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

Taiju Suzuki

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arXiv · 2609.21266

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 $ρ\colonπ_1(M)\to G$ 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.

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Taiju Suzuki. 2026-09-18. Chern-Simons invariants and volumes of representations in Nil, Sol, and Euclidean geometries. https://arxiv.org/abs/2609.21266

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