Local Mirror Symmetry for One-Legged Topological Vertex

TL;DR

This paper proves the Bouchard-Mariño Conjecture for the one-legged topological vertex by deriving recursion relations from Hodge integrals, establishing local mirror symmetry for a specific Calabi-Yau geometry with a D-brane.

Contribution

It introduces a proof of the Bouchard-Mariño Conjecture for the one-legged topological vertex using recursion relations derived from Hodge integrals, linking to local mirror symmetry.

Findings

Derived Eynard-Orantin recursion relations from cut-and-join equations.

Confirmed the conjecture for the one-legged topological vertex.

Established local mirror symmetry for the local C^3 geometry with a D-brane.

Abstract

We prove the Bouchard-Mari\~no Conjecture for the framed one-legged topological vertex by deriving the Eynard-Orantin type recursion relations from the cut-and-join equation satisfied by the relevant triple Hodge integrals. This establishes a version of local mirror symmetry for the local $C^3$ geometry with one $D$-brane.