Quantum geometry and critical temperature enhancement in MgB$_2$ superconductivity
MgB$_2$, a phonon-mediated superconductor with record-high critical temperature $T_c\simeq 39$ K, is revisited to obtain a comprehensive theory of electrons, phonons, and their coupling with minimal ab initio input. We construct compact analytic models for the electronic structure, phonons, and electron-phonon coupling (EPC) of MgB$_2$. We show that strong in-plane B $sp^2$ bonding realizes an obstructed band structure whose natural description is a bond-centered kagome lattice, yielding small quasi-2D $\sigma$-band Fermi-surface cylinders and pronounced quantum-geometric effects. The phonon spectrum is found to closely track that of a graphene-like boron layer, but the heavy intercalated Mg atoms dominate the three acoustic branches and rigidly lift the boron modes into the optical sector, while the in-plane B-B bond-stretching mode exhibits a pronounced softening along $\Gamma$-A. By symmetry, this $\Gamma$-point bond-stretching mode is the only $\Gamma$ phonon that can couple to the $\sigma$ Fermi surface, explaining its dominant contribution to the EPC. Upon electron doping toward the doubly degenerate band edge of the $\sigma$ sheets, we find that a reduced density of states competes with enhanced EPC matrix elements. At light electron doping, ab initio calculations show that the EPC enhancement dominates, leading to an increase in $T_c$ (within the clean doping limit without disorder effects). Using the Gaussian approximation for the EPC tensor, we further show that this enhancement is overwhelmingly quantum geometric in origin, arising from a geometric EPC contribution of the small $\sigma$ Fermi surface peaked at $\Gamma$. Overall, our results provide a transparent, symmetry-based account of superconductivity in MgB$_2$ and suggest that quantum-geometric effects can be essential for shaping doping trends in phonon-mediated superconductors.