Coherent states in quantum $\mathcal{W}_{1+\infty}$ algebra and   qq-character for 5d Super Yang-Mills
  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. 
Jean-Emile Bourgine, Masayuki Fukuda, Yutaka Matsuo, Hong Zhang and   Rui-Dong Zhu
hep-th math-ph math.AG math.MP math.QA math.RT