Holonomy Reconstruction and Character Varieties of Lorentzian Curved Tetrahedra
Abstract
We give a recognition and reconstruction theorem for finite strictly convex tetrahedra in dS3 and AdS3 with spacelike, timelike, or null faces. The input consists of four nontrivial based SO+(1,2) holonomies satisfying the closure relation. We construct their Gram data and give a global condition ensuring that the reconstructed tetrahedron lies within the required geometric domain. This condition is strict copositivity of the signed inverse Gram form on the outward branch. Together with nondegeneracy of the Gram data, it guarantees a unique finite tetrahedron up to ambient isometry, with exactly the prescribed Levi--Civita face holonomies. We express this criterion as finitely many polynomial inequalities in trace coordinates, leading to an identification of the finite tetrahedra and a subset of the relative SL(2,ℝ) character varieties of the four-holed sphere. We also discuss the degenerate Gram strata, vector closure in the curvature-zero limit, and the projective dual tetrahedra, which include ideal and hyperideal tetrahedral sectors. The results provide classical geometric input for quantizing Lorentzian curved tetrahedra and for quantum gravity models with a nonzero cosmological constant.
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BibTeX
@misc{liu2026holonomy,
title = {{Holonomy Reconstruction and Character Varieties of Lorentzian Curved Tetrahedra}},
author = {Hongguang Liu and Qiaoyin Pan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.26410},
eprint = {2605.26410},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.26410},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Holonomy Reconstruction and Character Varieties of Lorentzian Curved Tetrahedra
AU - Hongguang Liu
AU - Qiaoyin Pan
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2605.26410
UR - https://arxiv.org/abs/2605.26410
AB - We give a recognition and reconstruction theorem for finite strictly convex tetrahedra in dS^3 and AdS^3 with spacelike, timelike, or null faces. The input consists of four nontrivial based SO^+(1,2) holonomies satisfying the closure relation. We construct their Gram data and give a global condition ensuring that the reconstructed tetrahedron lies within the required geometric domain. This condition is strict copositivity of the signed inverse Gram form on the outward branch. Together with nondegeneracy of the Gram data, it guarantees a unique finite tetrahedron up to ambient isometry, with exactly the prescribed Levi--Civita face holonomies. We express this criterion as finitely many polynomial inequalities in trace coordinates, leading to an identification of the finite tetrahedra and a subset of the relative SL(2,ℝ) character varieties of the four-holed sphere. We also discuss the degenerate Gram strata, vector closure in the curvature-zero limit, and the projective dual tetrahedra, which include ideal and hyperideal tetrahedral sectors. The results provide classical geometric input for quantizing Lorentzian curved tetrahedra and for quantum gravity models with a nonzero cosmological constant.
ER -