Superconformal anomaly and its compensating action

Volume 251, number 1 PHYSICS LETTERS B 8 November 1990

Superconformal anomaly and its compensating action

L. Bau l i eu , M. Be l lon

Laboratoire de Physique Th~orique et Hautes Energies l, Universit~ Pierre et Marie Curie, Tour 16, ler ~tage, 4, place Jussieu, F- 75252 Paris Cedex 05, France

a n d

R. G r i m m

lnstitut t~r Theoretische Physik, Universitdt Fridericiana, Kaiserstrafle 12, P f 6980, 14I-7500 Karlsruhe, FRG

Received 14 August 1990

We construct the superconformal anomaly, and its compensating action, using BRST cohomological methods in superspace. We determine the anomaly under the equivalent expressions of a Lorentz-Weyl anomaly and a diffeomorphism and supersym- merry anomaly. We construct the counterterms which relate both expressions. All fields which are relevant in the anomaly compensating mechanism, in particular the "Liouville" superfield, are related to the two-dimensional geometry. Factorization is explicit at all steps of the construction, due to our introduction of superBeltrami differentials.

1. Introduction

In previous papers, we have related the field var iables relevant in conformal theories [ 1-4 ] to the two-dimensional geometry based on Lorentz and Weyl invariances. In ref. [ 3 ] we have shown that the fields which permi t the compensa t ion of conformal anomalies , possibly lef t - r ight asymmetr ic , can be extracted from the connec- t ions which gauge the Lorentz and Weyl symmetr ies in two dimensions. We have also shown that the non-local W e s s - Z u m i n o two-dimensional gravity act ion [ 5 ], ob ta ined by the e l iminat ion of the compensat ing fields, can be expressed solely in terms o f the Bel t rami differentials, i.e., the sources o f the holomorphic and ant iholo- morphic par ts o f the s t ress-energy tensor. Here we shall extend our results from the bosonic to the supersymmetr ic case, by using the superspace techniques in t roduced in ref. [ 2 ] and reviewed in ref. [ 4 ].

Our approach differs f rom others in that we make consequent use of the Beltrami paramet r iza t ion [2,4] o f two-dimensional superspace geometry [ 6 ]. This allows in par t icular for a concise descr ipt ion o f the factorized superd i f feomorphism anomaly and its compensa t ing mechanism in terms o f BRST cohomological superspace methods. Moreover , componen t field expressions are easily ob ta ined in a construct ive way. We emphasize that our approach is purely geometrical . The basic pr inciples o f the construct ion in the supersymmetr ic case are the same as those o f the bosonic case. Concise use of superspace geometry allows to control efficiently the technical compl ica t ions arising from supersymmetr iza t ion .

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Volume 251, number 1

2. The bosonic case

PHYSICS LETTERS B 8 November 1990

Let us consider a two-dimensional geometry, based on Weyl, Lorentz and diffeomorphism transformations. We choose complex coordinates z and g. Let/25 and/Zz z be the Beltrami differential and its conjugate. The zweibein e has components

eZ=exp(q5 R) (dz+/z~-dg), e~=exp(qh.) (dg+/z~dz) . (1)

exp q~ R and exp eL are the conformal factors in the z (right) and Z (left) directions respectively. The connections w and A for local Lorentz and Weyl transformations are combined into to a = A - w and co L =A + oJ. The curva- tures are

TZ=DeZ=deZ+eZo~ a , TZ=DeZ=DeZ+e*oJ L, (2)

F a = d o J a , FL=doJ L. (3)

d = dx z 8~ + dx z 0z is the exterior derivative. We define it in the same way as in the superspace [ 4 ] for compati- bility with the next sections, and this explains the unusual position of the connection in (2).

To introduce the BRST operator, let g2 g and O L be the ghosts of Lorentz and Weyl transformations, and c the ghost vector field of two-dimensional diffeomorphisms. We define

(_7)R = o)R-F. ~"~ R ' o ~ L = o.)L -F~/~ L , (4)

f i Z = d z + # § d g + c Z ' f iZ=dz+l zZdz+cZ (5)

a n d a = d + s . We have shown in ref. [2 ] that the BRST equations can be written as

fi2 a~== 0oa~z+fi= 0zZZ=0, (6)

~ a = OzfiZ_a(~a+fi2X~" (7)

One has the conjugate equations with z ~ Z and R ~ L . Taking (7) at ghost number 0, one sees that ; ~ parametrizes the part ofoJ a which is left undetermined by the

vanishing torsion condition (2). As a matter of fact, eqs. ( 1 ), ( 4 ) - ( 7 ) separate the variables of the right sector of two-dimensional geometry into

(i) the Lorentz-Weyl independent var iables /~ and cZ; (ii) the variable exp (O a) which intertwines Lorentz-Weyl and diffeomorphism transformations and the as-

sociated ghost ~ [from (7) at ghost number one, sOa= - Q R + 0z CZ+ CzX~ ]; (iii) Z~ which transforms by construction as a connection under holomorphic changes of coordinates.

With respect to these transformations, one has the following covariant derivative acting on any fields of conformal weights r and k

1 ( 0 ~ - # ~ 0 ~ + l ~ ~ ~ a Vz= 1 z z Ozl~z--r#zOz#z)+rxz" (.8)

We showed that X~ is the natural geometrical object from which one can generate a field, which is inert under Weyl and Lorentz transformations and transforms under diffeomorphisms in such a way that it can compensate a conformal anomaly in the R-sector. Let us repeat the argument, since we will use it also in the supersymmetric case. A conformal anomaly in the R-sector satisfies a consistency equation: it is a two-form A~ with ghost number one which is annihilated by s modulo d, s f~A~ = 0. It is therefore related to the generalized three-form

I7~ = (ozf iz)a(ozpO . (9)

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V o l u m e 251 , n u m b e r 1 P H Y S I C S L E T T E R S B 8 N o v e m b e r 1990

Indeed ~-~/~ = 0 and D 3R is not a-exact. D3 R only involves the Beltrami variables and f/72 R~ = 2fd2z c z 03p§. 1"I~

can be related to

z~R = (~ R a(~) R " ( 1 0 )

A3R is the obvious expression of the R-sector anomaly as a Chern-Simons form. The equivalence between the expressions (9) and (10) was shown in ref. [2] by noting that these two expressions only differ by a a-exact

term:

dR ads= (0~v) a (0z~7~) + a ( d R a0s _az Oza Zr). ( 11 )

The anomaly compensating mechanism means in the BRST approach that one introduces a new field such that/~3R, or equivalently z~3R due to ( 11 ), can be expressed as a a-exact term. It is obvious that a field l R with transformation laws defined by

GR=at ~_d~=gzG~ + g e G R , ( 12 )

satisfies

ZT3R= - - a ( d R ~l R) , (13)

and is therefore a candidate for such a field. One can go further and define from l R a field which does not transform under Weyl and Lorentz transformations, i.e.,

L R = I R + 0 R. (14)

One has thus from ( 7 ) a n d ( 12 )

d t R - OzftZ--flZZRz =ftZGRz + f teG~ .

At ghost number 0, one has

(15)

1 1 R [ ( 0 z ~ R , G z = l - - - ~zlz~ze - p z O ~ ) L +#~ Oz/ZZ] - Z ~ , Gff - - - 1 - /~e /~ z [ ( O z - / ~ 0z)LR-- Odt§] . (16,17)

The idea used in ref. [2] is to impose a covariant constraint, G R =0, similar to the constraint on the torsion T = 0. Being covariant, this constraint is compatible with the BRST equations. From G R = 0, Z a is no longer an independent field. Rather it can be expressed as a function of L R,/l~- and/Zz~:

1 #z 0~)L +lZz 0zg z] • (18)

1 z z - # e g ~

The Wess-Zumino action whose s-variation reproduces the anomaly f1-1~ is thus

f 1 _ z R ~ z 2 z J R z = d2z 1 z-----~ [ ( O z - - # ~ O ~ ) L R ( O z - - P ~ O z ) L --/Zz(0z/-te) --2(0z/t~)(0z--/ZzZ0z)L R] • (19)

Let us summarize: The field variables for defining a quantum theory on the worldsheet are/z~-, c z and Z R in the R-sector. The conformal anomaly, expressed in terms of pz and c z, can be compensated by the lagrangian ( 19 ), which involves the field L R extracted from the connection o9 R. The intertwining field 0 R and its ghost O R are spurious, since they can be gauged away by means of the inhomogeneous transformation law s0R= £2 g + .... stemming from (7). The advantage of this presentation is that all fields relevant at the quantum level have a geometrical origin. Moreover, since everything has been worked out in the framework of differential calculus, the extension to the supersymmetric case will be straightforward, via the superspace formalism. Notice finally that by integrating out the field L l~ from the action (19), one gets a non-local action, only depending on the

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Beltrami differential/t, which can be called the two-dimensional gravity action for the R-sector.

3. Supersymmetric Beltrami parametrization

Consider now the superspace with local coordinates xM= (z, g, 0, 0). The frame in superspace is defined as the super one-form E A = d x M E ~ ,,1

We still have two connections in superspace: o9 (Lorentz) and A (Weyl). In the helicity basis (with to R = A - o9, o)L=A + tO), the torsion constraints o f two-dimensional superspace geometry [ 6 ] ~2 can be written as [2,4 ]

dEZ + EZo)R= - 2 iE°E ° , dE°+ ½E °O~R =E~EaTR + ½iE~EZ D a T R , (20a,b)

dE~ + E~o)I'= 2 iE°E ° , dEa + ½EOo)L=EZE°TL + ½ iEZE~ DoTL . (20c,d)

The ghost superfields are O R and t2 L for the right and left transformations and ~ for the superdiffeomorphism transformations. Ghosts and gauge fields are combined into unified quantities which are for the right sector

ff~Z=E~+icEZ, (21)

ff~°= E° + icE ° , (22)

~ R = o ) R +/2R • (23)

We shall limit ourselves to the description of the right sector, the left sector can be obtained by reflection. The constraint equations (20a) and (20b) imply the following BRST equations:

~l f f z+p~&R= _2i/~0/~o, ~1/~0+ ½/~0&R =j~zj~0-TR + ½iff_zff, gdoTR " (24a,b)

These equations imply the analogues o f (6):

ff~Z~lEz+2iff~zff~°ff,°=O, / ~ ~1/~°+ ½/~° d / ~ + i/~°/~°/~° = 0 . (25a,b)

The frame in two-dimensional superspace is parametrized around rigid superspace, with moving frame E (°)A, as follows:

EA IIArC AA I~ ' (O)BIArCAA . . . . . . c . . . . . B-*c. (26)

In ref. [ 2 ], we showed that the matrices M and A can be parametrized to be

l, 0 0 ) M=W~ M~ M~ M°ol ' A= 0 4 ~ "

\M~ M~ M~ M~/ 0 0 ~ /

(27,28)

We call Beltrami variables the superfields M~ and M g defined in (26) and (27). They are inert under the structure group transformation: they are by construction only subject to superdiffeomorphism transformations.

The superspace horizontality equations (24a), (24b) take the form

MZ ~ Z + 2igQZM°ff4°=O , MZ ~lllTI° + ½ffl° ~ I z + iM°hTI°ffl°=O . (29,30)

~ z and )Qo are defined as follows:

f f IZ=MZ+ icMZ=MZ+~, z , f f I ° = M ° + i c M ° = - M ° + ~ ° . (31,32)

t~1 In refs. [2,4], we denoted the indices 0and/Tby 1 and 2, and the indices z and 2by + and - . ~2 Superspace methods have also been used in refs. [ 7,8 ], for a detailed review see ref. [ 9 ].

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Eqs. (31 ) and (32) define the Beltrami ghost variables ,~z and ~o in terms of which the BRST algebra is factorized, as in the bosonic case. They are superfields of ghost number one whose lowest components are respectively the ghosts for diffeomorphisms and supersymmetry transformations in component formalism.

From the last equations we obtain the supersymmetric generalization of (6)

~l~Z+~p§ + 2i~o~o=o, alfflo+½~°Pz + ~ P ° = O , (33a,b)

with the definitions

z z .~z FZz=E (°)A D~M~, Pz=Fz+Dz~ , (34a)

0 0 ~ 0 F ° = E (°)A D~MA ° , P ~ = F ~ + D ~ . (34b)

DA is the partial differential in rigid superspace, d = E (0)ADA. Eq. (7) is generalized into

~bR+d logA§ - P § =M~z R + 4iyhr°2o. (35a)

~12o- ½~Zz2o _~o =~HRo + ½~OzR z + (~O+ ~,2o)zR + ½i~WazR . (35b)

We have defined

z R- (A§A})I/ZT R, Voz R--(A§)I/zA~DoT R, 2o=_A°/x/~z, ) ta-Aaz/x~. (36)

In the following we shall use the combination

2 R = j~fzxR -~ 4i)Qo2 o " ( 37 )

4. Superconformal anomaly

As in the bosonic case, the superconformal anomaly is related to a closed but not exact three-form. The generalization of (9) is

ErR = p z ~:lP§ + 8i~rzpopo. ( 38 )

As a consequence of (33a), (33b), one has

a P z + 4 i M ° P ° + M ~ g z = 0 af f , o _ , p o p z . , - o - - ~ - o , ~ - ~ - ~M B ~ + M B~z = 0 . (39a,b)

It follows that aEr~ = 0. Solutions of the consistency equation are obtained by expanding in ghost number dEr~ =0, i.e., Er3 = H ° + H i +/72 +/73 (from now on we will skip the index R):

d/7~3=0, s H ° + d H ~ = 0 , sH2~+d/ /2=0 , sH2+dHo3=0, Silo3=0. (40a,b,c,d,e)

We define the components of these superforms with respect to the frame E (o)A of rigid superspace:

1 E(O)AE(O)SE(O)C(/7~3)csA Hi Iy;'(O)AK'(O)BIITlh 1-I 2 =E(°)A(H2)A (41) ~ ~ J t ~ L , I I J 2 I B A , . j . .

Let us show that fdZz dZ0(H~ )0-o is the consistent anomaly, i.e.,

s ~ dZz dZO(H~)oo = 0 . (42)

We work in the rigid superspace frame, with supercovariant derivatives Dc = E b °)M 0M and constant non-zero components of the torsion T~ °)z = - 4 i and T~)g=4 i . One has from eq. (40b)

E [Dc( H1 )Da + TbO-s)F(HIz)FA ] = S (//03) CBA- (43) cyclic p e r m u t a t i o n s o f C B A

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Volume 25 i, number 1 PHYSICS LETTERS B 8 November 1990

Thus

( H ~ ) ~ = ~ ~ 1 ' , - - ~DoDo ' ( / 72 ) oo-- 41{Dz(1-12)~7~+Dz(H2)oo-s[ (1-1~3)002- ½iDo(//~3)oaa] }. ( 4 4 )

Applying s to this equation and integrating on dz dg, the terms on the second line cancel and thus

s f dZz d20(/-/~)~ =s f d2z(H~)zel°=e=°" (45)

From eq. ( 40c ), sfd2z(/-/2 l )ze I o=o=o = 0, which proves (42). The ghost number one contribution to/~3 is

H~=3z(FZB~+8iI,OzFOz) .-o ~ o +41.-. FzFz - (Dz~ z) (MZBz + 4i-M°Fz °)

+ (DzDz~,OMZF§ - (Dz,~ °) ( 16iM~F°~+4iM°F§). (46)

Bz is defined from (39a) as Bz = E t°)a D~D~M§. From the ghost number zero part ofeqs. (33) (the Beltrami superspace constraints), M,~ and Ma ° can be

expressed as a function of the independent supertields Mb and M~. In a particular superspace gauge, it is further possible to impose M~ = 0 [4 ]. In this case, the expressions of the Beltrami superfields simplify. We obtain

M°o = 1, M~= ~i DoMe. (47,48)

This gauge is preserved by BRST transformations if we set

~ o = _ ~i D0Z ~ . (49)

Using this gauge, the expression of the consistent anomaly simplifies a lot:

(H~)oa=2(Dz~ODzDoM~. (50)

Let us show now that, as in the bosonic case, there is an other form of the anomaly, equivalent to (50) in the BRST sense. It is convenient to rewrite the BRST equations (24) as

aff.Z + EZ(7.)R= T z = - 2 i g o E ° , ~tff~o + l ff~otT)R = T ° =gzff~o , ~ l (7)R=FR=gZgz + 4iEOg o . (51a,b,c)

/~z and K0 are given by

g o =/~0-TR + ½i/~z Do T R , gz = J~z(i DoDo T R + 2 T R T L ) - - 2ffJDo T R . (52a,b)

Applying a to ~o yields

/~z (~( 'o - ½ g o ~ g + ½/~og~) = 0 . (53)

Using these equations, one finds that the following three-form is closed under 8:

ZJ R =0~ R a(~R"] - 8i~ZRoKo. (54)

One can check that ~R is not d-exact. Thus, expanding A~3 in ghost number and picking the term with ghost number one, one gets an other form of the consistent anomaly than the one given in eq. (46).

We now show that ~R and/-]t~ are equivalent, i.e., differ by a d-exact term. Keeping in mind the relations /~z=~tzAz and/~0= (~o+fffZ2o) (d~z)I/2, w e define

ff'°=ffl°+ ~z2o , ffR=fflz.~z +aiff'O,;[O, ~z=R~A~,

Thus one has

?i~.o= _ ½ ~,o(p~ +2R) +~'.~o.

Applying a to this equation gives

Ao =Ko(A§) I/2 (5 5a,b,c,d)

(56)

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As in the bosonic case, we have the identity

~ - z - R z - R _ ~ ~ ~ ~ + ~ ( P ~ + a P D d(dlogAzo9 - f i z z ) --ff z ~lJ~z -(~3R

Using

~ ~ z 0 ~ z 0 0 ~ z 0 ~ d ( M / ~ 2 o ) = M ff zPz + M ff zAo + ~ i ( d f f z ) 2 R , a ( J~IZz~o2o ) = mZ.4oA o'4" J~lZP°z~o- ~ iFR2 R ,

one gets the expected relation between A and H, which is analogous to eq. ( 11 ):

/-~ =z~R + d [d log ,4 §osR-- fl§)~ R - 8iM~2o(P ° --.7/o) ] .

(57)

(58)

(59a,b)

(60)

5. Construction of the effective action

Consider a covariant superfield p R of chiral weights r= - 1, l= 0 which satisfies the following equation:

a l o g p R ( ~ R = G R , ~ R = f f A G R ~ z R ~~. R = E Gz + E G~ +E°GR+E°G~ (61,62)

Now we define the field BR=pRA§ (a close conceptual analogy with the bosonic case becomes obvious if one defines superfields l R = log p R, L R = log B R, the role of ~ R being taken by log ,4 § ). Using

a log ,4§ + t~R =2R + if§, (63)

we obtain

log B R - fizz = (~a +)~R. (64)

It is convenient to reexpress (37 ) and (62) on the basis A~rz, 90, 3~tz, 9~ defined by

9o=~O+~Z2o=gO(A§) - , / 2 , 9~=A~O+A~tz2a=/~o(,4~ )-1/2 (65)

This gives

--R ~ z R ~ g R G = M ff~ + M ffz + 9 ° f o g + g o f e r , (66)

and

~ R = ~ z z R 31_ 4i9°2o, (67)

since 202 o = 0. Of course the coefficients in (62) and (66) are related through

~ R - - z R (£R - - I A Z h l/2t2_R R - - £ R - A z G z , ~o - t ~ z ) "-'o, f#g - A z G z , f f~=(d~) l /2G~. (68)

We choose to impose the covariant constraints Gz g = 0, Go R = 0, which can be enforced by redefinitions o f z R and 20, and thus

G R =~gfqR + 9 a f ~ • (69)

Remember that the superspace geometry determines f i r [2]:

~R=4igOgazR 29037Igz~+2~O~z R ~ z ~ g R V M Zo + M M ~ z z . (70)

Using this definition, the relation aGR+ ff R= 0 yields at ghost number zero the following equations:

VOfC~+4izR=o Vaf¢~ +2iffR =0 Z ~ + ] i V o V o f ~ + I " L ~ R = o ' ~ 2 ~ ~'~7 ,

,~R 1 R . . t _ ] V z ~ a = O , R R I" L R f ~z- Vz (~z + ~lzo (qa = 0 . (71)

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We now rewrite the BRST equations in a more convenient form for computing the superLiouville term

~/17IZ+~Z(pZ +2g) = _ 2 i ~ o 9 o , ~ o + ½ 9O(pz +2R) =)Qz/IR " (72)

Here/ t~ = V0TR+ 1: i"~g_R ~v, "a- One has of course the conjugate equations

~ r s + ~ r e ( / ~ + 2 L ) = + 2 i p a ~ a , a ~ a + ½ ~a(pff+~L) =~rz/i~, (73)

where .7/~r = O 0 . L . . t _ l ; ~"arz .L v L T ~ U v a s 0 .

By definition a superfield ~u of conformal weights r and I satisfies

a ~+ r ~( P§ + 2 R ) + l ~( P~ + 2 L ) = 2 ~ l Z V z ~[_.l_]. 2~1gV5 ~./.3L ~ O V o ~Y'/- Ji- ~ ' ° V 0- ~- / . ( 74 )

The covariant superfield f¢~ has weights r= 0 and l= - ½. One has thus

a f¢~ - ½ f¢~ (p~ +)~L) = _ 4 i /~ -- 2i( p a + / i ~ s V a ) f#~. (75)

We have defined

9 ~ f ~ 0 R 1 ; ~" ' / fz.R (76) = v T - - ~ v J s 0 .

As in the bosonic case, the important equation is the relation (60) between the two equivalent expressions of the anomaly. This equation can be written as

~ z - - z • ~ z 0 0 " ; R ~ ~ R • ~ z ~ R R z ~ R ' ~ z 0 R F~ ~lP2 + 81M PzP~=co dco + 8iM Ao A o +~l[--~R ~t log A§--~zX --8im 2o( ff 2-Ao ) ] • (77)

Combining this equation with the following relations:

cbRac~R=GR~I~R__a(~RalogpR) ( ~ R - - - - R " - - / - - R - - R -- 1 " - - 5 R - - , dG =8 iM B a B a - d ( ~ i M ~ad f g ~ + M ~ g v a ~ ) , (78a,b)

d (4)0"z/~zgz'g) = 8iJl/lZAoAo + 8 i m z / ~ / ~ , (78C)

one gets

z - - z " ~ z 0 0 - - - - R /-7 R =/~z ~dPz + 8iM/~z/~z = d Z 2 , (79)

2 ~ = G g d l o g B -PzX -8 iM,~o(Pz ~Rx l"~a, zrz?g - M if5 ~af¢~ +4J~zJ~eTRTR. (80) - - / l 0 ) - - ~ u v l z:l O

We have thus shown that by introducing the superfield B R, the anomaly z~ can be written as the variation of a counterterm ~R. We can simplify the form o f Z R. Using the relation

z ~ 0 ~ R M 2 o ( d 2 o - ~ - A o _ ½~r0)~R) = 0 , (81)

we obtain

2 R = ~ R ~ l o g B R + 8 i ~ 1 Z ~ o ~ L ~ o , . ~5 R - - R - - z - - 5 g R ~iM f~a df~a + 4 M M z z + 16iM~P°2o

z ~ R " - - z R ~ 0 ~ g R - - 0 ' - R -PzZ +4iM ZzM 2o-M fgz V f~a • (82)

Expanding this object in ghost number

2~ =ZO + Zl + Z~o , (83)

the term Z ° with ghost number zero yields the anomaly compensating action. One has indeed from (80)

1 0 1 H2 ~ ' ~ - s Z 2 +dZl . (84)

Z°=GR d log BR + 8iM~2od2o -- ~t~,,l~'~reraa d f ~ + 4M~M~zaZR

__ fZzXR " z R 0 5 R • +4~M YzM 2o-M f#5 Vaf~ (85)

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To get the Liouville superdensity as a superfield whose BRST variation reproduces the anomaly, we expand simply Z ° on the fiat superspace basis:

Z o = tw(o)Aw(o)Bt70~ ~ z . , .it., ~ . t . , 2 J B A . (86)

Indeed since sE to)A= 0, the s-variation of the component ( Z ° )oÜ integrated on d 20 is proportional to the anomaly. The mechanism is analogous to the one detailed between (42) and (46).

The supersymmetric Liouville action is therefore given as

~ v Z = ~ d 2 2 d 2 0 ( Z ° ) o o - . (87)

This ( Z °) oa can be identified in the explicit expression given in (86).

6. Projection in component formalism

The contact with ordinary fields is obtained by taking the lowest superfield values and projecting to space- time differentials:

dz+d~/~§=M~ll , dg/~°=M°ll , ~*=~Zll , ~°=-~°11. (88a,b,c,d)

The symbol II means that we take 0 = 0 = 0 , d 0 = d 0 = 0. Having introduced the superfield B R, the independent component fields are defined in the right sector as the

lowest components of the superfields B R, 2o, f#~ and z R. One has thus a multiplet structure:

2o

B R T R " (89)

We chose covariant constraints such that the lowest component of the superfield z R is not an independent field [see eqs. (69) - (71 )] . Similarly, the lowest component of the superfield Z R and ~R are determined by the constraints

1 z g [ ( 0 z - - / l z ~ 0 g ) l o g n R - - flz0-(~ 3 !- 4i/Zz*/l°2o + / Z ~ 0z/~ z] ,

- l t ~ Z z + 0~logB R - 0z~§-4i/1~2o,

or, equivalently

(90)

(91)

1 .~R + ~ t ~ - - [ z R z " 0 z 0- R (0z -g ,0z ) log B - 0dte--41/t ,20+#z#z f#a] • (92)

We arrive therefore at the following expression of the action which compensates the superconformal anomaly:

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o~vz = f d22 ( 1 1 -ltzlt~ (Oe-lt~ Oz)lOgBR(Oz - ~ 0z)log B R + 8 i 2 o ( 0 z - # ~ 0z))to

+ / i ( # ~ ( 0 z _ / t ~ 0 e ) f q ~ + 4 ( 1 z z R R 2 ( 0 ~ - z R a R ,ueO~)log B #zfaa -lz~12z)Z z - -16i2 0 0z/tz°-+ 1-#~#z

8 2 2 z Z + i z---~-~ ( 0 z - / t § 0~)logBR/t°)to+ 1 --/tz/z~ Ozlt~(Oz--btz 0z) logB R - ~ a

1 - #~/z~ 1 - / ~ z / ~

+ i 8 o , _ _ (,i.lZzOzflZ)(OzllZ)_l__flV)to].tz~O)]. ~ / . t ~ ; t o / l ~ 0z#§+ 1 . 8 o a R "X (93) l - / t z / ~ z 1 -- / . t ~/.tz z 1--~/tzZ ]

The BRST t ransformat ions o f the componen t fields B R, ;to, f¢~ and z R are obta ined f rom those of the superfields by using the techniques o f ref. [ 4 ]. WE do not d isplay them here.

We have therefore shown that the compensa t ing mechanism for the superconformal anomaly can be straight- forwardly extended f rom the bosonic case, still in the f ramework o f differential geometry.

7. C o n c l u s i o n

The emphas is in this let ter was on the algebraic descr ip t ion o f the factor ized conformal d i f feomorphism anomaly and the construct ion o f a compensa t ing action. The basic ideas are identical to those o f the purely bosonic case, shortly reviewed in section 2. The addi t iona l features encountered in the supersymmetr ic case are mere technica l complicat ions , which we solved by using a Bel t rami type paramet r iza t ion o f superspace. Since we have shown the comple te factor izat ion o f the BRST structure in the Bel t rami sector, it was sufficient to present the results in the right sector only, the left sector can be ob ta ined by reflection.

In the bosonic case, 2D quan tum gravi ty has been ob ta ined by choosing specific gauges [ 5,10,11 ]. Similarly, 2D quan tum supergravi ty [ 12-14] may be ob ta ined f rom the results presented in this paper by picking special gauge choices.

A c k n o w l e d g e m e n t

We would like to thank O. Thi immel for carefully reading the manuscr ipt .

R e f e r e n c e s

[ 1 ] L. Baulieu, M. Bellon and R. Grimm, Phys. Lett. B 198 (1987) 343. [ 2 ] L. Baulieu, M. Bellon and R. Grimm, Nucl. Phys. B 321 (1989) 697. [ 3 ] L. Baulieu, M. Bellon and R. Grimm, Phys. Lett. B 228 ( 1989 ) 325. [4] R. Grimm, Ann. Phys. (NY) 200 (1990) 49. [ 5 ] A. Polyakov, Phys. Lett. B 103 ( 1981 ) 207; B 103 ( 1981 ) 211; Mod. Phys. LeU. A 2 ( 1987 ) 893. [6] P.S. Howe, Phys. Lett. B 70 (1977) 453; J. Mod. Phys. A 12 (1979) 393. [7 ] M. Rocek, P. van Nieuwenhuizen and S.C. Zhang, Ann. Phys. (NY) 172 (1986) 348. [ 8 ] R. Garreis, J. Louis and B.A. Ovrut, Nucl. Phys. B 306 ( 1988 ) 567. [ 9 ] E. D'Hoker and D.H. Phong, Rev. Mod. Phys. 60 (1988) 917.

[ 10 ] K. Yoshida, Mod. Phys. Lett. A 4 ( 1989 ) 71. [ 11 ] J. Distler and H. Kawai, Nucl. Phys. B 321 (1989) 509. [ 12] M.R. Grisaru and R.-M. Xu, Phys. Lett. B 205 (1989) 486. [ 13] S. Aoyama, Phys. Lett. B 228 (1989) 355;

S. Aoyama and J. Julve, Padova preprint DFPD/90/TH8 (February 1990); T. Kuramoto, London preprint QMW/PH/90/02 (January 1990).

[ 14] J. Distler, Z. Hlousek and H. Kawai, Cornell preprint CLNS88/871 (1989).

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Superconformal anomaly and its compensating action