The paper under review addresses the fundamental question of finding geometric models for a Calabi-Yau 3-category associated with a quiver with a potential, focusing on a specific example.
Let \(k\) be a field. Consider a quiver \(Q\), which is a finite directed graph. A potential \(W\) on \(Q\) represents a formal \(k\)-linear combination of oriented cycles of the graph. Since an oriented cycle can be seen as a cyclic word on the arrows of \(Q\), one can take a cyclic derivative \(\partial_a W\) of a potential \(W\) with respect to a given arrow \(a\). The Jacobian algebra of \((Q, W)\) is defined as the quotient algebra \(\text{Path}(Q)/(\partial_a W)\), where \(\text{Path}(Q)\) denotes the path algebra generated by arrows and \((\partial_a W)\) is the two-sided ideal generated by \(\partial_a W\). Ginzburg introduced its derived enhancement, denoted as \(\mathcal{D}(Q, W)\), which is a Calabi-Yau 3-algebra now referred to as the Ginzburg DG algebra associated with \((Q, W)\).
The most natural setting for the appearance of such algebras is homological mirror symmetry. Consequently, it becomes important to find a geometric model for them within this context. Specifically, one could attempt to realize \(\mathcal{D}(Q, W)\text{-mod}\) as the Fukaya category of a 6-dimensional symplectic manifold or the derived category of coherent sheaves on a Calabi-Yau 3-fold. Alternatively, one could explore the same question for the Koszul dual algebra \(\mathcal{D}^!(Q, W)\).
In the Koszul dual description, if the quiver \(Q\) lacks self-loops, one has a spherical object \(\mathcal{F}_v\) for each vertex \(v\), meaning \(H^\bullet(\text{End}(\mathcal{F}_v))\cong H^\bullet(S^3)\). Moreover, each arrow from \(v\) to \(w\) in \(Q\) yields a basis element of \(\text{Hom}^1(\mathcal{F}_v, \mathcal{F}_w)\), which determines the other degrees based on the Calabi-Yau property. Finally, the potential \(W\) can be employed to identify an \(A_\infty\)-structure on \(\displaystyle\bigoplus_{a\colon v\to w}\text{Hom}^\bullet (\mathcal{F}_v, \mathcal{F}_w)\); each degree \(d\) component corresponds to the \((d-1)\)-ary operation \(\mu^{d-1}\) of the \(A\infty\)-structure.
The current paper presents geometric models for the simplest quiver (without a self-loop) with a nontrivial potential. That is, consider the quiver consisting of two vertices and arrows \(e\) and \(f\) forming a cycle together with the potential \(W(n) = (ef)^n\). It is worth noting that when \(n\geq 2\), no differentials are involved.
The first model is based on the A-side. The primary construction involves plumbing two copies of \(T^*S^3\) along \(S^1\), where the embedding \(S^1\hookrightarrow S^3\) is assumed to be unknotted. This construction can be viewed as the standard plumbing of 2-manifolds for each fiber of the normal bundle to \(S^1\hookrightarrow S^3\). The resulting manifolds depend on the choice of a trivialization of the normal bundle, which is parametrized by \(\mathbb{Z}\). Up to symplectomorphisms, they can be parametrized by \(\{W_0, W_1, \cdots \}\). Note that there are two core 3-spheres, denoted as \(Q_0\) and \(Q_1\). We denote the subcategory of the Fukaya category \(\text{Fuk}(W_i)\) generated by \(Q_0\) and \(Q_1\) as \(\mathcal{Q}_i\).
The second model utilizes the B-side geometry. Consider \(R_0=k[u,v,x,y]/(uv-xy^2)\) and \(R_i=k[u,v,x,y]/(uv-xy(x^i +y ))\) for \(i\geq 1\), which have isolated singularities at 0. The spectrum \(\text{Spec}(R_i)\) admits a crepant resolution \(Y_i\), resulting in two rational curves \(C_1\) and \(C_2\) intersecting at a point, or \(\mathcal{O}_{C_1}(-1)\) and \(\mathcal{O}_{C_2}(-1)\), both of which turn out to be spherical. We denote the subcategory of the derived category \(\text{Coh}(Y_i)\) generated by \(\mathcal{O}_{C_1}(-1)\) and \(\mathcal{O}_{C_2}(-1)\) as \(\mathcal{C}_i\).
The main results establish that these two categories, \(\mathcal{Q}_i\) and \(\mathcal{C}_i\), represent the given quiver with potential. Specifically, when \(k=\mathbb{C}\), the categories \(\mathcal{Q}_0\simeq \mathcal{C}_0\) represent the case of \(W=0\), and \(\mathcal{Q}_i\simeq \mathcal{C}_1\) represents the case of \(W=W(2) = (ef)^2\). Moreover, if \(k\) is a field with an odd prime characteristic \(p\), then \(\mathcal{Q}_p\simeq \mathcal{C}_p\) represents the case of \(W=W(p)\) (the potential doesn’t really behave as well as we would like in the finite characteristic, but there is a sense in which the result holds). These results also respect the natural symmetry of braid groups. Additionally, by employing the machinery of Koszul duality, the paper proves that the wrapped Fukaya category of \(W_i\) is equivalent to the singularity category \(\text{Coh}(Y_i)/\langle \mathcal{O}_{Y_i} \rangle\).