  Recent work has shown that certain integrable and conformal field theories in
two dimensions can be given a higher-dimensional origin from holomorphic
Chern-Simons in six dimensions. Along with anti-self-dual Yang-Mills and
four-dimensional Chern-Simons, this gives rise to a diamond correspondence of
theories. In this work we extend this framework to incorporate models realised
through gaugings. As well as describing a higher-dimensional origin of coset
CFTs, by choosing the details of the reduction from higher dimensions, we
obtain rich classes of two-dimensional integrable models including homogeneous
sine-Gordon models and generalisations that are new to the literature.
