 with Amit Dekel and we continued this to another project with took some of our results and wanted to do some more things with them but unfortunately both Michael and Amit quit the field and so I have I had to recruit other collaborators who are Diego Ciancanelli and Eduardo Vescovi but you know and we're all are curious to see whether they will meet the same fate but we have your counter example on the second line that not all of my collaborators leave the field. The there is a big invariant of of this topic and that is that I always promise that this paper is coming in the next month it has been an invariant for over a year now so let's let's try to break this invariance soon. So let me start with some background and I'll get to the topic and the most beautiful nice supersymmetric object and n equals four super young males is the half BPS circular Wilson loop. It preserves a subgroup of the conformal group which is the rigid conformal group in one dimensions and this is true for any conformal field theory but for n equals four supering males it also preserves half of the supercharges in particular this super group whose bosonic part is written here and its expectation value is well known it's given at finite n by this Laguerre polynomial and at large n by this Bessel function modified Bessel function some of the things I'm going to discuss today are valid for all n anything that you can get through localization some of the things I will discuss are valid only in the large n limit those that you get through integrability for example but for simplicity I will just work in the large n limit if you want to do things at large at finite n we can discuss it so now that we know and understand and love the circle very well and we already know it for many years we can ask what we can do about deformations of the circle so again as a restriction which is not necessarily for some of the things it's I'm doing it's necessary for some of the things it's not let me restrict the Wilson loop to be inside a two-dimensional plane and then I can write it in terms of one complex function of a parameter theta where it's e to the i theta plus a small function g of theta and I can assume this function is real because if it's imaginary I can absorb that in a reparameterization of theta so g of theta you should the log of sorry the exponent of g of theta is the radius of the deformed circle and we can write a g of theta in a Fourier expansion and B minus n is equal to the complex conjugate of Bn because it's a real function and now we can study what happens to the expectation value of the Wilson loop in a power series in these parameters B or more generally in a power series in this function g of theta and of course at lowest order we set g to 0 we just go back to the usual circle as I will review order B square is also known at all orders in the coupling it's related to the bremstrahlung function so the new questions to ask are about order 4 and higher in this expansion so this is one point of view another point of view is a the fact that you can look at a Wilson loop and insert operators into it and there is this double bracket notation for these expectation value of operators which are not really independent operators what you should think about is you have a path ordered exponential of this Wilson loop and taking here the BPS Wilson loop and inside it I'm inserting adjoint valued operators so you should think of an operator at position s1 an open Wilson loop connecting the point s1 and s2 another operator at s2 another open Wilson loop and so on up to here and then another open Wilson loop from sn back to s1 and this symbol just takes care of all of that the path ordering puts these insertions in the correct place inside this path ordered exponential or we can just write it like this the simplest case is when the operator O is just one of the adjoint scalar fields of our theory but it can be many other things including the field strength it should be clear that this has these are not operators in the theory on their own they are not gauging variant these are insertions this is another sector of the theory that we can study and really the operator in general is the entire thing but we can look at properties of these insertions now this you can do for gauging variants with any path and you don't even need this scalar piece here if you just worry about gauging variants in the case of the the case that I will study I will include it and I will take the path to be a circle and I told you that the circle preserves an sl2r a small conformal group and this means that two-point functions satisfy the usual a structure of two-point functions in a conformal field theory if they are primaries they will have to have the same dimension and it will be some number divided by the distance between them to the power of the dimension two of the dimension yes if they are primaries they have to but if they are not primaries if they are descendants then it's exactly the same as in a as with usual operators here I didn't assume primaries so descendants can can have that for but yeah it's when you talk about primaries it's always one over the distance to two delta here I wrote the distance on the on the circle you can do this both on the circle and on the line I will switch between these two pictures it doesn't make much of a difference when we talk about this this question the operators will have the same dimensions and normalizations and when we talk about three-point functions the same structure constants and you can go to four-point functions you already then will have one real cross-ratio and one can ask a one can study this defect cft and ask what are the what are the dimensions what are the normalization constants what are the structure constants and so on do everything that is done in the bootstrap so another question I want to ask is what can we say about these operators these insertions I will be casual in using the term operators instead of insertions but you should always know that they are not independent operators and how are they related to deformations of the circle so now that I introduced everything and I can tell you what I'm going to talk about so I will talk about first the relation between these last two transparencies deformations of the circle and the defect cft which is what we did in our published paper and I will tell you how you can study these the cft data the anomalous dimension and a structure constants and I will mention how this can be done then I will go to study the deformations of the circles so first at order a b squared or g squared I will review the Bremstein function and then I will discuss the case of order four and if I have time I will mention these points about a string theory which is part of the motivation for my work but it's a bit technical so when we get there I'll explain it any questions so far it goes here are eating me beware next speakers so a so we can represent deformations of the circle in terms of operator insertions normally in a usual Wilson loop if you add a small bump to the Wilson loop it's essentially insert inserting one plaquette and it's represented by a field strength and with one direction parallel and one direction normal to the Wilson loop where you make the bump if you take the circular mother center Wilson loop then instead of f you also get a piece involving this scalar so that is the first lowest dimension insertion you can get and this is actually also known as the displacement operate it's what you get by acting with a displacement operator a small geometric deformation of your defect more generally operators are classified by representations of the unbroken group and a well we wrote in our papers a few of these multiplets it's not very enlightening so I didn't copy them here at classical dimension one we have a the scalar fields so of course this is dimension two we have the six scalar fields phi i and they decompose into the singlet phi one which matches the scalar in the Wilson loop and a quintet of so five and the quintet is protected and it's actually a super partner of this operator and the singlet is not protected here are some operators of dimension two you see I have here many of the field strength they are all field strength but because we have the line that broke some of the Lorentz symmetry they sit in different representations of the preserved so three and you also have derivatives of the scalar fields either the singlet or the quintet and you have things made out of composite of the fields of course there are also fermionic insertions and all these things you can look at our paper and we to study the dimensions and normalizations and structure constants of these operators we have many tools in our disposal we have perturbation theory which you can do explicitly to any order that you want you can use ad s cft we can use integrability in some cases we can use localization and we can use the bootstrap so one of the points I want to emphasize in my talk is that this topic of the defect cft of the wilson loop is a rich laboratory where all these topics in modern theoretical physics come together and I will apply only a few of them in my talk I will not use bootstrap I will not use integrability really and in my talk I mainly focus on perturbation theory but all the others can also be implemented and there have been people studying them from the other have been studying this defect wilson loop cft from other points of view so let me mention the story of integrability it hasn't been applied in this case but it could be a cusp wilson loop you can calculate using integrability both Diego and I figured it out so a wilson loop in ad s cft is described by an open string and strings in ad s five are ad s five crosses five are integrable and the open string in ad s then becomes an open integrable model and you need to understand what are the boundary conditions and so on but the boundary conditions do preserve a integrability the boundary and buster equation so we're in good form and but that is the case when you have a cusp in this story that I discussed so far there was no cusp but the nice thing is that at the same at the apex of the cusp you can also introduce an insertion and for the question that I am studying now what I would really want to do is leave only the insertion and take away the cusp so I have a straight segment or a circle with some insertions into it the only insertions that were properly studied were z to the L's where z is one of the is a complex scalar field which does not involve phi one and this is a protected operator in the absence of the cusp and the presence of the cusp it's an interesting operator and its dimensions and so on were studied and they're quite involved but there is closed form formula for their expectation values in in the language this is the ground state of the spin chain if we want all other insertions would be excitations of this state so what I want to do really is study the the beta state for this system but this could be done integrability allows you to do it but actually nobody has has gone through this procedure to calculate what is the spectrum of these open string beta states and this clicks okay so that was about integrability I mentioned what could be done and what has been done but it's not exactly addressing our problems of the insertions into the circle what has been done and I will discuss a bit more is to apply usual perturbation theory and one can take any of these insertions and draw the relevant diagrams to calculate them and and this has been done what we did instead was try to use the relation between smooth deformations of the Wilson loop and insertions to calculate smooth deformations of Wilson loops and through them find the properties of the of the insertions so if you take an arbitrary curve which will be a deformation of the circle at one loop you just get an effective propagator which involves exchange of gauge fields and exchange of scalar fields here you divide by the distance squared and this is the one-loop expectation value of the Wilson loop if your curves are in R2 there is also a compact formula for the two-loop graphs which just is a this formula this is the result of the interacting graph these are a ladder like diagrams that give these terms and we have found efficient algorithms to calculate these integrals and what we want to do is rewrite them and extract the CFT data so generically if we have the curve of this form we can expand it in the formal power series in G and what we get at 0th order we get the circular Wilson loop at first order we get two G insertions and two two powers of G and two displacement operators then at order G cubed there is nothing really at order G to the fourth we have four displacement operators and then some contact terms when two of these G's are at the same point which essentially involve D of f so here is the expectation value of the Wilson loop written in a formal series which involves the CFT data four point functions three point functions two point functions and more two point functions and what we want to do is take this these expressions that we have here and repackage them in this form and through that read the information for these things now this involves quite a lot of work because it requires manipulating these the integrals that we have here it also involves these expressions that we have here that we get from this expansion may not be the most natural objects in terms of the representation theory so we may need to do some diagonalization and rewrite these guys in different way well that's why I have PhD students who did that and and then left the field and what we get is that for the displacement operator the normalization constant is given by this this was already known from explicit calculations before some three point functions that we found are the following and three point functions of a field strength all will have some tensorial structure because there are indices here so you get these complicated tensorial structures as far as I know this tensor has not been written before a this exact combination in principle you can extract it from group theory but I don't think this has been done and we found the structure constant which has not been known before for the unprotected skater we found the anomalous dimension which again is an agreement with previous calculations and for the triplet and quintant that I have here and we also have the same anomalous dimension as this which is consistent because they are actually in the same multiplet in the same super multiplet I like these results so much that I wrote them on a pot here you see the small deformation so if you don't know I'm a potter on the side and here are all the references and if you feel that you're missing a reference then do send me an email and I will not add it to this pot I normally bring one to show to the audience but that one is about this big and I so the same story can be done in a ADS CFT and this was done by these people and what they did was study the ADS 2 world sheet theory of a of the circular Wilson loop so the circle has a very very simple ADS dual where the string just takes the form of an ADS 2 inside ADS 5 and to calculate the fluctuation determinant you need to expand the green schwarz action to quadratic order a for their purpose they expanded it to the next order to quartic order and calculated a Whitton diagrams in this ADS 2 effective theory and interpreted it in terms of the bootstrap to extract some structure constants for two scalar fields combining into some composite two scalar insertions combining into a composite of two scalar fields and they found expressions like this and they have many many more of them so one can use also ADS CFT and the bootstrap to extract these anomalous dimension for the scalar field for the singlet and structure constants so there are some things to do here and other tools so this is what I wanted to say mostly about these operator insertions and I want to switch to the deformations of the Wilson loop the smooth deformations any questions so far okay so we have this expansion of the we have a general curve which we write in terms of the function g and I told you that at quadratic order you have two g's and two insertions of f now the operator f is special because it's a protected operator whose dimension is two it is in the same multiplet as the quintet of scalars who have protected dimension one so we know that this two point function will always be one of the one over the distance to the fourth power times a number and that number is the bremstrahlung function so I wrote here I replace this by b of lambda divided by the distance to the fourth power and then you have this integral which you can write in the Fourier basis like this in terms of b n remember b n are the Fourier coefficients of g now I will not tell you how but it's a very clever trick a on how you can find the bremstrahlung function by a different deformation of the circle that preserves supersymmetry and then related to a supersymmetric deformation of the circular Wilson loop whose expectation value you know exactly and that gives you the bremstrahlung function in this form is in the large n limit and you can take this for this arbitrary deformation that you have here and plug this function this coupling dependence into this formula and of course this is consistent with the explicit calculations that I wrote before this af that I write here is just the expansion of this bremstrahlung function to first two orders in lambda so we know the expression w two exactly the dependence on the Fourier coefficients and the dependence on the coupling is completely determined so now we want to go to order w to the fourth a b to the fourth and this is the expression which last week I think I found some mistakes in it but that's okay I mean they're correct in our mathematical code but in the way we translated the mathematical code to the draft we made some mistakes so so don't take a photo of this sorry oh okay mistakes are are allowed along the way it doesn't really matter um so this is at one loop this is order lambda and these are all b to the fourth power so you get see here b and cube b minus three n and you have b this is b and b minus n b m b minus m we actually did this for two loop order it's much more complicated again you get some you get the same b's appearing but the coefficients are harmonic numbers it's it's pretty nasty and we're still trying to study the structure of these expressions and correct them write them correctly but staring at them there are all kinds of structures in there so this is the expression at order b squared this is what we had which is much more compact and the entire lambda dependence just factors out into this expression here at the next order in lambda we don't get any more of this polynomial rational a natural number polynomial we get instead harmonic numbers so it gets very complicated aim so are there any structures at this order we do think that there are and let me give you some hints to that and that comes from studying the near circular Wilson loop in ADS CFT so I will not review it but you can well just this transparency to study a Wilson loops in R2 you need to look at strings propagating in ADS 3 do that in efficient ways to use the pomaya reduction and then you end up with a generalized Cauchy Gordon equation where f is equal to 1 this is just a Cauchy Gordon equation we're given a function f you need to solve for alpha a given f you solve for alpha and then you find a that the regular is action regularized action of the string is given by this integral you can evaluate and the shape of the Wilson loop is completely obscured in this description I just have these functions f of alpha but you can reproduce it there is an algorithm to also calculate the shape of the Wilson loop but I'm I'm not writing it for you and in the case of the circle f is equal to 0 alpha you immediately solve to be that and the case that I want to study is the near circle so I will take f to be small of order epsilon and study things in a power series in epsilon now you should note that these two formulas finding alpha does not involve the phase of f and the area also does not involve the phase of f but you should believe me that the shape of the Wilson loop depends on the phase of f if I construct it in this way and you need to choose one you need to choose a prescription but so we have a family of Wilson loops which are related to each other by phase by multiplying f by phase whose expectation value at strong coupling the area of the string world sheet is the same now you can use that and evaluate the string action as a power series in epsilon and what you find is the following expression at order epsilon squared you get some function of p which these are the Taylor coefficients of this function f and something with a p squared at order epsilon to the fourth you find a very compact formula with things of order a to the fourth and note that the overall phase doesn't appear here because I have here a and a bar and here everywhere I have two a's and two a bars so if I multiply f by a constant phase this will cancel so we can write the shape of the Wilson loop in a power series in epsilon relate f and b and at lowest order b is equal to this thing where phi is this phase so the shape of the Wilson loop will depend on the phase of f on this curly phi and now if you take this expression a you see here one over n times n squared minus one so in the expression for the a at order b squared we had everywhere we had a b times b bar times a n times n squared minus one if we want to translate it to an expression in terms of a what we will find is exactly this expression so this is consistent with this a universal behavior for the order b squared but we can also plug this and the higher order terms into the expression that I had the corrected expression that I had for a order b to the fourth and then we find an expression in at order a to the fourth which is this so this is at strong coupling this is at weak coupling I just plug this into my weak coupling expression and what I find a sorry this I plug I find this expression so this is at weak coupling order lambda we get this which reproduces the bremstrahlung function once we translate a to b and we get this expression and note that even though this is at weak coupling and this is at order a a to the fourth we don't get here any dependence on this curly phi that I had before so and this is remarkably simple similar to the strong calculate strong coupling calculation that I had here it's just it's the same denominator and just a different polynomial in the numerator this expression so this is at weak coupling at strong coupling we have a very similar expression and this is not related to the bremstrahlung function this is at order a to the fourth power so in a meets paper who studied this relation between a these kinds of relations he observed many examples that they're at weak coupling there was no dependence on this phi in the expectation value of the wilson loop up to order epsilon to the eighth not including it starts at order epsilon to the eighth here we proved that at order epsilon to the fourth there is no dependence on this phi so there is this extra secret symmetry whose origin we still don't understand which extends from strong coupling and is exists at one loop order and also at two loop orders but I didn't write the expression here there is no dependence on this a on this phase so this family of wilson loops were related by this phase phi have the same expectation values at these orders at strong coupling this is known as a symmetry at weak coupling this is not a symmetry it's an approximate symmetry whose origin we don't understand yet as I told you a meet didn't find examples with dependence at order epsilon to the sixth but we did his analysis was not a he was just looking at some examples looking at more examples we did find dependencies at order epsilon to the sixth so there you will already get expressions which don't involve just a a bar a bar a you're going to get combinations within different numbers of a's and a bars and they will have explicit dependence on the phase so for the new project I made also some pots and you see here this whether you take a photo of it or not this is already burnt and in the kiln so it has the wrong formulas on it my customers don't mind and it's supposed to represent a draft it's fine so let me conclude the circular wilson loop is an ideal lab to study defect cft one can use perturbation theory localization integrability the ope techniques and a bootstrap and a ads cft we are calculating the expectation values of the deformed circular wilson loops or alternatively insertions into the wilson loop and there is a explicit even if complicated map between them we have explicit results for anomalous dimensions and the general expression for the deformation at certain orders at one loop two loops and strong couplings we have dimensions structure constants and so on there is a surprising dependence on the spectro parameter a curly phi the phase at order epsilon to the fourth we don't understand and a the fact that this strong coupling expression weak coupling expression are so similar is also puzzling to us and we want to understand it and the nice thing is that I didn't do here really anything too novel or sophisticated and this calculation could have been done 20 years ago in the age when wilson loops and ads cft were invented and I was just using perturbation theory and very basic ads cft so we can still find surprises and basic ads cft calculations thank you