Topological strings and Wilson loops

TL;DR

This paper establishes a refined topological string framework to compute 5d Wilson loop expectation values, providing exact results and connecting to quantum geometry of local Calabi-Yau models.

Contribution

It introduces a refined topological string correspondence for 5d Wilson loops and develops methods for exact computations in both A- and B-models.

Findings

Exact 5d Wilson loop partition functions computed

Recovery of quantum periods for local Calabi-Yau models

Refined generalization of A-periods provided

Abstract

We propose the refined topological string correspondence to the expectation values of half-BPS Wilson loop operators in 5d $\mathcal{N}=1$ gauge theory partition function on the Omega-deformed background $\mathbb{R}^4_{\epsilon_{1,2}}\times S^1$. We provide the refined topological vertex method and the refined holomorphic anomaly equation method in the topological string theory, from which we have exact computations on the 5d Wilson loops partition functions in both A- and B-models. Finally, with the exact results we have in B-model, we recover the quantum periods of local $\mathbb{P}^1\times\mathbb{P}^1$ model and local $\mathbb{P}^2$ model in the study of quantum geometry and we further give a refined generalization of A-period.