 first of all thanks to the organizers for the opportunity to speak here. I also wanted to express my sincere regrets that I had to miss the first half. I had a previous obligation but I'm happy to hear that the videos are online so I can catch up on what I missed. Speaking of missing the first half I noticed that my abstract featured only the first half so I think that's fitting. So what I want to talk about is quantum cluster varieties as they were introduced by Falk Gantroth and how I've come to understand them as an instance of factorization homology as introduced by Ayala Francis and Tanaka. Just quickly before I get started I want to advertise a pair of workshops. So in happening in Edinburgh in June there's two weeks. There's a summer school which is happening the third through the tenth and then there's the conference week the next week happening the tenth through the fourteenth and so we have excellent lineup of speakers and I'm hoping that many of you can can can make it. So for the summer school there's some funded positions for students in postdocs and people who attend the school will also be funded to attend for the next week. So I hope you can help me advertise this. Okay so that's that I don't need the projector anymore. Okay so I want to start by just recalling Falk Gantroth's construction. I'm sorry first I'll start by reviewing character varieties and their quantizations. Okay and then I'll give a more detailed discussion of Falk Gantroth. Now so these cluster algebras quantum cluster algebras they're really complicated. There's a lot of a lot to say about them and in some sense part of the the point of this work is that I never really understood the axioms of cluster algebras and I want to bypass those. So my presentation of Falk Gantroth construction will be a bit biased and a bit abbreviated but I need to give you the main details and then and then I'll talk to you about factorization homology approach and parabolic induction. Okay so I think we know what is meant by the character variety of a surface. So we look so equivalently we can think about representations of pi one of the surface into our favorite group G. So G can be any reductive group for what I'll say but I'm especially gonna focus attention on GLN and even the case GL2 is interesting for today. So if we are algebras we can think about just maps from the fundamental group into G modulo conjugation action or if we're a bit more topological we can think about G local systems on S regarded up to isomorphism. Okay so that's the the character variety and so a basic problem is singular in general but so even regarded as a singular variety it carries a canonical Poisson bracket, Vitiabat and Goldman which I imagine people here are familiar with. Okay and so longstanding question that's been answered in many different ways is sort of what's the right way to quantize this Poisson structure, taking it into account the singularities and trying to do this in as natural a way as possible. So one way to deal with the singularities is just to pass to the character stack. So consider, I'll denote it with an underline, the character stack. Yeah I will consider both cases yeah it's interesting in all these cases. Yeah as you're alluding to it's much easier when the surface has at least one puncture but we can also consider this stack when it's close. Indeed when it's closed then even working with the stack doesn't quite deal with the singularities we also have to worry about taking derived intersections. Okay and so in this context sort of a motivating result, Lindsey Francis and Navler says that if we study the category of quasi-coherent sheaves on the character stack of a surface, again S is either closed or open, this is the factorization homology of the category of representations of our group G and so they prove this in the in the setting of infinity categories, the derived setting and in work I'll mention later we also prove that the same thing holds just at the abelian level. Okay so whether you take the derived category of the abelian level the point is that the classical character variety is computed just by the factorization homology of rep G. So with David Benzvi, Adrian Brochet and myself we introduced the quantization which I'll denote by ZQS and the definition of this is just we take the factorization homology of the category of representations of the quantum groups that's a braided tensor category and E2 algebra and so it makes perfect sense to compute factorization homology in this sense and we propose this as the sort of most natural quantization of the character stack. Okay so that's just a definition but so there are, this is not a talk about this construction but I'll just for context say so we described so for punctured surfaces which I'll indicate with a with a circ here we wrote these as modules of algebra of categories of modules for a certain algebra in rep QG and these algebras we identified these are isomorphic to certain very explicit algebras introduced in the 90s by Alexei Avgrosa and Shomerus. Okay so when your surface is punctured then the the pi one is just a free group and so the character variety the character stack is just a bunch of copies of G considered up to simultaneous conjugation action and using that presentation so first Falken Rossley for punctured surfaces they rewrote the Atia Bot Poisson bracket in a in a way using classical R matrices and then inspired by this Alexei Avgrosa and Shomerus they simply proposed by generators and relations an algebra called the which I call the AGS algebra which where you simply replace the R matrices in the relations with with quantum R matrices and so you define by generators and relations a quantization but you have to do it by hand and you have to make some choices on the surface okay so the nice thing about factorization homology is this is this category is completely canonical without any choices and what we showed is that if once you have this canonical thing then you go and make some choices then you recover these nice algebras that people introduced in the 90s okay so that was the sort of story for punctured surfaces for closed surfaces I'll just say I'm not going to really focus on this today but we describe zq of s for closed surfaces the quantum Hamiltonian reduction and I'll also say that work in progress of one of my PhD students relates these to the so-called skein skein algebra skein categories that people studied also in the 90s okay so the the upshot is that for ordinary character varieties these these quantizations are sort of the universal thing and because of the universal thing you can start connecting them to whatever you want okay so but before all this business about stacks and character stacks and factorization homology and so on there's a there's another way to deal with both the quantization question and the singularity question which was the idea of a fucking gun trough okay and what they what they did is they said let's consider a different a slightly different variety okay so instead of a surface s they consider what I'll call a parabolic surface and a parabolic surface for so in the fuck gun trough notation so in fuck gun trough notation what I'm calling a parabolic surface they would call I think a decorated surface you you have a surface it may have boundary components proper boundary components and it may have what they call punctures and the boundary components are required to have certain marked points and the punctures we also regard as sort of a marked point and they suggested to consider a different variety so just write it like this okay right and so what we consider is we consider g local systems on s together with a reduction to the borrel at the marked points and a t-framing at the marked points okay and we consider these up to isomorphism okay so if you're not a geometry then this reduction to borrel might be might require some explanation that there's only really two cases to consider basically this is some extra data that we that we attach so if you have a mark point on the boundary then this reduction to borrel is just the choice of a flag in in the in the fiber over the mark point okay and if your mark point is at a puncture then it's the same thing but there's a condition that it be preserved by the holonomie okay so what they said is all right I don't they didn't like the singularities that come from stabilizers and and they basically rigidified the problem they said well okay we're going to add some extra data of some of some flags at the mark points and some compatibility in the case of punctures and then so that's the that's the reduction to to borrel and then the t-framing so and so the way I've phrased it here the moduli space is g mod n and the t-framing just says we'll replace this by sorry is g mod b and you replace this by g mod n so you consider not just a flag but you pick a basis in each step of the flag which is just you know a choice of a single vector in each in each co-carnal and and so so that so so basically you're just replacing g mod b by g mod n but now it has a residual t t action a line sorry yes a line yeah well no sorry a vector here and and and if we quotient by that t action then we only consider the line okay so this so this would be a line and and in here we really we really pick a vector okay all right and so what's interesting about this this choice that they that they make is now so and now when we consider up to isomorphism I don't mean as a stack I really mean we take just we quotient in a geometric sense by isomorphisms and the thing that you get in this case because you've added these extra framings is still nice so this has the structure of a cluster variety and so roughly a cluster variety x is is a variety which is covered by a bunch of charts u alpha and each u alpha is is homeomorphic to just a an algebraic torus c star to the r and these are all the same that's the same dimension r every time and moreover there's a Poisson bracket on x which is quadratic in each in each chart so what that means so so I'm saying that the each chart functions on it is just a Laurent ring and several variables and the Poisson bracket this is this famous formulas the Poisson bracket is just x i times x j times some integer a i j and I'm gonna say a little bit more about how you extract these integers in a second so the Poisson bracket is is sort of sometimes called log canonical it's just it's the Poisson so this is what you would expect if you're try and do sort of canonical Poisson structure on a on a torus for some matrix a i j here and so right and so now if you have two of these different tori in and you glue them together the transition maps they have to preserve this Poisson bracket in the suitable sense and so that implies essentially so that's almost the sense I don't want to get into implies that the the transition maps are these cluster transformations okay so very specific ways that you change variables when you move between these charts and so combinatorial they minded people love to write down these formulas that I never understand but it's really just encoding the fact that this Poisson bracket needs to be globally defined all right so let me explain just a tiny piece of how they construct these charts and I want to do it just in the case of GL2 okay because I always found this procedure very mysterious and and I'm hoping that you will also find it mysterious and then I want to explain why it's just what happens when you read the Ayala Francis Tanaka manual on factorization homology I think that's interesting okay so what you do is you triangulate your surface so they say you first need to choose a triangulation okay this just show you how bad I am at this we'll pretend that's a triangulation okay so you cover the thing with triangles and the only thing that the only stipulation is that the end points of the triangle need to be at one of your marked points and that's it okay so the end points of the triangles are at your end points and and so the first thing they do is they tell you what to do with a triangle okay so let me show you I will show you quickly just GL3 so you can see the difference but then I want to move to GL2 for the rest of the talk that's simply just because otherwise you have to get into the like the conatorics of root systems and stuff and for SL2 you see all the basic ideas without all the without all the fuss okay so what they would do is they would draw a triangle like this and they draw some squares on the edges and on the inside they draw a circle and here for GL2 something nice which happens is that there are no internal ones okay and then there's a certain quiver that they draw it sort of looks like the triforce if you're a Zelda fan okay and and so here it's just like so and and okay so what you do to get these charts the C star star to the R the chart associated to this has C star to the number of box boxes okay plus the number of circles okay so you have a variable for each one of these things and then those AIJs are just coming from the adjacency matrix of the quiver that I've drawn here so they're like plus minus one or two depending or more so I can I've only drawn with without repeated edges but you're allowed to have multiple edges stacking okay and so so the AIJs are just this thing so they tell you that they just declare from the sky that this is what you should do to a triangle and then they say that when you glue two triangles together along some edge so let me again just do GL3 and GL2 what happens when you glue along two triangles along an edge is that these square ones that you've glued they now become circles and you still have the circles that you had before and then the edges that you and the the squares remain squares so so in SL2 already you get now a circle here and there's a there's a simple calculus for how you how you fill in a quiver here which I won't go into but so so you basically just glue these quivers together and so so in the in their terminology these squares are called frozen and the circles are I guess unfrozen all right so you keep doing this you keep gluing these triangles together and then in the end what you get is basically you get a quiver some complicated quiver that you've drawn on your thing it's somehow in some way related to your triangulation and then only on the boundaries do you ever get these frozen things because you've glued all the internal edges together and so you you just have these this boundary here okay half squares well talk to them alright and so okay and so so in the end what you get is each triangulation leads you to some some chart c star to some rank and what they say is that you can think of this as a chart this is some u alpha these actually they tell you that these actually should correspond to functions on the on the their the character variety for this parabolic surface okay and then there's an interesting so of course if you change triangulation you get different charts so you get you get one of these for each chart I should say also for SL2 that's that's basically it in other types there's some more data that you have to indicate so each of these triangles is colored by some some root data that I don't want to go into okay can you tell us what the open set is and the character it's a bit of a pain I mean yeah sorry yes I will get there at least for the triangles we'll start to see it as I as I do the rest of the analysis and and then basically as I will try to explain to you you just glue these open sets together and some stand away from triangles and so yeah that will be a major part of the talk okay so this always baffled me I mean why why should you do that why do those give functions that's a good question so you get this plus on bracket they claim as far as I know well I don't know what the status of this this claim is for general g but that is this plus on bracket here is the a tia bot somehow you should think of it as the a tia bot plus on bracket and well so so the rest of the talk I just want to come to an understanding of this construction and in particular its quantization using factorization homology so so again kind of gave myself an easy job because I wanted to convince you this was confusing so I just have to do a bad job explaining it and I think I've done well right I should say right before I go on though the point so that for them a very important point of this simple plus on bracket is that when you quantize such a plus on bracket you simply say that x i x j equals q to the a i j x j x i if you think about that that's a sort of obvious quantization of this thing and and so so their quantization of these character varieties that they proposed is what they call a quantum cluster ensemble you don't say sort of once and for all what the quantum gadget is you say what all these charts are and then you you quantize all the change of variables between the charts and you just declare that whole system of all these of all these quantum tori together with their changes of variables you declare that to be your quantum character variety okay so I would like to understand how to connect that to these to this character stack story and I just want to give one piece of motivation why this is not just me revisiting the 90s and so demafti and subsequently demafti govela and guntrov they proposed an algorithm to compute so-called quantum a polynomials of knots and and what they do is they say you should not just triangulate surfaces like we're doing here but you should learn how to triangulate three manifolds so if k is a hyperbolic knot that they explain that if you look at at the complement of this knot this can be tiled by tetra by ideal tetrahedron and an ideal tetrahedron tetrahedron is something like this so it's a tetrahedron but at the end there are these facets and and these so you can tile the complement by these by these facets and and the point is that the the union of all these facets union of these of the facets is a triangulation of the torus okay because they the the union of them is just the boundary of the three manifold and if you've constructed the three manifold by deleting a knot then you've just got a triangulation of the torus and then they say well we can now take this triangulated torus do fuck guntrov description and we get all these nice quantum tori and then they explain how to use these quantum tori to to compute the this so-called quantum a polynomial which is a very nice invariant of knots that's still quite mysterious okay but the problem with the prescription as as tutor says quite well in his in his first paper on the subject is that this choice of triangulation tetrahedral triangulation it's it's it's a choice that you need to make yeah and and so you don't know that the corresponding computation that you've done for the would be quantum a polynomial is well-defined and so as a first step I want to explain that the fact guntrov thing is not a choice it's a canonical construction and and these charts are the choices that we are allowed to make once we know it's canonical and then once we have a three-dimensional theory again using ideas from factorization of ology then we can actually prove start proving things about their their construction so we can prove for instance that there is a well-defined invariant and they're just making choices to compute it all right so let me get into the the story about factorization homology now okay so Francis Tanaka plus some some modification of Ben Ben's V Francis Nadler tells us that if we want to understand causing coherent sheaves on these parabolic surfaces we can also compute this so I'll stop using the integral notation just because it gets a bit cumbersome so we can compute this as factorization homology of our parabolic surface with coefficients and now this is going to need some explanation so before we only had to tell you a group G but now I need to tell you the group B and the group T so these are going to be our local coefficients for a factorization homology theory and and what this means is that whenever I see a a line defect so a parabolic surface up so there's one thing I need to say so so in fact Guntrov so I want to translate between these pictures so you had punctures and marked points okay and so the connective factorization homology we just have to do a sort of trivial change of perspective so if I see a marked point I'm just going to replace that by a line a contractable curve I'm going to label the curve by B I'm going to label the inside by T and the bulk of the surface is labeled by G and similarly when I see a marked point I'm going to grow that into a little line here and I'm again going to mark that by G that by B the bulk is always G and the inside is T okay so this is just the same data just a different way to think of it and now ordinarily in in sort of unstratified factorization homology we we have to specify some braided tensor category that we assign to the bulk of some surface so that's G we have another braided tensor category which is just rep T and then B is a one morphism in a suitable category and this is sort of explained in various places I think Claudius thesis there's a nice exposition of why you should think of these as one morphisms so you have B on the line you have G here and you have T here okay and as a sort of spoiler when we quantize we'll have rep QG here we'll have rep QB and here we'll have rep QT and and there's a there's something you need to check which is that rep QB has the structure to be an interface between rep QG and rep QT this is sort of classical in some some way the way you construct the r matrix for rep QG is a sort of quantum double of rep QB and that's that is you trace through that that tells you that this is a morphism you know between these two braided tensor categories it's the kind of thing that you can mark here all right what does what picture I mean this one yeah so in stratified factorization homology you consider a stratified let's say surface so this is a stratified surface and the principal strata I need to label by some local coefficients those local coefficients are some e2 algebra this is what we learned from our friends at Tanaka and the one-dimensional strata are there a locally what's called a locally constant factorization algebra on this stratified space so in the idea is that somehow okay the idea is like every point on this stratified surface either looks like this thing this thing or this thing so this thing is what tells us what to do with points like this it's not braided that's a good point but it's so let me say more specifically what we need from rep QB we have a braided tensor functor from rep QG tensor rep QT with the reverse braiding to the Drenfeld center of rep QB so this is what sometimes called a GT central manoidal category so it's not just a manoidal category it has to have these anchor maps down to rep QB and these so so the first map here this is just from rep QG to rep QB that's just the forgetful functor from rep QT to rep QB that's just the pullback under the projection and then I'm claiming that this has a canonical central lift which comes from the quantum double construction to give you a braided tensor functor like this and so I'm going to get in these questions this structure right here of a pair of braided tensor categories a manoidal category and a functor to the Drenfeld center that is what one unwinds to be the allowable local coefficients for a one-dimensional defect between two two-dimensional between two e2 algebras okay and so right I didn't give a formula for factorization of ology because I'm sort of taking that as given in this in this audience but I'll say that the idea is the same as you do for ordinary factorization of ology for ordinary factorization ology I would sort of cover this thing by discs and then I would correspondingly have some sort of co-limit that I need to compute and I would compute that in the category of categories to category of categories here as well we cover this thing by discs but now there are three types of discs and so we write some this this thing is just defined as some co-limit of all the ways of embedding these three kinds of discs and their district unions into s and then we have some term which is just which is just rep qg to the number of g discs rep qb to the number of v discs and rep qt to the number of t discs so basically these so every time that we would give a partial covering of this by these three different types of discs we would write down the corresponding categories and then we take a co-limit in categories and we we get some answer okay so if that sounds like hopelessly abstract somehow the whole point of my research program is that using tools from quantum groups you can actually unwind this and make it quite explicit that's what I'd like to do that's what we already did in the unmarked case with benzene brochure and that's what I'd sort of like to do now in the parabolic case okay no it is the same as a it has a different braiding it's the same as a minoidal category but it has the quadratic the standard quadratic pairing giving you the braiding and that's important I mean to get this structure you have to fix that particular one absolutely all right so it is sort of implicit so the definition with okay lay Schrader Shapiro myself just completely following what we did with benzene brochure is the quantum invariant we just defined to be the factorization homology of the parabolic surface with rep qg rep qb and rep qt I do apologize for sort of skipping over this this sort of calculus it would be a whole lecture in itself and we wouldn't get to the punch line in that case no no you start with so in factorization homology you start with an e2 algebra in whatever and you end up with just an object in whatever so here we started with a we started with a braided we braided center category which is an e2 algebra in categories and at the end we just get a category precisely the braiding comes from symmetries of the disk and if you take a general surface you you doesn't have the same symmetries okay so right so what I want to explain now is some examples so they're the most important examples so let's look at okay so this thing this is just rep qb by construction and I'm going to call this conf one so this is zq of conf one and in general the first class of examples I want to give you is is where we take a disk g and we put n of these little parabolic induction restriction things around the boundary so that's comf n and of course here I I'll stop worrying about g or rep qg I mean you can treat the classical on the quantum story in the same breath that's the whole point so this is comf conf n so I have a parabolic restriction restriction along each of these things and a t-framing all right so if you think about this all this is saying is that comf n as a stack is just g mod n cross g mod n so the number of of mark points that's just the fact that I have to fix a framed flag at each point by definition and then when we consider it up to isomorphism I want to quotient by the G action which is just simultaneously changing the trivialization on the left and by the t to the n action the torus acts on on the right rather I don't want to quotient by this I want to just remember it so in this business it's because I think you miss this so you wrote the reduction to be yes the reduction to be and a t-framing I think that was before you arrived yeah yeah yeah this is an important point it's not such a big deal but it's it's necessary this is I mean reduction to be in t-framing is the same as this reduction to be that's right yeah but I but I remember the t-action yeah that's right okay very good okay so all right so it so so sort of feature of these factorization homology theories is that the categories you get they always have a distinguished object and so in fact this distinguished object is the one which gave rise to these AGS algebras and that's somehow a big a big important point for me so and I want to understand is what is this distinguished object look like in these cases and and I'll just I think I'd just like to state that if I if I take the endomorphic internal endomorphisms of this distinguished object so I'll say what I mean by that in a sec what I get is isomorphic to a copy so I take OQG mod n tensor OQG mod n this is okay so this is the standard quantization sort of FRT style quantization of the the coordinate ring of this the standard affine space and this is the braided tensor product of algebras and this all takes place in rep QG tensor rep QT to the n okay and now I can explain what I mean by internal endomorphisms so see that this is where the T and the G action are so useful I can I can this category just carries an action of G rep QG just by inserting discs into the into the boundary here and rep QT by inserting discs into the boundary here so if you're familiar with how we proceeded in the unmarked case it's the same thing but now we have these additional sites and they're acted on by an even simpler gauge group rep QT okay so we can exploit that so that says that this is a module category for that braid tensor category and a fun thing to do in that case is compute internal endomorphisms and the claim is just like we got these AGS algebras by just doing some standard playing with adjoin functors and so on when you do it in this setting you exactly get the expected quantization of the standard affine spaces okay so that's a computation how is this statement related to the previous statement that quanth ends is equal to this product well when Q equals one then I'm saying that there kind of quotient by G right and here I'm working G equivalently and T equivalently so so it's I mean whether I so it's if I work G equivalently that's the same as considering the stack quotient mod G yeah this yeah sorry this is the braided tensor product of algebras so if I have algebras in a in a braid tensor category there's a canonical way to combine them and they don't commute they commute using the braiding it's the only thing you can write down and this says the standard FRT construction so if you don't know what that is it's just there's a canonical way to deform functions on G mod n let me make a remark actually about this this is I think useful so this is the same as the FRT quantization of G so it's just a Q deformation of functions on G and then we take n in variance and so if you think about payter biotherm that's just the direct sum of over lambdas I have irreducible representation V lambda that tells me how G acts on this thing and T acts with weight chi which is minus omega not lambda so that's just the lowest weight for the for the representation okay so the highest weight for the dual representation and so so if you forget about that for a second I'm just saying that this is a direct sum one copy of each irreducible representation of G it's quite a harmless thing and then I'm saying that there's a grading on it by the by T and that is given with this weight okay so this is like a very lovely thing and when you multiply coordinates it couldn't be easier V lambda tensor V mu has a canonical projection onto V lambda plus mu and that's how you multiply okay so note if you tensor V lambda tensor V mu you get a whole bunch of different things with different multiplicities but I'm saying that the multiplication in this algebra because you're fixated on highest weights it only preserves the thing of the incorrect weight all right so this is a very easy algebra to work with some how this is the same end yeah same end yeah yeah so did you say what the category was in this case this is the category that we take as a definition in this case is there an answer no yeah okay good question yeah excellent question so so here I'm talking about the coordinate algebra of functions you may be familiar that like g mod n is not an affine space so it's a little bit dodgy to start all of a sudden talking about this this dist this dist object is going to be like global functions but let me remind you that that's the same thing that happens in front gone trough they do not study the stack or indeed even the variety of these parabolic local systems they only ever study global functions on it so they they themselves are only ever working up to co-dimension to so indeed the correct sort of quantum invariant is this one but if I want to connect to fuck on trough I had better start studying the algebras of functions and so this is what justifies me looking at this so that so somehow the distinguished object corresponds to taking global sections that's yes there's a functor of global sections but just taking hams with this object yeah now note I should say one thing so this was the internal endomorphism so to get to what you're what you're saying I need to take the invariance so I want to take this thing and I need to take the G cross t to the n invariance so this thing is kind of the stack but but still global sections on you know so so the the structure she found the stack and if I want to go down to the variety which is where they work I need to take invariance for the group that just quotients out by the group action yeah these are really good questions thank you okay we do want to embarrass yeah yes that's right we want yeah sorry again we want to take G invariance but we want to remember the T action so that's gonna be important when we start amalgamating okay so what have we done we've so I should have said that this thing this algebra is what fuck on trough assign to the quantization of this thing so so far we're just following the a la francis tonic prescription and we get exactly what they do for for discs at least with many punctures but there's a few there's a few caveats even for discs so so I want to zoom in on comp three and and then come and then go back to comp two so so for comp three so this thing is not a quantum torus so so so I'm I'm misleading you somehow even when I take G invariance this is not a quantum torus it does not have these x i j coordinates even when I take G invariance so there's two steps that we need to do so first of all what we end up looking at is g mod n tensor g mod n okay and that's just the sum of the lambda tensor v mu tensor v new again so if for instance this this kind of calculus is in Chen and gun shove wasn't quite explicit in the original papers but it's certainly there so we look at a triple tensor product of irreducible representations and we want to know when are there g invariance and and what you find so so in type a I think this is Chen and gun shove and in general type this was due to Ian lay as far as I understand and there are certain triples so there exists triples lambda mu and new such that this is a one-dimensional space okay that turns out to be important okay since this is a one-dimensional space even when we quantize it means that these these things will Q commute so these guys this algebra is not a quantum torus but when we when we take G invariance and we consider these guys they Q they Q commute so that means there are some basis in which they just commute by powers of Q and so what we can do is we can invert these okay and and the claim is that once we invert these so that the two-step process is we invert and then we take G invariance and and the thing that we get at that stage is a quantum torus I mean we have some non-commutative algebra and these are some elements in that algebra and they don't have inverses in the algebra we formally take their aura localization it's very useful if you're considering aura localizations to have Q commuting things because then you can just not worry about fractions you just have sort of non-commutative fractions okay all right all right so right so classic there's some classical geometry here which is if I consider triples of flags so what have I got so sort of on this picture associated to each of these edges I'm supposed to each of these vertices really I'm supposed to imagine I have a flag F1 F2 and F3 and what's really happening in this once when I invert these things is I'm asking that pairwise each of F1 F2 and F3 are in generic position as flags and then there's a further generic generosity that I can ask I can reflect flag 1 through flag 2 and I want that to be generic with respect to flag 3 as well so there's some I'll just say biocombinatorics for producing these charts and it's a well understood thing okay and so what it allows us to do is in comp 3 there's a subcategory of comp 3 which is which is just generated by this algebra so I take that algebra let me call it a and I invert these elements I invert these these guys and and I consider modules for that thing in repqg of qt to the n which is to say I just look at a subcategory where these operators happen to act invertibly so it's a subcategory of here and I should have mentioned by the way already for comp 2 there's something similar going on so comp 2 contains let me just say classically so comp 2 is g mod n mod n and then I look at the the t cross t action and I have the g action and there's an open subset in here so this is comp 2 there's an open subset where the two flags are in generic position and and if I look at qc of comp 2 tilde as a stack this is just a manoidal category which is rep t and I want to stress something here in the formalism I was discussing with with Sasha Brouwerman that this can be rep qt for the disk but here this is only manoidal so I just mean rep t so I'm claiming that there's a not obvious equivalence there's a sub a manoidal subcategory of qc comp 2 which is just rep t and when we quantize that remains rep t so I'll write it this way zq comp t is a manoidal category rep t all right so that so there's that's comp 2 and that corresponds to this picture here all right so we're getting this is a subcategory it's a manoidal subcategory it corresponds to an open sub stack of the classical gadget okay I'm running a bit uh short on time where are the erasers oh here we go okay so I think I'll just have time to explain so okay so so just to summarize what I've tried to explain is that comp 3 we have this open this is the subcategory comp 3 and this is isomorphic to some quantum torus so modules over some quantum torus and and it's the same quantum torus that fuck on trough prescribed so that's quite nice and now what I need to explain is finally the amalgamation process right they define their invariant by gluing triangles together all right so this is uh actually I have to say we were stuck for quite a long time uh uh before Ian lay uh jumped onto the project and it was really this uh important observation so look suppose I look at comp 4 so that's you should think of that as a quadrilateral in fuck on trough picture and I want to decompose that into triangles so uh pictorially I want to draw a line like this okay and now uh I offer answers to not go tell us we have excision for uh gluing along uh cylinders and here the cylinder that we're gluing along so this is so here's an obvious statement this is a union of this triangle union over this triangle wide rank too long two earth fill lines uh this is the cylinder that we're going to glue this is the this is the interval direction and this is the the the direction and what are we gluing over we're gluing over this thing so that's conf 3 that's conf 3 and that's conf 2 okay so it's just a fact of life that the invariant say this is just one example z q comp 4 is by excision just z q of comp 3 tensor with z q comp 3 over z q comp 2 okay and this is hard this is a stack that we somehow not can be a stack that we don't understand but we can now uh look at the tildes okay and the tildes we do understand this is a quantum torus this is a quantum torus and this is just rep t so what that says in in words is that these frozen variables when we tensor over something which is just representations of a torus and then we take t invariants for that torus what it exactly prescribes is that we multiply these two things together and we force that we restrict to the the degree zero subalgebra and so taking uh this is this is rep t even when we quantize and uh and this uh procedure so excision from this from from excision follows the amalgamation prescription of front contrast so i'm i'm already out of time um so uh let me just say that for a general surface we just write down a canonical formula and when we compute it we get exactly the cluster charts that were predicted by fucking gantra thank you and what we do in three dimensions um right so um so okay without these restrictions without the parabolic markings we showed with uh brochet and noah snider uh using work of of uh claudias and uh of um runa haugsang um we showed that there's a 3d tft that extends factorization homology okay so you have this coborus and hypothesis you need to check some axioms and we check those um uh so the claim is uh that if you take any rigid braided tensor category it lives us canonically in some four category of such and it's three dualizable as soon as it's rigid rigid is all you need so this echoes some results of uh schoemer priest douglas schoemer presen snider for a few for tensor categories and it's the braided analog of that um so to define that four category that's where we need claudia and uh and theo johnson freed and uh and runa haugsang's work uh but then you just read the manual you check some axioms and you see that you have three dualizability since you have rigidity um so now as i've said uh rep qb serves as a one morphism from rep qg to rep qt in that uh in that same four category and so in order to sort of uh uh consider not just a three-dimensional tft but a three-dimensional tft with these defects we need to understand dualizability okay and if we could understand that dualizability then you could just sort of um and if uh some topologists do some work for us then you could uh define sort of tfts with with interfaces and then you could implement the uh the uh demofte gabella gone trough or at least some relative of it for for a given um triangulation um so one direction is that uh you'd like to to know that you can make sense of some sort of drawing where you have sort of g out in space t out in space and b along some along some plane and so there's some some hard technical uh work that needs to be done there um another approach i should say is that inside any parabolic surface you can just uh cut away all of the parabolic bits and then you just get a g surface of surface colored by g and uh and you can just try and do tft with that instead um so that would be a way to sort of bypass uh these uh these uh berels but um yeah it's this is a future hope but it's certainly not something that's uh that's in the works right now yeah so does it mean you can say that just surfaces without mark points or is that what no like um so i mean if you have so i'll just explain you the dictionary like if you have some fuck on trough surface like so okay so i've said that this is b t g and uh b t g well what i can do is i can just look at the sub stratified surface where uh where i just cut away that all the stuff that i don't like about t so there's just a sub surface in here that has no markings but i can regard it as a parabolicly marked surface just in a trivial way and this gives me functors from uh our our usual character variety that we uh studied with david and adrian it includes into the parabolic one so there's a functor an inclusion of functor okay and and it comes from in fact what it what it tells you is something that people knew in some cases classically which is that the ags algebras have cluster embeddings so this is something you can try to do by hand you can take some you know random non-commutative algebra and try and embed it as a sub algebra of a quantum torus and uh by by thinking about this uh this picture you get that the ags algebra associated this punctured surface it embeds into the corresponding cluster uh uh ensemble and so that's that's another thing you can do yeah