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

Abstract

An interesting deformation of Jackiw–Teitelboim (JT) gravity has been proposed by Witten by adding a potential term $$U(\phi )$$ U ( ϕ ) as a self-coupling of the scalar dilaton field. During calculating the path integral over fields, a constraint comes from integration over $$\phi $$ ϕ as $$R(x)+2=2\alpha \delta (\vec {x}-\vec {x}')$$ R ( x ) + 2 = 2 α δ ( x → - x → ′ ) . The resulting Euclidean metric suffered from a conical singularity at $$\vec {x}=\vec {x}'$$ x → = x → ′ . A possible geometry is modeled locally in polar coordinates $$(r,\varphi )$$ ( r , φ ) by $$\mathrm{d}s^2=\mathrm{d}r^2+r^2\mathrm{d}\varphi ^2,\varphi \cong \varphi +2\pi -\alpha $$ d s 2 = d r 2 + r 2 d φ 2 , φ ≅ φ + 2 π - α . In this letter we show that there exists another family of ”exact” geometries for arbitrary values of the $$\alpha $$ α . A pair of exact solutions are found for the case of $$\alpha =0$$ α = 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 $$\alpha \ne 0$$ α ≠ 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 $$x=x'$$ x = x ′ . We extended the study to the exact space of metrics satisfying the constraint $$R(x)+2=2\sum _{i=1}^{k}\alpha _i\delta ^{(2)}(x-x'_i)$$ R ( x ) + 2 = 2 ∑ i = 1 k α i δ ( 2 ) ( x - x i ′ ) as a modulus diffeomorphisms for an arbitrary set of deficit parameters $$(\alpha _1,\alpha _2,\ldots ,\alpha _k)$$ ( α 1 , α 2 , … , α k ) . The space is the moduli space of Riemann surfaces of genus g with k conical singularities located at $$x'_k$$ x k ′ , denoted by $$\mathcal {M}_{g,k}$$ M g , k .

Exact solutions and Birkhoff's theorem in Jackiw-Teitelboim Gravity

“…We mention here that a null coordinate transformation of the system of the equations presented in the eqs. (31,32) leads to a non linear PDE can be integrated even for a hyperbolic geometry with arbitrary genus n, this class of solutions along a possible type of AdS 2 → AdS 2 phase transition investigated recently by the author in [28]. In this work we complete our study about exact solutions in the JT gravity.…”

“…Free energy in JT gravity and the role of critically investigated in [27]. Classical defects of AdS geometry studied in [28]. The scalar mode propagation in JT gravity considered in [29].…”

2D holography beyond the Jackiw-Teitelboim model

“…First of all, we note that the argumentation by Jensen [10] is of rather general nature and is not restricted to just JT. The dualities to matrix models proposed [11] for the JT model were extended to more general dilaton gravities [12] (see also [13][14][15] for related study). Rather general dilaton gravities were encountered in the context of TT deformations [16][17][18] or if AdS 2 holography is approached from higher-dimensional theories [19][20][21][22].…”

“…13). Note that the combinations Sϕ 1 , Sϕ 2 and ϕ 1−a 1 have a-independent conformal weights and transform likeδ ξ (Sϕ 1 ) = (Sϕ 1 ) + (Sϕ 1 ), (9.25) δ ξ (Sϕ 2 ) = (Sϕ 2 ) + 2 (Sϕ 2 = δ ξ φ = φ − φ, (9.27)where we recognize a boundary current Sϕ 1 , a weight-2 quantity Sϕ 2 and a boundary vector φ.…”

Target space diffeomorphisms in Poisson sigma models and asymptotic symmetries in 2D dilaton gravities

“…Quite naturally, many works were dedicated to the search of extensions of the JT/SYK paradigm. In particular, matrix model boundary duals were suggested in [11] and studied in [12][13][14][15][16]. Generalized sets of asymptotic conditions leading to larger asymptotic symmetry algebras were suggested in [17,18].…”