Turbulent strings in AdS/CFT Takaaki Ishii (University of Crete) - PowerPoint PPT Presentation

Turbulent strings in AdS/CFT Takaaki Ishii (University of Crete) arXiv:1504.02190 with Keiju Murata 12 May 2015@Oxford

Contents 1. Introduction 2. Review of the static solution 3. Numerical setup 4. Results 5. Summary

What I will do Perturb holographic quark-antiquark potential AdS boundary We solve nonlinear time evolution

Motivation I think time-dependent dynamics in gauge/gravity duality is interesting - AdS thermalization Relation to real QGP? New BH dynamics? - AdS turbulent instability What is essential? AdS? Einstein? Nonlinearity? - c.f.) Dynamical meson melting Time evolution in thermalizing D3/D7 [TI-Kinoshita-Murata-Tanahashi]

Turbulent instability in D3/D7 Singularity formation after some wave reflections [Hashimoto-Kinoshita-Oka-Murata] strong redshift of a light ray x singularity Electric field quench: 0 → E - formation AdS boudary “Meson turbulence” - Pole Probably due to nonlinearity in DBI - brane fluctuation } Considering F1 would be simpler

Contents 1. Introduction 2. Review of the static solution 3. Numerical setup 4. Results 5. Summary

Time-like holographic Wilson loop [Maldacena, Rey-Yee] AdS 5 xS 5 Static gauge: ( τ , σ )=(t,z) Target space embedding: x 1 =X 1 (z) Solution for separation L 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 z 0 : string tip Γ 0 =0.599

A convenient parametrization Inverse function of F(z;k) is sn(x;k) 4.5 4 3.5 3 2.5 2 1.5 1 Polar-like coordinates (r, φ ) 0.5 0 -2 -1 0 1 2 where the static solution is r=z 0 A nice identity

Linearized perturbations Longitudinal fluctuations around r=z 0 [Callan-Guijosa, Klebanov-Maldacena-Thorn] Linearized EoM for eigenvalues/functions 45 1.2 1 40 0.8 0.6 35 0.4 0.2 0 30 -0.2 -0.4 -0.6 25 -0.8 -1 0 0.5 1 1.5 2 2.5 20 15 10 5 0 0 5 10 15 20 25

Contents 1. Introduction 2. Review of the static solution 3. Numerical setup 4. Results 5. Summary

Perturb the string endpoints Longitudinal Longitudinal Transverse Transverse circular quench one-sided Z 2 - symmetric linear quench quench quench Quench profile: a compact C ∞ function 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1

Worldsheet double null coordinates Induced metric Worldsheet: u,v Target space: T(u,v), Z(u,v), X 1,2,3 (u,v) Equations of motion Constraints

Discretization To solve EoMs, we use O(h 2 ) central finite di ff erential Compute N by using EWS data

Initial data Initial data satisfies the constraint Solution (gauge: φ =u when v=0) where we used Boundary quench is then added at 0<T bdry < Δ t

Contents 1. Introduction 2. Review of the static solution 3. Numerical setup 4. Results 5. Summary

Longitudinal one-sided quench ε =0.03, Δ t/L=2 Amplitude: ε = Δ x/L Duration: Δ t/L

Cusp formation 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.78 0.5 0.4 0.4 0.76 0.3 0.3 0.74 0.2 0.2 0.1 0.72 0.1 -0.244 -0.24 -0.236 -0.232 0 0 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 - Cusps are seen in target space (x,z)-coordinates - Fields on worldsheet (u,v)-coordinates are regular - Cusps are created in a pair (around t/L~5)

Analysis 1: Cusp detection The conditions satisfied at a cusp: Δ t/L=2 Cusp formation time Corresponding when ε is changed formation points

Critical amplitude There is a minimal amplitude for cusp formation An extrapolation to t cusp ~ ∞ : Scaling (in small Δ t/L) ε crit ~0.075 for Δ t/L=2 ε crit ~( Δ t/L) 3

Analysis 2: Energy spectrum Decompose nonlinear solutions in linear eigenmodes e n Log-log plot ε =0.005, Δ t/L=2 (no cusp) ε =0.01 (cusps T~27) ***Dashed lines: linearized computations

Energy cascade 1 1 0.1 0.1 0.01 0.01 0.001 0.001 0.0001 0.0001 linear theory 1e-005 1e-005 linear theory 1e-006 1e-006 1e-007 1e-007 1 2 5 10 20 50 1 2 5 10 20 50 ε =0.005, Δ t/L=2 (no cusp) ε =0.01 (cusps T~27) Cusp formation: direct energy cascade → power law No cusp: no clear power law

Analysis 3: Forces on the endpoints Force diverges when a cusp reaches the boundary 0.8 1e+010 0.6 1e+008 0.4 1e+006 0.2 10000 0 100 -0.2 1 -0.4 0.01 -0.6 -0.8 0.0001 0 20 40 60 80 100 120 140 160 0 5 10 15 20 25 30 ε =0.005, Δ t/L=2 (no cusp) ε =0.01 (cusps T~27) ***Red: x=L/2, green: x=-L/2

Z 2 -symmetric quench ε =0.025 Δ t/L=2 T~6.85 T~9.15

Z 2 -symmetric quench Δ t/L=2 - Formation times are discretized by wave collisions - First cusp formations by such collisions (red ● ). The cusps are pair-created and annihilated. - Traveling cusps can be formed first (green ▲ )

Transverse linear quench z x 2 x 1 ε =0.03, Δ t/L=2 ***Green arrows: forces String oscillates in 1+3 dim (t,z,x 1 ,x 2 )

Transverse linear quench 1 0.8 0.1 0.6 0.01 0.4 0.001 0.2 0 0.0001 0.03 linear theory 0 1e-005 -0.03 0.6 0.4 0.2 0 -0.06 1e-006 -0.6 -0.4 -0.2 1 2 5 10 20 50 - Cusps are formed at T~14.45 - The energy spectrum keeps a power law

Transverse circular quench ε =0.02, Δ t/L=2 x 3 x 2 x 1 z z x 2 x 3 x 1 x 1 String oscillates in all 1+4 dim (t,z,x 1 ,x 2 ,x 3 )

Energy spectrum (Log-log plot) No cusp: no sustaining power law c.f.) Probability of cusp formation is zero if dim>4

Transverse circular quench 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.02 0 0 0 0 -0.6 -0.4 -0.2 -0.6 -0.4 -0.2 -0.02 -0.02 0 0 0.2 0.2 0.4 0.6 -0.04 0.4 0.6 -0.04 Cuspy, but not real cusps 1 1 0.1 0.1 0.01 0.01 0.001 0.0001 0.001 linear theory 1e-005 0.0001 1e-006 1e-007 1e-005 1 2 5 10 20 50 1 2 5 10 20 50 Direct cascade → inverse cascade

Summary We computed nonlinear dynamics of the quark- antiquark fundamental string in AdS - Cusps and turbulent behavior in ≦ 1+3 dim - No cusp and direct/inverse cascades in 1+4 dim c.f.) Cosmic strings in flat space

Discussion Gravitational backreaction may be necessary - Curvature diverges at the cusps - AdS gravitational wave bursts? - Boundary interpretation: gluon bursts? Future works - Large amplitude/finite temperature - Non-conformal backgrounds - Application to drag force

This research has been co-financed by the European Union (European Social Fund, ESF) and Greek national funds through the Operational Program "Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF), under the grants schemes “Funding of proposals that have received a positive evaluation in the 3rd and 4th Call of ERC Grant Schemes” and the program “Thales".