Webs of integrable theories.

We present an intuitive diagrammatic representation of a new class of integrable $\sigma$-models. It is shown that to any given diagram corresponds an integrable theory that couples 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 $\sigma$-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 "propagators" having momenta $k_i$, with each of the propagators being associated with an asymmetrically gauged WZW model at an arbitrary integer level $k_i$. Gauge invariance of the full action is translated to momentum conservation at the vertices. We also show how to immediately read from the diagrams the corresponding $\sigma$-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 momentum conservation at the vertices is relaxed and the case where the deformation matrix is not diagonal in the space of integrable models.