{
  "id": "1803.06353",
  "version": "v1",
  "published": "2018-03-16T18:01:30.000Z",
  "updated": "2018-03-16T18:01:30.000Z",
  "title": "Potentials for Moduli Spaces of A_m-local Systems on Surfaces",
  "authors": [
    "Efim Abrikosov"
  ],
  "comment": "24 figures",
  "categories": [
    "math.RT",
    "math.AG"
  ],
  "abstract": "We study properties of potentials on quivers $Q_{\\mathcal{T},m}$ arising from cluster coordinates on moduli spaces of $PGL_{m+1}$-local systems on a topological surface with punctures. To every quiver with potential one can associate a $3d$ Calabi-Yau $A_\\infty$-category in such a way that a natural notion of equivalence for quivers with potentials (called \"right-equivalence\") translates to $A_\\infty$-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 $Q_{\\mathcal{T}, m}$. Then we provide a full description of the space of equivalence classes of all \\emph{generic} potentials for the case $m = 2$ (corresponds to $PGL_3$-local systems). In particular, we show that it is finite-dimensional. This claim extends results of Gei\\ss, Labardini-Fragoso and Schr\\\"oer who have proved analogous statement in $m=1$ case. In many cases $3d$ Calabi-Yau $A_\\infty$-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 $Q_{\\mathcal{T},m=1}$. 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-16T18:01:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "moduli spaces",
      "fukaya categories",
      "calabi-yau",
      "equivalence classes",
      "local systems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}