 What I will talk about today will be based on joint work with these four amazing collaborators, two of them are currently in the audience. So we call from yesterday that we were trying to use a class of theories known as 40 n equals 2 super conformal theories. Let's denote the theory by T to get some well-defined 3D T-cliff T such that you will have well-defined particle function on all three metaphors. And we realize that we could probably use an S1 asymmetry acting on its Coulomb branch. And indeed, this helps to regularize partition function on S1 times S2, sorry, S3 partition function. By turning this ill-defined partition function into a rational function in one variable. And we call that from last time, we have this simple example where T is the 3 U1 theory. This partition function now becomes something like 1 over 1 minus T, which is well-defined as long as T is not equal to 1. But we also remarked yesterday that you shouldn't expect to do this for all three metaphors in a way that you get a C-star family of T-cliff T's. So you cannot expect to get C-star family 3D T-cliff T's applying this to S1 times M3 with more general M3. So from the point of view of 3D T-cliff T, one should expect this because 3D T-cliff T are rigid. And there is nothing such as a C-star family of T-cliff T's that are not constant. And you also expect this from physics. Because here, you are using this S1 action. And S1 is known as an r-symmetry in the physics literature. And in general, it's not a good idea to use an r-symmetry. So what's exactly the geometric meaning of having an r-symmetry? Geometrically, this means that the S1 will act the hypercalor structure. So for example, you will act on complex structures. So you call that given a hypercalor metric, we have Cp1 worth of compatible complex structures. And this S1 will rotate this two-spher where that looks like this. And you have already seen this in Hirako's lecture. And this S1 also acts now trivially on the synthetic structures. There's no dry one, right? And to see this, you can first realize that for hypercalor manifolds, such as MT, once you fix a complex structure, you will become a holomorphic-symptatic manifolds. So you will have a homomorphic-symptatic form at omega. And this homomorphic-symptatic form also plays an important role in this. So MT is completely integrable, exactly with respect to this homomorphic-symptatic form. So you have, again, this map to BT, the U plane, and the fibers are going to be complex Lagrangians. This S1 action will rotate the holomorphic-symptatic form. So if I parametrize this S1 with an angle between 0 and 2 pi, then it will send omega i to e to i theta times n omega i. And here, if n is larger than 1, then you will actually have the n subgroup inside S1 that fix omega i. And in fact, it will fix the entire hypercalor structure. So what's the meaning of this dn from the physical point of view? So now you have, let me erase this. You have a short exact sequence that looks like the following. So you have your group S1. And then there is the dn subgroup. And the quotient is something that I will refer to as U1r. And this is exactly the U1r symmetry that appears in the n stock. So this is part of the r symmetry. And it acts the supersymmetry algebra in a natural way. In other words, the action of S on the supersymmetry algebra will factorize through this map. The corresponding geometric statement is that this U1r will act on hypercalor structures. And the action of S1 on the hypercalor structure will also factorize through this map. So although this U1r is constantly referred to as the U1r symmetry, the true symmetry of the theory is actually S1. So U1 is only realized projectively. This will be actually a symmetry, the theory t. And correspondingly, this U1r, although it acts on hypercalor structures, is now a symmetry of the Coulomb branch. And S1 acts as a geometry on the Coulomb branch. So then here, this is dn by definition. It's a symmetry of the theory that won't act on the supersymmetry algebra. And this is exactly what physics will refer to as a flavor symmetry. Now it's a discrete flavor symmetry. And from the geometric point of view, this is an isometry of m that fix the hypercalor structure. So it's trichromorphic isometry. And as I mentioned last time, if there is some canonical discrete flavor symmetry, then there might be some interesting thing that we can do. We can maybe attempt to use this to regularize the partition function. So then this n will be determined by the theory. And obviously, there are two possibilities. The first possibility is that n just equals 1. And then we cannot use this discrete symmetry for anything. The second possibility is that n is larger than 1. Then we actually can do something. And I've mentioned many examples that belong to the first class of theories. This includes all gauge theories, all theories of class S. And then for the second class of theory, this has to be non-gauge theories. And they cannot be of class S. But sometimes they are closely related to class S theories. Since all the examples that I've given last time belong to the first class of theories, you may wonder whether they are actually interesting theories that belong to the second class. So let me give you a few. And these examples are known as rank 1, address dog's theories. Since there is no gauge group, rank 1 is, of course, not referring to the rank of the gauge group. Instead, it's referring to the dimension of its Coulomb branch. And in this case, the Coulomb branch, or the Coulomb branches are going to be one dimensional. So the base will just look like a complex plane with one singularity. And over that, you will have an exceptional fiber. And there are three theories of this type. One is known as the A1A2 theory. The second one is known as the A1A3 theory. And the third one is known as the A1D4 theory. For the first theory, again, if you are away from the singularity, then the fiber is going to be a smooth illicit curve. And then if you move to this particular singular point, then now the fiber will look like this. And again, if you move away from this point, you get another smooth illicit curve. So this singular fiber in Kodaira's classification is known as type 2 singularity. And the monogamy around this singular fiber will be given by the following matrix. For this theory, n equals to 5. So it actually belongs to the second class. For the second theory, you will have a similar picture. The base is still one-dimensional. So it looks like a complex plane, but with a singularity of its special calor metric. And over this point, you will have a fiber that roughly looks like this. This is type 3 singularity in Kodaira's classification. And the monogamy is given by this particular element in S L to Z. And for this theory, n would equal to 3. For the last one, again, the base of the vibration looks like this. And over this special point, the singular fiber looks like the following. This is type 4 singularity. And the monogamy is going to be given by 0, 1, minus 1, minus 1. And for this theory, n equals to 2. So you see that there are actually interesting theories that belong to the second class. If you go to Herank, then there are many more. Continuous global symmetry. Well, the R symmetry is also global symmetry. It's continuous. But if you're asking for flavor symmetry, I think in this case it's discrete. So in this case, it's d5 and d3 and d2. Well, it's subgroup of S1 and the quotient dc1r. More questions? So you say that there is no way of obtaining these theories from n-fibrin with some type of. Oh, so it can be obtained by a combination using n-fibrin. But now it has to involve irregular singularity. And in the definition of class S, I kind of only allow regular singularity to make a slight distinction. And that's why I mentioned that it's actually some of them are closely related to class S theories. Any other question? Can it be equal to 1 or a theorem? So these are the only theory for rank 1 that has Herank. So for example, if you take Meenach and Lemchensky's E6, E7, or E8 theory, these are some special class S theories. And these all have n equals 1. But in Herank, you have a lot of theories with n equal and being larger than 1. So now you may wonder, well, these are discrete symmetries. How can you use discrete symmetry to do any kind of regularization? So to understand how to do regularization using discrete symmetry, let's again get back to the S theory, Huber space. You call that the Huber space of the theory S theory is identified with the space of regular functions, the 4D Coulomb branch, Bt. And this is already Twisted theory. So the full theory can contain more states in the Huber space. And then if you compute the S1, the character of the S1 action, then you obtain the following quantity. Let me use t again for the equivalent parameter. The S1 character will be given by the following rational function, 1 over products of 1 minus t to the d's power and to the n's power. Here, the i's are weights of S1 action, B. And n i's are going to be multiplicities of these weights. And then you may wonder whether you can do something like this. You now set t to be the n's root of unity and ask whether this quantity is well-defined. If this is well-defined, then you kind of use this discrete symmetry to regularize the S1 times S3 partition function. Turns out that this will be well-defined as long as your theory has an additional property. Mm-hmm. Oh, what's the meaning of this particular label? Yeah, this is something that I didn't mention and was not originally planning to mention. So maybe let me still make this remark. So there is a family, a Geostogla series, pyrimatrized by a pair of A.D. Dinghan diagrams. And the data for this pair of A.D. Dinghan diagram is translated into a singular Calabria 3-fold. This is some particular singular Calabria 3-fold that can be constructed as a singular hyper surface in C4. And you have some explicit equation that defines this hyper surface. And this equation depends on G1 and G2. And then the physics construction of this theory is that you can specify type 2B string theory on this particular singular Calabria 3-fold. And then you will see that for generic choice of G1, G2, you will actually get a theory with n larger than 1. So it will be like a Geostogla theory. Let's answer a question. And are there more questions? Over the number of different ways in this decomposition. So maybe let me make a separate remark, since I have already made this one. So there is another overlapping family. And this family of theories are not labeled by this data, but instead a different set of data is labeled by a single G, the algebra of type A, D. And then your surface sigma. But now it is topologically has to be just P1. But it can be decorated by some marked points. And one of the marked points corresponds to a irregular singularity. And one can also possibly allow another regular singularity, something like this. And then at the north pole of this P1, you have a regular singularity. You can perhaps include another regular singularity at the south pole. And the construction for this causal theory is indeed, as Francesco mentioned, you can compadify and fibering on this particular two manifolds. And this leads to another theory, T, in four dimensions. And because there are irregular singularities, if you look at the modular space of solution of some BPS equations, there will be stoke phenomena. And roughly n is related to the number of stoke rays. A regular singularity, that's some stoke rays. And n is correlated with the number of stoke rays. But let's get back to the question of whether this is well-defined. It turns out that if the quantum field theory satisfies another condition, then this will define it. So the condition is that this Zn action is a subgroup of S1. So it also acts some B. And the condition that this only has one fixed point. So it only fixes the super conformal point. And I will refer to this condition by the both letter F. We will say that the theory has poverty F if this is satisfied. So why this is necessary for this? Oh, for this we will define. This is actually necessary, I think. And later I will say a little bit more whether it's possible to relax this a little bit. So from the point of view of getting 3T-5T, they seem to be somewhat necessary. But if you relax this, you still have some kind of structure. Maybe not a full 3T-5T, but some other theory where you may not be able to define all part-in function, but you may still be able to define some not-invariance. So for now, let's just make an assumption that we'll be working with theory with poverty F. And now another remark, maybe we continue from here. Number three is that all this rank one adjusts our theories as this property. And more generally, for higher rank, if you consider an comma an theory, then they also have poverty F if m plus 1 and m plus 1 are co-prime. So you see that, in a sense, generically, this property is satisfied. And even if they are not co-prime, sometimes they also satisfy this property. So when this property is satisfied, then the claim is that now if you write the expression for this, then if you set e to be a root of unity. So here, zeta n is e to the 2 pi i over n. And the claim is that now this is regularized and is valued in the following field. And you can do actually slightly more than this. So you can allow, you can have an entire family by setting t to be a power of this particular root of unity. This still will define as long as your gamma is co-prime with n. So the n is the group of integers mod n. And the n start denotes the group of the multiplicative group of integers mod n. And elements in this group are going to be represented by integers that are co-prime with n mod n. So you see that you have an entire family of particular functions, at least on S3. And then you can ask whether this can help you to regularize any particular functions. And the conjecture is that, yes, it will. And this will be upgraded to an entire zn star family 3D t-craftings, whether it has any meaning. Yeah, that's an extremely good question. So far it seems that I have first assumed that t has absolute value smaller than 1 and obtained this expression. And then I just let it continue to a root of unity. And it's very interesting to understand this procedure from the physics. So it seems that this procedure is about collapsing infinite power of states into a single one. But what exactly? And this kind of operation is familiar in some other subject. But here, I don't think we have a complete set factor answer of what does this regularization process correspond to physically. So it's an extremely interesting question. I will say a little bit more about 3D t-craftings and then give you some examples. So for many people here, you're perhaps already very familiar with the statement about 2D t-craftings. It's a kind of a well-known fact that the data for 2D t-crafting is the same as the data that defines commutative for venous algebra. And there is a corresponding statement for 3D t-craftings. The corresponding statement is that the data that defines a certain class of 3D t-craftings is equivalent to the data of a modular tensor category. Modular tensor categories has already appeared in Pavius talk. And this line over here is, in a sense, a categorification of the previous line. So if you have some 3D t-crafting, you can always obtain a 2D t-crafting by reducing this on a circle. The corresponding operation on the side of a modular tensor category is that you take the Gorton-Dig group of the category. The Gorton-Dig of a modular tensor category is sometimes also referred to as the Walinger ring. And then you'll get back commutative for venous algebra. So if you already know a little bit about commutative for venous algebra, or just commutative algebra, you already have some insight into modular tensor categories. So what I will do now is that I will first state a conjecture relating supersymmetric quantum field theories to modular tensor categories. And then I will give you a more detailed list of data that define a modular tensor category. So conjecture. So if I'm given a theory t, and this is a super conformal theory with this property that I denote by both f, we conjecture that there will be a family of modular tensor categories that I will denote by c sub t gamma, where gamma is valued in z n. And the second part of the conjecture is that you can get, you can obtain at least a category for gamma equals 0. Sorry, gamma equals 1 from geometry of the cooling branch. So far, I have been a little bit vague about how to, well, I've been completely vague about how to get this category from geometry. Later, I will say a little bit more about the dictionary between the algebraic data that define a modular tensor category and the geometric data of the cooling branch. So if you manage to reconstruct this category from geometry of mt, then the third part of the conjecture state that you can also get ct with more general gamma by doing a Galois transformation on the modular tensor category, ct gamma equals 1. So now I will try to first say a little bit more about what are the tensor categories and the algebraic data that define them. And at the same time, give you an example for this particular theory, a1, a2. So you call that for a1, a2, this number n equal to 5. And the 5 star are going to be represented by interior numbers between 0 and 5 that are co-prime with 5. So they will be represented by these four elements. And there will be four different mtc's. So what is an mtc? Pepe already mentioned that a modus has a category is, so 0 does not help to, so you mean gamma equal to 0. 0 does not help to regularize. So you won't define 3D t-crack t because some part of the function is still divergent. But you may hope that this still define some category that has less data than mtc. But for now, we will assume that it's actually co-prime with n, so that you have a hope of regularizing all part of the function. So you have completely well defined 3D t-crack t's. So Pepe already mentioned that a modus has a category is a linear, a billion, semi-simple category with finally many simple objects. And the starting points of the algebraic data that define mtc will first consist of a set, a finite set, gamma, known as the label set, which is identified with the esomorphism classes of simple objects in this category. There is a particular special object known as the unit of the mtc. And indeed, on this mtc, there is a fusion product or fusion rule. And the unit is the identity. Under the fusion rule. So the fusion rule will be a map from products of two copies of lambda into non-negative linear combinations of lambda with integer coefficients. So here, what is lambda? It turns out that for all these four categories, the label set and the fusion rule are going to be the same because, well, Galois transformation does not change the label sets. And it will also now change the fusion rule because it only involves integer coefficients. So lambda will consist of two elements. One is this unit 1. And for the other elements, I would do it by phi. And then the fusion rule will be given by phi times phi equals 1 plus phi. So since we are in Italy, I would like to mention that sometimes this is referred to as the Fibonacci fusion rule. So if you take actually n copies of phi and then fuse them together, then you will have a decomposition into, again, 1 plus phi. But with coefficients being the Fibonacci numbers, a model times a category has a word modular in its name. And this is referring to the fact that there is an SL2D action, the vector space on the c-span of lambda. And this data can be represented by SNT matrices. So here, now, the SNT matrices will actually depends on which one of the four categories you are talking about. For gamma equals 1, S will be given by the following. So you have 2 over square root of 5 as the overall vector. And minus sine 2 pi over 5 sine pi over 5. And sine pi over 5 and sine 2 pi over 5. We all know that Fibonacci sequence has something to do with 5. And indeed, you see 5 everywhere. And for the T matrix, it's going to be diagonal in a basis given by the simple objects. And it will look like the following. e to the 11 pi i over 30 e to the minus pi i over 30. And 2 zeros. And there are some additional pieces of data that enter into the definition of MTC. These are known as the F and R matrices. These come from the fact that a modern tensor category is a fusion category. So it has some associator. And that's transformed roughly into the data of the F matrix. And then the R matrix comes from the fact that it's also MTC-spreaded. So we have already seen R matrices on Monday. But R matrices here will be constant R matrices, not the special R matrices that we have seen. That depends on additional parameters. So I won't write down all the F and R matrices here. But instead, I will just write a particular one. So an F matrix is labeled by quadruple of simple objects in the category. For this one, it's given by minus pi i square root of pi minus i pi R square root of pi and pi. Here, pi here is really the golden ratio. So why do I want to write down this F matrix? Well, it turns out that this S matrix is not unitary. Oh, this is e to 11 pi i over 30. So this particular one is non-unitary. In fact, for this category, it's not possible to make all of the STFR matrices unitary. Here, in this space, ST matrices are unitary. But this particular F matrix is now unitary. Therefore, we're actually having a non-unitary MTC. So this particular MTC has a name. It's sometimes referred to as the young MTC. And it can be identified with the category of modules of VOA. And the VOA is known as the young model. This is a non-unitary, minimal model, sometimes known as the 2 comma 5, minimal model. And this is for gamma equals 1. How about the other gammas? So let me make a table here. The allowed value of gamma is 1, 2, 3, and 4. And we have four water tensor categories. And for gamma equals 1, this is already identified. We see the young MTC. And for gamma equals 4, you will be something also simple. You will be the compact conjugate of the young MTC. However, for 2 and 3, we got something that is somewhat unexpected. For gamma equals 2, you have G2 at level 1. So G2 at level 1 has two integrable modules. And they form, again, a model tensor category. And it's exactly this category for gamma equals 2. For gamma equals 3, you will now instead have F4 at level 1. Strictly speaking, you'll have to answer this with two copies of E8 at level 1. And among these five MTCs, these two are going to be non-unitry. And, well, these two are actually unitary. So you see that you can get some interesting unitary and non-unitary MTCs. This is not terribly surprising, because, although Galois action acts nicely and preserves all the coherence conditions that F and R matrices need to satisfy, it does not preserve unitary of a matrix. So if you have some matrix, some constraints that look like this, this is not algebraic. And this is not going to be preserved by some Galois action. So you can get sometimes unitary theories. But we kind of expect that you, in general, get non-unitary theories. And that is kind of interesting. So recently there has been interest from the quantum topology community in non-unitary theories. And hopefully, using this approach, one can obtain some interesting non-unitary theories, not the known one like the DR model, but some more exotic ones. And hopefully that can shed some light into quantum topology. So I think I'm already out of time. But maybe let me just mention one more thing. So I promised to give you refinement of part two of the conjecture. The part two conjecture states that you can go from geometric data of the Coulomb branch to this data here. But we don't completely understand that yet. So we now have an incomplete dictionary. So let me just write three entries of the dictionary and then stop. So first, for this label set, we expect this to be identified with S1 fixed points on the Coulomb branch. It turns out that once this property F is satisfied, the fixed points are discrete. And it can perfectly be identified with the label set. And then we have the S matrix. And then this is related to the weight S1 action on the normal bundle, a particular fixed point lambda. Writing like this, I have already made use of the identification of fixed points and elements in the MTC. And for the T matrix, the T matrix is going to be diagonal in the basis given by simple objects. And this diagonal entry, T lambda lambda, is expected to be related to the value of the moment map, the S1 action. Obviously, this only gave you the first two parts of the data. And then you can ask the first, sorry, one, three of the list of data. And then you can ask maybe how to get the fusion rules, how to get the F matrix, how to get R matrices. And these are interesting questions that we currently cannot answer. But let's stop here and try for running over time.