 This has been a very interesting week for me. So along the course of this program, we have learned a lot about gauge theory in three dimensions and also in four dimensions. And we also learned about their interesting properties and the very rich invariants that we can define using them, such as the Gaussian invariants, cyber-witten invariants, and the Waffle-witten invariants. But this lecture and the next wouldn't be about gauge theories. Instead, we'll be talking about non-gauge theories. So what's wrong with gauge theories? What's the motivation to go beyond gauge theories? In my mind, gauge theories are grays. They are stepping stones to understand more general quantum field theories. But there are two minor shortcomings of gauge theories. First of all, we expect that from gauge theory, you are not going to be able to get stronger invariants of four manifolds than the cyber-witten invariants. Indeed, if you study gauge theories as of her rank or with more matter, somehow it seems that you can always relate this new invariants to the usual cyber-witten invariants. And another shortcoming is not about how powerful they are, but about the structure of these invariants. It's also not expected that gauge theory will give you full T-quive T's. So one manifestation of this is that for those invariants coming from gauge theory, sometimes it's not defined on some manifolds. In other words, although sometimes you can still consider interesting co-boardisms, you shouldn't expect that you can package all these invariants together into a nice T-quive T. Yeah, for example, for cyber-witten invariants, you would put conditions on B2 plus. And if it's equal to 1, then we know that you need to talk about this invariant in a particular chamber. And then if you want to study co-board them, it will be a nightmare. Well, sometimes you need a metric, and that's not the extra structure we want to use. We want to have some topological quantum field theory. Will gauge theory help with these two minor problems? From Paris' talk, and perhaps also his talk tomorrow, he will demonstrate how to use non-gauge theories in the sixth dimension to attack problem one. So today, we have already seen that these invariants, even from the simplest, sixth theory has a very good structure. For today, it's not exactly stronger than cyber-witten invariants. But since they have this structure, we expect that if I use more complicated sixth theory, one has a good hope of getting something actually stronger. And for this talk, and the next one tomorrow morning, the goal is instead try to cure the second problem. We'll be using non-gauge theory in four dimensions. The goal is to get some 3D T-quivities that's our actual T-quivities. So these are going to be well-defined or expected to be well-defined or three-manifers. Well, it's a 2 plus 1. You'll be cool if one can somehow leave this into 4D, as we don't have any very interesting 4D T-quivities. So we know that for gauge theory, you know how to start. You start with gauge group and the method representation. So what can you do for non-gauge theory? As you might expect, non-gauge theory also share a lot of properties with gauge theories. So we'll be considering 4D equals 2 quantifier theories. And we have already learned from Professor Megatron's lecture that we'll have a Coulomb branch for gauge theories. And this is true in general. For 4D equals 2 theory, whether it's gauge theory or not, you have Coulomb branches. And for a particular theory T, I would denote and T to be the Coulomb branch of the theory T s1 times r3. And for physics reasons, we expect that this is actually a haver-keller manifold, possibly with some singularities. And being haver-keller means that you will have a full s2 worth of complex structures compatible with the haver-keller metric. Here, I could have talked about how to define this Coulomb branch away from two special complex structures. I would denote one complex structure by i. And the other will be minus i. And for this lecture next, we'll be actually focusing on this complex structure i. In complex structure i, MT is still going to be a complete integral system. And the base will be denoted by BT. This will be Coulomb branch of the theory, but instead r4. And this is sometimes also referred to as the u-plane. And you have already heard a lot about it. The fibers now has a property that is always compact. And they are generically a billion varieties. And this is, in a sense, why this particular complex structure is special. And when the fiber, there are particular points on this B where the fiber becomes singular. And then you can observe this phenomenon from this space by looking at this special-keller metric. So the base B is instead equipped with a special-keller metric. And this metric can also have singularities. And the singularity of the special-keller metric indicates that the fiber over some particular points also becomes singular. So let me give you some examples. So let's start with the theory that we are very familiar with. For physicists, the theory is well known as u1, great theory. In hierarchical stock, this means that you start from the data, tree being c star and m being the zero representation. And then the Coulomb branch and t will, as was shown in the hierarchical lecture, topologically, be t2 times r squared. But we are going to put some complex structure on this, turning this to complex manifolds. It's easy to guess that this will be the second factor, will become a copy of the complex plane. But how about the first one? It turns out that to specify the complex structure here, you need some additional data. So the u1 gauge theory is not just specified by the gauge group and mass representation. It's, in fact, labeled elitic curve. I will use sigma for this elitic curve. And the first factor will be identified with Jacobian of this elitic curve. So to draw a cartoon, you will have something like this. Here, the second factor C is also identified with the base or the u plane. And the fibers are going to be Jacobians of sigma. So this is, perhaps, a simple example of the Coulomb branch. Any questions? So example number two, we can consider, again, a billion theory, but of high rank, you went to the G gauge theory. And then if you apply the BFN construction, you obtain G copies of this. So Mt will be given by T2 times R squared to the G's power. And what's the complex structure that you should put here? It turns out that, again, you need some additional data. You went to the G gauge theory. It's going to be labeled by a genius G human surface. I will, again, use sigma. This will be a genius G human surface. And the Coulomb branch, as a complex manifold, is identified, again, with Jacobian of sigma times C to the G. And the base B will be just C to the G. And you can interpret this as a space of homophobic differentials of sigma. And the fibers are going to be just these Jacobians. In these two examples, we see that this vibration in general, I will denote this vibration by pi. In these two cases, we see that the vibration is trivial. And this vibration being trivial is basically synonymous with the theory being free. The word free is used by physicists to describe a class of theories that are somewhat boring. And the way to see this geometrically is to see that this vibration is going to be trivial. This does not sound terribly interesting, but we know that all vibrations are trivial locally. So if you look at a small enough patch, this will be the geometry that you are going to see. So this has also a philis counterpart. So for philises, they will say that, in general, the theory T, non-singular points on a u-plane, can be approximated by a free u1 to the G gauge theory. And we have already seen in Yen's talk that this fact has been used to amazing effect. And this theory will be referred to by physicists as the low energy defective, or maybe L, E, E, T. So geometrically, this just means that if you look at a u-plane, you first see a lot of singularities of various cold dimensions. And then if you look at a small patch away from the singularity, you should be able to zoom in. And then you will recover the picture on the other side of the board. So this process of zooming in is also known by physicists as RG flows. But we know that sometimes there are singularities. What happens if you actually zoom in around a singularity? Then you will not get these free theories. You will get some interacting theories. So let me maybe give you some examples for interacting theories. And let me maybe keep this picture for example 1. So now, example 3. These are examples for interacting theories. Interacting is the opposite of being free. And example 3 is what I would call the monopole theory. And you have already seen this multiple times in this program. This theory is characterized by this data. First, 3 equals c star. And n equals 1 copy of c with the obvious c star action. And for physicists, probably these theories better refer to as QED with 1 electron. And for this theory, the Coulomb branch would look like the following. Again, you have a one-dimensional base. So it looks like the complex plane. But the special calor metric would have a singularity. And if you're away from this singularity, you will have, again, an elliptic curve. But over this special point, you will have a curve with a double point. And this kind of singularity in elliptic vibration was classified by Kodaira. And this is what he would call an I1 singularity. So if you go around this singularity, you'll get an SO2D monogamy. And that monogamy is going to be like this. Example number four. This example is, again, familiar. This is SU2 gauge theory. So the data is given by 3 equals to SO2C. And again, equals 0. And for this theory, if you are being very naive, you would guess that the Coulomb branch looks like the following, pretty much like the picture over there, but with only an I4 singularity. But it turns out that this is not entirely correct. Cyberand Witten pointed out that there are quantum effects that correct this picture to the following. Now, instead of having a single I4 singularity, you have two I1 singularity. The fibers will look like roughly like this and this. Away from these two singularities, you have, again, an utility curve. So now, the base, B, just look like the u-plane that you have encountered many times. So two singularities. And at infinity, you can view the infinity outside another singularity. And the monogamy around these two singularities are going to be given by 1, 0, minus 1, 0. And minus 1, 4, minus 1, 3. And oh, sorry, this is 1. And you can check that these are actually elements in SO2Z. And if you multiply them, you will get a monogamy that is compatible with I4 singularity. So one of these singularities is usually referred to, again, as monopole singularity. Well, the other is sometimes referred to as dion singularity, just to distinguish that from the monopole singularity. But locally, the picture will just look like this. And you have to change our basis for the utility curve. Exactly. If you multiply them, you will see that you get something that is conjugate to 1, 4, 0, 1. And that's the monogamy at infinity. So at infinity, here at infinity, you again have I4. And you can see that by multiplying them. Any more questions? And both theories on the blackboards give you an interesting four-manifold invariance. And for this, as you all know, that this leads to the cyber-witten invariance. And here, this geometry is sometimes referred to as cyber-witten geometry. But if you study invariance for this theory, you will get Donaldson invariance. And the fact that there are two copies of monopole theory on the u-plane is basically the physical reason why you would have the written conjecture, which states that there's a way to decompose Donaldson invariance into some of two copies of cyber-witten invariance. And to be more precise, there is, of course, another contribution from the u-plane, which is exactly the focus of Jan's talk. So for this geometry on the board, it seems that we start from the simplest one, free theory, free theory of higher rank, to this theory that looks slightly more interesting, and to this one, which seems to be the most interesting theory on the board. But actually, for the first three examples, this theory enjoys a special poverty that the first theory does not have. So let me erase this board and remark that for the first three different cases, the Coulomb branch empty and meet a nice S1 action. This S1 action here will just be the rotation of the complex plane, B. And in the fourth example, there's no way to cook up a similar S1 action. At least if you want to find the S1 action on B that acts as isometry, there's nothing that you can do. So the fourth example, there's no S1 action. And when physicists see something like this, they will say that these theories are, well, at least they look like conformal theories. And this theory looks like non-conformal theories. And here we see that this S1 is actually there in this naive picture. But due to quantum effects, it's somehow destroyed. This is sometimes known as quantum anomaly. And physics would say that this S1 is anonymous in SU2 gauge theory. So what's special about conformal theories? What kind of property do we expect for conformal theories? So since we are talking about theories supersymmetry, combining supersymmetry and conformal, we'll get super conformal theories. We expect the following properties that the Coulomb branch enjoy. We expect that there is S1 isometry acting on the Coulomb branch MT. And this S1 also acts on the base B in such a way that pi is actually S1 equivalent. Is this imagery that corresponds to this S1 action? Yes, so this is exactly this U1 r symmetry that was mentioned in Jan's talk. So remember that in the beginning, he talked about SU2l, SU2r times SU2 times U1. And S1 is related to this U1. It's not exactly equal, but it maps onto this U1. So I will say more about this later. So this is the expected property. And we also expect that if you look at S1 fixed points, the U plane, this has to be a point. And this special point is sometimes referred to as the super conformal point. And the super conformal point gives the U plane structure, the structure of vector space. So sometimes I will use 0 to denote this super conformal point. It seems that there are a lot of requirements. And so far, the theory that are super conformal looks a little bit boring. And you may want to ask whether there are interesting super conformal theories. And there are many of them. At level, well, T5Ts are automatically super conformal. But probably you are asking whether you feel at level of formative invariance. So there was this very long paper by Greg Moore, Moreno, and Plaza, I might pronounce the name correctly, where they discuss whether you can do something interesting with super conformal theories. And the invariance they get indeed enjoys some special properties. So they motivate them to formulate this super conformal simple type condition. So there are some interesting features for this invariant coming from super conformal theories. Any more questions? So the example for super conformal theory. So now an example, maybe 5. This is a B class of theories, known as class I theories, that has appeared in many previous talks. So for theory of type class S, it will be labeled the algebra of type A, D, E. And you can also allow a bimian U1 factors. And then we have a service. And for them, the Coulomb branch, as was also mentioned, will be identified with the modular space, g-higgs model, the Riemann surface sigma. So as Greg mentioned, if you apply Bfn construction, or if you expect somehow you can apply Bfn construction, you are going to get the same space, but with a slightly different complex structure, where the space will be viewed as the model space, will be viewed as the g-character writing on sigma. And for the combat structure that we are working in, we're actually getting the model space of g-higgs models. And then it's well known that for g-higgs models, indeed there is a map, it's a total space of a complete integral system. And the vibration is sometimes referred to as a kitchen vibration. And indeed you have an S1 action that has all these properties. And this S1 action is sometimes referred to as the kitchen action, or kitchen's S1 action. So this is a very big class of theories. And in fact, the example 1, 2 can be viewed as special cases of this theory. So if you take actually g to be u1, well, I mean, I think I'm not sure we can see them. Probably I should see. And then again use this sigma, then you can reproduce example 1, 2. And interesting fact about class S theory is that sometimes they are gate theories, but sometimes they are not gate theories. So to give an example of non-gate theory, but of type class S, for example 6, you will consider g to be S of 3 and sigma being p1. But now you allow this to have mark points. And you will have three mark points decorated by some ramification data. And the coolant branch will look like the following. Now the base is still one-dimensional. So it will still look like the compact plane. And because this is super conformal, there is a unique singularity on the base. And now over that, you will have a single fiber. Let's look like this. In Kodaira's classification, this is known as four-star, or IV-star singularity. Toys and fetuses, this is referred to as the E8 singularity, sorry, E6. E8 will correspond to two-star. The total space is smooth. It's a singularity in the elitic vibration. The fibers is a singularity. Yeah, the fibers are. And also, if you look at the spectrometer metric, you will have really a singularity in this metric. So remember the picture that Jan has drawn for us? You should really stretch this singularity to infinity to form some cast. Well, for gauge theory, for physics, we'll have a list of expectations. And in this example, one can check that one expectation is actually not satisfied. So maybe let me just say this. We know that for gauge theory, there are particular type of deformations known as exactly marginal deformations. And for this kind of theory, we are not observing that. So we know that it's not a gauge theory. There are also other arguments, but slightly more complicated. No, they are not no-guanguan. And therefore, they cannot be gauge theory, because gauge theories are all the grandeur. Other questions? So what is the effective theory? If I move to the singularity, the effective theory is known as Mino-Handem-Czanski's E6 theory. So Mino-Handem-Czanski's sometimes just noted as MN E6. And as you expect, there's also MN E7 and E8. These are all expected to be non-guage theories. Yeah, if you move away from this, you'll get free theory. That's right. No, if you zoom in at the singularity, yes. So you can imagine that there's some bigger theory that has this type of singularity. If you are away from this, you get free theory. But if you're really zooming in at the singularity, you get this theory, this theory is a conformity theory. So and then the goal that we have is to define some invariance of the four or three manifolds. But this kind of modern spaces looks somewhat problematic because there are non-compiled directions. But if you pay attention to all the previous talks, you would say that this is not a big problem as long as we have some symmetry so that we can do some equivalent localization. So now, for a super conformity theory, we know that there is S1 action. And now you can ask whether this S1 help. So if you look at some particular partition function, can somehow you use this S1 to do some equivalent integral and get some finite answer. Well, it can help a little bit, actually. Let me tell you in what kind of cases it actually helps. So if you have the theory T, and this theory is in four dimensions, and you can consider the Hilbert space of T. And here I actually mean the topologically twisted theory S3. So by Felix's argument, the Hilbert space of this topologically twisted theory on S3 can be identified with the space of regular functions, the U plane. And for U plane that looks like this, obviously the space of regular functions is going to be infinite dimensional. So if you want to compute partition function, the theory T of S1 times S3, this partition function is identified with the dimension of the Hilbert space. And then clearly this is infinite. It's U-defined. And this indicates that there is some problem. And then you can notice that, oh, there's S1 action. So you can actually, instead of look at partition function of T, you can look at S1 equivalent partition function of T. In other words, instead of computing the dimension, you now compute the holomorphic or the characteristic S1 equivalent of the structure shift of B. And then because by assumption, the fixed points of this S1 there's a unique fixed point. This will be well-defined and given by a very simple formula. So to be completely elementary and pedagogical, let's just take the example of the free theory, the U-plane is identified with a complex plane. And the space of functions, I will write H, T, S3 to denote the Hilbert space of the theory T, S3. This is now identified with, well, is spanned by polynomials in one variable. Let me use Z for that variable. And obviously, this is infinite dimensional. But then if you use this S1 action, that tells you that you should count this contribution from all these spaces in a slightly different way. So instead, you would get, instead of 1 plus 1 plus 1, you get 1 plus T plus T squared plus T cubed plus dot, dot, dot. And then if you sum them up, you get 1 over 1 minus T. You can, of course, directly reproduce this by using the most basic Euclidean localization technique. So you see that in this case it helped a little bit. So at least for S3, before we're having infinity, now we're having one, we're having some rational function in T. And this is well-defined as long as T is not 1. And then you can ask this question. Regularize S1 times M3 partition function. So given three manifolds, you will have, again, some Hilbert space. And then you, perhaps, would hope that somehow you can still use this S1 action to regularize this and maybe get, now, a C star family, 3D T-qualities, somehow topological with respect to M3. And it's a C star family because only for T equals 1 is not well-defined. So you may hope that with respect to log T, you will have a complete C star family. But this is too unrealistic, impossible. And a quick way to see that this is a complete pipe dream is to notice that there is a logothium. A well-known result in the theory of fusion category tell you that it's known as octanino rigidity, which basically imply that you cannot, you can never. So there won't exist one parameter deformation, 3D T-qualities. In other words, 3D T-qualities are rigid. It's not possible to just deform that by some T-parameter. So somehow this must fail. And indeed, for physics, you also see that this is not realistic because S1 is an r symmetry. And by definition, an r symmetry means that this acts on supercharges. And that can cause a problem because, in the definition of T-quad T, somehow you want to get some scalar supercharges then you want to look at this q-cormology. And if this S1 acts non-trivial way, somehow there will be a problem associated with this T-quad T. And this is roughly the physical reason to see the physical way to see that you cannot do this. So what can you do? So you don't want r symmetry. Instead, you want a flavor symmetry. And flavor symmetry means that you will not act on supercharges. So you will have a well-defined action on this kind of humor space. And but then there's still some tension between these rigidly results. For this, we expect that we should use actually a discrete flavor symmetry that acts on supercharges. Otherwise, if you compute, again, some equivalent index, you are, again, getting sister family. And you know that that's impossible. So to have some well-defined 3D T-quad T's is only possible if you manage to use some discrete flavor symmetries. But how to find discrete flavor symmetries? For different theory, they tend to have different kind of flavor symmetries. It turns out that for a large class of theories, there are some canonical ones. So S2. S3. S3. S1 cross S3. S1 times S3, yes. It will work for S1 times S3. Oh, you're asking why it works for S1 times S3? Yeah, it's actually a long story. So somehow, if you can verify on S3 or any cipher manifolds, in the end, for these quantum mechanics, there is going to be some flavor symmetry. It's closely related to this U1. And because there is a flavor symmetry, somehow you can use it. So this only work for S1 times roughly cipher manifolds. And cipher manifolds, although they still have gyronomy, there's some special limit where the gyronomy can become U1 if you pick some singular metric. That's kind of why you can have additional flavor symmetry that can help. But in general, you won't expect this. So let me just end with one remark about how to find such a discrete flavor symmetry. It turns out that very often, you have a Zn subgroup inside your S1. And n here is a positive integer. And therefore, it can be either 1 or larger than 1. If n equals 1, then this group is trivial. You cannot do anything. If n is larger than 1, then you can do something. And this n only depends on the theory. Yeah, there is largest possible n, such that you have a flavor Zn symmetry. There are theories such that the largest is just a trivial thing. And these include all previous examples and all gauge theories. And for a larger than 1, this theory has to be non-gauge theories. So there is something that non-gauge theory can do, but gauge theory can never do. And tomorrow, we will see that for a nice class of non-gauge theories that belong to the second class, actually you can or is expected to get a family of 3D TQFT. But now you won't be a C star family. So tomorrow, we will try to get another C star family, but a Zn star family of 3D TQFTs. Now, there's no contradiction or no gauge theorem preventing us from getting this Dn star family. So I will end here, and see you tomorrow.