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Ads Vacua, Attractor Mechanism and Generalized Geometries Institute for the Physics and Mathematics of the Universe (IPMU), University of Tokyo: Mar 05, 2009 AdS Vacua, Attractor Mechanism and Generalized Geometries based on arXiv:0810.0937 [hep-th] Tetsuji KIMURA Yukawa Institute for Theoretical Physics, Kyoto University Introduction We are looking for the origin of 4D physics Physical information Particle contents and spectra (Broken) symmetries Potential, vacuum and cosmological constant TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES What kind of 4D models come from String Theories? # What kind of Compactifications? 4 = 10 − 6 = 11 − 7 TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Q D 32 16 8 4 "12" F 11 M IIBIIA HE8 HSO I torus 10 S || {{{ 9 U M K3 ······ 8 U F Calabi-Yau 7 U M F S-duality 6 U M { (2,2) (2,0) (1,1) (1,0) 5 U IIB M FF 4 U S N=8N=4 N=2 N=1 B. de Wit and J. Louis, in the Proceedings \NATO Advanced Study Institute on Strings, Branes and Dualities (1997)" hep-th/9801132 TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Many Abelian Supergravities (SUGRA) in lower dimensions Compactifications on Tori, Calabi-Yaus, etc. Minkowski ground state, massless fields Global E7 symmetry (4D N = 8 case) Many Gauged SUGRA in lower dimensions Compactifications on group manifolds, torsionful manifolds, etc. Scalar potential generating masses [Moduli Stabilization] Non-trivial cosmological constant TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES There are various Gauged SUGRA which cannot be derived from String Theories compactified on conventional geometric backgrounds TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES There are various Gauged SUGRA which cannot be derived from String Theories compactified on conventional geometric backgrounds L99 We want to derive all Gauged SUGRA from String Theories TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES There are various Gauged SUGRA which cannot be derived from String Theories compactified on conventional geometric backgrounds L99 We want to derive all Gauged SUGRA from String Theories Compactify String Theories on non-conventional geometries: TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES There are various Gauged SUGRA which cannot be derived from String Theories compactified on conventional geometric backgrounds L99 We want to derive all Gauged SUGRA from String Theories Compactify String Theories on non-conventional geometries: Nongeometric String Backgrounds TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES What is a Nongeometric String Background? Structure group = Diffeo. (GL(d; R)) + |Duality transf.{z groups} " coming from String dualities GL(d; R) + duality transf. d-dim. internal space Md ' monodrofold TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES SUGRA on Nongeometric String Backgrounds ex.) Lower-dim. Gauged SUGRA compactified by Scherk-Schwarz mechanism c c [Za;Zb] = fab Zc + Habc X a b ab c abc \Kaloper-Myers" algebra: [X ;X ] = Q c X + R Zc a a c ac [X ;Zb] = f bc X − Q b Zc Various “fluxes” are involved N. Kaloper, R.C. Myers hep-th/9901045 J. Shelton, W. Taylor, B. Wecht hep-th/0508133, A. Dabholkar, C.M. Hull hep-th/0512005 M. Gra~na,R. Minasian, M. Petrini, D. Waldram arXiv:0807.4527 TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES ' $ String Theories compactified on Nongeometric Backgrounds # All(?) Gauged SUGRA & % Hitchin's Generalized Geometries to study vacua Hull's Doubled Formalism to find gauge symmetries IPMU Workshop \Supersymmetry in Complex Geometry" (January 2009) TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES 4D N = 1 supergravity 4D N = 1 action: Z ( ) 1 1 S = R ∗ 1 − F a ^ ∗F a − K rφM ^ ∗∇φN − V ∗ 1 2 2 MN ( ) K MN 2 1 a 2 V = e K DMW DN W − 3jWj + jD j 2 K : K¨ahlerpotential K c W : superpotential L99 δ µ = rµ " − e 2 W γµ " a L99 a a µν a D : D-term δχ = ImFµνγ " + iD " 10D string theory provides K, W, Da via compactifications: 10 = 4 + 6 TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Motivation and Results Search 4D SUSY vacua in type IIA theory compactified on generalized geometries Moduli stabilization We find SUSY AdS (or Minkowski) vacua Mathematical feature We obtain a powerful rule to evaluate vacua: Discriminants of superpotentials governing the cosmological constant Stringy effects We see α0 corrections in certain configurations TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Contents Introduction Killing Spinors and Fluxes Generalized (Complex) Geometries Exterior Derivatives and Flux Charges Setup in N = 1 Theory My Work: Search of SUSY AdS Vacua Summary and Discussions Contents Introduction Killing Spinors and Fluxes Generalized (Complex) Geometries Exterior Derivatives and Flux Charges Setup in N = 1 Theory My Work: Search of SUSY AdS Vacua Summary and Discussions Compactifications in 10D type IIA string Decompose 10D type IIA SUSY parameters: 1 ⊗ 1 c ⊗ 1 2 ⊗ 2 c ⊗ 2 = "1 (a η−) + "1 (a η+) ; = "2 (b η+) + "2 (b η−) TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Compactifications in 10D type IIA string Decompose 10D type IIA SUSY parameters: 1 ⊗ 1 c ⊗ 1 2 ⊗ 2 c ⊗ 2 = "1 (a η−) + "1 (a η+) ; = "2 (b η+) + "2 (b η−) δ(fermions) = 0 provide Killing spinor equations on the 6D internal space M: ( ) 1 ( ) ( ) δ A = @ + ! γab ηA + 3-form fluxes · η A + other fluxes · η A = 0 m m 4 mab 1 2 with a pair of SU(3) invariant Weyl spinors η+, η+: 2 1 0 m 1 − 0 m ≡ 1y m 2 η+ = ck(y)η+ + c?(y)(v + iv ) γm η− ; (v iv ) η+ γ η− TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Compactifications in 10D type IIA string Decompose 10D type IIA SUSY parameters: 1 ⊗ 1 c ⊗ 1 2 ⊗ 2 c ⊗ 2 = "1 (a η−) + "1 (a η+) ; = "2 (b η+) + "2 (b η−) δ(fermions) = 0 provide Killing spinor equations on the 6D internal space M: ( ) 1 ( ) ( ) δ A = @ + ! γab ηA + 3-form fluxes · η A + other fluxes · η A = 0 m m 4 mab 1 2 with a pair of SU(3) invariant Weyl spinors η+, η+: 2 1 0 m 1 − 0 m ≡ 1y m 2 η+ = ck(y)η+ + c?(y)(v + iv ) γm η− ; (v iv ) η+ γ η− ' $ Calabi-Yau three-fold # Information of SU(3)-structure manifold with torsion 1 2 6D SU(3) Killing spinors η+, η+: # Generalized Geometry & % TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES I Calabi-Yau three-folds 99K Fluxes are highly restricted 8 > <> type IIA : No fluxes − > type IIB : F3 τH (warped Calabi-Yau) :> heterotic : No fluxes I SU(3)-structure manifolds 99K Some components of fluxes can be interpreted as torsion Piljin Yi, TK \Comments on heterotic flux compactifications" JHEP 0607 (2006) 030, hep-th/0605247 TK \Index theorems on torsional geometries" JHEP 0708 (2007) 048, arXiv:0704.2111 I Generalized geometries 99K All (non)geometric fluxes can be introduced \Complete" classification of N = 1 SUSY solutions TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Compactifications in 10D type IIA −| j2 2 2A(y) µ ν 2 10D = 4D (Λcosmo. = µ ) + 6D: ds10 = e gµν dx dx + ds6 0 Consider polyforms Φ± on the internal space M which satisfy −2A+φ − ^ 2A−φ 0 − 0 e (d H )(e Φ+) = 2µ ReΦ− e−2A+φ(d − H^)(e2A−φΦ0 ) = −3i Im(µΦ0 ) + dA ^ Φ0 − h + − i 1 + eφ (jaj2 − jbj2)F + i(jaj2 + jbj2) ∗ λ(F ) 16 M. Gra~na,R. Minasian, M. Petrini, A. Tomasiello hep-th/0505212 TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Compactifications in 10D type IIA −| j2 2 2A(y) µ ν 2 10D = 4D (Λcosmo. = µ ) + 6D: ds10 = e gµν dx dx + ds6 0 Consider polyforms Φ± on the internal space M which satisfy −2A+φ − ^ 2A−φ 0 − 0 e (d H )(e Φ+) = 2µ ReΦ− e−2A+φ(d − H^)(e2A−φΦ0 ) = −3i Im(µΦ0 ) + dA ^ Φ0 − h + − i 1 + eφ (jaj2 − jbj2)F + i(jaj2 + jbj2) ∗ λ(F ) 16 M. Gra~na,R. Minasian, M. Petrini, A. Tomasiello hep-th/0505212 I 0 0 6 1 2 On Calabi-Yau (dΦ± = 0) or SU(3)-structure manifolds (dΦ± = 0) (η+ = η+): ' $ 0 −iJ 0 − Φ+ = e ; Φ− = Ω − y − y Jmn = 2i η+ γmn η+ ; Ωmnp = 2i η− γmnp η+ & % TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Compactifications in 10D type IIA −| j2 2 2A(y) µ ν 2 10D = 4D (Λcosmo. = µ ) + 6D: ds10 = e gµν dx dx + ds6 0 Consider polyforms Φ± on the internal space M which satisfy −2A+φ − ^ 2A−φ 0 − 0 e (d H )(e Φ+) = 2µ ReΦ− e−2A+φ(d − H^)(e2A−φΦ0 ) = −3i Im(µΦ0 ) + dA ^ Φ0 − h + − i 1 + eφ (jaj2 − jbj2)F + i(jaj2 + jbj2) ∗ λ(F ) 16 M. Gra~na,R. Minasian, M. Petrini, A. Tomasiello hep-th/0505212 I 0 0 6 1 2 On Calabi-Yau (dΦ± = 0) or SU(3)-structure manifolds (dΦ± = 0) (η+ = η+): ' $ 0 −iJ 0 − Φ+ = e ; Φ− = Ω − y − y Jmn = 2i η+ γmn η+ ; Ωmnp = 2i η− γmnp η+ & % I × 1 6 2 On SU(3) SU(3) generalized geometries (η+ = η+ at some points y): ' $ 0 0 −ij − ^ −iv^v 0 −ij ^ 0 Φ+ = (cke ic?w) e ; Φ− = (cke + ic?w) (v + iv ) J A = j ± v ^ v0 ; ΩA = w ^ (v ± iv0) & % TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Contents Introduction Killing Spinors and Fluxes Generalized (Complex) Geometries Exterior Derivatives and Flux Charges Setup in N = 1 Theory My Work: Search of SUSY AdS Vacua Summary and Discussions Generalized almost complex structures Introduce a generalized almost complex structure J on F ⊕ F ∗ s.t.
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