Quivers, Dimers, and Mutation IV: Quivers with Potential

Recall in the last post, we ran into issues with dimers admitting non-cluster quivers. This is because as purely combinatorial objects, quivers fail to take into account the topology of the surface they might be embedded into. For dimer quivers, there is actually a canonical way of encoding the embeddeding via a quiver with potential.

Quivers with potential, from what I can tell, arose in the work of theoretical physics, especially when dealing with mirror symmetry. In 2007, Derksen-Weyman-Zelevinsky developed a representation theoretic perspective of mutation in their paper, providing an algebraic description of quiver mutation. For us, there is a natural way to assign the potential using the bipartite property of dimer models and the orientation of the surface. From there, we will describe a (simplified) version of the Derksen-Weyman-Zelevinsky mutation of quivers with potential.

[definition] Let $Q$ be a quiver and $kQ$ be its path algebra. A potential W is an element of $kQ/[kQ, kQ]$ where $[kQ, kQ]$ is the subspace spanned by commutators. A tuple $(Q, W)$ is called a quiver with potential. [/definition]

Since any nonzero-element of $[kQ, kQ]$ is a non-cyclic path, it follows that $kQ/[kQ, kQ]$ gives us the cyclic paths in $kQ$. The reason why this perspective is useful from a representation theoretic point of view is that the category of finite-dimensional representations of $Q$ is equivalent to the category of finitely-generated modules over $kQ$. This is not our concern, but a curious reader might consider consulting Quiver Representations by Schiffler. For dimer models specifically, there is a canonical choice of potential for the corresponding dimer quivers:

\[\begin{aligned} W = \sum (\text{cycles of } Q \text{ about } \bullet) - \sum (\text{cycles of } Q \text{ about } \circ), \end{aligned}\]

where the cycles are determined up to cyclical equivalence. As an example, consider the following dimer model along with its dimer quiver:

Here we have $W = \alpha\gamma + \beta\delta - \alpha\beta - \delta\gamma$.

Recall that we made a big deal about quivers being void of $2$-cycles. Now that we have the language of quivers with potential, we can actually relax this constraint by requiring that the potential of the dimer quiver contains no $2$-cycles. This allows us to account for $2$-cycles occuring due to faces bordering each other along more than one edge.

[definition] A potential $W$ of $Q$ is reduced if $W$ has no terms of degree $2$. The reduced part of the pair $(Q, W)$ is defined to be the pair $(Q’, W’)$ obtained by removing both the edges forming any $2$-cycles in $W$ and repeating until no such $2$-cycles exist. [/definition]

To make defining the version of mutation for quivers with potential easier, we will introduce some more formalism into our definition of a quiver.

[definition] A quiver $Q = (Q_0, Q_1, s, t)$ is a direct multigraph consisting of a set $Q_0$ of vertices, a set $Q_1$ of arrows, and two functions $s, t: Q_1 \to Q_0$ that assign each arrow $\alpha\in Q$ its source vertex $s(\alpha) \in Q_0$ and its target vertex $t(\alpha) \in Q_0$. We view $\alpha$ as an arrow from vertex $s(\alpha)$ to vertex $t(\alpha)$. [/definition]

[definition] Let $(Q, W)$ be a quiver with potential. We define $\tilde{\mu}_k(Q)$ to be the quiver obtained by applying the first two steps of quiver mutation. We then define $[W]$ to be the potential on $\tilde{\mu}_k(Q)$ obtained by reversing the orientation of each of the arrows originating from or targetting $k$. We also define

\[\begin{aligned} \Delta_k(Q) = \sum_{\alpha, \beta \in Q_1\mid t(\alpha)=s(\beta)=k} [\alpha\beta]\beta^*\alpha^* \end{aligned}\]

and $\tilde{\mu}_k(W) = [W] + \Delta_k(Q)$, where $\beta^*$ and $\alpha^*$ are $\beta$ and $\alpha$ with reversed orientation. Finally, we define the QP mutation $\mu_k(Q, W)$ at $k$ to be the reduced part of $(\tilde{\mu}_k(Q), \tilde{\mu}_k(W))$. [/definition]

In short, this definition modifies the usual quiver mutation algorithm so that newly created cycles are added and $2$-cycles are deleted only when they are in the potential $W$. This allows us to control which $2$-cycles are allowed to be deleted and which are left alone. This is important because we can then introduce a notion of mutation of dimer quivers that works even in the presense of $2$-cycles. Notice that it is not obvious that the reduced part should always exist. It is guaranteed to exist by the Splitting Theorem of Derksen-Weyman-Zelevinsky.

All of this, of course, is a very simplified version of quivers with potential as well as QP mutation. The reader is encouraged to explore the gory details in Derksen, Weyman, and Zelevinsky’s paper.