Generalized Supergravity Equations for the WZW Model

Abstract

Generalized supergravity equations (GSEs) were originally proposed as a modification of the standard IIB supergravity equations to satisfy the background of $\hat{I}\cdot$-deformed $Ad{S}_5\times S^5$ 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 ${\mathcal{L}}_{{}_I}{G}_{{}_{\hat{I}¼\hat{I}½}}=0$ 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.

15 Pages, 1 Table, 2 Refs. added