Schwinger terms in Weyl-invariant and diffeomorphism-invariant 2-d scalar field theory

8 Pages·0.14 MB·English

by  Christoph Adam

Schwinger terms in Weyl-invariant and diffeomorphism-invariant 2-d scalar field
theory∗
Christoph Adama,b † and Gerardo L. Rossinia,c ‡
a Center for Theoretical Physics,
Laboratory for Nuclear Science and Department of Physics
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
b Inst. f. theoret. Physik d. Uni Wien
Boltzmanngasse 5, 1090 Wien, Austria
c Departamento de F´ısica, Universidad Nacional de La Plata
C.C. 67, 1900 La Plata, Argentina
(MIT-CTP-2687, hep-th/9710238)
8
9 WecomputetheSchwingertermsintheenergy-momentumtensorcommutatoralgebrafromthe
9 anomalies present in Weyl-invariant and diffeomorphism-invariant effective actions for two dimen-
1 sionalmasslessscalarfieldsinagravitationalbackground. WefindthattheSchwingertermsarenot
sensitive to the regularization procedure and that theyare independentof the background metric.
n
a
PACS numbers: 04.62.+v, 11.30.-j
J

6
1

I. INTRODUCTION
2
v
8 The theory of a (quantized) scalar field coupled to gravity has to follow an ad-hoc prescription: the functional
3 integration over the scalar field φ involves the evaluation of a determinant of the Laplace operator, which is ambigu-
2
ous. For massless scalar fields in two-dimensional space-time the standard prescription implements a diffeomorphism
0
invariant regularization that leads to the well known Polyakov action [1] ΓP[gµν], a functional of the background
1
7 metric gµν that is indeed diffeomorphism invariant but has an (equally well known) anomaly with respect to Weyl
9 transformations.
/ Recently an alternative evaluation of the theory has been given, where a Weyl invariant regularization has been
h
implemented [2–5]. The resulting effective actionΓˆ[gµν], while being Weyl invariant,doesnot remaininvariantunder
t
- general coordinate transformations,but only under those with unit Jacobian.
p
Gravitational and Weyl anomalies lead to anomalous contributions to the equal-time commutators of the energy-
e
h momentum tensor [6,7] (see also [8] for the analogous fact in current algebra). So the question arises whether these
: two versions of the theory lead to the same anomalous commutators. In this paper we investigate this question and
v
find that, indeed, the anomalous commutators coincide in both versions of the theory and lead to the well known
i
X result from Conformal Field Theory [9]. We do this calculation both for flat and curved space-time. In the latter
r case of general metric the computation is done without any gauge fixing; this is the proper procedure because gauge
a
fixingwouldbeinconflictwiththeWeyl-invariantregularization,thatbreaksdiffeomorphisminvariance. Theresults,
when properly interpreted, lead to the same Schwinger terms as in the flat space-time and, therefore, show that the
Schwinger terms do not depend on the curvature.
II. DIFFEOMORPHISM-INVARIANT AND WEYL-INVARIANT REGULARIZATIONS
First we have to fix our conventions. We use the flat Minkowskian metric η with signature (+,−). The metric
ab
g (x) is related to the zweibein via
µν
∗ThisworkissupportedinpartbyaSchr¨odingerStipendiumoftheAustrianFWF,anExternalScholarshipfromCONICET,
Argentina,andfundsprovidedbytheU.S.DepartmentofEnergy (D.O.E.)undercooperativeresearch agreement #DF-FC02-
94ER40818.
†E-mail address: [email protected], [email protected]
‡E-mail address: [email protected], [email protected]
1
g (x)=η ea(x)eb(x); (1)
µν ab µ ν
we also need the zweibein determinant
e(x):=detea(x)= |detg (x)| (2)
µ µν
q
and the inverse zweibein Eµ(x),
a
Eµ(x):=η gµν(x)eb(x). (3)
a ab ν
For the curvature we use the sign convention R = −∂ Γα +..., where R is the Ricci tensor and Γα is the
µν α µν µν µν
Christoffel connection.
Weyl transformations act like
g (x)→exp(2σ(x))g (x), ea(x)→exp(σ(x))ea(x). (4)
µν µν µ µ
When the effective action Γ is not invariant under Weyl transformations, an infinitesimal change δWg (x) =
σ µν
2σ(x)g (x) induces a Weyl anomaly GW(x):
µν
δWΓ:= d2xσ(x)GW(x), (5)
σ
Z
δΓ
GW(x)=−2gµν(x) =−e(x)gµν(x)T (x), (6)
δgµν(x) µν
where T is the v.e.v. of the energy momentum tensor Θ ,
µν µν
2 δΓ
T (x)=hΘ (x)i= . (7)
µν µν e(x)δgµν(x)
Under an infinitesimal coordinate transformation (diffeomorphism) δDxµ =−ξµ(x) the metric and zweibein trans-
ξ
form like
δDgµν(x)=−Dµξν(x)−Dνξµ(x), δDea(x)=ξλ∂ ea(x)+ea(x)∂ ξλ (8)
ξ ξ µ λ µ λ µ
and a diffeomorphism anomaly is given as
δDΓ:= d2xξν(x)GD(x), (9)
ξ ν
Z
1 δΓ
GD(x)=2e(x)Dµ =e(x)DµT (x). (10)
ν (cid:18)e(x)δgµν(x)(cid:19) µν
It will be convenient later on to use covariant derivatives acting on the combination eT , using the rule eD =
µν α
(D −Γλ )e. Thus we rewrite GD as
α αλ ν
GD(x)=(Dµ−gµρΓλ )(e(x)T (x)). (11)
ν ρλ µν
Further we will frequently use the following variational formulae,
δgµν(x)
=−η ec(x)(gµα(x)gνλ(x)+gνα(x)gµλ(x))δ(2)(x−y), (12)
δea(y) ac λ
α
δe(x) 1
=− e(x)g (x)δ(2)(x−y), (13)
δgµν(y) 2 µν
δR(x)
=[R (x)+(D D −g 2) ]δ(2)(x−y), (14)
δgµν(y) µν µ ν µν (x)
2
where R is the curvature scalar and R is the Ricci tensor.
µν
The classical action of the theory reads
e(x)
S = d2x gµν(x)∂ φ(x)∂ φ(x). (15)
µ ν
Z 2
When a diffeomorphism invariant path integration with respect to φ is chosen, one obtains the Polyakov effective
action [1]
1
ΓP[gµν]=− d2xd2ye(x)R(x)2−1(x,y)e(y)R(y), (16)
96π Z
where 2−1(x,y) is the scalar symmetric Green function of the covariant Laplacian (satisfying 2 2−1(x,y) =
(x)
e−1(x)δ(2)(x−y)). ΓP is diffeomorphism invariant,
GD(x)=0, (17)
ν
and possesesthe wellknown Weyl anomaly (for a comprehensivereview, see for instance [10]and references therein),
1
GW(x)=− e(x)R(x). (18)
24π
The alternative, Weyl invariant evaluation that was discussed in [2–5] relies on the observation that the classical
action (15) depends only on the Weyl invariant quantity γµν, where
1
γµν(x)=e(x)gµν(x), γ (x)= g (x). (19)
µν µν
e(x)
AsthebreakingoftheclassicalWeylinvarianceinPolyakov’spathintegrationmaybetracedbacktoadiffeomorphism-
invariant and Weyl non-invariant normalization for the path integral measure,
Dφexp(i d2xe(x)φ2(x))=1, (20)
Z Z
the Weyl invariant evaluation can be achieved by choosing instead
Dφexp(i d2xφ2(x))=1. (21)
Z Z
ThisleadstoaWeyl-invarianteffectiveactionΓˆ[gµν]whichdepends ongµν(x) onlythroughthe combinationγµν. By
construction the two effective actions ΓP and Γˆ coincide for metrics with unit determinant, therefore
1
Γˆ[gµν]≡ΓP[γµν]=− d2xd2yRˆ(x)2−1(x,y)Rˆ(y), (22)
96πZ
where Rˆ(x) is the curvature scalar evaluated from γµν (notice that Rˆ(x) is not a true scalar).
Γˆ is Weyl-invariant,butitacquiresananomalyundercoordinatetransformationswithJacobiannotequaltounity.
This anomaly may actually be easily computed from the Weyl anomaly of the Polyakov action. The v.e.v. of the
energy-momentum tensor computed from Γˆ is
2 δΓˆ δΓˆ δΓˆ
Tˆ (x)= =2 −γ γαβ
µν e(x)δgµν(x) δγµν(x) µν δγαβ(x)
1
=TP(γ)− γ γαβTP (γ). (23)
µν 2 µν αβ
Here TP(γ) is the energy-momentum tensor TP , as computed from the Polyakov action, evaluated at gµν = γµν.
µν µν
Obviously, there is no Weyl anomaly, gµνTˆ =0.
µν
In order to evaluate the diffeomorphism anomaly we need the identity D (gµνTˆ )= 1Dˆ (γµνTˆ ), which may be
µ να e µ να
easilyprovenbyusing the tracelessnessandsymmetryofTˆ (hereDˆ is the covariantderivativeforthe metricγµν).
µν µ
We then find for the diffeomorphism anomaly
3
GˆD =eD (gµνTˆ )
α µ να
1
=Dˆ γµνTP (γ)− γµνγ γβδTP(γ)
µ(cid:18) να 2 να βδ (cid:19)
1
=− Dˆ (γβδTP(γ))
2 α βδ
1
=− ∂ Rˆ. (24)
α
48π
Here we have used the vanishing of the diffeomorphismanomaly for ΓP and the fact that Dˆ reduces to the ordinary
α
derivative on scalars. The anomaly is a pure divergence because only the symmetry with respect to transformations
with non-unit Jacobian is broken (see [3]).
III. SCHWINGER TERMS
Inthissectionwewanttorelatetheanomaliesoftheprevioussectiontotheequal-timecommutators(ETCs)ofthe
energy-momentumtensor,bothinflatandcurvedspace-time. Herewewillfollowamethodthatwasdevelopedin[11]
and used there for the calculation of ETCs in the flat space-time limit. We want to find the Schwinger terms in the
generalcaseofa nonflatspace-time,too, whichmakesthings slightlymore complicated. We choosethe hypersurface
x0 =0 as a quantization surface. For ETCs we write
δ(x0−y0)[e(x)Θµ(x),e(y)Θν(y)]=Θµν(x,y)+Sµν(x,y), (25)
a b ab ab
where we have used the zweibein formalism in order to conform with [11] (i.e. µ, ν are space-time indices whereas a,
b are Lorentz indices). In eq. (25) Θµν is the canonical part, depending again on the regularized energy-momentum
ab
operatorsΘµ(x), whereasSµν arec-numbers(the Schwingerterms). Intheflatcaseregularizationmeansjustnormal
a ab
ordering, and therefore the v.e.v. of eq. (25) arises only from Sµν in the r.h.s. In the general case this is no longer
ab
true [12] but our knowledge of the flat case will still enable us to identify the individual pieces.
In the flat case it is well known that the canonical part is proportional to the first spatial derivative of the delta
function, e.g. Θ00(x,y) ∼ i(Θ0(x)+Θ0(y))δ(x0 −y0)δ′(x1 −y1), whereas the Schwinger term is proportional to a
01 0 0
triple spatial derivative, S00(x,y)∼cδ(x0−y0)δ′′′(x1−y1) (c is a constant).
01
In the generalcase both the expression for the classicalenergy-momentum tensor (see (15)) and the regularization
will introduce a dependence on the metric and its derivatives in eq. (25). However, we will assume that the number
of derivatives on the delta function remains unchanged, i.e. we will continue to identify the δ′′′ piece of the v.e.v.
of eq. (25) with the Schwinger term. By treating the deviation from the flat space-time action (15) as interaction,
S =S[gµν]−S[ηµν], Γ=−iln<0|T∗expiS |0>=−ilnZ =−iln<out|in> we find for the two point function
I I
δ2Γ
−i = <out|T∗(e(x)Θµ(x)e(y)Θν(y))|in>
δea(x)eb(y) a b
µ ν
− <out|e(x)Θµ(x)|in><out|e(y)Θν(y)|in>
a b
1δ((e(x)Θµ(x))
+ <out| a |in>
i δeb(y)
ν
:=Tµν(x,y)+Ωµν(x,y), (26)
ab ab
where Tµν(x,y) is the connected, time-ordered two-point function
ab
Tµν(x,y)=<out|T(e(x)Θµ(x)e(y)Θν(y))|in>
ab a b
−<out|e(x)Θµ(x)|in><out|e(y)Θν(y)|in> (27)
a b
and Ωµν contains the remaining pieces and is local (i.e. proportional to δ(x−y) and derivatives thereof).
ab
Now we want to relate this two-point function to functional derivatives of the anomalies in eqs. (6,10). Defining
these functional derivatives as
δGD(x)
Iα(x,y):=−iEµ µ , (28)
ab a δeb(y)
α
δGW(x)
Πα(x,y):=−i , (29)
b δeb(y)
α
4
we find the relations
−Iα(x,y)+Aα (x,y)=(D −Γλ ) (Tρα(x,y)+Ωρα(x,y))
ab ab ρ ρλ (x) ab ab
=S0α(x,y)+(D −Γλ ) Ωρα(x,y) (30)
ab ρ ρλ (x) ab
and
Πα(x,y)+Bα(x,y)=ea(x)Ωµα(x,y). (31)
b b µ ab
Here Aα (x,y) and Bα(x,y) stem from variations of the anomalies (6,10) that do not vary the one-point function
ab b
e(x)T (x) (e.g. Bα(x)(x,y) = − δgµν(x) e(x)T (x)). They produce δ functions and first derivatives thereof and
µν b δeb(y) µν
(cid:16) α (cid:17)
vanish in the flat limit. They are unimportant in the sequel. Further, we have assumed in eqs. (30,31) that the
anomalies of the Heisenberg operators Θa are themselves c-numbers. Under this assumption the anomalies do not
µ
contribute to the connected two-point function, e.g. < T ((DµΘ (x))Θ (y)) > = 0. (Here we slightly differ in
µν αβ c
the conventions from [11]. They treat the operator Θµ(x) as an interaction picture operator and, therefore, obtain
a
additional commutators [Θ0(x),L (x0)] in their relations.)
a I
As we use the zweibein formalism, we need the corresponding equation for the Lorentz anomaly, even though the
lattervanishesinbothregularizationsofourtheory. UnderinfinitesimalLorentztransformationsthezweibeinchanges
as
δLea =−αaeb, (32)
α µ b µ
inducing a variation of the effective action
δLΓ:= d2xαabGL (x) (33)
α Z ab
where
1 δ δ
GL (x)=− (η ec −η ec )Γ. (34)
ab 2 ac µδeb bc µδea
µ µ
Then, defining
δGL (x)
Lα (x,y):=−i ab (35)
cab δec(y)
α
we find a further set of equations
Lα (x,y)+Cα (x,y)=η edΩµα(x,y)−η edΩµα(x,y) (36)
cab cab cd µ ab ad µ cb
(where Cα is irrelevant, analogous to the above A and B).
cab
Next we need the explicit expressions for the functional derivatives of the anomalies (Lα being zero in both cases
cab
of interest). For the Polyakovaction ΓP we have GD =0 and
δGW(x) 1 δ(e(x)R(x))δgµν(z)
=− d2z (37)
δeb(y) 24πZ δgµν(z) δeb(y)
α α
1
= η ec(y)(gµαgνλ+gµλgνα) (D D −g 2) δ(2)(x−y),
24π bc λ (y) µ ν µν (x)
whereas for the Weyl-invariantly regularizedeffective action Γˆ we find GˆW =0 and
δGˆD(x) 1 δRˆ(x) δγρσ(z) δgβδ(z′)
λ =− ∂x d2zd2z′
δeb(y) 48π λZ δγρσ(z)δgβδ(z′) δeb(y)
α α
e(y) 1
= (δρδσ − g gρσ) η ec(y)(gβαgδǫ+gδαgβǫ)
48π µ ν 2 µν (y) bc ǫ (y)
×∂x(Dˆ Dˆ −γ 2ˆ) δ(2)(x−y). (38)
λ ρ σ ρσ (x)
Now the procedure of [11] for evaluating the Schwinger terms S0α consists in expanding all the local functions of
ab
eqs. (30,31,36) into derivatives of δ functions, e.g.
5
Iα(x,y)= Iα(k,n−k)(x)∂k∂n−kδ(2)(x−y). (39)
ab ab 0 1
Xn,k
The index k = 0,···,n counts the number of time derivatives, while n−k counts space derivatives. In particular,
S0α(x,y) has only spatial derivatives of δ functions,
ab
S0α(x,y)= S0α(n)∂nδ(2)(x−y). (40)
ab ab 1
Xn
Thus, one obtains a system of linear equations for the unknown coefficient functions S0α(n) and Ωµα(k,n−k).
ab ab
First let us briefly review the flat space-time computation that was done in [11] (they used it for chiral fermions,
too, where diffeomorphism and Weyl anomalies are present). In this case all derivatives only act on the δ functions.
Therefore the explicit expression analogous to (38) for Iα contains only terms with three derivatives, and the corre-
ab
sponding expression (37) for Πα only terms with two derivatives. Further, the covariant derivative in eq. (30) turns
a
into an ordinary derivative. As a consequence, the resulting system of equations may be solved separately for each
fixed number of derivatives (n derivatives for I, S and n−1 derivatives for Π, Ω); for each fixed n the number of
unknowns S0α(n) and Ωµα(k,n−k) equals the number of equations. As only Πα(k,2−k) and Iα(k,3−k) are non-zero, one
ab ab a ab
finds a non-zero result only for S0α(3), Ωµα(k,2−k) (even in the non-flat case, we will only consider the coefficient of
ab ab
the triple derivative of the Schwingerterm, therefore we dropthe superscript(3)). Eliminating the Ωs, one arrivesat
the flat space result
S0α =−Iα(0,3)−Iα(1,2)−Iα(2,1)−Iα(3,0)−Πα(1,1), (41)
0b 0b 1b 0b 1b b
S0α =−Iα(0,3)−Iα(1,2)−Iα(2,1)−Iα(3,0)−Πα(0,2)−Πα(2,0). (42)
1b 1b 0b 1b 0b b b
These equations we have to evaluate for the two versions ΓP and Γˆ of our theory in the flat limit. In the first case
only Πα are non-zero, in the second case only Iˆα. Both versions lead to the same Schwinger terms,
b ab
S00 =S00 =0, (43)
00 11
i
S00 =S00 = . (44)
01 10 12π
For the Weyl anomaly this result was in fact already computed in [11] (we differ in signs because of different metric
and curvature conventions). For the diffeomorphism anomaly we find the same result, showing that the Schwinger
terms are not sensitive to the regularizationprescription.
Nextwewanttodiscussthecaseofgeneralmetric. Inthiscaseonehascovariantderivativesineqs.(30,37,38),and
therefore the system of equations (30,31,36)mixes different number of derivatives. However, Iα and Πα still contain
ab b
at most three and two derivatives, respectively, acting on δ functions. If one also assumes that Ωαµ contains at most
ab
two derivatives (which is a very reasonable assumption, as all diagrams contributing to <T(e(x)Θa(x)e(y)Θb(y))>
µ ν
areat mostquadraticallydivergent),it stillholds that the subsystemofequations containingthe maximalnumber of
derivatives (three for I, S and two for Π, Ω) may be solved separately.
This system of equations is a little bit more complicated and leads again to the same solution for both the Weyl
anomaly of ΓP or the diffeomorphism anomaly of Γˆ. The coefficients of ∂3δ(2)(x−y) in the Schwinger terms read
1
i e0e1
S00 =S00 =− 1 1 ,
00 11 6π(g )2
11
i (e0)2+(e1)2
S00 =S00 = 1 1 , (45)
10 01 12π (g )2
11
and (defining κ= i )
12πe(g11)3
S01 =κ(−e0e0g g −e0e1eg +(e0)2((g )2+e2)+2e0e1eg ),
00 0 1 01 11 0 1 11 1 01 1 1 01
S01 =κ(e0e1g g +e1e1eg −e0e1((g )2+e2)−2(e1)2eg ),
01 1 0 01 11 0 1 11 1 1 01 1 01
S01 =κ(e0e1g g +e0e0eg −e0e1((g )2+e2)−2(e0)2eg ),
10 0 1 01 11 0 1 11 1 1 01 1 01
S01 =κ(−e1e1g g −e0e1eg +(e1)2((g )2+e2)+2e0e1eg ).
11 0 1 01 11 1 0 11 1 01 1 1 01
(46)
6
Although some components look rather ugly, this result is precisely what one expects, as we want to discuss now.
Let us transform Sµα to pure space-time indices via
ab
Sµ′α′ =EνEβgµ′µgα′αS . (47)
ab a b µναβ
Notice that we cannot invert this relation because we do not know all the components of Sµα. However, due to the
ab
symmetriesS =S =S ,S actuallyconsistsofsix independentcomponents. Theexpressions(45,46)
µναβ νµαβ αβµν µναβ
for Sµ′α′ lead to five independent equations for S . Therefore we are able to express all components of S in
ab µναβ µναβ
terms of one unknown function Λ, where the form of Λ is restricted by the requirement that all S tend to their
µναβ
well known Minkowski space version in the flat limit. We obtain
4ie3g g 8ie3(g )3 (g )5
S = 00 01 − 01 + 01 Λ,
0000 12π(g )3 12π(g )4 (g )4
11 11 11
ie3g 4ie3(g )2 (g )4
S = 00 − 01 + 01 Λ,
0001 12π(g )2 12π(g )3 (g )3
11 11 11
2ie3g (g )3
S =S =− 01 + 01 Λ,
0101 0011 12π(g )2 (g )2
11 11
ie3 (g )2
S =− + 01 Λ,
0111 12πg g
11 11
S =g Λ, (48)
1111 01
where Λ may be non-zero (but finite) in the flat limit.
For a proper interpretation of this result we need some basic facts about canonical quantization in curved space-
time. We chose the hypersurface x0 =const as a quantization surface. The direction of the (arbitrarily chosen) time
coordinate is not an intrinsic property of this surface, and, therefore, time components of tensors are not invariant
under coordinate transformationsthat do not change the coordinates on the hypersurface. Instead one has to choose
the projection of the time components onto the timelike vector lµ orthogonal to the surface, e.g. (T is a general
µν
tensor, i is the space index)
T →T , lµT , lνT , lµlνT (49)
µν ij µj iν µν
(see e.g. [13]). The vector lµ is given by
lµ =eg0µ. (50)
Here we chose the normalization lµl = −g , which is the proper normalization in order to obtain the correct
µ 11
commutator algebra on the quantization surface, see e.g. [6,14] (this normalization corresponds to the requirement
that lµ is a vector, not a vector density: for a general tangent vector bµ to the hypersurface, the orthogonalcovector
1
l is l = ǫ¯ bν, where ǫ¯ = eaebǫ = eǫ is a tensor. For our specific choice bµ = δµ one finds precisely (50) for
µ µ µν 1 µν µ ν ab µν 1 1
l ). Further we should remember that Sµν was defined as the commutator of [e(x)Θµ(x),e(y)Θν(y)] (see eq. (25)),
µ ab a b
i.e. to obtain the commutators of the Θµ themselves we still have to divide by e2. Doing so, and performing the
a
projections, we recover precisely the central extension of the Virasoro algebra [15]
lµlνlαlβS =lµlνS =lµlαS =0 (51)
µναβ µν11 µ1α1
i
e−2(x)lµlνlαS =e−2(x)lµS = (52)
µνα1 µ111 12π
and the arbitrary function Λ cancels out in all expressions (51, 52). The pure space component S =g Λ, which
1111 01
is not related to any symmetry generator, remains undetermined by our procedure.
IV. CONCLUSIONS
We have analyzed the anomalous Schwinger terms in the equal-time energy-momentum tensor algebra in two
different regularizations of 2-d scalar field theory in a curved background.
7
The usual computations make use of the conformal gauge, which is of course appropriate for the diffeomorphism-
invariant regularization. Once the metric is set to its conformally flat form, all the machinery of Conformal Field
Theory can be applied essentially as in flat space-time [16]. In contrast, the gauge fixing can not be performed in
the Weyl-invariant version of the theory. In order to compare both regularizations one then needs a more general
framework, in which no gauge fixing is made at any step.
Inthis frameworkwe haveachievedatwo-foldresult. Onthe one hand,wehaveshownthatthe energy-momentum
operatorscontinueto obey the Virasoroalgebrain the caseofa generalmetric, without using any gauge fixingfor the
computation. On the other hand, we have proven that both versions of the theory, eq. (16) and eq. (22), obey the
same commutation relations, regardless of the symmetries broken by the regularizationprocedures.
Acknowledgements: The authors are grateful to Prof. Roman Jackiw for suggesting the problem and for guidance
throughout the work. C.A. is supported by a Schr¨odinger stipendium of the Austrian FWF. G.L.R. is partially
supported by CONICET, Argentina. This work is supported in part by funds provided by the U.S. Department of
Energy (D.O.E.) under cooperative research agreement #DF-FC02-94ER40818.
[1] A.M. Polyakov, Phys.Lett. 103, 207 (1981).
[2] D.R.Karakhanyan,R.P. Manvelyan,and R.L. Mkrtchyan,Phys. Lett. B 329, 185 (1994).
[3] R.Jackiw, hep-th/9501016 (unpublished).
[4] G. Amelino-Camelia, D. Bak, and D. Seminara, Phys. Lett. B 354, 213 (1995).
[5] J. Navarro Salas, M. Navarro, and C.F. Talavera, Phys. Lett.B 356, 217 (1995).
[6] C. Teitelboim, Phys. Lett. B 126, 41 (1983).
[7] M. Tomiya, Phys. Lett. B 167, 411 (1986).
[8] R.Jackiw,in“CurrentAlgebraandAnomalies”,S.Treiman,R.Jackiw,B.Zumino,andE.Witten,(PrincetonUniversity
Press/World Scientific,Princeton, NJ/Singapore, 1985).
[9] P. Ginsparg, Introduction to Conformal Field Theory, in “Fields, Strings and Critical Phenomena”, Les Houches 1988
(Elsevier, Amsterdam 1989).
[10] M.J. Duff,Class. Quant.Grav. 11, 1387 (1994).
[11] M. Ebner, R.Heid, and G. Lopes Cardoso, Z. Phys. C 37, 85 (1987).
[12] M. Bos, Phys. Rev.D 34, 3750 (1986).
[13] P.A.M. Dirac, Phys.Rev. 114, 924 (1959).
[14] C. Teitelboim, in “Quantum theory of gravity”, ed.S.M. Christensen (Adam-Hilger Ltd.,Bristol, 1984).
[15] M.A. Virasoro, Phys. Rev.D 1, 2933 (1970).
[16] D.Friedan, E. Martinec and S.H. Shenker,Nucl. Phys. B 271, 93 (1986).
8