Resolved Calabi–Yau cones and flows from LabcLabc superconformal field theories

We discuss D3-branes on cohomogeneity-three resolved Calabi–Yau cones over LabcLabc spaces, for which a 2-cycle or 4-cycle has been blown up. In terms of the dual quiver gauge theory, this corresponds to motion along the non-mesonic, or baryonic, directions in the moduli space of vacua. In particular, a dimension-two and/or dimension-six scalar operator gets a vacuum expectation value. These resolved cones support various harmonic (2,1)(2,1)-forms which reduce the ranks of some of the gauge groups either by a Seiberg duality cascade or by Higgsing. We also discuss higher-dimensional resolved Calabi–Yau cones. In particular, we obtain square-integrable (2,2)(2,2)-forms for eight-dimensional cohomogeneity-four Calabi–Yau metrics.