  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.
