 So yesterday we started talking about what will eventually lead to the HGT conjecture. Well, by now it's not much of a conjecture anymore, it's HGT correspondence. And let me just remind you of what we're after. So first of all, I'm just going to claim that there exists 40 n equals 2 super conformal field theories labeled by some punctured Riemann surface. So give me some punctured Riemann surface I can produce 40 n equals 2 SCFT. Now give me a punctured Riemann surface and I can also try to compute two-dimensional CFT observables on this Riemann surface if I interpret the punctures as vertex operator insertions. So on the other hand we can compute 2D CFT correlators if I have some clear idea of which vertex operators I should insert at these punctures. And the HGT correspondence says that if I take this 40 n equals 2 super conformal field theory I compute this partition function on the force sphere or actually on the squashed force sphere then there is a relationship between these two elements. But before getting to this, let me come back to this first statement that there is a 40 n equals 2 SCFT associated to any Riemann surface. So yesterday we started discussing class S theories that's how these theories are named. Class S theories of type A1. So we started by considering four free hypermultiplets. We observed that this theory has a USB-8 flavor symmetry. We focused on a subgroup of this USB-8 symmetry which is an SU2 times SU2 times SU2 subgroup of this USB-8 and now we designed some picture. We said each of these subgroups of the flavor symmetry we're going to denote in some picture maybe as follows. So this picture is just a representation of a theory of four free hypermultiplets where pictorially you make clear that there is this SU2 times SU2 times SU2 subgroup. So each of these punctures corresponds to one of these flavor symmetry subgroups. Then we started talking about a next example of a theory with an SU2 gauge group. So we looked at an SU2 gauge theory with four fundamental flavors. So this was a super conformal field theory precisely because this matter content is such that the beta function vanishes. And we remember that if you have four fundamental hypermultiplets of SU2 because the fundamental of SU2 is pseudo-real, this theory will carry an SU8 flavor symmetry. So in a quiver description, the original theory looks like an SU2 gauge group with these four fundamental flavors. And then I decided to focus on some splitting of these four fundamental flavors. I'm going to group them instead of us in one set of four, in two sets of two. So here these four flavors carry this SU8 flavor symmetry. But now I split them in two groups. These are hypermultiplets in a pseudo-rerepresentation of SU2. These carry SU4 flavor symmetry. These carry SU4 flavor symmetry. And SU4 is, of course, SU2 times SU2. So I can design some new quiver notation where I have the SU2 gauge group and I make manifest these two SU2 flavor symmetries on each side of this quiver. So this we drew yesterday as well. We observed that this picture is really equivalent to a picture like this where I have taken two of these types of objects. I've taken two times four hypermultiplets and I gauge the diagonal SU2. So just thinking about this theory, before I decide to gauge, I really have eight hypermultiplets. And then I gauge an SU2 subgroup of the flavor symmetry of those eight hypermultiplets. That subgroup is precisely the diagonal subgroup of two such trillions. That's the name that these things are given. So I take two such trillions. I focus on an SU2 times SU2 times SU2 for each of these two trillions. And I gauge the diagonal subgroup of the SU2. That is precisely what gives you this theory and that is precisely represented in this picture. And here I should observe that the tube in this picture is nothing else than the gauge theory, than the gauging in the quantum field theory. Okay, that was our second example yesterday. So let's now do a new example just to get some more intuition. But we slowly start seeing Riemann surfaces appear which we can associate in some canonical way to four-dimensional theories with SU2 gauge groups. So let me do a third example. I'm going to start looking at a bit longer quivers. So let me look at this quiver. Now I have two SU2 gauge groups. I have two fundamental flavors charged or two flavors transforming in the fundamental of this gauge group. I have two flavors transforming in the fundamental of this gauge group. And in between I have a bifundamental, an object that is fundamental both under this gauge group and under this gauge group. So each of these two gauge groups effectively has four hypermultiplets. Two are manifest. The other two come from this bifundamental object. So both of these gauge groups have zero beta functions. So this is a super conformal field theory. And as before, we would like to get a bit of an understanding of what flavor symmetry this thing has. So again, two hypermultiplets charged under SU2 because the two of SU2 is pseudo-real, they carry an SO4 flavor symmetry. And the same over here. These guys carry an SO4 flavor symmetry. Now this bifundamental object, so this isn't hypermultiplet represented by this link here which transforms under the fundamental of both these SU2s. This bifundamental object is real. You can think of it as the vector of SO4 just decomposed into SU2 and SU2. So it is a real object and real representations, you remember from the first lecture, carry one real object charged under a real representation of the gauge group carries an USP flavor symmetry, namely a USP2 flavor symmetry. So this link carries a USP2 flavor symmetry which is of course the same as SU2. So in total we have two SU2s from this SO4. We have another two SU2s from this SO4 and we have a fifth SU2 under which this bifundamental is transforming. So in total this theory has five SU2 factors in this flavor symmetry group and let me represent it in a picture similar like this, making manifest all these five SU2s. So I'm going to break the SO4 just like we had over here. This SO4 was split in picture like this. So we have an SU2 and another SU2. I have my first SU2 gauge group. This bifundamental carried an SU2 flavor symmetry. I have my second SU2 gauge group and then again I have this fork of two SU2s. So this quiver I like to represent like this, just like we represented this quiver like this. Or if I'm essentially fattening the lines of this quiver and represent gauge groups by tubes, this thing can be represented as, well again the tubes here are the two gaugings we're performing. Okay, so this is fun. It seems that any reman surface we draw, any punctured reman surface we draw corresponds to some gauge theory with SU2 gauge groups. But this is really true. Is it any reman surface? So let's think about that for a second. Why do I draw something like this? This looks a little bit wilder. So we have this triple gauging of one single trinium. So in that picture, in a picture like that, I'm really doing something like this where I have one trinium sitting in the middle and I used all three of the flavor symmetries of that trinium, of that beast over here to gauge something else. So is this still an SCFT? And the answer is yes it is. Since for each of these gauge theories you can verify that they all have beta function equal to zero since they all carry four fundamental flavors which are manifest like, these pairs are the fundamental flavors carried under each of these SU2s. So it looks a little bit strange perhaps but even this theory is an honest to God SU2 gauge theory that you can in principle write down. Even though it's not a linear quiver I can impossibly draw this thing in a way like I had over here. I cannot take this thing and draw some kind of a more mundane quiver description. Is that clear to everyone? Since if I have mundane quiver descriptions I will always have at worst situations like this where two of the indices of one trinium are gauged but the third one is sticking out. In this case I really gauged all three and that is still a conformal operation but it's not some type of regular quiver. It's what people call a generalized quiver. So really it is a statement that seems to be true. If I take any reman surface with some number of punctures it can have a non-zero genus if you like. It will correspond to some four-dimensional Nx2 super conformal field theory. So this was all for SU2s. We had SU2 gauge groups. We had SU2 flavor symmetry sitting everywhere. That's where this claimer comes from that it's class S tier is of type A1. A1 is SU2 and we see SU2s all over the place. Sorry say again. Okay sorry, yes I should have said. You should make your reman surface by stitching together a pair of pants. So it's okay, you're right. It's not any reman surface. It should be decomposable in pair of pants where by pair of pants I just mean this thing, the trinium. So two punctures is too few. No, I mean I don't know what you call them unique. No, it's not unique. Let's give an example of, well let me get back to this question in a second. It's not unique if you keep track of a bit more information than I've been telling you so far. So let's first briefly look at the generalization to higher rank. Just to point out a few features, but in the rest of the talk I will not really care about the higher rank generalization. So let's instead of having four free hyper multiplets, let's look at nf squared, well let's just do n squared free hyper multiplets that are larger than two. So this thing, again it carries a USP2n squared flavor symmetry. We can choose to focus on a subgroup of that flavor symmetry which is SUn times SUn times U1 and we can decide to represent this object again by three punctured sphere or by some trinium like over there. But now we need to have two types of punctures just because I want to associate these flavor symmetries to the punctures but obviously U1 is not the same as SUn so I have two types of punctures. The dot corresponds to the U1 and the two circles correspond to the SUn factor. Okay so now if I have this pictorial representation I can just run the same game. I'm going to look at an SUn gauge theory with two n fundamental flavors and now observe that the flavor symmetry of this thing is not enhanced because of the n of SUn it's a complex representation so this just carries a U2n flavor symmetry. But I can still choose to consider a group of n and another group of n in a picture like this where we start getting some linear quivers and these two groups of n hyper multiplets each carry an Un flavor symmetry. Then still following exactly the same steps as up there I'm going to represent this as my SUn gauge group a U1 flavor symmetry and an SUn flavor symmetry so I'm splitting this Un in SUn times U1 the same on the other side and again just like over there I'm going to represent this by some punctured Riemann surface but now also involving this U1 punctures again because we see them sitting over here already. So this is the exact analog of that puncture of that picture the tube here again represents gauging since we're doing well I didn't write that this is type n minus 1 since we're doing type An minus 1 this gauging is an SUn gauging it's precisely this gauging we have the U1 puncture and the SUn flavor just like here on the other side the same and this picture really tells you what is happening we took two such objects twice n squared 3 hyper multiplets and we connected them by some tube where this tube is gauging we drew things a little bit more cleanly where this gauging is really you have this SUn flavor symmetry you take the diagonal subgroup and that is what you call your gauge group that gives you that picture in that operation I just described taking two n squared 3 hyper multiplets engaging this diagonal SUn is precisely the same as what I described over here so good I can draw more general linear quivers like this you will get pictures where I start connecting this tube with another of these types of three punctured spheres I can keep going but over here we noticed a slightly more funky object appear we noticed that an object like this appears and we're obviously not going to get an object like this from the ingredients I've described so far the only ingredient I really have is this type of a sphere two what goes on the name of maximal punctures two punctures that carry an SUn flavor symmetry one puncture that carries a U1 flavor symmetry I would like to draw a quiver like that I would really need an object I would really need a four-dimensional theory which I can associate to this object where I have three maximal punctures so okay which theory can we possibly assign to this now this was analyzed and studied in beautiful detail by Davide Gaiotto in his paper the answer is it's nothing Lagrangian some strongly coupled isolated super conformal field theory so the question marks here without telling you how you can ever be sure of that this is the correct answer is some strongly coupled isolated SCFT this is not supposed to be obvious and the CFT goes under the name of Tn maybe you've seen that somewhere so okay these type of pictures they really generalize and it's a beautiful framework to understand four-dimensional Nicos II super conformal field theories but as soon as you increase the rank you're going to have to deal with these types of theories associated with three punctures fear with three maximal punctures and then you're automatically inside the realm of strongly coupled and isolated super conformal field theories so for the case over there this distrillion theory these four free hyper-multiplets that's the only free case that you can associate to a three punctured sphere with three maximal punctures in any higher rank this beast is some strongly coupled theory just to be concrete maybe some people of you are familiar with this case so if you take A2 this is really the E6 theory of Minahan and Nemashansky okay this was my little detour just to indicate to you that this framework is more general than what I'm going to use it can be generalized to higher rank in fact it can be generalized to different gauge group you can do D type, you can do E type you can generalize even more to non-simply least if you have twist lines anyway it's a very rich story but that would stick to the A1 case for the rest of this talk sorry say again I'm going to engage part of gauge network I'm going to engage where? like we just keep SU2 and let SUn minus to be in the strongly coupled theory in which theory? in this framework oh then it won't be super conformal if you try to gauge in SU2 with the matter content that is described there no I mean there is SUn gauge network now I want to put SU2 as a gauge network and rest of others to be engaged right but that will not be conformal the matter content is just not right to make that ever conformal the only gauging you can do within the context I've described so far is between maximum punctures if you want to stay super conformal I mean if you don't care about super conformal properties then yeah you can do whatever almost whatever you want so let's there's a little bit more data than what I have done so far so I'm back to A1 I will stay in A1 for the rest of the talk so I had this picture to represent an SU2 theory with four flavors and I told you that this tube corresponds to gauging but there's a little bit more data in this Riemann surface than what I said namely that the length of this tube is directly proportional to 1 over G Young Mill squared we can impose that as an additional rule on this picture the longer the tube the weaker the gauge coupling and in fact in the limit where the gauge coupling is well infinitesimal or let's say even zero this picture becomes infinitely stretched and in fact it just pinches if you turn off the gauge coupling you just go back to your two free theories so that's it's then obvious that this extra rule that the length of the tube is proportional to 1 over G Young Mill squared sort of makes sense because if I separate them infinitely far it's essentially as if I have pinched off the two spheres and I get my two free theories back is that clear to everyone so if you have a sufficiently long tube you're really talking about weak gauging meaning that the Young Mill's coupling which since we're talking about super conformal field theories it's really a parameter it's not a scale as it would be in just regular quantum field theories so weak gauging corresponds to sufficiently long tubes but then what happens if I decide to well make the tube less long if I start dialing the gauge coupling to be larger and larger in other words if I start considering stronger and stronger more and more strongly coupled theories so I start having a picture like this this is somewhat stronger coupled I start having a picture like this this is very strongly coupled so I should keep track of which labels it were and a part of this story is that in fact at this point a new dual description of the same theory emerges so you may be familiar with S duality this is exactly what will happen now what will I do now is this is very strongly coupled but a new weakly coupled picture can emerge if I start pulling apart things in the other direction so this is a different type of gauging you see that well you see it from the labels we have different matter content in the two blobs I'm gluing together this is a different gauging different weak gauging and the procedure that took us from this description by dialing the gauge coupling we made it more and more strongly coupled the theory and then at one point there is a new dual description that emerges with new degrees of freedom where the gauge coupling becomes weakly coupled again and this is this sequence of pictures here we have the tube like this which is weakly gauged here we have the tube like this and these are two dual descriptions of the same theory so this duality it goes under the name of S-duality and it was studied first I guess by Cyberg and Witten and you see I mean if I turn this picture back like sideways it really looks like this picture but not quite because you see the labels changed here was A, B, C, D and here is A, C, B, D if I turn it the labels are still different so this duality action it produces a theory which looks exactly like this but the matter content got a little bit well, permuted said differently here in this type of quiver where we had this S08 flavor symmetry the hyper-multipleps really transformed in the vector representation of S08 but over here the hyper-multipleps they transformed now in one of the spinor representations of S08 so it really, it's the same theory like if you write like Rangin the same theory but you should remember that over here we had an 8V over here we have an 8S just because these labels got a bit permuted is this sort of clear? this is a linear triad instead of duality no I mean at the moment it's just duality and then I'm going to declare now what generalized S dualities are and then ok, I mean the word dual becomes a little bit moot it's like yeah I mean there will be infinitely many these types of degenerations you can consider which we're all going to call dual so let me emphasize a bit more what I mean given some theory of type A1 which corresponds to some reman surface with some number of punctures you can always decompose it in pairs of pants like we have over here we have here two pairs of pants over here we have two pairs of pants but notice they're not the same pairs of pants so they're different pairs of pants decompositions where again the pair of pants is you really need to think of it like like so the puncture sphere is a pair of pants because you can draw it like this with the three punctures so any pair of pants decomposition of a reman punctured reman surface which we associate to an A1 theory corresponds to some weekly coupled frame like the example over here here we had just two, well actually we have three choices as you were pointing out we have actually three choices because we can do yet another permutation of these labels to find the third description instead of the spinner we would get a cold spinner so any pair of pants decomposition of a reman surface corresponds to some weekly coupled frame and moving in between all these different descriptions corresponds to what goes under the name of generalized S-dualities so if you do procedures like this what it really means is you have a gauge coupling which you start tuning to be larger and larger and a dual description emerges in terms of a different gauge theory which is then again weekly coupled like we had in this picture is everyone okay with this statement? so this at first sight maybe useless pictures they already contain some more physical information than we thought at first sight in fact they contain a lot more information as soon as you start thinking about sideways within curves and all that kind of information that is what is available for any four dimensional but okay I will not discuss that kind of stuff what I do want to quickly point out is that if you are just studying Lagrangian theories say for a higher rank again just for a second let's again look at this theory we had just now this was just some Lagrangian theory that you engage group with n plus n hyper multiplets if you play this kind of a game we can go to its strongly coupled regime and then we can try to find a new weekly coupled description like so and now you see that these objects that we felt compelled to introduce these three punctures here with three maximal punctures they really arise even when something as simple as taking a Lagrangian theory and dialing its gauge coupling to be larger and larger so this is a dual description of this theory so let's do if we say do A2 this is just an SU3 gauge theory with six fundamental flavors so the gauge is this tube we have three plus three flavors but over here we find this strongly coupled beast that I just told you this E6 of Minahan Emissionski coupled to some other stuff which if you are familiar with Argyra cyber duality is precisely describing that duality if you're not familiar with this duality then the take-home message is just that if you take a Lagrangian theory and you try to apply dualities or even more simply you just turn its gauge coupling to be larger and larger and it will land on descriptions which are again weakly coupled but involved in strongly interacting SCFTs as building blocks okay this was really the last thing I have to say about higher rank well here we know the quiver this was just some Lagrangian theory with a linear quiver it was SU3 with in the way it's drawn here we find three fundamental hypers which was really coming from this picture we started off with that is this side now this side doesn't have a Lagrangian description obviously because we have this strongly interacting blocks sitting here so at best you can give a generalized quiver description just like we had the terminology earlier this is some generalized quiver which you can try to give some generalized quiver description in terms of an object that maybe looks like this where okay this piece I was not really planning to describe it to you but fine this is a description if you really want so this thing is a strongly interacting block this is this E6 Minnehane-Machanski theory it doesn't have a Lagrangian description so this is not a normal quiver you would typically encounter and then okay this extra block gauges in SU2 inside this SU3 you have one fundamental hyper of that SU2 so good let's go back to A1 and stay there now for A1 theories even though we have these generalized quivers there were still Lagrangian building blocks the thing associated to the three punctured sphere was just a bunch of four free hyper multiplets so all A1 theories are Lagrangian even though we draw quivers which you would typically not quite consider there's still all Lagrangian if you keep track of all the flavor symmetries properly so within that class of theories in principles in the Lagrangian theories we can crank the localization machine and we can compute of all these theories at least in principle their S4 partition function just because they're all Lagrangian we can compute their force-phere partition function so let's focus on a particular example the example we started off with describing so this is a theory I would like to consider and as you remember this is just an SU2 gauge theory with four flavors where the four flavors carried as a weight flavor symmetry of which I have made manifest an SU2 to the fourth now for each of these SU2 flavor symmetries I can turn on masses so you remember if you do any type of localization but in particular on the force-phere vector multiplets that couple to the flavor symmetries they're not dynamical they're background vector multiplets and for each of these background vector multiplets inside the localization computation you can turn on a BPS configuration that does not break super symmetry so you can do it and when you do it you turn on masses for these different flavor symmetries I guess you've heard that a few times in the school already so I'll just do it so let me write the masses corresponding to these four SU2s as pA where A is equal 1, 2, 3, 4 if you like in terms of perhaps language that is more familiar if you go back to this picture where you had the SU2 gauge group with the U4 flavor symmetry we can focus on the carton well, on the maximal torus inside here as a U1 to the fourth we can turn on masses for all of these U1s this is more canonical language than these four SU2s and the relation between the masses you would turn on for these U1s and the masses we have turned on for these cartons inside these SU2s is as follows if I call these masses Mi for I runs from 1 to 4 then I simply have that it's p1 plus p2 m2 is p1 minus p2 this is not really important if you understand that in this picture I can turn on these four masses then this piece of the blackboard is somewhat useless information anyway, so these are the standard masses you would turn on for the U1 to the fourth and they are related to these SU2 masses just by these linear combinations okay, beautiful so let's compute the four-sphere partition function of this object so what do I get? I will get an integral so the gauge group is SU2 it's carton subalgebra as U1 so we just get one real integral turns from minus infinity to plus infinity along the real line what do we get? we get in here this one loop determinant which I will specify in a second just to remind you of how it looks like it will depend on on this gauge parameter which I have also called p maybe I should call this one capital P you will notice why I'm writing everything in terms of letters p so we have this one loop determinant which depends on this gauge parameter and on the four mass parameters we have the classical action which I will write in terms of q's so remember this q and q bar someone asked me that last time as well they are canonically defined as e to the 2 pi i tau where tau was this combination which I always forget in terms of g and mil squared with some coefficient and the theta angle well, this with some coefficient let me try to find the coefficients back so in my conventions it's theta over 2 pi plus 4 pi i over g and mil squared so that's tau, q is e to the 2 pi i of that tau and here I have a q and a q complex conjugate raised to the power p squared this is clear to everyone so in principle you have in the exponent here you have trace of the object you are integrating over squared so the object that we are integrating over is an SU2 matrix in the carton so it looks like this square this, you get p squared p squared trace this, you get trace the squared and you get 2p squared and then there are some factors I hope I did them right, maybe there's some factor of 2 here so anyway, this is a classical action written in some slightly different way than I have done so far and then of course we have these instanton partition functions which we managed to write as sums over in this case just two tuples of young diagrams of stuff and we have another instanton partition function so this is for the instantons at the north pole, this is for the instantons at the south pole and the dot dot dot you can in principle compute with these matrix integrals we discussed two days ago okay this is the four-sphere partition function of this particular theory let me also specify z1 loop since it will be important the z1 loop depending on capital P and these mass parameters little p looks like in terms of these Upsilon functions 2ip minus 2ip divide by the product for j let's do a around some 1 to 4 of q over 2 plus ip plus ipa plus so okay I went back to these mus but these mus are still defined over here in terms of the piece where p is the mass associated to these four SU2s okay so this object we knew how to compute let's keep it and let's change topic let's change topic to conform of field theories so you see the Upsilon really has is a function of b you remember it's like a double infinite product of mb plus nb inverse plus the argument and then another factor n plus 1b plus n plus 1b inverse minus the argument so there's b's for all these Upsilon functions also this q, q is b plus b inverse and then in the instant partition function we had these two equivalent parameters for the rotations in the two orthogonal planes in R4 they also entered there so there's b and b inverses all over the place inside these instant partition functions they I had a minus so there's this minus sign okay so this is the one side of the correspondence beginning let's do the other side the CFT side the CFT we should be studying is Liouville so Liouville is some particular two-dimensional conform of field theory but before getting to Liouville let me just give some general give some generalities about CFTs so some quick quick reminder of CFT facts so you often hear the statement that the conform of field theory is completely determined as soon as you're giving its spectrum and its collection of three-point couplings it's true in any dimension any conform of field theory is well I'll tell you in a second what is actually determined by giving the spectrum and the three-point couplings of any three local operators so if you have that information you can compute any correlator of finitely many local operators so it doesn't say anything about defect operators so maybe I should call this the local CFT data if you're giving if you're giving this data you can compute any correlators of local operators so let's see why this is true this is the first aim to quickly remind you of why these two pieces of data are sufficient and the reason is of course because conform of field theory is very constraining it restricts correlators quite well and let's explore that restriction for a second so conformal invariance since Liouville is just it's just a regular CFT it's not super symmetric or anything I will restrict myself to just plain conformal invariance which means that in D larger than 2 we have symmetry group D plus 1 comma 1 so remember I wrote down an example of this before we have momentum we have rotations we have a dilatation we have special conformal transformations and that's it so these generators if you put them together in some neat matrix you will be able to verify that they satisfy the commutation relations of SO D plus 1 comma 1 in Euclidean signature but in D equals to 2 which is an example where Liouville lives actually this SO 3 comma 1 which is really SO 2 C but people are often somewhat sloppy they think of essentially two copies of SO 2 and they think of these as the left moving copy these are the right moving copy or the holomorphic and anti-holomorphic copy where these SO 2 acts as Möbius transformations on the complex coordinate Z these acts as Möbius transformations on the complex coordinate Z bar of SO 2 that the global conformal invariance of a two-dimensional CFT but really it enhances to a full Virasaurus symmetry I hope this is somewhat familiar so just quickly an SO 2 algebra is generated by L0 and L plus or minus 1 with the canonical commutation relations and these guys are generated by the barred versions each satisfying their own SO 2 algebra and the Virasaurus symmetry is really an infinite extension of these 3 plus 3 modes to all modes where N is an integer and the same for the barred version so in two dimensions we really have an infinite symmetry group that is acting on well that defines the conformal invariance okay just I wrote it down before but let me quickly do so again the conformal invariance is a symmetry of space such that it leaves the metric invariant up to some val factor where of course the transformation is all diffeomorphisms which leave the metric invariant up to a val factor and of course we know how diffeomorphisms act it's all transformations where you take a diffeomorphism of space the metric which we take to be flat space metric so this is really just the delta since we're doing Euclidean it transforms like this if this object is the same as the original metric which we chose to be delta up to the scale factor we're talking about conformal transformation and I showed you before that these are the generators in arbitrary D but as I just said in D equals 2 it gets enhanced so what does this conformal invariance tell us about correlation functions let's imagine we have a correlation function of some okay I will restrict myself to identical primary scalars so these are scalar operators maybe for that purpose I should denote them with a phi they're just scalar operators and they're primary okay technically speaking it means that the k's acting on these guys are zero never mind here that was not too clear so if you take a correlator of identical scalar operators located at points x prime 1, x prime n where these points are related to the points x1 through xn by a conformal transformation so let's take a configuration of n points apply a conformal transformation we end up with points x1 prime to xn prime and these two correlators must be equal up to a bunch of these types of vial factors is this clear to everyone? this is the statement of conformal invariance so in particular imagine that we're doing the two-point function and we study a configuration which is obtained by performing a dilatation then we should enforce that this thing is equal to itself before you applied the dilatation so you see this is this is not obviously true but at any function that would compute this correlation function if you want this property to be true that if you rescale the coordinates you pull out an overall factor but ok this was a dilatation you can do this more generally for all these types of generators it's a standard exercise which I'm sure you have done before if you have ever taken a CFT class if you ever studied CFT on your own and the result is by imposing these types of conditions is in particular that the two-point function is completely fixed once you choose the normalization of your operators the two-point functions are completely fixed so in particular let's do again the two-point function of primary scalars primary scalar operators then we find a phi of x not necessarily take them identical but conformal invariance in fact forces me to take them identical they have to be identical scalars in a suitable basis and the coordinate dependence is just given by this so in particular you can see that this property holds if I rescale these two coordinates I'm going to get an overall factor over here lambda to the minus 2 delta phi which is precisely this type of factor so two-point functions are completely fixed by conformal invariance once you choose their normalization and once you like fix your basis of operators and in that basis I can take them to be diagonal or I can choose basis such that they are diagonal but three-point functions of again let's do primary scalars as an example but more generally three-point functions are completely fixed up to the three-point couplings so this is where this conformal data starts showing up two-point functions we could completely fix up to choosing some basis of normalization which we can always freely do three-point functions though will depend on numerical coefficients which are the three-point couplings that I mentioned here as part of the CFD data so indeed if you do this for three scalar operators then you find the standard answer that again I'm sure most of you have seen before it's proportional to this coefficient which we cannot fix by pure symmetry thoughts this is some dynamical information but kinematics of this three-point function does fix the coordinate dependence completely okay so this is okay I always just restrict myself to scalars but more generally you can do tensors, you can do anything and then still these two statements are true it may be that you have different tensor structures that appear in the three-point function of non-trivial representations of the Lorentz group but then still they each come with their own three-point coupling and all the position dependence and that kind of information is fixed by conformal invariance for a three-point function now when we get to four-point functions this is where conformal invariance is not sufficiently powerful anymore to fix the full coordinate invariance or the coordinate dependence for us but instead it tells you that say again for scalar operators and to make my life easier for identical scalars conformal invariance is somewhat weak it just tells you that it must look like something like so so we have an explicit coordinate factor up front this factor soaks up the scaling behavior of this correlation function but we have some arbitrary function of the conformal cross ratios so this umv there are particular combinations of these four coordinates which are invariant under any conformal transformation so any function of stuff that is invariant under conformal transformations is of course invariant under conformal transformations and we cannot fix this by symmetry considerations so okay just to be concrete in four dimensions the conformal cross ratios are x1 minus oh I should have said when I write x12 or more generally xij I mean xi xi minus xj so x12 conformal cross here is x12 squared x34 squared divided by x13 squared x24 squared and v is the same as u where you x change 2 and 4 so in particular you see manifestly that this thing is invariant under translations because it's always differences of coordinates it's invariant under scale transformations because we have equally many x's in numerator and denominator it's invariant under Lorentz rotations because we have manifestly taken singlets under rotations the only thing that is not trivial to check about this thing is that it is invariant under special conformal transformations and well if you're bored you can check it for yourself but I assume that most of you have seen these things before so this is the 4d case in 2d the conformal cross ratios are a little bit simpler because the global conformal cross ratios at least are a bit simpler they don't need all these pesky squares if I write them in terms of the holomorphic coordinates of the four points so these four points x1, x2, 3, 4 in two dimensions I can write them as a pair of complex coordinates z1, z1 bar, z2, z2 bar and so forth then I can define a cross ratio as follows for z and complete the similar for z bar so good so far we have learned that conformal symmetry is pretty good at constraining 2-point functions 3-point functions are pretty good as well just one constant to be determined by other means 4-point functions we lose bit of steam because we have this function of conformal cross ratios which symmetry doesn't have anything to say about symmetry also doesn't have anything to say about these coefficients that's why they enter in the conformal data but ok I just claimed that over there the spectrum of operators so knowing which operators exist in the theory with some conformal dimensions some Lorentz quantum numbers together with these numbers is sufficient to fix all correlators of all local operators finally many at least and now I say I don't know how to fix this so there seems to be some contradiction in what I'm telling you and the resolution is that conformal symmetry gives you another tool which we haven't used yet and which will in fact guarantee us that at least in principle we know how to compute this arbitrary function in terms of the conformal data ok I don't want to lose that blackboard so I'll stick to this half so one tool that we haven't used is operator product expansion the OPE as it is often called so again restricting myself to scalar primaries I can look at the OPE of some scalar operator with some other operator so what does the OPE do takes as an input two operators or the product of two operators and it writes it for you as an infinite sum of local operators so we take two operators and we re-express it as a sum over local operators so we go from two to one operator that's essentially what the OPE does it re-expresses such a product as an expansion in all the operators in the theory in fact I say all the operators in the theory but you can do a little bit more of an organization in these operators you can split them in terms of primary operators where maybe I should really at least once write what I mean with primary so let's take an operator at the origin such an operator is a primary if it is annihilated by the action of the special conformal generators if you have such an operator you have two components of your multiplet and you can act arbitrarily many times with momenta in all kinds of different configurations on that operator to start building some type of a multiplet so all local operators are organized in multiplets of the conformal group and these multiplets at least for physics purposes they're always highest weight representations where the highest weight is a primary which by definition is killed by operators and then I can act arbitrarily many times with any combination of raising operators which in this case are the p's so why is K a lowering and P a raising operator that comes from these commutation relations with respect to the dilatation operator which tell us that K has a negative eigenvalue under dilatation whereas P has a positive eigenvalue so pretty much like the harmonic oscillator we have raising operators and lowering operators which have one unit of positive or negative energy the same for K and P K has a negative eigenvalue under dilatation so this case are the lowering operators the lowest state in your multiplet therefore is killed by these operators and then we start building anything on top of that operator by acting with arbitrarily many momentum operators so all the statements I made over there in terms of this correlation function of primary operators I always meant that this KLR satisfies such a relation so ok we can restrict ourselves to a sum over primary operators because all these descendant fields everything you can build by taking the primary and acting arbitrarily many times with momentum that will be organized thanks to conformal symmetry in some manageable structure that is something we know how to deal with just because this is a multiplet of our conformal group and we understand these multiplets so this OPE really takes a form of some numerical constant lambda 1 to O of some object which is completely fixed by conformal invariance so it looks like this so this is some object it takes as an input the position of this operator it also contains a derivative it acts on my primary so essentially this object takes the primary and builds the entire tower of descendants in such a way that conformal invariance is nicely preserved so it's some differential operator in general and over here we see a numerical coefficient I've already called it lambda again because you can verify that this lambda is necessarily that lambda how you can do that you just take this type of an OPE you take the product of two scalars in here you replace it with this entire expression then we get a bunch of two-point functions but two-point functions were diagonal so really at the end of the day you can convince yourself that these coefficients need to be the three-point functions and in the same way you can in fact by playing this game you can figure out what this object is when it acts on a scalar primary I will not play all these games I just want to remind you hopefully that the product of two operators in a CFT can be re-expressed as a sum of primaries of some differential object acting on another set of primary operators if you like I can at least give you the first term of this operator as you would expect the first term is just some exponential of the position just because we know that conformal invariance will require me that the scaling dimension works out and the only scaling dimension for position we have is the position x so on the left we have two times we have delta phi 1, delta phi 2 which in this case I took equal but we can generalize a bit we have delta phi 1 plus delta phi 2 minus delta of O so this is the first term in that thing it's 1 plus blah blah blah where the blah blah blah is some differential stuff which that makes descendants of this operator O so this thing acts on the operator O so here it makes the descendants I need to hurry a little bit so we have this OPE now for this puzzle let's now use it again let me simplify my life and consider four identical scalars so what can I do here I can use the OPE to expand this product of two local operators in terms of such a sum I can use the OPE here and expand this product of local operators in terms of that sum over local operators in other words I can use the product algebra twice I will have a sum over two primary operators I have a double sum over all primary operators of the three point coupling between phi phi and that local operator the three point coupling of phi phi and the second local operator I'm just using that thing twice so I'm going to write it twice the position that you need to insert now is of course this x2 position for the second OPE is 3 minus x4 and this thing acts on what is left so we started off with a four point function but we're replacing this with a sum over primary operators we're replacing this with a sum over primary operators so what's really left is just a two point function of primary operators acted on by these beasts and then we should set y is equal to y tilde is equal to zero now remember that two point functions of primary operators can always be chosen diagonal I can always choose my basis of local operators such that these guys are diagonal so what that really means is that this double sum over primary operators is really just a single sum just because that thing always is always diagonal now the two point function of any local operator we know how to compute conformity variance fixes it for us these complicated differential operators conformity variance fixes them for us although I haven't shown you quite how to do that but they are in principle fixed they just conformal kinematics that determines them so at the end of the day this entire object is something that conformity variance fixes for us there's nothing unknown in here from a kinematical point of view and this entire object goes under the name of a conformal partial wave so conformal partial wave and again it's a kinematical object it can be computed well just using symmetry properties of the conformal groups we're considering and if you pull out some useful factors it becomes the standard conformal block so essentially for my purpose in this talk conformal partial wave and conformal block is actually the same thing so now I achieved what I promised you I have computed the 4-point function in terms of information that I do know I do know the 3-point couplings this thing is completely determined by conformal invariance so indeed I can compute the 4-point function in terms of information I know and this story of course generalizes if I want to compute a 5-point function I can always use the OPE reduce it to a 4-point function well to a sum of 4-point functions to a sum of 3-point functions in principle at this point I'm done because 3-point functions I know but you can even go one step further and reduce it to a sum of 2-point functions as I did over here and 2-point functions again you know or if you really insist you do the OPE yet again reduce it to a sum of 1-point functions and in conformal field theory all 1-point functions are 0 except for the one of the identity operator that you can easily check by just imposing conformal invariance on a 1-point function you will see that the only possibility is to have the 1-point function of the identity operator all the other 1-point functions must be 0 ok so just to draw a picture people typically denote the operation I did here in a picture like this so here they have the operator at position x1 here the operator at position x2 here x3 and x4 so this is the picture people draw this is the conformal block or conformal block decomposition of this 4-point function internally we have this operator O and now maybe you start seeing the connection to what we have been doing over there because if I fatten these lines I get exactly that picture back ok so this is conformal block decomposition but if you're interested in the conformal bootstrap you should observe that there was no reason why we should group these operators first and then these operators I could as well have grouped these and these together to perform my OPE so I could as well have done something like this and the fact that these two expressions need to be equal is very non-trivial and these are the conformal bootstrap constraints anyway this is a localization school not a bootstrap school I'll just leave it I think I had a position dependence already but maybe you're right oh well ok maybe you're right ok in the last 10 minutes let's do some Liouville theory so let's become concrete about which TFT we're talking about we're talking about the two-dimensional conformal field theory as I said two-dimensional conformal field theories in fact have quite big symmetry algebra they have a Virazoro symmetry in the holomorphic sector a Virazoro symmetry in the anti-holomorphic sector so concretely it means that we have a conserved stress tensor which satisfies an operator product expansion as follows plus regular terms you see these numbers C appear here this is the central charge of the two-dimensional conformal field theory and in for Liouville it is given by 1 plus 6 times q squared where q is of course b plus b inverse and any b goes you choose any real b then you find some value for q you find your central charge so similarly for the Virazoro algebra acting on the anti-holomorphic dependence with the same central charge and in this CFT I should tell you what is the spectrum and what is what are the three-point couplings so let me just tell you a closed-up sector of the spectrum those are scaler there are scaler primary operators so they exist in these theory scalar primary operators they are labeled by is there a question so the resist scaler primary operators in this theory which are labeled by some continuous variable alpha where really you should think of alpha as q over 2 plus i times some Liouville momentum where p is some real number and their conformal weight conformal weight is equal to the right moving conformal weight is equal to the left moving conformal weight is given by alpha times q minus alpha so ok I haven't quite told you what conformal weights are so maybe I should say that the conformal dimension is equal to the sum of these two ok so we have as a sub-sector of the spectrum scalar primary operators which are labeled by a continuous parameter alpha which is imaginary but really has a truly free real parameter sitting here and their conformal dimension is twice this number so alpha times q minus alpha you can verify that if you plug that in you always get some real positive number as it should be now you should also notice that v alpha and v q minus alpha they have the exact same conformal dimension it's obviously the case and in fact these two operators they are really the same operator in the theory so that means that you can write one in terms of the other with some function in between these things called reflection amplitude it's not very important what it is it's just important to realize that only really only have the range that you can have in alpha is physical the other half goes back to itself upon using this kind of relation okay we know well we know the spectrum at least the spectrum we care about the 3-point couplings I will tell you in a second but let me first write the operator product expansion of these scalar primaries it's a little bit more complicated than what we had over there over there our OPE was just a sum over primary operators but you see here that we have a continuum of primary operators in the theory so instead of a sum we really end up with an integral so I put a half here precisely because of this reason I explained here only half of the range is physically relevant I have an integral sitting here over well over all these primaries I have some explicit position dependence which at one point I specify the very first term of this operator C this is the term I'm going to write so before I called this entire object this C acting on the operator now I have taken the first term of that operator which is just the overall scaling dimension taken care of with some position dependence of the primary at the origin and all these dot dot dot since now we're in a two-dimensional CFT they're just not just conformal invariance but they really view as overall conformal descendants but they really view as descendants but that shouldn't scare us the idea is exactly the same as up there but instead of being able to act with all kinds of momentum operators here we can act with all the raising operators in the mode expansion of the stress tensor so remember that the stress tensor has the mode expansion in terms of these modes ln where minus one zero and one generated the global part of the conformal group we can act on this beast with all the raising operators which in particular mean with all the N for N positive so we act with all these and we get fear as overall descendants actually not L0 sorry and of course in here I made one I forgot to write one thing the three point coupling so this is the OPE and now I should just tell you what is the three point coupling and I have given you enough information to at least compute correlators of these primary scalar operators so I deliberately put the index up here and this object with one index up is related to standard three point coupling which I denote as this thing with three with three inputs as follows where you do this reflection and this object is given by the DOZZ formula so this stands for three people who figured out this formula four people well the Zs are a bit degenerate it's two times same logic of Dolan and Osborn Dolan and Osborn Dolsenko and anyway so this object it has some overall pre-factor which we shouldn't be scared of since we can still decide to change the normalization of the operators times the derivative of the Upsilon function evaluated at zero an Upsilon function evaluated at two alpha one at two alpha two one at two alpha three divided by four more of these Upsilon functions where the symbol alpha is the sum of the three parameters this is just an expression that people have figured out I guess maybe I should put a T here because it was proved unambiguously by the T so okay this is a three point coupling of Liouville theory and you see very nicely you see a bunch of Upsilon functions up here which slowly start to make contact with what I have written over there so in the last one minute let me tie up things by computing the four point function in Liouville theory so I'm just going to use the OPE twice I use it here well no I actually only use it once OPE here and then I have a three point function and three point functions I know because I know the three point couplings so that's what I'm doing it's equivalent to what we did just now in terms of doing it twice so this is the result we get again this factor of half because of this reflection thing we get two three point couplings and we get a conformal block squared the square comes from that's everything is really well you can factorize everything in holomorphic and anti-holomorphic well you can separately study holomorphic dependence and anti-holomorphic dependence or it's a differently we have Virazora times Virazora bar symmetry acting so you get anyway you get two copies of the conformal block sitting here one for the holomorphic dependence one for the anti-holomorphic dependence so you get this mod squared and this is as I was mentioning earlier this is the Virazora conformal block so it contains all information about all Virazora descendants of these primaries which is of the primary which is labeled by P just like we had in our okay it's gone but anyway just like the C operators this differential operator it contains all information about conformal descendants now we have an object that is similarly carrying all information of the Virazora descendants if I act on it on this two-point function okay my time is really up so I should I would just like to mention now that this result looks very much like that result if you plug in these two three-point couplings but it doesn't quite look like that you need to do some clever renormalization of your of your vertex operators so we're going to renormalize these guys a little bit just change their normalization then the three-point couplings will change a little bit it's important to keep in mind that in the OPE we had this index up so if you do the normalization be a bit careful we get a normalization factor here normalization factor here but essentially here you get it as well absorb it all in here you get the inverse well you get inverses from here and you get the normalization factor itself from there so these two are not quite an equal footing when you start introducing these normalization factors but if you do it alright then you find that this object is identically equal to that object with parameters identified in the natural way if I haven't made mistakes in writing that object over there of course I used to have expressions for this they're gone by now but anyway it should match so this is the statement of the AGT correspondence we have computed a four-point correlator in Liouville theory which again is represented by a picture like this and we have shown that this four-point function is equal to the partition function of the force sphere of a theory labeled by precisely this type of picture or if you like me to to fatten it we're computing a Liouville correlator on the sphere with four punctures where at the punctures we have these vertex operator insertions so at the level of the three-point couplings it's relatively easy I was describing it earlier after you take care of some normalizations you will find exactly the one loop determinant and the real miracle of this story is that this Virazoro conformal block precisely matches with the instanton partition function if you do all the parameter identifications the way you should do and this generalizes to all theories of class S of type A1 it also generalizes to theories of class S of type An in that case we're not talking about Liouville theory but we're talking about Toda Toda is a bit well it's a lot more complicated so that's all I have to say about AGT correspondings I think