I guess we can get started.
So it's a real pleasure to have Hong Liu back at Rutgers again.
Kong. was supposed to stop here.
be followed pathway that is almost the best thing we could have imagined for a postdoc at Rutgers. We really wanted him to be a professor at Rutgers.
But we couldn't keep him. He became a professor at MIT, which is the second best thing.
He's gone on to do really quite amazing things in a variety of topics, a lot of them in the anti-desitter formal field theory correspondence, which is what he's here.
to tell us about today. He was. telling me this morning about.
little hints about what his talk is going to be about. It sounds like novel and amazing. We're going to take him to dinner. So anybody who's interested in.
Come in, please tell me after the seminar. And we'll be going to salt the seafood place in New Brunswick.
everybody to invite you. So take it away. Okay, great. Okay, thanks. Yeah, thanks, Tom, for the introduction. And yeah, it's a great pleasure to be back here again. Yeah, many very good memories here.
Uh, so yeah, so we'll be talking about some work which I just finished in the past December.
Oh, you click on the slides. Oh. Then you work with me.
Okay. Okay, so yeah, so the one of the central.
relation in the holographic dictionary, say, between Cft and ads gravity is this very simple relation between the Cft partition function.
on some Euclidean manifold M. And the gravitational partition function with M as the boundary of the space type.
And so here M can be something very general, just some compact Euclidean manifold. You can put your theory on, and it can be disconnected, can have disjoint component, for example, and can also have highly non-trivial topologies.
Okay. So one of the important parameters in the CFT is this n, which characterizes the number because freedom. So normally we write the total field theoretical degree of freedom. We write it as n square. And so this is related.
to the G. Newton, Newton constant on the gravity side by one of G. Newton. Yeah, up to some numerical factor.
So this is a relation which is supposed to be valid at full quantum gravitational level.
But, however, of course, though we cannot really. calculate this gravitational size in the quantum regime. Actually, we don't even know how to define it.
So what is mostly people do is to consider the semi-classical limit.
of this relation. Say, you take the G. Newton goes to 0 limit on the gravity side. So that's what is normally called semiclassical limit. And then from that relation between N and G. Newton, and that corresponding to the angle to infinity limit in the Cft.
So emphasize this limit here should not be understood as, say.
say the mathematical, just the point by limit, so they should be understood as in the following way.
So the limit should really be understood as the symptotic expansions in the respective parameters. So, for example, on the gravitational side.
we represent the gravitational partition function. Formally as a path integral.
And, uh, so… so here I define my action, gravitational action with the G Newton out as the prefactor.
And then… then, yeah, and then formally. This gravitational path integral should be viewed as a device.
to do the side point of approximation. So in the J. Newton goes to 0 limit, just sum over all possible side point.
And I ask? Yeah, obvious question. So Gene Newton has dimensions. Yeah. And so am I supposed to think about the way you take the dimensions out by taking the overall scale of the Mount Balderham, or that's right. That's right. So so here.
in in ads, it's easy. It's because there's a curvature scale. So essentially, you keep the curvature scale fixed. You take G Newton goes to 0. It means that you take the curvature radius to be fixed, but take the Planck length to go to 0.
Yeah. By gravity, do you mean strengthening? Here, by gravity of here, I can try to generalize to string theory. But the moment, let's just imagine just to Einstein gravity cross matter theory is the same as Feynman diagram expansion, and that's…
not renormalizing. Yeah, yeah, it's not normal either. So that's why this is only viewed as a device for generating this expansion.
So to generate this asymptotic expansion, you don't need the reliability. You can just treat it as effective field theory. And then you can generate effective field theory. Well, there's a cutoff there. Yeah, yeah, there's some kind of — yeah, yeah, you really that's an effective field theory. Part of the answer at all.
Yeah, yeah, you can generate. Yeah. So so this I is.
sum over all possible saddles means that the solution to a classical equation motion with this boundary condition.
And then this Z1 would be considered as a one loop contribution, say, for the matter contribute for the fluctuation around this saddle. And then here are the 2 loop, etc. Okay, so in principle, this is something you can carry out explicitly.
And yeah, so so so in order to indeed, since the graphs T theory is not normalizable, you have to treat it as effective field theory, and then do the your standard procedure.
And on the CFT side. So it's generally assumed that in the large N limit.
We can expand the CFT partition function. In one of n square again as a symptotic series.
So so here I. corresponding to sum over some log in saddles.
And then then then you will have disruptor. So you have some margins leading contribution. The the fluctuation around the margin cycle, and then the higher loop contributions.
Okay, sorry, you don't literally mean the 1st line of gravity. Do you? I mean, there's all sorts of things like stokes phenomena.
I mean, yeah, so this is Euclidean. Yeah, this is Euclidean. Euclidean is cleaner. Yeah, but not all sample points might contribute.
the confirm integration. Yeah, contours of integration. Yeah, yeah, yeah. For example, the conformal mode is supposed to do contour deformation, and not all contours might contribute right right. Yeah, yeah, you have all these subtleties. Yeah, you have all these subtlety, but just the.
Imagine you know the rule of doing it and then and then the gravity answer will be this way.
Is that the conjecture that there exists a contour such that the saddles contribute? Does every single saddle contributes? No, no, no, no, the bomb over all the sandals which contribute.
Yeah. But even even then, like the Z is the Z1 guaranteed to be?
Just the one loop. I thought there's all these like Stokes parameters and coefficients. Is it still? Is it always just c 1. They have high orders.
Yeah. Change parameters, you might have some saddles become unaccessible and accessible. Yeah. And then you'll be relating the two series by some kind of stokes. Here, I just saw, for example, if you do the area function.
The area integral is just sum of all possible cycles. And here I'm not only do the dominant side. You also sum of all subdominant saddles as far as they they they they contribute.
Yeah, the stock's phenomenon is about every example exactly in the airy example in one domain you don't could take the exponentially growing cell.
Yeah, yeah. Yeah. In some parameter space. Yeah, in some regional parameter space. So so so you assume. Yeah, of course there's a parameter space in defining the the M. But just for each parameter space, you write down the series.
I think it's fair to say that there are a huge number of subtleties in this field. Sure, sure, sure. So here, I'm just writing down a general structure.
And it's generally assumed that the Cft side, you have the same structure of the large N expansion.
Okay, folks. And then then what's this equation means?
just means you can match them. order by order. Okay? So so the saddle point should match.
The leading contribution in the sido should match one loop, etc. Okay, they should just match order by order.
So, so that's the general assumption.
But yes, in the. Cft calculation. Yeah, there were no renormalization that we knew it was right.
Um… There is some kind of, yeah, I think when you do one over n expansion, so the gravitational side, you're talking about the UV divergence here. Yeah. So the UV divergence on the gravity side.
It's a little bit tricky to translate on the CFT side, but one… One thing which could be related to this UV divergence on the gravity side in the CFT side is how you.
How you? You see, if you redefine your end a little bit.
And then you are going to reshuffle. those things. And so that's a little bit like the, yeah.
Or you say that's the yeah, it's related. Yeah. Yeah, yeah, yeah. So the ambiguity for yeah, for example, you can imagine. Yeah.
We don't have a precise translation of the UV divergence on the gravity side to the CFT side, but the one of the impact implications of divergence is the some kind of ambiguity we define the terms, and that's related to the some possible ambiguity you defined those terms.
Okay, good. And but there have been indications that this relation actually may not be correct.
Okay. So what? One indication is a very old problem.
called the factorization problem, which actually first was hinted by the Whit and Yau in 1999, and then Madison and Mouse sharpened it.
So the basic idea is the following. So let's consider putting your Cft. Take this M to be destroyed unit of 2.
Euclidean manifold. Okay, so when you put the Cft on the 2 disjoint and manifold Cftane function just factorize.
And the diffractionization does not depend on whether you take ngot's infinity limit or not. Okay, so so so I obstructing the view on this side. Yeah.
So, uh… the.
So you should have this factorization of the CFT amplitude of t partition function.
which then the gravity or partition function in the G. Newton goes to 0 limit should also factorize. Okay, so factorization should not be affected by this limit.
But, however. On the gravity side, you often find expressions which do not factorize due to wormholes.
So here is an example. So you have two boundaries, M1 and M2, then you can have a wormhole contribution which connects M1 and M2. And this contribution can be exponentially suppressed. It's OK compared to the leading contribution. But remember, we compare them order by order. As far as anything.
which is non-factorized on this side means something has to be non-factorized on the other side.
And so presence of such kind of Euclidean wormholes.
and then the spoil factorization. Such kind of amplitude certainly does not factorize.
And so, yeah, so this has been a puzzle. So how do you understand those wormholes?
So in some lower dimensional gravity theory, this question is understood better.
For example, in the Jt gravity, which is a very simple 2-dimensional gravity. So in 2 dimensional gravity, you can actually just define.
your your gravity using path integral. Okay. Because the lower dimension. There's no dynamic where every time everything is simple.
And you can just define your gravity using path integral, and then you can just carry out this calculation using passing the pool. And then you can show that the JT gravity defined that way is actually related to some ensemble average.
set of some CFTs. So if you do ensemble average, say, over the C of T's, then, of course, in general, they don't have to factorize. Okay? And if you have a product, then when you average that and then the average does not have to affect what.
And so there's some indication that for pure Ad 3, gravity may also be related to some random CFTs. Again, pure ADS 3 gravity. You can also define it in principle just using gravitational path.
And so the indication some random CFTs. And so people have devoted lots of discussion to it.
But for general higher dimensional theories. We know that the gravitational path integral cannot be well defined. Okay, they are just not well defined, and we know they're not enough. Okay, they don't capture the string degrees of freedom, etc. So so they just the entire dimension. The story is much more complicated.
And also those examples from screen theory and. Yeah, those explicit example from screen theory. We just don't know how to. Yeah, yeah, this kind of average does not seem to apply.
Anyway, so… So people have. So there are 2 logical possibilities here. Both have been discussed by by many, many people. So I will not give the references here.
So, for example, one logical possibility is that gravity expression should also factorize.
So now you ask, what's wrong on the gravity side? So should you just exclude the wormhole contribution?
Or should somehow the gravitational path integral in the semi-classical limit, maybe the saddle point is not enough, you have to do something else. Okay? So it's not clear.
And so also. Another possibility.
is some kind of average or cross-screening is needed to map the CFT to gravity result.
So the so in other words, you you imagine. Yeah, just.
As the analog of this Jt gravity. And you imagine somehow, rather than consider single CFT due to gravity, we actually can see the family of them, and then you average of them.
And then. But at the moment, there's no candidate what this average should be. Okay. So right now, people have been just mostly doing this gravity calculations.
Then they just interpret, and then they just define this object in terms of gravity.
And so currently people just use gravity a result to define this quantity. But what what this quantity is? Even in 3 dimensions. Yeah.
So right now, there's no intrinsic boundary definition. And they are also.
2 2 would mean that the original holographic conjecture is significant revising, I guess. You mean this one? Yeah, yeah, yeah, that's right. Yeah, yeah, yeah. So so yeah, this will mean that you have to.
revise your CF BDSFT proposal. I mean, people like to say that.
infinitely many successes of ABS-CFT. However, of course, grading is it better preserved all those exactly. Yeah, yeah, yeah. So that's something I'm going to mention. Yeah.
Good. So question. Yeah. Why isn't it possible that the thing you have to add is something analogous to the to the wormholes on the factorized CF.
So there's something that connects them. Yeah, it. Just on the CFT side. There's no candidate. People have not been able to identify such a candidate.
you'd have to change the fact that the partition function of CFTs on a disjoint unit of manifolds is the product of the participants. Exactly. That's what I was saying. So wouldn't that be a logical thought? No, no, if you.
If we use the standard definition of quantum field theory.
Maybe it's something on the left hand side. Right. Yeah, yeah, that's something I'm going to propose.
Yeah, it's something I'm going to propose. It's along that direction, but without changing that fundamental thing that… From basic definition of quantum field theory holds factor.
Good? So so people are actually, let me just mention elaborate on this second point a little bit more. People have come up with all kind of ideas of doing averages.
say they average over the Hamiltonians. maybe some CFT data, say the OP coefficients dimensions.
States, observables or M. Okay, so there are many ideas how to do this kind of averages.
Many of them have achieved some partial success means that in some subclass of series they can actually map to the gravity results. Okay.
and to certain regime. But there are many both conceptual and technical challenges.
For example, there's just no natural ensemble exists. On the gravity side, for example, we really understand. So yeah, sorry. So for the example, for example, in the duality between n equals super young meal theory and the type 2B.
string theory, there's just no ensemble. Okay? And furthermore, even if you want to do some kind of algorithm.
That procedure has to be universal. Okay, you cannot say.
This already worked for this series, this average works for that theory, and this average works for this observable, this average works for that observable.
for fundamental relation like this, if you want to propose an average has to be universal. Okay?
And somehow should apply to all theories, and we not know in the details. That's the right answer, but I mean, we could say that the Xenom logic of metric gives a natural measure.
on the space of conformal fields. Yeah, yeah, it's a mesh on the space of conformal field series. It's true. Yeah, but here is not about the it's not about the whether you have a.
the measures. It's about, for example, in the case of Ink for super young mercury due to type 2B string theory.
There's no families of including military due to type 2B solutions. There's no.
Yeah, you don't even know what kind of family. Well, there's a family part of a trip by Cal.
by which town. But tau is already mapped to the to the gravity side. It's one. So you cannot average that.
Yeah. You have to have some other things to average.
But but there's no other things. And also this, remember, on the gravity side, you have this asymptotic expansion.
And so this large asymptotic structure, which. We have already seen many successes in comparing these 2 series. Okay? And that structure you need to preserve.
And this kind of average can easily destroy that kind of structure.
And also we have had many precision success. Okay, Greg just mentioned earlier. There are many precision success. Say, for example, to map mapping the ink for super young calculations, Abgm calculations to gravity calculations.
They're all based on single series. Okay. And if you average over them, you are just going to destroy them. And so you have to really preserve all these kind of many precise match.
And finally. Conceptually, doing the average is actually a potentially.
So so if you do some kind of average.
So if you say that some average CFT to be mapped to the gravity side.
And then even if you start with a unitary Cft, when you do the average, you are going to destroy unitarity.
But one of the key advantage of the ADSFT or this kind of theory is that you want to use the utility of CFD to understand the utility of quantum gravity. I don't really understand that. So if you're in Euclidean signature instead of unitarity, you have Oster.
positivity. And so that says the reflection of an operator. An operator is a correlation function, which is positive definite right? If I average that statement.
with a positive measure. I'll still have something positive. Oh, that's just one. I think averaging should preserve Oster positivity. Yeah, that's that productivity with just one implication of the utilitarity. Right?
And in particular, if you want to map. If there's a fundamental relation to related Cft and gravity involving average.
You also have to do it for Lorentzian spacetime, too.
Okay? This is just the… you have to do for Lorenzing, you have to do it for everything.
You cannot say, oh, I only do the Euclidean partition function.
You have to do it for everything. There has to be a procedure. Statement was that if you satisfy master positivity in the Euclidean space, then you can construct a unitary theorem space.
Yeah. Yeah, I'm just saying there's a potential difficulties. I'm not saying… yeah, just it's a simple here, just a simple logical statement.
say, if I take a family of unitary theories. Then I average of them, and the reduction theory is not average statistical statistical average of unitary series is not unitary.
From that point of view, it totally but I think if you think it… reflection positivity is also obvious that that's the problem. Yeah, it's true. Yeah, yeah.
Good? Okay. Why is the the.
like cases that are known, not affected by this factor. Why did it not show up?
All of these examples of the dictionary working. Yeah. Why? But like, why is that not affected by the fact? Oh, you speak because people normally, when they do the checking, they only check the leading term.
Yeah. Yeah, yeah, it's easy to preserve the leasing term. But the key is you have to preserve the whole thing.
You have to preserve the whole thing, not only all the time also soft dopamine term, just the whole structure.
We're here by leading. You mean the leading saddle. Yeah, the yeah, normally people, when they do the average.
to compare with gravity side. they just compare.
They just compare this quantity. Yeah, this leading one. Sometimes they do this one. But you have the whole thing. Yeah, you just do the i equal to 0, the leading one, and then maybe the z equals 1 into that one. But you have the whole series, but then you have all the subdominant ones.
The subdominant ones are like non-perturbatively small. Yeah, yeah. But people do check or a lot of words for evasion theory, right equals force to be unreal, right?
Yeah, loads checking is not fun. Some average loss checking is for the specific theory. Yeah, yeah, yeah, yeah, they do. And that's for the lead. Those checks are for the leading sound. Yeah, that's checks are for the single series.
I'm guessing that kind of check, that kind of check for single theory can be destroyed when you average over them.
That requires symmetry, very delicate structure. Okay, so this is one problem.
But there's another problem. Which is much less emphasized.
which I call erratic end problem. So so this erratic end problem was hinted in the earlier paper by Schenneker and Whitten in the paper 2022, and then I emphasize a little bit in my September, in the earlier paper in September.
So the basic idea of this erratic M program is the following. So, as I mentioned, on the gravity side.
We have this formally, this limit is defined formally by doing the saddle point of this path integral.
Okay? And then you have this and the key about this path integral and about this structure.
Inside, there's no J-neuton dependence in the action. All in the variables. So the dependence only appears here. This is a very specific structure.
And so that means… The hosting has a very specific structure dependence on G. Newton.
Okay? And yeah, here, sorry. I missed the bound Q anyway. So so this is normally called. So I will refer to this specific structure smooth behavior in J. Newton.
Okay, this is just a name. It does not mean the smooth in the ordinary sense. Okay, just call this behavior smooth behavior. So mathematically, this is called the trans series.
Okay, mathematically called the transfer is essentially just a sum over many asymptotic series, including subdominant ones.
And the transfer series have… also specific structure, etc. Mathematicians has been studying them since the 80s. Of course, physicists has been studying them a much longer time.
So on the Cft side, as I mentioned, the standard assumption.
You said the CFT partition function, the ngot infinite limit also have this smooth behavior.
They have this kind of expansion. So, uh, but.
Cft partition function. Actually, if you look at Cfting function actually end dependence is very different.
The field depend on N. So field are n by n matrices.
And the traits depend on n in the long trivial way.
And so so in the Cft n actually enters in the partition function in the rather complicated way.
And so A in the CFT part in the boundary CFT is just not the coupling, but also characterize the number that you could read them.
Okay. So so there's no reason.
Why the CFT part didn't function when you take angle to infinity limits should have this specific structure.
Okay, I should have this specific structure. And in fact, that actually the fact that the CFT side.
as n goes too large, you can have rather erratic behavior on N. Yeah, was emphasized by Greg actually many years ago when when study the say black hole into be counting, etc. In the even though those are not.
say the partition function we are talking about here. And in the partition function, we have not seen explicit example. But but the agricult has pointed out in many other contexts this kind of erratic.
Yeah, some kind of independence. It's not say this kind of smooth dependence.
And so there's no reason. why this should always be true.
Okay. And so so if this is not always true, that means that cft side.
and the gravity side, they may have completely different structure.
Okay, in this limit, the graphic design, just by definition always have that smooth structure.
Okay, always have that kind of trans-theory structure. But there's no reason CFT side have this trans series dropped.
And so so we have been always making this assumption.
So based on the structure of the Feynman diagram, based on some simple matrix quantum mechanics or simple matrix integrals.
But I'm going to, yeah, if I have time, I'm going to mention nature. Actually.
If you look at Cftane function, there's reason to believe.
a much more complicated structure may exist. beyond that those can be seen from the matrix simple matrix integral or matrix quantum mechanics.
With those papers, the Greg were not simple matrix quantum mechanics. No, no, they are not simple matrix quantum mechanics. They're they're a complicated way to do this state counting.
Yeah, BPS indices. Yes, it must be the chops. Yeah, yeah, it must be coefficients of modular forms also.
Is this related to Bps indices? No, it's the.
It is that these things like arithmetic properties, and then.
Some arithmetic functions have a lot of scatter. They have an overall and they have an… if you smooth them out, they have a… maybe I don't want to get… and they smooth them out, they give… they have a 1 over N asymptotic expansion. But if you actually look at them as functions of N.
They have a lot of scattering. Yeah. So Greg have seen many such examples, but unfortunately, we haven't seen any explicit example in the partition function. You want to partition. That's right. Yeah, I want a partition function to make this the sharpest way is the partition function.
Okay, but but I want to raise doubt. Okay, a race question.
on this assumption. Okay? And I want to explore. Say, if you want to relax this assumption.
What can we get? Okay, so what you propose.
If we relax this assumption. Actually, we can gain a lot.
Okay, we can gain a lot just by some very simple proposal.
Okay, so I will make a simple proposal to address both problems.
So the idea is very simple. So let's imagine.
Let's take angle to infinity name it. Suppose the anchor infinite limit of this Cft partition function. For example, as Greg just mentioned, say, say, if you look at some other quantities, imagine you can separate them into a smooth part.
corresponding to a trans series. And then then something else. Okay, which may say depend on A in the erratic way does not have this smooth one way expansion or exponential dependence, etc.
Okay. And let's imagine such things exist. Okay. So so one possible. One possible since I've seen this.
is that you may say, have this kind of.
Say cosine say n behavior or n squared behavior.
Okay, so imagine this behavior which does not come from the saddle point. So some saddle point can give rise to this behavior. But you may imagine not all this kind of behavior can give rise to the can come from the cytop point.
So if you have this kind of behavior does not come from the saddle point.
And then that will be in this erratic part.
Okay, this will be this erratic part. And so, yeah, so I will call this the the trans series part, which I call it smoothing n.
And then at the moment, I don't know how to define this, the other part.
Right now, I just define what it is not. Okay, just the part cannot be written as a trans series. And as I call it erratic.
But this erratic part is also non-perturbative. Yeah. Yeah, it's by definition. Yeah, it should depend on n in the Hydian non-trivial way. Yeah.
So so then the new dictionary, which I will propose.
It's an image. Imagine how this splitting. And then I just identified the smooth part with the gravity, because they have the same structure.
Okay? And then this is identified with the gravitational partition function.
And then to match the CFT expression with the semiclassical gravity one.
This means we have to isolate this smooth pot.
So we can imagine isolate this loose part using a filter, what they call a filter. Just say, imagine you have a larger limit.
say of your CFT partition function, and then you just project out this erratic path, only keep the smooth part, as I call this a filter.
And then so and then then this filter is intrinsically defined, provided that the split is well defined.
Are the wormhole effects supposed to be the erratic? Good, good, good, that's what I'm going to talk about. Yeah.
So so it's not simply the erratic part. Yeah, it's come from the erratic part.
So so the filter is intimately provided this pretty is well defined. Okay.
So now. So in one step. So our proposal… You said you add this filter on the CFT partition function.
And then that is identified with the gravity one. So so this filter implicitly the large anonymity is taken. Okay, it's only defined in the larger limit.
What about the original proposal that doesn't have the semi-classical in it?
which proposal like Zcfd equals the gravity. Oh, that we don't change. Yeah, we only change in the larger limit.
The point is we don't know what's gravity is. Yeah, that's right.
That's why we still at that one, we don't have any clue at the moment, and we just assume it's still correct.
yeah, this is the only one you can actually calculate in. Yeah. But wait a second. If G and is one over n squared. Look at it.
you know, n equals 13, 14, 15, and 16. Yeah. Then on the CFT side, you're imagining there's some kind of scatter around some average behavior. Yeah. And so it better be there on the gravity side, too. Great, good, good. So, so, so yeah, we'll talk about this point later. Yeah.
So… so right now we only talk about the this smooth, this gravity part which defined through the path integral is because the wormhole just comes from the path integral. Okay, we just come from the saddle point. So this solve that problem first.
And so this provide the intrinsic boundary definition of what people write this safety bar. And now I just.
Okay? And so. And now, because of this filter.
So I will explain in the next slides when you act on this the disjoint manifold.
And then this normal factorize. Okay? So that will address non factorization problem.
Anyway, before doing that, let me just mention the properties of this F.
should satisfy from its definition. So by definition, the F should behave like a projector.
Okay, so if you have something, or it's already smooth. So, of course, when you act on F, it's still smooth.
And so we assume that the F should be a linear in the sense that if A is some trans series, Ab are some trans series, and you multiply it, they should not affect you do the filter. Okay, because A and B are already smooth.
And so you can just take them outside the field.
And the last which I don't know how to prove at the moment. It does not directly follow this definition.
Okay, I it will. If we know how to do the split, of course, we can deduce whether that's positive. But since here I'm just postulating it. So so I will postulate that that split.
also have the following structure that is positive definite means that if you it cannot take a positive quantity into a negative quantity.
So this is a very lateral assumption to make.
So yeah, so these are just these basic properties. So now with these properties.
Then this can solve all the difficulties I mentioned earlier regarding the averages.
because this is defined for your individual theory. And this is universally defined.
And you don't need to do average, etc. So it's all those difficulties you can address. If this object exists.
Okay, now let's see how do we understand wormholes?
Yeah, yeah. So this property warrant that the filtering does not suffer from the difficulties mentioned earlier for averages.
And now let's try to understand where this is the wormhole come from.
So so now it is concerning on this destroying the union. And then, of course, still factorize. And the one is the partition function. M. One and Z 2 is partition function on M.
Yeah, and then if you do the filter on the one, then you get the gravity at this boundary M1.
And if you do the filter on Z1 times Z 2, then you should do the gravity responder with M. 1 unit.
And now let me just define. Yeah, just introduce a slightly simpler rotation for this erratic part. So so this FD1 is the smooth part, right? So so when I subtract from the one, the smoothest part, I get the erratic part. I just call this erratic part delta z1.
So by definition, the f theta Z1 is equal to 0.
So naively, if you want to map the filter to the gravity side, naively you say, oh.
This theoretic path information is completely lost. Okay, somehow they got filtered out.
But actually, the story is more interesting here. So now let's just look at this structure a little bit more explicitly.
So, so, so let me just write here. So imagine you have Z.
y equal to. the Svd 1.
plus delta Z. One. Hi, this is the F. 2.
Delta Z. Okay? Now, if you look at this object.
Okay, if you look at this object. And then… then you have 4 terms here.
Yeah, photons here. So by definition. Yeah. So the so the fd1 times fd. 2. This goes to here.
and Fc. 1 times delta Z 2. just equal to 0 when you act on F.
It's because of this property, okay? It's because you can take the F out. And similarly with that cross term. And then you have this delta Z1 times delta Z. 2.
But now here is the key. Even though each delta z1 is erratic.
But we multiply them. Then in principle, you can generate smooth things.
Just, for example, imagine say the delta is e1 contains some kind of phase.
And then of each dependent n in a wildly way, and that is easy to depend on phase in the opposite way. When you multiply them, then they can cancel. And then the product.
They can again be smooths. Okay? So so that means that this thing in general is not vanishing. It does not have to vanish.
And so and so if this does not vanish, and then this does not factorize.
So now, if we identify the non-factorization part of the Cft. With non-factorization part of the gravity.
Then we have the identification that this thing is related to the wormhole.
And so to say that along the way. When we write this, we say, actually, secretly, delta EZ one delta EZ. 2. Even though each of them is erratic. But somehow they're erratic parts can have correlations that when you multiply them they can cancel. And so we say there's a correlations among erratic independence.
Okay. And then so so now the slogan. is that this wormhole. What wormhole captures this non factorization of the wormhole on the gravity side captures each such correlations.
So among the erratic part of the partition function on M one.
And the erratic part of the partition function on M2.
Okay. And those wormhole actually captures this kind of correlations.
And so yeah, and you can just trivially generalize it, say, to 3 lac wormholes. You can also generalize. Yeah. So yeah, for 3 lag. Actually, I don't need to put the connected ones automatically connected.
And you can also generalize it by including some operating insertions in addition to M. You can insert some operators. So m, the light operating insertions. And then you can say that — take the product of n such manifold.
And now let's consider the connected part of this filtering. Okay, again, this product needs to the smooth part, and then that should be related to the wormhole, a connected wormhole connected with this end manifold. Okay, this m bound.
And so this relation then provides intrinsic boundary definition, a boundary explanation of the original wormholes and the prediction of the amplitude.
So this is something in principle you can falsify. So I can try to calculate this kind of thing on the gravity side on the Cft side. Then I can compare with gravity.
And by this identification, even though we don't know how to calculate this.
object yet. Okay, even though we don't know how to calculate this object yet.
But that the mere existence of this structure. actually put constraints on the wormholes. Okay, put a constraint on this wormhole partition function.
So, so actually, if you have immediately implications is predict the infinite tower of inequalities for the wormhole amplitude.
Okay, so they'd see how this works. So yeah. So here I need to use a little bit abstract mathematics. The abstract algebra. But but hopefully you can just yeah, if you don't know it, I hope you can just accept it.
So let's imagine… yeah, let's just view. This erratic part has some kind of function on the space of all configurations. Remember, this X, just the devote the collections of all possible manifolds, Euclidean manifold plus operator insertions.
And then you can just imagine this erratic part of the partition function just viewed as a function.
of this X configurations. And then we have taken the product of this is manifolds with points with operators labeling points. That's right. Exactly. Yeah, that's the data.
And, uh, and then we can consider the product of them.
Together with complex conjugation. So there's a larger complex conjugation. You can put on this thing corresponding. Yeah, you you insert the conjugate operators. And also in some cases reverse the orientation of your manifold.
And so this thing is naturally defined with a complex configuration.
And together, they form an algebra. Okay, so commutative algebra, just the product of numbers.
And now… This F.
This filter is a mapping of this algebra. It's given any element of this algebra, which is a plot, which comes from the product.
And then that mapping to something smooth. Remember, the F map to some smooth. So so this is a map from the algebra to the space of trans series.
So one mathematical fact is that the space of trans series is actually a field, a mathematically, it's a field.
And it's also a space where the absolute ordering is defined. So there's absolute ordering in this. Yeah.
We say which is dominant, subdominant, etc. So this is the space which is defined with ordering.
So these are all the formal series, right? They don't converge. Yeah, they don't converge. But you can still define as a series in the operational way. Yeah.
Let's see. So you say it's a totally ordered field. Yeah. Field in the sense of algebra.
That's right. That's right. So. and the ordering is.
by the order of the lead intern. Yeah, by the order of the leading term, if the leading term is the same, you compare the powers. Yeah.
You compare. Yeah, if the leading mode is the same, you compare the power, and even the leading coefficient. Yeah, yeah, you compare the leading coefficient of the power and we do that? So the complex field is not totally wood.
So you mean the real life, the real field is. Oh, or you can define a modulus for it.
Well, if they have the same modulus, but different phase? Yeah, yeah. And then they're equal. Yeah.
Yeah, you can still have. Yeah, yeah, but it's field is not totally uttered as far as I know. Right, right, right. Yeah. By totally ordered, I mean, you can give absolute value and the absolute value you can assign order to them.
Good. So so this actually. Using the language of the algebra.
So this actually this mapping actually defines a state on on this algebra with the with the target given by this space of trans series.
And now, if you have algebra, and you have a state.
Then you can actually define a Hilbert space. So this is called the JS procedure.
And then now the Hilbert space, the vectors in this Hilbert space valued in the space of trans theories. Okay? It's not valid in the complex numbers valued in the space of 20. That's just a little bit space over that sphere. That's right. Yeah, it's a variable space over that field.
So you can define a Hilbert space. Why do you should even tell the artery?
Yeah, you just need… CNS construction, they have a state is a linear functional value to complex numbers. Yeah. You need positivity. Yeah, I need the positivity, which I do.
I need the ordering is because the complex numbers have.
orderings in terms of absolute value. At least the Cauchy swaths. Yeah.
So this defines a Hilbert space. Okay?
Well, yeah, go on. Okay. So this defines the Hilbert space.
And now, so heuristically. But this JNS procedure does in the heuristic way is that for each element of the algebra.
So any of such product, essentially, you can associate the vector.
Okay, this that element. So so each element of the algebra then corresponding to a vector in this Hilbert space. I guess there's something I'm I'm not getting about this Wilford space.
The norms of vectors in this Hilbert space. Yeah. Are they supposed to be by night positive numbers? No. The norm of the norm in this Hilbert space is valued in the trans series in this.
In this field. It's absorbed into that of positivity. I mean, I mean, the Hilbert space, you know, part of the definition of Hilbert space is norms.
North squared is not negative. Yeah. I guess that's why you need the total ordering, maybe, but… Yeah, I need the ordering some notion of that axiom. It's no negative.
Sorry, I'm not sure. It's the same question that Greg is asking. Okay. How in that field do I define positive things?
Oh, you, you can define the absolute value on the chance series. Yeah.
How would you say? Because every term is a complex number. Yeah, yeah, yeah. And so you just said, I said, that's right. Yeah.
Okay. Okay, so so heuristically, so so.
So, of course, this procedure is mathematically rigorous, but heuristically is that for each element of the algebra associated the vector could be 0. Yeah, it can be 0. But yeah, the part of the Hilbert space is that the you.
you factorize those off. Exactly. You factorize the node states. Yeah, yeah, you factor out the law states. Yeah, yeah, here, I'm just doing heuristic, I'm not doing very, uh, uh… So… so the overlap… Then of the vectors.
Then just given by this state, the overlap of the vectors just given by how the F acting on them. And for example, the overlap between the two just reduced to the V star times this and the F acting on it.
Okay, and then remember the intern, this is related to the wormhole connected and X, Y and X2.
So so so this way your your relation before didn't have a star, right?
relation before. Yeah. Before the relation to the wormhole. Yeah. I thought I don't remember. Oh, yeah, yeah, you can put the star there. Put a star there is your… Which one is the wormhole? No, no, no, wormhole.
So so the wormhole. So the so the yeah, let me just write this way. So earlier I just write generally. Yeah, let me go back.
So so here here just right generally. the MYM to M. 3. So MYMA 2, M. 3. They encode the data regarding, say, the orientation of the manifold, the operating insertion, etc.
And so they're just all hidden in this rotation.
Okay? But now. Here, I don't need to do the compressed conjugation. It just abstract symbol. So M here. So M. 1 here can be the the the yeah.
So here, the claim is that this. Delta E is ECFT. This star is equal to X1 bar. Okay? And the bar corresponding to the one you will reverse the orient. Yeah, is that what I meant that there's a natural compressed. Okay, so… Yeah.
Okay, so you're saying that there's one wormhole connecting the manifold from one to M. 2. Yeah. And another wormhole connecting from one to.
Yeah, yeah, yeah, yeah, yeah, yeah, yeah, that's right. I guess there is still okay. Yeah, yeah. Yeah.
Yeah, yeah. Good! So, so you can have this.
And then so now you can use coach Schwarz.
And so so since this Hilbert space. Since this space of trans-series field with ordering, which you can define for the long.
And then now you have Coach Schwartz. then give you this kind of inequality among worms.
Okay, so this is a highly long trivial inequity among the wormholes.
tells you that from X prime to X prime X to X and X to X prime.
they should satisfy this constraint. Okay. And this is something you can check.
And checking them is actually not that easy, okay? So I haven't checked them myself.
But actually, for completely different motivations, this group. They wanted to define some positivity using gravitational path integral. They have proposed the same kind of equality. And then, yeah, they are much more powerful. So they have checked many examples.
Which seems to work. What kind of examples? So the JT is trivially work, and then they look at various examples of ADSV. I think they also maybe have a look at some higher dimension examples. Yeah, their paper has not appears. I'm not sure all the details. That's what they told me.
Yeah, sorry, the absolute value and the less than or equal to is in terms of these trend series. Yeah, yeah, yeah. So the order range should be defined in terms of the ordering of the trans series. Yeah, because none of the thing is convergent. It's not convergent. Yeah, yeah.
Okay? But this also propose a infinite number of the but you can consider arbitrary number. If you have infinite power.
And you can, for example, consider, say, X1, X1, X2 to x3, and then this should be smaller than that one.
And yeah, and etc. So so so since Yang, she helped me check some of these actually worked in the in the Jt. Okay, at least it's a trivial check in the Jt. So so so Jt story is a little bit different. So so so.
It's because this is supposed to work with the duality with the CFT dual. So JT is not quite the — yeah, anyway. So you can also check within JT.
So so actually, this Hilbert space is actually not.
are some some just abstract mathematics. So we will actually interpret this Hilbert space, the semi-classical Hilbert space of all isolated closed universe. Okay, you can construct using the Cft.
So actually, this is related to the close universe cosmology.
Okay, so now let me just make some remarks regarding this identification.
First is that we start with this relation at full quantum level, which.
Let's assume still assume it's to be true. So I want to claim.
I want to say, when we look at the G, when we say in the G, Newton goes to 0 limit.
When we say the children goes to 0 limit, the gravity partition function.
It's given by. Yeah, yeah. So yeah, one second. Let me just go here. So what we are saying is that when you take the G. Newton 0 limit, we actually have this filtered relation.
And so fields of the state of T should be field to the gravity.
So I want to say the standard gravitational path integral.
When we define the semi-classical limit. Using the gravitational path integral and the center point approximation.
We have essentially implicitly done a filtering. That's what Z gravity means. Yeah, yeah, yeah, right? Because nobody tell us that gravitational path integral unit inside approximation should be.
the the the the right. Yeah, nobody has derived it from quantum gravity. Okay, it's postulate.
So so so I want to claim in that postulate, we have already done some filter.
Okay, and indeed in string theory. We have already seen some hints. Okay, or if you just believe the Pakistan Hawking entropy, which is the G. Newton not only controls the coupling, but also the number of freedom in quantum gravity.
So in StreamC, we have what's called stringent exclusion principle. And if you want to interpret the Pirkenstein-Hawking, etc.
So in claiming that the gravitational path integral. Describe the jutilton goes to zero limit. So we have implicitly already done the filter.
So right now I just trying to point out what's the filtering on the CFT side.
So now let's go back to this factorization issue.
So so again. So on the Cft side, when you take angle to infinite limit, you still should have factorization. Some people use the semi-classical.
gravitational path integral. to derive Beckenstein Hawking within this page curve.
You can derive. the value of the entropy.
But to interpret the entropy as the number of degrees of freedom, and to interpret really, that really has an entropy which characterizes normal degrees of freedom, you have to give G Newton a different meaning.
You can get that formula, but it's interpretation, you have to give the mean. Yeah.
So so on the. So when you take angle to infinity on the Cft side, you still have factorization. You should still have factorization when you include this erratic part.
So previously, it's not clear on the gravity side.
that we don't see factorization is because, say, the semi-classical gravitational path integral approximation might fail. Okay.
Or I should exclude the wormhole contribution. Okay?
So here, under this proposal. I would say the Ukrainian pathological is varied within its own domain of applicability. So if you want to extract some satter point approximation, it's perfectly valid. And the wormhole saddle should be included as your ordinary store.
But nevertheless, now we see. But now we need to. But now, in order to understand the factorization, then we need to understand the gravitational counterparts.
of this E-related, even in the Jeducing goes to zero limit.
Okay. So so so if we can identify such things on the CFT side, that will help us to understand what kind of things should happen on the gravity side.
So by the means we have to include more things.
Beyond the decider point in the path integral. Or, you actually have to include some other structure.
Okay, so it's not clear that's what you should do. It's because the passing logo by itself is not well defined beyond saddle point. There's no way you can define that integral mathematically in higher dimension.
So so. But at least tells you somehow in the gravity side, even in the GPU thing goes to 0 limit, there might be some kind of erratic behavior.
And also let me make some comment on the connection with the average people have been doing.
So so here is the proposal again, just so we have smooth part and erratic part.
So so this picture is actually consistent with the discussion of the various averages.
So you can imagine in specific problems. Okay?
So the gravity paradig related to this smooth part, okay? But in specific questions, it may be easier to use some specific average to extract this part. So you may imagine for some specific problems, you can just do some average to get rid of this part. Then, of course, you just get this part.
And then and then that put those averages just on the on the better clear conceptual ground. Okay? And now we understand what those averages means. It just it's just a device.
to extract this part. And it's perfect fine. There may exist multiple averages to extract this, and it's perfectly fine. You use different averages to for different observables. And it's because because we know this quantity.
exist in the fundamental way. So so this. So the average then is a little bit like what in statistical physics you do on some you do the you do the cross screening. Okay, and the details of cross screening is important as far as you know, their physical principle governing that the quantity you extract is universal.
And so here, what we want to do is to provide a fundamental and universal way to define this quantity. Once this is defined, and then in practice.
This may not be the best way to extract it. Okay. In practice, maybe it's much easier for specific question. You just design some average to extract that content.
Okay, so so now let's go to Lorentzian signature. So so far I've been talking about the Euclidean signature.
So so let me talk about very quickly mention what's happening in the Lorenzian signature.
So in the Lorentzian signature, let me just first remind you the conventional wisdom.
So in the Lorentzian signature, normally we do it in a somewhat different way. Normally, you start with a Bach Lorenzi manifold, say, M.
And then you say, Oh, since we already have a semi-classical Lorenzi manifold n, then they must exist.
Some CFT states, some family of CFT states. So wait, there's a space or a space time? A space-time. Yeah, yeah, it just Lorentian spacetime n.
say, say, black hole or some some say ads, etc. So if you have some Bach Lorenz manifold, you say that it must exist.
This family of Cft states. define for each N that when you take angle to infinity limit the physics of this data captured by this no random manifold.
In the sense that observables in the state psi n, when you take angle to infinity limit, it can be extract from the gravity side through semi-classical gravity quantities.
Okay, so that's our standard story for the Lorenzian signature.
and you can also ask the question in the converse way.
Suppose I give you a family of such kind of states. Okay, one for each end, and take n goes infinity. How do we know that corresponding to an emergent spark space time?
So… so yeah, keeping a sequence of boundary state, how do we know it has a semi-classical gravity distribution? Of course, this is a very difficult question related to the question of emergence.
So so so we have been trying to answer this question. Okay, in the somewhat precise way, mostly using this volume algebra stuff. So so one working definition is the following.
So take all your collection of single trace operator, which they are defined in the larger limit.
And then you look at all these correlation functions.
So of the single trace operators within these states and take angles infinity limit.
So if this limit is well defined, means that having the right factorization properties have well defined, say, for example, 2 point function is order 1, 3 point function factorized, etc, and have all this expectation for large N theory.
And then we say these states should have a semi-classical Lorentian description.
But now, given with our new Euclidean dictionary. Clearly, this Lorenian dictionary should also be modified.
And so in particular, this leads to a more general emerge of emergence of the space time.
So so let's imagine we have some Lorentzian observable Gn as a function of N and in some state psi n.
And again, we can separate and good infinity limit the smooth part and erratic part. And then we define the filter to identify the smooth part, and then to identify with a semi-classical graph.
And so and then you can also define a filtered Lorentzian emergence.
which is more general than the standard one. So so this look at this family of states.
We say this has a semi-classical gravity description. And if the filtered.
So if you look at this correlation function of single trace operator in this psi n, we don't require this object itself have a well-defined large limit. We just require the filtered part to have a well-defined logism.
and then and then then this will can define a more general notion.
Dr. Lorenzi emerges. So actually, you can show that the yeah, we can. We are going to argue that the closed universes.
and also black hole interiors are examples of such kind of a more general Lorenzian emergence.
So yeah, so I'm already reached the plan of my talk. So I can just conclude here. Let me just mention if I have infinite amount of time, what I would do.
So so now, if you agree with this proposal, if you based on this proposal.
And then we have accumulated many examples on gravity side of this wormholes. And now we can try to deduce what you can learn about the Cft site. And then you can try to check it.
Okay, so so so now from the gravity result, we can actually make conjectures on the erratic dependence of the Euclidean partition function.
And so let me just mention one thing here.
So one place. You can look for such an erratic partition function. Say, it's a two-dimensional CFT.
patina funky on the genus 2 manifolds. on genus 2 on genus one partition function. You don't see it.
You don't see it. It's guaranteed by a theorem of wheat and yao. But on GLS 2 you should be able to see it according to this proposal, or genus one, but with some defect operating system.
Okay? And then you should be able to see it. And the key is that that involves the you have to involve some kind of Ope coefficients of heavy operators.
And those Op equation of G. Those 2 partition function involving the the OP coefficient of heavy operators. And so that's a place which this erratic independence can come from in the partition problem.
And also, yeah, you can see how the closed universe arise from this field to the Lorentz emergence. We can separate.
Actually, there's two conceptually very different types of closed universe. One is purely isolated closed universes, means the full spacetime has no boundary.
And this kind of close universes you cannot assign them to a CFT states.
Okay, because there's no boundary. Okay, so so the emergence of them is different, conceptually different from the second kind of a close universe corresponding closed universe entangled with ideas.
So in this second class, actually, because we have ADS, you have a boundary, you can define a CFT states on the boundary to describe the whole space time, including the closed universe. But for the first one, actually, there's no… doesn't appear there's a CFT state describing anyway. So so they're conceptual differences between them, etc.
And also, you can argue that the black hole interior corresponding to the examples of this filtered emergence. Ok, I can just stop here. Yeah, thank you.
So a few more questions.
So Huang is this stuff discussed in your paper? Yeah, yeah, they were discussed in the paper. Yeah. Okay, so I won't ask any more questions if I did this before, so… Yes.
DPS State, counties, indices. you know, or like a microcanonical partition function. Yeah. But you don't accept that as being a partition function. I accept them as partition function. It just is the dependence on them. It's like a partition function, and I do some kind of Laplace transform to isolate.
Right, right, right. Yeah, yeah, I do, I do. It just that you to formulate them in the sharp puzzle is a little bit different in the sense that they are erratic dependents are erratic dependence on the charge.
And not directly on the on the end of the cage. Yeah, of the gauge group. It's erratic dependence on the charge. And so that dependence also dependence on the.
energy. I mean, let's consider the coefficients of them.
the coefficients in the four-year expansion of a modular form. Yeah, yeah.
You know, those are kind of Arabetic quantities, and there might be some.
you know, overall growth, like, you know, some end to something like that. Yeah, definitely. But if you actually look at the distribution of the coefficients, they might be jumping around with an average growth. Right, right, yeah. So the density of states.
definitely have erratic dependence. Yeah. But the standard assumption.
Yeah. Sub of C and Q to the n I could.
call that Z of tau. We could take a.
Laplace transform. Yeah, yeah, yeah. Yeah, but the key event. Yeah, but the key is that so so one assumption people have made all throughout.
Say, for example, let's just look at the end, I want to stress is the energy, not your end. Yeah, yeah, yeah. I think that's your complaint about the charge. Yeah, that's correct. So the average, I think, even if you ask people so so the energy.
People do agree there must be some kind of erratic dependence.
But people have always assumed whatever those erratic and dependence on energy.
Somehow they should always be averaged out. When you form partition functions. Now the average out in the leading term, but in all subdominant terms.
Not even exponentially small terms. So you would be happier if, say, we took.
Suppose we took the symmetric. Well, actually, I'll talk to you afterwards. Okay, great. Yeah, yeah, symmetrical orbifolds say.
If we can compute the genus 2 partition function, and then I think we should be able to learn a lot. Yeah.
I was going to say symmetric powers of K. 3 and take the elliptic jitter signals. Yeah, yeah, that's also that is, you know, Sim K of K. 3. Yeah, yeah, yeah. So why do I say that? Because, as you know, if you sum those all up, you could.
some automorphic form in three variables. Yeah, yeah. And so maybe there would be coefficients in there. Yeah, so yeah yeah certainly that's the object one could, you know.
Yeah, also like super symmetric indices. for higher dimensional gauge theories, ABGM, say some other topological indices. So people have been powerful enough. They calculated those topological indices at finite n.
So now you can take and go to infinity limit. We can search for this kind of structure.
Okay, let me remind you if you want to come to dinner, come talk to me after the talk, and let's thank God again.
Mm-hmm. Uh-huh.
Is there a way that you can filter out you have quantum or erratic parts? Is there a way to add some sort of like reconflections like sum of photographs of the bone, like 2? Oh, yeah, yeah, yeah, yeah, yeah. Yeah, yeah, yeah, yeah, you can have two point functions, but you can also have 3 connected. Yeah, yeah.
Oh, and the yeah, but what about 14. You can write as so 4. So the 4 will include 22. But we also include the connected. I see. Okay.
Okay.
Do not…