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Liouville Field Theory and 2D Gravity I Introduction

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Liouville Field Theory and 2D Gravity I Introduction Liouville Field Theory and d Gravity George Jorjadze Dept of Theoretical Physics Razmadze Mathematical Institute Tbilisi Georgia Abstract We investigate character of physical degrees of freedom for the mo dels of d gravity and study its dep endence on the top ology of spacetime manifold I Intro duction The EinsteinHilb ert action in dimensions do es not lead to dynamical equa tions for the metric tensor g X X x x a and b ab This degeneracy is related to the fact that the corresp onding Lagrangian p L g R g is a total derivative and the action is expressed only by ab the surface terms For the conformal gauge g X exp q X ab the scalar curvature R X takes the form q X R X e q X and the degeneracy of the EinsteinHilb ert Lagrangian b ecomes apparent email jorjrmiacnetge One can sp ecify a mo del of d gravity by requiring that the eld q X satises some dynamical equation The Liouville eld equation q X q X e where is a nonzero constant is usually considered as a mo del of dimensional general relativity The eld q X satisfying leads to the manifold with a constantcurvature R X see The Liouville equation arises in a large variety of problems of mathematical physics due to the conformal invariance of this equation General solution of has the form A x A x q x x log jj A x A x where x x x are the light cone co ordinates isasignof jj and A are any functions with A The standard action for the scalar eld in Minkowski space which leads to the Liouville equation has a noncovariant 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 in tro duce some additional auxiliary elds For a mo del 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 Construction of gauge invariant physical variables is the most imp ortant problem of dimensional Einstein general relativity as well In this note we study this problem for the simple mo dels of d gravity using the conformal invariance of the Liouville theory dilaton eld II d gravity with the We start with the action Z q ab g S d x g R g e a b ab ab where is a cosmological constant g det g g is the inverse matrix ab to the ducial metric tensor g R g denotes the corresp onding scalar ab ab curvature and is a scalar led which is known as the Liouville mo de or the dilaton eld The physical metric tensor g is related to g by ab ab g e g ab ab and the action takes the form Z p ab g S d x g R g a b ab where we have used the relation of scalar curvatures in dimensions for conformally related metrics R g e r R g ab ab p ab and we have neglected the surface term g g using the form of a b the Beltrami op erator r for the metric g ab Action denes the following dynamical equations S p r R g ab g S cd p g g g r r r a b ab c d ab a b ab g g Taking trace of and using we get that the dynamical system describ es a spacetime manifold with a constant curvature R g ab For the light cone co ordinates x x x the conformal metric takes the form q g g g g e and from we obtain q q e q q q q where q As it was exp ected the eld q X satises the Liouville equation while X is a free eld According the traceless energymomentum tensors of these two elds are equal to each other One can integrate equations and express the Liouville eld q X as a functional of the free eld X After integration we get F x F x q x x log jj F x F x F x F x where Z Z x x F x d exp d x x x x are the co ordinates of a xed point on the spacetime and the condition integration constants satisfy the jj The latter denes the SLR group manifold and the freedom of the Li ouville eld q X for a given free eld X is describ ed only by three parameters of SLR group The solution for can be written in the form x x B B q x x log where B F jj B x B x The case is a degenerated one and the corresp onding relation b etween the Liouville and free elds arises for the Backlund transformation see also Note that reduces to after the substitution B A Thus repro duces the general solution Integrating equations with resp ect to the free eld one can derive the map from the Liouville elds to the free elds G x G x x x b log jjG x a G x a which is also characterized by three parameters a b and the functions G have the form Z Z x x q x q x G x d e G x d e x x The conformal form of the metric tensor xes the gauge freedom only up to the conformal transformations x y f x by The corresp onding transformation of the eld q is given q x x qx x q f x f x logf x f x it is easy to check that the general solution can be obtained by the conformal transformations of the function q x x log jjx x which is a solution of Therefore all Liouville elds with xed are related to by the conformal transformations and hence by suitable choice of lo cal co or dinates one can x the eld q X After xing q X the dynamical freedom for the free eld X is describ ed only by the parameters a b see As a result the full gage xing leads to the nite dimensional system rather than the eld theory It should b e noted that all our construction of this section as well as the conjecture ab out the conformal form of the metric tensor is valid only lo cally The maps and contain singularities when the corresp onding denominators are equal to zero The domain of regularity of q and elds considered as a patches of the global spacetime manifold can be 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 has a nite number of physical degrees of freedom The numb er and dynamical character of these parameters are related to the global prop erties of the spacetime manifold III Hamiltonian reduction of d gravity In this section we analyze the same problem using the Hamiltonian approach This approach is noncovariant and wehave to separate the spacetime co or dinates We assume that the spacetime manifold M can be represented in the form M T X where T is isomorphic to R and X is a one dimen sional manifold The co ordinates x x T and x x X we interpret as the time and the space co ordinates resp ectively We also use the notations x t x x and the corresp onding derivatives we denote by dot eg and prime eg We assume g and g t x Due to the reparametrization invariance the Lagrangian of is sin gular and in the Hamiltonian description leads to a constrained dynamical system For the reduction to the physical gauge invariant variables we use the FaddeevJackiws metho d see also One can checkby direct calculation that the ab ove mentioned degeneracy of the EinsteinHilb ert Lagrangian in dimensions can b e written in the form p C C gRg C B B A B ab t x G C G C where A B and C are the comp onents of the metric tensor g ab A B g ab B C and p p g B AC G Then up to the surface terms the Lagrangian of takes the form L B C B C C B B C B B G C C C C C C G C C where we have excluded the variable A using relation Since the La grangian do es not contain the time derivatives of G and B elds these variables can playa role of Lagrange multipliers According to the equivalent system can be describ ed by the action Z h i S dx dt V C V L V C C where the V and V are auxiliary elds and the function L V C C is constructed from Lagrangian substituting the time derivatives and C by the velo cities V and V resp ectively C One can easily eliminate the variables V and V taking the corresp ond C ing variations and we arrive at Z G B S dx dt C C C C with C C C C C C C C C C C The elds G and B are indeed Lagrange multipliers and the corresp onding variations givetwo constraints and To analyze this constraint surface it is convenient to intro duce the new set of canonically conjugated variables q logC logC assuming C p C C Here we use the notations of SecI I taking into account that the eld logC in the conformal gauge coincides with the eld q of the previous section Note that the conformal gauge corresp onds to the choice of Lagrange multipliers G C B which provides the Liouville and the free equations for q and elds resp ectively For the new variables we nd U V U V where q U p q e q V U pq p V Note that the functions U and U are T and T comp onents of the traceless energymomentum tensor of the Liouville theory while V and V are the same comp onents for the free theory The next step of the reduction pro cedure is a calculation of the canon ical form Z dx px dq x x dx X on the constrained surface This surface can b e represented in the form U V where q U U q p q p e U V V V The canonical form denes the canonical commutation relations fpxqy g x y f xy g x y and we obtain the Magri brackets for the functions
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