Quantum cluster algebras via factorization homology
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Transcript: English(auto-generated)
00:17
First of all, thanks to the organizers for the opportunity to speak here.
00:22
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.
00:43
So I think that's fitting. What I want to talk about is quantum cluster varieties as they were introduced by Frank Gontroff and how I've come to understand them as an instance of factorization homology,
01:01
as introduced by Ayala, Francis and Tanaka. Just quickly before I get started, I want to advertise a pair of workshops. Happening in Edinburgh in June, there's a summer school which is happening the 3rd through the 10th,
01:20
and then there's the conference week the next week happening the 10th through the 14th. We have an excellent lineup of speakers, and I'm hoping that many of you can make it. For the summer school, there's some funded positions for students and post-docs,
01:45
and people who attend the school will also be funded to attend for the next week. I hope you can help me advertise this. So that's that. I don't need the projector anymore.
02:06
OK, so I want to start by just recalling Frank Gontroff's construction. First I'll start by reviewing character varieties and their quantizations,
02:32
and then I'll give a more detailed discussion of Frank Gontroff. Now, so these quantum cluster algebras, they're really complicated.
02:50
There's a lot to say about them, and in some sense part of the point of this work is that I never really understood the axioms of cluster algebras, and I wanted to bypass those.
03:02
So my presentation of Frank Gontroff's construction will be a bit biased and a bit abbreviated, but I need to give you the main details, and then I'll talk to you about factorization homology approach
03:21
and parabolic induction. So I think we know what is meant by the character variety of a surface, so equivalently we can think about representations of π1 of the surface into our favorite group G.
03:47
So G can be any reductive group for what I'll say, but I'm especially going to focus attention on GLn, and even the case GL2 is interesting for today. So if we are algebraic, we can think about just maps from the fundamental group into G,
04:04
modulo conjugation action, or if we're a bit more topological, we can think about G local systems on S regarded up to isomorphism.
04:22
Okay, so that's the character variety, and so a basic problem is singular in general,
04:45
but regardless of singular variety, it carries a canonical Poisson bracket, Vitiabat and Goldman, which I imagine people here are familiar with.
05:04
Okay, and so a long standing question that's been answered in many different ways is sort of what's the right way to quantize this Poisson structure, taking into account the singularities, and trying to do this in as natural a way as possible.
05:22
So one way to deal with the singularities is just to pass to the character stack.
05:40
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.
06:02
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 closed. 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.
06:23
Okay, and so in this context, sort of a motivating result, M.G., 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,
06:44
this is the factorization homology of the category of representations of our group G, and so they proved this in the setting of infinity categories, the derived setting,
07:01
and in work I'll mention later, we also proved that the same thing holds just at the abelian level. Okay, so whether you take the derived category at the abelian level, the point is that the classical character variety is computed just by the factorization homology of rep G. So with David Benz V, Adrian Brochet, and myself, we introduced a quantization,
07:33
which I'll denote by ZQS, and the definition of this is just we take the factorization homology
07:44
of the category of representations of the quantum groups. That's a braided tensor category, an 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.
08:01
Okay, so that's just a definition, but this is not a talk about this construction, but I'll just for context say, so we described, so for punctured surfaces,
08:20
which I'll indicate 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
08:42
introduced in the 90s by Alexeyev, Grossa, and Shomerus. Okay, so when your surface is punctured, then 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,
09:07
so first Fock and Rossely for punctured surfaces, they rewrote the Atiyah bot Poisson bracket in a way using classical R matrices,
09:27
and then inspired by this, Alexeyev, Grossa, and Shomerus, they simply proposed by generation relations an algebra, which I call the AGS algebra, where you simply replace the R matrices and the relations with quantum R matrices,
09:44
and so you define by generation 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 category is completely canonical without any choices, and what we showed is that once you have this canonical thing, then you go and make some choices,
10:03
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,
10:21
but we describe CQ of S for closed surfaces via quantum Hamiltonian reduction.
10:41
And I'll also say that work in progress of one of my Ph.D. students relates these to the so-called Skane algebra, Skane categories that people studied also in the 90s. Okay, so the upshot is that for ordinary character varieties, these quantizations are sort of the universal thing, and because they're the universal thing, you can start connecting them to whatever you want.
11:05
Okay, but before all this business about stacks and character stacks and factorization homology and so on, there's another way to deal with both the quantization question and the singularity question,
11:20
which was the idea of Fock and Gantroff. Okay, and what they did is they said let's consider a slightly different variety. Okay, so instead of a surface S, they consider what I'll call a parabolic surface.
11:44
And a parabolic surface for, so in the Fock-Gantroff notation, so in Fock-Gantroff notation, what I'm calling a parabolic surface,
12:03
they would call I think a decorated surface. 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.
12:23
And they suggested to consider a different variety, so I'll just write it like this. Okay, right, and so what we consider is we consider G-local systems on S
12:54
together with a reduction to the Borel at the marked points and a T-framing at the marked points.
13:17
Okay, and we consider these up to isomorphism.
13:21
Okay, so if you're not a geometer, then this reduction to Borel might require some explanation. There's only really two cases to consider. Basically, this is some extra data that we attach. So if you have a marked point on the boundary,
13:40
then this reduction to Borel is just the choice of a flag in the fiber over the marked point. Okay, and if your marked point is at a puncture, then it's the same thing,
14:07
but there's a condition that it be preserved by the holonomy around the puncture. Okay, so what they said is, all right, they didn't like the singularities that come from stabilizers,
14:24
and they basically rigidified the problem. They said, well, okay, we're going to add some extra data of some flags at the marked points and some compatibility in the case of punctures, so that's the reduction to Borel.
14:41
And then the T-framing, so the way I phrased it here, the moduli space is G mod N, and the T-framing just says we'll replace this by, sorry, it's 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,
15:00
which is just a choice of a single vector in each co-kernel, and so basically you're just replacing G mod B by G mod N, but now it has a residual T action.
15:21
What's a line? A line, sorry. Yes, a line, yeah. Well, no, sorry, a vector here, and if we quotient by that T action, then we only consider the line. Okay, so this would be a line, and in here we really pick a vector.
15:45
All right, so what's interesting about this choice that they make is now when we consider up to isomorphism,
16:03
I don't mean as a stack. I really mean we take, 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,
16:27
and so roughly a cluster variety X is a variety which is covered by a bunch of charts, U alpha, and each U alpha is homeomorphic to just an algebraic torus, C star to the R,
16:45
and these are all the same, it's the same dimension R every time, and moreover, there's a Poisson bracket on X, which is quadratic in each chart.
17:12
So what that means, so I'm saying that each chart functions on it is just a Laurent ring and several variables, and the Poisson bracket, this is this famous formula,
17:23
because the Poisson bracket is just X i times X j times some integer a i j, and I'm going to say a little bit more about how you extract these integers in a second. So the Poisson bracket is sometimes called log canonical. It's just the Poisson, so this is what you would expect if you're trying to do
17:47
a sort of canonical Poisson structure on a torus for some matrix a i j here. Right, and so now if you have two of these different tori and you glue them together,
18:03
the transition maps, they have to preserve this Poisson bracket in a suitable sense, and so that implies essentially, so that's almost, in a sense I don't want to get into,
18:21
implies that 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, the 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.
18:44
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,
19:00
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.
19:22
So they say you first need to choose a triangulation. Okay, just to 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 stipulation is that the endpoints of the triangle need to be at one of your marked points,
19:44
and that's it. Okay, so the endpoints of the triangles are at your endpoints. And so the first thing they do is they tell you what to do with the triangle. Okay, so let me show you. I will show you quickly just GL3 so you can see the difference,
20:03
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, like, the combinatorics of root systems and stuff, and for SL2 you see all the basic ideas without all the fuss. Okay, so what they would do is they would draw a triangle like this,
20:23
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.
20:40
Okay, and then there's a certain quiver that they draw that sort of looks like the Triforce if you're a Zelda fan. Okay, and so here it's just like so.
21:03
And, okay, so what you do to get these charts, the C star to the R, the chart associated to this has C star to the number of boxes. Okay, plus the number of circles.
21:24
Okay, so you have a variable for each one of these things, and then those aij's 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've only drawn without repeated edges,
21:42
but you're allowed to have multiple edges stacking, okay? And so the aij's are just this thing. So 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,
22:01
so let me again just do GL3 and GL2, what happens when you glue 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,
22:24
and then the edges that you, and the squares remain squares. So in SL2, already you get now a circle here. And there's a simple calculus for how you fill in a quiver here,
22:47
which I won't go into. So you basically just glue these quivers together, and so in their terminology, these squares are called frozen,
23:01
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
23:23
that'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 just have this boundary here. Okay? Which would be better to replace squares, like half squares.
23:49
Yeah, well, talk to them. All right, and so, okay, and so in the end what you get is each triangulation leads you to some chart, C star to some rank,
24:09
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
24:23
on their character variety for this parabolic surface. And then there's an interesting, so of course if you change triangulation, you get different charts, so you get one of these for each chart. I should say also for SL2, that's basically it.
24:44
In other types, there's some more data that you have to indicate, so each of these triangles is colored by some root data that I don't want to go into. Can you tell us what the open set is in the character variety?
25:01
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 do the rest of the analysis, and then basically, as I will try to explain to you, you just glue these open sets together in some standard way from triangles, and so yeah, that'll be a major part of the talk.
25:20
So this always baffled me. I mean, why should you do that? Why do those give functions? That's a good question. So you get this Poisson bracket. They claim, as far as I know, I don't know what the status of this claim is for General G, but that this Poisson bracket here is the Atiyah bot.
25:41
Somehow, you should think of it as the Atiyah bot Poisson bracket. 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 I kind of gave myself an easy job
26:11
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. I should say, before I go on, though,
26:23
for them, a very important point of this simple Poisson bracket is that when you quantize such a Poisson bracket, you simply say that xi xj equals q to the aij xj xi.
26:41
If you think about that, that's a sort of obvious quantization of this thing. So their quantization of these character varieties that they propose is what they call a quantum cluster ensemble. You don't say sort of once and for all what the quantum gadget is.
27:01
You say what all these charts are, and then you quantize all the change of variables between the charts, and you just declare that whole system of all these quantum tori together with their changes of variables, you declare that to be your quantum character variety. So I would like to understand how to connect that
27:23
to this character stack story. And I just want to give one piece of motivation why this is not just me revisiting the 90s. So Demofte and subsequently Demofte, Gabella, and Gontrov,
27:50
they proposed an algorithm to compute so-called quantum a-polynomials of knots.
28:05
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,
28:23
they explain that if you look at the complement of this knot, this can be tiled by ideal tetrahedron. And an ideal tetrahedron is something like this.
28:44
So it's a tetrahedron. But at the end, there are these facets. So you can tile the complement by these facets.
29:02
And the point is that the union of all these facets is a triangulation of the torus. Because the union of them is just the boundary of the three-manifold.
29:25
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, Dufort-Gontrov description, and we get all these nice quantum tori.
29:42
And then they explain how to use these quantum tori to compute 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 Tudor says quite well in his first paper on this subject,
30:03
is that this choice of tetrahedral triangulation, it's a choice that you need to make. And so you don't know that the corresponding computation that you've done for the would-be quantum a-polynomial is well-defined.
30:21
And so as a first step, I want to explain that the Dufort-Gontrov thing is not a choice. It's a canonical construction. 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 homology,
30:40
then we can actually start proving things about their construction. So we can prove, for instance, that there is a well-defined invariant, and they're just making choices to compute it. So let me get into the story about factorization homology now.
31:11
So Ayala Francis Tanaka plus some modification
31:29
of Benz v. Francis Nadler tells us that if we want to understand quasi-coherent sheaves
31:41
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.
32:01
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 what this means is that whenever I see a line defect, so a parabolic surface... So there's one thing I need to say.
32:22
So in Font-Gantra, I want to translate between these pictures. So you had punctures and marked points. So to connect to factorization homology, we just have to do a sort of trivial change of perspective.
32:41
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.
33:03
And I'm again going to mark that by B. The bulk is always G and the inside is T. So this is just the same data, just a different way to think of it. And now, ordinarily in sort of unstratified factorization homology,
33:21
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 a rep T. And then B is a 1-morphism in a suitable category. This is sort of explained in various places.
33:44
I think in Claudia's thesis there's a nice exposition of why you should think of these as 1-morphisms. So you have B on the line, you have G here, and you have T here. And as a sort of spoiler, when we quantize, we'll have rep QG,
34:05
here we'll have rep QB, and here we'll have rep QT. And there's something you need to check, which is that rep QB has the structure
34:20
to be an interface between rep QG and rep QT. This is sort of classical in some way. The way you construct the R matrix for rep QG is a sort of quantum double of rep QB. And you trace through that, that tells you that this is a morphism between these two braided tensor categories. It's the kind of thing that you can mark here.
34:42
All right. What does this picture mean? What does what picture mean? The one I have at the bottom. This one. So in stratified factorization homology, you consider a stratified, let's say, surface. So this is a stratified surface.
35:00
The principle strata I need to label by some local coefficients. Those local coefficients are some E2 algebra. This is what we learned from Francis Tanaka. And the one-dimensional strata are what's called a locally constant factorization algebra on this stratified space.
35:24
The idea is that 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 just monoid located. It's not braided.
35:42
It's not braided. That's a good point. So let me say more specifically what we need from RepQB. We have a braided tensor functor from RepQG tensor RepQT with the reverse braiding to the Drenfeld center of RepQB.
36:01
So this is what's sometimes called a GT central monoidal category. So it's not just a monoidal category. It has to have these anchor maps down to RepQB. So the first map here, this is just from RepQG to RepQB.
36:21
That's just the forgetful functor. From RepQT to RepQB, 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 getting these questions. This structure right here of a pair of braided tensor categories,
36:45
a monoidal 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 E2 algebras.
37:02
And so I didn't give a formula for factorization homology because I'm sort of taking that as given in this audience. But I'll say that the idea is the same as you do for ordinary factorization homology. For ordinary factorization homology, I would sort of cover this thing by disks.
37:22
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, two category of categories. Here, as well, we cover this thing by disks, but now there are three types of disks. And so we write some, this thing is just defined as some co-limit
37:41
of all the ways of embedding these three kinds of disks and their disjoint unions into S. And then we have some term which is just, repqg to the number of g-disks,
38:02
repqb to the number of b-disks, and repqt to the number of t-disks. So basically these, so every time that we would give a partial covering of this by these three different types of disks,
38:21
we would write down the corresponding categories. And then we take a co-limit in categories and we get some answer. 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.
38:41
That's what we already did in the unmarked case with Benzvi and Broche, and that's what I'd sort of like to do now in the parabolic case. No, it is the same as a, it has a different braiding. It's the same as a monoidal category, but it has the quadratic, the standard quadratic pairing
39:03
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 Le, Schrader, Shapiro, myself,
39:27
just completely following what we did with Benzvi and Broche is the quantum invariant we just defined to be the factorization homology of the parabolic surface
39:41
with rep qg, rep qb, and rep qt. I do apologize for sort of skipping over this sort of calculus. It would be a whole lecture in itself, and we wouldn't get to the punchline in that case.
40:01
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 tensor category, which is an E2 algebra in categories, and at the end we just get a category. Precisely the braiding comes from symmetries of a disk,
40:22
and if you take a general surface, it doesn't have the same symmetries. Okay, so what I want to explain now is some examples. They're the most important examples. So let's look at, okay, so this thing,
40:48
this is just rep qb by construction, and I'm going to call this conf1. So this is zq of conf1, and in general,
41:02
the first class of examples I want to give you is where we take a disk, g, and we put n of these little parabolic induction and restriction things around the boundary, so that's confn. And of course here, I'll stop worrying about g or rep qg.
41:23
I mean, you can treat the classical and the quantum story in the same breath. That's the whole point. So this is confn, so I have a parabolic restriction along each of these things and a t-framing. All right, so if you think about this,
41:41
all this is saying is that confn as a stack is just g mod n cross g mod n, so the number of mark points. That's just the fact that I have to fix a framed flag at each point. It's by definition.
42:00
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 the right. Rather, I don't want to quotient by this. I want to just remember it.
42:22
So in this business, it's... In case you might have a g mod n or a g mod b. Right, because I think you missed this, so... You wrote the reduction to b. Yes, the reduction to b and a t-framing. I think that was before you arrived. Yeah, this is an important point.
42:40
It's not such a big deal, but it's necessary. Reduction to b and t-framing is the same, it's just reduction to n, isn't it? That's right. Yeah, but I remember the t action. That's right. Okay. Very good.
43:00
Okay, so... All right, so... So a 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
43:22
is the one which gave rise to these AGS algebras, and that's somehow a big important point for me. So what I want to understand is what does this distinguished object look like in these cases? And I think I'd just like to state that if I take the internal endomorphisms
43:43
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.
44:00
Okay, so this is the standard quantization, sort of FRT-style quantization of the coordinate ring of the standard affine space. And this is the braided tensor product of algebras.
44:23
And this all takes place in repqg tensor repqt to the n. And now I can explain what I mean by internal endomorphisms.
44:47
So, see, this is where the t and the g action are so useful. This category just carries an action of repqg just by inserting disks into the boundary here,
45:03
and repqt by inserting disks 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, repqt. So we can exploit that. So that says that this is a module category for that braided tensor category,
45:21
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 adjoint 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.
45:41
How is this statement related to the previous statement that qn is equal to this problem? Well, when q equals 1, then I'm saying that... They're kind of quotient by g, isn't it? Right, and here I'm working g equivariantly and t equivariantly. So if I work g equivariantly,
46:00
that's the same as considering the stack quotient mod g. Yeah. Do you know what the labels on the arrows say? This? Yeah, sorry. This says the braided tensor product of algebras. So if I have algebras in a braided tensor category,
46:22
there's a canonical way to combine them, and when 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.
46:40
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 invariants. And so if you think about Peter Bial theorem, that's just the direct sum over lambdas.
47:02
I have the irreducible representation v lambda. That tells me how g acts on this thing, and t acts with weight chi, which is minus omega naught lambda. So that's just the lowest weight for the representation.
47:21
So the highest weight for the dual representation. And 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 t, and that is given with this weight. So this is a very lovely thing.
47:42
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. So note, if you tensor v lambda tensor v mu, you get a whole bunch of different things with different multiplicities.
48:01
But I'm saying that the multiplication in this algebra, because you're fixated on highest weights, it only preserves the thing of the correct weight. So this is a very easy algebra to work with somehow. This n is confusing. This has the same n.
48:22
Did you say what the category was in this case, or we didn't? This is the category that we take as a definition. In this m case, is there an explicit answer for the category? No, yeah, OK, good question. Excellent question. So here I'm talking about the coordinate algebra of functions.
48:40
You may be familiar that g mod n is not an affine space. So it's a little bit dodgy to start all of a sudden talking about 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-gantruff. They do not study the stack or indeed even the variety of these parabolic local systems.
49:01
They only ever study global functions on it. So they themselves are only ever working up to codimension 2. So indeed, the correct sort of quantum invariant is this one, but if I want to connect to front-gantruff, I had better start studying the algebras of functions, and so this is what justifies me looking at this.
49:20
So somehow the distinguished object corresponds to taking global sections. I was just saying that there's a factor from your category to modulus over that algebra. Yes, there's a functor of global sections. Just taking harms with this object. Now, note, I should say one thing. So this was the internal endomorphism. So to get to what you're saying, I need to take the invariance.
49:44
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 still global sections, so the structure sheaf on the stack,
50:03
and if I want to go down to the variety, which is where they work, I need to take invariance for the group. That's just quotients out by the group action. These are really good questions. Thank you. OK. Wait, do you want invariance with g and g? If you take invariance with g and g, you get an outcome.
50:23
Yes, that's right. Sorry, again. We want to take g invariance, but we want to remember the t action. So that's going to be important when we start amalgamating. OK. So what have we done? So I should have said that this thing, this algebra, is what Fock, Gontroff assign to the quantization of this thing.
50:42
So far, we're just following the Aiella-Francis-Tinnocchi prescription, and we get exactly what they do for disks at least with many punctures. But there's a few caveats even for disks. So I want to zoom in on conf3 and then go back to conf2.
51:10
So for conf3, this thing is not a quantum torus. So I'm misleading you somehow. Even when I take g invariance, this is not a quantum torus.
51:21
It does not have these xij 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.
51:44
And that's just the sum of the lambda, tensor v mu, tensor v nu. And so if, for instance, this kind of calculus is in Chen and Gontroff, it wasn't quite explicit in the original papers, but it's certainly there. So we look at a triple tensor product of irreducible representations,
52:03
and we want to know when are there g invariants. And what you find, so in type A, I think this is Chen and Gontroff, and in general type this was due to Ian Lay, as far as I understand. There are certain triples, so there exist triples lambda, mu, and nu,
52:24
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 things will q commute.
52:42
So these guys, this algebra is not a quantum torus, but when we take g invariants and we consider these guys, they q commute. So that means there's some basis in which they just commute by powers of q. And so what we can do is we can invert these.
53:07
And the claim is that once we invert these, so the two-step process is we invert, and then we take g invariants,
53:22
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 oral localization. It's very useful if you're considering oral localizations
53:41
to have q commuting things, because then you can just not worry about fractions. You just have non-commutative fractions. There's some classical geometry here, which is if I consider triples of flags,
54:04
so what have I got? Sort of on this picture, associated to each of these vertices, I'm supposed to imagine I have a flag, f1, f2, and f3. And what's really happening when I invert these things
54:21
is I'm asking that pair-wise each of f1, f2, and f3 are in generic position as flags. And then there's a further genericity that I can ask. I can reflect flag one through flag two, and I want that to be generic with respect to flag three as well. So there's some, I'll just say, biocominatorics
54:44
for producing these charts. And it's a well-understood thing. And so what it allows us to do is in conf3, there's a subcategory of conf3,
55:07
which is just generated by this algebra. So I take that algebra, let me call it a, and I invert these elements.
55:21
I invert these guys, and I consider modules for that thing in repqg, repqt, to the n, which is to say I just look at the subcategory
55:41
where these operators happen to act invertibly. So it's a subcategory of here. And I should have mentioned, by the way, already for conf2, there's something similar going on. So conf2 contains, let me just say classically, so conf2 is g mod n mod n,
56:04
and then I look at the t cross t action, and I have the g action. And there's an open subset in here. So this is conf2.
56:21
There's an open subset where the two flags are in generic position. And if I look at qc of conf2 tilde as a stack, this is just a monoidal category, which is rept. And I want to stress something here. In the formalism I was discussing with Sasha Braberman
56:44
that this can be repqt for the disk, but here this is only monoidal, so I just mean rept. So I'm claiming that there's a non-obvious equivalent There's a monoidal subcategory of qc conf2, which is just rept. And when we quantize, that remains a rept.
57:03
So I'll write it this way. zq conft is a monoidal category rept. All right, so that's conf2, and that corresponds to this picture here. All right, so we're getting... This is a subcategory. This is a subcategory. It's a monoidal subcategory. It corresponds to an open substack of the classical gadget.
57:26
Okay, I'm running a bit short on time. Where are the erasers? Oh, here we go.
57:46
Okay, so I think I'll just have time to explain. Okay, so just to summarize what I've tried to explain is that conf3, we have this open subcategory conf3,
58:05
and this is isomorphic to some quantum torus, so modules over some quantum torus, and it's the same quantum torus that Fuck, Untraff prescribes,
58:21
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... Actually, I have to say, we were stuck for quite a long time before Ian Lay jumped onto the project,
58:41
and it was really this important observation. Suppose I look at conf4. You should think of that as a quadrilateral in Fuck, Untraff picture, and I want to decompose that into triangles. Pictorially, I want to draw a line like this.
59:02
Now, Ayala and Francis Tanaka tell us we have excision for gluing along cylinders, and here, the cylinder that we're gluing along, so here's an obvious statement. This is a union of this triangle, union over this triangle.
59:22
Why are you drawing two vertical lines? This is the cylinder that we're going to glue. This is the interval direction, and this is the direction. And what are we gluing over? We're gluing over this thing. So that's conf3, that's conf3, and that's conf2.
59:46
So it's just a fact of life that the invariant... This is just one example. zq conf4 is, by excision, just zq of conf3.
01:00:01
tensor with zq-conf3 over zq-conf2, OK? And this is hard. This is a stack that we somehow, non-communicative stack, that we don't understand. But we can now look at the tildes, OK?
01:00:21
And the tildes, we do understand. This is a quantum torus. This is a quantum torus. And this is just repT. So what that says in words is that these frozen variables, when we tensor over something which is just representations of a torus, and then we take T invariance for that torus,
01:00:40
what it exactly prescribes is that we multiply these two things together, and we restrict to the degree 0 sub-algebra. And so this is repT, even when we quantize. And this procedure, so excision, from excision
01:01:03
follows the amalgamation prescription of Fock and Gontroff. So I'm already out of time. So 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 Fock and Gontroff.
01:01:21
Thank you. And what do you do in three dimensions? What do you do in three dimensions? Right, so OK, without these restrictions,
01:01:43
without the parabolic markings. We showed with Brochet and Noah Snyder, using work of Claudius and of Mrina Hauksang,
01:02:05
we showed that there's a 3D TFT that extends factorization homology. OK, so you have this coborism hypothesis. You need to check some axioms, and we check those. In what general, I mean, in surface relational homology,
01:02:22
or what? So the claim is that if you take any rigid braided tensor category, it lives 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 Douglas, Schomer, Preis, and Snyder for tensor categories.
01:02:41
And it's the braided analog of that. So to define that four category, that's where we need Claudia and Theo Johnson-Fried and Mrina Hauksang's work. But then you just read the manual, you check some axioms, and you see that you have three dualizabilities since you have rigidity. So now, as I've said, RepQB serves as a one morphism
01:03:04
from RepQG to RepQT in that same four category. And so in order to consider not just a three-dimensional TFT, but a three-dimensional TFT with these defects,
01:03:22
we need to understand dualizability. And if we could understand that dualizability, then you could just sort of, and if some topologists do some work for us, then you could define TFTs
01:03:43
with interfaces. And then you could implement the de Mofti, Gabella, Gontroff, or at least some relative of it, for a given triangulation. So one direction is that you'd like to know that you can make sense of some sort of drawing
01:04:02
where you have sort of G out in space, T out in space, and B along some plane. And so there's some hard technical work that needs to be done there. Another approach I should say is that inside any parabolic surface, you can just cut away all of the parabolic bits.
01:04:22
And then you just get a G surface, a surface colored by G. And you can just try and do TFT with that instead. So that would be a way to sort of bypass these barrels. This is a future hope, but it's certainly not something that's in the works right now.
01:04:41
It doesn't mean you can't say they're just surfaces without marked points. No, so I mean if you have, so I'll just explain you the dictionary. If you have some Gontroff surface like so, so I've said that this is B, T, G, and B, T, G.
01:05:05
Well, what I can do is I can just look at the sub-stratified surface 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 parabolically marked surface
01:05:21
just in a trivial way. And this gives me functors from our usual character variety that we studied with David and Adrian. It includes N to the parabolic one. So there's a functor, an inclusion of functor.
01:05:43
And it comes from, in fact, 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 random non-commutative algebra and try and embed it as a sub-algebra of a quantum
01:06:01
torus. And by thinking about this picture, you get that the AGS algebra associated to this punctured surface embeds into the corresponding cluster ensemble. And so that's another thing you can do.