{
  "abstract": "We construct a cohomology theory $H_A(-)$ canonically attached to any 1-dimensional commutative group scheme A over a connective $E_\\infty$-ring. The additive group in characteristic zero gives 2-periodic de Rham cohomology, the multiplicative group gives topological periodic cyclic homology, and an elliptic curve gives elliptic cohomology. The construction extends to the G-equivariant setting for an algebraic group G, yielding in the elliptic case a theory of equivariant elliptic cohomology on algebraic varieties with G-action that takes values in sheaves on the stack of G-bundles. Over the complex numbers, and for G connected reductive, this lifts Grojnowski's theory from the moduli space of G-bundles to the stack. The key player is the category of perfect complexes on A with 0-dimensional support, equipped with the convolution product induced by the group law of A. It plays the role of a categorified circle and is used to define a loop space $L_A^G(X)$, whose structure sheaf gives rise to $H_A^G(X)$ after periodization. Sheaves on $L_A^G(X)$ equivariant under loop rotation provide a category of coefficients for $H_A^G(-)$ in the same way that D-modules provide coefficients for de Rham cohomology. At the point X = *, the theory also gives rise to an $E_\\infty$-ring spectrum (or sheaf thereof) $K_A$, which recovers complex K-theory for $A = \\mathbf{G}_m$, and the sheaf $\\mathcal{O}^{top}$ of topological modular forms for the universal elliptic curve.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05707",
  "author_total": 2,
  "authors": [
    {
      "name": "Emile Bouaziz"
    },
    {
      "name": "Adeel A. Khan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5cdpvscdm36yz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05707",
  "fulltext_url": null,
  "id": "5cdpvscdm36yz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05707",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.AT"
    ],
    [
      "arxiv",
      "math.RT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.05707",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5cdpvscdm36yz.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.05707",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05707"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Equivariant elliptic cohomology via loop spaces",
  "title_text": "Equivariant elliptic cohomology via loop spaces",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:29Z",
  "url": "https://research.draconitic.com/paper/5cdpvscdm36yz",
  "venue": "arXiv",
  "versions": []
}