Spiralling branes and R-matrices

We extend the dictionary between Type IIB branes and representations of the Ding-Iohara-Miki (DIM) algebra to the case when one of the space directions is a circle. It is well-known that the worldvolume theory on branes wrapping the circle is a 5d $\mathcal{N}=1$ gauge theory with adjoint matter, or more generally of cyclic quiver type, and the corresponding intertwiners of the DIM algebra give their Nekrasov partition functions. However, we find that there exists a much wider natural class of intertwiners corresponding to branes spiralling around the compactified direction, with many interesting properties. We consider two examples, one corresponding to a spiralling D5 brane and another to a D3 brane. The former gives rise to the K-theoretic vertex function counting sheaves on $\mathbb{C}^3$ while the latter produces the "non-stationary elliptic Ruijsenaars wavefunctions" introduced recently by Shiraishi.