Lunin-Maldacena backgrounds from the classical Yang-Baxter equation — towards the gravity/CYBE correspondence

Takuya Matsumoto,Kentaroh Yoshida

doi:10.1007/jhep06(2014)135

Takuya Matsumoto, Kentaroh Yoshida

Open Access

https://doi.org/10.1007/jhep06(2014)135

Copy DOI

Abstract

We consider \gamma-deformations of the AdS_5xS^5 superstring as Yang-Baxter sigma models with classical r-matrices satisfying the classical Yang-Baxter equation (CYBE). An essential point is that the classical r-matrices are composed of Cartan generators only and then generate abelian twists. We present examples of the r-matrices that lead to real \gamma-deformations of the AdS_5xS^5 superstring. Finally we discuss a possible classification of integrable deformations and the corresponding gravity solution in terms of solutions of CYBE. This classification may be called the gravity/CYBE correspondence.

Highlights

Constructed from classical r-matrices play the central role in constructing deformed classical actions
An essential point is that the classical r-matrices are composed of Cartan generators only and generate abelian twists
An essential point is that the classical r-matrices are composed of Cartan generators only and do not satisfy the nilpotency condition in comparison to Jordanian deformations considered in [48]

Summary

Classical R-operators for abelian twists

In the previous work [48], we have studied classical r-matrices of Jordanian type, which satisfy the following properties: 1) solutions of the classical Yang-Baxter equation (CYBE), 2) the skew-symmetricity, 3) the nilpotency. We refer the r-matrices of this type as to abelian r-matrices because these generate abelian twists which are particular examples of the Drinfeld-Reshetikhin twists [12, 13, 55] These commute with each other and satisfy CYBE obviously. For a particular class of marginal deformations of N =4 SYM [56] called β-deformations, the gravitational duals were presented by Lunin and Maldacena [57] Their original construction is based on an SL(2, R) symmetry and a single parameter is contained. For the other field components, see [58] This solution is often called the three-parameter real γ-deformed AdS5×S5 background..

One-parameter case

Three-parameter case

Conclusion and discussion

A Our notation and convention

B Rewriting γ-deformed backgrounds

C Derivation of deformed actions