5d/6d Wilson loops from blowups

Abstract

We generalize Nakajima-Yoshioka's blowup formula to calculate the partition functions counting the spectrum of bound states to half-BPS Wilson loop operators in 5d (and 6d) supersymmetric field theories. The partition function in the presence of a Wilson loop operator on the $\Omega$-background is factorized when put on the blowup $\hat{\mathbb{C}}^2$ into two Wilson loop partition functions under the localization. This structure provides a set of blowup equations for Wilson loop operators. We explain how to formulate the blowup equations and solve them to compute the partition functions of Wilson loop operators. We test this idea by explicitly calculating the Wilson loop partition functions in various 5d/6d field theories and comparing them against known results and expected dualities.