Universal Non-Perturbative Effect in Quantum Gravity / String - - PowerPoint PPT Presentation

Universal Non-Perturbative Effect in Quantum Gravity / String Theory

• Dec. 12, 2018

at Osaka City Univ. Hikaru Kawai (Kyoto Univ.)

Two great discoveries by Prof. Nambu

Spontaneous Symmetry Breaking ⇒ Higgs mechanism ⇒ Standard Model String theory ⇒ Theory including Quantum Gravity ⇒ Theory of Everything

In these six years, 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.

Higgs potential with the running coupling

   

4

4 V     

mH mH = 1 125. 5.6 6 GeV

mHiggs =125.6 GeV

𝜒[GeV]

I will discuss the possibility that such behavior of the SM and Higgs field is understood as a consequence

• f 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

g G m

1 2 3 2

, , , .

S H S

g g g g g 

2 2 2

.

H S S

g m m 

2 4

.

S s

m g    

 

 

2 2 2 2 2 17

100GeV 10 GeV

H S S

m g m

 

 

4 4 4 17

2 ~ 3meV 10 GeV

S

m  unnatural ! → unnatural ! !→

tree

(cont’d)

The real values of the cosmological constant and Higgs mass are very unnatural: If we consider everything in the ordinary field theory, they are merely free parameters. In principle we can give any values to them by hand, but still we need fine tunings in

• rder 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 . H

m M

+ ⇒

0.  

⇒

2

0.

H

m 

⇒

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

• bserved in the LHC below 2 TeV.

We have to think about other possibilities.

Possible atitudes to the naturalness problem

• ther 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 Lagrangian:

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 universe.

• 3. The parameters are fixed by some mechanisms

that are not in the ordinary field theory.

I will discuss 3, and show that there is a

universal non-perturbative effect in quantum gravity / string theory that provides a mechanism to tune the parameters automatically.

(cont’d)

Nature does fine tunings!

Low energy effective theory

• f

QG / string theory

Co Cons nsider der the he low w ene nergy rgy effec ecti tive ve the heory ry of qu quan antum um grav avity ity / st string ing the heory ry obt btained ained af after er int ntegra gratin ting out ut the he sh short t di dist stance ance fluc uctuatio uations. ns.

 

eff

gauge matter ...

D

S d x g R       



Us Usua ually y pe people ple be believe ieves s tha hat it sh shoul uld d be be a a local al ac action:

• n:

Is Is it true rue?

Basic question

As As we we wi will di disc scuss, uss, the he low w ene nergy rgy effect ective ve the heory ry ma may ha have the he mu multi-local local form:

 

eff 1 2

, , , ( ) ( ).

i i i j i j i jk i j k i i j i jk D i i

S f S S c S c S S c S S S S d x g x O x      

   

He Here 𝑷𝒋 ar are gau auge e inv nvar arian iant t sc scala alar r local al

• pe

perator rators su such h as as 𝟐 , 𝑺 , 𝑺𝝂𝝃𝑺𝝂𝝃 , 𝑮𝝂𝝃 𝑮𝝂𝝃, 𝝎𝜹𝝂𝑬𝝂𝝎 , ⋯ . Pr Probab bably ly, , it is not s not su sufficient cient if we we tak ake the he fluc uctua uatio tion n of sp spac ace-tim time e topo pology logy int nto ac accou

• unt

nt.

Star art t wi with h Eu Euclidean dean pa path h int ntegral ral tha hat inv nvolve lves s the he sum ummation ion over r topo pologie logies and nd cons nsider ider the he low w ene nergy rgy effect ective ive the heory ry

• btaine

ined d after r int ntegrati grating ng out ut the he sh short- distanc nce e conf nfigu gurat ration ions. s.

 

 

topology

exp , dg S 

 

One ne way to un understa tand nd suc uch ac h action

• n is

to cons nsider ider the he sp spac ace-tim time e wo wormh mhole

• le like

ke conf nfig igur uratio ations ns as as Co Colem eman an di did. d.

Why multi-local ?

Am Among ng su such h sh short-dista distance nce conf nfig iguratio urations ns the here e sh shoul uld d be be a a sp spac ace-time ime wo wormh mhole

• le-like

like conf nfig igurat uration ion in wh n which h a t a thi hin n tub ube conn nnects cts two wo po point nts s on n the he sp spac ace-time.

• ime. He

Here, re, the he two wo po point nts s ma may be belon

• ng

g to either her the he sa same me un universe rse

• r di

different rent un universes. ses.

Do Do no not co conf nfuse use wi with h 3D D wo worm rmhole. hole. We ar are co consi nsideri dering ng 4D D co conf nfiguration iguration suc uch h tha hat a baby by un univ iverse rse is is cr created ated and nd absorbed. sorbed.

If we we se see su such h wo wormho mhole le-like like conf nfigu gurat ration ion from m the he si side de of the he lar arge e un universe(s), se(s), it looks ks like ke two wo sm smal all pu punc nctur ures es. . Bu But th the effect ct of ma maki king ng a a sm smal all pu punc ncture ure is s equ quivalent alent to an an ins nsertio rtion n of a l a local al ope perator rator. . Fu Furthe herm rmore,

• re, if we

we int ntegrate grate over er the he gau auge e field d an and t d the he me metric ric on t n the he tub ube, , eac ach h

• pe

perator rator is pr s projec jected ted to a a gau auge e inv nvar ariant iant sc scal alar ar ope perator rator.

in integrate 𝑩𝝂, 𝒉𝝂𝝃

 

 

4 4 ,

( ) ( ) ( ) ( ) exp .

i j i j i j

c d x d y g x g y O x O dg S y 

  

Th Therefo efore, re, a a wo wormho mhole le cont ntribut ributes es to the he pa path h int ntegral ral as as

x y

𝒅𝒋𝒌 ar are nu numb mbers s tha hat ar are de determ ermine ined d by by the he pa path h int ntegral ral over er the he tub ube, , an and e d expe pecte cted d to be be

• f orde

der r 1 1 in t n the he Pl Plan anck k uni unit. He Here 𝑷𝒋 ar are gau auge e inv nvar arian iant t sc scala alar r local al

• pe

perator rators s su such h as as 𝟐 , 𝑺 , 𝑺𝝂𝝃𝑺𝝂𝝃 , 𝑮𝝂𝝃 𝑮𝝂𝝃, 𝝎𝜹𝝂𝑬𝝂𝝎 , ⋯ .

𝒚 and nd 𝒛 are re po positi itions

• ns in

in the he la larg rge e un univ ivers rse(s). e(s).

If If we we sum um over r the he nu number ber of wo worm rmholes, holes, we we ha have  

4 4 ,

( ) ( ) ( ) ( ) exp .

i j i j i j

c d x d y g x g y O x g y S O d        

  

4 4 , 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 y n c d x d y g x g y O x O y

 

            

    

The he wo worm rmholes holes induce the “bi-local” act ction: ion:

bi bifur urca cated ted wo wormho mholes les ⇒ cubic terms, quartic terms, …

. ) ( ) ( ,

eff

x O x g x d S S S S c S S c S c S

i D i k j i k j i k j i j j i i j i i i i

   

     

“ The low energy effective theory of quantum grav avity ity tha hat inv nvolves lves topo pology logy cha hang nge is no s not si simp mply y local al bu but mu multi-local local.” Th Thus us we we ha have se seen n In n sh short, t, “topology change induces multi-loca local l terms ms in the low energy effective theory.”

In n the he cas ase e of Co Colem eman, an, he he cons nside idered red the he wo wormh mhole

• le as

as a a class assical ical so solutio ution of the he Eu Euclidea dean n 4D 4D the heory ry like ke ins nstan anton

• n.

Co Colem eman an ha had e d ess ssent entially ally the he sa same me resu sult lt in n 19 1989 89. .

Comment on Coleman’s

He Here e we we ar are consi nsiderin dering g a W a Wilsoni sonian an like ke effect ective ve the heory ry. . We int ntegra grate te over r wo wormho mhole le-like like conf nfig igurat uration ions s no no ma matter er wh whethe her r the hey ar are clas assical sical so solut ution ions s or no not.

We ha have se seen n tha hat we we ha have the he mu multi-local local ac action ion irresp espect ectively ively to the he existen stence ce of the he clas assical sical so solut ution ion. Pr Probab bably ly, , the heorie

• ries

s wi with h thi his s pr prope perty rty ar are mo more na natur ural al tha han t n tho hose se wi witho hout ut it. Th This is s is be becau ause se we we ha have to int ntrod

• duce

uce ad addi dition ional al cond nditio itions ns on t n the he pa path h int ntegra gral l if

• ne

ne wa want nts s to fix the he topo pology logy of the he sp spac ace-ti time. me. Wha hat we we ne need d is on s only the he pr prope perty rty tha hat topo pology logy cha hang nge of sp spac ace-time ime is inc s includ uded ed in n the he the heory

• ry.

Al Also so st string ng the heory ry se seems ms to cont ntain ain topo pology logy cha hang nge of sp spac ace-time ime au automa matica tically lly. In n thi his se s sens nse e the he mu multi-local local ac action

• n can

an be be regarded arded as as a un a universal sal form m of the he low w ene nergy gy effect ective ve the heory ry of qu quan antum um gr grav avit ity y / st string ing the heory ry.

for the he effect ective ive ac action ion

eff

, ( ) ( ),

i i i j i j i jk i j k i i j i jk D i i

S c S c S S c S S S S d x g x O x     

   

 

 

eff

exp Z d S   



We wa want nt to do do the he pa path i h int ntegral gral

Consequences of multi-local action

wh where e 𝒅𝒋 , 𝒅𝒋𝒌 , 𝒅𝒋𝒌𝒍 ⋯ ar are cons nsta tant nts s of O( O(1) 1) in n the he Pl Plan anck k un unit.

we we int ntrod

• duce

uce the he La Lapl plac ace e tran ansfo sform rm

 

    



eff

exp exp .

i i i

Z d S d w d S               

   

Co Coup upling ng cons nstant tants s ar are no not me merely ely cons nsta tant nts, s, bu but the hey sh shoul uld d be be int ntegra grated. ted.

 

 

 

eff 1 2 1 2

exp , , , , exp .

i i i

S S S d w S             

 

Th Then t n the he pa path i h int ntegral gral be becomes

• mes

Re Regar ardin ding g 𝑻𝐟𝐠𝐠 as as a f a fun unction

• n of 𝑻𝒋’s

 

eff eff 1 2

, , , S S S S 

pat ath integ egra ral l for r a lo a local al ac action

• n 𝝁𝒋 𝑻𝒋.

𝝁𝒋 ar are coupling pling consta stants nts of the ac action.

• n.

       

eff

exp ( ), ( ) exp .

i i i

Z d S d w Z Z d S                  

   

Thi his s cont ntradic radicts ts wi with h our ur expe perience rience. We kno know co w coup upling ing cons nstant tants s ar are jus ust cons nsta tant nts, s, an and we d we ne never r obse bserve rve a a su supe perposit position ion of di differe erent nt coup upling ing cons nsta tant nts. s. We ha have se seen n tha hat the he pa partit ition ion fun unction

• n is

s given en by by av averaging aging 𝒂(𝝁) with h weight ght 𝒙(𝝁). Ho Howe wever ver, , the he pr proble blem m is re s reso solved lved if 𝒂(𝝁) ha has s a s a sha harp p pe peak ak ar aroun und d so some me val alue ue 𝝁~𝝁(𝟏). . In n tha hat cas ase e we we can an sa say tha hat the he na natur ure e itse self lf tun une the he pa param amete eters rs to to 𝝁(𝟏).

An An in interes restin ting g po point nt of thi his sc s scenario nario is th s that at 𝒙(𝝁) is no s not imp mporta rtant nt if it is a sm s a smooth th fun unction

• n of 𝝁.

We ar are no not su sure e if it it is th s the cas ase e or no not, , bu but it it is po s poss ssible ible be because ause exp(−Seff) ma may be be a we a well- be beha haved ed fun unction

• n of 𝑻𝒋’s, and so is its Laplace

tran ansfo sform rm. In n tha hat cas ase e the he so so cal alled ed Bi Big Fix occurs curs: Al All the he coup upling lings s of the he low w ene nergy gy effect ective ive the heory ry ar are fixed d by by the hems mselve elves, s, an and w d we do do no not ne need d pr precise cise inf nformati rmation

• n ab

about ut the he sh short rt di dist stan ance ce ph physi sics. cs.

Ne Necessi essity ty of mu multiverse verse Ev Even n if we we st star art t from m a c a conn nnect ected ed un universe, rse, we we ha have di disc scon

• nne

necte cted d un universes rses af after r int ntegratin rating

• ut

ut the he sh short rt-dist distan ance ce fluc uctua uatio tions. ns.

   exp

i i i

Z d w d S            

  

   

 

single single

1 ( ) ! exp ( ) .

n n

d w Z n d w Z      

 

 

  

n

Th Therefo efore re we ha have to su sum ov

• ver

r the he nu numbe ber of uni universes rses in t n the he evalu aluat ation ion of the he pa partition tion fun unction

• n.

He Here e 𝒂𝐭𝐣𝐨𝐡𝐦𝐟 𝝁 is s the he pa partitio ition n fun unction

• n for a

a si sing ngle e un universe. se.

“solution” to to th the c cos

• smol
• log
• gic

ical al con

• nst

stan ant t pr prob

• ble

lem

 

 

 

single(

) exp . Z dg gR g    

  

4

S r

 

 

 

2 4

exp exp 1/ , no solution, dr r r            



r S

1  

domi minates nates irresp espective ectively y of f



 .

w 

Fo For si simp mplici icity ty, , we we ke keep p onl nly the he cosmo smologic logical al cons nsta tant nt 𝚳. . If the he un universe se is la s large, e, 𝒂𝐭𝐣𝐨𝐡𝐦𝐟(𝚳) is s evalu aluat ated ed by by cons nside idering ring 𝑻𝟓 geo eome metry try class assicall ically: y:  

 

single

exp ( ) . Z d w Z    



Ho Howe wever ver thi his arg s argum ument ent is pro s proble blematic matic.

WDW eq. ←wr wrong ng sign “Ground state” does not exist st. .

total

H  

total universe matter graviton 2 universe

1 2

a

H H H H H p             

Wick k rotatio ation n is no not we well define fined. d. t

matter,

H

universe

H

: radius of the universe a

matter

H

is is bo bounded nded fr from m be belo low. w.

universe

H

is bo bounded nded fr from m ab above. ve.

Pr Proble blem m of the he Wick k rotation ation for grav avity ty

 

 

 

single(

) exp . Z dg gR g    

  

 

 

 

2 4

exp exp 1/ , no solution, dr r r            



r S

1  

Ac Actua ually ly wh what at we we ha have cal alcula ulated ted is a wr wrong ng si sign n versi rsion

• n of the

he tun unne neling ng pr probab bability ility with h whi hich h un universe se po pops ps out ut from no nothi hing ng thr hroug ugh h the he po potent ntial ial ba barrier rier.

 

 

 

2 4

exp exp 1/ , no solution, dr r r             



r S

The he sy syst stem em wi with h grav avity ity sh shoul uld d be be expre pressed ssed in L n Lorent entzian zian si signa nature ture. . Th Therefo efore re the he ar argum umen ent t we we ha have emp mploye

• yed

d to get the he mu multi-local local ac action

• n is no

s not literall rally y true ue. Bu But we we expe pect ct tha hat the he low w ene nergy rgy effective ctive theory heory is sti is still giv ll given en by t by the he m mult ulti-loca local acti l action. n. In n fac act we we obt btain ain the he sa same me effect ective ive La Lagran angia gian in t n the he IIB IB ma matrix ix mo mode del l wi with h Lo Lorent entzian zian si signa nature ure.

Multi-local action in Lorentzian signature

We as assume sume tha hat t the he lo low w ene nergy rgy effective ective the heory

• ry

is is giv iven n by the he mul ulti ti-local local act ction ion:

 

eff 1 2

, , , ( ) ( ),

i i i j i j i jk i j k i i j i jk D i i

S f S S c S c S S c S S S S d x g x O x      

   

whe here re 𝑷𝒋 are gaug uge e in invariant riant sca calar lar lo loca cal l

• p
• per

erators ators suc uch h as 𝟐 , 𝑺 , 𝑺𝝂𝝃𝑺𝝂𝝃 , 𝑮𝝂𝝃 𝑮𝝂𝝃, 𝝎𝜹𝝂𝑬𝝂𝝎 , ⋯ .

Integrating coupling constants

We re repe peat at the he arg rguments uments in in t the he Lorentz rentzian ian sig igna nature. ture.

Becaus use 𝑻𝐟𝐠𝐠 is is a a fun uncti tion n of 𝑻𝒋’s , we can express 𝐟𝐲𝐪(𝒋𝑻𝐟𝐠𝐠) by a Fo Four urie ier trans nsfo form rm as

 

 

 

1 2 1 2

exp , , , , exp ,

eff i i i

iS S S d w i S           

 

whe here 𝝁𝒋’s are Fourier conjugate variables to 𝑻𝒋’s, and nd 𝒙 is is a f fun unctio tion n of 𝝁𝒋’s .

 

    



eff

exp exp .

i i i

d Z d iS S d w i             

   

Th Then t n the he path h in integral ral for 𝑻𝐟𝐠𝐠 becomes es

Because 𝑷𝒋 are local operators, 𝒋 𝝁𝒋 𝑻𝒋 is an ordinary local action where 𝝁𝒋 are regarded as the coupling constants. The heref refore

• re the

he system stem is is the he ord rdin inary ary fie ield ld the heory

• ry,

, but ut we we ha have e to in integrate rate over r the he co coup upli ling ng co const nstants ants wi with h some e we weig ight ht 𝒙(𝝁).

If a small region 𝝁~𝝁(𝟏)dominates the 𝝁 integral, it means that the coupling constants are fixed to 𝝁(𝟏).

Nature does fine tunings

         

eff

exp exp ( ).

i i i

Z d iS d w d i S d w Z                 

    

= 𝑎 𝜇 Ordinary field theory

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

Because our universe has been cooled down for long time. we may approximate 𝒂 𝝁 = 𝐟𝐲𝐪 −𝒋𝑾𝑭𝒘𝒃𝒅 𝝁 , where 𝑭𝒘𝒃𝒅 𝝁 is the vacuum energy density and 𝑾 is the space-time volume. 1) extremum If 𝑭𝒘𝒃𝒅 𝝁 is smooth and has an extremum at 𝝁𝒅 , 𝒂 𝝁 = 𝐟𝐲𝐪 −𝒋𝑾𝑭𝒘𝒃𝒅 𝝁 ~ 𝟑𝝆 𝒋 𝑾|𝑭𝒘𝒃𝒅

′′

𝝁𝒅 | 𝜺 𝝁 − 𝝁𝒅 + 𝑷(𝟐 𝑾) Thus 𝝁 is fixed to 𝝁𝑫 in the limit 𝑾 → ∞ .



vac

E

C



2) Kink (need not be an extremum) If If 𝑭𝒘𝒃𝒅 𝝁 has a kin ink (fi first order phase tr transition) at t 𝝁𝒅 , , 𝒂 𝝁 = 𝐟𝐲𝐪 −𝒋𝑾𝑭𝒘𝒃𝒅 𝝁 ~ 𝒋 𝑾 𝟐 𝑭𝒘𝒃𝒅′(𝝁𝒅 + 𝟏) − 𝟐 𝑭𝒘𝒃𝒅′(𝝁𝒅 − 𝟏) 𝜺 𝝁 − 𝝁𝒅 + 𝑷( 𝟐 𝑾𝟑) Thus 𝝁 is is fix ixed to 𝝁𝑫 in in th the lim limit it 𝑾 → ∞ . .

𝑏 𝑐 dx exp 𝑗𝑊𝑦 𝜒 𝑦

= 1 𝑗𝑊 exp 𝑗𝑊𝑦 𝜒 𝑦

a b

+ O( 1 V2)

𝑏 𝑐 dx exp 𝑗𝑊𝑔(𝑦) 𝜒 𝑦

= 1 𝑗𝑊 exp 𝑗𝑊𝑔(𝑦 ) 1 𝑔′(𝑦) 𝜒 𝑦

a b

+ O( 1 V2)

(𝑔is monotonic)



vac

E

C



monotonic

If we consider the time evolution of the universe, the definition of 𝒂(𝝁) is no longer clear. In particular, we need to specify the initial and final sates to define 𝒂(𝝁) .

Generalization

However, even if we do not know the precise form of 𝒂(𝝁), we can expect a similar mechanism to the previous case.

Actually some of the couplings can be determined without knowing the details of 𝒂(𝝁).

QCD

 Z

1.

• 1. It

It becom comes es im impo port rtan ant t only ly after ter the he QC QCD D ph phase se tra ransition nsition. 2. . T The he mass ss spe pect ctrum rum of ha hadr drons

• ns is

is in invariant riant un unde der 𝜾𝑹𝑫𝑬 → −𝜾𝑹𝑫𝑬 . ⇒ It It is is ex expe pected cted tha hat t 𝒂 is is alm lmost

• st ev

even en in in 𝜾𝑹𝑫𝑬. ⇒ 𝜾𝑹𝑫𝑬 is is tun uned ed to 0 if if the here re are re no no other her extrema. rema. (1) Symmetry examp mple le 𝜾𝑹𝑫𝑬

Nielse sen,N ,Ninomi miya

Conditions: 1. Physics changes drastically at some value of the coupling 𝝁𝑫 .

• 2. 𝒂 is monotonic elsewhere.

⇒ The coupling is tuned to 𝝁𝑫 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 par aramete ameters rs of n f nat ature ure ar are chosen sen such h that at the vac acuum uum is at at a a (m (multiple) iple) critical cality ity point int.”

Examples: (1) 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. (2) Higgs self coupling SM parameters seem to be chosen in such a way that the vacuum is marginally stable.

∞

finite

Higgs at Planck scale

• results from experiments-

Desert

LHC revealed that SM is very good up to a few TeV. At least theoretically, SM has no contradiction up to the string scale. It is natural to imagine that SM is directly connected to string theory without large modification.

SM SM

Str Strin ing g th theor

• ry

ms

RG analyses indicate that three quantities, become zero around the string scale.

Triple coincidence

 

, ,

B B B

m



  

Froggatt and Nielsen ’95. Multiple Point Criticality Principle (MPP)

The Higgs potential becomes flat (or zero) around the string scale.

V

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 16 I  

=

Higgs self coupling

[Hamada, Oda, HK,1210.2538, 1308.6651]

[1405.4781]

mHiggs =125.6 GeV

Higgs bare mass

Higgs potential

𝜒[GeV]

   

4

4 V     

mH mH = 1 125. 5.6 6 GeV

mHiggs =125.6 GeV

Cr Critica ical l Hi Higgs s inf nflat ation ion

Hamad mada, , Od Oda, , Park rk and nd HK, K,PhysRevLett.112.2413 PhysRevLett.112.241301 01

We trus ust the he flat at po poten entia tial l inc nclud uding ing the he inf nflectio ction n po point nt. We as assu sume me tha hat na natur ure do does s fine ne tun uning ngs s on n the he SM pa M param amete eters rs if ne necessar essary.

• y. In pa

n particula icular r we we tune une the he top p qu quar ark k ma mass ss in suc n such h a wa a way tha hat the he po poten ential ial be becomes

• mes flat

at ar aroun und d the he st string ing sc scal ale. e.

Hig Higgs s in infl flation

Higgs potential can be flat around the string scale. It suggests the possibility of Higgs inflation.

If If we we in introd roduce uce a no non-mini minimal mal co coup upli ling ng a re realisti listic c Hig iggs in inflat lation ion is is po possible. sible.

2

R  

 

2 2

, . 1 /

h h P

h V h M     

In the Einste stein in frame ame the effect ffective ive po poten tential tial be becomes

• mes

𝜊 can be as small as 10.

Cr Critica ical l Hi Higgs s inf nflat ation ion

• Desert

SM is valid to the string scale at least theoretically. SM might be directly connected to string theory without large modification.

• Marginal stability

Higgs field is near the stability bound.

• Zero bare mass

The bare Higgs mass is close to zero at the string scale. It implies that Higgs is a massless state of string theory.

• Flat potential and Higgs inflation

Higgs self coupling and its beta function become zero at the string scale. Higgs potential might be flat around the string scale. It suggests the possibility of Higgs inflation.

SM SM arou round Pla

lanck k sc scal ale

Multiverse in Lorentzian signature

 

    



eff

exp exp .

i i i

Z d i S d w d i S             

   

As As i in th the Eu Eucli lidean dean cas ase it it is is nat natur ural al to to consid nsider er the multiver verse. se.

   

1 1 single universe

1 exp ! exp

n i i i n i i i

d i S Z n Z d i S    

 

             

    

n

   

 

1

exp . d w Z     

Path i integra ral for r a univ iverse

 

 

 

 

 

 

   

1 1

exp exp ˆ exp ˆ

E E

Z d i S f dadpdN i dt pa NH i f dT iTH i f H i f i

  

    

   

      

   

i

f

2 3

1 1 1 ˆ ( ) 2 H p a U a a a

  



: radius of the universe a

Que uestion: stion: Is Is the here e a n a nat atur ural al cho hoice e for the hem? m?

T

If th the in init itia ial l an and fin inal al st stat ates s ar are giv iven, en, th the pat ath in inte tegr gral al is is ev evalu aluat ated ed as as usu sual al: : (mi mini i su supers rspace ace)

 

E E

E E   



  

2 3 4

1 ( )

matt rad

C C U a a a a     

S3 topology

Initial ial state ate

For th the in init itia ial l st stat ate, we as assu sume me th that at th the univ iverse erse emerges with a small size ε.

, : probability amplitude of a universe emerging. i a matter      

a  

Evolution of the universe

Λ～curvature ～energy density with

S3 topology

     

 

1

1 , sin , ,

z E

a da p a a p a     

 

  



 

4

, 2 ( ). p a a U a   

2 3 4

1 ( )

matt rad

C C U a a a a      WKB solution

 

4

S d x g R matter     



a

cr

  

cr

  

cr

  

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

Final state: case 1 Final state: case 1

For th the fi final al st stat ate, we hav ave tw two poss ssib ibili ilities ties. .

finite

The univ iverse erse is is cl closed sed. We as assu sume th the fin inal al st stat ate is is

. f a matter      

The pat ath in inte tegr gral al

 

 

 

1 2

ˆ .

E

Z f H i const



   





cr

  

Final state Final state: case 2 : case 2

∞

The univ iver erse se is is op

• pen.

It is t is n not t cle lear ar how to to de defin ine th the pat ath in inte tegr gral al for th the univ iverse: erse:

lim .

IR

IR IR a

f c a a matter



 

a

 

 

 

1

exp . Z d i S    

As As an an ad ad hoc as assu sump mpti tion n we consid sider er

cr

  

     

 

 

 

 

* 1 3 * 4 3 * 4

1 sin 1 sin .

IR E IR E IR IR E IR IR E

Z c a a c a a a c a             

   

        

Then th the par arti titi tion

• n function

tion be become

• mes

mat rad 2 3 4

1 ( ) C C U a a a a     

     

 

1

1 sin

z E

a da p a a p a  

 

  



The resu sult lt does s not d t depend on e excep cept t for th the ph phas ase whic ich come mes s from m th the cla lass ssical ical ac acti tion. n.

 

4

, 2 ( ) p a a U a   

IR

a

(cont’d)

Thus we we have ave the e pat ath integr egral al fo for a a univer verse se

∞

finite

 

1

Z 

cr

for   

• f order 1,

const

cr

for   

 

3 4

1 sin .

IR

const a     

Then n th the i inte tegration gration fo for th the mult ltiverse iverse

   

 

1

exp . Z d w Z     

has as a lar arge e peak ak at at , , wh which h means ans that at the cosmo mologi logical cal constant stant at at th the lat ate stag ages es of t f the unive iverse rse al almost st van anishes ishes. .



 

cr

  

Maximum entropy principle

Then th the mu mult ltiv iverse erse par arti titi tion

• n funct

ction ion is is gi given ven by by

cr rad

1/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 si simp mpli licit city y we as assu sume me th the to topolog logy y of th the sp spac ace an and th that at all all ma matt tters s decay ay to to ra radia iation tion at at th the lat late st stag ages. s.

rad 2 4

1 ( ) C U a a a    

   

 

 

 

1 4 rad 4

exp 1 exp const exp const .

cr

Z d w Z C              



3

S

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 m e may y un unde derstand rstand the he fla latnes ness s of

• f the

he Hig iggs gs po potential ntial as a co conseq nsequence uence of ME 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

𝜒[GeV]

   

4

4 V     

mH mH = 1 125. 5.6 6 GeV

mHiggs =125.6 GeV

Multi-local action from IIB matrix model

IIB matrix model

 

2 Matrix

1 1 , , . 4 2 S Tr A A A

  

                    

This is a candidate of non-perturbative definition of string theory. Formally, this is the dimensional reduction

• f 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

• f simultaneously diagonalizable matrices

𝑩𝝂

(𝟏) = 𝑸𝝂 corresponds to the flat space,

if the eigenvalues are uniformly distributed. In other words, the background 𝑩𝝂

(𝟏) = 𝒋𝝐𝝂

represents the flat space. How about curved space? Is it possible to consider some background like 𝑩𝝂

(𝟏) = 𝒋𝛂𝝂 ?

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 𝑵: any D-dimensional manifold, 𝝌𝜷: a regular representation field on 𝑵. Here the index 𝜷 stands for the components of the regular representation of the Lorentz group 𝑻𝑷(𝑬 − 𝟐, 𝟐). The crucial point is that for any representation 𝒔, its tensor product with the regular representation is decomposed into the direct sum of the regular representations:

.

r reg reg reg

V 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 reg

V V V V    

are written as

, ( )

, ( 1,.., ).

b a

C a D

 



Here 𝒄 and β are the dual of the vector and the regular representation indices on the LHS. (𝒃) indicates the 𝒃-th space of the regular reprezentation on the RHS, and 𝜷 is its index.

Then for each 𝒃 (𝒃 = 𝟐. . 𝑬)

, ( ) b a b

C

   

   

is a regular representation field on 𝑵. In other words, if we define 𝛂(𝒃) by

 

, ( ) ( )

,

b a a b

C

   

    

each 𝛂(𝒃) is an endomorphism on the space of the regular representation field on 𝑵.

Thus we have seen that the covariant derivatives on any D dimensional manifold can be expressed by D matrices.

Therefore any D-dimensional manifold 𝑵 with 𝑬 ≤ 𝟐𝟏 can be realized in the space

• f the IIB matrix model as

( ) ,

1, , , 0, 1, ,10

a a

a D A a D

where 𝛂(𝒃) is the covariant derivative on 𝑵 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 a

A i  

• n the classical EOM

 

, 0,

a a b

A A A     

we have

 

( ) ( ) ( )

, , , , ( ) 0 , .

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 𝑬 ≤ 𝟐𝟏 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 𝑻𝑽(𝑶) symmetry.

In fact, the infinitesimal diffeomorphism and local Lorentz transformation act on 𝝌𝜷 as 𝝌 → 𝟐 + 𝝄𝝂𝝐𝝂 𝝌 and 𝝌 → 𝟐 + 𝜻𝒃𝒄𝑷𝒃𝒄 𝝌 , respectively. Both of them are unitary because they preserve the norm of 𝝌𝜷 𝝌𝜷 𝟑 = 𝒆𝑬𝒚 𝒉 𝝌𝜷∗𝝌𝜷

Bad points Fluctuations around the classical solution 𝑩𝒃

(𝟏) = 𝒋𝛂(𝒃)

• 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 𝑩𝒃

(𝟏) = 𝒋𝝐𝒃 ⊗ 𝟐𝒔𝒇𝒉 ,

where 𝟐𝒔𝒇𝒉 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.

.

a a a

A 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 𝝔 to obtain the low energy effective action. Here we assume that the background 𝑩 𝒃

𝟏 contains

• nly the low energy modes, and 𝝔 contains the rest.

More precisely, we first decompose the matrices 𝑩𝒃 into the background 𝑩 𝒃

𝟏 and the fluctuation 𝝔 :

Substituting the decomposition into the action of the IIB matrix model, and dropping the linear terms in 𝝔, we obtain



     



2 2 2

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

 

2 ( ) ( )

1 , 4

a b

S 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: 𝑿 = 𝑳𝒃𝒄𝒅⋯ , 𝒒𝒓𝒔⋯ 𝑼𝒔 𝑩𝒃

𝟏𝑩𝒄 𝟏𝑩𝒅 𝟏 ⋯

𝑼𝒔(𝑩𝒒

𝟏𝑩𝒓 𝟏𝑩𝒔 𝟏 ⋯ )

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 actions

2-loop Planar eff , ,

1 , 6

i jk i j k i j k

S c S S S  

while non-planar diagrams give a local action

2-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

x O x g x d S S S S c S S c S c S

i D i k j i k j i k j i j j i i j i i i i

   

     

We have seen that

Although there is no precise correspondence, the loops in the IIB matrix model resemble the wormholes.

x y

𝑦 𝑧

≅

We may say that if the theory involves gravity and topology change, its low energy effective action becomes the multi-local action universally.

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 might 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

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 2

1 , . 2

ab NG a b

S 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 𝚻𝝂𝝃 that is constructed from the space-time coordinate 𝒀𝝂.

It is known that the Nambu-Goto action is equivalent to the Schild action





2 2 2 Schild

1 , , 2 4 2 S d g X X d g

 

       

 





: a volume density on the world shee 1 , , t

ab a b

X Y g X Y g    

which is nothing but the Poisson bracket if regard the worldsheet as a phase space.

   

2 Schild 2 2 Schild

1 2 1 . 2 2

ab a b ab a b

S g X X g S d X X

   

                    



The equivalence can be seen easily, by eliminating 𝒉 from the Scild action:

The crucial point is that the Schild action has a structure of phase space.





2

1 , . 2 4 2 X X

 

    

In fact it is given by the integration over the phase space

• f a quantity that is expressed in terme the

Poisson bracket

2

d g 



symplectic structure of the worldsheet

Not that we do not need Worldsheet metric, but what we need is just the volume density 𝒉.

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 B i 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 , 4 S Tr A A Tr

 

        

and the path integral is regularized like

 

   

Schild Matrix 1

exp exp . vol(Diff) ( )

n

dg dX dA Z 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 .

2

d g 



 

1 Tr 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 4 S 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

 

2

1 , . 4 S 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.

 

2

2 , 4

d N

S Tr A A

 

             

the large-N reduced model is equivalent to the d- dimensional Yang-Mills if the eigenvalues of 𝑩𝝂 are 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

N S Tr U U U U

     

  

 

  



Actually there are several ways to make the eigenvalues distribute uniformly.

quenching

We constrain the diagonal elements of 𝑩𝝂 to a uniform distribution by hand (𝑩𝝂)𝒋𝒋= 𝒒𝝂

(𝒋) .

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.

Bhan anot-Hell ller-Neu eube berger er, , Gr Gross-Kit itaz azaw awa

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

Go Gonzal alez-Arroyo yo, , Korthal als Alt ltes (‘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

 

 

2

2 , . 4

d N

S Tr A A i B

  

              

The lattice version of this is called the twisted reduced model:

 

† † reduced 1

.

d i

N S e Tr U U U U



      



 

 



Go Gonzal alez-Arroyo yo, Ok , Okaw awa a

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, Azeya yanag agi, , Han anad ada, , Ya Yacobi bi

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 1 2 2 X i i

       

                                

N=2 SUSY

1 1 SUSY 2 2 SUSY 1 1 2 2 SUSYX

i i

  

                

Gauge fixing for the κ-symmetry

1 2

    

2 2

1 2 , 2 .

ab GS a b ab a b

S d i X X X

    

                       



N=2 SUSY should be combined with 𝝀 symmetry so that the gauge condition is maintained

1 1 1 SUSY 2 2 2 SUSY 1 2 SUSY 1 2 1 1 2 2

2 2 X X X

     

                                     

N=2 SUSY becomes simple if we consider a combination

       

1 2 1 2 2

1 1 2 2 0. X i X

    

                   

1 2 1 2

2 . 2          

Then we have the following simple form:

Schild action

N=2 SUSY









Schild 2 2 2

1 , , , 2 4 2 2 S i d g X X X d g

   

                 

 

 





     

1 1 2 2

1 , 2 X X X i X

     

                

Everything is written in terms

• f 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 F A i A

    

                

 

 

2 Matrix

1 1 , , 1 . 4 2 S 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 2 S Tr A A A

  

                    

Drop the second term, and consider large-N Formally, this is the dimensional reduction

• f 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 2 F A i

   

          

 

2 Matrix

1 1 , , 4 2 S Tr A A A

  

                    

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

   

2 2

A      

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 models com

• mment

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

Mot

• tl, Dijkg

kgraaf-Ve Verlinde-Ve Verlinde de W Witt-Ho Hoppe-Ni Nicol

• lai,

Ba Banks ks-Fi Fisc schler-Shenke ker-Suss sski kind

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

    

   

 A A A Tr g S

gives a constructive definition of superstring. However there are some fundamental

• pen 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[ , ] . 2 4 1 [ , ] S A A A Tr Tr g

   

           

How about Lorentzian signature? If we simply apply the analytic continuation the path integral becomes unbounded:

10,

A iA  

 

 

 

2 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 A g Tr A A A A g

       

                           



From the point of view of the large-N reduction, it is natural to take

 

exp .

M

Z dAd i S  



Is IR cutoff necessary?

Be Beca caus use e of f th the e su super ersy symmet etry ry the the fo forc rce e bet etwe ween en ei eige genv nvalue ues s ca canc ncel els s bet etwe ween en boso sons ns and nd fe ferm rmions ns

 

 

 

2 (1 ) ( ) ( ) ,

2 log

loop i j eff F i j

S D d p p



    



It It se seem ems th that t we we h have e to to impos

• se

e an n inf nfra rare red cutoff ff by hand t to pre revent the eigenvalues fr from ru running away t to infi finity.

 

eigen l 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 j i j i j i j i j

S S S x tr p p S p p

    

    



               



Aoki, i, Iso, , Kit itaz azaw awa, a, Ta Tada da, , HK

Be Because se of

• f th

the fe fermio ionic ic de degr grees s of

• f fr

freedo dom, th there app ppears s a weak k att ttracti tive fo force be betw tween th the eig igenvalu lues, s, and d at t le least st th the parti titi tion fu functi tion

• n be

becom

• mes

s fi finit ite. Ho However it it is is not

• t cle

lear wheth ther all ll th the cor

• rrela

lati tion

• n fu

functi tion

• ns

s are fi finit ite or

• r not
• t.

Austin ing an and d Wheater, Krau auth, Ni , Nicola lai i an and d Stau auda dacher, Suya yama an and d Ts Tsuchiy iya, a, Ambj bjorn, , Anag agnostopo poulo los, , Bie ietenholz lz, , Ho Hotta an and d Nis ishim imura, a, Bia iala las, , Burda da, , Petersson an and d Ta Taba baczek, Gr Green an and d Gu Gutpe perle le, Mo Moore, , Ne Nekras asov an and d Shat atas ashvi vili li.

 

  



 

1-loop (1 loop) 16 ( ) eff

exp ,

i i

Z p d S p  



 

 

( , ) i j

S

   

i j

  

 

( , )

0, 8

n i j

S n  

Since is quadratic in , which has only 16 components, we have

an and

  



 

   

   

 

4 8 , , 1-loop eff 4 8 ( , ) ( , )

exp , exp 4 8 1 tr tr .

i j i j i j i j i j i j

S S S 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 , tr

i j i j

a S b S



16 ( ) 1 N i i

d 



 

 

   

 

   

 

 

 

(1 loop) various terms

exp .

i j i j

Z p f p p f p p O N

      

  



Th Therefo efore, re, for eac ach h pa pair of i an and j d j we we ha have 3 3 cho hoices ces whi hich h car arry ry the he po powers rs of 0, 0, 8, 8, 16 16 respe specti ctively vely. The herefo efore re the he nu number of factor

• rs

s othe her r tha han 1 n 1 sho houl uld be less s tha han o n or equa ual to 2N, , an and d we can an conc nclud lude e tha hat the he effective ective ac action ion ind nduc uced from m the he fermi mion

• nic

ic ze zero

• mo

mode des s is of s of 𝑷 𝑶 :

.

On On th the othe her r ha hand nd, we we ha have 16 16N N di dime mension nsional al fermi mion

• nic

ic int ntegra gral l . .

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

    

   

 A A A Tr g S

g

So, 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 have

1 2 Planck

. 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

    

   

 A A A Tr g S

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.

ˆ 1 , 0, ,3 . 0, 4, ,9

k

p A Here satisfy the CCR’s

ˆ p

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 universe

S.-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 𝑰𝒐

𝟓 and higher-spin in Yang-Mills

matrix models” Marcus Sperling, H. Steinacker, arXiv:1806.05907

• 4. Diffeomorphism invariance

and 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

• ne-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

• f simultaneously diagonalizable matrices

𝑩𝝂

(𝟏) = 𝑸𝝂 corresponds to the flat space,

if the eigenvalues are uniformly distributed. In other words, the background 𝑩𝝂

(𝟏) = 𝒋𝝐𝝂

represents the flat space. How about curved space? Is it possible to consider some background like 𝑩𝝂

(𝟏) = 𝒋𝛂𝝂 ?

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 𝑵: any D-dimensional manifold, 𝝌𝜷: a regular representation field on 𝑵. Here the index 𝜷 stands for the components of the regular representation of the Lorentz group 𝑻𝑷(𝑬 − 𝟐, 𝟐). The crucial point is that for any representation 𝒔, its tensor product with the regular representation is decomposed into the direct sum of the regular representations:

.

r reg reg reg

V 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 reg

V V V V    

are written as

, ( )

, ( 1,.., ).

b a

C a D

 



Here 𝒄 and β are the dual of the vector and the regular representation indices on the LHS. (𝒃) indicates the 𝒃-th space of the regular reprezentation on the RHS, and 𝜷 is its index.

Then for each 𝒃 (𝒃 = 𝟐. . 𝑬)

, ( ) b a b

C

   

   

is a regular representation field on 𝑵. In other words, if we define 𝛂(𝒃) by

 

, ( ) ( )

,

b a a b

C

   

    

each 𝛂(𝒃) is an endomorphism on the space of the regular representation field on 𝑵.

Thus we have seen that the covariant derivatives on any D dimensional manifold can be expressed by D matrices.

Therefore any D-dimensional manifold 𝑵 with 𝑬 ≤ 𝟐𝟏 can be realized in the space

• f the IIB matrix model as

( ) ,

1, , , 0, 1, ,10

a a

a D A a D

where 𝛂(𝒃) is the covariant derivative on 𝑵 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 a

A i  

• n the classical EOM

 

, 0,

a a b

A A A     

we have

 

( ) ( ) ( )

, , , , ( ) 0 , .

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 𝑬 ≤ 𝟐𝟏 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 𝑻𝑽(𝑶) symmetry.

In fact, the infinitesimal diffeomorphism and local Lorentz transformation act on 𝝌𝜷 as 𝝌 → 𝟐 + 𝝄𝝂𝝐𝝂 𝝌 and 𝝌 → 𝟐 + 𝜻𝒃𝒄𝑷𝒃𝒄 𝝌 , respectively. Both of them are unitary because they preserve the norm of 𝝌𝜷 𝝌𝜷 𝟑 = 𝒆𝑬𝒚 𝒉 𝝌𝜷∗𝝌𝜷

Comment: The norm is not positive definite for Lorentzian theory.

Bad points Fluctuations around the classical solution 𝑩𝒃

(𝟏) = 𝒋𝛂(𝒃)

• 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 𝑩𝒃

(𝟏) = 𝒋𝝐𝒃 ⊗ 𝟐𝒔𝒇𝒉 ,

where 𝟐𝒔𝒇𝒉 is the unit matrix on the space of the regular representation.

Because the unit matrix 𝟐𝒔𝒇𝒉 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

• f 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

x O x g x d S S S S c S S c S c S

i D i k j i k j i k j i j j i i j i i i i

   

     

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 𝑷𝒋 are local scalar operators such as 𝟐 , 𝑺 , 𝑺𝝂𝝃𝑺𝝂𝝃 , 𝑮𝝂𝝃 𝑮𝝂𝝃, 𝝎𝜹𝝂𝑬𝝂𝝎 , ⋯ . 𝑻𝒋 are parts of the conventional local actions. The point is that 𝑻𝐟𝐠𝐠 is a function of 𝑻𝒋’s. 𝑻𝐟𝐠𝐠 is not local, but it is not completely non-local in the sense that it is a function of local actions.

.

a a a

A 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 𝝔 to obtain the low energy effective action. Here we assume that the background 𝑩 𝒃

𝟏 contains

• nly the low energy modes, and 𝝔 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 𝑩𝒃 into the background 𝑩 𝒃

𝟏 and the fluctuation 𝝔 :

Substituting the decomposition into the action of the IIB matrix model, and dropping the linear terms in 𝝔, we obtain



     



2 2 2

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

 

2 ( ) ( )

1 , 4

a b

S 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: 𝑿 = 𝑳𝒃𝒄𝒅⋯ , 𝒒𝒓𝒔⋯ 𝑼𝒔 𝑩𝒃

𝟏𝑩𝒄 𝟏𝑩𝒅 𝟏 ⋯

𝑼𝒔(𝑩𝒒

𝟏𝑩𝒓 𝟏𝑩𝒔 𝟏 ⋯ )

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 actions

2-loop Planar eff , ,

1 , 6

i jk i j k i j k

S c S S S  

while non-planar diagrams give a local action

2-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

x O x g x d S S S S c S S c S c S

i D i k j i k j i k j i j j i i j i i i i

   

     

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

𝑦 𝑧

≅

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 interpretation
• f

multiverse wave function

Probabi abilistic tic inter terpretat tation (1 (1)

pos

• stul

tulate ate

T T : age of the he un univ iverse rse

 

2

probability of finding a universe of size z dz z  

   

E

z z     

 

2

1 ( ) 1

E

dz z dz z p z d z d z T     

  

2

1 2 H H z p z zp p               

   

 

1 exp ( )

z E

z i dz p z z p z    



me mean anin ing g of th this is me meas asure

z

T

 

2 2

z dz dT  

⇒

2

probability of a universe emerging in unit time  

the he tim ime e tha hat t ha has passed ed after er the he un univ iverse rse is is cr create ted

Probab abilisti tic inte terpreta tation ( (2)

 is is a s a super perpositi position

• n of

f th the unive iverse rse wi with th var arious ious ag age,

z

T

+

z

T

z

T

+ +…

 

T T dT 

gives es the prob

• babili

ability ty of f fi finding ing a univer verse se of ag f age .

 

2 2

z dz dT  

infrared cutoff We introduce

• duce an

an infr frar ared ed cutof

• ff

f fo for the size of u f univer verses. ses.

IR

z

cease ses s to to exi xist st bo bounces s ba back

• r

   

2 2 2

life time of the universe z dz dT     

 

dim imensi sionl

• nless

ess

∞

finite

Lifeti time of f the un universe se

Wa Wave Fu Functi tion

• n of
• f th

the mult ltiv iverse se (1 (1)

Mult ltivers iverse e ap appear ars s nat aturally ally in in qu quan antu tum m grav avity ity / st strin ing g th theory ry. . ma matr trix ix mo model ・

b b a

C 

  , ) (

b b a

C 

  , ) (

Each ch block lock repres esents ents a u a uni niver verse. se.

・ qu quan antu tum m gravity avity

Okad ada, a, HK

Wa Wave Fu Functi tion

• n of
• f th

the mult ltiv iverse se (2 (2)

The mu mult ltiv iverse erse sa sate te is is a su a superpo posit sition ion of N N-vers erses. es.

 

multi

,

N N

d w    

 

   

 

     

1 1

, , , , ,

N N N

z z z z       

,   ,    

 

multi N

d w    

 

  

exp

i i i

Z d w d i S            

  

Wa Wave Fu Functi tion

• n of
• f th

the mult ltiv iverse se (3 (3)

Proba babilist bilistic ic in inte terpret pretat ation ion

 

multi

,

N N

d w    

 

   

 

     

1 1

, , , , ,

N N N

z z z z       

1 1 1,

,

N N N

z z dz z z dz  

th the proba babi bility lity of fin indin ing g N u N univ iverse erses s wit ith si size

   

2 2 1 1

, , ,

N N N

z z dz dz w d    

an and fi findin ing g th the couplin ling g const stan ants ts in in

. d    

represen resents ts

Prob

• babi

bility ty di dist strib ibutio ion of

• f

     

1 1

, , , , ,

N N N

z z z z        



       

 

   

 

 

2 2 1 1 2 2 2 2

, , , ! exp , exp

N N N N

dz dz P z z w N dz z w w         

 

   

 

   

2

, (life time of the universe)

E

dz z    



 

E

      is is ch chosen sen in in su such h a wa a way th that at i is ma maxim imize ized,



 

  can an be be ve very ry lar arge. ge.

 

  irresp especti ectively vely of f  . w 

If If we we acc ccept ept the he pr probabi abilistic listic in interpretation rpretation of the he mul ultiv tiverse erse wa wave e fun unction ction, , the he co coup upli ling ng co const nstan ants ts are re ch chosen en in in su such ch a wa way tha hat t the he li lifetime ime of the he un univ iverse erse bec ecomes

• mes maximum.

ximum.

WKB sol with

Cosmological constant

Λ～curvature～energy density (extremely small) What value of Λ maximizes

 

?

S3 topology

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

IR

z

   

1 , ,

E

z z p z   



 

, 2 ( ). p z U z   

assum umin ing g all ll matters ters deca cay to ra radia iation ion

 

2

,

E

dz z  





 

cr

0    

cr

  

cr

  

The other couplings (Big Fix)

   

 

 

2 2

exp P w     

The expone nent nt is is div ivergent ergent, , an and regu gulated lated by by th the IR c R cuto toff ff :

 

2 rad cr

1 , log .

IR

E IR z

dz z C z z  





 

cr rad

1/C  

   

2

,

E

dz z    



  



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

 

rad

C 

as assu sumi ming g al all l ma matt tters s decay ay to to ra radia iati tion

• n