Adjoint Reidemeister torsion of 3-manifolds with torus boundary for semisimple algebraic groups
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| Format: | Preprint |
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2026
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| author | Ishibashi, Tsukasa Mizuno, Yuma |
| author_facet | Ishibashi, Tsukasa Mizuno, Yuma |
| contents | Let $M$ be a compact oriented $3$-manifold with boundary consisting of tori, and let $G$ be a semisimple algebraic group. We define the adjoint torsion function on the moduli stack of $G$-local systems on $M$ satisfying a certain regularity condition, extending the construction by Porti for $G = \mathrm{SL}_2$. When $M$ is a cusped hyperbolic manifold, we prove that the local system associated with the image of the complete hyperbolic structure via a principal embedding $\mathrm{PGL}_2 \to G$ satisfies the regularity condition. Moreover, we provide a formula expressing its adjoint torsion as a product of $\mathrm{PGL}_2$-torsions associated with the simple $\mathrm{PGL}_2$-modules with multiplicity given by the exponents of the Lie algebra of $G$.
We compute the adjoint $\mathrm{PGSp}_4$-torsions of the figure-eight knot complement for two boundary-unipotent local systems, one is arising from the complete hyperbolic structure via a principal embedding, and the other is defined over a number field of degree $6$ and not arising from any $\mathrm{PGL}_2$-local system via principal embeddings. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_00816 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Adjoint Reidemeister torsion of 3-manifolds with torus boundary for semisimple algebraic groups Ishibashi, Tsukasa Mizuno, Yuma Geometric Topology Algebraic Geometry Quantum Algebra Representation Theory 57K31 (Primary), 17B45, 57K32, 57Q10, Let $M$ be a compact oriented $3$-manifold with boundary consisting of tori, and let $G$ be a semisimple algebraic group. We define the adjoint torsion function on the moduli stack of $G$-local systems on $M$ satisfying a certain regularity condition, extending the construction by Porti for $G = \mathrm{SL}_2$. When $M$ is a cusped hyperbolic manifold, we prove that the local system associated with the image of the complete hyperbolic structure via a principal embedding $\mathrm{PGL}_2 \to G$ satisfies the regularity condition. Moreover, we provide a formula expressing its adjoint torsion as a product of $\mathrm{PGL}_2$-torsions associated with the simple $\mathrm{PGL}_2$-modules with multiplicity given by the exponents of the Lie algebra of $G$. We compute the adjoint $\mathrm{PGSp}_4$-torsions of the figure-eight knot complement for two boundary-unipotent local systems, one is arising from the complete hyperbolic structure via a principal embedding, and the other is defined over a number field of degree $6$ and not arising from any $\mathrm{PGL}_2$-local system via principal embeddings. |
| title | Adjoint Reidemeister torsion of 3-manifolds with torus boundary for semisimple algebraic groups |
| topic | Geometric Topology Algebraic Geometry Quantum Algebra Representation Theory 57K31 (Primary), 17B45, 57K32, 57Q10, |
| url | https://arxiv.org/abs/2603.00816 |