Yeah, thank you for hanging behind here. or he's correct.
Tell us today about aspects of locality, unitarity and symmetries.
We're going to be taking Ibrahima to dinner at Chef Tanz today. We'll be leaving early so he can make his train back.
So anybody who wants to come, please come see me after the talk.
Thank you for… nice introduction. It's always fun to come to Rutgers and talk to you guys. So this is a paper that came out end of last year, but it's part of.
long time discussion that I've had with Shlomo Revimat, his student is a former postdoc of mine, but this is continuing in various directions. So this talk will be a little bit.
unconventional from the usual way we at least think about symmetries in the modern world today. And the point is, I wanted to sort of.
step back a little bit to try to understand the following.
issue, which is that, as we all learn in quantum mechanics.
very early on. We have Wigner theorem.
which tells us that symmetry operators are either unitary.
or anti-unitry. Now, if you have either an anti.
Now, Unit 3 operators. 4.
and the unitary symmetry operators. and that is it. Okay. So from this point of view, there's a rather interesting question in the modern day when we talk about QFT, we expect to get a full high effusion category to describe symmetries, which is a very exciting topic.
that is near and dear to my heart. But from this, from the point of view of Wigner's theorem.
Um, there is a bit of a clash, which is that when we think about higher categorical symmetries, we typically get symmetry operators that are non that are non-invertible.
have an apparently seeming to be non-unitary and not compatible with this statement of Wigner.
So to to say this more sharply. In quantum mechanics, of course, if I have a system and I have some symmetry, it commutes with a Hamiltonian, and we'll stick to just unitary symmetry operators. It can act, and the system could be in any representation of such a group.
For example. and then I can imagine building a quantum field theory as taking different quantum systems, let's say, on a lattice, for example. And taking a tensor product and taking some continuum limit. Okay?
So from this point of view, quantum field theory is strictly a quantum system in some limit.
Okay. So from that point of view, Wigner's theorem should still apply that the symmetries of my system should be governed by unitary operators. But in the modern day, when we think about generalized symmetries in various contexts, what we see the phenomena of non-invertibility is ubiquitous.
So what? What give? What? What is the missing of the 2?
And I think. The heart of the issue, at least from my analysis, it will come come thin because of locality.
Okay? So let me try to explain a little bit more. What do I mean by this point?
So if I start with any. Let me take some system.
let's say a ladder system, and I have. quantum systems that are defined in some cells of this time.
Okay? Each one of these cells, I have a well defined quantum systems where I expect Wigner's theorem to apply in the usual sense.
However, when I go to the full cell and try to before I take any continuum, I'm expected to take some tensor product of the symmetry.
along the line, along the line. The key question you could ask is that if I want to think about the symmetries acting on the Hilbert space coming from the tensor product.
Whether such asymmetry, such an operator that you would construct that commute with that Hamiltonian, whether they would necessarily decompose into a tensor product of something acting locally.
Okay. So as soon as you extend. to some to a space or to a lattice, locality comes in and there's a natural question, in addition to Wigner's theorem, you have to ask the question of locality.
Okay? And immediately you should run into the conclusion that a generic operator that commutes with the Hamiltonian, a priori have no reason to act locally. Okay?
And I think this is at the heart of the issue of.
Why, when we talk about non-inverted non-invertible symmetry versus unitary symmetry acting.
Implicit to that, there is some choice being made, and the choice that is sort of implicitly being made is you prioritizing locality versus.
unitarity, okay? And this talk is going to be about that. And how do you see it in some sharp way? And what are the things you can actually compute? But of course, we can say all of these words, um… But they become meaningful if there is something we can compute in some reasonable way. So this talk is about how to define some observable that can tell you.
about the… that can measure the distinction of unitarity versus locality.
Okay. Any questions?
So to that point, first, st we should come back to this picture here.
and we should ask, suppose I have some group G.
with elements G alpha. acting in this way. And whenever we do this decomposition, there is an implicit assumption that the Hilbert space of my system decomposes into a tensor product.
of some local Hilbert spaces, HI. Okay? And there are N cells. We'll use this n.
So if you have some group element G, some symmetry that is acting. Is the size of this whole. Yes.
The fight. I will label fight. Okay, so I will.
For the time being, assume a very strict version of locality. There is various loosening we can do, and there are some questions you can ask about gauge theory, the theory of fermions. But let's pause for a moment.
And that has to do with when we can expect this factorization. Okay? But let's assume this very strict version where I have where locality means that I can decompose my physical system into cells.
And the five of the cells, we typically take it to be larger than the coherence length of the system.
Okay. So we configure some group element G. And what we typically want to ask, we have rho.
of G alpha, which is the representation over the whole fell. Okay? And.
In general, this row of G alpha that's acting on the full Hilbert space may or may not respect this factorization.
Okay. So there are 2 things that can happen.
One, you can have u alpha. factoriz, meaning u alpha can be expressed also as a densor product.
of you. alpha i where you alpha i is a representation of the of the group in each cell.
Okay. Or 2. You can have that you alpha does not factor out.
Okay, so these are the 2 possibilities. So you could. So then we'll focus on this one and see, does this buy me anything interesting for this system?
Okay. So this is where what what we want to explore. So in this rather strict version. Let's suppose that each cell is in some representation RI.
of this, then what I would like to compute is going to be.
The trace. of the representation rho of G alpha.
weighted with the phrase. of the identity operator.
the identity of the system. So does this quantity have rather nice properties here.
So I can split this identity to act on every single cell, and this quantity I can just simply rewrite as a product.
over I of rho. I, which I should write as.
trace of. you alpha i.
over trade of the identity element. I will just write. Yeah.
That's fine. Right? So this quantity, of course, you're assuming you're you're in the case. One. Yeah, I'm I'm in case one, which is the case that I'm going to talk about. And this, of course, is simply going to be.
the product of. Of the characters.
in the R representation of G Alpha over the dimension of the representation.
This is your thing.
The iceber space is the representation. Yes. Okay. What this? What I will say will actually not depend on the choice of R in a moment, as you will see.
This is slightly more detail.
Okay. So the question we want to think about here is that now, let me take this quantity.
by I I of Ri. of the group element G over the dimension of the representation. And then what I want to do is I want to know what happens as I take n to infinity here. I can take n to infinity. So this is taking.
some continuum limit where I have a large number of these states coming together.
So if this group acts faithfully, for example, we have the following fact, which is that.
IFRI. for all for the non-trivial element. G. It's strictly less than.
dimension of the representation. So this thing then is going to go to 0.
actually take n to infinity. Great. So there is something interesting that happens, which is that if the representation, if the symmetry operator is locally acting, meaning it factorizes in this way, you immediately observe that the character of the representation.
for the full Hilbert space vanishes. Well, that depends on your value conditions for how G alpha depends on the U alpha i depend as you go. But then that would violate the locality constraint.
Well, I mean, you cook with boundary conditions that they all do the warm.
Yeah, but boundary conditions are. you know, we assume everything falls off at infinity. Let's see. What does it mean?
The UI. Control David.
So if you have to say something about what you're going to assume.
about the U alpha i's as the sites i go out to infinity.
So right. So the the when I say here, I I can still take this to be finite, right? Which is that I'm working. So this is this is not an issue. And when I'm taking n to infinity, I'm just partitioning my my space finer and finer.
Oh, I see. Okay, so that's one way you can think about it.
So there is a… there are 2 limits you can take. There's a large end limit and the large volume limit. In fact, when we do this properly in a field theory, this is a question that we have to address in some suitable way.
But for this example, this is the strictest version of this picture that I could say, and there's already something to learn here.
Okay. So so what? What is the meaning of the fact that this is happening? The way you could? Did you assume representation was irreducible?
Uh, yes. Yeah, but that point itself is not going to be super important. The key thing that's important is whether it's faithful or not. If it contains the identity representation.
Then there's one term that doesn't go to zero.
Sorry, sorry, sorry. So G alpha is non-trivial. So let me get to this point, and I think I will when I write the next thing, it will. It will make you happy.
Okay. Very good. So to you can try to be more precise with this. So this quantity.
This trace here, I'm going to call it C alpha, which is the trace of the representation. I can write it.
in this way as a, I can try to decompose it into.
1 over, let's D. Let me should I do this?
I. For the sake of argument, I will take all representations at each cell to be the same.
So that's not going to have any bearing of the argument. And non-trivial. The only difference is that.
It's just going to change the details of the limit that I want to take. So I'll just take all representation to be the same, but I can try to compute.
decompose directly this quantity, since it's a character of the representation into characters.
of, uh… for each word A here.
is labeling reps. Okay, so I can try to compute this.
Because that's just the club for the decomposition of R to the end. More or less. Yes. So I can compute it explicitly. So I want to do this because it's instructive.
This is going to be. I can compute this from the character products.
the alpha. I mean, look at G Alpha.
I r raised to the nth power times G Alpha and compute this exactly.
So in computing this, I can pull out the identity element, which is what your question was about. But this, when I pull out the identity element, I can write this in this way, dr to the n.
over G times the dimension of the representation that I'm looking at.
buff a thumb. of G Alpha, not the identity.
mindful character. of G alpha times.
character representation raised to the R. of G alpha with the normal evation of dr.
VR and everything raised to inspire. It looks like you're assuming GS finite. Well, you could probably just generalize to compact rooms. Yes, I have answers for compact groups, where you can compute it. So now, if you stare at this formula with this statement here.
you immediately see that this terms. the cave fast. Okay? So what you obtain that this goes to simply dr. Of n.
where the order of the group times the a.
which then tells you this quantity. that I'm actually looking at at the very particular properties that in this large n limit, this quantity is simply 1 over G.
times the sum of. DAIA.
of G Alpha. Okay? So what this seems to say that when I take this.
large end limit, which is a continuum limit, it seems that the Hilder space of my system is just many copies or infinite copies, how many.
copy the factor of dr to the n of the regular representation.
which is a rather interesting fact. But if I, even though I can start with individual quantum systems.
And I think this larger sense with where each system could be in any representation that it wish to be.
In a continuum limit, it seems that if the symmetry is acting locally, meaning it is factorized, it factorizes in this way.
Right? Then it followed that the Hilbert space of the system is going to be some many copies of the regular representation. I don't see a word. Oh, I see. You're saying the Hilbert space factorizes as an overall Hilbert space times one copy of the regular representation.
I just do the overall. That's the character in the regular representation. I don't see many copies of it. Oh, because.
Because there is the… the normalization that I took.
I started off with some normalization here. Right? So the more proper way to say it, the more proper way to say it is that this computation you can say is that in if I take the ratio of some representation, MA.
Ma prime, and I take the large N. right? This goes to the ratio of the dimensions of the representation. I'm still confused. If I had n copies of the regular representation, I would have the thing that you call C of alpha. So the nth power.
It's looking more to me like you have a neutral Hilbert space tensor one copy of the regular representation. No, no, no, no. This. So if so here I have normalized. So so let and and I picked a very particular normalization.
So I can remove this normalization and drop it. I can still write this formula, but this formula is going to be formally infinite.
It would be large, because there would be a factor of D of R that's coming in. So the the correct statement to say is you take a ratio of 2 representations of the multiplicity of 2 representations, and ask, what does that go to at large N?
So in the large end, this goes to the ratio of their dimensions. It's not that you have one copy of the regular representation times something. It is that the whole spectrum that you have. If I ask.
What is the ratio of one representation to the other is always going to be DA over DA prime.
It's an infinite dimensional Hilbert space. But this quantity is meaningful.
which is similar. It intensity of the regular representation.
And the other is the female power of the regular representation.
That's what you just presented looks to me more like the former principle.
Maybe I'm misunderstanding here. Move on. Maybe I'll maybe I would be confused myself.
because I just took the trace. I just computed the character of the representation of the Hilbert space. That's all I've computed right? And I'd say that that character is proportional to the character of the regular representation. Yes, exactly. So the ratio of any to the ratio of the multiplicity between any 2 representation.
is that of the regular application. Totally agree. But that's also true. Take L 2 of R tends for the regular representation. Where L 2 of R is a singlet under G.
Ah, ah, okay, so what you're asking, can I have just a bunch of singlets around? That's different from taking the n-tensor power of the regular representation. That is completely correct. But your statement there, what you're saying is that I have a factor of my Hilbert space where the symmetry doesn't act.
right? But yeah, I agree with that. Yeah, but but but I started with the assumption that the symmetry acts non-trivially in the full thing. I don't have a sector.
But what you're saying is completely sensible. If I have a sector of the Hilbert space where the symmetry doesn't act, then, of course, I cannot make any statement about the representation of that factor. But for the factor that it acts faithfully, this is a true statement. I don't want to derail you, but maybe just one question.
So take the nth tensor power of the regular of this equation. Yes.
What would you say is the character? I would say it's your C alpha to the nth power. That's correct. Oh.
But then, if I… but the but the property of the regular representation, if I take the nth product and I decompose it, it also decomposes to a regular presentation.
It cannot decompose to anything else. Right? So so that statement is already implicit.
Okay. Right. Because the the point is that the regular representation is this sort of interesting central limit theorem for representation theory. If I take the nth power of the regular implementation, that also decomposes into.
copied on the regular implementation. But so so that maybe that's a better answer to your question.
Okay. Good. So the. Oh, we already are half time.
So what does this do? So this tells you that there is an interesting interplay of locality and symmetry, which is that in particular, if a symmetry is acting locally, it has this rather interesting property that the Hilbert space cannot be in some random representation. It has to have this structure.
And this is a property of locality. So this allows you then to try to think about a new object that you can define for your quantum system in quantum field theory, which is this function.
In general, which I'll define as the limit, as a as a regulator, and I'll explain why I need to define it in this way.
time U alpha e to the minus beta H.
beta goes to 0. Over trace.
of e to the minus beta H. So what am I doing? So in general, in the in the case where I started with some finite dimensional system and then take the limit, and so on. Of course, everything is well defined. Nothing is diverging. But if I go to a quantum field theory, I want to study the same sort of a property.
In a quantum field theory, if I just try to take a trace.
of some operator, even even when it is a symmetry operator, which are the nicest operators.
formally, that is not a well-defined thing to do. Okay, so instead I can define a quantity which I regulate in some way. I regulate it using the Hamiltonian since you commutes with the Hamiltonian.
So then you can ask in a given QFT.
If you have a symmetry that's acting locally. Does this function satisfy this interesting regularity observation that we observe by building from some finances?
Okay. And the answer here seems to be correct. Yes. If.
G-Alpha. is locally acting.
Then. Phi Alpha of G alpha of phi alpha here.
is equal to delta alpha identity. Okay, so this is a statement. It's… not exactly a theorem, because I haven't proved it. But this is something you can go and check as many examples as you would like, and see what it what it tells you, and I will discuss that. And in particular.
This is a place where you can now try to explore something that is special about non-invertible symmetry than generalized symmetries, right? Symmetry that fit in some categorical language. It turns out that when the symmetry operator that's acting is categorical is coming from some fusion category.
This statement seems to hold.
This? Yeah. This is G Alpha. If G alpha is locally active.
Any questions? So this, I'm not sure I understand the equation in the box.
This equation. What do you mean by delta alpha? Meaning it is 0 for any for every every element of the group that is not the identity.
They said what we got. Yes. Yes, the so this quantity is actually 0.
and this quantity is a trace is the character of the regular representation.
Right? So so this. So this was done to them. So so here this formula is actually 0. So this is one of the reasons why this is a computation that makes sense. What is the ratio of the multiplicity at large and the ratio of the multiplicity, the ratio.
of the dimensions, therefore you get a regular representation.
Well, plus 0 is delta alpha E. Right. Answering to the question I made the stipulation for the element does not the identity.
Great. Yeah. And the fact that that that quantity is delta alpha E also just follows from the character product. Yeah.
Okay, so this object actually has appeared in the literature in various contexts. It appeared in the context when people were studying asymptotic density matrices of, let's say, black holes.
Right? Where you have some charge in your system, and then you want to know what is the asymptotic distribution of for the density matrix. So this appeared first in a paper of Uri and Harlow, where they explored such a function.
We actually discovered this and then retroactively found out their papers were more or less the same. But in that context, they weren't trying to make a statement about the Hilbert space of system, which is of the QFT as we have, but they were trying to study is what is the asymptotic behavior of the density matrix.
And the fact that the asymptotic behavior of the density matrix was regular in their work is exactly the statement that asymptotically, the ratio of the multiplicities are that of the regular representation.
Okay, so there's a sequence of papers first by Carlo and Aguri, then Pal and Sun also studied this for 2D CFTs. And then this sort of statement or this asymptotic version.
of taking the traits of a symmetry operator. You can also ask what happens if this is a non-invertible symmetry. This was explored by Lynn Okada and Yuji Tajikawa.
Where the same sort of… Statements seem to be buried in their result that when you look at a quantum field theory and you have a symmetry that's acting locally, that somehow the Hilbert space has to organize in this way.
Okay. So I can show you a couple of examples.
But I'm not sure I have… quite a bit of time.
And, you know, some of these examples are actually very good things to work out for students. You can, for example, take Heisenberg spin chain, where you have a spin one half.
That's sitting at each place. So there's an SO3 that's acting. So this is an example with non-abeliant with continuous non-abelian symmetries. There's an SO3 that's acting on each site.
And then you can write the standard Hamiltonian where nearest neighborhood interactions are present. And then you have the diagonal So 3 that's acting everywhere. Okay? So in that case, you can, for example, ask.
In that case, for the Heisenberg spin chain.
Uh, you have, what is the multiplicity?
lucidity. of the eighth representation.
I'll echo it free. So this you can compute exactly, for example, and if you compute this, you find that n.
the multiplicity for the case representation where you have n spins.
It's an exact formula. It's going to be given by.
and factorial. 2k plus 1, which is important.
over n over 2. of K plus one.
And over to minus K. Authority. So when you fix N.
Sorry, when you fix K. and K prime like 2 2 representation. And then you take nth.
to infinity. You can explicitly check that. And K over.
and K prime is going to go to. 2k plus one over 2k prime plus one.
Okay, so you can explicitly check in this case and there that you do recover the ratio.
that would be present in the regular representation. You get this regularity property again. But this is the case where you have an SO3 symmetry that is acting on the Heisenberg spin chain, but it has a local action at each site.
Great. So it from that fact, you can. You can that allows you to compute this, and then you can see that the ratio tends to what you want it to do.
Okay, you can also compute it for free scalar with a U1 action and a free fermion. Those are examples where you can do it explicitly by hand. And the statement seems to hold.
to also highlight here, we've defined the trace using the Hamiltonian in simple symmetric theory. You can actually also do this right in super symmetric series. Let's say we take n equal to 2 degrees. So right here, I have to at some point say something about inequality theory.
as being the limit. of Q equals one, where Q is a standard Q that appears in indices. And, in fact, what you will obtain, you can regulate this.
In this way. minus 1 to the f times the element times Q.
to the delta minus R, where R is the SU 2 charge.
Then brace. minus one for the F.
So this quantity here for if you know, this is the ratio of the Shure index right of the refined sure index with the sure index itself. So this.
In this case, these quantities are actually computable for n equal to 2 theories explicitly, and in particular you can computer theory. You computer for class s theories, and so on. And you can take, for example, the SUR symmetry.
Right? So in the SU2R symmetry, you can compute.
You can take. So this quantity here is what we call the is a sure index, which has some function.
depends on Q. and V, which you can compute. So this index.
I, Q. be the refined index that you compute in this way you can decompose it into characters of the.
You can put this into the characters of the Z.
Yeah. Uh, you can dig more into the characters, where this is actually 2 characters.
with representation and and n over 2. Okay, you can compute this explicitly. And you can show that these quantities, which is the multiplicity that appears, turns out, indeed.
Cn of… uh, Q.
over Cm. Of Q. You take the limit as Q.
goes to one. This exactly goes to N plus 1 over M plus one.
And you can check this in all integral to theories where you where you know this thing is computed, and you see that this works out. You can also equally do something.
nice, which is that instead of decomposes into the S2 characters, let me decompose it into the carton U1 characters. In that case, the multiplicity that you would get, they go to… Everybody should know this answer.
one right because the multiplicity of a representation. Yes, yeah.
right? So you can just explicitly check that goes to one.
Which… which is actually quite… I mean, so it's sort of surprising that this feature is just there.
And, um… It hasn't been noted, at least in this way.
Okay, so… So there you can also explicitly check this when you have. So since I don't have time, I will not go through it. It is a very instructive exercise to consider rational Cft, wherein rational Cfts you have symmetries that are acting.
But the symmetries are given by Verlin. So those symmetries are characterized by some fusion category. And then you can check exclusively when you have, you can then take one of those Berlinde lines and taking the trace correspond to taking it on the spatial circle.
Okay? And then compute that that trace in that way, and ask what happens at large N. Well, sorry, and ask what happens in those case. In those cases. It's already a field theory. It's a 2d field theory. Everything works.
You can then argue from there that the representation of the Verlende lines for the Rcfd is going to be the regular module.
taking the traits of the action of the frontal line with e to the minus theta H. And then what you use in that case. So there in that case, this highlights. Then you want to take beta to 0, but then to take beta to 0, you use modul invariance.
Right? You get the quantum dimension. You get the Froggenius variant quantum dimension. Yes. You get the ratio of the Frobenius pair on quantum dimension.
And that argument could be generalized in a more physical way. But so if you're curious, I'll be happy to walk through that computation for you. But the way you can generalize that argument, at least for 2DCFT, is you want to compute.
the partition function. Uh, as a function of beta of, let's say, trace in the Hilbert space.
of G E to the minus beta H. Notice that when I say this, G could be any operator, invertible or non-invertible in 2d. You can use modular invariance, where this thing.
So this Z of beta. is going to map by using a cardi formula e to the minus one over beta times delta G. Okay? So what is delta G here? So delta G.
So first, st when when I do modular transformation.
here. So let me write this as this. And this is a modular.
transformation. When I do a modular transformation, taking here beta.
0 correspond to taking beta twiddle to infinity. Yeah.
Okay? So now I'm computing the partition function where I'm taking the infinite temperature limit. Sorry, the 0 temperature limit. So that should pick out the ground state.
Okay, so when I have G inserted right? Then doing this modular transformation takes the g, which was along the spatial circle. And then the modular transformation puts it along the temporal circle.
So then that's a deformation of the Hamiltonian. that I explicitly have. And then what you're computing is that the lowest.
state in the twisted sector of the Hamiltonian. Okay, so this delta G is exactly the energy that you get there.
And then you can take now the ratio of VG of beta over z of the identity of beta, because that's the normalization that you would get. So this quantity is going to go ask.
some number times e to the minus theta, uh… Delta G minus delta E, where this is the ground state in the untwisted sector.
an unfiltered factor. I know.
Mr. The zip quantity listed 0. Beta twiddle. Yes. So this goes to 0.
Okay? So this goes to 0 when I take beta twiddle to 0.
Sorry. You're right. Okay? And if.
G is ironically, this is 1. I mean, he was giving the facilities unity because it could be Yamily where they can probably get the theory I'm assuming unit.
Good. Thank you. Okay, so this is a a rather nice observation. This is the thing that tells you that this is a feature.
actually of the the 2D theories will have this feature, and we expect this to extend to higher dimensions, where you can check it in many in many, many, many settings.
So if you're if you want, I can show you many other examples that we do. But this also highlights something else that is interesting.
Which is related to. Uh, what I mean, this argument seems rather special to the two-dimensional case.
Yes. So let me say more about this. It's also pretty close to the original argument for calling the quantum dimension. Yes.
But where you take the trace, you know, you define the dimension of the ith representation.
It's the limit as Q goes to one of the character of the nth representation divided by the trivial representation, and then you get si0 over 00. Yes. So the lesson that I'm getting to here, the fact that that's true is inherent to what we… to when we talk about categorical symmetries.
Okay, so there there. One thing here that that is that is true. So let's go back to the initial setup where I have some quantum system, and then I have some symmetry that I say acts on it. Okay, there are 2 things that could happen in the standard quantum field theory. When we have a categorical symmetry.
We demand that, you know, whenever I have a symmetry operator acting.
In any system where I decompose into a time, direction, and space, where I define a Hilbert space, that there exists a weak rotation where such a symmetry operator becomes a defect. So this is a.
feature that is necessary to define categorical symmetries. Okay? And in fact, this is precisely the property that when I have a categorical symmetry, I talk about in a Euclidean.
Field theory, where I'm allowed to insert this in any way I like, right? And the fact that you have this exact duality or this exact equivalence between the.
symmetry operators that are acting at a Hilbert space at some fixed time, with the defect is because you can do this weak rotation, and this is guaranteed to you by categorical symmetries. And this is exactly the property that's used here.
Okay, this is exactly the property of this use here. So this is the thing that it highlights. So now, suppose I did not know that I have this full Lorentz symmetry. I didn't have all of the beauty of fusion categories and so on.
And I just say that, OK, I have some t equal to 0, and at t equal to 0, I have some Hilbert space.
and on this Hilbert page I have a set of operator view that can meet with the Hamiltonian.
and all of these operators form a symmetry. Okay?
So in general, from this point of view, of course, whenever I have something that commutes a Hamiltonian, I can sorry the u of group representation, or just more general. So let me come to this point, right? So if I have any. So what we're instructed to do in standard quantum mechanics.
If I have any operator that commutes with a Hamiltonian, I can make it Hermitian if I want.
And then I can exponentiate it. Okay, so that's what we're supposed to do from a Wigner point of view. And then when you exponentiate, you get a unitary operator, which makes sense.
to talk about. But the difference now is that when such a unitary operator acts locally, then this regularity condition is satisfied.
And such unitary operator does appear as a topological operator.
in the standard sense. But you can also have a unitary operator that is exponentiated that becomes highly non-local.
Those guys you cannot rotate. Okay? So this feature of being able to rotate operators from the from from fixed time to fixed space is in a sense, the magic that where locality is somehow entering in a way that I haven't made exactly sharp.
But we see it here play out explicitly in this calculation. So do you have an example of a non-locally acting synergy? Yes.
Yes. So the simplest I can. So I have many examples in many different directions that you might that that you might be interested.
So let me show a simple example of this.
Okay. And then I will try to tell you that this quantity is actually quite interesting, and that it encodes a lot more information.
Then it seems at face value.
So the simplest example that I can try to tell you is, let's consider the Ising fusion category. Right? The Ising fusion category is composed of 3 elements. You have the identity. You have a Z. 2 invertible line, and then you have D is not invertible.
Okay? And then we have the. Fusion rule that E squared is equal to E plus eta. So this is the Eisen fusion category.
You can, of course, by the discussion that we just have.
You can take explicit model. Take the term of transfer field Ising model at criticality. Okay, you. So you have an explicit model there, and you can explicitly check that you can compute trace of eta.
which is trace of D. Which is 0. Okay, so these are locally acting symmetries in the sense which we want our symmetries. So here, what I mean by locally acting for fusion, what I mean here, it has to mean that locally acting means that they satisfy.
The local property that we expect topological lines, topological field theory to do. This ultra locality that nothing interesting happens when they touch, and this locality property that is defined by the Etia Hitchen.
definition of a topological field theory. Thank you. All right. What? What is this fusion of category acting on?
What is this, Chase? What do you mean by the trace? So I can do it. So one example I can take the transverse field Eisen model.
at criticality, right? So I can have an explicit realization of this, and I can take a continuum if you like.
Okay, there's also two DCFTs where the IV infusion acts. So you can check in those cases.
So here is an example of a symmetry that is unitary, but it's going to be non-local.
Okay, so from this thing, I will do something that is illegal from the point of view of a fusion category, but it is completely legal from the point of view of decomposing my physics into one plus one spacetime. So if I treat these as operators that act.
On my Hilbert space, then I can define. new symmetry operators eta plus minus, which is 1 over root 2 times D.
plus minus E minus Ada over 2. So here, look, I'm taking. I'm defining a new operator which involves.
Plus and minuses and irrational numbers. This sort of a thing is actually illegal from the point of view of fusion category, because there is no sensible way by which I should add and subtract or multiply by irrational numbers of.
uh, when I'm talking about defects, because defects define Hilbert spaces. Okay?
But from the point of view of thinking about operators that are acting on a Hilbert space at a fixed time, this is a completely allowed operation.
And indeed, if I do this, then I have, instead of talking about this non-invertible nonsense. Some people would say non-invertible nonsense. I like non-invertible stuff. Then I would say my system instead admits.
V. 5 2 3.
Okay, so the first one is AIA, and the second one I can take you to the Eta plus eta minus, you can check if this is diagonal of T.
But the fact that D was not invertible gives you a constraint. It tells you that.
If I look at any Hilberg case where this is acting following projection has to be satisfied.
Okay? That if I multiply this projected times that projector, it must be 0.
acting on any acting on the Hilbert's place. And either plus or be unitary? Yes, by construction. Sorry. Right. So I should say this by construction, eta plus minus.
8 or plus minus dagger. Okay? So you do get a Z 2 times Z 2 symmetry with a constraint.
So what does this constraint tell you right? So Ada that's acting here. I can think about the Hilbert space, and you can check this in several examples. You can check this explicitly. But the Hilbert space.
For example, for Ada. All right, let me write this way.
If I have the state Ada, I will get plus and minus states.
Okay? So and this would be 50%.
You're here. And another 50% of the states would be here. So this would be the fact that Ada has this regular representation and it's a Z2 is an abelian, so the dimensions are all 1. So half of them are in the plus representation.
The other half are in the minus. Okay, the minus is the… so the plus is actually that entity.
But if you then look at this constraint, you can see that if I am in the plus representation, then I solve this constraint. Then the plus representation is going to split under Ada plus into 8 plus.
And I might half half. Okay? But then, if I am in the minus representation, then I have a constraint here which tells me that everything is in the plus.
Okay, so you can see exclusively. Here you have a 3 to 1 ratio in terms of the representations that are appearing right? So you can explicitly see that, um.
Now, so this operator makes complete sense on the lattice. You can compute it. You can construct it in the transverse realizing model. You can play with it. You can do all sorts of things that you want. But then you clearly see that when you write it, that the representation of the Hilbert space.
It's certainly not balanced in this way. Go ahead. Within the statement that the delta operator is not exactly defined for transverse realizing because it makes it translation. Good. So here I'm implicitly taking the continuum.
But you can define the appropriate operator. So. I can show you.
Yes. Okay. Question. Could this all be done in the CFT instead of the transverse realizing? Yes.
I haven't done the explicit computation in the CFT, but it's doable.
Great. So this is an explicit example where you can see this regularity playing out right? And and and so these.
Hey, the plus and minus don't act on the local operator.
Uh, good. They act on a local operator algebra, but they map it to a non-local operator. That's what I meant. Yeah, they mix local operators to non-local ones. So in fact, if you… in our paper, we completely computed the eigen operator of eta plus minus, and you can see that they're non-local in general.
They have to have some Ada lines attached to them.
Okay. Okay, good. So what is this going to be good for?
Let's try to see if this is good for something.
And I will be very fast in schematics.
Okay, so now you can define the following quantity. So you can take. You can define the following quantity.
B of G. which is equal to the norm.
of this trace. Okay? For any operator G. So why do I want to do this? I want to do this for the following reasons. Suppose.
Suppose you have some quantum field theory. some quantum system.
and I give you the Hamiltonian of the system, and then you hand it to the student, and then you say, go and find all the operators that commute with the Hamiltonian that are well defined.
Okay, that's a that's a that's a reasonable thing to do.
So in general, if you compute all session operators from this Hamiltonian point of view, you're supposed to construct unitary operators. How would you know that you have a fusion category? Is there a way to get… to go from computing all to the operators to figuring out that you have a fusion category or knowing which one acts locally, which one doesn't act locally?
But but the set of all operators is the set of all projections on the so in the talk, there is the part where I where I wanted to discuss about projectors, but I cut it out because of time. Good. So projectors inject. I'm glad you asked this question. So projectors, if I just take a random projector.
It's not generally well defined. If I… let's say I have some lattice system and I diagonalize the Hamiltonian, and I look at the projector on the nth state.
So that object actually is not. well defined. Certainly, when I tried to go to Continuum. So let's say then I go from N lattice to n plus 1.
Right? The change of that projector. Sure. It's by no means smooth energy interval is well defined. If yes, good. So if you smear, then you might make it well defined. But such a projector are highly non-local thing. Absolutely. They're highly non-local things. So there's still a question of what are the right adjectives from this this setup.
That I just give you. But I want to study this, and then I'll tell you some nice properties it has. Okay, before we get there.
So suppose I tell you that I have a bunch of operators, they're all well-defined in some way.
What you can do is you can take all of those operators. They're unitary, and let's consider the most general unitary operator that you can get. I'll call it G of some angles.
Beta I. Right? You can construct such a thing.
where this theta i is going to be a manifold, which basically maps out the directions corresponding to these different operators that you construct.
It turns out this quantity B of G. have very interesting properties. The first, this quantity is bounded between 1 and 0, okay?
at 0 is going to be all. uh, what we would call simple.
operators. that are unitary.
So if you have an operator where B of G is 0, then that's an operator. Basically, that is unitary, and that's also locally acting.
from the discussion that we just had. If it is 1, then this would be the identity.
Great. But then, in general, this this bead can take values along this whole thing.
So observe that this is a maximum, and this is a minimum.
Okay? And then you can ask, are there anything interesting in the intermediate regions?
Great. So. So you could. You can try to ask, are there anything interesting in intermediate reasons?
What you observe is that. Let's say I I started off with this Ising fusion category. So she uses the fusion as an object of the fusion together.
Yes, I can take. So so what I'm going to say now is going to be taking explicit examples of fusion categories, and then I can say my statement now.
So suppose I consider some fusion category C, right? You can think of, for example, the Isling.
Ada and B. Okay, it is clear that. So if I so I can construct this B right? It's going to be some unitary operators constructed from these 3 lines. So this would be. So B here would be something like a 1 times identity.
Plus, let's say a 2 times Ada plus a 3 times D.
Okay? And then I pick A1, A2, and A3, such that this is unitary. And in fact, the simplest way you can construct this — sorry, G.
In that you can show that G, you can write it as E to the i eta times e to the times theta times some phi times D. You can construct this operator where this defines for you.
a manifold of two variables. Okay. Do you expect to be able to do this just for Tambara Yamagami, or just I can do this for any any symmetry category. Yeah.
So here there is no I. There is a good reason for that, but you can take my word for now that this is going to be… so this is a function of two variables, and this defines a unitary operator. So I can compute.
Bog. of Theta 5.
Of course, when you compute B of theta phi, I mean, I have an explicit form of the answer, but it's not so interesting to write it. You can show that it has an extremum, which is 1, and at the extremum, when you evaluate it, exactly gives you the identity operator.
It had the minimum which is at 0, and at the minimum at 0 it gives you the Ada operator. Then you can ask, can it actually know about the non-invertible symmetry?
Then you find that B. of G have.
a saddle point.
Right? There's some value where B of G has a saddle point. And and and you can ask, what is happening at that saddle point at that saddle point. You find that the operator that it evaluates to it B. So G here.
is going to be given by some. Uh, let me call this eta c. And what eta c corresponds to.
Eta is. one minus some projector, which I will be more specific, where this projector is going to be equal to E.
plus. Ada plus.
Wait, sorry. Root 2 times D. Over the thumb.
of the dimensions court. as a function of the theta and phi. Yes.
They're the saddle point which exists, and at the faddle point B evaluates to this G. And that evaluates to this eta, where the.
So you need to essentially critical. Yeah, the critical point.
It had the critical point in it where the storms. Oh yeah, the B is right. And the symmetry there. There is from Z 2, and the Z 2 is 1 minus times 2, this project in this projector is given here. This projector knows about D.
And in particular, when you compute. B of G explicitly.
You get that this is 1 minus 2 over the sum.
over the dimensions of all the lines squared.
Sorry. Yeah, this. So you can show explicitly that B at the saddle point is this.
To B knows about the dimensions of the noninvertible.
symmetries. So you're sending over the simple objects of the fusion category, the quantum dimensions of the simple one. What is that? Yeah, the quantum dimension of simple objects. So you can. You can show that at the saddle point.
or at this extreme momentum. And a fee B is 1 minus 2 over this. And you can prove this exactly.
And in general, what you find. Uh, is that… If I take a random fusion category.
And I thought of just promote all the all the lines of being some operators that are acting. What you can show is that whenever you have an invertible symmetry operator, right? Sorry, you can take all of the symmetry. You can construct the most general.
Unitary object you can do. So this gives you some non-trivial manifold. And then this manifold has extremas.
So at the extremas where it's 0, right? So every extrema where it's 0 corresponds to a unitary.
locally acting surgery. So this would be the invertible symmetry operators, and you can see this explicitly.
And then what you could. But then also B will have some saddle points.
at other places. So at at the location of the saddle points you can show that for every saddle point.
You have some projector. which I'll call C prime.
which is going to be a gauging projector. And this is this we can make sure.
D of GA. A over.
Ga prime. D. Prime.
squared, where c prime is a subcategory in the general infusion category. And then whatever you can define this projector super a sub.
Sorry. sub into the focal fission where you have one plus 2.
Eoc minus. And then what you can show explicitly is that B of eta c prime is a saddle point.
with the solution explicitly being one minus 2 over some of D.
of GA. squared, where the sum is taken.
here. Right.
So the statement is then the saddle points correspond to symmetries.
different projectors, gauging projectors that you can construct. This is the gauging projector.
Great. And what? And and what we have been observed so far is that the number of saddle points plus the number of of of invertible things that you can make all sum up to actually be the number of operators that exist.
in diffusion category. So you're saying this PC prime is always a projector. It always first? Yep, you can prove this.
I have to know something about the future. That's right. So you can use the so this actually uses the facts of the fusion algebra that DA times DA satisfies.
Yeah, yeah. So then you can prove that. So this is a projector, and it's a gauging projector. This is P just correspond to different gauging projectors that you can construct.
Ah, because this project you into. the the states that are going to be.
uh uh. Sorry, either projects you into it projects you to the subspace, which is going to have projector. I get that it's a projector.
So then what? So because it's a projector, the A to C prime squares to one. Yes.
Okay. and it's unitary by construction.
But it's non-local. And you can show B of eta C is always going to look like this.
And moreover, the value where eta c is sitting is a saddle point. It's a special point.
If not any random place on this manifold. What do you assume? Good. So the thing that the only thing that I assume for the fusion category is the fusion algebra.
Was this special to 2 dimensions? So this was general so far the reliable way we can talk about all of these things, because we talk about 0 form symmetries, which where I can talk about them as operators, and I can make sense of it in this way.
or for two dimensions. But the question is, how does this generalize to higher D, and how to show it? And we expect that it should. But we don't have a paper yet worked out.
But anyways, I'm well over time, but the point that I want to highlight is that this quantity B that we proposed seemed to have a lot more information about the system that it had the right to do.
Okay, so first we started out as being a thing that can tell you whether symmetry operator is local or non-local. But then, when you when you evaluate it for actual fusion categories that you see that there is some version of a maximization that's emerging.
right? Which is the thing that we would like to try to make sharp in some way. The critical points of your B function for the set of unitary output.
unitary teeth, line defects. Yes. as these critical as critical points that.
What? Generate a subfusion category. They have critical points that get associated to its z2.
non-locally acting symmetries, which have their origin. from some from the projectors that you could construct.
And the number of those, it seems in examples that we checked, the number of those is always enough that you actually recover all the different lines.
all different. All the different simple operators, simple simple lines, invertible or not invertible.
Because once you… So you're saying the C prime like I'm confused. You're saying that you are.
But what I'm saying is that there's a. a small C prime of exhibit.
Yeah. You write AC program, PC fund. That's all C program.
He wrote C prime is a subset of C subcategory of C. Yes, you probably made a capital C prime. Yes, that should be a capital C. Okay, so now you're saying that the critical points of your functions.
select correspond to objects and generate some subcategories. Surely the thing.
The capital C frame is generated by this swallowsing frame.
Yes. Okay. Yes. So the the the statement is that.
If I take a fusion category and I take the most general unitary operator that I can construct, I get this function B. Okay?
then I can… then the statement is, how many saddles do you have for this? Right? You can count all the different saddles. So how many minima did it have to give you the number of simple operators.
Simple operator that are unitary, and they form a subgroup.
And then you ask, what are the different critical points that you have inside? If you have n critical points, the claim is that for each one of those values, the critical point, there is a projector associated with this.
and for and and then for every non-invertible symmetry, you can construct the projector.
The YFC primer subcategory. I think maybe I just abused language a bit. I just this is a selection of operators such that they define for you a projection. They generate the whole category.
If not, it's not a single C prime. Not a single. But if I know all of them, yeah, that's exactly why I'm complaining about the small C prime and the big C prime.
The small C prime is laid the critical points of this non-negative function. Yes. Okay. Now, then you can use select subset of objects. But asking, does the set of objects generate the entire fusion category? No.
It doesn't have to generate the entire fusion category. Yeah. Does it generate a future? Yeah, to generate a subfusion category within the system. Yeah, a subfusion category. Yes, which is which is how how close is. Yeah, that's right. That's right.
I mean, it could be the whole thing. But then there are things in the fusion category C, which corresponds to your.
your symmetries, your symmetry defect lines. that are not critical.
for the beef option? No, good. So the statement is that the critical points have all the information that you need. Why don't they generate the entire category? There are different critical points. No, but if I put them together. Yes, yes, that's right. That's a statement.
If I go and find three critical points for each critical point, I can associate it to it the projector. Yes. And then NYT linear combination. I take the category generated by those.
Yes. Does that is that the full category? Yes. That's a claim.
Do the projectors canoe. Do the projected commute doesn't that's a good question. In general, they should not. So it's not a decomposition. No, it is not a decomposition. It's a set of subcategories which together generate. Yes.
the full thing. Yes. So far, it's all an observation that, like, I haven't, like, what you can do is that you can ask and say, okay, B has a critical point. What's happening at the critical points? First, it was surprising that the critical points, like, when you evaluate the example of the critical points could only be written in this way.
the value. And then you can then ask, what is the symmetry? The symmetry looks like that associated to some projector. Right? So we've discovered this other way around. We haven't proved it the fully right? So in order for this to be a sort of like litmus test.
for obtaining fusion categories of a system that I don't know, you have to prove the other way.
So you've just done this by examples. Yes. And then you can see that whenever you have a fusion category that acting, this thing is true.
So I don't know yet what this function is and what is it doing? But it's very curious. But it sort of comes out. It's it's sort of surprising that trying to understand.
this sort of weird tension of locality of unitary implies that something like this exists.
This is why I said the talk was weird, and it's incomplete, but it's work in progress.
I think I'll stop here.
You said it's work in progress. Do you have something on the archive?
Yeah, there's one paper on the archive. Yeah, there's a paper on the archive. Currently, one of the things we're trying to do is see if we can actually use this as a machine to figure out fusion categories, where you can basically… you have a bunch of lattice models.
That you can study where you don't know what the fission category is, but people have to MPO constructions of the operator that commute the Hamiltonian, so you can actually go there and try to study whether, you know, so there are examples where a priori, you don't know what diffusion categories, can you figure it out?
Can you figure it out. And then we work. And that's something we're trying to do with some students. Yeah, that's in two dimensions.
So you've used this exponential formula to make unitary operators. In principle, there are other ways to put things together to make unitaries.
Would you get the same results if you did something like that? I mean, if just a coordinate transformation.
That's what I'm really asking. That's right, that's right. It is a coordinate transformation. Yeah. Oh, another way to think about it, you have to use the operator, they have some algebra among them, and what I'm really doing is computing a Lie algebra.
And then I have this function that's defined on that sorry, computing a Lie group. And I have this function that's defined on that Lie group, and I'm just parameterized. It gives coordinate from that point of view. So however you construct this B.
Then off you go. But but but but what some of the sort of subtleties that we are still trying to.
understand it, what are actually the… so if I'm starting from scratch, I don't know anything. What are the rules? Do projectors count in this game?
The answer should be no right? Because projectors themselves are not going to be things that are well defined when I take continuum. So so another question that we don't have complete answers of is that if I give you a random quantum system and ask you to go find all the symmetry that commutes.
With the Hamiltonian, what are the rules of what applies? If I take any function of the Hamiltonian and commit to the Hamiltonian and I have infinite version of them. So basically, there's… Questions like that we're trying to sort out in in some detail. Yeah.
What about integrative Hamiltonians? Correct.
You can have fusion categories acting on that. But the fact that they're integrable just means that you can solve it, but it's actually nice, right? Yeah.
Yeah, so good. So that's the case where you have continuous symmetries, completely continuous symmetries. Even fusion categories for continuous symmetry is not a settled point.
even before you get there. But you have to be careful.
Well, let's thank you very much. Well, thank you so much.
It was a kooky talk, and I hope you enjoy. It was really, really interesting. Thank you.