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.
Highlights
Refined topological string theory is an intriguing and mysterious creature
Unrefined amplitudes are related to the low energy effective action for superstring theory [2,3]; encode integral closed (Gopakumar-Vafa) and open (Ooguri-Vafa) BPS invariants [4,5]; for some class of manifolds they are related to Chern-Simons theory [6] and—in consequence of this relation—for toric CalabiYau threefolds can be computed in the formalism of the topological vertex [7]; for some particular CalabiYau manifolds, topological string amplitudes are captured by Nekrasov partition functions [8], and can be
We briefly summarize the formalism of refined topological vertex, geometric transitions, refined holonomies, and refined BPS states
Summary
Refined topological string theory is an intriguing and mysterious creature. It is expected to arise as a deformation (refinement) of an ordinary (unrefined) topological string theory. The third important aim of this work concerns threefolds without compact four-cycles: we show that for such manifolds open refined topological string amplitudes take form of generating series for symmetric quivers, and we identify corresponding quivers Such a relation to quivers was originally found in the context of knots-quivers correspondence [26,27], its links with topological strings were further elucidated in [28,29], and (still in the unrefined case) it was generalized to Aganagic-Vafa branes in strip geometries [23,30]; related results are discussed in [31,32]. In the Appendix we collect various identities useful in earlier calculations
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
