Supersymmetric AdS backgrounds and their moduli spaces Jan Louis, University Hamburg In collaboration with: S.de Alwis, P.Karndumri, S.L¨ust, L.McAllister, C.Muranaka, P.R¨uter, H.Triendl, A.Westphal, M.Zagermann [arXiv:1312.5659,1406.3363,1506.08040,1507.01623,1601.00482,1607.08249, 1611.02982, 1612.00301, and in progress] Milano, Februrary 2017 2 Motivation 1. AdS backgrounds often appear as intermediate step (before “uplifting”) in models of string phenomenology 2. AdS backgrounds feature prominently in AdS/CFT correspondence In both cases their moduli space C also is of interest 1. moduli stabilization/values of physical parameters 2. corresponds to conformal manifold of dual CFT ⇒ study (supersymmetric) AdS backgrounds and their moduli spaces in supergravity in arbitrary space-time dimensions with arbitrary number of supercharges 3 D = 4, N = 1 – basic formuli i chiral multiplet: (φ,χ) , gravitational multiplet: (gµν , ψµ) 1 i ¯¯j ¯ Lbosonic = 2 R − Gi¯j ∂µφ ∂µφ − V , i, j = 1,..., nc , where 2 2 Gi¯j = ∂i∂¯jK , V = |A1i| − 3|A0| , 1 2 K 1 A0 = e W(φ) , A1i = DiA0 , DiA0 = ∂iA0 + 2 (∂iK)A0 Supersymmetry transformation of the fermions δχi = A1i ǫ + ..., δψµ = Dµǫ + A0γµǫ + ... Supersymmetric AdS backgrounds: hδψµi = hδχii = 0 ⇒ hA1ii = 0 , hA0i 6= 0 , hA0i ∼ cos. const. / AdS-radius Classification for N = 1 difficult/impossible (?) 4 D = 4, N = 1 – supersymmetric moduli space 1. Minkowski background (hA0i = 0) generic W : no moduli space special W : holomorphic W (φ) implies: moduli space C is K¨ahler submanifold of original K¨ahler field space M (and protected by supersymmetry) 2. AdS background hA0i 6= 0 ¯ 1 • φ cannot drop out of A1φ = ∂φA0 + 2 (∂φK)A0 and hV i • no protection against quantum corrections • moduli space C is real submanifold of original K¨ahler 1 manifold M, at best with dim(C) = 2 dim(M) 5 D = 4, N = 1 – supersymmetric moduli space necessary condition: i ¯i δhA0i = hA1iiδφ + hA¯ 1iiδφ¯ = 0 , δhA¯ 0i = ... = 0 j 1 ¯¯j ¯ δhA1ii = h∂jA1iiδφ + 2 hKi¯jA0iδφ = 0 , δhA1ii = ... = 0 (1st eq. identically satisfied in supersymmetric backgrounds) Rewrite 2nd condition: j δφ mij hK ¯A0i M = 0 , with M = ij , ¯¯j ¯ δφ hK¯ijA0i ¯m¯i¯j where mij = h∇iA1ji = fermionic mass matrix ➪ generically M has full rank ⇒ δφj = 0 ⇒ no moduli space ➪ for special (tuned) W/A0: moduli space possible [DLMTW, ...] 6 Maximally supersymmetric backgrounds in arbitrary dim. D Distinguish two cases: backgrounds with/without fluxes 1. backgrounds without fluxes 7 Maximally supersymmetric backgrounds in arbitrary dim. D Distinguish two cases: backgrounds with/without fluxes 1. backgrounds without fluxes Supersymmetry transformation of fermions in gauged supergravities I I I J i i J δψµ = Dµǫ + A0J(φ)γµǫ + ..., δχ = A1J(φ)ǫ + ... 2 2 with V ∼|A1| − c(d,q)|A0| Supersymmetric backgrounds preserving all q supercharges: I I J hDµǫ + A0J(φ)γµǫ i = 0 , hA1i = 0 , ∀ I, J 2 Consistency condition: Rµνρσ ∼ trA0 (gµρgνσ − gµσgνρ) (D−d) Solutions: AdSD or Md × T , 1 ≤ d ≤ D (D−d) (note: AdSd × Y not possible) 8 Supersymmetric moduli spaces C of AdSD • supersymmetric moduli spaces C⊂M • necessary condition: δφhA0i =0= δφhA1i • Gradient-flow equations: [d’Auria, Ferrara] δhA0i = hA1i δφ~ = 0 , δhA1i = M δφ~ , M := mf + hA0i • necessary condition for moduli space: M has null eigenspace ∗ = 0 C⊂M has same geometry as M hA0i 6= 0 C⊂M has different geometry as M • q = 4: can tune superpotential/mf but C generically destroyed by quantum corrections • q = 8, 16, 32: no superpotential, can choose spectrum and gauged isometries of M, supersymmetric protection 9 Example: d = 5,q = 16 supercharges [JL, Triendl, Zagermann] ➪ multiplets and scalar geometry i 0,m i • gravity (gµν , ψµ, Aµ ,χ , Σ), m = 1,..., 5 ⇒ 6 graviphotons a a am • vector (Aµ, λ ,φ ), a = 1,..., nv scalar field space: M(Σ,φ)= R+ × SO(5,nv) SO(5)×SO(nv) ➪ supersymmetric AdS backgrounds can be “classified” gauge group: G = U(1) × Gˆ ⊂ SO(5, nv) where Gˆ ⊂ SO(3, nv) → SO(3) × H , H = compact subgroup ➪ moduli space possible: [G¨unaydin,Pilch,Warner,Zagermann] SU(1, p) SO(2, 2p) SO(5, n ) C = ⊂ ⊂ v U(1) × SU(p) SO(2) × SO(2p) SO(5) × SO(nv) 10 General story: AdS moduli space for M = G/H [JL, S. L¨ust, R¨uter] ➪ gauging isometries of M induces gauging of R-symmetry HR ➪ Maximally supersymmetric AdS backgrounds: g HR ⊂ HR gauged only by (subset of) graviphotons Aµ ➪ A0 and Aµ transform in specific rep. of HR depending on D,q ➪ l hA0i 6= 0 implies HR → HR (little group) ➪ additional constraint: g l HR ⊂ HR ⊂ HR of AdS-vacuum has to have Aµ in its adj. rep. ➪ flat directions of V : g g HR charged Goldstone bosons + HR-singlets = moduli 11 Example: d = 7,q = 32 supercharges [JL, S. L¨ust, P. R¨uter] ➪ I [MN] IJK gravity multiplet: (gµν , ψµ, Aµ ,χ ,φ) I, J, K = 1,..., 4, M,N = 1,..., 5 ➪ SL(5) scalar field space: M(φ)= SO(5) ➪ A0 ∈ 1 ⊕ 5, A1 ∈ 5 ⊕ 14 ⊕ 35 of HR = SO(5) ∼ USp(4) l g ⇒ HR = HR = HR = USp(4) ➪ δφ ∈ 14 ⇒ no GB, no singlets ⇒ no moduli 12 Results so far: AdS(d,q) M C [ ] (4,4) K¨ahler Real [DLMTW] (4,8) SK× QK Real×K¨ahler [DMLTW] (5,8) SR× QK K¨ahler [Tachikawa,LM] O(1,n ) (6,8) T × QK – [?] O(nT ) SK: Special K¨ahler QK: Quanternionic K¨ahler SR: Special Real [DMLTW]=[de Alwis, JL, McAllister, Triendl, Westphal] [LM]=[JL, Muranaka] 13 Results so far: g AdS(d,q) M HR C [ ] SO(6,6+n ) SU(1,1) (4,16) v × SO(4) AX [LT] SO(6)×SO(6+nv ) U(1) SO(5,5+nv ) + SU(1,p) (5,16) × R U(2) [GPWZ,LTZ] SO(5)×SO(5+nv ) SU(p)×U(1) SO(4,4+nv ) + (6,16) × R SU(2) × SU(2) AX [KL] SO(4)×SO(4+nv ) SO(3,3+nv ) + (7,16) × R USp(2) AX [LL] SO(3)×SO(3+nv ) E7,7 (4,32) SU(8) SO(8) AX [LLR] E6,6 SU(1,1) (5,32) USp(8) SU(4) U(1) [LLR] SO(5,5) (6,32) SO(5)×SO(5) XX XX [LLR] SL(5) (7,32) SO(5) USp(4) AX [LLR] AX: AdS background exists and can be classified but no moduli space, XX: AdS background does not exist [LT]=[JL,Triendl] [LLR]=[JL, L¨ust, R¨uter] [KL]=[Karndumri,JL] 14 AdS/CFT correspondence AdSd bulk with q supercharges ⇔ q q − SCFTd 1 on AdSd boundary with 2 + 2 super+sconformal charges AdS ⇔ SCFT (broken) gauge group (anomalous) flavour group scalars with mass m couplings of gauge invariant operators moduli exactly marginal couplings moduli space C conformal manifold C i S = S0 + ϕ Oi Pi R 2d gij(ϕ)= x hOi(x)Oj(0)iS [Zamolodchikov] 15 Compare AdS with SCFT-results AdS(d,q) C [ ] SCFT d− , q CSCFT [ ] ( 1 2 ) 1 (4,4) Real [DLMTW] (3,2) Real [CDI] (4,8) Real×K¨ahler [DMLTW] (3,4) K¨ahler [GKSTW] (5,8) K¨ahler [T,LM] (4,4) K¨ahler [Ansin] (6,8) – [?] (5,4) SXX SX: SCFT exists but no moduli space, SXX: SCFT does not exist [CDI]=[Cordova,Dumitrescu,Intriligator] [GKSTW]=[Green,Komargodski,Seiberg,Tachikawa,Wecht] 1 none known, no susy protection 16 Compare AdS with SCFT-results (Note: no SCFT in d> 6) AdS(d,q) C [ ] SCFT d− , q CSCFT [ ] ( 1 2 ) (4,16) AX [LT] (3,8) SX [CDI] SU(1,p) 2 (5,16) SU(p)×U(1) [GPWZ,LTZ] (4,8) K¨ahler (?) [P,GGK] (6,16) AX [KL] (5,8) SX [CDI] (7,16) AX [LL] (6,8) SX [LL] (4,32) AX [LLR] (3,16) SX [CDI] SU(1,1) SU(1,1) (5,32) U(1) [LLR] (4,16) U(1) [?] (6,32) XX [LLR] (5,16) SXX [Nahm] (7,32) AX [LLR] (6,16) SX [LL] [GPWZ]=[G¨unaydin,Pilch,Warner,Zagermann] [P,GGK]= [Papadodimas; Gerchkovitz,Gomis,Komargodski] 2 ∗ [GGK] only show K¨ahler, [P] shows tt -geometry 17 Supersymmetric backgrounds in arbitrary dimensions D Distinguish two cases: backgrounds with/without fluxes 2. backgrounds with fluxes 18 Supersymmetric backgrounds in arbitrary dimensions D Distinguish two cases: backgrounds with/without fluxes 2. backgrounds with fluxes [JL, S. L¨ust, ...] Supersymmetry transformation of the fermions I I I J I J i i J i J δψµ = Dµǫ + (F0µ)Jǫ + A0 J γµǫ , δχ = (F1)Jǫ + A1J ǫ , A A J A J δλ = (F2)J ǫ + A2J ǫ , F0,1/2 = sum of fluxes in gravitational multiplet/other multiplets supersymmetric backgrounds: F1 = F2 = A1 = A2 = 0 ⇒ F0 = 0 two exceptions: 1. chiral theories with selfdual fluxes 2. supergravities without χ’s in gravitational multiplet 19 Supersymmetric backgrounds in arbitrary dimensions D ➪ candidate supergravities/fluxes: dimension supersymmetry q possible flux classification of maximal supersymmetric solutions D = 11 N = 1 32 F (4) [Figueroa-O’Farrill,Papadopoulos] (5) D = 10 IIB 32 F [Figueroa-O’Farrill,Papadopoulos] + (3) D = 6 N = (2, 0) 16 5 × F [Chamseddine,Figueroa-O’Farrill,Sabra] + (3) D = 6 N = (1, 0) 8 F [Gutowski,Martelli,Reall] + D = 5 N = 2 8 F (2) [Gauntlett,Gutowski,Hull,Pakis,Reall] D = 4 N = 2 8 F (2) [Tod] ➪ study this list case by case to find • hA0i = hA1i = hA2i =0 • no flux for R-symmetry possible [Hristov,Looyestijn,Vandoren;Gauntlett,Gutowski;Akyol,Papadopoulos] ➪ background solutions have to coincide with ungauged solutions they are classified in ungauged case (see refs.