ar X iv : h ep - t h / 99 12 16 3 v 1 1 7 D ec 1 99 9 Asymptotic Zero Energy States for SU ( N ≥ 3 )

Some ideas are presented concerning the question which of the harmonic wavefunctions constructed in [hep-th/9909191] may be annihilated by all supercharges. In an attempt to extend our knowledge about the asymptotic form of zeroenergy wave functions of SU(N) invariant supersymmetric matrix models beyond N = 2, it was recently shown [1], for N = 3, how to construct Weyl ⊗ Spin(d) invariant asymptotic states out of the Cartan-subalgebra degrees of freedom. To find out which of these harmonic wavefunctions is annihilated by the asymptotic supercharges [2] Qβ = −iγ βα∇tkΘαk (1) t = 1, . . . , d = (2), 3, 5 or 9 α, β = 1, . . . , sd = 2, 4, 8 resp. 16 k = 1, 2 is non-trivial. The “guess” presented in this note will hopefully be a first step. Should the answer really be that for arbitraryN , already for this “free” problem, exactly one supersymmetric state exists for d = 9 (and none for d = 2, 3, 5), this should obviously have a “simple” (deep?) mathematical explanation, of more general relevance. Each harmonic state, constructed in [1], has the form Ψ = ∑ l,S,R,m rΨ̄ lm (x1,x2)|S ×R;m〉 (2) with l=0,1,. . . ,Ψ̄ lm a harmonic polynomial of degree l (in the variables x1,x2; r := √ x1 + x 2 2) transforming under the Weyl-group (the permutation group S3) and Spin(d) according to the irreducible representation S(= 1,∈ or ρ), cf. [1], A detailed calculation of the matrix element (13) is under investigation by J. Plefka.