 So this part is not going to be very pedagogical, but it's kind of a bigger value. But after that, I will start after this overview part, I will start more systematic presentation. So we want to kind of, the idea is to get some sort of universal perspective on the variants of manifolds from kind of quantum field series. And then one kind of can realize that six dimensions is quite specific in physics, especially for this purpose. So six is a maximal dimension where we can have interacting 50 with supersymmetry. So we need, so first of all, we need supersymmetry from kind of mathematical perspective. We need supersymmetry to construct topological quantum field series via written construction. So we need to produce co-homological TQTs. And so what is known about interacting quantum field series with supersymmetry in six dimensions? So usually in physics, people are interested in what is called theory, it's an infrared kind of, like one of the particular examples of series, which we can study our so-called conformal field series. And so one can get the notion of six dimensional super conformal field series. So this is a quantum field series. So QFTs in 6D with super symmetry. So super conformal theory is a certain extension of our, is a certain extension of the conformal symmetry where we extend our Lie algebra by some odd generators. So there is a proposed classification of such theories. So first of all, they are usually divided into two types, which are called 2,0 and 1,0. The corresponding Lie algebra of the super conformal symmetry with respect to which those series, those field series are invariant are following. So this is some sort of standard super group notation, OSP 8 slash 4 and OSP 8,2. So this is the exact meaning of this. It's not going to be important for us. What will be important for us is that, so how can we label? So what is the classification? So suppose we fix a certain super conformal field, super conformal symmetry is 2,0, 1,0. And then there is a statement that the series from this first class, they label it by simply g, which is a simply-laced Lie algebra. So those are known to be classified by Ad or U1. So the world of 1,0 or super conformal theory is more rich. So they're classified by, at least there is a proposed classifications, which says that they label it by y's. So let's say you know this y, which is a singularity in elliptically fibrous collabial c-fold. So in fact, so this is this symmetry is smaller than this one. So those guys should be actually embedded as a special class of these guys. And this embedding is realized when the singularity is just Ad singularity times elliptic f. Ad singularity corresponding to simply-laced the algebra g. So what else is known about? So this can be understood as a data. A data, so in principle, even those 6D super conformal fields here are not defined. But we know, mathematically not well defined, we know that they should be, in principle, they should be defined, expected to be defined by given this data, there is a particular theory. You can consider this data as an analog, for example, of the data of 3D n equals 4 theory, which was mentioned this morning. So there, the data was a choice of group and it's a presentation. So here, instead of that data, you have this input. But what's important is that they are not gauge series. So it's not really clear what to do with them. So in particular, there is no Lagrangian description in them. And so, roughly, so morally, how you can try to look at them and you can, this is a, you also can see where the problem lies. This is a contains, so those series, they contain a theory of non-Abelian germs. So instead of connection, in the usual theory, you can imagine you have a job, which is defined when you have an Abelian group. And moreover, there is a restriction that there is a, so this job has a three-form curvature and this curvature should be sold. If reduced on a swan, often can be described in terms of usual gauge in. So in particular, if I take 2,0 SCFT, so denoted tau, so this would be labored by some Lie algebra, simply Lie algebra G when reduced on a swan is equivalent to a certain supersymmetric version of gauge theory. This is five-dimensional gauge theory with gauge group G such that the Lie compact, such that the Lie algebra of G is, so there are certain ambiguities how exactly you choose a G because there can be different choices of big G, which satisfies this condition. But let me know not to go into those details. Any questions so far? So now let me give you some examples of how can one use. So OK, no, just before coming to examples, let me mention. So the idea of how we can use this was the second general idea, how we can use those guys to produce environs of molecules. So let me fix 6D SCFTT. So you can imagine fixing one of these, fixing by the data, which I listed before, either for 2,0 case you choose a simply Lie algebra for 1,0 case you choose a singularity in elliptically fiber collab of three-fold. And sigma 6 minus D, so this would be a 6 minus D dimensional manifold, possibly with some defects. So this may be defects you can understand some sort of puncture, some sort of puncture sensitive, some market points present, marked by some additional data. So now I'm being very generic. And so if I reduce my 6D theory on this manifold, this gives me and I do what is called a topological twist. So this is a general procedure, which starting from supersymmetric theory produces topological quantum field theory. This should produce me a dimensional topological quantum field theory. Again, this is a TFT. Here we consider TFTs of co-homological, right? Here, so in general what I want, I fix this guy. So this guy is some special thing. So I twist in remaining, in general what I do is just a twist in the remaining D dimensions. But I also can do, so in general, I can do both twisting, so in general I should do both twisting in the remaining D dimensions, but I also can do twisting here. So I get a D dimensional TFTs, label it by, which I have known by Z, label it by the choice of my 6D theory and this sigma 6 minus. So in particular what this means is that if I, for example, if I consider something like a partition function of the 6D theory T on a 6-manifold, which is a product of a D-manifold times the sigma 6 minus D, this is a partition function of this effective TFT. And so in general, so if this is a closed manifold, then this should be some number, but in general I want to understand this thing as usual as we are atia, axioms, symmetric, monoidal function from the category of D dimensional broadisms to the category of vector spaces. So I will always consider vector spaces over complex numbers. And more usually we want to consider vector spaces graded by some abelian group gamma. So in general, the vector spaces by themselves, they can be infinite dimensional, but what we want is that in fixed grading, it's a finite dimensional vector space. So again, this just means that if we take a D minus 1 dimensional manifold, we get a certain graded vector space. And if we take a broadism between those manifolds, we get a linear map between those vector spaces. And of course, for closed D dimensional manifold, we just get a complex number. Yes. So for example, in particular, so if this is a closed manifold, then this is just a complex number, which is a differential invariant. So by TQFT here, so broadisms here means smooth broadisms. So in general, we expect the quantum field theory. To define quantum field theory in general, we need to use smooth structure. So this is a general picture. And now I would like to mention some examples how we can, using this construction, produce some known or also some known differential variants and also predict some new invariants. And after that, I will focus on some particular cases. Well, it depends what kind of, so the grading can come from two parts. So first, it can come from the symmetry of this guy. So it can be some isometrics of this guy, which will give you, so this vector space will be representation of those isometrics. And the gradings will be weights of the isometrics. This is one source of grading. The other source of grading is the global symmetry of my six-dimensional theory, which is unbroken by this topological twist, which is residual R symmetry, what is called R symmetry. And so in general, this can come from global symmetries of my 6D theory, the grading. And isometrics of this sigma. So now one can play this game, pick some sigma and see what happens. But so let me give you some examples. So example, which probably many are familiar with. So let me take t to be of this 2,0 type, where I fix my g to be a c2, the algebra. Then so in particular, so if you can see the sigma, so d equals to 4 and sigma 2 to be t2 with complex structure tau, then this z, t, su2, t2, and 4. So let me write it in words. Is a generating function for buffer written environments and 4. So generating function with respect to variable q, which is related to tau as e to the 2 pi i. So buffer written environments, so I believe Lotter Getscher will tell you more about it. But this is some environments constructed from the instanton by studying solutions of essentially insta. Well, in general, this is an environment constructed from considered model space of solutions or so-called buffer written equations, which in certain cases just can be reduced to considering model spaces of instantons. So I keep t to be the same, but let me take d equals 5 and sigma 1 just this one. Then if I consider, so now I get a, I can understand this as some 5-dimensional, well, OK, 5-dimensional to give t up to some subtleties, but let me not talk about them. So this is the, I believe, S1. But what is important here is that, so for four manifold, this should give me some vector space. So this is a vector space, and it will be z-graded. And z-grading comes from the u1 symmetry which rotates S1. But moreover, one can show that it has a structure vertex operator. So this is one of the cases which I want to consider in more detail in my lectures. And so in a sense, this categorifies the case 1a because then this thing can be considered just a trace of the space, the character of the space with respect to this grading. And categorifies 1a and explains a modularity property. So it has a well-known conjecture that these generative functions for what written variants should be modular forms. You can also consider something going to actually lower dimension. So take d, well, let me take sigma 2 to be, again, t2, but consider this tq of t on a c manifold, close c manifold. Then this should be some vector space, and this should coincide with this SL2c floor, and there's some homology theory which categorifies Casone and variant for c manifold. So another interesting case. So let me keep t to be, again, of this 2,0 type. So it will be labelled by just a simple list, the algebra g. And then take case 2a, consider a sigma to be a disk times s1. So this is a solid torus, the boundary is a torus, and it will have a complex structure. So then if I consider my 6d series on a disk times s1 times c manifold, then this would give you invariance which there was recent, some interesting letters in this environment. So far, they mathematically well defined only for some subclass of c manifolds, and they are denoted by z hat. So it will depend on the choice of this algebra g and a c manifold, and it's valued in c racing q with integer coefficients. But they are related to a more familiar environment. So if I take q to be a limit of the series, so this is a convergent series. If I take a limit q to be a root of unity, so this should reduce to what is called beaten raciotichin to rife of m3. Yes, well, you have to, there's some choice of boundary condition involved here. And this choice of boundary condition will, so there is a choice of boundary conditions which, some natural choice of set of boundary conditions which are labelled by a set of spin-c structures on a 3-manifold. So this has additional label dependence on the spin-c structure of a 3-manifold. And so the beaten raciotichin to rife environment is reproduced by taking a certain linear combinations, of those guys with respect to the spin-c structures. So if one takes sigma to be just a disk, then for a 3-manifold, we now get a vector space which will be z2 graded. And this can be understood as an analog of Havanov-Rozansky homology for closed 3-manifolds. So the Havanov-Rozansky homologies is what categorifies John's polynomial of nodes. So the raciotichin to rife construction produces both invariants of links and closed 3-manifolds. So the same construction can produce invariants. The same construction which produces invariants, like John's polynomial invariants of links, produces beaten raciotichin to rife invariant of closed 3-manifolds. So this can be understood as the analog of this categorification by Havanov-Rozansky homology of the colored John's polynomial. Yes, example 1 and 2. So well, here it's a bit yes. So well, there's no kind of very direct relation, because here you have something like a disk. But in a sense, there is a natural. So in a sense, these z-hards can be understood as elements. So since the boundary of your solid torus is t2, for example, these z-hards can be understood as elements in this SL2C floor homology on a 3-manifold. But for example, I mean, so here, for example, you can see the 3-manifold and disk times s1. So you get some q-series. So in many ways, these q-series are kind of similar to q-series which generate function for work with environments. But they don't have model properties. But in some simple 3-manifolds, they have mock model properties. Well, it depends who you are, but there is some belief. So instead of DOA assigns for a 4-manifold, the analog here should be some logarithmic DOA assigned to a 3-manifold. At least this is what happens for some simple 3-manifold. So let me consider example. So I'm close to the end of the list examples. This is the important ones. So if I take sigma to be two-dimensional, and I take it to be s2 with two marked points. So there will be some additional data assigned at these points, which I'm not kind of some universal data assigned to this point, which I'm not going to talk about this now. And well, I take, so here, let me take g to be u1, something very simple. Again, in this 2.0 case. Then the statement that is extra. Then you can realize, in this way, is I work. So part of this data is actually one of the punctures is colored by spincy connections on the 4-manifold, which fixes the other. And the other marked point is universal. There is no color choice. And one can try to generalize this. So take sigma to be s2 is, say, n marked points. And this gives me what is called a multi-manipold invariance. So this is a generalization of a Zeppelin invariance of, for manifold way, instead of one Higgs field, you have multiple Higgs fields. So this is another case, which I will focus in my lectures. And moreover, so all this can be interpreted as an interpretation in terms of DOA vector-period algebra, which appeared before. This was vector-period algebra, which was obtained as a vector space. It was obtained by putting this theory on just this one. So as I mentioned before, this gives you some great vector space, which moreover can be cubed, is believed to be cubed with the structure of the vector space. And now, also let me mention a bit. So here in all these examples, I consider the case when 2 comma 0 case, when everything was labelled by simply-lazy algebra. But let me briefly mention what can happen when t is 1 comma 0 of this more general list. So it's labelled by a choice of singularity and elliptically fiber collabial threefold. Then, so one thing one can consider as before, to choose sigma to be t2 to tors with complex structure tau. And then, well, depending on choice t, this can be interpreted as in the case of a Baphavite environment, this will be some series in cube with integral coefficients. And they can be interpreted as a case-theoretic non-Abelian monopole. So there is certain, what happens is there is certain, for certain cases, the generating functions for such non-Abelian monopole environments will also have some modular properties, which come from the fact that this, from the symmetry, from the mapping class group action on these two tors. Well, the choice of collabial threefold means what exactly, so it tells you what gauge group you consider and what is the representation. So what is the kind of meta-choice? What is the meta-choice, but only you cannot get all possible choices of the gauge group and meta. So only particular combinations, and for this particular combination, this is expected to have a model of properties. Well, what I'm saying, you can construct this series for kind of arbitrary choice of gauge group and a meta. But for this arbitrary choice, they're not going to have any model of properties. Only for those choices, which come from collabiaus, you will have a model of properties. So there's, I'm going to say the conjecture form. Well, again, for some particular, I didn't explain this, but if I have time in the lecture side, I will also mention this thing in more detail. But moreover, there is a conjecture. The first part, the second part of this can be refined to an element of the coefficient ring of our generalized homology series, which is called Tmf, which is invented by Hopkins of a point. So it's known that as a, so the free part, if you tensor this with Q, this coefficient ring, you get a ring of model of forms. But there are also some torsion part, which is not captured just by this Q series. No, the torsion point is not, it should be constructed in a bit different way. But there is some prediction that it should be refined to include torsion. Well, torsion, I mean torsion in this ring. So the free part of this ring is a model of forms. But OK, this is the end of this big physical physics picture. Over here? Are there questions? Well, it's not the end of the lecture. Yes. So for this, you actually just need to use the casement G equals C1. So you can capture homology information of your manifold and things like error characteristic signature by just considering this T, which is associated to U1. Yes, yes. Well, in general, it's not, so the point, well, you cannot, well, let me say the following. There are not enough supersymmetries to define the full 6D topological quantum field theory. But if you produce 6-manifold, it has some special galonomy, then you may do something. If your vibration will have some particular special galonomy, then I think you can do something with this. But the galonomy on the tangent bundle, the galonomy, like is the? Yes. I mean, yeah, to define the theory, you can equip it with metric, but then you show that it doesn't actually depend. Yes, yeah. Well, which case? Yeah, this case. This case, well, so far, for something like product of Riemann surfaces. And, but, well, you can also try to do this. Yeah, I mean, so far, but, well, I believe you can, like, if you use this kind of definition, you should be able, probably, to compute for complex surfaces while using technique by Dutch and Coolman. But, yeah, in our work, we did this. So the work with, for example, this work by myself, Dupe, who is here, Kumrenwafo and Sergei Gukov, we did this, for example, for this, for in 1,0 case, in 1,0 case for product of Riemann surface. But for 2,0 case, you can do more. So, OK, so let me, so we have 10 minutes. So let me start a kind of more systematic expedition. So since not, I think not everyone is familiar with the notion of vertex operator algebras, I should briefly review this. And this is what I will start doing. Because later, we will need to use this notion. So what is the definition? So let me, first, define the vertex algebra. So what actually, so some people may call it, what I will define here, some might people call it, vertex broad. And so this contains the following data. So this will be vector space, which is z to graded. So this is the part where you can call it super. So the most classical definition doesn't have z to graded. So there is a, so I will denote using kind of physics notation. I will denote z to grading as f, the value of z to graded as by f. And so there is special, so the following data. So the following quadruple vector space v, a choice of what is called unit element in v, in physics called vacuum. And so it has z to grading. It's even. And then there is a translation called translation operator. So this is an element, so this is an endomorphism from v, or endomorphism on v. And what's most important, so probably I should write it here. So let me write it 1, 2, 3, is what is usually referred as state operator or state field correspondence. And this is often denoted in mass literature by y. And this is a, so one way is to define it as a linear map from v to endomorphisms of v, endomorphisms of v. And so you take a series, a formula around series in z, with coefficients being endomorphisms of v. So other way, equivalently, of course, you can understand as a map from v times or v to v itself, sorry, to series, to Lorentz series in z, with coefficients being endomorphisms of v. And moreover, so this is a, so vector, vector operator, vertex algebra is a quadruple, is a following quadruple. And moreover, they should satisfy the following axioms. So if I take the state field correspondence of 1, so this is, will be just constant series, this coefficient being identity operator. And the t, the translation operator, should satisfy the following property. So if I take the value of this state field correspondence of some element u, u, so for any u in v, so this gives me some element for endomorphism v, so series in z, this coefficient value in endomorphism of v. So you can take a commentator of this operator acting on v with t. And this should be the same as a derivative with respect to z, this formal parameter, why the third axiom is, sometimes it's referred to as locality or Jacobi identity. Is that, so it might be not very visible here, so let me continue on this part. So this locality is that for any pair uv from my vector space, there exists positive number n such that the following is satisfied. So if I take a commentator of y of u, this formal variable z, and y of v is from variable x and multiply by sufficiently high power of x minus z to the power n, this should be 0. So this was the definition of vertex algebra. Now what is a vertex? Oh, sorry, I forgot to mention. So of course, this commutator is a graded commutator. Of course, this commutator is, by definition, is plus minus 1. So here, f of u times f of v, y. So as usual, you take in this super world, as usual, you take commutator or anticommutator depending on the parity of u and v. And of course, this map y is even with respect to z to grade. So now let me state a definition of vertex operator algebra and I'll stop. So is a vertex algebra with additional data. So the data is a certain element omega of v of my vector space which satisfies the following condition. So if I take the state of field correspondence of this element, so let me denote this series by t of z so that it has an expansion of the following form. So those guys, by definition, those are some operators acting on my vector space v. And this omega is such that, so if those ln's satisfy the asteriological c, where c, so here c is just a particular complex number times identity on v. And moreover, v is z graded by l0, meaning that v can be decomposed into with respect to this z-grading such that l0 on each component acts just by multiplication by some integer. And l minus 1 should coincide with this translation operator which you had before. And omega itself should be element of degree 2. So one very brief thing is that modules of vertex algebra v are vector spaces equipped with the following map. So there's a map from v to now in the endomorphisms of m. And we take a series of this. For this, we take a series in, again, formula n series in z with a coefficient toilet in endomorphisms of m. Of course, v itself is a model of v. OK, here I'll stop.