| 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 Ainfinity-category in such a way that a natural notion of equivalence for quivers with potentials (called "right-equivalence") translates to Ainfinity-equivalence of associated categories [KS08, Section 8].;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 Geibeta, Labardini-Fragoso and Schroer ([LF08], [GLFS13]) who have proved analogous statement in m = 1 case.;In many cases 3d Calabi-Yau Ainfinity-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 ,1, see [B513], [Smil3]. 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. |
