 So this is about homological whole algebras and motivic invariance of crevice and so we have to first want to somehow explain the aim and scope of this series of lectures and we have to answer the questions What why and how and where So this is a school on enumerative geometry and we want to enumerate Things which don't seem too geometric namely quiver representations. Yeah, so we want to for how Upmost aim is to want to count quiver representations So why why do we want to count quiver representations? So we'll first see that quiver representations form a category which is similar to categories of Coherent sheaths on curves namely, it's a category. It's a nice abelian category of global dimension one So one point is that category of representations of Quiver Q is in some respect similar to coherent sheaths on a smooth projective curve Of course, there are essential differences in general, but there are some all similar for example because both are categories of global dimension equal to one Yeah, and so this is kind of a Non-commutative counterpart to curves or coherent sheaths on curves So we are doing Non-commutative geometry of curves and we want to enumerate something Was that me? Okay, I'll be careful Okay, so that's the first reason why we want to do this second reason is that quivers are something of like a toy model for more interesting things maybe for you more interesting things like Quivers with potential so three Calabi yaw situation so for two or three Calabi yaw situations like quivers with potentials But most of the time I will only treat the quiver case and really view it as a toy model and Understanding all the phenomena in the quiver case. You are then prepared for studying the real stuff Like cruise with potential Okay another reason for studying curve is is that If you are really interested in some category of sheaths, but you are interested in some local phenomena Phenomena like only involving sheaths which are filtered by a finite number of basic objects Then if you are lucky you can model everything by a quiver situation many other situations in categories of coherent sheaths or in enumerative geometry Can be modeled by quivers and I don't want to be too precise here at the moment We will see some instances of this principle later. Yeah, so that's why we look at at this quiver situation Although it seems quite far from the usual setup of enumerating things coming from algebraic geometry Okay, so that's the why and Now to the what and where Where do we count? Well, and you have certainly seen at the school already that Doing enumerative geometry counting things can be much much more sophisticated than just taking the cardinality of a finite set You can have counts which are negative which are even only rational or whatever. Yeah, so finding the right the right domain where to count things is highly non-trivial and This is also what happens here in this theory So well if you have an honest finite number of quiver representations We can just count the cardinality of this finite set, but that's not what usually happens You can have parametric families of quiver representations. So for example, you could have a p1 family so what is the count for a p1 family and so we need the right domain where to count and this will be the written decree of varieties and this I would introduce in a minute So all our counts will be in certain variants of Grotten-Deek rings of varieties So these are Very simple-minded motives and that's where this motivic invariant comes from yeah So motivic invariants are some invariants associated to quiver representations, which are counts But counts in the Grotten-Deek ring of varieties so that's the the where and Another question is What are quiver representations and this is also something I should tell you today? So today I will first explain the Grotten-Deek ring of varieties because that's where we will count things and Then I will give you some basic terminology On quiver representations in case you don't know it and then finally we will consider the Motivic generating function, which is something like a partition function So then finally we will see as the three third part today Aq a certain motivic generating function and this will be an element of a so-called motivic quantum torus Which we will introduce Okay, so that's the plan for today and let me just give you an outlook of how we continue. So today will be just mostly definitions and concepts yeah, so where do we count what do we count and it's just about writing down this so-called motivic partition function and exploring it in a few very very simple examples and then tomorrow we will continue and So tomorrow we will take this motivic generating function aq and factor it and trying to factor it will lead us to all sorts of Modulized spaces of quiver representations and their motivic invariance their motives. So factoring this leads to Modulized spaces and in a special case Hopefully I can do this tomorrow This will lead us to certain gamma-vitin invariance and the connection is why are the so-called tropical vertex and So maybe we will see this already tomorrow and then talk three and four Will be about taking all these concepts one first two talks which are really just Enumerating things lots of explicit identities and Categorifying them Categorify all the identities which will produce in an algebraic object called the cohomological Hall algebra of the quiver Yeah, but before we categorify we have to see the numerical stuff. So what is this counting really about? so That's my current idea of the outline of the four talks and let's see how this develops and I'm really happy to skip Most of this if you're interested in different things So please just tell me. All right so let's start with the first part of today's talk introducing this curtain degree of varieties and Okay, so Let's work over the base field of complex numbers That's enough For today so let's introduce the good degree of varieties over C and So what notion of variety do we do we take? Actually, that's not so important for defining this good degree of varieties this good degree of varieties is not too sensitive on the fine difference of reduced and non-reduced structures and One question is always do you view as a variety as something irreducible or not? Let's just say a quasi-projective not necessarily irreducible variety So really in the naive sense take complex projective space of some dimension and take some polynomial Equalities and inequalities so something locally closed in a complex projective space That's our notion of variety No Okay, then we introduce K zero of RC we first take something really really large namely we take the free abelian group in all Isomorphism classes of such varieties free abelian group in isoclasses of complex varieties and I hope we don't have any set theoretic issues here how to define such a variety where you first have to fix the number for the dimension of the projective space then you take finitely many polynomials and Defining the polynomial equations and inequalities and that's it. So that's maybe something we can do inside sets Okay, so it's mostly harmless to just take this free abelian group in the set of isomorphism classes And then we mod out a relation just one relation namely the cut-and-paste relation modulo the cut-and-paste relation, which is x equals a Plus you if A in x is a closed sub variety With open complement you That's the cut-and-paste relation. Yeah, so whenever you have a closed sub variety So whenever you have a decomposition of variety into a closed sub variety and it's open complement then you require additivity and This is a relation we force on on this free abelian group That's it. All right, so We will do a few computations in there But let's first continue to define a ring structure on it because at the moment it's just a group and the ring structure It's just given by multiplication multiplication. It's just given by the Cartesian product of varieties perfect What else do we need We need the so-called left sheds motive L like both style L which is defined as the motive of the Affine line. All right That's almost it for the moment We need some well two more definitions, but let's start with Okay, so please leave some space here at the end of the definition Exactly x of variety complex variety then the class of x called the motive of x so motives in a very naive sense just the classes in this in this group and well, okay, so there's there's a lot to say about this this ring and I will not give you all the things which are known about this ring For example how it relates to Biorationality of varieties. That's another very interesting story, but we don't need this and actually Let's just do some example calculations Yeah, so I will continue the definition a little bit, but let's first start with Examples What is the class of Affine space Affine n space Well affine n space you can Realize as the n-fold Cartesian product of the affine line Yeah, so Cartesian product is the product in this ring and the class of the affine line has a special name the left shits motive So this is just the nth power of the left shits motive wonderful Okay, that was our first computation in this ring another computation is projective space well projective space realized as well you have homogeneous coordinates from zero to n and You have a very nice Stratification of projective space by deciding whether the zeroth homogeneous coordinate is zero or non-zero if the zeroth Coordinate is non-zero you can normalize it to one and you get an affine space principal affine open and The complement of this is when the first when the zeroth homogeneous Coordinate is zero and this leaves projective space of one smaller dimension Yeah, okay, so this is just the set where the zero coordinate is non-zero And this is defined by zero coordinate being zero All right That's an open that's a closed complement so we see that the class of P n is the class of a n Plus class of P n minus one where we now have used the cut-and-paste relation for the first time and This is of course the basis for an induction because you can continue with P n minus one and Decompose it into a n minus one and P n minus two and so on so doing this inductively you arrive at a n plus a n minus one plus Plus a one plus A zero which is just the class of a point Okay, so since a n can be realized as powers of the of the left sheds motive We can realize this as a truncated geometric series So in other words the motive of projective space is one minus l to the n plus one divided by one minus l the truncated geometric series Aha, and you already see that Well, this is of course Completely formal expression at the moment, but it might be useful to allow denominators like one minus l No, and this we will continue with the next example. What's the class of the? The motive of the grass manian grass manian of k-dimensional subspace is of n-dimensional space Well, if k equals one, this is just the case of projective space because projective space just is the grass manian of lines Okay, P n is gr1 n plus one grass manian projective space just the grass manian of one-dimensional subspaces of n plus one dimensional space and This decomposition into the of the projective space generalizes using the the Schubert's decompositions yeah, so using the Schubert's cells Schubert varieties you can prove that this is the so-called Quantum binomial or Gaussian binomial coefficient evaluated at l the left its motive And if you haven't seen this in quantum group theory or wherever let me introduce it for you So let me define n the quantum number and q as 1 minus q to the n by 1 minus q let's then define the quantum factorial as product of quantum numbers and then we define the Quantum binomial as fraction of quantum factorials Okay, that's the the standard way for introducing these Gaussian Binomial coefficients. Yeah, so you define the binomial coefficient as n factorial by k factorial n minus k factorial You quantize this Factorials are just products of numbers So, okay, so you just have to quantize numbers and the quantization of the number n is 1 minus q to the n by 1 minus q and of course This reminds us of the truncated geometric series. We have encountered in the motive of projective space so exercise compute the motive of GLN We will do this later anyway when we introduce this motivic generating function of a quiver So hint is that well think about this is an invertible matrix. What are the possibilities for the first column? When the first column of an invertible matrix you can have any vector except zero Then fixing this in the second color column You can have any vector except anything in the line generated by the first vector and so on Yeah, and if you write this down carefully in terms of certain vibrations, then you can compute the motive of General linear group All right, okay, so this ends the examples for now we need some Some two ingredients For this golden ring of varieties which I cannot motivate easily at the moment and So I'm afraid at the moment you just have to believe that this will be relevant so first of all we will make this This thing here formal that sometimes we want to have denominators and in fact we want to have even more Potential denominators and then we need the notion of the virtual motive, which is a little twist But which will be very important for the motivic Donaldson Thomas theory. I can't explain this at the moment It's all about virtual fundamental classes of modular spaces and So at the moment this the next definition will be somewhat unmotivated and I just can say believe me it will be important later okay, we will work in the localization and now let me just call this the the motivic ring our mod For short so we take this groten-deck ring and we allow ourselves to first of all To invert the left sheds motive so we want to be able to invert the affine line And if you already know something about this groten-deck ring Then you will think that when you know that this is a terrible idea a priori and really have to see if anything survives I will tell you in a few minutes so we will invert the left sheds motive and Also for reasons of so-called virtual motifs and virtual fundamental classes. We need a square root of it so we will even Need a square root of the left sheds motive and I want to invert All these one minus L to the end where we have seen the very beginning here and calculating the motive of projective space Lumpter, there's no lambda. Sorry. Oh, sorry Yes One minus L to the end so It looks a bit arbitrary at the moment, but for example localizing at all these Brings us to the well. This is nothing I would really I will really use but it's the fact that this is then the groten-deck ring of Certain classes of stacks namely quotient stacks by all groups GL Yeah, if you form you want to define a groten-deck group of stacks where stacks are everything can form out of quotient stacks by GLN Then you get this localization. So it also appears in nature if you view stacks as nature And this is as I already said for these virtual motifs and This is now the final part of the definition the virtual motive is Defined as a little twist to the motive of X Namely we twist it to minus L one-half to minus the dimension of X for X irreducible and So just as a as a very simple example Let's just look at the virtual motive of Of P1 because the essential feature of these virtual motifs. I can already illustrate this example So we take the usual motive of P1 and we twist it by Minus square root of the left shits motive to minus dimension of the project of line Which is minus one so minus L one-half to the minus one times P1 and this is L one-half plus L to the Minus one-half yeah, because the motive of P1 was one plus L okay This fine, or did I make a sign mistake? I Certainly made a sign mistake here minus sorry so and just note that this is stable and under Switching L and L inverse and that's the important feature of this. I mean, okay, this is just by a pronger a duality In this example, and this is somehow the the main feature of these virtual motive that it makes things more symmetric Everything else will follow okay so Now we have to be very careful It might be that we are that in working in with this ring. We are just talking about the zero ring So is this ring non-trivial at all we could also ask this Before doing all these localizations I mean even if we just take all isomorphism classes of varieties and mod out by the cut and paste relation What's the wise what properties of varieties survive under this operation and Well, in fact something survives. This is actually non-trivial Because there is a non-trivial map to to another ring Yeah, and this is the virtual hodge polynomial so Okay, let me try to to motivate this a little bit with the notion of Motivic invariant Motivic invariant of varieties is some invariant of varieties which fulfills the cut and paste relation. Yeah, for example one Motivic invariant of varieties is its Euler characteristic The Euler characteristic satisfies the cut and paste relation Euler characteristic of x is additive on taking an open and the closed complement Or for example, if you have if you work over finite fields then counting point over finite fields is obviously a Motivic invariant because it satisfies cut and paste. Yeah, so these are so-called motific invariance and The classes in the Goten-Dekringer varieties these motives. These are some of the universal Motivic invariance. Yeah, that's universal domain where you have this cutting-paste relation. So improving that this Motivic generating series the this motivic ring is non-trivial We just have to exhibit an interesting enough Motivic invariant. So Euler characteristic doesn't do the trick Because well 1 minus L to the n well the Euler characteristic of the affine line of any affine space is 1 And so the Euler characteristic Evaluates this to 0 and we cannot invert it now So Euler characteristic is not a good example, but we need the virtual Hodge polynomial virtual Hodge polynomial gives actually a ring homomorphism from Goten-Dekringer varieties to Sorry So what target do we take? Let's just say Loro polynomials in two variables x and y namely you map x to Some of all P and q sum over all k Signed twist by minus 1 to the k and then it's HPq H k C X C I will explain So yes, so so so this is compactly supported homology homology with compact support, so that's the case Compactly supported homology and HPq. That's the Hodge delinear numbers of this Yeah, so it comes from the mixed Hodge structure on on this homology so These are the Hodge delinear numbers. Oh The dimension of course I take sorry, sorry the dimension No I'm fine. No, it's just really the Hodge delinear numbers in kth homology Hodge delinear numbers So these are already dimensions signed by k and Then I record P and q Okay, and so By deline's theory this invariant is Motivic it fulfills the cut-and-paste relation and it's compatible with the product Yeah, take the Cartesian product of two varieties then this behaves multiplicatively as Laurent polynomials and We can compute it for the affine line left sheds motive is sent to x y Yeah, and so it's sent to x y so And something survives after localization. Yeah, so localizing at L plus minus one-half just means introducing a Square width of x y in its inverse and these localizations just mean we need denominators one minus x y to the end Yeah, something survived So this this ring homomorphism is non-trivial. This is good news Finally it might seem like like this ring is That might seem like everything's fine with this ring this armot but it is not because Now the surprising thing is what is already known since 20 years ago that this got in the ring of varieties has zero devices and The situation is even worse. This is only known since a few years The left sheds motive the class of the affine line is a zero divisor Yeah, and in constructing this ring. We are localizing at a zero divisor Which is fine, but we are of course losing many many things and Let me just very briefly discuss this so This localized ring of motives is non-trivial because we have an interesting invariant, but L Is a zero divisor in case you? Of complex varieties There exists varieties x Non-isomorphic varieties x and y in fact there actually three Calabria varieties such that The difference of their motives Times six power of the left sheds motive is zero Which in particular means? Well Alice zero divisor, so if we localize at the zero divisor then we are automatically identifying x and y which are Different varieties. Yeah, so in localizing this we are making many many identifications but we are But we know that something survives because for example we can still detect the virtual hodge polynomial Me personally nothing. No, I don't know what is not about this I mean what what is known? This is this this birationality story if you if you take the Rootnik ring of varieties and just mod out the left sheds motive then what you get is the The monoid ring or up of varieties up to stable birationality For example, but about the kernel of this map I don't know luckily for us in the quiver situation all the calculations will happen in In the sub ring generated by the left sheds motive. Yeah We will only work But this is nothing we know a priori we only know it a posteriori we will only work in the subring generated by by L so everything will be just a Irrational expression in L everything we produce But we only know a posteriori Yeah, so we will make some computations in this ring and then at the end it will turn out that Actually everything is like a polynomial or is this like a rational function in the left sheds motive, but this we don't know from the beginning Okay, so this ends part one of the talk Yes, please Construction of the X and Y They mentioned How do you know they're not the same class given that if you take like Richard Hodge polynomial, they match the same thing Well, I looked this up yesterday evening just to make sure I I don't remember the details, but I mean so I can look up the reference for you and it's a quite a short paper and Not so terry difficult to read apparently And these varieties X and Y are really explicit, but I don't remember No, no so so just if you just take say the the Poincare polynomial, that's not a motivic invariant That's maybe another good example of what is not a motivic invariant. Yes, that's maybe good Poincare polynomial, it's not motivic namely look at a one single point and the complement a one without the point and that just over the complex numbers and let's just look at at comology yeah, and Then for the point Comology with compact support Single acromology with compact support then for the point we have Comological degree and then we have a one-dimensional zero of comology and that's it a one has second compactly supported comology But a one minus a point if you just pinch the then you can retract everything to an s1 and then you have Comology in degree zero and one and you see this doesn't add up Yeah What instead of the Poincare polynomial you can do and this is a specialization of the virtual hodge polynomial It's the virtual Poincare polynomial Which it's not the The Poincare polynomial for compactly supported comology But you take the Lin's weight filtration and the associated graded and that again does the trick Yeah, but that's the reason why why this doesn't work. I mean for for smooth projective varieties You can just take the usual Poincare polynomial usual hodge polynomial. Yeah But for arbitrary singular Non-projective varieties you have to take these virtual guys All right crevasse So now we have this good degree of varieties, which is not only useful for for this enumerative Geometry of crevasse, which we will do here But this is also very useful for the whole topic of for example, what if it integrations Motivic zeta functions of varieties Namely the motivic one possible motivic zeta function of variety was it a function usually is Something involving the arithmetic of variety Where you count point over finite fields But how do you do this with a with a complex variety? Well, the idea in motivic integration for motivic zeta functions is to take a complex variety and take its Its jet spaces the nth jet space So kx mod x to the n valued points take the motors of all of them and put them together into Generating series zeta function. Yeah, and this is also a whole area, which opens up when you know this Courtney bring of varieties crevasse So this is usually a Topic where you can spend lots of time because it involves lots of notation. So I Will just start somewhere and if I don't give enough notation cover notation then please tell me so in particular If I've never seen these and their representations this will be Quite difficult and I have to spend more time on that. So please just tell me if you want more details Okay, so quiver is a finite quiver q is a finite Directed graph with a set of vertices Which we call q zero and with arrows Which I will just write as as maps alpha from i to j is an arrow between Vertices i and j so that's a quiver and Examples that's a great quiver. That's the quiver. We have to study first just a single vertex Okay, a single vertex and a loop Also makes a very interesting quiver Two vertices and a single arrow something we'll study to arrow two vertices and two parallel arrows two vertices and two arrows in both directions Vertex with two loops. Well, and of course lots of complicated things Well, whatever. Yeah, so Loops at vertices multiple edges everything is allowed But we will actually be happy with studying these these very small quivers so that's a quiver and What else do we need Okay, we need representations of a quiver So I promised at the very beginning of the talk This will be a class of categories, which is quite similar to coherent sheets on on smooth projective curves So let me introduce this category of representations category of representations of such a quiver q Over a field, which is just for us just the complex numbers. So what are the objects? What are the representations the objects are? tuples Vi sorry vi index by i and F alpha Index by alpha and let me explain So the objects in this category are tuples where this is a complex finite dimensional complex vector space So you put a finite dimensional complex vector space at each vertex of your graph Okay, so if you put a vector space at each vertex then at each arrow you can put a linear map See linear map and that's it. That's a quiver representation. Yeah So a quiver representation is a configuration of vector spaces and linear maps along a certain graph So for example a representation of this quiver means you choose two vector spaces a map from the left to the right And from the right to the left linear maps and that's it Okay, that's the that's an ob that's the object and what are the morphisms? so morphism Phi from V to W where V is A collection of vector spaces and linear maps and W is also a collection of vector spaces and linear maps a morphism fry from V to W is a collection of linear maps Between the vector spaces which are involved Such that So the easiest way to remember this definition is such that all diagrams commute and then trying to figure out what these diagrams are You already see the definition Let's do this Such that all diagrams commute. What diagrams do we get from all this data? Okay, so here we have a quiver representation V and here we have a quiver representation W and now we have one arrow alpha from i to j in our graph Associated to this arrow. We have a linear map f alpha from the vector space vi to vj That's this data. We have something similar for the second representation Wi wj and a map say g alpha okay, so Now a morphism from V to W consists of morphism linear maps between the vector spaces So we have a linear map here phi i and the linear map here phi j That's the square diagram and you can ask it to commute and This should hold for all arrows alpha from i to j That's it. Okay That's the objects and now we need that's morphisms and we need a composition of morphisms Well, that's easy because we can compose such linear maps Component wise composition is component wise That's it so that's a category you you can define out of a graph and Now the question is why do you want to do this and? Let me tell you Well, first of all you might Define this category because you're interested in solving linear algebra problems. That's a quite natural problem You have a configuration of vector spaces and linear maps and you want to classify it up to base change And this is precisely what this definition of morphism does So just look at this diagram for a second and assume that these maps here are isomorphisms then this means that The second representation you get from the first one by a base change First you take a base change here in the origin then you take this map and then you perform base change in the target so note that isomorphism classes in This category rap cq, which I just defined Classify certain linear algebra classifier configurations classifier configurations of vector spaces and Linear maps namely configurations along a graph Along q up to base change So this is a category Where the isomorphism problem solves a linear algebra problem of classifying configurations of vector spaces and linear maps Just quite good. Okay, maybe not what we are interested in you are interested in or originally so fact is That this category Has all the nice properties you expect from categories of modules or sheeps namely it is equivalent to the category of modules Over an algebra non-commutative algebra called the path algebra of q so called path algebra of q So representations of a quiver form a category of modules in particular It is an abelian category It is a category of finite dimensional modules. So all objects have finite length Filtrations you have the Jordan-Hurther theorem. You have the Krull-Schmidt theorem You can do homological algebra because this category has enough projectives and injectives and so on very nice very nice abelian categories and And one important thing is that a Global dimension of this category of modules or of this algebra is just one Yeah, so and this is the similarity to coherent sheeps on smooth projective curves We're talking about hereditary categories. We only have to take care of x zero and one and All higher x vanish anyway, which is great news in particular for defining modular spaces Yeah, if you have ever worked with modular spaces of vector bundles on smooth projective curves You really appreciate vanishing of higher x Because for example, it makes the modular spaces smooth and so on okay that's this category and What else do we need before I can define All right, we need the so-called homological Euler form so Let me define the so-called dimension vector Dimension vector of a representation V. Well, what could be the the dimension of a representation V Well, the representation V in particular involves vector spaces namely one vector space for each vertex So let's just take the tuple of all the dimension of the vector spaces VI and this defines An element in n q zero. So that's the dimension vector of the representation and then I Will define the so-called Euler form And this is then the final ingredient for definitions. I think more or less the Euler form Acute brackets is defined. These are two vectors Defined as sum over all I di e I minus sum over all arrows Di e j So this is the Euler form of of the quiver q at the moment that that's just a combinatorially defined thing because it just involves the graph structure Yeah, it involves the structure of vertices and arrows of our graph But now we will see that this Euler form as you can already guess from the name is the homological Euler form of this category of representations so So this this formula Okay, write it down And then discuss it We take the space of homomorphisms from V to W namely homomorphisms of quiver representations Yeah, and the notion of morphisms of quiver representations as we defined it above Do we have this notion now? I Do hope so. Ah, here we go. These are morphisms They actually form a vector space This is a K linear category and so you can take the dimension of the space of homomorphisms We have enough projective and injective so we can define x1 So the next term in the homological Euler form would be x2, but this vanishes identically So that's the homological Euler form That's just the difference of x0 and x1 and this is given by evaluating this special form at the dimension vectors of V and W so This Euler form is really just given in terms of this of the graph structure because it just involves the structure of Vertices and arrows so this is something you can directly calculate and this is the homological Euler form So this is of the same Importance as Riemann-Roch is for the theory of of of sheaves on curves Yeah, all right That's why this form is so important. Okay, and now I should really come to the final definition Okay so the fancy way of Continuing is to say there is something like the called the modulized stack of Isomorphism classes of quiver representations and we just take the generating functions of the classes of all these stacks But this is not really fair because I haven't defined what the class the motive of a stack is So let me do this in a more elementary way so let me define something like the modulized stack of Representations But I don't really want to use stacks so I try to avoid it and do it in a more elementary way namely We write down a universal parameter space for quiver representations of a fixed dimension vector and Then we mod out by the equivalence relation of taking isomorphism classes And we'll see that this is a quotient stack by a product of groups GL And I think I can use these five five minutes five minutes definitely enough. Okay Okay So fix a Dimension vector D Then I define Rd of Q as the direct sum along all the arrows of The space of linear maps from a di dimensional to a dj dimensional vector space a Point in here is a quiver representation on the vector spaces C to the di point or C valued point a point in this space is a Quiver representation on the vector spaces C to the di and Since the vector spaces are fixed anyway up to isomorphism once we have fixed that I mentioned this is no restriction at all Yeah So instead of taking quiver representations on arbitrary vector spaces vi We can always assume that we have quiver representations on spaces C to the di so a point in this here is a quiver representation So when our two quiver representations isomorphic I already explained this the important keyword is base change and So I will define a group action on this By the group Gd and the group Gd is nothing else than a product over general linear groups and this acts on this variety by base change and The formula will actually resemble the actual action for a group just conjugation action Let me write down the formula for this action Because we should have it here on the blackboard Here's the formula for the action. So a tuple gi no a tuple phi i of invertible matrices acts on a tuple f alpha of linear maps by Gj f alpha g i inverse Where alpha is an arrow from i to j and the G is of course again a phi sorry Phi So this looks like an adjoint action Yeah, G times f times g phi times f times phi inverse Is that reasonable every G is assumed to be a phi. Sorry This is a base change action. You're doing the inverse in the origin then going f alpha then Phi j in the target Phi i inverse f alpha phi j Okay, so you have a group action on a variety and the observation is So and this is a total logical observation the Gd orbits in Rd of q are the isomorphism classes of representations of q of dimension vector d And we should have a 30 second meditation on this on this observation and really see that it is total logical So a point in this affine space here This is just affine space just a direct sum of spaces of linear maps Just an affine space a point in this affine space is Equivalent representations on these vector spaces and we can always assume without loss of generalities that the rect vector space is involved of the c to the di to such points Define isomorphic river representations By this definition of morphisms even only if they are related by this formula So even only they are conjugate under this group action So if and only if they are in the same orbit for the group action Yeah, so just from the very definition the Gd orbits are The isomorphism classes of representations Okay, and this brings us to the final definition for today and to the end of the talk because now we can Say what this modular stack should really be and what the counting of quiver representations should be This brings us back to the logic from the very beginning So this modular stack should be nothing else than the corrosion stack of The space by the group so these brackets now mean the corrosion stack But I don't want to use stacks anyway But just as a side remark the modular stack by definition is the corrosion stack of this space by this group Okay, so now we can say what is the count of quiver representations? So what's counting quiver representations? Now means the following Well, first of all we can count in any dimension in any dimension vector D in any fixed dimension vector D We should take the count of points of this quotient stack So we can define this namely we take the corresponding classes in the goten die group of varieties we take the motive of the space and divide by the motive of this group and If you solve this little exercise Computing the motive of GL of GL n then you will see that everything appearing here The denominator is something we localized at anyway Well for good reasons, which I can't explain explain at the moment. We will take the virtual motives and Then at the end we should form a huge partition function. So we should Sum up overall Dimension vectors keeping track of the dimensions because that's a numerical invariant and this is a function This is a motivic generating function aq Great sound showing the importance of this of this definition. So this is the the central definition So, okay, I haven't told you yet where this lives So what are these extra former variables t to the d this I will explain tomorrow and then we will try to Factor this. Well, this is something like a partition function or zeta function or whatever And of course the impulse is then to factor it into a product of something Yeah, like you do with any zeta function or partition function and this will do here and in trying to factor this All sorts of modular space is associated to quiver representations will pop up almost automatically and that's the the miracle Okay, that's enough for today. Thank you very much Thank you are there any question How do you okay like you do in commutative algebra Like you just do the formal localization in any ring Yes, exactly Exactly, yeah, so the the localization which which I which I considered is really in the sense of So it's really like you have an arbitrary commutative ring RS you really Join a form of variable t and then mod out by s times t equals 1 to to make t invertible and Yes, of course if this is zero divisor, then you make certain things zero, so in I wrote down this relation in the gotenick ring of varieties where you have two kalabi house Which after multiplying by the six power of the left sheds motive becomes zero so in particular in our Localized ring of motives this thing is zero. So these classes are the same So that just means this localized gotenick ring cannot distinguish many many varieties, which are definitely non isomorphic but Well, we know that at least something survives and that's enough for us at the moment Yeah There is one question in the room Is it possible to analytically continue the construction with respect to the dimension vector? analytically continue Sometimes Well, okay, so unfortunately, I didn't have time for for the first example so what we will do tomorrow as a start is taking this this series and Writing down a factorization in in these two rings. Let me let me just show you this this Yeah, let me give it as an exercise So If Q is the trivial quiver Well, that's the same as an algebraic geometry if you have a new concept Then you should first explore what does it mean for the variety a single point and there you should understand it Yeah before starting to do it for whatever variety And so if you have a concept for quiver representation, just take the quiver Which is just a single vertex the category of representations. It's just the category of vector spaces That's the category you really understand so we should understand this motivic generating series And even then this motivic generating series is nothing Completely trivial it admits a factorization and this we will see tomorrow As an infinite product over one Minus l to the i plus one-half times formal variable t so and that's kind of a typical phenomenon and Well, okay, then you can start exploring the analytic continuation properties of such infinite products Yes, how much of a Q can we recover from a Q? Ha ha So formally Well, okay Not that much not as much as you would expect because I was cheating a little bit It's not only about the series But it's also the ring the so-called motivic quantum torus in which we consider it and This ring in which we consider this aq and in which we'll factor it will contain more Information than aq itself. So for example, this aq is the same At the moment whether you consider this quiver or this quiver, but these quivers are drastically different in their representation theoretic behavior and It will depend on on on the motivic quantum torus in which we will consider and factor this thing And this is something we will only see tomorrow The aq is is not dependent on this but but the ring in which we'll view it because I the ring in which so we will view this in some In some form of power series ring which will depend on on Q really on the orientation Yeah, so there is some extra bit which I couldn't define yet There is one more question. Yes How the formula of the other form of dimension vectors which was stated is proved How it is proved with a with a standard resolution. Yeah, you write down a standard projective resolution This is quite classical and Wow, okay, so I can't do it now because I had to introduce Bit of notation yes, I can sketch it. Okay Maybe not in all details, but let me just give you the flair of how you write down this the standard resolution and Okay, either with a standard resolution or with the following exact sequence. Yes, okay, let me do it like this I'm almost done and then I can Explain so in the these two middle terms here are just spaces of linear maps yeah, and an element in here looks like a couple of these files and A couple of to this such tuple of the files I associate precisely the thing which I asked to commute namely this F alpha phi j minus phi i G alpha indexed by alpha from i to j Okay, so this is the thing which commutes in this diagram defining morphisms So in particular this means if phi i is mapped to zero under this map Then phi i defines a homomorphism of quiver representations Okay, on the other hand if you realize the x1 and In the naive sense namely as equivalent equivalence classes of exact sequences Then writing this down in all details you will see that x1 is actually elements in here Modulo the image of this map or you can see it by some standard Projective resolution of representations and this gives you such a four-term exact sequence now. Just compare compare dimensions Dimensions of this minus dimension of this is the homological Euler form It's the same as dimension of this space minus dimension of this space and this difference is precisely given by the Euler form, so Fancy would be to really work with standard resolutions, but it's just a linear algebra thing actually there was one more question in the audience so if you take a Representation value the difference of alien time within vector spaces. How much can we do the similar thing as in the vector space? Ah, so if you take quiver representations in some other abelian category and Korean sheeps for example, yeah Yes, there are relative versions of many of these constructions. So for example There is an analog of of of this standard resolution and of these homological properties which shows that the Global dimension of the category of Representations in a category a is one more than the global dimension of a yeah, so there exist relative versions of all this No No, this factorization And a very deep point uses the fact that everything is hereditary We are really in global dimension one and You can you can push this from Yeah, I mean all these factorizations really work smoothly only for a hereditary category And you can push it to the two kalabi yaw or three kalabi yaw situation in certain situations But that's it But if you just consider quiver representations in some arbitrary category of coherent sheeps I doubt that you get such nice factorizations. I'm afraid Okay Yes, there is a sergiality which classically is called the australian translation so there exists some australian translation tall Which you can realize which you can write down as you take the X one with With the path algebra itself, which is something like the regular representation in the second component and then dualizing But now I did I wrote down to our inverse That's now, okay, and then you have Formula X one MN is isomorphic to home and tower M Dualized and that's really quite literally sergiality in this one-dimensional setup Yeah, you have this duality from X one to X zero dualized by applying this the serf hunter And you have a question Is your virtual class definition of varying in any sense? I don't know what definition Situation I Don't know All right, no more questions. Let's thank Marcus again. Thank you