Webs of integrable theories

TL;DR

This paper introduces a diagrammatic approach to constructing a broad class of integrable sigma models by combining fundamental integrable theories at vertices and along edges, revealing new models with rich parameter spaces.

Contribution

It presents an intuitive diagrammatic representation linking fundamental integrable models to complex coupled theories, enabling systematic construction and analysis of new integrable sigma models.

Findings

Diagrammatic representation of integrable sigma models.

Explicit correspondence between diagrams and sigma model actions.

Identification of models with at least n^2+1 parameters.

Abstract

We present an intuitive diagrammatic representation of a new class of integrable $\s$-models. It is shown that to any given diagram corresponds an integrable theory that couples $N$ WZW models with a certain number of each of the following four fundamental integrable models, the PCM, the YB model, both based on a group $G$, the isotropic $\s$-model on the symmetric space $G/H$ and the YB model on the symmetric space $G/H$. To each vertex of a diagram we assign the matrix of one of the aforementioned fundamental integrable theories. Any two vertices may be connected with a number of lines having an orientation and carrying an integer level $k_i$. Each of these lines is associated with an asymmetrically gauged WZW model at an arbitrary level $k_i$. Gauge invariance of the full action is translated to level conservation at the vertices. We also show how to immediately read from the diagrams the corresponding $\s$-model actions. The most generic of these models depends on at least $n^2+1$ parameters, where $n$ is the total number of vertices/fundamental integrable models. Finally, we discuss the case where the level conservation at the vertices is relaxed and the case where the deformation matrix is not diagonal in the space of integrable models.