Quantum Liouville theory from topologically massive gravity: 1 + 1 cosmological constant as square of 2 + 1 graviton mass

NUCLEAR P HVSI C S B

Nuclear Physics B 375 (1992) 362—380 North-Holland

Quantum Liouville theory from topologically massive gravity: 1 + 1 cosmological constant as square of 2 + 1 graviton mass *

Ian I. Kogan ** Department of Physics, University of British Columbia, Vancouver, BC, Canada V6TJZJ Received 17 October 1991 Accepted for publication 6 February 1992

The connection between two-dimensional quantum gravity (Quantum Liouville theory) on the boundary ÔM of the three-dimensional manifold M and three-dimensional Topologically Massive Gravity (TMG) on the manifold M is considered. The two-dimensional cosmological constant is proportional to the square of the topological mass of the threedimensional graviton, which establishes a connection between world-sheet and bulk scales. Different physical applications are discussed.

1. Introduction Recently Carlip demonstrated in a very important paper [1] that topologically massive gravity (TMG) [21, i.e. the sum of Einstein and gravitational Chern—Simons terms, defined on the (2 + 1)-dimensional space-time M induces a two-dimensional gravitational (Liouville) action on the boundary ~M. This derivation confirms the earlier suggestion [3] that two-dimensional quantum gravity (2-d Liouville theory) should be related to 2 + 1 TMG and inverse topological mass of graviton which is proportional to the gravitational Chern—Simons coefficient, is a measure of the off-criticality. This connection between Liouville theory and TMG is very important for the Open Topological Membrane (OTM) approach to string theory [4—61where the string world-sheet is considered as boundary of the open membrane. The goal of this paper is to study the above-mentioned connection between the 1 + 1 Liouville theory and 2 + 1 TMG in more detail. It will be shown that TMG corresponds to Liouville theory in the phase with condensate of This work is supported in part by the Natural Sciences and Engineering Research Council of Canada. **On leave of ITEP, Moscow, USSR. *

0550-32l3/92/$05.00

© 1992 Elsevier Science Publishers B.V. All rights reserved —

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dreibeins e~,which can be normalized as (e,~)= ö~,i.e. with classical (2 + 1)dimensional space-time (the same phase we must get in our 3 + 1 gravity to describe nature). Let us note that for such a background we have soldered rotations in space-time (general coordinate transformations) and local S0(2,l) rotations, which gives us natural explanations why we have to solder the twodimensional rotations on the boundary and local S0(2,1) rotations the necessary step to transform the SO (2,1) Kac—Moody algebra into a Virasoro algebra [7]. It turns out that the two-dimensional cosmological constant in the Liouville theory is proportional to the square of the topological mass. This connection establishes the correspondence between a world-sheet (boundary) scale, which is determined by a cosmological constant ji, and (2 + 1)-dimensional (bulk) scale, which is defined by the graviton mass M. The graviton mass M is proportional to (2 + 1)-dimensional Planck mass K, which is the natural scale for (2 + 1)-dimensional space-time. It is also very interesting that a zero cosmological constant corresponds to TMG without Einstein term (with massless graviton) and the corresponding classical action obeys conformal invariance. We shall also discuss the stress-energy tensor in Liouville theory and demonstrate how the additional term (which is necessary to correctly reproduce the Liouville central charge) originates from the boundary part of the Einstein term. We shall also briefly discuss the renormalization of the gravitational Chern—Simons coefficient and compare it with the leading terms in the 1/k’ expansion of the exact results in Liouville theory KPZ scaling relations [7—10] We shall present some arguments in favor of a possible relevance of the phase without dreibein condensate to the description of 2-dimensional induced gravity in the case of matter central charges c in the region 1 < c < 25 (c = 1 barrier). In conclusion we briefly discuss the possible three-dimensional description of the c = 1 strings and Witten SL(2;D~)/U(l)black holes [11] as different phases of the U (1) Topologically Massive Gauge theory (TMGT) interacting with TMG. —

—

2. Quantum Liouville theory from topologically massive gravity Let us first of all establish correspondence between 2 + 1 TMG and 1 + 1

Quantum Liouville theory. To do this we start from Carlip’s original approach (see ref. [1]) and demonstrate that special properties of two terms in the TMG action, the Chern—Simons and Einstein terms, do not lead to the SO(2,l) WZW model, but to the restricted SO(2,1) model, which is known to be equivalent to Liouville theory. The key point is that the Einstein term naturally

364

1.1. Kogan / Quantum Liouville theory from TMG

leads to the constraint, which transforms the SO(2,1) WZW model into Liouville theory. Let us start from the TMG action in the first-order formalism considering spin-connections ~ab = _~ba = WabdX,1 and dreibeins e’~= e~dx’2as independent variables and introducing Lagrange multiplier fields )!~= 1(this 1~dx’ to representation was suggested in refs. [1,12 J). It will be more convenient use the following representation for the spin-connection: ~ab = abc~c. The bulk part of the TMG action is (1)

STMG=SE+SCS+SA,

where the 2 + 1 Einstein action is SE

=

KIM

A

(dwa +

fabcwb A

wc),

(2)

the gravitational Chern—Simons term =

81nfM~

+ 4EabcwaAwbAwc)

(3)

ec).

(4)

and the Lagrange multiplier term SA

=

IM

A

(dea +

A

The curvature for the spin-connection is =

R~

0dx~Adx~ = d~ +c~2~wbAwc,

(5)

and this action is invariant with respect to the reparametrization group (general 1, ~u = 1, 2, 3 covariance) with three local parameters e’ ôA~= a~(A~v) + ~ (6) where A~,is any covariant vector (one-form) ~ SO(2,1) group with gauge parameters ~a, a = 1,2,3 ôea

=

~abc~b~’c,

ö,ta

=

Eabc~btc,

öWa

=

and to the local

~dq~a + abccbbwc.

(7)

Defining new variables Pa

=

2a + KWa,

öIJa

= ~Kd4~,l5a

+ C abc~bPc,

(8)

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365

and integrating (1) by parts we get the Carlip—Deser—Xiang form [1,121 of the TMG action STMG

fM~

A

(dea + 2fabcWb

I

+

+ ~8ThJM ~

~

A ec)

—

Keabcea A 6AWC)

CWaAW

A wc}

+

Kf

dM

(9)

waAea

The last boundary term SOM comes from the integration by parts of the Einstein term and will be extremely important later. Let us note, that only the third, SO(2,l) Chern—Simons, term is not explicitly gauge invariant the boundary term plus the two first terms in (9) are invariant with respect to the local S0(2, 1) transformations (7). Now we shall get from S0(2,l) CS action to the WZW model on the boundary. There are a lot of papers devoted to this subject (see refs. [1,4,13— 16] and references therein). To get the WZW action let us find the general variation of 5TMG which is the sum of the bulk and boundary contributions. The bulk part gives the classical equations of motion, the boundary part takes the form —

ÔSIOM

=

.~_f 8R

8M

ów~AWa

+ Kf OM

(ôw°Aea + Wa Aôea)

—f

dM

pa Aöea.

(10)

To get a normal path integral we must add some boundary action to STMG in a such way that the total variation will be given only by mutually commuting coordinates. Let us note, that our conjugate variables are fl~and e±,/J~and e~and w~and w~.If one specifies w’~and e’~at the boundary, it is necessary to add the quadratic term

s

8Th8M w~w~+~cf aM w~e~_f dM JJ~e~, 2=~k_f

(11)

and the total boundary part of the variation is Ô(STMGIaM + S 2)

=

8xaM

A~öw~ +

f

(Kw~—fl~)ôe~,

(12)

where Aa=wa+Mea

(13)

are the boundary gauge fields, which transform under the SO (2,1) group (7) in the same way as the spin-connection, and M = 8irK/k’ is the mass of the 2 + 1 graviton [2].

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366

Let us note that the last two terms in (11) are SO (2,1) invariant, because the difference KW fi transforms as e, without derivative term (see eq. (8)). Adding 52 to the TMG action (9) we finally get —

~=L~~ ~f~ A

+

8m M k’ I

+~—/

(dea + 2EabcWb A ec) A dwa

+

~0~~a

I

W~W~+Kj

OltjöM

—

Kabcea

A

wb A wc}

A wb A wc)

I w~e~— / ~

JOM

(14)

JaM

As we have mentioned, the gauge dependence of the last two terms is cancelled by the first two terms in (14), so the only gauge dependence arises from the Chern—Simons term SCS and the first boundary WZW± term. It is easy to demonstrate, that these two terms induce at the boundary the chiral gauged SO(2,1) WZW model with action Scwzw(g,w)

=~f

~f

aM Tr(g’D~gg’O±g) + 12m

+

—J

Tr(w 2g’8~g)+

—J

3

OMTr(g’dg)

Trw~w~,

(15)

where the last two terms give us the interaction of the SO(2,1)-valued WZW field g with boundary values of the CS fields w~ = w~a,.As was pointed out by Carlip [1] if we would like to rewrite this SO(2,l) model as SL(2;EFl) model we are to change k’ —~ 4k’. This is connected with the fact that the trace in the fundamental representation of SO(2,l) is four times that in the fundamental representation of SL(2;ll~).Later we shall work with SL(2;l1) and shall use the normalization of the generators Tr aaab = 2grn’, instead of the standard ~ so it is not necessary to change k’ now. What is important in the case of the TMG action (14) is the boundary term Kfw~e~,

proportional to Planck mass K which we must add to our WZW action (15). (The fle term is irrelevant, we only need it to cancel the gauge dependence of the we term.) Then the boundary (1 + 1)-dimensional action takes the form k’ r k’ ~ 2~8M Trwz(Jz—Mez)-i-—J S=Swzw(~)+_J Trw~w~, (16) where J±= g’~g.

1.1. Kogan / Quantum Liouville theoryfrom TMG

367

To get Liouville theory from this model we must be in the proper phase of the TMG. There are two possible phases in the theory. The first, let us call it topological phase, exists when the vacuum expectation value of dreibeins is zero, (efl) = 0. There is no background space-time in this phase and it is even impossible to diagonalize the quadratic form of the bulk part of the action and find the propagators. To invert the quadratic part of the action we are to fix the gauge for all the three fields /3, w and e, but we have only two gauge groups reparametrizations and local SO(2,l) or SL(2;E!~) rotations. This means that the quadratic approximation does not exist in this phase, which is evident in the second-order formalism after resolving the constraint de + w A e = 0. The physical properties of this phase are unclear However, there is second phase with non-zero condensate of dreibeins (nonzero background metric) which can be chosen in the form —

~.

(17) It is this phase where the graviton becomes massive due to the Einstein action which becomes the mass term of the w field. In this phase local SO(2,1) symmetry is broken spontaneously and only simultaneous space-time and tangent rotations preserve the symmetry of the ground state. The same is the case on the boundary, where we have to solder the two-dimensional rotations on the boundary and SO(2,l) rotations the necessary step to transform the S0(2,1) Kac—Moody algebra into a Virasoro algebra [7]. In this broken phase with background space-time, we get from (16) the Liouville action. To do this we need two things. First of all, we shall neglect dreibein fluctuations near the classical value. Then, if e~= it is easy to see that the only non-zero component of e~is e~ = e~ ie~.Using the matrix notation e~= eta’ we get from (17) —

—

(00\ e~= ~l o)

‘

e~=

(01’\

~oo)

.

(18)

Under the gauge transformations the vacuum is not invariant, the transformation law is —~

g’ez,~g.

(19)

However, there is a residual local symmetry which preserves e~.To find it, it is convenient to use the Gauss decomposition for the SL(2;ll) group, which *In fact there are actually a number of different phases, depending on how degenerate e~is. The “intermediate phases” are found when e is non-zero, but has vanishing determinant; the number of ways this degeneration occurs depends on the topology [17].

1.1. Kogan / Quantum Liouville theoryfrom TMG

368

was originally suggested by Alekseev and Shatashvili [18] (1 0~(e~ 0 ~ (1 w l)~0e~)~0!

—

20

()

It is easy to see that e~has a symmetry group, which is the Bore! subgroup of the lower triangular matrices of the total SL (2;R) group. The symmetry group for e~is the upper triangular Borel subgroup. However, only e±enters the action (16), so the Bore! subgroup { U = exp (~.%a ) } gauge symmetry is compatible with the vacuum. Thus we get a natural explanation ofthe relevance of the Borel subgroup it is the residual symmetry of the holomorphic (or antiholomorphic) part of the dreibein condensate. Now we can immediately obtain the standard gauged WZW mode! from the action (16). It is evident that only the w~component enters the product Trw~e~ and because we would like to preserve the Borel subgroup local invariance our action transforms into —

—

S=Swzw(g)+~f 2ir

dM

w~(J~-M),

(21)

which is invariant under the Borel subgroup gauge transformations g

—~

Ug

=

eag,

W~—4W~+Tr(aUU’a~).

(22)

It is known [18—20] that this model has only one degree of freedom (the first one disappears due to the constraint J = M, the second one can be gauged away) and describes Liouville theory. The action (2!) takes the most simple form in the Gauss representation (20), the left- and right-currents are

/ a9~ ~e2~9’~ô~+ 2~Oq~~2e2~OA.\ ( j \ e~ô,1 —9ç/~+ ~e2~02 J —

g’8g

=

—

(23)

,

and

/

0gg~ =

f

a~—2e2~aw

e2~aw

\ .

8)L +

2~%8q~ A2e2~’aw —

—~

+

(24)

)~e2~8~ I

The constraint J~= M takes the form e2~O±~ = M,

(25)

II. Kogan / Quantum Liouville theory from TMG

369

and the action (21) is now =

-~-f

2~O~w +w~(e2~a~~j’—M), (26) ~ a2~a+e

47t

where the transformations (22) are (27)

(28) Using this gauge symmetry we can put ~l = 0, then integrating over gauge field w~one gets the constraint (25). Solving this constraint and considering the Liouvi!!e field q~as independent variable we shall get the Liouville kinetic term in conformal gauge. The light-cone action can be obtained if cii is an independent variable, as was shown in refs. [18,19]. However, we get only one chiral sector of the Liouville theory. To reproduce the complete Liouville action in DDK form [9,10] we can use left—right symmetric version of our gauged mode!, which was considered in ref. [20]. This left—right symmetric mode! can be obtained from the three-dimensional manifold with topology Ax D~,where A is the annulus after sewing left and right sectors. The new action can be obtained by gauging both upper Bore! subgroup for the left-moving sector and lower Borel subgroup for the right-moving sector (this means in some sense, that instead of eliminating one degree of freedom by constraint we use the second Bore! subgroup gauge symmetry to gauge it away) S

=

~f 4ir

8Z~0~

OM

+ e2~(8~2 + A~)(a~+ A~)—M(AZ + Ak), (29)

which is invariant under the transformation

Az—~Az+8~OL, A—*A~+O~O~.

(30)

Using this gauge symmetry we can fix the gauge by putting w = = 0 and then integrate over A. Due to the exponential factor e2~in the quadratic term there is a non-trivia! determinant factor fl~ e2~which gives the dilaton term QR~2~çb, where R~2~ is the world-sheet curvature [21]. It is possible to get this term in another way if we redefine fields A~,± A~exp(—~5)and get the usual quadratic term ~ But now the transformation law for the new field A will be —*

A~

—~

A~+ e~9 2OL,

A~ A~+ e~8±OR, —~

(31)

1.1. Kogan / Quantum Liouville theory from TMG

370

and after fixing the gauge and introducing Faddeev—Popov ghosts we get a non-trivial determinant depending on the Liouvi!!e field ~ (see ref. [22]for example)

-±~f ~

det[e~8~e~ô~] =

~

(32)

aM

and reproduce the same dilaton term as in ref. [21]. We shall not discuss here how to get the precise value of the renorma!ized parameters in the Liouvi!!e !agrangian, which can be determined from the conditions of conformal invariance [9,!0]. What is important is that after some renorma!ization of the field we get the Liouville action with cosmological constant ~u

=

LM

2~çb + jie~,

~

(33)

+ QR~

and this 1 + 1 cosmological constant turns out to be the square of the 2 + 1 graviton mass

(34) This is the main result of this section and it establishes an extremely unexpected connection between two- and three-dimensional physics. At the end of this section let us discuss some analogy (however, not complete) of the above mentioned connection between the cosmological constant a and the square of the graviton mass M2. The simplest example we would like to consider here is the Feynman path-integral representation for the relativistic propagator of a scalar particle with mass M (for simplicity let us consider the Wick-rotated propagator in euc!idean space, where d is the space-time dimension)

)

Gtx ‘

‘j’

r j

—J —

=J

,l

~

ip(x—y)

e

~

—J

(27r)dp2~~~M2 ~x(T)=y

p00

0

p —

dTe2lj x(0)=x

,-1

‘4’ ~ (2m)’~

1 Jo

dTe2+M2)T

( ~T Vx(r)exP~_~J ~ 0

) (35) J

It is possible to represent the integration over the proper time T as integration over one-dimensional metrics h (r) modulo the one-dimensional

II. Kogan / Quantum Liouville theoryfrom TMG

371

reparametrization group f(r) (see for example ref. [23]), so the relativistic scalar propagator takes the form G(x,y)

=

f

Vx(r)~~~T~ exp {_~f’dt~~) _M2f

dr (36)

and represents a one-dimensional universe with coordinate x E [0,1], where x(r) is some scalar field and h(t) is the one-dimensional metric. The proper time is the only reparametrization invariant characteristic of the metric the length of the path (hypervolume of the one-dimensional universe): —

f0’ v”iid~

= T. Now we see that the2 last term in the action cosmological is the oneis the one-dimensional dimensional cosmological term and M constant. Of course this is not a complete analog of our observation, but it seems quite interesting.

3. Correspondence between 1 + 1 and 2 + 1 scales. Zero 1 + 1 cosmological constant and 2 + 1 conformal invariance The obtained resu!t allows us to connect scaling properties on the worldsheet with the natural scale inside the membrane (in (2 + 1)-dimensional space-time). Let us discuss it in more details. The cosmological constant ~uis connected with the world-sheet scale because of reparametrization invariance, which means that regularization must be done with invariant interval ~ = g~~dz0dz~), which after fixing the conformal gauge takes the form ~ab (z) = e~ab, where ~~ab is some fiducial world-sheet metric (depending on moduli for higher-genus Riemann surfaces) and the Liouville field ~ (z) is the only physical degree of freedom with the quantum action in DDK form [9,10] SL(ç5)

=

f

~-

d2

~[

aac(~)0bcb(~) + QR(~)q~)+ (37)

with central charge CL = 1 + 3 Q2. The cosmological constant operator e~has conformal dimension 4 = 1. On!y in this case we can add the cosmological constant term to the Liouville action and consider the cosmological constant i~as a scaling parameter— A

2+~aQl,

a=~Q_~/Q2_8.

(38)

0~cs

The invariant ultraviolet cut-off A depends on the Liouvi!!e field, A = e~’2A, where A is the ultraviolet cut-off in the fiducial background. Thus a

I.!. Kogan / Quantum Liouville theoryfrom TMG

372

shift ~

~ + a changes the cut-off A and at the same time changes

~i.

Thus,

we can put =

A2~t(A= 1),

(39)

and we see that ji —~ 0 corresponds to the infrared limit A 0. It is easy to see how p~enters the correlation functions in Liouville theory —~

(40)

(~eMi~) = fv~ñefl~1e_sL~.

For simplicity we shall consider the statistical sum ZL

=

(1)

=

fv~e_s~

=

fdAe_/8Th~ZL(A),

(41)

where the statistical sum over random surfaces with fixed area A ZL(A)

=

(f

fVq~o

d2 ~

_A)exP[_SL(95;I1

=

0)],

(42)

does not depend on i~,which only enters the area integration measure in (41). Again we see that small i’ corresponds to the infrared limit, because ~u is the conjugate variable to the area the cosmological constant is the chemical potential for the hypervo!ume (area here) of the universe; small ~ucorresponds to large A, i.e. to the infrared region. The area dependence of ZL(A) can be obtained in different ways [7— 10,24,25] the result is —

ZL(A)

IA~_~Q10, csA

(43)

where h is the genus of the surface. The 1u-dependence is obtained after integration over A, (44) where the gamma function F (—s) is taken as analytical continuation in the parameter s = (h 1 )Q/ct. Let us note that in the case of the genera! correlation function (40) will be the as in eq.way (44), 1 [the (h au-dependence l)Q/ + >~,/i~].This is same one possible to with parameter s = cs regularize the area integration [25—27], which leads to poles at negative integer s, for example, s = —2 for C = 1 strings on the sphere (and 0 on the torus). In this case the fl-dependence may be more complicated, and terms —

—

1.1.

Kogan / Quantum Liouville theory from TMG

373

ln1u may appears: fl~F(—) = fl/f + flln1u. Another regularization was suggested in ref. [28], without poles, but with an imaginary part of Z, for which the fl-dependence will always be power-like. Thus we see that the 1 + ! cosmological constant fi, being the analog of the chemical potential for the area A, at the same time defines (or depends on) the world-sheet scale, i.e. the generally covariant UV cut-off. From eqs. (34) and (39) 2fl(A fl(A)

=

1),

=

~

=

M~= (k~)2

A

one gets the dependence of the 1 + 1 scale A on the graviton mass M, proportional to the P!anck mass K, which is the natural 2 + 1 scale A(M)

#Ao

=

=

-~--A 0,

(46)

where M0 is some normalization mass. Let us note that in the two-dimensional theory ~ = 0 corresponds to the critical situation when the dominant contribution in the area integration measure comes from the infrared region (infinitely large A). At negative /1 this measure is not norma!izab!e and a!! integrals diverge A thec~.It is 2 ~ 0atand critical amusing from TMGtowe non-negative = M of the TMG. Does this value /2 =that 0 corresponds thegetzero P!anck massfi limit limit have some special properties ? The answer is affirmative and it is known that in this limit the TMG action obeys conformal invariance. This was shown in ref. [2] in the second-order formalism, after resolving the constraint (4). Then the TMG action reduces to the CS action (3) only, but with spin-connection w(e), —~

v

a_~ —

~

a

~ab

rca

+ w~ej~ e~ i —

40e~= Wpab

=

~

+

(a~—~b),(47)

~

and the action is third-order in derivatives. It is convenient to decompose the variation of the spin-connection as a sum of the dreibein variation ôe and the Christoffel symbol variation of’ (which depends only on the metric —

5pv

aa

—

~

ab

a~pa —e jib e~ ~ —

—

4

a e,,bD ~)~e0.

Then the variation of the CS action will be fM ~~LA

=

f

6e~OI’~ e~DAOe~). (49) —

e,00AR~(e”

II. Kogan

374

/ Quantum Liouville theoryfrom TMG

It is easy to see that after integration by parts the last terms leads to the structure VA~~AR~e~ which is zero due to the Bianchi identity; the covariant derivative D~defined in eq. (47) acts on one space-time index and one tangent index (as must be the case for the dreibein), but multiplying curvature Rab by dreibein e~we precisely get the Bianchi identity in mixed representation for curvatures (one space-time and one tangent index). Only the metric part contributes to the variation and using the identity 0100

=

JjPgaAog~~+ ~gaP [öAOg~~

+

OvOgpA

—

ô~Og

1~],

(50)

one gets [2]

f~

os~

Og4~

~(R~

—

~g~R).

(5!)

Because of the general covariance this variation is zero under a general coordinate transformation, but it is also zero under the 2 + 1 local conformal transformation og~ = f(x)g~~ji.

(52)

But this means that TMG at K = 0 has an additional local symmetry, which is absent in the general case, so the number of the physical degrees of freedom must be smaller. Because we have only one bulk degree of freedom, the massive graviton, it means that when K = 0 and the graviton mass disappears this graviton also disappears from the physical spectrum it becomes a pure gauge degree of freedom (with respect to the local conformal group). It is easy to see in the first-order formalism, where the TMG action in the K = 0 limit is the sum of two terms: the gravitational CS term (3) and the Lagrange multiplier term (4). It is evident that the Lagrange multiplier )~and the dreibein e are completely symmetric now. The classical equations of motion have the form —

Radwa + dF~+

abc~

A

w~=

2c~~cWbAFC=

C

0,

abce A A~,

F’~= (ea,2c~).

(53)

We see that now the dreibein ea and Lagrange multiplier ~a are completely equivalent (this would not be the case if one takes into account the Einstein term) and using the symmetry of the problem we can demonstrate that both ea and I~can be gauged away after which the equation for w is the zero-curvature condition the equations for pure SO(2,!) CS theory without degrees of freedom. Thus we have proved (at least classically) that the zero-mass limit —

I.!. Kogan / Quantum Liouville theoryfrom TMG

375

of TMG is equivalent to pure SO(2,1) CS theory. Analogously in the zero-k’ limit it will be equivalent to pure Einstein gravity, i.e. ISO(2,l) CS theory [29,30]. Only in the presence of both terms shall we get the dynamical degree of freedom. Now we know that zero cosmological constant on the boundary corresponds to an additional local symmetry inside. It will be interesting to understand whether this correspondence is relevant to the cosmological constant problem in any case we see that in the TMG approach to Liouvi!!e theory we definitely have only positive /2, and zero cosmological constant corresponds to some critical point of the three-dimensional field theory the conforma!!y invariant point! In the last part of this section we discuss the possibility to get a fluctuating cosmological constant. This is very important for string theory applications, where the cosmological constant operator is the tachyon vertex and the value of the cosmological constant is the background tachyon field. If one would like to describe this phenomenon in the TMG picture it is necessary to consider a fluctuating P!anck constant. Is this possible? The answer is yes, if one gets the Einstein term after spontaneous symmetry breaking. The simplest mode! of this type was considered by Deser and Yang in ref. [3!]. The idea is to consider a conformally invariant scalar field ~b coupled to the pure gravitational Chern—Simons action (in the second-order formalism) —

—

—

~

(54)

This action has local conformal symmetry ~(x)

—~

~(x)’

=

g,~0(x) g~0(x)’= Q4(x)g,~~(x), S[g,1fl =S[g’,’I’]. —*

(55) (56)

If (1) = 0 we have one massless sca!ar particle, gravity has no physical degrees of freedom. However, if the scalar field acquires a non-zero vacuum expectation value (1) = ~ one can gauge away the sca!ar field if one takes Q(x) = (‘~I)/cI(x).After that the new scalar field will be constant i(x)’ = = ~ and one gets the TMG action for the metric ~ = S[G] where the Planck mass

K iS

=

Kf

~/~R(G) +Scs(G),

nothing but the v.e.v. of the sca!ar field.

(57)

376

II. Kogan / Quantum Liouville theoryfrom TMG

In this construction the P!anck mass may be fluctuating (because the v.e.v. of the scalar field is not a parameter, but a dynamical quantity) and we get a rather remarkable correspondence between the external tachyon field T (r) in the target space R, r E R, whose v.e.v. gives the two-dimensional cosmological constant (T) = /2, and the membrane conformally invariant sca!ar field P (x) (x E M, where M is the 2 + 1-dimensional manifold), whose v.e.v. (~) K gives the 2 + 1 Planck mass. Thus the background tachyon field leads to spontaneous conformal symmetry breaking in the 2 + 1 space (but not on the 1 + 1 boundary, until the tachyon is on-shell!). It will be extremely interesting to understand whether this correspondence exists for other string fields. 4. Liouville central charge and k’ renormalization. SL(2;EF!)/U(1) black holes and c = 1 strings as two phases of TMG It is known [8] that the stress-energy tensor in the Liouville theory is given by the KPZ formula [8] T(z)

=

k’ + 2Tr : Ja(z)Ja(z) : +O~-J°,

(58)

with central charge (we omit the ghost contributions) C

=

k’+2 + 6k’.

(59)

The additional term in the stress-energy tensor is necessary to change the anomalous dimension of the current J from 1 to 0 because only in this case the constraint J = M does not violate the conformal symmetry. Because we get Liouvi!le theory as the induced boundary contribution of the TMG, this modification of the stress-energy tensor must be natural. How to get eq. (58) from TMG? We need to consider the variation of the boundary action (16) with respect to variations of the metric, i.e. dreibeins ef 05=

1

T(z)Oe~,

(60)

JaM

which gives the stress-energy tensor (in mixed representation). Variation with respect to e~gives the T component. The first term in (16), i.e. the usual SL(2;Ell) WZW action gives the standard Sugawara-type stress-energy tensor, which is the first term in eq. (58). Variation of terms like we are irrelevant, because they do not contain WZW

II. Kogan / Quantum Liouville theoryfrom TMG

377

variables and from this point of view are not part of the two-dimensional stress-energy tensor. However, there is a second term in (16), the variation of which is Tr J~Ow~. If we remember the relation between spin-connection w~’j~~ and dreibein e~ (47), we immediately find that in the phase with condensate of dreibeins the spin-connection variation is =

~(O1’0ee—a’~oe~),

(61)

and the only non-zero component of the spin-connection variation is 0w2

=

0w~ = 8

2e~.

(62)

This component corresponds to the J°component of the current and we get the variation J9D~e~, which after integrating by parts gives us the second term in eq. (58). Thus we see that the reason for the additional contribution to the stressenergy tensor is the connection between w and e it is important that we have TMG, not pure Chern—Simons theory. After including the ghost contribution the total central charge of the Liouville theory is (k = —k’) [8] —

CL =

k—2 + 6k—28.

(63)

The total central charge (including matter) is C = CL + Cm. Let us remember that 2 + 1 P-odd matter theory (massive fermions or topologically massive gauge bosons, for example) induces the gravitational Chern— Simons term (the one-loop contributions were calculated in refs. [32,33]) which p!ays a very important role in coding the two-dimensional information it describes the central charge (induced Weyl anomaly) of the corresponding two-dimensional theory [1,3,5,13,34] k~d= C/6. It is interesting that the Liouvi!!e central charge has the form CL = 6kIR —

k1R=(k_2)_~+k’2,

(64)

and in the case k >> 1 we get ktR = k (1 + 0(1/k)). It will be interesting to understand how to reproduce the 1/k corrections from TMG perturbation theory (1/k is the expansion parameter in TMG) and along the way we reproduce the ghost contribution from gravity corrections such a situation seems natural, because we know that ghosts are connected with 2 + 1 Einstein gravity [351. —

378

I.!. Kogan / Quantum Liouville theoryfrom TMG

It is known that for matter with central charge 1 25 to the phase without space-time (and without usual particles) with (e) = 0. In this phase the constraint J = M disappears and we get some SL(2;E!1) model. Thus we see that depending on the phase of the TMG we can get either the Liouvi!!e field or the SL(2;!1) WZW model, presumably the ghosts contributions are the same. In the last case we have no Liouvi!!e field and conclude that we have a critical string. Let us consider the simplest example U(l) topologically massive gauge theory (abe!ian Chern—Simons theory in the lowenergy limit) and TMG. This is the most simple example of matter interacting with TMG. In the Liouville then phase we get the C = 1 string and the total central charge cancellation condition is (Cm = 1, but we use general notation) —

—

k—2 +6k+

(Cm2)260.

(65)

In the other phase ~e) = 0 we must neglect the 6k term and the new criticality condition will be k—2 +

(Cm

2)

26

=

0,

(66)

and for Cm = 1 we get the criticality condition for the gauged SL(2;ff~)/U(1) mode! [11], which was considered by Witten as the two-dimensional analog of the black hole. It was shown in ref. [111 (see also refs. [36—40])that there is a connection between this black hole and the C = 1 string. We see that the three-dimensional description makes this connection evident. The theory which describes both models is TMG plus U(1) TMGT, the full action is S

=

SsL(2;t~) +

5U(1)

+ SE + SA,

(67)

where the second term is the matter action and we denote the CS action (3) here as SSL(2;n). In the (e) ~ 0 phase we get the Liouville and ghost fields from the TMG part of the action and the matter action S,~ (1) induces C = 1 matter on the world-sheet. Thus we get the C = 1 string. In the unbroken (e) = 0 phase, that is as yet unclear to us, we get the whole SL(2;ll~)WZW model, which is non-unitary. However U(l) is the compact subgroup of SL(2;EI~) and we can consider the sum of the gravitational CS

1.1. Kogan / Quantum Liouville theory from TMG

379

action and matter action SSL(2;~) + Su(1) as the three-dimensional action for the coset mode! SL(2;R)/U(l). It is known [3,5,14] that coset models G/H can be obtained from the difference of the G- and H-valued Chern—Simons actions. Let us note that the Cm — 2 = —1 term in eq. (66) correctly describes the change of the central charge sign for the U (1) matter, which must be negative in the coset construction. Of course this picture is very approximate, however, such a natural explanation of the connection between SL(2;EP1)/U(!) black holes and C = 1 strings seems not to be accidental.

5.

Discussion and conclusion

In conclusion we summarize the obtained result. First of all, following Carlip’s original idea [1] we construct the worldsheet action induced by topologically massive gravity and found that in the phase of TMG with a dreibein condensate (e~)= we get the constrained SL(2;!~)WZW mode! which is equiva!ent to Liouville theory. Thus the natural explanation of the hidden SL(2;ER) symmetry in two-dimensional gravity is the connection with 2 + 1 gravity, where the SO(2,l) group, isomorphic to SL(2;EI~),is the local symmetry group. Using a left—right symmetric construction of the constrained SL(2~R)WZW mode! we demonstrate that the cosmological constant in the Liouvi!!e theory equals the square of the 2 + 1 graviton mass, /1 = M2. It is interesting that this connection makes the positivity of the two-dimensional cosmological constant very natural. Because /2 is connected with the world-sheet scale, there is a natural correspondence between the scale inside the membrane (the bulk scale), which is the P!anck mass K M = ~ and the world-sheet scale A It is amusing that a zero cosmological constant, from the three-dimensional point of view, corresponds to local conformal invariance and this idea might be relevant for the cosmological constant problem in higher dimensions. We also demonstrated that the TMG action leads to the correct Liouville stress-energy tensor and briefly discussed three-dimensional renormalization of the gravitational Chern—Simons coefficient k’. Finally we discussed the properties of the unbroken phase and conjectured that SL(2;R)/U( 1) black holes and C = 1 strings are described by different phases of a single theory topologica!!y massive gravity interacting with U (1) topo!ogically massive gauge theory. There are a lot of unsolved interesting questions and the author hopes to consider them in the future. —

380

1.1. Kogan / Quantum Liouville theory from TMG

I am deeply indebted to Steve Carlip for extremely stimulating discussions, constructive criticism and for informing me about his results [1] prior to pub!ication. I would like to thank Gordon Semenoff and Nathan Weiss for very interesting discussions and kind hospitality at the University of British Columbia. References [1] S. Carlip, Nucl. Phys. B362 (1991) 111 [2] S. Deser, R. Jackiw and S. Templeton, Phys. Rev. Lett. 48 (1982) 975; Ann. Phys. (N.Y.) 140 (1982) 372 [3] I. Kogan, Phys. Lett. B256 (1991) 369 [4] I. Kogan, Phys. Lett. B231 (1989) 377 [5] S. Carlip and I. Kogan, Phys. Rev. Lett. 64 (1990) 148; Mod. Phys. Lett. A6 (1991) 171 [6] P. Horava, Prague Institute of Physics preprint PRA-HEP-90/3 (1990) [7] A.M. Polyakov, Mod. Phys. Lett. A2 (1987) 893 [8] V.G. Knizhnik, A.M. Polyakov and A.B. Zamolodchikov, Mod. Phys. Lett. A3 (1988) 819 [9] F. David, Mod. Phys. Lett. A3 (1988) 1651 [10] J. Distler and H. Kawai, Nucl. Phys. B321 (1989) 509 [11] E. Witten, Phys. Rev D44 (1991) 314 [12] S. Deser and X. Xiang, Brandeis preprint BRX-314 (1990) [13] E. Witten, Commun. Math. Phys. 121 (1989) 351 [14] G. Moore and N. Seiberg, Phys. Lett. B220 (1989) 220 [15] 5. Elitzur, G. Moore, A. Schwimmer and N. Seiberg, NucI. Phys. B326 (1989)108 [16] W. Ogura, Phys. Lett. B229 (1989) 61 [17] S. Carlip, private communication [18] A. Alexeev and S. Shatashvili, Nucl. Phys. B323 (1989) 719 [19] M. Bershadsky and H. Ooguri, Commun. Math. Phys. 126 (1989) 49 [20] P. Forgacs et al., Phys. Lett. B201 (1988) 466 [21] T.H. Busher, Phys. Lett. B227 (1989) 214 [22] A. Gerasimov et al., Int. J. Mod. Phys. AS (1990) 2495 [23] A.M. Polyakov, Gauge fields and strings (Harwood Academic Publishers, New York, 1987) [24] A. Gupta, S.P. Trivedi and M.B. Wise, NucI. Phys. B340 (1990) 475 [25] N. Seiberg, prepnnt RU-90-29 (June 1990) [26] A.M. Polyakov, Mod. Phys. Lett. A6 (1991) 635 [27] M. Goulian and M. Li, preprint UCSBTH-90-6l (November 1990) [28] I. Kogan, Phys. Lett. B265 (1991) 269 [29] A. Achucaro and P.K. Townsend, Phys. Lett. B180 (1986) 89 [30] E. Witten, Nucl. Phys. B311 (1988) 46 [31] S. Deser and Z. Yang, Mod. Phys. Lett. A4 (1989) 2123 [32] J.J. Van der Bij, R.D. Pisarski and S. Rao, Phys. Lett. B179 (1986) 87 [33] I. Vuorio, Phys. Lett. B175 (1986) 176 [34] E. Witten, in Physics and mathematics of strings, Memorial Volume for Vadim Knizhnik, ed. L. Brink et al. (World Scientific, Singapore, 1990) [35] S. Carlip and I. Kogan, Phys. Rev. Lett. 67 (1991) 3647 [36] 5. Elitzur, A. Forge and E. Rabinovici, NucI. Phys. B359 (1991) 581 [37] G. Mandal, A. Sengupta and S. Wadia, lAS preprint IASSNS-HEP-9l/l0 (1991) [38] E. Martinec and S. Shatashvili, Chicago preprint EFI -91-22 (1991) [39] R. Dijkgraaf, E. Verlinde and H. Verlinde, Nuci. Phys. B371 (1992) 269 [40] M. Bershadsky and D. Kutasov, Phys. Lett. B266 (1991) 345