Free energy v.s. Sasaki-Einstein volume for infinite families of M2-brane theories

We investigate infinite families of 3d N = 2 superconformal Chern-Simons quivers with an arbitrarily large number of gauge groups arising on M2-branes over toric CY4's. These theories have the same matter content and superpotential of those on D3-branes probing cones over L-a,L-b,L-a Sasaki-Einstein manifolds. For all these infinite families, we explicitly show the correspondence between the free energy F on S-3 and the volume of the 7-dimensional base of the associated CY4, even before extremization. Symmetries of the toric diagram are exploited for reducing the dimensionality of the space over which the volume of the Sasaki-Einstein manifold is extremized. Similarly, the space of trial R-charges of the gauge theory is constrained using symmetries of the quiver. Our results add to those existing in the literature, providing further support for the correspondence. We develop a lifting algorithm, based on the Type IIB realization of these theories, that takes from CY3's to CY4's and we use it to efficiently generate the models studied in the paper. Finally, we show that in all the infinite families we consider F-2 can be expressed, even off-shell, as a quartic function in R-charges associated to certain 5-cycles. This suggests that a quartic formula on R-charges, analogous to a similar cubic function for the central charge a in 4d, exists for all toric toric CY4's and we present some ideas regarding its general form.