Real classical geometry with arbitrary deficit parameter(s) α(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(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 (x ->-x ->'). The resulting Euclidean metric suffered from a conical singularity at x ->=x ->'. A possible geometry is modeled locally in polar coordinates (r,phi) by ds2=dr2+r2d phi 2,phi 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 not equal 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 '. We extended the study to the exact space of metrics satisfying the constraint R(x)+2=2 Sigma i=1kalpha i delta (2)(x-xi ') as a modulus diffeomorphisms for an arbitrary set of 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 xk', denoted by Mg,k.

Automated summary

An interesting deformation of Jackiw-Teitelboim (JT) gravity, proposed by Witten, involves adding a potential term as a self-coupling for the scalar dilaton field. Exact solutions have been found for specific parameter values, which are used to construct Green functions for nonhomogeneous models. The study also addresses phase transitions between different patches of AdS due to discontinuity in the metric.