JHEP04(2020)050 4 Maximal Springer = 2 Super- 12 April 8, 2020

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JHEP04(2020)050 4 maximal Springer = 2 super- 12 April 8, 2020 . These pro- : 5 March 25, 2020 N S : n R February 28, 2020 × : R involving S-duality 5 , S Published SO(6)] Accepted × × Received 1 S 1) , × 4 d U(1) while preserving Published for SISSA by × https://doi.org/10.1007/JHEP04(2020)050 vacua with various amounts of supersym- exceptional ﬁeld theory, its uplift to a class 4 = 4 super-Yang-Mills theory. One such AdS 7(7) N [email protected] , and Mario Trigiante is presented. c 1 . 3 Colin Sterckx 2002.03692 a,b The Authors. Flux compactiﬁcations, String Duality, Supergravity Models, Superstring c

supersymmetric S-folds Multi-parametric families of AdS , [email protected] = 2 S-folds of type IIB supergravity of the form AdS = 2 N E-mail: [email protected] UniversitéLibre de Bruxelles (ULB) andService de International Physique Solvay Théoriqueet Institutes, Mathématique, Campus de la Plaine, CPDipartimento di 231, Fisica, B-1050, Politecnico Brussels, diI-10129 Belgium Torino, Turin, Corso Italy Duca and degli INFN, Abruzzi Sezione 24, di Torino, Italy Departamento de Universidad F´ısica, de Oviedo, Avda. Federico Garc´ıaLorca 18, 33007 Oviedo,Instituto Spain Universitario de CienciasCalle y de de Tecnolog´ıasEspaciales la Asturias Independencia (ICTEA), 13, 33004 Oviedo, Spain b c d a Open Access Article funded by SCOAP ArXiv ePrint: twists of hyperbolic type along S Keywords: Vacua supergravity that arises from thevide reduction natural of candidates type to IIB holographically describe supergravityCFT’s new on which strongly are coupled localised three-dimensional onvacua interfaces features of a symmetrysymmetry. enhancement Fetching to techniques SU(2) from theof E Abstract: metry and residual gauge symmetry are found in the [SO(1 Adolfo Guarino, N JHEP04(2020)050 ] 2 ] (see vacua 3 4 8] of the SO(6) , π/ 7 [0 → 3 ∈ ]. Other types of ], have also been 7 ) – 12 = 0 theory [ ic U(1) – 5 c ]. Setting the param- , 8 × 1 10 1 [ 1 SU(3) − SU(3) symmetry 2 = arg(1 + 0 limit [ → → √ ω 4 2 , → 0 vacua of the c ]. Also, for the new supersymmetric 4 9 U(1) U(1) symmetry ∈ U(1) 14 , c × × ] or black holes [ × maximal supergravity 13 9 , 8 12 17 9 SU(2) SU(2) SU(2) – 1 – n R 10 → → → 2 2 6 -EFT . The various AdS SO(6)] 7 7(7) × vacua 16 4 1) and, more importantly, new and genuinely dyonic AdS 5 , c 7 remains elusive and various no-go theorems have been stated against = 2 supersymmetry c = 0 vacua with SU(2) = 1 vacua with U(1) = 2 vacua with U(1) = 4 vacuum with SO(4) symmetry N 1 18 = 8 theory: gauging and scalar potential N symmetry N N N 6= 0, the question about the eleven-dimensional interpretation of the electro- N c invariant sector vacua of [SO(1 3 2 = 0 then the standard (electric) SO(8) supergravity of de Wit and Nicolai [ 2.3.2 2.3.3 2.3.4 Z 2.3.1 4 c ] for an updated encyclopedic reference) get generalised to one-parameter families 4 3.2 S-fold interpretation 3.3 Connection with Janus-like solutions 3.1 Type IIB uplift using E 2.1 The 2.2 2.3 New families of AdS investigated using instead aelectromagnetic phase-like deformation parameterisation parameter.solutions However, at and despite themagnetic parameter rich structure ofthe new existence of such a higher dimensional origin [ is recovered which issupergravity on known a to seven-sphere arisealso S [ upon dimensionalof reduction vacua of when turning eleven-dimensional on also appear which do notfour-dimensional have solutions, a well like defined domain-walls (electric) [ phenomenology as far as the existenceThe of prototypical new gaugings example and is vacuum the solutionselectromagnetic are dyonically-gauged concerned. duality SO(8) supergravity on where the thetheories gauging action parameterised of generates by a a continuous one-parametereter parameter family to of inequivalent 1 Introduction Electromagnetic duality in four-dimensional maximal supergravity has provided a very rich 4 Conclusions 3 S-folds with Contents 1 Introduction 2 AdS JHEP04(2020)050 ] , ) ], 1 k ( IIB 39 27 = 2 ) A U(1) ]. Z ] (see , N × 26 33 ] solutions ]. Various SL(2 32 16 ]. This four- . Within this ∈ ] respectively c vacua describe 16 k 32 4 vacua preserving 4 = 2&SU(2) S-folds with −ST 5 . N S ] and [ 1 = × -fold CFT’s in [ 27 = 1&SU(3) [ = 0 or 1 [ 1 J J S c ] and subsequently, in [ N of hyperbolic type along S -fold CFT which, in turn, is = 4 super-Yang-Mills theory × J . 28 4 k 1 U(1) symmetry remain missing. N ] solutions have been constructed × ] and diverges at infinity. Together with −ST 24 s 31 – g has been much less investigated in the 21 ) = c k correspondence. ( in the dual ], and has a holographic interpretation as 3 k M 25 = 1 cases, the gravity duals of the would be = 4&SO(4) symmetry, – 2 – N N = 0&SO(6) [ /CFT ] upon suitable gauging of flavour symmetries. A = 0 or 1 without loss of generality [ 4 ]. As for the ISO(7) theory, the electromagnetic -EFT). These S-folds involve S-duality twists 38 c S-fold backgrounds of type IIB supergravity using N 27 [ theory which arises from the reduction of type IIB 5 7(7) ], a hyperbolic monodromy S 5 12 S 33 ], and black hole [ × = 4 and monodromies × 1 20 n R S ]. From a holographic perspective, these AdS , N R )) theory [ IIB × 32 19 ) N ] where the string coupling 4 Z of a three-dimensional super-Chern-Simons dual theory [ U(1) symmetry and, as in the previous cases, involving S-duality , k = 4 case [ 30 SO(6)] (U( × , T × 29 N ]), which are localised on interfaces of = 1&SU(3) symmetry. While the S-folds in [ 1) 36 , 6= 0, the holographic interpretation of the deformation parameter remains N c of the ten-dimensional theory [ ]. In this case the electromagnetic deformation parameter is a discrete (on/off) 0 ], domain-wall [ ]) in correspondence to the various amounts of supersymmetry, as well as the ˆ ] and [ F ]. In the 15 19 40 [ exceptional field theory (E – -fold CFT’s localised on the interface with SU(2) 37 6 35 , J = 4&SO(4) solution, additional 17 = 4 and SO(4) symmetric solution was reported in [ vacua at [ 7(7) 3) that induce SL(2 34 On the other hand, a classification of interface SYM theories was performed in [ The role of the electromagnetic deformation Unlike for the SO(8) theory, much more is by now known about the dyonically-gauged 4 4 N N = 2 ≥ k match the symmetries ofN the In this work we fillsupersymmetry, this SU(2) gap and present atwists new that family induce of AdS monodromies of hyperbolic type along S diagram illustrating this type IIB construction is depicted in(see figure also [ largest possible global symmetry, preserved by theric interface operators. cases Three were supersymmet- identified:symmetry interfaces and with new strongly coupled three-dimensionalalso [ CFT’s, referred to(SYM) as [ was shown to introduce aconstructed from Chern-Simons the level and can be systematicallythe type constructed IIB as theory quotients [ the of degenerate Janus-like solutionshave been of found and upliftedhyperbolic monodromies to in similar [ S-fold backgrounds of type IIB supergravity with different amounts of supersymmetry asan well as of residualuplifted gauge to symmetry. a In classthe particular, of E AdS ( context of typecase IIB is supergravity. the [SO(1 supergravity The on relevant the dyonically-gauged product deformation supergravity is in again this adimensional discrete supergravity has (on/off) been shown deformation, to namely, contain various types of AdS AdS which necessarily require a non-zero electromagneticmassive deformation IIA parameter context, theparameter electromagnetic parameter isthe identified Chern-Simons with level the Romans mass obscure from the perspective of the AdS ISO(7) supergravity that arisessphere from S the reduction ofdeformation, massive namely, IIA it supergravity can on be a set six- to AdS JHEP04(2020)050 ] 28 U(1) (2.1) 2 or 4. × , vacuum 1 4 , 1 action = 0 and connection IIB ) 1 we present our N Z 4 , geometry along S , by implementing 3 5 ) -EFT involving 3 k S vacua with symmetry ( along S vacua with symmetry SL(2 = 4 SYM SU(4) at specific values × 7(7) A k 4 vacua with a symmetry 4 ∈ 1 N 4 ∼ S J −ST × . In section -fold CFT 1 4 . J with a localised interface ) = 12 k ( ], and further investigated in [ maximal supergravity. We find four M n R 31 maximal supergravity U(1). In section 12 3 -EFT, we uplift such an AdS × symmetric AdS we perform a study of multi-parametric 12 symmetric AdS 2 Uplift method : E hyperbolic twists n R 2 7(7) U(1) and SO(6) ] actually correspond to very special points 2 vacuum. 12 SO(6)]

/CFT n R ×

4 4 32 × , vacuum – 3 – n R = 2 supersymmetric and containing a vacuum 4 1) SO(6)]

AdS 31 , , N × = 0&SU(2) symmetric AdS 28 = 1&U(1) , SU(3) SO(6)] = 2&U(1) 2 1) N , SO(6)] N × vacua initiated in [ N = 2 S-folds of type IIB supergravity with SU(2) U(1) and SU(3) at specific values of the two arbitrary × U(1) 4 U(1) at a special value of the arbitrary parameter. 1) 5 × N 1) G = [SO(1 × , S , × Type IIB & S-fold with AdS vacua of [ 5 × S 4 R × [SO(1 -fold CFT’s. 1 J S Reduction on gauging with an AdS × 4 = 4&SO(4) symmetric AdS vacua in the [SO(1 4 N vacua of [SO(1 4 . Type IIB S-folds with hyperbolic monodromies = 3 = 4 ], for the dyonically-gauged maximal supergravity with non-abelian gauge group = 10 parameters. A one-parameter family of enhancement to SU(2) A single A three-parameter family of enhancements to SU(2) of the three arbitrary parameters. A two-parameter family of enhancements to SU(2) D D 32 D The paper is organised as follows. In section • • • • We will show how the AdS (featuring residual symmetry enhancements) withinEach multi-parametric of families these of solutions. families preservesMore a specifically given we amount find: supersymmetry, namely, 2 AdS We continue the studyand of [ AdS with a residual symmetrya enhancement generalised to Scherk-Schwarz SU(2) (S-S)to ansatz in a E class ofsymmetry AdS and a non-trivialconclusions hyperbolic and monodromy discuss along future S directions. with three-dimensional families of AdS families of vacua, one of them being Figure 1 JHEP04(2020)050 ) ) ), is ] 2.1 ], to V 2.2 = 0) AB U(1) (2.6) (2.4) (2.5) (2.2) (2.3) [ ) and , ] 27 × A ). For matrices. A MN t AB 2.1 [ M = ( , ABCD , ] ] SL(8) in ( t ] ] (with F . D F M D δ ⊂ δ B 1) and E , C A , ][ ][ t B 6 B , B A 0 ˜ η t ]. This tensor codifies η N S , 1 δ A C E A 1 [ [ [ [ 43 − Q R δ [ δ − δ 8 8 (and its inverse − − dM 7(7) The scalar potential in ( + 7 = = diag( 1 ∧ ∗ MN ] ] c RS . The coset representative of E M CD , M CD = [ ] ] dM 7(7) NQ 912 EF EF AB [ 7(7) ][ Tr duality group of maximal supergravity. ][ η M vacua, we set to zero all the vector and . For the gauging of G E SU(8) 1 96 AB AB 4 [ [ AB 7(7) ∈ ], which slightly differ from those of [ η X X t MP 1 + ] for the explicit form of the − − 32 ∗ M – 4 – 42 VV = = and ˜ S in the SL(8) basis. ] ] living in the = PQ , the scalar matrix =8 coupled to Einstein gravity in the presence of a scalar P AB EF EF decomposes into antisymmetric pairs 7(7) [ g η ] X ] N MN R V M MN ]). MN CD CD M [ ][ − ][ X 0) and ˜ M MN 45 , 2 vacua is new and we will uplift the solution with SU(2) R , AB AB 6 X [ [ 4 I 44 , 2 X X g 672 = the index . = =8 ] 3 generators of E 7(7) N ] 31 =8 E L ) within the maximal theory and has the following general form: = diag(0 N 56 being a fundamental index of E 8 denotes a fundamental index of SL(8). This implies that, for gaugings of ⊂ V 2.1 AB ABCD [ = 8 theory: gauging and scalar potential η electric : embedding tensor t ]) tensor fields of the theory, so that the bosonic Lagrangian reduces to the ,..., magnetic : ,..., = 2 family of AdS = N 41 = 1 = 1 N A M ABCD t Under SL(8) The We adopt the conventions in the appendix of [ 1 in terms of two symmetricthese are matrices subgroups of SL(8), the non-vanishingtensor electric and are magnetic given components by of [ the embedding Moreover, it also specifies thevector gauge fields connection transforming which involves underthe both the theory electric (for Sp(56) and reviews magnetic group see of [ electromagnetic transformationswhere of which depends on the gauge coupling on a constant how the gauge group G is embedded into the E with constructed by direct exponentiation of theand 70 non-compact generators which survives our truncationthe to group the G Einstein-scalar in sector, ( is induced by the gauging of which describes the scalar fields potential. The scalar fields serve as coordinates on the coset space of maximal supergravity describe the dyonically-gauged maximalthe supergravity purposes with of this gauge(auxiliary work, group [ i.e. G the in studyfollowing ( of one AdS enhanced residual symmetry to a newsupergravity and in analytic section family of S-fold backgrounds of type IIB 2.1 The We follow the conventions and notation of [ JHEP04(2020)050 ] with (2.9) (2.7) (2.8) (2.11) (2.10) i ABCD [ z t ) allows ]. 2.5 48 , , , . ) ) ) 8 8 47 8 x , x , x 7 7 − x x , 7 , , , − − 8 8 8 (scalars) and , , , , , , , x . 8 8 8 6 6 8 8 8 8 t t t 6 . ), we will now construct B 7 x x 8 8 8 8 x t t t t A − − − − − t − 2.4 , , 7 7 7 − − − − , 8246 5 5 7 7 7 t 5 7 7 7 7 t t t ,..., x . 7 7 7 7 , x t t t t c , x = ], and it originally appeared in − is given by 4 + + − 4 , 7 = 1 6 6 6 4 − − + + x ∝ 46 , x χ A 6 6 6 i x 6 6 6 6 3 t t t − . In addition, we will also require 6 6 6 6 maximal supergravity. This sector x , − t t t t AB + + − 3 , (2) 2 − η 3 5 5 5 12 , + + − − Z , x 5 5 5 2 5 5 5 5 t t t 2 , x ,, g , x 5 5 5 5 2 (1) 2 x t t t t = 1 supergravity coupled to seven chiral n R + − + − Z − , x , 4 4 4 4738 6358 2578 − + − + , 1 1 t t t 4 4 4 with 1 N so that ˜ 4 4 4 4 t t t x x x 4 4 4 4 ( ( = = = c t t t t and ( – 5 – + − + 5 6 4 I SO(6)] 3 3 3 → → + − + − χ χ χ → 3 3 3 3 3 3 3 ) ) t t t × ) 3 3 3 3 8 8 i 8 t t t t − + + ϕ 1) , x , x 2 2 2 − , , x + − + − 7 7 2 2 2 7 ,, g , g g 2 2 2 2 t t t ie 2 2 2 2 , x , x , x t t t t 6 6 − + + + = 1 supergravity coupled to seven chiral multiplets 6 1238 1458 1678 generator acting as i 1 1 1 + − − + , x , x t t t 1 1 1 , x χ ∗ 2 5 5 N 1 1 1 1 t t t 5 1 1 1 1 = = = Z − in the SL(8) basis. The former have associated generators of t t − − − t t , x , x , x 1 2 3 4 4 = 4 χ χ χ ======g g g i , x , x 1 2 3 6 7 4 5 7(7) , x z 3 3 ϕ ϕ ϕ ϕ ϕ ϕ ϕ 3 g g g g g g g , x , x , x invariant sector describes 2 2 ] and the purely electric SO(8) theory [ 2 3 2 , x , x 19 , x 1 1 Z 1 x x x :( :( :( 7. The same invariant sector has recently been explored in the dyonically-gauged invariant sector ∗ 2 (1) 2 (2) 2 3 2 Z Z Z Z invariant sector of the [SO(1 ,..., To describe this sector of the maximal theory, we first focus on a four-element Klein 3 2 Z = 1 the form The fourteen real spinless(pseudo-scalars) fields are of associated E with generators invariance under an extra The resulting multiplets (and no vector multiplets) together with the remaining generators can be recast asi a minimal ISO(7) theory [ the context of type II orientifold compactifications with generalisedsubgroup fluxes of [ SL(8). Its action on the fundamental index for an (on/off) electromagnetic parameter 2.2 In order to efficiently search fora extrema of the scalar potential ( As stated in the introduction, the magnetic part of the embedding tensor in ( whereas the latter correspond with generators given by JHEP04(2020)050 ) 2.15 (2.15) (2.16) (2.17) (2.12) (2.14) (2.13) ), and , 2.12 ) 7 z 6 z ), restricted to vacua. We find 5 z 4 ) and ( 4 2.4 for a set of seven z j 2.2 is the inverse of the − ¯ z j ). Importantly, all the d ¯ z (1 , i z , gc # ∧ ∗ ), the scalar potential, as 2.14 i K i i . subspace of the coset space ). ϕ g ¯ + 2 7 dz as i W 2.12 2 . ) 2.1 ) i ϕ c 7 i W χ K . z 3 j 7 ) =1 ¯ z ∂χ i )] 6 , X SU(4). The three supersymmetric i ( i − z SO(2)] 2 i ¯ z z 1 4 3 and / ϕ ∂ ∼ z ¯ " 2 ( i W − e j i ϕ + − ¯ z , z 5 + ( Exp i z = 1 model with seven chirals presented in = 2 [SL(2) 2 ) = 1 sector follow from ( # − = 0 i z WD i i ∈ N χ z kin t + N g 2 or 4 supersymmetry as well as various resid- to SO(6) ∂ϕ W i log[ – 6 – D L , i ( 4 j z 2 χ 1 z ¯ z VV ) and Kählerpotential ( 7 , 1 i =1 D i z z X 7 7 =1 =1 = i i X X K − = 0 2.15 h + ( 1 4 12 is the Kählerderivative and = 5 invariant sector entering ( K MN − N z e − 3 2 4 ) with coefficients " vacua M K = z W Z = . Note that only the last term in the superpotential ( ) 3 4 z K 2.11 K kin =1 j i + ¯ z z L , N i ∂ 6 = Exp 2 V z z ∂ 4 V = 1 formula z + ( 2 ) and ( 0 to a purely electric gauging of G in ( ≡ z ), can be recovered from a holomorphic superpotential N j W ¯ z i → + i z 2.10 2.4 z c 6 ∂ z K with Kählerpotential 5 ≡ i z z 1 z W i g z invariant sector, reveals a rich structure of (fairly) symmetric AdS D ). The kinetic terms in the resulting vacua we will present in this section are genuinely dyonic, namely, they disappear if 3 2 = 2 4 Z 2.3 W that follow from the superpotentialAdS ( taking the limit the four families of vacuaual preserving gauge symmetries rangingfamilies from are U(1) also supersymmetricthe within previous the section, and therefore satisfy the F-flatness conditions turns out to be sensitive to the electromagnetic parameter 2.3 New families ofA AdS thorough study of the structure of extrema of the scalar potential ( using the standard where Kählermetric Lastly, when restricted tocomputed the from ( chiral fields in ( are given by These match the standard kinetic terms yields a parameterisation of an Exponentiating ( JHEP04(2020)050 ] 29 6= 0 (2.22) (2.18) (2.19) (2.21) (2.20) 3 SO(6) χ = → , 2 χ factor appears i , 5) 2 √ = ), only 3 of them U(1) = = 0 enhances the i 7 1 3 , z × , thus resulting in a χ 2 , 2.20 3 , 1 , = (1 + 2 , χ 3 6 6 1 solutions that preserves , 1 z χ √ 2 4 , = SU(3) , = 5 3 z 7 , = 1 SU(3) symmetry , → z 2 , i 3 = , radius. This family of solutions 2 3 lying below the Breitenlohner- , 3 , i < j , = i < j . → 4 , 2 4 − z 6 2 , = 1 z , 28) , i U(1) 1 × = 2) = 1 , − U(1) = 1 2) 5 2) × i 0( c × ( z i 2 × 1) × 2 × ( g ( , ) × = 2 2 2 i 4) U(1). Lastly, setting ) and , ) 4 j χ 6( is the AdS 2) √ × j z 2 × ( χ 2 2) 4) . A further identification × χ 0 – 7 – 2 i 2) 2 , SU(2) ]. The computation of the vector masses yields − × × − χ 3( ( ( × SU(2) /V i 3 2 49 2 2 i ( 3) χ = 0 solutions that preserves SU(2) and is located at i = 1 supersymmetric AdS = 3 − [ 2 i χ ( → χ χ χ × ( 0 − U(1) 4 χ + → ( and 3 2 3 2 3 2 N 2 N , 3 2 V 1 4 3 2 9 2 × = √ + + + 2) i = 0( 2 4 3 4 3 4 3 3 + 6 3 + × 2 L + ! L − − − − − ]. 5 3 , 2 3 2 , √ = 6( 32 m 1 i and 2 χ 3 + L − 2 χ 3 SO(6). This SO(6) symmetric solution was originally studied in [ , 2 m , + c 1 ∼ 2 χ = χ − 3 , + 2

, 1 c 1 z χ = 0 vacua with SU(2) = 1 vacua with U(1) = being arbitrary (real) parameters. This family of solutions has a vacuum energy 3 3 ≡ , , 2 N symmetry N 2 , , χ 1 1 and is located at z χ 2 2.3.2 There is a two-parameterU(1) family of implies a symmetry enhancement tosymmetry SU(3) to SU(4) from a ten-dimensional perspectiveIIB and, S-fold backgrounds more in recently, [ connected with a family of type Note that a genericare solution generically in massless. this Therefore, family outcorrespond preserves to of physical an the directions SU(2) 28 in the massless symmetrywhen scalar scalars as potential. imposing in three An a ( additional vectors pairwise U(1) symmetry identification between enhancement the to free SU(2) axions Freedman bound for stability in AdS where is perturbatively unstable due to the mass eigenvalue given by and a spectrum of normalised scalar masses of the form There is a three-parameter family of with 2.3.1 JHEP04(2020)050 ]. U(1) 32 (2.28) (2.27) (2.24) (2.26) (2.23) (2.25) × . ) i (1 + 2 , 1 , 3 , √ , 3 3 2 , , = , 2 so that a total of four 2 , 7 , solutions that preserves 3 z , = 1 4 2 , = = 1 i 1 = 1 i 5 χ i z ), only 2 of them correspond to , as only two vectors are generically 2) 2) 2 2) , and 2.25 2) × × , ( . 3 i < j , × 1 × 2 i , , 2( ( i ( , − 2 i χ 2 2 i 3 1 c − U(1) symmetry , , χ , = = 0 χ 2 − 2 g c 6 i < j . 3 × , , 1) 2 z 5 = 1 2) χ g × 2) i √ 3 × = + = 1 ( 16 + 45 162 2( × 4 + 45 − 4 2 25 2 – 8 – i 64 + 45 z ) q , 6( χ 2) q − j = SU(2) 1 6 q 1 3 √ , × χ = 2 supersymmetric AdS 1) 0 + 1 6 ( 2) = V ic 2 1 × − → 0 × ) N 2 i χ 2) i ( 2 i 4 V j 6( = 2 2 i χ 2 i χ χ × χ 2 ( 5 4 χ ( 4 , χ , z 2 i 5 4 5 radius. The computation of the vector masses yields 5 4 − + 5 + χ 2) i , 4 + + 4 5 28) χ × 9 9 ( 14 14 2 + × 9 9 + 2 4 16 25 5 1 − − − 9 7 √ = 0( i = 0( 2 + 2 L solutions has a vacuum energy given by solutions has a vacuum energy given by 2 L is the AdS χ 4 4 2 m 0 − m /V c 3 − = = 2 vacua with U(1) 3 = ¯ z 2 N − L and is located at = 0 so that a total of eight vectors become massless. The SU(3) symmetric solution = 2 3 , 1 2 z , 1 This family of AdS 2.3.3 There is a one-parameterU(1) family of massless. Therefore, out ofphysical the directions 28 in massless the scalars potential. inwhen The ( imposing residual symmetry a gets pairwisevectors enhanced identification become to SU(2) between massless. the Finallyχ there axions is a symmetry enhancementwas to recently SU(3) uplifted when to setting a ten-dimensional family of type IIB S-fold backgrounds in [ Note that a generic solution in this family preserves U(1) where and a spectrum of normalised scalar masses of the form This family of AdS subject to the constraint JHEP04(2020)050 4 (3.1) (2.31) (2.32) (2.33) (2.34) (2.29) (2.30) at the AdS c . ) 6) ∝ i × 3 , ( 2 2 , , 1 (1 + χ z 2 , 11) 1 , ) it becomes clear that . √ × 4) ], and then uplifted to a 2 + 4 8) 8) = 2( × , 28 − × 2.31 × 7 − 2( ]. ¯ z , 15) − solution. Therefore, out of the ), only 4 of them correspond to 27 + 2( + 2( × , , 6) 4 as only two vectors are generically 2 = 2 2( × 2 χ 2) 6 χ 2.29 ) and ( 10) z 2( × p p × , , = 4( , 2.27 1 4( 5 , 7) − 2 z 2 4) c × χ ), ( 2 , × = , g = 1 6( 3 , χ 2) 4 2( c 1) z 2 + 2.22 − × − – 9 – 2) × solution that preserves SO(4) and is located at = , 6( = × 4 g , ), ( , 0 6) 10( V 2) × 2) , + 1( × 2.18 and × , 2 2) ( χ 2 = 0( × 17( χ radius. The computation of the vector masses yields radius. The computation of the vector masses yields 48) 2 16 4 √ 4 4 L × p ic 2 ), only 26 of them correspond to physical directions in the = 0( 3 2 m = 4 solution was first reported in [ = = 0 and two additional vectors become massless. This special = 0( L 2 3 2.33 2 , χ χ = 2 supersymmetry z 2 N m L = 2 30) N is the AdS is the AdS 2 m × 1 + 4 z 0 0 − = /V /V = 4 supersymmetric AdS 1 3 3 = 0( z 2 − − N U(1) when . = 4 vacuum with SO(4) symmetry solution has a vacuum energy given by L = = 2 amounts to a rescaling of the vacuum expectation values of 3 × 4 2 2 N c m L L vacuum will be uplifted to a ten-dimensional family of type IIB S-fold backgrounds 4 From this moment on we will set without loss of generality.varying From ( 48 massless scalarsscalar in potential. ( This ten-dimensional family of type IIB S-fold backgrounds in [ 3 S-folds with where thus reflecting the SO(4) residual symmetry at the AdS as for the previous solution, and a spectrum of normalised scalar masses of the form There is an This AdS physical directions in the scalarto potential. SU(2) However, the residualAdS symmetry gets enhanced in section 2.3.4 Note that a generic solution in thismassless. family Therefore, preserves U(1) out of the 30 massless scalars in ( where and a spectrum of normalised scalar masses of the form JHEP04(2020)050 ) ], is 56 Y 50 ( with (3.2) (3.3) . Let K 0 5 M S 2.3.3 /V U 3 × -EFT [ irreducible 1 − S vacua and, in 7(7) = × 912 4 2 ) and the scalar 4 L x ( µν g , ) via a generalised S-S K radius . 2.2 4 ) (vector and tensor fields vacuum to a class of ten- , ) MN x 4 ( M -EFT reduces to the external x, Y ρX ( ρϑ KL 1 7 are genuinely dyonic as they do 7(7) M to a ten-dimensional background ) MN = = 2 2.3 Y , N 3) and a 56-dimensional generalised ( M 2.3 ) 912 L M

x ) N ( ]. Here we are interested in the uplift of = 2 family of solutions in section 1 simply corresponds to a rescaling of the K 56) in the fundamental representation U ,..., − µν Q ) denotes projection onto the 51 g N g U U ) Y = 0, there is an enhancement of symmetry to . ( ( P 1 = 0 ρ Y ,..., 912 ∂ χ | ( K N 2 – 10 – Q -covariant generalised diffeomorphisms. In order µ ∂ M U(1) symmetric AdS ( − ), a runaway behaviour towards the boundary of = 1 N 1 ρ U ) − µ × 7(7) 1 vacua in section 2.9 ρ x M -EFT − 3 ( 4 ) = ) = U coordinates is then encoded in a twist matrix ( − M 7(7) P M N Y x, Y x, Y M ( ( Y M ) ) 1 µν 1 ], and implementing a generalised S-S ansatz in E − symmetry. It is a one-parameter family of AdS and thus to a redefinition of the AdS g MN ) satisfying − 0 to implement a purely electric gauging. In this limit one has = 2&SU(2) U 1 2 27 . Y U ( − M ( ( → c where the embedding tensor lives. N ρ 2 N c g ∂ ] as the natural framework to carry out this mission. → ∞ 7(7) ∝ vacuum amongst those in section 0 or, by virtue of ( 50 3 , 0 4 2 , has been set to unity, varying V ) and the internal generalised metric 1 → -EFT lives in an extended space-time that consists of an external four- is the embedding tensor specifying the gauging in the four-dimensional super- c ϕ ) of the four-dimensional maximal supergravity in ( ) 3 x , is a constant scaling tensor and K ( x, Y 2 7(7) , ( ] 1 vacuum of a four-dimensional gauged maximal supergravity, which thus selects M -EFT of [ z MN µν U(1). Following [ ϑ MN 51 4 , subject to the action of the E g X × M 7(7) The E Going back to the goal of this section, the 7(7) where gravity, representation of E The entire dependence on the and a scaling function are consistently set tofields zero). These areansatz connected [ with the metric internal space with coordinates of E to uplift an AdS of type IIB supergravity,metric the relevant field content of E IIB supergravity on spheresan and hyperboloids AdS [ the E dimensional space with coordinates an S-duality hyperbolic monodromy along S 3.1 Type IIBGeneralised uplift Scherk-Schwarz (S-S) using reductions E of exceptionala field theory very (EFT) efficient have method proved to perform consistent truncations of eleven-dimensional and type new and preserves a U(1) the special case of theSU(2) parameter vanishing we will uplift suchdimensional an S-fold backgrounds of type IIB supergravity of the form AdS vacuum energy us emphasise again that allnot the survive AdS the limit that Im( moduli space vacua. After JHEP04(2020)050 - ) is 7(7) (3.8) (3.9) (3.4) (3.5) (3.6) (3.7) (3.10) (3.11) 3.3 )–( ) reads 3.2 , 3.3 in ( . , )–( mβ N : five of them are      ! k ] ) M 3.2 M 2 ) . M 1 y AB Y [ ] − ˜ y kα , one has in ( AB 2 , ∈ U 0 0 m 2 Y − ρ , CD [ y ) (1 + ˜ ] ρ 7(7) 4 ˜ ∈ y U M D E , ] − i ρ 18 B . + vanishes identically. The gen- = y ) which is given in this case by ⊂ ) , i ] Y i , 1 0 y = 1 + ˜ y 4 M − ( 2 2 ˆ = ρ | CD 4 ϑ nβ ˆ [ lmn ˆ ρ , U ] ~y p K ρ mαnβ y p ( ) j y C y [ AB M ) involving generalised twist parame- − | (˜ M i [ M ) the non-vanishing components of the A ) ρ ˆ [ , )˚ K 1 αβ kp ) ; 1 mn i 3.2 ˆ j 1 − 2.1 Ky 0 0 ˚ y 1 2 G y mn ( − . Considering the electric-magnetic splitting 1 2 2 , U M mp + ρ y ρ 2 U ) under SL(8) G , M G and ˚ ] and ˜ ij 1 1 2 1 2 2 1 δ G – 11 – ) the scaling function AB ) = ˆ ˜ 1 6 y CD − G G ˜ ˜ y [ = 2( y , 1 ] 2 ,Y , i ] 10 0 0 ˆ 0 ˚ i = = = = F − y ) and the tensor SL(8) in ( 2 y AB 0 ( ρ ˚ [ β AB α ( CD 2 ] [ ) − | αβ ρ ] mn ⊂ 2.5 1 Y      ~y AB mn m − mn G = 2 1 AB Y B [ B U − | = ( ˆ ) ( K ] α ∈ 1 ˆ ρ ρ ˚ l ) are given by [

M − 7 27 i k = 1 Y U B = 3.5 Y ( 4 = ) also depends on a function ˆ ρ αβ N = B i M 3.9 6) and one is magnetic ˜ 3 2 A ) y ) 1 1 − − − U ,..., U ( ( ), one arrives at the final uplift formulae klmn ], together with the S-S ansatz ( = 2 C ). On the other hand, the generalised twist matrix ( i 3.9 i ( y 53 i ( , )–( y 52 ≡ 3.5 ~y Using the dictionary between the fields of type IIB supergravity and those of E For the dyonic gauging of G EFT [ ters ( The twist matrix ina ( hypergeometric function [ and with components and SL(8)-valued and possesses a block diagonal structure In terms of the physical coordinates ( where the two factors in ( embedding tensor were giveneralised in S-S ( ansatz dependselectric on six physicalof generalised coordinates coordinates ( JHEP04(2020)050 ) 6 6, R 3.11 (3.16) (3.13) (3.14) (3.15) (3.12) on , two- ,..., 7 18 m y mn Y Y = 2 G m , the metric y i 5 U(1) symme- 7 i i y × Y 2 , . − j and axion-dilaton GL(5) compatible ) y i 1 7) and = 1 4 7 × , ) by performing the y β ˜ C i, n y ]) for more details on α 6’ Y , − 7 Y i y αβ ij m 32 = ( + ( , δ Y mα x, y GL(1) 1 m entering the r.h.s. of ( = 2&SU(2) ( m y 27 ] to describe the geometry of = − y mn ) . → δ ij + 32 N 2 + iα ] iα MN , ˆ R n G y 6) to parameterise S Y 6 mαnβ Y M mα 2 × m y ]. ), with i [ + ( M ˜ y δ 7 32 IIB + ,..., 0 ij , i ij ) = Y y y 1 with ] (and also [ ] and , = 2 n mnp i and y Y SL(2) = ( 53 ( 20 0 , vacuum with 2, four-form potential k m i lmn [ + 0 × , m p y j mnp 4 52 + ( Y y y Y o j – 12 – 0 mα 1 2 M ∂ k ijk +1 0 = 1 y y , ij ) n 2 | ˆ 2 y + l GL(6) α G -EFT and their connection with the gauged maximal nβ , ~y ijkj y p m k ≡ mn 6’ y 7(7) y − | δ i M are constructed as jl 1 m , δ + ( 7 5 y mn β ) with ik We adopt the conventions of [ to a ten-dimensional background of type IIB supergravity α K 2 δ mn ≡ − +2 Y 7 ) y 7 1 1 ,C αβ M Y , 2 mα , + 6 i B 2.3.3 y . Using coordinates ( y 5 ij factorisation of the geometry we are behind of. The various mappings δ n ⊃ → → 5 iα = ( iα = S y = Y α × ij M m B 7(7) ) ˆ 56 G 1 Y Y E 7 18 y Y 7) of the form ). We have explicitly verified that the ten-dimensional equations of motion and (or S m -EFT: y R 7 i i ,..., 3.11 y 7(7) Y . The various blocks We now move to the uplift of the AdS = 2 We adopt the type IIB conventions in the appendix B of [ 2 αβ m so that the Killing vectors on S However it will also be( convenient to introduce a set of embedding coordinates Ten-dimensional metric. the round five-sphere S and its inverse are given by supergravities. try discussed in section using ( Bianchi identities of type IIB supergravity are satisfied. We refer the readerthe to generalised the S-S reductions original of works E [ The physical coordinates arewhich identified implies as a furtherwith group-theoretical the branching GL(6) between coordinates discussed above are summarised as form potentials m can be extracted fromgroup-theoretical decomposition the that is internal relevant for generalisedinto the metric embedding E of type IIB supergravity for the purely internal components of the type IIB fields: (inverse) metric JHEP04(2020)050 (3.21) (3.23) (3.18) (3.20) (3.22) (3.17) (3.19) . vacuum, and . ] , 4 π          2 , 2 2 γ [0 Y 2 + 6 7 6 6 ∈ + α Y Y Y Y 2 5 2 2 4 5 = 2 AdS , Y Y . , , γ N −Y −Y −Y + 1 4 ) ] ij sin 2 , − 4 L ∞ ) , π Y , 2 , 5 2 β + [0 ) 5 5 5 6 2 5 Y ij βdγ θ, Y Y Y Y ∈ −∞ 2 3 4 5 ˆ + G ( βdγ , − Y 2 4 sin cos sin ) ∈ 1 −Y −Y −Y −Y Y 2 = 0 in the θ φ α sin y , β η 4 4 5 6 2 4 χ ] , α , Y Y Y Y = ∆ ) π 1 + 2 3 4 4 2 − Y , = cos = sin + cos mn 2 = 0 βdγ 1 −Y −Y −Y −Y [0 1 3 5 + sin kl √ 2 7 18 ∈ of the form – 13 – M αdβ Y ij = with = 2∆(1 + ˜ j , 5 , y αdβ 7 4 5 7 2 + cos + M sin ρ mn Y Y Y Y k 2 5 1 2 , α 6 3 3 2 γ γ 1818 K − dα ij ] ) ], the internal part of the ten-dimensional metric has Y i Y ( (cos ( 2 + π 2 − K M η G kl 1 2 1 2 1 2 −Y −Y −Y 27 2 + 2 4 α , α ρ K ρ 2 4 = = = [0 ) is nowhere vanishing and reads Y 1 2 3 ∈ 2 σ σ σ 6 = ∆˚ = ∆ = ∆˚ = sinh cos cos Y 3.17 k ij ˜ y 11 1 6 7 4 5 ∆ = (det + G 2 G β G 2 Y Y Y Y β , φ 2 5 2 6 2 2 θ, y i Y ) given by 2 π −Y −Y −Y −Y + , sin cos sin 2 0 4 θ φ θ 3.11 h Y ∈ = 0 as a consequence of having set          θ 18 = = cos = cos = cos ij 4 6 2 ij y y y L M The six-dimensional internal metric becomes more transparent if first introducing a Following the derivation of [ with ranges given by In this manner, and upon introducing a set of SU(2) left-invariant one-forms and then a set of angular variables for S The warping factor ∆ in ( new variable for the magnetic coordinate where where we have defined components in ( JHEP04(2020)050 U(1) (3.28) (3.29) (3.25) (3.26) (3.30) (3.24) (3.27) , so the c × = 0. ∝ χ ) is 1) , 3.27 1 . . ) with , ) transform as a 1 ! 2 3 − 2.27 = 2&SU(2) η one finds ∆ σ η 3.11 c acts as a rotation on N + = ( . 2 sinh 1 σ ij cosh 3 σ , η ) in ( − 2 S γ 2 , 4 η m η vacuum in ( θds ,C 3 kl 2 2 S 2 4 B M sinh form + 8∆ β cosh θds , 2 2 5 − γ 2 ) = ( σ 1 4 S + cos 1

metrics to describe the deformation ) on the parameter part of the geometry, one obtains a frames for the metric ( α of type IIB supergravity. An explicit 2 − ) 3) and × = 2 S 3 B 4 , θ ≡ A 3 2 R 3.25 ( 2 S + cos ds IIB 3 β , ) γ 2 σ αβ ds ) + 2 S R 1 , 2 = = 1 m − ds 2 cos(2 i 9 symmetry, where U(1) dη ( Y A ˆ e + − j ( σ – 14 – twist matrix ∂ 2 + 6 undeformed k i 4 dη kl , θdφ IIB radius at the four-dimensional , = U(1) ) dx and K 2 1 2 AdS 4 ! 1 r R L σ ik − × , 6= 0, as for its associated AdS 3 and (squashed) S − η η ds ] shows that G c φ ∆ , = = sin = ) becomes pathological. In other words, the ten-dimensional solution 2 ∆ 1 1 2 27 i 6 8 ˆ − e ˆ ˆ e e SL(2 sinh cosh 2 3.27 = = 0 = ∆ U(1) ∆ ) and the external AdS 2 6 η ⊂ η 1 2 The two-form potentials α α × j θdφ ds 1 ij 1) = 2 , 3.24 , B B dr, sinh 2 cosh 1 r L dη dθ, σ ds

+ sin = = = = ≡ 2 = 1 being the AdS 0 4 5 7 . These metrics are explicitly given by vacuum. β 0 of the metric ( 5 e e e e 0 dθ 4 α → /V A = potentials. 3 c 2 : ˆ :: ˆ : ˆ ˆ 2 − 2 S )-plane. Finally, our choice of 4 2 3 3 R 2 C 2 S 2 S ds = , σ ds 2 ds ds 2 AdS 1 L σ Bringing together ( ds and Restoring the explicit dependence of the warping factor ( 3 2 (electric) limit is genuinely dyonic, namely, it requires in terms of a local SO(1 symmetric AdS B doublet under the global S-dualitycomputation group along SL(2 the lines of [ with This metric has anthe SU(2) ( ten-dimensional metric of the form and where we haveof introduced the S internal S with a warping factor the internal six-dimensional metric takes a simple JHEP04(2020)050 due 5 . Us- (3.36) (3.33) (3.34) (3.35) (3.32) (3.31) e , F η . , 3 factor due σ 3 3 σ ∧ φ σ 2 ∧ σ ∧ 1 ∧ 1 σ can be explicitly 1 σ ) σ 4 ) θ , ) θ ∧ . C , β ) θ δ 3 θ ) ! σ 1 ) is also broken by dφ − φ ) ∧ 2 cos(2 φ φ 2 cos(2 2 A φ 2 4 sin(2 ( − σ 4 sin(2 − sin 6 γδ ∧ sin(2 6 θ φ but break the U(1) m 1 θ φ 2 γ )sin(2 σ σ 2 α θ ) ∧ t sin + sin − 1 2 U(1) + cos 2 dφ A sin(2 , 2 σ can be obtained from the last equation − vacuum under consideration, and using ∧ × 1 2 β σ ) 1 + sin ∧ b = ( 4 φ − ∧ φ αβ β . 2 . α θdθ ! φ ) m ). Computing the associated (purely internal) dθ φ 3 ) ) A τ φ cos ) of the form φ φ sin(2 1 – 15 – = θ Re cos 3.11 θ 2 (sin θ α − 2 = 2 AdS 3.30 d (cos B ) cos(2 τ d θ 2 sin sin ) | N 2 θ 1 2 Re τ / ∧ | 1 + sin 3 − − ) preserve SU(2) sin(2

θdη 2∆ sin(2 1 2 2 3 τ -twist in ( 5 at the 3.32 1 2 √ 1 + M A γ cos Im = + β m θ 5 vol = 2∆ = kl φdθ H 2 M / φdθ sin ∧ 5 γδ M 4 αβ α and sin vol on the m B The axion-dilaton matrix m cos 2∆ Φ 4∆ The internal component of the four-form potential αβ − √ αβ θ − 6 ie m 1 2 θ ) + cos ? − 0 4 2 cos 1 C 2 √ dC 1 ). Transforming linearly under S-duality, a direct computation shows an explicit = √ − = = (1+ τ 5 = = potential. 3.11 e F 1 2 4 b b with in ( dependence of denotes the volume of theto deformed its five-sphere. explicit Note dependence that on U(1) the coordinate Axion-dilaton. where obtained from the third uplift formula infive-form ( field strength, and imposing ten-dimensional self-duality, one gets The two-form potentials in ( to the explicit dependence on the coordinate C with ing the scalar block differential form notation, one finds This matrix encodes the dependence of the two-form potentials on the direction JHEP04(2020)050 ] . ] 27 27 IIB ) (3.38) (3.40) (3.39) (3.37) R , -family k direction of ] within the ). This twist , ] S-folds with , R 36 , , namely, when      36 ! 3 . This concludes 3.30 , T 1 4 φ ≥ 32 1 0 1 1 4) ] and [ k 2 0

− , 35 ) to a discrete √ – 2 along the = 1 [ hyperbolic monodromies = k ! ( η 33 -twist in ( ). These monodromies can 3.30 , then the ten-dimensional N , T T T = 2 supersymmetry. IIB β A 1 4 1 4 ) U(1) symmetry discussed in α Z N with period 4) δ 4) 3.39 , × 2 IIB − sinh cosh ) on the angle k ) 2 ] and T = − √ k 2. Therefore, as discussed in [ R and T T 2 ! 27 β 2( , + k 1 α 3.36 > -twist in ( √ ( η A 1 0 ) A sinh cosh      k = 4 [ k − ( ! →

N η M ) = − ), i.e. k = ) = ( 0 1 0 -family of SL(2 g T k )

= 2 and SU(2) – 16 – 3.30 k ( + = T η N ( S + A ) η η log(2) and Tr ( ) 1 k ) being in the hyperbolic conjugacy class of SL(2 -twist in ( ( − − A A A ) 4) 3.37 η with = ( − ) 1 1 k 2 ) with − ( S vacuum with -family of S-duality monodromies ( k k k A ( 4 M √ ) gets generalised to a k coordinate to be periodic Ag -family of S-fold backgrounds. Moreover, the argument wielded in [ + ) = ) is totally encoded in the local SL(2 η k in the geometry. Generalising the k k = 3.37 3) of new ones ( 1 ) −ST S k 3.27 ≥ M ( is broken by the explicit dependence of ( k A to a ten-dimensional background of type IIB supergravity. It is worth empha- → ]), these backgrounds can be interpreted as locally geometric compactifications ) = φ involving a k R ( 32 5 ) = log( S k with M ( 2.3.3 × T N 1 Lastly, various holographic aspects of both ∈ k for the straightforward upliftrelied of on the four-dimensional the supersymmetries monodromyThis to is ( still ten our dimensions case, so the S-folds presentedhyperbolic here monodromies preserve have respectively been investigated in [ be written as and thus define a with (see also [ on S the monodromy ( ( when forcing the replacing 3.2 S-fold interpretation The dependence of thethe full geometry type ( IIB solutionmatrix on is the of coordinate hyperbolic type and thus induces a non-trivial monodromy the uplift ofsection the AdS sising that, if trivialisingequations the of motion of type IIB supergravity are no longer satisfied. Again U(1) JHEP04(2020)050 ∞ to ], and (3.41) (3.44) (3.42) (3.43) 38 . There- η . −∞ 2 3 − . Reading off σ ) presented in ∧ Φ IIB 2 ) , σ 3.39 → R , ∧ , )) theories [ 1 2 σ 3 N ) σ ∧ φ + (U( ) on the original solution 2 1 σ T dφ ) sin(2 4 3.41 φ θ 2 . = 4 ) φ +8∆ N 2 2 sin 2 )sin(2 . θ σ , transformation, equivalently a change − ! η sin(2 ) θ 2 1 θ 3 2 θ θ − IIB sin(2 σ − 2 2 ) e U(1) symmetry we just obtained can be ), namely, 1 2 ∧ 1 1 1 R 2 ∝ , × − cos(2

+cos σ transformation ( ) sin 2 sin 2 Φ that interpolates between the singular values 0 3.33 2 ∧ ) yields a shift of the form Φ φ − e dθ – 17 – 1 1 − ) s √ 5 SL(2 σ IIB g ) φ = θdφ ) θ ) transforms linearly under SL(2 2 ∧ 3.36 2 s ∈ R cos(2 g ∆ , = 1 SCFT’s. It would be interesting to extend these Λ = ) and ( dφ 1 2 cos(2 3.35 ∧ cos(2 +sin = 2&SU(2) N ) on ( 2 SL(2 3.27 dθ η ∧ − in ( ( 3 N dθ 5 1 = 2 S-folds with hyperbolic monodromies ( ∈ dη ) η + − + log αβ 2 ? 2 − N A η e singlets and are not affected by the transformation. Therefore, one finds m 2 2 dη τ − − (1+ + IIB θ 4 ) = 3 R 0 , a new type IIB background is generated. The metric and self-dual five- , Φ = 2 AdS C cos θ ds 3.1 1 sin − 4 ∆ upon performing a global Λ 1 2 η . = = 4∆ 2 5 ∞ e Upon performing the Λ F ds The axion-dilaton matrix the new components of and found in section form flux are SL(2 they take the same form as in ( fore, a degenerate Janus-likebecomes behaviour manifest with a linear dilaton Φ running from giving rise to a varying string coupling The composed action of Λ The type IIBmapped solution to with a newdinate (but equivalent) solutionof with duality a frame, based linear on dilaton the profile matrix element along the coor- their potential generalisation to holographic studies to the this work. 3.3 Connection with Janus-like solutions context of three-dimensional quiver theories involving JHEP04(2020)050 ]. 39 ] and (3.45) (3.46) 32 doublet whereas 2 one has an IIB C ) η R , 2 frame in which and σ 2 0 σ ∧ IIB C 0. On the contrary, ∧ ) = 1&SU(3) [ ] for the dyonically- = 2 supersymmetric R dφ , → N N 32 ) dφ , 2 φ ) ], C , φ 31 31 ) diverges (strong coupling) , , sin 2 2 28 − ψσ , . 3.42 + sin φ ψσ 0 and can be eliminated by means of a gauge φ cos 3 3 in ( 3 θ sin → σ σ . σ θ s 0 ) U(1). By implementing a generalised ) transform as an SL(2 ∧ ∧ g )(cos ∧ θ C θ 1 1 )(cos 1 × ) cos = 0&SO(6) [ θ vacua in [ θ σ σ σ ) cos ) ) 3.32 θ 4 φ φ N sin(2 sin(2 )–( 2 cos(2 sin(2 1 2 η sin(2 sin − η 1 2 − e e + + sin − 3.31 4 4 – 18 – ] S-folds, there is no SL(2 − φ φ one has that dθ 2∆ 2∆ = 6 . However, since these terms are generated by the generalised 32 ) dθ 4 π √ √ 4 φ ) ) in ( 2 2 2 − + φ )(cos )(cos d d ψ -angle) vanishes identically or becomes independent of ∆ , the solution becomes dominated by θ θ 0. At intermediate values of the coordinate = 1 [ ,C θ − + → −∞ sin 2 → + sin 2 maximal supergravity and found multi-parametric families 2 → η B N φ − B C φ 2 sin(2 sin(2 12 → ∞ , we will retain them here. φ which are proportional to B θ θ → → = ( η 2 2 2 α 3.1 (cos nR C C B vacua are shown to correspond to the points of largest symmetry ] and cos cos (cos B θ 4 4 4 27 4 θ and cos 2 cos SO(6)] B 1 2 + 2∆ + 2∆ ] AdS = 4 [ 1 2 × . η 28 η η − N 1) (and thus the dual e e vacua. Within one such families, all the solutions preserve the same amount , 0 4 = = C 2 2 C dominates over other gauge potentials, e.g., B 2 vacua and, more concretely, on the vacuum within this family featuring the largest 0 (weak coupling) and B In the second part of the paper we focused on the new family of In the asymptotic region at 4 The two terms in = 4&SO(4) [ → 4 s transformation of the form where we have shifted the coordinate S-S ansatz discussed in section symmetry breaking patterns of three-dimensional interface SYM theories presented inAdS [ possible residual symmetry, which turns to be SU(2) of new AdS of supersymmetry but, importantly,values residual of symmetry the parameters. enhancements occur TheN previously at known particular enhancement within their respective families. This is in line with the analysis of (global) the coordinate 4 Conclusions In this work we havegauged extended [SO(1 the study of AdS in the asymptotic region at g interpolating behaviour between these twounlike regimes. for Finally, the it isthe also axion worth noticing that, and The nowhere vanishing warping factor still reads and take the new form The two-form potentials JHEP04(2020)050 5 4 5 3) S S and ≥ × × , 2 ]. ] with ]. 1 5 k hyper- ]. S 36 – 55 × IIB ) 4 SPIRE 33 SPIRE Z (with IN , IN k [ ][ gauged U(1) symmetric Supergravity × −ST = 8 SO(8) N ) = vacua presented in this (1982) 323 k 4 ( ]. -fold CFT’s of [ M J ]. B 208 arXiv:1209.0760 SPIRE [ = 2 supersymmetry of the AdS = 2 and SU(2) IN SPIRE Supergravity and the Magic of Machine N N IN ][ is deformed into a product of S ][ 5 invariant critical points of maximal ] display SU(2) isometries in the internal Nucl. Phys. vacuum to a new family of AdS , 2 Evidence for a family of 32 4 G isometries and a warping factor, whereas the (2012) 201301 φ – 19 – All 109 ]. U(1) arXiv:1209.3003 ] that uplifts the factor explicitly by introducing a dependence on the ]), especially due to the non-trivial SL(2 [ Supergravity × 54 arXiv:1906.00207 . It would also be interesting to investigate holographic φ 32 SPIRE σ k [ U(1) symmetry and IN ), which permits any use, distribution and reproduction in = 8 [ × = 1 S-folds (in the spirit of the N U(1) −ST N × (2012) 108 Phys. Rev. Lett. ) = (2019) 057 , 12 k ( (1983) 169 CC-BY 4.0 08 M -EFT, we uplifted the AdS = 2 and This article is distributed under the terms of the Creative Commons Some New Extrema of the Scalar Potential of Gauged JHEP N , 7(7) 128B with SU(2) JHEP 3 , . In many aspects, the realisation of symmetries is much alike the AdS φ ), as well as to study holographic RG flows by explicitly constructing domain- k . The residual SU(2) 1 vacuum of the five-dimensional SO(6) maximal supergravity presented in [ −ST Phys. Lett. Learning supergravity supergravity theories Finally it would be interesting to investigate the brane setups underlying the families 5 I.M. Comsa, M. Firsching and T. Fischbacher, SO(8) A. Borghese, A. Guarino and D. Roest, G. Dall’Agata, G. Inverso and M. Trigiante, B. de Wit and H. Nicolai, N.P. Warner, = [4] [5] [1] [2] [3] Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. References We are gratefulSamtleben to for interesting Nikolay conversations.government Bobev, grant PGC2018-096894-B-100 The Gianluca and work bygrant Inverso, the of FC-GRUPIN-IDI/2018/000174. Principado Yolanda AG de The Asturias is Lozano research(convention through supported 4.4503.15). of the and CS by CS Henning is is the a supported Spanish Research by Fellow of IISN-Belgium the F.R.S.-FNRS (Belgium). new analytic type IIAfuture. backgrounds. We plan to address these andAcknowledgments related issues in the aspects of such J wall solutions interpolating betweenwork. the various Lastly, families since ofgeometry, the AdS it S-folds would also here be and interesting in to [ apply non-abelian T-duality in order to generate coordinate background by Pilch and WarnerAdS [ of S-folds presented here (andbolic in monodromies [ S-folds of type IIB supergravityalong with hyperbolic S monodromies vacuum are realised on(squashed) the S S-folds: thebackground internal S fluxes break the U(1) S-S ansatz in E JHEP04(2020)050 , 08 01 Phys. 04 12 ]. (2019) JHEP , , black , Gauged 4 Class. 11 JHEP JHEP , , (1986) 374 -Deformed (2014) 026 ]. , JHEP SPIRE ω JHEP 3 (2017) 141 , , IN 02 6 SO(8) ][ JHEP 09 -invariant sector 169B , supergravity SPIRE /CFT 4 IN JHEP ][ ]. = 8 SU(3) , JHEP , N (2017) 100 6 The ]. ]. ]. SPIRE 08 Phys. Lett. supergravity (2015) 276 ]. IN , ][ (7) SPIRE supergravity and supergravity in SPIRE SPIRE arXiv:1707.06884 JHEP IN Chern-Simons-matter theory IN IN SPIRE [ , B 746 ][ IN ][ ][ ]. arXiv:1211.5335 supergravity and the duality hierarchy ][ ]. = 3 SO(8) New Gaugings and Non-Geometry [ -defomed gauged N ω (7) SPIRE truncation of massive IIA on S mathrmISO SPIRE (2017) 190 IN ]. IN Symplectic Deformations of Gauged Maximal Phys. Lett. ][ Holographic microstate counting for AdS , Black hole entropy in massive Type IIA arXiv:1302.6219 = 8 10 ][ ]. ]. – 20 – [ Triality, Periodicity and Stability of (2013) 082 ]. ]. N deformation of gauged STU supergravity SPIRE Halving Flowing to Romans-mass-driven flows on the D2-brane 03 IN ω arXiv:1405.2437 arXiv:1302.6057 mathrmISO arXiv:1506.03457 SPIRE SPIRE JHEP ][ [ [ ]. SPIRE SPIRE Black holes in [ IN IN , arXiv:1707.06886 An IN IN [ ][ ][ ]. ]. ]. ][ ][ JHEP (2013) 077 ]. , SPIRE Dyonic Consistent BPS black holes from massive IIA on S Deformations of gauged 05 IN Electric/magnetic duality and RG flows in AdS arXiv:1311.7459 ][ (2014) 133 (2013) 107 SPIRE SPIRE SPIRE [ SPIRE arXiv:1508.04432 IN IN IN [ IN Thermodynamics of Static Dyonic AdS Black Holes in the 07 05 ][ ][ ][ JHEP ][ (2017) 1700049 , (2018) 035004 arXiv:1910.06866 65 Massive N=2a Supergravity in Ten-Dimensions 35 [ JHEP JHEP (2014) 137 BPS black hole horizons from massive IIA On new maximal supergravity and its BPS domain-walls , , arXiv:1605.09254 arXiv:1509.02526 arXiv:1311.2933 arXiv:1402.1994 [ [ [ [ (2016) 079 ]. 02 B 732 arXiv:1907.11681 [ (2020) 100 SPIRE IN arXiv:1703.10833 arXiv:1706.01823 arXiv:1505.00117 arXiv:1311.0785 holes in massive IIA supergravity Quant. Grav. [ [ [ 143 03 (2016) 168 Supergravity JHEP eleven dimensions Fortsch. Phys. (2015) 020 (2014) 175 Kaluza-Klein Gauged Supergravity Theory [ (2014) 071 Lett. of new maximal supergravity [ Supergravity F. Benini, H. Khachatryan and P. Milan, L.J. Romans, A. Guarino and J. Tarr´ıo, A. Guarino, S.M. Hosseini, K. Hristov and A. Passias, A. Guarino, J. Tarr´ıoand O. Varela, A. Guarino, J. Tarr´ıoand O. Varela, G. Dall’Agata, G. Inverso and A. Marrani, A. Guarino and O. Varela, A. Guarino, J. Tarr´ıoand O. Varela, K. Lee, C. Strickland-Constable and D. Waldram, A. Guarino and O. Varela, S.-Q. Wu and S. Li, B. de Wit and H. Nicolai, A. Anabalon and D. Astefanesei, H. Lü,Y. Pang and C.N. Pope, J. Tarr´ıoand O. Varela, A. Borghese, A. Guarino and D. Roest, A. Guarino, A. Borghese, G. Dibitetto, A. Guarino, D. Roest and O. Varela, [9] [7] [8] [6] [24] [25] [21] [22] [23] [19] [20] [16] [17] [18] [14] [15] [12] [13] [10] [11] JHEP04(2020)050 , ]. , 09 01 ] , , 12 06 ] ] ] SPIRE IN JHEP JHEP ][ , Supergravity , JHEP JHEP , , ]. Class. Quant. (2018) 019 = 8 , N 06 ]. Super Yang-Mills SPIRE arXiv:0804.2907 and its field theory -fold CFTs [ IN 5 S = 4 ][ arXiv:1907.11132 ]. arXiv:1508.05055 arXiv:1504.08009 SPIRE JHEP [ [ [ Janus and J-fold Solutions , hep-th/0407073 supergravities N IN AdS [ ][ ]. SPIRE = 4 (2010) 097 ]. IN D supersymmetric AdS vacua in ]. ]. Supersymmetric Indices of 3d ][ SPIRE 2 Dual of the Janus solution: An 06 Gauged Supergravity in 4 Dimensions (2015) 501 ]. IN > ]. SPIRE ][ (2019) 081901 (2015) 091601 SPIRE SPIRE = 8 N IN (2005) 066003 IN IN arXiv:0807.3720 ][ SPIRE JHEP N [ B 900 ][ ][ , SPIRE IN 115 The moduli spaces of The IN arXiv:1410.0711 ][ Type II supergravity origin of dyonic gaugings D 100 Magnetic charges in local field theory The maximal D 71 [ ][ ]. ]. String Theory Origin of Dyonic – 21 – ]. Interface Yang-Mills, supersymmetry and Janus Ten-dimensional supersymmetric Janus solutions ]. arXiv:1905.07183 (2009) 721 A dilatonic deformation of [ Nucl. Phys. SPIRE SPIRE 13 SPIRE , IN IN arXiv:1612.05123 (2014) 174 Phys. Rev. SPIRE Phys. Rev. IN [ , ][ ][ , On the Vacua of IN hep-th/0603013 hep-th/0603012 arXiv:1112.3345 ]. ][ Phys. Rev. Lett. 12 [ [ [ ][ Holographic duals of 3d S-fold CFTs , S-folds and (non-)supersymmetric Janus solutions hep-th/0304129 S-duality of Boundary Conditions In Janus Configurations, Chern-Simons Couplings, And The [ (2019) 008 arXiv:0808.4076 SPIRE [ JHEP 08 Super Yang-Mills Theory IN , ][ (2006) 16 (2006) 79 (2012) 70 superpotentials (2017) 066020 = 4 c (2003) 072 JHEP N Lectures on Gauged Supergravity and Flux Compactifications , CSO 05 B 753 B 757 B 859 hep-th/0507289 arXiv:0705.2101 arXiv:1907.04177 arXiv:1810.12323 D 95 [ [ [ [ (2008) 214002 ]. ]. ]. ]. Adv. Theor. Math. Phys. , 25 JHEP , SPIRE SPIRE SPIRE SPIRE -fold SCFTs IN IN IN arXiv:1804.06419 IN [ (2007) 049 Grav. theta-Angle in [ (2005) 016 Theory Nucl. Phys. from Sasaki-Einstein Manifolds [ interface conformal field theory [ (2019) 046 S Nucl. Phys. (2019) 113 maximal supergravity dual Nucl. Phys. and Its Chern-Simons Duals [ Phys. Rev. B. de Wit, H. Samtleben and M. Trigiante, H. Samtleben, B. de Wit, H. Samtleben and M. Trigiante, A. Guarino, D. Gaiotto and E. Witten, E. D’Hoker, J. Estes and M. Gutperle, D. Gaiotto and E. Witten, N. Bobev, F.F. Gautason, K. Pilch, M. Suh and J. Van Muiden, A.B. Clark, D.Z. Freedman, A. Karch and M. Schnabl, I. Garozzo, G. Lo Monaco and N. Mekareeya, I. Garozzo, G. Lo Monaco, N. Mekareeya and M. Sacchi, G. Dall’Agata and G. Inverso, A. Guarino and C. Sterckx, B. Assel and A. Tomasiello, D. Bak, M. Gutperle and S. Hirano, E. D’Hoker, J. Estes and M. Gutperle, G. Inverso, H. Samtleben and M. Trigiante, A. Gallerati, H. Samtleben and M. Trigiante, A. Guarino, D.L. Jafferis and O. Varela, [43] [44] [41] [42] [38] [39] [40] [36] [37] [34] [35] [31] [32] [33] [29] [30] [27] [28] [26] JHEP04(2020)050 , , ] ]. (2014) SPIRE ]. vacuum of D 89 IN (2009) 21 ]. 4 ]. ][ ]. SPIRE supergravity in SPIRE IN SPIRE B 814 IN ][ = 1 AdS IN = 8 ][ arXiv:1609.09745 Phys. Rev. SPIRE ][ N [ , N IN [ 7(7) ]. ]. More dual fluxes and moduli fixing Nucl. Phys. (2017) 1 , arXiv:1506.01065 SPIRE IN SPIRE (1982) 197 680 ]. IN ][ Flux algebra, Bianchi identities and ][ Exceptional Field Theory: Review and arXiv:1506.01385 hep-th/9812035 (2015) [ [ [ arXiv:1909.10969 [ 115B SPIRE 6(6) IN E Consistent Type IIB Reductions to Maximal 5D – 22 – Properties of the new New vacua of gauged ][ Positive Energy in anti-de Sitter Backgrounds and Phys. Rept. ]. (2000) 14 , (2020) 099 Phys. Lett. (2015) 065004 arXiv:1410.8145 SPIRE , hep-th/0002192 01 Exceptional field theory. II. E Consistent Kaluza-Klein Truncations via Exceptional Field [ ]. [ B 487 IN A new supersymmetric compactification of chiral IIB supergravity ][ D 92 PoS(CORFU2014)133 , SPIRE JHEP hep-th/0602089 , [ IN (2000) 22 (2015) 131 ][ Phys. Lett. 01 , Gauged Supergravities Phys. Rev. , B 487 (2006) 070 ]. JHEP arXiv:1312.4542 05 , [ SPIRE arXiv:0811.2900 IN five-dimensions Embedding of Type IIB Supergravity Phys. Lett. 066017 Theory Freed-Witten anomalies in F-theory compactifications [ Gauged Extended Supergravity JHEP [ maximal supergravity A. Khavaev, K. Pilch and N.P. Warner, A. Baguet, O. Hohm and H. Samtleben, K. Pilch and N.P. Warner, O. Hohm and H. Samtleben, O. Hohm and H. Samtleben, A. Baguet, O. Hohm and H. Samtleben, G. Aldazabal, P.G. Camara and J.A. Rosabal, P. Breitenlohner and D.Z. Freedman, N. Bobev, T. Fischbacher and K. Pilch, G. Aldazabal, P.G. Camara, A. Font and L.E. Ibáñez, M. Trigiante, [55] [53] [54] [50] [51] [52] [48] [49] [46] [47] [45]