Quantum Stability at One Loop for BPS Membranes in a Lorentz-Covariant RVPD Matrix Model

We establish the first rigorous one-loop proof of quantum stability for BPS membranes in the Lorentz-covariant M2-brane matrix model with Restricted Volume-Preserving Deformations (RVPD). Exploiting the closure of restricted $\kappa$-symmetry with RVPD ensures that the BRST complex terminates without an infinite ghost tower, keeping the gauge-fixed measure analytically controllable. Our main result establishes that the 2D, 4D, 6D, and 8D noncommutative membranes remain stable, while the 10D configuration inevitably develops a tachyonic mode. Our analysis unifies the treatment of zero-modes, connects the effective action to central charges, and clarifies relations to BFSS, BLG/ABJM, and prospective M5-brane matrix models, providing a roadmap for RVPD-based extensions.