Volume 258, numbcr 1,2

PHYSICS LETTERS B

4 April 1991

Superstrings in arbitrary dimensions A.H. Chamseddine lnstitute fiJr Theoretical Physics, University of Ziirich, Schrnberggasse 9, CH-8001 Ziirich, Switzerland Received 21 January. 1991

A supersymmetric extension of the recently found solution to two-dimensional gravity is presented. The classical supergravity action implies the constant supercurvature constraint, which takes the induced super Liouville action on-shell. The path integral is evaluated without the need to impose a restriction on the dimension of coupled matter. The superconforrnal algebra is shown to be non-anomalous.

In the past m a n y attempts wcre m a d e to quantize two-dimcnsional gravity coupled to arbitrary m a t t e r systems [ 1 ]. In the absence o f a purely geometrical c a n d i d a t e for two d i m e n s i o n a l gravity, it was assumed that the metric couples to m a t t e r as an external field. At the classical level only massless m a t t e r could be coupled without difficulty. At the q u a n t u m level one degree o f freedom associated with the Weyl invariance becomes dynamical, except in the critical dimension d = 26 for bosonic matter, and d = 10 for s u p e r s y m m e t r i c matter, where this invariance is preserved. The challenge was to quantize the system in d i m e n s i o n s other than the critical ones. F o r such systems, the quantization o f the Liouville m o d e becomes necessary. Such quantization was performed and leads to consistent non-critical strings, but only in d i m e n s i o n s d~< 1 [ 2 ]. To o v e r c o m e these difficulties it is essential to first understand classical gravity. The difficulty to include massive m a t t e r systems, or even a cosmological constant, suggests that a classical gravity action restricting the geometry, in analogy with highcr dimensions, must be included. After all, if we think o f two-dimensional systems as a d i m e n sional reduction and truncation o f three-dimensional systems, no such difficulties should occur at the classical level. The two-dimensional action obtained with this reduction and truncation is o f the form [3 ]

Supported by the Swiss National Foundation (SNF).

•

d2~ w/-gt~(R + A ) ,

(1)

M

where ~ is the dilaton field, and A is a cosmological constant. It is usual in the reduction to lower d i m e n sions to decouple the dilaton field by rescaling the metric, so that the dilaton field acquires a kinetic energy, and can be truncated if so wished. However, this is not possible in two dimensions and the dilaton field cannot be dccoupled. The policy a d o p t e d was to set the two-dimensional gravity action to zero. In a recent paper [ 4 ], I have shown that when the action in eq. ( 1 ) is taken as the classical gravity action, then the coupling to massless matter systems can be quantized without difficulty. The reason that the usual difficulty associated with the induced Liouville action can be avoided, is that the dilaton field imposes a constraint that the curvature is constant. This makes it possible to reduce the Liouville action to its value on-shell, and this is a constant factor. W i t h this modification it was also shown that there is no conformal anomaly. This analysis was performed for bosonic systems and it is natural to extend it to the supersymmetric case [ 5 ]. This is most easily done using superfields. In this lctter I shall show that all steps performed in the bosonic case can be generalized. The relevant constrained supergeometry is well known [6]. I will first s u m m a r i z e the essential results, then give the action. Throughout this paper I shall use the notation and conventions o f D ' H o k e r

0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )

97

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and Phong [7 ]. The coordinates of superspace are denoted by ZM= (~". 0~). Equivalently, the complex coordinates (~, ~ 0, 0) could be used. The coordinates in the tangent space are denoted by A = (z, g, + , - ) . The supergeometr3, is constrained so that all the components of the torsion and curvature are expressed in terms of a simple scalar superfield R+_. The superzweibein LM ~''~ connects curved indices to flat ones. The torsion constraints T .c b = T ~ = 0 ,

Vc

T,~-2(, c

--

).1~,

(2)

imply that the components of E ~ are

4 April 1991

R+_ + K = 0 ,

(8)

whose component form is A=-K,

A,=0,

R = ~iKE'a'Z,,ysZ, , -- ½A 2.

The component form of the supergravity action (7) is then

Af 2to

dZ~et~R+ F(A + K ) - ~OKA - ]ie'"Z,,7.sz, OK

- 275 DmZ, ~ m,__ ~i27 mZ,,,A] ,

/z'~ =e~, + ()7~Z,~- ½i06e~A,

here

Em ~ = - ~ Xt, , ,. _½i(0.%),~A_½ (~4.)~o~.,

cb( Z M) = ~ + 0~'2~, + i061 ,",

+i0(5[t

, ~ -~7.,,,A] 3 ~(7~A) ,

(9)

(3)

( 10 )

and the component form of E has been used: E=e[l+½OT"Z,-½iOO(A+'i¢""4 Zm,sZ,,]v" ~ . To thc supergravity action, we couple the matter action:

E ~ - - 8 ~-( 1 + ]i06A),

where (-Ore =

--e-

1 a em£

p q .x a Opeq --

I~

I,.=~

,~ . 2 p ~ , Ap ,

~Zm/5,

A = - i 7 s e"~ D m z . - {7'~X,,¢t,

d2zED_XUD+X.+~IZ(M),

(11)

where

D,.Z. = O.Z. + ½~om75Z~ .

(4)

The components of the superconnection -QM can be expressed in terms of the superzweibein E~:

d2z E R . _ - 2re

z(M) = ~ M

M

m+ =2iEM+(O~tE~+ )E,+~ , ~ = E ' ~ t O.ug2+ .

is the Euler characteristic. The covariant derivatives D_ and D+ can be read from

The component form of the supercurvature is

IkA = E ~ t DM =E~t(Og +tOM) •

R+ _ =A + O"A, +iOOC ,

(5)

where C = R + ½iZ,7"A + ~i~"bZ~7571,A + ½A 2

,

R=e'"~ 0.,oJ,,.

(6)

The conjectured supergravity action is ib l~g= ~

d2zEq2(R+

+K) ,

(7)

where the volume element is given by d2~:E = d 2~dO d0-sdet E A , and • is the dilaton superfield. The role of • is to impose the constraint 98

(12)

Quantizing this theory is carried out by integrating exp[ - (lm+Isg) ] over all supergeometries E ~ satisfying the torsion constraints (2) [8], over the dilaton supcrfield q~, an over all superficlds X u. The sum is also taken over all possible topologies of the super Riemann surfaces. The fixed genus h partition function is Zh= f DE~D~DXUexp[-(lm+I~)]

'

(13)

where the integration over E.,g4 is restricted to the independent components only. However, because of all this has been extensively treated in the literature [ 58 ], and my main interest is in the modification caused by the scalar superfield ~, I will take the liberty to use known results.

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PHYSICS LETTERSB

The space of supergeometries is parametrizcd by E A = e x p ( V ) exp(Z) exp(L) L,vtTA,

4 April 1991

(sde,' o

(14)

(sdet'ao

= k ~ j

exp[-&L(2;) ].

(19)

wherc V, V and L are the fields for supcr diffeomorphism, super Weyl and super Lorentz transformations. The transformed 1 ~ is in a slice S of dimension ( 6 h - 6 ) transversal to the action of sDiff(M) within the space of supergeometries. The measured D E ~ decomposes as

The integration of the dilaton superfield • implies the delta function constraint

D E ~ = sdet (P~ P~ ) ~/2 sdet r:

Zh=£2 S ~ ° m J s d e t < ~ s l q b K >

~( R+_ + K) . Making use of the above relations, we obtain the result . sMh

sdet< q'j I q~,~>)/2

× DZDL DV 'a I-I dins,

s d e t ( / 2 , I qb~ >

(15)

× (sdet P"~PI),/2 {8zr2sdet'Z~o'] -a/2

\ ifd2zE J

J

where ms arc coordinates for the slice S, q~s is a basis for Ker(P] ), ~ts arc dual super Beltrami differentials and qbs=exp(3Z)~j. The action of the operator P~ is given by

(P~rV) b =

-

(ycyb)

'aCxDog V".

The super determinants in (15) are evaluated with respect to the superzweibein Ea..~4, and the effect of super Weyl scaling on them is [ 5-8 ] sdet P[ Pt sdet < q~j I qsx > sdet/~ Pl - sdet ( ~ j I ~¢ ) exp [ - 10&L (Z) ] ,

(20) As in the bosonic case, the X integration can now be performed without difficulty, because the constraint of constant supercurvature employed through the delta function can be substituted into the super Liouville action. The simplification occurs as the equation of motion of the super Liouville action is compatible with the constant supercurvature constraint. Denoting by

Y=R+_ +K, ( 16 )

where

& , = ~1

× 1- I)Xg(R+_ + K ) cxp[~ ( d - 1 0 ) & L ( S ) ] .

the Z integration can be changed to Y integration: DX=

I

d2zE(f)_Xf)+X+il~÷_X).

(17) =2

As the main concern in this paper is the superstring, any global or local Lorentz anomaly is cancelled between left- and right-movers on the world sheet. For heterotic strings, the absence of the local and global Lorentz anomaly will be a constraint on thc matter interactions, but for now I will not bc concerned with that. Going back to the path integral, the X ~' integration is immediate, and gives the contribution

Q(kifd2z E8zt2 sdet'Ao) -a/2,

(18)

where £2 is the volume of space-time, ,5o = D + D_. The transformation of (18) under the super Weyl scaling is read from the relation [ 5-8 ]

(21)

DY sdet' 16 Y/SSI DY sdet' [Ao+ ½iKexp( - S ) [ '

(22)

where the relation R+_ = e x p ( - S )

(/~+_ - 2 i I)+ D_X)

(23)

has been used. To isolate the Zdependence in the denominator ofeq. (22), the following property is used: In sdet' Iz~ + ~ i K e x p ( - X ) [ =In sdct'A o +ln sdet'[l + ~iKAff I e x p ( - Z ) ] . In the heat kernel expansion of the second term in the last equation, the leading divergence cancels and will only contribute to the Euler characteristic. It then can be absorbed in the coefficient q in the action ( 11 ). After taking the zero modes into account, the L- integrand reduces to 99

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PItYSICS LETTERS B

F(K)

- --d-8lnK(if

-1

=L\ i-T~2 } (8z2sdet'~o'] - - \ ifd2z/~ ]

4 April 1991

exp[~(d-10)SsL(X)]]v= °

-I

=-~(d-8)(lnK)z.

Collecting all terms, the fixed genus h partition function becomes

( 8]~2sdct'~) ~ -(d+2)/2

{exp [ ~( d - 8 )SsL(S) 1}Y=o.

Zh(K) = Q e x p ( - r / Z ) \ ~

-]

(24) The explicit solution of the Y = 0 equation must now be substituted. By performing a super Weyl scaling to a geometry where R" t + _ =0, this equation reduces to the super Liouville equation D+ D _ S ' + -~iKexp ( Z ' ) = 0 ,

(25)

where D+ = or7 + 0

'

xK-I(d-s)/4Iz I

sdet (#v I $~< )

sMh

= K-f (a-s)/41zZh .

(28)

To be consistent with the Gauss-Bonnet theorem, the constant supercurvature condition forces the length L to be constant when h ~ 1:

Z= ~i f dZzER . . . .

0g"

•

omg sdet ( Cg I S x )

~i K f d 2 z E = - K 27tL . (29)

The general solution ofeq. (25) is given by [9] 2i D + a D_c~ exp(X')=

K F-F-a6~ '

(26)

where F=F(z, O) and a=a(z, O) arescalar and spinor holomorphic supeffields, while F and a are antiholomorphic. These are subject to the conditions

D+F=aD+a,

Z~°t~'=

~ f dL f dK 5( KL + 21tz)Zh( K )

= ~;dL~fdKd(K+2rc~)Zh(K)

D _ P = ( ~ D_ ~ ,

.

(30)

The delta function in eq. (30) makes the K integration immediate:

whose explicit solution is

F=f(z) +O~,(z) , + 1

The total fixed genus partition function is obtained by integrating over the parameter K and the length L, taking the topological restriction into account,

, .

Zt°ta' =

~fdLLV-3Zh,

(31)

h=O

The functions F, F, a and c~ can be thought of as the superconformal transforms of z, g, 0 and 6~ For a fixed background geometry, Z' is related to the Z used in (17) by i

exp(X' ) = e x p (X)

z - 2 - 0~0-"

(27)

After substituting the solution of eq. (27) into (24), it is easily seen that the k; I7, a and c~ contributions cancel except for the constant term proportional to K in the linear part I],'XI~+_ in the super Liouville action (17). The remaining contribution is then given by I00

where y= 2 + 4t ( d - 8)Z is the string susceptibility, and where some constant factors have been absorbed in -(2. Eq. (31) is in agreement with the semi-classical approximation [ I0]. It is thus seen that the super Liouville equation can be integrated out completely, as the constant supercurvature constraint reduces its contributions to a dependence on the length L. It should be clear by now that thc rolc of the dilaton superfield is extremely important in avoiding the super Weyl anomaly. The fact that the super Weyl anomaly was encountered bcfore is then a reflcction of the fact that the classical supergravity action (8) was missing. The conclusion is that it must be included and cannot be ignored. It is also

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clear that in the presence of this classical action all possible matter interactions can be included provided the dilaton superfield is properly coupled. It is now appropriate to investigate the reason behind the absence of anomalies from the superconformal field theory point of view. It was already seen that the measure D E ~ is transformed to the super Weyl scaled measure D E ~ through eq. ( 15 ), (16). It was argued by Distler and Kawai [2 ] that this relation is renormalized, and the coefficients in the super Liouville action should be allowed to become parameters. The values of these parameters arc fixed by the requirements of the cancellation of the superconformal anomaly, and the scale factor in the metric transforms as a proper superconformal field. It was shown, in the absence of the modifications proposed here, that it is possible to fix the parameters in the measure to satisfy the above requirements, provided that d ~ 1 [2,5]. I will now show that this procedure leads to exactly the same equation (28) and the formulas appearing in the path integral formulation will not be modified. To start, we require the path integral measure to transform as

Fz=DXDBDX"=F2exp[-S(Z,E')]

,

(32)

where B and C are the superghosts resulting from the super diffeomorphism gauge fixing [ 11 ]. The action S(X, fig) is assumed to be of the same form as the super Liouville action, but with renormalized coefficients. The coefficients in S are normalized so that it takes the form S=

Lf lr

d2z ~ [ a ( f ) _ X I3+ Z+i/~+ _X)

+/t~ cxp(c~.S) ] .

4 April 1991

]] =/%. exp[ -S'°'a~(_r, ~, --.)1,

(34)

where the ... refers to entries which are not effected by the scaling. The total action is given by

St°t"'( S, L ~, B, C, X) =S(Z', t?) + / + S gh°~' where I = lm+ l~g, and the ghost action is sgho~t= 2-~ 1 f B D _ C+c.c. As the supcrconformal anomaly of the ghost and matter parts arc known, it is sufficient to determine the anomaly of the supergravity part in the action given by

S(X, 175)+l~g +igt f d2z E .

(35)

For simplicity I shall set/~,/q and Kto zero, since the superconformal anomaly is independent of these parameters. The action to be quantized is now

_a g

f d2z fi(D_ X I)+ S + i/~+ _ )

+~

f d2zEq)R+- •

(36)

The stress tensor corresponding to this action is obtained from the variation (33)

The parameters a and a will be determined by the requirements that the superconformal anomaly vanishes and that exp (ceX) is a superconformal tensor of scaling dimensions ( ~, ~ ). The theory being dependent only on E~,~ should be invariant under the simultaneous shifts /~-,exp(p)/~, X - , X - l p , where EAJ,u=exp(c~X) fi~ and p is a constant superfield. The invariance of the theory under the above shifts imply the identity

8I- ~

d2z IZ'HT ,

where I t = E M_8E~. Using eq. (23) and the relations

[8] 8 (E/~,+ _ ) = F.(i D3+H ) ,

8(£ ~_ z i5+ x) = - £ i i ~+ x i5+ x, 8 [ fi(f)_ • f)+s+ f)+ 4, f)_ s) l = -1~11(0+ ~ f)2+X+ D2+ ~ 13+X),

(37)

we deduce that 101

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T S g = 2 a ( - f) + Z I7)2 L-+ I7)3+X)

+b(f)~+ ~ - I 5 + q~I5%S- f)% q~f)+ 2.) .

(38)

This equation for the stress tensor can also be obtained by working in the component formalism. The kinetic operator in the action (36) is a mixture of XX and zq~ terms and is of the form

4 April 1991

and the requirement that exp(~.X) is of weight ½ it is seen that the parameter a must be fixed to a = 1. Thus the action S coincides exactly with the super Liouville action (17) obtained in the path integral analysis, with identical and not renormalized coefficients. It is also possible to use scaling arguments to deduce the dependence of the partition function on the length L. By considering the constant rescaling Z - , exp(p)~', the shift in the total action is

S,Ot~l ( K ) From this expression it is easy to find all the propagators, which after rescaling q)~ ( 1 / b ) ~ , are given by

= 0 , ( 39 )

where z t , _ = z ~ - z 2 - O t O z . With these quantization conditions, it is possible to calculate the operator product expansion of the stress tensors T Sg [ 1 1 ]: 2 a + ~ + 30,2 T"S(z,,O,)T~g(zz,02) - z32 -~2 TSg(zz,02)

+ I/2 Dz T ~ (zz, 02) + 0,2 Oz T ~g(zz, 02) + .... ZI2

(40) From the first term in eq. (40), we deduce that the superconformal anomaly of the supergravity part (classical and induced) is ~'sg=4(2a+ l ). By adding the contributions of the ghost and matter ~gh+ m = d - 10, and requiring that the total supercontbrmal anomaly vanishes, the parameter a is determined: or

a=-~(d-8).

(41)

Also from the operator product expansion [ 1 1 ]

T~g(zl, Oi ) exp [a_Z(z2, 02) ] a 012

- ~ z 2 exp[cxZ(z2, 02)]

+

1/2 2"12

D2 exp[aZ(z2, 02)]

+ 01.2 02 exp[aZ(z2, 02) ] , .7-12

102

d2zEl~+_

S'°"~'(Kexp(p)).

From this it is seen that the partition function scales

< q)(zl, 0, ) q~(z2, 02 ) > = a In zl 2,

U°'"~=8a+d-8=0,

8-~- p i

as

= - ~ In z,2,

ZI2

-

(42)

Z[K]=Z[Kexp(p)]

exp[~ ( d - 8 ) Z o ] ,

which can be solved by

Z[ K] = C K -

[ (d--g)/4]Z ,

a relation identical in the K dependence to eq. (28) obtained by direct evaluation. To conclude, 1 have shown that, just as in the bosonic case [4], when a classical supergravity action is taken into account, the usual super Weyl anomalies do not pose a problem because of the constant supercurvature constraint introduced by the dilaton superfield equation of motion. (A similar constraint was introduced in ref. [ 12 ] within the random matrix approach.) Indeed, the classical action does not have the super Weyl invariance, so the super Liouville mode must always be integrated. By examining the partition function it is clear that the dilaton superfield and the super Liouville mode contribute the equivalent of two free scalar superfields. This is why the familiar ten-dimensional superstring will now correspond to d = 8. Because now the coupling to arbitraD' matter is allowed non-critical superstrings are always possible. In fact the notion of criticality does not apply here because the super Weyl invariance is neither present at the classical nor at the quantum level. The most interesting possibility is, as usual, the heterotic one. In this case the possible matter coupling will be subject to the constraints of the absence of global and local Lorentz anomalies [13]. I will leave the detailed analysis to a different publication where a general formalism will be given and the spec-

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PHYSICS LETTERS B

t r u m will be a n a l y s e d in detail. H e r e I will s i m p l y q u o t e s o m e o f the results c o n c e r n i n g the d i m e n s i o n s o f the c o u p l e d m a t t e r for h c t e r o t i c strings. A q u i c k way to d e t e r m i n e thc c o n s t r a i n t s on the h e t e r o t i c dim e n s i o n s is to r e q u i r e the m o d u l a r i n v a r i a n c e o f the p a r t i t i o n function. A s s u m i n g that the h e t e r o t i c coordinates h a v e shifted o r twisted b o u n d a r y c o n d i t i o n s , the restriction c o m i n g f r o m the c a n c e l l a t i o n o f the global phase is

d L- - ~3 d R = 0

,

m o d 12,

w h e r e d~ a n d dR are, respectively, the d i m e n s i o n s o f the left- a n d r i g h t - m o v e r s . A n o t h e r c o n d i t i o n c o m e s f r o m level m a t c h i n g and r e q u i r e s dE - - d R = 0 ,

mod4.

B o t h c o n d i t i o n s can be satisfied p r o v i d e d that d R is a m u l t i p l e o f 8 and dE is a m u l t i p l e o f 12. T h e m i n i m a l h c t e r o t i c m o d e l has 8 r i g h t - m o v e r s and 12 l e f t - m o v ers, while the f a m i l i a r critical m o d e l has 8 r i g h t - m o v ers and 24 left-movers. With m i n o r modifications, the analysis o f the m o d u l a r i n v a r i a n t c o n s t r u c t i o n o f twisted strings as g i v e n in ref. [ 14] can be generalized. A difficult and i m p o r t a n t p r o b l e m that m u s t be tackled is h o w to i n c o r p o r a t e the m o d i f i c a t i o n s presented here in the m a t r i x models, a n d r a n d o m surfaces a p p r o a c h [ 15,16 ]. I w o u l d like to t h a n k Jfirg F r 6 h l i c h for v e r y useful discussions.

References [ 1 ] A.M. Polyakov, Phys. Lctt. B 163 ( 1981 ) 207; J.-L. Gervais and A. Ncveu, Nucl. Phys. B 199 (1982) 50; E. D'Hoker and R. Jackiw, Phys. Rev. D 26 (1982) 3517; E. Witten, Commun. Math. Phys. 117 (1988) 353; J. Labastida, M. Pernici and E. Winen, Nucl. Phys. B 310 (1988)611; J. Polchinski, Nucl. Phys. B 324 (1989) 123. [2] A.M. Polyakov, Mod. Phys. Lcn. A 2 (1987) 899; V.G. Knizhnik, A.M. Polyakov and A.A. Zamolodchikov, Mod. Phys. Left. A 3 (1988) 819;

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F. David, Mod. Phys. Lctt. A 3 (1988) 1651; J. Distler and H. Kawai, Nucl. Phys. B 321 (1989) 509. [ 3 ] R. Jackiw, in: Quantum theory ofgravity, ed. S. Christensen (Adam Hilger, Bristol, 1984) p. 403; C. Teitelboim, Quantum theory &gravity, ed. S. Christensen (Adam Hilger, Bristol, 1984) p. 327; A.H. Chamseddine and D. Wyler, Phys. Lctt. B 228 ( 1989 ) 75; Nucl. Phys. B 340 (1990) 595; A.H. Chamseddine, Nucl. Phys. B 346 (1990) 213. [4] A.H. Chamseddine, Phys. Lcn. B 256 ( 1991 ) 379. [ 5 ] M. Grisaru and R. Xu, Phys. Left. B 205 ( 1988 ) 486; A.M. Polyakov and A.B. Zamolodchikov, Mod. Phys. Len. A 3 (1988) 1213; J. Distler, Z. Hlousek and H. Kawai, Intern. J. Mod. Phys. A2 (1990) 391. [6] P. Howe, J. Phys. A 12 (1979) 393; E. Martinec, Phys. Rcv. D 23 (1983) 2604. [7] For a review see: E. I)'Hoker and D.H. Phong, Rev. Mod. Phys. 60 (1988) 917. [ 8 ] E. D'Hoker and D.H. Phong, Nucl. Phys. B 278 (1986) 225; G. Moore, P. Nelson and J. Polchinski, Phys. Lett. B 169 (1986) 47; S. Chaudhuri, H. Kawai and S.-H. Tye, Phys. Rev. D 36 (1987) 1148. [9] J. Avris, Nucl. Phys. B 212 (1983) 151. [ 10] A.B. Zamolodchikov, Phys. Lett. B 117 (1982) 87. [ 11 ] D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B 271 (1986) 93. [ 12] B. Durhuus, J. Fr~hlich and T. Jonsson, Nucl. Phys. B 257 (1985) 779. [ 13 ] E. Wittcn, in: Symp. on Anomalics, geometry, topology, eds. W.A. Bardeen and A.R. White (World Scientific, Singapore) p. 61. [ 14] A.H. Chamseddine, J.-P. Derendingcr and M. Quiros, Nucl. Phys. B 326 ( 1989 ) 497. [15] J. Frrhlich, in: Lecture Notes in Physics, Vol. 216, ed. L. Garrido (Springer, Berlin, 1985 ) p. 32; J. Ambjorn, B. Durhuus and J. Fr6hlich, Nucl. Phys. B 257 (1985) 433. [16] D.J. Gross and A.A. Migdal, Phys. Rev. Left. 64 (1990) 717; M. Douglas and S. Shenkcr, Nucl. Phys. B 335 (1990) 635; E. Brrzin and V. Kazakov, Phys. Lett. B 236 (1990) 144; for a general review see R. Fernandez, J. Frrhlich and A. Sokal, Random walks, critical phenomena and triviality in quantum field theory (Springer, Berlin) Ch. 7, to be published.

103