最後の課題は、ADHCFTの課題について説明します。リミットで、バルクフィールのエクサイティションをパターバトブリーに応じることができます。バルクフィールのスムーズのジュアメトリーに応じることができます。バルクフィールのエクサイティションに応じることができます。その後、バルクフィールのエクサイティションをパターバトブリーに応じることができます。その場合、バルクフィールのエクサイティションのエクサイティションをパターバトブリーに応じることができます。 そして、バルクフィールのゲックナ反発のモアレーターにローカルはデレッグ npdcb4va2d Dat erad バルクフィールのエクサイティションをパターバトブリーに応じることはできます。、バルクフィールのエクサイティションのエクサイティションに応じることができます。デレッグ hoch fragile の性能セフトービーに応じて図成することができます。ため、アビオ建築を確認します。私はそれを短い説明するためにADS-D-1を追加するためにアソリッドスリッドを追加するためにアンジュエータスペースの部分をローカルオペレーターにアンジュエータスペースの特徴のポジションを追加するためにローカルオペレーターにマイク bodies 瓜にディジェティのパ時間とストローカルオペレーターを同時に滑り漂亮にと呪 먹을今お pickled事実而已BXX prime all of X prime plus collection due to bulk interaction and other effects.The question is the domain.So where are you going to integrate?So let me denote that subspace.What did I denote that subspace?I guess I didn't denote it anything.So let me just call it as C.So I'm going to specify this subspace.Let me first open up.So boundary correspond to Z equals 0.So let me open this up.And then consider Z equals 0 boundary of ADS, P plus 1.So this is the d-dimensional space on the boundary.What I'm going to do is specify some subspace of the space-like section of the boundary.And associated to that, there is a domain of dependence that I have defined.This square region.This is a domain of dependence.So for associated to this section A, you have this domain of dependence.Now this is another picture of ADS.So here I'm projecting anti-itter space on this two-dimensional surface.The boundary is here.So I'm suppressing other dimension of ADS.So in that case, in this picture, this domain of dependence here.So this is a domain of dependence.It's just I'm suppressing all the spatial direction on the boundary.This is the time direction on the boundary.So this is my domain of dependence.And according to HKLL construction, what this she is, is the intersection of future and the past of this domain of dependence in the park.So this shaded region here is she.So this is interesting.So namely that you have more sort of refined version of the correspondence between the bulk local operator and the boundary operator, operator of the conformal fuselage.Apparently ADS-CFT correspondence tells you that any operator that you can construct in anti-itter space gravitational theory should be constructable in terms of boundary operator because ADS-CFT correspondence saysThere is a correspondence between the Hilbert space of ADS and Hilbert space for CFT and isomorphism of the operator algebra.But this tells you more restricted state.This makes a more stronger statement.This makes stronger statement that you don't have to actually consider superposition of all the operator.You just restrict yourself to this domain of dependence.Now this leads to a puzzle.And the one of you, I guess pointed out that puzzle yesterday.And I'm going to repeat the puzzle and then try to resolve it.Explain why something like that is possible.So there are two paradoxes.Okay, the first paradox is this.So let's just consider some space.So here I have, let me just give you some, let me see if I can draw this in a nice way.So we have this domain of dependence on the boundary.So this is A and then this is DA, excuse me.So on the boundary you have D of A.And then I don't think I can do justice to that, but there is some region C that you extend into the bulk.I don't think I'm doing a good job on that.So you have some region in the bulk that this domain of dependence extend to.Let's consider a space like section of this ADS.And then draw this over here.So what I'm doing is that you have this.So this is a boundary.So suppose this is T called 0 section of ADS D plus1.Then we have some subspace A on the boundary.And then there is a space like section of this domain C.So the claim of HKLL construction is that if you are interested in constructing operator right here,then you can basically do the constructed by considering super position of operator in the domain of dependence of A.OK?Yes please.Excuse me.Yes yes yes.You are quite right.This is D of A and Z and X is in C.Which is actually what is called-I denoted it as C because it's actually called causal wedge.ウェッジのような環境に関しては、Aの値段に関係されています。これを解説してくれてありがとう。今、私はこれを確認しました。インテグラルは、バウンドに関係されています。そして、ローカルオペレーターに関係されています。ありがとうございます。私の詳書は、勸いが結びましたけど、私が言った問題は、これくらいだけは敗因していなかったですか? fancy婚 scaredこれに関係されています。私は次のことを考え、これは広岡國製のお蔵を調整しました。そして2つの方法は、カーサルウェジを選びます。この点でカーサルウェジを選びます。しかし、上乗りの last one の結果は、上乗りの last one の結果は、この雰囲気が生まれ、この雰囲気の雰囲気が生まれ、ここの雰囲気が生まれ、どうなるのかというのです。これは1パーゼルですもう1パーゼルですまた、 ファンを見ながら真ん中のADSに何かを組み立てたいとまた、ファンを見ながらクラウド表現をしますここから進めるように1つの両半を使って1つのコンバイトにとっており、ここから研究をしてみます1つのコンバイトを見てみましょうA1、A2、A3、A4、A5、A68-6、7HKL construction では、このスーパーポジションを利用するために、このウエッジをコントロールするために、このウエッジをコントロールするために、このウエッジ、このウエッジ、このウエッジ、このウエッジ、このウエッジの使い方を行います。3.3.3-4-4-3.10.3-4-4-2.4.3-4-4.6.3-4-4.3.5-4-4-4-4.コントンフューセリのアクセウムのシューアズ・レンマはローカルオペレーターをコミュートしてローカルオペレーターをスペシャルに分けてそのオペレーターはアイデンティーのトリビアでこの期待を繰り返すことができますはい、問題ありますもっと悪くなりますネイティマルのリージョンは真ん中に何もできませんそれはパズルです私はパラドクスをプロポーズしています全リージョンを選ぶとリコンストラクトを選ぶと私はこのオディオンに成功したことができますそれはこのオディオンについてコンフュージョンを選ぶことができますそれはこのパラドクスの目的ですそしてこのオディオンについて私はこのローカルオペレーターの特性とも違いがあるのですそしてこのパズルはローカルオペレーターの中にあることができますしかし、このパラドクスを1つずつ取り組んでいなければならないこのパラドクスを1つずつ取り組むことができますこれはアルム・ヘリのアルム、ヘリ、ドーン、ハロー、7041、すみません。そして、サブセクエントリーがドーン、ハロー、アラン・ウォールでドーン、ハロー、ハローを自身で解決しました。それについて、その点は、コンフォーマルフィルセリーでどちらかを考えていません。ジェネリックステーションはコンフォーマルフィルセリーで素晴らしいジュオメトリーのバークやローカルエクサイテーションは必要です。コンフォーマルフィルセリーの特徴のステーションはバークジュオメトリーのコンフォーマルフィルセリーでサブセクエントリーでコンフォーマルフィルセリーのスペースが全てのスペースを考えています。スペースはジェネリックステーションを考えています。今回はコンフォーマルフィルセリーでサブセクエントリーを考えています。自身を自身にこれを使って他のコンテクセットで20年間の時代に、コントローマーの中で、人々は、コントローマーの中で、コミュニケーションのエロアコレクションを行っています。コードサブスペースのアイデアを使って、コントローマーの中で、ローバースタイルを使っています。そして、そのコントローマーの中で特にような結果がある。この専門家のもともに、私はこのコントローマーの中で nutritional painも説明します。エロアクコレクションの致命応答は、對幕を制御する必要はありますまず、実際に言いたいのですが例え、せっかく、言いたいという1,2,3を更新するもの dashそれがクラスティカルデータの仕事になることができるかもしれませんしかし、コンタムメカニックで仕事はできませんコンタムスタイトのコピースはできませんコロニングのセオリムと呼ばれていませんフォローランスを見てみると、ユニタリオペライトのアクティングを取得することはありませんこの場合、フォローランスのアクティングを取得してくださいそして、プサイでアクティングを取得してくださいフォローランスのアクティングを取得してくださいここで、「プサイ、テンサー、プサイとも合わないのを作って if you can do that, you can copy the state."その際、プサイを作るために、プサイを信用できた場合、プサイ、テンサー・プサイ、 Pci, can send Pci, tencer Pci,そして、楽しんでいくと、プサイを、プサイと合っても、プサイ、テンサーと合っても、プサイ、 Pci, perseverance to Pci,そして、プサイにも、プサイを、プサイを使っても、プサイは多くのステージを中身に書くことが出来ることもできることもできるプサイは、プサイの長さの任せをたくさん下げていくコミュニケーションを確認することができます。幸いですが、このノークローニングの理由はそれに対する必要がありません。どうしようとしていますか?この方法は、実際にこの方法ではありませんが、それに対する必要があります。これがコードサーブスペースの意味です。では、ノーションを説明します。ここは例えです。もう一度考えてみましょう。前に2個のステープシステムを考えます。3個のステープシステムを考えましょう。1 Chewna生姜3よりも ent Strkosh ر 1と 2 h12 Chewna生姜3よりも ent Strkosh R明芽 forces 3よりも ent Strkosh Ksh systems RM1形などです。��를取り出すという 2 Chewna生姜3の感覚をマスクアップにこのようにそしてそれについてI definestate 0 tildewhich is1 over square root of 3of 0000plus 111plus 2221 tildeis 1 over square root of 3of 012plus 120plus 201and 2 childrenis 1 over square root of 3of 021plus 102plus 210so those belong to the tensor product of the3-identical Hilbert spacesright?so this is h1h2h3so I have some kind of redundancyso there are nineso there are threeso this Hilbert space is 29 dimensionsand in this 29-dimensional Hilbert spaceI pick three states27 excuse me2727-dimensional Hilbert spaceand I pick three states out ofthis Hilbert spaceok so now I'm going to encode dataI want to send one of these three statesbut instead I'm going to send one of these three statesok and then the data may get corruptedso how can you correct itso the idea is the followingthat onh1 tensor h2you can actually definesome unitary operatorthere is some unitary operator ulet's call that u12ok so that it has the following propertythat u12 tensoridentity acting on the third entryand if I pick any one of thisso I is 012then this is going to be I tensorqr root of3001122so this is an interesting operatorso namely that if you give me one statethis give you so this I is 012so if you want to send 0for example what you can do is actyou start with that act inverse of thisand it gives you this stateand then if you get that stateyou can act on it and you can reverse itso you can do this operationso this isthis is actually a very useful operatorbecause this acts only on the first two entriesso that means that you can recover thissuppose you send this operatorand suppose during the processthe third entrythe third hillbill space gets corruptedso you lose the informationbut since this you can still act on thisand then you can recover this informationbecause this acts trivially on the third entryso we can say it more specificallyin the following wayso suppose you can utilize this operatorin the following fashionso suppose you have any operatorall acting on the first hillbill spacesay suppose you have some anyso this is the operatorand this is the matrix elementso suppose you have any operatoracting on the first hillbill spacethen I can upgrade thatinto unitary operator acting onthis three subspacesso three dimensional subspacesin the following wayso I can define a new operatorall12 hatas u12 inverseall u12so the way that it actsis that first of alland then tensor i3so this one acts triviallythis one acts triviallyon the third entryit mixes upfirst and second entryin such a waythat all12ishilderis equal to jof all ijactually it's gettingI shouldI should write it over hereso ishilder acting on thisis like thisso if you havethis operatorall in the first hillbill spaceI can do this operationto construct operatoracting on thisby the wayso this code subspaceh codeis given by thesethree statesso what this procedureis doing is that3 operatorin the first hillbill spaceyou can do this operationto define operatorin this code subspacewhich acts exactly in the same waybut this procedureis tolerantagainst the corruptionof the third hillbill spacebecause this is not acting on the third hillbill spaceand so even ifso for example during this procedurethe third hillbill space is corruptedyou want to get affectedyou can always make correctionsby doing thisso this is the idea of code subspaceso this is very close torepeating yourselfintroducing redundancy in the classicalerror correctionexcept thatyou are avoiding this programof quantum no-croning theoremok?yeah that's rightso you have to know where the correction happensand what you can possibly do isdo it three timesand then they make correctionsyeah that's rightso this you need toso for examplein this procedureif the corruption happensin the first entrythey won't help youso this will help youif corruption happens in the third entryyes they agreebut the point I wanted to make hereis thatthis structureisand these structures are very similarand in factyou can make this more precisenamely what is happening hereso let's comparethis storyso here the storyso I hopethe situationis clear ok so maybeI should make one more pointthe important point is thatI can do this procedurefor any one of the three hillbell spacesso namelythatso let me write it over hereso for anyohutacting on thiswhich goes like thisyou can haveall one twoall two we cando that for all hillbell spacesof two three or three oneso you can have two threeall three onesuch thatall one two actson thislike thisand then two three acts the same wayand three one acts the same wayso all these three operatorsexactly the same wayexcept that all one twoacts only non-triviallyon the first entriesnon-triviallyand then it actstrivially on the third entryand all two threeacts non-triviallyon the second and the third entryetc.so lookwe have the situationwhere you have actuallythree different operatorsacting exactly the same wayif they are restrictedto the core subspaceso the point is thatthis is exactly likewhat is happening herenamely so theproposal that has been put forthis that in thistotal conformal hillbell spacethere is someparticular subspacethis sort of has geometricdescriptionwhich now accordingto this ideacan be regarded as some kind ofcored subspacein such a waythatwe can havethe following situationso the conformal field theory hillbell spaceas we discussedin a couple of lectures agowe assume can be decomposed into hillbell spaceassociated to region Aand it's complementsoand then similarly forcored subspacethere is a subspaceassociated to this region Alet me call this regionAand then correspondingcored domain assmall Aso this is a bulk regionassociated to the capital Aso such thatif I pick any statefromthiscored subspacethen there isand then if you pick any operatoracting on thecored subspaceassociatedto this regionso for example thislocal operator right hereis an example of thisthenthere is some operatorA acting onthis regionsuch thatOAactsexactly likeOover hereso this is what is happening hereso for eachfor any boundary regionAfor any bulk operatorassociated to ityou can choose operatorspacewhich reproduces this operatorbut if youjust require thatfor some cold subspacethenas I demonstrated over hereyou can have such a situationwithout contradictionandso this is actuallymore sort ofthis has more contentthan just analogythe fact that you can have this constructionmeans that the way thatregionAregionB for exampleentangledwithin thishirber spaceis very similar to the way thatthese states are entangledamong thesehirber spaceso in the next two lectures including this onenext one I would like toexhibit how tocharacterize such entanglementyes you have a questionso H of capital Ais thesubspace of thehirber space on the boundaryCFTsmall Ais thehirber space associated to thisso we are assumingthat the bulk can havepartavative quantizationof the gravitational theoryso certainly you have a focus spaceyou can considerjust like you can decomposethe boundary subspaceinto segmentyou can decompose the bulkspace into segmentso you have adecomposition of this cold subspaceinto tensor productofthat associated to this domainand it's complementyeah so what I'm doing is thatI'm interested in constructing operatoracting onacting onhereso this is a subspace of H coldno no no I'm not sayingthis is a subspace of thisor because thisboth of them belong to HCFTyou are worried that whetherthis is a trivialitythat if this is identical to thisthen this statement is a trivialitybut that's notthat's not the caseso bulk operator can be reconstructedbut that does not meanthat Hilbert spaces are the sameok soso this is an analog ofthis is an analog of phiso I'm takingthisI'm taking this phiis an example of thisso phi isoperator which say createslocal state in the bulkwhich is still I mean any operatoris operator on conformalfinsery so we cansay it in this way butphi is an operator acting on this cold subspaceand I'm saying thatthat operationcan be reconstructedby operator actingonly on this subspaceyeah well becausethis is a subspace ofHilbert spaceso H coldso this belongs to herebut I'm not saying that this belongs to hereit's a different decompositionso let me seethis equation is exactlythe same kind of equationthat I'm sort of exhibiting over hereyeahso here I have in mindH cold to beso supposeso I assume that bulk hassmooth geometric descriptionI'm a part of it very quantizing itso you havesome folk spacewith small perturbation of this geometryand that's my H coldwhich is a subspace ofHCFT because this contains much biggerpartavation including black holeand even non-geometrickind of configurationsI'm not saying that these correspond to hereI'm saying that this isclearly subspace of thisyeah so associated to this choiceI choose domain of dependenceon the boundary and I choosecozal wedge and that's moreyes yes yes yes yesso associated to this choiceI have this decompositionand then I'm saying that thecozal wedge so this type ofcozal wedge reconstructionsometimes called ADS renderer reconstructionis a claim thatany localexcitation in this cold subspaceassociated to this regioncan be reconstructedin this way and sort of this isformalization of this statementI'm saying that this formalizationif you formalize it in this waythen these are exactly the same kind offeature as thisquantum error correcting proceduresorryHyeah hillbill space ofconformal field theory on the boundaryso the statement of ADS CFT correspondenceI repeat is that hillbillspace of full hillbillspace of the gravitational theory in ADSis isomorphicto the full hillbill space ofconformal field theory on the boundaryand in itin itthere are states which can be describedin terms of geometry and small excitationsand that's I'm callingcozal subspaceI assume that according to ADS CFT correspondencethatthe bulk geometry in the large endsituation can be described assmooth geometry with somesmall excitation around itand this is a formalization of that statementyesso just like herethere are three operatorsand that's exactly you asked that questionwho asked that questionat the end of the last timethere was some redundancy in the definition of operatorssomebody asked whether this definitionof local operator is unique or notandfor example here is an exampleyou have three operatorsO12, O23, O31which acts exactly the same wayon the core subspacethere is a redundancy in the definitionand this redundancyis exactly what you need for errorcollectionhere you are seeing exactlythe same kind of redundancyin the bulk local descriptionthat this same operatorcan be described in a multitudeof waydepending on wherewhich interval you choose over herewhich part of the boundaryyou chooseam I answering your questionI pick any statethat is sort of either grand stateof this geometry or some small excitationyesso some OAtakes yououtside of the core subspacebut I'm saying that you can chooseokay so that its action on core subspacestays within the core subspacethis is an examplethese operatorsso if you consider a generic operatorin the tensor Hilbert space27x23matrixthen that will take you outside of the core subspacebut these three operatorskeeps you within the core subspaceso I'm saying thatthere are analogously hereyou have such operatoryeah so it stays operationstays within the core subspaceyes what's 1,2why this is 1,2,1what do you mean by 1,2,1yeah so in the case of this operationOA is this onewhy it's non-localwho told you that OA is a local operatorI didn't tell you thatmaybe I wasn'tO is only associated to capital Aso herein this constructionOA is a linear superpositionover thisspaceby the way this integralis over the causal domainbut since this is causal domaineven if you have an operatorhere you can actually bring this backto here by the Hamiltonian evolutionsothis can be written as integralover the operatorrestricted to this Hilbert space over hereokaysonowso this is sort of resolvingthis kind of paradoxbut then you can askso this is okay good storybut why should I believe thatthis is actually the way that the entanglementis happeningso I would like to tell you a little bit moredetail information abouthow geometric states are entangledand try to quantify thatso that will be thepoint ofmy lecture for the remainingset of lecturessolet mefirstmotivateso I'm going to tell you aboutone way to quantifyentanglement propertyof the holographic statesthe states that belong to this coldsubspaceor in terms ofholographic entanglementbut beforeI get into this subjectlet me motivate thatby discussing entanglementof a particular type of statesand in particulareternal black holein ADSso let me first tell you somegeneral thing aboutmixed state in conformalso there is a notion called purificationwhich is the followingnotionso in my first lectureI told you aboutensemble of quantum statesand the density matrixassociated to itso suppose you have an ensembleof statesso you have a statewith probabilitypiso in order to describe thisso on the Hilbert spacein order to describe thisyou consider some density matrixwhich is given bythis expressionand the purificationis a following conceptso if somebody give you density matrixyou can actually constructpure statebut not on this Hilbert spacebut in a bigger Hilbert spaceso what you dois you consider another Hilbert spaceso I denote ash-childerh-childer can be isomorphic to hor biggerso that it has an orthogonal basispsi-i-childerexcuse meso these arepsi-ipsi-iand thenso these are orthogonal basisand then I considera statewhich can be expressed in this wayand this belongs obviouslyto h-childertensorhokso the point of thisconstruction is that you can recoverthis density matrixrowas a partial traceover this density matrixso this newHilbert spaceso I introduce this new Hilbert spacebut immediately after thatI trace it overand go back to the original Hilbert spaceso this procedure is called purificationnamely density matrix in generalis a mixed statebut then there is a pure stateassociated to thatin such a way that partial traceget you back to the original density matrixso this is called purificationnow note thatpurification procedure is not uniquethere are lots of redundanciesthere are a lot of arbitrarinessbecause I haven't told youwhat kind of basis you chooseand as far as these are orthogonal basisthen you can purify itso that means thatfor exampleyou can haveunitary operatoron the new Hilbert spaceand you can actthis onthis thingso where u is acting onthe first onethis is as good as this onenamely thatyou can consider this as a newpure stateand then if you take partial tracegives you the samedensity matrixso here is an exampleso supposewe consider canonical ensembleso the canonical ensemblehasfree energythecanonical ensemblecan be expressedin this waywhere you have HamiltonianH hatand better is the inverse temperatureof this distributionand thenz is a normalizationso that trace of this is going to be 1and you can write it as a sumover all theenergy eigenstateEI is an eigenvalue of H hatfor this stateso now I canpurify itfor simplicity I take H tildeto be the same Hilbert space as this oneand I use the same basisso in that casewe have what is called the sum of fieldsdoubleso we can purify this in the same waysonot that this is better over 2just like I had the square root hereI had the square root here becausethat's howhere I site squareso I had to have the square root hereto get the right normalizationso similarly I have beta over 2 hereand then we haveI tensor Iso this thing belongs toH tensor Hand this is called the sum fielddouble stateso making the mix stateinto pure stateby addingextra Hilbert space is not somethingthat is sort of artificialit's a very natural thingbecause if you think abouthow you learn the thermodynamicsin undergraduatesthe way we learn thermodynamics is that where you havea thermal statewith temperature Tbut the way that you think aboutthermal state is that in fact it isin some you have some extra stateswhich is heat busand then total spaceso you have a density matrixbut the total spacestate can be purebut then thismy targetstate and heat buscan be entangledin such a way that if you forgetabout heat buswhich is the same as taking partial trace herethen you reproducethis thermal ensembleso considering this purificationis something that is verysort of natural from this kindof considerationnamely the relation betweenmicro canonical ensemblehere and the canonicalensemble hereoknowit is well establishedthat if you have a thermal ensemblelike that thenin the conformal field theorythen ads dual of thatis a black holenamely more preciselyadsdualof this thermal state1 over z times e to theminus beta h hatisthermal gasin adswhen the temperatureis lowdimensionand then it is actually theads dual sealed geometrythe temperature is highnot that beta is animbursed temperatureso beta below this meansthat you are in high temperaturethis means that you are in low temperatureit's kind of counterintuitivebetter is 1 over td is a dimensionof the conformal field theoryyou have a questionads scale to be 1and I am alsosetting so in the CFT sidethe scale is setbecause I am quantizing the systemon the spherethe sphere has a radius so that isset the scale and that gives you thescale that measures the high temperatureand the low temperature scalethat will be a CFT sidein the ads side you have an ads scaleok so you measure the temperature againstyeahok so the thing is thatso if you have a final temperature thingthen what we mean byfinite temperature is that if you analyticallycontinue time in eukaryan directionit will be periodicwith period 2 pi beta2 pi beta orperiod beta I guessand so the geometryso you have to have asymptotic ads geometrywith eukaryan signaturetime direction is periodicby betathere are two types of solutionto Einstein equationwith cosmological constantwith that property one is thatjust takes a pure ads spaceeukaryanize itand periodically identify itthat's one solutionand another solution is that you putblack hole in itand there are just a mass of the black holeso that hooking temperaturegives you this temperatureso you have two solutionsso you have to choose which oneand then you have to choose the onefor which the action is smallerbecause you havetwo possible quantum stateand then which one dominatesdepends on which action is smallso this was studiedby hooking and pageand they pointed outthat depending on the temperaturewhen the temperature is lowthen it's more advantageadvantagesnot to make black holebut just have some thermal statebut then as you raise the temperaturethe actionfor the black hole becomes smallerat some temperature whereone over tis equal to two pi over d minus oneso there is a critical temperaturethere is a phase transitionso what happens is thatsuppose you have adswhen the temperatureso you start to turn on the rangeand you heat upyou start creating some particleyou turn on the temperature moreyou create more particleand eventually at this critical temperaturethese gas particles collapse into black holeso this is a critical temperaturefor formation of black hole in adsso that's why you have two phasesokso this is howyou pay attention to this phaseso in Laurentian signaturethe black holethe penrose diagramof black hole in ads looks like thisyou have singularityhereso this is maximally extendedblack hole geometrywherethis isone boundary whereyou have time directiontimes d minus onewhereand then you have anothertime directionand then d minus one sphereso you haveactually three regions so let me denote itone two threefourso you have thesethree regions and here is the singularityand these are singularitythese are horizonsokso but the interesting thing is thatif you consider Laurentian signature adsyou have actually two asymptotic ads regionwith two boundaryboth of them has geometryall times spherewhereas in the original adsyou only had one boundaryso if you have a pure adsyou have only one boundaryand you associate conformal few series on itbut it turns outthat if you go to high temperature phaseand if the spacetime collapse into black holesuddenly you discover that actuallyyou have two copiesof the boundariesboth of them have the same geometryso what's going onso Marder Sennain 2001proposed thatthis geometry actually describesthe purificationof the finite temperaturegeometrynamely that since you have two boundariesit's natural that this statebelongs to tensor productof the conformal few seriesand this therefore isdual to conformal few seriesstate which is the subfield doublewhich is the tensor productof CFD stateso I should probably stopvery soonbut so these states correspondto this stateand this statesomorepreciselyeach one of them correspond toso for example the first entry heredescribed this stateso naturallythis region one is under controlof this sideand this region two is under controlof this sideand somehow these two Hilbert spaces are entangledeven though you have twoindependent conformal few seriesHilbert spaces they are entangledand the fact that they are entangledis represented in the dualgeometry dual ads gravityat black hole geometryI should end here but I justwanted to make one commentand for you to think abouttonight and then we can discuss tomorrowwhich is that I told you herethat there are actuallychoices you can makefor the basisnamely when you do the purificationyou can choose any orthogonal basis on the other sideso that's what it meansthat you can perform unitary transformationon this sideand you can still make the thermal stateso that means that if you are interestedin constructing a thermal stateand then if you are only interestedin constructing pure statewhose partial trace reproducesa thermal state herewe can perform unitary transformation heresuch unitary transformationinclude operation to create statesinthis side because these areoperate acting only on this sideso you can shoot lots ofenergy in these directionsfor example you still havethe same thermal stateso if you just know that you havea thermal state here you don't knowwhat's going on hereso you think that ads dualof a thermal stateis black holeand you create this stateand you think thatthis is the ads black holethe geometry near horizon is smoothso you can go inside of the black holeand before you hit the boundaryyou'll be safebut then you get surprised because suddenlyyou got hit by all these energiesbecause you didn't know what kind of basisthe other person is choosingsomehow this is related to the fire orparadox i thinkbut this is something you can think aboutok so thank you