Conformal manifolds for the conifold and other toric field theories

In the space of couplings of the 4D N = 1 gauge theory associated to D3 branes probing Calabi-Yau singularities, there is a manifold over which superconformal invariance is preserved. The AdS/CFT correspondence is valid precisely for this "conformal manifold". We identify the conformal manifold for all the Y-p,Y-q toric singularities, paying special attention to the case of the conifold, Y-1,Y-0. For a general Y-p,Y-q the conformal manifold is three dimensional, while for the conifold it is five dimensional. There is always an exactly marginal deformation, analogous to the beta-deformation of N = 4 SYM, which involves fluxes in the dual gravity description. This beta-deformation exists for any toric Calabi-Yau singularity.