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

AdS boundary

Perturb holographic quark-antiquark potential We solve nonlinear time evolution

Motivation

I think time-dependent dynamics in gauge/gravity duality is interesting

• AdS thermalization

What is essential? AdS? Einstein? Nonlinearity?

• AdS turbulent instability

Relation to real QGP? New BH dynamics?

• c.f.) Dynamical meson melting

[TI-Kinoshita-Murata-Tanahashi]

Time evolution in thermalizing D3/D7

Turbulent instability in D3/D7

Singularity formation after some wave reflections

[Hashimoto-Kinoshita-Oka-Murata]

AdS boudary Pole

}

x

singularity formation strong redshift

• f a light ray

brane fluctuation

• Electric field quench: 0→E
• “Meson turbulence”
• Probably due to nonlinearity in DBI

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

Static gauge: (τ,σ)=(t,z) Target space embedding: x1=X1(z) AdS5xS5

[Maldacena, Rey-Yee]

0.5 0.4 0.3 0.2 0.1

• 0.1
• 0.2
• 0.3
• 0.4
• 0.5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

z0: string tip Γ0=0.599

Solution for separation L

A convenient parametrization

0.5 1 1.5 2 2.5 3 3.5 4 4.5

• 2
• 1

1 2

Polar-like coordinates (r,φ) where the static solution is r=z0 Inverse function of F(z;k) is sn(x;k) A nice identity

Linearized perturbations

5 10 15 20 25 30 35 40 45 5 10 15 20 25

• 1
• 0.8
• 0.6
• 0.4
• 0.2

Longitudinal fluctuations around r=z0 Linearized EoM for eigenvalues/functions

[Callan-Guijosa, Klebanov-Maldacena-Thorn]

Contents

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

Perturb the string endpoints

Longitudinal Z2 - symmetric quench Longitudinal

• ne-sided

quench Transverse linear quench Transverse circular quench

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Quench profile: a compact C∞ function

Worldsheet double null coordinates

Worldsheet: u,v Target space: T(u,v), Z(u,v), X1,2,3(u,v) Equations of motion Induced metric Constraints

Discretization

Compute N by using EWS data To solve EoMs, we use O(h2) central finite differential

Initial data

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

Contents

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

Longitudinal one-sided quench

ε=0.03, Δt/L=2 Amplitude: ε=Δx/L Duration: Δt/L

Cusp formation

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

• 0.6
• 0.4
• 0.2

0.2 0.4 0.6

0.72 0.74 0.76 0.78

• 0.244
• 0.24
• 0.236
• 0.232
• 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:

Cusp formation time when ε is changed Δt/L=2 Corresponding formation points

Critical amplitude

There is a minimal amplitude for cusp formation

An extrapolation to tcusp~∞: εcrit~0.075 for Δt/L=2 Scaling (in small Δt/L) εcrit~(Δt/L)3

Analysis 2: Energy spectrum

ε=0.005, Δt/L=2 (no cusp) ε=0.01 (cusps T~27) ***Dashed lines: linearized computations

Decompose nonlinear solutions in linear eigenmodes en Log-log plot

Energy cascade

linear theory

1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1 1 1 2 5 10 20 50

Cusp formation: direct energy cascade → power law No cusp: no clear power law

ε=0.01 (cusps T~27) ε=0.005, Δt/L=2 (no cusp)

linear theory

1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1 1 1 2 5 10 20 50

Analysis 3: Forces on the endpoints

• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 20 40 60 80 100 120 140 160 0.0001 0.01 1 100 10000 1e+006 1e+008 1e+010 5 10 15 20 25 30

Force diverges when a cusp reaches the boundary

ε=0.01 (cusps T~27) ε=0.005, Δt/L=2 (no cusp) ***Red: x=L/2, green: x=-L/2

Z2-symmetric quench

ε=0.025 Δt/L=2 T~6.85 T~9.15

Z2-symmetric quench

• 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 ▲)

Δt/L=2

Transverse linear quench

String oscillates in 1+3 dim (t,z,x1,x2)

x1 x2 z

***Green arrows: forces ε=0.03, Δt/L=2

Transverse linear quench

• 0.6 -0.4 -0.2

0.2 0.4 0.6

• 0.06
• 0.03

0.03 0.2 0.4 0.6 0.8 linear theory

1e-006 1e-005 0.0001 0.001 0.01 0.1 1 1 2 5 10 20 50

• Cusps are formed at T~14.45
• The energy spectrum keeps a power law

Transverse circular quench

String oscillates in all 1+4 dim (t,z,x1,x2,x3)

x1 x1 z z x2 x2 x3 x3 x1

ε=0.02, Δt/L=2

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

• 0.6 -0.4 -0.2

0.2 0.4 0.6 -0.04

• 0.02

0.02 0.2 0.4 0.6 0.8

• 0.6 -0.4 -0.2

0.2 0.4 0.6

• 0.04
• 0.02

0.2 0.4 0.6 0.8

1e-005 0.0001 0.001 0.01 0.1 1 1 2 5 10 20 50

Direct cascade → inverse cascade Cuspy, but not real cusps

linear theory

1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1 1 1 2 5 10 20 50

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

Future works

• Large amplitude/finite temperature
• Non-conformal backgrounds
• Application to drag force

Gravitational backreaction may be necessary

• Curvature diverges at the cusps
• AdS gravitational wave bursts?
• Boundary interpretation: gluon bursts?

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".