  We consider type IIB string theory with $N$ D3 branes and various
configurations of sevenbranes, such that the string coupling $g_s$ is fixed to
a constant finite value. These are the simplest realizations of F-theory, and
are holographically dual to rank $N$ Argyres-Douglas conformal field theories
(CFTs) with $SU(2)$ and $SU(3)$ flavor groups, and Minahan-Nemeschansky CFTs
with $E_6, E_7$ and $E_8$ flavor groups. We use the Seiberg-Witten curves of
these theories to compute the mass deformed sphere free energy $F(m)$ at large
$N$ in terms of novel matrix models with non-polynomial potentials. We show how
$F(m)$ can be used along with the analytic bootstrap to fix the large $N$
expansion of flavor multiplet correlators in these CFTs, which are dual to
scattering of gluons on $AdS_5\times S^3$, and in the flat space limit
determine the effective theory of sevenbranes in F-theory. As a first step in
this program, we use the matrix models to compute the $\log N$ term in $F(m)$
and thereby fix the logarithmic threshold in the $AdS_5\times S^3$ holographic
correlator, which matches the flat space prediction.
