More on 5d Wilson Loops in Higher-Rank Theories and Blowup Equations

TL;DR

The paper develops and validates a comprehensive framework for Wilson loop observables in higher-rank 5d ${\cal N}=1$ gauge theories, using both Chern-character insertions and qq-characters to access loops in diverse representations including the exceptional group $G_2$. It introduces a systematic method to formulate blowup equations for Wilson loops by leveraging one-form symmetry constraints and low-instanton data, enabling bootstrap computations of loop VEVs on the original spacetime from the blown-up geometry. A central finding is the universal one-instanton structure of Wilson-loop free energies, which exhibits a Hilbert-series-like form in many cases and a $v$-dependent, $x$-independent pattern when Coulomb parameters are turned off. The results connect refined BPS enumerations, topological strings, and defect partition functions, and establish concrete checks across SU$(N)$, SO$(5)$, Sp$(2)$, and $G_2$, with explicit blowup equations and isomorphism checks (e.g., $\mathfrak{so}(5)\cong\mathfrak{sp}(2)$ and $\mathfrak{su}(2)\cong\mathfrak{sp}(1)$) up to several instantons, paving the way for broader analytical control of Wilson-loop observables in 5d gauge theories.

Abstract

In this article, we further explore the construction and computation of expectation values for Wilson loops in higher-rank 5d $\mathcal{N} = 1$ gauge theories on $\mathbb{C}_2 \times S_1$, by explicitly computing the Wilson loops via Chern-character insertion and qq-characters, including cases with the exceptional gauge group $G_2$. In particular, we propose a systematic way to write down the general blowup equations for Wilson loops by using the constraints from the one-form symmetry and low-instanton data from the instanton partition function. In addition, for one-instanton contributions in a large family of Wilson loop representations, we observe that they admit a $q_1q_2$-expansion, similar to the Hilbert-series structure of instanton partitions in pure gauge theories.

Figures (1)

• Figure 1: Adding flavor branes and taking the mass parameters to infinity, one can realize the Wilson loops in tensor product of fundamental or anti-fundamental representations and read off their contributions from the asymptotic behavior.