Counting conifolds and Dijkgraaf–Vafa matrix models for three matrices

We study superpotential perturbations of q   deformed N=4N=4 Yang–Mills for q a root of unity. This is a special case whose geometry is associated to an orbifold with three lines of codimension two singularities meeting at the origin. We perform field theory perturbations that leave only co-dimension three singularities of conifold type in the geometry. We show that there are two “fractional brane” solutions of the F-term equations for each singularity in the deformed geometry, and that the number of complex deformations of that geometry also matches the number of singularities. This proves that for this case there are no local or non-local obstructions to deformation. We also show that the associated Dijkgraaf–Vafa matrix model has a solvable sector, and that the loop equations in this sector encode the full deformed geometry of the theory.