Fermi gas formalism for D-type quiver Chern-Simons theory with non-uniform ranks

We construct the Fermi gas formalism for the partition function of supersymmetric Chern-Simons theories with affine $D$-type quiver diagrams with non-uniform ranks of the gauge groups and Fayet-Illiopoulos parameters by two different approaches: the open string formalism and the closed string formalism. In the closed string formalism approach, we find a novel connection between the partition function of this theory and the partition function of a four-nodes circular quiver supersymmetric Chern-Simons theory. We also studied a symmetry of a density matrix appeared in the closed string formalism. We further calculate the exact values of the partition function for finite $N$, with which we identified the exponent of the leading non-perturbative effect in $1\mathrm{/}N$ corresponding to the worldsheet instantons in the circular quiver supersymmetric Chern-Simons theories.