  In this work we verify consistency of refined topological string theory from
several perspectives. First, we advance the method of computing refined open
amplitudes by means of geometric transitions. Based on such computations we
show that refined open BPS invariants are non-negative integers for a large
class of toric Calabi-Yau threefolds: an infinite class of strip geometries,
closed topological vertex geometry, and some threefolds with compact
four-cycles. Furthermore, for an infinite class of toric geometries without
compact four-cycles we show that refined open string amplitudes take form of
quiver generating series. This generalizes the relation to quivers found
earlier in the unrefined case, implies that refined open BPS states are made of
a finite number of elementary BPS states, and asserts that all refined open BPS
invariants associated to a given brane are non-negative integers in consequence
of their relation to (integer and non-negative) motivic Donaldson-Thomas
invariants. Non-negativity of motivic Donaldson-Thomas invariants of a
symmetric quiver is therefore crucial in the context of refined open
topological strings. Furthermore, reinterpreting these results in terms of webs
of five-branes, we analyze Hanany-Witten transitions in novel configurations
involving lagrangian branes.
