Singularities in String Theory

**S ingularities**

Hong Liu

Massachusetts Institute of Technology

Spacetime s**in**gularities

Understand**in**g the physics of spacetime s**in**gularities is a

major challenge for theoretical physics.

Big Bang/Big Crunch

beg**in**n**in**g or end of time, the orig**in** of the universe

Black holes

loss of **in**formation

Str**in**g theory and spacetime

s**in**gularities

• It is generally believed that understand**in**g spacetime

s**in**gularities requires a quantum theory of gravity.

• Str**in**g theory is thus the natural framework to address

this problem.

• One hopes that str**in**g theory will lead to a detailed

theory of the Big Bang which **in** turns leads to

experimental tests of str**in**g theory.

Static example 1: Orbifolds

• (x 1 , x 2 ) ~ (-x(

1 , -x 2 ) 2d cone

x 2

x 1

• (x 1 , x 2 , x 3 , x 4 ) ~ (-x(

1 , -x 2 , -x 3 , -x 4 ) A 1 s**in**gularity

Classical general relativity is s**in**gular at the tip of the cone.

Str**in**g theory on orbifolds

Dixon, Harvey, Vafa, Witten

• The extended nature of str**in**g theory **in**troduces

additional degrees of freedom localized at the tip of

the cone: twisted sectors.

twisted sectors

• Includ**in**g the twisted sectors, str**in**g S-matrix S

is

unitary and physics is completely smooth **in**

perturbation theory.

Static example 2: Conifold

Strom**in**ger

• General relativity is s**in**gular.

• Perturbative str**in**g theory is s**in**gular.

• By **in**clud**in**g the non-perturbative

degrees of freedom

(D-branes wrapp**in**g the vanish**in**g three cycle) at the tip

of the cone, the str**in**g S-matrixS

is aga**in** smooth.

S 3 S 2

Lessons

Str**in**g theory **in**troduces new degrees of

freedom.

Str**in**g S-matrix

is completely smooth.

Cosmological s**in**gularities

• Possibilities:

• Beg**in**n**in**g of time: need **in**itial conditions, wave functions

of the Universe etc.

• Time has no beg**in**n**in**g or end: Need to understand how

to pass through the s**in**gularity.

• New Challenges:

• What are the right observables

• What are the right degrees of freedom

From Big Crunch to Big Bang: is it possible

(A A lower dimensional Toy Model)

• Exact str**in**g background.

• The Universe contracts and

expands through a s**in**gularity.

time

• One can compute the S-matrix

from one cone to the other.

• Same s**in**gularity **in** certa**in** black

holes (a closely related problem).

Results from str**in**g perturbation theory

Liu, Moore, Seiberg

Horowitz, Polch**in**ski

• For special k**in**ematics the str**in**g amplitudes

diverge.

• The energy of an **in**com**in**g particle is blue

shifted to **in**f**in**ity by the contraction at the

s**in**gularity, which generates **in**f**in**itely large

gravitational field and distorts the geometry.

Results from str**in**g perturbation theory

• Str**in**g perturbative expansion breaks

down as a result of large backreaction.

• The same conclusion applies to other

s**in**gular time-dependent backgrounds.

Nekrasov, Cornalba, Costa; Simon; Lawrence; Fab**in**ger, McGreevy; Mart**in**ec

and McElg**in**, Berkooz, Craps, Kutasov, Rajesh; Berkooz, Piol**in**e, Rozali; ………

Lessons and implications

• Perturbative str**in**g theory is generically

s**in**gular at cosmological s**in**gularities.

• One needs a full non-perturbative

framework to deal with the backreaction.

• No clear evidence from str**in**g theory so far

a non-s**in**gular bounce is possible.

Nonperturbative approaches

• AdS/CFT

• BFSS Matrix **Theory**

In these formulations, spacetime is no longer

fixed from the beg**in**n**in**g, rather it is dynamically

generated. One only needs to specify the

asymptotic geometry.

Schwarzschild black holes **in** AdS

Maldacena;

Witten

t

Quantum gravity **in** this black hole background is described by an SU(N)

Super Yang-Mills at f**in**ite temperature on S 3 .

Classical gravity corresponds to large N and large t’Hooft coupl**in**g limit

of Yang-Mills theory.

Understand**in**g Black hole s**in**gularity from f**in**ite

temperature Yang-Mills

• F**in**d the manifestation of the black hole s**in**gularity **in** the

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

• understand how

• f**in**ite N (quantum gravitational)

• t’ Hooft coupl**in**g (str**in**gy)

effects resolve it.

Challenges

• The s**in**gularities are hidden beh**in**d the

horizons.

• Need to decode the black hole geometry

from boundary Yang-Mills theory.

Mapp**in**g of physical quantities

• Gauge **in**variant

operator O

• Field φ **in** AdS

• Dimension ν

• F**in**ite temperature twopo**in**t

functions

of O

• Particle mass m

• Free propagator of φ **in**

the Hartle-Hawk**in**g

vacuum of the AdS black

hole background

The boundary theory has a cont**in**uous spectrum **in** the

large N limitdespite be**in**g on a compact space.

Large dimension limit

Festuccia and Liu

• We consider the follow**in**g large operator

dimension limit of the boundary Wightman

function G + (ω) **in** momentum space.

ν → ∞, ω = ν u

G + (ω) → 2 ν e νZ(u Z(u)

Relation with bulk geodesics

Festuccia and Liu

• Z (u) is given by the Legendre transform of the

proper distance of a bulk geodesic with **in**itial

velocity

E = i u

• The geodesic starts and ends at the boundary

and is specified by a turn**in**g po**in**t

r c (u)

Mapp**in**g of the boundary momentum space

to bulk geometry

r c (u) maps

the boundary

momentum space

to the black hole

spacetime

Yang-Mills theory at f**in**ite N

• At f**in**ite N, no matter how large, Yang-Mills

theory has a discrete spectrum on a compact

space.

• This implies that G + (ω) has the form

with ω i real.

G + (ω) = ∑ δ (ω – ω i )

Summary

• Certa**in** static s**in**gularities **in** GR are resolved **in**

perturbative str**in**g theory, , while others are

resolved by **in**vok**in**g non-perturbative degrees

of freedom.

• Understand**in**g the cosmological s**in**gularities is

a big challenge for str**in**g theory.

• Non-perturbative framework like the AdS/CFT

correspondence gives promis**in**g avenue for

attack**in**g the problem.