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Resolution of the singularity in Gregory-Laflamme transition through Matrix Model. Takeshi Morita Tata Institute of Fundamental Research. based on collaboration with M. Mahato, G. Mandal and S. Wadia. (work in progress). Introduction and Motivation.

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slide1

Resolution of the singularity in Gregory-Laflamme transition

through Matrix Model

Takeshi Morita

Tata Institute of Fundamental Research

based on collaboration with M. Mahato, G. Mandal and S. Wadia

(work in progress)

slide2

Introduction and Motivation

・ Dynamical time evolutions of gravity and naked singularities.

Big ban singularity, Black hole evaporation, Gregory-Laflamme transition, ・・・

General relativity cannot describe the process beyond the singularity.

cf. Cauchy problem, Initial value problem

・ Conjecture:

Quantum effects of gravity will make smooth these singularities.

Q. How can string theory answer this problem?

In this study, we considered this problem

in the Gregory-Laflamme transition

by using the gauge/gravity correspondence.

cf. Big ban singularity, A Matrix Big Ban, Craps-Sethi-Verlinde (2005)

slide3

Gregory-Laflamme transition

Gregory-Laflamme (1993)

時空が コンパクト化しているときの2つの重力解

Black string Black hole

・ Stability of the solutions

Transition from unstable BS to BH is called Gregory-Laflamme transition

(We are considering the near extremal limit.)

slide4

Gregory-Laflamme transition

Gregory-Laflamme (1993)

・Horowitz-Maeda conjecture (2001)

If we assume that there are no singularities outside the horizon,

the classical event horizon cannot pinch off at any finite affine time.

Start from unstable BS ( )

Infinite affine time

near extremal case

Infinite affine time

Infinite asymptotic time

(natural time for the gauge theory)

slide5

Gauge/Gravity correspondence

cf. 疋田さんのレビュー

Gravity

Gauge theory(2D SYM)

Gregory-Laflamme transition

Gross-Witten-Wadia transition

1/N effect

quantum effect

By considering the time evolution of the gauge theory,

we evaluate the time evolution of the Gregory-Laflamme transition

and show how quantum effects resolve the naked singularity.

slide6

Gauge/Gravity correspondence

Susskind (1998), Aharony, Marsano, Minwalla, Wiseman (2004) + Papadodimas, Raamsdonk (2005)

Gravity

Gauge theory(2D SYM)

IIA gravity on

1d SYM (Matrix theory) with

T-dual on the direction

N D-particles on

2d U(N) SYM(D1-branes) on

radius:

radius:

slide7

Gauge/Gravity correspondence

Aharony, Marsano, Minwalla, Wiseman (2004) + Papadodimas, Raamsdonk (2005)

・ Phases of 2d SYM

Order parameter: Eigen value density of

the Wilson loop along the .

GWW type transition

ungapped phase

gapped phase

BS/BH transition

BS

BH

依存性も一致

slide8

Matrix Model description

・ c=1 matrix model (SYMは解くのが難しいので、、、)

According to the universality, the effective theory near the critical point will be

described by c=1 matrix model with the inverse harmonic potential.

: We fix L and N.

Fermi energy

slide9

Matrix Model description

・ c=1 matrix model

Eigen value density:

i-th Eigen value of M:

slide10

Matrix Model description

・ c=1 matrix model

Eigen value density:

i-th Eigen value of M:

BS/BH transition

slide11

Forcing and time evolution

If we replace the constant a as a time dependent increasing function a(t),

then we can naively expect the transition happen when t=t*.

slide12

Forcing and time evolution

・ Forcing and time evolution

It is natural to guess that this gauge theory process corresponds to

the following gravity process

However, through the argument in the Horowitz-Maeda conjecture:,

the transition doesn’t happen in the classical gravity, because of the naked singularity.

On the other hand, if we consider quantum effects, something will happen around t=t*.

slide14

Classical time evolution of the matrix

Universal late time behavior

: model dependent constant.

: universal part

: non-universal parts

We found that in case the transition happens, this behavior is universal!!

Independent of the detail of a(t), forcing (for example )

and unitary matrix models behave like this too.

The transition (the gap arises) happens at

Consistent!!

Horowitz-Maeda conjecture

slide16

Quantum time evolution of the matrix

・ a=constant

We can solve Schrodinger equations by using parabolic cylinder function. (Moore 1992)

・ Eigen value density:

infinite N matrix model

finite N matrix model

The densities are made smooth through the 1/N effects.

slide17

Quantum time evolution of the matrix

Since the energy spectrum is discrete,

we can apply adiabatic approximation if a(t) satisfies:

Then we obtain the wave function is obtained by

Time evolution of eigen value density

slide18

Quantum time evolution of the matrix

・ Comparison to the infinite N

Time evolution of finite N matrix model

Time evolution of infinite N matrix model

slide19

Quantum time evolution of the matrix

・ Conjecture on the gravity

Classical gravity

Infinite asymptotic time

Quantum gravity

slide21

Start from unstable BH ( )

Similar to the two black hole collision

Non-singular process → finite time

slide22

Classical time evolution of the matrix

・ gapped to ungapped transition

slide23

Classical time evolution of the matrix

・ gapped to ungapped transition

finite time

The transition happens always in finite time.

But we have not found the equation describing the transition.

It seems that this result is consistent with BH to BS transition.

slide24

Summary of our Result

・ Gauge theory

gapped phase

ungapped phase

1. If N is infinite, infinite time is necessary for the transition.

2. If N is finite, the transition happens in finite time.

3. If we consider the opposite process (gapped to ungapped), the transition happen in finite time.

Quantum effect 1/N effect

・ Known facts in gravity

1. Horowitz-Maeda conjecture:

In classical general relativity, infinite time is necessary for the GL transition

to avoid the naked singularity.

2. Quantum effect would make smooth the naked singularity.

3. The transition from BH to BS happens without any singularity.

slide25

このモデルの問題点

  • M~U (Wilson loog)低エネルギーでは adjoint scalarもmasslessで効いてくる。cf. Azeyanagi-Hanada-Hirata-Shimada
  • 1 Matrix → Multi-Matrix?
  • c=1 Matrix modelの相転移は3次。GL transitionの次数は次元による。
  • 10次元ではおそらく一次相転移
  • ( 2D SYMも1st order transitionの可能性が高い)高次元(D>10)の重力のtoy model?
  • 時間発展の定性的な性質は相転移の次数によるか?
slide27

Toward the derivation of the effective matrix model from 2d SYM

Our matrix model

Fundamental theory: 2d SYM on S1

  • Is it possible to derive the matrix model from 2d SYM?
  • What is ‘a’?

We attempt to solve this problem

in weak coupling analysis and mean field approximation.

Condensation of the adjoint scalars may give the potential.

(Work in progress…)

slide28

Conclusion

  • We found that the one matrix model can reproduce the time evolution of the classical gravity qualitatively in the large N limit.
  • The singular behavior is resolved by 1/N effect.
  • The matrix model may be derived from 2d SYM through the condensation.

Future Work

  • Complete the derivation of the matrix model from 2d SYM.
  • Construction of more realistic model (including interaction, adjoint scalar)
  • Calculation in gravity side and comparison to matrix model result
  • Quantum evolution without the adiabatic approximation.
  • Including Hawking Radiation.
  • Application to other singular system. black hole evaporation.