Magnetically-charged supersymmetric flows of gauged $\mathcal{N}\,=\,8$ supergravity in five dimensions

We study magnetically-charged supersymmetric flow equations in a consistent truncation of gauged $\mathcal{N}\,=\,8$ supergravity in five dimensions. This truncation gives gauged $\mathcal{N}\,=\,2$ supergravity coupled to two vector multiplets and two hypermultiplets. We derive magnetically-charged flow equations of scalar fields from vector and hypermultiplets. It turns out that there could be only up to two nontrivial scalar fields out of eight from the hypermultiplets. Along the way, we recover some known $AdS_3$ solutions of the flow equations.


Introduction
Supergravity has provided a simple and important venue in that the AdS/CFT correspondence [1] could be tested in concrete examples. One of many notable examples is the so-called STU model. There are STU models in four-and five-dimensional gauged supergravity theories. To be specific, in five dimensions, we have an STU model in gauged N = 2 supergravity coupled to two vector multiplets. When we turn on a magnetic field, as it breaks the Lorentz invariance of the background, solutions cannot flow to AdS 5 , but to AdS 3 ×Σ k where Σ k is a two-dimensional surface with constant curvature. A number of such black string solutions are known in this model [2,3,4,5,6,7,8].
Moreover, magnetically-charged flow solutions of the five-dimensional STU model enabled us to study renormalization group flows across dimensions holographically. Maldacena and Nunez found solutions interpolating AdS 5 and AdS 3 ×H 2 which are dual to RG flows flowing from 4d N = 4 super Yang-Mills theory to 2d superconformal field theories of various supersymmetries [9]. Their solutions are recently generalized and employed to holographically test c-extremization of 2d superconformal field theories [10,11,12]. See also [13].
So far, we have briefly reviewed achievements from studying the five-dimensional STU model which is gauged N = 2 supergravity coupled to two vector multiplets. One short coming of the STU model is that this model has only one critical point which is maximally supersymmetric and dual to 4d N = 4 super Yang-Mills theory. In order to study other critical points dual to various 4d superconformal field theories, one has to include other multiplets to the STU model, e.g. vector or hypermultiplets. Indeed, in [14], magnetically-charged flows interpolating AdS 5 and AdS 3 ×H 2 were obtained with vector and hypermultiplets. However, only one scalar field out of eight from hypermultiplets is nontrivial there. In this paper, we try to generalize their work and turn on four or less non-trivial scalar fields from hypermultiplets in the gauged N = 8 supergravity setting. As a main and final result, we will show that only up to two nontrivial scalar fields from hypermultiplets can be turned on. Along the way, we rediscover a number of AdS 3 ×H 2 solutions found in [9,10,11]. Moreover, in [15], the generic flow equations with vector and hypermultiplets were obtained. Our work could be also seen as a particular and concrete realization of their flow equations.
In section 2 we review a consistent truncation of gauged N = 8 supergravity. In section 3, as a warm up exercise, we rederive magnetically-charged flow equations with scalar fields only from vector multiplets. We recover some known solutions of the flow equations. In section 4, finally, we derive magnetically-charged flow equations with scalar fields from vector multiplets and also hypermultiplets. In section 5 concluding remarks are offered. In appendix A we briefly review gauged N = 8 supergravity in five dimensions. In appendix B we present the equations of motion for the consistent truncation.

A consistent truncation 2.1 The truncation
We consider a consistent truncation of gauged N = 8 supergravity in five dimensions, [18,19,20], previously studied in [16] and more recently in [17].
The 42 scalar fields of gauged N = 8 supergravity in five dimensions live on the coset manifold E 6(6) /U Sp (8). The basic structure of the coset manifold explained in [20] is summarized in appendix A. Fundamental representation of E 6(6) is real and 27-dimensional. The infinitesimal E 6(6) transformation in the SL(6, R)×SL(2, R) basis, (z IJ , z Iα ), is [20] where Λ I J and Λ α β are real and traceless generators of SL(6, R) and SL(2, R) respectively, and Σ IJKα is real and antisymmetric in IJK.
The truncation of our interest is obtained by a Z 4 -invariant subtruncation of Z 2 ×Z 2 -invariant sector [16,17] We have coset generators, Σ IJKα , With the complex coordinates, where We also have two SL(6, R) generators, 3 There are total ten real scalar fields, and they parametrize the coset manifold, This truncation gives gauged N = 2 supergravity in five dimensions coupled to two vector multiplets and two hypermultiplets. The scalar fields, α and β, parametrizing the SO(1, 1) factors, belong to two N = 2 vector multiplets. The scalar fields, ϕ j and θ j , j = 1, . . . , 4, parametrizing the SU (1, 1)/U (1) factors, belong to two N = 2 hypermultiplets. 4 The Lagrangian of the truncation is given by There are three vector fields: two are from two N = 2 vector multiplets and one is the graviphoton from an N = 2 gravity multiplet. The scalar potential is where the superpotential is (2.11) The scalar potential is independent of the phases, θ j . Refer to [16,17] for the critical points of the scalar potential and more details of the truncation.

The supersymmetry equations
For magnetically-charged flows, we consider the background, where and Σ k = R 2 , S 2 , H 2 , for k = 0, +1, −1, respectively, for the x-y plane. The only non-zero component of the gauge field, A µIJ , is given by (2.14) Hence, the field strength is We consider the superpotential and the spinors in five dimensions. The superpotential, W , is obtained as one of the eigenvalues of W ab tensor [21], where k = 1, 2 and W is given in (2.11). The eigenvectors, η a (1) , η a (2) , are and they are related to each other by 2) , where Ω ab is the U Sp(8) symplectic form in e.g. [21]. We employ the gamma matrix conventions in [21]. The spinors are whereˆ 1 andˆ 2 are spinors with four complex components. The supersymmetry equations are obtained by setting the supersymmetry variations of fermionic fields, i.e. the spin-3/2 and spin-1/2 fields, to zero. The bosonic parts of the variations are [20] For the consistent truncation we have, we have Hence, the only non-zero component is It is convenient to define the quantities, H, Q, along with the superpotential, W , Hence, we have

Magnetically-charged flow equations: vector multiplets
In the consistent truncation we have, there are ten scalar fields, α, β from N = 2 vector multiplets, and ϕ j , θ j , j = 1, . . . , 4 from N = 2 hypermultiplets. In this section, as a warm up exercise and to set up the conventions, we rederive magnetically-charged flow equations only with the scalar fields from vector multiplets. Similar equations were obtained in gauged N = 2 supergravity coupled to vector multiplets in five dimensions in [2,3,4] and [7,8]. In the next section, we will consider inclusion of the scalar fields from hypermultiplets.

The flow equations
In this section, we only turn on the scalar fields from vector multiplets, α and β. We set the scalar fields from hypermultiplets, ϕ j = 0, θ j = 0, j = 1, . . . , 4. The superpotential in (2.11) reduces to This superpotential does not have any critical point beside the maximally supersymmetric one. Now we consider the spin-3/2 field variations in (2.22). For µ = t, x, y, r, respectively, we obtain and the prime denotes the derivative with respective to r. The equation for µ = z is identical to the one for µ = t. We employ the projection conditions, From (3.2) and (3.3), we obtain and we obtain a condition, Now we consider the spin-1/2 field variations in (2.23). They reduce to Employing the projection conditions, (3.8), we obtain Now let us summarize the supersymmetric flow equations obtained in (3.9) and (3.14), where with a condition, 17) and the x-y plane is Σ k = R 2 , S 2 , H 2 , for k = 0, +1, −1, respectively.

Some known solutions
In this subsection, we recover some known analytic and numerical solutions of the supersymmetric flow equations obtained in (3.15) with (3.16) and (3.17).

Supersymmetric domain wall solutions of AdS 5 and AdS 3 × R 2
We numerically solve the supersymmetric flow equations, and obtain a magnetically-charged domain wall solution interpolating AdS 5 and AdS 3 × R 2 , which was previously studied in [8].
In [6,7], an AdS 3 ×R 2 solution was found which is a solution of the supersymmetric flow equations. The solution is given by the metric, 18) and the scalar fields, where α * and β * are some constants. The constant scalar fields give the values of and and they satisfy the condition, (3.17), with k = 0. Now we numerically solve the supersymmetric flow equations and recover the domain wall solution of [8], that interpolate between an AdS 5 in the UV and the AdS 3 × R 2 solution of (3.18) in the IR. We choose g = 2, α * = (log 2)/3, β * = 0, and a numerical solution is depicted in Figure 1.

Supersymmetric black string solutions with trivial scalar fields
The Chamseddine-Sabra solution in (23) of [2] and the Klemm-Sabra solution in (30) of [3] satisfy the supersymmetric flow equations. The solutions are given by the metric, (3.22) and trivial scalar fields, with a condition, where l = g/2 and k = +1, −1 for S 2 and H 2 , respectively. In the near horizon limit, the solution asymptotes to AdS 3 ×S 2 [2] or AdS 3 ×H 2 [3]. Asymptotically AdS 3 ×H 2 solution is a black string solution with a regular horizon at r = 1/( √ 3l), but asymptotically AdS 3 ×S 2 has a naked singularity.

Supersymmetric black string solutions with nontrivial scalar fields
The Cacciatori-Klemm-Sabra black string solution in (6.4) of [4] satisfies the supersymmetric flow equations. The solution is given by the metric, and nontrivial scalar fields, with a condition, where l = g/2. In the near horizon limit, the solutions asymptote to AdS 3 ×S 2 or AdS 3 ×H 2 for k = +1, −1, respectively. In the consistent truncation we have, there are ten scalar fields, α, β from N = 2 vector multiplets, and ϕ j , θ j , j = 1, . . . , 4 from N = 2 hypermultiplets. In the previous section, we obtained the flow equations only with the scalar fields from N = 2 vector multiplets. In this section, we finally derive magnetically-charged flow equations with inclusion of the scalar fields from N = 2 hypermultiplets. In this section, we will fix p(r) = 1 in the background metric of (2.12). We will assume that the scalar fields, θ j , j = 1, . . . , 4, are functions of the y-coordinate only, θ j = θ j (y). The rest of the scalar fields and functions depend on the radial r-coordinate only. The reason will be clear below.
First we consider the spin-3/2 field variations in (2.22). For µ = t, x, y, r, respectively, we obtain U γ 4 a − and the prime and the dot denote the derivatives with respective to r and y, respectively. The equation for µ = z is identical to the one for µ = t. We employ the projection conditions, From (4.1) and (4.2), we obtain The difference of (4.2) and (4.3) gives Now we consider the spin-1/2 field variations in (2.23). We employ the projection conditions, (4.7), and the spin-1/2 field variations reduce to Recall two equations from the spin-3/2 field variations in (4.8), and These are the complete supersymmetric flow equations. However, as we will see in the next subsection, these flow equations do not satisfy the equations of motion.

The flow equations against the equations of motion
In this subsection, we check the first-order flow equations against the second-order equations of motion. The equations of motion are given in appendix B. The flow equations, (4.10) and (4.11), satisfy all the equations of motion, but not the tt-and rr-components of the Einstein equation, respectively, and the hypermultiplet scalar equations of motion, Therefore, we conclude that the flow equations do not satisfy the equations of motion. In order to make the flow equations to satisfy the equations of motion, by observing (4.13) and (4.14), we find that we have to turn off some scalar fields from hypermultiplets. In the next subsection, we will consider all possible conditions on the scalar fields from (4.13) and (4.14).

The correct flow equations and their AdS 3 × Σ k solutions
In the previous subsection, we have seen that the flow equations we obtained do not satisfy the equations of motion. In this subsection, by considering the equations of motion, we obtain conditions on the scalar fields from hypermultiplets to make the flow equations to satisfy the equations of motion. Our analysis covers previously studied flow equations in [14] in section 4.3.1, but, we also present new flow equations with two nontrivial scalar fields from hypermultiplets in section 4.3.2.

Flow equations with one nontrivial scalar field from hypermultiplets
First, we consider keeping only one scalar field, ϕ 1 , and turn off rest of the scalar fields from hypermultiplets, To satisfy the equations of motion, (4.13) and (4.14), we should have 16) and they give where a is a constant. As we will see below, this case gives precisely the flow equations studied in [14]. The flow equations reduce to and where q 1 , q 2 , q 3 are given in (4.17), and the scalar fields, θ j , j = 1, . . . , 4, are constant. These flow equations indeed satisfy the equations of motion. We find AdS 3 × Σ k solutions of the flow equations. We set the scalar fields to be constant and where U 0 and V 0 are constants. Then the flow equations are algebraic, and we obtain AdS 3 × Σ k solutions, where |a| < 1/4. We see that this solution only makes sense when k = −1. After a proper normalization of the r-coordinate and taking k = −1, this is indeed the AdS 3 × H 2 solution found in (4.2) of [14], where the scalar field, ϕ 1 , was denoted by χ. Their flow equations look different from ours, but it is due to the different coordinates of the background metric from ours.
In [14], the dual field theory of this solution was proposed to have 2d N = (0, 2) supersymmetry, and central charge of the dual field theory was calculated and matched with the holographic calculation.

Flow equations with two nontrivial scalar fields from hypermultiplets
Now we consider turning on ϕ 1 and one of ϕ 2 , ϕ 3 , ϕ 4 .
• First, we consider To satisfy the equations of motion, (4.13) and (4.14), we should have and they give Hence, one of the gauge fields, q 1 , is turned off, and there is no free parameter left like a in the previous subsection. The flow equations reduce to and where q 1 , q 2 , q 3 are given in (4.24), and the scalar fields, θ j , j = 1, . . . , 4, are constant. These flow equations indeed satisfy the equations of motion. These flow equations are new and our main result. We find AdS 3 × Σ k solutions of the flow equations. We set the scalar fields to be constant and set as (4.20) again. Then the flow equations are algebraic, and we obtain AdS 3 × Σ k solutions, We see that this solution only makes sense when k = −1. This AdS 3 × H 2 solution is not new. This solution was first considered in section 3.2 of [9], and recently discussed further in [11] and (4.5) of [14]. The dual field theory of this solution was proposed to have 2d N = (2, 2) supersymmetry, and central charge of the dual field theory was calculated and matched with the holographic calculation.
• Second, we consider To satisfy the equations of motion, (4.13) and (4.14), we should have and they give Hence, one of the gauge fields, q 2 , is turned off, and there is no free parameter left. The flow equations reduce to the identical form of (4.25) with (4.26), but now q 1 , q 2 and q 3 are given by (4.30). These flow equations also satisfy the equations of motion.
We can also set the scalar fields to be constant and the warp factors as in (4.20), and obtain AdS 3 × Σ k solutions, • Third, we consider To satisfy the equations of motion, (4.13) and (4.14), we should have but there is no q 1 , q 2 and q 3 satisfying these conditions.

Flow equations with three nontrivial scalar fields from hypermultiplets
Lastly, we consider turning on ϕ 1 and two of ϕ 2 , ϕ 3 , ϕ 4 . By observing the equations of motion, (4.13) and (4.14), we find it not possible. We conclude that there could be up to two nontrivial scalar fields out of four, ϕ j , j = 1, . . . , 4, with constant θ j , j = 1, . . . , 4, from hypermultiplets in our truncation.

Conclusions
In this paper, we obtained the magnetically-charged supersymmetric flow equations for scalar fields from two N = 2 vector multiplets and two N = 2 hypermultiplets by a consistent truncation of gauged N = 8 supergravity in five dimensions. We concluded that there could be only up to two nontrivial scalar fields out of eight from the two hypermultiplets in the flow equations. As we increase the number of nontrivial scalar fields from hypermultiplets, some U (1) gauge fields had to be turned off. This should be related to the Higgs mechanism as scalar fields couple to gauge fields, gauge fields become massive. 5 Along the way, we have rediscovered a number of known AdS 3 solutions. Our work could be understood as an extension of the flow equations presented in [14] where only one scalar field from hypermultiplets were turned on.
In this paper, we have encountered a rare situation where the first-order supersymmetry equations were in conflict with the second-order equations of motion. This trouble could have been resulted from the nontrivial dependence of a scalar field on a coordinate other than the holographic radial direction, r, i.e. θ j (y). 6 Our flow equations provide a particular and concrete realization of the generic flow equations for general N = 2 matters in [15]. It would be interesting to further generalize our result to include more N = 2 matter multiplets by employing larger truncation of gauged N = 8 supergravity in five dimensions. Otherwise, it could be possible to rederive our results in the N = 2 supergravity formalism by employing and extending the scalar field parametrizations of [22,23] we obtain the coset representatives in the SL(6, R)×SL(2, R) basis, U IJ KL , U IJKα and U Iα Jβ .
We also have the coset representatives in the U Sp(8) basis, The inverse coset representatives are Now we consider the action of the theory [20]. The bosonic part of the Lagrangian is where the covariant derivative is defined by with the U Sp(8) connection, (A.9) The kinetic term for scalar fields is defined by The scalar potential is defined by (A.14) We also define for the last three terms of Lagrangian. We adopt the gamma matrix convention of [20], with where η ij = diag (+, −, −, −, −), and γ 0 , γ 1 , γ 2 , γ 3 are pure imaginary as in four-dimensions and γ 4 = i γ 5 is pure real. The matrices γ 0 and γ 5 are antisymmetric and γ 1 , γ 2 , γ 3 are symmetric.