GW/DT invariants and 5D BPS indices for strips from topological recursion
Sibasish Banerjee, Alexander Hock, Olivier Marchal
TL;DR
The paper establishes a direct link between topological recursion and GW/DT invariants plus 5D BPS indices for strip toric Calabi–Yau geometries by leveraging Log-TR and the $x$-$y$ duality. It derives closed-form genus-$g$ free energies $F_g$ from the mirror curve without resorting to full TR recursions, and shows how these encode Donaldson-Thomas invariants and 5D BPS data through a GV/DT correspondence. The authors validate the framework on several strip geometries (four-, five-, and six-punctured spheres), explicitly extracting $oldsymbol{PS}$-type charges $oldsymbol{}$ and comparing to known results from Ooguri–Szendrői, Banerjee, Mozgovoy and related formalisms, reinforcing mirror symmetry links. The work lays groundwork for open/ refined invariants, gluing via $x$-$y$ duality, and future exploration of nonperturbative TR regimes in this setting.
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.