DFT on Calabi-Yau threefolds and Axion Inflation

Transcription

1 ETH Zürich, p.1/42 DFT on Calabi-Yau threefolds and Axion Inflation Ralph Blumenhagen Max-Planck-Institut für Physik, München (RB, Font, Fuchs, Herschmann, Plauschinn, arxiv: ) (RB, Font, Fuchs, Herschmann, Plauschinn, Sekiguchi, Wolf, arxiv: ) (RB, Font, Plauschinn, arxiv: )

2 Introduction ETH Zürich, p.2/42

3 ETH Zürich, p.2/42 Introduction Moduli stabilization in string theory: Race-track scenario KKLT LARGE volume scenario Based on instanton effects exponential hierarchies can generate M susy M Pl Experimentally: Supersymmetry not found at LHC with M < 1TeV. Not excluded large field inflation: M inf M GUT Contemplate scenario of moduli stabilization with only polynomial hierarchies string tree-level with fluxes

4 Introduction ETH Zürich, p.3/42

5 ETH Zürich, p.3/42 Introduction PLANCK 2015, BICEP2 results: upper bound: r < 0.07 spectral index: n s = ±0.004 and its running α s = 0.002± amplitude of the scalar power spectrum P = (2.142±0.049) 10 9 Lyth bound and φ r O(1) M pl 0.01 ( M inf = (V inf ) 1 r ) GeV

6 Introduction ETH Zürich, p.4/42

7 ETH Zürich, p.4/42 Introduction Inflationary mass scales: Hubble constant during inflation: H GeV. mass scale of inflation: V inf = M 4 inf = 3M2 Pl H2 inf M inf GeV mass of inflaton during inflation: M 2 Θ = 3ηH2 M Θ GeV Large field inflation with Φ > M pl : Makes it important to control Planck suppressed operators (eta-problem) Invoking a symmetry like the shift symmetry of axions helps

8 Axion inflation ETH Zürich, p.5/42

9 ETH Zürich, p.5/42 Axion inflation Axions are ubiquitous in string theory so that many scenarios have been proposed Natural inflation with a potential V(θ) = Ae S E (1 cos(θ/f)). Hard to realize in string theory, as f > 1 lies outside perturbative control. (Freese,Frieman,Olinto) Aligned inflation with two axions, f eff > 1. (Kim,Nilles,Peloso) N-flation with many axions and f eff > 1. (Dimopoulos,Kachru,McGreevy,Wacker) Comment: These models have come under pressure by the weak gravity conjecture, which for instantons was proposed to be f S E < 1. (Montero,Uranga,Valenzuela),(Brown,Cottrell,Shiu,Soler)

10 Axion monodromy inflation ETH Zürich, p.6/42

11 ETH Zürich, p.6/42 Axion monodromy inflation Monodromy inflation: Shift symmetry is broken by branes or fluxes unwrapping the compact axion polynomial potential for θ. (Kaloper, Sorbo), (Silverstein,Westphal) non-pert. fluxes Discrete shift symmetry acts also on the fluxes, i.e. one gets different branches tunneling à la Coleman-de Lucia

12 Axion monodromy inflation ETH Zürich, p.7/42

13 ETH Zürich, p.7/42 Axion monodromy inflation Recent proposal: Realize axion monodromy inflation via the F-term scalar potential induced by background fluxes. (Marchesano.Shiu,Uranga),(Hebecker, Kraus, Wittkowski),(Bhg, Plauschinn) Advantages Generating the inflaton potential, supersymmetry is broken spontaneously by the very same effect by which usually moduli are stabilized Generic, as the field strengths F p+1 = dc p +H C p 2 involves the gauge potentials C p 2.

14 Objective ETH Zürich, p.8/42

15 ETH Zürich, p.8/42 Objective For a controllable single field inflationary scenario, all moduli need to be stabilized such that M Pl > M s > M KK > M inf > M mod > H inf > M Θ Aim: Systematic study of realizing single-field fluxed F-term axion monodromy inflation, taking into account the interplay with moduli stabilization. Continues the studies from (Bhg,Herschmann,Plauschinn), (Hebecker, Mangat, Rombineve, Wittkowsky) by including the Kähler moduli. Note: There exist a no-go theorem for having an unconstrained axion in supersymmetric minima of N = 1 supergravity models (Conlon)

16 Introduction ETH Zürich, p.9/42

17 Introduction Realistic string model building = break N=2 N=1 via orientifold projection Scalar potential splits V DFT = V F +V D +V NS tad Fits into 4d N = 1 SUGRA, i.e. V F can be computed via a Kähler and superpotential Apply this to SPHENO, i.e. a high moduli stabilization scheme and inflationary cosmology Lit: Type IIB orientifolds on CY threefolds with geometric and non-geometric fluxes. (Shelton,Taylor,Wecht), (Aldazabal,Camara,Ibanez,Font), (Grana, Louis, Waldram), (Benmachiche, Grimm), (Micu, Palti, Tasinato) Recently: (RB, Font, Fuchs, Herschmann, Plauschinn, Sekiguchi, Wolf, arxiv: ) based on ideas from (Marchesano.Shiu,Uranga),(Hebecker, Kraus, Wittkowski),(Bhg, Plauschinn) ETH Zürich, p.9/42

18 Framework ETH Zürich, p.10/42

19 ETH Zürich, p.10/42 Framework Framework: Type IIB orientifolds on CY threefolds with geometric and non-geometric fluxes. (Shelton,Taylor,Wecht), (Aldazabal,Camara,Ibanez,Font), (Grana, Louis, Waldram), (Benmachiche, Grimm), (Micu, Palti, Tasinato) Dimensional reduction of DFT on Calabi-Yau manifold with (small) fluxes, treated as perturbations.(rb, Font, Plauschinn, arxiv: )

20 Framework ETH Zürich, p.11/42

21 ETH Zürich, p.11/42 Framework Generalization of the dimensional reduction a la Taylor/Vafa SUGRA with H 3,F 3 -form on CY (Taylor,Vafa, hep-th/ ) Technical problem: DFT action contains the unknown metric g ii on the CY Solution: Rewrite the action (on CY) such that only Ω, J and -op. appear, like L = e 2φ 2 d10 x GH ijk H i j k gii g jj g kk = e 2φ 2 H H, DFT contains many more terms that are not of this simple type.

22 Flux formulation of DFT ETH Zürich, p.12/42

23 ETH Zürich, p.12/42 Flux formulation of DFT (Siegel; Hohm, Kwak; Aldazabal, Geissbuhler, Marques, Nunez, Penas;...) In a generalized frame formalism with N-bein ( ) E A e i a e k a B I = ki 0 e a, i the action in the NS-NS sector is given by S NSNS = 1 d D X e [F 2d ABC F A B C 2κ 2 10 ( 1 4 SAA η BB η CC 1 12 SAA S BB S CC 1 6 ηaa η BB η CC +F A F A (η ] AA S AA ) )

24 Dimensional Reduction ETH Zürich, p.13/42

25 ETH Zürich, p.13/42 Dimensional Reduction We perform the dimensional reduction such that The (unknown) metric of the Calabi-Yau three-fold g ij (x) only depends on the usual coordinates. On a CY there are no non-trivial homological one-cycles, and hence F A = 0. The term F ABC F A B C η AA η BB η CC vanishes via Bianchi identities. We give constant expectation values to the fluxes F IJK and assume that they are only subject to quadratic constraints (Bianchi identities). We are only interested in terms contributing to the scalar potential.

26 Flux formulation of DFT ETH Zürich, p.14/42

27 ETH Zürich, p.14/42 Flux formulation of DFT Thus, we consider the part of the DFT action in the flux formulation S NSNS = 1 2κ 2 d D X e 2d F IJK F I J K 10 ( 1 4 HII η JJ η KK 1 12 HII H JJ H +...) KK and perform a further dimensional reduction on a CY. The generalized flux F IJK splits into components F ijk = H ijk, F i jk = F i jk, F i jk = Q i jk, F ijk = R ijk.

28 Fluxes ETH Zürich, p.15/42

29 ETH Zürich, p.15/42 Fluxes Geometric fluxes H,F- and F-flux and non-geometric fluxes Q,R are operators acting on p-forms Twisted differential D H = 1 3! H ijk dx i dx j dx k, F = 1 2! Fk ij dx i dx j ι k, Q = 1 2! Q i jk dx i ι j ι k, R = 1 3! Rijk ι i ι j ι k. D = d H F Q R. Nilpotency, D 2 = 0, leads to a set of quadratic constraints on the fluxes.

30 Bianchi identities ETH Zürich, p.16/42

31 ETH Zürich, p.16/42 Bianchi identities These constraints correspond to Bianchi identities 0 = H m[ab F m cd], 0 = F m [bcf d a]m +H m[ab Q md c], 0 = F m [ab]q [cd] m 4F [c m[aq d]m b] +H mab R mcd, 0 = Q [bc m Q a]m d +R m[ab F c] md, 0 = R m[ab Q cd] m, 0 = R mn[a F b] mn, 0 = R amn H bmn F a mnq mn b, 0 = Q mn [a H b]mn, 0 = R mnl H mnl.

32 DFT on CY ETH Zürich, p.17/42

33 ETH Zürich, p.17/42 DFT on CY With χ := e B D ( e B+iJ) and Ψ := e B D ( e B Ω ), the DFT action in the flux formulation S NSNS = 1 2κ 2 10 d D X e 2d F IJK F I J K ( 1 4 HII η JJ η KK 1 12 HII H JJ H KK +...) can be rewritten as (very tedious computation) S NSNS = 1 [ 1 2κ 2 e 2φ 10 2 χ χ Ψ Ψ 1 ( ) ( ) 1( ) ( ) ] Ω χ Ω χ Ω χ Ω χ. 4 4

34 R-R sector ETH Zürich, p.18/42

35 ETH Zürich, p.18/42 R-R sector The DFT action in the R-R sector can be expressed as L RR = 1 2 G G. with G = F (3) +e B D ( e B C ). We have managed to reexpress the DFT action compactified on a CY just in terms of quantities that do not explicitly contain the CY metric. now can further dimensionally reduce the Lagrangian along Taylor/Vafa

36 Further Reduction on CY ETH Zürich, p.19/42

37 ETH Zürich, p.19/42 Further Reduction on CY On CY X introduce a symplectic basis as {α Λ,β Λ } H 3 (X), Λ = 0,...,h 2,1. For the even cohomology of X we introduce bases of the form {ω A } H 1,1 (X), {σ A } H 2,2 (X) with A = 1,...,h 1,1. It is convenient to extend this with the zero- and six-form {ω A } = { g V dx6, ω A }, {σ A } = { 1, σ A}, A = 0,...,h 1,1.

38 Further Reduction on CY ETH Zürich, p.20/42

39 ETH Zürich, p.20/42 Further Reduction on CY One defines the constant fluxes via the relations Dα Λ = q A Λ ω A +f ΛA σ A, Dβ Λ = q ΛA ω A + f Λ Aσ A, Dω A = f Λ Aα Λ +f ΛA β Λ, Dσ A = q ΛA α Λ q A Λ β Λ. with f Λ0 = r Λ, fλ 0 = r Λ, q 0 Λ = h Λ, q Λ0 = h Λ.

40 Further Reduction on CY ETH Zürich, p.21/42

41 ETH Zürich, p.21/42 Further Reduction on CY Using then the combined basis {ω A,σ A }, we can define a complex (2h 1,1 +2)-dimensional vector V 1 in the following way V 1 = 1 6 κ ABCJ A J B J C J A κ ABC J B J C, We introduce a (2h 2,1 +2)-dimensional vector as ( ) V 2 = X Λ F Λ.

42 Further Reduction on CY ETH Zürich, p.22/42

43 ETH Zürich, p.22/42 Further Reduction on CY V = 1 2 ( F T +C T O T) M 1 (F+O C ) with + e 2φ 2 V T 1 O T M 1 O V 1 + e 2φ 2 V T 2 Õ M 2 ÕT V 2 e 2φ 4V V T 2 C O M 1,2 = ( V 1 V T 1 +V 1 V T 1 ) O T C T V 2. ( )( )( ) 1 ReN ImN ImN 1 ReN 1 related to the period matrices of the CY.

44 Further Reduction on CY ETH Zürich, p.23/42

45 ETH Zürich, p.23/42 Further Reduction on CY This scalar potential was shown to agree with the one of N=2 gauged supergravity. (D Auria, Ferrara, Trigiante, hep-th/ ) Scalar potential of DFT on fluxed CY Scalar potential of N=2 GSUGRA Orientifold projection: N=2 N=1 susy

46 Orientifold projection ETH Zürich, p.24/42

47 ETH Zürich, p.24/42 Orientifold projection Perform an orientifold projection σ : J J, σ : Ω Ω, with O7- and O3-planes. Splits the cohomology into even and odd parts H p,q (X) = H p,q + (X) Hp,q (X). The fields of the ten-dimensional theory transform as Ω P ( 1) F L = { g, φ, C (0), C (4) even, B, C (2) odd.

48 Orientifold projection ETH Zürich, p.25/42

49 ETH Zürich, p.25/42 Orientifold projection The holomorphic three-form Ω is expanded in H 3 (X) Ω = X λ α λ F λ β λ. The Kähler form J and the components of the ten-dimensional form fields are combined into the complex fields τ = C (0) +ie φ, G a = c a +τb a, T α = i 2 κ αβγˆt βˆt γ +ρ α κ αabc a b b i 4 eφ κ αab G a (G G) b,

50 Orientifold projection ETH Zürich, p.26/42

51 Orientifold projection These moduli can be encoded in a complex, even multi-form Φ ev c as Φ ev c = τ +G a ω a +T α σ α. with ω a H 11 and σ α H+ 22. One finds the orientifold even fluxes F (3) : F λ, Fλ, H : h λ, hλ, F : fˆλα, fˆλα, f λa, fλ a, Q : qˆλa, qˆλa, q λ α, q λα, R : rˆλ, rˆλ. α λ H 21 and αˆλ H ETH Zürich, p.26/42

52 F-term potential ETH Zürich, p.27/42

53 ETH Zürich, p.27/42 F-term potential Kähler potential: K = log [ i(τ τ) ] 2log ˆV log [ i X Ω Ω ], Implicite in the T α and G a moduli. The superpotential in the presence of all fluxes is ( ) W = F (3) +DΦ ev c Ω. X

54 F-term potential ETH Zürich, p.28/42

55 ETH Zürich, p.28/42 F-term potential After a tedious computation, one can rewrite V F as V F = M4 Pl eφ 2ˆV 2 X ( 1 4 e 2φ [ 1 2 (Re ˇχ) (Re ˇχ)+ 1 2 ˇΨ ˇΨ ( Ω ˇχ ) ( Ω ˇχ ) 1 4 ( Ω ˇχ ) ( Ω ˇχ ) ] + 1 2Ǧ Ǧ ) [(Imτ) (ImG a )ω a +(ImT α )σ α] DF (3). Prefactor= M 4 s so that the first two lines match with the orientifold projected DFT action The third line corresponds to NS-NS tadpole terms, which will be cancelled by local sources.

56 D-term potential ETH Zürich, p.29/42

57 D-term potential Only (Re ˇχ) appeared, as (Im ˇχ) H 3 +(X). Its contribution in the DFT Lagrangian is Defining L H 3 + = 1 2 e 2φ (Im ˇχ) (Im ˇχ). Dˆλ = rˆλ ( V 1 2 κ αabt α b a b b) + qˆλa κ aαb t α b b fˆλ α t α, Dˆλ = rˆλ( V 1 2 κ αab t α b a b b) qˆλa κ aαb t α b b +fˆλα t α. one can compute (Robins,Wrase, arxiv: ) L H 3 + = 1 ) T ( D 2 e 2φ ˇM 1 D ) ( D D ETH Zürich, p.29/42

58 D-term potential ETH Zürich, p.30/42

59 ETH Zürich, p.30/42 D-term potential Transforming into a basis where rˆλ = qˆλa = fˆλ α = 0, this can be written as a D-term for the h 21 + abelian gauge fields V D = M4 Pl 2 [ (Ref) 1] ab Da D b. Summary: Performing an orientifold projection, the even part of the DFT scalar potential splits into three pieces V = V F +V D +V NS-tad, with the NS-NS tadpole cancelled after R-R tadpole cancellation.

60 Objective ETH Zürich, p.31/42

61 Objective Scheme of moduli stabilization such that the following aspects are realized: There exist non-supersymmetric minima stabilizing the saxions in their perturbative regime. All mass eigenvalues are positive semi-definite, where the massless states are only axions. For both the values of the moduli in the minima and the mass of the heavy moduli one has parametric control in terms of ratios of fluxes. One has either parametric or at least numerical control over the mass of the lightest (massive) axion, i.e. the inflaton candidate. The moduli masses are smaller than the string and the Kaluza-Klein scale. ETH Zürich, p.31/42

62 ETH Zürich, p.32/42

63 ETH Zürich, p.32/42 results could be presented in the discussion session Thank You!

64 A representative model ETH Zürich, p.33/42

65 ETH Zürich, p.33/42 A representative model Kähler potential is given by K = 3log(T +T) log(s +S). Fluxes generate superpotential W = i f+ihs +iqt, with f,h,q Z. Resulting scalar potential V = (hs+ f) 2 16sτ 3 6hqs 2q f 16sτ 2 5q2 48sτ + θ2 16sτ 3

66 A representative model ETH Zürich, p.34/42

67 ETH Zürich, p.34/42 A representative model Non-supersymmetric, tachyon-free minimum with τ 0 = 6 f 5q, s 0 = f h, θ 0 = 0. Mass eigenvalues M 2 mod,i = µ i hq 3 MPl 2 16 f 2 4π, with µ i > 0. Gravitino-mass scale: M 3 2 p M mod

68 Mass scales ETH Zürich, p.35/42

69 ETH Zürich, p.35/42 Mass scales Cosmological constant in AdS minimum: V 0 = µ C hq 3 MPl 4 16 f 2 4π Uplift: It is possible to find (new) scaling type minima with V 0 0 by including an uplifting D3-brane (D-term) V = V F + A V 4 3 +(V D ) (Bhg, Damian, Font, Fuchs, Herschmann, Sun, arxiv: ) Relation of mass scales: M s p M KK p M mod

70 Axion inflaton ETH Zürich, p.36/42

71 ETH Zürich, p.36/42 Axion inflaton Generate a non-trivial scalar potential for the massless axion Θ by turning on additional fluxes f ax and deform This quite generically leads to W inf = λw +f ax W. M mod p M Θ = M mod p M KK Toy model with uplifted scalar potential ( (hs+ f) V = λ sτ 3 6hqs 2q f 16sτ 2 5q2 48sτ )+ θ2 16sτ 3 +V up.

72 Toy model ETH Zürich, p.37/42

73 ETH Zürich, p.37/42 Toy model Backreaction of the other moduli adiabatically adjusting during the slow-roll of θ flattens the potential (Dong,Horn,Silverstein,Westphal) V back θ

74 Effective potential ETH Zürich, p.38/42

75 ETH Zürich, p.38/42 Effective potential Large field regime: θ/λ f. The potential in the large-field regime becomes V back (Θ) = hq 3 λ 2 f 2 ( 1 e γθ). with γ 2 = 28/(14+5λ 2 ) (similar to Starobinsky-model). For θ/λ f: 60 e-foldings from the quadratic potential Intermediate regime: linear inflation For θ/λ f: Starobinsky inflation

76 Tensor-to-scalar ratio ETH Zürich, p.39/42

77 ETH Zürich, p.39/42 Tensor-to-scalar ratio r With decreasing λ the model changes from chaotic to Starobinsky-like inflation. λ

78 Parametric control ETH Zürich, p.40/42

79 ETH Zürich, p.40/42 Parametric control From UV-complete theory point of view, large-field inflation models require a hierarchy of the form M Pl > M s > M KK > M inf > M mod > H inf > M Θ, where neighboring scales can differ by (only) a factor of O(10). Main observation the larger λ, the more difficult it becomes to separate the high scales on the left for small λ, the smaller (Hubble-related) scales on the right become difficult to separate.

80 Conclusions ETH Zürich, p.41/42

81 ETH Zürich, p.41/42 Conclusions Systematically investigated the flux induced scalar potential for non-supersymmetric minima, where we have parametric control over moduli and the mass scales. All moduli are stabilized at tree-level the framework for studying F-term axion monodromy inflation. Since the inflaton gets its mass from a tree-level effect, one gets a high susy breaking scale. As all mass scales are close to the Planck-scale, it is difficult to control all hierarchies. Does large field inflation necessarily must include stringy/kk effects? The (MS)SM could arise on a set of intersecting D7-branes mutual constraints between fluxes and branes (Freed/Witten anomalies). Is sequestering, M soft M3, possible? 2

82 ETH Zürich, p.42/42

83 Thank You! ETH Zürich, p.42/42