Geometric free energy of toric AdS4/CFT3 models

We study the supersymmetric free energy of three dimensional Chern-Simons-matter theories holographically dual to AdS$_4$ times toric Sasaki-Einstein seven-manifolds. In the large $N$ limit, we argue that the square of the free energy can be written as a quartic polynomial of trial R-charges. The coefficients of the polynomial are determined geometrically from the toric diagrams. We present the coefficients of the quartic polynomial explicitly for generic toric diagrams with up to 6 vertices, and some particular diagrams with 8 vertices. Decomposing the trial R-charges into mesonic and baryonic variables, and eliminating the baryonic ones, we show that the quartic polynomial reproduces the inverse of the Martelli-Sparks-Yau volume function. On the gravity side, we explore the possibility of using the same quartic polynomial as the prepotential in the AdS gauged supergravity. Comparing Kaluza-Klein gravity and gauged supergravity descriptions, we find perfect agreement in the mesonic sector but some discrepancy in the baryonic sector.