Deformed Integrable Models from Holomorphic Chern-Simons Theory

Abstract

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 four-dimensional Chern-Simons theory, while under solving along fibres it leads to four-dimensional(4D) integrable theory, the anti-self-dual Yang-Mills or its generalizations. From both four-dimensional 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 4D integrable model approach, even though they could be obtained from 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 $\lambda$-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.

References in corpus (4)

• Supersymmetric gauge theory and the Yangian
• The Chern-Simons Origin of Superstring Integrability
• The Faddeev-Reshetikhin model from a 4D Chern-Simons theory
• Comments on $\lambda$--deformed models from 4D Chern-Simons theory

Cited by in corpus (6)

• 4-dimensional Chern-Simons theory and integrable field theories
• Integrable Deformations from Twistor Space
• Geometry of the spectral parameter and renormalisation of integrable $\sigma$-models
• Holographic Complexity and Phase Transition for AdS Black Holes
• Topological 4D gravity and gravitational defects
• Four-Dimensional Chern-Simons and Gauged Sigma Models