The Statistics of Vacuum Geometry

We investigate the vacuum moduli space of supersymmetric gauge theories en masse by probing the space of such vacua from a statistical standpoint. Using quiver gauge theories with ${\cal N}=1$ supersymmetry as a testing ground, we sample over a large number of vacua as algebraic varieties, computing explicitly their dimension, degree and Hilbert series. We study the distribution of these geometrical quantities, and also address the question of how likely it is for the moduli space to be Calabi-Yau.