  We construct local operators in short representations of supersymmetry
algebras from polyvector fields on the quantum moduli space of vacua of
supersymmetric gauge theories. These operators form a super Lie algebra under a
natural bracket operation with structure constants determined by terms in the
operator product expansion of the corresponding operators. We propose a formula
for the superconformal index in terms of an index over polyvector fields on the
moduli space of vacua.
  Along the way, we construct several models with moduli space of vacua
corresponding to affine cones over smooth bases using the classical geometry of
Severi varieties and the Landsberg-Manivel projective geometries corresponding
to the Freudenthal magic square of exceptional Lie algebras. Curiously, we
relate the Landsberg-Manivel projective geometries to the exceptional enhanced
symmetry surprises of Dimofte and Gaiotto. Finally, we determine Beasley-Witten
higher-derivative F-terms in new examples arising from Severi varieties and
remark on their origin in classical projective duality.
