https://doi.org/10.1140/epjc/s10052-021-08985-1
Regular Article - Theoretical Physics
Real classical geometry with arbitrary deficit parameter(s) 
in deformed Jackiw–Teitelboim gravity
Department of Physics, College of Science, Sultan Qaboos University, P.O. Box 36, Al-Khodh 123, Muscat, Sultanate of Oman
Received:
12
November
2020
Accepted:
16
February
2021
Published online:
1
March
2021
An interesting deformation of Jackiw–Teitelboim (JT) gravity has been proposed by Witten by adding a potential term 
as a self-coupling of the scalar dilaton field. During calculating the path integral over fields, a constraint comes from integration over 
as 
. The resulting Euclidean metric suffered from a conical singularity at 
. A possible geometry is modeled locally in polar coordinates 
by 
. In this letter we show that there exists another family of ”exact” geometries for arbitrary values of the 
. A pair of exact solutions are found for the case of 
. 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 
. 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 
. We extended the study to the exact space of metrics satisfying the constraint 
as a modulus diffeomorphisms for an arbitrary set of deficit parameters 
. The space is the moduli space of Riemann surfaces of genus g with k conical singularities located at 
, denoted by 
.
© The Author(s) 2021
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