2018/03/16 by Efim Abrikosov, Abrikosov, Efim · 1 citation
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1803.06353
openalex publication_date 2018/03/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study properties of potentials on quivers QT,m arising from cluster coordinates on moduli spaces of PGLm+1-local systems on a topological surface with punctures. To every quiver with potential one can associate a 3d Calabi-Yau A_∞-category in such a way that a natural notion of equivalence for quivers with potentials (called "right-equivalence") translates to A_∞-equivalence of associated categories. For any quiver one can define a notion of a "primitive" potential. Our first result is the description of the space of equivalence classes of primitive potentials on quivers QT, m. Then we provide a full description of the space of equivalence classes of all generic potentials for the case m = 2 (corresponds to PGL3-local systems). In particular, we show that it is finite-dimensional. This claim extends results of Geiß, Labardini-Fragoso and Schröer who have proved analogous statement in m=1 case. In many cases 3d Calabi-Yau A_∞-categories constructed from quivers with potentials are expected to be realized geometrically as Fukaya categories of certain Calabi-Yau 3-folds. Bridgeland and Smith gave an explicit construction of Fukaya categories for quivers QT,m=1. We propose a candidate for Calabi-Yau 3-folds that would play analogous role in higher rank cases, m > 1. We study their (co)homology and describe a construction of collections of 3-dimensional spheres that should play a role of generating collections of Lagrangian spheres in corresponding Fukaya categories.