Exact Solutions of (deformed) Jackiw-Teitelboim Gravity
aa r X i v : . [ h e p - t h ] S e p Exact Solutions of (deformed) Jackiw-Teitelboim Gravity
Davood Momeni ∗ and Phongpichit Channuie
2, 3, 4, 5, † Department of Physics, College of Science, Sultan Qaboos University, P.O. Box 36,Al-Khodh 123, Muscat, Sultanate of Oman College of Graduate Studies, Walailak University, Thasala,Nakhon Si Thammarat, 80160, Thailand School of Science, Walailak University, Thasala,Nakhon Si Thammarat, 80160, Thailand Research Group in Applied, Computational and Theoretical Science (ACTS),Walailak University, Thasala, Nakhon Si Thammarat, 80160, Thailand Thailand Center of Excellence in Physics, Ministry of Higher Education, Science,Research and Innovation, Bangkok 10400, Thailand
It is well known that Jackiw-Teitelboim (JT) gravity posses the simplest theory on 2-dimensional gravity. The model has been fruitfully studied in recent years. In the presentwork, we investigate exact solutions for both JT and deformed JT gravity recently proposedin the literature. We revisit exact Euclidean solutions for Jackiw-Teitelboim gravity us-ing all the non-zero components of the dilatonic equations of motion using proper integraltransformation over Euclidean time coordinate. More precisely, we study exact solutionsfor hyperbolic coverage, cusp geometry and another compact sector of the AdS spacetimemanifold. We also introduce a nonminimal derivative coupling term to the original JT theoryfor the novel deformation and quantify its solutions. PACS numbers:
I. INTRODUCTION
The discovery of exactly solvable models of quantum gravity, starting with Kitaev [1] andsubsequent investigations [2, 3] posses one of the most exciting developments over the past fewyears. Since then, finding other models solvable to the same extent would be highly valuable, inparticular to test the robustness of the ideas. Especially, in holographic duality, we describe aboundary theory on a manifold M via a bulk description on a manifold N whose boundary is ∗ Electronic address: [email protected] † Electronic address: [email protected] governed by M . As mentioned in Ref.[4], the theory on a manifold M of the boundary theory isassumed to be a random hermitian matrix theory, and the theory on N is arguably the simplestpossibility of the theory on 2 d gravity. This is well known as Jackiw-Teitelboim (JT) gravity. Itscorrelation functions in various incarnations were later studied by many authors, see for example[5–13] including higher genus and random matrix descriptions [4]. We noticed that all classicalsolutions for all dilaton gravities and the spherical reductions of higher D gravities to D = 2 havebeen studied so far, see e.g.[14]. Especially, classical and quantum integrability of two-dimensionaldilaton gravities was found in [15] in Euclidean space.Concretely, JT gravity is a simple model of a real scalar field φ coupled to gravity in twodimensions [16, 17]. Pure JT gravity is the limiting case D → tricky conformal transformations [18]. Onestarts from a general D -dimensional Einstein Hilbert-action including arbitrary D -dimensionalmatter action L M given by S = − κ D Z d D x √− gR + Z d D x √− g L M , (1)where κ D denotes a D -dimensional gravitational coupling. It was found that when taking D → D → κ D ∼ O (1 − D ) and lim D → G µν ∼ O (1 − D ).We then subtract a term (total derivative) from the action in the limiting regime. After applyinga conformal transformation on the second subtracted term and taking the limit D →
2, we endup with a simple scalar field theory where the scalar field (dilaton) coupled to the scalar curvatureanalogous to φR . There is a possibility to redefine the gravitational coupling in this limiting caseto remove any ambiguity in the theory. Furthermore, JT gravity action can be obtained froma suitable dimensional reduction of a general four-dimensional GR when the spacetime is timeindependent, spherically symmetric. The resulting theory is a subclass of the dilaton theories asan attempt to define effective theory for the gravity at large distances [19] .For the case of negative cosmological constant where one can add L M = − , Λ = − /L AdS ≡−
1, the bulk action in Euclidean signature is governed by S = − Z N d x √ gφ ( R + 2) + ... , (2)where ellipses denote a topological term, e.g. Euler characteristic, R is the scalar curvature of themetric tensor g . Evidently, the model has been fruitfully studied in recent years. These includefor example the works done by Refs.[20–23, 25, 26, 30]. The solutions within JT gravity in thepresence of nontrivial couplings between the dilaton and the Abelian 1-form were constructed in[24]. Recently, it was found by Witten [25] that a simple correspondence of JT gravity with arandom matrix is possible in part since JT gravity is simple. Moreover, the results of Ref.[4]showed that JT gravity in two dimensions is dual not to any particular quantum system but to arandom ensemble of quantum systems. The author of [26] later explained the simple deformationof JT gravity by adding a self-coupling of the scalar field. He found that the resulting model isstill dual to a random ensemble of quantum systems, with a different density of energy levels.To generalize JT gravity to a large class of models, let us emphasize a particular example ofa self-coupling of the scalar field proposed in Ref.[26]. Consider an action for a scalar field φ anda metric g with each term having at most two derivatives. The most general possible action maytake the form S = − Z N d x √ g (cid:16) F ( φ ) R + G ( φ ) |∇ φ | + V ( φ ) (cid:17) , (3)where F ( φ ) , G ( φ ) and V ( φ ) are functions depending on the (dilaton) field φ . It was noticed thatquantum aspects in the theory governed by the action (3) have been studied in detail in Ref.[27].However, the above action can be simplified using a Weyl transformation of the metric togetherwith a reparametrization of the scalar field φ . With this transformation, we find that two of thethree functions can be eliminated. Hence we can end up with the case of just single function: S = − Z N d x √ g (cid:16) φR + U ( φ ) (cid:17) . (4)In order to recover the result present in [26], we can introduce a function U ( φ ) = 2 φ + W ( φ )to Eq.(4), where W ( φ ) denotes the departure from original JT gravity. We will investigate thesolutions for deformed JT gravity in Sec.IV. However, adding a self-coupling of the scalar field isnot the only possibility to deform JT gravity. We consider a nonminimal derivative coupling termto the original JT theory for the deformation. We present this new deformation in Sec.V.In the present work, we investigate exact solutions for both JT and deformed JT gravity recentlyproposed in the literature. We revisit exact Euclidean solutions for Jackiw-Teitelboim gravity usingall the non-zero components of the dilatonic equations of motion using proper integral transforma-tion over Euclidean time coordinate in Sec.II. More precisely, in Sec.III, we study exact solutionsfor hyperbolic coverage, cusp geometry and another compact sector of the AdS spacetime man-ifold. In addition, we study dJT gravity and examine its solution in Sec.IV. Moreover, in Sec.V,we also introduce a nonminimal derivative coupling term to the original JT theory for the newdeformation and quantify its solutions. We finally conclude our findings in the last section. II. JACKIW-TEITELBOIM GRAVITY
Let us analyze the equations of motion of the Jackiw-Teitelboim theory. We now carry outcorresponding calculations in JT gravity by starting with the JT action of the form [21] S = − h Z √ gφ ( R + 2) d x + 2 Z ∂ φ bdy K i , (5)where g, R, K, φ , and φ bdy refer to the metric, scalar curvature, extrinsic curvature, dilaton fieldand its value on the boundary, respectively. Note here that we are working with units κ = − / , L AdS ≡
1. Using the action (5), Einstein field equations can be directly derived to obtain R + 2 = 0 , (6) ∇ µ ∇ ν φ − g µν ∇ α ∇ α φ + g µν φ = 0 . (7)Note that in two dimensions, Einstein tensor follows G µν ≡
0. The second EoM (7) gives theenergy-momentum tensor T µν = 0. The first equation (6) gives us the scalar curvature for pureAdS . However it cannot fix any type of the geometry. Basically, we are working in the probe limitbecause the dilaton field doesn’t backreact on the geometry. As a result, we can use any type ofthe AdS metric and invoke an Euclidean form in the coordinates ( t, z ), where z = 0 is an AdSboundary. Because of the linear coupling to the dilaton field, the metric in the bulk is localized toAdS so that we can write ds = dt + dz z . (8)Therefore, the non-vanishing components of Eq.(7) for ( tt ) , ( tz ) , ( zz ) for a dilaton profile φ = φ ( t, z ) read ( tt ) : φ ′′ − φz = 0 (9)( tz ) : ˙ φ ′ + ˙ φz − ¨ φ − φ ′′ + φz = 0 (10)( zz ) : ¨ φ − φz = 0 , (11)where “primes” represent derivative with respect to t , and “dots” denote derivative with respectto z . In Ref.[21], one solution of the above system takes the form φ ( t, z ) = α + γt + δ ( t + z ) z . (12)The authors used this particular solution to study boundary term in particular to define boundaryfield value φ bdy = φ ( t, ǫ ) | ǫ → . However, we are going to quantify a more general solution. Here wewill show that the above solution is just a particular solution and it is a member of a more generalfamily of the exact solutions. In addition to the system of the PDEs given in Eqs.(9-11), we havea trace equation of the EoM (7) as a simple Helmholtz equation on AdS , ∇ α ∇ α φ − φ = 0 . (13)If we rewrite the above trace equation in the background (8) we simply obtain¨ φ + φ ′′ − φz = 0 . (14)We stress here that any solution (including a particular solution ) should satisfy all the fieldequations Eqs.(9-11) and (14). We will obtain eq. (14) just by adding Eqs.(9,11) together as well.Let us do the following: if we substitute φ from Eq. (9) into Eq.(10) and using Eq.(11), we findthe following PDE : z ( z ˙ φ ) ′ − φ = 0 . (15)We will find an exact dilaton profile φ ( t, z ). Particularly, we want to find φ bdy = φ ( t, z = 0). Notethat any exact solution for (15) is also an exact solution for all Eqs.(9-11) and (14). III. EXACT SOLUTIONS
In the present section, we quantify all possible solutions for Eqs.(9-11) and (14) by consider-ing the hyperbolic coverage, cusp geometry and another compact sector of the AdS spacetimemanifold. A. Hyperbolic space
In the first case, we consider space when Ω = { z ∈ [0 , ∞ ] × t ∈ [ −∞ , ∞ ] } . Using the standardFourier transformation of the dilaton profile φ ( t, z ) as φ ( t, z ) = Z ∞−∞ ˜ φ ( ω, z ) e iωt dt . (16)One can show that the Fourier amplitude function ˜ φ ( ω, z ) satisfies the following simple ODE :˜ φ ( ω, z ) − iωz ( z ˜ φ ( ω, z )) ′ = 0 . (17)We can simply solve Eq.(17) to obtain an exact solution for the Fourier amplitude and find that˜ φ ( ω, z ) = c z e izω , (18)where c is a constant of integration. By inserting this Fourier amplitude into the Fourier integraltransformation Eq.(16), we obtain φ ( t, z ) = c √ iπ | zt | / (cid:16) ( | t | + it ) ker ( 2 p | t |√ iz ) + ( t − i | t | )kei ( 2 p | t |√− iz )+( t + i | t | )(ker ( 2 p | t |√− iz ) + i kei ( 2 p | t |√ iz )) (cid:17) , (19)where ker and kei are Kelvin functions. It is worth noting that the solution given above covers awhole hyperbolic spacetime. The AdS boundary value for the dilaton field can be simply obtainedwhen setting z = ǫ →
0. Using the series expansion of the Fourier amplitude in the vicinity of z = ǫ , we can evaluate the following inverse Fourier integral: φ ( t, ǫ ) = c ǫ Z ∞−∞ e i ( ωt + ωǫ ) dω + O ( z − ǫ ) . (20)It is not possible to analytically solve the above integration. Basically an integration gives us alengthy expression of transcendental functions. There is a term of the integral which depends ofthe UV cutoff ǫ − and that term is given by 2 πδ ( t ). The leading terms can be expanded in seriesof orders of O ( ǫ n ) , n >
1. Finally, the boundary value for a dilaton profile is approximately givenby φ bdy ( t ) ≈ c √ πǫ δ ( t ) + O ( ǫ n ) . (21)In comparison to the general AdS/CFT program, see e.g. the discussion around Sec.4.6 of Ref.[32],the boundary value for the field is scaled as ǫ − . This implies that the renormalized field is φ r ∝ δ ( t ) yielding the existence of a boundary shock field amplitude. In the language of the CFT,the conformal dimension is fixed as ∆ − = − classical Euclidean background. B. Hyperbolic cusp geometry
We consider another interesting case for which Ω = { z ∈ [0 , ∞ ] × t ∈ [0 , } . In this cuspgeometry, it is convenient to expand the dilaton field in terms of an orthonormal basis of thefunctions as follows: φ ( t, z ) = ψ ( z )2 + ∞ X n =1 (cid:16) ϕ n ( z ) sin( nπt ) + ψ n ( z ) cos( nπt ) (cid:17) . (22)By substituting the Fourier form given in Eq.(22) into the dilaton field equation (15), one can showthat ψ ( z ) ≡ ϕ n ( z ) , ψ n ( z ) satisfy the following set of the operatorequation for each mode with n ≥ O n ( z ) { ϕ n ( z ) } = ψ n ( z ) , ˆ O n ( z ) { ψ n ( z ) } = − ϕ n ( z ) . (23)Here the differential operator ˆ O n ( z ) is defined asˆ O n ( z ) ≡ nπz (1 + z ddz ) , z ∈ [0 , ∞ ] . (24)Rewritten the above system of ODEs given in Eqs.(23) in Hamiltonian form, this equations forman infinite-dimensional dynamical system which can be written in matrix form as follows: O n ( z )ˆ O n ( z ) − ϕ n ( z ) ψ n ( z ) = 0 . (25)It is worth noting that the authors of [31] reformulated the evolution of the quantum state asan infinite-dimensional dynamical system with mathematical features different from the standardtheory of infinite-dimensional dynamical systems.Using a change of the variables as z → u = ln z , we find that the differential operator transformsas ˆ O n ( u ) ≡ nπe u (1 + ddu ) , u ∈ [ −∞ , ∞ ] . (26)Here it is written in terms of a variable u in which the conversion can be straightforwardly done.
1. Exact dilaton profile
We can integrate out the ODEs given in Eqs.(23) by using the auxiliary operator [ ˆ O n ( u )] .Using this, we can get an exact solution which can be solved from the resulting equation: ϕ n ( u ) ′′ + ϕ ( u ) ′ + e u n π ϕ ( u ) = 0 . (27)Once an exact solution for ϕ ( u ) is quantified, then the other dilaton Fourier amplitudes can beobtained using an operation ψ n ( u ) = ˆ O n ( u ) ϕ n ( u ). We solve for an exact solution for Eq.(27) toyield ϕ n ( u ) = r nπ e − u (cid:0) c n cos( e u nπ ) + c n sin( e u nπ ) (cid:1) , (28) ψ n ( u ) = r nπ e − u (cid:0) c n cos( e u nπ ) − c n sin( e u nπ ) (cid:1) , (29)and the dilaton profile reduces to the following expression: φ ( t, u ) = e − u ∞ X n =1 r nπ (cid:16) c n cos( e u nπ − nπt ) − c n sin( e u nπ − nπt ) (cid:17) , (30)with u ∈ ( −∞ , ∞ ). Invoking the AdS boundary (bdy) value φ bdy ( t ) as an initial condition where φ bdy ( t ) = φ ( t, −∞ ) and a UV cutoff as Λ = lim u →−∞ e − u , then the Fourier coefficients c n , c n areobtained c n = Λ − r nπ Z φ bdy ( t ) sin( nπt ) , (31) c n = 2Λ − r nπ Z φ bdy ( t ) cos( nπt ) . (32)By specifying the initial bdy value of the fields, we can write the full dilaton profile using theFourier series (30). Some special classes of the bdy models are listed below: • Static boundary φ bdy ( t ) = φ bdy (0) = φ bdy : the integrals can be simplified and we obtain c n = 0 , c n = Λ − φ bdy s n + 1) π ) , n = 0 , , ... ∞ . (33)Notice that the dilaton profile in this case cannot be easily written in a closed form, but itis given by φ ( t, u ) = − − φ bdy e − u ∞ X n =0 sin( e u (2 n +1) π − (2 n + 1) πt ) π (2 n + 1) . (34) • Shock boundary regime φ bdy ( t ) = φ c δ ( t − t c ) , < t c <
1: In this case, we have c n = Λ − φ c r nπ sin( nπt c ) , (35) c n = 2Λ − φ c r nπ cos( nπt c ) . (36)We find that the full dilaton profile in this case has a closed form as φ ( t, u ) = 2Λ − φ c e − u ∞ X cos( e u nπ − nπ ( t − t c )) . (37) C. Geometry with
Ω = { z ∈ [0 , × t ∈ [0 , ∞ ) } We first assume the Euclidean time t ∈ [0 , ∞ ). One of the effective methods to solve it is touse a Laplace transform along a suitable boundary conditions for the domain Ω = [0 , × [0 , ∞ ).A Laplace transform of Eq.(15) is given by φ ( t, z ) = Z ∞ ˜ φ ( s, z ) e − st dt. (38)Substituting it into Eq.(15), we obtain sz ∂∂z ˜ φ ( s, z ) + ( sz −
1) ˜ φ ( s, z ) + ( − zφ (0 , z ) − z φ ′ (0 , z )) = 0 . (39)Using the standard procedure, we can find exact solutions for the field amplitude ˜ φ ( s, z ) in theLaplace plane s and then by performing an inverse Laplace transform one can obtain the generaldilaton profile φ ( s, z ). We need initial condition (IC) to solve it. In particular the followingboundary conditions are introduced: φ ( t, z ) = t → + ∞ f ( z ) if t = 0 . (40)As we see, the ˜ φ ( s, z ) depends on the IC function, f ( z ). An exact solution for ˜ φ ( s, z ) is obtainedas follows: ˜ φ ( s, z ) = c z e − sz + f ( z ) s − f (1) sz e − − zsz − e − sz s z Z z f ( w ) e sw w dw. (41)Note that z ∈ [0 , ≤
1. In particular, we can check that the value of the field amplitude ˜ φ ( s, z )at the point z = 1 is given by ˜ φ ( s,
1) = c e − s . (42)Hence one can use inverse Laplace transform to obtain φ ( t,
1) = J (2 √ t ) √ t . Now, the complete solutionusing inverse Laplace transform of the above expression can be obtained and we find φ ( t, z ) = f (0) − f (1) z J (2 r tz ) − c ( tz ) / J (2 r tz )+ c δ ( t ) z − √ t Z z dwf ( w ) J (2 q t ( w − z ) wz ) √− wz + w z , (43)where f (0) , f (1) , c are arbitrary parameters. • Evaluation of φ ( z, t ) for a class of critical f ( z ): We can evaluate the integral appeared inthe exact dilaton profile given in Eq.(43) for the following simple but physically significantcases: – If f ( w ) = δ ( w − w ) is a localized dilaton profile at the initial time, the exact profiletakes the following form: φ ( t, z ) = f (0) − f (1) z J (2 r tz ) − c ( tz ) / J (2 r tz )+ c δ ( t ) z − tw z θ (1 − w , w ) F (cid:0) t (cid:0) w − z (cid:1)(cid:1) , (44)0where w ∈ R and θ ( x , x ) defines the multivariables Heaviside theta function, whichis equal 1 only if all of the arguments x i are positive and F (cid:0) t (cid:0) w − z (cid:1)(cid:1) is thegeneralized hypergeometric function. – If f ( w ) = θ ( w − w ) is a step function as a shock wave at initial time, φ ( t, z ) = f (0) − f (1) z J (2 r tz ) − c ( tz ) / J (2 r tz ) + c δ ( t ) z , (45)with ℜ ( w ) > , ℑ ( w ) = 0 and J i ( x ) are Bessel functions. • Dilaton on AdS boundary: In Ref. [21], it is assumed that φ bdy ( t ) ≡ φ ( t, z = 0) is constant,but here we keep it as an arbitrary function of the Euclidean time t . In the limiting case, z → ǫ , we find φ ( t, z → ǫ ) = φ bdy ( t ). We can simply do series expansion up to any order of ǫ as well as study the φ bdy ( t, z → ǫ ) for both regimes, i.e. initial time t = 0 and late time t = ∞ , using the exact profile given in Eq.(43). Furthermore, we can quantify the boundaryvalue via the Laplace amplitude ˜ φ ( s, z ). Basically, we have˜ φ ( s, z → ǫ ) = c ǫ e − sǫ + f (0) s − f (1) sǫ e − sǫ − e − sǫ s ǫ Z f ( w ) e sw w dw. (46)By performing series expansion, we can show that the boundary value of the dilaton field isgiven approximately by the following expression: φ bdy ≈ f (0) − β √ ǫ − f (1) ǫ / + ζǫ / + 1 ǫ . (47) • Initial value of the field on the AdS boundary: At initial time t = 0, we have to take thelimit of the expression Eq.(46) and keep ǫ ≪ φ bdy (0 , z → ǫ ) = f (0) − f (1) ǫ + c δ (0) ǫ . (48)In the CFT language, the conformal dimension here is ∆ − = − • Late time value of the field on the AdS boundary: For a late time, when t → ∞ , we have φ bdy ( ∞ , z → ǫ ) ≈ f (0) − √ t Z ǫ dwf ( w ) cos(2 q t ( ǫ − w ) − π ) w . (49)We can show that for all w ∈ [ ǫ,
1] it provides that | cos(2 q t ( ǫ − w ) − π ) | ≤
1. Consequently,we have | φ bdy ( ∞ , z → ǫ ) | ≤ | f (0) | + √ t Z ǫ | f ( w ) | d | w | . (50)1If f ( w ) ∝ w n , n = −
1, we then obtain | φ bdy ( ∞ , z → ǫ ) | ≤ | f (0) | + √ tn + 1 . (51)However, for the case with f ( w ) ∝ w , then we findlim t →∞ | f (0) | − | φ bdy ( t, ǫ ) | log ǫ √ t ≥ . (52)Note that the last case is a generic case of the AdS in two dimensions and it doesn’t obeythe power-law behavior for boundary value of the φ in other dimensions (see for example thediscussion around Sec.4.6 of Ref.[32] for boundary value of a generic scalar field in dimensionsdifferent from two). IV. DEFORMED JACKIW-TEITELBOIM GRAVITY
We next consider deformed JT (dJT) gravity investigated recently in Ref.[25]. The model isdescribed easily by including a potential term to the original JT action. It takes the form S = − Z d x √ g (cid:16) φ ( R + 2) + W ( φ ) (cid:17) . (53)We take W as a general function of φ and later we will consider its perturbation to compare withWitten’s results. Notice that the generalization of JT gravity without scalars has been proposedso far in Refs.[28, 29]. The field equations derived from action Eq.(53) are simply the vanishingEinstein tensor and a type of the Klein-Gordon equation for the dilaton field φ : R + 2 + W ′ ( φ ) = 0 , (54) ∇ µ ∇ ν φ − g µν ∇ α ∇ α φ + (cid:16) φ + W ( φ )2 (cid:17) g µν = 0 . (55)The pure AdS solution exist as an exact solution if and only if the potential function satisfies thefollowing constraints: W ′ ( φ c ) = 0 , W ( φ c ) = − φ c , (56)with a uniform dilaton profile φ = φ c . For example with the potential function as W ( φ ) = −
2Λ (aneffective negative cosmological constant term), we have an exact AdS for a uniform profile φ = φ c .If φ = φ c , there is also a possibility to still have pure AdS if and only if W ( φ ) ≡ const., which weneed to investigate very carefully in detail in the next section.2 A. Exact pure AdS in dJT for non uniform dilaton φ = φ c For a constant scalar profile, there is a trivial class of the AdS solutions. But if φ = const., tohave AdS we need W ( φ ) to satisfy the following relations: W ′ ( φ ) = 0 , (57) ∇ µ ∇ ν φ − g µν ∇ α ∇ α φ + (cid:16) φ + W ( φ )2 (cid:17) g µν = 0 . (58)We adapt once the Euclidean metric Eq.(8) and the non-vanishing components of the Einsteinequation read ( tt ) : φ ′′ − φz − W ( φ )2 z = 0 , (59)( tz ) : ˙ φ ′ + ˙ φz − ¨ φ − φ ′′ + φz + W ( φ )2 z = 0 , (60)( zz ) : ¨ φ − φz − W ( φ )2 z = 0 , (61)with W ′ ( φ ) = 0. From this constraint, we obtain, W ( φ ) = W , similar to the pure JT. In thiscase, the trace equation reads ∇ µ ∇ µ φ − (2 φ + W ) = 0 which can be simply derived using theabove EoMs. By eliminating φ ′′ in Eqs.(59,60) and by adding it to the Eq.(61), we end up withthe following PDE: z ( z ˙ φ ) ′ − φ − W . (62)This is a modified version of the JT equation given in Eq.(15). We can investigate all types of thesolutions as those we already discussed in the pure JT gravity. We are interested in the Euclideanhyperbolic coverage of the AdS where one can use Fourier transformation. Therefore the Fourieramplitude can be obtained as ˜ φ ( ω, z ) − iωz ( z ˜ φ ( ω, z )) ′ + W δ ( ω ) = 0 . (63)An exact solution for the Fourier amplitude cam be determined to obtain˜ φ ( ω, z ) = c z e izω + iW e izω ωz δ ( ω ) Ei ( − iωz ) , (64)where c is an integration constant and Ei ( z ) = − R ∞− z e − t t dt gives the exponential integral function.By inserting this Fourier amplitude into the Fourier integral transformation Eq.(16), we find φ ( t, z ) = c √ iπ | zt | / (cid:16) ( | t | + it ) ker ( 2 p | t |√ iz ) + ( t − i | t | )kei ( 2 p | t |√− iz )+( t + i | t | )(ker ( 2 p | t |√− iz ) + i kei ( 2 p | t |√ iz )) (cid:17) + 12 π Z ∞−∞ iW e izω − ωt ωz δ ( ω ) Ei ( − iωz ) dω . (65)3The integral in the last line can be computed using the counter integral by finding the residue forthe integrand at pole ω = 0. The result contains a divergence term δ (0) . B. Non AdS solutions for the dJT
Consider the system of the EOMs given in the Eqs.(54,55). We assume that there may be somenon AdS black hole solutions (e.g. solitons, topological defects, and so forth) where R = −
2. Inthis case we find R + 2 + U ′ ( φ ) = 0 , (66) ∇ µ ∇ ν φ − g µν ∇ α ∇ α φ + ( φ + U ( φ )2 ) g µν = 0 . (67)Here we can adapt a kind of light-cone coordinate system ( u, v ) in the spacetime, but we have tofirst show that such metric is accessible. To do this, we start with a simple general Euclidean timeindependent (stationary) metric in the Poincar´e coordinates ( t, z ): ds = f ( z )( dt + h ( z ) f ( z ) dz ) − h ( z ) + k ( z ) f ( z ) f ( z ) dz . (68)One can define ξ = t + R h ( z ) f ( z ) dz and therefore the metric reduces to ds = f ( z ) dξ − h ( z ) + k ( z ) f ( z ) f ( z ) dz . (69)Now if one define the light-cone coordinate system as u = ξ − Z p h ( z ) + k ( z ) f ( z ) f ( z ) dz , (70) v = ξ + Z p h ( z ) + k ( z ) f ( z ) f ( z ) dz , (71)we can finally write a general metric as ds = f ( u, v ) dudv, f ( u, v ) = f ( z ( u, v )) . (72)Rewriting Eqs.(66, 67) and using the metric (72), we then obtain ∂ u ∂ v log f ( u, v ) + (1 + U ′ ( φ )2 ) f ( u, v )2 = 0 , (73)as well as ( uu ) : ∂ u φ ( u, v ) − ∂ u log f ( u, v ) ∂ u φ ( u, v ) = 0 , (74)( uv ) : ( φ + U ( φ )2 ) f ( u, v ) − ∂ u ∂ v φ ( u, v ) = 0 , (75)( vv ) : ∂ v φ ( u, v ) − ∂ v log f ( u, v ) ∂ v φ ( u, v ) = 0 . (76)4We can integrate out Eq.(76) to yield ∂ v φ ( u, v ) = α ( u ) f ( u, v ) , . (77)where α ( u ) denotes an arbitrary function on u . Similarly using Eq.(74), we obtain ∂ u φ ( u, v ) = β ( v ) f ( u, v ) . (78)A consistency relation for Eqs(77, 78) provides us with ∂ u ( α ( u ) f ( u, v )) = ∂ v ( β ( v ) f ( u, v )) . (79)Fortunately it can be simplified to obtain the following PDE for f ( u, v ): β ( v ) ∂ v log f ( u, v ) − α ( u ) ∂ u log f ( u, v ) = α ′ ( u ) − β ′ ( v ) . (80)If we change the variables from v → V = R β − ( v ) dv, u → U = R α − ( u ) du , we can simplify PDEgiven in Eq.(80) as follows: ∂ V log f ( U, V ) − ∂ U log f ( U, V ) = ∂ U log α ( U ) − ∂ V log β ( V ) . (81)Notice that this is an inhomogenous first-order PDE and the solution can be easily found in thefollowing cases: • Case I: α ( U ) = U M +1 , β ( V ) = V N +1 , M, N = −
2: We find f ( U, V ) = exp (cid:0) P ( U + V ) − (1 + M ) log U − (1 + N ) log V (cid:1) , (82)where P is an arbitrary function. Now we can integrate out the PDE given in Eq.(75). Weneed to transform back to the coordinates from U, V → u, v up to some arbitrary integrationconstants to obtain V = − v − N N , U = − u − M M . (83)Therefore the metric function changes to the following equivalent form in the original nullcoordinates system: f ( u, v ) = exp (cid:16) P ( u − M M + v − N N ) + u − M (1 + M ) M + v − N (1 + N ) N (cid:17) . (84)Note that the solution given in Eq.(84) is a family of the exact solutions for the theory andit is different from the AdS. It is worth mentioning that a soliton solution for the PDE givenin Eq.(75) provides us an exact profile for the dilaton field. Basically one can solve Eq.(75)along the exact metric presented in Eq.(84) to yield φ ( u, v ) = − v − N ( N +1) ( − N ) N +1 Z ∂uf ( u, v ) . (85)5 • Case II: α ( U ) = U − , β ( V ) = V − : We have f ( U, V ) = Q ( U + V ) + log( − U V ) . (86)We can transform back to the original coordinates ( u, v ) and use u = p | U | , v = p | V | . Asa result, the metric function reads f ( u, v ) = Q ( u + v uv . (87)In this simple case, the dilaton profile can be obtained by integrating the following (exact)Pfaffian form dφ ( u, v ) = f ( u, v )( duu + dvv ) , (88)which can be integrated easily to give φ ( u, v ) = − v Z ∂uf ( u, v ) . (89) V. NONMINIMAL-COUPLED DEFORMATION OF JT GRAVITY
Adding a potential term U ( φ ) to the action of the JT gravity is not the only possibility forthe modification. It is possible to consider a kinetic coupling term as well, where the full theoryincluding the JT gravity and a correction, is written as follows: S = − Z √ g (cid:16) φ ( R + 2) + αR µν φ µ φ ν (cid:17) d x . (90)Here R µν ≡ Rg µν is the Ricci tensor in two dimensions. Our main motivation consideringthis theory is the reductions of 4-dimensional gravity using the conformal tricky limit D → φ µ (cid:3) φ or ( φ µ ) ≡ ( φ µ φ µ ) , and so on, to the original JT gravity action. The only viability is to includethe coupling term in the form of φ G GB . The Gauss-Bonnet term G GB vanishes trivially in alleven dimensions, i.e, when D = 2 , , , ... (including our interesting JT gravity proposal). Butthe minimal toy model which remains safely is the coupling term between Ricci tensor and thekinetic term , i.e, R µν φ µ φ ν . We mention here that the four dimensional coupling term where theRicci tensor is replaced by Einstein tensor doesn’t give any new solution because the Einsteintensor vanishes identically. The coupling constant α is an arbitrary and in our discussion is a tinyparameter where we can expand all the solutions (a metric function and a dilaton profile) in terms6of it in a perturbative form. To find the EoMs, the effective way is to reduce the action to the minisuperspace by considering an arbitrary form for the metric as ds = e ψ ( u,v ) dudv , (91)in the Euclidean null coordinates u, v = z ± t . Here t is the Euclidean time and the mini superspaceLagrangian takes the following form L = φe ψ ( u,v ) − ψ uv ( φ + 2 αφ u φ v ) . (92)From the above mini superspace Lagrangian, the Euler-Lagrange equations can be derived to obtain ∂ u ( ∂ L ∂φ u ) + ∂ v ( ∂ L ∂φ v ) − ∂ L ∂φ = 0 , (93) ∂ u ( ∂ L ∂ψ uu ) + ∂ v ( ∂ L ∂ψ vv ) + ∂ u ∂ v ( ∂ L ∂ψ uv ) − ∂ u ( ∂ L ∂ψ u ) − ∂ v ( ∂ L ∂ψ v ) + ∂ L ∂ψ = 0 . (94)Using the above equations, we can write the following system of the PDEs as a set of the EoMs − φ uv + φe ψ − α ∂ u ∂ v ( φ u φ v ) = 0 , (95) e ψ − ψ uv + α (cid:16) ∂ u ( φ v ψ uv ) + ∂ v ( φ u ψ uv ) (cid:17) = 0 . (96)If we set α = 0, it is illustrative to mention here that the second equation is just R + 2 = 0, i.e. theEoM for φ in the pure JT. It is worth noting that for the conformal metric (91), the Ricci scalartakes the form R = − e − ψ ψ uv . It is difficult to find exact solutions of ( φ, ψ ) for an arbitrary α . However, if we consider α as a tiny parameter of the original JT, it is possible to developperturbative solutions. A. Perturbative solutions
Let us consider a tiny dimensionless parameter α and we consider perturbation of the field asfollows: φ = ∞ X n =0 α n φ n , ψ = ∞ X n =0 α n ψ n , (97)Performing the expansion of a tiny dimensionless parameter α order by order in the expansion, thesolutions can be obtained. Therefore, we can find φ n +1 , ψ n +1 using the iteration technique.7
1. Zero’th order solutions
If we consider the EoMs up to order O ( α ), the zeroth order (pure JT) fields are given by − φ | uv + φ e ψ = 0 , (98) e ψ − ψ | uv = 0 , (99)where we have defined Φ i | u, v, . . . , . | {z } n -times = ∂ n Φ i ∂u∂v . . . , . | {z } n -times , . The metric function ψ can be simply obtainedby considering a solution in the form ψ = A ( u + v ) n . Let us choose n = − e ψ = −
14 ( u + v ) − . (100)It is possible to find φ using the same technique. Considering a power law form φ = B ( u + v ) m ,one can show that the particular (not general) dilaton profile takes the form φ = X ± C ± ( u + v ) m ± , m ± = 12 ± √ . (101)Note that there exist other exact solutions for φ . However, we anticipate not to quantify them hereand leave these for our future investigation. In comparison to the discussion around the boundaryvalue of the field in Ref.[32], here at zeroth order approximation, one can say that the boundaryvalue of the dilaton field φ bdy can be expressed near the AdS boundary region where in the nullcoordinates the region given by u + v → φ bdy ≈ ǫ m − φ r (102)gives us the meaning of the m − as the dominant conformal dimension and one can suggest that φ r ∝ C − and fully independent from the spatial coordinates. Basically in AdS when one tryto reach the AdS boundary region, will end to a singular/degenerate metric where the boundayinformation can be coded only on one degree of the freedom, probably a parametrization of the time.The boundary values of the dilaton field can be used to estimate the leading term in the JT gravityaction, i.e, the term appeared as boundary term (an analogouds to the GibbonsHawkingYorkboundary term in ordinary four dimensional GR).8
2. First order solutions
Let us next consider the first-order perturbation in O ( α ), the following system of the PDEs canbe obtained ˆ O φ + b ( u, v ) ψ = c ( u, v ) , (103)ˆ O ψ = d ( u, v ) , (104)where ˆ O ≡ − ∂ u ∂ v + e ψ , (105)and the functions b ( u, v ) , c ( u, v ) , d ( u, v ) are defined as b ( u, v ) = φ e ψ , (106) c ( u, v ) = 14 ∂ u ∂ v ( φ | u φ | v ) , (107) d ( u, v ) = − ∂ u ( φ | v ψ | uv ) + ∂ v ( φ | u ψ | uv ) . (108)Therefore, the solutions for φ , ψ take the form φ = c ( u, v ) − b ( u, v ) ˆ O − d ( u, v ) , (109) ψ = ˆ O − d ( u, v ) . (110)They are all algebraic expressions. The inverse of the hyperbolic operator ˆ O can be obtained easilyby solving a Poisson equation over the ( u, v ) plane. The operator ˆ O can be decomposed yieldingto an interpretation of the dilaton field as a doublet field as follows:ˆ O = ( − √ ∂ u + ˆ A ( u, v ))( 12 √ ∂ v + ˆ B ( u, v )) , (111)where the operators ˆ A, ˆ B are ˆ A = h e − ψ ∂ v + ˆ Ce − ψ i − , (112)ˆ B = h e ψ (2 ψ | v + ∂ v ) + ˆ C i . (113)In the above expressions, ˆ C is a constant (uniform) operator, i.e, ˆ C ( u, v ) = C ( u, v ). Additionally,the ladder operators can be defined asˆ O + = − √ ∂ u + ˆ A ( u, v ) , (114)ˆ O − = 12 √ ∂ v + ˆ B ( u, v ) , (115)which satisfy the following commutation relation: (cid:2) ˆ O + , ˆ O − (cid:3) − = − √ (cid:16) e ψ ψ | u ˆ A − + ( ∂ u ∂ v − ψ | u ( ˆ C + ∂ v )) ˆ A (cid:17) . (116)9
3. More about an inverse operator ˆ O − In seems that any information about the deformed dilaton profile as well as the backreactedmetric depends on the inverse operator ˆ O − . The hyperbolic operator O is a non singular operatorand can be expressed explicitly using the zeroth order metric function asˆ O = − ∂ u ∂ v − u + v ) (117)For sake of simplicity just let us turn back to the Poincare coordinates, where we can write it asˆ O = −
132 ( ∂ zz − ∂ tt + 2 z ) (118)If we seeking for inverse of the operator , it will be just the inverse of the inhomogenous wave opera-tor on two dimensions and an alternative simple representation for it is given as the following(usingbinomial theorem): ˆ O − = ∞ X n =0 n X k =0 ( − n − k +1 n ! z k +2 n − k !( n − k )! ( ∂ tt ) k ( z ∂ zz ) n − k (119)where the term ( z ∂ zz ) n − k can be computed using the Leibniz formula. In the above derivationwe assumed that the norm of the differential operators doesn’t diverge in the boundary. VI. CONCLUSIONS
In this work, we investigated exact solutions for both JT and deformed JT gravity recentlyproposed by E. Witten [26]. We revisited the Jackiw-Teitelboim theory and examined its exactEuclidean solutions using all the non-zero components of the dilatonic equations of motion usingproper integral transformation over Euclidean time coordinate. More precisely, we solved thesystem of field equations to find exact solutions for hyperbolic coverage, cusp geometry and anothercompact sector of the AdS spacetime manifold. Then we considered its deformation called dJTgravity and figured out the solutions for both AdS and non AdS cases. The deformation can besimply done by adding a potential term U ( φ ) to the action of the JT gravity. However, addingsuch a potential term to the action is not the only possible option for the modification. We thenintroduced a nonminimal derivative coupling term to the original JT theory for the new deformationand quantified its solutions.By following the conformal method, the authors of Ref.[18] showed that when considering thelimit of D → D -dimensional ordinary GR, the resulting action yields a special case0of 2D dilaton gravity, JT gravity. If one take the limit in even dimensional spacetimes (generallyincluding the four dimensions), the Gauss-Bonnet term will appear along with other kinetic terms.Here we only considered the first minimal theory where the Ricci tensor could be coupled to thescalar field via a kinetic term. We constructed perturbative solutions for this deformed JT theory.Regarding our present work, it is possible to extend this study to account of higher-order correctionsto JT gravity from the phenomenological point of the view to explore dual boundary theory. Acknowledgments