In the physics literature, dimers show up particularly in the study of mirror symmetry as well as statistical
mechanics. Dimer models offer an interesting source of balanced quivers (every vertex has in-degree equal to out-degree).
Since quivers arising from triangulations are also balanced (with the way we defined them and excluding obvious
counterexamples like ideal triangulations on bordered surfaces with punctures), it may be worthwhile studying
dimer models to see if there are any interesting properties that arise.
Let $S$ be a compact oriented Riemann surface. A bipartite tiling $T$ of $S$ is polygonal cell decomposition
of $S$ whose nodes and edges form a bipartite graph $\Gamma$. We also define
- $\Gamma_0$ to be the set of nodes of $\Gamma$. Since $\Gamma$ is bipartite, we have
$\Gamma_0 = \Gamma_0^\bullet \cup \Gamma_0^\circ$, where we call $\Gamma_0^\bullet$ the black nodes and
- $\Gamma_1$ to be the set of edges of $\Gamma$
- $\Gamma_2$ to be the set of faces of $\Gamma$
If $\Gamma$ is finite, then we call $\Gamma$ a dimer model.
Very forwardly put, quivers are directed multigraphs. One relatively modern interest of quivers
is the idea of quiver mutation. There are a lot of good ways to motivate quiver mutation, all
of which (to my knowledge, at least) involve cluster algebras in some way or another. The approach
we will take is to motivate them through triangulations of a regular $n$-gon (which is rooted in
cluster algebra theory, but we will ignore it).
In this post (and for the rest of the sequence of posts on this topic), we will define a triangulation
(of an $n$-gon) as a subdivision of the $n$-gon up into triangles. This subdivision is obtained by taking
the sides of the $n$-gon along with a maximal collection of non-intersecting diagonals (except possibly
at the endpoints). Here is an example of a triangulation of the regular hexagon.
This is the first post in a sequence of posts I will release regarding some of the topics I covered
in my slides for JMM2024. This is intended to be
largely expository and not much is really new mathematics.
Ultimately, we would like to build up towards a connection between quivers and dimer models obtained
through a variant of quiver mutation known as QP mutation. This is done by taking a dimer model
$\Gamma$ and orienting its graph dual to obtain the associated dimer quiver. It is easy to come
up with dimers where the dimer quiver consists of at least one $2$-cycle. When talking about quiver
mutation specifically, we assume that the quiver has no loops or $2$-cycles. Accordingly, this means
that many dimer quivers are not allowed to be mutated by definition.
This issue can be worked around by introducing a potential. Potentials are elements of $kQ/[kQ, kQ]$
where $kQ$ is the path algebra of the quiver and $[kQ, kQ]$ is the subspace of $kQ$ spanned by the
commutators. Then we use a variant of quiver mutation known as QP mutation as described by Derksen,
Weymann, and Zelevinsky. QP mutation uses the potential to control which $2$-cycles in the graph are
removed during quiver mutation. In our case, this can be exploited to sensibly mutate dimer quivers,
leading to a new kind of transformation on dimer models known as $n$-face urban renewal.
The significance of $n$-face urban renewal is that it corresponds to mutation of quivers with potential
in the case of $n = 4$. Furthermore, it also shares various properties with QP mutation (such as
being an involution).
Today I gave a talk at the Joint Mathematics Meetings. I think I did okay, but I believe that I
hardly did the subject justice as it was necessary to try to fit everything into a 10-minute timewindow.
This personally disastisfies me, so I decided to begin writing a bit of (really) elementary exposition
to some of the ideas I presented on and then giving relevant papers as references here on this blog.
The way I intend to go about this is to release separate posts (whenever I want) regarding the subject.
New content coming soon!
I decided to finally update my blog with something for once. I will be going to the 2024 Joint
Mathematics Meeting in San Francisco and presenting. Feel free to reach out to me at my email
[email protected] or Discord (itzsomebody) if you would like to meet up!