Topological Strings on Toric geometries in the presence of Lagrangian branes

TL;DR

This paper develops formulas for refined open topological string partition functions on specific non-compact Calabi-Yau 3-folds with Lagrangian branes, linking them to equivariant indices on quiver moduli spaces.

Contribution

It introduces new conjectures connecting open topological string partition functions with equivariant indices on quiver moduli spaces in a partially compactified setting.

Findings

Proposed explicit expressions for refined open topological string partition functions.

Established conjectural equivalences with generating functions of equivariant indices.

Demonstrated the role of partial compactification and mass deformation in the gauge theory context.

Abstract

We propose expressions for refined open topological string partition function on certain non-compact Calabi Yau 3-folds with topological branes wrapped on the special lagrangian submanifolds. The corresponding web diagrams are partially compact and a lagrangian brane is inserted on one of the external legs. Partial compactification introduces a mass deformation in the corresponding gauge theory. We propose conjectures that equate these open topological string partition functions with the generating function of equivaraint indices on certain quiver moduli spaces. To obtain these conjectures we use the identification of topological string partition functions with equivariant indices on the instanton moduli spaces.