Conifunneling : An Extreme Path on the String Landscape

Transcription

1 Conifunneling : An Extreme Path on the String Landscape I-Sheng Yang ISCAP and Physics Department Columbia University, NY with Pontus Ahlqvist, Brian Greene, David Kagan, Eugene A. Lim and Saswat Sarangi String Theory Seminar, National Taiwan University

2 Conifunneling

3 Conifunneling

4 Conifunneling

5 Conifunneling Extreme Geometry Response to Vacuum Transitions. Why? How? Generic? Good news?

6 Outline 1 Multifield Vacuum Transition Simplest Nontrivial Stringy Flux Landscape 3 Conifunneling

7 From One to Many V φ t φ t φ f φ f φ 1 Both technically and physically nontrivial. M. Cvetic, H. H. Soleng ; P. M. Saffin, A. Mazumdar, E. J. Copeland 9853; M. Johnson, M. Larfors 89.64; ISY ; A. Aguirre, M. Johnson, M. Larfors ; A. Brown, A. Dahlen

8 Single Field Instanton V φ f φ f φ t φ f φ t φ S E = r 3 dr [ 1 φ f ( ) dφ + V (φ)] dr φ f Critical point with exactly one negative mode. d φ dr + 3 dφ r dr = dv dφ

9 Multifield in Thin Wall φ t φ f φ f φ t f φ 1 φ f φ f S E = V T (volume inside) + 4π 3 r 3 σ + V F (volume outside) The only negative mode is in r. [ ( ) ( ) ] Gij dφi dφj σ = dr + V ( φ) dr dr

10 Relaxation To minimize σ = dr [ Gij ( dφi dr Introduce dynamics, L = G ij E = G ij d δl Add friction to the EoM, dt ) ( ) dφj dr + V ( φ) ], ( φi φj φ i j) φ V ( φ), ( φ i φ j + φ iφ j) + V (φ) δ φ i d dr δl δφ i + f φ i = V φ i. Check pitfalls: multiple local minima, runaway situation. Show Example!! A. Aguirre, M. Johnson, M. Larfors J. Giblin Jr, L. Hui, E. Lim, ISY

11 Follow a Valley? φ i = f i (ψ j ), j = 1 m, m < n, l = 1 (n m), k = 1 m, K ij θi l f j ψ k =, K ij θi lθl j = 1, V θ =, V >. θ 4 4 f K kl = K i f j ij ψ k ψ l, V (ψ j ) = V (f i (ψ j )). y 4 x 4 6

12 Global Path 4 V 4 V 4 V y 4 x y 4 x y 4 x 4 (a) 6 4 (b) 6 4 (c) 6 A Valley = a mountain ridge. σ = V G ij dφ i dφ j Good news? Look for global properties instead of 1 1 valleys.

13 Multifield Vacuum Transition Simplest Nontrivial Stringy Flux Landscape Freezing Light Fields σ= Z p V Gij dφi dφj Minimize path length Minimize V Conifunneling

14 Multifield Summary Thin wall : find the path that minimizes tension σ. Freeze light fields. Relaxation method. Hope to find global paths.

15 How Many Fields? 4 fields model (z, τ) of flux-stabilized Calabi-Yau threefolds. 1 complex structure modulus, 1 axio-dilaton, many Kahler moduli Assumption: At large volume, the dynamics of Kahler moduli are at low energy scales (including KKLT stabilization), so we can freeze them in the tunneling. Mirror Quintic and its 13 cousins. Doran and Morgan, math/557.

16 Nonconanical Kinetic Terms Complex structure modulus z pairs of intersecting cycles {π (z), π 3 (z)}, {π 1 (z), π (z)}. π π 3 π π 1 K z z µ z µ z, K z z = z z K, K = log (i (π 3 π π 3 π + π 1 π π 1 π ))

17 Potential from the Fluxes π π 3 π π 1 Choose R-R flux F = (f, f 1, f, f 3 ) and NS-NS flux H = (h, h 1, h, h 3 ). W (z, τ) = Ω ( ) F (3) τh (3) = F Π τh Π, M V = e K ( K z z D z WD z W + K τ τ D τ WD τ W ) no scale potential, GKP, hep-th/1597

18 Monodromy Transformation π π 3 π π

19 Multiple Layers π π + π 3, (π 3 π π 3 π + π 1 π π 1 π ), W (z, τ) = F Π τh Π, f f + f 3, h h + h 3. Picture stolen from Danielsson, Johansson and Larfors, hep-th/61.

20 Vacuum Hunting F = ( 3, 6, 9, 1), H = ( 1,, 7, ) No apparent rules. Monte-Carlo A. Giryavets, S. Kachru, P.K. Tripathy, hep-th/4443 At most one per sheet. Similar behavior near special point. Vacuum chains. SUSY subchains.

21 Bulk Behavior 4 fields relaxation. Getting unboundedly close to the conifold point.

22 Near Conifold K z z = c 1 log(1 z) c S S 3 R + 1 z 4/3 Klebanov & Strassler 7191; Shiu, Torroba, Underwood & Douglas ; Douglas & Torroba

23 Closer Look σ = dx (V + K τ τ τ τ + K z z ṙ + K z z r θ )

24 Compromise σ = σ 1 + σ m + σ r1 = VKz z dr + r c π r VKz z rdθ + VKz z dr r c

25 Conifuneling = The Shotest Path 1 L = 1 ( ṙ + r θ ) V inverse (r, θ), θ max = π. 1

26 All About the Metric L = K(r) ( ṙ + r θ ), K(r) = r β, r = r β+1 β + 1, L = 1 ( r + r (β + 1) θ ), θ max = π β + 1. z β = /3 θ max = 3π

27 Special Case : BPS Paths σ = dx ( K zz żż + K ττ τ τ + e K ( K zz D z W D z W + K ττ D τ W D τ W )) ( = Z + dx K zz ż K zz z Z + K ττ τ K ττ τ Z ). Z = e K/ W, ż = ±K zz z Z, τ = ±K ττ τ Z.

28 BPS Conifunneling V = Z, paths follow Z Focal points Z = V =. BPS paths are monotonic in Z All SUSY vacua in a chain are minima in Z. Unless going through somewhere special. LCS point is infinite distance away, and Z = is forbidden by the string coupling constant. The only option is the conifold point.

29 Geometric Picture D5 brane wraping a 3-cycle σ = σ 1 + σ + σ 5 brane 1D (V shrinking 3 cycle )(volume factor)(warp factor).

30 Outlooks Explicit Geometric Construction Strongly warped conifolds are special cases in string theory where we have more detailed geometric descriptions. Knowing (partially) the CY metric, we can try to explicitly construct the geometry. General Rules of Tunnelings General tunnelings might go through special points. This can provide a semi-analytical description for tunnelings and provide an overview of connectivity of the landscape. For example, our θ max implies that in a tunneling, f can only be changed up to 3f 3 /. Topological Transitions Collisions