Liouville Field Theory and d

George Jorjadze

Dept of Theoretical 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 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

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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 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

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 energy

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 U and V

fU xU y g U xU y x y x y

fV xV y g V xV y x y x y

and also

fU xU y g fV xV y g

 

As a result the functions are the rst class constraints since they satisfy

the commutation relations of the following Lie algebra

x y g x y x y f

The two rst class constraints x usually need two gauge xing con

ditions for each p oint x Since at this stage we have only four elds

p q it is exp ected that the system has no physical degrees of freedom

for the eld variables Then after the full Hamiltonian reduction the sys

tem with only a nite number of degrees of freedom can remain Below we

demonstrate this fact explicitly using again the metho d

Here it is useful to intro duce the functions

Z

x

q 

g q p e with x d

x

where x is a xed point from X x X Up to the total derivative we

obtain

q  q  q  q 

pq g e g e e g e g

t  t 

For the restriction of on the constrained surface weintegrate

these equations with resp ect to p and q elds After integration we get a

result which is similar to

f f f f f f f f

 

  

p

f f D



f f



q log with D f f f f

  



jjD

where

Z Z

x

d exp d f x

x x

and the integration constants satisfy the condition

Using we nd that the restriction of the functions g on

the constrained surface is given by

q

jjf g

and takes the form

h i

pq f f f f f f

  



D

After integration over X the co ecients of and give the corresp onding

canonical conjugated variables The only eld theory term is

f f f f





D f f



but it again gives the surface terms since it is related to the closed 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 repro duce the result of SecI I

To analyze the character of nite dimensional system we need the global

description of the mo del Such description can be related to the gluing of

Liouville elds with singularities see for example and and we are

going to discuss this approach elsewhere

Acknowledgments

One of the authors GJ is very grateful to the organizers of the seminar

q for the invitation and kind hospitality

This work was supp orted by the grants from INTAS RFBR

and the Georgian Academy of Sciences

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