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 uh 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 um motivic generating series for quiver q and then we saw two different kinds of factorizations factorization one was if you uh well okay we

also computed examples if you take a logarithmic q derivative or more precisely a logarithmic l derivative of aq then you get a generating function for actual moduli spaces namely the

Hilbert schemes of representations of the quiver log l derivative of aq gives generating series of i will not make this precise again because we'll take more a look closer at the

second factorization we got a generating series of virtual motives of Hilbert schemes and actually in the question and answer session yesterday afternoon i almost gave the complete proof for this so then we saw the factorization two and so today i will mainly discuss factorization three

factorization two was the wall crossing formula and i will repeat the definition wall crossing formula aq admits a factorization as an ordered product over the reals in descending

order of local series aq x sst that's the wall crossing formula and this local

motivic generating series that's defined as the global motivic generating series but just looking at semi-stable representations of a fixed slope so let me repeat this aq x sst is defined as one plus some overall dimension vectors of slope x 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 motives 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 q0 the number of vertices you have a

co-dimensional one subspace because that the the slope of d is a fixed real x that's just one linear condition so i have a co-dimensional one subspace all right so and everything here depends on the choice of a stability function or a 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 defined this term and these terms looks like um 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 invariance today i cannot discuss this

this whole stability space okay so that that was the that was the second result and i also uh 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 representations

okay so that was a summary of yesterday and now to somehow motivate the dt invariance let me make one remark um in this factorization one the surprising thing was that we were doing

something very formal with the motivic generating function namely we're 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

somehow then the motivation for defining the t invariance which we'll do in the next 10 minutes the 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 u co-prime that is for all proper non-zero sub-dimension vectors the slope is different

yes yes yes so this means ei less than equal di for all i but they are 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 moduli 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 this local series for the slope equals

the motive of an actual variety more or less so the motive of an actual moduli 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 moduli space of mu stable representations of q of dimension vector d we're taking the virtual motif and we have to decorate this by one of these annoying little factors uh 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 moduli stack in my 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 oh 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 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 a 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 the 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 uh switch back to gd but i mean this geometric quotient is a pgd principal 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 the 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 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 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 motive of Hilbert schemes and here we get as the lowest order terms of these series

we get more or less the motives of moduli spaces parameterizing stables but only under this special co-primality assumption which is not too bad well we know this phenomenon from moduli spaces of vector bundles the good theorems are always about the co-prime case where

and degree of the bundle are co-prime and in general you get singular moduli spaces things are much more complicated but anyway we are here in the quiver situation which is supposed to be much simpler than moduli 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 yeah so this c 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 a 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

um 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

efficiently do this factorization we need a bit of notation because otherwise we will have lots and lots of very ugly threefold 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 platystic exponential

so to simplify so this is a very u 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 platystic exponential well from the term exponential you can already guess it's something what what is an exponential an exponential is

is a transformation which maps sums to products the exponential is more less defined by the equations we want to do this for such formal series formal power series it should be it 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 platystic 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 motivic quantum space and i define the exponent the platystic exponential of this as the general as the geometric series okay and the other thing is it should convert

sums into products exp of x plus g should be exp of f times exp of g like you expect from

an exponential huh so it's really like an exponential but the initial condition is transform monomials into a geometric series and now this is a wonderful tool for making

huge infinite pure if f and g can do excuse me should commit um 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

um group let me be precise this defines a continuous group homomorphism from um well

now i have to be really careful so i um because i don't know what what motives i have in this

good degree of varieties i will just take the sub ring generated by l and properly localized yeah this is a sub ring of our motivic ring and then i join all these t to the d and i'm considering the maximal ideal in there where of all series with constant term

zero instead of all f in here such that f of zero is zero so which is just the maximal ideal and i map it 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 a 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 we are allowed to take in this formal power series right can you extend the predictive exponential your whole motivic ring

haha oh there is a there's lots of work on this um yes yes basically you can do this you can get these um well you can get uh 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 uh this motivic ring and um so you have to define these adams operations and they are defined using taking symmetric powers of varieties yeah so there's uh there's lots of

of work on doing this in in the generality and luckily for us we only have to work with everything which is motivated by the left chest motive yeah so we can do this this uh simple thing here okay and i'm not absolutely precise here so this defines a continuous group

homomorphism in case um that this ring f of zero is equal to one on the right f of not f of one thank you very much f of zero is one yes no i better not plug in one in this series in case uh that this ring is commutative

or actually we will now um work in sub rings which are commutative okay so that's the general oh yes sorry yeah so we so we take l but you said there

defined x also for l to the i over two so do we uh yeah yeah to to everything to everything which is generated by l in here yeah l and and all the localizations we already have okay

yes well the problem is doing this doing this formally somehow um disguises the very simple nature of of this construction yeah so um let's better do an example yeah namely 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 uh plastic exponential terminology to simplify this so we computed 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 the 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 somehow popped

up naturally in the geometry above there this is just dividing something by the virtual motif of the multiplicative group so this is the first indication that this x is something reasonable and now we have seen that this somehow 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 we have a quiver

and we also fix the stability as before because we want to consider these local series mu 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 a 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 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 d t d 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 a rational function in the left in the half lebschitz motive so a priori is only a rational function in the half lebschitz 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 d t d mu of of q while of this of q i don't like um let me omit it so

the quiver is not a variable the variable is still the lebschitz motive somehow just like this d to the d okay okay so now that's the central definition which we will explore for

the rest of the talk and let's uh try to digest this and let's see what is the logic of of this definition okay so first of all okay so we take our motivic generating function yeah question

about notation no it's about the uh the t to the d i mean we want do you also include the mutual element i mean like what's what d equal to um i mean the buff yeah i guess so yes yeah

yeah okay um i just want efficient notation take i don't have any idea so let me just

formally allow zero it somehow worked better with the definition of this local series where just put this one plus in front right here it just doesn't work okay so our logic is using

the hard and our simon filtration 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

moduli 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 platystic exponential to have this place platystic exponential well

defined we need some commutativity and this commutativity this local commutativity is contained here in the definition that the restriction to the euler form uh 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 lebschitz 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 you have seen here so this hopefully motivates this very brave

definition of motivic dt invariance of a quiver and okay so now we have something to explore the question about the specific explanation yes please i want to be confused about this where does one hide the factorial the factorial well because it's exponential you know you have over n factorial yeah okay see i didn't want to talk about lambda rings

so i always um prefer this this ad hoc definition of the plastic exponential by saying it has the functional equation of an exponential and this initial condition monomial goes to geometric series um 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 adams operations so psi is the sum over all i 1 over i and then psi i in a lambda ring with adams operations psi i so in a lambda

ring you have this these these lambda i operations and this gives the whole ring

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 sum functions and these are the animals operations so it's all this formal lambda ring stuff yeah and so this is the way you you can do this in arbitrary lambda rings

and there you have the honest exponential yeah okay which also easily proves so in this way you can easily prove that you have an inverse a plastic logarithm so log is then just psi inverse composed with the with the usual log

from a power series log where psi inverse arises from this by um i by i times adams operation psi i

thanks for the question because we will come back to the moebius inversion anyway in a minute okay so okay so at length i try to explain the logic of why we define these dt invariants

and now let's see some examples and to compute any single example is quite hard but at least we yes please i didn't know that such a polarization is possible yes you are right okay i forgot to tell you something um i forgot to tell you that uh

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 log

and uh and this in 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 um yeah so exp has an inverse

log so you can just apply log here and this defines ddt invariants 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 create 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 are zero dtd is zero for all d strictly 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 an exp and then you can read off the dt invariance for this namely the dt in dimension one is the virtual motif

of the affine line and all other dt are zero aha now this is not too bad this already tells us

something about or 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 and dimension one for the loop quiver we are looking at matrices up to conjugation thank john canonical form and we have a discrete we have a continuous classification parameter involved namely eigenvalues and this is the a1 encoding

eigenvalues so this is the moduli space for possible for possible eigenvalue of a matrix aha so um 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 q 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 here

the motif of the virtual motif of the stable moduli space m d mu s t of q virtual motif 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 this final theorem is that the

dt invariant in fact is always of geometric origin it's not the motif of of an actual moduli space at least not for mathematicians for physicists it is because physicists believe

in a certain moduli space of quiver representations with which mathematicians can't define but um okay so theorem is dt is geometric that's the slogan more precisely

under all the assumptions under all the assumptions which we need to define we have the following dtd mu equals okay so what's the geometry the geometry is the moduli space of semi-stables and now for 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 moduli space of semi-stables that is in general so this is typically a very singular moduli 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 orbifold

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 uh 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 cohomology theory which is perfectly well suited for singular varieties which is intersection cohomology yeah and so the final result is you take compactly supported intersection cohomology with rational coefficients of this and we take the Poincare polynomial of all these

and where our polynomial summation parameter is of course again our our minus square root of left checks so we take the Poincare polynomial of local compactly supported intersection

homology um better take a dimension take the sum over all i and that's almost it except for a little twist factor why 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 same stable point this might happen you might have um only properly semi-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 uh if it's uh 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 uh it's zero and uh so this is the interpretation of spain meinard and myself of something which was conjectured or believed in in physics by uh manschott 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 moduli space and let me call it the physics moduli 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 well this is just a vague dream it's not a theorem and the precise

mathematics statement is that it is the Poincare polynomial in uh intersection homology if we can find a small resolution of yes of course once you have a small resolution you are done

because the cohomology of a small resolution is the intersection cohomology of the variety and if we are lucky that the whole cohomology is just pure tate then it's the um Poincare polynomial is the same as as the virtual motive but um i mean think about moduli spaces of vector bundles on curves only in only in rank two uh

there are known de-singularizations of the singular moduli spaces yeah so rank two degree uh even then you know a de-singularization of the moduli space with only orbit fault singularities there is this general procedure of francis carven to de-singularize singular

moduli spaces but the bookkeeping is terrible and nobody knows what the final outcome would be so this is unsolved so i 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 magic yes yeah yeah so the assumption which i used to uh actually define the dt invariance this continues to hold otherwise dt is not not even defined i mean you 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 is it very clear that this should live in the ring generated by the left no no that's only a posteriori so this this theorem is by the way proof of the so-called integrality conjecture for all dt invariants that they are really polynomial in the left

shots or in the half left shots motive yeah and 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 portion stack of the same stable locus by g of d yes i think so so um 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 a quiver which is a bunch of loops say m loops where m is at least two and then i would say even for dimension three one can easily compute the dt invariant

just from its definition but nobody knows a small desingularization of the moduli space so i mean even dimension three pairs of matrices i would say this is unknown then if it's if it

is the actual motive of something yeah um 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 shots motive we can specialize l at one this wasn't clear a priori because a priori 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 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 um we take the specialization of l to one and this is then given by now i'm not sure about the

convention so let me say plus or minus yeah because i'm always mixing up the signs um it's a moebius inversion namely it's one over d squared sum over all divisors of d moebius

function of d by e an ugly sign minus one to the m minus one times d minus e and then a huge binomial coefficient m times e minus one over e minus one so that's the sample formula how dt invariants look like and this is almost the only case where you have such an explicit formula

we have 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 leading term like a binomial coefficient m d over d which is pretty large um in general it's very complicated because you have this moebius 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 moebius inversion over binomial coefficients should be divisible by d squared this is terrible this is i mean it is an integer it is it is 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 geometric geometricity statement you just take the euler characteristic and compactly support intersection cohomology

so it's really just an euler characteristic of some ic intersection homology euler characteristic of of a modular space yes actually actually there is a general purity result so actually uh only even uh intersection

cohomology 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 uh let me give you three pictures relating everything to the huge topic of scattering diagrams oh yes please

so does this formula appear in some geometric dt theory too there are a few other places where such a formula appears i have seen one or two references

where this appears this also appears in some setups as certain uh copacuma wafer invariants which look similar um i have have to check these sources again but i mean expressions like this such a moebius inversion over binomial coefficients appears

somewhere here and there and the universal source somehow is always dt invariance for the 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 um scattering diagrams so i just want to show

you what happens for a 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 uh 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 euler form is symmetric and then um 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 invariants appearing for dimension vectors one zero one one and zero one which is accidentally just the positive roots for an a2 system for m equals two you have two infinite series and a special case here here the dt

invariant is two uh the numerical dt invariant is two by the way um so here it is always one and here it is the dt invariant is the virtual motif of p1 but you don't have the dt invariant

for any multiples so this is basically like an affine a1 tilde root system or 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 invariants are non-zero and they actually explode the numerical dt invariants grow exponentially in this whole cone and that's uh that corresponds to the hyperbolic rank two

root systems somehow yeah so there's an intricate relation to two root systems this has to do with uh with katz's theory of quiver rips and of any composable quiver representations and so these are lattice points of some hyperbola and if you just uh take the rays through all

these points then you recover certain scattering diagrams which appear in all sorts of chroma fit theory in the tropical vertex and uh cluster theory and where else okay well okay that's enough for today thank you very much any question

from general view is also related to the root system yeah but um well the the connection to to um infinite root systems is weaker in general it's definitely weaker this rank two this this two vertex quiver is exceptional for that

um but i mean that there is still there is still okay so in general so for a general quiver q let me just remark that if the dt invariant is non-zero

then you require that the uh 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 the reason why this is not sufficient is that you have to make this choice

of of stability and um 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 um dd is less than or equal one and the d is a so-called shurian root and uh and if you make a sufficiently generic choice of the stability

so you avoid finitely many hyperplanes and stability space then you can conclude that um that the dt invariant is non-zero

yes if dt is non-zero then d is definitely now yeah this is only for hyperbolic root systems yeah yeah yeah okay yeah yeah um yes yes even even the root okay yeah yeah but okay

okay yes okay um so we have to ask maxime

okay um well dt invariants are the mathematical precision for for bps state counts and uh 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 so but no we we have a question in the q a is it possible to decorate a x cube by chain glasses of bundles over the ah okay the semi-stable locus are the ssd yeah okay so yeah on these

moduli spaces of semi-stables or in case stability and semi-stability coincide you have tautological bundles and you have some you know

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 R-mod. So not working just with

with this pretty coarse information of the motif of the variety but working in some... well, bringing some k-theory into the game using Schoen classes or using classes of bundles or sheaves. That's lots of things to do, I don't know.

Yeah, but okay, yeah, you could think about reasonable replacements for for this base ring R-mod where you can do all this. Any further question? Is there any relation with this DT invariance and the virtual motif or

river scheme that you described? Yes, okay, so that's the the DTPT story which I have briefly mentioned. Yes, okay, so what you can do is... 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. We have factorized this as the x of 1 over l to minus one half minus l to one half sum over all d dT d 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 for the slope x Hilbert schemes of the quiver. Now combine these two, so plug in this

factorization of the local motific generating series into the numerator and denominator here and then you have a relation between the DT invariance and these local Hilbert schemes. And this is where DTPT for

quivers comes from. And this is great for computing these numerical DT invariants. That's the way you you compute the numerical DT invariants because if you combine these two formulas then you can directly specialize l to one.

Well what is the PT side in this equation? Well okay this is, no, no this terminology is very formal. Yeah no no there is no explicit, no there is no explicit PT side.

So I have a different question. So how computable are these DT invariants for general quiver let's say? Is there a way to go from one quiver to another if they are somewhat related? 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 recursions starting from the motific generating function which is explicit and a computer can easily do the calculations for you. But there's no way 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. 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 moduli at all then the classification is discrete. But the underlying graph being ADE-Dinkin

is absolutely not stable under any arrow insertion or deletion or 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.

So yeah unfortunately you can't do things inductively over the quiver. You have to do everything separately for for any quiver. Unfortunately let's postpone everything else for now. Also there was a question that I will show you later. Okay we're going to do the exercise session that will take place in an hour anyway. So let's thank Marcus again.