Hong Liu - Euclidean Wormholes, Quantum Chaos, and the Factorization Puzzle

Okay. Good. Good. Yeah. Uh yeah, thanks for the invitation. Yeah, it's a great pleasure to be at IS and uh so um yeah mostly I will talk about content related to these two papers uh which appeared recently. Um yeah so in ADSFT duality we have this fundamental relation between the gravity partition function and the CTF partition function. So the gravity partition function is evaluated say um with m as the boundary say boundary of quantum gravity and and then yeah then equal to the c of t partition function on m. So on the gravity side you have the important parameter which is 2 Newton which is inversely related to say uh the number of degrees of freedom on the CFT side which can be normally written as 1 / n² and so of course this relation is supposed to be a fully quantum relation so with finite gton and finite n etc. But so far we don't really know how to formulate quantum gravity as finite gton. So we normally just consider the semiclassical limit and in the semiclassical limit we can try to expand the gravity partition function in powers of G newton as what we normally do in semiclassical expansion and then on this side to told us how to do the large CFT then we can expand in one square and then these two sides should agree and this has been a very powerful relation and has been extremely successful. ful so many many checks and also uh give us lots of insight but there's a fundamental problem with this relation so this is a fun so-called factoriization problem which was already long in the early days of the duality so if you consider M which consists of disjoint uline say of two manifold on the CFT side the story is trivial if you have a CFT and disjoint uline and then just become the factorized into the safety partition function on the two manifold and if you take the uh yeah it it doesn't matter what value of n you take any value of n they still factoriize okay and including n to be very large but on the gravity side if you use the semiclassical description of this gravity partition function which say for example you sum over all the cycle point and then you find include In addition to factorized contributions, there are also such kind of wormhole contributions. Okay. And such wormhole contributions which connects this m1 m2 and then does not factoriize. Okay. And uh so the gravity expression yeah does not factoriize due to wormholes. Then that means these two cannot be equal. Okay. And also one particularly puzling thing about this wormhole is that the existence of the wormhole tells you somehow there's some kind of correlation between the partition functions of CFT1 on M1 and the CFT on M2. But how do you correlate two partition functions are two two different manifolds? Okay. So what kind of correlations that captures and uh and again this also has been the yeah mystery and so let me just elaborate on this relation a little bit more precisely. Okay. So what do we normally mean by this relation okay so so this relation should not be understood say as a mathematical limit. Okay. We should really uh understand it as identification of so-called trend series. So if you don't know the lame trans series it's okay it's very easy to explain. So so on the gravity side from gibbons hawking say we can approximate the semiclassical gutton goes to zero limit by the semiclassical evaluation of the gravitational path integral. Okay, in particular uh uh you can do this in uklidian signature and uh so and the gravitational path integral should be understood just as a device for you to do cider point approximation or or slightly a generalization of the cider point approximation. Say say uh uh uh you can in the situation for example in the lower dimensional gravity which you can reduce this path integral to finite dimensional uh integration over finite dimens finite number of parameters and then you can try to evaluate this precisely okay but but of course this part integral is not defined in higher dimensions and so so in general because of the gutton only appears you can always put the gton ahead of the in front of the gravitational action and then such kind of evaluation will get in general give you this kind of structure. Okay. So you just sum over say distinct settle point or constraint settle point and then then this give you the the the leading order contribution exponent then you have one loop two loop etc and all the of course all the configuration should have m as uh uh as its boundary. Okay. So this sum over all so so each term in this sum is actually infinite asymtotic series and so here you see essentially have a sum over uh the symotic series and this is called the trans series. Okay and uh yeah so this I sum over a different asymtotic series and on the CFT side so following tot we we have the similar structure. So from to from the evaluation of the uh general uh uh matrix integrals you have the similar structure and you can sum over large and set point and then you have the leading order contribution and then say a Taurus genus one and the higher order genus and then the identification of these two means you have to identify these two trans series term by term. Okay. And you really have to identify the sum. You have to identify exponent. You have one loop. So, so you really have to identify them term by term. And for this reason, okay, it doesn't matter. So, sometimes people say, oh, this wormhole does not matter because it's often subleading or subdomant especially small. So, it doesn't matter. Okay. As far as this wormhole exists, no matter how small it is, always spoils uh the uh factoriization. Okay. It's because you have to uh match all the expansially small contributions. Okay. And uh so so there are several logical possibilities people have discussed uh uh uh to address this uh uh factoriization problem and the most straightforward one just exclude the wormholes. Okay. But there's no good rational actually for doing that is because the yeah wormholes are often perfect semiclassical solutions and there's no reason to exclude them. And the second possibility is that there currently unknown contributions to the gravitational path integral which make this semiclonical gravity expression factoriize. Okay. And so there have been attempt to uh uh uh in this approach for example these so-called half wormholes are some kind of uh uh some kind of settle point additional set points in say in very special models for example in the in the in the SYK with fixed coupling and then uh uh uh people have been arguing so exist some additional saddles which can help restore the factoriization. Okay, but how this works in general is far from being clear at the moment. And the third possibility is ensemble average as mentioned by Toria and Dian yesterday. And so in this case, you just assume CFT depend on some parameters. So in addition to our standard definition of the CFT, you assume somehow there's some additional hidden parameters alpha and then then the gravitational part integral then should be identified with the uh the average of the alphas say with some measure. Okay. And so this obviously uh address the factoriization problem is because when you do the average and the the product they no longer factoriize. Okay. And the one prime example which has motivated lots of discussion is because of the JT and there you can show explicitly the left hand side is actually related to random matrix integrals and but there are also many challenges in this ensemble average approach. For example uh uh for for for the duality coming from string theory say for example in foria m or type 2 b string um um say ads3 time s3 or or k3 there's no such kind of lateral ensemble okay and uh in particular many precise spark and the boundary matchings relied on non-averaged series okay and so if you don't average carefully you just destroy those kind of success phenomenal success we have established so far and also yeah there's also some other uh uh uh technical or conceptual uh challenges which I will not mention but let me just mention one aspect is that you can say oh maybe the gravitational semiclassical limit you should do average and now question is do you do average also at finite n at finite gton and it seems unnatural So you only do this say when you take G and go to zero limit okay and uh and then uh and then suddenly you don't do this uh when you go to finite end but when if you go to finite n you still equate the gravity partition function with some ensemble average and then we have potential difficulty with unitarity and one of the say attraction of the uh holography is that we can understand the box duality from the boundary duality a B utilarity from the boundary utilarity. But now if you do ensemble a then you destroy that kind of utilarity. Okay. And so so here I will propose a new idea which works with a single boundary dual. So you can just works with a standard formulation say over the duality coming from string theory and without doing any ensemble average. Okay. So, so before doing that, let me just uh before stating my proposal, let me just motivate using a familiar uh and closely related example. Okay. So let's consider say not the larger limit but consider let's consider some just ordinary quantum mechanical system say in particular chaotic quantum mechanical system just look at the semiclassical limit okay and uh so yeah just look at h bar limit normally we do wkb or you do yeah and uh so let's consider just a few body chaotic system say it can be a threebody system or bill in some some stadium And now let's consider it some partition function or equivalently the density of state they're just related by laplas transform in the in the h bar go zero limit okay standard semiclassical limit you teach in your quantum mechanics class. So, so in this case we have the celebrated goods formula for the density of state which goodwiller formula said the density of state for such a system in the semiclassical limit can be written uh uh uh in two type of terms. So one term which is called smooth terms they they for example the they can be written as expansion of say h bar. Okay so the leading term just the standard wild term. So you look at the the volume the face space volume with this energy and then yeah then just divided by uh one of h bar to the power. So a is a number of degrees freedom. So this is we familiar with in the undergraduate quantum mechanics and then you can systematically correct it. Okay. And so so this leading term only depend on the volume of the phase space uh uh the energy shell and then the higher order can depend on the shape uh and etc. Okay. And uh so then there's oscilly terms and the oscilly terms consists of in general infinite sum. So this sum over a is a sum of all periodic orbits [clears throat] with energy e. Okay. So all periodic orbits with energy e. So this is a classical semiclassical description. Okay. Using the classical periodic orbit. And this s a here is just a classical action of the orbit A. Okay. And the and this capital A is some amplitude. Okay. And then you have this famous goodwiller term which have this structure. Okay. And then then the full density of states is given by the sum of the two terms. So now let's look at these two terms more carefully. Uh these two sets of terms more carefully. So this smooth term and indeed this can be written as expansion h bar uh uh uh say power series expansion h bar you can also uh uh include say uh some nonpertive terms okay something one over h bar uh you express one h bar say if you have instant uh uh yeah etc and uh so so in general you can write it as a trans series in h bar okay yeah you can just uh think of this as a time series in H bar and this is smooth. It's because it has the standard say h bar expansion even though strictly mathematically h bar goes to zero limits uh say yeah yeah yeah you get infinity uh but we still call this smooth in the sense that it has a well definfined h bar expansion and it's smooth in both h bar and in the energy okay and essentially captures some kind of depend only on microscopic and cross grain data okay And this oscilly terms are very different. And then they consider this sum of all periodic orbits and the periodic orbits actually highly microscopic information. Okay, they actually say at the most the mthoscopic and highly erratic. Okay, if you look at the the periodic orbit of a chaotic system and then it can be yeah have a very intricate structure just completely erratic. Yeah. That's because of this yeah a chaotic system and then the dependence on h bar is erratic even though each term have this ordinary uh w kb form say expansion one h bar but here you can see it's of infinite sum and the coefficient in h bar for each term in the sum is highly erratic okay so the sa and the the periodic orbit is highly erratic so so there's no rule you can give to to this SA. Okay. So, so this is a infinite sum of highly erratic coefficient of exponential value of H bar. Okay. And you cannot really write this sum as a asic expansion. Okay. So this is not a trans series in our conventional sense. Okay. And then depend on energy is also erratic and this is actually very important. It's because the you know we know that the the density of states of a chaotic system is highly erratic and if you just look at the wild term and it corrections that does not capture that kind of erratic features that kind of chaotic features and then this term written in this way actually can capture the chaotic nature of the density of state. Okay. And this sum is hugely important actually uh uh uh uh uh this formula is very powerful. You can actually evaluate numerically uh uh to find the density of states and to uh to deduce the energy levels of a chaotic system is actually highly effective. >> So does the first imply the second >> uh uh uh the first uh I think they they're related. They're related. Yeah. I won't say they implies but certainly they are related. Yeah. Sorry, the correction in the first line is it minus 10 + uh >> the correction O. >> Oh yeah yeah yeah that's right. Yeah well yeah thank you. >> And will this formula be able to see show you that you have some of delta functions in the end? >> Uh yeah so uh in general not but for some special systems you can. Yeah. >> Yeah. So uh so so some special systems say for example for a particle in the hyperbolic manifold and actually this reduced to the so-called the the sbook s and in that case actually even though this looks like a continuous formula but actually encode the discrete data. Yeah all the discrete data is encoded here. >> Yes. Uh so about the writing dependence on energy it's a bit counterintuitive because there are many periodic orbits and if h is very very small they're all defased so naively I would expect those contributions in the sum over eight to cancel each other. Yeah, it's you see the cancer in very different way. You see the thing is that if you change year a little bit and because the forotic system if you change year a little bit your orbit change a lot and so this a this aa coion actually change a lot and so you get completely different s and uh and so yeah there indeed some kind of cancellations etc but still you have to incorporate that kind of chaotic feature of the spectrum. Yeah, this one uh uh this one won't give you that. Okay, that one won't give you that. >> Yes, >> related to gradation to question. I mean I understand what you want to say about the dependence on E being erratic but the dependence on H bar being erratic is a bit confusing to me in in the following sense. If you fix a energy level then the you know if you fix your configuration space then the periodic orbits that contribute are fixed and your coefficients in your second oscilly expansion are what they are. Oh yeah, it's about but the it's a infinite sum with the cofficient which you cannot control with the with the SA you cannot control >> but SA doesn't depend on H bar right SA only depends on >> yeah yeah yeah but but that determines this that determine say if you evaluate this numerically >> with a fixed energy >> and you vary H bar and this will be highly erratic it is because SA is highly highly random >> [laughter] >> for different a. Yeah. Okay. Good. And what this equation, what this formula says, okay, is that h bar goes to the zero limit is singular in the sense it cannot be captured by this kind of smooth expansion h bar. Okay, it's singular in the sense that it cannot be captured by this kind of regular uh expansion H bar as we normally assume when we do WKB. Okay. And then this goodwiller term this theory part precisely reflect the singular nature using the periodic orbit and captures those singular behavior. >> Sorry I'm just confused by your answer to Sh's question. I I thought the idea was that you reproduce the delta function peak through this uh say then what is missing? I mean you said in some systems it is true but uh in the other system so what is missing? >> Yeah we don't know. Yeah I think it's >> known that in some cases >> yeah yeah it's in some system it's missing. Yeah. >> So is it that the oscilly piece is incomplete? >> Yeah I think there's still something incomplete here. Okay. But it's highly successful in the sense you can you can use this to uh uh to to deduce energy levels actually can compare with exact calculation and it's a very good approximation but it's believed to be missing still missing something and how to capture what is missing here I don't I think it's open question I don't know at least I don't know the answer yeah yeah I haven't seen anywhere a systematic discussion yeah >> and from the uh the remon za [clears throat] function when you do this There it captures it. >> Yeah. Yeah. There. Yeah. There is like the sailboard case. Yeah. There there is actually Yeah. Right. Good. So, so now let's go back to our uh ADSFT problem. Okay. So, so when we do this standard matching, so there's a very fundamental assumption here is that both series are regular. Okay. So both limits when you take geon goes to zero limit and n go to infinity limits are smooth in the sense as the h bar expansion in the in the uh uh in the semiclassical quantum mechanics case okay they can expand it in either gton or in n square okay so now I want to relax this assumption okay that the gutton z go to zero limit is smooth or in c of t the angle to infinity an image is smooth somehow there's no reason we should make the this kind of assumptions okay even though we always make them but seem yeah it seems no reason we should make them okay we should explore what happens when we relax those assumptions okay even though for large theory we don't know we don't have the analog of the giller formula and also we don't have the similar go formula for the for the g for the quantum gravity Okay. And but but somehow we should uh uh uh explore this possibility and then what I will show is that if we take this limit being singular means there's something else. Okay. And then actually that leads to dramatic implications and in particular it can lead this to the resolution of the factoriization puzzle and can provide intrinsic boundary description wormholes and to explain what kind of correlations which wormhole really capture. Okay. And also the hol also can provide hoquenzian signature can provide the uh holographic string of closed universes in ads. Also I should add here also the the physics uh in the interior of a black hole and also can predict highly hyper lumpic effect in gravity means the kind of lumpic effect with double exponential in Newton. Okay. And uh yeah, but for today's talk uh uh uh I will be uh uh yeah uh I don't have time to discuss all these and maybe I will discuss only the first two. Okay. Okay. So so now let's try to relax assumption. So uh so let's think about on the CTF side let's re relax the assumption that hope expansion always works. Okay. So, so we have imagine we have the CFT partition function which the one one of yeah one description is let's imagine you can do it exactly for any fat end okay so this is the so if you're powerful enough you can do this and then what we normally do oh what we normally do when we take infinity limit we assume that the uh you can expand in one n square okay you can expand in one square but Now I want to add another step. I want to say when you uh take n large okay [snorts] uh this limit again you should not understand it as a mathematical limit just means n take n to be large and I want to postulate that the s partition function as in the good will case you can separate into two type of terms. One is the smooth term which you can expand in one way any expansion and then you have erratic term okay which you cannot which capture the singular nature of the larger limit okay and then when you go to the smooth part then you have to do another uh uh operation which we call the t filter to get rid of this erratic part and so so I will call this three tier descriptions so tier one is the exact description and the tier two is that you you still you can see the envir uh you it's a symptomic description but still contains some kind of microscopic information. Okay. And yeah, one second. And then you have this tier three description which you only can uh remain uh keep the smooth part and this will only be only microscopic data. Okay. Yes. >> In your erratic piece there's e to the minus n and e to the minus n^ squ pieces. >> They can have those things. >> Yeah. In principle they can have those things. Yeah. So erratic part can in principle can have those things. Yeah. Just like in the go formula the radic part can have exponential I / h bar and and here you can in principle have that just this part you cannot have intrinsic expansion uh involvement n square okay >> is the splitting between the two terms ambiguous >> we don't know this is a postulate >> okay right now I just want to relax this assumption that this is the only structure and and so this is just the simplest way to uh to relax that assumption. You just have something else. Yeah. Uh uh uh uh uh yeah. So there's some there's one complication on the CTF side which is not at least apparent on the gravity side which is that n is an integer. uh and when n is an integer and then you analytically continue it to n not an integer there are ambiguities and then you have to specify fall off behavior and so so are you implicitly assuming a certain particular there exists some choice of falloff behavior or something which would make that analytic continuation unambiguous to even compare it meaningfully with the gravity side >> yeah I think that's a very interesting question which we don't know the precise answer at the moment. So here I just say whatever is some additional contribution whether you can treat an analytically or not etc and somehow there's something else here. Yeah, I just want to partulate there's something else here and whether it's due to the integer or or just due to Yeah. Yeah. >> Because that's an additional ambiguity may be presently present. If we if we believe the duality really I think that ambiguity also present on the gravitational sides in the sense that somehow G Newton the G Newton say for example in the unit for example in ads in the unit of cosmod constant is still quantized >> and so there's still some kind of quantization also on the gravity side >> the time gravitons >> yeah exactly limits and so on right >> so that's right >> yeah yeah yeah so the similar >> but there's still some prescription you have to make on both sides that's That's right. >> To match. >> Yeah, that's right. Exactly. >> Good. >> Yeah. I think I think this is [clears throat] actually in my mind this is very important for the general point that this is trying to get at which is in good swer like h bar is a parameter in what you could rightfully call a single theory here. When you upgrade to the space of theories parameterized by n, it's really a family level generalization. [clears throat] And that kind of thing is done in other mathematical physical settings and but it's an extra structure. It's a it's a different type of problem and that's the kind of problem that this is. >> Yeah. Yeah. Also I think indeed but but there's also a lot of way to think about it is that in tier two you instead of consider a family of series you always consider only a oh what's happening >> so you only consider a single theory and then you try you treat the right hand side just as approximation to a very large Yeah, I just treated this uh uh Oh, let's see. I think my computer is going crazy. >> Yeah, exactly. It's a It's a chaotic talk. [laughter] Oh >> computers good question. So when we calculate the partition function large n we have ways to do it. >> Yeah. >> We only get smooth pieces. >> What are we missing in our standard hot expansion technique? >> Yeah. Yeah. I think it's the same kind of question you say if you just naively do do this while expansion of the density of state say of the say some system you don't do good goer and then then somehow you only see those wire pieces and yeah I think it's a it's a key question it's a key question whether when we do larger expansion whether there are additional pieces and how we find them yeah >> but the states for example is a change in change structure completely I mean you know >> yeah exactly at the state the state of the >> and also sometimes for example if you look at the d5 system and d5 the numbers are both even for example then the center charge by a factor of two so you have very many phenomena which I mean you know if you increase m but you know you really have very ric physics happening >> that's right that's right that's right yeah that's right >> I don't know if there is many windows open at the same time [laughter] >> oh you I think I should close all the numbers. >> One second. [laughter] Somehow just maybe we'll do it again. >> Maybe you just move to the blackboard. [laughter] >> Right. Right. Okay. So maybe I will just move to the blackboard. Um [snorts] so uh yeah. Yeah. Let's uh yeah. So so now let's just relax this. Now let's just relax this. Let's just imagine you have CTF and then you have CFT smooth and then CFT erratic. Okay. So, so now let's imagine so, so I will yeah here I always take N to be very large. Okay. interview very large and now >> can you just close the computer because it's very distracting to see the screen >> right >> okay so so we have this structure >> someone is still doing something >> maybe just unplug the projector >> okay so [laughter] yeah yeah just Okay, good. So, [laughter] some maybe it's not life over maybe it's there for Okay. Anyway, so so let's imagine we have this structure and now let's just say imagine we have a way to extract the smooth piece. Okay. So let me define a filter which when you take the act on the CFT and then extract this smooth part and which filter out this erratic part. Okay. So by definition it act on the erratic part to be zero. Sorry in this in this case I have a question you say if you say smooth you mean smooth as a function of of the parameter n or as a function of energy no >> so so it's smooth so it's smooth means that you have a by smooth just means can be written as a trans series in in one square and uh so so I don't have a of course this is a partial at moment just from what we understand normally that kind smooth piece they typically depend on energy in a smooth way and uh and for example uh uh they can be uh uh yeah yeah yeah typically depend energy in the smooth way and uh and and in the gravity side they corresponding to say the the microscopic geometric data >> okay so so in the whole setup we assuming that we we don't have just a single CFT but we we we'll always have a have a have a family of CFS parameterized by something like M and we We ask about dependence on this parameter n. >> Yeah. So you all right I guess I'm asking if there's a statement to be made in a single cf. >> Yeah. Yeah. Yeah. Yeah. We always have a family of cft depend on n but you should view this relation as a statement about the single c and this n expansion is approximation. Say for example this yeah just like what you do in QED E just one alpha is just one over 137. But we always expand alpha in some analytic parameter. Okay. So you always say let's just imagine n is 10,000 and then here when we do this approximation we treat n as some parameter and then you can treat it analytically. Yeah. And and here you on the left hand side always view it as a single safety. Okay. And not a family of safety. >> Yeah. You understand what this filter means? I mean if somebody gives me a function I can separate from this function. the smooth part. If somebody just gives me a point, a single CFT, what does it mean to separate it into smooth plus? It means nothing. >> Yeah, this is a statement about the No, this is another statement. This is a a statement not about it's a statement about the right hand side. >> But there has to be a family of CTFs otherwise there's nothing to talk about. Yeah. The right hand side depend on n as a parameter >> which means there is a family of safeties. >> Yeah. Yeah. There is a family of safeties. But you should view this formula as approximation to a single safety at some particular value. >> Not a single if there is just a safety at 10 equals 100 there is nothing to talk about. >> No no no no. I think uh you have to separate expansion into exact theory. So the exact Q is 100 say one over 137. But when we do the do ptopic expansion in QD we we always expand in E treat treat as some parameter here just the same thing just the same thing and and just right hand side you treat as a parameter and the left hand side you can either view it as a family of CFTs or you can treat as a single CFT. It's up to you. Okay. It's up to you. You want to treat the QED as a family of series or a single series. It's up to you. Yeah. And then then we can just imagine you have a filter and then then that extract the smooth part and then eratic part is zero. Okay. And so there's some so this filter so if this split exist then the filter should exist. Okay. The filter should exist. And in particular this filter should act as a a projection. Say if you act twice uh uh yeah I just for any quantity here you just spit out the trans series part. Okay you spit out the trans series part and you throw everything else away. And then you also have the uh um and the u yeah you also have the property say if I have a z1 plus bz2 then should be equal to a fd1 and plus b fd2 say if a and b are smooth in the sense that if a and b are ordinary trend series okay and so so the last property of this uh uh filter is that the uh we want it to be a a positive in the sense that if you have a positive quantity uh we want the uh uh the filter to be positive. Okay. So, so now with this filter and now the new dictionary which we will propose is that the Z CFT smooth. Okay, which is the same as you the uh uh you filter the CFT uh angle to infinity. Okay. And then this should be identified with the uh uh uh uh uh uh the gravitational pass integral expression. Okay. So so this evalore m and so that should be related to gravitational passing expression. And uh so this is a new dictionary. So this is a new dictionary. Okay. >> Yes. Where does this positivity condition come from? If you purely look at a decomposition of transient pieces, where does the positivity condition come from or is it just is that extra physical input? >> Yeah, I think it's the actual physical input is the actual physical input say in the unitary series and uh uh uh and this f I think yeah this is the postulate at the moment. So these two can just follow from the definition uh of the split and here uh uh uh uh I would say it's a physical requirement. >> So Z star is your complex conjugate. What parameters like all parameters? I mean >> yeah Z is a number right Z is just some kind of trans series. >> Uh yeah yeah this is a number and and you can just treat this as some ordinary functions and then you take the star. So it's a Laurenian part I mean I'm just wondering if if you're considering a uklidian part function then >> yeah it's a number >> is real >> yeah fun is a number >> is a real function of beta I mean if I just think of >> oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh here I can also include the operating insertions and when you have operating insertions excited then doesn't have to be real >> and your you take them to be complex numbers as a source. >> And and what goes physically wrong if you violate a condition? Um I yeah I think uh um yeah just uh I find to be a desirable condition to impose is because the um uh somehow you want your microscopic data to preserve the udarity structure say of your say microscopic data. Yeah. Yeah. Just this is a heristic physical motivation. Yeah. Okay, good. So now let me just explain how this actually can uh uh uh uh can address this factoriization problem. And so now it's obvious is because the um now now let's consider the Z CFT on M1 and on M2. So this factoriize into the Z1 on M1 and Z2 on M2 just Z value on M1 and M2. Okay. And now to pro uh to compare with the gravity side I have to do a filter Z1 and Z2 with the gravity one. Okay. I have to uh do a filter. So now let's imagine how we do the filter. So if we have D1 then you have D1 smooth plus D1 erratic and similarly you have D2. And now if you look at the product of Z1 and Z2, then you can have Z1 smooth, Z2 smooth and then you can have Z1 erratic and Z2 erratic and then you can have cross terms. Okay, you have two cross terms. Yeah, smooth times erratic. Okay, so let me just save time. And now when we do the filter and this term will just go to zero. Okay, it's because it's the smooth times erratic and that acts on erratic should go to zero. When f act on erratic should go to zero and then you just have f acting on this but so the smooth part by definition factoriize. So this just uh because f does not do anything to the smooth part. But now the key thing is that you take the two erratic piece you take the product that can in principle generate a smooth piece. Okay. Say for example imagine this have some kind of erratic phase in this Iran uh Z1 and you have some erratic face of opposite uh uh sign in Z2 when you take the product of them and in principle that can cancel then generate something smooth. Okay. So now the key thing is that this Z1 erratic and Z2 erratic in general does not have to be zero. Okay. And then this quantity is not uh no longer factorized. And so now this is mapped to the gravity side which the gravity m1 m2 which you have factorized piece which will identify with this ident factorized piece and then you have the wormhole piece and then now we can just identify the wormhole piece with this piece. Okay. Then we have a equality f1 erratic z2 erratic is equal to the wormhole. Okay. So we have this key formula. So that tells you where does this wormhole arise on the CFT site and what kind of correlation which this wormhole captures. So Who captures the following kind of correlation is that if you look at the partition function one M1 and partition function M2 even though they indeed they are uncorrelated but they can be correlated in the way that if certain part here times certain part here can lead to the smooth part and then that leads to a correlation. Okay and that kind of correlation is captured by this wormhole. Okay, that correlation is captured by the Wall. So now let me give you example of this kind of a phenomenon using the good formula. So in the good middle formula say let's now focus on say remember this erratic this oscilly part of the good will formula which I can just now identify as my erratic part. Okay. Okay. And then if you do a laplas transform then you get the uh you get the erratic part say of your thermal partition function. And now let's look at the product of two erratic part. Okay remember this ratic part can be written as sum over periodic orbit. Okay. And now let's look at the uh uh the product say divided by two of two erratic part separated by some small energy. Okay. uh uh separated by some small energy and that's will comes in uh yeah uh uh uh yeah the reason we look at that is because this is related to the spectral form factor which I will uh uh uh we will see in a moment. So, so if you look at these two products and then essentially you have the uh this for this one and this for this one and then you take the product together and now you can expand yeah because now let's take the epsilon the energy difference be much smaller than the energy and in this case then you can expand this exponential this SA in power series in epsilon etc. Anyway, when you take the product together and then their terms in in here in here and here this face they can cancel each other. So you have CC here some of them appear as a positive face some appears negative face and turns out so the product will contain many things. Okay. And and it turns out include a universal term which is smooth. Okay. And plus some other things minus 1 / 2 pi square epsilon square. So this term is smooth in epsilon. Okay. It's a it's a it's just some some regular function. And it's a this term is all the h bar to the power zero. Okay. Okay, it's also smooth in H bar [cough] in J bar and of course there also contains many other erratic term etc. Okay. And can also contain some higher power some other powers of epsilon beta regular anyway. But there are no exponentially suppressed ones. >> Um we there might be I don't know but but this is the easiest one you can isolate. >> No I meant the analog of a wormhole like exponential suppression. >> Oh. Oh this is a wormhole. Uh this is expensially surprised. It's because this order uh uh h bar to the zero and and is suppressed compared to the leading order which is one / h bar to some power. >> This is the ramp right? >> Yeah this is the ramp. So so this when you free a transform to coordinate space. So this give you the ramp and when you free transform to coordinate space and then this one / epsilon square behavior precisely give you give you this precisely this 2 / 2 pi okay the linear ramp. So so we see that the linear ramp can be understood as you do the projection of this kind of product. Okay. And then and then that will give you this smooth piece. The leading order term is a smooth piece as epsilon goes to zero. And uh uh and then then then then then that give you the ramp. Okay. And in holography in ads the ramp come from a wormhole. Okay. The reason the w should come from a wormhole sorry it should be a double trace. uh uh I do a filter uh uh uh this is a double trace uh uh since this is a double trace and this will involve two boundaries and in holography this is the e [laughter] described by a so-called double cone structure a wormhole uh uh like this so this is the uh uh uh uh s shank and stamford uh doublecom wormhole okay so This is example that this kind of wormhole say in gravity indeed can be considered as uh uh uh extracting uh the the u the the um the smooth part of the product of erratic piece. Okay. So uh uh uh uh Eric also has written uh papers in the two dimensional CTF and argue the same uh uh uh the same phenomena. So so yeah so I think I can uh u stop here and yeah so so here I just give you an illustration that yeah one second. Yeah. Yeah. I just give you illustration that the uh uh somehow when you relax this idea that the larger limit can uh uh uh is smooth actually the potential lot of things uh the there can be many new phenomena. Okay. Uh there are many more things I can say and uh yeah but um yeah I think it's um yeah I can stop here. Maybe people can ask questions. Yes. [applause] Yes, >> it seems like it was important in this good example that the well the erratic pieces coming from the sum over semiclass orbits periodic semi I mean do we expect that I don't see where we would get classical periodic orbits in for like a black hole ads black hole >> yeah so um yeah indeed so You see um so in order say let's think about how would we generalize this to a larger system. Okay. So, so if you want to generalize to the Nigian system, you see the key is that you want to see uh uh you need to probe this kind of level spacing which in the context of the large system that's what good will formula uh probes it's probe this kind of level spacing and in order to probe this kind of scale which is the black hole micro state scale you have to go to you have to be able you access deconfined data in the in the in the in the uh uh uh in the significant way. So previously when we look look at n limit we all look at the confined data uh uh we look at the single trace correlation functions we look at the just the partition function it's all confined data and somehow in order to understand this analog of the periodic orbit you have to look at this the deconfined data in the in the CTF and similarly you have to be able to somehow go to the gravit you have to by definition the gravitational phase space and uh defined And from Einstein gravity is all confined data. Somehow you have to go beyond that. Yeah. Yeah. >> Uh are solvable matrix large matrix models too simple to show this phenomen. >> So I I'm not sure it's too simple. I just so this phenomenon you you can easily find them. You should be able to find them when you have a black hole sector. And somehow when you have a black hole sector, you have this uh uh uh sector of a very chaotic state. The OP can be chaotic etc. But whether that's a necessary condition, I don't know. Yeah. >> So one could test it out with solvable large end matrix models. One could try. >> Yeah. Yeah. That's right. Yeah. Yeah. I think definitely one should try and then for example I think this h bar goes to zero limit even though the singular nature even though it's most clear when you look at the chaotic system by looking at go formula but for integral system maybe this phenomena is also there and they just less less visible yeah less visible so so we don't really yeah I think we should just explore yeah He just not assume you can always have a smooth expansion. Yeah. >> Yeah. >> Yeah. I was I was curious in your response to the previous two questions. >> Yeah. >> I mean I I understand the motivation and and I like this filter function but our prior it wasn't clear that this required you to have something in the spectrum of black holes because this was a statement about larger limits >> in families of theories. >> Yeah. which didn't particularly distinguish whether you know you talked about giant gravitons or black holes and so I understand from the gravity side black holes are the clue and that's where the wormholes come from but like Shiran was asking there should be something if it's true should be visible in models where you may not necessarily have the confined phase but nevertheless there should be this uh dichotomy visible >> right you see uh uh I I don't know uh uh where you will look for such kind of behavior. I'm just saying uh uh uh in the chaotic system when you have a chaotic sector then then maybe it's easier and maybe it's easier maybe it's more visible and but I don't know the full extent say h bar goes to zero limit in in ordinary quantum mechanics and maybe it's more visible in chaotic system but maybe also happens in integral system yeah so so I'm not excluding it I'm just saying that's maybe it's the easier place to find it and you can also Imagine the situation the examples which such erratic piece is not there and if you look at the partition function of the yeah if you look at the partition function of harmonic oscillator then you don't see it and if the system is too simple you may not be able to see it >> super conformal index is probably something like that >> that's right yeah super conformal index if it's say one half bps state say for the uh um for the income of super you may not see that >> but I think even 116 people about you that these wormholes don't exist that the gravity side >> that's right >> it's probably too simple for this >> yeah I think that's actually a very interesting question whether I my own feel is that 160 may not be too simple just there because somehow you have to go around the super symmetry a little bit you see the reason you don't have a wormhole is because the the zero mode of a gravitino somehow if you find some way to destroy those zero mode etc and then You may be able to Yeah. Yeah. >> Yes. >> Uh don't you expect this cancellation that smooth piece to be finally tuned? >> Sorry. >> Don't you expect that this will be finally tuned as smooth to piece? >> Yeah, it they don't have to be fine-tuned. It just say um so certainly it's not arbitrary and so so they need some kind of correlation. So that's what we mean by correlation. And in the sense that somehow the erratic piece here somehow erratic piece here they can secretly cancel somehow that define some correlation. Yeah. >> But say I take M1 M2 and then I deform M2 a little bit. This will change very the the second piece. Um yeah, not necessarily that kind of correlation may not you see you only need even though the uh even though the detailed behavior dependence on n or those things are erratic but the structure uh uh the fact that you can cancel them uh uh uh that may not be uh too too erratic. Yeah. Yeah. Uh uh certainly you need to satisfy certain conditions and wormholes you don't say uh uh always find them. Yeah. sometime. Yeah, they need to satisfy certain conditions. >> Toulate what Joseph said. Now you moved the puzzle from what it was to exhibiting this filter. >> Yeah. >> Which you specified in an extremely vague way. There is gazillion possibilities. [snorts] You should find one natural which will satis which will resolve this puzzle. Otherwise, this is not a resolution of the puzzle. This is just visual thinking. >> Yeah. Yeah. Yeah. No, no, I'm not saying this is violation. I I think this is a proposal for revolution >> and one possible filter is just to average in a small window of n and so we are back to the previous resolution and then this is not even a new resolution. No, no, actually it's not clear uh you see you see uh uh the point I think the the important point is whether you resolve this I think of course we want to resolve this puzzle but I would like to explore the question the nature of the larger limit and the nature of G and goes to zero limit okay so previously we just assumed certain structure and now I want to question that structure I want to be all I want to be don't take that structure for granted. That's the only thing I'm proposing. Okay. And then I say and based on that thing and then we may be able to solve this factoriization puzzle. Okay. uh and it's fine you say uh I don't have a clear uh uh uh uh yeah we need example uh uh to see this kind of a split or example to exhibit this kind of erratic structure uh uh which I don't have at the moment but I want to raise the question >> so perhaps um we can continue our discussion over the coffee break Thank you. [applause] [music] >> [music]