GW/DT invariants and 5D BPS indices for strips from topological recursion

TL;DR

This paper establishes a direct link between topological recursion and topological string theory for strip geometries, deriving free energies and exploring the role of $x$-$y$ duality in connecting GW, DT invariants, and 5D BPS indices.

Contribution

It extends topological recursion to Logarithmic Topological Recursion and applies it to derive free energies for strip geometries, clarifying the $x$-$y$ duality's significance.

Findings

Derived all free energies from topological recursion for strip geometries.

Clarified the role of $x$-$y$ duality in topological string theory.

Connected GW, DT invariants, and 5D BPS indices through new recursion techniques.

Abstract

Topological string theory partition function gives rise to Gromov-Witten invariants, Donaldson-Thomas invariants and 5D BPS indices. Using the remodeling conjecture, which connects Topological Recursion with topological string theory for toric Calabi-Yau threefolds, we study a more direct connection for the subclass of strip geometries. In doing so, 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 these techniques, our main result in this paper is a direct derivation of all free energies from topological recursion for general strip geometries. In analyzing the expression of free energy, we shed some light on the meaning and the influence of the $x$-$y$ duality in topological string theory and its interconnection to GW and DT invariants as well as the 5D BPS index.