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The moduli space of partially broken N = 2 supergravities Jan Louis a Universit¨t Hamburg in collaboration with Paul Smyth & Hagen Triendl based on 0911.5077, 1008.1214 Vicente Cort´s, Paul Smyth & Hagen Triendl in preparation e a “Advances in string theory, wall crossing, and quaternion-K¨hler geometry” Paris, September 2010 2 History of spontaneous N = 2 → N = 1 breaking Á ’84: [Cecotti,Girardello,Porrati] showed: “Two into one won’t go” Minkowski vacua of 4d, N = 2 supergravities either have full N = 2 or N = 0 supersymmetry. Á ’95 way out: include magnetic FI-term/magnetic charges [Antoniadis,Partouche,Taylor; Ferrara,Girardello,Porrati] few explicit examples constructed Á recently systematic analysis [Smyth,Triendl,JL] using embedding tensor formalism of [deWit,Samtleben,Trigiante] 3 Outline of the talk 1. brief review of N = 2 supergravity and the embedding tensor formalism 2. formulate the conditions of partial supersymmetry breaking in the embedding tensor formalism 3. discuss solutions 4. derive N = 1 eﬀective action and discuss its moduli space 5. apply to type II string theory 4 N = 2 supergravity Á spectrum gravity multiplet: gµν , ΨµA , A0 , µ A=1,2 vector multiplets: Ai , λiA , ti , µ i=1,...,nv hypermultiplets: ζα , qu , α=1,...,2nh , u=1,...4nh Á scalar ﬁeld space M = MSK (t) × MQK (q) v h 5 a Special K¨hler manifolds MSK (t): special K¨hler manifold v a [de Wit,Van Proeyen] a • K¨hler metric determined by holomorphic prepotential F ¯ t ¯ t Kv = − ln i XI (¯)FI (t) − XI (t)FI (¯) , FI ≡ ∂I F , I,=0,...,nv special coordinates: XI = (1, ti ) • Sp(2nv + 2) electric-magnetic duality rotations act on ˜ symplectic vectors (AI , AµI ) and VΛ = (XI , FI ) , Λ=0,...,2nv µ 6 a Quaternionic K¨hler manifold MQK (q): quaternionic K¨hler manifold h a [Bagger, Witten,...] • triplet of SU (2)-covariantly closed almost complex structures Jx (q) exists ∇Jx = 0 , Jx Jy = −δ xy Id + ǫxyz Jz , x=1,2,3 a associated triplet of K¨hler-forms Kx = dω x + 1 ǫxyz ω y ∧ ω z , 2 ω : SU (2) − connection • Killing vectors kλ determined in terms of triplet of Killing prepotentials Px (moment maps) kλ · Jx = ∇ Px λ 7 Embedding Tensor formalism [de Wit,Samtleben,Trigiante] introduce simultaneously into the action: Á electric gauge ﬁeld: AI , µ I=0,...,nv ˜ Á dual magnetic gauge ﬁeld: AµI ˜ Á embedding tensor: Θλ = (Θλ , ΘIλ ) , Λ=0,...,2nv Λ I ˜ ˜ Dµ q = ∂µ q−AI Θλ kλ (q)−AµI ΘIλ kλ (q) , µ I kλ ≡ Killing vectors Θλ is constrained by: Λ - N = 2 supersymmetry - mutual locality of charges Á two-forms Bµν with only topological couplings, algebraic ﬁeld eqs. and appropriate gauge transformations to keep d.o.f. intact 8 N = 2 supersymmetry variations of the fermions gravitino : δǫ ΨµA = Dµ ǫA − SAB γµ ǫB + . . . , A,B=1,2 gaugino : δǫ λiA = WiAB ǫB + . . . , i=1,...,nv hyperino : δǫ ζα = NA ǫA + . . . , α α=1,...,2nh where SAB ∼ VΛ Θλ Px σAB , Λ λ x x=1,2,3 WiAB ¯ ∼ (∇i VΛ ) Θλ Px σ xAB , Λ λ I=0,...,nv NA α ¯ A ∼ VΛ Θλ Uαu ku , Λ λ A (Uαu is vielbein of quaternionic metric) Recall: kλ · Jx = ∇ Px , λ VΛ = (XI , FI ) 9 Spontaneous N = 2 → N = 1 supersymmetry breaking preserving N = 1 corresponds to δǫ1 λiA = 0, δǫ1 ζα = 0 , δǫ1 ΨµA = 0 δǫ2 λiA = 0, δǫ2 ζα = 0 , δǫ2 ΨµA = 0 which implies (dropped ) WiAB ǫB 1 = 0, NαA ǫA = 0 , 1 SAB ǫB = 1 1 2 µǫ∗ A 1 , WiAB ǫB 2 = 0, NαA ǫA = 0 , 2 SAB ǫB = 2 1 2 µǫ∗ A 2 , where Dµ ǫ1 A = 2 µγµ ǫ∗ A , 1 1 µ ∼ cosmological constant This implies: N = 1 vacua are AdS or Minkowskian and automatically stable 10 Super-Higgs eﬀect Á recall N = 2 spectrum: gravity multiplet s = [2, 3 , 3 , 1] 2 2 vector multiplet s = [1, 1 , 1 , 0, 0] 2 2 hyper multiplet s = [ 2 , 1 , 0, 0, 0, 0] 1 2 Á reorganizes after partial breaking into N = 1 multiplets 3 1 massive gravitino multiplet s = [ 2 , 1, 1, 2 ] 3 massless gravity multiplet s = [2, 2 ] vector multiplets s = [1, 1 ] 2 1 chiral multiplets s = [ 2 , 0, 0] 11 Super-Higgs eﬀect Á recall N = 2 spectrum: gravity multiplet s = [2, 3 , 3 , 1] 2 2 vector multiplet s = [1, 1 , 1 , 0, 0] 2 2 hyper multiplet s = [ 2 , 1 , 0, 0, 0, 0] 1 2 Á reorganizes after partial breaking into N = 1 multiplets 3 1 massive gravitino multiplet s = [ 2 , 1, 1, 2 ] Á minimal ingredients for partial supersymmetry breaking: 1. one vector multiplet 2. one charged hyper ⇒ 2 Goldstone bosons 3. two commuting Killing vectors in hyper-sector 12 gravitino & gaugino variations vanishing gravitino & gaugino variations implies ˜ Θλ − FIJ ΘJλ Px σAB ǫB = 0 , I λ x 1 λ=1,2 ˜ Á “no” N = 1 solution for ΘJλ = 0 [Cecotti,Girardello,Porrati] Á two Killing vectors k1,2 ⇒ choose convenient SU (2)-frame P3 = P3 ≡ 0 , 1 2 and deﬁne P± = P1 ± i P2 1,2 1,2 1,2 Á N = 1 solution in terms of complex vector CI [Smyth,Triendl,JL] Θ1 = − Im(P+ FIJ CJ ) , I ˜ ΘI1 = −Im(P+ CI ) , 2 2 Θ2 = I Im(P+ FIJ CJ ) , ˜ ΘI2 = Im(P+ CI ) , 1 1 ¯ Á locality of charges implies CI (ImF)IJ CJ = 0 Note: no constraint on MSK v 13 hyperino variation Recall: need 2 commuting Killing vectors k1,2 and A ˆ ˆ ¯ NαA ǫA = 0 , 1 NαA ǫA = 0 , 2 NA ∼ Uαu ku , α k u ≡ V Λ Θλ k u Λ λ this implies ˆ ˆ k · J3 = i k , ˆ k · (J1 − iJ2 ) = 0 ˆ ⇒ MQK is constrained by existence of holomorphic k h 14 hyperino variation Recall: need 2 commuting Killing vectors k1,2 and A ˆ ˆ ¯ NαA ǫA = 0 , 1 NαA ǫA = 0 , 2 NA ∼ Uαu ku , α k u ≡ V Λ Θλ k u Λ λ this implies ˆ ˆ k · J3 = i k , ˆ k · (J1 − iJ2 ) = 0 ˆ ⇒ MQK is constrained by existence of holomorphic k h N = 1 moduli space solution of: δΘλ = 0 , Λ ˆ ˆ δ k · J3 − i k = 0 generically stabilizes large number of moduli! 15 Outline for the rest of the talk 1. assume MQK with holomorphic Killing vector exists h and derive N = 1 eﬀective action 2. construct explicit solution for type II string theory 16 N = 1 eﬀective action steps: Á integrate out massive gravitino multiplet ( 2 , 1, 1, 1 ) 3 2 together with all multiplets at the same scale Á compute LN=1 (K, W, f ) in terms of “N = 2 data” eﬀ Á N = 1 constraints: a a • scalar ﬁeld space is K¨hler (with K¨hler potentials K) • W, f are holomorphic (None of these constraints hold in N = 2.) 17 N = 1 scalar ﬁeld space Recall N = 2 scalar ﬁeld space M = MSK (t) × MQK (q) v h integrating out massive multiplets takes: Á subspace MSK ⊂ MSK N=1 v a – automatically K¨hler Á quotient MQK ⊂ MQK /(k1 , k2 ) N=1 h e [Cort´s,Smyth,Triendl,JL] ˆ ˆ with K¨hler metric huv and K¨hlerform K a a ˆ huv = huv − k1u k1v + k2u k2v /k2 , 1 ˆ K = dω 3 (Note: MQK can be as large as 4nh − 2) N=1 Á resulting N = 1 scalar ﬁeld space is indeed K¨hler a MN=1 = MQK × MSK N=1 N=1 18 N = 1 holomorphic couplings 1 Á superpotential: W= e 2 K VΛ Θλ P− , ¯ ∂W = 0 Λ λ Á gauge kinetic function: fIJ = FIJ |N=1 , ¯ ∂f = 0 19 N = 1 solutions in type II string theory Á consider type II compactiﬁed on 6-dim. manifolds with SU (3) × SU (3)-structure [Grana,Waldram,JL,...] Á d = 4 low energy eﬀective action is an N = 2 supergravity Á MQK is in the image of c-map h [Cecotti,Girardello,Ferrara,Sabharwal] This implies: a • (2nh − 2)-dim. special K¨hler subspace with RR-scalars ﬁbered over it • (2nh − 1) isometries associated with RR-scalars exist Á explicit N = 1 solution can be constructed 20 N = 1 solutions in type II string theory Á N = 1 solution in terms of “doubly symplectic” charge matrix eAI pI A Θ= , A=0,...,nh −1, I=0,...,nv mA I q AI with ¯ ¯ eAI = Re(FIJ C J GAB DB ) , ¯ pI = (C J GAB DB ) , A I ¯ ¯ mA = Re(FIJ C J DA ) , ¯ qAI = Re(C I DA ) where ¯ DA Im(GAB )DB = 0, a (G is prepotential of special K¨hler base) Á Remarks: • solution is mirror symmetric • solution only exists for non-geometric ﬂuxes/torsion • analogous solution for AdS background 21 N = 1 solutions in type II string theory Á N = 1 solution in terms of “doubly symplectic” charge matrix eAI pI A Θ= , A=0,...,nh −1, I=0,...,nv mA I q AI with ¯ ¯ eAI = Re(FIJ C J GAB DB ) , ¯ pI = (C J GAB DB ) , A I ¯ ¯ mA = Re(FIJ C J DA ) , ¯ qAI = Re(C I DA ) where ¯ DA Im(GAB )DB = 0, a (G is prepotential of special K¨hler base) Á N = 1 K¨hler potential: a K = KSK + 2φ can be expressed in terms of holomorphic coordinates given in [Rocek,Vafa,Vandoren] 22 Conclusion Á N = 2 → N = 1 generically possible if magnetic charges present – both in Minkowski and AdS backgrounds Á no constraint on MSK v Á MQK is constraint by existence of holomorphic k h Á N = 1 eﬀective action can be constructed a with K¨hlerian scalar ﬁeld space and holomorphic W, f Á for type II compactiﬁed on manifolds with SU (3) × SU (3)-structure N = 1 solutions exist if non-geometric ﬂuxes/torsion are present

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posted: | 1/25/2012 |

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