Deformed Integrable Models from Holomorphic Chern-Simons Theory

TL;DR

This paper explores how two-dimensional integrable models and their deformations can be derived from six-dimensional holomorphic Chern-Simons theory, revealing limitations and new coupled models through twistor space reductions.

Contribution

It demonstrates the derivation of certain 2D integrable deformations from 6D theory and identifies obstacles in realizing others, proposing a coupled deformation model via boundary conditions.

Findings

$ ext{lambda}$-deformation cannot be obtained from 4D integrable models.

Coupled $ ext{lambda}$ and $ exteta$ deformations are achievable from 6D theory.

Obstacles arise from incompatibility between symmetry reduction and boundary conditions.

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.