Refined open topological strings revisited

TL;DR

This paper verifies the consistency of refined topological string theory by computing refined open amplitudes, establishing non-negativity of refined open BPS invariants, and relating them to quiver generating series and motivic Donaldson-Thomas invariants.

Contribution

It introduces a new method for computing refined open amplitudes via geometric transitions and demonstrates their relation to quiver series and motivic invariants.

Findings

Refined open BPS invariants are non-negative integers for various toric Calabi-Yau threefolds.

Refined open string amplitudes can be expressed as quiver generating series.

Non-negativity of motivic Donaldson-Thomas invariants underpins the invariants' positivity.

Abstract

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.