Geometric modeling of a line of alternating disclinations: Application to grain boundaries in graphene

We develop a conformal geometric model for grain boundaries in graphene based on a periodic line of alternating disclinations. Within the framework of $(2+1)$-dimensional gravity, we solve a reduced form of the Einstein equations to determine the conformal factor, from which the induced metric, scalar curvature, and holonomy are obtained analytically. Each pentagon–heptagon pair is modeled as a disclination dipole, forming a continuous distribution that captures the geometric signature of experimentally observed 5$|$7 grain boundaries. We show that the curvature is localized near the defect line and that the geometry becomes asymptotically flat, with trivial holonomy at large distances. This construction provides a tractable and physically consistent realization of the Katanaev–Volovich framework, connecting topological defect theory with atomistic features of graphene.