Integrable deformations of the AdS$_5\times$S$^5$ superstring and the classical Yang-Baxter equation -- Towards the gravity/CYBE correspondence --

TL;DR

This paper explores how classical r-matrices satisfying the CYBE can systematically generate integrable deformations of the AdS$_5\times$S$^5$ superstring, establishing a gravity/CYBE correspondence with concrete examples.

Contribution

It introduces the gravity/CYBE correspondence linking superstring deformations to classical r-matrices and provides explicit examples including Lunin-Maldacena backgrounds and non-commutative gauge theory duals.

Findings

Established the gravity/CYBE correspondence.

Constructed explicit deformations for known backgrounds.

Extended the framework to non-integrable backgrounds.

Abstract

Based on the formulation of Yang-Baxter sigma models developed by Klimcik and Delduc-Magro-Vicedo, we explain that various deformations of type IIB superstring on AdS$_5\times$S$^5$ can be characterized by classical $r$-matrices satisfying the classical Yang-Baxter equation (CYBE). The relation may be referred to as `the gravity/CYBE correspondence.' We present non-trivial examples of the correspondence including Lunin-Maldacena backgrounds for $\beta$-deformations of the ${\cal N} = 4$ super Yang-Mills theory and the gravity duals for non-commutative gauge theories. We also discuss non-integrable backgrounds such as AdS$_5\times T^{1,1}$ as a generalization.