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21 July 1994

Physics Letters B 332 (1994) 329-335

PHYSICS LETTERS B

Liouville and Toda W3 gravity potentials via matter-dependent anomaly cancellation

Jonathan Benn

Blackett Laboratory, Imperial College, Prince Consort Road, London SW7 2BZ, UK

Received 9 April 1994 Editor: P.V. Landshoff

Abstract

I determine the allowed Liouville sector potentials of BRST quantized non-critical strings and W 3 strings. The potentials, previously derived from conformal-field-theory techniques, are obtained via the cancellation of matter-dependent (potential- dependent) anomalies in the Slavnov-Taylor-Ward identities. These anomalies arise as a consequence of adding potentials to an existing 2D gravity or renormalized chiral W 3 gravity action. Higher order "miracles" and Wess-Zumino consistency are considered as checks on the results.

1. Introduct ion

That the potential terms which arise in standard Liouville theory suffer renormalizations in a full quantum treatment has been known for a long time.

Previous techniques [ 1-3 ] have used the conformal-field-theory method of requiring that the potentials transform as ( 1, 1 ) conformal weight fields under the free-field Virasoro transformations and as total O/~z derivatives under W3 transformations (in the Toda extension of Liouville theory). This idea is not strictly consistent since Liouville theory is not a free-field theory but an interacting one. A fully consistent analysis for the ordinary string case was carried out some years ago [4]. Here the authors required that the modes of the stress tensor derived from the full, covariant action

I f ( 4/Zo 2 ~) I = ~ dZo'vl~ lgiJoiq:,c3jcp+Q~oR(g)+ --~-5- e a (1)

should (in a flat gauge) form a centrally-extended Virasoro algebra. This method, though, is not easily extendable to the W3 case, not only because the full, covariant action is not known, but because the current algebra is non- linear.

An equivalent scheme more natural to a field-theorist would be to require the maintenance of the Slavnov- Taylor-Ward identities when the quantum effects of the potential term are taken into account. Such a term would throw up new matter-dependent (Liouville-field-dependent) anomalies, which by their nature are BRST-trivial and hence can be cancelled by appropriate local counterterms or renormalizations of the BRST transformations. I will

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330 J. Benn / Physics Letters B 332 (1994) 329-335

show that the counterterms are essentially already present in the sense that cancellation of the new matter-dependent anomalies simply requires a renormalization of A in ( 1 ).

The motivation for the method comes from simple ideas in the field-theory of (1). Classically, (1) exhibits Virasoro covariance, that is if the following transformation is carried out

P g o ~ e p g u , q ~ q ~ - ~ , (2)

then, provided A takes on its classical value 1/Q, the action is invariant up to a ~independent residue, namely f d2o - R (~)p [ 2 ]. This is the analogue of the familiar universal gravitational anomaly f d2zcO 3h in B RST language ~. Now, the one-loop contribution to the effective action involving the potential term is not invariant and so provides a one- loop anomaly. This may be cancelled by a renormalization of the ~ transformation. The covariance of the one-loop effective action under the new transformations provides the familiar relation Q = 1/A + hA~2.

This result will be derived more rigorously in the rest of this paper using BRST techniques and the Ward identities. Higher-loop effects will be considered as well as Wess-Zumino consistency. The treatment will involve ( 1 ) in the Polyakov gauge where ( 1 ) reduces to a chiral action.

The full covariant Liouville action may be treated BRST style but my main aim is to treat the two-scalar realization of the W3 Liouville (Toda) action for which no geometric action is known. I simply adopt a background charge renormalized chiral W3 gravity action for two scalars and introduce a potential term and carry out the anomaly analysis in a natural extension of the simple string case.

2. Liouville anomaly analysis

I employ the pared down version of the Liouville action provided by the Polyakov gauge and with the infra-red regulator factor set equal to one (plus some other trivial redefinitions). Normal ordering is assumed, which amounts in the case of the potential to the fact that one neglects tadpole diagrams involving the exponential. The full classical action is (in familiar coordinates (o-, r) --* (z, g)) using BV notation

1; /'o = _ d2z-½"~q~Oq~-b~c+Trb(h-hback)+h(½Oq~aq~+QO2~p+2bOc+Obc)+ea~+K~(cOq~-QOc) (3)

71"

q- Kh( ~C q- ¢Oh - ~ch ) q- KccOc + Kb 7rb . (4)

Anomalies are strictly anomalies in the Slavnov-Taylor-Ward identities, given in the antibracket formalism by (F,

F) = 2A.F i . e .

~T ~T A~. f , (5)

with the usual notation and @i representing all the matter fields, gauge fields, ghosts and A u is the composite universal

anomaly operator. One expands the effective action F i n loops (and implicitly in antifield number)

r = F o + + r , + r,o + . . . . (6)

where the first term is the classical action without the potential and subscripts 0e, 1, le label the classical potential term, one-loop potential-independent terms and one-loop potential-dependent terms respectively.

Note that it will not be the aim of this letter to cancel such "universal" anomalies i.e. non-BRST-trivial anomalies that depend on the background gauge-fields only [5-7].

J. Benn / Physics Letters B 332 (1994) 329-335 331

The new one-loop matter-dependent anomaly that results from the potential-dependent term is given by (Fo,/ '~ ~) i.e.

BF 0 BFle hA 2 f 1 ea~(w) A~ -- gKh ~ - -- + ~ d2z d2w 8h(z) (z_w)---------- ~

•x2f 0z. hA2/ - 2~r d2z dzW ~c(z) -~£ 8 2 ( z - w ) e a~(w) + nonlocal 2 ~ d2z 0c(z)e a~(~) + nonlocal . (7)

The important point to note is that all higher-loop diagrams contribute only non-local anomalies. So, one can consider the potential which we insert in the string action to contain an already renormalized A. Then the anomaly calculated above implicitly contains an infinite sequence of higher order h terms. The condition that the local anomalies be cancelled to all order in h is thus

(Fo, rOe) q- (/%0,/'le) =0. (8) So, using

(Fo, Foe) = 7rl- f d2z(l + AQ)Oc(z)ea~(~) (9)

(8) immediately yields the familiar result 1 + AQ + hA2/2 = 0 and so the renormalized A is

1 A= ~ ( - a + ~ ) . (10)

The plus solution gives a smooth classical limit as h ~ 0. Strictly, the analysis should go as follows. First one assumes the classical value for A, A (°), then the genuine

O(h) anomaly is (7) with A equal to A (°). Then an O(h ) correction to A (°) is introduced, namely A ~'). This is determined by the requirement that in the classical variation of the potential it produces an O ( h ) term which cancels the genuine O(h ) anomaly. From here one repeats the procedure, computing now the new O ( h 2) anomaly arising from the A ~ ~) term in the potential. This one cancels by a further renormalization of A. Thus A can be perturbatively constructed

A = a ( ° ) +hA( l ) + h 2 a (2) .... a (°)= - 1 / Q . (11)

In this way one can reconstruct the above exact result (10) to arbitrary order. This has been performed up to O ( h 3) as a simple check.

It is perhaps more important to check higher-loop "miraculous" anomaly cancellation 2. Although all these anomalies are non-local they should still vanish once all the local anomalies are cancelled. The most basic type of two-loop anomaly is in the three-point contribution to/ '2

h2A (°)4 f ~z Ow 47r _ dZz d2u dzw h(z) -~z 6Z(z-w)ea~(w) 0-~-- 6 2 ( w - u ) h ( u ) (12)

This gives a non-local ( 0 - 1 ) term

~]~0 ~F2 h2A(°)4 f Oz. A~ - ~Kh ~h 4 ~ _ d2z 0c(z)e a~(~) ~ h(z) . (13)

plus extra non-local (~ 2) anomalies. This two-loop anomaly is cancelled by the non-local anomaly of (7) with

2 By "'miracle" I mean the vanishing of higher-order anomalies which can occur in spite of the fact that all the parameters of the theory have been fixed in order to cancel the lowest order anomalies.

332 J. Benn / Physics Letters B 332 (1994) 329-335

O(h) contribution in A 2 i.e. the one-loop non-local contribution in (7) with an O(h) counterterm vertex

h2QA ~1)A¢°)2 f ~z 27r dZz h(z) -~ [ea~Z)~c(z) ] . (14)

Provided A ¢1) = -A<°)2/2Q these two terms cancel in the Ward identity. The same should also be true of similar higher loop terms i.e. with a series ofn q>loops attached to the potential and each loop with an external gauge-field. This O(h") (at lowest order in A) term has two contributions to make to the Ward identity--varying it with respect to h yields a non-local term of order n - 1 (i.e. ~- <" ~)) with n - 1 gauge-fields and with respect to q~ it yields a non-local term of order n with n external gauge-fields. The first should be cancelled by O(h) counterterm vertices in the n - 1 loop varied with respect to q~. The second should help in cancelling an appropriate n + 1 loop diagram, and so on. The diagrams become increasingly more complex and one may also obtain new diagrams by inserting external gauge-fields into internal ~loops. These, however, are of a different type to the above "bare" diagrams and should cancel amongst themselves when one bears in mind that an insertion into any diagram could be countered by making a similar insertion into the "bare" diagram which cancels the first. The analysis is tedious but could in principle be carried out to any order.

The Wess-Zumino consistency conditions for the above matter-dependent anomalies are manifest since at each perturbative step one constructs the anomaly as (minus) the classical variation of some expression. This expression is precisely the potential, renormalized to the appropriate order in h. After this, consistency of the local matter- dependent anomaly A'~ n (up to O(h") )

(Fo, A m) --o (15)

simply reduces to

(F, (F, F)) =0, (Fo,/7o) = 0 (16)

i.e. the Jacobi identity for the antibracket and the invariance (up to universal anomalies) of the classical action respectively.

3. Toda W3 anomaly analysis

As mentioned in the introduction I will now repeat the above analysis for a chiral two-scalar renormalized W3 gravity action with a potential term exp (A i q~i).

Here, in contrast with the one-scalar Virasoro action, there are already matter-dependent anomalies present. These are cancelled by the addition of background-charge terms [6,7]. If one is only considering matter-dependent anomalies then the values of these charges remain undetermined, save for the requirement Q1 = ~/3Qz (which amounts to requiring that the spin-2 and spin-3 currents in the action form a realization of a W3 algebra). The background-charges only take on explicit values if one is also considering universal anomaly cancellation. The potential produces a new set of one- and two-loop local anomalies peculiar to the W3 action in addition to those in the Virasoro symmetry (which are simple extensions of the Liouville anomaly above with A 2 ~ E A~ z). I quote the full, renormalized action for completeness

J. Benn / Physics Letters B 332 (1994) 329-335 333

I f d2z - 10q:0~o + + ~/'hai 0z~oi + 2b0c + Obc + 3 h( Oq~i # o e + 20fly)

+ B( ldij k 0~i O~j 0~0 k "JF ~/heij Oq~ i 02~j + hfi 03~9i + Oj~C "~- 3 flOc)

+ B[ - O(byOq~i 0~o,) - bOyO~o i 0~o i -- 2v/-ha~ O(by02~oi) - 2~/haibOy02~o,]

+ B~h(2",/03b + 90y02b + 1502y0b + 1003Tb) - bOc - / 3 0 T + ffb(h - hback ) + ~',e(B -- Bba~k)

2i- K~i [ c0~o i ~- ")ldok O~j 0~0 k "~- 2( Kh + b) y0y0¢,] + ff-hK~,f~ 02") /

+ vChK~ { - ai Oc +eij - eji y02~j -- eji O"yO~j - - 2ol i 0[ (Kh + b) ya'y] }

+ Kh[ OC + cOh - Och - ( TOB - OTB) - 21/~( T0B - OTB) a, 02~o~]

- ~4hKh(2yO3B- 30TOZB + 302y0B- 203yB)

+ KB ( 03' + c OB - 20cB + 2 yOh - ayh) + Kr( c 0 y - 20c y)

+ Kc(COc - yayo~ 0~i - 2 v / ~ a i "YO'yO 26Oi)

- ~ h K c ( 2 y 0 3 y - 30y02y) + Kb'rrh + K,e'n'a (17)

here dil l= 1, d122 = - - 1 = d 2 1 2 , e l l = a l = - - e 2 2 , e l 2 = - - 2 0 1 2 , e21 = 0 , f l = 1 2 ~c~ 1 , f2 = -cqc~ 2. Note that this action contains quadratic source terms due to the non-closure of the algebra off-shell. The action contains the combination Kh + b which is derived from the antifield h * in the BV formalism [ 8 ] signifying that the action is gauge-fixed. The action goes up to order 2 in the antifields and the anomalies too may be expanded in an antifield perturbation series (in fact the anomaly linear in the antifields can be determined by the consistency conditions [ 8 ] ).

The action above includes V~ and h background-charge terms [6,7]. It is true that this is not the most general chiral W3 gravity action free of matter-dependent anomalies and in fact I will view the background-charge terms as different and often absorb the h into the background-charges themselves ~ --* Qi (in part to compare with others [ 3 ] ). Recall that the ordinary one-scalar Virasoro action does not possess matter-dependent anomalies. There, no such confusion arises and I included no V~ factor in the background-charge term.

One proceeds as in the Liouville case, recognising the fact that all anomalies at three-loops and above are non- local. Therefore, one allows the A~'s to take on their fully-renormalized form. Imposition of the Ward identities allows one to immediately spot the analogue of the LiouviIle/Virasoro constraint i.e.

1 +Q,A; + ½h ~ A 2 = 0 . (18) i

I now give the remaining non-trivial local anomalies. There are essentially three such anomalies. Two must be cancelled simultaneously and supply a cubic equation for the A~'s. The third is a first-order antifield anomaly which is cancelled "miraculous ly" . I will derive one as an example, it is given by

~Fo ~FIeB 9 ~Fo ~FleK,~, A m = 8KB ~B + 8 ~ 8Ko

f Oz 6Z(z_w)ea .¢(w) = _ h~r d2zd2w[(A~e-A2)O~Pl-2A'A20cpz](z)OY(z)

_h~r d2z d2w[(A12 -,~)Nq~ -2~N~](z)~, (z) ge ( z - w ) e " ~ ( w ~

~ ~ d~z[(,~-a~)a(~,0~o~) 2,\,a~a(~,a~o~)](z)e ~ ~ (19) "77"

334 J. Benn / Physics Letters B 332 (1994) 329-335

The subscripts on the F ' s signify which part of the one-loop contribution is being considered. The other two local anomalies are

m _ m m A 1 T - - A 2 y ' 4 - A I . j ,

1 f dEz[h2Al( I 2 h ( A 2 Q 1 _ A 2 Q I _ 2 A I A 2 Q E ) ] O 2 T ( z ) e a . ~ ( z ) ( 2 0 ) = - - ~;~, - A 2 ) + 2rr

and the antifield anomaly

h (AT +Az z) f d2z O[VOV(Kh + b ) ] (z)e a~(z~ (21) AK% ¢r

These new anomalies must now be compared with the other new terms in the Ward identity namely the tree-level term which is simply the variation of the potential under the classical (and "half-classical") transformations i.e. one requires

- - - + 3 , % ~ + A 1 % + A~b = 0 , ( 2 2 ) 8K~ 8q~,

where Fo is the whole of (17) except the potential and/'o~ is the potential part. From here on the analysis matches precisely that of the conformal-field-theory OPE requirement f dw

W(z)ea¢¢w) = 0, where W(z) is the standard expression for the W3 current. After a small amount of integration by parts one can compare the two sets of expression in (22). Actually one does not need to solve a cubic equation, the two branches of the solution can deduced by comparing just the TO 2qh and 3'02q~z terms and employing the Virasoro constraint (18). For the above two sets of terms in (22) to cancel one obtains

- - A I A 2 Q 1 Q 2 +Q1al " 1 1 2 / " 1 2 = - - 1 A I A (23)

- 2A1 02 q- ½A1 Q2A2 - 01 A2 - Ol Q2 A2 = - 1a2A, (24)

where

A = [hEAt( 'gA12-A 2) +h(A2Q, _A2Q1 -2AIAEQ2)] • (25)

The first has an obvious solution A t = 0 and substituting this into the second or into the Virasoro constraint yields

the first branch solution

1 A2 = ~ ( - Q z + ~ / Q 2 - 2 h ) , A , = 0 . (26)

It is straightforward to check that this solution ensures cancellation of all other terms in (22) i.e. 3,0~o~0q~ i terms. The other branch is obtained by noticing that the above equations are inconsistent unless A ~Q2 + A zQ~ = O.

Substitution into the Virasoro constraint yields (using Q1 = v/-3Q2)

1 A2 = ~ (Q2 -vFQ~2-2h), A, = -V/-3A2. (27)

As before the above solution implies the vanishing of all other terms in (22). Substitution of the familiar values of the background-charge needed for universal anomaly cancellation then

provides the form of the potentials given in [ 1,3]. The problem is, of course, reminiscent of the physical state requirements of the tachyon in two-scalar W3 strings [9,10]. In the present letter, however, the conditions are more stringent (owing to the fact that the potential is not viewed as acting on a ghost vacuum). In the Toda W3 theory outlined here one obtains four allowed momenta (two of which must be discarded on physical grounds). In [9,10] one finds more allowed discrete tachyon momenta, six to be exact.

J. Benn / Physics Letters B 332 (1994) 329-335 335

A more careful analysis of the problem where one pertubatively constructs the A/'s order by order in h should be performed. The above results above have been checked to the classical and h order after which the calculations become enormous.

An interesting check is that the first-order local antifield anomaly cancels "free of charge" in the sense that it simply requires the Virasoro constraint necessary for the antifield-independent anomaly cancellation. This is as expected since the antifield anomalies are in some sense trivial, as mentioned above.

Higher-loop non-local anomaly cancellation has been studied too, as in the Liouville case. However, here the number of diagrams becomes prohibitive unless one examines the first branch of the solution where A 1 vanishes to all orders. Wess-Zumino consistency is again manifest in the pertubative analysis of the anomalies.

The connection between this letter and previous work should be mentioned. In [4] it was important to maintain a particular structure in the presence of a potential term. In that case the structure was a Virasoro algebra (and hence a nilpotent BRST operator up to universal anomalies). This, of course, is simply a way of saying one requires the preservation of a current algebra when the stress-tensor now includes the potential.

My approach has been to maintain the Slavnov-Taylor-Ward identities for an action modified by a potential. Owing to a lack of understanding of W3-geometry, a geometric action of the form (1) cannot be constructed at present. One constructs a minimalist Toda W3 action, the analogue of the Liouville action in chiral form (where the potential no longer appears in the stress-tensor), before maintaining BRST nilpotence in the sense of preserving the Ward identities.

Acknowledgement

I would like to thank K.S. Stelle for the many illuminating discussions we had during the course of this work.

References

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