Alright so speaker to these are the real who speak about cause a local photography from Panama.
Then, so I want to start thinking
Yeah.
And
So without further ado, I start
So what is the goal of these products. Well, he's trying to understand the logic from the TV from a number of the revenue side. So what mark a safety feature of these. He said, first of all the forces us to think about fighting boundaries, instead of us at once and
also impose Watson is greater than
Or semi classical
Well of course title fight in the following slides.
But any very conversation.
To have some questions. So the first one that I will try to answer is what are the dual degrees of freedom. So if you thought about blogger for you. There's a feeling about in one the
Theory, the boundaries that Nicole exactly the same amount of information. But what are these what are the reasons, even in the case of crime.
And what is the numbers of course then
Later on,
We would also like to understand how this framework converse more standard holographic a more connected yes 50 and to semi classical computation. So you see circle.
quantities are additional did you buy bread, how to score and in any culture Quality Framework. Right. So he is ready to calculus and anything Jacoby approaches have been
Studying in seeing up many different frameworks and sit up straight.
And students
So,
To be able to talk about these and to explore these questions, we actually need. If you're going to run into that that we know to be to be complete and an under control.
We don't have one enforcement actions, unfortunately. But when we come to a lot, and in particular the accelerator model for pigeon 3D gravity and financial
Counseling provides us a provide a framework for gravity, which is mathematically it unfolded in facts are simple, which is an advantage.
And it is a pretty clear quantum dramatic interpretation. The which you can leverage and, in particular, these offers a complete make centralization on democracy even reading the quantum real. So this is, of course, related to the fact of the
Matter is no welcome to therefore we can
We have a date termination invited to this realization, buy a book about driving and these fights. Interestingly, hosting the boat, but not quite the boundary which is exactly work of the interest of savages framework as the time all the problems of
Continually and the the colonization in in for quantum gravity Magellan
So I want to mention that, as I said, this work is done in collaboration with Christopher color and he his advisors can today. Christopher's defending this disease. So I just wanted to encourage him and congratulating case there that already done.
So they take a messages for these are out of the for the whole framework to work, then the technical content of the following
So to get bunch of graffiti is an ideal framework to investigate boundaries in the locker fee.
For quantum geometries and gravity theories. And the reason is that can full control of the theory and many different levels. Quantum classical at the dramatic level with financial Wrexham talking boundary conditions and so on. So it's really a fun
Playground to try and test ideas about tomography edge modes continuum gives a semi classical limit so normalization and criticality and so on.
And in particular, these allows us to compare and use tools designed for different fields like it yes 50 or statistically statistically model interoperability and condensed matter techniques.
In what we need. So far, which is pretty exploratory, to a large extent.
Rather than we don't have a full picture yes something
But we see we see many things coming together and, for example, we have the force.
That you may have been a little graphic jollity from a a pumpkin geometric perspective.
We computed a for the, for the first time in Spanish conform up with arbitrarily large.
Amounts of building blocks. And this was interesting to check for example, what happens to the orientation fluctuation problem, which is actually not a problem in so we didn't run into any difficulty because of it.
And we were also able to match via this quantum gravitational computation some previous
Computation done for 3D graphics
And these came with some
Kind of discreet cut off. Which one is intrinsic to the framework, but also some corrections to turn on performative backgrounds. So the resolve to do, and also to clarify and ethnically to play with. I want also to mention that for
These. He is also part of a larger, larger project aim to understand the local degrees of freedom of speech theories and privacy.
In the presence of boundary. So in finance and banking regions. So this is of course not only my interest. My personal interest, but there are many people working on these topics, both in the LGBT community and in the outer world.
So,
I have to admit that my I prepared a lot of slides. So probably a we go through the first couple of
Parts of the talk, but a one, make it through the last one which is not too bad, in the sense that really the basis of the framework or explain at the beginning. So let's see how far I can, I can
So let me start with some
preliminaries about really pants and gravity. So it will be fast because I assume most people are quite familiar with this, if you if third question. So, don't, don't hesitate.
So 3D quantum gravity in the first third of the formal is can be expressed as an extra 2pm topological quantum field theory.
And the price to pay, is that the metric is replaced by the trailer, which is actually
Which is allowed to be non editable and can fluctuate its, its overall orientation point by point of
The questions of motion of these of these theory and the code flatness of the connection and portion freakiness of the connection.
And the solutions of these equations of motion are pure gauge configurations, where the connection is as we see if your gauge connection and the triad is itself up to up to gauge
These equations of motion pulled back on a crush initial surface. They are, of course, related to the cemeteries of the fury. The, the time constraints.
Which generates respectively. The F shift symmetry and loaded symmetry. So we see that they not only the connection is a pure gauge connection, but also the trial is a pure gauge trial with respect to this ship cemetery which exists because of the young kid entities have
satisfied by the curvature
Now we have shipped symmetry is extremely important in this setup, because at the quantum level interrupt is represents the
Different model is is what encodes different margins symmetry in the console retro model so that residual
Before more physical action of the few modifications in the center retro model is actually better because of of the presence of his shift symmetry, I will try to be more explicit in when it will come up in in the following slides.
And other things interesting aspect is that these two symmetries or not they are independently from each other, but there is an interplay between the two.
So the organized in unit. What's on the
Group, which has extra structure on the top of the group structure, which means that the conversation. The cemetery this total cemetery group is actually before to a greenfield Apple of Esther to
Symmetry in the quantum theory. This was shown in many ways within the term Simon sportsman is or within the loop four minutes by Kareem and Catherine was work.
Many years about 10 years ago. So what is interesting about this for us is that we will see that these two symmetries. In fact, will give two different representations of judgment and they are not
Which do not appear at the same time, but that right there. Conjugate representations each other.
So what is the setup for our computations. Well, we consider a
Good portion of a cushy surface sigma which is funded by a, a one, the boundary. See the corner the corner surface and we consider the evolution of these fixed in time well will be Legion time
Up to the final state.
Here for yeah so along these evolution this evolution will be controlled by some boundary conditions that we willing post through some boundaries, the network. And so really we see that the boundary condition will be fixing some one metric
At the boundary and not some not to classical
This be known as we have seen the solutions of the theory or your gauge in the bug, which means that the financial state in the bulk of the days of this financial same will actually be essentially irrelevant will be trivialized.
Once we gone shell and the only relevant information which city is fun.
We will confuse the transition of interest between the initial state for you it to the final stage for. Yeah. Even the boundary condition side be by using the concern of ratchet dynamics in in the bulk of this of the cinema.
Now, in fact, in most practical application when we will not conclude this transition amplitude between
When the initial in the final stages, but rather a partition function for the manifold would be foundry, which is a solid totals manifold, which is obtained by identifying the final state with the initial state up to some some twists. So this twist means that
The final disk is rotated by an angle gamma before being identified with initial gifts. The reason for this is to have a partition functions that depend on one more variable. So to make the results more
Information rich, but in fact that the actual reason to do this is the fact that these these gamma, in a sense, and freezes well allows us to produce some both degrees of freedom.
Some which represent essentially the curvature around the no one more than the curvature Delano me around the non contract double cycle of the stores. These
So you can think of violence transporting effect for from the bottom to the top.
Slice and they will look rotated to each other. Upon intensification up to gamma because of the curvature about another the curvature Southern Yolanda me around them.
Which can be
So,
We will, in the last part of the stone to which I'm not really sure we get to
We will be actually comparing
Quantum Computation Dan on this Taurus to semi classical ones.
So,
We want also to identify the holographic theory and the reason the way this is done.
Is through the recycling through the following formula we have a buffet integral in the bulk for 3D privacy with fixed boundary condition at the boundary of the seven chakras.
Which has to be compared to into the partition function for a partition function for it to the field theory living on the boundary coupled to these same
Background metric. Ah, so of course
These, these answers is pretty empty unless one has conditions on the
On the boundary on the boundary door theory which, for example, has to be local and so on. But the idea is the following. What is it boundary condition for the gravity theory becomes a background structure for the tool.
Okay, this is just the genetics of thought here is a bunch of references. I'm not going to go through them. They're just for those of you who are interested
In looking into this. Okay, so
Having
The structure of, of the, of this atop that me if there are no questions, let me go into the presentation of how to build the edge that it's
So,
Let me start by describing a standard network.
Let's ignore for the moment, the fact that there is a coordinator, like a boundary to PS portion of the cushy surface. So if you completely ignored that what we have is a
Is a description about what I per surface was to will is given by some craft gamma, which is the support for our speed network functional and dysfunctional our functions of the
alarmist alarm the links of the graph and bye
And as usual we impose gauging violence for these functional and which is expressed as these environments under against transformations that the vertices of the of the graph.
Okay, this is old standard. Now what happens if we introduce a boundary. So if we introduce a boundary, we see that the presence of the boundary actually breaks the gauge environments for the links that cross the boundary
If we don't want these links to be trivialized the dependence on the Ilana means on these links to be triggered life. We have which would amount to set the spins on these links crossing the boundary to zero, we have to
We have introduced compensating what I'm calling compensating gauge fields at the at the nodes at the one violent nodes sitting on the boundary
Which carry this group of you the degrees of freedom. The pink. Geez, which we see transform
Me on
On one side, because that means the box seats on the on one side of them.
So this choice of compensating fields of course focuses on larger and symmetry were kind of ignoring the shift symmetry part which we
Invite later on. Now this choice of degrees of freedom to an electric central prescription in the language of Cassini where the and rows of all and which means also that this beans on the
The edge is crossing the boundaries are have to be considered as fixed as support at least the super selected so we cannot take quantum superposition on the bandwidth only statistical ones.
So I just, I'm just saying this because in the following. I will keep these links these spins fixed his electric clocks is through the bunker fixed
And the reason an underlying reason is the choice of abandoning degrees of freedom ain't here. The two go hand in hand.
Okay, so to see how to represent these
Two pro the ways in which we can represent this boundary degrees of freedom. Let me
Just revenue quickly. The way that is financial state is usually
Can be usually encoding, both in the group picture and in the spin picture. So these happens, thanks to a sort of groups will transform that changes our
Functions of the group element into some function of some speed sending corner and of course the business. We're expanding nine is this business. Now if we add this boundary degrees of freedom is which are provided. We see that
We see that the new structures come up in particular, we have the super
Super selected or better fixed boundary screens, which are our boundary conditions. So we have essentially were forced to fix the metric on the boundary of the disk. Then use metric on the boundaries of the disk by fixing spins.
We and we have also these group elements in there which are
Fully transformed into magnetic indices. And so what's the matter with function will depend on bug, spin, spin which encode the quantum bulk metric something refiners which had the same use in and some magic in this is for the boundaries.
So just as an analogy. When can think of the group elements sitting on the boundary as analogous to the best amino fields for Simon's is not exactly
Right, because those live in in the big book g. So to start off as to why this one is just an extra two but just as a mental analogy can be useful. Just think of them that way and the ends at this point or present somewhat the
The, the dual picture, which is now a
One time thing discrete states.
About as far as games and this happens at fixed JS. So this is a welding finally
There is another way to think of to present this boundary degrees of freedom, which is in terms of spinners. This is like going to an over complete basis for
Magnetic indices and I am seeing these because it can be useful to make a contact with both guns watchman bill and framework where he he considers to be no cannot abandon the degrees of freedom. The mapping is not really accept, but I think he, it can be
It can be studying more and it could be useful to to make it more precise. There are other differences, he's he's working that direction framework without boundaries, mostly, so it's not exactly the same but
He I think morally speaking the the mapping
Instead, so if one looks at what these fingers represent the
Essentially are present. The same thing kind of decrease
So what are these degrees of freedom geometrically speaking they are the
Present the orientations of the of the boundary. So the intrinsic metric is fixed by us fixing the spins the boundary spins.
One of the shape of the boundary. So the extrinsic shape of the boundary is encoded into these elements spinners or magnetic etc. Depending on how you want think
Okay.
Now, if we consider the dynamics of the theory we know that's everything must be flat and different motivation invited
So essentially, whatever is happening in the bulk of the sprint network doesn't really matter because these trigger lies, but the nomics and so all then on treatment information will actually be encoded into these hundred degrees of freedom.
This is true as long as we do not unfreeze somebody degree of freedom by means of someone triggers the publisher, which is what the titles partition function actually is helpful for because it starts thinking out some topological degrees of freedom that exists in the bulk
Of the disk, but they're not proclaiming these simple sentence.
Okay, so now that we understood the, the, the, the nature of the boundary degrees of freedom, we can focus on there under the dynamics.
So let's go again for a moment neglect the presence of a boundary. So we have an initial states evolving on to the final stage.
If you want to compute the transition of pitcher in 3D quantum gravity. What we're going to do is to consider interpolated revelation of Saint with some coloring in terms of spins. Some over all possible colorings and waiting. All of these
Different Santa Roger states prescription. So in particular, we assign 16 symbols to that degree and so
We see that the boundary spins out a way to buy their
Their fun by by the boundary function functional
The initial and final stages.
So this provides the transition period between just just it
Here we are working in the spinner presentations.
By the. We can also work in the group or presentation and by using recovering theory. Of course, these can be written as a position of flatness on what everywhere in the bulb and therefore also on the bulk of the boundaries things
Now it's it's good to appreciate that.
The second
The first the first expression that one in terms of some of her spins.
Can be felt as a group, averaging four shift symmetries. So this is, in a way, imposing different models mean body until about by brute force group averaging over over it. So this is like taking the vertices in the boat and grow leveraging them Aaron, there are all possible
Positions in in the bud flat space in the second of the presentations that very imposing flatness in on group elements which are best tailored to talk about
Lauren symmetry. So yes, or two environments.
So why should symmetry is completely hidden in this picture. So here again we see that
The metric and the connection representation of the state, some
Emphasize either the shift symmetry or the lot of symmetry of view.
So now let's reintroduce the boundary
So how do we treat this time like boundary is not really time like because he's he's implementing gravity, but just to fix. So not get confused. Let me call this the time direction.
So what we do is to simply post boundary conditions by means of an understated say be which is another spin per state fixed on
The display network. So now, in this case the partition function in the group presentation will be just a bunch of delta function associated to all about faces of the script ization
Constructed well which is waiting. What which it's weighing the initial and finance states and also the boundary. The boundary speech, our boundary conditions. So we see that
The initial in final state or soldered together to the boundary state by some over these boundary with this corner degrees of freedom.
Here we are sitting at the at the two corners of these of these a solid cylinder
Now we can since the dynamic essentially trivial, we can in if the topology is trivial. Then, so we are considering the city that are not the Tigris we can completely sold in one line.
These, these integral over the bug connection which just gives us the evaluation of these funders network built house the glowing of the emotional and final state and the boundary state. So all of these is evaluated on on the identity.
So this looks quite trivial but secretly is actually encoding our boundary theory as I'm going to discuss next
So this boundary connector state.
Is in in this picture.
Prescribed that fixed pins so he's actually fixing the boundary. The intrinsic boundary metric of the
Of the cylinder along longer, longer about the time like MongoDB.
Now,
If you think one layer of the network at the time. So we look at the transition between one this and the next and the next is what we see is that
We have any initial state for the edge of degrees of freedom. So we see that the bump degrees of freedom, have been trivialized your fixed one so they have
Kind of disappears from the description and we are only left with the way function for the boundary. The degrees of freedom is magnetic in this is
Which is evolved by by our boundary stage as it is can be describing now plays the role of a transition matrix between the initial states on to some final state and is contracted again with the
The edge stage. The final edge it over here. So the book has been trivialize we've left with the boundary theory for this magnetic in disease, which takes the form of a quantum spin chain.
And the boundary condition can now be interpreted the boundary stage is now giving us a transition matrix that allows us to evolve the
This being defined can spin chain from one step to the next up to the final stage.
So this is how
The simple spin natural evaluation over here can actually be reinterpreted rewrap as a tool boundaries here.
So to make things more concrete, let me consider the case where all the boundaries in suffix to one half. In this case, the boundary theory is given by an internal system which is in user tropic Heisenberg's pinching
And
This is best understood in it's convenient.
In the alternative provided for in terms of a two dimensional statistical mechanical model. So we see that
If we have the extended boundary
We can, we are essentially assigning
Magnetic indices to all the links of this of these two d
Triangulation or what regulation.
And we're coming over them anger. Wait, when them by the, by some waves which are fixed paying the interest miners are we have a slide.
With these explain it better. But for here, I just want to say that these two days that boundary statistical model is actually six very experimental units because they were
Usually this is 636 month and it has three parameters. But here, only to survive because there is no relation between those which can be understood.
As the fact that the entertainer space. He's just two dimensional. In this case, I think, Well, I had a picture.
But essentially, one can see this A, B, and C as the three different recoupling channels for these interest Weiner between for skin one of states and the result in a dependence between these three channels because interest minor spaces, just looking
Hi. Hello. Can you hear me.
Yes.
This is a journey. I have a question.
Can you can you explain why your boundary me Tony and his spin chain Amazonian is localized. There are easy way to see that the boundary today should be local. You have first neighbors interaction.
Yeah. So thanks so this is at this level is a bit of a
Garage, in the sense that I'm not saying that this transfer matrix is actually given by the exponential of that time etonian
I'm saying that
As an integral role model.
It can be described by that is, I'm etonian so the transfer matrix of the spin chain.
Is actually some very well balanced superposition of not just the Newtonian but also the translation operator, the admin and all the other infinity tower of conserve charges that are both put together into one single object which is this transfer metrics.
And so it's not really true.
That for a small step of evolution. The spin chain is about with the Smithsonian, is it more subtle than that so
He's my
Explain why you're not really
Yeah.
correct to say that the boundary theory is more local
Boundary me Tony and is no local
I think, I think, you know, in a certain sense, yes. So there are there are some
I think the question is to be qualified to the extent of what a non local means in this context. So I can think of ways in which it can be seen as local in the sense that in the continuum when actually
I would say, yeah, in general, you would get no local models.
No local interactions and in some limits to expect them to become local, for example, just syntactically me. However, the fact that both theories topological
Should also tell you that if you choose properly your boundary conditions, you might find Welcome to our theories
And it's not
It's not a victory because the boundary dynamic is actually
Think you know just continuing on the same question. If you just go back to slide 22
Yes, I'm just clarification. So going from 22 to 23. Are you assuming that the number of vertices on the mission or have you already identified
The initial in the final surface.
Yeah, so they they have the same amount of
3D sweater squares be natural on the boundary. Yes, I said so.
Okay, good. So you're already have identified and so up to that twist, but for the number of vertices in the initial and the final. I mean, no matter what is it just fixed year now because ideally
Yes.
I, I haven't started generalizations, but there are of course ways to do that.
It was just trying to have a framework which is enough hundred control and and I can find it there justifying this choice for what comes next. But we threw that in principle, one would be interested in the in
In the issue of how the boundaries, the topology of the boundaries fanatics, the connectivity of the boundaries connectors influence the our statements and I think this is an important problem.
To be investigated and this is actually an interesting Chicago where to do it because we can do computations. Okay.
And since I have you in answer to your genius question. You said that the transfer metric metrics is not just directly. Your mother that Hamiltonian. So what is the purpose of introducing the Hamiltonian. I mean, since the passport is the fundamental thing.
Yeah, I agree. It's just
It's just, it's something easy to be true to visualize and since I was talking about in chains. It was it's an easy way to characterize the Heisenberg pinching is to give you some etonian
So as an internal system that's system is actually an SMS pinching, what is its evolution of the rate or whether it is just age range or some combination of age and it's other conserve charges is a
Is a different is a slightly different question. So yeah, this is not the boundary Hamiltonian, strictly speaking, but is a full characterization of the boundary gives a full characterization of the boundary
Few
Okay, thank you.
So in a in a completely exact way you want can map the boundary system onto the six vertex model and think of it just quietly and then it is what it is in a sentence.
With ours in that case of transfer money from one line to the next is just the transfer month weeks of the six best customer in here that there is no
subtext to the presentation.
We literally met each configuration around the vertex of the six vertex model on to one intertwined or configuration.
Well, one entry in the intertwining of the moon and the poles one ways are actually
encoding the choice of interest liner. So A and B, if I fix DMV this equation, etc. And this is a and b, our release planning. The Entertainer space. So changing a and be changing the the boundary metric by the change of the of the interest minor at that time I sign in future.
This is also know my genius choice about my boundary. Of course I can introduce you know much a 90s and so on. But then it becomes harder to just
So what I want to add
So one reason to talk about the six of ethics model is that it is a model that has been studied for many decades. And there are lots of tools to explore it. And it would be interesting to revisit these tools to
To address quantum gravitational questions or to clarify what side of the differences between a lot of speed limit your Susan many speed limits which have
Come up in in the quantum gravity. For example, when thinking about Michaels and other situations. So she hadn't we have creating models where those ask those questions can be asked and explore
Okay, now I want to go back to shift symmetry. And I want to show how these
Boundary vertex models. So the statistical models on the boundary. And as you see here, I'm looking, not just the square
Boundaries, the network, but just some arbitrary one. So this is a game. It's been network for my boundary state and not for the initial state. So it gives us the boundary beat the time like one
So I can define the vertex model a statistical vertex model on this network on the pink for
And by manipulating you tapped into that can translate it into a face mode.
And in this translation. I will be translating between Lauren symmetries boundary degree of freedom. So she's symmetry baba degree of freedom.
So how does this work. So we have a much better we assign it magnetic index which link. And this is because I have implicitly fixed Delano me along that link to be just an entity. So there is just one one magnetic index for
Better. Better edge for link and the weight of my my statistical models are given by the the entries of the interest pointer at teach physics. So my statistical model is a some over configurations for this magnetic in this is weighted by the entries of the interest wonders.
Which are the Boltzmann wins in these interviews.
So, but now instead of just evaluating my boundaries be natural ever introduced the group elements age, which I'm calling age and also introduce delta functions that impose all the faces to be to be evaluated on the flat on a flat configuration and reflect on them.
So I'm introducing boxes that are presenting the gration over a group element associate you can link and delta function to represented by dotted lines.
Now I can expand this delta function by using the divine theory and this introduces jays for rich some BJ some new spins associated to each face now not so the links beans are faced
Here these days that have some in over our new our new objects. And as we see this is I'm kind of transforming to a new set of variables and by using recoupling theory I cannot express this as some some over my boundary over the speech is this new variable DJ way to buy some
recapping theory object like a 60 day simple, in this case, but can be more general than that and my statistical model is obtained by
summing over all possible assignment of these new J variables which are now associated to the center of the supernatural faces that for to the boundary vertices
And these levels are skilled couple to my bedroom spins little Jane, which I've never, I've never ever number change.
So in these way we have translated from a what is called the vertex model to a interaction round face kind of model and districts are in fact that boundary degrees of freedom.
Or that it would have obtained if I were trying to compensate for the breaking of the shift symmetry, rather than the lines.
To see more explicitly is to how these two models look at everything done here in one case is just the population of the boundaries in that one written as a contraction of into fighters and interpreting that magnetic indices as my
Degrees of freedom. And in the other case is a more complicated object, but in some in over these new JS, which are associated to spin that for faces, but it's been a little faces on the background of some little jays and interpreters, which are fixed vomiting.
Okay.
Oops. This was a glitch.
So now we know that the DM. So there's the magnetic the boundary magnetic in this is, in a sense, represents our contest Lawrence frames. So the orientation of the boundaries so its shape. What about these new Jace.
So the new jays have been obtained by free transforming the boundaries financial evaluation. And this is exactly what one does when one goes from a connection the presentation of the
Oh, I forgot to mention that this duality between here between a vertex model and the face model is well known in statistical physics and it's called face first equality.
And the process essentially go along the same steps, a bit more formal because the
The US often quantum groups and so on. But in the internal system. One of these points of what non duality. What is inhabited here is probably is the fact that its source is in the in is we see it as a consequence of the of the cemetery structure of the theory in the box.
And so exactly this point emphasizes how this new jays are shifting between compensating fields which dramatically represents the distance of these boundary vertex boundary face so vertex in the sense of the democratization from some traditional central point in about
This I've done this transformation between the two set up without ever
Mentioning the bulk. So in that sense, the bulky, it's not really necessary to do this, but if I go back and think of these from a 3D gravity perspective.
These is like choosing a very specific choice of bulk regulation, which has a unique about ethics and everything is connected to these bumper stickers directly from the boundary and these chaser exactly those particles.
Interestingly,
These type of representation of yet modes in of the of the boundary dual degrees of freedom in terms of distance from individual center was forced explored many years ago by catalyst in the context of
The PS3, and he was using completely different techniques and he was up was interested in understanding how the new vehicle boundary theory in 3D.
That one financing 3D gravity actually originates in something geometric at these ideas of what begins degrees of freedom and identified them in terms of the normal difference which are so these
This narrative, although the technical the technical details are very different. The narrative actually matches quite well.
I want also to mention that these type of
Representation of the degrees of freedom was first exploring by a young can been done.
In the first paper on the stopping from a
From a revenue perspective.
And what they were doing was still consider a amplitude of a of the Toros in quantum graduate outcomes they need arise from from venture capitalists and they entered it was quite clear that the boundaries degrees of freedom that we're surviving that's that's the top where these
These, these beans, peas.
Transfer speeds to the bunker.
Okay, so, of course, you and I have talked about a very restrictive set of
Abundance mistakes that are many others. Not to mention, changing the supernatural graph. And so I'm just being on something simple even embed see
Like square lattice, even in that situation, what can do much more. So fix spins. So if your speech beings in
But I'm going to screens to something that is larger than one of them, the interstellar space becomes larger and larger. So when has control is a such a subset of these entertainers and before starting a sudden
In particular, this was done for semi it for the Venus patella kind of entertainers and we need the semi classical analysis that's I can go Vedic basically fiddle
And then more recently with Dr. Also, other states where also the bond the responses that are some Dover. So these are more coherence spin networks. And in fact, when we use our display network generating functions.
And in which are very interesting property and among all the properties. What was particularly nice was the fact that it was it was given enough to to that is exactly computer. So this might sound like something that
Is not that interesting, is more like a curiosity freak. But in fact, if you're looking for some interval boundary theories, it's a good thing that we start
To twist to explore a boundary states that are that allow us to to compute things explicitly and not relying on approximations in these way.
We have the whole i'm not saying we got them yet, but it's a step towards trying to understand where there are some other
There are other types of boundary states.
Was geometric properties have to be fully understood and explore and and so on. But that can allow us to do some
Math and mapping between between theories and other and other form one is known for that. And it's something that is computed will as much as possible.
And in fact in 3D gravity in the continuum is it yes your business. And so there are many, many cases, one finds boundary theories which are
Which are
Beautiful mathematical properties and are essentially computer exactly computer. So it's a step in that direction. I'm not. This is part of the more exploratory
Line. So just very briefly. And since our time is almost over. Let me just say a couple of words of what we did in the computation of the petition functions. So in this case, as I said, we focused on the on the aptitude for the tardiness for the twisted pelvis.
And in this case, we see that the amplitude is not anymore devaluation of this network on the end of the boundaries for natural identity. But the reason the leftover about the above degree of freedom, which is the line me around the no contract doubles.
So in this way, the boundary theories that there was described before. Get coupled to the degree of freedom and the interaction can have very interesting effects.
So in this pinching case, we see that these causes the, what we're doing is to evolve this beam chain by empty step that turning it is is the you
And before going. And this last integral is actually a hard projector at the low end point and from the spin chain perspective, this is protecting our spin chain on it's been zero
So if we had a defect in the bulk
Fees, we couldn't project on two different on two different sectors of the of the suspension. So these third sounding like the presence of it but particle or in quotes, black hole which has a spin and disease readable from the boundary perspective.
So I don't want to get into too many details we started explored it also the company to bug.
Which aligns with the students with an 11 so
So just to to quickly give a glimpse of what we were looking for when quantum computing these boundary amplitude. Let me just flash
The thermal twisty thermal as partition function, which can be completed, which can be computed in many different ways. For example, with a few theory computation.
In that case, the classical action is given just by these pre exponent exponential term. So, these, these just evaluation of the action with counter terms and boundary terms.
On to the d s solutions.
And here are the boundaries as some topic.
And then the second piece is that the product is actually the one loop determinant of that computation. So in fact, if the rising says an alternate ID.
A product in the sense that the denominator numerator and denominator of functional determinants for the to spin degree of freedom, the one
Spin the ghost and the seamless pin ghost. The ghost and so on. So, if these kind of torsion structure that appears and gives rise to these however in different
From the boundary perspective, this was computed by alone in Wheaton and revisited. More recently, and these quantities actually been a sort of character for some specific state. So these would be
The pre factor is the contribution from the ground state and the product is the contribution from the descendant.
Now if we take the last cosmological constantly me which namely should take us on to thermal binkowski space. Everything is Euclidean, by the way.
What we find is that
We end up on this kind of expression, which
Is related with which can be completed independently again from the gravity theory and from the DMS three character seen as a limit of the sort of characters. So there is a consistent picture.
And the question is whether we can find recover something like these from a full blown quantum gravity population. The first step in this in this part was taken by a violent phantom Yonkers, I said. And what they did was to compute the
Quantum Roger colliculus amplitude of a of a cylinder. Now the difference that the senior sitting there has a finite boundaries.
And they impose like justice three cylinder geometry boundary geometry there.
And also, since he's just a Lena riser edge of calculus. What they're doing what they need to pick a background solution. So there is about the party speaks that not not fixed, but is
At zero of the respects the reserve background and then they consider oscillations around these these regular background. So these mimics in the discrete the computation done for the customer to feel theory by bottom each Gonzales Maloney, and not look down here.
And we see that the two calculations match in the limit of a very fine boundary
In the, in the continuum limits of this computation, where the boundary sense to
To continuously was funny.
That bulk. It doesn't matter whether it is refined or not because the young introduce the regulation by
Now the question is whether we are able to do the same infants undergraduate and they don't have the time to go through the details. The short answer is yes. And we find
These formula plus corrections due to other non-conservative backgrounds to do the computation of us live in hospitality. And what is interesting is that interesting is that we don't introduce
It background for the bulk the budget is completely integrated out exactly or at most for the remaining
Known three month degree of freedom. We do a subtle point approximation, but it's not really a we don't fix a boundary around which we return we integrate the boundary out. Exactly. And in fact,
In them in the more recent computation that is has not appeared yet we consider these generating function.
bongani states. And in that case, we can compute the partition function. Exactly. Even for all the degrees of freedom books boundary and bulk and actually in the formula looks, it looks a bit
looks different, but in fact, we see that there is kind of the same structure and is
Is the DMS three character in a sense regularize in a very peculiar way. We don't know how to interpret it, we would we dream that this is some signature of some new symmetry group for that exists for the final boundary
And so we are trying to understanding how to make sense of these of these wish.
And I think, oh, I want to mention that this type of regularization is somewhat similar to the one that happens if your regular is the DMS character a lot. Yes, yes. In that case,
Is also like putting the theory in a box, except that we also before the dynamic of the theory to fit the box nicely disease. The everything is Kirk and the radius of curvature is also fixing the distance of the boundary in some sense.
But when the hour about that we use is in fact it's straight. It kind of a final box in flat space. So it's not exactly the same but we see that the
That there are similarities. So we still introduce the overall structure is the same and but we introduced some offset for the polls that mimics with the
One that exists yet. Yes, kids. So this is all hints. They're pretty vague and I there is really a lot to do to to understand better what is going on.
What is going on. So let me summarize in part one. The Caribbean 3D graphics and cemeteries and in particular how Lawrence and shift cemeteries are somewhat conjugated which was surrounded by of later to understand the face perfect quality of the to the statistical models.
In part two, we discussed the nature of the degrees of freedom within define them as the shape of the foundry, which again can be expressed either
In terms of the experience of curvature or as the distance from the center. And again, these are the two points of your given by either Lawrence boundary compensating degrees of freedom or shift on the decomposition
Input. Input free which looked at the dynamics in terms of this, the physical model and in chain models and discuss the dualities between the theories and in very briefly, in part, for I summarize some results about the petition functions.
So, as I say, there is a lot to do. We can consider a DS by going to, to repeat, oh, we can play around with different types of boundary conditions to try to understand classical interesting thought that uses continuum limits and so on.
And compared to other to all these other approaches to a library, which are much more developed the numbers and
And in fact it's the, I think one of them that the most interesting things to do right now is to try to understand the parables interesting strive for establishing have a dictionary and holographic dictionary between circles in the back end on the mound.
We can also generalize to 40 by focusing on some topological theory of flex based time which
Was introduced so the white when these are not something more complicated because
I think it has this theory, which was introduced by Lauren and I Steve and later and recently revived by younger myself said assignment in Florida internally and and
And picture what we what we did was to conceive. Well, what this model does is a nice extension in natural extension to four d of of the 3D Roger calculus. So everything is the flex
But we have an isometric interpretation which doesn't really happen for four dimensional BF theory. So it's kind of an intermediate ground to play so sorry for the delay and thank you all for this.
Questions.
They have a question.
Yeah yeah
Can you go back to page 30
Yes.
Yeah. So on the right hand side you have the transfer matrix. Yes. Yeah, having tea. So, can you can you clarify.
Our the transfer matrix is defining if there's a relationship with that Tommy Tony and others punching before. And the second question is, are you then defined in D. We, the inverse temperature meta or not.
Oh, OK. So for 14 the dress reminder just excited. What was thrown up here. Here is the same thing. I have just and instead of this evolution. So this takes me from the, from the state on the bottom box down here to the state to the top, then what
I introduce is a twisting operator, which is just the translation of the foundry spin chain of some and gamma unit.
And in principle, it would just close this, but in fact see the carbon copies been to this degree of freedom tells me that they have to introduce the project. So this is how to read this is correspondence. So this should be compared to what I've written down here in small, which is a
Trace of the boundary theory.
For the boundary field for the most indies this monetary theory of you to the minus beta age and gamma JJ. Well, it's just the name that gives to these patients presentation.
So in a sense, yes, I would like to interpret empty or better empty time to the spin associated to that direction as the, as Peter as the inverse temperature, but how much the you
Yeah, how far you can go with these analogy depends on the details of your, of your boundary state and boundary theory. So, and unfortunately I don't have
Yes, very, very nice way to to make that correspondence, but that's a wish for sure.
And the expression. You were leaving at the end, the one to appear.
In the send this as
The calculation of
Concern or Asia or or it's a different there are there are different ingredients.
So which one
The one that you have on page.
36
These one. So what this is an exact computation of their concern or ratchet partition function for a specific boundary states that
Is this one. So it looks quite horrible if you've never looked at it before, but in fact is is quite simple structure. So, these are the Venus pity on a
Boundary states. So the usual ones. And one thing, super positions by thinking it's kind of a sudden distribution for the spins weighted by these London tower factors.
So the only thing we do is to take these London tower homogeneous. So this is not any more real generating function because you would need them to to be different at every vertex. But once you take them homogeneous, we can compute the partition function. Exactly. And this is the result.
So,
It is that right
Yes.
So that boundary state as many nice properties. I think we still have to fully understand
The geometry of this boundary stage, especially in the limits that are most interesting to look at, but it's doable. It's not that
It's just
For the one that you were showing before the one page 30. Do you know the scaling of the partition function we're in deep, the number of layers.
Here.
Is the question is a
Temperature in the same way as the
same temperature as you as in the six of ethics model. No, that is not true.
Because they are the temperature is read that are finished by your choice of interest.
So index. So when has to be careful when doing this translation. So of
How to interpret physically so
I'm
In the six vertex model taken as such.
Which is here. So these Boltzmann wait a be a be a fixed by your choice of interest Weiner and the Bozeman, wait, somehow secretly are fixing your temperature. In fact,
It's essentially the only you, the system is only one parameter which is the temperature because the rest is normalization. The second one, so
This temperature here has nothing to do with the size of the lattice is a different is a different
Yeah, I can do this differently. So from the six of ethics bother perspective, you see that the size of the lock. This is not really the temperature that are looking the Moto X and the tool can be played with independently.
And also these kind of model doesn't really have a nice the
Face position is kind of a the stochastic version of it. It's kind of at the gym. The generic version which is freaking so would be, yeah.
Okay, thank you.
More questions.
Okay. I mean, I'm much less
Tech. I mean, must not as technical issues is basically, it seemed to me that I was little bit confused because they're they're simply big disconnect between the
First three parts which are coherent and it had to do with lambda equals zero, as far as I understood it. So, and then suddenly you brought brought in the US in the EDS and
And the negative cosmological constant here. Right. And so when you wrote, so the statement, but it's not clear to me is that in the last part than this, you don't have
Much of a control between the bulk theory and the and the boundary theory or what if you can say anything about that that will be helpful, and also because you met some remarks or the game is group and
So,
I'm gonna get. Yeah.
I
Need to. So, eds. It was just the regional motivations, he's worried. The first computations were done and and we have started
These, these fields have a log of in 3D was started there so it seemed a natural place.
To start from two to make comparisons. And so he started really, this was the first place this partition function was computed and then
flatly meet was investigated in different ways. Later on, both on its own say so, as a independent theory to check the computation as an as a limit of the IDs. So in that case,
What people did was to compute the partition function as such as a Q amp D one loop partition function. And what they found is that the it was mentioning some degenerate limit of the yes computation. And so it was natural to ask whether these was reflected in by some
So since the it yes computation as a clear interpretation in terms of the Venus auto character he was not able to ask whether there was some senior group behind
Regulating the shape that the structure of this petition and the answer.
Yet MS MS character. Just a second is in fact the A is just a secretly is just a degenerate version of the 001 so it's quite different from the BMS four dimensional DMS a group know
Were studied the three dimensional beam is group, long time ago. Right. And that is what
What Barney started out with. I mean, Ben Schmidt and the jack and I but but that was all in a lot is in signature is this BMS group that you're talking about the same group in which comes from the origin signature or is this something else which has to do Euclidean space or
These essentially a different one for a for the dish cross some super translation what they call super the translation which is
It is it is what we found. First, and that was just the North Indian three dimensional
Yeah, so this is an extension of
So that, because they're super rotation.
extended version which is
Yeah, it's not, it's
Not button super business or maximum three mentors is not an extended version three dimensions. Super techy does come up. I mean, you just get the first one you don't get the Lord in this group on the bass, bass.
Okay, so, so, so this is that group that you're talking about, which is a lot is Andrew, so there is no geometrical understanding of what this happens is just an observation is that correct
It's just, it's just an observation I wish.
I think a. It's been a while since I've looked into that. And I don't know if there are bunches but people were trying to build theories for the boundaries that had that group as a symmetry group.
But I know the status of the correspondence. So they were doing this a little bit by hand. It seemed to me. So trying to brute force some theories with that symmetry group, but I don't know if they had sense how they would arise independent
And then final remark is the question is really the very philosophy, transparency is that usually was referring to like this transparency 36 and so on your back to lambda equals zero cases that correct
Yes, yes, he's loving zero case. Okay.
And again, the statement here is that there's no
Obvious geometrical interpretation of the mystery arises, it just a group that happens to be so.
This is not really DMS three anymore. It's not that the character anymore and we don't know if it is a character of a symmetric wrote for real. We have other ways to write these which connected to the way the DMS character are computed I it would take a while to explain how that works.
You don't know exactly how those words. Okay, thank you.
And when they say they extended the mastery and meant central extended not not that it was an extension of the of the Lawrence. So they take the usual PMS three and the central expanded and that's what they said okay
Thank you.
Okay, there are no more questions, let's thank the speaker again.