(p,q)-webs of DIM representations, 5d N=1 instanton partition functions and qq-characters

Understand

Instanton partition functions of $\mathcal{N}=1$ 5d Super Yang-Mills reduced on $S^1$ can be engineered in type IIB string theory from the $(p,q)$-branes web diagram.

• To this diagram is superimposed a web of representations of the Ding-Iohara-Miki (DIM) algebra that acts on the partition function.
• In this correspondence, each segment is associated to a representation, and the (topological string) vertex is identified with the intertwiner operator constructed by Awata, Feigin and Shiraishi.
• We define a new intertwiner acting on the representation spaces of levels $(1,n)\otimes(0,m)\to(1,n+m)$, thereby generalizing to higher rank $m$ the original construction.