A Note on q-Deformed Two-Dimensional Yang-Mills and Open Topological Strings

In this note we make a test of the open topological string version of the OSV conjecture in the toric Calabi-Yau manifold X=O(-3)→ P² with background D4-branes wrapped on Lagrangian submanifolds. The D-brane partition function reduces to an expectation value of some inserted operators of a q-deformedYang-Mills theory living on a chain of P¹'s in the base P² of X. At large $N$ this partition function can be written as a sum over squares of chiral blocks, which are related to the open topological string amplitudes in the local P² geometry with branes at both the outer and inner edges of the toric diagram. This is in agreement with the conjecture.