Multiverse Partition function and

Maximum Entropy Principle

Dec. 18, 2018

at KEK

Hikaru Kawai

(Kyoto Univ.)

Special features of SM and the Higgs field have been revealed by LHC.

In particular, they have found that(i) SM can be valid up to the Planck

scale.(ii) The effective potential of the Higgs

field is flat around the Planck scale.⇒ Critical Higgs inflation

Bare couplings as a function of the cutoff

SM is valid up to the Planck scale.

Top mass from[1405.4781][Hamada, Oda, HK ,1210.2538, 1308.6651]

mHiggs =125.6 GeV

2

1 2

1

16I

=

Higgs potential with

the running coupling 4

4V

mH = 125.6 GeV

mHiggs =125.6 GeV

𝜑[GeV]

MPP

MEP

MEP: Maximal entropy principleMPP: Multiple Point Criticality Principle

If we introduce a non-minimal coupling

a realistic Higgs inflation is possible.

2R

2 2

, .1 /

h h

P

hV

h M

In the Einstein frame the

effective potential becomes

𝜉 can be as small as 10.

Critical Higgs inflation

I will discuss the possibility thatsuch behavior of the Higgs field is understood as a consequence of quantum gravity / string theory in the context of the naturalness problem.

Naturalness problem

Suppose the underlying fundamental theory, such as string theory, has the momentum scale mS and the coupling constant gS .

The naturalness problem

Then, by dimensional analysis and the power counting of the couplings, the parameters of the low energy effective theory are given as follows:

dimension 2 (Higgs mass)

dimension 4

(vacuum energy or cosmological constant)

dimension 0

(gauge and Higgs couplings)

dimension -2 (Newton constant)

2

2.S

N

S

gG

m

1 2 3

2

, , ,

.

S

H S

g g g g

g

2 22 0 .H S Sg mm

2 40 .Ss mg

222 2 2 17100GeV 10 GeVH S Sm g m

44 4 172 ~ 3meV 10 GeVSm

unnatural ! →

unnatural ! !→

tree

(cont’d)

The real values of the cosmological constant and Higgs mass are very unnatural:

If we consider everything only in the ordinary field theory, they are free parameters. In principle we can give any values to them by hand, but still we need fine tunings in order to get our universe.

Or, if we assume the existence of the underlying fundamental theory like string theory, we need to explain the values of the parameters that look extremely unnatural from the point of view of the low energy effective local field theory.

SUSY as a solution to the naturalness problem

Bosons and fermions cancel the UV divergences:

However, SUSY must be spontaneously broken at some momentum scale MSUSY , below which the cancellation does not work.

+

bosons fermions

⇒2 2

SUSY .Hm M

+ ⇒ 0.

⇒

2 0.Hm

⇒4

SUSY .M

(cont’d)

Therefore, if MSUSY is close to mH , the Higgs mass is naturally understood, although the cosmological constant is still a big problem.

However, no signal of new particles is observed in the LHC below 2 TeV.

We have to think about other possibilities.

Possible attitudes to the naturalness problem

other than SUSY

1. We do not have to mind. We should simply take the parameters as they are.

2. Anthropic principle. (one of the variations)

In some models, the wave function of the universe is a superposition of various worlds, each of which obeys different low energy effective theory:

1 2 3 .

We are sitting in one of them, where the parameters should be such that human beings can appear in the history of the universe.

3. The parameters are fixed by some mechanisms

that are not in the ordinary field theory.

There are several attempts which are related:

• asymptotic safetyShaposhnikov, Wetterich

• multiple point criticality principleFroggatt, Nielsen.

• classical conformalityMeissner, Nicolai,

Foot, Kobakhidze, McDonald, Volkas

Iso, Okada, Orikasa.

• baby universe and multi-local actionColeman

Okada, Hamada, Kawana, Sakai, HK

I will discuss the last one, and show that there is a universal non-perturbative effect in quantum gravity / string theory that provides a mechanism to tune the parameters automatically.

“Nature does fine tunings!”

Low energy effective theoryof

QG / string theory

Consider the low energy effective theory of

quantum gravity / string theory obtained after

integrating out the short distance fluctuations.

eff gauge matter ...DS d x g R

Usually people believes that it should be a

local action:

Is it true?

Basic question

As we will see, the low energy effective

theory has the multi-local form:

eff 1 2, ,

,

( ) ( ).

i i i j i j i j k i j k

i i j i j k

D

i i

S f S S

c S c S S c S S S

S d x g x O x

Here 𝑶𝒊 are gauge invariant scalar local

operators such as

𝟏 , 𝑹 , 𝑹𝝁𝝂𝑹𝝁𝝂 , 𝑭𝝁𝝂 𝑭

𝝁𝝂, 𝝍𝜸𝝁𝑫𝝁𝝍 ,⋯ .

It is not sufficient if we take the fluctuation of

space-time topology into account.

Sugawara ‘82 Nambu ‘84 Linde ‘87 Coleman ‘89

Start with Euclidean path integral that

involves the summation over topologies

and consider the low energy effective theory

obtained after integrating out the short-

distance configurations.

topology

exp ,dg S

One way to understand such action is

to consider the space-time wormhole like

configurations as Coleman did.

Why multi-local ?

Among such short-distance configurations

there should be a space-time wormhole-like

configuration in which a thin tube connects

two points on the space-time. Here, the two

points may belong to either the same universe

or different universes.

Do not confuse with 3D

wormhole.

We are considering 4D

configuration such that

a baby universe is

virtually created and

absorbed.

If we see such wormhole-like configuration

from the side of the large universe(s), it looks

like two small punctures.

But the effect of making a small puncture is

equivalent to an insertion of a local operator.

Furthermore, if we integrate over the gauge

field and the metric on the tube, each

operator is projected to a gauge invariant

scalar operator.

integrate𝑨𝝁, 𝒈𝝁𝝂

4 4

,

( ) ( ) ( ) ( ) exp .i j i j

i j

c d x d y g x g y O x Odg Sy

Therefore, a wormhole contributes to the

path integral as

x

y

𝒄𝒊𝒋 are numbers that are determined by the

path integral over the tube, and expected to be

of order 1 in the Planck unit.

Here 𝑶𝒊 are gauge invariant scalar local

operators such as

𝟏 , 𝑹 , 𝑹𝝁𝝂𝑹𝝁𝝂 , 𝑭𝝁𝝂 𝑭

𝝁𝝂, 𝝍𝜸𝝁𝑫𝝁𝝍 ,⋯ .

𝒙 and 𝒚 are

positions in the

large universe(s).

If we sum over the number of wormholes, we have

4 4

,

( ) ( ) ( ) ( )exp .i j

i j

i j

c d x d y g x g y O xg yS Od

4 4

0 ,

4 4

,

1( ) ( ) ( ) ( )

!

exp ( ) ( ) ( ) ( ) .

n

i j

i j

n i j

i j

i j

i j

c d x d y g x g y O x O yn

c d x d y g x g y O x O y

The wormholes induce the “bi-local” action:

bifurcated wormholes

⇒ cubic terms, quartic terms, …

.)()(

,eff

xOxgxdS

SSScSScScS

i

D

i

kji

kji

kjij

ji

iji

i

ii

“ The low energy effective theory of quantum

gravity that involves topology change is not

simply local but multi-local.”

Thus we have seen

In short,

“topology change induces multi-local terms

in the low energy effective theory.”

In the case of Coleman, he considered the

wormhole as a classical solution of the

Euclidean 4D theory like instanton.

Coleman had essentially the same result in

1989.

Comment on Coleman’s

Here we are considering a Wilsonian

effective theory.

We integrate over wormhole-like

configurations no matter whether they are

classical solutions or not.

We have seen that we have the multi-local

action irrespectively to the existence of the

classical solution.

Probably, theories with this property are more

natural than those without it.

This is because we have to introduce

additional conditions on the path integral if

we want to fix the topology of the space-time.

What we need is only the property that

topology change of space-time is included in

the theory.

Also string theory seems to contain topology

change of space-time automatically.

In this sense the multi-local action can be

regarded as a universal form of the low energy

effective theory of quantum gravity / string

theory.

for the multi-local action

eff ,

( ) ( ),

i i i j i j i j k i j k

i i j i j k

D

i i

S c S c S S c S S S

S d x g x O x

effexpZ d S

We want to do the path integral

Consequences of multi-local action

where 𝒄𝒊 , 𝒄𝒊𝒋 , 𝒄𝒊𝒋𝒌 ⋯ are constants of O(1) in

the Planck unit.

we introduce the Laplace transform

effexp exp .i i

i

Z d S d w d S

Coupling constants are not merely constants,

but they should be integrated.

eff 1 2 1 2exp , , , , exp .i i

i

S S S d w S

Then the path integral becomes

Regarding 𝑺𝐞𝐟𝐟 as a function of 𝑺𝒊’s

eff eff 1 2, , ,S S S S

path integral for a local action 𝝀𝒊 𝑺𝒊.𝝀𝒊 are coupling constants of the action.

effexp ( ),

( ) exp .i i

i

Z d S d w Z

Z d S

This contradicts with our experience.

We know coupling constants are just

constants, and we never observe a

superposition of different coupling constants.

We have seen that the partition function is

given by averaging 𝒁(𝝀) with weight 𝒘(𝝀).

However, the problem is resolved if 𝒁(𝝀) has

a sharp peak around some value 𝝀~𝝀(𝟎). In that case we can say that the nature itself

tune the parameters to 𝝀(𝟎).

An interesting point of this scenario is that

𝒘(𝝀) is not important if it is a smooth

function of 𝝀.

We are not sure if it is the case or not, but it

is possible because exp(−Seff) may be a well-

behaved function of 𝑺𝒊’s, and so is its Laplace

transform.

In that case the so called Big Fix occurs:

All the couplings of the low energy effective

theory are fixed by themselves, and we do

not need precise information about the short

distance physics.

Necessity of multiverse

Even if we start from a connected universe, we

have disconnected universes after integrating

out the short-distance fluctuations.

exp i i

i

Z d w d S

single

0

single

1( )

!

exp ( ) .

n

n

d w Zn

d w Z

n

Therefore we have to sum over the number of

universes in the evaluation of the partition

function.

Here 𝒁𝐬𝐢𝐧𝐠𝐥𝐞 𝝀 is the partition function for a

single universe.

“solution” to the cosmological constant problem

single ( ) exp .Z dg gR g 4Sr

2 4exp

exp 1/ , 0

no solution, 0

dr r r

r

S

1

dominates irrespectively of 0 .w

For simplicity, we keep only the cosmological

constant 𝚲.

If the universe is large, 𝒁𝐬𝐢𝐧𝐠𝐥𝐞(𝚲) is evaluated

by considering 𝑺𝟒 geometry classically:

singleexp ( ) .Z d w Z

However this argument is problematic.

WDW eq.

←wrong sign

“Ground state” does not exist.

total 0H

total universe matter graviton

2

universe

1

2a

H H H H

H p

Wick rotation is not well defined. t

matter ,H

universeH

: radius of the universea

matterH is bounded from below.

universeH is bounded from above.

Problem of the Wick rotation for gravity

single ( ) exp .Z dg gR g

2 4exp

exp 1/ , 0

no solution, 0

dr r r

r

S

1

Actually what we have calculated

is a wrong sign version of the tunneling

probability with which universe pops out from

nothing through the potential barrier.

2 4exp

exp 1/ , 0

no solution, 0

dr r r

r

S

Systems with gravity should be expressed in

Lorentzian signature.

Therefore the argument we have employed

to get the multi-local action is not literally

true.

But we expect that the low energy effective

theory is still given by the multi-local action.

In fact we obtain the same effective

Lagrangian in the IIB matrix model with

Lorentzian signature.

Multi-local action in Lorentzian signature

Okada, HK

Hmada, Kawana, HK

We start with the multi-local action:

eff 1 2, ,

,

( ) ( ),

i i i j i j i j k i j k

i i j i j k

D

i i

S f S S

c S c S S c S S S

S d x g x O x

where 𝑶𝒊 are gauge invariant scalar local

operators such as

𝟏 , 𝑹 , 𝑹𝝁𝝂𝑹𝝁𝝂 , 𝑭𝝁𝝂 𝑭

𝝁𝝂, 𝝍𝜸𝝁𝑫𝝁𝝍 ,⋯ .

Integrating coupling constants

We repeat the arguments in the Lorentzian

signature.

We express 𝐞𝐱𝐩(𝒊𝑺𝐞𝐟𝐟) by a Fourier transform as

1 2 1 2exp , , , , exp ,eff i i

i

iS S S d w i S

where 𝝀𝒊’s are Fourier conjugate variables to 𝑺𝒊’s,

and 𝒘 is a function of 𝝀𝒊’s .

1 2 1 2exp , , , , exp ,eff i i

i

iS S S d w i S

Then the path integral for 𝑺𝐞𝐟𝐟 becomes

𝒊𝝀𝒊 𝑺𝒊 is an ordinary local action where 𝝀𝒊 are regarded

as the coupling constants.

Therefore the system is the ordinary field theory,

but we have to integrate over the coupling constants

with some weight 𝒘(𝝀).

If a small region 𝝀~𝝀(𝟎)dominates the 𝝀 integral,

it means that the coupling constants are fixed to 𝝀(𝟎).

Warming up before considering multiverse

A simplified analysis ’14 ’15 Hamada, Kawana, HK

Because our universe has been cooled down for long time.

we may approximate 𝒁 𝝀 = 𝐞𝐱𝐩 −𝒊𝑽𝑬𝒗𝒂𝒄 𝝀 ,

where 𝑬𝒗𝒂𝒄 𝝀 is the vacuum energy density and 𝑽 is the

space-time volume.

1) extremum

If 𝑬𝒗𝒂𝒄 𝝀 is smooth and has an extremum at 𝝀𝒄 ,

𝒁 𝝀 = 𝐞𝐱𝐩 −𝒊𝑽𝑬𝒗𝒂𝒄 𝝀

~𝟐𝝅

𝒊 𝑽|𝑬𝒗𝒂𝒄′′ 𝝀𝒄 |

𝜹 𝝀 − 𝝀𝒄 +𝑶(𝟏

𝑽)

Thus 𝝀 is fixed to 𝝀𝑪 in the limit 𝑽 → ∞ .

vacE

C

Nature does fine tunings

2) Kink (need not be an extremum)

If 𝑬𝒗𝒂𝒄 𝝀 has a kink (first order phase transition) at 𝝀𝒄 ,

𝒁 𝝀 = 𝐞𝐱𝐩 −𝒊𝑽𝑬𝒗𝒂𝒄 𝝀

~𝒊

𝑽

𝟏

𝑬𝒗𝒂𝒄′(𝝀𝒄 + 𝟎)−

𝟏

𝑬𝒗𝒂𝒄′(𝝀𝒄 − 𝟎)𝜹 𝝀 − 𝝀𝒄 + 𝑶(

𝟏

𝑽𝟐)

Thus 𝝀 is fixed to 𝝀𝑪 in the limit 𝑽 → ∞ .

𝑎𝑏dx exp 𝑖𝑉𝑥 𝜑 𝑥

=1

𝑖𝑉exp 𝑖𝑉𝑥 𝜑 𝑥

a

b

+ O(1

V2)

𝑎𝑏dx exp 𝑖𝑉𝑓(𝑥) 𝜑 𝑥

=1

𝑖𝑉exp 𝑖𝑉𝑓(𝑥 )

1

𝑓′(𝑥)𝜑 𝑥

a

b

+ O(1

V2)

(𝑓is monotonic)

vacE

C

monotonic

If we consider the time evolution of the universe

and multiverse, 𝒁(𝝀) is not so simple.

Generalization

However, even if we do not know the precise form of

𝒁(𝝀), we can expect a similar mechanism to the

simplified cases.

Actually in some cases the couplings can be

determined without knowing the details of 𝒁(𝝀).

QCD

Z

1. It becomes important only after the QCD phase

transition.

2. The mass spectrum of hadrons is invariant under

𝜽𝑸𝑪𝑫 → −𝜽𝑸𝑪𝑫 .

⇒ It is expected that 𝒁 is almost even in 𝜽𝑸𝑪𝑫.

⇒ 𝜽𝑸𝑪𝑫 is tuned to 0 if there are no other extrema.

(1) Symmetry example 𝜽𝑸𝑪𝑫 Nielsen,Ninomiya

Conditions:

1. Physics changes drastically at some

value of the coupling 𝝀𝑪 .

2. 𝒁 is monotonic elsewhere.

⇒ The coupling is tuned to 𝝀𝑪 as we have discussed.

(2) Edge or the point of drastic change

Z

C

This explains the Multiple Point criticality Principle

proposed by Froggatt and Nielsen:

“The parameters of nature are tuned such that the

vacuum is at a (multiple) criticality point.”

Examples:

(i) Cosmological constant

The behavior of the universe

at the late stages changes

drastically when the cosmological constant becomes

zero.

⇒ The cosmological constant is tuned close to 0.

(ii) Higgs self coupling

SM parameters seem to be chosen in such a way that

the vacuum is marginally stable.

However, there is a competition with the flat potential.

∞finite

Multiverse in Lorentzian signature

effexp exp .i i

i

Z d i S d w d i S

As in the Euclidean case it is natural to consider

the multiverse.

1

0

1 single universe

1exp

!

exp

n

i i

i n

i i

i

d i S Zn

Z d i S

n

1exp .d w Z

Path integral for a universe

2 31 1 1ˆ .2

universe aH p a aa a

: radius of the universea

mini-superspace -Born Oppenheimer approx.-

universe m-g ( ) ,a

S3 topology for space total universe m-gˆ ˆ ˆ ,H H aH

adiabatic wave function for matter-graviton

m-g m-g m-gˆ ( ) ( ) ( )H a a E a a

total Hamiltonian:

changes adiabatically with 𝒂 .m-g ( )a

universe m-g ( ) ,a

2 3 4

1( ) matt radC C

U aa a a

m-g m-g m-gˆ ( ) ( ) ( ) .H a a E a a

total univers

3

3

2

e

2

ˆ ˆ ( )

1 1 1

2

.1 1 1 ˆ( )2

radmatt

H H E a

Cp a a C

aa a

p a U a Ha a

( ) radmatt

CE a C

a

⇒2 31 1 1ˆ .

2universe aH p a a

a a

acts on universe astotal universe m-gˆ ˆ ˆH H H

i

f

2 31 1 1ˆ ( )2

H p a U aa a

Question:

Is there a natural choice for them?

T

Now the path integral is evaluated as usual, if the

initial and final states are given:

ˆE E

E E

H E

E E

2 3 4

1( ) matt radC C

U aa a a

1

1

0

0 0

exp

exp

ˆexp

ˆ

E E

Z d i S

f dadpdN i dt pa NH i

f dT iTH i

f H i

f i

Initial state

For the initial state, we assume that the universe

emerges from nothing with a small size ε.

,

: probability amplitude of a universe emerging.

i a matter

a

Λ～curvature ～energy density

S3 topology

a

cr cr

cr

( )U a ( )U a ( )U a

mat rad

2 3 4

1( )

C CU a

a a a

For the final state, we have two possibilities.

Final state: case 1

finite

The universe is closed.

We assume the final state is

.f a matter

The path integral

1 0 0

2

0

ˆ

.

E E

E

Z f H i f i

const

cr

Final state: case 2

∞

The universe is open.

It is not clear how to define the

path integral for the universe:

m-glim ( ) . IR

IR IR IRa

f c a a a

a

1 exp .Z d i S

As an ad hoc assumption we consider

cr

*

1 0 0 0 0

3 *

04

3 *

04

1sin

1sin .

E E IR E IR E

IR IR E

IR

IR E

Z f i c a a

c a aa

c a

Then the partition function becomes

The result does not depend on except for the phase

which comes from the classical action.IRa

(cont’d)

mat rad

2 3 4

1( ) ,

C CU a

a a a

01 0

1sin ,

z

E a da p aa p a

4, 2 ( ).p a a U a

WKB solution

2 31 1 1ˆ ( ).2

H p a U aa a

Thus we have

the path integral for a universe

∞

finite

1Z

crfor of order 1,const

crfor 3

4

1sin .IRconst a

Then the integration for the multiverse

1exp .Z d w Z

has a large peak at , which means that

the cosmological constant at the late stages of the

universe almost vanishes.

cr

Maximum entropy principle

Then the multiverse partition function is given by

cr rad1/C

Maximum entropy principle (MEP)

The low energy couplings are determined in such a way that the entropy at the late stages of the universe is maximized.

For simplicity we assume the topology of the space and

that all matters decay to radiation at the late stages.

rad

2 4

1( )

CU a

a a

1

4rad

4

exp

1exp const exp const .

cr

Z d w Z

C

3S

cr

( )U a

There are many ways to obtain MEP:

Suppose that we pic up a universe randomly from the multiverse. Then the most probable universe is expected to be the one that has the maximum entropy.

Okada and HK ’11

We may understand the flatness of the Higgs

potential as a consequence of MEP.

If we accept the inflation scenario in which universe

pops out from nothing and then inflates, most of the

entropy of the universe is generated at the stage of

reheating just after the inflation stops.

Therefore the potential of the inflaton should be tuned

in such a way that inflation occurs as long as possible.

Furthermore, if the Higgs field plays the role of inflaton,

the above analysis asserts that the SM parameters are

tuned such that the Higgs potential becomes flat at high

energy scale.

Flatness of the Higgs potential

Higgs potential with

the running coupling 4

4V

mH = 125.6 GeV

mHiggs =125.6 GeV

𝜑[GeV]

MPP

MEP

In wide classes of quantum gravity or string theory, the

low energy effective action has the multi local form:

We need to give a good definition of the path integral

for such action.

The fine tuning problem may be solved by the

dynamics of such action.

In the most optimistic case, the Big Fix occurs, and

all the low energy coupling constants would be

determined even if we do not know the detail of the

short distance physics.

eff .i i i j i j i jk i j k

i i j i jk

S c S c S S c S S S

Summary

Multi-local action from

IIB matrix model

IIB matrix model

2

Matrix

1 1, , .

4 2S Tr A A A

This is a candidate of non-perturbative

definition of string theory.

Formally, this is the dimensional reduction

of the 10D super YM theory to 0D.

Ishibashi, HK,

Kitazawa, Tsuchiya

Y. Kimura, M. Hanada and HK

The basic question :

In the large-N reduced model, a background

of simultaneously diagonalizable matrices

𝑨𝝁(𝟎)= 𝑷𝝁 corresponds to the flat space,

if the eigenvalues are uniformly distributed.

In other words, the background 𝑨𝝁(𝟎)= 𝒊𝝏𝝁

represents the flat space.

How about curved space?

Is it possible to consider some background

like

𝑨𝝁(𝟎)= 𝒊𝛁𝝁 ?

Covariant derivatives as matrices

Actually, there is a way to express the covariant

derivatives on any D-dim manifold by D matrices.

More precisely, we consider

𝑴: any D-dimensional manifold,

𝝋𝜶: a regular representation field on 𝑴.

Here the index 𝜶 stands for the components of

the regular representation of the Lorentz group

𝑺𝑶(𝑫 − 𝟏, 𝟏).

The crucial point is that for any representation 𝒓,its tensor product with the regular representation

is decomposed into the direct sum of the regular

representations:

.r reg reg regV V V V

In particular the Clebsh-Gordan coefficients for

the decomposition of the tensor product of the

vector and the regular representaions

vector reg reg regV V V V

are written as ,

( ) , ( 1,.., ).b

aC a D

Here 𝒃 and β are the dual of the vector and the

regular representation indices on the LHS.

(𝒂) indicates the 𝒂-th space of the regular

reprezentation on the RHS, and 𝜶 is its index.

Then for each 𝒂 (𝒂 = 𝟏. . 𝑫),

( )

b

a bC

is a regular representation field on 𝑴.

In other words, if we define 𝛁(𝒂) by

,

( ) ( ) ,b

a a bC

each 𝛁(𝒂) is an endomorphism on the space of the

regular representation field on 𝑴.

Thus we have seen that the covariant

derivatives on any D dimensional manifold

can be expressed by D matrices.

Therefore any D-dimensional manifold 𝑴with 𝑫 ≤ 𝟏𝟎 can be realized in the space

of the IIB matrix model as

0 ( ) , 1, ,,

0, 1, ,10

a

a

a DA

a D

where 𝛁(𝒂) is the covariant derivative on 𝑴

multiplied by the C-G coefficients.

Good point 1

Good points and bad points

Einstein equation is obtained at the classical level.

In fact, if we impose the Ansatz ( )a aA i

on the classical EOM , 0,a a bA A A

we have

( ) ( ) ( ), 0

0 , ,

, ( )

0 , 00 .

a a b

a a b

cd cd ca

a ab cd a ab cd ab c

a

d

a b ab b

c

a

R O R O R

R R R

Any Ricci flat space with 𝑫 ≤ 𝟏𝟎 is a

classical solution of the IIB matrix model.

Good point 2

Both the diffeomorphism and local Lorentz

invariances are manifestly realized as a part of

the 𝑺𝑼(𝑵) symmetry.

In fact, the infinitesimal diffeomorphism and

local Lorentz transformation act on 𝝋𝜶 as

𝝋 → 𝟏 + 𝝃𝝁𝝏𝝁 𝝋 and

𝝋 → 𝟏 + 𝜺𝒂𝒃𝑶𝒂𝒃 𝝋 , respectively.

Both of them are unitary because they preserve

the norm of 𝝋𝜶𝝋𝜶𝟐 = 𝒅𝑫𝒙 𝒈 𝝋𝜶∗𝝋𝜶

Bad points

Fluctuations around the classical solution

𝑨𝒂(𝟎)= 𝒊𝛁(𝒂)

1. contain infinitely many massless states.

2. Positivity is not guaranteed.

This can be seen by considering the

fluctuations around the flat space.

In this case the background is equivalent to

𝑨𝒂(𝟎)= 𝒊𝝏𝒂⊗𝟏𝒓𝒆𝒈 ,

where 𝟏𝒓𝒆𝒈 is the unit matrix on the space of

the regular representation.

A. Tsuchiya, Y. Asano and HK

Low energy effective action

We have seen that any D-dim manifold is

contained in the space of D matrices.

Therefore in principle the IIB matrix model

contains and describes the effects of the

topology change of the space-time.

It is interesting to consider the low energy

effective action of the IIB matrix model.

As we will see, we indeed obtain the multi-

local action.

0 .a a aA A

The multi-local action is the consequence of the well-

known fact that the effective action of a matrix model

contains multi trace operators.

Then we integrate over 𝝓 to obtain the low energy

effective action.

Here we assume that the background 𝑨 𝒂𝟎 contains

only the low energy modes, and 𝝓 contains the rest.

More precisely, we first decompose the matrices 𝑨𝒂into the background 𝑨 𝒂

𝟎 and the fluctuation 𝝓 :

Substituting the decomposition into the action of

the IIB matrix model, and dropping the linear

terms in 𝝓, we obtain

20 0 0 0

0

0

2

20

0

1

4

2 , , , 2 , ,

4 , , , fermion

,

.

a b a b a b a b b a

a b a

a

b

b

a b

S Tr

A A

A

A

A

A A

A

In principle, the 0-th order term 20 0

0 ( ) ( )

1,

4a bS Tr A A

can be evaluated with some UV regularization,

which should give a local action.

The one-loop contribution is obtained by the Gaussian

integral of the quadratic part.

Then the result is given by a double trace operator as

usual:

𝑾 = 𝑲𝒂𝒃𝒄⋯ , 𝒑𝒒𝒓⋯ 𝑻𝒓 𝑨𝒂𝟎𝑨𝒃𝟎𝑨𝒄𝟎⋯ 𝑻𝒓(𝑨𝒑

𝟎𝑨𝒒𝟎𝑨𝒓𝟎⋯)

The crucial assumption here is that both of the

diffeomorphism and the local Lorentz invariance are

realized as a part of the SU(N) symmetry.

Then each trace should give a local action that is

invariant under the diffeomorphisms and the local

Lorentz transformations:

1-loop

eff

1, ( ) ( ).

2

D

i j i j i i

i j

S c S S S d x g x O x

In the two loop order, from the planar

diagrams we have a cubic form of local

actions2-loop Planar

eff

, ,

1,

6i jk i j k

i j k

S c S S S

while non-planar diagrams give a local

action2-loop NP

eff .i i

i

S c S

z y

x

x

Similar analyses can be applied for higher loops.

the low energy effective theory of the IIB matrix

model is given by the multi-local action:

.)()(

,eff

xOxgxdS

SSScSScScS

i

D

i

kji

kji

kjij

ji

iji

i

ii

We have seen that

Although there is no precise correspondence, the

loops in the IIB matrix model resemble the

wormholes.

x

y

𝑥

𝑦 ≅

We may say that if the theory involves gravity and

topology change, its low energy effective action

becomes the multi-local action universally.

Review on IIB matrix model

1. Definition of IIB matrix model

The basic idea

For simplicity, we start with bosonic string.

In fact, in the Schild action, the worldsheet

can be regarded as a symplectic manifold,

and the action is given by the integration of

a quantity that is expressed in terms of the

Poisson bracket.

“Worldsheet of string has a structure of

phase space.”

This situation becomes manifest when we

express the string in terms of the Schild

action.

Basically, bosonic string is described by the

Nambu-Goto action

2 21, .

2

ab

NG a bS d X X

Schild action

The Nambu-Goto action is nothing but the area of

the worldsheet, which is expressed in terms of an

anti-symmetric tensor 𝚺𝝁𝝂 that is constructed from

the space-time coordinate 𝑿𝝁.

It is known that the Nambu-Goto action is

equivalent to the Schild action

2

2 2

Schild

1, ,

2 4 2S d g X X d g

: a volume density on the world shee

1, ,

t

ab

a bX Y

g

X Yg

which is nothing but the Poisson bracket if

regard the worldsheet as a phase space.

2

Schild

22

Schild

10

2

1.

2 2

ab

a b

ab

a b

S g X Xg

S d X X

The equivalence can be seen easily, by eliminating

𝒈 from the Scild action:

The crucial point is that the Schild action

has a structure of phase space.

21

, .2 4 2

X X

In fact it is given by the integration over the

phase space

of a quantity that is expressed in terme the

Poisson bracket

2d g

symplectic structure of the worldsheet

Not that we do not need Worldsheet metric, but

what we need is just the volume density 𝒈.

Matrix regularization

Then we want to discretize the worldsheet

in order to define the path integral.

2

-symm

function matrix

1, ,

1

etry ( )-symmet y

2

r

A B A Bi

d g A T

W U N

rA

A natural discretization of phase space is

the “quantization”.

If we quantize a phase space, it becomes

the state-vector space, and we have the

following correspondence:

Then the Schild action becomes

2

Matrix

1, 1 ,

4S Tr A A Tr

and the path integral is regularized like

Schild Matrix

1

exp exp .vol(Diff ) ( )n

dg dX dAZ iS iS

SU n

Here we have used the fact that the phase

space volume is diff. invariant and

becomes the matrix size after the

regularization.

Therefor the path integral over

becomes summation over .

2d g 1Tr n

g

n

Multi-string states

One good point of the matrix regularization is

that all topologies of the worldsheet are

automatically included in the matrix integral.

Disconnected worldsheets are also included as

block diagonal configurations as

Furthermore the sum over the size of the

matrix is automatically included, if the

worldsheet is imbedded in a larger matrix

as a sub matrix.

If we take this picture that all the worldsheets

emerge as sub matrices of a large matrix, the

second term of

2

Matrix

1, 1

4S Tr A A Tr

can be regarded as describing the chemical

potential for the block size.

Thus we expect that the whole universe is

described by a large matrix that obeys

21, .

4S Tr A A

This is nothing but the large-N reduced model,

which I will explain in a moment.

On the other hand, if we start from type IIB

superstring, we will get the reduced model for

supersymmetric gauge theory. In this case

eigenvalues do not collapse, and we can have

non-trivial space-time.

From the analyses of the large-N reduced

model, it is known that in this model the

eigenvalues collapse to one point, and it can

not describe an extended space-time.

This might be related to the instability of

bosonic string by tachyons.

The Large-N reduction

The basic statement is

“The large-N gauge theory with periodic

boundary condition does not depend on the

volume of the space-time.”

In particular, the theory in the infinite

space-time is equivalent to that on one point.

The space-time emerges from the internal

degrees of freedom of the reduced model.

Here I will summarize the notion of the large-N reduction briefly.

22

,4

dN

S Tr A A

the large-N reduced model

is equivalent to the d-

dimensional Yang-Mills

if the eigenvalues of 𝑨𝝁are uniformly distributed.

However, it is not automatically realized.

It is known that the eigenvalues collapse

to one point unless we do something.

In fact by analyzing the planar Feynman diagrams, we can show that

Strong coupling

If the coupling is sufficiently strong,

the quantum fluctuation may overwhelm

the attractive force.

It actually happens at least for the lattice

version of the reduced model.

† †

reduced

1

, .d

c

NS Tr U U U U

Actually there are several ways to make

the eigenvalues distribute uniformly.

quenching

We constrain the diagonal elements of 𝑨𝝁to a uniform distribution by hand

(𝑨𝝁)𝒊𝒊= 𝒑𝝁(𝒊)

.

Then the perturbation series reproduce

that of the D-dimensional gauge theory.

However, this is rather formal, and the

gauge invariance is no longer manifest.

A lattice version of the quenching that

keeps the manifest gauge invariance was

proposed, but now it is known that it does

not work. Bhanot-Heller-Neuberger, Gross-Kitazawa

twisting

(0) ˆ

ˆ ˆ, ( ),

A p

p p i B B

the theory is equivalent to gauge theory

in a non-commutative space-time.

If we expand around the non-

commutative back ground

A

Gonzalez-Arroyo, Korthals Altes (‘83)

Because the equation of motion of the

reduced model is given by

, , 0,A A A

the non-commutative back ground

is a classical solution.

But it is not the absolute minimum of the

action.

One way to make it stable is to modify the

model to

22

, .4

dN

S Tr A A i B

The lattice version of this is called the

twisted reduced model:

† †

reduced

1

.d

iNS e Tr U U U U

Gonzalez-Arroyo, Okawa

Several MC analyses have been made on the

twisted reduced model, and they found some

discrepancy from the infinite volume theory,

which is related to the UV-IR correspondence.

Heavy adjoint fermions

They have introduced

additional heavy adjoint

fermions.

Kovtun-Unsal-Yaffe (2007),Bringoltz-Sharpe (2009),Poppitz, Myers, Ogilvie, Cossu, D’Elia, Hollowood, Hietanen, Narayanan,Azeyanagi, Hanada, Yacobi

Then the collapse of the eigenvalues can

be avoided without changing the long

distance physics.

Schild action of IIB string

We consider the Schild action of the type

IIB superstring.

Green-Schwarz action

2 2

1 1 2 2

1 1 2 2

1 1 2 2

1

2

,

,

GS

ab

a b b

ab

a b

ab

a b

a a a a

S d

i X

X i i

1 2

( 0 ~ 9)

, : 10D Mayorana-Weyl

X

κ-symmetry

1 1

2 2

1 1 2 2

1 1

2 2

2

1

1 ,

1

12

2

X i i

N=2 SUSY1 1

SUSY

2 2

SUSY

1 1 2 2

SUSY X i i

Gauge fixing for the κ-symmetry 1 2

2 212 ,

2

.

ab

GS a b

ab

a b

S d i X

X X

N=2 SUSY should be combined with 𝜿 symmetry

so that the gauge condition is maintained1 1 1

SUSY

2 2 2

SUSY

1 2

SUSY

1 21

1 22

2

2

X X X

N=2 SUSY becomes simple if we consider a

combination

1

2

1

2

2

1

12

2

0.

X i

X

1 2

1 2

2

.2

Then we have the following simple form:

Schild action

N=2 SUSY

Schild

22 21

, , ,2 4 2 2

S

id g X X X d g

1

1

2

2

1,

2

0

X X

X i

X

Everything is

written in terms

of the Poisson

bracket.

Then we can convert the action to the Schild

action as in the case of bosonic string:

Matrix regularization

1

1

2

2

1

2

0

F

A i

A

2

Matrix

1 1, , 1 .

4 2S Tr A A A Tr

N=2 SUSY

Applying the matrix regularization, we have

,F i A A

IIB matrix model

2

Matrix

1 1, , .

4 2S Tr A A A

Drop the second term, and consider large-N

Formally, this is the dimensional reduction

of the 10D super YM theory to 0D.

A good point is that the N=2 SUSY is

maintained after the discretization.

IIB matrix model

Ishibashi, HK,

Kitazawa, Tsuchiya

N=2 SUSY

One of the N=2 SUSY is nothing but the

supersymmety of the 10D super YM theory.

1

1

1

2F

A i

2

Matrix

1 1, ,

4 2S Tr A A A

Even so, they form non trivial N=2 SUSY:

2

20A

The other one is almost trivial.

1 1 2 2 1 2, 0, , 0, , .Q Q Q Q Q Q P

IIB matrix model is nothing but the

dimensional reduction of the 10D super YM

theory to 0D.

The other matrix modelscomment

It is natural to think about the other possibilities.

In fact, they have considered the dimensional

reduction to various dimensions:

0D ⇒ IIB matrix model

1D ⇒Matrix theory

2D ⇒Matrix string

4D ⇒AdS/CFT

Motl, Dijkgraaf-Verlinde-Verlinde

de Witt-Hoppe-Nicolai,Banks-Fischler-Shenker-Susskind

From the viewpoint of the large-N reduction,

they are equivalent if we quench the diagonal

elements of the matrices.

However the dynamics of the diagonal elements

are rather complicated.

At present the relations among them are not

well-understood.comment end

Open questions

We expect that the IIB matrix model

)],[2

1],[

4

1(

1 2

2 AAATr

gS

gives a constructive definition of superstring.

However there are some fundamental

open questions:

Is an infrared cutoff necessary?

How the large-N limit should be taken?

How does the space-time emerge?

Does diff. invariance exist rigorously?

2. Ambiguities in the definition

• Euclidean or Lorentzian• Necessity of the IR cutoff• How to take the (double) scaling limit

Euclidean or Lorentzian?

In general, systems with gravity do not

allow a simple Wick rotation, because the

kinetic term of the conformal mode (the

size of the universe) has wrong sign.

On the other hand, the path integral of the

IIB matrix model seems well defined for the

Euclidean signature, because the bosonic

part of the action is positive definite:

2

2

1 1[ , ] .

24

1[ , ]S A AA TrTr

g

How about Lorentzian signature?

If we simply apply the analytic continuation

the path integral becomes unbounded:

0 10 ,A iA

2

0 2 2

2

exp ,

1 1 1[ , ][ , ] [ , ]

4 2

1 1 1[ , ] [ , ] .

2 4

M

M

i i j

Z dAd S

S Tr A A A A Tr Ag

Tr A A A Ag

From the point of view of the large-N

reduction, it is natural to take

exp .MZ dAd i S

Is IR cutoff necessary?

Because of the supersymmetry the force between eigenvalues cancels between bosons and fermions

2

(1 ) ( ) ( )

,

2 log 0loop i j

eff F

i j

S D d p p

It seems that we have to impose an infrared cutoff by hand to prevent the eigenvalues from running away to infinity.

eigenl A l

(1)

( )N

p

A

p

But there is a subtlety.

Because the diagonal elements of fermions are

zero modes of the quadratic part of the action,

we should keep them when we consider the

effective Lagrangian.

The one-loop effective Lagrangian for the

diagonal elements is given by

(1)

( )N

p

A

p

1

N

4 8

, ,1-loop

eff

2, 2,

, ,4 8

.

i j i j

i j

i ji j i j

i ji j

S SS x tr

p pS

p p

Aoki, Iso, Kitazawa, Tada, HK

Because of the fermionic degrees of freedom, there appears a weak attractive force between the eigenvalues, and at least the partition function becomes finite. However it is not clear whether all the correlation functions are finite or not.

Austing and Wheater,Krauth, Nicolai and Staudacher,Suyama and Tsuchiya,Ambjorn, Anagnostopoulos, Bietenholz, Hotta and Nishimura,Bialas, Burda, Petersson and Tabaczek,Green and Gutperle,Moore, Nekrasov and Shatashvili.

1-loop(1 loop) 16 ( )

effexp ,i

i

Z p d S p

( , )i jS i j

( , ) 0, 8n

i jS n

Since is quadratic in ,

which has only 16 components, we have

and

4 8

, ,1-loop

eff

4 8

( , ) ( , )

exp , exp4 8

1 tr tr .

i j i j

i j

i j i j

i j

S SS p tr

a S b S

Let’s estimate the order of this interaction.

We first integrate out the fermionic variables

4 8

( , ) ( , )1, tr , tri j i ja S b S

16 ( )

1

Ni

i

d

(1 loop)

various terms

exp .

i j i jZ p f p p f p p

O N

Therefore, for each pair of i and j we have 3

choices

which carry the powers of

0, 8, 16 respectively.

Therefore the number of factors other than 1

should be less than or equal to 2N, and we

can conclude that the effective action induced

from the fermionic zero modes is of 𝑶 𝑵 :

.

On the other hand, we have 16N dimensional

fermionic integral .

2N

.

This should be compared to the bosonic case

2(1 loop)

2

exp 2 log

exp ( )

i j

i j

Z p D p p

O N

SUSY reduces the attractive force by at least

a factor 1/N.

So, in the naïve large-N limit, simultaneously

diagonal backgrounds are stable.

However, it is not clear what happens in the

double scaling limit.

How to take the large-N limit

In the IIB matrix model , we usually regard

A as the space-time coordinates.

)],[2

1],[

4

1(

1 2

2 AAATr

gS

gSo, has dimensions of length squared.

How is the Planck scale expressed?

If it does not depend on the IR cutoff l, as

we normally guess, we should have1

2Planck .l N g

In other words, we should take the large-N

limit keeping this combination finite.

?

At present we have no definite answer.

3. Meaning of the matrices and

Emergence of the space-time

What do the matrices stand for?

If we regard the IIB matrix model

)],[2

1],[

4

1(

1 2

2 AAATr

gS

as the matrix regularization of the Schild

action, Aμ are space-time coordinates.

On the other hand if we regard it as the

large-N reduced model, the diagonal

elements of Aμ represent momenta.

It is not a priori clear how the space-time

emerges from the matrix degrees of

freedom.

0ˆ 1 , 0, ,3

.0, 4, ,9

kpA

Here satisfy the CCR’sp̂

Another interesting possibility is to consider a

non-commutative back ground such as

ˆ ˆ, ( ),p p i B B

There are many possibilities to realize the

space-time.

and is the unit matrix.

Then we have a 4D noncommutative flat

space with SU(k) gauge theory.

1k k k

Actually various models that are close to the

standard model can be constructed by choosing

an appropriate background.

“Intersecting branes and a standard model realization

in matrix models.”

A. Chatzistavrakidis, H. Steinacker, and G. Zoupanos.

JHEP09(2011)115

(ex.)

“An extended standard model and its Higgs geometry

from the matrix model,”

H. Steinacker and J. Zahn,

PTEP 2014 (2014) 8, 083B03

Expandind universeS.-Kim, J. Nishimura, A. Tsuchiya

Phys.Rev.Lett. 108 (2012) 011601

can be regarded as

an expanding 3+1

D universe.

(ex.)

Expandind universe 2

Recently another interesting picture of the expanding

universe has been obtained by considering fuzzy

manifolds.

(ex.)

“Quantized open FRW cosmology from Yang-Mills matrix models”

H. Steinacker,

Phys.Lett. B782(2018) 176-180

“The fuzzy 4-hyperboloid 𝑯𝒏𝟒 and higher-spin in Yang-Mills

matrix models”

Marcus Sperling, H. Steinacker,

arXiv:1806.05907

4. Diffeomorphism invarianceand

Gravity

Diff. invariance and gravity

Because we have exact N=2 SUSY, it is

natural to expect to have graviton in the

spectrum of particles.

Actually there are some evidences.

(1) Gravitational interaction appears from

one-loop integral.

(2) Emergent gravity by Steinacker. Gravity

is induced on the non-commutative back

ground.

However, it would be nicer, if we can

understand how the diffeomorphism

invariance is realized in the matrix model.

Y. Kimura, M. Hanada and HK

The basic question :

In the large-N reduced model, a background

of simultaneously diagonalizable matrices

𝑨𝝁(𝟎)= 𝑷𝝁 corresponds to the flat space,

if the eigenvalues are uniformly distributed.

In other words, the background 𝑨𝝁(𝟎)= 𝒊𝝏𝝁

represents the flat space.

How about curved space?

Is it possible to consider some background

like

𝑨𝝁(𝟎)= 𝒊𝛁𝝁 ?

Covariant derivatives as matrices

Actually, there is a way to express the covariant

derivatives on any D-dim manifold by D matrices.

More precisely, we consider

𝑴: any D-dimensional manifold,

𝝋𝜶: a regular representation field on 𝑴.

Here the index 𝜶 stands for the components of

the regular representation of the Lorentz group

𝑺𝑶(𝑫 − 𝟏, 𝟏).

The crucial point is that for any representation 𝒓,its tensor product with the regular representation

is decomposed into the direct sum of the regular

representations:

.r reg reg regV V V V

In particular the Clebsh-Gordan coefficients for

the decomposition of the tensor product of the

vector and the regular representaions

vector reg reg regV V V V

are written as ,

( ) , ( 1,.., ).b

aC a D

Here 𝒃 and β are the dual of the vector and the

regular representation indices on the LHS.

(𝒂) indicates the 𝒂-th space of the regular

reprezentation on the RHS, and 𝜶 is its index.

Then for each 𝒂 (𝒂 = 𝟏. . 𝑫),

( )

b

a bC

is a regular representation field on 𝑴.

In other words, if we define 𝛁(𝒂) by

,

( ) ( ) ,b

a a bC

each 𝛁(𝒂) is an endomorphism on the space of the

regular representation field on 𝑴.

Thus we have seen that the covariant

derivatives on any D dimensional manifold

can be expressed by D matrices.

Therefore any D-dimensional manifold 𝑴with 𝑫 ≤ 𝟏𝟎 can be realized in the space

of the IIB matrix model as

0 ( ) , 1, ,,

0, 1, ,10

a

a

a DA

a D

where 𝛁(𝒂) is the covariant derivative on 𝑴

multiplied by the C-G coefficients.

Good point 1

Good points and bad points

Einstein equation is obtained at the classical level.

In fact, if we impose the Ansatz ( )a aA i

on the classical EOM , 0,a a bA A A

we have

( ) ( ) ( ), 0

0 , ,

, ( )

0 , 00 .

a a b

a a b

cd cd ca

a ab cd a ab cd ab c

a

d

a b ab b

c

a

R O R O R

R R R

Any Ricci flat space with 𝑫 ≤ 𝟏𝟎 is a

classical solution of the IIB matrix model.

Good point 2

Both the diffeomorphism and local Lorentz

invariances are manifestly realized as a part of

the 𝑺𝑼(𝑵) symmetry.

In fact, the infinitesimal diffeomorphism and

local Lorentz transformation act on 𝝋𝜶 as

𝝋 → 𝟏 + 𝝃𝝁𝝏𝝁 𝝋 and

𝝋 → 𝟏 + 𝜺𝒂𝒃𝑶𝒂𝒃 𝝋 , respectively.

Both of them are unitary because they preserve

the norm of 𝝋𝜶𝝋𝜶𝟐 = 𝒅𝑫𝒙 𝒈 𝝋𝜶∗𝝋𝜶

Comment: The norm is not positive definite for

Lorentzian theory.

Bad points

Fluctuations around the classical solution

𝑨𝒂(𝟎)= 𝒊𝛁(𝒂)

1. contain infinitely many massless states.

2. Positivity is not guaranteed.

This can be seen by considering the

fluctuations around the flat space.

In this case the background is equivalent to

𝑨𝒂(𝟎)= 𝒊𝝏𝒂⊗𝟏𝒓𝒆𝒈 ,

where 𝟏𝒓𝒆𝒈 is the unit matrix on the space of

the regular representation.

Because the unit matrix 𝟏𝒓𝒆𝒈 is infinite

dimensional, we have infinite degeneracy, and

in particular we have infinitely many massless

states. → bad point 1

In general, the regular representation contains

infinite tower of higher spins, and we have

many negative norm states. It is not clear

whether we have sufficiently many symmetries

to eliminate those negative norm states.

→ bad point 2

One possible way out is to consider a non-

commutative version.

What we have done is to regard the matrices

as endomorphisms on the space of the regular

representation fields.

It is easy to show that this space is equivalent

to the space of the functions on the frame

bundle of the spin bundle.

If we can construct a non-commutative version

of such bundle, we can reduce the degrees of

freedom significantly without breaking the

diffeomorphism and local Lorentz invariance.

5. Topology change of the space-time and

Low energy effective action

A. Tsuchiya, Y. Asano and HK

Low energy effective action

We have seen that any D-dim manifold is

contained in the space of D matrices.

Therefore in principle the IIB matrix model

contains and describes the effects of the

topology change of the space-time.

As was pointed out by Coleman some years

ago, such effects give significant corrections

to the low energy effective action.

It is interesting to consider the low energy

effective action of the IIB matrix model.

.)()(

,eff

xOxgxdS

SSScSScScS

i

D

i

kji

kji

kjij

ji

iji

i

ii

Actually we can show that if we integrate out the heavy

states in the IIB matrix model, the remaining low

energy effective action is not a local action but has a

special form, which we call the multi-local action:

Here 𝑶𝒊 are local scalar operators such as

𝟏 , 𝑹 , 𝑹𝝁𝝂𝑹𝝁𝝂 , 𝑭𝝁𝝂 𝑭

𝝁𝝂, 𝝍𝜸𝝁𝑫𝝁𝝍 ,⋯ .

𝑺𝒊 are parts of the conventional local actions.

The point is that 𝑺𝐞𝐟𝐟 is a function of 𝑺𝒊’s.

𝑺𝐞𝐟𝐟 is not local, but it is not completely non-local

in the sense that it is a function of local actions.

0 .a a aA A

This is essentially the consequence of the well-

known fact that the effective action of a matrix

model contains multi trace operators.

Then we integrate over 𝝓 to obtain the low energy

effective action.

Here we assume that the background 𝑨 𝒂𝟎 contains

only the low energy modes, and 𝝓 contains the rest.

We also assume that this decomposition can be

done in a SU(N) invariant manner.

More precisely, we first decompose the matrices 𝑨𝒂into the background 𝑨 𝒂

𝟎 and the fluctuation 𝝓 :

Substituting the decomposition into the action of

the IIB matrix model, and dropping the linear

terms in 𝝓, we obtain

20 0 0 0

0

0

2

20

0

1

4

2 , , , 2 , ,

4 , , , fermion

,

.

a b a b a b a b b a

a b a

a

b

b

a b

S Tr

A A

A

A

A

A A

A

In principle, the 0-th order term 20 0

0 ( ) ( )

1,

4a bS Tr A A

can be evaluated with some UV regularization,

which should give a local action.

The one-loop contribution is obtained by the Gaussian

integral of the quadratic part.

Then the result is given by a double trace operator as

usual:

𝑾 = 𝑲𝒂𝒃𝒄⋯ , 𝒑𝒒𝒓⋯ 𝑻𝒓 𝑨𝒂𝟎𝑨𝒃𝟎𝑨𝒄𝟎⋯ 𝑻𝒓(𝑨𝒑

𝟎𝑨𝒒𝟎𝑨𝒓𝟎⋯)

The crucial assumption here is that both of the

diffeomorphism and the local Lorentz invariance are

realized as a part of the SU(N) symmetry.

Then each trace should give a local action that is

invariant under the diffeomorphisms and the local

Lorentz transformations:

1-loop

eff

1, ( ) ( ).

2

D

i j i j i i

i j

S c S S S d x g x O x

In the two loop order, from the planar

diagrams we have a cubic form of local

actions2-loop Planar

eff

, ,

1,

6i jk i j k

i j k

S c S S S

while non-planar diagrams give a local

action2-loop NP

eff .i i

i

S c S

z y

x

x

Similar analyses can be applied for higher loops.

the low energy effective theory of the IIB matrix

model is given by the multi-local action:

.)()(

,eff

xOxgxdS

SSScSScScS

i

D

i

kji

kji

kjij

ji

iji

i

ii

We have seen that

This reminds us of the theory of baby universes

by Coleman.

Although there is no precise correspondence, the

loops in the IIB matrix model resemble the

wormholes.

x

y

𝑥

𝑦 ≅

Probably this phenomenon occurs quite universally.

We may say that if the theory involves gravity and

topology change, its low energy effective action

becomes the multi-local action.

End (review on IIB matrix model)

3. Probabilistic interpretationof

multiverse wave function

Probabilistic interpretation (1)

postulate

T : age of the universe

2

probability of finding a universe of size z dz z

0Ez z

2

0

1

( )

1

Edz z dzz p

z d

z

dz

T

21

2

HH z p z zp

p

0

1exp

( )

z

E z i dz p zz p z

meaning of this measure

z

T

2 2

z dz dT ⇒

2 probability of a universe emerging in unit time

the time that has passed

after the universe is created

Probabilistic interpretation (2)

is a superposition of the universe with various age,

z

T

+

z

T

z

T+ +…

T T dTgives the probability of finding a

universe of age .

2 2z dz dT

infrared cutoffWe introduce an infrared cutoff

for the size of universes.

IRz

ceases to exist

bounces back

or

2 2 2

life time of the universez dz dT

dimensionless

∞finite

Lifetime of the universe

Wave Function of the multiverse (1)

Multiverse appears naturally in quantum gravity / string theory.

matrix model

・

b

b

aC

,

)(

b

b

aC

,

)(

Each block

represents

a universe.

・

quantum gravity

Okada, HK

Wave Function of the multiverse (2)

The multiverse sate is a superposition of N-verses.

multi

0

,N

N

d w

1 1, , , , ,N N Nz z z z

, ,

multi

N

d w

exp i i

i

Z d w d i S

Wave Function of the multiverse (3)

Probabilistic interpretation

multi

0

,N

N

d w

1 1, , , , ,N N Nz z z z

1 1 1, , N N Nz z dz z z dz

the probability of finding N universes with size

2 2

1 1, , ,N N Nz z dz dz w d

and finding the coupling constants in

.d

represents

Probability distribution of

1 1, , , , ,N N Nz z z z

2 21

1

0

2 2

2 2

, , ,!

exp ,

exp

NN N

N

dz dzP z z w

N

dz z w

w

2

0 , (life time of the universe)Edz z

0E

is chosen in such a way that is maximized,

can be very large.

irrespectively of .w

If we accept the probabilistic interpretation of the

multiverse wave function, the coupling constants

are chosen in such a way that the lifetime of the

universe becomes maximum.

WKB sol with

Cosmological constant

Λ～curvature～energy density

(extremely small)

What value of Λ maximizes

0

?

S3 topology

The cosmological constant in the far future is predicted to be very small.

IRz

0

1,

,E z

z p z

, 2 ( ).p z U z

assuming all matters decay to radiation

2

0 ,Edz z

0 cr0 cr cr

The other couplings (Big Fix)

2 2

expP w

The exponent is divergent, and regulated by the IR cutoff :

2

0 rad0

cr

1, log .

IR

E IR

z

dz z C zz

cr rad1/C

2

0 ,Edz z

MEP

are determined in such a way that is maximized.Again we have MEP.

radC

assuming all matters decay to radiation