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1 ( ) Flux Compactifications, and Gauged Supergravities ( )

2

3 ( ) Gauged supergravity : ( ) ( R )

4 Nucl. Phys. B258 (1985) 46 10D M 6 4D N = 1 10D = 4D Minkowski + 6D internal space M 6 : ds 2 10D = η µν dx µ dx ν 4D + g mn (x, y) dy m dy n M 6 4D N = 1 vacuum : η = on M 6 with 0 = δψ m = m η +... Trivial background on M 6 : H 3 = 0 = dϕ Anomaly cancellation condition : 0 = Tr { R 2 R 2 F 2 F 2 } M 6 = Calabi-Yau Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA - 4 -

5 Calabi-Yau Calabi-Yau M CY Ricci Kähler SU(3) SU(4) SO(6) Levi-Civita 2 (J) 3 (Ω) dj = [m J np] = 0 dω = [m Ω npq] = 0 Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA - 5 -

6 Calabi-Yau compactification in type IIA NS-NS jump 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) R-R C 1 (x, y) = A 0 1(x) C 3 (x, y) = A a 1(x) ω a (y) + ξ I (x)α I (y) ξ I (x)β I (y) 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 dα I = 0 = d ω Λ = 0 = dβ I Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA - 6 -

7 D Q "12" F M IIB IIA HE8 HSO I S torus 9 U M K3 8 U F Calabi-Yau 7 U M U M F (2,2) (2,0) (1,1) (1,0) U IIB M F F U S N=8 N=4 N=2 N=1 S-duality in hep-th/ ? No! / type II abelian Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA - 7 -

8 SU(3)-structure manifolds non-cy manifold M 6 Ricci 2-form (SU(3)-structure manifold) 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 W 1, W 2, W 3, W 4, W 5 : intrinsic torsion classes G 2 /SU(3) cosets Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA - 8 -

9 Heterotic string on SU(3)-structure manifolds if dh = 0 : smooth M 6 with fluxes is reduced to M CY without fluxes Piljin Yi and TK, hep-th/ Index theorem on torsionful manifolds ω 1 3 H relation among ω H ω + H Fluxes, α corrections and SUSY vacua (Dirac eq.) (SUSY variation) (invariant polynomial) TK, arxiv: hep-th/ , arxiv: Intersecting five-branes in heterotic string S. Mizoguchi and TK, arxiv: , arxiv: Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA - 9 -

10 Calabi-Yau abelian (type II strings ) Kaluza-Klein (NSNS-sector ) : truncated out (32-SUSY 8-SUSY) (NSNS-sector ) (RR-sector ) Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

11 Non-abelian non-abelian type II non-abelian (heterotic/type II duality in lower-dimensions)

12 (I) (II) (III)?? (IV),(V),(VI)?? Calabi-Yau SU(3)-structure manifolds Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

13 Generalized Geometry and Doubled Formalism

14 jump 4D N = 8 SUGRA with CSO(p, q, r) gauge symmetry, etc. jump (Non)geometric String Backgrounds (?) Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

15 (Non)geometric String Backgrounds? = Diffeo (GL(d, R)) (O(d, d), U- ) }{{} GL(d, R) duality transf. d M d Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

16 Generalized Geometries and Doubled Formalism math/ hep-th/ N.J. Hitchin C.M. Hull Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

17 An extension of geometrical structure g mn geometry associated with g mn geometry associated with g mn, B mn geometry associated with g mn, B mn, C (p) Conventional geometry (manifold) O(6) global symmetry Generalized geometry O(6, 6) T-duality symmetry Exceptional generalized geometry E 7(7) U-duality symmetry Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

18 (almost) complex geometry almost complex structure J m n on T M 6 s.t. J m n : T M 6 T M 6 J 2 = 1 6 O(6) invariant metric η, s.t. J T ηj = η Structure group on T M 6 : η mn GL(6) O(6) η mn, ε m1 m 6 O(6) SO(6) η mn, ε m1 m 6, J mn SO(6) U(3) η mn, ε m1 m 6, J mn, Ω mnp U(3) SU(3) Connection between geometry and physics : J mn = 2i η ± γ mn η ±, Ω mnp = 2i η γ mnp η + Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

19 Generalized (almost complex) geometry a generalized almost complex structure J Π Σ on T M 6 T M 6 s.t. J Π Σ : T M 6 T M 6 T M 6 T M 6 J 2 = 1 12 O(6, 6) invariant metric L, s.t. J T LJ = L Structure group on T M 6 T M 6 : Φ + = k=0 L GL(12) O(6, 6) J 2 = 1 12 O(6, 6) U(3, 3) J 1, J 2 U 1 (3, 3) U 2 (3, 3) U(3) U(3) integrable J 1,2 U(3) U(3) SU(3) SU(3) J ±ΠΣ = ReΦ ±, Γ ΠΣ ReΦ ± 1 k! η +γ mk m 1 η + γ m 1 m k e ij, Φ = k=0 1 k! η γ mk m 1 η + γ m 1 m k iω Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

20 Exceptional generalized geometry generalized geoemetry SU(3) SU(3) E 7(7) T M 6 T M 6 momentum winding (GCT) (B 2 gauge) 6 6 T M 6 T M 6 Λ 5 T M 6 Λ 5 T M 6 Λ even T M 6 momentum winding NS5 KK5 D0, D2, D4, D6 (GCT) (B 2 gauge) (B 6 gauge) (GCT of dual vielbein) (C (p), C (8 p) gauges) db 6 = 10 db 2 ; dc (8 p) = 10 dc (p) (p = 1, 3) in type IIA = 12 : O(6, 6) ( ) = 56 : E 7(7) hep-th/ , arxiv: , arxiv: , arxiv: , etc. jump Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

21 Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

22 Doubled formalism: Doubled geometry and double field theory general coord. trsf. GL(12, Z) sigma model doubled/polarised on M 12 Û = U T 12 doubled/polarised sigma model on M 6 U = U T 6 sigma model on M 6 Ũ = U T 6 general coord. trsf. GL(6,Z) Z 15 T-duality trsf. O(6,6;Z) general coord. trsf. GL(6,Z) Z 15 generalized geometry geometry M 6 M 12 Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

23 Sigma model action on a doubled space M 12 with scalar moduli matrix M IJ : 1 S = Σ 4 M MN dy M dy N ( ) gmn B mp g pq B qn B mp g pn M MN = g mp B qn g mn Y M = ( ) Y m Ỹ m M MN takes value in coset O(6, 6) global symmetry by g O(6, 6) : Y M O(6, 6) O(6) O(6) Y M = g M NY N fractional transformation on M mn = g mn + B mn : g = ( A β Θ Θ : A : β : A T ) : M ( A T M + Θ )( βm + A ) 1 B B B + Θ (Θ = dθ) g mn g mn B mn T-duality generalized geometry generalized vector V = (v, ξ) T (v T M 6, ξ T M 6 ) compact spaces M 6, M 12 double field theory Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

24 ungauged SUGRA gauged SUGRA

25 B. de Wit H. Samtleben M. Trigiante and many guys.. Embedding Tensor Formalism from hep-th/ : X M = Θ M α t α : µ D µ = µ ga M µ X M # SUSY 9D 8D 7D 6D 5D 4D 3D 32 arxiv: arxiv: hep-th/ arxiv: hep-th/ arxiv: hep-th/ hep-th/ arxiv: arxiv: (arxiv: ) Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

26 Embedding tensor formalism isometry group G G 0 A M µ ( G 0 ) X M = Θ M α t α ( G) [X M, X N ] = T MN P X P (T MN P Θ M α (t α ) N P ) δa M µ = µ Λ M +ga P µ T P Q M Λ Q D µ Λ M full covariance F µν g 1 [D µ, D ν ] M = µ A M ν ν A M µ + gt [NP ] M A N µ A P ν δf M µν = gλ P T NP M F M µν 2g T (P Q) M A P [µ δaq ν] not covariant! δa M µ = D µ Λ M gt M (P Q) (P Q) Ξ µ, Hµν M F M µν + gt M (P Q) B (P Q) µν Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

27 Embedding tensor Θ M α Exceptional generalized geometry E 7(7) Flux compactifications (non)geometric fluxes (?) Non-abelianity (!?) Heterotic string compactifications

28 (I) (II) (III) (IV),(V),(VI) Calabi-Yau SU(3)-structure manifolds / generalized geometries doubled formalism / exceptional generalized geometries embedding tensor formalism Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

29 non-abelianity

30 AdS arxiv: , arxiv: F Λ 2 = da Λ 1 + m Λ B 2 Embedding tensor formalism electric frame magnetic frame ( ) (work in progress??) (P Q) B µν arxiv: in { vector multiplets vector-tensor multiplets? hep-th/ hypermultiplets hyper-tensor multiplets? hep-th/ Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

31 AdS in 4D N = 2 gauged SUGRA with B-field from massive type IIA on a nearly-kähler coset TK, arxiv: , arxiv:

32 AdS Reissner-Nordström AdS N = 2 gauged SUGRA AdS/CMT AdS Einstein + Λ c.c. + Maxwell + charged matters Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

33 AdS vs vs (4D) vs (10D) Romans mass (type IIA) Maxwell vs (4D) field strength Romans mass AdS/CMT type IIA (?) Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

34 a nearly-kähler coset G 2 /SU(3) D. Cassani and A.K. Kashani-Poor [arxiv: ] NSNS-sector : torsion and H-flux RR-sector : 2-, 4-form and Romans mass (0-form) dj = 3 2 Im(W 1Ω), dω = W 1 J J jump dω Λ = e Λ α, dα = 0, dβ = e Λ ω Λ, d ω Λ = 0 jump e Λ m Λ R = 0 1 vector multiplet with cubic prepotential F = X1 X 1 X 1 X 0 1 universal hypermultiplet (no other HMs) hyper-tensor multiplet Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

35 4D supersymmetric multiplets 4D multiplets from NSNS from RR fermions supergravity multiplet g µν A 0 µ ψ i µ 1 vector multiplet t A 1 µ λ i 1 hyper-tensor multiplet φ, B µν ξ, ξ ζ α A 0 µ : from RR one-form C 1 t : from complexified Kähler modulus t = X 1 /X 0 = b + iv A 1 µ : from RR three-form C 3 ξ, ξ : from RR three-form C 3 jump F Λ 2 = da Λ 1 + m Λ RB 2 Stückelberg-type deformation Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

36 in arxiv: in arxiv: jump N = 1 AdS N = 0 AdS jump Λ N c.c. =0 < Λ N c.c. =1 < Λ N c.c. =0 [NOTE] type IIA 4D N = 1 AdS vacua Romans mass D. Lüst and D. Tsimpis, hep-th/ Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

37 RN-AdS Reissner-Nordström AdS ds 2 = e 2A(r) dt 2 + e 2A(r) dr 2 + r 2 dω 2 e 2A(r) = 1 2η r + Z2 r 2 + r2 l 2, Z2 = Q 2 + P 2, Λ c.c. = 3 l 2 vector fields A Λ µ pλ = F Λ 2, q Λ = F Λ θϕ f Λ (θ, ϕ) sin θ, F Λ tr e 2C(r) r 2 g Λ (θ, ϕ) F 2 Λ A Λ µ, B µν, g µν 0 = µ t = µ φ = D µ ξ = D µ ξ = [µ B νρ] = F Λ µν Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

38 RN-AdS Massive type IIA on G 2 /SU(3) 4D N = 2 gauged SUGRA with B-field Reissner-Nordström AdS (Z 2 = 0 ) (...) Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

39

40 type II non-abelian gauge symmetries generalized geometry, doubled formalism embedding tensor formalism embedding tensor ) AdS-BH, embedding tensor Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

41 Embedding tensor formalism in 4D local N = 2 [arxiv: ] Conformal supergravity with off-shell Weyl multiplet off-shell vector multiplets on-shell hypermultiplets ( Poincaré supergravity) Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

42 Lie LiE: a computer algebra package for Lie group computations (?) cf. R. Slansky, Phys. Rept. 79 (1981) 1 E 7(7) SU(8) E 7(7) SL(2) SO(6, 6) E 7(7) ( ) S = = = = = = = = (2, 12) + (1, 32) 133 = (1, 66) + (3, 1) + (2, 32 ) 912 = (2, 220) + (2, 12) + (1, 352 ) + (3, 32) 8645 = (1, 66) + (1, 2079) + (3, 66) + (3, 495) + (3, 1) + (4, 462 ) + (2, 32) + (2, 352) + (2, 1728 ) + (4, 32 ) Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

43 Appendix Type IIA on CY Intrinsic torsion Flux charges 4D N = 2 gauged SUGRA Coset spaces Gauged SUGRA from type IIA on G 2 /SU(3) Duality groups CSO(p, q, r) Truncation from N = 8 to N = 4, 2, 1 4D N = 2 gauged conformal SUGRA

44 4D N = 2 SUGRA from type IIA on Calabi-Yau 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 2 F Σ ReN ΛΣF Λ 2 F Σ 2 } 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) HM = Special QG App.top {q u } 4n H + 4 = {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 App.top Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

45 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 special hermitian W 1 = W 2 = W 4 = W 5 = 0 Kähler W 1 = W 2 = W 3 = W 4 = 0 jump App.top Calabi-Yau 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 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 jump App.top Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

46 Flux charges 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 ] F(k) ) R-R 1 2 Ĝ = (G Λ 0 + G Λ 2 + G Λ 4 ) ω Λ ( G 0 Λ + G 2 Λ + G 4 Λ ) ω Λ +(G I 1 + G I 3) α I ( G 1 I + G 3 I ) β I G Λ 0 p Λ, G0 Λ q Λ ξ I e ΛI + ξ I e Λ I App.top c (p Λ, q Λ ) T : R-R flux charges (p 0 : Romans mass) App.top Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

47 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 and Gauged SUGRA

48 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 and Gauged SUGRA

49 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: App.top Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

50 Coset spaces with SU(3)-structure D. Cassani and A.K. Kashani-Poor, arxiv: jump App.top 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 and Gauged SUGRA

51 4D N = 2 gauged SUGRA from type IIA on G 2 /SU(3) 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 and Gauged SUGRA

52 Equations of motion 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 and Gauged SUGRA

53 (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 Note: m 0 R > 0 ; ξ0 is not fixed ; Λ II c.c. < Λ I c.c. < Λ III c.c. arxiv: jump App.top ] 1/3 Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

54 Duality groups scalar field space : M = G/H D maximal SUGRA half-maximal SUGRA G H G H 9 GL(2) SO(2) GL(1) SO(1, 1 + n) SO(1 + n) 8 SL(2) SL(3) SO(2) SO(3) GL(1) SO(2, 2 + n) SO(2) SO(2 + n) 7 SL(5) SO(5) GL(1) SO(3, 3 + n) SO(3) SO(3 + n) 6 SO(5, 5) SO(5) SO(5) GL(1) SO(4, 4 + n) SO(4) SO(4 + n) 5 E 6(6) USp(8) GL(1) SO(5, 5 + n) SO(5) SO(5 + n) 4 E 7(7) SU(8) SL(2) SO(6, 6 + n) SO(6) SO(6 + n) U-duality S-duality T-duality see, for instance, arxiv: App.top Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

55 CSO(p, q, r) Left-invariant one-form E = g 1 dg with g G de = E E or de A = 1 2 f BC A E B E C with E = E A T A, [T A, T B ] = f AB C T C This is written as de AB θ CD E AC E DB with E AB = E C (T C ) AB θ CD = diag.(+1, }. {{.., +1 }, 1, }. {{.., 1 }, 0, }. {{.., 0 } ) p q r CSO(p, q, 0) = SO(p, q) CSO(p, q, 1) = ISO(p, q) etc. CSO(p, q, r) = SO(p, q) R (p+q)r etc. jump App.top This terminology is intended by C.M. Hull, Phys. Lett. B148 (1984) 297 Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

56 Truncations from N = 8 to N = 4, 2, 1 N = 8 : E 7(7) SU(8) vectors 56 scalars 133 N = 2 : ( ) SL(2) G 2(2) SO(2) T SO(4) vectors (4, 1) scalars (3, 1) + (1, 14) N = 2 : ( ) SL(2) SO(2) T vectors (4, 1) scalars (3, 1) + (1, 8) SU(2, 1) SU(2) U(1) U emb tens 912 N = 4 : ( ) SL(2) SO(2) emb tens (2, 1) + (4, 14) + (2, 7) emb tens (2, 1) + (4, 8) + (2, 1) S vectors (2, 12) SO(6, 6) SO(6) SO(6) scalars (3, 1) + (1, 66) emb tens (2, 12) + (2, 220) N = 1 : Φ=S,T,U vectors ( ) SL(2) SO(2) scalars (3, 1, 1) + (1, 3, 1) + (1, 1, 3) emb tens (2, 2, 2) + (2, 4, 4) Φ N = 1 : Φ=S,T ( ) SL(2) SO(2) vectors Φ R U scalars (1, 1) + (3, 1) + (1, 3) emb tens (2, 4) + (2, 4) Starting from gauged maximal supergravity, one can move step by step downwards or towards the right by performing group-theoretical truncations. The labels S, T and U are introduced in order to keep track of the different group factors along the truncations. G. Dibitetto, A. Guarino and D. Roest, arxiv: jump App.top Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

57 4D N = 2 conformal gauged supergravity L = L (1) kin + L (2) kin + L aux + L conf + L H,conf + L top + L g + L g 2 ] e 1 L (1) kin = iω MN D µ X M D µ X N + i 4 Ω MN [Ω im /DΩ N i Ω M i /DΩ in e 1 L (2) kin = i 4 [ F ΛΣ H Λ µν H µνσ F ΛΣ H +Λ µν H +µνσ ] + i 2 Ω MN [ψ i µ /DX M γ µ Ω N i ψ µi /DX M γ µ Ω in ] [ ] O µνλ H µνλ N ΛΣ O µνλ O µν Σ + (h.c.) e 1 L aux = 1 8 [N N ΛΣ ΛΓ Yij Γ 2( i FΛΓΠ Ω Γ i ΩΠ j F ) ][ ( ΛΓΠΩ kγ Ω lπ ε ik ε jl N ΣΞ Y ijξ i 2 F ΣΞ Ω iξ Ω j F ΣΞ Ω Ξ mω n ε im ε jn)] e 1 L conf = ( )] ] 1 6 [R K + e 1 ε µνρσ ψ i µγ ν D ρ ψ σi ψ i µψνt j µν ij + (h.c.) K [D + 12 ψi µγ µ χ i + 12 ψ µiγ µ χ i ( ) ] ( ) ] 1 [K Λ 4 e 1 ε µνρσ ψ µi γ ν ψρd i σ X Λ ψ µiγ µ γ ρσ Ω Λ j T ρσij 1 + (h.c.) [K Λ 3 ΩΛ i γµν D µ ψν i Ω Λ i χi + (h.c.) e 1 L H,conf = ( )] ] 1 6 [R χ + e 1 ε µνρσ ψ i µγ ν D ρ ψ σi 1 4 ψi µψνt j µν ij + (h.c.) + [D 12 χ + 12 ψi µγ µ χ i + 12 ψ µiγ µ χ i 1 2 G ( αβd µ A β i D µ A iα G αβ ζ α /Dζ β + ζ β /Dζ α) ( ) ] 1 4 W αβγδ ζ α γ µ ζ β ζ γ γ µ ζ δ χ A [γ iα A 2 3 ζα γ µν D µ ψν i + ζ α χ i 1 6 ζα γ µ ψ νi T µνij + (h.c.) [ Ω ( αβζ α γ µν T µνij ε ij ζ β 1 2 ζα γ µ γ ν ψ µi ψ i ν G αβ ζ β + ε ij Ω αβ ψ νj ζ β) ] + G αβ ζ β γ µ /DA iα ψ µi 1 4 e 1 ε µνρσ G αβ ψ i µγ ν ψ ρj A β i D σ A jα + (h.c.) e 1 L top = i 8 ge 1 ε µνρσ( Θ Λa B µνa + Θ Λm B µνm ) ( 2 ρ W σλ + gt MNΛ W M ρ W N σ 1 4 gθ Λ b B ρσb 1 4 gθ Λ n B ρσn ) + i 3 ge 1 ε µνρσ T MNΛ W M µ W N ν e 1 L g = 1 2 g [iω MQ T P N Q ε ij X N Ω M i ( ) ρ Wσ Λ gt P Q Λ Wρ P Wσ Q + i 6 ge 1 ε µνρσ T Λ MN Wµ M Wν N ( Ω P j + γ µ ψ µj X P ) + (h.c.)] [ + 2g k AM γiα A εij ζ α( Ω M j ( ) ρ W σλ gt P QΛWρ P Wσ Q + γ µ ψ µj X M) ] [ ( + (h.c.) + g µ ij Mψ µi γ µ Ω M j + γ µν ψ νj X M) ] + (h.c.) ] ] [ ] +2g [X M γ T M α Ω βγ ζ α ζ β + X M γ T M α Ω βγ ζ α ζ β [F 14 g ΛΣΓ µ ijλ Ω Σ i ΩΓ j + F ΛΣΓµ Λ ij Ω iσ Ω jγ + gy ijλ µ ijλ (F Σ ΛΣ + F ΛΣ )µ ij e 1 L g 2 = ig 2 Ω MN ( TP Q M X P X Q)( T RS N X R X S) 2g 2 k A Mk B Ng AB X M X N 1 2 g2 N ΛΣ µ ij Λ µ ijσ B. de Wit and M. van Zalk, arxiv: jump App.top Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

58 torsionful manifolds : hep-th/ index theorem with torsion : arxiv: intersecting five-branes : arxiv: , arxiv: type II doubled geometries, generalized geometries : arxiv: , arxiv: , arxiv: AdS gauged SUGRA via flux compactifications : arxiv: , arxiv: , arxiv: App.top Tetsuji KIMURA : Flux Compactifications and Gauged SUGRA

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