• 沒有找到結果。

# String theory and

N/A
N/A
Protected

Share "String theory and "

Copied!
22
0
0

(1)

## Singularities in String Theory Singularities in String Theory

### Hong Liu Hong Liu

Massachusetts Institute of Technology

(2)

### Spacetime singularities singularities

Understanding the physics of

Understanding the physics of spacetimespacetime singularities is a singularities is a major challenge for theoretical physics.

major challenge for theoretical physics.

Big Bang/Big Crunch Big Bang/Big Crunch

beginning or end of time, the origin of the universe?

beginning or end of time, the origin of the universe?

Black holes

loss of information?

loss of information?

(3)

### singularities

 It is generally believed that understanding spacetimeIt is generally believed that understanding spacetime singularities requires a quantum theory of gravity.

singularities requires a quantum theory of gravity.

 String theory is thus the natural framework to address String theory is thus the natural framework to address this problem.

this problem.

 One hopes that string theory will lead to a detailed One hopes that string theory will lead to a detailed theory of the Big Bang which in turns leads to

theory of the Big Bang which in turns leads to experimental tests of string theory.

experimental tests of string theory.

(4)



11

22

11

22



11

22

33

44

11

2 2

33

44

11

x1 x2

### singularity

Classical general relativity is singular at the tip of the cone.

(5)

### String theory on orbifolds String theory on orbifolds

Dixon, Harvey, Vafa, Witten

 The The extended natureextended nature of string theory introduces of string theory introduces additional degrees of freedom

additional degrees of freedom localizedlocalized at the tip of at the tip of the cone:

the cone: twisted sectors.twisted sectors.

 Including the twisted sectors, Including the twisted sectors, string Sstring S--matrix is matrix is unitary

unitary and physics is completely smoothand physics is completely smooth inin perturbation theory.

twisted sectors

perturbation theory.

(6)

### Static example 2: Conifold Conifold

Strominger

 General relativity is singularGeneral relativity is singular..

 Perturbative string theory is singularPerturbative string theory is singular..

 By including the By including the nonnon--perturbative perturbative degrees of freedom degrees of freedom (D(D--branes wrapping the vanishing three cycle) at the tip branes wrapping the vanishing three cycle) at the tip of the cone, the

of the cone, the string Sstring S--matrixmatrix is again is again smoothsmooth..

S3

S2

(7)

(8)

### Cosmological singularities Cosmological singularities

 Possibilities:Possibilities:



 Beginning of time: need initial conditions, wave functions Beginning of time: need initial conditions, wave functions of the Universe etc.

of the Universe etc.



 Time has no beginning or end:Time has no beginning or end: Need to understand how Need to understand how to pass through the singularity.

to pass through the singularity.

 New Challenges:New Challenges:

 What are the rightWhat are the right observables?observables?



 What are the right degrees of freedom?What are the right degrees of freedom?

(9)

### ( (

A lower dimensional Toy ModelA lower dimensional Toy Model

### ) )

• Exact string background.

• The Universe contracts and expands through a singularity.

•• One can compute the One can compute the SS--matrix matrix from one cone to the other.

from one cone to the other.

• Same singularity in certain black holes (a closely related problem).

time

(10)

### Results from string perturbation theory Results from string perturbation theory

Liu, Moore, Seiberg Horowitz, Polchinski





(11)





### singular time - - dependent backgrounds. dependent backgrounds.

Nekrasov, Cornalba, Costa; Simon; Lawrence; Fabinger, McGreevy; Martinec and McElgin, Berkooz, Craps, Kutasov, Rajesh; Berkooz, Pioline, Rozali; ………

(12)







(13)

## Nonperturbative approaches Nonperturbative approaches





(14)

### Schwarzschild black holes in

Maldacena;

Witten

t

Quantum gravity in this black hole background is described by an SU(N) Super Yang-Mills at finite temperature on S3.

Classical gravity corresponds to large N and large t’Hooft coupling limit of Yang-Mills theory.

(15)

### temperature Yang - - Mills ? Mills ?

 Find the manifestation of the black hole singularity in the Find the manifestation of the black hole singularity in the large N and large t

large N and large t’’ HooftHooft limit of Yang-limit of Yang-Mills theory.Mills theory.

 understand how understand how

 finite N (quantum gravitational) finite N (quantum gravitational)

 tt’’ HooftHooft coupling (stringy)coupling (stringy) effects resolve it.

effects resolve it.

(16)





(17)

### Mapping of physical quantities Mapping of physical quantities

 Gauge invariant operator

### O

 Dimension νν

 Finite temperature two- point functions

of

### O

 Particle mass m

 Free propagator of φ in the Hartle-Hawking

vacuum of the AdS black hole background

The boundary theory has a continuous spectrum in the large N limitdespite being on a compact space.

(18)

## Large dimension limit Large dimension limit

Festuccia and Liu



+

+

νZ(uνZ(u))

(19)

## Relation with bulk geodesics Relation with bulk geodesics

Festuccia and Liu



cc

(20)

cc

(21)

## Yang Yang - - Mills theory at finite N Mills theory at finite N





+

+

i

i

(22)







### attacking the problem.

The principal chiral model has two conserved currents corresponding to the G × G symmetry of the action.. These currents are

introduction to continuum and matrix model formulation of non-critical string theory.. They typically describe strings in 1+0 or 1+1 dimensions with a

◆ Understand the time evolutions of the matrix model to reveal the time evolution of string/gravity. ◆ Study the GGE and consider the application to string and

It should be stressed that the four eigenvalues obtained here do not change even if we include other field outside KBc subalgebra or outside the dressed B 0 gauge, since such fields

Normalizable moduli (sets of on‐shell vacua in string theory) Scale

[CIY4], Analytic Study for the String Theory Landscapes via Matrix Models, Phys.Rev... THEME:

At least one can show that such operators  has real eigenvalues for W 0 .   Æ OK. we  did it... For the Virasoro

If the best number of degrees of freedom for pure error can be speciﬁed, we might use some standard optimality criterion to obtain an optimal design for the given model, and