Modern Physics Letters A. Vol. 3. No. 17 (1988) 1651-1656
© World Scientiljc Publishing Company
CONFORMAL FIELD THEORIES COUPLED TO 2-D GRAVITY
IN THE CONFORMAL GAUGE
F. DAVfD*
Service de Physique Théorique' de Saclay, 91 190-Gif sur Yvette, Cedex, France
Received 20 July 1988
The coupling of conformai field theories 10 2-d gravit y may he studied in the conformai gauge.
As an appücation, the resulls of Knizhnik. Polyakov and ZamoJodchikov for the scaling dimensions of conformai fields are derived in a simple way. Tbeir conjecture for the susceptibility
exponent y of strings is proven and extended 10 arbitrary genus surfaces. The result agrees with
exact results from randem lattice models.
Recently, Polyakov proposed a new treatment of 2-dimensional gravityi that relies
upon a light cone gauge fixing procedure for dealing with the invariance of the theory
under local diffeomorphisms. The emerging effective theory contains a SL(2, ~) current
algebra. Knizhnik, Polyakov and Zamolodchikov developed this approach 2 to study
conformaI field theories coupled to 2-d gravity, and to derive a relation between the
conformai dimension Il. of any (primary and spinless) field qJ coupIed to gravity and
its original conformaI dimension 6(0) (with no coupling to gravity). They also conjectured a formula for the "string susceptibility" exponent y related to the large area
behaviour of the partition function. The SL(2,~) current algebra plays a central role
in the derivation of those results. They are in striking agreement with exact results for
discretized models of 2-d gravity, where the internal metric is described by random
triangulations, both for "pure string models" in d = 0 and d = - 2 dimensions 3 - ' and
for critical statistical systems on random iallices, such as Q states Polis models for
Q = 2 (Ising), 6 Q --> 1 (percolation), 7 Q --> 0 (tree iike polymers), 8 and self-avoiding
polymers. 9 The purpose of this letter is to show that the coupling of 2-d conformaI
theories to gravity may be studied in a quite simpie way in the conformai gauge. 10 The
resulting effective action is indeed the Liouville field theoryll At the critical point
where the physical cosmological constant vanishes, the Liouville theory is nothing but
a free field theory. This makes the calculation of the scaling dimensions straightforward
and allows one to recover in a simple way the results ofRef. 2. ln addition, this method
allows us to compute the susceptibility exponent y as a function of the topology of the
2·d "universe", and thus to prove and to extend the conjecture of Ref. 2.
Let us start from the action for 2-d gravity on a manifold M
• Physique Théorique CNRS.
f Laboratoire de l'Institut de Recherche Fondamentale du Commissariat à l'Energie Atomique.
1651
1652 F. Douid
S[g••] = K
Ld2,,~
(1)
coupled in a minimal way to sorne 2-d conformai field theory characterized in particular
by its central charge C and its set of primary fields 'P" As in the seminal paper,ll we
fix a conformaI gauge by choosing a family (g~.(m)} of conformally inequivalent
metrics labelled by the moduli m of the surface M, and by writing a general metric as
(2)
Integrating out the conformai fields and computing the Faddeev-Popov determinant,
one gets the Liouville efTective action for 11.12 the conformai degrees of freedom <Il
+ (K
- K.)
L "pe"
d2
+ function (moduli)
(3)
with
(4)
À=26-C.
Ka is a (ultraviolet divergent) non-universal term. Writing K - Ka =
~ p2,
one
481t
recovers the standard Liouville action with mass Il. At K = Ka (p = 0) there is no mass
scale. Thus Ka corresponds to the critical value of K where the area of the 2-d universe
is expected to diverge. Then (3) becomes a free field theory.
When the action (3) is quantized and <Il treated as a dynamical field, both À and the
interaction term 1l2 e" are renormalized. This has been first noticed 13 • 14 in the full
quantization of the interacting theory (Il > 0) but may be seen in a much simpler way
by considering the massless theory (Il = 0) as an effective theory and by using the fact
that the background metric g~. is a gauge fixing parameter." Therefore the effective
theory must satisfy the consistency condition that there is no Weyl anomaly when a
classical conformai transformation
(5)
is performed. The most general effective action involving <Il, the conformai field theory
and the ghosts is simply lS
ConformaI Field Theories Coupled to 2-d Gravity .. .
1653
where SCFT and So contain the minimal coupling of the CFT and of the ghosts to the
classical background metric g~,. This action is in general not Weyl invariant even at
the classicalleve!. Indeed the classical equations of motion for <1> are
(7)
and under (5)
~cl
-4
(S)
c:J)d - qJo.
This gives a classical contribution proportionalto the coupling constant À to the trace
of the stress energy tensor which writes
a
<T.. (0'»
_
5r
_
Ctot r:o
0
- 8'Po(0') - - 4S" V 9 (O')R (0').
(9)
A standard calculation gives
C..,=À+1+C-26
(10)
where 1 is the contribution of the quantum nuctuations of <1>, which is a free bosonic
field, C that ofthe CFT, - 26 that of the ghosts l l
Thus the consistency condition C.., = 0 fixes'o
À=25-C.
(Il )
In order to move away from the crilical point, one has to add to the action (6)
interaction terros "'iS obtained by integrating sorne scaling fields 'Pi over the space M.
The most natural ansatz is
(12)
Classically (À = 00),
(13)
where "'1°) is the scaling dimension of the field 'Pi (as in Refs. 1 and 2, we consider only
spinless primary fields). The term corresponding to the unit operator
qJo =
1,
is nothing but the Liouville interaclion term
(14)
JHe"'. In the quantized theory, Ai must
1654
F. David
differ from (13). Indeed tJ.iS has to be Weyl invariant and the conformaI weight of the
operator 'PieA,,,, has to be one. This weight is easily obtained by calculating the gO
dependence ofthe vacuum expectation value ofthis operator with action (6). This gives
the constraint
(15)
Ar
where Ai cornes from the c1assical part of <l> and
from the quantum fluctuations. In
the c1assicallimit, one must recover (13) and this fixes the solution of (15) to be
A.' 1252- C [1 _ JI + 25 24- C (tJ.!O)' - I)J
(16)
A o =-M25 - C - J(25 - C)(1 - C)].
(17)
=
and in particular
Let us now consider the Liouville theory away from the critical point, with action
00
S[h,] = SL(<l»
+L
h,tJ.,S(<l»
(18)
i=O
where ho oc p' is the renormalized cosmological constant and the h,'s (i > 0) source
terms for the scaling fields 'Pi' From the equations of motion for ho '" 0, the natural
ground states are spaces with non-zero curvature. Let us therefore take for g2. a curved
compact metric with Euler characteristics
(19)
From (6) and (12), a shift on the Liouville field
1
<l> -+ <l> - -ln(h o)
Ao
(20)
leads to the crucial relation for the effective action
(21)
From (21), we can easily derive the scaling dimension of any field 'Pi coupled to gravity.
Indeed from the above considerations, ho is proportional to (K - K o) where the
Conformai Field Theories Coupled 10 2·dGravÎty...
1655
cosmological constant K has dimension Oength)-z. Thus the scaling dimension of ho
is 1. This is confirmed by considering the average area of the surface. From (21), for
hi = 0 (i > 0), we have
(22)
From (21), the scaling dimension of hi is AJA o and therefore the scaling dimension of
the field 'Pi is
Ai
A
Ô·= 1 - -
•
(23)
o
One can check that (23), with (16) and (17), gives the correct solution of the equations
of Knizhnik, Polyakov and Zamolodchikov2 Finally, (21) allows us to obtain the
susceptibility exponent y. Indeed from (22), the total vacuum energy diverges with the
area A as
J.X
roc--InA.
12A o
(24)
From the definition lo of the susceptibility y (r '" (2 - y)lnA), we get
J.
1
y = 2 - X12A = (2 - X) 24 [(25 - C)
o
+ J(1
- C)(25 -
C)].
This formula agrees with exact results for random lattices obtained for C =
(25)
- 2
and
0,16 and with the semi-classical calculations of Rer. 17. For the sphere (X = 2), it agrees
•
with the value conjectured by Knizhnik, Polyakov and Zamolodchikov.z
Finally, let us discuss briefly the case 1 5 C 5 25. This is the domain where tachyons
appear in the Liouville theory 14 and it has heen argued by various authors that this
is related to the branching of surfaces'- S • 1 5.18 For the sphere (X = 2), in the weak
coupling phase C < 1 p. > 24), y is negative and it vanishes for C = 1 where (25)
becomes singular. This fact is in striking similarity with the important result of
Durhuus, Fr6h1ich and Jonsson. 19 They showed that in a class of models of random
surfaces on hypercubic lattices, if y > 0, then the surface is a branched po1ymer.
Although there is no rigorous proof that this result is valid for other models of random
surfaces, one may expect that this is a somewhat universal property. In sllch a situation,
for J. < 24, a typical surface wou1d be a branched polymer made of"blobs" with typical
width of the order of the ultraviolet cut-off a, and the Liouville theory, which is an
effective theory describing the metric fluctuations of a 2-d object at scales much larger
than this cut-olT, would he inadequate to descrihe such an object. On the contrary, for
J. > 24, the fact that y < 0 means that the average area of a planar surface (x = 2) at
1656 F. David
the critieal point is fmite, and therefore of the arder of the (squared) ultraviolet cut-olT.
Ta ohtain a large surface is possihle either hy fIxing the total area ta be large
(microcanonical ensemble), as in Ref. 17, or by enforcing sorne special houndary
conditions, as in Ref. 14. For C = 25 (J. = 0) and Ji' = 0, the Liouville fIeld <t> decouples
classieally from the background metric g~b' one recovers the usual bosonic string and
<t> plays the role of the missing 26th dimension." The special values of À (18,12 and
6) where sorne consistent string models may he constructed from the interacting
Liouville theory (Ji' > 0) have also tachyons. '0 As for the usual bosonic string, it is
not clear whether they are related ta "physical" models of random surfaces.
Acknowledgments
J thank A. Bilai, B. Duplantier, E. Guitter, J. Kostov and S. Leibler for their interest
and their comments, and especially B. Duplantier for a careful reading of the manuscript.
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