Year: 2026
Type: preprint
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 $1\toΠ\toΓ\to P\to1$. For a principal $P$-bundle $Q$, the configuration bundle is $Y_Π=Q\times_P T_Π$. We identify two distinct global conditions associated with the same extension class: smooth configurations exist precisely when the pulled-back class $f_Q^*[c]\in H^2(X;Π_Q)$ 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 $\mathbb Z^5\oplus(\mathbb Z/2\mathbb Z)^2$. 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.
Source: arXiv (Cornell University)
Author: Prots, Aleksey
Institution: Kuban State University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7160931060