 OK. Thank you for the invitation. And it's my great pleasure to give a talk here. Today I'm going to talk about two Calabiol categories and the Motivic Donut Centromes invariants associated to such kind of categories for three-dimensional Calabiol categories. The Motivic Donut Centromes invariants was introduced by a conservative sobo man via the following two approaches. Namely, the first one is Motivic Hall Algebra. And the second one is the homological Hall Algebra. And through the kind of integration map into the so-called quantum torus, of course, the coefficient ring of these two kinds of quantum torus are different. But anyways, via these two approaches, they get the Motivic Donut Centromes invariants. This is the three Calabiol case. And today I'm going to talk about some analog story of two Calabiol categories. This is a joint work with a sobo man. OK. First of all, I'm going to introduce the notion of inconstructible A infinity d-dimensional Calabiol categories. I'm going to abbreviate it into d-Calabiol categories. In fact, there is a notion of inconstructible A infinity categories. There is a long list of data and axioms for this kind of A infinity categories. And I'm not going to write them down here. But the key point is that we can associate stack of objects to this kind of category such that, I mean, by the stack of objects, I mean the following thing. There are, I mean, there is a countable collection of algebraic varieties, which we denoted done by yi. Here, i is a countable set with action of the group GLN, GLNI. Here, NI is any positive integer. Depends only on this i. Such that the following condition is satisfied. Well, I've got to mention that we fixed ground field k. Such that the character of k is 0. OK, now for this stack of objects, we say that there exists such kind of varieties with this group action such that for any field extension, k prime, there exists equivalence. Equivalence of group points, the isomorphism classes of triangulated infinity k prime linear category, which is denoted by c of k prime. And the countable union of the quotient stack. Objects. Yes, objects. Assets of objects. And the quotient stack, which we denoted done by yi GL. OK, this is called the stack of objects. All right, this is the triangulated infinity category. And for now, let me introduce the notion of d-caliber categories. Category dimension d is a weakly unit, unit k, of course, over k. k linear category, which is triangulated infinity, triangulated such that all the graded vector spaces, I mean the home space, is finite dimension for each pair of objects, e and f. And furthermore, we have the following data. The first one is a non-degenerate pairing on the home space. We denoted the pairing by this notation. It's on the home of the f, tensored with, is mapped to k shifted by negative d. And this pairing is symmetric with interchanging the two elements. And the second piece of data is the potential, namely for any integer which is bigger than or equal to 2. And any objects, n objects, up to en in the objects of this category, there exists a polynomial linear cyclic invariant, namely z mod nc invariant on the following space. And we denoted this map by w. It's on the tensor of the following home spaces, home from ei to ei plus 1. Here we need a degree shifting by 1. And it's mapped to k shifted by 3 minus d. Here, of course, we suppose that en plus 1 is equal to e1. OK. Then the third part is the compatibility of this data, namely the potential wn of a sequence of homes, a1 up to an, is equal to the non-degenerate pairing of the following thing. n minus 1, a1 up to minus 1 paired with an. Here, n minus 1 is the higher composition map of this infinity structure. All right, this is the notion of the dimensional Calabria category. And now, if we assume that d is equal to 2, then we will prove that a certain type of two Calabria categories is 1 to 1 corresponds to a certain type of quivers. OK, so let's state the following theorem. Now let's denote c by a two-dimensional Calabria category, which is generated by the following spherical collections, namely by a finite collection, which is denoted by e. It consists of finite number of objects. This set of objects satisfies the following condition. The first one is x0. Each object is one-dimensional. Probably here I've got to mention that is the homology home space. OK, then the second assumption is x0 of ei ej is equal to 0, where i is not equal to j. And the third part, the x space where the degree is negative of any of these objects is equal to 0, no matter i and j are equal to each other or not. OK, this is the assumption of our category. Now we state the following thing. The equivalence classes of such categories with respect to the infinity equivalence, preserving the two Calabria structure and that set of generators. OK, one-to-one correspondence with finite symmetric quivers. Here by symmetric quiver, I mean the number of arrows from one vertex i to j is equal to the number of the arrows in the opposite direction. Yes, with even number of loops. Even number of loops. OK, now let me probably give you a sketch. Can you guess if you have some relations in the quiver? Relations? No, no relations so far. OK, suppose that we are given such kind of two Calabria category, we want to construct a quiver q in the following way. For each of this spherical generator, we give a vertex. Well, maybe let's denote the set of vertex of this quiver by q0 and the set of arrows by q1. We have a vertex i corresponds to each of these spherical generator. And spherical is kind of misleading curve because you have loops. Well, OK, this generator. OK, all right, then the number of arrows from the vertex i to j is equal to the dimension probably there is some subtlety here. How to say it in a clear way? You mentioned the fixed one doesn't work? Yeah, I mean, because of that non-genetic degenerate pairing on the home space, we can see that the dimension of, wait a minute, that's OK. Now, because dimension of E i E j is equal to dimension of x1 E j E i by our definition, we can see that the number from i to j is equal to the number from j to j. This proves that the quiver is symmetric. And again, because of this non-generate pairing, we can see that dimension of x1 is even, which means that the number of loops at each vertex is even. OK, this is one direction. And the other direction from a quiver, we want to construct the collabial category. OK, to illustrate this kind of construction, for simplicity, we just assume that this quiver only has one vertex and a bunch of loops, and the number of loops is equal to 2n. OK, then we introduce the following graded vector space, which is denoted by a. It's equal to k shifted by 1. Direct sum with k2n with k negative 1. All right, then we want to introduce graded coordinates on this vector space, namely, coordinates, namely on the first piece. Let's denote it by alpha. Which has degree 1. And on the middle part, k to 2n, let's denote the coordinate by cosine i. i is from 1 to n. There are 2n coordinates. And on the third piece, let's denote, I've got to mention that the degree here is 0. And the third piece, the coordinates, is denoted by beta. And the degree is equal to negative 1. Now to construct this collabial category, we only need to construct potential on this graded vector space. OK, so let's denote the following potential. w canonical i alpha squared beta plus the sum over i from 1 to n of alpha xi psi i minus alpha xi xi. Of course, up to cyclic permutation. OK, so this is the function on this graded vector space, A. It's easy to see that with respect to the following Poisson bracket, which is the partial differential with respect to xi, partial differential with respect to xi i acted on f and g plus the second part is the partial differentials with respect to alpha and beta respectively. And it's easy to see that w canonical with itself is 0. And moreover, this potential gives a product on this graded vector space by the third piece of data of the definition of our collabial category. All right, so this is the construction from the quiver k to the collabial category. And if we want to specify this one to one correspondence, we need to prove that any deformation of this canonical potential, I mean to prove that this canonical potential cannot be deformed. So we need to consider the following differential graded the algebra g canonical, which is a graded sum over all the integers of g canonical to n. OK, here where g canonical n is the following thing. It's belong to g hat n. I will introduce g hat n later such that the c-click degree of this w is bigger than or equal to 2. By c-click degree, I mean the number of all these coordinates appear in each terms of this w. And that g hat n is the following thing. Well, maybe let me introduce the following notation. Here, c-click means c-click words on this space, namely c-click words in terms of these coordinates. OK, so here g hat n is all the w's with cohomology degree equal to n. OK, so the deformation of this canonical Calabria structure is controlled by this differential graded the algebra. Of course, I've got to give you the differential. The differential here is given by the Poisson bracket with respect to the canonical potential. And it turns out that the cohomology of degree bigger than or equal to 1 of this g canonical is equal to 0, which means that the deformation, I mean, this canonical Calabria structure cannot be deformed. So we proved the one-to-one correspondence of these two Calabria categories, I mean, the infinity equivalence categories of classes of these categories and these symmetric equivalents. This is a big class of two-dimensional Calabria categories. Now I'm going to introduce an approach to Donald St. Thomas's invariance of two Calabria categories. First, I need to briefly recall the motivic step function. Suppose that we have the following pairs, x and g, where x is an algebraic variety and g is a fine algebra group and g acts on x. OK, then there is a notion of a map of this between these two pairs of quotient stack. Let's denote the source by y, h, and the target by this kind of map is given by correspondence. And let's consider the following group generated by the isomorphism classes of this map of quotient stacks, isomorphism classes of such kind of maps, namely, it's generated by the following symbol. Isomorphism class is denoted in this way and we need to add some relationship, namely, for the disjoint union, y1, h1, disjoint union with y2, h2, mapped to xg. It's equal to the sum of these two parts, y1, g1, xg, with y2, g2, sorry, h. This is the first relation and the second relation is the following thing, y2, h mapped to xg is equal to y1 cross with a fine space here, y2, y1, yes, y1, is an h equivalent equivalent, h equivalent, not the bundle of rank d. OK, this is the motivic stack function. Then we consider the following space, consider the following space, namely, well, let's denote this group by motivic stack function with target xg. Let's consider the following motivic stack function on spec k module, which is the direct sum of all the motivic stack functions with the target given by yi, gl, ni. Here, yi, gl, ni is our decomposition of the stack of objects of our Tokalabial category. Of course, here we need to add some other things, namely, add l to the nth, where n is negative. Here, l is the class of the offline line. OK, then we can introduce whole multiplication on this space, which I'm not going to write down in detail here. Otherwise, it will eat up all the time. And this whole multiplication, let's call this algebra, motivic whole algebra. And it turns out that this product is associative. OK, after defining this motivic whole algebra, we can approach to the definition of Donaldson-Thomas invariance. But before that, we need to fix the following data. The first part is a triple gamma and the pairing and the quadratic form q, where gamma is an abelian group of finite rank, namely, it's esomorphic to the n. And this is a bilinear form on this gamma. And q is a quadratic form on gamma tensored with the real numbers. And the second part is the class map from the stack of objects of our category to this group gamma, such that they induce the map from the objects of our category, I mean the k-bar linear category. Factors, through the following group homomorphism, the group homomorphism, which is denoted by class k-bar, is from k-node of this k-linear category to the group gamma. And it's also compatible with the Euler form of the category and the bilinear form given on this gamma in the following way. The Euler form on the category is equal to the non-degenerate form on the gamma of the class of these two objects. And the third part is the construct of stability condition compatible with our quadratic form in the following way. The form restricted on the kernel of the central charge. Did I mention the central charge? Central charge is an additive map from gamma to the complex numbers. The quadratic form on the class of our object E is non-negative, where this object is semi-stable, semi-stable with respect to our stability condition. All right, after fixing this stability condition, we can define the following quantum torus, the commutative r, is defined in the following way. It's noted by r sub gamma r is equal to the direct sum of gamma generated by the following symbol, e gamma hat, where this generator e gamma hat satisfies the following relationship. e gamma 1 hat, e gamma 2 hat is equal to l to 1 half of gamma 1, gamma 2 times e gamma 1 plus gamma 2 hat. And e naught hat is equal to 1. This is the definition of quantum torus with a coefficient ring r. And suppose we are given a stability condition and a strict sector v inside of the complex numbers, we can also define the following quantum torus r sub v associated to this sector in the following way. Here, of course, gamma is inside of gamma intersects with something here are many details. But roughly speaking, the class of gamma is mapped to this sector v. This is the definition of quantum torus. And the value of our Donaldson-Thomas invariance will be in this quantum torus. OK, now let's see to be a two-caliber category of our class introduced previously over k. And let's let the coefficient ring of our quantum torus r to be the following thing. It's the motivic function over spec k. And the following symbols are formally added to the ring of our quantum torus. And since our category is two-caliber, this quantum torus is commutative. Of course, it's because the non-degenerate form on the lattice gamma is symmetric. OK, now let's define the motivic weight omega, which belongs to the motivic stack function over the stack of objects of this two-caliber category. It's defined to be l to the 1 half of the Euler form of any object e of the category. OK, after fixing this data, we have the following theorem. The following integration map, which is denoted by phi from the motivic whole algebra of this two-caliber category to the quantum torus we just introduced is it could be defined in the following way, where suppose we are given such a stack function over the object, it's mapped by phi to the following motivic integration of that motivic weight l to 1 half kind of e. OK, this map satisfies the following condition phi of mu 1, mu 2, here mu 1, mu 2 are two elements of our motivic whole algebra is equal to phi of mu 1 times phi of mu 2 for the argument of gamma 1 bigger than the argument of gamma 2. OK, here, of course, our mu i belongs to hc of. Now let's define the following element in our quantum torus. Suppose l inside of the complex plan is a ray. Then let's define the following generating function, which is denoted by Al mode is equal to the sum over all the esomorphism classes. Here, e belongs to the subcategory of our two-caliber category cl, which is generated by the semi-stable, generated by semi-stable objects with class belongs to our ray. It's defined as this generating function. This is called the motivic Donaldson-Thomas series of our category corresponds to this chosen ray l. Of course, if we choose any strict sector v inside of the complex plan, we can also define generating function a sub v mode in a similar way. And it turns out that these different generating series corresponds to different sectors satisfies the following factorization property, namely, Av mode is equal to Av1 mode times Av2 mode. Here, our v is a strict sector. This is v. And it's decomposed into the following two parts, v1 and v2. And the product is taken in the clockwise direction, OK? So these are all called the motivic Donaldson-Thomas invariance. I guess I can stop here in time. Thank you. Any questions, please? What's the difference between three-dimensional parallel parallel and two-dimensional parallel category in essence, essentially? I think the relationship, I have a precise statement in terms of cohomological whole algebra, rather than motivic whole algebra. But in some way, the motivic whole algebra is related to cohomological whole algebra. So maybe I can tell you something in terms of cohomological whole algebra. I will give you an example in terms of quiver with vertices q0 and arrows q1. And we construct a double quiver q bar, where for any arrow in the original quiver, we add an inverse arrow. Well, it is an original arrow. We add an inverse arrow a star from j to i. And then we construct a triple quiver by additionally add loops li to each of the vertices. And in this way, we get the following pair, the triple quiver and the potential here potential is defined in the following way. Some over all the arrows and the added loops is the commutator of the arrows with its dual times li. OK, then for such a pair quiver with potential, we can define the cohomological whole algebra, which is called critical, cohomological whole algebra defined by conservative sobelman. It's defined using the sheaf of vanishing cycles. And from this double quiver, we can have the pre-projective algebra, pi q, which is actually equal to c q bar, modded by the relations given by the commutators. OK, this is three-dimensional. And for two calabiol, let's consider this pre-projective algebra. We can also define the cohomological whole algebra of this pre-projective algebra. And the product could be induced from this critical cohomological whole algebra in the upper star. So because of the following equation, the critical cohomology of the wrap q hat with respect to this potential is esomorphic to the compact spot, to the compactly supported cohomology of the representations of this pre-projective algebra, of course, up to some degree shifting. This is a relation between 3 and 2. And from this cohomology of 3, you have Donaldson-Thomas invariance for three-dimensional. And for this cohomology of the pre-projective algebra, you have the Donaldson-Thomas invariance into two calabiol. So that's the relationship. I have one more question. So there are multiple whole algebra with some group structure of an algebra group. But there, I cannot see the group structure. Here, the representations of the q of dimension gamma is acted on by gl gamma. So from what you just said, in your situation, you could maybe interpret this as a calabiol of three phenomenon, but maybe you've just expressed differently. So if I give you a general two calabiol category, is there a reason to think that you can build a cohomological whole algebra like this? You mean using this reduction? No, no. Just if you don't assume that it just comes from a finite quiver. Well, the definition works as well. It doesn't matter if you add this kind of restriction or not. I think for this decay, you have to construct some kind of functions, values, and goodness class of motives from your latest, yeah? And this should be something like God's polynomial, yeah, popular? Yeah, yes, yes. For our two calabiol categories with those generators, there is evidence for the existence of this God's polynomial. Yeah, yeah, but do you have actual vector spaces whose dimension says this? Because it should be construction of quads from the algebra. Unfortunately, I don't have any precise statement so far. Your speaker? Thank you.