# $O(d,d)$ transformations preserve classical integrability

## TL;DR

This paper demonstrates that $O(d,d)$ transformations maintain the classical integrability of two-dimensional non-linear sigma models by constructing their Lax pairs, including for deformations like $Jar{J}$ and TsT transformations.

## Contribution

It shows that all $O(d,d)$ deformations preserve integrability and provides explicit Lax pairs for various deformations within the doubled formalism.

## Key findings

- $O(d,d)$ transformations preserve integrability.
- Lax pairs become non-local due to winding modes.
- Explicit Lax pairs for $Jar{J}$ and TsT deformations.

## Abstract

In this note, we study the action of $O(d,d)$ transformations on the integrable structure of two-dimensional non-linear sigma models via the doubled formalism. We construct the Lax pairs associated with the $O(d,d)$-transformed model and find that they are in general non-local because they depend on the winding modes. We conclude that every $O(d,d;\mathbb{R})$ deformation preserves integrability. As an application we compute the Lax pairs for continuous families of deformations, such as $J\bar{J}$ marginal deformations and TsT transformations of the three-sphere with $H$-flux.

---
Source: https://tomesphere.com/paper/1907.03759