 Okay, I would like to start by summarizing the general point of view that I advocated in the last two lectures. So the point of view that they advocated, by the way, there were very many good questions after the session, very, very many of them. And I think it would be of a great benefit if these questions were asked during the session because many other people must have had the same questions. So please ask questions which sound like good questions during the session, not afterwards. So let me just summarize. So there are two general facts that you can say about renormalization group flows. One general fact is basically what Hooft has discovered in the late 70s which can be phrased in the language that I'm using in the following way. So for what Hooft has discovered you need the following ingredients. You need a conserved current. So you need this equation to hold as an operatorial equation in the sense that I've been careful to define in the previous lectures. So you need a conserved current. And then you have to study what happens at coincident points. This is what Hooft tells us to do. If you can respect this equation at coincident points then Hooft has an argument that says that if this is true in the UV then this must be true in the infrared. So it's like a conservation law. So Hooft's argument is that if D mu, J mu can be respected or can be satisfied at coincident points then this must also be a property of the infrared. So the same must be true in the infrared because the infrared emerges from the ultraviolet. And therefore you get some anomaly matching. And we've seen one example of how this logic works. Whereby we prove that Delta K UV is the same as Delta K infrared using this idea. Well, I've given you a completely independent self-contained proof by analyzing current correlation functions. But you could also say that this is a Hooft anomaly matching condition. I just gave you a very concrete proof. But you can understand it from this general point of view. So this is the Hooft principle. This is from the early 80s. This is when it has been understood that this is true. A more recent idea, a more recent idea is that there could be symmetries which are actually not conserved. There could be some charges that are not actually conserved along the RG flow but just approximately conserved. So we could have approximate currents that they satisfy d mu j mu equals zero in the deep UV and in the deep infrared. Okay, this is like what a phenomenologist would call an accidental symmetry. Only that phenomenologists call accidental symmetries only those symmetries that are satisfied in the infrared. For phenomenologists, it would be crazy to call a symmetry that is emergent in the ultraviolet and accidental symmetry. That's not how effective filter it works. But it is true that in many situations, like the situation that I've explained with the axial current in electrodynamics, there is an accidental symmetry in the UV and an accidental symmetry in the infrared. When there is such an approximate symmetry, you can introduce spurions. You can introduce something like these pions which more generally you would call spurions. So we can more generally call them spurions to restore the symmetry everywhere. Except for anomalies, you can go through this anomaly-matching idea, derive some effective action and sometimes prove inequalities like this. So after some work, this kind of setup leads to interesting inequalities between fixed points of the renormalization group. So this is an old idea that leads to equalities and this is a newer idea which uses less. It uses approximate symmetries and it leads to less, namely inequalities. But also that has lots of interesting consequences because it allows you to say something about what possible renormalization group flows are allowed. Okay, so this is a rough summary of the two different things that we know are true in general about renormalization group flows that there are these approximate symmetries, exact symmetries and they have different consequences. Now what I would do now is to, well, I have given you a proof of this business with K, UV and K infrared just from current correlation functions. So in this case, I gave you two proofs, right? One was a direct brute force analysis of the current correlation functions and the other was more abstract using this accidental symmetry and the Spurion field. Yes, 2D, yeah, sorry, I'm talking about 2D here. No, no, this general picture is true in any D but in 2D I gave you a proof of this. I gave you two proofs. One was a direct conventional approach using current correlation functions and one was a more abstract approach that was probably much harder to understand but it's more general. It's more abstract and also more general. Okay, so what I would like to do now is to, you probably know more or less the consequences of Tuft anomaly matching, I mean it's in the books. Now I would like to go over the consequences of this idea. This can be applied not just for this gamma in 2D but many other cases. I could go over them now just to give you some scan of the literature and you could look at my lecture notes where there is a proof of each of these cases that I'm going to list. I just gave you the proof in this case but they're all the same. Is this the same idea again and again and again? You get the same kind of results. So are there any questions about the general picture? Okay, so let's go over the cases. This, let's go over, let me list just a few cases where the second approach is useful. So in both cases, sorry, in both cases I have given you two proofs. I have given you a direct correlation function analysis which has no hidden assumptions and it's very clear and it's very neat. And then there is this more abstract arguments which give the same results and they're more general. So in most cases, I want to emphasize that in most cases the direct analysis of correlation functions is way too hard. In many cases, nobody has ever managed to do it properly and the only arguments that exist are of this more abstract type of anomaly matching and so on. So in many cases we don't have two different proofs. We have just one which is more abstract. We don't have the direct argument. So I'm just going to make a list of four examples, five examples in which this idea has been implemented and gave some results. One is a KUV bigger than K infrared. That's in two dimensions for U1 symmetries. That's the case that I've given you two different proofs of. The next one is about the central charge. So whenever you see this kind of inequalities you have to ask yourself what is the enhanced symmetry? So I told you that these inequalities come from having an enhanced symmetry. So the nice thing about our GIF flows is that there is an enhanced symmetry for this axial current in U1, if you have a U1 symmetry. But even if you don't have a U1 symmetry there is always an enhanced symmetry at the fixed points by definition. This is the conformal group, okay? So this has to do with the enhanced symmetry here or the accidental symmetry. The enhanced symmetry in this case is the conformal group. So you just go over the same type of argument with pulleons you try to somehow make the conformal group manifest along the flow and so on. Now there is an argument of that sort in four dimensions where A is some anomaly and C is some anomaly here. So this is some anomaly in two dimensions. This is some anomaly in four dimensions. Now there is another setup in which you can make progress. So let's imagine a three-dimensional space but with boundary. This appears in many applications in condensed matter physics that you have a three-dimensional sample with a boundary. And so you can consider renormalization group flows on the boundary but they are not purely two-dimensional renormalization group flows because they get affected by what's happening in the bulk. So in this case situation there is no available analysis of correlation functions at least not of now and the only proofs like in this case are through this more abstract arguments and you can prove again that there is some notion of K that you can establish. There is some notion of C that you can establish. And also there is some or six-dimensional RG flows. It's almost proven but not quite that this is true. There is still, I mean, there is no proof but it's not very far, it seems. So there is a bunch of cases that were out of reach but now they can be solved using this a little bit more refined idea. The advantage of this example that they gave you is that it captures the essence of all of the other examples. So K was some anomaly in some axial current, C some anomaly in conformal transformations, A some anomaly in conformal transformations and all of these transformations are exact at the fixed points, they are like in hand symmetries and they get violated by the crossover physics. So all these ideas are the same. They are all based on the same idea and they're very, very similar. So if you go over this example again and again and again and you reproduce all the details, you will be able to do all the other examples too. Okay, so now are there any questions about it? So now I'm going to discuss a subject that is slightly different but not very much. Right, so this is a non-dimensional sample. So like two plus one and this boundary would be one plus one. Or you could have a three plus one dimensional thing with some one plus one dimensional defect. Then you could also carry out this procedure. So this is about our G flows on defects that are embedded in a higher dimensional space. This appears in many applications in condensed matter physics and in statistical physics. Yeah, so this is about defects and this is about bulk theories which have the full point correct group. Here you don't even have the full point correct group because you have some boundary, okay? Any other questions about this? Okay, so I want to discuss a new topic which is conformal perturbation theory, conformal manifolds and many other related things. So I'll just try to give you some summary of some fundamental results in the field with some applications, recent applications. And well, the way in which this is related to the previous discussion is through some anomalies that will appear. This anomalies will be slightly different than the anomalies that we've analyzed so far. So there is a precise sense that we can discuss later in which all the anomalies that we've discussed so far are something that's called type A, but the anomaly that we're going to discuss now in this context of conformal perturbation theory is something that's called type B. So there are two distinct types of anomalies that appear in various applications. So now we'll see a type B anomaly. Okay, so let me just define the setup. So we have a conformal field theory which I'm just calling this conformal field theory P. P is a CFT. You've learned about CFTs. Now I'm going to assume, this is an assumption, that there exists an operator whose dimension is D. In fact, I'll assume that there is a family of such operators, the dimensions of which are D. And we're going to study what happens to the theory if we deform the action by a sum over lambda i, O, i, D, DX. That's the starting point of what the theory of conformal perturbations. Because naively speaking, these perturbations don't introduce a new scale. The dimension of this operator is D. This, the dimension of this thing is minus D. So the lambda i are the naively dimensionless. So the lambda i are naively dimensionless, classically dimensionless. Naively means classically, okay? So first order in lambda, the i dimensionless. And then the main concern of conformal perturbation theory is what happens at higher orders in lambda. So the dimension of O i is D in the original CFT P where lambda vanishes. But you can imagine that at higher order in lambda, let's say a second order in lambda, O would pick up some anomalous dimension and it would either be relevant or irrelevant. And the main concern of conformal perturbation theory is to understand whether it's relevant or irrelevant. So we can try to write down the beta function for lambda i. So D lambda i, D log mu is beta i, which is equal to, let's say, beta one, i, i, jk, lambda j, lambda k plus something of order lambda cubed. So this is the first term that may appear in the beta function for the coupling constants lambda. There is no linear term because the naive classical dimension of this perturbation is zero. So our mission to compute this beta function. So I'm going to review for you how this is done and then we'll discuss the situation in which this beta function vanishes, which appears in some applications. So let's try to compute beta i, jk, beta one, i, jk. What do I mean by compute? By compute, I mean that we have the CFT at P and the CFT at P is characterized by OP coefficients and by anomalous dimensions of all the operators that exist in that CFT. And the idea is to try to relate this beta function to this data that is encoded at the CFT at point P. So is the problem clear? Okay, so let's do it. So what we're going to do is we're going to expand the partition function, which is some function of the lambda i in the coupling constants, lambda i. So we would have one. So z of lambda i equal to z at lambda i equals to zero. And then this multiplies one minus lambda i O i plus a half of lambda i, lambda j. Let's call it lambda j, lambda k, lambda j, lambda k. And there is d x of O j at x, d dy, d dy of O k at y. There is some, well, this is all in some correlation, inside a correlation function as well as this, okay? This is just an expansion of the partition function to second order. Now the idea is that you can get from that, you can, the idea is that this integral, there is an integral over x and y. So the idea is that when x is very close to y, there is a correlation function. So this is a correlation function, the idea is that when x is very close to y, there may be a logarithmic singularity due to the operator product expansion. And this logarithmic singularity would renormalize that term in the partition function. So in particular, if we consider these two operators, O j at x, O k at zero, from general principles, there may be a non-zero component, C i j k, O i in the operator product expansion. And this would have to be divided by x to the power d, just by dimensional analysis. So such a term may exist in the operator product expansion of these operators. Now this guy could lead to a logarithmic singularity when you integrate over x. So when you integrate over x minus y, this could lead to a logarithmic singularity and thereby renormalize this term. So we can compute it. So let's quickly, let's compute it. So we can compute this logarithm. So how do we compute it? The idea is that we imagine integrating over x and y, which are very, very close. So we pick up a logarithmic divergence here and the theory has some UV cut of, let's say mu UV. So what we would find is that the contribution from this logarithm is a half times the volume of an SD minus one sphere, just from the angle, just because the integral over d dx over x to the d is a log times the volume of a SD minus one sphere. Then we have a logarithm of mu UV over mu, which is the scale down to which we integrate. So we imagine doing the integral for x minus y, much, much smaller than mu UV to the power minus one, but much, much bigger than mu to the power minus one. This is some shell of high energy modes that we're integrating out. This is where this logarithm would come from, from this shell of high energy modes. So this is the contribution that we get from this. Let's say if we plug d equals to four, which is a four dimensional CFT, just for the sake of it, then the volume of the three sphere is two pi squared. So we would get pi squared times log mu UV over mu. Okay, so there is a logarithmic divergence at second order in conformal perturbation theory, if these OP coefficients are non-vanishing. So that means that we have to renormalize the coupling constant lambda i, and it leads to a beta function. So we would find that d lambda i over d log mu is exactly given by, let's say in four dimensions, okay? This is when d is equal to four, you can easily write this formula in any d one. So it's gonna be pi squared times ci jk, where ci jk are now computable OP coefficients. Lambda j, lambda k plus order lambda cubed. So this is the simplest kind of computation that you can do in conformal perturbation theory, where we computed the beta function for some marginal operators, and the beta function turns out to be proportional to the OP coefficients. Okay, now you could take as a homework exercise to compute these guys, the lambda cubed terms. It's more complicated, but it's doable. You can develop this theory to higher orders and see that everything is consistent. You can also take, for example, the phi to the four theory. This is the usual, the most famous marginal deformation in four dimensions. And you can, there is the usual computation for the conformal dimension, sorry, for the beta function for lambda using Feynman diagrams, but you can just compute the OP coefficient in free field theory, use this formula and see that you get the same result. So here there is no, you don't need to do any loops in this approach. It's just the OP coefficient in some free field theory. So it's a conceptually simpler computation. Okay, so what I'm going to be, what I'm interested in, so what I would like to tell you about is that there are some situations. So there are situations where the beta functions for this lambda I are identically zero to all orders. Yes, yeah, so Marco asked a very nice question of what the hell am I doing these one point function vanishes? That's correct, so you could, so the proper way of doing this computation, which is what's in the notes, is that you compute the overlap of the unit operator with some string of operators that you inserted at infinity. Just some string of operators. And then you expand this unit operator. This is the formal expansion of the unit operator. And so you have to add some dots here for some possible insertions. And you add some dots here for some possible insertions. But there's still some UV divergence from the double integral. So the logic carries through. So you need to put some, yeah, to make these arguments formally correct, you need to add some insertions very far away from X and Y. Yeah, okay. So there are some situations in which the beta functions for such marginal operators vanish identically. I want to list three examples where this happens. And we'll try to study these cases in a little bit more detail. Mostly because they represent a new anomaly that is interesting. When this happens, there is an associated new anomaly which is of B type and it leads to interesting consequences. So the first case in two dimensions, this was discovered in statistical physics in sequence one models. This is what people in statistical physics called the Ashkin-Teller model, which has a parameter lambda that is classically dimensionless, but it's actually dimensionless to all orders, okay? So this is the first instance in which it was discovered. It's a two-dimensional model. Now, there are also many examples where this takes place in some large n theories. There are also large n models in three and four dimensions where this happens. And another big, huge class of examples are supersymmetric theories. Super symmetric theories in various dimensions. It equals one in four dimensions and equals two in four dimensions and equals one in three dimensions. You name it. Actually, I'm going to concentrate on n equals two in four dimensions, which is eight supercharges. This, the application that I'm going to discuss is going to be for this theory, which is a cyber-guitand theory. The concrete example that I'll study would be some cyber-guitand theory. But there are many other examples where this happens and it's interesting to try to understand this situation a little bit better when the beta function vanishes identically. So the nice thing when the beta function vanishes identically is that you remember that we started from some conformal field theory at some point P. But now that we have some beta function that vanishes identically, we have a space M of conformal filters, right? Because we can deform the conformal filter by any of these exactly marginal couplings. So the terminology is that if the beta function vanishes identically, then the corresponding couplings are called exactly marginal. This is in contra-distinction with marginally irrelevant or marginally irrelevant that appear in QCD or five to the four. Here they would be called exactly marginal. So if there are exactly marginal couplings, then there is a manifold M. This is a manifold of conformal filters. So if we have a manifold of conformal filters, we can do geometry. We can ask, is there a nice metric? We can start measuring distances in theory space. You can take some point P and you can ask what is the geodesic distance between two conformal filters? So you can introduce ideas from geometry into quantum filter in this situation. You can also do it, I mean, you can do it more generally, but in this situation it's particularly clean. You have a manifold of theories and you can ask if there are Romanian structure or mission structure. What are the geodesic distances? What is the metric? What is the Riemann curvature? You can start asking questions in geometry about the space of conformal filters. So for that, we first need to define a metric. And the definition of a metric was given by Zamologikov. So Zamologikov gave a definition for a metric, which at first sight looks very confusing, but I'm going to explain why it makes sense. So Zamologikov said, look at these operators, OI, which are exactly marginal. Put it at X, put another one at zero. This is going to look like one over X to the power 2D because they're exactly marginal, so their dimensions are D. But then there's going to be some normalization factor, which depends on P. So this is computed at some point P on the conformal manifold. And so you would have some tensor, Gij of P, that depends on P. So the claim is that Gij defines a Riemannian metric. This is the claim, the Gij defines a Riemannian metric. You could ask, why does this claim, why does this make sense? You might be used to the fact that two point functions are always normalized to one, right, that they're chosen to be or to normal, that Oij is just chosen to be delta ij. So you could just choose it to be delta ij at every point, and so the metric is trivial. Why is this definition true? So the point is that you can't do it at every point. You can choose the operators such that their normalization is delta ij at some given point, but you can't do it everywhere. It's like choosing normal coordinates in differential geometry. You can do it at the point, but you can't do it globally. In fact, you can argue that, so you can ask, where are the deformorphisms that allow us to choose Riemannian normal coordinates? So where are the deformorphisms? The deformorphisms are just rare definitions of coupling constants. So the coordinates on this manifold, the coordinates on this manifolds are the lambda i. So the lambda i are coordinates, and if we do lambda i going to some function i of lambda j, if we do a general coordinate transformation in the space of coupling constants, then this will take the metric as usual, then gij will transform as like the f over the lambda, you know this formula. It will transform like a tensor. So by doing general coordinate transformations in the space of theories, you can at most set the metric to be one at one point, like to be the unit metric in one point, and you can also set the derivatives at that point to be zero. Maybe you can choose the Christoffel symbols to vanish like in Riemannian geometry. So we can, so let me summarize, you can choose Riemann normal coordinates by rare definitions in coupling constants space, in the space of couplings. Good, so the environment information in differential geometry, so in differential geometry there is a nice quantity that tells you why you can't, that measures the amount to the extent to which you are unable to set Riemann normal coordinates everywhere. This quantity is known as the Riemann tensor. So if the Riemann tensor is non-vanishing, you can't choose normal coordinates everywhere, you can just choose it at one point. So one may try to compute the Riemann tensor in this situation. This is a hard computation, which for the interested students is left as an exercise, but one can give a closed form expression for the Riemann tensor on the space of theories. So this is the expression, I'm giving you the expression, it's an integral over some region A, which I'll define. It's true in any number of dimensions, but the integral, sorry, let me hold. It's true in any number of dimensions. So we have O, I at zero, O, K, O, L at one, O, J, infinity. Then I subtract K. Now I just have to define the region A. This is just, I'm not expecting you to understand this formula. It's just that you know that one can write down a completely sensible formula for the Riemann tensor in the space of theories. So the region A is defined to be Z smaller than one minus Z, Z smaller than one. So it's a funny region in the dimensional space which looks like the intersection of some sphere with a half plane. I mean the interior of this intersection. This is computable. So once you have a conformal filter region, just compute this integral and you'll be able to know the Riemann tensor, yes. No, this is exactly the space of the couplings or the coordinates in this space. So different CFTs are parametrized by the coupling constants. Yeah, the space of couplings in this case is the conformal. This is what we call the conformal manifold. It's the manifold of CFTs. So let's say again, what's the question? On this point, oh sorry, on this manifold, each point is a CFT. The lambda is coordinates. So once you choose the couplings, you get a CFT. Yeah, this is just, I mean, there could also be coupling constants that correspond to relevant operators that take you out of the CFT to some other CFT. But I'm not focusing on those. I'm just focusing on the space of exactly marginal couplings. In general, the picture could be that you have some space of CFTs, but then there is some relevant coupling that takes you out maybe to another space of CFTs. But I'm just focusing on the space of CFTs that is spent given a point P. The INJ coordinates are, no, no. So we have a CFT P, which has a bunch of operators, OI, which are marginal, in fact, exactly marginal. So we add them to the action with coefficients lambda I. And that spends the space of CFTs, okay? So one can give some, okay. Now I'll give you an example, the simplest example. This is Dash Kinteler model. It's the first one that was discovered that has this property. In high energy physics, or due to BPZ, we would call this model the C equals one model. In any case, in this model, in this model, which is the simplest of this kind, the space of exactly marginal couplings is two-dimensional. So we have two exactly marginal couplings, there are two of them. And so we have a two-dimensional space of conformal field theories. And the space looks like that. Looks like the fundamental domain of a torus. But this is now a space of conformal field theories. So don't get confused. So the space of conformal field theories is some thing like that, where this is e to the i pi over three. This point is e to the i pi over three. And this is two i pi over three. And we know the metric exactly. These models are solvable, and people have been able to compute the Riemann metric and the Riesch scalar. So one finds that in this case, the metric in theory space is given by, let's call this coordinate here z. This is a complex coordinate in theory space. So the metric here is given by the z, the z bar over z minus z bar squared. It's an exact result that is known from two-dimensional statistical physics. So this line is identified with this line, and this arc is identified with this semi arc. So this semi arc is identified with this semi arc. Right, and this is also known as the Poincare metric. This is the canonical metric, the constant curvature metric on the upper half plane. But now this is not geometry. This is the geometry in theory space. Okay, so this is a space of theories which have exactly marginal couplings, and people have been able to compute the Riemann metric, the Riemann curvature and the metric exactly. So, and there are many other cases where we have exact results of that sort. Okay, so what I would like to discuss is some interesting synergy between this ideas and entanglement entropy and sphere partition functions that allowed to compute this metric in a new class of examples. Okay, so getting some information about the space of theories is an interesting problem. And I would like to report some ideas to review some ideas that allow to solve the following problem. So there is this, let me tell you what's the problem that has one is trying to solve, and how it was solved using some ideas about B type anomalies and various other related things. So this is the setup, and let me tell you what's the problem. The problem is that one can consider, for example, this is just an example, but there are many other applications. One can consider N equals to conformal filters in four dimensions. So these are some conformal filters with eight supercharges. Now, these theories typically have lots of exactly marginal parameters. They typically have lots of exactly marginal parameters. And an interesting problem with lots of connections to mathematical physics and to various questions, general questions in quantum filters to compute the metric in the space of theories. So let me just review something simple that is a known about this. Okay, so a special case, a very special case that you should know about. I'm going, this is just a review of a very, very special case, is maximally supersymmetric young mill theory, which is N equals four, the one that appears in the idea safety correspondence. N equals four, there is again, one complex exactly marginal parameter. So tau in C, some exactly marginal parameter. There is just one in N equals four. Tau is equal to theta over two pi plus i four pi over G young mill squared. So the G young mills, so the G young mills coupling and the theta angle together furnish a two dimensional space, one complex dimensional space of exactly marginal couplings. And again, the space of theories is given by this fundamental domain. It has been long known that in N equals four, the metric in the space is again the Poincare metric. So it's very similar to the Ashken Taylor model in statistical physics. So the metric in the space of N equals four theories is this. This has been in fact useful in checking the idea safety correspondence. That's one of the first checks that people have done to check that, you know, this metric is reproduced by the ADS dual. So this is an exact result about maximally supersymmetric young mill theory that has been long known. Now, if you take more slightly more interesting quantum filters such as Cyberg-Witton theory, I'm just reviewing some literature. So if we take Cyberg-Witton theory, which is let's say SU two with four fundamental representations, we have an SU two gauge theory with four fundamental representations. This is a conformal fuel theory. And again, there is one exactly marginal coupling. Again, just one complex exactly marginal coupling, which is given by this very same formula. And again, the space of theory looks like a fundamental domain. The space of theory looks like a fundamental domain. But in this case, the metric is not known. And for various applications, it's very interesting to try to compute. People have done perturbative computations. So it has been known for some time that the metric is d tau, d tau bar, times one over tau, tau minus bar. And then there's some, there's lots of corrections. I wouldn't remember the coefficients that are in the notes, but there is something like zeta three over tau minus tau bar to the four. There is an infinite series of corrections if you just do a weak coupling expansion. This is some expansion around weak coupling where the imaginary part of tau is very, very large. So this is a good series. And so it has been an interesting challenge to try to compute this series. And then you can examine various ideas about QFT, such as resurgence, whether the series converges, whether it's an asymptotic series, what are the geodesic distances, what is the volume of the space. You can try to ask lots of interesting questions if you have that. And some of information is available about the first few perturbative terms. So in the remaining 10 minutes, I want to explain the general idea that allows to connect this computation to an anomaly and how this anomaly can be solved. And now we have the full series expansion. So this problem has been solved using some ideas about anomalies that I want to explain. But so far this is just the, I mean, I've just defined the problem. Is the problem clear? Good. So I want to explain why this computation is related to computing an anomaly and how this anomaly can be computed. By the way, before I might not have time to discuss it at the end, so I'll just say it quickly now. Now that the full series is available, one can, for example, analyze the Borel properties of these perturbative expansions. It turns out that it's an asymptotic expansion like Dyson has predicted. There are some poles on the Borel plane. There are some various other interesting facts about this expansion. But the main message is that it turns out to be an asymptotic series, not a convergence series, very much like Dyson predicted. Okay, so a type B anomaly. So I want to explain this very subtle anomaly that allows to address this problem. So let's consider the two point functions of OI again. I'm just considering the same object that Zamalochik have considered when he defined his metric. I'm just writing the same formula that I wrote down again, but only now I'm going to discuss it in four dimensions. Since I'm interested in four dimensional theories, I'm going to specify to D equals four. So I have one over X to the eighth, I, J. So in position space, this looks like a perfectly conformal correlator. This is completely consistent with everything we know about conformal filters. But there is some funny thing that happens if you transform it to Fourier space. So let's try to transform it to Fourier space. In Fourier space, this is where we discover anomalies, right? Position space is good for making sure that we understand what we're doing, but in Fourier space, it's much easier to discover anomalies. So let's just do the Fourier transform and see what happens. So what is the Fourier transform of one over X to the eighth? D for X, X to the eighth, E to the I, K, X. Okay, so what is this? This is obviously divergent, so it doesn't make any sense, you could say. But of course it does, Fourier transforms exist, and one has to just know how to regulate it. So, namely, if you just do dimensional analysis, this should have been K to the fourth. So it would seem that this correlation function is given by G, I, J at P times P to the power four. But this cannot be right. Why this cannot be right? Because if it were P to the fourth, this is a polynomial in momentum space. What did we learn about polynomials in momentum space in position space? There are delta functions. This is not a delta function. This has support at separated points. So this cannot be right. So what's the right answer? Okay, good. So we have to stick some log. That's how we do Fourier transforms, right? So this is the right answer. What, you could ask, why does this formula make sense? Well, on face value, it doesn't make any sense. But one has to treat correlation. One has to treat these objects that have quantum filter as a distribution, as you know. Not as a function as a distribution. And in the sense of distributions, this makes perfect sense. So there is a log and there is a scale. Weird, because it's a conformal filter, right? But this scale is not so harmful. It's not so harmful, because if you change the scale, you get a contact term, because it's pure polynomial, right? So if we change mu to mu prime, we get P to the fourth, which is just a box squared of a delta function in four dimensions. So it's a contact term. So this seems to be okay. There is some scale that we had to, there is some spurious scale that we had to introduce, but this spurious scale is kind of unimportant, because if you change it, you just change something at coincident points. And this is, but these coincident points correspond to anomalies, right? But we'll learn that if you can't satisfy some symmetry at coincident points, then it means that there is an anomaly. This anomaly is very different from the anomalies that we've discussed in the previous sessions, because in the previous sessions, there were no logarithms. Here there are logarithms. When there are logarithms, we say that these anomalies are type B. When there are no logarithms, we say that it's type A, roughly speaking. So these anomalies are of a different mathematical nature, but there are still anomalies. So let's try to think about this. Let me try to explain the consequences of this anomaly. So imagine that these coupling constants are promoted to functions. This is just to explain the consequences of these anomalies. So just imagine for a second, the these coupling constants are transferred, are promoted into functions. So what we add to the action is lambda ix, o ix, d dx, or d4x, for the sake of concreteness. So in this case, if we now make, okay. So in this case, the logarithm that we found, the logarithm of p squared over mu squared, leads to a trace, leads to a trace anomaly, which is that t mu mu is equal to lambda x squared lambda. And this is i, sorry, this is, let me just put the indices and then I'll explain where this comes from. This is the fundamental equation. This is the fundamental equation that I claim that there is a new anomaly as a result of this logarithm. And if you promote the coupling constants into functions, then it really, then it shows up in the trace of the energy momentum tensor with this coefficient, which is the zomologic of metric. Okay, so the zomologic of metric appears in some anomaly. This is the zomologic of metric. So I would like to explain why this formula is true. Why this formula is true? So classically, lambda is dimensionless. It shouldn't have led to a violation of conformal invariance. So t mu mu had to be zero classically. But quantum mechanically, because of this scale that we need to introduce, if we make a scale transformation, which we have to add to the action some contact term. So that means that as we make scale transformations, we have to add some local terms to the action. Now, t mu mu is the generator for scale transformations. So when we make a scale transformation, what this formula says is that when we make a scale transformation, we have to add something to the action. And this something is exactly the object that you see here. If you make a scale transformation, you have to add a box squared of some delta function, some contact term. So this is what this equation says, mathematically, that if we make a scale transformation, we have to add the box squared of something. So very roughly speaking, this equation is equivalent to this type B anomaly. So type B anomalies always work like that, that you find some logarithm, and then you can write it as t mu mu equals something. Okay, so we learned that this homological metric appears inside some anomaly. Now, the point is that, the next point is that if your theory is also supersymmetric, then you have to, if your theory is also supersymmetric, you have to make sure that the right hand side is consistent with supersymmetry. If the theory is supersymmetric, then there would be many other logarithms, not just this one. So this would be just one logarithm, there would be many others, because there are many other operators in the same supersymmetric multiple. And so you need to make sure that, so we need to make sure that the dots are consistent with suiting. Now, technically, this is a hard exercise to make sure this formula is consistent with supersymmetry, but it's technical exercise. It's more less clear what to do. And so what you find after you do that is that you have to add like five or six more terms here, so that the whole thing is consistent with supersymmetry. But one of the terms that you find after you do that is the following. So after doing this, you find the following term. I'll just write the final answer. So one finds the following term, t mu mu equals lots of lots of things, and then there is box of some function of the couplings times the reaches scalar. This is one of the terms that you find if you supersymmetrize it. Now, what is this function K? K is a certain function from which the zomological metric can be derived by taking two derivatives. Oh, sorry. By taking two derivatives in some way, one can derive the zomological metric. I'm a little bit fast here because there are some indices which are called holomorphic, and some indices which are called autoholomorphic, but it's not important for the main message. The main message that I wanted to make to deliver is that this anomaly always exists even in these C equals one models, and it's interesting to investigate it. But in the particular case of supersymmetric theory, it's also accompanied by this kind of term. And this term leads to the following result. So this is a general result about this kind of supersymmetric theories in four dimensions, and it's actually also true in two dimensions. So if you put the theory on a sphere, could be a two-sphere or a four-sphere in two and four-dimensions respectively, you get e to the k of this coupling constants. So the idea is that, let me explain now why this is true. So the sphere, basically the same as flat space up to a conformal transformation. This is the stereographic map that you know. The sphere and flat space for conformal filters are very, very similar objects because they just differ by adding one point at infinity, which is a conformal transformation. So you can put a conformal filter in the sphere without much effort. It's a canonical transformation. But due to this anomaly, we due to this term in T mu mu, you can make a few steps and prove that the partition function on the sphere computes the exponential of the metric of the zoological metric essentially. Yes, you mean integrability with respect to what? Right, right. Yeah, I'll mention that in a second. So the rough point, the idea that I wanted to explain is that there are anomalies which are of the sort that we discussed in the previous session, but there are also anomalies which are all completely different kind. They come from logarithms and they lead to this kind of result in one situation. So they allow to compute the metric on theory space by doing the sphere partition function at every point. So the idea is that, let me just make a final picture and then I'll answer the question briefly. So this is the space of theories, right? This is some point that we're interested in. So the idea is that our probe is going to be a sphere partition function, which could be a two-sphere or a four-sphere if we're in four or two dimensions. We just take that and measure it at every point, okay? We compute the sphere partition function at every point and the claim is that if we do that at every point, we get an exponential of the zoological metric. This is a result that can be proven by these anomaly arguments that I outlined. Now, it turns out that the sphere partition functions can be computed by some ideas from localization. Some of them were computed long ago, some of them much more recently, but because there is this relation between the two objects, which are completely different. One is the metric in theory space and the other is the sphere partition function that you measure at every point. But because there is this relation between the sphere partition function and the metric, you can use results about localization to compute the metric in theory space exactly. And so in essence, this kind of relation leads to a solution of lots of problems in mathematical physics ranging from mirror symmetry to various metrics on modulate spaces that are related in the end of the day to this anomaly. Now, to your question. Conformal manifolds only have a remaining structure. For example, in this Ashkin-Teller model, there is only a remaining structure. There isn't much more than that. But in supersymmetric theories, usually these spaces are keller. Usually they are keller. And so there is an integrable keller structure. So usually they are complex and even further they are keller. That's not always the case. This is in the vast majority of cases. But there are some exceptions. For example, if you study a Sigma model on K3, this is one example that appears in the literature a lot. Then it's actually not keller. It's not even complex. It's a quaternionic keller manifold. So it's not keller. It's not even complex. It's more complicated. I don't think that there is a general picture of when, what happens when. But I think that maybe it's something that can be resolved. But yeah, in general, in this example, it's even equals two that I've discussed. You get an integrable keller structure. And so there is a relation between sphere partition functions and an exponential of the keller metric. And that allows to solve a very nice list of problems that remained from the 90s about cyber good and theory, this relation. More generally, these type B anomalies that come from logarithms lead to various other results that are in the same spirit that people are investigating now. So I could talk about it for much longer. But yeah, any other questions? Yes, you're asking what happens if this space of theories has a singular point. So indeed, even in the Ashkin-Teller model, there's this, you remember that there was this casp singularities. So it's not an actual manifold. There could be some singular points. In fact, in essentially all the examples we know, there is at least one casp. So you can actually prove something. I'll tell you some result that without proof, this proof is also related to anomalies, but it will be too long to explain. There's actually a global result about singularities or some sort of result in that direction, which is that, you know, that these spaces are keller. So it turns out that the space, turns out that this manifold M, let me write it in words, M cannot be compact without singularities. So you can prove some structural result that at least in one direction, you have to have infinite distance, or you have to have some singularities. This is a result that you can prove rather rigorously, that the space of theories cannot be a compact space. And indeed, in all the cases that have ever been studied, there's some way to run away to infinity. This is just one example that I kept repeating, but there are many others. And in all of them, there seems to be a way to run to infinity. This seems to be a very general fact about the spaces of theories that they cannot be compact, which a priori could be, but it seems to be impossible. There is no good classification of the possible singularities in the space of theories, but there are many examples in which you can try to study. Yeah, for example, this is a singular point. That usually means that you can maybe restrict, you can divide the theory by some orbit fold, but there is some enhanced symmetry. Yeah, in this point, there is a slightly enhanced symmetry, which is some combination of the fundamental group of SL2Z, leave this point invariant. So there is a slightly bigger symmetry. So I think that's usually the case, that singular points are associated to extra symmetries, but I don't know if that's always the case. Other questions? Yes? No, you see, if we just take the original point P and we add perturbation, which is some exponential of some couplings times some integrated marginal operators, this just describes some new point P prime. It's not like a dynamical process that you go from here to here. Like every CFT corresponds to some choice of coupling constants. There is no dynamical evolution from one. There is no time in which you flow from here to here. This is just a space of conformal field theories. Yeah, so the coordinates on this space are the coupling constants. So I've given you arguments why this space is a Riemannian space. You can compute the Riemann curvature and you can measure geodesic distances. That makes sense. You can also measure total volume. By the way, this is an interesting fact that even though this space looks non-compact, it's volume is finite. So the volume of this thing, if I remember correctly, is a pi squared over three, or something of that sort. Maybe pi squared over eight. So even though there is a non-compact direction, the volume is finite. And there is a conjecture that the volume of the spaces is always finite, sorry. Yeah, there is a caspect infinity, but the volume is still finite. So I think there is a conjecture that in all the space of conformal filters has a finite volume, but it's non-compact. You see? It's never compact because of this theorem, but it may have always finite volume. That's, as far as we can say, that's true. Yes? What isometry? Can you? Yeah, yeah, so they look similar, right? I mean, the modulate space of n equals four and c equals one looks similar, but we also know many examples where it's not the same. So if you take cyber-guitare, it looks like this thing. But if you take SU3 gauge theory, which is some generalization of cyber-guitare, actually the modulate space is bigger. It's like this. Related? Yeah, I mean, this is a deep question, but I think there may be a sense in which they're related, they're very closely related. There is this idea of linking two dimensional theories and four dimensional theories via compactifications from higher dimensions. And in that context, it's actually precisely correct that you can think about n equals four, some of the modulate space of some torus. There may be a precise sense in which they're related. Any other questions? Yes? Right, so what I discussed in the first two lectures, sorry, we probably smoothly transformed into a discussion session? Oh, it's okay, but is my here? Yeah, so it's true. What I discussed in the first and second lecture was about the renormalization group flows in flat space. You could also discuss curved space and there may be interesting things to say. In fact, this was about, I mean, this last session had some connection to curved space because to determine the metric in the theory space, one used the sphere to compute the partition function at every point. So this already had some connection to curved space. The rule of thumb, this is a rule of thumb that works in essentially every example, is that A type anomalies, that anomalies that don't have logs, they have interesting consequences for RG flows in flat space. While B type anomalies, those that have logs, they usually have consequences only in curved space. Interesting consequences only in curved space. I don't know of situations where this is not true. So B type anomalies lead to some interesting results sometimes about spheres or other manifolds, but A type anomalies have consequences in flat space, for RG flows in flat space.