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 ψ 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 QβΨ = 0 , Qβ = ( −i ∂ ∂qtA γ βα + 1 2 fABCqsBqtCγ st βα ) ΘαA (1) A,B,C = 1, ..., N − 1, s, t = 1, ..., d (d = 2, 3, 5 or 9), α, β = 1, ..., 2(d− 1) look like; fABC are real, totally antisymmetric structure constants of SU(N), the hermitian fermionic operators ΘαA satisfy canonical anticommutation relations, {ΘαA,ΘβB} = δαβδAB, and the real symmetric γ-matrices satisfy γ αβγ st αβ + γ s α′βγ st αβ + (β ↔ β ′) = 2 ( δαα′γ t ββ − δββ′γ αα )