Yang-Baxter deformation of WZW model based on Lie supergroups: The cases of $GL(1|1)$ and $(C^3 +A)$

TL;DR

This paper generalizes Yang-Baxter deformations of WZW models to Lie supergroups, classifies solutions for specific superalgebras, and demonstrates that certain deformed models remain conformal at one-loop order, offering new insights into supergravity solutions.

Contribution

It extends YB deformation techniques to Lie supergroups and classifies solutions for specific superalgebras, analyzing their conformal invariance.

Findings

YB deformations classified for $gl(1|1)$ and ${ m C}^3 + { m A}$ superalgebras.

Some models retain conformal invariance after deformation.

Deformations alter B-field components while preserving the metric in certain cases.

Abstract

We proceed to generalize the Yang-Baxter (YB) deformation of Wess-Zumino-Witten (WZW) model to the Lie supergroups case. This generalization enables us to utilize various kinds of solutions of the (modified) graded classical Yang-Baxter equation ((m)GCYBE) to classify the YB deformations of WZW models based on the Lie supergroups. We obtain the inequivalent solutions (classical r-matrices) of the (m)GCYBE for the $gl(1|1)$ and $({\cal C}^3 +{\cal A})$ Lie superalgebras in the non-standard basis, in such a way that the corresponding automorphism transformations are employed. Then, the YB deformations of the WZW models based on the $GL(1|1)$ and $(C^3 + A)$ Lie supergroups are specified by skew-supersymmetric classical r-matrices satisfying (m)GCYBE. In some cases for both families of deformed models, the metrics remain invariant under the deformation, while the components of $B$-fields are changed. After checking the conformal invariance of the models up to one-loop order, it is concluded that the $GL(1|1)$ and $(C^3 + A)$ WZW models are conformal theories within the classes of the YB deformations preserving the conformal invariance. However, our results are interesting in themselves, but at a constructive level, may prompt many new insights into (generalized) supergravity solutions.