The main result of this paper is the construction of a conformally covariant
operator in two dimensions acting on scalar fields and containing fourth order
derivatives. In this way it is possible to derive a class of Lagrangians
invariant under conformal transformations. They define conformal field theories
satisfying equations of the biharmonic type. Two kinds of these biharmonic
field theories are distinguished, characterized by the possibility or not of
the scalar fields to transform non-trivially under Weyl transformations. Both
cases are relevant for string theory and two dimensional gravity. The
biharmonic conformal field theories provide higher order corrections to the
equations of motion of the metric and give a possibility of adding new terms to
the Polyakov action.Comment: 11 pages, LaTeX, no figure

Ambjorn

Antoniadis

Bluman

Boulatov

Buchbinder

Elizalde

Ferrari

Franco Ferrari

Gusynin

Harvey

Hawking

Ichinose

Jackiw

Kawasaki

Michell

Myers

Pagani

Polyakov

Stelle

Teitelboim

Physics Letters B

English

arXiv

The main result of this paper is the construction of a conformally covariant operator in two dimensions acting on scalar fields and containing fourth order derivatives. In this way it is possible to derive a class of Lagrangians invariant under conformal transformations. They define conformal field theories satisfying equations of the biharmonic type. Two kinds of these biharmonic field theories are distinguished, characterized by the possibility or not of the scalar fields to transform non-trivially under Weyl transformations. Both cases are relevant for string theory and two dimensional gravity. The biharmonic conformal field theories provide higher order corrections to the equations of motion of the metric and give a possibility of adding new terms to the Polyakov action

Ferrari, F

CERN Document Server

Biharmonic Conformal Field Theories
Franco Ferrari
Dipartimento di Fisica, Universita di Trento,
38050 Povo (TN), Italy and
INFN, Gruppo Collegato di Trento
July 1995
Preprint UTF 353
Abstract
The main result of this paper is the construction of a conformally
covariant operator in two dimensions acting on scalar elds and con-
taining fourth order derivatives. In this way it is possible to derive a
class of Lagrangians invariant under conformal transformations. They
dene conformal eld theories satisfying equations of the biharmonic
type. Two kinds of these biharmonic eld theories are distinguished,
characterized by the possibility or not of the scalar elds to transform
non-trivially under Weyl transformations. Both cases are relevant for
string theory and two dimensional gravity. The biharmonic conformal
eld theories provide in fact higher order corrections to the equations
of motion of the metric and give a possibility of adding new terms to
the Polyakov action.
1
1 Introduction
The main result of this paper is the construction of an operator in two di-
mensions with fourth order derivatives. In this way it is possible to derive
a class of Lagrangians invariant under the Virasoro group [3]. They dene
conformal eld theories (CFT’s) satisfying equations of the biharmonic type.
The biharmonic CFT’s discussed here are interesting under several points of
view, from string theory [4] to two dimensional gravity [5]. The point of view
chosen in this paper starts from the two dimensional (2− d) Maxwell eld
theory. In the Lorentz gauge, after eliminating the longitudinal degrees of
freedom, the Maxwell theory becomes a biharmonic scalar eld theory, satis-
fying biharmonic equations of motion 42’ = 0. Here 4 = @@ denotes the
2− d Laplacian and ’ is a scalar eld. Of course, the biharmonic equation
is not conformally invariant. This is another way to state that conformal
transformations are not a symmetry of 2−d Quantum Electrodynamics. Ex-
ploiting an old theorem [1], it is however possible to make the biharmonic
equation invariant under the SL(2;C) subgroup of the Virasoro group, allow-
ing for point transformations in the scalar elds ’. In this way one obtains
a curious example of SL(2;C) invariant gauge eld theory, without the need
of introducing vector elds [2].
Much more dicult is the task of extending the set of symmetries of
the biharmonic equation to the whole Virasoro group. One may think for
example to construct a conformal invariant operator of the fourth order K
adding to the biharmonic operator 42 a suitable linear combination of lower
order operators, built out of the metric g , the covariant derivativesr and
the Ricci tensor R [6]:
K = 42 + aRrr + bR4 +cg
(rR)r + dR
2+
eRR + fR
R + g4R
Unfortunately this approach fails in two dimensions, since the coecients
a− g, chosen in such a way that K becomes invariant under Weyl rescalings
of the metric g , diverge.
To solve this problem, we dene here a generalized Laplacian D[A ; a]
which transforms as follows:
D’0 = eD’ (1)
2
under a rescaling of the elds of the kind ’0 = e’. D is constructed using
the same strategy of the covariant derivatives in gauge theories but here,
besides a vector gauge eld a, also a spin two gauge eld A is needed in
order to fulll 1. For a conformal transformation  = (z), where  and z
are complex coordinates, the real function  appearing in eq. 1 is dened by
 = −log
dz
d
.
Starting from the generalized Laplacian D, we are able to construct a set
of interacting conformal eld theories, called here of type I, with equations
of motion of the biharmonic type
D2’ =
V (’)
’
(2)
Here V (’) represents an interaction potential. The above equations of motion
are invariant under the Virasoro group. However, Weyl rescalings of the
metric are a symmetry of the theory only in the free case, i.e. when V (’) =
0. The action of type I eld theories is renormalizable if the gauge elds
A and a are external. The free energy{momentum tensor can be easily
computed and it is traceless. The problem of introducing kinetic terms also
for the gauge elds without spoiling renormalizability, locality or conformal
invariance seems instead without solution.
In a slight variation of the biharmonic CFT’s presented here, called type
II for convenience, Weyl rescalings are admitted for the scalar elds, i.e.
’0 = e’ if the metric undergoes a Weyl transformation of the kind g0 =
e2g . Clearly, the generalized Laplacian D is covariant with respect to this
transformation but, as we will see, it is no longer possible to dene Weyl
covariant equations of motion also at the free level. Despite of this fact, if
a conformal metric g = h(x) is chosen, the type II biharmonic CFT’s
continue to remain invariant under a conformal change of variables  = (z).
The energy{momentum tensor is however not traceless. The second type
of biharmonic CFT’s plays an important role in 2 − d gravity, representing
possible corrections to Liouville eld theory with higher order derivatives.
The second point of view from which the biharmonic CFT’s can be con-
sidered is that of string theory and conformal eld theory. Let us in fact
consider the string action in the Polyakov formulation [7]:
Sstr =
Z
d2x [@X@
X +RX] (3)
3
How much can this action be generalized, maintaining the properties of con-
formal invariance and locality? This is a physically important question,
since the coupling of string theory with other CFT’s yields remarkable conse-
quences, as it has been shown for instance in the case of the Thirring model
[8] discussed in ref. [9], where a new method of spontaneous symmetry
breaking was found in this way. Unfortunately, the set of CFT’s admitting
a Lagrangian formulation and which may be coupled to string theory is very
limited. The biharmonic CFT’s introduced in this paper enlarge this set to a
new class of conformal eld theories. Some interesting possibilities of letting
both type I and type II CFT’s interact with string theory will be given below.
A third and nal point of view which will be only briefly discussed here
is that of 2 − d gravity. In the second type of biharmonic CFT’s, where
the scalar elds ’ are allowed to rescale according to a Weyl transforma-
tion of the metric explained above, it is possible to perform the following
identication: ’ = jgj−
1
4 , where g is the determinant of the metric. Thus
the type II biharmonic CFT’s provide local equations of motion in jgj−
1
4 and
consequently in the metric. As mentioned before, the type II equations of
motion are no longer Weyl invariant. Therefore, conversely to what happens
in string theory, also the conformal factor of the metric can be determined
adding biharmonic CFT’s to the Polyakov action 3. With respect to the
Liouville eld theory, the biharmonic corrections to 2− d gravity have the
advantage of providing simple equations, e.g. in the free case, in which the
potential V (’) is set to zero.
In the following, the biharmonic CFT’s will be treated in details, both in
the Weyl invariant and Weyl noninvariant versions. Possible ways to couple
them to string theory will be discussed, together with their interpretation
as generalizations of the action for 2 − d gravity containing higher order
derivatives.
2 Biharmonic Conformal Field Theories
Let us start by considering the action of 2 − d Quantum Electrodynamics
(QED). In complex coordinates z = x1 + ix2, z = x1 − ix2, we have:
SMaxwell =
Z
d2zF 2zz (4)
4
with Fzz = @zAz − @zAz. Complex indices will be denoted with the rst
letters of the Greek alphabet, for instance  = z; z. Exploiting the Hodge
decomposition for the gauge elds A:
Az = @z+ @z’ Az = @z− @z’ (5)
one obtains:
SMaxwell =
Z
d2z(4’)2 (6)
Thus the QED action is not conformally invariant in 2− d. To see this more
in details, let us consider the biharmonic equations of motion coming from
6:
4(4’) = 0 (7)
Performing a conformal change of variables  = (z), the Laplacian trans-
forms as follows: 4@z@z =
4
J2
@@ , where J =
dz
d
. Therefore, the biharmonic
equation 7 becomes in the new coordinates:
1
J2
@@

1
J2
@@’

= 0
and it is clearly not conformally invariant unless J is a constant.
One way to extend the set of transformations leaving the biharmonic
equation invariant is to allow for point transformations in the eld ’ of the
kind [1]:
’ (z; z) = J’0

; 

(8)
If ’ obeys the above requirement, the action 6 becomes invariant under the
Mo¨bius group of transformations  = az+b
cz+d
, with a; b; c; d being arbitrary com-
plex numbers satisfying the condition ad− bc 6= 0. As pointed out in ref. [2],
this is a remarkable theory, having an SL(2,C) group of local symmetries and
no gauge elds. On a sphere, the gauge group coincides with the SL(2,C)
group of automorphisms admitted on a genus zero Riemann surface. Other
symmetries of the biharmonic equations have been obtained in [10]. In or-
der to obtain invariance under the whole set of conformal transformations,
however, the introduction of gauge elds is necessary.
To this purpose, we dene a new Laplacian Dzz , requiring that
Dzz’
0(z; z) = e(z;z)Dzz’(z; z) (9)
5
if ’ is rescaled by an arbitrary real factor:
’0 (z; z) = e(z;z)’ (z; z) (10)
To fulll requirement 9, let us try the following ansatz:
Dzz’ = [@z@z + aAzz + b (az@z + az@z) + c (azaz)]’ (11)
where Azz and a are gauge elds undergoing suitable transformations in
order to reabsorb all the unwanted terms in  (z; z) which arise in the rhs
of eq. 9 after exploiting the derivatives @z@z. In the case of a conformal
change of variables  = (z), the scalar eld ’ transforms as in eq. 8.
Therefore, putting (z; z) = − log J , with J =
dz
d
 in eqs. 9 and 10, it is
easy to realize that the new Laplacian Dzz is covariant also under conformal
transformations:
D
h
A; a ; a
i
’0

; 

= JDzz [Azz; az]’ (z; z) (12)
Eqs. 9 and 12 are veried performing in eq. 11 the substitutions:
a = −1 b = −1 c = 1 (13)
Moreover, the elds Azz and a should obey the following transformations:
Add = (Azz + @z@z (z; z)) dzdz (14)
ad + ad = (az + @z (z; z)) dz + (az + @z (z; z)) dz (15)
Starting from the modied Laplacian 11, we can easily build the free action
of type I biharmonic CFT’s:
Sconf =
Z
d2z (Dzz’)
2 (16)
Apparently, the above Lagrangian density
Lzzzz = (Dzz’)
2
has the wrong tensorial properties. Proper Lagrangians in complex 2− d
coordinates should be in fact (1; 1) tensors Tzz with one complex and a com-
plex conjugate indices. For instance, the Lagrangian of massless scalar elds
6
X(z; z) is given by Lzz = @zX@zX. However, due to eq. 8, the scalar elds
’ behave as ’  1p
gzz
, i.e. they may be regarded as

1
2
; 1
2

dierentials. This
fact assures the covariance of Sconf also under general dieomorphisms, as we
will see below. Let us notice that if we insist in the interpretation of ’ as a
spinor, no ambiguity with the spin structures arises, because ’ transforms as
an element of

1
2
; 0

⊗

0; 1
2

. As a consequence, the multivaluedness of the
1
2
; 0

component induced by the presence of an hypothetical spin structure
cancels against the phase of the

0; 1
2

component. Thus Sconf describes a
well dened conformal eld theory with higher order derivatives. As in string
theory, the dependence on the metric disappears once conformal coordinates
are chosen. In 2− d this is a great advantage, since every Riemannian man-
ifold is conformally flat. Let us now discuss possible kinetic terms for the
gauge elds Azz and a. A simple way to introduce a dynamics for the gauge
elds is suggested by the transformation rules 14{15 and consists in rewriting
Azz and a in terms of two other scalar elds A and a:
Azz = @z@zA
az = @za az = @za
For consistency with eqs. 14{15, A and a should transform as follows under
a conformal change of variables:
A0 = A+ (z; z) a0 = a+ (z; z)
It is now easy to dene a conformally invariant kinetic term for A and a:
Skin =
Z
d2z@z(A− a)@z(A− a)
In this way, however, the action S = Sconf + Skin becomes no longer renor-
malizable. Other attempts to build local kinetic terms for the gauge elds
either lead to non-renormalizable actions or break the conformal symmetry.
Hence, we will assume in the following that the elds Azz and a are exter-
nal elds. This is not a serious limitation. Eective Lagrangians without the
presence of those gauge elds can be always constructed solving the classical
equations of motion in Azz and a. Alternatively, one may integrate out the
external elds in the path integral using suitable weights in order to assure
convergence.
7
At this point, we compute the free energy momentum tensor of the con-
formal eld theories of type I dened by the action Sconf . Using a general, not
conformally flat metric, we obtain in real coordinates the following expression
for the functional Sconf :
Sconf =
Z
d2x g [4’− g (A + a@ − aa)’]
2 (17)
where g = jdet g j and 4’ = 1p
g
@
p
gg@’

. In eq. 17 the determinant
g of the metric and not its square root appears. The reason is the presence of
the eld ’ with the transformation rule 8, suitably generalized to the case of
general dieomorphisms y = y(x) putting J = det
dx
dy
. Clearly the action
17 is invariant under general dieomorphisms and also under a Weyl rescaling
of the metric g0 = e
(x)g . Setting l’ = − (A + a@ − aa)’, the
energy momentum tensor (Tconf ) 
Sconf
g
can be written as follows:
(Tconf ) = 2
p
g (4’+ g l’) 
@

1
2
p
ggg@’−
p
ggg@’

+

1
2
p
ggg −
p
ggg

l’

(18)
From eq. 18 it turns out that (Tconf ) is traceless, as it should be in a
conformal eld theory.
From the transformation law 8, one can also give to the eld ’ an inter-
pretation as the inverse of the fourth root of the metric. Let us discuss this
case, performing the identication
’ = g−
1
4 (19)
Since g is a semipositive denite quantity, (we allow also for degenerate met-
rics), there is no ambiguity in taking its fourth root. Moreover, as explained
before, all the problems with the spin structures are absent, despite of the
fact that now ’ has spinorial indices. Substituting eq. 19 in eq.16, Sconf
becomes
Sconf =
Z
d2z
h
Dzz g
−1
4
i2
(20)
and it is local in g−
1
4 . Thus Sconf may be considered in this case as a possible
correction of the 2−d gravity action, characterized by the presence of higher
8
order derivatives. Remarkably, after the identication 19, Sconf remains con-
formally invariant, but clearly the invariance under Weyl rescalings of the
metric is lost. Therefore, contrary to the Polyakov formulation of string the-
ory, the equations of motion coming from Sconf are also able to determine
the conformal factor of the metric. In principle, we can dene an action for
’ = (g)−
1
4 that satises both Weyl and conformal symmetries:
S 0conf =
Z
d2x g
1
4
h
(4− g l) g
1
4
i
(21)
In a conformally flat metric:
S 0conf =
Z
d2z g−
1
4

Dzz g
1
4

As we see, S 0conf is unfortunately non{local in the eld ’ = (g)
−1
4 and there-
fore we will neglect it. In general, for a eld ’ transforming as (g)−
1
4 , we will
have that the free action of type II biharmonic CFT’s coincides with Sconf .
Self-interactions of the metric eld can be added for both type I and type
II theories introducing the following functional:
Sint =
Z
d2zVn(’) (22)
Substituting
Gzz = Azz −
1
2
(@zaz + @zaz)
the most general possible potential Vn(’) is given by:
Vn(’) = a−1’
−2 + a0Gzz + a1G
2
zz’
2 + a2G
3
zz’
4 + : : :+ anG
n
zz’
2n (23)
with a1 : : : an denoting a set of real coupling constants. Eqs. 22{23 describe a
well dened conformal action, invariant under the transformations 8, 12 and
14{15. However, in both cases of type I and II biharmonic CFT’s, the action
Sint is not invariant for Weyl rescalings. The above potential is interesting
also because it yields several possibilities of coupling the biharmonic CFT’s
with the basic elds of string theory, i.e. the scalar elds X, the metric g
and, in the super case , the spin currents of the kind J =   .
9
3 Conclusions
Concluding, let us briefly summarize our results. Starting from the new
Laplacian dened by eqs. 9, 11, 13 and 14{15, we are able to construct
conformal eld theories with biharmonic equations of motion. Type I bi-
harmonic CFT’s are characterized by the free action 16. After a conformal
transformation, the scalar elds ’ undergo the point transformation dened
in eq. 10. The type I free biharmonic CFT is both Weyl and conformal
invariant. Its energy{momentum tensor is traceless. The Weyl invariance is
however lost if a self{interacting Lagrangian is added (eqs. 22{23).
In type II biharmonic CFT’s ’ is allowed to rescale under Weyl transfor-
mations. The action is again provided by eq. 16, but in this case the Weyl
invariance is lost. It is however possible to dene an action with second
order derivatives which is both invariant under Weyl and conformal transfor-
mations (eq. 21). Unfortunately, this action is non-local in the scalar elds.
After the identication 19, the type II biharmonic CFT’s become particu-
larly relevant in 2−d gravity and string theory. Also for type II theories one
can add to the free action an interacting term Sint of the kind given in eqs.
22{23.
Finally, many interesting issues could not be treated here because they
are outside of the aims of this letter. For instance the relation, if it exists,
between the biharmonic CFT’s and the usual ones [11]. Also the problem of
their quantization and several aspects connected to possible consequences in
string theory could not be fully developed due to the lack of space. We hope
however to have drawn with this letter the attention to the interesting class
of biharmonic conformal eld theories. They provide in fact local and simple
equations of motion for the metric in two dimensions (see e.g. eq. 20) and
can be easily coupled with the elds entering superstring theory. In this way
it seems possible to extend the Polyakov action introducing new terms and
new free parameters. Hopefully this will provide more degrees of freedom,
that will at least be sucient to settle some of the remaining problems of
string theories.
10
4 Acknowledgements
It is a pleasure to thank Guido Cognola for useful discussion and for pointing
out ref. [6].
References
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(1901), 134-142.
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bridge University Press, Vols. 1 and 2, London and New York (1987).
[5] C. Teitelboim, Phys. Lett. 126B (1983), 41; R. Jackiw, Nucl. Phys. B252
(1985), 343; A. M. Polyakov, Mod. Phys. Lett. A2 (1987), 893.
[6] V. P. Gusynin and V. V. Romankov, Sov. J. nucl. Phys. 46 (6) (1987),
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11


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