Integrable deformed $T^{1,1}$ sigma models from 4D Chern-Simons theory

TL;DR

This paper derives integrable deformed $T^{1,1}$ sigma models from 4D Chern-Simons theory, providing a geometric framework and explicit examples, including an anisotropic model and a $ ext{G/H}$ $ ext{lambda}$-model, expanding understanding of integrable 2D models.

Contribution

The paper introduces a method to obtain integrable deformed $T^{1,1}$ sigma models from 4D Chern-Simons theory, linking gauge theory boundary conditions to sigma model backgrounds.

Findings

Derived the ABL model from 4D Chern-Simons theory.

Explicitly constructed target-space metric and B-field for the models.

Presented examples including an anisotropic $T^{1,1}$ and a $ ext{G/H}$ $ ext{lambda}$-model.

Abstract

Recently, a variety of deformed $T^{1,1}$ manifolds, with which 2D non-linear sigma models (NLSMs) are classically integrable, have been presented by Arutyunov, Bassi and Lacroix (ABL) [arXiv:2010.05573]. We refer to the NLSMs with the integrable deformed $T^{1,1}$ as the ABL model for brevity. Motivated by this progress, we consider deriving the ABL model from a 4D Chern-Simons (CS) theory with a meromorphic one-form with four double poles and six simple zeros. We specify boundary conditions in the CS theory that give rise to the ABL model and derive the sigma-model background with target-space metric and anti-symmetric two-form. Finally, we present two simple examples 1) an anisotropic $T^{1,1}$ model and 2) a $G/H$ $\lambda$-model. The latter one can be seen as a one-parameter deformation of the Guadagnini-Martellini-Mintchev model.