Testing R-parity with geometry

We present a complete classification of the vacuum geometries of all renormalizable superpotentials built from the fields of the electroweak sector of the mssm. in addition to the severi and affine calabi-yau varieties previously found, new vacuum manifolds are identified; we thereby investigate the geometrical implication of theories which display a manifest matter parity (or r-parity) via the distinction between leptonic and higgs doublets, and of the lepton number assignment of the right-handed neutrino fields. we find that the traditional r-parity assignments of the mssm more readily accommodate the neutrino see-saw mechanism with non-trivial geometry than those superpotentials that violate r-parity. however there appears to be no geometrical preference for a fundamental higgs bilinear in the superpotential, with operators that violate lepton number, such as $$ nu hoverline{h} $$ , generating vacuum moduli spaces equivalent to those with a fundamental bilinear.