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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
language:English
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