A new family of AdS_4 S-folds in type IIB string theory
Igal Arav, K. C. Matthew Cheung, Jerome P. Gauntlett, Matthew M. Roberts, Christopher Rosen
IImperial/TP/2021/JG/01ICCUB-20-XXX
A new family of
AdS S-foldsin type IIB string theory
Igal Arav , K. C. Matthew Cheung , Jerome P. Gauntlett Matthew M. Roberts and Christopher Rosen Institute for Theoretical Physics, University of Amsterdam,Science Park 904, PO Box 94485,1090 GL Amsterdam, The Netherlands Blackett Laboratory, Imperial CollegeLondon, SW7 2AZ, U.K. Departament de F´ısica Qu´antica i Astrof´ısica and Institut de Ci`encies del Cosmos (ICC),Universitat de Barcelona, Mart´ı Franqu`es 1, ES-08028,Barcelona, Spain
Abstract
We construct infinite new classes of
AdS × S × S solutions of type IIBstring theory which have non-trivial SL (2 , Z ) monodromy along the S direction. The solutions are supersymmetric and holographically dual,generically, to N = 1 SCFTs in d = 3. The solutions are first constructedas AdS × R solutions in D = 5 SO (6) gauged supergravity and thenuplifted to D = 10. The solutions all arise as limiting cases of Janussolutions of d = 4, N = 4 SYM theory which are supported both by adifferent value of the coupling constant on either side of the interface, aswell as by fermion and boson mass deformations. As special cases, theconstruction recovers three known S-fold constructions, which preserve N = 1 , N = 1 AdS × S × S solution (not S-folded). We also present some novel “one-sided Janus” solutions that are non-singular. a r X i v : . [ h e p - t h ] J a n ontents D = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Janus solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 AdS × R solutions and S-folds . . . . . . . . . . . . . . . . . . . . . 133.4 Free energy of the S-folds . . . . . . . . . . . . . . . . . . . . . . . . . 16 N = 1 ∗ equal mass, SO (3) invariant model 17 N = 1 S-fold . . . . . . . . . . . . . 184.2 New S-fold solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 SU (2) invariant 24 N = 1 S-fold . . . . . . . . . . . . . 265.2 New S-fold solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 N = 4 supersymmetry . . . . . . . . 316.2 Other constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 A.1 The 10-scalar model in maximal gauged supergravity . . . . . . . . . 37A.2 The uplift to type IIB supergravity . . . . . . . . . . . . . . . . . . . 41A.2.1 The SO (3) invariant 4-scalar model . . . . . . . . . . . . . . . 42A.2.2 The SO (3) × SO (3) invariant 3-scalar model . . . . . . . . . . 43A.2.3 The SU (2) invariant 5-scalar model . . . . . . . . . . . . . . . 45A.3 The SL (2 , R ) action in five and ten dimensions . . . . . . . . . . . . 46A.4 The N = 4 one-sided Janus solution in type IIB . . . . . . . . . . . . 491 Introduction
The landscape of non-geometric solutions of string/M-theory which are associatedwith the AdS/CFT correspondence is still largely unexplored territory. By definition,such solutions are patched together using duality symmetries and hence they are notordinary solutions of the low-energy supergravity approximation. Nevertheless, infavourable situations one can still utilise supergravity constructions to obtain valuableinsights.Within the context of type IIB string theory, which is the focus of this paper, wecan consider S-folds i.e. non-geometric solutions that are patched together using the SL (2 , Z ) symmetry. For AdS/CFT applications we are interested in solutions of typeIIB supergravity of the form AdS × M with, in general, the axion-dilaton, the three-forms and the self dual five-form all active on M . The S-fold construction impliesthat M will have monodromies in SL (2 , Z ), which act on the axion-dilaton and thethree-forms. If these monodromies involve contractible loops in M then, in general,one is led to the presence of brane singularities and regions where the supergravityapproximation breaks down. However, one can hope to make further progress if thesolutions lie within the context of F-theory as in the AdS solutions discussed in [1,2],for example.We can also consider AdS × M solutions of type IIB supergravity where the SL (2 , Z ) monodromies do not involve contractible loops. In this case, provided thatthe fields are all varying slowly on M , we can expect the type IIB supergravityapproximation to be valid, and that such solutions do indeed correspond to dualCFTs. Examples of such solutions were presented in [3] and further discussed in [4]:the spacetime is of the form AdS × S × S with non-trivial SL (2 , Z ) monodromyjust around the S direction. The solutions preserve the supersymmetry associatedwith N = 4 SCFTs in d = 3, and we shall refer to them as N = 4 S-folds. Thesesolutions can be constructed as a certain limit of a class of N = 4 Janus solutions [5]which describe N = 4, d = 3 superconformal interfaces of d = 4, N = 4 SYM theory.Using this perspective, and the results of [6, 7], a specific conjecture for the SCFTdual to these N = 4 S-folds was given in [4]. N = 1 and N = 2 S-fold solutions of the form AdS × S × S have also beenconstructed in [8–10]. In particular, it was shown in [10] how they can be obtainedas limiting solutions of N = 1 [11–13] and N = 2 [5] Janus solutions, also describinginterfaces of d = 4, N = 4 SYM theory. Furthermore, the N = 1 AdS × S × S S-folds have been generalised to N = 1 AdS × S × SE S-folds, where SE is anarbitrary five-dimensional Sasaki-Einstein manifold [14].2n the Janus solutions that are used to construct the S-folds just mentioned [4,10],the complex gauge coupling τ of N = 4 SYM theory takes different values on eitherside of the interface. It was recently pointed out that this is not necessarily the caseand it is possible to have interfaces in N = 4 SYM with the same value of τ on eitherside of the interface which are supported by spatially dependent fermion and bosonmass deformations, while preserving d = 3 conformal symmetry [15]. The associatedsupersymmetric Janus solutions of type IIB supergravity which are holographicallydual to such interfaces were also constructed in [15] by first constructing them in D = 5 SO (6) gauged supergravity. Furthermore, there is a particularly interesting AdS × R solution that can be obtained as a limit of this class of Janus solutionswhich is periodic in the R direction and uplifts to give a smooth AdS × S × S solution of type IIB supergravity (i.e. with no S-folding) [15].The constructions of [15] can be immediately generalised to give Janus solutionswhich have spatially dependent masses and varying τ . It is therefore natural to askif there are limiting classes of such Janus solutions which can be utilised to constructnew S-fold solutions and/or periodic solutions. While we have not found any moreperiodic solutions, we have found infinite new classes of AdS × R solutions of D = 5 SO (6) gauged supergravity that give rise to infinite new classes of S-fold solutions ofthe form AdS × S × S , generically preserving N = 1 supersymmetry in d = 3.Our new construction will utilise various consistent sub-truncations of D = 5 SO (6) gauged supergravity all lying within the 10-scalar truncation of [16] which, notsurprisingly, just keeps 10 of the 42 scalars as well as the metric. One of these scalars,the D = 5 dilaton ϕ , which for the vacuum AdS solutions is dual to the couplingconstant of N = 4 SYM theory, plays a privileged role as we expand upon below .Within this truncation we numerically construct families of AdS × R solutions thatarise as certain limits of Janus solutions with N = 4 SYM on either side of theinterface. We then uplift these to obtain AdS × R × S of type IIB supergravity,using the results of [17, 18]. Additional AdS × R × S solutions in D = 10 can thenbe generated using SL (2 , R ) transformations. Finally, within this larger family ofsolutions of type IIB supergravity one can find discrete examples where we can S-foldleading to supersymmetric AdS × S × S S-fold solutions of type IIB string theory.The D = 5 metric for the solutions we discuss in this paper are all of the form ds = e A ( r ) [ ds ( AdS ) − dr ] , (1.1) As far as we are aware this is the first example of a supersymmetric
AdS × M solution of typeIIB supergravity, with compact M that is smooth i.e. without sources. We note that, in general, the type IIB dilaton of the uplifted solutions is not the same as the D = 5 dilaton, as explained in appendix A. D = 5 scalar fields just a function of the radial coordinate. The ansatztherefore preserves d = 3 conformal invariance. The D = 5 solutions associated withthe known N = 1 , AdS × R withconstant warp factor A and with all of the D = 5 scalars constant, except for the D = 5 dilaton, ϕ , which varies linearly in the radial coordinate.The new AdS × R solutions involve several novel features. Firstly, the metric on AdS × R is no longer a direct product but a warped product, since the warp factornow has non-trivial dependence on the radial direction. Thus, the metric no longeradmits a Killing vector associated with translations in the R direction. In particularthe new solutions cannot be constructed in the D = 4 gauged supergravity theoriesthat were used to construct the known S-fold solutions as in [3, 8, 9]. Secondly, thewarp factor A ( r ) and all of the D = 5 scalars are now periodic in the R direction,with the same period ∆ r , except for ϕ which is now a “linear plus periodic” (LPP)function of r . In figure 1 we have illustrated how these solutions arise as limitingcases of Janus solutions of N = 4 SYM which, generically, have the N = 4 SYMcoupling taking different values on either side of the interface, as well as additionalfermion and boson mass deformations. - - -
10 0 10 20 30 - - � φ � /L - - -
10 0 10 20 30 - - � ϕ � α � /L Figure 1: A D = 5 Janus solution that is approaching the new AdS × R solutions forthe SO (3) invariant model. As ¯ r → ±∞ , the solution is approaching AdS on eitherside of the interface: the warp factor is behaving as A → ± ¯ r/L , the D = 5 dilaton isapproaching two different constants ϕ → ϕ ± , while the remaining scalar fields φ , α and φ (not displayed) are going to zero. In the intermediate regime we see the buildup of a periodic structure for the warp factor and the scalar fields, with ϕ having, inaddition, a dependence linear in ¯ r i.e. ϕ is a “linear plus periodic” (LPP) function.In the new limiting AdS × R solutions the intermediate structure extends all theway out to infinity. Note that we have used the proper distance radial coordinate ¯ r given in (3.2). 4he plan of the paper is as follows. We begin in section 2 by discussing the10-scalar truncation of maximal D = 5 SO (6) gauged supergravity given in [16]as well as various sub-truncations. In section 3 we discuss the general frameworkfor constructing the new AdS × R solutions in D = 5 and the procedure for thenobtaining AdS × S × S S-folds solutions of type IIB string theory.In sections 4 and 5 we discuss in more detail the constructions for two particularsub-truncations: an SO (3) ⊂ SU (3) ⊂ SO (6) invariant model involving four scalarfields and an SU (2) ⊂ SU (3) ⊂ SO (6) invariant model involving five scalar fields.The SO (3) invariant model, called the N = 1 ∗ equal mass model in [15], includes the AdS × R solutions associated with the known N = 1 and N = 4 S-fold solutionsas well as the periodic AdS × R solution found in [15]. We note that figure 1is associated with this model. The SU (2) invariant model includes the AdS × R solutions associated with the known N = 2 S-fold solutions and it also includes thoseassociated with the known N = 1 S-fold solutions. In both truncations, our newfamily of S-fold solutions includes the previous known solutions. Furthermore, inboth cases one can identify the existence of some of our new family of solutions by aperturbative construction about the known N = 1 S-fold solution (but, interestingly,not around the N = 2 , AdS vacuum on one side and either a known S-fold solution, an LPPdilaton solution or the periodic D = 5 solution of [15] on the other. Unlike otherone-sided Janus solutions, they are non-singular. In the case that it approaches the N = 4 AdS × R S-fold solution we are able to construct the solution analyticallyand we show how, after uplifting to type IIB supergravity, it fits into the generalclass of
AdS × S × S × Σ solutions preserving N = 4 supersymmetry that werestudied in [19,20] (see also [21–23] for some later developments). We also discuss howthe solution is related to solutions describing D3-branes ending on D5-branes. Inappendix A we have included some useful results concerning how to uplift solutionsof the 10-scalar model in D = 5 to type IIB supergravity. We are interested in a truncation of N = 8, SO (6) gauged supergravity in D = 5,discussed in [16], that involves the metric and ten scalar fields which parametrise thecoset M = SO (1 , × SO (1 , × (cid:104) SU (1 , U (1) (cid:105) . (2.1)5he SO (1 , × SO (1 ,
1) is parametrised by two scalars β , β while the remainingeight scalars of this truncation, parametrising four copies of the Poincar´e disc, canbe packaged into four complex scalar fields z A via z = tanh (cid:104) (cid:0) α + α + α + ϕ − iφ − iφ − iφ + iφ (cid:1)(cid:105) ,z = tanh (cid:104) (cid:0) α − α + α − ϕ − iφ + iφ − iφ − iφ (cid:1)(cid:105) ,z = tanh (cid:104) (cid:0) α + α − α − ϕ − iφ − iφ + iφ − iφ (cid:1)(cid:105) ,z = tanh (cid:104) (cid:0) α − α − α + ϕ − iφ + iφ + iφ + iφ (cid:1)(cid:105) . (2.2)Schematically, these 10 scalar fields are dual to the following Hermitian operators in N = 4 SYM theory:∆ = 4 : ϕ ↔ tr F µν F µν , , ∆ = 3 : φ i ↔ tr( χ i χ i + cubic in Z i ) + h.c. , i = 1 , , ,φ ↔ tr( λλ + cubic in Z i ) + h.c. , ∆ = 2 : α i ↔ tr( Z i ) + h.c. , i = 1 , , ,β ↔ tr( | Z | + | Z | − | Z | ) ,β ↔ tr( | Z | − | Z | ) . (2.3)The operators of d = 4, N = 4 SYM appearing on the right hand side of (2.3)have been written in an N = 1 language, with Z i and χ i the bosonic and fermioniccomponents of the associated three chiral superfields Φ i while λ is the gaugino ofthe vector multiplet. Thus, the D = 5 dilaton ϕ is dual to the coupling constant of N = 4 SYM theory, while φ i , φ are fermionic mass terms and α i , β , β are bosonicmass terms.The action is given by S Bulk = 14 πG (5) (cid:90) d x (cid:112) | g | (cid:104) − R + 3( ∂β ) + ( ∂β ) + 12 K A ¯ B ∂ µ z A ∂ µ ¯ z ¯ B − P (cid:105) , (2.4)and we work with a (+ − − − − ) signature convention. Here K is the K¨ahler potentialgiven by K = − (cid:88) A =1 log(1 − z A ¯ z A ) . (2.5)6he scalar potential P can be conveniently derived from a superpotential-like quantity W ≡ L e β +2 β (cid:0) z z + z z + z z + z z + z z + z z + z z z z (cid:1) + 1 L e β − β (cid:0) − z z + z z − z z − z z + z z − z z + z z z z (cid:1) + 1 L e − β (cid:0) z z − z z − z z − z z − z z + z z + z z z z (cid:1) , (2.6)via P = 18 e K (cid:20) ∂ β W ∂ β W + 12 ∂ β W ∂ β W + K ¯ BA ∇ A W∇ ¯ B W − WW (cid:21) , (2.7)where K ¯ BA is the inverse of K A ¯ B and ∇ A W ≡ ∂ A W + ∂ A KW .The model is invariant under Z × S discrete symmetries, which leave W invariant.First, it is invariant under the Z symmetry z A → − z A , ⇔ { φ i , φ , α i , ϕ } → −{ φ i , φ , α i , ϕ } . (2.8)Second, it is invariant under an S permutation symmetry which acts on ( − z , − z , z )as well as β , β and is generated by two elements: { z ↔ − z ⇔ φ ↔ φ , α ↔ α } , β → −
12 ( β + β ) , β →
12 ( β − β ) , { z ↔ − z ⇔ φ ↔ φ , α ↔ α } , β → − β . (2.9)There is also an invariance under the interchange of pairs of the z A : z ↔ z , z ↔ z , ⇔ ( φ , φ ) → − ( φ , φ ) , ( α , α ) → − ( α , α ) ,z ↔ z , z ↔ z , Z ⇐ = ⇒ ( φ , φ ) → − ( φ , φ ) , ( α , α ) → − ( α , α ) ,z ↔ z , z ↔ z , Z ⇐ = ⇒ ( φ , φ ) → − ( φ , φ ) , ( α , α ) → − ( α , α ) , (2.10)where the equivalence in the last two uses (2.8). Together (2.8)-(2.10) generate Z × S as observed in [24].The model is also invariant under shifts of the dilaton ϕ → ϕ + c . (2.11)For later use, we note that this shift symmetry is generated by the following holo-morphic Killing vector l = 12 (cid:88) A =1 ( − s ( A ) (cid:0) − ( z A ) (cid:1) ∂∂z A , (2.12)7here s ( A ) = 0 for A = 1 , s ( A ) = 1 for A = 2 ,
3. Furthermore, if we define (cid:101)
K ≡ K + log W + log W , (2.13)we have l A ∂ A (cid:101) K + l ¯ A ∂ ¯ A (cid:101) K = 0 , (2.14)and the corresponding moment map µ = µ ( z A , ¯ z A ), satisfying µ = il A ∂ A (cid:101) K = K A ¯ B ∂ ¯ B µ ∂ ¯ A (cid:101) K , (2.15)is given by µ = − i (cid:88) A =1 ( − s ( A ) z A − ¯ z A − z A ¯ z A . (2.16)In terms of the fields given in (2.2) we find that the moment map only depends on φ i , φ and takes the form µ = 12 [tan( − φ − φ − φ + φ ) − tan( − φ + φ − φ − φ ) − tan( − φ − φ + φ − φ ) + tan( − φ + φ + φ + φ )] . (2.17)Expanding about φ i = 0 we have to lowest order µ ∼ φ .The 10-scalar truncation is not a supergravity theory. However, the conditions fora solution of the 10-scalar model to preserve supersymmetry as a solution of D = 5 SO (6) gauged supergravity were written down in [16] and also used in [15]. Here weuse exactly the same conventions as [15].There are a number of different consistent sub-truncations of the 10-scalar modelwhich were also discussed in [16], that we summarise in figure 2. The figure alsodisplays where one can find the three known AdS × R solutions with a linear D = 5dilaton ϕ which are associated with S-folds preserving N = 1 , SO (6) that is preserved by the truncation. Thesesub-truncations preserve various subsets of the Z × S discrete symmetries given in(2.8)-(2.10). All of the sub-truncations preserve the Z symmetry (2.8) as well asshifts of the dilaton (2.11) when the dilaton ϕ is present in the truncation. In thispaper we will be mostly interested in two cases: the N = 1 ∗ equal mass, SO (3)invariant model, with SO (3) ⊂ SU (3) ⊂ SU (4) and involving four scalar fields;and the 5-scalar SU (2) invariant model, with SU (2) ⊂ SU (3) ⊂ SU (4). While the SO (3) invariant model does not preserve any additional symmetries, the SU (2) modelpreserves a further Z that is contained in (2.10).8 = − z , β = 0 z = ¯ z z = − z z = z , β = 0 z = z , z = 0 z = − z z = ¯ z = z ,β = 0 z = ¯ z z = z z = − z z = − z ,β = 0 z = ¯ z SO (2) invariant φ = φ , φ , φ α = α , α , ϕ, β U (1) × U (1) invariant φ = φ , φ = − φ α = α , α , ϕ, β SU (2) invariant φ , φ α , ϕ, β N = 1 ∗ equal-mass truncation SO (3) invariant φ = φ = φ , φ α = α = α , ϕ N = 2 ∗ truncation φ = φ , α = α , β SU (2) × U (1) invariant φ = − φ , α , ϕ, β contains N = 2 S-fold 3-scalar truncation SO (3) × SO (3) invariant φ = φ = φ = − φ α = α = α , ϕ contains N = 4 S-fold N = 1 ∗ one-mass truncation φ , α , β contains LS point 2-scalar truncation SU (3) invariant φ , ϕ contains N = 1 S-fold Figure 2: Various sub-truncations of the ten scalar model. In this paper we focus onthe N = 1 ∗ equal mass, SO (3) invariant truncation and the 5-scalar SU (2) invarianttruncation, marked by red boxes, as well as their associated sub-truncations in thebottom line. The boxes with the blue outline are truncations that contain known AdS × R S-fold solutions discussed in [10]. The boxes with the green outline aretruncations which were used in [15].
The construction of the S-fold solutions starts with solutions of D = 5 supergravity.These are then uplifted to type IIB, where additional solutions are generated usingthe SL (2 , R ) symmetry of type IIB supergravity. Finally, the S-folding procedure,using the SL (2 , Z ) symmetry of type IIB string theory, is made. D = 5 We consider solutions of D = 5 supergravity of the form ds = e A ds ( AdS ) − N dr , (3.1)where ds ( AdS ) is the metric on AdS , which we take to have unit radius, and A , N as well as the scalar fields β , β , z A are functions of r only. Clearly this ansatzpreserves d = 3 conformal invariance. There is still some freedom in choosing theradial coordinate. In this paper we will either use the “conformal gauge” with N = e A ,9s in (1.1), or the “proper distance gauge” with N = 1conformal gauge: N = e A , radial coordinate: r , proper distance gauge: N = 1 , radial coordinate: ¯ r , (3.2)with d ¯ r = e A dr .We are interested in supersymmetric configurations which, generically, are associ-ated with N = 1 supersymmetry in d = 3 (i.e. two Poincar´e plus two superconformalsupercharges). As shown in [15], we obtain such solutions provided that we satisfythe following BPS equations (in the conformal gauge), ∂ r A − i = 2 B r ,∂ r B r = 2 F B r ¯ B r , (3.3)where F is a real quantity just depending on W , K given by F ≡ −
32 1 |W| ∇ A WK A ¯ B ∇ ¯ B ¯ W − | ∂ β log W| − | ∂ β log W| , (3.4)as well as ∂ r z A = − K A ¯ B ∇ ¯ B WW ¯ B r ,∂ r β = − ∂ β log W ¯ B r ,∂ r β = − ∂ β log W ¯ B r . (3.5)In these equations the quantity B r is defined as B r ≡ e iξ + A + K / W where ξ ( r ) isa phase that appears in the Killing spinors. It is helpful to point out that the BPSequations are left invariant under the transformation r → − r, z A → ¯ z A , ξ → − ξ + π . (3.6)The BPS equations are also invariant under the discrete Z × S symmetries in (2.8)-(2.10) and this will also be the case for any of the sub-truncations in figure 2 forwhich they are still present. Additional general aspects of the space of solutions tothese BPS equations were discussed in section 5 of [15].It will also be useful to notice that the dilaton shift symmetry (2.11) of the 10-scalar model gives rise to a conserved quantity for the BPS equations. Specifically, With essentially no loss of generality, the parameter κ = ± κ = +1. E ≡ L e A µ ( z, ¯ z ) , (3.7)where the moment map was given in (2.16) or (2.17). This result can be derived viathe Noether procedure as follows. The Killing vector l A generating the symmetry(2.11), gives rise to a conserved current for the full equations of motion. For ourradial ansatz we deduce that the radial component of this current, given by E ∝ √ gg rr (cid:16) K A ¯ B ∂ r ¯ z ¯ B l A + K B ¯ A ∂ r z B l ¯ A (cid:17) , (3.8)is a conserved quantity, independent of r . Using the BPS equations we then obtain E ∝ e A (cid:16) ∂ A (cid:101) K B r l A + ∂ ¯ A (cid:101) K ¯ B r l ¯ A (cid:17) , = e A (cid:20) ( l A ∂ A (cid:101) K + l ¯ A ∂ ¯ A (cid:101) K ) Re( B r ) − i (cid:16) l A ∂ A (cid:101) K − l ¯ A ∂ ¯ A (cid:101) K (cid:17)(cid:21) , = − e A (cid:16) il A ∂ A (cid:101) K (cid:17) = − e A µ . (3.9)where to get to the second line we wrote B r = Re( B r ) − i , and to get to the thirdline we used (2.14) and (2.15). We now briefly summarise some aspects of the Janus solutions constructed in [15].We first recall that the
AdS vacuum solution, dual to d = 4, N = 4 SYM, has awarp factor given by e A = L cosh ¯ rL , (3.10)with all of the scalars vanishing, z A = 0.Janus solutions, describing superconformal interfaces of d = 4, N = 4 SYM, canbe obtained by solving the BPS equations and imposing boundary conditions so thatthey approach the AdS vacuum solution (3.10) at ¯ r = ±∞ , with suitable falloffs forthe scalar fields, associated with supersymmetric sources for the dual operators. Adetailed analysis of holographic renormalisation for such Janus solutions was carriedout in [15] (using the proper distance gauge). The focus in [15] was to construct Janussolutions that are dual to interfaces of N = 4 SYM that are supported by fermion andboson masses that have a non-trivial spatial dependence on the direction transverseto the interface. These solutions were constructed within the following truncations,shown in green boxes in figure 2: the N = 2 ∗ truncation (three scalar fields), the11 = 1 ∗ one-mass truncation (three scalar fields) and the N = 1 ∗ equal-mass, SO (3)invariant truncation (four scalar fields).Within the Janus solutions of the N = 1 ∗ equal-mass, SO (3) invariant truncation(green and red box in figure 2) a special limiting AdS × R solution was found withthe warp factor A and all of the scalar fields periodic in the R direction. As such,this solution can be compactified on the R direction and after uplifting to type IIB,one obtains a regular AdS × S × S solution (without S-folding). In the sequel wewill present new AdS × R solutions which are no longer periodic in the R directionthat can also be found as limiting classes of Janus solutions. In the new solutionsthe D = 5 dilaton, ϕ , is a LPP function while the remaining scalars and warp factorare periodic in the R direction; an illustration is given in figure 1. All of our newS-fold solutions arise as limits of D = 5 Janus solutions with ϕ ( s ) , which parametrisesthe source for the operator dual to ϕ , taking different values on either side of theinterface. In other words the Janus solutions are interfaces of d = 4, N = 4 SYMwith the coupling constant taking different values on either side of the interface.It will also be helpful to recall that for the N = 1 ∗ one-mass truncation, inaddition to the AdS vacuum solution dual to d = 4, N = 4 SYM, there are alsotwo other AdS solutions, LS ± , which are both dual to the Leigh-Strassler N = 1SCFT. In [15, 25] interesting limiting solutions of the Janus solutions associated withinterfaces involving the LS SCFT were found. In particular we found solutions dualto an RG interface with N = 4 SYM on one side of the interface and the LS theory onthe other, as well as Janus solutions with the LS theory on either side of the interface.In this paper we also construct solutions within the 5-scalar SU (2) truncation in figure2 (red box), which contain the LS ± fixed points. In addition to the new LPP solutionswe also find limiting Janus solutions that involve Janus interfaces for the LS ± fixedpoints themselves i.e. solutions with LS ± on either side of the interface with a linear D = 5 dilaton.Finally, as somewhat of an aside, we note that the conserved quantity E given in(3.7) implies a constraint amongst the sources and expectation values of operators of N = 4 SYM theory for the Janus configurations. Following the holographic renor-malisation carried out in [15], which used the proper distance gauge, the expansionat, say, the ¯ r → ∞ end of the interface is given by φ i = φ i, ( s ) e − ¯ r/L + · · · + φ i, ( v ) e − r/L + · · · , α i = α i, ( s ) ¯ rL e − r/L + α i, ( v ) e − r/L + · · · ,β i = β i, ( s ) ¯ rL e − r/L + β i, ( v ) e − r/L + · · · , ϕ = ϕ ( s ) + · · · + ϕ ( v ) e − r/L + · · · ,A = ¯ rL + · · · + A ( v ) e − r/L + · · · . (3.11)12ere φ i, ( s ) , α i, ( s ) , ... give the source terms of the dual operators, while φ i, ( v ) , α i, ( v ) , ... can be used to obtain the expectation values, explicitly given in [15]. Using thisexpansion as well the conditions on sources and expectation values imposed by theBPS conditions, we find that the integral of motion is given by E = 1 L (2 φ , ( v ) − φ , ( s ) φ , ( s ) φ , ( s ) ) . (3.12) AdS × R solutions and S-folds Our principal interest in this paper concerns a new class of solutions to the BPSequations of the form (in conformal gauge): ds = e A [ ds ( AdS ) − dr ] ,ϕ = kr + f ( r ) , (3.13)where k is a constant and A, f and all other scalars satisfy A ( r ) = A ( r + ∆ r ) , f ( r ) = f ( r + ∆ r ) , z A ( r ) = z A ( r + ∆ r ) . (3.14)Notice that, in general, the D = 5 dilaton ϕ is an LPP function, while the warpfactor and the remaining scalar fields are all periodic functions of r , with period ∆ r .Over one period ϕ changes by an amount ∆ ϕ given by∆ ϕ ≡ ϕ ( r + ∆ r ) − ϕ ( r ) = k ∆ r . (3.15)Although we have defined ∆ ϕ in the conformal gauge, importantly (and unlike k, ∆ r )it is invariant under coordinate changes of the form r → ρ with dρ = G ( r ) dr where G ( r ) is a periodic function, G ( r + ∆ r ) = G ( r ). We can also define the proper distanceof a period ∆¯ r , which is given by ∆¯ r = (cid:90) ∆ r e A dr . (3.16)For the special case when k = 0, when ϕ is also periodic, these solutions are peri-odic in the r direction and we can then immediately compactify the radial directionto obtain an AdS × S solution. In this case, if we identify after just one period, ∆¯ r is the length of the S . We presented one such solution in [15] and this will appear inour new constructions. For this purely periodic solution the period of the warp factor After integrating we can write ρ = cr + H ( r ) with H ( r + ∆ r ) = H ( r ) and H having no zeromode. Inverting this, we can write r = (1 /c ) ρ + ˜ H ( ρ ) with ˜ H ( ρ + ∆ ρ ) = ˜ H ( ρ ), where ∆ ρ = c ∆ r .In this gauge we can then write ϕ = ( k/c ) ρ + ˜ f ( ρ ) with ˜ f ( ρ + ∆ ρ ) = ˜ f ( ρ ) and ∆ ϕ = k ∆ r .
13s half of that of the scalar fields. Another special case is when k (cid:54) = 0 and f = 0, sothat ϕ is purely linear in r , as well as A and all other scalar fields being constant.These AdS × R solutions are associated with the known AdS S-fold solutions: wecan periodically identify the radial direction after uplifting to type IIB supergravityand making a suitable identification with an SL (2 , Z ) transformation, as we outlinein more generality below.We now continue with the more general class of LPP solutions of the form (3.13)with both k (cid:54) = 0 and f (cid:54) = 0 and show that these too can give rise to new classes of AdS S-fold solutions. We begin by noting, as explained in appendix A (see also [10]),that the dilaton-shift symmetry (2.11) of the D = 5 theory, ϕ → ϕ + c , acts as aspecific SL (2 , R ) transformation in D = 10. If the type IIB dilaton, Φ and axion C are parametrised as m αβ = e Φ C + e − Φ − e Φ C − e Φ C e Φ , (3.17)then the transformation is given by m → ( S − ) T m S − where S ∈ SL (2 , R ), in thehyperbolic conjugacy class, is given by S ( c ) = e c e − c , (3.18)Equivalently, we have Φ → Φ + 2 c and C → e − c C .To carry out the S-fold procedure, we next note that starting from the uplifted D = 5 solutions we can obtain a family of uplifted type IIB solutions after actingwith a general element P ∈ SL (2 , R ). For example, the axion and dilaton in thislarger family will be of the form ˜ m ( ϕ ) = ( P − ) T m ( ϕ ) P − , where we have includedthe dependence on the D = 5 dilaton for emphasis. Within this larger family of typeIIB solutions we then look for solutions that we can periodically identify along theradial direction with period q ∆ r i.e. q ∈ N times the fundamental period ∆ r , up tothe action of an M ∈ SL (2 , Z ) transformation. Recalling that as we translate by ∆ r in the radial direction in the conformal gauge (3.13) we have ϕ → ϕ + ∆ ϕ , and hencewe require that ˜ m ( ϕ + q ∆ ϕ ) = ( M − ) T ˜ m ( ϕ ) M − , (3.19)which can be achieved provided that P ∈ SL (2 , R ) is such that M = ± P S ( q ∆ ϕ ) P − . (3.20)The different S-folded solutions which can be obtained in this way are labelled bythe conjugacy classes of M in SL (2 , Z ). A discussion of such classes can be found14n [26, 27] (see also [28]). For any conjugacy class M , we have that −M and ±M − also represent conjugacy classes. Clearly from the form of S in (3.18) we must be inthe hyperbolic conjugacy class with | T r ( M ) | >
2. We have the following possibilitiesfor M (as well as the conjugacy classes −M and ±M − ): we can have M = n − , n ≥ , (3.21)with trace n , as well as “sporadic cases” M ( t ) of trace t . For example for 3 ≤ t ≤ M (8) = , M (10) = , M (12) = . (3.22)For these cases, in order to find solutions to (3.19) (focussing on the upper sign in(3.20)) we must have q ∆ ϕ = arccosh n , for n ≥ , q ≥ . (3.23)For example, for the S-folds that are identified using M in SL (2 , Z ) given in (3.21)we have P = − √ n − ( − n + √ n − (1 + n √ n − ) . (3.24)Interestingly, the S-folding procedure preserves the supersymmetry as we now ex-plain. If we translate the D = 5 solution by ∆ r then we have ϕ → ϕ + ∆ ϕ . Such ashift in the dilaton is equivalently obtained by carrying out a K¨ahler transformation K → K + f + ¯ f and W → e − f W with f = f ( z A ). Under this transformation thepreserved supersymmetries, a symplectic Majorana pair, transform as ε → e ( f − ¯ f ) / ε and ε → e − ( f − ¯ f ) / ε as noted in [15]. Now, as we explained above, this transforma-tion is implemented on the bosonic fields as an element of S ∈ SL (2 , R ). In appendixA we show that this is also true for the preserved supersymmetries. Thus, as wetranslate by ∆ r , the solution and the preserved supersymmetries get transformed bythe same element of SL (2 , R ). This will also be true after uplifting to D = 10 andhence, after conjugating by P ∈ SL (2 , R ), the S-fold procedure will not break anysupersymmetry. Note that writing M n for the matrix in (3.21), we can also write M (8) − = −M M − , M (10) − = −M · M − and M (12) − = −M · M − . .4 Free energy of the S-folds The
AdS × S × S S-fold solutions of the kind we have just described should be dual,in general, to N = 1 SCFTs in d = 3. A key observable is F S , the free energy of theSCFT on S . This can be calculated holographically after a dimensional reduction on S × S to a four-dimensional theory of gravity and then evaluating the regularisedon-shell action for the AdS vacuum solution of this theory. With a four-dimensionaltheory that has an AdS vacuum solution with unit radius we have F S = π G (4) . (3.25)Here G (4) is the four-dimensional Newton’s constant which can be obtained from thefive-dimensional Newton’s constant via1 G (4) = 1 G (5) (cid:90) q ∆ r dre A . (3.26)Here we remind the reader that the radial coordinate, r , is associated with the D = 5conformal gauge, as in (3.13), and also that in the construction of the S-fold solutionwe made the S-fold identification after going along q periods of the periodic functions.Recalling that the AdS vacuum with radius L solves the equations of motion and isdual to d = 4, N = 4SYM with gauge group SU ( N ), we have the standard result116 πG (5) = N π L . (3.27)Putting this together we get our final formula for the free energy: F S = N L q (cid:90) ∆ r dre A , = N L arccosh n ∆ ϕ (cid:90) ∆ r dre A . (3.28)The first expression is valid for all solutions, including the periodic solution (forwhich it is natural to take q = 1), while the second expression is valid for the S-folded solutions. In the special case of the known N = 1 , , D = 5 dilaton (i.e. ϕ = kr in (3.13)) and A is constant, we can rewritethis as F S = N L e A k arccosh n . (3.29)Finally, following the arguments in [4], at fixed n the type IIB supergravity ap-proximation should be valid in the large N limit since higher derivative correctionswill be suppressed by terms of order 1 / √ N .16 N = 1 ∗ equal mass, SO (3) invariant model This model is obtained from the 10-scalar model by setting z = z = − z , orequivalently α = α = α and φ = φ = φ , as well as β = β = 0. This four-scalarmodel is parametrised by the two complex fields z = tanh (cid:2) (cid:0) α + ϕ − iφ + iφ (cid:1)(cid:3) , z = tanh (cid:2) (cid:0) α − ϕ − iφ − iφ (cid:1)(cid:3) . (4.1)The integral of motion (3.7) for this truncation is given by E = 1 L e A
12 [ − tan(3 φ − φ ) + 3 tan( φ + φ )] . (4.2)This model has two further sub-truncations as illustrated in figure 2, and in par-ticular contains the known N = 1 and N = 4 AdS × R S-fold solutions. Firstly,if we set z = − z , equivalently, α = φ = 0, then we obtain a two-scalar SU (3)invariant model depending on ϕ, φ that overlaps with the truncation considered inthe context of N = 1 S-folds in section 4 of [10].The N = 1 AdS × R S-fold solution is given (in conformal gauge) by ϕ = √ r, φ = cos − (cid:114) , e A = 5 L , α = φ = 0 , (4.3)and we have E = √ . There is another N = 1 S-fold solution obtained from thesymmetry (2.8), with opposite sign for E . The free energy of these solutions can beobtained from (3.29) and is given by F S = 25 √ n N . (4.4)in agreement with [10].On the other hand if we further set z = ¯ z , or equivalently φ = − φ , thenwe obtain a three-scalar SO (3) × SO (3) invariant model depending on α , φ , ϕ thatoverlaps with the truncation considered in the context of N = 4 S-folds in section 2of [10]. The N = 4 S-fold solution is given (in conformal gauge) by ϕ = 1 √ r , φ = − φ = −
12 cot − √ , e A = L √ , α = 0 , (4.5) They consider a model with four scalars: ( ϕ, χ, c, ω ). One should set c = ω = 0 and then identifysin φ = tanh χ as well as g = 2 /L . They consider a model with five scalars: ( ϕ, χ, α, c, ω ). One should set c = ω = 0 and thenidentify α = α and sin 4 φ = − tanh 4 χ . We also note that setting z = ¯ z in the BPS equations(3.5) leads to an additional algebraic reality constraint. The compatibility of imposing this constraintwith the BPS equations can be verified as in section 5 of [15] for a similar issue associated with thereality of the scalar fields β , β . E = . Again there is another N = 4 S-fold solution obtained from the sym-metry (2.8), with opposite sign for E . From (3.29) the free energy of these solutionsis given by F S = 12 arccosh n N . (4.6)in agreement with [4, 10].The model also contains a single periodic AdS × R solution that was foundnumerically in [15] which has E = 0. In this solution the warp factor e A and all thescalar fields, including ϕ , are periodic in the radial direction. Thus, it can immediatelybe compactified to give an AdS × S solution of D = 5 supergravity and then upliftedto an AdS × S × S solution of type IIB using the results of appendix A. From thenumerical results we can calculate the free energy (3.28) and we find F S ≈ q × . N , (4.7)where q is the number of periods over which we have compactified.The periodic solution was found as a limiting case of a class of Janus solutionsin [15]. The focus in [15] was Janus solutions that approach the N = 4 SYM vacuumwith the same value of ϕ ( s ) on either side of the interface, corresponding to the samevalue of τ of N = 4 SYM on either side of the interface. It is straightforward togeneralise these Janus solutions to allow ϕ ( s ) to take different values on either side ofthe interface. As already noted, taking limits of these solutions leads to new familiesof AdS × R solutions with ϕ an LPP function of the radial coordinate, r , whichparametrises R . Before summarising these new solutions, all found numerically, wediscuss how some of the new family of solutions can also be seen by perturbing the AdS × R solution associated with the N = 1 S-fold solution. N = 1 S-fold
Within the N = 1 ∗ equal mass model, we consider linearised perturbations of theBPS equations about the AdS × R solution (4.3), associated with the N = 1 S-fold.There are zero modes associated with shifts of ϕ , A and there is also a freedom toshift the coordinate r . There are two linearised modes that depend exponentially on r . Of most interest is that there is also a linearised periodic mode of the form δα = sin √ r , δφ = −√ √ r . (4.8)18ith a little effort we can use this periodic mode to construct a perturbativeexpansion in a parameter (cid:15) , that takes the form α = ∞ (cid:88) m,p =1 a ( α ) m,p (cid:15) m sin pKr , φ = φ zm ( (cid:15) ) + ∞ (cid:88) m,p =1 a ( φ ) m,p (cid:15) m cos pKr ,φ = φ zm ( (cid:15) ) + ∞ (cid:88) m,p =1 a ( φ ) m,p (cid:15) m cos pKr , ϕ = k ( (cid:15) ) r + ∞ (cid:88) m,p =1 a ( ϕ ) m,p (cid:15) m sin pKr ,A = A zm ( (cid:15) ) + ∞ (cid:88) m,p =1 a ( A ) m,p (cid:15) m cos pKr , (4.9)where all functions are periodic in the radial direction with period ∆ r ≡ πK , with ϕ having an extra linear piece, and hence an LPP function, exactly as in (3.13)-(3.15).The wavenumber K is itself given by the following series in (cid:15) : K ≡ π ∆ r = √ − √ (cid:15) − √ (cid:15) − √ (cid:15) + · · · , (4.10)which we notice is decreasing as we move away from the N = 1 S-fold solution.Interestingly, we notice that α has vanishing zero mode in this expansion, while thezero modes of the remaining periodic functions are explicitly given by φ zm = − √ (cid:15) − √ (cid:15) − √ (cid:15) + · · · ,φ zm = cos − (cid:114) − √ (cid:15) − √ (cid:15) − √ (cid:15) + · · · ,A zm = log 5 L − (cid:15) − (cid:15) − (cid:15) + · · · . (4.11)In addition the slope of ϕ takes the form k = √ − √ (cid:15) − √ (cid:15) − √ (cid:15) + · · · . (4.12)Furthermore we also have ∆ ϕ ≡ k ∆ r is given by∆ ϕ = 3 π + 1305 π (cid:15) + 95032143 π (cid:15) + 11893037855571 π (cid:15) + · · · . (4.13)The integral of motion (4.2) is given by E = 25 √ (cid:0) − (cid:15) − (cid:15) − (cid:15) + · · · (cid:1) . (4.14)One finds that all of the expansion parameters a ( ∗ ) m,p appearing in (4.9) are onlynon-zero when m + p is even. This implies the following property of the perturbative19olution under a half period shift in the radial coordinate. Specifically, let Ψ = { A, α , φ , φ } denote the periodic functions so that the whole solution is specified byΨ( (cid:15), r ) and ϕ ( (cid:15), r ). We then findΨ( (cid:15), r + π/K ) = Ψ( − (cid:15), r ) , ϕ ( (cid:15), r + π/K ) = ϕ ( − (cid:15), r ) + constant , (4.15)where the constant can be removed by (2.11). This means that changing the sign of (cid:15) gives, essentially, the same solution (i.e up to a shift in the radial direction plus ashift of ϕ ).Finally, after uplifting to type IIB, using the results of appendix A, and carryingout the S-fold procedure, as described in section 3.3, we obtain new S-folds of typeIIB provided that we can solve (3.23). The free energy for the S-folded solutions canthen be obtained from (3.28) and is given by F S = 25 √ (cid:18) − (cid:15) − (cid:15) + . . . (cid:19) arccosh n N . (4.16)To solve (3.23) we first note that 2 cosh 3 π ∼ .
6. Thus, the smallest value of n that can be reached in (3.23) is n = 12392, which occurs for q = 1 and (cid:15) ∼ . q , which, for a given n , havesmaller values of (cid:15) . Thus, we can find S-fold solutions with arbitrarily small (cid:15) . Wealso note that while these AdS × R solutions are perturbatively connected with the N = 1 AdS × R S-fold solution, they are not as S-folds of type IIB string theory.This is clear when we recall that for the latter we can solve (3.23) for any n ≥ r over which we S-fold, while for the perturbativesolutions, as just noted, we have n ≥ N = 1 ∗ equal mass, SO (3) invariant truncation we are considering alsocontains the known N = 4 AdS × R S-fold solution (4.5). If we consider the linearisedperturbations of the BPS equations about this solution we again find zero modesassociated with shifts of ϕ , A and there is also a freedom to shift the coordinate r .The remaining modes all depend exponentially on the radial coordinate. In particular,there is no longer a linearised periodic mode and this feature will manifest itself inthe family of new solutions we discuss in the next section. The new
AdS × R solutions, with ϕ a LPP function, can be constructed as limitingcases of Janus solutions. A convenient way to numerically solve the BPS equations(3.3)-(3.5) is to set initial conditions for the scalar fields at a turning point of themetric warp function, A , which corresponds to Re( B r ) = 0 along with the values20f the scalar fields at the turning points. Some general comments concerning thisprocedure were made in sections 5 and 6 of [15].In more detail we consider Janus solutions with the turning point of A located at r = r tp . Since the BPS equations are unchanged by shifting the radial coordinate bya constant, we can take r tp = 0. We can also use the the shift symmetry (2.11) tochoose ϕ ( r tp ) = 0. We can then focus on solutions that are invariant under the Z symmetry, obtained by combining (2.8) and (3.6), r → − r, z A → − ¯ z A , ξ → − ξ + π . (4.17)This implies that φ i , φ are even functions of r and α i , ϕ are odd functions. Inparticular, at the turning point we can take α i ( r tp ) = 0 as part of our initial valuedata. For the SO (3) invariant model, these Janus solutions are therefore fixed by thevalues of φ ( r tp ) and φ ( r tp ). By suitably tuning the values of the scalar field at theturning points we are able to construct the limiting cases of solutions associated withthe S-folds.The space of solutions that we have found in this way is summarised by thecoloured curve in figure 3, with the colour giving the value of |E | , given by (4.2). Ifone starts with turning point data that lies anywhere within the curve, one obtains aJanus solution of N = 4 SYM theory with fermion and boson masses and a couplingconstant that varies as one crosses the interface. For example, the Janus solutiondepicted in figure 1 corresponds to the black cross inside the curve in figure 3. Onthe other hand if one starts outside the curve then one finds that the solution becomessingular on both sides of the interface as in the solutions discussed in [15], for example.Observe that the figure is symmetric under changing the signs of both φ ( r tp )and φ ( r tp ), as a result of the symmetry (2.8). The associated AdS × R solutionsobtained by this symmetry, which is a discrete R -symmetry combined with an S -duality transformation for the associated Janus solutions, are physically equivalent.The value of E is positive for the upper part of the curve between the two red dotsand negative for the lower part. We next point out that the blue dots correspond tothe two N = 1 AdS × R S-fold solutions, with ϕ a linear function of r , as in (4.3).The red dots correspond to the fully periodic AdS × R solution found in [15]. We willcome back to the green dots and squares in a moment. The remaining points on the If we relax the condition that the initial data is invariant under the Z symmetry, then we donot find any LPP solutions of the type we are interested in for constructing S-folds. Instead we findsome interesting “one-sided” Janus solutions that we discuss in section 6. We also note that thegeneral periodic perturbative solution (4.9) did not assume invariance under the Z symmetry, yetit is in fact invariant. ℰ | ���������������� Figure 3: Turning point initial data for the
AdS × R solutions of the N = 1 ∗ equalmass SO (3) invariant model. Red dots correspond to the exactly periodic solution,blue dots correspond to the N = 1 linear dilaton solutions, green dots to the N = 4linear dilaton solutions and green squares to the bounce solutions. The remainingpoints correspond to AdS × R solutions with ϕ a LPP function of r . All points insidethe curve correspond to Janus solutions of N = 4 SYM theory (the black cross is theJanus solution in figure 1), while points outside the curve have singularities. Pointson the curve with the same colour represent the same solution, up to shifts of ϕ andthe discrete symmetry (2.8).curve all correspond to AdS × R solutions with ϕ an LPP function of r . Also, if onestarts at the N = 1 S-fold solution at the top of the curve, then one can match on tothe perturbative family of solutions that we constructed in the previous subsectionand there is a similar story for the N = 1 S-fold solution at the bottom of the curve.Points on the curve with the same colour have the same value of |E | and represent,essentially, the same solution, up to dilaton shifts (2.11) and the discrete symmetry(2.8) if E has the opposite sign. Indeed if we move to the right from the blue dot atthe top all the way to the red dot at the right, the LPP solutions (all of which have E positive) are essentially the same as those as one moves to the left; although theturning point data at r = r tp is different, the data of one of the solutions at r = r tp agrees with the turning point data of the other solution at r = r tp +∆ r/
2, after makinga suitable shift of ϕ using (2.11). One can explicitly check this feature analyticallyfor the perturbative solution (4.9). We also note that this feature is consistent withthe fact that there is just a single periodic solution which has the property that ifone uses (2.11) to have no zero mode for ϕ , then the solution is invariant under a22alf period shift combined with a Z symmetry transformation (2.8).We now return to the green dots and squares in figure 3. The green dots, locatedat |E | = 1 / N = 4 linear dilaton solutions given in (4.5), while thegreen squares represent “bounce” solutions that involve those solutions, as we nowexplain. We first consider the limiting class of the LPP solutions as we move alongthe coloured curve in figure 3 towards the upper green dot to the left. To illustrate,in the left panel of figure 4 we have displayed the behaviour of one of the periodicfunctions, φ ( r ), as one approaches the critical initial data associated with the greendot, which has φ ( r tp ) = − / − √ ∼ − . N = 4 linear dilaton solution (4.5) forall values of r . In the right panel of figure 4 we have also displayed the approach tothe upper green square to the right. In this case the solution develops a region thatapproaches the N = 4 linear dilaton solution (4.5) as one moves away from r = 0in either direction. Exactly at the initial values associated with the green square thesolution will no longer be an LPP solution but degenerates into a “bounce solution”which approaches the N = 4 linear dilaton solution (4.5) at both ¯ r/L → ±∞ , witha kink in the middle. We also see that these degenerations of the LPP solutions splitthe whole family of solutions into two branches of LPP solutions: one that includesthe perturbative solutions built using the N = 1 linear dilaton solutions and anotherthat contains the periodic solution. - - - - - � / � ϕ � - - - - - � / � ϕ � Figure 4: Family of LPP solutions for the N = 1 ∗ equal mass SO (3) invariant modelwith turning point data illustrating the approach to the green dots and squares asin figure 3, with |E | = 1 /
2. The figures display just the periodic behaviour of φ forclarity and just one period. The left panel shows that the limiting solutions associatedwith the green dots degenerate into the N = 4 linear dilaton solution, marked witha dashed green line. The right panel shows the limiting solution associated with thegreen square becomes a bounce solution which approaches the N = 4 linear dilatonsolution, at both ¯ r → ±∞ , with a kink in φ centred at ¯ r = 0.23n order to obtain S-fold solutions of type IIB string theory we also need to imposethe quantisation condition (3.23). In figure 5 we have plotted some of these discretesolutions as well as F S given in (3.28). The discrete set of vertical points colouredblue and green correspond to the N = 1 and N = 4 S-fold solutions with lineardilatons, respectively, and n increasing from 3 to infinity as one goes up; for theseS-folds we can obtain all values n ≥ r over whichwe S-fold. The red dots correspond to the periodic solution for different values of thenumbers of period, q , that are used in making the S compactification. The remainingdiscrete points correspond to N = 1 S-fold solutions with ϕ an LPP function, forrepresentative values of q = 1 , ,
3. Starting from the left, for a given q , we have n = 3 at the left and then rising to infinity as one approaches the bounce solutionor the N = 4 S-fold solution at E = 1 /
2, where the free energy diverges. Movingfurther to the right the value of n decreases from infinity down to a bounded value[2 cosh q π ], at the intersection with the N = 1 solutions on the blue line, which canbe deduced from the perturbative analysis (4.13). ��� ��� ��� ��� ��� ��������� ℰ ℱ � � � � �������� = � = � � = � � = � � = � Figure 5: Plot of the discrete S-folded solutions and the associated free energy of thedual field theory, F S , for the N = 1 ∗ equal mass SO (3) invariant model as in figure3. The discrete points rapidly become indistinguishable from continuous lines. SU (2) invariant This model is obtained from the 10-scalar model by setting z = − z , z = − z , orequivalently α = α = 0, φ = φ = 0, β = 0. This model involves five scalar fields24arametrised by β , z = tanh (cid:2) (cid:0) α + ϕ − iφ + iφ (cid:1)(cid:3) , z = tanh (cid:2) (cid:0) α − ϕ − iφ − iφ (cid:1)(cid:3) . (5.1)In addition to the symmetry (2.8), this model is also invariant under the symmetry φ → − φ , α → − α , (5.2)with β , φ , ϕ unchanged, which is a remnant of the discrete transformations given in(2.10) for the 10-scalar truncation. This additional symmetry will clearly manifestitself in the set of solutions we construct. The integral of motion (3.7) for thistruncation is now given by E = 1 L e A [ − tan( φ − φ ) + tan( φ + φ )] . (5.3)If we further set z = − z , equivalently, α = φ = 0, as well as β = 0 then wethen we obtain a two-scalar model depending ϕ, φ that overlaps with the truncationconsidered in the context of N = 1 S-folds in section 4 of [10], which we also discussedin the previous section. In particular the AdS × R solution associated with the N = 1S-folds is given by ϕ = √ r, φ = cos − (cid:114) , e A = 5 L ,β = α = φ = 0 , (5.4)with E = √ . There is another N = 1 S-fold solution that can be obtained fromthe symmetry (2.8), with opposite sign for E .On the other hand if we set z = ¯ z or equivalently φ = − φ then we obtain a four-scalar model depending on φ , α , ϕ, β that overlaps with the truncation consideredin the context of N = 2 S-folds in section 3 of [10]. Also note that after utilisingthe symmetry (5.2) we can also truncate to a 4-scalar model by taking z = ¯ z , orequivalently φ = + φ . The N = 2 S-fold solution, with φ = − φ , can be written ϕ = r , φ = − φ = − π , β = −
112 log 2 , e A = L / , α = 0 , (5.5)with E = . After using the symmetries (2.8) and (5.2) there are now a total of four N = 2 AdS × R S-fold solutions with ϕ linear in r . From (3.29) the free energy ofthese solutions is given by F S = 12 arccosh n N . (5.6) They consider a model with seven scalars: ( ϕ, χ, α, λ, c, ω, ψ ). One should set c = ω = ψ = 0and then identify α = β , λ = α , sin 2 φ = − tanh 2 χ as well as g = 2 /L .
25n agreement with [10].Finally, if we set z = z or equivalently φ = ϕ = 0 then we obtain the N = 1 ∗ one-mass truncation used in [15], which contains three scalars β , φ , α and retainsthe symmetry (5.2). This truncation also contains two LS AdS fixed point solutions,LS ± , which are related by (5.2) and given by β = −
16 log 2 , φ = ± π , α = 0 , ˜ L = 32 / L , (5.7)where ˜ L is the radius of the AdS . N = 1 S-fold
Much as in the last section, within the 5-scalar truncation we can build a perturbativesolution about the N = 1 S-fold solution given in (5.4). The key point is that thereis now a periodic linearised perturbation of the form δα = sin √ r , δφ = −√ √ r . (5.8)With some effort we can use this to construct a perturbative expansion in aparameter (cid:15) , that takes the form α = ∞ (cid:88) m,p ∈ odd a ( α ) m,p (cid:15) m sin pKr , φ = ∞ (cid:88) m,p ∈ odd a ( φ ) m,p (cid:15) m cos pKr ,φ = φ zm ( (cid:15) ) + ∞ (cid:88) m,p ∈ even a ( φ ) m,p (cid:15) m cos pKr , ϕ = k ( (cid:15) ) r + ∞ (cid:88) m,p ∈ even a ( ϕ ) m,p (cid:15) m sin pKr ,β = β zm ( (cid:15) ) + ∞ (cid:88) m,p ∈ even a ( β ) m,p (cid:15) m cos pKr , A = A zm ( (cid:15) ) + ∞ (cid:88) m,p ∈ even a ( A ) m,p (cid:15) m cos pKr , (5.9)where the the sums over odd integers start from 1 and the sums over even integersstart from 2. All functions, except ϕ are periodic in the radial direction with period∆ r = πK , with ϕ an LPP function, exactly as in (3.13)-(3.15). The wavenumber K is itself given by the following series in (cid:15) : K ≡ π ∆ r = √ − √ (cid:15) − √ (cid:15) − √ (cid:15) + · · · , (5.10)which we notice is decreasing as we move away from the N = 1 S-fold solution.26otice that both α and φ have vanishing zero mode in this expansion. The zeromodes of the remaining periodic functions are explicitly given by φ zm = cos − (cid:32)(cid:114) (cid:33) − √ (cid:15) − √ (cid:15) − √ (cid:15) + · · · ,β zm = − (cid:15) − (cid:15) − (cid:15) + · · · ,A zm = log (cid:18) L (cid:19) − (cid:15) − (cid:15) − (cid:15) + · · · , (5.11)In addition the slope of ϕ takes the form k = √ − √ (cid:15) − √ (cid:15) − √ (cid:15) + · · · . (5.12)Furthermore, we also have ∆ ϕ ≡ k ∆ r is given by∆ ϕ = 3 π + 175 π (cid:15) + 5295375 π (cid:15) + 153607091549 π (cid:15) + · · · . (5.13)The integral of motion (5.3) is given by E = 25 √ (cid:0) − (cid:15) + 4598507 (cid:15) + 96057473771147 (cid:15) + · · · (cid:1) . (5.14)We now write the periodic functions collectively as Ψ = { A, φ , β } and Ψ = { α , φ } so that the whole solution is specified by Ψ ( (cid:15), r ), Ψ ( (cid:15), r ) and ϕ ( (cid:15), r ). Wethen find Ψ ( (cid:15), r + π/K ) = Ψ ( − (cid:15), r ) = +Ψ ( (cid:15), r ) , Ψ ( (cid:15), r + π/K ) = Ψ ( − (cid:15), r ) = − Ψ ( (cid:15), r ) ,ϕ ( (cid:15), r + π/K ) = ϕ ( − (cid:15), r ) + constant , (5.15)where the constant can be removed by (2.11) and we note that the last equalities inthe first two lines are associated with the symmetry (5.2).After uplifting to type IIB and carrying out the S-fold procedure as describedin section 3.3, we obtain new S-folds of type IIB provided that we can solve (3.23).This can be done as in the discussion following (4.16) and, in particular, the smallestvalue of n that can be reached in (3.23) is n = 12392, which occurs for q = 1 and (cid:15) ∼ . F S = 25 √ (cid:18) − (cid:15) − (cid:15) + . . . (cid:19) arccosh n N . (5.16)This truncation also contains the known AdS × R N = 2 S-fold solutions, butthere is no longer a linearised periodic mode within this truncation in which to buildan analogous solution. This is similar to the known AdS × R N = 4 S-fold solutionsin the SO (3) invariant truncation that we considered in the previous section.27 .2 New S-fold solutions The new
AdS × R solutions, with ϕ a LPP function, can be constructed as limitingcases of Janus solutions, much as in the last section. We again start by constructingJanus solutions with turning point of A at r = r tp , with r tp = 0. We can use the theshift symmetry (2.11) to choose ϕ ( r tp ) = 0. We then focus on solutions that areinvariant under the Z symmetry, obtained by combining (2.8) and (3.6), r → − r, z A → − ¯ z A , ξ → − ξ + π . (5.17)This implies that φ , φ are even functions of r and α , ϕ are odd functions. Thus,we again take α ( r tp ) = 0 as part of our initial value data for the solutions. From(3.3)-(3.5), and as explained in section 5 of [15], the solutions are now specified by thevalues of φ ( r tp ) and φ ( r tp ), with the value of β ( r tp ) fixed by this data. By suitablytuning the values of the scalar field at the turning points we are able to construct thelimiting cases of solutions associated with the S-folds.The space of solutions we have found in this way is summarised by the curveshown in figure 6. If one starts with turning point data that lies anywhere within thecurve, one obtains a Janus solution of N = 4 SYM theory with fermion and bosonmasses and a coupling constant that varies as one crosses the interface. On the otherhand if one starts outside the curve then one finds that the solution becomes singularon both sides of the interface.Observe that the figure is symmetric under changing the signs of either φ ( r tp ) or φ ( r tp ). This is a result of the symmetries (2.8) and (5.2). The associated AdS × R solutions obtained using these symmetries, which for the Janus solutions are acombination of a discrete R -symmetry and an S -duality transformation (in the caseof (2.8)), are physically equivalent. The value of E is positive for the upper partof the curve and negative for the lower part. We next point out that the blue dotscorrespond to the N = 1 AdS × R S-fold solutions which have ϕ a linear functionof r . The green dots represent the N = 2 AdS × R S-fold solutions as well asassociated “soliton” solutions that we discuss further below. The remaining pointson the coloured, solid lines all correspond to
AdS × R solutions with ϕ an LPPfunction of r . Also, if one starts at the N = 1 S-fold solution at the top of the curve,then one can match on to the perturbative family of solutions that we constructed inthe previous subsection. As in the previous section, if we relax the condition that the initial data is invariant under the Z symmetry, then we only find limiting solutions that are in the “one-sided” Janus class discussedin section 6. We also note that the perturbative solution (5.9) is invariant under this symmetry. ℰ | Figure 6: Turning point initial data for the
AdS × R solutions of the 5-scalar SU (2)invariant model. The blue dots correspond to the N = 1 linear dilaton solutionswhile the green dots correspond to the N = 2 linear dilaton solutions, as well asassociated soliton solutions. The red dots correspond to the two LS AdS solutions,LS ± . The remaining points on the solid lines correspond to AdS × R solutions with ϕ a LPP function of r , with the same colour representing the same physical solution.All points inside the curve correspond to Janus solutions of N = 4 SYM theory whilepoints outside the curve have singularities. The dashed lines correspond to an LS ± to LS ± Janus solution.Points on the solid curve with the same colour represent, essentially, the sameLPP solution, up to dilaton shifts and possible discrete symmetries. Moving fromthe right of the blue dot at the top all the way to the green dot at the right onefinds LPP solutions that are essentially the same as those as one moves to the left;although the turning point data at r = r tp is different, the data of one of the solutionsat r = r tp agrees with the turning point data of the other solution at r = r tp + ∆ r/ ϕ using (2.11). Note that the two sets of turning pointdata are also related by (5.2). One can explicitly check these features analytically forthe perturbative solution (5.9).In the limit of approaching the green dots in figure 6 along the solid curve, theLPP solutions degenerate into the AdS × R N = 2 S-fold solutions as illustrated inthe left panel in figure 7 for one of the periodic functions, φ ( r ). As one approachesthe critical initial data associated with the green dot which has φ = π ∼ .
39, thesolution degenerates into the N = 2 S-fold solution, with the region around ¯ r = 0extending out all the way to infinity. Interestingly, essentially using the same family29f solutions, one can construct another limiting solution which is a kind of “soliton”solution that approaches one of the AdS × R N = 2 S-fold solutions as ¯ r → −∞ and a different AdS × R N = 2 S-fold solution, related by flipping the sign of φ , as¯ r → ∞ . This limiting solution is illustrated in the right panel of figure 7. - - - - � / � ϕ � - - - - � / � ϕ � Figure 7: Limiting families of solutions for the 5-scalar SU (2) invariant model, withjust the periodic behaviour of φ displayed for clarity. The left panel illustratesthe approach to the green dots in figure 3, along the coloured curve; one finds thatthe solution will approach the N = 2 linear dilaton solution associated with theupper green dashed line for all ¯ r . In the right panel we display a different limitingsolution, obtained by fixing φ (0) = 0, which degenerates into a soliton solution thatapproaches one N = 2 linear dilaton solution, at ¯ r → −∞ and another N = 2 lineardilaton solution at ¯ r → ∞ with opposite sign of φ (related by (5.2)).We next turn to the remaining points in figure 6. The red dots are the two LS AdS fixed points given in (5.7), which we refer to as LS ± . Moving along the class of Janussolutions on the horizontal axis towards the red dots at the right, say, one finds thatthe Janus solutions degenerate into three components; a Poincar´e invariant RG flowsolution that starts off at the AdS vacuum and then approaches the LS + AdS fixedpoint, the LS + fixed point solution itself and then another Poincar´e invariant RG flowsolution going between LS + and the AdS vacuum. The dashed curves correspond toanother interesting degeneration of the Janus solutions. As one approaches the dashedcurve on the right side of the figure one again finds three components: there is thesame two Poincar´e invariant components on the outside and the middle componentis now an LS Janus solution that moves between LS + and LS + on either side of theinterface, with ϕ linear in ¯ r . There is similar behaviour as one approaches the reddot or the dashed line on the left side of the figure with LS − replacing LS + .To obtain S-fold solutions of type IIB string theory we also need to impose thequantisation condition (3.23). In figure 8 we have plotted some of these discrete so-lutions as well as F S given in (3.28). The discrete set of vertical points coloured30lue and green correspond to the N = 1 and N = 2 S-fold solutions with linear dila-tons, respectively, and n increasing from 3 to infinity as one goes up. The remainingdiscrete points correspond to N = 1 S-fold solutions with ϕ an LPP function, forrepresentative values of q = 1 ,
2. Starting from the right at the blue dots, for a given q , we have n starting from [2 cosh q π ], which can be deduced from the perturba-tive analysis (5.13), and then rising to infinity as one approaches the N = 2 S-foldsolution at E = 1 /
2, where the free energy diverges. ℰ ℱ � � � � = � = � � = � � = � Figure 8: Plot of the discrete S-folded solutions and the associated free energy ofthe dual field theory, F S , for the 5-scalar SU (2) invariant model as in figure 6. Thediscrete points rapidly become indistinguishable from a continuous line. In this section we discuss a novel class of D = 5 solutions within the ansatz (3.1),that at one end of R approach the AdS vacuum, while at the other end approachan AdS × R solution with the D = 5 dilaton, ϕ , either a linear function or an LPPfunction of r . We can also construct solutions that approach the periodic AdS × R solution at the other end. We refer to these solutions as “one-sided Janus” solutions.In contrast to other one sided Janus solutions that have been previously constructed,for example in [15, 29–31], remarkably these new solutions are free from singularities. N = 4 supersymmetry We first consider an analytic solution that lies within the SO (3) × SO (3) invarianttruncation that involves 3 scalar fields, φ = φ = φ = − φ , α = α = α and ϕ .Using the proper distance gauge with radial coordinate ¯ r , we find the following31olutiontan 4 φ = − √ e − r/L (1 + e − r/L ) / , cosh 4 α = 1 + e − r/L ( e − r/L ) (cid:114) e − r/L ( e − r/L ) ,e ϕ − ϕ ( s ) = 1 + 3 e − r/L (1 + e − r/L ) (1 + 3 e − r/L ) , e A = Le ¯ r/L √ e − r/L √ . (6.1)For these solutions, in which the warp factor A does not have a turning point, wefind that that the integral of motion is given by E = . Recall that the N = 4 S-foldsolution with a linear dilaton given in (4.5) also had E = . In other words, takingthe limit E → in the family of Janus solutions in this truncation can either givethe N = 4 S-fold solution or this new solution, which describes a one-sided Janussolution.At the ¯ r → + ∞ end these solutions approach the AdS vacuum solution, dual to N = 4 SYM theory. After shifting the radial coordinate ¯ r → ¯ r − L log L √ , so we caneasily compare with [15], we find that as ¯ r → ∞ we have the asymptotic expansion φ = − L e − r/L + · · · , α = L e − r/L − L e − r/L + · · · ,ϕ = ϕ ( s ) − L e − r/L + · · · , A = ¯ rL + L e − r/L − L e − r/L + · · · . (6.2)From the results given in [15] we can immediately deduce that all sources for theoperators dual to the scalar fields vanish. Furthermore, we can also determine theone point functions. As explained in detail [15], we can determine the one-pointfunctions that are associated with N = 4 SYM theory on flat spacetime withcoordinates ( t, y i ); we find that the one-point functions having spatial dependence onone of the spatial directions, say y , with L (cid:104)O α (cid:105) = N π y , L (cid:104)O φ (cid:105) = − N π y , (cid:104)O ϕ (cid:105) = − N π y , (6.3)where we used (3.27). These expressions display the appropriate dependence on y that is consistent with d = 3 conformal invariance with respect to the ( t, y , y ) fordual operators of scaling dimension ∆ = 2 , r → −∞ , again after shifting ¯ r → ¯ r − L log L √ , the asymptotic As opposed to
AdS to which it is related by a Weyl transformation. Note that the operators have not been canonically normalised, which explains the factors of L appearing on the left hand side. φ = −
14 tan − (cid:16) √ (cid:17) + 13 √ L e r/L + 118 √ L e r/L + · · · ,α = 13 L e r/L − L e r/L + · · · ,ϕ = ¯ rL + ϕ ( s ) − log L √ − L e r/L + 59 L e r/L + · · · ,A = log L √ L e r/L − L e r/L + · · · . (6.4)This shows that the solution at this end is precisely approaching the N = 4 AdS × R S-fold solution with ϕ a linear function of r , which was given in (4.5).The solution solves the BPS equations (3.3)-(3.5) and hence it preserves at least N = 1 supersymmetry. However, since it is a solution that lies within the SO (3) × SO (3) invariant truncation it actually preserves N = 4 supersymmetry. Furthermore,after uplifting the solution to type IIB, using the formulae in appendix A.2.2, weobtain a D = 10 metric of the form ds = f ds Ads + f d Ω + f d ˜Ω + ds (Σ) , (6.5)where d Ω and d ˜Ω are metrics on round two-spheres and f , f and f are functionsof the coordinates on Σ. A full classification of such solutions which preserve N = 4supersymmetry can be found in [19, 20]. In appendix A.4 we explicitly show that ouruplifted solution lies within this framework. In particular, the Riemann surface istaken to be an infinite strip with complex coordinate w with w = x + iψ , (6.6)where −∞ < x < ∞ and ψ ∈ [0 , π/ h = − i L √ e ϕ ( s ) (sinh w − sinh ¯ w ) = L √ e ϕ ( s ) cosh x sin ψ ,h = e ϕ ( s ) L √ e w + e ¯ w ) = e ϕ ( s ) L √ e x cos ψ , (6.7)and in comparing with (6.1) we should identify x = ¯ r/L .It is interesting to compare this solution with the supergravity solutions associatedwith the near horizon limit of a collection of N D3-branes ending on N coincidentD5-branes. More specifically, we want N = KN where K ∈ Z , the linking number,is the same for all D5-branes. From the results of [19–23] we can write the harmonic33unctions for such solutions as h = √ πN (cid:96) s √ g s (cid:18) e x sin ψ + √ g s N / √ πK log (cid:20) g s N e x + πK + 2 / √ πg s N K sin ψe x g s N e x + πK − / √ πg s N K sin ψe x (cid:21)(cid:19) ,h = √ πg s N (cid:96) s √ e x cos ψ , (6.8)where g s = e ϕ ( s ) is the string coupling constant and (cid:96) s is the string length. In thelarge x limit, as we approach the N = 4 SYM end, this solution behaves as h = √ πN (cid:96) s √ g s (cid:18) cosh x sin ψ + πK g s N e − x sin 3 ψ + O ( e − x ) (cid:19) ,h = √ πg s N (cid:96) s √ e x cos ψ . (6.9)Thus, after identifying the Einstein frame AdS curvature √ πN (cid:96) s = L , as x → ∞ we see that this solution has the same asymptotic form as (6.7), with sub-leadingcorrections. Moreover, note that we also obtain the expansion (6.9) by taking thelimit N → ∞ while holding the linking number K fixed. It is straightforward to construct additional one-sided Janus solutions numerically.In fact we have found no obstruction to constructing solutions that approach the
AdS vacuum at one end and any of the AdS × R solutions that we have discussedin the previous sections at the other end; namely the N = 1 , ϕ a linear function, the more general S-fold solutions with ϕ an LPP function or theperiodic solution. The one-sided Janus solutions approaching the S-folds with lineardilaton do not have any turning points. The solutions approaching the S-folds witheither ϕ an LPP function or the periodic solution do have turning points, but theturning point data is not symmetric under the Z symmetry as we imposed for thesolutions summarised in figures 3 and 6. All of these one-sided Janus solutions areregular.To illustrate we have displayed in figure 9 a solution constructed in the N = 1 ∗ equal mass SO (3) invariant model of section 4 that approaches the AdS vacuum at¯ r → −∞ and the periodic AdS × R solution at ¯ r → + ∞ . Notice that this particularJanus solution has the feature that the dilaton ϕ is bounded.34 - - r φ Janus A Janus φ periodic A periodic - - - - - r ϕ Janus α Janus ϕ periodic α periodic Figure 9: A one sided Janus solution (solid lines) for the N = 1 ∗ equal mass SO (3)invariant model that approaches AdS as ¯ r → −∞ and approaches, very rapidly, theexactly periodic AdS × R solution (dashed lines) as ¯ r → + ∞ . The left panel plotsthe behaviour of the warp factor A and the D = 5 dilaton ϕ and the right panel plotsthe scalar fields α , φ (we have not plotted φ for clarity). We have constructed a rich set of new S-fold solutions of type IIB string theory ofthe form
AdS × S × S which are dual to N = 1 SCFTs in d = 3. The solutions arepatched together along the S direction using a non-trivial SL (2 , Z ) transformationin the hyperbolic conjugacy class. The solutions are first constructed in D = 5gauged supergravity and then uplifted to D = 10. In the previously known AdS × R solutions associated with S-folds preserving N = 1 , , D = 5dilaton is a linear function of a coordinate on the R direction. Crucially, in the newsolutions the D = 5 dilaton is now a linear plus periodic (LPP) function. We alsoshowed that some of the new families of LPP AdS × R solutions can be seen in aperturbative expansion about the N = 1 S-fold solution with a linear dilaton. Inaddition, for the SO (3) invariant model the numerical construction of such solutionsrevealed additional branches of LPP AdS × R solutions, not perturbatively connectedwith any known S-fold solutions.An interesting feature of the new AdS × S × S solutions is that we can makethe size of the S parametrically larger than the size of the S , by carrying out theS-folding procedure after multiple periods with respect to the underlying periodicstructure. This will gives rise to an an interesting hierarchy of scaling dimensions inthe N = 1 d = 3 SCFT.A proposal for the N = 4 SCFT in d = 3 dual to the N = 4 S-folds of [3]was given in [4]. One takes the the strongly coupled [ T U ( N )] theory of [7] and thengauges the global U ( N ) × U ( N ) global symmetry using an N = 4 vector multiplet. Inaddition one adds a Chern-Simons term at level n , where n is the integer that is used35o make the S-folding identifications (see (3.23)). Proposals for the N = 4 SCFT in d = 3 dual to the N = 2 S-folds of [9] were also discussed in [10]. It would be veryinteresting to identify the N = 1 SCFTs in d = 3 that are dual to the S-fold solutionsof [8], the new constructions in this paper, as well as the periodic AdS × S × S solution of [15]. The small amount of supersymmetry makes this challenging, but onecan hope that the connection with Janus solutions which we have highlighted in thispaper, as well as in [15], will allow progress to be made.We have seen that the periodic AdS × R solution found in [15], which uplifts tosmooth AdS × S × S of type IIB supergravity, is a rather exceptional solution in thegeneral constructions of this paper. It would be very interesting to know whether ornot there are additional such solutions of the form AdS d × T n × M k either in D = 10or D = 11 supergravity.We have focussed on constructing supersymmetric S-fold solutions, but one canalso investigate non-supersymmetric possibilities. In fact a non-supersymmetric AdS × R × M solution of type IIB supergravity was discussed long ago in [32] and [33]. Thesesolutions are associated with the D = 10 dilaton linear in the R direction, and havebeen subsequently rediscovered several times [8, 34–36]. However, in [8, 35, 36] it wasargued that these solutions are unstable (in contrast to the claim in [32]) and henceare not of interest for S-folds with CFT duals.Our constructions have also revealed a novel class of non-singular “one-sidedJanus” solutions preserving N =1,2 or 4 supersymmetry. These regular solutionsapproach the AdS vacuum on one side and an AdS × R solution with the D = 5dilaton a linear function of the radial coordinate or an LPP function. We also con-structed a solution that approaches the periodic AdS × R solution of [15] on theother side, which is both regular and has bounded dilaton. For the solution that ap-proaches the N = 4 S-fold solution with linear dilaton we were able to construct ananalytic solution. Using the results of [19–23] we interpreted this solution as arisingfrom D3-branes ending on D5-branes and it will be worthwhile to investigate this inmore detail.It seems likely that it will be possible to construct additional LPP AdS × R and one-sided Janus solutions within the 10 scalar truncation and more generallywithin the full SO (6) gauged supergravity with 42 scalars. It may also be possibleto construct new type IIB solutions of the form AdS × S × SE , where SE is aSasaki-Einstein manifold, generalising the work of [14]. More generally, one can tryto construct non-geometric solutions of the form AdS d × T n × M k , where T n is an n -dimensional torus and the solutions are patched together in the T n directions usingU-duality transformations [37]. 36 cknowledgments We thank Alessandro Tomasiello for helpful discussions. This work was supported bySTFC grant ST/T000791/1. KCMC is supported by an Imperial College President’sPhD Scholarship. JPG is supported as a KIAS Scholar and as a Visiting Fellow atthe Perimeter Institute. The work of CR is funded by a Beatriu de Pin´os Fellowship.
A Uplifting to type IIB supergravity
A.1 The 10-scalar model in maximal gauged supergravity
We first discuss how the 10-scalar model is obtained from maximal SO (6) gaugedsupergravity in D = 5. The 42 scalars of SO (6) gauged supergravity parametrise thecoset E /U Sp (8), with U Sp (8) the maximal compact subgroup of E . To describethis coset space, it is convenient to work in a basis for E that is adapted to itsmaximal subgroup SL (6) × SL (2 , R ), recalling that the gauge group SO (6) ⊂ SL (6).Following [38], we write the generators of E in the fundamental representationin this basis as X = − [ I [ M δ J ] N ] √ IJP β √ MNKα Λ P K δ β α + Λ β α δ P K , (A.1)where the indices I, J, ... = 1 , , . . . ,
6, raised and lowered with δ IJ , label the fun-damental of SL (6), while the indices α, β, ... = 1 ,
2, raised and lowered with (cid:15) αβ ,are SL (2 , R ) indices. It is often convenient to consider X as a 27 ×
27 matrix as-sociated with the branching of the fundamental of E under SL (6) × SL (2 , R ),like → ( , ) + ( , ). From this perspective, a fundamental index of E , A = 1 , , . . .
27 splits according to { A } = { [ IJ ] , Iα } , where [ IJ ] are the 15 antisym-metric pairs of SL (6) indices.The non-compact part of this algebra is generated by the 20 symmetric, tracelessΛ I J ∈ SL (6), the 2 symmetric, traceless Λ α β ∈ SL (2 , R ) and the 20 Σ IJKα anti-symmetric in
IJ K and satisfying Σ
IJKα = (cid:15) IJKLMN (cid:15) αβ Σ LMNβ . It is possible tochoose a gauge for the coset element such that these 42 non-compact generators arein one-to-one correspondence with the scalar fields of the gauged supergravity.In this gauge, the truncation to the 10-scalar model discussed [16], retains the37etric and the ten scalar fields { β , β , ¯ α , ¯ α , ¯ α , ¯ φ , ¯ φ , ¯ φ , ¯ φ , ¯ ϕ } defined byΛ I J = diag ( ¯ α + β + β , − ¯ α + β + β , ¯ α + β − β , − ¯ α + β − β , ¯ α − β , − ¯ α − β ) , Λ α β = diag ( ¯ ϕ, − ¯ ϕ ) , (A.2)and Σ = − Σ = 12 √ (cid:0) ¯ φ + ¯ φ + ¯ φ − ¯ φ (cid:1) , Σ = − Σ = 12 √ (cid:0) − ¯ φ + ¯ φ + ¯ φ + ¯ φ (cid:1) , Σ = − Σ = 12 √ (cid:0) ¯ φ − ¯ φ + ¯ φ + ¯ φ (cid:1) , Σ = − Σ = 12 √ (cid:0) ¯ φ + ¯ φ − ¯ φ + ¯ φ (cid:1) . (A.3)These barred scalar fields are non-linearly related to the unbarred scalar fields thatwe use in (2.2), however they do agree at linear order. It is straightforward todemonstrate that the generators associated with this truncation generate SO (1 , × SU (1 , ⊂ E . Specifically, if we let B (1) , B (2) each generate an SO (1 , S ( A )1 , , for A = 1 , , , SU (1 ,
1) satisfying (cid:2) S ( A )1 , S ( A )2 (cid:3) = 2 S ( A )3 , (cid:2) S ( A )1 , S ( A )3 (cid:3) = 2 S ( A )2 , (cid:2) S ( A )2 , S ( A )3 (cid:3) = − S ( A )1 , (A.4)then we can explicitly identify the generators using table 1.The ten scalar fields which are retained in the truncated theory parametrise thecoset SO (1 , × [ SU (1 , /U (1)] . It is convenient to parametrise this coset interms of two real scalars β , and four complex scalars z A , which are functions ofthe remaining scalars { ¯ α , ¯ α , ¯ α , ¯ φ , ¯ φ , ¯ φ , ¯ φ , ¯ ϕ } , with the z A transforming linearlyunder the U (1) ⊂ SU (1 , SU (1 , U (1) charge, by defining the generators E ( A ) = 12 (cid:0) S ( A )1 + iS ( A )2 (cid:1) , and F ( A ) = 12 (cid:0) S ( A )1 − iS ( A )2 (cid:1) . (A.5)The desired parametrisation of the coset is then given by V = e β B (1) + β B (2) · (cid:89) a e s ( | z A | ) ( z A E ( A ) +¯ z A F ( A ) ) , (A.6)where s ( | z A | ) = 1 | z A | arcsech (cid:112) − | z A | . (A.7)38 (1) B (2) S (1)1 S (1)2 S (2)1 S (2)2 S (3)1 S (3)2 S (4)1 S (4)2 ¯ α α − − α − − ϕ − − φ ¯ φ − − ¯ φ − − ¯ φ − − β β SO (1 , × SU (1 , ⊂ E algebra inthe that are associated with the ten scalar truncation can be obtained from thistable and (A.2),(A.3).We will work with right cosets, in which V transforms from the left under globalelements of SO (1 , × SU (1 , and from the right under local U (1) rotations.The U (1) invariant tensor defined by M = V · V † , (A.8)can then be used to construct the kinetic terms for the scalar fields of the D = 510-scalar model via L ( k )10 = 196 tr (cid:0) ∂ µ M ∂ µ M − (cid:1) , (A.9)as given in (2.4). It will also play a distinguished role in the uplift of this model toten dimensions as we discuss below.The scalar potential P of the 10-scalar model appearing in (2.4) can be obtainedfrom this coset representative using the general results for the form of the scalar po-tential in the SO (6) gauged supergravity given in [38]. To do this, and following [38],it is helpful to change to a basis adapted to U Sp (8) ⊂ E using the antisymmetrichermitian gamma matrices of Cliff(7). An explicit representation is provided by the39et of 8 × , Γ I ) given byΓ = − σ ⊗ σ ⊗ σ , Γ = σ ⊗ σ ⊗ σ , Γ = σ ⊗ σ ⊗ σ , Γ = − σ ⊗ σ ⊗ , Γ = 1 ⊗ σ ⊗ , Γ = σ ⊗ σ ⊗ σ , Γ = − ⊗ σ ⊗ σ , (A.10)where the σ , , are Pauli matrices. From these one constructsΓ IJ = 12 [Γ I , Γ J ] and Γ Iα = (Γ I , i Γ I Γ ) , (A.11)whose “spinor” indices a, b are U Sp (8) indices. In particular (Γ IJ ) ab transforms in the of U Sp (8), indexed by the symplectic traceless index pairs [ ab ]. The symplectictrace is taken with respect to the invariant tensorΩ ab = − Ω ab = − i (Γ ) ab . (A.12)Introducing the notation V A ab = (cid:0) V IJ ab , V
Iαab (cid:1) , and V = U IJ P Q U IJ,Rβ U Kα,P Q U Kα Rβ , (A.13)for the coset representative in the U Sp (8) and SL (6) × SL (2 , R ) bases, respectively,one can use (A.11) to relate the two: V P Q ab = 18 (cid:2) (Γ IJ ) ab U P Q IJ + 2 (cid:0) Γ Iα (cid:1) ab U P Q,Iα (cid:3) ,V Kαab = 14 √ (cid:2) (Γ IJ ) ab U Kα,IJ + 2 (cid:0) Γ Iβ (cid:1) ab U Kα Iβ (cid:3) . (A.14)The W tensors in [38] are then given by W abcd = δ IJ (cid:15) αβ V Iαa (cid:48) b (cid:48) V Jβc (cid:48) d (cid:48) Ω aa (cid:48) Ω bb (cid:48) Ω cc (cid:48) Ω dd (cid:48) , W ab = Ω dc W cadb , (A.15)and the scalar potential of the SO (6) gauged supergravity is P = − g (cid:0) W ab W ab − W abcd W abcd (cid:1) , (A.16)where U Sp (8) indices are raised and lowered with the symplectic invariant (A.12)according to the rules implicit in (A.15). After substituting (A.6), using g = 2 L , (A.17)and some calculation we obtain (2.7) for the 10-scalar truncation.40 .2 The uplift to type IIB supergravity
The uplift of the bosonic sector of the maximal gauged supergravity to type IIBsupergravity is given in [18]. The D = 10 Einstein metric can be written in the formd s = ∆ − / (cid:0) d s + G mn d θ m d θ n (cid:1) , (A.18)where ds is the D = 5 metric, θ m , m = 1 , , ...,
5, parametrise S and the metric G mn and the warp factor ∆ are defined below. The type IIB dilaton, Φ, and axion, C , parametrise the coset SL (2 , R ) /SO (2) and can be packaged in terms of a two-dimensional matrix via m αβ = e Φ C + e − Φ − e Φ C − e Φ C e Φ , (A.19)with det m = 1. The remaining type IIB fields consist of two-form potentials ( A , A ),which transform as an SL (2 , R ) doublet and from which we identify the NS-NS two-form B (2) and the RR two-form C (2) via B (2) = A , C (2) = A , (A.20)as well as the four-form potential C (4) that is associated with the self-dual five-formflux as in [18].We focus on uplifting the gravity-scalar sector of the D = 5 theory for which thescalar matrix M introduced in (A.8) plays a key role. In the SL (6) × SL (2 , R ) basiswe can write the components of M and its inverse M − as M = M IJ,P Q M IJ Rβ M Kα P Q M Kα,Rβ , M − = M IJ,P Q M IJ Rβ M Kα P Q M Kα,Rβ . (A.21)We also introduce the round metric on the five-sphere, ˚ G mn , with inverse ˚ G mn . Wecan write the Killing vectors of the round metric in terms of constrained coordinates Y I on S , satisfying Y I Y I = 1, via K IJ m = − L ˚ G mn Y [ I ∂ n Y J ] . (A.22)In term of these quantities, the ten-dimensional fields of the uplifted D = 5gravity-scalar sector are given by G mn = K IJ m K P Q n M IJ,P Q ,m αβ = ( m αβ ) − = ∆ / Y I Y J M Iα,Jβ ,A αmn = − L(cid:15) αβ G nk K kIJ M IJ P β ∂ m Y P ,C mnkl = L (cid:16)(cid:112) ˚ G(cid:15) mnklp ˚ G pq ∆ / m αβ ∂ q (cid:0) ∆ − / m αβ (cid:1) + ˚ ω mnkl (cid:17) , (A.23)41here d˚ ω = 16vol S . Note that the D = 10 warp factor ∆ is defined implicitly usingthe fact that the axio-dilaton matrix (A.23) satisfies det m = 1.Restricting now to the 10-scalar model, we can illustrate the above formulae bywriting down the components of the axion and dliaton matrix:∆ − / m = e β +2 β (cid:16) (1 + z )(1 + ¯ z )(1 + z )(1 + ¯ z )(1 − z ¯ z )(1 − z ¯ z ) ( Y ) + (1 − z )(1 − ¯ z )(1 − z )(1 − ¯ z )(1 − z ¯ z )(1 − z ¯ z ) ( Y ) (cid:17) + e β − β (cid:16) (1 + z )(1 + ¯ z )(1 − z )(1 − ¯ z )(1 − z ¯ z )(1 − z ¯ z ) ( Y ) + (1 − z )(1 − ¯ z )(1 + z )(1 + ¯ z )(1 − z ¯ z )(1 − z ¯ z ) ( Y ) (cid:17) + e − β (cid:16) (1 + z )(1 + ¯ z )(1 − z )(1 − ¯ z )(1 − z ¯ z )(1 − z ¯ z ) ( Y ) + (1 − z )(1 − ¯ z )(1 + z )(1 + ¯ z )(1 − z ¯ z )(1 − z ¯ z ) ( Y ) (cid:17) (A.24)∆ − / m = e β +2 β (cid:16) ( z − ¯ z )( z − ¯ z )(1 − z ¯ z )(1 − z ¯ z ) − ( z − ¯ z )( z − ¯ z )(1 − z ¯ z )(1 − z ¯ z ) (cid:17) Y Y + e β − β (cid:16) ( z − ¯ z )( z − ¯ z )(1 − z ¯ z )(1 − z ¯ z ) − ( z − ¯ z )( z − ¯ z )(1 − z ¯ z )(1 − z ¯ z ) (cid:17) Y Y + e − β (cid:16) ( z − ¯ z )( z − ¯ z )(1 − z ¯ z )(1 − z ¯ z ) − ( z − ¯ z )( z − ¯ z )(1 − z ¯ z )(1 − z ¯ z ) (cid:17) Y Y (A.25)∆ − / m = e β +2 β (cid:16) (1 + z )(1 + ¯ z )(1 + z )(1 + ¯ z )(1 − z ¯ z )(1 − z ¯ z ) ( Y ) + (1 − z )(1 − ¯ z )(1 − z )(1 − ¯ z )(1 − z ¯ z )(1 − z ¯ z ) ( Y ) (cid:17) + e β − β (cid:16) (1 + z )(1 + ¯ z )(1 − z )(1 − ¯ z )(1 − z ¯ z )(1 − z ¯ z ) ( Y ) + (1 − z )(1 − ¯ z )(1 + z )(1 + ¯ z )(1 − z ¯ z )(1 − z ¯ z ) ( Y ) (cid:17) + e − β (cid:16) (1 + z )(1 + ¯ z )(1 − z )(1 − ¯ z )(1 − z ¯ z )(1 − z ¯ z ) ( Y ) + (1 − z )(1 − ¯ z )(1 + z )(1 + ¯ z )(1 − z ¯ z )(1 − z ¯ z ) ( Y ) (cid:17) (A.26)There are a number of additional sub-truncations of the 10-scalar model as sum-marised in figure 2. In this paper we are particularly interested in the SO (3) in-variant 4-scalar model as well as the SU (2) invariant 5-scalar model and their sub-truncations. A.2.1 The SO (3) invariant 4-scalar model This truncation is obtained from the 10-scalar model by taking β = β = 0 and z = − z = − z . The truncation is invariant under SO (3) ⊂ SU (3) ⊂ SO (6).42imilar to [39] a useful parametrisation of the five-sphere adapted to this isometry isgiven by Y + iY Y + iY Y + iY = e iα cos χ R + ie iα sin χ R . (A.27)Here 0 ≤ α ≤ π , 0 ≤ χ ≤ π/ R = e ξ g e ωg e ξ g is an SO (3) rotation matrixparametrised by three Euler angles ω, ξ , ξ where g , g are the 3 × g = e − e , and g = e − e , (A.28)with e ij having a unit in the i, j position and zeroes elsewhere. In this parametrisation,the round metric on the five-sphere is written as a U (1) fibration over C P asd˚Ω = d s C P + (d α − sin 2 χτ ) , (A.29)where d s C P = d χ + sin χ τ + cos χ τ + cos χ τ , (A.30)and the τ , , are locally left-invariant one-forms for SO (3) given by τ = − sin ξ d ω + cos ξ sin ω d ξ ,τ = cos ξ d ω + sin ξ sin ω d ξ ,τ = d ξ + cos ω d ξ . (A.31)This parametrisation of C P is cohomogeneity-one with principle orbits actually givenby SO (3) / Z ⊂ SU (3) (rather than SO (3)). The singular orbits are an RP at χ = 0and an S at χ = π/ SO (3) invariant model, the ten dimensional metricwill, in general have non-trivial dependence on α and more general dependence on χ than that given in (A.31) and the symmetry will be the SO (3) / Z associated withthe τ i . For the further truncation to the SU (3) invariant model in figure 2, the χ dependence will be as in (A.30), giving rise to SU (3) symmetry associated with C P ,but there will be non-trivial dependence on α . A.2.2 The SO (3) × SO (3) invariant 3-scalar model The SO (3) × SO (3) invariant sector has three scalars, and can be obtained form the SO (3) invariant model just discussed by setting z = ¯ z . Specifically, we have z = tanh (cid:104) (cid:0) α + ϕ − iφ (cid:1)(cid:105) , z = tanh (cid:104) (cid:0) α − ϕ (cid:1)(cid:105) , (A.32)43ith β = β = 0. For this case we can parametrise the five-sphere using the coordi-nates Y = cos ψ sin θ cos ξ, Y = cos ψ sin θ sin ξ, Y = cos ψ cos θ ,Y = sin ψ sin ˜ θ cos ˜ ξ, Y = sin ψ sin ˜ θ sin ˜ ξ, Y = sin ψ cos ˜ θ , (A.33)with 0 ≤ θ, ˜ θ ≤ π , 0 ≤ ξ, ˜ ξ ≤ π and 0 ≤ ψ ≤ π/
2. In these coordinates the roundmetric on the five-sphere is given byd˚Ω = d ψ + cos ψ dΩ + sin ψ d ˜Ω , (A.34)with dΩ = d θ + sin θ d ξ and d ˜Ω = d˜ θ + sin ˜ θ d ˜ ξ . The SO (3) × SO (3) symmetryof the gauged supergravity model is generated by the Killing vectors for each of theround two-spheres.For this model it will be useful to write down some additional uplifting formulae.The D = 10 metric takes the form ds = ∆ − / (cid:104) ds + L (cid:0) dψ + d Ω e α sec 4 φ + tan ψ + d ˜Ω e − α sec 4 φ + cot ψ (cid:1)(cid:105) , (A.35)with the D = 10 warp factor given below. The axion-dilaton matrix is diagonal with m =∆ / (cid:104) cos ψ (1 + z )(1 + ¯ z )(1 − z ) (1 − | z | )(1 − ( z ) ) + sin ψ (1 − z ) (1 − ( z ) ) (cid:105) , =∆ / (cid:2) e ϕ − α sin ψ + e α +2 ϕ sec 4 φ cos ψ (cid:3) ,m =∆ / (cid:104) sin ψ (1 − z )(1 − ¯ z )(1 + z ) (1 − | z | )(1 − ( z ) ) + cos ψ (1 + z ) (1 − ( z ) ) (cid:105) , =∆ / (cid:2) e − ϕ − α sec 4 φ sin ψ + e α − ϕ cos ψ (cid:3) , (A.36)and m = m = 0, where the D = 10 warp factor is given by∆ / = e α sec ψ (cid:112) ( e α sec 4 φ + tan ψ ) ( e α + tan ψ sec 4 φ ) . (A.37)Thus, we have vanishing axion, C = 0, and e Φ = m .The NS-NS and R-R two-forms are found to be B (2) = L i sin ψ ( z − z − ¯ z )Π vol ˜ S ,C (2) = − L i cos ψ ( z + 1)( z − ¯ z )Π vol S , (A.38)44hereΠ = z (cid:2) ( z −
1) sin ψ − ¯ z ( z + cos 2 ψ ) (cid:3) + z cos 2 ψ + ¯ z ( z −
1) sin ψ + 1 , Π = z (cid:2) ( z + 1) cos ψ + ¯ z ( z + cos 2 ψ ) (cid:3) + z cos 2 ψ + ¯ z ( z + 1) cos ψ + 1 , (A.39)and vol S = sin θ dθ ∧ dξ , vol ˜ S = sin ˜ θ d ˜ θ ∧ d ˜ ξ . Finally, the four-form potential isgiven by C (4) = L ω − L ψ (cid:16) z ( z + 2¯ z −
2) + z (¯ z − − z − − Π + z (1 + z − ¯ z z ) + ( z + 1)¯ z − z ( z + 2¯ z + 2) + z (¯ z + 4) + 2¯ z Π + 3Π − z z (¯ z + 1) + z − z ¯ z + ¯ z + 4 (cid:17) vol S ∧ vol ˜ S , (A.40)where the four-form ˆ ω is given byˆ ω = (cid:0) ψ −
12 sin 4 ψ (cid:1) vol S ∧ vol ˜ S , (A.41)and satisfies d ˆ ω = 16vol S , where the volume form is with respect to the round metric(A.34). A.2.3 The SU (2) invariant 5-scalar model This truncation is obtained from the 10-scalar model by taking β = 0, z = − z and z = − z . The resulting truncation is invariant under SU (2) ⊂ SU (3) ⊂ SO (6). Toparametrise the five-sphere so that this symmetry is manifest, similar to [10] one candefine Y + iY = e i ( ξ + ξ ) sin ρ cos( ω/ ,Y + iY = e i ( − ξ + ξ ) sin ρ sin( ω/ ,Y + iY = e iα cos ρ , (A.42)with ω, ξ , ξ Euler angles of SU (2) with0 ≤ ω ≤ π, ≤ ξ ≤ π, ≤ ξ < π , (A.43)and 0 ≤ ρ ≤ π/
2, 0 ≤ α ≤ π . In these coordinates the metric on the round spheretakes the form d˚Ω = d ρ + cos ρ d α + 14 sin ρ (cid:0) τ + τ + τ (cid:1) , (A.44)where the τ i are SU (2) left-invariant forms given in (A.31). The SU (2) symmetrythen corresponds to the Killing vector fields associated with the SU (2) action. In45eneral ∂ α will not be a Killing vector of the uplifted solutions of the SU (2) invariant5-scalar model and furthermore, the coefficients of the τ i will differ from that of(A.44).We can also write ξ = 2 α + γ so that Y + iY = e iα + i ξ + i γ sin ρ cos( ω/ ,Y + iY = e iα − i ξ + i γ sin ρ sin( ω/ ,Y + iY = e iα cos ρ . (A.45)We then have d˚Ω = d s C P + (cid:0) d α + 12 sin ρτ (cid:1) , (A.46)where d s C P = d ρ + 14 sin ρ (˜ τ + ˜ τ ) + 116 sin ρ ˜ τ , (A.47)and the ˜ τ , , are left-invariant one-forms for SU (2)˜ τ = − sin γ d ω + cos γ sin ω d ξ , ˜ τ = cos γ d ω + sin γ sin ω d ξ , ˜ τ = d γ + cos ω d ξ . (A.48)For the uplift of the SU (2) invariant 5-scalar model, the metric will in general dependon α and moreover the extra U (1) associated with rotating ˜ τ into ˜ τ that is manifestin (A.47) will no longer be present. Moving to the SU (3) truncation in figure 2 theuplifted metric will have a C P factor, as in (A.47), giving rise to the SU (3) symmetrybut there will be dependence on α , in general. Moving instead to the SU (2) × U (1)invariant truncation in figure 2 the uplifted metric will in general have dependenceon α , and the U (1) associated with rotating ˜ τ into ˜ τ that is manifest in (A.47) willbe present. A.3 The SL (2 , R ) action in five and ten dimensions Both the D = 5 maximal gauged supergravity and the type IIB supergravity areinvariant under global SL (2 , R ) transformations. Focussing on the gravity and scalarsector of the D = 5 theory the relationship between the two SL (2 , R ) transformationscan be made explicit using uplift formulae in (A.23).Consider first the D = 5 theory in which the SL (2 , R ) ⊂ E can be generatedby the X of (A.1) with Λ α β a linear combination of the three matrices (Λ i ) α β givenby (cid:0) Λ (cid:1) α β = ( σ ) αβ , (cid:0) Λ (cid:1) α β = ( σ ) αβ , (cid:0) Λ (cid:1) α β = ( − iσ ) αβ , (A.49)46xplicitly, in terms of the 27 dimensional representation the SL (2 , R ) generators arethus X i (cid:12)(cid:12) SL (2 , R ) = × (Λ i ) α β (Λ i ) α β (Λ i ) α β (Λ i ) α β (Λ i ) α β (Λ i ) α β . (A.50)A finite SL (2 , R ) transformation in the D = 5 theory, using the i th generator, canthen be written S i (5) = e c X i | SL (2 , R ) where c is constant. This transformation acts onthe scalar matrix M given in (A.8) via M → M (cid:48) = S i (5) · M · S i (5) T . (A.51)From this one can infer the corresponding transformation of the scalars parametrisingthe coset which, in general, is non-linear. For the specific case of the transformationassociated with the i = 3 generator, one finds the following action on the ten-scalarmodel: β → β , β → β ,z → z + tanh c c z , z → z − tanh c − tanh c z ,z → z − tanh c − tanh c z , z → z + tanh c c z . (A.52)From (2.2) one can conclude that this transformation is equivalent to a simple shiftin the five dimensional field ϕ → ϕ + c . Also note that the SL (2 , R ) transformationsassociated with the i = 1 , SL (2 , R ) action in D = 10. From (A.23) we can concludethat the D = 5 transformation by the element S i (5) is equivalent to a transformationby S i (10) = e c ( Λ i ) α β , (A.53)in the D = 10 theory. For example, and of most interest, the transformation associ-ated with the i = 2 generator gives rise to m − → m (cid:48) − = S · m − · S T , (A.54)47his transformation is equivalent to m αβ → m (cid:48) αβ = e − c m m m e c m , (A.55)and translates, in turn, into the following simple transformation of the D = 10 dilatonand axion: Φ → Φ + 2 c and C → e − c C . (A.56)The transformation by S plays a key role for our solutions, as it allows one to S-foldthe D = 5 solutions, as we discuss in the text (note that we call this transformationsimply S in (3.18)).In checking that the S-fold procedure we employ does not break supersymmetryit is also useful to see how an S ∈ SL (2 , R ) transformation acts on the D = 5supersymmetry parameters. A transformation by any element of the E globalsymmetry group is associated with a local compensating U Sp (8) transformation, H ,which acts on the fermions. For the action of S we find that H ∈ U (1) ⊂ U Sp (8),in the fundamental representation, is explicitly given by H = k k ¯ k − k k k ¯ k − k k k ¯ k − k
00 0 0 k k ¯ k − k k − k k k ¯ k − k k k ¯ k − k k k
00 0 0 ¯ k − k k k (A.57)with k = (cid:16) g g g g ¯ g ¯ g ¯ g ¯ g (cid:17) / , k = (cid:16) ¯ g g ¯ g g g ¯ g g ¯ g (cid:17) / ,k = (cid:16) ¯ g ¯ g g g g g ¯ g ¯ g (cid:17) / , k = (cid:16) g ¯ g ¯ g g ¯ g g g ¯ g (cid:17) / , (A.58)and g = 1 + tanh ( c/ z , g = 1 − tanh ( c/ z ,g = 1 − tanh ( c/ z , g = 1 + tanh ( c/ z . (A.59)The action on the supersymmetry parameters ε can be seen by diagonalising the W -tensor W ab of D = 5 gauged supergravity (A.15) and restricting ε a to lie within thespace spanned by the eigenvectors of W ab with eigenvalues e K / W (1st) and e K / W (5th). In this basis the U Sp (8) transformation is found to beˆ H = diag (cid:0) k , k , k , k , ¯ k , ¯ k , ¯ k , ¯ k (cid:1) . (A.60)48he dilaton shift action can also be seen as a K¨ahler transformation acting in the D = 5 theory, as noted in [15]. Under ϕ → ϕ + c we have K → K + f + ¯ f and W → e − f W with f = f ( z A ) given by e f = cosh ( c/ g g g g . (A.61)Under this transformation the preserved supersymmetries of the BPS equations trans-form as ε → e ( f − ¯ f ) / ε and ε → e − ( f − ¯ f ) / ε i.e. ε → k ε and ε → ¯ k ε . Thisshows that the dilaton shift is realised by an SL (2 , R ) transformation that is alsoacting as an SL (2 , R ) transformation on the preserved supersymmetries. This allowsus to conclude that the S-folding procedure will preserve the supersymmetry of the D = 5 solutions as noted in the text. A.4 The N = 4 one-sided Janus solution in type IIB Here we show that the one-sided Janus solution (6.1), after being uplifted to D = 10,can be cast into the form of the general AdS solutions of type IIB which preserve N = 4 supersymmetry [19, 20].In [19, 20] they consider the type IIB Einstein metric written in the formd s = f d s AdS + f dΩ + f d ˜Ω + ds (Σ) , (A.62)where ds (Σ) is the metric on a Riemann surface. Introducing a complex coordinate w on Σ we write ds (Σ) = 4 ρ d w d ¯ w , (A.63)where ρ as well as f , f , f are functions of w, ¯ w . To specify a solution in the languageof [19], it is sufficient to provide two harmonic functions on the Riemann surface, h , h . To do so, as in [41], one can introduce the real functions W ≡ ∂ w h ∂ ¯ w h + ∂ w h ∂ ¯ w h ,N ≡ h h | ∂ w h | − h W ,N ≡ h h | ∂ w h | − h W . (A.64)Then, for example, the D = 10 dilaton Φ is given by e = N N , (A.65)49hile the metric functions have the form ρ = W h h N N , f = 2 e Φ2 h (cid:114) − WN ,f = 2 e − Φ2 h (cid:114) − WN , f = 2 e − Φ2 (cid:114) − N W . (A.66)To connect with the uplifted one sided Janus solution (6.1) we take the Riemannsurface to be an infinite strip and write w = ¯ rL + iψ , (A.67)with −∞ < ¯ r < ∞ and ψ ∈ [0 , π/ h = − i e − ϕ ( s ) L √ w − sinh ¯ w ) = e − ϕ ( s ) L √ rL sin ψ ,h = e ϕ ( s ) L √ e w + e ¯ w ) = e ϕ ( s ) L √ e ¯ r/L cos ψ , (A.68)and hence W = − L
16 sin 2 ψ ,N = e − ϕ ( s ) L
256 sin 2 ψ (cid:0) e − r/L (cid:1) (cid:0) e r/L cos ψ + e r/L − cos 2 ψ (cid:1) ,N = e ϕ ( s ) L e r/L sin 2 ψ (cid:0) e r/L + cos 2 ψ (cid:1) . (A.69)With a little effort we can show that this agrees with the uplift of (6.1) after usingthe results in section A.2.2. For example, in both cases the D = 10 dilaton is givenby e = e ϕ ( s ) e r/L (cid:0) e r/L + cos 2 ψ (cid:1) (1 + e r/L ) (2 + 2 e r/L cos ψ + e r/L − cos 2 ψ ) . (A.70)Notice that as as ¯ r → ∞ , where the solution approaches the AdS vacuum, we have e → e ϕ ( s ) while as ¯ r → −∞ we have e → References [1] C. Couzens, C. Lawrie, D. Martelli, S. Schafer-Nameki, and J.-M. Wong,“F-theory and AdS /CFT ,” JHEP (2017) 043, arXiv:1705.04679[hep-th] . In fact to get an exact match with the metric and also for the two-form and four-form potentialsin (A.33) we should relabel ˜ θ → π − θ , ˜ ξ → ξ + π as well as θ → ˜ θ , ξ → ˜ ξ , so that d Ω ↔ d ˜Ω andvol ˜ S → − vol S and vol S → vol ˜ S . /CFT (2,0),” JHEP (2018) 008, arXiv:1712.07631 [hep-th] .[3] G. Inverso, H. Samtleben, and M. Trigiante, “Type II supergravity origin ofdyonic gaugings,” Phys. Rev. D no. 6, (2017) 066020, arXiv:1612.05123[hep-th] .[4] B. Assel and A. Tomasiello, “Holographic duals of 3d S-fold CFTs,” JHEP (2018) 019, arXiv:1804.06419 [hep-th] .[5] E. D’Hoker, J. Estes, and M. Gutperle, “Interface Yang-Mills, supersymmetry,and Janus,” Nucl. Phys.
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