continuous spacetime symmetries. Take it away, Hinyaki.
Thank you very much. Thank you very much for inviting me to give a talk. Thanks everybody for coming.
Um, I want to discuss this paper that we.
put out now almost half a year ago with Fabio, Nicola, Hotat, and Sakura.
And it's about how to understand in the language of symmetry TFTs, which I will introduce.
spacetime symmetries.
So let me start by discussing what the symmetry TFT is. It's a kind of topological theory.
That's useful for understanding symmetries in general categorical or generalized symmetries.
Um, so let me try to introduce what those are.
Let's go back to the standard case that we learned in textbooks or standard symmetry generators. Imagine that you have an.
A billion continuous symmetric. And then you have a generator for the symmetry, which in the path integral language you can write as the integral of a sum d minus one surface. So, for example.
In the standard case, you will choose this to be a space, or a space like slice.
And then you integrate the nether current. In my conventions, the nether current is a D-1 form. If you want to translate this to more familiar statements.
This would be something like the integral over a space of the charge. So when a specific component, which is typically written as J zero.
PM implicitly acting with the star just because you'll do Macomb.
So when you put these symmetry generators in the path integral.
you can generate the action of symptoms already know how that works.
So here J is another current. And inside the path integral, the another current satisfies the word identity.
So, it's not quite conserved, it's almost conserved. Except that it gets some contribution.
from the insertions of charged operators. So imagine that you put operators OK with charge QK.
at points xk. So Dj fails to be closed precisely at this point by delta function.
And how much it fails to be closed at those points is the amount of chart that you have at those points.
So one of the fundamental observations in this field is that you can think of symmetry operators as topological operators.
At least away from this delta functions, away from the insertion of the charged operators.
So in mind that you have one such integral, something like this.
And you deform sigma a little bit to some sigma prime.
Now I want to compare the symmetry generator at Sigma and the symmetry generator at sigma prime, so I'm going to multiply by the inverse.
Um, just because of this exponential form, you get the integral of… So, the exponential of alpha times the integral over sigma prime minus sigma in J.
Now, because sigma and sigma prime are related via slight deformation, there will be some.
D-dimensional manifold B. Such the boundary of B.
is sigma prime minus sigma. So if you imagine taking sigma and deforming tilt a bit, it will be the kind of the inside with two boundary sigma prime and sigma.
Now you write this as… Boundary of B, your price talks, you get DJ, DJ is this delta functions because of the word identity, and if you assume that none of the points, or you have inserted the operators are in B.
Then you get the identity. So this means that in the path integral, this operator here is the identity or equivalently u alpha and sigma and u alpha in sigma prime.
They give you the same results in the partition.
So there was a starting point. This is a very.
all observation, but Gyotoka pushed in Cyber and Willett 12 years ago.
argue that this viewpoint of topological operators generalizes in many useful ways.
Um, so the picture you described was something like this. We have a charged operator. Let's now put it inside some UK, and we surround it by the charge generator and we do the same calculations before.
We act with UAlpha, which has this form. Now, sigma written here, is the boundary of some B.
We get the integral of… e to the alpha… Integral on B. We get e to the alpha integral on B of dj.
And because of the word identity, you can precisely in this phase, which is what you expect. The presence of a charged operator.
is detected by the symmetry generator by a sin interface.
Just put a portion out to the church.
argued is that it's very useful to do this for any sort of, uh.
operator, not just point-like operators. Here I'm drawing a… P-dimensional operator.
So, if you want to do this game of surrounding the operator, and then filling it in and applying the word identities.
The thing that surrounds and links the B-dimensional surface, it's a D minus p minus one.
surface, where D is the total dimension of the spacetime.
In the previous case, we have a D minus 1 surface, something that surrounds a point. In this case, when this is not the point, but p-dimensional, we get a D minus p minus 1 surface.
You can apply the same logic other than this, there's a word identity that assigns a phase when you contract this.
which is e to the alpha. Same SKU, the charge for… I'm going relative quick because I assume that many of you have heard a variation of this, uh.
Discussion in many contexts, but if you have questions, please do interrupt.
I'm more than happy to stop and…
Okay. So… We saw that symmetries are… Any symmetry and standard symmetry gives a topological operator.
The realization in the last few years has been that.
It's useful to sort of make this connection stronger.
And… define symmetry to be any topological operator in your theories, and then you have a topological operator, we say that we have a symbol.
Now, the symmetry that you get out of this construction, not all topological operators are group-like.
Not all topological operators are as standard as. The ones we are used to in the textbooks, but this happens to be useful. This is a useful organization that we can extend previous results.
So, the ones I just described, the political operators of co-dimension P plus 1. So like the ones.
Here, this could mention p plus 1, B dimensions.
We call p-form symmetries. And that's because the objects that they measure are the charged objects are p-dimensional. So here I'm following the language in the original paper by Gayo Kapustin, Chaver.
In particular, organized images I'm going to call zero form symmetries, because they measure point like operators that is 0 dimensional.
All right, so that was fairly standard. And I imagine many of you have heard of the variation of this introduction. That's why I was being quick. What I'm going to describe now is something perhaps less well known, but I think very useful also.
Which is that if you're going to talk out the set of topological operators in any quantum field theory.
We have realized, since the last five years or so, 5, 6 years.
that it's very, very useful to take some sort of holographic, if you let me use this word.
perspective on these topological operators, and instead of talking about topological operators in D dimensions.
We're going to talk about the topological field theory in one higher dimension. And we call this topological theory, the Sim TFT or symmetric theory.
So there have been many papers that have discussed this symmetry theory.
And here's a selection. One thing I want to… Highlight, because I thought it was a nice convergence, is that some of these papers are condensed matter papers, some of these papers are string theory papers, and some of these papers are not papers.
And everybody sort of agreeing. sort of reaching this… Description, which seems to be really useful. So what is MTF?
Well, it's a description of the symmetries of a theory.
In which, instead of working with a d-dimensional theory, you build a sandwich, like the one I'm describing here.
So, I'm extending the D-dimensional picture to D plus 1 dimensional picture. I'm adding one extra dimension, which is the horizontal one in this picture.
And the local degrees of freedom. Live on the right boundary, in this picture, which I'm calling P tiller.
So you should think of this as the local degrees of freedom of your quantum field theory. Imagine that you are doing n equal to 4, so this would be the genomes and the.
Gage bosons and the scalars are equal to 4.
And these all live in this gapless bundle. Then there is a five-dimensional topological theory.
which is known as the symmetry theory, which admits these local degrees of freedom as edge modes.
And this symmetry theory also admits gapped boundaries, topological boundaries.
Yeah.
All right, Yaki, the T tilt. The right boundary doesn't have… the right boundary theory doesn't have to be gapless.
It doesn't have to be, no, sorry. I have in mind construction of Gabbler's theories through this construction, a complete write. I could do the whole thing up.
It could be like, yeah, the whole thing could be cut.
It could very well be gapped. For example, you can understand the different Yang Mills theories for.
for different. global forms of the gauge group, just using this picture.
Correct.
Yeah, yeah, yeah, definitely.
Um, bye. So… As Gurya said, for example, if you want to understand the different global forms of many theories, I will give an example in a second, but Dan Mills, for example, the thing you would do in this picture.
is to choose the left boundary gap, although there is completely right, the red one covers a bigger gap.
So you can choose in different ways and all these different choices, they give you the different global structures for the quantum field theory that you are.
I think it's also important to say that rho is topological.
Mm-hmm. Yes. Sorry, I… Yeah, yeah, sorry. Yeah, Andrew.
Alright. So, in this kind of description, if we wanted to construct the four dimensional theory, the language that I'm drawing is a 5-dimensional picture. There's one extra dimension, which is the horizontal one.
And we typically speak of… collapsing the sandwich. This is the topical theory that's not really very meaningful, but I think it's a good way of.
Get an intuition for what's happening. So we bring these two boundaries together. And we obtain an effective 4-dimensional material, three dimensional image.
This topography theory, we have some topological operators in them. And once they hit the left boundary.
They may or may not survive, depending on the elephant. It might become trivial or not.
So some of them will become trivial, some of them will survive faster.
So in this way, you can generate topical operators in a d-dimensional theory, starting from a set of topological parameters in d plus 1 dimension.
So here's one example. If you take n equal to four superjam means in four dimensions with gauge algebra SUN.
This completely fixes the Lagrangian, and once you fix, I get tau, the complexity of the coupling.
But as a quantum theory, you still have choices to make.
Typically, this is described as the choice of global form for the gauge group, so the algebra is UN and meets a global form factor gauge group, which is just UN. It could also be PSUN.
And here, I should also mention that there's an extra set of data, some sort of discrete theta angles that were fully classified by 107 Tachikawa.
Will not need to go into those, but… There's a bit of extra exercise to do sometimes, even if you fix the global form of the screen.
But once you specify this finite set of data, you have specified what the theory is.
One way of classifying or to understand the distinction here is that.
Different choices of global form have different one forms of interest.
In this picture. Um, the way you obtain the different forms, so SDN or PSUN.
is by choosing different roles. different topological boundaries for the symmetry, topography.
In this particular case, if you want to classify all the different global forms for SGN, the thing you need to pick for the symmetry TFT is some discrete.
Gage theory, in particular, I said N gauge theory.
I'll give the Lagrangian for this theory in a few slides.
But what you would do if you want to understand this theory, is put the local degrees of freedom for an equal to 4 on the right.
put the set engage theory, and then you classify all the possible boundaries, all the possible topological boundaries for symmetry. And in this way, you will reproduce precisely the classification of harmonies.
Any questions?
Okay. So… This construction might remind you of most of the talk will be to try to connect this closer to holography for a specific class of cynicals.
So far, I think it's fair to say that we understand.
quite well, I would say. In the case of discrete symmetries, like this.
Set in symmetries that would appear in this case. In this case, these theories differ by.
The one-form symmetry, which are set to 10 symmetries, typically.
We simply understand things better, but in the last couple of years, we have started understanding how this picture generalizes to.
Um, to continuous symmetries. And the same sort of picture is still true, is believed to be true if you complicate the symmetry theory a little bit, and the way I will describe it.
But the same picture seems to… seems to…
Now, everything, pretty much, that has been understood has been so far on internal symmetries.
And what I want to discuss is our paper, which is a first attempt, trying to understand how continuous spacetime syntries, such as translations of rotations.
Could fit into this picture. It's not a given that you could do this, but I think it does, at least.
And here I want to emphasize that I'm dealing with continuous space-time symmetries. In particular, I want to think about the park.
spacetime symmetries that are connected to the identity. So translations.
I connect to entity or location connectivity. That still leaves the problem of incorporating discrete spacetime symmetries, such as term reversals or reflections, into the same TFT, and here I'm much more lost. Here, I don't know what to do.
I think it's a very important problem. But it's not clear to me that this… discrete spacetime symmetries, such as time traversal.
How they fit in the same picture, if they fit at all, certainly not.
Okay. So yeah, that's what we do in the paper.
The sympia, one of the things I didn't emphasize, but one of the things that makes it quite useful.
is that even if it's reminiscent of holography. It does not really need to have a holographic tool.
topography here in the bulk is, in some sense, an abstract representation of symmetries, and it does not necessarily have to come from.
But whenever you have a holographic dual, there should be a way of relating sim TFT.
with the holography, the holography picture. In holography, you can still talk about symmetries in the park.
And this symmetry should be there. in a way that's compatible with MPL.
So I'm going to start with cases that do have equivalent.
Right, so I was just speaking of the SUN and equal to 4 theory, 4 dimensions.
And as I said, Haroun7 classified all the possible global forms, so SGN and PSUN, and many others.
Um, so how would you understand all these different global forms, holographic?
And this was answered very soon after the finished paper by Whitten in 98.
And what he observed was that if you look at the action of type 2B on S5, so ADS5 versus 5 with 10 units of plaques on the S5.
If you reduce this action on VS5. You have to really be careful because of the shows like this and whatnot, but the effective action that you get in.
In ADS 5 is of this one, two pi i n.
times the integral of ADS5, which I'm writing as X5, simply because… Unit topology to extinguish the global form, so you probably want a small generalization of ADS5.
Or B2H3C2. He argued that these effective action dominates the analysis of large differences. Now, there's an important subtlety here that was pointed out by Belva Moore later.
So you have to be quite careful with single stones.
I'm assuming that we are forgetting all the singleton.
I'm assuming the existence of boundary conditions. Uh, for which is a singleton dash notification.
This hasn't been done in complete details. You know, if you look at the boundary conditions that you have to put in ADS.
In this system, with this Ladrangia. You'll realize that, um… The choice of boundary conditions is in one-to-one correspondence with the classification of global forms in a hierogeneous hybrid and particle. So within the SUN and PSUN case and in this paper.
Years later, we reproduce the full classification upon gene SUM, but you know.
Classical knee transplant.
Um, and the way you do this is you look at the boundary state, and the boundary conditions in ABS.
By quantizing this BF theory. using the radial direction as time, so you… the Hamiltonian quantization of this theory, and the boundary condition will be a specific state in the Hilpert space of this theory.
So what you can argue with this paper in detail.
Is that the Hilbert space of this theory precisely gives you the choices between SUN and PSU.
So… One of the things you will notice and one of the things that motivated the Syn-PF deconstruction was that.
This Lagrangian here is precisely a discrete setting gauge theories, a representation of the discrete set 10 theory.
So, if you forget about everything else about holography and just keep this Lagrangian and classify it to political boundary conditions, you will still be able to reproduce all the global forms.
And in doing this, you can… You can obtain a construction that is valid also when n is small. We don't need n large for being able to quantize all the.
to quantize this theory. So the sympia fees in a way distillation of the aspects of ADS/CFT.
Which, um… To have to deal with global structures, I think, as one way from standpoint.
In the cases where you have a holographic motion.
So… What we then argued is that…
If somebody wants to ask, please let me know, because I cannot actually… I can actually introduce it.
So Witten's argument shows that the sim TFT comes from the near boundary behavior of B2 and that's what he argued. Again, this is effective hasn't been very carefully resolved.
Let's proceed. You might ask you might ask in other examples where you have a holographic description.
Um, can you sort of have the same picture? That is, can you extract the same TFT from the holographic data in some way?
And in other examples that we know, although we don't have a general proof.
The same TST seems to arise from the limiting behavior of the Park theory near the conformal bundle.
That is, if you look at ADS very, very close to the conformal boundary of ADS, and you keep the… Extremely large distance behavior, which tends to be some sort of TFK.
Then that thing should be the simplicity. And there has been checked in many examples, apart from Wittens. Here's two that I.
find particularly nice. certainly the only ones.
Um, so one of the things that makes these categorical symmetries store interesting is that the symmetries that we obtain don't have to be group-like.
Um, and whenever we have symmetries that are not group-like, typically we call them non-invertible, so this is a… An exciting new class of symmetries that we have been playing with.
And you might ask, where are… This non-invertible symmetry that people have found in quantum field theory in the holographic tool. So you can find.
For example, 4-dimensional theories with a holographic dual that do have non-invertible symmetries.
How did these things appear in the holographic one?
These papers, for example, it was argued. that non-invertebral symmetries appear in the holographic dual, that is in string theory.
From the physics of the brain in… In certain geometric structures. So D brains are, uh.
And when you put them close to evaluate. Um, one of the things that made me interested in this whole story is that I was trying to… Understand what it was the symmetry structure for certain theories for which no Lagrangian is known, in particular.
for the National Superconformity theories. n equal to 3, so torque supercharges.
And… Any quantity?
And for this series, we don't have a Lagrangian, but we do know what the holographic dual is.
So, by sort of refining this idea of extending this idea of Whitten.
to the horrific duals of n equal to 3. We could work out what the symmetry structure of these theories was.
Which is quite interesting because we know very little about this theory. They seem to exist.
But they are quite mischief, so we don't have a good handle on them.
So, I think it's fair to summarize that for all the cases that we understand.
Uh, whenever the… You know, you have an anomaly-free zero-form internal symmetry that's sort of the most vanilla thing that you can construct.
given by a group G. And then the same TFT is on BF here.
In the case of discrete symmetries, this is something you can prove in the case of continuous ones. I think there are good arguments for this.
Here, A is a connection in G. bees and form valued in the joint of G.
So this is a very simple topological theory with Group G.
And in all the cases that we know, this is the form of the BFC.
Again, here I'm simplifying some things. So if you have anomalies, so… Your symmetries are more complicated, you have to complicate this a little bit, but that's okay.
So, when we started thinking about the space isometries, the first thing that you could imagine trying.
Is, well, every other example that we know of this type, we want to analyze some sort of BFU.
So let's try to do BF theory for the spacetime symmetric.
And let's see if this works in the way that that was.
It was going to be the conjecture I'm going to test.
And surprisingly. This seems to work well. The extent that we can test it, it seems to make sense.
Quick question. Are you…
Going to ignore the global form of the spacetime symmetry group?
Since you're representing it in this…
Yeah, yeah, yeah, exactly. So everything I'm going to say is going to be about the algebra, if you want to. I like to think of it as a.
Transformations continuously connected today. And that's what I was mentioning before, I just don't know how to think about time traversal in this framework, for example.
It's a very good question.
Okay, so then… I think the first.
thing that comes to mind when you're faced with this proposal is that, well, if I'm talking about the space-time symmetries in holography.
I'm supposed to know how space-time symmetries act in holography. It has something to do with gravity in the box.
Maybe you can think of ephemorphisms that don't vanish at infinity.
This does not quite look like a BF tier. So where's this laying? Where's this hiding in gravity?
So… Let me explain how that works in the one example that we understand.
As usual in Adscft, you want to relate correlators.
In… in the CFT, for example, correlators of operators are in this form.
to partition function of a string theory with some boundary conditions.
And you would ideally want to make sense of formula like this after some suitable regularizations, something that connects the result you will calculate in CFT for some observable with the partition functionality.
So this is the basic form. In particular, for the case of zero-form symmetries, we know that the bulk.
representation of a zero form symmetry is one form gauge field. So an A mu dynamical A mu.
And, if you want to compute the correlation function CFT of.
This kind of operator, so some nether current. A couple to some background for the symmetry, which I'm calling A0. So here, A0 is the background for the symmetry. The zero point symmetry.
What you would like to do is compute the partition function of a string theory on ABS.
with boundary values. So BD here is the boundary. The boundary values, which are given by the background of the plane.
Alright, so let's translate this to the language I have been using of symmetry operators. A symmetry operator is an insertion.
on the CFT at integral of e to the alpha integral of some d minus one dimensional manifold of the current. So it pretty much of this form.
If I choose my boundary. value for H0, the background field, if you like.
to be the form of a delta function.
So this is already a bit different from the kind of thing one does normally in holography. We tend to prefer to work with a smooth backgrounds, because.
From the point of view of topical field theory, this comes singular backgrounds appear non-natural.
Anyway, otherwise it's the same… same… thing you would like to compute the partition function of a string theory with limiting value of this one.
Um, so let's try to understand this sort of background in a… in a way that's useful for applications to gravity.
Um, so if you choose this manifold where you're putting.
Then the symmetry narrator, Sigma. To be the boundary of some D-dimensional manifold, so the boundary of some ball, for example.
Then the background field, which is a delta function, you can understand as acting on the trivial background with a gauge transformation.
Which is AC equals to a0 plus D lambda, as you saw. With the gauge parameter being constant in the ball and zero outside.
I just want this essentially like a heavy side function. I'm going to take the derivative, you're going to get the delta function precisely where you want it.
Um, so we're acting on the boundary conditions for A with the generator of gauge transformations.
is the Gaul's law generator wrapped on. So this point of view is going to become useful in a minute.
And here, when I'm talking about Gauss' law, again, I'm thinking about this Hamiltonian picture I was mentioning before. I'm thinking already or quantization.
And there's a Hebert Space, which states are the boundary conditions.
Alright, so to understand this in a way that's useful to my analysis of gravity in a minute.
I'm going to represent Maxwell theory in the following way, which is perhaps.
Not the… it's definitely not the standard Maxwell way, but it's equivalent.
So I claim that in this theory, with two form field B and a one connection A, is equivalent to a standard Maxwell unit.
And the argument is probably familiar to most of you.
Um, here you notice that B appears quadratically, so you can integrate it out.
And up to a one-loop term that discussed in detail in this paper, for example, you get Maxwell's action, the standard one for eight.
Another thing that you could do, you don't have to, but you could.
is integrate out A, which appears as Lagrange multiplier here. This imposes that B is pure gauge. Let me call that B is DC, where C is a one-form connection.
And then what you get after integrated the Lagrange multiplier, so this term is gone, you get VC, which starts DC. So this is Maxwell theory with coupling 1 over e squared. So this is the standard argument for duality going back to…
Okay, so I'm going to think of this representation of Maxwell here.
In this formulation, the symmetry generator, the one that acts by shifting my background A0.
Well, how do you shift the field by acting with the conjugate momentum?
And with the conjugate momentum for A here, well, that's B.
So… The momentum generator, in other words.
the better that, you know, shifts of A. Is simply the integral of B on the manifold sigma, or you're shifting a.
Is it okay? This is just a statement that P generates a shift of X.
quantum mechanics. Um, here, assuming as before that sigma is the boundary of some ball, so you could also write this as the integral over D of dB.
And this is useful to think in this way, because here you go back to the statement about the case transformations. In this representation of Maxwell.
B is the filler strength for the dual of H.
I'm writing that as FD. And in standard coordinates, so in terms of A, we will say something slightly impressive that fd is something like a star f.
So you could read this operator if you want as the action of integral over d of D of star f. And D of star f is precisely thing as gauge transformations.
Doesn't the A equation of motion say that db is 0?
Yeah, yeah, but that's a… I mean, yes, but at the United States, you are imposing the cash flow, so Hilbert.
The stage in the Hibbert space are annihilated by the… by the Gauss law generic.
But it does still united gauge transformation. Many gauge representative you you do work.
But in your second equation, you have the integral of db, which is 0.
Oh, sure, yeah. Um…
Yeah. I mean, you can do that. You can do that exactly if you integrate out A, but then I don't think I'm allowed to represent the state in terms of A wave functions, right?
So I can choose to… parameterize the state in terms of A.
In which case, my wave function has to be gauge invariant.
But A, itself can shift. So A itself communicates variant.
But if I integrate out A to impose BB, then… I'm sorry.
Alpha is going from identified with alpha plus 2 pi i, right?
That's true. Yeah.
Okay.
So if I replace B by star F. Then, if I change the metric, I change the normalization of the integral.
That's true. Um, that's why… Yeah.
So that and alpha can't be identified by 2 pi i 2 pi.
You're correct, yeah. So this wiggle here, Drew, is imprecise, right? It's the usual thing that you can quantize and question properly. You cannot quantize f and start F at the same time.
Uh, so this was a little heuristic.
I can choose to work in this language as I did here.
And then B is not… quantized.
Yeah, and then you'll say…
Okay, just go on.
Yeah, yeah. You're right, because if I work in this representation, it would be… B is not necessarily quantized in any way.
I think I agree. Can we discuss after the talk? I think I know how to answer that.
I don't know, just go on, don't worry about it.
Thank you. Thank you. Alright, so… In fact, I think the answer is related to what I'm going to say.
So, in this description of Maxwell theory, the beach HSRB term is not invariant.
under gauge transformations of B. So this theory just does not have these gauge invariance.
And this operator, e to the i alpha integral of B.
There's no need to be gauging variant under this transformation. It has to be gauging variant under the rest of the digital transformations.
Okay, clearly it is. But it's no gauge invariant and shift away, but it doesn't have to be because the action isn't equal.
Um, so… This operator, it's a well-defined operator in this theory.
Without having to attach an open shortage. And it's well defined for any alpha, for any value.
When we're studying just the BF theory, so omitting this term here completely.
Uh, we often discard that operator because the gauge theory has… in which B going to B plus D beta is a gate invariance.
But in the Maxwell theory, we don't need to… So the operator needs to be included, because we want to discuss Maxwell theory.
But otherwise…
The analysis… Apart from the fact that we want to include visuals.
If I need an integer. Otherwise, everything we need from this action is what's being given to us from the last term, the 2 pi i.
B128. Okay, the limit of this action when e equals to 0 gives us the right action on the boundary conditions of A.
A different way of saying that is all I want from this action is to shift.
A, and for that, I just need the B field, which is the momentum.
whether I have the EA squared term or not.
So I could take the limit of Maxwell theory and the limit in which e goes to 0, and obtain this topological field theory here.
With the provision that if I really am talking about Maxwell in this limit, I probably want to talk about this opinion.
Um, so at this point of view was emphasized in this paper idea, Wang, and Chang.
And it's a bit of a certainty. It might seem like, um… Uh, you know, been a bit too careful here. But the problem has to do with the fact that the reason I want to be careful.
Is that when you think of this theory, the gauge invariance of the theory changes.
And quite a bit whether E is a small but non-zero or equal to 0.
The theory with E identically 0 has more gauge invariance than the theory with E is small but larger than 0.
And you're trying to do the quantization carefully, as this group of people did, for example.
Um, then you run into subjects.
purposes of my talk, I… I'm interview… I'm not going to dwell on this more, but it's an important thing.
Right. So… That's been understood, the simplicity for continuous symmetries.
Then should be some version of this. BF theory that allows you to write the operators e to the i alpha integral.
And this can be reached by taking the limit on which e goes to 0 in the faculty.
So this was argued in a series of papers, and many of the things that you would expect the same TFT to do in the case of continuous symmetries, in this case you want to do work, and it also works.
Bono Navilion International Centers.
So, that was enough, I think Max was here. Let me try to understand what happens with the space-time symmetry.
Let's say that we have a d-dimensional conformity theory.
In principle, my conjecture is broader. I was claiming that it should be BF or spacetime integral, but I'm going to test this in the case of conformity.
I think I'm going to better control and also.
or graphic dualities are going to be easier when you have conformative.
So, omitting discrete factors, the space-time symmetry group is the conformity, so d plus 1 component.
And the conformal group of the boundary arises from boundary behavior for a gravitational receptor in the case of.
Please leave a favor. Thank you. So, in general, can we write gravity as a BF theory for… The conformal group.
Plus some corrections, perhaps. These things like this term here, which is not a BF theory.
By something that becomes an important ad.
Okay. Um… So, see, we can… For even D bigger than 0, when the boundary dimension is even.
Positive. There's a conformal anomaly. Um, because as a conformal anomaly, you will have to write something more complicated than a PF theory. As I mentioned before.
So you know, you have a BF theory assumes that you don't have anomalies with this.
Um, so certainly what we have to require more.
For example, something like James Simons. So, in the paper, we say more about what happens for boundary.
when there is of even dimension. But we don't understand the situation fully. So the summary of what we understand is that we have, I think, a decent handle.
On the A-type anomalies. Um, but not of receipt type of need.
Okay.
I have a quick question.
Yeah.
Do you want something other than just the first-order formulation of Euclidean conformal gravity with…
Some topological terms in it?
Because you always have that in any dimension in principle, right?
Um… To clear and confirm account.
I mean, you're working with the Euclidean conformal group, so…
Yes, but the…
the Euclidean aspect is not that important, but I'm just curious. That seems to be the way you're setting it up.
But that's not presumably what you were…
I have in mind.
You're right. There was a possibility, yes. I think all I can say, yeah.
Because that will… that has a Simon's term in it, uh…
So…
Yeah, you're right, but… The bulk we want to keep as the bulk, so as a standard gravity in the bulk. So when we are doing holography.
Yeah.
In August 5, for example, you want the standard supergravity.
And you can write it in a first-order formalism, which is what you're after, presumed.
And I don't know. Yeah, I'm definitely going to go for the first order formation, but it's not going to be conformant.
It's going to be a standard double number.
So let me go through the other… Uh, cases, so for D equal to 1, so one-dimensional boundaries.
This has been a topic of a little research in the last few years. I could think of citations, but… there anyway. You have a bug described by JT gravity, which is already BF theory, used to the Lagrangian level.
Um, so… Well, what I'm saying seems reasonable.
But let me try to describe the case that I think is closer to.
What we would like to do, which is, uh… high arbitrary image.
And in particular, I would like to understand. Gravity in cases where it's not just a polarity.
So the first thing that happens is when the boundary is three-dimensional.
And then the bulk is four-dimensional. So the bulk will be described by gravity in ADS4.
And as Vivek was saying, would like to formulate things in a first order way and a formulation of areas for gravity in this space exists.
I should say that everything here, I need to say for the… financial life is classical, and it's a random claims of quantization here. But classically, at least.
You can write Einstein gravity in the following way.
or is the gravity plus a couple of times each other.
In the following way. It's a Bf wave. So roughly, it's a BF plus b squared, as we had before.
But let me explain the details of what happened.
So B here is a 2 form in my 4 manifold could be.
foreign for manifold, taking values in the adjoint. of SO41, which is the relevant conformal group for 3 inch.
So that's be the… you know, doing value field as before for this gate.
And F is the fill strength for a connection in the same.
This group. So, SO4, 1. The indices?
Here ijfen. The whole vector indices of SO41, so for example, these two will be antisymmetric, given me by joints.
And, in addition, I have this VM here. Which is a dimensionless fixed SO4, 1 vector.
So, in terms of a space-time indices, B is a true form. F is a 2 form, and B is a scalar.
And in terms of SO41 indices, which you should think as internal gauge indices.
This is a joint, this is a joint, and this.
Now if I put an epsilon here, you can construct something which is, uh… The top form, which is also a gauge inlet, and that is a formula.
Okay, so this theory that was written down to my knowledge in these papers here.
reforming some other ideas of McDoug Mansory. The claim shows that is.
Classically equivalent to Einstein gravity. So how does this go?
Well, first of all, you have to choose V in a specific way.
So by Gauge invariance, it doesn't matter. very much in which direction you pick it up, and just doing a case transformation, put anything in.
This direction. And if you choose the value of V to be of the non-zero component to be lambda Gn over 6.
This is a specific value. Then you're going to get a B that appears quadratically, so you can integrate it out.
And you get this action in here. So let me explain what happened.
He has only a non-zero diff component, so the epsilon here will pick i, j, and k.
Um, to address that to the IJKM. So, this Bijkl.
Um, to have indices in the first four indices.
So this is what this expression is doing. We integrate out, they still satisfy the problem.
And now, because everything takes values in the first four indices, which are in SO4.
Then I can reduce my index structure to SO4.
VGS notation, what I mean by putting the hat is that.
F has MN, which is a parentheses, is f and n were now the same effect as special for comma 1.
Is that clear? So, I'm just taking the… The indecision is also for Kuma want to be restricted to the foreign.
Alright, so these are basically doing the… the Gaussian integral, because in classically, the… integrating out of B.
And it's important here that I'm choosing a B, which is non-zero, and in this particular direction.
One very interesting question in this game is what happens if V happens to be null or zero, and I don't know.
I'm going to pick this one and claim it was the same as…
So to see that, what you do is you expand A in a way that's very similar to within Transimus in three dimensions.
So, your A is, um… connection in that joint, and you're going to separate them to separate.
the indices that involve 5, which is this one's here, from the.
The ones that involve five, you're going to call BL vines or VR vines in this case.
And the ones that don't, you're going to think of as a spin connection.
So if you expand A in this way and you work out what f hat is, and you plug it in this expression.
bit of algebra, you end up with. This Lagrangian here, or this action here.
Um, which is familiar, is the first-order formulation of, uh.
and then gravity, so this is the… term and this is the cosmological constant.
And a moment in a term, which is something like R, which is proportional to the Eular density, so it's a topological matter.
So, at least classically in four dimensions. I can write Einstein gravity if I introduce the presence of a cosmological constant, and if I allow myself to put the Euler term.
I can write it in terms of a PDF here.
So, quick, quick comment about the previous slide. I think you cannot choose a B2B…
Uh, space-like or null, because then the stabilizer group is not going to be the Lorentz group, right?
I think you want to get to four-dimensional gravity.
Yeah.
Yep. Sorry, yeah, so that's what I mean. You're right that if I pick V2B now, I don't get gravity. I just don't know what I get is.
Because the stabilizer group of SO41 for V… stabilizer group of V is SO3-1, right?
Got it. Yeah, so the… depends on how I choose B, I get the stabilization could be so poor or so 3 comma 1.
And the interpolating thing goes through a null vector. I don't know what you mean.
I was just throwing the question there. This is a question of this theory that I haven't seen analysed in the paper. I'm not clear how.
In the case that V is not.
Well, in the case that V is of this form, then you get the Einstein gravity.
20 minutes before my project.
All right. So this is… quite close to the kind of thing I wanted. I have a BF theory.
coupled to some perturbation, if you want. I haven't justified visual perturbation, but coupled to some extra term, which is B squared times one parameter, which is lambda times t.
Maybe I should mention that one of the motivations for these papers.
Were the realization that lambda times z in our universe is extremely tiny. So we wanted to think of this as a small perturbation, but… Because of the same reasons that I was mentioning before, it's not clear how to do the gauge fiction and so on, it's not clear how good there.
trouble that works. I think it's interesting.
So… Right, so we have a BF here. Let's try to see those corrections. Let's try to see how this BF theory couples to.
to a boundary, how this gravity be a form couples to the boundary.
And here I'm doing something similar to the video I was mentioning before, I think. If I take the boundary value of A.
Uh, to be some… sorry, if I take laboratory conditions of A.
That means that my CFT is coupled to some specific background.
Which has to do with the… precise form of A.
So, if I split A in this way. That is I split the generators of SO4, 1.
into the algebra of S03ism, SO3s of algebra, which is the relevant one in the three-dimensional boundary.
Then, the different components of A can be arranged in this way. You have something which is essentially the spring connection multiplying the generator of rotations times something which is a real drive.
Uh, multiplying the generative momentum and F and B, which have to do with the component.
So, this is how a digitally boundary value for A.
Would the couple to the CFT. It would be a CFT coupled to these conformal data.
This is very different to the way that you will couple Northern CFT to.
to an internal symmetry. But that doesn't really matter. We have a background.
for the CFC, that we know how to manipulate using topographic operators, because we know how to do topographic operators in the.
I'll just do that. We'll see how acting with the topological operator in the bulk.
BF theory shifts this data in K.
So I'm going to… because I can write my action as a BF3 plus a small correction.
And in the paper, we have an argument that this small correction could be not close to the boundary. I'm going to… focus on the BF edits you get if you ignore that less than the B squared.
And let's see if I get something that acts as the CMTFT for spacetime symmetries.
Oh, yeah. I think I have like 3 minutes, right?
you do have 5 minutes, but I guess we can continue for a few minutes unless people have to leave, so…
Gotcha, yeah.
Okay, I'm not at least give the punchline and then I will try to be self-contained in 5 minutes, and if anybody has other questions, I'm happy.
All right, so… I'll give evidence in a minute for why this BF theory behaves as you would expect.
generators of symmetry to behave. Um, but there's a couple of remarks I think you've learned from now, especially because I might be a little bit fast when I'll discuss them later.
One is that we're working in a first-order formulation of gravity, and the famous issue with first order information of gravity is not clear why the turbines must be non-degenerated.
In the first order formation, there's no obvious reason for that.
And that gives problems, and those problems will also come for us.
Later. And everyone should have taken on your test.
So let me try to give the… The punchline, how this behaves, the track definition time.
So… We have a BA function for the conformal group, trace of B which F.
And there's an equation of motion, which is that DA of B is equal to 0.
As I mentioned, we postulate that the way this couples to the vendor is as it couples in the gravitational case.
the composition turbines has been connected.
Now, B can also be expanded in a similar way, but it lives on the dual algebra, so I'm putting the stars.
Their star is the dual with respect to the object.
But otherwise, you can expand B.
No, because you have an equation of motion, which is DA of B equal to 0.
If you express it in components, you get this system of equations, which.
No particular transparenta, it was flashing to.
But if you go to Flood Background. Choose your boundary to have a flat metric to start with.
Uh, then the question simplify quite a lot, and they all become D or something equal to C.
That is the equations of motion for… The BF theory implies that you have conserved covers.
Not only that, with some… if you integrate this conserved currents.
You get some generators that are the familiar generators of the conformal algorithm.
So expanding this BF on particular components of B.
In the dual algebra, and integrating with the standard expressions that you find in any review of conformatics.
Here, the only thing I'm doing, which is in trivia, is instead writing the start of T, which is what you will find typically, and write in TA, which is…
So, in contraction with Gerda. So in this way, you can identify with the generators of the conformal algebra should be, if this whole picture works is p, j, k, and b.
And what we did, and with this, I will finish the following sort of experiment. Imagine that you have some operator here.
And you act with a symmetry generator. Which is the integral of T, which… Through my previous identification, it's essentially the energy-momentum tensor, so the technology is translations.
So if I do this operator, which, remember, you say hologrami or some B field in the back, which I'm pushing to the boundary.
Does it generate translations of this object? And indeed, it does, and the way it does is kind of interesting.
So, if you act with this. B, which is in the dual of translations.
Remember that B and A are canonically conjugate, in the BF theory, so it will act by shifting the canonically conjugate field, which is the VL1.
So by acting with the generator translations. You will shift the build line precisely at the position where you have inserted this generative translations.
So the pictures that you put with generative translations.
And between two insurgents of operators. And when you put the generator of translations, what has happened.
is that the Virbine, which was constant here, suddenly gets a delta function.
And then it goes back to the present. So…
There is a way of… You have a beer vine, which is a delta function.
Localize here. You can undo this modification of the verbine by acting with the different morphism.
we can construct. And the deformorphism is such that it's non-zero on the right of the line and zero on the left of the line.
And so we have a proof that you can always construct these diplomorphisms for regular scan.
these transformations and so on. Um, sorry.
We have a partial proof, and there's some details of the proof that we have quite worked out. But in this case, at least.
Uh, you can do, you can act with a deformorphism that.
Brings back the metric for the Virvine to the flat one, but separates.
The two points in particular, I suppose this point amount proportional to the momentum that you have at it.
So in this way, by undoing the action of this, uh… 2 metre narrator that we identified abstractly with P.
We obtain what the translation should do, which is to move this point.
It generates a deformorphism which is of.
Do I understand correctly that you're moving all the points with X positive?
Yes, correct. Yeah. So, I'm putting the… Symmetry in it, or on this slide.
So X equal to 0. And the interpretation, you could write either the diffamorphism marks by shifting the X positive once a fixed amount.
all the narrative ones, 15 minus X. It separates your points and amount X.
Okay. So I think that the… sort of main lesson, there's subtlety is that… And they're important, so I'm happy to stay and discuss.
But let me conclude.
So, we wanted to construct a simplistic description of continuous space-time symmetries.
The necessary condition for that was to have the same TFT description of continuous internal symmetries and non-avilion ones.
And luckily, there has been lots of progress on that in the last couple of years, and now we have technology.
space-time centers bring their own conceptual complications. I think more than technical, these are conceptual points that we have to solve.
But we have made some progress on those. In particular, I would use the connection to gravity and the holographic duo. I think we have reasonably satisfactory answer in four dimensions, bulk of four dimensions.
And also, what is the connection of operators in SMTF2 to deformorphisms? Again, we have, uh.
or some beginning of an understanding.
Um, and there are lots of open problems. Here I list a few. Let me just highlight the two main ones that I think are… much confusion?
Um, maybe 3-1. So, we should strengthen the connection to gravity.
In all dimensions, so everything I told you was about four-dimensional gravity, because there's this formulation by Antonio Mansouri.
Um… I'm not… I'm not sure how to do this in D bigger than 4, for example.
Also, in even dimensions, we have the issue of the conformed anomaly that we haven't fully captured, and… I think it's important to control.
Something I had to skip was that when you act with localized operators, this kind of gauge transformations with a delta function profile.
This background fields with a delta function profile, which are written naturally in gravity. You do get unnatural.
boundary metrics. In particular, with some sort of the generate or even… It's not clear what.
You should do in those cases. Probably the most confusing bit of the whole story.
And there are other things you could do. And the other thing that's to me very much mysterious is how to incorporate finite space-time symmetries in this story.
Um, which is, I think, an entirely different. Right. Thank you. Napoleon.
Alright, let's thank Inyaki for any nice talk, and if there are questions, please feel free to unmute and ask.
Can you ask a question?
So, uh, the SIM TFT theory is a topological theory in the conventional case.
In case of, uh, gravity, is it topological gravity, or that's
Yes.
conventional gravity with a dynamical graviton.
The way I like to think of it, so…
So, this action here, this is the standard gravity.
At least on the classical level. And the same TFT that we are proposing is the first term, yeah.
So, standard gravity is not topical in… They're saying that it doesn't have, um…
And it only becomes topological, in the sense that you don't have propagating.
The limit in which you take Vn to 0. So this is the analog.
of taking E equal to zero with Maxwell.
So I went a little bit fast, but… The standard gravity is this expression here, or at least classically this expression here.
And the sympia extract out of that is, like, in the Maxwell case.
The limit in which this theory becomes topological, in particular topological limit of gravity.
But you might want to call those imprecise, that lambda times unit goes to 0.
You might want to think of that as unit and go to zero, which is imperative.
Unitone has dimensions, but sort of heuristically, it's… Um, so I'd like to think of this is the behavior of gravity close to the… boundary of issues.
You with respect close to the boundary of ideas.
gravity to become non-dynamical. Something to freeze out, and… This BFA usually should be what remains of gravity.
Very, very close to a synthetic power.
Do you expect something like that to happen in the conventional holographic settings?
Uh, yeah, I don't think there's anything… Yes, I would expect that to happen.
For Radius 4, at least. So, of course, the histories are different, because the bug is already topological.
Can we discuss alternatives? But in four and higher dimensions.
Gravity is not too political. There's a propulating graviton.
I would still respect in every dimension that close enough to the boundary at infinity.
give us some sort of BF theory, but we have only been able to argue for it. It's on care in major states.
We have this formation on gravity. But we sort of know what happens.
Um… It's very important, maybe…
What about the derivation of the conformal anomaly of a boundary, of the boundary CFT?
Doesn't that depend on the coupling of the graviton to the.
Team, you know. what you need?
Uh, that's… Yeah, so the… the sea… Yes, on the derivation service.
Addington, Skanderas, and all that.
Yeah, yeah, exactly. So yeah, indeed the. The more careful way of doing what I'm doing, but unfortunately I have some data to run that.
is… will be to do the holographic randomization program.
And in a first-order formation of gravity, it's very important.
They will work in first order for us, because we want to have a BF structure. I think we know how to deal with CFTs otherwise.
And unfortunately, I found very little about holographic normalization to first order.
So, in first order formulation. Maybe a couple of papers.
So yeah, the careful way of proving what I was trying to argue.
So, what I'm doing here is a conjecture, but it… One word to do more carefully, you will have to… It started writing first-order versions of gravity, which are of BF type. I only know how to do that in.
Core dimensions, this kind of… version generalized to D bigger than 4.
But assuming that you have that, then you'll want to do the secondary holographic randomization.
And then try to argue that, uh. The holographic counter terms somehow.
Don't care about this last term.
Let's see, if I'm… this is the SIM TFT, right? So that you're describing.
That's it.
translations of three-dimensional field theories. But… So going back to an earlier part of your talk, what are in the.
You mentioned this. VF theory for B. 2 and C. 2.
Mm-hmm.
And type 2B supergravity. you know. So what are the boundary conditions that? Yeah, that one.
One of the boundary conditions that killed the singletons?
Um… They're typically written, but there's not very… yeah, so the short answer is I know, because it's a connoral system.
We typically say that we take something like BSLEF or B2, and take Neumann for C2.
You can't do both.
Okay. So what I meant is, if you take for this theory, if you take divisile for B2.
When you are taking Neumann for C2. They're not independent.
Well. In the quantizations of this theory that I'm familiar with, you have a singleton mode.
I would say…
and actually the singleton. The singleton mode is important because type 2B string theory is s duality invariant.
and the Langlands dual of SUN is PSUN. And so the singleton mode restores the U1.
That's true, um…
Yes. Um…
So you really have the dual. My picture has always been that the dual type 2B supergravity.
his UN Super Yang mills. and that nicely restores the s duality of type 2B supergravity.
That's true, but I think one other thing you could say is that if you quantize this theory without including a singleton.
And look at the boundary modes to the… The boundary conditions for this B2 with DC2 theory.
The boundary conditions don't have to respect edge duality.
They have to transform covariantly, and there is value.
Well, that's why I'm asking. I'm not sure how you get rid of the singleton mode.
They have to be Norwich. And the way the set of…
Yeah, the short answer is I don't know either. Nobody has understand this carefully, that's what I know.
But I offer a smoke?
This… excuse me.
So the singleton mode appears.
I mean, I understand that if you. If you take the sandwich.
you have B2 and C. 2, which are in differential colomology, then that's equivalent to a ZN gauge theory.
But. I think it's different in holography.
of her…
Yeah, I think… I think the status is in your paper with Belo.
Yes, in that case, if you took a 5-dimensional, as you said yourself, if you take the 5-dimensional Zm gauge theory, then you can explain the different global forms of.
any young mil. So it doesn't have to be an equals force to be young mouse.
any young mills there. You can explain the different global forms of the gauge group by different choices of row boundary, topological theories for the.
the VF theory. take the conversation on ideas. 4 is id 5 is different.
Yeah, you're completely correct.
You're completely right. I don't know the boundary conditions that will kill the singleton in the UN case. I know that if you do the analysis quantization this theory, you reproduce the… which gives me a bit of confidence.
And I also know that in the case that you put orienty force that kill the singleton mode for you.
Then, by this same sort of analysis. you look at the whole set of global problems.
So the whole picture is…
I mean, you know, I could put in the kinetic terms for B. 2 and 8 and C.
Yeah, yeah, correct. Um…
And then and then then it's a much less subtle quantization.
And the only the only one that I understand gives you a singleton mug.
Could I offer… could I offer a small comment?
So I'm not understanding this claim here.
I don't disagree, but… Yeah.
If we do what Greg's suggested at connections for beta NC2,
Then, of course, taking Beto the boundary will not impose the boundary condition
Which… which we want, which will give SUN or PSUN.
In fact, the wave function will be some holomorphic function of a combination of B2 and C2.
But the space of these functions will be complete in the sense that this will be conformal blocks, which
form a basis in the Hilbert space of this topological theory.
Which means that there is a particular way to impose boundary conditions by integrating over
this columnorphic combination of B2 and C2 at the boundary.
It's not just taking B2 to 0, but if you take this holomorphic combination and you integrate over this holomorphic combination with a particular weight,
That should be sufficient because of the completeness. To project on a particular conformal block you want.
So that would give you… yeah.
And that will be a way… and there will be a way to cook a particular SUN or PSUN theory in the bulk.
That's very interesting. I don't think I've seen that. Is that written down somewhere, or something?
Yeah, I'll send the reference.
Please, thank you.
Yeah, um, if I want to make these statements a bit more carefully, they are more completely technically, but I could put an identity fold, then there's no issue with the.
With a single-tone mode. Then everything else is still closed. So you get the set of boundary conditions matching the set of lower for the… So I'm speeds, and so on things.
We do more computing.
Thank you for the communication. You can send me a paper.
Thank you.
Okay, well, thanks for the talk, and Yoki.
Thank you. Thanks for the questions. Thank you. If anyone has more questions.
So, your only motivation for the…
McDowell Mansouri was to get this BF termed cleanly, as of in the first order?
Yeah, because the way I'm thinking of these things is the way I can think of these things is by.
By asking myself what's the limit? or the dynamic of you in the park.
In the limiting which the coupling goes to zero. That's how I tend to think of these questions.
And I only know how to answer this question for gravity.
If it has any sort of magic reform, so B squared.
plus BF.
Right, so my only point was that if you were to write first-order conformal gravity in that form, then the expression you wrote for A as a linear combination of the generators and the conformal algebra,
is kind of, uh, fairly natural, right? Because those things are coming from the…
Oh, yeah, yeah. No, those things are coming from exactly from coupling component gravity to CFT.
Yeah. And in fact, the point is that Euclidean gravity… sorry,
ordinary gravity or other theories of gravity can, in principle, be obtained from, uh…
conformal gravity is more or less the same way as
forms of supergravity, as you know very well, are… can be obtained from conformal supergravity, so by using compensators or something, the equivalent of the compensators here would be auxiliary
fields that you're adding, just like your B field.
Because what's happening in, as I understood, was that for a fixed value of the vector V, F is basically proportional to Hodge star of B,
Where the hot star is determined by a choice of V.
Right? So, effectively, that's kind of the flavor of what you're doing. So, that kind of first-order structure can be replicated by conformant gravity. I'm not sure how useful that is, but it seems like a more natural way to think about
uh… adding extra terms that
already exist. Like, here, adding the Euler curvature or other topological terms seems like something one has to do by hand.
And in conformal gravity, one has a…
Mm-hmm.
More or less systematic way of constructing those curvature invariants.
that's one point. And the other thing that I didn't understand was, uh, it's probably not…
Mm-hmm.
Probably related to Anatoly's question,
So you, in any dimension, in principle, you can write down topological gravity the way Witten did.
And the minimal, uh, version of it is… I mean, the famous one is in two dimensions, which was in the…
Consevich, uh, Witten Koncevic,
story, but in four dimensions, it would just be, like, a metric, a gravitino, and an odd vector field.
Okay. Mm-hmm.
Like, uh, sorry, and even, uh, bosonic vector field.
And they satisfy a simple algebra. Basically, it says that Q on G is the gravitino,
Q on the gravitino is, uh…
del mu phi nu plus del nu phi mu, the symmetrized derivative, and Q on phi is 0. So that's just a simple model for topological gravity, and one can write down an explicit
action, which is fairly simple.
Mm-hmm.
I imagine you want to do that in the first order, so why would that… could that not be a candidate for the BF?
that you have. I'm saying gravity, but I really just mean the conformal, the full group that you want. And I know how to get that out of conformative gravity.
So that's also another… I'm positing this not as an alternative, but more like
to say that the issues you have with the first-order formulation are a little more transparent in that way of thinking about gravity.
Mm-hmm.
I'm not saying that the conformal anomaly can necessarily be solved at all. I'm not making any deep statement. I'm simply saying that
Mm-hmm.
Uh, those formulations of conforming gravity seem somehow more natural to get the topological bulk theory.
I'm curious if there is some connection there.
So I was confused, I don't know much about topological gravity in higher dimensions. What's the relation of that to.
Ah, well, uh, so there is a paper, uh, by Witten on just
Standard gravity.
topological gravity. That's the one where he writes down the model. I think his model, uh…
is inspired by the 2D model, but it's actually supposed to be a four-dimensional model.
And what's the relation to standard gravity? Is this a limit or is it?
Restriction?
Well, uh, that's a good question. So, the relation to standard gravity that I understand is that you can get both out of limit of conformal supergravity, or conformal gravity, let's say.
Mm-hmm.
The written model can be obtained by doing BRST on conformal supergravity.
The super part is not that important, I'm just using super because there are extra fields that I can play with.
Mm-hmm.
The final theory is just essentially a model for topological gravity.
Mm-hmm.
So, that's the relation. I don't know if it's useful, but it's definitely a topological theory.
So, uh, to the extent that
you want to study a theory that's naturally coupled to gravity in the bulk, uh, and must be a bulk.
Mm-hmm.
Topological theory just occurs to me that that's also an interesting, uh…
structure that comes out of…
Basically, what I'm trying to say is that I think the McDowell-Mansuri, as I understood it,
you get, in a particular way, as a first-order formulation, which is very natural, and it replicates 4D gravity, but that's…
that's the only, uh, motivation for you studying it, right?
Right.
Yeah, I wanted to stay with the standard gravity ideas for just written in a way that made the limit more transparent.
Yeah, but because the conformal gravity would…
reproduce the symmetries of ADS4 somewhat more systematically. That's why I was curious why you weren't starting from conformal gravity and…
and writing it in the first order formation.
Well, that's…
This quick push was just my ignorance. I looked into conformal gravity and… I didn't get much out of it, but I'm not very familiar with it either. So, could you get, um…
Yeah, I can… I can probably send you something offline. I'm not sure how helpful it is, but I'm sure you already know those papers by Fredkin and Zeitlin and so, so…
I did. Yeah, I did look at it even. I think I said them at some point, right? But… I was always thinking of those as a prescription for coupling CFTs to gravity in a nice way.
Right.
Yeah, this one here.
Anyway, this is just, uh, this expression that you have on the slide right now somehow seems suggestive that if I were to work with conformal supergravity, this is a very natural
way of getting, uh, expansion like this. Instead of, uh, saying that this is…
a coupling of the boundary CFT to this decomposition. That looks, like, more like a choice that I have to make, but somehow
if I already had a gauge theory with generators of corresponding to these symmetries as having been gauged, which is… which conformal gravity would do for me, then
these couplings would be very natural.
Mm-hmm.
And so with the couplings in the action, and so would the limits of the couplings in the action. It's just that remark. I'm not saying anything more deeper than that.
No, no, that was only how these guys derive these couplings.
The thing always confuses me, I'm happy to have this discussion if you have time.
Sure. Maybe we can stop the recording, otherwise the file will probably get too long. Not that we're…
The thing… Uh, is this… recorded.
Yeah. Yeah. Christina, can we stop the recording, probably?