Superconductivity in MgB2 Revisited: Analytic Solution of Electrons, Phonons, and Their Coupling in an sp2 Bonded System

MgB2, a superconductor with Tc=39 K, is revisited from a symmetry-driven viewpoint. We construct minimal analytic models for its electrons, phonons, and electron–phonon coupling (EPC) with minimal ab initio input. Strong in-plane B sp2 bonding yields an obstructed effective kagome lattice centered on B–B bonds, naturally producing a small sigma-band Fermi surface with pronounced quantum geometry. The phonon spectrum mirrors that of graphene, with the intercalated Mg hosting low-frequency modes that lift the overall B branches. By symmetry, the boron bond-stretching modes are the only GM-point phonon that couple to the sigma Fermi surface, and thus dominate the EPC. Upon electron doping towards the 2-fold degenerate band edge of the sigma Fermi surface, a decreasing DOS competes with increasing EPC matrix elements. At low doping, the EPC enhancement prevails and raises Tc. Within a Gaussian approximation, we trace this to the quantum-geometric contribution that peaks at GM. Our framework offers a transparent, symmetry-based account of high-Tc phonon-mediated superconductivity in MgB2 and provides quantum-geometry predictors for doping trends.