Sequences of Type IIB String Vacua IIB String Vacua Magdalena - PowerPoint PPT Presentation

Sequences of Type Sequences of Type IIB String Vacua IIB String Vacua Magdalena Larfors Background and motivation Magdalena Larfors Type IIB compactifications Vacuum sequences Ludwig-Maximilians Universit¨ at, M¨ unchen Finiteness and D-limits “New Ideas at the Interface of Cosmology and String Theory” Statistical studies Finiteness and Warping UPenn, 17.03.2012. Conclusions and outlook

Background and motivation Sequences of Type IIB String Vacua Magdalena Larfors Background and String motivation compactifications Type IIB compactifications 10D supergravity Vacuum sequences Finiteness and M 10 = M 4 × w M 6 D-limits Fluxes and branes Statistical studies Finiteness and ... Warping Conclusions and outlook (Scientific American)

Background and motivation Topology of landscape: How many vacua? Sequences of Type IIB String Vacua Distribution of vacua? Magdalena Larfors Barriers between vacua? Background and motivation Type IIB Cosmological questions: compactifications Cosmological constant? Vacuum sequences Finiteness and Inflation? D-limits Statistical studies Vacuum stability (classical/quantum)? Finiteness and Warping Conclusions and Type IIB on warped CY manifolds. outlook Mathematically tractable. Moduli stabilisation. Sequences of connected vacua.

Type IIB compactifications Sequences of Type IIB String Vacua Some CY geometry Candelas, de la Ossa:91, ... Magdalena Larfors Complex structure ∼ Holomorphic 3-form Ω ∼ 3-cycles Background and K¨ ahler structure ∼ Real 2-form J ∼ 2,4-cycles motivation Type IIB compactifications Vacuum sequences Finiteness and D-limits Statistical studies Finiteness and Warping Conclusions and outlook

Type IIB compactifications Some CY geometry Candelas, de la Ossa:91, ... � � Periods: Π I ( z ) = C I Ω( z ) = M C I ∧ Ω( z ) Sequences of Type IIB String Vacua Magdalena Larfors   Π N ( z ) Background and Π N − 1 ( z ) motivation   collected in vector: Π( z ) = .   Type IIB .   compactifications .   Vacuum sequences Π 0 ( z ) Finiteness and D-limits Statistical studies � � M C I ∧ C J Intersection matrix: Q IJ = C I C J = Finiteness and Warping Conclusions and outlook CS moduli space is (special) K¨ ahler � � i Π † · Q − 1 · Π Ω ∧ ¯ � � K cs = − ln � = − ln i Ω M

Type IIB compactifications Fluxes Giddings, Kachru, Polchinski hep-th/0105097 Break SUSY: N = 2 → N = 1 Sequences of Type Warp geometry IIB String Vacua Magdalena Larfors Background and Flux vector: G = F − τ H motivation Type IIB compactifications Vacuum sequences Gukov–Vafa–Witten superpotential: Finiteness and D-limits W ( z , τ ) = G · Π Statistical studies Finiteness and K¨ ahler potential: Warping Conclusions and outlook K = − ln ( − i ( τ − ¯ τ )) + K cs ( z , ¯ z ) − 3 ln ( − i ( ρ − ¯ ρ )) Scalar potential:  ¯ τ ¯ g i ¯ W + g τ ¯ V = e K �  D i WD ¯ τ D τ WD ¯ � W

Type IIB compactifications K¨ ahler moduli Sequences of Type IIB String Vacua Not stabilised at classical level. Magdalena Larfors W : non-perturbative corrections. Background and K : perturbative and non-perturbative corrections. motivation Type IIB = ⇒ SUSY and non-SUSY vacua: compactifications Vacuum sequences KKLT Kachru et. al. hep-th/0301240 LARGE volume scenarios Finiteness and Balasubramanian et. al. hep-th/0502058, D-limits Conlon et. al. hep-th/0505076 ... Statistical studies Finiteness and Warping Warping Conclusions and outlook Suppressed at large volume. Important around special points in moduli space.

Sequences of Type IIB String Vacua Magdalena Larfors Background and motivation Landscape topography Type IIB compactifications Vacuum sequences Finiteness and D-limits Statistical studies Finiteness and Warping Conclusions and outlook

Vacuum sequences Danielsson, Johansson, ML hep-th/0612222 Chialva et. al. 0710.0620 Sequences of Type z IIB String Vacua Magdalena Larfors Background and motivation Monodromies: Type IIB compactifications Π( z ) → T · Π( z ) 0 1 Vacuum sequences T T Finiteness and LCS C D-limits Statistical studies Finiteness and Warping Conclusions and W = G · Π → G · T · Π K cs → K cs outlook Dual description: Π fixed, G → G · T .

Vacuum sequences Danielsson, Johansson, ML hep-th/0612222 Johnson, ML 0805.3705 Sequences of Type IIB String Vacua Magdalena Larfors Background and motivation Type IIB compactifications Vacuum sequences Finiteness and D-limits Statistical studies Finiteness and Warping Conclusions and outlook

Vacuum sequences Braun, Johansson, ML, Walliser 1108.1394 Is there a bound on the sequence length? Sequences of Type IIB String Vacua � 2.5 Magdalena Larfors � � � Background and motivation 2.0 � � Type IIB compactifications � Vacuum sequences 1.5 � Finiteness and Im( z ) D-limits � Statistical studies 1.0 � � Finiteness and � Warping � � Conclusions and 0.5 � outlook � � � � � � � � 0.0 � � � � � � � � � 5 � 4 � 3 � 2 � 1 0 1 Re( z ) Ahlqvist et. al. 1011.6588

Finiteness and D-limits Sequences of Type No-go theorem IIB String Vacua Ashok, Douglas hep-th/0307049 Magdalena Larfors ISD vacua: D τ W = D i W = 0 ⇔ ∗ G (3) = iG (3) Background and motivation Tadpole condition: Type IIB � � F (3) , H (3) � = M F (3) ∧ H (3) ≤ L max compactifications Vacuum sequences Finiteness and D-limits 2 Im τ � ¯ i � F (3) , H (3) � = G (3) , G (3) � Statistical studies N T · ( G τ ⊗G z ) · ˆ Finiteness and 2 Im τ � ¯ G (3) , ∗ G (3) � = ˆ 1 = N ≥ 0 Warping Conclusions and where ˆ N = ( F , H ). outlook

Finiteness and D-limits N T · ( G τ ⊗ G z ) · ˆ 0 ≤ ˆ Tadpole condition, ISD vacua: N ≤ L max Sequences of Type IIB String Vacua Magdalena Larfors If bounded ( G τ ⊗ G z ) eigenvaules: Λ i ( z , τ ) > ǫ Background and motivation Type IIB N 2 ≤ L max /ǫ ⇒ Admissible ˆ N : ˆ = Finite number of vacua. compactifications Vacuum sequences Finiteness and D-limits Evade no-go: find ( z , τ ) such that Λ i ( z , τ ) = 0 D-limit. Statistical studies Finiteness and Warping D-limits Conclusions and outlook G τ : Decoupling limit Im τ → ∞ G z : LCS and conifold loci.

Finiteness and D-limits: One-parameter models Refined no-go theorem: LCS (w. bounded G τ -eigenvalues) Let t ∼ − i log z , LCS point is at t 2 = Im t → ∞ Infinite sequence: Sequences of Type IIB String Vacua N n · G τ n ⊗ G t n · ˆ ˆ Magdalena Larfors lim n →∞ ( t 2 ) n = ∞ N T n � = ∞ Background and motivation � � Type IIB ⇒ F n · w n λ n H n · w n λ n = j = O (1 / j ) j = O (1 / j ) compactifications Vacuum sequences LCS limit: can compute G t n -eigenvalues and -vectors λ n j , w n j Finiteness and D-limits Statistical studies Finiteness and t − 2 t − 4 t − 6 λ 1 = a 11 t 3 wT � � � � � � �� 2 + O � t 2 � , = 1 , O , O , O Warping 1 2 2 2 � t − 1 � wT � � t − 2 � � t − 2 � � t − 4 �� Conclusions and λ 2 = a 22 t 2 + O = O , 1 , O , O , 2 2 2 2 2 outlook a 33 � t − 2 � wT � � t − 4 � � t − 2 � � t − 2 �� λ 3 = + O = O , O , 1 , O , 3 2 2 2 2 t 2 a 44 t − 5 wT t − 6 t − 4 t − 2 � � � � � � � � � � λ 4 = + O , = O , O , O , 1 2 4 2 2 2 t 3 2 ⇒ F 0 n = F 1 n = H 0 n = H 1 = n = 0 for large n = ⇒ � F (3) , H (3) � = 0

Finiteness and D-limits: One-parameter models 0.05 Sequences of Type IIB String Vacua Magdalena Larfors 0 Background and motivation � 0.05 Type IIB compactifications Im(z) Vacuum sequences � 0.1 Finiteness and D-limits Statistical studies � 0.15 Finiteness and Warping � 0.2 Conclusions and outlook � 0.25 � 0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Re(z)

Finiteness and D-limits Sequences of Type IIB String Vacua One-parameter models: decoupling limit and conifold Magdalena Larfors Background and No infinite sequences: motivation Decoupling limit Im τ → ∞ (also for more cs moduli) Type IIB compactifications Conifold locus (disclaimer: warping neglected) Vacuum sequences Finiteness and D-limits Two-parameter model Statistical studies Finiteness and Checked particular LCS limit: no infinite sequences. Warping Conclusions and outlook

Statistical studies Ashok, Douglas hep-th/0307049, Denef, Douglas hep-th/0404116, Giryavets et. al. hep-th/0404243, Eguchi, Tachikawa hep-th/0510061, Acharya, Douglas hep-th/0606212, Torroba hep-th/0611002 Sequences of Type IIB String Vacua Statistical distribution of flux vacua: Magdalena Larfors dN vac ( z ) ∼ det ( R ( z ) + ω ( z )) Background and motivation Type IIB compactifications 0.04 Vacuum sequences ρ Finiteness and D-limits 0.02 Statistical studies Finiteness and Warping -0.04 -0.02 0.02 0.04 Conclusions and outlook -0.02 -0.04