Minimal $(D,D)$ conformal matter and generalizations of the van Diejen model

We consider supersymmetric surface defects in compactifications of the $6d$ minimal $(D_{N+3},D_{N+3})$ conformal matter theories on a punctured Riemann surface. For case $N=1$ such are introduced into index computations by an action $BC_1\,(\sim A_1\sim C_1)$ van Diejen model. (re)derive this fact using three different field theoretic descriptions four dimensional models. The naturally associated with algebras $A_{N=1}$, $C_{N=1}$, and $(A_1)^{N=1}$. indices these $4d$ give rise to Kernel functions for $BC_1$ then generalizations $N>1$. operators introducing certain $A_{N}$, $C_N$, $(A_1)^{N}$ directly related effective gauge theory one can obtain compactifying circle five dimensions. explicitly compute $A_N$ case, derive various properties have satisfy as consequence dualities following from geometric setup. In some cases we able verify which turn serve checks said dualities. As by-product our constructions also discuss simple Lagrangian description corresponding compactification sphere maximal punctures $(D_5,D_5)$ explicit 6d SCFT arbitrary surfaces.