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An Outline of String Theory

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- Contents
- Background
- Elements of string theory
- Branes in string theory
- Black holes in string theory-holography-Maldacena’s conjecture

- I. Background
- The world viewed by a reductionist
- Let’s start from where Feynman’s lecture starts
- A drop of water enlarged 10^9
- times

H

O

- Feynman was able to deduce a lot of things
- from a single sentence:
- All forms of matter consist of atoms.
- Qualitative properties of gas, liquid…
- Evaporation, heat transport (to cool your
- Soup, blow it)
- 3. Understanding of sounds, waves…

Atomic structure

H:

10^{-8}cm

Theory: QED (including Lamb shift)

Interaction strength:

Nucleus 10^{-13} cm

QED explains all of chemistry and most of

physics.

Periodic table of elements, chemical reactions,

superconductors, some of biology.

Nucleus of H=proton

u=2/3 U(1), d=-1/3 U(1), in addition, colors of SU(3)

u

u

d

Size of proton=Compton length of quark/

So the strong interactions are truly strong,

perturbative methods fail.

QCD is Still unsolved

Another subatomic force: weak interaction

β-decay

How strong (or how weak) is weak interaction?

Depends on the situation. For quarks:

-mass of u-quark

-mass of W-boson

To asses the possibility of unification, let’s

Take a look at

2. A brief history of amalgamation of physical

theories.

Movement of earthly bodies.

Movement of celestial bodies.

Newtonian mechanics + universal gravitation.

17th century.

Heat, thermodynamics

Atomic theory, statistical mechanics of

Maxwell, Boltzmann, Gibbs, 19th century.

Electrodynamics

Magnetism

Light, X-rays, γ-rays

Faraday, Maxwell, 19th century.

Weak interaction

Semi-unification, Weinberg-Salam model.

The disparity between 10^{-2} and 10^{-6}

is solved by symmetry breaking in gauge

theory.

1960’s-1970’s

(`t Hooft, Veltman, Nobel prize in 1999, total

Five Nobel medals for this unification.)

Although eletro-weak, strong interaction

appear as different forces, they are governed

by the same universal principle:

Quantum mechanics or better

Qantum field theory

valid up to

- Further, there is evidence for unification of
- 3 forces:
- In 4 dimensions,
- goes up with E
- goes down with E
- (b) runs as powers of E if there are large compact dimensions ( )

Gravity, the first ever discovered interaction,

has resisted being put into the framework of

quantum field theory.

So, we have a great opportunity here!

Why gravity is different?

There are many aspects, here is a few.

(a) The mediation particle has spin 2.

(b) According to Einstein theory, gravity is geometry. If geometry fluctuates violently,

causal structure is lost.

(c) The existence of black holes.

(c1) The failure of classical geometry.

singularity

(c2) A black hole has a finite entropy, or a state of a black hole can not be specified by

what is observed outside.

Hawking radiation, is quantum coherence lost?

Curiously, the interaction strength at the

horizon is not .

The larger the BH, the weak the interaction.

GR predicts the surface gravity be black hole can not be specified by

Curiously,

Size of black hole=Compton length/

or

To summarize, the present day black hole can not be specified by ’s accepted

picture of our fundamental theory is

4. The emergence of string theory black hole can not be specified by

A little history

Strong interaction is described by QCD,

however, the dual resonance model was

invented to describe strong interaction first,

and eventually became a candidate of theory

of quantum gravity.

Initially, there appeared infinitely many

resonant states ( π,ρ,ω…)

None of the resonant states appears more black hole can not be specified by

fundamental than others. In calculating an

amplitude, we need to sum up all intermediate

states:

ππππ

= Σ

n

π πππ

Denote this amplitude by A(s,t) :

(a)

(b) Analytically extend black hole can not be specified by A(s,t) to the complex plane of s, t, we must have

Namely

Σn = Σ n

This is the famous s-t channel duality.

A simple formula satisfying (a) and (b) is the black hole can not be specified by

famous Veneziano amplitude

polynomial in t: Σ t^J, J-spin of the intermediate state

linear trajectory

This remarkable formula leads us to black hole can not be specified by

String theory

For simplicity, consider open strings (to which

Veneziano amplitude corresponds)

Ground state v=c

v=c

An excited state

v=c v=c

To calculate the spectrum of the excited states, black hole can not be specified by

We look at a simple situation (Neuman->Dirichlet)

x

σ

x

σ

Let the tension of the string be T, according to black hole can not be specified by

Heisenberg uncertainty relation

Now

or

If , then black hole can not be specified by

Casimir effect

The above derivation ignores factors such as

2’s, π’s. More generally, there can be

We discovered the linear trajectory.

- Morals: black hole can not be specified by
- There are infinitely many massive states resulting from a single string (Q.M. is essential)
- If we have only “bosonic strings”, no internal colors, we can have only integral spins.
- spin 1: gauge bosons
- spin 2: graviton
- To have a massless gauge boson, a=-1. To have a
- massless graviton, a=-2 (need to use closed strings).

- II. Elements of string theory black hole can not be specified by
- First quantized strings, Feynman rules
- Particle analogue
- Action

A classical particle travels along the shortest black hole can not be specified by

path, while a quantum particle can travel

along different paths simultaneously, so we

would like to compute

Generalization to a string black hole can not be specified by

T tension of the string

dS Minkowski area element

dS

Curiously, string can propagate consistently black hole can not be specified by

only when the dimension of spacetime is

D=26

Why is it so?

We have the string spectrum

Each physical boson on the world sheet black hole can not be specified by

contributes to the Casimir energy an amount

a=-1/24.

When n=1, we obtain a spin vector field with

# of degrees D-2

For

A tachyon! This breaks Lorentz invariance, so

only for D=26, Lorentz invariance is

maintained.

But there is a tachyon at black hole can not be specified by n=0, bosonic string

theory is unstable.

Unstable mode if E is complex

For a closed string

(There are two sets of D-2 modes, left moving and right moving:

)

For black hole can not be specified by n=2, we have a spin 2 particle, there are

however only ½ D(D-3) such states, it ought to

be massless to respect Lorentz invariance,

again D=26.

Interactions

In case of particles, use Feynman diagram to

describe physical process perturbatively:

++

+ …

Associated to each type of vertex black hole can not be specified by

more legs

there is a coupling constant

The only constraint on these couplings is

renormalizability.

Associated with each propagator

=

Or black hole can not be specified by

By analogy, for string interaction

+ +…

The remarkable fact is that for each topology

there is only one diagram.

While for particles, this is not the case, for black hole can not be specified by

example

= +

+ +

+…

Surely, this is the origin of s-t channel duality. black hole can not be specified by

One can trace this back to the fact that there

is unique string interaction vertex:

=

Rejoining or splitting

The contribution of a given diagram is black hole can not be specified by

n=# of vertices = genus of the world sheet.

In case of the closed strings

+

Again, there is a unique diagram for each black hole can not be specified by

topology, the vertex is also unique

=

The open string theory must contain closed

Strings

=

The intermediate state is a closed string, black hole can not be specified by

unitarity requires closed strings be in the

spectrum.

There is a simple relation between the open

string and the closed string couplings.

Emission vertex=

2. black hole can not be specified by Gauge interaction and gravitation

= massless open strings

= massless closed strings

Define the string scale

Yang-Mills coupling black hole can not be specified by

=

by dimensional analysis.

Gravitational coupling

So black hole can not be specified by

If there is a compact space

D=4+d =volume of the compactspace

We have

Since in 4 dimensions , we have black hole can not be specified by

Phenomenologically, at the unification

scale, so .

We see that in order to raise the string scale,

say , we demand . With the

advent of D-branes, in the T-dual picture this

Implies

Large extra dimensions

3. Introducing fermions, supersymmetry black hole can not be specified by

In order to incorporate spin ½ etc into the

string spectrum, one is led to introducing

fermions living on the world sheet.

Again, the particle analogue is

The same as what Dirac did.

( )

Similarly, one introduces on the black hole can not be specified by

world sheet.

This led to the discovery of supersymmetry for

the first time in the western world (2D)

(independent of Golfand and Lihktman)

Two sectors

(a) Ramond sector

(b) Neveu-Schwarz sector black hole can not be specified by

The Ramond sector contains spacetime

fermions

Zero mode

The Neveu-Schwarz sector contains bosons

Now the on-shell condition black hole can not be specified by

is modified to (open string)

n-integer in R sector

n-half integer in NS sector

D=10: NS: n=1/2, massless gauge bosons

R: n=0, massless fermions

- 8 bosons + 8 fermions =supermultiplet black hole can not be specified by
- in 10D.
- Spacetime supersymmetry is a consequence.
- In a way, we can say the following
- Bosonic strings are strings moving in the
- ordinary spacetime , but quantum
- mechanics disfavors pure bosons, they are
- unstable.

(b) Superstrings move in superspace , black hole can not be specified by

or , no way to avoid SUSY!

4. Five different string theories in 10

dimensions.

Consistency conditions allow for only 5

different string theories (it appears that we

have a complete list, thanks to duality)

4.1 open superstring or type I string theory

Characteristics:

- There are open strings, whose massless black hole can not be specified by
- modes are super Yang-Mills in 10D.
- (b) As we said, there must be closed strings
- (unitarity). The massless modes are N=1
- SUGRA in 10D.
- One can associate a charge to an end of
- an open string.
- fundamental representation of G, anti-fundamental rep of G

Combined, they form the adjoint rep of black hole can not be specified by G.

G can be U(N), Sp(N), SO(N).

For U(N), the two ends are different, therefore

one may label the orientation of the string.

For Sp(N) and SO(N), the two ends are identical,

thus the string is un-oriented.

(d) Further, anomaly cancellation

G=SO(32)

Type I theory is also chiral. black hole can not be specified by

4.2 Closed superstring, type IIA

For a closed string: and

The left movers are independent of the right

movers.

or superposition of them.

two sets of matrices. black hole can not be specified by

Therefore, two basic choices

One choice:

chiral

anti-chiral

We have type IIA superstring theory, no

chirality. Thus, it appears that it has nothing

to do with the real world.

The massless modes = type IIA SUGRA.

4.3 Type IIB superstring theory black hole can not be specified by

If

chiral

chiral

We have type IIB string theory, it is chiral.

Although type IIB theory is chiral, it has no

gauge group, it appears to be ruled out by

Nature too.

4.4 Two heterotic string theories black hole can not be specified by

L: 10D superstring

R: 26D bosonic string

26=10+16

Naively, it leads to gauge group , but the

Gauge symmetry is enhanced:

or

- In the heterotic theory, there is only one , black hole can not be specified by
- the theory is chiral.
- Remarkably, the low energy sector of the
- SO(32) heterotic theory is identical to that of
- type I theory, is this merely coincidence?
- Some lessons we learned before the summer
- of 1994:
- String theory is remarkably rigid, it must have SUSY, it
- must live in 10D. There are only 5 different theories. Even
- the string coupling constant is dynamical.

2. It has too many consistent vacuum solutions, to pick up black hole can not be specified by

one which describes our world, we have to develop

nonperturbative methods.

3. It tells us that some concepts of spacetime are illusion, for

instance T-duality tells us that a circle of radius R is

equivalent to a circle of radius 1/R (in string unit). Sometimes,

even spaces of different topologies are equivalent.

4. The theory is finite. The high energy behavior is extremely

soft.

The more the energy, the larger the area S. is small.

- 5. There are a lot of things unknown to us, we must be black hole can not be specified by
- modest (such as, what about the cosmological constant?)
- What we could not do before 1994:
- Any nonpertubative calculation.
- What happens to black holes, what happens to singularities.
- No derivation of the standard model.
- …

- III. Branes in String/M theory black hole can not be specified by
- Why branes?
- In the past, it was often asked that if one can
- replace particles by strings, why not other
- branes such as membranes?
- The answer to this question were always:
- We know how to quantize particles and
- strings, while we inevitably end up with
- inconsistency in quantizing other objects.

- (b) Perturbative string theory is unitary, no black hole can not be specified by
- need to add to the spectrum other things.
- Thus (a) and in particular (b) sounds like a
- no-go theorem.
- To avoid this no-go theorem, we need to look
- up no other than quantum field theory.
- In some QFT, there are solitons, these
- objects can be quantized indirectly by
- quantizing fluctuations of original fields in
- the soliton background.

(b) A theory may be unitary perturbatively, black hole can not be specified by

but nonperturbatively the S-matrix may not

be unitary (showing up in resummation of a

divergent series).

Such inconsistency arises in particular when

new stable particles exist, their masses are

heavy when g is small.

Some stable particles can be associated with black hole can not be specified by

conservation of charge.

For example, when there is an Abelian gauge

field

Happily, for a oriented closed string there is

also a gauge field

Of course, when the space has a simple black hole can not be specified by

topology, there is no conserved charge

string

If there is a circle and the string is wrapped

on it, there is a charge.

This is just conservation of winding number.

In a string theory, there is a variety of other black hole can not be specified by

high rank gauge fields, for instance, the so

called Ramond-Ramond tensor field:

But the perturbative states, strings, are not

coupled to them directly. Are these fields

wasted?

There is a plausible argument for the

existence of p-brane coupled to C .

One can always find a black-brane solution black hole can not be specified by

with a long-ranged

p+1

horizon

r

When , there is no apparent function black hole can not be specified by

source for . In other words, the

source is the smeared fields carried by the BH

solution.

This avoids the apparent paradox that

perturbative fields carry no charge.

If , , black brane decays, but it

will stop at

A soliton charged under , stable.

- The stability is due to black hole can not be specified by
- is conserved.
- implies naked singularity.
- The p-brane will be called D-brane, or
- multiple D-branes. Their tension is large when
- g small.
- They can be viewed as a “collective”
- excitation of strings, but there is another
- beautiful interpretation!

2. Emergence of D-branes black hole can not be specified by

D is shorthand for Dirichlet. In a closed string

theory, the ends of a open string are stuck on

a D-brane. Namely, these ends are confined in

the bulk. (The brane is like a defect in a

superconductor.)

+-

We argued that there must be fundamental black hole can not be specified by

branes saturating the BPS bound .

If is continuous, as the classical solution

suggests, we have the trouble for accounting

a continuous spectrum.

Fortunately, some time ago, it was proven

that must be quantized, according to a

generalized Dirac quantization condition.

Denote dual to black hole can not be specified by

rank=8-prank=p+2

Thus

Some unit

Both and are quantized.

We said that the microscopic description of a black hole can not be specified by

fundamental p-brane is D-brane. We now

follow the route that Polchinski originally

followed to see how this description emerges

in string theory.

2.1 T-duality

To understand the logic behind D-branes, we

need to review T-duality.

There are waves on a circle: black hole can not be specified by

There are also winding states on a circle:

Define a new radius such that black hole can not be specified by

Then

That is, wave modes winding modes.

We cannot distinguish a string theory on a

circle of radius R from another string theory

on a circle of radius . T-duality.

2.2 T-duality for open strings black hole can not be specified by

Starting with an open string theory which

contains closed strings automatically.

How do we map open string wave modes?

An open string can couple to a gauge field

tangent to a circle:

Thus, an open string wave mode is mapped to black hole can not be specified by

a winding mode with ends attached to

something: D-branes.

Boundary conditions on the ends of the string

are Dirichlet. In the original theory

momentum is conserved, thus in the dual

theory winding number is conserved, the ends

stick to branes.

In the original theory, winding is not

conserved, no such quantum number.

2.3 Brane tension black hole can not be specified by

emissionabsorption

Open string channel

Closed string channel

The old idea of black hole can not be specified by s-t channel duality:

=

one-loop tree-level

From the open string perspective, the

interaction between 2 D-branes :

Amplitude= vacuum fluctuations, independent of g

From the closed string perspective black hole can not be specified by

amplitude =

But

Exact formula is

2.4 Effective theory on D-branes black hole can not be specified by

Open string fluctuations longitudinal to D-

branes: gauge fields;

Open string fluctuations traverse to D-branes:

scalar fields;

Fermions = Goldstone modes.

The position of a D-brane black hole can not be specified by = vev of scalars

A geometric interpretation of the Higgs

mechanism:

massless

massive

3. Branes as solitonic solutions black hole can not be specified by

Back to the field.

(generalization of )

We use the action

Postulate a solution breaking black hole can not be specified by

Breaking

Further, black hole can not be specified by

The solution is

When black hole can not be specified by r large

so

When r small

There is no pt-like source for . That is, the

all non-linear structure of fields serve as a

smeared source-just like the monopole

solution in a broken gauge theory.

The mass, or rather the tension black hole can not be specified by

While

It is interesting to note that there is a formal

horizon:

But there is no entropy black hole can not be specified by

So this “black brane” is more or less a pure

state.

We know that it is the ground state of N

coincident D-branes.

4. Implications for string dualities black hole can not be specified by

• In type IIA string theory, there is pt-like

soliton with mass

so

How to understand the theory when ?

There is an additional circle of radius

so is a K-K mode of graviton.

• black hole can not be specified by Type IIB theory, there is

D-string

Bound states of D-strings + F-strings:

(p,q)-dyonic strings. This is implied by the

SL(2,Z) duality.

• In type I theory, there is also

Another kind of D-string, this is the heterotic

string.

The list continues …

Type I black hole can not be specified by SO(32) or

Heterotic SO(32)

heterotic string

32 free fermions

16 bosons

- IV. Black holes in string theory black hole can not be specified by
- Basics
- In real world, only a very massive collapsing
- body can form a black hole
- due to the fact that the basic matter
- constituents are fermions.
- Small black holes could (and perhaps did)
- form in early universe.

In an ideal situation, such as a free scalar field, black hole can not be specified by

any mass of black hole can form.

The typical black hole (in 4D)

No signal can escape from the horizon.

• black hole can not be specified by Black hole no hair theorem

Outside a black hole, one can measure only a

few conserved quantities, associated to long

range fields:

Mass, angular momentum, charge

Gravitational field, EM field

• black hole can not be specified by Bekenstein-Hawking entropy

Due to the no-hair theorem and the second

law of thermodynamics, a black hole must

have entropy.

State 1, state 2, state 3, … state 1 billion

The same black hole

An interesting theorem proven in 60 black hole can not be specified by ’s and

70’s:

A = area of black hole never decreases.

Thus, S of the black hole must be ~A

So, Bekenstein reasoned

S=αA

But, what isα?

Bekenstein argued, using an infalling massive black hole can not be specified by

spin particle, that . This differs

from the correct value ¼.

Hawking discovered Hawking radiation and

computed

Use

• black hole can not be specified by Thermodynamics

Zero-th law: there is a temperature.

First law:

Second law：

Third law: T=0 is impossible.

2. Black holes in string theory black hole can not be specified by

Pre D-brane era

Almost no string theorits believed in the claim

of Hawking, that QM breaks down, and

Einstein wins anyway.

Perturbative string theory is important in

dealing with such a situation, to quote

Susskind:

String theory perhaps has to solve itself before solving the

information loss paradox-Scientific American.

- There were a few proposals. An incomplete black hole can not be specified by
- list:
- It appears that some nonlocality must be involved in order for the radiation carries away information. String theoy has some nonlocality built in.
- (b)
- Strikingly similar to D-branes.

(c) Susskind-Horowitz-Polchinski correspondence principle black hole can not be specified by

For a massive string

oscillation level

So

But for a bh

Horowitz-Polchinski suggested (post-D-brane) black hole can not be specified by

that in order to form a bh, G must be tuned on.

But in 4D:

or

for

The correspondence point: for we

have string and for we have a bh.

Schematically black hole can not be specified by

lng

BH phase

String phase

lnN

Phase transition line?

3. Black holes in string theory-D-brane age black hole can not be specified by

3.1 Near extremal black D-branes

The pure D-brane solution

There is no entropy on the pure branes. black hole can not be specified by

Exciting the branes

hot gas

Specified to black hole can not be specified by p=3

is independent of

Counting the entropy of a free Yang-Mills gas,

one finds

The discrepancy is due to the large effective

coupling on the black brane:

p=3 black hole can not be specified by is called non-dilaton black brane, since

In general

For 6>p>3, theories are sufficient complex.

For p=2, not much research exists

For p=1, Hashimoto-Izthaki

For p=0, ML

3.2 Extremal black holes (branes) black hole can not be specified by

Strominger-Vafa

A black hole in 5D

T5:D5-branes

waves

D1-branesT4

Physical picture: black hole can not be specified by

D5-D1 open strings

species

The classical solutions

and other gauge fields, where black hole can not be specified by

The horizon volume

fixed at r=0 expands at r=0

To compute entropy, we also need black hole can not be specified by

So

Exact result:

Thus the # of states is

Microscopic origin: black hole can not be specified by

A 1D gas of open strings

In the weak coupling limit:

For a boson or a fermion:

The exact formula (Cardy) is black hole can not be specified by

For a boson c=1, for a fermion c=1/2. For the

system of the D1-D5 strings

- This result is valid even in case of the large : black hole can not be specified by
- by extrapolating BPS states.
- Further develoments:
- 4 charged BH in 4D.
- Near extremal BH by adding left moving modes.
- Hawking radiation.

The idea of Hawking radiation viewed in D- black hole can not be specified by

brane picture is simple:

D-brane calculation reproduces Hawking’s

formula (Das-Mathur)

(d) Grey-body factor black hole can not be specified by

. .

Potential due to the background

Maldacena-Strominger, complete agreement.

Are there magic nonrenormalization theorem?

Maldacea conjecture: black hole can not be specified by

The supergravity (or string theory) is dual to

the CFT on the branes. The fact that the near

horizon geometry is AdS is the initial strong

motivation for this conjecture.

In the D1-D5 case

Need large to have semi-classical black hole can not be specified by

Geometry:

Need small :

Another much-studied case is D3-branes,

AdS5XS5:

4. Beyond D-branes black hole can not be specified by

4.1 Horowitz-Polchinski’s correspondence

Curvature ~

String states

or brane states BH’s

Entropy matches ~ O(1)coefficient.

No need of D-brane charges.

4.2 Matrix BH black hole can not be specified by

…

……

….

boost Gas of D0-branes

Qualitatively understood:

Banks et al., Horowitz-Martinec, ML,

ML&Martinec

But in order to compute exact coefficient, need to

solve many body problem accurately.

4.3 AdS black hole can not be specified by

Can study near extremal BH only ( c>0 ).

But provides an opportunity to study

formation and evaporation of BH accurately.

One may also study singularity.

Technically unlikely to be solved in the near

future.

Both 4.2 and 4.3 are under the influence of D-

branes.

- 5. BH problem is unsolved black hole can not be specified by
- Counting entropy for Schwarzschild BH
- honestly, accurately.
- (b) Dynamic process of formation of BH in D-
- brane picture or AdS/CFT , information puzzle
- (c) Counting entropy for near-extremal BH
- accurately for p<3.
- (d) For p=3, understand ¾.

(e) Prove the existence of gas BH phase black hole can not be specified by

transition.

(d) Matrix BH need to be studied further

……

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