A New Integrable Model of (1+1)-Dimensional Dilaton Gravity Coupled to Toda Matter
aa r X i v : . [ h e p - t h ] M a r A New Integrable Model of (1+1)-DimensionalDilaton Gravity Coupled to Toda Matter
A.T. Filippov ∗ + Joint Institute for Nuclear Research, Dubna, Moscow Region RU-141980
November 7, 2018
Abstract
A new class of integrable two-dimensional dilaton gravity theories, in which scalar matterfields satisfy the Toda equations, is proposed. The simplest case of the Toda system is consideredin some detail, and on this example we outline how the general solution can be obtained. Wealso demonstrate how the wave-like solutions of the general Toda systems can be simply derived.In the dilaton gravity theory, these solutions describe nonlinear waves coupled to gravity. Aspecial attention is paid to making the analytic structure of the solutions of the Toda equationsas simple and transparent as possible, with the aim to apply the idea of the separation ofvariables to non-integrable theories.
The theories of (1+1)-dimensional dilaton gravity coupled to scalar matter fields are known tobe reliable models for some aspects of higher-dimensional black holes, cosmological models, andwaves. The connection between higher and lower dimensions was demonstrated in different contextsof gravity and string theory and, in several cases, has allowed finding the general solution orspecial classes of solutions in high-dimensional theories . A generic example is the sphericallysymmetric gravity coupled to Abelian gauge fields and massless scalar matter fields. It exactlyreduces to a (1+1)-dimensional dilaton gravity and can be explicitly solved if the scalar fields areconstants independent of the coordinates. These solutions can describe interesting physical objects– spherical static black holes and simplest cosmologies. However, when the scalar matter fields,which presumably play a significant cosmological role, are nontrivial, not many exact analyticalsolutions of high-dimensional theories are known . Correspondingly, the two-dimensional modelsof dilaton gravity that nontrivially couple to scalar matter are usually not integrable.To obtain integrable models of this sort one usually has to make serious approximations, in otherwords, to deform the original two-dimensional model obtained by direct dimensional reductions ofrealistic higher-dimensional theories . Nevertheless, the deformed models can qualitatively describe ∗ [email protected] See, e.g., [1]-[27] for a more detailed discussion of this connection, references, and solution of some integrabletwo-dimensional and one-dimensional models of dilaton gravity. See, e.g., [8], [11], [12], [17]-[23]; a review and further references can be found in [25], [26] and [23] We note that several important four-dimensional space-times with symmetries defined by two commuting Killingvectors may also be described by two-dimensional models of dilaton gravity coupled to scalar matter. For example,cylindrical gravitational waves can be described by a (1+1)-dimensional dilaton gravity coupled to one scalar field[28]-[30], [22]. The stationary axially symmetric pure gravity ([31], [11]) is equivalent to a (0+2)-dimensional dilatongravity coupled to one scalar field. Similar but more general dilaton gravity models were also obtained in stringtheory. Some of them can be solved by using modern mathematical methods developed in the soliton theory (see e.g.[1], [2], [11], [19]).
The effective Lagrangian of the (1+1)-dimensional dilaton gravity coupled to scalar fields ψ n ob-tainable by dimensional reductions of a higher-dimensional spherically symmetric (super)gravitycan usually be (locally) transformed to the form (see [20] - [23] for a detailed motivation and specificexamples): L = √− g " ϕR ( g ) + V ( ϕ, ψ ) + X mn Z mn ( ϕ, ψ ) g ij ∂ i ψ m ∂ j ψ n . (1)Here g ij ( x , x ) is the (1+1)-dimensional metric with the signature (-1,1), g ≡ det | g ij | , and R isthe Ricci curvature of the two-dimensional space-time, ds = g ij dx i dx j , i, j = 0 , . (2)The effective potentials V and Z mn depend on the dilaton ϕ ( x , x ) and on N − ψ n ( x , x ) (we note that the matrix Z mn should be negative definite to exclude the so called ‘phan-tom’ fields). They may depend on other parameters characterizing the parent higher-dimensionaltheory (e.g., on charges introduced in solving the equations for the Abelian fields). Here we con-sider the ‘minimal’ kinetic terms with the diagonal and constant Z -potentials, Z mn ( ϕ, ψ ) = δ mn Z n .This approximation excludes the important class of the sigma - model - like scalar matter discussed,e.g., in [27]; such models can be integrable if V ≡ Z mn ( ϕ, ψ ) satisfy certain rather stringentconditions. In (1) we also used the Weyl transformation to eliminate the gradient term for thedilaton.To simplify derivations, we write the equations of motion in the light-cone metric, ds = − f ( u, v ) du dv . By first varying the Lagrangian in generic coordinates and then passing to the light-cone coordinateswe obtain the equations of motion ( Z n are constants!) ∂ u ∂ v ϕ + f V ( ϕ, ψ ) = 0 , (3)2 ∂ i ( ∂ i ϕ/f ) = X Z n ( ∂ i ψ n ) , i = u, v . (4)2 Z n ∂ u ∂ v ψ n + f V ψ n ( ϕ, ψ ) = 0 , (5) ∂ u ∂ v ln | f | + f V ϕ ( ϕ, ψ ) = 0 , (6)where V ϕ = ∂ ϕ V , V ψ n = ∂ ψ n V . These equations are not independent. Actually, (6) follows from(3) − (5). Alternatively, if (3), (4), and (6) are satisfied, one of the equations (5) is also satisfied.The higher-dimensional origin of the Lagrangian (1) suggests that the potential is the sum ofthe exponentials of linear combinations of the scalar fields, q (0) n , and of the dilaton ϕ . In ourprevious work [23] we studied the constrained Liouville model, in which the system of the equationsof motion (3), (5) and (6) is equivalent to the system of the independent Liouville equations for thelinear combinations of the fields q n ≡ F + q (0) n , where F ≡ ln | f | . The easily derived solutions of theseequations should satisfy the constraints (4), which was the most difficult part of the problem. Thesolution of the whole problem revealed an interesting structure of the moduli space of the solutionsthat allowed us to easily identify static, cosmological and wave-like solutions and effectively embedthese essentially one-dimensional (in some broad sense) solutions into the set of all two-dimensionalsolutions and study their analytic and asymptotic properties.Here we propose a natural generalization of the Liouville model to the model in which the fieldsare described by the Toda equations (or by nonintegrable deformations of them). To demonstratethat the model shares many properties with the Liouville one and to simplify a transition fromthe integrable models to nonintegrable theories we suggest a different representation of the Todasolutions, which is not directly related to their group - theoretical background.Consider the theory defined by the Lagrangian (1) with the potential: V = N X n =1 g n exp q (0) n , Z n = − , (7)where q (0) n ≡ a n ϕ + N X m =3 ψ m a mn . (8)In what follows we also use q n ≡ F + q (0) n ≡ N X m =1 ψ m a mn , (9)where ψ + ψ ≡ ln | f | ≡ F ( f ≡ εe F , ε = ± ψ − ψ ≡ ϕ and hence a n = 1 + a n , a n = 1 − a n .Rewriting the equations of motion in terms of ψ n , we find that Eqs. (3) - (6) are equivalent to N equations of motion for N functions ψ n , ∂ u ∂ v ψ n = ε N X m =1 ǫ n a nm g m e q m ; ǫ = − , ǫ n = +1 , if n ≥ , (10)and two constraints, ∂ i ϕ ≡ ∂ i ( ψ − ψ ) = − N X n =1 ǫ n ( ∂ i ψ n ) , i = u, v . (11)With arbitrary parameters a mn , these equations of motion are not integrable. But as proposed in[16] - [18], [20] [23], Eqs.(10) are integrable and constraints (11) can be solved if the N -componentvectors v n ≡ ( a mn ) are pseudo-orthogonal. Actually, the potential V usually contains terms non exponentially depending on ϕ (e.g., linear in ϕ ), and thenthe exponentiation of ϕ is only an approximation, see the discussion in [23]. a mn and define the new scalar fields x n : x n ≡ N X m =1 a − nm ǫ m ψ m , ψ n ≡ N X m =1 ǫ n a nm x m . (12)In terms of these fields, Eqs.(10) read as ∂ u ∂ v x m ≡ εg m exp N X k,n =1 ǫ n a nm a nk x k ≡ exp N X k =1 A mk x k , (13)and we see that the symmetric matrix A ≡ a T ǫa , ǫ mn ≡ ǫ m δ mn , (14)defines the main properties of the model.If A is a diagonal matrix, we return to the N -Liouville model. If A is the Cartan matrix ofa Lie algebra, the system (13) coincides with the corresponding Toda system, which is integrableand can be more or less explicitly solved (see, e.g., [32], [33] ). Here we mostly consider the A N Toda systems having very simple solutions. However, the solutions have to satisfy the constraintsthat in terms of x n are:2 N X n =1 a n ∂ i x n = − N X n,m =1 ∂ i x m A mn ∂ i x n , i = u, v . (15)In the N -Liouville model the most difficult problem was to solve the constraints (15) but thisproblem was eventually solved. In the general nonintegrable case of an arbitrary matrix A we donot know even how to approach this problem. We hope that in the Toda case the solution canbe somehow derived but this problem is not addressed here. Instead, in Section 4 we introduce asimplified model that can be completely solved.Now, let us write the general equations in the form that is particularly useful for the Todasystems. Introducing notation X n ≡ exp( − A nn x n ) , ∆ ( X ) ≡ X ∂ u ∂ v X − ∂ u X ∂ v X, α mn ≡ − A mn /A nn , (16)it is easy to rewrite Eqs.(13) in the form:∆ ( X n ) = − ε g n A nn Y m = n X α nm m . (17)The multiplier | − ε g n A nn | can be removed by using the transformation x n x n + δ n and thefinal (standard) form of the equations of motion is∆ ( X n ) = ε n Y m = n X α nm m , (18)where ε n ≡ ± It can easily be seen that, due to the special structure of a mn ( a n = 1 + a n , a n = 1 − a n ), the Cartan matricesof the simple algebras of rank 2 and 3 cannot be represented in the form (14). Further analysis shows that thisprobably is also true for any rank. As will be shown in a forthcoming publication, any symmetric matrix A mn , whichis the direct sum of a diagonal L × L -matrix γ − n δ mn and of an arbitrary symmetric matrix ¯ A mn , can be representedin form (14). If ¯ A mn is a Cartan matrix, the system (13) reduces to L independent Liouville (Toda A ) equationsand the higher-rank Toda system. A mn are the Cartan matrices,they simplify to integrable equations (see [32]). For example, for the Cartan matrix of A N , onlythe near-diagonal elements of the matrix α mn are nonvanishing, α n − ,n +1 = 1. This allows one tosolve Eq.(18) for any N . The parameters α mn are invariant w.r.t. transformations x n λ n x n + δ n and hence A mn can be made non-symmetric while preserving the standard form of the equations(recall that the Cartan matrices of B N , C N , G , and F are not symmetric). In this sense, α mn arethe fundamental parameters of the equations of motion. From this point of view, the characteristicproperty of the Cartan matrices is the simplicity of Eqs.(18) which allows one to solve them bya generalization of separation of variables. As is well known, when A mn is the Cartan matrix ofany simple algebra, this procedure gives the exact general solution (see [32]). In next Section weshow how to construct the exact general solution for the A N Toda system and write a convenientrepresentation for the general solution that differs from the standard one given in [32]. A N Toda system
The A N equations are extremely simple,∆ ( X n ) = ε n X n − X n +1 , X , X N +1 , n = 1 , ..., N, (19)and can be reduced to one equation for X by using the relation between ∆ ( X ) and higherdeterminants, ∆ n ( X ) (see [32]):∆ (∆ n ( X )) = ∆ n − ( X ) ∆ n +1 ( X ) , ∆ ( X ) ≡ X, n ≥ . (20)From Eqs.(19), (20) we find that ∆ N +1 ( X ) = Y n ε n . (21)This equation looks horrible but is known to be soluble.Let us start with the Liouville ( A Toda) equation ∆ ( X ) = g (see [34], [35], [32], [23]).Calculating the derivatives of ∆ ( X ) w.r.t. u and v , we find that ∂ u ( X − ∂ v X ) = 0 , ∂ v ( X − ∂ u X ) = 0 . (22)It follows that if X satisfies (22) then there exist some ‘potentials’ U ( u ), V ( v ) such that ∂ u X − U ( u ) X = 0 , ∂ v X − V ( v ) X = 0 . (23)Thus the Liouville solution can be written as ([23]) X ( u, v ) = X a µ ( u ) C µν b ν ( v ) , (24)where a µ ( u ) and b ν ( u ) ( µ, ν = 1 ,
2) are linearly independent solutions of the equations a ′′ ( u ) − U ( u ) a ( u ) = 0 , b ′′ ( v ) − V ( v ) b ( v ) = 0 . (25)and C µν is a nonsingular matrix. As the potentials are unknown, the solutions a , b can be takenarbitrary while a , b then may be defined by the Wronskian first-order equations W [ a ( u ) , a ( u )] = 1 , W [ b ( v ) , b ( v )] = 1 . (26)The matrix C µν should obviously satisfy the normalization condition det C = g .5e have repeated this well known derivation at some length because it is completely applicableto the A N Toda equation (21). By similar but rather cumbersome derivations it can be shown that X satisfy the equations ∂ N +1 u X + N − X n =0 U n ( u ) ∂ nu X = 0 , ∂ N +1 v X + N − X n =0 V n ( v ) ∂ nv X = 0 . (27)Thus the solution of (21) can be written in the same ‘separated’ form (24), where now a µ ( u ) and b ν ( v ) satisfy the ordinary linear differential equations of the order N +1 (corresponding to Eqs.(27)),with the unit Wronskians, W [ a ( u ) , ..., a N +1 ( u )] = 1 , W [ b ( v ) , ..., b N +1 ( v )] = 1 , (28)and det C = Q ε n .As an exercise, we suggest the reader to prove these statements for N = 2. The key relationthat follows from the condition ∂ u ∆ ( X ) = 0 is the partial integral ∂ v (cid:20) ∂ v (cid:18) X∂ u X (cid:19) / ∂ v (cid:18) ∂ u X∂ u X (cid:19)(cid:21) = 0 . (29)It follows that the expression in the square brackets is equal to an arbitrary function A ( u ) andthus we have ∂ v (cid:20)(cid:18) X∂ u X (cid:19) + A ( u ) (cid:18) ∂ u X∂ u X (cid:19)(cid:21) = 0 . (30)Denoting the expression in the square bracket by − A ( u ) and introducing the notation U ( u ) = A /A and U ( u ) = A − , we get Eq.(27) with N = 2.Let us return to the general solution of Eq.(21). In fact, considering Eqs.(28) as inhomogeneousdifferential equations for a N +1 ( u ), b N +1 ( v ) with arbitrary chosen functions a n ( u ), b n ( v ) (1 ≤ n ≤ N ), it is easy to write the explicit solution of this problem: a N +1 ( u ) = N X n =1 a n ( u ) Z u d ¯ u W − N (¯ u ) M N, n (¯ u ) . (31)Here W N is the Wronskian of the arbitrary chosen functions a n and M N, n are the complementaryminors of the last row in the Wronskian. Replacing a by b and u by v we can find the expressionfor b N +1 ( v ) from the same formula (31). To complete the solution we should derive the expressionsfor all X n in terms of a n and b n . This can be done with simple combinatorics that allows one toexpress X n in terms of the n -th order minors. For example, it is very easy to derive the expressionsfor X : X = ε ∆ ( X ) = ε X i 0) and denotedΦ u ( u ) ≡ N X ( ∂ u ψ n ( u, , Φ v ( v ) ≡ N X ( ∂ v ψ n ( v, . Now, to get integrable equations for ψ we take the potential (7) with q (0) n given by the r.h.s. ofEq.(9). Then, we can use for the scalar fields the equations (10) and (12) – (14). If we take thepotential for which the ψ equations of motion can be reduced to integrable Toda equations we findan explicit solution for the nontrivial class of dilaton gravity minimally coupled to scalar matterfields. This model is a very complex generalization of the well studied CGHS model in which thescalar fields are free and V = g . In future, we plan a detailed study of the A N case. The easiestcase is N = 1 (the Liouville equation for one ψ ). The first really interesting but simple theory isthe case of two scalar fields satisfying the A Toda equations. Taking, for example, V = exp ( √ ψ − ψ ) + exp (2 ψ ) , we find the simplest realization of the A Toda dilaton gravity model the complete solution of whichcan be obtained by use of the above derivations.As a simple exercise one may consider the reduction from dimension (1+1) both to the dimension(1+0) (‘cosmological’ reduction) and to the dimension (0+1) (‘static’ or ‘black hole’ reduction) aswell as the moduli space reduction to waves. One of the most interesting problems for futureinvestigations is the connection between these three objects. It was discovered in the N -Liouvilletheory but now we see that it can be found in a much more complex theory described by theToda equations. It is not impossible that the connection also exists (in a weaker form?) in somenonintegrable theories, say, in theories close to the Toda models.Note in conclusion, that the one-dimensional Toda equations were earlier employed mostly inconnection with the cosmological and black hole solutions (see, e.g. [36] - [38]). To include intoconsideration the waves one has to step up at least on dimension higher. The principal aim of thepresent paper was to make the first step and explore this problem in a simplest two-dimensionalToda environment.A more detailed presentation will be published elsewhere. Acknowledgment: The author appreciates financial support from the Department of Theoretical Physics of theUniversity of Turin and INFN (Turin Section) and of the Theory Division of CERN, where someresults were obtained. The useful discussions with V. de Alfaro and A. Sorin are kindly acknowl-edged. 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