4d N=2 superconformal linear quivers with type IIA duals

We discuss the gravity duals of 4d N=2 superconformal field theories (SCFTs) arising from the low-energy limit of brane configurations of D4-branes stretched between and intersecting NS5-branes and D6-branes. This gives rise to a product of SU(N_i) groups, with bi-fundamental matter between adjacent groups, and extra fundamental hypermultiplets. The most general configuration in 11d (or type IIA) supergravity that is dual to a 4d N=2 SCFT (when the dual of this SCFT is a weakly curved background) was written down by Gaiotto and Maldacena, but finding it explicitly involves solving a complicated Toda equation. This equation simplifies only when the solution can be reduced to type IIA supergravity, so we ask for which SCFTs of this type is there a type IIA dual that is weakly coupled and weakly curved (away from NS5-branes and D6-branes). We find that such solutions (a special case of which was analyzed by Reid-Edwards and Stefanski) exist when there is a large number of gauge groups, with large ranks, and with large 't Hooft couplings for all but a finite number of groups. The general solution of this type is given by solving an axially symmetric Laplace equation in three dimensions, with specific boundary conditions. We match the parameters of the 4d SCFTs, including the exactly marginal coupling constants, with the boundary conditions for the Laplace equation.