Published for SISSA by Springer Received: February 28, 2020 Accepted: March 25, 2020 Published: April 8, 2020 N = 2 supersymmetric S-folds JHEP04(2020)050 Adolfo Guarino,a;b Colin Sterckxc and Mario Trigianted aDepartamento de F´ısica, Universidad de Oviedo, Avda. Federico Garc´ıaLorca 18, 33007 Oviedo, Spain bInstituto Universitario de Ciencias y Tecnolog´ıasEspaciales de Asturias (ICTEA), Calle de la Independencia 13, 33004 Oviedo, Spain cUniversit´eLibre de Bruxelles (ULB) and International Solvay Institutes, Service de Physique Th´eoriqueet Math´ematique, Campus de la Plaine, CP 231, B-1050, Brussels, Belgium dDipartimento di Fisica, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Turin, Italy and INFN, Sezione di Torino, Italy E-mail: [email protected], [email protected], [email protected] Abstract: Multi-parametric families of AdS4 vacua with various amounts of supersym- metry and residual gauge symmetry are found in the [SO(1; 1) × SO(6)] n R12 maximal supergravity that arises from the reduction of type IIB supergravity on R × S5. These pro- vide natural candidates to holographically describe new strongly coupled three-dimensional CFT's which are localised on interfaces of N = 4 super-Yang-Mills theory. One such AdS4 vacua features a symmetry enhancement to SU(2) × U(1) while preserving N = 2 super- symmetry. Fetching techniques from the E7(7) exceptional field theory, its uplift to a class 1 5 of N = 2 S-folds of type IIB supergravity of the form AdS4 × S × S involving S-duality twists of hyperbolic type along S1 is presented. Keywords: Flux compactifications, String Duality, Supergravity Models, Superstring Vacua ArXiv ePrint: 2002.03692 Open Access, c The Authors. https://doi.org/10.1007/JHEP04(2020)050 Article funded by SCOAP3. Contents 1 Introduction1 12 2 AdS4 vacua of [SO(1; 1) × SO(6)] n R maximal supergravity3 2.1 The N = 8 theory: gauging and scalar potential4 Z3 2.2 2 invariant sector5 2.3 New families of AdS4 vacua6 JHEP04(2020)050 2.3.1 N = 0 vacua with SU(2) ! SU(2) × U(1)2 ! SU(3) × U(1) ! SO(6) symmetry7 2.3.2 N = 1 vacua with U(1)2 ! SU(2) × U(1) ! SU(3) symmetry7 2.3.3 N = 2 vacua with U(1)2 ! SU(2) × U(1) symmetry8 2.3.4 N = 4 vacuum with SO(4) symmetry9 3 S-folds with N = 2 supersymmetry9 3.1 Type IIB uplift using E7(7)-EFT 10 3.2 S-fold interpretation 16 3.3 Connection with Janus-like solutions 17 4 Conclusions 18 1 Introduction Electromagnetic duality in four-dimensional maximal supergravity has provided a very rich phenomenology as far as the existence of new gaugings and vacuum solutions are concerned. The prototypical example is the dyonically-gauged SO(8) supergravity where the action of electromagnetic duality on the gauging generates a one-parameter family of inequivalent p theories parameterised by a continuous parameter c 2 0; 2 − 1 [1]. Setting the param- eter to c = 0 then the standard (electric) SO(8) supergravity of de Wit and Nicolai [2] is recovered which is known to arise upon dimensional reduction of eleven-dimensional 7 supergravity on a seven-sphere S . The various AdS4 vacua of the c = 0 theory [3] (see also [4] for an updated encyclopedic reference) get generalised to one-parameter families of vacua when turning on c and, more importantly, new and genuinely dyonic AdS4 vacua also appear which do not have a well defined (electric) c ! 0 limit [1,5 {7]. Other types of four-dimensional solutions, like domain-walls [8,9 ] or black holes [10{12], have also been investigated using instead a phase-like parameterisation ! = arg(1 + ic) 2 [0; π=8] of the electromagnetic deformation parameter. However, and despite the rich structure of new solutions at c 6= 0, the question about the eleven-dimensional interpretation of the electro- magnetic parameter c remains elusive and various no-go theorems have been stated against the existence of such a higher dimensional origin [13, 14]. Also, for the new supersymmetric { 1 { AdS4 vacua at c 6= 0, the holographic interpretation of the deformation parameter remains obscure from the perspective of the AdS4/CFT3 correspondence. Unlike for the SO(8) theory, much more is by now known about the dyonically-gauged ISO(7) supergravity that arises from the reduction of massive IIA supergravity on a six- sphere S6 [15]. In this case the electromagnetic deformation parameter is a discrete (on/off) deformation, namely, it can be set to c = 0 or 1 without loss of generality [16]. Various AdS4 [17{19], domain-wall [19, 20], and black hole [21{24] solutions have been constructed which necessarily require a non-zero electromagnetic deformation parameter c. Within this massive IIA context, the electromagnetic parameter is identified with the Romans mass JHEP04(2020)050 parameter F^0 of the ten-dimensional theory [25], and has a holographic interpretation as the Chern-Simons level k of a three-dimensional super-Chern-Simons dual theory [26]. The role of the electromagnetic deformation c has been much less investigated in the context of type IIB supergravity. The relevant dyonically-gauged supergravity in this case is the [SO(1; 1) × SO(6)] n R12 theory which arises from the reduction of type IIB supergravity on the product R × S5 [27]. As for the ISO(7) theory, the electromagnetic deformation is again a discrete (on/off) deformation, namely, c = 0 or 1 [16]. This four- dimensional supergravity has been shown to contain various types of AdS4 vacua preserving different amounts of supersymmetry as well as of residual gauge symmetry. In particular, an N = 4 and SO(4) symmetric solution was reported in [28] and subsequently, in [27], 1 5 uplifted to a class of AdS4 × S × S S-fold backgrounds of type IIB supergravity using the E7(7) exceptional field theory (E7(7)-EFT). These S-folds involve S-duality twists A(k) k 1 (k ≥ 3) that induce SL(2; Z)IIB monodromies M(k) = −ST of hyperbolic type along S , and can be systematically constructed as quotients of degenerate Janus-like solutions of the type IIB theory [29, 30] where the string coupling gs diverges at infinity. Together with the N = 4&SO(4) solution, additional N = 0&SO(6) [31] and N = 1&SU(3) [32] solutions have been found and uplifted to similar S-fold backgrounds of type IIB supergravity with hyperbolic monodromies in [32]. From a holographic perspective, these AdS4 vacua describe new strongly coupled three-dimensional CFT's, referred to as J-fold CFT's in [33] (see also [34, 35] and [36]), which are localised on interfaces of N = 4 super-Yang-Mills theory k (SYM) [37]. In the N = 4 case [33], a hyperbolic monodromy J = −ST 2 SL(2; Z)IIB was shown to introduce a Chern-Simons level k in the dual J-fold CFT which, in turn, is constructed from the T (U(N)) theory [38] upon suitable gauging of flavour symmetries. A diagram illustrating this type IIB construction is depicted in figure1 . On the other hand, a classification of interface SYM theories was performed in [39] (see also [40]) in correspondence to the various amounts of supersymmetry, as well as the largest possible global symmetry, preserved by the interface operators. Three supersymmet- ric cases were identified: interfaces with N = 4&SO(4) symmetry, N = 2&SU(2) × U(1) symmetry and N = 1&SU(3) symmetry. While the S-folds in [27] and [32] respectively match the symmetries of the N = 4 and N = 1 cases, the gravity duals of the would be N = 2 J-fold CFT's localised on the interface with SU(2)×U(1) symmetry remain missing. 1 5 In this work we fill this gap and present a new family of AdS4 × S × S S-folds with N = 2 supersymmetry, SU(2) × U(1) symmetry and, as in the previous cases, involving S-duality twists that induce monodromies of hyperbolic type along S1. { 2 { 1 5 D = 10 Type IIB & S-fold with AdS4 × S × S geometry Reduction Uplift method : E7(7)-EFT involving 5 1 on R × S hyperbolic twists A(k) along S [SO(1; 1) × SO(6)] n R12 N = 4 SYM D = 4 gauging with an AdS4 vacuum with a localised interface Z AdS J 2 SL(2; )IIB action 4/CFT D = 3 3 J-fold CFT3 JHEP04(2020)050 Figure 1. Type IIB S-folds with hyperbolic monodromies M(k) = −ST k along S1 and connection with three-dimensional J-fold CFT's. The paper is organised as follows. In section2 we perform a study of multi-parametric 12 families of AdS4 vacua in the [SO(1; 1) × SO(6)] n R maximal supergravity. We find four families of vacua, one of them being N = 2 supersymmetric and containing a vacuum with a residual symmetry enhancement to SU(2) × U(1). In section3 , by implementing a generalised Scherk-Schwarz (S-S) ansatz in E7(7)-EFT, we uplift such an AdS4 vacuum 1 5 to a class of AdS4 × S × S N = 2 S-folds of type IIB supergravity with SU(2) × U(1) symmetry and a non-trivial hyperbolic monodromy along S1. In section4 we present our conclusions and discuss future directions. 12 2 AdS4 vacua of [SO(1; 1) × SO(6)] n R maximal supergravity We continue the study of AdS4 vacua initiated in [31], and further investigated in [28] and [32], for the dyonically-gauged maximal supergravity with non-abelian gauge group G = [SO(1; 1) × SO(6)] n R12 : (2.1) We will show how the AdS4 vacua of [28, 31, 32] actually correspond to very special points (featuring residual symmetry enhancements) within multi-parametric families of solutions.