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In the space of couplings of the 4D N=1 gauge theory associated to D3 branesprobing Calabi-Yau singularities, there is a manifold over which superconformalinvariance 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 exactlymarginal deformation, analogous to the beta-deformation of N=4 SYM, whichinvolves fluxes in the dual gravity description. This beta-deformation existsfor any toric Calabi-Yau singularity.Comment: 30 pages, 5 figures; V2: References added, minor change

Conformal manifolds for the conifold and other toric field theories