Controlled Theory of Skyrmion Chern Bands in Moir\'e Quantum Materials: Quantum Geometry and Collective Dynamics

Recent experiments in moir\'e quantum materials exhibit quantized Hall states without an external magnetic field, motivating continuum mechanisms based on smooth moir\'e-periodic pseudospin textures. We present a controlled theory of skyrmion Chern bands generated by such textures. An exact local $SU(2)$ transformation reveals an emergent non-Abelian gauge field; for large branch splitting we perform an operator-level Schrieffer-Wolff expansion, yielding a single-branch Hamiltonian together with systematically dressed physical operators that define the projected interacting theory beyond strict adiabaticity. The leading dynamics is governed by a $U(1)$ Berry connection whose flux is set by the skyrmion density, while controlled non-adiabatic corrections are fixed by the texture's real-space quantum geometric tensor. In a Landau-level representation built from the averaged emergent field, moir\'e-periodic modulations induce Umklapp-resolved deformations of Girvin-MacDonald-Platzman kinematics and microscopic sources of excess optical quantum weight above the topological lower bound. Assuming a gapped Hall phase, we further derive a skyrmion-crystal effective field theory with a universal Berry-phase term and a noncommutative magnetophonon. Our results provide experimentally accessible signatures for twisted transition-metal dichalcogenide homobilayers and rhombohedral graphene aligned with hexagonal boron nitride.