The $\mathcal{N}=2$ Prepotential and the Sphere Free Energy

We study the mass-deformed sphere free energy of three-dimensional $\mathcal{N}=2$ superconformal field theories with holographic duals. Building on previous observations, we conjecture a proportionality relation between the sphere free energy on the boundary and the prepotential of the four-dimensional $\mathcal{N}=2$ supergravity theory in the bulk. We verify this formula by explicit computation in several examples of supergravity theories with vector multiplets and hypermultiplets.