Asymptotic Factorisation of the Ground-State for SU(N)-invariant Supersymmetric Matrix-Models

We give a simple - straightforward and rigorous - derivation that when the eigenvalues of one of the d = 9 (5,3,2) matrices in the SU(N) invariant supersymmetric matrix model become large (and well separated from each other) the ground-state wavefunction (resp. asymptotic zero-energy solution of the corresponding differential equation) factorizes, for all N > 1, into a product of supersymmetric harmonic oscillator wavefunctions (involving the ‘off-diagonal’ degrees of freedom) and a wavefunction that is annihilated by the free supercharge formed out of all ‘diagonal’ (Cartan sub-algebra) degrees of freedom. During the past few years, zero-energy states in supersymmetric matrix-models have been widely investigated [1-13]. In this paper, we will derive the asymptotic form of the ground-state wavefunction, for arbitrary N > 1. While the result we obtain may not be surprising (A. Smilga has previously stated the emergence of effectively free asymptotic supercharges, referring to work of E. Witten, and himself (cf.[14][15]); and the free Laplacian in [10] should come from an effective, hence also free supercharge) it is perhaps worth giving an explicit proof of how asymptotic solutions of