Beltrami parametrization and string theory

Volume 196, number 2 PHYSICS LETTERS B 1 October 1987

BELTRAMI PARAMETRIZATION AND STRING THEORY

L. BAULIEU and M. BELLON Laboratoire de Physique Th4orique et Hautes Energies, Universitb Pierre et Marie Curie, Tour 16, 4, place Jussieu, F- 75252 Paris Cedex 05, France

Received 22 June 1987

By introducing Beltrami differentials to describe the conformal structure of the worldsheet, we prove the factorization of two- dimensional diffeomorphisms and of the corresponding BRS symmetry which is relevant for the string theory. Weyl invariance appears as a fundamental symmetry of the problem. Factorization of the connected tree diagrams is proven.

In a previous work [ 1 ], a class of gauge fixed action for the tree bosonic string has been built from the requirement of the existence of a nilpotent Slavnov symmetry. In this approach, Weyl symmetry is sup- posed to be a part of the basic gauge symmetry on the parameter space as well as bidimensional diffeomorphisms. This yields a consistent gauge fixing of the scale part of the metric. The properties of the corresponding theory and of its stress-energy tensor are then determined by the Slavnov identity stemming from the underlying Weylxdiffeomorphism symmetry [ 1,2].

An important feature ofconformally invariant two- dimensional field theories in view of their applica- tion to string theory is the so-called factorization property [3]. This property has been mainly con- sidered in the conformal gauge. In this note we show that factorization is more geometrical in nature and can be proven for non-trivial backgrounds. A key ingredient in our work is the use of Beltrami differentials to parametrize the conformal classes of the metric. By using this parametrization, a surprisingly simple description of the BRS structure for bidimensional diffeomorphisms appears, A rather ele- gant construction of gauge fixed actions is obtained for the string theory in the critical dimension as well as for the Liouville theory [ 4], using the postulates of locality on the wordsheet together with local Weyl and diffeomorphism invariance. The factorization is explicit at every steps of our construction, despite the fact that we work in non-trivial backgrounds. It

is convenient to use a zweibein formalism which allows for the extension to the fermionic case.

The gauge symmetries we want to explore are the Weyl one- and two-dimensional diffeomorphisms, which leave the classical string action invariant:

J c l - = - f d 2 x d ~ g a f l O a x ' O f l X , ( 1 ) z

Z can be parametrized by local complex coordinates z and Z ~1 to make it into a Riemann surface. We will freely use + (respectively-) to denote the index z (respectively Z). Let e ± denote both components of the zweibein. The two-dimensional orthogonal algebra has only one generator and thus the spin connection is described by the single one-form o2. The torsion is defined to be T e = de e + o2 e ±. Imposing the constraint of zero torsion determines o2 as a function of e ± and thus also the curvature R=dco(e) =R(e)e+e -. However, we must keep in mind that the zweibein is not globally defined as a one-form. The zweibeins on two patches of the surface are related by a local rotation.

The action of the gauge symmetries can be expressed in term of the nilpotent BRS operators, which can be geometrically built [ 5 ]

sX=~O~X, s ¢ ~ = ~ 0 e ~ ~ , (2)

~ We shall use the euclidian frame work throughout this paper. It is straightforward to convert it to the lorentzian framework, with the orthogonal group replaced by the Lorentz group.

142 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)


s ~ = ~ 0 ~ £ 2 , sC=~'~O,~C,

se ± =£2e ± + ce ± + L ¢ e ± . (2 cont'd)

By assumption, s anticommutes with d and dx. L~ = i¢d- die is the Lie derivative along the anticommuting ghost vector field ~. (2 is the scalar ghost for Weyl symmetry and c the one for orthogonal trans- formations. From the condition T-+= 0, the transformation law of og(e) is

so(e) = *dI2+dc + L¢o)( e) . (3)

The operator *d means * d = e + @ + - e - ~ _ , if d is expressed as d = e + ~ + + e 9_ .

The Beltrami parametrization consists into the following expression for the zweibein:

e ± = 2 ± E ± , (4)

with

(EE_+)--(It 1_ # i ) ( c ~ x + ) , (5)

It is called a Beltrami differential. One has therefore

1 _ - i t + ) ( E + ) . (6) ( x + ) , , This yields the following factorized expression of the line element ds2:

ds 2 = 2 + 2 - E + " E -

= 2 + 2 - ( d x + + i t + d x - ) • (dx + +It+ d x - ) . (7)

The symbol • stands for the symmetrical product of forms. When this symbol is omitted, the product is the exterior product. In order for the metric to be real, 2 + and 2- must be complex conjugate of each other, as well as It+ and It . The metric is non-singular if and only if IIt]2~ 1. With the additional hypothesis of the orientability of the surface/7, It must be restricted to the unit disk, I It 12 < 1. This change of variable leads to the introduction of the operators D_+ from

d = E + D + + E - D . (8a)

This yields

( D + ) = 1 ( 1 - i t ) ( ) - 0+ (8b) D_ 1 - i t + i t - - I t + 1 0_ "

In the ghost sector, the ghost fields ~ must be rede- fined into ,~, with

(~_+) = (#1_ I t ; ) ( ~ _ + ) , (9)

so that ~-0 = S . D. Observe that one has ~ + = ice ±. This leads to the compatibility of the Beltrami parametrization with the general BRS formalism of ref. [51.

In terms of these new variables, the condition of vanishing torsion means that

dE ± + [d in2 +- +o) (e ) ]E ± =0. (10)

This implies

o ) ( e ) = o ) ( E ) + E + D + l n 2 - - E - D _ l n 2 + , (11)

where re(E) is defined by T ± ( E ) = d E ± + o ) ( E ) E ± =0.

The BRS equations for the zweibein and the diffeomorphism ghost ~ can be set under the unified form

( s + d ) ( e ± +ice ±) + ( +co + c+£2)(e +- + ice +) = 0 .

If one expresses this equation in term of the Beltrami variables It and ~, one finds

( d x - O_ + s ) ( # + d x - + ~ + )

- ½[#+dx- + 3 + , # + d x - + ~ + ] + O,

(dx+O+ + s ) ( # - dx + + S - )

- ½[#- dx+ + 3 - , I t - d x + + .V-]_ = 0 . (12)

The graded brackets [ , ] ± are defined by [a, b] ± = a O ± b - b O ± a i f a and b commute, and by [a, b ] ± = a O ± b + b O ± a if a and b anticommute. The expansion of eq. (12)' in ghost number yields the following transformation laws for It and ~:

sg + =O_Z, + + [ Z + , # + ] +

= ( 0 - # + 0 + ) Z + + Z + 0 + I t + ,

sg- =0 +3 , - + [ ~ - , i t ]_

= ( 0 + - # - 0 _ ) 3 - + ~ - 0 # - ,

s-V + = ½ [ ~ + , S + ] + = . V + O + Z + ,

s Z - = l [ Z - , z - ] _ = z - o _ z . (13)

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It is convenient to define q~ =ln , ~ , although it is not a globally defined scalar:

q5 =ln 2+2 - + l n ( 1 - I#l 2) . (14)

sX, s ~ and sO can be rewritten as follows:

s X = ~ - D X , s ~ = 2 0 + 0 - ~ + ~ . D q b ;

s O = S . D O . (15)

The nilpotency of s is obvious from eqs. (12), (15). Eq. (13) shows that the BRS symmetry of bidimensional diffeomorphisms can be split in a Weyl invariant way in two independent sectors. Technically this has been achieved by the change of variables that we have called Beltrami parametrization. As we shall show, this implies that left movers remain independent of the right movers in any background gauge. It is also a consequence of the fact that # is a para- metrisation of the null directions of the metric g.

In order to quantize the Polyakov action, we must complete the BRS symmetry by the introduction of the antighosts ~ and ~ associated respectively with E and O and the auxiliary fields b + and b:

s ~ + = b -+, sg]=b , sb -+=0 , s b = 0 . (16)

To describe the geometry of the worldsheet, we have thus been led to introduce the following quartets of fields:

#-+ q~ ." \ ." \

E +- ~± ~ /2 . \ . . " \ . . "

b -+ b

(17)

The BRS symmetry is defined in eqs. (13), (15), (16). When writing orthogonal invariant quantities, the ghost for local rotations c will disappear and it will not be relevant for our purpose.

To define our gauge fixing in the Weyl sector and eventually build the consistent anomaly, we need a reference metric go. We choose go compatible with the complex structure

go =exp[q)o(X + , x - ) ] dx+ d x - , (18)

and invariant under the BRS symmetry (sq~o=0). The classical string action on ~ can be rewritten

a s

Jc, = f d Z x ( 1 - # + # - ) D + X-D_ X= f X'cf*dX X X

= f d 2 x ( 1 - # + # - ) -I

× ( 0 + - # - 0 ) X . ( 0 _ - # + 0 + ) X , (19)

while both non-vanishing components of the classical stress-energy tensor are

O~ =8~¢~1/~#- = D+ X 'D+ X

= 1 ( 1 - # + # - ) 1(0-+ -#_v O-v)Xl 2 (20)

Following Polyakov's approach and assuming tha t the fundamental gauge invariance is Weyl N Diff, the gauge fixing procedure amounts to give arbitrary values for the components of the metric. This must be done in a BRS-invariant way. Background gauges are obtained by adding to J d the following BRS invariant action:

Y~v = f d 2x s [~+ (# + - # ~ ) + ~,- (/2- -#6- ) 27

+f2(4,- q,o)]

= f d 2 x [ b + ( # + - # ~ - ) + b - ( # - - # o - )

+ b ( ~ - 40) + g ] ( 2 0 + 0 - ~ + ~ . D ~ )

+ ~ + ( 0 _ - # + 0 + ) Z + + ~ + ~ + 0 + # +

+,~ ( 0 + - # - 0 _ ) ~ - + ~ . - ~ - 0 _ # - 1 . (21)

#+ and q~o determine a background that can be chosen at will. In order for the gauge fixing term to be globally defined, the Lagrange multiplier b + is nec- essarily a quadratic differential. The antighost ~ + must share this property. The equations of motion deduced from the variation of b -+, b, O, g] are algebraic and yield

# - + = # ~ , q~=qOo, t ) = O ,

O= - ½~'Dq)o - ½0.~. (22)

Eliminating these fields through their equations of motion, we end up with the following action:

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y(x,/tg =+ 3 ~) f , ~ - , = d2x(1 - / t~ / t f f ) l X

× (0+ - / t ~ 0 _ ) x . (0 - / t+ 0 + ) x

-} - fd2x[3 + ( O - / t ~ O + ) 3 +-1-3+.- ~+O+/to ~]

q - f d 2 x [ 3 ( 0 + - / t r - 0 _ ) ~ ' , - + 3 - ~ - O _ / t 6 - ] . X

(23)

This action is invariant under the modified nilpotent Slavnov symmetry

s X = ~ ' D o X ,

s/tg = 0 _ ~ + + [ 2 + , / t ~ ] + ,

s / t c = O + ~ + [ ~ - , / t r ] ,

[7 ]. In what follows, we shall skip on this problem. In order to write the Slavnov identity which

expresses the BRS invariance of the action, source terms for the BRS variations of the fields must be added to J :

~'to, = J + J s ,

Ys =~ d2x(X'sX+( +$3+ + ~ - s ~ ) . (25) X

/G is the source for the stress-energy tensor. I f we denote F the generating functional of 1-PI vertices, the Slavnov identity reads

0 = ( "2 / (~P (~F J a x a x) ax(x) X

8F 8F 8F 8F q - - - - i 8 ~ + ( x ) 8¢+(x) 8 ~ - ( x ) 8 ¢ - ( x )

sZ + ~[=+ =+ = - _ , _ 1+ , s Z - = ½ [ S - , ~ - ] . (24)

Do is equal to D, with/t replaced by/to. It is this form of the BRS symmetry which will be used to control the computation of the partition function. Since/to is an external field coupled to the stress-energy tensor, the BRS symmetry (24) defines the corresponding current algebra identities. For/to = 0, the well- known expression of the string action in the conformal gauge is recovered. The remarkable feature is the factorized form of the ghost action and its BRS symmetry, which holds independently of the choice of the background/to-

The equation of motion for 3 + states that 3 + is a holomorphic differential with respect to the complex structure defined by/to. When X is compact of genus g> 1, the Riemann-Roch theorem ~2 shows that there are 3 g - 3 linearly independent solutions for this equation. In order to define the ghost prop- agator in this case, and thereby the functional inte- gral, we must impose that 3 + remains orthogonal to these solutions. In a further work, we shall show that this constraint can be realized by a suitable gauge fixing on the antighost which preserves a BRS invariance and automatically insures that we are inte- grating on the moduli space with the proper measure

~2 See ref. [ 6] for a demonstration.

+ ( o _ z + + [~+ , / t g ])(x) - - 8F

a / t J (x )

_ 8F + ( 0 + S - + [ = - , / t f f ])(x) ~ ) . (26)

The antighost equation of motion can be turned into the functional identity

8FI83 +- = 0 T ~-+ + [.E -+ , / t~ ] + . (27)

By a procedure analogous to that used for ordinary gauge theories (see e.g. ref. [8]), it is possible to invert the Slavnov identity (25 ) together with (27) to recover the action (23) and the symmetry equations (24) up to trivial overall coefficients. In this process dimensionality and covariance under orthogonal symmetry must be used, as well as the invariance under holomorphic reparametrization of X.

Since we have a background type gauge, we remain with a background gauge symmetry acting on the background/to and all the fields. This symmetry is a classical one, and is consistent with the BRS symmetry, i.e. the corresponding Ward identity operator commutes with the Slavnov operator stemming from the BRS symmetry. The most compact way to express this symmetry is by mean of a new anticommuting

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operator 0- and a vector ghost pa ramete r v. One defines

V + = v + +/ t~ v - , V - = v - +/tC v + , (28)

and generalizes eq. (12) into

(dx- 0_ +s+0-)(/t~ dx- +~+ + V +)

- ½ [/t~ d x - + S + + V +, ~ dx- +.~+ + V + ] +

= 0 ,

(dx+ 0+ +s+a)( / t~ dx + +.7- + V-)

- ½ [ / t & d x + + ~ - + V - , / t&dx++. -7 -+V-]_

= 0 . (29)

By expansion in ghost number , this yields the same act ion of s on/to, X, Z, and 2 as in eqs. (13), (15), whereas the act ion of 0- is given by

a X = V . D o X ,

0-/t~ =0_ v + + [ v + , / t ~ ] +

= ( 0 _ - / t ~ 0 + ) v + + v + 0 + / t ~ ,

0-/26- = 0 + V- + [ V - , / t & ] _

= (0+ - / z ~ 0 ) V - + V- 0_/t& ,

0 - V + = V + O + V + , 0 - V - = V - O _ V - ,

a.7 + + s V + = [.7 + , V + ] +

= ~ ~+ 0+ V + + V+ O+~ +

0- and s an t i commute and 0"2=0. Moreover for a suitable choice of the act ion o f 0- on 2 , the act ion J can be proven to be invar iant under 0-. In the case s V= 0 for instance, one must have

0-2 + = V + 0 + 2 + - 2 2 + 0 + V + ,

0 - 2 - = V - 0 2 - - 2 2 - 0 _ V - , (31)

and one can verify that 0-22=0 and (0 - s+s0- )~=0 . It is plausible that the Ward ident i ty associated to the 0--invariance can be used as a subst i tute for the ant ighost equat ion o f mot ion in order to solve com- pletely the Slavnov ident i ty (25) . The determina- t ion o f the theory f rom the only ident i t ies associated with symmetr ies seems preferable.

In our approach based on the pr inciple o f locality, the local anomal ies are the possible obstruct ions to the Slavnov identity. The anomal ies are thus to be de te rmined from the consistency equat ions

s A l = 0 , A21 ,-~zJl 1 + s C ° + d C l . (32)

A 1 is a local functional o f the fields with ghost number 1 which is a two-form. One expects a single solu- t ion o f eq. (32) as a consequence o f the existence o f the existence of a mixed orthogonal × Weyl invar iant four- form 14 = do) dA. A is def ined to be an abel ian gauge field for Weyl symmetry: A is in t roduced only for the purpose o f describing the anomaly since we do not need any gauge field to have local Weyl symmetry. One can, however, give several locally equivalent forms of the anomaly candida te 32 ~ , which are ei ther p ropor t iona l to the Weyl ghost g2 or to the dif- f eomorph ism ghost ~ :

A{ =g2R(e) =g?dco(e) ,

Ai 1 = a R ( E ' ) = a d r o ( E ' ) , (33a)

0 - 3 - + s V - = [ ~ - , v ]_ a = ½ ( a + + a - )

= ~ - O _ V + V - O _ ~ - . (30)

S V can be chosen at will with the only restr ic t ion that s2V=0. There are two natural choices: s V = 0 and sv = 0. The former yields 0-~ -+ = V -+ 0 + S -+ + ~ + 0 + V -+ and the lat ter sV+-=v s/t~ and thus 0 - ~ = v ~ O ~ f f ' + ~Opv ~. The symmetry corresponding to sv= 0 has been used in ref. [2] . B y hypothesis, a commutes with 0 and one can easily check from eq. (28) that

= - ½ ( ¢ - O ~ o + 0 . ¢ + / t + 0+ ¢ - + / t+ 0+ ¢ - ) . (33b)

It is impor tan t that these two expressions of the anomaly are globally defined as two-forms on 22. This is the reason why we make use of the Weyl inert zweibein E ' -+ which is defined by E ' -+ = exp (½ q~o) × E ±. q~o has been in t roduced in eq. (18). Since S ~ o = 0 , one has that sE' -+ =a+-E ' +- +L~E ' +-. On two

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different patches, the different E ' are related by a local rotation: a + and a - are not globally defined, but so is their arithmetic mean a. To verify that A~ and A; ~ are consistent anomalies, one uses that

sR(e) = L c R ( e ) + d *dO,

sR(E ' ) = L c R ( E ' ) + d *da. (34)

To prove that A~ and A~ ~ are equivalent anomalies, we observe that e -+ and E ' -+ are related by a dila- tation, e -+ =e~u+E ' +, so that

½s0,u + + ! u - ) = g 2 - a + ½L~(!u + + ~ u - ) ,

R ( e ) = R ( E ' ) + ½ d * d ( ~ + +~/-).

It follows that

A~ =A~' + I s [ ~ R ( E ' ) - lgtd *dq/] + d C l ,

~u=~t+ + ~ - . (35)

A ~ is useful when we deal with the theory prior to the elimination of the Weyl ghost, A~ ~ is relevant after its elimination.

Inserting either A~ or A' 1 at the right-hand side of the Slavnov identity, one determines the possible anomalous vertices. The coefficient of the anomaly can then be computed by standard techniques (see e.g. ref. [9 ]), through the use of the Feynman rules stemming from the action (25), and is as expected proportional to D - 2 6 (D is the space-time dimension). This means that the Slavnov identity can only be ensured for D = 26.

A simple form of the anomaly can be obtained which depends only on the Beltrami variables. Start- ing from the BRS equations under the unified form (12) and using the identity for an anticommuting c, O(cOc02c) = cOc03c, one gets the following equations:

7~ (8+fi+ O+f i+)=O+[,a+(O+fi+)(O+f i+)] ,

?t + ( o yz - o L Fz - ) = o _ [ ~ - ( ,9 _ yz - ) ( o "-_ Fz - ) ] ,

(36)

where f i - + = i t ± d x ~ + Z + and d + d x ± 0 + + s . The expansion of eq. (36) in ghost number yields

s / d 2 x ( O + 3 +ozI t + ) = 0 , Z

S f d 2 x ( 0 _ ~ - 0z_it-) ---0, 27

s f d x + ('O + 3 + O~+2+ ) = O ,

F

s f d x ( 0 _ 2 - 0 2 _ Z ) = 0 . (37) F

O+,.7,+02it++O_3-O2_it - is locally equivalent to A~ 1 , but is not globally defined. This expression of r the anomaly can be made globally defined by co- variantizing all derivatives 0+ with respect to the holomorphic changes of variables. 0 + ~ + 02+~+ plays a role analogous to the one of the two-cocycle in Yang-Mills theory and can be used to relate the anomaly to a Schwinger-like term.

Factorization is the property that the connected Green funtions with insertions of the stress-energy tensor O ± = 8/6 Itff d only differ from zero when the O -+ have all the same indices. This property can be expressed in term of the generating functional of the connected Green functions Zo(Jx , I t ~ , l d , J_=+, J~.+, J z - , J_=- ), where the J+ stand for the source of the field ~b. Differentiation of Zc with respect to the background/to yields insertions of the stress-energy tensor in connected Green functions. The proof of factorization amounts to demonstrate that

Z~( Jx, t t+ , Itff , J z + , J z + , J s - , J.~- )

= Zc( Jx, it+ , J s + , J s + )

+ Zc( Jx, itff ,J_--= , J z - ) . (38)

Since we deal with a free field theory, it is obvious from the structure of the action (23) that Zc can be separated as follows:

Zc =3( i t~ , Itff ) + Z x ( J x , It+, It~ )

+ Z~.o~,(it~-, ]_--+, J~+) + z~,,o~,(it0-, J__-_, J__-_ ) , (39)

where Z x (respectively Zghos~) is quadratic in the sources ,Ix (respectively Jz). To prove the factorization equation (38), we will have to prove it for the

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part quadratic in ,Ix and the determinant term A(g~, #C ). Zx can be explicitly computed by gaus- sian integration:

Zx=I d2xJx(Z) 22

1 (1 -g~-g f f ) 1 ) × O_ -ggO+ o+ -O_g~ J'~ (z).

(40)

In the course of the computation, it is crucial to take advantage of the identity

(0+ - 0 g - )(1 - g + g - ) - l ( 0 - g + 0+ )

= ( 0 -0+g+)(l-g+g-)- '(0+ -g-o_) . (41)

Differentiating eq. (40) once with respect to g~, and using once more the identity (41), one gets the following expression for 8/SgJ-Zx:

8 - - Z x 8gg(z)

=(0_ t jx)(Z)(o - l Jx)(Z). -O+gff -O+gff

(42)

This expression is independent of gff, and therefore

5 2 - - Z x = 0 . (43) 8g;~Sgff

This equation shows that all tree diagrams involving the stress-energy tensor and string fields are factorized. This property is independent of the space-time dimension.

Now, using either the Slavnov identity stemming from the BRS invariance or the Ward identities stemming from the diffeomorphism invariance a (30) and setting all the sources but go equal to zero, one gets

- g d 0 + - 2 ( 0 + g J - ) ] ~ A = 0 . (44) [0_

This identity implies that (8/8g + )A is a quadratic differential. In particular, eq. (44) means that A is invariant under diffeomorphisms and therefore can be reduced to a function on Teichmtiller space, a

property which is therefore true only if D = 26. As a consequence of the Riemann-Roch theorem, a finite- dimensional basis for a quadratic differential on {~i} can be given, with i running from 1 to 3 g - 3, g being the genus of Z. (~2 /~g~-Sg0- )Z~ can then be expressed as a finite sum:

82 8#g (z)Sg~ (w) A

3g-- 3 = ~ c,j(gff,#ff)q)'(z)CrJ(w). (45)

i j = 1

This expression shows that the breaking of the factorization through the source-independent term of Zc cannot be singular for z--. w, and therefore cannot modify the short-distance expansion of the product O+(z)O-(w).

I f we want to define a consistent theory, i.e. one such that the Slavnov identity (44) can be enforced, for D#26 , we can follow the general method con- sisting in the introduction of an additional field, the gauge variation of which allows for an anomaly-com- pensating counterterm. This is possible for the string theory since the anomaly (33) corresponds to the mixed Weyl × orthogonal invariant four-form/4 = do) × dA. One introduces the Liouville field L charac- terized by the BRS variation

sL=2f2+ LeL=2£2+~.OL . (46)

With the Liouville field L, we can express the consistent anomaly A as an s-exact term:

A~ = O R ( e ) = ½s[LR(e) - ¼Ld *dL] + d C l . (47)

Postulating that the classical action is a function of X and L invariant under the BRS symmetry (15), (17), (46), its expression is as in eq. (19) but can include a new s-invariant term $xe+e-exp(-L):

~J~-Liouville = f d2x( 1 - - g + g - )D+ X'D_X X

+pfe+e e x p ( - L ) , (48)

where p is an arbitrary constant. Owing to eq. (47), there is no candidate for the anomaly and we can consistently fix the Weyl gauge symmetry from the BRS-invariant action (21). After the elimination of all the fields with algebraic equations of motion, we

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get the following action:

~ = f d2x(1 -/Z+/Z6-) -~ X

× (o+ - /z~ o _ ) x . (o_ - / z j o + ) x

+ p f E' +E ' - e x p ( - L ) %-

+ f d2x[.g+ (0_ -/Z~- 0 + ) 3 + +2+-E,+ O+/Z~ - X

+ $ - ( 0 + -/zo 0_)-E- + ~ - S - O _ / Z j ] . (49)

This action, with the only dynamical fields X, L, -E and ~ is still BRS invariant under the symmetry (25), completed by

sL' =2a+ LcL' =2a+~.OL' . (50)

This result is easier to prove if we start from the action (23), where the Weyl sector has been elimi- nated so that the anomaly takes the form (33b). In this case, we introduce a field L ' with the BRS variation (50), with L replaced by L ' . The relevant Wess-Zumino term is then given by

Swz = ½(D-25 ) j" [L'R(E') - ¼L'd *dL] x

= ½ ( D - 2 5 ) f [L 'R (E ' I -~d2x (1 - /Z~ /Z j ) -~ X

from below and for a classical solution for L ' to exist. Besides, L ' contributes in loop computations as one extra component of X, which explains the factor D - 2 5 and not D - 2 6 in front of the Wess-Zumino term.

Finally we can take advantage from the fact that the 2D diffeomorphism BRS symmetry (12) is for- mally identical to that of the Yang-Mills theory with # as a connection and ~ as an ordinary Yang-Mills ghost. This allows a straightforward derivation of the corresponding local BRS structure, which is useful to explore the string BRS-current algebra [ 10,11 ]. One can indeed introduce a commuting one-form o4 with commuting ghost 7. The BRS symmetry for o~ and 7 is defined by s a = d y and sy=0. to define the action of the local BRS on the fields, we just substitute in eq. (12) /z -+ by /Z'-+=#++o~-E -+ and -E-+ by -E'+ =7-E -+. By expansion in ghost number, one gets

s/z + =y(0 -E + -/Z+ O+-E + +-E+O+/Z +

-a_-E+O+-E + ) -/Z+-E+ 0+ 7 ,

s,E + = 7 ~ + 0+-E + . (52)

The local BRS variation of the string field X is obtained by substituting/Z' and 3 ' to/Z and ~ in the expression of sX=-E-DuX. The nilpotency of this local BRS symmetry is obvious by construction. The invariant classical action is obtained by changing/Z into/Z' in Jcl [ 10]. The gauge fixing term is

Jo~ =s[~g+ (# + -/zo ~ +a+/z+-E+)] +c.c.

× ( 0 + - / Z j 0 _ ) L ' - ( 0 - / Z ~ 0 + ) L ' ] . (51)

L ' can b e defined by L'=L-~u . the term rE' +E' -exp( - L ' ) is an invariant term, and we get back the action (49). The elimination of the Weyl sector can be done before or after the introduction of the Liouville field. For/zo = 0, this is the action derived by Polyakov [4]. Our derivation has the advantage to show that the Liouville term fe + e- exp ( - L), although not Weyl invariant, has been built from a Weyl invariant term. In this sense, Weyl invariance is truly an invariance of the Liou- ville theory.

The Liouville term e x p ( - L ) plays no role in the anomaly compensation mechanism. In view of the linear term L 'R(E ' ) from the Wess-Zumino term, it is, however, necessary for the action to be bounded

= b + ( # + - # o ~ +a+/Z+-E +)

- y $ + (o_ 3 + - i/z+, s + ] + - ~ _ 3+ o+-E +

+a+-E+ 0_-E + -2a+ /z+ ,E+ 0+S+) +c.c. (53)

As a conclusion, it is remarkable that Beltrami parametrization allows for a geometrical factorization of the 2D diffeomorphism BRS symmetry. The

introduction of the Weyl symmetry as a part of the basic gauge symmetries of the string theory is important to carry out a Polyakov approach of a sum- mation over metrics. This yields a particularly convenient gauge fixing of string theory for arbitrary background gauge choices and an easy proof of the factorization of the propagating modes. Using argu- ments based on locality, the origin of the anomaly as

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a or thogonalxWeyl anomaly is obvious as well as the derivation of the Liouville action for D ~ 26. After the consistent e l iminat ion of the Weyl sector, one recovers the usual results, the interpretation of which is of course less t ransparent from an algebraic point of view. We have left to further publications a BRS- invariant t reatment of the gauge fixing of the global zero modes of the antighost, as well as the extension of these results to the superstring theory.

We are grateful to O. Babelon and M. Talon for helpful discussions. Most of the material has been elaborated in collaboration with R. Stora. We thank C. Becchi for the communica t ion of his results prior to publication. One of us (L.B.) thanks the CERN theoretical division, where part of this work has been done, for its kind and generous hospitality.

References

[ 1 ] L. Baulieu, C. Becchi and R. Stora, Phys. Lett. B 180 (1986) 55.

[2] C. Becchi, to be published. [3] D. Friedan, E. Martinec and S. Shenker, Phys. Lett. B 160

(1985) 55; Nucl. Phys. B 271 (1986) 93; A. Belavin and V. Knizhnik, Phys. Lett. B 168 (1986) 201; A. Belavin, A. Polyakov and A. Zamolodchikov, Nucl. Phys. B241 (1984) 333.

[4] A.M. Polyakov, Phys. Lett. B 103 (1981) 207. [5] L. Baulieu and M. Bellon, Nucl. Phys. B 266 (1986) 75. [6] E.g.O. Forster, Lectures on Riemann surfaces (Springer,

Berlin, 1981). [ 7 ] L. Baulieu and M. Bellon, in preparation. [8] L. Baulieu and J. Thierry-Mieg, Nucl. Phys. B 197 (1982)

477; L. Baulieu, Phys. Rep. 129 (1985) 1.

[9] L. Baulieu and A. Bilal, Phys. Lett. B 192 (1987) 339. [ 10] L. Baulieu, B. Grossman and R. Stora, Phys. Lett. B 180

(1986) 95. [ 11 ] J.P. Ader and J.C. Wallet, Phys. Lett. B 192 (1987) 103.

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Beltrami parametrization and string theory