Generalized N = 1 orientifold compactifications
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1 Generalized N = 1 orientifold compactifications Thomas W. Grimm University of Wisconsin, Madison based on: [hep-th/ ] Iman Benmachiche, TWG [hep-th/ ] TWG Madison, Wisconsin, November 2006
2 2 Introduction and Motivation Phenomenology String Theory Four-dimensional N = 1 Supergravity Address moduli problem: generate potential for massless scalar fields due to background fluxes and non-calabi-yau geometries Explore generic features of N = 1 compactifications Landscape of String vacua? Complete dualities in the presence of background flux Four-dimensional gauge theory and specific models
3 3 A realization in Type II String theory: - minimal supersymmetry: background M 1,3 M 6 M 6 special manifold - moduli stabilization: background fluxes non-calabi-yau geometry - non-abelian gauge groups: space-time filling D-branes consistency: orientifold planes
4 4 Outline of the Talk Conditions on supersymmetric theories and vacua Some basics on generalized complex geometry Orientifold projection and N = 1 spectrum N = 1 chiral field space: Kähler potential and superpotential Mirror symmetry / T-duality with fluxes: a conjecture
5 5 Conditions on supersymmetric theories and vacua
6 6 Condition for supersymmetric D = 4 theory: (Type IIB example) metric ansatz: ds 2 10 = e A(y) η µν dx µ dx ν + g ij (y) dy i dy j Two ten-dimensional gravitinos Ψ 1 M, Ψ2 M decompose as: Ψ i µ = ψ i +µ η i + + ψ i µ η i ψ 1 µ, ψ 2 µ become four-dimensional gravitinos Ψ i m = four-dimensional fermionic modes for other multiplets Effective four-dimensional theory possesses N = 2 susy two globally defined spinors η 1 and η 2 on M 6 exist, they locally or globally coincide Graña,Louis,Waldram η 1, η 2 reduce structure group SO(6) of T M 6 : η 1 : SO(6) SU(3) 1 η 2 : SO(6) SU(3) 2
7 7 Supersymmetry conditions on D = 10 flux-background differential conditions relating η 1 and η 2 to the NS-NS and R-R background flux Behrndt,Cvetic,Liu; Lüst,Tsimpis; Witt; Graña,Minasian,Petrini,Tomasiello vanishing of the gravitino and dilatino variations: δψ i M = M ɛ i + (Flux) M ɛ i = 0 δλ i = ( φ) ɛ i + (Flux)ɛ i = 0 Susy spinors ɛ i decompose as: ɛ i = ζ i + η i + + ζ i η i (Type IIB example) Susy conditions imply: m η 1 = (Flux) 1 m, m η 2 = (Flux) 2 m Examples for M 6 : Calabi-Yau manifolds: η = η 1 = η 2 and m η = 0 Manifolds with SU(3) structure: η = η 1 = η 2, but m η 0 Manifolds with η 1 = η 2 only locally: manifolds with SU(3) SU(3) structure new framework: Generalized complex geometry Hitchin,Gualtieri,Witt
8 8 Some basics on generalized complex geometry
9 9 Central object: T T T M 6 T M 6 generalized tangent bundle T T takes the role of tangent bundle T in standard geometry: recall: metric g ij on M 6 is a map T T R almost complex structure I j i on M 6 defines a split T C = T (1,0) T (0,1) generalized metric G IJ as a map (T T ) (T T ) R ( ) g 1 B g 1 G IJ = g Bg 1 B Bg 1 generalized almost complex structure J J I defines split (T T ) C = L (1,0) L (0,1) J L (1,0) = il (1,0) J L (0,1) = il (0,1) J 2 = 1
10 10 T T has more structure: natural inner product on T T : V, W T η, ζ T (V + η, W + ζ) = ( V η ) ( ) ( generalized structure group of T T is SO(6, 6) W ζ ) = η(w ) + ζ(v ) SO(6, 6) induces a Clifford action on differential forms Φ: (V + η) Φ = V Φ + η Φ Indeed we have {V + η, W + ζ} Φ = (V + η, W + ζ)φ. Differential forms Φ on M 6 are spinors of SO(6, 6): S ev = Λ ev T Λ 6 T S odd = Λ odd T Λ 6 T
11 11 SO(6) spinors η 1, η 2 vs. SO(6, 6) spinors Φ Gualtieri; Witt; Graña,Minasian,Petrini,Tomasiello Globally defined even and odd forms naturally associated to η 1 and η 2 : Φ ev = Φ odd = 6 i=1 6 i=1 η 2 + γ m1...m i η 1 + dx m 1... dx m i η 2 γ m1...m i η 1 + dx m 1... dx m i Conditions on η 1, η 2 translate into conditions on forms Φ ev/odd
12 12 pure spinors: (V + η) Φ ev/odd = 0 V + η L ev/odd T + T L ev/odd are of dimension six and isotropic L (1,0) = L Φ ev/odd define two generalized almost complex structures non-degeneracy: Φ ev, Φ ev 0 Φ odd, Φ odd 0, are the Mukai pairings: e.g. for Φ ev = Φ 0 + Φ 2 + Φ 4 + Φ 6 Φ ev, Φ ev = Φ 0 Φ 6 Φ 2 Φ 4 + Φ 4 Φ 2 Φ 6 Φ 0 compatibility: Φ ev, Φ ev = 3 4 Φ odd, Φ odd and Φ odd, (V + η) Φ ev = 0 Φ ev/odd reduce structure group of T T to SU(3) SU(3) M 6 is generalized almost complex manifold with SU(3) SU(3) structure flux background: dφ ev 0 dφ odd 0
13 13 Important example: SU(3) structure manifolds Φ ev = e ij Φ odd = Ω where: J is a globally defined (1, 1) form Ω is a globally defined (3, 0) form J, Ω define the SU(3) structure: J J J = 3i 4 Ω Ω J Ω = 0 Remark on NS-NS B-field: note that world-sheet coupling is B + ij include by a B-transform: Φ Φ B e B Φ B-transform is a symmetry of Mukai pairings: ΦB, Ψ B = Φ, Ψ B-transform is SO(6, 6) rotation on T T : maps a pure spinor Φ into new pure spinor Φ B B has a non-trivial background flux H = db Φ B obtained by twisting with a gerbe
14 14 Orientifold projection and N = 1 spectrum
15 15 The orientifold projection (Type IIA example): Acharya,AganagicBrunner,Hori,Vafa Bosonic Type IIA spectrum NS-NS: φ, g MN, B 2 R-R: C odd = C 1 + C 3 + C 5 + C 7 + C 9 mod out (gauge-fix) discrete symmetries of the string theory: O = ( ) F L Ω p σ 1) world sheet parity Ω p 2) geometric symmetry σ of M 6 : σ 2 = 1 (identity on M 3,1 ) demand N = 1 supersymmetry λ(ω 2n ) = ( 1) n ω 2n λ(ω 2n 1 ) = ( 1) n ω 2n 1 σ Φ odd = λ( Φ odd ) σ Φ ev = λ(φ ev ) Benmachiche,TWG Calabi-Yau case: σ is anti-holomorphic and isometric involution O6 planes. truncate spectrum such that: O(Field) = Field σ φ = φ σ B 2 = B 2 σ C odd = λ(c odd )
16 16 Four-dimensional spectrum: M 6 generalized complex manifold: two-grading of forms Λ ev T = S ev Λ odd T = S odd M 6 possesses orientifold symmetry λσ : four-grading of forms Λ ev T = Λ ev + Λ ev Λ odd T = Λ odd + Λ odd Orientifold: restrict ten-dimensional fields to appropriate eigenspaces of σ R-R sector: four eigenspaces correspond to four fields in D = 4: scalars vectors C odd (0) = C odd Λ odd + two-forms C odd (2) = C odd Λ odd C(1) odd = C odd Λ ev three-forms C(3) odd = C odd Λ ev + So far: Infinite set of scalars, two-forms as well as vectors, three-forms related by duality condition on the ten-dimensional field strengths: F ev = λ(f ev ) F ev = dc odd + H C odd
17 17 NS-NS sector We define ϕ odd = e φ e B 2 Φ odd ϕ ev = e B 2 Φ ev Four-dimensional graviton and ϕ ev/odd encode all degrees of freedom in the NS-NS sector. Not all degrees of freedom in ϕ ev/odd are independent: ϕ odd pure spinor and non-degeneracy Im(ϕ odd ) is function of Re(ϕ odd ) Hitchin ϕ ev complicated function ϕ ev (t) of the true scalar deformations t of the generalized complex manifold (e.g. complex rescalings of ϕ ev are unphysical) (compare with Calabi-Yau case ϕ ev = e B+iJ = e ta ω a ) Gualtieri; Graña,Louis,Waldram
18 18 Performing the Kaluza-Klein reduction What are the light modes on SU(3) or SU(3) SU(3) structure manifold? determine a finite set of forms on M 6 used in the Kaluza-Klein expansion: finite = ev odd finite often constructed to match mirrors of Calabi-Yau compacitifications with fluxes Gurrieri,Louis,Micu,Waldram; Tomasiello; Graña,Louis,Waldram; Benmachiche,TWG How to perform a Kaluza-Klein reduction with a warp factor? restrict to regime of constant warping, large volume limit Giddings,Kachru,Polchinski
19 19 N = 1 chiral field space: Kähler potential and superpotential
20 20 1. How do the the scalars combine into N = 1 chiral multiplets? correct D-brane couplings: combine ϕ odd c e B 2 C odd + i Re(ϕ odd ) odd + linear in the N = 1 complex scaler fields ϕ ev (t) is holomorphic function of complex scalars t a 2. What is the metric on the scalar field space? N = 1 susy Kähler metric, i.e. G AB = A B K Kähler potential: K(t, ϕ odd c ) = ln ϕ ev, ϕ ev 2 ln ϕ odd, ϕ odd M 6 M 6 Benmachiche,TWG first term: as in N = 2 scale invariant functional on even forms Hitchin; Graña, Louis, Waldram second term: Kähler space inside the N = 2 quaternionic manifold, metric encoded by generalized Hitchin functional of Re(ϕ ev )
21 21 The N = 1 superpotential: The background fluxes and non-calabi-yau geometry induce a potential for the scalar fields. We allow for non-trivial NS-NS background flux H 3 and R-R fluxes F ev on M 6 H 3 = db 2 M 6, F ev = dc odd M 6 The manifold with SU(3) SU(3) structure generically has dϕ odd c deviation from a Calabi-Yau manifold contributes to the potential. 0 and dϕ ev 0. This The induced superpotential can be derived by a fermionic reduction. W (t, ϕ odd c ) = F ev + d H ϕ odd c, ϕ ev M 6 Benmachiche,TWG Here d H = d + H 3 denotes the H-twisted differential. This superpotential reduces to the known cases on general Calabi-Yau and SU(3) structure orientifolds. TWG,Louis; Villadoro,Zwirner; Graña,Louis,Waldram A similar analysis can be performed for type IIB set-ups: essentially exchanges the role of even and odd forms
22 22 Comments on the Hitchin functionals let ρ be a real even or odd form on M 6 (ρ S ev/odd ) Hitchin define invariant quartic form of ρ q(ρ) = Trµ(ρ) 2 = Γ MN ρ, ρ Γ MN ρ, ρ µ(ρ)(a) = σ(a)ρ, ρ is moment map of SO(6, 6), σ(a) representation of so(6, 6) Γ M are Gamma matrices of SO(6, 6) Proposition: q(ρ) < 0 ρ = ReΦ for pure spinor Φ and Φ, Φ 0 ˆρ = ImΦ is unique up to ordering on the open set of non-degenerate pure spinors define Hitchin functional H[ρ] = q(ρ) = i Φ, Φ M 6 M 6 can express ImΦ(ReΦ) ρ H[ρ] = ˆρ
23 23 Some other applications of the Hitchin functionals Variational problem: Extrema of H[ρ] for variations in a cohomology class correspond to generalized Calabi-Yau manifolds with dφ = 0 Hitchin H[ρ] is entropy functional for an N = 2 black hole: on harmonic forms it is the Legendre transform of the pre-potential F OSV; Dijkgraaf,Gukov,Neitzke,Vafa H[ρ] might serve as a space-time action for topological string theory on M 6 Dijkgraaf,Gukov,Neitzke,Vafa; Pestun,Witten
24 24 Mirror symmetry / T-duality with fluxes: a conjecture
25 25 The Question: What is the mirror dual of type IIB Calabi-Yau O3/O7 orientifolds with background fluxes? Mirror symmetry/t-duality is believed to map H-flux to a non-trivial mirror geometry. In type IIB Calabi-Yau orientifolds fluxes induce the Gukov-Vafa-Witten superpotential W GV W = (F 3 τh 3 ) Ω(z) Y Let us restrict to a simple case: one complex structure modulus z F 3 = 0 and H 3 = mα 1 + eβ 1 In large complex structure limit: W GV W = τ(e z + m z 2 ) Has the H-flux e and m an SU(3) SU(3) mirror geometry?
26 26 Perform mirror symmetry (T-dulality in three directions SYZ): H 3 has maximally two legs into the T-dualized directions (the Q-space Shelton,Taylor,Wecht) Mirror deformation due to electric flux e: The complex three-form Ω 3 is not anymore closed. SU(3) structure mirror ( half-flat ) Kachru,Schulz,Tripathy,Trivedi; Gurrieri,Louis,Micu,Waldram; Fidanza,Minasian,Tomasiello Φ odd = Ω 3 dre(ω 3 ) e Mirror deformation due to magnetic flux m: A non-trivial one-form Ω 1 on the mirror space is needed: Φ odd = Ω 1 + Ω 3 + Ω 5 dre(ω 1 ) m Such a one-form is present on an appropriate SU(3) SU(3) manifold! Can use the superpotential calculated above W SU(3) SU(3) = dϕ odd c, ϕ ev (t) = N 0 (e t + m t 2 ) Origin of Ω 1? Recall so(6, 6) = Λ 2 T Λ 2 T EndT β Λ 2 T two-vector Ω 1 + Ω 3 = e β Ω 3 : β correspond to non-commutativity of M 6 for open strings Kapustin,Mathai,Rosenberg
27 27 Conclusions discussed compactification of type II supergravity on orientifolds of SU(3) SU(3) manifolds determined four-dimensional spectrum without finite truncation, NS-NS and R-R sector is encoded by specific odd and even forms on M 6 Kähler potential consists of the two Hitchin functionals on M 6 holomorphic superpotentials for the geometry due to fluxes and non-calabi-yau geometry commented on mirror symmetry/t-duality of flux compactifications mirror spaces of Calabi-Yau compactifications with H-flux are generically generalized SU(3) SU(3) manifolds open questions: determination of light modes for non-calabi-yau compactifications effective four-dimensional theories for warped compactifications interesting compact examples with space-time filling D-branes
28 28
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