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Talk -- Factorization Homology and Quantum Character Stacks

20 Feb 2026 - Tags: my-talks

Today Yesterday in the Representation Theory Seminar at UCR I gave a talk about Factorization Homology and how it lets us compute a “Quantum Character Stack”. This is all based on a great paper, Integrating Quantum Groups Over Surfaces by Ben-Zvi, Brochier, and Jordan, which I’ve been reading and rereading for the last few years. It’s been a while since I’ve written up my thoughts after a talk, so I figured I’d do that here to take a break from thesis writing.

I have an old post going through a talk I gave on factorization homology almost exactly 2 years ago back in March 2024, which might give a longer perspective on these ideas. I’m going to be fairly terse here because I want to get to the fun computation (which I’ll put in a sister post), and I also want to write this in just a few hours.

The talk was kind of a whirlwild, haha. Especially for my audience, I needed to explain some basics about stacks and the rough idea of factorization homology before I could even hope to get to the actual definition of the quantum character stack! That’s a big ask for an hour long talk, but I think I did alright. I asked my friend Shane how he thought it went, and he very graciously said that I did a good job telling a story and showing that some could, in the abstract, compute things like this… but I didn’t actually show the audience how they can compute with it. I think that’s a fair review, and is pretty consistent with my experience writing and giving the talk. Every professor that I talked to said that it was really good, though, which made me happy. Thankfully that’s also pretty consistent with my experience giving the talk, haha.

I’ve given other really dense talks before, and I remember coming off a bit… energetic, lol. I was pleased that I think I managed to fit a lot of material into this talk while still appearing somewhat collected at the white board. If nothing else, I didn’t end the talk out of breath, haha.

Anyways, enough about my thoughts, let’s get to the talk itself!

The beginning of the talk was meant to motivate stacks to the audience – particularly some younger grad students who have asked me about them before. I actually have a looooong post about stacks in the works, where I talk about how to think of them, how to compute with them, and why you might care. I’ve had to put it on the back burner while I work on my thesis, but hopefully some day I’ll finish it up, since I have a lot of Thoughts™.

Given a surface $\Sigma$ and a reductive group $G$, we would like to have a space whose points are representations of $\pi_1 \Sigma$ valued in $G$. To do this we can look at $\text{Hom}(\pi_1 \Sigma, G)$ and then quotient out by “change of basis” given by conjugation in $G$. This has the extra benefit of removing the reliance of $\pi_1 \Sigma$ on a choice of base point, since a change of base point leads to a conjugate representation.

There are a few things you could mean by the quotient $\text{Hom}(\pi_1 \Sigma, G) \big / G$. The first and most naive is to literally take the space and quotient out by the orbit equivalence relation. This gives a space that isn’t even Hausdorff (and it makes a nice exercise to see why!) so this isn’t great. The more subtle approaches are both based on the observation that a function on $X \big / G$ should be the same thing as a $G$-equivariant function on $X$. If you haven’t seen this before it’s worth taking a second to think about why this should be true!

If you’re a 20th century algebraic geometer you would define the Character Variety $\text{Ch}(\Sigma,G)$ as \(\text{Spec} \big (\mathcal{O}(\text{Hom}(\pi_1 \Sigma, G))^G \big )\). This literally means “the space whose ring of functions is $G$-equivariant functions on $\text{Hom}(\pi_1 \Sigma,G)$”.

If you’re a 21st century geometer, you’re likely to de-emphasize $\mathbb{C}$-valued functions on $X$ (like $\mathcal{O}(X)$) for $\mathsf{Vect}_\mathbb{C}$-valued functions. These assign a vector space to every point in a way that “varies smoothly”, and the way to make this precise is via sheaves! So you find yourself interested in something like $\text{QCoh}(X)$. In this case, you might want to define the Character Stack $\underline{\text{Ch}}(\Sigma,G)$ to be “the space whose category of quasicoherent sheaves is $G$-equivariant sheaves on $\text{Hom}(\pi_1 \Sigma, G)$”.

It turns out that these two spaces are generally not the same! Let’s look at the simplest case where $\Sigma$ is just a disk. Then $\pi_1 \Sigma$ is the trivial group, so $\text{Hom}(\pi_1 \Sigma, G)$ is a point, with ring of functions given by $\mathbb{C}$. Then the $G$-action on this space (and this on the ring of functions) is trivial, so that the $G$-equivariant functions are still $\mathbb{C}$ and $\text{Ch}(\text{Disk},G) = \star$ is a point. In particular, its category of quasicoherent sheaves is just $\mathsf{Vect}$.

But what about the character stack $\underline{\text{Ch}}(\text{Disk},G)$? Well now we define its category of quasicoherent sheaves to be $G$-equivariant sheaves on $\text{Hom}(\pi_1 \Sigma, G) = \star$. So this is $\mathsf{Vect}^G$, which is not $\mathsf{Vect}$! Indeed, when we say that a vector space is $G$-equivariant, what do we mean? We mean that $g \cdot V$ should be “the same as $V$” for every $g \in G$, but the notion of sameness for vector spaces is isomorphism! So saying that $g \cdot V$ is “the same as $V$” is saying we have isomorphisms $\varphi_g : g \cdot V \cong V$. Of course, $G$ is still acting trivially on $\text{Hom}(\pi_1 \Sigma, G) = \star$, so $g \cdot V = V$ and so $\varphi_g$ is an isomorphism of $V$ with itself! These isomorphisms are supposed to be compatible, so we find that $\mathsf{Vect}^G$, the category of $G$-equivariant vector spaces, is actually the category $\text{Rep}(G)$ of vector spaces equipped with a $G$-action! The space whose category of sheaves in $\text{Rep}(G)$ is usually called $\mathsf{B}G$, and we’ll do this too1.

The character stack is better behaved in certain ways. It’s smooth (in a stacky sense) while the character variety usually isn’t2 (this is why it’s common to restrict to a “smooth locus” containing most representations). Moreover, the character stack for $\Sigma$ can be computed by gluing together the character stacks on an open cover for $\Sigma$, while the character variety has no such nice local-to-global property. The character variety also relies crucially on $G$ being reductive, while the character stack works for all groups $G$.

It’s a famous result of Goldman that the smooth locus of the character variety admits a symplectic structure, which quantizes to the ($G$-)skein algebra for $\Sigma$! It turns out the character stack also admits a symplectic structure (in a stacky sense) and it’s natural to want to quantize this too. It should have something to do with skein theory… But what?


Let’s change topic for a moment and talk about Factorization Homology. Again, we’ll be much more terse here than we probably should be.

The notion of an $E_n$-algebra in a monoidal $k$-category $\mathcal{C}$ interpolates between noncommutative algebras ($E_1$) and commutative algebras ($E_\infty$). When $n \gt k$ these stabilize so that $E_{k+1}$ algebras are already “fully commutative”. Since we spend a lot of time working in $1$-categories (like $\mathsf{Set}$ and $\mathsf{Vect}$) we only really see the distinction between $E_1$ (noncommutative algebras) and $E_2 = E_\infty$ (commutative algebras).

However, if we work in a familiar $2$-category like $\mathsf{Cat}$ then we can see a bit further! Now an $E_1$-algebra is a monoidal category, an $E_2$-algebra is a braided monoidal category, and an $E_3 = E_\infty$-algebra is a symmetric monoidal category.

At this point in the talk I said some words introducing braided monoidal categories and why they might be called that, but it’s starting to get late so I think I won’t say those words now. You can read all about this somewhere like here.

Precisely, let $\mathsf{Disk}^n$ be the ($\infty$-)category whose objects are disjoint unions of $n$-disks and whose morphisms are (spaces of) smooth embeddings. Then a functor from $\mathsf{Disk}^n \to \mathcal{C}$ which sends disjoint union to the tensor product in $\mathcal{C}$ is exactly an $E_n$-algebra in $\mathcal{C}$!

Since $\text{Disk}^n$ is a full subcategory of the category of all $n$-manifolds with smooth embeddings, we can try to extend a functor $\text{Disk}^n \to \mathcal{C}$ (read: an $E_n$-algebra $A$) to a functor $\text{Man}^n \to \mathcal{C}$. The free way to do this is via left Kan extension, and this is how we define factorization homology!

The “factorization homology of $M$ with coefficients in $A$”, denoted by $\int_M A$, is defined to be the value of the left Kan extension $\text{Lan}(A)$ on $M$. This admits a “pointwise” formula to compute it, but it’s much much better to use excision! Like any good homology theory, factorization homology has a notion of Mayer-Vietoris for computation.

At this point I included a computation of $\int_{S^1} A$ for an algebra $A$ in $\mathsf{Vect}$ and showed that it recovers the Hochschild homology $HH_0(A)$. Even though I was starting to run out of time, I couldn’t help but mention one of my favorite facts about this too! If you view $A$ as an algebra in chain complexes which happens to be concentrated in degree $0$ then $\int_{S^1} A$ instead computes a derived enhancement, which happens to be $CHH_\bullet(A)$ – the entire complex of Hochschild chains! Since factorization homology is functorial and $S^1$ acts on itself, we get an induced $S^1$-action on $CHH_\bullet(A)$. An $S^1$-action on a chain complex is the data of a new differential on that complex, and it’s natural to ask what differential we get on Hochschild homology from this game! Well the HKR theorem says that $HH_\bullet(A)$ is the algebraic de Rham complex on $\text{Spec}(A)$ (when $A$ is commutative), and the differential coming from this $S^1$-action is exactly the de Rham differential!

Again, I think I want to finish this post up quickly, so I won’t include a copy of this computation here… I feel bad about it, though, so I’ll say that this is done in Hiro Tanaka’s fantastic series of talks starting here.


At this point we’re finally ready to bring the threads together!

One of the main ideas in the Integrating Quantum Groups Over Surfaces paper is that

\[\text{QCoh}(\underline{\text{Ch}}(\Sigma,G)) = \int_\Sigma \text{Rep}(G)\]

The computation I’m including in the sister post is a very explicit very special case of this computation where you can really get your hands on everything. Well… I at least sketch it, haha. You can see if you read that post. I originally planned to include this computation in the talk, but at this point I only had about 5 minutes left and I wanted to make sure I said something about the quantum character stacks in the title of the talk!

The point is that $\text{Rep}(G)$ is symmetric monoidal, so is an $E_\infty$ algebra, but we’re only integrating it over the measly $2$-manifold $\Sigma$! So we could get by with an $E_2$-algebra, which is less commutative! Working in $\mathsf{Cat}$ this means we want a braided monoidal category, and my favorite example is the category $\text{Rep}_q(G)$ of representations of a quantum group!

So now we see what to do:

Generalizing the above formula, we want to say that

\[\text{QCoh}(\underline{\text{Ch}}_q(\Sigma,G) = \int_\Sigma \text{Rep}_q(G)\]

But what does this really mean?

Remember earlier when I said that the 21st century approach to geometry is to focus on the (derived) category of sheaves? Well just like Grothendieck said that every (commutative) ring should count as functions on a space, we might bravely hope that every dg-category should be sheaves on a space! It turns out that one can push this idea very far, and this is one of the modern approaches to Noncommutative Geometry. See, for example, Kontsevich’s fantastic article Geometry in dg-Categories from the equally fantastic book New Spaces in Mathematics – every chapter is a banger.

This “noncommutative” perspective on geometry is what will let us make sense of the quantum character stack as a geometric object, even though we really only have access to what its category of sheaves should be.


At this point I basically had to stop the talk, but I rushed to say a few last minute things that I hoped would convince the audience that this is something that you can get your hands on.

Obviously I mentioned Juliet Cooke’s thesis, where she shows that there’s a concrete skein category defined in terms of tangles in the thickened $\Sigma \times I$ modulo local relations coming from the quantum group $G_q$. This should be compared to the classical skein algebra which is defined in terms of links in $\Sigma \times I$ modulo those same local relations. It turns out that this skein category presents the quantum character stack in the sense that the factorization homology $\int_\Sigma \text{Rep}_q(G)$ is the cocompletion of the skein category.

Also, the Barr-Beck yoga says that any category which looks like a category of algebras should be one, and indeed there’s an algebra object3 \(A_\Sigma\) in \(\text{Rep}_q(G)\) so that \(\int_\Sigma \text{Rep}_q(G)\) is “just” a category of modules over \(A_\Sigma\) (internal to \(\text{Rep}_q(G)\), of course) and from a combinatorial presentation of $\Sigma$ Ben-Zvi, Brochier, and Jordan are able to compute explicit presentations of this internal algebra!

Alright, it’s a quick epilogue today. I would normally put the title, abstract, and slides here, but because it was an internal seminar and I gave a chalk talk I actually have none of those things, haha4.

Thanks for hanging out, everyone! It feels good to write about something that’s not my thesis, and I’m excited to go and write the sister post with this computation! That will have to wait a bit, though, since now it’s dinner time (I succeeded in writing this post in about three hours) and then I’m going climbing with some friends.

Stay safe, and we’ll chat soon ^_^.

  1. You might be familiar with a different notion of $\mathsf{B}G$ from homotopy theory. In that world you take a contractible space with a free $G$-action and then quotient by it to get a “classifying space” where maps from $X$ to $\mathsf{B}G$ are principal $G$-bundles on $X$.

    Up to homotopy a contractible space is a point, so this homotopy-theoretic $\mathsf{B}G$ is also “a point quotiented by $G$” just like our algebro-geometric example. Much of the same intuition goes into thinking about these two notions of $\mathsf{B}G$, but you have to remember that their implementations are different!

    Since a lot of my readers are familiar with topos theory, I’ll say here that the category $G\text{-}\mathsf{Set}$ is a topos, and we often denote it by $\mathsf{B}G$ for this same reason. Indeed, the topos $\mathsf{B}G$ thinks its category of vector spaces is $\text{Rep}(G)$ so this is secretly the algebro-geometric example again. ↩

  2. It’s a fun (but possibly tricky) exercise to compute the dimension of the tangent space at a generic point, then at the trivial representation. Remember that the tangent space to $\text{Ch}(\Sigma,G)$ at a representation $\rho : \pi_1 \Sigma \to G$ is given by the group cohomology

    \[T_\rho \text{Ch}(\Sigma,G) = H^1(\pi_1 \Sigma, \mathfrak{g})\]

    where $\mathfrak{g}$ is a $\pi_1 \Sigma$-module by composing $\rho$ with the adjoint action of $G$ on $\mathfrak{g}$.

    For example let’s take $\Sigma$ to be a punctured torus, whose fundamental group is free on two generators $a$ and $b$, and let’s take $G$ to be $SL_2(\mathbb{C})$. Then a point in $\text{Ch}(\Sigma,G)$ is a representation $\rho$ up to conjugation, is a pair of matrices $A,B \in SL_2$ up to simultaneous conjugation (these are the images of $a$ and $b$ under $\rho$).

    Now $\mathfrak{g} = \mathfrak{sl}_2(\mathbb{C})$ is the space of $2 \times 2$ trace $0$ matrices, and the adjoint action of $G$ on $\mathfrak{g}$ is conjugation! So $\mathfrak{sl}_2$ becomes a $F_2$-module (read: a $\pi_1 \Sigma$-module) by $a \cdot M = A^{-1} M A$ and $b \cdot M = B^{-1} M B$. From here you can compute the group cohomology $H^1(F_2, \mathfrak{sl}_2)$ explicitly, and you’ll see that the generic dimension is not the dimension when $A = B = \text{Id}$. ↩

  3. In fact this algebra comes as a kind of $\text{Rep}q(G)$-valued endomorphism object of the quantum structure sheaf… But I didn’t have time to say any of that at the end of the talk. See the _Integrating Quantum Groups Over Surfaces paper for more. ↩

  4. Actually, writing this has reminded me that I never wrote a talk debrief for my JMM talk about Fukaya categories and my thesis work… Maybe I’ll write a very belated post about that, especially since it does have a title, an abstract, and slides!

    We’ll see, though. I’ve been ridiculously busy lately. ↩