So thank you very much. It's a great honor to be able to be here and wish

Maxime a happy birthday. It wasn't your contract. Yeah, so well, what I'd like to talk about today, what I'd like to say, I mean, Maxime, as you all have known and said many times so far, has been at the origin of a number of conceptual

revolutions, I think, in a lot of different subjects, in fact. And what I'd like to talk about today is, you know, another one of these examples on the subject of basically character varieties, spaces of representations.

So, you know, the space of representations, so let's let X over C be basically a curve.

Maxime, he's asking me, you know, what about higher dimensional case? Okay, let's, for these questions, these questions are already mostly happening for curves, and we don't really know what to say further, I don't think, for higher dimensions yet.

Anyway, so let's take a curve, and so we should think of it as basically being either a compact curve or some kind of open curve, but we keep some control over what's happening at the puncture points and so on. And as I think many of you know, an important part of what's going on is,

also happens for irregular connections and stuff like that but I'm maybe not gonna make that much of a distinction for that here. And so the space of representations, let's say into SLN, let's say R,

this is an affine variety, usually I've been called, also the topologists call this the character variety.

So in particular it's open, and so it's non-compact. And so I think there's recently been some kind of conceptual revolution

in how we can look at the points at infinity. Unfortunately I don't actually understand this very well.

And also I'd like to thank Tom Bridgeland because I think in his talk yesterday he said a lot of things which one should say here. So but let me just, first before getting started, write down some of the people. So obviously Koncevich, Tsoiboman is,

they have a series of papers which are really at the origin, I think, of this. Also Gaiotto and Mornitzky.

So let me apologize, especially since I'm on film. Yeah, okay. I'm gonna be forgetting lots of people here so I'd like to apologize to all the people who I might forget here, which are numerous. Especially as I said, being on film.

Then I'd also like to write Bridgeland Smith. Then there's also people which are basically coming from 3D topology or low dimensional topology

and other similar types of directions. But for example, Richard Wentworth. And there's a whole group of people working roughly speaking around Francois Labouie.

Maybe I'll say Lofton and other people working with them. Ian Lee, for example.

And well, their work is based on stuff which we've actually been using a lot, which is a work by Parrot. And Parrot, I think, was working with for Derek Poulin using Kleiner-Lieb theory.

And this whole thing also goes back to Morgan-Chaland compactifications and so on. And I'd also like to mention Thomas Hausel

and his coworkers, Rodriguez, Villegas, Letelier, and so on. And then there's also a number of recent people which I'm sorry I don't know. For example, Tom mentioned in his talk

a certain number of people, Ikeda and some other people. But there's been several different recent reprints on archive. I'm sorry, I didn't collect all this information.

And I'd also like to say that what I'll be talking about here, aside from the sort of global picture, which is maybe older, but the actual theorems or observations, theorems, conjectures, and stuff like that, are joint work with Luen Mill, Alex Noll,

and Panof Pandit. And I'd also like to say, I mean, this was on some visits to Vienna,

so there's also a number of other people in the group in Vienna which we were discussing these things with. And I think we should particularly put Fabian Haydn, who was helping some discussion.

Okay, and we have a preprint on archive, so I'll be trying to draw some pictures here, but you can also see the much better versions of the pictures in the preprint. Okay, so let me just get started. So to get started, I'd just like to draw some pictures.

These are not the pictures in the preprint, but I'd just like to draw the sort of global picture of the structure that you have on modulite spaces of representation. So let's sort of put that over here. So I'd like to make a box,

but instead of making a box, I'll just leave this top open, okay? So this box is representation. Okay, the next box is connections,

more precisely vector bundles with integrable holomorphic connection, okay? The next box, which I'll put over here, is Higgs bundle.

Now I'd like to connect these together a little bit. So this is the Riemann-Hilbert correspondence. And here, well, we have the solutions

of the hitchen equations, which give the correspondence here. But in fact, what I'd like to draw here is actually more the modulite space of lambda connections. Let's draw it this way.

Okay, so we sort of have a coordinate here, which is lambda. So let me put it in quotes that lambda is sort of equal to H-bar, but I'm not really sure. I mean, you guys know this much better than I do. Sometimes people put an H-bar in here, and I guess the lambda coordinate is often somewhat related to H-bar as it occurs in quantum mechanics.

I'm not sure whether it's really reasonable to think that this parameter lambda, which is kind of a universal parameter that's showing up in math, whether this is really equal to H-bar, or is it just that it plays a similar role in some cases? And so this is lambda equals zero,

and this is lambda equal one. We have a GM action on the space of lambda connections, which scales lambda, and that's why I'm drawing these like this, but the images of the orbits here, the images of the points here, their orbits all go to the sort of ground level state in Higgs-Mundles, which is the nilpotent cone.

And so I left a little bit of room here for the Hitchin map. The Hitchin map maps the modulite space of Higgs-Mundles

to a vector space. So let's just call this A. A-N. I would call it, I guess you could call it B as the base of the Hitchin vibration. B is gonna show up a little later for a different thing, so, and this map is proper,

and for example, these guys go to zero. The nilpotent cone here is just the fiber over zero in the Hitchin map. Whereas this space is affine.

So let me just say with respect to what Richard was saying in the previous talk, I gather that this is a little bit of an analogy which is considered by you guys, maybe, which is the fact that all through,

this is a complex analytic isomorphism. So if you have an affine variety, it can't actually have any compact sub-variety. And that's preserved over to here because it's just the same complex analytic sub-variety. So up until here, there's no compact sub-varieties. So I gather that there's a little bit of an analogy

with this transversal section to the algebraic, to the nodal left-hand locus that Richard was talking about, in the sense that here we have something which has a lot of compact sub-varieties because in fact, it has a proper map, so all the fibers are compact sub-varieties.

But it sort of deforms to a guy which has no compact sub-variety. The nearby fiber here is just the same as this space. So I'm not sure what to do with that further, but I think that's an analogy

which you need to take into consideration here. So the fact that this is an affine variety means that we have, that it's open. That's why I left this open up here. And so we can think of trying to compactify. Okay, now what's a compactification gonna look like? So I'll draw this.

This is actually an example which you can actually calculate. The only one which I know how to actually calculate. In an example, the compactification looks something like this, for example. So the compactification, as I said with a normal crossings divisor, is gonna have a lot of different components

and they're gonna meet in sub-components and so on. Whereas on this side, we can also compactify, but in a different way. So in the moduli space of Higgs bundles, we have a C star action. This is the trace on the fiber lambda equals zero of the C star action, which trivializes

the rest of the space of lambda connections. And so we can just take throughout this piece and divide everybody else by C star. So we can just add at infinity here, a guy which is just the quotient of everything except the bottom here, by the action of C star.

If you think about it a little bit, that's actually not a, it's a smooth, I mean, if this space is smooth, which it happens in many cases, then that's basically gonna be a smooth divisor. It's not actually exactly smooth because it has some orbifold points. The orbifold points are actually interesting too, because they correspond to Higgs bundles,

which are preserved, which are invariant by some finite subgroup of C star. And those include things called cyclic Higgs bundles, which are being used by some people now. But if you think of it as an orbifold compactification, then the divisor here is actually smooth.

Okay, and this just goes to the, sorry, to the sort of PN minus one at infinity of this affine space. Maybe that's cheating a little bit. This affine space actually has, the action of C star is actually weighted here. So this is sort of a weighted, maybe, projective space.

Now it turns out that you can use this picture here to get the same kind of compactification here. In fact, the compactification has exactly the same divisor. So these guys are the same,

which is to say the points here are just, again, points in the space here up to C star invariance. And we can actually see how that works as follows. So if we choose a point out here

which is not on the double-button cone, then we take a section going out to this point. Then we translate this section back by the C star action. So you can see what's gonna happen here from this picture. When you start translating back this section to here,

you're gonna get a curve which goes out to infinity. Okay, maybe I'll, do we have some color here? Let's sort of shade it as it gets closer and closer to infinity.

And that corresponds to the shading here by going out to here. And so not only do we get a curve here, but we actually get a parameter too. Let me call this parameter T, which is basically one over lambda. So if we're calling lambda H-bar,

it's one over H-bar. Well, that's in quotes. So a point at infinity, if we think, we can think of a point at infinity here

on this divisor as being a point in the moduli space of Higgs bundles and going towards that point inside here corresponds to just going towards this point sort of horizontally in the moduli space of lambda connection. So from that, I mean, of course, you have to do the geometrical argument,

but from that, you can see that the divisor is gonna be just the same here and here. So from this point of view, we'd like to think of the behavior at infinity in these two different spaces as being roughly speaking the same. That's obviously a vast oversimplification,

but we can think of it as being roughly speaking the same. I'd like to draw furthermore a sort of a cylinder around this guy. So let's think of taking a neighborhood of infinity.

Now, the cylinder over this guy is gonna go to a neighborhood of infinity in the hitch and base. So let me just say my picture is actually, I'm actually drawing the picture in the case where this is A1, it's just the affine line.

So for the picture, this is actually a disk, okay? So I don't really wanna, this, let's think of the projective line as being sort of in the middle of the disk. In any case, the boundary of this neighborhood at infinity

is actually going to be a sphere. It's a sphere with a kind of hop vibration to Pn minus one in the general case. It's just a circle in the case of one of the base has dimension one. And again, this picture is also a picture.

This is actually a relatively accurate picture in this case of where the hitch and base has dimension one, where the spaces have dimension two, complex dimension two. And it turns out that this is, I learned this in a paper from Goldman and Toledo. This is a triangle in a cubic surface. Okay, so that's a very classical thing,

Frick and Klein or something like that. But if we think of taking the neighborhood of this guy here, it doesn't look a lot like the neighborhood here. But actually, the point is that if you think about it a little bit, it really does look a lot like this neighborhood in the sense that here we have,

if we look at this configuration of divisors at infinity, we see a circle, okay? If we look over here, we also see a circle, okay? So just from this picture, you can see that what's probably happening here is that that circle there is gonna go to this circle here. So when you go once around a point

in a really a little disk in this picture, you're going around an entire divisor of stuff over here. How is, for instance two, there's a certain bound of the vector space. Is that what you're talking about there or? I'm not sure, but I'm gonna be getting to that. I mean, getting to say something more,

somewhat closer to that. Before getting there, let me just say a couple of things. So one is that we can conjecture that, we can conjecture that the incidence complex

of the divisor at infinity in the character,

the divisor at infinity at the character variety is a sphere and that the correspondence, this maps by some kind of homotopy equivalence to the sphere at infinity in the hitchen base.

So Maxime keeps telling me that this is actually a theorem. So I'm not quite sure what the hypotheses of the theorem version of this statement are.

But any case, I think, and it's stated in a phrase and Jan pointed out, it's stated in a phrase in their paper, at least something like this. There's also a theorem which says something a little bit like that in a paper by Richard Wentworth, I think.

And I think that what you said about the Thurston boundary and so on is fairly close to this. This conjecture is only just motivated really by this picture, in fact. Yeah, so if you think, okay, we can think about this a little bit more clearly,

but how are you doing for time? You can make this correspondence a little bit more, a little bit more precise in this case, in just in this example, which is that if you think about. When that one's a triangle, what's the boundary of the middle one, the one with the connections? Well, as I said, this is pretty much the same

as this guy, I mean, roughly speaking. Multiple P1 or what? Well, no, no. How do the lectures hide different attributes? Well, okay, the point is, in the hitch and moduli space, the fibers of this map are elliptic curves in the two-dimensional case. So we're gonna have these, but in general, the general fiber is an abelian variety in any case.

So the fiber is basically gonna be some, let's call it abelian variety, some kind of Jacobian. In fact, it's more of a prim or something, but it's a torus, okay? It's a complex torus,

but in this one-dimensional case, there's no problems with discriminant loci, and in fact, it's always the same elliptic curve, and indeed, when you go once around, that does a minus one on that elliptic curve. I was telling you that yesterday, actually. When you go around here once, that does a minus one on this elliptic curve, okay?

And now, if you look over here, the fiber of, think of squishing the neighborhood of these three lines to the lines themselves, okay? The neighborhood of a point here is, it's a delta star across delta star, basically. So homotopically, the corner neighborhood

of the polydisc that goes around this point is S1 cross S1, and that's gonna correspond to the elliptic curve, in fact. And what about pieces inside here? Pieces inside here, you have to remember that this line is actually a P1. So between these two points, we have a GM. So there's already a one-dimensional circle direction there,

and then there's the other dimensional circle direction of the neighborhood of the disc, okay? So those sort of combine together to again make an elliptic curve, and you can sort of see in this example that this whole thing follows around, and if you follow along these coordinates, you exactly get a minus one also on the torus.

If you identify tori, tori, tori, tori, tori, tori, tori, you get minus one, I mean, something like, I forget why I didn't do this calculation. You have to do this calculation. There's a matrix with two ones, a minus one,

and a zero in it whose cube is minus one. That's going around this guy in that example. I didn't want to do too much detail on this example for several reasons, one of which is that this example is, I think, totally subsumed as a trivial baby case of what Jan and Maxim do.

But okay, so maybe just to continue with the picture a little bit, so we can see here there's some interesting regions which are the neighborhoods of crossing points, okay,

and some other interesting regions which are the neighborhoods of things which are not the crossing points. Now, you can actually calculate, again, in this example, you can actually just explicitly calculate with the equation for certain values. This example is the case of P1 minus four points with a rank two system and some conjugacy classes

at the four points. For some special values of the conjugacy classes, you can actually calculate the map and see what happens. What actually happens is that these regions here, which are small regions here, go to entire sectors over here. More precisely, there's three maps, like there's three walls, so these are really the walls.

Let's look at the pullback of those walls in the cylinder and it's the regions which are not on the wall here which correspond to regions like this here. So basically, this map from one side to the other

has this property that it's exchanging big regions and small regions, okay? I think that's what I would say, I mean, at least as a first epsilon approximation is really the conceptual, new conceptual thing which appears in Koncevich-Sobelman. So they basically have a picture like this in general

and these are the walls of which the wall crossing formula is talking about. So let me just mention here, as far as this is concerned, which is that this conjecture can also be viewed

as a geometric version of the P equals W,

of the P equals W conjecture of Thomas Hauswel and his coworkers. Now, why do we say that? Because basically, the map from the, from the device, the map from the neighborhood of infinity onto just the real incidence graph on this side, okay?

So take the incidence variety and just take a real simplicial complex, okay? That's the thing which we're conjecturing is a sphere here. So that's supposed to go to the map here which just maps you to the sphere at infinity and the hitch and base, okay?

So it's some kind of vague statement that's saying sort of weight stuff because, sorry, this map to this real incidence complex sort of corresponds to some top or bottom piece of the weight filtration, okay? The map that that induces on cohomology is gonna be the highest or lowest piece of the weight filtration in the cohomology of the,

maybe of a neighborhood infinity of the character variety. Whereas this piece, this sphere, if you're looking at the sphere at infinity but in the hitch and base, it's clearly, the cohomology class there pulls back to something which is clearly somehow related to the Larray structure for this vibration, okay?

And Haussel's conjecture is a more precise conjecture saying relating the Larray, some perverse Larray filtration on this side with the weight filtration on this side. So I don't know whether you can, this is only a tiny piece of both sides, if at all.

So I don't know, one could ask, can we really geometrically say that the rest of this conjecture corresponds to some geometric picture? I think that's probably actually feasible once we really understand the wall crossing picture, here I mean. You might say that maybe wall crossing

should actually imply this P equals W conjecture or something like that. Let's see, so what did I want to say next? So this is kind of the global picture here. Now, the next thing I'd like to talk about, maybe I'll leave that up here since the tradition here seems to be to use the side boards.

So let me just leave that up here. Now, what I'd like to talk about now is, well, we were just in Vienna, we were just trying to sort of understand this picture

and understand the relationship with things like stability conditions and stuff like that. Oh yeah, maybe I should say that before. So as I said, I don't really understand this, and I think Tom actually gave a good, a better discussion than I could here,

but what's the kind of zeroth order output of the papers by Koncevich-Sobelman? Says basically that the hitch and base

should be considered as corresponding to a space of stability conditions.

This is really more of a philosophical. Yeah, I don't know, well, I mean,

we've been working on trying to figure out how to get the category and everything from this picture, which we haven't found yet, but maybe you guys know how to do it, but maybe. We didn't see how to prove that, but.

I mean, in any case, I think, I'm not sure about, okay, maybe considered as corresponding to or is equal to. What I'd like to point out, the point I'd like to make here is that, in fact, the idea is that we have these structures. So on some kind of stab of D,

we have a structure of walls. Also, maybe one thing to say here, which Tom said yesterday, is that we have to, we have to integrate this whole thing

over the family, over the moduli space of curves, okay? We're not supposed to just fix a single curve, but we're supposed to put those in together in a family. And so, in some cases, this is the result in the paper of Bridgeland-Smith.

It's not necessary, it's good, but it's, for the large group, it's, you add small percentage of parameters in such a variant curve. It's very important that this is much larger. You never get to pull on the spaces. You can pull the blocks in the circle. Ah, ah, okay.

So you mean there's, there'll be other stability conditions which are not covered by this? Yeah, okay. But in any case, in the space of stability conditions, we have some walls, so if this guy is sitting inside the space of stability conditions, we'll, in any case, have walls which are the intersection of the walls into here. So we're gonna have some walls, okay? And these walls are the things which I drew in the example,

and the idea is that, is that these should be, maybe, let's say, the walls, maybe let me just say the chambers, in fact. The chambers should correspond to points

in the character variety,

in the compactification of the character variety, so points over here. So chambers over there should go to points over here, and the walls, I gather, are supposed to correspond to P1, basically.

So in general, I don't think anybody knows how to write down equations for this in any good way in general. Maybe there I'm talking about the compact case, case of compact Riemann surfaces. In the open case, we have the cluster coordinates, so we can do a little better. I'm not sure whether it's very clear what the higher dimensional pieces look like in here.

Maybe you know. But in any case, what Maxime points out is that we have these P1s which join together points. So part of the idea, I think, is that on a given P1, there's only gonna be two points. Is that accurate? So basically, if we look at the stratification

that you get from compactifying the space of representations, you'll have some points which will be the lowest dimensional strata. Between the points, they'll be connected by some P1s. Those P1s are supposed to correspond to real codimension one walls over here.

When we go from here to here, we do half the dimension. So things which are complex dimension one there could well correspond to real codimension one pieces here. Yeah, so now the SL2 case,

the case of some kind of character varieties with SL2 coefficients has been treated by a number of people and in lots of different ways. So it's a, you know, let's say well treated, among other things by Bridgeland Smith.

And in this case, so this is kind of the start of our discussions that we had with in Vienna. In that case, the hitch and base

is equal to a space of quadratic differentials. A quadratic differential Q just corresponds

to the spectral curve Y squared equals Q. In some kind of cotangent bundle of X. And a big role is played, and it's an important role is played by the foliation,

say a real part of Q equals zero, okay?

And this actually goes back to Thurston theory and so on. Let me just explain maybe, so, so underlying this situation, there's something called a WKB question.

So we can draw the WKB question here basically. So here I drew a curve, here I took a horizontal curve going out to a point in the space of Higgs bundles. If we translate that back into this picture, we get a nice algebraic curve in the moduli space of connections,

which goes out and has a nice transversal structure to the divisor at infinity. And in fact, we have this canonical parameter. So at the divisor at infinity, there's sort of a good way of measuring how close we are to the divisor at infinity. Now, if we take that and look at the monodromy representation,

I think this is what corresponds in Tom's talk to the place where he said, okay, let's take a complex projective structure and look at its monodromy. So we look at the monodromy, but here we can just say, let's take a connection and look at its monodromy, then this is gonna give some kind of weird path inside here, okay?

So this is gonna give some kind of path, let's call this row of T. And the map, so we have a map from basically C or some neighborhood infinity of C into T mapping to row of T.

And this is actually what's known as the WKB problem. And so it turns out that this function has some very nice properties, which I think we could sum up as saying it's a example of,

I mean, it's the example of Voros resurgence. Maybe without writing this down on the board, I can say more precisely. Here we can, even though we don't really know the equations, we can write down some coordinates, the Protez coordinates, which are just the traces of the representation applied to some group element.

And so we can think of, I mean, this is an affine variety, we can embed it by some coordinates. If we look at those coordinates as functions of T, then we can take the Laplace transform of those functions of T. The Laplace transform of those functions of T has a very nice analytic continuation property,

and that's Voros resurgence, basically. So this is a very exponential kind of a curve, but it does have some sort of hidden regularity properties. And what Gaiotto and Moronitsky say

is that they analyze. What's the curve in the connections? You're just scaling the connection, aren't you? We're taking a curve here. So I can be more precise about this. I mean, let's say a typical example, let me write that down here. A typical example of the connection is E, say V T is just, say, a trivial bundle.

Now T is equal to D plus some, so let's take some kind of initial connection,

where if we take the trivial bundle here and we just add a diagonal thing here, all right, well, okay, let's not have a trivial bundle, just some bundle, okay. Let's just choose some connection, nabla zero on the bundle E, and then we add a multiple of the Higgs field, okay?

So E, so the limit over here is E phi. This is one typical example. I mean, there's different ways of,

if you've got a point here, there's different ways of getting a horizontal section that goes out to that point. But roughly speaking, we expect that the behavior over there is sort of the same. That's gonna give you a number of different ways of having a family here that approach this point at infinity here. One, sort of the easiest conceptually easiest way

to write one down is to just take a bundle. Assume that we have a Higgs bundle whose underlying vector bundle admits a connection, which is not always true, in fact. But assuming that it's true, then we can just choose some connection. Then add a large multiple of some of the Higgs field. So this is what's called singular perturbation,

and this is where this WKB thing comes in. We're taking a connection which has the property that it has a large algebraic term. In terms of differential equations, that's what the same thing that you're gonna get if you do something like, you know, this H bar squared d by dz squared plus the potential.

If you turn this guy into a matrix form, to a two-by-two matrix form, and then divide by H bar, then you're gonna exactly get a one over H bar here, basically.

And so the WKB problem is just in general what happens when you have a system of ODEs like that that has a large parameter in it. So then there's the whole question of, I mean, Voron's resurgence is sort of what happens when you try to do exact WKB approximations. Well, so what happens in this SL2 case is the following,

which is that you can actually say what's going on a lot more clearly for this WKB problem. So the WKB problem in the SL2 case, which is, okay,

I'll just draw this picture. We have the foliation defined by the quadratic differential. So in the case of a generic quadratic differential,

this foliation is gonna have just triple points here like that. So these are points where the quadratic differential looks like ZDZ squared. Now suppose we have a point here and a point here, and we'd like to calculate the transport for our connection from the point P to the point Q.

Then, am I gonna succeed here? Let me do this with this color. Then we take a path joining P to Q. If we take a path which is transverse to the, oh, what's gonna happen here, this is not good.

If we take a path like that, which is transverse to the foliation, so gamma. If we take a path which is transverse to the foliation, then the transport for the connection, so TPQ of T,

it's kind of, it's rho of T applied to a group element, but you should actually think of it as a, you should look at the fundamental group wide rather than the fundamental group. So we should look at the transport for the connection going from the point P to the point Q, but the connection depends on T.

Then this is gonna look basically like E to the, some constant times T, and the constant is gonna be the length of the path transverse to the foliation, okay?

So the exponent of the transport matrix is the path length transverse to the foliation. This is not true if you take a bad path which is not transverse to the foliation.

If you took a path like this, then if you try to calculate the length of the path transverse to the foliation, then this real part of Q changes sign as you go past this point. You can see from differential equations, you're supposed to be taking the absolute value

of this guy. But this is gonna give you the wrong answer because we sort of went too far up and then too far back down again, okay? So this doesn't give you the right answer. It just means that if you try to integrate the ODE along this path, it looks like it would have this exponent which is the length along this path, but actually a whole bunch of stuff is canceling out

due to moving the path from here to here, okay? Maybe that's what's called quantum tunneling possibly. Okay, so should be stopping pretty soon.

I think all this, this is, you know, I'm not an expert on this by any means at all. I think we can say that what's going on in Koncevich-Sobelman and which is now, you know, this paper of Ridgeland-Smith and so on. There's this notion of BPS states. I sort of purposely drew this foliation as to be close to a foliation

which is gonna have a BPS state. So if you rotate things a little bit, you can see that there's gonna be a stage where the leaf, the coming out of this singular point equals the leaf going into this singular point. When you go through that stage, then this blue path, you have to sort of change the direction of the blue path as you go across that state, okay?

So our idea was to say that somehow or other the geometry, but we don't know how to make this precise in fact, but that somehow or other the geometry of the BPS states and so on.

So we'd like to say that the geometry of the BPS states corresponds in some way. And this is supposed to be the thing which is governing in some way which I don't understand the wall crossing formulas which fit into this global picture.

The geometry of the BPS states should correspond to looking at the tree which is the quotient by the foliation

which is another way of saying it's the space of leaves of the foliation, okay? So let's call that T. And we have a map, just the projection from X to T. So this guy is a tree and this map, let's call this H. H is what's known as a harmonic map, okay?

And in fact, it's a harmonic phi map where phi in this case is the quadratic differential but in general is the spectral curve, just phi one.

So it's sort of the square. In this case, it's just the square root of Q,

this quadratic differential. So if we take the square root, that's a differential but it's only defined up to sign, okay? And the differential, the differential of H is equal to the real part of phi basically. That's well-defined because there's no well-defined direction on the tree just up to plus or minus one, okay?

Now we have a situation where in this SL2 case, the quadratic differential itself leads to this map from X to T, okay? So let's call this T phi.

This tree only depends on the quadratic differential itself, okay? Now, yeah, yeah, yeah, yeah, okay, sorry. And this whole thing fits in, which I don't have time to talk about this too much but fits into the theory of Perot and so on

which is really going back to the Morgan shale and compactification in fact which is gonna say that if we look at a sequence of connections,

then we're gonna get a limiting harmonic phi map X tilde into some tree T.

So let's call this T omega because it depends on the choice of ultra filter or something like that. And this is for those of you who might wanna look where that is in our paper or know what it is yourself or something like that. It's sort of the cone, it's cone omega.

It's the asymptotic cone underlying here. Again, you'd have to write this down but as many of you know, the underlying, especially the hitch in correspondence, there's this notion of harmonic map to a symmetric space. And if you're thinking, oh, can we think of this tree as being the asymptotic tree

on some kind of Gromov boundary of harmonic maps to a symmetric spaces, the answer is yes. And in fact, that's the way you get it. And that's what Perot's theory says. So, but now, what's the relationship between this guy and this guy? It's just that this guy has an obvious universal property which is anytime you have a harmonic phi map

from X tilde into anything, into any tree, it obviously has to factor through the space of leaves just because phi is constant on the real part of phi, sorry, the real part of phi is zero on leaves so the harmonic map has to be constant on leaves. So this has a universal property like that.

So now, this is what we wanted to look at in the higher rank case. Why do we wanna look at this? Basically, for a number of many of these reasons here. Notably, I guess the idea is that the stability condition is supposed to somehow just depend on the choice

of point and hitch and base. So whereas this limiting tree here might depend on more than just the point and hitch and base. So we'd like to get a structure which only depends on the point and hitch and base.

So now, in our work, we've been looking at how to extend this to the case of just SL3, okay? So SLR or, we don't actually have anything,

we don't know what to say here. No, sorry, I guess some parts of the argument work for anything. So basically, Perrault's very nice paper

extended to group voids extended in a relatively simple way to group voids. Yields the following thing that if we have a phi, which is a, sorry, if we have a WKB question,

so let's say delta T equals delta nabla zero plus T times phi. So if we have a Higgs field phi and we look at a moduli space for connections and I think this should also work for a more general family where the bundles allowed to vary as a function of T and stuff like that. Um, but we just wrote this down in this case.

Yields a construction which corresponds to this situation, limiting, and also, you have to choose an ultra filter, limiting harmonic phi map.

So I'm using this kind of capital phi here for the spectral, for the Higgs field. This smaller phi is the multivalue differential, which is the spectral curve of the Higgs field. I mean, the values of the differential are just all the different eigenforms of capital phi here. And you just get a limiting map to some building,

to some kind of R building. And in the SL2, in the SL3 case, this is modeled on the apartment system, it's just R2 with the S3 group of symmetries and so on.

But I mean, in general, I think this should work for any kind of group or something like that. Um, you know, in general, you'll get the, the apartment system is gonna be the one which, which happens for the Bruatitz buildings for that, for that group, basically.

So we get a limiting harmonic phi map. That, but that's just kind of a formal thing where it actually depends on the ultra filter to take the limits on. Where, roughly speaking, the distance, distance from H of P, H of Q, is some kind of limit from,

it's just that it has to take the ultra filter limit, but it's the limit of one over T times the, the log of the absolute value of rho T. So it's, it's basically the scaling factor C here. It's just that we need to take a limit, and you might want to take a limb soup or something like that.

If the limit equals the, if the limit exists, then any ultra filter limit is gonna be equal to that limit. If it's just a limb soup, but not necessarily a limit, which actually happens in, in fact, notably at the boundaries along the wall, then the ultra filter limit might be different from the limb soup, for example.

But, you know, it's roughly speaking some good approximation to the WKB exponent. This is just the distance in this building. Okay, this is just kind of a formal thing, which I think basically sort of puts into, you know, puts into detail what was always kind of a hope, I think, ever since the theory

of harmonic maps to buildings came about. I mean, this is sort of what you think of them as corresponding to, in terms of, as related to, say, harmonic maps to symmetric spaces. But then the question is, what about, let me try this operation here.

I should stop.

So then the, the statement which we like to show is just that we have the same universal property. So, but let me put this as a conjecture. So the conjecture is that there exists a universal, in fact, for some reasons which I can't explain,

there exists a versatile harmonic phi map to a building from X tilde into some B, some building depending on phi.

So this is only supposed to depend on the spectral curve, so it's only supposed to depend on this multivalued differential, but it's supposed to have this universal property, but except it's only a versatile property, which is that if we have an actual map to a building obtained by that type of a condition, then it's supposed to factor.

The difficulty here is that X has a real two dimension. This guy, for bigger groups, has sort of an arbitrarily big dimension, so this can definitely not surject onto this guy. So on the points here, the many points here which don't come from points here, there's no uniqueness here, and you can actually construct situations where you can see that there.

you shouldn't expect a uniqueness of this map. So that's why it's a versatile guy. And so we could just show that in one particular case in our preprint. So we think this may be, let's say, may be a pre-theorem in progress,

maybe a three-theorem in progress in the SL3 case. In the SL3 case, the idea is to kind of cheat anyway. Here we're cheating a lot, you might say, because we're using the fact that the tree is going to be a quotient of x. So the points in the tree, we know what they are.

They're leaves inside x. Here we maybe cheat a little bit, which is that in the case of SL3, this has dimension 2, and this also has dimension 2. So this guy is actually going to cover some percentage of the points here, you might say. It's going to cover some open sectors inside here.

Let me just finish by maybe the drawing of what this actually looks like. So this is a theorem in the BNR in the example, so CR preprint. But let me just say what the harmonic map looks like in this example.

In this example, the spectral curve has two different ramification points. And there's what's called a caustic line joining these two guys. This region inside here, and these

are the Giodo-Mornitzky spectral network curves. This region here, bounded by the Giodo-Mornitzky spectral curves, maps into here. The rest of the picture actually maps down here.

And this whole thing maps into sort of an initial piece of a building. And the idea is that you can fill it out to make an actual building. But the initial piece of the building is actually pretty easy to think of. You just take two sheets of paper

and glue them together along this region. We'll just take two sheets of paper and glue them together along a region that looks like that. So just put some glue here and glue your two pieces of paper together. So they're not glued together down here.

And this region here, it sort of folds over along this caustic line. And the stuff above here goes to the back sheet, and the stuff below here goes to the front sheet. So that's the example we know how to look at. And then based on what's going on here,

trying to redo the proof a zillion times, we think we may know how to see what to do in the general case for SL3. Is this compactification compatible with mirror symmetry

in the sense that before compactification, we can view Higgs bundles for a group and Langlands dual as mirror pairs? And what happens after compactification? Maxime. I think mirror just changed to be the correct way to do it, but it doesn't change the base.

Yeah. I don't understand this at all, obviously, but my vague understanding is that we should actually somehow or other think, but I don't know how this is supposed to work. But somehow or other, more think of this guy as being mirror dual to this guy. No, I think mirror dual is representation of by one to mirror dual, that's it.

Oh, so you think that if you do mirror dual here, then you should get a mirror dual here? I don't do mirror dual mirror symmetry at all, and Higgs says, and Higgs says, but then just on the backing. But then what happens over here? No, mirror symmetry section, it is clear here.

You must go through this term, and then you put it in 0. But this layer case on the surface is compacted, you can puncture the stories with a more complicated map. No, it doesn't matter. Punches, you don't play your role. It's easy, and it does affect the tension. Well, the question is if mirror symmetry is spoiled by adding divisor to infinity.

So that's it. Ah, ah, ah. What happened to it? I think when you, I mean, I'm not sure, I think, at least in some way, you can think of going from a symbolic guy to a Poisson guy, right? Is that true? When you add the divisor and infinity? Maybe at least in this picture.

Yeah, I remember, so just some time ago, I noticed, because you have, for the last thing, you have, probably the curve is still low dimensional, yeah? Is it possible to extend to some kind of high dimension

of this variety in your familiar connections? Ah, sure, we'll find it. If you try to replace x by a higher dimensional variety. Yeah, or in the neighborhood of curve is high dimension variety. Pure null equation to get familiar connections, can you? I mean, try to do a, try to look at it in just an analytic neighborhood of the curve.

I don't know about that. I mean, yeah, I see. I think I understand your remark a little better. I'm not sure, I mean, you know, the real problem, I mean, certainly the real problem with this whole thing is that we don't have any way of saying what the points in the building,

which are not touched by points of curves, should actually correspond to. And that's the real kind of question here, which, you know, in the tree case, you can say the points of the tree correspond to leaves, so there's some geometrical thing. So that's a good question, I don't know, I don't know. Yeah, and I think it's also the other part,

just to mark it right, to understand what is, how to calculate DT invariance through this character variety, it's actually, it's a very neat equation. We should start with algebraic maps, so C start with character variety. Yeah, and this has the same, it gets some, you have the break variety, it has the same dimensions. Original character variety, and you get some kind of

break change of coordinates. So you think that, like, sort of counting C stars inside here? Oh, oh, the space of C stars inside here. You group the numbers, the numbers, and the variety has the same dimension as your character variety. Ah, okay. And it impacts in kind of two different ways, it gives some break change of coordinates, and it's a little bit DT invariance in terms of A.

You mean the two different ways, or the two end points, some rather, or? Yeah, kind of, yeah, two end points, you mean for all, like, two devices, two-pity, yeah. Yeah, so. You're looking at C stars, so there's gonna be two different points here. So you're thinking of this. No, no, C star will, yeah, we'll connect two different points, but the interior of C star will be

exactly inside character variety. Yeah, yeah, okay, but so that's gonna give you, like, a correspondence between the divisor and itself somehow there. Like, maybe a Hecke-type correspondence or something. Yeah, no, just it's kind of neat question here, too. Yeah, yeah, okay. Depending on the conditions, it's just a little bit different.

Okay, good, so we have a, now we have a correspondence between this divisor and itself, which is the moduli space of C stars, mapping in two different ways to, yeah, okay.