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

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.

💡 Research Summary

This paper investigates the computation of Wilson loop expectation values (VEVs) in five‑dimensional 𝒩=1 supersymmetric gauge theories of arbitrary rank, focusing on both classical gauge groups and the exceptional group G₂. The authors work on the Ω‑background S¹×ℂ² with deformation parameters ε₁, ε₂, where Wilson loops are introduced as ½‑BPS line defects wrapping the S¹ circle. Two complementary computational frameworks are developed.

The first method inserts the Chern character of a heavy static charged particle into the ADHM quantum mechanics that describes instanton contributions. By treating the Wilson loop as a source of a massive fundamental hypermultiplet and applying supersymmetric localization, the path integral reduces to a finite‑dimensional matrix model whose integrand is modified by the Chern character. This approach yields explicit formulas for Wilson loops in fundamental, antisymmetric, and higher tensor representations.

The second method employs qq‑characters, originally introduced via D4′ brane insertions in type IIA brane webs. A qq‑character is a generating function of line defects that can be expressed as a linear combination of Wilson loops in various antisymmetric representations. The authors show how to construct these characters for any gauge group, and how the resulting expressions match those obtained from the Chern‑character technique.

Using both methods, the paper provides detailed calculations for SU(N) (including fundamental and higher‑rank tensor representations), SO(5) and its isomorphic partner Sp(2), and the exceptional group G₂. The results are cross‑checked against each other and against known low‑instanton data, confirming the consistency of the two approaches.

A central observation is that the one‑instanton contribution to a large class of Wilson loops admits a universal expansion in the combination v = √(q₁q₂), where q₁ and q₂ are the Ω‑background fugacities. The expansion takes the form

 F^{(1)}