Singularities in String Singularities in String TheoryTheory

Hong LiuHong Liu

Massachusetts Institute of Technology

ICHEP 04 Beijing

Spacetime singularitiesSpacetime singularities

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

Big Bang/Big CrunchBig Bang/Big Crunch

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

Black holes

loss of information? loss of information?

String theory and spacetime String theory and spacetime singularitiessingularities

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

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

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

Static example 1: OrbifoldsStatic example 1: Orbifolds

(x(x11, x, x22) ~ (-x) ~ (-x11, -x, -x22) ) 2d cone2d cone

(x(x11, x, x22, x, x33, x, x44) ~ (-x) ~ (-x11, -x, -x2 2 , -x, -x33, -x, -x44))

AA11 singularity singularity

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

String theory on orbifoldsString theory on orbifolds

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 at the tip of the cone: of the cone: twisted sectors.twisted sectors.

Including the twisted sectors, Including the twisted sectors, string S-matrix is string S-matrix is unitary and physics is completely smooth.unitary and physics is completely smooth.

twisted sectors

Dixon, Harvey, Vafa, Witten

Static example 2: ConifoldStatic example 2: Conifold

General relativity is General relativity is singularsingular..

Perturbative string theory is Perturbative string theory is singularsingular..

By including the By including the non-perturbative non-perturbative degrees of degrees of freedom (D-branes wrapping the vanishing three freedom (D-branes wrapping the vanishing three cycle) at the tip of the cone, the cycle) at the tip of the cone, the string S-matrixstring S-matrix is is again again smoothsmooth..

S3

S2

Strominger

LessonsLessons

String theory introduces String theory introduces new new degrees of freedomdegrees of freedom. .

StringString S-matrix S-matrix is completely is completely smoothsmooth..

Cosmological singularitiesCosmological singularities

Possibilities:Possibilities:

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

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

New Challenges:New Challenges: What are the rightWhat are the right observables?observables?

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

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

(A Toy Model)(A Toy Model)

time

• Exact string background.

• The Universe contracts and expands through a singularity.

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

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

Results from string perturbation Results from string perturbation theorytheory

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

The energy of an incoming particle is The energy of an incoming particle is blue shifted toblue shifted to infinity infinity by the by the contraction at the singularity, which contraction at the singularity, which generates infinitely large gravitational generates infinitely large gravitational field and field and distorts the geometrydistorts the geometry..

String perturbative expansion String perturbative expansion breaks breaks downdown as a result of large backreaction. as a result of large backreaction.

The same conclusion applies to other The same conclusion applies to other singular time-dependent backgrounds. singular time-dependent backgrounds.

Liu, Moore, SeibergHorowitz, Polchinski

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

Lessons and implicationsLessons and implications

Perturbative string theory is generically Perturbative string theory is generically singularsingular at cosmological singularities. at cosmological singularities.

One needs a full One needs a full non-perturbativenon-perturbative framework to deal with the framework to deal with the backreaction.backreaction.

No clear evidence from string theory so No clear evidence from string theory so far a non-singular bounce is possible. far a non-singular bounce is possible.

Nonperturbative approachesNonperturbative approaches

AdS/CFT AdS/CFT

BFSS Matrix TheoryBFSS 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 AdSSchwarzschild black holes in AdS

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

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

Maldacena;Witten

t

Black hole singularity from Yang-Black hole singularity from Yang-Mills ?Mills ?

The correspondence gives, in principle, an explicit The correspondence gives, in principle, an explicit Hamiltonian description of black hole formation and decay, Hamiltonian description of black hole formation and decay, including the fate of the black hole singularity in a quantum including the fate of the black hole singularity in a quantum gravitational theory. gravitational theory.

The challenge is to decode the physics of black hole from The challenge is to decode the physics of black hole from Yang-Mills theory.Yang-Mills theory.

One would like to first find the signature of black hole singularity in One would like to first find the signature of black hole singularity in the large N and large t’ Hooft limit of Yang-Mills theorythe large N and large t’ Hooft limit of Yang-Mills theory

understand how finite N and t’ Hooft coupling effects resolve it.understand how finite N and t’ Hooft coupling effects resolve it.

Translate the effects into language of black hole physics Translate the effects into language of black hole physics

Draw general lessons from it. Draw general lessons from it.

Bouncing null geodesics Bouncing null geodesics Fidkowski, Hubeny, Kleban, Shenker

Complex temperature ? (I)Complex temperature ? (I)Festuccia and Liu

δw

In the strong coupling limit, one finds the retarded propagatorsfor arbitrary scalar operators have two lines of poles in lower half complex frequency plane.

Starinets, Nunez,…

Complex temperature ? (II)Complex temperature ? (II)

The momentum space retarded propagators also exhibit some universal behavior in the high frequency limit.

For certain operators:

Black hole singularity and complex Black hole singularity and complex temperaturetemperature

t=0

t=iB/4

SummarySummary

Certain Certain staticstatic singularities in GR are singularities in GR are resolved in resolved in perturbative string theoryperturbative string theory, while others are , while others are resolved by invoking non-perturbative degrees of resolved by invoking non-perturbative degrees of freedom.freedom.

Understanding the cosmological singularities is a Understanding the cosmological singularities is a big challenge for string theory. Non-perturbative big challenge for string theory. Non-perturbative framework like the framework like the AdS/CFT correspondenceAdS/CFT correspondence gives gives promising avenue for attacking the problem.promising avenue for attacking the problem.

String theory has the potential to make important String theory has the potential to make important progress in cosmology by addressing this progress in cosmology by addressing this question. question.