Courant Events Right: Yes? Can you hear me?
Courant Events Right: Okay, good.
Courant Events Right: What's the CEO?
Courant Events Right: Okay.
Courant Events Right: My pleasure to introduce, Jessica.
Courant Events Right: Jackbox in from ENS in France.
Courant Events Right: Yeah, we talked about correlation function with the keeper.
Courant Events Right: More than love for them.
Courant Events Right: Thank you very much, Denise!
Courant Events Right: So, my probabilistic path to probability theory goes via correlation functions in two-dimensional critical loop models.
Courant Events Right: I'm going to present the work done with a number of collaborators that you will see on the screen. There are a few non-permanent
Courant Events Right: people that I would like specifically to commit to your attention, Max Downing, my postdoc, soon moving to Trieste.
Courant Events Right: Paul Ru, who is going to defend his thesis, also moving to Trieste after the summer, Ron Goram Nieves Vivad, who is at Abu Dhabi, and Linea Kent Samuelson, who is a postdoc in Oxford. So, the models that I'm interested in.
Courant Events Right: Our lattice models…
Courant Events Right: So they live on hexagonal latches, like this, and what you do is you draw closed loops on that latches.
Courant Events Right: So loops cannot intersect, cross, or anything, and each loop has a statistical weight of N.
Courant Events Right: And they also come with a control parameter K that gives the weight per piece of step.
Courant Events Right: Which is adjusted in such a way that the model in the limit of a very large lattice, or a very fine mesh, is conformally invariant. So there are two possible values for this critical pay. One is small.
Courant Events Right: Relatively small, and that's the dilute critical point. And one is larger, and it's the so-called dense critical point.
Courant Events Right: And so whenever this loop weight is between minus 2 and 2, those loop models have a conformally invariant continuum limit.
Courant Events Right: Although, this is…
Courant Events Right: Has not been proved for more than sporadic values of N, but it's believed to be true.
Courant Events Right: So in that continuum limit, they are characterized as a conformal field theory with a certain central charge. So the central charge is the master parameter of the conformal field theory that basically governs the whole content of critical exponents.
Courant Events Right: And alternatively, it's possible to describe the same continuum limit starting directly from a probabilistic construction that's called the conformal loop ensemble.
Courant Events Right: And it also comes with a parameter, it's called kappa, and that kappa is related to the central charge, and the central charge is related to N, so each of those three ways of looking at things, they have one continuous parameter.
Courant Events Right: And, here's a… there's a beautiful picture.
Courant Events Right: On the circle, or the sphere, maybe, in which you see some nested loops, with the color showing the nesting level, so you can see that they are very complicated fractal objects. And our task is to compute correlation functions for those models.
Courant Events Right: So, there's some motivation coming from statistical physics.
Courant Events Right: So in statistical physics, you would define this partition function as it.
Courant Events Right: Where you sum, as I said, over the loops with those weights, k and n. And, for specific values, you get very well-known problems.
Courant Events Right: So, for example, in the case n equals to 1, where you don't even count how many loops you have, you will get side percolation in the dense phase, and you will get the easing model and the dilute phase.
Courant Events Right: If you send N to 0, so that essentially you get one loop, you get a self-avoiding walk in the dilute phase, and if you take n equals to 2, you get a Gaussian-free field.
Courant Events Right: also in physics, called an XY model.
Courant Events Right: And finally, if you,
Courant Events Right: Hmm, I should have said N, I think. N goes to 0 in the last line, if you do that in the dense phase, you get something that is equivalent to a cube goes to zero state parts model that gives you uniform spanning trees or forests via correlation functions.
Courant Events Right: So all those conformal field theories, they are actually really logarithmic conformal field theories.
Courant Events Right: But this aspect can be attenuated a bit by taking the value of n to be generic, so that we try to avoid actually taking straight away these integer values of n, but…
Courant Events Right: We take, sort of, you should imagine that it is some generic parameter living in this critical range.
Courant Events Right: So, the conformal field theory underlying this model has been elucidated a long time ago, at the level of the central charge and the critical exponents.
Courant Events Right: So, if you, parameterize your loop weight in this way, with N…
Courant Events Right: With a parameter beta, then, you're going to get
Courant Events Right: The central charge 13 minus written there. And then you will get, various kinds of critical exponents.
Courant Events Right: That I list here, so there are… it's important for… to follow the talk to, learn these three notations here. So, the first one with a D, that's called a degenerate field.
Courant Events Right: You should think of it as the energy operator in this model, the operator that finds out if there's a monomware somewhere.
Courant Events Right: And its generalizations.
Courant Events Right: So that's the 1-3 field, and…
Courant Events Right: There are 1, odd nodes in general, so they have a singular vector at level S,
Courant Events Right: So they are the kind of fields that,
Courant Events Right: You… you see in conventional conformal field theory.
Courant Events Right: But the other two kinds of fields, those are the real motivation for studying this model further.
Courant Events Right: So, there's a field called, diagonal field.
Courant Events Right: It comes with a general complex parameter.
Courant Events Right: It's a… it's a spinless field, it has the same right and left moving conformal dimensions.
Courant Events Right: And the last one is non-diagonal, so it has different left and right moving parts.
Courant Events Right: And, it has this strange,
Courant Events Right: property that these indices R and S,
Courant Events Right: that are usually thought of as cat's labels, and in conventional conformal field theory, they are integers. Well, they are not really integers here. R is half an integer, and S can be a more complicated fraction, just so that R times S is an integer.
Courant Events Right: R times S is the spin of this non-diagonal field. So, no.
Courant Events Right: Here is a bit more about this interpretation. So, as I said, we have this energy operator. The role of the diagonal operator is that if you insert it at some point, you're going to modify the weight of any loops that encircles this insertion point.
Courant Events Right: So it's going to be modified to a way that depends on this.
Courant Events Right: parameter P, that was, in general, a complex number, or prefer you can just think of real values.
Courant Events Right: And the last kind of field, the one that is called non-diagonal, it has this interpretation that it inserts 2R open paths.
Courant Events Right: In the model, and S determines a certain phase factor whenever some of these paths try to turn around their insertion point.
Courant Events Right: So, in the physics literature, they are called watermelon operators, because if you draw two of them next to each other, then the propagating lines, they look like a kind of
Courant Events Right: Design of a watermelon.
Courant Events Right: And
Courant Events Right: Yeah, so the reason that r times h should be an integer that's for single-valuedness correlation functions.
Courant Events Right: Also, the sum of the R labels should be an integer, because otherwise you cannot match up these inserted open curves. There's some handshake lemma going on here.
Courant Events Right: So, the game here, the name of the game, is to compute correlation functions.
Courant Events Right: So, for example, here on this slide, there's a configuration of site percolation.
Courant Events Right: So you take hexagons, and you color them black or white by flipping a coin.
Courant Events Right: And the loops then emerge as the contrasts that separate black and white.
Courant Events Right: And the kind of question that you might want to ask, and which has not been solved until quite recently, is, for example, what's the probability that given three points, say, for example, those three green points, what's the probability that those three green points belong to one same loop?
Courant Events Right: So, what happened locally in those points is that they each inject… Two open paths.
Courant Events Right: And the only way that two open paths coming out of each of three points can join up
Courant Events Right: Without immediately self-contracting, which will give zero by the definition of those non-diagonal operators, is that actually all those three points will then belong to one same group.
Courant Events Right: And that's,
Courant Events Right: the probability for that happening by a global component invariance has this dependence here on coordinates. I should mention here that when I use coordinates, you should think of them as complex numbers.
Courant Events Right: So, the two-dimensional space here is parameterized by… by complex coordinates, real and imaginary parts of those sets.
Courant Events Right: And the behavior in those sets is completely fixed by global conformed invariance and the knowledge of this critical exponent, which is 1A.
Courant Events Right: But the thing that was not known until recently is what
Courant Events Right: What constant should sit on top of this expression here?
Courant Events Right: Now, you might ask… you might ask, so why don't I just normalize operators so that its constant is 1?
Courant Events Right: And the answer to that is that there are already two-point functions that have been normalized in this way, so you cannot normalize any further.
Courant Events Right: So, so there's a non-trivial probability-related thing that is related to this, numerator.
Courant Events Right: So, this particular question here, What's actually solved,
Courant Events Right: Rigorously, but, side… 24.
Courant Events Right: And,
Courant Events Right: one of the results that I will… I would present is to extend this to arbitrary values of these R and S, so I recall once again that
Courant Events Right: non-diagonal field, they come with R and less parameters.
Courant Events Right: where R can control the number of legs, so I can… I can give you an infinite family of such, of such things.
Courant Events Right: But there's only one that has been proved by mathematicians.
Courant Events Right: So, where do those critical exponents come from?
Courant Events Right: Well, they come from this old work,
Courant Events Right: which is almost 40 years old. What you do is you take a torus.
Courant Events Right: And then you try to compute the modular invariant partition function on the torus, so this was done by…
Courant Events Right: It's by the Francesco Salado in 87.
Courant Events Right: So, on a torus, you would have a modular parameter. Tau is the aspect ratio of the torus, then…
Courant Events Right: by ITO.
Courant Events Right: And, then what you find… Is that,
Courant Events Right: The partition function in the continuum limit, it's expanded in terms of these characters.
Courant Events Right: Eta is the dedicated eta function, so it, sort of generates with multiplicities all the descendant fields that come from a given primary field.
Courant Events Right: And, you see here, in the first term here, you have all these… the degenerate fields that participate, and they all come with multiplicity 1, which shows that they are normal fields, the fields that one is used to in CMT.
Courant Events Right: But the others, they come with some non-trivial multiplicities, L,
Courant Events Right: And that shows you that they are quite special fields which don't have the
Courant Events Right: Maybe the properties that you would usually expect if your take on conformal field theory goes via minimal models.
Courant Events Right: So, they come with a character that's tied here.
Courant Events Right: And one thing that you can notice is that the first kind here, it has a subtraction, this is why it's degenerate fields, there's a singular vector at level S that's being subtracted. But here, there is no subtraction at all, so those fields are not degenerate.
Courant Events Right: And that is… makes it considerably more complicated to evaluate their correlation functions.
Courant Events Right: Of course, the usual,
Courant Events Right: 1980s take on conformal field theories is that you should use this degeneracy to compute correlation functions by moving to differential equations.
Courant Events Right: For specific values of R and S, when they are both integers, actually the character of these non-diagonal fields is more complicated. It contains a rank 2 Jordan cell, so they are indiecomposable characters.
Courant Events Right: So even when the loop weight, or the central charge, or the kappa parameter takes generic values, there's actually still something not logarithmic in those CFTs.
Courant Events Right: The multiplicities, they can be written in this, outlandish way here, also… Horizon Sams.
Courant Events Right: And it turns out that once everything has been done for any permissible R and S, they are polynomials in N.
Courant Events Right: Via this complicated formula.
Courant Events Right: Okay, now,
Courant Events Right: The LRS, actually, they are the smoking gun, that there's something very interesting going on in this kind of model.
Courant Events Right: Because if you look at,
Courant Events Right: You do a more complicated computation than just the torus partition function. If you compute the twisted torus partition function, you will realize that the state of space
Courant Events Right: of this model here has some bimodule structure, where on the right-hand side here, you have the thing that comes from conformal field theory, and on the left-hand side, you have something that reveals that the model has an ON global symmetry.
Courant Events Right: And,
Courant Events Right: The role of, this left-hand part here, this gamma is that the dimension of this is LRS that you saw just before.
Courant Events Right: So on the level of the stage space, this is what it looks like before you just take dimensions. Since you take dimensions, you get… you get characters from the right-hand side.
Courant Events Right: And you get multiplicities from the left-hand side, and then you are back to your Taurus partition book.
Courant Events Right: What is what you call a…
Courant Events Right: Well, so what you do is you sort of, you take a group element of ON, and then when you compute your torus partition function, there's kind of a twist when you go across the periodic boundary condition, you pick up… you pick up some group element, and if you keep this group element, generic, then you can…
Courant Events Right: It gets such an expression.
Courant Events Right: So, now you can take these representations and you can decompose them into OM.
Courant Events Right: So those gammas, lambdas here, probably with Greek letters, lambdas, I guess. And what happens is that in the beginning, for the small representations, you just get one term. So, for example, you see here that the thing that is supposed to just insert one piece of loop, that's the one part 0,
Courant Events Right: That just comes with, with the representation 1, and…
Courant Events Right: in general notation of ON, and this one comes with a symmetric and anticipated representation on two boxes. But starting from three lines, things become more complicated, because now you get some sort several representations.
Courant Events Right: So you see, there's a general formula here, I just show the first few examples, but you see, it gets worse and worse as you go along.
Courant Events Right: Do you get the same irreducible enforcement, or is it for no? Yeah, so, for example, yeah, I see that it's 22, for example, it comes in several places. Yeah, but on the same line, do you get 2-2 twice, or three times? Yes, you can.
Courant Events Right: Later, yeah, just if I had continued two more lines, you would see that they start coming with multiplicity tests, which are integers.
Courant Events Right: Okay, so the actual global symmetry is bigger than ON, And, one thing…
Courant Events Right: One thing that is complicated is that this bigger representation, this bigger symmetry, it does not come with a co-product structure, so it's, it's difficult to, to somehow to deal with it.
Courant Events Right: Good slip and hope.
Courant Events Right: Okay, so, now…
Courant Events Right: Let's start our quest into correlation functions by looking at the geometry of a sphere, or of the infinite plane. So, two-point functions, they are given just by the conformal dimensions that you have seen, and
Courant Events Right: the field which is normalized, three-point functions, I discussed a little bit. If you go to four-point functions.
Courant Events Right: They will start depending on some, PROF ratio.
Courant Events Right: And,
Courant Events Right: If we wanted to take four-point functions, say, where there was a degenerate field entering in the correlation function, then we would find that they would obey differential equations of the PPZ type.
Courant Events Right: And by that, we could solve for them, but those are not the cases that we're interested in. We're interested in the cases where there are no differential equations around.
Courant Events Right: Nevertheless, this degenerate field still gives us some useful information.
Courant Events Right: Because it gives us a so-called decentralized shift equation.
Courant Events Right: That I will discuss a little bit, you know?
Courant Events Right: And, there exist other models.
Courant Events Right: In which there are degenerate fields not only of 1 comma S type, but also of an type.
Courant Events Right: For example, in minimal models and in UVL field theory, that would be the case. And if you have those two types of shift equations, that would imply solvability. So those theories are easy, but this one is more complicated, because you are somehow missing one shift equation.
Courant Events Right: Therefore, we will need to do something more to find our correlation functions. We will need to constrain them by a principle that's called the conformal bootstrap.
Courant Events Right: And the conformal bootstrap starts at four-point correlation functions, and the idea is that you take your four points and you bring them together in pairs.
Courant Events Right: by Drew, and when you do that, you try to look at the short distance expansion.
Courant Events Right: And, you require that all three possible ways of bringing together the pairs in…
Courant Events Right: the pairs of point, that they will give you, the analytic expression of the same thing.
Courant Events Right: And that poses constraints that are called the bootstrap equations.
Courant Events Right: However, unlike…
Courant Events Right: bootstrap results that you might have seen elsewhere. We can do a little bit better, because it turns out that this degenerative field, the 1,3, it generates a so-called interchiral symmetry.
Courant Events Right: So we are using this degenerative shift equation to simplify things. I will show you why. Also, the global ON symmetry turns out to be helpful, but as I already mentioned, the actual symmetry is actually larger than ON, and in the very end, that will turn out to be a crucial
Courant Events Right: piece of information.
Courant Events Right: Okay, so now, still in the sphere, let's take four fields.
Courant Events Right: Let's take that correlation function.
Courant Events Right: And let's bring them together two by two, in what we would call the S-channel, and then let's look at what we get. So we are going to get an expansion in terms of certain amplitudes, Ts.
Courant Events Right: And, Gs, which are products of left and right conformal blocks.
Courant Events Right: The conformal blocks are known functions.
Courant Events Right: They are determined by Zemologikov's recursion relation, and they depend only on the central charge and on the dimension. But now, the crucial role of this 1,3 field is that it determines, actually, ratios of the amplitudes
Courant Events Right: D, where S has been shifted by two units.
Courant Events Right: So that means that you can somehow, you can sum up this expression here.
Courant Events Right: In such a way that you regroup everything that is shifted by units of 2.
Courant Events Right: And then you will get this kind of expression here, where now the H's are interchiral conformal blocks.
Courant Events Right: And S is now restricted to an interval of next 2.
Courant Events Right: So that's very helpful.
Courant Events Right: You can also produce the…
Courant Events Right: interchiral variants of those representations, R and W that we saw just before, they will then acquire a tilde to show that they are interchiral.
Courant Events Right: And then, the idea, as I said, is that now you should bring together points in three different ways, and you should claim that all of this is the expansion of the same correlation function
Courant Events Right: So… We know the spectrum of upscaling dimensions.
Courant Events Right: And we can find help by studying the solutions in view of this ON symmetry.
Courant Events Right: It's possible to constrain the solution space a bit by looking at something that's called a combinatorial map that will appear on the next slide.
Courant Events Right: And in favorable cases, at least numerically, this will give us a unique solution, a solution that describes a correlation function, but it remains
Courant Events Right: For me to say, what does this correlation function really mean?
Courant Events Right: Okay, this will come in a short moment.
Courant Events Right: When you write this, this relation to two correlations, you know that they are, well, the final… This one. No, one of the previous slides. You connect relations. Yes.
Courant Events Right: You are operating, right? So, to know that they are already defined things, let's go and continue our service action.
Courant Events Right: So they are already in the…
Courant Events Right: Yeah, so the component blocks, they are… they are convergent in a…
Courant Events Right: Okay, well, the correlation functions will not have singularities, because the conformal blocks will behave… they will be combined in a very specific way that will turn out to be such that in the end, there are no,
Courant Events Right: There are no similarities in the correlation function.
Courant Events Right: No. This you have to find by computation.
Courant Events Right: It's very important, because this determines the structure of it.
Courant Events Right: Yes?
Courant Events Right: Yes.
Courant Events Right: What's presented here is… Probably so that other organizations.
Courant Events Right: Portland.
Courant Events Right: Okay, so this, this is…
Courant Events Right: Those results describe what would be the continuum limit, actually, of any model of non-intersecting loops having this OM symmetry.
Courant Events Right: So the specific lattice model on this hexagonal lattice, this will have this continuum limit, but if you wanted to start with a slightly different model on a square lattice, or whatever lattice, you would get the same continuum limit.
Courant Events Right: No? There's a lot of work for mathematicians to do if you really want to prove, I mean, the…
Courant Events Right: for example, percolation, that's a case where things have been reasonably proved, okay? Well, have been proved. But, as long as you take more general values of N, I don't think any of this is more than…
Courant Events Right: Extremely well-established conjectures… conjectures, so… But we have a whole collaboration for 5 years, so… You can work.
Courant Events Right: Okay, good. So, let's get a little bit back to this legit model here.
Courant Events Right: So the lattice model comes also with a relation to diagram algebras, which gives another entry point in mathematics, but this time not in probability theory, but in algebra.
Courant Events Right: So if you look at what it takes to move past a couple of vertices.
Courant Events Right: You will see that all possible ways of covering two vertices, they are given by these eight diagrams, and what you should imagine is that if you take the product of
Courant Events Right: this update R, both on all human sites, and then on all odd sites. Then you're going to construct your whole lattice models, and you are going to construct all possible configurations of it.
Courant Events Right: So this, object, the linear operator that does that, that's called the transfer matrix, and it belongs to a certain cellular algebra that's called the unoriented Jones-Temper-Lieb algebra, some kind of variant of the usual Temple-Lieb algebra with periodic boundary conditions.
Courant Events Right: And, as anything in algebra, it acts on certain things, so it acts on…
Courant Events Right: It has representations, which are standard modules, and those standard modules, they are so-called link patterns, they are pairwise connections between points.
Courant Events Right: with certain defects, and the defects, they will sort of represent these lagged operators that I… that I showed you.
Courant Events Right: And the magic thing that happens here is that those R and S labels that you saw first in the conformal field theory, they have a very precise definition also in the lattice model, so that you can consider it both ways.
Courant Events Right: So it corresponds to draw defects, as before, and some momentum variable S whenever something spirals around your periodic boundary condition in the lattice model.
Courant Events Right: So, now… the role of the Owen symmetry here, comes from the fact
Courant Events Right: that, Owen, has a commutant, which is the Bravo algebra. The Brava algebra, that's what you would get if you pairwise connected things without regarding, the constraint of non-self intersections.
Courant Events Right: And so, when you, sort of.
Courant Events Right: use the branching rule to take the brow algebra back to this temporary deep algebra without the self-interceptions, then you will find that the stage space will decompose in this way.
Courant Events Right: And that looks extremely familiar, because this is precisely the kind of decomposition that you already saw in the conformal limit, where now these representations over here, they are those standard modules, and these guys, Lambda RS, they are exactly the same as before.
Courant Events Right: So that means that the structure of the stage space and the conformal limit is seen exactly at finite size.
Courant Events Right: For a cylinder of finite circumference L, L sites. It seemed precisely, provided just that L is big enough that it can accommodate these two R defect lines.
Courant Events Right: So it's completely stable with the size, and this is very fortunate, because this is a crucial thing that will help us analyze this model.
Courant Events Right: So, the conjecture is now…
Courant Events Right: that, the continuum limit, so to speak, the critical limit of this unoriented Jones-Timberly deep algebra, that's my conformal field theory, and the continuum limit of my standard module representation, that's those interchiral, representation that you saw before.
Courant Events Right: Which is another thing that you might want to prove.
Courant Events Right: Let's get back to correlation functions.
Courant Events Right: Let's imagine I have 4 points.
Courant Events Right: Let's imagine that I represent those four points, just 5 small dots, and let's imagine that I look at a case where one point ejects
Courant Events Right: Three lines?
Courant Events Right: And the other points they take is online, Ben,
Courant Events Right: I claim that there is something called a combinatorial map that is in projection with correlation functions.
Courant Events Right: So, the idea is that each point here that throws out legs, it comes with a cyclic order, and if you start turning around this point, for example, if you turn around the first point.
Courant Events Right: and you would like to connect now things like this. This is the same combinatorial map, but if you start ruining the cyclic order, you will get a different combinatorial map. So, this is one correlation function, and this is a different correlation function.
Courant Events Right: So, in the case of three points, it turns out that whenever you take a choice of fields.
Courant Events Right: You will just get one single combinatorial map, but as you saw here on the example, even if you take the same fields, say three legs, one leg, one leg, one leg, you can actually get different
Courant Events Right: combinatorial maps for four points. So there are going to be more than one combinatorial map.
Courant Events Right: So here are some examples with three points.
Courant Events Right: 3, 2 legs, one, one legs. You can also sometimes have zero legs, then you need those diagonal operators, the ones that were called BP, but in that case, you are not allowed to contract,
Courant Events Right: Points that come out of
Courant Events Right: one black dot, if they don't do anything globally, so they have to move around
Courant Events Right: Save one of those points here, otherwise it's not a valid combinatorial mapping.
Courant Events Right: So I have a much more formal definition of a combinatorial map, but I just wanted to give you first the idea.
Courant Events Right: So, now, let's now move to four points.
Courant Events Right: And, let's do the bootstrap.
Courant Events Right: So, the idea was to bring together the points two by two, as I said, and there's a useful characteristic that we shall need.
Courant Events Right: Imagine that you draw some close contour that will relate two points.
Courant Events Right: Two points will be on the inside of this contour, and the other two points on the outside.
Courant Events Right: If you draw this contour in such a way that it intersects the least possible number of legs.
Courant Events Right: Then, half of that number of legs, that's called the signature.
Courant Events Right: And, this signature is super useful for, for setting up the bootstrap, because it's going to constrain
Courant Events Right: The spectrum of the fields that can propagate between those two contractions of points.
Courant Events Right: In fact, the R-index that can propagate is bigger than or equal to this
Courant Events Right: To this signature in the corresponding channel in all three channels.
Courant Events Right: So, here's another conjecture. The dimension of the bootstrap solution space is just the number of combinatorial maps.
Courant Events Right: So this does not necessarily mean that if you give me a combinatorial map, and if you give me some numerical bootstrap solution, that I can find the correlation function that corresponds to that combinatorial map, because in the solution space, for example.
Courant Events Right: Is, sufficiently… has a sufficiently high dimension, and can be difficult to disentangle which one is which one.
Courant Events Right: And I think there's a slide now that will illustrate this point.
Courant Events Right: For example, take this… Case here with three legs, one leg, two legs, two legs.
Courant Events Right: You can check that there are 5 different combinatorial maps. You can compute their signatures.
Courant Events Right: And you can see that the third, fourth, and the fifth, they are each distinguished by their signature. There's one C map for each signature, but the first two guys, unfortunately, they have the same signature, so that means that on the bootstrap side, you will just find a two-dimensional solution space.
Courant Events Right: And it will be, in general, difficult to say which one corresponds to the first, and which becomes… belongs to the second combinatorial map. I thought the signature was just a single number. It's a triple of numbers, because there are three possible ways of contracting the points.
Courant Events Right: Right? So if you respect this point with this point, and then this point with this point, then you should look at this S channel, and then there's a T channel, and a U channel for the other two ways of doing it.
Courant Events Right: Okay.
Courant Events Right: Now let's talk about magic.
Courant Events Right: So,
Courant Events Right: In general, in this problem, there are going to be different kinds of loops, which will all have, in general, different statistical weights.
Courant Events Right: So, for example, suppose that you use only diagonal fields, those fields that did not have any legs.
Courant Events Right: then, you could have a loop that's around points, or some other two points, or some other two points, and those loops, I could give them weight W, S, T, and U for the S, T, and U channel.
Courant Events Right: I could also have loops that surround just one point, and those loops, they would have weight 1, 2, 3, and 4, probably 1, 2, 3, and 4, and I call them, external fields.
Courant Events Right: If you had the X, then of course you would have only a subset of those possibilities, but in general, you should think that you can have
Courant Events Right: WI fields, so for those external fields, and you can have WX field with X equals STRU for those channel fields.
Courant Events Right: And the idea would be to compute the correlation function with all those different statistical weights, in addition to the weight n of a trivial loop that doesn't surround anything.
Courant Events Right: Now, we have observed.
Courant Events Right: that if you set up this problem here on Alexis.
Courant Events Right: of… on a cylinder of circumference, Capitol Hill.
Courant Events Right: If you take the two pair of points to be separated by round L lattice spacings, infuse this…
Courant Events Right: Aramita K.
Courant Events Right: not necessarily the critical K, but your monomer capacity.
Courant Events Right: And you lose all those loop weights.
Courant Events Right: Well, the first thing I can say is that you can resolve that into the eigenvalues of the transfer matrix that I defined.
Courant Events Right: to the L power, and that comes with some… Avenue troops. Hey.
Courant Events Right: Nope.
Courant Events Right: On the next slide… The key observation is that
Courant Events Right: If I take that eigenvalue amplitude.
Courant Events Right: And if I take a ratio of it for different values of the channel field X,
Courant Events Right: Then it turns out that this is precisely the ratio between the four-point structure constants that appear in the continuum limit.
Courant Events Right: So this is… Truly remarkable, because on the left-hand side.
Courant Events Right: you are on a lattice model, you are in finite size, and you're maybe not even at the critical point, because you would find the same result for any value of k.
Courant Events Right: On the right-hand side, you're in a conformant field theory.
Courant Events Right: So somehow, there is some information in this ratio that does not speak at all to the conformal field theory. It must speak to something else, and since I already showed you this bimodule decomposition, you will see what they should speak to is the global symmetry.
Courant Events Right: So, the message here is… that,
Courant Events Right: You cannot expect to solve this kind of theory if you use only CMT techniques.
Courant Events Right: You should also take into account this global symmetry.
Courant Events Right: Because otherwise, there seems to be no room for understanding this kind of phenomenon.
Courant Events Right: Now, what we have found… on the CFT side.
Courant Events Right: Is that if you divide the four-point function by a certain
Courant Events Right: preference version of the four-point function, four-point structure constant that I will define shortly, then the result, has this form here.
Courant Events Right: This part here that is actually some kind of answer to your question about analyticity, because there's a certain ball here that is adjusted in such a way that
Courant Events Right: That you, you have a, a holomorphic dependence of the four-point function on each momentum variable. And,
Courant Events Right: This function here, F, is the known function, but the thing that is not known is the small D.
Courant Events Right: So the conformal bootstrap cannot predict
Courant Events Right: predict the small d, but it turns out to be a polynomial. And it turns out to be the same polynomial that you can get from this ratio here in the lattice model.
Courant Events Right: The ratio in the data small is all of this, so you can isolate the small div.
Courant Events Right: So, the end game is to identify this small d if you want to have complete understanding of your correlation function.
Courant Events Right: So, first, the first thing we did was we used the numerical conformal bootstrap, and we were able to guess what those polynomials were just by evaluating them in a certain number of points until we could understand that indeed it was a polynomial.
Courant Events Right: But now, starting now, but not yet written up, we can do better, because we can actually compute those small ds systematically using ideas of lattice algebras.
Courant Events Right: of the template type.
Courant Events Right: Now, in what follows.
Courant Events Right: There's a special function that will appear all the time, it's called the bounds double gamma function, it's defined by this horrible integral here, and it satisfies shift equations, so whenever the integral does not converge, you can just shift so that it will converge. And,
Courant Events Right: It turns out that…
Courant Events Right: If you thought that four-point structure constants would just factorize as a product of three-point structure constants.
Courant Events Right: The object that can do that is… is the DREP. So that's a product of two normalization by two-point constants of two three-point structure constants.
Courant Events Right: But this is, as we have seen, not really the four-point function, structure constant, excuse me, because it has to be multiplied again by this polynomial's all t.
Courant Events Right: But still, it signals out a particular role of these three-point structure constant C here.
Courant Events Right: And,
Courant Events Right: Okay, and this is… is known, so I will skip it. So this three-point structure constant, here.
Courant Events Right: We observe that, If you insert.
Courant Events Right: The case 101010, you get exactly the,
Courant Events Right: the probability that 3 points belong to the same loop, the thing that was solved by our mathematician Prince.
Courant Events Right: And so what we will then do is that we will conjecture that this formula here actually gives the three-point structure constant for any combinatorial map.
Courant Events Right: You just have to normalize things correctly by…
Courant Events Right: point functions, so that you're independent on field normalizations. And then, what you will get if you confront this with numerics is that, basically, for finite size, you get this cloud of points here. The redder it gets, the bigger it is. To extrapolate, you get those black points.
Courant Events Right: And the analytic result, it just… it's spot on.
Courant Events Right: And this we have verified for 15 or 20 different three-point correlation functions, so we are pretty confident that this conjecture is true, but it has not been proven yet, so this is your 27th task, if you accept the assignment.
Courant Events Right: Now, let me move to a paper that will come out in… 4 hours on archive?
Courant Events Right: I don't know what XXX is yet. We can also play this game here on a Taurus. On a Taurus, we look at one-point functions.
Courant Events Right: So, think of putting one point on a torus that spills out a certain number of legs. Legs are not allowed to contract unless they do something useful, which is to wrap around, periodic directions.
Courant Events Right: Then, the sphere four-point function, it looks like this. The torus one-point function, it looks like this, in full scrap notation, and it turns out that parameters somehow are…
Courant Events Right: A little bit related, some things get multiplied by 2, the central charge is not the same here and here, but there's a relation between torus and sphere correlation functions.
Courant Events Right: That we can put to good use.
Courant Events Right: But, you see, what happens on this sphere?
Courant Events Right: If you take three diagonal fields, is that the lagged field that sits here on the corner, it can spit out a certain number of cycles that turn around each point.
Courant Events Right: And on a thoroughs, if you insert one point, what it can do is that it can insert a certain number of loop ends that will wind around one or the other, or both periodic directions on the thoroughs in this way.
Courant Events Right: And there's sort of a correspondence between one and the other, but the thing that goes,
Courant Events Right: awry is that you… you are not allowed to have, on the torus, something that turns just around one point, so this has to be completely different.
Courant Events Right: So, the way we deal with this is to take place…
Courant Events Right: this operator on the, on the sphere, such that the corresponding group weight is zero.
Courant Events Right: So in that way, we can make torus and sphere correlating functions, coincide?
Courant Events Right: And we can look at various kinds of combinatorial maps.
Courant Events Right: That correspond to…
Courant Events Right: Rowing our legs, moving around certain periodic directions. And also, for all of this, we can work out these reference pieces.
Courant Events Right: And it turns out also that there are polynomials around, But,
Courant Events Right: The message is that those polynomials, we now know how to, to compute them, because they come from diagram algebra.
Courant Events Right: So, we can actually figure out, from the numerical bootstrap the small d, but then in an analytic computation, using lattice algebras, we can actually compute what those small g's are, and in those two lines here, which are not supposed to be really understandable, but which I…
Courant Events Right: bring up there, but nevertheless, I tell you how to do it.
Courant Events Right: Okay, so 4 points on a sphere, 1 point on a torus,
Courant Events Right: Suppose we want to have boundary conditions. What could we do then? Well, there are two big families of boundary conditions in group models, some that preserve the ON symmetry, and some that break down ON symmetry to a subgroup.
Courant Events Right: So, for example, one kind of diagonal boundary condition that you can have is three boundary conditions.
Courant Events Right: loops just come to the boundary, and then they reflect off it, and they do nothing particular. But you can also project
Courant Events Right: Sprint, that's on Hogue.
Courant Events Right: come together on the boundary on a… on a spin S-half representation, and this is a more general construction.
Courant Events Right: But that's still diagonal?
Courant Events Right: If you want non-diagonal boundary conditions, you could modify the weight of loops that come close to the boundary, or you could do tear-lay boundary conditions, where open-loop strands can come out of the boundary.
Courant Events Right: What I want to focus on is the diagonal boundary conditions.
Courant Events Right: And now I will move to a geometry that has a boundary, that could be a disk or an upper half plane. You could conformally map between one and the other.
Courant Events Right: So, in the upper half plane.
Courant Events Right: There are some very nice, known results.
Courant Events Right: that are actual theorems,
Courant Events Right: There is a crossing probability, conjectured by Carly, proved by Smirnoff for percolation. The probability that a percolation cluster connects to disjoint boundary arcs.
Courant Events Right: So that's a four-point function of boundary fields.
Courant Events Right: This is a super interesting and very nice correlation function, but it's not the one that we're interested in, because boundary fields, they are all degenerate.
Courant Events Right: And so, comes out satisfying differential equations, and this is not what we want here. We want something more complicated.
Courant Events Right: There's a percolation formula by Strong.
Courant Events Right: It's also sometimes called the left passage probability. You take caudal SLE in the upper half plane, and the probability that caudal SLE goes to the left of a bulk point.
Courant Events Right: That's a correlation function between one bulk field and two boundary fields, but those boundary fields are also degenerate, so also this satisfies the differential equation, and as a consequence, both of them are given by a hypergeometric function.
Courant Events Right: But, what we want to do?
Courant Events Right: Is to study one- and two-point functions of bulk fields that sit in the upper half plane, so they don't sit on the boundary, and in that case, we have no differential equations.
Courant Events Right: Also, we don't get just one single conformal block, we get an infinite sum over conformal blocks, a bit like you have seen in this interchiral business.
Courant Events Right: So, here is, what we can do about…
Courant Events Right: Two-point functions on the upper half plane.
Courant Events Right: So, if we use complex coordinates, the coordinate dependence is completely known, but what is not known is this factor in G sitting up here, a non-trivial function of the cross ratio.
Courant Events Right: So, to find out what it is, we need to use the bulk boundary bootstrap equations. The boundary is this horizontal segment here. There are two ways of bringing together the points. Either you take the two ball points together.
Courant Events Right: And then you bring the result to the boundary, so that gives the bulk channel, or S channel expansion. Or you take each of them to the boundary directly, then you get the T channel, or boundary channel expression.
Courant Events Right: So, I mean, it means that 2 minus the bar 2, the same 2.
Courant Events Right: That's not… Yes, yes.
Courant Events Right: I, I have to check, the strange. If it is real, it's one of them. There are no normal, there is no strange. Maybe, maybe it's a misprint, right?
Courant Events Right: No, no, I think it's all right. So, no, it was fantastic.
Courant Events Right: I get some pills.
Courant Events Right: Thanks, Dale. So, okay,
Courant Events Right: So, one-point functions are non-zero only for diagonal or degenerate fields, and the boundary spectrum consists of diagonal fields, because the boundary conditions are diagonal, and so what we can find
Courant Events Right: Is, for example, if we can find the two-point connectivity, what's the probability that,
Courant Events Right: that those two points, they sit in the same FK cluster.
Courant Events Right: And, that turns out to take the following structure for free and wired boundary conditions, this plus-minus combination of these two Fs, and each of the Fs, they are a sum of an infinite number of conformal blocks with some coefficients.
Courant Events Right: And it checks excellently against numerics.
Courant Events Right: And now, I'm getting close to the end, so now the…
Courant Events Right: Best game we can play now in this, and it's,
Courant Events Right: In this context here is now to take two bulk fields, such that we have four different types of loops.
Courant Events Right: groups that surround none of them points, they have a weight n.
Courant Events Right: oops, that's the right one… that run around one of the points has weight W1, the other point is W2, or W12 if they run around both of them. So that's the most…
Courant Events Right: General?
Courant Events Right: Two-point function of diagonal operators.
Courant Events Right: And it turns out, in that case, that all this conformal stuff, the reference pieces that they can work out, and the only thing that remains unknown
Courant Events Right: is… That in the boundary channel, we still get those polynomials.
Courant Events Right: Still another… Problem with polynomials.
Courant Events Right: And in this brain box here is… that is a Jones-Wenter projector in the diagram algebra. It turns out that the polynomial is equal to this diagram.
Courant Events Right: And this diagram can be evaluated by some recursion relations, so this problem actually, by putting together all those pieces, is completely solved.
Courant Events Right: So once again, the take-home message here is that you cannot solve these problems just by doing CFT. You have to do CFT plus diagram algebra.
Courant Events Right: if you don't put together those two pieces of information, I don't think you can really get there.
Courant Events Right: So now I arrive at my conclusions.
Courant Events Right: So we have,
Courant Events Right: We have analytic control, or we are very close to having analytic control of correlation functions of 3 and 4 points on the spheral plane. One point functions on the torus, the annulus we can deal with, I think, but…
Courant Events Right: We need to work a bit more, and 1 and 2 point functions on the disk are the upper half playing.
Courant Events Right: And, the next few years.
Courant Events Right: I'm planning to study operator product expansion in loops model… in loop models.
Courant Events Right: So, you see here that the relation between 3 and 4 point
Courant Events Right: Structure constants were such that there was no factorization, so there is something strange going on on the level of operator product expansions that needs to be figured out.
Courant Events Right: In geometries with boundaries, I only talked about diagonal fields, but non-diagonal fields are also interesting.
Courant Events Right: And, my big dream is to do this kind of thing here on any Riemann surface for any kind of correlation function, and for that, what I would need to do
Courant Events Right: is to develop some kind of Siegel-like approach that will work within the lattice algebra, so that I can keep track of all this, of all these polynomials.
Courant Events Right: And finally, since I…
Courant Events Right: basically characterized or self-characterized as a physicist. I would also like to find nice physics problems, where this approach here can be taken down to Earth, and it will give some amazing,
Courant Events Right: hopefully, predictions for some actual problems that could be related to physics. So, thank you for your…
Courant Events Right: Thank you, ZipReilly.
Courant Events Right: In equals.
Courant Events Right: Thank you, Vice President.
Courant Events Right: About the last moment.
Courant Events Right: Yes? They do see a card from your previous mention about it?
Courant Events Right: I agree.
Courant Events Right: I live very well in two dimensions.
Courant Events Right: You lose a lot of structure if you go to three dimensions.
Courant Events Right: Bye.
Courant Events Right: So there are other things you can do in three dimensions, which are…
Courant Events Right: Slava is a big, big specialist in.
Courant Events Right: But,
Courant Events Right: So you see, for example, all this, the extended symmetry of the lattice model, it came in particular from the fact that things were not allowed to cross.
Courant Events Right: In three dimensions, you don't have such a principle, so… so this you are going to lose.
Courant Events Right: Also, in two dimensions, we started right from slide number two, we started by knowing the spectrum of critical exponents. We know all critical exponents.
Courant Events Right: of those that enter those correlation functions here. Of course, Shin knows other critical exponents.
Courant Events Right: But we don't study correlation functions based on that yet, but maybe that could be done. In three dimensions, you have only approximate, albeit very accurate, knowledge of critical exponents.
Courant Events Right: So you cannot, like we do here, do those bootstrap computations.
Courant Events Right: By taking all conformal fields up to conformal weight 100, and compute everything with 2,000 digits of numerical precision to be really sure about what we do, in three dimensions, things would be much more approximate.
Courant Events Right: So I think… Oh, sorry, what was the question?
Courant Events Right: three-dimensional knowing the processing, dude.
Courant Events Right: Oh, but they will not go. I mean, they will call language.
Courant Events Right: Of course!
Courant Events Right: I was just, explaining that in three dimensions, the topological things that do can do will do in many of the things that we use,
Courant Events Right: Brandon, thank you for that.
Courant Events Right: So it's a much more complicated problem.
Courant Events Right: It's a bit of difficult.
Courant Events Right: So, showing us
Courant Events Right: The more he looks at this, the more he thinks that molded models are basically exactly solving
Courant Events Right: Well, various influencers.
Courant Events Right: Okay, and…
Courant Events Right: Shouldn't be cleared over here.
Courant Events Right: Oh, it's not…
Courant Events Right: Right, so the message is that in two dimensions, we believe that these models are exactly solvable, and we are finding the exact solutions.
Courant Events Right: So it seems that… There's some hope.
Courant Events Right: that for any combinatorial map on any Riemann surface in two dimensions, you would someday be able to hand out an analytic result, maybe even backed up with rigorous proofs.
Courant Events Right: Whereas in three dimensions, it looks, pretty…
Courant Events Right: It looks not as hopeful in three dimensions.
Courant Events Right: Okay. The easing correlations dissatisfy inequalities like the greatest inequality, like the four-point function is founded below by the product of this two-point function and these two points, then this…
Courant Events Right: Another one, do you think one could, examine if there are such correlation inequalities with your models? I mean, have you not…
Courant Events Right: good enough description of the correlations to be able to examine these kind of inequalities? I don't… I don't… honestly, I don't know those inequalities, so maybe it's a basic thing in your trade you should explain.
Courant Events Right: Okay.
Courant Events Right: You were mentioning about the Haven't seen results in your Slack channels. Yes, for their general partners?
Courant Events Right: Yes, but it's not so severe that… Okay, so…
Courant Events Right: If R and S are both integers.
Courant Events Right: Then, these two component blocks, they combine.
Courant Events Right: And, become a logarithmic conformal block.
Courant Events Right: So there are actually things within this combined block here that depend logarithmically on
Courant Events Right: Which is, Jordan… Jordan Partner.
Courant Events Right: Yes, yes.
Courant Events Right: This is… this is perfectly under control, and it's just ranked 2.
Courant Events Right: It does not get worse.
Courant Events Right: So, the reason that in the very beginning, I kept telling you that I wanted to take…
Courant Events Right: generic parameter values of N.
Courant Events Right: is that I know that if I said, for example, n equals to 1 to get percolation.
Courant Events Right: I know that, there will be,
Courant Events Right: There will be Jordan Tills of arbitrarily high rank.
Courant Events Right: Okay, so maybe, maybe, so I think correlation functions still, they can be handled by continuity.
Courant Events Right: But inside, there is a very complicated structure.
Courant Events Right: And that structure can be used to produce Larithms, in some cases.
Courant Events Right: And this is the… Highly non-trivial to study, but it's sort of also somewhere on the to-do list.
Courant Events Right: Okay, that was good.
Courant Events Right: A little more feature icons?
Courant Events Right: Thank you.
Courant Events Right: Cool.
Courant Events Right: Okay, thank you, Mr. Smith.
Courant Events Right: Excuse me.
Courant Events Right: Nice to spoiled for a while.
Courant Events Right: Hopefully.