  The instanton partition functions of $\mathcal{N}=1$ 5d super Yang-Mills are
built using elements of the representation theory of quantum
$\mathcal{W}_{1+\infty}$ algebra: Gaiotto state, intertwiner, vertex operator.
This algebra is also known under the names of Ding-Iohara-Miki and quantum
toroidal $\widehat{\mathfrak{gl}}(1)$ algebra. Exploiting the explicit action
of the algebra on the partition function, we prove the regularity of the 5d
qq-characters. These characters provide a solution to the Schwinger-Dyson
equations, and they can also be interpreted as a quantum version of the
Seiberg-Witten curve.
