{
  "abstract": "We define a derived stack $\\mathscr{Rep}_n(X)$ which generalizes the assignment $A\\leadsto \\operatorname{Rep}_n(A)$ to an algebra of its derived $GL_n$-representation variety of arXiv:1112.1449 from algebras $A$ to perfect stacks $X$ over characteristic 0 fields. In the case of a quasi-projective classical scheme $X$, we show that $\\mathscr{Rep}_n(X)$ admits a subfunctor $\\operatorname{Quot}^{n,\\text{fr}}_{\\mathscr{O}_X^n}(X)\\subset \\mathscr{Rep}_n(X)$, which is in fact represented by a derived scheme almost of finite type. We further construct a Fourier-Mukai integral transform between the derived categories of quasi-coherent sheaves on $X$ and $\\operatorname{Quot}^{n,\\text{fr}}_{\\mathscr{O}_X^n}(X)$ which induces a trace map at the level of Hochschild homology generalizing the trace morphism constructed in arXiv:1112.1449 (for finitely presented commutative algebras). We show that the subfunctor $\\operatorname{Quot}^{n,\\text{fr}}_{\\mathscr{O}_X^n}(X)$ is a derived enhancement of the framed locus of the Quot scheme of points and that this derived scheme is a $GL_n$-torsor over the stack of coherent length $n$ torsion sheaves. Hence, this stack is an analog of the derived character stack for quasi-projective schemes, and we show that for smooth, Calabi-Yau $X$, it inherits a shifted symplectic structure in the sense of arXiv:1111.3209 from the one constructed in arXiv:1812.11913 on the moduli stack of perfect complexes with proper support. This structure is shown to give a generalization of the classical symplectic structure on character varieties of surfaces of genus 1 constructed by Goldman \\cite{gold}.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09633",
  "author_total": 1,
  "authors": [
    {
      "name": "Jacob Erlikhman"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/40m3yrkkzc99n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09633",
  "fulltext_url": null,
  "id": "40m3yrkkzc99n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09633",
  "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.AG"
    ],
    [
      "arxiv",
      "math.KT"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "arxiv",
      "math.SG"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7212139434",
  "pdf_url": "https://arxiv.org/pdf/2609.09633",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/40m3yrkkzc99n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.09633",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09633"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.SG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Representation Varieties of Stacks and Trace Maps",
  "title_text": "Representation Varieties of Stacks and Trace Maps",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/40m3yrkkzc99n",
  "venue": "arXiv",
  "versions": []
}