Factorization of the 3d superconformal index

We prove that 3d superconformal index for general $\mathcal{N}=2$ U(N) gauge group with fundamentals and anti-fundmentals with/without Chern-Simons terms is factorized into vortex and anti-vortex partition function. We show that for simple cases, 3d vortex partition function coincides with a suitable topological open string partition function. We provide much more elegant derivation at the index level for $\mathcal{N}=2$ Seiberg-like dualities of unitary gauge groups with fundamantal matters and $\mathcal{N}=4$ mirror symmetry