Absence of Zero Energy States in the Simplest d=3 (d=5?) Matrix Models

Abstract

The method introduced in [hep-th/9805020] is simplified, and used to calculate the asymptotic form of all SU(2) \times SO(d=3, resp. 5) invariant wave functions satisfying $Q_{\hat{\beta}} \Psi = 0, \hat{\beta} = 1 ... 4$ resp. 8, where $Q_{\hat{\beta}}$ are the supercharges of the SU(2) matrix model related to supermembranes in d+2=5 (resp. 7) space-time dimensions. For d=3, there exist 2 asymptotic solutions, both of which are constant (hence non-normalizable) in the flat directions, confirming previous arguments that gauge-invariant zero energy states should not exist for d<9. For d=5, however, out of 4 asymptotic singlet solutions (3 with orbital angular momentum $l=0$, one having $l=1$) the one with $l=1$ does fall off fast enough to be asymptotically normalizable, hence requiring further analysis to be excluded as being extendable to a global solution.