Left-right asymmetric conformal anomalies

Volume 228, number 3 PHYSICS LETTERS B 21 September 1989

LEFT-RIGHT ASYMMETRIC CONFORMAL ANOMALIES

L. BAULIEU, M. BELLON Laboratowe de Phys:que l'hdor:que et Hautes Energws ~, Umversttb Pwrre et Marw Curte, Tour 16, 1 ~" etage, 4. Place Jusswu. F-75252 Parts Cedex 05, France

and

R. GRIMM lnstttut fur Theorettsche Phystk Umversttat FrMer:c:ana, Katserstrafle 12, Postfach 6980, D-7500 Karlsruhe. FRG

Received 10 July 1989

Using Beltraml variables, the shift between local Lorentz-Weyl anomahes and &ffeomorphlsm anomahes is exphcated. Left- right symmetric as well as asymmetric conformal anomahes are cancelled by introducing various types of compensating fields

1. Introduction

The Beitrami parametnzation of the world-sheet metric and the diffeomorphism ghosts provides a powerful tool for describing two-dimensional gravity with a natural generalization to the supersymmetric case [ 1 ]. An important property of this parametri- zatton is the factonzation of the complete BRST algebra into independent structures for the right (R) and left (L) sectors [1-3] . The (super)conformal anomaly can be parametrized solely in terms of Beltrami variables and splits into independent contributions for the left and right sectors. Each of these two contributions sausfies its own consistency equation.

In this paper we continue our study of conformal invanant theories using the Beltrami parametriza- uon. We introduce two fields which permit to com- pensate independently left and right conformal anomalies. We present a straightforward construction of the stress-energy tensor and of an invariant interaction term mvolvmg both supplementary fields. We also show that a given, possibly left-right asymmetric anomaly can be compensated using a single field (the Llouville theory corresponds to the left-

Uml6 assoc~de au CNRS UA 280.

right symmetric case), finally we discuss the relation- ship between our results and those presented in refs. [4,5].

Since we only use the differential calculus of 2D geometry, equipped with a Lorentz-Weyl structure, all our results have a quite straightforward supersymmetric generahzatlon which will be presented in a separate publication.

2. Beltrami parametrization and eonformal anomaly

In ref. [ 1 ] the factorized BRST algebra involving the Beltramt fields has been constructed. The local geometry of the world-sheet has been described in terms of a local frame (zweibein) and one-form po- tentmls for both Lorentz and Weyl gauge transformations. The usual algebraic BRST techniques were used to construct the BRST transformations of all classical fields and ghosts.

Let us recall the construction of ref. [ 1 ] : we para- metrize the world-sheet in terms of a complex coordinate z and its conjugate g. e : = d z e . Z + d g e : - and e : = d z e . Z + d Y e / d e f i n e the zweibem one forms m terms of which the metric is e=®e-'. In addiuon, the connecuons co and A for local Lorentz and Weyl transformauons, respectively, are combined into

0370-2693/89/$ 03.50 © Elsevier Science Pubhshers B.V. ( North-Holland Physics Publishing Division )

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cog =A -co and coL=A +CO. This allows us to write the torsion equations in the form

T : = D e~=de~ + e-'coa ,

T:=De-~=de-~+e~co t- , ( 1 )

with d = dz 0.+ d2 0z the usual exterior derivative. The Beltraml parametrization of the zweibein consists in defining

eZ=g--e: =, e e = l t e e , "- , (2)

with

g " = d z + d~ lte: , ge=dT. + dz g~ e (3)

In the following we shall consider the case of vanishing torsion, i.e. T== T : = O , which amounts simply to a covariant redefinition of the gauge potentials coa and coL.

In order to introduce the BRST operator s, let ~ a and ~L denote suitable combinations of the Lorentz and Weyl ghosts and e ~, e" the ghost vector field of 2D diffeomorphisms. We define:

(~)R COR..~.QR , o~L ~--'-:-- COL"t- Q L , (4)

U = e - - + l ~ e z , ~e=ee+t~e : . (5)

The decomposition (2) generalizes to

~:=~:e::, ~=u~+C ~,

U = [ t - ' e : ' , [ z : = u ~ + C -: , (6)

with the Beltrami ghost fields given as C : = c ~ + c ~lt.. -',

C : = c - ' + c : g j . Introducing the generalized operator a = d + s allows one to define the structure equations for the BRST symmetry as follows:

]P-= ~tU+ u~bR= 0 , ]P '= ag,-'+ ~e~ L = 0 , (7)

and

P a = ao5 a = U~'-~FL, P L = a ~ e = U U F ~ e . (8)

As usual in the BRST formalism these equations, once expanded in ghost number, determine the action ofs on all fields.

The consistency of this construction of s, i.e. the property ~] 2 = s 2 = 0, is due to the fact that constraints ( 7 ), ( 8 ) are compatible with the Bianchi xdentiues:

8P"=0, aff'-=o. (9)

From now on, we shall work essentially on the (z, R) sector, the other equations being generally obtainable by the substitution (z, R) ,--* (g, L).

The information about the transformation laws of Beltrami f i e l d s p Z = d z + d g / t / a n d C ~ (and in particular their decoupling from the Weyl-Lorentz depen- dent variable as well as the factorization properties) are simply obtained by multiplying the equation :P==0 by/~z. Due t o ~ g V = 0 one obtains

~a/~--=0. (lO)

Expanded in form degrees and ghost numbers, this equation yields the factorized BRST transformations s~..-" = 0.. C z + C ~ 0- / t / - / t / 0 z C ~ and sC: = - C z 0 - C ~.

Observe that we can express eq. (10) in the following equivalent form:

a/7~ +/~:/~z : = 0 , (11)

with

/~ : = 0 z / i - - dg Oz Ue" + 0., C ~ • (12)

Eq. ( 11 ) obviously implies eq. (10) because/ig~z= 0. The converse follows from the fact that, if a differential form A(/2, ~b,...) satisfies A/i:=0, then A must be of the form A = A z ~ L This can easily be verified by expansion in ghosts and forms. Applying this property to the a derivative ofeq. ( 11 ) and using a 2 = o one gets ( - a f i : ) ~ + f i ~ a p . - - = 0 . Since ~ : ~ : = 0 , one finds ~-" ?=l/~z -" = 0 and therefore

aff~ z+~: /~ = 0 , (13)

with

/~ = 0~/~==dg 0~p.:+02C ~ . (14)

The same procedure can be applied to extract the properties of (b R which derive from the condition T~=0. Using the definition U=/~--e~ ~ in eq. (6), we obtain [ t - - ( C o R + ~ t l o g e J - - ~ ) = O . It follows that o5 R can be written as

cb R = / ~ , - a log e/+fi-')C~. ( 1 5 )

At ghost number one, eq. ( 15 ) determines the transformation laws of e / , i.e.,

s log e = ~ = O = C : - ~ 2 a + C ' - x ~ . ( 1 6 )

The basic set of independent variables in the R sector consists of:

(1) the Lorentz-Weyl invariant Beltraml variables

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/a_--" and C z which transform among themselves under BRST transformations;

(ii) e/" which transforms fields with Weyl-Lorentz weights into differentials and I2 R the corresponding ghost field;

(iii) Z_ R which transforms by construction as a connection under holomorphic coordinate transformations (see refs. [ 1,6] for more details).

We can now derive the relevant equations for studying the conformal anomaly in the context of this 2D geometry. Using locality principles [ 3 ] we know that the conformal anomaly A~ is given as a 2-form, ghost number one solution of the consistency equation of the BRST symmetry:

sd~ + d d ~ = 0 . (17)

A~ is a local functional of the fields and defined mod- ulo s- and d-exact terms. It is possible to construct solutions to eq. ( 17 ) in terms of a generalized 3-form ,~3 which satisfies

d d 3 = 0 . (18)

Expanding m ghost number, e.g. J3 =A~ +zl~ +~/~, eq. ( 18 ) generates a tower of consistency equations, with eq. (17) emerging at ghost number two.

The obvious candidates for ,~3 correspond to the second chern classes related to fiR and /~L through the Chern-Simons formula [ 1,7 ]

~ ~L~t , CbRpR 3 =CO , ' ~ = • (19)

The anomaly corresponding to eq. (19) can be interpreted as a Weyl-Lorentz anomaly [7], depending on the variables of the Weyl-Lorentz sector.

As shown in refs. [ 1,3], the anomaly can also be expressed in terms of Beltraml variables, by means of

a (¢'__--aP. :) =0. (20)

This equation is obvious since aff_:+/2--/~:~0, /~--/~:= 0, but/~_--a/~.: is not a exact. Thus,

.a~ =p::ap:: (2~) is a candidate for a consistent anomaly in the R sector. As a matter of fact expanding/~R in ghost number yields at ghost number one flT~J=fO_.,uS x 0~C-- d2z, which is the consistent anomaly found in refs. [ 1,3].

The anomalies LJ Rt and H R~ can be related by a counterterm

CRa~R=~ " aPz--+a (~ . a log e.:+ Pz : ~ z ~ ) •

(22)

This equation follows directly from eq. ( 15 ). It shows how the diffeomorphism anomaly of the R-sector /7 Rt can be shifted into the Weyl-Lorentz anomaly d Rt of the structure group.

The above analysis can be repeated in the L-sector by substituting (z, R) ,--, (f, L). Therefore, the anomaly of a general conformal theory can be written as CL/7 L~ --CR/7~ j with two independent coefficients CL and CR.

3. Independent cancellation of left and right anomalies

Our analysis has shown that in the R-sector the conformal anomaly can be expressed either in terms of pure Beltrami variables (eq. (21)) or in terms of the Weyl and Lorentz gauge fields (eq. (19)) . Eq. (22) shows the equivalence between these two expressions for the anomaly.

The mechanism for compensating the anomaly can be interpreted as a particular case of the anomaly compensating mechanism involving a p-form gauge field coupled to a Chern-Slmons form [8]. More specifically, the 2-form curvature F R = d c o R is related to the first chern class, with coR the correspond- mg Chern-Simons l-form. It is thus natural to introduce a 0-form ¢R coupled to the Chern-Simons form COR through the definition of its field strength GR:

GR=d~ aR-coR • (23)

The Bianchi identity for G R is d G R = - d c o R. The BRST symmetry for ~pR is obtained as usual by pro- moting eq. (23) to a horizontality equation which completes eqs. ( 7 ), ( 8 )

GR=a~R--~R=~--G~ +~G~. (24)

One sees from this equation that ~a R transforms non- homogeneously, in more detail, SCOR=I2R+CzG~ + CeG R. This means that the field pR=exp(tpR) is a scalar with respect to general coordinate transformations and has weights r = 1, 1--0 with respect to the structure group transformations.

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Introducing the new field ~R makes the anomaly d~ ~ trivial. It is obtained as the BRST variation o f a local functional,

(~R a ~ R = __~R ao~R,gc.a(~R a 0 R ) . (25)

According to the BRST equations (~R a~R tS a cubic expression in the two nilpotent objects fi: and fi-" and is therefore zero. jR is thus d-exact, which means that the anomaly f A R~ is the variation o f a counterterm: f A y ' =Sf (wR d~oR).

We have seen that the anomaly H~ ~ , which is expressed in terms of Beltrami variables differs from A R~ by an s- and d-exact counterterm. With the mtro- ductlon o f the field {o R, H2 a~ is thus also a spurious anomaly, as a consequence ofeq. (22).

We proceed now to construct explicitly the local functional which reproduces the anomaly H R~ upon action o f the BRST transformattons. To this end we observe that the field

B R = e : Z p R (26)

is inert under structure group transformations but transforms as B R ( 2 ) = ( O z ' / O z ) B ' R(Z ' ) under holomorphic coordinate transformations. The properties of B R are most easily described if eqs. (24) and ( 15 ) are combined into

a log B R - I ~ : : - - f i z z R = f i : G ~ + f i :GR: . (27)

At ghost number zero, we obtain

l G . R = ~--ff V :BR= V. log B R ,

l G R = f f ~ V = B R = V: log B R . (28)

The derivatives

VzB R = --AvRB R

1 + ~ (0 : - - f l : ' * 0 : + f l z ~ O:f lzZ)B R ,

1 - - # : : g : -

1 V - B R = ( O e - - f l - z O: -- Ozf lzZ)B R . (29)

1 - p : : g : :

are covariant with respect to holomorphic changes o f coordinates, as explained in detail in refs. [ 1,6]. At ghost number one, eq. (27) yields the transformation laws o f B a

s log B R = OzCZ..FCZC.~. z l o g M R

+ C : <J.: log B R , ( 30 )

with the definitions [ 1,6 ]

log B R

1 -- 1 - -p : - -p : : [ (0:-- /1: : 0e) log B a + / t : = 0 :p: : ] ,

LP~ log B R

1 - [ ( 0 ~ - - p : : 0 _ - ) l o g B R - - o 4 t S ] . (31)

From the BRST equations we have (~Ra(~R=0. Combining this relation with (27) yields

/~:-" aP~ ~= a((7 R a log B a + f i - - f f : : Z R) . (32)

This equation shows that the diffeomorphism anomaly of the R-sector is trivial. In close analogy with the vanishing torsion condition we impose the covariant constraint

1 G R = ~--~ VzBR=0 . (33)

The constraint (33) is compatible with the BRST symmetry and thus consistent. Eq. (33) expresses 7.. R in function o f B R, i.e.

Z R =£(: log M R . (34)

Taking into account eq. (33), the ghost number one component ofeq. (32) reads

H~ ~ = S~Rz (mod d-exact t e rms) , ( 35 )

with

2"Rwz =/~:~e log B R d log B R ..~ U :'/~z : '~ log B R. ( 36 )

By construction, the Wess-Zumino lagrangian den- sity ~Rw z depends only on/1::,/~:: and B R

1 ~ w z = d z ' 1 -/t---It- -~ (O: _ a e 0:) log B R

× ( 0 . - - / t j 0:) logB R

1 - d z 2 l _ p :,u elz:"(O:,tte'-)2

- 2 ( O.iJ:: )O:log B R . (37)

The component with ghost number two in eq. (32)

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gives the counter term for compensat ing the second cocycle # dz C-- 0~ C z, i.e. the Schwinger term.

The same discussmn can of course be repeated for the L-sector by introducing another field tp L and its associated conformal field B L = e J exp (~L).

The anomaly cancellation mechanisms in the R- sector and in the L-sector are completely independent of each other and allow the el imination of any anomaly cLHt¢ ~ --CRH~ ~ with independent coefficients CL and CR- This mechanism involves two compensating fields.

It is clear f rom eq. (37) that the fields L R = l o g B R and L L= log BL are propagating fields. Once they have been introduced for compensat ing the anomaly, one must check the possibility of having invariant counter terms function of these fields. The following ac- non (Liouvdle interaction [ 9 ] ) is invariant under BRST transformations:

~ , , = ( 1 -U: :U: z) exp (L R ) e x p ( L L ) . (38a)

One has indeed

s ~ . , = 0 : ( c " ~ . , ) + 0~(c :~ .~ ) . (38b)

The total Liouville action is therefore

'ff~L ...... lie = ~ R z + ,~- ~vz + a ~ , t . (39)

The contr ibution to the stress-energy tensor is simply obtained by differentiation with respect to the Beltrami differential U-:"

~*~LIou,',lle T z z - - - -

8U..:

= 20:(~_L R ) + (_~L R ) 2+ (~__L L) 2

--aU:-" exp (L R) exp (L L) . (40)

In our approach, it is thus immedia te to find that in the conformal gauge (U= 0) the Liouville interaction does not contribute to the stress-energy tensor. Tee is given by the equation conjugated to (40) .

4. A n o m a l y c a n c e l l a t i o n w i t h a s i n g l e c o m p e n s a t i n g f i e ld

Given a general anomaly CL/-/2 El --CR/-/Rl we shall now exhibit another anomaly compensat ing mechanism which involves a single compensating field B and

reproduces the usual Liouvi i le-Polyakov action for C R ~ C L.

W e first observe that the anomaly can be written as

CI(~OLO(~oL--cR(~oR~](~t)R=2A' a~b' +2o5' aA-', (41)

where we defined rotated connect ions,4 ' and ~b' by

.4' = ½ (ULwL +UR09 R ) , (42a)

rb' = ~ (UL~OL-- UR~O R ) . (42b)

We now mtroduce the covariant constraint

dA' = 0 . (43)

Due to this constraint, the anomaly becomes trivial since we get from eq. (41)

CL(~)L O(.DL-- CR(.~) R dtbR = 2a(.4 '(b ' ) . (44)

Moreover the constraint dA' = 0 imphes the existence of a field p with

.'1' = ½d l o g p . (45)

In order to express the d i f feomorphism anomaly CL/'~ ,f ~ z__ CR/~z z d/~z z as the d-variat ion of a local functional, we first rescale the field p by conformal factors so that it is inert under structure group t ransformations

B = p ( e : Z ) , / ~ ( e f ) . / ~ . (46)

From eq. ( 15 ), we have

v/~R tbR = x/~R ~ -'-- x/~R a log ez - '+/7- 'Z~, (47)

X/~Lt-bL = X//-~L/~:-'-- X/~L a loge . 2+fteZ't-, (48)

where we have defined ) (R=v/~RZ~ and z':L=

Combining eqs. ( 4 5 ) - (48) , we get

d log B = V/~R/~: : + X/c~L/~ -" +/~ :z~R +/~-~Z "L . (49)

Eq. (49) at ghost number zero determines Z': R and Z "L as funcnons of B

1 x "R - [ ( o : - u.--" o: ) log B

1 -- # - :U::

- , f ( O:U:z ) + U: O:U/ I , (50)

l Z "L= - - [ ( O : - U - ~ 0-) log B

1 - U: ~U: =

a_-u:-3. (51)

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At ghost number one, it determines the s-variation of B

s log B

=C'-z'R +c"x~ '+x/~RO:C~+x/~LO:C-~. (52)

Eq. ( 52 ) shows that B has weights x/~R and x/~L. Using eqs. (22), (44), (45) and (46), we get

el. L 5 d L s_ CR L.Z d L :

(53)

Due to eqs. (47) and (48) we may replace aoS' by ~d(x/~L/~-e--X//~R/~-"+Cb-'z'L--fi-'Z~). We finally obtain

=c,L: a/",:- c, L: a/'_ = a [ ½/7-z'Y a log B-½/7=Z~ a logB

:( ½a log B

- x/~L/~--'( ½d log B+fi-'z~L) l . (54)

At ghost number one, this equation determines the anomaly compensating action ~wz such that

f n;=s f dz2. wz. (55) 1

*~WZ 1 - ,u: e ~ e :

× [ (0_- - #:: 0_-) log B(0: - ~_- 0.) log B

- -2x/~R(O:U::)(O:--U:: Oe) logB

--2V/~L (O:/~:-')(0_-- U:-- O:) log B

+ 2 cx~,.c~ ( O.4c- :)

--CR/t: ~(O:lz-:)Z--CL/Z._ ~(O:/Z::) : ] . ( 56 )

From this equation we see that our compensating field log B is a propagating field. It couples linearly to the Beltrami differential/t. The contribution of this field to the stress-energy tensor is given by:

T . - - - - -- 8~wz ~].,lz z

- - (Z'R)Z+2X/~R 0_-(X~) • (57)

Since Z~ transforms as a connection under holo- morphlc changes of coordinates, eq. (57) shows that

the stress-energy tensor transforms as --CR times a projective connection.

B is a (x/~R, x/~L) differential. If ~ =x/~L, i.e. for a left-right symmetric anomaly, an invariant action is given by f/~ ;~ 'exp (L), with L = (CR) - I/2 log b, and can appear through renormalization, if ~ ¢ x/~L, an lnvariant action must involve the covarlant derivatives V: and Vf. We may consider the following action:

• . ¢ , o , = f u - - u z ( V . . ) I + " ' ~ ( V : ) ' ' " ~ B -'" . (58)

In order for this action to be defined as a local functional, a should be chosen so that 1 + ax/~R and 1 + ax/~L are integers, which is only possible if the ratio of v/~R and ~ is rational. If we compare this pro- posal to eq. ( 38 ), we see that an invariant local interaction is more difficult to devise when we make use of a single compensating field.

In the symmetrical case (v/~R=x/~L), e.g. the string theory case, the constraint dA' = 0 simply means that Weyl connection is flat and is locally de- rived from the Llouville field. The asymmetrical case is formally very similar. However, the mixing between the Weyl and Lorentz connections which oc- curs in this case, although clearly quite locally, needs a better understanding at the global level.

5. Anomaly cancellation through chiral fields

Let us now go back to the case where the diffeomorphism anomalies are expressed independently through the use of two new fields B a and B E (section 3).

We can compare the expression we have found with the proposals of Polyakov and Yoshida [4,5 ]. If we set gF '=0 in the iagrangian (37), we get the functional given in ref. [5]:

~ R z = O z L R ( O e - - l l = : O = ) L R - 2 ( O : B e : ) O : L R . (59)

Furthermore, the condition/1:-'=0 is consistent only if we simultaneously set C :=0 . Both conditions re- duce the BRST variation of the field B a to

s L R = O z C Z + C Z O z L R . (60)

This is precisely the gauge variation which is given m refs. [4,5].

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This reduction of the BRST symmetry is however incompatible w~th the reality condi t ion for the metric in the euclidean case: a real metric is given by #='= (p_-')*. We could however consider this reduction just as a trick to verify the consistency of the BRST var iauon eq. (60) and the proper BRST van- ation of the lagrangian (59) . However, the lack of a dtrect geometrtc interpretat ion for this BRST symmetry could be a problem.

Another way to obtain the BRST vartatton (60) is to suppose that the field L R satisfies the constraint

1 ~_L R_ [ (0 e - p . . : 0:)L R - 0://~ :] = 0 .

1 - - g : -'p: :

(61)

In thts case the horizontali ty equation (27) becomes

aLR--[~:"=g--(GR +x R) . (62)

From eq. (61) , we have at ghost number zero G~ + X R -= OzL R and, at ghost number one, we recover the

BRST variation (60) . Eq. (62) ~s equivalent to fl- '(aLS+ff- :) =0, an equatton s tmdar to (10) .

Heuristically, a solution of the constraint (61) can be seen as a "half-field". The existence of a solution of this constraint can be deduced from a solutton of the Beltrami equat ion

0~f= #: - Off. (63)

log(0ff) is a solution to the constraint (61) and a non-local functional of the Beltrami differential p:--.

When the constraint (61) lS enforced, the Wess- Zumino action of the R-sector ts obtained from eq. (62). Indeed one gets

~:: ~[1~..-"= (tic O.__LR ~_IL R)~_Ip z. (64)

Using ~--~5/~-==0 (see section 2), we get /~=--d/~=-" = - d ( d L R ~ : ) . The W e s s - Z u m m o lagrangian is thus dLRF_: which is equivalent to (59) when the constramt (61) ts used. However, the main objec- tion to the constrained fields ~s that locality is lost.

Further results based on these approaches can be found in refs. [10,11 ].

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

Usmg the techniques of 2D geometry we have ana- lyzed the algebratc structure ofconformal anomalies.

We have shown how to shift the local anomalies related to i nvanan t Chern polynomials of the Lorentz- Weyl curvatures into the two-dimensional diffeomorphism anomalies expressed in terms of confor- mally invariant variables. For general L-R asymmetric anomalies we have exhlbtted several ways of cancelling the anomaly either by mean of a single compensat ing field or a pair of compensating fields. Dependmg on the choice one gets different kinds of possible Llouville mteractions. Our work should be completed by specifying the global properties of the various expressmns that we have obtained.

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

One of us (M.B.) ts grateful to ITP for extended hospitality during which part of this work has been

done.

R e f e r e n c e s

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Left-right asymmetric conformal anomalies