Quantum Curve for strip geometries, Topological Recursion and open GW/DT invariants

Open topological string partition function gives rise to open Gromov-Witten invariants, open Donaldson-Thomas invariants and 3D-5D BPS indices. Utilizing the remodelling conjecture which connects topological recursion and topological string theory, in this paper we study open topological string theory for the subclass of toric Calabi-Yau threefold known as strip geometries. For this purpose, certain new developments in the theory of topological recursion are applied as its extension to Logarithmic Topological Recursion (Log-TR) and the universal $x$–$y$ duality. Through this we derive the open topological string partition function and also the associated quantum curve. We also explain how this is related to the open Donaldson-Thomas partition function associated with certain symmetric quivers, exponential networks and $q$-Barnes type integrals. In the process, we also connect how 3D-5D wall crossing affects these partition functions as one varies $x$, in examples.