Non-integrability of the $\Omega$-deformation

TL;DR

This paper investigates the classical integrability of string dynamics in the supergravity dual of the $ abla$-deformed $ ext{AdS}_5 imes ext{S}^5$ background, finding that the deformation breaks integrability.

Contribution

It provides a detailed analysis showing that the $ abla$-deformation leads to non-integrable string dynamics, indicating a loss of classical integrability.

Findings

String fluctuations are non-integrable in the $ abla$-deformed background.

The $ abla$-deformation breaks the geometric isometries crucial for integrability.

Classical integrability is not preserved under the $ abla$-deformation.

Abstract

We study integrability on the supergravity vacuum dual to the field theoretical $\Omega$-deformation of $\mathcal{N}=4$ super Yang-Mills theory. The deformation manifests itself as turning on a Kalb-Ramond field on the (Euclidean) AdS$_5\times$S$^5$, while the associated $H_3$ flux ignores half of the geometric isometries. By constructing appropriate string embeddings that incorporate the essential $H_3$ flux contribution on this background, we study their fluctuations through the associated Hamiltonian systems. Each and every case demonstrates that the string exhibits non-integrable dynamics, which in turn suggests that the $\Omega$-deformation does not preserve classical integrability.