 Okay, on behalf of the organizing committee, I would like to welcome all of you for this year's spring school. I just wanted to mention one aspect of the spring school. This year, there are two special events. The one is the ICTP Award Ceremony, which is today. And the other one will be Dirac Medal. Both of these medals are very prestigious awards. So Dirac Medal will be next week. So this will be part of the spring school program. The other thing is that, as you will notice from the program, that most of the days we have a discussion session. And this is really an integral part of the school. In fact, I mean, during the lectures many times, if there are questions which require detailed answers, lectures won't have time to answer the questions during the lectures. And during the discussion sessions is the right time and you can discuss in detail the various questions related to the lectures and also in general. So please try to come to all the discussion sessions. Okay, so our first speaker is Professor Aninda Sinha from IIC Bangalore. And he's gonna talk about conformal bootstraps in marine spaces. Can you hear me? So first let me begin by thanking the organizers for inviting me to lecture here. I've been to Italy several times. To 3ST, this is my third time. And as always, it's a pleasure. So the title of my lectures is conformal bootstrap in marine space. Can you see this from the back? Or should I write bigger? I essentially have the way that I'm counting two and a half lectures. So I'll give a broad overview this afternoon. But today I'm going to talk about background and motivation. So this will cover a quick review of basics. So the basics will cover some background of CFTs. So which is basically in position space and the way that bootstrap is usually done in position space in modern times. Then I will also say why it may be in some sense natural to think about CFTs in melon space. I'll introduce what melon space is. And that needs me to quickly tell you about what it's called the Simon's star formula and what are called Mac polynomials. So in the second lecture, which will be today afternoon, I will give a broad overview about where this program is going. And then in my third lecture, which is tomorrow, I will give you more technical details about how melon space bootstrap is implemented up to what we understand now. So this is lecture one and this is lecture two, lecture three actually. And this program is being developed with, in collaboration with a bunch of people. There is Rajesh Gopakumar, who's at ICTS in Bangalore and two of my students, Apratin Kaviraj, who's finishing this year and who's going to work with Slava Rijkov in Paris and Kallol Sen, who's a postdoc at IPMU. And also, there's some work that I'll mention with Parijat Dev, who's there in the audience, who's a postdoc with me in ISC. And the main papers in this topic are, so there's a famous paper, well, by now famous paper by Alexander Polyakov in 1974. And the experts in this subject will tell you that, although in modern literature, everybody cites this paper, nobody really understood it. We, with Kallol Sen, I looked at this work in 2010 and we made some inroads into understanding what this content was, but really the main understanding arose in a set of papers that we wrote last year. So these are already on the archive and if you wanted, you could have a look at it. This is a short paper which actually summarizes what the key ideas are. But the Mellon Bootstrap that I'm going to describe both today, later on, and also tomorrow is based on some work in progress with Rajesh Gopakumar. Okay, so let me begin by telling you about what this program is, a bit about background and motivation. So what is it that we are trying to do here? So what we want is a Lagrangian-free framework that can enable us to study and hopefully classify conformal field theories in dimensions greater than two. We want to derive known and new results analytically and effectively and it'll become clear what the kind of results that we've been able to derive using the Bootstrap approach as we go along. So we want to make contact with whatever has been done in the diagrammatic approach using the Feynman diagram expansion and then we will see how to do similar calculations in a diagram-free manner and derive the known results and also derive new results using this Bootstrap program. So the known and new results primarily that I'm going to talk about in these lectures are going to be for the Wilson-Fischer fix point in D equals to four minus epsilon in an epsilon expansion but without having any Feynman diagrams in the intermediate stages and as will become clear that the intermediate steps that I'll be using are going to be manifestly finite but in the end they will still produce the same results that Feynman diagram approach can produce and new results with the Feynman diagram approach cannot produce. And more ambitiously in the future we want to use this framework to address the issue about how does string theory tie in all this? So maybe I should say we want to explore contact with string theory. So the canonical example is second order phase transitions. The canonical example that I will discuss is the critical point and this is the phase diagram, this is the pressure temperature, so this is the phase diagram of water. Yes, I think it's more than, I mean, I don't have anything. This comment wasn't meant for something specific but then of course holography, so for example we could address, is there in the large gen whether there is a large gen will we always have a weakly coupled string theory in that sense or is there something more? So for example, even for the 2D ising model can we have a string theory, can we specify the background where we can define string theory? The canonical example is a pressure temperature phase diagram of water and you know that there is a triple point here but there's a critical point here and then at the critical point we have enhanced symmetry, we have scale invariance and it's conjectured that we also have conformal invariance and if we use the conformal invariance of the critical point, can we make progress? This point, so this is the phase diagram of water but then there are a lot of other substances which have similar phase diagrams as a second-order phase transition which is associated with the critical point and there are conformal field theories which describe the critical points. The universality class of this is supposed to be the 3D ising model and in field theory we start off with a very simple scale of field theory which is the phi to the fourth but instead of being in four dimensions or in three dimensions, we pretend that we are in four minus epsilon dimensions and we do this because when we calculate the beta function by pretending that we are in four minus epsilon dimension we get a fixed point which is the famous Wilson-Fischer fixed point so the beta function of this takes on this form and when you calculate the value of the coupling where this is zero using this one loop beta function this is the one loop beta function you get lambda star to be equal to 16 pi square epsilon by three you want to be in three dimensions so you want to set epsilon to be one so that you get the 3D or get the 3D ising model but if you do that straight away then you notice that the coupling constant is 16 pi square by three which is a very big number so this is strongly coupled and really if you started off directly in three dimensions you wouldn't be able to use any perturbative techniques to make progress, correct? Thank you very much, yes so what Wilson can Wilson taught us or the way that they actually made some progress is that you pretend that you are in four minus epsilon dimension treat epsilon to be a small parameter and you calculate whatever you want to calculate so for example, you can calculate the scaling dimension of the field phi and so this will, because it's an interacting theory it is not going to be the free field value but if you do it using the techniques of quantum field theory using the Golan-Simonze equation and the like then you are going to get an expansion in epsilon so if you worked hard enough and this result is already there in the original paper by Wilson and this is covered in the Wilson-Cogart review you're going to get this result in up to three loop order so up to three loops you get rational numbers in front and there are irrational functions that start appearing from epsilon to the four so you get things like zeta three and the like from epsilon to the four order if you set epsilon equal to one in this expansion then the value that you get is 0.518604 but if you compare with Monte Carlo or nowadays the most precise estimate of this number comes from numerical bootstrap which I will allude to later on then this number is 0.518151 so you can see that retaining the three loops and doing this apparently illegal step of setting it equal to one you get reasonable agreement with numerical techniques these numbers have also been measured experimentally and the agreement with experiments for a large class of these are pretty good so this was a scaling dimension of phi you can go ahead and calculate the scaling dimension of anything that you want that are composite operators so for example phi square is an example of a composite operator you can calculate the scaling dimension of this using the same approach and there also if you retain up to three loop results the agreement with numerical stuff is pretty good so phi square is not the only composite operator you can actually consider composite operators with spin as well so for example you can calculate phi with a bunch of derivatives symmetrized and traceless so L here tells us a spin of this operator so for example if L was 2 then this would be the stress tensor which will not have any anomalous dimensions by the way the whatever is so this is the this is the bare dimension so this will be D minus 2 by 2 and whatever is the deviation from the bare dimension is what we will call the anomalous dimension and all these in the work of Wilson and Wilson and Fisher which are summarized in Wilson and Covert Review are known so for scaling dimension of phi the original calculation was up to epsilon cubed for these operators the original calculations were up to epsilon square and people have over the many so many years have gone ahead and calculated up to many loops so I think the current record is up to 6 loop order which involves hundreds of diagrams so which will obviously be tedious and any graduate student trying to reproduce them by hand good luck the drawback of this diagrammatic approach so calculating these will require evaluating a bunch of Feynman diagrams the drawback is that the Feynman diagram approach does not use the enhanced symmetry at the critical point so you first locate the fixed point by computing the beta function then you go and look at the Cullen-Simonsik equation and so on and also so the drawbacks are that does not use the conformal symmetry and also this approach it is very difficult to calculate what are called the operator product expansion coefficients and I will explain, I will define all these things a little bit better in a bit so the conformal bootstrap approach is a Lagrangian-free method that relies on conformal symmetry so the bootstrap uses conformal symmetry and it's an old idea that was pioneered by in the 1970s by Ferrara, Tato and Grillo by Polyakov and a lot of other people in the 1970s at least the principles were there although the equations that they managed to write down were quite hard to solve at that point and as a result of which the program in the 1970s did not proceed that far until the 1980s came and you have this famous work by Belovin, Polyakov and Zamologikov which addressed this issue in D equal to 2 so this in higher dimensions this program actually lay dormant until the famous work in 2008 by Ratatzi, Rychkov, Vicky and Tony and Vicky which essentially revived this program in higher dimensions but using a numerical approach there are lots of nice reviews of this numerical approach that exists in the literature in particular by Rychkov himself and also by Simons Duffin which are available online let me begin by summarizing very quickly some basics are there any questions so far so far as the background is concerned, motivations please feel free to stop me and ask me I will only be covering materials that I will need so there are lots of of course more rigorous reviews that exist and in particular the big fat yellow book which is the canonical book that we start off with that may be a good place to start so the conformal field theory, the conformal algebra in addition to the Pochery algebra also consists of additional generators the finite version of the transformation so this consists of performing a very specific kind of transformation where the new coordinates are allowed to transform in the following manner so these are the usual translations there are rigid rotations if you are in Euclidean signature or these are the Lorentz transformations so these are the usual transformations that we have in flat space but these are the new ones which are dilations and the special conformal transformations both of these are new so this is called the special conformal transformation and this is very important this will land up enabling us to fix a lot of things that would otherwise be not possible so this is dilations scale transformations are constant rescaling alpha is a constant parameter this is special conformal transformations so the generators of this are called P mu these are the l mu news the dilation generator is D and the special conformal transformations are called K mu they have a closed algebra among themselves the two important ones that we will need are the commutator of D and P's so that gives you I P mu and D with K which gives you K with a minus sign so just bear this in mind so the D with P has a plus I and D with K has a minus I and that will be important so with these transformations the first thing that we try to write down are conformal invariance or functions of positions which are left invariant by these transformations all these transformations so it turns out and you can convince yourself this special conformal transformation is responsible for what I am about to say well all of them are irresponsible but the key player is the special conformal transformation that in order to build conformal invariance non-trivial conformal invariance trivial ones are constants but non-trivial conformal invariance you will require at least four points and you get what are called conformal cross ratios so these x12 or xij in general square is xi minus xj mu times xi minus xj mu contracted with eta mu nu eta is a Minkowski metric or the Euclidean metric so most of the time I will be working in Euclidean signature so this could be just delta mu nu so there is one which is called u and the other one which is called v and there are for four points there are only two independent ones that you can write down and you can if you've never seen this before then you can take it as an exercise to use all these transformations to construct these conformal cross ratios and to show that these are the two independent ones that you can write down for four points you need four points at least four points when you have higher points you can get more invariance so if you have n points then the number of independent ones turn out to be n into n minus 3 by 2 so if you put n equal to 4 you get two which are these unvs and these unvs are going to play an important role in what I am going to say there are some transformations for these unvs which are useful to make a note of and these are easy to see so the first transformation is if you interchange 1 and 3 if you interchange 1 and 3 just interchange 1 and 3 here or you interchange 2 and 4 then unv exchange themselves that's easy to see if you interchange 1 and 3 you get 2 3 square and this becomes 1 4 square so that's what you get here if you interchange 1 and 3 this does not change so unv interchange positions the other one is if you interchange 1 and 4 or 2 and 3 then you get u goes to 1 over u v goes to v over u and the last one is if you interchange 1 and 2 or 3 and 4 then u goes to u over v and v goes to 1 over v the other thing that we will need are what are called primaries and descendants so you define an operator to be a primary operator of dimension delta so that it's eigenvalue under the dilation operator is i times delta and if you remember or if you see the algebra of d with k comes with a minus sign so if I start off with an operator whose dimension is delta and I consider a new operator whose dimension is k mu then using that algebra you can easily verify that the dimension of this new operator is going to be delta minus 1 and so by repeatedly acting on o with k I can constantly reduce the dimension which is not desirable because I don't want the dimension to be arbitrarily negative so any operator which is annihilated by this k in this case this operator is called a primary operator and then again using the algebra you can see that if I start off with a primary so now if o is a primary operator so let's call this primary and if I repeatedly act with the p's then I will get a new operator whose dimension or the eigenvalue of d is going to be i times delta so delta is the dimension of this operator plus j and so the dimension of this new operator is fixed in terms of the dimension of this old operator this is just an integer it labels the number of times that I am acting with p so these kind of operators are called descendants their dimensions are fixed in terms of the original operator on which you are acting with the p's is that clear ok the other thing that we need to know is that for a conformal field theory in any dimensions the conformal symmetry is powerful enough to fix the functional form of the two and three point functions and this is a nice exercise to do so you have to use the symmetry transformations that I have written down there and the algebra of generators so suppose I consider scalar primary operators I have got two of them phi 1 and phi 2 then the two point function of phi 1 with phi 2 just from the symmetries just from scale invariance and the first three transformations the translation, rotations and dilations you can convince yourself that this will be so c12 is some number the position dependence is fixed so if the dimension of phi 1 is delta 1 dimension of phi 2 is delta 2 then this is what you are supposed to get but if you use, so this follows from just using the momentum l mu nu and d but if you use special conformal transformations then what you will find is that it will demand that the conformal dimension of phi 1 and conformal dimension of phi 2 have to be the same so you get some number times the fact that delta 1 and delta 2 are the same so if I write delta 1 is equal to delta 2 equals to delta then that's what you get this magic of fixing the form of this two point functions to three point functions three point functions are also fixed three point functions are also fixed if you use just just use scaling then you will get weaker constraint but if you use scaling and special conformal transformation then you can show that this has to be some number c123 divided by x122 x232 raised to a raised to b and x132 raised to c where a b and c are all fixed in terms of the conformal dimensions of these so a is just for completeness I am writing them out but I don't really require them in anything that I am going to see in the future so this is fixed and although I have told you the story only for scalar primary operators this is actually true also for operators that carry spin so instead of this what will happen there will be more structures but the x dependence of those structures are all fixed up to these numbers so the fixing actually ends at the level of the three point functions for four point functions conformal invariance alone is not enough to fix the functional form at the level of four point functions so suppose I consider four identical scalars then what you will get is now identical scalars have dropped the index one two three four at four different points then conformal invariance tells you that the form is this so the conformal dimension of these identical scalar is phi delta phi delta phi for anything that I am going to say in the future we will refer to the conformal dimension of this scalar primary operator and I will for the rest of the lectures only focus on identical scalar four point functions and you can actually with a bit of algebra you can actually show that for the four point function you can fix the form to be of this kind where you have pulled out a particular position dependence and up to some unknown function of u and v so these were the conformal cross ratios that we defined earlier on this is the conformal symmetry of this kind can only fix it up to this form this function of u and v is referred to as the reduced correlator so if you pull this factor out which depends on position explicitly this factor cannot be written in terms of u and v this is some factor that you can pull out the rest of the stuff is going to be only a function of u and v is that clear now we also need to talk about the product expansion because that plays a key role in setting up the bootstrap equations the operator product expansion the operator product expansion is that a product of local operators may be local quantum operators can be expressed as a linear combination of well-defined local operators you can take this to be an axiom and this is frequently taken to be an axiom in the axiomatic formulation of quantum field theory specifically in our context this means that if I have a scalar operator say phi i and another scalar operator say phi j then as you take x goes to y then this is an operator statement this can be written as c i j k which is a function of x minus y times phi k y so this c i j k is a c number function and inbuilt inside the c number function the leading term up to normalizations that we specify includes what are called the OP coefficients remember that I said that there are these primary operators but also descendant operators the descendants are written in terms of primaries by acting on them with derivatives which are the p's so we can actually rewrite this equation a little bit so there are some operators here are defined in terms of the primary operators in terms of derivatives so we can rewrite this by only summing over so here we are summing over all possible operators but here we can use the fact that descendants are related to primaries by summing only over primaries and there is some functional which will involve some known functions involving the derivatives of y but now the operators that appear on the right hand side are only primary operators exercise you can try to show that if I have identical scalars then you can write this in this way so I am taking one of them at x the other one at 0 and in the limit that x approaches 0 so c12k's are just numbers which are called the OP coefficients so you can convince yourself that up to these numbers you can fix it in this form times x mu 1 dot x mu L so these are just products of x mu 1 mu 2 x mu 1 x mu 2 x mu 3 etc times some operator which will soak up these indices at 0 plus descendants so that is the form delta k that is right it should be delta k it is the dimension of this operator plus L over 2 and also because these are identical scalars this is the form that it will take and you can also convince yourself and it is easy enough to see that the indices mu 1 to mu L will be totally symmetrized because of the way that they are multiplying of these x's L at this stage it is some it just follows from the number of x's that I can pull out so I have pulled out a certain number of x's and so this index that is there on this x has to be soaked up with another index it will turn out to be related to the spin of this operator so L here actually is the spin of this operator and also this particular combination let me just introduce that here itself delta k minus L delta minus L this combination is referred to as the twist okay now this functional forms of these p's are actually fixed the way to fix them are that you first consider the 3-point function so when you consider the 3-point function which is basically you take this form and you put in an O on the right hand side so that will be O and there will be another O here that will give you the expectation value that will give you this guy so this will be equal to suppose O is a scalar operator just for concreteness then this will be simply equal to phi c phi phi O times the 2-point function of O, y and O, z now you know that the 2-point function is fixed so the right hand side is fixed so this is fixed by conformal symmetry so you get one form, one expression that irrespective of this operator product expansion this 3-point function is also fixed by conformal symmetry so this is fixed by conformal symmetry this follows from the OPE now you can compare the 2 expressions that you get and when you compare you can fix this p so that's one way of fixing the p's normalize, yes I'm assuming that it is diagonalized you're right, yes so in a conformal field theory we frequently refer to as these objects so which are the conformal dimension of all the operators that are there in the theory with their spins and the OPE coefficients these are what are called OPE coefficients these numbers so these OPE coefficients these are referred to as the CFT data and when we say that we want to solve it means that we want to get all the numbers all the delta i's and these numbers to solve the theory and the reason why we say that this solves the theory is because by repeatedly using the OPE if I have an endpoint function I can fully reconstruct the endpoint function with that data we just have to recursively use the OPE questions? no, we want to solve for it so we have to impose some additional constraints so it will turn out and that's what I'm going to come to in a bit that this is not the only structure only restrictions on a conformal field theory there'll be other things which we'll talk about in a bit yes so the other in region that we will need are what are called conformal blocks you start off with phi's and I will drop x1, x2, x3, x4 so whenever I'm writing this these operators are at x1, x2, x3 and x4 that is my ordering unless I don't specify otherwise so I can do the OPE twice, I can use these two plug that OPE and I can use these two plug that OPE and I will get CO squared it's squared because the two-point function because of conformal symmetry picks up only the operators which are same so that's why it's CO squared there is no cross-term the cross-term will arise if you have non-identical scalars you'll get CO squared times two products of these guys so I've just used that OPE that I've written there in this form and I've done it twice you remember that when I said that conformal symmetry fixes the form of the four-point function up to a function of UNV using this I can actually define the conformal blocks so I should put a fact that this is the definition of the conformal blocks using this we have CO squared times some function of UNV divided by this function, the position-dependent function that we pull out so this conformal blocks depends on the operator O and the language that we're going to use that it depends on the operator that is getting exchanged and this GUV which is labeled by O is called a conformal block it also is sometimes referred to as a conformal partial wave or conformal partial wave yeah okay yes that's true in papers like Dolan and Osborne they refer to this as the conformal partial wave and I just now mentioned the names of two people from Cambridge Dolan and Osborne did a lot of work in the early 2000s on these conformal partial waves so they have got a lot of important contributions trying to simplify the forms of these conformal partial waves a lot is actually known so these conformal partial waves actually satisfy a differential equation so these GOs UV depend on delta L they satisfy a second order and a fourth order differential equation the order is in UNV closed form solutions by that I mean that I can write it in terms of known functions are only known in D equal to 2 and D equal to 4 and also it's not in terms of explicitly in terms of V but implicitly in terms of UNV the actual dependence are in terms of these new variables Z and Z bar or X and Y it's not really a complex conjugate but you can define new variables Z and Z bar through these equations the closed form expressions which look like product of hyper geometric functions are in terms of the variables of these hyper geometric functions are Z and Z bar the even dimensional conformal blocks are actually built recursively out of the lower dimensional conformal blocks in some sense even those are known no useful closed form expressions for D equal to 3 or odd dimensions are known but nevertheless there are explicit forms of these conformal blocks that are known in any dimensions not closed form but explicit forms are still known in terms of sums over various things and I will tell you a little bit more about that later on but that is enough of background material before I tell you about bootstrap so let me just pause there and ask are there any questions, technicalities, confusions also if you want to know about literature where to study these more I can refer to them refer you to them in the future okay so I think bootstrap this is a good place to introduce bootstrap you have a four point function and I could have done this let me just explicitly write out the position dependence and as before we are going to consider again identical scalar operators this can be easily generalized to non-erynical scalars generalizing this well in principle I could generalize this to any operators but useful blocks need to exist in order for us to make use of this bootstrap conditions so let's focus only on identical scalars for now so I did the OPE by taking these two operators close together and these two operators close together and if this was a Euclidean correlator then it doesn't matter where I place these operators I could move things around so in particular I could have done the OPE by taking the first and third and the second and third so these arrows means taking close together and doing the OPE or I could have done it the other way taking the first and fourth and the second and third so there are three different ways it's convenient to refer to these as different channels so if I take if I do the OPE in this manner then I will call this the S channel or the direct channel if I do one and three close together and two and four close together this I will call the T channel or the crossed channel and one and four and two and three is the U channel or it's again known as the cross channel now it does not matter in which order I do the OPE all these should give rise to the same result so one two three four should be the same thing as one three two four in terms of blocks we can rewrite this condition and this is where we will make use of what happens when I swap two and four so V and U will interchange positions so when I do this when I write to the right hand side I will get this G O but now instead of U and V it will be V and U and this will be X one four two delta phi X two three two delta phi and this is an additional condition the four point function has to satisfy which follows from associativity of the algebra and this is a non-trivial constraint so this is supposed to hold at each point each value of U and V and of course if you do it at a random value of U and V one of the sums so there's a sum here over the spectrum so this is your sum overall operators there are OPE coefficients that are sitting here and there is no guarantee that the sum is going to convert so you have to do it carefully and if it does not convert you will have to resort to resummations or analytic continuations and so it's not completely trivial to solve that condition yeah except that one of them has U and V and the other one is V and U so you have to interchange the coordinates the U and V coordinates yeah well yes it's it's simply the statement that I can move things around and the order in which I'm doing things does not matter this is also the statement of crossing symmetry yes yes this is crossing symmetry crossing symmetry well I mean it just follows from the fact that these are bosonic operators and it does not matter where I'm placing them yeah in the Lorentz signature there will be phase factors important phase factors which we we are not going to talk about yeah in time-ordered correlation functions it's trickier yes okay so this is non-trivial to solve and what you do when you do when you solve it numerically it actually makes use of typically makes use of numerical methods to solve this equation or to look at these equations I should say look at these equations rather than solve because what will land up getting a bounce we won't get all operators unless and until there's even further input that are specified so numerical techniques there are essentially of two kinds one of them is the original one by Rathar Zirichkov Sonny and Vicky and there's also more less developed kind which is by Gliotzi and collaborator which deals with the truncated version but this also appears to be quite promising but it's less developed this version actually gives rise to bounds and this version actually gives rise to equalities not bounds yeah so in D greater than 2 typically there are infinite number of operators it's not a finite number so truncated means that you put a cut-off somewhere region of convergence yes that is important as well and as I said that you have to locate a suitable region of convergence where actually you are allowed to truncate it and people have analyzed those rigorously so you just say that there are no operators beyond a certain dimension in the spectrum just by hand you say that or maybe you are more rigorously saying that is that those converge to a small tail or something numerical techniques depend on these two methods the RRTV gives rise to bounds this essentially looks at this equation around the symmetric point u equal to quarter v equal to quarter 1 over 4 1 over 4 and you expand these equations so you look at this there are rigorous analysis of convergence done by Richkov and collaborators there is a 2012 paper and a 2015 paper which looks at the issue of convergence here and they further use what is called linear programming and there are sophisticated versions sophisticated extensions of this which are which which look at instead of the same operators all identical scalars that allows us to non identical scalars as well which depends on what is called semi-definite sorry semi-definite programming algorithm SDPA so SDPA there are lots of people who are instrumental in developing this this is explained well in Simmons Duffin's review of last year basically the output of this is that in the original version you look at what is the allowed leading order scalar in the OPE so this is a leading scalar in the OPE this is the dimension of the external scalar and you also assume that this external scalar which you are which appears in the correlator does not appear in the OPE so there is some kind of Z2 symmetry and what you land up getting are allowed regions so if you plot delta OS OS is the leading scalar appearing in the OPE 555 OPE and you get an allowed region so typically this region here is allowed this is allowed and everywhere above is not allowed so this is the kind of plot that if you open this 2008 paper you are going to see but the more interesting thing actually happens in D equal to 3 views of notation are let's let's look at delta phi square versus delta phi or in the Ising model case this would be what is called the epsilon operator and this is what is called the sigma operator then in D equal to 3 something interesting happens if you are looking at D equal to 4 or D equal to 4 and above you get some smooth curve but actually for dimensions lower than 4 this is quite true even for 4 minus something this could need not be an integer this could also be fractional you can do this analysis in fractional dimensions as well then what you get are kinks there are kinks in the plot so there is a kink here and quite remarkably this kink actually appears to be very close to the sort of 3D Ising model values for delta phi and delta phi square so this kink here it appears at roughly 0.518 and this is 1.412 I think and this kink appears very close to that value and so these people conjectured and because of the fact that if you look at study the kink very closely in two dimensions that agrees very very closely to the 2D Ising model which is known analytically but if you look at the 3D case and you make a conjecture that the kink is where the 3D Ising model lives then you can zoom into the kink use other correlators and make the allowed region here so I said that you get allowed regions when you start looking at multiple correlators the allowed region becomes islands so maybe some islands of this kind will start showing up that does not happen in four dimensions or above but in three dimensions these become islands so small islands and this island that you get by doing this analysis is actually much smaller than the error bars or error bands that follow from Monte Carlo analysis in that sense the numerical bootstrap is more precise than than any other known numerical techniques how long we have yeah I mean the next one is going to be a little involved but I am going to talk about MacPolynomia's next but I think that is a good place to stop yeah I will stop there and if there are some questions I mean short questions because long questions can be postponed or today there is no discussion session so perhaps you can ask any questions you have okay if not let us meet say at 1020