Mass-Flow Invariance of $Q$-Cohomology in BMN Matrix Quantum Mechanics

Abstract

We study the dependence of the dynamical supercharges of BMN matrix quantum mechanics on the mass parameter $\mu$. Taking the $\mu$-derivative at fixed canonical matrix variables, we show that the sixteen-component supercharge evolves by the adjoint action of a Hermitian quadratic bosonic operator $\mathcal{K}$, together with the spinor-space factor $i\gamma^{123}$. After projection to a $\gamma^{123}$-eigenspace, this flow integrates to a finite similarity transformation. For the nilpotent component $Q(\mu)=\mathcal Q^4_-(\mu)$, one obtains $Q(\mu)=M(\mu,\mu_0)Q(\mu_0)M(\mu,\mu_0)^{-1}$, giving an algebraic mass-flow non-renormalization statement for the $Q$-cohomology. The corresponding Hilbert-space statement has an analytic qualification, parallel to Witten's argument for supersymmetric quantum mechanics: $M$ is non-unitary and unbounded, so its action on the normalizable domain must be controlled. We formulate a small-step criterion by comparing the quadratic growth of $M$ with the Gaussian falloff of BMN oscillator wavefunctions within each component $\mu>0$ or $\mu<0$. As a concrete check, we evaluate this condition in the $N=2$ theory, whose two vacuum sectors are built on the trivial vacuum and the irreducible fuzzy-sphere vacuum. We also compute the induced $Q_{\rm BPS}$-action on the corresponding BPS letters: in the trivial sector it agrees with the standard BMN-sector BPS-letter differential of $\mathcal{N}=4$ SYM, while in the irreducible sector it vanishes.