# Double and cyclic $\lambda$-deformations and their canonical equivalents

## TL;DR

This paper demonstrates that doubly lambda-deformed sigma-models are canonically equivalent to two single lambda-deformed models, explaining their identical beta-functions and anomalous dimensions, and extends the framework to multi-matrix integrable deformations.

## Contribution

It proves the canonical equivalence of doubly lambda-deformed models to two single models and constructs multi-matrix integrable deformations of WZW models.

## Key findings

- Doubly lambda-deformed models are canonically equivalent to two single lambda models.
- Exact beta-functions and anomalous dimensions are identical for these models.
- Extension to multi-matrix integrable deformations of WZW models.

## Abstract

We prove that the doubly lambda-deformed sigma-models, which include integrable cases, are canonically equivalent to the sum of two single lambda-deformed models. This explains the equality of the exact beta-functions and current anomalous dimensions of the doubly lambda-deformed sigma-models to those of two single lambda-deformed models. Our proof is based upon agreement of their Hamiltonian densities and of their canonical structure. Subsequently, we show that it is possible to take a well defined non-Abelian type limit of the doubly-deformed action. Last, but not least, by extending the above, we construct multi-matrix integrable deformations of an arbitrary number of WZW models.