Real classical geometry with arbitrary deficit parameter(s) $\alpha$(${}_I$) in deformed Jackiw-teitelboim gravity

An interesting deformation of Jackiw-Teitelboim (JT) gravity has been proposed by Witten by adding a potential term $U\left(\varphi \right)$ as a self-coupling of the scalar dilaton field. During calculating the path integral over fields, a constraint comes from integration over $\varphi$ as $R\left(x\right)+2=2\alpha \delta (\overrightarrow{x}-{\overrightarrow{x}}^{\prime })$. The resulting Euclidean metric suffered from a conical singularity at $\overrightarrow{x}={\overrightarrow{x}}^{\prime }$. A possible geometry is modeled locally in polar coordinates $(r,\phi )$ by $d{s}^2=d{r}^2+r^{2}d{\phi }^2,\phi \cong \phi +2\pi -\alpha$. In this letter we show 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 $\alpha \ne 0$. We address a type of phase transition between different patches of the AdS in theory because of the discontinuity in the first derivative of the metric at $x=x^{\prime }$. We extended the study to the exact space of metrics satisfying the constraint $R\left(x\right)+2=2\sum i=1k{\alpha }_i{\delta }^{\left(2\right)}(x-x_i^{\prime })$ as a modulus diffeomorphisms for an arbitrary set of deficit parameters $({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_k)$. The space is the moduli space of Riemann surfaces of genus g with k conical singularities located at $x_k^{\prime }$, denoted by $M_{g,k}$.