# Conformal Twists, Yang-Baxter $\sigma$-models & Holographic Noncommutativity

## TL;DR

This paper explores integrable deformations of AdS$_5$ using Yang-Baxter solutions, revealing a holographic noncommutativity structure that links boundary conditions to bulk properties, with implications for dual noncommutative gauge theories.

## Contribution

It provides a comprehensive analysis of $\sigma$-models related to integrable deformations of AdS$_5$, connecting conformal twists, gravity backgrounds, and noncommutative gauge theories, including novel holographic noncommutativity insights.

## Key findings

- Open string metric remains undeformed AdS$_5$ in all cases.
- The noncommutative structure $ heta$ is determined by the boundary twist.
- Holographic noncommutativity links bulk and boundary properties.

## Abstract

Expanding upon earlier results [arXiv:1702.02861], we present a compendium of $\sigma$-models associated with integrable deformations of AdS$_5$ generated by solutions to homogenous classical Yang-Baxter equation. Each example we study from four viewpoints: conformal (Drinfeld) twists, closed string gravity backgrounds, open string parameters and proposed dual noncommutative (NC) gauge theory. Irrespective of whether the deformed background is a solution to supergravity or generalized supergravity, we show that the open string metric associated with each gravity background is undeformed AdS$_5$ with constant open string coupling and the NC structure $\Theta$ is directly related to the conformal twist. One novel feature is that $\Theta$ exhibits "holographic noncommutativity": while it may exhibit non-trivial dependence on the holographic direction, its value everywhere in the bulk is uniquely determined by its value at the boundary, thus facilitating introduction of a dual NC gauge theory. We show that the divergence of the NC structure $\Theta$ is directly related to the unimodularity of the twist. We discuss the implementation of an outer automorphism of the conformal algebra as a coordinate transformation in the AdS bulk and discuss its implications for Yang-Baxter $\sigma$-models and self-T-duality based on fermionic T-duality. Finally, we comment on implications of our results for the integrability of associated open strings and planar integrability of dual NC gauge theories.