Generalized Supergravity Equations for the WZW Model

TL;DR

This paper derives and solves generalized supergravity equations for WZW models on Lie groups, identifying conditions under which solutions exist, especially focusing on the role of light-like Killing vectors.

Contribution

It formulates simplified GSEs for WZW models on Lie groups and demonstrates solutions exist only when a light-like Killing vector is present.

Findings

Solutions found for Lie groups up to dimension five.

Two groups do not admit solutions due to lack of light-like vectors.

The existence of solutions depends on the light-like property of the Killing vector.

Abstract

Generalized supergravity equations (GSEs) were originally proposed as a modification of the standard IIB supergravity equations to satisfy the background of $\eta$-deformed $AdS_5 \times S^5$ introduced by Arutyunov {\it et al}. In this study, we proceed to write down the GSEs for the Wess-Zumino-Witten (WZW) models based on Lie groups. First, we simplify the GSEs for the WZW model by imposing the conditions for vanishing of the one-loop beta function equations. Then, by introducing an {\it Ansatz} for the Killing vector field $I$, it is shown that the Killing equation ${\cal L}_{_{I}} G_{_{\mu \nu}}=0$ is held. In addition, we introduce a generalized Killing vector such that the existence of a solution to the GSEs requires that this vector be light-like. In this way, we solve the simplified GSEs for the WZW models constructed on Lie groups up to dimension five. Unfortunately, two of the groups do not have light-like vectors and therefore do not admit the GSEs.