Holography. What is the desirable thing?
How… how… Are people familiar with authority code?
Yes, well, some people don't. People on that side are very familiar. People on this side feel less…
Let me maybe quickly go through the Tory code. Tori code is a… 2 plus 1 dimensional physical system, which is topological, and there are.
Annuals, electric and magnetic, and if you put one on top of each other, you'll get a fermion anion.
And then you can make this anion goal on a closed loop along one of the cycles of the torus, and there are two cycles, a red one and a blue one.
And then E is electric anion, and M is a magnetic anion.
Ground state of authority code has dimension 4. It's actually two qubits. In all possible senses, you think about it as a two qubits.
Because this, uh, topological lines. They act as power operators on those qubits, okay? And that's all we need to know.
To realize that if you want to condense electric anion, what does it mean? You condense electric anion, it means you create a state.
Which doesn't care about electric anion, on which electric line, close the electric line, will act by identity. Well, that's exactly the stabilizer state. You say that you want a state on which both of these guys act by identity. This is a stabilizer state.
There are only two anions which can be condensed, electric and magnetic. F cannot be condensed, it's a fermion.
It's very well known that Tori code has two topological boundary conditions, Dirichlet and Neumann, and they are classified by two classical codes, and corresponding, uh, corresponding states, stabilizer states, and if you think about this back in terms of Chern Simons.
This will be two Nurion CFTs, which are actually Orbifolds to each other, and that's… that's the same TFT picture, that if you have.
Toric code as a SIM TFT, those two theories will be orbitals of each other.
Now, what is the topology? Topology is when you say, I want any possible operator along a particular cycle to act trivially, because that cycle is a trivial cycle.
Okay, you'll have red cycle, and you can consider those two operators, the stabilizer state is the vacuum.
I have the blue cycle, the corresponding state is the stabilizer state as well. You have a combination of E and M cycle, the most general cycle is Na plus MB.
But everything is mod 2 in this case, because any anion stuck with itself will be an identity, so there are only three possible choices of cycle, and you have three stabilizer states.
And then there are also underlying classical codes in this case, and as you can see, those classical codes basically tell you which cycle is.
Contracting. A, B, or A plus B? So there was a nice visualization which actually tells you, if you think about it, then it tells you that this code actually tells you the lattice of holonis.
What about holography? Well, holography is very simple in case of the Tory code. You have electric and magnetic anions. These are the boundary CFTs.
You have… oh. You have… You have a sum over all possible topologies. In this case, there are only 3 topologies, because there are only 3 non-trivial cycles.
You can shrink. You're talking about just the handle body… Only handlebars, only solid core eyes, only solidarity. The different solid torque. Right. A separate statement is that in this simple theory, 3 solitori.
Our representatives of three equivalence classes with respect to this equivalence, which we discussed a few minutes ago. And therefore, in this case, it's just enough to sum over solitore, and then you have a holographic identity.
A boundary as an ensemble is equal to the sum over all possible topologies. In this case, an equivalent… equivalence classes of topologies.
And then you have an overall coefficient, but this is nevertheless a non-trivial statement.
Okay, which you can already visualize in case of a Tori code.
Okay, now, the rest of the talk is holography, that will be this story on steroids.
It will be, like, this story, but just… much more for…
Complicated. Uh, so… First, what I wanted to say is that if you have topological boundary condition in a three-dimensional theory, you have two-dimensional CFT.
If you have the same in five-dimensional theory, you have 4-dimensional CFD. A standard choice for me would be Narayan theory, or Maxwell theory. You can try to put different boundaries here. My story is not dependent on the choice of the boundary. My story is about a statement.
Uh, in the Hilbert space of the gift. So, you can put any boundary you want, but I will talk about the standard boundary, which corresponds to holomorphic quantization.
And then, here is a precise formula, which we derived a year ago, how you're supposed to sum over all possible topologies.
I will not go into length how this formula is derived, that will be a story for a different discussion.
But basically, the formula tells you the following, that the ensemble of.
Cft partition functions, weighted with a particular positive weights. Is given by a Ren normalized.
Sum over all possible manifolds with a given boundary. Okay? I was already explaining that this thing is a manifold with a boundary, it's exactly the same manifold, and then you sum over all possible mapping class group elements.
And you take the genius of this initial handlebody to infinity, and therefore you have all possible handlebodies appearing in the sum. And what is the script D?
Script G, the D to the G prime. D is the total quantum dimension of your TK of T.
And this formula is not restricted to a billion theories, it's actually a general formula for three-dimensional TQFT.
Uh, which has finite… Hilbert space. Let's see, so, um, so when you say you're subbing, I mean, you're looking… looks like a sum over the subplectic group, not over topologies. I'm summing over all mapping class group elements.
But what I'm having here is this Higger splitting with the handle body carved out.
And therefore, I'm actually summoning over three-dimensional manifolds.
This is exactly the same state we discussed when I was talking about manipuled with a boundary. I just put B here because I wanted to have not a state here, but a series.
You could give that as an interpretation, as a sum over some three manifolds.
Which have a HEGARD decalpass. Ah, you say… I will go to Infinite Genius, and I will sum over the whole mapping class group, and therefore.
I will cover all remaining. Okay, um, what is little g on the right-hand side?
Little John the Rekhand side is a genus of what you want to get on the left-hand side. Left-hand side is a CFT partition function on a Riemann surface of genus G.
Some overseas. Omega is a modular parameter of that Riemann surface. C is a sum over topological boundary conditions, or in other words, is a sum over CFTs.
So, what class of CFTs do you have on the left?
Uh, it depends on your B. It really… it is really a statement about Hilbert space.
Of the ITQ team. If you strip B out of this, you will have a state produced by.
Tkf key on a manifold with a boundary, and here you have a linear combination of topological boundary conditions.
Um…
Out or see? See, and here you have the limit.
Into the barge, any size of the groove. Sum over gamma, and here you have 0 u gamma 0.
This is a very large juice. This is a large genus minus the finant part.
So that is a state. Reduced by… by my TK on…
This guy with the manifold with a boundary. So this is the state of the Hilbert space, and this is also a state of the Hilbert space. If I multiply both sides of this identity by a particular boundary, whatever boundary you want doesn't matter, because that's the statement I'm making.
But if you consider simple boundary of your liking, then you will have CFT partition function. And so technical things, what about the existence of the sum on gamma and the limit?
Right, so I claim that this is an identity on the left-hand side is finite, and therefore I claim that this is also finite, which might be not obvious.
And I want to mention your paper. Which basically says that if I try to sum over all topologies, the sum will define it, right? So… so… but here, the size of the group is also finite. I'm not summing over all topologies, I'm actually averaging overall topologies.
And that average will be zero. Therefore, I multiply it by something which is actually positive and very large.
To bring it back to something final. Now… Okay, so Anandy and I only talked about some of our two-dimensional topologies, and Daniel wrote a paper about.
Generalization to higher dimensions, but… Daniel… There he is.
Did you actually discuss the seller to follow it? You don't run in any concrete way, okay?
Yeah, it's present. You have an infinite sum of unitary operators. Yes. One would worry about.
Convergence. Uh… I have an average over a group.
And for any G prime, my Hilbert space is finally dimensional, and if I average over a group which is infinite, but in a finite dimensional space, I actually have a projector.
And that's how this identity is derived. Secretly average on a group just projects you on an environment space.
And this states, uh, these are… this is a basis in this environment spectrum. That's all… that's all I'm saying here.
That's all I'm saying here. And then I'm given an IS interpretation that this is a sum of all possible topologies.
Now, you can ask if there are any examples to show that this is convergent? Yes, I'll give you an example. A billion examples I understand very well. Non-a-billion may be more complicated.
But we claim that this is completely general, and then if your theory is a billion, then all of these coefficients will be 1.
And therefore, I'll just have an ensemble of CFTs, which are all weighted equally.
And then, in terms of the abilion story, it basically… it is basically an identity between stabilizer states.
Because topological boundary conditions of stabilizer states, and. Topologists, path integrals on those topologies are also stabilizer states. You only need to calculate those weights, and those weights are non-trivial because, as I said.
Uh, path integral gives you a stabilizer state up to sort and normalizations, up to sort and factor, which is non-trivial.
So, therefore, calculating that might not be very trivial, but in certain cases, we managed to do it very explicitly. There are papers.
What happens if you start with a Tory code, or in fact, I believe, if you start with any.
General Simons with finite level, and you take N to be large.
We basically reproduce the results of these papers, we basically reproduce the sickle veil formula.
Okay, that's basically what happens. I will… and you have exactly the same story in 5D. The math is, in fact, the same.
You again have identity bitren stabilizer states, and then you reproduce a slightly different sequel veil formula when it's an Eisenstein series of an orthogonal group.
And again, this is… this is in this paper from this morning, so if this is something interesting, you can look there.
I want to give you one more way to think about this identity in case of 3D or 5D.
One of your theories appeal in. Basically, what you have is that.
Uh, the partition function… of an Orion CFG, or a partition function of a Maxwell theory, that's basically an Orion-Siegel theta function.
But you also will have Narayan Siegel theta functions appearing from the sum over topologies, because corresponding states are also self-dual codes, and therefore the identity that I'm saying that I'm talking about, in most general case.
Is an identity between sums over, uh, single neuron theta functions.
And in one case, your average over 9 module, over the metric and the B field of the NORIN moduli.
They can be interpreted as Narayan theories in 3D, or they can be interpreted as geometric data over four manifold and 5D.
And on another side, you will have different modular parameters. And in 4D, there's a different coupling constants of the Maxwell theory, and in 2D, there's a different modeler parameters of the Riemann surface.
And now, if you represent the identity like this, with certain weights, which are… in simplest case, if n is square 3.
Those coefficients beta will be just identity, so you forget about them.
In the simplest case, this identity at the level of, um… Narayan Siegel theta functions are very difficult to understand. You have a bunch of theta functions on the left, you have a bunch of theta functions on the right.
How do you prove they're the same? I have no idea. But if you write that as an identity between stabilizer codes, stabilizer states.
It becomes an algebraic identity, which is much more doable and much more comprehensible.
So basically, reformulated everything in this language of quantum states. Works as a charm, and you can understand this identity very well.
If you take M to infinity, you will get a lot of points on the corresponding modular space. These are hacky points.
And therefore, they will… cover the modular space with a canonical measure, and you will obtain the single vein formula.
But I want to say that if we consider non-mbulent to QFT, there should be another version of this, which generalizes the signal waveform. And that, of course, hasn't been worked out.
Okay, I'm really over time. I… maybe a very quick thing is that if you…
Don't want to consider a partition function, you can actually sum things over closed manifolds.
And then you will simply calculate the number of topological boundary conditions in your theory, and this is called mass formula, because that part of the SQL veil formula is called the mass formal.
Whether the correct name is Siegel Minkowsky mass formula. And then, for a billion theories, we can readily calculate that.
And, uh, answering Rick's question, that will be a finite quantity. You can take that limit, and you can divide by the group.
Everything will be finite, you will have this kind of sum coming out of this infinite sum.
And this sum will give you the correct number of codes. That is a result for a particular Boolean theory, but you can do that pretty much for any three-dimensional abilion theory.
There is a classification of all possible three-dimensional billing theories, and you can… you can repeat the calculation for any single one of them.
Well, if you have enough energy. Okay, the conclusions. There are two parts.
There are two parts. These are more technical and what I spend more time talking about today. This is more conceptual.
The technical part is that codes are closely related and naturally emerge from topological field theories if those field theories are a billion.
You have classical codes, which parameterize. Non-anomalous subgroups of symmetries, or condensable anions.
And then, if you have maximal subgroups, then you have topological boundary conditions.
But if you think about corresponding states in the Hilbert space.
Then, the best language is not of classical codes, but of quantum stabilizer codes. And I think that's the main message, at least at the technical level, that's the main message of what we understood during the last 3 years.
If you want to think about those states, the right way to think about them in terms of stabilizer codes.
And then path integral on a boundary, that's another part of the story, that's also stabilizer code, and there is also underlying classical code behind that.
And then there is another part of the story, how this machinery can be used. It can be used.
To study holographic identity, when you have a 3- or 5-dimensional topological field theory, which you literally sum over all possible topologies.
And in the abilion case, this is not an incomprehensible thing, because you can parameterize everything in terms of codes.
If you sum over close three manifolds, you will get a mass formula. If you sum over manifolds with a fixed boundary, you have a generalization of a SQL veil formula. It's an ensemble holography.
Okay? But then, moving forward, that's a very nice story, it's all… pretty much well understood. Looking forward, it would be very nice to generalize that to the non-abilion setting, where you have to think what the analog of code would be, and what the analog of stabilizer code would be, and how to do all of these calculations. And that is… that is an ultimate goal.
Question. Thank you.
More questions.
Right? I assume everything is clear.
Well, it's