An Outline of An Outline of String TheoryString Theory

Miao Li

Interdisciplinary Center of Theoretical Study, USTC, Hefei, ChinaInstitute of Theoretical PhysicsBeijing, China

Contents

I. Background

II. Elements of string theory

III. Branes in string theory

IV. Black holes in string theory-holography-Maldacena’s conjecture

I. Background

1. The world viewed by a reductionist

Let’s start from where Feynman’s lecture starts

A drop of water enlarged 10^9times

H

O

Feynman was able to deduce a lot of things

from a single sentence:

All forms of matter consist of atoms. 1. Qualitative properties of gas,

liquid…

2. 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:

Electron, point-like

Nucleus 10^{-13} cm

Dirac:

QED explains all of chemistry and most of

physics.

Periodic table of elements, chemical reactions,

superconductors, some of biology.

Sub-atomic structure

Nucleus of H=proton

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

u

ud

Neutron:

Interaction strengths

QEDSize of H=Compton length of electron/α=

d

u d

Strong interaction

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

Finally, gravity, the weakest of all four interactions

-mass of proton -Planck mass

(so )

Summary:

Strong interaction-SU(3) Yang-Mills

Electromagnetic

Weak interaction SU(2)XU(1)

Gravity

To assess 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.

Mechanics Heat, thermodynamics

Atomic theory, statistical mechanics ofMaxwell, Boltzmann, Gibbs, 19th century.

Electrodynamics Magnetism Light, X-rays, γ-rays

Faraday, Maxwell, 19th century.

Quantum electrodynamics Weak interaction

Semi-unification, Weinberg-Salam model.The disparity between 10^{-2} and

10^{-6}is solved by symmetry breaking in gaugetheory.

1960’s-1970’s

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

Five Nobel medals for this unification.)

Although electro-weak, strong interaction

appear as different forces, they are governed

by the same universal principle:

Quantum mechanics or betterQantum field theory

valid up to

Further, there is evidence for unification of

3 forces:

(a) In 4 dimensions,

goes up with E goes down with E

(b) runs as powers of E if there are large compact dimensions ( )

3. Difficulty with gravity

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.

Thus

amplitude=

The next order to the Born approximation

amplitude=

(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 thehorizon is not .

The larger the BH, the weak the interaction.

GR predicts the surface gravity be

Curiously,

Size of black hole=Compton length/

or

To summarize, the present day’s accepted

picture of our fundamental theory is

4. The emergence of string theory

A little history

Strong interaction is described by QCD, however, the dual resonance model wasinvented to describe strong interaction

first, and eventually became a candidate of

theoryof quantum gravity.

Initially, there appeared infinitely many resonant states ( π,ρ,ω…)

None of the resonant states appears more

fundamental than others. In calculating an

amplitude, we need to sum up all intermediate

states:

π π π π = Σ n

π π π πDenote this amplitude by A(s,t) :

(a)

(b) Analytically extend A(s,t) to the complex plane of s, t, we must have

Namely

Σ n = Σ n

This is the famous s-t channel duality.

A simple formula satisfying (a) and (b) is the

famous Veneziano amplitude

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

state

linear trajectory

This remarkable formula leads us to

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,

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

x

σ

x

σ

Let the tension of the string be T, according to

Heisenberg uncertainty relation

Now

or

If , then

Casimir effect

The above derivation ignores factors such as

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

We discovered the linear trajectory.

Morals:

(a)There are infinitely many massive states resulting from a single string (Q.M. is essential)

(b)If we have only “bosonic strings”, no internal colors, we can have only integral spins.

spin 1: gauge bosons spin 2: graviton

(c) 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

1. First quantized strings, Feynman rules

Particle analogue Action

A classical particle travels along the shortest

path, while a quantum particle can travel along different paths simultaneously, so

we would like to compute

Generalization to a string

T tension of the string

dS Minkowski area element

dS

Curiously, string can propagate consistently

only when the dimension of spacetime isD=26

Why is it so?

We have the string spectrum

Each physical boson on the world sheetcontributes to the Casimir energy an

amounta=-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 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 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

more legs

there is a coupling constant

The only constraint on these couplings isrenormalizability.

Associated with each propagator

=

Or

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

example

= +

+ +

+…

Surely, this is the origin of s-t channel duality.

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

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

In case of the closed strings

+

Again, there is a unique diagram for each topology, the vertex is also unique

=

The open string theory must contain closed

Strings

=

The intermediate state is a closed string, unitarity requires closed strings be in

the spectrum.

There is a simple relation between the open

string and the closed string couplings.

Emission vertex=

Now

Emission vertex=

Thus,

2. Gauge interaction and gravitation

= massless open strings

= massless closed strings

Define the string scale

Yang-Mills coupling

=

by dimensional analysis.

Gravitational coupling

So

If there is a compact space

D=4+d =volume of the compact space

We have

Since in 4 dimensions , we have

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

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

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

The Ramond sector contains spacetimefermions

Zero mode

The Neveu-Schwarz sector contains bosons

Now the on-shell condition

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

in 10D.

Spacetime supersymmetry is a consequence.

In a way, we can say the following

(a) Bosonic strings are strings moving in the

ordinary spacetime , but quantum mechanics disfavors pure bosons, they

are unstable.

(b) Superstrings move in superspace ,

or , no way to avoid SUSY!

4. Five different string theories in 10dimensions.

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:

(a) There are open strings, whose massless

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. (c) 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 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.

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.

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

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

L: 10D superstringR: 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 ,

the theory is chiral.

Remarkably, the low energy sector of theSO(32) heterotic theory is identical to

that of type I theory, is this merely coincidence?

Some lessons we learned before the summer

of 1994:

1. 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

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

modest (such as, what about the cosmological constant?)

What we could not do before 1994:

1. Any nonperturbative calculation.

2. What happens to black holes, what happens to singularities.

3. No derivation of the standard model.

…

III. Branes in String/M theory

1. 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:

(a) 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

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.

(a) 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,

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

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 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

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

with a long-ranged

p+1 horizon

r

When , there is no apparent function

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

(a) is conserved.

(b) 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

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

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

rank=8-p rank=p+2

Thus Some unit

Both and are quantized.

We said that the microscopic description of a

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:

There are also winding states on a circle:

Define a new radius such that

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

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:

if

The natural interpretation is

θ

Thus, an open string wave mode is mapped to

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

emission absorption

Open string channel

Closed string channel

The old idea of 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

amplitude =

But

Exact formula is

2.4 Effective theory on D-branes

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 = vev of scalars

A geometric interpretation of the Higgs mechanism:

massless

massive

3. Branes as solitonic solutions

Back to the field.

(generalization of )

We use the action

Postulate a solution breaking

Breaking

Further,

The solution is

When 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

While

It is interesting to note that there is a formal

horizon:

But there is no entropy

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

• 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.

• 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 SO(32) or

Heterotic SO(32)

heterotic string

32 free fermions

16 bosons

IV. Black holes in string theory

1. 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,

any mass of black hole can form.

The typical black hole (in 4D)

No signal can escape from the horizon.

• 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

• Classical information loss

Black hole

• 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’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 is α?

Bekenstein argued, using an infalling massive

spin particle, that . This differs

from the correct value ¼.

Hawking discovered Hawking radiation and

computed

Use

• Thermodynamics

Zero-th law: there is a temperature.

First law:

Second law：

Third law: T=0 is impossible.

• Quantum information loss

Radiation, mixed state

2. Black holes in string theory

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

list:

(a)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

For a massive string

oscillation level

So

But for a bh

Horowitz-Polchinski suggested (post-D-brane)

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

lng

BH phase String phase

lnN Phase transition line?

3. Black holes in string theory-D-brane age

3.1 Near extremal black D-branes

The pure D-brane solution

There is no entropy on the pure branes.

Exciting the branes

hot gas

Near extremal black brane

Thus

At the horizon

Horizon area =

Specified to 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 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)

Strominger-Vafa

A black hole in 5D

T5: D5-branes

waves

D1-branes T4

Physical picture:

D5-D1 open strings

species

The classical solutions

and other gauge fields, where

The horizon volume fixed at r=0 expands at r=0

To compute entropy, we also need

So

Exact result:

Thus the # of states is

Microscopic origin:

A 1D gas of open strings

In the weak coupling limit:

For a boson or a fermion:

The exact formula (Cardy) is

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 :

by extrapolating BPS states.

Further develoments:

(a) 4 charged BH in 4D.

(b) Near extremal BH by adding left moving modes.

(c) Hawking radiation.

The idea of Hawking radiation viewed in D-

brane picture is simple:

D-brane calculation reproduces Hawking’s

formula (Das-Mathur)

(d) Grey-body factor

. .

Potential due to the background

Maldacena-Strominger, complete agreement.

Are there magic nonrenormalization theorem?

Maldacea conjecture:

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 Geometry:

Need small :

Another much-studied case is D3-branes, AdS5XS5:

4. Beyond D-branes

4.1 Horowitz-Polchinski’s correspondence

Curvature ~

String statesor brane states BH’s

Entropy matches ~ O(1) coefficient.

No need of D-brane charges.

4.2 Matrix BH … …… …. 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

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

(a) 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

transition.

(d) Matrix BH need to be studied further

……