Spatially modulated and supersymmetric mass deformations of $\mathcal{N}=4$ SYM

We study mass deformations of $\mathcal{N}=4$, $d=4$ SYM theory that are spatially modulated in one spatial dimension and preserve some residual supersymmetry. We focus on generalisations of $\mathcal{N}=1^*$ theories and show that it is also possible, for suitably chosen supersymmetric masses, to preserve $d=3$ conformal symmetry associated with a co-dimension one interface. Holographic solutions can be constructed using $D=5$ theories of gravity that arise from consistent truncations of $SO(6)$ gauged supergravity and hence type IIB supergravity. For the mass deformations that preserve $d=3$ superconformal symmetry we construct a rich set of Janus solutions of $\mathcal{N}=4$ SYM theory which have the same coupling constant on either side of the interface. Limiting classes of these solutions give rise to RG interface solutions with $\mathcal{N}=4$ SYM on one side of the interface and the Leigh-Strassler (LS) SCFT on the other, and also to a Janus solution for the LS theory. Another limiting solution is a new supersymmetric $AdS_4\times S^1\times S^5$ solution of type IIB supergravity.


Abstract
We study mass deformations of N = 4, d = 4 SYM theory that are spatially modulated in one spatial dimension and preserve some residual supersymmetry. We focus on generalisations of N = 1 * theories and show that it is also possible, for suitably chosen supersymmetric masses, to preserve d = 3 conformal symmetry associated with a co-dimension one interface. Holographic solutions can be constructed using D = 5 theories of gravity that arise from consistent truncations of SO(6) gauged supergravity and hence type IIB supergravity. For the mass deformations that preserve d = 3 superconformal symmetry we construct a rich set of Janus solutions of N = 4 SYM theory which have the same coupling constant on either side of the interface. Limiting classes of these solutions give rise to RG interface solutions with N = 4 SYM on one side of the interface and the Leigh-Strassler (LS) SCFT on the other, and also to a Janus solution for the LS theory. Another limiting solution is a new supersymmetric AdS 4 × S 1 × S 5 solution of type IIB supergravity.

Introduction
Mass deformations of N = 4 d = 4 SYM theory that preserve some supersymmetry have been extensively studied and are associated with rich dynamical features under RG flow (see e.g. [1][2][3][4][5][6][7][8][9][10][11]). In this paper we will explore mass deformations of N = 4 SYM theory that are spatially modulated in one of the three spatial dimensions and yet still preserve some supersymmetry. A particularly interesting sub-class of such deformations also preserve conformal symmetry with respect to the remaining three spacetime dimensions and describe codimension one superconformal interfaces. The investigations of this paper are somewhat analogous to those that have been carried out in the context of ABJM theory. It is known that the homogeneous (i.e. spatially independent) mass deformations of ABJM theory [12,13] can be generalised to mass deformations that depend on one of the two spatial coordinates and preserve 1/2 of the supersymmetry [14]. Further generalisations, preserving less supersymmetry, were subsequently analysed in [15]. Holographic descriptions of such deformations, preserving 1/4 of the supersymmetry of D = 11 supergravity, were first constructed in [16] using a Q-lattice construction [17]. The results of [16] included novel solutions that are holographically dual to boomerang RG flows which flow from ABJM theory in the UV back to ABJM theory in the IR. The Q-lattice construction of [16] was substantially generalised in [18], where it was shown that there is a novel class of D = 11 supergravity solutions, again preserving 1/4 of the supersymmetry, which can be obtained by simply solving the Helmholtz equation on the complex plane. In addition to presenting a new set of solutions describing boomerang RG flows, the construction of [18] also included the Janus solutions of [19]. Finite temperature generalisations, using the Q-lattice construction, have been discussed in [20,21].
Before continuing, we pause to note that there are various usages of "Janus" in the literature. In this paper it will refer to a co-dimension one, planar, conformal interface that has the same CFT on either side of the interface (or the same up to a discrete parity symmetry). This includes the rich set of examples associated with N = 4 SYM theory that are obtained by varying the coupling constant and theta angle as in, for example, [22][23][24][25][26][27][28][29][30][31]. For these Janus configurations the CFT is being deformed by exactly marginal operators away from the interface, and in some cases there are also additional sources for relevant operators located on the interface itself. For the Janus solutions of D = 11 supergravity in [19] the ABJM theory is deformed by relevant operators located on the interface, while for those in [18,32] the ABJM theory is also deformed by relevant scalar operators that, generically, have spatial dependence away from the interface (see also [33]. ) In this paper we will show that there are interesting new supersymmetric Janus configurations of N = 4 SYM theory that arise from spatially modulated fermion and boson mass deformations but with the same coupling constant and theta angle on either side of the interface. In addition to these Janus solutions, we will also construct novel holographic solutions dual to conformally invariant, co-dimension one interfaces, separating two different CFTs. In these configurations the two CFTs are related by Poincaré invariant RG flow, so we refer to these as "RG interfaces" (see [34,35]) and they are further discussed in [36].
To determine which spatially modulated mass deformations of N = 4 SYM theory preserve supersymmetry, we can deploy the technology developed in [38] (for theories with less supersymmetry, one can use the simpler approach of [37]). As in [39] we couple the theory to off-shell conformal supergravity and then take the Planck mass to infinity so that the supergravity fields become non-dynamical. In this limit one is left with an N = 4 supersymmetric field theory coupled to the supergravity fields, which are now viewed as background couplings. The background couplings which preserve supersymmetry can then be determined by analysing the supersymmetry transformations of the field theory coupled to the supergravity theory.
We will focus our investigations on generalising the class of homogeneous mass deformations known as the N = 1 * theories. Recall that the field content of N = 4 SYM, in terms of an N = 1 language, consists of a vector multiplet coupled to three chiral superfields Φ a . Deforming the theory by adding a superpotential of the form W ∼ 3 a=1 m a trΦ a Φ a , where m a are constant, complex mass parameters, defines the N = 1 * class of theories. Three cases of particular interest are (i) the "one mass model" with (say) m 1 = m 2 = 0, (ii) the "equal mass model", with m 1 = m 2 = m 3 , and (iii) the N = 2 * theory with (say) m 1 = m 2 and m 3 = 0.
We will show that all of these N = 1 * theories can be generalised so that the mass parameters depend on one of the three spatial coordinates while generically preserving N = 1 Poincaré supersymmetry with respect to the remaining d = 3 spacetime dimensions. For the case of the N = 2 * theory there is an enhancement to N = 2 Poincaré supersymmetry in d = 3. Furthermore, it is possible to suitably choose the mass parameters so that the N = 1 Poincaré supersymmetry is enhanced to an N = 1 or N = 2 superconformal supersymmetry in d = 3, respectively. This latter class of deformations thus defines a class of Janus configurations of N = 4 SYM theory, which have the novel feature that the coupling constant and the theta angle take the same value on either side of the interface, in contrast to previously constructed Janus configurations of N = 4 SYM. At this point it is worth emphasising that our field theory results concerning supersymmetric Janus configurations of N = 4 SYM with constant coupling across the interface are complementary to the classification carried out in [26], for which it was assumed that the coupling constant varies across the interface and that any additional deformations are proportional to the spatial derivative of the coupling constant.
The deformations we consider can also be studied holographically by constructing solutions of type IIB supergravity. A convenient way to construct such solutions is to first construct them within the context of D = 5 maximal gauged supergravity [40][41][42] and then uplift them to D = 10 using [43,44]. In fact, for the deformations we consider, we can utilise the consistent truncations of D = 5 maximal gauged supergravity discussed in [45,46], that couple the metric to a number of scalar fields. In particular, there is a consistent truncation model that is suitable for studying the mass deformations for each of the three N = 1 * theories mentioned above.
We will first derive the BPS equations that are relevant for spatially modulated mass deformations of N = 4 SYM theory that preserve ISO(1, 2) symmetry. In this case the BPS equations are partial differential equations in two variables. For this class of solutions we will also carry out a detailed analysis of holographic renormalisation which is rather involved. There is a set of finite counterterms that one needs to consider in order to have a supersymmetric renormalisation scheme. By demanding that the energy density of these BPS configurations is a total spatial derivative, thus leading to vanishing total energy, imposes some constraints on the counter terms (in fact the same constraints are obtained by the "Bogomol'nyi trick" of [45][46][47]). However, determining the full set of conditions required for a supersymmetric scheme is left to future work.
We will then focus on the BPS equations for the special subclass of solutions associated with Janus configurations. By writing the D = 5 metric ansatz using a foliation by AdS 4 slices, the BPS equations become a set of ODEs that we then numerically solve for each of the three consistent truncations. For each of the three models we find Janus solutions that approach the N = 4 SYM AdS 5 vacuum on either side of the interface. We also find solutions that approach the AdS 5 vacuum on one side and are singular on the other, as well as solutions that are singular on both sides, whose physical interpretation is unclear.
Additionally, for the one mass model we find new types of solutions that are further explored in [36]. Recall that homogeneous mass deformations in the one mass model induce an RG flow to the Leigh-Strassler (LS) fixed point [3]. From the gravity side, within the truncation we consider, in addition to the N = 4 SYM AdS 5 vacuum solution, there are two additional AdS 5 solutions, related by a Z 2 symmetry, which we will denote LS ± , and each dual to the LS fixed point. Here we will construct novel solutions that are dual to superconformal RG interfaces, approaching the N = 4 SYM AdS 5 solution on one side and one of the two LS AdS 5 solutions on the other. We will also construct solutions that approach LS + AdS 5 on one side of the interface and LS − AdS 5 on the other, giving rise to Janus solutions of the Leigh-Strassler SCFT.
Some of the new explicit solutions that we construct here, including both the Janus and the RG interface solutions, are analogous to the supergravity solutions found in [32] associated with ABJM theory. Here we will also obtain more detailed information on the sources and expectation values of various operators by using our holographic renormalisation results.
We also find a particularly interesting new feature for the equal mass model. This model is the most complicated to analyse since it has four real scalar fields instead of three. Furthermore, one of the scalar fields is the dilaton dual to the coupling constant of N = 4 SYM. While there are certainly rich Janus solutions for which the coupling constant is different on either side of the interface, we focus our attention on solutions where it has the same value. We find a novel class of Janus solutions that, surprisingly, approach a solution that is periodic in a bulk coordinate. By compactifying this coordinate one then obtains a supersymmetric AdS 4 ×S 1 solution. After uplifting to type IIB this is a new supersymmetric AdS 4 × S 1 × S 5 solution that will be further explored in [48].
The plan of the rest of the paper is as follows. In section 2 we determine the conditions for spatially dependent mass deformations of N = 4 SYM theory to preserve supersymmetry, focussing on the three classes of N = 1 * theories. In section 3 we introduce the supergravity truncation of D = 5 maximal gauged supergravity [45,46] that couples the metric to ten scalars, as well as three further truncations that are relevant for studying the three classes of N = 1 * theories. In sections 4 and 5 we will present the BPS equations relevant for spatially dependent mass deformations that preserve d = 3 superPoincaré and superconformal invariance, respectively. In section 6 we present and discus various new supergravity solutions, including the new Janus solutions as well as the solutions dual to superconformal RG interfaces involving the LS fixed point for the one mass model and the novel AdS 4 × S 1 solution for the equal mass model. In section 6 we conclude the paper with some discussion. We have also included three appendices. Appendix A contains some details on the derivation of the BPS equations. Appendix B discusses in some detail the holographic renormalisation scheme that we use and appendix C develops this further for the class of Janus solutions.

SYM
The coupling of N = 4 SYM to off-shell conformal supergravity [49] was carried out in [50][51][52]. In [39] it was highlighted that this setup can be utilised to study supersymmetric deformations of N = 4 SYM. Furthermore, some supersymmetric deformations of N = 4 SYM, including some known Janus configurations involving non-trivial profiles for the coupling constant and theta angle, were recovered in this language in [39]. Here we will use this formalism to study a new class of spatially dependent mass deformations that generalize the homogeneous N = 1 * mass deformations.
We emphasise that in this section we use a "mostly plus" (−, +, +, +) convention for the metric. This should be contrasted with our later usage of a "mostly minus" convention when we discuss supergravity solutions.
The possible bosonic deformations of N = 4 SYM are parametrised by the bosonic auxiliary fields of the off shell conformal supergravity theory, which transform in particular representations of SU (4) R , the global R-symmetry of the undeformed theory. The deformations transforming in the 1 of SU (4) R are associated with placing N = 4 SYM on a curved manifold as well as spatially modulating the gauge coupling and theta angle, encapsulated in the complex parameter τ ≡ θ 2π + i 4π g 2 . In addition there are deformations E ij in the 10 of the SU (4) R , D ij kl in the 20 , both Lorentz scalars, as well as a one-form V i µ j and a two-form T ij µν transforming in the 15 and 6, respectively. In this paper, our primary focus will concern spatially dependent mass deformations of the bosonic and fermionic fields involving E ij and D ij kl and so in the following we set We note that the components E ij and D ij kl are both complex and satisfy with i, j, · · · = 1, . . . , 4. To see how these background fields couple to N = 4 SYM we recall that the field content of the latter is given by gauge fields A µ , fermions ψ i , transforming in the 4 of the SU (4) R and bosons φ ij , satisfying (φ ij ) * = φ ij = 1 2 ijkl φ kl , transforming in the 6. The deformed action, in flat spacetime, is given by 1 Here the first two lines are essentially the undeformed action with In the third and fourth lines we have used the following mass matrices for the bosons and fermions: At this stage E ij and D ij kl can have arbitrary dependence on the the spacetime coordinates. The supersymmetry transformations of the matter fields for this deformed theory are given by 2 Here i , η i parametrise possible Poincaré and superconformal supersymmetries, respectively. Such supersymmetries will only be present provided that there are solu-1 Note that we largely follow the conventions and notation of [50,51]. Thus, ψ i is a chiral spinor satisfying γ 5 ψ i = +ψ i transforming in the4 of SU (4). The conjugate spinor, ψ i , defined by ψ i ≡ B(ψ i ) * (in contrast to the notation used in [39]) where B −1 γ a B = γ * a , has the opposite chirality, γ 5 ψ i = −ψ i , and transforms in the 4 of SU (4). Note that we have changed the sign of (M ψ ) ij in (2.4) compared with [39], in agreement with eq. (10) of [50]. In addition, the dependence on τ in (2.3) is obtained from [50,51] by writing φ a , which can be used to parametrise SU (1, 1)/U (1), , again in contrast to [39]. 2 Note that i , η i both transform in the4 of SU (4) and satisfy the chirality conditions γ 5 i = + i , We note that (2.5) can be obtained from eq. (5) of [50].
Notice that a complete basis of solutions to the last line of (2.6) is given by In the following, when we refer to solutions to the background equations (2.6) with a given i we will always mean solutions as in the first line of (2.7), which are the Poincaré supersymmetries. When referring to a solution with a given η i , we will mean a solution as in the second line with an associated i (y), which are the super conformal supersymmetries.
Here we will not attempt to find the most general solution to (2.6). Instead we will focus on generalising some known homogeneous (i.e. spatially independent) mass deformations that, moreover, can be studied holographically within the context of known truncations of N = 8, SO(6) gauged supergravity in D = 5. Specifically, we will consider the homogeneous N = 1 * deformations and then allow for an additional dependence on just one of the three spatial coordinates.
To cast the N = 1 * deformations in the present formalism we first recall that in terms of N = 1 language the field content of N = 4 SYM consists of a vector multiplet, which includes the gauge-field and the gaugino, coupled to three chiral superfieds Φ a which transform in the 3 of SU (3) in the decomposition SU (3)×U (1) ⊂ SU (4) R . The homogeneous N = 1 * deformations are obtained by adding to the superpotential the term ∆W ∼ 3 a=1 m a trΦ a Φ a , with m a complex. This deformation gives rise to masses for the bosons and fermions in the chiral multiplets, but there is no mass deformation for the gaugino, consistent with preserving N = 1 supersymmetry. In the present formalism these N = 1 * deformations are associated with fermion mass deformations of the form as well as associated bosonic mass deformations, parametrised by both E ij and certain components of D ij kl that are needed to preserve supersymmetry, as we shortly recall below.
Here we will generalise the N = 1 * deformations by allowing m a to depend on one of the spatial coordinates: m a = m a (y). Let us first analyse the generic case with distinct m a = 0, before discussing some special subcases. From the first line of (2.6) we see that we can, generically, preserve N = 1 Poincaré supersymmetry of the form = (0, 0, 0, 4 ) . (2.9) In the homogeneous case, with m a constant, it is not difficult to see that the middle equation of (2.6) can be satisfied by choosing If we now allow m a = m a (y), taking i = 1, j = 2 and k = 3 in (2.6), for example, we also need to satisfy and we recall here that 4 is the spinor conjugate to 4 : 4 = B( 4 ) * . This can be solved after imposing the following projection on the Poincaré supersymmetry parameters where σ is a constant. Indeed, we find that these and in fact all components of (2.6) are satisfied by choosing as well as (2.10), with all other components zero. The projection condition (2.12) breaks half of the supersymmetry of the N = 1 * theories leaving two supercharges. As the deformation only depends on one of the three spatial dimensions, we have preserved Poincaré invariance in the remaining d = 3 spacetime dimensions. Thus, generically, the above deformation preserves N = 1 Poincaré supersymmetry in d = 3. For special choices of m a (y) we can preserve N = 1 conformal supersymmetry in d = 3. To see this, we take η i = (0, 0, 0, η 4 ) . (2.14) Then again considering, for example, i = 1, j = 2 and k = 3 in (2.6), as well as recalling the second line of (2.7), we are led to the condition for arbitrary complex constants λ a . Obviously these mass sources are singular at y = 0, which is the location of the interface. It is interesting to point out that we can choose different mass sources on either side of the interface and still preserve the same supersymmetry. In particular, for the conformally invariant case we can take with λ a andλ a independent constants. We will see such sources arise in the supergravity solutions that we construct later in the paper. Let us now further consider three special cases that we will focus on in the sequel.

N = 1 * one mass model
For this case we assume that only one of the m a is non-zero, say m 3 . We therefore consider a fermion mass matrix E ij of the form In the homogeneous case, with m independent of y, we can preserve N = 1 supersymmetry in d = 4 of the form (2.9) by also turning on Notice that the spinor (2.9) parametrising the N = 1 Poincaré supersymmetry is charged under this U (1) R so it is in fact an R-symmetry of the N = 1 * theory. When m = m(y), we can preserve N = 1 Poincaré supersymmetry in d = 3 satisfying (2.12) with Notice that when m = m(y) the U (1) R R-symmetry of the N = 1 * theory is broken and we are left only with an SU (2) global symmetry. If we choose m = λ y then, in addition, we preserve N = 1 superconformal supersymmetry.

N = 1 * equal-mass model
For this case, we assume m 1 = m 2 = m 3 so that the fermion mass matrix E ij takes the form In the homogeneous case, with m independent of y, taking D ij kl = 0 we preserve N = 1 supersymmetry in d = 4 of the form (2.9). By again considering the decomposition SU When m = m(y), we can preserve N = 1 Poincaré supersymmetry in d = 3 satisfying (2.12) with where, in this subsection, we have used the indices α, β, γ, · · · ∈ {1, 2, 3}. We observe that the spatially dependent deformations maintain the SO(3) global symmetry of the homogeneous case. If we choose m = λ y then, in addition, we preserve N = 1 superconformal supersymmetry.

N = 2 * model
For this case we assume that one of the masses is zero, say m 3 = 0, and the remaining two are equal m 1 = m 2 . Thus, we consider a fermion mass matrix E ij of the form We again first consider the homogeneous case with m independent of y. By choosing we find that there is now an enhancement to N = 2 supersymmetry of the form 1 and SU (2) 2 acting on the indices i, j ∈ {1, 2} and i, j ∈ {3, 4}, respectively. Then SU (2) R is SU (2) 2 , and clearly rotates the N = 2 supersymmetry parameters in (2.26). The U (1) ⊂ SU (2) 1 symmetry acts as an SO(2) rotation of the 1, 2 directions and leaves (2.26) inert. There can also be an enhancement of supersymmetry when m = m(y) compared with the previous cases. From (2.6) with (i, j) = (1, p), with p ∈ {3, 4}, and k = 2, we find the condition The tracelesness condition for D, in (2.2), now requires thatM is traceless (and therefore M is symmetric).M is therefore a traceless matrix in U (2). The remaining components of D can then be inferred from (2.2).
Note that the choice of the matrix M breaks the SU (2) R R-symmetry of the homogeneous deformations down to a U (1) R . This is expected since the spatially deformed solution preserves N = 2 Poincaré supersymmetry in d = 3 and so we expect an SO(2) = U (1) R-symmetry. The overall global symmetry is U (1) R × U (1). If we choose m = λ y then we preserve N = 2 superconformal symmetry in d = 3 with and (2.32)

Supergravity truncations
To study the spatially dependent mass deformations holographically we would like to construct suitable solutions of type IIB supergravity [53,54]. A convenient way to do this is to construct solutions of maximally supersymmetric SO(6) gauged supergravity in D = 5 and then uplift the solutions to D = 10 using the results of [43,44]. The D = 5 theory has 42 scalar fields, parametrising the coset E 6(6) /U Sp (8), which transform as 1+1, 10+10 and 20 with respect to SO (6). It is thus still rather unwieldy and so one seeks suitable consistent truncations of the D = 5 theory. In fact, for general constant, complex mass parameters m a , associated with the N = 1 * theories, there is an additional truncation that can be utilised, as discussed in [46], which can also be used when m a = m a (y). Specifically, one keeps the fields of SO(6) gauged supergravity that are invariant under a (Z 2 ) 3 symmetry of the SO(6) × SL(2, R) symmetry of the theory. This leads to an N = 2 D = 5 gauged supergravity theory coupled to two vector multiplets and four hypermultiplets. This theory contains eighteen scalar fields which parametrise the coset .
Schematically, these 18 scalar fields are dual to the following operators in N = 4 SYM theory: Here ϕ,φ are real and are the 1+1 irreps of SO(6) mentioned above. The fields φ i , φ 4 are complex and arise from the 10+10 irreps. The three complex scalar fields α i and the two real scalars β 1 , β 2 , which parametrise the SO(1, 1) × SO(1, 1) factors in the scalar manifold M 18 , arise from the 20 irrep 3 . For the N = 4 SYM operators appearing on the right hand side of (3.2), written in an N = 1 language, we note that Z i and χ i are the bosonic and fermionic components of the chiral superfields Φ i while λ is the gaugino of the vector multiplet. We also recall that the supergravity modes do not capture the Konishi operator tr(|Z 1 | 2 + |Z 2 | 2 + |Z 3 | 2 ). Having source terms for the three complex scalars φ i with i = 1, 2, 3 is dual to deforming N = 4 SYM by the the three fermion masses m a given in (2.8) (up to normalisation). Thus, allowing for spatially dependent sources for these φ i as well as suitable source terms for α i , β 1 and β 2 , dual to bosonic masses we can holographically study spatially dependent mass deformations with arbitrary complex m a (y) that we discussed in the last section. As far as we are aware, however, this N = 2 D = 5 gauged supergravity theory has not been explicitly constructed.
If we restrict to deformations for which the mass parameters m a (y) are all real, we can make further progress. Indeed, as discussed in [46], we can then utilise a further truncation that just keeps the metric and ten scalar fields which parametrise the coset This is achieved by truncating the N = 2 D = 5 gauged supergravity theory using an additional Z 2 symmetry, which lies in a certain [O(6) × SL ± (2, R)] /Z 2 subgroup, which is the actual symmetry group of N = 8 gauged supergravity [59]. Although not a supergravity theory, this truncation can be used to obtain supersymmetric solutions of SO(6) gauged supergravity and hence type IIB supergravity. The 10 real scalars consist of ϕ, φ i , φ 4 , α i and β 1 , β 2 all of which are now real and dual to the obvious Hermitian generalisations of the operators given in (3.2). We will refer to ϕ as the "dilaton". As already noted, the two scalars β 1 , β 2 parametrize the SO(1, 1)×SO(1, 1) factor in M 10 . The remaining eight scalars of this truncation, parametrising SU (1,1) , can 3 For reference, we note that β 1 , β 2 are the two real scalars that appear in the N = 2 gauged supergravity model coupled to two vector multiplets [55], often called the STU model. If we supplement the STU model with complex φ i , φ 4 we can obtain the charged cloud truncation considered in [56]; the scalars in this truncation parametrise the coset SO(1, 1) × SO(1, 1) × SU (1,1) , but it is a different set of scalars of SO(6) gauge supergravity than those kept in (3.3). It is also different to the truncation of [57,58], which has scalars parametrising the same coset, but does not contain any scalars in the 10 of SO(6), dual to fermion mass deformations. be packaged into four complex scalar fields z A via The gravity-scalar part of the Lagrangian can be written as where P is the scalar potential and K is the Kähler potential given by The scalar potential can be conveniently derived from a superpotential-like quantity where KB A is the inverse of K AB and ∇ A W ≡ ∂ A W + ∂ A KW. The ten scalar model is invariant under Z 2 × S 3 discrete symmetries acting on the bosonic fields (we will not make precise the discrete action on the fermions). First, it is invariant under the Z 2 symmetry 9) or equivalently, changing the sign of the eight real scalar fields, φ i , φ 4 , α i , ϕ. Second, it is also invariant under an S 3 permutation symmetry which acts on (z 2 , z 3 , z 4 ) as well as β 1 , β 2 and is generated by two elements. The first generator acts via and the second generator acts via The action on φ i , α i makes it clear that these generates S 3 . We also point out that the D = 5 theory is invariant under shifts of the dilaton where c is a real constant 4 .
Using the conventions of [46] (also see appendix A) a solution to the equations of motion of this model is supersymmetric provided that one can find a pair of symplectic Majorana spinors (ε 1 , ε 2 ), with ε 2 = −iγ 4 ε * 1 , that obey where we have defined We observe that the equations for preservation of supersymmetry given in (3.13) preserve the discrete symmetries (3.9)-(3.11). The AdS 5 vacuum solution dual to N = 4 SYM theory, is obtained when all of the scalars vanish. For this solution, the discrete symmetries (3.9)-(3.11) are associated with discrete R-symmetries of N = 4 SYM theory.
Following [46], we now identify additional truncations of the ten scalar model that can be used for real, spatially dependent mass deformations associated with each of the three cases considered in the last section.

N = 1 * one mass model
This model is obtained by taking the limit where two of the masses vanish, which we take to be m 1 = m 2 = 0 as in section 2.1 and now with m 3 real. Starting with the ten scalar model (3.3), we must have source terms for φ 3 and α 3 . It turns out to be consistent to set φ 1 = φ 2 = 0, α 1 = α 2 = 0 as well as ϕ = φ 4 = 0 and β 2 = 0: This results in a three-scalar model with fields z 1 , and β 1 , which we will use to construct supersymmetric Janus solutions later. The discrete symmetries (3.9)-(3.11) reduce to just the Z 2 symmetry generated by z 1 → −z 1 . An important feature of this model is that in addition to the AdS 5 vacuum solution with vanishing scalars, and dual to N = 4 SYM theory, there are also two other AdS 5 solutions, labelled LS ± . These two solutions are related by the Z 2 symmetry (3.9) and given by whereL the radius of the AdS 5 space for both LS ± solutions. When uplifted to type IIB these fixed point solutions preserve SU (2) × U (1) R global symmetry and are each holographically dual to the N = 1 SCFT found by Leigh and Strassler in [3]. By examining the linearised fluctuations of the scalar fields about the LS ± vacua, we find that α 3 is dual to an irrelevant operator O ∆=2+ The linearised modes involving φ 3 and β 2 mix, and after diagonalisation we find modes that are dual to one relevant and one irrelevant operator in the LS SCFT, which we label O ∆=1+ respectively. Note that if we set α 3 = 0, in this D = 5 model we obtain a model with two real scalars which is the same as that used to construct the homogeneous RG flows associated with the N = 1 * one mass model. These RG flows, which preserve SU (2)× U (1) R global symmetry, flow to the Leigh-Strassler fixed point [3] in the IR and were constructed in [4] and uplifted to type IIB in [59]. This gravitational model, with α 3 = 0, preserves the SU (2) × U (1) R global symmetry 5 and since the U (1) R is broken when the mass deformations are spatially modulated, as discussed in section 2.1, it cannot be used in this context.

N = 1 * equal-mass model
For this model we have m 1 = m 2 = m 3 ≡ m, as in section 2.2, and here we are considering m to be real. We must have φ 1 = φ 2 = φ 3 as well as α 1 = α 2 = α 3 and both non-zero, associated with the sources for the fermion and boson mass deformations. It turns out to be inconsistent to further set the gaugino condensate φ 4 or the scalar ϕ to zero. However, it is consistent to set β 1 = β 2 = 0. Thus, setting in the ten scalar model (3.3) leads to a model with four scalars, parametrised by (z 1 , z 2 ) with The discrete symmetries (3.9)-(3.11) reduce to the symmetry generated by (z 1 , z 2 ) → −(z 1 , z 2 ). This truncation is invariant under shifts of the dilaton (3.12). The Kähler potential (3.6) is now and an explicit expression for the potential P can be found in (3.8) of [11]. We use this four scalar model to construct supersymmetric Janus solutions later. We note that this four scalar model can be further truncated to give a theory with two real scalars, by setting α 1 = ϕ = 0. This theory keeps φ 1 , associated with real SO(3) ⊂ SU (3) R invariant fermion masses, and the gaugino condensate field φ 4 . This model is in fact the same model as that used by GPPZ [5] to construct RG flows associated with homogeneous SO(3) invariant mass deformations (and uplifted to type IIB in [9,10] extending [59]); see (3.12) of [11] for the explicit field redefinition. It cannot be used for spatially dependent masses, however.
For the equal mass model, with spatially dependent complex masses, there is in fact an alternative consistent truncation that can be used. By keeping an SO(3) ⊂ SU (3) ⊂ SO(6) invariant sector of SO(6) gauge supergravity, one can obtain an SO(6) gauged supergravity, one obtains an N = 2 supergravity coupled to one vector multiplet and one hypermultiplet [59]. The five scalars parametrise the coset With β 2 the SO(1, 1) factor, the remaining coset is obtained by supplementing φ 3 with a complex partner, associated with a complex fermion mass, and two more scalars ϕ,φ dual to operators as in (3.2). N = 2 supergravity coupled to two hypermultiplets [59,60]. The 8 scalars of this theory parametrise the quaternionic-Kähler manifold .
The scalar fields of this model consist of adding a complex partner to the four real scalars α 1 , ϕ, φ 1 , φ 4 of the four-scalar model (3.19). Although we will not utilise this truncation in this paper, it is a natural arena for additional investigations of spatially dependent masses for the equal mass model.

N = 2 * model
This model is obtained by setting two of the masses to be equal and one to be zero and specifically we consider m 1 = m 2 = 0 and m 3 = 0, as in section 2.3, now with m 1 real. To study this case we can consistently set φ 1 = φ 2 , α 1 = α 2 and β 1 = 0, while imposing α 3 = φ 3 = φ 4 = ϕ = β 2 = 0 in the ten scalar model (3.3). Equivalently, we set leading to a three-scalar model, parametrised by z 1 and β 1 , discussed in [45], which we will use to construct supersymmetric Janus solutions later. This model is invariant under the discrete symmetry generated by z 1 → −z 1 . Note that if we set α 1 = 0, we obtain a gravitational model with two real scalars which is the same 6 as that used to construct the RG flows associated with the homogeneous N = 2 * deformations in [61]. These RG flows preserve SU (2) R × U (1) global symmetry. Thus, this gravitational model cannot be used 7 to study spatially dependent mass deformations for the N = 2 * model since, as discussed in section 2.3, the spatial dependence breaks SU (2) R × U (1) down to U (1) R × U (1).

Supersymmetric mass deformations with ISO(1, 2)
symmetry In this section we will discuss the BPS equations that are associated with supersymmetric mass deformations with ISO(1, 2) symmetry. Furthermore, in appendix B we will analyse holographic renormalisation for this class of solutions, which will be useful in future studies of these solutions as well as when we discuss physical properties of the supersymmetric Janus solutions, as a special sub-class. We leave the details in the appendix B, but we highlight here that there are a number of interesting issues, including a large number of possible finite counter terms, subtleties in obtaining a supersymmetric renormalisation scheme and interesting source terms which appear in the conformal anomaly.
In the context of the ten scalar truncation discussed in section 3, we consider the ansatz with A, V, N and the scalars z A , β 1 , β 2 , functions of (x, r) only. This ansatz preserves ISO(1, 2) symmetry associated with the coordinates t, y 1 , y 2 . The coordinates r, x, together, parametrise both the remaining field theory direction, upon which the mass deformations depend, as well as the holographic radial coordinate. There is some residual gauge freedom in this ansatz, associated with reparametrising (r, x) and in practise we will find it convenient to fix this in different ways in the sequel. In appendix A we derive a set of BPS equations associated with the preservation, in general, of 1/4 of the supersymmetry. We use the orthonormal frame (e 0 , e 1 , e 2 , e 3 , e 4 ) = (e A dt, e A dy 1 , e A dy 2 , e V dx, N dr) and note that the supersymmetry transformations are parametrised by pair of symplectic Majorana spinors 1 and 2 . We find that the Killing spinors are independent of t, y 1 , y 2 and satisfy the following projection condition with κ = ±1, which implies γ 012 2 = iκ 2 as a result of the Majorana condition 2 = −iγ 4 * 1 , as well as where ξ is a function of (x, r). It is also worth noting that we then have * 1 = ie −iξ 1 .
The associated system of BPS equations are then given by where we recall the definition of A µ given in (3.14), as well as The dependence of the Killing spinor on (x, r) can be determined and we find that they are given by 1 = e A/2 e iξ/2 η 0 where η 0 is a constant spinor satisfying the projection given in (4.2). We note that these BPS equations are not all independent, and there is also an issue of consistency, given the reality of various functions entering these equations, a point we return to below. Observe that these BPS equations are invariant under It is interesting to point out that if we choose the gauge N = e V , then the equations can be written in a simplified form, analogous to what was seen in [18]. We introduce the complex coordinate w = r − iκx and the (1, 0) form B defined by The equations (4.4) can then be cast in the form where F is a real quantity just depending on W, K given by and ∂,∂ are the holomorphic and anti-holomorphic exterior derivatives. Similarly, Interestingly, as we show in appendix A, we can use this formulation of the BPS equations to show that the consistency of the BPS equations requires a non-trivial condition on W, which we give in (A.21). Furthermore, we can show that the specific W that appears in the ten scalar truncation, given in (3.7), does in fact satisfy this condition. We expect the underlying reason for this is that we are working with a theory that comes from a truncation of a supersymmetric theory.

BPS equations for Janus solutions
We now consider a particular sub-class of the BPS configurations that we considered in the last subsection. The ansatz for the D = 5 metric is given by where A J = A J (r), N = N (r) and we take the scalar fields β 1 , β 2 , z A to be functions of r only. Here ds 2 (AdS 4 ) is the metric on AdS 4 of radius , given, for example, in Poincaré coordinates by The factor of can be absorbed after redefining A J , but it is convenient 8 to keep it explicit. Notice that we recover the metric on AdS 5 with radius L if N = 1 and We can obtain the BPS equations for the Janus solutions as a special sub-class of those considered in the last section. Specifically, if we take Specifically, in the resulting BPS equations that we write down below, if we take → ∞ we obtain the BPS equations for ordinary Lorentz invariant RG flows with metric ds 2 5 = e 2A(r) ds 2 (R 1,3 ) − dr 2 . One can also make contact with the BPS flow equations in Euclidean signature of [46], for an ansatz ds 2 5 = e 2A(r) ds 2 (S 4 ) − dr 2 , after taking 2 → − 2 .
then the metric ansatz (4.1) precisely gives (5.1). From the first and third BPS equations in (4.4) we then get respectively, with the second equation in (4.4) implied by the first of these. Furthermore, from (4.5) we get the remaining BPS equations We can also obtain the Poincaré type Killing spinors for the Janus solutions directly from those given in the last section and we find where η 0 is a constant spinor, and 2 = e −iξ γ 4 1 . In addition there are also superconformal type Killing spinors of the form where η 0 is a constant spinor satisfying and again 2 = e −iξ γ 4 1 . Observe that the BPS equation (5.5), (5.6) are invariant under the transformation but note that the latter changes the projection on the Killing spinor. They are also invariant under Further insight into these BPS equations can be obtained by choosing the gauge N = e A J and then recasting them in a manner similar to what we did in the last section. Specifically, if we define then we obtain 9 the following BPS equations: where F is the real quantity just depending on W, K given in (4.9), as well as Now the right hand side of (5.14) is real, which implies that Im(B) is constant, in agreement with (5.13). To further examine the consistency of the equations, given the reality of β 1 and β 2 , we first note that for any function G(z A , β 1 , β 2 ) which depends only on the scalar fields and is anti-holomorphic in the four complex scalars z A , using (5.13)-(5.15) we can deduce whereÔ is a differential operator on the scalar manifold defined aŝ Then, taking the r derivative of the last two equations in (5.15), we obtain the following necessary conditions for these set of equations to be consistent with β i being real: Notice that these conditions do not involve B, just the scalar fields, and hence they are necessary conditions on K and W. One can explicitly check that these conditions are satisfied for (3.6) and (3.7) in the ten scalar model. It is also not difficult to see that if (5.18) is satisfied, then it is sufficient for a solution to exist, given a set of starting values for z A , β i , B satisfying the condition The equations below can be immediately obtained from those in (A.14)-(A.21) if we choose e V = N = e A J and e A = e A J −x/ , which also gives rise to the metric ansatz (5.1) with N = e A J , but, in contrast to (5.4) and (5.2), the metric for the AdS 4 sections are in horospherical coordinates rather than Poincaré type coordinates.
Indeed, by taking the taking the r derivative of the expression on the left hand side, using (5.16) and given that (5.18) holds, we see that (5.19) is guaranteed to be satisfied along the flow. Furthermore, for any starting values of z A , β i , one can always choose a starting value of B r that satisfies (5.19), and then solve the equations. From the above arguments, given (5.18) is satisfied one can also conclude the following: • If the starting values of z A , β i are such that ∂ β i log W = 0 or Im(∂ β i log W) = 0 (for i = 1, 2), then given a chosen value of κ l one can always find a starting value for Re(B r ) such that (5.19) is satisfied and solve the equations. It is then guaranteed from (5.19) that along each point in the flow, either ∂ β i log W = 0 or Im(∂ β i log W) = 0.
• Conversely, a choice of starting values with ∂ β i log W = 0 and real, for either β 1 or β 2 , is consistent with equations (5.15) and (5.14) but is incompatible with equation (5.13) since it requires Im(B r ) = 0.
• From the last two equations in (5.15) it is clear that the turning point for β i corresponds to a point in which ∂ β i log W = 0.
• For a turning point of A J , from (5.13) we have Re(B r ) = 0 and therefore (5.19) implies that at this point we must have Re(∂ β i log W) = 0. Thus, at this turning point we are just free to specify initial conditions for the z A which implies that the family of solutions is of dimension twice the number of z A that are active.
Note that we have proved these results for the flows using the gauge N = e A J . however, they are gauge invariant results for the flows and hence, are also valid for the gauge N = 1 that we use to numerically construct the solutions in the next section.

Supersymmetric Janus Solutions
In this section we present various solutions to the BPS equations (5.5), (5.6) that we derived in the previous section, including families of Janus solutions. More precisely, we will do this for each of three different consistent truncations of the ten scalar model that we discussed in section 3. Before doing that we make some general comments concerning how we obtain the field theory sources and expectation values, both with AdS 4 boundary metric and with flat boundary metric, using the holographic renormalisation scheme that we outline in detail in appendices B and C.

Preliminaries
Let us focus on the Janus solutions which describe a planar co-dimension one interface in N = 4 SYM that is supported by spatially dependent mass sources. These solutions have a metric of the form given in (5.1): where we have now set N = 1 for convenience, with Notice that in the gauge N = 1, the BPS equations are invariant under shifts of the radial coordinate It is illuminating to first recall that the N = 4 SYM AdS 5 vacuum solution with this AdS 4 slicing is given by with vanishing scalar fields. If we now employ the coordinate transformation we obtain the AdS 5 metric, with flat-slicing, given by In the (ρ, y 3 ) coordinates the conformal boundary is reached at ρ → ∞ and has a flat boundary metric with coordinates (t, y i ). On the other hand in the (r, x) coordinates the conformal boundary has three components: two half spaces r → ±∞ at x = 0, associated with y 3 > 0 and y 3 < 0, respectively, joined together at the planar interface at x = 0 and finite r, associated with y 3 = 0. As r → ±∞ we naturally obtain the AdS 4 metric on the two half spaces. A few more details are provided in appendix C and we have also illustrated the set-up there in figure 6.
6.1.1 Janus solutions: field theory on AdS 4 The Janus solutions of N = 4 SYM that we construct approach the N = 4 SYM AdS 5 vacuum as r → ±∞ but with additional mass sources. Analogous to the discussion for the AdS 5 vacuum solution itself, the conformal boundary of these Janus solutions consists of three components: two half spaces, with AdS 4 metrics, joined together at a planar interface along the boundary of the AdS 4 . Note that the boundary at x = 0 is not a standard asymptotically locally AdS 5 region, as the scalars are not approaching an extremum of the potential, but only at r = ±∞. Let us first consider the r → ∞ end of the interface, returning to the r → −∞ end in section 6.1.3. As r → ∞ we demand that we have the schematic expansion Recall that in the N = 1 gauge the BPS equations have the residual shift symmetry in r (6.3). By shifting the radial coordinate via r → r − A 0 L we can always remove the constant A 0 term and we shall do so in the following. In particular all the expressions for the expectation values and sources given below are obtained with The various other coefficients in this expansion, which are all constant, are constrained by the BPS equations, as we detail below. The constants φ i,(s) , α i,(s) , β i,(s) , ϕ (s) are associated with constant sources for the mass deformations of N = 4 SYM when placed on AdS 4 . Recalling from (3.2) that these are sources for operators of conformal dimension ∆ = 3, 2, 2, 4, respectively, it is useful to note that the field theory sources on AdS 4 that are invariant under a Weyl rescaling of the AdS 4 radius are given by φ i,(s) , 2 α i,(s) , 2 β i,(s) , ϕ (s) . In this paper we will not discuss deformations that involve the coupling constant of N = 4 SYM, and so we will always set The BPS equations then imply that these sources satisfy Notice that these relations respect the field theory scaling dimensions of the sources on AdS 4 that we just mentioned above. (6.7), with suitable contributions from the sources, give rise to the expectation values of the scalar operators. We will give explicit expressions for these in each of the specific truncations below. Here, to illustrate, we just highlight a simple example: using the renormalisation scheme discussed in appendix B, we find that for N = 4 SYM on AdS 4 we have Here δ α is an undetermined constant that parametrises a finite counter term, which we haven't fixed. As we will see below it is intimately connected with a novel feature of the expectation values of the operators in flat spacetime. We also note that due to the structure of the conformal anomaly 2 O α i is not invariant under rescalings of , as one might have expected, a point we return to below.

Janus solutions: field theory on flat spacetime
We are primarily interested in obtaining the sources and expectation values for operators of N = 4 SYM in flat spacetime, as in section 2. Now the metric on AdS 4 in (6.2) is conformal to flat spacetime. Thus, we can obtain the relevant quantities in flat spacetime from those on AdS 4 by simply performing a Weyl transformation with Weyl factor x 2 / 2 . However, while the sources transform covariantly under Weyl transformations, the expectation values do not due to the presence of source terms appearing in the conformal anomaly A (similar to [62,64]), schematically given by where the dots refer to extra terms involving finite counter terms (see (B.14),(B.15)). In fact we can obtain the relevant results within holography by carrying out a bulk coordinate transformation so that as we approach the r → ∞ with y 3 > 0. Substituting this into (6.7) then leads to an expansion of the bulk fields as ρ → ∞, which one can find in appendix C. Having done this, we can then employ 10 the holographic renormalisation scheme for ISO(1, 2) invariant configurations discussed in appendix B, in order to read off the sources and the expectation values with the field theory now on flat spacetime. The non-trivial sources for the dual scalar operators in N = 4 SYM theory now have the expected dependence on the spatial coordinate y 3 (still with y 3 > 0) that we saw in section 2: with φ 4,(s) = ϕ (s) = 0. Recalling that the numerators in these expressions are the scale invariant field theory sources on AdS 4 , we see that in flat spacetime these field theory sources have scaling dimensions 1, 2, 2 associated with operators that have dimensions ∆ = 3, 2, 2, respectively. Furthermore, when combined with the BPS relations (6.10), these expressions are in alignment with those that we derived in section 2 for each of the three different N = 1 * truncations. Due to the structure of the conformal anomaly, the expressions for the expectation values are more involved. To illustrate, here we just note that we have and give explicit expressions for the other expectation values for the three subtruncations below. We highlight the appearance of the novel log(y 3 ) term that appears in this expectation value. Notice that performing a scaling of the y 3 coordinate is associated with a shift in δ α , which parametrises a finite counter term. We can certainly choose a renormalisation scheme in which we set δ α = 0. However, there are additional similar finite counter terms that appear in expectation values of other operators, as we will see in each of the specific truncations below, and we have not been able to find a simple argument that would fix all of them in a way that is consistent with supersymmetry. Given the log terms appearing in the expectation values we expect that there will be at least one set of supersymmetric finite counterterms that one is free to add. We leave further investigation on this issue to future work. From the above results we can conclude that under a Weyl transformation of the AdS 4 boundary metric of the form h ab → Λ 2 h ab , with Λ = x/l, the sources transform covariantly with φ i(s) → Λ −1 φ i(s) , α i(s) → Λ −2 α i(s) and β i(s) → Λ −2 β i(s) . However, the expectation values do not transform covariantly due to the anomaly and, for example, 4πGL Λ −2 log Λ. The transformation properties for all the expectation values can be obtained from (B.19)-(B.21). It is worth emphasising that these results imply that some care is required in comparing expectation values of operators on AdS 4 for solutions with different values of the AdS 4 radius due to this non-covariant rescaling. In practice, in all of our numerics we have set = 1 (as well as L = 1).

The r → −∞ end of the conformal boundary
The analysis above concerned the component of the conformal boundary for the Janus solutions with AdS 4 slicing at r → ∞. There is a similar analysis for the component at r → −∞, which by assumption, is again approaching the N = 4 SYM AdS 5 vacuum. Firstly, we can consider the expansion given in (6.7) after replacing r → −r: ϕ =φ (s) + · · · +φ (v) e 4r/L + · · · , (6.16) and we can and will setÃ by shifting the radial coordinate 11 . As we show in appendix C.2, the BPS equations then imply that the coefficients are related as in the r → ∞ case, but after taking κ → −κ. We emphasise that we are not changing the projections on the preserved Killing spinors in doing this. So, for example, with this expansion at the r → −∞ 11 When one numerically constructs a solution, one generically finds that A 0 andÃ 0 in (6.7) and (6.16) are non-zero and not equal. In order to utilise our holographic renormalisation results with A 0 =Ã 0 = 0, one needs to shift the radial coordinate by different constants at r = ±∞.
end we now haveα i,(s) = +κ L φ i,(s) , i = 1, . . . , 3 , withφ 4,(s) =φ (s) = 0. Furthermore, to carry out the coordinate transformation back to flat space we can use (6.13) with r → −r and y 3 → −y 3 . This will then give the relevant quantities on the y 3 < 0 part of the conformal boundary, with flat boundary metric. Thus, to obtain the flat boundary results for y 3 < 0 from those for y 3 > 0, we need to make the replacements y 3 → −y 3 and κ → −κ.
As an illustration, we can consider the sub-class of explicit N = 4 SYM Janus solutions that are symmetric under the Z 2 symmetry given below in (6.20). For this class we have, for example, that φ i (r) = −φ i (−r), while α i (r) = +α i (−r). We then find for AdS 4 boundary metric if we have a source φ i(s) at the r → ∞ end, we will have a source φ i(s) = − φ i(s) at the r → −∞ end. Transforming to flat boundary coordinates we then find that for both y 3 > 0 and y 3 < 0 we have a source of the form φ i(s) /y 3 , i.e. antisymmetric in y 3 . Similarly we would have source 2 α i(s) at both r → ±∞ ends and transforming to flat space source 2 α i(s) /(y 3 ) 2 , i.e. symmetric in y 3 . Similar comments apply to expectation values.
We can similarly consider the sub-class of explicit Janus solutions that are symmetric under the Z 2 symmetry given below in (6.21). For this class we have, for example, that φ i (r) = +φ i (−r), while α i (r) = −α i (−r). We then find in AdS 4 slicing we will have source φ i(s) at both r → ±∞ ends and transforming to flat boundary coordinates we would have a source of the form φ i(s) /|y 3 |, i.e. symmetric in y 3 . Similarly, we would have a source 2 α i(s) at the r → +∞ end and 2α i(s) = − 2 α i(s) at the r → −∞ end, and transforming to flat space we get a source 2 α i(s) /(y 3 ) 2 for y 3 > 0 and a source − 2 α i(s) /(y 3 ) 2 for y 3 < 0 i.e. antisymmetric in y 3 . Similar results apply to the expectation values.

Constructing solutions
Having made some general comments on how we determine the sources and expectation values for the Janus solutions, in the next subsections we turn to summarising the solutions that we have found for the three different truncations. At this point it is worth recalling the various general constraints on the space of solutions that are itemised at the end of the last section, below (5.19).
It is also helpful to recall that the ten scalar model, and the three further truncations, are all invariant under the Z 2 symmetry that takes Furthermore, the BPS equations for the Janus solutions in (5.5),(5.6) are also invariant under the Z 2 symmetry that acts as Combining these two, we conclude that the BPS equations are also invariant under We have utilised various approaches to solving the BPS equations numerically. One approach is to start at, say, r → ∞, and then use the expansion (6.7) to set initial conditions to integrate in to smaller values of r and see where one ends up. As we will see, while some solutions end up at a similar asymptotic region at r → −∞, and hence are Janus solutions of N = 4 SYM, there are also solutions that run off to singular behaviour. Furthermore, there are also solutions which do not have an asymptotic region of the form (6.7) or (6.16). Another approach, and a more general one, is to start at a point in the bulk, for example a turning point of the function A J (r) at say r = 0 and then integrate out to smaller and larger values of r, and again see where one ends up. In the following we will summarise the main results of these constructions.
To simplify the discussion it will be helpful to first discuss the N = 2 * model, which is the simplest, before discussing the two N = 1 * models.

N = 2 * model
This model was summarised in section 3.3. There is one complex scalar field z 1 , which we write as z 1 = tanh[α 1 − iφ 1 ] and one real scalar field β 1 .
Consider solutions that approach N = 4 SYM with mass sources at, say r → ∞. Following the discussion in the last subsection and using the results of appendices B, C we can summarise the source and expectation values for the relevant operators that are active. All of the source terms are specified by φ 1,(s) with The field theory sources on AdS 4 are given by φ 1,(s) , α 1,(s) , β 1,(s) , with φ 1,(s) , 2 α 1,(s) , 2 β 1,(s) , invariant under Weyl scalings of , while those on flat spacetime are given by (6.14): and have scaling dimensions 1, 2, 2, respectively. For the associated expectation values of the operators in flat spacetime, we have which then, along with φ 1,(s) , determines the remaining expectation values where δ α , δ β are unspecified finite counterterms. An important aspect of the above summary, is that for a specific choice of finite counter terms, all of the scalar sources and expectation values of the dual field theory can be obtained by giving φ 1,(s) as well as 2 α 1,(v) . We now set = 1 (as well as L = 1) and also fix the sign arising in the BPS equations: κ = +1.
Following the discussion given at the end of section 5, we know that there is a two parameter family of solutions for this model. A useful way to parametrise them is to take one parameter to be the phase of the complex scalar z 1 at the turning point of the function A J (r). Due to the symmetries given in (6.19), (6.20) we can restrict to solutions for which this phase lies in the domain [0, π/2]. Then, fixing this phase we can construct a one parameter family of solutions that we can represent by parametric plots of the real and imaginary parts of the complex scalar field z 1 , as displayed in figure 1. In these plots the black squares correspond to turning points of the function A J (r) and, from left to right, the phase is equal to 0, π/4, π/2, respectively. The blue dot at the origin in each of the plots corresponds to the N = 4 SYM AdS 5 vacuum solution.
For each fixed value of the phase, there is a one parameter family of N = 4 SYM Janus solutions (blue curves) that approach the N = 4 SYM AdS 5 vacuum solution at r → ±∞ with, generically, spatially modulated mass terms that are parametrised by φ 1,(s) . Furthermore, focussing on the r → +∞ end we find 0 < φ 1,(s) < φ 1,(s) | crit and φ 1,(s) | crit = ∞. The exception to this occurs for the class of solutions in which the phase at the turning point is exactly π/2 (right plot in figure 1): for this class, remarkably, we find that it is source free, φ 1,(s) = 0, on both sides of the interface, a point we return to below. We also note that, somewhat surprisingly, for the generic solutions as the phase approaches π/2, the critical value of the source, φ 1,(s) | crit does not approach zero. Another interesting feature of this model is that for each Janus solution, with phase not equal to π/2, on either side of the interface at r → ±∞, we always find 12 thatφ 1,(s) = −φ 1,(s) . If we convert to sources in flat space, recalling that we have set = 1, this means we have a source of the form φ 1,(s) /y 3 , for all y 3 and where here φ 1,(s) is the expansion coefficient at r = +∞ (which we noted above is in the range 0 < φ 1,(s) < φ 1,(s) | crit ).
We can also determine the expectation values of various operators for the Janus solutions on each side of the interface at r = ±∞. With = 1, we just explain the behaviour of α 1,(v) which can be used to get all expectation values of scalar operators. 12 This suggests that there is some kind of conserved quantity for the BPS equations which we have yet to identify. In seeking such quantity it is important to note, as we state below, that the expectation values are not simply related on either side of the interface.
For the special case when the phase is equal to zero (left plot in figure 1), the solutions are invariant under the symmetry (6.20) and as explained in section 6.1.3, we find α 1,(v) is the same on each side of the interface. For this case we also find for the r = +∞ end with 0 < φ 1,(s) < φ 1,(s) | crit , that as φ 1,(s) goes from 0 to φ 1,(s) | crit , then α 1,(v) increases from 0, hits a maximum and then decreases to a finite negative value at φ 1,(s) | crit .
By contrast, for the class of Janus solutions when the phase is in the domain (0, π/2) we find that α 1,(v) andα 1,(v) do not have the same value at r = ±∞, respectively.
When the phase is exactly equal to π/2, there is a different picture. As we noted above there are no sources on either side of the interface. We also find for the two sides of the interface α 1,(v) = −α 1,(v) and the energy density (B.24) is zero off of the interface. The absence of sources on either side of the interface is noteworthy. It seems most likely that there is a distributional source that is located on the interface itself, otherwise we would have a configuration that spontaneously breaks translations, and it would be interesting to verify this in detail.
The plots given in figure 1 also reveal that there are other non-Janus solutions for this model. When the phase is in the open domain (0, π/2), there is also a one parameter family of solutions that approach N = 4 SYM as r → −∞, with −∞ <φ 1,(s) < −φ 1,(s) | crit . As one moves along the radial direction, at some finite value of the radial coordinate, past the turning point, one hits a singularity, with |z 1 | → 1. Such solutions, corresponding to the black curves in figure 1 are one-sided interfaces (of a type for which it has been suggested they describe BCFTs [65]). Finally, there are also solutions which approach singular behaviour at both ends of the radial domain, denoted by black dashed lines in figure 1. When the phase is equal to π/2, all solutions are regular Janus solutions except for the one solution in the right plot of figure 1 which would have a turning point with Im(z 1 ) = 1, a singular point in field space.

N = 1 * one-mass model
This model was summarised in section 3.1. There is again one complex scalar field z 1 , which we write as z 1 = tanh[ 1 2 α 3 − iφ 3 ] and one real scalar field β 1 . A particularly interesting feature of this model, that plays an important role in the solutions, is the presence of the two LS ± AdS 5 fixed point solutions given in (3.17).
Consider solutions that approach N = 4 SYM with mass sources at, say, r → ∞. Following the discussion in section 6.1 and using the results of appendices B, C we can summarise the source and expectation values for the relevant operators that are active. All of the source terms are specified by φ 3,(s) with The field theory sources on AdS 4 are given by φ 3,(s) , α 3,(s) , β 1,(s) , with φ 3,(s) , 2 α 3,(s) , 2 β 1,(s) , invariant under Weyl scalings of , while for those on flat spacetime the dimensionful quantities are given by (6.14): which then, with along with φ 3,(s) determines the remaining expectation values An important aspect of the above summary, is that for a specific choice of finite counter terms, all of the scalar sources and expectation values of the dual field theory can be obtained by giving φ 3,(s) as well as 2 α 3,(v) . We now set = κ = 1.
We next turn to the solutions which we have summarised in figure 2. As before each plot corresponds to a fixed phase of the scalar field z 1 at the turning point of A J (r). From left to right we again have the phase is 0, π/4 and π/2, respectively. The blue dot at the origin is the N = 4 SYM AdS 5 vacuum solution, while the two red dots correspond to the two LS ± AdS 5 fixed points given in (3.17), each dual to the N = 1 LS SCFT.
First consider the left panel in figure 2. There is a one parameter family of N = 4 SYM Janus solutions (blue curves) that approach the N = 4 SYM AdS 5 vacuum solution with spatially modulated mass terms. Since the phase is zero, these solutions are invariant under the symmetry (6.20) and, as discussed in section 6.1.3, we find that we have source φ 3,(s) on the r → +∞ side of the interface and sourceφ 3,(s) = −φ 3,(s) on the r → −∞ side. From the flat space perspective we therefore have (with = 1) a source of the form φ 3,(s) /y 3 , for all y 3 . Similarly, we find thatα 3,(v) = α 3,(v) on either side of the interface. These Janus solutions exist for 0 < φ 3,(s) < ∞.
As φ 3,(s) → ∞, we have α 3,(v) → ∞ and the Janus solutions approach a new type of solution (red curve): namely, a novel Janus solution with the LS + AdS 5 vacuum on one side of the interface and the LS − AdS 5 vacuum on the other. These solutions are discussed in more detail in [36]. Note that there are no source terms that are active on either side of this LS + /LS − interface; this actually follows from the fact that once we demand that there are no sources for the irrelevant scalar operators with ∆ = 2 + √ 7 and ∆ = 3 + √ 7, it is not possible to source the relevant scalar operator of the LS SCFT with dimension ∆ = 1 + √ 7 whilst preserving supersymmetry [36]. We also note that the irrational scaling dimensions for these operators seem to exclude the possibility of having distributional sources for these scalar operators on this interface while still preserving conformal symmetry. As explained in [36] the two sides of the LS + /LS − interface are related by a discrete automorphism. Beyond this novel LS Janus solution there is also a one parameter family of solutions that approach singular behaviour, with |z 1 | → 1 at finite values of r. Re z 1 Im z 1 Figure 2: The family of BPS solutions for the N = 1 * one mass model is summarised by parametrically plotting the real and imaginary parts of the scalar field z 1 . The black squares correspond to turning points of the function A J (r) and the three plots, from left to right, correspond to solutions where the phase of the complex scalar field at the turning point is 0, π/4 and π/2, respectively. The blue dot at the origin is the N = 4 SYM AdS 5 vacuum and the blue lines are Janus solutions. The two red dots are the two LS ± AdS 5 solutions, each dual to the Leigh-Strassler SCFT. In the middle plot the red curve is a conformal RG interface with N = 4 SYM on one side of the interface and the LS SCFT on the other. In the left plot the red curve is a conformal interface with LS on either side of the the interface. The boundary of field space is |z 1 | = 1 and the black curves are singular on one or both ends. As one moves from r = −∞ to r = +∞ one moves clockwise on the curves.
The middle panel of figure 2 shows the set of solutions when the phase is π/4 and this provides the generic picture for phases in the open domain (0, π/2). There is again a one parameter family of N = 4 SYM Janus solutions (blue curves) with, for the r = ∞ end, 0 < φ 3,(s) < φ 3,(s) | crit , with finite φ 3,(s) | crit . As φ 3,(s) → φ 3,(s) | crit , we have α 3,(v) approaching a finite value and the Janus solutions approach another new type of solution (red curve). Before discussing that, we note that φ 3,(s) at r = ∞ andφ 3,(s) at r = −∞ are not simply related in general and hence we have flat space sources as in (2.18). Returning to the new solution (red curve), we see that it approaches the N = 4 SYM AdS 5 vacuum at r → ∞ and the LS + AdS 5 solution at r → −∞. This describes a conformal RG interface, with N = 4 SYM on one side of the interface, with spatially dependent sources with φ 3,(s) = φ 3,(s) | crit , and the LS SCFT on the other. Once again there are no sources on the LS + side of the interface. This solution is also discussed in more detail in [36]. Beyond this solution, for φ 3,(s) | crit < φ 3,(s) < ∞ we obtain solutions which start off at the mass deformed N = 4 SYM AdS 5 vacuum at r → ∞ and then become singular at some finite value of r, as marked with the black lines in the middle panel of figure 2. There are also solutions that become singular at both r → ±∞ which are marked by black dashed lines in figure 2.
What we described in the previous paragraph was for the phase at the turning point equal to π/4 and applies for the phase in the range (0, π/2), with one small difference. Beyond some value of the phase, we find that φ 3,(s) is no longer always positive for the Janus solutions. In particular, this leads to a limiting red curve solution, with a specific value of the phase of the turning point, describing an RG interface where the source φ 3,(s) | crit vanishes on the N = 4 SYM side. This solution is also discussed in [36].
Finally, when the phase is π/2 (third plot in figure 2), there is a one parameter family of N = 4 SYM Janus solutions that exist for −∞ < φ 3,(s) < 0. These solutions are invariant under the symmetry (6.21) and, as discussed in section 6.1.3, we find that the source on either side of the interface at r = ±∞ takes the same valueφ 3,(s) = φ 3,(s) . From the flat space perspective we therefore have (with = 1) a source of the form φ 3,(s) /|y 3 |, for all y 3 . There is also a one parameter family of solutions that are singular at finite values of the radial coordinate in each direction and are marked by the dashed black lines in the right plot in figure 2.

N = 1 * equal-mass model
This model was summarised in section 3.2. There are two independent complex fields z 1 , z 2 which we write Consider solutions that approach N = 4 SYM with mass sources at, say r → ∞. As we have mentioned several times, in this paper we focus on solutions for which the source terms for the coupling constant and the gaugino mass vanish: All of the source terms for BPS configurations are then specified by φ 1,(s) with The field theory sources on AdS 4 are given by φ 1,(s) , α 1,(s) , with φ 1,(s) , 2 α 1,(s) invariant under Weyl scalings of , while those on flat spacetime are given by (6.14): , For BPS configurations the remaining expectation values are determined by these expressions, along with φ 1,(s) , via Note that δ α , δ 4(5) parametrise finite counterterms which we have not fixed. We now set = κ = 1. Following the discussion given at the end of section 5, we know that there is a four parameter family of solutions for this model. Here we will just study a one parameter family of solutions, leaving a more complete exploration for future work. We also note the following technical point in solving the numerical equations. If we construct a solution with, say, the N =4 SYM dilaton source non-vanishing at the r → ∞ end, ϕ (s) = 0, then we can obtain a solution with ϕ (s) = 0 by using the shift symmetry of the dilaton (3.12).
In figure 3 we have summarised a one-parameter family of N = 4 SYM Janus solutions for this model (with ϕ (s) = 0 on both sides), for which the phase of both scalars is zero at the turning point and so the solutions are invariant under the symmetry (6.20). In contrast to previous models it is convenient to label this family of solutions not by the values of z i at the turning point but instead in terms of the value of α 1 at the turning point which we label as (α 1 ) tp . In particular, we note that this is invariant under the dilaton shift. For a fixed value of (α 1 ) tp there is a one-parameter family of solutions for which z i tp are real, all related by shifts of the dilaton and so for regular solutions we can use this symmetry to fix ϕ (s) = 0 for each value of (α 1 ) tp (and using (6.20) we find it is set to zero on both sides). We find that regular solutions exist for −α crit < (α 1 ) tp < α crit with α crit ≈ 0.447. In figure  3 we have displayed a series of Janus solutions as blue curves, for various values in the range (α 1 ) tp ∈ [0, α crit ). Interestingly, as (α 1 ) tp increases the solutions start to develop a sequence of more and more loops in the scalar field parameter space and, surprisingly, as (α 1 ) tp → α crit we obtain a new solution which is exactly periodic in the radial coordinate r (the red curve), which we return to below.
Note that in figure 3 we have just plotted z 1 ; the behaviour of z 2 is broadly similar. We also note in addition to the Janus solutions, there are also a host of solutions that are singular at both ends. The last panel in figure 3 illustrates a few such solutions. In particular there are solutions that can wind several times around, before hitting the singularity.
We increases we see the appearance of more and more loops, asymptoting to the red curve, in figure (g), which describes a solution periodic in the radial direction. In figure (h) we have exactly the same solutions as figure (g) but with the addition of some illustrative solutions (black dashed lines) that are singular at both ends (and without the red curve for clarity) . As one moves from r = −∞ to r = +∞ one moves clockwise on the curves. φ 4,(v) for a given φ 1,(s) instead of two. We now return to the limiting periodic solution corresponding to the red curve in figure 3. As (α 1 ) tp → α crit all of the functions develop more and more periods in the radial direction, with the period and shape changing very little as the limit is taken. In figure 5 we have plotted the metric function A J as well as the scalar functions z 1 , z 2 as a function of r for a solution close to α crit . For any (α 1 ) tp < α crit we have a Janus solution, so A J → ±r/L and all the scalars go to zero as r → ±∞. The region in between, however, approaches a solution that has periodic behaviour in the entire region r ∈ (−∞, ∞). By compactifying the radial direction for this limiting periodic solution, we obtain a new AdS 4 × S 1 solution that will be further explored in [48]. Note that we can also approach this critical solution from above, (α 1 ) tp > α crit , where solutions develop more and more periods before becoming singular (see figure 3(h)).  Figure 5: For the N = 1 * equal mass model as (α 1 ) tp → α crit , approaching the red curve in figure 3, the N = 4 SYM Janus solutions have a radial region approaching a solution that is periodic in the radial coordinate. For any −α crit < (α 1 ) tp < α crit , the solution is a Janus solution and so A J → ±r/L and z A → 0 as r → ±∞. Both the period and shape of the middle region is essentially unchanged as we approach the critical solution, with just more periods appearing, and clearly reveals the functional form of the periodic solution. The blue and orange curves are the real and imaginary components of z 1 , z 2 , respectively.
We conclude this section with one further comment regarding a possibly confusing feature of figure 3. As argued above, there exists a one parameter family of periodic solutions related by the shift symmetry of the dilaton. When constructing Janus solutions, we fix this shift symmetry by requiring our solutions have vanishing dilaton on the boundaries, but for different values of (α 1 ) tp this corresponds to a different average value of the dilaton in the periodic intermediate region. Thus the enveloping ellipses of the blue curves differ for the different curves in figure 3, though they are all related by a dilaton shift (3.12). Said another way, when ϕ (s) = 0 on the boundaries for each value of (α 1 ) tp , only the fields φ 1 , α 1 and φ 4 have a well-defined limit when (α 1 ) tp → α crit , whereas ϕ does not.

Discussion
In this paper we have analysed mass deformations of N = 4 SYM theory that depend on one of the three spatial dimensions and preserve some amount of supersymmetry. We have focussed on configurations with constant coupling constant. We have also explored these deformations within the context of holography, studying both configurations that preserve ISO(1, 2) symmetry as well those that in addition preserve conformal symmetry. For the latter class of deformations we have also constructed a number of interesting new classes of explicit Janus solutions.
In section 2 we analysed the supersymmetric mass deformations of N = 4 SYM from a field theory perspective. We achieved this by coupling N = 4 SYM to off shell conformal supergravity and then taking the Planck mass to infinity as in [39]. For configurations that have constant τ (i.e. constant coupling constant and theta angle) as well as no deformations in the 15 and 6, parametrised by V i µ j and T ij µν , respectively, we reduced the problem to solving the equations given in (2.6). We then focussed on deformations that generalised the homogeneous N = 1 * mass deformations, studying in some detail three particular examples: the one mass model, the equal mass model and the N = 2 * model. It would be interesting to further investigate other possible solutions to (2.6). It would also be interesting to analyse more general deformations that also allow τ to depend on the spatial coordinates: this will include the classification of [26], which considered deformations with varying coupling constant combined with other deformations all proportional to spatial derivatives of the coupling constant 13 , but additional cases are possible by relaxing the latter condition. More generally, one can also explore supersymmetry preserving deformations that also involve V i µ j and T ij µν . In the remainder of the paper we analysed the supersymmetric mass deformations, with constant τ , from a holographic perspective. We utilised a consistent truncation of D = 5 gauged supergravity that involves 10 real scalar fields which allowed us 13 The supersymmetric Janus supergravity solutions corresponding to [26] have recently been discussed in [31]. From [31] one can check that the there are no source terms for the dimension ∆ = 2, 3 operators away from the interface, consistent with [26].
to obtain BPS equations preserving ISO(1, 2) symmetry for real mass deformations. The natural arena to analyse complex mass deformations would be to utilise an N = 2 D = 5 gauged supergravity theory coupled to two vector multiplets and four hypermutiplets, with scalar manifold as in (3.1). However, this supergravity theory has not yet been explicitly constructed, but has been explored recently in [63].
For the ISO(1, 2) preserving configurations associated with real mass deformations we carried out in some detail the holographic renormalisation procedure. We saw that the model admits a large number of finite counter terms. We managed to reduce this number a little by demanding that supersymmetric configurations have vanishing energy density. It would be desirable to identify a fully supersymmetric scheme along the lines of [66], but this could be a challenging task. Our results indicate that there will not be a unique supersymmetric scheme due to the possibility of adding finite supersymmetric invariants; perhaps it is easier to explicitly determine these invariants first. For the Janus configurations, our holographic renormalisation allowed us to clearly identify sources and expectation values of operators viewing the interface as describing N = 4 SYM on flat spacetime with spatially modulated mass sources or N = 4 SYM on AdS 4 spacetime with constant mass sources.
We showed that the deformed N = 4 SYM theory has a conformal anomaly that includes terms that are quadratic and quartic in the scalar source terms similar to [62,64]. For Janus solutions we showed that while the sources for the scalar operators on either side of the interface transform covariantly with respect to Weyl transformations, the expectation values for the corresponding operators do not. In particular, the expectation values of the operators for interfaces of N = 4 SYM on flat spacetime contained novel terms logarithmic in the coordinate transverse to the interface as well as the usual terms expected from conformal invariance.
In this paper we have focussed on constructing Janus solutions of supergravity, with d = 3 conformal invariance. However, it would be interesting to further study the more general class of solutions that just preserve ISO(1, 2) symmetry. What would be most desirable is if the BPS equations can be suitably manipulated to give a simpler system set of equations, as was seen for the analogous constructions of [18] in D = 4.
In section 6, we discussed various explicit Janus solutions of N = 4 SYM for the N = 2 * theory as well as one-mass and equal mass models. For all cases, our constructions also revealed solutions that approach the N = 4 SYM AdS 5 as r → ∞ (or r → −∞ in some cases) and then become singular at some finite value of r. As such, these solutions have a conformal boundary dual to N = 4 SYM with mass deformations on a half space that ends at a singularity. It would be interesting to examine these solutions in more detail, including elucidating the precise nature of the singularities in type IIB supergravity, and see if they can be interpreted as BCFTs, as suggested in [65]. Perhaps they can also be interpreted as a kind of RG flow for N = 4 SYM on AdS 4 . It seems even more challenging to find any physical interpretation for the singular solutions that do not have any conformal boundary.
For the one mass model we also found some interesting special solutions which involve the two LS ± AdS 5 fixed points that this model admits, each dual to the LS SCFT. We found examples of both RG interface solutions, with N = 4 SYM on one side of the interface, and the LS SCFT on the other, as well as a novel LS + /LS − Janus solution dual to a novel conformal interface of the LS SCFT. Both of these are further discussed in [36].
The equal mass model also revealed solutions with interesting new features. In this model we constructed a class of N = 4 SYM Janus solutions that develop a periodic structure in the bulk radial coordinate, and in the critical limit we find solutions which are exactly periodic. After compactifying the radial direction, we obtained a new supersymmetric AdS 4 × S 1 solution that uplifts to a new AdS 4 × S 1 × S 5 solution of type IIB supergravity, which will be further explored in [48]. This solution is somewhat reminiscent of the interesting AdS 4 × S 1 solutions in [31]. An important difference, however, is that while our new solutions are simply periodic in the S 1 direction, the solutions of [31] have non-trivial SL(2, Z) monodromy. One might anticipate that there are many more Janus solutions that can be constructed in gauged supergravity that have the axion and dilaton activated as well as the mass sources that we have focussed on. It seems likely that this will also lead to a host of new AdS 4 × S 1 solutions for which there is non-trivial SL(2, Z) monodromy along the S 1 direction, as in the solutions of [31]. Indeed we have already constructed some specific examples that will be reported in [48].
Consider the ansatz with A, V, N and the scalars z A , β 1 , β 2 , functions of (x, r) only. We use the orthonormal frame (e 0 , e 1 , e 2 , e 3 , e 4 ) = (e A dt, e A dy 1 , e A dy 2 , e V dx, N dr). We assume that the Killing spinor is independent of t, y 1 , y 2 and begin by imposing the projection and hence γ 012 1 = −iκ 1 . Using the Majorana condition 2 = −iγ 4 * 1 , we also have γ 34 2 = iκ 2 and γ 012 2 = iκ 2 . From the t, y 1 , y 2 components of the gravitino equations we get while from the x, r components we get, respectively, Taking the complex conjugate of (A.3) and using the Majorana condition we deduce that We therefore introduce a phase ξ(x, r) via and solve (A.3) by imposing the projection We also note that (A.6) implies the integrability condition −∂ r κe V e K/2 Im(e −iξW ) = ∂ x N e K/2 Re(e −iξW ) . (A.8) We can now rewrite (A.4) in the form By taking the complex conjugate of these two equations and using * 1 = ie −iξ 1 , we deduce that in each expression, the left and right hand sides each separately vanish. We thus conclude that the Killing spinor takes the form where η 0 is a constant spinor satisfying γ 012 η 0 = −iκη 0 .
The combined system of BPS equations are thus given by as well as the following equations from the remaining fermion variations We note that these equations are not all independent. We also observe that these equations are invariant under Next we can rewrite them in a simplified manner if we choose the gauge N = e V . We can define the complex coordinate w = r − iκx so that∂ = dw 1 2 (∂ r − iκ∂ x ) and also the (1, 0) form B as where After fixing the gauge-freedom, by fixing N , the BPS equations (A.11) and (A.12) are a set of 16 real equations for 13 real functions, A, V, ξ, z A , β 1 , β 2 with A = 1, . . . , 4, in the ten scalar truncation, and therefore naively seem to be over constrained. To analyse the consistency of these equations it is convenient to work in the gauge N = e V , and analyse (A.15), (A.17). We first observe that F is a manifestly real quantity that depends only on the scalar fields. The BPS equations are constrained due to the fact that A, β 1 and β 2 are all real. If one takes the holomorphic exterior derivative∂ of the equations for these functions in (A.15),(A.17) one obtains necessary conditions for the equations to be satisfied. For A this condition is given by: which is automatically satisfied from (A.15). We are therefore left with two constraints to check.
To do so, it is useful to first prove the following result. Consider any function G(z A , β 1 , β 2 ) which depends only on the scalar fields and is anti-holomorphic in the four complex scalars z A . Using the BPS equations (A.15),(A.17) we deduce whereÔ is a differential operator on the scalar manifold defined aŝ Then, taking the ∂ derivative of the last two equations in (A.17), we obtain the following necessary conditions for these set of equations to be consistent with β i being real: Notice that these conditions do not involve B, just the scalar fields, and hence they are conditions on K and W. One can explicitly check that these conditions are satisfied for (3.6) and (3.7) in the ten scalar model. We expect that these conditions are sufficient conditions for consistency; while we have not proven this in general, we did for the sub-class of Janus solutions as discussed below (5.18). It would be interesting if there is a way to understand these consistency conditions more directly from the underlying N = 2 supergravity theory.

B Holographic Renormalisation
In this appendix we provide some details on the holographic renormalisation that we use and, in particular, give expressions for various one point functions. We will focus on configurations that preserve ISO(1, 2) symmetry, first considering general configurations before restricting to BPS configurations. In appendix C we will specialise to those that, in addition, preserve conformal symmetry.

B.1 General case
We consider the class of solutions that are general enough to describe sources which depend on one of the spatial directions and preserve ISO(1, 2) symmetry. Specifically, we consider metrics of the form with all scalar fields functions of (r, x) only. The conformal boundary is located at r → ∞ and there we have the expansion where h ab (x) is the metric for the spacetime where the dual field theory lives, which we write as where the function f (x) is included for convenience (it can be useful in utilising different gauge choices in numerically solving the equations). Two cases of particular interest are firstly, when Ω(x) is constant, associated with a flat boundary metric. Secondly, when e Ω = /x and f (x) = constant, associated with an AdS 4 boundary metric, with radius (more precisely, this gives a component of the boundary for the Janus solutions as we elaborate further in appendix C). The full action can be written as the sum of four terms: The first two terms are the bulk action and the boundary Gibbons-Hawking term, given by where the the bulk Lagrangian L for the ten scalar model is given in (3.5) and the trace of extrinsic curvature for the outward pointing normal one-form n = dr is given by K = − 1 2 γ ab ∂ r γ ab . As in [46] we also have with the AdS 5 vacuum solution, with vanishing scalar fields, dual to SU (N ) N = 4 SYM theory. The boundary action S ct that is required to remove divergences takes the form where all quantities are evaluated with respect to γ ab evaluated in the limit r → ∞. Finally, the finite counterterms that we shall consider are given by which depends on 13 coefficients {δ R 2 , δ ∆R 2 , . . . , }, which we take to be constants, and again we have utilised the boundary metric γ ab evaluated in the limit r → ∞.
There are a number of comments concerning this choice of finite counter terms, which defines a renormalisation scheme. We first note S f inite respects the discrete symmetries (3.9)-(3.11) of the D = 5 theory. Next, we recall that the scalar field ϕ is dual to a marginal operator in N = 4 SYM theory, as in (3.2), and its boundary value can be identified with changing the coupling constant of the theory. Being a source for a marginal operator there are many additional finite counter terms that one might consider, including allowing the δ coefficients appearing in S f inite to be functions of ϕ as well as including terms with derivatives of ϕ. These additional counterterms could be significantly simplified if we impose that they respect the shift symmetry ϕ → ϕ + const. of the bulk D = 5 gravitational theory, but we are not sure that this is a justified assumption. However, for the purpose of this paper, where will assume that there are no sources active for ϕ, as we make precise below, and the fact that we will only be calculating one point functions, S f inite is in fact general enough. For example, if any of the δ's did depend on ϕ it would only be the terms linear in ϕ that could affect the one-point functions, and we exclude such terms using the symmetry 14 (3.9) Finally, demanding that this renormalisation scheme is supersymmetric places certain restrictions on {δ R 2 , ..., } as we discuss below.
Using the bulk equations of motion, we can develop the following, schematic, 14 When considering backgrounds with nontrivial dilaton, observe that (3.9) is not a symmetry of the field theory since it also involves an SL(2, Z) duality transformation; here we are simply demanding that our scheme does respect (3.9).
asymptotic expansion as r → ∞: where all coefficients, except for ϕ (s) , can depend on the coordinate x. In this expansion φ i,(s) , α i,(s) , β i,(s) and ϕ (s) provide sources for the corresponding dual operators in N = 4 SYM given in (3.2). Our interest in this paper is spatially dependent mass deformations and hence we allow φ i,(s) , α i,(s) , β i,(s) to depend on x but we take Note however, that in general ϕ (v) does depend on x and is related to the operator dual to ϕ acquiring a spatially dependent expectation value. We also note that in developing the asymptotic expansion, there is one algebraic and one differential constraint relating A (v) and V (v) which ensure the Ward identities in the boundary theory, given below, are satisfied. The expectation value for the stress tensor is given by where the former expression is for ϕ, φ i , φ 4 , dual to operators with ∆ = 4, 3, 3 and the latter for α i , β i , dual to operators with ∆ = 2. We will not present these expressions for general finite counterterms as the expressions are lengthy. Instead we just note that we have checked that the following Ward identity is satisfied where here the covariant derivative is defined with respect to the field theory metric h ab in (B.2),(B.3) and this metric has been used to raise the index on T a b . We also recall here that we have assumed that the source ϕ (s) vanishes.

B.2 BPS configurations
We now restrict to ISO(1, 2) configurations which satisfy the BPS equations in (4.4), (4.5). Continuing to assume that ϕ (s) = 0 we find the following constraints on the sources In particular, we see that the sources α i,(s) , β i,(s) are determined by φ i,(s) with i = 1, . . . , 3. We also find an additional set of relations amongst the expansion functions which provide relations between the expectation values of the various dual scalar operators as well as the stress tensor. As they are rather long, we will not record them here, but we will below for each of the sub-truncations that we study.
It is now illuminating to use these results to calculate the energy density for flat field theory metric, h ab = η ab (i.e. Ω = f = 0 in (B.3)). Firstly, we find that stress energy tensor itself takes the form where the matrix σ ab = diag(1, −1, −1, 3) satisfies η ab σ ab = 0 and hence does not contribute to the conformal anomaly. Using this, we obtain the following expression for the local energy density for BPS configurations in flat space: For a supersymmetric renormalisation scheme, we demand that the finite counter terms are such that the right hand side is a total spatial derivative, in order that the total energy for spatially modulated supersymmetric sources (with compact support) is exactly zero. This implies the following conditions must be satisfied These conditions are the same as one would get if one used the "Bogomol'nyi trick" used in [45][46][47], but we note that the analysis of [45,46] did not include the possibility of δ α and δ β . Since we would like to work with a scheme that preserves supersymmetry we will impose (B.25). We do not know of any other simple consequences of supersymmetry that allows to impose additional constraints on the 19 coefficients appearing in the finite counterterm action. Ideally, one would like to implement a fully supersymmetric holographic renormalisation scheme, along the lines of [66], but we leave that for future work. We will explicitly see that the terms δ α , δ β , in particular, appear in novel contributions to the expectation values of operators for Janus solutions (e.g. see (6.15)).

B.2.3 N = 2 * model
This model is obtained from the 10-scalar model by setting φ 1 = φ 2 , α 1 = α 2 and β 1 = 0, while imposing α 3 = φ 3 = φ 4 = ϕ = β 2 = 0. Thus, we set The expansion for the general ISO(1, 2) configurations is given by where φ 1,(s) , α 1,(s) and β 1,(s) are the source terms for the scalar operators. The one point functions are given by where 2 and R refer to the field theory metric h ab in (B.3), along with the expected results Turning to the supersymmetric ISO(1, 2) BPS configurations satisfying (4.4), (4.5), the relation between the sources is given by The BPS equations also impose relations between the coefficients with "(v)" subscript in (B.44) given by Under the renormalisation scheme (B.25) these are equivalent to the following set of relationships between the one point functions of the scalar operators for the BPS where again R refer to the field theory metric h ab in (B.3).

C One point functions for Janus solutions
For orientation we first recall the metric for AdS 5 written in "Janusian" coordinates which makes manifest the foliation by AdS 4 spaces. We then discuss how the results of the previous appendix can be employed to obtain holographic data for the Janus solutions discussed in section 6.
with ρ ∈ (−∞, +∞) and the conformal boundary at ρ → ∞. We heavily utilise the (ρ, y 3 ) coordinates and the (r, x) coordinates in this paper, with the former associated with flat spacetime boundary metric and the latter associated with AdS 4 boundary metric.
C.2 BPS Janus solutions: field theory on AdS 4 The BPS Janus solutions discussed in sections 5 and 6 are special sub-classes of the ISO(1, 2) BPS solutions discussed in appendix B with and all scalar fields taken to be a function of r only. As r → ±∞ the N = 4 SYM Janus solutions approach the N = 4 SYM AdS 5 vacuum but with additional mass sources. Like the N = 4 SYM AdS 5 vacuum solution itself, the conformal boundary again consists of three components, with two half spaces (with AdS 4 metrics) that are joined at a planar interface. Let us first consider the r → ∞ end, returning to the r → −∞ end in section C.2.4. After recalling (B.9), as r → ∞ we have the schematic expansion of the BPS equations (5.5),(5.6) (with N = 1) given by A J = r L + A 0 + · · · + A (v) e −4r/L + · · · , φ i = φ i,(s) e −r/L + · · · + φ i,(v) e −3r/L + · · · , i = 1, . . . , 4 , α i = α i,(s) r L e −2r/L + α i,(v) e −2r/L + · · · , , i = 1, . . . , 3 , β i = β i,(s) r L e −2r/L + β i,(v) e −2r/L + · · · , i = 1, . . . , 2 , ϕ = ϕ (s) + · · · + ϕ (v) e −4r/L + · · · . (C.6) The various constant coefficients in this expansion are constrained by the BPS equations, as discussed below. We have highlighted a constant term A 0 that can appear in the expansion for A J . By shifting the radial coordinate via r → r − A 0 L we can always remove this term and we shall do so in the following. In particular all the expressions for the expectation values and sources given below are obtained with The terms φ i,(s) , α i,(s) , β i,(s) , ϕ (s) give rise to source terms for N = 4 SYM on this component of the conformal boundary with AdS 4 metric. Recalling that these are sources for operators of conformal dimension ∆ = 3, 2, 2, 4 respectively, it is helpful to note that the field theory sources on AdS 4 , that are invariant under Weyl scalings of , are given by φ i,(s) , 2 α i,(s) , 2 β i,(s) . We are always assuming that ϕ (s) = 0 and from (B.22), the BPS conditions relating the sources are given for the ten scalar model by In a similar manner φ i,(v) , α i,(v) , β i,(v) and ϕ (v) , with suitable contributions from the sources, give rise to the expectation values of the scalar operators. We can obtain these results for each of the three truncations considered in appendix B, after using e Ω = /x and f (x) = 0, as we summarise below. For BPS Janus configurations, from (B.33) we can then express the remaining nontrivial expectation values in terms of O α 3 along with φ 3,(s) as follows: (C.10) Notice that these expressions depend on the δ α , δ β , δ Rφ 2 (1) which parametrise finite counterterms that we haven't fixed. We also have independent of the counterterms. Notice that for a fixed choice of δ α , δ β , δ Rφ 2 (1) , we can therefore specify all of the scalar sources and expectation values of the dual field theory by giving φ 3,(s) and α 3,(v) .
Notice that these expressions depend on the δ α , δ β , δ Rφ 2 (1) which parametrise finite counterterms which we haven't fixed. We also have independent of the counterterms. Notice that for a fixed choice of δ α , δ β , δ Rφ 2 (1) , we can therefore specify all of the scalar sources and expectation values of the dual field theory by giving φ 1,(s) and α 1,(v) .

C.2.4 Results for the r → −∞ end, AdS 4 boundary
We now discuss analogous results, for the sources and expectation values, for the conformal boundary, with AdS 4 metric, at the r → −∞ end. Here we can develop an asymptotic expansion to the BPS equations (5.5),(5.6) (with N = 1) of the form A J = −r L +Ã 0 + · · · +Ã (v) e 4r/L + · · · , φ i =φ i,(s) e r/L + · · · +φ i,(v) e 3r/L + · · · , i = 1, . . . , 4 , α i =α i,(s) −r L e 2r/L +α i,(v) e 2r/L + · · · , , i = 1, . . . , 3 , β i =β i,(s) −r L e 2r/L +β i,(v) e 2r/L + · · · , i = 1, . . . , 2 , ϕ =φ (s) + · · · +φ (v) e 4r/L + · · · , (C.18) and we will always setÃ 0 = 0, which can be achieved by a shift of the radial coordinate. This has exactly the same form as in (C.6) after the interchange r → −r. The BPS equations will then relate various coefficients. We can easily deduce these relations using the following argument. We first recall that the BPS equations (5.5), (5.6) are invariant under the transformation r → −r, ξ → ξ + π and κ → −κ. Second, we want to use the result that if a solution has ξ = 0 at r = +∞ then necessarily it will have ξ = π at r = −∞. This can be seen from (5.13): at r → ±∞ the scalars are approaching zero so the phase of W is going to zero. Thus, the phase of B r at r → ±∞ is ξ and from (5.13) we see that ξ must change by π in going from r = +∞ to r = −∞. Taking these two results together, we can then deduce that all of the results that we obtained for the r → +∞ end can be taken over to the r → −∞ end provided that wherever κ appears in the former, it is replaced 15 with −κ in the latter. Thus, for example, we can conclude that the BPS equations in (5.5),(5.6) (with N = 1) imply that in the expansion (C.18) at r → −∞ we now havẽ 15 Recall that κ = ±1 enters the Killing spinor projections (5.7). To avoid possible confusion, we emphasise that we are holding this projection fixed in developing the asymptotic expansion (C.18) at r → −∞; the argument we have given is just a way of getting at the result. 16 Note that the results in this section can also obtained from our results (B.16)-(B.21).
with y 3 > 0. Substituting this into (6.7) then leads to an expansion as ρ → ∞ with the metric asymptoting to ds 2 → −dρ 2 + e 2ρ/L dt 2 − dy 2 1 − dy 2 2 − dy 2 3 , (C. 21) and recalling the discussion in section C.1, this component of the conformal boundary is for y 3 > 0. As ρ → ∞ we find that the expansion for the scalars given in (C.6) then becomes and we note that we have set φ 4,(s) = 0 as implied by the BPS relations (C.8).
To proceed, we now notice that this form of the solution is a special case of the ISO(1, 2) invariant configurations discussed in appendix B, with Ω(x) = f (x) = 0, provided that we replace the coordinates (r, x) in that appendix with (ρ, y 3 ). As a consequence we can immediately read off the sources and the expectation values for the various operators. The non-zero scalar sources in flat spacetime are of the form The BPS relations between the remaining expectation values are given by The BPS relations between the remaining expectation values are given by The above analysis concerning sources and expectation values was for the conformal boundary end located at r → ∞ (AdS 4 boundary metric) or y 3 > 0 (flat boundary metric). In section C.2.4 we discussed the asymptotic expansion of the solution, with AdS 4 boundary, for the conformal boundary end located at r → −∞. For this end we can then employ the coordinate transformation to flat space, as given in (C.20) but switching r → −r and y 3 → −y 3 . This will then give the relevant quantities on the y 3 < 0 part of the conformal boundary, with flat boundary metric. Recalling the discussion in section C.2.4, we can therefore obtain the flat boundary results for y 3 < 0 from those for y 3 > 0, by making the replacements y 3 → −y 3 and κ → −κ.