{
  "abstract": "In this paper, we realize volumes of representations as real-valued Chern-Simons invariants in Nil, Sol, and Euclidean geometries. To this end, we formulate a Chern-Simons invariant of a pair of connections on a principal bundle that need not be trivial. For a connected closed oriented 3-manifold $M$ and a representation $ρ\\colonπ_1(M)\\to G$ into the identity component $G$ of the isometry group of one of these geometries, we construct an auxiliary connection on the associated flat $G$-bundle. We show that, for a suitably normalized invariant polynomial, the Chern-Simons invariant of the auxiliary and flat connections equals the volume of the representation. For the holonomy representation of a geometric structure, this invariant recovers the Riemannian volume. We also compute the Chern-Simons invariant of the Levi-Civita connection for representative closed manifolds in each of these geometries.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21266",
  "author_total": 1,
  "authors": [
    {
      "name": "Taiju Suzuki"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ctwqpa4acpw76.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21266",
  "fulltext_url": null,
  "id": "ctwqpa4acpw76",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21266",
  "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.DG"
    ],
    [
      "arxiv",
      "math.GT"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7213903464",
  "pdf_url": "https://arxiv.org/pdf/2609.21266",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ctwqpa4acpw76.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.21266",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21266"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Chern-Simons invariants and volumes of representations in Nil, Sol, and Euclidean geometries",
  "title_text": "Chern-Simons invariants and volumes of representations in Nil, Sol, and Euclidean geometries",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/ctwqpa4acpw76",
  "venue": "arXiv",
  "versions": []
}