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
11, x , x
22) ~ ( ) ~ ( - - x x
11, , - - x x
22) ) 2d cone 2d cone
(x (x
11, x , x
22, x , x
33, x , x
44) ~ ( ) ~ ( - - x x
11, , - - x x
2 2, , - - x x
33, , - - x x
44) A ) A
11singularity
x1 x2
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..
S3
S2
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 S3.
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(uνZ(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
cc(u) (u)
Mapping of the boundary momentum space Mapping of the boundary momentum space
to bulk geometry to bulk geometry
r r
cc(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
ireal.
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.