Fluxes and gaugings. Marios Petropoulos. CPHT Ecole Polytechnique CNRS. The European Superstring Theory Network.

Transcription

1 Fluxes and gaugings Marios Petropoulos CPHT Ecole Polytechnique CNRS The European Superstring Theory Network Ayia Napa, September 2008

2 Historical perspective: the early years of antisymmetric-tensor backgrounds The reduction à la Scherk Schwarz (1979) Toroidal reduction with fields continuous modulo symmetry transformations The torus is endowed with a constant spin connection the vev of an antisymmetric tensor: a geometric flux

3 Consequences of the Green Schwarz mechanism (1984) Construction of D = 10 N = 1 (16 supercharges) supergravity coupled to E 8 E 8 or SO(32) Yang Mills as a low-energy limit of heterotic string theory [Gross, Harvey, Martinec, Rohm; Thierry-Mieg 84] Reduction of this theory to D = 4 N = 1 (4 supercharges) by explicit supersymmetry breaking at the Lagrangian level with Minkowski space time without dilaton or antisymmetric-tensor vevs with internal 6-D space be Ricci-flat, Kähler with SU(3) holonomy: Calabi Yau [Candelas, Horowitz, Strominger, Witten 84] allowing thereby to incorporate chiral fermions protect the hierarchy

4 Important issues to be addressed Break N = 1 N = 0 at the TeV scale Treat the massless scalars (moduli) and the Abelian vectors that accompany any compactification Very hard not yet fully satisfactory understanding Must accommodate new data: more general space times anti-de Sitter (role in string theory) de Sitter (potential role in cosmology)

5 The D = 10 N = 1 action Gravitational multiplet: graviton e, dilaton Φ, two-form B MN, right Majorana Weyl dilatino λ, left gravitino ψ Q Vector multiplet: vector, left Majorana Weyl gaugino χ Antisymmetric-tensor gravitino bilinear coupling: 2 e g 2 Φ H MNP ψ Q Γ MNP ψ Q 8M 2 Pl An antisymmetric-tensor vev can trigger a super-higgs mechanism Generate a gravitino mass m3/2 = 1 4M 2 Pl g 2 Φ H mnp Γ mnp Give masses to the gauginos

6 Original idea for the H mnp [Derendinger, Ibañez, Nilles; Dine, Rohm, Seiberg, Witten 85; Kounnas, Porrati 87] created dynamically by its coupling to the gauginos 1 8M 2 Pl e g 2 Φ H mnptr χγ mnp χ whenever a source term Tr χγ mnp χ appears (use eoms) possibly due to a gaugino condensate χχ = Λ 3 leads to m3/2 Λ3 /16M 2 Pl Assuming a confining scale Λ GeV one gets m3/2 M w Realistic within E 8 better than advocating large compactification scales R L Pl

7 Modern perspective The existence of non-trivial cycles inside the internal space allow for H mnp = 0: perturbative fluxes and flux compactifications Plethora of available antisymmetric tensors: H 3, F n, ω 2 Rich interplay with their sources (branes) that allows to tackle simultaneously model building (orientifold models) Wide palette of desirable effects Fluxes generate scalar potentials: stabilize unwanted massless scalars and generate desired new scales Fluxes introduce non-abelian couplings and can trigger spontaneous symmetry breaking (e.g. generate Stückelberg masses) Fluxes allow for more general maximally symmetric space times: V effective cosmological constant

8 Searching for string flux vacua [reviews: Graña 05; Blumenhagen, Körs, Lüst, Stieberger 06] Exact string vacua: 2-D conformal sigma-models Capture only NS backgrounds Limited number of constructions free-fermionic Gepner orbifold slight variations Advantage: clear picture of the full spectrum (e.g. 3-family requirement leads to models) Supergravity solutions: leading order in α two-derivative Include all possible antisymmetric tensors Allow for two complementary approaches: in 10 or in 4 dimensions

9 The 10-D approach top-down Solve 10-D equations and find N = 1 backgrounds with compact (but not too small with respect to l s = 2π α ) internal 6-D space Analyze the 4-D effective action: extract the massless spectrum and integrate/truncate the massive modes subtle task easy for toroidal models tractable for CYs very hard otherwise usually further simplifications/approximations are needed The closest to a genuine string background with elegant geometrical tools without guaranty for phenomenological application [Giddings, Kachru, Polchinski 02; huge literature since]

10 The 4-D approach bottom-up Work directly with a 4-D theory: gauged supergravity Interpret it as an effective theory of a compactification with antisymmetric-tensor backgrounds translate the gauging parameters into flux numbers reconstruct the fundamental theory difficult task Tailor-made for phenomenological applications [Derendinger, Kounnas, Petropoulos, Zwirner 04; Villadoro, Zwirner 05]

11 Note: the reduction from 10 D may require extra 4-D data, the oxidation to 10 D is not always possible 4-D gaugings are the most general deformations of N = 4, 8 sugras N = 1, 2 cases are less universal and require extra parameters for completing the 10-D flux dictionary Not all 4-D gauged sugras are heterotic, type-i, type-ii or M-theory geometric vacua handle to understand the non-geometric string backgrounds A good picture of the landscape of gauged sugras is nevertheless necessary

12 Aim of the present lectures: give a flavor of the techniques and achievements of these two approaches to flux compactification Leaves aside many aspects either by lack of time or of material:... Systematic scan of 3 D 7 gaugings, including relevance for M2 dynamics Systematic scan of N = 1 SU(3) group-structure spaces Non-geometric backgrounds M-theory compactifications Beyond maximally symmetric space-times (e.g. in presence of domain walls) Non-perturbative effects Soft-supersymmetry breaking terms, D-terms in N = 1 Phenomenology of the scalar potentials String theory in general has been evolving over the years according to the top-down and bottom-up lines of thought...

13 Menu Equations: forms, sources and constraints The Calabi Yau tale Introducing antisymmetric-tensor backgrounds Effective 4-D theory and moduli stabilization Gauged half-maximal supergravities and the embedding tensor 4-D gauged N = 4 supergravities as 10-D flux compactifications 4-D gauged supergravities with N = 1 supersymmetry 10-D origin: T6 /Z 2 Z 2 type II orientifolds with fluxes General conclusions and criticisms Appendix 1: More general gaugings and non-geometric backgrounds Appendix 2: Trading fluxes for branes and building domain walls

14 Equations: forms, sources and constraints The Calabi Yau tale Introducing antisymmetric-tensor backgrounds Effective 4-D theory and moduli stabilization Gauged half-maximal supergravities and the embedding tensor 4-D gauged N = 4 supergravities as 10-D flux compactifications 4-D gauged supergravities with N = 1 supersymmetry 10-D origin: T6 /Z 2 Z 2 type II orientifolds with fluxes General conclusions and criticisms Appendix 1: More general gaugings and non-geometric backgrounds Appendix 2: Trading fluxes for branes and building domain walls

15 The 10-D action Bosonic part (κ 2 4 = κ2 10/l 6 s = 1 /M 2 Pl = 8πG) S = 1 2κ10 2 d 10 x g [L NSNS + L RR + L CS ] Neveu Schwarz and Ramond sectors string frame L NSNS = e 2Φ ( R + 4g MN M Φ N Φ 1 12 H MNK H MNK ) L RR = 1 2 p 1 p! F M 1...M p F M 1...M p The Chern Simons term Absent in the democratic formulation (F p = ± F 10 p ) In the unconstraint formulation (only F 5 = F 5 ): - L IIA = 1 2 B 2 F 4 F 4 - L IIB = 1 2 C 4 H 3 F 3

16 The equations of motion eoms for the Einstein eom for the dilaton forms eoms for NSNS and p = 0,..., 5 RR Bianchi identities (eoms for p = 6,..., 10 RR in the democratic formulation) dh 3 = 0 H 3 = db 2 df 5 = H 3 F 3 F 5 = dc C 2 db B 2 dc 2.. modified in the presence of geometric fluxes: df = ω 2 F +

17 D-branes: sources for the RR fields Sources modify the equations of motion and the Bianchi identities S D brane = S DBI + S WZ The Dirac Born Infeld term S DBI = µ p W d p+1 ξ e Φ det (g ab + 2πα F ab ) (modify Einstein, axion, dilaton in type II µ p = 2π /l p+1 s ) The Wess Zumino (topological) term S WZ = µ p W cosh(2πα F) Â(RT ) Â(R N ) q C q (modify RR-form eoms and Bis democratic formulation)

18 Sources and compact spaces Inside a compact space the lines of the fields must close the total charge must vanish The integrated rhs of the eoms and Bis (d F =, df = ) do not vanish in general (depending on the homology due to non-trivial cycles inside X ): constraints appear tadpole (induced source vertex) cancellation conditions

19 How do tadpoles vanish? With brane/anti-brane sources: break supersymmetry With brane/orientifold planes: can keep half supersymmetry both are ingredients of modern model building both appear in flux compactifications Best paradigm: type II compactification on Calabi Yau space (e.g. T 6 /Z 2 Z 2 ) with O-planes supersymmetry is reduced to N = 1 D-branes/fluxes must be introduced for cancelling tadpoles Remarks At the full string-theory level tadpole conditions stem out of an UV-finiteness requirement Extra quantum conditions: the fluxes of antisymmetric-tensor fields through internal cycles must be quantized Σ p F p Z

20 Solving the equations General philosophy: search for M X backgrounds potentially warped This requires to introduce a 10-D ansatz and Solve eoms for Einstein, dilaton, antisymmetric tensors Bis for all forms including all source terms (D-branes/O-planes) Impose tadpole conditions and flux quantization (fluxes rational numbers) Verify the dilatino and gravitino variations (left-over supersymmetries) Check that the internal 6-D space is compact and compatible with the current approximations (use of two-derivative action)

21 The search for supersymmetric compactifications The requirement of supersymmetry simplifies the procedure because supersymmetry transformations relate the various background fields If the background is compatible with supersymmetry the eoms for the forms are satisfied the Bis are satisfied (including tadpole conditions) the sources are calibrated (i.e. supersymmetric balance DBI/WZ) Then Einstein and dilaton eoms are automatically solved (with the exception of E time space = 0 that must be checked independently whenever non identically zero) [literature on calibrations; Koerber, Tsimpis 07]

22 Equations: forms, sources and constraints The Calabi Yau tale Introducing antisymmetric-tensor backgrounds Effective 4-D theory and moduli stabilization Gauged half-maximal supergravities and the embedding tensor 4-D gauged N = 4 supergravities as 10-D flux compactifications 4-D gauged supergravities with N = 1 supersymmetry 10-D origin: T6 /Z 2 Z 2 type II orientifolds with fluxes General conclusions and criticisms Appendix 1: More general gaugings and non-geometric backgrounds Appendix 2: Trading fluxes for branes and building domain walls

23 Ansatz and equations The background fields Warped geometry M X M 4-D maximally symmetric space time X 6-D internal space ds 2 = e A(y) g µν (x) dx µ dx ν + g ij (y) dy i dy j (µ, ν = 0,..., 3; i, j = 4,..., 9; M, N = 0,..., 9) Constant dilaton No antisymmetric tensors

24 Supersymmetry variations ψ M = ( ψ 1 M ψ 2 M gravitino : δψ M = M ɛ dilatino : δλ = 0 ) : Majorana Weyl vector-spinors of opposite (IIA) or same (IIB) chirality ( ) λ 1 λ = λ 2 : Majorana Weyl spinors of opposite chirality ( ) ɛ 1 ɛ = : infinitesimal Majorana Weyl parameters ɛ 2 M : Levi Civita covariant derivative

25 Supersymmetry analysis: existence of covariantly-constant spinors Decompose all gamma matrices and spinors with respect to 4 and 6 dimensions in 4-D: introduce two x-dependent spinors with a choice of basis s.t. ξ 1,2 = ξ1,2 + (aim: N = 2) in 6-D: introduce one y-dependent spinor 4 of Spin(6) with a choice of basis s.t. η = η+ (c.c. matrix 1 ) { ɛ 1 = ξ+ 1 η + + ξ 1 η IIA ɛ 2 = ξ+ 2 η + ξ 2 η + IIB ɛ a = ξ+ a η + + ξ a η Insert in the supersymmetry requirement M ɛ = 0

26 The emergence of Calabi Yau Supersymmetry requirement: space time components Condition on the spinor: µ ɛ + 1 ( 2 γµ γ 5 / A ) ɛ = 0 Integrability condition: [ ] µ, ν ɛ = 1 2 ia i A γ µν ɛ for a maximally symmetric space time k + ( A) 2 = 0 on a compact internal space A = constant The space time is Minkowski, without warping and ξ a ± constant

27 Analyzing the internal equation Condition on the spinor: i η ± = 0 Integrability condition: R ij = 0 The internal manifold must be compact Ricci flat and admit a spinor nowhere vanishing and covariantly constant with respect to the Levi Civita connection The group structure must be reduced from Spin(6) to SU(3) (existence of the spinor) The holonomy group coincides with the SU(3) (covariant constancy of the spinor) The 6-D internal manifold is a Calabi Yau space

28 The two pillars of a CY Kähler (1, 1)-form J 2 dj 2 = 0 J ij = 2l 2 s iη + γ ij η Holomorphic (3, 0)-form Ω 3 dω 3 = 0 Ω ijk = 2l 3 s iη + γ ijk η + Properties: J 2 J 2 J 2 = 3i 4 Ω 3 Ω 3 J 2 Ω 3 = 0

29 Small tour in CY geometry Moduli h 1,1 Kähler moduli: t A h 2,1 complex-structure moduli: z k Kähler and holomorphic forms: bases and moduli-dependence ϖ A : h 1,1 harmonic (1, 1) forms h 1 1,1 l 2 J 2 = t A ϖ A s A=1 ( α Λ, β Λ) : dim H 3 = 2h 2,1 + 2 real symplectic forms h 1 2,1 l 3 Ω 3 = (X Λ (z) α Λ F Λ (z) β Λ) s Λ=0 Special coordinates: X Λ (1, u k )

30 Tool box Special dual cycles α Σ = β Λ = δ A Λ Σ Λ B Σ Non-vanishing integral over X X α Σ β Λ = δ Λ Σ Intersection numbers K ABC = ϖ A ϖ B ϖ C, K AB = 1 X l 2 s X ϖ A ϖ B J 2,...

31 Effective 4-D theory: general N = 2 N = 2 supergravity Graviton, vectors and interacting scalars from hyper and vector multiplets (one complex scalar each) The scalars define a Kähler manifold M They ( can) be chosen to define genuine holomorphic coordinates ϕ α, ϕ β s.t. the metric on M is G α β = 2 K ϕ α ϕ β with K (ϕ, ϕ) the Kähler potential The Kähler potential receives miscellaneous corrections

32 The scalars enter two other functions N = 1 language 1. The holomorphic superpotential W that allows to define - the F -auxiliary fields ( ) F α D α W = α + κ4 2 K ϕ α W - the F -term scalar potential 2. The gauge kinetic functions f ab (ϕ) V F = e κ2 4 K ( G α β D α W D β W 3κ 2 4 W 2) The superpotential receives only non-perturbative corrections

33 Effective 4-D theory: CY compactifications Multiplets IIA: h 2,1 + 1 hypermultiplets and h 1,1 vector multiplets IIB: h 1,1 + 1 hypermultiplets and h 2,1 vector multiplets The scalar manifold is factorized: M = M cs M K M S Kähler potential K = K cs + K K + K S [ κ4k 2 cs = ln i ] l 6 Ω 3 Ω 3 s X [ ] 1 κ4k 2 K = ln J 2 J 2 J 2 6l 6 s κ 2 4K S = ln [S + S] X The superpotential vanishes no scalar potential h 1,1 + h 2,1 + 1 moduli as advertised from the CY geometry

34 The six-torus T 6 36 real moduli h 1,1 = 9 complex Kähler moduli h 2,1 = 9 complex complex-structure moduli Forms Coordinates x i = x i + 1, y i = y i + 1, i = 1, 2, 3 dim H 3 = 2h 2,1 + 2 = 20 three-forms l 3 s α 0 = dx 1 dx 2 dx 3 l 3 s β 0 = dy 1 dy 2 dy 3 l 3 s α 1 = dy 1 dx 2 dx 3 l 3 s β 1 = dx 1 dy 2 dy 3. [2]. [2] l 3 s α 4 = dx 1 dy 1 dx 2 l 3 s β 4 = dy 2 dy 3 dx 3. [5]. [5]

35 Complex structures Complex-structure moduli: ( ρ i j, τi j ) i, j = 1, 2, 3 Complex coordinates: dz i = ρ i j dx j + τ i j dy j Complex basis for the three-forms: l 3 s ϖ A0 = dz 1 dz 2 dz 3 l 3 s ϖ B0 = d z 1 d z 2 d z 3 l 3 s ϖ A1 = d z 1 dz 2 dz 3 l 3 s ϖ B1 = dz 1 d z 2 d z 3. [2]. [2] l 3 s ϖ A4 = dz 1 d z 1 dz 2 l 3 s ϖ B4 = dz 1 d z 1 d z 2. [5]. [5] Holomorphic three-form: Ω 3 = dz 1 dz 2 dz 3 = l 3 s h 2,1 =9 ) (X Λ (ρ, τ) α Λ F Λ (ρ, τ) β Λ Λ=1

36 A concrete example: type IIB on T6 /Z 2 Z 2 Moduli T6 /Z 2 Z 2 is a singular point in the CY moduli space Kähler: h 1,1 untw = 3 and h 1,1 tw = 0 Complex-structure: h 2,1 untw = 3 and h 2,1 tw = 48 Complex structure untwisted moduli Complex-structure untwisted moduli: U i C, i = 1, 2, 3 Complex coordinates: dz j = dx j + iu j dy j dim H 3 = 2h 2,1 untw + 1 = 8 three-forms: l 3 s ϖ A0 = dz 1 dz 2 dz 3 l 3 s ϖ B0 = d z 1 d z 2 d z 3 l 3 s ϖ A1 = d z 1 dz 2 dz 3 l 3 s ϖ B1 = dz 1 d z 2 d z 3. [2]. [2]

37 Prepotential and Kähler potential untwisted moduli The holomorphic three-form is Ω 3 = dz 1 dz 2 dz 3 = l 3 ( s X Λ (U) α Λ F Λ (U) β Λ) with X 0 = 1 X j = iu j F 0 = iu 1 U 2 U 3 F i = ε ijk U j U k Holomorphic prepotential: F = X 1 X 2 X 3 X 0 = iu 1 U 2 U 2 Kähler potential (T i C are the Kähler moduli): κ 2 4K = ln { (S + S) 3 i=1 ( U i + Ū i ) ( T i + T i )} As in the general case, the perturbative superpotential vanishes

38 Equations: forms, sources and constraints The Calabi Yau tale Introducing antisymmetric-tensor backgrounds Effective 4-D theory and moduli stabilization Gauged half-maximal supergravities and the embedding tensor 4-D gauged N = 4 supergravities as 10-D flux compactifications 4-D gauged supergravities with N = 1 supersymmetry 10-D origin: T6 /Z 2 Z 2 type II orientifolds with fluxes General conclusions and criticisms Appendix 1: More general gaugings and non-geometric backgrounds Appendix 2: Trading fluxes for branes and building domain walls

39 Ansatz and procedure The background fields Warped geometry M X M 4-D maximally symmetric space time X 6-D internal space ds 2 = e A(y) g µν (x) dx µ dx ν + g ij (y) dy i dy j Dilaton profile Φ(y) Antisymmetric tensors: compatibility with maximal symmetry imposes 0 or 4 legs in the space time H 3 and F p 3 only through internal cycles sources only with p 3

40 The equations Impose and solve supersymmetry variations Solve form eoms and Bis Caution: fluxes are quantized and enter with the sources in the tadpole conditions Supersymmetry variations δψ M = M ɛ /H MPɛ eφ /F n Γ M P n ɛ n δλ = / Φɛ /HPɛ eφ ( 1) n (5 n)/f n P n ɛ n P, P n : projectors (combinations of Γs)

41 Requiring N = 1 in 4 dimensions ɛ 1 = ξ + η + + ξ η ɛ 2 = ξ + η + ξ η ± the choices of sign correspond to IIA/B Antisymmetric tensors alter the 4-D effective theory which may admit ground states with N = 1 only Tadpole cancellation requires in general orientifold planes (O6, O3/O7 or O5/O9) which project out one supersymmetry

42 Inserting the expression for ɛ makes δψ M and δλ vanish provided η is a spinor defined and non-vanishing everywhere, covariantly constant with respect to a connection with torsion defined in terms of the various antisymmetric tensors X must admit an SU(3) group structure i.e. J 2 and Ω 3 s.t. J 2 and Ω 3 given in terms of η J 2 J 2 J 2 = 3i 4 Ω 3 Ω 3 J 2 Ω 3 = 0 J ij = 2l 2 s iη + γ ij η Ω ijk = 2l 3 s iη + γ ijk η +

43 η is not constant with respect to the Levi-Civita connection Integrability conditions do not lead to Ricci flatness the curvature is now related to the various antisymmetric tensors The holonomy group is not SU(3) dj 2 = 0 and dω 3 = 0 and this is a measure of the torsion The actual expressions for dj 2 and dω 3 depend on the backgrounds These expressions must fit the SU(3) group structure and this constrains in turn the backgrounds (metric, dilaton, forms) Form eoms and Bis must still be satisfied

44 The advent of generalized geometry The SU(3) torsion classes The backgrounds must be s.t. (SU(3) decomposition) dj 2 = 3 2 Im ( W 1 Ω 3 ) + W 4 J 2 + W 3 dω 3 = W 1 J 2 J 2 + W 2 J 2 + W 5 Ω 3 with W 1 a complex scalar, W 2 a complex primitive 2-form, W 3 a real primitive (2, 1) (1, 2)-form, W 4 a real vector and W 5 a complex (1, 0)-form (primitive: vanishing contraction with J 2 ) Fluxes introduce torsion generalized geometry [math literature] W 1 = W 2 = 0: complex geometry W 1 = W 3 = W 4 = 0: symplectic geometry W 1 = W 2 = W 3 = W 4 = 0: complex & symplectic Kähler W m = 0: SU(3) holonomy no torsion CY space Not all cases admit an exhaustive classification not all classes exhibit genuine concrete solutions in terms of fluxes

45 Summary of the first lecture Search for M X with warped maximally symmetric M, dilaton profile, antisymmetric tensors and N = 1 supersymmetry X is endowed with SU(3) group structure (J 2, Ω 3 ) s.t. dj 2 = dω 3 = decomposed with respect to the SU(3): torsion classes The antisymmetric tensors must be compatible with the above decomposition solve eoms and Bis plus tadpole cancellation conditions create quantized fluxes through internal cycles The geometry must fit the above decomposition depends on the torsion classes and on the flux numbers Beyond N = 1 CY orientifolds

46 Type IIB with H 3 and F 3 The general SU(3) decomposition of dj 2 and dω 3 sets constraints on H 3 and F 3 Three possible classes with A dj 2 ± ih 3 B F 3 ie Φ H 3 C d ( e Φ J 2 ) ± if3 (2, 1) and primitive More work is needed to exhibit actual solutions (eoms, Bis, tadpole conditions) possible for the case B where there is an underlying CY geometry [many groups: Hamburg, London, Munich, Paris,... ]

47 Bianchi identities df 5 = H 3 F 3 : induced charge density with quantized charge 1 l 4 s X H 3 F 3 N flux Tadpole: explicit D3/O3 charges are needed 1 df 5 = H 3 F 3 + 2µ 3 κ10 2 πd Q 3 µ 3 κ10 2 πo3 6 N flux + N D3 + Q 3 N O3 = 0 Note: geometric-flux alternative (term ω 2 F 5 in Bis) 1 Q p = 32/2 9 p

48 Actual framework: N = 1 type II orientifold Ansatz for the self-dual form F 5 = (1 + )dα dx 0 dx 1 dx 2 dx 3 Form eoms α(y) in terms of H 3, F 3 (free constant numbers) B-class SU(3) decomposition Φ constant ds 2 = αη µν dx µ dx ν + 1 α gij CY dy i dy j Geometry: warped Minkowski space time plus conformal Calabi Yau [revival of fluxes: Giddings, Kachru, Polchinski 02; dots]

49 Equations: forms, sources and constraints The Calabi Yau tale Introducing antisymmetric-tensor backgrounds Effective 4-D theory and moduli stabilization Gauged half-maximal supergravities and the embedding tensor 4-D gauged N = 4 supergravities as 10-D flux compactifications 4-D gauged supergravities with N = 1 supersymmetry 10-D origin: T6 /Z 2 Z 2 type II orientifolds with fluxes General conclusions and criticisms Appendix 1: More general gaugings and non-geometric backgrounds Appendix 2: Trading fluxes for branes and building domain walls

50 Moduli stabilization: expectations and caveats The Holy Grail: a string compactification without moduli No trick to generate it! Chance to find it? Controllable bona fide backgrounds do have moduli: stabilization is a therapy bonus: generating new scales Protocol: find a solution with a clear moduli space and modify the parameters of the solution in order to reduce this moduli space This is a perturbative approach in the vicinity of the original vacuum leaving aside most classes of vacua Either the new background is a small perturbation of the original one Or the new is a finite modification but its moduli space is no longer under control

51 Usual pattern Starting point: Calabi Yau orientifolds Type II Calabi Yau (data: moduli space, cohomology,... ) T 6 /Z N T 6 /Z N Z M Orientifold projection O: σω( 1) F L in IIA O6 planes σω( 1) F L or σω in IIB O3/O7 or O5/O9 planes No longer a CY properties in IIB forms have a parity: h 1,1 = h+ 1,1 + h1,1, h2,1 = h+ 2,1 + h2,1 Kähler moduli: h+ 1,1 invariant (J 2) plus h 1,1 new (B 2, C 2 ) complex-structure moduli: h 2,1 (Ω 3) ( O3/O7, + O5/O9) dilaton S = exp Φ ic 0 modified intersection numbers

52 Note on the orientifold effective 4-D theory: N = 1 supergravity h 2,1 + h 1,1 + 1 chiral and h± 2,1 vector N = 1 multiplets Caution: the holomorphic fields are not the original CY moduli appropriate field redefinitions are needed the scalar manifold has a different geometry no longer fully factorized similarly for the Kähler potential Even more: D-branes must be added to cancel the O-plane tadpoles open string moduli appear with contributions to the scalar manifold, the Kähler potential,... : brane displacements in transverse space and Wilson-line moduli

53 Final step in the search of a flux compactification Given the orientifold one can add the allowed fluxes according to the various parity properties and search for new solutions The allowed fluxes exist as a matter of principle they truly exist only when a genuine solution that incorporates them is found

54 Example IIB on T6 /Z 2 Z 2 with O = I 6 Ω( 1) F L O3-planes Seed for MSSM-like model building not very successful per se The seven main moduli h 1,1 = 3 Kähler T i : diagonal metric components and orbifold-unprojected B 2 axions = 3 cs U i : diagonal metric components and orbifold-unprojected C 2 axions h 2,1 axion-dilaton S: φ and C 0 Kähler potential κ4 2K = ln { (S + S) 3 ( i=1 U i + Ū i ) ( T i + T i )}

55 Cohomology dim H 3 = 2h 2,1 + 2 = 8 three-forms all odd under the orientifold projection: l 3 s α 0 = dx 1 dx 2 dx 3 l 3 s β 0 = dy 1 dy 2 dy 3 l 3 s α 1 = dy 1 dx 2 dx 3 l 3 s β 1 = dx 1 dy 2 dy 3. [2]. [2] holomorphic form (see T6 /Z 2 Z 2 ) odd: Ω 3 = l 3 ( s α0 + iu j α j iu 1 U 2 U 3 β 0 + ε ijk U i U j β k)

56 Antisymmetric tensors H 3, F 3 (G 3 F 3 is H 3 ) odd under O (because of Ω or ( 1) F L) and both supported by the T6 /Z 2 Z 2 orientifold a Λ, b Λ, c Λ, d Λ, Λ = 0,..., h 2,1 = 3 are the 16 antisymmetric-tensor internal components: G 3 = l 2 s 3 0 [(a Λ + isc Λ) α Λ + (b Λ + isd Λ ) β Λ] F 135 = a 0, H 135 = c 0 F 246 = b 0, H 246 = d 0 F 235 = a 1, H 235 = c 1 F 146 = b 1, H 146 = d 1. these internal three-form components are quantized flux numbers and induce a charge N flux = a Λ d Λ b Λ c Λ.

57 The final flux compactification The geometry is no longer the T6 /Z 2 Z 2 orientifold The H 3, F 3 induce a backreaction and the new geometry is now a conformal CY (class B in the SU(3) decomposition)

58 The issue of the moduli Physical intermezzo: why fluxes should stabilize the moduli? Internal cycles have sizes/shapes related to the moduli (e.g. 3-cycles and cs moduli) If antisymmetric tensors are switched on, they produce fluxes through these cycles and bring energy in the balance (e.g. H 3, F 3 through 3-cycles in IIB) This energy may lift the degeneracy of some moduli (e.g. stabilize the cs moduli in IIB with H 3, F 3 )

59 How does this stabilization emerge? By counting the moduli of the new solution 10-D viewpoint: hard 4-D viewpoint: we need the effective theory 10-D flux numbers deformation parameters of the 4-D effective supergravity New 10-D solution vacuum of the deformed 4-D effective theory Moduli and would-be moduli of the new 10-D background scalars of the deformed 4-D theory, now with a potential Stabilized (would-be) moduli of the new 10-D background massive scalars of the 4-D vacuum with vevs matching their values in the 10-D flux solution i.e. its superpotential not straightforward in presence of sophisticated internal manifolds

60 Effective IIB theory with H 3, F 3 Internal 6-D space: conformal CY with H 3, F 3, F 5 An important approximation Problem: warp factor α(y) higher derivative corrections Way out: assume at first approximation α const. (in)famous CY with torsion does not solve the equations provides the computational framework of the CY orientifolds (X ) within a reasonably controlled approximation allows to determine the superpotential [Gukov, Taylor, Vafa, Witten 99, 00; Mayr 00] W IIB (S, U) = 1 κ10 2 G 3(S) Ω 3 (U) X No Kähler-moduli dependence potential no-scale structure

61 Back to the example: IIB on T6 /Z 2 Z 2 plus O3/D3 l 3 s 4π W = b 0 + isd 0 i ( a 0 + isc 0) U 1 U 2 U 3 +i (b j + isd j ) U j + ( a i + isc i ) ε ijk U j U k The flux numbers are free modulo the tadpole condition a Λ d Λ b Λ c Λ + N D3 + Q 3 N O3 = 0 Minimization of V reasonable values of the moduli no strong coupling, no high curvatures Set of allowed flux numbers ground state of the potential V some internal conformal CY (with conformal factor given in terms of the fluxes) Remaining moduli Kähler moduli of the original T6 /Z 2 Z 2 orientifold The would-be moduli (S and cs) are stabilized massive scalars in the 4-D effective theory with masses determined by flux numbers

62 Some remarks All this is valid within the current approximations The control of these approximations is worse in IIA or heterotic compactifications although the stabilization perspectives are better Pause: can we stabilize the moduli with fluxes and sets of intersecting D-branes that have some phenomenological value? Example: IIA on T6 /Z 2 Z 2 plus O6/D6 Search for vacua with the usual restrictions 3 chiral families SU(3) SU(2) U(1) U(1) Output billions of models with extra chiral massless matter none without! Situation improves with other orientifolds (e.g. Z 6II ) Superiority of search for exact backgrounds?

63 IIA with non-vanishing NSNS or RR fluxes Internal geometries turn out to be in general exotic animals in the bestiary of SU(3) group structure: almost Kähler, half-flat,... Unknown moduli space Unapplicable CY familiar techniques Difficulties to reduce to 4-D

64 In practice: back to the CYs with torsion less accurately than in IIB with H 3, F 3 Minkowski or AdS 4 space-time Constant dilaton Internal CY orientifold (with O6) X F 0, F 2, H 3 obeying F 2 = dc 1 + F 0 B 2 df 2 = ω 2 F 2 + F 0 H 3 Q 6 µ 6 κ 2 10 π O6 3 F 2p, p > 1 unconstrained Form eoms and Bis can be satisfied but this set-up does not solve NS-sector eoms or equivalently does not fit the SU(3) group structure supersymmetry requirement

65 This rough approximation makes possible the 4-D reduction and the computation of the corresponding superpotential Term due to the addition of fluxes [Grimm, Louis 04] [ ] W IIA NSR flux (S, T, U) = 1 3 κ10 2 Ω c H 3 + F 2p e J c X ( ) with Ω c = C 3 + irec Ω 3, C = exp φ + K cs 2κ4 2 J c = B 2 + ij 2 p=0 and Plausible correction resulting from the backreaction of the geometry (captured by the torsion class: dω 3 = W 2 J 2 ) W IIA geom (S, T, U) = 1 κ10 2 Ω c dj c X Depends both on the complex-structure and Kähler moduli of X

66 Minimizing the corresponding IIA scalar potential is expected to exhibit supersymmetric 4-D vacua presumably corresponding to the exotic 6-D manifolds with more stabilized moduli (with respect to the original CY orientifold i.e. before the addition of fluxes) Remarks Better stabilization can also be achieved in IIB with more fluxes (e.g. geometrical) at a similar price less control over the genuine compactification space, its moduli and the corresponding reduction (note: mirror symmetry in presence of fluxes brings systematically torsion) Concrete examples of 6-D internal manifolds (half-flat, nilmanifolds) accompanying AdS 4 with fluxes were recently discussed [Tomasiello 07; Caviezel, Koerber, Körs, Lüst, Tsimpis, Zagermann 08]

67 Intermediate conclusions Top-down approach to flux compactifications Nice and rich mathematical framework for characterizing solutions of the two-derivative 10-D equations [Gualtieri, Hitchin] Concrete results corresponding to genuine compact manifolds with N = 1 supersymmetry are hard to exhibit [Graña, Minasian, Petrini, Tomasiello 06] The corresponding 4-D reductions and effective theories require further approximations like the Calabi Yau with torsion with some exceptions though [see Cassani s talk and works 07, 08 for N = 2]

68 What would be a bottom-up alternative? Start with a 4-D effective theory Parameterize all possible deformations: gaugings,... Translate the gauging,... parameters into flux numbers Reconstruct the microscopic 10-D theory Indeed: from the low-energy viewpoint N = 8 or N = 4 ungauged sugras with neutral scalars without potential toroidal compactifications N = 8 or N = 4 (or less) gauged sugras with charged scalars under (non-)abelian gauge groups and moduli-dependent superpotential (and potential) flux compactifications

69 Equations: forms, sources and constraints The Calabi Yau tale Introducing antisymmetric-tensor backgrounds Effective 4-D theory and moduli stabilization Gauged half-maximal supergravities and the embedding tensor 4-D gauged N = 4 supergravities as 10-D flux compactifications 4-D gauged supergravities with N = 1 supersymmetry 10-D origin: T6 /Z 2 Z 2 type II orientifolds with fluxes General conclusions and criticisms Appendix 1: More general gaugings and non-geometric backgrounds Appendix 2: Trading fluxes for branes and building domain walls

70 We focus on 4-D N = 4 theories Remind the basics of ungauged N = 4 sugra that describes e.g. toroidal compactification of heterotic theory (or type II orientifolds) Analyze the gauging procedure using the embedding tensor outstanding tool that captures all consistency constraints describes exhaustively the gauge algebras allows for reconstructing the scalar potential

71 Ungauged 4-D N = 4 supergravity Origin Reduction on a flat torus T 6 of 10-D heterotic supergravity (16 supercharges) 1 10-D gravitational multiplet l 10-D vector multiplets Spectrum 1 gravitational and n vector multiplets (6 + l) Bosonic on-shell content gravitational multiplet: 1 graviton, 6 graviphotons, 2 real scalars combined into the axion-dilaton τ = χ + i exp 2φ vector multiplet: 1 vector, 6 real scalars Fermionic on-shell content gravitational multiplet: 4 gravitinos, 4 Weyl fermions (including 3 gauginos) vector multiplet: 4 Weyl fermions (including 1 gaugino)

72 Interactions and local symmetries Bosonic Lagrangian: e 1 L bos = R + G ΛΣ (ϕ) µ ϕ Λ µ ϕ Σ 1 4 N MN(ϕ) F M µν F Nµν Λ, Σ = 1,..., 2 + 6n (scalars) M, N = 1,..., 6 + n (vectors) Gauge group: U(1) 6+n (Abelian local symmetries) All scalars are neutral and non-minimally coupled to the vectors: interaction terms of the type N (ϕ) F 2 There is no scalar potential The elimination of the auxiliary fields generates sigma-model interactions captured by the scalar manifold: M = SL(2, R) U(1) SO(6, n) SO(6) SO(n)

73 Global symmetries (some on-shell only) The Lagrangian depends on the vector potentials through their field strengths only [Gaillard, Zumino 81] electric magnetic duality symmetry Sp(12 + 2n, R) broken to the SL(2, R) SO(6, n) Sp(12 + 2n, R) (due to the other fields) with the embedding n (2, 6 + n) G = SL(2, R) SO(6, n) is the U-duality group: symmetry of the full theory acting linearly on all fields except for the scalars (e.g. the vectors belong to the fundamental 6 + n of SO(6, n)) H = U(1) SO(6) SO(n) is its maximal compact subgroup ( no ghosts in the sigma model)

74 Representation of the 2 + 6n scalars The axion dilaton τ parameterizes the coset SL(2,R) U(1) ( ) a b SL(2, R) acts as Möbius transformation τ c d cτ+d aτ+b ( ) M αβ are the components of 1 1 Reτ Imτ Reτ τ 2 which transforms in the bi-fundamental of SL(2, R) The remaining 6n scalars (from the vector multiplets) that parameterize the coset can be represented with a SO(6,n) SO(6) SO(n) (6 + n) (6 + n) symmetric matrix M with entries M KL : V = ( VM m, V N a ) m = 1,..., 6 a = 1,..., n are the coset representatives (they belong to SO(6, n) and satisfy η MN = m VM mv N m a VM a V N a = diag(6+, n ) ) M VV T transforms in the bi-fundamental of SO(6, n)

75 The bosonic Lagrangian: scalars, vectors and manifest symmetry Kinetic term for the SO(6,n) SO(6) SO(n) scalars 1 8 µm MN µ M MN Kinetic term for the SL(2,R) U(1) axion dilaton 1 2(Imτ) 2 µτ µ τ = 1 4 µm αβ µ M αβ Kinetic term for the Abelian electric vectors 1 4 ImτM MN F M µν F Nµν Reτ η MN ε µνκλ F M µν F N κλ

76 Gauging: deformation compatible with supersymmetry [de Wit, Nicolai 82; Hull 84; ] Promotion of a subgroup of the U-duality group to a local gauge symmetry supported by (part of) the existing U(1) n+6 vectors

77 Summary of the previous two lectures Top-down method: search for 10-D M X with internal antisymmetric-tensor vevs and 4 supercharges X is endowed with SU(3) group structure Extraction of the 4-D N = 1 data (W ) we need a basis which is often approximated to the closest CY one better treatment in some cases [see Cassani s talk for N = 2] Elaborating around the bottom-up: now N = 4 later N = 1 Analyze the possible deformations of a supersymmetric 4-D theory and promote them to 10 D Start with N = 4 gauged supergravities [de Wit, Nicolai 82; Hull 84]: the gauging procedure captures all deformations Idea: promote a subgroup of the global group of U-duality to a local gauge symmetry supported by part of the existing vectors Tool: embedding tensor

78 The SL(2, R) SO(6, n) algebra The generators of the duality group are 1. T MN = T NM, M,... = 1,..., 6 + n generate the SO(6, n) 2. S βγ = S γβ, β,... = +, generate the SL(2, R) ( ) S 0 0 = generates the electric magnetic duality 2 0 ( ) S = generates the axionic shifts τ τ + b 0 0 ( ) S = generates the axionic rescalings τ a τ {S ++, S + } generate the non-semi-simple subalgebra A 2,2 SL(2, R) of axionic rescalings and axionic shifts

79 Only A 2,2 SO(6, n) is realized off-shell in heterotic theory The A 2,2 does not mix electric and magnetic gauge fields genuine electric magnetic duality transformations relate (non locally) different Lagrangians with equivalent field equations Aim: gauge any subgroup of SL(2, R) SO(6, n) Method: put electric and magnetic duals in a unique setting [de Wit, Samtleben, Trigiante 02 ] Advantage: gauge groups which are not manifest invariance of the original Lagrangian important in flux compactifications

80 A Lagrangian exists that includes electric and magnetic duals without altering the number of propagating degrees of freedom { } The 2 (6 + n) vectors A M+ µ, A M µ form a (2, Vec) of SL(2, R) SO(6, n) (Vec of Sp(12 + 2n, R)) both appear in the Lagrangian { } without kinetic terms for A M µ (choice of symplectic frame) { } Extra 2-form auxiliary fields Cµν MN, Cµν αβ dual to the scalars { } { } eoms set the duality relation among A M+ µ and A M µ { } duality relation with the scalars set by the eoms of the Note: hierarchies of forms often appear in implementing gauge invariance A M µ

81 The embedding tensor: a primer Gauging: µ D µ = µ ga Lα µ Ξ αl with Ξ αl some generators The generators of the gauge algebra are linear combinations of the U-duality ones: Ξ αl = 1 2 (Θ αlmn T MN + Θ αlβγ S βγ) where {Θ αlmn, Θ αlβγ } (2, Vec Adj) + (2 3, Vec) of SL(2, R) SO(6, n) is the embedding tensor At most 6 + n Ξ s are independent: Θ is subject to constraints reducing its rank

82 Gauge invariance, supersymmetry,... [linear constraints] γn Ξ(αL βm Ω δq)γn = 0 This reduces the embedding tensor to (2, Ant 3 ) + (2, Vec): Ξ αl = 1 ) (f αlmn T MN + η LQ ξ αp T QP + ε γβ ξ βl S γα 2 (mysterious constraint easy to solve) Final gauging parameters: f αlmn, ξ αl

83 The fundamental of SL(2, R) SO(6, n) must contain the adjoint of the gauge algebra and the latter must close [quadratic constraints] ] [Ξ αl, Ξ βm = Ξ (i) η MN ξ αm ξ βn = 0 (ii) η MN ξ (αm f β)nij = 0 γn Ξ [αl βm] γn (iii) ε αβ ( ξ αi ξ βj + η MN ξ αm f βnij ) = 0 (iv) η MN f αmi [J f βkl]n 1 2 ξ α[j f βkl]i 1 6 ε αβ ε γδ ξ γi f δjkl ηmn ξ αm f βn[jk η L]I f αjkl ξ βi = 0 (Jacobi-like)

84 Important remarks Introducing naive covariant derivatives is not enough to γn guarantee gauge invariance because Ξ = 0 (αl βm) - auxiliary 2-foms are necessary - enter the field strengths, the gauge transformations and the covariant derivatives f s and ξ s are the gauging parameters which determine - the algebra and its commutators - the charges and covariant derivatives - the scalar potential - the mass matrices f αjkl are not necessarily structure constants of some algebra

85 Lagrangian formulation including electric and magnetic An explicit Lagrangian is associated with any consistent gauging and its bosonic sector has three parts [Schön and Weidner 06] L kin kinetic terms for graviton, electric vectors and scalars: as previously, with covariant derivatives and appropriately modified gauge field strengths that include now the 2-forms L top auxiliary-field contributions (magnetic vectors and 2-forms) necessary to maintain gauge invariance and the correct number of propagating fields ( L pot = e 4 f αmnp f βqrs M αβ( 1 3 MMQ M NR M PS + ( 2 3 ηmq M MQ) η NR η PS) 4 9 f αmnp f βqrs ε αβ M MNPQRS + ) 3ξα M ξ N β Mαβ M MN the scalar potential Rich phenomenology: supersymmetry breaking, spontaneous symmetry breaking, Stückelberg masses, bounded, unbounded, runaway potentials,...

86 Equations: forms, sources and constraints The Calabi Yau tale Introducing antisymmetric-tensor backgrounds Effective 4-D theory and moduli stabilization Gauged half-maximal supergravities and the embedding tensor 4-D gauged N = 4 supergravities as 10-D flux compactifications 4-D gauged supergravities with N = 1 supersymmetry 10-D origin: T6 /Z 2 Z 2 type II orientifolds with fluxes General conclusions and criticisms Appendix 1: More general gaugings and non-geometric backgrounds Appendix 2: Trading fluxes for branes and building domain walls

87 Central problem Flux gauging correspondence: {f αlmn, ξ αl } 4-D { flux numbers} 10-D Here in theories with 16 supercharges

88 A particular class of gaugings Pure SO(6, n) gaugings: ξ αl = 0... One consistency constraint [(iv)]: η MN f αmi [J f βkl]n = 0 This generalized Jacobi is solved by introducing the de Roo Wagemans phases [de Roo, Wagemans 85]... involving electric vectors only: f LMN = 0 One set of ( 1 + n 6 ) (5 + n)(4 + n) parameters: f+lmn f LMN One constraint: η MN f MI [J f KL]N = 0

89 One solution We focus on n = 6: 12 vectors in total Light-cone-like convention: {I } {i, i }, η = ( 0 I6 I sets of (220 in total) parameters: f ijk, f ijk, f ij k, f i j k further choice: f ij k = f i j k = parameters: f ijk 3β ijk and f k ij 2γ k ij constraints: γ k [ij γn l]k = 0 = γk [ij β ln]k )

90 The gauge algebra At most 12 independent generators out of 2 12 Ξ α,l - Ξ +i = 3 2 β ijk T jk + 2γ k ij T j k Ξ i - Ξ +i = γ i jk T jk Ξ i - Ξ i,i = 0 Commutation relations for {Ξ i, Ξ j } SO(6, 6) - [ Ξ i, Ξ j ] = 0 ] - [Ξ j, Ξ i = 2γ i jk Ξk ] - [Ξ i, Ξ j = 2γ k ij Ξ k + 3β ijk Ξk

91 Properties Algebra generally non-semi-simple and non-compact (note: non-compact/non-semi-simple groups can be gauged provided their non-positive/degenerate Cartan metrics are not used) {Ξ i } spans an Abelian subalgebra: translations inside the MASA of SO(6, 6) appear as central charges Unimodular gaugings: γ j ij = 0 not compulsory, required for flat or semi-simple algebras, more exotic if relaxed

92 The dynamics of the electric gaugings ( h The 36 scalars M MN ij h = ik ) b kj b ik h kj h ij b ik h kl acquire a b lj potential Unbounded for semi-simple algebras Bounded with zero cosmological constant without tachyons for flat algebras Provides masses for some scalars The dilaton is runaway

93 The 12 vectors: embedded in SO(6, 6) Generators of local symmetries enter in covariant derivatives The scalars h, b, τ are charged under {Ξ i } and/or {Ξ i } Gauged translation symmetries {Ξ i } generate Stückelberg couplings axions are gauged away and the translation vectors become massive

94 Origin: heterotic 10-D reduction 10-D pure N = 1 supergravity Action for the bosonic sector (up to 1 /2κ 2 10) M dx X dy g e Φ ( R + g MN M Φ N Φ 1 12 H MNK H MNK ) Bosonic symmetries diffeomorphisms (internal) SO(1, 1) rescalings (external): Φ Φ + 4λ, {g MN, B MN } e λ {g MN, B MN }

95 Dimensional reduction X is compact: infinitude of 4-D modes Reduction: effective theory on M for a finite subset consistent provided the pattern of selection guarantees the decoupling of these modes Data: X plus a decomposition plus an ansatz for the y-dependance of all fields dictates the selection pattern converts some 10-D symmetries into 4-D gauge invariance Necessary consistency condition: L independent of y

96 Standard reduction on flat torus: no y-dependence 1 graviton, Abelian vectors, scalars, 1 dilaton, 1 axion (dual to the NSNS form) all massless and neutral Scherk Schwarz reduction: twisted torus [Scherk, Schwarz 79; long literature] Toroidal reduction with spin connection based on a symmetry subgroup e.g. diffeomorphisms Fields acquire a non-trivial y-profile (not L) Univalueness of the fields is lost (twists) but the symmetries guarantee continuity for the fiber bundles The torus resembles locally a group manifold The selected symmetries act as isometries and are converted into (non-abelian) gauge symmetries in the reduced theory

97 Twisted torus with internal NSNS background [Maharana, Schwarz 92; Kaloper, Myers 99] Metric ansatz ds 2 = g µν (x) dx µ dx ν ) ) + h ij (x) (θ i (y) + A i µ(x) dx (θ µ j (y) + A j µ(x) dx ν Vielbein: θ i = e i j (y) dy j Spin connection: ω i dθ i = γ i jk θj θ k ] Dual tangent vectors e i : [e i, e j = 2γ k ij e k Consistency: local group structure γ k ij constant internal spin-connection vevs i.e. geometrical fluxes Bianchi identity: dω i = 0 γ i j[k γj lm] = 0 Isometric invariance of the internal volume form: γ i ij = 0 (unimodularity)

98 Antisymmetric-tensor ansatz H = 1 2 [µb νλ] (x) dx µ dx ν dx λ + [µ B ν]i (x) dx µ dx ν θ ( ) i µ b ij (x) B µk (x)γ k ij dx µ θ i θ j β ijkθ i θ j θ k Consistency demands β ijk be constant (eoms) internal Kalb Ramond vevs i.e. NSNS fluxes Bianchi identity: dh = 0 γ k [ij β ln]k = 0 Dilaton ansatz Φ(x) = φ(x) log det h

99 Reduced action After various field redefinitions, the 4-D action is found to match that of the electric gauging and the transformation properties under 10-D local symmetries reveal the same gauge group and equal charges The Kaluza Klein vectors Ξ i The Kalb Ramond vectors Ξ i (Abelian) The correspondence gaugings fluxes is encoded in f ijk 3β ijk and f k ij 2γ k ij for the NSNS and geometrical fluxes respectively they satisfy the same quadratic constraints Note: there are non-unimodular gaugings with f j ij = 0 while the above reduction leads to γ j ij = 0

100 There are many more gaugings! With ξ αl = 0 Gaugings embedded in SL(2, R) SO(6, 6) involving electric magnetic or axionic symmetries Axionic gaugings (ξ i = ξ +i ) correspond to Scherk Schwarz reductions involving the 10-D duality shift under SO(1, 1): external twisted tori non-unimodular reductions [Derendinger, Petropoulos, Prezas 07] Towards a general correspondence {f αlmn, ξ αl } 4-D {flux numbers} 10-D generalized Jacobi identites full Bianchi identities

101 Further gaugings further fluxes f +IJK, ξ +L : 232 electric parameters - f +ijk NSNS, f +ijk, ξ +i spin-connection [Kaloper, Myers 99; Andrianopoli, Lledo, Trigiante 05 in the unimodular case; Derendinger, Petropoulos, Prezas 07 in the non-unimodular;... ] - f +ij k T-dual NSNS, f +i j k, ξ +i T-dual spin-connection: non-geometric [Hull et al. 05; Shelton, Taylor, Wecht 05; Dall Agata, Prezas, Samtleben, Trigiante 07;... ] f IJK, ξ L : 232 magnetic-dual parameters - f ijk NSNS, f ijk, ξ i spin-connection - f ij k T-dual NSNS, f i j k, ξ i T-dual spin-connection The number of degrees of freedom does not change the algebra, its SL(2, R) SO(6, n) embedding and the higher-dimensional setup do What is precisely the higher-dimensional setup? Non-geometric backgrounds?

102 Equations: forms, sources and constraints The Calabi Yau tale Introducing antisymmetric-tensor backgrounds Effective 4-D theory and moduli stabilization Gauged half-maximal supergravities and the embedding tensor 4-D gauged N = 4 supergravities as 10-D flux compactifications 4-D gauged supergravities with N = 1 supersymmetry 10-D origin: T6 /Z 2 Z 2 type II orientifolds with fluxes General conclusions and criticisms Appendix 1: More general gaugings and non-geometric backgrounds Appendix 2: Trading fluxes for branes and building domain walls

103 The role of supersymmetry In maximal or half-maximal supergravity Universal duality group & scalar manifold: organizing pattern for the gaugings presented here for N = 4, equally good for N = 8 with M = E 7,7 SU(8) In N = 2 and N = 1 supergravities No universal structure available the scalar manifold can be involved: not factorized, not isotropic,... In N = 1 supergravity the gauging procedure does not capture all possible deformations compatible with supersymmetry (e.g. D-terms are related to gauge kinetic functions not to the superpotential) Gaugings are studied systematically in specific classes of N = 1 theories obtained by truncation of N = 4 supergravity

104 Z 2 Z 2 truncated N = 4 gauged supergravity The action of Z 2 Z 2 Defined directly on the N = 4 4-D fields so that (the original number of vector multiplets is n = 6) M N=4 = SL(2,R) U(1) SO(6,6) SO(6) SO(6) M N=1 = SL(2,R) U(1) 3 i=1 SO(2,2) SO(2) SO(2) Ti,U i Respects an SL(2, R) SO(2, 2) SO(3) global symmetry under which the scalars have well-defined charges SO(6, 6) SO(2, 2) SO(3) with 12 (4, 3) SO(3) is a plane-intrechange symmetry

105 The gaugings in the truncated theory are organized by the projected embedding tensor Part of the parameters {f αlmn, ξ αl } (2, 220) + (2, 12) of SL(2, R) SO(6, 6) are projected out by the Z 2 Z 2 Specific case (studied in N = 4 in relation with fluxes) ξ αl = 0: pure SO(6, 6) gaugings f LMN = 0: only with electric vectors 220 parameters satisfying η MN f MI [J f KL]N = 0 (f LMN f +LMN ) The 220 parameters are reduced to 24 according to the embedding 220 (20, 1) + (3, 1) + (1, 1): Λ abc Λ (abc), Λ d, a, b, c, d = 1,..., 4 with η df Λ abd Λ cef = η df Λ bcd Λ afe = η df Λ cad Λ bfe

106 Superpotential in gauged Z 2 Z 2 -truncated supergravities The machinery of the embedding tensor delivers the most general N = 1 superpotential depending on 24 parameters A redefinition of the N = 4 fields is required to respect holomorphicity, fit chiral N = 1 multiplets and recover the scalar manifold M Z2 Z 2 depends on the underlying theory, heterotic or type IIA/B In type IIA the procedure leads to a superpotential depending on all 7 moduli at hand (case presented: plane-symmetric) l 3 s 4π W = Λ iλ 4 S + iλ 112 (T 1 + T 2 + T 3 ) + iλ 114 (U 1 + U 2 + U 3 ) Λ 1 S (T 1 + T 2 + T 3 ) Λ 122 (T 1 T 2 + T 2 T 3 + T 3 T 1 ) + Λ 113 (T 1 U 1 + T 2 U 2 + T 3 U 3 ) Λ 124 (T 1 U 2 + T 1 U 3 + T 2 U 1 + T 2 U 3 + T 3 U 1 + T 3 U 2 ) iλ 222 T 1 T 2 T 3 9 parameters

107 Equations: forms, sources and constraints The Calabi Yau tale Introducing antisymmetric-tensor backgrounds Effective 4-D theory and moduli stabilization Gauged half-maximal supergravities and the embedding tensor 4-D gauged N = 4 supergravities as 10-D flux compactifications 4-D gauged supergravities with N = 1 supersymmetry 10-D origin: T6 /Z 2 Z 2 type II orientifolds with fluxes General conclusions and criticisms Appendix 1: More general gaugings and non-geometric backgrounds Appendix 2: Trading fluxes for branes and building domain walls

108 Type IIA CY orientifolds with O6 planes The T6 /Z 2 Z 2 orbifold with O6 planes Z 2 projection: (x 1, y 1, x 2, y 2 ) (x 1, y 1, x 2, y 2 ) Z 2 projection: (x 2, y 2, x 3, y 3 ) (x 1, y 1, x 2, y 2 ) σω( 1) F L involution: y i y i Good handle on the effect of fluxes within the usual approximations and on the exploration beyond the orbifold point (using unstabilized moduli)