Four-dimensional Chern-Simons theory and integrable models

In this review, we introduce the connections between the four-dimensional Chern-Simons theory and integrable models, including the relationship with integrable lattice models and integrable field theories. Due to the connection between the four-dimensional Chern-Simons theory and the Yang-Baxter equation, the partition function of the integrable lattice model can be naturally obtained from the four-dimensional Chern-Simons theory. By introducing the two-dimensional surface defects into the four-dimensional Chern-Simons theory, two-dimensional classical integrable field theories can also be constructed. We review two ways to construct integrable field theories using either ordered or disordered defects, and provide a detailed exposition on extracting the Lax connection and the action of the two-dimensional integrable field theories from the four-dimensional Chern-Simons theory. Finally, we discuss the subsequent research developments related to the four-dimensional Chern-Simons theory and offer an outlook on future directions.