Maximally Supersymmetric Solutions in Supergravity

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1 Maximally Supersymmetric Solutions in Supergravity Severin Lüst Universität Hamburg arxiv: , , and in progress in collaboration with J. Louis November 24, / 17

2 Introduction Supersymmetric solutions in supergravity are well studied, many are classified. They are believed to uplift to solutions of the full string theory and are related to string solitons, i.e. non-perturbative string theory. Moreover anti-de Sitter solutions feature prominently in AdS/CFT correspondence. Here: Maximally supersymmetric solutions. 1. Classification of all background space-times for all gauged and deformed supergravity theories in D 3 space-time dimensions. 2. AdS solutions and their moduli spaces. 2 / 17

3 Outline / Results A) Set the stage to discuss solutions in a generic framework. B) Solutions without fluxes: Only two possible cases: Mink D and AdS D (only gauged / deformed SUGRAs) C) Solutions with (non-trivial) fluxes: Exist only for a small class of theories. Solutions coincide with those of the corresponding ungauged theories. For these theories all solutions are known and classified. Exhaustive list of solutions. D) AdS D solutions Characterize gaugings. Moduli spaces for theories with coset scalar field space. 3 / 17

4 Maximally supersymmetric backgrounds Classical solutions / backgrounds for which δ ɛ B = δ ɛ F = 0. SUSY variations of bosonic / fermionic fields Here: bosonic solutions, i.e. F = 0 δ ɛ B F = 0. only remaining condition: δ ɛ F = 0 indep. supersymmetry paramters ɛ preserved supercharges 4 / 17

5 Supersymmetry variations gravitini: δψ i µ = D µ ɛ i + (F 0µ ) i j ɛj + A i 0 j γ µ ɛ j covariant derivative: fluxes from the gravity multiplet gaugings / deformations D µ ɛ i = µ ɛ i + (Q µ ) i j ɛj spin-1/2 fermions from the gravity multiplet ( dilatini ): δχ a = (F 1 ) a i ɛ i + A a 1 i ɛ i spin-1/2 fermions from other multiplets ( gaugini, hyperini,...): δλ s = (F 2 ) s i ɛ i + A s 2 i ɛ i fluxes from other multiplets 5 / 17

6 Solutions without fluxes (1) Easiest case: all fluxes vanish: F 0µ = F 1 = F 2 = 0 Supersymmetric solutions of gauged supergravity without background fluxes need to satisfy the Killing spinor equation: δψ i µ = µ ɛ i + A i 0 jγ µ ɛ j = 0 Integrability condition: ( ) 1 4 R µν αβ δk i + 2A i 0 ja j 0 k δα µδν β γ αβ ɛ k = 0 Unbroken supersymmetry (without fluxes): Mink D and AdS D are the only possible solutions. 6 / 17

7 Solutions without fluxes (2) Spin-1/2 variations: δχ a = A a 1i ɛ i = 0, δλ s = A s 2i ɛ i = 0 Algebraic conditions Λ A 2 0 = 2(D 1)(D 2) 1, A 1 = A 2 = 0 Compare with the potential: V = c 0 tr(a 0 A 0) + c 1 tr(a 1 A 1) + c 2 tr(a 2 A 2), Mink D (Λ = 0) solutions exist for all theories. For D 7 most gauged theories admit AdS D (Λ < 0) solutions. (See second half of this talk.) [de Alwis,Louis,McAllister,Triendl,Westphal][Louis,Triendl][Louis,SL][Louis,Triendl,Zagermann][Louis,Muranaka] 7 / 17

8 Solutions with non-trivial flux (1) For more interesting solutions: Allow for non-vanishing flux. Firstly: spin-1/2 variations: δχ a = (F 1 ) a i ɛ i + A a 1i ɛ i = 0, δλ s = (F 2 ) s i ɛ i + A s 2i ɛ i = 0 Unbroken supersymmetry: F 1 = F 2 = A 1 = A 2 = 0 F 0µ = 0 Generically no non-trivial fluxes possible! Two exceptions 1. theories without χ s in the gravitational multiplet. 2. chiral theories with selfdual fluxes. 8 / 17

9 Solutions with non-trivial flux (2) Secondly: gravitino variations: δψ i µ = µ ɛ i + (Q µ ) i j + (F 0µ) i j ɛj + A i 0 jγ µ ɛ j = 0 Integrability condition: ( 1 4 R µνρσγ ρσ δ i j (H µν ) i j +... ) ɛ j = 0, where H µν is the field strength corresponding to Q µ. Unbroken supersymmetry: H µν = 0 Q µ = A 0 = 0 (See also [Hristov,Looyestijn,Vandoren] [Gauntlett,Gutowski] [Akyol,Papadopoulos]) The supersymmetry variations take the same form as in the ungauged / undeformed case. 9 / 17

10 Solutions with non-trivial flux (3) SUGRAs with non supersymmetry breaking flux: dimension supersymmetry q possible flux classification D = 11 N = 1 32 F (4) [Figueroa-O Farrill,Papadopoulos] D = 10 IIB 32 F (5) + [Figueroa-O Farrill,Papadopoulos] D = 6 N = (2, 0) 16 5 F (3) + [Chamseddine,Figueroa-O Farrill,Sabra] D = 6 N = (1, 0) 8 F (3) + [Gutowski,Martelli,Reall] D = 5 N = 2 8 F (2) [Gauntlett,Gutowski,Hull,Pakis,Reall] D = 4 N = 2 8 F (2) [Tod] The maximally supersymmetric solutions are classified: AdS p S (D p) and AdS (D p) S p, for p-form flux. Hpp-wave as Penrose-limit of AdS S solutions. [Penrose;Gueven;Blau,Figueroa-O Farrill,Hull,Papadopoulos] Exceptional solutions in D = 5. [Gauntlett,Gutowski,Hull,Pakis,Reall] 10 / 17

11 All maximally supersymmetric solutions (with non-trival flux) dim. SUSY q AdS S Hpp-wave others D = 11 N = 1 32 AdS 4 S 7 KG 11 - AdS 7 S 4 D = 10 IIB 32 AdS 5 S 5 KG 10 - D = 6 N = (2, 0) 16 N = (1, 0) 8 AdS 3 S 3 KG 6 - D = 5 N = 2 8 AdS 2 S 3 KG 5 Gödel-like, AdS 3 S 2 NH-BMPV* D = 4 N = 2 8 AdS 2 S 2 KG 4 - * = near-horizon limit of the BMPV [Breckenridge,Myers,Peet,Vafa] black hole 11 / 17

12 AdS solutions So far: classification of background space-time geometries. Now: focus on algebraic conditions and target space geometry. Most promising: AdS D (here: D 4): Conditions A and A 1 = A 2 = 0 require gauging. May have non-trivial moduli spaces. Motivation: AdS/CFT correspondence AdS D solution with q supercharges SCFT in (D 1) dim. with q/2 supercharges gauge symmetry global symmetry moduli space conformal manifold 12 / 17

13 The gauged R-symmetry The conditions A and A 1 = A 2 = 0 require Q µ 0. The R-symmetry group H R must be gauged by H g R H R, where H g R needs to be generated by the vector fields from the gravity multiplet, i.e. the graphiphotons. H g R is uniquely determined to be the maximal subgroup of H R, s.t. a) A 0 is H g R invariant, i.e. [h g R, A0] = 0 b) The decomposition of the H R -representation r GP of the graviphotons w.r.t. H g R contains the adjoint representation of Hg R, i.e. r GP ad H g R / 17

14 Coset spaces From now on: Focus on theories where the scalar field space is a coset: M scal = G H, H = H R H mat Gauge group: G g G, where G g = G g R G g mat, s.t. H g R is the maximal compact subgroup of G g R. Lie algebra of G: g = h k Notice: [h, k] k, i.e. k transformes in some h representation. Decompose k into h g R irreps: k = i k i, [h g R, k i] k i 14 / 17

15 The moduli space Flat directions of the potential V (φ): f = {δφ k : V ( φ + δφ) = V ( φ )} = {δφ k : [δφ, g g R ] gg R } (f is the non-compact part of the normalizer N g (g g R )) Observation: non-compact part of g g R Goldstone bosons f = k g R k 0 h g R singlets in k, i.e. [h g R, k 0] = 0 span the moduli space The moduli k 0 are the non-compact part of g 0 = h 0 k 0 (which is the centralizer C g (g g R )). M AdS = G 0 H 0 15 / 17

16 All AdS D solutions for D 4 dim. SUSY H g R M AdS D = 7 N = 4 USp(N ) N = 2 [Louis,SL] [Pernici,Pilch,van Nieuwenhuizen] [Louis,SL,Rüter (to app.)] D = 6 N = (1, 1) SU(2) SU(2) [Romans] SU(1, 1) N = 8 SU(4) U(1) [Louis,SL,Rüter] D = 5 N = 6 N = 4 U(N /2) SU(1, p) U(1) SU(p) N = 2 Kähler [Louis,Muranaka] N = 8 [Günaydin,Romans,Warner] [Ferrara,Porrati,Zaffaroni] [Corrado,Günaydin,Warner,Zagermann] [Louis,Triendl,Zagermann] [de Wit, Nicolai][Louis,SL,Rüter] N = 5, 6 [de Wit,Nicolai] D = 4 N = 4 [Louis,Triendl] SO(N ) N = 3 N = 2 Kähler [de Alwis, et al.] N = 1 real [de Alwis, et al.] Perfect agreement with marginal deformations of SCFTs [Cordova,Dumitrescu,Intriligator] 16 / 17

17 Summary Systematic classification of all maximally supersymmetric supergravity backgrounds: no fluxes: only flat space-time and anti-de Sitter. fluxes: - generically not possible - in the gauged case: same solutions as for ungauged theories - all solutions are known and classified General recipe for the computation of AdS D moduli spaces for theories with coset scalar manifold. 17 / 17

18 Summary Systematic classification of all maximally supersymmetric supergravity backgrounds: no fluxes: only flat space-time and anti-de Sitter. fluxes: - generically not possible - in the gauged case: same solutions as for ungauged theories - all solutions are known and classified General recipe for the computation of AdS D moduli spaces for theories with coset scalar manifold. Thank You! 17 / 17