Partition functions of 3d $\hat D$-quivers and their mirror duals from 1d free fermions

We study the matrix models calculating the sphere partition functions of 3d gauge theories with $\mathcal{N}=4$ supersymmetry and a quiver structure of a $\hat D$ Dynkin diagram (where each node is a unitary gauge group). As in the case of necklace ($\hat A $) quivers, we can map the problem to that of free fermion quantum mechanics whose complicated Hamiltonian we find explicitly. Many of these theories are conjectured to be dual under mirror symmetry to certain unitary linear quivers with extra Sp nodes or antisymmetric hypermultiplets. We show that the free fermion formulations of such mirror pairs are related by a linear symplectic transformation. We then study the large N expansion of the partition function, which as in the case of the $\hat A$-quivers is given to all orders in 1/N by an Airy function. We simplify the algorithm to calculate the numerical coefficients appearing in the Airy function and evaluate them for a wide class of $\hat D$-quiver theories.