Real Classical Geometry with arbitrary deficit parameter(s) $\alpha(_{I})$ in Deformed Jackiw-Teitelboim Gravity

An interesting deformation of the Jackiw-Teitelboim (JT) gravity has been proposed by Witten by adding a potential term $U(\phi)$ as a self-coupling of the scalar dilaton field. During calculating the path integral over fields, a constraint comes from integration over $\phi$ as $R(x)+2=2\alpha \delta(\vec{x}-\vec{x}')$. The resulting Euclidean metric suffered from a conical singularity at $\vec{x}=\vec{x}'$. A possible geometry modeled locally in polar coordinates $(r,\varphi)$ by $ds^2=dr^2+r^2d\varphi^2,\varphi \cong \varphi+2\pi-\alpha$. In this letter we showed that there exists another family of "exact" geometries for arbitrary values of the $\alpha$. A pair of exact solutions are found for the case of $\alpha=0$. One represents the static patch of the AdS and the other one is the non static patch of the AdS metric. These solutions were used to construct the Green function for the inhomogeneous model with $\alpha\neq 0$. We address a type of the phase transition between different patches of the AdS in theory because of the discontinuity in the first derivative of the metric at $x=x'$. We extended the study to the exact space of metrics satisfying the constraint $R(x)+2=2\sum_{i=1}^{k}\alpha_i\delta^{(2)}(x-x'_i)$ as a modulo diffeomorphisms for an arbitrary set of the deficit parameters $(\alpha_1,\alpha_2,..,\alpha_k)$. The space is the moduli space of Riemann surfaces of genus $g$ with $k$ conical singularities located at $x'_k$ denoted by $\mathcal{M}_{g,k}$.