Flux compactifications, N=2 gauged supergravities and black holes

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: 2011 11 2 Flux Compactifications, N = 2 Gauged

1 : Flux Compactifications, N = 2 Gauged Supergravities and Black Holes based on arxiv: [hep-th]

2 Introduction 4 10 Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes - 2 -

3 Introduction ( ) 10 = 4 (A)dS Minkowski 6 N = 2... Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes - 3 -

4 Introduction WHY N = 2? N 4 ( ) N = 1 ( ) N = 2 ( ) = controllable!! N = 2 N = 1 Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes - 4 -

5 Introduction HOW 4D N = 2 from 10D? II ( IIA ) 6 vs 1/4 (ex: Calabi-Yau 3-fold, etc.) (cf: T 6 4 N = 8 )... Calabi-Yau Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes - 5 -

6 Introduction Beyond Calabi-Yau Calabi-Yau? CY CY (back reactions) 1. CY with fluxes 4D ungauged SUGRA break 10D Eqs. of Motion 2. non-cy with fluxes 4D gauged SUGRA non-cy: SU(n)-structure with torsion, generalized geometry, etc. gauge fields, matter fields, gauge coupling const., mass parameters... Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes - 6 -

7 Calabi-Yau compactification in type IIA Calabi-Yau 3-fold M CY Ricci Kähler ( ) ( SU(3) SU(4) SO(6)) ds 2 10D = η µν (x) dx µ dx ν 4D + g mn (x, y) dy m dy n CY Levi-Civita 2 (J) 3 (Ω) dj = [m J np] = 0 dω = [m Ω npq] = 0 Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes - 7 -

8 Flux compactifications in type IIA non-cy 3-fold M 6 Ricci 2-form ( ) (SU(3)-structure ) dj 0 and/or dω 0 CY dj = 3 2 Im(W 1Ω) + W 4 J + W 3, dω = W 1 J J + W 2 J + W 5 Ω Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes - 8 -

9 Intrinsic torsion classes of SU(3)-structure manifolds dj = 3 2 Im(W 1Ω) + W 4 J + W 3, dω = W 1 J J + W 2 J + W 5 Ω hermitian W 1 = W 2 = 0 balanced W 1 = W 2 = W 4 = 0 complex (1/4-SUSY Minkowski 1,3 ) special hermitian W 1 = W 2 = W 4 = W 5 = 0 Kähler W 1 = W 2 = W 3 = W 4 = 0 CY W 1 = W 2 = W 3 = W 4 = W 5 = 0 conformally CY W 1 = W 2 = W 3 = 3W 4 + 2W 5 = 0 symplectic W 1 = W 3 = W 4 = 0 nearly Kähler W 2 = W 3 = W 4 = W 5 = 0 almost complex (1/4-SUSY AdS 4 ) almost Kähler W 1 = W 3 = W 4 = W 5 = 0 quasi Kähler W 3 = W 4 = W 5 = 0 semi Kähler W 4 = W 5 = 0 half-flat ImW 1 = ImW 2 = W 4 = W 5 = 0 Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes - 9 -

10 Flux compactifications in type IIA (dj, dω) ( ) ( )( ) β I I eλ m ΛI ω Λ d NS-NS α I e ΛI m Λ I Σ Q T e I 0, e 0I : H-flux charges (H fl = e I 0 α I + e 0I β I ) e I a, e ai : geometric flux charges ( ) m ΛI, m Λ I: nongeometric flux charges (e I Λ, e ΛI ) ω Λ Σ + F F 0 + F F 10 e B Ĝ ( F = λ( F), λ( F k ) ( ) [k+1 2 ] Fk ) R-R 1 2 Ĝ = (G Λ 0 + G Λ 2 + G Λ 4 ) ω Λ ( G 0Λ + G 2Λ + G 4Λ ) ω Λ +(G I 1 + G I 3) α I ( G 1I + G 3I ) β I G Λ 0 p Λ, G0Λ q Λ ξ I e ΛI + ξ I e Λ I c (p Λ, q Λ ) T : R-R flux charges (p 0 : Romans mass) Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes

11 Flux compactifications in type IIA 10 IIA (democratic formulation) S (10D) IIA = S NS + S R : S NS + S R = 1 2 e 2ϕ{ 1 } R 1 + 4dϕ dϕ 2Ĥ3 Ĥ3 1 8 [ F F] 10 F = λ( F) (d + Ĥ ) F = 0 (d Ĥ ) F = 0 4 N = 2 ( ) ( ) 3 isometry group special Kähler geometry (SKG) quaternionic geometry (QG) QG SKG Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes

12 Flux compactifications in type IIA = 2 g 2 e 2φ[ Π T Q H T M V Q ΠH + Π T V Q M H Q T Π V + 4Π T H C T H Q T( Π V Π T V + Π V Π T ) ] V Q CH Π H V R = 1 2 g2 e 4φ( c + Qξ )T ( ) M V c + Qξ V NS V NS + V R = V 4D = g 2[ 4h uv k u k v + Π V = e K V/2 (X Λ, F Λ ) T t a = X a /X 0 a = 1,..., n V SKG V of vector-moduli M V,H ( 1 ReN 0 1 P + P P 3 ) ( ImN 0 3 x=1 0 (ImN ) 1 ( G ab D a P x D b P x 3 P x 2)] (abelian : k a Λ = 0) = 2e φ Π T V Q C H Π H = 2e φ Π T V Q C H Π H = e 2φ Π T V C V (c + Qξ) ) ( ) 1 0 ReN 1 V,H C V,H : symplectic invariant metrics where Π H = e K H/2 (Z I, G I ) T z i = Z i /Z 0 i = 1,..., n H SKG H of hyper-moduli F Λ N ΛΣ X Σ Q C T V Q C H Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes

13 4D N = 2 SUGRA N = 2 ( ) N = 2 ( ) 0 = m ΛI = m Λ I = p Λ Standard Gauged SUGRA n V n H [hep-th/ ] 3 0 = m ΛI = m Λ I Gauged SUGRA n V n H 1 [hep-th/ ] generic Gauged SUGRA n V ñ H n T [hep-th/ ] {a, ξ I, ξ I } {p Λ, m Λ I, m ΛI } [hep-th/ ], [arxiv: ] Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes

14 Black hole as an application Anti-de Sitter Black Holes in 4D N = 2 Gauged SUGRA Comments AdS-BH with naked singularity in pure AdS SUGRA L.J. Romans [hep-th/ ], M.M. Caldarelli and D. Klemm [hep-th/ ], etc. SUSY solution of rotating AdS black hole with regular horizon AdS-BH with regular horizon in Gauged SUGRA with VMs (no HMs) [hep-th/ ], [arxiv: ], [arxiv: ], etc. Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes

15 Question Question How can we obtain AdS-BH solutions with hypermultiplets? Setup and Result 4D N = 2 gauged SUGRA with VM and UHM from 10D type IIA on non-cy with fluxes Ansatze for matter fields Regular solution (AdS Black Hole) Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes

16 Contents Introduction 4D N = 2 Gauged SUGRA from 10D type IIA Vacua Black holes Summary and Discussions

17 Contents Introduction 4D N = 2 Gauged SUGRA from 10D type IIA Vacua Black holes Summary and Discussions

18 Example of Non-CY: coset space G 2 /SU(3) Coset space G 2 SU(3) D. Cassani and A.K. Kashani-Poor [arxiv: ] nearly-kähler (almost complex geometry) NSNS-sector : torsion and H-flux RR-sector : 2-, 4-form and Romans mass (0-form) 1 VM with cubic prepotential F = X1 X 1 X 1 X 0 1 UHM (no other HMs) nilmanifolds and solvmanifolds: M. Graña, R. Minasian, M. Petrini and A. Tomasiello [hep-th/ ] coset spaces with SU(3)- or SU(2)-structure: P. Koerber, D. Lüst and D. Tsimpis [arxiv: ] a pair of SU(3)-structures with (m ΛI, m Λ I): D. Gaiotto and A. Tomasiello [arxiv: ] Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes

19 Geometric flux compactification in type IIA 10D type IIA on G 2 SU(3) with fluxes S = 4D N = 2 abelian gauged SUGRA with B-field (Λ = 0, 1 and ξ 0 (ξ 0, ξ 0 ) T ) [ 1 2 R ( 1) µ ΛΣ F Λ F Σ ν ΛΣ F Λ F Σ g tt dt dt ) dφ dφ e 4φ e2φ db db (Dξ 0 Dξ 0 + D ξ 0 D ξ 0 + db ξ 0 d ξ db ( e RΛ e Λ0 ξ 0) A Λ 1 ] 2 mλ R e RΛ B B V ( 1) g µν, t, B µν, φ; (e 0 Λ, e Λ0 ) : NS-NS sector A Λ µ, ξ 0, ξ 0 ; (m Λ R, e RΛ) : R-R sector GM : (g µν, A 0 µ), VM : (A a µ, t), UHM TM : (φ, B µν, ξ 0, ξ 0 ) Precise data on G 2 SU(3) : e 10 0, m 0 R 0, e R0 0 Dξ 0 = dξ 0 e 0 Λ A Λ 1, D ξ 0 = d ξ 0 e Λ0 A Λ 1 e 0 Λ = 0 = e 00 F2 Σ = da Σ 1 + m Σ R B 2 m 1 R = 0 = e R1 V (t, φ, ξ 0 ) = V NS (t, φ) + V R (t, φ, ξ 0 ) µ ΛΣ ImN ΛΣ, ν ΛΣ ReN ΛΣ D. Cassani [arxiv: ] Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes

20 Equations of motion in gauged SUGRA with B-field R µν 1 2 R g µν = 1 4 g µν µ ΛΣ F Λ ρσf Σρσ µ ΛΣ F Λ µρf Σ νσ g ρσ g µν g tt ρ t ρ t + 2g tt µ t ν t g µν ρ φ ρ φ + 2 µ φ ν φ e 4φ 24 g µν H ρσλ H ρσλ + e 4φ 4 H ρσ µρσh ν (δg µν ) ) e2φ 2 g µν (D ρ ξ 0 D ρ ξ 0 + D ρ ξ0 D ρ ξ0 + e 2φ( ) D µ ξ 0 D ν ξ 0 + D µ ξ0 D ν ξ0 g µν V, 0 = 1 ( g µ µλσ F Σµσ) ( ) ϵµνρσ g 2 g µ ν ΛΣ Fνρ Σ + ϵµνρσ 2 g µb νρ (e RΛ ξ 0 e Λ0 ) e 2φ Q Λ0 D σ ξ 0, (δa Λ µ) 0 = 1 ( g ) µ gtt g µν ν t + 1 g 4 t(µ ΛΣ )FµνF Λ Σµν ϵµνρσ 8 g t(ν ΛΣ )FµνF Λ ρσ Σ t g tt µ t µ t t V, (δt) 0 = 2 ( g ) µ g µν ν φ + e4φ g 6 H µνρh µνρ e 2φ( ) D µ ξ 0 D µ ξ 0 + D µ ξ0 D µ ξ0 φ V, (δφ) 0 = 1 ( µ e 4φ gh µρσ) ] + ϵµνρσ [D µ ξ 0 (C H ) 00 D ν ξ 0 + (e RΛ ξ 0 e Λ0 )F Λ g g µν (δb µν ) +2m Λ Rµ ΛΣ F Σρσ ϵµνρσ g m Λ Rν ΛΣ F Σ µν, 0 = 2 g µ ( g e 2φ g µν D ν ξ 0) + V ξ 0 ϵµνρσ 2 g µb νρ D σ ξ 0 (C H ) 00. (δξ 0 ) Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes

21 Contents Introduction 4D N = 2 Gauged SUGRA from 10D type IIA Vacua Black holes Summary and Discussions

22 (Non)-SUSY AdS vacua Vacuum I : N = 1 t = ±1 + i 15 2 Vacuum II : N = 0 [ 3 5 (e 10 ) 2 t = ( ± 1 i 3 ) [ 3 5 (e 10 ) 2 Vacuum III : N = 0 t e R0 m 0 R ] [ 1/3, ξ 0 = m 0 R (e R0) e 10 [ 5 (e 10 ) 4 V = e R0 m 0 R ] 1/3, ξ 0 [ V = m 0 R (e R0) 5 = [ 9 m 0 R (e R0 ) 2 25 e 10 ] 1/3 [, exp(φ ) = 4 ] 1/3 5 e m 0 R (e R0 ) 2 ] 1/3 Λ I c.c. < 0 25 (e 10 ) 4 3 m 0 R (e R0 ) 5 ] 1/3 Λ II c.c. < 0 [ ] 12 e = i R0 1/3 5 (e10 ) 2 m 0, ξ 0 = 0, exp(φ ) = [ 5 R V = 25 [ 5 5 (e 10 ) 4 ] 1/ m 0 R (e R0) 5 Λ III c.c. < 0 ] 1/3 [ ] 1/3, exp(φ ) = 2 25 e m 0 R (e R0 ) 2 5 e m 0 R (e R0) 2 ] 1/3 Note: m 0 R > 0 ; ξ0 is not fixed ; Λ II c.c. < Λ I c.c. < Λ III c.c. D. Cassani and A.K. Kashani-Poor [arxiv: ] Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes

23 Contents Introduction 4D N = 2 Gauged SUGRA from 10D type IIA Vacua Black holes Summary and Discussions

24 Ansatze Consider spacetime metric (extremal, static, spherically symmetric AdS 2 S 2 ) ds 2 = e 2A(r) dt 2 + e 2A(r) dr 2 + e 2C(r) r 2( dθ 2 + sin 2 θ dϕ 2) Define electromagnetic charges p Λ 1 4π I S 2 F Λ 2, q Λ 1 4π S 2 FΛ2 [ p Λ µ ΛΣ p Σ + (q Λ ν ΛΓ p Γ )(µ 1 ) ΛΣ (q Σ ν Σ p ) ] Impose (covariantly) constant condition F Λ2 ν ΛΣ F Σ 2 + µ ΛΣ ( F Σ 2 ) 0 µ t, 0 µ φ, 0 D µ ξ 0, 0 D µ ξ0, 0 [µ B νρ] Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes

25 Equations of motion The equation of motion for g µν : δg tt δg rr : e 2C(r) = e2c 2 r 2 ( c1 r + 1 ) 2 δg θθ, δg ϕϕ : e 2A(r) = e 4c 6 I 2 1 e 4c 2(c 1 r + 1) [ 3 c 2 1 (c 1r + 1) 2 (c 1 r + 1) 3 { V + 6 c 1 a1 c 1 a 2 (c 1 r + 1) }] δg rr : a 2 = 1 2(c 1 ) 2 e2c 2 C(r) and A(r) are expressed in terms of I 1, V and constants of integration {a i, c i } e 2A(r) = e 2c 2 (c 1 ) 2 2a 1 c 1 (c 1 r + 1) + e 4c2I 1 (c 1 ) 2 (c 1 r + 1) 2 V 3(c 1 ) 2(c 1r + 1) 2 1 2η r new + Z2 r 2 new Choosing c 1 r + 1 r new (and c 1 1), we can read the Black Hole information: Λ c.c. 3 r2 new e 2c 2 1 : scalar curvature of S 2 a 1 η : mass parameter I 1 Z 2 : square of charges V Λ c.c. : cosmological constant Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes

26 Equations of motion The equations of motion for t, φ, ξ 0 : δt : 0 = e 4C I 1 r 4 t + V t V t = 0 and I 1 t = 0 δφ : 0 = 2 V NS + 4 V R V = V NS + V R = 1 2 V NS = V R δξ 0 : 0 = V ξ 0 = V NS ξ 0 + V ( R ξ 0 0 = V ξ 0 ) is trivial { t, ξ 0, φ ; V } BHs Regular Solution = { t, ξ 0, φ ; Λ c.c. } constant in whole region Vacua Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes

27 Equations of motion The equations of motion for t, B µν : δt : δb µν : 0 = e 4C I 1 r 4 t + V t 0 = m Λ R µ ΛΣ ( g 2 V t = 0 and I 1 t = 0 ϵ µνρσ F Σρσ) + m Λ R ν ΛΣ F Σ µν ( e RΛ e Λ0 ξ 0) F Λ µν and 0 = D µ ξ0 0 = [ µ, ν ] ξ 0 = e Λ0 F Λ µν with Fθϕ Λ = p Λ sin θ, Ftr Λ = 1 rnew 2 (µ 1 ) ΛΣ( q Σ ν ΣΓ p Γ) Solve them to find an appropriate BH charge configuration Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes

28 AdS black holes It turns out that all the charges are zero : p Λ = 0 = q Λ (highly non-trivial) I 1 Z 2 = 0, F Λ µν = 0 0 = Fµν 1 = 2 [µ A 1 ν] + m 1 R B µν A 1 µ = µ λ 0 (gauge-fixing) 0 = F 0 µν = 2 [µ A 0 ν] + m0 R B µν 2 [µ A 0 ν] = m 0 R B µν = (constant) 0 = D µ ξ0 = µ ξ0 e 00 A 0 µ e 10 A 1 µ = µ ξ0 ( e 00 = 0 = A 1 µ) Λ c.c. V < 0 η a 1 is still arbitrary Schwarzschild-AdS Black Holes Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes

29 Message 1 Black holes from CY : I 1 t = 0 Value of vector modulus t is not fixed at infinity attractor mechanism BH charges govern value of fields at horizon BH mass is given by BH charges Black holes from non-cy : V t = 0 and I 1 t = 0 Value of vector modulus t is (mostly) fixed at infinity moduli stabilization BH charges are governed by geometric- and RR-flux charges BH mass is arbitrary Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes

30 Message 2 (covariantly) constant solutions all the charges are zero! In order to find a black hole with non-trivial charges, the covariantly constant condition must be (partially) relaxed. cf.) Gauged SUGRA without hypermultiplets Fayet-Iliopoulos parameters can be involved There are charged black hole solutions with constant scalars. Bellucci, Ferrara, Marrani and Yeranyan [arxiv: ] Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes

31 Contents Introduction 4D N = 2 Gauged SUGRA from 10D type IIA Vacua Black holes Summary and Discussions

32 Summary and Discussions Studied : 4D N = 2 gauged SUGRA with VMs and TM(UHM) via flux compactification. Reconfirmed : Romans mass is inevitable. Imposed : covariantly constant condition. Found : Schwarzschild-AdS BHs. Different from cases of Calabi-Yau Find charged AdS-BH solutions. Consider a stationary AdS-BH. Various directions! Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes

33 Thanks for your attention

34 Appendix Terminology Calabi-Yau compactifications in type IIA Geometric flux compactifications in type IIA Scalar potential

35 Terminology and formulae on special Kähler geometry Prepotential : F is a holomorphic function of X Λ of degree two (F Λ = F/ X Λ ) Kähler potential : K V = log [ i ( X Λ F Λ X Λ F Λ )] ( ) ( Symplectic section : Π V e K V/2 X Λ L Λ F Λ M Λ Kähler metric : g ab = t a t b K V, Kähler covariant derivative : D a Π V = ( t a + 1 K ) V 2 t a Π V ), 1 = i ( L Λ M Λ L Λ M Λ ) t a = Xa X 0 ( f Λ a h Λa Period matrix : N ΛΣ = F ΛΣ + 2i (ImF) ΛΓX Γ (ImF) Σ X X Π (ImF) ΠΞ X Ξ ) Formulae : M Λ = N ΛΣ L Σ, h Λa = N ΛΣ f Σ a (Symplectic matrix) : (M V ) ΛΣ = ( 1 ReN 0 1 ) ( ImN 0 0 (ImN ) 1 ) ( 1 0 ReN 1 ) ( In a similar way... Π H e K H/2 Z I G I ), z i = Zi Z 0, K H = log [ i ( Z I G I Z I G I )], etc. Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes

36 Calabi-Yau Compactification in Type IIA Calabi-Yau 3-fold M CY Ricci-flat, torsionless, (compact) Kähler manifold with SU(3) holonomy group Invariant two-form J and three-form Ω on CY w.r.t. Levi-Civita connection: dj = [m J np] = 0, dω = [m Ω npq] = 0 This is suitable for 1/4-SUSY condition with vanishing background fields δ SUSY ψ m± = m ε (10D) ± = 0 ε (10D) + = ε (4D) 1+ η1 + + (c.c.), ε (10D) (ε (4D) 1,2+ )c = ε (4D) 1,2, (η1,2 ) = η 1,2 + = : = ε (4D) 2+ η2 + (c.c.) SU(3)-invariant SU(4) SO(6) η 1 ± = η 2 ± η ±, J mn i η ± γ mn η ±, Ω i η γ mnp η + Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes

37 Calabi-Yau Compactification in Type IIA NS-NS fields in 10D are expanded around CY : R-R fields : ϕ(x, y) = φ(x) ( ) (χȷ g mn (x, y) = iv a (x) (ω a ) mn (y), g mn (x, y) = i z ȷ ) mpq Ω pq n (x) Ω 2 (y) B 2 (x, y) = B 2 (x) + b a (x)ω a (y) t a (x) b a (x) + iv a (x) Ĉ 1 (x, y) = A 0 1(x) Ĉ 3 (x, y) = A a 1(x) ω a (y) + ξ I (x)α I (y) ξ I (x)β I (y) cohomology class on CY basis degrees H (1,1) ω a a = 1,..., h (1,1) H (0) H (1,1) ω Λ = (1, ω a ) Λ = 0, 1,..., h (1,1) H (2,2) H (6) ω Λ = ( ω a, vol. vol. ) H (2,1) χ i i = 1,..., h (2,1) H (3) (α I, β I ) I = 0, 1,..., h (2,1) dω Λ = 0 = d ω Λ dα I = 0 = dβ I Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes

38 Calabi-Yau Compactification in Type IIA 10D Type IIA action S (10D) IIA = S NS + S R + S CS : S NS = 1 e 2ϕ{ 1 } R 1 + 4dϕ dϕ 2 2Ĥ3 Ĥ3 S R + S CS = 1 { F2 4 F 2 + ( F4 Ĉ1 F ) ( 3 F4 Ĉ1 F ) } B 2 F 4 F 4 4D N = 2 ungauged SUGRA: Neither gauge couplings, Nor scalar potential S (4D) = {1 2 R 1 G ab dta dt b h uv dq u dq v ImN ΛΣF Λ F Σ ReN ΛΣF Λ F Σ} gravitational multiplet g µν, A 0 1 vector multiplet (VM) A a 1, t a, t b t a SKG V hypermultiplet (HM) z i, z ȷ, ξ i, ξj z i SKG H universal hypermultiplet (UHM) φ, a, ξ 0, ξ0 a B 2 (Hodge dual) {q u } 4n H + 4 HM = Special QG = {z i, z ȷ } + {ξ i, ξ j } + {φ, a, ξ 0, ξ 0 } 2n H (SKG H ) 2n H 4 (UHM) = {z i, z ȷ } SKG H + {φ} + {a, ξ I, ξ J } Heisenberg Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes

39 Geometric flux compactification in type IIA 10D type IIA action S (10D) S NS = 1 2 IIA = S NS + S R = S NS + S R + S CS : (democratic form) e 2ϕ{ 1 } R 1 + 4dϕ dϕ 2Ĥ3 Ĥ3, SR = 1 8 [ F F] 10 with constraint F = λ( F) and EoM (Bianchi) (d + Ĥ ) F = 0 (d Ĥ ) F = 0 non-cy with SU(3)-structure with m Λ R = 0 non-cy with SU(3)-structure with mλ R = 0 4D N = 2 abelian gauged SUGRA (with ξ I (ξ I, ξ I ) T ): S (4D) = d 4 x g [ 1 2 R ImN ΛΣF Λ µνf Σµν ϵµνρσ 8 g ReN ΛΣF Λ µνf Σ ρσ g ab µ t a µ t b g iȷ µ z i µ z ȷ µ φ µ φ + e2φ 2 (M H) IJ D µ ξ I D µ ξ J e2φ 4 ( Dµ a ξ I (C H ) IJ D µ ξ J) 2 V (t, t, q) ] (e Λ I, e ΛI ) : geometric flux charges & e RΛ : RR-flux charges (with constraints e Λ I e ΣI e ΛI e Σ I = 0) t a SKG V and z i SKG H HM are ungauged (in general) non-cy data D µ ξ I = µ ξ I e Λ I A Λ µ & D µ ξi = µ ξi e ΛI A Λ µ D µ a = µ a (2e RΛ ξ I e ΛI + ξ I e Λ I )A Λ µ V (t, t, q): scalar potential D. Cassani [arxiv: ] Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes

40 Generic form of 4D N = 2 gauged SUGRA with B-field Non-vanishing m Λ R dualizes the axion field a in standard SUGRA to B-field. 4D gauged action is different from the standard one: S (4D) = [ 1 2 R( 1) ImN ΛΣF Λ 2 F Σ ReN ΛΣF Λ 2 F Σ 2 g ab dt a dt b g iȷ dz i dz ȷ dφ dφ e 4φ 4 H 3 H 3 e2φ 2 (M H) IJ Dξ I Dξ J V ( 1) + 1 [ξ 2 db I (C H ) IJ Dξ J + ( 2e RΛ ξ I e ΛI + ξ I I e ) ] Λ A Λ 1 1 ] 2 mλ Re RΛ B 2 B 2 Constraints among flux charges: e Λ I e ΣI e ΛI e Σ I = 0, m Λ R e Λ I = 0 = m Λ R e ΛI Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes

41 Scalar potential Scalar potential from (non)geometric flux compactifications: V = g 2[ 4h uv k u k v + 3 x=1 V NS = g ab D a P + D b P + + g iȷ D i P + D ȷ P + 2 P + 2 ( g ab D a P x D b P x 3 P x 2)] =... V NS + V R (abelian: k a Λ = 0) = 2 g 2 e 2φ[ Π T Q H T M V Q ΠH + Π T V Q M H Q T Π V + 4Π T H C T H Q T( Π V Π T V + Π V Π T ) ] V Q CH Π H V R = g ab D a P 3 D b P 3 + P 3 2 = 1 2 g2 e 4φ( e RΛ e ΛI ξ I + e Λ I ξi ) (ImN ) 1 ΛΣ ( e RΣ e ΣI ξ I + e Σ I ξi ) Π V = e KV/2 (X Λ, F Λ ) T t a = X a /X 0 a = 1,..., n V SKG V of vector-moduli ( 0 1 C V,H = 1 0 ) P + P 1 + ip 2 = 2e φ Π T V Q C H Π H P P 1 ip 2 = 2e φ Π T V Q C H Π H ; Q = P 3 m ΛI m Λ I = e 2φ Π T V C V (c R + Qξ) ( ) I eλ e ΛI, Q = C T H Q C V c R = Π H = e KH/2 (Z I, G I ) T z i = Z i /Z 0 i = 1,..., n H SKG H of hyper-moduli ( ) m Λ R e RΛ Cassani et.al. [arxiv: ], [arxiv: ] Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes

42 Flux Compactifications in Type IIA Coset spaces with SU(3)-structure: D. Cassani and A.K. Kashani-Poor [arxiv: ] M 6 G 2 SU(3) = S6 Sp(2) S(U(2) U(1)) = CP 3 SU(3) U(1) U(1) = F(1, 2; 3) SM = SKG V SU(1, 1) U(1) ( SU(1, 1) ) 2 : t 3 : st 2 U(1) ( SU(1, 1) ) 3 : stu U(1) HM = SQG SU(2, 1) U(2) : UHM SU(2, 1) U(2) : UHM SU(2, 1) U(2) : UHM SKG H HM matters 1 VM + 1 UHM 2 VM + 1 UHM 3 VM + 1 UHM Each SKG V has a cubic prepotential: F = 1 3! d abc X a X b X c X 0 nilmanifolds and solvmanifolds: M. Graña, R. Minasian, M. Petrini and A. Tomasiello [hep-th/ ] coset spaces with SU(3)- or SU(2)-structure: P. Koerber, D. Lüst and D. Tsimpis [arxiv: ] a pair of SU(3)-structures with (m ΛI, m Λ I): D. Gaiotto and A. Tomasiello [arxiv: ] Tetsuji KIMURA : Flux Compactifications, N=2 Gauged SUGRA and Black Holes

43 Fin.

3 exotica

3 exotica ( / ) 2013 2 23 embedding tensors (non)geometric fluxes exotic branes + D U-duality G 0 R-symmetry H dim(g 0 /H) T-duality 11 1 1 0 1 IIA R + 1 1 1 IIB SL(2, R) SO(2) 2 1 9 GL(2, R) SO(2) 3 SO(1, 1) 8