(1+1)-dimensional gauge symmetric gravity model and related exact black hole and cosmological solutions in string theory

We introduce a four-dimensional extension of the Poincar\'{e} algebra $(\mathcal{N})$ in (1+1)-dimensional space-time and obtain a (1+1)-dimensional gauge symmetric gravity model using the algebra $\mathcal{N}$. We show that the obtained gravity model is dual (canonically transformed) to the (1+1)-dimensional anti de Sitter ($AdS$) gravity. We also obtain some black hole and Friedmann-Robertson-Walker (FRW) solutions by solving its classical equations of motion. Then, we study $\frac{\mathbf{A}_{\mathbf{4,8}}}{\mathbf{A}_{\mathbf{1}}\otimes \mathbf{A}_{\mathbf{1}}}$ gauged Wess-Zumino-Witten (WZW) model and obtain some exact black hole and cosmological solutions in string theory. We show that some obtained black hole and cosmological metrics in string theory are same as the metrics obtained in solutions of our gauge symmetric gravity model.