{
  "abstract": "We study formal deformations of the natural $\\osp(n|2)$-action, $n\\geq 3$, on the superspace $\\Sc^n_d=\\bigoplus_{k\\geq 0}\\Fc^n_{d-\\frac{k}{2}}$ of symbols of linear differential operators on weighted densities over $\\R^{1|n}$. Starting from the first cohomology space computed in \\cite{10}, we compute the cup-product $\\Hd^1\\vee \\Hd^1\\to \\Hd^2$ which carries the quadratic obstructions. The answer is governed by the $\\osp(n|2)$-invariant operators $A_k=η_1\\cdotsη_n\\partial_x^{k-1}$: the two cocycles $h_k$ and $\\widetilde{h}_k$ spanning the off-diagonal part of $\\Hd^1$ are exactly the two derivatives of the coboundary of $A_k$ with respect to the two weights. Consequently, all the products of two off-diagonal classes and all the products of two diagonal classes vanish, and the whole obstruction is carried, for each $k$, by a single non-trivial 2-cocycle $Ω_k$. If $2d\\notin\\N$ the space $\\Hd^1\\vee\\Hd^1$ is identically zero, so every infinitesimal deformation is integrable. If $2d=m\\in\\N$ we obtain exactly $m$ quadratic integrability conditions, $τ_{2-n-k}(t_k-\\widetilde{t}_k)+τ_k\\widetilde{t}_k=0$, $1\\leq k\\leq m$, and we prove that they are also sufficient: no condition of order $\\geq 3$ occurs and the versal deformation is of degree one in the parameters. In particular every integrable formal deformation is equivalent to its infinitesimal part.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02426",
  "author_total": 2,
  "authors": [
    {
      "name": "Imed Basdouri"
    },
    {
      "name": "Mabrouk Ben Ammar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fjt54kr2y088t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02426",
  "fulltext_url": null,
  "id": "fjt54kr2y088t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02426",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2602|Advanced Topics in Algebra"
    ]
  ],
  "openalex": "W7207718235",
  "pdf_url": "https://arxiv.org/pdf/2609.02426",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fjt54kr2y088t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.02426",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02426"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Deformations of the $\\mathfrak{osp}(n|2)$-action on the superspace of symbols of differential operators on $\\R^{1|n}$",
  "title_text": "Deformations of the osp(n|2)-action on the superspace of symbols of differential operators on R^1|n",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/fjt54kr2y088t",
  "venue": "arXiv",
  "versions": []
}