# Integrable Lambda Models And Chern-Simons Theories

## TL;DR

This paper establishes a connection between lambda models and double Chern-Simons theories, explaining the origin of non-ultralocality in Maillet brackets and proposing a new perspective on integrable string sigma models.

## Contribution

It reveals a novel relationship between lambda models and Chern-Simons theories, providing insights into their phase spaces and non-ultralocality issues, with implications for quantization.

## Key findings

- Lambda models' phase space linked to double Chern-Simons theories.
- Non-ultralocality arises as a boundary algebra in these theories.
- Potential to avoid non-ultralocal Poisson algebras in lattice regularizations.

## Abstract

In this note we reveal a connection between the phase space of lambda models on $S^{1}\times \mathbb{R}$ and the phase space of double Chern-Simons theories on $D\times \mathbb{R}$ and explain in the process the origin of the non-ultralocality of the Maillet bracket, which emerges as a boundary algebra. In particular, this means that the (classical) $AdS_{5}\times S^{5}$ lambda model can be understood as a double Chern-Simons theory defined on the Lie superalgebra $\mathfrak{psu}(2,2|4)$ after a proper dependence of the spectral parameter is introduced. This offers a possibility for avoiding the use of the problematic non-ultralocal Poisson algebras that preclude the introduction of lattice regularizations and the application of the QISM to string sigma models. The utility of the equivalence at the quantum level is, however, still to be explored.