EMERGENT QUANTUM GRAVITY FROM SYM
D a v i d b e r e n s t e i n , U C S B 2 0 0 6
Based on :hep-th/0403110,hep-th/0507203 hep-th/0509015 with D. Correa, S. Vazquez hep-th/0511104 with D. Correa Work in progress Outline
AdS/CFT (and it’s dictionary)
Quantum hall droplets in AdS/CFT
Dual supergravity description
Beyond half BPS
All loop string energies The AdS/CFT correspondence
Conjectured exact quantum duality between:
N=4 SYM on a sphere Type IIB Superstring on AdS S5 5 ×
(0) O ∼8 |O#
dr2 ds2 = r2 + dΩ2 r2 3 ! " s s | #SY M ≡ | #AdS
ds2 = cosh2 ρdt2 + sinh2 ρdΩ2 + dρ2 − 3
tr(Zn)
1 1 = dt (D Z)2 Z2 L 2 t − 2 #
χR(a†)
Y ZZZY Z . . . ZZY n , n , n , . . . , n | # ∼ | 1 2 3 k#
1 We have a lot of reasons why the correspondence should work.
The big question is: how does it work?
More precisely: where does geometry come from? O(0) ∼ |O"
2 GENERALITIES2 2 dr 2 ds = r + dΩ3 Map states to states ! r2 "
|s"SY M ≡ |s"AdS
Map operators to operators For example: Superconformal algebra
Want to identify superconformal multiplets
1 O(0) ∼ |O"
2 2 2 dr 2 ds = r + dΩ3 ! r2 " In the gravity side the SYM time corresponds to global time |s"coordinatesSY M ≡ |s"AdS
2 2 2 2 2 2 ds = − cosh ρdt + sinh ρdΩ3 + dρ
Notice gravitational potential: Discrete spectrum Supergravity spectrum is known. The supergravity multiplet is BPS (protected by SUSY): This means we can compare with free field theory.
1 O(0) ∼ |O" Choose highest weight state of given representation 2 Highest weight half BPS states are2 build2 outdr of 2 ds = r + dΩ3 only one of the scalar modes ! r2 " of one complex field of the N=4 SYM theory. Let us call it Z |s"SY M ≡ |s"AdS Use complex basis for oscillators, so that that
2 2 2 2 2 2 a† Z a ∂ Z ∼ds =Z −∼ cZosh ρdt + sinh ρdΩ3 + dρ AdS S5 5 × All single particle (graviton) states tr(Zn) are descendants(0) of (Witten, Gubser, Klebanov,O Polyakov)∼ |O#
dr2 ds2 = r2 + dΩ2 r2 3 ! " s s | #SY M ≡ | #AdS
ds2 = cosh2 ρdt2 + sinh2 ρdΩ2 + dρ2 − 3
tr(Zn)
1 1 = dt (D Z)2 Z2 L 2 t − 2 #
χR(a†)
Y ZZZY Z . . . ZZY n , n , n , . . . , n | # ∼ | 1 2 3 k#
1
1 Want to study all BPS states, not just single particle.
At large quantum numbers some BPS configurations are semiclassical solutions to the equations of motion (coherent states for example) a† Z a ∂ Z ∼ Z ∼ Z AdS S5 5 ×
(0) O ∼ |O# Look at all states that are half BPS dr2 ds2 = r2 + dΩ2 r2 3 All of these half BPS states have! to be (descendants" of) multi-traces of the complex scalar field Z s s | #SY M ≡ | #AdS
2 2 2 2 2 2 ds = cosh ρdt + sinh ρdΩ3 + dρ Because these states are− supersymmetric, there are no quantum corrections to theirtr (effectiveZn) dynamics
1 1 = dttr (D Z) 2 Z 2 L 2| t | − 2| | #
χR(Z)
Y ZZZY Z . . . ZZY n , n , n , . . . , n | # ∼ | 1 2 3 k#
1 a† Z a ∂ Z ∼ Z ∼ Z AdS S5 5 ×
a† Z a ∂ Z ∼ (0)Z ∼ Z AdSO S∼5 |O# 5 × 2 2 2 dr 2 ds =(0)r + dΩ O ∼ |Or#2 3 aZ† !Z aZ ∂Z " ∼ 2 ∼ 2 2 dr 52 ds = r AdS+5 dΩS s SY Mr2 ×s 3AdS | # ! ≡ | # " (0) s O s ∼ |O# 2 | #SY2M ≡2| #AdS 2 2 2 ds = cosh ρdt + sinh ρdΩ3 + dρ dr2 − ds2 = r2 + dΩ2 ds2 = cosh2 ρdt2 + sinhr22 ρdΩ23 + dρ2 − ! 3 " n strSY(MZ )s AdS tr| (#Zn) ≡ | # 2 2 2 2 2 2 ds = cosh ρdt + sinh ρdΩ3 + dρ − 1 2 1 2 = dttr1 (D 2Z)1 2 Z = dttr (DtZ) t Z We needLL classical solutions2| 2| of |then − form|2| −| 2| | ## tr(Z )
Z(t) = Z0 exp(1 it) 1 Z=(t)dt=tr Z(0Dexp(Z) 2 it) Z 2 L 2| t | − 2| | These saturate the classical# BPS bound.
We also need solutionsZ of(t) the= ZEOM0 exp( ofit )the gauge field. Gauss’ constraint: momentχR(Z map) of generator of gauge [transformationsπ , Z] [π vanishes.¯, Z¯] = 0 Z − Z [π , Z] [π ¯, Z¯] = 0 Y ZZZY Z . . . ZZYZ −n1,Zn2, n3, . . . , nk | Leads to the following# constraint∼ | on the set of BPS# configurations: the matrix Z is normal. [Z, Zχ¯R](=Z) 0
Y ZZZY Z . . . ZZY n , n , n , . . . , n | # ∼ | 1 2 3 k# χR(Z)
Y ZZZY Z . . . ZZY n , n , n , . . . , n | # ∼ | 1 2 3 k#
1 1
1 Two simple ways to solve quantum system: Method 1: Take axial gauge, solve system (free matrix Impose Gauss’ law 5 D-branes on AdS5 S : giant gravitons. harmonic oscillators) × 5 Are there interesting half BPS states in AdS5 S ? × Fix N large, but do not take the infinite N limit Spectrum• is set of multitraces of Z Notice that traces are algebraically independent up to order N Algebraically • T r(ZN+1) = Polynomial in T r(Zk) independent • up to order N This upper limit on the gravitons is called a stringy exclusion principle. • This basis in notSus orthogonalskinds idea: strings(it is approximatelywith a lot of energy grow transversely to their direction orthogonal in theof motion. large NAlso limit,holographic for fixedar gumenergy)ents suggest that this is necessary to avoid making a black hole.
8 a† Z a ∂ Z ∼ Z ∼ Z AdS S5 5 ×
(0) O ∼ |O#
dr2 ds2 = r2 + dΩ2 r2 3 ! " s s | #SY M ≡ | #AdS ds2 = cosh2 ρdt2 + sinh2 ρdΩ2 + dρ2 − 3
n This is provitrded(Zby)Schur polynomials.
Z is a (operator valued) matrix in the adjoint of the group U(N). One can Aconside more rcomplicatedZ as generic waymat torix makeof GL gauge(N, C). invariantsThe trace makesof Z can a be taken in a recompletepresentat orthogonali1on of the group2basis.1 whicThis2h isis vianot charactersthe fundamental. of U(N). = dt (DtZ) Z L(Corely,Jevicki,Irreps are give Ramgoolam)2 n by Young−ta2bleaux with columns of length less than or equal to N#
χR(Z)
The label R is over irreducibles of U(N) (Young tableaux) Y ZZZY ZThe. .se. ZfoZrmYa completen1, nba2sis, nof3,half. . . BPS, nk states, one for each tableaux ( | Jevicki et al). Energy# ∼ | is the number of # boxes of young tableaux
S- Giants are columns AdS giants are rows. (CJR) 12 (Balasubramanian, Berkooz, Naqvi, Strassler)
1 Method 2: Choose a gauge where Z is diagonal Write everything in terms of eigenvalues
Classically we obtain N free particles in harmonic oscillator.
Permutation of eigenvalues is Two important a residual gauge symmetry effects: Change of variables implies change of measure After all of this is taken into account, we obtain N free fermions in the harmonic oscillator
(Brezin, Itzykson, Parisi, Zuber)
The hamiltonian is therefore
1 2 2 2 µ− ∂z µ ∂z + mz 2 − i i i !" # Transform Consider describing the system in terms of the following wave-functions ψ˜ 1 2 2 2 ψ = ∆− ψ˜ µ = (zi zj) = ∆(Z) − i 19 20 Basis of Slater determinants coincides exactly with character basis (Schur functions) Go to the phase space of the system, this is a two dimensional plane. (Berenstein) hep-th/0403110 Fermi sea is a circle centered around origin. Fermi statistics and uncertainty principle make these behave as an incompressible fluid (QHE droplet) Basic Dictionary between QHE and BPS states. Edge excitations are small gravitons (follows from work of M. Stone and others) 0ARTICLE Giants growing$ BRANES into AdS are electrons away (OLE from droplet ' % R D A V G I E T O S N T A Giants growing TEinto S are holes away from edge. Beyond linearized supergravity Suggestive that one can describe arbitrary droplet shapes in supergravity. Coherent states What objects are usually well of gravitons described by supergravity? Large stacks of coinciding D- branes Indeed, there is a formalism that can reconstruct a geometry associated to any (macroscopic) droplet configuration. (Lin, Lunin, Maldecena) 2 2 i 2 2 2 i i 2 2 ds = h− (dt + Vidx ) + h (dy + dx dx ) + yexp(G)dΩ3 + yexp( G)dΩ˜ 3 2 − − h− = 2ycosh(G) y∂yVi = "ij∂jz, y(∂iVj ∂jVi) = "ij∂yz 1 − z = tanh G 2 One canThesho solutionw that i tisall indep termsends ofon az .3dAnd PDEfrom withthe tboundaryhird line it follo ws that z satisfies y = 0 is special. It correconditionssponds at y=0to degenerations of the spheres. To have singularity free solutions z has to take special values. Other values lead to ∂yz ∂i∂iz + y∂y = 0 cone-singularities. y ! " There a(Aregenetwralizedo spLaplaceeRegularitycial-likvaluee equation) impliess of z = 1/2 that correspnd to only one of the spheres degenerating in such a way±that the degeneration looks just like spherical cThisoo rdinatgives a twoes. coloringGiven ofz thein theplaneb whichounda is identifiedry it ca nwithbe rebuilt in the interior the droplet picture of the QHE 36 of the x1, x2, y geometry. One also finds quantization of area THe boundary y = 0(Diracis tw o-diquantizationmensional condition)(x1, x2), and it is painted in two colors These configurations correspond exactly with the quantum plane description I gave before. 37 Quantum mechanically, we know that in principle all possible configurations on phase space aQreuvanatliudmwmitehchsaonmiceallpyr,owbeakbniloiwtythdaetnisnitpyridnecitpelremalilnpeodssbibylethcoenwfiaguvreaftuionncstoionnp.hWasehy willshfoourldmswpeaacpe iarcrkee avthasliiosd nowniatehbosvoleymreasplrhloobatahpbeileritsdy? dWeonbesitwjyeadncetttetrmowiuninetddhebrysattahnecdwoapnvretefciuinnsecultyionhu.oWws htdyheisnsity profile ρ("x), so that sumdsescaripptsphiooenualdforwliloenwpgsickiinnthdiesβtoanHile forvoecmraanall motbhiceerros?sacoWppepicwrdaonetxsctiromiputniaodnteresotdfantdhbepyrseycsistneelmytehvgoiwaratsholims ethat depend on ρ, plus a mathemdaestcirciapltifoonrmfoullloawtsioinn.dTethaiisl firsomwhaatmwicerowsciolpl idcodeinscrtihpitsiosnecotfiothne. system via some coHownstraT tohinme t aigettdhefeaom rainfermionticttahl efeoremntduolati tsdensity?iaosnli.mnTphuleis.misTbwahekaert wtoheefwmiplludalotri-itnpiatcrhtlisiecslsectwtiooanv.ebfeuncetqioun aofl tthoe N which is a very large ground staTteheanideacoimn ptuheteeintd’sisqsuimarpel.e.ThTeakwe atvhee fmunulctti-iopnartiisclgeivweanvebyfunction of the number.grTounhdisstatiesangdicvoemnputbe yit’stsqhuearef. oTlhleowwavienfugnction is given by ψF = (zi zj) exp( zz¯/2) ψF = −(zi zj) ex−p( zz¯/2) i – 13 – rings. These are the ones that correspond to energy eigenstates of the Hamiltonian, and therefore have extra symmetry. The expectation is that in general for other droplet configurations we can use the coarse grained dynamics to describe the salient features of the distribution of particles, and that this is the essence of what semiclassical gravity is capturing: the leading asymptotic expansion about a coarse grained saddle point configuration, which in our case depends on the wave function of the CFT dual in a very precise way. It is also important to notice that in this coarse grained description we have given, configurations of droplets of density one can arise from very few pure states in the quantum system. This is because we need to pack the Fermions as much as is possible in phase space, and this leaves no degrees of freedom left over that could account for various micro-states having the same coarse grained description. This suggests a reason why some half-BPS geometries have singularities. These singular willgfeoormmetarierseacsoorrneaspbolyndshtoapreegdioonbsjoefctpowsiittihveadcenosnittyinbueoluloswdtehnesictryitipcarol fivaleluρe(["x1)5,] s(ao that more recent discussion can be found in [23], where the structure of the singularities is sums appearing in βH can be approximated by integrals that depend on ρ, plus a clarified), like the half-BPS superstars [22]. In the coarse grained description we have consgtirvaenintabfoovre,thoneetcoatnalfinnudmabloert ooff wpaavretifculnecstitoonsbwe heiqchuadliffteor Nby wmhinicuhteivsaariavteiornys large numobf etrh.eTlohciasl idsegnisviteyn, wbhyicthheisfoalllloowweidngbecause the packing of fermions is not tight. These variations nevertheless can give rise to orthogonal states, so there are many z z¯ d2xρ(x)"x2 (2.13) pure states which end up corresipion→ding to the same saddle point: the coarse grained i description would not det!ect these va"riations. This suggests that the the curvature singularities in this case are related to entro2py.2It is also possible that some special 2 log( zi zj ) d xd yρ("x)ρ("y) log( "x "y ) (2.14) micro-states for these p|ack−ings |mi→ght have a nice large N descr|ipt−ion |(for example i The idea now is to write a variat2ional principle for βH to find the most likely ψ Zgravity (2.21) configuration of particles (the one w| it|h∼least energy will be sharply peaked, because of tahnedlcaoragresen-gurmainbOneeedr o ocanbfsed rshowevagbrel eethatssaos coherentfafdriecetdioo nstatesmar)y, o boffy ttraceshveafroyl linionw gtheinHg fermionsowritth respect to ρ and picture produce setting that variation to zero. This w∗ill produce the following integral equation deformations of the ψfermionψ density˜ Z which match the (2.22) supergravity deformationsO of the∼ "functionO# z at the boundary 2 ! 2 where ˜ can be related"xto+anCex=te2nsivedqyuaρn(tyi)tyl.ogT(h"xe lef"yt h)and side is the square(2.16) O | − | of the microscopic wave function of "the Universe (in terms of the dual CFT wave which is valid only for those x where ρ(x) = 0, as where ρ(x) vanishes the variation function), which exists in a minisuperspace# approximation. This can be considered is coanssatraminineidsu.pIenrspthaceeawbaovvee,fuCncitsioan LaalgaraHnagretlem-Hualtwipkliinegr [e2n4f]o, recxincegptthtehactoninstorauirnt on the cnausme bweerhoafvfeerΛm – 13 – Contents 1. Introduction 1 where th2e.nk dGenaoutegmedultim-inadticreisx. Tqhuesaenatrue mconjmecetucrhedantoicbse eoigfencsotamtems ouf tthine g matrices 1 Hamiltonian of energy n , (2.22) 3. A saddle point appr|ojx| imation to BMN state energies 6 j ! above the ground state, which are moreover approximately orthogonal in the large N 4. Conclusion 9 limit [3]. This follows from identifying these states with the corresponding graviton states in the = 4 SYM theory. These give an approximate Fock space of oscillators, N one for each multi-index, on which one can take coherent states. These coherent states can be analyzed using similar techniques as those used above, and they give wave-like shape deformations of the five sphere, also with singular support in the embedding space R6. 1. Introduction 3. A saddle point approximation to BMN state energies Now we2w.ant Gto ausue gtheedresmultsaotfrtihxe laqstuseacntiotnutmo calmculaetce henaerngiiecs sof ostrfincgyommuting matri- modes in thecCeFsT. For this, we need an explanation of how the other modes of the SYM theory decouple to obtain the matrix model of commuting matrices. To do this we need to begin with the = 4 SYM theory compactified on a round S3. We Let us consider a Nmatrix quantum mechanics model for 2d Hermitian matrices of obtain the following action for the scalars rank N which commute with each other (we can equally consider it as a matrix 6 6 BEYOND m1/2ode BPSl for d nor1mal ma a2tric1esaw2hich co1m2mutae wb itbh eaach other). As argued in [?] such Ssc = dΩ3 dt tr (Dµφ ) (φ ) gY M [φ , φ ][φ , φ ] . (3.1) 3 2 − 2 − 4 a Smodel re#sua=lt1s from consideringa,beither 1/2, 1/4 or $1/8 BPS states in = 4 SYM States" that! respect 1/8 SUSY!=1 N compactified on a sphere, where d = 1, 2, 3 respectively. The BPS states near the The mass term for the scalars is induced by the conformal coupling of the scalars to the curvEnergyvataucrue uomf thae r=Se3 ,Angularmwahidche ios fchmo sumomentumenlttipo lheavgerraavdiituasteiqounaal ltoqounaen. tTah,issaolsothsetys can be described in a the scalepfuor etlimy egdeeorimvaettivreisc. Wfasithhiothnis. nIonrmtahliizsatisoenc,ttihoenvowluemwe iolfl tdheeaSl3 wis i2tπh2.the systems where the To smtuadytrBicPeSs coonmfigmuruattieonws,itohnoeunteeddesstcoricboinncegnthroatwe otnhethoetchoenrstdanetgrmeoedseos f freedom of the = 4 In free ofieldf the φ alimits, while cankeepin beg ev ewrittenry other mo inde intermsthe vac uofum . 3T hs-waveis gives an effective N SYM decouple. reduction to a gauged matrix quantum mechanical model of six Hermitian matrices. complex scalar components: X, Y, Z i This model, aLfteetr uressclaalbinegl tthheemHaetrrimcesittioanhamveaatrkicineestibc yanXd q,uiad=rat1i,c.p.o.t2endt,iaalnd the complex normal term as min athterilcaesst sbeyctiZonj, i=s oXf t2hje−f1ol+lowiiXng2jfo,rmfor j = 1, . . . 2d. We require moreover that 6 6 1 1 1 i j S = dt tr (D Xa)2 (Xa)2 g[X2 ,[XXa, X] =b][X0b, Xa] . (3.2) (2.1) sc 2 t − 2 − 8π2 Y M #a=1 a,b $ " ! !=1 We will workTwhiethmthoids edlimheanssiaongalalyugreeduincevdarmiaondecle(uslnigdhtelry SmUod(iNfied))tirnanwshfaotrmations, where we act i i −1 follows. by conjugation on all the matrices simultaneously X UX U . It is easy to see → ContributesThetmhattrictehseatcot hntoisstpr aoEiinn ttbuta2re.1n onotits rienq vutoairrei daJntot cuonmdmeurtet.hIefsweetdriaangosnfoarlimzeaXti1o, ns. and under thNe oaswsumwpetiwonanthtattoitsoelivgenvaalGueasuasrseiaonf omrdaertr√ixNm(aosdceallcquulaatendtuinm mechanics associated Setthe p ritev iotous szeroection, byand ahandlso as ex p(BPSected fro mcondition)usual matrix integrals), we find that by puttintog vtehvessien mthae tinritcereasc.tiWonetehrmavceomtwinog ofrpomtiotnhes choemrme:utwateorcs,atnhecoeffnescitdiveer a first order dynamics where Z¯j and Zj are canonically conjugate variables (this is equivalent to stating that X2i−1 and X2i are canonically conjugate, or we can consider a second order – 8 – – 1 – Contents 1. Introduction 1 2. Gauged matrix quantum mechanics of commuting matrices 1 3. A saddle point approximation to BMN state energies 6 dynamics where we also include the time derivatives of the matrices (this doubles 4. Cotnhcelnuusmiobner of matrices effectively). 9 The Hamiltonian will look as follows 1 1 H = tr(Π2) + tr((Xj)2) (2.2) dynamics where we also include t2he timje der2ivatives of the matrices (this doubles thBeencuamusbeerwoef hmaavteriaceSs Ueff(eNct)ivaeclyti)o.n which leaves the model invariant, we can gauge this SUT(hNe )Haacmtioltno,naianndwwilel cloaonkaasskfoalbloowust the singlet sector of the matrix model. This 1 1 is the model we will concern Hou=rselvtre(sΠw2)it+h. tr((Xj)2) (2.2) 1. Introduction 2 j 2 Because we have this SU(N) action on the matrices, we can exploit the fact that Because wie have a SU(N) action which leaves the model invariant, we can gauge thethmisaStUric(Nes) Xactioanre, ahnedrwmeitciaannatsok aubsoeutatSheUs(inNgl)ettrseacntsofroormf tahteiomnattroixdmiaogdoeln.aTlihzeis any 2. Gauged matrix quantum m1 echanics of commuting matri- oneisotfhethmeoXdelmwaetwriiclel sc,onlecterunsosuaryselXves. wBitehc.ause the matrices commute, they can be ces 1 diagonBaleiczaeudsesiwmeuhltaavneetohuisslSy,Us(oNi)f awcetiodniaognonthaelimzeatXric,esw, ewedicaagnoenxapllioziet athlleoftahctertshatt the samtheetmimaetr.ices Xi are hermitian to use a SU(N) transformation to diagonalize any Let us coonnTseihdoiesfrtrheade uXmceamstrtaihtxreicnqeuus,malnebteurumsofsadmyeegXcrhe1e.asnBoiecfcsfaruemseedoodthmeel tmfooartthri2ecdeesigHceoenmrvmmaliuutteieas,nothfmetyhaectarmnicaebtsreicoefs. dynamics where we also include the time derivatives of the matrices (this doubles rank NInwddehieaidgco,hnfoacrliozemeadcmhsiumdtiuealgtwaonnietaohlucseolyam,csphooinofetwhneterdoifa(tgwhoeenamcliazanetrXiec1qe,suwaXellidyiawcgeooncaslniidzaesraslolitcoitaahtseersaa2mtdtahvtercitxor the number of matrices effectiveljyj). model foofresadgmenneovtarimlmuea.sl matrices which commute with each other). As argued in [?] such This reduces the numTbehreofHdaemgrielteosnoifafnreewdiiollmlotookthaesefioglelnovwaslues of the matrices. a model results from considering either 1!x/j2, 1(/X4joj)r 1/8 BPS states in = 4 SY(2M.3) ! i 1 1 N Indeed, for each diagonal component of the matrices Xjj we can a2ssociate a 2dj v2ector compactified on a sphere, where d = 1, 2, 3 respectivHely=. Tthre(ΠBj )P+S sttar(t(eXs n)e)ar the (2.2) In otfheisgefnovramluewse have removed all of the infinitesimal ga2uge transf2ormations on the vacuumXa. rHeomweavdeer, otfhemreulatrieplgelogbraalvtirtaantsiofonramlaqtiuoaninstwa,hiscoh tpheermy uctaenthbeeedigeesncvraiblueeds oinf tahe Because we!xjhave(XajjS)U(N) action which leaves the m(2o.3d)el invariant, we can gauge purely mgeaotrmiceetsriact tfahsehsiaomn.e tIinmet.hiTshseescetgioanugwe!etrawnisllfodrmeaaltiwonitshpetrhmeustyesttheemvsecwtohresre!x tihneto In this form we havethreims SovUed(Na)ll aocfttiohne,iannfinditweesimcaanl agsakugaebtoruatnstfhoermsiantgiolentssoencttohrjeof the matrix model. This matriceesacchomotmheurt.e wBeitchaouuset dofestchriisbifnacgt,howwavtehefuontchtieorndsehgarveestoofbfreeesydmommeotfritcheunde=r t4he X. However, there arise tghloebmalotdraenlswfoermwailtliocnosnwcehrinchopuerrsmeluvtees twheithei.genvalues ofNthe SYM dpeecromuuptlea.tions of the vectors !xj. matrices at the same timeB. eTchaeusseegwauegheatvreantshfiosrmSUat(ioNn)s paecrtmiounteonthtehveemctoartsri!xcjesi,ntwoe can exploit the fact that i Let uesaTlcahhbeoetlshytehsrt.eemHBeeccraamunsitettiahhounfestmhbiasettfrraiiincctee,essrwpXbaryvieetXaefrdue,nahciste=irosmne1sti,thi.oaa.fnv.eN2tdoto,buabosnseeodsnaytsmhSoemUnce(toNarmic)spputlarneacdxneesnrfwootirhrtmehaa2tlidon to diagonalize any j Effective2j−1 dynamics2j is a gauged Matrix1 matricedsimbpeyenrmZsiuotna=sti(oXonrs aof2tdh+edivioXmenceetono,rssfifoo!xtnhrja.ejl Xp=hma1s,ae.t.srpi.c2aedcse,.)l.WeteursesqauyirXe m. oBreecoavuesretthhaetmatrices commute, they can be If wTquantumehetrseyasttetmhe cs aymechanicsnstdetihmaugsoclnbaaeslsiiizcneatd eloflryspi,r mcommutingwetueeldtcaaannseuosuseetslayo,fdsiN oamatricesgifobnwoaseolndasinasoganotnzaatlosipzfieancXed1ws,oiwtluheti2dodinasgoonf alize all others at the i j thedidmyennasmioincsal(osryast2edmd.saimUmneendsteiiormntae[hXl.epseh,aXassesu]spm=acp0et)i.ons we find N free harmonic osci(ll2a.t1o)rs inCan2d d Idiagonalizeifmweentsrieoantst,hwe shyi scallthems hmatricesTcolhauislssdircebadleluytc,re wesimultaneously,saettcheaednnauussmeNabedirdiaeognfotdniceagal rla enbypesasarto tzgaugefictfolreefisend(dbo mososlotunotsiot)hnoesnoefiage2ndvalues of the matrices. Thdeimmtheoendsdeiyolnnhaamal sihcaarlmgsyaosuntgeimce .oinsUcvnialdlraeitraontrhc.eeseuansdsuemr pStUion(Ns w)etfirnadnsNfofrrmeeahtaiormnso,niwc hoescriellaiwtoersact Indeed, for each diagonal component of the matrices Xjj we can associate a 2d vector in 2d dimensions, whichtransformationsshould be treated as .N ideintical paritic−le1s (bosons) on a 2d by conjugaQtiuoanntounmalml etchheanmoicafatelrlgiyec,newvsaesluicmeasnulntaontedooustlhyatXimmedUiaXtelUy. T.hiIst iissbeeacsayusteotsheeere dimensional harmonic oscillator. → that thaerecomnesatrsauirnetfa2c.t1oriss itnhvaatraiarinste ufrnodmerthtehevsoelutmraenosfotrhme agtaiuo!xgnes.orb(Xit,i a)nd which affect (2.3) QAssociateuantum mec haa n6-vectorically, we c apern no eigenvaluet do that immediately. jThis isjjbecause there the dynamics of the system. This measure factor has been co!mputed in [?], so that Now awree mweaansturteofascotlovres taIhnaGttahauirsisssefoiafrrnmommwtaehtehriavxvoelummroeemdoeoflvteqhdueaagnlaltuuogfme tohrmebieticn,hfiaannditiewcsshimiachsaslaogffcaeiuacttgeedtransformations on the Measure term from 2going to eigenvalue2 basis to these mthaetrdiycneasm. iWcseofhtahveesXtyws.toeHmoo.pwTteivhoeinsµrs,mth=eheaersruee:reawr!xfaeeicgtcolaor!xnbhjaaclsotnbrsaeiendnsefcorormampfiautrtisoetdnosirnwd[he?ri]c,dhsoypnteharamt(ui2ct.se4)the eigenvalues of the ¯j j | − | where Z and Z are canonmiactarlilcyescaotntju2hgeastaie – 2 – dynamics where we also include the time derivatives of the matrices (this doubles the number of matrices effectively). The Hamiltonian will look as follows 1 1 H = tr(Π2) + tr((Xj)2) (2.2) 2 j 2 Because we have a SU(N) action which leaves the model invariant, we can gauge this SU(N) action, and we can ask about the singlet sector of the matrix model. This is the model we will concern ourselves with. Because we have this SU(N) action on the matrices, we can exploit the fact that the matrices Xi are hermitian to use a SU(N) transformation to diagonalize any one of the X matrices, let us say X1. Because the matrices commute, they can be diagonalized simultaneously, so if we diagonalize X1, we diagonalize all others at the same time. This reduces the number of degrees of freedom to the eigenvalues of the matrices. i Indeed, for each diagonal component of the matrices Xjj we can associate a 2d vector of egenvalues !x (Xi ) (2.3) j ! jj In this form we have removed all of the infinitesimal gauge transformations on the X. However, there are global transformations which permute the eigenvalues of the matrices at the same time. These gauge transformations permute the vectors !xj into each other. Because of this fact, wave functions have to be symmetric under the permutations of the vectors !xj. The system can thus be interpreted as set of N bosons on a space with 2d dimensioisnasn(eoirgeanfu2ndctdioimn oefntshioenaablovpehHasaemislptoanciean).. Since it is real and positive it is very If wleikterlyeatthattheitsryesptreesmenctslatshseicgarloluyn, dwsetactaenouf stheeasydsitaegmo.nTalhaisnwsailtlzbteoorfitnhdogsoonlaul ttioons of 2 ˜ the dynaotmheicrawlasvyesftuenmct.ioUnsnodfedrifftehreesnet eansesrugmy pbytiuosninsgwtehefimndeasNurefrµee. hNaarmmeolyn, ilcetoψscbilelators another eigenstate of H with different energy. Then is an eigenfunicnt2idondimoefnstihones,awbhoicvheshHoualdmbielttroenatieadna.s NSiindecneticiatl ipsarrtiecalels (abnosdonps)oosnitaiv2de it is very dimensional harmonic oscillator. i 2 ˜∗ likely that it represents the ground state odxfjµthψ eψ =sy0stem. This will b(2e.7)orthogonal to Quantum mechanically, we can! n"ot do that immediately. This is because there other wave fuanrce tmioeanAsuslsroeo, fvaecvdtsoioffrfseotrbhseaenrtvtaabrlieessenwefrirlolgmbyetehbveaylvuoautluesdmiwneigtohf tµth2h,eeagnadmutgeheiasosirnubigrtee,nearµnadl2m.wahkNiecshaitmaffeeclty, let ψ˜ be the dynahamrdictsoodfotahcealscyuslatteimon.. TIthisiscomnveeansiuenrtetfoapcterofrorhmasa sbimeeinlarcitoymtrpauntsfeodrminati[o?n],asnod that another eigenstate aobfsoHrb a wfacittorhofdµiiffnteortehenwtaveenfuenrctgioyn.s, sTo thhaetn µ2 = !x !x 2 (2.4) | i − j| i<ψˆj = µψ (2.8) i!2 ˜∗ and the reduced hamiltonian is givdexn bµy ψ ψ = 0 i (2.7) and the measure factor associated tjo ψ is the usual dxj, which is N copies of the measure for a single eigenvalue. Notice that µ is the square root of a function which ! 1 2 1# 2 H =" iµ i + !xi (2.5) is symOnemetr icanc in tfindhe ex cgroundhange of a −state,ll t2hµe2v∇ eandctors∇ absorb"xi. So2i|f t hsquare|e particl eroots are bofos ons with i 2 Also, vevs of obserrvesapebctlettheosth wemeasuremilealsubreeµ ,inethv wavee"aplaurtiac lfunctions:tesegdivenwbiytthh copiede wµave, fu afromnnctdion ψtfreeˆharies alsionbogsoenns, eral makes it We wairteh nthoewuisnutael rmesetaesudreinfosrteuadcyhibnfermionsogsotnh.eTghriosurnegdulsatraittye, wwhaevree wfuenhcatvioenNoifdtenhteicsayl stem hard to do a caanldcsuollvaciotnpigioestnho.ef tsIhyetstmiesemascuinorentfhavecetotnrhieoerfmnanotdintydonivaimpduieaclrlfbiomosroitmn., Imtaatkuessrinmits poiolusatsirbtilhetaytto ttrreatnthseformation and system thermodynamically, because we can place all bosons on the same phase space absorb a factor of µ into the wave functions, so t2hat and ask about the distributiψon0s ofepxaprt(icles. !x /2) (2.6) ∼ − i Now we want to study the large N lim"it of the distribution of these bosons for the wave function ψˆ. If we squaψrˆe ψ=ˆ, wµe gψet a probability distribution on the phase (2.8) space of the 2N particles. This is given by – 2 – i and the measure factor assψˆo2 ciaµt2 eexdp( to ψx2)i=s etxphe usu"xa2 +l 2 dloxg "x, w"xhich is(2N.9) copies of the | 0| ∼ − i − i |j i − j| % i – 3 – ψˆ2 µ2 exp( x2) = exp "x2 + 2 log "x "x (2.9) | 0| ∼ − i − i | i − j| % i ψˆ2 exp d2dxρ(x)"x2 + d2dxd2dyρ(x)ρ(y) log("x "y) (2.10) | 0| ∼ − i − ' ! ! ( – 3 – is an eigenfunction of the above Hamiltonian. Since it is real and positive it is very likely that it represents the ground state of the system. This will be orthogonal to other wave functions of different energy by using the measure µ2. Namely, let ψ˜ be another eigenstate of H with different energy. Then i 2 ˜∗ dxjµ ψ ψ = 0 (2.7) ! " Also, vevs of observables will be evaluated with µ2, and this in general makes it hard to do a calculation. It is convenient to perform a similarity transformation and absorb a factor of µ into the wave functions, so that ψˆ = µψ (2.8) i and the measure factor associated to ψ is the usual dxj, which is N copies of the measure for a single eigenvalue. Notice that µ is the square root of a function which # is symmetric in the exchange of all the vectors "xi. So if the particles are bosons with respect to the measure µ, the particles given by the wave function ψˆ are also bosons, with the usual measure for each boson. This regularity, where we have N identical copies Interpretof the measu rcollectione factor of an ofind ieigenvaluesvidual boson, m aaskes positionsit possible t ooftr eat the system thermodynamically, because we can place all bosons on the same phase space and ask about the distparticlesributions of pinar t6dicle sphase. space. Now we want to study the large N limit of the distribution of these bosons for the Nwa vBosonse function inψˆ. I6df w ewithsquar elogarithmicψˆ, we get a pro brepulsiveability distri binteractions.ution on the phase space of the 2N particles. This is given by ψˆ2 µ2 exp( x2) = exp "x2 + 2 log "x "x (2.9) | 0| ∼ − i − i | i − j| % i ψˆ2 exp d2dxρ(x)"x2 + d2dxd2dyρ(x)ρ(y) log("x "y) (2.10) | 0| ∼ − i − ' ! ! ( – 3 – ρ is positive (the density of bosons) and the total number of particles is N. This is ρ = N. Now, we want to evaluate the distribution ρ by a saddle point approxi- mation. The idea is to treat the problem as a variational problem for ρ where we ! want to maximize the value of ψˆ2 which is our probability density. We impose the | 0| condition of the number of particles as a constraint with a Lagrange multiplier. We find that on the support of ρ #x2 + C = 2 d2dyρ(y) log( #x #y ) (2.11) | − | " In general, one can show that for even numbers of dimensions the function log(#x #y) − is proportional to the Green’s function for the operator ( 2)d, so that operating on ∇ both sides of the equation with this operator one finds that for d > 1, ρˆ vanishes. This is incompatible with the constraint that ρ = N. This is what we find under the assumption that ρˆ is a smooth function. ! DensityWhat we offin dbosonsthis wa yis ias tsingularhat ρˆ has configuration.singular support. SymmetriesBecause of sp hofer ical symmeensembletry, one ca nsuggestmake a stheimp lfollowinge ansatz for densityρ which hofas “eigenvalues”singular support. One takes a singular spherically symmetric distribution at uniform radius r0 δ( #x r ) ρ = N | | − 0 (2.12) 2d−1 2d−1 r0 V ol(S ) which has been properlWey no getrma liaz eroundd. One fivesees spherethis by transforming the integral d2dxρ(x) to spherical coordinates. Now we substitute this ansatz into 2.10, and minimize with respect to r0. Since ! 2 all Canpartic alsoles en daddup acoherentt the same rstatesadius r 0by, th tracese term w ofith theρ( xcomplex)#x is easy to evaluate. We find that this is equal to Nr2. The second term is harder to evaluate. This matrices. These keep the 0singular aspect of! density, but reqdeformuires int ethegrat ishape.ng over Thisrelativ eisa veryngles. similarThe ter mtow whatith th ehappenslogarithm is equal to log(r (1 cos θ)) = log( for the relative angle between two points on the sphere. This 0 in− gravity, we therefore identify it with gravity. term can be written as follows T (r ) = N 2 d2dxd2dyρ(x)ρ(y) log( #x #y ) = (2.13) 2 0 | − | " δ(r r ) δ(r" r ) dΩ2d−1 dΩ"2d−1drdr" − 0 − 0 (log(r ) + log(1 cos θ))(2.14) Vol(S2d−1) Vol(S2d−1) 0 − " Notice that in the above equation, only the first term of the sum will depend on r0, while the complicated angular integral will be in the second term of the sum. Thus we find that T2(r0) is equal to 2 2 T2(r0) = N log(r0) + N c (2.15) where c is a constant, independent of r0. From here, we need to minimize the function f(r ) = Nr2 N 2 log(r ) N 2c (2.16) 0 0 − 0 − – 4 – where the nk denote multi-indices. These are conjectured to be eigenstates of the Hamiltonian of energy n , (2.22) | j| j ! above the ground state, which are moreover approximately orthogonal in the large N limit [3]. This follows from identifying these states with the corresponding graviton states in the = 4 SYM theory. These give an approximate Fock space of oscillators, N one for each multi-index, on which one can take coherent states. These coherent states can be analyzed using similar techniques as those used above, and they give wave-like shape deformations of the five sphere, also with singular support in the embedding space R6. It is suggested that one treat the above operator as the following state: It is suggested that one treat the above operator as the !following state: 3. A saddle point approximatlion†jtoJ−Bl M†Nj ˆstate energies ψk,l exp(ikl/J) zjYj! zj! Xj ψ0 0 od (3.5) ! | ! ∼ ! l †j J−l †j | ! Now ψwe want tol useextph(eikrels/uJlts) ofj,tjhze lYast! zsec! tioXn to ψcˆalc0ulate energies of strin(g3y.5) | k,l! ∼ ! ! j j j j 0| !od modes in the CFlT. For this, we nje,ejd! an explanation of how the other modes of the where in the abSoYvMe nthoetoarytido!encouwpele htoavoebtaeixnp!thliecimtlaytrixthmeodweal vofecfoumnmcuttiiongnminatrtichees. cTooodrdo inate bawsihserfeorinththeedaiabthgoivos enwaenlnoetceaodtmtiopnboewgniennwhtiasthvoetfheetxhpel=icci4tolSmyYmMthuethteiwnoargyvecpoamfurpntaccotiiffioetndhoienn XathrioeumncdoaoStrr3d.iciWnesae,teand N whbearseis wfoer hthaveedoitabhgteaoinoaffthl-edcfoioamllgopwooinnnagelancmttisoonodfeorsthtwheercsicotatmleamrns uatsinfgrepeaortscoifllathtoerXs ai cmtiantgricoens, tahnedoff- where we have the off-diagonal m6odes written as free 6oscillators acting on the off- diagonal vacuum 0 od. As we have1 argua e2 d 1befao2re, thi1s 2is reaasbonbabale because the S|sc!= dΩ3 dt tr (Dµφ ) (φ ) gY M [φ , φ ][φ , φ ] . (3.1) diagonal vacuum 0 . 3 As we have2 argued−b2efore,−this 4is reasonable because the off-diagonal modes aroed Sheavy. #a=1 a,b $ | ! " ! !=1 off-Ndoiawgownealwmaondtetsoaerevahleuaavtye. the energy of the above state. We do this as follows. Now we waTnhte tmoasesvtaelrumaftoer thheesceanlaerrsgiys iondfutchede baybtohveecosntfaotrme.alWcoeupdliongthofisthaesscfaolallroswtos. the curvature of the S3, which is chosen to have radius equal to one. This also sets the scale for time derivatives.ψWk,ilthHthtoistanl oψrmka,llization, the volume of the S3 is 2π2. E #ψk,l H| total ψ| k,l ! (3.6) To study BPS coEnfig∼ura#tions|ψ, okn,le ψne|ke,dl s t!o concentrate on the constant mod(3es.6) a ∼ #ψk,l ψ| k,l ! Approximationsof the φ s ,thatwhile leadkeepin gtoev ecommutingry#other| mod!e in matricesthe vacuum. improveThis gives a nateff ective Now we can intredgurcatitoen toouatgatuhgeedomffa-tdriixaqguoanntaulmmmoecdheasn,icaalnmdodeelvoaflsuixatHeermthiteiainr meanterircgesy. over Now westrongcan int couplingegrate out!t Off-diagonalhe off-diagonal mmodesodes, a becomend evalua heavyte their. energy over This model, after rescaling the matrices to have a kinetic and quadratic potential thtehegrgoruonunddstsatattee aassssuummiinngg thatt tthheessee aarreefrfereeosocsicllialltaotrosrsjoijnoiinngintghetheeigenigveanluveasl.ues. term as in the last section, is of the followi"ng form ThTeheoffo-ffd-idaigagoonnaallmmooddeess bbeettwween eeiiggeennvvaalulueessj,jj, "jaraeretotobebteretareteadteodrtohrotghoongaolntaol tthoethe 6 6 ! 1 1 1 vevcetcotror"xj"xj "x"xj j,! ,bSbeecc=aauusseedtaatr ccomppo(oDnneeXnnatt)2aalolonn(gXgat)ht2hatatdidriercetcigot2inon[iXs aio,sXbotb]ab[Xitnabe,idXneab]dy b.ay(g3aa.2u)ggaeuge −− sc 2 t − 2 − 8π2 Y M #a=1 a,b $ trtarnasnfsofromrmataitoionnoonntthhee"ccoommmmu!tinngg mmaattrriciceess.. !=1 TThihsisgigvivesesuWusseaawnnilleenwneoerrkggyywiftohrthtthihseedoiomsscecniilslialoanttoaolrlysrswrewhduhiccihecdhismiosedqeeulqau(lsalitglohttoly modified) in what follows. The matrices at this point ar1e1not required to commute. If we diagonalize X1, 22 ! 2 2 EEosc = 22 11++ gg "x"xj "xj"x ! (3.7()3.7) and under the oasscsumption that22iπtπs22eiYgYeMnMv|a|luej−s −are |ojf|order √N (as calculated in the previous section, and""also as expected from usual matrix integrals), we find that AdAddidnigngththeeeneneerbrgygypyuototffinttghhveeevddsiiiangtohnne aainllteppriaiecectcieoenbbtyeyrumusicsnoinmggin2g.22f.3r2o3mwewthegeecgtoemtthmtauhttaattohrest,htteohettaeolffteeacntlieverengyergy isigsigvievnenbyby Etotal = J + Eosc (3.8) Etotal = J +# Eos!c (3.8) –# 8 – ! whwehreerewweehhavaveettooeevvaalluuaattee tthe avveerraaggeeeenneerrggyyoof fthteheosocsicllialltaotrofrorfotrhethweawveavfuenfcutniocntion we considered. This results into a multiple integral we considered. This results into a multiple integral i 2 l J−l 2 1 2 ! ! ! 2 dx ψ0 l exp(ikl/J) j,j zjzj 2 1 + 4π2 gY M "xj "xj i 2 l J−l 2 1 2 ! 2 dx |ψ0 | | exp(ikl/J) ! z z ! | 2 1 + 2 g | "x−j |"xj Eosc = l j,j j j l J4−πl Y M (3.9) | | | i 2 | & ! 2 | − | E#osc != # $ %dx ψ0 l %exp(2πikl/J) j,j! zjzj (3.9) i| | 2| & l J|−l 2 # ! # $ %dx ψ0 l %exp(2πikl/J) j,j! zjzj! Now we can evalu#at$e this|int|eg|%ral by a saddle p%oint approxim| ation, similar to whNatowwewdeidcianntheevaplrue#avtio$eutshsiesctiinotne.g%rTahlisbyis daosnaedidnletwpo%ositnetpsa.pFpirrosxt,imthaetinotne,grsaiml wiliallr to whbaetdwome idniadteind tbhyecponrefivgiuoruastisoencstiwohni.chTmhiasxiismdiozeneψin 2t,wwohsicthepssu.gFgeirststt,htahte"xintaengdra"xl !will | 0| j j beshdooumldinbaetelodcbatyedcoenxfiagcutlryatoinontshewshpihcheremwaxeimfouizned ψin t2h,ewphriecvhiosuusggseecsttiotnh.atT"xhusanwde "x ! | 0| j j shroeudludcebethleocinatteegdraelxtaoctalny ionntetghraelsopvheerrerelwaetivfeouanndgleins athsseocpiraetvedioutos tshecetpioons.itiTonhsusj we readnudcej"tohnetihnetespghraelret,owahnereinwteeghraavleoivdeerntriefileadtilvoecaatniognlsesonastshoecsipahteedretwoitthheindpiovsiidtuioanl s j anpdarjt"icolnestihneosuprhenres,emwbhleereofwtehehapvreviidoeunstsifiecetdiolno.cations on the sphere with individual particSleescoindolyu,rweensseeme bthleatofinthteheprliemvitouosf sJecltairogne. we can improve the saddle point appSreocxoinmdaltyi,onweduseeetotthhaet eixntrtahpeolwimerist ooffzJj, zlaj!rigne twhee ncuamneirmatporroavnedtdheensoamdidnlaetopr.oint approximation due to the extra powers of zj, zj! in the numerator and denominator. – 8 – – 8 – Can also suggest origin of string scale: off-diagonal modes are massive and can be represented by lines joining eigenvalues (points on the sphere): STRING BITS. Need to dress them with gravity (eigenvalues). One can also verify string tension which has been properly normalized. One sees this by transforming the integral 2d d xρ0(x) to spherical coordinates. Now we substitute this ansatz into (2.10), and minimize with respect to r0. ! 2 Since all particles end up at the same radius r0, the term with ρ0(x)"x is easy to evaluate. We find that this is equal to Nr2. The second term is harder to evaluate. 0 ! This requires integrating over relative angles. The term with the logarithm is equal to log[r (1 cos θ)] for θ the relative angle between two points on the sphere. This 0 − term can be written as follows T (r ) = N 2 d2dxd2dyρ(x)ρ(y) log "x "y (2.13) 2 0 | − | " " dΩ d− dΩ = N 2 2 1 2d−1 drdr"δ(r r )δ(r" r ) [log(r ) + log(1 cos θ)] . Vol(S2d−1)2 − 0 − 0 0 − " Notice that in the above equation, only the first term of the sum will depend on r0, while the complicated angular integral will be in the second term of the sum. Thus we find that T2(r0) is equal to 2 2 T2(r0) = N log(r0) + N c , (2.14) where c is a constant, independent of r0. From here, we need to minimize the function f(r ) = Nr2 N 2 log(r ) N 2c , (2.15) 0 0 − 0 − One can calculate the radius of the sphere exactly from where we find that N r0 = . (2.16) # 2 Notice that this result is independent of d. At first, this seems puzzling, but one can argue that this is the correct result by calculating the force particle i exerts on particle j in the directiSizeon n iso rindependentmal to the ofsp hnumberere. of matrices Looking at the figure 1, if the angle between the particles is 2θ, then the distance between them is l = 2r0 sin θ. The net force is then 2/l pointed along the straight line joining particles i and j. The normal to the sphere and this line meet at an angle of π/2 θ, and the force normal to the sphere from the particle at angle 2θ − (this force is pointing in the vertical direction in the figure) is then ij cos(π/2 θ) 1 Fv = − = , (2.17) 2r0 sin θ 2r0 which is independent of the angle that particles i, j subtend on the sphere, so long as they are both located on the sphere. This is why the result does not depend on d: the angular distribution of particles (how many particles reside at angle 2θ) does not matter to calculate the net force exerted on particle j. The upshot of the above calculation is that the distribution of eigenvalues is a singular distribution of particles. They form a thin shell of a sphere with radius r0 independent of d > 1. The radius is exactly N/2. $ – 6 – nent along that direction is obtained by a gauge transformation on the commuting matrices. This is explained also in [3]. The above Hamiltonian can be regarded as a small perturbation in the large N ’t Hooft limit as long as g2 !x !x 2 stays finite. Y M | i − j| Our approximation in what follows is that we will treat the off-diagonal modes as free fields, while we keep the information of the distribution of eigenvalues for the slow degrees of freedom exactly. For this approximation, the dimensional reduction of the matrix model is a reasonable description of the system for the degrees of freedom we are considering, since we are ignoring interactions between off-diagonal modes. When we include the fact that the matrix model is gauged, we need to be careful j about gauge invariance. This means that for each off-diagonal Xk mode arriving at l eigenvalue j, we need a second off-diagonal mode leaving it Xj. This means that the off-diagonal modes form a closed path between various points on the five-sphere. This is how we would like to think of a closed string state, where the off-diagonal modes can be labeled string bits. We will use this convention in what follows. We now want to calculate the approximate energy of a state consisting of a single string bit joining two eigenvalues. First, we need a way to identify which pairs of eigenvalues we are considering, and we also want to control the total angular momentum of the string bit on the S5, J. To obtain states with the correct value of J, one also needs to turn on the diagonal matrices to a state that is not the vacuum. This procedure can be considered as a gravitational dressing of the state to impart it with momentum. For large J, this will give us precisely the BMN limit [4]. We also need to make sure that we consider physical states of the gauge theory. Gauge invariance forces us to turn on at least two such oscillators. One from eigen- values i to eigenvalues j, and the other from eigenvalue j to eigenvalue i. From our results in the previous sections, eigenvalues are going to be associated to positions on the five sphere. Let us now consider a typical BMN-type operator J O exp(ikl/J)tr(ZlY ZJ−lX) . (3.5) k ∼ l One can calculate!=0 string energies for BMN states: For k = 0 this is very similar to the operator # J O exp(ikl/J)tr(Zl−1[Y, Z]ZJ−l−1[X, Z]) . (3.6) k ∼ l !=0 Using the operator state correspondence, we are supposed to relate the diagonal components of Z to the eigenvalues we found in the last section for the matrix Z. Let us call these eigenvaluDiagonales zi. We d omodesthis because the state is almost BPS with energy approximately equal to J. The presence of the commutators means we are Off-diagonal modes – 10 – turning on off-diagonal components of the fields Y and X. These are to be treated †i †k as raising operators in the quantum mechanical model (3.3); call them Yj and Xl for the corresponding matrix modes. We suggest that one treat the above operator as the following state in the reduced matrix model: J ! l †j J−l †j ψ exp(ikl/J) z Y ! z ! X ψˆ 0 , (3.7) | k! ∼ j j j j 0| !od l j,j! One !treats=0 the off-diagonal! modes as free fields. where in the aboOneve ncalculatesotation thewe energyhave e ofxp thelic iBMNtly th statee w aassumingve func ttheion off-in the coordinate basis for the diagdiagonalonal co mmodespon edon’tnts oaffectf th etheco diagonalmmutin onesg p atort firstof order.the X a matrices, and where we have the off-diagonal modes written as free oscillators acting on the off- The calculation of energies can be done in a saddle point diagonal vacuum 0 . | !od approximation. Now we want to evaluate the energy of the above state. We do this as follows: ψ Htotal ψ E # k| | k! . (3.8) ∼ ψ ψ # k| k! From the Hamiltonian (3.3) we see that, after subtracting the ground state energy, each oscillator will carry an energy 2 osc gY M ! ! 2 Ejj = 1 + 2 #xj #xj . (3.9) " 2π | − | Adding the energy of the diagonal piece by using (2.22) we get that the total energy is given by Etotal = J + Eosc , (3.10) # ! where we have to evaluate the average energy of the oscillator for the wave function we considered. This results into a multiple integral 2 gY M i 2 l J−l 2 ! 2 dx ψ0 j,j! l exp(ikl/J)zjzj! 2 1 + 2 #xj #xj Eosc = | | | | 2π | − | . (3.11) i 2 l J−l 2 # ! dx ψ ! exp(ikl/J&)z z ! # $ % |%0| j,j | l j j | In the above, we h#av$e done the%contra%ction between raising and lowering opera- tors of the off-diagonal modes as follows ai a†k = δiδk (3.12) # j l ! l l This is what makes the sums run over a single pair of eigenvalues in (3.11). The extra factor of 2 in the above equation is due to the fact that we have two string bits between the same pair of eigenvalues in the problem. Now we can evaluate this integral by a saddle point approximation, similar to what we did in the previous section. This is done in two steps. First, the integral will – 11 – The geometrical interpretation of this result is as follows. We have two eigenval- The gueeosmonetthreiscpahlerienattearparretitcaultairodniamoeftetrh, iwsherreestuheltBiPsSanusllfgoeloldoewsicss. aWssoeciahteadvteo two eigenval- ues on thetshpe BhMerNeliamtitafopr tahretciocrurelsapronddiinagmcoentfiegru,rawtihonearree ltohcaeteBd.PTShenquulalsig-meomdeenstuicms associated to the BMN loinmthite fBoMrNthsterincgo,rcrheasrapcotenridzeidnbgycko/n2Jfi,gtruarnasltatieosntoatrhee laoncglaetoendt.heTsphheerqe ubea-si-momentum tween the two eigenvalues. The energy of the BMN impurities (the off-diagonal on the BMmNodesst)riisncga,lcuclhataerdabcytethreizEeudclibdeyankd/is2tJan,cetraassnocsilaatetdestottohetehmebeadndginlgeoof nthethe sphere be- tween thefivtew-sopheereiginetnovaaflluateEs.uclidTeahne6-ednimeerngsiyonaolfgetohmeetrBy. MThNis isimshpowunriintifiegsur(et2h. e off-diagonal i modes) is calculated by the Euclidean distajnce associated to the embedding of the 2 r0 sin ! five-sphere into a flat Euclidean 6-dimensional geometry. This is shown in figure 2. r0 2! i j 2 r0 sin ! r0 The geometrical interpretation of this result is as follows. We have two eigenval- 2! ues on the sphere at a particular diameter, where the BPS null geodesics associated to the BMN limit for the corresponding configuration are located. The quasi-momentum Figure 2: Geometry of the string bit between two eigenvalues, string bit shown in red. on the BMN string, characterized by k/2J, translates to the angle on the sphere be- tween the two eigenvalues. TTheheenanergglye boeftwtheeenBtMhNe eimgepnuvrailtuieess (isthθe=offk-/d2iJagonal modes) is calculated by the Euclidean distance associated to the embedding of the five-sphere into a flat Euclidean 6-dTimheenseinoenraglygeoofmtehtery.stTrihnigs ibsisthcoawnn bineficghuarrea2c.terized in terms of its kinetic energy (coming from the free field = 4 SYM result), plus the strong coupling mass term i j N from the i2n r tsineraction with the eigenvalue condensate. This mass term is proportional 0 ! Result localizes to a string bit to !xi !xj and not to the angle that the two eigenvalues on the sphere form. In this r | − | 0 of particular fixed length Figure 2:seGnsee,otmhee2!sttrryingobfitthleeavesstrtihnegspbheirte,bbeuttwfoeresnmatwll oangeliegθenthveaeluffeecst,issntreginliggibblei.t shown in red. Now we use the value of r calculated in the previous section, and we find a The angle between the eigenvalues 0is θ = k/2J result which is equal to g2 N The energy of the stringEobscit=c2an1 b+e YcMharsainc2t(ke/r2iJz)e.d in terms of (i3t.1s6)kinetic energy " # π2 (coming from the free field = 4 !SYM result), plus the strong coupling mass term Figure 2: Geometry of the sMatchestrinIng btihteb eBBMNtwMeeNn ltiw mstringoiteigwenh vecalculationarleueJs, isstrtinagkeb nitot ts ohallobw enorderslianrrgeed,. wine thenee d‘t tHoofto also take k 2πn with N ∼ The angle betweefnrothme eigtehnvealcouplinguiens tifisexθre=ad.ck /tTexactly2ihJoenabwo,vi ebythre takingstuhltereed iuJgc eelargesntvoa lku fixed.e con (Smalldensa tangles)e. Th is mass term is proportional The energytoof th!xe istring!xbjit acanndbenchoatratctoeritzhedeinatnergmlseofthitsaktintethiceentewrgoy eignen2 values on the sphere form. In this | − | Eosc = 2 1 + g2 N . (3.17) (coming from the free field = 4DoesSYM notresu lmatcht), plus tstringhe stro nsigmag coup lmodeling mas:s approximationtYerMm J of sense, thNe string bit leaves th"e sp#here!, but for small angle θ the effect is negligible. from the interaction with the eigenvalue condensate. This mass term is proportional " # free string bits is naive, but not in BMN limit. 1 to !x !x and not toNthoe wanglwewthehiacthutshmeeattwtcohheeesigevnxvaaaclltuleyes tohonefthBerM0spNhcelraiemlfcioturmtloa. Itanelltdhoirsdinerstihn epeprtruerbvaitoiounsthseeorcytio. nT,hias nd we find a | i − j| sense, the stringrbeistulelatveswtheiscshphohueirlesd, beueqttufaoakrelsnmtaolsl eavnigdleenθctehetheafftectthies anpegplrigoixbilme.ations done above are reasonable for Now we use the value of rt0hecsaelcsutlatteeds. in the previous section, and we find a result which is equal to Now we can try to compare our result (3.16) with some other conjectures in the g2 N literature. 2 Indeed, we fionsdc that the above aYnaMlytic form2ula is exactly the result of osc gY M N 2 E = 2 1 + sin (k/2J) . (3.16) E = 21 1 + sin (k/2J2 ) . (3.16)2 The norma2lization "of gY M a#bove is to be identifieπd with 4πg in [4] " # ! π ! In the BMN limit where J is taken to be large, we need to also take k 2πn with In the BMN limit where J is taken t∼o be large, we need to also take k 2πn with n fixed. The above result reduces to ∼ n fixed. The above result reduces to n 2 – 13 – Eosc = 2 1 + g2 N . (3.17) " # Y M J ! " # n 2 all orders osc 1 2 which matches exactly the BMN limit to in perEturbatio=n th2eory 1. +ThgisY M N . (3.17) should be taken as evidence that the approximations don"e above#are rea!sonable for J these states. " # Now we canwtrhy itcohcommpaartecohurersesueltx(a3c.1t6l)ywitthhseomBe oMtheNr colnijmectiutretsoin athlel orders in perturbation theory 1. This literature. Indeed, we find that the above analytic formula is exactly the result of 1 shoul2d be taken as evidence that the approximations done above are reasonable for The normalization of gY M above is to be identified with 4πg in [4] these states. Now we can try to compare our result (3.16) with some other conjectures in the literature. Inde–e1d3,– we find that the above analytic formula is exactly the result of 1 2 The normalization of gY M above is to be identified with 4πg in [4] – 13 – Conclusion • New understanding of AdS/CFT • Gravity appears as hydrodynamics (collective phenomena). Locality can be addressed • Background independence: can address many topologies in one CFT (didn’t show it in detail here) • Picture of the origin of strings and calculate some non- BPS energies exactly (BMN limit) • Can do a lot of other type IIB near horizon geometries and explain AdS/CFT for all of these. (Work in progress) Open problems • Radial direction of AdS still not understood • String sigma model not understood • How one recovers gravitational interactions not understood (SUGRA action): we have matched on- shell solutions • Small black holes can be addressed • Need to understand 3-form fluxes (seems to be related to mild noncommutativity). • ...... Lot’s more......