Yang-Baxter deformations of WZW model on the Heisenberg Lie group

TL;DR

This paper explores Yang-Baxter deformations of the WZW model on the Heisenberg group, classifying solutions, analyzing resulting backgrounds, and demonstrating their conformal invariance up to two loops.

Contribution

It provides a classification of all nonequivalent YB deformations of the $H_4$ WZW model and analyzes their geometric and conformal properties.

Findings

Identified ten nonequivalent YB deformed backgrounds.

Some deformed metrics are isometric to the original WZW metric.

All deformed backgrounds are conformally invariant up to two loops.

Abstract

The Yang-Baxter (YB) deformations of Wess-Zumino-Witten (WZW) model on the Heisenberg Lie group ($H_4$) are examined. We proceed to obtain the nonequivalent solutions of (modified) classical Yang-Baxter equation ((m)CYBE) for the $ h_4$ Lie algebra by using its corresponding automorphism transformation. Then we show that YB deformations of $H_4$ WZW model are split into ten nonequivalent backgrounds including metric and $B$-field such that some of the metrics of these backgrounds can be transformed to the metric of $H_{4}$ WZW model while the antisymmetric $B$-fields are changed. The rest of the deformed metrics have a different isometric group structure than the $H_{4}$ WZW model metric. As an interesting result, it is shown that all new integrable backgrounds of the YB deformed $ H_4$ WZW model are conformally invariant up to two-loop order. In this way, we obtain the general form of the dilaton fields satisfying the vanishing beta-function equations of the corresponding $\sigma$-models.