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).