  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,q}
toric singularities, paying special attention to the case of the conifold,
Y^{1,0}. For a general Y^{p,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.
