The generalized geometry of Calabi-Yau orientifolds with fluxes Thomas W. Grimm University of Wisconsin, Madison based on: [hep-th/0507153] (to appear in Fortsch.Phys.) Nucl. Phys. B718, 2005 [hep-th/0412277] with J. Louis Nucl. Phys. B699, 2004 [hep-th/0403067] with J. Louis Madison, September 2005

2 Introduction and Motivation ➪ From String theory to D = 10 supergravity low String Theory − − − − − − − − → D = 10 Supergravity energies effective theory for massless string modes: “weak coupling” • concentrate on the two maximal supersymmetric theories in D = 10 Type IIA and IIB String Theory ↓ D = 10 Type IIA and IIB Supergravity with N = 2 phenomenology: Four-dimensional setups with gauge theory and N=1 supersymmetry

3 ➪ Four-dimensional setups: Compactification space-time background: M 1 , 3 × Y 6 • minimal supersymmetry in D = 4 : Y 6 is special manifold – Calabi-Yau ➪ Gauge theory: Type II string theories allow for D-branes • extended objects with gauge-theory on their world-volume • boundaries for open strings • supersymmetric D-branes break half of SUSY on their world-volume

4 ➪ Brane-world setups: necessity of orientifolds - minimal supersymmetry: Y 6 – compact Calabi-Yau manifold non-Abelian gauge groups: - space-time filling D-branes ⇒ consistency: orientifold planes ↓ Kaluza-Klein reduction Effectiv four-dimensional N = 1 Supergravity Theory Problem: many moduli fields – flat directions of the potential example: size of Y 6 v(x) corresponds to four-dimensional field

5 ➪ Our goal: 1) Determine effective N = 1 supergravity theory for these moduli fields in general Calabi-Yau orientifolds of type IIA and IIB string theory 2) Discuss geometry of N = 1 moduli space 3) Include mechanism to generate a potential: Background fluxes ⇒ moduli stabilization

6 Outline of the talk 1) Effective action of Type II Calabi-Yau orientifolds – Type IIB Calabi-Yau orientifolds – O3/O7 example – Type IIB Calabi-Yau orientifolds with several linear multiplets 2) Type IIA Calabi-Yau orientifolds – K¨ ahler potential and generalized geometry of moduli space 3) Fluxes in Type II orientifolds

7 1. Effective action of type II Calabi-Yau orientifolds ➪ d = 10 N = 2 massless (bosonic) spectrum: Type IIA Type IIB φ, ˆ ˆ G MN , ˆ φ, ˆ ˆ G MN , ˆ NS-NS: B 2 NS-NS: B 2 C 1 , ˆ ˆ C 0 , ˆ ˆ C 2 , ˆ R-R: C 3 R-R: C 4 ➪ compactification on compact Calabi-Yau Y 6 : Calabi-Yau manifold ≡ exists globally defined two-form J and (3 , 0) -form Ω s.t. dJ = 0 , d Ω = 0 Ω – holomorphic three-form: J – K¨ ahler form: δ Ω complex structure deformations δJ K¨ ahler structure deformations ≡ shape moduli z K ≡ size moduli v A

8 ➪ Defining the orientifold Acharya,Aganagic,Brunner,Hori,Vafa • mod out (gauge-fix) discreate symmetries of the string theory • focus on Type IIB 1) world sheet parity Ω p ‘orienti-’ – allow for non-orientable world-sheets: e.g. Klein bottle, M¨ obius strip 2) geometric symmetry σ of M 10 = M 4 × Y 6 , involution σ ( σ 2 = 1 ) ‘-fold’ – like in orbifold ⇒ orientifold planes – fix-points of σ • demand N = 1 supersymmetry σ is holomorphic and isometric involution: σ ∗ J = J σ ∗ Ω = − Ω σ ∗ Ω = +Ω orientifolds with O3/O7 planes orientifolds with O5/O9 planes O = ( − ) F L Ω p σ ∗ O = Ω p σ ∗ • supergravity: truncate spectrum such that: O ( Field ) = Field

9 ➪ Four-dimensional N = 1 Spectrum Type IIB O 3 /O 7 orientifolds • Kaluza-Klein reduction: expand fields in zero modes of Y 6 consistent with orientifold projection H ( p,q ) = H ( p,q ) ⊕ H ( p,q ) involution splits cohomologies + − B 2 = b a ω a , ω a ∈ H (1 , 1) ˆ ˆ ˆ φ = φ , C 0 = C 0 − c a ω a , ω α + V λ α λ , ω α ∈ H (2 , 2) , α λ ∈ H (3) ˆ ˆ C 2 = C 4 = ρ α ˜ ˜ + + h (2 , 1) z k z k shape moduli − h (1 , 1) ( v α , ρ α ) T α size moduli + chiral multiplets h (1 , 1) ( b a , c a ) G a − 1 ( φ, C 0 ) τ h (2 , 1) V λ vector multiplets + gravity multiplet 1 g µν

10 ➪ Four-dimensional N = 1 effective action • Kaluza-Klein reduction: determine effective action consistent with orientifold projection • N = 1 , D = 4 effective action in standard form: Wess,Bagger J − V ¯ J DM I D ¯ − 1 L = 2 R − K I ¯ M 2 Re f λκ ( F λ ) µν ( F κ ) µν − F κ ) µν , 2 Im f λκ ( F λ ) µν ( ˜ − 1 1 K I ¯ J ¯ e K � J D I WD ¯ W − 3 | W | 2 � + 1 2 (Re f ) − 1 λκ D λ D κ . V = M I ≡ ( z k , T α , G a , τ ) : all scalar fields, F λ = dV λ J = ∂ I ¯ ∂ J K ( M, ¯ K¨ ahler metric: K I ¯ M ) holomorphic superpotential: W ( M ) , D I W = ∂ I W + ( ∂ I K ) W holomorphic gauge kinetic function: f ( M ) ⇒ determine K (and f ) from orientifold effective action later: determine W, D α due to background flux

11 ➪ The K¨ ahler potential Type IIB orientifolds with O 3 /O 7 planes TWG,Louis chiral moduli fields � B 2 + iJ � C = τ + G a ω a + T α ˜ e − φ e − ˆ − i e − ˆ B 2 ∧ ˆ ω α Ω( z ) , Re • τ, G a , T α – complicated def of complex coordinates on N = 1 moduli space K¨ ahler potential � � Ω( z ) ∧ ¯ τ ) − 2 ln e − 3 2 φ K = − ln Ω(¯ z ) − ln ( τ − ¯ J ∧ J ∧ J , Y 6 Y 6 • general form of K¨ ahler potential in terms of topological data of Y 6 • of no-scale type: positive potential V ≥ 0 (if no superpotential for T α ) • last term in K: implicit function of real parts of τ, G a , T α ⇒ Im T α admits shift symmetry: Im T α → Im T α + c ⇒ K becomes explicit in the linear multiplet picture

12 ➪ Type II orientifolds with several linear multiplets – O3/O7 example replace chrial multiplets T α with linear multiplets ( L α , D α idea: 2 ) (Im T α possess shift symmetries) Dual picture 2 ) coupled to chiral multiplets N I = z k , τ, G a linear multiplets ( L α , D α ⇒ standard effective action for chiral/linear multiplet system Binetruy,Girardi,Grimm • kinetic terms and couplings encoded by kinetic potential ˜ K ( N, L ) = K ( N, L ) − 3 F ( N, L ) ahler potential K ( N, T ) and chiral coordinates T α + ¯ • K¨ T α are Legendre transform of ˜ K ( N, L ) and L α : T α + ¯ T α = ˜ K ( N, T ) = ˜ K − ˜ K L α L α K L α ,

13 O 3 /O 7 orientifold example � Ω( z ) ∧ ¯ τ ) + ln( K αβγ L α L β L γ ) K ( z, τ, G, L ) = − ln Ω(¯ z ) − ln( τ − ¯ τ ) − 1 K αab L α ( G − ¯ G ) a ( G − ¯ G ) b F ( τ, G, L ) = − i ( τ − ¯ The scalar potential potential in the presence of linear multiplets V = e K � K N A ¯ N B D N A W D N B W − (3 − L α K L α ) | W | 2 � ˜ L α K L α = 3 • since in O 3 /O 7 orientifolds ⇒ trivially V ≥ 0 ➪ similar analyis for O 5 /O 9 orientifolds possible

14 What about type IIA orientifolds? Just the mirror of both O3/O7 and O5/O9 setups?

15

16 2. Type IIA Calabi-Yau orientifolds TWG,Louis ➪ orientifold projection O = ( − 1) F L Ω p σ σ is anti-holomorphic, isometric involution of Y 6 σ ∗ Ω = e 2 iθ ¯ σ ∗ J = − J Ω ⇒ O 6 planes wrap special Lagrangian cycles in Y 6 calibrated with Re ( e − iθ Ω) ➪ chiral moduli fields h (1 , 1) J c = B 2 + iJ = t a ω a chiral multiplets − • coupling to the string world-sheet h (2 , 1) + 1 chiral multiplets Ω c = C 3 + i Re ( C Ω) = N k α k + T κ β κ • C ∝ e − φ − iθ ⇒ Ω c coupling to wrapping supersymmetric D 2 branes • gauge-couplings for space-time filling D 6 branes Blumenhagen, Braun, K¨ ors, L¨ ust

17 ➪ K¨ ahler potential � � K SK ( t ) + K Q ( N, T ) = − ln J ∧ J ∧ J − 2 ln C Ω ∧ C Ω Y 6 Y 6 • K Q calculated by using Legendre transformation (linear multiplet formalism) � C Ω = CZ K α K − C F K β K define: V = Y 6 C Ω ∧ C Ω , chiral picture: orientifolded Hitchin’s generalized geometry q k = Re ( CZ k ) K Q ( q k , q λ ) = − 2 ln V � � q , q λ = Re ( C F λ ) Legendre transformation dual picture: orientifolded N = 2 special geometry ❄ ❄ + 1 π λ = 1 q k = Re ( CZ k ) ˜ K Q ( q k , π λ ) = − 2 ln V V Im ( CZ λ ) � � � � q, π V F q, π ,

18 ➪ Generalized complex geometry (very brief) Hitchin,Gualtieri • differential geometry on T ≡ TY 6 ⊕ T ∗ Y 6 instead on TY 6 alone: ⇒ generalized metric on T ∼ = metric, B-field and dilaton on Y ⇒ generalized complex structure on T ∼ = complex/symplectic structure, B-field and dilaton on Y • TY 6 ⊕ T ∗ Y 6 has natural SO (6 , 6) structure ⇒ spinors of Spin (6 , 6) ? two Weyl representations S ev = Λ even T ∗ Y S odd = Λ odd T ∗ Y special complex spinors define generalized complex structure on T examples: e − φ e B 2 + iJ ∈ S ev , C Ω ∈ S odd We used the real parts of these forms in defining the N = 1 K¨ ahler coordinates on the truncated quaternionic space M Q ! • Hitchin defines a functional V ( ρ ) on the real parts of these forms ahler potentials on M Q ⇒ evaluated on light modes of the theory: K¨

19 The spinors e − φ e − B 2 + iJ and C Ω are the special cases corresponding to Calabi-Yau orientifolds. The mathematical framework is much more powerful It can incorporate orientifolds of non-Calabi Yau spaces. ’Generalized complex orientifolds’

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