The character varieties are a remarkable family of spaces that lie at the center of many different active strands of research of the past 30 or 40 years. The basic definition is actually quite simple. We start with a group , and consider the space of representations
This space is an algebraic variety, possibly with singularities, and there is an action of given by conjugation. We can think of this as a change of basis in the representation. The character variety is the quotient
There are a few things to note here. First, we can easily replace by any Lie group in the definition, giving what we can call the -character variety. Second, the group is usually the fundamental group of a manifold . In this case, the Riemann-Hilbert correspondence says that the character variety is (analytically) isomorphic to the moduli space of flat connections on . Roughly, the flat connections are certain kinds of differential equations on the manifold, and by solving them we obtain monodromy representations of the fundamental group. These give the corresponding points of the character variety.
Arguably, the most interesting examples occur when the manifold has dimensions two or three. For example, in dimension two the character variety is a Poisson manifold, whereas, in dimension three, it is closely related to things like Chern-Simons theory and knot invariants. Recently, I’ve been trying to learn a bit more about the relations to low-dimensional topology: things like the Goldman-Turaev Lie bialgebra, string topology, and skein algebras. Below I’ll try to summarize a bit of what I’ve understood so far, along with some questions and confusion.
Warning: In what follows I will be pretty hand-wavy and imprecise, and there are likely mistakes, although I hopefully got the main ideas generally correct.
The Goldman bracket on the character variety.
Let’s start with the character variety of a surface . In this case, the character variety is a Poisson manifold. This goes back to the work of Atiyah and Bott, who constructed a symplectic structure on the moduli space of flat connections over a compact Riemann surface. The way they did this was by constructing the moduli space as the symplectic reduction of the infinite-dimensional space of all connections. This construction is definitely worth learning, but I want to talk instead about a different description of the Poisson structure which is due to Goldman. This description has the advantage of being very geometric.
The first thing we need to do is describe a class of functions on the character variety which are associated to loops. In the physics parlance, these are often called Wilson loop observables. Here’s how it works. Given an element of the fundamental group and a representation , we can take the trace of the evaluation to produce a complex number
Because the trace is invariant under conjugation, this assignment descends to a function on the character variety
But now observe that this allows us to assign a function to each `free homotopy class’ of loops on the surface. Indeed, any loop on the surface can be made into an element of the fundamental group by using a path to attach it to the basepoint. There isn’t a canonical way to do this, but the difference in attaching paths only changes the element up to conjugation. Therefore, when we pass to the function , this choice goes away and we get something well-defined.
With that out of the way, we can describe the Poisson bracket of the functions associated with two loops. Here’s the formula
Let’s unpack what this means. First of all, the sum is taken over the points in the intersection of the two loops. For this, we need to make sure that the loops intersect transversely at finitely many points. Second, is simply the function associated to the loop , which is obtained by concatenating and at the point . The signs are determined by the orientations, but I don’t want to get into this.
Here’s a picture of two loops on a surface:
And here’s a picture of the loops showing up in the Goldman bracket:
Notice that this description of the Poisson bracket only involved talking about the loops, and not the functions they give rise to. In other words, our description of the Poisson bracket is completely agnostic about the representation showing up in the definition of the character variety. What this turns out to mean is that we’ve actually described a universal Poisson bracket that works for many character varieties at once. This is called the Goldman Lie algebra, and formally, it is defined on the vector space spanned by all free homotopy classes of loops in . This is actually only one part of a more elaborate structure, which is called the Goldman-Turaev Lie bialgebra. But I won’t tell you about this today.
At this point, you might object and say that I haven’t fully described the Poisson bracket. I’ve only told you about the bracket between a special class of functions, and I need to tell you about the rest. Well, it turns out that the functions associated to loops generate the entire algebra of functions on character variety. In other words, we get all the functions by adding and multiplying the Wilson loops. Using the chain rule for Poisson brackets, we can then deduce the Poisson bracket for all functions starting from the Goldman Lie bracket of loops.
Given that the character variety is a Poisson manifold, many people feel compelled to quantise it. There are many many many approaches to doing this, and some words that I’ve come by are: cluster theory, quantum groups, skein algebras, factorisation homology, etc. I haven’t really understood what these are, but I have spent a bit of time recently looking into skein algebras. In the first instance, these give a deformation quantization of the character varieties. This was shown in an old paper of Turaev from 1991, which is called ‘Skein quantisation of Poisson algebras of loops on surfaces’. There is also a way of generalising Skein algebras using quantum groups (or ribbon categories) in order to quantise more general character varieties. But I think I’ll stick with the simplest case and only tell you about ordinary skein algebras.
The first step is to thicken our surface so that we have a 3-manifold . Now instead of considering arbitrary loops in , which could have a bunch of self-crossings, we consider the ways of lifting them to nice embedded knots or links in the thickening. We should also think of these not as knots made out of strings, but ribbons. The formal way to state this is that our knots and links are framed.
Okay, now we want to build an algebra out of these links. We start by taking the vector space generated by isotopy classes of framed links. This has a product given by stacking one link above another: given two links and , we think of as living inside and as living inside . To take the product , we stack the thickened surfaces on top of each other by gluing the intervals. Here is a picture:
To obtain the Skein algebra from this, we impose two relations. The first one is about the unknot and I’m going to ignore it. The second is the Skein relation and it’s about what to do with crossings. It is given by the following picture, which is supposed to be about what goes on in a small ball. In this picture we are viewing the surface from above.
The here is a non-zero complex number and we are free to vary it, giving a whole family of algebras. We can also treat it as a formal variable in order to get an algebra over the ring of Laurent polynomials . The image of a link in this algebra is called a skein.
Typically, the resulting algebra is non-commutative, but there is a special case where it is commutative and I want to start by mentioning it. This is the case where the surface is a disc, so that the thickening is isomorphic to . The reason the algebra is commutative in this case is that we can simply horizontally slide knots past each other, in order to interchange them. In fact, the skein algebra in this case turns out to be isomorphic to the ring of Laurent polynomials . The image of a link under this isomorphism is a famous invariant called the Kauffman bracket.
Let’s go back to the case of a general surface and ponder the failure of commutativity. Since the product involves stacking skeins, the difference between and should ultimately come down to the difference between an over-crossing and an under-crossing. Rotating the above Skein identity and taking the difference, we get the following:
Looking at this picture, we see that the algebra becomes commutative when the parameter or . Essentially, this is because the right hand side vanishes, and we can then pass one strand through the other. Intuitively, this is telling us that the interval direction in the thicked surface doesn’t matter, and so we’re back to the case of algebras of loops on a surface. In fact, when the Skein algebra recovers the algebra of functions on the character variety, and when we get a twisted version. Hence, the Skein algebra is a `non-commutative’ character variety.
By looking at the behaviour of the family of algebras near the commutative point , we get an induced Poisson structure on the character variety. It is defined in the following way: given two functions and , their Poisson bracket is given by the following formula involving the commutator
If we apply this to the above picture, we see that the Poisson bracket between two loops is defined in terms of the concatenation of the loops at their intersection points. This is exactly what goes on in the Goldman bracket! The proof that the Skein algebra actually quantises the Goldman bracket in this way is a bit more tricky, but at least intuitively, it isn’t so hard to believe.
Before ending this section, I’ll link to this paper which I found very informative.
String topology and 3-manifold character varieties
Let’s change gears slightly. So far, everything I’ve talked about was for surfaces. When is a 3-manifold the character variety is also interesting. In this case, it isn’t symplectic, but rather, if considered properly as a derived stack, the character variety has a -1-shifted symplectic structure. I won’t say what this means, but its related to words and letters such as Chern-Simons, BV, PTVV, and so on.
Now, what about the Goldman bracket? In the late 90s or early 2000s, Chas and Sullivan discovered that it generalizes to manifolds of any dimension, and they called the resulting collection of structures String topology. I understand this even less, but I’ll make an attempt to explain some of it.
Let’s first go back to the case of a surface , and think about the Goldman bracket a little longer. First, recall that it is defined on , the vector space of free homotopy classes of loops in . In this expression is the loop space of the surface, meaning the space of maps from a circle into our surface. So when we take , we are considering the vector space spanned by the connected components of , and this is supposed to represent the space of free homotopy classes of loops.
In order to generalize this to a higher dimensional manifold , we have to consider all the homology groups of its loop space. In fact, Chas and Sullivan consider the -equivariant homology of the loop space
I’m not exactly sure why they do this, but it means that they are taking the quotient of the loop space by the action of rotating the loops. In effect, this just means that they want to consider loops which dont have a distinguished base point.
One of the things Chas and Sullivan discovered is that the equivariant homology groups have the structure of a (shifted) graded Lie bracket, which specializes to the Goldman bracket in the case of surfaces. I’ll try to describe this in the case of a 3-manifold . For this dimension, the Lie bracket is defined on , which is the graded vector space where we’ve shifted degrees so that lives in degree 0. This means in particular that we have the following two brackets:
I’ll try to describe these, starting with the second bracket, which is a bit simpler. A vector is just a single loop in our 3-manifold . On the other hand, a vector is a path of loops in . In other words, a cylinder. So now what about the bracket . Well, defines a 2-dimensional submanifold of , whereas defines a 1-dimensional submanifold. So, after wiggling things a bit if necessary, these two manifolds will intersect at finitely many points. Each of these points corresponds to a ‘time’ in the path direction of , and at each of these special times, we have two loops intersecting at some points. So, we mimic the Goldman bracket and concatenate these loops. The result is a bunch of loops in , which we add up with some signs to get an element of . Here is a picture:
From here, describing the first bracket is not so difficult. We now have two paths of loops in : . Let’s take to be the path coordinate in . For each fixed value of , we are in the previous situation: a cylinder and a loop meeting at finitely many points. So we can just repeat the previous construction. Now we do this construction as varies, and each of these concatenated loops sweeps out a cylinder. This gives us .
Final questions and confusion
Okay, so how is this related to the character variety? Basically, I don’t know, but I would expect that the string topology bracket gives rise to the shifted Poisson bracket on the character variety just as the Goldman bracket gives rise to the Poisson structure on the character variety of a surface. It certainly has the right degrees to work. There are some papers pointing in this direction. One by Cattaneo, Froehlich, and Pedrini called ‘Topological Field theory interpretation of String topology’, and one by Abbaspour and Zeinalian called ‘String bracket and flat connections’. I haven’t yet managed to understand them, but both of these were written before the recent papers on shifted symplectic and Poisson geometry. So maybe there is now an update to the story.
Another question is whether string topology and Skein algebras are related: I would expect (maybe naively) that the String topology deforms into some sort of Skein algebra. In fact, there is a paper from 2003 by Kaiser called ‘Deformation of string topology into homotopy skein modules’ which seems to be doing just that. Again, I don’t really understand what’s going on here.