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

TL;DR

This paper tests the open topological string version of the OSV conjecture in a specific toric Calabi-Yau setting, linking D-brane partition functions to q-deformed Yang-Mills theory and confirming the conjecture's predictions.

Contribution

It provides a novel test of the open topological string OSV conjecture in a toric Calabi-Yau manifold using q-deformed Yang-Mills theory.

Findings

Partition function matches the sum over squares of chiral blocks.

Agreement with the conjecture at large N.

Connection between D-brane partition functions and open topological string amplitudes.

Abstract

In this note we make a test of the open topological string version of the OSV conjecture, proposed in hep-th/0504054, in the toric Calabi-Yau manifold $X= O(-3)\to\mathbf{P}^2$ 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-deformed Yang-Mills theory living on a chain of $\mathbf{P}^1$'s in the base $\mathbf{P}^2$ 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 $\mathbf{P}^2$ geometry with branes at both the outer and inner edges of the toric diagram. This is in agreement with the conjecture.