Deformed integrable models from holomorphic Chern-Simons theory

We study the approaches to two-dimensional integrable field theories via a six-dimensional (6D) holomorphic Chern-Simons theory defined on twistor space. Under symmetry reduction, it reduces to a 4D Chern-Simons theory, while under solving along fibres it leads to a four-dimensional (4D) integrable theory, the anti-self-dual Yang-Mills or its generalizations. From both 4D theories, various two-dimensional integrable field theories can be obtained. In this work, we try to investigate several two-dimensional integrable deformations in this framework. We find that the lambda-deformation, the rational eta-deformation, and the generalized lambda-deformation can not be realized from the 4D integrable model approach, even though they could be obtained from the 4D Chern-Simons theory. The obstacle stems from the incompatibility between the symmetry reduction and the boundary conditions. Nevertheless, we show that a coupled theory of the lambda-deformation and the eta-deformation in the trigonometric description could be obtained from the 6D theory in both ways, by considering the case that (3,0)-form in the 6D theory is allowed to have zeros.

Automated summary

In this study, we investigate the approaches to two-dimensional integrable field theories using a six-dimensional holomorphic Chern-Simons theory. We find that certain two-dimensional integrable deformations cannot be obtained using a four-dimensional integrable model approach, but can be derived from a four-dimensional Chern-Simons theory. This limitation arises from the incompatibility between symmetry reduction and boundary conditions. Nevertheless, we demonstrate that a coupled theory of the lambda-deformation and the eta-deformation in the trigonometric description can be obtained from the six-dimensional theory by allowing the (3,0)-form to have zeros.