Courant Events Right: It's fine.
Courant Events Right: Okay, so maybe we start, this afternoon. Let's start the afternoon session.
Courant Events Right: So, we have the pleasure to have, Diane Mazakh from Sioux and HES.
Courant Events Right: And he's going to talk about the confirmation, more equal to 3.
Courant Events Right: Okay, fair enough.
Courant Events Right: Thanks, thanks very much, Colin, for the introduction, and thanks also to Nina and Roland for bringing us
Courant Events Right: Altogether, and asking me to speak.
Courant Events Right: So, as the… as the title of the talk suggests, some of the main… main topic of the talk is going to be conformability in more than two dimensions.
Courant Events Right: Which, it's probably fair to say, has been traditionally more a part of
Courant Events Right: Physics, and it hasn't really penetrated into, like, mainstream pure mathematics.
Courant Events Right: And maybe this kind of state of affairs can be nicely explained by a quote by David Kashidan, a great mathematician who said that physics is very interesting, there are many great theorems, but unfortunately, there are no definitions.
Courant Events Right: So, but today, I would like to try to convince you that, at least in the case of CFTD, this is not quite fair to use this quote.
Courant Events Right: Because there is something rather close to a definition, which is this… goes under the name of Control Bootstrap, which is some kind of axiomatic approach to control filter in… which works in general dimension.
Courant Events Right: So the goal of the talk
Courant Events Right: is to firstly explain what a conform bootstrap is, in this context of general dimensions.
Courant Events Right: And because we are in a probabilistic conference, I try to connect it as much as possible with a probabilistic approach.
Courant Events Right: And finally, as a bonus, there's going to be an analogy, an appearance of speculative theory of automorphic forms.
Courant Events Right: Somehow, yeah, some of my… the goal of the talk is to somehow explain… maybe explain to the probabilists in the audience what Conform Bootstrap is about in general dimension, but also to the experts on the conform bootstrap on how it actually relates to probability.
Courant Events Right: Although, I should say I'm not really an expert on probabilistic approaches, but I try to study a little bit about it to make sure I can give a more coherent talk.
Courant Events Right: So,
Courant Events Right: The plan… corresponding plan of the talk is here, so I'll start by talking about conform facilitaries as measures on spaces of distributions.
Courant Events Right: Then, to bring in the bootstrap, we need reflection positivity, so I'll review what reflection positivity is, and how it allows us to formulate this axiomatic bootstrap approach.
Courant Events Right: And finally, somewhat more speculatively.
Courant Events Right: I will talk about how the bootstrap can be implemented even without reflection positivity, how some of its great successes can optimistically be replicated for non-reflection positive models.
Courant Events Right: And I should say that I'm happy to take questions at any time, so please don't hesitate to interrupt me.
Courant Events Right: Okay, very good, so…
Courant Events Right: Of course, conform field theories arise in many different ways, in many different contexts in physics, but maybe the closest to the heart
Courant Events Right: of this audience is how they arise, sort of classically from statistical lattice models, and the fundamental example that people usually start with, and I will do the same, is the Ising model in D dimension. So the Ising model in the ZD lattice, where D is greater than or equal to 2,
Courant Events Right: Which, we can define as a probability measure on some maps to minus 1 and 1 from the lattice, or let's say from the truncated lattice, in this box, and the measure takes this form, so there's some normalization, and there is an interaction Hamiltonian, which, in the simplest case, is just the nearest neighbor… nearest neighbor interaction, sigma X, sigma Y.
Courant Events Right: Now, if you want to be completely rigorous, you should put some boundary condition on this box, and then when you take the size of the box to infinity, you get a probability measure on the space of maps from the vertex set to this set of minus 1, minus 1, and 1.
Courant Events Right: Okay, and now…
Courant Events Right: Various interesting things start happening once you study the properties of this measure. So there is a critical inverse temperature, such that above this critical temperature, the two-point functions decay exponentially at large distances.
Courant Events Right: And exactly at this critical temperature, we expect that the correlations decay like a power law. So there's some overall constant.
Courant Events Right: And there is a parallel 1 over X minus y to sum 2 delta, so this is the convention that's usual in physics, where delta is the scaling exponent of this field sigma.
Courant Events Right: And, well, this has been completely resolved in two dimensions, basically goes back to the exact solution of Onsager, a two-dimensional easing model, and you find that it is delta is equal to 1 over 8.
Courant Events Right: I mean, three dimensions, there's been a lot of, lot of interesting work, and currently the best estimate, comes from, comes from a bootstrap, is this, it's this number.
Courant Events Right: known to many, many decimal places, but I should say, this number does not come from analyzing the statistic clottis model, it comes from this
Courant Events Right: Method which works directly in the continuum, which I'll review.
Courant Events Right: In the main part of the talk. And then, above 3 dimensions, it's just a free field with dimension being equal to D minus 2 over 2.
Courant Events Right: that weedsome.
Courant Events Right: In three dimensions, yes, yes, yeah, I mean, that's right. There have been various previous works on this, but…
Courant Events Right: Performance episodes on expansion is certainly one of the most important ones.
Courant Events Right: Yeah, so I… yeah, I apologize if the references are not going to be complete.
Courant Events Right: Okay, so… Now, it turns out that we can take a…
Courant Events Right: You can take a nice continuum limit, exactly at its critical point, so exactly at its critical temperature,
Courant Events Right: we can define this random distribution, so for A being some lattice spacing.
Courant Events Right: can define this distribution on the lattice phi of A, to be basically some normalization factor, which makes sure that everything converges less than the continuum limit, times this sum of delta masses weighted by the… by the sigma.
Courant Events Right: And so, defined in this way, phi of A is a random distribution in D dimensions.
Courant Events Right: And now, the fundamental conjectures in the field can be formulated as follows. So, most importantly… well, maybe not most important, but this limit should exist, so this… you can take the limit A goes to 0, and you get some probability distribution on the… on the space of distributions.
Courant Events Right: And rather remarkably, this resulting measure is somewhat very universal. It doesn't depend whether you start with a ZD lattice or some other lattice, it doesn't depend on whether you add some extra couplings in the Hamiltonian. You always pretty much get the same thing, so there is some kind of nice, rigid, beautiful mathematical object, which is this continuum measure.
Courant Events Right: Arising from the Ising model in D-dimensions.
Courant Events Right: And one of the manifestations of its beauty is that it's conformally invariant.
Courant Events Right: So what is… what is this conformal invariant?
Courant Events Right: So this, idea… well…
Courant Events Right: this starts with the RG, with the idea that the model at critical point is somewhat similar on the overall rescaling.
Courant Events Right: And then Polyakov realized that, or had a great insight, that when you have a scaling variant system that's somehow local, where all the interactions are local, then actually the symmetry should be enhanced to conformal invariance, so some transformations which are scaling locally.
Courant Events Right: So let's try to be precise about it. So what's conformal invariance? Definition, we have two remaining manifolds, M and tilde, and the deformorphism between them is conformal. If the pullback of the sort of target space metric is just the original metric times some vial prefecture.
Courant Events Right: This, factor omega.
Courant Events Right: And, unlike in two dimensions, where confound transformations are basically holomorphic or anti-holomorphic maps.
Courant Events Right: In three dimensions and higher, there is this rigidity theorem of Lieuval, which says that
Courant Events Right: Any conformal deformos in between domains in RD actually extends to
Courant Events Right: to the… to the sphere. So now we are thinking of sphere as the strographic projection of RD. So we are just adding… adding a point to RD,
Courant Events Right: at infinity, making it into SD, and then,
Courant Events Right: any conformap always extends. And furthermore, there is a nice, explicit description of what the group of all of these transformations is.
Courant Events Right: It's simply the… well, if you want them to be orientation-preserving, then it's this,
Courant Events Right: Special orthogonal group, 1 comma d plus 1.
Courant Events Right: So in the group, in the indefinite signature of D plus 2 dimensions. Okay, so that's conformal transformation in D dimensions, and well…
Courant Events Right: Conjecturally, for the 3D-ising model, for example, the resulting measure on the space of distributions is invariant under this group, so let me try to be precise about what this means.
Courant Events Right: So let's, let's take F to be some test function on the sphere. I mean, it doesn't really matter if it's on the sphere or on RD, but to be completely precise, it should be on the sphere, because there are point… there are transformations in the
Courant Events Right: conformal group which map infinity to finite points, so let's F to be smooth on the sphere, alpha be a conformal transformation, and phi to be a distribution.
Courant Events Right: And now, how does confound resolution act on a distribution? Well, it simply moves around the test function, multiplies by this prefactor. This is where the delta appears, so somehow the definition includes… well, delta is a part of the definition.
Courant Events Right: And then we can say that if we have some measure on a space of distributions, then it's conformal invariant with this parameter delta if the measure is preserved by this transformation, right? So the distribution
Courant Events Right: Has a conform… has come from a covariant, but the measure should be conformal invariant with this delta.
Courant Events Right: Conjecturally, the scaling limit
Courant Events Right: of the Ising model, at the critical temperature should be then conformal invariant in all dimensions, and this is actually a theorem in two dimensions. In three dimensions, it's certainly open, and in four dimensions, it's also a theorem, because the scalar limit is a Gaussian free field.
Courant Events Right: Okay, so that's… That's it. Now… So… Now… Confirm what… yeah.
Courant Events Right: What about Gaussian free field? So the… why is the Gaussian free field conformed invariant?
Courant Events Right: Well, what is a cost-in-free field? It's a…
Courant Events Right: measure on spatial distribution such that this, phi of F is a Gaussian random variable.
Courant Events Right: With a zero mean and a…
Courant Events Right: Covariance being given by this formula.
Courant Events Right: And then the proof of conformal invariance is just the proof that this, this covariance kernel is, is conformal… is conformally covariant, with this, delta being B minus 2 over 2. Okay, so for Gaussian free field, it's… it's easy. For a 3D Ising model, it's very hard.
Courant Events Right: Okay, so that's the… that's the first example.
Courant Events Right: Which I'll talk about. A second example, which I will talk about, is percolation.
Courant Events Right: Let me… let me quickly review it.
Courant Events Right: For the non-experts.
Courant Events Right: So what is percolation? Let's do percolation on the… on the square lattice and the dimensions.
Courant Events Right: Now, the edges are either open or closed, so they are independent, identical distributed random variables. With probability P, they are open. With probability 1 minus P, they are closed.
Courant Events Right: And let me denote, by this, in this notation, X to the Y, the event that, two vertices
Courant Events Right: are connected by a sequence of open edges.
Courant Events Right: And then it turns out that there is a critical probability
Courant Events Right: Such that below this critical probability,
Courant Events Right: The… any… any vertex is… is, certainly… not connected to Infinita.
Courant Events Right: So that there is no infinite,
Courant Events Right: Open cluster, and above the critical probability, there is some positive probability that
Courant Events Right: Any finite vertex actually is connected to infinity.
Courant Events Right: So this is how we define the probability in this case.
Courant Events Right: And the corresponding conjectures, which express
Courant Events Right: Conformal invariants are the following, so this critical probability, the two-point connectivity.
Courant Events Right: Should, again, scale like a parallel.
Courant Events Right: as… at large separations, now with some different critical… with some different critical exponent, which is, I put this prime.
Courant Events Right: And the resultant scaling limit should also be conformal invariant in any dimensions.
Courant Events Right: Now, in two dimensions, this is a… this is a theorem.
Courant Events Right: Where delta prime is, 5 over 48. It, in physics, it goes back to… to the work of Cardi, and…
Courant Events Right: others, and in mathematics, it was resolved by mapping the problem to SLE6.
Courant Events Right: However, in more than… well, in 3, 4, and 5 dimensions, it's basically completely open. It should define some interacting CFT, where, for example, in three dimensions, the corresponding critical exponent is about 0.477.
Courant Events Right: And now sixth dimension is the upper critical dimension for this, for this model, and again, we get a Gaussian-free field in, in more than…
Courant Events Right: More than 5 dimensions with, delta prime. Yep.
Courant Events Right: So, in… to be easy to define the formula variance of the scale, doing it very nicely in terms of the properties of this random
Courant Events Right: Distribution. What is the corresponding mechanical logics for the population?
Courant Events Right: Well, you can formulate in different ways. Usually, it's formulated by looking at the boundaries of per equation clusters, but actually, I prefer to think about distribution, so to answer your question.
Courant Events Right: There's this, this, this slide.
Courant Events Right: Yeah, so one would like to formulate conformal invariance
Courant Events Right: In this class of models, just as a conformal inherents of a measure in a space of distributions, and you can do it.
Courant Events Right: In the following way. So this,
Courant Events Right: this, maybe goes under the name of divide and Color Models, which were studied by Heckstrom, and then Kamiya actually
Courant Events Right: Discuss this, this idea in the context of the continuum limit.
Courant Events Right: And the definition is as follows. So first, you sample the edges according to critical percolation, and then you define a field living on the vertices, just like in the Ising model, in the following way. So you basically color all of the connected vertex clusters with either plus or minus 1,
Courant Events Right: With equal probability.
Courant Events Right: and all… all independently. So for each cluster, you independently decide whether it's going to be plus one or minus 1, and this defines a field on the… on the vertices. Now, you also have… you can make some… some other choices. For example, instead of plus or minus 1, you can choose basically any… any set of complex numbers with arbitrary.
Courant Events Right: Distribution on them, and this is going to define some random field.
Courant Events Right: And the point of introducing this random field is that now you can represent the connectivity probabilities as expectation values of products of these fields. So, for example, the two-point function in this specific case when we do plus or minus 1 vehicle probability, is just the connectivity, because
Courant Events Right: If X and Y are in different clusters, their color is independent, so you get 0. If they're in the same cluster.
Courant Events Right: You all, you get,
Courant Events Right: always one, so this is a… this is just a true identity. So, now the two-point function literally, should have this… should have this scaling. And,
Courant Events Right: We can, then talk… we can talk about random distribution, which is just the corresponding scaling limit with the percolation critical exponent delta prime of this, of this distribution.
Courant Events Right: And the conjecture conform invariance is that this object should be a conform invariant random element of the distributions on SD. So, that'll be this.
Courant Events Right: So, how does this… yeah, first time I see this definition, very nice, but how does this relate to physics, we usually… in order to come up with some field circulation, we can see that the sphere clean is, like, some…
Courant Events Right: OEM model, and deliver agreement to one, and then put…
Courant Events Right: But this doesn't seem to make much sense mathematically as well. How is this related to this?
Courant Events Right: Well, to be… to be perfectly honest, I… I don't… so, right, so usually the way…
Courant Events Right: We can describe field theory… well, this is described per question, theoretically, as the Q-stay possible where Q goes to 1, but what you said doesn't really make sense, because in the limit where Q goes to 1, there is no field left. The field has lost all the components. That's somewhat related to, like, if you… I think it's related to the fact, maybe, that
Courant Events Right: The edge field is basically boring, it's completely uncorrelated.
Courant Events Right: But this…
Courant Events Right: this works. I mean, this is… well, as far… of course, there's… there's no proof that actually this object is conformal invariant, that it even converges, as far as I know, but I would like to talk to the experts here to explain to me whether this is true, but it seems to be… seems to make sense. It was actually…
Courant Events Right: discussed, rigorously by Kamiya in the context of two dimensions. But I think if it works in two dimensions, there is no reason why it shouldn't work in higher dimensions as well.
Courant Events Right: Alright, so…
Courant Events Right: So, so far, from the way that I've been talking about CFTs, one may conclude that we can try to define CFT in D-dimensions as the study of conformal invariant measures on the space of D-dimensional distributions.
Courant Events Right: And then you can ask, what does this, you know, what does this twiddle That would all mean?
Courant Events Right: And I don't really know what it means, but I can tell you what it doesn't mean. So, for example, it doesn't mean this inclusion, because there are definitely some,
Courant Events Right: some CFTs, which do not come from ordinary measures.
Courant Events Right: For example, when you have fermions or conformal glitch theories, like, all of these things,
Courant Events Right: Certainly makes sense, but they didn't come in this way. And the opposite inclusion also doesn't really work.
Courant Events Right: Because they're…
Courant Events Right: direct… somehow, this definition is too general. There are conformal invariant measures on spaces of distributions which do not look like CFDs at all, and we are going to see some examples at the end of the talk coming from automorphic forms.
Courant Events Right: So somehow, to address these points, one would like to inject into the definition some more properties that we know are true.
Courant Events Right: about CFDs, like, for example, some more properties of these conforming measures,
Courant Events Right: And it… it actually… it turns out that once you do that, you arrive at this nice bootstrap definition of a CFP, at least in the reflection-positive case, and then you can kind of try to apply this bootstrap definition also more generally, and this will address this, this first point as well. So, maybe let me… let me now explain all of this,
Courant Events Right: In more detail.
Courant Events Right: All right, so… Basically, the new constraint that leads to the bootstrap definition is reflection positivity.
Courant Events Right: And to define reflection positivity, let me first introduce the Schrinker functions, or correlation functions. So, in the context of a CFT,
Courant Events Right: Let's take, some measure, some conformal invariant measure with this parameter delta on the space of distributions.
Courant Events Right: And then the nth, Schwinger function.
Courant Events Right: is a distribution on endpoints in RDM, which is just the… which is just the expectation
Courant Events Right: The validation volume of a product of these distributions,
Courant Events Right: measured against the… against these test functions, so it's just the endpoint correlation function.
Courant Events Right: And because the measure is conformal invariant, the district… these Schringer functions are also conformal invariant with the same… the same parameter delta. And as a simple corollary, you get, for example, at one point functions.
Courant Events Right: Vanish, and two-point functions take this nice conformal invariant form.
Courant Events Right: Where the parameter delta appears, appears here in the… in the denominator.
Courant Events Right: Okay, so now… Given British Winger functions, we can define what reflection positivity means.
Courant Events Right: And intuitively, it's just the statement that when you take an expectation value of a… of a bunch of fields in lower half plane and the reflected configuration in the upper half plane, then this is a positive number.
Courant Events Right: And this somehow, at least at the intuitive level of rigor, follows when you have a positive measure.
Courant Events Right: Which is translation variant, and which is some kind of locality, or sometimes the locality is called the Markov property, I think. Basically, this allows you to cut space-time in half, and you can do the statistic of some over the upper half plane and over the lower half plane, and both, okay, both have some real numbers.
Courant Events Right: But you… basically, you get a square because of this Markov property, you're just summing over whatever happens in the middle.
Courant Events Right: I think intuitively, reflection positivator should follow from somehow these three things.
Courant Events Right: But, okay, let's try to make it,
Courant Events Right: The personalized definition of reflection positive is here in terms of the Schwinger function, so let's take XD to be this D… well, you're going to do the reflection in the XD coordinate.
Courant Events Right: And the measure is virtually positive. If you… if you somehow measure it against the test functions in… with endpoints.
Courant Events Right: So FJ is a… it's like a J-point test function.
Courant Events Right: and it has support in the lower half plane, in all of this J core, all of this J coordinates, then this, object is positive. So this object is just the expectation value, basically what's here in the upper picture. Expectation value of the observable in the lower half plane times its reflected,
Courant Events Right: friend in the upper half plane, whereas theta is just a reflection, XD, X equals to minus XD.
Courant Events Right: Okay, like…
Courant Events Right: So, and for isinc, reflection positivity is a theorem, but for percolation, so for this measure of percolation, reflection positivity fails, basically, because percolation doesn't have this locality, locality or Markov property.
Courant Events Right: The measure, you defined that that's not reflection positive, just IID, basically.
Courant Events Right: No, I don't think it's… I don't think it's reflection positive. I mean, it…
Courant Events Right: as… at least, I wouldn't know how to… I wouldn't know how to prove it. I mean, it cannot be reflection positive, because the scaling dimension is saying 3 dimensions is below the unitarity bound, so…
Courant Events Right: It's got an image like this.
Courant Events Right: Well, the distributions can't define a non-local way, because you start from
Courant Events Right: You put the same color in each cl… in each connected cluster?
Courant Events Right: Oops, yeah.
Courant Events Right: Alright, so…
Courant Events Right: Now, to sort of continue in this review of constructive field theory, the nice thing that happens
Courant Events Right: which was explained by Osterwal and Schrader, is that if you have a reflection positive theory, then you can naturally define a Hilbert space, which is another physical Hilbert space for your quantum field theory. And the way this goes is that you start with this vector space, A0, which is just a vector space of the observables put
Courant Events Right: into… Sorry.
Courant Events Right: Just wondering here…
Courant Events Right: what happens at the coinciding points, and I'm never quite sure, right? So, define this as measures on mass prime.
Courant Events Right: So the correlation function sort of makes sense. Yeah. The points come together, but…
Courant Events Right: I mean, in principle, if you're from a QFP, remember the QFT direction, and we look at lots of other product papers that says.
Courant Events Right: cut out the diagonal, so that you sort of don't stay right, for example.
Courant Events Right: So, in these cases, you don't have to do it, because the scaling dimension is sufficiently small. This delta is always less than D over 2, which means that the singularity is actually integrable at all… on all pairs, so as far as I can tell, this… this defines, these are continuous maps on the space of smooth functions and J variables,
Courant Events Right: Without any problem, yeah. But it's because the delta is sufficiently small.
Courant Events Right: enters that…
Courant Events Right: It's experimentally the case in the probabilistic models, like in the Ising models, in percolation, it's, it's not true in gauge theories, but in that case, we don't really have the probabilistic description anyway, so for the mills, the lowest operator has dimension 2, I think, which is basically exactly equal to the Uber.
Courant Events Right: For the over tools. There, it doesn't work.
Courant Events Right: Yep.
Courant Events Right: Yeah, it's a great question. I don't… I can't say I fully understand
Courant Events Right: It's as if the probabilistic description forces upon you to the existence of an operator of dimension less than D over 2, but I… I really wouldn't know how to prove it. Maybe there are some counterexamples, like there are these… there are these fractional Gaussian fields.
Courant Events Right: Which you can apparently define also where delta is greater than D over 2.
Courant Events Right: Something to be discussed.
Courant Events Right: Alright, so there's… then we have this,
Courant Events Right: So, the Hilbert space comes from observables supported in the lower half plane.
Courant Events Right: They carry natural semi-norm, which is just the one coming from reflection positivity, so it's… it's not negative. And now you can define the helper space as the completion of this vector space, modded out by the wide null vectors.
Courant Events Right: I mean, in general, there can be… there can be many null vectors, so that's fine, and you just get… you still get a nice Hilbert space coming from these observables supported in the lower half plane. But this is kind of the usual KFD story, and what Lucia and Mac explained is how it gets enhanced in the presence of conformal symmetry.
Courant Events Right: So if we start from a measure which is conformal invariant, then this Hilbert space is a positive energy unitary representation of SO2 comedy, let's say the Lie Algebra SO2 comedy.
Courant Events Right: So, let me… let me explain quickly why this is the case.
Courant Events Right: So, firstly, why SO2 comedy?
Courant Events Right: So we are somehow going from Euclidean signature, in which case the
Courant Events Right: In which case, the relevant group was SO1 comedy plus 1 to the Lorentzn signature, where the relevant group was SO2 comedy.
Courant Events Right: And that's the group of confirmed automorphisms of the Lorentzian cylinder. So you have R times SD-1, and they have an opposite sign metric on the R factor, and the SD-1 factor.
Courant Events Right: And the way to go to Lawrence and Cylinder from the original picture with the upper half plane, lower half plane, is to go. You first map is Euclidean space.
Courant Events Right: Cut by a co-dimensional hyperplane to the… to this unit disc, so the… the lower half plate goes to the unit ball.
Courant Events Right: This becomes SD-1, and now you can map this to the Euclidean cylinder.
Courant Events Right: where the interior of the unit ball just becomes the bottom half of this Euclidean cylinder. So all of these are conformal maps.
Courant Events Right: And the… the unit… sorry, the un…
Courant Events Right: Well, the unit sphere becomes just the unit sphere at a constant Equity in time.
Courant Events Right: And the reason… the reason is the Lorenson signature becomes relevant.
Courant Events Right: is, because… so if you start from a generator of the Euclidean conformer Group, which is,
Courant Events Right: Which is invariant under this reflection theta.
Courant Events Right: it becomes… still becomes anti-Hermitian, but if you start with one which is odd under this theta, it becomes Hermitian. So you need to put an extra i to make it anti-Hermitian, to give you a unitary operator, and this effectively switches from SO1 commodity plus 1 to SO2 commodity. So it's because of this theta, this reflection.
Courant Events Right: And then, okay, positive energy…
Courant Events Right: is the statement that this Hermitian generator of SO2, which is just the translation.
Courant Events Right: around this Euclidean time direction, or maybe I should say… oh, the Hermitian generator translates in Euclidean time, and the entire mission into Lorentzian time.
Courant Events Right: This needs to be positive, and this was… this was proven already in the case of the usual Hamiltonian in the original Stravaldan Schrader-Baker bias, so nice,
Courant Events Right: iterative argument.
Courant Events Right: Okay, so we have a… for NA…
Courant Events Right: conformed filtering, or a conform measure, which is reflection positive, we have this Hilbert space, and it's a unit representation of SO2 commaD, this positive energy. So, the first thing that we should do with it
Courant Events Right: If we can.
Courant Events Right: Is to decompose it into irreucibles of this, of this group, of this algebra.
Courant Events Right: And the irreducible switch appear.
Courant Events Right: well, they've been classified. They are the irreducible positive energy unitary representations of this algebra, and they are… they carry two labels, so the scaling dimension and spin.
Courant Events Right: Scaling dimension is, basically the eigenvalue of the lowest… so the lowest weight representation, scaling dimension is the eigenvalue of the lotations.
Courant Events Right: under the rotations of the lowest weight, and rho is the representation of SOD, in which this lowest weight is transformed.
Courant Events Right: And under the restriction of SO2 commodity to SO2 times SOD, you get this kind of decomposition, where you have a…
Courant Events Right: The SO2 generators Generator acts like delta plus positive integers.
Courant Events Right: Where the lowest one is the primary, and all the higher ones are descendants.
Courant Events Right: So this… each of these is a fin-dimensional representation of… of SOD.
Courant Events Right: So the representation becomes more and more complicated, but it stays finite-dimensional.
Courant Events Right: Okay, so that's… that's the…
Courant Events Right: there's the states, and now, just to… to look at a concrete example, let's look at the Gaussian free field. So, in Gaussian free field, you can do… one can do everything explicitly, and this
Courant Events Right: Osterwalter Schroeder reconstructed Hilbert space, can be shown to be… Equal to this… Infinite direct sum.
Courant Events Right: of any symmetric power of one of these representations. So R, D minus 2 over 2 comma 0 is a scalar representation, so 0 means it's a trivial representation of SOD, so scalar, every dimension D-2 over 2, and you take a sum of all symmetric powers.
Courant Events Right: Where… well, what's the… what's the idea?
Courant Events Right: The idea is that the asymmetric power is just generated by normal order products of fields in the lower half plane.
Courant Events Right: So, some of these normal order… normal order product is just something that's chosen to be such that all the… all the Enfield normal product are orthogonal.
Courant Events Right: to all of the lower-end fields. So it's kind of the analog of Hermit polynomials for the Gaussian measure on R.
Courant Events Right: So for n equals 0, you just get the vacuum, which is the constant function.
Courant Events Right: Run equals 1, you get a…
Courant Events Right: all of these objects, which just give rise to a single irre usable representation. So some of all of the single field observables are
Courant Events Right: You know, are permuted.
Courant Events Right: Among each other by the action of the conformer group.
Courant Events Right: But for N equals 2,
Courant Events Right: For n equals 2, you get this, you have to subtract, and you get something which is no longer irreucible, decomposes into infinitely many
Courant Events Right: lowest weight representations of this type, with, various spins, so these… all the… basically, all of the even, spins, symmetric traceless representations of SOD.
Courant Events Right: Okay, so that's the Gaussian free field, and of course, it would be great to understand what exactly is the spectrum.
Courant Events Right: for something like the 3Dizing model, what we know is that some… typically some infinite discrete sum of representations. I should say, for Leo Bill, of course, this is continuous.
Courant Events Right: It's a special case where the spectrum is continuous.
Courant Events Right: Let me ask a question? Yep. So, at this level of the theory that you're developing, do you see any interesting reason which would suggest that in higher dimensions, as we know.
Courant Events Right: experimentally, You usually see discrete subs, while the two-dimensional centers will not discrete.
Courant Events Right: There's a possibility it will continue to grow as we do things.
Courant Events Right: I don't know, but perhaps it's because we just haven't found the right unlock of legal in higher dimensions of it.
Courant Events Right: I don't have a… Well, there is an analog, but it's not unitary, right?
Courant Events Right: 140.
Courant Events Right: Right, is this just the Laplace N squared theory?
Courant Events Right: Yeah, I don't… I don't think I have anything smart to say.
Courant Events Right: Okay, so much about states, but to formulate a bootstrap, we need operators, some operators.
Courant Events Right: Operators come in a rather canonical infinite set, Which… which arise from the…
Courant Events Right: somehow I need to start personalized with reconstruction, then it's… now you have the… you have the conformer group, so you can act… you can look at the…
Courant Events Right: eigenstates of D, so you should always look at the eigenstates of the dilutation, and…
Courant Events Right: They are… they have to be…
Courant Events Right: smearings of the local fields, which are concentrated around the point which is fixed by the rotation. So the rotation is this
Courant Events Right: Generator.
Courant Events Right: We'll go like that, and they are all concentrated over here. So, naturally.
Courant Events Right: That's the beauty of conformal symmetry, that automatically, the eigenstates are localized.
Courant Events Right: And now you can just use them.
Courant Events Right: Use this natural basis, Given to you by diagonalizing the rotation, and act with it on the states.
Courant Events Right: So in this way, you get… this is… this is how you get state-operator correspondence. I mean, this is… this is, of course, not… not rigorous, what I just explained, but this is the intuitive reason why there is state operator correspondence.
Courant Events Right: So… and it means that this helper space that we talked about comes with a natural canonical infinite set of Weitemann fields, T of J, where this J is the same label as the… this J of irreleusible representations.
Courant Events Right: Which are… which are operator value distributions, so distributions on…
Courant Events Right: 1 comma d minus 1, valued in some linear operators on some dense domain. So there… all of these operators have some common
Courant Events Right: dense domain inside a Hilbert space. They're unbounded operators, so it's not the entire… not the entirety of H, but
Courant Events Right: That's okay. And they satisfy some properties, which is basically the Weiteman axioms adapted to the conformal case. So they are all Hermitian, well, up to some charge conjugation matrix for non-trivial spin. When you act with that… when you act with such a local operator on the vacuum.
Courant Events Right: So this point number two is the new ingredient that you get in the conformal case, that if you act on a vacuum.
Courant Events Right: you generate something inside this irreducible representation of your Hilbert space.
Courant Events Right: It's still the case that spatial-separated operators commute, just like general Weitman theory.
Courant Events Right: And the conformal symmetry is a statement that for any element of this Lorentzian conformant group.
Courant Events Right: So this is the universe… this is… this is the group that acts
Courant Events Right: some of the exponentiation of the algebra is the universal cover which acts on the Lorentzian cylinder. If you conjugate, you just get the,
Courant Events Right: The field with the test function moved around by the action of this generator.
Courant Events Right: Okay, so this is… this is what I would…
Courant Events Right: Okay, called somehow the Weiteman axioms for…
Courant Events Right: for CFT, and in principle, they should be derivable from the… from having a conformed variant probability measure, but I don't think, as far as I know, that hasn't been done.
Courant Events Right: Okay, now that you have these axioms, basically everything will follow.
Courant Events Right: And the first thing that follows is this uniqueness of three-point functions.
Courant Events Right: So we… we want to… we want to talk about these operators, phi of j.
Courant Events Right: And the operators are just specified by giving all of the matrix elements between any pair of vectors.
Courant Events Right: And these matrix elements are also called the OP coefficients, or three-point functions. So, the key proposition here
Courant Events Right: Is to look at the… when you look at the matrix element.
Courant Events Right: of this Weidman field between a pair of vectors, where the vectors are living in the irreducible representations.
Courant Events Right: then this matrix element is completely determined by conformal symmetry as a function of the two vectors inside these erads, and this function F.
Courant Events Right: And it's determined up to finitely many constants. So these are the… these are the three-point function OPU coefficients CIJK. In two dimensions, their number is always… always one.
Courant Events Right: But in higher dimensions, there can be some finite number of them, which… where the number S…
Courant Events Right: Kind of grows with the growing complexity of the three spin representations.
Courant Events Right: And the idea of the proof is just that they come from a group acts transitively on all triplets of points, so you can always bring them to a canonical position, and it's what specifies your matrix elements.
Courant Events Right: So this is just a consequence of the… of the axioms from the previous slide.
Courant Events Right: And so what is the… what is the conform bootstrap?
Courant Events Right: Well, it's the… the idea going back to the work of these people, but is the… is the idea to… to impose,
Courant Events Right: Comidativity, space-like separation.
Courant Events Right: as a matrix element. So we have this oper… we have this operator, which is the commodator, and we just look at the matrix elements of this commodator, and the matrix element should be zero. So let's test it in a pair of vectors, VI and VM.
Courant Events Right: in… in some errips.
Courant Events Right: inside a Hilbert space, let's assume that we have two test functions which are space-like, and then the commodator vanishes, so the matrix element vanishes.
Courant Events Right: Now, trivially, you can rewrite this as the…
Courant Events Right: this, no, I should say, this is just the inner product in the Hilbert space, I probably didn't say that.
Courant Events Right: So this inner product is equal to this inner product.
Courant Events Right: And now to derive the conform bootstrap equation, you just insert the specular resolution
Courant Events Right: Of the identity here in the middle.
Courant Events Right: When you do that, you basically get an infinite sum.
Courant Events Right: of products of matrix elements of the individual Weiteman fields.
Courant Events Right: So on the left-hand side, you get this kind of sum, where you have, like, CIJN, you're summing over N, which is the sum of the irreducible representations.
Courant Events Right: All the contributions of the descendants get packaged into a conformer block.
Courant Events Right: All the non-trivial information about these Weiteman fields is contained in the CIJ case, and they kind of appear bilinearly.
Courant Events Right: And on the right-hand side, you get the same thing with some permutation of the… of the labels.
Courant Events Right: And by construction, and by this proposition about three-point functions, the conformal blocks are fixed… uniquely fixed by conformal symmetry.
Courant Events Right: So this is… this is how you derive the control bootstrap equation in the… in this kind of manifestly Lorentzian unitary setting.
Courant Events Right: So…
Courant Events Right: This is not a comparable booster equation I'm familiar with, because in the usual booster relation, several blocks are dependent on the coordinates, and usually we discuss when those coordinates have to be chosen.
Courant Events Right: some positions where, you know, some freedom, kinematic relation was satisfied, so then things converge. Here.
Courant Events Right: Do something which looks similar superficially, but…
Courant Events Right: But… but it's so… somewhat different, so…
Courant Events Right: It's, is the… is this somehow… well, the claim… all of the Euclidean boot… come from bootstrap equations are special cases of this one. This is somehow the one that contains everything, just by… by choosing appropriate F1 and F2 and VI and VM, so these are somehow the…
Courant Events Right: These are, like, the metro… Well, say when…
Courant Events Right: when F1 and F2 are smearing a long time.
Courant Events Right: Then these would give you functionals which are, like, integrals around the boundary of the… of the convergent region.
Courant Events Right: So, you can engineer your test function in such a way that they pick up
Courant Events Right: the NF Derivate at the crossing symmetric point, if you want.
Courant Events Right: I'm not, you know, I'm not worried about things like the world.
Courant Events Right: swappability and so on, but the swappability ultimately… It boils down to…
Courant Events Right: So deciding what are the allowed… what is the allowed space of those functions.
Courant Events Right: But, okay.
Courant Events Right: I just wanted to kind of explain the conform bootstrap in a way which really starts from the manifestly Lorentz invariant unitary, you know, in a product with
Courant Events Right: which is unit with respect to Lorentzian conformal group, and the conformal block arises at respect to resolution in this Lorentzian setting. So, the words Lorentzian are somewhat important, and…
Courant Events Right: we can… we can recover the Euclidean-looking story, By choosing appropriate test functions.
Courant Events Right: produce.
Courant Events Right: Very simple question. Special resolution of identity, like, what do you mean here? You just use some of all these representations in your habit space? Yeah, all the, the whole… yeah, just…
Courant Events Right: It's an inner product of two vectors, and you express the inner product of two vectors as a, you know, as a sum over inner products with a basis.
Courant Events Right: Okay, so it's an infinite set of equations for infinible unknowns, and the beauty of it is that, you know, we kind of…
Courant Events Right: want to reduce this huge, uncountable, infinite complexity of cono field theory into the complexity which is somehow much more manageable in numbers. It's like a countably discrete set of data, subject to countably many very concrete equations.
Courant Events Right: And, some comments one can make is that even though we started from some measure.
Courant Events Right: This measure now completely disappeared from the description now. Everything is formulated in terms of this Hilbert space and Weiteman fields acting on the Hilbert space.
Courant Events Right: And therefore, it's kind of natural to generalize and say that, well, the same axioms were also true, even… they'll be true even more generally, even if there is no measure, like when there are fermions or gauge fields.
Courant Events Right: Now, another comment is that, for example, for the 3Dizing model and for a variety of other conformal field theories, these equations imply
Courant Events Right: very good bounds on the spectral data, on the scaling dimensions and the CIJ case.
Courant Events Right: And so what I… the kind of logic that I tried to follow.
Courant Events Right: was… was motivated by this being probably the conference, so I wanted to start from the measure.
Courant Events Right: using Osterwalde Schrader, going to the Hilbert, going to Whitemann, and then driving the bootstrap, but it's… it's also very interesting, maybe even more interesting, to ask the opposite question, whether we can start with the bootstrap accents, which are these very concrete, on a workable equations, and derive from them the…
Courant Events Right: older axiomatizations of quantum filters, such as the Weiteman axioms, and this was actually done in the context of… done for the four-point function in these papers.
Courant Events Right: Alright, so…
Courant Events Right: now we can kind of say that we've axiom hadized reflection-positive CFDs, and they are something like solutions of the crossing equations, but it is also not quite true, because there are other things which happen in reflection-positive CFDs, such as some non-local operators, defects, or putting them on various geometries, which are not
Courant Events Right: not SDs, I was basically just working in the… in this flat space, and it's an… it's an open problem to see whether considering
Courant Events Right: These other settings, will put additional constraints on the same spectral data.
Courant Events Right: beyond… beyond the bootstrap equations. I mean, so far, as far as I know, there are no such examples, but it's… it's possible.
Courant Events Right: So that's… I guess that's the end of the reflection positive part.
Courant Events Right: So you basically advocate that if
Courant Events Right: Mathematically, this conformally invariant measure is shown to exist. Everything else will follow, like, one magic curve.
Courant Events Right: Because usually people think, oh, well, this measure, it's like an after problem. Two dimensions, it comes at the very end, and nobody cares even much about it. People want to know that.
Courant Events Right: But you say it's the key.
Courant Events Right: Well, I don't know if it's the key, I just… I wanted to… Let me just say that all this crossing properties is very deep, I mean, even if you prove this measure, you'll still have to work very, very hard to show the crossing property. But you say no, once you show the measure, everything else is just some technicalities, followed by general nonsense.
Courant Events Right: Well, I don't know, I think that somehow the ideal… Can you repeat the question?
Courant Events Right: Yeah, the… Slava… Slava Slava is saying that, I'm advocating.
Courant Events Right: That, we should always start with the measure, and then everything else, such as the bootstrap equations, will follow.
Courant Events Right: I guess that's… well, it's true that it should be possible to prove along, like, these steps that I basically went through, that this is… this is the case.
Courant Events Right: But I'm not… okay, I'm not trying to say this is the right fundamental point of view. Maybe somehow a slightly more fundamental point of view is this point of view of having the Hilbert space and having the Weiteman fields acting on it, because it also generalizes to gauge fields and so on.
Courant Events Right: But, yeah. That's the first time I've seen this, because these are the people who are supposed to show these things, and I think… No, no, no. I think you have to show these axioms, it looks very scary, because how are we going to show them? But if you have to just show that there is some conformity variant probability field, it seems like a much more approachable problem.
Courant Events Right: Yeah, no, I think, I think it costs cheaper. That's so easy.
Courant Events Right: No, I mean, don't get me wrong, I think…
Courant Events Right: I think it'll be… it'll be great if somebody makes what I'm saying rigorous.
Courant Events Right: Yeah.
Courant Events Right: Depending.
Courant Events Right: Alright.
Courant Events Right: Oh, and I… yeah. What question? So, so…
Courant Events Right: If I understood it correctly, like, for this bootstrap, like, the… out of the data you need is you need to know the spectrum, you need to know the representation. Well, that's what you're trying to construct. It's a part of data which…
Courant Events Right: It was… if prior undetermined, but you're trying to determine it.
Courant Events Right: Okay, the goods of equations will tell you what, what you, what the minimum dimensions are answered, or… Yes. Yes.
Courant Events Right: I mean, they would know…
Courant Events Right: the scaling dimensions enter as a prior undetermined parameters into the conform bootstrap equations, and then,
Courant Events Right: Empirically, it looks like the composers are very powerful at constraining for which bodies of these parameters equations are true, and it's an open problem whether this is, like, a complete maximization or not.
Courant Events Right: I'll talk a bit more about that, so… Alright, so… Yep.
Courant Events Right: You'll displace the associated domain interaction.
Courant Events Right: Yeah, you… yeah, you just… it has to be a sphere, or… I mean…
Courant Events Right: And a reflection, yeah.
Courant Events Right: I mean, they're all… they're all related by a conformal map.
Courant Events Right: around the corner.
Courant Events Right: Well, in this case, it doesn't… it doesn't matter. You get the same… I mean, no matter what sphere you choose, you get the same hybrid space.
Courant Events Right: Yeah, but it's more serious.
Courant Events Right: Yeah, but there… well, in the case of a CFT, there is a, you know, there's, like, a unitary map between them, a canonical unitary map between them
Courant Events Right: construction.
Courant Events Right: Right? Lots of stuff in the spaces.
Courant Events Right: Yeah? Small base of small balls.
Courant Events Right: Are we doing this place, actually.
Courant Events Right: Yes?
Courant Events Right: Yes. Senator Bork, excuse me.
Courant Events Right: Yeah, I… I agree. Okay, maybe let's… should we just…
Courant Events Right: You can, yeah, that's… there are various ways for using the bootstrap. I decided to do it this way, where there is only one reflection, and this OP is a statement about Weiteman fields, but…
Courant Events Right: Okay, so all this was reflection positivity, but can we do confirm bootstrap without reflection positivity? So, for example, to try to study percolation or self-avoiding walks.
Courant Events Right: Other things. So, the… the way that reflection positivity was used is basically there is some norm.
Courant Events Right: Well, there's a Hilbert space, a norm which is preserved by this group, and this gives us some spectral resolution with some positive coefficients whose coefficients are known by conformal symmetry, so the conformal blocks.
Courant Events Right: And… we can try to mimic this.
Courant Events Right: Without reflection positivity.
Courant Events Right: Where we… we can try to replace reflection positivity by this probabilistic positivity, so just the statement that, the measure is positive.
Courant Events Right: Which is still going to be true for percolation, or also avoiding welcome.
Courant Events Right: So, thanks.
Courant Events Right: So, here's kind of…
Courant Events Right: more concretely, and this is… this is based on some, mostly the discussions about this… on discussion with Peter Kravchuk, so some work in progress. So, we can try to replace the role played by this
Courant Events Right: reflection positive Hilbert space by the space of L2 observables. So now we are going back to the probabilistic picture, the picture where we have some measure on a space of distributions.
Courant Events Right: And the nice thing is that now there's a canonical norm, which is just the, you know, L2 norm.
Courant Events Right: On the space of observables, so it's a very big space, but…
Courant Events Right: That's okay. And this norm is preserved by the Euclidean conformant group, this SO1 commaD plus 1. So basically, what we are proposing is to do the bootstrap, where we replace the role played by SO2 comma dunitarity with SO1 comma D plus 1 unitarity.
Courant Events Right: So now, now it's the one common E plus 1 which acts on this… that's… that's the definition of the… of conformal invariance in this probabilistic setting.
Courant Events Right: And nice, you know, nice thing is that it's still a semi-simple E group, so it has a very nice and work-out representation theory of infinite-dimensional unit representations. So you can talk about the spectrum.
Courant Events Right: the spectrum of, apparently in the Hilbert space, and again, you find the vacuum, which now… so in the reflection positive Hilbert space, I'm like, oh…
Courant Events Right: And also in this one, the vacuum comes from just the constant observables, so just the one-dimensional space of constants.
Courant Events Right: But now…
Courant Events Right: there is a different kind of representation which plays a role, which is this complementary series. So it's… complementary… complementary series basically means precisely this constraint that
Courant Events Right: delta of D is less than D over 2.
Courant Events Right: It's an infinite-dimensional unit representation of SO1 comma D plus 1.
Courant Events Right: And it comes from these single-field observables. You're just smearing phi against those functions. But now, the rest is some continuous spectrum. Or, in principle, there could be several
Courant Events Right: complementary series. I think discrete… usually discretely managed. For example, for the 3Dizing model, there are two of them, the sigma and the epsilon. For a 3Dizing model, there is just one, just the sigma.
Courant Events Right: But on top of that, there is a… Large continuous spectrum.
Courant Events Right: But the nice thing is that somehow this continuous spectrum is gapped from the complementary series.
Courant Events Right: So you can kind of nicely distinguish that.
Courant Events Right: And now you can run the usual bootstrap machinery by replacing the role of these Weiteman fields with, with these nice commutative, observe… well, just, just,
Courant Events Right: distributions weighed against test functions.
Courant Events Right: And… so, for example, the OPE, follows by…
Courant Events Right: Well, the OP is always some statement of a spectral decomposition of a product.
Courant Events Right: So in this case, this… the product is the product of two random variables, phiF1, phiF2.
Courant Events Right: As long as it's in L2.
Courant Events Right: which, which is basically a statement that the four-point function integrated against F1, F2, F3, F4 is finite, which is… which is the case. In models of interest, then you can decompose it.
Courant Events Right: using the… using the spectrum of infinite dimensional unitary representations in DL2. So, for example, the… the vacuum gives you just the two-point function, then the… this complementary series gives you
Courant Events Right: Sort of a discrete contribution in the sum.
Courant Events Right: And then there's some continuous spectra.
Courant Events Right: Let's say, in the case of percolation, this is basically the three-point function, so it's the, you know, it's proportional to the triple connectivity that the Jesper was talking about in the two-dimensional case.
Courant Events Right: And again, the OP is somehow fixed by conformal symmetry, so this F1 star F2 is some convolution, the kind of conformal invariant map.
Courant Events Right: Between three, you know, from 2 to 1 representation.
Courant Events Right: Okay, and now the bootstrap equation you just get by,
Courant Events Right: By considering the 4-point function, where now we know everything is commutative, so it's just a product of 4,
Courant Events Right: random variables, And you insert this OPE.
Courant Events Right: and you get the product of two-point function from the vacuum, then you get the contribution of the complementary series, which comes with the C555, the triple connectivity squared, and then you get the contribution of the continuous spectrum.
Courant Events Right: And permutation symmetry is not manifest by this, so it's a constraint
Courant Events Right: It's a constraint on data, just like in the…
Courant Events Right: conform bootstrap, it's a constraint on the spectral data, here's also a constraint on the spectral data, and we are… we believe that this will lead to… to bounce on the CP555, for example, in… in percolation, so…
Courant Events Right: What we found is…
Courant Events Right: is that in the… in the 3D icing model, we can play the same game, so we can kind of forget that the 3D icing model is reflection positive, and just use its probabilistic positivity. And in that case, there is an upper bound on the analog of this three-point function, which is now the three-point function between sigma, sigma, and epsilon, so… but in…
Courant Events Right: In percolation, we have to work a bit harder, so we haven't quite, quite gotten there, but it's a problem.
Courant Events Right: I think this is going to work. Why do you have to work hard?
Courant Events Right: Actually, how do you input here? You don't… yeah, it's a… it's a kind of a… yeah, you don't… so this is…
Courant Events Right: This is a problem. There's no ill.
Courant Events Right: There's no way to input image theory working, so you're just getting some… you're just getting some general bounds, which are probably not going to be saturated by any particular theory, but they're going to be valid bounds.
Courant Events Right: Assuming… conformal invariance, and it's probabilistic positivity. So, actually.
Courant Events Right: Yeah, the last, in the last few minutes.
Courant Events Right: Yeah, I want to mention… actually, how much time do I have?
Courant Events Right: 6 minutes?
Courant Events Right: Yeah, I guess so.
Courant Events Right: Yeah, in the last part, I want to… somehow provide…
Courant Events Right: both evidence that probabilistic positivity is very useful in the context of the bootstrap, but also evidence that there are probably many other solutions of this bootstrap which do not come from CFDs.
Courant Events Right: And that's this analogy… with… between CFTs and the automorphic forms.
Courant Events Right: So here's… here's how it goes. So you start from some hyperbolic D plus 1 manifold, so D plus 1 because
Courant Events Right: the… isometric group of the high probability DP plus 1 space is the same as the Euclidean controlling group.
Courant Events Right: And a hyperbolic D plus 1 manifold is just some quotient.
Courant Events Right: of the hyperbolic D plus 1 space.
Courant Events Right: And so the main observation is that this cosette space is somehow a good toy model.
Courant Events Right: for the… for the measure space that's being acted upon by the conformal group. So it's a toy model for the space of distributions with the
Courant Events Right: But it comes from an invariant measure.
Courant Events Right: And that's because, well, it's a measure space, there's this… there's a harm measure on SO1 commaD plus 1,
Courant Events Right: And so it's preserved.
Courant Events Right: that there's a natural action of S1 commodity plus 1 by right multiplication on the space.
Courant Events Right: So it's basically the setting in which the bootstrap is formulated, a measure space with an action of SO1 commaD plus 1.
Courant Events Right: And… but now, in this case, the…
Courant Events Right: the notion of L2 observables, which I was kind of advocating, looking at in a previous slide, is an object which is kind of central to large parts of mathematics, mainly number theory, because this is the space where automorphic forms live. So this is somehow… if you decompose.
Courant Events Right: this L2 space into reusable representations. You get exactly what's known as automorphic forms, so these are somehow eigenfunctions or differential operators.
Courant Events Right: Living on the…
Courant Events Right: Things like hyperbolic manifolds, so for example, modular forms or mass forms are special cases of this, and number theorists like these automorphic forms because they can construct L functions out of them, sort of generalized Riemann Zeta function in various directions.
Courant Events Right: Actually, yes.
Courant Events Right: That's cool.
Courant Events Right: Of the… of the group. So the… the advantage of working with the group is that this… this thing has the action, right action of SO comma D plus 1.
Courant Events Right: The manifold doesn't. I mean, no, this quotient… this quotient has no symmetry, generically, because you can quotient both on the left and on the right.
Courant Events Right: Yeah.
Courant Events Right: And so, for example, you can now talk about the CIJKs, which are somehow integrals of triple products of automorphic forms in this case, and they have been… they are related to L functions.
Courant Events Right: Special, sort of, degree 8 o functions, at least in the case when these equal to 1.
Courant Events Right: And bootstrap in this setting can be thought of as imposing associativity of multiplication of these sort of random variables living in C… living in smooth functions on this quotient.
Courant Events Right: But this is actually something that has been explored before by some number theorists.
Courant Events Right: Especially Bernstein and Reznikov, to bounce some L functions, but so we, we explain how
Courant Events Right: what exactly is the connection with Conformal Bootstrap, and did various other things with it. So, in particular, what we did is,
Courant Events Right: So, it's to derive some bounds on the Laplace spectra.
Courant Events Right: Of hyperbolic manifolds, some of these slide, the spectral gaps.
Courant Events Right: And also, some bounds on L function. This is work from last year.
Courant Events Right: work. Basically, the reason why we can bound these L functions is because they're related to the CIJ case.
Courant Events Right: And, we have some bound when k goes to infinity, which is relative to the Lindel Law hypothesis for certain traporectal functions.
Courant Events Right: And one of my collaborators, Anshul Adwe, also approved a remarkable sort of converse theorem for the bootstrap in this setting.
Courant Events Right: So what he proved… you can kind of ask, is the bootstrap complete, right? That's something we would like to understand for the conformal filters. Is the… is the conformal bootstrap a complete axiomatization of the subject? And it turns out to be the case in this… in this toy model. So what exactly he proved is that…
Courant Events Right: Whoa.
Courant Events Right: So, of course, we know that if you give me a hyperbolic manifold, you get a solution of the bootstrap, but what he proved is that it's the converse. So, if you have a solution of the bootstrap with SO1, 2, with this probabilistic positivity, and a discrete spectrum, then it's… there is guaranteed to be a hyperbolic manifold
Courant Events Right: Which gives rise to the spectrum.
Courant Events Right: In the… at least the discrete spectrum kind of maps to the fact that this group is co-compact.
Courant Events Right: Okay.
Courant Events Right: So I, I just want to…
Courant Events Right: We'll follow up on that with some final slide about goals and dreams for the future, so…
Courant Events Right: I already talked about this bootstrap definition for the reflection-positive CFT, but it wasn't completely concrete, like, in particular.
Courant Events Right: I wasn't, precise about this, this domain. I think that's somehow the main missing… missing thing, to really identify what exactly is this domain inside a Hilbert space.
Courant Events Right: On which, on which the fees act.
Courant Events Right: So, yeah, it would be nice to have a precise definition.
Courant Events Right: of CFTD in this reflection-positive bootstrap sense, So that's…
Courant Events Right: probably doable, and what's more in the world of dreams is to, like, try to identify the analog of this Gmod Gamma, which is an object which provides examples, right? So in this
Courant Events Right: If you give me a hyperbolic manifold, you get a solution of the bootstrap, it's toy model bootstrap, and it'll be great if there exists
Courant Events Right: Kind of harmonic analytic objects, which are guaranteed to provide spectra, that are… You know, that's a…
Courant Events Right: that solved the conformal bootstrap equation. Maybe it's… maybe the answer is just conformal measure spaces, or maybe the answer is something more general.
Courant Events Right: More non-commutative, in a sense, so the white man.
Courant Events Right: axioms, but who knows? And then, okay, once we have these two points, then we can try to prove the analog of the converse theorem that I've approved in this setting, you know, to prove that somehow all of the solutions of this definition actually arise, from this… from this kind of, explicit setting. So, yeah, that's,
Courant Events Right: That's what I wanted to say. Thanks.
Courant Events Right: The last abstraction between the space, which improved looks like the full space in the final recession.
Courant Events Right: Since it was into the funding.
Courant Events Right: Which, sorry, which construction? This construction of the L2… L2 of G mod gamma?
Courant Events Right: Or which one?
Courant Events Right: This one. Oh, yes, yes, yes, yes, yes, yes, yes, because here the functions are living in D dimensions as opposed to on a slice. Yeah.
Courant Events Right: Yeah, yeah, yeah, that's true, yes.
Courant Events Right: Yeah, the group acts on the SDE.
Courant Events Right: I mean, it's a… yeah, it's a… somehow, that's the reason also why this, H is D plus 1-dimensional.
Courant Events Right: That's right.
Courant Events Right: So you mentioned the point model, but is there, like, this means here, a way of precise construction of a measure of the type of ND starting probability?
Courant Events Right: We assume that this is kind of like a single multiple open space, the default space? Yeah.
Courant Events Right: Nope.
Courant Events Right: Yeah, yeah, so when you… there is a construction.
Courant Events Right: Which is basically… so when… when you have a mass form, Upon the hyperbolic management, so…
Courant Events Right: metabolic manifold leak, HP plus 1, and you have a second functional Laplacean.
Courant Events Right: With an eigenbody which is in the complementary series again.
Courant Events Right: So this delta is less than D over 2.
Courant Events Right: Oh, so this is… this is a bad notation. So this is a… this is the Laplacian, and this is the scaling dimension.
Courant Events Right: Then, this kind of amounts to the fact that there is a… there is a map.
Courant Events Right: from the complementary series. So, the complementary is some space of… space of functions.
Courant Events Right: So… You can… From gene variant map from here to here.
Courant Events Right: And, if you just look at the smooth functions over here, This gives you,
Courant Events Right: It gives you a map from…
Courant Events Right: Gives me a map from this space to… to distributions.
Courant Events Right: on our leader.
Courant Events Right: options.
Courant Events Right: And then you can just take the harm measure.
Courant Events Right: Over here, and do a push forward.
Courant Events Right: Here.
Courant Events Right: And you get a gene brain measure on a spatial distribution, but somehow this measure has a very, very small support. It's basically the… it's a finite dimensional support, which is somehow the distribution which is defined by the mass form on the boundary of hyperbolic space.
Courant Events Right: and translate it
Courant Events Right: by… by all the elements of SO and Comedy Boss 1, so it's, you know, it's way too small. The support is way too small to… to be, like, a CFT measure, I think.
Courant Events Right: Yeah. If you try to, see if this works yet, and this is still, you know, the, measured here in Park Central?
Courant Events Right: Yeah, well, I… we thought about it a little bit, but we didn't…
Courant Events Right: Yeah. Yeah, we thought about it a little bit, it's, it's, it's, it's interesting.
Courant Events Right: Oh, yeah.
Courant Events Right: I guess in this… in the setting of, you know, quotients like G mod gamma.
Courant Events Right: There's… there's a concrete realization, right, so these, regular graphs.
Courant Events Right: Should provide examples.
Courant Events Right: But yeah, I'm not sure what would be the analog on the quantum filter side of that story, maybe.
Courant Events Right: For questions?
Courant Events Right: Me too.
Courant Events Right: they explained for one strike also.
Courant Events Right: But I don't know if I don't want to say that one day, like,
Courant Events Right: Yeah, so there's some kind of coarse assumptions you can make to more or less zoom out exactly on his ink, which is to say that there are only two
Courant Events Right: scalar operators, which are relevant. So it's a small assumption about this, specular decomposition.
Courant Events Right: This one, right? It's basically say that in the sector where rho is trivial, There are only two deltas
Courant Events Right: less than 3.
Courant Events Right: So it's some kind of a sparsity of the light spectrum.
Courant Events Right: And just with that, What more or less special musters.
Courant Events Right: you got…
Courant Events Right: you get the estimate on the isink that I quoted on one of the previous slides. It's a very… it's a very mild extra assumption that allows you to zoom in kind of close enough
Courant Events Right: to the… on the async in the a prioribic… space of, like, CST data.
Courant Events Right: Okay, no other questions, let's send the speaker again.