Geometric and Topological Aspects of Quadrupoles of Disclinations: Conformal Metrics and Self-Forces

We study the geometric and physical effects of quadrupolar configurations of disclinations using a conformal metric approach in $(2+1)$ dimensions. Two cases are considered: a linear quadrupole, inducing anisotropic curvature with a $\cos (2\theta )$ as profile, and a square quadrupole, yielding a more isotropic field with higher angular harmonics. We solve the Poisson equation for the conformal factor and compute the corresponding Green functions. Using these configurations, we evaluated the electrostatic and magnetostatic self-energies and self-forces for linear sources. The results reveal how symmetry and curvature influence self-interaction effects, with the magnetostatic self-force exhibiting a sign reversal compared to the electrostatic case. Connections with previous models of dislocations and cosmic strings are discussed, with potential applications in graphene, nematics, and gravitational analogs.