Magdalena Larfors. New Ideas at the Interface of Cosmology and String Theory. UPenn, Ludwig-Maximilians Universität, München

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1 Ludwig-Maximilians Universität, München New Ideas at the Interface of Cosmology and String Theory UPenn,

2 String 10D supergravity M 10 = M 4 w M 6 Fluxes and branes... (Scientific American)

3 Topology of landscape: How many vacua? Distribution of vacua? Barriers between vacua? Cosmological questions: Cosmological constant? Inflation? Vacuum stability (classical/quantum)? on warped CY manifolds. Mathematically tractable. Moduli stabilisation. Sequences of connected vacua.

4 Some CY geometry Candelas, de la Ossa:91,... Complex structure Holomorphic 3-form Ω 3-cycles Kähler structure Real -form J,4-cycles

5 Some CY geometry Candelas, de la Ossa:91,... Periods: Π I (z) = C I Ω(z) = M C I Ω(z) Π N (z) Π N 1 (z) collected in vector: Π(z) =. Π 0 (z) Intersection matrix: Q IJ = C I C J = M C I C J CS moduli space is (special) Kähler ( K cs = ln i ) Ω Ω = ln ( iπ Q 1 Π ) M

6 Fluxes Break SUSY: N = N = 1 Warp geometry Giddings, Kachru, Polchinski hep-th/ Flux vector: G = F τh Gukov Vafa Witten superpotential: W (z, τ) = G Π Kähler potential: K = ln ( i(τ τ)) + K cs (z, z) 3 ln ( i(ρ ρ)) Scalar potential: V = e K ( g i j D i WD j W + g τ τ D τ WD τ W )

7 Kähler moduli Not stabilised at classical level. W : non-perturbative corrections. K: perturbative and non-perturbative corrections. = SUSY and non-susy vacua: KKLT Kachru et. al. hep-th/ LARGE volume scenarios Balasubramanian et. al. hep-th/050058, Conlon et. al. hep-th/ Suppressed at large volume. Important around special points in moduli space.

8 Landscape topography

9 Danielsson, Johansson, ML hep-th/061 Chialva et. al z Monodromies: Π(z) T Π(z) 0 T LCS 1 T C W = G Π G T Π K cs K cs Dual description: Π fixed, G G T.

10 Danielsson, Johansson, ML hep-th/061 Johnson, ML

11 Braun, Johansson, ML, Walliser Is there a bound on the sequence length? Im(z) Re(z) Ahlqvist et. al

12 No-go theorem Ashok, Douglas hep-th/ ISD vacua: D τ W = D i W = 0 G (3) = ig (3) Tadpole condition: F (3), H (3) = i Im τ Ḡ (3), G (3) where ˆN = (F, H). F (3), H (3) = M F (3) H (3) L max = 1 Im τ Ḡ(3), G (3) = ˆN T (G τ G z ) ˆN 0

13 Tadpole condition, ISD vacua: 0 ˆN T (G τ G z ) ˆN L max If bounded (G τ G z ) eigenvaules: Λ i (z, τ) > ɛ = Admissible ˆN : ˆN L max /ɛ Finite number of vacua. Evade no-go: find (z, τ) such that Λ i (z, τ) = 0 G τ : Decoupling limit Im τ G z : LCS and conifold loci. D-limit.

14 : One-parameter models Refined no-go theorem: LCS Let t i log z, LCS point is at t = Im t Infinite sequence: = F n w n j (w. bounded G τ -eigenvalues) lim n (t ) n = ˆN n G τn G tn ˆN T n = O(1/ λ n j ) H n w n j = O(1/ λ n j ) LCS limit: can compute G tn -eigenvalues and -vectors λ n j, w n j λ 1 = a 11 t 3 + O ( ) [ ( T t, w 1 = 1, O t ( λ = a t + O λ 3 = a 33 t λ 4 = a 44 t 3 ( + O ( + O t 5 ) (, O t 4 ) (, O t 6 )] t 1 ), w T ( [O = t ) (, 1, O t ) (, O t 4 )] t ), w T ( 3 [O = t 4 ) (, O t ) (, 1, O t )] ), w4 T ( [O = t 6 ) (, O t 4 ) (, O t ) ], 1 = F 0 n = F 1 n = H 0 n = H 1 n = 0 for large n = F (3), H (3) = 0

15 : One-parameter models Im(z) Re(z)

16 One-parameter models: decoupling limit and conifold No infinite sequences: Decoupling limit Im τ (also for more cs moduli) Conifold locus (disclaimer: warping neglected) Two-parameter model Checked particular LCS limit: no infinite sequences.

17 µ =(π) 5 d 1, µ 3 = i(π) 6 (c 1a 1 a 1c 1 + d 1b 1 b 1d 1). One finds the following expression for the Kähler metric ( g xx = µ µ1 ln x ) µ + µ3 + + O( x ln x ). (5.5) µ 0 Then the curvature form is 1 4 x (ln x + C), (5.6) where the constant C is determined to be Ashok, Douglas hep-th/ , Denef, Douglas hep-th/ , µ Giryavets et. al. hep-th/040443, 0 µ 0 Eguchi, Tachikawa hep-th/ , Acharya, Douglas hep-th/06061, R xx = 1 Torroba hep-th/ C =1 µ1 + µ (5.7) µ 0µ µ In computing Kähler covariantized derivatives with respect to ψ, itisalsousefultonote Statistical distribution of flux vacua: that dn vac (z) det (R(z) + ω(z)) 5.. Distribution of flux vacua xk ψ = µ1 µ x ln x + O(x). (5.8) µ 0 µ ρ Fig. 3: Each point is a vacuum on the x =1 ψ complex plane. The monte carlo simulation data is: number of random fluxes N = ;randomfluxinterval f,h ( 100, 100); complex structure ψ space region x < There are 1149 vacua, but 6306 of them arise at x < and have been removed from the plot

18 Ahlqvist et. al , Giryavets et. al. hep-th/ Giddings Maharana hep-th/ , Douglas et. al , log ρ Fig. 4: The plot of a numerical evaluation of πic1(ψ). Figure : A comparison between numerical log ρ and analytical distributions. Red circles The monte carlo M simulation data for each point is: number of random fluxes N =10 7 numerical data while the blue curve is the integrated analytical distribution. Distance ;randomflux interval f,h D( 60, 60). The data is fit by the curve conifold ξ is plotted on a log scale on the horizontal axis, while the vacuum count is p,whereρ =lnr. The ρ+c conifold point r =0isatρ ρw = 0 = = ( for this coordinate. ) the Dρ w vertical W w axis. = ( 0 = ) A(F,H) C(F,H) ρ exp B(F,H) ρ Consider exp equation (88), which V now only depends on θ. Undertheassumptionthatthe CY A(F,H) one near conifold vacuum for each set of fluxes, the left hand side must either start out and go negative or vice versa. To find the zero-crossing, we divide the region [0, π] equally pieces and then determine in which region (if any) equation (88) changes sign. region is found, we apply the same method to that region, splitting it into two smaller Vacua pushed away from conifold. continuing in this way until we reach a predetermined level of accuracy. There are two comments. First, in equation (91) it is not clear that the value of ρ is real, or even pos must therefore exclude the regions where ρ is either negative or complex. Fortunately, if it is never negative since W (x) must have the same sign as x. Anecessaryandsufficient for ρ to be real is that the argument of the Lambert W function is greater than or equal This means that the relevant region to begin with may not be the entireinterval[0, π]

19 Conformal CY of type IIB string theory. Infinite sequences of vacua only possible in One-parameter CY: LCS and decoupling limit: finite Conifold: finite (w/w.o. warping) Agrees with statistical result. To do-list and open questions: Multiple, CY with more parameters... Vacuum properties: stability, CC,... Landscape dynamics Inflation Yang Quantum stability Johnson, ML ,..., Ahlqvist et. al Kähler moduli