Real classical geometry with arbitrary deficit parameter(s) alpha (_{I}) in deformed Jackiw\u2013Teitelboim gravity

Davood Momeni

doi:10.1140/epjc/s10052-021-08985-1

Abstract

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=2alpha delta (vec {x}-vec {x}'). The resulting Euclidean metric suffered from a conical singularity at vec {x}=vec {x}'. A possible geometry is modeled locally in polar coordinates (r,varphi ) by mathrm{d}s^2=mathrm{d}r^2+r^2mathrm{d}varphi ^2,varphi cong varphi +2pi -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'. We extended the study to the exact space of metrics satisfying the constraint R(x)+2=2sum _{i=1}^{k}alpha _idelta ^{(2)}(x-x'_i) as a modulus diffeomorphisms for an arbitrary set of deficit parameters (alpha _1,alpha _2,ldots ,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}.

Highlights

The resulting Euclidean metric suffered from a conical singularity at x = x
We investigated deformed geometries for deformed JT gravity recently proposed by Witten [28]
There is a discontinuity in the metric derivative and that implies a type of phase transition between the AdS and AdS metrics

Summary

Introduction

JT gravity is a family of scalar field theories where the scalar field (dilaton) φ coupled to gravity in two dimensions is a minimal theory of gravity in 2d [18,19]. [26] the authors studied defects in JT gravity holographically by studying the deformation of the Schwarzian theory as the dual quantum boundary action. Witten investigated a simple deformation of JT gravity by adding a potential term. A remarkable program for the qunatization of the JT gravity like theories and viable higher order corrections to it widely studied in the in Refs. Some new exact solutions for dJT were studied recently in [35] in favor of Maldacena’s duality conjecture and boundary Schwarzian theories. [35] we showed that how pure AdS seed metric for pure JT gravity will be deformed in the dJT. The problem we want to address here is how this perturbative potential (3) will deform the pure JT gravity bulk geometry. 2 we formulate the problem of the deformed singular metrics with a single deficit parameter α.

Problem statement

The integral balance method and the meaning of α

Constructing Green function

Black hole solutions

On singular manifold with time-dependent metrics

Conclusion