Torsion, valley holonomy, and Berry phases of quasiparticles encircling dislocations and defect complexes in graphene

We investigate the geometric phase acquired by low-energy quasiparticles encircling defects in graphene. Following Anandan, this phase is the holonomy of the Poincare connection; its translational part couples momentum to torsion, giving $P_a\int T^a$. In the Katanaev-Volovich theory, dislocation torsion is confined to the defect core. For graphene, the relevant momentum is \hbar(τ_3\KK+\mathbf q). We show that an isolated dislocation yields a topological valley factor $e^{\mp2πi/3}$ when its Burgers vector \bb lies outside the $\sqrt3\times\sqrt3$ superlattice, together with a dispersive factor e^{-i\mathbf q\cdot\bb} for every nonzero \bb. The topological factor recovers the lattice result of Mesaros et al. as a torsion-flux phase; 4-8 dislocations retain only the dispersive contribution. Composing Volterra elements gives the holonomy of defect complexes. At zero net Frank angle, it depends only on the net Burgers vector. The valley flux obeys a three-coloring rule: pentagons and heptagons contribute according to their Kekule site color, whereas squares and octagons do not. Atomic models confirm that ring content alone does not fix the phase: a 5-8-5 core is invisible as a divacancy or as a dislocation with |\bb|=\sqrt3a, but carries flux $\pm1/3$ for \bb=2\mathbf a_1. The torsion trace produces no geometric phase: its vector coupling violates time-reversal symmetry, while the allowed axial coupling causes only anisotropic short-range scattering.