## Singularities in String Theory Singularities in String Theory

### Hong Liu Hong Liu

Massachusetts Institute of Technology

### Spacetime

### 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?

### String theory and

### String theory and spacetime spacetime singularities

### 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.

### Static example 1: Orbifolds Static example 1: Orbifolds

### (x (x

_{1}

_{1}

### , x , x

_{2}

_{2}

### ) ~ ( ) ~ ( - - x x

_{1}

_{1}

### , , - - x x

_{2}

_{2}

### ) ) 2d cone 2d cone

### (x (x

_{1}

_{1}

### , x , x

_{2}

_{2}

### , x , x

_{3}

_{3}

### , x , x

_{4}

_{4}

### ) ~ ( ) ~ ( - - x x

_{1}

_{1}

### , , - - x x

_{2 }

_{2 }

### , , - - x x

_{3}

_{3}

### , , - - x x

_{4}

_{4}

### ) A ) A

_{1}

_{1}

### singularity

x_{1}
x_{2}

### singularity

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

### 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.

### Static example 2:

### 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..

S^{3}

S^{2}

### Lessons Lessons

### String theory introduces

### String theory introduces new degrees of new degrees of freedom

### freedom . . String

### String S S - - matrix matrix is completely is completely smooth smooth . .

### 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?

### From Big Crunch to Big Bang: is it possible?

### From Big Crunch to Big Bang: is it possible?

### ( (

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

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

Liu, Moore, Seiberg Horowitz, Polchinski

### For special kinematics the string amplitudes For special kinematics the string amplitudes diverge.

### diverge.

### The energy of an incoming particle is blue The energy of an incoming particle is blue shifted to

### shifted to infinity infinity by the contraction at the by the contraction at the singularity, which generates infinitely large singularity, which generates infinitely large

### gravitational field and

### gravitational field and distorts the geometry distorts the geometry . .

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

### String perturbative expansion String perturbative expansion breaks breaks

### down down as a result of large as a result of large backreaction backreaction . .

### The same conclusion applies to other The same conclusion applies to other singular time

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

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

### Lessons and implications Lessons and implications

### Perturbative string theory is generically Perturbative string theory is generically singular

### singular at cosmological singularities. at cosmological singularities.

### One needs a full One needs a full non non - - perturbative perturbative framework to deal with the

### framework to deal with the backreaction backreaction . .

### No clear evidence from string theory so far No clear evidence from string theory so far a non

### a non - - singular bounce is possible. singular bounce is possible.

## Nonperturbative approaches Nonperturbative approaches

### AdS AdS /CFT /CFT

### BFSS Matrix Theory BFSS Matrix Theory

### In these formulations, spacetime is no longer fixed from the beginning, rather it is dynamically generated. One only needs to specify the

### asymptotic geometry.

### Schwarzschild black holes in

### Schwarzschild black holes in AdS AdS

Maldacena;

Witten

t

Quantum gravity in this black hole background is described by an SU(N)
Super Yang-Mills at finite temperature on S^{3}.

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

### Understanding Black hole singularity from finite Understanding Black hole singularity from finite

### temperature Yang

### 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.

### Challenges Challenges

### The singularities are hidden behind the The singularities are hidden behind the horizons.

### horizons.

### Need to decode the black hole geometry Need to decode the black hole geometry from boundary Yang

### from boundary Yang- - Mills theory. Mills theory.

### Mapping of physical quantities Mapping of physical quantities

Gauge invariant operator

### O

Dimension νν

Finite temperature two- point functions

of

### O

Field φ in AdS

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.

## Large dimension limit Large dimension limit

Festuccia and Liu

### We consider the following large operator We consider the following large operator dimension limit of the boundary Wightman dimension limit of the boundary Wightman

### function

### function G

_{+}

### (ω) in momentum space. in momentum space.

### ν ν → → ∞ ∞ , ω = , ν ν u u

### G

_{+}

### (ω) → → 2 2 ν ν e e

^{νZ(u}

^{ν}

^{Z(u}

^{)}

^{)}

## Relation with bulk geodesics Relation with bulk geodesics

Festuccia and Liu

### Z (u) is given by the Z (u) is given by the Legendre Legendre transform of the transform of the proper distance of a bulk geodesic with initial proper distance of a bulk geodesic with initial

### velocity velocity

### E = i u E = i u

### The geodesic starts and ends at the boundary The geodesic starts and ends at the boundary and is specified by a turning point

### and is specified by a turning point

### r r

_{c}

_{c}

### (u) (u)

### Mapping of the boundary momentum space Mapping of the boundary momentum space

### to bulk geometry to bulk geometry

### r r

_{c}

_{c}

### (u) maps (u) the boundary

### momentum space

### to the black hole

### spacetime

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

### At finite N, no matter how large, Yang At finite N, no matter how large, Yang - - Mills Mills theory has a discrete spectrum on a compact theory has a discrete spectrum on a compact

### space.

### space.

### This implies that G This implies that

_{+}

### (ω) has the form G

_{+}

### (ω) = ∑ δ (ω – ω

_{i }

### )

### with ω

_{i }

### real.

### Summary Summary

### Certain Certain static static singularities in GR are singularities in GR are resolved in resolved in perturbative string theory

### perturbative string theory , while others are , while others are resolved by invoking non

### resolved by invoking non- - perturbative degrees perturbative degrees of freedom.

### of freedom.

### Understanding the cosmological singularities is Understanding the cosmological singularities is a big challenge for string theory.

### a big challenge for string theory.