Crystallographic Torus Bundles: Obstructions, Elastic Sectors, and Global Acoustic Modes
Abstract
We formulate the global translational order parameter of a crystal with a fixed finite orientational background as a section of an affine torus bundle induced by the crystallographic extension 1→Π→Γ→ P→1. For a principal P-bundle Q, the configuration bundle is YΠ=Q×P TΠ. We identify two distinct global conditions associated with the same extension class: smooth configurations exist precisely when the pulled-back class fQ*[c]∈ H2(X;ΠQ) vanishes, whereas parallel configurations exist precisely when the restriction of [c] to the monodromy subgroup vanishes. Thus a global translational configuration may exist even when a parallel reference configuration is topologically obstructed. For phase-independent quadratic first-order theories, the affine torsor structure yields an exact global fluctuation identity and a background-independent Hessian about every stationary configuration, without requiring a parallel reference state. We realize the distinction explicitly on a compact flat three-manifold with half-turn holonomy and a nonsymmorphic C2-extension. Although no parallel configuration exists, smooth strained equilibria do. Their homotopy sectors form a torsor for ℤ5⊕(ℤ/2ℤ)2. We determine the minimizers and minimum energy in every sector, derive a strictly positive lower energy bound over all configurations of the quadratic model, and compute the complete isotropic elastic spectrum, including the kernel and first positive eigenvalue. For fixed macroscopic quotient, flat linear bundle, and constitutive coefficients, changing the microscopic affine extension can therefore change the global configuration space and sector energies while leaving the quadratic fluctuation operator unchanged.
Related
- Crystallographic quotients of Artin braid groups and torus bundles over flat $3$-manifolds
- Regular bundles on orbifolds II: ranks and obstructions
- Standard Representations of Crystallographic Coxeter Groups
- The variance of the number of lattice points in narrow sectors
- Counterexamples to torsion-freeness in the K-theory of crystallographic group C*-algebras
BibTeX
@misc{prots2026crystallographic,
title = {{Crystallographic Torus Bundles: Obstructions, Elastic Sectors, and Global Acoustic Modes}},
author = {Aleksey Prots},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.09323},
eprint = {2605.09323},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.09323},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Crystallographic Torus Bundles: Obstructions, Elastic Sectors, and Global Acoustic Modes
AU - Aleksey Prots
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2605.09323
UR - https://arxiv.org/abs/2605.09323
AB - We formulate the global translational order parameter of a crystal with a fixed finite orientational background as a section of an affine torus bundle induced by the crystallographic extension 1→Π→Γ→ P→1. For a principal P-bundle Q, the configuration bundle is Y_Π=Q×_P T_Π. We identify two distinct global conditions associated with the same extension class: smooth configurations exist precisely when the pulled-back class f_Q^*[c]∈ H^2(X;Π_Q) vanishes, whereas parallel configurations exist precisely when the restriction of [c] to the monodromy subgroup vanishes. Thus a global translational configuration may exist even when a parallel reference configuration is topologically obstructed. For phase-independent quadratic first-order theories, the affine torsor structure yields an exact global fluctuation identity and a background-independent Hessian about every stationary configuration, without requiring a parallel reference state. We realize the distinction explicitly on a compact flat three-manifold with half-turn holonomy and a nonsymmorphic C_2-extension. Although no parallel configuration exists, smooth strained equilibria do. Their homotopy sectors form a torsor for ℤ^5⊕(ℤ/2ℤ)^2. We determine the minimizers and minimum energy in every sector, derive a strictly positive lower energy bound over all configurations of the quadratic model, and compute the complete isotropic elastic spectrum, including the kernel and first positive eigenvalue. For fixed macroscopic quotient, flat linear bundle, and constitutive coefficients, changing the microscopic affine extension can therefore change the global configuration space and sector energies while leaving the quadratic fluctuation operator unchanged.
ER -