Okay, so let's first summarize what we did yesterday also to somehow refix notation we had this

Motivic Ring, which will be our coefficient ring for all the computations we will perform today and this I wrote as R mod, which is the Grothemic ring of complex algebraic varieties

So free beating group in all varieties up twice amorphism modulo cut and paste relation and then localized at motif of the affine line Which we want to have invertible and we want to have a square root of it and we want to have

Naturally natural denominators 1 minus L to the I inverse for a greater equal 1 and this is a ring where we have Virtual motives of varieties That's where the virtual motives live

Okay, that was the one thing I explained yesterday. And the second thing was if Q is a quiver We can attach to it a category of finite dimensional representations

Which has very nice homological properties one homological property was even something like a saduality Which are only briefly mentioned but what the most important thing was that we have an explicitly computable homological Euler form

Category of representations it is hereditary So of global dimension 1 and the homological Euler form Is something we know Exclusively Maybe I will redefine it today. And then finally we consider the stacks of

of isomorphism classes so the moduli stack of isomorphism classes in this thing is Nothing else then. Well, we have seen a certain affine space

With an action of a nice algebraic group and we can take From the quotient stack and take the disjoint union over all dimension vectors. So this is the quotient stack

this was an affine space and this was a product of general linear groups okay, and Using this I then defined this motivic generating function, which I will define in a in a minute But that's the summary of of what we did yesterday and This is always an irritating moment for me in a lecture series

That I summarized what I did in the previous talk and it's just like five minutes and then I always think like okay I could have done this in five minutes, but it took me an hour. So What was going wrong? I arcade was the explanations missing and Trying to give you a feeling for what all this is. Yeah. Okay, so I hope this this

Whole hour yesterday wasn't useless for you Okay. So anyway now we will come up with a formal definition of the motivic generating series

in the motivic quantum space So it's a it's a generating series some of feels like a zeta function

But very important is also the coefficient and then the ring in which this this generating series actually lives and Okay, so I will define the following Let me So this will be the motivic quantum space and this is the following

Well, this is a form of power. It's like a formal power series ring. Yeah with as many variables as you have word he sees in the quiver and The coefficient ring is our Motivate ring are not and everything is slightly twisted by the quiver structure and this we will see now is

the Complete our mod Algebra Complete because we are looking at form the power series where the base is

T to the D Where D is a dimension vector? So this is this one discrete invariant of a quiver representation the dimension vector encoding the dimensions of the vector space is involved and multiplication and the multiplication

Is defined as follows. So this is something like a quantum version of a formal power series ring we take minus L to the 1 half Antisymmetrized Euler form T to the

D plus E That's our base ring for all the computations which will follow Yeah, so it's formal power series in as many variables as we have vertices in the quivers, okay, and

Instead of writing down TI to the DI Product overall TI to the DI reformally work with such monomials T to the D, which is just short Yes It's a minus it's it's the anti-symmetrization of the Euler form it's the anti-symmetrization. Yes

Exactly exactly the anti-symmetrized Euler form and this we take as as multiplication. So it's like Excuse me and it's associative There's the associative unit algebra. The unit is just T to the zero

And it's complete. Yeah with respect to the while the augmentation ideal is the ideal everything Generated by T to the D for D nonzero and everything is complete with respect to this ideal We have formal power series, which we will use

Haha, can I want you well, I think Yes Yes Well, the the most tricky point is somehow why do you anti-symmetrized the Euler form and

Well, then this anti-symmetrized Euler form is like the honest Euler form of some three-color be our situation So somehow imitating kind of imitating a three-color be our situation with a quiver Which is hereditary of global dimension one. That's somehow some At some of philosophically the explanation but the practical feature is that

Just the formulas are As smooth as possible with this twist Yeah This is Yeah, I call it the motivic quantum space because for quantum torus, I would like to have

Inverses of the T to the I maybe yeah, so that would should be over some Laurent polynomial ring, but motivic quantum space Okay, so and inside this Yeah, so, okay. So very pragmatically the motivation is the formulas are very smooth when you work in this ring

For me personally for many years. I work with a different ring I just worked with a twist just by the Euler form not by the anti-symmetrized Euler form and I always had problems with with the formulas which were not really nice and but this is the way the formulas are easier to remember

Although you always have this ugly little minus square root of the left. That's motive inside this you cannot get rid of Okay, so and now let me finally introduce this motivic generating function and then explain why it's a more Knows the river so a Q is then defined as sum over all dimension vectors and then we take the

Virtual motive of this space of representations divided by the virtual motive of the base change group and form a variable T to the D So this is the motivic quantum space and the serious is the motivic generating series

Instead of an a you could also write is that so there really feels like well, whatever you prefer is it a function or a

partition function It's really some of this kind as we will now see because that's the series we will try to fact

Excuse me the other the virtual motive. Yes. Okay, the original motive was you normalize The motive by an appropriate half power of the left. It's motive for X irreducible

So that for example Poincare duality Manifests itself as invariance under exchanging L and L inverse. This was the example we have seen for projective space Yeah, that was the the twist which is necessary. We will actually use this twist now and make this

motivic generating series more explicit Yes Let's do this That's maybe the Perfect point to formulate a little lemma and you know the proof of this lemma when you have done this exercise

Which I gave you yesterday. The exercise was to compute the motive of GLN of the group GLN. Yeah, and So let's compute this thing here. Let's first Okay

Okay, so you will see after this calculation that the function depends on the symmetrized Euler form and The ring depends on the anti symmetrized Euler form So altogether the datum of this series in this ring recovers the quiver. We will see this in a second

Now the the multiplication in the ring the multiplication in the ring it really depends on the anti symmetrized Euler form Yeah, and this this function only depends on the symmetrized Euler form, which we will now see if I formulate this lemma. Okay, so

The lemma just is an explicit calculation of that So this RD of Q if you recall the definition from yesterday, maybe I should recall it. Why not? This RD of Q is just an affine space. Yeah RD of Q is just direct sum over all the arrows in the quiver of the space of linear maps from a DI dimensional to a DJ dimensional space

So this is just affine space of some dimension and its motive is just a power of the Lefschet's motive The group GD which acts on this is a product over general linear groups

for the vertices So its motive is a product of the motives of general linear groups Which we have calculated yesterday in this exercise Okay, and if you just plug in the result from yesterday's exercise into this then you see that we can reformulate

this AQ just as sum over all D and to the q0 and Then the numerator is minus L 1 half to the power minus the Euler form applied to DD

T to the D and then the Denominator is just a product of Pochhammer symbols where you plug in the Lefschet's motive Or the inverse of the Lefschet's motive so it's just a product over the Pochhammer 1 minus L inverse times times 1 minus L to the

minus DI So this is something like a L hyper geometric series Yeah, like in Q hyper geometric series, except that our quantum parameter is now always the Lefschet's motive

Yeah, so this really feels like a L hyper geometric series Okay, and So this lemma is just really direct calculation

with the with the motives of this affine space and this group and From this we can now draw a remark that The datum of this series and the ring Knows the quiver. So this motivic ring

So this motivic quantum space knows The anti-symmetrized Euler form because it's really in the definition of the multiplication

Okay, the motivic generating series only involves the quadratic form defined by The Euler form it's only about the quadratic form But from the quadratic form you can recover the symmetrized bilinear form by polarization the usual polarization trick

Yeah so a Q knows About the quadratic forms and then after polarizing it knows about the symmetrization So both together know about the whole Euler form

so both together Know about the Euler form of the quiver Because well, we can stand at the decomposed into a symmetric and anti-symmetric part So we know the Euler form and from the Euler form We can directly recover the quiver because the Euler form just encodes the vertices and the arrows. Yeah

Okay So this answer this question from yesterday of how much from of the quiver is seen by this so a Q itself only sees the symmetrization but The important thing is to have it in this ring and and then we really recover the whole Euler form

Okay, so time for first examples namely for examples where you really don't see all this quiver machinery, yes Yes

How do you how do you get this out of it well take the coefficient of T to the D and develop it into a

Formal power series in L inverse then the lowest degree coefficient is is this one and from the exponent? You can read off the the Euler form. They can read off the quadratic form for example. Yeah Yeah, I mean No, this really means well, you can just read it off from the definition

Yeah, so how much of the quiver is contained in the definition of the motivic quantum space and the motivic generating series Okay. Now we have to do two classical examples. There's a question. Yes, please. Is there some difference equation? Yes, yes there is

I once used it in another version, so We have different twists, but yes, yes, you can characterize it by some nice Q difference equation and We will see other Other Q analysis business popping up in a minute for example taking the logarithmic Q derivative

Is something which plays a role here and leads to Hilbert schemes? Yes, we'll see this in a few minutes Yeah, yeah, yeah, so I mean this this is some of the right keyword Yeah So this brings you into Q or L analysis and you can can think about all possible tools from Q or L analysis

Which you might want to want to apply to the series to study it. Yeah Okay, so example But I think I need a larger blackboard for this So in this example, I will do the case of the quiver, which is just a vertex

And a vertex with a single loop Because these correspond to classical things namely. Let us recall The so-called Q binomial theorem and it's a theorem which exists in two versions

luckily for us one version is Let me first write down the classical version so the you take the the following hyper geometric series sum of all D T to the D divided by the

Pohama symbol 1 minus Q times times 1 minus Q to the D and this summation factors into a nice product product I from 0 to infinity 1 over 1 minus Q to the I

T this is usually proved using just partition combinatorics Yeah, so on the left hand side you see generating functions for partitions by by weight and This is one way you can prove this And another nice identity, which also goes under the name of Q binomial theorem is if you have some quadratic form

In The numerator which is Q to the D D minus 1 half T to the D and then again the Pohama and

This is product Over all I from 0 to infinity 1 plus Q to the I T and now if you think well, okay this might generalize maybe I can put some other Q to the D

Squared here in the numerator or not Then number theory people will tell you no That's a completely different story If you just if you don't put Q to the D squared half here in the numerator But Q to the D squared then you are in the realm of Rogers Ramanujan identities

And all sorts of really difficult number theory stuff Yeah, so they are just precisely these two values for the numerators where you have such a nice factorization Okay, so from this it follows quite easily

That we can factor the generating series for the quiver, which is just a single vertex Into I have to read this off because I have to be extremely careful about signs In this whole theory Product over all I from 0 to infinity 1 over 1 minus L to the I plus 1 half

times T This corresponds to the first identity and corresponding to the second identity Which is the quiver with just one loop you have a Q equals

Product over I from 1 to infinity 1 minus L to the I T Okay, so um, yeah a good advice to get into this motivic invariant business is to

Do the calculation? I mean everything's here. So for example Okay, let's let's look at this the second case here for this one loop quiver. The Euler form is actually

identically 0 because you are taking a Contribution D squared for the vertex minus D squared for the loop so the other form is identically 0 so this term vanishes and Then you have just this quotient and then you rearrange bringing So I'm pulling enough powers of L out and then you see you arrive here and it's I

It's really recommended to do the calculation and it's completely elementary But you will see that you have to be extremely careful about all the signs and this is a typical Phenomenon for this theory of motivic invariance that all the signs and these ugly little half powers

really Are really important so you should do this calculation. I mean, it's completely elementary, but one should do this. We will need this later There's one more case of a connected quiver for which you can write down such a product factorization of the motivic generating function in an elementary fashion namely the third case is

This quiver and if you know about quiver representations These three the trivial quiver the one loop and this one. These are the only three

symmetric connected Quivers which are not wild Yeah, because these are this is an a0 tilde. This is an a1 tilde and this is an a0 and All other symmetric connected quivers are wild and it's really a tame wild thing that you can write down the factorization

So there exists an explicit factorization for this Which I don't want to recall now but I will recall the categorification of this maybe in the last talk and Okay, so these are the only elementary examples of these motivic generating functions

Why What does white mean okay, yeah, okay Good point. This is minus

Surprisingly, it's minus. Yeah, although here in this identity you have this distinction between the minus and the plus but here it is minus Both times due to this funny little twist by minus square root of left Yeah Whiteness, okay. Um Let's just check for the time. Okay

a category C a billion finite length Very nice category like the modules over some algebra It's called wild

well To Be precise strongly white, but let me not explain this distinction if you can embed the category of representations of a free algebra in it if There exists an embedding of the category of representations of a free algebra in two variables

Faithfully into C yeah, so it means the classification of isomorphism classes in the category is as least as difficult as classifying representations of the free algebra and

It is known for example that if a is any finite dimensional algebra you can embed its representation category Inside the representation of a free algebra. Yeah, so having solved the classification problem for the free algebra You have solved the classification problem for arbitrary algebras

Yeah, so this is some kind of recursive thing and that's that's what it's supposed white geometrically you could say Representations appear objects appear in arbitrarily high dimensional moduli so equivalently Moduli are of arbitrary

arbitrarily high Dimension well not moduli of all representation. So for example if you if you look at Smooth projective curves and the category of coherent sheaves Then you have two cases which are not wired namely p1 and elliptic curves

Well because for p1 the classification of of coherent sheaves is discrete everything is direct sum of torsion sheaves and and O ends and for elliptic curves you take the atia classification and the moduli are harmless the moduli of

Bundles are only the the elliptic curve itself, basically Yeah, but starting in genus 2 you have moduli spaces of coherent sheaves of arbitrary high dimension. That's called what? yeah, and for example such an explicit embedding of of representations of the free algebra into coherent sheaves on a smooth projective curve that

Was really used by Seshadri in studying the bi-rational geometry of the moduli spaces So this Implicitly really appeared in vector bundle theory Microsoft we have a place in the chapter for a thing category is the dimension representation. So is bounded by one

the dimension of well in decomposable or stable representations the moduli So yeah, okay. Well, you have to be precise about what moduli you consider usually you consider Stable stable objects. Yeah, and if you restrict to stables then for a tame algebra, you only get one-dimensional moduli

Yes, indeed That is the famous. I mean in finite dimensional algebra theory. There's the famous tame wild dichotomy Either your algebra is wild moduli Of arbitrarily high dimension and you can embed the free algebra or you are tame Which means moduli spaces are only one-dimensional maybe with many many different irreducible components, but only one-dimensional

Sorry Haha, okay I will try to find a good reference

Yeah, I mean there are many technicalities around this There is really a distinction of being wild and being strongly wild and what I call wild here is really strongly wild But okay, I'll check for some nice references

Okay Well, it's it's an accident. Actually. That's there's nothing behind this I think so namely What you can do is you can exchange L and L inverse

and in the generating function, yeah, and this changes changes the Euler form to To what To the diagonal form Delta minus the Euler form

Yeah, so usually if you take the Euler form of a quiver and instead compute the diagonal one minus the Euler form You're no longer have the Euler form of a quiver because negative numbers of arrows don't exist The only exception is that you interchange the quiver trivial quiver and the one loop quiver So it's an accident. Yeah

Yeah, so yeah, I was always hoping for some boson fermion principle Well, we will see this boson fermion thing also when we categorify All this in the common logical whole algebras because then we will get free bosons free fermions as common logical whole algebras But it's it's just a coincidence in this very special case. No more questions

Okay Check the time. Yes, okay Wonderful. So let's start with actually working with this generating function a of Q and

Yes, it's it's always very interesting to see for me that Usually you meaning the audience of some lecture series is more interested in the qualitative aspects like this this whiteness thing Whereas I'm really into this quantitative aspects I really want to do some hardcore calculations with this series this what I really like what the audience usually doesn't like so much

So that's always a conflict and we have to navigate through this But I'm always happy if you if you ask lots of questions and I can explain different things So that's actually fine and if I then have to skip some of the dirty calculations then that's fine

Okay Let me give you something which is somewhere halfway between qualitative and quantitative Let me give you Okay. Now this is all about factorization part one Factorization just means I want to do some multiplicative things with with a Q

So for example, I want to factor it into a product this we have seen in this in these two examples Then we can factor this into an infinite product We will later see how to do this in general, but this I first want to motivate we can also do something else This is a form of power series whose constant term is one

The constant term is just for T to the zero and for T to the zero just have a constant term one any form of power series with constant term one is Invertible Yeah, so a Q is not only an element in this

Motivic quantum space. It's even an invertible element so we can look at its inwards. And so here's a theorem and Later when we have seen how to define DT invariance then this theorem is Is exactly on a very formal level the DT PT correspondence?

Although I can't call it DT PT at the moment. So it just says, okay, so let's take this Motivic Which way around let's take this motivic serious not in the variable L, but let's twist the variable L

slightly little bit I Will explain this notation in a second. Let me twist The variable T the form of variable T by a minus L 1 half to the N Where N is a vector and have to explain what this notation is and let me multiply it by

It's inwards with a slightly different twist Minus L 1 half minus NT Inwards, I don't want to write this down as a fraction Because I'm working I'm still working in a non commutative ring, yeah

This motivic ring is still non commutative because we are working with this anti-symmetrized Euler form there Yeah, so I shouldn't write it as a fraction if you calculate this fraction Then you get a motivic generating series for Hilbert schemes

virtual motive of Hilbert scheme of the quiver T To the D. So that's the formula and now I have to explain all the terms And actually I will explain the idea of the proof which is

essentially really simple and the rest is just dirty calculations in this motivic ring Okay, if you know Tom Bridgeland's papers about motivic Hall algebras, this is what he calls the quotient identity

In the motivic Hall algebra, so let me first explain the terms. What does it mean to plug in? Not T, but minus L 1 half to the NT into this

well, try to guess the definition and Your first guess is the correct one f of Minus L 1 half to the N times T

Is defined as Okay Let's assume f of T is a serious with coefficients ad to the D Just some form of power series and then plugging in this This we define as sum over D and then we take the coefficient ad and then we take minus L 1 half

to the N times D T to the D where The series is defined like this, okay Let's see if this makes sense

Our usual variable minus L to the 1 half that's always the standard variable here so assume f of T is a formal power series sum over all D a coefficient ad t to the D and

This this D is always a vector of length q0 and now I want to define what it means to have this This twist in the variable T and This twist in the variable T means you take the coefficient you take T to the D But then you twist to minus L 1 half and times D and and D are both vectors of length q0

And this is just the Neve scalar product some of the product of the entries Yeah, okay. So that's a way to twist a formal power series in many variables Okay, and that's what we do twice In defining the left-hand side and we use the fact that aq is actually invertible in this ring and then we get

The generating series for Hilbert schemes and I have to explain what Hilbert schemes are now think about again think about coherent sheaves on a curve and Think about how you define the moduli spaces of a vector bundles on curves

This you usually do by GIT So first you find a large enough Hilbert scheme Which parameterizes all your semi stable bundles of a fixed rank and degree and then you mod out by some further structure group SL And do the geometric invariant theory. Yeah, and this kind of Hilbert scheme is precisely what we need here

so To define the Hilbert scheme if we have such a dimension vector and we have an associated Projective representation Projective representation of our quiver Q

More precisely this quiver Q has one Projective in the composable representation for each vertex Which is usually called pi and then you just take the direct sum over all pi to the ni and that's what you call Well We're not see these projectives and we don't really need to see them

So but there is a projective representation This will somehow play the role of O of 1 in the vector bundle case and now in the vector bundle case to form to Realize all semi stable representations of fixed rank and slope you present them all as a quotient of some large

On a large power of some o1 or on or whatever and so this is what we'll do here. I define help dn of Q as a set as follows

It is the set of all presentations of a representation of dimension vector V as a quotient of a fixed projective representation up to Well up to a natural notion of equivalence, which you can already guess

Yeah, namely two representations are two such presentations are equivalent if You have a diagram intertwining the two representations of the two presentations Well, that's what you would call a Hilbert scheme also in the coherent sheaf setup

and that's what you call a Hilbert scheme here in the in the quiver setup and This exists as as an algebraic variety using GIT

so you can realize this as a set of stable points in some large vector space and and then you mod out by the structure group and Well now we have such a nice identity so this is a full quadski Yes

Yes, yes the full quadski, yes, yes. Yes. Yeah. We're not taking care of any more numerical invariance like Hilbert's. Yeah Yeah, okay. Well, okay to call it the quadski would somewhat be the better analogy. Yes, that's right Yes, okay. So we're performing a very simple algebraic operation here and

in terms of Q analysis, you would call this a logarithmic Logarithmic L derivative that would be the Sloane well in Q analysis

The logarithmic derivative Would be F of Q times T divided by F of T Yeah, it takes some twisted version of your former series and divided by that. That's someone an analogue of that Okay

Okay, so that's our first identity using this this ring and Actually, it tells us that Well, what is important about this is that the left-hand side is something very formal AQ is defined in terms of

Crocean stacks so they don't have a direct geometric meaning but the right-hand side is something Totally explicit. It's really about motives of actual varieties in particular I Notice this just one To finish the sentence on the left-hand side

You are not a priori allowed to specialize L to 1 we have talked about these motivic measures yesterday one Motivic measure was the virtual Hodge polynomial what we are this is always okay, but to take Euler characteristic requires Specializing L to 1 because the other characteristic of the affine line is 1

And a priori are not allowed to do this if you localize like we do in our motivic ring For localized by terms like 1 minus L to the I so we are usually not allowed to take Euler characteristic and this you can really see from this explicit form of the Motivic general rating series better not specialize L to 1 here

Yeah, because of these denominators but Taking this logarithmic L derivative Someone mysteriously cancels out all the denominators and what you're left with is these Hilbert schemes And that's the honest

Motive of a variety of which you can take the order characteristic and these are very interesting numbers Yeah So the right-hand side left-hand side is completely formal and The right-hand side is something concrete and that's the surprising thing that you're doing very formal things in this motivic ring

but you get out some actual geometry and That we will now see in factorization part 2 in another case, but first Fabian's question Okay, this this this notation which I defined here so this is really the formal definition yeah

So so this this power only makes sense in this formal substitution of the variable T Which is also nonsense. There is no single variable T. Yeah, there are variables TI for the vertices of the quiver but this is defined as twisting the coefficients of the formal series by Minus square root of left chance to n times D where this is the naive scalar product of two-dimension vectors. Yeah

So this is really just a formal definition to arrive at smooth formulas and to make the Analogy to to Q analysis some of it clear Okay so that was this first instance of this

Factorization thing and Let's do a second one So this black one will reappear in a second

So usually when you want to define some moduli spaces you have to choose a notion of stability In this factorization part 1 we were quite lucky that we didn't have to choose a stability. It's somehow implicit in

defining the silver scheme, but So there's no stability involved here and now in factorization part 2 we will involve a stability factorization part 2 and

So the usual modern formulation of stability would be with a central charge function I will Do the old-fashioned slope stability because this may be closer to the to the coherent sheaf Intuition which some of you might have in mind so let me just

Assume we are given a slope function mu from Non-zero dimension vectors to the reals and this should be

Axiomatically should be something which looks like the slope function for for vector bundles. So it's of the form degree divided by rank so mu of D should be theta of D divided by Kappa of D and

This should somehow play the role of degree and rank for vector bundles So it should be linear functions on the group should be linear on dimension vectors So theta and Kappa should be real valued linear functionals on the on the quiver and

Of course, you need some more condition namely this denominator should never be zero for a non-zero dimension vector so we require that Kappa of Nq0 Except zero is contained in the positive reals

Okay, so whatever is formally necessary to define something which behaves like a slope function So that's our abstract abstraction of slope function. And if you just take Theta and Kappa as the real and imaginary part of a complex valued function function, then you have a central charge

Yeah, so if you take theta plus I times Kappa, which is a complex valued function Then this is what is usually called central charge function and this is some of the the modern formulation of stability which usually

Works with central charge, but I will keep this old-fashioned slopes slope function Okay Just as an aside, okay. Now we are given a slope function. We have a notion of semi stability of quiver representations as

usual a Representation V of Q is called stable or semi stable if the slope decreases or

weakly decreases on sub representations if the slope of any sub representation is less than or equal the slope of the representation For all non-zero

Proper Sub representations This is exactly the the notion of semi stability which you have seen from in vector bundles or coherent sheaves in whatever setup

just take the abstraction of Not choosing the slope function a priori and of course Well, if we make the wrong choice for theta and Kappa then this notion might be trivial It might happen that any representation is semi stable or just no representation at all is semi stable

Yeah, so all the subtlety is in choosing these linear functions for a given quiver, that's a completely different story and We'll not not touch upon this and just say We want to prove a result Which does not use any special feature of this of the slope function?

should be just true for for whatever Slope Function you have even if it produces a trivial notion of same stability All right Okay

So finally then inside our D inside the space of all representations You have a semi stable locus this is the set of all representations which are semi stable and because semi stability is a

generosity condition This is a risky open subset No, you're just trying to avoid certain bad dimension vectors of sub representations

You're trying to avoid dimension vectors of sub representations Which do not fulfill this property and avoiding certain sub representations is a generosity condition. So it's open Okay final ingredient is that So now that we have this relative version

of this representation space We can define a relative version of the motivic generating function So if X is any real slope we define the X semi stable motivic generating function now this deserves a

Larger space. Sorry, so if X is a fixed slope

We Can now define a relative or semi stable generating function stable motivic

generating function which is a Q Now X seems stable of T and this is well You can already guess what it is. Namely you take the virtual motive not of this whole representation space

but just of its semi stable part and You divide by the same structure group Okay, but now we have to bring the concrete slope into the business. So we are doing the summation only over those dimension vectors

Where the slope is X so where D is In mu inverse X so where the slope of the dimension vector D is X or 0, all right

and Then the result is what is now called

wall crossing formula and this is somehow the It's the prototype of all wall crossing formulas you You can encounter in this theory and I will tell you what it Formally amounts to namely almost nothing

I Yes, okay, so okay So I should be I should write this in different way equivalent but different because this was not nicely nicely written So I would like to take the sum only over those dimension vectors

Whose slope is X because I want to do everything local in the slope But then I have a problem namely the slope is only defined for dimension vectors, which are non-zero and

I definitely want to have my motivic generating series having constant term one because I want to do Multiplicative things but okay. So let me just formally write one plus Yeah, one plus all the dimension vectors which are non-zero but half this fixed slope So that's maybe a more honest way of writing this down

Okay, and then the result is the wall crossing formula and this is a big advantage of this

Of this motivic ring that this has an extremely smooth formulation Aq is the product over all reals of these local functions, okay and

Ordered it's an order product and it's ordered by a decreasing slope yeah, and Well, you have to think a few minutes about well defining ordered products over the reals

obviously, yeah, but the important thing is that all these are power series on the right hand side all our power series starting with one and Everything is graded by the dimension vector Yeah, and then if you think about it for for a minute You can easily see that such an ordered product over the reals to send in descending order is actually well defined

Yes No, because I allow theta and kappa to be real valued a priori Yeah

Yes, because later on I would like to make a genericity assumption and Sometimes to make things as generic as possible. I want this to be real valued So that for example, the the fiber for a fixed slope is just one one way Yeah, and if I just make these similar

Reals which are independent over the rationals then then that's definitely fine. So I gained a little bit Okay time is almost over but let me just show you the proof of this That's not my terminology Well, okay, so it's okay. Yeah before explaining the proof. Let me let me show you what is the wall crossing

aspect the wall crossing aspect is not this identity, but Well on this fixed category of representations of the quiver, you can look at different notions of stability Yeah, so you could have two different

slope functions Two different slope functions giving you really really different Stables or semi stables, but this formula tells you that this order product is always the same so the order product Over this series

For with respect to mu is the same as this order product With respect to mu prime so instability space in the space of all possible stabilities you have many walls defining the transitions where changing the stability function really changes the set of stable or semi stable objects and it means that

Crossing crossing a wall in stability space might drastically change the class of semi stable representations But there are certain things which are unchanged namely these order products because they just evaluate to the same motivic generating function

so that's where the name wall crossing comes from and Well formally two minutes. Yeah, the proof formally is formally equivalent To the existence of the hardenera simon filtration

Yeah, so every object V Admits a unique So-called hardenera simon filtration, and this is incredibly strong. Yeah, I mean if you look at things like the yorden holder filtration

They are not unique, the sub-factors of the Jordan-Höller filtration, they are unique up to isomorphism and permutation. But here it is, the filtration itself which is unique, and if you define it correctly, if you index the terms in the following filtration by the reals, in a tricky way, then it is even functorial, you get a functorial filtration.

There exists a unique HN filtration, v equals 0 equals v0 in v1, in vs equals v. It's defined by two properties, namely, first property, all sub-factors are semi-stable

for your fixed slope function, for some slope, and second property, the slopes are decreasing.

So this already appears in work of Harter and Narasimhan on calculating the number of points of moduli spaces of vector models on curves. And it's really an incredibly rigid structure on the category because it's a filtration which you can functorially attach to representation.

And this basically proves this moduli identity, namely these decreasing slopes you see here in this decreasing order of the product. And everything you have to do is you have to stratify your representation space.

So your representation space R d is a union of strata of a fixed Harter and Narasimhan type.

Where Harter and Narasimhan type just means you record the dimension vectors of the sub-corrosions. This is a finite filtration, a finite stratification by locally closed. So you get a corresponding identity in the Gottingde Kringer varieties.

The motive of this is just the sum of the motives of these. And these are, well, they are isomorphic to base change from the group GD to a certain parabolic group. And then you have a vector bundle over a product of semi-stable loci for smaller dimension vectors.

Vector bundle over a product of semi-stable loci for certain dimension vectors, smaller dimension vectors.

I don't want to be too technical about the notation here, but that's the principle. You stratify this affine space into locally closed things. Each of these locally closed things is a vector bundle over the product of semi-stable loci up to some change of group.

You induce from a parabolic group to our base change group. And this is a very simple geometric fact which is a consequence of the existence of the Harter and Narasimhan filtration. And writing down what this means in the Gottingde Kringer variety immediately gives you this product identity. So that's basically the whole proof.

And, well, the most important feature is buried in here, the fact that this is a vector bundle that's equivalent to the category which you're considering to be hereditary. So this is more or less equivalent to the global dimension of your category of representations to be one.

So that's the limitation of this. The global dimension of this category of representations is one. The dimension of rep Cq equals one. So here it means global dimension.

Yes, it means global dimension of this one. Exactly. Okay, so that's basically enough for today. And so next time we will start with this identity and explore these local contributions, these functions, and relate them to moduli spaces of semi-stable representations.

And then finally define the DT invariance. Okay, thank you very much. Are there any questions? Yes, yes. Of course, it is a standard Kringer scheme for the plane.

Of course it is a Kringer notation, but it is a Kringer. Zeta is just ideal for giving the dimension. Yeah, yeah. So that's not related to a quiver, but to a quiver with relations. Okay, so you can consider this Hilbert scheme which I introduced for the quiver with one vertex and two loops.

And the dimension is d and this extra datum m is just one. So this parameterizes left ideals of finite codimension in the free algebra in two variables.

Of codimension of... exactly, thank you. Of codimension d in the free algebra.

And now inside this you can consider those points. I mean, formally this is represented as equivalence classes of two linear operators a and b and a cyclic vector v.

v is cyclic. And then you can consider the closed subset which is defined by the equation a b equals b a, commuting linear operators. And that shows that inside here as a closed subset you find the usual Hilbert scheme of points in the plane.

This has actually much simpler structure than this one. You made it the same as your second definition?

Really quick? Ah, okay. The reason why this is the same as my definition is, well, the free algebra

is one of these algebras which has the property that any projective representation is already free. So as projective representation we just take the free algebra itself and then let's have a look at what is the surjection from the free algebra to v.

Well, let's take the image of one. And the image of one is then a cyclic vector and x and y are mapped to two linear operators a and b such that v is a cyclic vector for this. And that's the connection to this other way of writing the Hilbert scheme.

There are some questions online. Yes. Can you please give a bit more detail on what the hard analysis is defined to be? Hard analysis and type. Type, yes, okay. Yeah, okay. For time reasons I tried to avoid this. Thanks for the question. And now I can give you the definition.

Okay. Okay, so I'll just elaborate on this thing here and do the precise notation. So, okay, so the union you take is over all decompositions of your

dimension vector into non-zero dimension vectors for which the slopes are strictly decreasing. Yeah, because that's the one condition which appears here. And then we define R dq, let me just say H n d dot.

d dot is this decomposition. And H n d dot, that's the set of all representations v such that the dimension vector of the ith sub quotient in the hard and narrow cement filtration is precisely d i for i from 1 to s.

Yeah, and this is the ith hard and narrow cement sub factor, sub quotient. Okay, so from the hard and narrow cement filtration I just record the dimension vectors of the sub quotients.

They have necessarily have to fulfill this condition. And this is in fact locally closed subset. Yeah, so existence of such a filtration, of such an H n filtration, that's in fact a locally closed condition. That's not difficult to see.

And then what you do is, you have a, well, okay, and then you identify this with an associated fibre bundle. Namely inside the group Gd you find a parabolic subgroup corresponding to this d dot with respect to this decomposition.

That's like the standard parabolic you find in a GL for any decomposition of your dimension. And then you have a certain vector bundle of known rank, so the rank is easily computable but if I write it down I make a sign mistake.

Of known rank over the product i from 1 to s of the semi stable locus in the representation variety of i. And the basic idea is that if you have something in here then to the representation v you attach the tuple of the hard and narrow cement sub quotients.

That's the basic idea or that's what you want to do. But you can't do this literally on this level, you can do this literally on the stack level, literally but only tautologically.

So you cannot attach to the representation its sub-corrosions because the sub-corrosions do not have a preferred base, linear basis. And that's why you have to perform some base change here, but that's essentially what's going on. And from this you just compute the motive. Motive of rd is the sum over all these.

Motive of gd divided by motive of the parabolic times the motive of this product times a power of the Lefschet's motive for the vector bundle. And if you just write this down and simplify everything then you get to this product formula.

Another question. Is there a connection between the variation of stability x and the monodromy of Aqx? The monodromy of Aqx. Sounds great. I have no idea what this could mean, but sounds great.

Would be wonderful if it's true. So if the one who asked this question could give me some details of what this monodromy could be. He's an anonymous participant. Okay, sorry. So if this anonymous participant can hear me now, if you could just send me an email, anonymous if you want, of what this monodromy could be, I would be happy to think about it because it sounds like a great suggestion.

There is one more. Could you explain how to obtain the answer for Aq for the one point quiver without an arrow directly from the definition of rdq? Is an rdq trivial to the quiver?

Yes, it is. Let's do this calculation. I was hoping for something like this for the question and answer session this afternoon, but if we have time I can do this calculation now. Then maybe it's also a good idea to do this. Yeah? Let's do one or two of these tricky calculations using all these signs and half powers of l in the Q&A session this afternoon.

I have myself one question. Are you going to prove the first factorization? No, if you're interested I can try to squeeze it in. Yeah? Fine. Okay, can do it. I have a philosophical question. So can you do this for the stack of quiver sheets on the MOOC protected card instead of the quiver?

Maybe define the generated series and the… No, there are tons of convergence issues. I mean, just in defining the moduli space you have all these problems of boundedness.

Because the moduli stack is not a finite type. Exactly, because it's not a finite type. All these boundedness problems which you have will appear here. And things which you write down are usually not convergent because things are not a finite type.

So it has to be very careful and just first approximate it by restricting to bounded values of the slopes and then study the convergence for a slope going to plus minus infinity. But it has to be very careful.

Any other questions? Yeah, just like about the first formula. So the second formula, geometrically, means that you have to start a schema specification. And does the first one have some geometric meaning? Yeah, okay. That's also more or less your question. I don't know if I should do it now or…

Do you think you can squeeze this in the question? I will do it in the Q&A session. I will give the proof for this and the calculation for the trivial quiver and maybe some other calculation. We will do this, yes. Thank you very much.