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Chern-Simons invariants and volumes of representations in Nil, Sol, and Euclidean geometries

  • Taiju Suzuki
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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:π1(M)→ 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.

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Cite

BibTeXRIS
BibTeX
@misc{suzuki2026chernsimons,
  title = {{Chern-Simons invariants and volumes of representations in Nil, Sol, and Euclidean geometries}},
  author = {Taiju Suzuki},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.21266},
  eprint = {2609.21266},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.21266},
}
RIS
TY  - UNPB
TI  - Chern-Simons invariants and volumes of representations in Nil, Sol, and Euclidean geometries
AU  - Taiju Suzuki
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.21266
UR  - https://arxiv.org/abs/2609.21266
AB  - 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:π_1(M)→ 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.
ER  -