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The instanton partition functions of $\mathcal{N}=1$ 5d super Yang-Mills arebuilt 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 quantumtoroidal $\widehat{\mathfrak{gl}}(1)$ algebra. Exploiting the explicit actionof the algebra on the partition function, we prove the regularity of the 5dqq-characters. These characters provide a solution to the Schwinger-Dysonequations, and they can also be interpreted as a quantum version of theSeiberg-Witten curve.

Coherent states in quantum $\mathcal{W}_{1+\infty}$ algebra and qq-character for 5d Super Yang-Mills