Refined open topological strings revisited

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.

3d $\mathcal{N}=2$ Brane Webs and Quiver Matrices

“…Fortunately, there are special cases without this crossing, namely performing HW transitions vertically. This kind of HW transitions give rise to equivalent non-toric 3d brane webs with the same vortex partition functions up to some extra contributions from singlets, see [18] for more details. In the following subsection, we compute the vortex partition functions of some brane webs to verify the assignment rule for real mass parameters and check the equivalence of seemingly different brane webs.…”

“…The open topological string partition function (2.1) can be written in the form of a mathematical quiver generating function, as discussed in e.g. [18,32,35],…”

“…where Ω d 1 ,...,d N ;j are integer motivic Donaldson-Thomas (DT) invariants [25] and can be lifted to refined version Ω (s,r) d 1 ,...,d N by shifting x i → q a i t b i x i properly [18]. The quiver form (2.4) arises in the context of the 3d N = 2 theories in view of their relations to knot theory [33,34] and open topological strings on strip Calabi-Yau manifolds [18,32,35].…”

“…where Ω d 1 ,...,d N ;j are integer motivic Donaldson-Thomas (DT) invariants [25] and can be lifted to refined version Ω (s,r) d 1 ,...,d N by shifting x i → q a i t b i x i properly [18]. The quiver form (2.4) arises in the context of the 3d N = 2 theories in view of their relations to knot theory [33,34] and open topological strings on strip Calabi-Yau manifolds [18,32,35]. These quiver matrices C ij can be physically interpreted as mixed Chern-Simons levels k eff ij of the mirror dual theories that contain gauge group U (1) ⊗N and N fundamental multiplets F. This equivalence C ij = k eff ij is derived in [24] using 3d mirror transformations implemented on sphere partition functions.…”

“…Comparing the terms and factors in vortex partition functions of various brane webs, one could see how the brane webs encode these physical quantities. We implement refined topological vertex method to compute the vortex partition functions of 3d N = 2 gauge theories that can be interpreted as open topological string amplitudes that can be obtained by geometric transitions from closed topological string amplitudes [8,18]. For recent developments in topological vertex and 5d N = 1 theories see e.g.…”