Five-brane webs, 3d $\mathcal{N}=2$ theories and quantum curves

We propose a relation between the brane configurations consisting of D3-branes and 5-brane webs which realize 3d $\mathcal{N}=2$ supersymmetric Chern-Simons theories and quantum curves by focusing on the $S^{3}$ partition functions. In particular, we conjecture that the Newton polygons of the quantum curves are equal to the toric diagrams which are dual to the 5-brane webs. For brane configurations whose worldvolume theories have Lagrangian descriptions, we show an explicit derivation of the relation by using the supersymmetric localization and the Fermi gas formalism. We also provide some evidence of the conjecture for non-Lagrangian theories. We see that our conjecture gives us new matrix models for 5-brane webs including a $\left(p,q\right)$5-brane with arbitrary $p$. This leads to explicit relations between the brane configurations, the matrix models and genus one quantum curves.