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

Comment on Coleman’s Colem Co eman an ha had e d ess ssent entially ally the he sa same me resu sult lt in n 19 1989 89. . In n the he cas ase e of Co Colem eman, an, he he cons nside idered red the he wormh wo mhole ole 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 on. 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. 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 ory. Pr Probab bably ly, , the heorie ories 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. This is Th s is be becau ause se we we ha have to int ntrod oduce uce ad addi dition ional al cond nditio itions ns on t n the he pa path h int ntegra gral l if one ne wa want nts s to fix the he topo pology logy of the he sp spac ace-ti time. me.

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 on 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.

Consequences of multi-local action We wa want nt to do do the he pa path i h int ntegral gral         Z d exp S eff for the he effect ective ive ac action ion        S c S c S S c S S S , eff i i i j i j i jk i j k i i j i jk   D S d x g x O x ( ) ( ), i i wh where e 𝒅 𝒋 , 𝒅 𝒋𝒌 , 𝒅 𝒋𝒌𝒍 ⋯ ar are cons nsta tant nts s of O( O(1) 1) in n the he Pl Plan anck k un unit.

Re Regar ardin ding g 𝑻 𝐟𝐠𝐠 as as a f a fun unction on of 𝑻 𝒋 ’s    S S S S , , , eff eff 1 2 we int we ntrod oduce uce the he La Lapl plac ace e tran ansfo sform rm                   exp S S S , , d w , , exp  S . eff 1 2 1 2 i i   i Th Then t n the he pa path i h int ntegral gral be becomes omes                          Z d exp S d w d exp S . eff i i   i on 𝝁 𝒋 𝑻 𝒋 . pat ath integ egra ral l for r a lo a local al ac action 𝝁 𝒋 ar are coupling pling consta stants nts of the ac action. on. Co Coup upling ng cons nstant tants s ar are no not me merely ely cons nsta tant nts, s, but the bu hey sh shoul uld d be be int ntegra grated. ted.

We ha have se seen n tha hat the he pa partit ition ion fun unction on is s given en by by av averaging aging 𝒂(𝝁) with h weight ght 𝒙(𝝁) .                Z d exp S d w Z ( ), eff            ( ) exp   . Z d S i i   i 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 supe su perposit position ion of di differe erent nt coup upling ing cons nsta tant nts. s. Ho Howe wever ver, , the he pr proble blem m is re s reso solved lved if 𝒂(𝝁) ha has s ue 𝝁~𝝁 (𝟏) . a s a sha harp p pe peak ak ar aroun und d so some me val alue . In n tha hat cas ase e we we can an sa say tha hat the he na natur ure e itse self lf to 𝝁 (𝟏) . tun une the he pa param amete eters rs to

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 on 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(−S eff ) ma may be be a we a well- be beha haved ed fun unction on of 𝑻 𝒋 ’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 on 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 onne necte cted d un universes rses af after r int ntegratin rating out ut the he sh short rt-dist distan ance ce fluc uctua uatio tions. ns.

Th Therefo efore re we ha have to su sum ov over 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 on. n       exp          Z d w d  S  i i   i  1         n d w Z ( ) single n !  0 n          d w exp Z ( ) . single He Here e 𝒂 𝐭𝐣𝐨𝐡𝐦𝐟 𝝁 is s the he pa partitio ition n fun unction on for a a si sing ngle e un universe. se.

“solution” to to th the c cos osmol olog ogic ical al con onst stan ant t pr prob oble lem For si Fo 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, 𝒂 𝐭𝐣𝐨𝐡𝐦𝐟 (𝚳) is s evalu aluat ated ed by by cons nside idering ring 𝑻 𝟓 geo eome metry try class assicall ically: y:          4     S single ( ) exp . Z dg gR g r          S 2 4 dr exp r r r       exp 1/ , 0    1  no solution, 0     . w   domi minates nates irresp espective ectively y of f 0          Z d w exp Z ( ) . single

Ho Howe wever ver thi his arg s argum ument ent is pro s proble blematic matic. Proble Pr blem m of the he Wick k rotation ation for grav avity ty   WDW eq. H 0 total     H H H H total universe matter graviton   1    ← wr wrong ng sign 2   H p universe a   2 a : radius of the universe “Ground state” does not exist st. . H universe Wick k rotatio ation n is no not we well define fined. d. t is is bo bounded nded fr from m be belo low. w. H H matter , matter is bo bounded nded fr from m ab above. ve. H universe

Ac Actua ually ly wh what at we we ha have cal alcula ulated ted              Z single ( ) dg exp gR g .          S 2 4 dr exp r r   r     exp 1/ , 0     1 no solution, 0   is a wr wrong ng si sign n versi rsion on 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. S          2 4 dr exp r r        r exp 1/ , 0     no solution, 0

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

Integrating coupling constants We as assume sume tha hat t the he lo low w ene nergy rgy effective ective the heory ory is giv is iven n by the he mul ulti ti-local local act ction ion:    S f S S , , eff 1 2        c S c S S c S S S , i i i j i j i jk i j k i i j i jk   D ( ) ( ), S d x g x O x i i whe here re 𝑷 𝒋 are gaug uge e in invariant riant sca calar lar lo loca cal l op oper erators ators suc uch h as 𝟐 , 𝑺 , 𝑺 𝝂𝝃 𝑺 𝝂𝝃 , 𝑮 𝝂𝝃 𝑮 𝝂𝝃 , 𝝎𝜹 𝝂 𝑬 𝝂 𝝎 , ⋯ . We re repe peat at the he arg rguments uments in in t the he Lorentz rentzian ian sig igna nature. ture.

Becaus use 𝑻 𝐟𝐠𝐠 is is a a fun uncti tion n of 𝑻 𝒋 ’s , we can express 𝐟𝐲𝐪(𝒋𝑻 𝐟𝐠𝐠 ) by a Fo Four urie ier trans nsfo form rm as                 exp iS S S , , d w , , exp  i S , eff 1 2 1 2 i i   i whe here 𝝁 𝒋 ’s are Fourier conjugate variables to 𝑻 𝒋 ’s, and nd 𝒙 is is a f fun unctio tion n of 𝝁 𝒋 ’s . Then t Th n the he path h in integral ral for 𝑻 𝐟𝐠𝐠 becomes es                        Z d exp iS d w d exp i S . eff i i   i Because 𝑷 𝒋 are local operators, 𝒋 𝝁 𝒋 𝑻 𝒋 is an ordinary local action where 𝝁 𝒋 are regarded as the coupling constants. The heref refore ore the he system stem is is the he ord rdin inary ary fie ield ld the heory ory, , 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) .

Nature does fine tunings                       Z d exp iS d w d exp  i S eff i i   i        = 𝑎 𝜇 d w Z ( ). Ordinary field theory If a small region 𝝁~𝝁 (𝟏) dominates the 𝝁 integral, it means that the coupling constants are fixed to 𝝁 (𝟏) .

A simplified analysis ’14 ’15 Hamada, Kawana, HK Because our universe has been cooled down for long time. we may approximate 𝒂 𝝁 = 𝐟𝐲𝐪 −𝒋𝑾𝑭 𝒘𝒃𝒅 𝝁 , where 𝑭 𝒘𝒃𝒅 𝝁 is the vacuum energy density and 𝑾 is the space-time volume. 1) extremum If 𝑭 𝒘𝒃𝒅 𝝁 is smooth and has an extremum at 𝝁 𝒅 , 𝒂 𝝁 = 𝐟𝐲𝐪 −𝒋𝑾𝑭 𝒘𝒃𝒅 𝝁 E 𝟑𝝆 𝜺 𝝁 − 𝝁 𝒅 + 𝑷( 𝟐 vac ~ 𝑾 ) ′′ 𝒋 𝑾|𝑭 𝒘𝒃𝒅 𝝁 𝒅 |   Thus 𝝁 is fixed to 𝝁 𝑫 in the limit 𝑾 → ∞ . C

2) Kink (need not be an extremum) If 𝑭 𝒘𝒃𝒅 𝝁 has a kin If ink (fi first order phase tr transition) at t 𝝁 𝒅 , , 𝒂 𝝁 = 𝐟𝐲𝐪 −𝒋𝑾𝑭 𝒘𝒃𝒅 𝝁 ~ 𝒋 𝟐 𝟐 𝜺 𝝁 − 𝝁 𝒅 + 𝑷( 𝟐 𝑭 𝒘𝒃𝒅 ′(𝝁 𝒅 + 𝟏) − 𝑾 𝟑 ) 𝑾 𝑭 𝒘𝒃𝒅 ′(𝝁 𝒅 − 𝟏) Thus 𝝁 is is fix ixed to 𝝁 𝑫 in in th the lim limit it 𝑾 → ∞ . . 𝑐 dx exp 𝑗𝑊𝑦 𝜒 𝑦 𝑏 b 1 + O( 1 E monotonic = 𝑗𝑊 exp 𝑗𝑊𝑦 𝜒 𝑦 V 2 ) vac a 𝑐 dx exp 𝑗𝑊𝑔(𝑦) 𝜒 𝑦 𝑏 b 1 1 + O( 1  = 𝑗𝑊 exp 𝑗𝑊𝑔(𝑦 ) 𝑔′(𝑦) 𝜒 𝑦 V 2 )  a C ( 𝑔 is monotonic)

Generalization 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) . 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) . (1) Symmetry examp mple le 𝜾 𝑹𝑫𝑬 Nielse sen,N ,Ninomi miya 1. It 1. 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 ons is is in invariant riant un unde der 𝜾 𝑹𝑫𝑬 → −𝜾 𝑹𝑫𝑬 . ⇒ It It is is ex expe pected cted tha hat t 𝒂 is is alm lmost ost ev even en in in 𝜾 𝑹𝑫𝑬 . ⇒ 𝜾 𝑹𝑫𝑬 is is tun uned ed to 0 if if the here re are re no no other her extrema. rema. Z  QCD

(2) Edge or the point of drastic change Z Conditions: 1. Physics changes drastically at some value of the coupling 𝝁 𝑫 .   2. 𝒂 is monotonic elsewhere. C ⇒ The coupling is tuned to 𝝁 𝑫 as we have discussed. 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 ∞ finite 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.

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. m s SM SM Str Strin ing g th theor ory

Triple coincidence RG analyses indicate that three quantities,      , , m  B B B become zero around the string scale. The Higgs potential becomes flat (or zero) around the string scale. V Froggatt and Nielsen ’95. Multiple Point Criticality Principle (MPP)

Bare couplings as a function of the cutoff SM is valid up to the Planck scale. m Higgs =125.6 GeV 1  2 = I  1 2 16 Top mass from [Hamada, Oda, HK ,1210.2538, 1308.6651] [1405.4781]

Higgs self coupling Higgs bare mass [1405.4781] m Higgs =125.6 GeV [Hamada, Oda, HK,1210.2538, 1308.6651]

         Higgs potential 4 V 4 m Higgs =125.6 GeV mH mH = 1 125. 5.6 6 GeV 𝜒[GeV]

Higgs Hig s in infl flation Higgs potential can be flat around the string scale. It suggests the possibility of Higgs inflation. Critica Cr 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 poten po ential ial be becomes omes flat at ar aroun und d the he st string ing sc scal ale. e.

Critica Cr ical l Hi Higgs s inf nflat ation ion If we If we in introd roduce uce a no non-mini minimal mal co coup upli ling ng   2 R a re realisti listic c Hig iggs in inflat lation ion is is po possible. sible. 𝜊 can be as small as 10. In the Einste stein in frame ame the effect ffective ive po poten tential tial be becomes omes h      V , . h h   2 2 1 h / M P

SM arou SM round Pla lanck k sc scal ale • 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.

Multiverse in Lorentzian signature

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.                      Z d exp i S d w d exp  i S  . eff i i   i     1        n d exp  i S  Z i i 1   n !  i n 0          Z d exp i S   1 i i single universe   i n            exp . d w Z 1

Path i integra ral for r a univ iverse S 3 topology 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) f           Z d exp i S 1       1     T f dadpdN exp i dt pa NH i  0     ˆ   f dT exp iTH i i     ˆ 1 1 1   ˆ     f H i 2 3 H p a U a ( )  2 a a    f i   E 0 E 0 1 C C      matt rad U a ( )   2 3 4  a a a      E E  E E a : radius of the universe Que uestion: stion: Is Is the here e a n a nat atur ural al cho hoice e for the hem? m?

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

Evolution of the universe S 3 topology U a ( ) U a ( ) U a ( ) a *          cr cr cr Λ～ curvature ～ energy density WKB solution         4   S d x g R matter 1     z         a a , sin da p a ,    E 0   1 0 a p a ,      with 4 p a , 2 a U a ( ). 1 C C      matt rad U a ( ) 2 3 4 a a a

For th the fi final al st stat ate, we hav ave tw two poss ssib ibili ilities ties. .    Final state: case 1 Final state: case 1 cr The univ iverse erse is is cl closed sed. We as assu sume th the fin inal al st stat ate is is finite       f a matter . The pat ath in inte tegr gral al     ˆ    Z f H i  1   2   const .  0 E

   Final state: case 2 Final state : case 2 cr The univ iver erse se is is op open. 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:           Z d exp i S . 1 As As an an ad ad hoc as assu sump mpti tion n we consid sider er a   f lim c a a matter . IR IR  a IR

(cont’d) Then th the par arti titi tion on function tion be become omes             * Z c a a   1 IR E 0 IR E 0   1          3 * sin c a a   IR IR E 0 4 a IR   1 sin          3 * c a .   IR E 0 4   1     z       a sin da p a    E 0  1 0 a p a 1 C C        mat rad U a ( )    4 p a , 2 a U a ( ) 2 3 4 a a a The resu sult lt does s not d t depend on e excep cept t for th the ph phas ase a IR whic ich come mes s from m th the cla lass ssical ical ac acti tion. n.

Thus we we have ave   the e pat ath integr egral al fo for a a univer verse se Z  1 finite for    const of order 1, cr   1 sin for         3 const a .  cr IR 4 ∞  Then n th the i inte tegration gration fo for th the mult ltiverse iverse            Z d w exp Z . 1   has as a lar arge e peak ak at at , , wh which h means ans that at    cr 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. .

Maximum entropy principle 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 3 S 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. U a ( ) 1 C     rad U a ( ) 2 4 a a  1/ C cr rad    cr Then th the mu mult ltiv iverse erse par arti titi tion on funct ction ion is is gi given ven by by            exp Z d w Z 1     1      exp const exp const C . 4    rad   4 cr 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.

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

Flatness of the Higgs potential We m e may y un unde derstand rstand the he fla latnes ness s of of 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.

         Higgs potential 4 V 4 m Higgs =125.6 GeV mH mH = 1 125. 5.6 6 GeV 𝜒[GeV]

Multi-local action from IIB matrix model

Ishibashi, HK, IIB matrix model Kitazawa, Tsuchiya     1 1 2  A              S Tr A A , , .        Matrix   4 2 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.

Y. Kimura, Covariant derivatives as matrices M. Hanada and HK The basic question : In the large-N reduced model, a background of simultaneously diagonalizable matrices (𝟏) = 𝑸 𝝂 corresponds to the flat space, 𝑩 𝝂 if the eigenvalues are uniformly distributed. (𝟏) = 𝒋𝝐 𝝂 In other words, the background 𝑩 𝝂 represents the flat space. How about curved space? Is it possible to consider some background like (𝟏) = 𝒋𝛂 𝝂 ? 𝑩 𝝂

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, 𝝌 𝜷 : 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:     V V V V . r reg reg reg

In particular the Clebsh-Gordan coefficients for the decomposition of the tensor product of the vector and the regular representaions     V V V V vector reg reg reg   b , are written as C , ( a 1,.., D ).  ( ) a 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 , C    ( ) a b is a regular representation field on 𝑵 . In other words, if we define 𝛂 (𝒃) by         , b C ,   ( ) ( ) a a b  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 of the IIB matrix model as ( ) , a 1, , D 0 a , A a 0, a D 1, ,10 where 𝛂 (𝒃) is the covariant derivative on 𝑵 multiplied by the C-G coefficients.

Good points and bad points Good point 1 Einstein equation is obtained at the classical level.   A i In fact, if we impose the Ansatz a ( ) a      on the classical EOM A A A , 0,   a a b we have         , 0     ( ) a ( ) a ( ) b          0 , ,   a a b         cd cd ca , R O ( R ) O R   a ab cd a ab cd ab c       c d R 0 , R 0 R 0 . a a b ab a b 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 𝝌 𝜷 as 𝝌 → 𝟐 + 𝝄 𝝂 𝝐 𝝂 𝝌 and 𝝌 → 𝟐 + 𝜻 𝒃𝒄 𝑷 𝒃𝒄 𝝌 , respectively. Both of them are unitary because they preserve the norm of 𝝌 𝜷 𝝌 𝜷 𝟑 = 𝒆 𝑬 𝒚 𝒉 𝝌 𝜷∗ 𝝌 𝜷

Bad points Fluctuations around the classical solution (𝟏) = 𝒋𝛂 (𝒃) 𝑩 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 (𝟏) = 𝒋𝝐 𝒃 ⊗ 𝟐 𝒔𝒇𝒉 , 𝑩 𝒃 where 𝟐 𝒔𝒇𝒉 is the unit matrix on the space of the regular representation.

Low energy effective action A. Tsuchiya, Y. Asano and HK 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.

The multi-local action is the consequence of the well- known fact that the effective action of a matrix model contains multi trace operators. More precisely, we first decompose the matrices 𝑩 𝒃 𝟏 and the fluctuation 𝝔 : into the background 𝑩 𝒃    0 A A . a a a 𝟏 contains Here we assume that the background 𝑩 𝒃 only the low energy modes, and 𝝔 contains the rest. Then we integrate over 𝝔 to obtain the low energy effective action.

Substituting the decomposition into the action of the IIB matrix model, and dropping the linear terms in 𝝔 , we obtain  1 2    0 0 , S Tr A A   a b 4   2                 0 0 0 0 0 2 , , , 2 , , A A A A A         a b a b a b a b b a               2  0 4 A , , , fermion .   a b a b a b   1 2    0 0 In principle, the 0-th order term S Tr A , A   0 ( ) a ( ) b 4 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 = 𝑳 𝒃𝒄𝒅⋯ , 𝒒𝒓𝒔⋯ 𝑼𝒔 𝑩 𝒃 𝟏 ⋯ 𝟏 ⋯ ) 𝟏 𝑩 𝒅 𝟏 𝑩 𝒄 𝟏 𝑩 𝒓 𝟏 𝑩 𝒔 𝑼𝒔(𝑩 𝒒 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     1-loop D S c S S , S d x g x O x ( ) ( ). eff i j i j i i 2 i j

Similar analyses can be applied for higher loops. In the two loop order, from the planar diagrams we have a cubic form of local y z actions   1 2-loop Planar S c S S S , x eff i jk i j k 6 i j k , , while non-planar diagrams give a local action    2-loop NP S c S . eff i i i x

We have seen that the low energy effective theory of the IIB matrix model is given by the multi-local action :         S c S c S S c S S S , eff i i i j i j i j k i j k i i j i j k   D S d x g ( x ) O ( x ) . i i

Although there is no precise correspondence, the loops in the IIB matrix model resemble the wormholes. x 𝑦 ≅ 𝑧 y We may say that if the theory involves gravity and topology change, its low energy effective action becomes the multi-local action universally.

Summary In wide classes of quantum gravity or string theory, the low energy effective action has the multi local form:        . S c S c S S c S S S eff i i i j i j i jk i j k i i j i jk 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.

Review on IIB matrix model

1. Definition of IIB matrix model

The basic idea “ Worldsheet of string has a structure of phase space .” This situation becomes manifest when we express the string in terms of the Schild action. 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. For simplicity, we start with bosonic string.

Schild action Basically, bosonic string is described by the Nambu-Goto action 1                2 2 ab , . S d X X NG a b 2 The Nambu-Goto action is nothing but the area of the worldsheet, which is expressed in terms of an anti-symmetric tensor 𝚻 𝝂𝝃 that is constructed from the space-time coordinate 𝒀 𝝂 .

It is known that the Nambu-Goto action is equivalent to the Schild action     1 2         2 2 S d g X , X d g ,   Schild 2 4 2 : a volume density on the world shee t g  1      ab , , X Y X Y a b g which is nothing but the Poisson bracket if regard the worldsheet as a phase space. The equivalence can be seen easily, by eliminating 𝒉 from the Scild action:     1 2          ab S 0 g X X   Schild a b 2 g    1  2           2 ab S d X X .  Schild a b 2 2

symplectic structure of the worldsheet The crucial point is that the Schild action has a structure of phase space. In fact it is given by the integration over the phase space   2 d g of a quantity that is expressed in terme the Poisson bracket     1 2    X , X .   2 4 2 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. 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:  function matrix  1     A B , A B , i 1    2 d g A T rA  2  W -symm etry U N ( )-symmet y r 

Then the Schild action becomes   1   2        S Tr A , A Tr 1 ,     Matrix 4 and the path integral is regularized like      dg dX dA        Z exp iS exp iS . Schild Matrix vol(Diff) SU n ( )  n 1 Here we have used the fact that the phase   2 space volume is diff. invariant and d g    becomes the matrix size after the 1 Tr n regularization. Therefor the path integral over g becomes summation over . 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   1   2        S Tr A , A Tr 1     Matrix 4 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   1 2      S Tr A , A .     4 This is nothing but the large-N reduced model, which I will explain in a moment.

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. 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.

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

In fact by analyzing the planar Feynman diagrams, we can show that the large-N reduced model d N      2 2      S Tr A , A         4 is equivalent to the d- dimensional Yang-Mills if the eigenvalues of 𝑩 𝝂 are uniformly distributed. However, it is not automatically realized. It is known that the eigenvalues collapse to one point unless we do something.