The complex geometry of holographic flows of quiver gauge theories

We argue that the complete Klebanov-Witten flow solution must be described by a Calabi-Yau metric on the conifold, interpolating between the orbifold at infinity and the cone over T(1,1) in the interior. We show that the complete flow solution is characterized completely by a single, simple, quasi-linear, second order PDE, or ``master equation,'' in two variables. We show that the Pilch-Warner flow solution is almost Calabi-Yau: It has a complex structure, a hermitian metric, and a holomorphic (3,0)-form that is a square root of the volume form. It is, however, not Kahler. We discuss the relationship between the master equation derived here for Calabi-Yau geometries and such equations encountered elsewhere and that govern supersymmetric backgrounds with multiple, independent fluxes.