Classical and quantum (2+1)-dimensional spatially homogeneous string cosmology

We introduce three families of classical and quantum solutions to the leading order of string effective action on spatially homogeneous $(2+1)$-dimensional space-times with the sources given by the contributions of dilaton, antisymmetric gauge $B$-field, and central charge deficit term $\Lambda$. At the quantum level, solutions of Wheeler-DeWitt equations have been enriched by considering the quantum versions of the classical conditional symmetry equations. Concerning the possible applications of the obtained solutions, the semiclassical analysis of Bohm's mechanics has been performed to demonstrate the possibility of avoiding the classical singularities at the quantum level.


General considerations
Studies on (2+1)-dimensional gravity dating back to 1963 [1], have received growing interest since Deser, Jackiw, and 't Hooft surveyed the classical and quantum dynamics of point sources [2,3,4], and Witten demonstrated the Chern-Simons theory representation of (2 + 1)-dimensional gravity [5,6,7]. Motivations to consider this simpler model compared to the known (3 + 1)-dimensional gravity is that, besides sharing fundamental features with general relativity, it avoids some of the difficulties that general relativity is usually facing, such as the nature of singularities, cosmic censorship, and the conceptual foundations of quantum gravity. In (2 + 1)dimensions the gravitational constant G has dimensions of length. However, the theory is renormalizable and the appearing divergences in its perturbation theory can be canceled via field redefinitions [8]. Classical and quantum solutions in this dimension have been widely investigated, for instance, in [9,10,11,12].
In this work, we are going to present cosmological solutions at classical and quantum levels on (2 + 1)dimensional space-times where the two-dimensional space part has the symmetries of two-dimensional Lie algebra and admits a homogeneous metric. In (3 + 1)-dimensions, the homogeneous space-times, usually referred as Bianchi type space-times, which are assumed to possess the symmetry of spatial homogeneity and defined based on the simply-transitive three-dimensional Lie groups classification [13], have been widely used to generate cosmological and black hole solutions [14,15,16,17,18,19,20]. An interesting approach to find classical and quantum cosmological solutions on these homogeneous space-times has been using the Lie symmetries [17], and the symmetries of the supermetrics [21]. A commonly used application of symmetries in gravitational theories is selecting particular solutions of the field equations. These symmetries include, in particular, the well-known Noether symmetry which has been applied in several models including scalar-tensor cosmology and higher derivative theories of gravity [22,23,24], the Hojman symmetry [25], and the conditional symmetries of the configuration space which have been used to construct solutions on spherical and homogeneous Bianchi type space-times [21,26,27,28,29].
Here, we start with low-energy string effective action whose equations of motion are equivalent to the oneloop β-function equations of σ-model. These equations are conformal invariance conditions of the σ-model and, on the other hand, are equivalent to the Einstein field equations [30]. Solutions of these equations and their application in studying the evolution of the universe are called string cosmology [15,18,19,31]. We are going to find the solutions on (2 + 1)-dimensional spatially homogeneous space-time taking into account the contributions of dilaton, B-field, and the central charge deficit term Λ, which is equivalent to a dilaton potential in the Einstein frame string effective action V (φ) = −Λe −2φ . Then, we will continue in the quantum cosmology context focusing on the Hamiltonian approach of gravity introducing the Wheeler-DeWitt equations. To obtain the final solutions of Wheeler-DeWitt equations on the (2 + 1)-dimensional homogeneous space-times, the conditional symmetries approach will be adopted and imposed on the wave functions by promoting the generators of the conditional symmetries to quantum operators. Also, following the Bohm's approach [32,33], the quantum potentials and semiclassical solutions will be obtained to check whether the classical singularities can be avoided at the quantum level. In the following, we add some introductory remarks on string effective action equations of motion and the conditional symmetries approach.
1.2 Low energy string effective action equations of motion and a spatially homo- For a σ-model with background fields of metric g µν , dilaton φ, and antisymmetric tensor gauge B-field, the requirement of the conformal invariance of the theory is vanishing of the β-function equations, given at one-loop order by [34,35] where, the H is the field strength tensor of B-field defined by is the central charge deficit of D-dimensional bosonic theory [30]. The g µν is the string frame metric and describes physics from the string viewpoint. Alternatively, the Einstein frame metricg µν is introduced byg and the Einstein frame effective action for bosonic string is given by [34,35] in which∇ indicates the covariant derivative with respect tog, and κ 2 , where λ p is the Planck length and G D is the D-dimensional gravitational Newton constant. In this frame, the one-loop β-functions (1)-(3) recast the following forms that can also be obtained by the variation of the action (5) with respect tog µν [35] where T µν is the effective energy-momentum tensor defined by In this paper, we focus on (2 + 1)-dimensional space-times where the t constant hypersurface is given by a homogeneous space corresponding to the 2-dimensional Lie group with real two-dimensional Lie algebra [T 1 , T 2 ] = T 2 . In this regard, we start with the string frame metric ansatz where N and g ij are functions of time t, and {σ i , i = 1, 2} are left invariant basis 1-forms on the lie group, obeying σ 2 = − 1 2 σ 1 ∧ σ 2 . The relation between coordinate and non-coordinate basis is given by We will present three families of solution with diagonal and non-diagonal metric g ij .

Wheeler-DeWitt equation and Semiclassical approximation
With S = dt L, the Lagrangian L is given in the general form of whereÑ is the lapse function in Einstein frame metric, and G αβ (q) is the supermetric defined on minisuperspace as the configuration space with variables q α . Here and hereafter the dot symbol stands for derivation with respect to t. Basically, this Lagrangian is singular since ∂L ∂Ṅ = 0. However, if one considersÑ =Ñ (ã 1 ,ã 2 , φ) or N (t) = 1, then L becomes regular. Defining the conjugate momentum p α = ∂L ∂q α , the corresponding Hamiltonian to (11) is given by Taking advantage of the freedom provided by the time parametrization invariance, a constant potential lapse parametrization n =Ñ V ,Ḡ αβ = V G αβ , andV = 1 can be chosen in such a way that the Lagrangian reads [21] where q α and n are dependent dynamical variables. In this case, the symmetry generators ξ, i.e. L ξḠαβ = 0, satisfying a Lie algebra of the form [ξ i , ξ j ] = c k ij ξ k , correspond to conserved quantities Q i = ξ α i p α [21,26]. Quantization of this system can be carried out by promoting q α , their conjugate momenta, and Hamiltonian density to quantum operators. Then, imposing the classical constraint as a condition on the wave function Ψ leads to the Wheeler-DeWitt equationĤ whose solutions describe the dynamics of the system. Also, promoting the conditional symmetry generators Q i to operators and imposing them on the wave function yields the eigenvalue problem where η i are the classical charges such that Q i = η i . Since the classical algebra {Q i , Q j } = c k ij Q k is isomorphic to quantum algebra with the same structure constants, a consistency condition is required which is given by the following integrability conditions [21,26] c k ij η k = 0.
Practically, this condition determines the allowed elements of certain subalgebras to be applied on the wave function. In fact, the additional equations (15) select particular solutions of Wheeler-DeWitt equation and give rise to wave functions containing no arbitrary functions. Also, to determine some of the physical features of the Wheeler-DeWitt equation solutions, the Bohm's mechanics can be performed on the obtained wave functions to identify the quantum potential and semiclassical geometries [32,33]. Specifically, if the wave function is given in the polar form where Ω(q) is the amplitude and ω(q) is the phase of the wave function, substituting Ψ in the Wheeler-DeWitt equation yields the modified Hamilton-Jacobi equation [21,27] κ 4 which is of the form H(q α , p α ) + Q = 0 where is the quantum potential. Also, from the semiclassical point of views, the equations of motion are given by [27] ∂ω ∂q When the quantum potential Q vanishes, the solutions of (20) coincide with the classical ones obtained in WKB approximation, if and only if B is a solution of the corresponding Hamilton-Jacobi equation. In this work, we wish to use the solutions of the Wheeler-DeWitt equation to determine the quantum potential and semiclassical geometries. The singularity behavior of these geometries and their matter content will be also investigated. The paper is organized as follows: In section 2, the low energy string equations of motion are solved to obtain cosmological solutions on (2 + 1)-dimensional space-time where the space part corresponds to the 2-dimensional Lie group with diagonal metric. Then, transforming to the Einstein frame where the (2 + 1)-dimensional spacetime is minimally coupled to massless dilaton and B-field, we present the corresponding formalism of classical canonical gravity for the considered space-time and solve the Wheeler-DeWitt equation taking into account the conditional symmetries. In section 3, the same procedure will be applied to the case of (2 + 1)-dimensional model coupled to dilaton, in the absence of the B-field. Also, in section 4, we obtain the classical and quantum solutions choosing a non-diagonal metric ansatz with including the contributions of B-field, dilaton, and the central charge deficit term Λ which appears as a dilaton potential V (φ) = −Λe −2φ . Finally, some concluding remarks are presented in section 5.
where the one-forms σ i are given by (10), and a i are functions of time t. We choose N = e φ a 1 a 2 . For this (2 + 1)-dimensional space-time with the contribution of B-field of the form B = 1 2 A(t) σ 1 ∧ σ 2 , the field strength tensor H is given by The one-loop β-function equations (1)-(3) now lead to the following equationṡ where H i = d dt ln a i . Also (x 1 , t) component of (1) gives the following constraint equation We have set Λ = 0 in this analysis. Now, adding (23) and (24) to (25) gives Also, using (23) and (28) (1), which is given by (24), yields the following initial value equation Solutions of the set of equations (23), (26), (27) and (28) give the dilaton and components of metric and B-field by where k 1 , and k 2 are constants. These solutions satisfy the initial value equation (29). In order to obtain the Einstein frame metric, after performing the conformal transformation (4), one can make a time redefinition dt = e φ N dt, i.e. t = 3 4(t−t0) − k 1 , to obtain the line element in Einstein frame where we haveã The origins of two time coordinates t andt can be coincided with each other by settingt 0 = − 3k1 4 . For this class of solutions, considering the effective energy (7) and (8), the pressures are zero where the energy density and Einstein frame scalar curvature are given by ρ = 2R = 1 4(t−t0) 2 . This solution describes a matter-dominant expanding universe with decreasing curvature and string coupling g s = e −φ . The singularity of this space-time appears whenã → 0, i.e.t →t 0 .

Solutions of Wheeler-DeWitt equation
To quantize this system we start with the Einstein frame metric 1 and the field strength tensor of type (22). Then, given the effective action (5), the Lagrangian reads Defining p φ = ∂L ∂φ , p A = ∂L ∂Ȧ , and pã = ∂L ∂ȧ , the Hamiltonian is constructed via applying the Legendre transformation H = pãȧ + p φφ + p AȦ − L, and we have Accordingly, the supermetric of the configuration space is The generators of the superspace symmetries are then given by 1 Inspired by the classical solutions (32), we considered an isotropic ansatz for g ij .
which satisfy the Lie bracket algebra [ξ 1 , ξ 2 ] = ξ 2 . For these generators we have the following system of the integrals of motion in configuration space The obtained solutions in (32) and (33) satisfy the Q i = η i equations with Here, the the integrability condition (16) requires η 2 = 0, which is not acceptable. Hence, the only admissible subalgebra to be applied on wave function is Q 1 . Now, using the standard canonical quantization p q →p q = i ∂ ∂q to quantize this classical system, we obtain the Wheeler-DeWitt equation We will consider this equation along with the conditionQ 1 Ψ = η 1 Ψ, which reads Solving this set of equations yields the wave function Here and hereafterĥ = κ 2 3 . The λ 1 and λ 2 are integrating constants, µ 1 can be either 1 or zero, , and J and Y are the Bessel functions of the first and second kinds. On the other hand, adopting the WKB approximation [36] with the wave function where the Wheeler-DeWitt equation (41) leads to Hamilton-Jacobi equation and symmetry equation (42) gives we obtain the following solution where s = ±1 and c 1 is a constant. Since pã = 2ȧ, and noting that we have pãΨ = sΨ, the s = 1 corresponds toȧ > 0 or expanding universe where the s = −1 corresponds toȧ < 0 or contracting one. The solutions with s = 1 are then acceptable where atã → −∞ limit which is forbidden classically, the Ψ disappears. It is worth mentioning that at small and large values of Bessel functions arguments, using the approximated forms of Bessel functions given in the appendix, the wave function (43) takes the following forms, respectively The corresponding quantum potentials at early and late times are then, respectively, given by Q sm = 4µ 2 2 −1 32ã and Q la = − 1 2ĥ . When µ 2 = 1 2 , or equivalently η 1 = 0, the quantum potential Q sm vanishes and the wave function (43) becomes equivalent to the WKB approximated solutions (47). It is worth reminding that the classical solutions (32) show a singularity atã → 0 limit, where A and e 2φ tend to zero, as well. With µ 2 = 1 2 , the divergence of the approximated wave function (48) at the origin can be removed, in such a way that for µ 1 = 0 case, setting λ 2 = − 2πĥ −1 , the Hartle and Hawking's no-boundary proposal of Ψ = 1 atã = 0 [37] can be admitted, while for µ 1 = 1, the wave function vanishes at the origin, being consistent with DeWitt's boundary condition at the singularity [38].
The quantum potentials do not vanish in general, and hence the semiclassical geometry is not the same as the classical one. The phase function is ω = 2η 1ĥ −1 φ. Then, noting (20), the solutions of semiclassical equations with respect to (ã, φ) areã where c is a constant which is however not an essential constant of space-time and can be reabsorbed. Independent of the chosen gauge forÑ , the Ricci scalar is constant, i.e.R = − 2 c , as well as all the other curvature scalars. Also, all of the higher derivatives of Riemann tensor vanish. Therefore, the semiclassical solutions show no curvature and/or higher derivative curvature singularities. Also, checking (6) shows that, similar to the classical ones, the matter content of this semiclassical solution is a dust with energy density ρ = 1 c .

Cosmological solutions
Considering the same metric ansatz (21) for the case of vanishing strength tensor of B-field, i.e. H = 0, the solutions of equations (23), (27) and (28) give the string frame scalar factor and dilaton field as following where p 1 , p 2 , k, φ 0 are real constants. Then, the initial value equation (29) gives the following constraint on constants Performing the conformal transformation (4) and the time redefinition t = k p1 − 1 2p1 ln 1 + 2 p1 t−t0−p1 with choosingt 0 = −p 1 coth(k) to have the origin of two time coordinates coincided, we get the following Einstein frame metric of the form (32) with scalar factorã 2 = (t −t 0 ) 2 − p 2 1 , where we havẽ where the prime symbol stands for derivative with respect tot, and q is the deceleration parameter. Also, the effective pressure, energy density, and Einstein frame Ricci scalar are given by This class of solutions describes a decelerated expanding universe with decreasing absolute value of deceleration parameter. The space-time is negatively curved where |R | is decreasing and diverges atã = 0.

Solutions of Wheeler-DeWitt equation
With H = 0, the A-dependent terms in the Lagrangian (35) and Hamiltonian (36) are absent, where the supermetric of the configuration space is given by Accordingly, the generators of the superspace symmetries are whose Lie bracket Algebra is The system of the first integrals of motion expressed in the configuration space variables generated by corresponding ξ i are given by Noting the classical solutions obtained in the previous subsection, we have the following classical charges The quantum integrability condition (16) requires η 1 = η 2 = 0, but leaves η 3 arbitrary. Hence, the only admissible subalgebras to be imposed on wave function is Q 3 . Noting (41), the Wheeler-DeWitt equation is given here bŷ The solution of (62) and (63) is . On the other hand in WKB approximation, where the Wheeler-DeWitt equation and conditional symmetry equation (63) lead to the following equations we find the solution where c 1 is a constant and s = ±1. Asymptotic behavior of the solution (64) is similar to that of the wave function (43). At the small and large limits of the Bessel functions argument the quantum potentials are, respectively, given by Q sm = 4µ 2 −1 16ã 2 and Q la = 1 h 2 . The Q sm vanishes when µ = 1 2 or equivalently p 1 = 0 which is not classically an interesting case. The semiclassical solutions are given here again by (50). Although the classical solutions were described by a perfect fluid with characteristic given by (54), the matter content of semiclassical solutions is dust.

.1 Cosmological solutions
In the line element (9), if we consider a non-diagonal setting for the two-metric on spacelike hypersurface by where N = N (t), a = a(t), and choose the lapse function N = e φ a 2 , then the one loop β-function equations (1)-(3) with field strength tensor of the form (22) give the following equationṡ where H = d dt ln(a). We included the contribution of Λ = 0, which yields the solutions called non-critical string cosmology [19,30]. Now, by adding twice of (69) to (70) we get Also, adding half of (70) to (69) gives the initial value equatioṅ Solutions of (68), (72) and (71) give the string frame scalar factor, dilaton and field strength tensor (22) as following where p 1 , p 2 , k 1 , k 2 , and b are real constants and the initial value equation (73) leads to the following constraint on constants Performing the conformal transformation (4) on the above solutions gives the Einstein frame line element in which scalar factor and lapse function are given bỹ .
Also, the energy momentum tensors (7) and (8) give the effective pressure and energy density as following Both pressure and energy density are positive for this class of solutions, but the dominant energy condition ρ > P is violated. The cosmological timet can be defined here by dt =Ñ dt. However, noting (77), integrating of this expression and transforming from t tot is not straightforward. In this case, to investigate the behavior of solutions, the time derivatives in the physical quantities in Einstein frame can be rewritten in terms of t-derivatives such as followingã Accordingly, the given solutions describe a decelerated expanding universe with decreasing positive valued decelerating parameter.

Solutions of Wheeler-DeWitt equation
To obtain the Hamiltonian formalism for this model we start with a metric ansatz of the form (76), where we get the following Lagrangian Now, by choosingÑ = ne 2φã−2 the Lagrangian can be brought into the form of (13) where the potential V is constant Then, the Hamiltonian will be where the supermetric of the configuration space is In this case the generators of the superspace symmetries are given by whose Lie bracket algebra is given by The obtained solutions (76) and (77) are not consistent with Q i = η i equations for these Killing vectors. However, if we employ a time redefinition dT =ã 4 e −2φ dt, the transformed version of (76) and (77) satisfies Q i = η i equations via On the other hand, the integrability condition (16) implies η 1 = η 2 = η 3 = 0. Accordingly, only the one dimensional subalgebra Q 2 is admissible here to be imposed on the wave function.
To quantize this classical system, noting the a 2 and e 2φ factors coming, respectively, with p 2 a and p φ in (82), we need to include the Hartle-Hawking ordering parameters [37] B and C as following The B and C will be assumed to be real constants. The Wheeler-DeWitt equation is then given bŷ Also, we have theQ 2 Ψ = η 2 Ψ condition which reads The solution of this set of equations is where µ 1 can be either 1 or zero, It is worth mentioning that at classical singularity whereã → 0, noting (74) and (77), the u function has large but finite value. Hence, in µ 1 = 0 case the DeWitt's boundary condition Ψ = 0 [38] is satisfied at the singularity, while in the µ 1 = 1 case we have a finite valued wave function at the origin for which the Hartle and Hawking's no-boundary proposal Ψ = 1 [37] can be admitted.
Also, in WKB approximation with the wave function of the form Ψ = e î h S we have the following solutions where D 1 is a constant. The Q sm vanishes if c 1 = 1 4 2 (2c 2 2 − B 2 − 2B − 1) where the µ 1 constant becomes imaginary given by µ 1 = i 4 6(B 2 + 2B + 1). Also, at the large limit of u function the approximated wave function and corresponding quantum potential are given by With the non-vanishing quantum potentials, the semiclassical space-time will differ from the classical one.
Solving (20) for this system we obtainã 2 =Ñ = Le c 2 κ 2 3 2 t and A = −k 1 k −2 2 e −2k2κ 2 3 t . With a time redefining dt =Ñ dt, and assuming k 2 > 0, the line element is given by for whichR = − 1 2t 2 . The Ricci scalar, diverging att = 0, does not contain any parameter to make it vanish. Also, the matter content of this semiclassical solution is a perfect fluid with ρ = P = 1 4t 2 .

Conclusion
In this paper, we constructed cosmological solutions at classical and quantum levels for low energy string effective action on (2 + 1)-dimensional spatially homogeneous space-time in three examples including the contributions of (i) dilaton and B-field, (ii) only the dilaton field, and (iii) dilaton, B-field, and the central charge deficit term Λ which plays the role of a negative dilaton potential V (φ) = −Λe −2φ in the Einstein frame. In doing so, for each case, we first obtained the solutions for one-loop β-function equations, which are the conformal invariance condition of the corresponding σ-model and equivalent to the equations of motion of string effective action. Then, using the symmetries of supermetric, the classical integrals of motion are obtained for each class of solutions. At the quantum level, we proceed by implementing the canonical quantization procedure, taking into account the conditional symmetries. Turning the Hamiltonian constraint H and conditional symmetries Q i into operators, we used an integrability condition to determine the admissible conditional symmetry operators that can be applied on the wave functions. In each class of solutions, solving the Wheeler-DeWitt equations supplemented with the conditional symmetry equations, we obtained the wave functions describing the quantum dynamics of systems. Concerning the applications of the solutions, we took advantageous of the Bohm's approach [27,32,33] to determine the quantum potential and semiclassical geometries. For the (2 + 1)-dimensional model coupled to dilaton and B-field we found solutions describing a dust model. At the quantum level, employing the conditional symmetries we obtained solutions where the classical singularity can be avoided at the semiclassical level where the matter content is again dust. Its singularity behavior is similar to that of the Friedmann-Lemaître-Robertson-Walker (FLRW) case in [21], where the semiclassical solutions are non-singular. In the absence of B-field, we obtained the second family of solutions which classically behave as perfect fluid. At its quantum level, similar to the space-time with dilaton and B-field, solving the Wheeler-DeWitt and conditional symmetry equations, the Bohm's analysis has been performed and the obtained semiclassical solutions showed a non-singular behavior where the matter content, being different from the classical one, was dust.
Also, we constructed non-critical string cosmology solutions including contributions of dilaton, B-field, and the central charge deficit term Λ. The quantum cosmological solutions have been obtained in two parametrizations of the lapse function. In a constant potential parametrization, the integrals of motion and solutions of the Wheeler-DeWitt equation along with the conditional symmetry equation have been obtained. This class included in particular the solutions in agreement with the Hartle-Hawking no-boundary proposal and DeWitt boundary condition at the classical singularity. Considering another parameterize in which the additional conditional symmetry equations can be ignored, the solutions for the Wheeler-DeWitt equation have been found in the polar form, for which the Bohm's approach was applied to find the quantum potentials and semiclassical geometry. We arrived at a singular space-time at semiclassical level whose matter content, being different from that of the classical solutions which was an imperfect fluid, is given by a perfect fluid.

A Asymptotic forms of Bessel functions of the first and second kind
Bessel function of the first kind J µ (u) is defined as [21] J µ (u) = u 2 where the Bessel function of the second kind Y µ (u) is defined as At large argument of Bessel function we have where at small limit of Bessel function argument we have