using the formula given in [37] and the mapping between seven-and eleven-dimensional fields given here. The truncation of eleven-dimensional supergravity on S 4 leading to N = 2 SO(4) seven-dimensional gauged supergravity is described by the metric ansatz with the following definitions Dµ α = dµ α + gA αβ (1) µ β and ∆ = cos 2 ξXT αβ µ α µ β + X −4 sin 2 ξ . (142) µ α , α = 1, 2, 3, 4, are coordinates on S 3 satisfying µ α µ α = 1.
Together with the four-form ansatz given in [37], the Lagrangian for the resulting N = 2 gauged supergravity, after multiplied by 1 2 , reads with the scalar potential given by A symmetric scalar matrixT αβ , α, β = 1, 2, 3, 4 with unit determinant describes nine scalars in SL(4, R)/SO(4) coset. This is equivalent to SO(3, In term of the SL(4, R)/SO(4) coset representative V α R with SO(4) indices R, S, . . . = 1, 2, 3, 4, we have the relatioñ The SO(3, 3)/SO(3)×SO(3) coset representative L I A is related to that of SL(4, R)/SO(4) by the relation and Γ Iαβ = (Γ i αβ , −Γ i+3 αβ ), i = 1, 2, 3, see more detail in [32]. Note also that η A RS also satisfy similar relations which we will not repeat them here. We use the following choice of Γ I αβ Γ 1 = −iσ 2 ⊗ σ 1 , All these ingredients lead to the following identification of the fields and parameters in seven and eleven dimensions g 2 = g 1 = 16h = 2g, X = e − σ 2 , With this identification, it can also be easily verified that the scalar matrix for the gauge kinetic terms also match For convenience, we explicitly give the SL(4, R)/SO(4) coset representative V α R and SO(4) gauge fields A αβ as follow.
• SO(3) diag singlet scalar: We have used the relation A i = g 2 g 1 A i+3 with g 2 = g 1 .
• SO(2) × SO(2) singlet scalar: • SO(2) diag singlet scalars: In all cases, it can be verified using the relation (146) that the above V α R give precisely L I A in the main text.