TTbar and root-TTbar deformations in four-dimensional Chern-Simons theory

The four-dimensional Chern-Simons (CS) theory provides a systematic procedure for realizing two-dimensional integrable field theories. It is therefore a natural question to ask whether integrable deformations of the theories can be realized in the four-dimensional CS theory. In this work, we study TT and root-TT deformations of two-dimensional integrable field theories, formulated in terms of dynamical coordinate transformations, within the framework of four-dimensional CS theory coupled to disorder defects. We illustrate our procedure in detail for the degenerate E-model, a specific construction that captures and unifies a broad range of integrable systems, including the principal chiral model.