{
  "annotation_id": "1675fdaa-b553-4881-ad14-86baaf8f6379",
  "date_created": "2026-02-17T05:53:12.645000Z",
  "date_modified": "2026-02-17T05:53:12.645000Z",
  "file_hash": "ce1b270c3ae20ee65d69ce909d526a64f45df917dd71268337248f715ba3508a",
  "private": false,
  "record": {
    "abstract": "The space Loc(m,S) of rank m flat bundles on a closed surface S is K_2-symplectic. A threefold M bounding S gives rise a K_2-Lagrangian in Loc(m,S) given by the flat bundles on S extending to M. We generalize this, replacing the zero section in the cotangent bundle to M by certain singular Lagrangians.\n  First, we introduce Q-diagrams in threefolds. They are collections Q of smooth cooriented surfaces, intersecting transversally everywhere but in a finite set of quadruple crossing points. We require that shifting any surface of the collection from such a point in the direction of its coorientation creates a simplex with the cooriented out faces. The Q-diagrams are 3d analogs of bipartite ribbon graphs.\n  Let L be the Lagrangian in the cotangent bundle to M given by the union of the zero section and the conormal bundles to the cooriented surfaces of Q. Let X(L) be the stack of admissible dg-sheaves on M with the microlocal support in L, whose microlocalization at the conormal bundle to each cooriented surface of Q is a rank one local system.\n  We introduce the boundary dL of L. It is a singular Lagrangian in a symplectic space, providing a symplectic stack X(dL), and a restriction functor from X(L) to X(dL). The image of the latter is Lagrangian. We show that, under mild conditions on Q, this Lagrangian has a cluster description, and so it is a K_2-Lagrangian. It also has a simple description in the non-commutative setting.",
    "arxiv_id": "2601.07538",
    "authors": [
      "Alexander B. Goncharov",
      "Maxim Kontsevich"
    ],
    "categories": [
      "math.AG"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "title": "Non-commutative cluster Lagrangians",
    "url": "https://arxiv.org/abs/2601.07538",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "eb737a24-20f4-45b9-b687-37bf537b183c",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}