Defects and type D relativistic Toda lattice for some 5d gauge theories

We perform folding on the ADHM construction of the instanton moduli space from $SU$ to $SO$ group. A Young diagram description for the $SO$ instanton is obtained after modifying the real and complex moment maps of the ADHM data. We study the Bethe gauge correspondence between type D relativistic Toda lattice and 5d $\mathcal{N}=1$ folded theory. In particular we prove that the regular monodromy defect in the folded gauge theory is the stationary wavefunction of the type D relativistic Toda lattice.