 Thank you very much. So due to practical reasons, we have a little delay and my talk will be a bit shorter because I want to definitely stop in time for Richard's talk, which is still scheduled for three to four and So let's directly dive into what we did yesterday. So again a very short summary We first introduced this MOTIVIC generating series in the MOTIVIC quantum space This was this ring with a slightly twisted multiplication MOTIVIC Generating series for quiver Q and then we saw two different kinds of factorizations factorization one was if you, well, okay, we also computed examples if you take a logarithmic Q derivative or Or precisely a logarithmic and a derivative of a Q Then you get a generating function for actual modular spaces namely the Hilbert schemes of representations of the quiver log L derivative of a Q gives Generating series of I will not make this precise again because we'll take more Look closer at the second factorization We got a generating series of virtual motifs of Hilbert schemes and actually in the question and answer session Yesterday afternoon, I almost gave the complete proof of this. So then we saw the factorization 2 and So today I will mainly discuss factorization 3 factorization 2 was the wall crossing formula and I will repeat the definition wall crossing formula a Q Admits a factorization as an ordered product over the reels in descending order of local series a Qx sst that's the wall crossing formula and This local Motific generating series that's defined as the global Motific generating series, but just looking at semi stable representations of a fixed slope So let me repeat this a Qx sst is defined as 1 plus some overall dimension vectors of slope acts and Then we take the Semi stable locus inside the representation variety and We still act with the structure group. We take the quotient of virtual emotives and t to the d So but this only involves dimension vectors of a fixed slope. That's important and So that means in the space of all dimension vectors, which is a space of dimension Cardinality of Q zero the number of vertices you have a co-dimensional a co-dimension one subspace Because that the slope of D is a fixed real X That's just one linear condition. So I will co-dimension one subspace All right, so and everything here depends on the choice of a stability function or slope function We had a slope function mu of D defined as theta of D by kappa of D where theta and kappa are Just linear functions real valued linear functions on the space of all dimension vectors And there's one question of what the space of stability structures Define this term in these terms looks like This is something I cannot really discuss in a nice way today But I will keep it in mind for tomorrow but I'm afraid since I want to concentrate on Factorization property 3 and DT in variance today. I cannot discuss this this whole stability space okay, so that that was the that was the second result and I also Almost gave you all the details of the proof namely This is formally equivalent to the existence of this unique filtration this harder narrow cement filtration of of rips tations Okay, so that was a summary of yesterday and now to somehow Motivate the DT in variance. Let me make one remark In this factorization one the surprising thing was that we were doing something very formal with the motoric generating function Then we were just taking a formal logarithmic derivative And out of this very procedure We got something very concrete and geometric namely the motives of Hilbert schemes And the same happens here namely if you look at these local series. They also contain geometric information so fact is and This is some other than the motivation for defining DT in variance, which we'll do in the next 10 minutes fact is that if You have a dimension vector, which is co-prime if D is Co-prime for this slope function, I will tell you what it is new co-prime that is for all proper non-zero Subdimension vector The slope is different Yes, yes, yes, so this means II less than equal DI for all I but they're not equal So component-wise, yeah so for all smaller dimension vectors the slope is different and If you work this out for example for two vertex quiver Then this is really like a co-prime like the co-primality Which you know from the theory of modular spaces of vector bundles Yeah, the co-prime case there means rank and degree are co-prime and this is something similar here It basically means that more or less well For a generic choice of of mu this co-primality means that the entries of D are co-prime, so the GCD of all the entries is one It's not a proper multiple of anything else if this holds this co-primality then The following holds The t to the d coefficient in this series Then the t to the d coefficient in In this local series for slope equals The motive of an actual variety more or less So the motive of an actual modular space Okay, and now I have to decorate this this motive a little bit to to make this Exact so first of all we're taking the Motive of the modular space of mu stable representations of q of dimension vector D We're taking the virtual motive and we have to decorate this by one of these annoying little factors One over L to the minus one-half minus L to the one-half Okay, where now I will explain what this is where this is defined as Taking the the semi stable locus and Taking a geometric quotient by the group action So geometric quotient, so this is now a moduli Space no longer a modular stack space. Okay So that's the precise formulation of this fact and now let me Discuss it a little bit. Yes, please Stable or yes, okay, so let me put brackets around here because it doesn't matter. That's that's the first point Which I would like to explain It actually doesn't matter Okay, so I defined Stability and semi stability semi stability means the slope is weakly decreasing on sub-representations and Stability means the slope is strictly decreasing on sub-representations Now if we have a dimension vector with this special property then Asking for the slope to decrease or to weakly decrease is equivalent Yeah, because there is no proper sub-representation which can have the same slope are wonderful This means that stable and semi stable locus are exactly the same and Now this has a wonderful consequence namely if you look if you consider geometric invariant theory It tells you that for the semi stable locus. You always have a very nice quotient and For the stable quotient for for the stable locus. This quotient is in fact geometric Yeah, so a geometric quotient here exists. It doesn't matter whether we write semi stable or stable It's the same and the geometric quotient exists The group action of GD on here is not free. There is always the group of scalars So scalars diagonally embedded which acts trivially Because if you remember the group action on this space of quiver representations this group action was given by some kind of conjugation and Conjugation is of course trivial if you take scalars. Yeah, so the scalars act trivially so what you are honestly taking here is a quotient by the projectivized group But I haven't introduced this so let me just Switch back to GD but I mean this geometric quotient is a PGD principle bundle and Taking this projectivization amounts to factoring a multiplicative group of the field out of out of this group and So this ugly little factor here. That's just the virtual motive of the multiplicative group of the field Yeah, so this explains this factor. We always have we have an annoying annoying little factor coming from virtual motive of the multiplicative group of the field Because we don't have a free action of the group GD Okay, but anyway in This very special situation if you look at this the quotient of the motives or say the motive of the stack is then more or less the motive of this quotient space and That's yeah, so this is some of the it's the lowest order term in this series Aha, this is great news and really formally compares to factorization number one Doing a certain formal manipulation with this generating series Suddenly an honest geometric information pops up the actual motive of a space here It was motive of Hilbert schemes and here we get as the lowest order terms of these series We get more or less the motives of modular spaces parameterizing stables But only under this special co-primality assumption Which is not too bad Well, we know this phenomenon from modular spaces of vector bundles The good theorems are always about the co-prime case where rank and degree of the bottom bundle are co-prime and in general You get singular modular spaces things are much more complicated But anyway, we are here in the quiver situation Which is supposed to be much simpler than modular spaces of vector bundles because well at the end of the day We're just doing linear algebra So we can be very brave and ask What is the meaning of the other coefficients in here? Yeah, so the lowest order coefficients give the motive So this is e to the d coefficient. That's the lowest order lowest order coefficients in This local series. So what about the other coefficients? Okay, and now comes the extremely brave Idea of taking this local series under certain Technical hypothesis and just brute force factoring it into an infinite product And then the exponents appearing in this brute force factorization should have a meaning. These are the dt invariants Okay now we factor Now we factor this local series Yeah, so in the first step we are taking the factorization of the of the global series into these Slope local series and now we want to factor this series to efficiently do this factorization We need a bit of notation because otherwise we will have Lots and lots of a very ugly three-fold Infinite products and we just want to get rid of them in the from the notation and that's why we introduce the so-called plastic exponential So to simplify So this is a very you a very small Notational intermission before we come back to the geometry To simplify a huge infinite product We make the following Definition of the so-called plastic exponential Well from the term exponential you can already guess it's something What what is an exponential an exponential is is a transformation with maps sums to products Now the exponential is more or less defined by the functional equations We want to do this for such formal series formal power series It should be it could should convert sums to products and Then the very simple-minded ideal is to convert a monomial to a geometric series Yeah, so let me give you the axioms for defining the exp the plastic exponential if you apply exp to a Monomial and a monomial for us would be something like L to the i half t to the d yeah where d is a dimension vector and I is an integer These are our monomials in our localized Motivic ring motivate quantum space and I define the exponent the plastic exponential of this as the general as the geometric series okay, and The other thing is it should convert sums into products X of X plus G should be X of F Times X of G like you expect from an exponential So it's really like an exponential, but the initial condition is transform monomials into geometric series and now this is a wonderful tool for making huge infinite Yes, they should commute actually so we will only do this under a certain technical assumption which I will Introduce in the next definition. Yes So this defines continuous Group that we reprecise this defines a continuous group homomorphism from Well now I have to be really careful so I Because I don't know what what motives I have in this good degree of varieties I will just take the suffering generated by L and properly localized now So this is a suffering of our motivic ring and then I adjoin all these T to the D and I'm considering the maximal ideal in there where of all series with constant Term zero the set of all F in here such that F of zero is zero so which is just the maximal ideal and I'm a bit to the set of all F in here where the constant term is one Because all the constant terms of these geometric series are one Yeah, so I take power series with constant term zero and I map it map them to Power series with constant term one and it is a group homomorphism where here you take the additive structure and Here you take the multiplicative structure, which is expressed in this Functional equation. Yeah, and it's continuous because well we want to continue this to infinite sums To all the infinite sums which are we are allowed to take in this formal power series Can you extend the predictive exponential your whole motivic ring? Ha ha oh There is a there's lots of work on this Yes, yes, basically you can do this and you can get these Well, you can get Okay, so the plastic exponential is defined whenever you have a lambda ring and so you have to find the right lambda ring structure on this motivic ring and So you have to define these atoms operations and they are defined using taking symmetric powers of varieties So there's there's lots of work on on doing this in in the generality and Luckily for us. We only have to work with everything which is motivated by the lashes motive Yeah, so we can do this this simple thing here Okay, and I'm not absolutely precise here. So this defines a continuous group of office in case That this ring F of not f of one. Thank you very much f of zero is one. Yes No, I better don't plug in one in these series in case that this ring is commutative Actually, we will now work in sub rings which are commutative Okay, so that's the general. Oh, yes, sorry So we take L, but you said there Define X also for L to the I over two. So do we Yeah, yeah, yeah To to everything to everything which is generated by L in here L and and all the localizations we already have Okay Yes, well the problem is doing this doing this formally somehow Disguises the very simple nature of this construction. Yeah, so Let's better do an example Yeah, namely it's let's come back to to the motivic Generating series of the trivial quiver without any arrows which we computed yesterday as An exercise and let's use this Plastic exponential terminology to simplify this so we compute that this yesterday as product over I from zero to infinity One over one minus L to the I plus one half t Okay So now we want to rewrite this as a plastic exponential of something and now the rule is really simple this product Converts into a sum that's what X is good for and here you see a geometric series and this transforms into a monomial and that's it Yeah, it's precisely these two axioms You transform form this product into a sum and you transform the geometric series into a simple monomial But now you see that you can simplify this infinite sum here inside because it gives you again a geometric series So this you can rewrite as X of Okay, so I have the sum over all I L to the I and then a constant part so it's L to one half times t divided by one minus L and for good reasons I somehow normalize this and Multiply numerator and denominator by L to the minus one half to finally arrive at L to the minus one half minus L to the one half and Out of a sudden we see this denominator here, which some are popped up naturally in the geometry above there This is just dividing something by the virtual motive of the multiplicative group. Aha uh-huh, so this is the first indication that this X is something reasonable and Now we have seen that this summer magically pops up here in Factoring the geometric the motivic generating series for the trivial quiver And it also appears in these lowest order terms of these of these local motivic generating series now Let's unify everything Into one central definition and Yes Formal definition so assume so have a quiver and We also fix the stability as before because we want to consider these local series new stability X is a fixed slope and Now we make the assumption such that the Euler form the Euler form of the quiver when restricted To all dimension vectors of the sixth slope This is the co-dimension one space of dimension vectors. This should be symmetric So this is this is the important technical assumption Yeah, so the Euler form when restricted to this co-dimension one space of dimension vectors of a fixed slope should be symmetric Because this implies that a certain part of the motivic ring Namely the part of the motivic ring where we only consider monomials of the slope is Commutative yeah, because the twist in this motivic ring was defined using the anti-symmetrized Euler form Let's note this so that implies the part of the motivic ring X X Which is the span of all t to the d? Where d is of slope x is Commutative that's the reason why we make this assumption. Yeah, so a certain local part of this form of power series ring is commutative and Then we define certain rational functions dTd mu of q So it depends on the dimension vector of slope x it depends on the stability and on the quiver and a priori this is just a Rational function in the left sets in the half left sets motive So a priori it's only a rational function in the half left sets motive and this you define by Factoring the local series. So you just take this local series and You factor it into your product means you write it as an exponential of the following form Well first a standard term l to the minus one half minus l to the one half This now doesn't come as a surprise and then the sum over all dimension vectors of slope x dTd mu of Of q while of this of q. I don't like So Quiver is not a variable the variable is still the left as motive somehow Okay Okay, so now that's the central definition which we will explore for the rest of the talk and let's Try to digest this and let's see what is the logic of this definition Okay, so first of all, okay, so we take our more typical generating function Yeah question about notation No, it's about the T to the D. I mean we want do you also include the Mutual element. I mean like what's what D Um Yeah, I guess so yes. Yeah. Yeah, okay I Just one efficient notation Take I don't have any idea. So let me just formally Allow zero it somehow work better with the definition of these local series where I just put this one plus in front Okay so Our logic is using the hard on our sim infiltration We can factor the the whole motivic generating series into local contributions from the slopes The proof of concept is that at least the lowest order terms of this series Have a geometric meaning because they encode the motive of actual modulite spaces Aha, so it's what it was a good idea to do this factorization Now we want more we want to factor this whole thing and the universal tool for writing down such Factorizations into huge products is the pletistic exponential To have this place a pletistic exponential well defined We need some commutativity and this commutativity this local commutativity is Contained here the definition that the restriction to the Euler form To a fixed slope is symmetric and then we just take this series and factor it Into an infinite product and a priori the guys appearing there Could have all sorts of crazy denominators So let let's just say it's irrational functions in the half left its motive to be on the safe side And here we always have a standard term which doesn't come as a surprise because we already have it When we just factor do the factorization for the trivial For the trivial quiver, so that's a factor which we cannot avoid First of all and it is also reasonable to expect this factor from the geometry Which we have seen here So this hopefully motivates this very brave definition of motility in variants of a quiver and Okay, so now we have something to explore Yes, please Where does one hide the factorial The factorial because exponentially, you know you have over and yeah, okay See I didn't want to talk about lambda rings. So I always Prefer this this ad hoc definition of the plus this plastic exponential by saying it has the function of the equation of an Exponential and this initial condition monomial goes to geometric series When you define this in a general lambda ring you define It as follows So you can define the plastic exponential as the usual exponential Composed with psi which is the generating series of all atoms operations So psi is the sum over all I one over I and then psi I in a lambda ring with Adams operations Psi I So in a in lambda ring you have this these these lambda I operations and this gives the whole ring a structure of a module over our Symmetric functions and the lambda I correspond to the elementary symmetric functions and then you do the base change to the Power series to the power some functions and these are the elements operations So it's all this formal lamp during stuff. Yeah, and so this is the way you can do this in Arbitrary lambda rings and there you have the honest exponential Okay, which also easily proves so in this way you can easily prove that you have an inverse a Plastic logarithm. So lock is then just psi inverse Composed with the with the usual lock from a power series lock where psi inverse Arises from this by Mirbius inversion Mirbius function of I by I times Adams operations psi I Thanks for the question because we will come back to the Mirbius inversion anyway in a minute. Okay, so Well, okay, so at length I try to explain the logic of why we define these DT invariance and Now let's see some examples And to compute any single example is quite hard, but at least we Yes, please Yes, you are right. Okay, I forgot to tell you something I forgot to tell you that Yes, I Forgot to tell you this that this Not only defines a continuous group homomorphism, but actually an isomorphism of groups because the exp has an inverse lock and And this particular tells you you can factor any series with constant term one in this way Sorry for that. That's of course the important point Yeah, so exp has an inverse lock so you can just apply lock here and this defines the DT invariance So what kind of what if examples do we have? Aha, so one example is on the blackboard Namely the trivial quiver Yeah, then we have this factorization and so for the career a trivial quiver We find that the DT one we don't need any stability We can just take the trivial stability here that the DT one is one and all other DT I zero DT D is zero for all D Certainly greater than one. Okay. That's the first example. The second example is The one loop quiver We can also do the one loop quiver because we have seen yesterday How to factor the motivic generating function for the one loop quiver into an infinite product now take this infinite product from yesterday Reinterpret it as as an exp and then you can read off the DT invariance for this namely the DT in dimension one is the virtual emotive of the affine line and All other DT are zero now This is not too bad. This already tells us something about Gives us a hint of the DT invariant being of geometric origin Yeah, because well, what is the classification of quiver representations for this quiver? That's the classification of vector spaces every vector space is a direct sum of a unique one-dimensional space Which explains the invariant one in dimension one For the loop quiver. We are looking at matrices up to conjugation think John canonical form and we have a discreet We have a continuous Classification parameter involved namely eigenvalues and this is the a1 encoding eigenvalues So this is the modulate space for possible for possible eigenvalue of a matrix aha so third example Which is also on the blackboard is Where we have seen these lowest order terms Here in the co-prime case and This is also something we should notate so third example you Is now arbitrary and We need this assumption that D is co-prime D co-prime for the chosen stability and then the DT invariant is As we have seen Yeah, the motive of the virtual motive of the stable modulate space MD new ST of Q virtual mode motive Yeah, because this is precisely the lowest order term in this local Generating series and if you then rewrite this as a as an infinite product then these lowest order terms survive anyway Yeah, so that's not difficult to see that this is true. Aha. So in all cases apparently the DT invariant carries some geometric information and I hope this prepares us all for the Final theorem yes Final theorem for today and The final theorem is that the DT invariant in fact is always of Geometric origin. It's not the motive of of an actual modulate space at least not for mathematicians for physicists it is Because physicists believe in a certain Modulate space of quiver representations with which mathematicians can't define but okay, so Theorem is DT is Geometric that's the Sloan more precisely under all the assumptions under all the assumptions which we need to define We have the following DT D mu equals Okay, so what's the geometry the geometry is the modulate space of See me stable And now for a general D. We have to be careful in the co-prime case I convinced you that stable and semi-stable is the same anyway, so we don't have to care But in general we have to take care and we are taking the modulate space of semi-stables that is in general So this is typically a Very singular modulate space Yeah, because it's really a gIT quotient, but you have different types of stabilizers and so in general you're only I mean It's you know, well, you have severe singularities, not just all before singularities. These are really severe singularities vertices of cones for example At least normal singularities, but that's it so it's a singular space and Well, you can already guess that taking the motive of a singular space It's not so well behaved like motive of something smooth projective. So what can we do better? Well, there is a coromology theory, which is perfectly well suited for singular varieties, which is intersection coromology Yeah, and so the final result is you take Compactly supported intersection coromology with rational coefficients of this and We take the Bronca-Rae polynomial of all these And where our polynomial summation parameter is of course again our our minus square root of left sheds So take the Bronca-Rae polynomial of local compactly supported intersection coromology Better take a dimension Take the sum over all i and that's almost it except for a little twist factor Well, you guess what? So again, you have to twist by minus square root of left sheds to Euler form of the of minus one so this holds if if there is at least one stable point and If there is no stable point this might happen you might have only properly see we stable points Then the dt invariant refuses to exist. So it's just zero Okay, so this is the precise result that the dt invariant is geometric and If it's it's non-zero non-empty So if there is a stable point, yeah, then this is the formula if there's no stable point then it's zero and So this is the interpretation of Sven-Mainat and myself of something which was conjectured or believed in in physics by Manschalt-Piolin and Zen for example, namely they say the dt invariant is the actual motive of some modular space so physically We really want to have something like the dt invariant is the virtual motive of some Modular space and let me call it the physics modular space, but nobody can define this mathematically Yeah But this is generally believed to be true that the dt invariant is really an actual motive of some space Which we just can't define mathematically and the replacement for this So this is just a vague dream. It's not a theorem And the precise mathematics statement is that it is the Poincare polynomial in intersection homology If we can find a small resolution Once you have a small resolution you are done because the homology of a small resolution is the intersection homology of a variety and if we are lucky that the whole homology is just puretate then Poincare polynomial is the same as as the virtual motive, but I mean think about modular spaces of vector bundles on curves only in only in rank 2 There are known Dissingularizations of the singular modular spaces. Yeah, so rank 2 degree even then you know a Dissingularization of the modular space with only all default singularities There is this general procedure of Francis Carvin to Dissingularize singular modular spaces, but the bookkeeping is terrible and Nobody knows what the final outcome would be. So this is unsolved I'm not sure that the small resolution exists. I was looking for this for years, but It's not very promising We have a question in the Q&A. Yes, do you always need to assume that the other form is a magic? Yes Yeah, yeah, so the assumption which I used to actually define the dt invariance this continues to hold Otherwise dt is not not even defined. I mean you could define it, but it's just nonsense. It's just not interesting Yeah, so still under this under this assumption that you have this local symmetry symmetry Yes No, no, that's only a posteriori So this this theorem is by the way proof of the so-called integrity conjecture for dt invariance that they are Really polynomial in the left sets oil in a half left sets motive Yeah We have another question in the last theorem Is it easy to see an example where the dt invariant is not the class of a quotient stack of the semi-stable locus by GOD Yes, I think so so I Yeah, it's a bit more time. I mean lots of interesting examples already happen so very interesting class of examples is if you just take Quiver, which is a bunch of loops say m loops where m is at least two and Then I would say even for dimension 3 One can easily compute the dt invariant Just from its definition but Nobody knows a small desingularization of the modular space So, I mean even dimension 3 pairs of matrices. I Would say this is unknown then if it's if it is the actual motive of something Oh, by the way, this is this is a good example anyway So now that we know that the dt invariant behaves polynomially in the half left sets motive we can Specialize L at one This wasn't clear a priori because I priori it was only a rational function in L We can specialize it to L equals one and then what we get is the so-called numerical dt invariant and at least for loop quivers There is a closed formula I think it's open almost the only case where you have a closed formula for these dt invariants numerical dt invariants And let me show you the formula. So this is the m loop quiver and The dt invariant We take the specialization of L to 1 and this is then given by Now I'm not sure about the convention. So let me say plus or minus because I'm always mixing up the science It's a Mobius inversion. It's 1 over d squared some over all Devices of D Mobius function of D by e an ugly sign minus 1 to the m minus 1 Times d minus e and then a huge binomial coefficient m times e minus 1 over e minus 1 so that's a sample formula how dt invariants look like and This is almost the only case where you have such an explicit formula that two or three more and Well, you see first of all that the dt invariant somehow the numerical dt invariants grow pretty fast grow exponentially because Well, if you grow in D, then you have this a leading term like an binomial coefficient md over D. She's pretty large In general, it's very complicated because you have this Mobius inversion and I don't want to give you as an exercise That this sum is a priori divisible by d squared Which I claim here. Yeah, so I claim that this is integral So in particular this this Mobius inversion over binomial coefficients should be divisible by D squared. This is terrible. This is I mean It is an integer. It is and it's actually It's actually an integer No, I mean it's well my proof of that is number theoretic Yes, if you look here at the at this geometry Geometricity statement you just take the Euler characteristic and compactly support intersection so it's really just an Euler characteristic of some I See intersection homology Euler characteristic of of a modular space Yes, actually actually there is a general purity result so actually Only even intersection homology spaces are can be non-zero Yeah, I haven't claimed this here, but there's some purity going on Yeah, so this is how they typically look and then finally to finish the proof. Let me give you three pictures Relating everything to the huge topic of scattering diagrams. Oh Oh, yes, please There are a few other places where such a formula appears I Have seen one or two references where where this appears this also appears in some setups as certain Gopakumar Waffa invariance which looks similar I have to check these sources again But yeah, I mean expressions like this such a murbius inversion over binomial coefficients appears Somewhere here and there and the universal source somehow is always DT invariance for the multiple loop river But this is really strange. Nobody really knows why this happens Just a final time for a final picture. Is that okay? Which relates everything to? scattering diagrams So I just want to show you what happens for Quiver with two vertices so final example Q is the quiver with two vertices which are connected by M arrows and There's a distinction whether M is one two or at least three and we need a stability function So the stability function is the slope of D1 D2 Is defined as well D1 minus D2 by D1 plus D2 and then you can check that locally The order of form is symmetric and then I will show you a picture of The support of the DT invariance. So the set of all dimension vectors Where the DT invariant is non-zero Not the actual value just where it is non-zero and we have to make a distinction whether M equals one M equals two or M equals greater equal three and for M equals one you have precisely three DT invariance appearing for dimension vectors one zero one one and zero one Which is accidentally just the positive roots for an A2 root system For M equals two you have two infinite series and a special case here Here the DT invariant is two the numerical DT invariant is two by the way So here it is always one and here it is the DT invariant is the virtual motive of P1 But you don't have the DT invariant for any multiple. So this is basically like an affine A1 tilde root system positive roots where you have the real positive roots and here is the imaginary root But you don't take its multiples And for M equals three you have the lattice points of a certain of a certain hyperbola and Then you have a region a whole cone, which is completely dense So all the DT invariance are Non-zero and they actually explode the numerical DT invariance grow exponentially in this whole cone and that's with that corresponds to the Hyperbolic rank two root systems somehow. Yeah, so that's an intricate relation to two root systems this has to do with the with Katz's theory of quiver ribs and of in the composable quiver representations and These are lattice points of some hyperbola And if you just take the rays through all these points Then you recover certain scattering diagrams which appear in all sorts of grammar fit and theory in the tropical vertex and cluster theory and Wells Okay Well, okay, that's enough for today. Thank you very much Yeah, but well the the connection to to infinite root systems is weaker in general Definitely weaker this rank to this this two vertex quiver is exceptional for that But I mean that there is still there is still Okay, so in general For a general quiver Q, let me just remark that if the DT invariant is non-zero then You require that the the quadratic form associated to the Euler form has a value less than or equal one So I have a root for the corresponding infinite root system But you don't have a converse Yeah, the the the reason why this is not sufficient is that you have to make this choice of stability and Okay, there is a partial converse. Yes. Oh This is yes, okay Thanks for the question because I think this is actually not written anywhere, but there is a partial converse namely if DD is less than equal one and the D is in so-called shurian root and and If you make a sufficiently generic choice of the stability So you avoid finally many hyperplanes and stability space then you can conclude that That the DT invariant is non-zero Yes, if DT is non-zero then D is definitely Now, you know, this is only for hyperbolic root systems. Yeah. Yeah. Okay. Yeah Yes, yes, even even the root. Okay. Yeah. Yeah Yes, okay, so we have to ask Maxim Okay Well DT invariants are the mathematical precision for for BPS state counts and There are string theorists who to any quiver associate some Kind of a of a string theory for which you can do reasonable BPS state counts But We we have a question in the Q&A is it possible to decorate a x-q by chain classes of bundles of Okay, the semi-stable locus are the SSD. Yeah Okay, so yeah on these modular spaces of semi-stables Or in case stability and see me stability coincide you have tautological bundles and you have some talk. Yeah, they have the joint classes of these tautological bundles and You want to Okay, I don't know So the general answer would be there are of course many many possibilities of somehow Decorating this whole theory by taking another base ring then our mod so not working just with with this Pretty course information of the motive of the variety, but working in some Bring some good to bring some some case theory into the game using shown classes or using classes of bundles or sheets That's lots of things to do. I don't know Yeah, but I would okay. Yeah, you could think about reasonable replacements for for the space ring our mod Where you can do all this? Any further question? Is there any relation with these DT variants and the virtual motive of Hiebert Hiebert scheme that you describe? Yes, okay, so that's the the DTPT story which I Which I have briefly mentioned. Yes, okay, so what you can do is Huh So you can figure it out as an easy exercise. I can also give you a reference, but Okay, so we have this we have factored this local series they factorize this as the x of One over L2 minus one half minus L2 one half sum over all D DTD Times T to the D. So that was our factorization. We also have a local analog of the thing I explained Yesterday of factorization one namely you can define this logarithmic q derivative This is this logarithmic q derivative We have seen yesterday, but now we define it on the local level and what we get is indeed a generating function of kind of local local for the slope x Herbert schemes of The quiver now combine these two so plug in this factorization of the Motivic general local motivic generating series into the numerator and denominator here and then you have a relation between the DT invariance and these local Herbert schemes and This is this is where DTP T for quivers comes from and this is great for computing these numerical DT invariance That's the way you to compute the numerical DT invariance because If you combine these two formulas, then you can directly specialize L to one But what is the PT side in this equation? Well, okay, this is No Now this terminology is very formal. Yeah. No, no, there is no explicit No, there is no explicit PT side Yeah I have a different question. So how computable are these DT invariance in it for general quiver? Let's say is there a way to go from one quiver to another? No No, you have to do it really separately for for every quiver and I mean everything is like nested recursions doubly or triply nested recursion starting from the Motivic generating function Which is explicit and a computer can easily do the calculations for you but there's no way to To pass between different quivers because if you do some easy operation just on the quiver adding a vertex adding an Arrow deleting an arrow you completely change the representation theory of the quiver Yeah so I mean for example Gabriel's theorem tells us that if The underlying graph of the quiver is an ADE-Dinkin diagram Then we have no modular at all then the classification is discreet but the underlying graph being ADE-Dinkin is Absolutely not stable under any error insertion or deletion as stable under deletion but not under insertion You can easily take an an ADE-Dinkin diagram add a single arrow, and it is completely out of the dinkin or affine classification Yeah, so Yeah, unfortunately, you can't do things inductively over the quiver. You have to do everything separately for any quiver Unfortunately, let's postpone everything else. Also, there was a question that I'll show you later Okay, we will respond to the exercise session that will take place in an hour anyway, so let's thank Amarkus again