Regular $(2+1)$-dimensional spatially homogeneous $\alpha'$-corrected black holes in string theory
We present the solution of string effective action on spatially homogeneous $(2+1)$-dimensional spacetime with the source given by dilaton and antisymmetric $B$-field in the presence of central charge deficit term $\Lambda$. We consider the two-dimensional space part to be Weyl-related to a surface of arbitrary negative constant Gaussian curvature with symmetries of two-dimensional Lie algebra. It is shown that the geometry is a Lobachevsky-type geometry that can be reduced to hyperbolic type if the negative Gaussian curvature of the surface equals $-\frac{1}{2}$. At leading order string effective action, we obtained a black hole solution whose line element is Weyl-related to this homogeneous spacetime with arbitrary negative Gaussian curvature. The solution can be reduced to the BTZ black hole by choosing a special set of parameters. Also, it is shown that the central charge of the CFT coincides with that of the AdS spacetime related to the Chern-Simons model with gauge group $SO(2,2)$. The solutions appear to be in the high curvature limit $R\alpha'\gtrsim1$, with emphasis on including the higher order $\alpha'$ corrections. Considering the two-loop (first order $\alpha'$) $\beta$-function equations of $\sigma$-model, we present the $\alpha'$-corrected black hole solutions.