Reflection Operators in Integrable CFT
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Transcript: English(auto-generated)
00:15
So I met Samsung more than 30 years ago, first time in Leningrad.
00:20
35. 35, right, in Leningrad now since Peterborough, right? And we have a close contact in Rutgers, 91, 92, right? A lot of discussions, a lot of, well, I learn a lot from Samsung. It's actually many problems we discussed at the time.
00:42
I'm still thinking about the problems, and fortunately some of them were solved. But yeah, that was a great time. And it's my pleasure to participate in this conference in honor of Samsung. And I would like to thank the organizer
01:01
for this great opportunity. OK, I'm going to talk today about the recent work with my former great student, Gleb Kitausov. And I'm going to discuss the certain aspect
01:20
of integrable quantum field theory. And this sound, the integrable conformal field theory, conformal field theory, actually it's a little bit strange. Because in general, conformal field theory doesn't give many integrable structures. Conformal structure may be some extent in conformal symmetries, but I will
01:42
focus on some additional input, which is sometimes possible to introduce for the conformal field theory. So here is the outline of my talk. First, I will discuss most of the time of this kind of introduction, discuss
02:01
the integrable structures. First, what does it mean, at least what's meaning of this. And then we, among the problem which occurs in studies of integrable structures, problem with the organization of certain set of commutant operators, usually integrability,
02:22
we have an infinite commutant set of operators. And the problem is how to diagonalize it, how to construct the common spectrum. Then the central subject of my talk is reflection operators.
02:41
And that can be introduced in many conformal field theory. But we will focus on some particular simplest possible, simplest situations of so-called Kedivy integrable structures for conformal field theory. Well, in some application, maybe a few words. All right. So let me start with a rather general introduction.
03:05
So we are considering that we are doing the two-dimensional integrable quantum field theory at this point. So integrability means the existence of the infinite number of continuity equations
03:22
of these forms, the t. It's a tensor of densities. And since we are in two dimensions, that makes sense to introduce the Lorentz invariance and introduce the like-con variables in the Euclidean versions, the Galomorhic-Pintingalomorhic coordinates. And then these equations, continuity equations, become somewhat simpler.
03:44
And so this is local equations. But assuming the certain boundary conditions imposing the system, it's simply set up. It's just a system with the space coordinate occupied on the circle radius r.
04:03
So our world-sheet geometry is just a cone, just a cylinder. Then having these continuity equations, assuming the density of periodic functions along the space directions.
04:20
Then clearly, such a combination, such integrals taken over the time slice would be integral of motion. They're conserved. You can freely move the integration control along the cylinder, right? And it will not change the value of such an integral. So this is integral of motions. But this is not enough for integrability
04:43
because having just the classical Liouville integrability, it makes sense to require that this operator mutually commute, right? Mutually commute. And then all this picture works for conformal field theory.
05:03
But now the idea is, let's assume that the ultraviolet behavior of this conformal field theory controlled by the certain conformal field theory. So originally, we have a theory with a correlation lens, let's say, with the mass. And then consider the limit.
05:21
So the only dimensional parameters here in the problem is the size of the cylinder measured in the unit of correlation lens. So we assume that this dimension parameter goes to 0. Then the equations, original equations, this one, the conservation equations, it turns out
05:41
to the condition that the field city and this density became galomorphic in this limit because the left-hand side actually vanished in this limit. So we end up with a set of Cauchy-Riemann equations
06:04
for the densities. And for the densities to the infinite set of densities, S levels the S plus 1, it's Lorentz spin of the density. All right, and this way, this way,
06:23
these equations are rather actually shows that the infinite number of integral of motions, if, for example, we expand from these conditions, assuming that it is periodic functions, we can expand in the Fourier modes.
06:41
And these conditions mean that all expansion coefficient actually integral of motion. So a lot of integral of motion we can build this way. But they're not commute. Nevertheless, we can construct the commuting family by taking the limit, again, taking the limit of the integral of motion from conformal field theory to quantum field theory.
07:01
And to consider the limit, we get something which would be not only conserved, but also commuting. All right, and in these situations, so the problems in this way, it becomes rather mathematically well-defined problem. The problem of finding the common spectrum
07:23
of these operators, I call it local integral of motions, local because we have integrals over the local densities. And this operator, since the conformal field theory can be classified, the space of states
07:41
of conformal field theory classifieds in terms of inducible representations of the conformal algebra, the raster algebra, maybe other algebra of extended symmetry, then this operator can be thought of as an operator acting in the presentations of such an algebra. And the problem of diagonalization
08:02
such an operator becomes well-defined. A mathematical problem can be formulated just in the problem in representation theory for such an algebra. Sorry, how did you define iS there? iS? The next one. This one? No, no, no, it's anti-holomorphic in the Z
08:21
problem, and it is holomorphic in Z. So how do you? Yeah, we actually, our starting point was this integral of motions. And this limit of this integral of motions, you have in the limit when the correlation goes to zero, it became the conformal integral.
08:42
So this is a formula, just what this part actually in the integral disappeared. It's proportional to the, it's in the scale, in the massless limit, and only we end up on the integral like this. So why did it do it?
09:03
You're asking me something? No, because I didn't understand this. This one? This equation, how does it give the second integral of motions? You see, since we have a, we assume the p invariance, we can always replace z by z bar.
09:22
So clearly, if you cover this series with this integral of motion, that you can always flip the directions, the space directions, which is the place the spin minus spin, the Lorentz spin. So that, OK, OK, so that is, so, and let me just,
09:45
my basic example would be the Cinch Gordon model. So this is classical, this model, the actions given by this famous formula. And this classical integrable systems, it's a massive theory,
10:05
the correlation length related to the parameter mu in front of the cost. And well, in classical, there's a set of local integral of motions of the form we discussed, just where local densities, even local densities, even spins.
10:23
They are commutes, and this formula, all this formula has a classical analog and then quantize, and we quantize the simplest, so the sum of the first non-trivial integral of motion, e1 and e1 bar, gives the Hamiltonian and the differences at the moment.
10:43
All right, so that is an example. And now, the questions we, so this theory is massive, but the questions, we are interested in the conformal limit, and this is a little bit tricky, because we, well, mu seems to be small, right?
11:02
And the limit, one correlation length goes to 0. Then it's mu goes to infinity, that mu becomes small. In the first glance, we can neglect these terms, and if you have a free theory, but it's not quite true, because this potential is unbounded.
11:21
Is there anything, kind of, the hand, or two, or R, and then there's a hand? Can you move the hand? Unfortunately, but you move mouse. Ah, OK. Yes, yes, yes, yes. Oh, yeah, it's hidden, right, yeah. The nothing, this is not secret, you know, any secret.
11:40
It's hidden by this. All right, but that's not quite true, because in the configuration space of the theory, so when there is a domain, right, or the field phi becomes large, and it doesn't matter what small you choose mu, this term may be important, right? So in particular, it means that we should actually,
12:03
the picture looks like this. It's like a quantum mechanical picture, so the quantum field theory effect here, it's not quite important for understanding what's going on. So potential is always, if the potential, if in the conformal limit, the potential is almost zero
12:21
everywhere, and within the certain segment, which depends on this ratio, the dimensions parameter. But then, because the exponential barrier, we cannot neglect these terms, right? And we have a two symmetric exponential barrier, but in domain, in configuration space where phi is large,
12:42
we cannot neglect the potential term, but we can replace it by the exponent, and the result, we get the Liouville actions. All right, so in this sense, the ultraviolet limit of the Cinz-Gordan model is described
13:02
in terms of the Liouville theory. In the Liouville theory, it's indeed this conformal field theory. It possesses the energy momentum tensors. It's Galeromorphic, all right, if it has this famous form. It's due to the equations of motions.
13:25
It's due to the equations of motions. It's a Galeromorphic functions, and we can expand this function in the, since we can set the periodic boundary conditions for,
13:42
at least for energy momentum tensor, the coefficients generate the famous Verasoro algebra, and each, as I said before, each terms of these expansions, all right. So actually, each generator is trying
14:02
to be integral of motions. It's conserved, but they're not commute. This is not commuting set of integral of motions. Nevertheless, we, as I said, we can, using this generator LN, we can build the operators which commute with the, mutually commute, form the mutual commuting set.
14:23
And how it works, so to build such an operators, we can consider the limit of the integral of motions in Cinz-Gordan model, and as a result, we get the following operators, right,
14:40
which is built out of energy momentum tensor, Galeromorphic component of energy momentum tensor. So they look like t, the density, the first density is very simple, t, t squared, or certain composite field built out of the energy momentum tensor. And if you, this gives you a set
15:05
of mutually commuting operators. And in the limit, well, when central charge goes to infinity, the limit C goes to infinity. Remember that C involves Q and which involves
15:21
B plus one or B, so C goes to infinity means that B goes to zero. So it's a kind of classical limit. So this set actually can set with the set of the integral of motion, local integral of motion for kdV theory, it's not surprising because they classically kdV and Cinz-Gordan model
15:43
and the same integrable hierarchy. So that is rather predictable. But anyway, we have these operators. And as I said, the density, the corresponding density,
16:01
built, composed, well, it can be constructed using the t through the operator product expansion, actually regularized operator product expansions. For example, t squared is a regular part of the operator product expansion t and t and which appeared in first, if you subtract the singular part,
16:20
you get the operators and this is definition of t squared. Similarly, you can define t cubed. And this is rather well-defined procedure this is all the densities and ambiguously defined here. And they can be expressed in terms of the Verasoro generators, right? And well, expansion modes of the energy momentum tensor.
16:44
The first one is just L0 up to constant and then we have the expression like this, which involve the infinite sum. Well, so formally it's not an element of universal and open algebra, right? But nevertheless, maybe that looks
17:03
maybe a little bit confusing because maybe we may have the problem in this thing in the infinite sum when we're doing the operators. But in this situation, actually, this is not a problem because of the following.
17:21
So now we even consider the representations of Verasoro algebra, the highest representation characterized by the conformal dimensions delta and try to use the basis, for example, and try to use the basis built out of these modes, its standard definitions of construction,
17:43
the basis in the highest weight representations. Then in each level subspaces, in level subspaces means that it's graded, that it's a graded space. So the number of more, the sum of this and more cell and actually is fixed by the level.
18:04
And in spite that we have infinite sum here, right? The ones that this operator acts on the level subspaces, the sum actually truncated. And so the operator each of the this operator acts irreducibly and acts in within
18:21
invariantly within the level subspaces. So in other words, that at each level subspaces, the I, it's just the matrix. The size of the matrix is equal to the dimensions of the level subspaces, which is in general, it's a number of partition of integer n.
18:42
So that is for any fixed level, this is just a finite matrix. Maybe large, right? Of course, if n is large, then of course, number of partition very large. But it's still a matrix and we have a problem of, we can explicitly construct these matrices using the definition and we have a...
19:03
So this is just an example illustration. So this for level n, we have the only one state, right? So there's no, a matrix is one dimensional. The space is one dimensional, then at level one,
19:21
we have only also only one state, right? It's again, the matrix is trivial. So the, but in the level two, the first non-trivial example, it's a level two. When you have two states, L minus one squared and L minus two, so the integral of motions, it's basically, as I said,
19:41
for calculation of this matrix on the second level, we can drop all the terms which involve L minus three higher than in this sum, in this infinite sum, this is why only the few two terms and all other actually kill the any state at this level.
20:04
So this is just a two by two matrix and we can diagonalize this matrix. And so here's two eigenvectors you see and two eigenvalues. So the eigenvalues, as you see, the eigenvalues are kind of algebraic functions of the central church dimensions and dimensions, right?
20:25
And now those are immediate questions occurs in this point. So what, how do you gonna write this matrix is for the higher level, because matrix is very large. And so that is clearly problem.
20:40
And so the questions, so the first questions was natural questions. So for given the integrable structure, how to calculate the spectrum of local interval motions. And I should emphasize at this point, so we consider the particular integrable structure of conformal field theory. In conformal field theory, there is no integrable structures from,
21:01
and the one which we discussed, which appeared through the certain integrable structure, originally introduces the massive theory and doing the special, short distance limit of this theory,
21:21
we get some particular basis of state, which diagonalized all the integral motions. But the matter of facts, we can introduce in the same, in the same folks, in the same grassroots highest weight representation, we can introduce it different integrable structure. For example,
21:42
another well-known example of the integrable system, integrable field theory two dimensions, it's called bullet dot model, which looks slightly different than the Cinich Gordon, because here we have two exponent, but with the different two, but with the different exponents here,
22:01
two B and minus four B, right? And again, this is integrable systems. Again, we can consider the conformal limit. And as a result, we get the set of commuting operator, which acts in the same space. But now the set is different
22:20
than the set of equations, that the quantum integral motions, they looks like these, even spins, well, spins are different. So you see integral, the density, this density is two, but in TV, we have density T squared. There is no density T squared, only the T cube appear the first.
22:43
And so this is what I should then say, that this operator acts in the same space, in the same space of conformal field theory. And, but it's another, it's a different integrable structure. And we, for this problem, we also have a, we can address the questions of the signalization of this set.
23:02
But you fix the representation, right? Yeah, we chose to consider the- C and- We fixed delta and C, right? Anything special happens if they generate one? Yes, of course. Yes, yes. There are some, but at this point, we did the integral of motions, but you see it's algebraic function.
23:20
You can, in principle, no, no, no. Well, at least for the four, it appears of course, in the context of the field theory, right? On the particular field theory we're talking about. Here, we're talking about just some basic property of representations, one representations.
23:41
Yes, so that is a, because all the operator acts on the level of space invariantly, we can forget about the right neuralities and all other steps. Okay, so this is some principle. So what we should, the main message from this, my intro, it's actually almost half of my time,
24:02
but there are a variety of integrable structure, not just one. So we just mentioned pdv, bullet dot, there is another one with a k and s, which include the famous non-linear Schrodinger equation. There is another so-called paper, people's sausage,
24:20
integrable structure. But what's important then, once you choose these integrable structures, then with the presence, because of the presence of infinite dimensional algebra of extended symmetry, the problem of a dignitization of local integral motions, it can be, admit the mathematical satisfactory construction,
24:43
formulations, and in terms of representation theory, it's a well-defined problem. And now I just explained how to solve, how the problem of a dignitization of a calculation
25:01
of spectrum of local integrable functions can be solved. Actually I explained the solutions, and this solution was given by, in our work with Sasha Zamoluchik-Volodyi-Bazhanov a long time ago, and principle, so descriptions of the spectrum based on,
25:21
remarkably, it's related to the spectral theory, which Leon discussed today. And I'm lucky that he gave a nice introduction in the spectral theory of this rate. And actually, many I'm discussing here, it's actually related,
25:41
deeply related to what Leon said. And so basically the key ingredient here is differential equations, Schrodinger equations, right? If in all this is usual Schrodinger equation, but the potentials you x, first, if you, what's important in this potential,
26:01
forget about this term first. This is a clearly that this potential is just in the harmonic three-dimensional oscillators, parameter alpha is in the case, alpha is equal to one, we have a potential like x squared plus L,
26:20
L plus one over x squared, right? Which is centripetal potentials, three-dimensional oscillators, right? Another case, we can consider here that alpha is equal to infinity. For example, then the potential will look like,
26:42
well, infinite well potential, which is also an example, which Leon mentioned in his talk. But anyway, this alpha, anyway, so this is, this terms is clear, so from the formal,
27:01
from the mathematical point of view, we have differential equations with one regular singularity at x equals zero and irregular singularity at x equals infinity. But I also would like to introduce some additional singularities, but very special way, right? So it will contain, so clearly there would be set of singularity
27:21
dependent on the set VA. And we impose certain condition restrictions on the position of the singularities. And the condition looks like this, it's a set of algebraic equations and imposing this equation, the meaning of this equations is such that
27:42
once this position satisfy this kind of equation, then the solution, any solution of differential equation, actually be single valued in the vicinity of the singularities. It's very strong restrictions, you see, once you introduce the singularity in the potential, then in general,
28:01
you should get some non-trivial monodromic property of solution, you can, if the potential has a singularity, even regular singularity, right? The functions, the wave functions, not single valued in the vicinity of the singularity, but imposing this condition
28:20
that the singularities are single valued, we fix, they cannot, it require the fixed fine tuning of the parameters of the position of such singularities. And this is equation, so of course it's this equation, which described the condition that singular apparent,
28:43
sometimes it's such a singularity, it's called apparent singularities. You see the delta here, it's related to L, to angular momentum term. Well, and the next step, it's a spectral determinant, which was a subject of Leon talk, right?
29:03
So clearly for the potential like this, so this is u x. So if you consider the potential like this, the potential look like this. So we have only discrete spectrum, right? You can clearly formulate the problem, spectral problem with the e g,
29:22
it's a spectrum, it's energy spectrum for the Schrodinger equation. Well, this is without this additional term, but including this term actually not change. So it is self adjourned, right? I think so.
29:40
No, no, no, no, no, for, for, for, for, for, for, for, for, for, for, for, for, for, for, for, for this terms. Because you won't pay attention. Yeah, yeah. But for complex P, well, for, for, for real L, of course, imposing by the conditions, where one that certain real values of L
30:02
certain domain will be self adjourned. But actually the, the, the spectrum, I'm not quite for, for general, the V is complex numbers here, right? It's not, doesn't look self adjourned. But anyway, the problem can be introduced and because of the presence,
30:21
all the singularity, apparent singularity, the monodromy, you can forget about this. The additional insertion still formulate the same type of problem, like unusual and germanic oscillators. So the central work would be the spectral determinant. Well, and actually this product can work only in the case
30:44
and alpha greater than one. But then if alpha is not greater than, is smaller than one, then they need some regularizations. But again, the most important would be the trace identities, which is Leon mentioned. And basically it's useful to consider
31:04
the large E asymptotic. I consider the energy goes to large, actually it's negative. Well, at least as was mentioned in Leon talk, right?
31:20
The spectral determinant admit the expansion of this form and the coefficients of this expansion, the first leading coefficient doesn't depend on the presence of apparent singularities, actually the numbers of the number. Then if you focus on subleading terms of these expansions, right? The structure of general structure looks like this.
31:42
Then we should actually focus on the coefficient like this, which appeared in the certain powers over the expansion. So the spectral determinant, inverse powers. And these quantities actually can be, for such a quantity, the trace identity can be written
32:02
and that they can be calculated in terms of, systematically in terms of using the WKB approximations. Some and other coefficients here, but we are actually at this point, we're interested in these coefficients like these, because the reason we're interested in these coefficients because they're exactly the insight
32:23
with the general value of local integral of motions. So the statement, which has come from our paper, is that if you take these algebraic systems that identify the delta, the same delta as the highest rate representation,
32:41
the central charge related to this parameter alpha. Alpha is two alpha. So this is related to, and the monicity of the potential and n, it's a level, number of apparent singularities, it's a level subspaces.
33:01
Then we checked explicitly for just the solving this equation for small n, one, two, three, up to five, I guess, that the number of solutions of this equation
33:21
coincide with the number of partitions of integers, up to, of course, the action of the symmetric group. Sorry, so this is an equation for what? For apparent singularities, for V, right. And this condition just guarantees the potential for the apparent singularities. And so, and that's actually the statement
33:43
that number of solutions coincide with number of partitions was proven just recently by David Mazayeva. OK, and it means, so the number of solutions coincide with the number of states at the level subspaces.
34:01
And we can use these sets to label the states and the level subspaces. And so the integral of motions can be considered as the functions of this set, or of the set, solutions of this set. And the statement was that this coefficient
34:22
exactly gives the eigenvalues of local integral of motions, yes. Possibly one more stupid question. So it continues one of the questions as before. What if your verbal module is degenerate? So if delta is special, then what happens with the number?
34:41
Right, then probably you need a small number of solutions. No, solution would be the same. But something different would be, maybe I will mention. But the number of solutions remain the same. But something, of course, interesting happened. But it's a different story. So here are some explicit formula.
35:01
For example, e1, and by the way, each of these, as a function of this set, it's a symmetric polynomial, of course. It's a symmetric polynomial for order of m minus one. So in the case of the first integral of motions,
35:21
the order of the symmetric polynomial is zero. So it's just the number of apparent singularities. For i3, it would be just a linear, symmetric, it would be linear functions of v. For the next integral of motion would be quadratic one, that this one, linear quadratic terms, and so on.
35:42
And so in other words, without, so in order to find at least the spectrum of the local integral of motions, we need to solve these systems, and then it's done, right? So it's a symmetric polynomial, give the eigenvalue. Of course, this can be checked, direct calculations.
36:00
By the organizations of matrix. But together with the local integral of motions, there is another interesting integral of motions, with another integral structure, right? Which, another structure which is included in this integral structure, it's a non-local integral of motion. It was discussed in our paper a long time ago,
36:23
again with Sasha and Volodya. And the simplest example of non-local integral of motion is in so-called reflection operators. It's commuted, it's among the commuting families, commute with the local integral of motions. And it has many applications,
36:41
and what's interesting that this reflection operators, reflection operator for the KDB integrable structure is related to the reflection as matrix in real theory. Okay, and the matrix features, well, it was discussed in many works before, but in probably the best explanations of this object
37:02
was given by Sasha and Alosha in 96. Yes. This correspondence that you had before works for any central charge? Yes, it worked for any central charge. Like I said, the top is bigger than one, so then. It's actually, it's algebraic equations. And you see, when you diagonalize your matrix, it doesn't matter what domain.
37:20
It can be applied for any values of central charge dimensions. And that's the beauty of this, because you don't need to worry about the domain of your parameters. So, and let me remind you how the Liouville matrix works.
37:42
So again, we return to the Liouville theory, energy momentum. Again, the energy momentum expand the Verasoro algebras. We have the, this is our representation in terms of field, phi. But now, let's think about this again. Once you finish all these things,
38:01
and you diagonalize everything, you have to find the right one. At the end, you get back your mamodo, right? Yes, yes. We are talking about, I should emphasize, actually the problem in this formulation is a pure algebraic representation theory. We don't play, we are not going to glue these different chiralities. We just focus on one chirality,
38:21
actually one level, when everything is well-defined problem in linear algebra. So that is, so then let's think about this a little bit, what happened with the Liouville theory. And in the case when the phi, again, we have considered the domain of configuration space
38:42
where phi becomes very large, but negative. If it's large and negative, then the storms, it's negligible, right? And we have just the free actions, the theory of free body field, and which should mean the function d phi, right?
39:03
Would be galomorphic function, which can be explained in the Fourier series of this form. And from the action, we read the canonical commutation relation for these oscillators, creating and annihilating oscillators. And so in this asymptotic domain,
39:23
the energy momentum tensors can be expressed in terms of the creating and annihilating operators, right? And it looks like this. It's a famous business relation, business relation formula, for example, that's an effective business relation formula for the grassroots algebra.
39:45
All right, and but remember that, but we can build representations of the algebra now. It's a folk space. And as a folk space, right? So this formula actually provide the certain conformal
40:05
as a structural representation of various sort of algebra or in the folk space, right? So, but this folk space actually, as a linear space is amorphic to the Verma model, or there are sort of algebra. So, and the zero mode momentum of the folk space
40:23
related to the dimensions of delta. But now it's interesting, the important feature that delta is even function of p. So in other words, we have two choices which correspond to the same delta with the folk space with positive p and the negative p.
40:41
And clearly this can be interpreted just in the spirit of quantum mechanics, we can interpret it as each of these folks, or the basis of the folk spaces, the space of asymptotic states, right? So for positive p, we have a set of in-state which considered as a propagation.
41:02
So like mini super space approximation, we consider the propagation. So the way we just get on the financial barrier, right? And this folk space would respond to the in-state and the opposite sign is a negative p would be reflected, out-state, right?
41:21
Once we have such a two spaces in and out, spaces within and out-state, we can introduce this matrix. And so it's a reflectionless matrix, we will theory because this form, there's some structures. It contains the some normalization factor, which is famous normalization factors, which contains a lot of informations about Liouville theory.
41:46
But for us, it would be important how these operators, how the reflection, this matrix is operated to be precise, it's between the different folk spaces. And it's interesting and important, which was emphasized by Sasha and Alosha,
42:01
that this intertwined means that first of all, factorized on the right and left, chirality, an absolutely independent operator, S bar and S. And each of these operator acts in between, it's intertwined between folk spaces. Chiral, chiral part of the S matrix.
42:21
We normalize such a way to make this one, then all the normalization goes to this factor, right? And what's important that this reflection, chiral reflection operators is chiral S matrix is fully determined by conformal symmetry. It is quite evident because we, as I said in this theory,
42:45
the energy momentum tensor, this issue of these components is conserved charges, right? So in classical, just think, if you think about the classical, that this quantity would conserve. In principle, what we need to do is solve in a classical level to construct this reflection.
43:01
As metric, we need to express the oscillators, A in terms of ln, inverse this relation, this formulas. But in other words, we should solve this equation with respect to phi, this basically Riccati equations. And the quantum level,
43:22
this term can be formulated to the statement that this operators unambiguously defined and the constructions and then the next idea that having this, if you have this S matrix, reflection S matrix, then we can build the operator which commute with the local integral of motions.
43:41
And it works the following way. Let me emphasize first that this operators X between the different space, right? Different folks space, it doesn't have any sense to diagonalize this operators, right? It's intertwined. But instead of, but we also can introduce the intertwine and it's another intertwine which acts similarly,
44:01
but very simple way just to flip the sign of oscillators. This assignments are called conjugations. And then if you think about this picture with the Liouville, returning to the Liouville theory and see what happened to be the wave, which is, we firstly forget about the left barrier.
44:23
We consider the scattering on the right barrier, then under the scattering, we get the phase, which is a Liouville matrix dictated by Liouville matrix that then we should go to the right body,
44:42
left body here again and find this way, the total scattering. If you describe the whole scattering, we need to put the scattering on the two barriers. But after the scattering, because of identification of space, we need to introduce this additional set conjugations
45:03
because it's flipped the sign of phi. But anyway, that principle, that the combination like scattering C, and this barrier, then we should inverse identify if phi, if minus phi and if phi should insert the reflections.
45:22
Check conjugations, then scattering about this barrier give us this operator and then back. So the result, we may expect that such a combinations of operator, which now acts in the folks space, commute with the local integral of motion. Sorry, just a question.
45:41
This is already an extended kind of folks space? No, no, it's just a chiral space. We are focusing, because we are focused on the one chirality, and we focus on one chirality and we'll do this operator, which is, well, you see, this is squared of operators, right? Clearly it's a squared. For this reason, I introduce CS and actually for just historical reason,
46:01
I put minus one, but doesn't. But what's important that these separators is now X in the folks space. Given folks spaces in this operator, I call it reflection operator. So it's additional operators. As a matrix, it's very similar to the reflection matrix,
46:21
but we need to flip the sign somewhere. Okay, and so construction of reflection operator is very simple. Indeed, it follows the following way. So we have a business asians formula like this. What we want, we have a virostral construction, virostral algebra will be the same central stretch,
46:42
but if you flip the sign of the background charge Q, right? If you just flip minus here, but of course it would be different oscillator basis. In other words, in the folks space, we can introduce three different basis. First of all, Virostral basis,
47:01
then the one basis, folk basis built out of operator AM and other base built out of operator AM bar. But of course, this is a base in the same space. So there is a matrix omega which relate the Gaussian-Berg basis and Virostral basis, this one, there is a certain matrix
47:22
and similar for the tilde basis. Sorry, but what happens to the labeling? For the what? Labeling. Labeling? Labeling, yeah. What labeling? And how would you relate that like? It doesn't matter, you know, it's just a basis. Let me choose this. This is a set of the basis in the space.
47:42
Okay, this is the basis that you choose for the Verasoro-Jubilator block and... You see, it doesn't matter. Actually, what would be interesting for me, it's a basis which relate, not this matrix relate the base Verasoro one with the Gaussian-Berg operators.
48:00
And, but also using the same matrix again, but also applying the Sisk conjugations, I can relate, build the matrix which relate the basis, the tilde basis with the Verasoro one. And what I'm going to do is just because they're the same I need the relation between in and out,
48:22
the tilde and the un-tilde basis. Okay, probably he wants to say that you have to order them, say, and one places and two. Oh yes, yes, of course. If you order, this should be the basis, right? But finally, what base you use, it doesn't matter. Because finally, we're having this,
48:41
now I'm making the quality here. And as a result, I get the matrix which relate the tilde basis. So if Verasoro basis here, it's just intermediate step and you know the matrix finally, it's hidden in the summation indexes here. And so the lesson is that reflection operator
49:02
is expressed in terms of single matrices. What you need to do is calculate these matrices, choose the sum basis, calculate this matrix and then build the reflection operator in this way. Yeah, this is the ordering here, it's not, well, it should be just the basis somehow, somebody is calling for this. Okay, and it's not difficult to show now
49:23
that this period indeed commute with the local interval of motion and the main observation based, if you write the energy momentum tensors, it looks like this, t for example, right? It contains the quadratic terms, which is invariant with respect to flip of the sign. But it contains also the terms, but this linear term,
49:41
but this terms is second derivative, which is actually disappeared when you integrate. And it happened, this miracle happened for all integrals of motions, all of them when expressing terms of field phi actually, even function of phi, so they invariant with respect to this flipping the sign. So it doesn't matter the form of the integral of motion,
50:00
doesn't matter how you, what fields you used, phi or phi for tilde. And this means actually the operators, the operator, which is relate the phi bar fields and phi actually would commute with the local integral of motions.
50:21
All right, and here is explicit formula, so just illustration for this. And it goes this way. So for example, in the first non-trivial level, it's a second level. And the second level, we have two states, oscillatory state, right? A minus one squared, A minus two. And this basis, in the Verasoro base,
50:41
we have L minus one squared and L minus two. And we can express this test in terms of the oscillatory huge inter-superlative formula, right? So this gives us the matrix omega. And once you have the matrix omega, you build the matrix of reflection operators.
51:01
And this reflection is this matrix indeed commute with the local integral of motion, metric of reflection operators. So this way, we have this problem of, that since it's commute, it's equal to zero, right? They can simultaneously diagonalize the reflection operators with the local integral of motions.
51:23
And since we label these states by these apparent singularities, then the immediate questions, what would be the spectrum of, what would be the engine value of the reflection operators, yes. What would be the engine value of reflection operators? And the answer is rather, again, we should return to the differential equations,
51:42
differential equations, which we start with. But now, the subject of our interest would not the subleading coefficient, but first subleading coefficient, these two ones. You see, in the expansion, we have something which doesn't depend on state, but the first non-trivial coefficients in the spectral determinant is the one,
52:04
which is also the one, which is probably the main subleading asymptotic, which depend on the position of apparent singularities, these coefficients. So there is no such,
52:21
some rules such as trace identities, like for local integral of motions from the theory of differential equations. So it's more complicated problem, but nevertheless, it's remarkably that, well, so this problem was for a long time, but remarkably, very recently, it was work of Irvin and Tarasov, just about the theory of fixing the differential equation
52:45
with the three regular singularities and any number of apparent singularities, and they give some beautiful result about the solutions, which present explicit solutions to such as equations, and using the result of Irvin and Tarasov
53:02
in performing the certain limit, we can actually extract these coefficients. And in terms and express, and now it's of course, it's a certain symmetric function, but well, you know, the results are so simple, but it is given by a certain determinant involving the apparent singularities,
53:23
but of course, the formula is not particularly transparent, but finally, there we have explicit formula for the eigenvalues of reflection operators in any states, right? And again, they express in terms of the apparent singularities.
53:43
Okay, so probably I don't have a time to discuss. Okay, the final, probably the final stuff I would like to mention, that if you understand what's going on here, that you can easily construct the reflection operators for many integrable system, for example, for bullet dots, what should we do? In this case, we have a tool that looks like
54:02
a cinch board by the different exponent. And then in other words, to construct the reflection operator, we need to take the reflection operator corresponding to this exponent, right? Reflection matrix, but now we should glue it with the reflection operator corresponding to the exponent with a different background charge.
54:21
And this combination indeed commutes with the local integral of bullet dot and still conformalize the problem of the categorization of this. Another example, for example, we can consider the model, this one. This model is a famous, this part of the model
54:40
is a famous dual representation for black hole, Euclidean black hole in dual fields. And you can think about the Lundreich complex centigrade model as perturbations of Euclidean black hole by the exponential parietal, like a Louisville field. And the same story happened here.
55:02
So for this conformal field theory, there's analog of the reflection operator, the reflection S matrix, the S matrix, I call it SIGAR S matrix. And so then to build the reflection operator for this integrable systems,
55:20
we need to accompany this by the reflection operator of the Louisville theory. And then indeed you can check that reflection operators commute with the local integral of motion for this model. And interesting that a long time ago, Samsung discard this model in the presentation, in black hole.
55:46
And there was some manipulations with the fast integrals and finally the conclusion that this model should be somehow equivalent to the Louisville theory. That was some weird, you know, but remarkably, that finally one will come out,
56:03
which is of our papers, which I'm not going to discuss, that this reflection S matrix for this model, for the black hole, actually can be expressed, indeed expressed in terms of Louisville theory. There is explicit formula which relate this to S matrix. It was basically up to some factors
56:21
that almost the same. This is kind of some interesting problem, some interesting long, this problem has a long story. And now we see that indeed some, the signals that the theory somehow very deeply relate.
56:41
All right, okay. Okay, I don't have a time to discuss the applications. And so basically, so that my conclusion, I've discussed some integrable structures, reflection operators. And also, all of this actually has interest in application to study the scale and limit,
57:03
to scale and limit of critical latest systems, integrable spin chains, and the reflection operator is very useful and actually allows one to classify the state in the latest system, the better states. Actually, the states which they analyze all integral of motion, the analog of better states on the latest.
57:22
And that gives you a very powerful method to track the scale and limit of each individual better states. And there are some interesting ideas that maybe the scale and limit of the theory describe the non-compaxible models of the story.
57:41
But it's a different story. Okay, anyway, so once again, happy birthday. Sounds so nice. Questions, comments? Yes. Yeah, I have possibly made a question. How is it on that picture that you showed with the C.H. Gordon potential, right?
58:02
You have some long but some finite distance between those two walls. So then I don't quite understand this formula that you have that the total product of reflection matrices commutes with local integrals. Is it approximate in- The basically-
58:21
Is it exact or what? Good, that's a good point, yes. That is actually, that was in principle in my transparency as a part of my discussions. But you see, indeed, you consider that just apply this quantization condition for this mode, zero mode. And naively, in the zero order approximation,
58:42
we just can replace this potential by the infinite well potential. So the quantization, I'm sorry. So the quantization condition, like the total length of the well, multiplied by P, which is a multiplied by P, is proportional to the integers, right? It's quantization condition for the allowed values of P.
59:03
And then you should take into account the effect of scattering, right? The phases which is gained on the tuning points. And Sasha and Alosha Zamolodchik of 95 proposed the following equations, asymptotic equations,
59:22
which involve basically the reflection operators, right? Actually, there was some kind of mistake in the papers, because they say that S-matrix commute with, but this is not S-matrix reflection operator, because it's S-matrix X in different spaces.
59:40
But basically, quantization condition for P, it looks like this, right? And in principle now, we can take the in the short distance to provide the behavior, you can explain the quantization condition. If you know the spectrum reflection operator, you can write this quantization for each level.
01:00:00
Now, what next? Suppose we solve these equations. Then when actually we should remember that the kdV equation would be polynomial with respect to p. And now we should substitute to solutions of the functions of the distances of the r. And what we get, again, the value of integral of motions
01:00:20
are out up to the power law correction. So this quantization conditions actually count all logarithms. But the power law, of course, is out of the program. It's some approximations. So it's some kind of WK, kind of well-developed WK. Yes, but sometimes, in the case of the similar
01:00:43
can be done for modified C.G. Gordon equations. And Leon taught us about the quantization condition in this case. But in that case, it was exact. But here, we have a power law. But accuracy of this approximation
01:01:01
is rather good, and Fede knows very well, because he studied for many integrals. I thought I spoke well. Yeah, because it's actually, numerically, it's very good. It gives a rather good accuracy. Professor? So in the case of cigar, the actual full thing
01:01:24
So they were asking restrictions on numbers to get central charge 26 or something. So it was a specific coefficient. Our meters were fixed, right? Yes, yes. So does it matter? You see, at this point, as I said,
01:01:40
if you take a look on the formulas here, we should be going to write this matrix. This parameter q you're talking about, right? It should fix somehow. But you see, this is functions, which is a algebraic function. Actually, it's a function of q. It's a meromorphic. It's a function of b. It's a meromorphic function.
01:02:01
You can set any numbers there. This matters. Of course, there's some miracle, some interesting. I've seen the maps that we did with Marcin was specific to what? Yeah, that is another point, because I talk about the general relations. And for these particular values, I don't know what happened. But actually, the relation exists for any background
01:02:24
charge, right? And this is just algebraic relation, because it's a relation between the matrices. You check it level by level, and you see that there's a. I have a question. Bulldog.
01:02:41
And this is another, is it again, karepin? The harmonic is 10 by 9. And construction for algebraic build on this is a headache. Can you simplify the construction for algebraic build on this? It's a good point. You're talking about the Zirkin-Karepin model, probably. Bullet dot? No, bullet dot on like a Cinz Gordon.
01:03:02
You know, this is a complex. This is the complex. You mentioned spin chains. Yeah, I mentioned spin chain. Certainly, I don't know the answer for this problem, but now there is some certain progress in this.
01:03:26
But the full answer is still, well, at least for me, it's I don't know.