One can specify a model of 2d gravity by requiring that the eld q(x) satises some dynamical equation. The Liouville eld equation [3] ) q(

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1 Liouville Field Theory and 2d Gravity George Jorjadze Dept. of Theoretical Physics, Razmadze Mathematical Institute Tbilisi, Georgia Abstract We investigate character of physical degrees of freedom for the models of 2d gravity and study it's dependence on the topology of space-time manifold. I Introduction The Einstein-Hilbert action in 2-dimensions does not lead to dynamical equations for the metric tensor g ab (X) (X := (x 0 x ) (a and b = (0 )) []. This degeneracy is related to the fact that the corresponding Lagrangian L 0 = p ;g R(g ab ) is a `total derivative' and the action is expressed only by the `surface terms'. For the conformal gauge [2] g ab (X) = exp q(x) the scalar curvature R(X) takes the form 0 0 ;! : (.) R(X) =e ;q(x) (@ 2 2 0)q(X): (.2) and the degeneracy of the Einstein-Hilbert Lagrangian becomes apparent. jorj@rmi.acnet.ge

2 One can specify a model of 2d gravity by requiring that the eld q(x) satises some dynamical equation. The Liouville eld equation [3] (@ ) q(x)+e q(x) =0 (.3) (where is a non-zero constant) is usually considered as a model of 2- dimensional general relativity [2], [4]. The eld q(x) satisfying (.3) leads to the manifold with a constant curvature R(X) = (see (.2)). The Liouville equation arises in a large variety of problems of mathematical physics due to the conformal invariance of this equation. General solution of (.3) has the form [3] q(x + x ; 8A + 0 (x + )A ; 0 (x ; ) ) = log (.4) jj [A + (x + )+A ; (x ; )] 2 where x = x x 0 are the light cone coordinates, isasignof( ==jj) and A are any functions with A 0 > 0. The standard action for the scalar eld in Minkowski space, which leads to the Liouville equation (.3) has a non-covariant form from the point of view of general relativity. To obtain the Liouville equation (as the equation for the metric tensor in the conformal gauge) from the covariant action, it is necessary to introduce some additional auxiliary elds. For a model with a reparametrization invariant action and auxiliary elds, usually it is dicult to guess what kind of physical degrees of freedom the system has and what are its physical variables [5, 6]. Construction of gauge invariant physical variables is the most important problem of 4-dimensional Einstein general relativity as well. In this note we study this problem for the simple models of 2d gravity, using the conformal invariance of the Liouville theory. II 2d gravity with the dilaton eld We start with the action [4] Z S = q d 2 x ;^g 2 b + R(^g ab) ; e (2.) where is a `cosmological constant', ^g := det ^g ab, ^g ab is the inverse matrix to the ducial metric tensor ^g ab, R(^g ab ) denotes the corresponding scalar 2

3 curvature and is a scalar led, which is known as the Liouville mode, or the dilaton eld. The physical metric tensor g ab is related to ^g ab by and the action (2.) takes the form Z S = p d 2 x ;g g ab = e ^g ab (2.2) ; 2 b + R(g ab ) ; (2.3) where we have used the relation of scalar curvatures in 2-dimensions for conformally related metrics (2.2) R(^g ab )=e [r 2 + R(g ab )] (2.4) and we have neglected the surface a ( p ;g g b ), using the form of the Beltrami operator r 2 for the metric g ab. Action (2.3) denes the following `dynamical' equations p ;g S = r2 + R(g ab )=0 (2.5) ; p 2 S = ;g a@ ab b + g ab ; 2 d ; ; 2(g ab r 2 ;r a r b ) =0: (2.6) Taking trace of (2.6) and using (2.5) we get that the dynamical system (2.3) describes a space-time manifold with a constant curvature R(g ab )=. For the light cone coordinates x = x x 0 the conformal metric (.) takes the form: g + + = g ; ; =0 g + ; = g ; + = ; 2 eq and from (2.5)-(2.6) we 2 + ; ' =0 4@2 + ; q =eq (2.7) (@ + ') 2 ; 2@ ' =(@ +q) 2 ; 2@ q (@ ;') 2 ; 2@ 2 ; ; ' =(@ ;q) 2 ; 2@ 2 ; ; q (2.8) 3

4 where ' = q ;. As it was expected the eld q(x) satises the Liouville equation (.3), while '(X) is a free eld. According (2.8), the traceless energy-momentum tensors of these two elds are equal to each other [2, 7, 8]. One can integrate equations (2.8) and express the Liouville eld q(x) as a functional of the free eld '(X). After integration we get q(x + x ; ) = log 8F 0 +(x + )F 0 ;(x ; ) jj [F + (x + )F ; (x ; )+ + F + (x + )+ ; F ; (x ; )+] 2 (2.9) where F (x )= x 0 d exp x 0 '(! (2.0) (x + 0 x; 0 ) are the coordinates of a xed point on the space-time, and the integration constants ( ) satisfy the condition + ; ; = ( := =jj): (2.) The latter denes the SL(2:R) group manifold, and the freedom of the Liouville eld q(x) (for a given free eld '(X)) is described only by three parameters of SL(2:R) group. The solution (2.9) for 6= 0canbe written in the form q(x + x ; )=log 8B 0 +(x + )B 0 ;(x ; ) jj [B + (x + )B ; (x ; ) ; ()] 2 where B = F + (2.2) The case = 0 is a degenerated one and the corresponding relation between the Liouville and free elds arises for the Backlund transformation [9] (see also [8]). 7! A. Note that (2.2) reduces to (.4) after the substitution B Thus, (2.9) reproduces the general solution (.4). Integrating equations (2.8) with respect to the free eld, one can derive the map from the Liouville elds to the free elds '(x + x ; )=log 8G 0 +(x + )G 0 ;(x ; ) + b (2.3) jj[(g + (x + )+a + )(G ; (x ; 2 )+a ; )] 4

5 which is also characterized by three parameters (a, b) and the functions G have the form G + (x + )= + x + 0 d + e q(+ x + 0 ) G ; (x ; )= ; x ; 0 d ; e q(x+ 0 ;) : (2.4) The conformal form of the metric tensor (.) xes the gauge freedom only up to the conformal transformations x 7! y = f (x ) (2.5) The corresponding transformation of the eld q is given by q(x + x ; ) 7! ~q(x + x ; )=q(f + (x + ) f ; (x ; ))+logf + 0 (x + )f ; 0 (x ; ): (2.6) it is easy to check that the general solution (.4) can be obtained by the conformal transformations of the function q(x + x ; )=log 8 jj(x + + x ; ) 2 (2.7) which is a solution of (.3). Therefore, all Liouville elds (with xed ) are related to (2.7) by the conformal transformations (2.6), and hence, by suitable choice of local coordinates one can x the eld q(x). After xing q(x) the dynamical freedom for the free eld '(X) is described only by the parameters a b (see (2.3)). As a result, the full gage xing leads to the nite dimensional system rather than the eld theory. It should be noted that all our construction of this section (as well as the conjecture about the conformal form of the metric tensor (.)) is valid only locally. The maps (2.9) and (2.3) contain singularities when the corresponding denominators are equal to zero. The domain of regularity of q and ' elds can be considered as a patches of the global space-time manifold. For the global description of the system one has to glue the Liouville (and free) elds given on dierent patches. In this way we can conclude that the system (2.) has a nite number of physical degrees of freedom. The number and dynamical character of these parameters are related to the global properties of the space-time manifold. 5

6 III Hamiltonian reduction of 2d gravity In this section we analyze the same problem using the Hamiltonian approach. This approach is non-covariant and we have to separate the space-time coordinates. We assume that the space-time manifold M can be represented in the form M = T X, where T is isomorphic to R and X is a one dimensional manifold. The coordinates x 0 (x 0 2T) and x (x 2X) we interpret as the time and the space coordinates respectively. We also use the notations x 0 := t, x := x and the corresponding derivatives we denote by `dot' t := _ ) and `prime' x := 0 ). We assume g 00 > 0andg < 0. Due to the reparametrization invariance the Lagrangian of (2.3) is singular and in the Hamiltonian description leads to a constrained dynamical system [5, 6]. For the reduction to the physical (gauge invariant) variables we use the Faddeev-Jackiw's method [0] (see also []). One can check by direct calculation that the above mentioned degeneracy of the Einstein-Hilbert Lagrangian in 2-dimensions can be written in the form p ;gr(gab )=@ t " G _C ; 2B 0 + B C 0 C!# x " G A0 ; B!# _C C (3.) where A B and C are the components of the metric tensor g ab g ab = A B B C! (3.2) and G := p ;g = p B 2 ; AC: (3.3) Then, up to the `surface terms', the Lagrangian of (2.3) takes the form L = 2G ;G C _ 2 ;2B _ 0 + B2 C _ C; 0 C C 2 2C 2B0 ; C _ ; B C0 C +2 0 C B _ ; B 2 0 C 20 C (3.4) where we have excluded the variable A, using relation (3.3). Since the Lagrangian (3.4) does not contain the time derivatives of G and B `elds', these variables can play a role of Lagrange multipliers. 6

7 According to [0] the equivalent system can be described by the action Z S = dx dt h ( _ ; V )+ C ( _ C ; V C )+L( V :::) i (3.5) where the C V and V C are auxiliary elds and the function L( V :::) is constructed from Lagrangian (3.4), substituting the time derivatives _ and _C by the `velocities' V and V C respectively. One can easily eliminate the variables V and V C (taking the corresponding variations) and we arrive at Z S = dx dt _ +C _C + G 0 + B (3.6) C C with 0 = C C + 2 C 2 2 C ; C ; ; C 0 = C C 0 ; 0 +2C 0 C : (3.7) C The `elds' G and B are indeed Lagrange multipliers and the corresponding variations givetwo constraints 0 =0and =0. To analyze this constraint surface it is convenient to introduce the new set of canonically conjugated variables q := log(;c) ' := log(;c) ; (assuming C < 0) p := C C + := ; : (3.8) Here we use the notations of Sec.II, taking into account that the eld log(;c) in the conformal gauge (.) coincides with the eld q of the previous section. Note that the conformal gauge (.) corresponds to the choice of Lagrange multipliers (G = ;C B = 0), which provides the Liouville and the free equations for q and ' elds respectively. For the new variables we nd where 0 = U 0 ; V 0 =0 = U + V =0 (3.9) U 0 = 2 (p2 + q 0 2 )+e q ; 2q 00 V 0 = 2 (2 + ' 0 2 ) ; 2' 00 U = ;pq 0 +2p 0 V = ;' (3.0) 7

8 Note that the functions U 0 and U are T 00 and T 0 components of the traceless energy-momentum tensor of the Liouville theory [2, 7, 8], while V 0 and V are the same components for the free theory. The next step of the reduction procedure [0] is a calculation of the canonical -form Z = dx [p(x) dq(x) +(x) d'(x)] (3.) X on the constrained surface (3.9). This surface can be represented in the form := U ; V =0,where U := 2 (U 0 U )= 4 (q0 p) 2 ; (q 0 p) eq V := 2 (V 0 V )= 4 ('0 ) 2 ; (' 0 ) 0 : (3.2) The canonical -form (3.) denes the canonical commutation relations fp(x) q(y)g = (x ; y) f(x) '(y)g = (x ; y) (3.3) and we obtain the Magri brackets [2] for the functions U and V and also fu (x) U (y)g = [(U (x)+u (y)) 0 (x ; y) ; (x ; y)] fv (x) V (y)g = [(V (x)+v (y)) 0 (x ; y) ; (x ; y)] (3.4) fu + (x) U ; (y)g =0 fv + (x) V ; (y)g =0: (3.5) As a result, the functions are the rst class constraints, since they satisfy the commutation relations of the following Lie algebra f (x) (y)g = [ (x)+ (y)] 0 (x ; y): (3.6) The two rst class constraints (x) usually need two gauge xing conditions for each point x [6]. Since, at this stage, we have only four elds (p q '), it is expected that the system has no physical degrees of freedom for the eld variables. Then, after the full Hamiltonian reduction, the system with only a nite number of degrees of freedom can remain. Below we demonstrate this fact explicitly using again the method [0]. Here it is useful to introduce the functions g =(q 0 p ; ' 0 ) e ;(q)=2 with (x) = 8 x 0 d () (3.7)

9 where x 0 obtain is a xed point from X (x 0 2 X ). Up to the total derivative, we p _q + _' ' g t e (q;)=2 ; g t e (q+)=2 ' e (q+)=2 _g + ; e (q;)=2 _g ; : (3.8) For the restriction of (3.8) on the constrained surface (3.9) we integrate these equations with respect to p and q elds. After integration we get a result, which is similar to (2.9): where p = f 00 + f 0 + ; f 00 ; f 0 ; ; 2 f ;f 0 + ; f + f 0 ; + + f 0 + ; ; f 0 ; D q = log 8f 0 + f 0 ; jjd 2 with D := f +f ; + + f + + ; f ; + (3.9) f (x) = x 0 d exp 2 " '() Z x 0 d () # (3.20) and the integration constants ( ) satisfy the condition (2.). Using (3.7) - (3.20), we nd that the restriction of the functions g on the constrained surface (3.9) is given by and (3.8) takes the form p _q + _' ' 4 D g = ; q 2jj(f + ) (3.2) h f 0 +( _ f ; + _f ; + _ + ) ; f 0 ;( _ f + + _f + + _ ; ) i : (3.22) After integration over X, the coecients of _ and _ give the corresponding canonical conjugated variables. The only `eld theory' term is 4(f 0 + _ f ; ; f 0 ; _ f + ) D(f + f ; ) (3.23) but it again gives the `surface terms', since it is related to the closed 2-form in f space. Thus, we get that after the full Hamiltonian reduction there are no degrees of freedom for the eld variables. In this waywe reproduce the result of Sec.II. To analyze the character of nite dimensional system we need the global description of the model. Such description can be related to the gluing of Liouville elds with singularities (see, for example, [7] and [8]) and we are going to discuss this approach elsewhere. 9

10 Acknowledgments One of the authors (G.J.) is very grateful to the organizers of the seminar `q-98' for the invitation and kind hospitality. This work was supported by the grants from: INTAS ( ), RFBR ( ) and the Georgian Academy of Sciences. References [] B. A. Dubrovin, S.P. Novikov and A.T. Fomenko, Modern Geometry, (Nauka, Moscow, 979). [2] R. Jackiw, Progress in Quantum Field Theory, editedby H. Ezawa and S. Kamefuchi (Elsevier Science Publishers B.V., 986), pp [3] J. Liouville, J. Math. Pures Appl. 8, 7 (853). [4] N. Seiberg, Random Surfaces and Quantum Gravity, edited by O. Alvarez et al. (Plenum Press, New York, 99) pp [5] P.A.M. Dirac, Lectures on Quantum Mechanics. Belfer Graduate School of Science, (Yeshive University, New York, 964). [6] M. Henneaux and C. Teitelboim, Quantization of Gauge Systems. (Princeton University Press, Princeton, 992). [7] G. P. Jorjadze, A. K. Pogrebkov and M. C. Polivanov. Teor. Mat. Fiz., 40, , 979. [8] G. P. Jorjadze. Teor. Mat. Fiz. 65, , (985). [9] G. L. Lamb, Lect. Notes in Math., 55, 69-79, (976). [0] L. Faddeev and R. Jackiw, Phys. Rev. Lett.60, 692 (988). R. Jackiw (Constrained) Quantization Without Tears. Montepulciano, Italy, (993). [] G. Chechelashvili, G. Jorjadze and N. Kiknadze. Teor. Mat. Fiz., 09, 90, (996), or TMF, , (996). [2] F. Magri. J.Math. Phys., 9, 56-62, (978). 0